Optimal. Leaf size=60 \[ \frac {i e^{i n \text {ArcTan}(a x)} (1-i a n x) \left (c+a^2 c x^2\right )^{-\frac {n^2}{2}}}{a^3 c n \left (1-n^2\right )} \]
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Rubi [A]
time = 0.08, 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 = {5187}
\begin {gather*} \frac {i (1-i a n x) e^{i n \text {ArcTan}(a x)} \left (a^2 c x^2+c\right )^{-\frac {n^2}{2}}}{a^3 c n \left (1-n^2\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 5187
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 = 0.02, size = 55, normalized size = 0.92 \begin {gather*} -\frac {e^{i n \text {ArcTan}(a x)} (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 {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.11, size = 62, normalized size = 1.03
method | result | size |
gosper | \(\frac {\left (-a x +i\right ) \left (a x +i\right ) \left (n a x +i\right ) {\mathrm e}^{i n \arctan \left (a x \right )} \left (a^{2} c \,x^{2}+c \right )^{-1-\frac {n^{2}}{2}}}{a^{3} n \left (n^{2}-1\right )}\) | \(62\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 7.03, size = 78, normalized size = 1.30 \begin {gather*} -\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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \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 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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