Optimal. Leaf size=83 \[ \frac{3 \text{PolyLog}\left (2,e^{2 (a+b x)}\right )}{2 b^4}-\frac{3 x^2 \coth (a+b x)}{2 b^2}+\frac{3 x \log \left (1-e^{2 (a+b x)}\right )}{b^3}-\frac{x^3 \text{csch}^2(a+b x)}{2 b}-\frac{3 x^2}{2 b^2} \]
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Rubi [A] time = 0.163233, antiderivative size = 83, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {5419, 4184, 3716, 2190, 2279, 2391} \[ \frac{3 \text{PolyLog}\left (2,e^{2 (a+b x)}\right )}{2 b^4}-\frac{3 x^2 \coth (a+b x)}{2 b^2}+\frac{3 x \log \left (1-e^{2 (a+b x)}\right )}{b^3}-\frac{x^3 \text{csch}^2(a+b x)}{2 b}-\frac{3 x^2}{2 b^2} \]
Antiderivative was successfully verified.
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Rule 5419
Rule 4184
Rule 3716
Rule 2190
Rule 2279
Rule 2391
Rubi steps
\begin{align*} \int x^3 \coth (a+b x) \text{csch}^2(a+b x) \, dx &=-\frac{x^3 \text{csch}^2(a+b x)}{2 b}+\frac{3 \int x^2 \text{csch}^2(a+b x) \, dx}{2 b}\\ &=-\frac{3 x^2 \coth (a+b x)}{2 b^2}-\frac{x^3 \text{csch}^2(a+b x)}{2 b}+\frac{3 \int x \coth (a+b x) \, dx}{b^2}\\ &=-\frac{3 x^2}{2 b^2}-\frac{3 x^2 \coth (a+b x)}{2 b^2}-\frac{x^3 \text{csch}^2(a+b x)}{2 b}-\frac{6 \int \frac{e^{2 (a+b x)} x}{1-e^{2 (a+b x)}} \, dx}{b^2}\\ &=-\frac{3 x^2}{2 b^2}-\frac{3 x^2 \coth (a+b x)}{2 b^2}-\frac{x^3 \text{csch}^2(a+b x)}{2 b}+\frac{3 x \log \left (1-e^{2 (a+b x)}\right )}{b^3}-\frac{3 \int \log \left (1-e^{2 (a+b x)}\right ) \, dx}{b^3}\\ &=-\frac{3 x^2}{2 b^2}-\frac{3 x^2 \coth (a+b x)}{2 b^2}-\frac{x^3 \text{csch}^2(a+b x)}{2 b}+\frac{3 x \log \left (1-e^{2 (a+b x)}\right )}{b^3}-\frac{3 \operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{2 (a+b x)}\right )}{2 b^4}\\ &=-\frac{3 x^2}{2 b^2}-\frac{3 x^2 \coth (a+b x)}{2 b^2}-\frac{x^3 \text{csch}^2(a+b x)}{2 b}+\frac{3 x \log \left (1-e^{2 (a+b x)}\right )}{b^3}+\frac{3 \text{Li}_2\left (e^{2 (a+b x)}\right )}{2 b^4}\\ \end{align*}
Mathematica [C] time = 6.1293, size = 228, normalized size = 2.75 \[ \frac{3 \text{csch}(a) \text{sech}(a) \left (-b^2 x^2 e^{-\tanh ^{-1}(\tanh (a))}+\frac{i \tanh (a) \left (i \text{PolyLog}\left (2,e^{2 i \left (i \tanh ^{-1}(\tanh (a))+i b x\right )}\right )-b x \left (-\pi +2 i \tanh ^{-1}(\tanh (a))\right )-2 \left (i \tanh ^{-1}(\tanh (a))+i b x\right ) \log \left (1-e^{2 i \left (i \tanh ^{-1}(\tanh (a))+i b x\right )}\right )+2 i \tanh ^{-1}(\tanh (a)) \log \left (i \sinh \left (\tanh ^{-1}(\tanh (a))+b x\right )\right )-\pi \log \left (e^{2 b x}+1\right )+\pi \log (\cosh (b x))\right )}{\sqrt{1-\tanh ^2(a)}}\right )}{2 b^4 \sqrt{\text{sech}^2(a) \left (\cosh ^2(a)-\sinh ^2(a)\right )}}+\frac{3 x^2 \text{csch}(a) \sinh (b x) \text{csch}(a+b x)}{2 b^2}-\frac{x^3 \text{csch}^2(a+b x)}{2 b} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.032, size = 177, normalized size = 2.1 \begin{align*} -{\frac{{x}^{2} \left ( 2\,bx{{\rm e}^{2\,bx+2\,a}}+3\,{{\rm e}^{2\,bx+2\,a}}-3 \right ) }{{b}^{2} \left ({{\rm e}^{2\,bx+2\,a}}-1 \right ) ^{2}}}-3\,{\frac{{x}^{2}}{{b}^{2}}}-6\,{\frac{ax}{{b}^{3}}}-3\,{\frac{{a}^{2}}{{b}^{4}}}+3\,{\frac{\ln \left ( 1+{{\rm e}^{bx+a}} \right ) x}{{b}^{3}}}+3\,{\frac{{\it polylog} \left ( 2,-{{\rm e}^{bx+a}} \right ) }{{b}^{4}}}+3\,{\frac{\ln \left ( 1-{{\rm e}^{bx+a}} \right ) x}{{b}^{3}}}+3\,{\frac{\ln \left ( 1-{{\rm e}^{bx+a}} \right ) a}{{b}^{4}}}+3\,{\frac{{\it polylog} \left ( 2,{{\rm e}^{bx+a}} \right ) }{{b}^{4}}}-3\,{\frac{a\ln \left ({{\rm e}^{bx+a}}-1 \right ) }{{b}^{4}}}+6\,{\frac{a\ln \left ({{\rm e}^{bx+a}} \right ) }{{b}^{4}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.53725, size = 176, normalized size = 2.12 \begin{align*} \frac{3 \, x^{2} -{\left (2 \, b x^{3} e^{\left (2 \, a\right )} + 3 \, x^{2} e^{\left (2 \, a\right )}\right )} e^{\left (2 \, 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{3 \, x^{2}}{b^{2}} + \frac{3 \,{\left (b x \log \left (e^{\left (b x + a\right )} + 1\right ) +{\rm Li}_2\left (-e^{\left (b x + a\right )}\right )\right )}}{b^{4}} + \frac{3 \,{\left (b x \log \left (-e^{\left (b x + a\right )} + 1\right ) +{\rm Li}_2\left (e^{\left (b x + a\right )}\right )\right )}}{b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.35122, size = 2545, normalized size = 30.66 \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^{3} \cosh \left (b x + a\right ) \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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