Integrand size = 95, antiderivative size = 27 \[ \int \frac {52-26 x+4 x^2+e^{1+2 x^2+x^4} \left (-4+2 x+16 x^2-4 x^3+16 x^4-4 x^5\right )}{169+e^{2+4 x^2+2 x^4}-26 x^2+x^4+e^{1+2 x^2+x^4} \left (-26+2 x^2\right )} \, dx=\frac {(4-x) x}{13-e^{\left (1+x^2\right )^2}-x^2} \] Output:
(4-x)*x/(13-x^2-exp((x^2+1)^2))
Time = 2.40 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.78 \[ \int \frac {52-26 x+4 x^2+e^{1+2 x^2+x^4} \left (-4+2 x+16 x^2-4 x^3+16 x^4-4 x^5\right )}{169+e^{2+4 x^2+2 x^4}-26 x^2+x^4+e^{1+2 x^2+x^4} \left (-26+2 x^2\right )} \, dx=\frac {(-4+x) x}{-13+e^{\left (1+x^2\right )^2}+x^2} \] Input:
Integrate[(52 - 26*x + 4*x^2 + E^(1 + 2*x^2 + x^4)*(-4 + 2*x + 16*x^2 - 4* x^3 + 16*x^4 - 4*x^5))/(169 + E^(2 + 4*x^2 + 2*x^4) - 26*x^2 + x^4 + E^(1 + 2*x^2 + x^4)*(-26 + 2*x^2)),x]
Output:
((-4 + x)*x)/(-13 + E^(1 + x^2)^2 + 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.
\(\displaystyle \int \frac {4 x^2+e^{x^4+2 x^2+1} \left (-4 x^5+16 x^4-4 x^3+16 x^2+2 x-4\right )-26 x+52}{x^4-26 x^2+e^{2 x^4+4 x^2+2}+e^{x^4+2 x^2+1} \left (2 x^2-26\right )+169} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {4 x^2+e^{x^4+2 x^2+1} \left (-4 x^5+16 x^4-4 x^3+16 x^2+2 x-4\right )-26 x+52}{\left (-x^2-e^{\left (x^2+1\right )^2}+13\right )^2}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {4 x^2}{\left (x^2+e^{\left (x^2+1\right )^2}-13\right )^2}-\frac {26 x}{\left (x^2+e^{\left (x^2+1\right )^2}-13\right )^2}+\frac {52}{\left (x^2+e^{\left (x^2+1\right )^2}-13\right )^2}+\frac {16 e^{x^4+2 x^2+1} x^4}{\left (-x^2-e^{\left (x^2+1\right )^2}+13\right )^2}+\frac {16 e^{x^4+2 x^2+1} x^2}{\left (-x^2-e^{\left (x^2+1\right )^2}+13\right )^2}+\frac {2 e^{x^4+2 x^2+1} x}{\left (-x^2-e^{\left (x^2+1\right )^2}+13\right )^2}-\frac {4 e^{x^4+2 x^2+1}}{\left (-x^2-e^{\left (x^2+1\right )^2}+13\right )^2}-\frac {4 e^{x^4+2 x^2+1} x^5}{\left (-x^2-e^{\left (x^2+1\right )^2}+13\right )^2}-\frac {4 e^{x^4+2 x^2+1} x^3}{\left (-x^2-e^{\left (x^2+1\right )^2}+13\right )^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \text {Subst}\left (\int \frac {e^{x^2+2 x+1}}{\left (-x-e^{(x+1)^2}+13\right )^2}dx,x,x^2\right )-2 \text {Subst}\left (\int \frac {e^{x^2+2 x+1} x}{\left (-x-e^{(x+1)^2}+13\right )^2}dx,x,x^2\right )-2 \text {Subst}\left (\int \frac {e^{x^2+2 x+1} x^2}{\left (-x-e^{(x+1)^2}+13\right )^2}dx,x,x^2\right )-13 \text {Subst}\left (\int \frac {1}{\left (x+e^{(x+1)^2}-13\right )^2}dx,x,x^2\right )+52 \int \frac {1}{\left (x^2+e^{\left (x^2+1\right )^2}-13\right )^2}dx+4 \int \frac {x^2}{\left (x^2+e^{\left (x^2+1\right )^2}-13\right )^2}dx-4 \int \frac {e^{x^4+2 x^2+1}}{\left (-x^2-e^{\left (x^2+1\right )^2}+13\right )^2}dx+16 \int \frac {e^{x^4+2 x^2+1} x^2}{\left (-x^2-e^{\left (x^2+1\right )^2}+13\right )^2}dx+16 \int \frac {e^{x^4+2 x^2+1} x^4}{\left (-x^2-e^{\left (x^2+1\right )^2}+13\right )^2}dx\) |
Input:
Int[(52 - 26*x + 4*x^2 + E^(1 + 2*x^2 + x^4)*(-4 + 2*x + 16*x^2 - 4*x^3 + 16*x^4 - 4*x^5))/(169 + E^(2 + 4*x^2 + 2*x^4) - 26*x^2 + x^4 + E^(1 + 2*x^ 2 + x^4)*(-26 + 2*x^2)),x]
Output:
$Aborted
Time = 0.22 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.78
method | result | size |
risch | \(\frac {\left (x -4\right ) x}{x^{2}+{\mathrm e}^{\left (x^{2}+1\right )^{2}}-13}\) | \(21\) |
parallelrisch | \(\frac {x^{2}-4 x}{x^{2}+{\mathrm e}^{x^{4}+2 x^{2}+1}-13}\) | \(27\) |
norman | \(\frac {-{\mathrm e}^{x^{4}+2 x^{2}+1}-4 x +13}{x^{2}+{\mathrm e}^{x^{4}+2 x^{2}+1}-13}\) | \(38\) |
Input:
int(((-4*x^5+16*x^4-4*x^3+16*x^2+2*x-4)*exp(x^4+2*x^2+1)+4*x^2-26*x+52)/(e xp(x^4+2*x^2+1)^2+(2*x^2-26)*exp(x^4+2*x^2+1)+x^4-26*x^2+169),x,method=_RE TURNVERBOSE)
Output:
(x-4)*x/(x^2+exp((x^2+1)^2)-13)
Time = 0.07 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.96 \[ \int \frac {52-26 x+4 x^2+e^{1+2 x^2+x^4} \left (-4+2 x+16 x^2-4 x^3+16 x^4-4 x^5\right )}{169+e^{2+4 x^2+2 x^4}-26 x^2+x^4+e^{1+2 x^2+x^4} \left (-26+2 x^2\right )} \, dx=\frac {x^{2} - 4 \, x}{x^{2} + e^{\left (x^{4} + 2 \, x^{2} + 1\right )} - 13} \] Input:
integrate(((-4*x^5+16*x^4-4*x^3+16*x^2+2*x-4)*exp(x^4+2*x^2+1)+4*x^2-26*x+ 52)/(exp(x^4+2*x^2+1)^2+(2*x^2-26)*exp(x^4+2*x^2+1)+x^4-26*x^2+169),x, alg orithm="fricas")
Output:
(x^2 - 4*x)/(x^2 + e^(x^4 + 2*x^2 + 1) - 13)
Time = 0.09 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.81 \[ \int \frac {52-26 x+4 x^2+e^{1+2 x^2+x^4} \left (-4+2 x+16 x^2-4 x^3+16 x^4-4 x^5\right )}{169+e^{2+4 x^2+2 x^4}-26 x^2+x^4+e^{1+2 x^2+x^4} \left (-26+2 x^2\right )} \, dx=\frac {x^{2} - 4 x}{x^{2} + e^{x^{4} + 2 x^{2} + 1} - 13} \] Input:
integrate(((-4*x**5+16*x**4-4*x**3+16*x**2+2*x-4)*exp(x**4+2*x**2+1)+4*x** 2-26*x+52)/(exp(x**4+2*x**2+1)**2+(2*x**2-26)*exp(x**4+2*x**2+1)+x**4-26*x **2+169),x)
Output:
(x**2 - 4*x)/(x**2 + exp(x**4 + 2*x**2 + 1) - 13)
Time = 0.09 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.96 \[ \int \frac {52-26 x+4 x^2+e^{1+2 x^2+x^4} \left (-4+2 x+16 x^2-4 x^3+16 x^4-4 x^5\right )}{169+e^{2+4 x^2+2 x^4}-26 x^2+x^4+e^{1+2 x^2+x^4} \left (-26+2 x^2\right )} \, dx=\frac {x^{2} - 4 \, x}{x^{2} + e^{\left (x^{4} + 2 \, x^{2} + 1\right )} - 13} \] Input:
integrate(((-4*x^5+16*x^4-4*x^3+16*x^2+2*x-4)*exp(x^4+2*x^2+1)+4*x^2-26*x+ 52)/(exp(x^4+2*x^2+1)^2+(2*x^2-26)*exp(x^4+2*x^2+1)+x^4-26*x^2+169),x, alg orithm="maxima")
Output:
(x^2 - 4*x)/(x^2 + e^(x^4 + 2*x^2 + 1) - 13)
Time = 0.13 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.96 \[ \int \frac {52-26 x+4 x^2+e^{1+2 x^2+x^4} \left (-4+2 x+16 x^2-4 x^3+16 x^4-4 x^5\right )}{169+e^{2+4 x^2+2 x^4}-26 x^2+x^4+e^{1+2 x^2+x^4} \left (-26+2 x^2\right )} \, dx=\frac {x^{2} - 4 \, x}{x^{2} + e^{\left (x^{4} + 2 \, x^{2} + 1\right )} - 13} \] Input:
integrate(((-4*x^5+16*x^4-4*x^3+16*x^2+2*x-4)*exp(x^4+2*x^2+1)+4*x^2-26*x+ 52)/(exp(x^4+2*x^2+1)^2+(2*x^2-26)*exp(x^4+2*x^2+1)+x^4-26*x^2+169),x, alg orithm="giac")
Output:
(x^2 - 4*x)/(x^2 + e^(x^4 + 2*x^2 + 1) - 13)
Time = 1.94 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.07 \[ \int \frac {52-26 x+4 x^2+e^{1+2 x^2+x^4} \left (-4+2 x+16 x^2-4 x^3+16 x^4-4 x^5\right )}{169+e^{2+4 x^2+2 x^4}-26 x^2+x^4+e^{1+2 x^2+x^4} \left (-26+2 x^2\right )} \, dx=-\frac {4\,x-x^2}{{\mathrm {e}}^{x^4+2\,x^2+1}+x^2-13} \] Input:
int((exp(2*x^2 + x^4 + 1)*(2*x + 16*x^2 - 4*x^3 + 16*x^4 - 4*x^5 - 4) - 26 *x + 4*x^2 + 52)/(exp(4*x^2 + 2*x^4 + 2) + exp(2*x^2 + x^4 + 1)*(2*x^2 - 2 6) - 26*x^2 + x^4 + 169),x)
Output:
-(4*x - x^2)/(exp(2*x^2 + x^4 + 1) + x^2 - 13)
Time = 0.77 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.48 \[ \int \frac {52-26 x+4 x^2+e^{1+2 x^2+x^4} \left (-4+2 x+16 x^2-4 x^3+16 x^4-4 x^5\right )}{169+e^{2+4 x^2+2 x^4}-26 x^2+x^4+e^{1+2 x^2+x^4} \left (-26+2 x^2\right )} \, dx=\frac {-e^{x^{4}+2 x^{2}} e -4 x +13}{e^{x^{4}+2 x^{2}} e +x^{2}-13} \] Input:
int(((-4*x^5+16*x^4-4*x^3+16*x^2+2*x-4)*exp(x^4+2*x^2+1)+4*x^2-26*x+52)/(e xp(x^4+2*x^2+1)^2+(2*x^2-26)*exp(x^4+2*x^2+1)+x^4-26*x^2+169),x)
Output:
( - e**(x**4 + 2*x**2)*e - 4*x + 13)/(e**(x**4 + 2*x**2)*e + x**2 - 13)