Optimal. Leaf size=46 \[ -\frac{8}{3} b x^{3/2} \tanh ^{-1}(\tanh (a+b x))+2 \sqrt{x} \tanh ^{-1}(\tanh (a+b x))^2+\frac{16}{15} b^2 x^{5/2} \]
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Rubi [A] time = 0.0208972, antiderivative size = 46, 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} \[ -\frac{8}{3} b x^{3/2} \tanh ^{-1}(\tanh (a+b x))+2 \sqrt{x} \tanh ^{-1}(\tanh (a+b x))^2+\frac{16}{15} b^2 x^{5/2} \]
Antiderivative was successfully verified.
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Rule 2168
Rule 30
Rubi steps
\begin{align*} \int \frac{\tanh ^{-1}(\tanh (a+b x))^2}{\sqrt{x}} \, dx &=2 \sqrt{x} \tanh ^{-1}(\tanh (a+b x))^2-(4 b) \int \sqrt{x} \tanh ^{-1}(\tanh (a+b x)) \, dx\\ &=-\frac{8}{3} b x^{3/2} \tanh ^{-1}(\tanh (a+b x))+2 \sqrt{x} \tanh ^{-1}(\tanh (a+b x))^2+\frac{1}{3} \left (8 b^2\right ) \int x^{3/2} \, dx\\ &=\frac{16}{15} b^2 x^{5/2}-\frac{8}{3} b x^{3/2} \tanh ^{-1}(\tanh (a+b x))+2 \sqrt{x} \tanh ^{-1}(\tanh (a+b x))^2\\ \end{align*}
Mathematica [A] time = 0.0311231, size = 40, normalized size = 0.87 \[ \frac{2}{15} \sqrt{x} \left (-20 b x \tanh ^{-1}(\tanh (a+b x))+15 \tanh ^{-1}(\tanh (a+b x))^2+8 b^2 x^2\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.04, size = 47, normalized size = 1. \begin{align*}{\frac{2\,{b}^{2}}{5}{x}^{{\frac{5}{2}}}}+{\frac{ \left ( 4\,{\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) -4\,bx \right ) b}{3}{x}^{{\frac{3}{2}}}}+2\, \left ({\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) -bx \right ) ^{2}\sqrt{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.02751, size = 49, normalized size = 1.07 \begin{align*} \frac{16}{15} \, b^{2} x^{\frac{5}{2}} - \frac{8}{3} \, b x^{\frac{3}{2}} \operatorname{artanh}\left (\tanh \left (b x + a\right )\right ) + 2 \, \sqrt{x} \operatorname{artanh}\left (\tanh \left (b x + a\right )\right )^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.99867, size = 62, normalized size = 1.35 \begin{align*} \frac{2}{15} \,{\left (3 \, b^{2} x^{2} + 10 \, a b x + 15 \, a^{2}\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}^{2}{\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 = 1.1664, size = 32, normalized size = 0.7 \begin{align*} \frac{2}{5} \, b^{2} x^{\frac{5}{2}} + \frac{4}{3} \, a b x^{\frac{3}{2}} + 2 \, a^{2} \sqrt{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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