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 = 0.0127803, antiderivative size = 71, normalized size of antiderivative = 1., 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 5061
Rule 69
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*}
Mathematica [A] time = 0.0291691, 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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Maple [F] time = 0.067, size = 0, normalized size = 0. \begin{align*} \int{{\rm e}^{in\arctan \left ( ax \right ) }}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int e^{\left (i \, n \arctan \left (a x\right )\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{1}{\left (-\frac{a x + i}{a x - i}\right )^{\frac{1}{2} \, n}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int e^{i n \operatorname{atan}{\left (a x \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int e^{\left (i \, n \arctan \left (a x\right )\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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