Optimal. Leaf size=41 \[ \frac {x^{m+1} F_1\left (-m-1;\frac {1}{8},-\frac {1}{8};-m;\frac {1}{a x},-\frac {1}{a x}\right )}{m+1} \]
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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} \[ \frac {x^{m+1} F_1\left (-m-1;\frac {1}{8},-\frac {1}{8};-m;\frac {1}{a x},-\frac {1}{a x}\right )}{m+1} \]
Antiderivative was successfully verified.
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Rule 133
Rule 6173
Rubi steps
\begin {align*} \int e^{\frac {1}{4} \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} \sqrt [8]{1+\frac {x}{a}}}{\sqrt [8]{1-\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )\right )\\ &=\frac {x^{1+m} F_1\left (-1-m;\frac {1}{8},-\frac {1}{8};-m;\frac {1}{a x},-\frac {1}{a x}\right )}{1+m}\\ \end {align*}
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Mathematica [F] time = 0.69, size = 0, normalized size = 0.00 \[ \int e^{\frac {1}{4} \coth ^{-1}(a x)} x^m \, dx \]
Verification is Not applicable to the result.
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fricas [F] time = 0.62, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (a x + 1\right )} x^{m} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {7}{8}}}{a x - 1}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{m}}{\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{8}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.04, size = 0, normalized size = 0.00 \[ \int \frac {x^{m}}{\left (\frac {a x -1}{a x +1}\right )^{\frac {1}{8}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{m}}{\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{8}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {x^m}{{\left (\frac {a\,x-1}{a\,x+1}\right )}^{1/8}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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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.
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