\(\int \frac {(-2+16 e^{\frac {1}{5} (5-2 x)} x-2 x \log (x)) \log (\frac {1}{4} (-1+20 e^{\frac {1}{5} (5-2 x)}-x+(1+x) \log (x)))}{-x+20 e^{\frac {1}{5} (5-2 x)} x-x^2+(x+x^2) \log (x)} \, dx\) [959]

Optimal result

Integrand size = 84, antiderivative size = 30 \[ \int \frac {\left (-2+16 e^{\frac {1}{5} (5-2 x)} x-2 x \log (x)\right ) \log \left (\frac {1}{4} \left (-1+20 e^{\frac {1}{5} (5-2 x)}-x+(1+x) \log (x)\right )\right )}{-x+20 e^{\frac {1}{5} (5-2 x)} x-x^2+\left (x+x^2\right ) \log (x)} \, dx=5-\log ^2\left (5 e^{1-\frac {2 x}{5}}+\frac {1}{4} (1+x) (-1+\log (x))\right ) \] Output:

5-ln(5*exp(-2/5*x+1)+1/4*(1+x)*(ln(x)-1))^2
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.03 \[ \int \frac {\left (-2+16 e^{\frac {1}{5} (5-2 x)} x-2 x \log (x)\right ) \log \left (\frac {1}{4} \left (-1+20 e^{\frac {1}{5} (5-2 x)}-x+(1+x) \log (x)\right )\right )}{-x+20 e^{\frac {1}{5} (5-2 x)} x-x^2+\left (x+x^2\right ) \log (x)} \, dx=-\log ^2\left (\frac {1}{4} \left (-1+20 e^{1-\frac {2 x}{5}}-x+(1+x) \log (x)\right )\right ) \] Input:

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]
 

Output:

-Log[(-1 + 20*E^(1 - (2*x)/5) - x + (1 + x)*Log[x])/4]^2
 

Rubi [F]

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

\(\displaystyle -2 \int \frac {\left (e^{2 x/5} \log (x) x-8 e x+e^{2 x/5}\right ) \log \left (\frac {1}{4} \left (-x+20 e^{1-\frac {2 x}{5}}+(x+1) \log (x)-1\right )\right )}{x \left (-e^{2 x/5} (x+1)+e^{2 x/5} \log (x) (x+1)+20 e\right )}dx\)

Input:

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]
 

Output:

$Aborted
 

Maple [F]

\[\int \frac {\left (-2 x \ln \left (x \right )+16 x \,{\mathrm e}^{-\frac {2 x}{5}+1}-2\right ) \ln \left (\frac {\ln \left (x \right ) \left (1+x \right )}{4}+5 \,{\mathrm e}^{-\frac {2 x}{5}+1}-\frac {x}{4}-\frac {1}{4}\right )}{\left (x^{2}+x \right ) \ln \left (x \right )+20 x \,{\mathrm e}^{-\frac {2 x}{5}+1}-x^{2}-x}d x\]

Input:

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)
 

Output:

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)
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.83 \[ \int \frac {\left (-2+16 e^{\frac {1}{5} (5-2 x)} x-2 x \log (x)\right ) \log \left (\frac {1}{4} \left (-1+20 e^{\frac {1}{5} (5-2 x)}-x+(1+x) \log (x)\right )\right )}{-x+20 e^{\frac {1}{5} (5-2 x)} x-x^2+\left (x+x^2\right ) \log (x)} \, dx=-\log \left (\frac {1}{4} \, {\left (x + 1\right )} \log \left (x\right ) - \frac {1}{4} \, x + 5 \, e^{\left (-\frac {2}{5} \, x + 1\right )} - \frac {1}{4}\right )^{2} \] Input:

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= 
"fricas")
 

Output:

-log(1/4*(x + 1)*log(x) - 1/4*x + 5*e^(-2/5*x + 1) - 1/4)^2
 

Sympy [A] (verification not implemented)

Time = 0.63 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.97 \[ \int \frac {\left (-2+16 e^{\frac {1}{5} (5-2 x)} x-2 x \log (x)\right ) \log \left (\frac {1}{4} \left (-1+20 e^{\frac {1}{5} (5-2 x)}-x+(1+x) \log (x)\right )\right )}{-x+20 e^{\frac {1}{5} (5-2 x)} x-x^2+\left (x+x^2\right ) \log (x)} \, dx=- \log {\left (- \frac {x}{4} + \frac {\left (x + 1\right ) \log {\left (x \right )}}{4} + 5 e^{1 - \frac {2 x}{5}} - \frac {1}{4} \right )}^{2} \] Input:

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)
 

Output:

-log(-x/4 + (x + 1)*log(x)/4 + 5*exp(1 - 2*x/5) - 1/4)**2
 

Maxima [F]

\[ \int \frac {\left (-2+16 e^{\frac {1}{5} (5-2 x)} x-2 x \log (x)\right ) \log \left (\frac {1}{4} \left (-1+20 e^{\frac {1}{5} (5-2 x)}-x+(1+x) \log (x)\right )\right )}{-x+20 e^{\frac {1}{5} (5-2 x)} x-x^2+\left (x+x^2\right ) \log (x)} \, dx=\int { -\frac {2 \, {\left (8 \, x e^{\left (-\frac {2}{5} \, x + 1\right )} - x \log \left (x\right ) - 1\right )} \log \left (\frac {1}{4} \, {\left (x + 1\right )} \log \left (x\right ) - \frac {1}{4} \, x + 5 \, e^{\left (-\frac {2}{5} \, x + 1\right )} - \frac {1}{4}\right )}{x^{2} - 20 \, x e^{\left (-\frac {2}{5} \, x + 1\right )} - {\left (x^{2} + x\right )} \log \left (x\right ) + x} \,d x } \] Input:

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= 
"maxima")
 

Output:

-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)
 

Giac [F]

\[ \int \frac {\left (-2+16 e^{\frac {1}{5} (5-2 x)} x-2 x \log (x)\right ) \log \left (\frac {1}{4} \left (-1+20 e^{\frac {1}{5} (5-2 x)}-x+(1+x) \log (x)\right )\right )}{-x+20 e^{\frac {1}{5} (5-2 x)} x-x^2+\left (x+x^2\right ) \log (x)} \, dx=\int { -\frac {2 \, {\left (8 \, x e^{\left (-\frac {2}{5} \, x + 1\right )} - x \log \left (x\right ) - 1\right )} \log \left (\frac {1}{4} \, {\left (x + 1\right )} \log \left (x\right ) - \frac {1}{4} \, x + 5 \, e^{\left (-\frac {2}{5} \, x + 1\right )} - \frac {1}{4}\right )}{x^{2} - 20 \, x e^{\left (-\frac {2}{5} \, x + 1\right )} - {\left (x^{2} + x\right )} \log \left (x\right ) + x} \,d x } \] Input:

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= 
"giac")
 

Output:

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)
 

Mupad [B] (verification not implemented)

Time = 3.35 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.87 \[ \int \frac {\left (-2+16 e^{\frac {1}{5} (5-2 x)} x-2 x \log (x)\right ) \log \left (\frac {1}{4} \left (-1+20 e^{\frac {1}{5} (5-2 x)}-x+(1+x) \log (x)\right )\right )}{-x+20 e^{\frac {1}{5} (5-2 x)} x-x^2+\left (x+x^2\right ) \log (x)} \, dx=-{\ln \left (5\,{\mathrm {e}}^{-\frac {2\,x}{5}}\,\mathrm {e}-\frac {x}{4}+\ln \left (x^{1/4}\right )\,\left (x+1\right )-\frac {1}{4}\right )}^2 \] Input:

int((log(5*exp(1 - (2*x)/5) - x/4 + (log(x)*(x + 1))/4 - 1/4)*(2*x*log(x) 
- 16*x*exp(1 - (2*x)/5) + 2))/(x - 20*x*exp(1 - (2*x)/5) + x^2 - log(x)*(x 
 + x^2)),x)
 

Output:

-log(5*exp(-(2*x)/5)*exp(1) - x/4 + log(x^(1/4))*(x + 1) - 1/4)^2
 

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.67 \[ \int \frac {\left (-2+16 e^{\frac {1}{5} (5-2 x)} x-2 x \log (x)\right ) \log \left (\frac {1}{4} \left (-1+20 e^{\frac {1}{5} (5-2 x)}-x+(1+x) \log (x)\right )\right )}{-x+20 e^{\frac {1}{5} (5-2 x)} x-x^2+\left (x+x^2\right ) \log (x)} \, dx=-\mathrm {log}\left (\frac {e^{\frac {2 x}{5}} \mathrm {log}\left (x \right ) x +e^{\frac {2 x}{5}} \mathrm {log}\left (x \right )-e^{\frac {2 x}{5}} x -e^{\frac {2 x}{5}}+20 e}{4 e^{\frac {2 x}{5}}}\right )^{2} \] Input:

int((-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)
 

Output:

 - log((e**((2*x)/5)*log(x)*x + e**((2*x)/5)*log(x) - e**((2*x)/5)*x - e** 
((2*x)/5) + 20*e)/(4*e**((2*x)/5)))**2