Integrand size = 84, antiderivative size = 30 \[ \int \frac {\left (-2+16 e^{\frac {1}{5} (5-2 x)} x-2 x \log (x)\right ) \log \left (\frac {1}{4} \left (-1+20 e^{\frac {1}{5} (5-2 x)}-x+(1+x) \log (x)\right )\right )}{-x+20 e^{\frac {1}{5} (5-2 x)} x-x^2+\left (x+x^2\right ) \log (x)} \, dx=5-\log ^2\left (5 e^{1-\frac {2 x}{5}}+\frac {1}{4} (1+x) (-1+\log (x))\right ) \]
Time = 0.09 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.03 \[ \int \frac {\left (-2+16 e^{\frac {1}{5} (5-2 x)} x-2 x \log (x)\right ) \log \left (\frac {1}{4} \left (-1+20 e^{\frac {1}{5} (5-2 x)}-x+(1+x) \log (x)\right )\right )}{-x+20 e^{\frac {1}{5} (5-2 x)} x-x^2+\left (x+x^2\right ) \log (x)} \, dx=-\log ^2\left (\frac {1}{4} \left (-1+20 e^{1-\frac {2 x}{5}}-x+(1+x) \log (x)\right )\right ) \]
Integrate[((-2 + 16*E^((5 - 2*x)/5)*x - 2*x*Log[x])*Log[(-1 + 20*E^((5 - 2 *x)/5) - x + (1 + x)*Log[x])/4])/(-x + 20*E^((5 - 2*x)/5)*x - x^2 + (x + x ^2)*Log[x]),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.
\(\displaystyle \int \frac {\left (16 e^{\frac {1}{5} (5-2 x)} x-2 x \log (x)-2\right ) \log \left (\frac {1}{4} \left (-x+20 e^{\frac {1}{5} (5-2 x)}+(x+1) \log (x)-1\right )\right )}{-x^2+\left (x^2+x\right ) \log (x)+20 e^{\frac {1}{5} (5-2 x)} x-x} \, dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {8 e \left (-2 x^2+2 x^2 \log (x)-2 x+7 x \log (x)+5\right ) \log \left (\frac {1}{4} \left (-x+20 e^{1-\frac {2 x}{5}}+(x+1) \log (x)-1\right )\right )}{x (x+1) (\log (x)-1) \left (-e^{2 x/5} x-e^{2 x/5}+e^{2 x/5} x \log (x)+e^{2 x/5} \log (x)+20 e\right )}-\frac {2 (x \log (x)+1) \log \left (\frac {1}{4} \left (-x+20 e^{1-\frac {2 x}{5}}+(x+1) \log (x)-1\right )\right )}{x (x+1) (\log (x)-1)}\right )dx\) |
\(\Big \downarrow \) 7239 |
\(\displaystyle \int \frac {2 \left (8 e x-e^{2 x/5}-e^{2 x/5} x \log (x)\right ) \log \left (\frac {1}{4} \left (-x+20 e^{1-\frac {2 x}{5}}+(x+1) \log (x)-1\right )\right )}{x \left (-e^{2 x/5} (x+1)+e^{2 x/5} (x+1) \log (x)+20 e\right )}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle 2 \int -\frac {\left (e^{2 x/5} \log (x) x-8 e x+e^{2 x/5}\right ) \log \left (\frac {1}{4} \left (-x+20 e^{1-\frac {2 x}{5}}+(x+1) \log (x)-1\right )\right )}{x \left (-e^{2 x/5} (x+1)+e^{2 x/5} \log (x) (x+1)+20 e\right )}dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -2 \int \frac {\left (e^{2 x/5} \log (x) x-8 e x+e^{2 x/5}\right ) \log \left (\frac {1}{4} \left (-x+20 e^{1-\frac {2 x}{5}}+(x+1) \log (x)-1\right )\right )}{x \left (-e^{2 x/5} (x+1)+e^{2 x/5} \log (x) (x+1)+20 e\right )}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle -2 \int \left (\frac {(x \log (x)+1) \log \left (\frac {1}{4} \left (-x+20 e^{1-\frac {2 x}{5}}+(x+1) \log (x)-1\right )\right )}{x (x+1) (\log (x)-1)}-\frac {4 e \left (2 \log (x) x^2-2 x^2+7 \log (x) x-2 x+5\right ) \log \left (\frac {1}{4} \left (-x+20 e^{1-\frac {2 x}{5}}+(x+1) \log (x)-1\right )\right )}{x (x+1) (\log (x)-1) \left (-e^{2 x/5} x+e^{2 x/5} \log (x) x-e^{2 x/5}+e^{2 x/5} \log (x)+20 e\right )}\right )dx\) |
\(\Big \downarrow \) 7239 |
\(\displaystyle -2 \int \frac {\left (e^{2 x/5} \log (x) x-8 e x+e^{2 x/5}\right ) \log \left (\frac {1}{4} \left (-x+20 e^{1-\frac {2 x}{5}}+(x+1) \log (x)-1\right )\right )}{x \left (-e^{2 x/5} (x+1)+e^{2 x/5} \log (x) (x+1)+20 e\right )}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle -2 \int \left (\frac {(x \log (x)+1) \log \left (\frac {1}{4} \left (-x+20 e^{1-\frac {2 x}{5}}+(x+1) \log (x)-1\right )\right )}{x (x+1) (\log (x)-1)}-\frac {4 e \left (2 \log (x) x^2-2 x^2+7 \log (x) x-2 x+5\right ) \log \left (\frac {1}{4} \left (-x+20 e^{1-\frac {2 x}{5}}+(x+1) \log (x)-1\right )\right )}{x (x+1) (\log (x)-1) \left (-e^{2 x/5} x+e^{2 x/5} \log (x) x-e^{2 x/5}+e^{2 x/5} \log (x)+20 e\right )}\right )dx\) |
\(\Big \downarrow \) 7239 |
\(\displaystyle -2 \int \frac {\left (e^{2 x/5} \log (x) x-8 e x+e^{2 x/5}\right ) \log \left (\frac {1}{4} \left (-x+20 e^{1-\frac {2 x}{5}}+(x+1) \log (x)-1\right )\right )}{x \left (-e^{2 x/5} (x+1)+e^{2 x/5} \log (x) (x+1)+20 e\right )}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle -2 \int \left (\frac {(x \log (x)+1) \log \left (\frac {1}{4} \left (-x+20 e^{1-\frac {2 x}{5}}+(x+1) \log (x)-1\right )\right )}{x (x+1) (\log (x)-1)}-\frac {4 e \left (2 \log (x) x^2-2 x^2+7 \log (x) x-2 x+5\right ) \log \left (\frac {1}{4} \left (-x+20 e^{1-\frac {2 x}{5}}+(x+1) \log (x)-1\right )\right )}{x (x+1) (\log (x)-1) \left (-e^{2 x/5} x+e^{2 x/5} \log (x) x-e^{2 x/5}+e^{2 x/5} \log (x)+20 e\right )}\right )dx\) |
\(\Big \downarrow \) 7239 |
\(\displaystyle -2 \int \frac {\left (e^{2 x/5} \log (x) x-8 e x+e^{2 x/5}\right ) \log \left (\frac {1}{4} \left (-x+20 e^{1-\frac {2 x}{5}}+(x+1) \log (x)-1\right )\right )}{x \left (-e^{2 x/5} (x+1)+e^{2 x/5} \log (x) (x+1)+20 e\right )}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle -2 \int \left (\frac {(x \log (x)+1) \log \left (\frac {1}{4} \left (-x+20 e^{1-\frac {2 x}{5}}+(x+1) \log (x)-1\right )\right )}{x (x+1) (\log (x)-1)}-\frac {4 e \left (2 \log (x) x^2-2 x^2+7 \log (x) x-2 x+5\right ) \log \left (\frac {1}{4} \left (-x+20 e^{1-\frac {2 x}{5}}+(x+1) \log (x)-1\right )\right )}{x (x+1) (\log (x)-1) \left (-e^{2 x/5} x+e^{2 x/5} \log (x) x-e^{2 x/5}+e^{2 x/5} \log (x)+20 e\right )}\right )dx\) |
\(\Big \downarrow \) 7239 |
\(\displaystyle -2 \int \frac {\left (e^{2 x/5} \log (x) x-8 e x+e^{2 x/5}\right ) \log \left (\frac {1}{4} \left (-x+20 e^{1-\frac {2 x}{5}}+(x+1) \log (x)-1\right )\right )}{x \left (-e^{2 x/5} (x+1)+e^{2 x/5} \log (x) (x+1)+20 e\right )}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle -2 \int \left (\frac {(x \log (x)+1) \log \left (\frac {1}{4} \left (-x+20 e^{1-\frac {2 x}{5}}+(x+1) \log (x)-1\right )\right )}{x (x+1) (\log (x)-1)}-\frac {4 e \left (2 \log (x) x^2-2 x^2+7 \log (x) x-2 x+5\right ) \log \left (\frac {1}{4} \left (-x+20 e^{1-\frac {2 x}{5}}+(x+1) \log (x)-1\right )\right )}{x (x+1) (\log (x)-1) \left (-e^{2 x/5} x+e^{2 x/5} \log (x) x-e^{2 x/5}+e^{2 x/5} \log (x)+20 e\right )}\right )dx\) |
\(\Big \downarrow \) 7239 |
\(\displaystyle -2 \int \frac {\left (e^{2 x/5} \log (x) x-8 e x+e^{2 x/5}\right ) \log \left (\frac {1}{4} \left (-x+20 e^{1-\frac {2 x}{5}}+(x+1) \log (x)-1\right )\right )}{x \left (-e^{2 x/5} (x+1)+e^{2 x/5} \log (x) (x+1)+20 e\right )}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle -2 \int \left (\frac {(x \log (x)+1) \log \left (\frac {1}{4} \left (-x+20 e^{1-\frac {2 x}{5}}+(x+1) \log (x)-1\right )\right )}{x (x+1) (\log (x)-1)}-\frac {4 e \left (2 \log (x) x^2-2 x^2+7 \log (x) x-2 x+5\right ) \log \left (\frac {1}{4} \left (-x+20 e^{1-\frac {2 x}{5}}+(x+1) \log (x)-1\right )\right )}{x (x+1) (\log (x)-1) \left (-e^{2 x/5} x+e^{2 x/5} \log (x) x-e^{2 x/5}+e^{2 x/5} \log (x)+20 e\right )}\right )dx\) |
\(\Big \downarrow \) 7239 |
\(\displaystyle -2 \int \frac {\left (e^{2 x/5} \log (x) x-8 e x+e^{2 x/5}\right ) \log \left (\frac {1}{4} \left (-x+20 e^{1-\frac {2 x}{5}}+(x+1) \log (x)-1\right )\right )}{x \left (-e^{2 x/5} (x+1)+e^{2 x/5} \log (x) (x+1)+20 e\right )}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle -2 \int \left (\frac {(x \log (x)+1) \log \left (\frac {1}{4} \left (-x+20 e^{1-\frac {2 x}{5}}+(x+1) \log (x)-1\right )\right )}{x (x+1) (\log (x)-1)}-\frac {4 e \left (2 \log (x) x^2-2 x^2+7 \log (x) x-2 x+5\right ) \log \left (\frac {1}{4} \left (-x+20 e^{1-\frac {2 x}{5}}+(x+1) \log (x)-1\right )\right )}{x (x+1) (\log (x)-1) \left (-e^{2 x/5} x+e^{2 x/5} \log (x) x-e^{2 x/5}+e^{2 x/5} \log (x)+20 e\right )}\right )dx\) |
\(\Big \downarrow \) 7239 |
\(\displaystyle -2 \int \frac {\left (e^{2 x/5} \log (x) x-8 e x+e^{2 x/5}\right ) \log \left (\frac {1}{4} \left (-x+20 e^{1-\frac {2 x}{5}}+(x+1) \log (x)-1\right )\right )}{x \left (-e^{2 x/5} (x+1)+e^{2 x/5} \log (x) (x+1)+20 e\right )}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle -2 \int \left (\frac {(x \log (x)+1) \log \left (\frac {1}{4} \left (-x+20 e^{1-\frac {2 x}{5}}+(x+1) \log (x)-1\right )\right )}{x (x+1) (\log (x)-1)}-\frac {4 e \left (2 \log (x) x^2-2 x^2+7 \log (x) x-2 x+5\right ) \log \left (\frac {1}{4} \left (-x+20 e^{1-\frac {2 x}{5}}+(x+1) \log (x)-1\right )\right )}{x (x+1) (\log (x)-1) \left (-e^{2 x/5} x+e^{2 x/5} \log (x) x-e^{2 x/5}+e^{2 x/5} \log (x)+20 e\right )}\right )dx\) |
\(\Big \downarrow \) 7239 |
\(\displaystyle -2 \int \frac {\left (e^{2 x/5} \log (x) x-8 e x+e^{2 x/5}\right ) \log \left (\frac {1}{4} \left (-x+20 e^{1-\frac {2 x}{5}}+(x+1) \log (x)-1\right )\right )}{x \left (-e^{2 x/5} (x+1)+e^{2 x/5} \log (x) (x+1)+20 e\right )}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle -2 \int \left (\frac {(x \log (x)+1) \log \left (\frac {1}{4} \left (-x+20 e^{1-\frac {2 x}{5}}+(x+1) \log (x)-1\right )\right )}{x (x+1) (\log (x)-1)}-\frac {4 e \left (2 \log (x) x^2-2 x^2+7 \log (x) x-2 x+5\right ) \log \left (\frac {1}{4} \left (-x+20 e^{1-\frac {2 x}{5}}+(x+1) \log (x)-1\right )\right )}{x (x+1) (\log (x)-1) \left (-e^{2 x/5} x+e^{2 x/5} \log (x) x-e^{2 x/5}+e^{2 x/5} \log (x)+20 e\right )}\right )dx\) |
\(\Big \downarrow \) 7239 |
\(\displaystyle -2 \int \frac {\left (e^{2 x/5} \log (x) x-8 e x+e^{2 x/5}\right ) \log \left (\frac {1}{4} \left (-x+20 e^{1-\frac {2 x}{5}}+(x+1) \log (x)-1\right )\right )}{x \left (-e^{2 x/5} (x+1)+e^{2 x/5} \log (x) (x+1)+20 e\right )}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle -2 \int \left (\frac {(x \log (x)+1) \log \left (\frac {1}{4} \left (-x+20 e^{1-\frac {2 x}{5}}+(x+1) \log (x)-1\right )\right )}{x (x+1) (\log (x)-1)}-\frac {4 e \left (2 \log (x) x^2-2 x^2+7 \log (x) x-2 x+5\right ) \log \left (\frac {1}{4} \left (-x+20 e^{1-\frac {2 x}{5}}+(x+1) \log (x)-1\right )\right )}{x (x+1) (\log (x)-1) \left (-e^{2 x/5} x+e^{2 x/5} \log (x) x-e^{2 x/5}+e^{2 x/5} \log (x)+20 e\right )}\right )dx\) |
\(\Big \downarrow \) 7239 |
\(\displaystyle -2 \int \frac {\left (e^{2 x/5} \log (x) x-8 e x+e^{2 x/5}\right ) \log \left (\frac {1}{4} \left (-x+20 e^{1-\frac {2 x}{5}}+(x+1) \log (x)-1\right )\right )}{x \left (-e^{2 x/5} (x+1)+e^{2 x/5} \log (x) (x+1)+20 e\right )}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle -2 \int \left (\frac {(x \log (x)+1) \log \left (\frac {1}{4} \left (-x+20 e^{1-\frac {2 x}{5}}+(x+1) \log (x)-1\right )\right )}{x (x+1) (\log (x)-1)}-\frac {4 e \left (2 \log (x) x^2-2 x^2+7 \log (x) x-2 x+5\right ) \log \left (\frac {1}{4} \left (-x+20 e^{1-\frac {2 x}{5}}+(x+1) \log (x)-1\right )\right )}{x (x+1) (\log (x)-1) \left (-e^{2 x/5} x+e^{2 x/5} \log (x) x-e^{2 x/5}+e^{2 x/5} \log (x)+20 e\right )}\right )dx\) |
\(\Big \downarrow \) 7239 |
\(\displaystyle -2 \int \frac {\left (e^{2 x/5} \log (x) x-8 e x+e^{2 x/5}\right ) \log \left (\frac {1}{4} \left (-x+20 e^{1-\frac {2 x}{5}}+(x+1) \log (x)-1\right )\right )}{x \left (-e^{2 x/5} (x+1)+e^{2 x/5} \log (x) (x+1)+20 e\right )}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle -2 \int \left (\frac {(x \log (x)+1) \log \left (\frac {1}{4} \left (-x+20 e^{1-\frac {2 x}{5}}+(x+1) \log (x)-1\right )\right )}{x (x+1) (\log (x)-1)}-\frac {4 e \left (2 \log (x) x^2-2 x^2+7 \log (x) x-2 x+5\right ) \log \left (\frac {1}{4} \left (-x+20 e^{1-\frac {2 x}{5}}+(x+1) \log (x)-1\right )\right )}{x (x+1) (\log (x)-1) \left (-e^{2 x/5} x+e^{2 x/5} \log (x) x-e^{2 x/5}+e^{2 x/5} \log (x)+20 e\right )}\right )dx\) |
\(\Big \downarrow \) 7239 |
\(\displaystyle -2 \int \frac {\left (e^{2 x/5} \log (x) x-8 e x+e^{2 x/5}\right ) \log \left (\frac {1}{4} \left (-x+20 e^{1-\frac {2 x}{5}}+(x+1) \log (x)-1\right )\right )}{x \left (-e^{2 x/5} (x+1)+e^{2 x/5} \log (x) (x+1)+20 e\right )}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle -2 \int \left (\frac {(x \log (x)+1) \log \left (\frac {1}{4} \left (-x+20 e^{1-\frac {2 x}{5}}+(x+1) \log (x)-1\right )\right )}{x (x+1) (\log (x)-1)}-\frac {4 e \left (2 \log (x) x^2-2 x^2+7 \log (x) x-2 x+5\right ) \log \left (\frac {1}{4} \left (-x+20 e^{1-\frac {2 x}{5}}+(x+1) \log (x)-1\right )\right )}{x (x+1) (\log (x)-1) \left (-e^{2 x/5} x+e^{2 x/5} \log (x) x-e^{2 x/5}+e^{2 x/5} \log (x)+20 e\right )}\right )dx\) |
\(\Big \downarrow \) 7239 |
\(\displaystyle -2 \int \frac {\left (e^{2 x/5} \log (x) x-8 e x+e^{2 x/5}\right ) \log \left (\frac {1}{4} \left (-x+20 e^{1-\frac {2 x}{5}}+(x+1) \log (x)-1\right )\right )}{x \left (-e^{2 x/5} (x+1)+e^{2 x/5} \log (x) (x+1)+20 e\right )}dx\) |
Int[((-2 + 16*E^((5 - 2*x)/5)*x - 2*x*Log[x])*Log[(-1 + 20*E^((5 - 2*x)/5) - x + (1 + x)*Log[x])/4])/(-x + 20*E^((5 - 2*x)/5)*x - x^2 + (x + x^2)*Lo g[x]),x]
3.10.59.3.1 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]]
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
\[\int \frac {\left (-2 x \ln \left (x \right )+16 x \,{\mathrm e}^{-\frac {2 x}{5}+1}-2\right ) \ln \left (\frac {\ln \left (x \right ) \left (1+x \right )}{4}+5 \,{\mathrm e}^{-\frac {2 x}{5}+1}-\frac {x}{4}-\frac {1}{4}\right )}{\left (x^{2}+x \right ) \ln \left (x \right )+20 x \,{\mathrm e}^{-\frac {2 x}{5}+1}-x^{2}-x}d x\]
int((-2*x*ln(x)+16*x*exp(-2/5*x+1)-2)*ln(1/4*ln(x)*(1+x)+5*exp(-2/5*x+1)-1 /4*x-1/4)/((x^2+x)*ln(x)+20*x*exp(-2/5*x+1)-x^2-x),x)
int((-2*x*ln(x)+16*x*exp(-2/5*x+1)-2)*ln(1/4*ln(x)*(1+x)+5*exp(-2/5*x+1)-1 /4*x-1/4)/((x^2+x)*ln(x)+20*x*exp(-2/5*x+1)-x^2-x),x)
Time = 0.25 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.83 \[ \int \frac {\left (-2+16 e^{\frac {1}{5} (5-2 x)} x-2 x \log (x)\right ) \log \left (\frac {1}{4} \left (-1+20 e^{\frac {1}{5} (5-2 x)}-x+(1+x) \log (x)\right )\right )}{-x+20 e^{\frac {1}{5} (5-2 x)} x-x^2+\left (x+x^2\right ) \log (x)} \, dx=-\log \left (\frac {1}{4} \, {\left (x + 1\right )} \log \left (x\right ) - \frac {1}{4} \, x + 5 \, e^{\left (-\frac {2}{5} \, x + 1\right )} - \frac {1}{4}\right )^{2} \]
integrate((-2*x*log(x)+16*x*exp(-2/5*x+1)-2)*log(1/4*log(x)*(1+x)+5*exp(-2 /5*x+1)-1/4*x-1/4)/((x^2+x)*log(x)+20*x*exp(-2/5*x+1)-x^2-x),x, algorithm= \
Time = 0.63 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.97 \[ \int \frac {\left (-2+16 e^{\frac {1}{5} (5-2 x)} x-2 x \log (x)\right ) \log \left (\frac {1}{4} \left (-1+20 e^{\frac {1}{5} (5-2 x)}-x+(1+x) \log (x)\right )\right )}{-x+20 e^{\frac {1}{5} (5-2 x)} x-x^2+\left (x+x^2\right ) \log (x)} \, dx=- \log {\left (- \frac {x}{4} + \frac {\left (x + 1\right ) \log {\left (x \right )}}{4} + 5 e^{1 - \frac {2 x}{5}} - \frac {1}{4} \right )}^{2} \]
integrate((-2*x*ln(x)+16*x*exp(-2/5*x+1)-2)*ln(1/4*ln(x)*(1+x)+5*exp(-2/5* x+1)-1/4*x-1/4)/((x**2+x)*ln(x)+20*x*exp(-2/5*x+1)-x**2-x),x)
\[ \int \frac {\left (-2+16 e^{\frac {1}{5} (5-2 x)} x-2 x \log (x)\right ) \log \left (\frac {1}{4} \left (-1+20 e^{\frac {1}{5} (5-2 x)}-x+(1+x) \log (x)\right )\right )}{-x+20 e^{\frac {1}{5} (5-2 x)} x-x^2+\left (x+x^2\right ) \log (x)} \, dx=\int { -\frac {2 \, {\left (8 \, x e^{\left (-\frac {2}{5} \, x + 1\right )} - x \log \left (x\right ) - 1\right )} \log \left (\frac {1}{4} \, {\left (x + 1\right )} \log \left (x\right ) - \frac {1}{4} \, x + 5 \, e^{\left (-\frac {2}{5} \, x + 1\right )} - \frac {1}{4}\right )}{x^{2} - 20 \, x e^{\left (-\frac {2}{5} \, x + 1\right )} - {\left (x^{2} + x\right )} \log \left (x\right ) + x} \,d x } \]
integrate((-2*x*log(x)+16*x*exp(-2/5*x+1)-2)*log(1/4*log(x)*(1+x)+5*exp(-2 /5*x+1)-1/4*x-1/4)/((x^2+x)*log(x)+20*x*exp(-2/5*x+1)-x^2-x),x, algorithm= \
-2*integrate((8*x*e^(-2/5*x + 1) - x*log(x) - 1)*log(1/4*(x + 1)*log(x) - 1/4*x + 5*e^(-2/5*x + 1) - 1/4)/(x^2 - 20*x*e^(-2/5*x + 1) - (x^2 + x)*log (x) + x), x)
\[ \int \frac {\left (-2+16 e^{\frac {1}{5} (5-2 x)} x-2 x \log (x)\right ) \log \left (\frac {1}{4} \left (-1+20 e^{\frac {1}{5} (5-2 x)}-x+(1+x) \log (x)\right )\right )}{-x+20 e^{\frac {1}{5} (5-2 x)} x-x^2+\left (x+x^2\right ) \log (x)} \, dx=\int { -\frac {2 \, {\left (8 \, x e^{\left (-\frac {2}{5} \, x + 1\right )} - x \log \left (x\right ) - 1\right )} \log \left (\frac {1}{4} \, {\left (x + 1\right )} \log \left (x\right ) - \frac {1}{4} \, x + 5 \, e^{\left (-\frac {2}{5} \, x + 1\right )} - \frac {1}{4}\right )}{x^{2} - 20 \, x e^{\left (-\frac {2}{5} \, x + 1\right )} - {\left (x^{2} + x\right )} \log \left (x\right ) + x} \,d x } \]
integrate((-2*x*log(x)+16*x*exp(-2/5*x+1)-2)*log(1/4*log(x)*(1+x)+5*exp(-2 /5*x+1)-1/4*x-1/4)/((x^2+x)*log(x)+20*x*exp(-2/5*x+1)-x^2-x),x, algorithm= \
integrate(-2*(8*x*e^(-2/5*x + 1) - x*log(x) - 1)*log(1/4*(x + 1)*log(x) - 1/4*x + 5*e^(-2/5*x + 1) - 1/4)/(x^2 - 20*x*e^(-2/5*x + 1) - (x^2 + x)*log (x) + x), x)
Time = 14.27 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.87 \[ \int \frac {\left (-2+16 e^{\frac {1}{5} (5-2 x)} x-2 x \log (x)\right ) \log \left (\frac {1}{4} \left (-1+20 e^{\frac {1}{5} (5-2 x)}-x+(1+x) \log (x)\right )\right )}{-x+20 e^{\frac {1}{5} (5-2 x)} x-x^2+\left (x+x^2\right ) \log (x)} \, dx=-{\ln \left (5\,{\mathrm {e}}^{-\frac {2\,x}{5}}\,\mathrm {e}-\frac {x}{4}+\ln \left (x^{1/4}\right )\,\left (x+1\right )-\frac {1}{4}\right )}^2 \]