Optimal. Leaf size=43 \[ -\frac{1}{2} x \text{PolyLog}\left (2,-e^x\right )+\frac{1}{2} x \text{PolyLog}\left (2,e^x\right )+\frac{1}{2} \text{PolyLog}\left (3,-e^x\right )-\frac{1}{2} \text{PolyLog}\left (3,e^x\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0429293, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 6, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.667, Rules used = {6213, 2531, 2282, 6589} \[ -\frac{1}{2} x \text{PolyLog}\left (2,-e^x\right )+\frac{1}{2} x \text{PolyLog}\left (2,e^x\right )+\frac{1}{2} \text{PolyLog}\left (3,-e^x\right )-\frac{1}{2} \text{PolyLog}\left (3,e^x\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 6213
Rule 2531
Rule 2282
Rule 6589
Rubi steps
\begin{align*} \int x \tanh ^{-1}\left (e^x\right ) \, dx &=-\left (\frac{1}{2} \int x \log \left (1-e^x\right ) \, dx\right )+\frac{1}{2} \int x \log \left (1+e^x\right ) \, dx\\ &=-\frac{1}{2} x \text{Li}_2\left (-e^x\right )+\frac{x \text{Li}_2\left (e^x\right )}{2}+\frac{1}{2} \int \text{Li}_2\left (-e^x\right ) \, dx-\frac{1}{2} \int \text{Li}_2\left (e^x\right ) \, dx\\ &=-\frac{1}{2} x \text{Li}_2\left (-e^x\right )+\frac{x \text{Li}_2\left (e^x\right )}{2}+\frac{1}{2} \operatorname{Subst}\left (\int \frac{\text{Li}_2(-x)}{x} \, dx,x,e^x\right )-\frac{1}{2} \operatorname{Subst}\left (\int \frac{\text{Li}_2(x)}{x} \, dx,x,e^x\right )\\ &=-\frac{1}{2} x \text{Li}_2\left (-e^x\right )+\frac{x \text{Li}_2\left (e^x\right )}{2}+\frac{\text{Li}_3\left (-e^x\right )}{2}-\frac{\text{Li}_3\left (e^x\right )}{2}\\ \end{align*}
Mathematica [A] time = 0.0263989, size = 71, normalized size = 1.65 \[ \frac{1}{4} \left (-2 x \text{PolyLog}\left (2,-e^x\right )+2 x \text{PolyLog}\left (2,e^x\right )+2 \text{PolyLog}\left (3,-e^x\right )-2 \text{PolyLog}\left (3,e^x\right )+x^2 \log \left (1-e^x\right )-x^2 \log \left (e^x+1\right )+2 x^2 \tanh ^{-1}\left (e^x\right )\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.036, size = 62, normalized size = 1.4 \begin{align*}{\frac{{x}^{2}{\it Artanh} \left ({{\rm e}^{x}} \right ) }{2}}-{\frac{{x}^{2}\ln \left ({{\rm e}^{x}}+1 \right ) }{4}}-{\frac{x{\it polylog} \left ( 2,-{{\rm e}^{x}} \right ) }{2}}+{\frac{{\it polylog} \left ( 3,-{{\rm e}^{x}} \right ) }{2}}+{\frac{{x}^{2}\ln \left ( 1-{{\rm e}^{x}} \right ) }{4}}+{\frac{x{\it polylog} \left ( 2,{{\rm e}^{x}} \right ) }{2}}-{\frac{{\it polylog} \left ( 3,{{\rm e}^{x}} \right ) }{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] time = 0.962817, size = 80, normalized size = 1.86 \begin{align*} \frac{1}{2} \, x^{2} \operatorname{artanh}\left (e^{x}\right ) - \frac{1}{4} \, x^{2} \log \left (e^{x} + 1\right ) + \frac{1}{4} \, x^{2} \log \left (-e^{x} + 1\right ) - \frac{1}{2} \, x{\rm Li}_2\left (-e^{x}\right ) + \frac{1}{2} \, x{\rm Li}_2\left (e^{x}\right ) + \frac{1}{2} \,{\rm Li}_{3}(-e^{x}) - \frac{1}{2} \,{\rm Li}_{3}(e^{x}) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [C] time = 1.60316, size = 375, normalized size = 8.72 \begin{align*} \frac{1}{4} \, x^{2} \log \left (-\frac{\cosh \left (x\right ) + \sinh \left (x\right ) + 1}{\cosh \left (x\right ) + \sinh \left (x\right ) - 1}\right ) - \frac{1}{4} \, x^{2} \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) + 1\right ) + \frac{1}{4} \, x^{2} \log \left (-\cosh \left (x\right ) - \sinh \left (x\right ) + 1\right ) + \frac{1}{2} \, x{\rm Li}_2\left (\cosh \left (x\right ) + \sinh \left (x\right )\right ) - \frac{1}{2} \, x{\rm Li}_2\left (-\cosh \left (x\right ) - \sinh \left (x\right )\right ) - \frac{1}{2} \,{\rm polylog}\left (3, \cosh \left (x\right ) + \sinh \left (x\right )\right ) + \frac{1}{2} \,{\rm polylog}\left (3, -\cosh \left (x\right ) - \sinh \left (x\right )\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x \operatorname{atanh}{\left (e^{x} \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x \operatorname{artanh}\left (e^{x}\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]