Optimal. Leaf size=49 \[ 2 i a^3 \log (x)-2 i a^3 \log (-a x+i)+\frac {2 a^2}{x}+\frac {i a}{x^2}-\frac {1}{3 x^3} \]
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Rubi [A] time = 0.03, antiderivative size = 49, 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, 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 77
Rule 5062
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*}
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Mathematica [A] time = 0.02, size = 49, normalized size = 1.00 \[ 2 i a^3 \log (x)-2 i a^3 \log (-a x+i)+\frac {2 a^2}{x}+\frac {i a}{x^2}-\frac {1}{3 x^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 47, normalized size = 0.96 \[ \frac {6 i \, a^{3} x^{3} \log \relax (x) - 6 i \, a^{3} x^{3} \log \left (\frac {a x - i}{a}\right ) + 6 \, a^{2} x^{2} + 3 i \, a x - 1}{3 \, x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.14, size = 77, normalized size = 1.57 \[ 2 \, a^{3} i \log \left (-i + \frac {i}{a i x + 1}\right ) + \frac {\frac {24 \, a^{3} i^{2}}{a i x + 1} + 10 \, a^{3} - \frac {15 \, a^{3} i^{2}}{{\left (a i x + 1\right )}^{2}}}{3 \, {\left (i - \frac {i}{a i x + 1}\right )}^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 55, normalized size = 1.12 \[ -\frac {1}{3 x^{3}}+\frac {i a}{x^{2}}+2 i a^{3} \ln \relax (x )+\frac {2 a^{2}}{x}+2 a^{3} \arctan \left (a x \right )-i a^{3} \ln \left (a^{2} x^{2}+1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 57, normalized size = 1.16 \[ -2 i \, a^{3} \log \left (i \, a x + 1\right ) + 2 i \, a^{3} \log \relax (x) + \frac {6 i \, a^{3} x^{3} + 3 \, a^{2} x^{2} + 2 i \, a x - 1}{3 i \, a x^{4} + 3 \, x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.43, size = 33, normalized size = 0.67 \[ 4\,a^3\,\mathrm {atan}\left (2\,a\,x-\mathrm {i}\right )+\frac {2\,a^2\,x^2+a\,x\,1{}\mathrm {i}-\frac {1}{3}}{x^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.21, size = 54, normalized size = 1.10 \[ - 2 a^{3} \left (- i \log {\left (4 a^{4} x \right )} + i \log {\left (4 a^{4} x - 4 i a^{3} \right )}\right ) - \frac {- 6 a^{2} x^{2} - 3 i a x + 1}{3 x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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