Integrand size = 106, antiderivative size = 31 \[ \int \frac {e^{2 x}+e^{2 e^{25-10 x+x^2}} \left (4+4 e^x\right )+e^{e^{25-10 x+x^2}} \left (-4 e^x+e^{25-8 x+x^2} (20-4 x)\right )}{8 e^{2 e^{25-10 x+x^2}}+2 e^{2 x}-8 e^{e^{25-10 x+x^2}+x}} \, dx=\frac {e^x}{2-e^{-e^{(5-x)^2}+x}}+\frac {x}{2} \]
Time = 0.24 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.13 \[ \int \frac {e^{2 x}+e^{2 e^{25-10 x+x^2}} \left (4+4 e^x\right )+e^{e^{25-10 x+x^2}} \left (-4 e^x+e^{25-8 x+x^2} (20-4 x)\right )}{8 e^{2 e^{25-10 x+x^2}}+2 e^{2 x}-8 e^{e^{25-10 x+x^2}+x}} \, dx=\frac {1}{2} \left (e^x+\frac {e^x}{-1+2 e^{e^{25-10 x+x^2}-x}}+x\right ) \]
Integrate[(E^(2*x) + E^(2*E^(25 - 10*x + x^2))*(4 + 4*E^x) + E^E^(25 - 10* x + x^2)*(-4*E^x + E^(25 - 8*x + x^2)*(20 - 4*x)))/(8*E^(2*E^(25 - 10*x + x^2)) + 2*E^(2*x) - 8*E^(E^(25 - 10*x + x^2) + x)),x]
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.
\(\displaystyle \int \frac {e^{2 e^{x^2-10 x+25}} \left (4 e^x+4\right )+e^{e^{x^2-10 x+25}} \left (e^{x^2-8 x+25} (20-4 x)-4 e^x\right )+e^{2 x}}{8 e^{2 e^{x^2-10 x+25}}-8 e^{e^{x^2-10 x+25}+x}+2 e^{2 x}} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {e^{2 e^{x^2-10 x+25}} \left (4 e^x+4\right )+e^{e^{x^2-10 x+25}} \left (e^{x^2-8 x+25} (20-4 x)-4 e^x\right )+e^{2 x}}{2 \left (2 e^{e^{(x-5)^2}}-e^x\right )^2}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{2} \int \frac {4 e^{2 e^{x^2-10 x+25}} \left (1+e^x\right )+e^{2 x}-4 e^{e^{x^2-10 x+25}} \left (e^x-e^{x^2-8 x+25} (5-x)\right )}{\left (2 e^{e^{(x-5)^2}}-e^x\right )^2}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \frac {1}{2} \int \left (\frac {4 e^{2 e^{(x-5)^2}}+e^{2 x}-4 e^{x+e^{(x-5)^2}}+4 e^{x+2 e^{(x-5)^2}}}{\left (2 e^{e^{(x-5)^2}}-e^x\right )^2}-\frac {4 e^{x^2-8 x+e^{(x-5)^2}+25} (x-5)}{\left (-2 e^{e^{(x-5)^2}}+e^x\right )^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{2} \left (8 \int \frac {e^{3 e^{x^2-10 x+25}}}{\left (2 e^{e^{(x-5)^2}}-e^x\right )^2}dx+20 \int \frac {e^{x^2-8 x+e^{(x-5)^2}+25}}{\left (2 e^{e^{(x-5)^2}}-e^x\right )^2}dx+4 \int \frac {e^{2 e^{x^2-10 x+25}}}{-2 e^{e^{(x-5)^2}}+e^x}dx-4 \int \frac {e^{x^2-8 x+e^{(x-5)^2}+25} x}{\left (2 e^{e^{(x-5)^2}}-e^x\right )^2}dx+x\right )\) |
Int[(E^(2*x) + E^(2*E^(25 - 10*x + x^2))*(4 + 4*E^x) + E^E^(25 - 10*x + x^ 2)*(-4*E^x + E^(25 - 8*x + x^2)*(20 - 4*x)))/(8*E^(2*E^(25 - 10*x + x^2)) + 2*E^(2*x) - 8*E^(E^(25 - 10*x + x^2) + x)),x]
3.1.95.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Time = 0.49 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.94
method | result | size |
risch | \(\frac {x}{2}+\frac {{\mathrm e}^{x}}{2}-\frac {{\mathrm e}^{2 x}}{2 \left ({\mathrm e}^{x}-2 \,{\mathrm e}^{{\mathrm e}^{\left (-5+x \right )^{2}}}\right )}\) | \(29\) |
parallelrisch | \(\frac {2 \,{\mathrm e}^{x} x -4 \,{\mathrm e}^{{\mathrm e}^{x^{2}-10 x +25}} x -4 \,{\mathrm e}^{x} {\mathrm e}^{{\mathrm e}^{x^{2}-10 x +25}}}{4 \,{\mathrm e}^{x}-8 \,{\mathrm e}^{{\mathrm e}^{x^{2}-10 x +25}}}\) | \(53\) |
int(((4*exp(x)+4)*exp(exp(x^2-10*x+25))^2+((-4*x+20)*exp(x)^2*exp(x^2-10*x +25)-4*exp(x))*exp(exp(x^2-10*x+25))+exp(x)^2)/(8*exp(exp(x^2-10*x+25))^2- 8*exp(x)*exp(exp(x^2-10*x+25))+2*exp(x)^2),x,method=_RETURNVERBOSE)
Leaf count of result is larger than twice the leaf count of optimal. 71 vs. \(2 (23) = 46\).
Time = 0.27 (sec) , antiderivative size = 71, normalized size of antiderivative = 2.29 \[ \int \frac {e^{2 x}+e^{2 e^{25-10 x+x^2}} \left (4+4 e^x\right )+e^{e^{25-10 x+x^2}} \left (-4 e^x+e^{25-8 x+x^2} (20-4 x)\right )}{8 e^{2 e^{25-10 x+x^2}}+2 e^{2 x}-8 e^{e^{25-10 x+x^2}+x}} \, dx=\frac {2 \, {\left (x + e^{x}\right )} e^{\left ({\left (x e^{\left (2 \, x\right )} + e^{\left (x^{2} - 8 \, x + 25\right )}\right )} e^{\left (-2 \, x\right )}\right )} - x e^{\left (2 \, x\right )}}{2 \, {\left (2 \, e^{\left ({\left (x e^{\left (2 \, x\right )} + e^{\left (x^{2} - 8 \, x + 25\right )}\right )} e^{\left (-2 \, x\right )}\right )} - e^{\left (2 \, x\right )}\right )}} \]
integrate(((4*exp(x)+4)*exp(exp(x^2-10*x+25))^2+((-4*x+20)*exp(x)^2*exp(x^ 2-10*x+25)-4*exp(x))*exp(exp(x^2-10*x+25))+exp(x)^2)/(8*exp(exp(x^2-10*x+2 5))^2-8*exp(x)*exp(exp(x^2-10*x+25))+2*exp(x)^2),x, algorithm=\
1/2*(2*(x + e^x)*e^((x*e^(2*x) + e^(x^2 - 8*x + 25))*e^(-2*x)) - x*e^(2*x) )/(2*e^((x*e^(2*x) + e^(x^2 - 8*x + 25))*e^(-2*x)) - e^(2*x))
Time = 0.15 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00 \[ \int \frac {e^{2 x}+e^{2 e^{25-10 x+x^2}} \left (4+4 e^x\right )+e^{e^{25-10 x+x^2}} \left (-4 e^x+e^{25-8 x+x^2} (20-4 x)\right )}{8 e^{2 e^{25-10 x+x^2}}+2 e^{2 x}-8 e^{e^{25-10 x+x^2}+x}} \, dx=\frac {x}{2} + \frac {e^{x}}{2} + \frac {e^{2 x}}{- 2 e^{x} + 4 e^{e^{x^{2} - 10 x + 25}}} \]
integrate(((4*exp(x)+4)*exp(exp(x**2-10*x+25))**2+((-4*x+20)*exp(x)**2*exp (x**2-10*x+25)-4*exp(x))*exp(exp(x**2-10*x+25))+exp(x)**2)/(8*exp(exp(x**2 -10*x+25))**2-8*exp(x)*exp(exp(x**2-10*x+25))+2*exp(x)**2),x)
Time = 0.23 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.29 \[ \int \frac {e^{2 x}+e^{2 e^{25-10 x+x^2}} \left (4+4 e^x\right )+e^{e^{25-10 x+x^2}} \left (-4 e^x+e^{25-8 x+x^2} (20-4 x)\right )}{8 e^{2 e^{25-10 x+x^2}}+2 e^{2 x}-8 e^{e^{25-10 x+x^2}+x}} \, dx=\frac {x e^{x} - 2 \, {\left (x + e^{x}\right )} e^{\left (e^{\left (x^{2} - 10 \, x + 25\right )}\right )}}{2 \, {\left (e^{x} - 2 \, e^{\left (e^{\left (x^{2} - 10 \, x + 25\right )}\right )}\right )}} \]
integrate(((4*exp(x)+4)*exp(exp(x^2-10*x+25))^2+((-4*x+20)*exp(x)^2*exp(x^ 2-10*x+25)-4*exp(x))*exp(exp(x^2-10*x+25))+exp(x)^2)/(8*exp(exp(x^2-10*x+2 5))^2-8*exp(x)*exp(exp(x^2-10*x+25))+2*exp(x)^2),x, algorithm=\
Leaf count of result is larger than twice the leaf count of optimal. 936 vs. \(2 (23) = 46\).
Time = 0.31 (sec) , antiderivative size = 936, normalized size of antiderivative = 30.19 \[ \int \frac {e^{2 x}+e^{2 e^{25-10 x+x^2}} \left (4+4 e^x\right )+e^{e^{25-10 x+x^2}} \left (-4 e^x+e^{25-8 x+x^2} (20-4 x)\right )}{8 e^{2 e^{25-10 x+x^2}}+2 e^{2 x}-8 e^{e^{25-10 x+x^2}+x}} \, dx=\text {Too large to display} \]
integrate(((4*exp(x)+4)*exp(exp(x^2-10*x+25))^2+((-4*x+20)*exp(x)^2*exp(x^ 2-10*x+25)-4*exp(x))*exp(exp(x^2-10*x+25))+exp(x)^2)/(8*exp(exp(x^2-10*x+2 5))^2-8*exp(x)*exp(exp(x^2-10*x+25))+2*exp(x)^2),x, algorithm=\
1/2*(4*x^3*e^(2*x^2 - 9*x + 50) - 16*x^3*e^(2*x^2 - 10*x + e^(x^2 - 10*x + 25) + 50) + 16*x^3*e^(2*x^2 - 11*x + 2*e^(x^2 - 10*x + 25) + 50) - 8*x^2* e^(2*x^2 - 9*x + e^(x^2 - 10*x + 25) + 50) - 40*x^2*e^(2*x^2 - 9*x + 50) + 16*x^2*e^(2*x^2 - 10*x + 2*e^(x^2 - 10*x + 25) + 50) + 160*x^2*e^(2*x^2 - 10*x + e^(x^2 - 10*x + 25) + 50) - 160*x^2*e^(2*x^2 - 11*x + 2*e^(x^2 - 1 0*x + 25) + 50) - 4*x^2*e^(x^2 + x + 25) - 16*x^2*e^(x^2 - x + 2*e^(x^2 - 10*x + 25) + 25) + 16*x^2*e^(x^2 + e^(x^2 - 10*x + 25) + 25) + 80*x*e^(2*x ^2 - 9*x + e^(x^2 - 10*x + 25) + 50) + 100*x*e^(2*x^2 - 9*x + 50) - 160*x* e^(2*x^2 - 10*x + 2*e^(x^2 - 10*x + 25) + 50) - 400*x*e^(2*x^2 - 10*x + e^ (x^2 - 10*x + 25) + 50) + 400*x*e^(2*x^2 - 11*x + 2*e^(x^2 - 10*x + 25) + 50) + 8*x*e^(x^2 + x + e^(x^2 - 10*x + 25) + 25) + 20*x*e^(x^2 + x + 25) + 80*x*e^(x^2 - x + 2*e^(x^2 - 10*x + 25) + 25) - 16*x*e^(x^2 + 2*e^(x^2 - 10*x + 25) + 25) - 80*x*e^(x^2 + e^(x^2 - 10*x + 25) + 25) + x*e^(11*x) - 4*x*e^(10*x + e^(x^2 - 10*x + 25)) + 4*x*e^(9*x + 2*e^(x^2 - 10*x + 25)) - 200*e^(2*x^2 - 9*x + e^(x^2 - 10*x + 25) + 50) + 400*e^(2*x^2 - 10*x + 2* e^(x^2 - 10*x + 25) + 50) - 40*e^(x^2 + x + e^(x^2 - 10*x + 25) + 25) + 80 *e^(x^2 + 2*e^(x^2 - 10*x + 25) + 25) - 2*e^(11*x + e^(x^2 - 10*x + 25)) + 4*e^(10*x + 2*e^(x^2 - 10*x + 25)))/(4*x^2*e^(2*x^2 - 9*x + 50) - 16*x^2* e^(2*x^2 - 10*x + e^(x^2 - 10*x + 25) + 50) + 16*x^2*e^(2*x^2 - 11*x + 2*e ^(x^2 - 10*x + 25) + 50) - 40*x*e^(2*x^2 - 9*x + 50) + 160*x*e^(2*x^2 -...
Time = 11.26 (sec) , antiderivative size = 87, normalized size of antiderivative = 2.81 \[ \int \frac {e^{2 x}+e^{2 e^{25-10 x+x^2}} \left (4+4 e^x\right )+e^{e^{25-10 x+x^2}} \left (-4 e^x+e^{25-8 x+x^2} (20-4 x)\right )}{8 e^{2 e^{25-10 x+x^2}}+2 e^{2 x}-8 e^{e^{25-10 x+x^2}+x}} \, dx=\frac {x}{2}+\frac {{\mathrm {e}}^x}{2}+\frac {{\mathrm {e}}^{3\,x}+10\,{\mathrm {e}}^{x^2-7\,x+25}-2\,x\,{\mathrm {e}}^{x^2-7\,x+25}}{2\,\left (2\,{\mathrm {e}}^{{\mathrm {e}}^{-10\,x}\,{\mathrm {e}}^{x^2}\,{\mathrm {e}}^{25}}-{\mathrm {e}}^x\right )\,\left (10\,{\mathrm {e}}^{x^2-9\,x+25}+{\mathrm {e}}^x-2\,x\,{\mathrm {e}}^{x^2-9\,x+25}\right )} \]
int((exp(2*x) - exp(exp(x^2 - 10*x + 25))*(4*exp(x) + exp(2*x)*exp(x^2 - 1 0*x + 25)*(4*x - 20)) + exp(2*exp(x^2 - 10*x + 25))*(4*exp(x) + 4))/(8*exp (2*exp(x^2 - 10*x + 25)) + 2*exp(2*x) - 8*exp(exp(x^2 - 10*x + 25))*exp(x) ),x)