Optimal. Leaf size=35 \[ -\frac{i}{2 a (a x+i)^2}-\frac{2}{3 a (a x+i)^3} \]
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Rubi [A] time = 0.0384182, antiderivative size = 35, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {5073, 43} \[ -\frac{i}{2 a (a x+i)^2}-\frac{2}{3 a (a x+i)^3} \]
Antiderivative was successfully verified.
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Rule 5073
Rule 43
Rubi steps
\begin{align*} \int \frac{e^{5 i \tan ^{-1}(a x)}}{\left (1+a^2 x^2\right )^{3/2}} \, dx &=\int \frac{1+i a x}{(1-i a x)^4} \, dx\\ &=\int \left (\frac{2}{(i+a x)^4}+\frac{i}{(i+a x)^3}\right ) \, dx\\ &=-\frac{2}{3 a (i+a x)^3}-\frac{i}{2 a (i+a x)^2}\\ \end{align*}
Mathematica [A] time = 0.0160125, size = 24, normalized size = 0.69 \[ -\frac{1+3 i a x}{6 a (a x+i)^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.049, size = 20, normalized size = 0.6 \begin{align*}{\frac{1}{ \left ( ax+i \right ) ^{3}} \left ( -{\frac{i}{2}}x-{\frac{1}{6\,a}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.51073, size = 80, normalized size = 2.29 \begin{align*} \frac{-24 i \, a^{4} x^{4} - 80 \, a^{3} x^{3} + 96 i \, a^{2} x^{2} + 48 \, a x - 8 i}{48 \,{\left (a^{7} x^{6} + 3 \, a^{5} x^{4} + 3 \, a^{3} x^{2} + a\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.91439, size = 84, normalized size = 2.4 \begin{align*} \frac{-3 i \, a x - 1}{6 \, a^{4} x^{3} + 18 i \, a^{3} x^{2} - 18 \, a^{2} x - 6 i \, a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.647313, size = 44, normalized size = 1.26 \begin{align*} - \frac{3 i a^{7} x + a^{6}}{6 a^{10} x^{3} + 18 i a^{9} x^{2} - 18 a^{8} x - 6 i a^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.11667, size = 26, normalized size = 0.74 \begin{align*} -\frac{3 \, a i x + 1}{6 \,{\left (a x + i\right )}^{3} a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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