Optimal. Leaf size=65 \[ \frac{\text{PolyLog}\left (2,e^{2 (a+b x)}\right )}{b^3}+\frac{2 x \log \left (1-e^{2 (a+b x)}\right )}{b^2}-\frac{x^2 \coth (a+b x)}{b}-\frac{x^2}{b}+\frac{x^3}{3} \]
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Rubi [A] time = 0.124658, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {3720, 3716, 2190, 2279, 2391, 30} \[ \frac{\text{PolyLog}\left (2,e^{2 (a+b x)}\right )}{b^3}+\frac{2 x \log \left (1-e^{2 (a+b x)}\right )}{b^2}-\frac{x^2 \coth (a+b x)}{b}-\frac{x^2}{b}+\frac{x^3}{3} \]
Antiderivative was successfully verified.
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Rule 3720
Rule 3716
Rule 2190
Rule 2279
Rule 2391
Rule 30
Rubi steps
\begin{align*} \int x^2 \coth ^2(a+b x) \, dx &=-\frac{x^2 \coth (a+b x)}{b}+\frac{2 \int x \coth (a+b x) \, dx}{b}+\int x^2 \, dx\\ &=-\frac{x^2}{b}+\frac{x^3}{3}-\frac{x^2 \coth (a+b x)}{b}-\frac{4 \int \frac{e^{2 (a+b x)} x}{1-e^{2 (a+b x)}} \, dx}{b}\\ &=-\frac{x^2}{b}+\frac{x^3}{3}-\frac{x^2 \coth (a+b x)}{b}+\frac{2 x \log \left (1-e^{2 (a+b x)}\right )}{b^2}-\frac{2 \int \log \left (1-e^{2 (a+b x)}\right ) \, dx}{b^2}\\ &=-\frac{x^2}{b}+\frac{x^3}{3}-\frac{x^2 \coth (a+b x)}{b}+\frac{2 x \log \left (1-e^{2 (a+b x)}\right )}{b^2}-\frac{\operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{2 (a+b x)}\right )}{b^3}\\ &=-\frac{x^2}{b}+\frac{x^3}{3}-\frac{x^2 \coth (a+b x)}{b}+\frac{2 x \log \left (1-e^{2 (a+b x)}\right )}{b^2}+\frac{\text{Li}_2\left (e^{2 (a+b x)}\right )}{b^3}\\ \end{align*}
Mathematica [C] time = 4.88174, size = 163, normalized size = 2.51 \[ \frac{-\text{PolyLog}\left (2,e^{-2 \left (\tanh ^{-1}(\tanh (a))+b x\right )}\right )-b^2 x^2 e^{-\tanh ^{-1}(\tanh (a))} \coth (a) \sqrt{\text{sech}^2(a)}+2 b x \log \left (1-e^{-2 \left (\tanh ^{-1}(\tanh (a))+b x\right )}\right )+2 \tanh ^{-1}(\tanh (a)) \left (\log \left (1-e^{-2 \left (\tanh ^{-1}(\tanh (a))+b x\right )}\right )-\log \left (i \sinh \left (\tanh ^{-1}(\tanh (a))+b x\right )\right )+b x\right )+i \pi b x-i \pi \log \left (e^{2 b x}+1\right )+i \pi \log (\cosh (b x))}{b^3}+\frac{x^2 \text{csch}(a) \sinh (b x) \text{csch}(a+b x)}{b}+\frac{x^3}{3} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.056, size = 156, normalized size = 2.4 \begin{align*}{\frac{{x}^{3}}{3}}-2\,{\frac{{x}^{2}}{b \left ({{\rm e}^{2\,bx+2\,a}}-1 \right ) }}-2\,{\frac{{x}^{2}}{b}}-4\,{\frac{ax}{{b}^{2}}}-2\,{\frac{{a}^{2}}{{b}^{3}}}+2\,{\frac{\ln \left ( 1+{{\rm e}^{bx+a}} \right ) x}{{b}^{2}}}+2\,{\frac{{\it polylog} \left ( 2,-{{\rm e}^{bx+a}} \right ) }{{b}^{3}}}+2\,{\frac{\ln \left ( 1-{{\rm e}^{bx+a}} \right ) x}{{b}^{2}}}+2\,{\frac{\ln \left ( 1-{{\rm e}^{bx+a}} \right ) a}{{b}^{3}}}+2\,{\frac{{\it polylog} \left ( 2,{{\rm e}^{bx+a}} \right ) }{{b}^{3}}}-2\,{\frac{a\ln \left ({{\rm e}^{bx+a}}-1 \right ) }{{b}^{3}}}+4\,{\frac{a\ln \left ({{\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 = 1.40041, size = 146, normalized size = 2.25 \begin{align*} -\frac{2 \, x^{2}}{b} + \frac{b x^{3} e^{\left (2 \, b x + 2 \, a\right )} - b x^{3} - 6 \, x^{2}}{3 \,{\left (b e^{\left (2 \, b x + 2 \, a\right )} - b\right )}} + \frac{2 \,{\left (b x \log \left (e^{\left (b x + a\right )} + 1\right ) +{\rm Li}_2\left (-e^{\left (b x + a\right )}\right )\right )}}{b^{3}} + \frac{2 \,{\left (b x \log \left (-e^{\left (b x + a\right )} + 1\right ) +{\rm Li}_2\left (e^{\left (b x + a\right )}\right )\right )}}{b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.17313, size = 1211, normalized size = 18.63 \begin{align*} -\frac{b^{3} x^{3} -{\left (b^{3} x^{3} - 6 \, b^{2} x^{2} + 6 \, a^{2}\right )} \cosh \left (b x + a\right )^{2} - 2 \,{\left (b^{3} x^{3} - 6 \, b^{2} x^{2} + 6 \, a^{2}\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) -{\left (b^{3} x^{3} - 6 \, b^{2} x^{2} + 6 \, a^{2}\right )} \sinh \left (b x + a\right )^{2} + 6 \, a^{2} - 6 \,{\left (\cosh \left (b x + a\right )^{2} + 2 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + \sinh \left (b x + a\right )^{2} - 1\right )}{\rm Li}_2\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right ) - 6 \,{\left (\cosh \left (b x + a\right )^{2} + 2 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + \sinh \left (b x + a\right )^{2} - 1\right )}{\rm Li}_2\left (-\cosh \left (b x + a\right ) - \sinh \left (b x + a\right )\right ) - 6 \,{\left (b x \cosh \left (b x + a\right )^{2} + 2 \, b x \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + b x \sinh \left (b x + a\right )^{2} - b x\right )} \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) + 1\right ) + 6 \,{\left (a \cosh \left (b x + a\right )^{2} + 2 \, a \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + a \sinh \left (b x + a\right )^{2} - a\right )} \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) - 1\right ) - 6 \,{\left ({\left (b x + a\right )} \cosh \left (b x + a\right )^{2} + 2 \,{\left (b x + a\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) +{\left (b x + a\right )} \sinh \left (b x + a\right )^{2} - b x - a\right )} \log \left (-\cosh \left (b x + a\right ) - \sinh \left (b x + a\right ) + 1\right )}{3 \,{\left (b^{3} \cosh \left (b x + a\right )^{2} + 2 \, b^{3} \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + b^{3} \sinh \left (b x + a\right )^{2} - b^{3}\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 )^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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