Optimal. Leaf size=20 \[ -x-\frac {2 i \log (i-a x)}{a} \]
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Rubi [A]
time = 0.01, antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {5169, 45}
\begin {gather*} -x-\frac {2 i \log (-a x+i)}{a} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 5169
Rubi steps
\begin {align*} \int e^{-2 i \tan ^{-1}(a x)} \, dx &=\int \frac {1-i a x}{1+i a x} \, dx\\ &=\int \left (-1-\frac {2 i}{-i+a x}\right ) \, dx\\ &=-x-\frac {2 i \log (i-a x)}{a}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 30, normalized size = 1.50 \begin {gather*} -x+\frac {2 \text {ArcTan}(a x)}{a}-\frac {i \log \left (1+a^2 x^2\right )}{a} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.07, size = 19, normalized size = 0.95
method | result | size |
default | \(-x -\frac {2 i \ln \left (-a x +i\right )}{a}\) | \(19\) |
risch | \(-x -\frac {i \ln \left (a^{2} x^{2}+1\right )}{a}+\frac {2 \arctan \left (a x \right )}{a}\) | \(30\) |
meijerg | \(\frac {i \left (\frac {i a x \left (3 i a x +6\right )}{3 i a x +3}-2 \ln \left (i a x +1\right )\right )}{a}+\frac {x}{i a x +1}\) | \(51\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 16, normalized size = 0.80 \begin {gather*} -x - \frac {2 i \, \log \left (i \, a x + 1\right )}{a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.56, size = 21, normalized size = 1.05 \begin {gather*} -\frac {a x + 2 i \, \log \left (\frac {a x - i}{a}\right )}{a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.05, size = 14, normalized size = 0.70 \begin {gather*} - x - \frac {2 i \log {\left (a x - i \right )}}{a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 65 vs. \(2 (16) = 32\).
time = 0.40, size = 65, normalized size = 3.25 \begin {gather*} a^{2} {\left (\frac {i \, {\left (i \, a x + 1\right )}}{a^{3}} + \frac {2 i \, \log \left (\frac {1}{\sqrt {a^{2} x^{2} + 1} {\left | a \right |}}\right )}{a^{3}} - \frac {i}{{\left (i \, a x + 1\right )} a^{3}}\right )} + \frac {i}{{\left (i \, a x + 1\right )} a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.41, size = 19, normalized size = 0.95 \begin {gather*} -x-\frac {\ln \left (x-\frac {1{}\mathrm {i}}{a}\right )\,2{}\mathrm {i}}{a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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