Optimal. Leaf size=27 \[ -\frac{2 \tanh ^{-1}(\tanh (a+b x))}{3 x^{3/2}}-\frac{4 b}{3 \sqrt{x}} \]
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Rubi [A] time = 0.0086571, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {2168, 30} \[ -\frac{2 \tanh ^{-1}(\tanh (a+b x))}{3 x^{3/2}}-\frac{4 b}{3 \sqrt{x}} \]
Antiderivative was successfully verified.
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Rule 2168
Rule 30
Rubi steps
\begin{align*} \int \frac{\tanh ^{-1}(\tanh (a+b x))}{x^{5/2}} \, dx &=-\frac{2 \tanh ^{-1}(\tanh (a+b x))}{3 x^{3/2}}+\frac{1}{3} (2 b) \int \frac{1}{x^{3/2}} \, dx\\ &=-\frac{4 b}{3 \sqrt{x}}-\frac{2 \tanh ^{-1}(\tanh (a+b x))}{3 x^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.0181129, size = 21, normalized size = 0.78 \[ -\frac{2 \left (\tanh ^{-1}(\tanh (a+b x))+2 b x\right )}{3 x^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.037, size = 20, normalized size = 0.7 \begin{align*} -{\frac{2\,{\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) }{3}{x}^{-{\frac{3}{2}}}}-{\frac{4\,b}{3}{\frac{1}{\sqrt{x}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.986474, size = 26, normalized size = 0.96 \begin{align*} -\frac{4 \, b}{3 \, \sqrt{x}} - \frac{2 \, \operatorname{artanh}\left (\tanh \left (b x + a\right )\right )}{3 \, x^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.02828, size = 35, normalized size = 1.3 \begin{align*} -\frac{2 \,{\left (3 \, b x + a\right )}}{3 \, x^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 21.2364, size = 27, normalized size = 1. \begin{align*} - \frac{4 b}{3 \sqrt{x}} - \frac{2 \operatorname{atanh}{\left (\tanh{\left (a + b x \right )} \right )}}{3 x^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15382, size = 15, normalized size = 0.56 \begin{align*} -\frac{2 \,{\left (3 \, b x + a\right )}}{3 \, x^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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