Optimal. Leaf size=41 \[ \frac {x^{1+m} F_1\left (-1-m;-\frac {1}{4},\frac {1}{4};-m;\frac {1}{a x},-\frac {1}{a x}\right )}{1+m} \]
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Rubi [A]
time = 0.03, 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 = {6308, 138}
\begin {gather*} \frac {x^{m+1} F_1\left (-m-1;-\frac {1}{4},\frac {1}{4};-m;\frac {1}{a x},-\frac {1}{a x}\right )}{m+1} \end {gather*}
Antiderivative was successfully verified.
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Rule 138
Rule 6308
Rubi steps
\begin {align*} \int e^{-\frac {1}{2} \coth ^{-1}(a x)} x^m \, dx &=-\left (\left (\left (\frac {1}{x}\right )^m x^m\right ) \text {Subst}\left (\int \frac {x^{-2-m} \sqrt [4]{1-\frac {x}{a}}}{\sqrt [4]{1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )\right )\\ &=\frac {x^{1+m} F_1\left (-1-m;-\frac {1}{4},\frac {1}{4};-m;\frac {1}{a x},-\frac {1}{a x}\right )}{1+m}\\ \end {align*}
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Mathematica [F]
time = 0.39, size = 0, normalized size = 0.00 \begin {gather*} \int e^{-\frac {1}{2} \coth ^{-1}(a x)} x^m \, dx \end {gather*}
Verification is not applicable to the result.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int x^{m} \left (\frac {a x -1}{a x +1}\right )^{\frac {1}{4}}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int x^m\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{1/4} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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