Integrand size = 121, antiderivative size = 28 \[ \int \frac {8-32 e^{2 x}-32 x+e^x (32+32 x)}{100+e^{16}+4 e^{4 x}-20 x-16 e^{3 x} x+41 x^2-4 x^3+4 x^4+e^8 \left (20-2 x+4 x^2\right )+e^{2 x} \left (40+4 e^8-4 x+24 x^2\right )+e^x \left (-80 x-8 e^8 x+8 x^2-16 x^3\right )} \, dx=1+\frac {4}{5+\frac {1}{2} \left (e^8-x\right )+\left (e^x-x\right )^2} \] Output:
4/(5-1/2*x+1/2*exp(8)+(exp(x)-x)^2)+1
Time = 0.06 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.07 \[ \int \frac {8-32 e^{2 x}-32 x+e^x (32+32 x)}{100+e^{16}+4 e^{4 x}-20 x-16 e^{3 x} x+41 x^2-4 x^3+4 x^4+e^8 \left (20-2 x+4 x^2\right )+e^{2 x} \left (40+4 e^8-4 x+24 x^2\right )+e^x \left (-80 x-8 e^8 x+8 x^2-16 x^3\right )} \, dx=\frac {8}{10+e^8+2 e^{2 x}-x-4 e^x x+2 x^2} \] Input:
Integrate[(8 - 32*E^(2*x) - 32*x + E^x*(32 + 32*x))/(100 + E^16 + 4*E^(4*x ) - 20*x - 16*E^(3*x)*x + 41*x^2 - 4*x^3 + 4*x^4 + E^8*(20 - 2*x + 4*x^2) + E^(2*x)*(40 + 4*E^8 - 4*x + 24*x^2) + E^x*(-80*x - 8*E^8*x + 8*x^2 - 16* x^3)),x]
Output:
8/(10 + E^8 + 2*E^(2*x) - x - 4*E^x*x + 2*x^2)
Time = 0.58 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.07, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.025, Rules used = {7239, 27, 7237}
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.
\(\displaystyle \int \frac {-32 x-32 e^{2 x}+e^x (32 x+32)+8}{4 x^4-4 x^3+41 x^2+e^8 \left (4 x^2-2 x+20\right )+e^{2 x} \left (24 x^2-4 x+4 e^8+40\right )+e^x \left (-16 x^3+8 x^2-8 e^8 x-80 x\right )-16 e^{3 x} x-20 x+4 e^{4 x}+e^{16}+100} \, dx\) |
\(\Big \downarrow \) 7239 |
\(\displaystyle \int \frac {8 \left (-4 x-4 e^{2 x}+4 e^x (x+1)+1\right )}{\left (2 x^2-4 e^x x-x+2 e^{2 x}+10 \left (1+\frac {e^8}{10}\right )\right )^2}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle 8 \int \frac {-4 x-4 e^{2 x}+4 e^x (x+1)+1}{\left (2 x^2-4 e^x x-x+2 e^{2 x}+e^8+10\right )^2}dx\) |
\(\Big \downarrow \) 7237 |
\(\displaystyle \frac {8}{2 x^2-4 e^x x-x+2 e^{2 x}+e^8+10}\) |
Input:
Int[(8 - 32*E^(2*x) - 32*x + E^x*(32 + 32*x))/(100 + E^16 + 4*E^(4*x) - 20 *x - 16*E^(3*x)*x + 41*x^2 - 4*x^3 + 4*x^4 + E^8*(20 - 2*x + 4*x^2) + E^(2 *x)*(40 + 4*E^8 - 4*x + 24*x^2) + E^x*(-80*x - 8*E^8*x + 8*x^2 - 16*x^3)), x]
Output:
8/(10 + E^8 + 2*E^(2*x) - x - 4*E^x*x + 2*x^2)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(u_)*(y_)^(m_.), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Si mp[q*(y^(m + 1)/(m + 1)), x] /; !FalseQ[q]] /; FreeQ[m, x] && NeQ[m, -1]
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Time = 0.63 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00
method | result | size |
norman | \(\frac {8}{2 \,{\mathrm e}^{2 x}-4 \,{\mathrm e}^{x} x +2 x^{2}+{\mathrm e}^{8}-x +10}\) | \(28\) |
risch | \(\frac {8}{2 \,{\mathrm e}^{2 x}-4 \,{\mathrm e}^{x} x +2 x^{2}+{\mathrm e}^{8}-x +10}\) | \(28\) |
parallelrisch | \(\frac {8}{2 \,{\mathrm e}^{2 x}-4 \,{\mathrm e}^{x} x +2 x^{2}+{\mathrm e}^{8}-x +10}\) | \(28\) |
Input:
int((-32*exp(x)^2+(32*x+32)*exp(x)-32*x+8)/(4*exp(x)^4-16*x*exp(x)^3+(4*ex p(8)+24*x^2-4*x+40)*exp(x)^2+(-8*x*exp(8)-16*x^3+8*x^2-80*x)*exp(x)+exp(8) ^2+(4*x^2-2*x+20)*exp(8)+4*x^4-4*x^3+41*x^2-20*x+100),x,method=_RETURNVERB OSE)
Output:
8/(2*exp(x)^2-4*exp(x)*x+2*x^2+exp(8)-x+10)
Time = 0.08 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.96 \[ \int \frac {8-32 e^{2 x}-32 x+e^x (32+32 x)}{100+e^{16}+4 e^{4 x}-20 x-16 e^{3 x} x+41 x^2-4 x^3+4 x^4+e^8 \left (20-2 x+4 x^2\right )+e^{2 x} \left (40+4 e^8-4 x+24 x^2\right )+e^x \left (-80 x-8 e^8 x+8 x^2-16 x^3\right )} \, dx=\frac {8}{2 \, x^{2} - 4 \, x e^{x} - x + e^{8} + 2 \, e^{\left (2 \, x\right )} + 10} \] Input:
integrate((-32*exp(x)^2+(32*x+32)*exp(x)-32*x+8)/(4*exp(x)^4-16*x*exp(x)^3 +(4*exp(8)+24*x^2-4*x+40)*exp(x)^2+(-8*x*exp(8)-16*x^3+8*x^2-80*x)*exp(x)+ exp(8)^2+(4*x^2-2*x+20)*exp(8)+4*x^4-4*x^3+41*x^2-20*x+100),x, algorithm=" fricas")
Output:
8/(2*x^2 - 4*x*e^x - x + e^8 + 2*e^(2*x) + 10)
Time = 0.12 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.93 \[ \int \frac {8-32 e^{2 x}-32 x+e^x (32+32 x)}{100+e^{16}+4 e^{4 x}-20 x-16 e^{3 x} x+41 x^2-4 x^3+4 x^4+e^8 \left (20-2 x+4 x^2\right )+e^{2 x} \left (40+4 e^8-4 x+24 x^2\right )+e^x \left (-80 x-8 e^8 x+8 x^2-16 x^3\right )} \, dx=\frac {4}{x^{2} - 2 x e^{x} - \frac {x}{2} + e^{2 x} + 5 + \frac {e^{8}}{2}} \] Input:
integrate((-32*exp(x)**2+(32*x+32)*exp(x)-32*x+8)/(4*exp(x)**4-16*x*exp(x) **3+(4*exp(8)+24*x**2-4*x+40)*exp(x)**2+(-8*x*exp(8)-16*x**3+8*x**2-80*x)* exp(x)+exp(8)**2+(4*x**2-2*x+20)*exp(8)+4*x**4-4*x**3+41*x**2-20*x+100),x)
Output:
4/(x**2 - 2*x*exp(x) - x/2 + exp(2*x) + 5 + exp(8)/2)
Time = 0.12 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.96 \[ \int \frac {8-32 e^{2 x}-32 x+e^x (32+32 x)}{100+e^{16}+4 e^{4 x}-20 x-16 e^{3 x} x+41 x^2-4 x^3+4 x^4+e^8 \left (20-2 x+4 x^2\right )+e^{2 x} \left (40+4 e^8-4 x+24 x^2\right )+e^x \left (-80 x-8 e^8 x+8 x^2-16 x^3\right )} \, dx=\frac {8}{2 \, x^{2} - 4 \, x e^{x} - x + e^{8} + 2 \, e^{\left (2 \, x\right )} + 10} \] Input:
integrate((-32*exp(x)^2+(32*x+32)*exp(x)-32*x+8)/(4*exp(x)^4-16*x*exp(x)^3 +(4*exp(8)+24*x^2-4*x+40)*exp(x)^2+(-8*x*exp(8)-16*x^3+8*x^2-80*x)*exp(x)+ exp(8)^2+(4*x^2-2*x+20)*exp(8)+4*x^4-4*x^3+41*x^2-20*x+100),x, algorithm=" maxima")
Output:
8/(2*x^2 - 4*x*e^x - x + e^8 + 2*e^(2*x) + 10)
Time = 0.26 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.96 \[ \int \frac {8-32 e^{2 x}-32 x+e^x (32+32 x)}{100+e^{16}+4 e^{4 x}-20 x-16 e^{3 x} x+41 x^2-4 x^3+4 x^4+e^8 \left (20-2 x+4 x^2\right )+e^{2 x} \left (40+4 e^8-4 x+24 x^2\right )+e^x \left (-80 x-8 e^8 x+8 x^2-16 x^3\right )} \, dx=\frac {16}{2 \, x^{2} - 4 \, x e^{x} - x + e^{8} + 2 \, e^{\left (2 \, x\right )} + 10} \] Input:
integrate((-32*exp(x)^2+(32*x+32)*exp(x)-32*x+8)/(4*exp(x)^4-16*x*exp(x)^3 +(4*exp(8)+24*x^2-4*x+40)*exp(x)^2+(-8*x*exp(8)-16*x^3+8*x^2-80*x)*exp(x)+ exp(8)^2+(4*x^2-2*x+20)*exp(8)+4*x^4-4*x^3+41*x^2-20*x+100),x, algorithm=" giac")
Output:
16/(2*x^2 - 4*x*e^x - x + e^8 + 2*e^(2*x) + 10)
Time = 3.32 (sec) , antiderivative size = 71, normalized size of antiderivative = 2.54 \[ \int \frac {8-32 e^{2 x}-32 x+e^x (32+32 x)}{100+e^{16}+4 e^{4 x}-20 x-16 e^{3 x} x+41 x^2-4 x^3+4 x^4+e^8 \left (20-2 x+4 x^2\right )+e^{2 x} \left (40+4 e^8-4 x+24 x^2\right )+e^x \left (-80 x-8 e^8 x+8 x^2-16 x^3\right )} \, dx=-\frac {\frac {16\,x^2}{{\mathrm {e}}^8+10}+\frac {16\,{\mathrm {e}}^{2\,x}}{{\mathrm {e}}^8+10}-\frac {8\,x}{{\mathrm {e}}^8+10}-\frac {32\,x\,{\mathrm {e}}^x}{{\mathrm {e}}^8+10}}{2\,{\mathrm {e}}^{2\,x}-x+{\mathrm {e}}^8-4\,x\,{\mathrm {e}}^x+2\,x^2+10} \] Input:
int(-(32*x + 32*exp(2*x) - exp(x)*(32*x + 32) - 8)/(4*exp(4*x) - 20*x + ex p(16) - 16*x*exp(3*x) - exp(x)*(80*x + 8*x*exp(8) - 8*x^2 + 16*x^3) + exp( 8)*(4*x^2 - 2*x + 20) + exp(2*x)*(4*exp(8) - 4*x + 24*x^2 + 40) + 41*x^2 - 4*x^3 + 4*x^4 + 100),x)
Output:
-((16*x^2)/(exp(8) + 10) + (16*exp(2*x))/(exp(8) + 10) - (8*x)/(exp(8) + 1 0) - (32*x*exp(x))/(exp(8) + 10))/(2*exp(2*x) - x + exp(8) - 4*x*exp(x) + 2*x^2 + 10)
\[ \int \frac {8-32 e^{2 x}-32 x+e^x (32+32 x)}{100+e^{16}+4 e^{4 x}-20 x-16 e^{3 x} x+41 x^2-4 x^3+4 x^4+e^8 \left (20-2 x+4 x^2\right )+e^{2 x} \left (40+4 e^8-4 x+24 x^2\right )+e^x \left (-80 x-8 e^8 x+8 x^2-16 x^3\right )} \, dx=-32 \left (\int \frac {e^{2 x}}{4 e^{4 x}-16 e^{3 x} x +4 e^{2 x} e^{8}+24 e^{2 x} x^{2}-4 e^{2 x} x +40 e^{2 x}-8 e^{x} e^{8} x -16 e^{x} x^{3}+8 e^{x} x^{2}-80 e^{x} x +e^{16}+4 e^{8} x^{2}-2 e^{8} x +20 e^{8}+4 x^{4}-4 x^{3}+41 x^{2}-20 x +100}d x \right )+32 \left (\int \frac {e^{x}}{4 e^{4 x}-16 e^{3 x} x +4 e^{2 x} e^{8}+24 e^{2 x} x^{2}-4 e^{2 x} x +40 e^{2 x}-8 e^{x} e^{8} x -16 e^{x} x^{3}+8 e^{x} x^{2}-80 e^{x} x +e^{16}+4 e^{8} x^{2}-2 e^{8} x +20 e^{8}+4 x^{4}-4 x^{3}+41 x^{2}-20 x +100}d x \right )+32 \left (\int \frac {e^{x} x}{4 e^{4 x}-16 e^{3 x} x +4 e^{2 x} e^{8}+24 e^{2 x} x^{2}-4 e^{2 x} x +40 e^{2 x}-8 e^{x} e^{8} x -16 e^{x} x^{3}+8 e^{x} x^{2}-80 e^{x} x +e^{16}+4 e^{8} x^{2}-2 e^{8} x +20 e^{8}+4 x^{4}-4 x^{3}+41 x^{2}-20 x +100}d x \right )-32 \left (\int \frac {x}{4 e^{4 x}-16 e^{3 x} x +4 e^{2 x} e^{8}+24 e^{2 x} x^{2}-4 e^{2 x} x +40 e^{2 x}-8 e^{x} e^{8} x -16 e^{x} x^{3}+8 e^{x} x^{2}-80 e^{x} x +e^{16}+4 e^{8} x^{2}-2 e^{8} x +20 e^{8}+4 x^{4}-4 x^{3}+41 x^{2}-20 x +100}d x \right )+8 \left (\int \frac {1}{4 e^{4 x}-16 e^{3 x} x +4 e^{2 x} e^{8}+24 e^{2 x} x^{2}-4 e^{2 x} x +40 e^{2 x}-8 e^{x} e^{8} x -16 e^{x} x^{3}+8 e^{x} x^{2}-80 e^{x} x +e^{16}+4 e^{8} x^{2}-2 e^{8} x +20 e^{8}+4 x^{4}-4 x^{3}+41 x^{2}-20 x +100}d x \right ) \] Input:
int((-32*exp(x)^2+(32*x+32)*exp(x)-32*x+8)/(4*exp(x)^4-16*x*exp(x)^3+(4*ex p(8)+24*x^2-4*x+40)*exp(x)^2+(-8*x*exp(8)-16*x^3+8*x^2-80*x)*exp(x)+exp(8) ^2+(4*x^2-2*x+20)*exp(8)+4*x^4-4*x^3+41*x^2-20*x+100),x)
Output:
8*( - 4*int(e**(2*x)/(4*e**(4*x) - 16*e**(3*x)*x + 4*e**(2*x)*e**8 + 24*e* *(2*x)*x**2 - 4*e**(2*x)*x + 40*e**(2*x) - 8*e**x*e**8*x - 16*e**x*x**3 + 8*e**x*x**2 - 80*e**x*x + e**16 + 4*e**8*x**2 - 2*e**8*x + 20*e**8 + 4*x** 4 - 4*x**3 + 41*x**2 - 20*x + 100),x) + 4*int(e**x/(4*e**(4*x) - 16*e**(3* x)*x + 4*e**(2*x)*e**8 + 24*e**(2*x)*x**2 - 4*e**(2*x)*x + 40*e**(2*x) - 8 *e**x*e**8*x - 16*e**x*x**3 + 8*e**x*x**2 - 80*e**x*x + e**16 + 4*e**8*x** 2 - 2*e**8*x + 20*e**8 + 4*x**4 - 4*x**3 + 41*x**2 - 20*x + 100),x) + 4*in t((e**x*x)/(4*e**(4*x) - 16*e**(3*x)*x + 4*e**(2*x)*e**8 + 24*e**(2*x)*x** 2 - 4*e**(2*x)*x + 40*e**(2*x) - 8*e**x*e**8*x - 16*e**x*x**3 + 8*e**x*x** 2 - 80*e**x*x + e**16 + 4*e**8*x**2 - 2*e**8*x + 20*e**8 + 4*x**4 - 4*x**3 + 41*x**2 - 20*x + 100),x) - 4*int(x/(4*e**(4*x) - 16*e**(3*x)*x + 4*e**( 2*x)*e**8 + 24*e**(2*x)*x**2 - 4*e**(2*x)*x + 40*e**(2*x) - 8*e**x*e**8*x - 16*e**x*x**3 + 8*e**x*x**2 - 80*e**x*x + e**16 + 4*e**8*x**2 - 2*e**8*x + 20*e**8 + 4*x**4 - 4*x**3 + 41*x**2 - 20*x + 100),x) + int(1/(4*e**(4*x) - 16*e**(3*x)*x + 4*e**(2*x)*e**8 + 24*e**(2*x)*x**2 - 4*e**(2*x)*x + 40* e**(2*x) - 8*e**x*e**8*x - 16*e**x*x**3 + 8*e**x*x**2 - 80*e**x*x + e**16 + 4*e**8*x**2 - 2*e**8*x + 20*e**8 + 4*x**4 - 4*x**3 + 41*x**2 - 20*x + 10 0),x))