Optimal. Leaf size=110 \[ -\frac{\tanh ^{-1}(\tanh (a+b x))^{5/2}}{x}+\frac{5}{3} b \tanh ^{-1}(\tanh (a+b x))^{3/2}-5 b \left (b x-\tanh ^{-1}(\tanh (a+b x))\right ) \sqrt{\tanh ^{-1}(\tanh (a+b x))}+5 b \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^{3/2} \tan ^{-1}\left (\frac{\sqrt{\tanh ^{-1}(\tanh (a+b x))}}{\sqrt{b x-\tanh ^{-1}(\tanh (a+b x))}}\right ) \]
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Rubi [A] time = 0.0689825, antiderivative size = 110, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {2168, 2159, 2161} \[ -\frac{\tanh ^{-1}(\tanh (a+b x))^{5/2}}{x}+\frac{5}{3} b \tanh ^{-1}(\tanh (a+b x))^{3/2}-5 b \left (b x-\tanh ^{-1}(\tanh (a+b x))\right ) \sqrt{\tanh ^{-1}(\tanh (a+b x))}+5 b \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^{3/2} \tan ^{-1}\left (\frac{\sqrt{\tanh ^{-1}(\tanh (a+b x))}}{\sqrt{b x-\tanh ^{-1}(\tanh (a+b x))}}\right ) \]
Antiderivative was successfully verified.
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Rule 2168
Rule 2159
Rule 2161
Rubi steps
\begin{align*} \int \frac{\tanh ^{-1}(\tanh (a+b x))^{5/2}}{x^2} \, dx &=-\frac{\tanh ^{-1}(\tanh (a+b x))^{5/2}}{x}+\frac{1}{2} (5 b) \int \frac{\tanh ^{-1}(\tanh (a+b x))^{3/2}}{x} \, dx\\ &=\frac{5}{3} b \tanh ^{-1}(\tanh (a+b x))^{3/2}-\frac{\tanh ^{-1}(\tanh (a+b x))^{5/2}}{x}-\frac{1}{2} \left (5 b \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )\right ) \int \frac{\sqrt{\tanh ^{-1}(\tanh (a+b x))}}{x} \, dx\\ &=-5 b \left (b x-\tanh ^{-1}(\tanh (a+b x))\right ) \sqrt{\tanh ^{-1}(\tanh (a+b x))}+\frac{5}{3} b \tanh ^{-1}(\tanh (a+b x))^{3/2}-\frac{\tanh ^{-1}(\tanh (a+b x))^{5/2}}{x}+\frac{1}{2} \left (5 b \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^2\right ) \int \frac{1}{x \sqrt{\tanh ^{-1}(\tanh (a+b x))}} \, dx\\ &=5 b \tan ^{-1}\left (\frac{\sqrt{\tanh ^{-1}(\tanh (a+b x))}}{\sqrt{b x-\tanh ^{-1}(\tanh (a+b x))}}\right ) \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^{3/2}-5 b \left (b x-\tanh ^{-1}(\tanh (a+b x))\right ) \sqrt{\tanh ^{-1}(\tanh (a+b x))}+\frac{5}{3} b \tanh ^{-1}(\tanh (a+b x))^{3/2}-\frac{\tanh ^{-1}(\tanh (a+b x))^{5/2}}{x}\\ \end{align*}
Mathematica [A] time = 0.0530697, size = 106, normalized size = 0.96 \[ \sqrt{\tanh ^{-1}(\tanh (a+b x))} \left (\frac{14}{3} b \left (\tanh ^{-1}(\tanh (a+b x))-b x\right )-\frac{\left (\tanh ^{-1}(\tanh (a+b x))-b x\right )^2}{x}+\frac{2 b^2 x}{3}\right )-5 b \tanh ^{-1}\left (\frac{\sqrt{\tanh ^{-1}(\tanh (a+b x))}}{\sqrt{\tanh ^{-1}(\tanh (a+b x))-b x}}\right ) \left (\tanh ^{-1}(\tanh (a+b x))-b x\right )^{3/2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.116, size = 193, normalized size = 1.8 \begin{align*} 2\,b \left ( 1/3\, \left ({\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) \right ) ^{3/2}+2\,a\sqrt{{\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) }+2\, \left ({\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) -bx-a \right ) \sqrt{{\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) }+{\frac{ \left ( -1/2\,{a}^{2}-a \left ({\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) -bx-a \right ) -1/2\, \left ({\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) -bx-a \right ) ^{2} \right ) \sqrt{{\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) }}{bx}}-5/2\,{\frac{{a}^{2}+2\,a \left ({\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) -bx-a \right ) + \left ({\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) -bx-a \right ) ^{2}}{\sqrt{{\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) -bx}}{\it Artanh} \left ({\frac{\sqrt{{\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) }}{\sqrt{{\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) -bx}}} \right ) } \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{artanh}\left (\tanh \left (b x + a\right )\right )^{\frac{5}{2}}}{x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.1134, size = 309, normalized size = 2.81 \begin{align*} \left [\frac{15 \, a^{\frac{3}{2}} b x \log \left (\frac{b x - 2 \, \sqrt{b x + a} \sqrt{a} + 2 \, a}{x}\right ) + 2 \,{\left (2 \, b^{2} x^{2} + 14 \, a b x - 3 \, a^{2}\right )} \sqrt{b x + a}}{6 \, x}, \frac{15 \, \sqrt{-a} a b x \arctan \left (\frac{\sqrt{b x + a} \sqrt{-a}}{a}\right ) +{\left (2 \, b^{2} x^{2} + 14 \, a b x - 3 \, a^{2}\right )} \sqrt{b x + a}}{3 \, x}\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 = 1.14152, size = 120, normalized size = 1.09 \begin{align*} \frac{\sqrt{2}{\left (\frac{15 \, \sqrt{2} a^{2} b^{2} \arctan \left (\frac{\sqrt{b x + a}}{\sqrt{-a}}\right )}{\sqrt{-a}} + 2 \, \sqrt{2}{\left (b x + a\right )}^{\frac{3}{2}} b^{2} + 12 \, \sqrt{2} \sqrt{b x + a} a b^{2} - \frac{3 \, \sqrt{2} \sqrt{b x + a} a^{2} b}{x}\right )}}{6 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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