Optimal. Leaf size=63 \[ -16 b^2 x^{3/2} \tanh ^{-1}(\tanh (a+b x))+12 b \sqrt{x} \tanh ^{-1}(\tanh (a+b x))^2-\frac{2 \tanh ^{-1}(\tanh (a+b x))^3}{\sqrt{x}}+\frac{32}{5} b^3 x^{5/2} \]
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Rubi [A] time = 0.0359327, antiderivative size = 63, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {2168, 30} \[ -16 b^2 x^{3/2} \tanh ^{-1}(\tanh (a+b x))+12 b \sqrt{x} \tanh ^{-1}(\tanh (a+b x))^2-\frac{2 \tanh ^{-1}(\tanh (a+b x))^3}{\sqrt{x}}+\frac{32}{5} b^3 x^{5/2} \]
Antiderivative was successfully verified.
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Rule 2168
Rule 30
Rubi steps
\begin{align*} \int \frac{\tanh ^{-1}(\tanh (a+b x))^3}{x^{3/2}} \, dx &=-\frac{2 \tanh ^{-1}(\tanh (a+b x))^3}{\sqrt{x}}+(6 b) \int \frac{\tanh ^{-1}(\tanh (a+b x))^2}{\sqrt{x}} \, dx\\ &=12 b \sqrt{x} \tanh ^{-1}(\tanh (a+b x))^2-\frac{2 \tanh ^{-1}(\tanh (a+b x))^3}{\sqrt{x}}-\left (24 b^2\right ) \int \sqrt{x} \tanh ^{-1}(\tanh (a+b x)) \, dx\\ &=-16 b^2 x^{3/2} \tanh ^{-1}(\tanh (a+b x))+12 b \sqrt{x} \tanh ^{-1}(\tanh (a+b x))^2-\frac{2 \tanh ^{-1}(\tanh (a+b x))^3}{\sqrt{x}}+\left (16 b^3\right ) \int x^{3/2} \, dx\\ &=\frac{32}{5} b^3 x^{5/2}-16 b^2 x^{3/2} \tanh ^{-1}(\tanh (a+b x))+12 b \sqrt{x} \tanh ^{-1}(\tanh (a+b x))^2-\frac{2 \tanh ^{-1}(\tanh (a+b x))^3}{\sqrt{x}}\\ \end{align*}
Mathematica [A] time = 0.0309749, size = 57, normalized size = 0.9 \[ \frac{2 \left (-40 b^2 x^2 \tanh ^{-1}(\tanh (a+b x))+30 b x \tanh ^{-1}(\tanh (a+b x))^2-5 \tanh ^{-1}(\tanh (a+b x))^3+16 b^3 x^3\right )}{5 \sqrt{x}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.043, size = 64, normalized size = 1. \begin{align*} -2\,{\frac{ \left ({\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) \right ) ^{3}}{\sqrt{x}}}+12\,b \left ( 1/5\,{b}^{2}{x}^{5/2}+2/3\, \left ({\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) -bx \right ) b{x}^{3/2}+ \left ({\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) -bx \right ) ^{2}\sqrt{x} \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.04928, size = 74, normalized size = 1.17 \begin{align*} 12 \, b \sqrt{x} \operatorname{artanh}\left (\tanh \left (b x + a\right )\right )^{2} - \frac{2 \, \operatorname{artanh}\left (\tanh \left (b x + a\right )\right )^{3}}{\sqrt{x}} + \frac{16}{5} \,{\left (2 \, b^{2} x^{\frac{5}{2}} - 5 \, b x^{\frac{3}{2}} \operatorname{artanh}\left (\tanh \left (b x + a\right )\right )\right )} b \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.92586, size = 78, normalized size = 1.24 \begin{align*} \frac{2 \,{\left (b^{3} x^{3} + 5 \, a b^{2} x^{2} + 15 \, a^{2} b x - 5 \, a^{3}\right )}}{5 \, \sqrt{x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{atanh}^{3}{\left (\tanh{\left (a + b x \right )} \right )}}{x^{\frac{3}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.11838, size = 47, normalized size = 0.75 \begin{align*} \frac{2}{5} \, b^{3} x^{\frac{5}{2}} + 2 \, a b^{2} x^{\frac{3}{2}} + 6 \, a^{2} b \sqrt{x} - \frac{2 \, a^{3}}{\sqrt{x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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