Integrand size = 284, antiderivative size = 36 \[ \int \frac {10 x-15 x^2-18 x^3+21 x^4-2 x^5-4 x^6+x^7+e^x \left (-10 x+5 x^2+8 x^3-6 x^4+x^5\right )+\left (5 x^2-4 x^4+x^5\right ) \log \left (\frac {1+x}{5-5 x+x^2}\right )}{5-30 x+41 x^2+15 x^3-12 x^4+17 x^5-21 x^6+6 x^7-4 x^8+x^9+e^{2 x} \left (5-4 x^2+x^3\right )+e^x \left (-10+30 x+8 x^2-16 x^3+6 x^4-8 x^5+2 x^6\right )+\left (10 x-30 x^2-8 x^3+16 x^4-6 x^5+8 x^6-2 x^7+e^x \left (-10 x+8 x^3-2 x^4\right )\right ) \log \left (\frac {1+x}{5-5 x+x^2}\right )+\left (5 x^2-4 x^4+x^5\right ) \log ^2\left (\frac {1+x}{5-5 x+x^2}\right )} \, dx=\frac {x}{-3+\frac {1-e^x-x^3}{x}+\log \left (\frac {1+x}{5+(-5+x) x}\right )} \] Output:
x/((1-x^3-exp(x))/x-3+ln((1+x)/(5+(-5+x)*x)))
Time = 0.07 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.06 \[ \int \frac {10 x-15 x^2-18 x^3+21 x^4-2 x^5-4 x^6+x^7+e^x \left (-10 x+5 x^2+8 x^3-6 x^4+x^5\right )+\left (5 x^2-4 x^4+x^5\right ) \log \left (\frac {1+x}{5-5 x+x^2}\right )}{5-30 x+41 x^2+15 x^3-12 x^4+17 x^5-21 x^6+6 x^7-4 x^8+x^9+e^{2 x} \left (5-4 x^2+x^3\right )+e^x \left (-10+30 x+8 x^2-16 x^3+6 x^4-8 x^5+2 x^6\right )+\left (10 x-30 x^2-8 x^3+16 x^4-6 x^5+8 x^6-2 x^7+e^x \left (-10 x+8 x^3-2 x^4\right )\right ) \log \left (\frac {1+x}{5-5 x+x^2}\right )+\left (5 x^2-4 x^4+x^5\right ) \log ^2\left (\frac {1+x}{5-5 x+x^2}\right )} \, dx=\frac {x^2}{1-e^x-3 x-x^3+x \log \left (\frac {1+x}{5-5 x+x^2}\right )} \] Input:
Integrate[(10*x - 15*x^2 - 18*x^3 + 21*x^4 - 2*x^5 - 4*x^6 + x^7 + E^x*(-1 0*x + 5*x^2 + 8*x^3 - 6*x^4 + x^5) + (5*x^2 - 4*x^4 + x^5)*Log[(1 + x)/(5 - 5*x + x^2)])/(5 - 30*x + 41*x^2 + 15*x^3 - 12*x^4 + 17*x^5 - 21*x^6 + 6* x^7 - 4*x^8 + x^9 + E^(2*x)*(5 - 4*x^2 + x^3) + E^x*(-10 + 30*x + 8*x^2 - 16*x^3 + 6*x^4 - 8*x^5 + 2*x^6) + (10*x - 30*x^2 - 8*x^3 + 16*x^4 - 6*x^5 + 8*x^6 - 2*x^7 + E^x*(-10*x + 8*x^3 - 2*x^4))*Log[(1 + x)/(5 - 5*x + x^2) ] + (5*x^2 - 4*x^4 + x^5)*Log[(1 + x)/(5 - 5*x + x^2)]^2),x]
Output:
x^2/(1 - E^x - 3*x - x^3 + x*Log[(1 + x)/(5 - 5*x + x^2)])
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {x^7-4 x^6-2 x^5+21 x^4-18 x^3-15 x^2+\left (x^5-4 x^4+5 x^2\right ) \log \left (\frac {x+1}{x^2-5 x+5}\right )+e^x \left (x^5-6 x^4+8 x^3+5 x^2-10 x\right )+10 x}{x^9-4 x^8+6 x^7-21 x^6+17 x^5-12 x^4+15 x^3+41 x^2+e^{2 x} \left (x^3-4 x^2+5\right )+\left (x^5-4 x^4+5 x^2\right ) \log ^2\left (\frac {x+1}{x^2-5 x+5}\right )+e^x \left (2 x^6-8 x^5+6 x^4-16 x^3+8 x^2+30 x-10\right )+\left (-2 x^7+8 x^6-6 x^5+16 x^4-8 x^3-30 x^2+e^x \left (-2 x^4+8 x^3-10 x\right )+10 x\right ) \log \left (\frac {x+1}{x^2-5 x+5}\right )-30 x+5} \, dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {x \left (x^6-4 x^5-2 x^4+21 x^3-18 x^2+\left (x^3-4 x^2+5\right ) x \log \left (\frac {x+1}{x^2-5 x+5}\right )+e^x \left (x^4-6 x^3+8 x^2+5 x-10\right )-15 x+10\right )}{\left (x^3-4 x^2+5\right ) \left (-x^3+x \log \left (\frac {x+1}{x^2-5 x+5}\right )-3 x-e^x+1\right )^2}dx\) |
| \(\Big \downarrow \) 2463 |
| \(\displaystyle \int \left (\frac {x \left (x^6-4 x^5-2 x^4+21 x^3-18 x^2+\left (x^3-4 x^2+5\right ) x \log \left (\frac {x+1}{x^2-5 x+5}\right )+e^x \left (x^4-6 x^3+8 x^2+5 x-10\right )-15 x+10\right )}{11 (x+1) \left (-x^3+x \log \left (\frac {x+1}{x^2-5 x+5}\right )-3 x-e^x+1\right )^2}+\frac {(6-x) x \left (x^6-4 x^5-2 x^4+21 x^3-18 x^2+\left (x^3-4 x^2+5\right ) x \log \left (\frac {x+1}{x^2-5 x+5}\right )+e^x \left (x^4-6 x^3+8 x^2+5 x-10\right )-15 x+10\right )}{11 \left (x^2-5 x+5\right ) \left (-x^3+x \log \left (\frac {x+1}{x^2-5 x+5}\right )-3 x-e^x+1\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {x \left (x^6-4 x^5-2 x^4+21 x^3-18 x^2+\left (x^3-4 x^2+5\right ) x \log \left (\frac {x+1}{x^2-5 x+5}\right )+e^x \left (x^4-6 x^3+8 x^2+5 x-10\right )-15 x+10\right )}{\left (x^3-4 x^2+5\right ) \left (-x^3+x \log \left (\frac {x+1}{x^2-5 x+5}\right )-3 x-e^x+1\right )^2}dx\) |
| \(\Big \downarrow \) 2463 |
| \(\displaystyle \int \left (\frac {x \left (x^6-4 x^5-2 x^4+21 x^3-18 x^2+\left (x^3-4 x^2+5\right ) x \log \left (\frac {x+1}{x^2-5 x+5}\right )+e^x \left (x^4-6 x^3+8 x^2+5 x-10\right )-15 x+10\right )}{11 (x+1) \left (-x^3+x \log \left (\frac {x+1}{x^2-5 x+5}\right )-3 x-e^x+1\right )^2}+\frac {(6-x) x \left (x^6-4 x^5-2 x^4+21 x^3-18 x^2+\left (x^3-4 x^2+5\right ) x \log \left (\frac {x+1}{x^2-5 x+5}\right )+e^x \left (x^4-6 x^3+8 x^2+5 x-10\right )-15 x+10\right )}{11 \left (x^2-5 x+5\right ) \left (-x^3+x \log \left (\frac {x+1}{x^2-5 x+5}\right )-3 x-e^x+1\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {x \left (x^6-4 x^5-2 x^4+21 x^3-18 x^2+\left (x^3-4 x^2+5\right ) x \log \left (\frac {x+1}{x^2-5 x+5}\right )+e^x \left (x^4-6 x^3+8 x^2+5 x-10\right )-15 x+10\right )}{\left (x^3-4 x^2+5\right ) \left (-x^3+x \log \left (\frac {x+1}{x^2-5 x+5}\right )-3 x-e^x+1\right )^2}dx\) |
| \(\Big \downarrow \) 2463 |
| \(\displaystyle \int \left (\frac {x \left (x^6-4 x^5-2 x^4+21 x^3-18 x^2+\left (x^3-4 x^2+5\right ) x \log \left (\frac {x+1}{x^2-5 x+5}\right )+e^x \left (x^4-6 x^3+8 x^2+5 x-10\right )-15 x+10\right )}{11 (x+1) \left (-x^3+x \log \left (\frac {x+1}{x^2-5 x+5}\right )-3 x-e^x+1\right )^2}+\frac {(6-x) x \left (x^6-4 x^5-2 x^4+21 x^3-18 x^2+\left (x^3-4 x^2+5\right ) x \log \left (\frac {x+1}{x^2-5 x+5}\right )+e^x \left (x^4-6 x^3+8 x^2+5 x-10\right )-15 x+10\right )}{11 \left (x^2-5 x+5\right ) \left (-x^3+x \log \left (\frac {x+1}{x^2-5 x+5}\right )-3 x-e^x+1\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {x \left (x^6-4 x^5-2 x^4+21 x^3-18 x^2+\left (x^3-4 x^2+5\right ) x \log \left (\frac {x+1}{x^2-5 x+5}\right )+e^x \left (x^4-6 x^3+8 x^2+5 x-10\right )-15 x+10\right )}{\left (x^3-4 x^2+5\right ) \left (-x^3+x \log \left (\frac {x+1}{x^2-5 x+5}\right )-3 x-e^x+1\right )^2}dx\) |
| \(\Big \downarrow \) 2463 |
| \(\displaystyle \int \left (\frac {x \left (x^6-4 x^5-2 x^4+21 x^3-18 x^2+\left (x^3-4 x^2+5\right ) x \log \left (\frac {x+1}{x^2-5 x+5}\right )+e^x \left (x^4-6 x^3+8 x^2+5 x-10\right )-15 x+10\right )}{11 (x+1) \left (-x^3+x \log \left (\frac {x+1}{x^2-5 x+5}\right )-3 x-e^x+1\right )^2}+\frac {(6-x) x \left (x^6-4 x^5-2 x^4+21 x^3-18 x^2+\left (x^3-4 x^2+5\right ) x \log \left (\frac {x+1}{x^2-5 x+5}\right )+e^x \left (x^4-6 x^3+8 x^2+5 x-10\right )-15 x+10\right )}{11 \left (x^2-5 x+5\right ) \left (-x^3+x \log \left (\frac {x+1}{x^2-5 x+5}\right )-3 x-e^x+1\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {x \left (x^6-4 x^5-2 x^4+21 x^3-18 x^2+\left (x^3-4 x^2+5\right ) x \log \left (\frac {x+1}{x^2-5 x+5}\right )+e^x \left (x^4-6 x^3+8 x^2+5 x-10\right )-15 x+10\right )}{\left (x^3-4 x^2+5\right ) \left (-x^3+x \log \left (\frac {x+1}{x^2-5 x+5}\right )-3 x-e^x+1\right )^2}dx\) |
| \(\Big \downarrow \) 2463 |
| \(\displaystyle \int \left (\frac {x \left (x^6-4 x^5-2 x^4+21 x^3-18 x^2+\left (x^3-4 x^2+5\right ) x \log \left (\frac {x+1}{x^2-5 x+5}\right )+e^x \left (x^4-6 x^3+8 x^2+5 x-10\right )-15 x+10\right )}{11 (x+1) \left (-x^3+x \log \left (\frac {x+1}{x^2-5 x+5}\right )-3 x-e^x+1\right )^2}+\frac {(6-x) x \left (x^6-4 x^5-2 x^4+21 x^3-18 x^2+\left (x^3-4 x^2+5\right ) x \log \left (\frac {x+1}{x^2-5 x+5}\right )+e^x \left (x^4-6 x^3+8 x^2+5 x-10\right )-15 x+10\right )}{11 \left (x^2-5 x+5\right ) \left (-x^3+x \log \left (\frac {x+1}{x^2-5 x+5}\right )-3 x-e^x+1\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {x \left (x^6-4 x^5-2 x^4+21 x^3-18 x^2+\left (x^3-4 x^2+5\right ) x \log \left (\frac {x+1}{x^2-5 x+5}\right )+e^x \left (x^4-6 x^3+8 x^2+5 x-10\right )-15 x+10\right )}{\left (x^3-4 x^2+5\right ) \left (-x^3+x \log \left (\frac {x+1}{x^2-5 x+5}\right )-3 x-e^x+1\right )^2}dx\) |
| \(\Big \downarrow \) 2463 |
| \(\displaystyle \int \left (\frac {x \left (x^6-4 x^5-2 x^4+21 x^3-18 x^2+\left (x^3-4 x^2+5\right ) x \log \left (\frac {x+1}{x^2-5 x+5}\right )+e^x \left (x^4-6 x^3+8 x^2+5 x-10\right )-15 x+10\right )}{11 (x+1) \left (-x^3+x \log \left (\frac {x+1}{x^2-5 x+5}\right )-3 x-e^x+1\right )^2}+\frac {(6-x) x \left (x^6-4 x^5-2 x^4+21 x^3-18 x^2+\left (x^3-4 x^2+5\right ) x \log \left (\frac {x+1}{x^2-5 x+5}\right )+e^x \left (x^4-6 x^3+8 x^2+5 x-10\right )-15 x+10\right )}{11 \left (x^2-5 x+5\right ) \left (-x^3+x \log \left (\frac {x+1}{x^2-5 x+5}\right )-3 x-e^x+1\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {x \left (x^6-4 x^5-2 x^4+21 x^3-18 x^2+\left (x^3-4 x^2+5\right ) x \log \left (\frac {x+1}{x^2-5 x+5}\right )+e^x \left (x^4-6 x^3+8 x^2+5 x-10\right )-15 x+10\right )}{\left (x^3-4 x^2+5\right ) \left (-x^3+x \log \left (\frac {x+1}{x^2-5 x+5}\right )-3 x-e^x+1\right )^2}dx\) |
| \(\Big \downarrow \) 2463 |
| \(\displaystyle \int \left (\frac {x \left (x^6-4 x^5-2 x^4+21 x^3-18 x^2+\left (x^3-4 x^2+5\right ) x \log \left (\frac {x+1}{x^2-5 x+5}\right )+e^x \left (x^4-6 x^3+8 x^2+5 x-10\right )-15 x+10\right )}{11 (x+1) \left (-x^3+x \log \left (\frac {x+1}{x^2-5 x+5}\right )-3 x-e^x+1\right )^2}+\frac {(6-x) x \left (x^6-4 x^5-2 x^4+21 x^3-18 x^2+\left (x^3-4 x^2+5\right ) x \log \left (\frac {x+1}{x^2-5 x+5}\right )+e^x \left (x^4-6 x^3+8 x^2+5 x-10\right )-15 x+10\right )}{11 \left (x^2-5 x+5\right ) \left (-x^3+x \log \left (\frac {x+1}{x^2-5 x+5}\right )-3 x-e^x+1\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {x \left (x^6-4 x^5-2 x^4+21 x^3-18 x^2+\left (x^3-4 x^2+5\right ) x \log \left (\frac {x+1}{x^2-5 x+5}\right )+e^x \left (x^4-6 x^3+8 x^2+5 x-10\right )-15 x+10\right )}{\left (x^3-4 x^2+5\right ) \left (-x^3+x \log \left (\frac {x+1}{x^2-5 x+5}\right )-3 x-e^x+1\right )^2}dx\) |
| \(\Big \downarrow \) 2463 |
| \(\displaystyle \int \left (\frac {x \left (x^6-4 x^5-2 x^4+21 x^3-18 x^2+\left (x^3-4 x^2+5\right ) x \log \left (\frac {x+1}{x^2-5 x+5}\right )+e^x \left (x^4-6 x^3+8 x^2+5 x-10\right )-15 x+10\right )}{11 (x+1) \left (-x^3+x \log \left (\frac {x+1}{x^2-5 x+5}\right )-3 x-e^x+1\right )^2}+\frac {(6-x) x \left (x^6-4 x^5-2 x^4+21 x^3-18 x^2+\left (x^3-4 x^2+5\right ) x \log \left (\frac {x+1}{x^2-5 x+5}\right )+e^x \left (x^4-6 x^3+8 x^2+5 x-10\right )-15 x+10\right )}{11 \left (x^2-5 x+5\right ) \left (-x^3+x \log \left (\frac {x+1}{x^2-5 x+5}\right )-3 x-e^x+1\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {x \left (x^6-4 x^5-2 x^4+21 x^3-18 x^2+\left (x^3-4 x^2+5\right ) x \log \left (\frac {x+1}{x^2-5 x+5}\right )+e^x \left (x^4-6 x^3+8 x^2+5 x-10\right )-15 x+10\right )}{\left (x^3-4 x^2+5\right ) \left (-x^3+x \log \left (\frac {x+1}{x^2-5 x+5}\right )-3 x-e^x+1\right )^2}dx\) |
| \(\Big \downarrow \) 2463 |
| \(\displaystyle \int \left (\frac {x \left (x^6-4 x^5-2 x^4+21 x^3-18 x^2+\left (x^3-4 x^2+5\right ) x \log \left (\frac {x+1}{x^2-5 x+5}\right )+e^x \left (x^4-6 x^3+8 x^2+5 x-10\right )-15 x+10\right )}{11 (x+1) \left (-x^3+x \log \left (\frac {x+1}{x^2-5 x+5}\right )-3 x-e^x+1\right )^2}+\frac {(6-x) x \left (x^6-4 x^5-2 x^4+21 x^3-18 x^2+\left (x^3-4 x^2+5\right ) x \log \left (\frac {x+1}{x^2-5 x+5}\right )+e^x \left (x^4-6 x^3+8 x^2+5 x-10\right )-15 x+10\right )}{11 \left (x^2-5 x+5\right ) \left (-x^3+x \log \left (\frac {x+1}{x^2-5 x+5}\right )-3 x-e^x+1\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {x \left (x^6-4 x^5-2 x^4+21 x^3-18 x^2+\left (x^3-4 x^2+5\right ) x \log \left (\frac {x+1}{x^2-5 x+5}\right )+e^x \left (x^4-6 x^3+8 x^2+5 x-10\right )-15 x+10\right )}{\left (x^3-4 x^2+5\right ) \left (-x^3+x \log \left (\frac {x+1}{x^2-5 x+5}\right )-3 x-e^x+1\right )^2}dx\) |
| \(\Big \downarrow \) 2463 |
| \(\displaystyle \int \left (\frac {x \left (x^6-4 x^5-2 x^4+21 x^3-18 x^2+\left (x^3-4 x^2+5\right ) x \log \left (\frac {x+1}{x^2-5 x+5}\right )+e^x \left (x^4-6 x^3+8 x^2+5 x-10\right )-15 x+10\right )}{11 (x+1) \left (-x^3+x \log \left (\frac {x+1}{x^2-5 x+5}\right )-3 x-e^x+1\right )^2}+\frac {(6-x) x \left (x^6-4 x^5-2 x^4+21 x^3-18 x^2+\left (x^3-4 x^2+5\right ) x \log \left (\frac {x+1}{x^2-5 x+5}\right )+e^x \left (x^4-6 x^3+8 x^2+5 x-10\right )-15 x+10\right )}{11 \left (x^2-5 x+5\right ) \left (-x^3+x \log \left (\frac {x+1}{x^2-5 x+5}\right )-3 x-e^x+1\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {x \left (x^6-4 x^5-2 x^4+21 x^3-18 x^2+\left (x^3-4 x^2+5\right ) x \log \left (\frac {x+1}{x^2-5 x+5}\right )+e^x \left (x^4-6 x^3+8 x^2+5 x-10\right )-15 x+10\right )}{\left (x^3-4 x^2+5\right ) \left (-x^3+x \log \left (\frac {x+1}{x^2-5 x+5}\right )-3 x-e^x+1\right )^2}dx\) |
| \(\Big \downarrow \) 2463 |
| \(\displaystyle \int \left (\frac {x \left (x^6-4 x^5-2 x^4+21 x^3-18 x^2+\left (x^3-4 x^2+5\right ) x \log \left (\frac {x+1}{x^2-5 x+5}\right )+e^x \left (x^4-6 x^3+8 x^2+5 x-10\right )-15 x+10\right )}{11 (x+1) \left (-x^3+x \log \left (\frac {x+1}{x^2-5 x+5}\right )-3 x-e^x+1\right )^2}+\frac {(6-x) x \left (x^6-4 x^5-2 x^4+21 x^3-18 x^2+\left (x^3-4 x^2+5\right ) x \log \left (\frac {x+1}{x^2-5 x+5}\right )+e^x \left (x^4-6 x^3+8 x^2+5 x-10\right )-15 x+10\right )}{11 \left (x^2-5 x+5\right ) \left (-x^3+x \log \left (\frac {x+1}{x^2-5 x+5}\right )-3 x-e^x+1\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {x \left (x^6-4 x^5-2 x^4+21 x^3-18 x^2+\left (x^3-4 x^2+5\right ) x \log \left (\frac {x+1}{x^2-5 x+5}\right )+e^x \left (x^4-6 x^3+8 x^2+5 x-10\right )-15 x+10\right )}{\left (x^3-4 x^2+5\right ) \left (-x^3+x \log \left (\frac {x+1}{x^2-5 x+5}\right )-3 x-e^x+1\right )^2}dx\) |
| \(\Big \downarrow \) 2463 |
| \(\displaystyle \int \left (\frac {x \left (x^6-4 x^5-2 x^4+21 x^3-18 x^2+\left (x^3-4 x^2+5\right ) x \log \left (\frac {x+1}{x^2-5 x+5}\right )+e^x \left (x^4-6 x^3+8 x^2+5 x-10\right )-15 x+10\right )}{11 (x+1) \left (-x^3+x \log \left (\frac {x+1}{x^2-5 x+5}\right )-3 x-e^x+1\right )^2}+\frac {(6-x) x \left (x^6-4 x^5-2 x^4+21 x^3-18 x^2+\left (x^3-4 x^2+5\right ) x \log \left (\frac {x+1}{x^2-5 x+5}\right )+e^x \left (x^4-6 x^3+8 x^2+5 x-10\right )-15 x+10\right )}{11 \left (x^2-5 x+5\right ) \left (-x^3+x \log \left (\frac {x+1}{x^2-5 x+5}\right )-3 x-e^x+1\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {x \left (x^6-4 x^5-2 x^4+21 x^3-18 x^2+\left (x^3-4 x^2+5\right ) x \log \left (\frac {x+1}{x^2-5 x+5}\right )+e^x \left (x^4-6 x^3+8 x^2+5 x-10\right )-15 x+10\right )}{\left (x^3-4 x^2+5\right ) \left (-x^3+x \log \left (\frac {x+1}{x^2-5 x+5}\right )-3 x-e^x+1\right )^2}dx\) |
| \(\Big \downarrow \) 2463 |
| \(\displaystyle \int \left (\frac {x \left (x^6-4 x^5-2 x^4+21 x^3-18 x^2+\left (x^3-4 x^2+5\right ) x \log \left (\frac {x+1}{x^2-5 x+5}\right )+e^x \left (x^4-6 x^3+8 x^2+5 x-10\right )-15 x+10\right )}{11 (x+1) \left (-x^3+x \log \left (\frac {x+1}{x^2-5 x+5}\right )-3 x-e^x+1\right )^2}+\frac {(6-x) x \left (x^6-4 x^5-2 x^4+21 x^3-18 x^2+\left (x^3-4 x^2+5\right ) x \log \left (\frac {x+1}{x^2-5 x+5}\right )+e^x \left (x^4-6 x^3+8 x^2+5 x-10\right )-15 x+10\right )}{11 \left (x^2-5 x+5\right ) \left (-x^3+x \log \left (\frac {x+1}{x^2-5 x+5}\right )-3 x-e^x+1\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {x \left (x^6-4 x^5-2 x^4+21 x^3-18 x^2+\left (x^3-4 x^2+5\right ) x \log \left (\frac {x+1}{x^2-5 x+5}\right )+e^x \left (x^4-6 x^3+8 x^2+5 x-10\right )-15 x+10\right )}{\left (x^3-4 x^2+5\right ) \left (-x^3+x \log \left (\frac {x+1}{x^2-5 x+5}\right )-3 x-e^x+1\right )^2}dx\) |
| \(\Big \downarrow \) 2463 |
| \(\displaystyle \int \left (\frac {x \left (x^6-4 x^5-2 x^4+21 x^3-18 x^2+\left (x^3-4 x^2+5\right ) x \log \left (\frac {x+1}{x^2-5 x+5}\right )+e^x \left (x^4-6 x^3+8 x^2+5 x-10\right )-15 x+10\right )}{11 (x+1) \left (-x^3+x \log \left (\frac {x+1}{x^2-5 x+5}\right )-3 x-e^x+1\right )^2}+\frac {(6-x) x \left (x^6-4 x^5-2 x^4+21 x^3-18 x^2+\left (x^3-4 x^2+5\right ) x \log \left (\frac {x+1}{x^2-5 x+5}\right )+e^x \left (x^4-6 x^3+8 x^2+5 x-10\right )-15 x+10\right )}{11 \left (x^2-5 x+5\right ) \left (-x^3+x \log \left (\frac {x+1}{x^2-5 x+5}\right )-3 x-e^x+1\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {x \left (x^6-4 x^5-2 x^4+21 x^3-18 x^2+\left (x^3-4 x^2+5\right ) x \log \left (\frac {x+1}{x^2-5 x+5}\right )+e^x \left (x^4-6 x^3+8 x^2+5 x-10\right )-15 x+10\right )}{\left (x^3-4 x^2+5\right ) \left (-x^3+x \log \left (\frac {x+1}{x^2-5 x+5}\right )-3 x-e^x+1\right )^2}dx\) |
| \(\Big \downarrow \) 2463 |
| \(\displaystyle \int \left (\frac {x \left (x^6-4 x^5-2 x^4+21 x^3-18 x^2+\left (x^3-4 x^2+5\right ) x \log \left (\frac {x+1}{x^2-5 x+5}\right )+e^x \left (x^4-6 x^3+8 x^2+5 x-10\right )-15 x+10\right )}{11 (x+1) \left (-x^3+x \log \left (\frac {x+1}{x^2-5 x+5}\right )-3 x-e^x+1\right )^2}+\frac {(6-x) x \left (x^6-4 x^5-2 x^4+21 x^3-18 x^2+\left (x^3-4 x^2+5\right ) x \log \left (\frac {x+1}{x^2-5 x+5}\right )+e^x \left (x^4-6 x^3+8 x^2+5 x-10\right )-15 x+10\right )}{11 \left (x^2-5 x+5\right ) \left (-x^3+x \log \left (\frac {x+1}{x^2-5 x+5}\right )-3 x-e^x+1\right )^2}\right )dx\) |
Input:
Int[(10*x - 15*x^2 - 18*x^3 + 21*x^4 - 2*x^5 - 4*x^6 + x^7 + E^x*(-10*x + 5*x^2 + 8*x^3 - 6*x^4 + x^5) + (5*x^2 - 4*x^4 + x^5)*Log[(1 + x)/(5 - 5*x + x^2)])/(5 - 30*x + 41*x^2 + 15*x^3 - 12*x^4 + 17*x^5 - 21*x^6 + 6*x^7 - 4*x^8 + x^9 + E^(2*x)*(5 - 4*x^2 + x^3) + E^x*(-10 + 30*x + 8*x^2 - 16*x^3 + 6*x^4 - 8*x^5 + 2*x^6) + (10*x - 30*x^2 - 8*x^3 + 16*x^4 - 6*x^5 + 8*x^ 6 - 2*x^7 + E^x*(-10*x + 8*x^3 - 2*x^4))*Log[(1 + x)/(5 - 5*x + x^2)] + (5 *x^2 - 4*x^4 + x^5)*Log[(1 + x)/(5 - 5*x + x^2)]^2),x]
Output:
$Aborted
Time = 18.68 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.00
| method | result | size |
| parallelrisch | \(-\frac {x^{2}}{x^{3}-x \ln \left (\frac {1+x}{x^{2}-5 x +5}\right )+3 x +{\mathrm e}^{x}-1}\) | \(36\) |
| risch | \(-\frac {2 x^{2}}{i \pi x \,\operatorname {csgn}\left (i \left (1+x \right )\right ) \operatorname {csgn}\left (\frac {i}{x^{2}-5 x +5}\right ) \operatorname {csgn}\left (\frac {i \left (1+x \right )}{x^{2}-5 x +5}\right )-i \pi x \,\operatorname {csgn}\left (i \left (1+x \right )\right ) \operatorname {csgn}\left (\frac {i \left (1+x \right )}{x^{2}-5 x +5}\right )^{2}-i \pi x \,\operatorname {csgn}\left (\frac {i}{x^{2}-5 x +5}\right ) \operatorname {csgn}\left (\frac {i \left (1+x \right )}{x^{2}-5 x +5}\right )^{2}+i \pi x \operatorname {csgn}\left (\frac {i \left (1+x \right )}{x^{2}-5 x +5}\right )^{3}+2 x^{3}-2 x \ln \left (1+x \right )+2 x \ln \left (x^{2}-5 x +5\right )+6 x +2 \,{\mathrm e}^{x}-2}\) | \(177\) |
Input:
int(((x^5-4*x^4+5*x^2)*ln((1+x)/(x^2-5*x+5))+(x^5-6*x^4+8*x^3+5*x^2-10*x)* exp(x)+x^7-4*x^6-2*x^5+21*x^4-18*x^3-15*x^2+10*x)/((x^5-4*x^4+5*x^2)*ln((1 +x)/(x^2-5*x+5))^2+((-2*x^4+8*x^3-10*x)*exp(x)-2*x^7+8*x^6-6*x^5+16*x^4-8* x^3-30*x^2+10*x)*ln((1+x)/(x^2-5*x+5))+(x^3-4*x^2+5)*exp(x)^2+(2*x^6-8*x^5 +6*x^4-16*x^3+8*x^2+30*x-10)*exp(x)+x^9-4*x^8+6*x^7-21*x^6+17*x^5-12*x^4+1 5*x^3+41*x^2-30*x+5),x,method=_RETURNVERBOSE)
Output:
-x^2/(x^3-x*ln((1+x)/(x^2-5*x+5))+3*x+exp(x)-1)
Time = 0.10 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.97 \[ \int \frac {10 x-15 x^2-18 x^3+21 x^4-2 x^5-4 x^6+x^7+e^x \left (-10 x+5 x^2+8 x^3-6 x^4+x^5\right )+\left (5 x^2-4 x^4+x^5\right ) \log \left (\frac {1+x}{5-5 x+x^2}\right )}{5-30 x+41 x^2+15 x^3-12 x^4+17 x^5-21 x^6+6 x^7-4 x^8+x^9+e^{2 x} \left (5-4 x^2+x^3\right )+e^x \left (-10+30 x+8 x^2-16 x^3+6 x^4-8 x^5+2 x^6\right )+\left (10 x-30 x^2-8 x^3+16 x^4-6 x^5+8 x^6-2 x^7+e^x \left (-10 x+8 x^3-2 x^4\right )\right ) \log \left (\frac {1+x}{5-5 x+x^2}\right )+\left (5 x^2-4 x^4+x^5\right ) \log ^2\left (\frac {1+x}{5-5 x+x^2}\right )} \, dx=-\frac {x^{2}}{x^{3} - x \log \left (\frac {x + 1}{x^{2} - 5 \, x + 5}\right ) + 3 \, x + e^{x} - 1} \] Input:
integrate(((x^5-4*x^4+5*x^2)*log((1+x)/(x^2-5*x+5))+(x^5-6*x^4+8*x^3+5*x^2 -10*x)*exp(x)+x^7-4*x^6-2*x^5+21*x^4-18*x^3-15*x^2+10*x)/((x^5-4*x^4+5*x^2 )*log((1+x)/(x^2-5*x+5))^2+((-2*x^4+8*x^3-10*x)*exp(x)-2*x^7+8*x^6-6*x^5+1 6*x^4-8*x^3-30*x^2+10*x)*log((1+x)/(x^2-5*x+5))+(x^3-4*x^2+5)*exp(x)^2+(2* x^6-8*x^5+6*x^4-16*x^3+8*x^2+30*x-10)*exp(x)+x^9-4*x^8+6*x^7-21*x^6+17*x^5 -12*x^4+15*x^3+41*x^2-30*x+5),x, algorithm="fricas")
Output:
-x^2/(x^3 - x*log((x + 1)/(x^2 - 5*x + 5)) + 3*x + e^x - 1)
Time = 0.22 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.86 \[ \int \frac {10 x-15 x^2-18 x^3+21 x^4-2 x^5-4 x^6+x^7+e^x \left (-10 x+5 x^2+8 x^3-6 x^4+x^5\right )+\left (5 x^2-4 x^4+x^5\right ) \log \left (\frac {1+x}{5-5 x+x^2}\right )}{5-30 x+41 x^2+15 x^3-12 x^4+17 x^5-21 x^6+6 x^7-4 x^8+x^9+e^{2 x} \left (5-4 x^2+x^3\right )+e^x \left (-10+30 x+8 x^2-16 x^3+6 x^4-8 x^5+2 x^6\right )+\left (10 x-30 x^2-8 x^3+16 x^4-6 x^5+8 x^6-2 x^7+e^x \left (-10 x+8 x^3-2 x^4\right )\right ) \log \left (\frac {1+x}{5-5 x+x^2}\right )+\left (5 x^2-4 x^4+x^5\right ) \log ^2\left (\frac {1+x}{5-5 x+x^2}\right )} \, dx=- \frac {x^{2}}{x^{3} - x \log {\left (\frac {x + 1}{x^{2} - 5 x + 5} \right )} + 3 x + e^{x} - 1} \] Input:
integrate(((x**5-4*x**4+5*x**2)*ln((1+x)/(x**2-5*x+5))+(x**5-6*x**4+8*x**3 +5*x**2-10*x)*exp(x)+x**7-4*x**6-2*x**5+21*x**4-18*x**3-15*x**2+10*x)/((x* *5-4*x**4+5*x**2)*ln((1+x)/(x**2-5*x+5))**2+((-2*x**4+8*x**3-10*x)*exp(x)- 2*x**7+8*x**6-6*x**5+16*x**4-8*x**3-30*x**2+10*x)*ln((1+x)/(x**2-5*x+5))+( x**3-4*x**2+5)*exp(x)**2+(2*x**6-8*x**5+6*x**4-16*x**3+8*x**2+30*x-10)*exp (x)+x**9-4*x**8+6*x**7-21*x**6+17*x**5-12*x**4+15*x**3+41*x**2-30*x+5),x)
Output:
-x**2/(x**3 - x*log((x + 1)/(x**2 - 5*x + 5)) + 3*x + exp(x) - 1)
Time = 0.23 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.97 \[ \int \frac {10 x-15 x^2-18 x^3+21 x^4-2 x^5-4 x^6+x^7+e^x \left (-10 x+5 x^2+8 x^3-6 x^4+x^5\right )+\left (5 x^2-4 x^4+x^5\right ) \log \left (\frac {1+x}{5-5 x+x^2}\right )}{5-30 x+41 x^2+15 x^3-12 x^4+17 x^5-21 x^6+6 x^7-4 x^8+x^9+e^{2 x} \left (5-4 x^2+x^3\right )+e^x \left (-10+30 x+8 x^2-16 x^3+6 x^4-8 x^5+2 x^6\right )+\left (10 x-30 x^2-8 x^3+16 x^4-6 x^5+8 x^6-2 x^7+e^x \left (-10 x+8 x^3-2 x^4\right )\right ) \log \left (\frac {1+x}{5-5 x+x^2}\right )+\left (5 x^2-4 x^4+x^5\right ) \log ^2\left (\frac {1+x}{5-5 x+x^2}\right )} \, dx=-\frac {x^{2}}{x^{3} + x \log \left (x^{2} - 5 \, x + 5\right ) - x \log \left (x + 1\right ) + 3 \, x + e^{x} - 1} \] Input:
integrate(((x^5-4*x^4+5*x^2)*log((1+x)/(x^2-5*x+5))+(x^5-6*x^4+8*x^3+5*x^2 -10*x)*exp(x)+x^7-4*x^6-2*x^5+21*x^4-18*x^3-15*x^2+10*x)/((x^5-4*x^4+5*x^2 )*log((1+x)/(x^2-5*x+5))^2+((-2*x^4+8*x^3-10*x)*exp(x)-2*x^7+8*x^6-6*x^5+1 6*x^4-8*x^3-30*x^2+10*x)*log((1+x)/(x^2-5*x+5))+(x^3-4*x^2+5)*exp(x)^2+(2* x^6-8*x^5+6*x^4-16*x^3+8*x^2+30*x-10)*exp(x)+x^9-4*x^8+6*x^7-21*x^6+17*x^5 -12*x^4+15*x^3+41*x^2-30*x+5),x, algorithm="maxima")
Output:
-x^2/(x^3 + x*log(x^2 - 5*x + 5) - x*log(x + 1) + 3*x + e^x - 1)
Time = 0.36 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.97 \[ \int \frac {10 x-15 x^2-18 x^3+21 x^4-2 x^5-4 x^6+x^7+e^x \left (-10 x+5 x^2+8 x^3-6 x^4+x^5\right )+\left (5 x^2-4 x^4+x^5\right ) \log \left (\frac {1+x}{5-5 x+x^2}\right )}{5-30 x+41 x^2+15 x^3-12 x^4+17 x^5-21 x^6+6 x^7-4 x^8+x^9+e^{2 x} \left (5-4 x^2+x^3\right )+e^x \left (-10+30 x+8 x^2-16 x^3+6 x^4-8 x^5+2 x^6\right )+\left (10 x-30 x^2-8 x^3+16 x^4-6 x^5+8 x^6-2 x^7+e^x \left (-10 x+8 x^3-2 x^4\right )\right ) \log \left (\frac {1+x}{5-5 x+x^2}\right )+\left (5 x^2-4 x^4+x^5\right ) \log ^2\left (\frac {1+x}{5-5 x+x^2}\right )} \, dx=-\frac {x^{2}}{x^{3} - x \log \left (\frac {x + 1}{x^{2} - 5 \, x + 5}\right ) + 3 \, x + e^{x} - 1} \] Input:
integrate(((x^5-4*x^4+5*x^2)*log((1+x)/(x^2-5*x+5))+(x^5-6*x^4+8*x^3+5*x^2 -10*x)*exp(x)+x^7-4*x^6-2*x^5+21*x^4-18*x^3-15*x^2+10*x)/((x^5-4*x^4+5*x^2 )*log((1+x)/(x^2-5*x+5))^2+((-2*x^4+8*x^3-10*x)*exp(x)-2*x^7+8*x^6-6*x^5+1 6*x^4-8*x^3-30*x^2+10*x)*log((1+x)/(x^2-5*x+5))+(x^3-4*x^2+5)*exp(x)^2+(2* x^6-8*x^5+6*x^4-16*x^3+8*x^2+30*x-10)*exp(x)+x^9-4*x^8+6*x^7-21*x^6+17*x^5 -12*x^4+15*x^3+41*x^2-30*x+5),x, algorithm="giac")
Output:
-x^2/(x^3 - x*log((x + 1)/(x^2 - 5*x + 5)) + 3*x + e^x - 1)
Time = 2.59 (sec) , antiderivative size = 216, normalized size of antiderivative = 6.00 \[ \int \frac {10 x-15 x^2-18 x^3+21 x^4-2 x^5-4 x^6+x^7+e^x \left (-10 x+5 x^2+8 x^3-6 x^4+x^5\right )+\left (5 x^2-4 x^4+x^5\right ) \log \left (\frac {1+x}{5-5 x+x^2}\right )}{5-30 x+41 x^2+15 x^3-12 x^4+17 x^5-21 x^6+6 x^7-4 x^8+x^9+e^{2 x} \left (5-4 x^2+x^3\right )+e^x \left (-10+30 x+8 x^2-16 x^3+6 x^4-8 x^5+2 x^6\right )+\left (10 x-30 x^2-8 x^3+16 x^4-6 x^5+8 x^6-2 x^7+e^x \left (-10 x+8 x^3-2 x^4\right )\right ) \log \left (\frac {1+x}{5-5 x+x^2}\right )+\left (5 x^2-4 x^4+x^5\right ) \log ^2\left (\frac {1+x}{5-5 x+x^2}\right )} \, dx=-\frac {25\,x^3\,{\mathrm {e}}^x+x^9\,\left ({\mathrm {e}}^x+33\right )-x^8\,\left (9\,{\mathrm {e}}^x-19\right )-x^2\,\left (25\,{\mathrm {e}}^x-25\right )-x^6\,\left (6\,{\mathrm {e}}^x-61\right )-x^5\,\left (50\,{\mathrm {e}}^x-70\right )+x^4\,\left (40\,{\mathrm {e}}^x-90\right )+x^7\,\left (24\,{\mathrm {e}}^x-106\right )-16\,x^{10}+2\,x^{11}}{\left (3\,x+{\mathrm {e}}^x+x^3-x\,\ln \left (\frac {x+1}{x^2-5\,x+5}\right )-1\right )\,\left (40\,x^2\,{\mathrm {e}}^x-25\,{\mathrm {e}}^x-50\,x^3\,{\mathrm {e}}^x-6\,x^4\,{\mathrm {e}}^x+24\,x^5\,{\mathrm {e}}^x-9\,x^6\,{\mathrm {e}}^x+x^7\,{\mathrm {e}}^x+25\,x\,{\mathrm {e}}^x-90\,x^2+70\,x^3+61\,x^4-106\,x^5+19\,x^6+33\,x^7-16\,x^8+2\,x^9+25\right )} \] Input:
int((10*x + log((x + 1)/(x^2 - 5*x + 5))*(5*x^2 - 4*x^4 + x^5) - 15*x^2 - 18*x^3 + 21*x^4 - 2*x^5 - 4*x^6 + x^7 + exp(x)*(5*x^2 - 10*x + 8*x^3 - 6*x ^4 + x^5))/(exp(2*x)*(x^3 - 4*x^2 + 5) - 30*x + log((x + 1)/(x^2 - 5*x + 5 ))^2*(5*x^2 - 4*x^4 + x^5) + exp(x)*(30*x + 8*x^2 - 16*x^3 + 6*x^4 - 8*x^5 + 2*x^6 - 10) - log((x + 1)/(x^2 - 5*x + 5))*(30*x^2 - 10*x + 8*x^3 - 16* x^4 + 6*x^5 - 8*x^6 + 2*x^7 + exp(x)*(10*x - 8*x^3 + 2*x^4)) + 41*x^2 + 15 *x^3 - 12*x^4 + 17*x^5 - 21*x^6 + 6*x^7 - 4*x^8 + x^9 + 5),x)
Output:
-(25*x^3*exp(x) + x^9*(exp(x) + 33) - x^8*(9*exp(x) - 19) - x^2*(25*exp(x) - 25) - x^6*(6*exp(x) - 61) - x^5*(50*exp(x) - 70) + x^4*(40*exp(x) - 90) + x^7*(24*exp(x) - 106) - 16*x^10 + 2*x^11)/((3*x + exp(x) + x^3 - x*log( (x + 1)/(x^2 - 5*x + 5)) - 1)*(40*x^2*exp(x) - 25*exp(x) - 50*x^3*exp(x) - 6*x^4*exp(x) + 24*x^5*exp(x) - 9*x^6*exp(x) + x^7*exp(x) + 25*x*exp(x) - 90*x^2 + 70*x^3 + 61*x^4 - 106*x^5 + 19*x^6 + 33*x^7 - 16*x^8 + 2*x^9 + 25 ))
Time = 0.21 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.00 \[ \int \frac {10 x-15 x^2-18 x^3+21 x^4-2 x^5-4 x^6+x^7+e^x \left (-10 x+5 x^2+8 x^3-6 x^4+x^5\right )+\left (5 x^2-4 x^4+x^5\right ) \log \left (\frac {1+x}{5-5 x+x^2}\right )}{5-30 x+41 x^2+15 x^3-12 x^4+17 x^5-21 x^6+6 x^7-4 x^8+x^9+e^{2 x} \left (5-4 x^2+x^3\right )+e^x \left (-10+30 x+8 x^2-16 x^3+6 x^4-8 x^5+2 x^6\right )+\left (10 x-30 x^2-8 x^3+16 x^4-6 x^5+8 x^6-2 x^7+e^x \left (-10 x+8 x^3-2 x^4\right )\right ) \log \left (\frac {1+x}{5-5 x+x^2}\right )+\left (5 x^2-4 x^4+x^5\right ) \log ^2\left (\frac {1+x}{5-5 x+x^2}\right )} \, dx=-\frac {x^{2}}{e^{x}-\mathrm {log}\left (\frac {x +1}{x^{2}-5 x +5}\right ) x +x^{3}+3 x -1} \] Input:
int(((x^5-4*x^4+5*x^2)*log((1+x)/(x^2-5*x+5))+(x^5-6*x^4+8*x^3+5*x^2-10*x) *exp(x)+x^7-4*x^6-2*x^5+21*x^4-18*x^3-15*x^2+10*x)/((x^5-4*x^4+5*x^2)*log( (1+x)/(x^2-5*x+5))^2+((-2*x^4+8*x^3-10*x)*exp(x)-2*x^7+8*x^6-6*x^5+16*x^4- 8*x^3-30*x^2+10*x)*log((1+x)/(x^2-5*x+5))+(x^3-4*x^2+5)*exp(x)^2+(2*x^6-8* x^5+6*x^4-16*x^3+8*x^2+30*x-10)*exp(x)+x^9-4*x^8+6*x^7-21*x^6+17*x^5-12*x^ 4+15*x^3+41*x^2-30*x+5),x)
Output:
( - x**2)/(e**x - log((x + 1)/(x**2 - 5*x + 5))*x + x**3 + 3*x - 1)