\(\int \frac {25 x^2+e^x (-100 e^{4/x}+96 x^2)+(10 x^2+e^x (-40 e^{4/x}+39 x^2)) \log (x)+(x^2+e^x (-4 e^{4/x}+4 x^2)) \log ^2(x)}{-25 x^2+e^x (25 e^{4/x} x^2+95 x^3)+(-10 x^2+e^x (10 e^{4/x} x^2+39 x^3)) \log (x)+(-x^2+e^x (e^{4/x} x^2+4 x^3)) \log ^2(x)} \, dx\) [363]

Optimal result

Integrand size = 175, antiderivative size = 28 \[ \int \frac {25 x^2+e^x \left (-100 e^{4/x}+96 x^2\right )+\left (10 x^2+e^x \left (-40 e^{4/x}+39 x^2\right )\right ) \log (x)+\left (x^2+e^x \left (-4 e^{4/x}+4 x^2\right )\right ) \log ^2(x)}{-25 x^2+e^x \left (25 e^{4/x} x^2+95 x^3\right )+\left (-10 x^2+e^x \left (10 e^{4/x} x^2+39 x^3\right )\right ) \log (x)+\left (-x^2+e^x \left (e^{4/x} x^2+4 x^3\right )\right ) \log ^2(x)} \, dx=\log \left (e^{4/x}-e^{-x}+4 x-\frac {x}{5+\log (x)}\right ) \] Output:

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

Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.93 \[ \int \frac {25 x^2+e^x \left (-100 e^{4/x}+96 x^2\right )+\left (10 x^2+e^x \left (-40 e^{4/x}+39 x^2\right )\right ) \log (x)+\left (x^2+e^x \left (-4 e^{4/x}+4 x^2\right )\right ) \log ^2(x)}{-25 x^2+e^x \left (25 e^{4/x} x^2+95 x^3\right )+\left (-10 x^2+e^x \left (10 e^{4/x} x^2+39 x^3\right )\right ) \log (x)+\left (-x^2+e^x \left (e^{4/x} x^2+4 x^3\right )\right ) \log ^2(x)} \, dx=-x-\log (5+\log (x))+\log \left (5-5 e^{\frac {4}{x}+x}-19 e^x x+\log (x)-e^{\frac {4}{x}+x} \log (x)-4 e^x x \log (x)\right ) \] Input:

Integrate[(25*x^2 + E^x*(-100*E^(4/x) + 96*x^2) + (10*x^2 + E^x*(-40*E^(4/ 
x) + 39*x^2))*Log[x] + (x^2 + E^x*(-4*E^(4/x) + 4*x^2))*Log[x]^2)/(-25*x^2 
 + E^x*(25*E^(4/x)*x^2 + 95*x^3) + (-10*x^2 + E^x*(10*E^(4/x)*x^2 + 39*x^3 
))*Log[x] + (-x^2 + E^x*(E^(4/x)*x^2 + 4*x^3))*Log[x]^2),x]
 

Output:

-x - Log[5 + Log[x]] + Log[5 - 5*E^(4/x + x) - 19*E^x*x + Log[x] - E^(4/x 
+ x)*Log[x] - 4*E^x*x*Log[x]]
 

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.

\(\displaystyle \int \frac {25 x^2+e^x \left (96 x^2-100 e^{4/x}\right )+\left (x^2+e^x \left (4 x^2-4 e^{4/x}\right )\right ) \log ^2(x)+\left (10 x^2+e^x \left (39 x^2-40 e^{4/x}\right )\right ) \log (x)}{-25 x^2+e^x \left (95 x^3+25 e^{4/x} x^2\right )+\left (e^x \left (4 x^3+e^{4/x} x^2\right )-x^2\right ) \log ^2(x)+\left (e^x \left (39 x^3+10 e^{4/x} x^2\right )-10 x^2\right ) \log (x)} \, dx\)

\(\Big \downarrow \) 7292

\(\displaystyle \int \frac {-25 x^2-e^x \left (96 x^2-100 e^{4/x}\right )-\left (x^2+e^x \left (4 x^2-4 e^{4/x}\right )\right ) \log ^2(x)-\left (10 x^2+e^x \left (39 x^2-40 e^{4/x}\right )\right ) \log (x)}{x^2 (\log (x)+5) \left (-19 e^x x-5 e^{x+\frac {4}{x}}-4 e^x x \log (x)-e^{x+\frac {4}{x}} \log (x)+\log (x)+5\right )}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {96 x^2+4 x^2 \log ^2(x)+39 x^2 \log (x)-100 e^{4/x}-4 e^{4/x} \log ^2(x)-40 e^{4/x} \log (x)}{x^2 (\log (x)+5) \left (19 x+5 e^{4/x}+4 x \log (x)+e^{4/x} \log (x)\right )}+\frac {95 x^3+4 x^3 \log ^2(x)+39 x^3 \log (x)+25 e^{4/x} x^2+96 x^2+e^{4/x} x^2 \log ^2(x)+4 x^2 \log ^2(x)+10 e^{4/x} x^2 \log (x)+39 x^2 \log (x)-100 e^{4/x}-4 e^{4/x} \log ^2(x)-40 e^{4/x} \log (x)}{x^2 \left (19 x+5 e^{4/x}+4 x \log (x)+e^{4/x} \log (x)\right ) \left (19 e^x x+5 e^{x+\frac {4}{x}}+4 e^x x \log (x)+e^{x+\frac {4}{x}} \log (x)-\log (x)-5\right )}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {-96 e^x x^2-25 x^2-\left (4 e^x x^2+x^2-4 e^{x+\frac {4}{x}}\right ) \log ^2(x)-\left (39 e^x x^2+10 x^2-40 e^{x+\frac {4}{x}}\right ) \log (x)+100 e^{x+\frac {4}{x}}}{x^2 (\log (x)+5) \left (-19 e^x x-5 e^{x+\frac {4}{x}}-\left (4 e^x x+e^{x+\frac {4}{x}}-1\right ) \log (x)+5\right )}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {96 x^2+4 x^2 \log ^2(x)+39 x^2 \log (x)-100 e^{4/x}-4 e^{4/x} \log ^2(x)-40 e^{4/x} \log (x)}{x^2 (\log (x)+5) \left (19 x+5 e^{4/x}+4 x \log (x)+e^{4/x} \log (x)\right )}+\frac {95 x^3+4 x^3 \log ^2(x)+39 x^3 \log (x)+25 e^{4/x} x^2+96 x^2+e^{4/x} x^2 \log ^2(x)+4 x^2 \log ^2(x)+10 e^{4/x} x^2 \log (x)+39 x^2 \log (x)-100 e^{4/x}-4 e^{4/x} \log ^2(x)-40 e^{4/x} \log (x)}{x^2 \left (19 x+5 e^{4/x}+4 x \log (x)+e^{4/x} \log (x)\right ) \left (19 e^x x+5 e^{x+\frac {4}{x}}+4 e^x x \log (x)+e^{x+\frac {4}{x}} \log (x)-\log (x)-5\right )}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {-96 e^x x^2-25 x^2-\left (4 e^x x^2+x^2-4 e^{x+\frac {4}{x}}\right ) \log ^2(x)-\left (39 e^x x^2+10 x^2-40 e^{x+\frac {4}{x}}\right ) \log (x)+100 e^{x+\frac {4}{x}}}{x^2 (\log (x)+5) \left (-19 e^x x-5 e^{x+\frac {4}{x}}-\left (4 e^x x+e^{x+\frac {4}{x}}-1\right ) \log (x)+5\right )}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {96 x^2+4 x^2 \log ^2(x)+39 x^2 \log (x)-100 e^{4/x}-4 e^{4/x} \log ^2(x)-40 e^{4/x} \log (x)}{x^2 (\log (x)+5) \left (19 x+5 e^{4/x}+4 x \log (x)+e^{4/x} \log (x)\right )}+\frac {95 x^3+4 x^3 \log ^2(x)+39 x^3 \log (x)+25 e^{4/x} x^2+96 x^2+e^{4/x} x^2 \log ^2(x)+4 x^2 \log ^2(x)+10 e^{4/x} x^2 \log (x)+39 x^2 \log (x)-100 e^{4/x}-4 e^{4/x} \log ^2(x)-40 e^{4/x} \log (x)}{x^2 \left (19 x+5 e^{4/x}+4 x \log (x)+e^{4/x} \log (x)\right ) \left (19 e^x x+5 e^{x+\frac {4}{x}}+4 e^x x \log (x)+e^{x+\frac {4}{x}} \log (x)-\log (x)-5\right )}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {-96 e^x x^2-25 x^2-\left (4 e^x x^2+x^2-4 e^{x+\frac {4}{x}}\right ) \log ^2(x)-\left (39 e^x x^2+10 x^2-40 e^{x+\frac {4}{x}}\right ) \log (x)+100 e^{x+\frac {4}{x}}}{x^2 (\log (x)+5) \left (-19 e^x x-5 e^{x+\frac {4}{x}}-\left (4 e^x x+e^{x+\frac {4}{x}}-1\right ) \log (x)+5\right )}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {96 x^2+4 x^2 \log ^2(x)+39 x^2 \log (x)-100 e^{4/x}-4 e^{4/x} \log ^2(x)-40 e^{4/x} \log (x)}{x^2 (\log (x)+5) \left (19 x+5 e^{4/x}+4 x \log (x)+e^{4/x} \log (x)\right )}+\frac {95 x^3+4 x^3 \log ^2(x)+39 x^3 \log (x)+25 e^{4/x} x^2+96 x^2+e^{4/x} x^2 \log ^2(x)+4 x^2 \log ^2(x)+10 e^{4/x} x^2 \log (x)+39 x^2 \log (x)-100 e^{4/x}-4 e^{4/x} \log ^2(x)-40 e^{4/x} \log (x)}{x^2 \left (19 x+5 e^{4/x}+4 x \log (x)+e^{4/x} \log (x)\right ) \left (19 e^x x+5 e^{x+\frac {4}{x}}+4 e^x x \log (x)+e^{x+\frac {4}{x}} \log (x)-\log (x)-5\right )}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {-96 e^x x^2-25 x^2-\left (4 e^x x^2+x^2-4 e^{x+\frac {4}{x}}\right ) \log ^2(x)-\left (39 e^x x^2+10 x^2-40 e^{x+\frac {4}{x}}\right ) \log (x)+100 e^{x+\frac {4}{x}}}{x^2 (\log (x)+5) \left (-19 e^x x-5 e^{x+\frac {4}{x}}-\left (4 e^x x+e^{x+\frac {4}{x}}-1\right ) \log (x)+5\right )}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {96 x^2+4 x^2 \log ^2(x)+39 x^2 \log (x)-100 e^{4/x}-4 e^{4/x} \log ^2(x)-40 e^{4/x} \log (x)}{x^2 (\log (x)+5) \left (19 x+5 e^{4/x}+4 x \log (x)+e^{4/x} \log (x)\right )}+\frac {95 x^3+4 x^3 \log ^2(x)+39 x^3 \log (x)+25 e^{4/x} x^2+96 x^2+e^{4/x} x^2 \log ^2(x)+4 x^2 \log ^2(x)+10 e^{4/x} x^2 \log (x)+39 x^2 \log (x)-100 e^{4/x}-4 e^{4/x} \log ^2(x)-40 e^{4/x} \log (x)}{x^2 \left (19 x+5 e^{4/x}+4 x \log (x)+e^{4/x} \log (x)\right ) \left (19 e^x x+5 e^{x+\frac {4}{x}}+4 e^x x \log (x)+e^{x+\frac {4}{x}} \log (x)-\log (x)-5\right )}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {-96 e^x x^2-25 x^2-\left (4 e^x x^2+x^2-4 e^{x+\frac {4}{x}}\right ) \log ^2(x)-\left (39 e^x x^2+10 x^2-40 e^{x+\frac {4}{x}}\right ) \log (x)+100 e^{x+\frac {4}{x}}}{x^2 (\log (x)+5) \left (-19 e^x x-5 e^{x+\frac {4}{x}}-\left (4 e^x x+e^{x+\frac {4}{x}}-1\right ) \log (x)+5\right )}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {96 x^2+4 x^2 \log ^2(x)+39 x^2 \log (x)-100 e^{4/x}-4 e^{4/x} \log ^2(x)-40 e^{4/x} \log (x)}{x^2 (\log (x)+5) \left (19 x+5 e^{4/x}+4 x \log (x)+e^{4/x} \log (x)\right )}+\frac {95 x^3+4 x^3 \log ^2(x)+39 x^3 \log (x)+25 e^{4/x} x^2+96 x^2+e^{4/x} x^2 \log ^2(x)+4 x^2 \log ^2(x)+10 e^{4/x} x^2 \log (x)+39 x^2 \log (x)-100 e^{4/x}-4 e^{4/x} \log ^2(x)-40 e^{4/x} \log (x)}{x^2 \left (19 x+5 e^{4/x}+4 x \log (x)+e^{4/x} \log (x)\right ) \left (19 e^x x+5 e^{x+\frac {4}{x}}+4 e^x x \log (x)+e^{x+\frac {4}{x}} \log (x)-\log (x)-5\right )}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {-96 e^x x^2-25 x^2-\left (4 e^x x^2+x^2-4 e^{x+\frac {4}{x}}\right ) \log ^2(x)-\left (39 e^x x^2+10 x^2-40 e^{x+\frac {4}{x}}\right ) \log (x)+100 e^{x+\frac {4}{x}}}{x^2 (\log (x)+5) \left (-19 e^x x-5 e^{x+\frac {4}{x}}-\left (4 e^x x+e^{x+\frac {4}{x}}-1\right ) \log (x)+5\right )}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {96 x^2+4 x^2 \log ^2(x)+39 x^2 \log (x)-100 e^{4/x}-4 e^{4/x} \log ^2(x)-40 e^{4/x} \log (x)}{x^2 (\log (x)+5) \left (19 x+5 e^{4/x}+4 x \log (x)+e^{4/x} \log (x)\right )}+\frac {95 x^3+4 x^3 \log ^2(x)+39 x^3 \log (x)+25 e^{4/x} x^2+96 x^2+e^{4/x} x^2 \log ^2(x)+4 x^2 \log ^2(x)+10 e^{4/x} x^2 \log (x)+39 x^2 \log (x)-100 e^{4/x}-4 e^{4/x} \log ^2(x)-40 e^{4/x} \log (x)}{x^2 \left (19 x+5 e^{4/x}+4 x \log (x)+e^{4/x} \log (x)\right ) \left (19 e^x x+5 e^{x+\frac {4}{x}}+4 e^x x \log (x)+e^{x+\frac {4}{x}} \log (x)-\log (x)-5\right )}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {-96 e^x x^2-25 x^2-\left (4 e^x x^2+x^2-4 e^{x+\frac {4}{x}}\right ) \log ^2(x)-\left (39 e^x x^2+10 x^2-40 e^{x+\frac {4}{x}}\right ) \log (x)+100 e^{x+\frac {4}{x}}}{x^2 (\log (x)+5) \left (-19 e^x x-5 e^{x+\frac {4}{x}}-\left (4 e^x x+e^{x+\frac {4}{x}}-1\right ) \log (x)+5\right )}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {96 x^2+4 x^2 \log ^2(x)+39 x^2 \log (x)-100 e^{4/x}-4 e^{4/x} \log ^2(x)-40 e^{4/x} \log (x)}{x^2 (\log (x)+5) \left (19 x+5 e^{4/x}+4 x \log (x)+e^{4/x} \log (x)\right )}+\frac {95 x^3+4 x^3 \log ^2(x)+39 x^3 \log (x)+25 e^{4/x} x^2+96 x^2+e^{4/x} x^2 \log ^2(x)+4 x^2 \log ^2(x)+10 e^{4/x} x^2 \log (x)+39 x^2 \log (x)-100 e^{4/x}-4 e^{4/x} \log ^2(x)-40 e^{4/x} \log (x)}{x^2 \left (19 x+5 e^{4/x}+4 x \log (x)+e^{4/x} \log (x)\right ) \left (19 e^x x+5 e^{x+\frac {4}{x}}+4 e^x x \log (x)+e^{x+\frac {4}{x}} \log (x)-\log (x)-5\right )}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {-96 e^x x^2-25 x^2-\left (4 e^x x^2+x^2-4 e^{x+\frac {4}{x}}\right ) \log ^2(x)-\left (39 e^x x^2+10 x^2-40 e^{x+\frac {4}{x}}\right ) \log (x)+100 e^{x+\frac {4}{x}}}{x^2 (\log (x)+5) \left (-19 e^x x-5 e^{x+\frac {4}{x}}-\left (4 e^x x+e^{x+\frac {4}{x}}-1\right ) \log (x)+5\right )}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {96 x^2+4 x^2 \log ^2(x)+39 x^2 \log (x)-100 e^{4/x}-4 e^{4/x} \log ^2(x)-40 e^{4/x} \log (x)}{x^2 (\log (x)+5) \left (19 x+5 e^{4/x}+4 x \log (x)+e^{4/x} \log (x)\right )}+\frac {95 x^3+4 x^3 \log ^2(x)+39 x^3 \log (x)+25 e^{4/x} x^2+96 x^2+e^{4/x} x^2 \log ^2(x)+4 x^2 \log ^2(x)+10 e^{4/x} x^2 \log (x)+39 x^2 \log (x)-100 e^{4/x}-4 e^{4/x} \log ^2(x)-40 e^{4/x} \log (x)}{x^2 \left (19 x+5 e^{4/x}+4 x \log (x)+e^{4/x} \log (x)\right ) \left (19 e^x x+5 e^{x+\frac {4}{x}}+4 e^x x \log (x)+e^{x+\frac {4}{x}} \log (x)-\log (x)-5\right )}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {-96 e^x x^2-25 x^2-\left (4 e^x x^2+x^2-4 e^{x+\frac {4}{x}}\right ) \log ^2(x)-\left (39 e^x x^2+10 x^2-40 e^{x+\frac {4}{x}}\right ) \log (x)+100 e^{x+\frac {4}{x}}}{x^2 (\log (x)+5) \left (-19 e^x x-5 e^{x+\frac {4}{x}}-\left (4 e^x x+e^{x+\frac {4}{x}}-1\right ) \log (x)+5\right )}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {96 x^2+4 x^2 \log ^2(x)+39 x^2 \log (x)-100 e^{4/x}-4 e^{4/x} \log ^2(x)-40 e^{4/x} \log (x)}{x^2 (\log (x)+5) \left (19 x+5 e^{4/x}+4 x \log (x)+e^{4/x} \log (x)\right )}+\frac {95 x^3+4 x^3 \log ^2(x)+39 x^3 \log (x)+25 e^{4/x} x^2+96 x^2+e^{4/x} x^2 \log ^2(x)+4 x^2 \log ^2(x)+10 e^{4/x} x^2 \log (x)+39 x^2 \log (x)-100 e^{4/x}-4 e^{4/x} \log ^2(x)-40 e^{4/x} \log (x)}{x^2 \left (19 x+5 e^{4/x}+4 x \log (x)+e^{4/x} \log (x)\right ) \left (19 e^x x+5 e^{x+\frac {4}{x}}+4 e^x x \log (x)+e^{x+\frac {4}{x}} \log (x)-\log (x)-5\right )}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {-96 e^x x^2-25 x^2-\left (4 e^x x^2+x^2-4 e^{x+\frac {4}{x}}\right ) \log ^2(x)-\left (39 e^x x^2+10 x^2-40 e^{x+\frac {4}{x}}\right ) \log (x)+100 e^{x+\frac {4}{x}}}{x^2 (\log (x)+5) \left (-19 e^x x-5 e^{x+\frac {4}{x}}-\left (4 e^x x+e^{x+\frac {4}{x}}-1\right ) \log (x)+5\right )}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {96 x^2+4 x^2 \log ^2(x)+39 x^2 \log (x)-100 e^{4/x}-4 e^{4/x} \log ^2(x)-40 e^{4/x} \log (x)}{x^2 (\log (x)+5) \left (19 x+5 e^{4/x}+4 x \log (x)+e^{4/x} \log (x)\right )}+\frac {95 x^3+4 x^3 \log ^2(x)+39 x^3 \log (x)+25 e^{4/x} x^2+96 x^2+e^{4/x} x^2 \log ^2(x)+4 x^2 \log ^2(x)+10 e^{4/x} x^2 \log (x)+39 x^2 \log (x)-100 e^{4/x}-4 e^{4/x} \log ^2(x)-40 e^{4/x} \log (x)}{x^2 \left (19 x+5 e^{4/x}+4 x \log (x)+e^{4/x} \log (x)\right ) \left (19 e^x x+5 e^{x+\frac {4}{x}}+4 e^x x \log (x)+e^{x+\frac {4}{x}} \log (x)-\log (x)-5\right )}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {-96 e^x x^2-25 x^2-\left (4 e^x x^2+x^2-4 e^{x+\frac {4}{x}}\right ) \log ^2(x)-\left (39 e^x x^2+10 x^2-40 e^{x+\frac {4}{x}}\right ) \log (x)+100 e^{x+\frac {4}{x}}}{x^2 (\log (x)+5) \left (-19 e^x x-5 e^{x+\frac {4}{x}}-\left (4 e^x x+e^{x+\frac {4}{x}}-1\right ) \log (x)+5\right )}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {96 x^2+4 x^2 \log ^2(x)+39 x^2 \log (x)-100 e^{4/x}-4 e^{4/x} \log ^2(x)-40 e^{4/x} \log (x)}{x^2 (\log (x)+5) \left (19 x+5 e^{4/x}+4 x \log (x)+e^{4/x} \log (x)\right )}+\frac {95 x^3+4 x^3 \log ^2(x)+39 x^3 \log (x)+25 e^{4/x} x^2+96 x^2+e^{4/x} x^2 \log ^2(x)+4 x^2 \log ^2(x)+10 e^{4/x} x^2 \log (x)+39 x^2 \log (x)-100 e^{4/x}-4 e^{4/x} \log ^2(x)-40 e^{4/x} \log (x)}{x^2 \left (19 x+5 e^{4/x}+4 x \log (x)+e^{4/x} \log (x)\right ) \left (19 e^x x+5 e^{x+\frac {4}{x}}+4 e^x x \log (x)+e^{x+\frac {4}{x}} \log (x)-\log (x)-5\right )}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {-96 e^x x^2-25 x^2-\left (4 e^x x^2+x^2-4 e^{x+\frac {4}{x}}\right ) \log ^2(x)-\left (39 e^x x^2+10 x^2-40 e^{x+\frac {4}{x}}\right ) \log (x)+100 e^{x+\frac {4}{x}}}{x^2 (\log (x)+5) \left (-19 e^x x-5 e^{x+\frac {4}{x}}-\left (4 e^x x+e^{x+\frac {4}{x}}-1\right ) \log (x)+5\right )}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {96 x^2+4 x^2 \log ^2(x)+39 x^2 \log (x)-100 e^{4/x}-4 e^{4/x} \log ^2(x)-40 e^{4/x} \log (x)}{x^2 (\log (x)+5) \left (19 x+5 e^{4/x}+4 x \log (x)+e^{4/x} \log (x)\right )}+\frac {95 x^3+4 x^3 \log ^2(x)+39 x^3 \log (x)+25 e^{4/x} x^2+96 x^2+e^{4/x} x^2 \log ^2(x)+4 x^2 \log ^2(x)+10 e^{4/x} x^2 \log (x)+39 x^2 \log (x)-100 e^{4/x}-4 e^{4/x} \log ^2(x)-40 e^{4/x} \log (x)}{x^2 \left (19 x+5 e^{4/x}+4 x \log (x)+e^{4/x} \log (x)\right ) \left (19 e^x x+5 e^{x+\frac {4}{x}}+4 e^x x \log (x)+e^{x+\frac {4}{x}} \log (x)-\log (x)-5\right )}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {-96 e^x x^2-25 x^2-\left (4 e^x x^2+x^2-4 e^{x+\frac {4}{x}}\right ) \log ^2(x)-\left (39 e^x x^2+10 x^2-40 e^{x+\frac {4}{x}}\right ) \log (x)+100 e^{x+\frac {4}{x}}}{x^2 (\log (x)+5) \left (-19 e^x x-5 e^{x+\frac {4}{x}}-\left (4 e^x x+e^{x+\frac {4}{x}}-1\right ) \log (x)+5\right )}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {96 x^2+4 x^2 \log ^2(x)+39 x^2 \log (x)-100 e^{4/x}-4 e^{4/x} \log ^2(x)-40 e^{4/x} \log (x)}{x^2 (\log (x)+5) \left (19 x+5 e^{4/x}+4 x \log (x)+e^{4/x} \log (x)\right )}+\frac {95 x^3+4 x^3 \log ^2(x)+39 x^3 \log (x)+25 e^{4/x} x^2+96 x^2+e^{4/x} x^2 \log ^2(x)+4 x^2 \log ^2(x)+10 e^{4/x} x^2 \log (x)+39 x^2 \log (x)-100 e^{4/x}-4 e^{4/x} \log ^2(x)-40 e^{4/x} \log (x)}{x^2 \left (19 x+5 e^{4/x}+4 x \log (x)+e^{4/x} \log (x)\right ) \left (19 e^x x+5 e^{x+\frac {4}{x}}+4 e^x x \log (x)+e^{x+\frac {4}{x}} \log (x)-\log (x)-5\right )}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {-96 e^x x^2-25 x^2-\left (4 e^x x^2+x^2-4 e^{x+\frac {4}{x}}\right ) \log ^2(x)-\left (39 e^x x^2+10 x^2-40 e^{x+\frac {4}{x}}\right ) \log (x)+100 e^{x+\frac {4}{x}}}{x^2 (\log (x)+5) \left (-19 e^x x-5 e^{x+\frac {4}{x}}-\left (4 e^x x+e^{x+\frac {4}{x}}-1\right ) \log (x)+5\right )}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {96 x^2+4 x^2 \log ^2(x)+39 x^2 \log (x)-100 e^{4/x}-4 e^{4/x} \log ^2(x)-40 e^{4/x} \log (x)}{x^2 (\log (x)+5) \left (19 x+5 e^{4/x}+4 x \log (x)+e^{4/x} \log (x)\right )}+\frac {95 x^3+4 x^3 \log ^2(x)+39 x^3 \log (x)+25 e^{4/x} x^2+96 x^2+e^{4/x} x^2 \log ^2(x)+4 x^2 \log ^2(x)+10 e^{4/x} x^2 \log (x)+39 x^2 \log (x)-100 e^{4/x}-4 e^{4/x} \log ^2(x)-40 e^{4/x} \log (x)}{x^2 \left (19 x+5 e^{4/x}+4 x \log (x)+e^{4/x} \log (x)\right ) \left (19 e^x x+5 e^{x+\frac {4}{x}}+4 e^x x \log (x)+e^{x+\frac {4}{x}} \log (x)-\log (x)-5\right )}\right )dx\)

Input:

Int[(25*x^2 + E^x*(-100*E^(4/x) + 96*x^2) + (10*x^2 + E^x*(-40*E^(4/x) + 3 
9*x^2))*Log[x] + (x^2 + E^x*(-4*E^(4/x) + 4*x^2))*Log[x]^2)/(-25*x^2 + E^x 
*(25*E^(4/x)*x^2 + 95*x^3) + (-10*x^2 + E^x*(10*E^(4/x)*x^2 + 39*x^3))*Log 
[x] + (-x^2 + E^x*(E^(4/x)*x^2 + 4*x^3))*Log[x]^2),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 291.12 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.86

method	result	size
parallelrisch	\(-\ln \left (5+\ln \left (x \right )\right )+\ln \left (x \,{\mathrm e}^{x} \ln \left (x \right )+\frac {{\mathrm e}^{\frac {4}{x}} {\mathrm e}^{x} \ln \left (x \right )}{4}+\frac {19 \,{\mathrm e}^{x} x}{4}+\frac {5 \,{\mathrm e}^{\frac {4}{x}} {\mathrm e}^{x}}{4}-\frac {\ln \left (x \right )}{4}-\frac {5}{4}\right )-x\)	\(52\)
risch	\(\ln \left (4 x +{\mathrm e}^{\frac {4}{x}}-{\mathrm e}^{-x}\right )+\ln \left (\ln \left (x \right )+\frac {19 \,{\mathrm e}^{x} x +5 \,{\mathrm e}^{\frac {x^{2}+4}{x}}-5}{4 \,{\mathrm e}^{x} x +{\mathrm e}^{\frac {x^{2}+4}{x}}-1}\right )-\ln \left (5+\ln \left (x \right )\right )\)	\(69\)

Input:

int((((-4*exp(4/x)+4*x^2)*exp(x)+x^2)*ln(x)^2+((-40*exp(4/x)+39*x^2)*exp(x 
)+10*x^2)*ln(x)+(-100*exp(4/x)+96*x^2)*exp(x)+25*x^2)/(((x^2*exp(4/x)+4*x^ 
3)*exp(x)-x^2)*ln(x)^2+((10*x^2*exp(4/x)+39*x^3)*exp(x)-10*x^2)*ln(x)+(25* 
x^2*exp(4/x)+95*x^3)*exp(x)-25*x^2),x,method=_RETURNVERBOSE)
 

Output:

-ln(5+ln(x))+ln(x*exp(x)*ln(x)+1/4*exp(4/x)*exp(x)*ln(x)+19/4*exp(x)*x+5/4 
*exp(4/x)*exp(x)-1/4*ln(x)-5/4)-x
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 105 vs. \(2 (26) = 52\).

Time = 0.12 (sec) , antiderivative size = 105, normalized size of antiderivative = 3.75 \[ \int \frac {25 x^2+e^x \left (-100 e^{4/x}+96 x^2\right )+\left (10 x^2+e^x \left (-40 e^{4/x}+39 x^2\right )\right ) \log (x)+\left (x^2+e^x \left (-4 e^{4/x}+4 x^2\right )\right ) \log ^2(x)}{-25 x^2+e^x \left (25 e^{4/x} x^2+95 x^3\right )+\left (-10 x^2+e^x \left (10 e^{4/x} x^2+39 x^3\right )\right ) \log (x)+\left (-x^2+e^x \left (e^{4/x} x^2+4 x^3\right )\right ) \log ^2(x)} \, dx=-x + \log \left (4 \, x + e^{\frac {4}{x}}\right ) + \log \left (\frac {{\left (19 \, x + 5 \, e^{\frac {4}{x}}\right )} e^{x} + {\left ({\left (4 \, x + e^{\frac {4}{x}}\right )} e^{x} - 1\right )} \log \left (x\right ) - 5}{{\left (4 \, x + e^{\frac {4}{x}}\right )} e^{x} - 1}\right ) + \log \left (\frac {{\left (4 \, x + e^{\frac {4}{x}}\right )} e^{x} - 1}{4 \, x + e^{\frac {4}{x}}}\right ) - \log \left (\log \left (x\right ) + 5\right ) \] Input:

integrate((((-4*exp(4/x)+4*x^2)*exp(x)+x^2)*log(x)^2+((-40*exp(4/x)+39*x^2 
)*exp(x)+10*x^2)*log(x)+(-100*exp(4/x)+96*x^2)*exp(x)+25*x^2)/(((x^2*exp(4 
/x)+4*x^3)*exp(x)-x^2)*log(x)^2+((10*x^2*exp(4/x)+39*x^3)*exp(x)-10*x^2)*l 
og(x)+(25*x^2*exp(4/x)+95*x^3)*exp(x)-25*x^2),x, algorithm="fricas")
 

Output:

-x + log(4*x + e^(4/x)) + log(((19*x + 5*e^(4/x))*e^x + ((4*x + e^(4/x))*e 
^x - 1)*log(x) - 5)/((4*x + e^(4/x))*e^x - 1)) + log(((4*x + e^(4/x))*e^x 
- 1)/(4*x + e^(4/x))) - log(log(x) + 5)
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 60 vs. \(2 (20) = 40\).

Time = 2.88 (sec) , antiderivative size = 60, normalized size of antiderivative = 2.14 \[ \int \frac {25 x^2+e^x \left (-100 e^{4/x}+96 x^2\right )+\left (10 x^2+e^x \left (-40 e^{4/x}+39 x^2\right )\right ) \log (x)+\left (x^2+e^x \left (-4 e^{4/x}+4 x^2\right )\right ) \log ^2(x)}{-25 x^2+e^x \left (25 e^{4/x} x^2+95 x^3\right )+\left (-10 x^2+e^x \left (10 e^{4/x} x^2+39 x^3\right )\right ) \log (x)+\left (-x^2+e^x \left (e^{4/x} x^2+4 x^3\right )\right ) \log ^2(x)} \, dx=- x + \log {\left (\frac {4 x \log {\left (x \right )} + 19 x}{\log {\left (x \right )} + 5} + e^{\frac {4}{x}} \right )} + \log {\left (\frac {- \log {\left (x \right )} - 5}{4 x \log {\left (x \right )} + 19 x + e^{\frac {4}{x}} \log {\left (x \right )} + 5 e^{\frac {4}{x}}} + e^{x} \right )} \] Input:

integrate((((-4*exp(4/x)+4*x**2)*exp(x)+x**2)*ln(x)**2+((-40*exp(4/x)+39*x 
**2)*exp(x)+10*x**2)*ln(x)+(-100*exp(4/x)+96*x**2)*exp(x)+25*x**2)/(((x**2 
*exp(4/x)+4*x**3)*exp(x)-x**2)*ln(x)**2+((10*x**2*exp(4/x)+39*x**3)*exp(x) 
-10*x**2)*ln(x)+(25*x**2*exp(4/x)+95*x**3)*exp(x)-25*x**2),x)
 

Output:

-x + log((4*x*log(x) + 19*x)/(log(x) + 5) + exp(4/x)) + log((-log(x) - 5)/ 
(4*x*log(x) + 19*x + exp(4/x)*log(x) + 5*exp(4/x)) + exp(x))
 

Maxima [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.54 \[ \int \frac {25 x^2+e^x \left (-100 e^{4/x}+96 x^2\right )+\left (10 x^2+e^x \left (-40 e^{4/x}+39 x^2\right )\right ) \log (x)+\left (x^2+e^x \left (-4 e^{4/x}+4 x^2\right )\right ) \log ^2(x)}{-25 x^2+e^x \left (25 e^{4/x} x^2+95 x^3\right )+\left (-10 x^2+e^x \left (10 e^{4/x} x^2+39 x^3\right )\right ) \log (x)+\left (-x^2+e^x \left (e^{4/x} x^2+4 x^3\right )\right ) \log ^2(x)} \, dx=\log \left (\frac {{\left ({\left (\log \left (x\right ) + 5\right )} e^{\left (x + \frac {4}{x}\right )} + {\left (4 \, x \log \left (x\right ) + 19 \, x\right )} e^{x} - \log \left (x\right ) - 5\right )} e^{\left (-x\right )}}{\log \left (x\right ) + 5}\right ) \] Input:

integrate((((-4*exp(4/x)+4*x^2)*exp(x)+x^2)*log(x)^2+((-40*exp(4/x)+39*x^2 
)*exp(x)+10*x^2)*log(x)+(-100*exp(4/x)+96*x^2)*exp(x)+25*x^2)/(((x^2*exp(4 
/x)+4*x^3)*exp(x)-x^2)*log(x)^2+((10*x^2*exp(4/x)+39*x^3)*exp(x)-10*x^2)*l 
og(x)+(25*x^2*exp(4/x)+95*x^3)*exp(x)-25*x^2),x, algorithm="maxima")
 

Output:

log(((log(x) + 5)*e^(x + 4/x) + (4*x*log(x) + 19*x)*e^x - log(x) - 5)*e^(- 
x)/(log(x) + 5))
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 55 vs. \(2 (26) = 52\).

Time = 0.17 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.96 \[ \int \frac {25 x^2+e^x \left (-100 e^{4/x}+96 x^2\right )+\left (10 x^2+e^x \left (-40 e^{4/x}+39 x^2\right )\right ) \log (x)+\left (x^2+e^x \left (-4 e^{4/x}+4 x^2\right )\right ) \log ^2(x)}{-25 x^2+e^x \left (25 e^{4/x} x^2+95 x^3\right )+\left (-10 x^2+e^x \left (10 e^{4/x} x^2+39 x^3\right )\right ) \log (x)+\left (-x^2+e^x \left (e^{4/x} x^2+4 x^3\right )\right ) \log ^2(x)} \, dx=-x + \log \left (4 \, x e^{x} \log \left (x\right ) + 19 \, x e^{x} + e^{\left (\frac {x^{2} + 4}{x}\right )} \log \left (x\right ) + 5 \, e^{\left (\frac {x^{2} + 4}{x}\right )} - \log \left (x\right ) - 5\right ) - \log \left (\log \left (x\right ) + 5\right ) \] Input:

integrate((((-4*exp(4/x)+4*x^2)*exp(x)+x^2)*log(x)^2+((-40*exp(4/x)+39*x^2 
)*exp(x)+10*x^2)*log(x)+(-100*exp(4/x)+96*x^2)*exp(x)+25*x^2)/(((x^2*exp(4 
/x)+4*x^3)*exp(x)-x^2)*log(x)^2+((10*x^2*exp(4/x)+39*x^3)*exp(x)-10*x^2)*l 
og(x)+(25*x^2*exp(4/x)+95*x^3)*exp(x)-25*x^2),x, algorithm="giac")
 

Output:

-x + log(4*x*e^x*log(x) + 19*x*e^x + e^((x^2 + 4)/x)*log(x) + 5*e^((x^2 + 
4)/x) - log(x) - 5) - log(log(x) + 5)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {25 x^2+e^x \left (-100 e^{4/x}+96 x^2\right )+\left (10 x^2+e^x \left (-40 e^{4/x}+39 x^2\right )\right ) \log (x)+\left (x^2+e^x \left (-4 e^{4/x}+4 x^2\right )\right ) \log ^2(x)}{-25 x^2+e^x \left (25 e^{4/x} x^2+95 x^3\right )+\left (-10 x^2+e^x \left (10 e^{4/x} x^2+39 x^3\right )\right ) \log (x)+\left (-x^2+e^x \left (e^{4/x} x^2+4 x^3\right )\right ) \log ^2(x)} \, dx=-\int \frac {{\ln \left (x\right )}^2\,\left ({\mathrm {e}}^x\,\left (4\,{\mathrm {e}}^{4/x}-4\,x^2\right )-x^2\right )+\ln \left (x\right )\,\left ({\mathrm {e}}^x\,\left (40\,{\mathrm {e}}^{4/x}-39\,x^2\right )-10\,x^2\right )+{\mathrm {e}}^x\,\left (100\,{\mathrm {e}}^{4/x}-96\,x^2\right )-25\,x^2}{\ln \left (x\right )\,\left ({\mathrm {e}}^x\,\left (10\,x^2\,{\mathrm {e}}^{4/x}+39\,x^3\right )-10\,x^2\right )+{\mathrm {e}}^x\,\left (25\,x^2\,{\mathrm {e}}^{4/x}+95\,x^3\right )-25\,x^2+{\ln \left (x\right )}^2\,\left ({\mathrm {e}}^x\,\left (x^2\,{\mathrm {e}}^{4/x}+4\,x^3\right )-x^2\right )} \,d x \] Input:

int(-(log(x)^2*(exp(x)*(4*exp(4/x) - 4*x^2) - x^2) + log(x)*(exp(x)*(40*ex 
p(4/x) - 39*x^2) - 10*x^2) + exp(x)*(100*exp(4/x) - 96*x^2) - 25*x^2)/(log 
(x)*(exp(x)*(10*x^2*exp(4/x) + 39*x^3) - 10*x^2) + exp(x)*(25*x^2*exp(4/x) 
 + 95*x^3) - 25*x^2 + log(x)^2*(exp(x)*(x^2*exp(4/x) + 4*x^3) - x^2)),x)
 

Output:

-int((log(x)^2*(exp(x)*(4*exp(4/x) - 4*x^2) - x^2) + log(x)*(exp(x)*(40*ex 
p(4/x) - 39*x^2) - 10*x^2) + exp(x)*(100*exp(4/x) - 96*x^2) - 25*x^2)/(log 
(x)*(exp(x)*(10*x^2*exp(4/x) + 39*x^3) - 10*x^2) + exp(x)*(25*x^2*exp(4/x) 
 + 95*x^3) - 25*x^2 + log(x)^2*(exp(x)*(x^2*exp(4/x) + 4*x^3) - x^2)), x)
 

Reduce [B] (verification not implemented)

Time = 2.94 (sec) , antiderivative size = 66, normalized size of antiderivative = 2.36 \[ \int \frac {25 x^2+e^x \left (-100 e^{4/x}+96 x^2\right )+\left (10 x^2+e^x \left (-40 e^{4/x}+39 x^2\right )\right ) \log (x)+\left (x^2+e^x \left (-4 e^{4/x}+4 x^2\right )\right ) \log ^2(x)}{-25 x^2+e^x \left (25 e^{4/x} x^2+95 x^3\right )+\left (-10 x^2+e^x \left (10 e^{4/x} x^2+39 x^3\right )\right ) \log (x)+\left (-x^2+e^x \left (e^{4/x} x^2+4 x^3\right )\right ) \log ^2(x)} \, dx=-\mathrm {log}\left (-\mathrm {log}\left (x \right )-5\right )+\mathrm {log}\left (\frac {-e^{\frac {x^{2}+4}{x}} \mathrm {log}\left (x \right )-5 e^{\frac {x^{2}+4}{x}}-4 e^{x} \mathrm {log}\left (x \right ) x -19 e^{x} x +\mathrm {log}\left (x \right )+5}{x}\right )+\mathrm {log}\left (x \right )-x \] Input:

int((((-4*exp(4/x)+4*x^2)*exp(x)+x^2)*log(x)^2+((-40*exp(4/x)+39*x^2)*exp( 
x)+10*x^2)*log(x)+(-100*exp(4/x)+96*x^2)*exp(x)+25*x^2)/(((x^2*exp(4/x)+4* 
x^3)*exp(x)-x^2)*log(x)^2+((10*x^2*exp(4/x)+39*x^3)*exp(x)-10*x^2)*log(x)+ 
(25*x^2*exp(4/x)+95*x^3)*exp(x)-25*x^2),x)
 

Output:

 - log( - log(x) - 5) + log(( - e**((x**2 + 4)/x)*log(x) - 5*e**((x**2 + 4 
)/x) - 4*e**x*log(x)*x - 19*e**x*x + log(x) + 5)/x) + log(x) - x