Integrand size = 383, antiderivative size = 30 \[ \int \frac {-2 x^3+2 x^2 \log ^2(2)+\left (-5 x^2-6 x^3-x^4+\left (5 x^2+x^3\right ) \log ^2(2)\right ) \log \left (\frac {3}{5+x}\right )+\left (10 x^2+2 x^3+\left (-10 x-2 x^2\right ) \log ^2(2)\right ) \log \left (\frac {3}{5+x}\right ) \log \left (-x+\log ^2(2)\right )+\left (-10 x^2-2 x^3+\left (10 x+2 x^2\right ) \log ^2(2)\right ) \log \left (\frac {3}{5+x}\right ) \log \left (\log ^2\left (\frac {3}{5+x}\right )\right )}{\left (-5 x^3-x^4+\left (5 x^2+x^3\right ) \log ^2(2)\right ) \log \left (\frac {3}{5+x}\right )+\left (10 x^2+2 x^3+\left (-10 x-2 x^2\right ) \log ^2(2)\right ) \log \left (\frac {3}{5+x}\right ) \log \left (-x+\log ^2(2)\right )+\left (-5 x-x^2+(5+x) \log ^2(2)\right ) \log \left (\frac {3}{5+x}\right ) \log ^2\left (-x+\log ^2(2)\right )+\left (\left (-10 x^2-2 x^3+\left (10 x+2 x^2\right ) \log ^2(2)\right ) \log \left (\frac {3}{5+x}\right )+\left (10 x+2 x^2+(-10-2 x) \log ^2(2)\right ) \log \left (\frac {3}{5+x}\right ) \log \left (-x+\log ^2(2)\right )\right ) \log \left (\log ^2\left (\frac {3}{5+x}\right )\right )+\left (-5 x-x^2+(5+x) \log ^2(2)\right ) \log \left (\frac {3}{5+x}\right ) \log ^2\left (\log ^2\left (\frac {3}{5+x}\right )\right )} \, dx=\frac {x^2}{x-\log \left (-x+\log ^2(2)\right )+\log \left (\log ^2\left (\frac {3}{5+x}\right )\right )} \] Output:
x^2/(x+ln(ln(3/(5+x))^2)-ln(ln(2)^2-x))
Time = 0.08 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.00 \[ \int \frac {-2 x^3+2 x^2 \log ^2(2)+\left (-5 x^2-6 x^3-x^4+\left (5 x^2+x^3\right ) \log ^2(2)\right ) \log \left (\frac {3}{5+x}\right )+\left (10 x^2+2 x^3+\left (-10 x-2 x^2\right ) \log ^2(2)\right ) \log \left (\frac {3}{5+x}\right ) \log \left (-x+\log ^2(2)\right )+\left (-10 x^2-2 x^3+\left (10 x+2 x^2\right ) \log ^2(2)\right ) \log \left (\frac {3}{5+x}\right ) \log \left (\log ^2\left (\frac {3}{5+x}\right )\right )}{\left (-5 x^3-x^4+\left (5 x^2+x^3\right ) \log ^2(2)\right ) \log \left (\frac {3}{5+x}\right )+\left (10 x^2+2 x^3+\left (-10 x-2 x^2\right ) \log ^2(2)\right ) \log \left (\frac {3}{5+x}\right ) \log \left (-x+\log ^2(2)\right )+\left (-5 x-x^2+(5+x) \log ^2(2)\right ) \log \left (\frac {3}{5+x}\right ) \log ^2\left (-x+\log ^2(2)\right )+\left (\left (-10 x^2-2 x^3+\left (10 x+2 x^2\right ) \log ^2(2)\right ) \log \left (\frac {3}{5+x}\right )+\left (10 x+2 x^2+(-10-2 x) \log ^2(2)\right ) \log \left (\frac {3}{5+x}\right ) \log \left (-x+\log ^2(2)\right )\right ) \log \left (\log ^2\left (\frac {3}{5+x}\right )\right )+\left (-5 x-x^2+(5+x) \log ^2(2)\right ) \log \left (\frac {3}{5+x}\right ) \log ^2\left (\log ^2\left (\frac {3}{5+x}\right )\right )} \, dx=\frac {x^2}{x-\log \left (-x+\log ^2(2)\right )+\log \left (\log ^2\left (\frac {3}{5+x}\right )\right )} \] Input:
Integrate[(-2*x^3 + 2*x^2*Log[2]^2 + (-5*x^2 - 6*x^3 - x^4 + (5*x^2 + x^3) *Log[2]^2)*Log[3/(5 + x)] + (10*x^2 + 2*x^3 + (-10*x - 2*x^2)*Log[2]^2)*Lo g[3/(5 + x)]*Log[-x + Log[2]^2] + (-10*x^2 - 2*x^3 + (10*x + 2*x^2)*Log[2] ^2)*Log[3/(5 + x)]*Log[Log[3/(5 + x)]^2])/((-5*x^3 - x^4 + (5*x^2 + x^3)*L og[2]^2)*Log[3/(5 + x)] + (10*x^2 + 2*x^3 + (-10*x - 2*x^2)*Log[2]^2)*Log[ 3/(5 + x)]*Log[-x + Log[2]^2] + (-5*x - x^2 + (5 + x)*Log[2]^2)*Log[3/(5 + x)]*Log[-x + Log[2]^2]^2 + ((-10*x^2 - 2*x^3 + (10*x + 2*x^2)*Log[2]^2)*L og[3/(5 + x)] + (10*x + 2*x^2 + (-10 - 2*x)*Log[2]^2)*Log[3/(5 + x)]*Log[- x + Log[2]^2])*Log[Log[3/(5 + x)]^2] + (-5*x - x^2 + (5 + x)*Log[2]^2)*Log [3/(5 + x)]*Log[Log[3/(5 + x)]^2]^2),x]
Output:
x^2/(x - Log[-x + Log[2]^2] + Log[Log[3/(5 + 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.
| \(\displaystyle \int \frac {-2 x^3+2 x^2 \log ^2(2)+\left (2 x^3+10 x^2+\left (-2 x^2-10 x\right ) \log ^2(2)\right ) \log \left (\frac {3}{x+5}\right ) \log \left (\log ^2(2)-x\right )+\left (-2 x^3-10 x^2+\left (2 x^2+10 x\right ) \log ^2(2)\right ) \log \left (\frac {3}{x+5}\right ) \log \left (\log ^2\left (\frac {3}{x+5}\right )\right )+\left (-x^4-6 x^3-5 x^2+\left (x^3+5 x^2\right ) \log ^2(2)\right ) \log \left (\frac {3}{x+5}\right )}{\left (-x^2-5 x+(x+5) \log ^2(2)\right ) \log \left (\frac {3}{x+5}\right ) \log ^2\left (\log ^2(2)-x\right )+\left (-x^2-5 x+(x+5) \log ^2(2)\right ) \log \left (\frac {3}{x+5}\right ) \log ^2\left (\log ^2\left (\frac {3}{x+5}\right )\right )+\left (2 x^3+10 x^2+\left (-2 x^2-10 x\right ) \log ^2(2)\right ) \log \left (\frac {3}{x+5}\right ) \log \left (\log ^2(2)-x\right )+\left (\left (2 x^2+10 x+(-2 x-10) \log ^2(2)\right ) \log \left (\log ^2(2)-x\right ) \log \left (\frac {3}{x+5}\right )+\left (-2 x^3-10 x^2+\left (2 x^2+10 x\right ) \log ^2(2)\right ) \log \left (\frac {3}{x+5}\right )\right ) \log \left (\log ^2\left (\frac {3}{x+5}\right )\right )+\left (-x^4-5 x^3+\left (x^3+5 x^2\right ) \log ^2(2)\right ) \log \left (\frac {3}{x+5}\right )} \, dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {x \left (2 x \left (x-\log ^2(2)\right )+(x+5) \log \left (\frac {3}{x+5}\right ) \left (x \left (x+1-\log ^2(2)\right )-2 \left (x-\log ^2(2)\right ) \log \left (\log ^2(2)-x\right )+2 \left (x-\log ^2(2)\right ) \log \left (\log ^2\left (\frac {3}{x+5}\right )\right )\right )\right )}{(x+5) \left (x-\log ^2(2)\right ) \log \left (\frac {3}{x+5}\right ) \left (x-\log \left (\log ^2(2)-x\right )+\log \left (\log ^2\left (\frac {3}{x+5}\right )\right )\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {x^2 \left (x^2 \left (-\log \left (\frac {3}{x+5}\right )\right )+2 x-4 x \left (1-\frac {\log ^2(2)}{4}\right ) \log \left (\frac {3}{x+5}\right )+5 \left (1+\log ^2(2)\right ) \log \left (\frac {3}{x+5}\right )-2 \log ^2(2)\right )}{(x+5) \left (x-\log ^2(2)\right ) \log \left (\frac {3}{x+5}\right ) \left (x-\log \left (\log ^2(2)-x\right )+\log \left (\log ^2\left (\frac {3}{x+5}\right )\right )\right )^2}+\frac {2 x}{x-\log \left (\log ^2(2)-x\right )+\log \left (\log ^2\left (\frac {3}{x+5}\right )\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {\log ^4(2) \left (4-\log ^2(2)\right ) \int \frac {1}{\left (x-\log \left (\log ^2(2)-x\right )+\log \left (\log ^2\left (\frac {3}{x+5}\right )\right )\right )^2}dx}{5+\log ^2(2)}+\frac {25 \left (4-\log ^2(2)\right ) \int \frac {1}{\left (x-\log \left (\log ^2(2)-x\right )+\log \left (\log ^2\left (\frac {3}{x+5}\right )\right )\right )^2}dx}{5+\log ^2(2)}-\frac {\log ^6(2) \int \frac {1}{\left (x-\log \left (\log ^2(2)-x\right )+\log \left (\log ^2\left (\frac {3}{x+5}\right )\right )\right )^2}dx}{5+\log ^2(2)}-\frac {125 \int \frac {1}{\left (x-\log \left (\log ^2(2)-x\right )+\log \left (\log ^2\left (\frac {3}{x+5}\right )\right )\right )^2}dx}{5+\log ^2(2)}+5 \left (1+\log ^2(2)\right ) \int \frac {1}{\left (x-\log \left (\log ^2(2)-x\right )+\log \left (\log ^2\left (\frac {3}{x+5}\right )\right )\right )^2}dx-\frac {\log ^4(2) \int \frac {x}{\left (x-\log \left (\log ^2(2)-x\right )+\log \left (\log ^2\left (\frac {3}{x+5}\right )\right )\right )^2}dx}{5+\log ^2(2)}+\frac {25 \int \frac {x}{\left (x-\log \left (\log ^2(2)-x\right )+\log \left (\log ^2\left (\frac {3}{x+5}\right )\right )\right )^2}dx}{5+\log ^2(2)}-\left (4-\log ^2(2)\right ) \int \frac {x}{\left (x-\log \left (\log ^2(2)-x\right )+\log \left (\log ^2\left (\frac {3}{x+5}\right )\right )\right )^2}dx-\int \frac {x^2}{\left (x-\log \left (\log ^2(2)-x\right )+\log \left (\log ^2\left (\frac {3}{x+5}\right )\right )\right )^2}dx-\frac {125 \left (1+\log ^2(2)\right ) \int \frac {1}{(x+5) \left (x-\log \left (\log ^2(2)-x\right )+\log \left (\log ^2\left (\frac {3}{x+5}\right )\right )\right )^2}dx}{5+\log ^2(2)}-\frac {125 \left (4-\log ^2(2)\right ) \int \frac {1}{(x+5) \left (x-\log \left (\log ^2(2)-x\right )+\log \left (\log ^2\left (\frac {3}{x+5}\right )\right )\right )^2}dx}{5+\log ^2(2)}+\frac {625 \int \frac {1}{(x+5) \left (x-\log \left (\log ^2(2)-x\right )+\log \left (\log ^2\left (\frac {3}{x+5}\right )\right )\right )^2}dx}{5+\log ^2(2)}+\frac {5 \log ^4(2) \left (1+\log ^2(2)\right ) \int \frac {1}{\left (x-\log ^2(2)\right ) \left (x-\log \left (\log ^2(2)-x\right )+\log \left (\log ^2\left (\frac {3}{x+5}\right )\right )\right )^2}dx}{5+\log ^2(2)}+\frac {\log ^6(2) \left (4-\log ^2(2)\right ) \int \frac {1}{\left (\log ^2(2)-x\right ) \left (x-\log \left (\log ^2(2)-x\right )+\log \left (\log ^2\left (\frac {3}{x+5}\right )\right )\right )^2}dx}{5+\log ^2(2)}+\frac {\log ^8(2) \int \frac {1}{\left (\log ^2(2)-x\right ) \left (x-\log \left (\log ^2(2)-x\right )+\log \left (\log ^2\left (\frac {3}{x+5}\right )\right )\right )^2}dx}{5+\log ^2(2)}+\frac {2 \log ^4(2) \int \frac {1}{\log \left (\frac {3}{x+5}\right ) \left (x-\log \left (\log ^2(2)-x\right )+\log \left (\log ^2\left (\frac {3}{x+5}\right )\right )\right )^2}dx}{5+\log ^2(2)}-\frac {50 \int \frac {1}{\log \left (\frac {3}{x+5}\right ) \left (x-\log \left (\log ^2(2)-x\right )+\log \left (\log ^2\left (\frac {3}{x+5}\right )\right )\right )^2}dx}{5+\log ^2(2)}-2 \log ^2(2) \int \frac {1}{\log \left (\frac {3}{x+5}\right ) \left (x-\log \left (\log ^2(2)-x\right )+\log \left (\log ^2\left (\frac {3}{x+5}\right )\right )\right )^2}dx+2 \int \frac {x}{\log \left (\frac {3}{x+5}\right ) \left (x-\log \left (\log ^2(2)-x\right )+\log \left (\log ^2\left (\frac {3}{x+5}\right )\right )\right )^2}dx+\frac {50 \log ^2(2) \int \frac {1}{(x+5) \log \left (\frac {3}{x+5}\right ) \left (x-\log \left (\log ^2(2)-x\right )+\log \left (\log ^2\left (\frac {3}{x+5}\right )\right )\right )^2}dx}{5+\log ^2(2)}+\frac {250 \int \frac {1}{(x+5) \log \left (\frac {3}{x+5}\right ) \left (x-\log \left (\log ^2(2)-x\right )+\log \left (\log ^2\left (\frac {3}{x+5}\right )\right )\right )^2}dx}{5+\log ^2(2)}+2 \int \frac {x}{x-\log \left (\log ^2(2)-x\right )+\log \left (\log ^2\left (\frac {3}{x+5}\right )\right )}dx\) |
Input:
Int[(-2*x^3 + 2*x^2*Log[2]^2 + (-5*x^2 - 6*x^3 - x^4 + (5*x^2 + x^3)*Log[2 ]^2)*Log[3/(5 + x)] + (10*x^2 + 2*x^3 + (-10*x - 2*x^2)*Log[2]^2)*Log[3/(5 + x)]*Log[-x + Log[2]^2] + (-10*x^2 - 2*x^3 + (10*x + 2*x^2)*Log[2]^2)*Lo g[3/(5 + x)]*Log[Log[3/(5 + x)]^2])/((-5*x^3 - x^4 + (5*x^2 + x^3)*Log[2]^ 2)*Log[3/(5 + x)] + (10*x^2 + 2*x^3 + (-10*x - 2*x^2)*Log[2]^2)*Log[3/(5 + x)]*Log[-x + Log[2]^2] + (-5*x - x^2 + (5 + x)*Log[2]^2)*Log[3/(5 + x)]*L og[-x + Log[2]^2]^2 + ((-10*x^2 - 2*x^3 + (10*x + 2*x^2)*Log[2]^2)*Log[3/( 5 + x)] + (10*x + 2*x^2 + (-10 - 2*x)*Log[2]^2)*Log[3/(5 + x)]*Log[-x + Lo g[2]^2])*Log[Log[3/(5 + x)]^2] + (-5*x - x^2 + (5 + x)*Log[2]^2)*Log[3/(5 + x)]*Log[Log[3/(5 + x)]^2]^2),x]
Output:
$Aborted
Time = 73.72 (sec) , antiderivative size = 60, normalized size of antiderivative = 2.00
| method | result | size |
| parallelrisch | \(\frac {-250 x^{2} \ln \left (2\right )^{4}+50 \ln \left (2\right )^{6} x^{2}}{50 \left (x +\ln \left (\ln \left (\frac {3}{5+x}\right )^{2}\right )-\ln \left (\ln \left (2\right )^{2}-x \right )\right ) \ln \left (2\right )^{4} \left (\ln \left (2\right )^{2}-5\right )}\) | \(60\) |
| risch | \(\frac {2 x^{2}}{-i \pi \operatorname {csgn}\left (i \left (\ln \left (3\right )-\ln \left (5+x \right )\right )\right )^{2} \operatorname {csgn}\left (i \left (\ln \left (3\right )-\ln \left (5+x \right )\right )^{2}\right )+2 i \pi \,\operatorname {csgn}\left (i \left (\ln \left (3\right )-\ln \left (5+x \right )\right )\right ) \operatorname {csgn}\left (i \left (\ln \left (3\right )-\ln \left (5+x \right )\right )^{2}\right )^{2}-i \pi \operatorname {csgn}\left (i \left (\ln \left (3\right )-\ln \left (5+x \right )\right )^{2}\right )^{3}+2 x -2 \ln \left (\ln \left (2\right )^{2}-x \right )+4 \ln \left (\ln \left (3\right )-\ln \left (5+x \right )\right )}\) | \(124\) |
Input:
int((((2*x^2+10*x)*ln(2)^2-2*x^3-10*x^2)*ln(3/(5+x))*ln(ln(3/(5+x))^2)+((- 2*x^2-10*x)*ln(2)^2+2*x^3+10*x^2)*ln(3/(5+x))*ln(ln(2)^2-x)+((x^3+5*x^2)*l n(2)^2-x^4-6*x^3-5*x^2)*ln(3/(5+x))+2*x^2*ln(2)^2-2*x^3)/(((5+x)*ln(2)^2-x ^2-5*x)*ln(3/(5+x))*ln(ln(3/(5+x))^2)^2+(((-2*x-10)*ln(2)^2+2*x^2+10*x)*ln (3/(5+x))*ln(ln(2)^2-x)+((2*x^2+10*x)*ln(2)^2-2*x^3-10*x^2)*ln(3/(5+x)))*l n(ln(3/(5+x))^2)+((5+x)*ln(2)^2-x^2-5*x)*ln(3/(5+x))*ln(ln(2)^2-x)^2+((-2* x^2-10*x)*ln(2)^2+2*x^3+10*x^2)*ln(3/(5+x))*ln(ln(2)^2-x)+((x^3+5*x^2)*ln( 2)^2-x^4-5*x^3)*ln(3/(5+x))),x,method=_RETURNVERBOSE)
Output:
1/50*(-250*x^2*ln(2)^4+50*ln(2)^6*x^2)/(x+ln(ln(3/(5+x))^2)-ln(ln(2)^2-x)) /ln(2)^4/(ln(2)^2-5)
Time = 0.09 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.00 \[ \int \frac {-2 x^3+2 x^2 \log ^2(2)+\left (-5 x^2-6 x^3-x^4+\left (5 x^2+x^3\right ) \log ^2(2)\right ) \log \left (\frac {3}{5+x}\right )+\left (10 x^2+2 x^3+\left (-10 x-2 x^2\right ) \log ^2(2)\right ) \log \left (\frac {3}{5+x}\right ) \log \left (-x+\log ^2(2)\right )+\left (-10 x^2-2 x^3+\left (10 x+2 x^2\right ) \log ^2(2)\right ) \log \left (\frac {3}{5+x}\right ) \log \left (\log ^2\left (\frac {3}{5+x}\right )\right )}{\left (-5 x^3-x^4+\left (5 x^2+x^3\right ) \log ^2(2)\right ) \log \left (\frac {3}{5+x}\right )+\left (10 x^2+2 x^3+\left (-10 x-2 x^2\right ) \log ^2(2)\right ) \log \left (\frac {3}{5+x}\right ) \log \left (-x+\log ^2(2)\right )+\left (-5 x-x^2+(5+x) \log ^2(2)\right ) \log \left (\frac {3}{5+x}\right ) \log ^2\left (-x+\log ^2(2)\right )+\left (\left (-10 x^2-2 x^3+\left (10 x+2 x^2\right ) \log ^2(2)\right ) \log \left (\frac {3}{5+x}\right )+\left (10 x+2 x^2+(-10-2 x) \log ^2(2)\right ) \log \left (\frac {3}{5+x}\right ) \log \left (-x+\log ^2(2)\right )\right ) \log \left (\log ^2\left (\frac {3}{5+x}\right )\right )+\left (-5 x-x^2+(5+x) \log ^2(2)\right ) \log \left (\frac {3}{5+x}\right ) \log ^2\left (\log ^2\left (\frac {3}{5+x}\right )\right )} \, dx=\frac {x^{2}}{x - \log \left (\log \left (2\right )^{2} - x\right ) + \log \left (\log \left (\frac {3}{x + 5}\right )^{2}\right )} \] Input:
integrate((((2*x^2+10*x)*log(2)^2-2*x^3-10*x^2)*log(3/(5+x))*log(log(3/(5+ x))^2)+((-2*x^2-10*x)*log(2)^2+2*x^3+10*x^2)*log(3/(5+x))*log(log(2)^2-x)+ ((x^3+5*x^2)*log(2)^2-x^4-6*x^3-5*x^2)*log(3/(5+x))+2*x^2*log(2)^2-2*x^3)/ (((5+x)*log(2)^2-x^2-5*x)*log(3/(5+x))*log(log(3/(5+x))^2)^2+(((-2*x-10)*l og(2)^2+2*x^2+10*x)*log(3/(5+x))*log(log(2)^2-x)+((2*x^2+10*x)*log(2)^2-2* x^3-10*x^2)*log(3/(5+x)))*log(log(3/(5+x))^2)+((5+x)*log(2)^2-x^2-5*x)*log (3/(5+x))*log(log(2)^2-x)^2+((-2*x^2-10*x)*log(2)^2+2*x^3+10*x^2)*log(3/(5 +x))*log(log(2)^2-x)+((x^3+5*x^2)*log(2)^2-x^4-5*x^3)*log(3/(5+x))),x, alg orithm="fricas")
Output:
x^2/(x - log(log(2)^2 - x) + log(log(3/(x + 5))^2))
Time = 0.39 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.73 \[ \int \frac {-2 x^3+2 x^2 \log ^2(2)+\left (-5 x^2-6 x^3-x^4+\left (5 x^2+x^3\right ) \log ^2(2)\right ) \log \left (\frac {3}{5+x}\right )+\left (10 x^2+2 x^3+\left (-10 x-2 x^2\right ) \log ^2(2)\right ) \log \left (\frac {3}{5+x}\right ) \log \left (-x+\log ^2(2)\right )+\left (-10 x^2-2 x^3+\left (10 x+2 x^2\right ) \log ^2(2)\right ) \log \left (\frac {3}{5+x}\right ) \log \left (\log ^2\left (\frac {3}{5+x}\right )\right )}{\left (-5 x^3-x^4+\left (5 x^2+x^3\right ) \log ^2(2)\right ) \log \left (\frac {3}{5+x}\right )+\left (10 x^2+2 x^3+\left (-10 x-2 x^2\right ) \log ^2(2)\right ) \log \left (\frac {3}{5+x}\right ) \log \left (-x+\log ^2(2)\right )+\left (-5 x-x^2+(5+x) \log ^2(2)\right ) \log \left (\frac {3}{5+x}\right ) \log ^2\left (-x+\log ^2(2)\right )+\left (\left (-10 x^2-2 x^3+\left (10 x+2 x^2\right ) \log ^2(2)\right ) \log \left (\frac {3}{5+x}\right )+\left (10 x+2 x^2+(-10-2 x) \log ^2(2)\right ) \log \left (\frac {3}{5+x}\right ) \log \left (-x+\log ^2(2)\right )\right ) \log \left (\log ^2\left (\frac {3}{5+x}\right )\right )+\left (-5 x-x^2+(5+x) \log ^2(2)\right ) \log \left (\frac {3}{5+x}\right ) \log ^2\left (\log ^2\left (\frac {3}{5+x}\right )\right )} \, dx=\frac {x^{2}}{x - \log {\left (- x + \log {\left (2 \right )}^{2} \right )} + \log {\left (\log {\left (\frac {3}{x + 5} \right )}^{2} \right )}} \] Input:
integrate((((2*x**2+10*x)*ln(2)**2-2*x**3-10*x**2)*ln(3/(5+x))*ln(ln(3/(5+ x))**2)+((-2*x**2-10*x)*ln(2)**2+2*x**3+10*x**2)*ln(3/(5+x))*ln(ln(2)**2-x )+((x**3+5*x**2)*ln(2)**2-x**4-6*x**3-5*x**2)*ln(3/(5+x))+2*x**2*ln(2)**2- 2*x**3)/(((5+x)*ln(2)**2-x**2-5*x)*ln(3/(5+x))*ln(ln(3/(5+x))**2)**2+(((-2 *x-10)*ln(2)**2+2*x**2+10*x)*ln(3/(5+x))*ln(ln(2)**2-x)+((2*x**2+10*x)*ln( 2)**2-2*x**3-10*x**2)*ln(3/(5+x)))*ln(ln(3/(5+x))**2)+((5+x)*ln(2)**2-x**2 -5*x)*ln(3/(5+x))*ln(ln(2)**2-x)**2+((-2*x**2-10*x)*ln(2)**2+2*x**3+10*x** 2)*ln(3/(5+x))*ln(ln(2)**2-x)+((x**3+5*x**2)*ln(2)**2-x**4-5*x**3)*ln(3/(5 +x))),x)
Output:
x**2/(x - log(-x + log(2)**2) + log(log(3/(x + 5))**2))
Time = 0.30 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.03 \[ \int \frac {-2 x^3+2 x^2 \log ^2(2)+\left (-5 x^2-6 x^3-x^4+\left (5 x^2+x^3\right ) \log ^2(2)\right ) \log \left (\frac {3}{5+x}\right )+\left (10 x^2+2 x^3+\left (-10 x-2 x^2\right ) \log ^2(2)\right ) \log \left (\frac {3}{5+x}\right ) \log \left (-x+\log ^2(2)\right )+\left (-10 x^2-2 x^3+\left (10 x+2 x^2\right ) \log ^2(2)\right ) \log \left (\frac {3}{5+x}\right ) \log \left (\log ^2\left (\frac {3}{5+x}\right )\right )}{\left (-5 x^3-x^4+\left (5 x^2+x^3\right ) \log ^2(2)\right ) \log \left (\frac {3}{5+x}\right )+\left (10 x^2+2 x^3+\left (-10 x-2 x^2\right ) \log ^2(2)\right ) \log \left (\frac {3}{5+x}\right ) \log \left (-x+\log ^2(2)\right )+\left (-5 x-x^2+(5+x) \log ^2(2)\right ) \log \left (\frac {3}{5+x}\right ) \log ^2\left (-x+\log ^2(2)\right )+\left (\left (-10 x^2-2 x^3+\left (10 x+2 x^2\right ) \log ^2(2)\right ) \log \left (\frac {3}{5+x}\right )+\left (10 x+2 x^2+(-10-2 x) \log ^2(2)\right ) \log \left (\frac {3}{5+x}\right ) \log \left (-x+\log ^2(2)\right )\right ) \log \left (\log ^2\left (\frac {3}{5+x}\right )\right )+\left (-5 x-x^2+(5+x) \log ^2(2)\right ) \log \left (\frac {3}{5+x}\right ) \log ^2\left (\log ^2\left (\frac {3}{5+x}\right )\right )} \, dx=\frac {x^{2}}{x - \log \left (\log \left (2\right )^{2} - x\right ) + 2 \, \log \left (-\log \left (3\right ) + \log \left (x + 5\right )\right )} \] Input:
integrate((((2*x^2+10*x)*log(2)^2-2*x^3-10*x^2)*log(3/(5+x))*log(log(3/(5+ x))^2)+((-2*x^2-10*x)*log(2)^2+2*x^3+10*x^2)*log(3/(5+x))*log(log(2)^2-x)+ ((x^3+5*x^2)*log(2)^2-x^4-6*x^3-5*x^2)*log(3/(5+x))+2*x^2*log(2)^2-2*x^3)/ (((5+x)*log(2)^2-x^2-5*x)*log(3/(5+x))*log(log(3/(5+x))^2)^2+(((-2*x-10)*l og(2)^2+2*x^2+10*x)*log(3/(5+x))*log(log(2)^2-x)+((2*x^2+10*x)*log(2)^2-2* x^3-10*x^2)*log(3/(5+x)))*log(log(3/(5+x))^2)+((5+x)*log(2)^2-x^2-5*x)*log (3/(5+x))*log(log(2)^2-x)^2+((-2*x^2-10*x)*log(2)^2+2*x^3+10*x^2)*log(3/(5 +x))*log(log(2)^2-x)+((x^3+5*x^2)*log(2)^2-x^4-5*x^3)*log(3/(5+x))),x, alg orithm="maxima")
Output:
x^2/(x - log(log(2)^2 - x) + 2*log(-log(3) + log(x + 5)))
Leaf count of result is larger than twice the leaf count of optimal. 1059 vs. \(2 (30) = 60\).
Time = 15.22 (sec) , antiderivative size = 1059, normalized size of antiderivative = 35.30 \[ \int \frac {-2 x^3+2 x^2 \log ^2(2)+\left (-5 x^2-6 x^3-x^4+\left (5 x^2+x^3\right ) \log ^2(2)\right ) \log \left (\frac {3}{5+x}\right )+\left (10 x^2+2 x^3+\left (-10 x-2 x^2\right ) \log ^2(2)\right ) \log \left (\frac {3}{5+x}\right ) \log \left (-x+\log ^2(2)\right )+\left (-10 x^2-2 x^3+\left (10 x+2 x^2\right ) \log ^2(2)\right ) \log \left (\frac {3}{5+x}\right ) \log \left (\log ^2\left (\frac {3}{5+x}\right )\right )}{\left (-5 x^3-x^4+\left (5 x^2+x^3\right ) \log ^2(2)\right ) \log \left (\frac {3}{5+x}\right )+\left (10 x^2+2 x^3+\left (-10 x-2 x^2\right ) \log ^2(2)\right ) \log \left (\frac {3}{5+x}\right ) \log \left (-x+\log ^2(2)\right )+\left (-5 x-x^2+(5+x) \log ^2(2)\right ) \log \left (\frac {3}{5+x}\right ) \log ^2\left (-x+\log ^2(2)\right )+\left (\left (-10 x^2-2 x^3+\left (10 x+2 x^2\right ) \log ^2(2)\right ) \log \left (\frac {3}{5+x}\right )+\left (10 x+2 x^2+(-10-2 x) \log ^2(2)\right ) \log \left (\frac {3}{5+x}\right ) \log \left (-x+\log ^2(2)\right )\right ) \log \left (\log ^2\left (\frac {3}{5+x}\right )\right )+\left (-5 x-x^2+(5+x) \log ^2(2)\right ) \log \left (\frac {3}{5+x}\right ) \log ^2\left (\log ^2\left (\frac {3}{5+x}\right )\right )} \, dx=\text {Too large to display} \] Input:
integrate((((2*x^2+10*x)*log(2)^2-2*x^3-10*x^2)*log(3/(5+x))*log(log(3/(5+ x))^2)+((-2*x^2-10*x)*log(2)^2+2*x^3+10*x^2)*log(3/(5+x))*log(log(2)^2-x)+ ((x^3+5*x^2)*log(2)^2-x^4-6*x^3-5*x^2)*log(3/(5+x))+2*x^2*log(2)^2-2*x^3)/ (((5+x)*log(2)^2-x^2-5*x)*log(3/(5+x))*log(log(3/(5+x))^2)^2+(((-2*x-10)*l og(2)^2+2*x^2+10*x)*log(3/(5+x))*log(log(2)^2-x)+((2*x^2+10*x)*log(2)^2-2* x^3-10*x^2)*log(3/(5+x)))*log(log(3/(5+x))^2)+((5+x)*log(2)^2-x^2-5*x)*log (3/(5+x))*log(log(2)^2-x)^2+((-2*x^2-10*x)*log(2)^2+2*x^3+10*x^2)*log(3/(5 +x))*log(log(2)^2-x)+((x^3+5*x^2)*log(2)^2-x^4-5*x^3)*log(3/(5+x))),x, alg orithm="giac")
Output:
(x^3*log(3)*log(2)^2*log(3/(x + 5)) - x^3*log(2)^2*log(x + 5)*log(3/(x + 5 )) - x^4*log(3)*log(3/(x + 5)) + 5*x^2*log(3)*log(2)^2*log(3/(x + 5)) + x^ 4*log(x + 5)*log(3/(x + 5)) - 5*x^2*log(2)^2*log(x + 5)*log(3/(x + 5)) - 4 *x^3*log(3)*log(3/(x + 5)) - 2*x^2*log(2)^2*log(3/(x + 5)) + 4*x^3*log(x + 5)*log(3/(x + 5)) + 2*x^3*log(3/(x + 5)) + 5*x^2*log(3)*log(3/(x + 5)) - 5*x^2*log(x + 5)*log(3/(x + 5)))/(x^2*log(3)*log(2)^2*log(3/(x + 5)) - x*l og(3)*log(2)^2*log(log(2)^2 - x)*log(3/(x + 5)) + x*log(3)*log(2)^2*log(lo g(3/(x + 5))^2)*log(3/(x + 5)) - x^2*log(2)^2*log(x + 5)*log(3/(x + 5)) + x*log(2)^2*log(log(2)^2 - x)*log(x + 5)*log(3/(x + 5)) - x*log(2)^2*log(lo g(3/(x + 5))^2)*log(x + 5)*log(3/(x + 5)) - x^3*log(3)*log(3/(x + 5)) + 5* x*log(3)*log(2)^2*log(3/(x + 5)) + x^2*log(3)*log(log(2)^2 - x)*log(3/(x + 5)) - 5*log(3)*log(2)^2*log(log(2)^2 - x)*log(3/(x + 5)) - x^2*log(3)*log (log(3/(x + 5))^2)*log(3/(x + 5)) + 5*log(3)*log(2)^2*log(log(3/(x + 5))^2 )*log(3/(x + 5)) + x^3*log(x + 5)*log(3/(x + 5)) - 5*x*log(2)^2*log(x + 5) *log(3/(x + 5)) - x^2*log(log(2)^2 - x)*log(x + 5)*log(3/(x + 5)) + 5*log( 2)^2*log(log(2)^2 - x)*log(x + 5)*log(3/(x + 5)) + x^2*log(log(3/(x + 5))^ 2)*log(x + 5)*log(3/(x + 5)) - 5*log(2)^2*log(log(3/(x + 5))^2)*log(x + 5) *log(3/(x + 5)) - 2*x*log(3)*log(2)^2 + 2*log(3)*log(2)^2*log(log(2)^2 - x ) - 2*log(3)*log(2)^2*log(log(3/(x + 5))^2) + 2*x*log(2)^2*log(x + 5) - 2* log(2)^2*log(log(2)^2 - x)*log(x + 5) + 2*log(2)^2*log(log(3/(x + 5))^2...
Timed out. \[ \int \frac {-2 x^3+2 x^2 \log ^2(2)+\left (-5 x^2-6 x^3-x^4+\left (5 x^2+x^3\right ) \log ^2(2)\right ) \log \left (\frac {3}{5+x}\right )+\left (10 x^2+2 x^3+\left (-10 x-2 x^2\right ) \log ^2(2)\right ) \log \left (\frac {3}{5+x}\right ) \log \left (-x+\log ^2(2)\right )+\left (-10 x^2-2 x^3+\left (10 x+2 x^2\right ) \log ^2(2)\right ) \log \left (\frac {3}{5+x}\right ) \log \left (\log ^2\left (\frac {3}{5+x}\right )\right )}{\left (-5 x^3-x^4+\left (5 x^2+x^3\right ) \log ^2(2)\right ) \log \left (\frac {3}{5+x}\right )+\left (10 x^2+2 x^3+\left (-10 x-2 x^2\right ) \log ^2(2)\right ) \log \left (\frac {3}{5+x}\right ) \log \left (-x+\log ^2(2)\right )+\left (-5 x-x^2+(5+x) \log ^2(2)\right ) \log \left (\frac {3}{5+x}\right ) \log ^2\left (-x+\log ^2(2)\right )+\left (\left (-10 x^2-2 x^3+\left (10 x+2 x^2\right ) \log ^2(2)\right ) \log \left (\frac {3}{5+x}\right )+\left (10 x+2 x^2+(-10-2 x) \log ^2(2)\right ) \log \left (\frac {3}{5+x}\right ) \log \left (-x+\log ^2(2)\right )\right ) \log \left (\log ^2\left (\frac {3}{5+x}\right )\right )+\left (-5 x-x^2+(5+x) \log ^2(2)\right ) \log \left (\frac {3}{5+x}\right ) \log ^2\left (\log ^2\left (\frac {3}{5+x}\right )\right )} \, dx=\int \frac {\ln \left (\frac {3}{x+5}\right )\,\left (5\,x^2-{\ln \left (2\right )}^2\,\left (x^3+5\,x^2\right )+6\,x^3+x^4\right )-2\,x^2\,{\ln \left (2\right )}^2+2\,x^3-\ln \left (\frac {3}{x+5}\right )\,\ln \left ({\ln \left (2\right )}^2-x\right )\,\left (10\,x^2-{\ln \left (2\right )}^2\,\left (2\,x^2+10\,x\right )+2\,x^3\right )+\ln \left (\frac {3}{x+5}\right )\,\ln \left ({\ln \left (\frac {3}{x+5}\right )}^2\right )\,\left (10\,x^2-{\ln \left (2\right )}^2\,\left (2\,x^2+10\,x\right )+2\,x^3\right )}{\ln \left (\frac {3}{x+5}\right )\,\left (5\,x^3-{\ln \left (2\right )}^2\,\left (x^3+5\,x^2\right )+x^4\right )+\ln \left ({\ln \left (\frac {3}{x+5}\right )}^2\right )\,\left (\ln \left (\frac {3}{x+5}\right )\,\left (10\,x^2-{\ln \left (2\right )}^2\,\left (2\,x^2+10\,x\right )+2\,x^3\right )-\ln \left (\frac {3}{x+5}\right )\,\ln \left ({\ln \left (2\right )}^2-x\right )\,\left (10\,x-{\ln \left (2\right )}^2\,\left (2\,x+10\right )+2\,x^2\right )\right )-\ln \left (\frac {3}{x+5}\right )\,\ln \left ({\ln \left (2\right )}^2-x\right )\,\left (10\,x^2-{\ln \left (2\right )}^2\,\left (2\,x^2+10\,x\right )+2\,x^3\right )+\ln \left (\frac {3}{x+5}\right )\,{\ln \left ({\ln \left (2\right )}^2-x\right )}^2\,\left (5\,x-{\ln \left (2\right )}^2\,\left (x+5\right )+x^2\right )+\ln \left (\frac {3}{x+5}\right )\,{\ln \left ({\ln \left (\frac {3}{x+5}\right )}^2\right )}^2\,\left (5\,x-{\ln \left (2\right )}^2\,\left (x+5\right )+x^2\right )} \,d x \] Input:
int((log(3/(x + 5))*(5*x^2 - log(2)^2*(5*x^2 + x^3) + 6*x^3 + x^4) - 2*x^2 *log(2)^2 + 2*x^3 - log(3/(x + 5))*log(log(2)^2 - x)*(10*x^2 - log(2)^2*(1 0*x + 2*x^2) + 2*x^3) + log(3/(x + 5))*log(log(3/(x + 5))^2)*(10*x^2 - log (2)^2*(10*x + 2*x^2) + 2*x^3))/(log(3/(x + 5))*(5*x^3 - log(2)^2*(5*x^2 + x^3) + x^4) + log(log(3/(x + 5))^2)*(log(3/(x + 5))*(10*x^2 - log(2)^2*(10 *x + 2*x^2) + 2*x^3) - log(3/(x + 5))*log(log(2)^2 - x)*(10*x - log(2)^2*( 2*x + 10) + 2*x^2)) - log(3/(x + 5))*log(log(2)^2 - x)*(10*x^2 - log(2)^2* (10*x + 2*x^2) + 2*x^3) + log(3/(x + 5))*log(log(2)^2 - x)^2*(5*x - log(2) ^2*(x + 5) + x^2) + log(3/(x + 5))*log(log(3/(x + 5))^2)^2*(5*x - log(2)^2 *(x + 5) + x^2)),x)
Output:
int((log(3/(x + 5))*(5*x^2 - log(2)^2*(5*x^2 + x^3) + 6*x^3 + x^4) - 2*x^2 *log(2)^2 + 2*x^3 - log(3/(x + 5))*log(log(2)^2 - x)*(10*x^2 - log(2)^2*(1 0*x + 2*x^2) + 2*x^3) + log(3/(x + 5))*log(log(3/(x + 5))^2)*(10*x^2 - log (2)^2*(10*x + 2*x^2) + 2*x^3))/(log(3/(x + 5))*(5*x^3 - log(2)^2*(5*x^2 + x^3) + x^4) + log(log(3/(x + 5))^2)*(log(3/(x + 5))*(10*x^2 - log(2)^2*(10 *x + 2*x^2) + 2*x^3) - log(3/(x + 5))*log(log(2)^2 - x)*(10*x - log(2)^2*( 2*x + 10) + 2*x^2)) - log(3/(x + 5))*log(log(2)^2 - x)*(10*x^2 - log(2)^2* (10*x + 2*x^2) + 2*x^3) + log(3/(x + 5))*log(log(2)^2 - x)^2*(5*x - log(2) ^2*(x + 5) + x^2) + log(3/(x + 5))*log(log(3/(x + 5))^2)^2*(5*x - log(2)^2 *(x + 5) + x^2)), x)
Time = 0.23 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.00 \[ \int \frac {-2 x^3+2 x^2 \log ^2(2)+\left (-5 x^2-6 x^3-x^4+\left (5 x^2+x^3\right ) \log ^2(2)\right ) \log \left (\frac {3}{5+x}\right )+\left (10 x^2+2 x^3+\left (-10 x-2 x^2\right ) \log ^2(2)\right ) \log \left (\frac {3}{5+x}\right ) \log \left (-x+\log ^2(2)\right )+\left (-10 x^2-2 x^3+\left (10 x+2 x^2\right ) \log ^2(2)\right ) \log \left (\frac {3}{5+x}\right ) \log \left (\log ^2\left (\frac {3}{5+x}\right )\right )}{\left (-5 x^3-x^4+\left (5 x^2+x^3\right ) \log ^2(2)\right ) \log \left (\frac {3}{5+x}\right )+\left (10 x^2+2 x^3+\left (-10 x-2 x^2\right ) \log ^2(2)\right ) \log \left (\frac {3}{5+x}\right ) \log \left (-x+\log ^2(2)\right )+\left (-5 x-x^2+(5+x) \log ^2(2)\right ) \log \left (\frac {3}{5+x}\right ) \log ^2\left (-x+\log ^2(2)\right )+\left (\left (-10 x^2-2 x^3+\left (10 x+2 x^2\right ) \log ^2(2)\right ) \log \left (\frac {3}{5+x}\right )+\left (10 x+2 x^2+(-10-2 x) \log ^2(2)\right ) \log \left (\frac {3}{5+x}\right ) \log \left (-x+\log ^2(2)\right )\right ) \log \left (\log ^2\left (\frac {3}{5+x}\right )\right )+\left (-5 x-x^2+(5+x) \log ^2(2)\right ) \log \left (\frac {3}{5+x}\right ) \log ^2\left (\log ^2\left (\frac {3}{5+x}\right )\right )} \, dx=\frac {x^{2}}{\mathrm {log}\left (\mathrm {log}\left (\frac {3}{x +5}\right )^{2}\right )-\mathrm {log}\left (\mathrm {log}\left (2\right )^{2}-x \right )+x} \] Input:
int((((2*x^2+10*x)*log(2)^2-2*x^3-10*x^2)*log(3/(5+x))*log(log(3/(5+x))^2) +((-2*x^2-10*x)*log(2)^2+2*x^3+10*x^2)*log(3/(5+x))*log(log(2)^2-x)+((x^3+ 5*x^2)*log(2)^2-x^4-6*x^3-5*x^2)*log(3/(5+x))+2*x^2*log(2)^2-2*x^3)/(((5+x )*log(2)^2-x^2-5*x)*log(3/(5+x))*log(log(3/(5+x))^2)^2+(((-2*x-10)*log(2)^ 2+2*x^2+10*x)*log(3/(5+x))*log(log(2)^2-x)+((2*x^2+10*x)*log(2)^2-2*x^3-10 *x^2)*log(3/(5+x)))*log(log(3/(5+x))^2)+((5+x)*log(2)^2-x^2-5*x)*log(3/(5+ x))*log(log(2)^2-x)^2+((-2*x^2-10*x)*log(2)^2+2*x^3+10*x^2)*log(3/(5+x))*l og(log(2)^2-x)+((x^3+5*x^2)*log(2)^2-x^4-5*x^3)*log(3/(5+x))),x)
Output:
x**2/(log(log(3/(x + 5))**2) - log(log(2)**2 - x) + x)