Optimal. Leaf size=72 \[ \frac{2 \tanh ^{-1}(\tanh (a+b x))^{5/2}}{7 x^{7/2} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )}+\frac{4 b \tanh ^{-1}(\tanh (a+b x))^{5/2}}{35 x^{5/2} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^2} \]
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Rubi [A] time = 0.0330791, antiderivative size = 72, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {2171, 2167} \[ \frac{2 \tanh ^{-1}(\tanh (a+b x))^{5/2}}{7 x^{7/2} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )}+\frac{4 b \tanh ^{-1}(\tanh (a+b x))^{5/2}}{35 x^{5/2} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^2} \]
Antiderivative was successfully verified.
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Rule 2171
Rule 2167
Rubi steps
\begin{align*} \int \frac{\tanh ^{-1}(\tanh (a+b x))^{3/2}}{x^{9/2}} \, dx &=\frac{2 \tanh ^{-1}(\tanh (a+b x))^{5/2}}{7 x^{7/2} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )}+\frac{(2 b) \int \frac{\tanh ^{-1}(\tanh (a+b x))^{3/2}}{x^{7/2}} \, dx}{7 \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )}\\ &=\frac{4 b \tanh ^{-1}(\tanh (a+b x))^{5/2}}{35 x^{5/2} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^2}+\frac{2 \tanh ^{-1}(\tanh (a+b x))^{5/2}}{7 x^{7/2} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )}\\ \end{align*}
Mathematica [A] time = 0.040538, size = 48, normalized size = 0.67 \[ \frac{2 \left (7 b x-5 \tanh ^{-1}(\tanh (a+b x))\right ) \tanh ^{-1}(\tanh (a+b x))^{5/2}}{35 x^{7/2} \left (\tanh ^{-1}(\tanh (a+b x))-b x\right )^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.129, size = 59, 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{5}{2}}}{x}^{-{\frac{7}{2}}}}+{\frac{4\,b}{35\, \left ({\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) -bx \right ) ^{2}} \left ({\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) \right ) ^{{\frac{5}{2}}}{x}^{-{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.46936, size = 46, normalized size = 0.64 \begin{align*} \frac{2 \,{\left (2 \, b^{2} x^{2} - 3 \, a b x - 5 \, a^{2}\right )}{\left (b x + a\right )}^{\frac{3}{2}}}{35 \, a^{2} x^{\frac{7}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.18267, size = 105, normalized size = 1.46 \begin{align*} \frac{2 \,{\left (2 \, b^{3} x^{3} - a b^{2} x^{2} - 8 \, a^{2} b x - 5 \, a^{3}\right )} \sqrt{b x + a}}{35 \, a^{2} 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 = 1.25709, size = 80, normalized size = 1.11 \begin{align*} \frac{\sqrt{2}{\left (\frac{2 \, \sqrt{2}{\left (b x + a\right )} b^{7}}{a^{2}} - \frac{7 \, \sqrt{2} b^{7}}{a}\right )}{\left (b x + a\right )}^{\frac{5}{2}} b}{35 \,{\left ({\left (b x + a\right )} b - a b\right )}^{\frac{7}{2}}{\left | b \right |}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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