Optimal. Leaf size=68 \[ \frac{2}{\sqrt{x} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right ) \sqrt{\tanh ^{-1}(\tanh (a+b x))}}-\frac{4 b \sqrt{x}}{\left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^2 \sqrt{\tanh ^{-1}(\tanh (a+b x))}} \]
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Rubi [A] time = 0.032766, antiderivative size = 68, 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 = {2171, 2167} \[ \frac{2}{\sqrt{x} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right ) \sqrt{\tanh ^{-1}(\tanh (a+b x))}}-\frac{4 b \sqrt{x}}{\left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^2 \sqrt{\tanh ^{-1}(\tanh (a+b x))}} \]
Antiderivative was successfully verified.
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Rule 2171
Rule 2167
Rubi steps
\begin{align*} \int \frac{1}{x^{3/2} \tanh ^{-1}(\tanh (a+b x))^{3/2}} \, dx &=\frac{2}{\sqrt{x} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right ) \sqrt{\tanh ^{-1}(\tanh (a+b x))}}-\frac{(2 b) \int \frac{1}{\sqrt{x} \tanh ^{-1}(\tanh (a+b x))^{3/2}} \, dx}{-b x+\tanh ^{-1}(\tanh (a+b x))}\\ &=-\frac{4 b \sqrt{x}}{\left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^2 \sqrt{\tanh ^{-1}(\tanh (a+b x))}}+\frac{2}{\sqrt{x} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right ) \sqrt{\tanh ^{-1}(\tanh (a+b x))}}\\ \end{align*}
Mathematica [A] time = 0.0467405, size = 43, normalized size = 0.63 \[ -\frac{2 \left (\tanh ^{-1}(\tanh (a+b x))+b x\right )}{\sqrt{x} \sqrt{\tanh ^{-1}(\tanh (a+b x))} \left (\tanh ^{-1}(\tanh (a+b x))-b x\right )^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.146, size = 59, normalized size = 0.9 \begin{align*} -2\,{\frac{1}{ \left ({\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) -bx \right ) \sqrt{x}\sqrt{{\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) }}}-4\,{\frac{b\sqrt{x}}{ \left ({\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) -bx \right ) ^{2}\sqrt{{\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) }}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.49918, size = 43, normalized size = 0.63 \begin{align*} -\frac{2 \,{\left (2 \, b^{2} x^{2} + 3 \, a b x + a^{2}\right )}}{{\left (b x + a\right )}^{\frac{3}{2}} a^{2} \sqrt{x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.03001, size = 78, normalized size = 1.15 \begin{align*} -\frac{2 \,{\left (2 \, b x + a\right )} \sqrt{b x + a} \sqrt{x}}{a^{2} b x^{2} + a^{3} x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.17775, size = 68, normalized size = 1. \begin{align*} -\frac{2 \, b \sqrt{x}}{\sqrt{b x + a} a^{2}} + \frac{4 \, \sqrt{b}}{{\left ({\left (\sqrt{b} \sqrt{x} - \sqrt{b x + a}\right )}^{2} - a\right )} a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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