Optimal. Leaf size=23 \[ 4 b \sqrt{x}-\frac{2 \tanh ^{-1}(\tanh (a+b x))}{\sqrt{x}} \]
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Rubi [A] time = 0.0082348, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {2168, 30} \[ 4 b \sqrt{x}-\frac{2 \tanh ^{-1}(\tanh (a+b x))}{\sqrt{x}} \]
Antiderivative was successfully verified.
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Rule 2168
Rule 30
Rubi steps
\begin{align*} \int \frac{\tanh ^{-1}(\tanh (a+b x))}{x^{3/2}} \, dx &=-\frac{2 \tanh ^{-1}(\tanh (a+b x))}{\sqrt{x}}+(2 b) \int \frac{1}{\sqrt{x}} \, dx\\ &=4 b \sqrt{x}-\frac{2 \tanh ^{-1}(\tanh (a+b x))}{\sqrt{x}}\\ \end{align*}
Mathematica [A] time = 0.0196906, size = 20, normalized size = 0.87 \[ \frac{4 b x-2 \tanh ^{-1}(\tanh (a+b x))}{\sqrt{x}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.034, size = 20, normalized size = 0.9 \begin{align*} -2\,{\frac{{\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) }{\sqrt{x}}}+4\,b\sqrt{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.981791, size = 26, normalized size = 1.13 \begin{align*} 4 \, b \sqrt{x} - \frac{2 \, \operatorname{artanh}\left (\tanh \left (b x + a\right )\right )}{\sqrt{x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.0859, size = 28, normalized size = 1.22 \begin{align*} \frac{2 \,{\left (b x - a\right )}}{\sqrt{x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.64159, size = 22, normalized size = 0.96 \begin{align*} 4 b \sqrt{x} - \frac{2 \operatorname{atanh}{\left (\tanh{\left (a + b x \right )} \right )}}{\sqrt{x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15908, size = 18, normalized size = 0.78 \begin{align*} 2 \, b \sqrt{x} - \frac{2 \, a}{\sqrt{x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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