Optimal. Leaf size=48 \[ -2 i a^3 \log (x)+2 i a^3 \log (a x+i)+\frac {2 a^2}{x}-\frac {i a}{x^2}-\frac {1}{3 x^3} \]
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Rubi [A] time = 0.03, antiderivative size = 48, 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 = 0.01, size = 48, 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 = 0.41, size = 47, normalized size = 0.98 \[ \frac {-6 i \, a^{3} x^{3} \log \relax (x) + 6 i \, a^{3} x^{3} \log \left (\frac {a x + i}{a}\right ) + 6 \, a^{2} x^{2} - 3 i \, a x - 1}{3 \, x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.14, size = 42, normalized size = 0.88 \[ 2 \, a^{3} i \log \left (a x + i\right ) - 2 \, a^{3} i \log \left ({\left | x \right |}\right ) + \frac {6 \, a^{2} x^{2} - 3 \, a i x - 1}{3 \, x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 55, normalized size = 1.15 \[ -\frac {1}{3 x^{3}}-2 i a^{3} \ln \relax (x )-\frac {i a}{x^{2}}+\frac {2 a^{2}}{x}+i a^{3} \ln \left (a^{2} x^{2}+1\right )+2 a^{3} \arctan \left (a x \right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.42, size = 51, normalized size = 1.06 \[ 2 \, a^{3} \arctan \left (a x\right ) + i \, a^{3} \log \left (a^{2} x^{2} + 1\right ) - 2 i \, a^{3} \log \relax (x) + \frac {6 \, a^{2} x^{2} - 3 i \, a x - 1}{3 \, x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.07, size = 34, normalized size = 0.71 \[ 4\,a^3\,\mathrm {atan}\left (2\,a\,x+1{}\mathrm {i}\right )-\frac {-2\,a^2\,x^2+a\,x\,1{}\mathrm {i}+\frac {1}{3}}{x^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.21, size = 54, normalized size = 1.12 \[ - 2 a^{3} \left (i \log {\left (4 a^{4} x \right )} - i \log {\left (4 a^{4} x + 4 i a^{3} \right )}\right ) - \frac {- 6 a^{2} x^{2} + 3 i a x + 1}{3 x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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