Optimal. Leaf size=27 \[ -\frac {1}{x}-2 i a \log (x)+2 i a \log (i-a x) \]
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Rubi [A]
time = 0.02, antiderivative size = 27, 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 = {5170, 78}
\begin {gather*} -2 i a \log (x)+2 i a \log (-a x+i)-\frac {1}{x} \end {gather*}
Antiderivative was successfully verified.
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Rule 78
Rule 5170
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 = 27, normalized size = 1.00 \begin {gather*} -\frac {1}{x}-2 i a \log (x)+2 i a \log (i-a x) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.10, size = 25, normalized size = 0.93
method | result | size |
default | \(-\frac {1}{x}-2 i a \ln \left (x \right )+2 i a \ln \left (-a x +i\right )\) | \(25\) |
risch | \(-\frac {1}{x}-2 i a \ln \left (x \right )-2 a \arctan \left (a x \right )+i a \ln \left (a^{2} x^{2}+1\right )\) | \(34\) |
meijerg | \(\frac {a^{2} x}{i a x +1}+i a \left (\frac {3 i a x}{3 i a x +3}+2 \ln \left (i a x +1\right )-1-2 \ln \left (x \right )-2 \ln \left (i a \right )+\frac {i}{x a}\right )\) | \(66\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 34, normalized size = 1.26 \begin {gather*} 2 i \, a \log \left (i \, a x + 1\right ) - 2 i \, a \log \left (x\right ) - \frac {a x - i}{a x^{2} - i \, x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.19, size = 26, normalized size = 0.96 \begin {gather*} \frac {-2 i \, a x \log \left (x\right ) + 2 i \, a x \log \left (\frac {a x - i}{a}\right ) - 1}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.08, size = 32, normalized size = 1.19 \begin {gather*} - 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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.41, size = 34, normalized size = 1.26 \begin {gather*} -2 i \, a \log \left (\frac {i}{i \, a x + 1} - i\right ) - \frac {a}{\frac {i}{i \, a x + 1} - i} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.42, size = 17, normalized size = 0.63 \begin {gather*} -4\,a\,\mathrm {atan}\left (2\,a\,x-\mathrm {i}\right )-\frac {1}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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