Optimal. Leaf size=49 \[ -\frac {1}{3 x^3}+\frac {i a}{x^2}+\frac {2 a^2}{x}+2 i a^3 \log (x)-2 i a^3 \log (i-a x) \]
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Rubi [A]
time = 0.02, 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 = {5170, 78}
\begin {gather*} 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} \end {gather*}
Antiderivative was successfully verified.
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Rule 78
Rule 5170
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 = 49, normalized size = 1.00 \begin {gather*} -\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 {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.10, size = 44, normalized size = 0.90
method | result | size |
default | \(-\frac {1}{3 x^{3}}+\frac {i a}{x^{2}}+\frac {2 a^{2}}{x}+2 i a^{3} \ln \left (x \right )-2 i a^{3} \ln \left (-a x +i\right )\) | \(44\) |
risch | \(\frac {2 a^{2} x^{2}+i a x -\frac {1}{3}}{x^{3}}+2 i a^{3} \ln \left (-x \right )+2 a^{3} \arctan \left (a x \right )-i a^{3} \ln \left (a^{2} x^{2}+1\right )\) | \(56\) |
meijerg | \(i a^{3} \left (\frac {3 i a x}{3 i a x +3}+2 \ln \left (i a x +1\right )-1-2 \ln \left (x \right )-2 \ln \left (i a \right )+\frac {i}{x a}\right )-i a^{3} \left (\frac {5 i a x}{5 i a x +5}+4 \ln \left (i a x +1\right )-1-4 \ln \left (x \right )-4 \ln \left (i a \right )-\frac {i}{3 x^{3} a^{3}}-\frac {1}{a^{2} x^{2}}+\frac {3 i}{x a}\right )\) | \(123\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 57, normalized size = 1.16 \begin {gather*} -2 i \, a^{3} \log \left (i \, a x + 1\right ) + 2 i \, a^{3} \log \left (x\right ) + \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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.74, size = 47, normalized size = 0.96 \begin {gather*} \frac {6 i \, a^{3} x^{3} \log \left (x\right ) - 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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.12, size = 54, normalized size = 1.10 \begin {gather*} - 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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.41, size = 67, normalized size = 1.37 \begin {gather*} 2 i \, a^{3} \log \left (\frac {i}{i \, a x + 1} - i\right ) - \frac {10 \, a^{3} - \frac {24 \, a^{3}}{i \, a x + 1} + \frac {15 \, a^{3}}{{\left (i \, a x + 1\right )}^{2}}}{3 \, {\left (\frac {i}{i \, a x + 1} - i\right )}^{3}} \end {gather*}
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 \begin {gather*} 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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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