Optimal. Leaf size=56 \[ \frac{x \left (b x-\coth ^{-1}(\tanh (a+b x))\right )}{b^2}+\frac{\left (b x-\coth ^{-1}(\tanh (a+b x))\right )^2 \log \left (\coth ^{-1}(\tanh (a+b x))\right )}{b^3}+\frac{x^2}{2 b} \]
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Rubi [A] time = 0.0346625, antiderivative size = 56, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308, Rules used = {2159, 2158, 2157, 29} \[ \frac{x \left (b x-\coth ^{-1}(\tanh (a+b x))\right )}{b^2}+\frac{\left (b x-\coth ^{-1}(\tanh (a+b x))\right )^2 \log \left (\coth ^{-1}(\tanh (a+b x))\right )}{b^3}+\frac{x^2}{2 b} \]
Antiderivative was successfully verified.
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Rule 2159
Rule 2158
Rule 2157
Rule 29
Rubi steps
\begin{align*} \int \frac{x^2}{\coth ^{-1}(\tanh (a+b x))} \, dx &=\frac{x^2}{2 b}-\frac{\left (-b x+\coth ^{-1}(\tanh (a+b x))\right ) \int \frac{x}{\coth ^{-1}(\tanh (a+b x))} \, dx}{b}\\ &=\frac{x^2}{2 b}+\frac{x \left (b x-\coth ^{-1}(\tanh (a+b x))\right )}{b^2}+\frac{\left (-b x+\coth ^{-1}(\tanh (a+b x))\right )^2 \int \frac{1}{\coth ^{-1}(\tanh (a+b x))} \, dx}{b^2}\\ &=\frac{x^2}{2 b}+\frac{x \left (b x-\coth ^{-1}(\tanh (a+b x))\right )}{b^2}+\frac{\left (-b x+\coth ^{-1}(\tanh (a+b x))\right )^2 \operatorname{Subst}\left (\int \frac{1}{x} \, dx,x,\coth ^{-1}(\tanh (a+b x))\right )}{b^3}\\ &=\frac{x^2}{2 b}+\frac{x \left (b x-\coth ^{-1}(\tanh (a+b x))\right )}{b^2}+\frac{\left (b x-\coth ^{-1}(\tanh (a+b x))\right )^2 \log \left (\coth ^{-1}(\tanh (a+b x))\right )}{b^3}\\ \end{align*}
Mathematica [A] time = 0.0383789, size = 55, normalized size = 0.98 \[ -\frac{x \left (\coth ^{-1}(\tanh (a+b x))-b x\right )}{b^2}+\frac{\left (\coth ^{-1}(\tanh (a+b x))-b x\right )^2 \log \left (\coth ^{-1}(\tanh (a+b x))\right )}{b^3}+\frac{x^2}{2 b} \]
Antiderivative was successfully verified.
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Maple [C] time = 1.134, size = 28786, normalized size = 514. \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 [C] time = 1.82435, size = 69, normalized size = 1.23 \begin{align*} \frac{b x^{2} +{\left (i \, \pi - 2 \, a\right )} x}{2 \, b^{2}} - \frac{{\left (\pi ^{2} + 4 i \, \pi a - 4 \, a^{2}\right )} \log \left (-i \, \pi + 2 \, b x + 2 \, a\right )}{4 \, b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.66004, size = 225, normalized size = 4.02 \begin{align*} \frac{4 \, b^{2} x^{2} - 8 \, a b x - 16 \, \pi a \arctan \left (-\frac{2 \, b x + 2 \, a - \sqrt{4 \, b^{2} x^{2} + 8 \, a b x + \pi ^{2} + 4 \, a^{2}}}{\pi }\right ) -{\left (\pi ^{2} - 4 \, a^{2}\right )} \log \left (4 \, b^{2} x^{2} + 8 \, a b x + \pi ^{2} + 4 \, a^{2}\right )}{8 \, b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{2}}{\operatorname{acoth}{\left (\tanh{\left (a + b x \right )} \right )}}\, dx \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 \frac{x^{2}}{\operatorname{arcoth}\left (\tanh \left (b x + a\right )\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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