Integrand size = 259, antiderivative size = 35 \[ \int \frac {200 x^2+80 x^3+\left (-50 x^3-20 x^4\right ) \log (x)+\left (-400 x-160 x^2+\left (100 x^2+40 x^3\right ) \log (x)\right ) \log \left (x^2\right )+\left (200+80 x+\left (-50 x-20 x^2\right ) \log (x)\right ) \log ^2\left (x^2\right )+\left (\frac {4-x \log (x)}{x}\right )^{\frac {x}{-2 x+2 \log \left (x^2\right )}} \left (-100 x-85 x^2-5 x^3+10 x^3 \log (x)+\left (100+125 x+5 x^2-20 x^2 \log (x)\right ) \log \left (x^2\right )+(-40+10 x \log (x)) \log ^2\left (x^2\right )+\left (200+40 x+\left (-50 x-10 x^2\right ) \log (x)+\left (-100-20 x+\left (25 x+5 x^2\right ) \log (x)\right ) \log \left (x^2\right )\right ) \log \left (\frac {4-x \log (x)}{x}\right )\right )}{-8 x^2+2 x^3 \log (x)+\left (16 x-4 x^2 \log (x)\right ) \log \left (x^2\right )+(-8+2 x \log (x)) \log ^2\left (x^2\right )} \, dx=5 (5+x) \left (-x+\left (\frac {4}{x}-\log (x)\right )^{\frac {x}{2 \left (-x+\log \left (x^2\right )\right )}}\right ) \]
Time = 0.33 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.26 \[ \int \frac {200 x^2+80 x^3+\left (-50 x^3-20 x^4\right ) \log (x)+\left (-400 x-160 x^2+\left (100 x^2+40 x^3\right ) \log (x)\right ) \log \left (x^2\right )+\left (200+80 x+\left (-50 x-20 x^2\right ) \log (x)\right ) \log ^2\left (x^2\right )+\left (\frac {4-x \log (x)}{x}\right )^{\frac {x}{-2 x+2 \log \left (x^2\right )}} \left (-100 x-85 x^2-5 x^3+10 x^3 \log (x)+\left (100+125 x+5 x^2-20 x^2 \log (x)\right ) \log \left (x^2\right )+(-40+10 x \log (x)) \log ^2\left (x^2\right )+\left (200+40 x+\left (-50 x-10 x^2\right ) \log (x)+\left (-100-20 x+\left (25 x+5 x^2\right ) \log (x)\right ) \log \left (x^2\right )\right ) \log \left (\frac {4-x \log (x)}{x}\right )\right )}{-8 x^2+2 x^3 \log (x)+\left (16 x-4 x^2 \log (x)\right ) \log \left (x^2\right )+(-8+2 x \log (x)) \log ^2\left (x^2\right )} \, dx=\frac {5}{2} \left (-10 x-2 x^2+2 (5+x) \left (\frac {4}{x}-\log (x)\right )^{-\frac {x}{2 \left (x-\log \left (x^2\right )\right )}}\right ) \]
Integrate[(200*x^2 + 80*x^3 + (-50*x^3 - 20*x^4)*Log[x] + (-400*x - 160*x^ 2 + (100*x^2 + 40*x^3)*Log[x])*Log[x^2] + (200 + 80*x + (-50*x - 20*x^2)*L og[x])*Log[x^2]^2 + ((4 - x*Log[x])/x)^(x/(-2*x + 2*Log[x^2]))*(-100*x - 8 5*x^2 - 5*x^3 + 10*x^3*Log[x] + (100 + 125*x + 5*x^2 - 20*x^2*Log[x])*Log[ x^2] + (-40 + 10*x*Log[x])*Log[x^2]^2 + (200 + 40*x + (-50*x - 10*x^2)*Log [x] + (-100 - 20*x + (25*x + 5*x^2)*Log[x])*Log[x^2])*Log[(4 - x*Log[x])/x ]))/(-8*x^2 + 2*x^3*Log[x] + (16*x - 4*x^2*Log[x])*Log[x^2] + (-8 + 2*x*Lo g[x])*Log[x^2]^2),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.
| \(\displaystyle \int \frac {80 x^3+200 x^2+\left (\left (-20 x^2-50 x\right ) \log (x)+80 x+200\right ) \log ^2\left (x^2\right )+\left (-20 x^4-50 x^3\right ) \log (x)+\left (-5 x^3+10 x^3 \log (x)-85 x^2+(10 x \log (x)-40) \log ^2\left (x^2\right )+\left (5 x^2-20 x^2 \log (x)+125 x+100\right ) \log \left (x^2\right )+\left (\left (-10 x^2-50 x\right ) \log (x)+\left (\left (5 x^2+25 x\right ) \log (x)-20 x-100\right ) \log \left (x^2\right )+40 x+200\right ) \log \left (\frac {4-x \log (x)}{x}\right )-100 x\right ) \left (\frac {4-x \log (x)}{x}\right )^{\frac {x}{2 \log \left (x^2\right )-2 x}}+\left (-160 x^2+\left (40 x^3+100 x^2\right ) \log (x)-400 x\right ) \log \left (x^2\right )}{2 x^3 \log (x)-8 x^2+(2 x \log (x)-8) \log ^2\left (x^2\right )+\left (16 x-4 x^2 \log (x)\right ) \log \left (x^2\right )} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {-80 x^3-200 x^2-\left (\left (-20 x^2-50 x\right ) \log (x)+80 x+200\right ) \log ^2\left (x^2\right )-\left (-20 x^4-50 x^3\right ) \log (x)-\left (\left (-5 x^3+10 x^3 \log (x)-85 x^2+(10 x \log (x)-40) \log ^2\left (x^2\right )+\left (5 x^2-20 x^2 \log (x)+125 x+100\right ) \log \left (x^2\right )+\left (\left (-10 x^2-50 x\right ) \log (x)+\left (\left (5 x^2+25 x\right ) \log (x)-20 x-100\right ) \log \left (x^2\right )+40 x+200\right ) \log \left (\frac {4-x \log (x)}{x}\right )-100 x\right ) \left (\frac {4-x \log (x)}{x}\right )^{\frac {x}{2 \log \left (x^2\right )-2 x}}\right )-\left (-160 x^2+\left (40 x^3+100 x^2\right ) \log (x)-400 x\right ) \log \left (x^2\right )}{2 (4-x \log (x)) \left (x-\log \left (x^2\right )\right )^2}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{2} \int -\frac {5 \left (-\left (\left (-2 \log (x) x^3+x^3+17 x^2+20 x+2 (4-x \log (x)) \log ^2\left (x^2\right )-\left (-4 \log (x) x^2+x^2+25 x+20\right ) \log \left (x^2\right )-\left (8 x-2 \left (x^2+5 x\right ) \log (x)-\left (4 x-\left (x^2+5 x\right ) \log (x)+20\right ) \log \left (x^2\right )+40\right ) \log \left (\frac {4-x \log (x)}{x}\right )\right ) \left (\frac {4-x \log (x)}{x}\right )^{-\frac {x}{2 \left (x-\log \left (x^2\right )\right )}}\right )+16 x^3+40 x^2+2 \left (8 x-\left (2 x^2+5 x\right ) \log (x)+20\right ) \log ^2\left (x^2\right )-2 \left (2 x^4+5 x^3\right ) \log (x)-4 \left (8 x^2+20 x-\left (2 x^3+5 x^2\right ) \log (x)\right ) \log \left (x^2\right )\right )}{(4-x \log (x)) \left (x-\log \left (x^2\right )\right )^2}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {5}{2} \int \frac {-\left (\left (-2 \log (x) x^3+x^3+17 x^2+20 x+2 (4-x \log (x)) \log ^2\left (x^2\right )-\left (-4 \log (x) x^2+x^2+25 x+20\right ) \log \left (x^2\right )-\left (8 x-2 \left (x^2+5 x\right ) \log (x)-\left (4 x-\left (x^2+5 x\right ) \log (x)+20\right ) \log \left (x^2\right )+40\right ) \log \left (\frac {4-x \log (x)}{x}\right )\right ) \left (\frac {4-x \log (x)}{x}\right )^{-\frac {x}{2 \left (x-\log \left (x^2\right )\right )}}\right )+16 x^3+40 x^2+2 \left (8 x-\left (2 x^2+5 x\right ) \log (x)+20\right ) \log ^2\left (x^2\right )-2 \left (2 x^4+5 x^3\right ) \log (x)-4 \left (8 x^2+20 x-\left (2 x^3+5 x^2\right ) \log (x)\right ) \log \left (x^2\right )}{(4-x \log (x)) \left (x-\log \left (x^2\right )\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {5}{2} \int \left (\frac {\left (2 \log (x) x^3-x^3-4 \log (x) \log \left (x^2\right ) x^2+\log \left (x^2\right ) x^2-2 \log (x) \log \left (\frac {4}{x}-\log (x)\right ) x^2+\log (x) \log \left (x^2\right ) \log \left (\frac {4}{x}-\log (x)\right ) x^2-17 x^2+2 \log (x) \log ^2\left (x^2\right ) x+25 \log \left (x^2\right ) x-10 \log (x) \log \left (\frac {4}{x}-\log (x)\right ) x+5 \log (x) \log \left (x^2\right ) \log \left (\frac {4}{x}-\log (x)\right ) x-4 \log \left (x^2\right ) \log \left (\frac {4}{x}-\log (x)\right ) x+8 \log \left (\frac {4}{x}-\log (x)\right ) x-20 x-8 \log ^2\left (x^2\right )+20 \log \left (x^2\right )-20 \log \left (x^2\right ) \log \left (\frac {4}{x}-\log (x)\right )+40 \log \left (\frac {4}{x}-\log (x)\right )\right ) \left (\frac {4}{x}-\log (x)\right )^{-\frac {x}{2 \left (x-\log \left (x^2\right )\right )}}}{(4-x \log (x)) \left (x-\log \left (x^2\right )\right )^2}+\frac {2 (2 x+5) \log ^2\left (x^2\right )}{\left (x-\log \left (x^2\right )\right )^2}-\frac {4 x (2 x+5) \log \left (x^2\right )}{\left (x-\log \left (x^2\right )\right )^2}-\frac {16 x^3}{(x \log (x)-4) \left (x-\log \left (x^2\right )\right )^2}-\frac {40 x^2}{(x \log (x)-4) \left (x-\log \left (x^2\right )\right )^2}+\frac {2 x^3 (2 x+5) \log (x)}{(x \log (x)-4) \left (x-\log \left (x^2\right )\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7299 |
| \(\displaystyle -\frac {5}{2} \int \left (\frac {\left (2 \log (x) x^3-x^3-4 \log (x) \log \left (x^2\right ) x^2+\log \left (x^2\right ) x^2-2 \log (x) \log \left (\frac {4}{x}-\log (x)\right ) x^2+\log (x) \log \left (x^2\right ) \log \left (\frac {4}{x}-\log (x)\right ) x^2-17 x^2+2 \log (x) \log ^2\left (x^2\right ) x+25 \log \left (x^2\right ) x-10 \log (x) \log \left (\frac {4}{x}-\log (x)\right ) x+5 \log (x) \log \left (x^2\right ) \log \left (\frac {4}{x}-\log (x)\right ) x-4 \log \left (x^2\right ) \log \left (\frac {4}{x}-\log (x)\right ) x+8 \log \left (\frac {4}{x}-\log (x)\right ) x-20 x-8 \log ^2\left (x^2\right )+20 \log \left (x^2\right )-20 \log \left (x^2\right ) \log \left (\frac {4}{x}-\log (x)\right )+40 \log \left (\frac {4}{x}-\log (x)\right )\right ) \left (\frac {4}{x}-\log (x)\right )^{-\frac {x}{2 \left (x-\log \left (x^2\right )\right )}}}{(4-x \log (x)) \left (x-\log \left (x^2\right )\right )^2}+\frac {2 (2 x+5) \log ^2\left (x^2\right )}{\left (x-\log \left (x^2\right )\right )^2}-\frac {4 x (2 x+5) \log \left (x^2\right )}{\left (x-\log \left (x^2\right )\right )^2}-\frac {16 x^3}{(x \log (x)-4) \left (x-\log \left (x^2\right )\right )^2}-\frac {40 x^2}{(x \log (x)-4) \left (x-\log \left (x^2\right )\right )^2}+\frac {2 x^3 (2 x+5) \log (x)}{(x \log (x)-4) \left (x-\log \left (x^2\right )\right )^2}\right )dx\) |
Int[(200*x^2 + 80*x^3 + (-50*x^3 - 20*x^4)*Log[x] + (-400*x - 160*x^2 + (1 00*x^2 + 40*x^3)*Log[x])*Log[x^2] + (200 + 80*x + (-50*x - 20*x^2)*Log[x]) *Log[x^2]^2 + ((4 - x*Log[x])/x)^(x/(-2*x + 2*Log[x^2]))*(-100*x - 85*x^2 - 5*x^3 + 10*x^3*Log[x] + (100 + 125*x + 5*x^2 - 20*x^2*Log[x])*Log[x^2] + (-40 + 10*x*Log[x])*Log[x^2]^2 + (200 + 40*x + (-50*x - 10*x^2)*Log[x] + (-100 - 20*x + (25*x + 5*x^2)*Log[x])*Log[x^2])*Log[(4 - x*Log[x])/x]))/(- 8*x^2 + 2*x^3*Log[x] + (16*x - 4*x^2*Log[x])*Log[x^2] + (-8 + 2*x*Log[x])* Log[x^2]^2),x]
3.10.9.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Timed out.
\[\int \frac {\left (\left (\left (\left (5 x^{2}+25 x \right ) \ln \left (x \right )-20 x -100\right ) \ln \left (x^{2}\right )+\left (-10 x^{2}-50 x \right ) \ln \left (x \right )+40 x +200\right ) \ln \left (\frac {-x \ln \left (x \right )+4}{x}\right )+\left (10 x \ln \left (x \right )-40\right ) \ln \left (x^{2}\right )^{2}+\left (-20 x^{2} \ln \left (x \right )+5 x^{2}+125 x +100\right ) \ln \left (x^{2}\right )+10 x^{3} \ln \left (x \right )-5 x^{3}-85 x^{2}-100 x \right ) {\mathrm e}^{\frac {x \ln \left (\frac {-x \ln \left (x \right )+4}{x}\right )}{2 \ln \left (x^{2}\right )-2 x}}+\left (\left (-20 x^{2}-50 x \right ) \ln \left (x \right )+80 x +200\right ) \ln \left (x^{2}\right )^{2}+\left (\left (40 x^{3}+100 x^{2}\right ) \ln \left (x \right )-160 x^{2}-400 x \right ) \ln \left (x^{2}\right )+\left (-20 x^{4}-50 x^{3}\right ) \ln \left (x \right )+80 x^{3}+200 x^{2}}{\left (2 x \ln \left (x \right )-8\right ) \ln \left (x^{2}\right )^{2}+\left (-4 x^{2} \ln \left (x \right )+16 x \right ) \ln \left (x^{2}\right )+2 x^{3} \ln \left (x \right )-8 x^{2}}d x\]
int((((((5*x^2+25*x)*ln(x)-20*x-100)*ln(x^2)+(-10*x^2-50*x)*ln(x)+40*x+200 )*ln((-x*ln(x)+4)/x)+(10*x*ln(x)-40)*ln(x^2)^2+(-20*x^2*ln(x)+5*x^2+125*x+ 100)*ln(x^2)+10*x^3*ln(x)-5*x^3-85*x^2-100*x)*exp(x*ln((-x*ln(x)+4)/x)/(2* ln(x^2)-2*x))+((-20*x^2-50*x)*ln(x)+80*x+200)*ln(x^2)^2+((40*x^3+100*x^2)* ln(x)-160*x^2-400*x)*ln(x^2)+(-20*x^4-50*x^3)*ln(x)+80*x^3+200*x^2)/((2*x* ln(x)-8)*ln(x^2)^2+(-4*x^2*ln(x)+16*x)*ln(x^2)+2*x^3*ln(x)-8*x^2),x)
int((((((5*x^2+25*x)*ln(x)-20*x-100)*ln(x^2)+(-10*x^2-50*x)*ln(x)+40*x+200 )*ln((-x*ln(x)+4)/x)+(10*x*ln(x)-40)*ln(x^2)^2+(-20*x^2*ln(x)+5*x^2+125*x+ 100)*ln(x^2)+10*x^3*ln(x)-5*x^3-85*x^2-100*x)*exp(x*ln((-x*ln(x)+4)/x)/(2* ln(x^2)-2*x))+((-20*x^2-50*x)*ln(x)+80*x+200)*ln(x^2)^2+((40*x^3+100*x^2)* ln(x)-160*x^2-400*x)*ln(x^2)+(-20*x^4-50*x^3)*ln(x)+80*x^3+200*x^2)/((2*x* ln(x)-8)*ln(x^2)^2+(-4*x^2*ln(x)+16*x)*ln(x^2)+2*x^3*ln(x)-8*x^2),x)
Time = 0.28 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.80 \[ \int \frac {200 x^2+80 x^3+\left (-50 x^3-20 x^4\right ) \log (x)+\left (-400 x-160 x^2+\left (100 x^2+40 x^3\right ) \log (x)\right ) \log \left (x^2\right )+\left (200+80 x+\left (-50 x-20 x^2\right ) \log (x)\right ) \log ^2\left (x^2\right )+\left (\frac {4-x \log (x)}{x}\right )^{\frac {x}{-2 x+2 \log \left (x^2\right )}} \left (-100 x-85 x^2-5 x^3+10 x^3 \log (x)+\left (100+125 x+5 x^2-20 x^2 \log (x)\right ) \log \left (x^2\right )+(-40+10 x \log (x)) \log ^2\left (x^2\right )+\left (200+40 x+\left (-50 x-10 x^2\right ) \log (x)+\left (-100-20 x+\left (25 x+5 x^2\right ) \log (x)\right ) \log \left (x^2\right )\right ) \log \left (\frac {4-x \log (x)}{x}\right )\right )}{-8 x^2+2 x^3 \log (x)+\left (16 x-4 x^2 \log (x)\right ) \log \left (x^2\right )+(-8+2 x \log (x)) \log ^2\left (x^2\right )} \, dx=-\frac {5 \, {\left ({\left (x^{2} + 5 \, x\right )} \left (-\frac {x \log \left (x\right ) - 4}{x}\right )^{\frac {x}{2 \, {\left (x - 2 \, \log \left (x\right )\right )}}} - x - 5\right )}}{\left (-\frac {x \log \left (x\right ) - 4}{x}\right )^{\frac {x}{2 \, {\left (x - 2 \, \log \left (x\right )\right )}}}} \]
integrate((((((5*x^2+25*x)*log(x)-20*x-100)*log(x^2)+(-10*x^2-50*x)*log(x) +40*x+200)*log((-x*log(x)+4)/x)+(10*x*log(x)-40)*log(x^2)^2+(-20*x^2*log(x )+5*x^2+125*x+100)*log(x^2)+10*x^3*log(x)-5*x^3-85*x^2-100*x)*exp(x*log((- x*log(x)+4)/x)/(2*log(x^2)-2*x))+((-20*x^2-50*x)*log(x)+80*x+200)*log(x^2) ^2+((40*x^3+100*x^2)*log(x)-160*x^2-400*x)*log(x^2)+(-20*x^4-50*x^3)*log(x )+80*x^3+200*x^2)/((2*x*log(x)-8)*log(x^2)^2+(-4*x^2*log(x)+16*x)*log(x^2) +2*x^3*log(x)-8*x^2),x, algorithm=\
-5*((x^2 + 5*x)*(-(x*log(x) - 4)/x)^(1/2*x/(x - 2*log(x))) - x - 5)/(-(x*l og(x) - 4)/x)^(1/2*x/(x - 2*log(x)))
Exception generated. \[ \int \frac {200 x^2+80 x^3+\left (-50 x^3-20 x^4\right ) \log (x)+\left (-400 x-160 x^2+\left (100 x^2+40 x^3\right ) \log (x)\right ) \log \left (x^2\right )+\left (200+80 x+\left (-50 x-20 x^2\right ) \log (x)\right ) \log ^2\left (x^2\right )+\left (\frac {4-x \log (x)}{x}\right )^{\frac {x}{-2 x+2 \log \left (x^2\right )}} \left (-100 x-85 x^2-5 x^3+10 x^3 \log (x)+\left (100+125 x+5 x^2-20 x^2 \log (x)\right ) \log \left (x^2\right )+(-40+10 x \log (x)) \log ^2\left (x^2\right )+\left (200+40 x+\left (-50 x-10 x^2\right ) \log (x)+\left (-100-20 x+\left (25 x+5 x^2\right ) \log (x)\right ) \log \left (x^2\right )\right ) \log \left (\frac {4-x \log (x)}{x}\right )\right )}{-8 x^2+2 x^3 \log (x)+\left (16 x-4 x^2 \log (x)\right ) \log \left (x^2\right )+(-8+2 x \log (x)) \log ^2\left (x^2\right )} \, dx=\text {Exception raised: TypeError} \]
integrate((((((5*x**2+25*x)*ln(x)-20*x-100)*ln(x**2)+(-10*x**2-50*x)*ln(x) +40*x+200)*ln((-x*ln(x)+4)/x)+(10*x*ln(x)-40)*ln(x**2)**2+(-20*x**2*ln(x)+ 5*x**2+125*x+100)*ln(x**2)+10*x**3*ln(x)-5*x**3-85*x**2-100*x)*exp(x*ln((- x*ln(x)+4)/x)/(2*ln(x**2)-2*x))+((-20*x**2-50*x)*ln(x)+80*x+200)*ln(x**2)* *2+((40*x**3+100*x**2)*ln(x)-160*x**2-400*x)*ln(x**2)+(-20*x**4-50*x**3)*l n(x)+80*x**3+200*x**2)/((2*x*ln(x)-8)*ln(x**2)**2+(-4*x**2*ln(x)+16*x)*ln( x**2)+2*x**3*ln(x)-8*x**2),x)
\[ \int \frac {200 x^2+80 x^3+\left (-50 x^3-20 x^4\right ) \log (x)+\left (-400 x-160 x^2+\left (100 x^2+40 x^3\right ) \log (x)\right ) \log \left (x^2\right )+\left (200+80 x+\left (-50 x-20 x^2\right ) \log (x)\right ) \log ^2\left (x^2\right )+\left (\frac {4-x \log (x)}{x}\right )^{\frac {x}{-2 x+2 \log \left (x^2\right )}} \left (-100 x-85 x^2-5 x^3+10 x^3 \log (x)+\left (100+125 x+5 x^2-20 x^2 \log (x)\right ) \log \left (x^2\right )+(-40+10 x \log (x)) \log ^2\left (x^2\right )+\left (200+40 x+\left (-50 x-10 x^2\right ) \log (x)+\left (-100-20 x+\left (25 x+5 x^2\right ) \log (x)\right ) \log \left (x^2\right )\right ) \log \left (\frac {4-x \log (x)}{x}\right )\right )}{-8 x^2+2 x^3 \log (x)+\left (16 x-4 x^2 \log (x)\right ) \log \left (x^2\right )+(-8+2 x \log (x)) \log ^2\left (x^2\right )} \, dx=\int { \frac {5 \, {\left (16 \, x^{3} - 2 \, {\left ({\left (2 \, x^{2} + 5 \, x\right )} \log \left (x\right ) - 8 \, x - 20\right )} \log \left (x^{2}\right )^{2} + 40 \, x^{2} - 4 \, {\left (8 \, x^{2} - {\left (2 \, x^{3} + 5 \, x^{2}\right )} \log \left (x\right ) + 20 \, x\right )} \log \left (x^{2}\right ) - 2 \, {\left (2 \, x^{4} + 5 \, x^{3}\right )} \log \left (x\right ) + \frac {2 \, x^{3} \log \left (x\right ) - x^{3} + 2 \, {\left (x \log \left (x\right ) - 4\right )} \log \left (x^{2}\right )^{2} - 17 \, x^{2} - {\left (4 \, x^{2} \log \left (x\right ) - x^{2} - 25 \, x - 20\right )} \log \left (x^{2}\right ) + {\left ({\left ({\left (x^{2} + 5 \, x\right )} \log \left (x\right ) - 4 \, x - 20\right )} \log \left (x^{2}\right ) - 2 \, {\left (x^{2} + 5 \, x\right )} \log \left (x\right ) + 8 \, x + 40\right )} \log \left (-\frac {x \log \left (x\right ) - 4}{x}\right ) - 20 \, x}{\left (-\frac {x \log \left (x\right ) - 4}{x}\right )^{\frac {x}{2 \, {\left (x - \log \left (x^{2}\right )\right )}}}}\right )}}{2 \, {\left (x^{3} \log \left (x\right ) + {\left (x \log \left (x\right ) - 4\right )} \log \left (x^{2}\right )^{2} - 4 \, x^{2} - 2 \, {\left (x^{2} \log \left (x\right ) - 4 \, x\right )} \log \left (x^{2}\right )\right )}} \,d x } \]
integrate((((((5*x^2+25*x)*log(x)-20*x-100)*log(x^2)+(-10*x^2-50*x)*log(x) +40*x+200)*log((-x*log(x)+4)/x)+(10*x*log(x)-40)*log(x^2)^2+(-20*x^2*log(x )+5*x^2+125*x+100)*log(x^2)+10*x^3*log(x)-5*x^3-85*x^2-100*x)*exp(x*log((- x*log(x)+4)/x)/(2*log(x^2)-2*x))+((-20*x^2-50*x)*log(x)+80*x+200)*log(x^2) ^2+((40*x^3+100*x^2)*log(x)-160*x^2-400*x)*log(x^2)+(-20*x^4-50*x^3)*log(x )+80*x^3+200*x^2)/((2*x*log(x)-8)*log(x^2)^2+(-4*x^2*log(x)+16*x)*log(x^2) +2*x^3*log(x)-8*x^2),x, algorithm=\
-25*x - 10*integrate(x, x) + 5/2*integrate((x^(7/2)*(2*log(x) - 1) - (2*lo g(x)^3 + 6*log(x)^2 - 2*log(x) + 17)*x^(5/2) - 2*(log(x)^3 - 9*log(x)^2 - 21*log(x) + 10)*x^(3/2) + 8*sqrt(x)*log(x)^2 + 2*((log(x)^2 - log(x))*x^(5 /2) + (5*log(x)^2 - 9*log(x) + 4)*x^(3/2) - 20*sqrt(x)*(log(x) - 1))*log(- x*log(x) + 4))*e^(-log(-x*log(x) + 4)*log(x)/(x - 2*log(x)) + log(x)^2/(x - 2*log(x)))/((x^3*log(x) - 4*(log(x)^2 + 1)*x^2 + 4*(log(x)^3 + 4*log(x)) *x - 16*log(x)^2)*sqrt(-x*log(x) + 4)), x)
\[ \int \frac {200 x^2+80 x^3+\left (-50 x^3-20 x^4\right ) \log (x)+\left (-400 x-160 x^2+\left (100 x^2+40 x^3\right ) \log (x)\right ) \log \left (x^2\right )+\left (200+80 x+\left (-50 x-20 x^2\right ) \log (x)\right ) \log ^2\left (x^2\right )+\left (\frac {4-x \log (x)}{x}\right )^{\frac {x}{-2 x+2 \log \left (x^2\right )}} \left (-100 x-85 x^2-5 x^3+10 x^3 \log (x)+\left (100+125 x+5 x^2-20 x^2 \log (x)\right ) \log \left (x^2\right )+(-40+10 x \log (x)) \log ^2\left (x^2\right )+\left (200+40 x+\left (-50 x-10 x^2\right ) \log (x)+\left (-100-20 x+\left (25 x+5 x^2\right ) \log (x)\right ) \log \left (x^2\right )\right ) \log \left (\frac {4-x \log (x)}{x}\right )\right )}{-8 x^2+2 x^3 \log (x)+\left (16 x-4 x^2 \log (x)\right ) \log \left (x^2\right )+(-8+2 x \log (x)) \log ^2\left (x^2\right )} \, dx=\int { \frac {5 \, {\left (16 \, x^{3} - 2 \, {\left ({\left (2 \, x^{2} + 5 \, x\right )} \log \left (x\right ) - 8 \, x - 20\right )} \log \left (x^{2}\right )^{2} + 40 \, x^{2} - 4 \, {\left (8 \, x^{2} - {\left (2 \, x^{3} + 5 \, x^{2}\right )} \log \left (x\right ) + 20 \, x\right )} \log \left (x^{2}\right ) - 2 \, {\left (2 \, x^{4} + 5 \, x^{3}\right )} \log \left (x\right ) + \frac {2 \, x^{3} \log \left (x\right ) - x^{3} + 2 \, {\left (x \log \left (x\right ) - 4\right )} \log \left (x^{2}\right )^{2} - 17 \, x^{2} - {\left (4 \, x^{2} \log \left (x\right ) - x^{2} - 25 \, x - 20\right )} \log \left (x^{2}\right ) + {\left ({\left ({\left (x^{2} + 5 \, x\right )} \log \left (x\right ) - 4 \, x - 20\right )} \log \left (x^{2}\right ) - 2 \, {\left (x^{2} + 5 \, x\right )} \log \left (x\right ) + 8 \, x + 40\right )} \log \left (-\frac {x \log \left (x\right ) - 4}{x}\right ) - 20 \, x}{\left (-\frac {x \log \left (x\right ) - 4}{x}\right )^{\frac {x}{2 \, {\left (x - \log \left (x^{2}\right )\right )}}}}\right )}}{2 \, {\left (x^{3} \log \left (x\right ) + {\left (x \log \left (x\right ) - 4\right )} \log \left (x^{2}\right )^{2} - 4 \, x^{2} - 2 \, {\left (x^{2} \log \left (x\right ) - 4 \, x\right )} \log \left (x^{2}\right )\right )}} \,d x } \]
integrate((((((5*x^2+25*x)*log(x)-20*x-100)*log(x^2)+(-10*x^2-50*x)*log(x) +40*x+200)*log((-x*log(x)+4)/x)+(10*x*log(x)-40)*log(x^2)^2+(-20*x^2*log(x )+5*x^2+125*x+100)*log(x^2)+10*x^3*log(x)-5*x^3-85*x^2-100*x)*exp(x*log((- x*log(x)+4)/x)/(2*log(x^2)-2*x))+((-20*x^2-50*x)*log(x)+80*x+200)*log(x^2) ^2+((40*x^3+100*x^2)*log(x)-160*x^2-400*x)*log(x^2)+(-20*x^4-50*x^3)*log(x )+80*x^3+200*x^2)/((2*x*log(x)-8)*log(x^2)^2+(-4*x^2*log(x)+16*x)*log(x^2) +2*x^3*log(x)-8*x^2),x, algorithm=\
integrate(5/2*(16*x^3 - 2*((2*x^2 + 5*x)*log(x) - 8*x - 20)*log(x^2)^2 + 4 0*x^2 - 4*(8*x^2 - (2*x^3 + 5*x^2)*log(x) + 20*x)*log(x^2) - 2*(2*x^4 + 5* x^3)*log(x) + (2*x^3*log(x) - x^3 + 2*(x*log(x) - 4)*log(x^2)^2 - 17*x^2 - (4*x^2*log(x) - x^2 - 25*x - 20)*log(x^2) + (((x^2 + 5*x)*log(x) - 4*x - 20)*log(x^2) - 2*(x^2 + 5*x)*log(x) + 8*x + 40)*log(-(x*log(x) - 4)/x) - 2 0*x)/(-(x*log(x) - 4)/x)^(1/2*x/(x - log(x^2))))/(x^3*log(x) + (x*log(x) - 4)*log(x^2)^2 - 4*x^2 - 2*(x^2*log(x) - 4*x)*log(x^2)), x)
Time = 10.20 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.06 \[ \int \frac {200 x^2+80 x^3+\left (-50 x^3-20 x^4\right ) \log (x)+\left (-400 x-160 x^2+\left (100 x^2+40 x^3\right ) \log (x)\right ) \log \left (x^2\right )+\left (200+80 x+\left (-50 x-20 x^2\right ) \log (x)\right ) \log ^2\left (x^2\right )+\left (\frac {4-x \log (x)}{x}\right )^{\frac {x}{-2 x+2 \log \left (x^2\right )}} \left (-100 x-85 x^2-5 x^3+10 x^3 \log (x)+\left (100+125 x+5 x^2-20 x^2 \log (x)\right ) \log \left (x^2\right )+(-40+10 x \log (x)) \log ^2\left (x^2\right )+\left (200+40 x+\left (-50 x-10 x^2\right ) \log (x)+\left (-100-20 x+\left (25 x+5 x^2\right ) \log (x)\right ) \log \left (x^2\right )\right ) \log \left (\frac {4-x \log (x)}{x}\right )\right )}{-8 x^2+2 x^3 \log (x)+\left (16 x-4 x^2 \log (x)\right ) \log \left (x^2\right )+(-8+2 x \log (x)) \log ^2\left (x^2\right )} \, dx=-5\,\left (x+5\right )\,\left (x-\frac {1}{{\left (-\frac {x\,\ln \left (x\right )-4}{x}\right )}^{\frac {x}{2\,x-2\,\ln \left (x^2\right )}}}\right ) \]
int((exp(-(x*log(-(x*log(x) - 4)/x))/(2*x - 2*log(x^2)))*(10*x^3*log(x) - 100*x + log(x^2)^2*(10*x*log(x) - 40) + log(x^2)*(125*x - 20*x^2*log(x) + 5*x^2 + 100) + log(-(x*log(x) - 4)/x)*(40*x - log(x^2)*(20*x - log(x)*(25* x + 5*x^2) + 100) - log(x)*(50*x + 10*x^2) + 200) - 85*x^2 - 5*x^3) - log( x^2)*(400*x - log(x)*(100*x^2 + 40*x^3) + 160*x^2) - log(x)*(50*x^3 + 20*x ^4) + 200*x^2 + 80*x^3 + log(x^2)^2*(80*x - log(x)*(50*x + 20*x^2) + 200)) /(2*x^3*log(x) + log(x^2)^2*(2*x*log(x) - 8) - 8*x^2 + log(x^2)*(16*x - 4* x^2*log(x))),x)