Optimal. Leaf size=82 \[ \frac{\text{PolyLog}\left (2,e^{2 (a+b x)}\right )}{2 b^2}-\frac{\coth (a+b x)}{2 b^2}+\frac{x \log \left (1-e^{2 (a+b x)}\right )}{b}-\frac{x \coth ^2(a+b x)}{2 b}+\frac{x}{2 b}-\frac{x^2}{2} \]
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Rubi [A] time = 0.12246, antiderivative size = 82, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.7, Rules used = {3720, 3473, 8, 3716, 2190, 2279, 2391} \[ \frac{\text{PolyLog}\left (2,e^{2 (a+b x)}\right )}{2 b^2}-\frac{\coth (a+b x)}{2 b^2}+\frac{x \log \left (1-e^{2 (a+b x)}\right )}{b}-\frac{x \coth ^2(a+b x)}{2 b}+\frac{x}{2 b}-\frac{x^2}{2} \]
Antiderivative was successfully verified.
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Rule 3720
Rule 3473
Rule 8
Rule 3716
Rule 2190
Rule 2279
Rule 2391
Rubi steps
\begin{align*} \int x \coth ^3(a+b x) \, dx &=-\frac{x \coth ^2(a+b x)}{2 b}+\frac{\int \coth ^2(a+b x) \, dx}{2 b}+\int x \coth (a+b x) \, dx\\ &=-\frac{x^2}{2}-\frac{\coth (a+b x)}{2 b^2}-\frac{x \coth ^2(a+b x)}{2 b}-2 \int \frac{e^{2 (a+b x)} x}{1-e^{2 (a+b x)}} \, dx+\frac{\int 1 \, dx}{2 b}\\ &=\frac{x}{2 b}-\frac{x^2}{2}-\frac{\coth (a+b x)}{2 b^2}-\frac{x \coth ^2(a+b x)}{2 b}+\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 b}-\frac{x^2}{2}-\frac{\coth (a+b x)}{2 b^2}-\frac{x \coth ^2(a+b x)}{2 b}+\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 b}-\frac{x^2}{2}-\frac{\coth (a+b x)}{2 b^2}-\frac{x \coth ^2(a+b x)}{2 b}+\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 = 6.1231, size = 232, normalized size = 2.83 \[ \frac{\text{csch}(a) \text{sech}(a) \left (-b^2 x^2 e^{-\tanh ^{-1}(\tanh (a))}+\frac{i \tanh (a) \left (i \text{PolyLog}\left (2,e^{2 i \left (i \tanh ^{-1}(\tanh (a))+i b x\right )}\right )-b x \left (-\pi +2 i \tanh ^{-1}(\tanh (a))\right )-2 \left (i \tanh ^{-1}(\tanh (a))+i b x\right ) \log \left (1-e^{2 i \left (i \tanh ^{-1}(\tanh (a))+i b x\right )}\right )+2 i \tanh ^{-1}(\tanh (a)) \log \left (i \sinh \left (\tanh ^{-1}(\tanh (a))+b x\right )\right )-\pi \log \left (e^{2 b x}+1\right )+\pi \log (\cosh (b x))\right )}{\sqrt{1-\tanh ^2(a)}}\right )}{2 b^2 \sqrt{\text{sech}^2(a) \left (\cosh ^2(a)-\sinh ^2(a)\right )}}+\frac{\text{csch}(a) \sinh (b x) \text{csch}(a+b x)}{2 b^2}-\frac{x \text{csch}^2(a+b x)}{2 b}+\frac{1}{2} x^2 \coth (a) \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.083, size = 164, normalized size = 2. \begin{align*} -{\frac{{x}^{2}}{2}}-{\frac{2\,bx{{\rm e}^{2\,bx+2\,a}}+{{\rm e}^{2\,bx+2\,a}}-1}{{b}^{2} \left ({{\rm e}^{2\,bx+2\,a}}-1 \right ) ^{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 [B] time = 1.35998, size = 201, normalized size = 2.45 \begin{align*} -x^{2} + \frac{b^{2} x^{2} e^{\left (4 \, b x + 4 \, a\right )} + b^{2} x^{2} - 2 \,{\left (b^{2} x^{2} e^{\left (2 \, a\right )} + 2 \, b x e^{\left (2 \, a\right )} + e^{\left (2 \, a\right )}\right )} e^{\left (2 \, b x\right )} + 2}{2 \,{\left (b^{2} e^{\left (4 \, b x + 4 \, a\right )} - 2 \, b^{2} e^{\left (2 \, b x + 2 \, a\right )} + b^{2}\right )}} + \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 = 2.52685, size = 2569, normalized size = 31.33 \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 )^{3} \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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