Optimal. Leaf size=45 \[ \frac{\text{PolyLog}\left (2,-e^{2 (a+b x)}\right )}{2 b^2}+\frac{x \log \left (e^{2 (a+b x)}+1\right )}{b}-\frac{x^2}{2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0824533, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {3718, 2190, 2279, 2391} \[ \frac{\text{PolyLog}\left (2,-e^{2 (a+b x)}\right )}{2 b^2}+\frac{x \log \left (e^{2 (a+b x)}+1\right )}{b}-\frac{x^2}{2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3718
Rule 2190
Rule 2279
Rule 2391
Rubi steps
\begin{align*} \int x \tanh (a+b x) \, dx &=-\frac{x^2}{2}+2 \int \frac{e^{2 (a+b x)} x}{1+e^{2 (a+b x)}} \, dx\\ &=-\frac{x^2}{2}+\frac{x \log \left (1+e^{2 (a+b x)}\right )}{b}-\frac{\int \log \left (1+e^{2 (a+b x)}\right ) \, dx}{b}\\ &=-\frac{x^2}{2}+\frac{x \log \left (1+e^{2 (a+b x)}\right )}{b}-\frac{\operatorname{Subst}\left (\int \frac{\log (1+x)}{x} \, dx,x,e^{2 (a+b x)}\right )}{2 b^2}\\ &=-\frac{x^2}{2}+\frac{x \log \left (1+e^{2 (a+b x)}\right )}{b}+\frac{\text{Li}_2\left (-e^{2 (a+b x)}\right )}{2 b^2}\\ \end{align*}
Mathematica [C] time = 3.52623, size = 149, normalized size = 3.31 \[ \frac{1}{2} \left (x^2 \tanh (a)+\frac{-\text{PolyLog}\left (2,e^{-2 \left (\tanh ^{-1}(\coth (a))+b x\right )}\right )-b^2 x^2 \tanh (a) \sqrt{-\text{csch}^2(a)} e^{-\tanh ^{-1}(\coth (a))}+2 b x \log \left (1-e^{-2 \left (\tanh ^{-1}(\coth (a))+b x\right )}\right )+2 \tanh ^{-1}(\coth (a)) \left (\log \left (1-e^{-2 \left (\tanh ^{-1}(\coth (a))+b x\right )}\right )-\log \left (i \sinh \left (\tanh ^{-1}(\coth (a))+b x\right )\right )+b x\right )+i \pi b x-i \pi \log \left (e^{2 b x}+1\right )+i \pi \log (\cosh (b x))}{b^2}\right ) \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.026, size = 70, normalized size = 1.6 \begin{align*} -{\frac{{x}^{2}}{2}}-2\,{\frac{ax}{b}}-{\frac{{a}^{2}}{{b}^{2}}}+{\frac{x\ln \left ( 1+{{\rm e}^{2\,bx+2\,a}} \right ) }{b}}+{\frac{{\it polylog} \left ( 2,-{{\rm e}^{2\,bx+2\,a}} \right ) }{2\,{b}^{2}}}+2\,{\frac{a\ln \left ({{\rm e}^{bx+a}} \right ) }{{b}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.1975, size = 54, normalized size = 1.2 \begin{align*} -\frac{1}{2} \, x^{2} + \frac{2 \, b x \log \left (e^{\left (2 \, b x + 2 \, a\right )} + 1\right ) +{\rm Li}_2\left (-e^{\left (2 \, b x + 2 \, a\right )}\right )}{2 \, b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [C] time = 1.91707, size = 427, normalized size = 9.49 \begin{align*} -\frac{b^{2} x^{2} + 2 \, a \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) + i\right ) + 2 \, a \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) - i\right ) - 2 \,{\left (b x + a\right )} \log \left (i \, \cosh \left (b x + a\right ) + i \, \sinh \left (b x + a\right ) + 1\right ) - 2 \,{\left (b x + a\right )} \log \left (-i \, \cosh \left (b x + a\right ) - i \, \sinh \left (b x + a\right ) + 1\right ) - 2 \,{\rm Li}_2\left (i \, \cosh \left (b x + a\right ) + i \, \sinh \left (b x + a\right )\right ) - 2 \,{\rm Li}_2\left (-i \, \cosh \left (b x + a\right ) - i \, \sinh \left (b x + a\right )\right )}{2 \, b^{2}} \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 \sinh{\left (a + b x \right )} \operatorname{sech}{\left (a + b 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{sech}\left (b x + a\right ) \sinh \left (b x + a\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]