Optimal. Leaf size=41 \[ \frac {x^{m+1} F_1\left (-m-1;-\frac {3}{4},\frac {3}{4};-m;\frac {1}{a x},-\frac {1}{a x}\right )}{m+1} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.04, antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {6173, 133} \[ \frac {x^{m+1} F_1\left (-m-1;-\frac {3}{4},\frac {3}{4};-m;\frac {1}{a x},-\frac {1}{a x}\right )}{m+1} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 133
Rule 6173
Rubi steps
\begin {align*} \int e^{-\frac {3}{2} \coth ^{-1}(a x)} x^m \, dx &=-\left (\left (\left (\frac {1}{x}\right )^m x^m\right ) \operatorname {Subst}\left (\int \frac {x^{-2-m} \left (1-\frac {x}{a}\right )^{3/4}}{\left (1+\frac {x}{a}\right )^{3/4}} \, dx,x,\frac {1}{x}\right )\right )\\ &=\frac {x^{1+m} F_1\left (-1-m;-\frac {3}{4},\frac {3}{4};-m;\frac {1}{a x},-\frac {1}{a x}\right )}{1+m}\\ \end {align*}
________________________________________________________________________________________
Mathematica [F] time = 0.61, size = 0, normalized size = 0.00 \[ \int e^{-\frac {3}{2} \coth ^{-1}(a x)} x^m \, dx \]
Verification is Not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.62, size = 0, normalized size = 0.00 \[ {\rm integral}\left (x^{m} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{4}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{m} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.05, size = 0, normalized size = 0.00 \[ \int x^{m} \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{4}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{m} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int x^m\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/4} \,d x \]
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]
________________________________________________________________________________________