Integrand size = 206, antiderivative size = 38 \[ \int \frac {20-12 e^{10-8 x}-40 x-8 x^2+24 x^3-12 x^4+e^{5-4 x} \left (-76-8 x+24 x^2\right )}{25-10 x+31 x^2-36 x^3+15 x^4-18 x^5+9 x^6+e^{10-8 x} \left (e^6-6 e^3 x+9 x^2\right )+e^6 \left (x^2-2 x^3+x^4\right )+e^3 \left (-10 x+12 x^2-8 x^3+12 x^4-6 x^5\right )+e^{5-4 x} \left (30 x-6 x^2+18 x^3-18 x^4+e^6 \left (2 x-2 x^2\right )+e^3 \left (-10+2 x-12 x^2+12 x^3\right )\right )} \, dx=\frac {4}{-e^3+3 x-\frac {5-x}{-e^{5-4 x}-x+x^2}} \]
[Out]
\[ \int \frac {20-12 e^{10-8 x}-40 x-8 x^2+24 x^3-12 x^4+e^{5-4 x} \left (-76-8 x+24 x^2\right )}{25-10 x+31 x^2-36 x^3+15 x^4-18 x^5+9 x^6+e^{10-8 x} \left (e^6-6 e^3 x+9 x^2\right )+e^6 \left (x^2-2 x^3+x^4\right )+e^3 \left (-10 x+12 x^2-8 x^3+12 x^4-6 x^5\right )+e^{5-4 x} \left (30 x-6 x^2+18 x^3-18 x^4+e^6 \left (2 x-2 x^2\right )+e^3 \left (-10+2 x-12 x^2+12 x^3\right )\right )} \, dx=\int \frac {20-12 e^{10-8 x}-40 x-8 x^2+24 x^3-12 x^4+e^{5-4 x} \left (-76-8 x+24 x^2\right )}{25-10 x+31 x^2-36 x^3+15 x^4-18 x^5+9 x^6+e^{10-8 x} \left (e^6-6 e^3 x+9 x^2\right )+e^6 \left (x^2-2 x^3+x^4\right )+e^3 \left (-10 x+12 x^2-8 x^3+12 x^4-6 x^5\right )+e^{5-4 x} \left (30 x-6 x^2+18 x^3-18 x^4+e^6 \left (2 x-2 x^2\right )+e^3 \left (-10+2 x-12 x^2+12 x^3\right )\right )} \, dx \]
[In]
[Out]
Rubi steps \begin{align*} \text {integral}& = \int \frac {4 \left (-3 e^{10}-e^{5+4 x} \left (19+2 x-6 x^2\right )-e^{8 x} \left (-5+10 x+2 x^2-6 x^3+3 x^4\right )\right )}{\left (e^8-3 e^5 x-e^{3+4 x} (-1+x) x+e^{4 x} \left (-5+x-3 x^2+3 x^3\right )\right )^2} \, dx \\ & = 4 \int \frac {-3 e^{10}-e^{5+4 x} \left (19+2 x-6 x^2\right )-e^{8 x} \left (-5+10 x+2 x^2-6 x^3+3 x^4\right )}{\left (e^8-3 e^5 x-e^{3+4 x} (-1+x) x+e^{4 x} \left (-5+x-3 x^2+3 x^3\right )\right )^2} \, dx \\ & = 4 \int \left (\frac {e^{10} (5-x) \left (-15-19 e^3+e^6+2 \left (30-e^3+e^6\right ) x-\left (3+12 e^3+4 e^6\right ) x^2+6 \left (3+4 e^3\right ) x^3-36 x^4\right )}{\left (5-\left (1+e^3\right ) x+\left (3+e^3\right ) x^2-3 x^3\right )^2 \left (e^8-5 e^{4 x}-3 e^5 x+e^{4 x} \left (1+e^3\right ) x-3 e^{4 x} \left (1+\frac {e^3}{3}\right ) x^2+3 e^{4 x} x^3\right )^2}+\frac {5-10 x-2 x^2+6 x^3-3 x^4}{\left (5-\left (1+e^3\right ) x+\left (3+e^3\right ) x^2-3 x^3\right )^2}+\frac {e^5 \left (5 \left (19-2 e^3\right )+\left (21+e^3\right ) x-7 \left (5-3 e^3\right ) x^2-\left (57+4 e^3\right ) x^3+12 x^4\right )}{\left (5-\left (1+e^3\right ) x+\left (3+e^3\right ) x^2-3 x^3\right )^2 \left (e^8-5 e^{4 x}-3 e^5 x+e^{4 x} \left (1+e^3\right ) x-3 e^{4 x} \left (1+\frac {e^3}{3}\right ) x^2+3 e^{4 x} x^3\right )}\right ) \, dx \\ & = 4 \int \frac {5-10 x-2 x^2+6 x^3-3 x^4}{\left (5-\left (1+e^3\right ) x+\left (3+e^3\right ) x^2-3 x^3\right )^2} \, dx+\left (4 e^5\right ) \int \frac {5 \left (19-2 e^3\right )+\left (21+e^3\right ) x-7 \left (5-3 e^3\right ) x^2-\left (57+4 e^3\right ) x^3+12 x^4}{\left (5-\left (1+e^3\right ) x+\left (3+e^3\right ) x^2-3 x^3\right )^2 \left (e^8-5 e^{4 x}-3 e^5 x+e^{4 x} \left (1+e^3\right ) x-3 e^{4 x} \left (1+\frac {e^3}{3}\right ) x^2+3 e^{4 x} x^3\right )} \, dx+\left (4 e^{10}\right ) \int \frac {(5-x) \left (-15-19 e^3+e^6+2 \left (30-e^3+e^6\right ) x-\left (3+12 e^3+4 e^6\right ) x^2+6 \left (3+4 e^3\right ) x^3-36 x^4\right )}{\left (5-\left (1+e^3\right ) x+\left (3+e^3\right ) x^2-3 x^3\right )^2 \left (e^8-5 e^{4 x}-3 e^5 x+e^{4 x} \left (1+e^3\right ) x-3 e^{4 x} \left (1+\frac {e^3}{3}\right ) x^2+3 e^{4 x} x^3\right )^2} \, dx \\ & = -\frac {4 x^2}{5-\left (1+e^3\right ) x+\left (3+e^3\right ) x^2-3 x^3}+\frac {4}{3} \int \frac {15-3 \left (3+e^3\right ) x^2+18 x^3}{\left (5-\left (1+e^3\right ) x+\left (3+e^3\right ) x^2-3 x^3\right )^2} \, dx+\left (4 e^5\right ) \int \frac {5 \left (19-2 e^3\right )+\left (21+e^3\right ) x-7 \left (5-3 e^3\right ) x^2-\left (57+4 e^3\right ) x^3+12 x^4}{\left (5-\left (1+e^3\right ) x+\left (3+e^3\right ) x^2-3 x^3\right )^2 \left (e^8-3 e^5 x-e^{3+4 x} (-1+x) x+e^{4 x} \left (-5+x-3 x^2+3 x^3\right )\right )} \, dx+\left (4 e^{10}\right ) \int \frac {(5-x) \left (-15-19 e^3+e^6+2 \left (30-e^3+e^6\right ) x-\left (3+12 e^3+4 e^6\right ) x^2+6 \left (3+4 e^3\right ) x^3-36 x^4\right )}{\left (5-\left (1+e^3\right ) x+\left (3+e^3\right ) x^2-3 x^3\right )^2 \left (e^8-3 e^5 x-e^{3+4 x} (-1+x) x+e^{4 x} \left (-5+x-3 x^2+3 x^3\right )\right )^2} \, dx \\ & = \frac {4 x}{5-\left (1+e^3\right ) x+\left (3+e^3\right ) x^2-3 x^3}-\frac {4 x^2}{5-\left (1+e^3\right ) x+\left (3+e^3\right ) x^2-3 x^3}+\left (4 e^5\right ) \int \left (\frac {2 \left (5 \left (2-e^3\right )+4 \left (7+2 e^3\right ) x-\left (42-e^3\right ) x^2\right )}{\left (5-\left (1+e^3\right ) x+\left (3+e^3\right ) x^2-3 x^3\right )^2 \left (e^8-5 e^{4 x}-3 e^5 x+e^{4 x} \left (1+e^3\right ) x-3 e^{4 x} \left (1+\frac {e^3}{3}\right ) x^2+3 e^{4 x} x^3\right )}+\frac {15-4 x}{\left (5-\left (1+e^3\right ) x+\left (3+e^3\right ) x^2-3 x^3\right ) \left (e^8-5 e^{4 x}-3 e^5 x+e^{4 x} \left (1+e^3\right ) x-3 e^{4 x} \left (1+\frac {e^3}{3}\right ) x^2+3 e^{4 x} x^3\right )}\right ) \, dx+\left (4 e^{10}\right ) \int \left (\frac {-5 \left (42+e^3-e^6\right )+\left (72-2 e^3-9 e^6\right ) x-3 \left (14-9 e^3\right ) x^2}{\left (5-\left (1+e^3\right ) x+\left (3+e^3\right ) x^2-3 x^3\right )^2 \left (e^8-5 e^{4 x}-3 e^5 x+e^{4 x} \left (1+e^3\right ) x-3 e^{4 x} \left (1+\frac {e^3}{3}\right ) x^2+3 e^{4 x} x^3\right )^2}+\frac {9 \left (3-2 e^3\right )+2 \left (27+2 e^3\right ) x-12 x^2}{\left (5-\left (1+e^3\right ) x+\left (3+e^3\right ) x^2-3 x^3\right ) \left (e^8-5 e^{4 x}-3 e^5 x+e^{4 x} \left (1+e^3\right ) x-3 e^{4 x} \left (1+\frac {e^3}{3}\right ) x^2+3 e^{4 x} x^3\right )^2}\right ) \, dx \\ & = \frac {4 x}{5-\left (1+e^3\right ) x+\left (3+e^3\right ) x^2-3 x^3}-\frac {4 x^2}{5-\left (1+e^3\right ) x+\left (3+e^3\right ) x^2-3 x^3}+\left (4 e^5\right ) \int \frac {15-4 x}{\left (5-\left (1+e^3\right ) x+\left (3+e^3\right ) x^2-3 x^3\right ) \left (e^8-5 e^{4 x}-3 e^5 x+e^{4 x} \left (1+e^3\right ) x-3 e^{4 x} \left (1+\frac {e^3}{3}\right ) x^2+3 e^{4 x} x^3\right )} \, dx+\left (8 e^5\right ) \int \frac {5 \left (2-e^3\right )+4 \left (7+2 e^3\right ) x-\left (42-e^3\right ) x^2}{\left (5-\left (1+e^3\right ) x+\left (3+e^3\right ) x^2-3 x^3\right )^2 \left (e^8-5 e^{4 x}-3 e^5 x+e^{4 x} \left (1+e^3\right ) x-3 e^{4 x} \left (1+\frac {e^3}{3}\right ) x^2+3 e^{4 x} x^3\right )} \, dx+\left (4 e^{10}\right ) \int \frac {-5 \left (42+e^3-e^6\right )+\left (72-2 e^3-9 e^6\right ) x-3 \left (14-9 e^3\right ) x^2}{\left (5-\left (1+e^3\right ) x+\left (3+e^3\right ) x^2-3 x^3\right )^2 \left (e^8-5 e^{4 x}-3 e^5 x+e^{4 x} \left (1+e^3\right ) x-3 e^{4 x} \left (1+\frac {e^3}{3}\right ) x^2+3 e^{4 x} x^3\right )^2} \, dx+\left (4 e^{10}\right ) \int \frac {9 \left (3-2 e^3\right )+2 \left (27+2 e^3\right ) x-12 x^2}{\left (5-\left (1+e^3\right ) x+\left (3+e^3\right ) x^2-3 x^3\right ) \left (e^8-5 e^{4 x}-3 e^5 x+e^{4 x} \left (1+e^3\right ) x-3 e^{4 x} \left (1+\frac {e^3}{3}\right ) x^2+3 e^{4 x} x^3\right )^2} \, dx \\ & = \frac {4 x}{5-\left (1+e^3\right ) x+\left (3+e^3\right ) x^2-3 x^3}-\frac {4 x^2}{5-\left (1+e^3\right ) x+\left (3+e^3\right ) x^2-3 x^3}+\left (4 e^5\right ) \int \frac {15-4 x}{\left (5-\left (1+e^3\right ) x+\left (3+e^3\right ) x^2-3 x^3\right ) \left (e^8-3 e^5 x-e^{3+4 x} (-1+x) x+e^{4 x} \left (-5+x-3 x^2+3 x^3\right )\right )} \, dx+\left (8 e^5\right ) \int \frac {5 \left (2-e^3\right )+4 \left (7+2 e^3\right ) x-\left (42-e^3\right ) x^2}{\left (5-\left (1+e^3\right ) x+\left (3+e^3\right ) x^2-3 x^3\right )^2 \left (e^8-3 e^5 x-e^{3+4 x} (-1+x) x+e^{4 x} \left (-5+x-3 x^2+3 x^3\right )\right )} \, dx+\left (4 e^{10}\right ) \int \frac {-5 \left (42+e^3-e^6\right )+\left (72-2 e^3-9 e^6\right ) x-3 \left (14-9 e^3\right ) x^2}{\left (5-\left (1+e^3\right ) x+\left (3+e^3\right ) x^2-3 x^3\right )^2 \left (e^8-3 e^5 x-e^{3+4 x} (-1+x) x+e^{4 x} \left (-5+x-3 x^2+3 x^3\right )\right )^2} \, dx+\left (4 e^{10}\right ) \int \frac {9 \left (3-2 e^3\right )+2 \left (27+2 e^3\right ) x-12 x^2}{\left (5-\left (1+e^3\right ) x+\left (3+e^3\right ) x^2-3 x^3\right ) \left (e^8-3 e^5 x-e^{3+4 x} (-1+x) x+e^{4 x} \left (-5+x-3 x^2+3 x^3\right )\right )^2} \, dx \\ & = \frac {4 x}{5-\left (1+e^3\right ) x+\left (3+e^3\right ) x^2-3 x^3}-\frac {4 x^2}{5-\left (1+e^3\right ) x+\left (3+e^3\right ) x^2-3 x^3}+\left (4 e^5\right ) \int \left (\frac {15}{\left (5-\left (1+e^3\right ) x+\left (3+e^3\right ) x^2-3 x^3\right ) \left (e^8-5 e^{4 x}-3 e^5 x+e^{4 x} \left (1+e^3\right ) x-3 e^{4 x} \left (1+\frac {e^3}{3}\right ) x^2+3 e^{4 x} x^3\right )}+\frac {4 x}{\left (-5+\left (1+e^3\right ) x-\left (3+e^3\right ) x^2+3 x^3\right ) \left (e^8-5 e^{4 x}-3 e^5 x+e^{4 x} \left (1+e^3\right ) x-3 e^{4 x} \left (1+\frac {e^3}{3}\right ) x^2+3 e^{4 x} x^3\right )}\right ) \, dx+\left (8 e^5\right ) \int \left (\frac {5 \left (2-e^3\right )}{\left (5-\left (1+e^3\right ) x+\left (3+e^3\right ) x^2-3 x^3\right )^2 \left (e^8-5 e^{4 x}-3 e^5 x+e^{4 x} \left (1+e^3\right ) x-3 e^{4 x} \left (1+\frac {e^3}{3}\right ) x^2+3 e^{4 x} x^3\right )}+\frac {4 \left (7+2 e^3\right ) x}{\left (5-\left (1+e^3\right ) x+\left (3+e^3\right ) x^2-3 x^3\right )^2 \left (e^8-5 e^{4 x}-3 e^5 x+e^{4 x} \left (1+e^3\right ) x-3 e^{4 x} \left (1+\frac {e^3}{3}\right ) x^2+3 e^{4 x} x^3\right )}+\frac {\left (-42+e^3\right ) x^2}{\left (5-\left (1+e^3\right ) x+\left (3+e^3\right ) x^2-3 x^3\right )^2 \left (e^8-5 e^{4 x}-3 e^5 x+e^{4 x} \left (1+e^3\right ) x-3 e^{4 x} \left (1+\frac {e^3}{3}\right ) x^2+3 e^{4 x} x^3\right )}\right ) \, dx+\left (4 e^{10}\right ) \int \left (\frac {5 \left (-42-e^3+e^6\right )}{\left (5-\left (1+e^3\right ) x+\left (3+e^3\right ) x^2-3 x^3\right )^2 \left (e^8-5 e^{4 x}-3 e^5 x+e^{4 x} \left (1+e^3\right ) x-3 e^{4 x} \left (1+\frac {e^3}{3}\right ) x^2+3 e^{4 x} x^3\right )^2}+\frac {\left (72-2 e^3-9 e^6\right ) x}{\left (5-\left (1+e^3\right ) x+\left (3+e^3\right ) x^2-3 x^3\right )^2 \left (e^8-5 e^{4 x}-3 e^5 x+e^{4 x} \left (1+e^3\right ) x-3 e^{4 x} \left (1+\frac {e^3}{3}\right ) x^2+3 e^{4 x} x^3\right )^2}+\frac {3 \left (-14+9 e^3\right ) x^2}{\left (5-\left (1+e^3\right ) x+\left (3+e^3\right ) x^2-3 x^3\right )^2 \left (e^8-5 e^{4 x}-3 e^5 x+e^{4 x} \left (1+e^3\right ) x-3 e^{4 x} \left (1+\frac {e^3}{3}\right ) x^2+3 e^{4 x} x^3\right )^2}\right ) \, dx+\left (4 e^{10}\right ) \int \left (\frac {9 \left (3-2 e^3\right )}{\left (5-\left (1+e^3\right ) x+\left (3+e^3\right ) x^2-3 x^3\right ) \left (e^8-5 e^{4 x}-3 e^5 x+e^{4 x} \left (1+e^3\right ) x-3 e^{4 x} \left (1+\frac {e^3}{3}\right ) x^2+3 e^{4 x} x^3\right )^2}+\frac {2 \left (27+2 e^3\right ) x}{\left (5-\left (1+e^3\right ) x+\left (3+e^3\right ) x^2-3 x^3\right ) \left (e^8-5 e^{4 x}-3 e^5 x+e^{4 x} \left (1+e^3\right ) x-3 e^{4 x} \left (1+\frac {e^3}{3}\right ) x^2+3 e^{4 x} x^3\right )^2}+\frac {12 x^2}{\left (-5+\left (1+e^3\right ) x-\left (3+e^3\right ) x^2+3 x^3\right ) \left (e^8-5 e^{4 x}-3 e^5 x+e^{4 x} \left (1+e^3\right ) x-3 e^{4 x} \left (1+\frac {e^3}{3}\right ) x^2+3 e^{4 x} x^3\right )^2}\right ) \, dx \\ & = \frac {4 x}{5-\left (1+e^3\right ) x+\left (3+e^3\right ) x^2-3 x^3}-\frac {4 x^2}{5-\left (1+e^3\right ) x+\left (3+e^3\right ) x^2-3 x^3}+\left (16 e^5\right ) \int \frac {x}{\left (-5+\left (1+e^3\right ) x-\left (3+e^3\right ) x^2+3 x^3\right ) \left (e^8-5 e^{4 x}-3 e^5 x+e^{4 x} \left (1+e^3\right ) x-3 e^{4 x} \left (1+\frac {e^3}{3}\right ) x^2+3 e^{4 x} x^3\right )} \, dx+\left (60 e^5\right ) \int \frac {1}{\left (5-\left (1+e^3\right ) x+\left (3+e^3\right ) x^2-3 x^3\right ) \left (e^8-5 e^{4 x}-3 e^5 x+e^{4 x} \left (1+e^3\right ) x-3 e^{4 x} \left (1+\frac {e^3}{3}\right ) x^2+3 e^{4 x} x^3\right )} \, dx+\left (48 e^{10}\right ) \int \frac {x^2}{\left (-5+\left (1+e^3\right ) x-\left (3+e^3\right ) x^2+3 x^3\right ) \left (e^8-5 e^{4 x}-3 e^5 x+e^{4 x} \left (1+e^3\right ) x-3 e^{4 x} \left (1+\frac {e^3}{3}\right ) x^2+3 e^{4 x} x^3\right )^2} \, dx-\left (12 e^{10} \left (14-9 e^3\right )\right ) \int \frac {x^2}{\left (5-\left (1+e^3\right ) x+\left (3+e^3\right ) x^2-3 x^3\right )^2 \left (e^8-5 e^{4 x}-3 e^5 x+e^{4 x} \left (1+e^3\right ) x-3 e^{4 x} \left (1+\frac {e^3}{3}\right ) x^2+3 e^{4 x} x^3\right )^2} \, dx+\left (36 e^{10} \left (3-2 e^3\right )\right ) \int \frac {1}{\left (5-\left (1+e^3\right ) x+\left (3+e^3\right ) x^2-3 x^3\right ) \left (e^8-5 e^{4 x}-3 e^5 x+e^{4 x} \left (1+e^3\right ) x-3 e^{4 x} \left (1+\frac {e^3}{3}\right ) x^2+3 e^{4 x} x^3\right )^2} \, dx+\left (40 e^5 \left (2-e^3\right )\right ) \int \frac {1}{\left (5-\left (1+e^3\right ) x+\left (3+e^3\right ) x^2-3 x^3\right )^2 \left (e^8-5 e^{4 x}-3 e^5 x+e^{4 x} \left (1+e^3\right ) x-3 e^{4 x} \left (1+\frac {e^3}{3}\right ) x^2+3 e^{4 x} x^3\right )} \, dx-\left (8 e^5 \left (42-e^3\right )\right ) \int \frac {x^2}{\left (5-\left (1+e^3\right ) x+\left (3+e^3\right ) x^2-3 x^3\right )^2 \left (e^8-5 e^{4 x}-3 e^5 x+e^{4 x} \left (1+e^3\right ) x-3 e^{4 x} \left (1+\frac {e^3}{3}\right ) x^2+3 e^{4 x} x^3\right )} \, dx+\left (32 e^5 \left (7+2 e^3\right )\right ) \int \frac {x}{\left (5-\left (1+e^3\right ) x+\left (3+e^3\right ) x^2-3 x^3\right )^2 \left (e^8-5 e^{4 x}-3 e^5 x+e^{4 x} \left (1+e^3\right ) x-3 e^{4 x} \left (1+\frac {e^3}{3}\right ) x^2+3 e^{4 x} x^3\right )} \, dx+\left (8 e^{10} \left (27+2 e^3\right )\right ) \int \frac {x}{\left (5-\left (1+e^3\right ) x+\left (3+e^3\right ) x^2-3 x^3\right ) \left (e^8-5 e^{4 x}-3 e^5 x+e^{4 x} \left (1+e^3\right ) x-3 e^{4 x} \left (1+\frac {e^3}{3}\right ) x^2+3 e^{4 x} x^3\right )^2} \, dx+\left (4 e^{10} \left (72-2 e^3-9 e^6\right )\right ) \int \frac {x}{\left (5-\left (1+e^3\right ) x+\left (3+e^3\right ) x^2-3 x^3\right )^2 \left (e^8-5 e^{4 x}-3 e^5 x+e^{4 x} \left (1+e^3\right ) x-3 e^{4 x} \left (1+\frac {e^3}{3}\right ) x^2+3 e^{4 x} x^3\right )^2} \, dx-\left (20 e^{10} \left (42+e^3-e^6\right )\right ) \int \frac {1}{\left (5-\left (1+e^3\right ) x+\left (3+e^3\right ) x^2-3 x^3\right )^2 \left (e^8-5 e^{4 x}-3 e^5 x+e^{4 x} \left (1+e^3\right ) x-3 e^{4 x} \left (1+\frac {e^3}{3}\right ) x^2+3 e^{4 x} x^3\right )^2} \, dx \\ & = \frac {4 x}{5-\left (1+e^3\right ) x+\left (3+e^3\right ) x^2-3 x^3}-\frac {4 x^2}{5-\left (1+e^3\right ) x+\left (3+e^3\right ) x^2-3 x^3}+\left (16 e^5\right ) \int \frac {x}{\left (-5+\left (1+e^3\right ) x-\left (3+e^3\right ) x^2+3 x^3\right ) \left (e^8-3 e^5 x-e^{3+4 x} (-1+x) x+e^{4 x} \left (-5+x-3 x^2+3 x^3\right )\right )} \, dx+\left (60 e^5\right ) \int \frac {1}{\left (5-\left (1+e^3\right ) x+\left (3+e^3\right ) x^2-3 x^3\right ) \left (e^8-3 e^5 x-e^{3+4 x} (-1+x) x+e^{4 x} \left (-5+x-3 x^2+3 x^3\right )\right )} \, dx+\left (48 e^{10}\right ) \int \frac {x^2}{\left (-5+\left (1+e^3\right ) x-\left (3+e^3\right ) x^2+3 x^3\right ) \left (e^8-3 e^5 x-e^{3+4 x} (-1+x) x+e^{4 x} \left (-5+x-3 x^2+3 x^3\right )\right )^2} \, dx-\left (12 e^{10} \left (14-9 e^3\right )\right ) \int \frac {x^2}{\left (5-\left (1+e^3\right ) x+\left (3+e^3\right ) x^2-3 x^3\right )^2 \left (e^8-3 e^5 x-e^{3+4 x} (-1+x) x+e^{4 x} \left (-5+x-3 x^2+3 x^3\right )\right )^2} \, dx+\left (36 e^{10} \left (3-2 e^3\right )\right ) \int \frac {1}{\left (5-\left (1+e^3\right ) x+\left (3+e^3\right ) x^2-3 x^3\right ) \left (e^8-3 e^5 x-e^{3+4 x} (-1+x) x+e^{4 x} \left (-5+x-3 x^2+3 x^3\right )\right )^2} \, dx+\left (40 e^5 \left (2-e^3\right )\right ) \int \frac {1}{\left (5-\left (1+e^3\right ) x+\left (3+e^3\right ) x^2-3 x^3\right )^2 \left (e^8-3 e^5 x-e^{3+4 x} (-1+x) x+e^{4 x} \left (-5+x-3 x^2+3 x^3\right )\right )} \, dx-\left (8 e^5 \left (42-e^3\right )\right ) \int \frac {x^2}{\left (5-\left (1+e^3\right ) x+\left (3+e^3\right ) x^2-3 x^3\right )^2 \left (e^8-3 e^5 x-e^{3+4 x} (-1+x) x+e^{4 x} \left (-5+x-3 x^2+3 x^3\right )\right )} \, dx+\left (32 e^5 \left (7+2 e^3\right )\right ) \int \frac {x}{\left (5-\left (1+e^3\right ) x+\left (3+e^3\right ) x^2-3 x^3\right )^2 \left (e^8-3 e^5 x-e^{3+4 x} (-1+x) x+e^{4 x} \left (-5+x-3 x^2+3 x^3\right )\right )} \, dx+\left (8 e^{10} \left (27+2 e^3\right )\right ) \int \frac {x}{\left (5-\left (1+e^3\right ) x+\left (3+e^3\right ) x^2-3 x^3\right ) \left (e^8-3 e^5 x-e^{3+4 x} (-1+x) x+e^{4 x} \left (-5+x-3 x^2+3 x^3\right )\right )^2} \, dx+\left (4 e^{10} \left (72-2 e^3-9 e^6\right )\right ) \int \frac {x}{\left (5-\left (1+e^3\right ) x+\left (3+e^3\right ) x^2-3 x^3\right )^2 \left (e^8-3 e^5 x-e^{3+4 x} (-1+x) x+e^{4 x} \left (-5+x-3 x^2+3 x^3\right )\right )^2} \, dx-\left (20 e^{10} \left (42+e^3-e^6\right )\right ) \int \frac {1}{\left (5-\left (1+e^3\right ) x+\left (3+e^3\right ) x^2-3 x^3\right )^2 \left (e^8-3 e^5 x-e^{3+4 x} (-1+x) x+e^{4 x} \left (-5+x-3 x^2+3 x^3\right )\right )^2} \, dx \\ \end{align*}
Time = 10.13 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.61 \[ \int \frac {20-12 e^{10-8 x}-40 x-8 x^2+24 x^3-12 x^4+e^{5-4 x} \left (-76-8 x+24 x^2\right )}{25-10 x+31 x^2-36 x^3+15 x^4-18 x^5+9 x^6+e^{10-8 x} \left (e^6-6 e^3 x+9 x^2\right )+e^6 \left (x^2-2 x^3+x^4\right )+e^3 \left (-10 x+12 x^2-8 x^3+12 x^4-6 x^5\right )+e^{5-4 x} \left (30 x-6 x^2+18 x^3-18 x^4+e^6 \left (2 x-2 x^2\right )+e^3 \left (-10+2 x-12 x^2+12 x^3\right )\right )} \, dx=-\frac {4 \left (e^5-e^{4 x} (-1+x) x\right )}{e^8-3 e^5 x-e^{3+4 x} (-1+x) x+e^{4 x} \left (-5+x-3 x^2+3 x^3\right )} \]
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Time = 1.39 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.74
method | result | size |
norman | \(\frac {-4 x^{2}+4 x +4 \,{\mathrm e}^{-4 x +5}}{x^{2} {\mathrm e}^{3}-3 x^{3}-x \,{\mathrm e}^{3}-{\mathrm e}^{3} {\mathrm e}^{-4 x +5}+3 x^{2}+3 \,{\mathrm e}^{-4 x +5} x -x +5}\) | \(66\) |
parallelrisch | \(\frac {-12 x^{2}+12 x +12 \,{\mathrm e}^{-4 x +5}}{3 x^{2} {\mathrm e}^{3}-9 x^{3}-3 x \,{\mathrm e}^{3}-3 \,{\mathrm e}^{3} {\mathrm e}^{-4 x +5}+9 x^{2}+9 \,{\mathrm e}^{-4 x +5} x -3 x +15}\) | \(67\) |
risch | \(-\frac {4}{{\mathrm e}^{3}-3 x}-\frac {4 \left (-5+x \right )}{\left ({\mathrm e}^{3}-3 x \right ) \left (x^{2} {\mathrm e}^{3}-3 x^{3}-x \,{\mathrm e}^{3}-{\mathrm e}^{-4 x +8}+3 x^{2}+3 \,{\mathrm e}^{-4 x +5} x -x +5\right )}\) | \(70\) |
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Time = 0.28 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.55 \[ \int \frac {20-12 e^{10-8 x}-40 x-8 x^2+24 x^3-12 x^4+e^{5-4 x} \left (-76-8 x+24 x^2\right )}{25-10 x+31 x^2-36 x^3+15 x^4-18 x^5+9 x^6+e^{10-8 x} \left (e^6-6 e^3 x+9 x^2\right )+e^6 \left (x^2-2 x^3+x^4\right )+e^3 \left (-10 x+12 x^2-8 x^3+12 x^4-6 x^5\right )+e^{5-4 x} \left (30 x-6 x^2+18 x^3-18 x^4+e^6 \left (2 x-2 x^2\right )+e^3 \left (-10+2 x-12 x^2+12 x^3\right )\right )} \, dx=\frac {4 \, {\left (x^{2} - x - e^{\left (-4 \, x + 5\right )}\right )}}{3 \, x^{3} - 3 \, x^{2} - {\left (x^{2} - x\right )} e^{3} - {\left (3 \, x - e^{3}\right )} e^{\left (-4 \, x + 5\right )} + x - 5} \]
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Leaf count of result is larger than twice the leaf count of optimal. 94 vs. \(2 (22) = 44\).
Time = 0.34 (sec) , antiderivative size = 94, normalized size of antiderivative = 2.47 \[ \int \frac {20-12 e^{10-8 x}-40 x-8 x^2+24 x^3-12 x^4+e^{5-4 x} \left (-76-8 x+24 x^2\right )}{25-10 x+31 x^2-36 x^3+15 x^4-18 x^5+9 x^6+e^{10-8 x} \left (e^6-6 e^3 x+9 x^2\right )+e^6 \left (x^2-2 x^3+x^4\right )+e^3 \left (-10 x+12 x^2-8 x^3+12 x^4-6 x^5\right )+e^{5-4 x} \left (30 x-6 x^2+18 x^3-18 x^4+e^6 \left (2 x-2 x^2\right )+e^3 \left (-10+2 x-12 x^2+12 x^3\right )\right )} \, dx=\frac {4 x - 20}{- 9 x^{4} + 9 x^{3} + 6 x^{3} e^{3} - x^{2} e^{6} - 6 x^{2} e^{3} - 3 x^{2} + 15 x + x e^{3} + x e^{6} + \left (9 x^{2} - 6 x e^{3} + e^{6}\right ) e^{5 - 4 x} - 5 e^{3}} + \frac {12}{9 x - 3 e^{3}} \]
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Time = 0.40 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.55 \[ \int \frac {20-12 e^{10-8 x}-40 x-8 x^2+24 x^3-12 x^4+e^{5-4 x} \left (-76-8 x+24 x^2\right )}{25-10 x+31 x^2-36 x^3+15 x^4-18 x^5+9 x^6+e^{10-8 x} \left (e^6-6 e^3 x+9 x^2\right )+e^6 \left (x^2-2 x^3+x^4\right )+e^3 \left (-10 x+12 x^2-8 x^3+12 x^4-6 x^5\right )+e^{5-4 x} \left (30 x-6 x^2+18 x^3-18 x^4+e^6 \left (2 x-2 x^2\right )+e^3 \left (-10+2 x-12 x^2+12 x^3\right )\right )} \, dx=-\frac {4 \, {\left ({\left (x^{2} - x\right )} e^{\left (4 \, x\right )} - e^{5}\right )}}{3 \, x e^{5} - {\left (3 \, x^{3} - x^{2} {\left (e^{3} + 3\right )} + x {\left (e^{3} + 1\right )} - 5\right )} e^{\left (4 \, x\right )} - e^{8}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 100 vs. \(2 (33) = 66\).
Time = 0.36 (sec) , antiderivative size = 100, normalized size of antiderivative = 2.63 \[ \int \frac {20-12 e^{10-8 x}-40 x-8 x^2+24 x^3-12 x^4+e^{5-4 x} \left (-76-8 x+24 x^2\right )}{25-10 x+31 x^2-36 x^3+15 x^4-18 x^5+9 x^6+e^{10-8 x} \left (e^6-6 e^3 x+9 x^2\right )+e^6 \left (x^2-2 x^3+x^4\right )+e^3 \left (-10 x+12 x^2-8 x^3+12 x^4-6 x^5\right )+e^{5-4 x} \left (30 x-6 x^2+18 x^3-18 x^4+e^6 \left (2 x-2 x^2\right )+e^3 \left (-10+2 x-12 x^2+12 x^3\right )\right )} \, dx=\frac {16 \, {\left ({\left (4 \, x - 5\right )}^{2} + 24 \, x - 16 \, e^{\left (-4 \, x + 5\right )} - 25\right )}}{3 \, {\left (4 \, x - 5\right )}^{3} - 4 \, {\left (4 \, x - 5\right )}^{2} e^{3} + 33 \, {\left (4 \, x - 5\right )}^{2} - 24 \, {\left (4 \, x - 5\right )} e^{3} - 48 \, {\left (4 \, x - 5\right )} e^{\left (-4 \, x + 5\right )} + 484 \, x - 20 \, e^{3} + 64 \, e^{\left (-4 \, x + 8\right )} - 240 \, e^{\left (-4 \, x + 5\right )} - 770} \]
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Timed out. \[ \int \frac {20-12 e^{10-8 x}-40 x-8 x^2+24 x^3-12 x^4+e^{5-4 x} \left (-76-8 x+24 x^2\right )}{25-10 x+31 x^2-36 x^3+15 x^4-18 x^5+9 x^6+e^{10-8 x} \left (e^6-6 e^3 x+9 x^2\right )+e^6 \left (x^2-2 x^3+x^4\right )+e^3 \left (-10 x+12 x^2-8 x^3+12 x^4-6 x^5\right )+e^{5-4 x} \left (30 x-6 x^2+18 x^3-18 x^4+e^6 \left (2 x-2 x^2\right )+e^3 \left (-10+2 x-12 x^2+12 x^3\right )\right )} \, dx=\int -\frac {40\,x+12\,{\mathrm {e}}^{10-8\,x}+{\mathrm {e}}^{5-4\,x}\,\left (-24\,x^2+8\,x+76\right )+8\,x^2-24\,x^3+12\,x^4-20}{{\mathrm {e}}^6\,\left (x^4-2\,x^3+x^2\right )-10\,x+{\mathrm {e}}^{10-8\,x}\,\left (9\,x^2-6\,{\mathrm {e}}^3\,x+{\mathrm {e}}^6\right )+{\mathrm {e}}^{5-4\,x}\,\left (30\,x+{\mathrm {e}}^6\,\left (2\,x-2\,x^2\right )+{\mathrm {e}}^3\,\left (12\,x^3-12\,x^2+2\,x-10\right )-6\,x^2+18\,x^3-18\,x^4\right )-{\mathrm {e}}^3\,\left (6\,x^5-12\,x^4+8\,x^3-12\,x^2+10\,x\right )+31\,x^2-36\,x^3+15\,x^4-18\,x^5+9\,x^6+25} \,d x \]
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