Optimal. Leaf size=49 \[ 2 \sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{\tanh ^{-1}(\tanh (a+b x))}}\right )-\frac{2 \sqrt{\tanh ^{-1}(\tanh (a+b x))}}{\sqrt{x}} \]
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Rubi [A] time = 0.0285149, antiderivative size = 49, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {2168, 2165} \[ 2 \sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{\tanh ^{-1}(\tanh (a+b x))}}\right )-\frac{2 \sqrt{\tanh ^{-1}(\tanh (a+b x))}}{\sqrt{x}} \]
Antiderivative was successfully verified.
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Rule 2168
Rule 2165
Rubi steps
\begin{align*} \int \frac{\sqrt{\tanh ^{-1}(\tanh (a+b x))}}{x^{3/2}} \, dx &=-\frac{2 \sqrt{\tanh ^{-1}(\tanh (a+b x))}}{\sqrt{x}}+b \int \frac{1}{\sqrt{x} \sqrt{\tanh ^{-1}(\tanh (a+b x))}} \, dx\\ &=2 \sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{\tanh ^{-1}(\tanh (a+b x))}}\right )-\frac{2 \sqrt{\tanh ^{-1}(\tanh (a+b x))}}{\sqrt{x}}\\ \end{align*}
Mathematica [A] time = 0.0381369, size = 52, normalized size = 1.06 \[ 2 \sqrt{b} \log \left (\sqrt{b} \sqrt{\tanh ^{-1}(\tanh (a+b x))}+b \sqrt{x}\right )-\frac{2 \sqrt{\tanh ^{-1}(\tanh (a+b x))}}{\sqrt{x}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.144, size = 149, normalized size = 3. \begin{align*} -2\,{\frac{ \left ({\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) \right ) ^{3/2}}{ \left ({\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) -bx \right ) \sqrt{x}}}+2\,{\frac{b\sqrt{x}\sqrt{{\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) }}{{\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) -bx}}+2\,{\frac{\sqrt{b}\ln \left ( \sqrt{b}\sqrt{x}+\sqrt{{\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) } \right ) a}{{\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) -bx}}+2\,{\frac{\sqrt{b}\ln \left ( \sqrt{b}\sqrt{x}+\sqrt{{\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) } \right ) \left ({\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) -bx-a \right ) }{{\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) -bx}} \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 = 2.06312, size = 243, normalized size = 4.96 \begin{align*} \left [\frac{\sqrt{b} x \log \left (2 \, b x + 2 \, \sqrt{b x + a} \sqrt{b} \sqrt{x} + a\right ) - 2 \, \sqrt{b x + a} \sqrt{x}}{x}, -\frac{2 \,{\left (\sqrt{-b} x \arctan \left (\frac{\sqrt{b x + a} \sqrt{-b}}{b \sqrt{x}}\right ) + \sqrt{b x + a} \sqrt{x}\right )}}{x}\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{\operatorname{atanh}{\left (\tanh{\left (a + b x \right )} \right )}}}{x^{\frac{3}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16806, size = 77, normalized size = 1.57 \begin{align*} -\sqrt{b} \log \left ({\left (\sqrt{b} \sqrt{x} - \sqrt{b x + a}\right )}^{2}\right ) + \frac{4 \, a \sqrt{b}}{{\left (\sqrt{b} \sqrt{x} - \sqrt{b x + a}\right )}^{2} - a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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