\(\int \frac {30-20 e^4+e^{2 x} (10-40 e^4)+10 x+e^{3 x} (10-40 e^4-x^3)+e^x (30-20 e^4+10 x-3 x^3-x^4)+(6 e^{4+3 x} x^2+e^{4+x} (18 x^2+6 x^3)) \log (3+e^{2 x}+x)+(-12 e^{8+3 x} x+e^{8+x} (-36 x-12 x^2)) \log ^2(3+e^{2 x}+x)+(8 e^{12+3 x}+e^{12+x} (24+8 x)) \log ^3(3+e^{2 x}+x)}{-3 x^3-e^{2 x} x^3-e^{3 x} x^3-x^4+e^x (-3 x^3-x^4)+(6 e^{4+2 x} x^2+6 e^{4+3 x} x^2+e^4 (18 x^2+6 x^3)+e^{4+x} (18 x^2+6 x^3)) \log (3+e^{2 x}+x)+(-12 e^{8+2 x} x-12 e^{8+3 x} x+e^8 (-36 x-12 x^2)+e^{8+x} (-36 x-12 x^2)) \log ^2(3+e^{2 x}+x)+(8 e^{12+2 x}+8 e^{12+3 x}+e^{12} (24+8 x)+e^{12+x} (24+8 x)) \log ^3(3+e^{2 x}+x)} \, dx\) [702]

Optimal result

Integrand size = 404, antiderivative size = 29 \[ \int \frac {30-20 e^4+e^{2 x} \left (10-40 e^4\right )+10 x+e^{3 x} \left (10-40 e^4-x^3\right )+e^x \left (30-20 e^4+10 x-3 x^3-x^4\right )+\left (6 e^{4+3 x} x^2+e^{4+x} \left (18 x^2+6 x^3\right )\right ) \log \left (3+e^{2 x}+x\right )+\left (-12 e^{8+3 x} x+e^{8+x} \left (-36 x-12 x^2\right )\right ) \log ^2\left (3+e^{2 x}+x\right )+\left (8 e^{12+3 x}+e^{12+x} (24+8 x)\right ) \log ^3\left (3+e^{2 x}+x\right )}{-3 x^3-e^{2 x} x^3-e^{3 x} x^3-x^4+e^x \left (-3 x^3-x^4\right )+\left (6 e^{4+2 x} x^2+6 e^{4+3 x} x^2+e^4 \left (18 x^2+6 x^3\right )+e^{4+x} \left (18 x^2+6 x^3\right )\right ) \log \left (3+e^{2 x}+x\right )+\left (-12 e^{8+2 x} x-12 e^{8+3 x} x+e^8 \left (-36 x-12 x^2\right )+e^{8+x} \left (-36 x-12 x^2\right )\right ) \log ^2\left (3+e^{2 x}+x\right )+\left (8 e^{12+2 x}+8 e^{12+3 x}+e^{12} (24+8 x)+e^{12+x} (24+8 x)\right ) \log ^3\left (3+e^{2 x}+x\right )} \, dx=\log \left (1+e^x\right )+\frac {5}{\left (-x+2 e^4 \log \left (3+e^{2 x}+x\right )\right )^2} \] Output:

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

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00 \[ \int \frac {30-20 e^4+e^{2 x} \left (10-40 e^4\right )+10 x+e^{3 x} \left (10-40 e^4-x^3\right )+e^x \left (30-20 e^4+10 x-3 x^3-x^4\right )+\left (6 e^{4+3 x} x^2+e^{4+x} \left (18 x^2+6 x^3\right )\right ) \log \left (3+e^{2 x}+x\right )+\left (-12 e^{8+3 x} x+e^{8+x} \left (-36 x-12 x^2\right )\right ) \log ^2\left (3+e^{2 x}+x\right )+\left (8 e^{12+3 x}+e^{12+x} (24+8 x)\right ) \log ^3\left (3+e^{2 x}+x\right )}{-3 x^3-e^{2 x} x^3-e^{3 x} x^3-x^4+e^x \left (-3 x^3-x^4\right )+\left (6 e^{4+2 x} x^2+6 e^{4+3 x} x^2+e^4 \left (18 x^2+6 x^3\right )+e^{4+x} \left (18 x^2+6 x^3\right )\right ) \log \left (3+e^{2 x}+x\right )+\left (-12 e^{8+2 x} x-12 e^{8+3 x} x+e^8 \left (-36 x-12 x^2\right )+e^{8+x} \left (-36 x-12 x^2\right )\right ) \log ^2\left (3+e^{2 x}+x\right )+\left (8 e^{12+2 x}+8 e^{12+3 x}+e^{12} (24+8 x)+e^{12+x} (24+8 x)\right ) \log ^3\left (3+e^{2 x}+x\right )} \, dx=\log \left (1+e^x\right )+\frac {5}{\left (-x+2 e^4 \log \left (3+e^{2 x}+x\right )\right )^2} \] Input:

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]
 

Output:

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

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.

\(\displaystyle \int \frac {e^{3 x} \left (-x^3-40 e^4+10\right )+\left (e^{x+8} \left (-12 x^2-36 x\right )-12 e^{3 x+8} x\right ) \log ^2\left (x+e^{2 x}+3\right )+e^x \left (-x^4-3 x^3+10 x-20 e^4+30\right )+\left (6 e^{3 x+4} x^2+e^{x+4} \left (6 x^3+18 x^2\right )\right ) \log \left (x+e^{2 x}+3\right )+10 x+\left (10-40 e^4\right ) e^{2 x}+\left (e^{x+12} (8 x+24)+8 e^{3 x+12}\right ) \log ^3\left (x+e^{2 x}+3\right )-20 e^4+30}{-x^4-e^{2 x} x^3-e^{3 x} x^3-3 x^3+\left (e^{x+8} \left (-12 x^2-36 x\right )+e^8 \left (-12 x^2-36 x\right )-12 e^{2 x+8} x-12 e^{3 x+8} x\right ) \log ^2\left (x+e^{2 x}+3\right )+e^x \left (-x^4-3 x^3\right )+\left (6 e^{2 x+4} x^2+6 e^{3 x+4} x^2+e^{x+4} \left (6 x^3+18 x^2\right )+e^4 \left (6 x^3+18 x^2\right )\right ) \log \left (x+e^{2 x}+3\right )+\left (e^{x+12} (8 x+24)+e^{12} (8 x+24)+8 e^{2 x+12}+8 e^{3 x+12}\right ) \log ^3\left (x+e^{2 x}+3\right )} \, dx\)

\(\Big \downarrow \) 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

\(\displaystyle -10 \left (1-4 e^4\right ) \int \frac {1}{\left (x-2 e^4 \log \left (x+e^{2 x}+3\right )\right )^3}dx-100 e^4 \int \frac {1}{\left (x+e^{2 x}+3\right ) \left (x-2 e^4 \log \left (x+e^{2 x}+3\right )\right )^3}dx-40 e^4 \int \frac {x}{\left (x+e^{2 x}+3\right ) \left (x-2 e^4 \log \left (x+e^{2 x}+3\right )\right )^3}dx+\log \left (e^x+1\right )\)

Input:

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]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 3.56 (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	\(\frac {4 \ln \left ({\mathrm e}^{x}+1\right ) \ln \left ({\mathrm e}^{2 x}+3+x \right )^{2} {\mathrm e}^{8}-4 \ln \left ({\mathrm e}^{x}+1\right ) \ln \left ({\mathrm e}^{2 x}+3+x \right ) {\mathrm e}^{4} x +\ln \left ({\mathrm e}^{x}+1\right ) x^{2}+5}{4 \,{\mathrm e}^{8} \ln \left ({\mathrm e}^{2 x}+3+x \right )^{2}-4 \ln \left ({\mathrm e}^{2 x}+3+x \right ) {\mathrm e}^{4} x +x^{2}}\)	\(87\)

Input:

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)
 

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 130 vs. \(2 (26) = 52\).

Time = 0.13 (sec) , antiderivative size = 130, normalized size of antiderivative = 4.48 \[ \int \frac {30-20 e^4+e^{2 x} \left (10-40 e^4\right )+10 x+e^{3 x} \left (10-40 e^4-x^3\right )+e^x \left (30-20 e^4+10 x-3 x^3-x^4\right )+\left (6 e^{4+3 x} x^2+e^{4+x} \left (18 x^2+6 x^3\right )\right ) \log \left (3+e^{2 x}+x\right )+\left (-12 e^{8+3 x} x+e^{8+x} \left (-36 x-12 x^2\right )\right ) \log ^2\left (3+e^{2 x}+x\right )+\left (8 e^{12+3 x}+e^{12+x} (24+8 x)\right ) \log ^3\left (3+e^{2 x}+x\right )}{-3 x^3-e^{2 x} x^3-e^{3 x} x^3-x^4+e^x \left (-3 x^3-x^4\right )+\left (6 e^{4+2 x} x^2+6 e^{4+3 x} x^2+e^4 \left (18 x^2+6 x^3\right )+e^{4+x} \left (18 x^2+6 x^3\right )\right ) \log \left (3+e^{2 x}+x\right )+\left (-12 e^{8+2 x} x-12 e^{8+3 x} x+e^8 \left (-36 x-12 x^2\right )+e^{8+x} \left (-36 x-12 x^2\right )\right ) \log ^2\left (3+e^{2 x}+x\right )+\left (8 e^{12+2 x}+8 e^{12+3 x}+e^{12} (24+8 x)+e^{12+x} (24+8 x)\right ) \log ^3\left (3+e^{2 x}+x\right )} \, dx=\frac {4 \, x e^{4} \log \left ({\left ({\left (x + 3\right )} e^{24} + e^{\left (2 \, x + 24\right )}\right )} e^{\left (-24\right )}\right ) \log \left (e^{12} + e^{\left (x + 12\right )}\right ) - 4 \, e^{8} \log \left ({\left ({\left (x + 3\right )} e^{24} + e^{\left (2 \, x + 24\right )}\right )} e^{\left (-24\right )}\right )^{2} \log \left (e^{12} + e^{\left (x + 12\right )}\right ) - x^{2} \log \left (e^{12} + e^{\left (x + 12\right )}\right ) - 5}{4 \, x e^{4} \log \left ({\left ({\left (x + 3\right )} e^{24} + e^{\left (2 \, x + 24\right )}\right )} e^{\left (-24\right )}\right ) - 4 \, e^{8} \log \left ({\left ({\left (x + 3\right )} e^{24} + e^{\left (2 \, x + 24\right )}\right )} e^{\left (-24\right )}\right )^{2} - x^{2}} \] Input:

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="fricas")
 

Output:

(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)
 

Sympy [A] (verification not implemented)

Time = 0.28 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.52 \[ \int \frac {30-20 e^4+e^{2 x} \left (10-40 e^4\right )+10 x+e^{3 x} \left (10-40 e^4-x^3\right )+e^x \left (30-20 e^4+10 x-3 x^3-x^4\right )+\left (6 e^{4+3 x} x^2+e^{4+x} \left (18 x^2+6 x^3\right )\right ) \log \left (3+e^{2 x}+x\right )+\left (-12 e^{8+3 x} x+e^{8+x} \left (-36 x-12 x^2\right )\right ) \log ^2\left (3+e^{2 x}+x\right )+\left (8 e^{12+3 x}+e^{12+x} (24+8 x)\right ) \log ^3\left (3+e^{2 x}+x\right )}{-3 x^3-e^{2 x} x^3-e^{3 x} x^3-x^4+e^x \left (-3 x^3-x^4\right )+\left (6 e^{4+2 x} x^2+6 e^{4+3 x} x^2+e^4 \left (18 x^2+6 x^3\right )+e^{4+x} \left (18 x^2+6 x^3\right )\right ) \log \left (3+e^{2 x}+x\right )+\left (-12 e^{8+2 x} x-12 e^{8+3 x} x+e^8 \left (-36 x-12 x^2\right )+e^{8+x} \left (-36 x-12 x^2\right )\right ) \log ^2\left (3+e^{2 x}+x\right )+\left (8 e^{12+2 x}+8 e^{12+3 x}+e^{12} (24+8 x)+e^{12+x} (24+8 x)\right ) \log ^3\left (3+e^{2 x}+x\right )} \, dx=\log {\left (e^{x} + 1 \right )} + \frac {5}{x^{2} - 4 x e^{4} \log {\left (x + e^{2 x} + 3 \right )} + 4 e^{8} \log {\left (x + e^{2 x} + 3 \right )}^{2}} \] Input:

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)
 

Output:

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)
 

Maxima [A] (verification not implemented)

Time = 0.92 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.48 \[ \int \frac {30-20 e^4+e^{2 x} \left (10-40 e^4\right )+10 x+e^{3 x} \left (10-40 e^4-x^3\right )+e^x \left (30-20 e^4+10 x-3 x^3-x^4\right )+\left (6 e^{4+3 x} x^2+e^{4+x} \left (18 x^2+6 x^3\right )\right ) \log \left (3+e^{2 x}+x\right )+\left (-12 e^{8+3 x} x+e^{8+x} \left (-36 x-12 x^2\right )\right ) \log ^2\left (3+e^{2 x}+x\right )+\left (8 e^{12+3 x}+e^{12+x} (24+8 x)\right ) \log ^3\left (3+e^{2 x}+x\right )}{-3 x^3-e^{2 x} x^3-e^{3 x} x^3-x^4+e^x \left (-3 x^3-x^4\right )+\left (6 e^{4+2 x} x^2+6 e^{4+3 x} x^2+e^4 \left (18 x^2+6 x^3\right )+e^{4+x} \left (18 x^2+6 x^3\right )\right ) \log \left (3+e^{2 x}+x\right )+\left (-12 e^{8+2 x} x-12 e^{8+3 x} x+e^8 \left (-36 x-12 x^2\right )+e^{8+x} \left (-36 x-12 x^2\right )\right ) \log ^2\left (3+e^{2 x}+x\right )+\left (8 e^{12+2 x}+8 e^{12+3 x}+e^{12} (24+8 x)+e^{12+x} (24+8 x)\right ) \log ^3\left (3+e^{2 x}+x\right )} \, dx=-\frac {5}{4 \, x e^{4} \log \left (x + e^{\left (2 \, x\right )} + 3\right ) - 4 \, e^{8} \log \left (x + e^{\left (2 \, x\right )} + 3\right )^{2} - x^{2}} + \log \left (e^{x} + 1\right ) \] Input:

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="maxima")
 

Output:

-5/(4*x*e^4*log(x + e^(2*x) + 3) - 4*e^8*log(x + e^(2*x) + 3)^2 - x^2) + l 
og(e^x + 1)
 

Giac [F(-1)]

Timed out. \[ \int \frac {30-20 e^4+e^{2 x} \left (10-40 e^4\right )+10 x+e^{3 x} \left (10-40 e^4-x^3\right )+e^x \left (30-20 e^4+10 x-3 x^3-x^4\right )+\left (6 e^{4+3 x} x^2+e^{4+x} \left (18 x^2+6 x^3\right )\right ) \log \left (3+e^{2 x}+x\right )+\left (-12 e^{8+3 x} x+e^{8+x} \left (-36 x-12 x^2\right )\right ) \log ^2\left (3+e^{2 x}+x\right )+\left (8 e^{12+3 x}+e^{12+x} (24+8 x)\right ) \log ^3\left (3+e^{2 x}+x\right )}{-3 x^3-e^{2 x} x^3-e^{3 x} x^3-x^4+e^x \left (-3 x^3-x^4\right )+\left (6 e^{4+2 x} x^2+6 e^{4+3 x} x^2+e^4 \left (18 x^2+6 x^3\right )+e^{4+x} \left (18 x^2+6 x^3\right )\right ) \log \left (3+e^{2 x}+x\right )+\left (-12 e^{8+2 x} x-12 e^{8+3 x} x+e^8 \left (-36 x-12 x^2\right )+e^{8+x} \left (-36 x-12 x^2\right )\right ) \log ^2\left (3+e^{2 x}+x\right )+\left (8 e^{12+2 x}+8 e^{12+3 x}+e^{12} (24+8 x)+e^{12+x} (24+8 x)\right ) \log ^3\left (3+e^{2 x}+x\right )} \, dx=\text {Timed out} \] Input:

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="giac")
 

Output:

Timed out
 

Mupad [B] (verification not implemented)

Time = 0.39 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.83 \[ \int \frac {30-20 e^4+e^{2 x} \left (10-40 e^4\right )+10 x+e^{3 x} \left (10-40 e^4-x^3\right )+e^x \left (30-20 e^4+10 x-3 x^3-x^4\right )+\left (6 e^{4+3 x} x^2+e^{4+x} \left (18 x^2+6 x^3\right )\right ) \log \left (3+e^{2 x}+x\right )+\left (-12 e^{8+3 x} x+e^{8+x} \left (-36 x-12 x^2\right )\right ) \log ^2\left (3+e^{2 x}+x\right )+\left (8 e^{12+3 x}+e^{12+x} (24+8 x)\right ) \log ^3\left (3+e^{2 x}+x\right )}{-3 x^3-e^{2 x} x^3-e^{3 x} x^3-x^4+e^x \left (-3 x^3-x^4\right )+\left (6 e^{4+2 x} x^2+6 e^{4+3 x} x^2+e^4 \left (18 x^2+6 x^3\right )+e^{4+x} \left (18 x^2+6 x^3\right )\right ) \log \left (3+e^{2 x}+x\right )+\left (-12 e^{8+2 x} x-12 e^{8+3 x} x+e^8 \left (-36 x-12 x^2\right )+e^{8+x} \left (-36 x-12 x^2\right )\right ) \log ^2\left (3+e^{2 x}+x\right )+\left (8 e^{12+2 x}+8 e^{12+3 x}+e^{12} (24+8 x)+e^{12+x} (24+8 x)\right ) \log ^3\left (3+e^{2 x}+x\right )} \, dx=\ln \left ({\mathrm {e}}^x+1\right )+\frac {5}{{\left (x-2\,{\mathrm {e}}^4\,\ln \left (x+{\mathrm {e}}^{2\,x}+3\right )\right )}^2} \] Input:

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)
 

Output:

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

Reduce [B] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 194, normalized size of antiderivative = 6.69 \[ \int \frac {30-20 e^4+e^{2 x} \left (10-40 e^4\right )+10 x+e^{3 x} \left (10-40 e^4-x^3\right )+e^x \left (30-20 e^4+10 x-3 x^3-x^4\right )+\left (6 e^{4+3 x} x^2+e^{4+x} \left (18 x^2+6 x^3\right )\right ) \log \left (3+e^{2 x}+x\right )+\left (-12 e^{8+3 x} x+e^{8+x} \left (-36 x-12 x^2\right )\right ) \log ^2\left (3+e^{2 x}+x\right )+\left (8 e^{12+3 x}+e^{12+x} (24+8 x)\right ) \log ^3\left (3+e^{2 x}+x\right )}{-3 x^3-e^{2 x} x^3-e^{3 x} x^3-x^4+e^x \left (-3 x^3-x^4\right )+\left (6 e^{4+2 x} x^2+6 e^{4+3 x} x^2+e^4 \left (18 x^2+6 x^3\right )+e^{4+x} \left (18 x^2+6 x^3\right )\right ) \log \left (3+e^{2 x}+x\right )+\left (-12 e^{8+2 x} x-12 e^{8+3 x} x+e^8 \left (-36 x-12 x^2\right )+e^{8+x} \left (-36 x-12 x^2\right )\right ) \log ^2\left (3+e^{2 x}+x\right )+\left (8 e^{12+2 x}+8 e^{12+3 x}+e^{12} (24+8 x)+e^{12+x} (24+8 x)\right ) \log ^3\left (3+e^{2 x}+x\right )} \, dx=\frac {4 \,\mathrm {log}\left (e^{3 x}+e^{2 x}+e^{x} x +3 e^{x}+x +3\right ) \mathrm {log}\left (e^{2 x}+x +3\right )^{2} e^{8}-4 \,\mathrm {log}\left (e^{3 x}+e^{2 x}+e^{x} x +3 e^{x}+x +3\right ) \mathrm {log}\left (e^{2 x}+x +3\right ) e^{4} x +\mathrm {log}\left (e^{3 x}+e^{2 x}+e^{x} x +3 e^{x}+x +3\right ) x^{2}-4 \mathrm {log}\left (e^{2 x}+x +3\right )^{3} e^{8}+4 \mathrm {log}\left (e^{2 x}+x +3\right )^{2} e^{4} x -\mathrm {log}\left (e^{2 x}+x +3\right ) x^{2}+5}{4 \mathrm {log}\left (e^{2 x}+x +3\right )^{2} e^{8}-4 \,\mathrm {log}\left (e^{2 x}+x +3\right ) e^{4} x +x^{2}} \] Input:

int(((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*ex 
p(4)*exp(x)^2+(6*x^3+18*x^2)*exp(4)*exp(x)+(6*x^3+18*x^2)*exp(4))*log(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)
 

Output:

(4*log(e**(3*x) + e**(2*x) + e**x*x + 3*e**x + x + 3)*log(e**(2*x) + x + 3 
)**2*e**8 - 4*log(e**(3*x) + e**(2*x) + e**x*x + 3*e**x + x + 3)*log(e**(2 
*x) + x + 3)*e**4*x + log(e**(3*x) + e**(2*x) + e**x*x + 3*e**x + x + 3)*x 
**2 - 4*log(e**(2*x) + x + 3)**3*e**8 + 4*log(e**(2*x) + x + 3)**2*e**4*x 
- log(e**(2*x) + x + 3)*x**2 + 5)/(4*log(e**(2*x) + x + 3)**2*e**8 - 4*log 
(e**(2*x) + x + 3)*e**4*x + x**2)