Optimal. Leaf size=24 \[ \frac{a x^{m+2}}{m+2}+\frac{x^{m+1}}{m+1} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0799803, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {6150, 43} \[ \frac{a x^{m+2}}{m+2}+\frac{x^{m+1}}{m+1} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 6150
Rule 43
Rubi steps
\begin{align*} \int e^{\tanh ^{-1}(a x)} x^m \sqrt{1-a^2 x^2} \, dx &=\int x^m (1+a x) \, dx\\ &=\int \left (x^m+a x^{1+m}\right ) \, dx\\ &=\frac{x^{1+m}}{1+m}+\frac{a x^{2+m}}{2+m}\\ \end{align*}
Mathematica [A] time = 0.0132547, size = 20, normalized size = 0.83 \[ x^{m+1} \left (\frac{a x}{m+2}+\frac{1}{m+1}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.027, size = 27, normalized size = 1.1 \begin{align*}{\frac{{x}^{1+m} \left ( amx+ax+m+2 \right ) }{ \left ( 2+m \right ) \left ( 1+m \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 2.13452, size = 66, normalized size = 2.75 \begin{align*} \frac{{\left ({\left (a m + a\right )} x^{2} +{\left (m + 2\right )} x\right )} x^{m}}{m^{2} + 3 \, m + 2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 0.277716, size = 82, normalized size = 3.42 \begin{align*} \begin{cases} a \log{\left (x \right )} - \frac{1}{x} & \text{for}\: m = -2 \\a x + \log{\left (x \right )} & \text{for}\: m = -1 \\\frac{a m x^{2} x^{m}}{m^{2} + 3 m + 2} + \frac{a x^{2} x^{m}}{m^{2} + 3 m + 2} + \frac{m x x^{m}}{m^{2} + 3 m + 2} + \frac{2 x x^{m}}{m^{2} + 3 m + 2} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.13644, size = 55, normalized size = 2.29 \begin{align*} \frac{a m x^{2} x^{m} + a x^{2} x^{m} + m x x^{m} + 2 \, x x^{m}}{m^{2} + 3 \, m + 2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]