Optimal. Leaf size=63 \[ \frac{x \text{PolyLog}\left (2,e^{2 (a+b x)}\right )}{b^2}-\frac{\text{PolyLog}\left (3,e^{2 (a+b x)}\right )}{2 b^3}+\frac{x^2 \log \left (1-e^{2 (a+b x)}\right )}{b}-\frac{x^3}{3} \]
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Rubi [A] time = 0.132667, antiderivative size = 63, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {3716, 2190, 2531, 2282, 6589} \[ \frac{x \text{PolyLog}\left (2,e^{2 (a+b x)}\right )}{b^2}-\frac{\text{PolyLog}\left (3,e^{2 (a+b x)}\right )}{2 b^3}+\frac{x^2 \log \left (1-e^{2 (a+b x)}\right )}{b}-\frac{x^3}{3} \]
Antiderivative was successfully verified.
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Rule 3716
Rule 2190
Rule 2531
Rule 2282
Rule 6589
Rubi steps
\begin{align*} \int x^2 \coth (a+b x) \, dx &=-\frac{x^3}{3}-2 \int \frac{e^{2 (a+b x)} x^2}{1-e^{2 (a+b x)}} \, dx\\ &=-\frac{x^3}{3}+\frac{x^2 \log \left (1-e^{2 (a+b x)}\right )}{b}-\frac{2 \int x \log \left (1-e^{2 (a+b x)}\right ) \, dx}{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{\int \text{Li}_2\left (e^{2 (a+b x)}\right ) \, dx}{b^2}\\ &=-\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{\operatorname{Subst}\left (\int \frac{\text{Li}_2(x)}{x} \, dx,x,e^{2 (a+b x)}\right )}{2 b^3}\\ &=-\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}\\ \end{align*}
Mathematica [A] time = 0.0098819, size = 66, normalized size = 1.05 \[ \frac{x \text{PolyLog}\left (2,e^{2 a+2 b x}\right )}{b^2}-\frac{\text{PolyLog}\left (3,e^{2 a+2 b x}\right )}{2 b^3}+\frac{x^2 \log \left (1-e^{2 a+2 b x}\right )}{b}-\frac{x^3}{3} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.023, size = 166, normalized size = 2.6 \begin{align*} -{\frac{{x}^{3}}{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 = 1.29407, size = 130, normalized size = 2.06 \begin{align*} -\frac{1}{3} \, x^{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 = 2.10502, size = 489, normalized size = 7.76 \begin{align*} -\frac{b^{3} x^{3} - 3 \, b^{2} x^{2} \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) + 1\right ) - 6 \, b x{\rm Li}_2\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right ) - 6 \, b x{\rm Li}_2\left (-\cosh \left (b x + a\right ) - \sinh \left (b x + a\right )\right ) - 3 \, a^{2} \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) - 1\right ) - 3 \,{\left (b^{2} x^{2} - a^{2}\right )} \log \left (-\cosh \left (b x + a\right ) - \sinh \left (b x + a\right ) + 1\right ) + 6 \,{\rm polylog}\left (3, \cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right ) + 6 \,{\rm polylog}\left (3, -\cosh \left (b x + a\right ) - \sinh \left (b x + a\right )\right )}{3 \, b^{3}} \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 ) \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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