Optimal. Leaf size=19 \[ -x+\frac{2 i \log (a x+i)}{a} \]
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Rubi [A] time = 0.0091863, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {5061, 43} \[ -x+\frac{2 i \log (a x+i)}{a} \]
Antiderivative was successfully verified.
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Rule 5061
Rule 43
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*}
Mathematica [A] time = 0.0092926, size = 30, normalized size = 1.58 \[ \frac{i \log \left (a^2 x^2+1\right )}{a}+\frac{2 \tan ^{-1}(a x)}{a}-x \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.036, size = 30, normalized size = 1.6 \begin{align*} -x+{\frac{i\ln \left ({a}^{2}{x}^{2}+1 \right ) }{a}}+2\,{\frac{\arctan \left ( ax \right ) }{a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.50846, size = 38, normalized size = 2. \begin{align*} -x + \frac{2 \, \arctan \left (a x\right )}{a} + \frac{i \, \log \left (a^{2} x^{2} + 1\right )}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.54591, size = 45, normalized size = 2.37 \begin{align*} -\frac{a x - 2 i \, \log \left (\frac{a x + i}{a}\right )}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.311358, size = 12, normalized size = 0.63 \begin{align*} - x + \frac{2 i \log{\left (a x + i \right )}}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.1037, size = 22, normalized size = 1.16 \begin{align*} -x + \frac{2 \, i \log \left (a x + i\right )}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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