Integrand size = 165, antiderivative size = 33 \[ \int \frac {48+12 x-48 x^2-9 x^3-9 x^4+e^{-4+x} \left (-24-24 x+6 x^2-12 x^3-6 x^4\right )+\left (-48+24 x+60 x^2+18 x^3+e^{-4+x} \left (24+48 x+30 x^2+6 x^3\right )\right ) \log (x)}{64 x^2-32 x^3-44 x^4+12 x^5+9 x^6+e^{-8+2 x} \left (16 x^2+16 x^3+4 x^4\right )+e^{-4+x} \left (-64 x^2-16 x^3+32 x^4+12 x^5\right )} \, dx=\frac {3 \left (\frac {x^2}{2+x}-\log (x)\right )}{x \left (-4+2 e^{-4+x}+3 x\right )} \]
[Out]
\[ \int \frac {48+12 x-48 x^2-9 x^3-9 x^4+e^{-4+x} \left (-24-24 x+6 x^2-12 x^3-6 x^4\right )+\left (-48+24 x+60 x^2+18 x^3+e^{-4+x} \left (24+48 x+30 x^2+6 x^3\right )\right ) \log (x)}{64 x^2-32 x^3-44 x^4+12 x^5+9 x^6+e^{-8+2 x} \left (16 x^2+16 x^3+4 x^4\right )+e^{-4+x} \left (-64 x^2-16 x^3+32 x^4+12 x^5\right )} \, dx=\int \frac {48+12 x-48 x^2-9 x^3-9 x^4+e^{-4+x} \left (-24-24 x+6 x^2-12 x^3-6 x^4\right )+\left (-48+24 x+60 x^2+18 x^3+e^{-4+x} \left (24+48 x+30 x^2+6 x^3\right )\right ) \log (x)}{64 x^2-32 x^3-44 x^4+12 x^5+9 x^6+e^{-8+2 x} \left (16 x^2+16 x^3+4 x^4\right )+e^{-4+x} \left (-64 x^2-16 x^3+32 x^4+12 x^5\right )} \, dx \]
[In]
[Out]
Rubi steps \begin{align*} \text {integral}& = \int \frac {3 e^4 \left (-2 e^x \left (4+4 x-x^2+2 x^3+x^4\right )-e^4 \left (-16-4 x+16 x^2+3 x^3+3 x^4\right )+2 (2+x)^2 \left (e^x (1+x)+e^4 (-2+3 x)\right ) \log (x)\right )}{x^2 (2+x)^2 \left (2 e^x+e^4 (-4+3 x)\right )^2} \, dx \\ & = \left (3 e^4\right ) \int \frac {-2 e^x \left (4+4 x-x^2+2 x^3+x^4\right )-e^4 \left (-16-4 x+16 x^2+3 x^3+3 x^4\right )+2 (2+x)^2 \left (e^x (1+x)+e^4 (-2+3 x)\right ) \log (x)}{x^2 (2+x)^2 \left (2 e^x+e^4 (-4+3 x)\right )^2} \, dx \\ & = \left (3 e^4\right ) \int \left (\frac {e^4 (-7+3 x) \left (x^2-2 \log (x)-x \log (x)\right )}{x (2+x) \left (-4 e^4+2 e^x+3 e^4 x\right )^2}-\frac {4+4 x-x^2+2 x^3+x^4-4 \log (x)-8 x \log (x)-5 x^2 \log (x)-x^3 \log (x)}{x^2 (2+x)^2 \left (-4 e^4+2 e^x+3 e^4 x\right )}\right ) \, dx \\ & = -\left (\left (3 e^4\right ) \int \frac {4+4 x-x^2+2 x^3+x^4-4 \log (x)-8 x \log (x)-5 x^2 \log (x)-x^3 \log (x)}{x^2 (2+x)^2 \left (-4 e^4+2 e^x+3 e^4 x\right )} \, dx\right )+\left (3 e^8\right ) \int \frac {(-7+3 x) \left (x^2-2 \log (x)-x \log (x)\right )}{x (2+x) \left (-4 e^4+2 e^x+3 e^4 x\right )^2} \, dx \\ & = -\left (\left (3 e^4\right ) \int \left (\frac {4+4 x-x^2+2 x^3+x^4-4 \log (x)-8 x \log (x)-5 x^2 \log (x)-x^3 \log (x)}{4 x^2 \left (-4 e^4+2 e^x+3 e^4 x\right )}-\frac {4+4 x-x^2+2 x^3+x^4-4 \log (x)-8 x \log (x)-5 x^2 \log (x)-x^3 \log (x)}{4 x \left (-4 e^4+2 e^x+3 e^4 x\right )}+\frac {4+4 x-x^2+2 x^3+x^4-4 \log (x)-8 x \log (x)-5 x^2 \log (x)-x^3 \log (x)}{4 (2+x)^2 \left (-4 e^4+2 e^x+3 e^4 x\right )}+\frac {4+4 x-x^2+2 x^3+x^4-4 \log (x)-8 x \log (x)-5 x^2 \log (x)-x^3 \log (x)}{4 (2+x) \left (-4 e^4+2 e^x+3 e^4 x\right )}\right ) \, dx\right )+\left (3 e^8\right ) \int \left (-\frac {7 \left (x^2-2 \log (x)-x \log (x)\right )}{2 x \left (-4 e^4+2 e^x+3 e^4 x\right )^2}+\frac {13 \left (x^2-2 \log (x)-x \log (x)\right )}{2 (2+x) \left (-4 e^4+2 e^x+3 e^4 x\right )^2}\right ) \, dx \\ & = -\left (\frac {1}{4} \left (3 e^4\right ) \int \frac {4+4 x-x^2+2 x^3+x^4-4 \log (x)-8 x \log (x)-5 x^2 \log (x)-x^3 \log (x)}{x^2 \left (-4 e^4+2 e^x+3 e^4 x\right )} \, dx\right )+\frac {1}{4} \left (3 e^4\right ) \int \frac {4+4 x-x^2+2 x^3+x^4-4 \log (x)-8 x \log (x)-5 x^2 \log (x)-x^3 \log (x)}{x \left (-4 e^4+2 e^x+3 e^4 x\right )} \, dx-\frac {1}{4} \left (3 e^4\right ) \int \frac {4+4 x-x^2+2 x^3+x^4-4 \log (x)-8 x \log (x)-5 x^2 \log (x)-x^3 \log (x)}{(2+x)^2 \left (-4 e^4+2 e^x+3 e^4 x\right )} \, dx-\frac {1}{4} \left (3 e^4\right ) \int \frac {4+4 x-x^2+2 x^3+x^4-4 \log (x)-8 x \log (x)-5 x^2 \log (x)-x^3 \log (x)}{(2+x) \left (-4 e^4+2 e^x+3 e^4 x\right )} \, dx-\frac {1}{2} \left (21 e^8\right ) \int \frac {x^2-2 \log (x)-x \log (x)}{x \left (-4 e^4+2 e^x+3 e^4 x\right )^2} \, dx+\frac {1}{2} \left (39 e^8\right ) \int \frac {x^2-2 \log (x)-x \log (x)}{(2+x) \left (-4 e^4+2 e^x+3 e^4 x\right )^2} \, dx \\ & = -\left (\frac {1}{4} \left (3 e^4\right ) \int \left (-\frac {1}{-4 e^4+2 e^x+3 e^4 x}+\frac {4}{x^2 \left (-4 e^4+2 e^x+3 e^4 x\right )}+\frac {4}{x \left (-4 e^4+2 e^x+3 e^4 x\right )}+\frac {2 x}{-4 e^4+2 e^x+3 e^4 x}+\frac {x^2}{-4 e^4+2 e^x+3 e^4 x}-\frac {5 \log (x)}{-4 e^4+2 e^x+3 e^4 x}-\frac {4 \log (x)}{x^2 \left (-4 e^4+2 e^x+3 e^4 x\right )}-\frac {8 \log (x)}{x \left (-4 e^4+2 e^x+3 e^4 x\right )}-\frac {x \log (x)}{-4 e^4+2 e^x+3 e^4 x}\right ) \, dx\right )+\frac {1}{4} \left (3 e^4\right ) \int \left (\frac {4}{-4 e^4+2 e^x+3 e^4 x}+\frac {4}{x \left (-4 e^4+2 e^x+3 e^4 x\right )}-\frac {x}{-4 e^4+2 e^x+3 e^4 x}+\frac {2 x^2}{-4 e^4+2 e^x+3 e^4 x}+\frac {x^3}{-4 e^4+2 e^x+3 e^4 x}-\frac {8 \log (x)}{-4 e^4+2 e^x+3 e^4 x}-\frac {4 \log (x)}{x \left (-4 e^4+2 e^x+3 e^4 x\right )}-\frac {5 x \log (x)}{-4 e^4+2 e^x+3 e^4 x}-\frac {x^2 \log (x)}{-4 e^4+2 e^x+3 e^4 x}\right ) \, dx-\frac {1}{4} \left (3 e^4\right ) \int \left (\frac {4}{(2+x)^2 \left (-4 e^4+2 e^x+3 e^4 x\right )}+\frac {4 x}{(2+x)^2 \left (-4 e^4+2 e^x+3 e^4 x\right )}-\frac {x^2}{(2+x)^2 \left (-4 e^4+2 e^x+3 e^4 x\right )}+\frac {2 x^3}{(2+x)^2 \left (-4 e^4+2 e^x+3 e^4 x\right )}+\frac {x^4}{(2+x)^2 \left (-4 e^4+2 e^x+3 e^4 x\right )}-\frac {4 \log (x)}{(2+x)^2 \left (-4 e^4+2 e^x+3 e^4 x\right )}-\frac {8 x \log (x)}{(2+x)^2 \left (-4 e^4+2 e^x+3 e^4 x\right )}-\frac {5 x^2 \log (x)}{(2+x)^2 \left (-4 e^4+2 e^x+3 e^4 x\right )}-\frac {x^3 \log (x)}{(2+x)^2 \left (-4 e^4+2 e^x+3 e^4 x\right )}\right ) \, dx-\frac {1}{4} \left (3 e^4\right ) \int \left (\frac {4}{(2+x) \left (-4 e^4+2 e^x+3 e^4 x\right )}+\frac {4 x}{(2+x) \left (-4 e^4+2 e^x+3 e^4 x\right )}-\frac {x^2}{(2+x) \left (-4 e^4+2 e^x+3 e^4 x\right )}+\frac {2 x^3}{(2+x) \left (-4 e^4+2 e^x+3 e^4 x\right )}+\frac {x^4}{(2+x) \left (-4 e^4+2 e^x+3 e^4 x\right )}-\frac {4 \log (x)}{(2+x) \left (-4 e^4+2 e^x+3 e^4 x\right )}-\frac {8 x \log (x)}{(2+x) \left (-4 e^4+2 e^x+3 e^4 x\right )}-\frac {5 x^2 \log (x)}{(2+x) \left (-4 e^4+2 e^x+3 e^4 x\right )}-\frac {x^3 \log (x)}{(2+x) \left (-4 e^4+2 e^x+3 e^4 x\right )}\right ) \, dx-\frac {1}{2} \left (21 e^8\right ) \int \left (\frac {x}{\left (-4 e^4+2 e^x+3 e^4 x\right )^2}-\frac {\log (x)}{\left (-4 e^4+2 e^x+3 e^4 x\right )^2}-\frac {2 \log (x)}{x \left (-4 e^4+2 e^x+3 e^4 x\right )^2}\right ) \, dx+\frac {1}{2} \left (39 e^8\right ) \int \left (\frac {x^2}{(2+x) \left (-4 e^4+2 e^x+3 e^4 x\right )^2}-\frac {2 \log (x)}{(2+x) \left (-4 e^4+2 e^x+3 e^4 x\right )^2}-\frac {x \log (x)}{(2+x) \left (-4 e^4+2 e^x+3 e^4 x\right )^2}\right ) \, dx \\ & = \text {Too large to display} \\ \end{align*}
Time = 0.13 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.27 \[ \int \frac {48+12 x-48 x^2-9 x^3-9 x^4+e^{-4+x} \left (-24-24 x+6 x^2-12 x^3-6 x^4\right )+\left (-48+24 x+60 x^2+18 x^3+e^{-4+x} \left (24+48 x+30 x^2+6 x^3\right )\right ) \log (x)}{64 x^2-32 x^3-44 x^4+12 x^5+9 x^6+e^{-8+2 x} \left (16 x^2+16 x^3+4 x^4\right )+e^{-4+x} \left (-64 x^2-16 x^3+32 x^4+12 x^5\right )} \, dx=-\frac {3 e^4 \left (-x^2+(2+x) \log (x)\right )}{x (2+x) \left (2 e^x+e^4 (-4+3 x)\right )} \]
[In]
[Out]
Time = 0.38 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.30
method | result | size |
risch | \(-\frac {3 \ln \left (x \right )}{x \left (3 x -4+2 \,{\mathrm e}^{x -4}\right )}+\frac {3 x}{\left (2+x \right ) \left (3 x -4+2 \,{\mathrm e}^{x -4}\right )}\) | \(43\) |
parallelrisch | \(\frac {6 x^{2}-6 x \ln \left (x \right )-12 \ln \left (x \right )}{2 x \left (2 x \,{\mathrm e}^{x -4}+3 x^{2}+4 \,{\mathrm e}^{x -4}+2 x -8\right )}\) | \(46\) |
[In]
[Out]
none
Time = 0.28 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.27 \[ \int \frac {48+12 x-48 x^2-9 x^3-9 x^4+e^{-4+x} \left (-24-24 x+6 x^2-12 x^3-6 x^4\right )+\left (-48+24 x+60 x^2+18 x^3+e^{-4+x} \left (24+48 x+30 x^2+6 x^3\right )\right ) \log (x)}{64 x^2-32 x^3-44 x^4+12 x^5+9 x^6+e^{-8+2 x} \left (16 x^2+16 x^3+4 x^4\right )+e^{-4+x} \left (-64 x^2-16 x^3+32 x^4+12 x^5\right )} \, dx=\frac {3 \, {\left (x^{2} - {\left (x + 2\right )} \log \left (x\right )\right )}}{3 \, x^{3} + 2 \, x^{2} + 2 \, {\left (x^{2} + 2 \, x\right )} e^{\left (x - 4\right )} - 8 \, x} \]
[In]
[Out]
Time = 0.13 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.27 \[ \int \frac {48+12 x-48 x^2-9 x^3-9 x^4+e^{-4+x} \left (-24-24 x+6 x^2-12 x^3-6 x^4\right )+\left (-48+24 x+60 x^2+18 x^3+e^{-4+x} \left (24+48 x+30 x^2+6 x^3\right )\right ) \log (x)}{64 x^2-32 x^3-44 x^4+12 x^5+9 x^6+e^{-8+2 x} \left (16 x^2+16 x^3+4 x^4\right )+e^{-4+x} \left (-64 x^2-16 x^3+32 x^4+12 x^5\right )} \, dx=\frac {3 x^{2} - 3 x \log {\left (x \right )} - 6 \log {\left (x \right )}}{3 x^{3} + 2 x^{2} - 8 x + \left (2 x^{2} + 4 x\right ) e^{x - 4}} \]
[In]
[Out]
none
Time = 0.26 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.67 \[ \int \frac {48+12 x-48 x^2-9 x^3-9 x^4+e^{-4+x} \left (-24-24 x+6 x^2-12 x^3-6 x^4\right )+\left (-48+24 x+60 x^2+18 x^3+e^{-4+x} \left (24+48 x+30 x^2+6 x^3\right )\right ) \log (x)}{64 x^2-32 x^3-44 x^4+12 x^5+9 x^6+e^{-8+2 x} \left (16 x^2+16 x^3+4 x^4\right )+e^{-4+x} \left (-64 x^2-16 x^3+32 x^4+12 x^5\right )} \, dx=\frac {3 \, {\left (x^{2} e^{4} - {\left (x e^{4} + 2 \, e^{4}\right )} \log \left (x\right )\right )}}{3 \, x^{3} e^{4} + 2 \, x^{2} e^{4} - 8 \, x e^{4} + 2 \, {\left (x^{2} + 2 \, x\right )} e^{x}} \]
[In]
[Out]
none
Time = 0.30 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.70 \[ \int \frac {48+12 x-48 x^2-9 x^3-9 x^4+e^{-4+x} \left (-24-24 x+6 x^2-12 x^3-6 x^4\right )+\left (-48+24 x+60 x^2+18 x^3+e^{-4+x} \left (24+48 x+30 x^2+6 x^3\right )\right ) \log (x)}{64 x^2-32 x^3-44 x^4+12 x^5+9 x^6+e^{-8+2 x} \left (16 x^2+16 x^3+4 x^4\right )+e^{-4+x} \left (-64 x^2-16 x^3+32 x^4+12 x^5\right )} \, dx=\frac {3 \, {\left (x^{2} e^{4} - x e^{4} \log \left (x\right ) - 2 \, e^{4} \log \left (x\right )\right )}}{3 \, x^{3} e^{4} + 2 \, x^{2} e^{4} + 2 \, x^{2} e^{x} - 8 \, x e^{4} + 4 \, x e^{x}} \]
[In]
[Out]
Time = 9.72 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.12 \[ \int \frac {48+12 x-48 x^2-9 x^3-9 x^4+e^{-4+x} \left (-24-24 x+6 x^2-12 x^3-6 x^4\right )+\left (-48+24 x+60 x^2+18 x^3+e^{-4+x} \left (24+48 x+30 x^2+6 x^3\right )\right ) \log (x)}{64 x^2-32 x^3-44 x^4+12 x^5+9 x^6+e^{-8+2 x} \left (16 x^2+16 x^3+4 x^4\right )+e^{-4+x} \left (-64 x^2-16 x^3+32 x^4+12 x^5\right )} \, dx=-\frac {3\,\left (2\,\ln \left (x\right )+x\,\ln \left (x\right )-x^2\right )}{x\,\left (x+2\right )\,\left (3\,x+2\,{\mathrm {e}}^{x-4}-4\right )} \]
[In]
[Out]