Optimal. Leaf size=73 \[ -\frac{\tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{b x-\tanh ^{-1}(\tanh (a+b x))}}\right )}{b^{3/2} \sqrt{b x-\tanh ^{-1}(\tanh (a+b x))}}-\frac{\sqrt{x}}{b \tanh ^{-1}(\tanh (a+b x))} \]
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Rubi [A] time = 0.0341577, antiderivative size = 73, 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 = {2168, 2162} \[ -\frac{\tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{b x-\tanh ^{-1}(\tanh (a+b x))}}\right )}{b^{3/2} \sqrt{b x-\tanh ^{-1}(\tanh (a+b x))}}-\frac{\sqrt{x}}{b \tanh ^{-1}(\tanh (a+b x))} \]
Antiderivative was successfully verified.
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Rule 2168
Rule 2162
Rubi steps
\begin{align*} \int \frac{\sqrt{x}}{\tanh ^{-1}(\tanh (a+b x))^2} \, dx &=-\frac{\sqrt{x}}{b \tanh ^{-1}(\tanh (a+b x))}+\frac{\int \frac{1}{\sqrt{x} \tanh ^{-1}(\tanh (a+b x))} \, dx}{2 b}\\ &=-\frac{\tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{b x-\tanh ^{-1}(\tanh (a+b x))}}\right )}{b^{3/2} \sqrt{b x-\tanh ^{-1}(\tanh (a+b x))}}-\frac{\sqrt{x}}{b \tanh ^{-1}(\tanh (a+b x))}\\ \end{align*}
Mathematica [A] time = 0.0598, size = 70, normalized size = 0.96 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{\tanh ^{-1}(\tanh (a+b x))-b x}}\right )}{b^{3/2} \sqrt{\tanh ^{-1}(\tanh (a+b x))-b x}}-\frac{\sqrt{x}}{b \tanh ^{-1}(\tanh (a+b x))} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.135, size = 61, normalized size = 0.8 \begin{align*} -{\frac{1}{b{\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) }\sqrt{x}}+{\frac{1}{b}\arctan \left ({b\sqrt{x}{\frac{1}{\sqrt{ \left ({\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) -bx \right ) b}}}} \right ){\frac{1}{\sqrt{ \left ({\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) -bx \right ) b}}}} \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 = 2.12579, size = 277, normalized size = 3.79 \begin{align*} \left [-\frac{2 \, a b \sqrt{x} + \sqrt{-a b}{\left (b x + a\right )} \log \left (\frac{b x - a - 2 \, \sqrt{-a b} \sqrt{x}}{b x + a}\right )}{2 \,{\left (a b^{3} x + a^{2} b^{2}\right )}}, -\frac{a b \sqrt{x} + \sqrt{a b}{\left (b x + a\right )} \arctan \left (\frac{\sqrt{a b}}{b \sqrt{x}}\right )}{a b^{3} x + a^{2} 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{\sqrt{x}}{\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 = 1.13166, size = 49, normalized size = 0.67 \begin{align*} \frac{\arctan \left (\frac{b \sqrt{x}}{\sqrt{a b}}\right )}{\sqrt{a b} b} - \frac{\sqrt{x}}{{\left (b x + a\right )} b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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