Integrand size = 83, antiderivative size = 35 \[ \int \frac {80-3 e^{2 x}+1040 x-768 x^2+192 x^3-16 x^4+e^x \left (192-112 x+16 x^2\right )}{4032+e^{2 x}-4112 x+1536 x^2-256 x^3+16 x^4+e^x \left (-128+64 x-8 x^2\right )} \, dx=3+e^2-x+\log \left (\frac {5}{4-\left (-\frac {e^x}{4}+(4-x)^2\right )^2+x}\right ) \] Output:
ln(1/(1/5*x+4/5-1/5*((4-x)^2-1/4*exp(x))^2))+exp(2)+3-x
Time = 4.13 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.46 \[ \int \frac {80-3 e^{2 x}+1040 x-768 x^2+192 x^3-16 x^4+e^x \left (192-112 x+16 x^2\right )}{4032+e^{2 x}-4112 x+1536 x^2-256 x^3+16 x^4+e^x \left (-128+64 x-8 x^2\right )} \, dx=-x-\log \left (4032-128 e^x+e^{2 x}-4112 x+64 e^x x+1536 x^2-8 e^x x^2-256 x^3+16 x^4\right ) \] Input:
Integrate[(80 - 3*E^(2*x) + 1040*x - 768*x^2 + 192*x^3 - 16*x^4 + E^x*(192 - 112*x + 16*x^2))/(4032 + E^(2*x) - 4112*x + 1536*x^2 - 256*x^3 + 16*x^4 + E^x*(-128 + 64*x - 8*x^2)),x]
Output:
-x - Log[4032 - 128*E^x + E^(2*x) - 4112*x + 64*E^x*x + 1536*x^2 - 8*E^x*x ^2 - 256*x^3 + 16*x^4]
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.
| \(\displaystyle \int \frac {-16 x^4+192 x^3-768 x^2+e^x \left (16 x^2-112 x+192\right )+1040 x-3 e^{2 x}+80}{16 x^4-256 x^3+1536 x^2+e^x \left (-8 x^2+64 x-128\right )-4112 x+e^{2 x}+4032} \, dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (-\frac {8 \left (-4 x^4+72 x^3+e^x x^2-480 x^2-10 e^x x+1412 x+24 e^x-1522\right )}{16 x^4-256 x^3-8 e^x x^2+1536 x^2+64 e^x x-4112 x-128 e^x+e^{2 x}+4032}-3\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 12176 \int \frac {1}{16 x^4-256 x^3-8 e^x x^2+1536 x^2+64 e^x x-4112 x-128 e^x+e^{2 x}+4032}dx-192 \int \frac {e^x}{16 x^4-256 x^3-8 e^x x^2+1536 x^2+64 e^x x-4112 x-128 e^x+e^{2 x}+4032}dx-11296 \int \frac {x}{16 x^4-256 x^3-8 e^x x^2+1536 x^2+64 e^x x-4112 x-128 e^x+e^{2 x}+4032}dx+80 \int \frac {e^x x}{16 x^4-256 x^3-8 e^x x^2+1536 x^2+64 e^x x-4112 x-128 e^x+e^{2 x}+4032}dx+3840 \int \frac {x^2}{16 x^4-256 x^3-8 e^x x^2+1536 x^2+64 e^x x-4112 x-128 e^x+e^{2 x}+4032}dx-8 \int \frac {e^x x^2}{16 x^4-256 x^3-8 e^x x^2+1536 x^2+64 e^x x-4112 x-128 e^x+e^{2 x}+4032}dx-576 \int \frac {x^3}{16 x^4-256 x^3-8 e^x x^2+1536 x^2+64 e^x x-4112 x-128 e^x+e^{2 x}+4032}dx+32 \int \frac {x^4}{16 x^4-256 x^3-8 e^x x^2+1536 x^2+64 e^x x-4112 x-128 e^x+e^{2 x}+4032}dx-3 x\) |
Input:
Int[(80 - 3*E^(2*x) + 1040*x - 768*x^2 + 192*x^3 - 16*x^4 + E^x*(192 - 112 *x + 16*x^2))/(4032 + E^(2*x) - 4112*x + 1536*x^2 - 256*x^3 + 16*x^4 + E^x *(-128 + 64*x - 8*x^2)),x]
Output:
$Aborted
Time = 0.45 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.29
| method | result | size |
| risch | \(-x -\ln \left ({\mathrm e}^{2 x}+\left (-8 x^{2}+64 x -128\right ) {\mathrm e}^{x}+16 x^{4}-256 x^{3}+1536 x^{2}-4112 x +4032\right )\) | \(45\) |
| norman | \(-x -\ln \left (16 x^{4}-8 \,{\mathrm e}^{x} x^{2}-256 x^{3}+64 \,{\mathrm e}^{x} x +1536 x^{2}-128 \,{\mathrm e}^{x}+{\mathrm e}^{2 x}-4112 x +4032\right )\) | \(48\) |
| parallelrisch | \(-x -\ln \left (-257 x +252+x^{4}-16 x^{3}+96 x^{2}-\frac {{\mathrm e}^{x} x^{2}}{2}+4 \,{\mathrm e}^{x} x -8 \,{\mathrm e}^{x}+\frac {{\mathrm e}^{2 x}}{16}\right )\) | \(48\) |
Input:
int((-3*exp(x)^2+(16*x^2-112*x+192)*exp(x)-16*x^4+192*x^3-768*x^2+1040*x+8 0)/(exp(x)^2+(-8*x^2+64*x-128)*exp(x)+16*x^4-256*x^3+1536*x^2-4112*x+4032) ,x,method=_RETURNVERBOSE)
Output:
-x-ln(exp(2*x)+(-8*x^2+64*x-128)*exp(x)+16*x^4-256*x^3+1536*x^2-4112*x+403 2)
Time = 0.10 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.23 \[ \int \frac {80-3 e^{2 x}+1040 x-768 x^2+192 x^3-16 x^4+e^x \left (192-112 x+16 x^2\right )}{4032+e^{2 x}-4112 x+1536 x^2-256 x^3+16 x^4+e^x \left (-128+64 x-8 x^2\right )} \, dx=-x - \log \left (16 \, x^{4} - 256 \, x^{3} + 1536 \, x^{2} - 8 \, {\left (x^{2} - 8 \, x + 16\right )} e^{x} - 4112 \, x + e^{\left (2 \, x\right )} + 4032\right ) \] Input:
integrate((-3*exp(x)^2+(16*x^2-112*x+192)*exp(x)-16*x^4+192*x^3-768*x^2+10 40*x+80)/(exp(x)^2+(-8*x^2+64*x-128)*exp(x)+16*x^4-256*x^3+1536*x^2-4112*x +4032),x, algorithm="fricas")
Output:
-x - log(16*x^4 - 256*x^3 + 1536*x^2 - 8*(x^2 - 8*x + 16)*e^x - 4112*x + e ^(2*x) + 4032)
Time = 0.12 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.20 \[ \int \frac {80-3 e^{2 x}+1040 x-768 x^2+192 x^3-16 x^4+e^x \left (192-112 x+16 x^2\right )}{4032+e^{2 x}-4112 x+1536 x^2-256 x^3+16 x^4+e^x \left (-128+64 x-8 x^2\right )} \, dx=- x - \log {\left (16 x^{4} - 256 x^{3} + 1536 x^{2} - 4112 x + \left (- 8 x^{2} + 64 x - 128\right ) e^{x} + e^{2 x} + 4032 \right )} \] Input:
integrate((-3*exp(x)**2+(16*x**2-112*x+192)*exp(x)-16*x**4+192*x**3-768*x* *2+1040*x+80)/(exp(x)**2+(-8*x**2+64*x-128)*exp(x)+16*x**4-256*x**3+1536*x **2-4112*x+4032),x)
Output:
-x - log(16*x**4 - 256*x**3 + 1536*x**2 - 4112*x + (-8*x**2 + 64*x - 128)* exp(x) + exp(2*x) + 4032)
Time = 0.07 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.23 \[ \int \frac {80-3 e^{2 x}+1040 x-768 x^2+192 x^3-16 x^4+e^x \left (192-112 x+16 x^2\right )}{4032+e^{2 x}-4112 x+1536 x^2-256 x^3+16 x^4+e^x \left (-128+64 x-8 x^2\right )} \, dx=-x - \log \left (16 \, x^{4} - 256 \, x^{3} + 1536 \, x^{2} - 8 \, {\left (x^{2} - 8 \, x + 16\right )} e^{x} - 4112 \, x + e^{\left (2 \, x\right )} + 4032\right ) \] Input:
integrate((-3*exp(x)^2+(16*x^2-112*x+192)*exp(x)-16*x^4+192*x^3-768*x^2+10 40*x+80)/(exp(x)^2+(-8*x^2+64*x-128)*exp(x)+16*x^4-256*x^3+1536*x^2-4112*x +4032),x, algorithm="maxima")
Output:
-x - log(16*x^4 - 256*x^3 + 1536*x^2 - 8*(x^2 - 8*x + 16)*e^x - 4112*x + e ^(2*x) + 4032)
Time = 0.13 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.34 \[ \int \frac {80-3 e^{2 x}+1040 x-768 x^2+192 x^3-16 x^4+e^x \left (192-112 x+16 x^2\right )}{4032+e^{2 x}-4112 x+1536 x^2-256 x^3+16 x^4+e^x \left (-128+64 x-8 x^2\right )} \, dx=-x - \log \left (16 \, x^{4} - 256 \, x^{3} - 8 \, x^{2} e^{x} + 1536 \, x^{2} + 64 \, x e^{x} - 4112 \, x + e^{\left (2 \, x\right )} - 128 \, e^{x} + 4032\right ) \] Input:
integrate((-3*exp(x)^2+(16*x^2-112*x+192)*exp(x)-16*x^4+192*x^3-768*x^2+10 40*x+80)/(exp(x)^2+(-8*x^2+64*x-128)*exp(x)+16*x^4-256*x^3+1536*x^2-4112*x +4032),x, algorithm="giac")
Output:
-x - log(16*x^4 - 256*x^3 - 8*x^2*e^x + 1536*x^2 + 64*x*e^x - 4112*x + e^( 2*x) - 128*e^x + 4032)
Time = 3.48 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.34 \[ \int \frac {80-3 e^{2 x}+1040 x-768 x^2+192 x^3-16 x^4+e^x \left (192-112 x+16 x^2\right )}{4032+e^{2 x}-4112 x+1536 x^2-256 x^3+16 x^4+e^x \left (-128+64 x-8 x^2\right )} \, dx=-x-\ln \left (\frac {{\mathrm {e}}^{2\,x}}{16}-257\,x-8\,{\mathrm {e}}^x-\frac {x^2\,{\mathrm {e}}^x}{2}+4\,x\,{\mathrm {e}}^x+96\,x^2-16\,x^3+x^4+252\right ) \] Input:
int((1040*x - 3*exp(2*x) + exp(x)*(16*x^2 - 112*x + 192) - 768*x^2 + 192*x ^3 - 16*x^4 + 80)/(exp(2*x) - 4112*x - exp(x)*(8*x^2 - 64*x + 128) + 1536* x^2 - 256*x^3 + 16*x^4 + 4032),x)
Output:
- x - log(exp(2*x)/16 - 257*x - 8*exp(x) - (x^2*exp(x))/2 + 4*x*exp(x) + 9 6*x^2 - 16*x^3 + x^4 + 252)
\[ \int \frac {80-3 e^{2 x}+1040 x-768 x^2+192 x^3-16 x^4+e^x \left (192-112 x+16 x^2\right )}{4032+e^{2 x}-4112 x+1536 x^2-256 x^3+16 x^4+e^x \left (-128+64 x-8 x^2\right )} \, dx=-3 \left (\int \frac {e^{2 x}}{e^{2 x}-8 e^{x} x^{2}+64 e^{x} x -128 e^{x}+16 x^{4}-256 x^{3}+1536 x^{2}-4112 x +4032}d x \right )+192 \left (\int \frac {e^{x}}{e^{2 x}-8 e^{x} x^{2}+64 e^{x} x -128 e^{x}+16 x^{4}-256 x^{3}+1536 x^{2}-4112 x +4032}d x \right )-16 \left (\int \frac {x^{4}}{e^{2 x}-8 e^{x} x^{2}+64 e^{x} x -128 e^{x}+16 x^{4}-256 x^{3}+1536 x^{2}-4112 x +4032}d x \right )+192 \left (\int \frac {x^{3}}{e^{2 x}-8 e^{x} x^{2}+64 e^{x} x -128 e^{x}+16 x^{4}-256 x^{3}+1536 x^{2}-4112 x +4032}d x \right )-768 \left (\int \frac {x^{2}}{e^{2 x}-8 e^{x} x^{2}+64 e^{x} x -128 e^{x}+16 x^{4}-256 x^{3}+1536 x^{2}-4112 x +4032}d x \right )+16 \left (\int \frac {e^{x} x^{2}}{e^{2 x}-8 e^{x} x^{2}+64 e^{x} x -128 e^{x}+16 x^{4}-256 x^{3}+1536 x^{2}-4112 x +4032}d x \right )-112 \left (\int \frac {e^{x} x}{e^{2 x}-8 e^{x} x^{2}+64 e^{x} x -128 e^{x}+16 x^{4}-256 x^{3}+1536 x^{2}-4112 x +4032}d x \right )+1040 \left (\int \frac {x}{e^{2 x}-8 e^{x} x^{2}+64 e^{x} x -128 e^{x}+16 x^{4}-256 x^{3}+1536 x^{2}-4112 x +4032}d x \right )+80 \left (\int \frac {1}{e^{2 x}-8 e^{x} x^{2}+64 e^{x} x -128 e^{x}+16 x^{4}-256 x^{3}+1536 x^{2}-4112 x +4032}d x \right ) \] Input:
int((-3*exp(x)^2+(16*x^2-112*x+192)*exp(x)-16*x^4+192*x^3-768*x^2+1040*x+8 0)/(exp(x)^2+(-8*x^2+64*x-128)*exp(x)+16*x^4-256*x^3+1536*x^2-4112*x+4032) ,x)
Output:
- 3*int(e**(2*x)/(e**(2*x) - 8*e**x*x**2 + 64*e**x*x - 128*e**x + 16*x**4 - 256*x**3 + 1536*x**2 - 4112*x + 4032),x) + 192*int(e**x/(e**(2*x) - 8*e **x*x**2 + 64*e**x*x - 128*e**x + 16*x**4 - 256*x**3 + 1536*x**2 - 4112*x + 4032),x) - 16*int(x**4/(e**(2*x) - 8*e**x*x**2 + 64*e**x*x - 128*e**x + 16*x**4 - 256*x**3 + 1536*x**2 - 4112*x + 4032),x) + 192*int(x**3/(e**(2*x ) - 8*e**x*x**2 + 64*e**x*x - 128*e**x + 16*x**4 - 256*x**3 + 1536*x**2 - 4112*x + 4032),x) - 768*int(x**2/(e**(2*x) - 8*e**x*x**2 + 64*e**x*x - 128 *e**x + 16*x**4 - 256*x**3 + 1536*x**2 - 4112*x + 4032),x) + 16*int((e**x* x**2)/(e**(2*x) - 8*e**x*x**2 + 64*e**x*x - 128*e**x + 16*x**4 - 256*x**3 + 1536*x**2 - 4112*x + 4032),x) - 112*int((e**x*x)/(e**(2*x) - 8*e**x*x**2 + 64*e**x*x - 128*e**x + 16*x**4 - 256*x**3 + 1536*x**2 - 4112*x + 4032), x) + 1040*int(x/(e**(2*x) - 8*e**x*x**2 + 64*e**x*x - 128*e**x + 16*x**4 - 256*x**3 + 1536*x**2 - 4112*x + 4032),x) + 80*int(1/(e**(2*x) - 8*e**x*x* *2 + 64*e**x*x - 128*e**x + 16*x**4 - 256*x**3 + 1536*x**2 - 4112*x + 4032 ),x)