Optimal. Leaf size=52 \[ -\frac {4 i a^2}{a x+i}-8 a^2 \log (x)+8 a^2 \log (a x+i)-\frac {4 i a}{x}-\frac {1}{2 x^2} \]
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Rubi [A] time = 0.04, antiderivative size = 52, 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} \[ -\frac {4 i a^2}{a x+i}-8 a^2 \log (x)+8 a^2 \log (a x+i)-\frac {4 i a}{x}-\frac {1}{2 x^2} \]
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^3} \, dx &=\int \frac {(1+i a x)^2}{x^3 (1-i a x)^2} \, dx\\ &=\int \left (\frac {1}{x^3}+\frac {4 i a}{x^2}-\frac {8 a^2}{x}+\frac {4 i a^3}{(i+a x)^2}+\frac {8 a^3}{i+a x}\right ) \, dx\\ &=-\frac {1}{2 x^2}-\frac {4 i a}{x}-\frac {4 i a^2}{i+a x}-8 a^2 \log (x)+8 a^2 \log (i+a x)\\ \end {align*}
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Mathematica [A] time = 0.04, size = 52, normalized size = 1.00 \[ -\frac {4 i a^2}{a x+i}-8 a^2 \log (x)+8 a^2 \log (a x+i)-\frac {4 i a}{x}-\frac {1}{2 x^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.47, size = 78, normalized size = 1.50 \[ \frac {-16 i \, a^{2} x^{2} + 7 \, a x - {\left (16 \, a^{3} x^{3} + 16 i \, a^{2} x^{2}\right )} \log \relax (x) + {\left (16 \, a^{3} x^{3} + 16 i \, a^{2} x^{2}\right )} \log \left (\frac {a x + i}{a}\right ) - i}{2 \, a x^{3} + 2 i \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.12, size = 47, normalized size = 0.90 \[ 8 \, a^{2} \log \left (a x + i\right ) - 8 \, a^{2} \log \left ({\left | x \right |}\right ) - \frac {16 \, a^{2} i x^{2} - 7 \, a x + i}{2 \, {\left (a x + i\right )} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 60, normalized size = 1.15 \[ -\frac {1}{2 x^{2}}-\frac {4 i a}{x}-8 a^{2} \ln \relax (x )-\frac {4 i a^{2}}{a x +i}+4 a^{2} \ln \left (a^{2} x^{2}+1\right )-8 i a^{2} \arctan \left (a x \right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.42, size = 69, normalized size = 1.33 \[ -8 i \, a^{2} \arctan \left (a x\right ) + 4 \, a^{2} \log \left (a^{2} x^{2} + 1\right ) - 8 \, a^{2} \log \relax (x) + \frac {-16 i \, a^{3} x^{3} - 9 \, a^{2} x^{2} - 8 i \, a x - 1}{2 \, {\left (a^{2} x^{4} + x^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.47, size = 43, normalized size = 0.83 \[ -a^2\,\mathrm {atan}\left (2\,a\,x+1{}\mathrm {i}\right )\,16{}\mathrm {i}+\frac {8\,a^2\,x^2+\frac {a\,x\,7{}\mathrm {i}}{2}+\frac {1}{2}}{x^2\,\left (-1+a\,x\,1{}\mathrm {i}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.35, size = 60, normalized size = 1.15 \[ 8 a^{2} \left (- \log {\left (16 a^{3} x \right )} + \log {\left (16 a^{3} x + 16 i a^{2} \right )}\right ) + \frac {16 i a^{2} x^{2} - 7 a x + i}{- 2 a x^{3} - 2 i x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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