Optimal. Leaf size=83 \[ -\frac{3 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{b x-\tanh ^{-1}(\tanh (a+b x))}}\right ) \sqrt{b x-\tanh ^{-1}(\tanh (a+b x))}}{b^{5/2}}-\frac{x^{3/2}}{b \tanh ^{-1}(\tanh (a+b x))}+\frac{3 \sqrt{x}}{b^2} \]
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Rubi [A] time = 0.0536394, antiderivative size = 83, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {2168, 2159, 2162} \[ -\frac{3 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{b x-\tanh ^{-1}(\tanh (a+b x))}}\right ) \sqrt{b x-\tanh ^{-1}(\tanh (a+b x))}}{b^{5/2}}-\frac{x^{3/2}}{b \tanh ^{-1}(\tanh (a+b x))}+\frac{3 \sqrt{x}}{b^2} \]
Antiderivative was successfully verified.
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Rule 2168
Rule 2159
Rule 2162
Rubi steps
\begin{align*} \int \frac{x^{3/2}}{\tanh ^{-1}(\tanh (a+b x))^2} \, dx &=-\frac{x^{3/2}}{b \tanh ^{-1}(\tanh (a+b x))}+\frac{3 \int \frac{\sqrt{x}}{\tanh ^{-1}(\tanh (a+b x))} \, dx}{2 b}\\ &=\frac{3 \sqrt{x}}{b^2}-\frac{x^{3/2}}{b \tanh ^{-1}(\tanh (a+b x))}-\frac{\left (3 \left (-b x+\tanh ^{-1}(\tanh (a+b x))\right )\right ) \int \frac{1}{\sqrt{x} \tanh ^{-1}(\tanh (a+b x))} \, dx}{2 b^2}\\ &=\frac{3 \sqrt{x}}{b^2}-\frac{3 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{b x-\tanh ^{-1}(\tanh (a+b x))}}\right ) \sqrt{b x-\tanh ^{-1}(\tanh (a+b x))}}{b^{5/2}}-\frac{x^{3/2}}{b \tanh ^{-1}(\tanh (a+b x))}\\ \end{align*}
Mathematica [A] time = 0.0769652, size = 81, normalized size = 0.98 \[ -\frac{3 \sqrt{\tanh ^{-1}(\tanh (a+b x))-b x} \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{\tanh ^{-1}(\tanh (a+b x))-b x}}\right )}{b^{5/2}}-\frac{x^{3/2}}{b \tanh ^{-1}(\tanh (a+b x))}+\frac{3 \sqrt{x}}{b^2} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.135, size = 160, normalized size = 1.9 \begin{align*} 2\,{\frac{\sqrt{x}}{{b}^{2}}}+{\frac{a}{{b}^{2}{\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) }\sqrt{x}}+{\frac{{\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) -bx-a}{{b}^{2}{\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) }\sqrt{x}}-3\,{\frac{a}{{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 ) }-3\,{\frac{{\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) -bx-a}{{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 = 2.03279, size = 300, normalized size = 3.61 \begin{align*} \left [\frac{3 \,{\left (b x + a\right )} \sqrt{-\frac{a}{b}} \log \left (\frac{b x - 2 \, b \sqrt{x} \sqrt{-\frac{a}{b}} - a}{b x + a}\right ) + 2 \,{\left (2 \, b x + 3 \, a\right )} \sqrt{x}}{2 \,{\left (b^{3} x + a b^{2}\right )}}, -\frac{3 \,{\left (b x + a\right )} \sqrt{\frac{a}{b}} \arctan \left (\frac{b \sqrt{x} \sqrt{\frac{a}{b}}}{a}\right ) -{\left (2 \, b x + 3 \, a\right )} \sqrt{x}}{b^{3} x + a 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}^{2}{\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 = 1.13556, size = 62, normalized size = 0.75 \begin{align*} -\frac{3 \, a \arctan \left (\frac{b \sqrt{x}}{\sqrt{a b}}\right )}{\sqrt{a b} b^{2}} + \frac{a \sqrt{x}}{{\left (b x + a\right )} b^{2}} + \frac{2 \, \sqrt{x}}{b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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