Optimal. Leaf size=26 \[ 2 i a \log (x)-2 i a \log (a x+i)-\frac {1}{x} \]
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Rubi [A] time = 0.02, antiderivative size = 26, 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, 77} \[ 2 i a \log (x)-2 i a \log (a x+i)-\frac {1}{x} \]
Antiderivative was successfully verified.
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Rule 77
Rule 5062
Rubi steps
\begin {align*} \int \frac {e^{2 i \tan ^{-1}(a x)}}{x^2} \, dx &=\int \frac {1+i a x}{x^2 (1-i a x)} \, dx\\ &=\int \left (\frac {1}{x^2}+\frac {2 i a}{x}-\frac {2 i a^2}{i+a x}\right ) \, dx\\ &=-\frac {1}{x}+2 i a \log (x)-2 i a \log (i+a x)\\ \end {align*}
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Mathematica [A] time = 0.01, size = 26, normalized size = 1.00 \[ 2 i a \log (x)-2 i a \log (a x+i)-\frac {1}{x} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 26, normalized size = 1.00 \[ \frac {2 i \, a x \log \relax (x) - 2 i \, a x \log \left (\frac {a x + i}{a}\right ) - 1}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.13, size = 23, normalized size = 0.88 \[ -2 \, a i \log \left (a x + i\right ) + 2 \, a i \log \left ({\left | x \right |}\right ) - \frac {1}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 34, normalized size = 1.31 \[ -\frac {1}{x}+2 i a \ln \relax (x )-i a \ln \left (a^{2} x^{2}+1\right )-2 a \arctan \left (a x \right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.44, size = 31, normalized size = 1.19 \[ -2 \, a \arctan \left (a x\right ) - i \, a \log \left (a^{2} x^{2} + 1\right ) + 2 i \, a \log \relax (x) - \frac {1}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 17, normalized size = 0.65 \[ -4\,a\,\mathrm {atan}\left (2\,a\,x+1{}\mathrm {i}\right )-\frac {1}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.16, size = 32, normalized size = 1.23 \[ - 2 a \left (- i \log {\left (4 a^{2} x \right )} + i \log {\left (4 a^{2} x + 4 i a \right )}\right ) - \frac {1}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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