Integrand size = 404, antiderivative size = 29 \[ \int \frac {30-20 e^4+e^{2 x} \left (10-40 e^4\right )+10 x+e^{3 x} \left (10-40 e^4-x^3\right )+e^x \left (30-20 e^4+10 x-3 x^3-x^4\right )+\left (6 e^{4+3 x} x^2+e^{4+x} \left (18 x^2+6 x^3\right )\right ) \log \left (3+e^{2 x}+x\right )+\left (-12 e^{8+3 x} x+e^{8+x} \left (-36 x-12 x^2\right )\right ) \log ^2\left (3+e^{2 x}+x\right )+\left (8 e^{12+3 x}+e^{12+x} (24+8 x)\right ) \log ^3\left (3+e^{2 x}+x\right )}{-3 x^3-e^{2 x} x^3-e^{3 x} x^3-x^4+e^x \left (-3 x^3-x^4\right )+\left (6 e^{4+2 x} x^2+6 e^{4+3 x} x^2+e^4 \left (18 x^2+6 x^3\right )+e^{4+x} \left (18 x^2+6 x^3\right )\right ) \log \left (3+e^{2 x}+x\right )+\left (-12 e^{8+2 x} x-12 e^{8+3 x} x+e^8 \left (-36 x-12 x^2\right )+e^{8+x} \left (-36 x-12 x^2\right )\right ) \log ^2\left (3+e^{2 x}+x\right )+\left (8 e^{12+2 x}+8 e^{12+3 x}+e^{12} (24+8 x)+e^{12+x} (24+8 x)\right ) \log ^3\left (3+e^{2 x}+x\right )} \, dx=\log \left (1+e^x\right )+\frac {5}{\left (-x+2 e^4 \log \left (3+e^{2 x}+x\right )\right )^2} \]
Time = 0.16 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00 \[ \int \frac {30-20 e^4+e^{2 x} \left (10-40 e^4\right )+10 x+e^{3 x} \left (10-40 e^4-x^3\right )+e^x \left (30-20 e^4+10 x-3 x^3-x^4\right )+\left (6 e^{4+3 x} x^2+e^{4+x} \left (18 x^2+6 x^3\right )\right ) \log \left (3+e^{2 x}+x\right )+\left (-12 e^{8+3 x} x+e^{8+x} \left (-36 x-12 x^2\right )\right ) \log ^2\left (3+e^{2 x}+x\right )+\left (8 e^{12+3 x}+e^{12+x} (24+8 x)\right ) \log ^3\left (3+e^{2 x}+x\right )}{-3 x^3-e^{2 x} x^3-e^{3 x} x^3-x^4+e^x \left (-3 x^3-x^4\right )+\left (6 e^{4+2 x} x^2+6 e^{4+3 x} x^2+e^4 \left (18 x^2+6 x^3\right )+e^{4+x} \left (18 x^2+6 x^3\right )\right ) \log \left (3+e^{2 x}+x\right )+\left (-12 e^{8+2 x} x-12 e^{8+3 x} x+e^8 \left (-36 x-12 x^2\right )+e^{8+x} \left (-36 x-12 x^2\right )\right ) \log ^2\left (3+e^{2 x}+x\right )+\left (8 e^{12+2 x}+8 e^{12+3 x}+e^{12} (24+8 x)+e^{12+x} (24+8 x)\right ) \log ^3\left (3+e^{2 x}+x\right )} \, dx=\log \left (1+e^x\right )+\frac {5}{\left (-x+2 e^4 \log \left (3+e^{2 x}+x\right )\right )^2} \]
Integrate[(30 - 20*E^4 + E^(2*x)*(10 - 40*E^4) + 10*x + E^(3*x)*(10 - 40*E ^4 - x^3) + E^x*(30 - 20*E^4 + 10*x - 3*x^3 - x^4) + (6*E^(4 + 3*x)*x^2 + E^(4 + x)*(18*x^2 + 6*x^3))*Log[3 + E^(2*x) + x] + (-12*E^(8 + 3*x)*x + E^ (8 + x)*(-36*x - 12*x^2))*Log[3 + E^(2*x) + x]^2 + (8*E^(12 + 3*x) + E^(12 + x)*(24 + 8*x))*Log[3 + E^(2*x) + x]^3)/(-3*x^3 - E^(2*x)*x^3 - E^(3*x)* x^3 - x^4 + E^x*(-3*x^3 - x^4) + (6*E^(4 + 2*x)*x^2 + 6*E^(4 + 3*x)*x^2 + E^4*(18*x^2 + 6*x^3) + E^(4 + x)*(18*x^2 + 6*x^3))*Log[3 + E^(2*x) + x] + (-12*E^(8 + 2*x)*x - 12*E^(8 + 3*x)*x + E^8*(-36*x - 12*x^2) + E^(8 + x)*( -36*x - 12*x^2))*Log[3 + E^(2*x) + x]^2 + (8*E^(12 + 2*x) + 8*E^(12 + 3*x) + E^12*(24 + 8*x) + E^(12 + x)*(24 + 8*x))*Log[3 + E^(2*x) + x]^3),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.
\(\displaystyle \int \frac {e^{3 x} \left (-x^3-40 e^4+10\right )+\left (e^{x+8} \left (-12 x^2-36 x\right )-12 e^{3 x+8} x\right ) \log ^2\left (x+e^{2 x}+3\right )+e^x \left (-x^4-3 x^3+10 x-20 e^4+30\right )+\left (6 e^{3 x+4} x^2+e^{x+4} \left (6 x^3+18 x^2\right )\right ) \log \left (x+e^{2 x}+3\right )+10 x+\left (10-40 e^4\right ) e^{2 x}+\left (e^{x+12} (8 x+24)+8 e^{3 x+12}\right ) \log ^3\left (x+e^{2 x}+3\right )-20 e^4+30}{-x^4-e^{2 x} x^3-e^{3 x} x^3-3 x^3+\left (e^{x+8} \left (-12 x^2-36 x\right )+e^8 \left (-12 x^2-36 x\right )-12 e^{2 x+8} x-12 e^{3 x+8} x\right ) \log ^2\left (x+e^{2 x}+3\right )+e^x \left (-x^4-3 x^3\right )+\left (6 e^{2 x+4} x^2+6 e^{3 x+4} x^2+e^{x+4} \left (6 x^3+18 x^2\right )+e^4 \left (6 x^3+18 x^2\right )\right ) \log \left (x+e^{2 x}+3\right )+\left (e^{x+12} (8 x+24)+e^{12} (8 x+24)+8 e^{2 x+12}+8 e^{3 x+12}\right ) \log ^3\left (x+e^{2 x}+3\right )} \, dx\) |
\(\Big \downarrow \) 7239 |
\(\displaystyle \int \frac {e^{3 x} \left (x^3-10\right )-6 e^{x+4} x^2 \left (x+e^{2 x}+3\right ) \log \left (x+e^{2 x}+3\right )+e^x \left (x^4+3 x^3-10 x-30\right )+20 e^{x+4}+40 e^{3 x+4}-10 (x+3)-10 \left (1-4 e^4\right ) e^{2 x}-8 e^{x+12} \left (x+e^{2 x}+3\right ) \log ^3\left (x+e^{2 x}+3\right )+12 e^{x+8} x \left (x+e^{2 x}+3\right ) \log ^2\left (x+e^{2 x}+3\right )+20 e^4}{\left (e^x+1\right ) \left (x+e^{2 x}+3\right ) \left (x-2 e^4 \log \left (x+e^{2 x}+3\right )\right )^3}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {x^3-6 e^4 x^2 \log \left (x+e^{2 x}+3\right )-8 e^{12} \log ^3\left (x+e^{2 x}+3\right )+12 e^8 x \log ^2\left (x+e^{2 x}+3\right )-10 \left (1-4 e^4\right )}{\left (x-2 e^4 \log \left (x+e^{2 x}+3\right )\right )^3}-\frac {1}{e^x+1}+\frac {20 e^4 (2 x+5)}{\left (x+e^{2 x}+3\right ) \left (2 e^4 \log \left (x+e^{2 x}+3\right )-x\right )^3}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -10 \left (1-4 e^4\right ) \int \frac {1}{\left (x-2 e^4 \log \left (x+e^{2 x}+3\right )\right )^3}dx-100 e^4 \int \frac {1}{\left (x+e^{2 x}+3\right ) \left (x-2 e^4 \log \left (x+e^{2 x}+3\right )\right )^3}dx-40 e^4 \int \frac {x}{\left (x+e^{2 x}+3\right ) \left (x-2 e^4 \log \left (x+e^{2 x}+3\right )\right )^3}dx+\log \left (e^x+1\right )\) |
Int[(30 - 20*E^4 + E^(2*x)*(10 - 40*E^4) + 10*x + E^(3*x)*(10 - 40*E^4 - x ^3) + E^x*(30 - 20*E^4 + 10*x - 3*x^3 - x^4) + (6*E^(4 + 3*x)*x^2 + E^(4 + x)*(18*x^2 + 6*x^3))*Log[3 + E^(2*x) + x] + (-12*E^(8 + 3*x)*x + E^(8 + x )*(-36*x - 12*x^2))*Log[3 + E^(2*x) + x]^2 + (8*E^(12 + 3*x) + E^(12 + x)* (24 + 8*x))*Log[3 + E^(2*x) + x]^3)/(-3*x^3 - E^(2*x)*x^3 - E^(3*x)*x^3 - x^4 + E^x*(-3*x^3 - x^4) + (6*E^(4 + 2*x)*x^2 + 6*E^(4 + 3*x)*x^2 + E^4*(1 8*x^2 + 6*x^3) + E^(4 + x)*(18*x^2 + 6*x^3))*Log[3 + E^(2*x) + x] + (-12*E ^(8 + 2*x)*x - 12*E^(8 + 3*x)*x + E^8*(-36*x - 12*x^2) + E^(8 + x)*(-36*x - 12*x^2))*Log[3 + E^(2*x) + x]^2 + (8*E^(12 + 2*x) + 8*E^(12 + 3*x) + E^1 2*(24 + 8*x) + E^(12 + x)*(24 + 8*x))*Log[3 + E^(2*x) + x]^3),x]
3.8.2.3.1 Defintions of rubi rules used
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Time = 6.47 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.93
method | result | size |
risch | \(\frac {5}{{\left (2 \ln \left ({\mathrm e}^{2 x}+3+x \right ) {\mathrm e}^{4}-x \right )}^{2}}+\ln \left ({\mathrm e}^{x}+1\right )\) | \(27\) |
parallelrisch | \(\frac {4 \,{\mathrm e}^{8} \ln \left ({\mathrm e}^{x}+1\right ) \ln \left ({\mathrm e}^{2 x}+3+x \right )^{2}-4 \,{\mathrm e}^{4} \ln \left ({\mathrm e}^{x}+1\right ) \ln \left ({\mathrm e}^{2 x}+3+x \right ) x +\ln \left ({\mathrm e}^{x}+1\right ) x^{2}+5}{4 \,{\mathrm e}^{8} \ln \left ({\mathrm e}^{2 x}+3+x \right )^{2}-4 \,{\mathrm e}^{4} \ln \left ({\mathrm e}^{2 x}+3+x \right ) x +x^{2}}\) | \(87\) |
int(((8*exp(4)^3*exp(x)^3+(8*x+24)*exp(4)^3*exp(x))*ln(exp(x)^2+3+x)^3+(-1 2*x*exp(4)^2*exp(x)^3+(-12*x^2-36*x)*exp(4)^2*exp(x))*ln(exp(x)^2+3+x)^2+( 6*x^2*exp(4)*exp(x)^3+(6*x^3+18*x^2)*exp(4)*exp(x))*ln(exp(x)^2+3+x)+(-40* exp(4)-x^3+10)*exp(x)^3+(-40*exp(4)+10)*exp(x)^2+(-20*exp(4)-x^4-3*x^3+10* x+30)*exp(x)-20*exp(4)+10*x+30)/((8*exp(4)^3*exp(x)^3+8*exp(4)^3*exp(x)^2+ (8*x+24)*exp(4)^3*exp(x)+(8*x+24)*exp(4)^3)*ln(exp(x)^2+3+x)^3+(-12*x*exp( 4)^2*exp(x)^3-12*x*exp(4)^2*exp(x)^2+(-12*x^2-36*x)*exp(4)^2*exp(x)+(-12*x ^2-36*x)*exp(4)^2)*ln(exp(x)^2+3+x)^2+(6*x^2*exp(4)*exp(x)^3+6*x^2*exp(4)* exp(x)^2+(6*x^3+18*x^2)*exp(4)*exp(x)+(6*x^3+18*x^2)*exp(4))*ln(exp(x)^2+3 +x)-x^3*exp(x)^3-exp(x)^2*x^3+(-x^4-3*x^3)*exp(x)-x^4-3*x^3),x,method=_RET URNVERBOSE)
Leaf count of result is larger than twice the leaf count of optimal. 130 vs. \(2 (26) = 52\).
Time = 0.26 (sec) , antiderivative size = 130, normalized size of antiderivative = 4.48 \[ \int \frac {30-20 e^4+e^{2 x} \left (10-40 e^4\right )+10 x+e^{3 x} \left (10-40 e^4-x^3\right )+e^x \left (30-20 e^4+10 x-3 x^3-x^4\right )+\left (6 e^{4+3 x} x^2+e^{4+x} \left (18 x^2+6 x^3\right )\right ) \log \left (3+e^{2 x}+x\right )+\left (-12 e^{8+3 x} x+e^{8+x} \left (-36 x-12 x^2\right )\right ) \log ^2\left (3+e^{2 x}+x\right )+\left (8 e^{12+3 x}+e^{12+x} (24+8 x)\right ) \log ^3\left (3+e^{2 x}+x\right )}{-3 x^3-e^{2 x} x^3-e^{3 x} x^3-x^4+e^x \left (-3 x^3-x^4\right )+\left (6 e^{4+2 x} x^2+6 e^{4+3 x} x^2+e^4 \left (18 x^2+6 x^3\right )+e^{4+x} \left (18 x^2+6 x^3\right )\right ) \log \left (3+e^{2 x}+x\right )+\left (-12 e^{8+2 x} x-12 e^{8+3 x} x+e^8 \left (-36 x-12 x^2\right )+e^{8+x} \left (-36 x-12 x^2\right )\right ) \log ^2\left (3+e^{2 x}+x\right )+\left (8 e^{12+2 x}+8 e^{12+3 x}+e^{12} (24+8 x)+e^{12+x} (24+8 x)\right ) \log ^3\left (3+e^{2 x}+x\right )} \, dx=\frac {4 \, x e^{4} \log \left ({\left ({\left (x + 3\right )} e^{24} + e^{\left (2 \, x + 24\right )}\right )} e^{\left (-24\right )}\right ) \log \left (e^{12} + e^{\left (x + 12\right )}\right ) - 4 \, e^{8} \log \left ({\left ({\left (x + 3\right )} e^{24} + e^{\left (2 \, x + 24\right )}\right )} e^{\left (-24\right )}\right )^{2} \log \left (e^{12} + e^{\left (x + 12\right )}\right ) - x^{2} \log \left (e^{12} + e^{\left (x + 12\right )}\right ) - 5}{4 \, x e^{4} \log \left ({\left ({\left (x + 3\right )} e^{24} + e^{\left (2 \, x + 24\right )}\right )} e^{\left (-24\right )}\right ) - 4 \, e^{8} \log \left ({\left ({\left (x + 3\right )} e^{24} + e^{\left (2 \, x + 24\right )}\right )} e^{\left (-24\right )}\right )^{2} - x^{2}} \]
integrate(((8*exp(4)^3*exp(x)^3+(8*x+24)*exp(4)^3*exp(x))*log(exp(x)^2+3+x )^3+(-12*x*exp(4)^2*exp(x)^3+(-12*x^2-36*x)*exp(4)^2*exp(x))*log(exp(x)^2+ 3+x)^2+(6*x^2*exp(4)*exp(x)^3+(6*x^3+18*x^2)*exp(4)*exp(x))*log(exp(x)^2+3 +x)+(-40*exp(4)-x^3+10)*exp(x)^3+(-40*exp(4)+10)*exp(x)^2+(-20*exp(4)-x^4- 3*x^3+10*x+30)*exp(x)-20*exp(4)+10*x+30)/((8*exp(4)^3*exp(x)^3+8*exp(4)^3* exp(x)^2+(8*x+24)*exp(4)^3*exp(x)+(8*x+24)*exp(4)^3)*log(exp(x)^2+3+x)^3+( -12*x*exp(4)^2*exp(x)^3-12*x*exp(4)^2*exp(x)^2+(-12*x^2-36*x)*exp(4)^2*exp (x)+(-12*x^2-36*x)*exp(4)^2)*log(exp(x)^2+3+x)^2+(6*x^2*exp(4)*exp(x)^3+6* x^2*exp(4)*exp(x)^2+(6*x^3+18*x^2)*exp(4)*exp(x)+(6*x^3+18*x^2)*exp(4))*lo g(exp(x)^2+3+x)-x^3*exp(x)^3-exp(x)^2*x^3+(-x^4-3*x^3)*exp(x)-x^4-3*x^3),x , algorithm=\
(4*x*e^4*log(((x + 3)*e^24 + e^(2*x + 24))*e^(-24))*log(e^12 + e^(x + 12)) - 4*e^8*log(((x + 3)*e^24 + e^(2*x + 24))*e^(-24))^2*log(e^12 + e^(x + 12 )) - x^2*log(e^12 + e^(x + 12)) - 5)/(4*x*e^4*log(((x + 3)*e^24 + e^(2*x + 24))*e^(-24)) - 4*e^8*log(((x + 3)*e^24 + e^(2*x + 24))*e^(-24))^2 - x^2)
Time = 0.22 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.52 \[ \int \frac {30-20 e^4+e^{2 x} \left (10-40 e^4\right )+10 x+e^{3 x} \left (10-40 e^4-x^3\right )+e^x \left (30-20 e^4+10 x-3 x^3-x^4\right )+\left (6 e^{4+3 x} x^2+e^{4+x} \left (18 x^2+6 x^3\right )\right ) \log \left (3+e^{2 x}+x\right )+\left (-12 e^{8+3 x} x+e^{8+x} \left (-36 x-12 x^2\right )\right ) \log ^2\left (3+e^{2 x}+x\right )+\left (8 e^{12+3 x}+e^{12+x} (24+8 x)\right ) \log ^3\left (3+e^{2 x}+x\right )}{-3 x^3-e^{2 x} x^3-e^{3 x} x^3-x^4+e^x \left (-3 x^3-x^4\right )+\left (6 e^{4+2 x} x^2+6 e^{4+3 x} x^2+e^4 \left (18 x^2+6 x^3\right )+e^{4+x} \left (18 x^2+6 x^3\right )\right ) \log \left (3+e^{2 x}+x\right )+\left (-12 e^{8+2 x} x-12 e^{8+3 x} x+e^8 \left (-36 x-12 x^2\right )+e^{8+x} \left (-36 x-12 x^2\right )\right ) \log ^2\left (3+e^{2 x}+x\right )+\left (8 e^{12+2 x}+8 e^{12+3 x}+e^{12} (24+8 x)+e^{12+x} (24+8 x)\right ) \log ^3\left (3+e^{2 x}+x\right )} \, dx=\log {\left (e^{x} + 1 \right )} + \frac {5}{x^{2} - 4 x e^{4} \log {\left (x + e^{2 x} + 3 \right )} + 4 e^{8} \log {\left (x + e^{2 x} + 3 \right )}^{2}} \]
integrate(((8*exp(4)**3*exp(x)**3+(8*x+24)*exp(4)**3*exp(x))*ln(exp(x)**2+ 3+x)**3+(-12*x*exp(4)**2*exp(x)**3+(-12*x**2-36*x)*exp(4)**2*exp(x))*ln(ex p(x)**2+3+x)**2+(6*x**2*exp(4)*exp(x)**3+(6*x**3+18*x**2)*exp(4)*exp(x))*l n(exp(x)**2+3+x)+(-40*exp(4)-x**3+10)*exp(x)**3+(-40*exp(4)+10)*exp(x)**2+ (-20*exp(4)-x**4-3*x**3+10*x+30)*exp(x)-20*exp(4)+10*x+30)/((8*exp(4)**3*e xp(x)**3+8*exp(4)**3*exp(x)**2+(8*x+24)*exp(4)**3*exp(x)+(8*x+24)*exp(4)** 3)*ln(exp(x)**2+3+x)**3+(-12*x*exp(4)**2*exp(x)**3-12*x*exp(4)**2*exp(x)** 2+(-12*x**2-36*x)*exp(4)**2*exp(x)+(-12*x**2-36*x)*exp(4)**2)*ln(exp(x)**2 +3+x)**2+(6*x**2*exp(4)*exp(x)**3+6*x**2*exp(4)*exp(x)**2+(6*x**3+18*x**2) *exp(4)*exp(x)+(6*x**3+18*x**2)*exp(4))*ln(exp(x)**2+3+x)-x**3*exp(x)**3-e xp(x)**2*x**3+(-x**4-3*x**3)*exp(x)-x**4-3*x**3),x)
log(exp(x) + 1) + 5/(x**2 - 4*x*exp(4)*log(x + exp(2*x) + 3) + 4*exp(8)*lo g(x + exp(2*x) + 3)**2)
Time = 0.90 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.48 \[ \int \frac {30-20 e^4+e^{2 x} \left (10-40 e^4\right )+10 x+e^{3 x} \left (10-40 e^4-x^3\right )+e^x \left (30-20 e^4+10 x-3 x^3-x^4\right )+\left (6 e^{4+3 x} x^2+e^{4+x} \left (18 x^2+6 x^3\right )\right ) \log \left (3+e^{2 x}+x\right )+\left (-12 e^{8+3 x} x+e^{8+x} \left (-36 x-12 x^2\right )\right ) \log ^2\left (3+e^{2 x}+x\right )+\left (8 e^{12+3 x}+e^{12+x} (24+8 x)\right ) \log ^3\left (3+e^{2 x}+x\right )}{-3 x^3-e^{2 x} x^3-e^{3 x} x^3-x^4+e^x \left (-3 x^3-x^4\right )+\left (6 e^{4+2 x} x^2+6 e^{4+3 x} x^2+e^4 \left (18 x^2+6 x^3\right )+e^{4+x} \left (18 x^2+6 x^3\right )\right ) \log \left (3+e^{2 x}+x\right )+\left (-12 e^{8+2 x} x-12 e^{8+3 x} x+e^8 \left (-36 x-12 x^2\right )+e^{8+x} \left (-36 x-12 x^2\right )\right ) \log ^2\left (3+e^{2 x}+x\right )+\left (8 e^{12+2 x}+8 e^{12+3 x}+e^{12} (24+8 x)+e^{12+x} (24+8 x)\right ) \log ^3\left (3+e^{2 x}+x\right )} \, dx=-\frac {5}{4 \, x e^{4} \log \left (x + e^{\left (2 \, x\right )} + 3\right ) - 4 \, e^{8} \log \left (x + e^{\left (2 \, x\right )} + 3\right )^{2} - x^{2}} + \log \left (e^{x} + 1\right ) \]
integrate(((8*exp(4)^3*exp(x)^3+(8*x+24)*exp(4)^3*exp(x))*log(exp(x)^2+3+x )^3+(-12*x*exp(4)^2*exp(x)^3+(-12*x^2-36*x)*exp(4)^2*exp(x))*log(exp(x)^2+ 3+x)^2+(6*x^2*exp(4)*exp(x)^3+(6*x^3+18*x^2)*exp(4)*exp(x))*log(exp(x)^2+3 +x)+(-40*exp(4)-x^3+10)*exp(x)^3+(-40*exp(4)+10)*exp(x)^2+(-20*exp(4)-x^4- 3*x^3+10*x+30)*exp(x)-20*exp(4)+10*x+30)/((8*exp(4)^3*exp(x)^3+8*exp(4)^3* exp(x)^2+(8*x+24)*exp(4)^3*exp(x)+(8*x+24)*exp(4)^3)*log(exp(x)^2+3+x)^3+( -12*x*exp(4)^2*exp(x)^3-12*x*exp(4)^2*exp(x)^2+(-12*x^2-36*x)*exp(4)^2*exp (x)+(-12*x^2-36*x)*exp(4)^2)*log(exp(x)^2+3+x)^2+(6*x^2*exp(4)*exp(x)^3+6* x^2*exp(4)*exp(x)^2+(6*x^3+18*x^2)*exp(4)*exp(x)+(6*x^3+18*x^2)*exp(4))*lo g(exp(x)^2+3+x)-x^3*exp(x)^3-exp(x)^2*x^3+(-x^4-3*x^3)*exp(x)-x^4-3*x^3),x , algorithm=\
Timed out. \[ \int \frac {30-20 e^4+e^{2 x} \left (10-40 e^4\right )+10 x+e^{3 x} \left (10-40 e^4-x^3\right )+e^x \left (30-20 e^4+10 x-3 x^3-x^4\right )+\left (6 e^{4+3 x} x^2+e^{4+x} \left (18 x^2+6 x^3\right )\right ) \log \left (3+e^{2 x}+x\right )+\left (-12 e^{8+3 x} x+e^{8+x} \left (-36 x-12 x^2\right )\right ) \log ^2\left (3+e^{2 x}+x\right )+\left (8 e^{12+3 x}+e^{12+x} (24+8 x)\right ) \log ^3\left (3+e^{2 x}+x\right )}{-3 x^3-e^{2 x} x^3-e^{3 x} x^3-x^4+e^x \left (-3 x^3-x^4\right )+\left (6 e^{4+2 x} x^2+6 e^{4+3 x} x^2+e^4 \left (18 x^2+6 x^3\right )+e^{4+x} \left (18 x^2+6 x^3\right )\right ) \log \left (3+e^{2 x}+x\right )+\left (-12 e^{8+2 x} x-12 e^{8+3 x} x+e^8 \left (-36 x-12 x^2\right )+e^{8+x} \left (-36 x-12 x^2\right )\right ) \log ^2\left (3+e^{2 x}+x\right )+\left (8 e^{12+2 x}+8 e^{12+3 x}+e^{12} (24+8 x)+e^{12+x} (24+8 x)\right ) \log ^3\left (3+e^{2 x}+x\right )} \, dx=\text {Timed out} \]
integrate(((8*exp(4)^3*exp(x)^3+(8*x+24)*exp(4)^3*exp(x))*log(exp(x)^2+3+x )^3+(-12*x*exp(4)^2*exp(x)^3+(-12*x^2-36*x)*exp(4)^2*exp(x))*log(exp(x)^2+ 3+x)^2+(6*x^2*exp(4)*exp(x)^3+(6*x^3+18*x^2)*exp(4)*exp(x))*log(exp(x)^2+3 +x)+(-40*exp(4)-x^3+10)*exp(x)^3+(-40*exp(4)+10)*exp(x)^2+(-20*exp(4)-x^4- 3*x^3+10*x+30)*exp(x)-20*exp(4)+10*x+30)/((8*exp(4)^3*exp(x)^3+8*exp(4)^3* exp(x)^2+(8*x+24)*exp(4)^3*exp(x)+(8*x+24)*exp(4)^3)*log(exp(x)^2+3+x)^3+( -12*x*exp(4)^2*exp(x)^3-12*x*exp(4)^2*exp(x)^2+(-12*x^2-36*x)*exp(4)^2*exp (x)+(-12*x^2-36*x)*exp(4)^2)*log(exp(x)^2+3+x)^2+(6*x^2*exp(4)*exp(x)^3+6* x^2*exp(4)*exp(x)^2+(6*x^3+18*x^2)*exp(4)*exp(x)+(6*x^3+18*x^2)*exp(4))*lo g(exp(x)^2+3+x)-x^3*exp(x)^3-exp(x)^2*x^3+(-x^4-3*x^3)*exp(x)-x^4-3*x^3),x , algorithm=\
Time = 8.91 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.83 \[ \int \frac {30-20 e^4+e^{2 x} \left (10-40 e^4\right )+10 x+e^{3 x} \left (10-40 e^4-x^3\right )+e^x \left (30-20 e^4+10 x-3 x^3-x^4\right )+\left (6 e^{4+3 x} x^2+e^{4+x} \left (18 x^2+6 x^3\right )\right ) \log \left (3+e^{2 x}+x\right )+\left (-12 e^{8+3 x} x+e^{8+x} \left (-36 x-12 x^2\right )\right ) \log ^2\left (3+e^{2 x}+x\right )+\left (8 e^{12+3 x}+e^{12+x} (24+8 x)\right ) \log ^3\left (3+e^{2 x}+x\right )}{-3 x^3-e^{2 x} x^3-e^{3 x} x^3-x^4+e^x \left (-3 x^3-x^4\right )+\left (6 e^{4+2 x} x^2+6 e^{4+3 x} x^2+e^4 \left (18 x^2+6 x^3\right )+e^{4+x} \left (18 x^2+6 x^3\right )\right ) \log \left (3+e^{2 x}+x\right )+\left (-12 e^{8+2 x} x-12 e^{8+3 x} x+e^8 \left (-36 x-12 x^2\right )+e^{8+x} \left (-36 x-12 x^2\right )\right ) \log ^2\left (3+e^{2 x}+x\right )+\left (8 e^{12+2 x}+8 e^{12+3 x}+e^{12} (24+8 x)+e^{12+x} (24+8 x)\right ) \log ^3\left (3+e^{2 x}+x\right )} \, dx=\ln \left ({\mathrm {e}}^x+1\right )+\frac {5}{{\left (x-2\,{\mathrm {e}}^4\,\ln \left (x+{\mathrm {e}}^{2\,x}+3\right )\right )}^2} \]
int((20*exp(4) - 10*x + log(x + exp(2*x) + 3)^2*(exp(8)*exp(x)*(36*x + 12* x^2) + 12*x*exp(3*x)*exp(8)) + exp(x)*(20*exp(4) - 10*x + 3*x^3 + x^4 - 30 ) - log(x + exp(2*x) + 3)^3*(8*exp(3*x)*exp(12) + exp(12)*exp(x)*(8*x + 24 )) - log(x + exp(2*x) + 3)*(exp(4)*exp(x)*(18*x^2 + 6*x^3) + 6*x^2*exp(3*x )*exp(4)) + exp(3*x)*(40*exp(4) + x^3 - 10) + exp(2*x)*(40*exp(4) - 10) - 30)/(x^3*exp(2*x) + x^3*exp(3*x) - log(x + exp(2*x) + 3)^3*(8*exp(2*x)*exp (12) + 8*exp(3*x)*exp(12) + exp(12)*(8*x + 24) + exp(12)*exp(x)*(8*x + 24) ) + exp(x)*(3*x^3 + x^4) + log(x + exp(2*x) + 3)^2*(exp(8)*(36*x + 12*x^2) + exp(8)*exp(x)*(36*x + 12*x^2) + 12*x*exp(2*x)*exp(8) + 12*x*exp(3*x)*ex p(8)) + 3*x^3 + x^4 - log(x + exp(2*x) + 3)*(exp(4)*(18*x^2 + 6*x^3) + exp (4)*exp(x)*(18*x^2 + 6*x^3) + 6*x^2*exp(2*x)*exp(4) + 6*x^2*exp(3*x)*exp(4 ))),x)