Optimal. Leaf size=48 \[ -\frac{2 \tanh ^{-1}(\tanh (a+b x))^2}{3 x^{3/2}}-\frac{8 b \tanh ^{-1}(\tanh (a+b x))}{3 \sqrt{x}}+\frac{16 b^2 \sqrt{x}}{3} \]
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Rubi [A] time = 0.0230305, antiderivative size = 48, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {2168, 30} \[ -\frac{2 \tanh ^{-1}(\tanh (a+b x))^2}{3 x^{3/2}}-\frac{8 b \tanh ^{-1}(\tanh (a+b x))}{3 \sqrt{x}}+\frac{16 b^2 \sqrt{x}}{3} \]
Antiderivative was successfully verified.
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Rule 2168
Rule 30
Rubi steps
\begin{align*} \int \frac{\tanh ^{-1}(\tanh (a+b x))^2}{x^{5/2}} \, dx &=-\frac{2 \tanh ^{-1}(\tanh (a+b x))^2}{3 x^{3/2}}+\frac{1}{3} (4 b) \int \frac{\tanh ^{-1}(\tanh (a+b x))}{x^{3/2}} \, dx\\ &=-\frac{8 b \tanh ^{-1}(\tanh (a+b x))}{3 \sqrt{x}}-\frac{2 \tanh ^{-1}(\tanh (a+b x))^2}{3 x^{3/2}}+\frac{1}{3} \left (8 b^2\right ) \int \frac{1}{\sqrt{x}} \, dx\\ &=\frac{16 b^2 \sqrt{x}}{3}-\frac{8 b \tanh ^{-1}(\tanh (a+b x))}{3 \sqrt{x}}-\frac{2 \tanh ^{-1}(\tanh (a+b x))^2}{3 x^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.0442962, size = 40, normalized size = 0.83 \[ \frac{2 \left (-4 b x \tanh ^{-1}(\tanh (a+b x))-\tanh ^{-1}(\tanh (a+b x))^2+8 b^2 x^2\right )}{3 x^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.041, size = 38, normalized size = 0.8 \begin{align*} -{\frac{2\, \left ({\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) \right ) ^{2}}{3}{x}^{-{\frac{3}{2}}}}+{\frac{8\,b}{3} \left ( -{{\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ){\frac{1}{\sqrt{x}}}}+2\,b\sqrt{x} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.04449, size = 49, normalized size = 1.02 \begin{align*} \frac{16}{3} \, b^{2} \sqrt{x} - \frac{8 \, b \operatorname{artanh}\left (\tanh \left (b x + a\right )\right )}{3 \, \sqrt{x}} - \frac{2 \, \operatorname{artanh}\left (\tanh \left (b x + a\right )\right )^{2}}{3 \, x^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.9352, size = 55, normalized size = 1.15 \begin{align*} \frac{2 \,{\left (3 \, b^{2} x^{2} - 6 \, a b x - a^{2}\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 = 22.5462, size = 48, normalized size = 1. \begin{align*} \frac{16 b^{2} \sqrt{x}}{3} - \frac{8 b \operatorname{atanh}{\left (\tanh{\left (a + b x \right )} \right )}}{3 \sqrt{x}} - \frac{2 \operatorname{atanh}^{2}{\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 = 1.13166, size = 31, normalized size = 0.65 \begin{align*} 2 \, b^{2} \sqrt{x} - \frac{2 \,{\left (6 \, a b x + a^{2}\right )}}{3 \, x^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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