Optimal. Leaf size=60 \[ \frac{i (1-i a n x) \left (a^2 c x^2+c\right )^{-\frac{n^2}{2}} e^{i n \tan ^{-1}(a x)}}{a^3 c n \left (1-n^2\right )} \]
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Rubi [A] time = 0.111918, antiderivative size = 60, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.029, Rules used = {5079} \[ \frac{i (1-i a n x) \left (a^2 c x^2+c\right )^{-\frac{n^2}{2}} e^{i n \tan ^{-1}(a x)}}{a^3 c n \left (1-n^2\right )} \]
Antiderivative was successfully verified.
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Rule 5079
Rubi steps
\begin{align*} \int e^{i n \tan ^{-1}(a x)} x^2 \left (c+a^2 c x^2\right )^{-1-\frac{n^2}{2}} \, dx &=\frac{i e^{i n \tan ^{-1}(a x)} (1-i a n x) \left (c+a^2 c x^2\right )^{-\frac{n^2}{2}}}{a^3 c n \left (1-n^2\right )}\\ \end{align*}
Mathematica [A] time = 0.024067, size = 55, normalized size = 0.92 \[ -\frac{(a n x+i) \left (a^2 c x^2+c\right )^{-\frac{n^2}{2}} e^{i n \tan ^{-1}(a x)}}{a^3 c n \left (n^2-1\right )} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.06, size = 62, normalized size = 1. \begin{align*}{\frac{ \left ( -ax+i \right ) \left ( ax+i \right ) \left ( nax+i \right ){{\rm e}^{in\arctan \left ( ax \right ) }}}{n{a}^{3} \left ({n}^{2}-1 \right ) } \left ({a}^{2}c{x}^{2}+c \right ) ^{-1-{\frac{{n}^{2}}{2}}}} \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{\left (a^{2} c x^{2} + c\right )}^{-\frac{1}{2} \, n^{2} - 1} x^{2} 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 [A] time = 2.24294, size = 162, normalized size = 2.7 \begin{align*} -\frac{{\left (a^{3} n x^{3} + i \, a^{2} x^{2} + a n x + i\right )}{\left (a^{2} c x^{2} + c\right )}^{-\frac{1}{2} \, n^{2} - 1}}{{\left (a^{3} n^{3} - a^{3} n\right )} \left (-\frac{a x + i}{a x - i}\right )^{\frac{1}{2} \, n}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.14228, size = 487, normalized size = 8.12 \begin{align*} -\frac{a^{3} n x^{3} e^{\left (\frac{1}{2} \, \pi i n - \frac{1}{2} \, n^{2} \log \left (a x + i\right ) - \frac{1}{2} \, n^{2} \log \left (a x - i\right ) - \frac{1}{2} \, n^{2} \log \left (c\right ) - \frac{1}{2} \, n \log \left (a x + i\right ) + \frac{1}{2} \, n \log \left (a x - i\right ) - \log \left (a x + i\right ) - \log \left (a x - i\right ) - \log \left (c\right )\right )} + a^{2} i x^{2} e^{\left (\frac{1}{2} \, \pi i n - \frac{1}{2} \, n^{2} \log \left (a x + i\right ) - \frac{1}{2} \, n^{2} \log \left (a x - i\right ) - \frac{1}{2} \, n^{2} \log \left (c\right ) - \frac{1}{2} \, n \log \left (a x + i\right ) + \frac{1}{2} \, n \log \left (a x - i\right ) - \log \left (a x + i\right ) - \log \left (a x - i\right ) - \log \left (c\right )\right )} + a n x e^{\left (\frac{1}{2} \, \pi i n - \frac{1}{2} \, n^{2} \log \left (a x + i\right ) - \frac{1}{2} \, n^{2} \log \left (a x - i\right ) - \frac{1}{2} \, n^{2} \log \left (c\right ) - \frac{1}{2} \, n \log \left (a x + i\right ) + \frac{1}{2} \, n \log \left (a x - i\right ) - \log \left (a x + i\right ) - \log \left (a x - i\right ) - \log \left (c\right )\right )} + i e^{\left (\frac{1}{2} \, \pi i n - \frac{1}{2} \, n^{2} \log \left (a x + i\right ) - \frac{1}{2} \, n^{2} \log \left (a x - i\right ) - \frac{1}{2} \, n^{2} \log \left (c\right ) - \frac{1}{2} \, n \log \left (a x + i\right ) + \frac{1}{2} \, n \log \left (a x - i\right ) - \log \left (a x + i\right ) - \log \left (a x - i\right ) - \log \left (c\right )\right )}}{a^{3} n^{3} - a^{3} n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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