Integrand size = 197, antiderivative size = 31 \[ \int \frac {-5 x^2-20 x^5-15 x^4 \log ^2(3)+e^{\frac {16-8 x+2 x^2}{x^2}} \left (160 x-40 x^2-10 x^3+\left (160-40 x-5 x^2\right ) \log ^2(3)\right )}{x^4+2 x^7+x^{10}+\left (2 x^6+2 x^9\right ) \log ^2(3)+x^8 \log ^4(3)+e^{\frac {2 \left (16-8 x+2 x^2\right )}{x^2}} \left (x^6+2 x^5 \log ^2(3)+x^4 \log ^4(3)\right )+e^{\frac {16-8 x+2 x^2}{x^2}} \left (2 x^5+2 x^8+\left (2 x^4+4 x^7\right ) \log ^2(3)+2 x^6 \log ^4(3)\right )} \, dx=\frac {5}{x+x \left (e^{1+\frac {(-4+x)^2}{x^2}}+x^2\right ) \left (x+\log ^2(3)\right )} \]
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\[ \int \frac {-5 x^2-20 x^5-15 x^4 \log ^2(3)+e^{\frac {16-8 x+2 x^2}{x^2}} \left (160 x-40 x^2-10 x^3+\left (160-40 x-5 x^2\right ) \log ^2(3)\right )}{x^4+2 x^7+x^{10}+\left (2 x^6+2 x^9\right ) \log ^2(3)+x^8 \log ^4(3)+e^{\frac {2 \left (16-8 x+2 x^2\right )}{x^2}} \left (x^6+2 x^5 \log ^2(3)+x^4 \log ^4(3)\right )+e^{\frac {16-8 x+2 x^2}{x^2}} \left (2 x^5+2 x^8+\left (2 x^4+4 x^7\right ) \log ^2(3)+2 x^6 \log ^4(3)\right )} \, dx=\int \frac {-5 x^2-20 x^5-15 x^4 \log ^2(3)+e^{\frac {16-8 x+2 x^2}{x^2}} \left (160 x-40 x^2-10 x^3+\left (160-40 x-5 x^2\right ) \log ^2(3)\right )}{x^4+2 x^7+x^{10}+\left (2 x^6+2 x^9\right ) \log ^2(3)+x^8 \log ^4(3)+e^{\frac {2 \left (16-8 x+2 x^2\right )}{x^2}} \left (x^6+2 x^5 \log ^2(3)+x^4 \log ^4(3)\right )+e^{\frac {16-8 x+2 x^2}{x^2}} \left (2 x^5+2 x^8+\left (2 x^4+4 x^7\right ) \log ^2(3)+2 x^6 \log ^4(3)\right )} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {5 e^{8/x} \left (-e^{8/x} \left (x^2+4 x^5+3 x^4 \log ^2(3)\right )-e^{2+\frac {16}{x^2}} \left (2 x^3-32 \log ^2(3)+8 x \left (-4+\log ^2(3)\right )+x^2 \left (8+\log ^2(3)\right )\right )\right )}{x^4 \left (e^{2+\frac {16}{x^2}} \left (x+\log ^2(3)\right )+e^{8/x} \left (1+x^3+x^2 \log ^2(3)\right )\right )^2} \, dx \\ & = 5 \int \frac {e^{8/x} \left (-e^{8/x} \left (x^2+4 x^5+3 x^4 \log ^2(3)\right )-e^{2+\frac {16}{x^2}} \left (2 x^3-32 \log ^2(3)+8 x \left (-4+\log ^2(3)\right )+x^2 \left (8+\log ^2(3)\right )\right )\right )}{x^4 \left (e^{2+\frac {16}{x^2}} \left (x+\log ^2(3)\right )+e^{8/x} \left (1+x^3+x^2 \log ^2(3)\right )\right )^2} \, dx \\ & = 5 \int \left (-\frac {e^{8/x} \left (1+4 x^3+3 x^2 \log ^2(3)\right )}{x^2 \left (1+x^3+x^2 \log ^2(3)\right ) \left (e^{8/x}+e^{2+\frac {16}{x^2}} x+e^{8/x} x^3+e^{2+\frac {16}{x^2}} \log ^2(3)+e^{8/x} x^2 \log ^2(3)\right )}+\frac {e^{2+\frac {16}{x^2}+\frac {8}{x}} \left (2 x^6+32 \log ^2(3)-8 x^5 \left (1-\frac {\log ^2(3)}{2}\right )+32 x \left (1-\frac {\log ^2(3)}{4}\right )-8 x^2 \left (1-4 \log ^4(3)\right )+32 x^4 \left (1+\frac {1}{16} \log ^2(3) \left (-8+\log ^2(3)\right )\right )-x^3 \left (1+8 \log ^2(3) \left (-8+\log ^2(3)\right )\right )\right )}{x^4 \left (1+x^3+x^2 \log ^2(3)\right ) \left (e^{8/x}+e^{2+\frac {16}{x^2}} x+e^{8/x} x^3+e^{2+\frac {16}{x^2}} \log ^2(3)+e^{8/x} x^2 \log ^2(3)\right )^2}\right ) \, dx \\ & = -\left (5 \int \frac {e^{8/x} \left (1+4 x^3+3 x^2 \log ^2(3)\right )}{x^2 \left (1+x^3+x^2 \log ^2(3)\right ) \left (e^{8/x}+e^{2+\frac {16}{x^2}} x+e^{8/x} x^3+e^{2+\frac {16}{x^2}} \log ^2(3)+e^{8/x} x^2 \log ^2(3)\right )} \, dx\right )+5 \int \frac {e^{2+\frac {16}{x^2}+\frac {8}{x}} \left (2 x^6+32 \log ^2(3)-8 x^5 \left (1-\frac {\log ^2(3)}{2}\right )+32 x \left (1-\frac {\log ^2(3)}{4}\right )-8 x^2 \left (1-4 \log ^4(3)\right )+32 x^4 \left (1+\frac {1}{16} \log ^2(3) \left (-8+\log ^2(3)\right )\right )-x^3 \left (1+8 \log ^2(3) \left (-8+\log ^2(3)\right )\right )\right )}{x^4 \left (1+x^3+x^2 \log ^2(3)\right ) \left (e^{8/x}+e^{2+\frac {16}{x^2}} x+e^{8/x} x^3+e^{2+\frac {16}{x^2}} \log ^2(3)+e^{8/x} x^2 \log ^2(3)\right )^2} \, dx \\ & = -\left (5 \int \frac {e^{8/x} \left (1+4 x^3+3 x^2 \log ^2(3)\right )}{x^2 \left (1+x^3+x^2 \log ^2(3)\right ) \left (e^{2+\frac {16}{x^2}} \left (x+\log ^2(3)\right )+e^{8/x} \left (1+x^3+x^2 \log ^2(3)\right )\right )} \, dx\right )+5 \int \left (-\frac {8 e^{2+\frac {16}{x^2}+\frac {8}{x}}}{x^2 \left (e^{8/x}+e^{2+\frac {16}{x^2}} x+e^{8/x} x^3+e^{2+\frac {16}{x^2}} \log ^2(3)+e^{8/x} x^2 \log ^2(3)\right )^2}-\frac {e^{2+\frac {16}{x^2}+\frac {8}{x}}}{x \left (e^{8/x}+e^{2+\frac {16}{x^2}} x+e^{8/x} x^3+e^{2+\frac {16}{x^2}} \log ^2(3)+e^{8/x} x^2 \log ^2(3)\right )^2}+\frac {32 e^{2+\frac {16}{x^2}+\frac {8}{x}} \log ^2(3)}{x^4 \left (e^{8/x}+e^{2+\frac {16}{x^2}} x+e^{8/x} x^3+e^{2+\frac {16}{x^2}} \log ^2(3)+e^{8/x} x^2 \log ^2(3)\right )^2}-\frac {8 e^{2+\frac {16}{x^2}+\frac {8}{x}} \left (-4+\log ^2(3)\right )}{x^3 \left (e^{8/x}+e^{2+\frac {16}{x^2}} x+e^{8/x} x^3+e^{2+\frac {16}{x^2}} \log ^2(3)+e^{8/x} x^2 \log ^2(3)\right )^2}+\frac {e^{2+\frac {16}{x^2}+\frac {8}{x}} \left (3 x^2+5 x \log ^2(3)+2 \log ^4(3)\right )}{\left (1+x^3+x^2 \log ^2(3)\right ) \left (e^{8/x}+e^{2+\frac {16}{x^2}} x+e^{8/x} x^3+e^{2+\frac {16}{x^2}} \log ^2(3)+e^{8/x} x^2 \log ^2(3)\right )^2}\right ) \, dx \\ & = -\left (5 \int \frac {e^{2+\frac {16}{x^2}+\frac {8}{x}}}{x \left (e^{8/x}+e^{2+\frac {16}{x^2}} x+e^{8/x} x^3+e^{2+\frac {16}{x^2}} \log ^2(3)+e^{8/x} x^2 \log ^2(3)\right )^2} \, dx\right )+5 \int \frac {e^{2+\frac {16}{x^2}+\frac {8}{x}} \left (3 x^2+5 x \log ^2(3)+2 \log ^4(3)\right )}{\left (1+x^3+x^2 \log ^2(3)\right ) \left (e^{8/x}+e^{2+\frac {16}{x^2}} x+e^{8/x} x^3+e^{2+\frac {16}{x^2}} \log ^2(3)+e^{8/x} x^2 \log ^2(3)\right )^2} \, dx-5 \int \left (\frac {e^{8/x}}{x^2 \left (e^{8/x}+e^{2+\frac {16}{x^2}} x+e^{8/x} x^3+e^{2+\frac {16}{x^2}} \log ^2(3)+e^{8/x} x^2 \log ^2(3)\right )}+\frac {e^{8/x} \left (3 x+2 \log ^2(3)\right )}{\left (1+x^3+x^2 \log ^2(3)\right ) \left (e^{8/x}+e^{2+\frac {16}{x^2}} x+e^{8/x} x^3+e^{2+\frac {16}{x^2}} \log ^2(3)+e^{8/x} x^2 \log ^2(3)\right )}\right ) \, dx-40 \int \frac {e^{2+\frac {16}{x^2}+\frac {8}{x}}}{x^2 \left (e^{8/x}+e^{2+\frac {16}{x^2}} x+e^{8/x} x^3+e^{2+\frac {16}{x^2}} \log ^2(3)+e^{8/x} x^2 \log ^2(3)\right )^2} \, dx+\left (160 \log ^2(3)\right ) \int \frac {e^{2+\frac {16}{x^2}+\frac {8}{x}}}{x^4 \left (e^{8/x}+e^{2+\frac {16}{x^2}} x+e^{8/x} x^3+e^{2+\frac {16}{x^2}} \log ^2(3)+e^{8/x} x^2 \log ^2(3)\right )^2} \, dx+\left (40 \left (4-\log ^2(3)\right )\right ) \int \frac {e^{2+\frac {16}{x^2}+\frac {8}{x}}}{x^3 \left (e^{8/x}+e^{2+\frac {16}{x^2}} x+e^{8/x} x^3+e^{2+\frac {16}{x^2}} \log ^2(3)+e^{8/x} x^2 \log ^2(3)\right )^2} \, dx \\ & = -\left (5 \int \frac {e^{8/x}}{x^2 \left (e^{8/x}+e^{2+\frac {16}{x^2}} x+e^{8/x} x^3+e^{2+\frac {16}{x^2}} \log ^2(3)+e^{8/x} x^2 \log ^2(3)\right )} \, dx\right )-5 \int \frac {e^{8/x} \left (3 x+2 \log ^2(3)\right )}{\left (1+x^3+x^2 \log ^2(3)\right ) \left (e^{8/x}+e^{2+\frac {16}{x^2}} x+e^{8/x} x^3+e^{2+\frac {16}{x^2}} \log ^2(3)+e^{8/x} x^2 \log ^2(3)\right )} \, dx-5 \int \frac {e^{\frac {2 \left (8+4 x+x^2\right )}{x^2}}}{x \left (e^{2+\frac {16}{x^2}} \left (x+\log ^2(3)\right )+e^{8/x} \left (1+x^3+x^2 \log ^2(3)\right )\right )^2} \, dx+5 \int \frac {e^{\frac {2 \left (8+4 x+x^2\right )}{x^2}} \left (3 x^2+5 x \log ^2(3)+2 \log ^4(3)\right )}{\left (1+x^3+x^2 \log ^2(3)\right ) \left (e^{2+\frac {16}{x^2}} \left (x+\log ^2(3)\right )+e^{8/x} \left (1+x^3+x^2 \log ^2(3)\right )\right )^2} \, dx-40 \int \frac {e^{\frac {2 \left (8+4 x+x^2\right )}{x^2}}}{x^2 \left (e^{2+\frac {16}{x^2}} \left (x+\log ^2(3)\right )+e^{8/x} \left (1+x^3+x^2 \log ^2(3)\right )\right )^2} \, dx+\left (160 \log ^2(3)\right ) \int \frac {e^{\frac {2 \left (8+4 x+x^2\right )}{x^2}}}{x^4 \left (e^{2+\frac {16}{x^2}} \left (x+\log ^2(3)\right )+e^{8/x} \left (1+x^3+x^2 \log ^2(3)\right )\right )^2} \, dx+\left (40 \left (4-\log ^2(3)\right )\right ) \int \frac {e^{\frac {2 \left (8+4 x+x^2\right )}{x^2}}}{x^3 \left (e^{2+\frac {16}{x^2}} \left (x+\log ^2(3)\right )+e^{8/x} \left (1+x^3+x^2 \log ^2(3)\right )\right )^2} \, dx \\ & = -\left (5 \int \frac {e^{\frac {2 \left (8+4 x+x^2\right )}{x^2}}}{x \left (e^{2+\frac {16}{x^2}} \left (x+\log ^2(3)\right )+e^{8/x} \left (1+x^3+x^2 \log ^2(3)\right )\right )^2} \, dx\right )-5 \int \frac {e^{8/x}}{x^2 \left (e^{2+\frac {16}{x^2}} \left (x+\log ^2(3)\right )+e^{8/x} \left (1+x^3+x^2 \log ^2(3)\right )\right )} \, dx-5 \int \frac {e^{8/x} \left (3 x+2 \log ^2(3)\right )}{\left (1+x^3+x^2 \log ^2(3)\right ) \left (e^{2+\frac {16}{x^2}} \left (x+\log ^2(3)\right )+e^{8/x} \left (1+x^3+x^2 \log ^2(3)\right )\right )} \, dx+5 \int \left (\frac {3 e^{\frac {2 \left (8+4 x+x^2\right )}{x^2}} x^2}{\left (1+x^3+x^2 \log ^2(3)\right ) \left (e^{8/x}+e^{2+\frac {16}{x^2}} x+e^{8/x} x^3+e^{2+\frac {16}{x^2}} \log ^2(3)+e^{8/x} x^2 \log ^2(3)\right )^2}+\frac {5 e^{\frac {2 \left (8+4 x+x^2\right )}{x^2}} x \log ^2(3)}{\left (1+x^3+x^2 \log ^2(3)\right ) \left (e^{8/x}+e^{2+\frac {16}{x^2}} x+e^{8/x} x^3+e^{2+\frac {16}{x^2}} \log ^2(3)+e^{8/x} x^2 \log ^2(3)\right )^2}+\frac {2 e^{\frac {2 \left (8+4 x+x^2\right )}{x^2}} \log ^4(3)}{\left (1+x^3+x^2 \log ^2(3)\right ) \left (e^{8/x}+e^{2+\frac {16}{x^2}} x+e^{8/x} x^3+e^{2+\frac {16}{x^2}} \log ^2(3)+e^{8/x} x^2 \log ^2(3)\right )^2}\right ) \, dx-40 \int \frac {e^{\frac {2 \left (8+4 x+x^2\right )}{x^2}}}{x^2 \left (e^{2+\frac {16}{x^2}} \left (x+\log ^2(3)\right )+e^{8/x} \left (1+x^3+x^2 \log ^2(3)\right )\right )^2} \, dx+\left (160 \log ^2(3)\right ) \int \frac {e^{\frac {2 \left (8+4 x+x^2\right )}{x^2}}}{x^4 \left (e^{2+\frac {16}{x^2}} \left (x+\log ^2(3)\right )+e^{8/x} \left (1+x^3+x^2 \log ^2(3)\right )\right )^2} \, dx+\left (40 \left (4-\log ^2(3)\right )\right ) \int \frac {e^{\frac {2 \left (8+4 x+x^2\right )}{x^2}}}{x^3 \left (e^{2+\frac {16}{x^2}} \left (x+\log ^2(3)\right )+e^{8/x} \left (1+x^3+x^2 \log ^2(3)\right )\right )^2} \, dx \\ & = -\left (5 \int \frac {e^{\frac {2 \left (8+4 x+x^2\right )}{x^2}}}{x \left (e^{2+\frac {16}{x^2}} \left (x+\log ^2(3)\right )+e^{8/x} \left (1+x^3+x^2 \log ^2(3)\right )\right )^2} \, dx\right )-5 \int \frac {e^{8/x}}{x^2 \left (e^{2+\frac {16}{x^2}} \left (x+\log ^2(3)\right )+e^{8/x} \left (1+x^3+x^2 \log ^2(3)\right )\right )} \, dx-5 \int \left (\frac {3 e^{8/x} x}{\left (1+x^3+x^2 \log ^2(3)\right ) \left (e^{8/x}+e^{2+\frac {16}{x^2}} x+e^{8/x} x^3+e^{2+\frac {16}{x^2}} \log ^2(3)+e^{8/x} x^2 \log ^2(3)\right )}+\frac {2 e^{8/x} \log ^2(3)}{\left (1+x^3+x^2 \log ^2(3)\right ) \left (e^{8/x}+e^{2+\frac {16}{x^2}} x+e^{8/x} x^3+e^{2+\frac {16}{x^2}} \log ^2(3)+e^{8/x} x^2 \log ^2(3)\right )}\right ) \, dx+15 \int \frac {e^{\frac {2 \left (8+4 x+x^2\right )}{x^2}} x^2}{\left (1+x^3+x^2 \log ^2(3)\right ) \left (e^{8/x}+e^{2+\frac {16}{x^2}} x+e^{8/x} x^3+e^{2+\frac {16}{x^2}} \log ^2(3)+e^{8/x} x^2 \log ^2(3)\right )^2} \, dx-40 \int \frac {e^{\frac {2 \left (8+4 x+x^2\right )}{x^2}}}{x^2 \left (e^{2+\frac {16}{x^2}} \left (x+\log ^2(3)\right )+e^{8/x} \left (1+x^3+x^2 \log ^2(3)\right )\right )^2} \, dx+\left (25 \log ^2(3)\right ) \int \frac {e^{\frac {2 \left (8+4 x+x^2\right )}{x^2}} x}{\left (1+x^3+x^2 \log ^2(3)\right ) \left (e^{8/x}+e^{2+\frac {16}{x^2}} x+e^{8/x} x^3+e^{2+\frac {16}{x^2}} \log ^2(3)+e^{8/x} x^2 \log ^2(3)\right )^2} \, dx+\left (160 \log ^2(3)\right ) \int \frac {e^{\frac {2 \left (8+4 x+x^2\right )}{x^2}}}{x^4 \left (e^{2+\frac {16}{x^2}} \left (x+\log ^2(3)\right )+e^{8/x} \left (1+x^3+x^2 \log ^2(3)\right )\right )^2} \, dx+\left (10 \log ^4(3)\right ) \int \frac {e^{\frac {2 \left (8+4 x+x^2\right )}{x^2}}}{\left (1+x^3+x^2 \log ^2(3)\right ) \left (e^{8/x}+e^{2+\frac {16}{x^2}} x+e^{8/x} x^3+e^{2+\frac {16}{x^2}} \log ^2(3)+e^{8/x} x^2 \log ^2(3)\right )^2} \, dx+\left (40 \left (4-\log ^2(3)\right )\right ) \int \frac {e^{\frac {2 \left (8+4 x+x^2\right )}{x^2}}}{x^3 \left (e^{2+\frac {16}{x^2}} \left (x+\log ^2(3)\right )+e^{8/x} \left (1+x^3+x^2 \log ^2(3)\right )\right )^2} \, dx \\ & = -\left (5 \int \frac {e^{\frac {2 \left (8+4 x+x^2\right )}{x^2}}}{x \left (e^{2+\frac {16}{x^2}} \left (x+\log ^2(3)\right )+e^{8/x} \left (1+x^3+x^2 \log ^2(3)\right )\right )^2} \, dx\right )-5 \int \frac {e^{8/x}}{x^2 \left (e^{2+\frac {16}{x^2}} \left (x+\log ^2(3)\right )+e^{8/x} \left (1+x^3+x^2 \log ^2(3)\right )\right )} \, dx-15 \int \frac {e^{8/x} x}{\left (1+x^3+x^2 \log ^2(3)\right ) \left (e^{8/x}+e^{2+\frac {16}{x^2}} x+e^{8/x} x^3+e^{2+\frac {16}{x^2}} \log ^2(3)+e^{8/x} x^2 \log ^2(3)\right )} \, dx+15 \int \frac {e^{\frac {2 \left (8+4 x+x^2\right )}{x^2}} x^2}{\left (1+x^3+x^2 \log ^2(3)\right ) \left (e^{2+\frac {16}{x^2}} \left (x+\log ^2(3)\right )+e^{8/x} \left (1+x^3+x^2 \log ^2(3)\right )\right )^2} \, dx-40 \int \frac {e^{\frac {2 \left (8+4 x+x^2\right )}{x^2}}}{x^2 \left (e^{2+\frac {16}{x^2}} \left (x+\log ^2(3)\right )+e^{8/x} \left (1+x^3+x^2 \log ^2(3)\right )\right )^2} \, dx-\left (10 \log ^2(3)\right ) \int \frac {e^{8/x}}{\left (1+x^3+x^2 \log ^2(3)\right ) \left (e^{8/x}+e^{2+\frac {16}{x^2}} x+e^{8/x} x^3+e^{2+\frac {16}{x^2}} \log ^2(3)+e^{8/x} x^2 \log ^2(3)\right )} \, dx+\left (25 \log ^2(3)\right ) \int \frac {e^{\frac {2 \left (8+4 x+x^2\right )}{x^2}} x}{\left (1+x^3+x^2 \log ^2(3)\right ) \left (e^{2+\frac {16}{x^2}} \left (x+\log ^2(3)\right )+e^{8/x} \left (1+x^3+x^2 \log ^2(3)\right )\right )^2} \, dx+\left (160 \log ^2(3)\right ) \int \frac {e^{\frac {2 \left (8+4 x+x^2\right )}{x^2}}}{x^4 \left (e^{2+\frac {16}{x^2}} \left (x+\log ^2(3)\right )+e^{8/x} \left (1+x^3+x^2 \log ^2(3)\right )\right )^2} \, dx+\left (10 \log ^4(3)\right ) \int \frac {e^{\frac {2 \left (8+4 x+x^2\right )}{x^2}}}{\left (1+x^3+x^2 \log ^2(3)\right ) \left (e^{2+\frac {16}{x^2}} \left (x+\log ^2(3)\right )+e^{8/x} \left (1+x^3+x^2 \log ^2(3)\right )\right )^2} \, dx+\left (40 \left (4-\log ^2(3)\right )\right ) \int \frac {e^{\frac {2 \left (8+4 x+x^2\right )}{x^2}}}{x^3 \left (e^{2+\frac {16}{x^2}} \left (x+\log ^2(3)\right )+e^{8/x} \left (1+x^3+x^2 \log ^2(3)\right )\right )^2} \, dx \\ & = -\left (5 \int \frac {e^{\frac {2 \left (8+4 x+x^2\right )}{x^2}}}{x \left (e^{2+\frac {16}{x^2}} \left (x+\log ^2(3)\right )+e^{8/x} \left (1+x^3+x^2 \log ^2(3)\right )\right )^2} \, dx\right )-5 \int \frac {e^{8/x}}{x^2 \left (e^{2+\frac {16}{x^2}} \left (x+\log ^2(3)\right )+e^{8/x} \left (1+x^3+x^2 \log ^2(3)\right )\right )} \, dx+15 \int \frac {e^{\frac {2 \left (8+4 x+x^2\right )}{x^2}} x^2}{\left (1+x^3+x^2 \log ^2(3)\right ) \left (e^{2+\frac {16}{x^2}} \left (x+\log ^2(3)\right )+e^{8/x} \left (1+x^3+x^2 \log ^2(3)\right )\right )^2} \, dx-15 \int \frac {e^{8/x} x}{\left (1+x^3+x^2 \log ^2(3)\right ) \left (e^{2+\frac {16}{x^2}} \left (x+\log ^2(3)\right )+e^{8/x} \left (1+x^3+x^2 \log ^2(3)\right )\right )} \, dx-40 \int \frac {e^{\frac {2 \left (8+4 x+x^2\right )}{x^2}}}{x^2 \left (e^{2+\frac {16}{x^2}} \left (x+\log ^2(3)\right )+e^{8/x} \left (1+x^3+x^2 \log ^2(3)\right )\right )^2} \, dx-\left (10 \log ^2(3)\right ) \int \frac {e^{8/x}}{\left (1+x^3+x^2 \log ^2(3)\right ) \left (e^{2+\frac {16}{x^2}} \left (x+\log ^2(3)\right )+e^{8/x} \left (1+x^3+x^2 \log ^2(3)\right )\right )} \, dx+\left (25 \log ^2(3)\right ) \int \frac {e^{\frac {2 \left (8+4 x+x^2\right )}{x^2}} x}{\left (1+x^3+x^2 \log ^2(3)\right ) \left (e^{2+\frac {16}{x^2}} \left (x+\log ^2(3)\right )+e^{8/x} \left (1+x^3+x^2 \log ^2(3)\right )\right )^2} \, dx+\left (160 \log ^2(3)\right ) \int \frac {e^{\frac {2 \left (8+4 x+x^2\right )}{x^2}}}{x^4 \left (e^{2+\frac {16}{x^2}} \left (x+\log ^2(3)\right )+e^{8/x} \left (1+x^3+x^2 \log ^2(3)\right )\right )^2} \, dx+\left (10 \log ^4(3)\right ) \int \frac {e^{\frac {2 \left (8+4 x+x^2\right )}{x^2}}}{\left (1+x^3+x^2 \log ^2(3)\right ) \left (e^{2+\frac {16}{x^2}} \left (x+\log ^2(3)\right )+e^{8/x} \left (1+x^3+x^2 \log ^2(3)\right )\right )^2} \, dx+\left (40 \left (4-\log ^2(3)\right )\right ) \int \frac {e^{\frac {2 \left (8+4 x+x^2\right )}{x^2}}}{x^3 \left (e^{2+\frac {16}{x^2}} \left (x+\log ^2(3)\right )+e^{8/x} \left (1+x^3+x^2 \log ^2(3)\right )\right )^2} \, dx \\ \end{align*}
Time = 0.18 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.68 \[ \int \frac {-5 x^2-20 x^5-15 x^4 \log ^2(3)+e^{\frac {16-8 x+2 x^2}{x^2}} \left (160 x-40 x^2-10 x^3+\left (160-40 x-5 x^2\right ) \log ^2(3)\right )}{x^4+2 x^7+x^{10}+\left (2 x^6+2 x^9\right ) \log ^2(3)+x^8 \log ^4(3)+e^{\frac {2 \left (16-8 x+2 x^2\right )}{x^2}} \left (x^6+2 x^5 \log ^2(3)+x^4 \log ^4(3)\right )+e^{\frac {16-8 x+2 x^2}{x^2}} \left (2 x^5+2 x^8+\left (2 x^4+4 x^7\right ) \log ^2(3)+2 x^6 \log ^4(3)\right )} \, dx=\frac {5 e^{8/x}}{x \left (e^{2+\frac {16}{x^2}} \left (x+\log ^2(3)\right )+e^{8/x} \left (1+x^3+x^2 \log ^2(3)\right )\right )} \]
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Time = 1.31 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.81
method | result | size |
risch | \(\frac {5}{x \left (x^{2} \ln \left (3\right )^{2}+\ln \left (3\right )^{2} {\mathrm e}^{\frac {2 x^{2}-8 x +16}{x^{2}}}+x^{3}+x \,{\mathrm e}^{\frac {2 x^{2}-8 x +16}{x^{2}}}+1\right )}\) | \(56\) |
parallelrisch | \(\frac {5}{x \left (x^{2} \ln \left (3\right )^{2}+\ln \left (3\right )^{2} {\mathrm e}^{\frac {2 x^{2}-8 x +16}{x^{2}}}+x^{3}+x \,{\mathrm e}^{\frac {2 x^{2}-8 x +16}{x^{2}}}+1\right )}\) | \(56\) |
norman | \(\frac {5}{x \left (x^{2} \ln \left (3\right )^{2}+\ln \left (3\right )^{2} {\mathrm e}^{\frac {2 x^{2}-8 x +16}{x^{2}}}+x^{3}+x \,{\mathrm e}^{\frac {2 x^{2}-8 x +16}{x^{2}}}+1\right )}\) | \(58\) |
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Time = 0.25 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.35 \[ \int \frac {-5 x^2-20 x^5-15 x^4 \log ^2(3)+e^{\frac {16-8 x+2 x^2}{x^2}} \left (160 x-40 x^2-10 x^3+\left (160-40 x-5 x^2\right ) \log ^2(3)\right )}{x^4+2 x^7+x^{10}+\left (2 x^6+2 x^9\right ) \log ^2(3)+x^8 \log ^4(3)+e^{\frac {2 \left (16-8 x+2 x^2\right )}{x^2}} \left (x^6+2 x^5 \log ^2(3)+x^4 \log ^4(3)\right )+e^{\frac {16-8 x+2 x^2}{x^2}} \left (2 x^5+2 x^8+\left (2 x^4+4 x^7\right ) \log ^2(3)+2 x^6 \log ^4(3)\right )} \, dx=\frac {5}{x^{3} \log \left (3\right )^{2} + x^{4} + {\left (x \log \left (3\right )^{2} + x^{2}\right )} e^{\left (\frac {2 \, {\left (x^{2} - 4 \, x + 8\right )}}{x^{2}}\right )} + x} \]
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Time = 0.22 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.26 \[ \int \frac {-5 x^2-20 x^5-15 x^4 \log ^2(3)+e^{\frac {16-8 x+2 x^2}{x^2}} \left (160 x-40 x^2-10 x^3+\left (160-40 x-5 x^2\right ) \log ^2(3)\right )}{x^4+2 x^7+x^{10}+\left (2 x^6+2 x^9\right ) \log ^2(3)+x^8 \log ^4(3)+e^{\frac {2 \left (16-8 x+2 x^2\right )}{x^2}} \left (x^6+2 x^5 \log ^2(3)+x^4 \log ^4(3)\right )+e^{\frac {16-8 x+2 x^2}{x^2}} \left (2 x^5+2 x^8+\left (2 x^4+4 x^7\right ) \log ^2(3)+2 x^6 \log ^4(3)\right )} \, dx=\frac {5}{x^{4} + x^{3} \log {\left (3 \right )}^{2} + x + \left (x^{2} + x \log {\left (3 \right )}^{2}\right ) e^{\frac {2 x^{2} - 8 x + 16}{x^{2}}}} \]
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Time = 0.37 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.71 \[ \int \frac {-5 x^2-20 x^5-15 x^4 \log ^2(3)+e^{\frac {16-8 x+2 x^2}{x^2}} \left (160 x-40 x^2-10 x^3+\left (160-40 x-5 x^2\right ) \log ^2(3)\right )}{x^4+2 x^7+x^{10}+\left (2 x^6+2 x^9\right ) \log ^2(3)+x^8 \log ^4(3)+e^{\frac {2 \left (16-8 x+2 x^2\right )}{x^2}} \left (x^6+2 x^5 \log ^2(3)+x^4 \log ^4(3)\right )+e^{\frac {16-8 x+2 x^2}{x^2}} \left (2 x^5+2 x^8+\left (2 x^4+4 x^7\right ) \log ^2(3)+2 x^6 \log ^4(3)\right )} \, dx=\frac {5 \, e^{\frac {8}{x}}}{{\left (x^{3} \log \left (3\right )^{2} + x^{4} + x\right )} e^{\frac {8}{x}} + {\left (x e^{2} \log \left (3\right )^{2} + x^{2} e^{2}\right )} e^{\left (\frac {16}{x^{2}}\right )}} \]
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Time = 0.76 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.77 \[ \int \frac {-5 x^2-20 x^5-15 x^4 \log ^2(3)+e^{\frac {16-8 x+2 x^2}{x^2}} \left (160 x-40 x^2-10 x^3+\left (160-40 x-5 x^2\right ) \log ^2(3)\right )}{x^4+2 x^7+x^{10}+\left (2 x^6+2 x^9\right ) \log ^2(3)+x^8 \log ^4(3)+e^{\frac {2 \left (16-8 x+2 x^2\right )}{x^2}} \left (x^6+2 x^5 \log ^2(3)+x^4 \log ^4(3)\right )+e^{\frac {16-8 x+2 x^2}{x^2}} \left (2 x^5+2 x^8+\left (2 x^4+4 x^7\right ) \log ^2(3)+2 x^6 \log ^4(3)\right )} \, dx=\frac {5}{x^{3} \log \left (3\right )^{2} + x^{4} + x e^{\left (\frac {2 \, {\left (x^{2} - 4 \, x + 8\right )}}{x^{2}}\right )} \log \left (3\right )^{2} + x^{2} e^{\left (\frac {2 \, {\left (x^{2} - 4 \, x + 8\right )}}{x^{2}}\right )} + x} \]
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Time = 12.54 (sec) , antiderivative size = 383, normalized size of antiderivative = 12.35 \[ \int \frac {-5 x^2-20 x^5-15 x^4 \log ^2(3)+e^{\frac {16-8 x+2 x^2}{x^2}} \left (160 x-40 x^2-10 x^3+\left (160-40 x-5 x^2\right ) \log ^2(3)\right )}{x^4+2 x^7+x^{10}+\left (2 x^6+2 x^9\right ) \log ^2(3)+x^8 \log ^4(3)+e^{\frac {2 \left (16-8 x+2 x^2\right )}{x^2}} \left (x^6+2 x^5 \log ^2(3)+x^4 \log ^4(3)\right )+e^{\frac {16-8 x+2 x^2}{x^2}} \left (2 x^5+2 x^8+\left (2 x^4+4 x^7\right ) \log ^2(3)+2 x^6 \log ^4(3)\right )} \, dx=\frac {5\,{\left (x^5+2\,{\ln \left (3\right )}^2\,x^4+{\ln \left (3\right )}^4\,x^3\right )}^2\,\left (32\,x+64\,x^3\,{\ln \left (3\right )}^2+32\,x^2\,{\ln \left (3\right )}^4-16\,x^4\,{\ln \left (3\right )}^2-8\,x^3\,{\ln \left (3\right )}^4+4\,x^5\,{\ln \left (3\right )}^2+2\,x^4\,{\ln \left (3\right )}^4-8\,x\,{\ln \left (3\right )}^2+32\,{\ln \left (3\right )}^2-8\,x^2-x^3+32\,x^4-8\,x^5+2\,x^6\right )}{x^4\,\left ({\mathrm {e}}^{\frac {16}{x^2}-\frac {8}{x}+2}+\frac {x^3+{\ln \left (3\right )}^2\,x^2+1}{x+{\ln \left (3\right )}^2}\right )\,{\left (x+{\ln \left (3\right )}^2\right )}^3\,\left (96\,x^5\,{\ln \left (3\right )}^2+96\,x^4\,{\ln \left (3\right )}^4-24\,x^6\,{\ln \left (3\right )}^2+32\,x^3\,{\ln \left (3\right )}^6-24\,x^5\,{\ln \left (3\right )}^4-2\,x^7\,{\ln \left (3\right )}^2-8\,x^4\,{\ln \left (3\right )}^6-x^6\,{\ln \left (3\right )}^4+128\,x^8\,{\ln \left (3\right )}^2+192\,x^7\,{\ln \left (3\right )}^4-32\,x^9\,{\ln \left (3\right )}^2+128\,x^6\,{\ln \left (3\right )}^6-48\,x^8\,{\ln \left (3\right )}^4+8\,x^{10}\,{\ln \left (3\right )}^2+32\,x^5\,{\ln \left (3\right )}^8-32\,x^7\,{\ln \left (3\right )}^6+12\,x^9\,{\ln \left (3\right )}^4-8\,x^6\,{\ln \left (3\right )}^8+8\,x^8\,{\ln \left (3\right )}^6+2\,x^7\,{\ln \left (3\right )}^8+32\,x^6-8\,x^7-x^8+32\,x^9-8\,x^{10}+2\,x^{11}\right )} \]
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