Integrand size = 249, antiderivative size = 26 \begin {dmath*} \int \frac {-x^2-x^3+x^4+x^5+\left (-2 x-2 x^2+2 x^3+2 x^4\right ) \log (4)+\left (-1-x+x^2+x^3\right ) \log ^2(4)+e^x \left (-2 x^2-4 x^3-2 x^4+\left (-4 x-8 x^2-4 x^3\right ) \log (4)+\left (-2-4 x-2 x^2\right ) \log ^2(4)\right )+\left (2 x^2+6 x^3+4 x^4+\left (2 x+8 x^2+6 x^3\right ) \log (4)+\left (2 x+2 x^2\right ) \log ^2(4)+e^x \left (4 x^2+12 x^3+8 x^4+\left (4 x+16 x^2+12 x^3\right ) \log (4)+\left (4 x+4 x^2\right ) \log ^2(4)\right )\right ) \log \left (e^{-x} \left (x+2 e^x x\right )\right )}{\left (x+2 e^x x\right ) \log ^2\left (e^{-x} \left (x+2 e^x x\right )\right )} \, dx=\frac {(1+x)^2 (x+\log (4))^2}{\log \left (2 x+e^{-x} x\right )} \end {dmath*}
Time = 2.70 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.92 \begin {dmath*} \int \frac {-x^2-x^3+x^4+x^5+\left (-2 x-2 x^2+2 x^3+2 x^4\right ) \log (4)+\left (-1-x+x^2+x^3\right ) \log ^2(4)+e^x \left (-2 x^2-4 x^3-2 x^4+\left (-4 x-8 x^2-4 x^3\right ) \log (4)+\left (-2-4 x-2 x^2\right ) \log ^2(4)\right )+\left (2 x^2+6 x^3+4 x^4+\left (2 x+8 x^2+6 x^3\right ) \log (4)+\left (2 x+2 x^2\right ) \log ^2(4)+e^x \left (4 x^2+12 x^3+8 x^4+\left (4 x+16 x^2+12 x^3\right ) \log (4)+\left (4 x+4 x^2\right ) \log ^2(4)\right )\right ) \log \left (e^{-x} \left (x+2 e^x x\right )\right )}{\left (x+2 e^x x\right ) \log ^2\left (e^{-x} \left (x+2 e^x x\right )\right )} \, dx=\frac {(1+x)^2 (x+\log (4))^2}{\log \left (\left (2+e^{-x}\right ) x\right )} \end {dmath*}
Integrate[(-x^2 - x^3 + x^4 + x^5 + (-2*x - 2*x^2 + 2*x^3 + 2*x^4)*Log[4] + (-1 - x + x^2 + x^3)*Log[4]^2 + E^x*(-2*x^2 - 4*x^3 - 2*x^4 + (-4*x - 8* x^2 - 4*x^3)*Log[4] + (-2 - 4*x - 2*x^2)*Log[4]^2) + (2*x^2 + 6*x^3 + 4*x^ 4 + (2*x + 8*x^2 + 6*x^3)*Log[4] + (2*x + 2*x^2)*Log[4]^2 + E^x*(4*x^2 + 1 2*x^3 + 8*x^4 + (4*x + 16*x^2 + 12*x^3)*Log[4] + (4*x + 4*x^2)*Log[4]^2))* Log[(x + 2*E^x*x)/E^x])/((x + 2*E^x*x)*Log[(x + 2*E^x*x)/E^x]^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.
| \(\displaystyle \int \frac {x^5+x^4-x^3-x^2+\left (x^3+x^2-x-1\right ) \log ^2(4)+e^x \left (-2 x^4-4 x^3-2 x^2+\left (-2 x^2-4 x-2\right ) \log ^2(4)+\left (-4 x^3-8 x^2-4 x\right ) \log (4)\right )+\left (4 x^4+6 x^3+2 x^2+\left (2 x^2+2 x\right ) \log ^2(4)+\left (6 x^3+8 x^2+2 x\right ) \log (4)+e^x \left (8 x^4+12 x^3+4 x^2+\left (4 x^2+4 x\right ) \log ^2(4)+\left (12 x^3+16 x^2+4 x\right ) \log (4)\right )\right ) \log \left (e^{-x} \left (2 e^x x+x\right )\right )+\left (2 x^4+2 x^3-2 x^2-2 x\right ) \log (4)}{\left (2 e^x x+x\right ) \log ^2\left (e^{-x} \left (2 e^x x+x\right )\right )} \, dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {(x+1) (x+\log (4)) \left ((x+1) \left (x-2 e^x-1\right ) (x+\log (4))+2 \left (2 e^x+1\right ) x (2 x+1+\log (4)) \log \left (\left (e^{-x}+2\right ) x\right )\right )}{\left (2 e^x x+x\right ) \log ^2\left (e^{-x} x+2 x\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {(x+1) \left (-x^2+4 x^2 \log \left (\left (e^{-x}+2\right ) x\right )+2 x (1+\log (4)) \log \left (\left (e^{-x}+2\right ) x\right )-x (1+\log (4))-\log (4)\right ) (x+\log (4))}{x \log ^2\left (e^{-x} x+2 x\right )}+\frac {(x+1)^2 (x+\log (4))^2}{\left (2 e^x+1\right ) \log ^2\left (e^{-x} x+2 x\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \int \frac {x^4}{\left (1+2 e^x\right ) \log ^2\left (e^{-x} x+2 x\right )}dx-\int \frac {x^3}{\log ^2\left (e^{-x} x+2 x\right )}dx+2 (1+\log (4)) \int \frac {x^3}{\left (1+2 e^x\right ) \log ^2\left (e^{-x} x+2 x\right )}dx+4 \int \frac {x^3}{\log \left (e^{-x} x+2 x\right )}dx-2 (1+\log (4)) \int \frac {x^2}{\log ^2\left (e^{-x} x+2 x\right )}dx+\left (1+\log ^2(4)+\log (256)\right ) \int \frac {x^2}{\left (1+2 e^x\right ) \log ^2\left (e^{-x} x+2 x\right )}dx+6 (1+\log (4)) \int \frac {x^2}{\log \left (e^{-x} x+2 x\right )}dx-2 \log (4) (1+\log (4)) \int \frac {1}{\log ^2\left (e^{-x} x+2 x\right )}dx+\log ^2(4) \int \frac {1}{\left (1+2 e^x\right ) \log ^2\left (e^{-x} x+2 x\right )}dx-\log ^2(4) \int \frac {1}{x \log ^2\left (e^{-x} x+2 x\right )}dx-(1+\log (4))^2 \int \frac {x}{\log ^2\left (e^{-x} x+2 x\right )}dx-2 \log (4) \int \frac {x}{\log ^2\left (e^{-x} x+2 x\right )}dx+2 \log (4) (1+\log (4)) \int \frac {x}{\left (1+2 e^x\right ) \log ^2\left (e^{-x} x+2 x\right )}dx+2 \log (4) (1+\log (4)) \int \frac {1}{\log \left (e^{-x} x+2 x\right )}dx+2 (1+\log (4))^2 \int \frac {x}{\log \left (e^{-x} x+2 x\right )}dx+4 \log (4) \int \frac {x}{\log \left (e^{-x} x+2 x\right )}dx\) |
Int[(-x^2 - x^3 + x^4 + x^5 + (-2*x - 2*x^2 + 2*x^3 + 2*x^4)*Log[4] + (-1 - x + x^2 + x^3)*Log[4]^2 + E^x*(-2*x^2 - 4*x^3 - 2*x^4 + (-4*x - 8*x^2 - 4*x^3)*Log[4] + (-2 - 4*x - 2*x^2)*Log[4]^2) + (2*x^2 + 6*x^3 + 4*x^4 + (2 *x + 8*x^2 + 6*x^3)*Log[4] + (2*x + 2*x^2)*Log[4]^2 + E^x*(4*x^2 + 12*x^3 + 8*x^4 + (4*x + 16*x^2 + 12*x^3)*Log[4] + (4*x + 4*x^2)*Log[4]^2))*Log[(x + 2*E^x*x)/E^x])/((x + 2*E^x*x)*Log[(x + 2*E^x*x)/E^x]^2),x]
3.11.73.3.1 Defintions of rubi rules used
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Leaf count of result is larger than twice the leaf count of optimal. \(74\) vs. \(2(27)=54\).
Time = 1.57 (sec) , antiderivative size = 75, normalized size of antiderivative = 2.88
| method | result | size |
| parallelrisch | \(\frac {16 x^{2} \ln \left (2\right )^{2}+16 x^{3} \ln \left (2\right )+4 x^{4}+32 x \ln \left (2\right )^{2}+32 x^{2} \ln \left (2\right )+8 x^{3}+16 \ln \left (2\right )^{2}+16 x \ln \left (2\right )+4 x^{2}}{4 \ln \left (x \left (2 \,{\mathrm e}^{x}+1\right ) {\mathrm e}^{-x}\right )}\) | \(75\) |
| risch | \(\frac {8 x^{2} \ln \left (2\right )^{2}+8 x^{3} \ln \left (2\right )+2 x^{4}+16 x \ln \left (2\right )^{2}+16 x^{2} \ln \left (2\right )+4 x^{3}+8 \ln \left (2\right )^{2}+8 x \ln \left (2\right )+2 x^{2}}{2 \ln \left (2\right )+2 \ln \left (x \right )-2 \ln \left ({\mathrm e}^{x}\right )+2 \ln \left (\frac {1}{2}+{\mathrm e}^{x}\right )-i \pi \operatorname {csgn}\left (i {\mathrm e}^{-x} \left (\frac {1}{2}+{\mathrm e}^{x}\right )\right )^{3}-i \pi {\operatorname {csgn}\left (i x \left (\frac {1}{2}+{\mathrm e}^{x}\right ) {\mathrm e}^{-x}\right )}^{3}+i \pi \operatorname {csgn}\left (i {\mathrm e}^{-x} \left (\frac {1}{2}+{\mathrm e}^{x}\right )\right )^{2} \operatorname {csgn}\left (i {\mathrm e}^{-x}\right )-i \pi \,\operatorname {csgn}\left (i \left (\frac {1}{2}+{\mathrm e}^{x}\right )\right ) \operatorname {csgn}\left (i {\mathrm e}^{-x} \left (\frac {1}{2}+{\mathrm e}^{x}\right )\right ) \operatorname {csgn}\left (i {\mathrm e}^{-x}\right )-i \pi \,\operatorname {csgn}\left (i {\mathrm e}^{-x} \left (\frac {1}{2}+{\mathrm e}^{x}\right )\right ) \operatorname {csgn}\left (i x \left (\frac {1}{2}+{\mathrm e}^{x}\right ) {\mathrm e}^{-x}\right ) \operatorname {csgn}\left (i x \right )+i \pi {\operatorname {csgn}\left (i x \left (\frac {1}{2}+{\mathrm e}^{x}\right ) {\mathrm e}^{-x}\right )}^{2} \operatorname {csgn}\left (i x \right )+i \pi \,\operatorname {csgn}\left (i {\mathrm e}^{-x} \left (\frac {1}{2}+{\mathrm e}^{x}\right )\right ) {\operatorname {csgn}\left (i x \left (\frac {1}{2}+{\mathrm e}^{x}\right ) {\mathrm e}^{-x}\right )}^{2}+i \pi \,\operatorname {csgn}\left (i \left (\frac {1}{2}+{\mathrm e}^{x}\right )\right ) \operatorname {csgn}\left (i {\mathrm e}^{-x} \left (\frac {1}{2}+{\mathrm e}^{x}\right )\right )^{2}}\) | \(289\) |
int((((4*(4*x^2+4*x)*ln(2)^2+2*(12*x^3+16*x^2+4*x)*ln(2)+8*x^4+12*x^3+4*x^ 2)*exp(x)+4*(2*x^2+2*x)*ln(2)^2+2*(6*x^3+8*x^2+2*x)*ln(2)+4*x^4+6*x^3+2*x^ 2)*ln((2*exp(x)*x+x)/exp(x))+(4*(-2*x^2-4*x-2)*ln(2)^2+2*(-4*x^3-8*x^2-4*x )*ln(2)-2*x^4-4*x^3-2*x^2)*exp(x)+4*(x^3+x^2-x-1)*ln(2)^2+2*(2*x^4+2*x^3-2 *x^2-2*x)*ln(2)+x^5+x^4-x^3-x^2)/(2*exp(x)*x+x)/ln((2*exp(x)*x+x)/exp(x))^ 2,x,method=_RETURNVERBOSE)
1/4*(16*x^2*ln(2)^2+16*x^3*ln(2)+4*x^4+32*x*ln(2)^2+32*x^2*ln(2)+8*x^3+16* ln(2)^2+16*x*ln(2)+4*x^2)/ln(x*(2*exp(x)+1)/exp(x))
Leaf count of result is larger than twice the leaf count of optimal. 56 vs. \(2 (27) = 54\).
Time = 0.26 (sec) , antiderivative size = 56, normalized size of antiderivative = 2.15 \begin {dmath*} \int \frac {-x^2-x^3+x^4+x^5+\left (-2 x-2 x^2+2 x^3+2 x^4\right ) \log (4)+\left (-1-x+x^2+x^3\right ) \log ^2(4)+e^x \left (-2 x^2-4 x^3-2 x^4+\left (-4 x-8 x^2-4 x^3\right ) \log (4)+\left (-2-4 x-2 x^2\right ) \log ^2(4)\right )+\left (2 x^2+6 x^3+4 x^4+\left (2 x+8 x^2+6 x^3\right ) \log (4)+\left (2 x+2 x^2\right ) \log ^2(4)+e^x \left (4 x^2+12 x^3+8 x^4+\left (4 x+16 x^2+12 x^3\right ) \log (4)+\left (4 x+4 x^2\right ) \log ^2(4)\right )\right ) \log \left (e^{-x} \left (x+2 e^x x\right )\right )}{\left (x+2 e^x x\right ) \log ^2\left (e^{-x} \left (x+2 e^x x\right )\right )} \, dx=\frac {x^{4} + 2 \, x^{3} + 4 \, {\left (x^{2} + 2 \, x + 1\right )} \log \left (2\right )^{2} + x^{2} + 4 \, {\left (x^{3} + 2 \, x^{2} + x\right )} \log \left (2\right )}{\log \left ({\left (2 \, x e^{x} + x\right )} e^{\left (-x\right )}\right )} \end {dmath*}
integrate((((4*(4*x^2+4*x)*log(2)^2+2*(12*x^3+16*x^2+4*x)*log(2)+8*x^4+12* x^3+4*x^2)*exp(x)+4*(2*x^2+2*x)*log(2)^2+2*(6*x^3+8*x^2+2*x)*log(2)+4*x^4+ 6*x^3+2*x^2)*log((2*exp(x)*x+x)/exp(x))+(4*(-2*x^2-4*x-2)*log(2)^2+2*(-4*x ^3-8*x^2-4*x)*log(2)-2*x^4-4*x^3-2*x^2)*exp(x)+4*(x^3+x^2-x-1)*log(2)^2+2* (2*x^4+2*x^3-2*x^2-2*x)*log(2)+x^5+x^4-x^3-x^2)/(2*exp(x)*x+x)/log((2*exp( x)*x+x)/exp(x))^2,x, algorithm=\
(x^4 + 2*x^3 + 4*(x^2 + 2*x + 1)*log(2)^2 + x^2 + 4*(x^3 + 2*x^2 + x)*log( 2))/log((2*x*e^x + x)*e^(-x))
Leaf count of result is larger than twice the leaf count of optimal. 73 vs. \(2 (22) = 44\).
Time = 0.14 (sec) , antiderivative size = 73, normalized size of antiderivative = 2.81 \begin {dmath*} \int \frac {-x^2-x^3+x^4+x^5+\left (-2 x-2 x^2+2 x^3+2 x^4\right ) \log (4)+\left (-1-x+x^2+x^3\right ) \log ^2(4)+e^x \left (-2 x^2-4 x^3-2 x^4+\left (-4 x-8 x^2-4 x^3\right ) \log (4)+\left (-2-4 x-2 x^2\right ) \log ^2(4)\right )+\left (2 x^2+6 x^3+4 x^4+\left (2 x+8 x^2+6 x^3\right ) \log (4)+\left (2 x+2 x^2\right ) \log ^2(4)+e^x \left (4 x^2+12 x^3+8 x^4+\left (4 x+16 x^2+12 x^3\right ) \log (4)+\left (4 x+4 x^2\right ) \log ^2(4)\right )\right ) \log \left (e^{-x} \left (x+2 e^x x\right )\right )}{\left (x+2 e^x x\right ) \log ^2\left (e^{-x} \left (x+2 e^x x\right )\right )} \, dx=\frac {x^{4} + 2 x^{3} + 4 x^{3} \log {\left (2 \right )} + x^{2} + 4 x^{2} \log {\left (2 \right )}^{2} + 8 x^{2} \log {\left (2 \right )} + 4 x \log {\left (2 \right )} + 8 x \log {\left (2 \right )}^{2} + 4 \log {\left (2 \right )}^{2}}{\log {\left (\left (2 x e^{x} + x\right ) e^{- x} \right )}} \end {dmath*}
integrate((((4*(4*x**2+4*x)*ln(2)**2+2*(12*x**3+16*x**2+4*x)*ln(2)+8*x**4+ 12*x**3+4*x**2)*exp(x)+4*(2*x**2+2*x)*ln(2)**2+2*(6*x**3+8*x**2+2*x)*ln(2) +4*x**4+6*x**3+2*x**2)*ln((2*exp(x)*x+x)/exp(x))+(4*(-2*x**2-4*x-2)*ln(2)* *2+2*(-4*x**3-8*x**2-4*x)*ln(2)-2*x**4-4*x**3-2*x**2)*exp(x)+4*(x**3+x**2- x-1)*ln(2)**2+2*(2*x**4+2*x**3-2*x**2-2*x)*ln(2)+x**5+x**4-x**3-x**2)/(2*e xp(x)*x+x)/ln((2*exp(x)*x+x)/exp(x))**2,x)
(x**4 + 2*x**3 + 4*x**3*log(2) + x**2 + 4*x**2*log(2)**2 + 8*x**2*log(2) + 4*x*log(2) + 8*x*log(2)**2 + 4*log(2)**2)/log((2*x*exp(x) + x)*exp(-x))
Leaf count of result is larger than twice the leaf count of optimal. 68 vs. \(2 (27) = 54\).
Time = 0.34 (sec) , antiderivative size = 68, normalized size of antiderivative = 2.62 \begin {dmath*} \int \frac {-x^2-x^3+x^4+x^5+\left (-2 x-2 x^2+2 x^3+2 x^4\right ) \log (4)+\left (-1-x+x^2+x^3\right ) \log ^2(4)+e^x \left (-2 x^2-4 x^3-2 x^4+\left (-4 x-8 x^2-4 x^3\right ) \log (4)+\left (-2-4 x-2 x^2\right ) \log ^2(4)\right )+\left (2 x^2+6 x^3+4 x^4+\left (2 x+8 x^2+6 x^3\right ) \log (4)+\left (2 x+2 x^2\right ) \log ^2(4)+e^x \left (4 x^2+12 x^3+8 x^4+\left (4 x+16 x^2+12 x^3\right ) \log (4)+\left (4 x+4 x^2\right ) \log ^2(4)\right )\right ) \log \left (e^{-x} \left (x+2 e^x x\right )\right )}{\left (x+2 e^x x\right ) \log ^2\left (e^{-x} \left (x+2 e^x x\right )\right )} \, dx=-\frac {x^{4} + 2 \, x^{3} {\left (2 \, \log \left (2\right ) + 1\right )} + {\left (4 \, \log \left (2\right )^{2} + 8 \, \log \left (2\right ) + 1\right )} x^{2} + 4 \, {\left (2 \, \log \left (2\right )^{2} + \log \left (2\right )\right )} x + 4 \, \log \left (2\right )^{2}}{x - \log \left (x\right ) - \log \left (2 \, e^{x} + 1\right )} \end {dmath*}
integrate((((4*(4*x^2+4*x)*log(2)^2+2*(12*x^3+16*x^2+4*x)*log(2)+8*x^4+12* x^3+4*x^2)*exp(x)+4*(2*x^2+2*x)*log(2)^2+2*(6*x^3+8*x^2+2*x)*log(2)+4*x^4+ 6*x^3+2*x^2)*log((2*exp(x)*x+x)/exp(x))+(4*(-2*x^2-4*x-2)*log(2)^2+2*(-4*x ^3-8*x^2-4*x)*log(2)-2*x^4-4*x^3-2*x^2)*exp(x)+4*(x^3+x^2-x-1)*log(2)^2+2* (2*x^4+2*x^3-2*x^2-2*x)*log(2)+x^5+x^4-x^3-x^2)/(2*exp(x)*x+x)/log((2*exp( x)*x+x)/exp(x))^2,x, algorithm=\
-(x^4 + 2*x^3*(2*log(2) + 1) + (4*log(2)^2 + 8*log(2) + 1)*x^2 + 4*(2*log( 2)^2 + log(2))*x + 4*log(2)^2)/(x - log(x) - log(2*e^x + 1))
Leaf count of result is larger than twice the leaf count of optimal. 69 vs. \(2 (27) = 54\).
Time = 0.37 (sec) , antiderivative size = 69, normalized size of antiderivative = 2.65 \begin {dmath*} \int \frac {-x^2-x^3+x^4+x^5+\left (-2 x-2 x^2+2 x^3+2 x^4\right ) \log (4)+\left (-1-x+x^2+x^3\right ) \log ^2(4)+e^x \left (-2 x^2-4 x^3-2 x^4+\left (-4 x-8 x^2-4 x^3\right ) \log (4)+\left (-2-4 x-2 x^2\right ) \log ^2(4)\right )+\left (2 x^2+6 x^3+4 x^4+\left (2 x+8 x^2+6 x^3\right ) \log (4)+\left (2 x+2 x^2\right ) \log ^2(4)+e^x \left (4 x^2+12 x^3+8 x^4+\left (4 x+16 x^2+12 x^3\right ) \log (4)+\left (4 x+4 x^2\right ) \log ^2(4)\right )\right ) \log \left (e^{-x} \left (x+2 e^x x\right )\right )}{\left (x+2 e^x x\right ) \log ^2\left (e^{-x} \left (x+2 e^x x\right )\right )} \, dx=\frac {x^{4} + 4 \, x^{3} \log \left (2\right ) + 4 \, x^{2} \log \left (2\right )^{2} + 2 \, x^{3} + 8 \, x^{2} \log \left (2\right ) + 8 \, x \log \left (2\right )^{2} + x^{2} + 4 \, x \log \left (2\right ) + 4 \, \log \left (2\right )^{2}}{\log \left ({\left (2 \, x e^{x} + x\right )} e^{\left (-x\right )}\right )} \end {dmath*}
integrate((((4*(4*x^2+4*x)*log(2)^2+2*(12*x^3+16*x^2+4*x)*log(2)+8*x^4+12* x^3+4*x^2)*exp(x)+4*(2*x^2+2*x)*log(2)^2+2*(6*x^3+8*x^2+2*x)*log(2)+4*x^4+ 6*x^3+2*x^2)*log((2*exp(x)*x+x)/exp(x))+(4*(-2*x^2-4*x-2)*log(2)^2+2*(-4*x ^3-8*x^2-4*x)*log(2)-2*x^4-4*x^3-2*x^2)*exp(x)+4*(x^3+x^2-x-1)*log(2)^2+2* (2*x^4+2*x^3-2*x^2-2*x)*log(2)+x^5+x^4-x^3-x^2)/(2*exp(x)*x+x)/log((2*exp( x)*x+x)/exp(x))^2,x, algorithm=\
(x^4 + 4*x^3*log(2) + 4*x^2*log(2)^2 + 2*x^3 + 8*x^2*log(2) + 8*x*log(2)^2 + x^2 + 4*x*log(2) + 4*log(2)^2)/log((2*x*e^x + x)*e^(-x))
Time = 19.19 (sec) , antiderivative size = 259, normalized size of antiderivative = 9.96 \begin {dmath*} \int \frac {-x^2-x^3+x^4+x^5+\left (-2 x-2 x^2+2 x^3+2 x^4\right ) \log (4)+\left (-1-x+x^2+x^3\right ) \log ^2(4)+e^x \left (-2 x^2-4 x^3-2 x^4+\left (-4 x-8 x^2-4 x^3\right ) \log (4)+\left (-2-4 x-2 x^2\right ) \log ^2(4)\right )+\left (2 x^2+6 x^3+4 x^4+\left (2 x+8 x^2+6 x^3\right ) \log (4)+\left (2 x+2 x^2\right ) \log ^2(4)+e^x \left (4 x^2+12 x^3+8 x^4+\left (4 x+16 x^2+12 x^3\right ) \log (4)+\left (4 x+4 x^2\right ) \log ^2(4)\right )\right ) \log \left (e^{-x} \left (x+2 e^x x\right )\right )}{\left (x+2 e^x x\right ) \log ^2\left (e^{-x} \left (x+2 e^x x\right )\right )} \, dx=x^3\,\left (12\,\ln \left (2\right )+6\right )+x^2\,\left (16\,\ln \left (2\right )+8\,{\ln \left (2\right )}^2+2\right )+\frac {4\,x^2\,{\ln \left (2\right )}^2+4\,x\,\ln \left (2\right )+8\,x\,{\ln \left (2\right )}^2+8\,x^2\,\ln \left (2\right )+4\,x^3\,\ln \left (2\right )+4\,{\ln \left (2\right )}^2+x^2+2\,x^3+x^4-\frac {2\,x\,\ln \left ({\mathrm {e}}^{-x}\,\left (x+2\,x\,{\mathrm {e}}^x\right )\right )\,\left (2\,{\mathrm {e}}^x+1\right )\,\left (x+1\right )\,\left (x+\ln \left (4\right )+x\,\ln \left (64\right )+4\,{\ln \left (2\right )}^2+2\,x^2\right )}{2\,{\mathrm {e}}^x-x+1}}{\ln \left ({\mathrm {e}}^{-x}\,\left (x+2\,x\,{\mathrm {e}}^x\right )\right )}+x\,\left (\ln \left (16\right )+8\,{\ln \left (2\right )}^2\right )+4\,x^4-\frac {2\,\left (8\,x^2\,{\ln \left (2\right )}^2+4\,x^3\,{\ln \left (2\right )}^2-4\,x^4\,{\ln \left (2\right )}^2+4\,x^2\,\ln \left (2\right )+14\,x^3\,\ln \left (2\right )+4\,x^4\,\ln \left (2\right )-6\,x^5\,\ln \left (2\right )+2\,x^3+5\,x^4+x^5-2\,x^6\right )}{\left (x-2\right )\,\left (2\,{\mathrm {e}}^x-x+1\right )} \end {dmath*}
int(-(exp(x)*(2*log(2)*(4*x + 8*x^2 + 4*x^3) + 4*log(2)^2*(4*x + 2*x^2 + 2 ) + 2*x^2 + 4*x^3 + 2*x^4) - log(exp(-x)*(x + 2*x*exp(x)))*(exp(x)*(2*log( 2)*(4*x + 16*x^2 + 12*x^3) + 4*log(2)^2*(4*x + 4*x^2) + 4*x^2 + 12*x^3 + 8 *x^4) + 2*log(2)*(2*x + 8*x^2 + 6*x^3) + 4*log(2)^2*(2*x + 2*x^2) + 2*x^2 + 6*x^3 + 4*x^4) + 4*log(2)^2*(x - x^2 - x^3 + 1) + 2*log(2)*(2*x + 2*x^2 - 2*x^3 - 2*x^4) + x^2 + x^3 - x^4 - x^5)/(log(exp(-x)*(x + 2*x*exp(x)))^2 *(x + 2*x*exp(x))),x)
x^3*(12*log(2) + 6) + x^2*(16*log(2) + 8*log(2)^2 + 2) + (4*x^2*log(2)^2 + 4*x*log(2) + 8*x*log(2)^2 + 8*x^2*log(2) + 4*x^3*log(2) + 4*log(2)^2 + x^ 2 + 2*x^3 + x^4 - (2*x*log(exp(-x)*(x + 2*x*exp(x)))*(2*exp(x) + 1)*(x + 1 )*(x + log(4) + x*log(64) + 4*log(2)^2 + 2*x^2))/(2*exp(x) - x + 1))/log(e xp(-x)*(x + 2*x*exp(x))) + x*(log(16) + 8*log(2)^2) + 4*x^4 - (2*(8*x^2*lo g(2)^2 + 4*x^3*log(2)^2 - 4*x^4*log(2)^2 + 4*x^2*log(2) + 14*x^3*log(2) + 4*x^4*log(2) - 6*x^5*log(2) + 2*x^3 + 5*x^4 + x^5 - 2*x^6))/((x - 2)*(2*ex p(x) - x + 1))