Integrand size = 293, antiderivative size = 31 \[ \int \frac {e^{-2+e^{5 e^x}} \left (-50 x^2+50 e^{-3+x} x^2\right )+e^{-4+2 e^{5 e^x}} \left (50 e^{-3+x} x-50 x^2+e^{5 e^x+x} \left (250 e^{-3+x} x^2-250 x^3\right )\right ) \log \left (e^{-3+x}-x\right ) \log \left (\log \left (e^{-3+x}-x\right )\right )+\left (-50 x^2+50 e^{-3+x} x^2+e^{-2+e^{5 e^x}} \left (100 e^{-3+x} x-100 x^2+e^{5 e^x+x} \left (250 e^{-3+x} x^2-250 x^3\right )\right ) \log \left (e^{-3+x}-x\right ) \log \left (\log \left (e^{-3+x}-x\right )\right )\right ) \log \left (\log \left (\log \left (e^{-3+x}-x\right )\right )\right )+\left (50 e^{-3+x} x-50 x^2\right ) \log \left (e^{-3+x}-x\right ) \log \left (\log \left (e^{-3+x}-x\right )\right ) \log ^2\left (\log \left (\log \left (e^{-3+x}-x\right )\right )\right )}{\left (e^{-3+x}-x\right ) \log \left (e^{-3+x}-x\right ) \log \left (\log \left (e^{-3+x}-x\right )\right )} \, dx=25 x^2 \left (e^{-2+e^{5 e^x}}+\log \left (\log \left (\log \left (e^{-3+x}-x\right )\right )\right )\right )^2 \]
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Time = 2.37 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.16, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.010, Rules used = {6820, 12, 6819} \[ \int \frac {e^{-2+e^{5 e^x}} \left (-50 x^2+50 e^{-3+x} x^2\right )+e^{-4+2 e^{5 e^x}} \left (50 e^{-3+x} x-50 x^2+e^{5 e^x+x} \left (250 e^{-3+x} x^2-250 x^3\right )\right ) \log \left (e^{-3+x}-x\right ) \log \left (\log \left (e^{-3+x}-x\right )\right )+\left (-50 x^2+50 e^{-3+x} x^2+e^{-2+e^{5 e^x}} \left (100 e^{-3+x} x-100 x^2+e^{5 e^x+x} \left (250 e^{-3+x} x^2-250 x^3\right )\right ) \log \left (e^{-3+x}-x\right ) \log \left (\log \left (e^{-3+x}-x\right )\right )\right ) \log \left (\log \left (\log \left (e^{-3+x}-x\right )\right )\right )+\left (50 e^{-3+x} x-50 x^2\right ) \log \left (e^{-3+x}-x\right ) \log \left (\log \left (e^{-3+x}-x\right )\right ) \log ^2\left (\log \left (\log \left (e^{-3+x}-x\right )\right )\right )}{\left (e^{-3+x}-x\right ) \log \left (e^{-3+x}-x\right ) \log \left (\log \left (e^{-3+x}-x\right )\right )} \, dx=\frac {25 x^2 \left (e^{e^{5 e^x}}+e^2 \log \left (\log \left (\log \left (e^{x-3}-x\right )\right )\right )\right )^2}{e^4} \]
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Rule 12
Rule 6819
Rule 6820
Rubi steps \begin{align*} \text {integral}& = \int \frac {50 x \left (e^{e^{5 e^x}}+e^2 \log \left (\log \left (\log \left (e^{-3+x}-x\right )\right )\right )\right ) \left (-e^5 x+e^{2+x} x+\left (e^x-e^3 x\right ) \log \left (e^{-3+x}-x\right ) \log \left (\log \left (e^{-3+x}-x\right )\right ) \left (e^{e^{5 e^x}} \left (1+5 e^{5 e^x+x} x\right )+e^2 \log \left (\log \left (\log \left (e^{-3+x}-x\right )\right )\right )\right )\right )}{e^4 \left (e^x-e^3 x\right ) \log \left (e^{-3+x}-x\right ) \log \left (\log \left (e^{-3+x}-x\right )\right )} \, dx \\ & = \frac {50 \int \frac {x \left (e^{e^{5 e^x}}+e^2 \log \left (\log \left (\log \left (e^{-3+x}-x\right )\right )\right )\right ) \left (-e^5 x+e^{2+x} x+\left (e^x-e^3 x\right ) \log \left (e^{-3+x}-x\right ) \log \left (\log \left (e^{-3+x}-x\right )\right ) \left (e^{e^{5 e^x}} \left (1+5 e^{5 e^x+x} x\right )+e^2 \log \left (\log \left (\log \left (e^{-3+x}-x\right )\right )\right )\right )\right )}{\left (e^x-e^3 x\right ) \log \left (e^{-3+x}-x\right ) \log \left (\log \left (e^{-3+x}-x\right )\right )} \, dx}{e^4} \\ & = \frac {25 x^2 \left (e^{e^{5 e^x}}+e^2 \log \left (\log \left (\log \left (e^{-3+x}-x\right )\right )\right )\right )^2}{e^4} \\ \end{align*}
Time = 0.27 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.16 \[ \int \frac {e^{-2+e^{5 e^x}} \left (-50 x^2+50 e^{-3+x} x^2\right )+e^{-4+2 e^{5 e^x}} \left (50 e^{-3+x} x-50 x^2+e^{5 e^x+x} \left (250 e^{-3+x} x^2-250 x^3\right )\right ) \log \left (e^{-3+x}-x\right ) \log \left (\log \left (e^{-3+x}-x\right )\right )+\left (-50 x^2+50 e^{-3+x} x^2+e^{-2+e^{5 e^x}} \left (100 e^{-3+x} x-100 x^2+e^{5 e^x+x} \left (250 e^{-3+x} x^2-250 x^3\right )\right ) \log \left (e^{-3+x}-x\right ) \log \left (\log \left (e^{-3+x}-x\right )\right )\right ) \log \left (\log \left (\log \left (e^{-3+x}-x\right )\right )\right )+\left (50 e^{-3+x} x-50 x^2\right ) \log \left (e^{-3+x}-x\right ) \log \left (\log \left (e^{-3+x}-x\right )\right ) \log ^2\left (\log \left (\log \left (e^{-3+x}-x\right )\right )\right )}{\left (e^{-3+x}-x\right ) \log \left (e^{-3+x}-x\right ) \log \left (\log \left (e^{-3+x}-x\right )\right )} \, dx=\frac {25 x^2 \left (e^{e^{5 e^x}}+e^2 \log \left (\log \left (\log \left (e^{-3+x}-x\right )\right )\right )\right )^2}{e^4} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(58\) vs. \(2(27)=54\).
Time = 0.76 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.90
\[25 x^{2} {\mathrm e}^{2 \,{\mathrm e}^{5 \,{\mathrm e}^{x}}-4}+50 x^{2} {\mathrm e}^{{\mathrm e}^{5 \,{\mathrm e}^{x}}-2} \ln \left (\ln \left (\ln \left ({\mathrm e}^{-3+x}-x \right )\right )\right )+25 {\ln \left (\ln \left (\ln \left ({\mathrm e}^{-3+x}-x \right )\right )\right )}^{2} x^{2}\]
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Leaf count of result is larger than twice the leaf count of optimal. 87 vs. \(2 (27) = 54\).
Time = 0.26 (sec) , antiderivative size = 87, normalized size of antiderivative = 2.81 \[ \int \frac {e^{-2+e^{5 e^x}} \left (-50 x^2+50 e^{-3+x} x^2\right )+e^{-4+2 e^{5 e^x}} \left (50 e^{-3+x} x-50 x^2+e^{5 e^x+x} \left (250 e^{-3+x} x^2-250 x^3\right )\right ) \log \left (e^{-3+x}-x\right ) \log \left (\log \left (e^{-3+x}-x\right )\right )+\left (-50 x^2+50 e^{-3+x} x^2+e^{-2+e^{5 e^x}} \left (100 e^{-3+x} x-100 x^2+e^{5 e^x+x} \left (250 e^{-3+x} x^2-250 x^3\right )\right ) \log \left (e^{-3+x}-x\right ) \log \left (\log \left (e^{-3+x}-x\right )\right )\right ) \log \left (\log \left (\log \left (e^{-3+x}-x\right )\right )\right )+\left (50 e^{-3+x} x-50 x^2\right ) \log \left (e^{-3+x}-x\right ) \log \left (\log \left (e^{-3+x}-x\right )\right ) \log ^2\left (\log \left (\log \left (e^{-3+x}-x\right )\right )\right )}{\left (e^{-3+x}-x\right ) \log \left (e^{-3+x}-x\right ) \log \left (\log \left (e^{-3+x}-x\right )\right )} \, dx=50 \, x^{2} e^{\left ({\left (e^{\left (x + 5 \, e^{x}\right )} - 2 \, e^{x}\right )} e^{\left (-x\right )}\right )} \log \left (\log \left (\log \left (-{\left (x e^{3} - e^{x}\right )} e^{\left (-3\right )}\right )\right )\right ) + 25 \, x^{2} \log \left (\log \left (\log \left (-{\left (x e^{3} - e^{x}\right )} e^{\left (-3\right )}\right )\right )\right )^{2} + 25 \, x^{2} e^{\left (2 \, {\left (e^{\left (x + 5 \, e^{x}\right )} - 2 \, e^{x}\right )} e^{\left (-x\right )}\right )} \]
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Timed out. \[ \int \frac {e^{-2+e^{5 e^x}} \left (-50 x^2+50 e^{-3+x} x^2\right )+e^{-4+2 e^{5 e^x}} \left (50 e^{-3+x} x-50 x^2+e^{5 e^x+x} \left (250 e^{-3+x} x^2-250 x^3\right )\right ) \log \left (e^{-3+x}-x\right ) \log \left (\log \left (e^{-3+x}-x\right )\right )+\left (-50 x^2+50 e^{-3+x} x^2+e^{-2+e^{5 e^x}} \left (100 e^{-3+x} x-100 x^2+e^{5 e^x+x} \left (250 e^{-3+x} x^2-250 x^3\right )\right ) \log \left (e^{-3+x}-x\right ) \log \left (\log \left (e^{-3+x}-x\right )\right )\right ) \log \left (\log \left (\log \left (e^{-3+x}-x\right )\right )\right )+\left (50 e^{-3+x} x-50 x^2\right ) \log \left (e^{-3+x}-x\right ) \log \left (\log \left (e^{-3+x}-x\right )\right ) \log ^2\left (\log \left (\log \left (e^{-3+x}-x\right )\right )\right )}{\left (e^{-3+x}-x\right ) \log \left (e^{-3+x}-x\right ) \log \left (\log \left (e^{-3+x}-x\right )\right )} \, dx=\text {Timed out} \]
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Leaf count of result is larger than twice the leaf count of optimal. 64 vs. \(2 (27) = 54\).
Time = 0.44 (sec) , antiderivative size = 64, normalized size of antiderivative = 2.06 \[ \int \frac {e^{-2+e^{5 e^x}} \left (-50 x^2+50 e^{-3+x} x^2\right )+e^{-4+2 e^{5 e^x}} \left (50 e^{-3+x} x-50 x^2+e^{5 e^x+x} \left (250 e^{-3+x} x^2-250 x^3\right )\right ) \log \left (e^{-3+x}-x\right ) \log \left (\log \left (e^{-3+x}-x\right )\right )+\left (-50 x^2+50 e^{-3+x} x^2+e^{-2+e^{5 e^x}} \left (100 e^{-3+x} x-100 x^2+e^{5 e^x+x} \left (250 e^{-3+x} x^2-250 x^3\right )\right ) \log \left (e^{-3+x}-x\right ) \log \left (\log \left (e^{-3+x}-x\right )\right )\right ) \log \left (\log \left (\log \left (e^{-3+x}-x\right )\right )\right )+\left (50 e^{-3+x} x-50 x^2\right ) \log \left (e^{-3+x}-x\right ) \log \left (\log \left (e^{-3+x}-x\right )\right ) \log ^2\left (\log \left (\log \left (e^{-3+x}-x\right )\right )\right )}{\left (e^{-3+x}-x\right ) \log \left (e^{-3+x}-x\right ) \log \left (\log \left (e^{-3+x}-x\right )\right )} \, dx=25 \, {\left (x^{2} e^{4} \log \left (\log \left (\log \left (-x e^{3} + e^{x}\right ) - 3\right )\right )^{2} + 2 \, x^{2} e^{\left (e^{\left (5 \, e^{x}\right )} + 2\right )} \log \left (\log \left (\log \left (-x e^{3} + e^{x}\right ) - 3\right )\right ) + x^{2} e^{\left (2 \, e^{\left (5 \, e^{x}\right )}\right )}\right )} e^{\left (-4\right )} \]
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\[ \int \frac {e^{-2+e^{5 e^x}} \left (-50 x^2+50 e^{-3+x} x^2\right )+e^{-4+2 e^{5 e^x}} \left (50 e^{-3+x} x-50 x^2+e^{5 e^x+x} \left (250 e^{-3+x} x^2-250 x^3\right )\right ) \log \left (e^{-3+x}-x\right ) \log \left (\log \left (e^{-3+x}-x\right )\right )+\left (-50 x^2+50 e^{-3+x} x^2+e^{-2+e^{5 e^x}} \left (100 e^{-3+x} x-100 x^2+e^{5 e^x+x} \left (250 e^{-3+x} x^2-250 x^3\right )\right ) \log \left (e^{-3+x}-x\right ) \log \left (\log \left (e^{-3+x}-x\right )\right )\right ) \log \left (\log \left (\log \left (e^{-3+x}-x\right )\right )\right )+\left (50 e^{-3+x} x-50 x^2\right ) \log \left (e^{-3+x}-x\right ) \log \left (\log \left (e^{-3+x}-x\right )\right ) \log ^2\left (\log \left (\log \left (e^{-3+x}-x\right )\right )\right )}{\left (e^{-3+x}-x\right ) \log \left (e^{-3+x}-x\right ) \log \left (\log \left (e^{-3+x}-x\right )\right )} \, dx=\int { \frac {50 \, {\left ({\left (x^{2} - x e^{\left (x - 3\right )}\right )} \log \left (-x + e^{\left (x - 3\right )}\right ) \log \left (\log \left (-x + e^{\left (x - 3\right )}\right )\right ) \log \left (\log \left (\log \left (-x + e^{\left (x - 3\right )}\right )\right )\right )^{2} + {\left (x^{2} + 5 \, {\left (x^{3} - x^{2} e^{\left (x - 3\right )}\right )} e^{\left (x + 5 \, e^{x}\right )} - x e^{\left (x - 3\right )}\right )} e^{\left (2 \, e^{\left (5 \, e^{x}\right )} - 4\right )} \log \left (-x + e^{\left (x - 3\right )}\right ) \log \left (\log \left (-x + e^{\left (x - 3\right )}\right )\right ) - {\left (x^{2} e^{\left (x - 3\right )} - x^{2}\right )} e^{\left (e^{\left (5 \, e^{x}\right )} - 2\right )} + {\left ({\left (2 \, x^{2} + 5 \, {\left (x^{3} - x^{2} e^{\left (x - 3\right )}\right )} e^{\left (x + 5 \, e^{x}\right )} - 2 \, x e^{\left (x - 3\right )}\right )} e^{\left (e^{\left (5 \, e^{x}\right )} - 2\right )} \log \left (-x + e^{\left (x - 3\right )}\right ) \log \left (\log \left (-x + e^{\left (x - 3\right )}\right )\right ) - x^{2} e^{\left (x - 3\right )} + x^{2}\right )} \log \left (\log \left (\log \left (-x + e^{\left (x - 3\right )}\right )\right )\right )\right )}}{{\left (x - e^{\left (x - 3\right )}\right )} \log \left (-x + e^{\left (x - 3\right )}\right ) \log \left (\log \left (-x + e^{\left (x - 3\right )}\right )\right )} \,d x } \]
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Timed out. \[ \int \frac {e^{-2+e^{5 e^x}} \left (-50 x^2+50 e^{-3+x} x^2\right )+e^{-4+2 e^{5 e^x}} \left (50 e^{-3+x} x-50 x^2+e^{5 e^x+x} \left (250 e^{-3+x} x^2-250 x^3\right )\right ) \log \left (e^{-3+x}-x\right ) \log \left (\log \left (e^{-3+x}-x\right )\right )+\left (-50 x^2+50 e^{-3+x} x^2+e^{-2+e^{5 e^x}} \left (100 e^{-3+x} x-100 x^2+e^{5 e^x+x} \left (250 e^{-3+x} x^2-250 x^3\right )\right ) \log \left (e^{-3+x}-x\right ) \log \left (\log \left (e^{-3+x}-x\right )\right )\right ) \log \left (\log \left (\log \left (e^{-3+x}-x\right )\right )\right )+\left (50 e^{-3+x} x-50 x^2\right ) \log \left (e^{-3+x}-x\right ) \log \left (\log \left (e^{-3+x}-x\right )\right ) \log ^2\left (\log \left (\log \left (e^{-3+x}-x\right )\right )\right )}{\left (e^{-3+x}-x\right ) \log \left (e^{-3+x}-x\right ) \log \left (\log \left (e^{-3+x}-x\right )\right )} \, dx=-\int \frac {\ln \left (\ln \left ({\mathrm {e}}^{x-3}-x\right )\right )\,\ln \left ({\mathrm {e}}^{x-3}-x\right )\,\left (50\,x\,{\mathrm {e}}^{x-3}-50\,x^2\right )\,{\ln \left (\ln \left (\ln \left ({\mathrm {e}}^{x-3}-x\right )\right )\right )}^2+\left (50\,x^2\,{\mathrm {e}}^{x-3}-50\,x^2+\ln \left (\ln \left ({\mathrm {e}}^{x-3}-x\right )\right )\,{\mathrm {e}}^{{\mathrm {e}}^{5\,{\mathrm {e}}^x}-2}\,\ln \left ({\mathrm {e}}^{x-3}-x\right )\,\left (100\,x\,{\mathrm {e}}^{x-3}-100\,x^2+{\mathrm {e}}^{x+5\,{\mathrm {e}}^x}\,\left (250\,x^2\,{\mathrm {e}}^{x-3}-250\,x^3\right )\right )\right )\,\ln \left (\ln \left (\ln \left ({\mathrm {e}}^{x-3}-x\right )\right )\right )+{\mathrm {e}}^{{\mathrm {e}}^{5\,{\mathrm {e}}^x}-2}\,\left (50\,x^2\,{\mathrm {e}}^{x-3}-50\,x^2\right )+\ln \left (\ln \left ({\mathrm {e}}^{x-3}-x\right )\right )\,{\mathrm {e}}^{2\,{\mathrm {e}}^{5\,{\mathrm {e}}^x}-4}\,\ln \left ({\mathrm {e}}^{x-3}-x\right )\,\left (50\,x\,{\mathrm {e}}^{x-3}-50\,x^2+{\mathrm {e}}^{x+5\,{\mathrm {e}}^x}\,\left (250\,x^2\,{\mathrm {e}}^{x-3}-250\,x^3\right )\right )}{\ln \left (\ln \left ({\mathrm {e}}^{x-3}-x\right )\right )\,\ln \left ({\mathrm {e}}^{x-3}-x\right )\,\left (x-{\mathrm {e}}^{x-3}\right )} \,d x \]
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