Optimal. Leaf size=115 \[ -\frac{2 x \text{PolyLog}\left (2,-e^{a+b x}\right )}{b^2}+\frac{2 x \text{PolyLog}\left (2,e^{a+b x}\right )}{b^2}+\frac{2 \text{PolyLog}\left (3,-e^{a+b x}\right )}{b^3}-\frac{2 \text{PolyLog}\left (3,e^{a+b x}\right )}{b^3}-\frac{2 x \sinh (a+b x)}{b^2}+\frac{2 \cosh (a+b x)}{b^3}+\frac{x^2 \cosh (a+b x)}{b}-\frac{2 x^2 \tanh ^{-1}\left (e^{a+b x}\right )}{b} \]
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Rubi [A] time = 0.123183, antiderivative size = 115, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 7, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.438, Rules used = {5450, 3296, 2638, 4182, 2531, 2282, 6589} \[ -\frac{2 x \text{PolyLog}\left (2,-e^{a+b x}\right )}{b^2}+\frac{2 x \text{PolyLog}\left (2,e^{a+b x}\right )}{b^2}+\frac{2 \text{PolyLog}\left (3,-e^{a+b x}\right )}{b^3}-\frac{2 \text{PolyLog}\left (3,e^{a+b x}\right )}{b^3}-\frac{2 x \sinh (a+b x)}{b^2}+\frac{2 \cosh (a+b x)}{b^3}+\frac{x^2 \cosh (a+b x)}{b}-\frac{2 x^2 \tanh ^{-1}\left (e^{a+b x}\right )}{b} \]
Antiderivative was successfully verified.
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Rule 5450
Rule 3296
Rule 2638
Rule 4182
Rule 2531
Rule 2282
Rule 6589
Rubi steps
\begin{align*} \int x^2 \cosh (a+b x) \coth (a+b x) \, dx &=\int x^2 \text{csch}(a+b x) \, dx+\int x^2 \sinh (a+b x) \, dx\\ &=-\frac{2 x^2 \tanh ^{-1}\left (e^{a+b x}\right )}{b}+\frac{x^2 \cosh (a+b x)}{b}-\frac{2 \int x \cosh (a+b x) \, dx}{b}-\frac{2 \int x \log \left (1-e^{a+b x}\right ) \, dx}{b}+\frac{2 \int x \log \left (1+e^{a+b x}\right ) \, dx}{b}\\ &=-\frac{2 x^2 \tanh ^{-1}\left (e^{a+b x}\right )}{b}+\frac{x^2 \cosh (a+b x)}{b}-\frac{2 x \text{Li}_2\left (-e^{a+b x}\right )}{b^2}+\frac{2 x \text{Li}_2\left (e^{a+b x}\right )}{b^2}-\frac{2 x \sinh (a+b x)}{b^2}+\frac{2 \int \text{Li}_2\left (-e^{a+b x}\right ) \, dx}{b^2}-\frac{2 \int \text{Li}_2\left (e^{a+b x}\right ) \, dx}{b^2}+\frac{2 \int \sinh (a+b x) \, dx}{b^2}\\ &=-\frac{2 x^2 \tanh ^{-1}\left (e^{a+b x}\right )}{b}+\frac{2 \cosh (a+b x)}{b^3}+\frac{x^2 \cosh (a+b x)}{b}-\frac{2 x \text{Li}_2\left (-e^{a+b x}\right )}{b^2}+\frac{2 x \text{Li}_2\left (e^{a+b x}\right )}{b^2}-\frac{2 x \sinh (a+b x)}{b^2}+\frac{2 \operatorname{Subst}\left (\int \frac{\text{Li}_2(-x)}{x} \, dx,x,e^{a+b x}\right )}{b^3}-\frac{2 \operatorname{Subst}\left (\int \frac{\text{Li}_2(x)}{x} \, dx,x,e^{a+b x}\right )}{b^3}\\ &=-\frac{2 x^2 \tanh ^{-1}\left (e^{a+b x}\right )}{b}+\frac{2 \cosh (a+b x)}{b^3}+\frac{x^2 \cosh (a+b x)}{b}-\frac{2 x \text{Li}_2\left (-e^{a+b x}\right )}{b^2}+\frac{2 x \text{Li}_2\left (e^{a+b x}\right )}{b^2}+\frac{2 \text{Li}_3\left (-e^{a+b x}\right )}{b^3}-\frac{2 \text{Li}_3\left (e^{a+b x}\right )}{b^3}-\frac{2 x \sinh (a+b x)}{b^2}\\ \end{align*}
Mathematica [A] time = 3.82008, size = 138, normalized size = 1.2 \[ \frac{-2 b x \text{PolyLog}(2,-\sinh (a+b x)-\cosh (a+b x))+2 b x \text{PolyLog}(2,\sinh (a+b x)+\cosh (a+b x))+2 \text{PolyLog}(3,-\sinh (a+b x)-\cosh (a+b x))-2 \text{PolyLog}(3,\sinh (a+b x)+\cosh (a+b x))+b^2 x^2 \cosh (a+b x)-2 b^2 x^2 \tanh ^{-1}(\sinh (a+b x)+\cosh (a+b x))-2 b x \sinh (a+b x)+2 \cosh (a+b x)}{b^3} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.059, size = 196, normalized size = 1.7 \begin{align*}{\frac{ \left ({x}^{2}{b}^{2}-2\,bx+2 \right ){{\rm e}^{bx+a}}}{2\,{b}^{3}}}+{\frac{ \left ({x}^{2}{b}^{2}+2\,bx+2 \right ){{\rm e}^{-bx-a}}}{2\,{b}^{3}}}-2\,{\frac{{a}^{2}{\it Artanh} \left ({{\rm e}^{bx+a}} \right ) }{{b}^{3}}}-{\frac{\ln \left ( 1+{{\rm e}^{bx+a}} \right ){x}^{2}}{b}}+{\frac{{a}^{2}\ln \left ( 1+{{\rm e}^{bx+a}} \right ) }{{b}^{3}}}-2\,{\frac{x{\it polylog} \left ( 2,-{{\rm e}^{bx+a}} \right ) }{{b}^{2}}}+2\,{\frac{{\it polylog} \left ( 3,-{{\rm e}^{bx+a}} \right ) }{{b}^{3}}}+{\frac{\ln \left ( 1-{{\rm e}^{bx+a}} \right ){x}^{2}}{b}}-{\frac{\ln \left ( 1-{{\rm e}^{bx+a}} \right ){a}^{2}}{{b}^{3}}}+2\,{\frac{x{\it polylog} \left ( 2,{{\rm e}^{bx+a}} \right ) }{{b}^{2}}}-2\,{\frac{{\it polylog} \left ( 3,{{\rm e}^{bx+a}} \right ) }{{b}^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.34865, size = 205, normalized size = 1.78 \begin{align*} \frac{{\left ({\left (b^{2} x^{2} e^{\left (2 \, a\right )} - 2 \, b x e^{\left (2 \, a\right )} + 2 \, e^{\left (2 \, a\right )}\right )} e^{\left (b x\right )} +{\left (b^{2} x^{2} + 2 \, b x + 2\right )} e^{\left (-b x\right )}\right )} e^{\left (-a\right )}}{2 \, b^{3}} - \frac{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 )})}{b^{3}} + \frac{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 )})}{b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] time = 2.20668, size = 1084, normalized size = 9.43 \begin{align*} \frac{b^{2} x^{2} +{\left (b^{2} x^{2} - 2 \, b x + 2\right )} \cosh \left (b x + a\right )^{2} + 2 \,{\left (b^{2} x^{2} - 2 \, b x + 2\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) +{\left (b^{2} x^{2} - 2 \, b x + 2\right )} \sinh \left (b x + a\right )^{2} + 2 \, b x + 4 \,{\left (b x \cosh \left (b x + a\right ) + b x \sinh \left (b x + a\right )\right )}{\rm Li}_2\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right ) - 4 \,{\left (b x \cosh \left (b x + a\right ) + b x \sinh \left (b x + a\right )\right )}{\rm Li}_2\left (-\cosh \left (b x + a\right ) - \sinh \left (b x + a\right )\right ) - 2 \,{\left (b^{2} x^{2} \cosh \left (b x + a\right ) + b^{2} x^{2} \sinh \left (b x + a\right )\right )} \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) + 1\right ) + 2 \,{\left (a^{2} \cosh \left (b x + a\right ) + a^{2} \sinh \left (b x + a\right )\right )} \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) - 1\right ) + 2 \,{\left ({\left (b^{2} x^{2} - a^{2}\right )} \cosh \left (b x + a\right ) +{\left (b^{2} x^{2} - a^{2}\right )} \sinh \left (b x + a\right )\right )} \log \left (-\cosh \left (b x + a\right ) - \sinh \left (b x + a\right ) + 1\right ) - 4 \,{\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )}{\rm polylog}\left (3, \cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right ) + 4 \,{\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )}{\rm polylog}\left (3, -\cosh \left (b x + a\right ) - \sinh \left (b x + a\right )\right ) + 2}{2 \,{\left (b^{3} \cosh \left (b x + a\right ) + b^{3} \sinh \left (b x + a\right )\right )}} \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^{2} \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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