Optimal. Leaf size=82 \[ -\frac{\text{PolyLog}\left (2,-e^{a+b x}\right )}{2 b^2}+\frac{\text{PolyLog}\left (2,e^{a+b x}\right )}{2 b^2}-\frac{\text{csch}(a+b x)}{2 b^2}-\frac{x \tanh ^{-1}\left (e^{a+b x}\right )}{b}-\frac{x \coth (a+b x) \text{csch}(a+b x)}{2 b} \]
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Rubi [A] time = 0.124972, antiderivative size = 82, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 5, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.312, Rules used = {5457, 4182, 2279, 2391, 4185} \[ -\frac{\text{PolyLog}\left (2,-e^{a+b x}\right )}{2 b^2}+\frac{\text{PolyLog}\left (2,e^{a+b x}\right )}{2 b^2}-\frac{\text{csch}(a+b x)}{2 b^2}-\frac{x \tanh ^{-1}\left (e^{a+b x}\right )}{b}-\frac{x \coth (a+b x) \text{csch}(a+b x)}{2 b} \]
Antiderivative was successfully verified.
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Rule 5457
Rule 4182
Rule 2279
Rule 2391
Rule 4185
Rubi steps
\begin{align*} \int x \coth ^2(a+b x) \text{csch}(a+b x) \, dx &=\int x \text{csch}(a+b x) \, dx+\int x \text{csch}^3(a+b x) \, dx\\ &=-\frac{2 x \tanh ^{-1}\left (e^{a+b x}\right )}{b}-\frac{\text{csch}(a+b x)}{2 b^2}-\frac{x \coth (a+b x) \text{csch}(a+b x)}{2 b}-\frac{1}{2} \int x \text{csch}(a+b x) \, dx-\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{x \tanh ^{-1}\left (e^{a+b x}\right )}{b}-\frac{\text{csch}(a+b x)}{2 b^2}-\frac{x \coth (a+b x) \text{csch}(a+b x)}{2 b}-\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{\int \log \left (1-e^{a+b x}\right ) \, dx}{2 b}-\frac{\int \log \left (1+e^{a+b x}\right ) \, dx}{2 b}\\ &=-\frac{x \tanh ^{-1}\left (e^{a+b x}\right )}{b}-\frac{\text{csch}(a+b x)}{2 b^2}-\frac{x \coth (a+b x) \text{csch}(a+b x)}{2 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{\operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{a+b x}\right )}{2 b^2}-\frac{\operatorname{Subst}\left (\int \frac{\log (1+x)}{x} \, dx,x,e^{a+b x}\right )}{2 b^2}\\ &=-\frac{x \tanh ^{-1}\left (e^{a+b x}\right )}{b}-\frac{\text{csch}(a+b x)}{2 b^2}-\frac{x \coth (a+b x) \text{csch}(a+b x)}{2 b}-\frac{\text{Li}_2\left (-e^{a+b x}\right )}{2 b^2}+\frac{\text{Li}_2\left (e^{a+b x}\right )}{2 b^2}\\ \end{align*}
Mathematica [A] time = 2.07128, size = 144, normalized size = 1.76 \[ -\frac{-4 \text{PolyLog}\left (2,-e^{-a-b x}\right )+4 \text{PolyLog}\left (2,e^{-a-b x}\right )-4 (a+b x) \left (\log \left (1-e^{-a-b x}\right )-\log \left (e^{-a-b x}+1\right )\right )-2 \tanh \left (\frac{1}{2} (a+b x)\right )+2 \coth \left (\frac{1}{2} (a+b x)\right )+b x \text{csch}^2\left (\frac{1}{2} (a+b x)\right )+b x \text{sech}^2\left (\frac{1}{2} (a+b x)\right )+4 a \log \left (\tanh \left (\frac{1}{2} (a+b x)\right )\right )}{8 b^2} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.064, size = 156, normalized size = 1.9 \begin{align*} -{\frac{{{\rm e}^{bx+a}} \left ( bx{{\rm e}^{2\,bx+2\,a}}+bx+{{\rm e}^{2\,bx+2\,a}}-1 \right ) }{{b}^{2} \left ({{\rm e}^{2\,bx+2\,a}}-1 \right ) ^{2}}}-{\frac{\ln \left ( 1+{{\rm e}^{bx+a}} \right ) x}{2\,b}}-{\frac{a\ln \left ( 1+{{\rm e}^{bx+a}} \right ) }{2\,{b}^{2}}}-{\frac{{\it polylog} \left ( 2,-{{\rm e}^{bx+a}} \right ) }{2\,{b}^{2}}}+{\frac{\ln \left ( 1-{{\rm e}^{bx+a}} \right ) x}{2\,b}}+{\frac{\ln \left ( 1-{{\rm e}^{bx+a}} \right ) a}{2\,{b}^{2}}}+{\frac{{\it polylog} \left ( 2,{{\rm e}^{bx+a}} \right ) }{2\,{b}^{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 = 1.59869, size = 167, normalized size = 2.04 \begin{align*} -\frac{{\left (b x e^{\left (3 \, a\right )} + e^{\left (3 \, a\right )}\right )} e^{\left (3 \, b x\right )} +{\left (b x e^{a} - e^{a}\right )} e^{\left (b x\right )}}{b^{2} e^{\left (4 \, b x + 4 \, a\right )} - 2 \, b^{2} e^{\left (2 \, b x + 2 \, a\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 )}{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 )}{2 \, b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.37047, size = 2294, normalized size = 27.98 \begin{align*} \text{result too large to display} \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 )^{3}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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