Optimal. Leaf size=31 \[ -2+e^4+\log (5)+\frac {4}{e^{2 x}-\log \left (-\frac {2}{5}+(2-x) x\right )} \]
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Rubi [A]
time = 0.29, antiderivative size = 25, normalized size of antiderivative = 0.81, number of steps
used = 3, number of rules used = 3, integrand size = 100, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.030, Rules used = {6820, 12,
6818} \begin {gather*} \frac {4}{e^{2 x}-\log \left (-x^2+2 x-\frac {2}{5}\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 6818
Rule 6820
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {8 \left (-5+5 x-e^{2 x} \left (2-10 x+5 x^2\right )\right )}{\left (2-10 x+5 x^2\right ) \left (e^{2 x}-\log \left (-\frac {2}{5}+2 x-x^2\right )\right )^2} \, dx\\ &=8 \int \frac {-5+5 x-e^{2 x} \left (2-10 x+5 x^2\right )}{\left (2-10 x+5 x^2\right ) \left (e^{2 x}-\log \left (-\frac {2}{5}+2 x-x^2\right )\right )^2} \, dx\\ &=\frac {4}{e^{2 x}-\log \left (-\frac {2}{5}+2 x-x^2\right )}\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.06, size = 25, normalized size = 0.81 \begin {gather*} \frac {4}{e^{2 x}-\log \left (-\frac {2}{5}+2 x-x^2\right )} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.04, size = 23, normalized size = 0.74
method | result | size |
risch | \(\frac {4}{{\mathrm e}^{2 x}-\ln \left (-x^{2}+2 x -\frac {2}{5}\right )}\) | \(23\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.52, size = 24, normalized size = 0.77 \begin {gather*} \frac {4}{e^{\left (2 \, x\right )} + \log \left (5\right ) - \log \left (-5 \, x^{2} + 10 \, x - 2\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.42, size = 22, normalized size = 0.71 \begin {gather*} \frac {4}{e^{\left (2 \, x\right )} - \log \left (-x^{2} + 2 \, x - \frac {2}{5}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.10, size = 17, normalized size = 0.55 \begin {gather*} \frac {4}{e^{2 x} - \log {\left (- x^{2} + 2 x - \frac {2}{5} \right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.44, size = 24, normalized size = 0.77 \begin {gather*} \frac {4}{e^{\left (2 \, x\right )} + \log \left (5\right ) - \log \left (-5 \, x^{2} + 10 \, x - 2\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 5.94, size = 22, normalized size = 0.71 \begin {gather*} \frac {4}{{\mathrm {e}}^{2\,x}-\ln \left (-x^2+2\,x-\frac {2}{5}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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