Optimal. Leaf size=40 \[ \frac {2 x}{a^2}-\frac {i x^2}{a}-\frac {x^3}{3}+\frac {2 i \log (i-a x)}{a^3} \]
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Rubi [A]
time = 0.02, antiderivative size = 40, 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*} \frac {2 i \log (-a x+i)}{a^3}+\frac {2 x}{a^2}-\frac {i x^2}{a}-\frac {x^3}{3} \end {gather*}
Antiderivative was successfully verified.
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Rule 78
Rule 5170
Rubi steps
\begin {align*} \int e^{-2 i \tan ^{-1}(a x)} x^2 \, dx &=\int \frac {x^2 (1-i a x)}{1+i a x} \, dx\\ &=\int \left (\frac {2}{a^2}-\frac {2 i x}{a}-x^2+\frac {2 i}{a^2 (-i+a x)}\right ) \, dx\\ &=\frac {2 x}{a^2}-\frac {i x^2}{a}-\frac {x^3}{3}+\frac {2 i \log (i-a x)}{a^3}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 40, normalized size = 1.00 \begin {gather*} \frac {2 x}{a^2}-\frac {i x^2}{a}-\frac {x^3}{3}+\frac {2 i \log (i-a x)}{a^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.07, size = 40, normalized size = 1.00
method | result | size |
default | \(-\frac {\frac {1}{3} a^{2} x^{3}+i a \,x^{2}-2 x}{a^{2}}+\frac {2 i \ln \left (-a x +i\right )}{a^{3}}\) | \(40\) |
risch | \(-\frac {x^{3}}{3}-\frac {i x^{2}}{a}+\frac {2 x}{a^{2}}+\frac {i \ln \left (a^{2} x^{2}+1\right )}{a^{3}}-\frac {2 \arctan \left (a x \right )}{a^{3}}\) | \(47\) |
meijerg | \(-\frac {i \left (\frac {i x a \left (-5 i a^{3} x^{3}+10 a^{2} x^{2}+30 i a x +60\right )}{15 i a x +15}-4 \ln \left (i a x +1\right )\right )}{a^{3}}+\frac {i \left (\frac {i a x \left (3 i a x +6\right )}{3 i a x +3}-2 \ln \left (i a x +1\right )\right )}{a^{3}}\) | \(95\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 35, normalized size = 0.88 \begin {gather*} -\frac {a^{2} x^{3} + 3 i \, a x^{2} - 6 \, x}{3 \, a^{2}} + \frac {2 i \, \log \left (i \, a x + 1\right )}{a^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.88, size = 37, normalized size = 0.92 \begin {gather*} -\frac {a^{3} x^{3} + 3 i \, a^{2} x^{2} - 6 \, a x - 6 i \, \log \left (\frac {a x - i}{a}\right )}{3 \, a^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.06, size = 31, normalized size = 0.78 \begin {gather*} - \frac {x^{3}}{3} - \frac {i x^{2}}{a} + \frac {2 x}{a^{2}} + \frac {2 i \log {\left (a x - i \right )}}{a^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.41, size = 58, normalized size = 1.45 \begin {gather*} \frac {i \, {\left (i \, a x + 1\right )}^{3} {\left (\frac {6}{i \, a x + 1} - \frac {15}{{\left (i \, a x + 1\right )}^{2}} - 1\right )}}{3 \, a^{3}} - \frac {2 i \, \log \left (\frac {1}{\sqrt {a^{2} x^{2} + 1} {\left | a \right |}}\right )}{a^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.41, size = 36, normalized size = 0.90 \begin {gather*} \frac {\ln \left (x-\frac {1{}\mathrm {i}}{a}\right )\,2{}\mathrm {i}}{a^3}+\frac {2\,x}{a^2}-\frac {x^3}{3}-\frac {x^2\,1{}\mathrm {i}}{a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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