Optimal. Leaf size=50 \[ \frac{4 i}{a (1-i a x)}-\frac{2 i}{a (1-i a x)^2}+\frac{i \log (a x+i)}{a} \]
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Rubi [A] time = 0.0427698, antiderivative size = 50, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {5073, 43} \[ \frac{4 i}{a (1-i a x)}-\frac{2 i}{a (1-i a x)^2}+\frac{i \log (a x+i)}{a} \]
Antiderivative was successfully verified.
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Rule 5073
Rule 43
Rubi steps
\begin{align*} \int \frac{e^{5 i \tan ^{-1}(a x)}}{\sqrt{1+a^2 x^2}} \, dx &=\int \frac{(1+i a x)^2}{(1-i a x)^3} \, dx\\ &=\int \left (\frac{4}{(1-i a x)^3}-\frac{4}{(1-i a x)^2}+\frac{1}{1-i a x}\right ) \, dx\\ &=-\frac{2 i}{a (1-i a x)^2}+\frac{4 i}{a (1-i a x)}+\frac{i \log (i+a x)}{a}\\ \end{align*}
Mathematica [A] time = 0.0222498, size = 42, normalized size = 0.84 \[ \frac{i \left (4 i a x+(a x+i)^2 \log (a x+i)-2\right )}{a (a x+i)^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.049, size = 45, normalized size = 0.9 \begin{align*}{\frac{1}{ \left ( ax+i \right ) ^{2}} \left ( -4\,x-{\frac{2\,i}{a}} \right ) }+{\frac{{\frac{i}{2}}\ln \left ({a}^{2}{x}^{2}+1 \right ) }{a}}+{\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 = 1.48925, size = 85, normalized size = 1.7 \begin{align*} -\frac{32 \, a^{3} x^{3} - 48 i \, a^{2} x^{2} - 16 i}{8 \,{\left (a^{5} x^{4} + 2 \, a^{3} x^{2} + a\right )}} + \frac{\arctan \left (a x\right )}{a} + \frac{i \, \log \left (a^{2} x^{2} + 1\right )}{2 \, a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.85058, size = 115, normalized size = 2.3 \begin{align*} -\frac{4 \, a x -{\left (i \, a^{2} x^{2} - 2 \, a x - i\right )} \log \left (\frac{a x + i}{a}\right ) + 2 i}{a^{3} x^{2} + 2 i \, a^{2} x - a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.589308, size = 41, normalized size = 0.82 \begin{align*} - \frac{4 a^{4} x + 2 i a^{3}}{a^{6} x^{2} + 2 i a^{5} x - a^{4}} + \frac{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 = 1.11795, size = 41, normalized size = 0.82 \begin{align*} \frac{i \log \left (a x + i\right )}{a} - \frac{2 \,{\left (2 \, a x + i\right )}}{{\left (a x + i\right )}^{2} a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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