Optimal. Leaf size=27 \[ \frac{2}{5} x^{5/2} \tanh ^{-1}(\tanh (a+b x))-\frac{4}{35} b x^{7/2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0098923, antiderivative size = 27, 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} \[ \frac{2}{5} x^{5/2} \tanh ^{-1}(\tanh (a+b x))-\frac{4}{35} b x^{7/2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2168
Rule 30
Rubi steps
\begin{align*} \int x^{3/2} \tanh ^{-1}(\tanh (a+b x)) \, dx &=\frac{2}{5} x^{5/2} \tanh ^{-1}(\tanh (a+b x))-\frac{1}{5} (2 b) \int x^{5/2} \, dx\\ &=-\frac{4}{35} b x^{7/2}+\frac{2}{5} x^{5/2} \tanh ^{-1}(\tanh (a+b x))\\ \end{align*}
Mathematica [A] time = 0.0255402, size = 23, normalized size = 0.85 \[ \frac{2}{35} x^{5/2} \left (7 \tanh ^{-1}(\tanh (a+b x))-2 b x\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.037, size = 20, normalized size = 0.7 \begin{align*} -{\frac{4\,b}{35}{x}^{{\frac{7}{2}}}}+{\frac{2\,{\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) }{5}{x}^{{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 0.977613, size = 26, normalized size = 0.96 \begin{align*} -\frac{4}{35} \, b x^{\frac{7}{2}} + \frac{2}{5} \, x^{\frac{5}{2}} \operatorname{artanh}\left (\tanh \left (b x + a\right )\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.99508, size = 46, normalized size = 1.7 \begin{align*} \frac{2}{35} \,{\left (5 \, b x^{3} + 7 \, a x^{2}\right )} \sqrt{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 26.3986, size = 26, normalized size = 0.96 \begin{align*} - \frac{4 b x^{\frac{7}{2}}}{35} + \frac{2 x^{\frac{5}{2}} \operatorname{atanh}{\left (\tanh{\left (a + b x \right )} \right )}}{5} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.13407, size = 18, normalized size = 0.67 \begin{align*} \frac{2}{7} \, b x^{\frac{7}{2}} + \frac{2}{5} \, a x^{\frac{5}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]