Integrand size = 302, antiderivative size = 30 \[ \int \frac {e^{-\frac {10}{2+e^x}} \left (-4 x^2+8 \log (4)+e^{2 x} \left (-x^2+2 \log (4)\right )+e^x \left (-4 x^2+10 x^3+\left (8+20 x-10 x^2\right ) \log (4)\right )+e^{\frac {10}{2+e^x}} \left (4 x^4+\left (16 x^2-8 x^3\right ) \log (4)+\left (16-16 x+4 x^2\right ) \log ^2(4)+e^{2 x} \left (x^4+\left (4 x^2-2 x^3\right ) \log (4)+\left (4-4 x+x^2\right ) \log ^2(4)\right )+e^x \left (4 x^4+\left (16 x^2-8 x^3\right ) \log (4)+\left (16-16 x+4 x^2\right ) \log ^2(4)\right )\right )\right )}{4 x^4+\left (16 x^2-8 x^3\right ) \log (4)+\left (16-16 x+4 x^2\right ) \log ^2(4)+e^{2 x} \left (x^4+\left (4 x^2-2 x^3\right ) \log (4)+\left (4-4 x+x^2\right ) \log ^2(4)\right )+e^x \left (4 x^4+\left (16 x^2-8 x^3\right ) \log (4)+\left (16-16 x+4 x^2\right ) \log ^2(4)\right )} \, dx=5+x+\frac {e^{-\frac {10}{2+e^x}} x}{x^2+(2-x) \log (4)} \] Output:
x/(2*(2-x)*ln(2)+x^2)/exp(5/(2+exp(x)))^2+5+x
\[ \int \frac {e^{-\frac {10}{2+e^x}} \left (-4 x^2+8 \log (4)+e^{2 x} \left (-x^2+2 \log (4)\right )+e^x \left (-4 x^2+10 x^3+\left (8+20 x-10 x^2\right ) \log (4)\right )+e^{\frac {10}{2+e^x}} \left (4 x^4+\left (16 x^2-8 x^3\right ) \log (4)+\left (16-16 x+4 x^2\right ) \log ^2(4)+e^{2 x} \left (x^4+\left (4 x^2-2 x^3\right ) \log (4)+\left (4-4 x+x^2\right ) \log ^2(4)\right )+e^x \left (4 x^4+\left (16 x^2-8 x^3\right ) \log (4)+\left (16-16 x+4 x^2\right ) \log ^2(4)\right )\right )\right )}{4 x^4+\left (16 x^2-8 x^3\right ) \log (4)+\left (16-16 x+4 x^2\right ) \log ^2(4)+e^{2 x} \left (x^4+\left (4 x^2-2 x^3\right ) \log (4)+\left (4-4 x+x^2\right ) \log ^2(4)\right )+e^x \left (4 x^4+\left (16 x^2-8 x^3\right ) \log (4)+\left (16-16 x+4 x^2\right ) \log ^2(4)\right )} \, dx=\int \frac {e^{-\frac {10}{2+e^x}} \left (-4 x^2+8 \log (4)+e^{2 x} \left (-x^2+2 \log (4)\right )+e^x \left (-4 x^2+10 x^3+\left (8+20 x-10 x^2\right ) \log (4)\right )+e^{\frac {10}{2+e^x}} \left (4 x^4+\left (16 x^2-8 x^3\right ) \log (4)+\left (16-16 x+4 x^2\right ) \log ^2(4)+e^{2 x} \left (x^4+\left (4 x^2-2 x^3\right ) \log (4)+\left (4-4 x+x^2\right ) \log ^2(4)\right )+e^x \left (4 x^4+\left (16 x^2-8 x^3\right ) \log (4)+\left (16-16 x+4 x^2\right ) \log ^2(4)\right )\right )\right )}{4 x^4+\left (16 x^2-8 x^3\right ) \log (4)+\left (16-16 x+4 x^2\right ) \log ^2(4)+e^{2 x} \left (x^4+\left (4 x^2-2 x^3\right ) \log (4)+\left (4-4 x+x^2\right ) \log ^2(4)\right )+e^x \left (4 x^4+\left (16 x^2-8 x^3\right ) \log (4)+\left (16-16 x+4 x^2\right ) \log ^2(4)\right )} \, dx \] Input:
Integrate[(-4*x^2 + 8*Log[4] + E^(2*x)*(-x^2 + 2*Log[4]) + E^x*(-4*x^2 + 1 0*x^3 + (8 + 20*x - 10*x^2)*Log[4]) + E^(10/(2 + E^x))*(4*x^4 + (16*x^2 - 8*x^3)*Log[4] + (16 - 16*x + 4*x^2)*Log[4]^2 + E^(2*x)*(x^4 + (4*x^2 - 2*x ^3)*Log[4] + (4 - 4*x + x^2)*Log[4]^2) + E^x*(4*x^4 + (16*x^2 - 8*x^3)*Log [4] + (16 - 16*x + 4*x^2)*Log[4]^2)))/(E^(10/(2 + E^x))*(4*x^4 + (16*x^2 - 8*x^3)*Log[4] + (16 - 16*x + 4*x^2)*Log[4]^2 + E^(2*x)*(x^4 + (4*x^2 - 2* x^3)*Log[4] + (4 - 4*x + x^2)*Log[4]^2) + E^x*(4*x^4 + (16*x^2 - 8*x^3)*Lo g[4] + (16 - 16*x + 4*x^2)*Log[4]^2))),x]
Output:
Integrate[(-4*x^2 + 8*Log[4] + E^(2*x)*(-x^2 + 2*Log[4]) + E^x*(-4*x^2 + 1 0*x^3 + (8 + 20*x - 10*x^2)*Log[4]) + E^(10/(2 + E^x))*(4*x^4 + (16*x^2 - 8*x^3)*Log[4] + (16 - 16*x + 4*x^2)*Log[4]^2 + E^(2*x)*(x^4 + (4*x^2 - 2*x ^3)*Log[4] + (4 - 4*x + x^2)*Log[4]^2) + E^x*(4*x^4 + (16*x^2 - 8*x^3)*Log [4] + (16 - 16*x + 4*x^2)*Log[4]^2)))/(E^(10/(2 + E^x))*(4*x^4 + (16*x^2 - 8*x^3)*Log[4] + (16 - 16*x + 4*x^2)*Log[4]^2 + E^(2*x)*(x^4 + (4*x^2 - 2* x^3)*Log[4] + (4 - 4*x + x^2)*Log[4]^2) + E^x*(4*x^4 + (16*x^2 - 8*x^3)*Lo g[4] + (16 - 16*x + 4*x^2)*Log[4]^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.
| \(\displaystyle \int \frac {e^{-\frac {10}{e^x+2}} \left (-4 x^2+e^{2 x} \left (2 \log (4)-x^2\right )+e^x \left (10 x^3-4 x^2+\left (-10 x^2+20 x+8\right ) \log (4)\right )+e^{\frac {10}{e^x+2}} \left (4 x^4+\left (4 x^2-16 x+16\right ) \log ^2(4)+\left (16 x^2-8 x^3\right ) \log (4)+e^{2 x} \left (x^4+\left (x^2-4 x+4\right ) \log ^2(4)+\left (4 x^2-2 x^3\right ) \log (4)\right )+e^x \left (4 x^4+\left (4 x^2-16 x+16\right ) \log ^2(4)+\left (16 x^2-8 x^3\right ) \log (4)\right )\right )+8 \log (4)\right )}{4 x^4+\left (4 x^2-16 x+16\right ) \log ^2(4)+\left (16 x^2-8 x^3\right ) \log (4)+e^{2 x} \left (x^4+\left (x^2-4 x+4\right ) \log ^2(4)+\left (4 x^2-2 x^3\right ) \log (4)\right )+e^x \left (4 x^4+\left (4 x^2-16 x+16\right ) \log ^2(4)+\left (16 x^2-8 x^3\right ) \log (4)\right )} \, dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {e^{-\frac {10}{e^x+2}} \left (-4 x^2+e^{\frac {10}{e^x+2}} \left (e^x+2\right )^2 \left (x^2-x \log (4)+\log (16)\right )^2-e^{2 x} \left (x^2-2 \log (4)\right )+e^x \left (10 x^3-2 x^2 (2+5 \log (4))+20 x \log (4)+8 \log (4)\right )+8 \log (4)\right )}{\left (e^x+2\right )^2 \left (x^2-x \log (4)+\log (16)\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {10 e^{-\frac {10}{e^x+2}} x}{\left (e^x+2\right ) \left (x^2-x \log (4)+\log (16)\right )}-\frac {20 e^{-\frac {10}{e^x+2}} x}{\left (e^x+2\right )^2 \left (x^2-x \log (4)+\log (16)\right )}+\frac {e^{-\frac {10}{e^x+2}} \left (e^{\frac {10}{e^x+2}} x^4-2 e^{\frac {10}{e^x+2}} x^3 \log (4)-x^2+e^{\frac {10}{e^x+2}} x^2 \log ^2(4) \left (1+\frac {\log (256)}{\log ^2(4)}\right )-e^{\frac {10}{e^x+2}} x \log ^2(16)+e^{\frac {10}{e^x+2}} \log ^2(16)+\log (16)\right )}{\left (x^2-x \log (4)+\log (16)\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {10 e^{-\frac {10}{e^x+2}} x}{\left (e^x+2\right ) \left (x^2-x \log (4)+\log (16)\right )}-\frac {20 e^{-\frac {10}{e^x+2}} x}{\left (e^x+2\right )^2 \left (x^2-x \log (4)+\log (16)\right )}+\frac {e^{-\frac {10}{e^x+2}} \left (e^{\frac {10}{e^x+2}} x^4-e^{\frac {10}{e^x+2}} x^3 \log (16)-x^2+e^{\frac {10}{e^x+2}} x^2 \log ^2(4) \left (1+\frac {\log (256)}{\log ^2(4)}\right )-e^{\frac {10}{e^x+2}} x \log ^2(16)+e^{\frac {10}{e^x+2}} \log ^2(16)+\log (16)\right )}{x^4-x^3 \log (16)+x^2 \left (\log ^2(4)+\log (256)\right )-x \log ^2(16)+\log ^2(16)}\right )dx\) |
| \(\Big \downarrow \) 7299 |
| \(\displaystyle \int \left (\frac {10 e^{-\frac {10}{e^x+2}} x}{\left (e^x+2\right ) \left (x^2-x \log (4)+\log (16)\right )}-\frac {20 e^{-\frac {10}{e^x+2}} x}{\left (e^x+2\right )^2 \left (x^2-x \log (4)+\log (16)\right )}+\frac {e^{-\frac {10}{e^x+2}} \left (e^{\frac {10}{e^x+2}} x^4-e^{\frac {10}{e^x+2}} x^3 \log (16)-x^2+e^{\frac {10}{e^x+2}} x^2 \log ^2(4) \left (1+\frac {\log (256)}{\log ^2(4)}\right )-e^{\frac {10}{e^x+2}} x \log ^2(16)+e^{\frac {10}{e^x+2}} \log ^2(16)+\log (16)\right )}{x^4-x^3 \log (16)+x^2 \left (\log ^2(4)+\log (256)\right )-x \log ^2(16)+\log ^2(16)}\right )dx\) |
Input:
Int[(-4*x^2 + 8*Log[4] + E^(2*x)*(-x^2 + 2*Log[4]) + E^x*(-4*x^2 + 10*x^3 + (8 + 20*x - 10*x^2)*Log[4]) + E^(10/(2 + E^x))*(4*x^4 + (16*x^2 - 8*x^3) *Log[4] + (16 - 16*x + 4*x^2)*Log[4]^2 + E^(2*x)*(x^4 + (4*x^2 - 2*x^3)*Lo g[4] + (4 - 4*x + x^2)*Log[4]^2) + E^x*(4*x^4 + (16*x^2 - 8*x^3)*Log[4] + (16 - 16*x + 4*x^2)*Log[4]^2)))/(E^(10/(2 + E^x))*(4*x^4 + (16*x^2 - 8*x^3 )*Log[4] + (16 - 16*x + 4*x^2)*Log[4]^2 + E^(2*x)*(x^4 + (4*x^2 - 2*x^3)*L og[4] + (4 - 4*x + x^2)*Log[4]^2) + E^x*(4*x^4 + (16*x^2 - 8*x^3)*Log[4] + (16 - 16*x + 4*x^2)*Log[4]^2))),x]
Output:
$Aborted
Time = 4.03 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.07
| method | result | size |
| risch | \(x -\frac {x \,{\mathrm e}^{-\frac {10}{{\mathrm e}^{x}+2}}}{2 x \ln \left (2\right )-x^{2}-4 \ln \left (2\right )}\) | \(32\) |
| parallelrisch | \(\frac {\left (-2 x -2 \,{\mathrm e}^{\frac {10}{{\mathrm e}^{x}+2}} x^{3}-16 \ln \left (2\right )^{2} {\mathrm e}^{\frac {10}{{\mathrm e}^{x}+2}}+4 \,{\mathrm e}^{x} \ln \left (2\right )^{2} {\mathrm e}^{\frac {10}{{\mathrm e}^{x}+2}} x -{\mathrm e}^{x} x -8 \,{\mathrm e}^{x} \ln \left (2\right )^{2} {\mathrm e}^{\frac {10}{{\mathrm e}^{x}+2}}+8 \ln \left (2\right )^{2} {\mathrm e}^{\frac {10}{{\mathrm e}^{x}+2}} x -4 \ln \left (2\right ) {\mathrm e}^{\frac {10}{{\mathrm e}^{x}+2}} {\mathrm e}^{x} x -{\mathrm e}^{\frac {10}{{\mathrm e}^{x}+2}} {\mathrm e}^{x} x^{3}-8 \ln \left (2\right ) {\mathrm e}^{\frac {10}{{\mathrm e}^{x}+2}} x \right ) {\mathrm e}^{-\frac {10}{{\mathrm e}^{x}+2}}}{\left (2 x \ln \left (2\right )-x^{2}-4 \ln \left (2\right )\right ) \left ({\mathrm e}^{x}+2\right )}\) | \(187\) |
Input:
int((((4*(x^2-4*x+4)*ln(2)^2+2*(-2*x^3+4*x^2)*ln(2)+x^4)*exp(x)^2+(4*(4*x^ 2-16*x+16)*ln(2)^2+2*(-8*x^3+16*x^2)*ln(2)+4*x^4)*exp(x)+4*(4*x^2-16*x+16) *ln(2)^2+2*(-8*x^3+16*x^2)*ln(2)+4*x^4)*exp(5/(exp(x)+2))^2+(4*ln(2)-x^2)* exp(x)^2+(2*(-10*x^2+20*x+8)*ln(2)+10*x^3-4*x^2)*exp(x)+16*ln(2)-4*x^2)/(( 4*(x^2-4*x+4)*ln(2)^2+2*(-2*x^3+4*x^2)*ln(2)+x^4)*exp(x)^2+(4*(4*x^2-16*x+ 16)*ln(2)^2+2*(-8*x^3+16*x^2)*ln(2)+4*x^4)*exp(x)+4*(4*x^2-16*x+16)*ln(2)^ 2+2*(-8*x^3+16*x^2)*ln(2)+4*x^4)/exp(5/(exp(x)+2))^2,x,method=_RETURNVERBO SE)
Output:
x-x/(2*x*ln(2)-x^2-4*ln(2))*exp(-10/(exp(x)+2))
Time = 0.11 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.67 \[ \int \frac {e^{-\frac {10}{2+e^x}} \left (-4 x^2+8 \log (4)+e^{2 x} \left (-x^2+2 \log (4)\right )+e^x \left (-4 x^2+10 x^3+\left (8+20 x-10 x^2\right ) \log (4)\right )+e^{\frac {10}{2+e^x}} \left (4 x^4+\left (16 x^2-8 x^3\right ) \log (4)+\left (16-16 x+4 x^2\right ) \log ^2(4)+e^{2 x} \left (x^4+\left (4 x^2-2 x^3\right ) \log (4)+\left (4-4 x+x^2\right ) \log ^2(4)\right )+e^x \left (4 x^4+\left (16 x^2-8 x^3\right ) \log (4)+\left (16-16 x+4 x^2\right ) \log ^2(4)\right )\right )\right )}{4 x^4+\left (16 x^2-8 x^3\right ) \log (4)+\left (16-16 x+4 x^2\right ) \log ^2(4)+e^{2 x} \left (x^4+\left (4 x^2-2 x^3\right ) \log (4)+\left (4-4 x+x^2\right ) \log ^2(4)\right )+e^x \left (4 x^4+\left (16 x^2-8 x^3\right ) \log (4)+\left (16-16 x+4 x^2\right ) \log ^2(4)\right )} \, dx=\frac {{\left ({\left (x^{3} - 2 \, {\left (x^{2} - 2 \, x\right )} \log \left (2\right )\right )} e^{\left (\frac {10}{e^{x} + 2}\right )} + x\right )} e^{\left (-\frac {10}{e^{x} + 2}\right )}}{x^{2} - 2 \, {\left (x - 2\right )} \log \left (2\right )} \] Input:
integrate((((4*(x^2-4*x+4)*log(2)^2+2*(-2*x^3+4*x^2)*log(2)+x^4)*exp(x)^2+ (4*(4*x^2-16*x+16)*log(2)^2+2*(-8*x^3+16*x^2)*log(2)+4*x^4)*exp(x)+4*(4*x^ 2-16*x+16)*log(2)^2+2*(-8*x^3+16*x^2)*log(2)+4*x^4)*exp(5/(2+exp(x)))^2+(4 *log(2)-x^2)*exp(x)^2+(2*(-10*x^2+20*x+8)*log(2)+10*x^3-4*x^2)*exp(x)+16*l og(2)-4*x^2)/((4*(x^2-4*x+4)*log(2)^2+2*(-2*x^3+4*x^2)*log(2)+x^4)*exp(x)^ 2+(4*(4*x^2-16*x+16)*log(2)^2+2*(-8*x^3+16*x^2)*log(2)+4*x^4)*exp(x)+4*(4* x^2-16*x+16)*log(2)^2+2*(-8*x^3+16*x^2)*log(2)+4*x^4)/exp(5/(2+exp(x)))^2, x, algorithm="fricas")
Output:
((x^3 - 2*(x^2 - 2*x)*log(2))*e^(10/(e^x + 2)) + x)*e^(-10/(e^x + 2))/(x^2 - 2*(x - 2)*log(2))
Time = 0.30 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.87 \[ \int \frac {e^{-\frac {10}{2+e^x}} \left (-4 x^2+8 \log (4)+e^{2 x} \left (-x^2+2 \log (4)\right )+e^x \left (-4 x^2+10 x^3+\left (8+20 x-10 x^2\right ) \log (4)\right )+e^{\frac {10}{2+e^x}} \left (4 x^4+\left (16 x^2-8 x^3\right ) \log (4)+\left (16-16 x+4 x^2\right ) \log ^2(4)+e^{2 x} \left (x^4+\left (4 x^2-2 x^3\right ) \log (4)+\left (4-4 x+x^2\right ) \log ^2(4)\right )+e^x \left (4 x^4+\left (16 x^2-8 x^3\right ) \log (4)+\left (16-16 x+4 x^2\right ) \log ^2(4)\right )\right )\right )}{4 x^4+\left (16 x^2-8 x^3\right ) \log (4)+\left (16-16 x+4 x^2\right ) \log ^2(4)+e^{2 x} \left (x^4+\left (4 x^2-2 x^3\right ) \log (4)+\left (4-4 x+x^2\right ) \log ^2(4)\right )+e^x \left (4 x^4+\left (16 x^2-8 x^3\right ) \log (4)+\left (16-16 x+4 x^2\right ) \log ^2(4)\right )} \, dx=x + \frac {x e^{- \frac {10}{e^{x} + 2}}}{x^{2} - 2 x \log {\left (2 \right )} + 4 \log {\left (2 \right )}} \] Input:
integrate((((4*(x**2-4*x+4)*ln(2)**2+2*(-2*x**3+4*x**2)*ln(2)+x**4)*exp(x) **2+(4*(4*x**2-16*x+16)*ln(2)**2+2*(-8*x**3+16*x**2)*ln(2)+4*x**4)*exp(x)+ 4*(4*x**2-16*x+16)*ln(2)**2+2*(-8*x**3+16*x**2)*ln(2)+4*x**4)*exp(5/(2+exp (x)))**2+(4*ln(2)-x**2)*exp(x)**2+(2*(-10*x**2+20*x+8)*ln(2)+10*x**3-4*x** 2)*exp(x)+16*ln(2)-4*x**2)/((4*(x**2-4*x+4)*ln(2)**2+2*(-2*x**3+4*x**2)*ln (2)+x**4)*exp(x)**2+(4*(4*x**2-16*x+16)*ln(2)**2+2*(-8*x**3+16*x**2)*ln(2) +4*x**4)*exp(x)+4*(4*x**2-16*x+16)*ln(2)**2+2*(-8*x**3+16*x**2)*ln(2)+4*x* *4)/exp(5/(2+exp(x)))**2,x)
Output:
x + x*exp(-10/(exp(x) + 2))/(x**2 - 2*x*log(2) + 4*log(2))
\[ \int \frac {e^{-\frac {10}{2+e^x}} \left (-4 x^2+8 \log (4)+e^{2 x} \left (-x^2+2 \log (4)\right )+e^x \left (-4 x^2+10 x^3+\left (8+20 x-10 x^2\right ) \log (4)\right )+e^{\frac {10}{2+e^x}} \left (4 x^4+\left (16 x^2-8 x^3\right ) \log (4)+\left (16-16 x+4 x^2\right ) \log ^2(4)+e^{2 x} \left (x^4+\left (4 x^2-2 x^3\right ) \log (4)+\left (4-4 x+x^2\right ) \log ^2(4)\right )+e^x \left (4 x^4+\left (16 x^2-8 x^3\right ) \log (4)+\left (16-16 x+4 x^2\right ) \log ^2(4)\right )\right )\right )}{4 x^4+\left (16 x^2-8 x^3\right ) \log (4)+\left (16-16 x+4 x^2\right ) \log ^2(4)+e^{2 x} \left (x^4+\left (4 x^2-2 x^3\right ) \log (4)+\left (4-4 x+x^2\right ) \log ^2(4)\right )+e^x \left (4 x^4+\left (16 x^2-8 x^3\right ) \log (4)+\left (16-16 x+4 x^2\right ) \log ^2(4)\right )} \, dx=\int { -\frac {{\left (4 \, x^{2} + {\left (x^{2} - 4 \, \log \left (2\right )\right )} e^{\left (2 \, x\right )} - 2 \, {\left (5 \, x^{3} - 2 \, x^{2} - 2 \, {\left (5 \, x^{2} - 10 \, x - 4\right )} \log \left (2\right )\right )} e^{x} - {\left (4 \, x^{4} + 16 \, {\left (x^{2} - 4 \, x + 4\right )} \log \left (2\right )^{2} + {\left (x^{4} + 4 \, {\left (x^{2} - 4 \, x + 4\right )} \log \left (2\right )^{2} - 4 \, {\left (x^{3} - 2 \, x^{2}\right )} \log \left (2\right )\right )} e^{\left (2 \, x\right )} + 4 \, {\left (x^{4} + 4 \, {\left (x^{2} - 4 \, x + 4\right )} \log \left (2\right )^{2} - 4 \, {\left (x^{3} - 2 \, x^{2}\right )} \log \left (2\right )\right )} e^{x} - 16 \, {\left (x^{3} - 2 \, x^{2}\right )} \log \left (2\right )\right )} e^{\left (\frac {10}{e^{x} + 2}\right )} - 16 \, \log \left (2\right )\right )} e^{\left (-\frac {10}{e^{x} + 2}\right )}}{4 \, x^{4} + 16 \, {\left (x^{2} - 4 \, x + 4\right )} \log \left (2\right )^{2} + {\left (x^{4} + 4 \, {\left (x^{2} - 4 \, x + 4\right )} \log \left (2\right )^{2} - 4 \, {\left (x^{3} - 2 \, x^{2}\right )} \log \left (2\right )\right )} e^{\left (2 \, x\right )} + 4 \, {\left (x^{4} + 4 \, {\left (x^{2} - 4 \, x + 4\right )} \log \left (2\right )^{2} - 4 \, {\left (x^{3} - 2 \, x^{2}\right )} \log \left (2\right )\right )} e^{x} - 16 \, {\left (x^{3} - 2 \, x^{2}\right )} \log \left (2\right )} \,d x } \] Input:
integrate((((4*(x^2-4*x+4)*log(2)^2+2*(-2*x^3+4*x^2)*log(2)+x^4)*exp(x)^2+ (4*(4*x^2-16*x+16)*log(2)^2+2*(-8*x^3+16*x^2)*log(2)+4*x^4)*exp(x)+4*(4*x^ 2-16*x+16)*log(2)^2+2*(-8*x^3+16*x^2)*log(2)+4*x^4)*exp(5/(2+exp(x)))^2+(4 *log(2)-x^2)*exp(x)^2+(2*(-10*x^2+20*x+8)*log(2)+10*x^3-4*x^2)*exp(x)+16*l og(2)-4*x^2)/((4*(x^2-4*x+4)*log(2)^2+2*(-2*x^3+4*x^2)*log(2)+x^4)*exp(x)^ 2+(4*(4*x^2-16*x+16)*log(2)^2+2*(-8*x^3+16*x^2)*log(2)+4*x^4)*exp(x)+4*(4* x^2-16*x+16)*log(2)^2+2*(-8*x^3+16*x^2)*log(2)+4*x^4)/exp(5/(2+exp(x)))^2, x, algorithm="maxima")
Output:
x - integrate((4*x^2 + (x^2 - 4*log(2))*e^(2*x) - 2*(5*x^3 - 2*x^2*(5*log( 2) + 1) + 20*x*log(2) + 8*log(2))*e^x - 16*log(2))*e^(-10/(e^x + 2))/(4*x^ 4 - 16*x^3*log(2) + 16*(log(2)^2 + 2*log(2))*x^2 - 64*x*log(2)^2 + (x^4 - 4*x^3*log(2) + 4*(log(2)^2 + 2*log(2))*x^2 - 16*x*log(2)^2 + 16*log(2)^2)* e^(2*x) + 4*(x^4 - 4*x^3*log(2) + 4*(log(2)^2 + 2*log(2))*x^2 - 16*x*log(2 )^2 + 16*log(2)^2)*e^x + 64*log(2)^2), x)
Leaf count of result is larger than twice the leaf count of optimal. 170 vs. \(2 (27) = 54\).
Time = 0.33 (sec) , antiderivative size = 170, normalized size of antiderivative = 5.67 \[ \int \frac {e^{-\frac {10}{2+e^x}} \left (-4 x^2+8 \log (4)+e^{2 x} \left (-x^2+2 \log (4)\right )+e^x \left (-4 x^2+10 x^3+\left (8+20 x-10 x^2\right ) \log (4)\right )+e^{\frac {10}{2+e^x}} \left (4 x^4+\left (16 x^2-8 x^3\right ) \log (4)+\left (16-16 x+4 x^2\right ) \log ^2(4)+e^{2 x} \left (x^4+\left (4 x^2-2 x^3\right ) \log (4)+\left (4-4 x+x^2\right ) \log ^2(4)\right )+e^x \left (4 x^4+\left (16 x^2-8 x^3\right ) \log (4)+\left (16-16 x+4 x^2\right ) \log ^2(4)\right )\right )\right )}{4 x^4+\left (16 x^2-8 x^3\right ) \log (4)+\left (16-16 x+4 x^2\right ) \log ^2(4)+e^{2 x} \left (x^4+\left (4 x^2-2 x^3\right ) \log (4)+\left (4-4 x+x^2\right ) \log ^2(4)\right )+e^x \left (4 x^4+\left (16 x^2-8 x^3\right ) \log (4)+\left (16-16 x+4 x^2\right ) \log ^2(4)\right )} \, dx=\frac {x^{3} e^{\left (\frac {x e^{x} + 2 \, x - 5 \, e^{x}}{e^{x} + 2} + 5\right )} - 2 \, x^{2} e^{\left (\frac {x e^{x} + 2 \, x - 5 \, e^{x}}{e^{x} + 2} + 5\right )} \log \left (2\right ) + 4 \, x e^{\left (\frac {x e^{x} + 2 \, x - 5 \, e^{x}}{e^{x} + 2} + 5\right )} \log \left (2\right ) + x e^{x}}{x^{2} e^{\left (\frac {x e^{x} + 2 \, x - 5 \, e^{x}}{e^{x} + 2} + 5\right )} - 2 \, x e^{\left (\frac {x e^{x} + 2 \, x - 5 \, e^{x}}{e^{x} + 2} + 5\right )} \log \left (2\right ) + 4 \, e^{\left (\frac {x e^{x} + 2 \, x - 5 \, e^{x}}{e^{x} + 2} + 5\right )} \log \left (2\right )} \] Input:
integrate((((4*(x^2-4*x+4)*log(2)^2+2*(-2*x^3+4*x^2)*log(2)+x^4)*exp(x)^2+ (4*(4*x^2-16*x+16)*log(2)^2+2*(-8*x^3+16*x^2)*log(2)+4*x^4)*exp(x)+4*(4*x^ 2-16*x+16)*log(2)^2+2*(-8*x^3+16*x^2)*log(2)+4*x^4)*exp(5/(2+exp(x)))^2+(4 *log(2)-x^2)*exp(x)^2+(2*(-10*x^2+20*x+8)*log(2)+10*x^3-4*x^2)*exp(x)+16*l og(2)-4*x^2)/((4*(x^2-4*x+4)*log(2)^2+2*(-2*x^3+4*x^2)*log(2)+x^4)*exp(x)^ 2+(4*(4*x^2-16*x+16)*log(2)^2+2*(-8*x^3+16*x^2)*log(2)+4*x^4)*exp(x)+4*(4* x^2-16*x+16)*log(2)^2+2*(-8*x^3+16*x^2)*log(2)+4*x^4)/exp(5/(2+exp(x)))^2, x, algorithm="giac")
Output:
(x^3*e^((x*e^x + 2*x - 5*e^x)/(e^x + 2) + 5) - 2*x^2*e^((x*e^x + 2*x - 5*e ^x)/(e^x + 2) + 5)*log(2) + 4*x*e^((x*e^x + 2*x - 5*e^x)/(e^x + 2) + 5)*lo g(2) + x*e^x)/(x^2*e^((x*e^x + 2*x - 5*e^x)/(e^x + 2) + 5) - 2*x*e^((x*e^x + 2*x - 5*e^x)/(e^x + 2) + 5)*log(2) + 4*e^((x*e^x + 2*x - 5*e^x)/(e^x + 2) + 5)*log(2))
Timed out. \[ \int \frac {e^{-\frac {10}{2+e^x}} \left (-4 x^2+8 \log (4)+e^{2 x} \left (-x^2+2 \log (4)\right )+e^x \left (-4 x^2+10 x^3+\left (8+20 x-10 x^2\right ) \log (4)\right )+e^{\frac {10}{2+e^x}} \left (4 x^4+\left (16 x^2-8 x^3\right ) \log (4)+\left (16-16 x+4 x^2\right ) \log ^2(4)+e^{2 x} \left (x^4+\left (4 x^2-2 x^3\right ) \log (4)+\left (4-4 x+x^2\right ) \log ^2(4)\right )+e^x \left (4 x^4+\left (16 x^2-8 x^3\right ) \log (4)+\left (16-16 x+4 x^2\right ) \log ^2(4)\right )\right )\right )}{4 x^4+\left (16 x^2-8 x^3\right ) \log (4)+\left (16-16 x+4 x^2\right ) \log ^2(4)+e^{2 x} \left (x^4+\left (4 x^2-2 x^3\right ) \log (4)+\left (4-4 x+x^2\right ) \log ^2(4)\right )+e^x \left (4 x^4+\left (16 x^2-8 x^3\right ) \log (4)+\left (16-16 x+4 x^2\right ) \log ^2(4)\right )} \, dx=\int \frac {{\mathrm {e}}^{-\frac {10}{{\mathrm {e}}^x+2}}\,\left (16\,\ln \left (2\right )+{\mathrm {e}}^{\frac {10}{{\mathrm {e}}^x+2}}\,\left ({\mathrm {e}}^x\,\left (2\,\ln \left (2\right )\,\left (16\,x^2-8\,x^3\right )+4\,{\ln \left (2\right )}^2\,\left (4\,x^2-16\,x+16\right )+4\,x^4\right )+2\,\ln \left (2\right )\,\left (16\,x^2-8\,x^3\right )+4\,{\ln \left (2\right )}^2\,\left (4\,x^2-16\,x+16\right )+{\mathrm {e}}^{2\,x}\,\left (4\,{\ln \left (2\right )}^2\,\left (x^2-4\,x+4\right )+2\,\ln \left (2\right )\,\left (4\,x^2-2\,x^3\right )+x^4\right )+4\,x^4\right )+{\mathrm {e}}^x\,\left (2\,\ln \left (2\right )\,\left (-10\,x^2+20\,x+8\right )-4\,x^2+10\,x^3\right )-4\,x^2+{\mathrm {e}}^{2\,x}\,\left (4\,\ln \left (2\right )-x^2\right )\right )}{{\mathrm {e}}^x\,\left (2\,\ln \left (2\right )\,\left (16\,x^2-8\,x^3\right )+4\,{\ln \left (2\right )}^2\,\left (4\,x^2-16\,x+16\right )+4\,x^4\right )+2\,\ln \left (2\right )\,\left (16\,x^2-8\,x^3\right )+4\,{\ln \left (2\right )}^2\,\left (4\,x^2-16\,x+16\right )+{\mathrm {e}}^{2\,x}\,\left (4\,{\ln \left (2\right )}^2\,\left (x^2-4\,x+4\right )+2\,\ln \left (2\right )\,\left (4\,x^2-2\,x^3\right )+x^4\right )+4\,x^4} \,d x \] Input:
int((exp(-10/(exp(x) + 2))*(16*log(2) + exp(10/(exp(x) + 2))*(exp(x)*(2*lo g(2)*(16*x^2 - 8*x^3) + 4*log(2)^2*(4*x^2 - 16*x + 16) + 4*x^4) + 2*log(2) *(16*x^2 - 8*x^3) + 4*log(2)^2*(4*x^2 - 16*x + 16) + exp(2*x)*(4*log(2)^2* (x^2 - 4*x + 4) + 2*log(2)*(4*x^2 - 2*x^3) + x^4) + 4*x^4) + exp(x)*(2*log (2)*(20*x - 10*x^2 + 8) - 4*x^2 + 10*x^3) - 4*x^2 + exp(2*x)*(4*log(2) - x ^2)))/(exp(x)*(2*log(2)*(16*x^2 - 8*x^3) + 4*log(2)^2*(4*x^2 - 16*x + 16) + 4*x^4) + 2*log(2)*(16*x^2 - 8*x^3) + 4*log(2)^2*(4*x^2 - 16*x + 16) + ex p(2*x)*(4*log(2)^2*(x^2 - 4*x + 4) + 2*log(2)*(4*x^2 - 2*x^3) + x^4) + 4*x ^4),x)
Output:
int((exp(-10/(exp(x) + 2))*(16*log(2) + exp(10/(exp(x) + 2))*(exp(x)*(2*lo g(2)*(16*x^2 - 8*x^3) + 4*log(2)^2*(4*x^2 - 16*x + 16) + 4*x^4) + 2*log(2) *(16*x^2 - 8*x^3) + 4*log(2)^2*(4*x^2 - 16*x + 16) + exp(2*x)*(4*log(2)^2* (x^2 - 4*x + 4) + 2*log(2)*(4*x^2 - 2*x^3) + x^4) + 4*x^4) + exp(x)*(2*log (2)*(20*x - 10*x^2 + 8) - 4*x^2 + 10*x^3) - 4*x^2 + exp(2*x)*(4*log(2) - x ^2)))/(exp(x)*(2*log(2)*(16*x^2 - 8*x^3) + 4*log(2)^2*(4*x^2 - 16*x + 16) + 4*x^4) + 2*log(2)*(16*x^2 - 8*x^3) + 4*log(2)^2*(4*x^2 - 16*x + 16) + ex p(2*x)*(4*log(2)^2*(x^2 - 4*x + 4) + 2*log(2)*(4*x^2 - 2*x^3) + x^4) + 4*x ^4), x)
Time = 0.23 (sec) , antiderivative size = 81, normalized size of antiderivative = 2.70 \[ \int \frac {e^{-\frac {10}{2+e^x}} \left (-4 x^2+8 \log (4)+e^{2 x} \left (-x^2+2 \log (4)\right )+e^x \left (-4 x^2+10 x^3+\left (8+20 x-10 x^2\right ) \log (4)\right )+e^{\frac {10}{2+e^x}} \left (4 x^4+\left (16 x^2-8 x^3\right ) \log (4)+\left (16-16 x+4 x^2\right ) \log ^2(4)+e^{2 x} \left (x^4+\left (4 x^2-2 x^3\right ) \log (4)+\left (4-4 x+x^2\right ) \log ^2(4)\right )+e^x \left (4 x^4+\left (16 x^2-8 x^3\right ) \log (4)+\left (16-16 x+4 x^2\right ) \log ^2(4)\right )\right )\right )}{4 x^4+\left (16 x^2-8 x^3\right ) \log (4)+\left (16-16 x+4 x^2\right ) \log ^2(4)+e^{2 x} \left (x^4+\left (4 x^2-2 x^3\right ) \log (4)+\left (4-4 x+x^2\right ) \log ^2(4)\right )+e^x \left (4 x^4+\left (16 x^2-8 x^3\right ) \log (4)+\left (16-16 x+4 x^2\right ) \log ^2(4)\right )} \, dx=\frac {x \left (2 e^{\frac {10}{e^{x}+2}} \mathrm {log}\left (2\right ) x -4 e^{\frac {10}{e^{x}+2}} \mathrm {log}\left (2\right )-e^{\frac {10}{e^{x}+2}} x^{2}-1\right )}{e^{\frac {10}{e^{x}+2}} \left (2 \,\mathrm {log}\left (2\right ) x -4 \,\mathrm {log}\left (2\right )-x^{2}\right )} \] Input:
int((((4*(x^2-4*x+4)*log(2)^2+2*(-2*x^3+4*x^2)*log(2)+x^4)*exp(x)^2+(4*(4* x^2-16*x+16)*log(2)^2+2*(-8*x^3+16*x^2)*log(2)+4*x^4)*exp(x)+4*(4*x^2-16*x +16)*log(2)^2+2*(-8*x^3+16*x^2)*log(2)+4*x^4)*exp(5/(2+exp(x)))^2+(4*log(2 )-x^2)*exp(x)^2+(2*(-10*x^2+20*x+8)*log(2)+10*x^3-4*x^2)*exp(x)+16*log(2)- 4*x^2)/((4*(x^2-4*x+4)*log(2)^2+2*(-2*x^3+4*x^2)*log(2)+x^4)*exp(x)^2+(4*( 4*x^2-16*x+16)*log(2)^2+2*(-8*x^3+16*x^2)*log(2)+4*x^4)*exp(x)+4*(4*x^2-16 *x+16)*log(2)^2+2*(-8*x^3+16*x^2)*log(2)+4*x^4)/exp(5/(2+exp(x)))^2,x)
Output:
(x*(2*e**(10/(e**x + 2))*log(2)*x - 4*e**(10/(e**x + 2))*log(2) - e**(10/( e**x + 2))*x**2 - 1))/(e**(10/(e**x + 2))*(2*log(2)*x - 4*log(2) - x**2))