Optimal. Leaf size=44 \[ 8 b \sqrt{x} \tanh ^{-1}(\tanh (a+b x))-\frac{2 \tanh ^{-1}(\tanh (a+b x))^2}{\sqrt{x}}-\frac{16}{3} b^2 x^{3/2} \]
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Rubi [A] time = 0.0217644, antiderivative size = 44, 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} \[ 8 b \sqrt{x} \tanh ^{-1}(\tanh (a+b x))-\frac{2 \tanh ^{-1}(\tanh (a+b x))^2}{\sqrt{x}}-\frac{16}{3} b^2 x^{3/2} \]
Antiderivative was successfully verified.
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Rule 2168
Rule 30
Rubi steps
\begin{align*} \int \frac{\tanh ^{-1}(\tanh (a+b x))^2}{x^{3/2}} \, dx &=-\frac{2 \tanh ^{-1}(\tanh (a+b x))^2}{\sqrt{x}}+(4 b) \int \frac{\tanh ^{-1}(\tanh (a+b x))}{\sqrt{x}} \, dx\\ &=8 b \sqrt{x} \tanh ^{-1}(\tanh (a+b x))-\frac{2 \tanh ^{-1}(\tanh (a+b x))^2}{\sqrt{x}}-\left (8 b^2\right ) \int \sqrt{x} \, dx\\ &=-\frac{16}{3} b^2 x^{3/2}+8 b \sqrt{x} \tanh ^{-1}(\tanh (a+b x))-\frac{2 \tanh ^{-1}(\tanh (a+b x))^2}{\sqrt{x}}\\ \end{align*}
Mathematica [A] time = 0.042224, size = 40, normalized size = 0.91 \[ -\frac{2 \left (-12 b x \tanh ^{-1}(\tanh (a+b x))+3 \tanh ^{-1}(\tanh (a+b x))^2+8 b^2 x^2\right )}{3 \sqrt{x}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.04, size = 37, normalized size = 0.8 \begin{align*} -2\,{\frac{ \left ({\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) \right ) ^{2}}{\sqrt{x}}}+8\,b \left ({\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) \sqrt{x}-2/3\,b{x}^{3/2} \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.01822, size = 49, normalized size = 1.11 \begin{align*} -\frac{16}{3} \, b^{2} x^{\frac{3}{2}} + 8 \, b \sqrt{x} \operatorname{artanh}\left (\tanh \left (b x + a\right )\right ) - \frac{2 \, \operatorname{artanh}\left (\tanh \left (b x + a\right )\right )^{2}}{\sqrt{x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.0064, size = 55, normalized size = 1.25 \begin{align*} \frac{2 \,{\left (b^{2} x^{2} + 6 \, a b x - 3 \, a^{2}\right )}}{3 \, \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}^{2}{\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 = 1.1522, size = 32, normalized size = 0.73 \begin{align*} \frac{2}{3} \, b^{2} x^{\frac{3}{2}} + 4 \, a b \sqrt{x} - \frac{2 \, a^{2}}{\sqrt{x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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