Optimal. Leaf size=33 \[ \frac{2 \sqrt{\tanh ^{-1}(\tanh (a+b x))}}{\sqrt{x} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )} \]
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Rubi [A] time = 0.0133372, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059, Rules used = {2167} \[ \frac{2 \sqrt{\tanh ^{-1}(\tanh (a+b x))}}{\sqrt{x} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )} \]
Antiderivative was successfully verified.
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Rule 2167
Rubi steps
\begin{align*} \int \frac{1}{x^{3/2} \sqrt{\tanh ^{-1}(\tanh (a+b x))}} \, dx &=\frac{2 \sqrt{\tanh ^{-1}(\tanh (a+b x))}}{\sqrt{x} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )}\\ \end{align*}
Mathematica [A] time = 0.0392659, size = 32, normalized size = 0.97 \[ -\frac{2 \sqrt{\tanh ^{-1}(\tanh (a+b x))}}{\sqrt{x} \left (\tanh ^{-1}(\tanh (a+b x))-b x\right )} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.17, size = 29, normalized size = 0.9 \begin{align*} -2\,{\frac{\sqrt{{\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) }}{ \left ({\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) -bx \right ) \sqrt{x}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.47852, size = 20, normalized size = 0.61 \begin{align*} -\frac{2 \, \sqrt{b x + a}}{a \sqrt{x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.06163, size = 41, normalized size = 1.24 \begin{align*} -\frac{2 \, \sqrt{b x + a}}{a \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{1}{x^{\frac{3}{2}} \sqrt{\operatorname{atanh}{\left (\tanh{\left (a + b x \right )} \right )}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14597, size = 41, normalized size = 1.24 \begin{align*} \frac{4 \, \sqrt{b}}{{\left (\sqrt{b} \sqrt{x} - \sqrt{b x + a}\right )}^{2} - a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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