Optimal. Leaf size=31 \[ \frac{4}{a (a x+i)}-\frac{4 i \log (a x+i)}{a}+x \]
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Rubi [A] time = 0.0140897, antiderivative size = 31, 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} \[ \frac{4}{a (a x+i)}-\frac{4 i \log (a x+i)}{a}+x \]
Antiderivative was successfully verified.
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Rule 5061
Rule 43
Rubi steps
\begin{align*} \int e^{4 i \tan ^{-1}(a x)} \, dx &=\int \frac{(1+i a x)^2}{(1-i a x)^2} \, dx\\ &=\int \left (1-\frac{4}{(i+a x)^2}-\frac{4 i}{i+a x}\right ) \, dx\\ &=x+\frac{4}{a (i+a x)}-\frac{4 i \log (i+a x)}{a}\\ \end{align*}
Mathematica [A] time = 0.0204097, size = 42, normalized size = 1.35 \[ -\frac{2 i \log \left (a^2 x^2+1\right )}{a}+\frac{4}{a (a x+i)}-\frac{4 \tan ^{-1}(a x)}{a}+x \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.043, size = 41, normalized size = 1.3 \begin{align*} x+4\,{\frac{1}{a \left ( ax+i \right ) }}-{\frac{2\,i\ln \left ({a}^{2}{x}^{2}+1 \right ) }{a}}-4\,{\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 = 1.52143, size = 61, normalized size = 1.97 \begin{align*} x + \frac{8 \, a x - 8 i}{2 \,{\left (a^{3} x^{2} + a\right )}} - \frac{4 \, \arctan \left (a x\right )}{a} - \frac{2 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 = 1.53931, size = 95, normalized size = 3.06 \begin{align*} \frac{a^{2} x^{2} + i \, a x - 4 \,{\left (i \, a x - 1\right )} \log \left (\frac{a x + i}{a}\right ) + 4}{a^{2} x + i \, a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.421508, size = 26, normalized size = 0.84 \begin{align*} \frac{4 a}{a^{3} x + i a^{2}} + x - \frac{4 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 = 1.12512, size = 35, normalized size = 1.13 \begin{align*} x - \frac{4 \, i \log \left (a x + i\right )}{a} + \frac{4}{{\left (a x + i\right )} a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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