Optimal. Leaf size=87 \[ \frac{3 x \text{PolyLog}\left (2,e^{2 (a+b x)}\right )}{b^3}-\frac{3 \text{PolyLog}\left (3,e^{2 (a+b x)}\right )}{2 b^4}+\frac{3 x^2 \log \left (1-e^{2 (a+b x)}\right )}{b^2}-\frac{x^3 \coth (a+b x)}{b}-\frac{x^3}{b}+\frac{x^4}{4} \]
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Rubi [A] time = 0.190154, antiderivative size = 87, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.583, Rules used = {3720, 3716, 2190, 2531, 2282, 6589, 30} \[ \frac{3 x \text{PolyLog}\left (2,e^{2 (a+b x)}\right )}{b^3}-\frac{3 \text{PolyLog}\left (3,e^{2 (a+b x)}\right )}{2 b^4}+\frac{3 x^2 \log \left (1-e^{2 (a+b x)}\right )}{b^2}-\frac{x^3 \coth (a+b x)}{b}-\frac{x^3}{b}+\frac{x^4}{4} \]
Antiderivative was successfully verified.
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Rule 3720
Rule 3716
Rule 2190
Rule 2531
Rule 2282
Rule 6589
Rule 30
Rubi steps
\begin{align*} \int x^3 \coth ^2(a+b x) \, dx &=-\frac{x^3 \coth (a+b x)}{b}+\frac{3 \int x^2 \coth (a+b x) \, dx}{b}+\int x^3 \, dx\\ &=-\frac{x^3}{b}+\frac{x^4}{4}-\frac{x^3 \coth (a+b x)}{b}-\frac{6 \int \frac{e^{2 (a+b x)} x^2}{1-e^{2 (a+b x)}} \, dx}{b}\\ &=-\frac{x^3}{b}+\frac{x^4}{4}-\frac{x^3 \coth (a+b x)}{b}+\frac{3 x^2 \log \left (1-e^{2 (a+b x)}\right )}{b^2}-\frac{6 \int x \log \left (1-e^{2 (a+b x)}\right ) \, dx}{b^2}\\ &=-\frac{x^3}{b}+\frac{x^4}{4}-\frac{x^3 \coth (a+b x)}{b}+\frac{3 x^2 \log \left (1-e^{2 (a+b x)}\right )}{b^2}+\frac{3 x \text{Li}_2\left (e^{2 (a+b x)}\right )}{b^3}-\frac{3 \int \text{Li}_2\left (e^{2 (a+b x)}\right ) \, dx}{b^3}\\ &=-\frac{x^3}{b}+\frac{x^4}{4}-\frac{x^3 \coth (a+b x)}{b}+\frac{3 x^2 \log \left (1-e^{2 (a+b x)}\right )}{b^2}+\frac{3 x \text{Li}_2\left (e^{2 (a+b x)}\right )}{b^3}-\frac{3 \operatorname{Subst}\left (\int \frac{\text{Li}_2(x)}{x} \, dx,x,e^{2 (a+b x)}\right )}{2 b^4}\\ &=-\frac{x^3}{b}+\frac{x^4}{4}-\frac{x^3 \coth (a+b x)}{b}+\frac{3 x^2 \log \left (1-e^{2 (a+b x)}\right )}{b^2}+\frac{3 x \text{Li}_2\left (e^{2 (a+b x)}\right )}{b^3}-\frac{3 \text{Li}_3\left (e^{2 (a+b x)}\right )}{2 b^4}\\ \end{align*}
Mathematica [B] time = 0.60123, size = 204, normalized size = 2.34 \[ -\frac{e^{2 a} \left (6 \left (1-e^{-2 a}\right ) \left (b x \text{PolyLog}\left (2,-e^{-a-b x}\right )+\text{PolyLog}\left (3,-e^{-a-b x}\right )\right )+6 \left (1-e^{-2 a}\right ) \left (b x \text{PolyLog}\left (2,e^{-a-b x}\right )+\text{PolyLog}\left (3,e^{-a-b x}\right )\right )+2 e^{-2 a} b^3 x^3-3 \left (1-e^{-2 a}\right ) b^2 x^2 \log \left (1-e^{-a-b x}\right )-3 \left (1-e^{-2 a}\right ) b^2 x^2 \log \left (e^{-a-b x}+1\right )\right )}{\left (e^{2 a}-1\right ) b^4}+\frac{x^3 \text{csch}(a) \sinh (b x) \text{csch}(a+b x)}{b}+\frac{x^4}{4} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.056, size = 198, normalized size = 2.3 \begin{align*}{\frac{{x}^{4}}{4}}-2\,{\frac{{x}^{3}}{b \left ({{\rm e}^{2\,bx+2\,a}}-1 \right ) }}+3\,{\frac{{a}^{2}\ln \left ({{\rm e}^{bx+a}}-1 \right ) }{{b}^{4}}}-6\,{\frac{{a}^{2}\ln \left ({{\rm e}^{bx+a}} \right ) }{{b}^{4}}}-2\,{\frac{{x}^{3}}{b}}+6\,{\frac{{a}^{2}x}{{b}^{3}}}+4\,{\frac{{a}^{3}}{{b}^{4}}}+3\,{\frac{\ln \left ( 1+{{\rm e}^{bx+a}} \right ){x}^{2}}{{b}^{2}}}+6\,{\frac{x{\it polylog} \left ( 2,-{{\rm e}^{bx+a}} \right ) }{{b}^{3}}}-6\,{\frac{{\it polylog} \left ( 3,-{{\rm e}^{bx+a}} \right ) }{{b}^{4}}}+3\,{\frac{\ln \left ( 1-{{\rm e}^{bx+a}} \right ){x}^{2}}{{b}^{2}}}-3\,{\frac{\ln \left ( 1-{{\rm e}^{bx+a}} \right ){a}^{2}}{{b}^{4}}}+6\,{\frac{x{\it polylog} \left ( 2,{{\rm e}^{bx+a}} \right ) }{{b}^{3}}}-6\,{\frac{{\it polylog} \left ( 3,{{\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 = 1.38485, size = 197, normalized size = 2.26 \begin{align*} -\frac{2 \, x^{3}}{b} + \frac{b x^{4} e^{\left (2 \, b x + 2 \, a\right )} - b x^{4} - 8 \, x^{3}}{4 \,{\left (b e^{\left (2 \, b x + 2 \, a\right )} - b\right )}} + \frac{3 \,{\left (b^{2} x^{2} \log \left (e^{\left (b x + a\right )} + 1\right ) + 2 \, b x{\rm Li}_2\left (-e^{\left (b x + a\right )}\right ) - 2 \,{\rm Li}_{3}(-e^{\left (b x + a\right )})\right )}}{b^{4}} + \frac{3 \,{\left (b^{2} x^{2} \log \left (-e^{\left (b x + a\right )} + 1\right ) + 2 \, b x{\rm Li}_2\left (e^{\left (b x + a\right )}\right ) - 2 \,{\rm Li}_{3}(e^{\left (b x + a\right )})\right )}}{b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] time = 2.16178, size = 1643, normalized size = 18.89 \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 )^{2} \operatorname{csch}\left (b x + a\right )^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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