Optimal. Leaf size=36 \[ \frac {x^{m+1} F_1\left (m+1;-\frac {1}{4},\frac {1}{4};m+2;i a x,-i a x\right )}{m+1} \]
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Rubi [A] time = 0.03, 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 = {5062, 133} \[ \frac {x^{m+1} F_1\left (m+1;-\frac {1}{4},\frac {1}{4};m+2;i a x,-i a x\right )}{m+1} \]
Antiderivative was successfully verified.
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Rule 133
Rule 5062
Rubi steps
\begin {align*} \int e^{-\frac {1}{2} i \tan ^{-1}(a x)} x^m \, dx &=\int \frac {x^m \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}} \, dx\\ &=\frac {x^{1+m} F_1\left (1+m;-\frac {1}{4},\frac {1}{4};2+m;i a x,-i a x\right )}{1+m}\\ \end {align*}
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Mathematica [F] time = 0.27, size = 0, normalized size = 0.00 \[ \int e^{-\frac {1}{2} i \tan ^{-1}(a x)} x^m \, dx \]
Verification is Not applicable to the result.
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fricas [F] time = 0.45, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {i \, \sqrt {a^{2} x^{2} + 1} x^{m} \sqrt {\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}}}{a x - i}, x\right ) \]
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 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.11, size = 0, normalized size = 0.00 \[ \int \frac {x^{m}}{\sqrt {\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}}}\, dx \]
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 \[ \int \frac {x^{m}}{\sqrt {\frac {i \, a x + 1}{\sqrt {a^{2} x^{2} + 1}}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {x^m}{\sqrt {\frac {1+a\,x\,1{}\mathrm {i}}{\sqrt {a^2\,x^2+1}}}} \,d x \]
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 \[ \int \frac {x^{m}}{\sqrt {\frac {i \left (a x - i\right )}{\sqrt {a^{2} x^{2} + 1}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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