Results of Elementary Functions

DLMF	Formula	Maple	Mathematica	Symbolic Maple	Symbolic Mathematica	Numeric Maple	Numeric Mathematica
4.2.E8	${\displaystyle{\displaystyle\operatorname{log}_{a}z=\ifrac{\ln z}{\ln a}}}$	`log[a](z)=(ln(z))/(ln(a))`	`Log[a,z]=Divide[Log[z],Log[a]]`	Successful	Successful	-	-
4.2.E9	${\displaystyle{\displaystyle\operatorname{log}_{a}z=\frac{\operatorname{log}_{% b}z}{\operatorname{log}_{b}a}}}$	`log[a](z)=(log[b](z))/(log[b](a))`	`Log[a,z]=Divide[Log[b,z],Log[b,a]]`	Successful	Successful	-	-
4.2.E10	${\displaystyle{\displaystyle\operatorname{log}_{a}b=\frac{1}{\operatorname{log% }_{b}a}}}$	`log[a](b)=(1)/(log[b](a))`	`Log[a,b]=Divide[1,Log[b,a]]`	Successful	Successful	-	-
4.2.E12	${\displaystyle{\displaystyle\ln e=1}}$	`ln(exp(1))= 1`	`Log[E]= 1`	Successful	Successful	-	-
4.2.E13	${\displaystyle{\displaystyle\int_{1}^{e}\frac{\mathrm{d}t}{t}=1}}$	`int((1)/(t), t = 1..exp(1))= 1`	`Integrate[Divide[1,t], {t, 1, E}]= 1`	Successful	Successful	-	-
4.2.E14	${\displaystyle{\displaystyle\operatorname{log}_{e}z=\ln z}}$	`log[exp(1)](z)= ln(z)`	`Log[E,z]= Log[z]`	Successful	Successful	-	-
4.2.E15	${\displaystyle{\displaystyle\operatorname{log}_{10}z=\ifrac{(\ln z)}{(\ln 10)}}}$	`log[10](z)=(ln(z))/(ln(10))`	`Log[10,z]=Divide[Log[z],Log[10]]`	Successful	Successful	-	-
4.2.E15	${\displaystyle{\displaystyle\ifrac{(\ln z)}{(\ln 10)}=(\operatorname{log}_{10}% e)\ln z}}$	`(ln(z))/(ln(10))=(log[10](exp(1)))* ln(z)`	`Divide[Log[z],Log[10]]=(Log[10,E])* Log[z]`	Successful	Successful	-	-
4.2.E16	${\displaystyle{\displaystyle\ln z=(\ln 10)\operatorname{log}_{10}z}}$	`ln(z)=(ln(10))* log[10](z)`	`Log[z]=(Log[10])* Log[10,z]`	Successful	Successful	-	-
4.2.E20	${\displaystyle{\displaystyle\exp\left(z+2\pi i\right)=\exp z}}$	`exp(z + 2PiI)= exp(z)`	`Exp[z + 2PiI]= Exp[z]`	Successful	Successful	-	-
4.2.E21	${\displaystyle{\displaystyle\exp\left(-z\right)=1/\exp\left(z\right)}}$	`exp(- z)= 1/ exp(z)`	`Exp[- z]= 1/ Exp[z]`	Successful	Successful	-	-
4.2.E22	${\displaystyle{\displaystyle\|\exp z\|=\exp\left(\Re z\right)}}$	`abs(exp(z))= exp(Re(z))`	`Abs[Exp[z]]= Exp[Re[z]]`	Successful	Successful	-	-
4.2.E23	${\displaystyle{\displaystyle\operatorname{ph}\left(\exp z\right)=\Im z+2k\pi}}$	`argument(exp(z))= Im(z)+ 2kPi`	`Arg[Exp[z]]= Im[z]+ 2kPi`	Failure	Failure	Fail `-18.84955592 <- {z = 2^(1/2)+I2^(1/2), k = 3}` `-18.84955592 <- {z = 2^(1/2)-I2^(1/2), k = 3}` `-18.84955592 <- {z = -2^(1/2)-I2^(1/2), k = 3}` `-18.84955592 <- {z = -2^(1/2)+I2^(1/2), k = 3}`	Fail `-18.84955592153876 <- {Rule[k, 3], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}` `-18.84955592153876 <- {Rule[k, 3], Rule[z, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}` `-18.84955592153876 <- {Rule[k, 3], Rule[z, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}` `-18.84955592153876 <- {Rule[k, 3], Rule[z, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}`
4.2.E24	${\displaystyle{\displaystyle\exp z=e^{x}\cos y+ie^{x}\sin y}}$	`exp(z)= exp(x)cos(y)+ Iexp(x)*sin(y)`	`Exp[z]= Exp[x]Cos[y]+ IExp[x]*Sin[y]`	Failure	Failure	Fail `-.8272584772+1.775573363I <- {z = 2^(1/2)+I2^(1/2), x = 1, y = 1}` `1.772639846+1.591201978I <- {z = 2^(1/2)+I2^(1/2), x = 1, y = 2}` `3.332514076+3.679324696I <- {z = 2^(1/2)+I2^(1/2), x = 1, y = 3}` `-3.350888586-2.154747662I <- {z = 2^(1/2)+I2^(1/2), x = 2, y = 1}` ... skip entries to safe data	Fail `Complex[-0.8272584783533998, 1.7755733643246545] <- {Rule[x, 1], Rule[y, 1], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}` `Complex[1.7726398453192989, 1.591201979498678] <- {Rule[x, 1], Rule[y, 2], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}` `Complex[3.332514075382279, 3.679324697962366] <- {Rule[x, 1], Rule[y, 3], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}` `Complex[-3.3508885868787868, -2.154747660864471] <- {Rule[x, 2], Rule[y, 1], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}` ... skip entries to safe data
4.2.E26	${\displaystyle{\displaystyle z^{a}=\exp\left(a\operatorname{Ln}z\right)}}$	`(z)^(a)= exp(a*ln(z))`	`(z)^(a)= Exp[a*Log[z]]`	Successful	Failure	-	Successful
4.2.E28	${\displaystyle{\displaystyle z^{a}=\exp\left(a\ln z\right)}}$	`(z)^(a)= exp(a*ln(z))`	`(z)^(a)= Exp[a*Log[z]]`	Successful	Successful	-	-
4.2.E29	${\displaystyle{\displaystyle\|z^{a}\|=\|z\|^{\Re a}\exp\left(-(\Im a)\operatorname% {ph}z\right)}}$	`abs((z)^(a))=(abs(z))^(Re(a))* exp(-(Im(a))* argument(z))`	`Abs[(z)^(a)]=(Abs[z])^(Re[a])* Exp[-(Im[a])* Arg[z]]`	Failure	Failure	Successful	Successful
4.2.E30	${\displaystyle{\displaystyle\operatorname{ph}\left(z^{a}\right)=(\Re a)% \operatorname{ph}z+(\Im a)\ln\|z\|}}$	`argument((z)^(a))=(Re(a))* argument(z)+(Im(a))* ln(abs(z))`	`Arg[(z)^(a)]=(Re[a])* Arg[z]+(Im[a])* Log[Abs[z]]`	Failure	Failure	Fail `-6.283185309 <- {a = 2^(1/2)+I2^(1/2), z = -2^(1/2)+I2^(1/2)}` `6.283185309 <- {a = 2^(1/2)-I2^(1/2), z = -2^(1/2)-I2^(1/2)}` `6.283185309 <- {a = -2^(1/2)-I2^(1/2), z = -2^(1/2)+I2^(1/2)}` `-6.283185309 <- {a = -2^(1/2)+I2^(1/2), z = -2^(1/2)-I2^(1/2)}`	Fail `-6.283185307179586 <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}` `6.283185307179586 <- {Rule[a, Times[Complex[1, -1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}` `6.283185307179586 <- {Rule[a, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}` `-6.283185307179586 <- {Rule[a, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}`
4.2#Ex2	${\displaystyle{\displaystyle\operatorname{ph}\left(z^{a}\right)=a\operatorname% {ph}z}}$	`argument((z)^(a))= a*argument(z)`	`Arg[(z)^(a)]= a*Arg[z]`	Failure	Failure	Fail `.980258143-1.110720734I <- {a = 2^(1/2)+I2^(1/2), z = 2^(1/2)+I2^(1/2)}` `.9802581426+1.110720734I <- {a = 2^(1/2)+I2^(1/2), z = 2^(1/2)-I2^(1/2)}` `.980258144+3.332162204I <- {a = 2^(1/2)+I2^(1/2), z = -2^(1/2)-I2^(1/2)}` `-5.302927166-3.332162204I <- {a = 2^(1/2)+I2^(1/2), z = -2^(1/2)+I2^(1/2)}` ... skip entries to safe data	Fail `Complex[0.9802581434685473, -1.1107207345395915] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}` `Complex[0.9802581434685472, 1.1107207345395915] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}` `Complex[0.9802581434685469, 3.332162203618774] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}` `Complex[-5.302927163711039, -3.332162203618774] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}` ... skip entries to safe data
4.2.E32	${\displaystyle{\displaystyle e^{z}=\exp z}}$	`exp(z)= exp(z)`	`Exp[z]= Exp[z]`	Successful	Successful	-	-
4.2.E33	${\displaystyle{\displaystyle e^{z}=(\exp z)\exp\left(2kz\pi\mathrm{i}\right)}}$	`exp(z)=(exp(z))* exp(2kzPiI)`	`Exp[z]=(Exp[z])* Exp[2kzPiI]`	Failure	Failure	Fail `.6414354628+4.062928650I <- {z = 2^(1/2)+I2^(1/2), k = 3}` `-.1544020768e13-.1710664597e12I <- {z = 2^(1/2)-I2^(1/2), k = 3}` `.8993679173e11+.1849553851e11I <- {z = -2^(1/2)-I2^(1/2), k = 3}` `.3791252193e-1+.2401424313I <- {z = -2^(1/2)+I2^(1/2), k = 3}`	Fail `Complex[0.6414354615731531, 4.062928651501303] <- {Rule[k, 3], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}` `Complex[-1.5440207807554412^12, -1.710664745395911^11] <- {Rule[k, 3], Rule[z, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}` `Complex[8.993679264986926^10, 1.8495537828408436^10] <- {Rule[k, 3], Rule[z, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}` `Complex[0.03791252182632387, 0.24014243117514714] <- {Rule[k, 3], Rule[z, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}`
4.2.E36	${\displaystyle{\displaystyle-\pi<=\Im\left(\frac{1}{a}\operatorname{Ln}w\right% )}}$	`- Pi < = Im((1)/(a)*ln(w))`	`- Pi < = Im[Divide[1,a]*Log[w]]`	Failure	Failure	Successful	Successful
4.2.E36	${\displaystyle{\displaystyle\Im\left(\frac{1}{a}\operatorname{Ln}w\right)<=\pi}}$	`Im((1)/(a)*ln(w))< = Pi`	`Im[Divide[1,a]*Log[w]]< = Pi`	Failure	Failure	Successful	Successful
4.4.E1	${\displaystyle{\displaystyle\ln 1=0}}$	`ln(1)= 0`	`Log[1]= 0`	Successful	Successful	-	-
4.4.E2	${\displaystyle{\displaystyle\ln\left(-1+\mathrm{i}0\right)=+\pi\mathrm{i}}}$	`ln(- 1 + I0)= + PiI`	`Log[- 1 + I0]= + PiI`	Successful	Successful	-	-
4.4.E2	${\displaystyle{\displaystyle\ln\left(-1-\mathrm{i}0\right)=-\pi\mathrm{i}}}$	`ln(- 1 - I0)= - PiI`	`Log[- 1 - I0]= - PiI`	Failure	Failure	Fail `6.283185308*I <- {}`	Fail `Complex[0.0, 6.283185307179586] <- {}`
4.4.E3	${\displaystyle{\displaystyle\ln\left(+\mathrm{i}\right)=+\tfrac{1}{2}\pi% \mathrm{i}}}$	`ln(+ I)= +(1)/(2)PiI`	`Log[+ I]= +Divide[1,2]PiI`	Successful	Successful	-	-
4.4.E3	${\displaystyle{\displaystyle\ln\left(-\mathrm{i}\right)=-\tfrac{1}{2}\pi% \mathrm{i}}}$	`ln(- I)= -(1)/(2)PiI`	`Log[- I]= -Divide[1,2]PiI`	Successful	Successful	-	-
4.4.E5	${\displaystyle{\displaystyle e^{+\pi\mathrm{i}}=-1}}$	`exp(+ Pi*I)= - 1`	`Exp[+ Pi*I]= - 1`	Successful	Successful	-	-
4.4.E5	${\displaystyle{\displaystyle e^{-\pi\mathrm{i}}=-1}}$	`exp(- Pi*I)= - 1`	`Exp[- Pi*I]= - 1`	Successful	Successful	-	-
4.4.E6	${\displaystyle{\displaystyle e^{+\pi\mathrm{i}/2}=+\mathrm{i}}}$	`exp(+ Pi*I/ 2)= + I`	`Exp[+ Pi*I/ 2]= + I`	Successful	Successful	-	-
4.4.E6	${\displaystyle{\displaystyle e^{-\pi\mathrm{i}/2}=-\mathrm{i}}}$	`exp(- Pi*I/ 2)= - I`	`Exp[- Pi*I/ 2]= - I`	Successful	Successful	-	-
4.4.E7	${\displaystyle{\displaystyle e^{2\pi k\mathrm{i}}=1}}$	`exp(2Pik*I)= 1`	`Exp[2Pik*I]= 1`	Successful	Failure	-	Successful
4.4.E8	${\displaystyle{\displaystyle e^{+\pi\mathrm{i}/3}=\frac{1}{2}+\mathrm{i}\frac{% \sqrt{3}}{2}}}$	`exp(+ PiI/ 3)=(1)/(2)+ I(sqrt(3))/(2)`	`Exp[+ PiI/ 3]=Divide[1,2]+ IDivide[Sqrt[3],2]`	Successful	Successful	-	-
4.4.E8	${\displaystyle{\displaystyle e^{-\pi\mathrm{i}/3}=\frac{1}{2}-\mathrm{i}\frac{% \sqrt{3}}{2}}}$	`exp(- PiI/ 3)=(1)/(2)- I(sqrt(3))/(2)`	`Exp[- PiI/ 3]=Divide[1,2]- IDivide[Sqrt[3],2]`	Successful	Successful	-	-
4.4.E9	${\displaystyle{\displaystyle e^{+2\pi\mathrm{i}/3}=-\frac{1}{2}+\mathrm{i}% \frac{\sqrt{3}}{2}}}$	`exp(+ 2PiI/ 3)= -(1)/(2)+ I*(sqrt(3))/(2)`	`Exp[+ 2PiI/ 3]= -Divide[1,2]+ I*Divide[Sqrt[3],2]`	Successful	Successful	-	-
4.4.E9	${\displaystyle{\displaystyle e^{-2\pi\mathrm{i}/3}=-\frac{1}{2}-\mathrm{i}% \frac{\sqrt{3}}{2}}}$	`exp(- 2PiI/ 3)= -(1)/(2)- I*(sqrt(3))/(2)`	`Exp[- 2PiI/ 3]= -Divide[1,2]- I*Divide[Sqrt[3],2]`	Successful	Successful	-	-
4.4.E10	${\displaystyle{\displaystyle e^{+\pi\mathrm{i}/4}=\frac{1}{\sqrt{2}}+\mathrm{i% }\frac{1}{\sqrt{2}}}}$	`exp(+ PiI/ 4)=(1)/(sqrt(2))+ I(1)/(sqrt(2))`	`Exp[+ PiI/ 4]=Divide[1,Sqrt[2]]+ IDivide[1,Sqrt[2]]`	Successful	Successful	-	-
4.4.E10	${\displaystyle{\displaystyle e^{-\pi\mathrm{i}/4}=\frac{1}{\sqrt{2}}-\mathrm{i% }\frac{1}{\sqrt{2}}}}$	`exp(- PiI/ 4)=(1)/(sqrt(2))- I(1)/(sqrt(2))`	`Exp[- PiI/ 4]=Divide[1,Sqrt[2]]- IDivide[1,Sqrt[2]]`	Successful	Successful	-	-
4.4.E11	${\displaystyle{\displaystyle e^{+3\pi\mathrm{i}/4}=-\frac{1}{\sqrt{2}}+\mathrm% {i}\frac{1}{\sqrt{2}}}}$	`exp(+ 3PiI/ 4)= -(1)/(sqrt(2))+ I*(1)/(sqrt(2))`	`Exp[+ 3PiI/ 4]= -Divide[1,Sqrt[2]]+ I*Divide[1,Sqrt[2]]`	Successful	Successful	-	-
4.4.E11	${\displaystyle{\displaystyle e^{-3\pi\mathrm{i}/4}=-\frac{1}{\sqrt{2}}-\mathrm% {i}\frac{1}{\sqrt{2}}}}$	`exp(- 3PiI/ 4)= -(1)/(sqrt(2))- I*(1)/(sqrt(2))`	`Exp[- 3PiI/ 4]= -Divide[1,Sqrt[2]]- I*Divide[1,Sqrt[2]]`	Successful	Successful	-	-
4.4.E12	${\displaystyle{\displaystyle{\mathrm{i}^{+\mathrm{i}}}=e^{-\pi/2}}}$	`(I)^(+ I)= exp(- Pi/ 2)`	`(I)^(+ I)= Exp[- Pi/ 2]`	Successful	Successful	-	-
4.4.E12	${\displaystyle{\displaystyle{\mathrm{i}^{-\mathrm{i}}}=e^{+\pi/2}}}$	`(I)^(- I)= exp(+ Pi/ 2)`	`(I)^(- I)= Exp[+ Pi/ 2]`	Successful	Successful	-	-
4.4.E13	${\displaystyle{\displaystyle\lim_{x\to\infty}x^{-a}\ln x=0}}$	`limit((x)^(- a)* ln(x), x = infinity)= 0`	`Limit[(x)^(- a)* Log[x], x -> Infinity]= 0`	Successful	Failure	-	Successful
4.4.E14	${\displaystyle{\displaystyle\lim_{x\to 0}x^{a}\ln x=0}}$	`limit((x)^(a)* ln(x), x = 0)= 0`	`Limit[(x)^(a)* Log[x], x -> 0]= 0`	Failure	Failure	Skip	Successful
4.4.E19	${\displaystyle{\displaystyle\lim_{n\to\infty}\left(\left(\sum^{n}_{k=1}\frac{1% }{k}\right)-\ln n\right)=\gamma}}$	`limit((sum((1)/(k), k = 1..n))- ln(n), n = infinity)= gamma`	`Limit[(Sum[Divide[1,k], {k, 1, n}])- Log[n], n -> Infinity]= EulerGamma`	Successful	Successful	-	-
4.5.E1	${\displaystyle{\displaystyle\frac{x}{1+x}<\ln\left(1+x\right)}}$	`(x)/(1 + x)< ln(1 + x)`	`Divide[x,1 + x]< Log[1 + x]`	Failure	Failure	Skip	Successful
4.5.E1	${\displaystyle{\displaystyle\ln\left(1+x\right)<x}}$	`ln(1 + x)< x`	`Log[1 + x]< x`	Failure	Failure	Skip	Successful
4.5.E2	${\displaystyle{\displaystyle x<-\ln\left(1-x\right)}}$	`x < - ln(1 - x)`	`x < - Log[1 - x]`	Failure	Failure	Skip	Successful
4.5.E2	${\displaystyle{\displaystyle-\ln\left(1-x\right)<\frac{x}{1-x}}}$	`- ln(1 - x)<(x)/(1 - x)`	`- Log[1 - x]<Divide[x,1 - x]`	Failure	Failure	Skip	Successful
4.5.E3	${\displaystyle{\displaystyle\|\ln\left(1-x\right)\|<\tfrac{3}{2}x}}$	`abs(ln(1 - x))<(3)/(2)*x`	`Abs[Log[1 - x]]<Divide[3,2]*x`	Failure	Failure	Error	Successful
4.5.E4	${\displaystyle{\displaystyle\ln x<=x-1}}$	`ln(x)< = x - 1`	`Log[x]< = x - 1`	Failure	Failure	Successful	Successful
4.5.E5	${\displaystyle{\displaystyle\ln x<=a(x^{1/a}-1)}}$	`ln(x)< = a*((x)^(1/ a)- 1)`	`Log[x]< = a*((x)^(1/ a)- 1)`	Failure	Failure	Successful	Successful
4.5.E6	${\displaystyle{\displaystyle\|\ln\left(1+z\right)\|<=-\ln\left(1-\|z\|\right)}}$	`abs(ln(1 + z))< = - ln(1 -abs(z))`	`Abs[Log[1 + z]]< = - Log[1 -Abs[z]]`	Failure	Failure	Successful	Successful
4.7.E1	${\displaystyle{\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}\ln z=\frac{1}{z}}}$	`diff(ln(z), z)=(1)/(z)`	`D[Log[z], z]=Divide[1,z]`	Successful	Successful	-	-
4.7.E2	${\displaystyle{\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}\operatorname{Ln}z=% \frac{1}{z}}}$	`diff(ln(z), z)=(1)/(z)`	`D[Log[z], z]=Divide[1,z]`	Successful	Successful	-	-
4.7.E3	${\displaystyle{\displaystyle\frac{{\mathrm{d}}^{n}}{{\mathrm{d}z}^{n}}\ln z=(-% 1)^{n-1}(n-1)!z^{-n}}}$	`diff(ln(z), [z$(n)])=(- 1)^(n - 1)factorial(n - 1)(z)^(- n)`	`D[Log[z], {z, n}]=(- 1)^(n - 1)(n - 1)!(z)^(- n)`	Failure	Failure	Successful	Successful
4.7.E4	${\displaystyle{\displaystyle\frac{{\mathrm{d}}^{n}}{{\mathrm{d}z}^{n}}% \operatorname{Ln}z=(-1)^{n-1}(n-1)!z^{-n}}}$	`diff(ln(z), [z$(n)])=(- 1)^(n - 1)factorial(n - 1)(z)^(- n)`	`D[Log[z], {z, n}]=(- 1)^(n - 1)(n - 1)!(z)^(- n)`	Failure	Failure	Successful	Successful
4.7.E7	${\displaystyle{\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}e^{z}=e^{z}}}$	`diff(exp(z), z)= exp(z)`	`D[Exp[z], z]= Exp[z]`	Successful	Successful	-	-
4.7.E8	${\displaystyle{\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}e^{az}=ae^{az}}}$	`diff(exp(az), z)= aexp(a*z)`	`D[Exp[az], z]= aExp[a*z]`	Successful	Successful	-	-
4.7.E9	${\displaystyle{\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}a^{z}=a^{z}\ln a}}$	`diff((a)^(z), z)= (a)^(z)* ln(a)`	`D[(a)^(z), z]= (a)^(z)* Log[a]`	Successful	Failure	-	Successful
4.7.E10	${\displaystyle{\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}z^{a}=az^{a-1}}}$	`diff((z)^(a), z)= a*(z)^(a - 1)`	`D[(z)^(a), z]= a*(z)^(a - 1)`	Successful	Successful	-	-
4.7.E14	${\displaystyle{\displaystyle\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}z}^{2}}=aw}}$	`diff(w, [z$(2)])= a*w`	`D[w, {z, 2}]= a*w`	Failure	Failure	Skip	Successful
4.8.E1	${\displaystyle{\displaystyle\operatorname{Ln}\left(z_{1}z_{2}\right)=% \operatorname{Ln}z_{1}+\operatorname{Ln}z_{2}}}$	`ln(z[1]*z[2])= ln(z[1])+ ln(z[2])`	`Log[Subscript[z, 1]*Subscript[z, 2]]= Log[Subscript[z, 1]]+ Log[Subscript[z, 2]]`	Failure	Failure	Fail `.4e-9+6.283185307I <- {z[1] = 2^(1/2)-I2^(1/2), z[2] = -2^(1/2)-I2^(1/2)}` `.4e-9+6.283185307I <- {z[1] = -2^(1/2)-I2^(1/2), z[2] = 2^(1/2)-I2^(1/2)}` `0.+6.283185307I <- {z[1] = -2^(1/2)-I2^(1/2), z[2] = -2^(1/2)-I2^(1/2)}` `0.-6.283185307I <- {z[1] = -2^(1/2)+I2^(1/2), z[2] = -2^(1/2)+I2^(1/2)}`	Fail `Complex[0.0, 6.283185307179586] <- {Rule[Subscript[z, 1], Times[Complex[1, -1], Power[2, Rational[1, 2]]]], Rule[Subscript[z, 2], Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}` `Complex[0.0, 6.283185307179586] <- {Rule[Subscript[z, 1], Times[Complex[-1, -1], Power[2, Rational[1, 2]]]], Rule[Subscript[z, 2], Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}` `Complex[0.0, 6.283185307179586] <- {Rule[Subscript[z, 1], Times[Complex[-1, -1], Power[2, Rational[1, 2]]]], Rule[Subscript[z, 2], Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}` `Complex[0.0, -6.283185307179586] <- {Rule[Subscript[z, 1], Times[Complex[-1, 1], Power[2, Rational[1, 2]]]], Rule[Subscript[z, 2], Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}`
4.8.E2	${\displaystyle{\displaystyle\ln\left(z_{1}z_{2}\right)=\ln z_{1}+\ln z_{2}}}$	`ln(z[1]*z[2])= ln(z[1])+ ln(z[2])`	`Log[Subscript[z, 1]*Subscript[z, 2]]= Log[Subscript[z, 1]]+ Log[Subscript[z, 2]]`	Failure	Failure	Skip	Fail `Complex[0.0, 6.283185307179586] <- {Rule[Subscript[z, 1], Times[Complex[1, -1], Power[2, Rational[1, 2]]]], Rule[Subscript[z, 2], Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}` `Complex[0.0, 6.283185307179586] <- {Rule[Subscript[z, 1], Times[Complex[-1, -1], Power[2, Rational[1, 2]]]], Rule[Subscript[z, 2], Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}`
4.8.E3	${\displaystyle{\displaystyle\operatorname{Ln}\frac{z_{1}}{z_{2}}=\operatorname% {Ln}z_{1}-\operatorname{Ln}z_{2}}}$	`ln((z[1])/(z[2]))= ln(z[1])- ln(z[2])`	`Log[Divide[Subscript[z, 1],Subscript[z, 2]]]= Log[Subscript[z, 1]]- Log[Subscript[z, 2]]`	Failure	Failure	Fail `0.+6.283185307I <- {z[1] = 2^(1/2)-I2^(1/2), z[2] = -2^(1/2)+I2^(1/2)}` `0.+6.283185307I <- {z[1] = -2^(1/2)-I2^(1/2), z[2] = 2^(1/2)+I2^(1/2)}` `0.+6.283185307I <- {z[1] = -2^(1/2)-I2^(1/2), z[2] = -2^(1/2)+I2^(1/2)}` `0.-6.283185307I <- {z[1] = -2^(1/2)+I2^(1/2), z[2] = -2^(1/2)-I2^(1/2)}`	Fail `Complex[0.0, 6.283185307179586] <- {Rule[Subscript[z, 1], Times[Complex[1, -1], Power[2, Rational[1, 2]]]], Rule[Subscript[z, 2], Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}` `Complex[0.0, 6.283185307179586] <- {Rule[Subscript[z, 1], Times[Complex[-1, -1], Power[2, Rational[1, 2]]]], Rule[Subscript[z, 2], Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}` `Complex[0.0, 6.283185307179586] <- {Rule[Subscript[z, 1], Times[Complex[-1, -1], Power[2, Rational[1, 2]]]], Rule[Subscript[z, 2], Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}` `Complex[0.0, -6.283185307179586] <- {Rule[Subscript[z, 1], Times[Complex[-1, 1], Power[2, Rational[1, 2]]]], Rule[Subscript[z, 2], Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}`
4.8.E4	${\displaystyle{\displaystyle\ln\frac{z_{1}}{z_{2}}=\ln z_{1}-\ln z_{2}}}$	`ln((z[1])/(z[2]))= ln(z[1])- ln(z[2])`	`Log[Divide[Subscript[z, 1],Subscript[z, 2]]]= Log[Subscript[z, 1]]- Log[Subscript[z, 2]]`	Failure	Failure	Skip	Fail `Complex[0.0, 6.283185307179586] <- {Rule[Subscript[z, 1], Times[Complex[1, -1], Power[2, Rational[1, 2]]]], Rule[Subscript[z, 2], Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}` `Complex[0.0, 6.283185307179586] <- {Rule[Subscript[z, 1], Times[Complex[-1, -1], Power[2, Rational[1, 2]]]], Rule[Subscript[z, 2], Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}`
4.8.E5	${\displaystyle{\displaystyle\operatorname{Ln}\left(z^{n}\right)=n\operatorname% {Ln}z}}$	`ln((z)^(n))= n*ln(z)`	`Log[(z)^(n)]= n*Log[z]`	Failure	Failure	Fail `0.+6.283185307I <- {z = -2^(1/2)-I2^(1/2), n = 3}` `0.-6.283185307I <- {z = -2^(1/2)+I2^(1/2), n = 3}`	Fail `Complex[0.0, 6.283185307179586] <- {Rule[n, 3], Rule[z, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}` `Complex[0.0, -6.283185307179586] <- {Rule[n, 3], Rule[z, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}`
4.8.E6	${\displaystyle{\displaystyle\ln\left(z^{n}\right)=n\ln z}}$	`ln((z)^(n))= n*ln(z)`	`Log[(z)^(n)]= n*Log[z]`	Failure	Failure	Skip	Successful
4.8.E7	${\displaystyle{\displaystyle\ln\frac{1}{z}=-\ln z}}$	`ln((1)/(z))= - ln(z)`	`Log[Divide[1,z]]= - Log[z]`	Failure	Failure	Skip	Successful
4.8.E8	${\displaystyle{\displaystyle\operatorname{Ln}\left(\exp z\right)=z+2k\pi% \mathrm{i}}}$	`ln(exp(z))= z + 2kPi*I`	`Log[Exp[z]]= z + 2kPi*I`	Failure	Failure	Fail `0.-18.84955592I <- {z = 2^(1/2)+I2^(1/2), k = 3}` `0.-18.84955592I <- {z = 2^(1/2)-I2^(1/2), k = 3}` `0.-18.84955592I <- {z = -2^(1/2)-I2^(1/2), k = 3}` `0.-18.84955592I <- {z = -2^(1/2)+I2^(1/2), k = 3}`	Fail `Complex[0.0, -18.84955592153876] <- {Rule[k, 3], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}` `Complex[0.0, -18.84955592153876] <- {Rule[k, 3], Rule[z, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}` `Complex[0.0, -18.84955592153876] <- {Rule[k, 3], Rule[z, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}` `Complex[0.0, -18.84955592153876] <- {Rule[k, 3], Rule[z, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}`
4.8.E9	${\displaystyle{\displaystyle\ln\left(\exp z\right)=z}}$	`ln(exp(z))= z`	`Log[Exp[z]]= z`	Failure	Failure	Skip	Successful
4.8.E10	${\displaystyle{\displaystyle\exp\left(\ln z\right)=\exp\left(\operatorname{Ln}% z\right)}}$	`exp(ln(z))= exp(ln(z))`	`Exp[Log[z]]= Exp[Log[z]]`	Successful	Successful	-	-
4.8.E10	${\displaystyle{\displaystyle\exp\left(\operatorname{Ln}z\right)=z}}$	`exp(ln(z))= z`	`Exp[Log[z]]= z`	Successful	Successful	-	-
4.8.E11	${\displaystyle{\displaystyle\operatorname{Ln}\left(a^{z}\right)=z\operatorname% {Ln}a+2k\pi\mathrm{i}}}$	`ln((a)^(z))= zln(a)+ 2kPiI`	`Log[(a)^(z)]= zLog[a]+ 2kPiI`	Failure	Failure	Fail `0.-18.84955592I <- {a = 2^(1/2)+I2^(1/2), z = 2^(1/2)+I2^(1/2), k = 3}` `0.-18.84955592I <- {a = 2^(1/2)+I2^(1/2), z = 2^(1/2)-I2^(1/2), k = 3}` `0.-18.84955592I <- {a = 2^(1/2)+I2^(1/2), z = -2^(1/2)-I2^(1/2), k = 3}` `0.-18.84955592I <- {a = 2^(1/2)+I2^(1/2), z = -2^(1/2)+I2^(1/2), k = 3}` ... skip entries to safe data	Fail `Complex[1.1102230246251565^-16, -18.84955592153876] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[k, 3], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}` `Complex[0.0, -18.84955592153876] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[k, 3], Rule[z, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}` `Complex[-1.3877787807814457^-16, -18.84955592153876] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[k, 3], Rule[z, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}` `Complex[4.440892098500626*^-16, -18.84955592153876] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[k, 3], Rule[z, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}` ... skip entries to safe data
4.8.E12	${\displaystyle{\displaystyle\ln\left(a^{z}\right)=z\ln a+2k\pi\mathrm{i}}}$	`ln((a)^(z))= zln(a)+ 2kPiI`	`Log[(a)^(z)]= zLog[a]+ 2kPiI`	Failure	Failure	Fail `0.-6.283185308I <- {a = 2^(1/2)+I2^(1/2), z = 2^(1/2)+I2^(1/2), k = 1}` `0.-12.56637062I <- {a = 2^(1/2)+I2^(1/2), z = 2^(1/2)+I2^(1/2), k = 2}` `0.-18.84955592I <- {a = 2^(1/2)+I2^(1/2), z = 2^(1/2)+I2^(1/2), k = 3}` `0.-6.283185308I <- {a = 2^(1/2)+I2^(1/2), z = 2^(1/2)-I2^(1/2), k = 1}` ... skip entries to safe data	Fail `Complex[1.1102230246251565^-16, -6.283185307179586] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[k, 1], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}` `Complex[1.1102230246251565^-16, -12.566370614359172] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[k, 2], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}` `Complex[1.1102230246251565*^-16, -18.84955592153876] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[k, 3], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}` `Complex[0.0, -6.283185307179586] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[k, 1], Rule[z, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}` ... skip entries to safe data
4.8.E13	${\displaystyle{\displaystyle\ln\left(a^{x}\right)=x\ln a}}$	`ln((a)^(x))= x*ln(a)`	`Log[(a)^(x)]= x*Log[a]`	Failure	Failure	Successful	Successful
4.10.E1	${\displaystyle{\displaystyle\int\frac{\mathrm{d}z}{z}=\ln z}}$	`int((1)/(z), z)= ln(z)`	`Integrate[Divide[1,z], z]= Log[z]`	Successful	Successful	-	-
4.10.E2	${\displaystyle{\displaystyle\int\ln z\mathrm{d}z=z\ln z-z}}$	`int(ln(z), z)= z*ln(z)- z`	`Integrate[Log[z], z]= z*Log[z]- z`	Successful	Successful	-	-
4.10.E3	${\displaystyle{\displaystyle\int z^{n}\ln z\mathrm{d}z=\frac{z^{n+1}}{n+1}\ln z% -\frac{z^{n+1}}{(n+1)^{2}}}}$	`int((z)^(n)* ln(z), z)=((z)^(n + 1))/(n + 1)*ln(z)-((z)^(n + 1))/((n + 1)^(2))`	`Integrate[(z)^(n)* Log[z], z]=Divide[(z)^(n + 1),n + 1]*Log[z]-Divide[(z)^(n + 1),(n + 1)^(2)]`	Successful	Successful	-	-
4.10.E4	${\displaystyle{\displaystyle\int\frac{\mathrm{d}z}{z\ln z}=\ln\left(\ln z% \right)}}$	`int((1)/(z*ln(z)), z)= ln(ln(z))`	`Integrate[Divide[1,z*Log[z]], z]= Log[Log[z]]`	Successful	Successful	-	-
4.10.E5	${\displaystyle{\displaystyle\int_{0}^{1}\frac{\ln t}{1-t}\mathrm{d}t=-\frac{% \pi^{2}}{6}}}$	`int((ln(t))/(1 - t), t = 0..1)= -((Pi)^(2))/(6)`	`Integrate[Divide[Log[t],1 - t], {t, 0, 1}]= -Divide[(Pi)^(2),6]`	Successful	Successful	-	-
4.10.E6	${\displaystyle{\displaystyle\int_{0}^{1}\frac{\ln t}{1+t}\mathrm{d}t=-\frac{% \pi^{2}}{12}}}$	`int((ln(t))/(1 + t), t = 0..1)= -((Pi)^(2))/(12)`	`Integrate[Divide[Log[t],1 + t], {t, 0, 1}]= -Divide[(Pi)^(2),12]`	Successful	Successful	-	-
4.10.E8	${\displaystyle{\displaystyle\int e^{az}\mathrm{d}z=\frac{e^{az}}{a}}}$	`int(exp(az), z)=(exp(az))/(a)`	`Integrate[Exp[az], z]=Divide[Exp[az],a]`	Successful	Successful	-	-
4.10.E9	${\displaystyle{\displaystyle\int\frac{\mathrm{d}z}{e^{az}+b}=\frac{1}{ab}(az-% \ln\left(e^{az}+b\right))}}$	`int((1)/(exp(az)+ b), z)=(1)/(ab)(az - ln(exp(a*z)+ b))`	`Integrate[Divide[1,Exp[az]+ b], z]=Divide[1,ab](az - Log[Exp[a*z]+ b])`	Failure	Successful	Skip	-
4.10.E10	${\displaystyle{\displaystyle\int\frac{e^{az}-1}{e^{az}+1}\mathrm{d}z=\frac{2}{% a}\ln\left(e^{az/2}+e^{-az/2}\right)}}$	`int((exp(az)- 1)/(exp(az)+ 1), z)=(2)/(a)ln(exp(az/ 2)+ exp(- a*z/ 2))`	`Integrate[Divide[Exp[az]- 1,Exp[az]+ 1], z]=Divide[2,a]Log[Exp[az/ 2]+ Exp[- a*z/ 2]]`	Failure	Failure	Skip	Fail `Complex[-4.442882938158366, -4.442882938158366] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}` `Complex[4.442882938158366, -4.442882938158366] <- {Rule[a, Times[Complex[1, -1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}` `Complex[4.442882938158366, 4.442882938158366] <- {Rule[a, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}` `Complex[-4.442882938158366, 4.442882938158366] <- {Rule[a, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}`
4.10.E11	${\displaystyle{\displaystyle\int_{-\infty}^{\infty}e^{-cx^{2}}\mathrm{d}x=% \sqrt{\frac{\pi}{c}}}}$	`int(exp(- c*(x)^(2)), x = - infinity..infinity)=sqrt((Pi)/(c))`	`Integrate[Exp[- c*(x)^(2)], {x, - Infinity, Infinity}]=Sqrt[Divide[Pi,c]]`	Successful	Failure	-	Skip
4.10.E12	${\displaystyle{\displaystyle\int_{0}^{\ln 2}\frac{xe^{x}}{e^{x}-1}\mathrm{d}x=% \frac{\pi^{2}}{12}}}$	`int((x*exp(x))/(exp(x)- 1), x = 0..ln(2))=((Pi)^(2))/(12)`	`Integrate[Divide[x*Exp[x],Exp[x]- 1], {x, 0, Log[2]}]=Divide[(Pi)^(2),12]`	Successful	Successful	-	-
4.10.E13	${\displaystyle{\displaystyle\int_{0}^{\infty}\frac{\mathrm{d}x}{e^{x}+1}=\ln 2}}$	`int((1)/(exp(x)+ 1), x = 0..infinity)= ln(2)`	`Integrate[Divide[1,Exp[x]+ 1], {x, 0, Infinity}]= Log[2]`	Successful	Successful	-	-
4.12.E6	${\displaystyle{\displaystyle\phi(x)=\ln\left(x+1\right)}}$	`phi*(x)= ln(x + 1)`	`\[Phi]*(x)= Log[x + 1]`	Failure	Failure	Skip	Successful
4.12.E9	${\displaystyle{\displaystyle\psi(x)=\ell+{\ln^{(\ell)}}x}}$	`psi*(x)= ell + subs( temp=x, diff( ln(temp), temp$(ell) ) )`	`\[Psi]*(x)= \[ScriptL]+ (D[Log[temp], {temp, \[ScriptL]}]/.temp-> x)`	Failure	Failure	Fail `.454653676+2.121320343I <- {psi = 2^(1/2)+I2^(1/2), ell = 1, x = 3/2}` `.565764787+2.121320343I <- {psi = 2^(1/2)+I2^(1/2), ell = 2, x = 3/2}` `-1.471272250+2.121320343I <- {psi = 2^(1/2)+I2^(1/2), ell = 3, x = 3/2}` `.454653676-2.121320343I <- {psi = 2^(1/2)-I2^(1/2), ell = 1, x = 3/2}` ... skip entries to safe data	Fail `Complex[0.45465367689297564, 2.1213203435596424] <- {Rule[x, Rational[3, 2]], Rule[ℓ, 1], Rule[ψ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}` `Complex[0.5657647880040868, 2.1213203435596424] <- {Rule[x, Rational[3, 2]], Rule[ℓ, 2], Rule[ψ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}` `Complex[-1.4712722490329502, 2.1213203435596424] <- {Rule[x, Rational[3, 2]], Rule[ℓ, 3], Rule[ψ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}` `Complex[0.45465367689297564, -2.1213203435596424] <- {Rule[x, Rational[3, 2]], Rule[ℓ, 1], Rule[ψ, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}` ... skip entries to safe data
4.13.E1	${\displaystyle{\displaystyle We^{W}=x}}$	`W*exp(W)= x`	`W*Exp[W]= x`	Failure	Failure	Fail `-5.838722068+6.652975529I <- {W = 2^(1/2)+I2^(1/2), x = 1}` `-6.838722068+6.652975529I <- {W = 2^(1/2)+I2^(1/2), x = 2}` `-7.838722068+6.652975529I <- {W = 2^(1/2)+I2^(1/2), x = 3}` `-5.838722068-6.652975529I <- {W = 2^(1/2)-I2^(1/2), x = 1}` ... skip entries to safe data	Fail `Complex[-5.838722072781763, 6.652975531039188] <- {Rule[W, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[x, 1]}` `Complex[-6.838722072781763, 6.652975531039188] <- {Rule[W, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[x, 2]}` `Complex[-7.838722072781763, 6.652975531039188] <- {Rule[W, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[x, 3]}` `Complex[-5.838722072781763, -6.652975531039188] <- {Rule[W, Times[Complex[1, -1], Power[2, Rational[1, 2]]]], Rule[x, 1]}` ... skip entries to safe data
4.13#Ex1	${\displaystyle{\displaystyle\mathrm{Wp}\left(-1/e\right)=\mathrm{Wm}\left(-1/e% \right)}}$	`LambertW(0, - 1/ exp(1))= LambertW(-1, - 1/ exp(1))`	`ProductLog[0, - 1/ E]= ProductLog[-1, - 1/ E]`	Successful	Successful	-	-
4.13#Ex1	${\displaystyle{\displaystyle\mathrm{Wm}\left(-1/e\right)=-1}}$	`LambertW(-1, - 1/ exp(1))= - 1`	`ProductLog[-1, - 1/ E]= - 1`	Successful	Successful	-	-
4.13#Ex2	${\displaystyle{\displaystyle\mathrm{Wp}\left(0\right)=0}}$	`LambertW(0, 0)= 0`	`ProductLog[0, 0]= 0`	Successful	Successful	-	-
4.13#Ex3	${\displaystyle{\displaystyle\mathrm{Wp}\left(e\right)=1}}$	`LambertW(0, exp(1))= 1`	`ProductLog[0, E]= 1`	Successful	Successful	-	-
4.13#Ex4	${\displaystyle{\displaystyle U+\ln U=x}}$	`U + ln(U)= x`	`U + Log[U]= x`	Failure	Failure	Fail `1.107360742+2.199611725I <- {U = 2^(1/2)+I2^(1/2), x = 1}` `.107360742+2.199611725I <- {U = 2^(1/2)+I2^(1/2), x = 2}` `-.892639258+2.199611725I <- {U = 2^(1/2)+I2^(1/2), x = 3}` `1.107360742-2.199611725I <- {U = 2^(1/2)-I2^(1/2), x = 1}` ... skip entries to safe data	Fail `Complex[1.1073607429330403, 2.199611725770543] <- {Rule[U, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[x, 1]}` `Complex[0.10736074293304043, 2.199611725770543] <- {Rule[U, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[x, 2]}` `Complex[-0.8926392570669596, 2.199611725770543] <- {Rule[U, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[x, 3]}` `Complex[1.1073607429330403, -2.199611725770543] <- {Rule[U, Times[Complex[1, -1], Power[2, Rational[1, 2]]]], Rule[x, 1]}` ... skip entries to safe data
4.13#Ex5	${\displaystyle{\displaystyle U=U(x)}}$	`U = U*(x)`	`U = U*(x)`	Failure	Failure	Fail `-1.414213562-1.414213562I <- {U = 2^(1/2)+I2^(1/2), x = 2}` `-2.828427124-2.828427124I <- {U = 2^(1/2)+I2^(1/2), x = 3}` `-1.414213562+1.414213562I <- {U = 2^(1/2)-I2^(1/2), x = 2}` `-2.828427124+2.828427124I <- {U = 2^(1/2)-I2^(1/2), x = 3}` ... skip entries to safe data	Fail `Complex[-1.4142135623730951, -1.4142135623730951] <- {Rule[U, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[x, 2]}` `Complex[-2.8284271247461903, -2.8284271247461903] <- {Rule[U, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[x, 3]}` `Complex[-1.4142135623730951, 1.4142135623730951] <- {Rule[U, Times[Complex[1, -1], Power[2, Rational[1, 2]]]], Rule[x, 2]}` `Complex[-2.8284271247461903, 2.8284271247461903] <- {Rule[U, Times[Complex[1, -1], Power[2, Rational[1, 2]]]], Rule[x, 3]}` ... skip entries to safe data
4.13#Ex5	${\displaystyle{\displaystyle U(x)=W\left(e^{x}\right)}}$	`U*(x)= LambertW(exp(x))`	`U*(x)= ProductLog[Exp[x]]`	Failure	Failure	Fail `.414213562+1.414213562I <- {U = 2^(1/2)+I2^(1/2), x = 1}` `1.271281525+2.828427124I <- {U = 2^(1/2)+I2^(1/2), x = 2}` `2.034700655+4.242640686I <- {U = 2^(1/2)+I2^(1/2), x = 3}` `.414213562-1.414213562I <- {U = 2^(1/2)-I2^(1/2), x = 1}` ... skip entries to safe data	Fail `Complex[0.41421356237309515, 1.4142135623730951] <- {Rule[U, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[x, 1]}` `Complex[1.2712815257485788, 2.8284271247461903] <- {Rule[U, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[x, 2]}` `Complex[2.0347006555499627, 4.242640687119286] <- {Rule[U, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[x, 3]}` `Complex[0.41421356237309515, -1.4142135623730951] <- {Rule[U, Times[Complex[1, -1], Power[2, Rational[1, 2]]]], Rule[x, 1]}` ... skip entries to safe data
4.13.E4	${\displaystyle{\displaystyle\frac{\mathrm{d}W}{\mathrm{d}x}=\frac{e^{-W}}{1+W}}}$	`diff(LambertW(x), =)*(exp(- LambertW($0)))/(1 + LambertW($0))`	`D[ProductLog[x], =]*Divide[Exp[- ProductLog[$0]],1 + ProductLog[$0]]`	Error	Error	-	-
4.13.E5	${\displaystyle{\displaystyle\mathrm{Wp}\left(x\right)=\sum_{n=1}^{\infty}(-1)^% {n-1}\frac{n^{n-2}}{(n-1)!}x^{n}}}$	`LambertW(0, x)= sum((- 1)^(n - 1)((n)^(n - 2))/(factorial(n - 1))(x)^(n), n = 1..infinity)`	`ProductLog[0, x]= Sum[(- 1)^(n - 1)Divide[(n)^(n - 2),(n - 1)!](x)^(n), {n, 1, Infinity}]`	Failure	Successful	Skip	-
4.13.E6	${\displaystyle{\displaystyle W\left(-e^{-1-(t^{2}/2)}\right)=\sum_{n=0}^{% \infty}(-1)^{n-1}c_{n}t^{n}}}$	`LambertW(- exp(- 1 -((t)^(2)/ 2)))= sum((- 1)^(n - 1)* c[n]*(t)^(n), n = 0..infinity)`	`ProductLog[- Exp[- 1 -((t)^(2)/ 2)]]= Sum[(- 1)^(n - 1)* Subscript[c, n]*(t)^(n), {n, 0, Infinity}]`	Failure	Failure	Skip	Successful
4.14.E1	${\displaystyle{\displaystyle\sin z=\frac{e^{\mathrm{i}z}-e^{-\mathrm{i}z}}{2% \mathrm{i}}}}$	`sin(z)=(exp(Iz)- exp(- Iz))/(2*I)`	`Sin[z]=Divide[Exp[Iz]- Exp[- Iz],2*I]`	Successful	Successful	-	-
4.14.E2	${\displaystyle{\displaystyle\cos z=\frac{e^{\mathrm{i}z}+e^{-\mathrm{i}z}}{2}}}$	`cos(z)=(exp(Iz)+ exp(- Iz))/(2)`	`Cos[z]=Divide[Exp[Iz]+ Exp[- Iz],2]`	Successful	Successful	-	-
4.14.E3	${\displaystyle{\displaystyle\cos z+i\sin z=e^{+iz}}}$	`cos(z)+ Isin(z)= exp(+ Iz)`	`Cos[z]+ ISin[z]= Exp[+ Iz]`	Successful	Successful	-	-
4.14.E3	${\displaystyle{\displaystyle\cos z-i\sin z=e^{-iz}}}$	`cos(z)- Isin(z)= exp(- Iz)`	`Cos[z]- ISin[z]= Exp[- Iz]`	Successful	Successful	-	-
4.14.E4	${\displaystyle{\displaystyle\tan z=\frac{\sin z}{\cos z}}}$	`tan(z)=(sin(z))/(cos(z))`	`Tan[z]=Divide[Sin[z],Cos[z]]`	Successful	Successful	-	-
4.14.E5	${\displaystyle{\displaystyle\csc z=\frac{1}{\sin z}}}$	`csc(z)=(1)/(sin(z))`	`Csc[z]=Divide[1,Sin[z]]`	Successful	Successful	-	-
4.14.E6	${\displaystyle{\displaystyle\sec z=\frac{1}{\cos z}}}$	`sec(z)=(1)/(cos(z))`	`Sec[z]=Divide[1,Cos[z]]`	Successful	Successful	-	-
4.14.E7	${\displaystyle{\displaystyle\cot z=\frac{\cos z}{\sin z}}}$	`cot(z)=(cos(z))/(sin(z))`	`Cot[z]=Divide[Cos[z],Sin[z]]`	Successful	Successful	-	-
4.14.E7	${\displaystyle{\displaystyle\frac{\cos z}{\sin z}=\frac{1}{\tan z}}}$	`(cos(z))/(sin(z))=(1)/(tan(z))`	`Divide[Cos[z],Sin[z]]=Divide[1,Tan[z]]`	Successful	Successful	-	-
4.14.E8	${\displaystyle{\displaystyle\sin\left(z+2k\pi\right)=\sin z}}$	`sin(z + 2kPi)= sin(z)`	`Sin[z + 2kPi]= Sin[z]`	Failure	Failure	Successful	Successful
4.14.E9	${\displaystyle{\displaystyle\cos\left(z+2k\pi\right)=\cos z}}$	`cos(z + 2kPi)= cos(z)`	`Cos[z + 2kPi]= Cos[z]`	Failure	Failure	Successful	Successful
4.14.E10	${\displaystyle{\displaystyle\tan\left(z+k\pi\right)=\tan z}}$	`tan(z + k*Pi)= tan(z)`	`Tan[z + k*Pi]= Tan[z]`	Failure	Failure	Successful	Successful
4.15.E1	${\displaystyle{\displaystyle\cos\left(x+iy\right)=\sin\left(x+\tfrac{1}{2}\pi+% iy\right)}}$	`cos(x + Iy)= sin(x +(1)/(2)Pi + I*y)`	`Cos[x + Iy]= Sin[x +Divide[1,2]Pi + I*y]`	Successful	Successful	-	-
4.15.E2	${\displaystyle{\displaystyle\cot\left(x+iy\right)=-\tan\left(x+\tfrac{1}{2}\pi% +iy\right)}}$	`cot(x + Iy)= - tan(x +(1)/(2)Pi + I*y)`	`Cot[x + Iy]= - Tan[x +Divide[1,2]Pi + I*y]`	Successful	Successful	-	-
4.15.E3	${\displaystyle{\displaystyle\sec\left(x+iy\right)=\csc\left(x+\tfrac{1}{2}\pi+% iy\right)}}$	`sec(x + Iy)= csc(x +(1)/(2)Pi + I*y)`	`Sec[x + Iy]= Csc[x +Divide[1,2]Pi + I*y]`	Successful	Successful	-	-
4.17.E1	${\displaystyle{\displaystyle\lim_{z\to 0}\frac{\sin z}{z}=1}}$	`limit((sin(z))/(z), z = 0)= 1`	`Limit[Divide[Sin[z],z], z -> 0]= 1`	Successful	Successful	-	-
4.17.E2	${\displaystyle{\displaystyle\lim_{z\to 0}\frac{\tan z}{z}=1}}$	`limit((tan(z))/(z), z = 0)= 1`	`Limit[Divide[Tan[z],z], z -> 0]= 1`	Successful	Successful	-	-
4.17.E3	${\displaystyle{\displaystyle\lim_{z\to 0}\frac{1-\cos z}{z^{2}}=\frac{1}{2}}}$	`limit((1 - cos(z))/((z)^(2)), z = 0)=(1)/(2)`	`Limit[Divide[1 - Cos[z],(z)^(2)], z -> 0]=Divide[1,2]`	Successful	Successful	-	-
4.18.E1	${\displaystyle{\displaystyle\frac{2x}{\pi}<=\sin x}}$	`(2*x)/(Pi)< = sin(x)`	`Divide[2*x,Pi]< = Sin[x]`	Failure	Failure	Skip	Successful
4.18.E1	${\displaystyle{\displaystyle\sin x<=x}}$	`sin(x)< = x`	`Sin[x]< = x`	Failure	Failure	Skip	Successful
4.18.E2	${\displaystyle{\displaystyle x<=\tan x}}$	`x < = tan(x)`	`x < = Tan[x]`	Failure	Failure	Skip	Successful
4.18.E3	${\displaystyle{\displaystyle\cos x<=\frac{\sin x}{x}}}$	`cos(x)< =(sin(x))/(x)`	`Cos[x]< =Divide[Sin[x],x]`	Failure	Failure	Skip	Successful
4.18.E3	${\displaystyle{\displaystyle\frac{\sin x}{x}<=1}}$	`(sin(x))/(x)< = 1`	`Divide[Sin[x],x]< = 1`	Failure	Failure	Skip	Successful
4.18.E4	${\displaystyle{\displaystyle\pi<\frac{\sin\left(\pi x\right)}{x(1-x)}}}$	`Pi <(sin(Pix))/(x(1 - x))`	`Pi <Divide[Sin[Pix],x(1 - x)]`	Failure	Failure	Successful	Successful
4.18.E4	${\displaystyle{\displaystyle\frac{\sin\left(\pi x\right)}{x(1-x)}<=4}}$	`(sin(Pix))/(x(1 - x))< = 4`	`Divide[Sin[Pix],x(1 - x)]< = 4`	Failure	Failure	Successful	Successful
4.18.E5	${\displaystyle{\displaystyle\|\sinh y\|<=\|\sin z\|\leq\cosh y}}$	`abs(sinh(y))< =abs(sin(z))<= cosh(y)`	`Abs[Sinh[y]]< =Abs[Sin[z]]<= Cosh[y]`	Failure	Failure	Error	Successful
4.18.E6	${\displaystyle{\displaystyle\|\sinh y\|<=\|\cos z\|\leq\cosh y}}$	`abs(sinh(y))< =abs(cos(z))<= cosh(y)`	`Abs[Sinh[y]]< =Abs[Cos[z]]<= Cosh[y]`	Failure	Failure	Error	Successful
4.18.E7	${\displaystyle{\displaystyle\|\csc z\|<=\operatorname{csch}\|y\|}}$	`abs(csc(z))< = csch(abs(y))`	`Abs[Csc[z]]< = Csch[Abs[y]]`	Failure	Failure	Fail `.4602792559 <= .2757205648 <- {z = 2^(1/2)+I2^(1/2), y = 2}` `.4602792559 <= .9982156967e-1 <- {z = 2^(1/2)+I2^(1/2), y = 3}` `.4602792559 <= .2757205648 <- {z = 2^(1/2)-I2^(1/2), y = 2}` `.4602792559 <= .9982156967e-1 <- {z = 2^(1/2)-I2^(1/2), y = 3}` ... skip entries to safe data	Successful
4.18.E8	${\displaystyle{\displaystyle\|\cos z\|<=\cosh\|z\|}}$	`abs(cos(z))< = cosh(abs(z))`	`Abs[Cos[z]]< = Cosh[Abs[z]]`	Failure	Failure	Successful	Successful
4.18.E9	${\displaystyle{\displaystyle\|\sin z\|<=\sinh\|z\|}}$	`abs(sin(z))< = sinh(abs(z))`	`Abs[Sin[z]]< = Sinh[Abs[z]]`	Failure	Failure	Successful	Successful
4.18#Ex1	${\displaystyle{\displaystyle\|\cos z\|<2}}$	`abs(cos(z))< 2`	`Abs[Cos[z]]< 2`	Failure	Failure	Successful	Successful
4.18#Ex2	${\displaystyle{\displaystyle\|\sin z\|<=\tfrac{6}{5}\|z\|}}$	`abs(sin(z))< =(6)/(5)*abs(z)`	`Abs[Sin[z]]< =Divide[6,5]*Abs[z]`	Failure	Failure	Successful	Successful
4.19.E7	${\displaystyle{\displaystyle\ln\left(\frac{\sin z}{z}\right)=\sum_{n=1}^{% \infty}\frac{(-1)^{n}2^{2n-1}B_{2n}}{n(2n)!}z^{2n}}}$	`ln((sin(z))/(z))= sum(((- 1)^(n)* (2)^(2n - 1) bernoulli(2n))/(nfactorial(2n))(z)^(2*n), n = 1..infinity)`	`Log[Divide[Sin[z],z]]= Sum[Divide[(- 1)^(n)* (2)^(2n - 1) BernoulliB[2n],n(2n)!](z)^(2*n), {n, 1, Infinity}]`	Failure	Failure	Skip	Successful
4.19.E8	${\displaystyle{\displaystyle\ln\left(\cos z\right)=\sum_{n=1}^{\infty}\frac{(-% 1)^{n}2^{2n-1}(2^{2n}-1)B_{2n}}{n(2n)!}z^{2n}}}$	`ln(cos(z))= sum(((- 1)^(n)* (2)^(2n - 1)((2)^(2n)- 1) bernoulli(2n))/(nfactorial(2n))(z)^(2*n), n = 1..infinity)`	`Log[Cos[z]]= Sum[Divide[(- 1)^(n)* (2)^(2n - 1)((2)^(2n)- 1) BernoulliB[2n],n(2n)!](z)^(2*n), {n, 1, Infinity}]`	Failure	Failure	Skip	Successful
4.19.E9	${\displaystyle{\displaystyle\ln\left(\frac{\tan z}{z}\right)=\sum_{n=1}^{% \infty}\frac{(-1)^{n-1}2^{2n}(2^{2n-1}-1)B_{2n}}{n(2n)!}z^{2n}}}$	`ln((tan(z))/(z))= sum(((- 1)^(n - 1)* (2)^(2n)((2)^(2n - 1)- 1) bernoulli(2n))/(nfactorial(2n))(z)^(2*n), n = 1..infinity)`	`Log[Divide[Tan[z],z]]= Sum[Divide[(- 1)^(n - 1)* (2)^(2n)((2)^(2n - 1)- 1) BernoulliB[2n],n(2n)!](z)^(2*n), {n, 1, Infinity}]`	Failure	Failure	Skip	Successful
4.20.E1	${\displaystyle{\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}\sin z=\cos z}}$	`diff(sin(z), z)= cos(z)`	`D[Sin[z], z]= Cos[z]`	Successful	Successful	-	-
4.20.E2	${\displaystyle{\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}\cos z=-\sin z}}$	`diff(cos(z), z)= - sin(z)`	`D[Cos[z], z]= - Sin[z]`	Successful	Successful	-	-
4.20.E3	${\displaystyle{\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}\tan z={\sec^{2}}z}}$	`diff(tan(z), z)= (sec(z))^(2)`	`D[Tan[z], z]= (Sec[z])^(2)`	Successful	Successful	-	-
4.20.E4	${\displaystyle{\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}\csc z=-\csc z\cot z}}$	`diff(csc(z), z)= - csc(z)*cot(z)`	`D[Csc[z], z]= - Csc[z]*Cot[z]`	Successful	Successful	-	-
4.20.E5	${\displaystyle{\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}\sec z=\sec z\tan z}}$	`diff(sec(z), z)= sec(z)*tan(z)`	`D[Sec[z], z]= Sec[z]*Tan[z]`	Successful	Successful	-	-
4.20.E6	${\displaystyle{\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}\cot z=-{\csc^{2}}z}}$	`diff(cot(z), z)= - (csc(z))^(2)`	`D[Cot[z], z]= - (Csc[z])^(2)`	Successful	Successful	-	-
4.20.E7	${\displaystyle{\displaystyle\frac{{\mathrm{d}}^{n}}{{\mathrm{d}z}^{n}}\sin z=% \sin\left(z+\tfrac{1}{2}n\pi\right)}}$	`diff(sin(z), [z$(n)])= sin(z +(1)/(2)nPi)`	`D[Sin[z], {z, n}]= Sin[z +Divide[1,2]nPi]`	Successful	Successful	-	-
4.20.E8	${\displaystyle{\displaystyle\frac{{\mathrm{d}}^{n}}{{\mathrm{d}z}^{n}}\cos z=% \cos\left(z+\tfrac{1}{2}n\pi\right)}}$	`diff(cos(z), [z$(n)])= cos(z +(1)/(2)nPi)`	`D[Cos[z], {z, n}]= Cos[z +Divide[1,2]nPi]`	Successful	Successful	-	-
4.20.E9	${\displaystyle{\displaystyle\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}z}^{2}}+a^{2}w% =0}}$	`diff(w, [z$(2)])+ (a)^(2)* w = 0`	`D[w, {z, 2}]+ (a)^(2)* w = 0`	Failure	Failure	Fail `-5.656854245+5.656854245I <- {a = 2^(1/2)+I2^(1/2), w = 2^(1/2)+I2^(1/2)}` `5.656854245+5.656854245I <- {a = 2^(1/2)+I2^(1/2), w = 2^(1/2)-I2^(1/2)}` `5.656854245-5.656854245I <- {a = 2^(1/2)+I2^(1/2), w = -2^(1/2)-I2^(1/2)}` `-5.656854245-5.656854245I <- {a = 2^(1/2)+I2^(1/2), w = -2^(1/2)+I2^(1/2)}` ... skip entries to safe data	Fail `Complex[-5.656854249492381, 5.656854249492381] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}` `Complex[5.656854249492381, 5.656854249492381] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[w, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}` `Complex[5.656854249492381, -5.656854249492381] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[w, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}` `Complex[-5.656854249492381, -5.656854249492381] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[w, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}` ... skip entries to safe data
4.20.E10	${\displaystyle{\displaystyle\left(\frac{\mathrm{d}w}{\mathrm{d}z}\right)^{2}+a% ^{2}w^{2}=1}}$	`(diff(w, z))^(2)+ (a)^(2)* (w)^(2)= 1`	`(D[w, z])^(2)+ (a)^(2)* (w)^(2)= 1`	Failure	Failure	Fail `-16.99999998+0.I <- {a = 2^(1/2)+I2^(1/2), w = 2^(1/2)+I2^(1/2)}` `14.99999998+0.I <- {a = 2^(1/2)+I2^(1/2), w = 2^(1/2)-I2^(1/2)}` `-16.99999998+0.I <- {a = 2^(1/2)+I2^(1/2), w = -2^(1/2)-I2^(1/2)}` `14.99999998+0.I <- {a = 2^(1/2)+I2^(1/2), w = -2^(1/2)+I2^(1/2)}` ... skip entries to safe data	Fail `-17.0 <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}` `15.0 <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[w, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}` `-17.0 <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[w, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}` `15.0 <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[w, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}` ... skip entries to safe data
4.20.E11	${\displaystyle{\displaystyle\frac{\mathrm{d}w}{\mathrm{d}z}-a^{2}w^{2}=1}}$	`diff(w, z)- (a)^(2)* (w)^(2)= 1`	`D[w, z]- (a)^(2)* (w)^(2)= 1`	Failure	Failure	Fail `14.99999998-0.I <- {a = 2^(1/2)+I2^(1/2), w = 2^(1/2)+I2^(1/2)}` `-16.99999998-0.I <- {a = 2^(1/2)+I2^(1/2), w = 2^(1/2)-I2^(1/2)}` `14.99999998-0.I <- {a = 2^(1/2)+I2^(1/2), w = -2^(1/2)-I2^(1/2)}` `-16.99999998-0.I <- {a = 2^(1/2)+I2^(1/2), w = -2^(1/2)+I2^(1/2)}` ... skip entries to safe data	Fail `15.0 <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}` `-17.0 <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[w, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}` `15.0 <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[w, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}` `-17.0 <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[w, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}` ... skip entries to safe data
4.20.E12	${\displaystyle{\displaystyle w=A\cos\left(az\right)+B\sin\left(az\right)}}$	`w = Acos(az)+ Bsin(az)`	`w = ACos[az]+ BSin[az]`	Failure	Failure	Skip	Skip
4.20.E13	${\displaystyle{\displaystyle w=(1/a)\sin\left(az+c\right)}}$	`w =(1/ a)* sin(a*z + c)`	`w =(1/ a)* Sin[a*z + c]`	Failure	Failure	Fail `-43.99146068+34.43827298I <- {a = 2^(1/2)+I2^(1/2), c = 2^(1/2)+I2^(1/2), w = 2^(1/2)+I2^(1/2), z = 2^(1/2)+I2^(1/2)}` `1.560620374+.384416402I <- {a = 2^(1/2)+I2^(1/2), c = 2^(1/2)+I2^(1/2), w = 2^(1/2)+I2^(1/2), z = 2^(1/2)-I2^(1/2)}` `-.552876601+4.108989171I <- {a = 2^(1/2)+I2^(1/2), c = 2^(1/2)+I2^(1/2), w = 2^(1/2)+I2^(1/2), z = -2^(1/2)-I2^(1/2)}` `2.401710418+1.589052846I <- {a = 2^(1/2)+I2^(1/2), c = 2^(1/2)+I2^(1/2), w = 2^(1/2)+I2^(1/2), z = -2^(1/2)+I2^(1/2)}` ... skip entries to safe data	Fail `Complex[-43.991460739515965, 34.43827305491785] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[c, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}` `Complex[1.5606203716754656, 0.38441640190707305] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[c, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}` `Complex[-0.5528766038746884, 4.1089891749071095] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[c, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}` `Complex[2.4017104180648507, 1.5890528479992119] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[c, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}` ... skip entries to safe data
4.20.E14	${\displaystyle{\displaystyle w=(1/a)\tan\left(az+c\right)}}$	`w =(1/ a)* tan(a*z + c)`	`w =(1/ a)* Tan[a*z + c]`	Failure	Failure	Fail `1.060642513+1.060651152I <- {a = 2^(1/2)+I2^(1/2), c = 2^(1/2)+I2^(1/2), w = 2^(1/2)+I2^(1/2), z = 2^(1/2)+I2^(1/2)}` `1.097992560+1.014214371I <- {a = 2^(1/2)+I2^(1/2), c = 2^(1/2)+I2^(1/2), w = 2^(1/2)+I2^(1/2), z = 2^(1/2)-I2^(1/2)}` `1.770353735+1.772853405I <- {a = 2^(1/2)+I2^(1/2), c = 2^(1/2)+I2^(1/2), w = 2^(1/2)+I2^(1/2), z = -2^(1/2)-I2^(1/2)}` `1.045061232+1.116024928I <- {a = 2^(1/2)+I2^(1/2), c = 2^(1/2)+I2^(1/2), w = 2^(1/2)+I2^(1/2), z = -2^(1/2)+I2^(1/2)}` ... skip entries to safe data	Fail `Complex[1.0606425136739976, 1.0606511525471942] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[c, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}` `Complex[1.0979925605963208, 1.0142143722877455] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[c, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}` `Complex[1.7703537351803704, 1.7728534052480869] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[c, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}` `Complex[1.0450612330665354, 1.11602492841073] <- {Rule[a, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[c, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[w, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[z, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}` ... skip entries to safe data
4.21.E1	${\displaystyle{\displaystyle\sin u+\cos u=\sqrt{2}\sin\left(u+\tfrac{1}{4}\pi% \right)}}$	`sin(u)+ cos(u)=sqrt(2)sin(u +(1)/(4)Pi)`	`Sin[u]+ Cos[u]=Sqrt[2]Sin[u +Divide[1,4]Pi]`	Successful	Successful	-	-
4.21.E1	${\displaystyle{\displaystyle\sin u-\cos u=\sqrt{2}\sin\left(u-\tfrac{1}{4}\pi% \right)}}$	`sin(u)- cos(u)=sqrt(2)sin(u -(1)/(4)Pi)`	`Sin[u]- Cos[u]=Sqrt[2]Sin[u -Divide[1,4]Pi]`	Successful	Successful	-	-
4.21.E1	${\displaystyle{\displaystyle\sqrt{2}\sin\left(u+\tfrac{1}{4}\pi\right)=+\sqrt{% 2}\cos\left(u-\tfrac{1}{4}\pi\right)}}$	`sqrt(2)sin(u +(1)/(4)Pi)= +sqrt(2)cos(u -(1)/(4)Pi)`	`Sqrt[2]Sin[u +Divide[1,4]Pi]= +Sqrt[2]Cos[u -Divide[1,4]Pi]`	Successful	Successful	-	-
4.21.E1	${\displaystyle{\displaystyle\sqrt{2}\sin\left(u-\tfrac{1}{4}\pi\right)=-\sqrt{% 2}\cos\left(u+\tfrac{1}{4}\pi\right)}}$	`sqrt(2)sin(u -(1)/(4)Pi)= -sqrt(2)cos(u +(1)/(4)Pi)`	`Sqrt[2]Sin[u -Divide[1,4]Pi]= -Sqrt[2]Cos[u +Divide[1,4]Pi]`	Successful	Successful	-	-
4.21.E2	${\displaystyle{\displaystyle\sin\left(u+v\right)=\sin u\cos v+\cos u\sin v}}$	`sin(u + v)= sin(u)cos(v)+ cos(u)sin(v)`	`Sin[u + v]= Sin[u]Cos[v]+ Cos[u]Sin[v]`	Successful	Successful	-	-
4.21.E2	${\displaystyle{\displaystyle\sin\left(u-v\right)=\sin u\cos v-\cos u\sin v}}$	`sin(u - v)= sin(u)cos(v)- cos(u)sin(v)`	`Sin[u - v]= Sin[u]Cos[v]- Cos[u]Sin[v]`	Successful	Successful	-	-
4.21.E3	${\displaystyle{\displaystyle\cos\left(u+v\right)=\cos u\cos v-\sin u\sin v}}$	`cos(u + v)= cos(u)cos(v)- sin(u)sin(v)`	`Cos[u + v]= Cos[u]Cos[v]- Sin[u]Sin[v]`	Successful	Successful	-	-
4.21.E3	${\displaystyle{\displaystyle\cos\left(u-v\right)=\cos u\cos v+\sin u\sin v}}$	`cos(u - v)= cos(u)cos(v)+ sin(u)sin(v)`	`Cos[u - v]= Cos[u]Cos[v]+ Sin[u]Sin[v]`	Successful	Successful	-	-
4.21.E4	${\displaystyle{\displaystyle\tan\left(u+v\right)=\frac{\tan u+\tan v}{1-\tan u% \tan v}}}$	`tan(u + v)=(tan(u)+ tan(v))/(1 - tan(u)*tan(v))`	`Tan[u + v]=Divide[Tan[u]+ Tan[v],1 - Tan[u]*Tan[v]]`	Successful	Successful	-	-
4.21.E4	${\displaystyle{\displaystyle\tan\left(u-v\right)=\frac{\tan u-\tan v}{1+\tan u% \tan v}}}$	`tan(u - v)=(tan(u)- tan(v))/(1 + tan(u)*tan(v))`	`Tan[u - v]=Divide[Tan[u]- Tan[v],1 + Tan[u]*Tan[v]]`	Successful	Successful	-	-
4.21.E5	${\displaystyle{\displaystyle\cot\left(u+v\right)=\frac{+\cot u\cot v-1}{\cot u% +\cot v}}}$	`cot(u + v)=(+ cot(u)*cot(v)- 1)/(cot(u)+ cot(v))`	`Cot[u + v]=Divide[+ Cot[u]*Cot[v]- 1,Cot[u]+ Cot[v]]`	Successful	Successful	-	-
4.21.E5	${\displaystyle{\displaystyle\cot\left(u-v\right)=\frac{-\cot u\cot v-1}{\cot u% -\cot v}}}$	`cot(u - v)=(- cot(u)*cot(v)- 1)/(cot(u)- cot(v))`	`Cot[u - v]=Divide[- Cot[u]*Cot[v]- 1,Cot[u]- Cot[v]]`	Successful	Successful	-	-
4.21.E6	${\displaystyle{\displaystyle\sin u+\sin v=2\sin\left(\frac{u+v}{2}\right)\cos% \left(\frac{u-v}{2}\right)}}$	`sin(u)+ sin(v)= 2sin((u + v)/(2))cos((u - v)/(2))`	`Sin[u]+ Sin[v]= 2Sin[Divide[u + v,2]]Cos[Divide[u - v,2]]`	Successful	Successful	-	-
4.21.E7	${\displaystyle{\displaystyle\sin u-\sin v=2\cos\left(\frac{u+v}{2}\right)\sin% \left(\frac{u-v}{2}\right)}}$	`sin(u)- sin(v)= 2cos((u + v)/(2))sin((u - v)/(2))`	`Sin[u]- Sin[v]= 2Cos[Divide[u + v,2]]Sin[Divide[u - v,2]]`	Successful	Successful	-	-
4.21.E8	${\displaystyle{\displaystyle\cos u+\cos v=2\cos\left(\frac{u+v}{2}\right)\cos% \left(\frac{u-v}{2}\right)}}$	`cos(u)+ cos(v)= 2cos((u + v)/(2))cos((u - v)/(2))`	`Cos[u]+ Cos[v]= 2Cos[Divide[u + v,2]]Cos[Divide[u - v,2]]`	Successful	Successful	-	-
4.21.E9	${\displaystyle{\displaystyle\cos u-\cos v=-2\sin\left(\frac{u+v}{2}\right)\sin% \left(\frac{u-v}{2}\right)}}$	`cos(u)- cos(v)= - 2sin((u + v)/(2))sin((u - v)/(2))`	`Cos[u]- Cos[v]= - 2Sin[Divide[u + v,2]]Sin[Divide[u - v,2]]`	Successful	Successful	-	-
4.21.E10	${\displaystyle{\displaystyle\tan u+\tan v=\frac{\sin\left(u+v\right)}{\cos u% \cos v}}}$	`tan(u)+ tan(v)=(sin(u + v))/(cos(u)*cos(v))`	`Tan[u]+ Tan[v]=Divide[Sin[u + v],Cos[u]*Cos[v]]`	Successful	Successful	-	-
4.21.E10	${\displaystyle{\displaystyle\tan u-\tan v=\frac{\sin\left(u-v\right)}{\cos u% \cos v}}}$	`tan(u)- tan(v)=(sin(u - v))/(cos(u)*cos(v))`	`Tan[u]- Tan[v]=Divide[Sin[u - v],Cos[u]*Cos[v]]`	Successful	Successful	-	-
4.21.E11	${\displaystyle{\displaystyle\cot u+\cot v=\frac{\sin\left(v+u\right)}{\sin u% \sin v}}}$	`cot(u)+ cot(v)=(sin(v + u))/(sin(u)*sin(v))`	`Cot[u]+ Cot[v]=Divide[Sin[v + u],Sin[u]*Sin[v]]`	Successful	Successful	-	-
4.21.E11	${\displaystyle{\displaystyle\cot u-\cot v=\frac{\sin\left(v-u\right)}{\sin u% \sin v}}}$	`cot(u)- cot(v)=(sin(v - u))/(sin(u)*sin(v))`	`Cot[u]- Cot[v]=Divide[Sin[v - u],Sin[u]*Sin[v]]`	Successful	Successful	-	-
4.21.E12	${\displaystyle{\displaystyle{\sin^{2}}z+{\cos^{2}}z=1}}$	`(sin(z))^(2)+ (cos(z))^(2)= 1`	`(Sin[z])^(2)+ (Cos[z])^(2)= 1`	Successful	Successful	-	-
4.21.E13	${\displaystyle{\displaystyle{\sec^{2}}z=1+{\tan^{2}}z}}$	`(sec(z))^(2)= 1 + (tan(z))^(2)`	`(Sec[z])^(2)= 1 + (Tan[z])^(2)`	Successful	Successful	-	-
4.21.E14	${\displaystyle{\displaystyle{\csc^{2}}z=1+{\cot^{2}}z}}$	`(csc(z))^(2)= 1 + (cot(z))^(2)`	`(Csc[z])^(2)= 1 + (Cot[z])^(2)`	Successful	Successful	-	-
4.21.E15	${\displaystyle{\displaystyle 2\sin u\sin v=\cos\left(u-v\right)-\cos\left(u+v% \right)}}$	`2sin(u)sin(v)= cos(u - v)- cos(u + v)`	`2Sin[u]Sin[v]= Cos[u - v]- Cos[u + v]`	Successful	Successful	-	-
4.21.E16	${\displaystyle{\displaystyle 2\cos u\cos v=\cos\left(u-v\right)+\cos\left(u+v% \right)}}$	`2cos(u)cos(v)= cos(u - v)+ cos(u + v)`	`2Cos[u]Cos[v]= Cos[u - v]+ Cos[u + v]`	Successful	Successful	-	-
4.21.E17	${\displaystyle{\displaystyle 2\sin u\cos v=\sin\left(u-v\right)+\sin\left(u+v% \right)}}$	`2sin(u)cos(v)= sin(u - v)+ sin(u + v)`	`2Sin[u]Cos[v]= Sin[u - v]+ Sin[u + v]`	Successful	Successful	-	-
4.21.E18	${\displaystyle{\displaystyle{\sin^{2}}u-{\sin^{2}}v=\sin\left(u+v\right)\sin% \left(u-v\right)}}$	`(sin(u))^(2)- (sin(v))^(2)= sin(u + v)*sin(u - v)`	`(Sin[u])^(2)- (Sin[v])^(2)= Sin[u + v]*Sin[u - v]`	Successful	Successful	-	-
4.21.E19	${\displaystyle{\displaystyle{\cos^{2}}u-{\cos^{2}}v=-\sin\left(u+v\right)\sin% \left(u-v\right)}}$	`(cos(u))^(2)- (cos(v))^(2)= - sin(u + v)*sin(u - v)`	`(Cos[u])^(2)- (Cos[v])^(2)= - Sin[u + v]*Sin[u - v]`	Successful	Successful	-	-
4.21.E20	${\displaystyle{\displaystyle{\cos^{2}}u-{\sin^{2}}v=\cos\left(u+v\right)\cos% \left(u-v\right)}}$	`(cos(u))^(2)- (sin(v))^(2)= cos(u + v)*cos(u - v)`	`(Cos[u])^(2)- (Sin[v])^(2)= Cos[u + v]*Cos[u - v]`	Successful	Successful	-	-
4.21.E21	${\displaystyle{\displaystyle\sin\frac{z}{2}=+\left(\frac{1-\cos z}{2}\right)^{% 1/2}}}$	`sin((z)/(2))= +((1 - cos(z))/(2))^(1/ 2)`	`Sin[Divide[z,2]]= +(Divide[1 - Cos[z],2])^(1/ 2)`	Failure	Failure	Fail `-1.637854044-1.167010648I <- {z = -2^(1/2)-I2^(1/2)}` `-1.637854044+1.167010648I <- {z = -2^(1/2)+I2^(1/2)}`	Fail `Complex[-1.6378540442140963, -1.1670106484252494] <- {Rule[z, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}` `Complex[-1.6378540442140963, 1.1670106484252494] <- {Rule[z, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}`
4.21.E21	${\displaystyle{\displaystyle\sin\frac{z}{2}=-\left(\frac{1-\cos z}{2}\right)^{% 1/2}}}$	`sin((z)/(2))= -((1 - cos(z))/(2))^(1/ 2)`	`Sin[Divide[z,2]]= -(Divide[1 - Cos[z],2])^(1/ 2)`	Failure	Failure	Fail `1.637854044+1.167010648I <- {z = 2^(1/2)+I2^(1/2)}` `1.637854044-1.167010648I <- {z = 2^(1/2)-I2^(1/2)}`	Fail `Complex[1.6378540442140963, 1.1670106484252494] <- {Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}` `Complex[1.6378540442140963, -1.1670106484252494] <- {Rule[z, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}`
4.21.E22	${\displaystyle{\displaystyle\cos\frac{z}{2}=+\left(\frac{1+\cos z}{2}\right)^{% 1/2}}}$	`cos((z)/(2))= +((1 + cos(z))/(2))^(1/ 2)`	`Cos[Divide[z,2]]= +(Divide[1 + Cos[z],2])^(1/ 2)`	Failure	Failure	Successful	Successful
4.21.E22	${\displaystyle{\displaystyle\cos\frac{z}{2}=-\left(\frac{1+\cos z}{2}\right)^{% 1/2}}}$	`cos((z)/(2))= -((1 + cos(z))/(2))^(1/ 2)`	`Cos[Divide[z,2]]= -(Divide[1 + Cos[z],2])^(1/ 2)`	Failure	Failure	Fail `1.916716266-.9972227728I <- {z = 2^(1/2)+I2^(1/2)}` `1.916716266+.9972227728I <- {z = 2^(1/2)-I2^(1/2)}` `1.916716266-.9972227728I <- {z = -2^(1/2)-I2^(1/2)}` `1.916716266+.9972227728I <- {z = -2^(1/2)+I2^(1/2)}`	Fail `Complex[1.916716265666014, -0.9972227733456656] <- {Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}` `Complex[1.916716265666014, 0.9972227733456656] <- {Rule[z, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}` `Complex[1.916716265666014, -0.9972227733456656] <- {Rule[z, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}` `Complex[1.916716265666014, 0.9972227733456656] <- {Rule[z, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}`
4.21.E23	${\displaystyle{\displaystyle\tan\frac{z}{2}=+\left(\frac{1-\cos z}{1+\cos z}% \right)^{1/2}}}$	`tan((z)/(2))= +((1 - cos(z))/(1 + cos(z)))^(1/ 2)`	`Tan[Divide[z,2]]= +(Divide[1 - Cos[z],1 + Cos[z]])^(1/ 2)`	Failure	Failure	Fail `-.8463685478-1.658064547I <- {z = -2^(1/2)-I2^(1/2)}` `-.8463685478+1.658064547I <- {z = -2^(1/2)+I2^(1/2)}`	Fail `Complex[-0.8463685477398917, -1.6580645472823399] <- {Rule[z, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}` `Complex[-0.8463685477398917, 1.6580645472823399] <- {Rule[z, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}`
4.21.E23	${\displaystyle{\displaystyle\tan\frac{z}{2}=-\left(\frac{1-\cos z}{1+\cos z}% \right)^{1/2}}}$	`tan((z)/(2))= -((1 - cos(z))/(1 + cos(z)))^(1/ 2)`	`Tan[Divide[z,2]]= -(Divide[1 - Cos[z],1 + Cos[z]])^(1/ 2)`	Failure	Failure	Fail `.8463685478+1.658064547I <- {z = 2^(1/2)+I2^(1/2)}` `.8463685478-1.658064547I <- {z = 2^(1/2)-I2^(1/2)}`	Fail `Complex[0.8463685477398917, 1.6580645472823399] <- {Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}` `Complex[0.8463685477398917, -1.6580645472823399] <- {Rule[z, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}`
4.21.E23	${\displaystyle{\displaystyle+\left(\frac{1-\cos z}{1+\cos z}\right)^{1/2}=% \frac{1-\cos z}{\sin z}}}$	`+((1 - cos(z))/(1 + cos(z)))^(1/ 2)=(1 - cos(z))/(sin(z))`	`+(Divide[1 - Cos[z],1 + Cos[z]])^(1/ 2)=Divide[1 - Cos[z],Sin[z]]`	Failure	Failure	Skip	Fail `Complex[0.8463685477398916, 1.6580645472823403] <- {Rule[z, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}` `Complex[0.8463685477398916, -1.6580645472823403] <- {Rule[z, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}`
4.21.E23	${\displaystyle{\displaystyle-\left(\frac{1-\cos z}{1+\cos z}\right)^{1/2}=% \frac{1-\cos z}{\sin z}}}$	`-((1 - cos(z))/(1 + cos(z)))^(1/ 2)=(1 - cos(z))/(sin(z))`	`-(Divide[1 - Cos[z],1 + Cos[z]])^(1/ 2)=Divide[1 - Cos[z],Sin[z]]`	Failure	Failure	Skip	Fail `Complex[-0.8463685477398916, -1.6580645472823403] <- {Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}` `Complex[-0.8463685477398916, 1.6580645472823403] <- {Rule[z, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}`
4.21.E23	${\displaystyle{\displaystyle\frac{1-\cos z}{\sin z}=\frac{\sin z}{1+\cos z}}}$	`(1 - cos(z))/(sin(z))=(sin(z))/(1 + cos(z))`	`Divide[1 - Cos[z],Sin[z]]=Divide[Sin[z],1 + Cos[z]]`	Successful	Successful	-	-
4.21.E24	${\displaystyle{\displaystyle\sin\left(-z\right)=-\sin z}}$	`sin(- z)= - sin(z)`	`Sin[- z]= - Sin[z]`	Successful	Successful	-	-
4.21.E25	${\displaystyle{\displaystyle\cos\left(-z\right)=\cos z}}$	`cos(- z)= cos(z)`	`Cos[- z]= Cos[z]`	Successful	Successful	-	-
4.21.E26	${\displaystyle{\displaystyle\tan\left(-z\right)=-\tan z}}$	`tan(- z)= - tan(z)`	`Tan[- z]= - Tan[z]`	Successful	Successful	-	-
4.21.E27	${\displaystyle{\displaystyle\sin\left(2z\right)=2\sin z\cos z}}$	`sin(2z)= 2sin(z)*cos(z)`	`Sin[2z]= 2Sin[z]*Cos[z]`	Successful	Successful	-	-
4.21.E27	${\displaystyle{\displaystyle 2\sin z\cos z=\frac{2\tan z}{1+{\tan^{2}}z}}}$	`2sin(z)cos(z)=(2*tan(z))/(1 + (tan(z))^(2))`	`2Sin[z]Cos[z]=Divide[2*Tan[z],1 + (Tan[z])^(2)]`	Successful	Successful	-	-
4.21.E28	${\displaystyle{\displaystyle\cos\left(2z\right)=2{\cos^{2}}z-1}}$	`cos(2z)= 2(cos(z))^(2)- 1`	`Cos[2z]= 2(Cos[z])^(2)- 1`	Successful	Successful	-	-
4.21.E28	${\displaystyle{\displaystyle 2{\cos^{2}}z-1=1-2{\sin^{2}}z}}$