Optimal. Leaf size=28 \[ \frac {x^{m+1} F_1\left (m+1;\frac {1}{6},-\frac {1}{6};m+2;x,-x\right )}{m+1} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.02, antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {6126, 133} \[ \frac {x^{m+1} F_1\left (m+1;\frac {1}{6},-\frac {1}{6};m+2;x,-x\right )}{m+1} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 133
Rule 6126
Rubi steps
\begin {align*} \int e^{\frac {1}{3} \tanh ^{-1}(x)} x^m \, dx &=\int \frac {x^m \sqrt [6]{1+x}}{\sqrt [6]{1-x}} \, dx\\ &=\frac {x^{1+m} F_1\left (1+m;\frac {1}{6},-\frac {1}{6};2+m;x,-x\right )}{1+m}\\ \end {align*}
________________________________________________________________________________________
Mathematica [F] time = 0.48, size = 0, normalized size = 0.00 \[ \int e^{\frac {1}{3} \tanh ^{-1}(x)} x^m \, dx \]
Verification is Not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.59, size = 0, normalized size = 0.00 \[ {\rm integral}\left (x^{m} \left (-\frac {\sqrt {-x^{2} + 1}}{x - 1}\right )^{\frac {1}{3}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{m} \left (\frac {x + 1}{\sqrt {-x^{2} + 1}}\right )^{\frac {1}{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.03, size = 0, normalized size = 0.00 \[ \int \left (\frac {1+x}{\sqrt {-x^{2}+1}}\right )^{\frac {1}{3}} x^{m}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{m} \left (\frac {x + 1}{\sqrt {-x^{2} + 1}}\right )^{\frac {1}{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.04 \[ \int x^m\,{\left (\frac {x+1}{\sqrt {1-x^2}}\right )}^{1/3} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________