Integrand size = 124, antiderivative size = 28 \[ \int \frac {e^{e^{e^x}} \left (30 x-15 x^2+e^x \left (-15-5 x+5 x^2\right )+e^{e^x} \left (e^{2 x} \left (15 x-5 x^2\right )+e^x \left (-15 x^2+5 x^3\right )\right )\right )}{9 x^4-6 x^5+x^6+e^{2 x} \left (9 x^2-6 x^3+x^4\right )+e^x \left (-18 x^3+12 x^4-2 x^5\right )} \, dx=\frac {5 e^{e^{e^x}}}{(3-x) \left (e^x-x\right ) x} \] Output:
5*exp(exp(exp(x)))/x/(exp(x)-x)/(3-x)
Time = 0.10 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.93 \[ \int \frac {e^{e^{e^x}} \left (30 x-15 x^2+e^x \left (-15-5 x+5 x^2\right )+e^{e^x} \left (e^{2 x} \left (15 x-5 x^2\right )+e^x \left (-15 x^2+5 x^3\right )\right )\right )}{9 x^4-6 x^5+x^6+e^{2 x} \left (9 x^2-6 x^3+x^4\right )+e^x \left (-18 x^3+12 x^4-2 x^5\right )} \, dx=-\frac {5 e^{e^{e^x}}}{\left (e^x-x\right ) (-3+x) x} \] Input:
Integrate[(E^E^E^x*(30*x - 15*x^2 + E^x*(-15 - 5*x + 5*x^2) + E^E^x*(E^(2* x)*(15*x - 5*x^2) + E^x*(-15*x^2 + 5*x^3))))/(9*x^4 - 6*x^5 + x^6 + E^(2*x )*(9*x^2 - 6*x^3 + x^4) + E^x*(-18*x^3 + 12*x^4 - 2*x^5)),x]
Output:
(-5*E^E^E^x)/((E^x - x)*(-3 + x)*x)
Leaf count is larger than twice the leaf count of optimal. \(100\) vs. \(2(28)=56\).
Time = 0.45 (sec) , antiderivative size = 100, normalized size of antiderivative = 3.57, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.008, Rules used = {2726}
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.
\(\displaystyle \int \frac {e^{e^{e^x}} \left (-15 x^2+e^x \left (5 x^2-5 x-15\right )+e^{e^x} \left (e^{2 x} \left (15 x-5 x^2\right )+e^x \left (5 x^3-15 x^2\right )\right )+30 x\right )}{x^6-6 x^5+9 x^4+e^x \left (-2 x^5+12 x^4-18 x^3\right )+e^{2 x} \left (x^4-6 x^3+9 x^2\right )} \, dx\) |
\(\Big \downarrow \) 2726 |
\(\displaystyle \frac {5 e^{e^{e^x}-x} \left (e^{2 x} \left (3 x-x^2\right )-e^x \left (3 x^2-x^3\right )\right )}{x^6-6 x^5+9 x^4-2 e^x \left (x^5-6 x^4+9 x^3\right )+e^{2 x} \left (x^4-6 x^3+9 x^2\right )}\) |
Input:
Int[(E^E^E^x*(30*x - 15*x^2 + E^x*(-15 - 5*x + 5*x^2) + E^E^x*(E^(2*x)*(15 *x - 5*x^2) + E^x*(-15*x^2 + 5*x^3))))/(9*x^4 - 6*x^5 + x^6 + E^(2*x)*(9*x ^2 - 6*x^3 + x^4) + E^x*(-18*x^3 + 12*x^4 - 2*x^5)),x]
Output:
(5*E^(E^E^x - x)*(E^(2*x)*(3*x - x^2) - E^x*(3*x^2 - x^3)))/(9*x^4 - 6*x^5 + x^6 + E^(2*x)*(9*x^2 - 6*x^3 + x^4) - 2*E^x*(9*x^3 - 6*x^4 + x^5))
Int[(y_.)*(F_)^(u_)*((v_) + (w_)), x_Symbol] :> With[{z = v*(y/(Log[F]*D[u, x]))}, Simp[F^u*z, x] /; EqQ[D[z, x], w*y]] /; FreeQ[F, x]
Time = 1.93 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00
method | result | size |
risch | \(\frac {5 \,{\mathrm e}^{{\mathrm e}^{{\mathrm e}^{x}}}}{x \left (x^{2}-{\mathrm e}^{x} x -3 x +3 \,{\mathrm e}^{x}\right )}\) | \(28\) |
parallelrisch | \(\frac {5 \,{\mathrm e}^{{\mathrm e}^{{\mathrm e}^{x}}}}{x \left (x^{2}-{\mathrm e}^{x} x -3 x +3 \,{\mathrm e}^{x}\right )}\) | \(28\) |
Input:
int((((-5*x^2+15*x)*exp(x)^2+(5*x^3-15*x^2)*exp(x))*exp(exp(x))+(5*x^2-5*x -15)*exp(x)-15*x^2+30*x)*exp(exp(exp(x)))/((x^4-6*x^3+9*x^2)*exp(x)^2+(-2* x^5+12*x^4-18*x^3)*exp(x)+x^6-6*x^5+9*x^4),x,method=_RETURNVERBOSE)
Output:
5/x/(x^2-exp(x)*x-3*x+3*exp(x))*exp(exp(exp(x)))
Time = 0.10 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00 \[ \int \frac {e^{e^{e^x}} \left (30 x-15 x^2+e^x \left (-15-5 x+5 x^2\right )+e^{e^x} \left (e^{2 x} \left (15 x-5 x^2\right )+e^x \left (-15 x^2+5 x^3\right )\right )\right )}{9 x^4-6 x^5+x^6+e^{2 x} \left (9 x^2-6 x^3+x^4\right )+e^x \left (-18 x^3+12 x^4-2 x^5\right )} \, dx=\frac {5 \, e^{\left (e^{\left (e^{x}\right )}\right )}}{x^{3} - 3 \, x^{2} - {\left (x^{2} - 3 \, x\right )} e^{x}} \] Input:
integrate((((-5*x^2+15*x)*exp(x)^2+(5*x^3-15*x^2)*exp(x))*exp(exp(x))+(5*x ^2-5*x-15)*exp(x)-15*x^2+30*x)*exp(exp(exp(x)))/((x^4-6*x^3+9*x^2)*exp(x)^ 2+(-2*x^5+12*x^4-18*x^3)*exp(x)+x^6-6*x^5+9*x^4),x, algorithm="fricas")
Output:
5*e^(e^(e^x))/(x^3 - 3*x^2 - (x^2 - 3*x)*e^x)
Time = 0.17 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.04 \[ \int \frac {e^{e^{e^x}} \left (30 x-15 x^2+e^x \left (-15-5 x+5 x^2\right )+e^{e^x} \left (e^{2 x} \left (15 x-5 x^2\right )+e^x \left (-15 x^2+5 x^3\right )\right )\right )}{9 x^4-6 x^5+x^6+e^{2 x} \left (9 x^2-6 x^3+x^4\right )+e^x \left (-18 x^3+12 x^4-2 x^5\right )} \, dx=\frac {5 e^{e^{e^{x}}}}{x^{3} - x^{2} e^{x} - 3 x^{2} + 3 x e^{x}} \] Input:
integrate((((-5*x**2+15*x)*exp(x)**2+(5*x**3-15*x**2)*exp(x))*exp(exp(x))+ (5*x**2-5*x-15)*exp(x)-15*x**2+30*x)*exp(exp(exp(x)))/((x**4-6*x**3+9*x**2 )*exp(x)**2+(-2*x**5+12*x**4-18*x**3)*exp(x)+x**6-6*x**5+9*x**4),x)
Output:
5*exp(exp(exp(x)))/(x**3 - x**2*exp(x) - 3*x**2 + 3*x*exp(x))
Time = 0.10 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00 \[ \int \frac {e^{e^{e^x}} \left (30 x-15 x^2+e^x \left (-15-5 x+5 x^2\right )+e^{e^x} \left (e^{2 x} \left (15 x-5 x^2\right )+e^x \left (-15 x^2+5 x^3\right )\right )\right )}{9 x^4-6 x^5+x^6+e^{2 x} \left (9 x^2-6 x^3+x^4\right )+e^x \left (-18 x^3+12 x^4-2 x^5\right )} \, dx=\frac {5 \, e^{\left (e^{\left (e^{x}\right )}\right )}}{x^{3} - 3 \, x^{2} - {\left (x^{2} - 3 \, x\right )} e^{x}} \] Input:
integrate((((-5*x^2+15*x)*exp(x)^2+(5*x^3-15*x^2)*exp(x))*exp(exp(x))+(5*x ^2-5*x-15)*exp(x)-15*x^2+30*x)*exp(exp(exp(x)))/((x^4-6*x^3+9*x^2)*exp(x)^ 2+(-2*x^5+12*x^4-18*x^3)*exp(x)+x^6-6*x^5+9*x^4),x, algorithm="maxima")
Output:
5*e^(e^(e^x))/(x^3 - 3*x^2 - (x^2 - 3*x)*e^x)
\[ \int \frac {e^{e^{e^x}} \left (30 x-15 x^2+e^x \left (-15-5 x+5 x^2\right )+e^{e^x} \left (e^{2 x} \left (15 x-5 x^2\right )+e^x \left (-15 x^2+5 x^3\right )\right )\right )}{9 x^4-6 x^5+x^6+e^{2 x} \left (9 x^2-6 x^3+x^4\right )+e^x \left (-18 x^3+12 x^4-2 x^5\right )} \, dx=\int { -\frac {5 \, {\left (3 \, x^{2} - {\left (x^{2} - x - 3\right )} e^{x} + {\left ({\left (x^{2} - 3 \, x\right )} e^{\left (2 \, x\right )} - {\left (x^{3} - 3 \, x^{2}\right )} e^{x}\right )} e^{\left (e^{x}\right )} - 6 \, x\right )} e^{\left (e^{\left (e^{x}\right )}\right )}}{x^{6} - 6 \, x^{5} + 9 \, x^{4} + {\left (x^{4} - 6 \, x^{3} + 9 \, x^{2}\right )} e^{\left (2 \, x\right )} - 2 \, {\left (x^{5} - 6 \, x^{4} + 9 \, x^{3}\right )} e^{x}} \,d x } \] Input:
integrate((((-5*x^2+15*x)*exp(x)^2+(5*x^3-15*x^2)*exp(x))*exp(exp(x))+(5*x ^2-5*x-15)*exp(x)-15*x^2+30*x)*exp(exp(exp(x)))/((x^4-6*x^3+9*x^2)*exp(x)^ 2+(-2*x^5+12*x^4-18*x^3)*exp(x)+x^6-6*x^5+9*x^4),x, algorithm="giac")
Output:
integrate(-5*(3*x^2 - (x^2 - x - 3)*e^x + ((x^2 - 3*x)*e^(2*x) - (x^3 - 3* x^2)*e^x)*e^(e^x) - 6*x)*e^(e^(e^x))/(x^6 - 6*x^5 + 9*x^4 + (x^4 - 6*x^3 + 9*x^2)*e^(2*x) - 2*(x^5 - 6*x^4 + 9*x^3)*e^x), x)
Time = 0.18 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.07 \[ \int \frac {e^{e^{e^x}} \left (30 x-15 x^2+e^x \left (-15-5 x+5 x^2\right )+e^{e^x} \left (e^{2 x} \left (15 x-5 x^2\right )+e^x \left (-15 x^2+5 x^3\right )\right )\right )}{9 x^4-6 x^5+x^6+e^{2 x} \left (9 x^2-6 x^3+x^4\right )+e^x \left (-18 x^3+12 x^4-2 x^5\right )} \, dx=-\frac {5\,{\mathrm {e}}^{{\mathrm {e}}^{{\mathrm {e}}^x}}}{x^2\,{\mathrm {e}}^x-3\,x\,{\mathrm {e}}^x+3\,x^2-x^3} \] Input:
int((exp(exp(exp(x)))*(30*x + exp(exp(x))*(exp(2*x)*(15*x - 5*x^2) - exp(x )*(15*x^2 - 5*x^3)) - exp(x)*(5*x - 5*x^2 + 15) - 15*x^2))/(exp(2*x)*(9*x^ 2 - 6*x^3 + x^4) - exp(x)*(18*x^3 - 12*x^4 + 2*x^5) + 9*x^4 - 6*x^5 + x^6) ,x)
Output:
-(5*exp(exp(exp(x))))/(x^2*exp(x) - 3*x*exp(x) + 3*x^2 - x^3)
Time = 0.15 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.18 \[ \int \frac {e^{e^{e^x}} \left (30 x-15 x^2+e^x \left (-15-5 x+5 x^2\right )+e^{e^x} \left (e^{2 x} \left (15 x-5 x^2\right )+e^x \left (-15 x^2+5 x^3\right )\right )\right )}{9 x^4-6 x^5+x^6+e^{2 x} \left (9 x^2-6 x^3+x^4\right )+e^x \left (-18 x^3+12 x^4-2 x^5\right )} \, dx=-\frac {5 e^{e^{e^{x}}}}{x \left (e^{x} x -3 e^{x}-x^{2}+3 x \right )} \] Input:
int((((-5*x^2+15*x)*exp(x)^2+(5*x^3-15*x^2)*exp(x))*exp(exp(x))+(5*x^2-5*x -15)*exp(x)-15*x^2+30*x)*exp(exp(exp(x)))/((x^4-6*x^3+9*x^2)*exp(x)^2+(-2* x^5+12*x^4-18*x^3)*exp(x)+x^6-6*x^5+9*x^4),x)
Output:
( - 5*e**(e**(e**x)))/(x*(e**x*x - 3*e**x - x**2 + 3*x))