\(\int \frac {e^{10} (1+4 x^2)+e^{5+x^2} (2 x^2+4 x^3-4 x^4)+e^{2 x^2} (-x^2+4 x^3-3 x^4)+e^{2 x^2} \log (x)}{e^{10} x+e^{5+x^2} (2 x^2-2 x^3)+e^{2 x^2} (x+x^3-2 x^4+x^5)+e^{2 x^2} x \log (x)} \, dx\) [2784]

Optimal result

Integrand size = 130, antiderivative size = 29 \[ \int \frac {e^{10} \left (1+4 x^2\right )+e^{5+x^2} \left (2 x^2+4 x^3-4 x^4\right )+e^{2 x^2} \left (-x^2+4 x^3-3 x^4\right )+e^{2 x^2} \log (x)}{e^{10} x+e^{5+x^2} \left (2 x^2-2 x^3\right )+e^{2 x^2} \left (x+x^3-2 x^4+x^5\right )+e^{2 x^2} x \log (x)} \, dx=\log \left (\frac {x}{1+\left (-e^{5-x^2}-x+x^2\right )^2+\log (x)}\right ) \] Output:

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

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(88\) vs. \(2(29)=58\).

Time = 0.08 (sec) , antiderivative size = 88, normalized size of antiderivative = 3.03 \[ \int \frac {e^{10} \left (1+4 x^2\right )+e^{5+x^2} \left (2 x^2+4 x^3-4 x^4\right )+e^{2 x^2} \left (-x^2+4 x^3-3 x^4\right )+e^{2 x^2} \log (x)}{e^{10} x+e^{5+x^2} \left (2 x^2-2 x^3\right )+e^{2 x^2} \left (x+x^3-2 x^4+x^5\right )+e^{2 x^2} x \log (x)} \, dx=2 x^2+\log (x)-\log \left (e^{10}+e^{2 x^2}+2 e^{5+x^2} x+e^{2 x^2} x^2-2 e^{5+x^2} x^2-2 e^{2 x^2} x^3+e^{2 x^2} x^4+e^{2 x^2} \log (x)\right ) \] Input:

Integrate[(E^10*(1 + 4*x^2) + E^(5 + x^2)*(2*x^2 + 4*x^3 - 4*x^4) + E^(2*x 
^2)*(-x^2 + 4*x^3 - 3*x^4) + E^(2*x^2)*Log[x])/(E^10*x + E^(5 + x^2)*(2*x^ 
2 - 2*x^3) + E^(2*x^2)*(x + x^3 - 2*x^4 + x^5) + E^(2*x^2)*x*Log[x]),x]
 

Output:

2*x^2 + Log[x] - Log[E^10 + E^(2*x^2) + 2*E^(5 + x^2)*x + E^(2*x^2)*x^2 - 
2*E^(5 + x^2)*x^2 - 2*E^(2*x^2)*x^3 + E^(2*x^2)*x^4 + E^(2*x^2)*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.

\(\displaystyle \int \frac {e^{10} \left (4 x^2+1\right )+e^{2 x^2} \log (x)+e^{x^2+5} \left (-4 x^4+4 x^3+2 x^2\right )+e^{2 x^2} \left (-3 x^4+4 x^3-x^2\right )}{e^{2 x^2} x \log (x)+e^{x^2+5} \left (2 x^2-2 x^3\right )+e^{2 x^2} \left (x^5-2 x^4+x^3+x\right )+e^{10} x} \, dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {-3 x^4+4 x^3-x^2+\log (x)}{x \left (x^4-2 x^3+x^2+\log (x)+1\right )}+\frac {e^5 \left (4 e^5 x^6-8 e^5 x^5+8 e^5 x^4-6 e^5 x^3+2 e^{x^2} x^2+6 e^5 x^2+4 e^{x^2} x^2 \log (x)+4 e^5 x^2 \log (x)-2 e^{x^2} x \log (x)-4 e^{x^2} x^8+12 e^{x^2} x^7-16 e^{x^2} x^6+14 e^{x^2} x^5-12 e^{x^2} x^4-4 e^{x^2} x^4 \log (x)+6 e^{x^2} x^3+4 e^{x^2} x^3 \log (x)+e^5\right )}{x \left (x^4-2 x^3+x^2+\log (x)+1\right ) \left (e^{2 x^2} x^2-2 e^{x^2+5} x^2+2 e^{x^2+5} x+e^{2 x^2}+e^{2 x^2} \log (x)+e^{2 x^2} x^4-2 e^{2 x^2} x^3+e^{10}\right )}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {-2 e^{x^2+5} \left (2 x^2-2 x-1\right ) x^2-e^{2 x^2} \left (3 x^2-4 x+1\right ) x^2+e^{10} \left (4 x^2+1\right )+e^{2 x^2} \log (x)}{x \left (-2 e^{x^2+5} (x-1) x+e^{2 x^2} \log (x)+e^{2 x^2} \left (x^4-2 x^3+x^2+1\right )+e^{10}\right )}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {-3 x^4+4 x^3-x^2+\log (x)}{x \left (x^4-2 x^3+x^2+\log (x)+1\right )}+\frac {e^5 \left (4 e^5 x^6-8 e^5 x^5+8 e^5 x^4-6 e^5 x^3+2 e^{x^2} x^2+6 e^5 x^2+4 e^{x^2} x^2 \log (x)+4 e^5 x^2 \log (x)-2 e^{x^2} x \log (x)-4 e^{x^2} x^8+12 e^{x^2} x^7-16 e^{x^2} x^6+14 e^{x^2} x^5-12 e^{x^2} x^4-4 e^{x^2} x^4 \log (x)+6 e^{x^2} x^3+4 e^{x^2} x^3 \log (x)+e^5\right )}{x \left (x^4-2 x^3+x^2+\log (x)+1\right ) \left (e^{2 x^2} x^2-2 e^{x^2+5} x^2+2 e^{x^2+5} x+e^{2 x^2}+e^{2 x^2} \log (x)+e^{2 x^2} x^4-2 e^{2 x^2} x^3+e^{10}\right )}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {-2 e^{x^2+5} \left (2 x^2-2 x-1\right ) x^2-e^{2 x^2} \left (3 x^2-4 x+1\right ) x^2+e^{10} \left (4 x^2+1\right )+e^{2 x^2} \log (x)}{x \left (-2 e^{x^2+5} (x-1) x+e^{2 x^2} \log (x)+e^{2 x^2} \left (x^4-2 x^3+x^2+1\right )+e^{10}\right )}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {-3 x^4+4 x^3-x^2+\log (x)}{x \left (x^4-2 x^3+x^2+\log (x)+1\right )}+\frac {e^5 \left (4 e^5 x^6-8 e^5 x^5+8 e^5 x^4-6 e^5 x^3+2 e^{x^2} x^2+6 e^5 x^2+4 e^{x^2} x^2 \log (x)+4 e^5 x^2 \log (x)-2 e^{x^2} x \log (x)-4 e^{x^2} x^8+12 e^{x^2} x^7-16 e^{x^2} x^6+14 e^{x^2} x^5-12 e^{x^2} x^4-4 e^{x^2} x^4 \log (x)+6 e^{x^2} x^3+4 e^{x^2} x^3 \log (x)+e^5\right )}{x \left (x^4-2 x^3+x^2+\log (x)+1\right ) \left (e^{2 x^2} x^2-2 e^{x^2+5} x^2+2 e^{x^2+5} x+e^{2 x^2}+e^{2 x^2} \log (x)+e^{2 x^2} x^4-2 e^{2 x^2} x^3+e^{10}\right )}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {-2 e^{x^2+5} \left (2 x^2-2 x-1\right ) x^2-e^{2 x^2} \left (3 x^2-4 x+1\right ) x^2+e^{10} \left (4 x^2+1\right )+e^{2 x^2} \log (x)}{x \left (-2 e^{x^2+5} (x-1) x+e^{2 x^2} \log (x)+e^{2 x^2} \left (x^4-2 x^3+x^2+1\right )+e^{10}\right )}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {-3 x^4+4 x^3-x^2+\log (x)}{x \left (x^4-2 x^3+x^2+\log (x)+1\right )}+\frac {e^5 \left (4 e^5 x^6-8 e^5 x^5+8 e^5 x^4-6 e^5 x^3+2 e^{x^2} x^2+6 e^5 x^2+4 e^{x^2} x^2 \log (x)+4 e^5 x^2 \log (x)-2 e^{x^2} x \log (x)-4 e^{x^2} x^8+12 e^{x^2} x^7-16 e^{x^2} x^6+14 e^{x^2} x^5-12 e^{x^2} x^4-4 e^{x^2} x^4 \log (x)+6 e^{x^2} x^3+4 e^{x^2} x^3 \log (x)+e^5\right )}{x \left (x^4-2 x^3+x^2+\log (x)+1\right ) \left (e^{2 x^2} x^2-2 e^{x^2+5} x^2+2 e^{x^2+5} x+e^{2 x^2}+e^{2 x^2} \log (x)+e^{2 x^2} x^4-2 e^{2 x^2} x^3+e^{10}\right )}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {-2 e^{x^2+5} \left (2 x^2-2 x-1\right ) x^2-e^{2 x^2} \left (3 x^2-4 x+1\right ) x^2+e^{10} \left (4 x^2+1\right )+e^{2 x^2} \log (x)}{x \left (-2 e^{x^2+5} (x-1) x+e^{2 x^2} \log (x)+e^{2 x^2} \left (x^4-2 x^3+x^2+1\right )+e^{10}\right )}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {-3 x^4+4 x^3-x^2+\log (x)}{x \left (x^4-2 x^3+x^2+\log (x)+1\right )}+\frac {e^5 \left (4 e^5 x^6-8 e^5 x^5+8 e^5 x^4-6 e^5 x^3+2 e^{x^2} x^2+6 e^5 x^2+4 e^{x^2} x^2 \log (x)+4 e^5 x^2 \log (x)-2 e^{x^2} x \log (x)-4 e^{x^2} x^8+12 e^{x^2} x^7-16 e^{x^2} x^6+14 e^{x^2} x^5-12 e^{x^2} x^4-4 e^{x^2} x^4 \log (x)+6 e^{x^2} x^3+4 e^{x^2} x^3 \log (x)+e^5\right )}{x \left (x^4-2 x^3+x^2+\log (x)+1\right ) \left (e^{2 x^2} x^2-2 e^{x^2+5} x^2+2 e^{x^2+5} x+e^{2 x^2}+e^{2 x^2} \log (x)+e^{2 x^2} x^4-2 e^{2 x^2} x^3+e^{10}\right )}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {-2 e^{x^2+5} \left (2 x^2-2 x-1\right ) x^2-e^{2 x^2} \left (3 x^2-4 x+1\right ) x^2+e^{10} \left (4 x^2+1\right )+e^{2 x^2} \log (x)}{x \left (-2 e^{x^2+5} (x-1) x+e^{2 x^2} \log (x)+e^{2 x^2} \left (x^4-2 x^3+x^2+1\right )+e^{10}\right )}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {-3 x^4+4 x^3-x^2+\log (x)}{x \left (x^4-2 x^3+x^2+\log (x)+1\right )}+\frac {e^5 \left (4 e^5 x^6-8 e^5 x^5+8 e^5 x^4-6 e^5 x^3+2 e^{x^2} x^2+6 e^5 x^2+4 e^{x^2} x^2 \log (x)+4 e^5 x^2 \log (x)-2 e^{x^2} x \log (x)-4 e^{x^2} x^8+12 e^{x^2} x^7-16 e^{x^2} x^6+14 e^{x^2} x^5-12 e^{x^2} x^4-4 e^{x^2} x^4 \log (x)+6 e^{x^2} x^3+4 e^{x^2} x^3 \log (x)+e^5\right )}{x \left (x^4-2 x^3+x^2+\log (x)+1\right ) \left (e^{2 x^2} x^2-2 e^{x^2+5} x^2+2 e^{x^2+5} x+e^{2 x^2}+e^{2 x^2} \log (x)+e^{2 x^2} x^4-2 e^{2 x^2} x^3+e^{10}\right )}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {-2 e^{x^2+5} \left (2 x^2-2 x-1\right ) x^2-e^{2 x^2} \left (3 x^2-4 x+1\right ) x^2+e^{10} \left (4 x^2+1\right )+e^{2 x^2} \log (x)}{x \left (-2 e^{x^2+5} (x-1) x+e^{2 x^2} \log (x)+e^{2 x^2} \left (x^4-2 x^3+x^2+1\right )+e^{10}\right )}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {-3 x^4+4 x^3-x^2+\log (x)}{x \left (x^4-2 x^3+x^2+\log (x)+1\right )}+\frac {e^5 \left (4 e^5 x^6-8 e^5 x^5+8 e^5 x^4-6 e^5 x^3+2 e^{x^2} x^2+6 e^5 x^2+4 e^{x^2} x^2 \log (x)+4 e^5 x^2 \log (x)-2 e^{x^2} x \log (x)-4 e^{x^2} x^8+12 e^{x^2} x^7-16 e^{x^2} x^6+14 e^{x^2} x^5-12 e^{x^2} x^4-4 e^{x^2} x^4 \log (x)+6 e^{x^2} x^3+4 e^{x^2} x^3 \log (x)+e^5\right )}{x \left (x^4-2 x^3+x^2+\log (x)+1\right ) \left (e^{2 x^2} x^2-2 e^{x^2+5} x^2+2 e^{x^2+5} x+e^{2 x^2}+e^{2 x^2} \log (x)+e^{2 x^2} x^4-2 e^{2 x^2} x^3+e^{10}\right )}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {-2 e^{x^2+5} \left (2 x^2-2 x-1\right ) x^2-e^{2 x^2} \left (3 x^2-4 x+1\right ) x^2+e^{10} \left (4 x^2+1\right )+e^{2 x^2} \log (x)}{x \left (-2 e^{x^2+5} (x-1) x+e^{2 x^2} \log (x)+e^{2 x^2} \left (x^4-2 x^3+x^2+1\right )+e^{10}\right )}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {-3 x^4+4 x^3-x^2+\log (x)}{x \left (x^4-2 x^3+x^2+\log (x)+1\right )}+\frac {e^5 \left (4 e^5 x^6-8 e^5 x^5+8 e^5 x^4-6 e^5 x^3+2 e^{x^2} x^2+6 e^5 x^2+4 e^{x^2} x^2 \log (x)+4 e^5 x^2 \log (x)-2 e^{x^2} x \log (x)-4 e^{x^2} x^8+12 e^{x^2} x^7-16 e^{x^2} x^6+14 e^{x^2} x^5-12 e^{x^2} x^4-4 e^{x^2} x^4 \log (x)+6 e^{x^2} x^3+4 e^{x^2} x^3 \log (x)+e^5\right )}{x \left (x^4-2 x^3+x^2+\log (x)+1\right ) \left (e^{2 x^2} x^2-2 e^{x^2+5} x^2+2 e^{x^2+5} x+e^{2 x^2}+e^{2 x^2} \log (x)+e^{2 x^2} x^4-2 e^{2 x^2} x^3+e^{10}\right )}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {-2 e^{x^2+5} \left (2 x^2-2 x-1\right ) x^2-e^{2 x^2} \left (3 x^2-4 x+1\right ) x^2+e^{10} \left (4 x^2+1\right )+e^{2 x^2} \log (x)}{x \left (-2 e^{x^2+5} (x-1) x+e^{2 x^2} \log (x)+e^{2 x^2} \left (x^4-2 x^3+x^2+1\right )+e^{10}\right )}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {-3 x^4+4 x^3-x^2+\log (x)}{x \left (x^4-2 x^3+x^2+\log (x)+1\right )}+\frac {e^5 \left (4 e^5 x^6-8 e^5 x^5+8 e^5 x^4-6 e^5 x^3+2 e^{x^2} x^2+6 e^5 x^2+4 e^{x^2} x^2 \log (x)+4 e^5 x^2 \log (x)-2 e^{x^2} x \log (x)-4 e^{x^2} x^8+12 e^{x^2} x^7-16 e^{x^2} x^6+14 e^{x^2} x^5-12 e^{x^2} x^4-4 e^{x^2} x^4 \log (x)+6 e^{x^2} x^3+4 e^{x^2} x^3 \log (x)+e^5\right )}{x \left (x^4-2 x^3+x^2+\log (x)+1\right ) \left (e^{2 x^2} x^2-2 e^{x^2+5} x^2+2 e^{x^2+5} x+e^{2 x^2}+e^{2 x^2} \log (x)+e^{2 x^2} x^4-2 e^{2 x^2} x^3+e^{10}\right )}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {-2 e^{x^2+5} \left (2 x^2-2 x-1\right ) x^2-e^{2 x^2} \left (3 x^2-4 x+1\right ) x^2+e^{10} \left (4 x^2+1\right )+e^{2 x^2} \log (x)}{x \left (-2 e^{x^2+5} (x-1) x+e^{2 x^2} \log (x)+e^{2 x^2} \left (x^4-2 x^3+x^2+1\right )+e^{10}\right )}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {-3 x^4+4 x^3-x^2+\log (x)}{x \left (x^4-2 x^3+x^2+\log (x)+1\right )}+\frac {e^5 \left (4 e^5 x^6-8 e^5 x^5+8 e^5 x^4-6 e^5 x^3+2 e^{x^2} x^2+6 e^5 x^2+4 e^{x^2} x^2 \log (x)+4 e^5 x^2 \log (x)-2 e^{x^2} x \log (x)-4 e^{x^2} x^8+12 e^{x^2} x^7-16 e^{x^2} x^6+14 e^{x^2} x^5-12 e^{x^2} x^4-4 e^{x^2} x^4 \log (x)+6 e^{x^2} x^3+4 e^{x^2} x^3 \log (x)+e^5\right )}{x \left (x^4-2 x^3+x^2+\log (x)+1\right ) \left (e^{2 x^2} x^2-2 e^{x^2+5} x^2+2 e^{x^2+5} x+e^{2 x^2}+e^{2 x^2} \log (x)+e^{2 x^2} x^4-2 e^{2 x^2} x^3+e^{10}\right )}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {-2 e^{x^2+5} \left (2 x^2-2 x-1\right ) x^2-e^{2 x^2} \left (3 x^2-4 x+1\right ) x^2+e^{10} \left (4 x^2+1\right )+e^{2 x^2} \log (x)}{x \left (-2 e^{x^2+5} (x-1) x+e^{2 x^2} \log (x)+e^{2 x^2} \left (x^4-2 x^3+x^2+1\right )+e^{10}\right )}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {-3 x^4+4 x^3-x^2+\log (x)}{x \left (x^4-2 x^3+x^2+\log (x)+1\right )}+\frac {e^5 \left (4 e^5 x^6-8 e^5 x^5+8 e^5 x^4-6 e^5 x^3+2 e^{x^2} x^2+6 e^5 x^2+4 e^{x^2} x^2 \log (x)+4 e^5 x^2 \log (x)-2 e^{x^2} x \log (x)-4 e^{x^2} x^8+12 e^{x^2} x^7-16 e^{x^2} x^6+14 e^{x^2} x^5-12 e^{x^2} x^4-4 e^{x^2} x^4 \log (x)+6 e^{x^2} x^3+4 e^{x^2} x^3 \log (x)+e^5\right )}{x \left (x^4-2 x^3+x^2+\log (x)+1\right ) \left (e^{2 x^2} x^2-2 e^{x^2+5} x^2+2 e^{x^2+5} x+e^{2 x^2}+e^{2 x^2} \log (x)+e^{2 x^2} x^4-2 e^{2 x^2} x^3+e^{10}\right )}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {-2 e^{x^2+5} \left (2 x^2-2 x-1\right ) x^2-e^{2 x^2} \left (3 x^2-4 x+1\right ) x^2+e^{10} \left (4 x^2+1\right )+e^{2 x^2} \log (x)}{x \left (-2 e^{x^2+5} (x-1) x+e^{2 x^2} \log (x)+e^{2 x^2} \left (x^4-2 x^3+x^2+1\right )+e^{10}\right )}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {-3 x^4+4 x^3-x^2+\log (x)}{x \left (x^4-2 x^3+x^2+\log (x)+1\right )}+\frac {e^5 \left (4 e^5 x^6-8 e^5 x^5+8 e^5 x^4-6 e^5 x^3+2 e^{x^2} x^2+6 e^5 x^2+4 e^{x^2} x^2 \log (x)+4 e^5 x^2 \log (x)-2 e^{x^2} x \log (x)-4 e^{x^2} x^8+12 e^{x^2} x^7-16 e^{x^2} x^6+14 e^{x^2} x^5-12 e^{x^2} x^4-4 e^{x^2} x^4 \log (x)+6 e^{x^2} x^3+4 e^{x^2} x^3 \log (x)+e^5\right )}{x \left (x^4-2 x^3+x^2+\log (x)+1\right ) \left (e^{2 x^2} x^2-2 e^{x^2+5} x^2+2 e^{x^2+5} x+e^{2 x^2}+e^{2 x^2} \log (x)+e^{2 x^2} x^4-2 e^{2 x^2} x^3+e^{10}\right )}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {-2 e^{x^2+5} \left (2 x^2-2 x-1\right ) x^2-e^{2 x^2} \left (3 x^2-4 x+1\right ) x^2+e^{10} \left (4 x^2+1\right )+e^{2 x^2} \log (x)}{x \left (-2 e^{x^2+5} (x-1) x+e^{2 x^2} \log (x)+e^{2 x^2} \left (x^4-2 x^3+x^2+1\right )+e^{10}\right )}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {-3 x^4+4 x^3-x^2+\log (x)}{x \left (x^4-2 x^3+x^2+\log (x)+1\right )}+\frac {e^5 \left (4 e^5 x^6-8 e^5 x^5+8 e^5 x^4-6 e^5 x^3+2 e^{x^2} x^2+6 e^5 x^2+4 e^{x^2} x^2 \log (x)+4 e^5 x^2 \log (x)-2 e^{x^2} x \log (x)-4 e^{x^2} x^8+12 e^{x^2} x^7-16 e^{x^2} x^6+14 e^{x^2} x^5-12 e^{x^2} x^4-4 e^{x^2} x^4 \log (x)+6 e^{x^2} x^3+4 e^{x^2} x^3 \log (x)+e^5\right )}{x \left (x^4-2 x^3+x^2+\log (x)+1\right ) \left (e^{2 x^2} x^2-2 e^{x^2+5} x^2+2 e^{x^2+5} x+e^{2 x^2}+e^{2 x^2} \log (x)+e^{2 x^2} x^4-2 e^{2 x^2} x^3+e^{10}\right )}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {-2 e^{x^2+5} \left (2 x^2-2 x-1\right ) x^2-e^{2 x^2} \left (3 x^2-4 x+1\right ) x^2+e^{10} \left (4 x^2+1\right )+e^{2 x^2} \log (x)}{x \left (-2 e^{x^2+5} (x-1) x+e^{2 x^2} \log (x)+e^{2 x^2} \left (x^4-2 x^3+x^2+1\right )+e^{10}\right )}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {-3 x^4+4 x^3-x^2+\log (x)}{x \left (x^4-2 x^3+x^2+\log (x)+1\right )}+\frac {e^5 \left (4 e^5 x^6-8 e^5 x^5+8 e^5 x^4-6 e^5 x^3+2 e^{x^2} x^2+6 e^5 x^2+4 e^{x^2} x^2 \log (x)+4 e^5 x^2 \log (x)-2 e^{x^2} x \log (x)-4 e^{x^2} x^8+12 e^{x^2} x^7-16 e^{x^2} x^6+14 e^{x^2} x^5-12 e^{x^2} x^4-4 e^{x^2} x^4 \log (x)+6 e^{x^2} x^3+4 e^{x^2} x^3 \log (x)+e^5\right )}{x \left (x^4-2 x^3+x^2+\log (x)+1\right ) \left (e^{2 x^2} x^2-2 e^{x^2+5} x^2+2 e^{x^2+5} x+e^{2 x^2}+e^{2 x^2} \log (x)+e^{2 x^2} x^4-2 e^{2 x^2} x^3+e^{10}\right )}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {-2 e^{x^2+5} \left (2 x^2-2 x-1\right ) x^2-e^{2 x^2} \left (3 x^2-4 x+1\right ) x^2+e^{10} \left (4 x^2+1\right )+e^{2 x^2} \log (x)}{x \left (-2 e^{x^2+5} (x-1) x+e^{2 x^2} \log (x)+e^{2 x^2} \left (x^4-2 x^3+x^2+1\right )+e^{10}\right )}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {-3 x^4+4 x^3-x^2+\log (x)}{x \left (x^4-2 x^3+x^2+\log (x)+1\right )}+\frac {e^5 \left (4 e^5 x^6-8 e^5 x^5+8 e^5 x^4-6 e^5 x^3+2 e^{x^2} x^2+6 e^5 x^2+4 e^{x^2} x^2 \log (x)+4 e^5 x^2 \log (x)-2 e^{x^2} x \log (x)-4 e^{x^2} x^8+12 e^{x^2} x^7-16 e^{x^2} x^6+14 e^{x^2} x^5-12 e^{x^2} x^4-4 e^{x^2} x^4 \log (x)+6 e^{x^2} x^3+4 e^{x^2} x^3 \log (x)+e^5\right )}{x \left (x^4-2 x^3+x^2+\log (x)+1\right ) \left (e^{2 x^2} x^2-2 e^{x^2+5} x^2+2 e^{x^2+5} x+e^{2 x^2}+e^{2 x^2} \log (x)+e^{2 x^2} x^4-2 e^{2 x^2} x^3+e^{10}\right )}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {-2 e^{x^2+5} \left (2 x^2-2 x-1\right ) x^2-e^{2 x^2} \left (3 x^2-4 x+1\right ) x^2+e^{10} \left (4 x^2+1\right )+e^{2 x^2} \log (x)}{x \left (-2 e^{x^2+5} (x-1) x+e^{2 x^2} \log (x)+e^{2 x^2} \left (x^4-2 x^3+x^2+1\right )+e^{10}\right )}dx\)

Input:

Int[(E^10*(1 + 4*x^2) + E^(5 + x^2)*(2*x^2 + 4*x^3 - 4*x^4) + E^(2*x^2)*(- 
x^2 + 4*x^3 - 3*x^4) + E^(2*x^2)*Log[x])/(E^10*x + E^(5 + x^2)*(2*x^2 - 2* 
x^3) + E^(2*x^2)*(x + x^3 - 2*x^4 + x^5) + E^(2*x^2)*x*Log[x]),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 2.76 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.86

method	result	size
risch	\(\ln \left (x \right )-\ln \left (x^{4}-2 x^{3}+x^{2}-2 x^{2} {\mathrm e}^{-x^{2}+5}+2 x \,{\mathrm e}^{-x^{2}+5}+1+{\mathrm e}^{-2 x^{2}+10}+\ln \left (x \right )\right )\)	\(54\)
parallelrisch	\(2 x^{2}-\ln \left ({\mathrm e}^{2 x^{2}} x^{4}-2 \,{\mathrm e}^{2 x^{2}} x^{3}-2 x^{2} {\mathrm e}^{5} {\mathrm e}^{x^{2}}+x^{2} {\mathrm e}^{2 x^{2}}+2 \,{\mathrm e}^{5} {\mathrm e}^{x^{2}} x +{\mathrm e}^{2 x^{2}} \ln \left (x \right )+{\mathrm e}^{10}+{\mathrm e}^{2 x^{2}}\right )+\ln \left (x \right )\)	\(83\)

Input:

int((exp(x^2)^2*ln(x)+(-3*x^4+4*x^3-x^2)*exp(x^2)^2+(-4*x^4+4*x^3+2*x^2)*e 
xp(5)*exp(x^2)+(4*x^2+1)*exp(5)^2)/(x*exp(x^2)^2*ln(x)+(x^5-2*x^4+x^3+x)*e 
xp(x^2)^2+(-2*x^3+2*x^2)*exp(5)*exp(x^2)+x*exp(5)^2),x,method=_RETURNVERBO 
SE)
 

Output:

ln(x)-ln(x^4-2*x^3+x^2-2*x^2*exp(-x^2+5)+2*x*exp(-x^2+5)+1+exp(-2*x^2+10)+ 
ln(x))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 66 vs. \(2 (28) = 56\).

Time = 0.08 (sec) , antiderivative size = 66, normalized size of antiderivative = 2.28 \[ \int \frac {e^{10} \left (1+4 x^2\right )+e^{5+x^2} \left (2 x^2+4 x^3-4 x^4\right )+e^{2 x^2} \left (-x^2+4 x^3-3 x^4\right )+e^{2 x^2} \log (x)}{e^{10} x+e^{5+x^2} \left (2 x^2-2 x^3\right )+e^{2 x^2} \left (x+x^3-2 x^4+x^5\right )+e^{2 x^2} x \log (x)} \, dx=-\log \left ({\left ({\left (x^{4} - 2 \, x^{3} + x^{2} + 1\right )} e^{\left (2 \, x^{2} + 10\right )} - 2 \, {\left (x^{2} - x\right )} e^{\left (x^{2} + 15\right )} + e^{\left (2 \, x^{2} + 10\right )} \log \left (x\right ) + e^{20}\right )} e^{\left (-2 \, x^{2} - 10\right )}\right ) + \log \left (x\right ) \] Input:

integrate((exp(x^2)^2*log(x)+(-3*x^4+4*x^3-x^2)*exp(x^2)^2+(-4*x^4+4*x^3+2 
*x^2)*exp(5)*exp(x^2)+(4*x^2+1)*exp(5)^2)/(x*exp(x^2)^2*log(x)+(x^5-2*x^4+ 
x^3+x)*exp(x^2)^2+(-2*x^3+2*x^2)*exp(5)*exp(x^2)+x*exp(5)^2),x, algorithm= 
"fricas")
 

Output:

-log(((x^4 - 2*x^3 + x^2 + 1)*e^(2*x^2 + 10) - 2*(x^2 - x)*e^(x^2 + 15) + 
e^(2*x^2 + 10)*log(x) + e^20)*e^(-2*x^2 - 10)) + log(x)
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 92 vs. \(2 (22) = 44\).

Time = 4.66 (sec) , antiderivative size = 92, normalized size of antiderivative = 3.17 \[ \int \frac {e^{10} \left (1+4 x^2\right )+e^{5+x^2} \left (2 x^2+4 x^3-4 x^4\right )+e^{2 x^2} \left (-x^2+4 x^3-3 x^4\right )+e^{2 x^2} \log (x)}{e^{10} x+e^{5+x^2} \left (2 x^2-2 x^3\right )+e^{2 x^2} \left (x+x^3-2 x^4+x^5\right )+e^{2 x^2} x \log (x)} \, dx=2 x^{2} + \log {\left (x \right )} - \log {\left (\frac {\left (- 2 x^{2} e^{5} + 2 x e^{5}\right ) e^{x^{2}}}{x^{4} - 2 x^{3} + x^{2} + \log {\left (x \right )} + 1} + e^{2 x^{2}} + \frac {e^{10}}{x^{4} - 2 x^{3} + x^{2} + \log {\left (x \right )} + 1} \right )} - \log {\left (x^{4} - 2 x^{3} + x^{2} + \log {\left (x \right )} + 1 \right )} \] Input:

integrate((exp(x**2)**2*ln(x)+(-3*x**4+4*x**3-x**2)*exp(x**2)**2+(-4*x**4+ 
4*x**3+2*x**2)*exp(5)*exp(x**2)+(4*x**2+1)*exp(5)**2)/(x*exp(x**2)**2*ln(x 
)+(x**5-2*x**4+x**3+x)*exp(x**2)**2+(-2*x**3+2*x**2)*exp(5)*exp(x**2)+x*ex 
p(5)**2),x)
 

Output:

2*x**2 + log(x) - log((-2*x**2*exp(5) + 2*x*exp(5))*exp(x**2)/(x**4 - 2*x* 
*3 + x**2 + log(x) + 1) + exp(2*x**2) + exp(10)/(x**4 - 2*x**3 + x**2 + lo 
g(x) + 1)) - log(x**4 - 2*x**3 + x**2 + log(x) + 1)
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 90 vs. \(2 (28) = 56\).

Time = 0.09 (sec) , antiderivative size = 90, normalized size of antiderivative = 3.10 \[ \int \frac {e^{10} \left (1+4 x^2\right )+e^{5+x^2} \left (2 x^2+4 x^3-4 x^4\right )+e^{2 x^2} \left (-x^2+4 x^3-3 x^4\right )+e^{2 x^2} \log (x)}{e^{10} x+e^{5+x^2} \left (2 x^2-2 x^3\right )+e^{2 x^2} \left (x+x^3-2 x^4+x^5\right )+e^{2 x^2} x \log (x)} \, dx=2 \, x^{2} - \log \left (x^{4} - 2 \, x^{3} + x^{2} + \log \left (x\right ) + 1\right ) + \log \left (x\right ) - \log \left (\frac {{\left (x^{4} - 2 \, x^{3} + x^{2} + \log \left (x\right ) + 1\right )} e^{\left (2 \, x^{2}\right )} - 2 \, {\left (x^{2} e^{5} - x e^{5}\right )} e^{\left (x^{2}\right )} + e^{10}}{x^{4} - 2 \, x^{3} + x^{2} + \log \left (x\right ) + 1}\right ) \] Input:

integrate((exp(x^2)^2*log(x)+(-3*x^4+4*x^3-x^2)*exp(x^2)^2+(-4*x^4+4*x^3+2 
*x^2)*exp(5)*exp(x^2)+(4*x^2+1)*exp(5)^2)/(x*exp(x^2)^2*log(x)+(x^5-2*x^4+ 
x^3+x)*exp(x^2)^2+(-2*x^3+2*x^2)*exp(5)*exp(x^2)+x*exp(5)^2),x, algorithm= 
"maxima")
 

Output:

2*x^2 - log(x^4 - 2*x^3 + x^2 + log(x) + 1) + log(x) - log(((x^4 - 2*x^3 + 
 x^2 + log(x) + 1)*e^(2*x^2) - 2*(x^2*e^5 - x*e^5)*e^(x^2) + e^10)/(x^4 - 
2*x^3 + x^2 + log(x) + 1))
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 120 vs. \(2 (28) = 56\).

Time = 0.21 (sec) , antiderivative size = 120, normalized size of antiderivative = 4.14 \[ \int \frac {e^{10} \left (1+4 x^2\right )+e^{5+x^2} \left (2 x^2+4 x^3-4 x^4\right )+e^{2 x^2} \left (-x^2+4 x^3-3 x^4\right )+e^{2 x^2} \log (x)}{e^{10} x+e^{5+x^2} \left (2 x^2-2 x^3\right )+e^{2 x^2} \left (x+x^3-2 x^4+x^5\right )+e^{2 x^2} x \log (x)} \, dx=2 \, x^{2} - \log \left (x^{4} e^{\left (2 \, x^{2}\right )} - 2 \, x^{3} e^{\left (2 \, x^{2}\right )} + x^{2} e^{\left (2 \, x^{2}\right )} - 2 \, x^{2} e^{\left (x^{2} + 5\right )} + 2 \, x e^{\left (x^{2} + 5\right )} + e^{\left (2 \, x^{2}\right )} \log \left (x\right ) + e^{10} + e^{\left (2 \, x^{2}\right )}\right ) + \log \left (x^{4} - 2 \, x^{3} + x^{2} + \log \left (x\right ) + 1\right ) - \log \left (-x^{4} + 2 \, x^{3} - x^{2} - \log \left (x\right ) - 1\right ) + \log \left (x\right ) \] Input:

integrate((exp(x^2)^2*log(x)+(-3*x^4+4*x^3-x^2)*exp(x^2)^2+(-4*x^4+4*x^3+2 
*x^2)*exp(5)*exp(x^2)+(4*x^2+1)*exp(5)^2)/(x*exp(x^2)^2*log(x)+(x^5-2*x^4+ 
x^3+x)*exp(x^2)^2+(-2*x^3+2*x^2)*exp(5)*exp(x^2)+x*exp(5)^2),x, algorithm= 
"giac")
 

Output:

2*x^2 - log(x^4*e^(2*x^2) - 2*x^3*e^(2*x^2) + x^2*e^(2*x^2) - 2*x^2*e^(x^2 
 + 5) + 2*x*e^(x^2 + 5) + e^(2*x^2)*log(x) + e^10 + e^(2*x^2)) + log(x^4 - 
 2*x^3 + x^2 + log(x) + 1) - log(-x^4 + 2*x^3 - x^2 - log(x) - 1) + log(x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {e^{10} \left (1+4 x^2\right )+e^{5+x^2} \left (2 x^2+4 x^3-4 x^4\right )+e^{2 x^2} \left (-x^2+4 x^3-3 x^4\right )+e^{2 x^2} \log (x)}{e^{10} x+e^{5+x^2} \left (2 x^2-2 x^3\right )+e^{2 x^2} \left (x+x^3-2 x^4+x^5\right )+e^{2 x^2} x \log (x)} \, dx=\int \frac {{\mathrm {e}}^{10}\,\left (4\,x^2+1\right )-{\mathrm {e}}^{2\,x^2}\,\left (3\,x^4-4\,x^3+x^2\right )+{\mathrm {e}}^{2\,x^2}\,\ln \left (x\right )+{\mathrm {e}}^{x^2+5}\,\left (-4\,x^4+4\,x^3+2\,x^2\right )}{x\,{\mathrm {e}}^{10}+{\mathrm {e}}^{2\,x^2}\,\left (x^5-2\,x^4+x^3+x\right )+{\mathrm {e}}^{x^2+5}\,\left (2\,x^2-2\,x^3\right )+x\,{\mathrm {e}}^{2\,x^2}\,\ln \left (x\right )} \,d x \] Input:

int((exp(10)*(4*x^2 + 1) - exp(2*x^2)*(x^2 - 4*x^3 + 3*x^4) + exp(2*x^2)*l 
og(x) + exp(x^2)*exp(5)*(2*x^2 + 4*x^3 - 4*x^4))/(x*exp(10) + exp(2*x^2)*( 
x + x^3 - 2*x^4 + x^5) + x*exp(2*x^2)*log(x) + exp(x^2)*exp(5)*(2*x^2 - 2* 
x^3)),x)
 

Output:

int((exp(10)*(4*x^2 + 1) - exp(2*x^2)*(x^2 - 4*x^3 + 3*x^4) + exp(2*x^2)*l 
og(x) + exp(x^2 + 5)*(2*x^2 + 4*x^3 - 4*x^4))/(x*exp(10) + exp(2*x^2)*(x + 
 x^3 - 2*x^4 + x^5) + exp(x^2 + 5)*(2*x^2 - 2*x^3) + x*exp(2*x^2)*log(x)), 
 x)
 

Reduce [B] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 102, normalized size of antiderivative = 3.52 \[ \int \frac {e^{10} \left (1+4 x^2\right )+e^{5+x^2} \left (2 x^2+4 x^3-4 x^4\right )+e^{2 x^2} \left (-x^2+4 x^3-3 x^4\right )+e^{2 x^2} \log (x)}{e^{10} x+e^{5+x^2} \left (2 x^2-2 x^3\right )+e^{2 x^2} \left (x+x^3-2 x^4+x^5\right )+e^{2 x^2} x \log (x)} \, dx=-\mathrm {log}\left (e^{4 x^{2}} \mathrm {log}\left (x \right )+e^{4 x^{2}} x^{4}-2 e^{4 x^{2}} x^{3}+e^{4 x^{2}} x^{2}+e^{4 x^{2}}-2 e^{3 x^{2}} e^{5} x^{2}+2 e^{3 x^{2}} e^{5} x +e^{2 x^{2}} e^{10}\right )+\mathrm {log}\left (x \right )+4 x^{2} \] Input:

int((exp(x^2)^2*log(x)+(-3*x^4+4*x^3-x^2)*exp(x^2)^2+(-4*x^4+4*x^3+2*x^2)* 
exp(5)*exp(x^2)+(4*x^2+1)*exp(5)^2)/(x*exp(x^2)^2*log(x)+(x^5-2*x^4+x^3+x) 
*exp(x^2)^2+(-2*x^3+2*x^2)*exp(5)*exp(x^2)+x*exp(5)^2),x)
 

Output:

 - log(e**(4*x**2)*log(x) + e**(4*x**2)*x**4 - 2*e**(4*x**2)*x**3 + e**(4* 
x**2)*x**2 + e**(4*x**2) - 2*e**(3*x**2)*e**5*x**2 + 2*e**(3*x**2)*e**5*x 
+ e**(2*x**2)*e**10) + log(x) + 4*x**2