Integrand size = 150, antiderivative size = 32 \[ \int \frac {-20-10 x-5 x^2+\left (8 x+8 x^2-2 x^3-2 x^4+e^{12 x} \left (-48-48 x+12 x^2+12 x^3\right )\right ) \log ^2\left (\frac {-4+x^2}{1+x}\right )}{\left (-20-20 x+5 x^2+5 x^3\right ) \log \left (\frac {-4+x^2}{1+x}\right )+\left (4 x^2+4 x^3-x^4-x^5+e^{12 x} \left (-4-4 x+x^2+x^3\right )\right ) \log ^2\left (\frac {-4+x^2}{1+x}\right )} \, dx=\log \left (\log (4) \left (e^{12 x}-x^2+\frac {5}{\log \left (x-\frac {4+x}{1+x}\right )}\right )\right ) \]
Time = 1.12 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.69 \[ \int \frac {-20-10 x-5 x^2+\left (8 x+8 x^2-2 x^3-2 x^4+e^{12 x} \left (-48-48 x+12 x^2+12 x^3\right )\right ) \log ^2\left (\frac {-4+x^2}{1+x}\right )}{\left (-20-20 x+5 x^2+5 x^3\right ) \log \left (\frac {-4+x^2}{1+x}\right )+\left (4 x^2+4 x^3-x^4-x^5+e^{12 x} \left (-4-4 x+x^2+x^3\right )\right ) \log ^2\left (\frac {-4+x^2}{1+x}\right )} \, dx=-\log \left (\log \left (\frac {-4+x^2}{1+x}\right )\right )+\log \left (5+e^{12 x} \log \left (\frac {-4+x^2}{1+x}\right )-x^2 \log \left (\frac {-4+x^2}{1+x}\right )\right ) \]
Integrate[(-20 - 10*x - 5*x^2 + (8*x + 8*x^2 - 2*x^3 - 2*x^4 + E^(12*x)*(- 48 - 48*x + 12*x^2 + 12*x^3))*Log[(-4 + x^2)/(1 + x)]^2)/((-20 - 20*x + 5* x^2 + 5*x^3)*Log[(-4 + x^2)/(1 + x)] + (4*x^2 + 4*x^3 - x^4 - x^5 + E^(12* x)*(-4 - 4*x + x^2 + x^3))*Log[(-4 + x^2)/(1 + x)]^2),x]
-Log[Log[(-4 + x^2)/(1 + x)]] + Log[5 + E^(12*x)*Log[(-4 + x^2)/(1 + x)] - x^2*Log[(-4 + x^2)/(1 + x)]]
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.
| \(\displaystyle \int \frac {-5 x^2+\left (-2 x^4-2 x^3+8 x^2+e^{12 x} \left (12 x^3+12 x^2-48 x-48\right )+8 x\right ) \log ^2\left (\frac {x^2-4}{x+1}\right )-10 x-20}{\left (5 x^3+5 x^2-20 x-20\right ) \log \left (\frac {x^2-4}{x+1}\right )+\left (-x^5-x^4+4 x^3+4 x^2+e^{12 x} \left (x^3+x^2-4 x-4\right )\right ) \log ^2\left (\frac {x^2-4}{x+1}\right )} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {5 x^2-\left (-2 x^4-2 x^3+8 x^2+e^{12 x} \left (12 x^3+12 x^2-48 x-48\right )+8 x\right ) \log ^2\left (\frac {x^2-4}{x+1}\right )+10 x+20}{\left (-x^3-x^2+4 x+4\right ) \log \left (\frac {x^2-4}{x+1}\right ) \left (x^2 \left (-\log \left (\frac {x^2-4}{x+1}\right )\right )+e^{12 x} \log \left (\frac {x^2-4}{x+1}\right )+5\right )}dx\) |
| \(\Big \downarrow \) 2463 |
| \(\displaystyle \int \left (-\frac {5 x^2-\left (-2 x^4-2 x^3+8 x^2+e^{12 x} \left (12 x^3+12 x^2-48 x-48\right )+8 x\right ) \log ^2\left (\frac {x^2-4}{x+1}\right )+10 x+20}{12 (x-2) \log \left (\frac {x^2-4}{x+1}\right ) \left (x^2 \left (-\log \left (\frac {x^2-4}{x+1}\right )\right )+e^{12 x} \log \left (\frac {x^2-4}{x+1}\right )+5\right )}+\frac {5 x^2-\left (-2 x^4-2 x^3+8 x^2+e^{12 x} \left (12 x^3+12 x^2-48 x-48\right )+8 x\right ) \log ^2\left (\frac {x^2-4}{x+1}\right )+10 x+20}{3 (x+1) \log \left (\frac {x^2-4}{x+1}\right ) \left (x^2 \left (-\log \left (\frac {x^2-4}{x+1}\right )\right )+e^{12 x} \log \left (\frac {x^2-4}{x+1}\right )+5\right )}-\frac {5 x^2-\left (-2 x^4-2 x^3+8 x^2+e^{12 x} \left (12 x^3+12 x^2-48 x-48\right )+8 x\right ) \log ^2\left (\frac {x^2-4}{x+1}\right )+10 x+20}{4 (x+2) \log \left (\frac {x^2-4}{x+1}\right ) \left (x^2 \left (-\log \left (\frac {x^2-4}{x+1}\right )\right )+e^{12 x} \log \left (\frac {x^2-4}{x+1}\right )+5\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {5 \left (x^2+2 x+4\right )-2 \left (6 e^{12 x}-x\right ) \left (x^3+x^2-4 x-4\right ) \log ^2\left (\frac {x^2-4}{x+1}\right )}{(2-x) (x+1) (x+2) \log \left (\frac {x^2-4}{x+1}\right ) \left (5-\left (x^2-e^{12 x}\right ) \log \left (\frac {x^2-4}{x+1}\right )\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (12-\frac {-5 x^2-40 x^2 \log ^2\left (\frac {x^2-4}{x+1}\right )+8 x \log ^2\left (\frac {x^2-4}{x+1}\right )-60 x^2 \log \left (\frac {x^2-4}{x+1}\right )+240 x \log \left (\frac {x^2-4}{x+1}\right )+240 \log \left (\frac {x^2-4}{x+1}\right )+12 x^5 \log ^2\left (\frac {x^2-4}{x+1}\right )+10 x^4 \log ^2\left (\frac {x^2-4}{x+1}\right )-50 x^3 \log ^2\left (\frac {x^2-4}{x+1}\right )-60 x^3 \log \left (\frac {x^2-4}{x+1}\right )-10 x-20}{(x-2) (x+1) (x+2) \log \left (\frac {x^2-4}{x+1}\right ) \left (x^2 \log \left (\frac {x^2-4}{x+1}\right )-e^{12 x} \log \left (\frac {x^2-4}{x+1}\right )-5\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {5 \left (x^2+2 x+4\right )-2 \left (6 e^{12 x}-x\right ) \left (x^3+x^2-4 x-4\right ) \log ^2\left (\frac {x^2-4}{x+1}\right )}{(2-x) (x+1) (x+2) \log \left (\frac {x^2-4}{x+1}\right ) \left (5-\left (x^2-e^{12 x}\right ) \log \left (\frac {x^2-4}{x+1}\right )\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (12-\frac {-5 x^2-40 x^2 \log ^2\left (\frac {x^2-4}{x+1}\right )+8 x \log ^2\left (\frac {x^2-4}{x+1}\right )-60 x^2 \log \left (\frac {x^2-4}{x+1}\right )+240 x \log \left (\frac {x^2-4}{x+1}\right )+240 \log \left (\frac {x^2-4}{x+1}\right )+12 x^5 \log ^2\left (\frac {x^2-4}{x+1}\right )+10 x^4 \log ^2\left (\frac {x^2-4}{x+1}\right )-50 x^3 \log ^2\left (\frac {x^2-4}{x+1}\right )-60 x^3 \log \left (\frac {x^2-4}{x+1}\right )-10 x-20}{(x-2) (x+1) (x+2) \log \left (\frac {x^2-4}{x+1}\right ) \left (x^2 \log \left (\frac {x^2-4}{x+1}\right )-e^{12 x} \log \left (\frac {x^2-4}{x+1}\right )-5\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {5 \left (x^2+2 x+4\right )-2 \left (6 e^{12 x}-x\right ) \left (x^3+x^2-4 x-4\right ) \log ^2\left (\frac {x^2-4}{x+1}\right )}{(2-x) (x+1) (x+2) \log \left (\frac {x^2-4}{x+1}\right ) \left (5-\left (x^2-e^{12 x}\right ) \log \left (\frac {x^2-4}{x+1}\right )\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (12-\frac {-5 x^2-40 x^2 \log ^2\left (\frac {x^2-4}{x+1}\right )+8 x \log ^2\left (\frac {x^2-4}{x+1}\right )-60 x^2 \log \left (\frac {x^2-4}{x+1}\right )+240 x \log \left (\frac {x^2-4}{x+1}\right )+240 \log \left (\frac {x^2-4}{x+1}\right )+12 x^5 \log ^2\left (\frac {x^2-4}{x+1}\right )+10 x^4 \log ^2\left (\frac {x^2-4}{x+1}\right )-50 x^3 \log ^2\left (\frac {x^2-4}{x+1}\right )-60 x^3 \log \left (\frac {x^2-4}{x+1}\right )-10 x-20}{(x-2) (x+1) (x+2) \log \left (\frac {x^2-4}{x+1}\right ) \left (x^2 \log \left (\frac {x^2-4}{x+1}\right )-e^{12 x} \log \left (\frac {x^2-4}{x+1}\right )-5\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {5 \left (x^2+2 x+4\right )-2 \left (6 e^{12 x}-x\right ) \left (x^3+x^2-4 x-4\right ) \log ^2\left (\frac {x^2-4}{x+1}\right )}{(2-x) (x+1) (x+2) \log \left (\frac {x^2-4}{x+1}\right ) \left (5-\left (x^2-e^{12 x}\right ) \log \left (\frac {x^2-4}{x+1}\right )\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (12-\frac {-5 x^2-40 x^2 \log ^2\left (\frac {x^2-4}{x+1}\right )+8 x \log ^2\left (\frac {x^2-4}{x+1}\right )-60 x^2 \log \left (\frac {x^2-4}{x+1}\right )+240 x \log \left (\frac {x^2-4}{x+1}\right )+240 \log \left (\frac {x^2-4}{x+1}\right )+12 x^5 \log ^2\left (\frac {x^2-4}{x+1}\right )+10 x^4 \log ^2\left (\frac {x^2-4}{x+1}\right )-50 x^3 \log ^2\left (\frac {x^2-4}{x+1}\right )-60 x^3 \log \left (\frac {x^2-4}{x+1}\right )-10 x-20}{(x-2) (x+1) (x+2) \log \left (\frac {x^2-4}{x+1}\right ) \left (x^2 \log \left (\frac {x^2-4}{x+1}\right )-e^{12 x} \log \left (\frac {x^2-4}{x+1}\right )-5\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {5 \left (x^2+2 x+4\right )-2 \left (6 e^{12 x}-x\right ) \left (x^3+x^2-4 x-4\right ) \log ^2\left (\frac {x^2-4}{x+1}\right )}{(2-x) (x+1) (x+2) \log \left (\frac {x^2-4}{x+1}\right ) \left (5-\left (x^2-e^{12 x}\right ) \log \left (\frac {x^2-4}{x+1}\right )\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (12-\frac {-5 x^2-40 x^2 \log ^2\left (\frac {x^2-4}{x+1}\right )+8 x \log ^2\left (\frac {x^2-4}{x+1}\right )-60 x^2 \log \left (\frac {x^2-4}{x+1}\right )+240 x \log \left (\frac {x^2-4}{x+1}\right )+240 \log \left (\frac {x^2-4}{x+1}\right )+12 x^5 \log ^2\left (\frac {x^2-4}{x+1}\right )+10 x^4 \log ^2\left (\frac {x^2-4}{x+1}\right )-50 x^3 \log ^2\left (\frac {x^2-4}{x+1}\right )-60 x^3 \log \left (\frac {x^2-4}{x+1}\right )-10 x-20}{(x-2) (x+1) (x+2) \log \left (\frac {x^2-4}{x+1}\right ) \left (x^2 \log \left (\frac {x^2-4}{x+1}\right )-e^{12 x} \log \left (\frac {x^2-4}{x+1}\right )-5\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {5 \left (x^2+2 x+4\right )-2 \left (6 e^{12 x}-x\right ) \left (x^3+x^2-4 x-4\right ) \log ^2\left (\frac {x^2-4}{x+1}\right )}{(2-x) (x+1) (x+2) \log \left (\frac {x^2-4}{x+1}\right ) \left (5-\left (x^2-e^{12 x}\right ) \log \left (\frac {x^2-4}{x+1}\right )\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (12-\frac {-5 x^2-40 x^2 \log ^2\left (\frac {x^2-4}{x+1}\right )+8 x \log ^2\left (\frac {x^2-4}{x+1}\right )-60 x^2 \log \left (\frac {x^2-4}{x+1}\right )+240 x \log \left (\frac {x^2-4}{x+1}\right )+240 \log \left (\frac {x^2-4}{x+1}\right )+12 x^5 \log ^2\left (\frac {x^2-4}{x+1}\right )+10 x^4 \log ^2\left (\frac {x^2-4}{x+1}\right )-50 x^3 \log ^2\left (\frac {x^2-4}{x+1}\right )-60 x^3 \log \left (\frac {x^2-4}{x+1}\right )-10 x-20}{(x-2) (x+1) (x+2) \log \left (\frac {x^2-4}{x+1}\right ) \left (x^2 \log \left (\frac {x^2-4}{x+1}\right )-e^{12 x} \log \left (\frac {x^2-4}{x+1}\right )-5\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {5 \left (x^2+2 x+4\right )-2 \left (6 e^{12 x}-x\right ) \left (x^3+x^2-4 x-4\right ) \log ^2\left (\frac {x^2-4}{x+1}\right )}{(2-x) (x+1) (x+2) \log \left (\frac {x^2-4}{x+1}\right ) \left (5-\left (x^2-e^{12 x}\right ) \log \left (\frac {x^2-4}{x+1}\right )\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (12-\frac {-5 x^2-40 x^2 \log ^2\left (\frac {x^2-4}{x+1}\right )+8 x \log ^2\left (\frac {x^2-4}{x+1}\right )-60 x^2 \log \left (\frac {x^2-4}{x+1}\right )+240 x \log \left (\frac {x^2-4}{x+1}\right )+240 \log \left (\frac {x^2-4}{x+1}\right )+12 x^5 \log ^2\left (\frac {x^2-4}{x+1}\right )+10 x^4 \log ^2\left (\frac {x^2-4}{x+1}\right )-50 x^3 \log ^2\left (\frac {x^2-4}{x+1}\right )-60 x^3 \log \left (\frac {x^2-4}{x+1}\right )-10 x-20}{(x-2) (x+1) (x+2) \log \left (\frac {x^2-4}{x+1}\right ) \left (x^2 \log \left (\frac {x^2-4}{x+1}\right )-e^{12 x} \log \left (\frac {x^2-4}{x+1}\right )-5\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {5 \left (x^2+2 x+4\right )-2 \left (6 e^{12 x}-x\right ) \left (x^3+x^2-4 x-4\right ) \log ^2\left (\frac {x^2-4}{x+1}\right )}{(2-x) (x+1) (x+2) \log \left (\frac {x^2-4}{x+1}\right ) \left (5-\left (x^2-e^{12 x}\right ) \log \left (\frac {x^2-4}{x+1}\right )\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (12-\frac {-5 x^2-40 x^2 \log ^2\left (\frac {x^2-4}{x+1}\right )+8 x \log ^2\left (\frac {x^2-4}{x+1}\right )-60 x^2 \log \left (\frac {x^2-4}{x+1}\right )+240 x \log \left (\frac {x^2-4}{x+1}\right )+240 \log \left (\frac {x^2-4}{x+1}\right )+12 x^5 \log ^2\left (\frac {x^2-4}{x+1}\right )+10 x^4 \log ^2\left (\frac {x^2-4}{x+1}\right )-50 x^3 \log ^2\left (\frac {x^2-4}{x+1}\right )-60 x^3 \log \left (\frac {x^2-4}{x+1}\right )-10 x-20}{(x-2) (x+1) (x+2) \log \left (\frac {x^2-4}{x+1}\right ) \left (x^2 \log \left (\frac {x^2-4}{x+1}\right )-e^{12 x} \log \left (\frac {x^2-4}{x+1}\right )-5\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {5 \left (x^2+2 x+4\right )-2 \left (6 e^{12 x}-x\right ) \left (x^3+x^2-4 x-4\right ) \log ^2\left (\frac {x^2-4}{x+1}\right )}{(2-x) (x+1) (x+2) \log \left (\frac {x^2-4}{x+1}\right ) \left (5-\left (x^2-e^{12 x}\right ) \log \left (\frac {x^2-4}{x+1}\right )\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (12-\frac {-5 x^2-40 x^2 \log ^2\left (\frac {x^2-4}{x+1}\right )+8 x \log ^2\left (\frac {x^2-4}{x+1}\right )-60 x^2 \log \left (\frac {x^2-4}{x+1}\right )+240 x \log \left (\frac {x^2-4}{x+1}\right )+240 \log \left (\frac {x^2-4}{x+1}\right )+12 x^5 \log ^2\left (\frac {x^2-4}{x+1}\right )+10 x^4 \log ^2\left (\frac {x^2-4}{x+1}\right )-50 x^3 \log ^2\left (\frac {x^2-4}{x+1}\right )-60 x^3 \log \left (\frac {x^2-4}{x+1}\right )-10 x-20}{(x-2) (x+1) (x+2) \log \left (\frac {x^2-4}{x+1}\right ) \left (x^2 \log \left (\frac {x^2-4}{x+1}\right )-e^{12 x} \log \left (\frac {x^2-4}{x+1}\right )-5\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {5 \left (x^2+2 x+4\right )-2 \left (6 e^{12 x}-x\right ) \left (x^3+x^2-4 x-4\right ) \log ^2\left (\frac {x^2-4}{x+1}\right )}{(2-x) (x+1) (x+2) \log \left (\frac {x^2-4}{x+1}\right ) \left (5-\left (x^2-e^{12 x}\right ) \log \left (\frac {x^2-4}{x+1}\right )\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (12-\frac {-5 x^2-40 x^2 \log ^2\left (\frac {x^2-4}{x+1}\right )+8 x \log ^2\left (\frac {x^2-4}{x+1}\right )-60 x^2 \log \left (\frac {x^2-4}{x+1}\right )+240 x \log \left (\frac {x^2-4}{x+1}\right )+240 \log \left (\frac {x^2-4}{x+1}\right )+12 x^5 \log ^2\left (\frac {x^2-4}{x+1}\right )+10 x^4 \log ^2\left (\frac {x^2-4}{x+1}\right )-50 x^3 \log ^2\left (\frac {x^2-4}{x+1}\right )-60 x^3 \log \left (\frac {x^2-4}{x+1}\right )-10 x-20}{(x-2) (x+1) (x+2) \log \left (\frac {x^2-4}{x+1}\right ) \left (x^2 \log \left (\frac {x^2-4}{x+1}\right )-e^{12 x} \log \left (\frac {x^2-4}{x+1}\right )-5\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {5 \left (x^2+2 x+4\right )-2 \left (6 e^{12 x}-x\right ) \left (x^3+x^2-4 x-4\right ) \log ^2\left (\frac {x^2-4}{x+1}\right )}{(2-x) (x+1) (x+2) \log \left (\frac {x^2-4}{x+1}\right ) \left (5-\left (x^2-e^{12 x}\right ) \log \left (\frac {x^2-4}{x+1}\right )\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (12-\frac {-5 x^2-40 x^2 \log ^2\left (\frac {x^2-4}{x+1}\right )+8 x \log ^2\left (\frac {x^2-4}{x+1}\right )-60 x^2 \log \left (\frac {x^2-4}{x+1}\right )+240 x \log \left (\frac {x^2-4}{x+1}\right )+240 \log \left (\frac {x^2-4}{x+1}\right )+12 x^5 \log ^2\left (\frac {x^2-4}{x+1}\right )+10 x^4 \log ^2\left (\frac {x^2-4}{x+1}\right )-50 x^3 \log ^2\left (\frac {x^2-4}{x+1}\right )-60 x^3 \log \left (\frac {x^2-4}{x+1}\right )-10 x-20}{(x-2) (x+1) (x+2) \log \left (\frac {x^2-4}{x+1}\right ) \left (x^2 \log \left (\frac {x^2-4}{x+1}\right )-e^{12 x} \log \left (\frac {x^2-4}{x+1}\right )-5\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {5 \left (x^2+2 x+4\right )-2 \left (6 e^{12 x}-x\right ) \left (x^3+x^2-4 x-4\right ) \log ^2\left (\frac {x^2-4}{x+1}\right )}{(2-x) (x+1) (x+2) \log \left (\frac {x^2-4}{x+1}\right ) \left (5-\left (x^2-e^{12 x}\right ) \log \left (\frac {x^2-4}{x+1}\right )\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (12-\frac {-5 x^2-40 x^2 \log ^2\left (\frac {x^2-4}{x+1}\right )+8 x \log ^2\left (\frac {x^2-4}{x+1}\right )-60 x^2 \log \left (\frac {x^2-4}{x+1}\right )+240 x \log \left (\frac {x^2-4}{x+1}\right )+240 \log \left (\frac {x^2-4}{x+1}\right )+12 x^5 \log ^2\left (\frac {x^2-4}{x+1}\right )+10 x^4 \log ^2\left (\frac {x^2-4}{x+1}\right )-50 x^3 \log ^2\left (\frac {x^2-4}{x+1}\right )-60 x^3 \log \left (\frac {x^2-4}{x+1}\right )-10 x-20}{(x-2) (x+1) (x+2) \log \left (\frac {x^2-4}{x+1}\right ) \left (x^2 \log \left (\frac {x^2-4}{x+1}\right )-e^{12 x} \log \left (\frac {x^2-4}{x+1}\right )-5\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {5 \left (x^2+2 x+4\right )-2 \left (6 e^{12 x}-x\right ) \left (x^3+x^2-4 x-4\right ) \log ^2\left (\frac {x^2-4}{x+1}\right )}{(2-x) (x+1) (x+2) \log \left (\frac {x^2-4}{x+1}\right ) \left (5-\left (x^2-e^{12 x}\right ) \log \left (\frac {x^2-4}{x+1}\right )\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (12-\frac {-5 x^2-40 x^2 \log ^2\left (\frac {x^2-4}{x+1}\right )+8 x \log ^2\left (\frac {x^2-4}{x+1}\right )-60 x^2 \log \left (\frac {x^2-4}{x+1}\right )+240 x \log \left (\frac {x^2-4}{x+1}\right )+240 \log \left (\frac {x^2-4}{x+1}\right )+12 x^5 \log ^2\left (\frac {x^2-4}{x+1}\right )+10 x^4 \log ^2\left (\frac {x^2-4}{x+1}\right )-50 x^3 \log ^2\left (\frac {x^2-4}{x+1}\right )-60 x^3 \log \left (\frac {x^2-4}{x+1}\right )-10 x-20}{(x-2) (x+1) (x+2) \log \left (\frac {x^2-4}{x+1}\right ) \left (x^2 \log \left (\frac {x^2-4}{x+1}\right )-e^{12 x} \log \left (\frac {x^2-4}{x+1}\right )-5\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {5 \left (x^2+2 x+4\right )-2 \left (6 e^{12 x}-x\right ) \left (x^3+x^2-4 x-4\right ) \log ^2\left (\frac {x^2-4}{x+1}\right )}{(2-x) (x+1) (x+2) \log \left (\frac {x^2-4}{x+1}\right ) \left (5-\left (x^2-e^{12 x}\right ) \log \left (\frac {x^2-4}{x+1}\right )\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (12-\frac {-5 x^2-40 x^2 \log ^2\left (\frac {x^2-4}{x+1}\right )+8 x \log ^2\left (\frac {x^2-4}{x+1}\right )-60 x^2 \log \left (\frac {x^2-4}{x+1}\right )+240 x \log \left (\frac {x^2-4}{x+1}\right )+240 \log \left (\frac {x^2-4}{x+1}\right )+12 x^5 \log ^2\left (\frac {x^2-4}{x+1}\right )+10 x^4 \log ^2\left (\frac {x^2-4}{x+1}\right )-50 x^3 \log ^2\left (\frac {x^2-4}{x+1}\right )-60 x^3 \log \left (\frac {x^2-4}{x+1}\right )-10 x-20}{(x-2) (x+1) (x+2) \log \left (\frac {x^2-4}{x+1}\right ) \left (x^2 \log \left (\frac {x^2-4}{x+1}\right )-e^{12 x} \log \left (\frac {x^2-4}{x+1}\right )-5\right )}\right )dx\) |
Int[(-20 - 10*x - 5*x^2 + (8*x + 8*x^2 - 2*x^3 - 2*x^4 + E^(12*x)*(-48 - 4 8*x + 12*x^2 + 12*x^3))*Log[(-4 + x^2)/(1 + x)]^2)/((-20 - 20*x + 5*x^2 + 5*x^3)*Log[(-4 + x^2)/(1 + x)] + (4*x^2 + 4*x^3 - x^4 - x^5 + E^(12*x)*(-4 - 4*x + x^2 + x^3))*Log[(-4 + x^2)/(1 + x)]^2),x]
3.9.99.3.1 Defintions of rubi rules used
Int[(u_.)*(Px_)^(p_), x_Symbol] :> With[{Qx = Factor[Px]}, Int[ExpandIntegr and[u, Qx^p, x], x] /; !SumQ[NonfreeFactors[Qx, x]]] /; PolyQ[Px, x] && Gt Q[Expon[Px, x], 2] && !BinomialQ[Px, x] && !TrinomialQ[Px, x] && ILtQ[p, 0]
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Time = 8.44 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.69
| method | result | size |
| parallelrisch | \(-\ln \left (\ln \left (\frac {x^{2}-4}{1+x}\right )\right )+\ln \left (\ln \left (\frac {x^{2}-4}{1+x}\right ) x^{2}-{\mathrm e}^{12 x} \ln \left (\frac {x^{2}-4}{1+x}\right )-5\right )\) | \(54\) |
| risch | \(\ln \left (-x^{2}+{\mathrm e}^{12 x}\right )+\ln \left (\ln \left (x^{2}-4\right )-\frac {i \left (\pi \,x^{2} \operatorname {csgn}\left (i \left (x^{2}-4\right )\right ) \operatorname {csgn}\left (\frac {i}{1+x}\right ) \operatorname {csgn}\left (\frac {i \left (x^{2}-4\right )}{1+x}\right )-\pi \,x^{2} \operatorname {csgn}\left (i \left (x^{2}-4\right )\right ) {\operatorname {csgn}\left (\frac {i \left (x^{2}-4\right )}{1+x}\right )}^{2}-\pi \,x^{2} \operatorname {csgn}\left (\frac {i}{1+x}\right ) {\operatorname {csgn}\left (\frac {i \left (x^{2}-4\right )}{1+x}\right )}^{2}+\pi \,x^{2} {\operatorname {csgn}\left (\frac {i \left (x^{2}-4\right )}{1+x}\right )}^{3}-\pi \,\operatorname {csgn}\left (i \left (x^{2}-4\right )\right ) \operatorname {csgn}\left (\frac {i}{1+x}\right ) \operatorname {csgn}\left (\frac {i \left (x^{2}-4\right )}{1+x}\right ) {\mathrm e}^{12 x}+\pi \,\operatorname {csgn}\left (i \left (x^{2}-4\right )\right ) {\operatorname {csgn}\left (\frac {i \left (x^{2}-4\right )}{1+x}\right )}^{2} {\mathrm e}^{12 x}+\pi \,\operatorname {csgn}\left (\frac {i}{1+x}\right ) {\operatorname {csgn}\left (\frac {i \left (x^{2}-4\right )}{1+x}\right )}^{2} {\mathrm e}^{12 x}-\pi {\operatorname {csgn}\left (\frac {i \left (x^{2}-4\right )}{1+x}\right )}^{3} {\mathrm e}^{12 x}-2 i x^{2} \ln \left (1+x \right )+2 i {\mathrm e}^{12 x} \ln \left (1+x \right )-10 i\right )}{2 \left (x^{2}-{\mathrm e}^{12 x}\right )}\right )-\ln \left (\ln \left (x^{2}-4\right )-\frac {i \left (\pi \,\operatorname {csgn}\left (i \left (x^{2}-4\right )\right ) \operatorname {csgn}\left (\frac {i}{1+x}\right ) \operatorname {csgn}\left (\frac {i \left (x^{2}-4\right )}{1+x}\right )-\pi \,\operatorname {csgn}\left (i \left (x^{2}-4\right )\right ) {\operatorname {csgn}\left (\frac {i \left (x^{2}-4\right )}{1+x}\right )}^{2}-\pi \,\operatorname {csgn}\left (\frac {i}{1+x}\right ) {\operatorname {csgn}\left (\frac {i \left (x^{2}-4\right )}{1+x}\right )}^{2}+\pi {\operatorname {csgn}\left (\frac {i \left (x^{2}-4\right )}{1+x}\right )}^{3}-2 i \ln \left (1+x \right )\right )}{2}\right )\) | \(433\) |
int((((12*x^3+12*x^2-48*x-48)*exp(12*x)-2*x^4-2*x^3+8*x^2+8*x)*ln((x^2-4)/ (1+x))^2-5*x^2-10*x-20)/(((x^3+x^2-4*x-4)*exp(12*x)-x^5-x^4+4*x^3+4*x^2)*l n((x^2-4)/(1+x))^2+(5*x^3+5*x^2-20*x-20)*ln((x^2-4)/(1+x))),x,method=_RETU RNVERBOSE)
Leaf count of result is larger than twice the leaf count of optimal. 66 vs. \(2 (32) = 64\).
Time = 0.26 (sec) , antiderivative size = 66, normalized size of antiderivative = 2.06 \[ \int \frac {-20-10 x-5 x^2+\left (8 x+8 x^2-2 x^3-2 x^4+e^{12 x} \left (-48-48 x+12 x^2+12 x^3\right )\right ) \log ^2\left (\frac {-4+x^2}{1+x}\right )}{\left (-20-20 x+5 x^2+5 x^3\right ) \log \left (\frac {-4+x^2}{1+x}\right )+\left (4 x^2+4 x^3-x^4-x^5+e^{12 x} \left (-4-4 x+x^2+x^3\right )\right ) \log ^2\left (\frac {-4+x^2}{1+x}\right )} \, dx=\log \left (-x^{2} + e^{\left (12 \, x\right )}\right ) + \log \left (\frac {{\left (x^{2} - e^{\left (12 \, x\right )}\right )} \log \left (\frac {x^{2} - 4}{x + 1}\right ) - 5}{x^{2} - e^{\left (12 \, x\right )}}\right ) - \log \left (\log \left (\frac {x^{2} - 4}{x + 1}\right )\right ) \]
integrate((((12*x^3+12*x^2-48*x-48)*exp(12*x)-2*x^4-2*x^3+8*x^2+8*x)*log(( x^2-4)/(1+x))^2-5*x^2-10*x-20)/(((x^3+x^2-4*x-4)*exp(12*x)-x^5-x^4+4*x^3+4 *x^2)*log((x^2-4)/(1+x))^2+(5*x^3+5*x^2-20*x-20)*log((x^2-4)/(1+x))),x, al gorithm=\
log(-x^2 + e^(12*x)) + log(((x^2 - e^(12*x))*log((x^2 - 4)/(x + 1)) - 5)/( x^2 - e^(12*x))) - log(log((x^2 - 4)/(x + 1)))
Time = 0.62 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.97 \[ \int \frac {-20-10 x-5 x^2+\left (8 x+8 x^2-2 x^3-2 x^4+e^{12 x} \left (-48-48 x+12 x^2+12 x^3\right )\right ) \log ^2\left (\frac {-4+x^2}{1+x}\right )}{\left (-20-20 x+5 x^2+5 x^3\right ) \log \left (\frac {-4+x^2}{1+x}\right )+\left (4 x^2+4 x^3-x^4-x^5+e^{12 x} \left (-4-4 x+x^2+x^3\right )\right ) \log ^2\left (\frac {-4+x^2}{1+x}\right )} \, dx=\log {\left (\frac {- x^{2} \log {\left (\frac {x^{2} - 4}{x + 1} \right )} + 5}{\log {\left (\frac {x^{2} - 4}{x + 1} \right )}} + e^{12 x} \right )} \]
integrate((((12*x**3+12*x**2-48*x-48)*exp(12*x)-2*x**4-2*x**3+8*x**2+8*x)* ln((x**2-4)/(1+x))**2-5*x**2-10*x-20)/(((x**3+x**2-4*x-4)*exp(12*x)-x**5-x **4+4*x**3+4*x**2)*ln((x**2-4)/(1+x))**2+(5*x**3+5*x**2-20*x-20)*ln((x**2- 4)/(1+x))),x)
Leaf count of result is larger than twice the leaf count of optimal. 97 vs. \(2 (32) = 64\).
Time = 0.27 (sec) , antiderivative size = 97, normalized size of antiderivative = 3.03 \[ \int \frac {-20-10 x-5 x^2+\left (8 x+8 x^2-2 x^3-2 x^4+e^{12 x} \left (-48-48 x+12 x^2+12 x^3\right )\right ) \log ^2\left (\frac {-4+x^2}{1+x}\right )}{\left (-20-20 x+5 x^2+5 x^3\right ) \log \left (\frac {-4+x^2}{1+x}\right )+\left (4 x^2+4 x^3-x^4-x^5+e^{12 x} \left (-4-4 x+x^2+x^3\right )\right ) \log ^2\left (\frac {-4+x^2}{1+x}\right )} \, dx=\log \left (x + e^{\left (6 \, x\right )}\right ) + \log \left (-x + e^{\left (6 \, x\right )}\right ) + \log \left (\frac {{\left (x^{2} - e^{\left (12 \, x\right )}\right )} \log \left (x + 2\right ) - {\left (x^{2} - e^{\left (12 \, x\right )}\right )} \log \left (x + 1\right ) + {\left (x^{2} - e^{\left (12 \, x\right )}\right )} \log \left (x - 2\right ) - 5}{x^{2} - e^{\left (12 \, x\right )}}\right ) - \log \left (\log \left (x + 2\right ) - \log \left (x + 1\right ) + \log \left (x - 2\right )\right ) \]
integrate((((12*x^3+12*x^2-48*x-48)*exp(12*x)-2*x^4-2*x^3+8*x^2+8*x)*log(( x^2-4)/(1+x))^2-5*x^2-10*x-20)/(((x^3+x^2-4*x-4)*exp(12*x)-x^5-x^4+4*x^3+4 *x^2)*log((x^2-4)/(1+x))^2+(5*x^3+5*x^2-20*x-20)*log((x^2-4)/(1+x))),x, al gorithm=\
log(x + e^(6*x)) + log(-x + e^(6*x)) + log(((x^2 - e^(12*x))*log(x + 2) - (x^2 - e^(12*x))*log(x + 1) + (x^2 - e^(12*x))*log(x - 2) - 5)/(x^2 - e^(1 2*x))) - log(log(x + 2) - log(x + 1) + log(x - 2))
Time = 0.77 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.66 \[ \int \frac {-20-10 x-5 x^2+\left (8 x+8 x^2-2 x^3-2 x^4+e^{12 x} \left (-48-48 x+12 x^2+12 x^3\right )\right ) \log ^2\left (\frac {-4+x^2}{1+x}\right )}{\left (-20-20 x+5 x^2+5 x^3\right ) \log \left (\frac {-4+x^2}{1+x}\right )+\left (4 x^2+4 x^3-x^4-x^5+e^{12 x} \left (-4-4 x+x^2+x^3\right )\right ) \log ^2\left (\frac {-4+x^2}{1+x}\right )} \, dx=\log \left (-x^{2} \log \left (\frac {x^{2} - 4}{x + 1}\right ) + e^{\left (12 \, x\right )} \log \left (\frac {x^{2} - 4}{x + 1}\right ) + 5\right ) - \log \left (\log \left (\frac {x^{2} - 4}{x + 1}\right )\right ) \]
integrate((((12*x^3+12*x^2-48*x-48)*exp(12*x)-2*x^4-2*x^3+8*x^2+8*x)*log(( x^2-4)/(1+x))^2-5*x^2-10*x-20)/(((x^3+x^2-4*x-4)*exp(12*x)-x^5-x^4+4*x^3+4 *x^2)*log((x^2-4)/(1+x))^2+(5*x^3+5*x^2-20*x-20)*log((x^2-4)/(1+x))),x, al gorithm=\
log(-x^2*log((x^2 - 4)/(x + 1)) + e^(12*x)*log((x^2 - 4)/(x + 1)) + 5) - l og(log((x^2 - 4)/(x + 1)))
Timed out. \[ \int \frac {-20-10 x-5 x^2+\left (8 x+8 x^2-2 x^3-2 x^4+e^{12 x} \left (-48-48 x+12 x^2+12 x^3\right )\right ) \log ^2\left (\frac {-4+x^2}{1+x}\right )}{\left (-20-20 x+5 x^2+5 x^3\right ) \log \left (\frac {-4+x^2}{1+x}\right )+\left (4 x^2+4 x^3-x^4-x^5+e^{12 x} \left (-4-4 x+x^2+x^3\right )\right ) \log ^2\left (\frac {-4+x^2}{1+x}\right )} \, dx=\int \frac {10\,x+{\ln \left (\frac {x^2-4}{x+1}\right )}^2\,\left ({\mathrm {e}}^{12\,x}\,\left (-12\,x^3-12\,x^2+48\,x+48\right )-8\,x-8\,x^2+2\,x^3+2\,x^4\right )+5\,x^2+20}{\left ({\mathrm {e}}^{12\,x}\,\left (-x^3-x^2+4\,x+4\right )-4\,x^2-4\,x^3+x^4+x^5\right )\,{\ln \left (\frac {x^2-4}{x+1}\right )}^2+\left (-5\,x^3-5\,x^2+20\,x+20\right )\,\ln \left (\frac {x^2-4}{x+1}\right )} \,d x \]
int((10*x + log((x^2 - 4)/(x + 1))^2*(exp(12*x)*(48*x - 12*x^2 - 12*x^3 + 48) - 8*x - 8*x^2 + 2*x^3 + 2*x^4) + 5*x^2 + 20)/(log((x^2 - 4)/(x + 1))*( 20*x - 5*x^2 - 5*x^3 + 20) + log((x^2 - 4)/(x + 1))^2*(exp(12*x)*(4*x - x^ 2 - x^3 + 4) - 4*x^2 - 4*x^3 + x^4 + x^5)),x)