Optimal. Leaf size=38 \[ -\frac {2 a x+i}{6 a^3 c^3 (1-i a x)^3 (1+i a x)} \]
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Rubi [A] time = 0.08, antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {5082, 81} \[ -\frac {2 a x+i}{6 a^3 c^3 (1-i a x)^3 (1+i a x)} \]
Antiderivative was successfully verified.
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Rule 81
Rule 5082
Rubi steps
\begin {align*} \int \frac {e^{2 i \tan ^{-1}(a x)} x^2}{\left (c+a^2 c x^2\right )^3} \, dx &=\frac {\int \frac {x^2}{(1-i a x)^4 (1+i a x)^2} \, dx}{c^3}\\ &=-\frac {i+2 a x}{6 a^3 c^3 (1-i a x)^3 (1+i a x)}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 36, normalized size = 0.95 \[ \frac {2 a x+i}{6 a^3 c^3 (a x-i) (a x+i)^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.54, size = 49, normalized size = 1.29 \[ \frac {2 \, a x + i}{6 \, a^{7} c^{3} x^{4} + 12 i \, a^{6} c^{3} x^{3} + 12 i \, a^{4} c^{3} x - 6 \, a^{3} c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.12, size = 48, normalized size = 1.26 \[ -\frac {1}{16 \, {\left (a x - i\right )} a^{3} c^{3}} + \frac {3 \, a^{2} x^{2} + 12 \, a i x - 5}{48 \, {\left (a x + i\right )}^{3} a^{3} c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 34, normalized size = 0.89 \[ \frac {\frac {x}{3 a^{2}}+\frac {i}{6 a^{3}}}{c^{3} \left (a x +i\right )^{3} \left (a x -i\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.42, size = 62, normalized size = 1.63 \[ \frac {16 \, a^{3} x^{3} - 24 i \, a^{2} x^{2} - 8 i}{48 \, {\left (a^{9} c^{3} x^{6} + 3 \, a^{7} c^{3} x^{4} + 3 \, a^{5} c^{3} x^{2} + a^{3} c^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.64, size = 47, normalized size = 1.24 \[ \frac {\frac {x}{3\,a^6\,c^3}+\frac {1{}\mathrm {i}}{6\,a^7\,c^3}}{\frac {x\,2{}\mathrm {i}}{a^3}-\frac {1}{a^4}+x^4+\frac {x^3\,2{}\mathrm {i}}{a}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.44, size = 53, normalized size = 1.39 \[ - \frac {2 a x + i}{- 6 a^{7} c^{3} x^{4} - 12 i a^{6} c^{3} x^{3} - 12 i a^{4} c^{3} x + 6 a^{3} c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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