Optimal. Leaf size=165 \[ -\frac{3 x^2 \text{PolyLog}\left (2,-e^{a+b x}\right )}{b^2}+\frac{3 x^2 \text{PolyLog}\left (2,e^{a+b x}\right )}{b^2}+\frac{6 x \text{PolyLog}\left (3,-e^{a+b x}\right )}{b^3}-\frac{6 x \text{PolyLog}\left (3,e^{a+b x}\right )}{b^3}-\frac{6 \text{PolyLog}\left (4,-e^{a+b x}\right )}{b^4}+\frac{6 \text{PolyLog}\left (4,e^{a+b x}\right )}{b^4}-\frac{3 x^2 \sinh (a+b x)}{b^2}-\frac{6 \sinh (a+b x)}{b^4}+\frac{6 x \cosh (a+b x)}{b^3}+\frac{x^3 \cosh (a+b x)}{b}-\frac{2 x^3 \tanh ^{-1}\left (e^{a+b x}\right )}{b} \]
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Rubi [A] time = 0.177773, antiderivative size = 165, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 8, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {5450, 3296, 2637, 4182, 2531, 6609, 2282, 6589} \[ -\frac{3 x^2 \text{PolyLog}\left (2,-e^{a+b x}\right )}{b^2}+\frac{3 x^2 \text{PolyLog}\left (2,e^{a+b x}\right )}{b^2}+\frac{6 x \text{PolyLog}\left (3,-e^{a+b x}\right )}{b^3}-\frac{6 x \text{PolyLog}\left (3,e^{a+b x}\right )}{b^3}-\frac{6 \text{PolyLog}\left (4,-e^{a+b x}\right )}{b^4}+\frac{6 \text{PolyLog}\left (4,e^{a+b x}\right )}{b^4}-\frac{3 x^2 \sinh (a+b x)}{b^2}-\frac{6 \sinh (a+b x)}{b^4}+\frac{6 x \cosh (a+b x)}{b^3}+\frac{x^3 \cosh (a+b x)}{b}-\frac{2 x^3 \tanh ^{-1}\left (e^{a+b x}\right )}{b} \]
Antiderivative was successfully verified.
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Rule 5450
Rule 3296
Rule 2637
Rule 4182
Rule 2531
Rule 6609
Rule 2282
Rule 6589
Rubi steps
\begin{align*} \int x^3 \cosh (a+b x) \coth (a+b x) \, dx &=\int x^3 \text{csch}(a+b x) \, dx+\int x^3 \sinh (a+b x) \, dx\\ &=-\frac{2 x^3 \tanh ^{-1}\left (e^{a+b x}\right )}{b}+\frac{x^3 \cosh (a+b x)}{b}-\frac{3 \int x^2 \cosh (a+b x) \, dx}{b}-\frac{3 \int x^2 \log \left (1-e^{a+b x}\right ) \, dx}{b}+\frac{3 \int x^2 \log \left (1+e^{a+b x}\right ) \, dx}{b}\\ &=-\frac{2 x^3 \tanh ^{-1}\left (e^{a+b x}\right )}{b}+\frac{x^3 \cosh (a+b x)}{b}-\frac{3 x^2 \text{Li}_2\left (-e^{a+b x}\right )}{b^2}+\frac{3 x^2 \text{Li}_2\left (e^{a+b x}\right )}{b^2}-\frac{3 x^2 \sinh (a+b x)}{b^2}+\frac{6 \int x \text{Li}_2\left (-e^{a+b x}\right ) \, dx}{b^2}-\frac{6 \int x \text{Li}_2\left (e^{a+b x}\right ) \, dx}{b^2}+\frac{6 \int x \sinh (a+b x) \, dx}{b^2}\\ &=-\frac{2 x^3 \tanh ^{-1}\left (e^{a+b x}\right )}{b}+\frac{6 x \cosh (a+b x)}{b^3}+\frac{x^3 \cosh (a+b x)}{b}-\frac{3 x^2 \text{Li}_2\left (-e^{a+b x}\right )}{b^2}+\frac{3 x^2 \text{Li}_2\left (e^{a+b x}\right )}{b^2}+\frac{6 x \text{Li}_3\left (-e^{a+b x}\right )}{b^3}-\frac{6 x \text{Li}_3\left (e^{a+b x}\right )}{b^3}-\frac{3 x^2 \sinh (a+b x)}{b^2}-\frac{6 \int \cosh (a+b x) \, dx}{b^3}-\frac{6 \int \text{Li}_3\left (-e^{a+b x}\right ) \, dx}{b^3}+\frac{6 \int \text{Li}_3\left (e^{a+b x}\right ) \, dx}{b^3}\\ &=-\frac{2 x^3 \tanh ^{-1}\left (e^{a+b x}\right )}{b}+\frac{6 x \cosh (a+b x)}{b^3}+\frac{x^3 \cosh (a+b x)}{b}-\frac{3 x^2 \text{Li}_2\left (-e^{a+b x}\right )}{b^2}+\frac{3 x^2 \text{Li}_2\left (e^{a+b x}\right )}{b^2}+\frac{6 x \text{Li}_3\left (-e^{a+b x}\right )}{b^3}-\frac{6 x \text{Li}_3\left (e^{a+b x}\right )}{b^3}-\frac{6 \sinh (a+b x)}{b^4}-\frac{3 x^2 \sinh (a+b x)}{b^2}-\frac{6 \operatorname{Subst}\left (\int \frac{\text{Li}_3(-x)}{x} \, dx,x,e^{a+b x}\right )}{b^4}+\frac{6 \operatorname{Subst}\left (\int \frac{\text{Li}_3(x)}{x} \, dx,x,e^{a+b x}\right )}{b^4}\\ &=-\frac{2 x^3 \tanh ^{-1}\left (e^{a+b x}\right )}{b}+\frac{6 x \cosh (a+b x)}{b^3}+\frac{x^3 \cosh (a+b x)}{b}-\frac{3 x^2 \text{Li}_2\left (-e^{a+b x}\right )}{b^2}+\frac{3 x^2 \text{Li}_2\left (e^{a+b x}\right )}{b^2}+\frac{6 x \text{Li}_3\left (-e^{a+b x}\right )}{b^3}-\frac{6 x \text{Li}_3\left (e^{a+b x}\right )}{b^3}-\frac{6 \text{Li}_4\left (-e^{a+b x}\right )}{b^4}+\frac{6 \text{Li}_4\left (e^{a+b x}\right )}{b^4}-\frac{6 \sinh (a+b x)}{b^4}-\frac{3 x^2 \sinh (a+b x)}{b^2}\\ \end{align*}
Mathematica [A] time = 3.90936, size = 202, normalized size = 1.22 \[ \frac{-3 b^2 x^2 \text{PolyLog}(2,-\sinh (a+b x)-\cosh (a+b x))+3 b^2 x^2 \text{PolyLog}(2,\sinh (a+b x)+\cosh (a+b x))+6 b x \text{PolyLog}(3,-\sinh (a+b x)-\cosh (a+b x))-6 b x \text{PolyLog}(3,\sinh (a+b x)+\cosh (a+b x))-6 \text{PolyLog}(4,-\sinh (a+b x)-\cosh (a+b x))+6 \text{PolyLog}(4,\sinh (a+b x)+\cosh (a+b x))-3 b^2 x^2 \sinh (a+b x)+b^3 x^3 \cosh (a+b x)-2 b^3 x^3 \tanh ^{-1}(\sinh (a+b x)+\cosh (a+b x))-6 \sinh (a+b x)+6 b x \cosh (a+b x)}{b^4} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.092, size = 246, normalized size = 1.5 \begin{align*}{\frac{ \left ({x}^{3}{b}^{3}-3\,{x}^{2}{b}^{2}+6\,bx-6 \right ){{\rm e}^{bx+a}}}{2\,{b}^{4}}}+{\frac{ \left ({x}^{3}{b}^{3}+3\,{x}^{2}{b}^{2}+6\,bx+6 \right ){{\rm e}^{-bx-a}}}{2\,{b}^{4}}}+6\,{\frac{{\it polylog} \left ( 4,{{\rm e}^{bx+a}} \right ) }{{b}^{4}}}-{\frac{\ln \left ( 1+{{\rm e}^{bx+a}} \right ){x}^{3}}{b}}-{\frac{{a}^{3}\ln \left ( 1+{{\rm e}^{bx+a}} \right ) }{{b}^{4}}}+{\frac{\ln \left ( 1-{{\rm e}^{bx+a}} \right ){a}^{3}}{{b}^{4}}}+2\,{\frac{{a}^{3}{\it Artanh} \left ({{\rm e}^{bx+a}} \right ) }{{b}^{4}}}-6\,{\frac{{\it polylog} \left ( 4,-{{\rm e}^{bx+a}} \right ) }{{b}^{4}}}-6\,{\frac{{\it polylog} \left ( 3,{{\rm e}^{bx+a}} \right ) x}{{b}^{3}}}-3\,{\frac{{\it polylog} \left ( 2,-{{\rm e}^{bx+a}} \right ){x}^{2}}{{b}^{2}}}+6\,{\frac{{\it polylog} \left ( 3,-{{\rm e}^{bx+a}} \right ) x}{{b}^{3}}}+{\frac{\ln \left ( 1-{{\rm e}^{bx+a}} \right ){x}^{3}}{b}}+3\,{\frac{{\it polylog} \left ( 2,{{\rm e}^{bx+a}} \right ){x}^{2}}{{b}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.4724, size = 278, normalized size = 1.68 \begin{align*} \frac{{\left ({\left (b^{3} x^{3} e^{\left (2 \, a\right )} - 3 \, b^{2} x^{2} e^{\left (2 \, a\right )} + 6 \, b x e^{\left (2 \, a\right )} - 6 \, e^{\left (2 \, a\right )}\right )} e^{\left (b x\right )} +{\left (b^{3} x^{3} + 3 \, b^{2} x^{2} + 6 \, b x + 6\right )} e^{\left (-b x\right )}\right )} e^{\left (-a\right )}}{2 \, b^{4}} - \frac{b^{3} x^{3} \log \left (e^{\left (b x + a\right )} + 1\right ) + 3 \, b^{2} x^{2}{\rm Li}_2\left (-e^{\left (b x + a\right )}\right ) - 6 \, b x{\rm Li}_{3}(-e^{\left (b x + a\right )}) + 6 \,{\rm Li}_{4}(-e^{\left (b x + a\right )})}{b^{4}} + \frac{b^{3} x^{3} \log \left (-e^{\left (b x + a\right )} + 1\right ) + 3 \, b^{2} x^{2}{\rm Li}_2\left (e^{\left (b x + a\right )}\right ) - 6 \, b x{\rm Li}_{3}(e^{\left (b x + a\right )}) + 6 \,{\rm Li}_{4}(e^{\left (b x + a\right )})}{b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] time = 2.18018, size = 1409, normalized size = 8.54 \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 )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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