Optimal. Leaf size=48 \[ -\frac{8}{63} b x^{9/2} \tanh ^{-1}(\tanh (a+b x))+\frac{2}{7} x^{7/2} \tanh ^{-1}(\tanh (a+b x))^2+\frac{16}{693} b^2 x^{11/2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0220721, 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} \[ -\frac{8}{63} b x^{9/2} \tanh ^{-1}(\tanh (a+b x))+\frac{2}{7} x^{7/2} \tanh ^{-1}(\tanh (a+b x))^2+\frac{16}{693} b^2 x^{11/2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2168
Rule 30
Rubi steps
\begin{align*} \int x^{5/2} \tanh ^{-1}(\tanh (a+b x))^2 \, dx &=\frac{2}{7} x^{7/2} \tanh ^{-1}(\tanh (a+b x))^2-\frac{1}{7} (4 b) \int x^{7/2} \tanh ^{-1}(\tanh (a+b x)) \, dx\\ &=-\frac{8}{63} b x^{9/2} \tanh ^{-1}(\tanh (a+b x))+\frac{2}{7} x^{7/2} \tanh ^{-1}(\tanh (a+b x))^2+\frac{1}{63} \left (8 b^2\right ) \int x^{9/2} \, dx\\ &=\frac{16}{693} b^2 x^{11/2}-\frac{8}{63} b x^{9/2} \tanh ^{-1}(\tanh (a+b x))+\frac{2}{7} x^{7/2} \tanh ^{-1}(\tanh (a+b x))^2\\ \end{align*}
Mathematica [A] time = 0.0438811, size = 40, normalized size = 0.83 \[ \frac{2}{693} x^{7/2} \left (-44 b x \tanh ^{-1}(\tanh (a+b x))+99 \tanh ^{-1}(\tanh (a+b x))^2+8 b^2 x^2\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.042, size = 38, normalized size = 0.8 \begin{align*}{\frac{2\, \left ({\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) \right ) ^{2}}{7}{x}^{{\frac{7}{2}}}}-{\frac{8\,b}{7} \left ({\frac{{\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) }{9}{x}^{{\frac{9}{2}}}}-{\frac{2\,b}{99}{x}^{{\frac{11}{2}}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.02176, size = 49, normalized size = 1.02 \begin{align*} \frac{16}{693} \, b^{2} x^{\frac{11}{2}} - \frac{8}{63} \, b x^{\frac{9}{2}} \operatorname{artanh}\left (\tanh \left (b x + a\right )\right ) + \frac{2}{7} \, x^{\frac{7}{2}} \operatorname{artanh}\left (\tanh \left (b x + a\right )\right )^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.96114, size = 74, normalized size = 1.54 \begin{align*} \frac{2}{693} \,{\left (63 \, b^{2} x^{5} + 154 \, a b x^{4} + 99 \, a^{2} x^{3}\right )} \sqrt{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.11666, size = 32, normalized size = 0.67 \begin{align*} \frac{2}{11} \, b^{2} x^{\frac{11}{2}} + \frac{4}{9} \, a b x^{\frac{9}{2}} + \frac{2}{7} \, a^{2} x^{\frac{7}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]