Optimal. Leaf size=62 \[ \frac{4 a^3}{a x+i}+\frac{8 a^2}{x}-12 i a^3 \log (x)+12 i a^3 \log (a x+i)-\frac{2 i a}{x^2}-\frac{1}{3 x^3} \]
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Rubi [A] time = 0.0390173, antiderivative size = 62, 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, 88} \[ \frac{4 a^3}{a x+i}+\frac{8 a^2}{x}-12 i a^3 \log (x)+12 i a^3 \log (a x+i)-\frac{2 i a}{x^2}-\frac{1}{3 x^3} \]
Antiderivative was successfully verified.
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Rule 5062
Rule 88
Rubi steps
\begin{align*} \int \frac{e^{4 i \tan ^{-1}(a x)}}{x^4} \, dx &=\int \frac{(1+i a x)^2}{x^4 (1-i a x)^2} \, dx\\ &=\int \left (\frac{1}{x^4}+\frac{4 i a}{x^3}-\frac{8 a^2}{x^2}-\frac{12 i a^3}{x}-\frac{4 a^4}{(i+a x)^2}+\frac{12 i a^4}{i+a x}\right ) \, dx\\ &=-\frac{1}{3 x^3}-\frac{2 i a}{x^2}+\frac{8 a^2}{x}+\frac{4 a^3}{i+a x}-12 i a^3 \log (x)+12 i a^3 \log (i+a x)\\ \end{align*}
Mathematica [A] time = 0.034766, size = 62, normalized size = 1. \[ \frac{4 a^3}{a x+i}+\frac{8 a^2}{x}-12 i a^3 \log (x)+12 i a^3 \log (a x+i)-\frac{2 i a}{x^2}-\frac{1}{3 x^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.052, size = 68, normalized size = 1.1 \begin{align*} 4\,{\frac{{a}^{3}}{ax+i}}+6\,i{a}^{3}\ln \left ({a}^{2}{x}^{2}+1 \right ) +12\,{a}^{3}\arctan \left ( ax \right ) -{\frac{1}{3\,{x}^{3}}}-12\,i{a}^{3}\ln \left ( x \right ) -{\frac{2\,ia}{{x}^{2}}}+8\,{\frac{{a}^{2}}{x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.51194, size = 104, normalized size = 1.68 \begin{align*} 12 \, a^{3} \arctan \left (a x\right ) + 6 i \, a^{3} \log \left (a^{2} x^{2} + 1\right ) - 12 i \, a^{3} \log \left (x\right ) + \frac{72 \, a^{4} x^{4} - 36 i \, a^{3} x^{3} + 46 \, a^{2} x^{2} - 12 i \, a x - 2}{6 \,{\left (a^{2} x^{5} + x^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.78426, size = 189, normalized size = 3.05 \begin{align*} \frac{36 \, a^{3} x^{3} + 18 i \, a^{2} x^{2} + 5 \, a x - 36 \,{\left (i \, a^{4} x^{4} - a^{3} x^{3}\right )} \log \left (x\right ) - 36 \,{\left (-i \, a^{4} x^{4} + a^{3} x^{3}\right )} \log \left (\frac{a x + i}{a}\right ) - i}{3 \,{\left (a x^{4} + i \, x^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.758275, size = 63, normalized size = 1.02 \begin{align*} 12 a^{3} \left (- i \log{\left (x \right )} + i \log{\left (x + \frac{i}{a} \right )}\right ) + \frac{36 a^{4} x^{3} + 18 i a^{3} x^{2} + 5 a^{2} x - i a}{3 a^{2} x^{4} + 3 i a x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.10326, size = 80, normalized size = 1.29 \begin{align*} 12 \, a^{3} i \log \left (a x + i\right ) - 12 \, a^{3} i \log \left ({\left | x \right |}\right ) + \frac{36 \, a^{3} x^{3} + 18 \, a^{2} i x^{2} + 5 \, a x - i}{3 \,{\left (a x + i\right )} x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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