Integrand size = 240, antiderivative size = 27 \[ \int \frac {-10+12 x-6 x^2+e^{20+x} \left (-6+12 x-6 x^2\right )+\left (-9+23 x-17 x^2+3 x^3+e^{20+x} \left (3-6 x+3 x^2\right )\right ) \log \left (\frac {36-72 x+36 x^2}{81-252 x+250 x^2-84 x^3+9 x^4+e^{40+2 x} \left (9-18 x+9 x^2\right )+e^{20+x} \left (-54+138 x-102 x^2+18 x^3\right )}\right )}{\left (-9+23 x-17 x^2+3 x^3+e^{20+x} \left (3-6 x+3 x^2\right )\right ) \log \left (\frac {36-72 x+36 x^2}{81-252 x+250 x^2-84 x^3+9 x^4+e^{40+2 x} \left (9-18 x+9 x^2\right )+e^{20+x} \left (-54+138 x-102 x^2+18 x^3\right )}\right )} \, dx=x+\log \left (\log \left (\frac {4}{\left (-3+e^{20+x}+\frac {2}{-3+\frac {3}{x}}+x\right )^2}\right )\right ) \]
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\[ \int \frac {-10+12 x-6 x^2+e^{20+x} \left (-6+12 x-6 x^2\right )+\left (-9+23 x-17 x^2+3 x^3+e^{20+x} \left (3-6 x+3 x^2\right )\right ) \log \left (\frac {36-72 x+36 x^2}{81-252 x+250 x^2-84 x^3+9 x^4+e^{40+2 x} \left (9-18 x+9 x^2\right )+e^{20+x} \left (-54+138 x-102 x^2+18 x^3\right )}\right )}{\left (-9+23 x-17 x^2+3 x^3+e^{20+x} \left (3-6 x+3 x^2\right )\right ) \log \left (\frac {36-72 x+36 x^2}{81-252 x+250 x^2-84 x^3+9 x^4+e^{40+2 x} \left (9-18 x+9 x^2\right )+e^{20+x} \left (-54+138 x-102 x^2+18 x^3\right )}\right )} \, dx=\int \frac {-10+12 x-6 x^2+e^{20+x} \left (-6+12 x-6 x^2\right )+\left (-9+23 x-17 x^2+3 x^3+e^{20+x} \left (3-6 x+3 x^2\right )\right ) \log \left (\frac {36-72 x+36 x^2}{81-252 x+250 x^2-84 x^3+9 x^4+e^{40+2 x} \left (9-18 x+9 x^2\right )+e^{20+x} \left (-54+138 x-102 x^2+18 x^3\right )}\right )}{\left (-9+23 x-17 x^2+3 x^3+e^{20+x} \left (3-6 x+3 x^2\right )\right ) \log \left (\frac {36-72 x+36 x^2}{81-252 x+250 x^2-84 x^3+9 x^4+e^{40+2 x} \left (9-18 x+9 x^2\right )+e^{20+x} \left (-54+138 x-102 x^2+18 x^3\right )}\right )} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {10+6 e^{20+x} (-1+x)^2-12 x+6 x^2-(-1+x) \left (9+3 e^{20+x} (-1+x)-14 x+3 x^2\right ) \log \left (\frac {36 (-1+x)^2}{\left (9+3 e^{20+x} (-1+x)-14 x+3 x^2\right )^2}\right )}{(1-x) \left (9+3 e^{20+x} (-1+x)-14 x+3 x^2\right ) \log \left (\frac {36 (-1+x)^2}{\left (9+3 e^{20+x} (-1+x)-14 x+3 x^2\right )^2}\right )} \, dx \\ & = \int \left (\frac {2 \left (-14+29 x-20 x^2+3 x^3\right )}{(-1+x) \left (9-3 e^{20+x}-14 x+3 e^{20+x} x+3 x^2\right ) \log \left (\frac {36 (-1+x)^2}{\left (9+3 e^{20+x} (-1+x)-14 x+3 x^2\right )^2}\right )}+\frac {-2+\log \left (\frac {36 (-1+x)^2}{\left (9+3 e^{20+x} (-1+x)-14 x+3 x^2\right )^2}\right )}{\log \left (\frac {36 (-1+x)^2}{\left (9+3 e^{20+x} (-1+x)-14 x+3 x^2\right )^2}\right )}\right ) \, dx \\ & = 2 \int \frac {-14+29 x-20 x^2+3 x^3}{(-1+x) \left (9-3 e^{20+x}-14 x+3 e^{20+x} x+3 x^2\right ) \log \left (\frac {36 (-1+x)^2}{\left (9+3 e^{20+x} (-1+x)-14 x+3 x^2\right )^2}\right )} \, dx+\int \frac {-2+\log \left (\frac {36 (-1+x)^2}{\left (9+3 e^{20+x} (-1+x)-14 x+3 x^2\right )^2}\right )}{\log \left (\frac {36 (-1+x)^2}{\left (9+3 e^{20+x} (-1+x)-14 x+3 x^2\right )^2}\right )} \, dx \\ & = 2 \int \left (\frac {12}{\left (9-3 e^{20+x}-14 x+3 e^{20+x} x+3 x^2\right ) \log \left (\frac {36 (-1+x)^2}{\left (9+3 e^{20+x} (-1+x)-14 x+3 x^2\right )^2}\right )}-\frac {2}{(-1+x) \left (9-3 e^{20+x}-14 x+3 e^{20+x} x+3 x^2\right ) \log \left (\frac {36 (-1+x)^2}{\left (9+3 e^{20+x} (-1+x)-14 x+3 x^2\right )^2}\right )}-\frac {17 x}{\left (9-3 e^{20+x}-14 x+3 e^{20+x} x+3 x^2\right ) \log \left (\frac {36 (-1+x)^2}{\left (9+3 e^{20+x} (-1+x)-14 x+3 x^2\right )^2}\right )}+\frac {3 x^2}{\left (9-3 e^{20+x}-14 x+3 e^{20+x} x+3 x^2\right ) \log \left (\frac {36 (-1+x)^2}{\left (9+3 e^{20+x} (-1+x)-14 x+3 x^2\right )^2}\right )}\right ) \, dx+\int \left (1-\frac {2}{\log \left (\frac {36 (-1+x)^2}{\left (9+3 e^{20+x} (-1+x)-14 x+3 x^2\right )^2}\right )}\right ) \, dx \\ & = x-2 \int \frac {1}{\log \left (\frac {36 (-1+x)^2}{\left (9+3 e^{20+x} (-1+x)-14 x+3 x^2\right )^2}\right )} \, dx-4 \int \frac {1}{(-1+x) \left (9-3 e^{20+x}-14 x+3 e^{20+x} x+3 x^2\right ) \log \left (\frac {36 (-1+x)^2}{\left (9+3 e^{20+x} (-1+x)-14 x+3 x^2\right )^2}\right )} \, dx+6 \int \frac {x^2}{\left (9-3 e^{20+x}-14 x+3 e^{20+x} x+3 x^2\right ) \log \left (\frac {36 (-1+x)^2}{\left (9+3 e^{20+x} (-1+x)-14 x+3 x^2\right )^2}\right )} \, dx+24 \int \frac {1}{\left (9-3 e^{20+x}-14 x+3 e^{20+x} x+3 x^2\right ) \log \left (\frac {36 (-1+x)^2}{\left (9+3 e^{20+x} (-1+x)-14 x+3 x^2\right )^2}\right )} \, dx-34 \int \frac {x}{\left (9-3 e^{20+x}-14 x+3 e^{20+x} x+3 x^2\right ) \log \left (\frac {36 (-1+x)^2}{\left (9+3 e^{20+x} (-1+x)-14 x+3 x^2\right )^2}\right )} \, dx \\ \end{align*}
Time = 0.18 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.22 \[ \int \frac {-10+12 x-6 x^2+e^{20+x} \left (-6+12 x-6 x^2\right )+\left (-9+23 x-17 x^2+3 x^3+e^{20+x} \left (3-6 x+3 x^2\right )\right ) \log \left (\frac {36-72 x+36 x^2}{81-252 x+250 x^2-84 x^3+9 x^4+e^{40+2 x} \left (9-18 x+9 x^2\right )+e^{20+x} \left (-54+138 x-102 x^2+18 x^3\right )}\right )}{\left (-9+23 x-17 x^2+3 x^3+e^{20+x} \left (3-6 x+3 x^2\right )\right ) \log \left (\frac {36-72 x+36 x^2}{81-252 x+250 x^2-84 x^3+9 x^4+e^{40+2 x} \left (9-18 x+9 x^2\right )+e^{20+x} \left (-54+138 x-102 x^2+18 x^3\right )}\right )} \, dx=x+\log \left (\log \left (\frac {36 (-1+x)^2}{\left (9+3 e^{20+x} (-1+x)-14 x+3 x^2\right )^2}\right )\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. \(74\) vs. \(2(26)=52\).
Time = 0.73 (sec) , antiderivative size = 75, normalized size of antiderivative = 2.78
method | result | size |
norman | \(x +\ln \left (\ln \left (\frac {36 x^{2}-72 x +36}{\left (9 x^{2}-18 x +9\right ) {\mathrm e}^{40+2 x}+\left (18 x^{3}-102 x^{2}+138 x -54\right ) {\mathrm e}^{20+x}+9 x^{4}-84 x^{3}+250 x^{2}-252 x +81}\right )\right )\) | \(75\) |
parallelrisch | \(4+\ln \left (\ln \left (\frac {36 x^{2}-72 x +36}{18 \,{\mathrm e}^{20+x} x^{3}+9 x^{4}-102 \,{\mathrm e}^{20+x} x^{2}+9 \,{\mathrm e}^{40+2 x} x^{2}-84 x^{3}+138 \,{\mathrm e}^{20+x} x -18 \,{\mathrm e}^{40+2 x} x +250 x^{2}-54 \,{\mathrm e}^{20+x}+9 \,{\mathrm e}^{40+2 x}-252 x +81}\right )\right )+x\) | \(97\) |
risch | \(x +\ln \left (\ln \left (x^{2}+\left ({\mathrm e}^{20+x}-\frac {14}{3}\right ) x -{\mathrm e}^{20+x}+3\right )+\frac {i \left (\pi \operatorname {csgn}\left (i \left (-1+x \right )\right )^{2} \operatorname {csgn}\left (i \left (-1+x \right )^{2}\right )-2 \pi \,\operatorname {csgn}\left (i \left (-1+x \right )\right ) \operatorname {csgn}\left (i \left (-1+x \right )^{2}\right )^{2}+\pi \operatorname {csgn}\left (i \left (-1+x \right )^{2}\right )^{3}+\pi \,\operatorname {csgn}\left (i \left (-1+x \right )^{2}\right ) \operatorname {csgn}\left (\frac {i}{\left (x^{2}+\left ({\mathrm e}^{20+x}-\frac {14}{3}\right ) x -{\mathrm e}^{20+x}+3\right )^{2}}\right ) \operatorname {csgn}\left (\frac {i \left (-1+x \right )^{2}}{\left (x^{2}+\left ({\mathrm e}^{20+x}-\frac {14}{3}\right ) x -{\mathrm e}^{20+x}+3\right )^{2}}\right )-\pi \,\operatorname {csgn}\left (i \left (-1+x \right )^{2}\right ) \operatorname {csgn}\left (\frac {i \left (-1+x \right )^{2}}{\left (x^{2}+\left ({\mathrm e}^{20+x}-\frac {14}{3}\right ) x -{\mathrm e}^{20+x}+3\right )^{2}}\right )^{2}-\pi \,\operatorname {csgn}\left (\frac {i}{\left (x^{2}+\left ({\mathrm e}^{20+x}-\frac {14}{3}\right ) x -{\mathrm e}^{20+x}+3\right )^{2}}\right ) \operatorname {csgn}\left (\frac {i \left (-1+x \right )^{2}}{\left (x^{2}+\left ({\mathrm e}^{20+x}-\frac {14}{3}\right ) x -{\mathrm e}^{20+x}+3\right )^{2}}\right )^{2}-\pi {\operatorname {csgn}\left (i \left (x^{2}+\left ({\mathrm e}^{20+x}-\frac {14}{3}\right ) x -{\mathrm e}^{20+x}+3\right )\right )}^{2} \operatorname {csgn}\left (i \left (x^{2}+\left ({\mathrm e}^{20+x}-\frac {14}{3}\right ) x -{\mathrm e}^{20+x}+3\right )^{2}\right )+2 \pi \,\operatorname {csgn}\left (i \left (x^{2}+\left ({\mathrm e}^{20+x}-\frac {14}{3}\right ) x -{\mathrm e}^{20+x}+3\right )\right ) {\operatorname {csgn}\left (i \left (x^{2}+\left ({\mathrm e}^{20+x}-\frac {14}{3}\right ) x -{\mathrm e}^{20+x}+3\right )^{2}\right )}^{2}-\pi {\operatorname {csgn}\left (i \left (x^{2}+\left ({\mathrm e}^{20+x}-\frac {14}{3}\right ) x -{\mathrm e}^{20+x}+3\right )^{2}\right )}^{3}+\pi \operatorname {csgn}\left (\frac {i \left (-1+x \right )^{2}}{\left (x^{2}+\left ({\mathrm e}^{20+x}-\frac {14}{3}\right ) x -{\mathrm e}^{20+x}+3\right )^{2}}\right )^{3}+4 i \ln \left (-1+x \right )+4 i \ln \left (2\right )\right )}{4}\right )\) | \(435\) |
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Leaf count of result is larger than twice the leaf count of optimal. 73 vs. \(2 (28) = 56\).
Time = 0.26 (sec) , antiderivative size = 73, normalized size of antiderivative = 2.70 \[ \int \frac {-10+12 x-6 x^2+e^{20+x} \left (-6+12 x-6 x^2\right )+\left (-9+23 x-17 x^2+3 x^3+e^{20+x} \left (3-6 x+3 x^2\right )\right ) \log \left (\frac {36-72 x+36 x^2}{81-252 x+250 x^2-84 x^3+9 x^4+e^{40+2 x} \left (9-18 x+9 x^2\right )+e^{20+x} \left (-54+138 x-102 x^2+18 x^3\right )}\right )}{\left (-9+23 x-17 x^2+3 x^3+e^{20+x} \left (3-6 x+3 x^2\right )\right ) \log \left (\frac {36-72 x+36 x^2}{81-252 x+250 x^2-84 x^3+9 x^4+e^{40+2 x} \left (9-18 x+9 x^2\right )+e^{20+x} \left (-54+138 x-102 x^2+18 x^3\right )}\right )} \, dx=x + \log \left (\log \left (\frac {36 \, {\left (x^{2} - 2 \, x + 1\right )}}{9 \, x^{4} - 84 \, x^{3} + 250 \, x^{2} + 9 \, {\left (x^{2} - 2 \, x + 1\right )} e^{\left (2 \, x + 40\right )} + 6 \, {\left (3 \, x^{3} - 17 \, x^{2} + 23 \, x - 9\right )} e^{\left (x + 20\right )} - 252 \, x + 81}\right )\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 71 vs. \(2 (22) = 44\).
Time = 0.63 (sec) , antiderivative size = 71, normalized size of antiderivative = 2.63 \[ \int \frac {-10+12 x-6 x^2+e^{20+x} \left (-6+12 x-6 x^2\right )+\left (-9+23 x-17 x^2+3 x^3+e^{20+x} \left (3-6 x+3 x^2\right )\right ) \log \left (\frac {36-72 x+36 x^2}{81-252 x+250 x^2-84 x^3+9 x^4+e^{40+2 x} \left (9-18 x+9 x^2\right )+e^{20+x} \left (-54+138 x-102 x^2+18 x^3\right )}\right )}{\left (-9+23 x-17 x^2+3 x^3+e^{20+x} \left (3-6 x+3 x^2\right )\right ) \log \left (\frac {36-72 x+36 x^2}{81-252 x+250 x^2-84 x^3+9 x^4+e^{40+2 x} \left (9-18 x+9 x^2\right )+e^{20+x} \left (-54+138 x-102 x^2+18 x^3\right )}\right )} \, dx=x + \log {\left (\log {\left (\frac {36 x^{2} - 72 x + 36}{9 x^{4} - 84 x^{3} + 250 x^{2} - 252 x + \left (9 x^{2} - 18 x + 9\right ) e^{2 x + 40} + \left (18 x^{3} - 102 x^{2} + 138 x - 54\right ) e^{x + 20} + 81} \right )} \right )} \]
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Time = 7.16 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.56 \[ \int \frac {-10+12 x-6 x^2+e^{20+x} \left (-6+12 x-6 x^2\right )+\left (-9+23 x-17 x^2+3 x^3+e^{20+x} \left (3-6 x+3 x^2\right )\right ) \log \left (\frac {36-72 x+36 x^2}{81-252 x+250 x^2-84 x^3+9 x^4+e^{40+2 x} \left (9-18 x+9 x^2\right )+e^{20+x} \left (-54+138 x-102 x^2+18 x^3\right )}\right )}{\left (-9+23 x-17 x^2+3 x^3+e^{20+x} \left (3-6 x+3 x^2\right )\right ) \log \left (\frac {36-72 x+36 x^2}{81-252 x+250 x^2-84 x^3+9 x^4+e^{40+2 x} \left (9-18 x+9 x^2\right )+e^{20+x} \left (-54+138 x-102 x^2+18 x^3\right )}\right )} \, dx=x + \log \left (-\log \left (3\right ) - \log \left (2\right ) + \log \left (3 \, x^{2} + 3 \, {\left (x e^{20} - e^{20}\right )} e^{x} - 14 \, x + 9\right ) - \log \left (x - 1\right )\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 97 vs. \(2 (28) = 56\).
Time = 5.63 (sec) , antiderivative size = 97, normalized size of antiderivative = 3.59 \[ \int \frac {-10+12 x-6 x^2+e^{20+x} \left (-6+12 x-6 x^2\right )+\left (-9+23 x-17 x^2+3 x^3+e^{20+x} \left (3-6 x+3 x^2\right )\right ) \log \left (\frac {36-72 x+36 x^2}{81-252 x+250 x^2-84 x^3+9 x^4+e^{40+2 x} \left (9-18 x+9 x^2\right )+e^{20+x} \left (-54+138 x-102 x^2+18 x^3\right )}\right )}{\left (-9+23 x-17 x^2+3 x^3+e^{20+x} \left (3-6 x+3 x^2\right )\right ) \log \left (\frac {36-72 x+36 x^2}{81-252 x+250 x^2-84 x^3+9 x^4+e^{40+2 x} \left (9-18 x+9 x^2\right )+e^{20+x} \left (-54+138 x-102 x^2+18 x^3\right )}\right )} \, dx=x + \log \left (-\log \left (9 \, x^{4} + 18 \, x^{3} e^{\left (x + 20\right )} - 84 \, x^{3} + 9 \, x^{2} e^{\left (2 \, x + 40\right )} - 102 \, x^{2} e^{\left (x + 20\right )} + 250 \, x^{2} - 18 \, x e^{\left (2 \, x + 40\right )} + 138 \, x e^{\left (x + 20\right )} - 252 \, x + 9 \, e^{\left (2 \, x + 40\right )} - 54 \, e^{\left (x + 20\right )} + 81\right ) + \log \left (36 \, x^{2} - 72 \, x + 36\right )\right ) \]
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Time = 10.03 (sec) , antiderivative size = 74, normalized size of antiderivative = 2.74 \[ \int \frac {-10+12 x-6 x^2+e^{20+x} \left (-6+12 x-6 x^2\right )+\left (-9+23 x-17 x^2+3 x^3+e^{20+x} \left (3-6 x+3 x^2\right )\right ) \log \left (\frac {36-72 x+36 x^2}{81-252 x+250 x^2-84 x^3+9 x^4+e^{40+2 x} \left (9-18 x+9 x^2\right )+e^{20+x} \left (-54+138 x-102 x^2+18 x^3\right )}\right )}{\left (-9+23 x-17 x^2+3 x^3+e^{20+x} \left (3-6 x+3 x^2\right )\right ) \log \left (\frac {36-72 x+36 x^2}{81-252 x+250 x^2-84 x^3+9 x^4+e^{40+2 x} \left (9-18 x+9 x^2\right )+e^{20+x} \left (-54+138 x-102 x^2+18 x^3\right )}\right )} \, dx=x+\ln \left (\ln \left (\frac {36\,x^2-72\,x+36}{250\,x^2-252\,x-84\,x^3+9\,x^4+{\mathrm {e}}^{20}\,{\mathrm {e}}^x\,\left (18\,x^3-102\,x^2+138\,x-54\right )+{\mathrm {e}}^{2\,x}\,{\mathrm {e}}^{40}\,\left (9\,x^2-18\,x+9\right )+81}\right )\right ) \]
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