Optimal. Leaf size=19 \[ \frac {i}{2 a (1+i a x)^2} \]
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Rubi [A]
time = 0.02, antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {5181, 32}
\begin {gather*} \frac {i}{2 a (1+i a x)^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 32
Rule 5181
Rubi steps
\begin {align*} \int \frac {e^{-3 i \tan ^{-1}(a x)}}{\left (1+a^2 x^2\right )^{3/2}} \, dx &=\int \frac {1}{(1+i a x)^3} \, dx\\ &=\frac {i}{2 a (1+i a x)^2}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 18, normalized size = 0.95 \begin {gather*} -\frac {i}{2 a (-i+a x)^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.07, size = 16, normalized size = 0.84
method | result | size |
risch | \(-\frac {i}{2 a \left (a x -i\right )^{2}}\) | \(15\) |
default | \(\frac {i}{2 a \left (i a x +1\right )^{2}}\) | \(16\) |
meijerg | \(\frac {x \left (i a x +2\right )}{2 \left (i a x +1\right )^{2}}\) | \(20\) |
gosper | \(\frac {-a x +i}{2 a \left (i a x +1\right )^{3}}\) | \(22\) |
norman | \(\frac {x -\frac {3}{2} i a \,x^{2}-\frac {1}{2} i a^{3} x^{4}}{\left (a^{2} x^{2}+1\right )^{2}}\) | \(31\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.25, size = 13, normalized size = 0.68 \begin {gather*} \frac {i}{2 \, {\left (i \, a x + 1\right )}^{2} a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.69, size = 21, normalized size = 1.11 \begin {gather*} -\frac {i}{2 \, {\left (a^{3} x^{2} - 2 i \, a^{2} x - a\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.09, size = 22, normalized size = 1.16 \begin {gather*} - \frac {i}{2 a^{3} x^{2} - 4 i a^{2} x - 2 a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.42, size = 13, normalized size = 0.68 \begin {gather*} \frac {i}{2 \, {\left (i \, a x + 1\right )}^{2} a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.05, size = 24, normalized size = 1.26 \begin {gather*} \frac {1{}\mathrm {i}}{2\,\left (-a^3\,x^2+a^2\,x\,2{}\mathrm {i}+a\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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