Optimal. Leaf size=22 \[ \frac {5 x^2}{2 \log ^4\left (-2+9 e^{-x^2}+x\right )} \]
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Rubi [F]
time = 2.81, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps
used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {}
\begin {gather*} \int \frac {-10 x^2+180 e^{-x^2} x^3+\left (-10 x+45 e^{-x^2} x+5 x^2\right ) \log \left (-2+9 e^{-x^2}+x\right )}{\left (-2+9 e^{-x^2}+x\right ) \log ^5\left (-2+9 e^{-x^2}+x\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {5 x \left (-\frac {2 \left (e^{x^2}-18 x\right ) x}{9+e^{x^2} (-2+x)}+\log \left (-2+9 e^{-x^2}+x\right )\right )}{\log ^5\left (-2+9 e^{-x^2}+x\right )} \, dx\\ &=5 \int \frac {x \left (-\frac {2 \left (e^{x^2}-18 x\right ) x}{9+e^{x^2} (-2+x)}+\log \left (-2+9 e^{-x^2}+x\right )\right )}{\log ^5\left (-2+9 e^{-x^2}+x\right )} \, dx\\ &=5 \int \left (\frac {18 x^2 \left (1-4 x+2 x^2\right )}{(-2+x) \left (9-2 e^{x^2}+e^{x^2} x\right ) \log ^5\left (-2+9 e^{-x^2}+x\right )}+\frac {x \left (-2 x-2 \log \left (-2+9 e^{-x^2}+x\right )+x \log \left (-2+9 e^{-x^2}+x\right )\right )}{(-2+x) \log ^5\left (-2+9 e^{-x^2}+x\right )}\right ) \, dx\\ &=5 \int \frac {x \left (-2 x-2 \log \left (-2+9 e^{-x^2}+x\right )+x \log \left (-2+9 e^{-x^2}+x\right )\right )}{(-2+x) \log ^5\left (-2+9 e^{-x^2}+x\right )} \, dx+90 \int \frac {x^2 \left (1-4 x+2 x^2\right )}{(-2+x) \left (9-2 e^{x^2}+e^{x^2} x\right ) \log ^5\left (-2+9 e^{-x^2}+x\right )} \, dx\\ &=5 \int \left (-\frac {2 x^2}{(-2+x) \log ^5\left (-2+9 e^{-x^2}+x\right )}+\frac {x}{\log ^4\left (-2+9 e^{-x^2}+x\right )}\right ) \, dx+90 \int \left (\frac {2}{\left (9-2 e^{x^2}+e^{x^2} x\right ) \log ^5\left (-2+9 e^{-x^2}+x\right )}+\frac {4}{(-2+x) \left (9-2 e^{x^2}+e^{x^2} x\right ) \log ^5\left (-2+9 e^{-x^2}+x\right )}+\frac {x}{\left (9-2 e^{x^2}+e^{x^2} x\right ) \log ^5\left (-2+9 e^{-x^2}+x\right )}+\frac {2 x^3}{\left (9-2 e^{x^2}+e^{x^2} x\right ) \log ^5\left (-2+9 e^{-x^2}+x\right )}\right ) \, dx\\ &=5 \int \frac {x}{\log ^4\left (-2+9 e^{-x^2}+x\right )} \, dx-10 \int \frac {x^2}{(-2+x) \log ^5\left (-2+9 e^{-x^2}+x\right )} \, dx+90 \int \frac {x}{\left (9-2 e^{x^2}+e^{x^2} x\right ) \log ^5\left (-2+9 e^{-x^2}+x\right )} \, dx+180 \int \frac {1}{\left (9-2 e^{x^2}+e^{x^2} x\right ) \log ^5\left (-2+9 e^{-x^2}+x\right )} \, dx+180 \int \frac {x^3}{\left (9-2 e^{x^2}+e^{x^2} x\right ) \log ^5\left (-2+9 e^{-x^2}+x\right )} \, dx+360 \int \frac {1}{(-2+x) \left (9-2 e^{x^2}+e^{x^2} x\right ) \log ^5\left (-2+9 e^{-x^2}+x\right )} \, dx\\ &=5 \int \frac {x}{\log ^4\left (-2+9 e^{-x^2}+x\right )} \, dx-10 \int \left (\frac {2}{\log ^5\left (-2+9 e^{-x^2}+x\right )}+\frac {4}{(-2+x) \log ^5\left (-2+9 e^{-x^2}+x\right )}+\frac {x}{\log ^5\left (-2+9 e^{-x^2}+x\right )}\right ) \, dx+90 \int \frac {x}{\left (9+e^{x^2} (-2+x)\right ) \log ^5\left (-2+9 e^{-x^2}+x\right )} \, dx+180 \int \frac {1}{\left (9+e^{x^2} (-2+x)\right ) \log ^5\left (-2+9 e^{-x^2}+x\right )} \, dx+180 \int \frac {x^3}{\left (9+e^{x^2} (-2+x)\right ) \log ^5\left (-2+9 e^{-x^2}+x\right )} \, dx+360 \int \frac {1}{(-2+x) \left (9-2 e^{x^2}+e^{x^2} x\right ) \log ^5\left (-2+9 e^{-x^2}+x\right )} \, dx\\ &=5 \int \frac {x}{\log ^4\left (-2+9 e^{-x^2}+x\right )} \, dx-10 \int \frac {x}{\log ^5\left (-2+9 e^{-x^2}+x\right )} \, dx-20 \int \frac {1}{\log ^5\left (-2+9 e^{-x^2}+x\right )} \, dx-40 \int \frac {1}{(-2+x) \log ^5\left (-2+9 e^{-x^2}+x\right )} \, dx+90 \int \frac {x}{\left (9+e^{x^2} (-2+x)\right ) \log ^5\left (-2+9 e^{-x^2}+x\right )} \, dx+180 \int \frac {1}{\left (9+e^{x^2} (-2+x)\right ) \log ^5\left (-2+9 e^{-x^2}+x\right )} \, dx+180 \int \frac {x^3}{\left (9+e^{x^2} (-2+x)\right ) \log ^5\left (-2+9 e^{-x^2}+x\right )} \, dx+360 \int \frac {1}{(-2+x) \left (9-2 e^{x^2}+e^{x^2} x\right ) \log ^5\left (-2+9 e^{-x^2}+x\right )} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.19, size = 22, normalized size = 1.00 \begin {gather*} \frac {5 x^2}{2 \log ^4\left (-2+9 e^{-x^2}+x\right )} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.13, size = 20, normalized size = 0.91
method | result | size |
risch | \(\frac {5 x^{2}}{2 \ln \left (9 \,{\mathrm e}^{-x^{2}}+x -2\right )^{4}}\) | \(20\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 76 vs.
\(2 (22) = 44\).
time = 0.41, size = 76, normalized size = 3.45 \begin {gather*} \frac {5 \, x^{2}}{2 \, {\left (x^{8} - 4 \, x^{6} \log \left ({\left (x - 2\right )} e^{\left (x^{2}\right )} + 9\right ) + 6 \, x^{4} \log \left ({\left (x - 2\right )} e^{\left (x^{2}\right )} + 9\right )^{2} - 4 \, x^{2} \log \left ({\left (x - 2\right )} e^{\left (x^{2}\right )} + 9\right )^{3} + \log \left ({\left (x - 2\right )} e^{\left (x^{2}\right )} + 9\right )^{4}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.40, size = 22, normalized size = 1.00 \begin {gather*} \frac {5 \, x^{2}}{2 \, \log \left (x + e^{\left (-x^{2} + 2 \, \log \left (3\right )\right )} - 2\right )^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.11, size = 19, normalized size = 0.86 \begin {gather*} \frac {5 x^{2}}{2 \log {\left (x - 2 + 9 e^{- x^{2}} \right )}^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.59, size = 19, normalized size = 0.86 \begin {gather*} \frac {5 \, x^{2}}{2 \, \log \left (x + 9 \, e^{\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 = 5.66, size = 2169, normalized size = 98.59 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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