Optimal. Leaf size=30 \[ e^2+\frac {\log (3) (i \pi +\log (4-\log (4)))}{e^2 \left (5+x^2\right )} \]
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Rubi [A]
time = 0.01, antiderivative size = 26, normalized size of antiderivative = 0.87, number of steps
used = 3, number of rules used = 3, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {12, 28, 267}
\begin {gather*} \frac {\log (3) (\log (4-\log (4))+i \pi )}{e^2 \left (x^2+5\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 28
Rule 267
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=-\frac {(2 \log (3) (i \pi +\log (4-\log (4)))) \int \frac {x}{25+10 x^2+x^4} \, dx}{e^2}\\ &=-\frac {(2 \log (3) (i \pi +\log (4-\log (4)))) \int \frac {x}{\left (5+x^2\right )^2} \, dx}{e^2}\\ &=\frac {\log (3) (i \pi +\log (4-\log (4)))}{e^2 \left (5+x^2\right )}\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.00, size = 29, normalized size = 0.97 \begin {gather*} \frac {i \log (3) (\pi -i \log (4-\log (4)))}{e^2 \left (5+x^2\right )} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.40, size = 22, normalized size = 0.73
method | result | size |
gosper | \(\frac {\ln \left (3\right ) \ln \left (2 \ln \left (2\right )-4\right ) {\mathrm e}^{-2}}{x^{2}+5}\) | \(22\) |
default | \(\frac {\ln \left (3\right ) \ln \left (2 \ln \left (2\right )-4\right ) {\mathrm e}^{-2}}{x^{2}+5}\) | \(22\) |
norman | \(\frac {\ln \left (3\right ) \left (\ln \left (2\right )+\ln \left (\ln \left (2\right )-2\right )\right ) {\mathrm e}^{-2}}{x^{2}+5}\) | \(23\) |
risch | \(\frac {\ln \left (3\right ) {\mathrm e}^{-2} \ln \left (2\right )}{x^{2}+5}+\frac {\ln \left (3\right ) {\mathrm e}^{-2} \ln \left (\ln \left (2\right )-2\right )}{x^{2}+5}\) | \(33\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.37, size = 19, normalized size = 0.63 \begin {gather*} \frac {e^{\left (-2\right )} \log \left (3\right ) \log \left (2 \, \log \left (2\right ) - 4\right )}{x^{2} + 5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 19, normalized size = 0.63 \begin {gather*} \frac {e^{\left (-2\right )} \log \left (3\right ) \log \left (2 \, \log \left (2\right ) - 4\right )}{x^{2} + 5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.07, size = 44, normalized size = 1.47 \begin {gather*} - \frac {- 2 \log {\left (2 \right )} \log {\left (3 \right )} - 2 \log {\left (3 \right )} \log {\left (2 - \log {\left (2 \right )} \right )} - 2 i \pi \log {\left (3 \right )}}{2 x^{2} e^{2} + 10 e^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.39, size = 19, normalized size = 0.63 \begin {gather*} \frac {e^{\left (-2\right )} \log \left (3\right ) \log \left (2 \, \log \left (2\right ) - 4\right )}{x^{2} + 5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.14, size = 17, normalized size = 0.57 \begin {gather*} \frac {\ln \left (\ln \left (4\right )-4\right )\,{\mathrm {e}}^{-2}\,\ln \left (3\right )}{x^2+5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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