Optimal. Leaf size=24 \[ -x+\frac {i \pi +3 (2+x)+\log (4)}{9 x^2} \]
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Rubi [A]
time = 0.02, antiderivative size = 27, normalized size of antiderivative = 1.12, number of steps
used = 3, number of rules used = 2, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {12, 14}
\begin {gather*} \frac {12+2 i \pi +\log (16)}{18 x^2}-x+\frac {1}{3 x} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 14
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{9} \int \frac {-12-3 x-9 x^3-2 (i \pi +\log (4))}{x^3} \, dx\\ &=\frac {1}{9} \int \left (-9-\frac {3}{x^2}-\frac {i (2 \pi -i (12+\log (16)))}{x^3}\right ) \, dx\\ &=\frac {1}{3 x}-x+\frac {12+2 i \pi +\log (16)}{18 x^2}\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.01, size = 24, normalized size = 1.00 \begin {gather*} \frac {12+2 i \pi +6 x-18 x^3+\log (16)}{18 x^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.14, size = 25, normalized size = 1.04
method | result | size |
risch | \(\frac {2 \ln \left (2\right )+i \pi +6+3 x}{9 x^{2}}-x\) | \(23\) |
default | \(-x -\frac {-2 i \pi -4 \ln \left (2\right )-12}{18 x^{2}}+\frac {1}{3 x}\) | \(25\) |
gosper | \(\frac {\left (9 i x^{3}-2 i \ln \left (2\right )-3 i x +\pi -6 i\right ) \left (9 x^{3}+2 i \pi +4 \ln \left (2\right )+3 x +12\right )}{9 x^{2} \left (-9 i x^{3}-4 i \ln \left (2\right )-3 i x +2 \pi -12 i\right )}\) | \(66\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 21, normalized size = 0.88 \begin {gather*} -x + \frac {i \, \pi + 3 \, x + 2 \, \log \left (2\right ) + 6}{9 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 22, normalized size = 0.92 \begin {gather*} \frac {i \, \pi - 9 \, x^{3} + 3 \, x + 2 \, \log \left (2\right ) + 6}{9 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.13, size = 22, normalized size = 0.92 \begin {gather*} - x - \frac {- 3 x - 6 - 2 \log {\left (2 \right )} - i \pi }{9 x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.39, size = 21, normalized size = 0.88 \begin {gather*} -x - \frac {-i \, \pi - 3 \, x - 2 \, \log \left (2\right ) - 6}{9 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.74, size = 21, normalized size = 0.88 \begin {gather*} -x+\frac {\frac {x}{3}+\frac {2\,\ln \left (2\right )}{9}+\frac {2}{3}+\frac {\Pi \,1{}\mathrm {i}}{9}}{x^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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