Optimal. Leaf size=62 \[ \frac {4 a^3}{a x+i}-12 i a^3 \log (x)+12 i a^3 \log (a x+i)+\frac {8 a^2}{x}-\frac {2 i a}{x^2}-\frac {1}{3 x^3} \]
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Rubi [A] time = 0.04, antiderivative size = 62, normalized size of antiderivative = 1.00, 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 88
Rule 5062
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*}
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Mathematica [A] time = 0.04, size = 62, normalized size = 1.00 \[ \frac {4 a^3}{a x+i}-12 i a^3 \log (x)+12 i a^3 \log (a x+i)+\frac {8 a^2}{x}-\frac {2 i a}{x^2}-\frac {1}{3 x^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.47, size = 86, normalized size = 1.39 \[ \frac {36 \, a^{3} x^{3} + 18 i \, a^{2} x^{2} + 5 \, a x - 36 \, {\left (i \, a^{4} x^{4} - a^{3} x^{3}\right )} \log \relax (x) - 36 \, {\left (-i \, a^{4} x^{4} + a^{3} x^{3}\right )} \log \left (\frac {a x + i}{a}\right ) - i}{3 \, a x^{4} + 3 i \, x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.14, size = 59, normalized size = 0.95 \[ 12 \, a^{3} i \log \left (a x + i\right ) - 12 \, a^{3} i \log \left ({\left | x \right |}\right ) + \frac {36 \, a^{3} x^{3} + 18 \, a^{2} i x^{2} + 5 \, a x - i}{3 \, {\left (a x + i\right )} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 68, normalized size = 1.10 \[ -\frac {1}{3 x^{3}}-12 i a^{3} \ln \relax (x )-\frac {2 i a}{x^{2}}+\frac {8 a^{2}}{x}+\frac {4 a^{3}}{a x +i}+6 i a^{3} \ln \left (a^{2} x^{2}+1\right )+12 a^{3} \arctan \left (a x \right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.42, size = 77, normalized size = 1.24 \[ 12 \, a^{3} \arctan \left (a x\right ) + 6 i \, a^{3} \log \left (a^{2} x^{2} + 1\right ) - 12 i \, a^{3} \log \relax (x) + \frac {72 \, a^{4} x^{4} - 36 i \, a^{3} x^{3} + 46 \, a^{2} x^{2} - 12 i \, a x - 2}{6 \, {\left (a^{2} x^{5} + x^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.13, size = 55, normalized size = 0.89 \[ 24\,a^3\,\mathrm {atan}\left (2\,a\,x+1{}\mathrm {i}\right )+\frac {\frac {5\,x}{3}+12\,a^2\,x^3+a\,x^2\,6{}\mathrm {i}-\frac {1{}\mathrm {i}}{3\,a}}{x^4+\frac {x^3\,1{}\mathrm {i}}{a}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.38, size = 71, normalized size = 1.15 \[ 12 a^{3} \left (- i \log {\left (24 a^{4} x \right )} + i \log {\left (24 a^{4} x + 24 i a^{3} \right )}\right ) + \frac {- 36 a^{3} x^{3} - 18 i a^{2} x^{2} - 5 a x + i}{- 3 a x^{4} - 3 i x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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