Optimal. Leaf size=27 \[ -\frac{2 \tanh ^{-1}(\tanh (a+b x))}{5 x^{5/2}}-\frac{4 b}{15 x^{3/2}} \]
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Rubi [A] time = 0.0087224, 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} \[ -\frac{2 \tanh ^{-1}(\tanh (a+b x))}{5 x^{5/2}}-\frac{4 b}{15 x^{3/2}} \]
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^{7/2}} \, dx &=-\frac{2 \tanh ^{-1}(\tanh (a+b x))}{5 x^{5/2}}+\frac{1}{5} (2 b) \int \frac{1}{x^{5/2}} \, dx\\ &=-\frac{4 b}{15 x^{3/2}}-\frac{2 \tanh ^{-1}(\tanh (a+b x))}{5 x^{5/2}}\\ \end{align*}
Mathematica [A] time = 0.020748, size = 23, normalized size = 0.85 \[ -\frac{2 \left (3 \tanh ^{-1}(\tanh (a+b x))+2 b x\right )}{15 x^{5/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.039, size = 20, normalized size = 0.7 \begin{align*} -{\frac{4\,b}{15}{x}^{-{\frac{3}{2}}}}-{\frac{2\,{\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) }{5}{x}^{-{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.979525, size = 26, normalized size = 0.96 \begin{align*} -\frac{4 \, b}{15 \, x^{\frac{3}{2}}} - \frac{2 \, \operatorname{artanh}\left (\tanh \left (b x + a\right )\right )}{5 \, x^{\frac{5}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.97809, size = 39, normalized size = 1.44 \begin{align*} -\frac{2 \,{\left (5 \, b x + 3 \, a\right )}}{15 \, x^{\frac{5}{2}}} \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 = 1.16366, size = 18, normalized size = 0.67 \begin{align*} -\frac{2 \,{\left (5 \, b x + 3 \, a\right )}}{15 \, x^{\frac{5}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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