\(\int \frac {24-4 x+100 x^2-117 x^3+12 x^4+12 x^5+e^{2 x} (12 x^2+3 x^3)+e^x (8+4 x+72 x^2-30 x^3-12 x^4)+(-36+31 x+10 x^2+e^x (-12-11 x-2 x^2)) \log (4 x+x^2)}{(108 x^3-117 x^4+12 x^5+12 x^6+e^{2 x} (12 x^3+3 x^4)+e^x (72 x^3-30 x^4-12 x^5)+(12 x-5 x^2-2 x^3+e^x (4 x+x^2)) \log (4 x+x^2)) \log (\frac {81 x^4+9 e^{2 x} x^4-108 x^5+36 x^6+e^x (54 x^4-36 x^5)+(18 x^2+6 e^x x^2-12 x^3) \log (4 x+x^2)+\log ^2(4 x+x^2)}{9 x^3+e^{2 x} x^3-12 x^4+4 x^5+e^x (6 x^3-4 x^4)})} \, dx\) [893]

Optimal result

Integrand size = 319, antiderivative size = 31 \[ \int \frac {24-4 x+100 x^2-117 x^3+12 x^4+12 x^5+e^{2 x} \left (12 x^2+3 x^3\right )+e^x \left (8+4 x+72 x^2-30 x^3-12 x^4\right )+\left (-36+31 x+10 x^2+e^x \left (-12-11 x-2 x^2\right )\right ) \log \left (4 x+x^2\right )}{\left (108 x^3-117 x^4+12 x^5+12 x^6+e^{2 x} \left (12 x^3+3 x^4\right )+e^x \left (72 x^3-30 x^4-12 x^5\right )+\left (12 x-5 x^2-2 x^3+e^x \left (4 x+x^2\right )\right ) \log \left (4 x+x^2\right )\right ) \log \left (\frac {81 x^4+9 e^{2 x} x^4-108 x^5+36 x^6+e^x \left (54 x^4-36 x^5\right )+\left (18 x^2+6 e^x x^2-12 x^3\right ) \log \left (4 x+x^2\right )+\log ^2\left (4 x+x^2\right )}{9 x^3+e^{2 x} x^3-12 x^4+4 x^5+e^x \left (6 x^3-4 x^4\right )}\right )} \, dx=\log \left (\log \left (x \left (3-\frac {\log (x (4+x))}{x^2 \left (-3-e^x+2 x\right )}\right )^2\right )\right ) \] Output:

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

Mathematica [A] (verified)

Time = 0.14 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.29 \[ \int \frac {24-4 x+100 x^2-117 x^3+12 x^4+12 x^5+e^{2 x} \left (12 x^2+3 x^3\right )+e^x \left (8+4 x+72 x^2-30 x^3-12 x^4\right )+\left (-36+31 x+10 x^2+e^x \left (-12-11 x-2 x^2\right )\right ) \log \left (4 x+x^2\right )}{\left (108 x^3-117 x^4+12 x^5+12 x^6+e^{2 x} \left (12 x^3+3 x^4\right )+e^x \left (72 x^3-30 x^4-12 x^5\right )+\left (12 x-5 x^2-2 x^3+e^x \left (4 x+x^2\right )\right ) \log \left (4 x+x^2\right )\right ) \log \left (\frac {81 x^4+9 e^{2 x} x^4-108 x^5+36 x^6+e^x \left (54 x^4-36 x^5\right )+\left (18 x^2+6 e^x x^2-12 x^3\right ) \log \left (4 x+x^2\right )+\log ^2\left (4 x+x^2\right )}{9 x^3+e^{2 x} x^3-12 x^4+4 x^5+e^x \left (6 x^3-4 x^4\right )}\right )} \, dx=\log \left (\log \left (\frac {\left (-3 x^2 \left (-3-e^x+2 x\right )+\log (x (4+x))\right )^2}{\left (3+e^x-2 x\right )^2 x^3}\right )\right ) \] Input:

Integrate[(24 - 4*x + 100*x^2 - 117*x^3 + 12*x^4 + 12*x^5 + E^(2*x)*(12*x^ 
2 + 3*x^3) + E^x*(8 + 4*x + 72*x^2 - 30*x^3 - 12*x^4) + (-36 + 31*x + 10*x 
^2 + E^x*(-12 - 11*x - 2*x^2))*Log[4*x + x^2])/((108*x^3 - 117*x^4 + 12*x^ 
5 + 12*x^6 + E^(2*x)*(12*x^3 + 3*x^4) + E^x*(72*x^3 - 30*x^4 - 12*x^5) + ( 
12*x - 5*x^2 - 2*x^3 + E^x*(4*x + x^2))*Log[4*x + x^2])*Log[(81*x^4 + 9*E^ 
(2*x)*x^4 - 108*x^5 + 36*x^6 + E^x*(54*x^4 - 36*x^5) + (18*x^2 + 6*E^x*x^2 
 - 12*x^3)*Log[4*x + x^2] + Log[4*x + x^2]^2)/(9*x^3 + E^(2*x)*x^3 - 12*x^ 
4 + 4*x^5 + E^x*(6*x^3 - 4*x^4))]),x]
 

Output:

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

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.

\(\displaystyle \int \frac {12 x^5+12 x^4-117 x^3+100 x^2+\left (10 x^2+e^x \left (-2 x^2-11 x-12\right )+31 x-36\right ) \log \left (x^2+4 x\right )+e^{2 x} \left (3 x^3+12 x^2\right )+e^x \left (-12 x^4-30 x^3+72 x^2+4 x+8\right )-4 x+24}{\left (12 x^6+12 x^5-117 x^4+108 x^3+e^{2 x} \left (3 x^4+12 x^3\right )+\left (-2 x^3-5 x^2+e^x \left (x^2+4 x\right )+12 x\right ) \log \left (x^2+4 x\right )+e^x \left (-12 x^5-30 x^4+72 x^3\right )\right ) \log \left (\frac {36 x^6-108 x^5+9 e^{2 x} x^4+81 x^4+\log ^2\left (x^2+4 x\right )+e^x \left (54 x^4-36 x^5\right )+\left (-12 x^3+6 e^x x^2+18 x^2\right ) \log \left (x^2+4 x\right )}{4 x^5-12 x^4+e^{2 x} x^3+9 x^3+e^x \left (6 x^3-4 x^4\right )}\right )} \, dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {-12 x^5-12 x^4+117 x^3-3 e^{2 x} (x+4) x^2-100 x^2-e^x \left (-12 x^4-30 x^3+72 x^2+4 x+8\right )+4 x+(x+4) \left (-10 x+e^x (2 x+3)+9\right ) \log (x (x+4))-24}{\left (-2 x+e^x+3\right ) x (x+4) \left (3 x^2 \left (2 x-e^x-3\right )-\log (x (x+4))\right ) \log \left (\frac {\left (\log (x (x+4))-3 x^2 \left (2 x-e^x-3\right )\right )^2}{\left (-2 x+e^x+3\right )^2 x^3}\right )}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (-\frac {2 (2 x-5)}{\left (-2 x+e^x+3\right ) \log \left (\frac {\left (\log (x (x+4))-3 x^2 \left (2 x-e^x-3\right )\right )^2}{\left (-2 x+e^x+3\right )^2 x^3}\right )}+\frac {1}{x \log \left (\frac {\left (\log (x (x+4))-3 x^2 \left (2 x-e^x-3\right )\right )^2}{\left (-2 x+e^x+3\right )^2 x^3}\right )}-\frac {2 \left (6 x^5+9 x^4-60 x^3-x^2 \log (x (x+4))+2 x-6 x \log (x (x+4))-8 \log (x (x+4))+4\right )}{x (x+4) \left (6 x^3-3 e^x x^2-9 x^2-\log (x (x+4))\right ) \log \left (\frac {\left (\log (x (x+4))-3 x^2 \left (2 x-e^x-3\right )\right )^2}{\left (-2 x+e^x+3\right )^2 x^3}\right )}\right )dx\)

\(\Big \downarrow \) 7299

\(\displaystyle \int \left (-\frac {2 (2 x-5)}{\left (-2 x+e^x+3\right ) \log \left (\frac {\left (\log (x (x+4))-3 x^2 \left (2 x-e^x-3\right )\right )^2}{\left (-2 x+e^x+3\right )^2 x^3}\right )}+\frac {1}{x \log \left (\frac {\left (\log (x (x+4))-3 x^2 \left (2 x-e^x-3\right )\right )^2}{\left (-2 x+e^x+3\right )^2 x^3}\right )}-\frac {2 \left (6 x^5+9 x^4-60 x^3-x^2 \log (x (x+4))+2 x-6 x \log (x (x+4))-8 \log (x (x+4))+4\right )}{x (x+4) \left (6 x^3-3 e^x x^2-9 x^2-\log (x (x+4))\right ) \log \left (\frac {\left (\log (x (x+4))-3 x^2 \left (2 x-e^x-3\right )\right )^2}{\left (-2 x+e^x+3\right )^2 x^3}\right )}\right )dx\)

Input:

Int[(24 - 4*x + 100*x^2 - 117*x^3 + 12*x^4 + 12*x^5 + E^(2*x)*(12*x^2 + 3* 
x^3) + E^x*(8 + 4*x + 72*x^2 - 30*x^3 - 12*x^4) + (-36 + 31*x + 10*x^2 + E 
^x*(-12 - 11*x - 2*x^2))*Log[4*x + x^2])/((108*x^3 - 117*x^4 + 12*x^5 + 12 
*x^6 + E^(2*x)*(12*x^3 + 3*x^4) + E^x*(72*x^3 - 30*x^4 - 12*x^5) + (12*x - 
 5*x^2 - 2*x^3 + E^x*(4*x + x^2))*Log[4*x + x^2])*Log[(81*x^4 + 9*E^(2*x)* 
x^4 - 108*x^5 + 36*x^6 + E^x*(54*x^4 - 36*x^5) + (18*x^2 + 6*E^x*x^2 - 12* 
x^3)*Log[4*x + x^2] + Log[4*x + x^2]^2)/(9*x^3 + E^(2*x)*x^3 - 12*x^4 + 4* 
x^5 + E^x*(6*x^3 - 4*x^4))]),x]
 

Output:

$Aborted
 

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 0.20 (sec) , antiderivative size = 2581, normalized size of antiderivative = 83.26

\[\text {Expression too large to display}\]

Input:

int((((-2*x^2-11*x-12)*exp(x)+10*x^2+31*x-36)*ln(x^2+4*x)+(3*x^3+12*x^2)*e 
xp(x)^2+(-12*x^4-30*x^3+72*x^2+4*x+8)*exp(x)+12*x^5+12*x^4-117*x^3+100*x^2 
-4*x+24)/(((x^2+4*x)*exp(x)-2*x^3-5*x^2+12*x)*ln(x^2+4*x)+(3*x^4+12*x^3)*e 
xp(x)^2+(-12*x^5-30*x^4+72*x^3)*exp(x)+12*x^6+12*x^5-117*x^4+108*x^3)/ln(( 
ln(x^2+4*x)^2+(6*exp(x)*x^2-12*x^3+18*x^2)*ln(x^2+4*x)+9*exp(x)^2*x^4+(-36 
*x^5+54*x^4)*exp(x)+36*x^6-108*x^5+81*x^4)/(exp(x)^2*x^3+(-4*x^4+6*x^3)*ex 
p(x)+4*x^5-12*x^4+9*x^3)),x)
 

Output:

ln(1/2*I*Pi-ln(x-1/2*exp(x)-3/2)+1/4*I*Pi*csgn(I/x^3*(Pi*csgn(I*x)*csgn(I* 
(4+x))*csgn(I*(4+x)*x)-Pi*csgn(I*x)*csgn(I*(4+x)*x)^2-Pi*csgn(I*(4+x))*csg 
n(I*(4+x)*x)^2+Pi*csgn(I*(4+x)*x)^3-12*I*x^3+6*I*exp(x)*x^2+18*I*x^2+2*I*l 
n(x)+2*I*ln(4+x))^2/(-x+1/2*exp(x)+3/2)^2)^3+1/4*I*Pi*csgn(I/(-x+1/2*exp(x 
)+3/2)^2)*csgn(I/(-x+1/2*exp(x)+3/2)^2*(Pi*csgn(I*x)*csgn(I*(4+x))*csgn(I* 
(4+x)*x)-Pi*csgn(I*x)*csgn(I*(4+x)*x)^2-Pi*csgn(I*(4+x))*csgn(I*(4+x)*x)^2 
+Pi*csgn(I*(4+x)*x)^3-12*I*x^3+6*I*exp(x)*x^2+18*I*x^2+2*I*ln(x)+2*I*ln(4+ 
x))^2)^2+1/4*I*Pi*csgn(I*(Pi*csgn(I*x)*csgn(I*(4+x))*csgn(I*(4+x)*x)-Pi*cs 
gn(I*x)*csgn(I*(4+x)*x)^2-Pi*csgn(I*(4+x))*csgn(I*(4+x)*x)^2+Pi*csgn(I*(4+ 
x)*x)^3-12*I*x^3+6*I*exp(x)*x^2+18*I*x^2+2*I*ln(x)+2*I*ln(4+x))^2)*csgn(I/ 
(-x+1/2*exp(x)+3/2)^2*(Pi*csgn(I*x)*csgn(I*(4+x))*csgn(I*(4+x)*x)-Pi*csgn( 
I*x)*csgn(I*(4+x)*x)^2-Pi*csgn(I*(4+x))*csgn(I*(4+x)*x)^2+Pi*csgn(I*(4+x)* 
x)^3-12*I*x^3+6*I*exp(x)*x^2+18*I*x^2+2*I*ln(x)+2*I*ln(4+x))^2)^2+1/4*I*Pi 
*csgn(I/x^3)*csgn(I/x^3*(Pi*csgn(I*x)*csgn(I*(4+x))*csgn(I*(4+x)*x)-Pi*csg 
n(I*x)*csgn(I*(4+x)*x)^2-Pi*csgn(I*(4+x))*csgn(I*(4+x)*x)^2+Pi*csgn(I*(4+x 
)*x)^3-12*I*x^3+6*I*exp(x)*x^2+18*I*x^2+2*I*ln(x)+2*I*ln(4+x))^2/(-x+1/2*e 
xp(x)+3/2)^2)^2+1/4*I*Pi*csgn(I/(-x+1/2*exp(x)+3/2)^2*(Pi*csgn(I*x)*csgn(I 
*(4+x))*csgn(I*(4+x)*x)-Pi*csgn(I*x)*csgn(I*(4+x)*x)^2-Pi*csgn(I*(4+x))*cs 
gn(I*(4+x)*x)^2+Pi*csgn(I*(4+x)*x)^3-12*I*x^3+6*I*exp(x)*x^2+18*I*x^2+2*I* 
ln(x)+2*I*ln(4+x))^2)*csgn(I/x^3*(Pi*csgn(I*x)*csgn(I*(4+x))*csgn(I*(4+...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 122 vs. \(2 (29) = 58\).

Time = 0.08 (sec) , antiderivative size = 122, normalized size of antiderivative = 3.94 \[ \int \frac {24-4 x+100 x^2-117 x^3+12 x^4+12 x^5+e^{2 x} \left (12 x^2+3 x^3\right )+e^x \left (8+4 x+72 x^2-30 x^3-12 x^4\right )+\left (-36+31 x+10 x^2+e^x \left (-12-11 x-2 x^2\right )\right ) \log \left (4 x+x^2\right )}{\left (108 x^3-117 x^4+12 x^5+12 x^6+e^{2 x} \left (12 x^3+3 x^4\right )+e^x \left (72 x^3-30 x^4-12 x^5\right )+\left (12 x-5 x^2-2 x^3+e^x \left (4 x+x^2\right )\right ) \log \left (4 x+x^2\right )\right ) \log \left (\frac {81 x^4+9 e^{2 x} x^4-108 x^5+36 x^6+e^x \left (54 x^4-36 x^5\right )+\left (18 x^2+6 e^x x^2-12 x^3\right ) \log \left (4 x+x^2\right )+\log ^2\left (4 x+x^2\right )}{9 x^3+e^{2 x} x^3-12 x^4+4 x^5+e^x \left (6 x^3-4 x^4\right )}\right )} \, dx=\log \left (\log \left (\frac {36 \, x^{6} - 108 \, x^{5} + 9 \, x^{4} e^{\left (2 \, x\right )} + 81 \, x^{4} - 18 \, {\left (2 \, x^{5} - 3 \, x^{4}\right )} e^{x} - 6 \, {\left (2 \, x^{3} - x^{2} e^{x} - 3 \, x^{2}\right )} \log \left (x^{2} + 4 \, x\right ) + \log \left (x^{2} + 4 \, x\right )^{2}}{4 \, x^{5} - 12 \, x^{4} + x^{3} e^{\left (2 \, x\right )} + 9 \, x^{3} - 2 \, {\left (2 \, x^{4} - 3 \, x^{3}\right )} e^{x}}\right )\right ) \] Input:

integrate((((-2*x^2-11*x-12)*exp(x)+10*x^2+31*x-36)*log(x^2+4*x)+(3*x^3+12 
*x^2)*exp(x)^2+(-12*x^4-30*x^3+72*x^2+4*x+8)*exp(x)+12*x^5+12*x^4-117*x^3+ 
100*x^2-4*x+24)/(((x^2+4*x)*exp(x)-2*x^3-5*x^2+12*x)*log(x^2+4*x)+(3*x^4+1 
2*x^3)*exp(x)^2+(-12*x^5-30*x^4+72*x^3)*exp(x)+12*x^6+12*x^5-117*x^4+108*x 
^3)/log((log(x^2+4*x)^2+(6*exp(x)*x^2-12*x^3+18*x^2)*log(x^2+4*x)+9*exp(x) 
^2*x^4+(-36*x^5+54*x^4)*exp(x)+36*x^6-108*x^5+81*x^4)/(exp(x)^2*x^3+(-4*x^ 
4+6*x^3)*exp(x)+4*x^5-12*x^4+9*x^3)),x, algorithm="fricas")
 

Output:

log(log((36*x^6 - 108*x^5 + 9*x^4*e^(2*x) + 81*x^4 - 18*(2*x^5 - 3*x^4)*e^ 
x - 6*(2*x^3 - x^2*e^x - 3*x^2)*log(x^2 + 4*x) + log(x^2 + 4*x)^2)/(4*x^5 
- 12*x^4 + x^3*e^(2*x) + 9*x^3 - 2*(2*x^4 - 3*x^3)*e^x)))
 

Sympy [F(-1)]

Timed out. \[ \int \frac {24-4 x+100 x^2-117 x^3+12 x^4+12 x^5+e^{2 x} \left (12 x^2+3 x^3\right )+e^x \left (8+4 x+72 x^2-30 x^3-12 x^4\right )+\left (-36+31 x+10 x^2+e^x \left (-12-11 x-2 x^2\right )\right ) \log \left (4 x+x^2\right )}{\left (108 x^3-117 x^4+12 x^5+12 x^6+e^{2 x} \left (12 x^3+3 x^4\right )+e^x \left (72 x^3-30 x^4-12 x^5\right )+\left (12 x-5 x^2-2 x^3+e^x \left (4 x+x^2\right )\right ) \log \left (4 x+x^2\right )\right ) \log \left (\frac {81 x^4+9 e^{2 x} x^4-108 x^5+36 x^6+e^x \left (54 x^4-36 x^5\right )+\left (18 x^2+6 e^x x^2-12 x^3\right ) \log \left (4 x+x^2\right )+\log ^2\left (4 x+x^2\right )}{9 x^3+e^{2 x} x^3-12 x^4+4 x^5+e^x \left (6 x^3-4 x^4\right )}\right )} \, dx=\text {Timed out} \] Input:

integrate((((-2*x**2-11*x-12)*exp(x)+10*x**2+31*x-36)*ln(x**2+4*x)+(3*x**3 
+12*x**2)*exp(x)**2+(-12*x**4-30*x**3+72*x**2+4*x+8)*exp(x)+12*x**5+12*x** 
4-117*x**3+100*x**2-4*x+24)/(((x**2+4*x)*exp(x)-2*x**3-5*x**2+12*x)*ln(x** 
2+4*x)+(3*x**4+12*x**3)*exp(x)**2+(-12*x**5-30*x**4+72*x**3)*exp(x)+12*x** 
6+12*x**5-117*x**4+108*x**3)/ln((ln(x**2+4*x)**2+(6*exp(x)*x**2-12*x**3+18 
*x**2)*ln(x**2+4*x)+9*exp(x)**2*x**4+(-36*x**5+54*x**4)*exp(x)+36*x**6-108 
*x**5+81*x**4)/(exp(x)**2*x**3+(-4*x**4+6*x**3)*exp(x)+4*x**5-12*x**4+9*x* 
*3)),x)
 

Output:

Timed out
 

Maxima [A] (verification not implemented)

Time = 64.11 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.32 \[ \int \frac {24-4 x+100 x^2-117 x^3+12 x^4+12 x^5+e^{2 x} \left (12 x^2+3 x^3\right )+e^x \left (8+4 x+72 x^2-30 x^3-12 x^4\right )+\left (-36+31 x+10 x^2+e^x \left (-12-11 x-2 x^2\right )\right ) \log \left (4 x+x^2\right )}{\left (108 x^3-117 x^4+12 x^5+12 x^6+e^{2 x} \left (12 x^3+3 x^4\right )+e^x \left (72 x^3-30 x^4-12 x^5\right )+\left (12 x-5 x^2-2 x^3+e^x \left (4 x+x^2\right )\right ) \log \left (4 x+x^2\right )\right ) \log \left (\frac {81 x^4+9 e^{2 x} x^4-108 x^5+36 x^6+e^x \left (54 x^4-36 x^5\right )+\left (18 x^2+6 e^x x^2-12 x^3\right ) \log \left (4 x+x^2\right )+\log ^2\left (4 x+x^2\right )}{9 x^3+e^{2 x} x^3-12 x^4+4 x^5+e^x \left (6 x^3-4 x^4\right )}\right )} \, dx=\log \left (\log \left (-6 \, x^{3} + 3 \, x^{2} e^{x} + 9 \, x^{2} + \log \left (x + 4\right ) + \log \left (x\right )\right ) - \frac {3}{2} \, \log \left (x\right ) - \log \left (-2 \, x + e^{x} + 3\right )\right ) \] Input:

integrate((((-2*x^2-11*x-12)*exp(x)+10*x^2+31*x-36)*log(x^2+4*x)+(3*x^3+12 
*x^2)*exp(x)^2+(-12*x^4-30*x^3+72*x^2+4*x+8)*exp(x)+12*x^5+12*x^4-117*x^3+ 
100*x^2-4*x+24)/(((x^2+4*x)*exp(x)-2*x^3-5*x^2+12*x)*log(x^2+4*x)+(3*x^4+1 
2*x^3)*exp(x)^2+(-12*x^5-30*x^4+72*x^3)*exp(x)+12*x^6+12*x^5-117*x^4+108*x 
^3)/log((log(x^2+4*x)^2+(6*exp(x)*x^2-12*x^3+18*x^2)*log(x^2+4*x)+9*exp(x) 
^2*x^4+(-36*x^5+54*x^4)*exp(x)+36*x^6-108*x^5+81*x^4)/(exp(x)^2*x^3+(-4*x^ 
4+6*x^3)*exp(x)+4*x^5-12*x^4+9*x^3)),x, algorithm="maxima")
 

Output:

log(log(-6*x^3 + 3*x^2*e^x + 9*x^2 + log(x + 4) + log(x)) - 3/2*log(x) - l 
og(-2*x + e^x + 3))
 

Giac [F]

\[ \int \frac {24-4 x+100 x^2-117 x^3+12 x^4+12 x^5+e^{2 x} \left (12 x^2+3 x^3\right )+e^x \left (8+4 x+72 x^2-30 x^3-12 x^4\right )+\left (-36+31 x+10 x^2+e^x \left (-12-11 x-2 x^2\right )\right ) \log \left (4 x+x^2\right )}{\left (108 x^3-117 x^4+12 x^5+12 x^6+e^{2 x} \left (12 x^3+3 x^4\right )+e^x \left (72 x^3-30 x^4-12 x^5\right )+\left (12 x-5 x^2-2 x^3+e^x \left (4 x+x^2\right )\right ) \log \left (4 x+x^2\right )\right ) \log \left (\frac {81 x^4+9 e^{2 x} x^4-108 x^5+36 x^6+e^x \left (54 x^4-36 x^5\right )+\left (18 x^2+6 e^x x^2-12 x^3\right ) \log \left (4 x+x^2\right )+\log ^2\left (4 x+x^2\right )}{9 x^3+e^{2 x} x^3-12 x^4+4 x^5+e^x \left (6 x^3-4 x^4\right )}\right )} \, dx=\int { \frac {12 \, x^{5} + 12 \, x^{4} - 117 \, x^{3} + 100 \, x^{2} + 3 \, {\left (x^{3} + 4 \, x^{2}\right )} e^{\left (2 \, x\right )} - 2 \, {\left (6 \, x^{4} + 15 \, x^{3} - 36 \, x^{2} - 2 \, x - 4\right )} e^{x} + {\left (10 \, x^{2} - {\left (2 \, x^{2} + 11 \, x + 12\right )} e^{x} + 31 \, x - 36\right )} \log \left (x^{2} + 4 \, x\right ) - 4 \, x + 24}{{\left (12 \, x^{6} + 12 \, x^{5} - 117 \, x^{4} + 108 \, x^{3} + 3 \, {\left (x^{4} + 4 \, x^{3}\right )} e^{\left (2 \, x\right )} - 6 \, {\left (2 \, x^{5} + 5 \, x^{4} - 12 \, x^{3}\right )} e^{x} - {\left (2 \, x^{3} + 5 \, x^{2} - {\left (x^{2} + 4 \, x\right )} e^{x} - 12 \, x\right )} \log \left (x^{2} + 4 \, x\right )\right )} \log \left (\frac {36 \, x^{6} - 108 \, x^{5} + 9 \, x^{4} e^{\left (2 \, x\right )} + 81 \, x^{4} - 18 \, {\left (2 \, x^{5} - 3 \, x^{4}\right )} e^{x} - 6 \, {\left (2 \, x^{3} - x^{2} e^{x} - 3 \, x^{2}\right )} \log \left (x^{2} + 4 \, x\right ) + \log \left (x^{2} + 4 \, x\right )^{2}}{4 \, x^{5} - 12 \, x^{4} + x^{3} e^{\left (2 \, x\right )} + 9 \, x^{3} - 2 \, {\left (2 \, x^{4} - 3 \, x^{3}\right )} e^{x}}\right )} \,d x } \] Input:

integrate((((-2*x^2-11*x-12)*exp(x)+10*x^2+31*x-36)*log(x^2+4*x)+(3*x^3+12 
*x^2)*exp(x)^2+(-12*x^4-30*x^3+72*x^2+4*x+8)*exp(x)+12*x^5+12*x^4-117*x^3+ 
100*x^2-4*x+24)/(((x^2+4*x)*exp(x)-2*x^3-5*x^2+12*x)*log(x^2+4*x)+(3*x^4+1 
2*x^3)*exp(x)^2+(-12*x^5-30*x^4+72*x^3)*exp(x)+12*x^6+12*x^5-117*x^4+108*x 
^3)/log((log(x^2+4*x)^2+(6*exp(x)*x^2-12*x^3+18*x^2)*log(x^2+4*x)+9*exp(x) 
^2*x^4+(-36*x^5+54*x^4)*exp(x)+36*x^6-108*x^5+81*x^4)/(exp(x)^2*x^3+(-4*x^ 
4+6*x^3)*exp(x)+4*x^5-12*x^4+9*x^3)),x, algorithm="giac")
 

Output:

integrate((12*x^5 + 12*x^4 - 117*x^3 + 100*x^2 + 3*(x^3 + 4*x^2)*e^(2*x) - 
 2*(6*x^4 + 15*x^3 - 36*x^2 - 2*x - 4)*e^x + (10*x^2 - (2*x^2 + 11*x + 12) 
*e^x + 31*x - 36)*log(x^2 + 4*x) - 4*x + 24)/((12*x^6 + 12*x^5 - 117*x^4 + 
 108*x^3 + 3*(x^4 + 4*x^3)*e^(2*x) - 6*(2*x^5 + 5*x^4 - 12*x^3)*e^x - (2*x 
^3 + 5*x^2 - (x^2 + 4*x)*e^x - 12*x)*log(x^2 + 4*x))*log((36*x^6 - 108*x^5 
 + 9*x^4*e^(2*x) + 81*x^4 - 18*(2*x^5 - 3*x^4)*e^x - 6*(2*x^3 - x^2*e^x - 
3*x^2)*log(x^2 + 4*x) + log(x^2 + 4*x)^2)/(4*x^5 - 12*x^4 + x^3*e^(2*x) + 
9*x^3 - 2*(2*x^4 - 3*x^3)*e^x))), x)
 

Mupad [B] (verification not implemented)

Time = 2.19 (sec) , antiderivative size = 119, normalized size of antiderivative = 3.84 \[ \int \frac {24-4 x+100 x^2-117 x^3+12 x^4+12 x^5+e^{2 x} \left (12 x^2+3 x^3\right )+e^x \left (8+4 x+72 x^2-30 x^3-12 x^4\right )+\left (-36+31 x+10 x^2+e^x \left (-12-11 x-2 x^2\right )\right ) \log \left (4 x+x^2\right )}{\left (108 x^3-117 x^4+12 x^5+12 x^6+e^{2 x} \left (12 x^3+3 x^4\right )+e^x \left (72 x^3-30 x^4-12 x^5\right )+\left (12 x-5 x^2-2 x^3+e^x \left (4 x+x^2\right )\right ) \log \left (4 x+x^2\right )\right ) \log \left (\frac {81 x^4+9 e^{2 x} x^4-108 x^5+36 x^6+e^x \left (54 x^4-36 x^5\right )+\left (18 x^2+6 e^x x^2-12 x^3\right ) \log \left (4 x+x^2\right )+\log ^2\left (4 x+x^2\right )}{9 x^3+e^{2 x} x^3-12 x^4+4 x^5+e^x \left (6 x^3-4 x^4\right )}\right )} \, dx=\ln \left (\ln \left (\frac {{\mathrm {e}}^x\,\left (54\,x^4-36\,x^5\right )+9\,x^4\,{\mathrm {e}}^{2\,x}+\ln \left (x^2+4\,x\right )\,\left (6\,x^2\,{\mathrm {e}}^x+18\,x^2-12\,x^3\right )+81\,x^4-108\,x^5+36\,x^6+{\ln \left (x^2+4\,x\right )}^2}{{\mathrm {e}}^x\,\left (6\,x^3-4\,x^4\right )+x^3\,{\mathrm {e}}^{2\,x}+9\,x^3-12\,x^4+4\,x^5}\right )\right ) \] Input:

int((exp(x)*(4*x + 72*x^2 - 30*x^3 - 12*x^4 + 8) - 4*x + exp(2*x)*(12*x^2 
+ 3*x^3) + 100*x^2 - 117*x^3 + 12*x^4 + 12*x^5 + log(4*x + x^2)*(31*x - ex 
p(x)*(11*x + 2*x^2 + 12) + 10*x^2 - 36) + 24)/(log((exp(x)*(54*x^4 - 36*x^ 
5) + 9*x^4*exp(2*x) + log(4*x + x^2)*(6*x^2*exp(x) + 18*x^2 - 12*x^3) + 81 
*x^4 - 108*x^5 + 36*x^6 + log(4*x + x^2)^2)/(exp(x)*(6*x^3 - 4*x^4) + x^3* 
exp(2*x) + 9*x^3 - 12*x^4 + 4*x^5))*(exp(2*x)*(12*x^3 + 3*x^4) - exp(x)*(3 
0*x^4 - 72*x^3 + 12*x^5) + log(4*x + x^2)*(12*x + exp(x)*(4*x + x^2) - 5*x 
^2 - 2*x^3) + 108*x^3 - 117*x^4 + 12*x^5 + 12*x^6)),x)
 

Output:

log(log((exp(x)*(54*x^4 - 36*x^5) + 9*x^4*exp(2*x) + log(4*x + x^2)*(6*x^2 
*exp(x) + 18*x^2 - 12*x^3) + 81*x^4 - 108*x^5 + 36*x^6 + log(4*x + x^2)^2) 
/(exp(x)*(6*x^3 - 4*x^4) + x^3*exp(2*x) + 9*x^3 - 12*x^4 + 4*x^5)))
 

Reduce [B] (verification not implemented)

Time = 0.28 (sec) , antiderivative size = 140, normalized size of antiderivative = 4.52 \[ \int \frac {24-4 x+100 x^2-117 x^3+12 x^4+12 x^5+e^{2 x} \left (12 x^2+3 x^3\right )+e^x \left (8+4 x+72 x^2-30 x^3-12 x^4\right )+\left (-36+31 x+10 x^2+e^x \left (-12-11 x-2 x^2\right )\right ) \log \left (4 x+x^2\right )}{\left (108 x^3-117 x^4+12 x^5+12 x^6+e^{2 x} \left (12 x^3+3 x^4\right )+e^x \left (72 x^3-30 x^4-12 x^5\right )+\left (12 x-5 x^2-2 x^3+e^x \left (4 x+x^2\right )\right ) \log \left (4 x+x^2\right )\right ) \log \left (\frac {81 x^4+9 e^{2 x} x^4-108 x^5+36 x^6+e^x \left (54 x^4-36 x^5\right )+\left (18 x^2+6 e^x x^2-12 x^3\right ) \log \left (4 x+x^2\right )+\log ^2\left (4 x+x^2\right )}{9 x^3+e^{2 x} x^3-12 x^4+4 x^5+e^x \left (6 x^3-4 x^4\right )}\right )} \, dx=\mathrm {log}\left (\mathrm {log}\left (\frac {9 e^{2 x} x^{4}+6 e^{x} \mathrm {log}\left (x^{2}+4 x \right ) x^{2}-36 e^{x} x^{5}+54 e^{x} x^{4}+\mathrm {log}\left (x^{2}+4 x \right )^{2}-12 \,\mathrm {log}\left (x^{2}+4 x \right ) x^{3}+18 \,\mathrm {log}\left (x^{2}+4 x \right ) x^{2}+36 x^{6}-108 x^{5}+81 x^{4}}{e^{2 x} x^{3}-4 e^{x} x^{4}+6 e^{x} x^{3}+4 x^{5}-12 x^{4}+9 x^{3}}\right )\right ) \] Input:

int((((-2*x^2-11*x-12)*exp(x)+10*x^2+31*x-36)*log(x^2+4*x)+(3*x^3+12*x^2)* 
exp(x)^2+(-12*x^4-30*x^3+72*x^2+4*x+8)*exp(x)+12*x^5+12*x^4-117*x^3+100*x^ 
2-4*x+24)/(((x^2+4*x)*exp(x)-2*x^3-5*x^2+12*x)*log(x^2+4*x)+(3*x^4+12*x^3) 
*exp(x)^2+(-12*x^5-30*x^4+72*x^3)*exp(x)+12*x^6+12*x^5-117*x^4+108*x^3)/lo 
g((log(x^2+4*x)^2+(6*exp(x)*x^2-12*x^3+18*x^2)*log(x^2+4*x)+9*exp(x)^2*x^4 
+(-36*x^5+54*x^4)*exp(x)+36*x^6-108*x^5+81*x^4)/(exp(x)^2*x^3+(-4*x^4+6*x^ 
3)*exp(x)+4*x^5-12*x^4+9*x^3)),x)
 

Output:

log(log((9*e**(2*x)*x**4 + 6*e**x*log(x**2 + 4*x)*x**2 - 36*e**x*x**5 + 54 
*e**x*x**4 + log(x**2 + 4*x)**2 - 12*log(x**2 + 4*x)*x**3 + 18*log(x**2 + 
4*x)*x**2 + 36*x**6 - 108*x**5 + 81*x**4)/(e**(2*x)*x**3 - 4*e**x*x**4 + 6 
*e**x*x**3 + 4*x**5 - 12*x**4 + 9*x**3)))