Optimal. Leaf size=66 \[ -\frac{\text{PolyLog}\left (2,-e^{a+b x}\right )}{b^2}+\frac{\text{PolyLog}\left (2,e^{a+b x}\right )}{b^2}-\frac{\sinh (a+b x)}{b^2}+\frac{x \cosh (a+b x)}{b}-\frac{2 x \tanh ^{-1}\left (e^{a+b x}\right )}{b} \]
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Rubi [A] time = 0.0611381, antiderivative size = 66, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.429, Rules used = {5450, 3296, 2637, 4182, 2279, 2391} \[ -\frac{\text{PolyLog}\left (2,-e^{a+b x}\right )}{b^2}+\frac{\text{PolyLog}\left (2,e^{a+b x}\right )}{b^2}-\frac{\sinh (a+b x)}{b^2}+\frac{x \cosh (a+b x)}{b}-\frac{2 x \tanh ^{-1}\left (e^{a+b x}\right )}{b} \]
Antiderivative was successfully verified.
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Rule 5450
Rule 3296
Rule 2637
Rule 4182
Rule 2279
Rule 2391
Rubi steps
\begin{align*} \int x \cosh (a+b x) \coth (a+b x) \, dx &=\int x \text{csch}(a+b x) \, dx+\int x \sinh (a+b x) \, dx\\ &=-\frac{2 x \tanh ^{-1}\left (e^{a+b x}\right )}{b}+\frac{x \cosh (a+b x)}{b}-\frac{\int \cosh (a+b x) \, dx}{b}-\frac{\int \log \left (1-e^{a+b x}\right ) \, dx}{b}+\frac{\int \log \left (1+e^{a+b x}\right ) \, dx}{b}\\ &=-\frac{2 x \tanh ^{-1}\left (e^{a+b x}\right )}{b}+\frac{x \cosh (a+b x)}{b}-\frac{\sinh (a+b x)}{b^2}-\frac{\operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{a+b x}\right )}{b^2}+\frac{\operatorname{Subst}\left (\int \frac{\log (1+x)}{x} \, dx,x,e^{a+b x}\right )}{b^2}\\ &=-\frac{2 x \tanh ^{-1}\left (e^{a+b x}\right )}{b}+\frac{x \cosh (a+b x)}{b}-\frac{\text{Li}_2\left (-e^{a+b x}\right )}{b^2}+\frac{\text{Li}_2\left (e^{a+b x}\right )}{b^2}-\frac{\sinh (a+b x)}{b^2}\\ \end{align*}
Mathematica [A] time = 0.132542, size = 131, normalized size = 1.98 \[ -\frac{-\text{PolyLog}\left (2,-e^{-a-b x}\right )+\text{PolyLog}\left (2,e^{-a-b x}\right )-a \log \left (1-e^{-a-b x}\right )-b x \log \left (1-e^{-a-b x}\right )+a \log \left (e^{-a-b x}+1\right )+b x \log \left (e^{-a-b x}+1\right )+\sinh (a+b x)-b x \cosh (a+b x)+a \log \left (\tanh \left (\frac{1}{2} (a+b x)\right )\right )}{b^2} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.061, size = 139, normalized size = 2.1 \begin{align*}{\frac{ \left ( bx-1 \right ){{\rm e}^{bx+a}}}{2\,{b}^{2}}}+{\frac{ \left ( bx+1 \right ){{\rm e}^{-bx-a}}}{2\,{b}^{2}}}-{\frac{\ln \left ( 1+{{\rm e}^{bx+a}} \right ) x}{b}}-{\frac{a\ln \left ( 1+{{\rm e}^{bx+a}} \right ) }{{b}^{2}}}-{\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}}}+2\,{\frac{a{\it Artanh} \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 = 1.29479, size = 127, normalized size = 1.92 \begin{align*} \frac{{\left ({\left (b x e^{\left (2 \, a\right )} - e^{\left (2 \, a\right )}\right )} e^{\left (b x\right )} +{\left (b x + 1\right )} e^{\left (-b x\right )}\right )} e^{\left (-a\right )}}{2 \, 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}} + \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 = 2.16253, size = 753, normalized size = 11.41 \begin{align*} \frac{{\left (b x - 1\right )} \cosh \left (b x + a\right )^{2} + 2 \,{\left (b x - 1\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) +{\left (b x - 1\right )} \sinh \left (b x + a\right )^{2} + b x + 2 \,{\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )}{\rm Li}_2\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right ) - 2 \,{\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )}{\rm Li}_2\left (-\cosh \left (b x + a\right ) - \sinh \left (b x + a\right )\right ) - 2 \,{\left (b x \cosh \left (b x + a\right ) + b x \sinh \left (b x + a\right )\right )} \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) + 1\right ) - 2 \,{\left (a \cosh \left (b x + a\right ) + a \sinh \left (b x + a\right )\right )} \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) - 1\right ) + 2 \,{\left ({\left (b x + a\right )} \cosh \left (b x + a\right ) +{\left (b x + a\right )} \sinh \left (b x + a\right )\right )} \log \left (-\cosh \left (b x + a\right ) - \sinh \left (b x + a\right ) + 1\right ) + 1}{2 \,{\left (b^{2} \cosh \left (b x + a\right ) + b^{2} \sinh \left (b x + a\right )\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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 )^{2} \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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