\(\int \frac {-\frac {e^{-4+4 x}}{x^4}+\frac {4 e^{-3+3 x}}{x^2}-17 x^2+34 x^3-x^4+\frac {e^{-2+2 x} (-30 x+26 x^2)}{x^2}+\frac {e^{-1+x} (-4 x^2-28 x^3)}{x}}{-4 e^{-3+3 x}+\frac {e^{-4+4 x}}{x^2}+x^4-2 x^5+x^6+\frac {e^{-2+2 x} (-2 x^3+6 x^4)}{x^2}+\frac {e^{-1+x} (4 x^4-4 x^5)}{x}} \, dx\) [311]

Optimal result

Integrand size = 157, antiderivative size = 31 \[ \int \frac {-\frac {e^{-4+4 x}}{x^4}+\frac {4 e^{-3+3 x}}{x^2}-17 x^2+34 x^3-x^4+\frac {e^{-2+2 x} \left (-30 x+26 x^2\right )}{x^2}+\frac {e^{-1+x} \left (-4 x^2-28 x^3\right )}{x}}{-4 e^{-3+3 x}+\frac {e^{-4+4 x}}{x^2}+x^4-2 x^5+x^6+\frac {e^{-2+2 x} \left (-2 x^3+6 x^4\right )}{x^2}+\frac {e^{-1+x} \left (4 x^4-4 x^5\right )}{x}} \, dx=1+\frac {1}{x}+4 \left (4+\frac {4}{x-\left (-\frac {e^{-1+x}}{x}+x\right )^2}\right ) \] Output:

17+1/x+16/(x-(x-exp(x-ln(x)-1))^2)
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 8.36 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.29 \[ \int \frac {-\frac {e^{-4+4 x}}{x^4}+\frac {4 e^{-3+3 x}}{x^2}-17 x^2+34 x^3-x^4+\frac {e^{-2+2 x} \left (-30 x+26 x^2\right )}{x^2}+\frac {e^{-1+x} \left (-4 x^2-28 x^3\right )}{x}}{-4 e^{-3+3 x}+\frac {e^{-4+4 x}}{x^2}+x^4-2 x^5+x^6+\frac {e^{-2+2 x} \left (-2 x^3+6 x^4\right )}{x^2}+\frac {e^{-1+x} \left (4 x^4-4 x^5\right )}{x}} \, dx=\frac {1}{x}-\frac {16 e^2 x^2}{e^{2 x}-2 e^{1+x} x^2+e^2 (-1+x) x^3} \] Input:

Integrate[(-(E^(-4 + 4*x)/x^4) + (4*E^(-3 + 3*x))/x^2 - 17*x^2 + 34*x^3 - 
x^4 + (E^(-2 + 2*x)*(-30*x + 26*x^2))/x^2 + (E^(-1 + x)*(-4*x^2 - 28*x^3)) 
/x)/(-4*E^(-3 + 3*x) + E^(-4 + 4*x)/x^2 + x^4 - 2*x^5 + x^6 + (E^(-2 + 2*x 
)*(-2*x^3 + 6*x^4))/x^2 + (E^(-1 + x)*(4*x^4 - 4*x^5))/x),x]
 

Output:

x^(-1) - (16*E^2*x^2)/(E^(2*x) - 2*E^(1 + x)*x^2 + E^2*(-1 + x)*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.

\(\displaystyle \int \frac {-x^4-\frac {e^{4 x-4}}{x^4}+34 x^3-17 x^2+\frac {4 e^{3 x-3}}{x^2}+\frac {e^{2 x-2} \left (26 x^2-30 x\right )}{x^2}+\frac {e^{x-1} \left (-28 x^3-4 x^2\right )}{x}}{x^6-2 x^5+x^4+\frac {e^{4 x-4}}{x^2}+\frac {e^{x-1} \left (4 x^4-4 x^5\right )}{x}+\frac {e^{2 x-2} \left (6 x^4-2 x^3\right )}{x^2}-4 e^{3 x-3}} \, dx\)

\(\Big \downarrow \) 7292

\(\displaystyle \int \frac {e^4 x^2 \left (-x^4-\frac {e^{4 x-4}}{x^4}+34 x^3-17 x^2+\frac {4 e^{3 x-3}}{x^2}+\frac {e^{2 x-2} \left (26 x^2-30 x\right )}{x^2}+\frac {e^{x-1} \left (-28 x^3-4 x^2\right )}{x}\right )}{\left (e^2 x^4-e^2 x^3-2 e^{x+1} x^2+e^{2 x}\right )^2}dx\)

\(\Big \downarrow \) 27

\(\displaystyle e^4 \int -\frac {x^2 \left (x^4-34 x^3+17 x^2+\frac {4 e^{x-1} \left (7 x^3+x^2\right )}{x}-\frac {4 e^{3 x-3}}{x^2}+\frac {2 e^{2 x-2} \left (15 x-13 x^2\right )}{x^2}+\frac {e^{4 x-4}}{x^4}\right )}{\left (e^2 x^4-e^2 x^3-2 e^{x+1} x^2+e^{2 x}\right )^2}dx\)

\(\Big \downarrow \) 25

\(\displaystyle -e^4 \int \frac {x^2 \left (x^4-34 x^3+17 x^2+\frac {4 e^{x-1} \left (7 x^3+x^2\right )}{x}-\frac {4 e^{3 x-3}}{x^2}+\frac {2 e^{2 x-2} \left (15 x-13 x^2\right )}{x^2}+\frac {e^{4 x-4}}{x^4}\right )}{\left (e^2 x^4-e^2 x^3-2 e^{x+1} x^2+e^{2 x}\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle -e^4 \int \left (\frac {16 \left (2 e x^3-6 e x^2-2 e^x x+3 e x+4 e^x\right ) x^3}{e \left (e^2 x^4-e^2 x^3-2 e^{x+1} x^2+e^{2 x}\right )^2}-\frac {32 (x-1) x}{e^2 \left (e^2 x^4-e^2 x^3-2 e^{x+1} x^2+e^{2 x}\right )}+\frac {1}{e^4 x^2}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -e^4 \left (\frac {64 \int \frac {e^x x^3}{\left (e^2 x^4-e^2 x^3-2 e^{x+1} x^2+e^{2 x}\right )^2}dx}{e}+48 \int \frac {x^4}{\left (e^2 x^4-e^2 x^3-2 e^{x+1} x^2+e^{2 x}\right )^2}dx-\frac {32 \int \frac {e^x x^4}{\left (e^2 x^4-e^2 x^3-2 e^{x+1} x^2+e^{2 x}\right )^2}dx}{e}+\frac {32 \int \frac {x}{e^2 x^4-e^2 x^3-2 e^{x+1} x^2+e^{2 x}}dx}{e^2}-\frac {32 \int \frac {x^2}{e^2 x^4-e^2 x^3-2 e^{x+1} x^2+e^{2 x}}dx}{e^2}+32 \int \frac {x^6}{\left (e^2 x^4-e^2 x^3-2 e^{x+1} x^2+e^{2 x}\right )^2}dx-96 \int \frac {x^5}{\left (e^2 x^4-e^2 x^3-2 e^{x+1} x^2+e^{2 x}\right )^2}dx-\frac {1}{e^4 x}\right )\)

Input:

Int[(-(E^(-4 + 4*x)/x^4) + (4*E^(-3 + 3*x))/x^2 - 17*x^2 + 34*x^3 - x^4 + 
(E^(-2 + 2*x)*(-30*x + 26*x^2))/x^2 + (E^(-1 + x)*(-4*x^2 - 28*x^3))/x)/(- 
4*E^(-3 + 3*x) + E^(-4 + 4*x)/x^2 + x^4 - 2*x^5 + x^6 + (E^(-2 + 2*x)*(-2* 
x^3 + 6*x^4))/x^2 + (E^(-1 + x)*(4*x^4 - 4*x^5))/x),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 0.60 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.03

method	result	size
risch	\(\frac {1}{x}-\frac {16}{\frac {{\mathrm e}^{-2+2 x}}{x^{2}}-2 \,{\mathrm e}^{-1+x}+x^{2}-x}\)	\(32\)
parallelrisch	\(\frac {x^{2}-2 x \,{\mathrm e}^{x -\ln \left (x \right )-1}+\frac {{\mathrm e}^{-2+2 x}}{x^{2}}-17 x}{x \left (\frac {{\mathrm e}^{-2+2 x}}{x^{2}}-2 x \,{\mathrm e}^{x -\ln \left (x \right )-1}+x^{2}-x \right )}\)	\(63\)

Input:

int((-exp(x-ln(x)-1)^4+4*x*exp(x-ln(x)-1)^3+(26*x^2-30*x)*exp(x-ln(x)-1)^2 
+(-28*x^3-4*x^2)*exp(x-ln(x)-1)-x^4+34*x^3-17*x^2)/(x^2*exp(x-ln(x)-1)^4-4 
*x^3*exp(x-ln(x)-1)^3+(6*x^4-2*x^3)*exp(x-ln(x)-1)^2+(-4*x^5+4*x^4)*exp(x- 
ln(x)-1)+x^6-2*x^5+x^4),x,method=_RETURNVERBOSE)
 

Output:

1/x-16/(1/x^2*exp(-2+2*x)-2*exp(-1+x)+x^2-x)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 65 vs. \(2 (27) = 54\).

Time = 0.09 (sec) , antiderivative size = 65, normalized size of antiderivative = 2.10 \[ \int \frac {-\frac {e^{-4+4 x}}{x^4}+\frac {4 e^{-3+3 x}}{x^2}-17 x^2+34 x^3-x^4+\frac {e^{-2+2 x} \left (-30 x+26 x^2\right )}{x^2}+\frac {e^{-1+x} \left (-4 x^2-28 x^3\right )}{x}}{-4 e^{-3+3 x}+\frac {e^{-4+4 x}}{x^2}+x^4-2 x^5+x^6+\frac {e^{-2+2 x} \left (-2 x^3+6 x^4\right )}{x^2}+\frac {e^{-1+x} \left (4 x^4-4 x^5\right )}{x}} \, dx=\frac {x^{2} - 2 \, x e^{\left (x - \log \left (x\right ) - 1\right )} - 17 \, x + e^{\left (2 \, x - 2 \, \log \left (x\right ) - 2\right )}}{x^{3} - 2 \, x^{2} e^{\left (x - \log \left (x\right ) - 1\right )} - x^{2} + x e^{\left (2 \, x - 2 \, \log \left (x\right ) - 2\right )}} \] Input:

integrate((-exp(x-log(x)-1)^4+4*x*exp(x-log(x)-1)^3+(26*x^2-30*x)*exp(x-lo 
g(x)-1)^2+(-28*x^3-4*x^2)*exp(x-log(x)-1)-x^4+34*x^3-17*x^2)/(x^2*exp(x-lo 
g(x)-1)^4-4*x^3*exp(x-log(x)-1)^3+(6*x^4-2*x^3)*exp(x-log(x)-1)^2+(-4*x^5+ 
4*x^4)*exp(x-log(x)-1)+x^6-2*x^5+x^4),x, algorithm="fricas")
 

Output:

(x^2 - 2*x*e^(x - log(x) - 1) - 17*x + e^(2*x - 2*log(x) - 2))/(x^3 - 2*x^ 
2*e^(x - log(x) - 1) - x^2 + x*e^(2*x - 2*log(x) - 2))
 

Sympy [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00 \[ \int \frac {-\frac {e^{-4+4 x}}{x^4}+\frac {4 e^{-3+3 x}}{x^2}-17 x^2+34 x^3-x^4+\frac {e^{-2+2 x} \left (-30 x+26 x^2\right )}{x^2}+\frac {e^{-1+x} \left (-4 x^2-28 x^3\right )}{x}}{-4 e^{-3+3 x}+\frac {e^{-4+4 x}}{x^2}+x^4-2 x^5+x^6+\frac {e^{-2+2 x} \left (-2 x^3+6 x^4\right )}{x^2}+\frac {e^{-1+x} \left (4 x^4-4 x^5\right )}{x}} \, dx=- \frac {16 x^{2}}{x^{4} - x^{3} - 2 x^{2} e^{x - 1} + e^{2 x - 2}} + \frac {1}{x} \] Input:

integrate((-exp(x-ln(x)-1)**4+4*x*exp(x-ln(x)-1)**3+(26*x**2-30*x)*exp(x-l 
n(x)-1)**2+(-28*x**3-4*x**2)*exp(x-ln(x)-1)-x**4+34*x**3-17*x**2)/(x**2*ex 
p(x-ln(x)-1)**4-4*x**3*exp(x-ln(x)-1)**3+(6*x**4-2*x**3)*exp(x-ln(x)-1)**2 
+(-4*x**5+4*x**4)*exp(x-ln(x)-1)+x**6-2*x**5+x**4),x)
 

Output:

-16*x**2/(x**4 - x**3 - 2*x**2*exp(x - 1) + exp(2*x - 2)) + 1/x
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 59 vs. \(2 (27) = 54\).

Time = 0.10 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.90 \[ \int \frac {-\frac {e^{-4+4 x}}{x^4}+\frac {4 e^{-3+3 x}}{x^2}-17 x^2+34 x^3-x^4+\frac {e^{-2+2 x} \left (-30 x+26 x^2\right )}{x^2}+\frac {e^{-1+x} \left (-4 x^2-28 x^3\right )}{x}}{-4 e^{-3+3 x}+\frac {e^{-4+4 x}}{x^2}+x^4-2 x^5+x^6+\frac {e^{-2+2 x} \left (-2 x^3+6 x^4\right )}{x^2}+\frac {e^{-1+x} \left (4 x^4-4 x^5\right )}{x}} \, dx=\frac {x^{4} e^{2} - 17 \, x^{3} e^{2} - 2 \, x^{2} e^{\left (x + 1\right )} + e^{\left (2 \, x\right )}}{x^{5} e^{2} - x^{4} e^{2} - 2 \, x^{3} e^{\left (x + 1\right )} + x e^{\left (2 \, x\right )}} \] Input:

integrate((-exp(x-log(x)-1)^4+4*x*exp(x-log(x)-1)^3+(26*x^2-30*x)*exp(x-lo 
g(x)-1)^2+(-28*x^3-4*x^2)*exp(x-log(x)-1)-x^4+34*x^3-17*x^2)/(x^2*exp(x-lo 
g(x)-1)^4-4*x^3*exp(x-log(x)-1)^3+(6*x^4-2*x^3)*exp(x-log(x)-1)^2+(-4*x^5+ 
4*x^4)*exp(x-log(x)-1)+x^6-2*x^5+x^4),x, algorithm="maxima")
 

Output:

(x^4*e^2 - 17*x^3*e^2 - 2*x^2*e^(x + 1) + e^(2*x))/(x^5*e^2 - x^4*e^2 - 2* 
x^3*e^(x + 1) + x*e^(2*x))
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 59 vs. \(2 (27) = 54\).

Time = 0.15 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.90 \[ \int \frac {-\frac {e^{-4+4 x}}{x^4}+\frac {4 e^{-3+3 x}}{x^2}-17 x^2+34 x^3-x^4+\frac {e^{-2+2 x} \left (-30 x+26 x^2\right )}{x^2}+\frac {e^{-1+x} \left (-4 x^2-28 x^3\right )}{x}}{-4 e^{-3+3 x}+\frac {e^{-4+4 x}}{x^2}+x^4-2 x^5+x^6+\frac {e^{-2+2 x} \left (-2 x^3+6 x^4\right )}{x^2}+\frac {e^{-1+x} \left (4 x^4-4 x^5\right )}{x}} \, dx=\frac {x^{4} e^{2} - 33 \, x^{3} e^{2} - 2 \, x^{2} e^{\left (x + 1\right )} + e^{\left (2 \, x\right )}}{x^{5} e^{2} - x^{4} e^{2} - 2 \, x^{3} e^{\left (x + 1\right )} + x e^{\left (2 \, x\right )}} \] Input:

integrate((-exp(x-log(x)-1)^4+4*x*exp(x-log(x)-1)^3+(26*x^2-30*x)*exp(x-lo 
g(x)-1)^2+(-28*x^3-4*x^2)*exp(x-log(x)-1)-x^4+34*x^3-17*x^2)/(x^2*exp(x-lo 
g(x)-1)^4-4*x^3*exp(x-log(x)-1)^3+(6*x^4-2*x^3)*exp(x-log(x)-1)^2+(-4*x^5+ 
4*x^4)*exp(x-log(x)-1)+x^6-2*x^5+x^4),x, algorithm="giac")
 

Output:

(x^4*e^2 - 33*x^3*e^2 - 2*x^2*e^(x + 1) + e^(2*x))/(x^5*e^2 - x^4*e^2 - 2* 
x^3*e^(x + 1) + x*e^(2*x))
 

Mupad [F(-1)]

Timed out. \[ \int \frac {-\frac {e^{-4+4 x}}{x^4}+\frac {4 e^{-3+3 x}}{x^2}-17 x^2+34 x^3-x^4+\frac {e^{-2+2 x} \left (-30 x+26 x^2\right )}{x^2}+\frac {e^{-1+x} \left (-4 x^2-28 x^3\right )}{x}}{-4 e^{-3+3 x}+\frac {e^{-4+4 x}}{x^2}+x^4-2 x^5+x^6+\frac {e^{-2+2 x} \left (-2 x^3+6 x^4\right )}{x^2}+\frac {e^{-1+x} \left (4 x^4-4 x^5\right )}{x}} \, dx=\int -\frac {{\mathrm {e}}^{4\,x-4\,\ln \left (x\right )-4}+{\mathrm {e}}^{2\,x-2\,\ln \left (x\right )-2}\,\left (30\,x-26\,x^2\right )+{\mathrm {e}}^{x-\ln \left (x\right )-1}\,\left (28\,x^3+4\,x^2\right )-4\,x\,{\mathrm {e}}^{3\,x-3\,\ln \left (x\right )-3}+17\,x^2-34\,x^3+x^4}{x^2\,{\mathrm {e}}^{4\,x-4\,\ln \left (x\right )-4}-4\,x^3\,{\mathrm {e}}^{3\,x-3\,\ln \left (x\right )-3}-{\mathrm {e}}^{2\,x-2\,\ln \left (x\right )-2}\,\left (2\,x^3-6\,x^4\right )+{\mathrm {e}}^{x-\ln \left (x\right )-1}\,\left (4\,x^4-4\,x^5\right )+x^4-2\,x^5+x^6} \,d x \] Input:

int(-(exp(4*x - 4*log(x) - 4) + exp(2*x - 2*log(x) - 2)*(30*x - 26*x^2) + 
exp(x - log(x) - 1)*(4*x^2 + 28*x^3) - 4*x*exp(3*x - 3*log(x) - 3) + 17*x^ 
2 - 34*x^3 + x^4)/(x^2*exp(4*x - 4*log(x) - 4) - 4*x^3*exp(3*x - 3*log(x) 
- 3) - exp(2*x - 2*log(x) - 2)*(2*x^3 - 6*x^4) + exp(x - log(x) - 1)*(4*x^ 
4 - 4*x^5) + x^4 - 2*x^5 + x^6),x)
 

Output:

int(-(exp(4*x - 4*log(x) - 4) + exp(2*x - 2*log(x) - 2)*(30*x - 26*x^2) + 
exp(x - log(x) - 1)*(4*x^2 + 28*x^3) - 4*x*exp(3*x - 3*log(x) - 3) + 17*x^ 
2 - 34*x^3 + x^4)/(x^2*exp(4*x - 4*log(x) - 4) - 4*x^3*exp(3*x - 3*log(x) 
- 3) - exp(2*x - 2*log(x) - 2)*(2*x^3 - 6*x^4) + exp(x - log(x) - 1)*(4*x^ 
4 - 4*x^5) + x^4 - 2*x^5 + x^6), x)
 

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 66, normalized size of antiderivative = 2.13 \[ \int \frac {-\frac {e^{-4+4 x}}{x^4}+\frac {4 e^{-3+3 x}}{x^2}-17 x^2+34 x^3-x^4+\frac {e^{-2+2 x} \left (-30 x+26 x^2\right )}{x^2}+\frac {e^{-1+x} \left (-4 x^2-28 x^3\right )}{x}}{-4 e^{-3+3 x}+\frac {e^{-4+4 x}}{x^2}+x^4-2 x^5+x^6+\frac {e^{-2+2 x} \left (-2 x^3+6 x^4\right )}{x^2}+\frac {e^{-1+x} \left (4 x^4-4 x^5\right )}{x}} \, dx=\frac {e^{2 x}-2 e^{x} e \,x^{2}+e^{2} x^{4}-17 e^{2} x^{3}}{x \left (e^{2 x}-2 e^{x} e \,x^{2}+e^{2} x^{4}-e^{2} x^{3}\right )} \] Input:

int((-exp(x-log(x)-1)^4+4*x*exp(x-log(x)-1)^3+(26*x^2-30*x)*exp(x-log(x)-1 
)^2+(-28*x^3-4*x^2)*exp(x-log(x)-1)-x^4+34*x^3-17*x^2)/(x^2*exp(x-log(x)-1 
)^4-4*x^3*exp(x-log(x)-1)^3+(6*x^4-2*x^3)*exp(x-log(x)-1)^2+(-4*x^5+4*x^4) 
*exp(x-log(x)-1)+x^6-2*x^5+x^4),x)
 

Output:

(e**(2*x) - 2*e**x*e*x**2 + e**2*x**4 - 17*e**2*x**3)/(x*(e**(2*x) - 2*e** 
x*e*x**2 + e**2*x**4 - e**2*x**3))