Optimal. Leaf size=36 \[ \frac {x^{1+m} F_1\left (1+m;-\frac {3}{4},\frac {3}{4};2+m;i a x,-i a x\right )}{1+m} \]
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Rubi [A]
time = 0.02, antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {5170, 138}
\begin {gather*} \frac {x^{m+1} F_1\left (m+1;-\frac {3}{4},\frac {3}{4};m+2;i a x,-i a x\right )}{m+1} \end {gather*}
Antiderivative was successfully verified.
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Rule 138
Rule 5170
Rubi steps
\begin {align*} \int e^{-\frac {3}{2} i \tan ^{-1}(a x)} x^m \, dx &=\int \frac {x^m (1-i a x)^{3/4}}{(1+i a x)^{3/4}} \, dx\\ &=\frac {x^{1+m} F_1\left (1+m;-\frac {3}{4},\frac {3}{4};2+m;i a x,-i a x\right )}{1+m}\\ \end {align*}
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Mathematica [F]
time = 0.19, size = 0, normalized size = 0.00 \begin {gather*} \int e^{-\frac {3}{2} i \text {ArcTan}(a x)} x^m \, dx \end {gather*}
Verification is not applicable to the result.
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Maple [F]
time = 0.00, size = 0, normalized size = 0.00 \[\int \frac {x^{m}}{\left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )^{\frac {3}{2}}}\, dx\]
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]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{m}}{\left (\frac {i \left (a x - i\right )}{\sqrt {a^{2} x^{2} + 1}}\right )^{\frac {3}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \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 = -0.03 \begin {gather*} \int \frac {x^m}{{\left (\frac {1+a\,x\,1{}\mathrm {i}}{\sqrt {a^2\,x^2+1}}\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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