Optimal. Leaf size=38 \[ -\frac{4 a}{a x+i}+4 i a \log (x)-4 i a \log (a x+i)-\frac{1}{x} \]
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Rubi [A] time = 0.030845, antiderivative size = 38, 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} \[ -\frac{4 a}{a x+i}+4 i a \log (x)-4 i a \log (a x+i)-\frac{1}{x} \]
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^2} \, dx &=\int \frac{(1+i a x)^2}{x^2 (1-i a x)^2} \, dx\\ &=\int \left (\frac{1}{x^2}+\frac{4 i a}{x}+\frac{4 a^2}{(i+a x)^2}-\frac{4 i a^2}{i+a x}\right ) \, dx\\ &=-\frac{1}{x}-\frac{4 a}{i+a x}+4 i a \log (x)-4 i a \log (i+a x)\\ \end{align*}
Mathematica [A] time = 0.0242619, size = 38, normalized size = 1. \[ -\frac{4 a}{a x+i}+4 i a \log (x)-4 i a \log (a x+i)-\frac{1}{x} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.049, size = 45, normalized size = 1.2 \begin{align*} -4\,{\frac{a}{ax+i}}-2\,ia\ln \left ({a}^{2}{x}^{2}+1 \right ) -4\,a\arctan \left ( ax \right ) -{x}^{-1}+4\,ia\ln \left ( x \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.53394, size = 72, normalized size = 1.89 \begin{align*} -4 \, a \arctan \left (a x\right ) - 2 i \, a \log \left (a^{2} x^{2} + 1\right ) + 4 i \, a \log \left (x\right ) - \frac{10 \, a^{2} x^{2} - 8 i \, a x + 2}{2 \,{\left (a^{2} x^{3} + x\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.66007, size = 131, normalized size = 3.45 \begin{align*} -\frac{5 \, a x + 4 \,{\left (-i \, a^{2} x^{2} + a x\right )} \log \left (x\right ) + 4 \,{\left (i \, a^{2} x^{2} - a x\right )} \log \left (\frac{a x + i}{a}\right ) + i}{a x^{2} + i \, x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.608917, size = 37, normalized size = 0.97 \begin{align*} 4 a \left (i \log{\left (x \right )} - i \log{\left (x + \frac{i}{a} \right )}\right ) - \frac{5 a^{2} x + i a}{a^{2} x^{2} + i a x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.10653, size = 50, normalized size = 1.32 \begin{align*} -4 \, a i \log \left (a x + i\right ) + 4 \, a i \log \left ({\left | x \right |}\right ) - \frac{5 \, a x + i}{a x^{2} + i x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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