Optimal. Leaf size=48 \[ -\frac{8}{35} b x^{7/2} \tanh ^{-1}(\tanh (a+b x))+\frac{2}{5} x^{5/2} \tanh ^{-1}(\tanh (a+b x))^2+\frac{16}{315} b^2 x^{9/2} \]
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Rubi [A] time = 0.0228219, 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} \[ -\frac{8}{35} b x^{7/2} \tanh ^{-1}(\tanh (a+b x))+\frac{2}{5} x^{5/2} \tanh ^{-1}(\tanh (a+b x))^2+\frac{16}{315} b^2 x^{9/2} \]
Antiderivative was successfully verified.
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Rule 2168
Rule 30
Rubi steps
\begin{align*} \int x^{3/2} \tanh ^{-1}(\tanh (a+b x))^2 \, dx &=\frac{2}{5} x^{5/2} \tanh ^{-1}(\tanh (a+b x))^2-\frac{1}{5} (4 b) \int x^{5/2} \tanh ^{-1}(\tanh (a+b x)) \, dx\\ &=-\frac{8}{35} b x^{7/2} \tanh ^{-1}(\tanh (a+b x))+\frac{2}{5} x^{5/2} \tanh ^{-1}(\tanh (a+b x))^2+\frac{1}{35} \left (8 b^2\right ) \int x^{7/2} \, dx\\ &=\frac{16}{315} b^2 x^{9/2}-\frac{8}{35} b x^{7/2} \tanh ^{-1}(\tanh (a+b x))+\frac{2}{5} x^{5/2} \tanh ^{-1}(\tanh (a+b x))^2\\ \end{align*}
Mathematica [A] time = 0.0401329, size = 40, normalized size = 0.83 \[ \frac{2}{315} x^{5/2} \left (-36 b x \tanh ^{-1}(\tanh (a+b x))+63 \tanh ^{-1}(\tanh (a+b x))^2+8 b^2 x^2\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.039, size = 38, normalized size = 0.8 \begin{align*}{\frac{2\, \left ({\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) \right ) ^{2}}{5}{x}^{{\frac{5}{2}}}}-{\frac{8\,b}{5} \left ({\frac{{\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) }{7}{x}^{{\frac{7}{2}}}}-{\frac{2\,b}{63}{x}^{{\frac{9}{2}}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.01492, size = 49, normalized size = 1.02 \begin{align*} \frac{16}{315} \, b^{2} x^{\frac{9}{2}} - \frac{8}{35} \, b x^{\frac{7}{2}} \operatorname{artanh}\left (\tanh \left (b x + a\right )\right ) + \frac{2}{5} \, x^{\frac{5}{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 = 2.0062, size = 73, normalized size = 1.52 \begin{align*} \frac{2}{315} \,{\left (35 \, b^{2} x^{4} + 90 \, a b x^{3} + 63 \, a^{2} x^{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 x^{\frac{3}{2}} \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 = 1.12977, size = 32, normalized size = 0.67 \begin{align*} \frac{2}{9} \, b^{2} x^{\frac{9}{2}} + \frac{4}{7} \, a b x^{\frac{7}{2}} + \frac{2}{5} \, a^{2} x^{\frac{5}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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