Optimal. Leaf size=56 \[ \frac{x \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )}{b^2}+\frac{\left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^2 \log \left (\tanh ^{-1}(\tanh (a+b x))\right )}{b^3}+\frac{x^2}{2 b} \]
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Rubi [A] time = 0.0346303, 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} \[ \frac{x \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )}{b^2}+\frac{\left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^2 \log \left (\tanh ^{-1}(\tanh (a+b x))\right )}{b^3}+\frac{x^2}{2 b} \]
Antiderivative was successfully verified.
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Rule 2159
Rule 2158
Rule 2157
Rule 29
Rubi steps
\begin{align*} \int \frac{x^2}{\tanh ^{-1}(\tanh (a+b x))} \, dx &=\frac{x^2}{2 b}-\frac{\left (-b x+\tanh ^{-1}(\tanh (a+b x))\right ) \int \frac{x}{\tanh ^{-1}(\tanh (a+b x))} \, dx}{b}\\ &=\frac{x^2}{2 b}+\frac{x \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )}{b^2}+\frac{\left (-b x+\tanh ^{-1}(\tanh (a+b x))\right )^2 \int \frac{1}{\tanh ^{-1}(\tanh (a+b x))} \, dx}{b^2}\\ &=\frac{x^2}{2 b}+\frac{x \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )}{b^2}+\frac{\left (-b x+\tanh ^{-1}(\tanh (a+b x))\right )^2 \operatorname{Subst}\left (\int \frac{1}{x} \, dx,x,\tanh ^{-1}(\tanh (a+b x))\right )}{b^3}\\ &=\frac{x^2}{2 b}+\frac{x \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )}{b^2}+\frac{\left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^2 \log \left (\tanh ^{-1}(\tanh (a+b x))\right )}{b^3}\\ \end{align*}
Mathematica [A] time = 0.0322665, size = 55, normalized size = 0.98 \[ -\frac{x \left (\tanh ^{-1}(\tanh (a+b x))-b x\right )}{b^2}+\frac{\left (\tanh ^{-1}(\tanh (a+b x))-b x\right )^2 \log \left (\tanh ^{-1}(\tanh (a+b x))\right )}{b^3}+\frac{x^2}{2 b} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.033, size = 111, normalized size = 2. \begin{align*}{\frac{{x}^{2}}{2\,b}}-{\frac{ax}{{b}^{2}}}-{\frac{ \left ({\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) -bx-a \right ) x}{{b}^{2}}}+{\frac{\ln \left ({\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) \right ){a}^{2}}{{b}^{3}}}+2\,{\frac{\ln \left ({\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) \right ) a \left ({\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) -bx-a \right ) }{{b}^{3}}}+{\frac{\ln \left ({\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) \right ) \left ({\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) -bx-a \right ) ^{2}}{{b}^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.76817, size = 39, normalized size = 0.7 \begin{align*} \frac{a^{2} \log \left (b x + a\right )}{b^{3}} + \frac{b x^{2} - 2 \, a x}{2 \, b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.50836, size = 68, normalized size = 1.21 \begin{align*} \frac{b^{2} x^{2} - 2 \, a b x + 2 \, a^{2} \log \left (b x + a\right )}{2 \, 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{atanh}{\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 [A] time = 1.13289, size = 41, normalized size = 0.73 \begin{align*} \frac{a^{2} \log \left ({\left | b x + a \right |}\right )}{b^{3}} + \frac{b x^{2} - 2 \, a x}{2 \, b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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