Integrand size = 85, antiderivative size = 31 \[ \int \frac {-1+16 x+6 x^2-2 x^3-x^4+e^{2 x} \left (-7+2 x^2\right )}{16+e^{4 x}-8 x-31 x^2+18 x^4+8 x^5+x^6+e^{2 x} \left (8-2 x-8 x^2-2 x^3\right )} \, dx=\frac {x}{x^2+x \left (3+\frac {5-e^{2 x}+x}{-3+x+x^2}\right )} \] Output:
x/(x*(3+(5+x-exp(2*x))/(x^2+x-3))+x^2)
Time = 1.82 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.87 \[ \int \frac {-1+16 x+6 x^2-2 x^3-x^4+e^{2 x} \left (-7+2 x^2\right )}{16+e^{4 x}-8 x-31 x^2+18 x^4+8 x^5+x^6+e^{2 x} \left (8-2 x-8 x^2-2 x^3\right )} \, dx=\frac {-3+x+x^2}{-4-e^{2 x}+x+4 x^2+x^3} \] Input:
Integrate[(-1 + 16*x + 6*x^2 - 2*x^3 - x^4 + E^(2*x)*(-7 + 2*x^2))/(16 + E ^(4*x) - 8*x - 31*x^2 + 18*x^4 + 8*x^5 + x^6 + E^(2*x)*(8 - 2*x - 8*x^2 - 2*x^3)),x]
Output:
(-3 + x + x^2)/(-4 - E^(2*x) + x + 4*x^2 + x^3)
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.
\(\displaystyle \int \frac {-x^4-2 x^3+6 x^2+e^{2 x} \left (2 x^2-7\right )+16 x-1}{x^6+8 x^5+18 x^4-31 x^2+e^{2 x} \left (-2 x^3-8 x^2-2 x+8\right )-8 x+e^{4 x}+16} \, dx\) |
\(\Big \downarrow \) 7239 |
\(\displaystyle \int \frac {-x^4-2 x^3+6 x^2+e^{2 x} \left (2 x^2-7\right )+16 x-1}{\left (-x^3-4 x^2-x+e^{2 x}+4\right )^2}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {2 x^5+7 x^4-7 x^3-30 x^2+9 x+27}{\left (-x^3-4 x^2-x+e^{2 x}+4\right )^2}-\frac {2 x^2-7}{x^3+4 x^2+x-e^{2 x}-4}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 27 \int \frac {1}{\left (-x^3-4 x^2-x+e^{2 x}+4\right )^2}dx-7 \int \frac {1}{-x^3-4 x^2-x+e^{2 x}+4}dx+9 \int \frac {x}{\left (x^3+4 x^2+x-e^{2 x}-4\right )^2}dx-30 \int \frac {x^2}{\left (x^3+4 x^2+x-e^{2 x}-4\right )^2}dx-7 \int \frac {x^3}{\left (x^3+4 x^2+x-e^{2 x}-4\right )^2}dx-2 \int \frac {x^2}{x^3+4 x^2+x-e^{2 x}-4}dx+2 \int \frac {x^5}{\left (x^3+4 x^2+x-e^{2 x}-4\right )^2}dx+7 \int \frac {x^4}{\left (x^3+4 x^2+x-e^{2 x}-4\right )^2}dx\) |
Input:
Int[(-1 + 16*x + 6*x^2 - 2*x^3 - x^4 + E^(2*x)*(-7 + 2*x^2))/(16 + E^(4*x) - 8*x - 31*x^2 + 18*x^4 + 8*x^5 + x^6 + E^(2*x)*(8 - 2*x - 8*x^2 - 2*x^3) ),x]
Output:
$Aborted
Time = 0.21 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.87
method | result | size |
norman | \(\frac {x^{2}+x -3}{x^{3}+4 x^{2}+x -{\mathrm e}^{2 x}-4}\) | \(27\) |
risch | \(\frac {x^{2}+x -3}{x^{3}+4 x^{2}+x -{\mathrm e}^{2 x}-4}\) | \(27\) |
parallelrisch | \(-\frac {-x^{2}-x +3}{x^{3}+4 x^{2}+x -{\mathrm e}^{2 x}-4}\) | \(32\) |
Input:
int(((2*x^2-7)*exp(2*x)-x^4-2*x^3+6*x^2+16*x-1)/(exp(2*x)^2+(-2*x^3-8*x^2- 2*x+8)*exp(2*x)+x^6+8*x^5+18*x^4-31*x^2-8*x+16),x,method=_RETURNVERBOSE)
Output:
(x^2+x-3)/(x^3+4*x^2+x-exp(2*x)-4)
Time = 0.09 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.84 \[ \int \frac {-1+16 x+6 x^2-2 x^3-x^4+e^{2 x} \left (-7+2 x^2\right )}{16+e^{4 x}-8 x-31 x^2+18 x^4+8 x^5+x^6+e^{2 x} \left (8-2 x-8 x^2-2 x^3\right )} \, dx=\frac {x^{2} + x - 3}{x^{3} + 4 \, x^{2} + x - e^{\left (2 \, x\right )} - 4} \] Input:
integrate(((2*x^2-7)*exp(2*x)-x^4-2*x^3+6*x^2+16*x-1)/(exp(2*x)^2+(-2*x^3- 8*x^2-2*x+8)*exp(2*x)+x^6+8*x^5+18*x^4-31*x^2-8*x+16),x, algorithm="fricas ")
Output:
(x^2 + x - 3)/(x^3 + 4*x^2 + x - e^(2*x) - 4)
Time = 0.07 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.71 \[ \int \frac {-1+16 x+6 x^2-2 x^3-x^4+e^{2 x} \left (-7+2 x^2\right )}{16+e^{4 x}-8 x-31 x^2+18 x^4+8 x^5+x^6+e^{2 x} \left (8-2 x-8 x^2-2 x^3\right )} \, dx=\frac {- x^{2} - x + 3}{- x^{3} - 4 x^{2} - x + e^{2 x} + 4} \] Input:
integrate(((2*x**2-7)*exp(2*x)-x**4-2*x**3+6*x**2+16*x-1)/(exp(2*x)**2+(-2 *x**3-8*x**2-2*x+8)*exp(2*x)+x**6+8*x**5+18*x**4-31*x**2-8*x+16),x)
Output:
(-x**2 - x + 3)/(-x**3 - 4*x**2 - x + exp(2*x) + 4)
Time = 0.07 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.84 \[ \int \frac {-1+16 x+6 x^2-2 x^3-x^4+e^{2 x} \left (-7+2 x^2\right )}{16+e^{4 x}-8 x-31 x^2+18 x^4+8 x^5+x^6+e^{2 x} \left (8-2 x-8 x^2-2 x^3\right )} \, dx=\frac {x^{2} + x - 3}{x^{3} + 4 \, x^{2} + x - e^{\left (2 \, x\right )} - 4} \] Input:
integrate(((2*x^2-7)*exp(2*x)-x^4-2*x^3+6*x^2+16*x-1)/(exp(2*x)^2+(-2*x^3- 8*x^2-2*x+8)*exp(2*x)+x^6+8*x^5+18*x^4-31*x^2-8*x+16),x, algorithm="maxima ")
Output:
(x^2 + x - 3)/(x^3 + 4*x^2 + x - e^(2*x) - 4)
Time = 0.12 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.84 \[ \int \frac {-1+16 x+6 x^2-2 x^3-x^4+e^{2 x} \left (-7+2 x^2\right )}{16+e^{4 x}-8 x-31 x^2+18 x^4+8 x^5+x^6+e^{2 x} \left (8-2 x-8 x^2-2 x^3\right )} \, dx=\frac {x^{2} + x - 3}{x^{3} + 4 \, x^{2} + x - e^{\left (2 \, x\right )} - 4} \] Input:
integrate(((2*x^2-7)*exp(2*x)-x^4-2*x^3+6*x^2+16*x-1)/(exp(2*x)^2+(-2*x^3- 8*x^2-2*x+8)*exp(2*x)+x^6+8*x^5+18*x^4-31*x^2-8*x+16),x, algorithm="giac")
Output:
(x^2 + x - 3)/(x^3 + 4*x^2 + x - e^(2*x) - 4)
Time = 2.29 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.84 \[ \int \frac {-1+16 x+6 x^2-2 x^3-x^4+e^{2 x} \left (-7+2 x^2\right )}{16+e^{4 x}-8 x-31 x^2+18 x^4+8 x^5+x^6+e^{2 x} \left (8-2 x-8 x^2-2 x^3\right )} \, dx=\frac {x^2+x-3}{x-{\mathrm {e}}^{2\,x}+4\,x^2+x^3-4} \] Input:
int((16*x + exp(2*x)*(2*x^2 - 7) + 6*x^2 - 2*x^3 - x^4 - 1)/(exp(4*x) - 8* x - exp(2*x)*(2*x + 8*x^2 + 2*x^3 - 8) - 31*x^2 + 18*x^4 + 8*x^5 + x^6 + 1 6),x)
Output:
(x + x^2 - 3)/(x - exp(2*x) + 4*x^2 + x^3 - 4)
\[ \int \frac {-1+16 x+6 x^2-2 x^3-x^4+e^{2 x} \left (-7+2 x^2\right )}{16+e^{4 x}-8 x-31 x^2+18 x^4+8 x^5+x^6+e^{2 x} \left (8-2 x-8 x^2-2 x^3\right )} \, dx=-7 \left (\int \frac {e^{2 x}}{e^{4 x}-2 e^{2 x} x^{3}-8 e^{2 x} x^{2}-2 e^{2 x} x +8 e^{2 x}+x^{6}+8 x^{5}+18 x^{4}-31 x^{2}-8 x +16}d x \right )-\left (\int \frac {x^{4}}{e^{4 x}-2 e^{2 x} x^{3}-8 e^{2 x} x^{2}-2 e^{2 x} x +8 e^{2 x}+x^{6}+8 x^{5}+18 x^{4}-31 x^{2}-8 x +16}d x \right )-2 \left (\int \frac {x^{3}}{e^{4 x}-2 e^{2 x} x^{3}-8 e^{2 x} x^{2}-2 e^{2 x} x +8 e^{2 x}+x^{6}+8 x^{5}+18 x^{4}-31 x^{2}-8 x +16}d x \right )+6 \left (\int \frac {x^{2}}{e^{4 x}-2 e^{2 x} x^{3}-8 e^{2 x} x^{2}-2 e^{2 x} x +8 e^{2 x}+x^{6}+8 x^{5}+18 x^{4}-31 x^{2}-8 x +16}d x \right )+2 \left (\int \frac {e^{2 x} x^{2}}{e^{4 x}-2 e^{2 x} x^{3}-8 e^{2 x} x^{2}-2 e^{2 x} x +8 e^{2 x}+x^{6}+8 x^{5}+18 x^{4}-31 x^{2}-8 x +16}d x \right )+16 \left (\int \frac {x}{e^{4 x}-2 e^{2 x} x^{3}-8 e^{2 x} x^{2}-2 e^{2 x} x +8 e^{2 x}+x^{6}+8 x^{5}+18 x^{4}-31 x^{2}-8 x +16}d x \right )-\left (\int \frac {1}{e^{4 x}-2 e^{2 x} x^{3}-8 e^{2 x} x^{2}-2 e^{2 x} x +8 e^{2 x}+x^{6}+8 x^{5}+18 x^{4}-31 x^{2}-8 x +16}d x \right ) \] Input:
int(((2*x^2-7)*exp(2*x)-x^4-2*x^3+6*x^2+16*x-1)/(exp(2*x)^2+(-2*x^3-8*x^2- 2*x+8)*exp(2*x)+x^6+8*x^5+18*x^4-31*x^2-8*x+16),x)
Output:
- 7*int(e**(2*x)/(e**(4*x) - 2*e**(2*x)*x**3 - 8*e**(2*x)*x**2 - 2*e**(2* x)*x + 8*e**(2*x) + x**6 + 8*x**5 + 18*x**4 - 31*x**2 - 8*x + 16),x) - int (x**4/(e**(4*x) - 2*e**(2*x)*x**3 - 8*e**(2*x)*x**2 - 2*e**(2*x)*x + 8*e** (2*x) + x**6 + 8*x**5 + 18*x**4 - 31*x**2 - 8*x + 16),x) - 2*int(x**3/(e** (4*x) - 2*e**(2*x)*x**3 - 8*e**(2*x)*x**2 - 2*e**(2*x)*x + 8*e**(2*x) + x* *6 + 8*x**5 + 18*x**4 - 31*x**2 - 8*x + 16),x) + 6*int(x**2/(e**(4*x) - 2* e**(2*x)*x**3 - 8*e**(2*x)*x**2 - 2*e**(2*x)*x + 8*e**(2*x) + x**6 + 8*x** 5 + 18*x**4 - 31*x**2 - 8*x + 16),x) + 2*int((e**(2*x)*x**2)/(e**(4*x) - 2 *e**(2*x)*x**3 - 8*e**(2*x)*x**2 - 2*e**(2*x)*x + 8*e**(2*x) + x**6 + 8*x* *5 + 18*x**4 - 31*x**2 - 8*x + 16),x) + 16*int(x/(e**(4*x) - 2*e**(2*x)*x* *3 - 8*e**(2*x)*x**2 - 2*e**(2*x)*x + 8*e**(2*x) + x**6 + 8*x**5 + 18*x**4 - 31*x**2 - 8*x + 16),x) - int(1/(e**(4*x) - 2*e**(2*x)*x**3 - 8*e**(2*x) *x**2 - 2*e**(2*x)*x + 8*e**(2*x) + x**6 + 8*x**5 + 18*x**4 - 31*x**2 - 8* x + 16),x)