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 = 0.11, antiderivative size = 60, normalized size of antiderivative = 1.00, 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*}
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Mathematica [A] time = 0.03, 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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fricas [A] time = 0.47, size = 78, normalized size = 1.30 \[ -\frac {{\left (a^{3} n x^{3} + i \, a^{2} x^{2} + a n x + i\right )} {\left (a^{2} c x^{2} + c\right )}^{-\frac {1}{2} \, n^{2} - 1}}{{\left (a^{3} n^{3} - a^{3} n\right )} \left (-\frac {a x + i}{a x - i}\right )^{\frac {1}{2} \, n}} \]
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 [A] time = 0.14, size = 62, normalized size = 1.03 \[ \frac {\left (-a x +i\right ) \left (a x +i\right ) \left (n a x +i\right ) {\mathrm e}^{i n \arctan \left (a x \right )} \left (a^{2} c \,x^{2}+c \right )^{-1-\frac {n^{2}}{2}}}{a^{3} n \left (n^{2}-1\right )} \]
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 \[ \int {\left (a^{2} c x^{2} + c\right )}^{-\frac {1}{2} \, n^{2} - 1} x^{2} e^{\left (i \, n \arctan \left (a x\right )\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {x^2\,{\mathrm {e}}^{n\,\mathrm {atan}\left (a\,x\right )\,1{}\mathrm {i}}}{{\left (c\,a^2\,x^2+c\right )}^{\frac {n^2}{2}+1}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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