Optimal. Leaf size=307 \[ -\frac{x \text{PolyLog}\left (3,-\frac{(-c-d+1) e^{2 a+2 b x}}{-c+d+1}\right )}{4 b^2}+\frac{x \text{PolyLog}\left (3,-\frac{(c+d+1) e^{2 a+2 b x}}{c-d+1}\right )}{4 b^2}+\frac{\text{PolyLog}\left (4,-\frac{(-c-d+1) e^{2 a+2 b x}}{-c+d+1}\right )}{8 b^3}-\frac{\text{PolyLog}\left (4,-\frac{(c+d+1) e^{2 a+2 b x}}{c-d+1}\right )}{8 b^3}+\frac{x^2 \text{PolyLog}\left (2,-\frac{(-c-d+1) e^{2 a+2 b x}}{-c+d+1}\right )}{4 b}-\frac{x^2 \text{PolyLog}\left (2,-\frac{(c+d+1) e^{2 a+2 b x}}{c-d+1}\right )}{4 b}+\frac{1}{6} x^3 \log \left (\frac{(-c-d+1) e^{2 a+2 b x}}{-c+d+1}+1\right )-\frac{1}{6} x^3 \log \left (\frac{(c+d+1) e^{2 a+2 b x}}{c-d+1}+1\right )+\frac{1}{3} x^3 \coth ^{-1}(d \tanh (a+b x)+c) \]
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Rubi [A] time = 0.464891, antiderivative size = 307, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 6, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {6244, 2190, 2531, 6609, 2282, 6589} \[ -\frac{x \text{PolyLog}\left (3,-\frac{(-c-d+1) e^{2 a+2 b x}}{-c+d+1}\right )}{4 b^2}+\frac{x \text{PolyLog}\left (3,-\frac{(c+d+1) e^{2 a+2 b x}}{c-d+1}\right )}{4 b^2}+\frac{\text{PolyLog}\left (4,-\frac{(-c-d+1) e^{2 a+2 b x}}{-c+d+1}\right )}{8 b^3}-\frac{\text{PolyLog}\left (4,-\frac{(c+d+1) e^{2 a+2 b x}}{c-d+1}\right )}{8 b^3}+\frac{x^2 \text{PolyLog}\left (2,-\frac{(-c-d+1) e^{2 a+2 b x}}{-c+d+1}\right )}{4 b}-\frac{x^2 \text{PolyLog}\left (2,-\frac{(c+d+1) e^{2 a+2 b x}}{c-d+1}\right )}{4 b}+\frac{1}{6} x^3 \log \left (\frac{(-c-d+1) e^{2 a+2 b x}}{-c+d+1}+1\right )-\frac{1}{6} x^3 \log \left (\frac{(c+d+1) e^{2 a+2 b x}}{c-d+1}+1\right )+\frac{1}{3} x^3 \coth ^{-1}(d \tanh (a+b x)+c) \]
Antiderivative was successfully verified.
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Rule 6244
Rule 2190
Rule 2531
Rule 6609
Rule 2282
Rule 6589
Rubi steps
\begin{align*} \int x^2 \coth ^{-1}(c+d \tanh (a+b x)) \, dx &=\frac{1}{3} x^3 \coth ^{-1}(c+d \tanh (a+b x))+\frac{1}{3} (b (1-c-d)) \int \frac{e^{2 a+2 b x} x^3}{1-c+d+(1-c-d) e^{2 a+2 b x}} \, dx-\frac{1}{3} (b (1+c+d)) \int \frac{e^{2 a+2 b x} x^3}{1+c-d+(1+c+d) e^{2 a+2 b x}} \, dx\\ &=\frac{1}{3} x^3 \coth ^{-1}(c+d \tanh (a+b x))+\frac{1}{6} x^3 \log \left (1+\frac{(1-c-d) e^{2 a+2 b x}}{1-c+d}\right )-\frac{1}{6} x^3 \log \left (1+\frac{(1+c+d) e^{2 a+2 b x}}{1+c-d}\right )-\frac{1}{2} \int x^2 \log \left (1+\frac{(1-c-d) e^{2 a+2 b x}}{1-c+d}\right ) \, dx+\frac{1}{2} \int x^2 \log \left (1+\frac{(1+c+d) e^{2 a+2 b x}}{1+c-d}\right ) \, dx\\ &=\frac{1}{3} x^3 \coth ^{-1}(c+d \tanh (a+b x))+\frac{1}{6} x^3 \log \left (1+\frac{(1-c-d) e^{2 a+2 b x}}{1-c+d}\right )-\frac{1}{6} x^3 \log \left (1+\frac{(1+c+d) e^{2 a+2 b x}}{1+c-d}\right )+\frac{x^2 \text{Li}_2\left (-\frac{(1-c-d) e^{2 a+2 b x}}{1-c+d}\right )}{4 b}-\frac{x^2 \text{Li}_2\left (-\frac{(1+c+d) e^{2 a+2 b x}}{1+c-d}\right )}{4 b}-\frac{\int x \text{Li}_2\left (-\frac{(1-c-d) e^{2 a+2 b x}}{1-c+d}\right ) \, dx}{2 b}+\frac{\int x \text{Li}_2\left (-\frac{(1+c+d) e^{2 a+2 b x}}{1+c-d}\right ) \, dx}{2 b}\\ &=\frac{1}{3} x^3 \coth ^{-1}(c+d \tanh (a+b x))+\frac{1}{6} x^3 \log \left (1+\frac{(1-c-d) e^{2 a+2 b x}}{1-c+d}\right )-\frac{1}{6} x^3 \log \left (1+\frac{(1+c+d) e^{2 a+2 b x}}{1+c-d}\right )+\frac{x^2 \text{Li}_2\left (-\frac{(1-c-d) e^{2 a+2 b x}}{1-c+d}\right )}{4 b}-\frac{x^2 \text{Li}_2\left (-\frac{(1+c+d) e^{2 a+2 b x}}{1+c-d}\right )}{4 b}-\frac{x \text{Li}_3\left (-\frac{(1-c-d) e^{2 a+2 b x}}{1-c+d}\right )}{4 b^2}+\frac{x \text{Li}_3\left (-\frac{(1+c+d) e^{2 a+2 b x}}{1+c-d}\right )}{4 b^2}+\frac{\int \text{Li}_3\left (-\frac{(1-c-d) e^{2 a+2 b x}}{1-c+d}\right ) \, dx}{4 b^2}-\frac{\int \text{Li}_3\left (-\frac{(1+c+d) e^{2 a+2 b x}}{1+c-d}\right ) \, dx}{4 b^2}\\ &=\frac{1}{3} x^3 \coth ^{-1}(c+d \tanh (a+b x))+\frac{1}{6} x^3 \log \left (1+\frac{(1-c-d) e^{2 a+2 b x}}{1-c+d}\right )-\frac{1}{6} x^3 \log \left (1+\frac{(1+c+d) e^{2 a+2 b x}}{1+c-d}\right )+\frac{x^2 \text{Li}_2\left (-\frac{(1-c-d) e^{2 a+2 b x}}{1-c+d}\right )}{4 b}-\frac{x^2 \text{Li}_2\left (-\frac{(1+c+d) e^{2 a+2 b x}}{1+c-d}\right )}{4 b}-\frac{x \text{Li}_3\left (-\frac{(1-c-d) e^{2 a+2 b x}}{1-c+d}\right )}{4 b^2}+\frac{x \text{Li}_3\left (-\frac{(1+c+d) e^{2 a+2 b x}}{1+c-d}\right )}{4 b^2}+\frac{\operatorname{Subst}\left (\int \frac{\text{Li}_3\left (-\frac{(-1+c+d) x}{-1+c-d}\right )}{x} \, dx,x,e^{2 a+2 b x}\right )}{8 b^3}-\frac{\operatorname{Subst}\left (\int \frac{\text{Li}_3\left (-\frac{(1+c+d) x}{1+c-d}\right )}{x} \, dx,x,e^{2 a+2 b x}\right )}{8 b^3}\\ &=\frac{1}{3} x^3 \coth ^{-1}(c+d \tanh (a+b x))+\frac{1}{6} x^3 \log \left (1+\frac{(1-c-d) e^{2 a+2 b x}}{1-c+d}\right )-\frac{1}{6} x^3 \log \left (1+\frac{(1+c+d) e^{2 a+2 b x}}{1+c-d}\right )+\frac{x^2 \text{Li}_2\left (-\frac{(1-c-d) e^{2 a+2 b x}}{1-c+d}\right )}{4 b}-\frac{x^2 \text{Li}_2\left (-\frac{(1+c+d) e^{2 a+2 b x}}{1+c-d}\right )}{4 b}-\frac{x \text{Li}_3\left (-\frac{(1-c-d) e^{2 a+2 b x}}{1-c+d}\right )}{4 b^2}+\frac{x \text{Li}_3\left (-\frac{(1+c+d) e^{2 a+2 b x}}{1+c-d}\right )}{4 b^2}+\frac{\text{Li}_4\left (-\frac{(1-c-d) e^{2 a+2 b x}}{1-c+d}\right )}{8 b^3}-\frac{\text{Li}_4\left (-\frac{(1+c+d) e^{2 a+2 b x}}{1+c-d}\right )}{8 b^3}\\ \end{align*}
Mathematica [A] time = 0.452205, size = 345, normalized size = 1.12 \[ \frac{-6 b^2 x^2 \text{PolyLog}\left (2,\frac{(c-d-1) (\sinh (2 (a+b x))-\cosh (2 (a+b x)))}{c+d-1}\right )+6 b^2 x^2 \text{PolyLog}\left (2,\frac{(c-d+1) (\sinh (2 (a+b x))-\cosh (2 (a+b x)))}{c+d+1}\right )-6 b x \text{PolyLog}\left (3,\frac{(c-d-1) (\sinh (2 (a+b x))-\cosh (2 (a+b x)))}{c+d-1}\right )+6 b x \text{PolyLog}\left (3,\frac{(c-d+1) (\sinh (2 (a+b x))-\cosh (2 (a+b x)))}{c+d+1}\right )-3 \text{PolyLog}\left (4,\frac{(c-d-1) (\sinh (2 (a+b x))-\cosh (2 (a+b x)))}{c+d-1}\right )+3 \text{PolyLog}\left (4,\frac{(c-d+1) (\sinh (2 (a+b x))-\cosh (2 (a+b x)))}{c+d+1}\right )+4 b^3 x^3 \log \left (\frac{(c-d-1) (\cosh (2 (a+b x))-\sinh (2 (a+b x)))}{c+d-1}+1\right )-4 b^3 x^3 \log \left (\frac{(c-d+1) (\cosh (2 (a+b x))-\sinh (2 (a+b x)))}{c+d+1}+1\right )}{24 b^3}+\frac{1}{3} x^3 \coth ^{-1}(d \tanh (a+b x)+c) \]
Antiderivative was successfully verified.
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Maple [C] time = 5.085, size = 5294, normalized size = 17.2 \begin{align*} \text{output too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 2.28858, size = 379, normalized size = 1.23 \begin{align*} \frac{1}{3} \, x^{3} \operatorname{arcoth}\left (d \tanh \left (b x + a\right ) + c\right ) - \frac{1}{18} \, b d{\left (\frac{4 \, b^{3} x^{3} \log \left (\frac{{\left (c + d + 1\right )} e^{\left (2 \, b x + 2 \, a\right )}}{c - d + 1} + 1\right ) + 6 \, b^{2} x^{2}{\rm Li}_2\left (-\frac{{\left (c + d + 1\right )} e^{\left (2 \, b x + 2 \, a\right )}}{c - d + 1}\right ) - 6 \, b x{\rm Li}_{3}(-\frac{{\left (c + d + 1\right )} e^{\left (2 \, b x + 2 \, a\right )}}{c - d + 1}) + 3 \,{\rm Li}_{4}(-\frac{{\left (c + d + 1\right )} e^{\left (2 \, b x + 2 \, a\right )}}{c - d + 1})}{b^{4} d} - \frac{4 \, b^{3} x^{3} \log \left (\frac{{\left (c + d - 1\right )} e^{\left (2 \, b x + 2 \, a\right )}}{c - d - 1} + 1\right ) + 6 \, b^{2} x^{2}{\rm Li}_2\left (-\frac{{\left (c + d - 1\right )} e^{\left (2 \, b x + 2 \, a\right )}}{c - d - 1}\right ) - 6 \, b x{\rm Li}_{3}(-\frac{{\left (c + d - 1\right )} e^{\left (2 \, b x + 2 \, a\right )}}{c - d - 1}) + 3 \,{\rm Li}_{4}(-\frac{{\left (c + d - 1\right )} e^{\left (2 \, b x + 2 \, a\right )}}{c - d - 1})}{b^{4} d}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] time = 2.06443, size = 2587, normalized size = 8.43 \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} \operatorname{arcoth}\left (d \tanh \left (b x + a\right ) + c\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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