\(\int \frac {-24 x^3+6 x^4+e^x (-4 x^3+x^4)+e^{4 x+2 x^2} (4 x^3+15 x^4+12 x^5-4 x^6)+(-4+x)^{2/x} (-24 x+6 x^2+e^{4 x+2 x^2} (4 x+15 x^2+12 x^3-4 x^4))+(-4+x)^{\frac {1}{x}} (-48 x^2+12 x^3+e^x (-5 x-3 x^2+x^3)+e^{4 x+2 x^2} (8 x^2+30 x^3+24 x^4-8 x^5)+e^x (-4+x) \log (-4+x))}{-4 x^3+x^4+(-4+x)^{2/x} (-4 x+x^2)+(-4+x)^{\frac {1}{x}} (-8 x^2+2 x^3)} \, dx\) [1525]

Optimal result

Integrand size = 234, antiderivative size = 29 \[ \int \frac {-24 x^3+6 x^4+e^x \left (-4 x^3+x^4\right )+e^{4 x+2 x^2} \left (4 x^3+15 x^4+12 x^5-4 x^6\right )+(-4+x)^{2/x} \left (-24 x+6 x^2+e^{4 x+2 x^2} \left (4 x+15 x^2+12 x^3-4 x^4\right )\right )+(-4+x)^{\frac {1}{x}} \left (-48 x^2+12 x^3+e^x \left (-5 x-3 x^2+x^3\right )+e^{4 x+2 x^2} \left (8 x^2+30 x^3+24 x^4-8 x^5\right )+e^x (-4+x) \log (-4+x)\right )}{-4 x^3+x^4+(-4+x)^{2/x} \left (-4 x+x^2\right )+(-4+x)^{\frac {1}{x}} \left (-8 x^2+2 x^3\right )} \, dx=x \left (6-e^{2 x (2+x)}+\frac {e^x}{(-4+x)^{\frac {1}{x}}+x}\right ) \] Output:

x*(exp(x)/(exp(ln(-4+x)/x)+x)+6-exp(x*(2+x))^2)
 

Mathematica [A] (verified)

Time = 0.15 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00 \[ \int \frac {-24 x^3+6 x^4+e^x \left (-4 x^3+x^4\right )+e^{4 x+2 x^2} \left (4 x^3+15 x^4+12 x^5-4 x^6\right )+(-4+x)^{2/x} \left (-24 x+6 x^2+e^{4 x+2 x^2} \left (4 x+15 x^2+12 x^3-4 x^4\right )\right )+(-4+x)^{\frac {1}{x}} \left (-48 x^2+12 x^3+e^x \left (-5 x-3 x^2+x^3\right )+e^{4 x+2 x^2} \left (8 x^2+30 x^3+24 x^4-8 x^5\right )+e^x (-4+x) \log (-4+x)\right )}{-4 x^3+x^4+(-4+x)^{2/x} \left (-4 x+x^2\right )+(-4+x)^{\frac {1}{x}} \left (-8 x^2+2 x^3\right )} \, dx=x \left (6-e^{2 x (2+x)}+\frac {e^x}{(-4+x)^{\frac {1}{x}}+x}\right ) \] Input:

Integrate[(-24*x^3 + 6*x^4 + E^x*(-4*x^3 + x^4) + E^(4*x + 2*x^2)*(4*x^3 + 
 15*x^4 + 12*x^5 - 4*x^6) + (-4 + x)^(2/x)*(-24*x + 6*x^2 + E^(4*x + 2*x^2 
)*(4*x + 15*x^2 + 12*x^3 - 4*x^4)) + (-4 + x)^x^(-1)*(-48*x^2 + 12*x^3 + E 
^x*(-5*x - 3*x^2 + x^3) + E^(4*x + 2*x^2)*(8*x^2 + 30*x^3 + 24*x^4 - 8*x^5 
) + E^x*(-4 + x)*Log[-4 + x]))/(-4*x^3 + x^4 + (-4 + x)^(2/x)*(-4*x + x^2) 
 + (-4 + x)^x^(-1)*(-8*x^2 + 2*x^3)),x]
 

Output:

x*(6 - E^(2*x*(2 + x)) + E^x/((-4 + x)^x^(-1) + x))
 

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.

\(\displaystyle \int \frac {6 x^4-24 x^3+e^x \left (x^4-4 x^3\right )+\left (6 x^2+e^{2 x^2+4 x} \left (-4 x^4+12 x^3+15 x^2+4 x\right )-24 x\right ) (x-4)^{2/x}+(x-4)^{\frac {1}{x}} \left (12 x^3-48 x^2+e^x \left (x^3-3 x^2-5 x\right )+e^{2 x^2+4 x} \left (-8 x^5+24 x^4+30 x^3+8 x^2\right )+e^x (x-4) \log (x-4)\right )+e^{2 x^2+4 x} \left (-4 x^6+12 x^5+15 x^4+4 x^3\right )}{x^4-4 x^3+\left (x^2-4 x\right ) (x-4)^{2/x}+\left (2 x^3-8 x^2\right ) (x-4)^{\frac {1}{x}}} \, dx\)

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 2009

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

Input:

Int[(-24*x^3 + 6*x^4 + E^x*(-4*x^3 + x^4) + E^(4*x + 2*x^2)*(4*x^3 + 15*x^ 
4 + 12*x^5 - 4*x^6) + (-4 + x)^(2/x)*(-24*x + 6*x^2 + E^(4*x + 2*x^2)*(4*x 
 + 15*x^2 + 12*x^3 - 4*x^4)) + (-4 + x)^x^(-1)*(-48*x^2 + 12*x^3 + E^x*(-5 
*x - 3*x^2 + x^3) + E^(4*x + 2*x^2)*(8*x^2 + 30*x^3 + 24*x^4 - 8*x^5) + E^ 
x*(-4 + x)*Log[-4 + x]))/(-4*x^3 + x^4 + (-4 + x)^(2/x)*(-4*x + x^2) + (-4 
 + x)^x^(-1)*(-8*x^2 + 2*x^3)),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 0.05 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.03

\[-{\mathrm e}^{2 x \left (2+x \right )} x +6 x +\frac {x \,{\mathrm e}^{x}}{\left (x -4\right )^{\frac {1}{x}}+x}\]

Input:

int((((-4*x^4+12*x^3+15*x^2+4*x)*exp(x^2+2*x)^2+6*x^2-24*x)*exp(ln(x-4)/x) 
^2+((x-4)*exp(x)*ln(x-4)+(-8*x^5+24*x^4+30*x^3+8*x^2)*exp(x^2+2*x)^2+(x^3- 
3*x^2-5*x)*exp(x)+12*x^3-48*x^2)*exp(ln(x-4)/x)+(-4*x^6+12*x^5+15*x^4+4*x^ 
3)*exp(x^2+2*x)^2+(x^4-4*x^3)*exp(x)+6*x^4-24*x^3)/((x^2-4*x)*exp(ln(x-4)/ 
x)^2+(2*x^3-8*x^2)*exp(ln(x-4)/x)+x^4-4*x^3),x)
 

Output:

-exp(2*x*(2+x))*x+6*x+x*exp(x)/((x-4)^(1/x)+x)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 62 vs. \(2 (27) = 54\).

Time = 0.11 (sec) , antiderivative size = 62, normalized size of antiderivative = 2.14 \[ \int \frac {-24 x^3+6 x^4+e^x \left (-4 x^3+x^4\right )+e^{4 x+2 x^2} \left (4 x^3+15 x^4+12 x^5-4 x^6\right )+(-4+x)^{2/x} \left (-24 x+6 x^2+e^{4 x+2 x^2} \left (4 x+15 x^2+12 x^3-4 x^4\right )\right )+(-4+x)^{\frac {1}{x}} \left (-48 x^2+12 x^3+e^x \left (-5 x-3 x^2+x^3\right )+e^{4 x+2 x^2} \left (8 x^2+30 x^3+24 x^4-8 x^5\right )+e^x (-4+x) \log (-4+x)\right )}{-4 x^3+x^4+(-4+x)^{2/x} \left (-4 x+x^2\right )+(-4+x)^{\frac {1}{x}} \left (-8 x^2+2 x^3\right )} \, dx=-\frac {x^{2} e^{\left (2 \, x^{2} + 4 \, x\right )} + {\left (x e^{\left (2 \, x^{2} + 4 \, x\right )} - 6 \, x\right )} {\left (x - 4\right )}^{\left (\frac {1}{x}\right )} - 6 \, x^{2} - x e^{x}}{{\left (x - 4\right )}^{\left (\frac {1}{x}\right )} + x} \] Input:

integrate((((-4*x^4+12*x^3+15*x^2+4*x)*exp(x^2+2*x)^2+6*x^2-24*x)*exp(log( 
-4+x)/x)^2+((-4+x)*exp(x)*log(-4+x)+(-8*x^5+24*x^4+30*x^3+8*x^2)*exp(x^2+2 
*x)^2+(x^3-3*x^2-5*x)*exp(x)+12*x^3-48*x^2)*exp(log(-4+x)/x)+(-4*x^6+12*x^ 
5+15*x^4+4*x^3)*exp(x^2+2*x)^2+(x^4-4*x^3)*exp(x)+6*x^4-24*x^3)/((x^2-4*x) 
*exp(log(-4+x)/x)^2+(2*x^3-8*x^2)*exp(log(-4+x)/x)+x^4-4*x^3),x, algorithm 
="fricas")
 

Output:

-(x^2*e^(2*x^2 + 4*x) + (x*e^(2*x^2 + 4*x) - 6*x)*(x - 4)^(1/x) - 6*x^2 - 
x*e^x)/((x - 4)^(1/x) + x)
 

Sympy [A] (verification not implemented)

Time = 0.95 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00 \[ \int \frac {-24 x^3+6 x^4+e^x \left (-4 x^3+x^4\right )+e^{4 x+2 x^2} \left (4 x^3+15 x^4+12 x^5-4 x^6\right )+(-4+x)^{2/x} \left (-24 x+6 x^2+e^{4 x+2 x^2} \left (4 x+15 x^2+12 x^3-4 x^4\right )\right )+(-4+x)^{\frac {1}{x}} \left (-48 x^2+12 x^3+e^x \left (-5 x-3 x^2+x^3\right )+e^{4 x+2 x^2} \left (8 x^2+30 x^3+24 x^4-8 x^5\right )+e^x (-4+x) \log (-4+x)\right )}{-4 x^3+x^4+(-4+x)^{2/x} \left (-4 x+x^2\right )+(-4+x)^{\frac {1}{x}} \left (-8 x^2+2 x^3\right )} \, dx=- x e^{2 x^{2} + 4 x} + 6 x + \frac {x e^{x}}{x + e^{\frac {\log {\left (x - 4 \right )}}{x}}} \] Input:

integrate((((-4*x**4+12*x**3+15*x**2+4*x)*exp(x**2+2*x)**2+6*x**2-24*x)*ex 
p(ln(-4+x)/x)**2+((-4+x)*exp(x)*ln(-4+x)+(-8*x**5+24*x**4+30*x**3+8*x**2)* 
exp(x**2+2*x)**2+(x**3-3*x**2-5*x)*exp(x)+12*x**3-48*x**2)*exp(ln(-4+x)/x) 
+(-4*x**6+12*x**5+15*x**4+4*x**3)*exp(x**2+2*x)**2+(x**4-4*x**3)*exp(x)+6* 
x**4-24*x**3)/((x**2-4*x)*exp(ln(-4+x)/x)**2+(2*x**3-8*x**2)*exp(ln(-4+x)/ 
x)+x**4-4*x**3),x)
 

Output:

-x*exp(2*x**2 + 4*x) + 6*x + x*exp(x)/(x + exp(log(x - 4)/x))
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 62 vs. \(2 (27) = 54\).

Time = 0.11 (sec) , antiderivative size = 62, normalized size of antiderivative = 2.14 \[ \int \frac {-24 x^3+6 x^4+e^x \left (-4 x^3+x^4\right )+e^{4 x+2 x^2} \left (4 x^3+15 x^4+12 x^5-4 x^6\right )+(-4+x)^{2/x} \left (-24 x+6 x^2+e^{4 x+2 x^2} \left (4 x+15 x^2+12 x^3-4 x^4\right )\right )+(-4+x)^{\frac {1}{x}} \left (-48 x^2+12 x^3+e^x \left (-5 x-3 x^2+x^3\right )+e^{4 x+2 x^2} \left (8 x^2+30 x^3+24 x^4-8 x^5\right )+e^x (-4+x) \log (-4+x)\right )}{-4 x^3+x^4+(-4+x)^{2/x} \left (-4 x+x^2\right )+(-4+x)^{\frac {1}{x}} \left (-8 x^2+2 x^3\right )} \, dx=-\frac {x^{2} e^{\left (2 \, x^{2} + 4 \, x\right )} + {\left (x e^{\left (2 \, x^{2} + 4 \, x\right )} - 6 \, x\right )} {\left (x - 4\right )}^{\left (\frac {1}{x}\right )} - 6 \, x^{2} - x e^{x}}{{\left (x - 4\right )}^{\left (\frac {1}{x}\right )} + x} \] Input:

integrate((((-4*x^4+12*x^3+15*x^2+4*x)*exp(x^2+2*x)^2+6*x^2-24*x)*exp(log( 
-4+x)/x)^2+((-4+x)*exp(x)*log(-4+x)+(-8*x^5+24*x^4+30*x^3+8*x^2)*exp(x^2+2 
*x)^2+(x^3-3*x^2-5*x)*exp(x)+12*x^3-48*x^2)*exp(log(-4+x)/x)+(-4*x^6+12*x^ 
5+15*x^4+4*x^3)*exp(x^2+2*x)^2+(x^4-4*x^3)*exp(x)+6*x^4-24*x^3)/((x^2-4*x) 
*exp(log(-4+x)/x)^2+(2*x^3-8*x^2)*exp(log(-4+x)/x)+x^4-4*x^3),x, algorithm 
="maxima")
 

Output:

-(x^2*e^(2*x^2 + 4*x) + (x*e^(2*x^2 + 4*x) - 6*x)*(x - 4)^(1/x) - 6*x^2 - 
x*e^x)/((x - 4)^(1/x) + x)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 6706 vs. \(2 (27) = 54\).

Time = 0.85 (sec) , antiderivative size = 6706, normalized size of antiderivative = 231.24 \[ \int \frac {-24 x^3+6 x^4+e^x \left (-4 x^3+x^4\right )+e^{4 x+2 x^2} \left (4 x^3+15 x^4+12 x^5-4 x^6\right )+(-4+x)^{2/x} \left (-24 x+6 x^2+e^{4 x+2 x^2} \left (4 x+15 x^2+12 x^3-4 x^4\right )\right )+(-4+x)^{\frac {1}{x}} \left (-48 x^2+12 x^3+e^x \left (-5 x-3 x^2+x^3\right )+e^{4 x+2 x^2} \left (8 x^2+30 x^3+24 x^4-8 x^5\right )+e^x (-4+x) \log (-4+x)\right )}{-4 x^3+x^4+(-4+x)^{2/x} \left (-4 x+x^2\right )+(-4+x)^{\frac {1}{x}} \left (-8 x^2+2 x^3\right )} \, dx=\text {Too large to display} \] Input:

integrate((((-4*x^4+12*x^3+15*x^2+4*x)*exp(x^2+2*x)^2+6*x^2-24*x)*exp(log( 
-4+x)/x)^2+((-4+x)*exp(x)*log(-4+x)+(-8*x^5+24*x^4+30*x^3+8*x^2)*exp(x^2+2 
*x)^2+(x^3-3*x^2-5*x)*exp(x)+12*x^3-48*x^2)*exp(log(-4+x)/x)+(-4*x^6+12*x^ 
5+15*x^4+4*x^3)*exp(x^2+2*x)^2+(x^4-4*x^3)*exp(x)+6*x^4-24*x^3)/((x^2-4*x) 
*exp(log(-4+x)/x)^2+(2*x^3-8*x^2)*exp(log(-4+x)/x)+x^4-4*x^3),x, algorithm 
="giac")
 

Output:

-(x - 4)^((x - 4)/x)*(x - 4)^(5/x)*(x - 4)^4*e^(2*(x - 4)^3/x + 28*(x - 4) 
^2/x + 80*(x - 4)/x + 48)/((x - 4)^(1/x)*(x - 4)^4 + (x - 4)^(1/x)*(x - 4) 
^3*log(x - 4) + (x - 4)^(2/x)*(x - 4)^3 + 7*(x - 4)^(1/x)*(x - 4)^3 + (x - 
 4)^(2/x)*(x - 4)^2*log(x - 4) + 4*(x - 4)^(1/x)*(x - 4)^2*log(x - 4) + 3* 
(x - 4)^(2/x)*(x - 4)^2 + 8*(x - 4)^(1/x)*(x - 4)^2 - 4*(x - 4)^(2/x)*(x - 
 4) - 16*(x - 4)^(1/x)*(x - 4)) + (x - 4)^(1/x)*(x - 4)^5*e^((x - 4)^2/x + 
 4*(x - 4)/x + 4)/((x - 4)^(1/x)*(x - 4)^4 + (x - 4)^(1/x)*(x - 4)^3*log(x 
 - 4) + (x - 4)^(2/x)*(x - 4)^3 + 7*(x - 4)^(1/x)*(x - 4)^3 + (x - 4)^(2/x 
)*(x - 4)^2*log(x - 4) + 4*(x - 4)^(1/x)*(x - 4)^2*log(x - 4) + 3*(x - 4)^ 
(2/x)*(x - 4)^2 + 8*(x - 4)^(1/x)*(x - 4)^2 - 4*(x - 4)^(2/x)*(x - 4) - 16 
*(x - 4)^(1/x)*(x - 4)) - (x - 4)^(1/x)*(x - 4)^5*e^x/((x - 4)^(1/x)*(x - 
4)^4 + (x - 4)^(1/x)*(x - 4)^3*log(x - 4) + (x - 4)^(2/x)*(x - 4)^3 + 7*(x 
 - 4)^(1/x)*(x - 4)^3 + (x - 4)^(2/x)*(x - 4)^2*log(x - 4) + 4*(x - 4)^(1/ 
x)*(x - 4)^2*log(x - 4) + 3*(x - 4)^(2/x)*(x - 4)^2 + 8*(x - 4)^(1/x)*(x - 
 4)^2 - 4*(x - 4)^(2/x)*(x - 4) - 16*(x - 4)^(1/x)*(x - 4)) - (x - 4)^((x 
- 4)/x)*(x - 4)^(5/x)*(x - 4)^3*e^(2*(x - 4)^3/x + 28*(x - 4)^2/x + 80*(x 
- 4)/x + 48)*log(x - 4)/((x - 4)^(1/x)*(x - 4)^4 + (x - 4)^(1/x)*(x - 4)^3 
*log(x - 4) + (x - 4)^(2/x)*(x - 4)^3 + 7*(x - 4)^(1/x)*(x - 4)^3 + (x - 4 
)^(2/x)*(x - 4)^2*log(x - 4) + 4*(x - 4)^(1/x)*(x - 4)^2*log(x - 4) + 3*(x 
 - 4)^(2/x)*(x - 4)^2 + 8*(x - 4)^(1/x)*(x - 4)^2 - 4*(x - 4)^(2/x)*(x ...
 

Mupad [F(-1)]

Timed out. \[ \int \frac {-24 x^3+6 x^4+e^x \left (-4 x^3+x^4\right )+e^{4 x+2 x^2} \left (4 x^3+15 x^4+12 x^5-4 x^6\right )+(-4+x)^{2/x} \left (-24 x+6 x^2+e^{4 x+2 x^2} \left (4 x+15 x^2+12 x^3-4 x^4\right )\right )+(-4+x)^{\frac {1}{x}} \left (-48 x^2+12 x^3+e^x \left (-5 x-3 x^2+x^3\right )+e^{4 x+2 x^2} \left (8 x^2+30 x^3+24 x^4-8 x^5\right )+e^x (-4+x) \log (-4+x)\right )}{-4 x^3+x^4+(-4+x)^{2/x} \left (-4 x+x^2\right )+(-4+x)^{\frac {1}{x}} \left (-8 x^2+2 x^3\right )} \, dx=\int -\frac {{\mathrm {e}}^{2\,x^2+4\,x}\,\left (-4\,x^6+12\,x^5+15\,x^4+4\,x^3\right )-{\mathrm {e}}^x\,\left (4\,x^3-x^4\right )+{\mathrm {e}}^{\frac {\ln \left (x-4\right )}{x}}\,\left ({\mathrm {e}}^{2\,x^2+4\,x}\,\left (-8\,x^5+24\,x^4+30\,x^3+8\,x^2\right )-48\,x^2+12\,x^3-{\mathrm {e}}^x\,\left (-x^3+3\,x^2+5\,x\right )+\ln \left (x-4\right )\,{\mathrm {e}}^x\,\left (x-4\right )\right )+{\mathrm {e}}^{\frac {2\,\ln \left (x-4\right )}{x}}\,\left ({\mathrm {e}}^{2\,x^2+4\,x}\,\left (-4\,x^4+12\,x^3+15\,x^2+4\,x\right )-24\,x+6\,x^2\right )-24\,x^3+6\,x^4}{{\mathrm {e}}^{\frac {\ln \left (x-4\right )}{x}}\,\left (8\,x^2-2\,x^3\right )+{\mathrm {e}}^{\frac {2\,\ln \left (x-4\right )}{x}}\,\left (4\,x-x^2\right )+4\,x^3-x^4} \,d x \] Input:

int(-(exp(4*x + 2*x^2)*(4*x^3 + 15*x^4 + 12*x^5 - 4*x^6) - exp(x)*(4*x^3 - 
 x^4) + exp(log(x - 4)/x)*(exp(4*x + 2*x^2)*(8*x^2 + 30*x^3 + 24*x^4 - 8*x 
^5) - 48*x^2 + 12*x^3 - exp(x)*(5*x + 3*x^2 - x^3) + log(x - 4)*exp(x)*(x 
- 4)) + exp((2*log(x - 4))/x)*(exp(4*x + 2*x^2)*(4*x + 15*x^2 + 12*x^3 - 4 
*x^4) - 24*x + 6*x^2) - 24*x^3 + 6*x^4)/(exp(log(x - 4)/x)*(8*x^2 - 2*x^3) 
 + exp((2*log(x - 4))/x)*(4*x - x^2) + 4*x^3 - x^4),x)
 

Output:

int(-(exp(4*x + 2*x^2)*(4*x^3 + 15*x^4 + 12*x^5 - 4*x^6) - exp(x)*(4*x^3 - 
 x^4) + exp(log(x - 4)/x)*(exp(4*x + 2*x^2)*(8*x^2 + 30*x^3 + 24*x^4 - 8*x 
^5) - 48*x^2 + 12*x^3 - exp(x)*(5*x + 3*x^2 - x^3) + log(x - 4)*exp(x)*(x 
- 4)) + exp((2*log(x - 4))/x)*(exp(4*x + 2*x^2)*(4*x + 15*x^2 + 12*x^3 - 4 
*x^4) - 24*x + 6*x^2) - 24*x^3 + 6*x^4)/(exp(log(x - 4)/x)*(8*x^2 - 2*x^3) 
 + exp((2*log(x - 4))/x)*(4*x - x^2) + 4*x^3 - x^4), x)
 

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 72, normalized size of antiderivative = 2.48 \[ \int \frac {-24 x^3+6 x^4+e^x \left (-4 x^3+x^4\right )+e^{4 x+2 x^2} \left (4 x^3+15 x^4+12 x^5-4 x^6\right )+(-4+x)^{2/x} \left (-24 x+6 x^2+e^{4 x+2 x^2} \left (4 x+15 x^2+12 x^3-4 x^4\right )\right )+(-4+x)^{\frac {1}{x}} \left (-48 x^2+12 x^3+e^x \left (-5 x-3 x^2+x^3\right )+e^{4 x+2 x^2} \left (8 x^2+30 x^3+24 x^4-8 x^5\right )+e^x (-4+x) \log (-4+x)\right )}{-4 x^3+x^4+(-4+x)^{2/x} \left (-4 x+x^2\right )+(-4+x)^{\frac {1}{x}} \left (-8 x^2+2 x^3\right )} \, dx=\frac {x \left (-e^{\frac {\mathrm {log}\left (x -4\right )+2 x^{3}+4 x^{2}}{x}}-e^{2 x^{2}+4 x} x +6 e^{\frac {\mathrm {log}\left (x -4\right )}{x}}+e^{x}+6 x \right )}{e^{\frac {\mathrm {log}\left (x -4\right )}{x}}+x} \] Input:

int((((-4*x^4+12*x^3+15*x^2+4*x)*exp(x^2+2*x)^2+6*x^2-24*x)*exp(log(-4+x)/ 
x)^2+((-4+x)*exp(x)*log(-4+x)+(-8*x^5+24*x^4+30*x^3+8*x^2)*exp(x^2+2*x)^2+ 
(x^3-3*x^2-5*x)*exp(x)+12*x^3-48*x^2)*exp(log(-4+x)/x)+(-4*x^6+12*x^5+15*x 
^4+4*x^3)*exp(x^2+2*x)^2+(x^4-4*x^3)*exp(x)+6*x^4-24*x^3)/((x^2-4*x)*exp(l 
og(-4+x)/x)^2+(2*x^3-8*x^2)*exp(log(-4+x)/x)+x^4-4*x^3),x)
 

Output:

(x*( - e**((log(x - 4) + 2*x**3 + 4*x**2)/x) - e**(2*x**2 + 4*x)*x + 6*e** 
(log(x - 4)/x) + e**x + 6*x))/(e**(log(x - 4)/x) + x)