Optimal. Leaf size=48 \[ \frac{2 a^2}{x}-2 i a^3 \log (x)+2 i a^3 \log (a x+i)-\frac{i a}{x^2}-\frac{1}{3 x^3} \]
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Rubi [A] time = 0.0304261, antiderivative size = 48, 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} \[ \frac{2 a^2}{x}-2 i a^3 \log (x)+2 i a^3 \log (a x+i)-\frac{i a}{x^2}-\frac{1}{3 x^3} \]
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^4} \, dx &=\int \frac{1+i a x}{x^4 (1-i a x)} \, dx\\ &=\int \left (\frac{1}{x^4}+\frac{2 i a}{x^3}-\frac{2 a^2}{x^2}-\frac{2 i a^3}{x}+\frac{2 i a^4}{i+a x}\right ) \, dx\\ &=-\frac{1}{3 x^3}-\frac{i a}{x^2}+\frac{2 a^2}{x}-2 i a^3 \log (x)+2 i a^3 \log (i+a x)\\ \end{align*}
Mathematica [A] time = 0.0124324, size = 48, normalized size = 1. \[ \frac{2 a^2}{x}-2 i a^3 \log (x)+2 i a^3 \log (a x+i)-\frac{i a}{x^2}-\frac{1}{3 x^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.043, size = 55, normalized size = 1.2 \begin{align*} i{a}^{3}\ln \left ({a}^{2}{x}^{2}+1 \right ) +2\,{a}^{3}\arctan \left ( ax \right ) -{\frac{1}{3\,{x}^{3}}}-2\,i{a}^{3}\ln \left ( x \right ) -{\frac{ia}{{x}^{2}}}+2\,{\frac{{a}^{2}}{x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.50283, size = 69, normalized size = 1.44 \begin{align*} 2 \, a^{3} \arctan \left (a x\right ) + i \, a^{3} \log \left (a^{2} x^{2} + 1\right ) - 2 i \, a^{3} \log \left (x\right ) + \frac{6 \, a^{2} x^{2} - 3 i \, a x - 1}{3 \, x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.71316, size = 119, normalized size = 2.48 \begin{align*} \frac{-6 i \, a^{3} x^{3} \log \left (x\right ) + 6 i \, a^{3} x^{3} \log \left (\frac{a x + i}{a}\right ) + 6 \, a^{2} x^{2} - 3 i \, a x - 1}{3 \, x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.508047, size = 39, normalized size = 0.81 \begin{align*} - 2 a^{3} \left (i \log{\left (x \right )} - i \log{\left (x + \frac{i}{a} \right )}\right ) + \frac{6 a^{2} x^{2} - 3 i a x - 1}{3 x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.0904, size = 57, normalized size = 1.19 \begin{align*} 2 \, a^{3} i \log \left (a x + i\right ) - 2 \, a^{3} i \log \left ({\left | x \right |}\right ) + \frac{6 \, a^{2} x^{2} - 3 \, a i x - 1}{3 \, x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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