Optimal. Leaf size=25 \[ 2 \sqrt{x} \tanh ^{-1}(\tanh (a+b x))-\frac{4}{3} b x^{3/2} \]
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Rubi [A] time = 0.007859, antiderivative size = 25, 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} \[ 2 \sqrt{x} \tanh ^{-1}(\tanh (a+b x))-\frac{4}{3} b x^{3/2} \]
Antiderivative was successfully verified.
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Rule 2168
Rule 30
Rubi steps
\begin{align*} \int \frac{\tanh ^{-1}(\tanh (a+b x))}{\sqrt{x}} \, dx &=2 \sqrt{x} \tanh ^{-1}(\tanh (a+b x))-(2 b) \int \sqrt{x} \, dx\\ &=-\frac{4}{3} b x^{3/2}+2 \sqrt{x} \tanh ^{-1}(\tanh (a+b x))\\ \end{align*}
Mathematica [A] time = 0.0189722, size = 23, normalized size = 0.92 \[ \frac{2}{3} \sqrt{x} \left (3 \tanh ^{-1}(\tanh (a+b x))-2 b x\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.04, size = 20, normalized size = 0.8 \begin{align*} -{\frac{4\,b}{3}{x}^{{\frac{3}{2}}}}+2\,{\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) \sqrt{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.989077, size = 26, normalized size = 1.04 \begin{align*} -\frac{4}{3} \, b x^{\frac{3}{2}} + 2 \, \sqrt{x} \operatorname{artanh}\left (\tanh \left (b x + a\right )\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.99092, size = 34, normalized size = 1.36 \begin{align*} \frac{2}{3} \,{\left (b x + 3 \, a\right )} \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}{\left (\tanh{\left (a + b x \right )} \right )}}{\sqrt{x}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13399, size = 18, normalized size = 0.72 \begin{align*} \frac{2}{3} \, b x^{\frac{3}{2}} + 2 \, a \sqrt{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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