Optimal. Leaf size=89 \[ \frac{2 \sqrt{x} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )}{b^2}-\frac{2 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{b x-\tanh ^{-1}(\tanh (a+b x))}}\right ) \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^{3/2}}{b^{5/2}}+\frac{2 x^{3/2}}{3 b} \]
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Rubi [A] time = 0.0556897, antiderivative size = 89, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {2159, 2162} \[ \frac{2 \sqrt{x} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )}{b^2}-\frac{2 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{b x-\tanh ^{-1}(\tanh (a+b x))}}\right ) \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^{3/2}}{b^{5/2}}+\frac{2 x^{3/2}}{3 b} \]
Antiderivative was successfully verified.
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Rule 2159
Rule 2162
Rubi steps
\begin{align*} \int \frac{x^{3/2}}{\tanh ^{-1}(\tanh (a+b x))} \, dx &=\frac{2 x^{3/2}}{3 b}-\frac{\left (-b x+\tanh ^{-1}(\tanh (a+b x))\right ) \int \frac{\sqrt{x}}{\tanh ^{-1}(\tanh (a+b x))} \, dx}{b}\\ &=\frac{2 x^{3/2}}{3 b}+\frac{2 \sqrt{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}{\sqrt{x} \tanh ^{-1}(\tanh (a+b x))} \, dx}{b^2}\\ &=\frac{2 x^{3/2}}{3 b}+\frac{2 \sqrt{x} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )}{b^2}-\frac{2 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{b x-\tanh ^{-1}(\tanh (a+b x))}}\right ) \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^{3/2}}{b^{5/2}}\\ \end{align*}
Mathematica [A] time = 0.0892655, size = 86, normalized size = 0.97 \[ -\frac{2 \sqrt{x} \left (\tanh ^{-1}(\tanh (a+b x))-b x\right )}{b^2}+\frac{2 \left (\tanh ^{-1}(\tanh (a+b x))-b x\right )^{3/2} \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{\tanh ^{-1}(\tanh (a+b x))-b x}}\right )}{b^{5/2}}+\frac{2 x^{3/2}}{3 b} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.126, size = 207, normalized size = 2.3 \begin{align*}{\frac{2}{3\,b}{x}^{{\frac{3}{2}}}}-2\,{\frac{a\sqrt{x}}{{b}^{2}}}-2\,{\frac{ \left ({\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) -bx-a \right ) \sqrt{x}}{{b}^{2}}}+2\,{\frac{{a}^{2}}{{b}^{2}\sqrt{ \left ({\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) -bx \right ) b}}\arctan \left ({\frac{b\sqrt{x}}{\sqrt{ \left ({\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) -bx \right ) b}}} \right ) }+4\,{\frac{a \left ({\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) -bx-a \right ) }{{b}^{2}\sqrt{ \left ({\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) -bx \right ) b}}\arctan \left ({\frac{b\sqrt{x}}{\sqrt{ \left ({\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) -bx \right ) b}}} \right ) }+2\,{\frac{ \left ({\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) -bx-a \right ) ^{2}}{{b}^{2}\sqrt{ \left ({\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) -bx \right ) b}}\arctan \left ({\frac{b\sqrt{x}}{\sqrt{ \left ({\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) -bx \right ) b}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.10917, size = 244, normalized size = 2.74 \begin{align*} \left [\frac{3 \, a \sqrt{-\frac{a}{b}} \log \left (\frac{b x + 2 \, b \sqrt{x} \sqrt{-\frac{a}{b}} - a}{b x + a}\right ) + 2 \,{\left (b x - 3 \, a\right )} \sqrt{x}}{3 \, b^{2}}, \frac{2 \,{\left (3 \, a \sqrt{\frac{a}{b}} \arctan \left (\frac{b \sqrt{x} \sqrt{\frac{a}{b}}}{a}\right ) +{\left (b x - 3 \, a\right )} \sqrt{x}\right )}}{3 \, b^{2}}\right ] \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^{\frac{3}{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 = 1.1186, size = 61, normalized size = 0.69 \begin{align*} \frac{2 \, a^{2} \arctan \left (\frac{b \sqrt{x}}{\sqrt{a b}}\right )}{\sqrt{a b} b^{2}} + \frac{2 \,{\left (b^{2} x^{\frac{3}{2}} - 3 \, a b \sqrt{x}\right )}}{3 \, b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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