Optimal. Leaf size=38 \[ \frac {i-2 a x}{6 a^3 c^3 (1-i a x) (1+i a x)^3} \]
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Rubi [A]
time = 0.06, 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 = {5190, 82}
\begin {gather*} \frac {-2 a x+i}{6 a^3 c^3 (1-i a x) (1+i a x)^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 82
Rule 5190
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)^2 (1+i a x)^4} \, dx}{c^3}\\ &=\frac {i-2 a x}{6 a^3 c^3 (1-i a x) (1+i a x)^3}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 36, normalized size = 0.95 \begin {gather*} \frac {-i+2 a x}{6 a^3 c^3 (-i+a x)^3 (i+a x)} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.11, size = 62, normalized size = 1.63
method | result | size |
risch | \(\frac {\frac {x}{3 a^{2}}-\frac {i}{6 a^{3}}}{c^{3} \left (a x -i\right )^{3} \left (a x +i\right )}\) | \(34\) |
norman | \(\frac {\frac {x^{3}}{3 c}-\frac {i a \,x^{4}}{2 c}-\frac {i a^{3} x^{6}}{6 c}}{\left (a^{2} x^{2}+1\right )^{3} c^{2}}\) | \(47\) |
gosper | \(-\frac {\left (-2 a x +i\right ) \left (a x +i\right ) \left (-a x +i\right )}{6 \left (a^{2} x^{2}+1\right )^{2} c^{3} \left (i a x +1\right )^{2} a^{3}}\) | \(49\) |
default | \(\frac {-\frac {i}{8 a^{3} \left (-a x +i\right )^{2}}-\frac {1}{12 a^{3} \left (-a x +i\right )^{3}}-\frac {1}{16 a^{3} \left (-a x +i\right )}-\frac {1}{16 a^{3} \left (a x +i\right )}}{c^{3}}\) | \(62\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.64, size = 49, normalized size = 1.29 \begin {gather*} \frac {2 \, a x - i}{6 \, {\left (a^{7} c^{3} x^{4} - 2 i \, a^{6} c^{3} x^{3} - 2 i \, a^{4} c^{3} x - a^{3} c^{3}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.23, size = 53, normalized size = 1.39 \begin {gather*} - \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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 80 vs. \(2 (30) = 60\).
time = 0.43, size = 80, normalized size = 2.11 \begin {gather*} -\frac {1}{32 \, a^{3} c^{3} {\left (\frac {2 i}{i \, a x + 1} - i\right )}} - \frac {-\frac {3 i \, a^{3} c^{6}}{i \, a x + 1} - \frac {6 i \, a^{3} c^{6}}{{\left (i \, a x + 1\right )}^{2}} + \frac {4 i \, a^{3} c^{6}}{{\left (i \, a x + 1\right )}^{3}}}{48 \, a^{6} c^{9}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.59, size = 65, normalized size = 1.71 \begin {gather*} \frac {2\,a^3\,x^3+a^2\,x^2\,3{}\mathrm {i}+1{}\mathrm {i}}{6\,a^9\,c^3\,x^6+18\,a^7\,c^3\,x^4+18\,a^5\,c^3\,x^2+6\,a^3\,c^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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