Optimal. Leaf size=36 \[ \frac{2 x \sqrt{\tanh ^{-1}(\tanh (a+b x))}}{b}-\frac{4 \tanh ^{-1}(\tanh (a+b x))^{3/2}}{3 b^2} \]
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Rubi [A] time = 0.0145474, antiderivative size = 36, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {2168, 2157, 30} \[ \frac{2 x \sqrt{\tanh ^{-1}(\tanh (a+b x))}}{b}-\frac{4 \tanh ^{-1}(\tanh (a+b x))^{3/2}}{3 b^2} \]
Antiderivative was successfully verified.
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Rule 2168
Rule 2157
Rule 30
Rubi steps
\begin{align*} \int \frac{x}{\sqrt{\tanh ^{-1}(\tanh (a+b x))}} \, dx &=\frac{2 x \sqrt{\tanh ^{-1}(\tanh (a+b x))}}{b}-\frac{2 \int \sqrt{\tanh ^{-1}(\tanh (a+b x))} \, dx}{b}\\ &=\frac{2 x \sqrt{\tanh ^{-1}(\tanh (a+b x))}}{b}-\frac{2 \operatorname{Subst}\left (\int \sqrt{x} \, dx,x,\tanh ^{-1}(\tanh (a+b x))\right )}{b^2}\\ &=\frac{2 x \sqrt{\tanh ^{-1}(\tanh (a+b x))}}{b}-\frac{4 \tanh ^{-1}(\tanh (a+b x))^{3/2}}{3 b^2}\\ \end{align*}
Mathematica [A] time = 0.0514767, size = 32, normalized size = 0.89 \[ \frac{2 \left (3 b x-2 \tanh ^{-1}(\tanh (a+b x))\right ) \sqrt{\tanh ^{-1}(\tanh (a+b x))}}{3 b^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.062, size = 56, normalized size = 1.6 \begin{align*} 2\,{\frac{1/3\, \left ({\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) \right ) ^{3/2}-a\sqrt{{\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) }- \left ({\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) -bx-a \right ) \sqrt{{\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) }}{{b}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.77387, size = 41, normalized size = 1.14 \begin{align*} \frac{2 \,{\left (b^{2} x^{2} - a b x - 2 \, a^{2}\right )}}{3 \, \sqrt{b x + a} b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.33754, size = 47, normalized size = 1.31 \begin{align*} \frac{2 \, \sqrt{b x + a}{\left (b x - 2 \, a\right )}}{3 \, b^{2}} \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}{\sqrt{\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 = 1.14714, size = 31, normalized size = 0.86 \begin{align*} \frac{2 \,{\left ({\left (b x + a\right )}^{\frac{3}{2}} - 3 \, \sqrt{b x + a} a\right )}}{3 \, b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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