Optimal. Leaf size=35 \[ \frac{2 \tanh ^{-1}(\tanh (a+b x))^{7/2}}{7 x^{7/2} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )} \]
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Rubi [A] time = 0.0131392, antiderivative size = 35, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059, Rules used = {2167} \[ \frac{2 \tanh ^{-1}(\tanh (a+b x))^{7/2}}{7 x^{7/2} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )} \]
Antiderivative was successfully verified.
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Rule 2167
Rubi steps
\begin{align*} \int \frac{\tanh ^{-1}(\tanh (a+b x))^{5/2}}{x^{9/2}} \, dx &=\frac{2 \tanh ^{-1}(\tanh (a+b x))^{7/2}}{7 x^{7/2} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )}\\ \end{align*}
Mathematica [A] time = 0.0422256, size = 34, normalized size = 0.97 \[ \frac{2 \tanh ^{-1}(\tanh (a+b x))^{7/2}}{x^{7/2} \left (7 b x-7 \tanh ^{-1}(\tanh (a+b x))\right )} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.121, size = 29, normalized size = 0.8 \begin{align*} -{\frac{2}{7\,{\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) -7\,bx} \left ({\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) \right ) ^{{\frac{7}{2}}}{x}^{-{\frac{7}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.47513, size = 20, normalized size = 0.57 \begin{align*} -\frac{2 \,{\left (b x + a\right )}^{\frac{7}{2}}}{7 \, a x^{\frac{7}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.10801, size = 100, normalized size = 2.86 \begin{align*} -\frac{2 \,{\left (b^{3} x^{3} + 3 \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3}\right )} \sqrt{b x + a}}{7 \, a x^{\frac{7}{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 = 1.24288, size = 45, normalized size = 1.29 \begin{align*} -\frac{2 \,{\left (b x + a\right )}^{\frac{7}{2}} b^{8}}{7 \,{\left ({\left (b x + a\right )} b - a b\right )}^{\frac{7}{2}} a{\left | b \right |}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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