Optimal. Leaf size=45 \[ \frac{\text{PolyLog}\left (2,e^{2 (a+b x)}\right )}{2 b^2}+\frac{x \log \left (1-e^{2 (a+b x)}\right )}{b}-\frac{x^2}{2} \]
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Rubi [A] time = 0.0806948, 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 = {3716, 2190, 2279, 2391} \[ \frac{\text{PolyLog}\left (2,e^{2 (a+b x)}\right )}{2 b^2}+\frac{x \log \left (1-e^{2 (a+b x)}\right )}{b}-\frac{x^2}{2} \]
Antiderivative was successfully verified.
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Rule 3716
Rule 2190
Rule 2279
Rule 2391
Rubi steps
\begin{align*} \int x \coth (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 [A] time = 0.0061351, size = 47, normalized size = 1.04 \[ \frac{\text{PolyLog}\left (2,e^{2 a+2 b x}\right )}{2 b^2}+\frac{x \log \left (1-e^{2 a+2 b x}\right )}{b}-\frac{x^2}{2} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.023, size = 122, normalized size = 2.7 \begin{align*} -{\frac{{x}^{2}}{2}}-2\,{\frac{ax}{b}}-{\frac{{a}^{2}}{{b}^{2}}}+{\frac{\ln \left ( 1+{{\rm e}^{bx+a}} \right ) x}{b}}+{\frac{{\it polylog} \left ( 2,-{{\rm e}^{bx+a}} \right ) }{{b}^{2}}}+{\frac{\ln \left ( 1-{{\rm e}^{bx+a}} \right ) x}{b}}+{\frac{\ln \left ( 1-{{\rm e}^{bx+a}} \right ) a}{{b}^{2}}}+{\frac{{\it polylog} \left ( 2,{{\rm e}^{bx+a}} \right ) }{{b}^{2}}}-{\frac{a\ln \left ({{\rm e}^{bx+a}}-1 \right ) }{{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.
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Maxima [A] time = 1.29799, size = 78, normalized size = 1.73 \begin{align*} -\frac{1}{2} \, x^{2} + \frac{b x \log \left (e^{\left (b x + a\right )} + 1\right ) +{\rm Li}_2\left (-e^{\left (b x + a\right )}\right )}{b^{2}} + \frac{b x \log \left (-e^{\left (b x + a\right )} + 1\right ) +{\rm Li}_2\left (e^{\left (b x + a\right )}\right )}{b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.07054, size = 336, normalized size = 7.47 \begin{align*} -\frac{b^{2} x^{2} - 2 \, b x \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) + 1\right ) + 2 \, a \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) - 1\right ) - 2 \,{\left (b x + a\right )} \log \left (-\cosh \left (b x + a\right ) - \sinh \left (b x + a\right ) + 1\right ) - 2 \,{\rm Li}_2\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right ) - 2 \,{\rm Li}_2\left (-\cosh \left (b x + a\right ) - \sinh \left (b x + a\right )\right )}{2 \, b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x \cosh{\left (a + b x \right )} \operatorname{csch}{\left (a + b x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x \cosh \left (b x + a\right ) \operatorname{csch}\left (b x + a\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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