Optimal. Leaf size=71 \[ \frac {i 2^{\frac {n}{2}+1} (1-i a x)^{1-\frac {n}{2}} \, _2F_1\left (1-\frac {n}{2},-\frac {n}{2};2-\frac {n}{2};\frac {1}{2} (1-i a x)\right )}{a (2-n)} \]
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Rubi [A] time = 0.01, antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {5061, 69} \[ \frac {i 2^{\frac {n}{2}+1} (1-i a x)^{1-\frac {n}{2}} \, _2F_1\left (1-\frac {n}{2},-\frac {n}{2};2-\frac {n}{2};\frac {1}{2} (1-i a x)\right )}{a (2-n)} \]
Antiderivative was successfully verified.
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Rule 69
Rule 5061
Rubi steps
\begin {align*} \int e^{i n \tan ^{-1}(a x)} \, dx &=\int (1-i a x)^{-n/2} (1+i a x)^{n/2} \, dx\\ &=\frac {i 2^{1+\frac {n}{2}} (1-i a x)^{1-\frac {n}{2}} \, _2F_1\left (1-\frac {n}{2},-\frac {n}{2};2-\frac {n}{2};\frac {1}{2} (1-i a x)\right )}{a (2-n)}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 53, normalized size = 0.75 \[ -\frac {4 i e^{i (n+2) \tan ^{-1}(a x)} \, _2F_1\left (2,\frac {n}{2}+1;\frac {n}{2}+2;-e^{2 i \tan ^{-1}(a x)}\right )}{a (n+2)} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.45, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {1}{\left (-\frac {a x + i}{a x - i}\right )^{\frac {1}{2} \, n}}, x\right ) \]
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 \[ \mathit {sage}_{0} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.12, size = 0, normalized size = 0.00 \[ \int {\mathrm e}^{i n \arctan \left (a x \right )}\, dx \]
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 \[ \int e^{\left (i \, n \arctan \left (a x\right )\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\mathrm {e}}^{n\,\mathrm {atan}\left (a\,x\right )\,1{}\mathrm {i}} \,d x \]
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 \[ \int e^{i n \operatorname {atan}{\left (a x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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