Optimal. Leaf size=48 \[ -\frac{8}{15} b x^{5/2} \tanh ^{-1}(\tanh (a+b x))+\frac{2}{3} x^{3/2} \tanh ^{-1}(\tanh (a+b x))^2+\frac{16}{105} b^2 x^{7/2} \]
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Rubi [A] time = 0.0217704, antiderivative size = 48, 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} \[ -\frac{8}{15} b x^{5/2} \tanh ^{-1}(\tanh (a+b x))+\frac{2}{3} x^{3/2} \tanh ^{-1}(\tanh (a+b x))^2+\frac{16}{105} b^2 x^{7/2} \]
Antiderivative was successfully verified.
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Rule 2168
Rule 30
Rubi steps
\begin{align*} \int \sqrt{x} \tanh ^{-1}(\tanh (a+b x))^2 \, dx &=\frac{2}{3} x^{3/2} \tanh ^{-1}(\tanh (a+b x))^2-\frac{1}{3} (4 b) \int x^{3/2} \tanh ^{-1}(\tanh (a+b x)) \, dx\\ &=-\frac{8}{15} b x^{5/2} \tanh ^{-1}(\tanh (a+b x))+\frac{2}{3} x^{3/2} \tanh ^{-1}(\tanh (a+b x))^2+\frac{1}{15} \left (8 b^2\right ) \int x^{5/2} \, dx\\ &=\frac{16}{105} b^2 x^{7/2}-\frac{8}{15} b x^{5/2} \tanh ^{-1}(\tanh (a+b x))+\frac{2}{3} x^{3/2} \tanh ^{-1}(\tanh (a+b x))^2\\ \end{align*}
Mathematica [A] time = 0.0482864, size = 40, normalized size = 0.83 \[ \frac{2}{105} x^{3/2} \left (-28 b x \tanh ^{-1}(\tanh (a+b x))+35 \tanh ^{-1}(\tanh (a+b x))^2+8 b^2 x^2\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.04, size = 38, normalized size = 0.8 \begin{align*}{\frac{2\, \left ({\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) \right ) ^{2}}{3}{x}^{{\frac{3}{2}}}}-{\frac{8\,b}{3} \left ({\frac{{\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) }{5}{x}^{{\frac{5}{2}}}}-{\frac{2\,b}{35}{x}^{{\frac{7}{2}}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.03271, size = 49, normalized size = 1.02 \begin{align*} \frac{16}{105} \, b^{2} x^{\frac{7}{2}} - \frac{8}{15} \, b x^{\frac{5}{2}} \operatorname{artanh}\left (\tanh \left (b x + a\right )\right ) + \frac{2}{3} \, x^{\frac{3}{2}} \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 = 2.00495, size = 70, normalized size = 1.46 \begin{align*} \frac{2}{105} \,{\left (15 \, b^{2} x^{3} + 42 \, a b x^{2} + 35 \, a^{2} x\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 \sqrt{x} \operatorname{atanh}^{2}{\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 = 1.12525, size = 32, normalized size = 0.67 \begin{align*} \frac{2}{7} \, b^{2} x^{\frac{7}{2}} + \frac{4}{5} \, a b x^{\frac{5}{2}} + \frac{2}{3} \, a^{2} x^{\frac{3}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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