Integrand size = 116, antiderivative size = 31 \[ \int \frac {e^x \left (-304+152 x^2-19 x^4+e^2 \left (-304+152 x^2-19 x^4\right )\right )+e^{2 x} \left (28-5 x^2+x^4+e^2 \left (28-5 x^2+x^4\right )\right )}{5776-2888 x^2+361 x^4+e^x \left (1064 x-418 x^3+38 x^5\right )+e^{2 x} \left (49 x^2-14 x^4+x^6\right )} \, dx=\frac {1+e^2}{-19 e^{-x}+x-x \left (2-\frac {3}{-4+x^2}\right )} \] Output:
(exp(2)+1)/(x-19/exp(x)-x*(2-3/(x^2-4)))
Time = 10.99 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.13 \[ \int \frac {e^x \left (-304+152 x^2-19 x^4+e^2 \left (-304+152 x^2-19 x^4\right )\right )+e^{2 x} \left (28-5 x^2+x^4+e^2 \left (28-5 x^2+x^4\right )\right )}{5776-2888 x^2+361 x^4+e^x \left (1064 x-418 x^3+38 x^5\right )+e^{2 x} \left (49 x^2-14 x^4+x^6\right )} \, dx=-\frac {e^x \left (1+e^2\right ) \left (-4+x^2\right )}{e^x x \left (-7+x^2\right )+19 \left (-4+x^2\right )} \] Input:
Integrate[(E^x*(-304 + 152*x^2 - 19*x^4 + E^2*(-304 + 152*x^2 - 19*x^4)) + E^(2*x)*(28 - 5*x^2 + x^4 + E^2*(28 - 5*x^2 + x^4)))/(5776 - 2888*x^2 + 3 61*x^4 + E^x*(1064*x - 418*x^3 + 38*x^5) + E^(2*x)*(49*x^2 - 14*x^4 + x^6) ),x]
Output:
-((E^x*(1 + E^2)*(-4 + x^2))/(E^x*x*(-7 + x^2) + 19*(-4 + x^2)))
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.
| \(\displaystyle \int \frac {e^x \left (-19 x^4+152 x^2+e^2 \left (-19 x^4+152 x^2-304\right )-304\right )+e^{2 x} \left (x^4-5 x^2+e^2 \left (x^4-5 x^2+28\right )+28\right )}{361 x^4-2888 x^2+e^x \left (38 x^5-418 x^3+1064 x\right )+e^{2 x} \left (x^6-14 x^4+49 x^2\right )+5776} \, dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\left (1+e^2\right ) e^x \left (e^x \left (x^4-5 x^2+28\right )-19 \left (x^2-4\right )^2\right )}{\left (e^x x \left (x^2-7\right )+19 \left (x^2-4\right )\right )^2}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \left (1+e^2\right ) \int -\frac {e^x \left (19 \left (4-x^2\right )^2-e^x \left (x^4-5 x^2+28\right )\right )}{\left (19 \left (4-x^2\right )+e^x x \left (7-x^2\right )\right )^2}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\left (\left (1+e^2\right ) \int \frac {e^x \left (19 \left (4-x^2\right )^2-e^x \left (x^4-5 x^2+28\right )\right )}{\left (19 \left (4-x^2\right )+e^x x \left (7-x^2\right )\right )^2}dx\right )\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\left (\left (1+e^2\right ) \int \left (\frac {19 e^x \left (x^7+x^6-15 x^5-9 x^4+72 x^3+48 x^2-112 x-112\right )}{x \left (x^2-7\right ) \left (e^x x^3+19 x^2-7 e^x x-76\right )^2}-\frac {e^x \left (x^4-5 x^2+28\right )}{x \left (x^2-7\right ) \left (e^x x^3+19 x^2-7 e^x x-76\right )}\right )dx\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\left (\left (1+e^2\right ) \left (304 \int \frac {e^x}{\left (e^x x^3+19 x^2-7 e^x x-76\right )^2}dx-171 \int \frac {e^x}{\left (\sqrt {7}-x\right ) \left (e^x x^3+19 x^2-7 e^x x-76\right )^2}dx+304 \int \frac {e^x}{x \left (e^x x^3+19 x^2-7 e^x x-76\right )^2}dx-38 \int \frac {e^x x}{\left (e^x x^3+19 x^2-7 e^x x-76\right )^2}dx-152 \int \frac {e^x x^2}{\left (e^x x^3+19 x^2-7 e^x x-76\right )^2}dx+19 \int \frac {e^x x^3}{\left (e^x x^3+19 x^2-7 e^x x-76\right )^2}dx+171 \int \frac {e^x}{\left (x+\sqrt {7}\right ) \left (e^x x^3+19 x^2-7 e^x x-76\right )^2}dx+3 \int \frac {e^x}{\left (\sqrt {7}-x\right ) \left (e^x x^3+19 x^2-7 e^x x-76\right )}dx+4 \int \frac {e^x}{x \left (e^x x^3+19 x^2-7 e^x x-76\right )}dx-\int \frac {e^x x}{e^x x^3+19 x^2-7 e^x x-76}dx-3 \int \frac {e^x}{\left (x+\sqrt {7}\right ) \left (e^x x^3+19 x^2-7 e^x x-76\right )}dx+19 \int \frac {e^x x^4}{\left (e^x x^3+19 x^2-7 e^x x-76\right )^2}dx\right )\right )\) |
Input:
Int[(E^x*(-304 + 152*x^2 - 19*x^4 + E^2*(-304 + 152*x^2 - 19*x^4)) + E^(2* x)*(28 - 5*x^2 + x^4 + E^2*(28 - 5*x^2 + x^4)))/(5776 - 2888*x^2 + 361*x^4 + E^x*(1064*x - 418*x^3 + 38*x^5) + E^(2*x)*(49*x^2 - 14*x^4 + x^6)),x]
Output:
$Aborted
Time = 0.40 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.42
| method | result | size |
| norman | \(\frac {\left (4 \,{\mathrm e}^{2}+4\right ) {\mathrm e}^{x}+\left (-{\mathrm e}^{2}-1\right ) x^{2} {\mathrm e}^{x}}{{\mathrm e}^{x} x^{3}-7 \,{\mathrm e}^{x} x +19 x^{2}-76}\) | \(44\) |
| parallelrisch | \(-\frac {x^{2} {\mathrm e}^{2} {\mathrm e}^{x}+{\mathrm e}^{x} x^{2}-4 \,{\mathrm e}^{2} {\mathrm e}^{x}-4 \,{\mathrm e}^{x}}{{\mathrm e}^{x} x^{3}-7 \,{\mathrm e}^{x} x +19 x^{2}-76}\) | \(48\) |
| risch | \(\frac {\left (-{\mathrm e}^{2}-1\right ) x^{2}+4+4 \,{\mathrm e}^{2}}{\left (x^{2}-7\right ) x}+\frac {19 x^{4} {\mathrm e}^{2}-152 x^{2} {\mathrm e}^{2}+304 \,{\mathrm e}^{2}+19 x^{4}-152 x^{2}+304}{\left (x^{2}-7\right ) x \left ({\mathrm e}^{x} x^{3}-7 \,{\mathrm e}^{x} x +19 x^{2}-76\right )}\) | \(88\) |
Input:
int((((x^4-5*x^2+28)*exp(2)+x^4-5*x^2+28)*exp(x)^2+((-19*x^4+152*x^2-304)* exp(2)-19*x^4+152*x^2-304)*exp(x))/((x^6-14*x^4+49*x^2)*exp(x)^2+(38*x^5-4 18*x^3+1064*x)*exp(x)+361*x^4-2888*x^2+5776),x,method=_RETURNVERBOSE)
Output:
((4*exp(2)+4)*exp(x)+(-exp(2)-1)*x^2*exp(x))/(exp(x)*x^3-7*exp(x)*x+19*x^2 -76)
Time = 0.09 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.16 \[ \int \frac {e^x \left (-304+152 x^2-19 x^4+e^2 \left (-304+152 x^2-19 x^4\right )\right )+e^{2 x} \left (28-5 x^2+x^4+e^2 \left (28-5 x^2+x^4\right )\right )}{5776-2888 x^2+361 x^4+e^x \left (1064 x-418 x^3+38 x^5\right )+e^{2 x} \left (49 x^2-14 x^4+x^6\right )} \, dx=-\frac {{\left (x^{2} + {\left (x^{2} - 4\right )} e^{2} - 4\right )} e^{x}}{19 \, x^{2} + {\left (x^{3} - 7 \, x\right )} e^{x} - 76} \] Input:
integrate((((x^4-5*x^2+28)*exp(2)+x^4-5*x^2+28)*exp(x)^2+((-19*x^4+152*x^2 -304)*exp(2)-19*x^4+152*x^2-304)*exp(x))/((x^6-14*x^4+49*x^2)*exp(x)^2+(38 *x^5-418*x^3+1064*x)*exp(x)+361*x^4-2888*x^2+5776),x, algorithm="fricas")
Output:
-(x^2 + (x^2 - 4)*e^2 - 4)*e^x/(19*x^2 + (x^3 - 7*x)*e^x - 76)
Leaf count of result is larger than twice the leaf count of optimal. 87 vs. \(2 (20) = 40\).
Time = 0.36 (sec) , antiderivative size = 87, normalized size of antiderivative = 2.81 \[ \int \frac {e^x \left (-304+152 x^2-19 x^4+e^2 \left (-304+152 x^2-19 x^4\right )\right )+e^{2 x} \left (28-5 x^2+x^4+e^2 \left (28-5 x^2+x^4\right )\right )}{5776-2888 x^2+361 x^4+e^x \left (1064 x-418 x^3+38 x^5\right )+e^{2 x} \left (49 x^2-14 x^4+x^6\right )} \, dx=\frac {19 x^{4} + 19 x^{4} e^{2} - 152 x^{2} e^{2} - 152 x^{2} + 304 + 304 e^{2}}{19 x^{5} - 209 x^{3} + 532 x + \left (x^{6} - 14 x^{4} + 49 x^{2}\right ) e^{x}} + \frac {x^{2} \left (- e^{2} - 1\right ) + 4 + 4 e^{2}}{x^{3} - 7 x} \] Input:
integrate((((x**4-5*x**2+28)*exp(2)+x**4-5*x**2+28)*exp(x)**2+((-19*x**4+1 52*x**2-304)*exp(2)-19*x**4+152*x**2-304)*exp(x))/((x**6-14*x**4+49*x**2)* exp(x)**2+(38*x**5-418*x**3+1064*x)*exp(x)+361*x**4-2888*x**2+5776),x)
Output:
(19*x**4 + 19*x**4*exp(2) - 152*x**2*exp(2) - 152*x**2 + 304 + 304*exp(2)) /(19*x**5 - 209*x**3 + 532*x + (x**6 - 14*x**4 + 49*x**2)*exp(x)) + (x**2* (-exp(2) - 1) + 4 + 4*exp(2))/(x**3 - 7*x)
Time = 0.09 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.19 \[ \int \frac {e^x \left (-304+152 x^2-19 x^4+e^2 \left (-304+152 x^2-19 x^4\right )\right )+e^{2 x} \left (28-5 x^2+x^4+e^2 \left (28-5 x^2+x^4\right )\right )}{5776-2888 x^2+361 x^4+e^x \left (1064 x-418 x^3+38 x^5\right )+e^{2 x} \left (49 x^2-14 x^4+x^6\right )} \, dx=-\frac {{\left (x^{2} {\left (e^{2} + 1\right )} - 4 \, e^{2} - 4\right )} e^{x}}{19 \, x^{2} + {\left (x^{3} - 7 \, x\right )} e^{x} - 76} \] Input:
integrate((((x^4-5*x^2+28)*exp(2)+x^4-5*x^2+28)*exp(x)^2+((-19*x^4+152*x^2 -304)*exp(2)-19*x^4+152*x^2-304)*exp(x))/((x^6-14*x^4+49*x^2)*exp(x)^2+(38 *x^5-418*x^3+1064*x)*exp(x)+361*x^4-2888*x^2+5776),x, algorithm="maxima")
Output:
-(x^2*(e^2 + 1) - 4*e^2 - 4)*e^x/(19*x^2 + (x^3 - 7*x)*e^x - 76)
Time = 0.13 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.52 \[ \int \frac {e^x \left (-304+152 x^2-19 x^4+e^2 \left (-304+152 x^2-19 x^4\right )\right )+e^{2 x} \left (28-5 x^2+x^4+e^2 \left (28-5 x^2+x^4\right )\right )}{5776-2888 x^2+361 x^4+e^x \left (1064 x-418 x^3+38 x^5\right )+e^{2 x} \left (49 x^2-14 x^4+x^6\right )} \, dx=-\frac {x^{2} e^{\left (x + 2\right )} + x^{2} e^{x} - 4 \, e^{\left (x + 2\right )} - 4 \, e^{x}}{x^{3} e^{x} + 19 \, x^{2} - 7 \, x e^{x} - 76} \] Input:
integrate((((x^4-5*x^2+28)*exp(2)+x^4-5*x^2+28)*exp(x)^2+((-19*x^4+152*x^2 -304)*exp(2)-19*x^4+152*x^2-304)*exp(x))/((x^6-14*x^4+49*x^2)*exp(x)^2+(38 *x^5-418*x^3+1064*x)*exp(x)+361*x^4-2888*x^2+5776),x, algorithm="giac")
Output:
-(x^2*e^(x + 2) + x^2*e^x - 4*e^(x + 2) - 4*e^x)/(x^3*e^x + 19*x^2 - 7*x*e ^x - 76)
Timed out. \[ \int \frac {e^x \left (-304+152 x^2-19 x^4+e^2 \left (-304+152 x^2-19 x^4\right )\right )+e^{2 x} \left (28-5 x^2+x^4+e^2 \left (28-5 x^2+x^4\right )\right )}{5776-2888 x^2+361 x^4+e^x \left (1064 x-418 x^3+38 x^5\right )+e^{2 x} \left (49 x^2-14 x^4+x^6\right )} \, dx=\int \frac {{\mathrm {e}}^{2\,x}\,\left ({\mathrm {e}}^2\,\left (x^4-5\,x^2+28\right )-5\,x^2+x^4+28\right )-{\mathrm {e}}^x\,\left ({\mathrm {e}}^2\,\left (19\,x^4-152\,x^2+304\right )-152\,x^2+19\,x^4+304\right )}{{\mathrm {e}}^{2\,x}\,\left (x^6-14\,x^4+49\,x^2\right )-2888\,x^2+361\,x^4+{\mathrm {e}}^x\,\left (38\,x^5-418\,x^3+1064\,x\right )+5776} \,d x \] Input:
int((exp(2*x)*(exp(2)*(x^4 - 5*x^2 + 28) - 5*x^2 + x^4 + 28) - exp(x)*(exp (2)*(19*x^4 - 152*x^2 + 304) - 152*x^2 + 19*x^4 + 304))/(exp(2*x)*(49*x^2 - 14*x^4 + x^6) - 2888*x^2 + 361*x^4 + exp(x)*(1064*x - 418*x^3 + 38*x^5) + 5776),x)
Output:
int((exp(2*x)*(exp(2)*(x^4 - 5*x^2 + 28) - 5*x^2 + x^4 + 28) - exp(x)*(exp (2)*(19*x^4 - 152*x^2 + 304) - 152*x^2 + 19*x^4 + 304))/(exp(2*x)*(49*x^2 - 14*x^4 + x^6) - 2888*x^2 + 361*x^4 + exp(x)*(1064*x - 418*x^3 + 38*x^5) + 5776), x)
Time = 0.19 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.48 \[ \int \frac {e^x \left (-304+152 x^2-19 x^4+e^2 \left (-304+152 x^2-19 x^4\right )\right )+e^{2 x} \left (28-5 x^2+x^4+e^2 \left (28-5 x^2+x^4\right )\right )}{5776-2888 x^2+361 x^4+e^x \left (1064 x-418 x^3+38 x^5\right )+e^{2 x} \left (49 x^2-14 x^4+x^6\right )} \, dx=\frac {e^{x} \left (-e^{2} x^{2}+4 e^{2}-x^{2}+4\right )}{e^{x} x^{3}-7 e^{x} x +19 x^{2}-76} \] Input:
int((((x^4-5*x^2+28)*exp(2)+x^4-5*x^2+28)*exp(x)^2+((-19*x^4+152*x^2-304)* exp(2)-19*x^4+152*x^2-304)*exp(x))/((x^6-14*x^4+49*x^2)*exp(x)^2+(38*x^5-4 18*x^3+1064*x)*exp(x)+361*x^4-2888*x^2+5776),x)
Output:
(e**x*( - e**2*x**2 + 4*e**2 - x**2 + 4))/(e**x*x**3 - 7*e**x*x + 19*x**2 - 76)