Optimal. Leaf size=18 \[ \frac {1}{4+e^{\frac {1}{5} \log ^2(2-x)}} \]
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Rubi [A] time = 1.10, antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 69, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.058, Rules used = {12, 6688, 6742, 6686} \begin {gather*} \frac {1}{e^{\frac {1}{5} \log ^2(2-x)}+4} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 6686
Rule 6688
Rule 6742
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=-\left (2 \int \frac {e^{\frac {1}{5} \log ^2(2-x)} \log (2-x)}{-160+80 x+e^{\frac {2}{5} \log ^2(2-x)} (-10+5 x)+e^{\frac {1}{5} \log ^2(2-x)} (-80+40 x)} \, dx\right )\\ &=-\left (2 \int \frac {e^{\frac {1}{5} \log ^2(2-x)} \log (2-x)}{5 \left (4+e^{\frac {1}{5} \log ^2(2-x)}\right )^2 (-2+x)} \, dx\right )\\ &=-\left (\frac {2}{5} \int \frac {e^{\frac {1}{5} \log ^2(2-x)} \log (2-x)}{\left (4+e^{\frac {1}{5} \log ^2(2-x)}\right )^2 (-2+x)} \, dx\right )\\ &=\frac {1}{4+e^{\frac {1}{5} \log ^2(2-x)}}\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.02, size = 18, normalized size = 1.00 \begin {gather*} \frac {1}{4+e^{\frac {1}{5} \log ^2(2-x)}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.62, size = 15, normalized size = 0.83 \begin {gather*} \frac {1}{e^{\left (\frac {1}{5} \, \log \left (-x + 2\right )^{2}\right )} + 4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.32, size = 15, normalized size = 0.83 \begin {gather*} \frac {1}{e^{\left (\frac {1}{5} \, \log \left (-x + 2\right )^{2}\right )} + 4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 16, normalized size = 0.89
method | result | size |
risch | \(\frac {1}{{\mathrm e}^{\frac {\ln \left (2-x \right )^{2}}{5}}+4}\) | \(16\) |
norman | \(-\frac {{\mathrm e}^{\frac {\ln \left (2-x \right )^{2}}{5}}}{4 \left ({\mathrm e}^{\frac {\ln \left (2-x \right )^{2}}{5}}+4\right )}\) | \(29\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.65, size = 15, normalized size = 0.83 \begin {gather*} \frac {1}{e^{\left (\frac {1}{5} \, \log \left (-x + 2\right )^{2}\right )} + 4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.57, size = 15, normalized size = 0.83 \begin {gather*} \frac {1}{{\mathrm {e}}^{\frac {{\ln \left (2-x\right )}^2}{5}}+4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.26, size = 12, normalized size = 0.67 \begin {gather*} \frac {1}{e^{\frac {\log {\left (2 - x \right )}^{2}}{5}} + 4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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