Optimal. Leaf size=52 \[ -\frac{4 i a^2}{a x+i}-8 a^2 \log (x)+8 a^2 \log (a x+i)-\frac{4 i a}{x}-\frac{1}{2 x^2} \]
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Rubi [A] time = 0.0353444, antiderivative size = 52, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {5062, 88} \[ -\frac{4 i a^2}{a x+i}-8 a^2 \log (x)+8 a^2 \log (a x+i)-\frac{4 i a}{x}-\frac{1}{2 x^2} \]
Antiderivative was successfully verified.
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Rule 5062
Rule 88
Rubi steps
\begin{align*} \int \frac{e^{4 i \tan ^{-1}(a x)}}{x^3} \, dx &=\int \frac{(1+i a x)^2}{x^3 (1-i a x)^2} \, dx\\ &=\int \left (\frac{1}{x^3}+\frac{4 i a}{x^2}-\frac{8 a^2}{x}+\frac{4 i a^3}{(i+a x)^2}+\frac{8 a^3}{i+a x}\right ) \, dx\\ &=-\frac{1}{2 x^2}-\frac{4 i a}{x}-\frac{4 i a^2}{i+a x}-8 a^2 \log (x)+8 a^2 \log (i+a x)\\ \end{align*}
Mathematica [A] time = 0.0358388, size = 52, normalized size = 1. \[ -\frac{4 i a^2}{a x+i}-8 a^2 \log (x)+8 a^2 \log (a x+i)-\frac{4 i a}{x}-\frac{1}{2 x^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.048, size = 60, normalized size = 1.2 \begin{align*}{\frac{-4\,i{a}^{2}}{ax+i}}+4\,{a}^{2}\ln \left ({a}^{2}{x}^{2}+1 \right ) -8\,i{a}^{2}\arctan \left ( ax \right ) -{\frac{1}{2\,{x}^{2}}}-{\frac{4\,ia}{x}}-8\,{a}^{2}\ln \left ( x \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.53399, size = 93, normalized size = 1.79 \begin{align*} -8 i \, a^{2} \arctan \left (a x\right ) + 4 \, a^{2} \log \left (a^{2} x^{2} + 1\right ) - 8 \, a^{2} \log \left (x\right ) + \frac{-16 i \, a^{3} x^{3} - 9 \, a^{2} x^{2} - 8 i \, a x - 1}{2 \,{\left (a^{2} x^{4} + x^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.71169, size = 171, normalized size = 3.29 \begin{align*} \frac{-16 i \, a^{2} x^{2} + 7 \, a x - 16 \,{\left (a^{3} x^{3} + i \, a^{2} x^{2}\right )} \log \left (x\right ) + 16 \,{\left (a^{3} x^{3} + i \, a^{2} x^{2}\right )} \log \left (\frac{a x + i}{a}\right ) - i}{2 \,{\left (a x^{3} + i \, x^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.685398, size = 51, normalized size = 0.98 \begin{align*} 8 a^{2} \left (- \log{\left (x \right )} + \log{\left (x + \frac{i}{a} \right )}\right ) - \frac{16 i a^{3} x^{2} - 7 a^{2} x + i a}{2 a^{2} x^{3} + 2 i a x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.09499, size = 63, normalized size = 1.21 \begin{align*} 8 \, a^{2} \log \left (a x + i\right ) - 8 \, a^{2} \log \left ({\left | x \right |}\right ) - \frac{16 \, a^{2} i x^{2} - 7 \, a x + i}{2 \,{\left (a x + i\right )} x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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