Optimal. Leaf size=93 \[ -\frac{6 x \text{PolyLog}\left (2,-e^{a+b x}\right )}{b^3}+\frac{6 x \text{PolyLog}\left (2,e^{a+b x}\right )}{b^3}+\frac{6 \text{PolyLog}\left (3,-e^{a+b x}\right )}{b^4}-\frac{6 \text{PolyLog}\left (3,e^{a+b x}\right )}{b^4}-\frac{6 x^2 \tanh ^{-1}\left (e^{a+b x}\right )}{b^2}-\frac{x^3 \text{csch}(a+b x)}{b} \]
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Rubi [A] time = 0.104649, antiderivative size = 93, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.312, Rules used = {5419, 4182, 2531, 2282, 6589} \[ -\frac{6 x \text{PolyLog}\left (2,-e^{a+b x}\right )}{b^3}+\frac{6 x \text{PolyLog}\left (2,e^{a+b x}\right )}{b^3}+\frac{6 \text{PolyLog}\left (3,-e^{a+b x}\right )}{b^4}-\frac{6 \text{PolyLog}\left (3,e^{a+b x}\right )}{b^4}-\frac{6 x^2 \tanh ^{-1}\left (e^{a+b x}\right )}{b^2}-\frac{x^3 \text{csch}(a+b x)}{b} \]
Antiderivative was successfully verified.
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Rule 5419
Rule 4182
Rule 2531
Rule 2282
Rule 6589
Rubi steps
\begin{align*} \int x^3 \coth (a+b x) \text{csch}(a+b x) \, dx &=-\frac{x^3 \text{csch}(a+b x)}{b}+\frac{3 \int x^2 \text{csch}(a+b x) \, dx}{b}\\ &=-\frac{6 x^2 \tanh ^{-1}\left (e^{a+b x}\right )}{b^2}-\frac{x^3 \text{csch}(a+b x)}{b}-\frac{6 \int x \log \left (1-e^{a+b x}\right ) \, dx}{b^2}+\frac{6 \int x \log \left (1+e^{a+b x}\right ) \, dx}{b^2}\\ &=-\frac{6 x^2 \tanh ^{-1}\left (e^{a+b x}\right )}{b^2}-\frac{x^3 \text{csch}(a+b x)}{b}-\frac{6 x \text{Li}_2\left (-e^{a+b x}\right )}{b^3}+\frac{6 x \text{Li}_2\left (e^{a+b x}\right )}{b^3}+\frac{6 \int \text{Li}_2\left (-e^{a+b x}\right ) \, dx}{b^3}-\frac{6 \int \text{Li}_2\left (e^{a+b x}\right ) \, dx}{b^3}\\ &=-\frac{6 x^2 \tanh ^{-1}\left (e^{a+b x}\right )}{b^2}-\frac{x^3 \text{csch}(a+b x)}{b}-\frac{6 x \text{Li}_2\left (-e^{a+b x}\right )}{b^3}+\frac{6 x \text{Li}_2\left (e^{a+b x}\right )}{b^3}+\frac{6 \operatorname{Subst}\left (\int \frac{\text{Li}_2(-x)}{x} \, dx,x,e^{a+b x}\right )}{b^4}-\frac{6 \operatorname{Subst}\left (\int \frac{\text{Li}_2(x)}{x} \, dx,x,e^{a+b x}\right )}{b^4}\\ &=-\frac{6 x^2 \tanh ^{-1}\left (e^{a+b x}\right )}{b^2}-\frac{x^3 \text{csch}(a+b x)}{b}-\frac{6 x \text{Li}_2\left (-e^{a+b x}\right )}{b^3}+\frac{6 x \text{Li}_2\left (e^{a+b x}\right )}{b^3}+\frac{6 \text{Li}_3\left (-e^{a+b x}\right )}{b^4}-\frac{6 \text{Li}_3\left (e^{a+b x}\right )}{b^4}\\ \end{align*}
Mathematica [A] time = 6.86935, size = 167, normalized size = 1.8 \[ -\frac{\text{csch}\left (\frac{1}{2} (a+b x)\right ) \text{sech}\left (\frac{1}{2} (a+b x)\right ) \left (6 b x \sinh (a+b x) \text{PolyLog}(2,-\sinh (a+b x)-\cosh (a+b x))-6 b x \sinh (a+b x) \text{PolyLog}(2,\sinh (a+b x)+\cosh (a+b x))-6 \sinh (a+b x) \text{PolyLog}(3,-\sinh (a+b x)-\cosh (a+b x))+6 \sinh (a+b x) \text{PolyLog}(3,\sinh (a+b x)+\cosh (a+b x))+6 b^2 x^2 \sinh (a+b x) \tanh ^{-1}(\sinh (a+b x)+\cosh (a+b x))+b^3 x^3\right )}{2 b^4} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.028, size = 174, normalized size = 1.9 \begin{align*} -2\,{\frac{{x}^{3}{{\rm e}^{bx+a}}}{b \left ({{\rm e}^{2\,bx+2\,a}}-1 \right ) }}-6\,{\frac{{a}^{2}{\it Artanh} \left ({{\rm e}^{bx+a}} \right ) }{{b}^{4}}}-3\,{\frac{\ln \left ( 1+{{\rm e}^{bx+a}} \right ){x}^{2}}{{b}^{2}}}+3\,{\frac{{a}^{2}\ln \left ( 1+{{\rm e}^{bx+a}} \right ) }{{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}}}+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 = 1.38132, size = 163, normalized size = 1.75 \begin{align*} -\frac{2 \, x^{3} e^{\left (b x + a\right )}}{b e^{\left (2 \, b x + 2 \, a\right )} - b} - \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 = 2.11485, size = 1461, normalized size = 15.71 \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 )^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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