Optimal. Leaf size=110 \[ -\frac{16 b^2 \sqrt{x}}{3 \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^3 \sqrt{\tanh ^{-1}(\tanh (a+b x))}}+\frac{2}{3 x^{3/2} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right ) \sqrt{\tanh ^{-1}(\tanh (a+b x))}}+\frac{8 b}{3 \sqrt{x} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^2 \sqrt{\tanh ^{-1}(\tanh (a+b x))}} \]
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Rubi [A] time = 0.0540588, antiderivative size = 110, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {2171, 2167} \[ -\frac{16 b^2 \sqrt{x}}{3 \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^3 \sqrt{\tanh ^{-1}(\tanh (a+b x))}}+\frac{2}{3 x^{3/2} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right ) \sqrt{\tanh ^{-1}(\tanh (a+b x))}}+\frac{8 b}{3 \sqrt{x} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^2 \sqrt{\tanh ^{-1}(\tanh (a+b x))}} \]
Antiderivative was successfully verified.
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Rule 2171
Rule 2167
Rubi steps
\begin{align*} \int \frac{1}{x^{5/2} \tanh ^{-1}(\tanh (a+b x))^{3/2}} \, dx &=\frac{2}{3 x^{3/2} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right ) \sqrt{\tanh ^{-1}(\tanh (a+b x))}}+\frac{(4 b) \int \frac{1}{x^{3/2} \tanh ^{-1}(\tanh (a+b x))^{3/2}} \, dx}{3 \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )}\\ &=\frac{8 b}{3 \sqrt{x} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^2 \sqrt{\tanh ^{-1}(\tanh (a+b x))}}+\frac{2}{3 x^{3/2} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right ) \sqrt{\tanh ^{-1}(\tanh (a+b x))}}-\frac{\left (8 b^2\right ) \int \frac{1}{\sqrt{x} \tanh ^{-1}(\tanh (a+b x))^{3/2}} \, dx}{3 \left (b x-\tanh ^{-1}(\tanh (a+b x))\right ) \left (-b x+\tanh ^{-1}(\tanh (a+b x))\right )}\\ &=-\frac{16 b^2 \sqrt{x}}{3 \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^3 \sqrt{\tanh ^{-1}(\tanh (a+b x))}}+\frac{8 b}{3 \sqrt{x} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^2 \sqrt{\tanh ^{-1}(\tanh (a+b x))}}+\frac{2}{3 x^{3/2} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right ) \sqrt{\tanh ^{-1}(\tanh (a+b x))}}\\ \end{align*}
Mathematica [A] time = 0.0491826, size = 64, normalized size = 0.58 \[ \frac{2 \left (-6 b x \tanh ^{-1}(\tanh (a+b x))+\tanh ^{-1}(\tanh (a+b x))^2-3 b^2 x^2\right )}{3 x^{3/2} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^3 \sqrt{\tanh ^{-1}(\tanh (a+b x))}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.148, size = 105, normalized size = 1. \begin{align*} -{\frac{2}{3\,{\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) -3\,bx}{x}^{-{\frac{3}{2}}}{\frac{1}{\sqrt{{\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) }}}}-{\frac{8\,b}{3\,{\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) -3\,bx} \left ( -{\frac{1}{{\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) -bx}{\frac{1}{\sqrt{x}}}{\frac{1}{\sqrt{{\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) }}}}-2\,{\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 ) }}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.50479, size = 61, normalized size = 0.55 \begin{align*} \frac{2 \,{\left (8 \, b^{3} x^{3} + 12 \, a b^{2} x^{2} + 3 \, a^{2} b x - a^{3}\right )}}{3 \,{\left (b x + a\right )}^{\frac{3}{2}} a^{3} x^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.05685, size = 104, normalized size = 0.95 \begin{align*} \frac{2 \,{\left (8 \, b^{2} x^{2} + 4 \, a b x - a^{2}\right )} \sqrt{b x + a} \sqrt{x}}{3 \,{\left (a^{3} b x^{3} + a^{4} x^{2}\right )}} \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 = 1.27649, size = 144, normalized size = 1.31 \begin{align*} \frac{2 \, b^{2} \sqrt{x}}{\sqrt{b x + a} a^{3}} - \frac{4 \,{\left (3 \, b^{\frac{3}{2}}{\left (\sqrt{b} \sqrt{x} - \sqrt{b x + a}\right )}^{4} - 12 \, a b^{\frac{3}{2}}{\left (\sqrt{b} \sqrt{x} - \sqrt{b x + a}\right )}^{2} + 5 \, a^{2} b^{\frac{3}{2}}\right )}}{3 \,{\left ({\left (\sqrt{b} \sqrt{x} - \sqrt{b x + a}\right )}^{2} - a\right )}^{3} a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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