Optimal. Leaf size=26 \[ 2 i a \log (x)-2 i a \log (a x+i)-\frac{1}{x} \]
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Rubi [A] time = 0.0236566, antiderivative size = 26, normalized size of antiderivative = 1., 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 = {5062, 77} \[ 2 i a \log (x)-2 i a \log (a x+i)-\frac{1}{x} \]
Antiderivative was successfully verified.
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Rule 5062
Rule 77
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*}
Mathematica [A] time = 0.0085793, size = 26, normalized size = 1. \[ 2 i a \log (x)-2 i a \log (a x+i)-\frac{1}{x} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.043, size = 34, normalized size = 1.3 \begin{align*} -ia\ln \left ({a}^{2}{x}^{2}+1 \right ) -2\,a\arctan \left ( ax \right ) -{x}^{-1}+2\,ia\ln \left ( x \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.48702, size = 42, normalized size = 1.62 \begin{align*} -2 \, a \arctan \left (a x\right ) - i \, a \log \left (a^{2} x^{2} + 1\right ) + 2 i \, a \log \left (x\right ) - \frac{1}{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.79971, size = 69, normalized size = 2.65 \begin{align*} \frac{2 i \, a x \log \left (x\right ) - 2 i \, a x \log \left (\frac{a x + i}{a}\right ) - 1}{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.427577, size = 20, normalized size = 0.77 \begin{align*} - 2 a \left (- i \log{\left (x \right )} + i \log{\left (x + \frac{i}{a} \right )}\right ) - \frac{1}{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.09735, size = 31, normalized size = 1.19 \begin{align*} -2 \, a i \log \left (a x + i\right ) + 2 \, a i \log \left ({\left | x \right |}\right ) - \frac{1}{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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