Optimal. Leaf size=64 \[ \frac{2 \sqrt{x}}{b}-\frac{2 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{b x-\tanh ^{-1}(\tanh (a+b x))}}\right ) \sqrt{b x-\tanh ^{-1}(\tanh (a+b x))}}{b^{3/2}} \]
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Rubi [A] time = 0.03323, antiderivative size = 64, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {2159, 2162} \[ \frac{2 \sqrt{x}}{b}-\frac{2 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{b x-\tanh ^{-1}(\tanh (a+b x))}}\right ) \sqrt{b x-\tanh ^{-1}(\tanh (a+b x))}}{b^{3/2}} \]
Antiderivative was successfully verified.
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Rule 2159
Rule 2162
Rubi steps
\begin{align*} \int \frac{\sqrt{x}}{\tanh ^{-1}(\tanh (a+b x))} \, dx &=\frac{2 \sqrt{x}}{b}-\frac{\left (-b x+\tanh ^{-1}(\tanh (a+b x))\right ) \int \frac{1}{\sqrt{x} \tanh ^{-1}(\tanh (a+b x))} \, dx}{b}\\ &=\frac{2 \sqrt{x}}{b}-\frac{2 \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^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.0429287, size = 62, normalized size = 0.97 \[ \frac{2 \sqrt{x}}{b}-\frac{2 \sqrt{\tanh ^{-1}(\tanh (a+b x))-b x} \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{\tanh ^{-1}(\tanh (a+b x))-b x}}\right )}{b^{3/2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.126, size = 112, normalized size = 1.8 \begin{align*} 2\,{\frac{\sqrt{x}}{b}}-2\,{\frac{a}{b\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{{\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) -bx-a}{b\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 = 2.08856, size = 189, normalized size = 2.95 \begin{align*} \left [\frac{\sqrt{-\frac{a}{b}} \log \left (\frac{b x - 2 \, b \sqrt{x} \sqrt{-\frac{a}{b}} - a}{b x + a}\right ) + 2 \, \sqrt{x}}{b}, -\frac{2 \,{\left (\sqrt{\frac{a}{b}} \arctan \left (\frac{b \sqrt{x} \sqrt{\frac{a}{b}}}{a}\right ) - \sqrt{x}\right )}}{b}\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{\sqrt{x}}{\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 = 1.133, size = 42, normalized size = 0.66 \begin{align*} -\frac{2 \, a \arctan \left (\frac{b \sqrt{x}}{\sqrt{a b}}\right )}{\sqrt{a b} b} + \frac{2 \, \sqrt{x}}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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