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 = 0.03, antiderivative size = 38, normalized size of antiderivative = 1.00, 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 88
Rule 5062
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*}
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Mathematica [A] time = 0.03, size = 38, normalized size = 1.00 \[ -\frac {4 a}{a x+i}+4 i a \log (x)-4 i a \log (a x+i)-\frac {1}{x} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 60, normalized size = 1.58 \[ -\frac {5 \, a x + 4 \, {\left (-i \, a^{2} x^{2} + a x\right )} \log \relax (x) + 4 \, {\left (i \, a^{2} x^{2} - a x\right )} \log \left (\frac {a x + i}{a}\right ) + i}{a x^{2} + i \, x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.14, size = 37, normalized size = 0.97 \[ -4 \, a i \log \left (a x + i\right ) + 4 \, a i \log \left ({\left | x \right |}\right ) - \frac {5 \, a x + i}{a x^{2} + i x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 45, normalized size = 1.18 \[ -\frac {1}{x}+4 i a \ln \relax (x )-\frac {4 a}{a x +i}-2 i a \ln \left (a^{2} x^{2}+1\right )-4 a \arctan \left (a x \right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.42, size = 53, normalized size = 1.39 \[ -4 \, a \arctan \left (a x\right ) - 2 i \, a \log \left (a^{2} x^{2} + 1\right ) + 4 i \, a \log \relax (x) - \frac {10 \, a^{2} x^{2} - 8 i \, a x + 2}{2 \, {\left (a^{2} x^{3} + x\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.45, size = 37, normalized size = 0.97 \[ -8\,a\,\mathrm {atan}\left (2\,a\,x+1{}\mathrm {i}\right )-\frac {5\,x+\frac {1{}\mathrm {i}}{a}}{x^2+\frac {x\,1{}\mathrm {i}}{a}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.30, size = 44, normalized size = 1.16 \[ 4 a \left (i \log {\left (8 a^{2} x \right )} - i \log {\left (8 a^{2} x + 8 i a \right )}\right ) + \frac {5 a x + i}{- a x^{2} - i x} \]
Verification of antiderivative is not currently implemented for this CAS.
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