Integrand size = 60, antiderivative size = 29 \[ \int \frac {e^{-x-8 x^2} \left (10 x-4 x^2-81 x^3-16 x^4-5 e^{x+8 x^2} \log ^4(9)\right )}{\left (25+10 x+x^2\right ) \log ^4(9)} \, dx=\frac {-x+\frac {e^{-x-8 x^2} x^2}{\log ^4(9)}}{5+x} \]
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Leaf count is larger than twice the leaf count of optimal. \(64\) vs. \(2(29)=58\).
Time = 1.13 (sec) , antiderivative size = 64, normalized size of antiderivative = 2.21, number of steps used = 42, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {12, 27, 6874, 2274, 2266, 2236, 2272, 2273} \[ \int \frac {e^{-x-8 x^2} \left (10 x-4 x^2-81 x^3-16 x^4-5 e^{x+8 x^2} \log ^4(9)\right )}{\left (25+10 x+x^2\right ) \log ^4(9)} \, dx=\frac {e^{-8 x^2-x} x}{\log ^4(9)}-\frac {5 e^{-8 x^2-x}}{\log ^4(9)}+\frac {25 e^{-8 x^2-x}}{(x+5) \log ^4(9)}+\frac {5}{x+5} \]
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Rule 12
Rule 27
Rule 2236
Rule 2266
Rule 2272
Rule 2273
Rule 2274
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \frac {\int \frac {e^{-x-8 x^2} \left (10 x-4 x^2-81 x^3-16 x^4-5 e^{x+8 x^2} \log ^4(9)\right )}{25+10 x+x^2} \, dx}{\log ^4(9)} \\ & = \frac {\int \frac {e^{-x-8 x^2} \left (10 x-4 x^2-81 x^3-16 x^4-5 e^{x+8 x^2} \log ^4(9)\right )}{(5+x)^2} \, dx}{\log ^4(9)} \\ & = \frac {\int \left (\frac {10 e^{-x-8 x^2} x}{(5+x)^2}-\frac {4 e^{-x-8 x^2} x^2}{(5+x)^2}-\frac {81 e^{-x-8 x^2} x^3}{(5+x)^2}-\frac {16 e^{-x-8 x^2} x^4}{(5+x)^2}-\frac {5 \log ^4(9)}{(5+x)^2}\right ) \, dx}{\log ^4(9)} \\ & = \frac {5}{5+x}-\frac {4 \int \frac {e^{-x-8 x^2} x^2}{(5+x)^2} \, dx}{\log ^4(9)}+\frac {10 \int \frac {e^{-x-8 x^2} x}{(5+x)^2} \, dx}{\log ^4(9)}-\frac {16 \int \frac {e^{-x-8 x^2} x^4}{(5+x)^2} \, dx}{\log ^4(9)}-\frac {81 \int \frac {e^{-x-8 x^2} x^3}{(5+x)^2} \, dx}{\log ^4(9)} \\ & = \frac {5}{5+x}-\frac {4 \int \left (e^{-x-8 x^2}+\frac {25 e^{-x-8 x^2}}{(5+x)^2}-\frac {10 e^{-x-8 x^2}}{5+x}\right ) \, dx}{\log ^4(9)}+\frac {10 \int \left (-\frac {5 e^{-x-8 x^2}}{(5+x)^2}+\frac {e^{-x-8 x^2}}{5+x}\right ) \, dx}{\log ^4(9)}-\frac {16 \int \left (75 e^{-x-8 x^2}-10 e^{-x-8 x^2} x+e^{-x-8 x^2} x^2+\frac {625 e^{-x-8 x^2}}{(5+x)^2}-\frac {500 e^{-x-8 x^2}}{5+x}\right ) \, dx}{\log ^4(9)}-\frac {81 \int \left (-10 e^{-x-8 x^2}+e^{-x-8 x^2} x-\frac {125 e^{-x-8 x^2}}{(5+x)^2}+\frac {75 e^{-x-8 x^2}}{5+x}\right ) \, dx}{\log ^4(9)} \\ & = \frac {5}{5+x}-\frac {4 \int e^{-x-8 x^2} \, dx}{\log ^4(9)}+\frac {10 \int \frac {e^{-x-8 x^2}}{5+x} \, dx}{\log ^4(9)}-\frac {16 \int e^{-x-8 x^2} x^2 \, dx}{\log ^4(9)}+\frac {40 \int \frac {e^{-x-8 x^2}}{5+x} \, dx}{\log ^4(9)}-\frac {50 \int \frac {e^{-x-8 x^2}}{(5+x)^2} \, dx}{\log ^4(9)}-\frac {81 \int e^{-x-8 x^2} x \, dx}{\log ^4(9)}-\frac {100 \int \frac {e^{-x-8 x^2}}{(5+x)^2} \, dx}{\log ^4(9)}+\frac {160 \int e^{-x-8 x^2} x \, dx}{\log ^4(9)}+\frac {810 \int e^{-x-8 x^2} \, dx}{\log ^4(9)}-\frac {1200 \int e^{-x-8 x^2} \, dx}{\log ^4(9)}-\frac {6075 \int \frac {e^{-x-8 x^2}}{5+x} \, dx}{\log ^4(9)}+\frac {8000 \int \frac {e^{-x-8 x^2}}{5+x} \, dx}{\log ^4(9)}-\frac {10000 \int \frac {e^{-x-8 x^2}}{(5+x)^2} \, dx}{\log ^4(9)}+\frac {10125 \int \frac {e^{-x-8 x^2}}{(5+x)^2} \, dx}{\log ^4(9)} \\ & = \frac {5}{5+x}-\frac {79 e^{-x-8 x^2}}{16 \log ^4(9)}+\frac {e^{-x-8 x^2} x}{\log ^4(9)}+\frac {25 e^{-x-8 x^2}}{(5+x) \log ^4(9)}-\frac {\int e^{-x-8 x^2} \, dx}{\log ^4(9)}+\frac {\int e^{-x-8 x^2} x \, dx}{\log ^4(9)}+\frac {81 \int e^{-x-8 x^2} \, dx}{16 \log ^4(9)}-\frac {10 \int e^{-x-8 x^2} \, dx}{\log ^4(9)}+\frac {10 \int \frac {e^{-x-8 x^2}}{5+x} \, dx}{\log ^4(9)}+\frac {40 \int \frac {e^{-x-8 x^2}}{5+x} \, dx}{\log ^4(9)}+\frac {800 \int e^{-x-8 x^2} \, dx}{\log ^4(9)}+\frac {1600 \int e^{-x-8 x^2} \, dx}{\log ^4(9)}-\frac {3950 \int \frac {e^{-x-8 x^2}}{5+x} \, dx}{\log ^4(9)}-\frac {6075 \int \frac {e^{-x-8 x^2}}{5+x} \, dx}{\log ^4(9)}-\frac {7900 \int \frac {e^{-x-8 x^2}}{5+x} \, dx}{\log ^4(9)}+\frac {8000 \int \frac {e^{-x-8 x^2}}{5+x} \, dx}{\log ^4(9)}+\frac {160000 \int e^{-x-8 x^2} \, dx}{\log ^4(9)}-\frac {162000 \int e^{-x-8 x^2} \, dx}{\log ^4(9)}-\frac {790000 \int \frac {e^{-x-8 x^2}}{5+x} \, dx}{\log ^4(9)}+\frac {799875 \int \frac {e^{-x-8 x^2}}{5+x} \, dx}{\log ^4(9)}-\frac {\left (4 \sqrt [32]{e}\right ) \int e^{-\frac {1}{32} (-1-16 x)^2} \, dx}{\log ^4(9)}+\frac {\left (810 \sqrt [32]{e}\right ) \int e^{-\frac {1}{32} (-1-16 x)^2} \, dx}{\log ^4(9)}-\frac {\left (1200 \sqrt [32]{e}\right ) \int e^{-\frac {1}{32} (-1-16 x)^2} \, dx}{\log ^4(9)} \\ & = \frac {5}{5+x}-\frac {5 e^{-x-8 x^2}}{\log ^4(9)}+\frac {e^{-x-8 x^2} x}{\log ^4(9)}+\frac {25 e^{-x-8 x^2}}{(5+x) \log ^4(9)}+\frac {403 \sqrt [32]{e} \sqrt {\frac {\pi }{2}} \text {erf}\left (\frac {1+16 x}{4 \sqrt {2}}\right )}{2 \log ^4(9)}-\frac {150 \sqrt [32]{e} \sqrt {2 \pi } \text {erf}\left (\frac {1+16 x}{4 \sqrt {2}}\right )}{\log ^4(9)}-\frac {\int e^{-x-8 x^2} \, dx}{16 \log ^4(9)}+\frac {10 \int \frac {e^{-x-8 x^2}}{5+x} \, dx}{\log ^4(9)}+\frac {40 \int \frac {e^{-x-8 x^2}}{5+x} \, dx}{\log ^4(9)}-\frac {3950 \int \frac {e^{-x-8 x^2}}{5+x} \, dx}{\log ^4(9)}-\frac {6075 \int \frac {e^{-x-8 x^2}}{5+x} \, dx}{\log ^4(9)}-\frac {7900 \int \frac {e^{-x-8 x^2}}{5+x} \, dx}{\log ^4(9)}+\frac {8000 \int \frac {e^{-x-8 x^2}}{5+x} \, dx}{\log ^4(9)}-\frac {790000 \int \frac {e^{-x-8 x^2}}{5+x} \, dx}{\log ^4(9)}+\frac {799875 \int \frac {e^{-x-8 x^2}}{5+x} \, dx}{\log ^4(9)}-\frac {\sqrt [32]{e} \int e^{-\frac {1}{32} (-1-16 x)^2} \, dx}{\log ^4(9)}+\frac {\left (81 \sqrt [32]{e}\right ) \int e^{-\frac {1}{32} (-1-16 x)^2} \, dx}{16 \log ^4(9)}-\frac {\left (10 \sqrt [32]{e}\right ) \int e^{-\frac {1}{32} (-1-16 x)^2} \, dx}{\log ^4(9)}+\frac {\left (800 \sqrt [32]{e}\right ) \int e^{-\frac {1}{32} (-1-16 x)^2} \, dx}{\log ^4(9)}+\frac {\left (1600 \sqrt [32]{e}\right ) \int e^{-\frac {1}{32} (-1-16 x)^2} \, dx}{\log ^4(9)}+\frac {\left (160000 \sqrt [32]{e}\right ) \int e^{-\frac {1}{32} (-1-16 x)^2} \, dx}{\log ^4(9)}-\frac {\left (162000 \sqrt [32]{e}\right ) \int e^{-\frac {1}{32} (-1-16 x)^2} \, dx}{\log ^4(9)} \\ & = \frac {5}{5+x}-\frac {5 e^{-x-8 x^2}}{\log ^4(9)}+\frac {e^{-x-8 x^2} x}{\log ^4(9)}+\frac {25 e^{-x-8 x^2}}{(5+x) \log ^4(9)}+\frac {12801 \sqrt [32]{e} \sqrt {\frac {\pi }{2}} \text {erf}\left (\frac {1+16 x}{4 \sqrt {2}}\right )}{64 \log ^4(9)}-\frac {100 \sqrt [32]{e} \sqrt {2 \pi } \text {erf}\left (\frac {1+16 x}{4 \sqrt {2}}\right )}{\log ^4(9)}+\frac {10 \int \frac {e^{-x-8 x^2}}{5+x} \, dx}{\log ^4(9)}+\frac {40 \int \frac {e^{-x-8 x^2}}{5+x} \, dx}{\log ^4(9)}-\frac {3950 \int \frac {e^{-x-8 x^2}}{5+x} \, dx}{\log ^4(9)}-\frac {6075 \int \frac {e^{-x-8 x^2}}{5+x} \, dx}{\log ^4(9)}-\frac {7900 \int \frac {e^{-x-8 x^2}}{5+x} \, dx}{\log ^4(9)}+\frac {8000 \int \frac {e^{-x-8 x^2}}{5+x} \, dx}{\log ^4(9)}-\frac {790000 \int \frac {e^{-x-8 x^2}}{5+x} \, dx}{\log ^4(9)}+\frac {799875 \int \frac {e^{-x-8 x^2}}{5+x} \, dx}{\log ^4(9)}-\frac {\sqrt [32]{e} \int e^{-\frac {1}{32} (-1-16 x)^2} \, dx}{16 \log ^4(9)} \\ & = \frac {5}{5+x}-\frac {5 e^{-x-8 x^2}}{\log ^4(9)}+\frac {e^{-x-8 x^2} x}{\log ^4(9)}+\frac {25 e^{-x-8 x^2}}{(5+x) \log ^4(9)}+\frac {10 \int \frac {e^{-x-8 x^2}}{5+x} \, dx}{\log ^4(9)}+\frac {40 \int \frac {e^{-x-8 x^2}}{5+x} \, dx}{\log ^4(9)}-\frac {3950 \int \frac {e^{-x-8 x^2}}{5+x} \, dx}{\log ^4(9)}-\frac {6075 \int \frac {e^{-x-8 x^2}}{5+x} \, dx}{\log ^4(9)}-\frac {7900 \int \frac {e^{-x-8 x^2}}{5+x} \, dx}{\log ^4(9)}+\frac {8000 \int \frac {e^{-x-8 x^2}}{5+x} \, dx}{\log ^4(9)}-\frac {790000 \int \frac {e^{-x-8 x^2}}{5+x} \, dx}{\log ^4(9)}+\frac {799875 \int \frac {e^{-x-8 x^2}}{5+x} \, dx}{\log ^4(9)} \\ \end{align*}
Time = 2.61 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.07 \[ \int \frac {e^{-x-8 x^2} \left (10 x-4 x^2-81 x^3-16 x^4-5 e^{x+8 x^2} \log ^4(9)\right )}{\left (25+10 x+x^2\right ) \log ^4(9)} \, dx=\frac {e^{-x (1+8 x)} x^2+5 \log ^4(9)}{(5+x) \log ^4(9)} \]
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Time = 0.52 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.10
method | result | size |
risch | \(\frac {5}{5+x}+\frac {x^{2} {\mathrm e}^{-x \left (8 x +1\right )}}{16 \left (5+x \right ) \ln \left (3\right )^{4}}\) | \(32\) |
parallelrisch | \(\frac {\left (80 \ln \left (3\right )^{4} {\mathrm e}^{x} {\mathrm e}^{8 x^{2}}+x^{2}\right ) {\mathrm e}^{-x} {\mathrm e}^{-8 x^{2}}}{16 \ln \left (3\right )^{4} \left (5+x \right )}\) | \(44\) |
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Time = 0.25 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.34 \[ \int \frac {e^{-x-8 x^2} \left (10 x-4 x^2-81 x^3-16 x^4-5 e^{x+8 x^2} \log ^4(9)\right )}{\left (25+10 x+x^2\right ) \log ^4(9)} \, dx=\frac {{\left (80 \, e^{\left (8 \, x^{2} + x\right )} \log \left (3\right )^{4} + x^{2}\right )} e^{\left (-8 \, x^{2} - x\right )}}{16 \, {\left (x + 5\right )} \log \left (3\right )^{4}} \]
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Time = 0.14 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.24 \[ \int \frac {e^{-x-8 x^2} \left (10 x-4 x^2-81 x^3-16 x^4-5 e^{x+8 x^2} \log ^4(9)\right )}{\left (25+10 x+x^2\right ) \log ^4(9)} \, dx=\frac {x^{2} e^{- 8 x^{2}}}{16 x e^{x} \log {\left (3 \right )}^{4} + 80 e^{x} \log {\left (3 \right )}^{4}} + \frac {5}{x + 5} \]
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Time = 0.27 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.28 \[ \int \frac {e^{-x-8 x^2} \left (10 x-4 x^2-81 x^3-16 x^4-5 e^{x+8 x^2} \log ^4(9)\right )}{\left (25+10 x+x^2\right ) \log ^4(9)} \, dx=\frac {\frac {80 \, \log \left (3\right )^{4}}{x + 5} + \frac {x^{2} e^{\left (-8 \, x^{2} - x\right )}}{x + 5}}{16 \, \log \left (3\right )^{4}} \]
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Time = 0.26 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.10 \[ \int \frac {e^{-x-8 x^2} \left (10 x-4 x^2-81 x^3-16 x^4-5 e^{x+8 x^2} \log ^4(9)\right )}{\left (25+10 x+x^2\right ) \log ^4(9)} \, dx=\frac {80 \, \log \left (3\right )^{4} + x^{2} e^{\left (-8 \, x^{2} - x\right )}}{16 \, {\left (x + 5\right )} \log \left (3\right )^{4}} \]
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Time = 0.25 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.79 \[ \int \frac {e^{-x-8 x^2} \left (10 x-4 x^2-81 x^3-16 x^4-5 e^{x+8 x^2} \log ^4(9)\right )}{\left (25+10 x+x^2\right ) \log ^4(9)} \, dx=\frac {x^2-16\,x\,{\mathrm {e}}^{8\,x^2+x}\,{\ln \left (3\right )}^4}{80\,{\mathrm {e}}^{8\,x^2+x}\,{\ln \left (3\right )}^4+16\,x\,{\mathrm {e}}^{8\,x^2+x}\,{\ln \left (3\right )}^4} \]
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