Optimal. Leaf size=126 \[ \frac{x \text{PolyLog}\left (2,e^{2 (a+b x)}\right )}{b^2}-\frac{\text{PolyLog}\left (3,e^{2 (a+b x)}\right )}{2 b^3}+\frac{\sinh ^2(a+b x)}{4 b^3}-\frac{x \sinh (a+b x) \cosh (a+b x)}{2 b^2}+\frac{x^2 \log \left (1-e^{2 (a+b x)}\right )}{b}+\frac{x^2 \sinh ^2(a+b x)}{2 b}+\frac{x^2}{4 b}-\frac{x^3}{3} \]
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Rubi [A] time = 0.196835, antiderivative size = 126, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 9, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {5450, 5372, 3310, 30, 3716, 2190, 2531, 2282, 6589} \[ \frac{x \text{PolyLog}\left (2,e^{2 (a+b x)}\right )}{b^2}-\frac{\text{PolyLog}\left (3,e^{2 (a+b x)}\right )}{2 b^3}+\frac{\sinh ^2(a+b x)}{4 b^3}-\frac{x \sinh (a+b x) \cosh (a+b x)}{2 b^2}+\frac{x^2 \log \left (1-e^{2 (a+b x)}\right )}{b}+\frac{x^2 \sinh ^2(a+b x)}{2 b}+\frac{x^2}{4 b}-\frac{x^3}{3} \]
Antiderivative was successfully verified.
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Rule 5450
Rule 5372
Rule 3310
Rule 30
Rule 3716
Rule 2190
Rule 2531
Rule 2282
Rule 6589
Rubi steps
\begin{align*} \int x^2 \cosh ^2(a+b x) \coth (a+b x) \, dx &=\int x^2 \coth (a+b x) \, dx+\int x^2 \cosh (a+b x) \sinh (a+b x) \, dx\\ &=-\frac{x^3}{3}+\frac{x^2 \sinh ^2(a+b x)}{2 b}-2 \int \frac{e^{2 (a+b x)} x^2}{1-e^{2 (a+b x)}} \, dx-\frac{\int x \sinh ^2(a+b x) \, dx}{b}\\ &=-\frac{x^3}{3}+\frac{x^2 \log \left (1-e^{2 (a+b x)}\right )}{b}-\frac{x \cosh (a+b x) \sinh (a+b x)}{2 b^2}+\frac{\sinh ^2(a+b x)}{4 b^3}+\frac{x^2 \sinh ^2(a+b x)}{2 b}+\frac{\int x \, dx}{2 b}-\frac{2 \int x \log \left (1-e^{2 (a+b x)}\right ) \, dx}{b}\\ &=\frac{x^2}{4 b}-\frac{x^3}{3}+\frac{x^2 \log \left (1-e^{2 (a+b x)}\right )}{b}+\frac{x \text{Li}_2\left (e^{2 (a+b x)}\right )}{b^2}-\frac{x \cosh (a+b x) \sinh (a+b x)}{2 b^2}+\frac{\sinh ^2(a+b x)}{4 b^3}+\frac{x^2 \sinh ^2(a+b x)}{2 b}-\frac{\int \text{Li}_2\left (e^{2 (a+b x)}\right ) \, dx}{b^2}\\ &=\frac{x^2}{4 b}-\frac{x^3}{3}+\frac{x^2 \log \left (1-e^{2 (a+b x)}\right )}{b}+\frac{x \text{Li}_2\left (e^{2 (a+b x)}\right )}{b^2}-\frac{x \cosh (a+b x) \sinh (a+b x)}{2 b^2}+\frac{\sinh ^2(a+b x)}{4 b^3}+\frac{x^2 \sinh ^2(a+b x)}{2 b}-\frac{\operatorname{Subst}\left (\int \frac{\text{Li}_2(x)}{x} \, dx,x,e^{2 (a+b x)}\right )}{2 b^3}\\ &=\frac{x^2}{4 b}-\frac{x^3}{3}+\frac{x^2 \log \left (1-e^{2 (a+b x)}\right )}{b}+\frac{x \text{Li}_2\left (e^{2 (a+b x)}\right )}{b^2}-\frac{\text{Li}_3\left (e^{2 (a+b x)}\right )}{2 b^3}-\frac{x \cosh (a+b x) \sinh (a+b x)}{2 b^2}+\frac{\sinh ^2(a+b x)}{4 b^3}+\frac{x^2 \sinh ^2(a+b x)}{2 b}\\ \end{align*}
Mathematica [A] time = 2.51488, size = 178, normalized size = 1.41 \[ \frac{\sinh (a) (\sinh (a)+\cosh (a)) \left (-48 b x \text{PolyLog}\left (2,-e^{-a-b x}\right )-48 b x \text{PolyLog}\left (2,e^{-a-b x}\right )-48 \text{PolyLog}\left (3,-e^{-a-b x}\right )-48 \text{PolyLog}\left (3,e^{-a-b x}\right )+24 b^2 x^2 \log \left (1-e^{-a-b x}\right )+24 b^2 x^2 \log \left (e^{-a-b x}+1\right )+6 b^2 x^2 \cosh (2 (a+b x))-6 b x \sinh (2 (a+b x))+3 \cosh (2 (a+b x))+8 b^3 x^3\right )}{12 \left (e^{2 a}-1\right ) b^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.072, size = 222, normalized size = 1.8 \begin{align*} -{\frac{{x}^{3}}{3}}+{\frac{ \left ( 2\,{x}^{2}{b}^{2}-2\,bx+1 \right ){{\rm e}^{2\,bx+2\,a}}}{16\,{b}^{3}}}+{\frac{ \left ( 2\,{x}^{2}{b}^{2}+2\,bx+1 \right ){{\rm e}^{-2\,bx-2\,a}}}{16\,{b}^{3}}}+{\frac{{a}^{2}\ln \left ({{\rm e}^{bx+a}}-1 \right ) }{{b}^{3}}}-2\,{\frac{{a}^{2}\ln \left ({{\rm e}^{bx+a}} \right ) }{{b}^{3}}}+2\,{\frac{{a}^{2}x}{{b}^{2}}}+{\frac{4\,{a}^{3}}{3\,{b}^{3}}}+{\frac{\ln \left ( 1+{{\rm e}^{bx+a}} \right ){x}^{2}}{b}}+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 = 1.41126, size = 231, normalized size = 1.83 \begin{align*} -\frac{2}{3} \, x^{3} + \frac{{\left (16 \, b^{3} x^{3} e^{\left (2 \, a\right )} + 3 \,{\left (2 \, b^{2} x^{2} e^{\left (4 \, a\right )} - 2 \, b x e^{\left (4 \, a\right )} + e^{\left (4 \, a\right )}\right )} e^{\left (2 \, b x\right )} + 3 \,{\left (2 \, b^{2} x^{2} + 2 \, b x + 1\right )} e^{\left (-2 \, b x\right )}\right )} e^{\left (-2 \, a\right )}}{48 \, 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 = 2.17355, size = 1858, normalized size = 14.75 \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^{2} \cosh \left (b x + a\right )^{3} \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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