Integrand size = 138, antiderivative size = 28 \[ \int \frac {e^{\frac {e^{4 e^{e^4}} \left (e^3 x+x^2\right )}{-e^3-x+3 e^{4 e^{e^4}} x}} \left (3 e^{8 e^{e^4}} x^2+e^{4 e^{e^4}} \left (-e^6-2 e^3 x-x^2\right )\right )}{e^6+2 e^3 x+x^2+9 e^{8 e^{e^4}} x^2+e^{4 e^{e^4}} \left (-6 e^3 x-6 x^2\right )} \, dx=e^{\frac {x}{-e^{-4 e^{e^4}}+\frac {3 x}{e^3+x}}} \]
Time = 0.38 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.36 \[ \int \frac {e^{\frac {e^{4 e^{e^4}} \left (e^3 x+x^2\right )}{-e^3-x+3 e^{4 e^{e^4}} x}} \left (3 e^{8 e^{e^4}} x^2+e^{4 e^{e^4}} \left (-e^6-2 e^3 x-x^2\right )\right )}{e^6+2 e^3 x+x^2+9 e^{8 e^{e^4}} x^2+e^{4 e^{e^4}} \left (-6 e^3 x-6 x^2\right )} \, dx=e^{-\frac {e^{4 e^{e^4}} x \left (e^3+x\right )}{e^3+x-3 e^{4 e^{e^4}} x}} \]
Integrate[(E^((E^(4*E^E^4)*(E^3*x + x^2))/(-E^3 - x + 3*E^(4*E^E^4)*x))*(3 *E^(8*E^E^4)*x^2 + E^(4*E^E^4)*(-E^6 - 2*E^3*x - x^2)))/(E^6 + 2*E^3*x + x ^2 + 9*E^(8*E^E^4)*x^2 + E^(4*E^E^4)*(-6*E^3*x - 6*x^2)),x]
Time = 3.03 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.43, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.022, Rules used = {6, 7292, 7257}
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.
\(\displaystyle \int \frac {\left (3 e^{8 e^{e^4}} x^2+e^{4 e^{e^4}} \left (-x^2-2 e^3 x-e^6\right )\right ) \exp \left (\frac {e^{4 e^{e^4}} \left (x^2+e^3 x\right )}{3 e^{4 e^{e^4}} x-x-e^3}\right )}{9 e^{8 e^{e^4}} x^2+x^2+e^{4 e^{e^4}} \left (-6 x^2-6 e^3 x\right )+2 e^3 x+e^6} \, dx\) |
\(\Big \downarrow \) 6 |
\(\displaystyle \int \frac {\left (3 e^{8 e^{e^4}} x^2+e^{4 e^{e^4}} \left (-x^2-2 e^3 x-e^6\right )\right ) \exp \left (\frac {e^{4 e^{e^4}} \left (x^2+e^3 x\right )}{3 e^{4 e^{e^4}} x-x-e^3}\right )}{\left (1+9 e^{8 e^{e^4}}\right ) x^2+e^{4 e^{e^4}} \left (-6 x^2-6 e^3 x\right )+2 e^3 x+e^6}dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {\left (-e^{4 e^{e^4}} \left (1-3 e^{4 e^{e^4}}\right ) x^2-2 e^{3+4 e^{e^4}} x-e^{6+4 e^{e^4}}\right ) \exp \left (\frac {e^{4 e^{e^4}} x \left (x+e^3\right )}{-\left (\left (1-3 e^{4 e^{e^4}}\right ) x\right )-e^3}\right )}{\left (1-3 e^{4 e^{e^4}}\right )^2 x^2+2 \left (e^3-3 e^{3+4 e^{e^4}}\right ) x+e^6}dx\) |
\(\Big \downarrow \) 7257 |
\(\displaystyle \exp \left (-\frac {e^{4 e^{e^4}} x \left (x+e^3\right )}{\left (1-3 e^{4 e^{e^4}}\right ) x+e^3}\right )\) |
Int[(E^((E^(4*E^E^4)*(E^3*x + x^2))/(-E^3 - x + 3*E^(4*E^E^4)*x))*(3*E^(8* E^E^4)*x^2 + E^(4*E^E^4)*(-E^6 - 2*E^3*x - x^2)))/(E^6 + 2*E^3*x + x^2 + 9 *E^(8*E^E^4)*x^2 + E^(4*E^E^4)*(-6*E^3*x - 6*x^2)),x]
3.6.86.3.1 Defintions of rubi rules used
Int[(u_.)*((v_.) + (a_.)*(Fx_) + (b_.)*(Fx_))^(p_.), x_Symbol] :> Int[u*(v + (a + b)*Fx)^p, x] /; FreeQ[{a, b}, x] && !FreeQ[Fx, x]
Int[(F_)^(v_)*(u_), x_Symbol] :> With[{q = DerivativeDivides[v, u, x]}, Sim p[q*(F^v/Log[F]), x] /; !FalseQ[q]] /; FreeQ[F, x]
Time = 1.02 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.07
method | result | size |
risch | \({\mathrm e}^{-\frac {x \left ({\mathrm e}^{3}+x \right ) {\mathrm e}^{4 \,{\mathrm e}^{{\mathrm e}^{4}}}}{-3 x \,{\mathrm e}^{4 \,{\mathrm e}^{{\mathrm e}^{4}}}+{\mathrm e}^{3}+x}}\) | \(30\) |
parallelrisch | \({\mathrm e}^{-\frac {x \left ({\mathrm e}^{3}+x \right ) {\mathrm e}^{4 \,{\mathrm e}^{{\mathrm e}^{4}}}}{-3 x \,{\mathrm e}^{4 \,{\mathrm e}^{{\mathrm e}^{4}}}+{\mathrm e}^{3}+x}}\) | \(30\) |
norman | \(\frac {{\mathrm e}^{3} {\mathrm e}^{\frac {\left (x \,{\mathrm e}^{3}+x^{2}\right ) {\mathrm e}^{4 \,{\mathrm e}^{{\mathrm e}^{4}}}}{3 x \,{\mathrm e}^{4 \,{\mathrm e}^{{\mathrm e}^{4}}}-{\mathrm e}^{3}-x}}+\left (-3 \,{\mathrm e}^{4 \,{\mathrm e}^{{\mathrm e}^{4}}}+1\right ) x \,{\mathrm e}^{\frac {\left (x \,{\mathrm e}^{3}+x^{2}\right ) {\mathrm e}^{4 \,{\mathrm e}^{{\mathrm e}^{4}}}}{3 x \,{\mathrm e}^{4 \,{\mathrm e}^{{\mathrm e}^{4}}}-{\mathrm e}^{3}-x}}}{-3 x \,{\mathrm e}^{4 \,{\mathrm e}^{{\mathrm e}^{4}}}+{\mathrm e}^{3}+x}\) | \(103\) |
int((3*x^2*exp(exp(exp(4)))^8+(-exp(3)^2-2*x*exp(3)-x^2)*exp(exp(exp(4)))^ 4)*exp((x*exp(3)+x^2)*exp(exp(exp(4)))^4/(3*x*exp(exp(exp(4)))^4-exp(3)-x) )/(9*x^2*exp(exp(exp(4)))^8+(-6*x*exp(3)-6*x^2)*exp(exp(exp(4)))^4+exp(3)^ 2+2*x*exp(3)+x^2),x,method=_RETURNVERBOSE)
Time = 0.26 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.25 \[ \int \frac {e^{\frac {e^{4 e^{e^4}} \left (e^3 x+x^2\right )}{-e^3-x+3 e^{4 e^{e^4}} x}} \left (3 e^{8 e^{e^4}} x^2+e^{4 e^{e^4}} \left (-e^6-2 e^3 x-x^2\right )\right )}{e^6+2 e^3 x+x^2+9 e^{8 e^{e^4}} x^2+e^{4 e^{e^4}} \left (-6 e^3 x-6 x^2\right )} \, dx=e^{\left (\frac {{\left (x^{2} + x e^{3}\right )} e^{\left (4 \, e^{\left (e^{4}\right )}\right )}}{3 \, x e^{\left (4 \, e^{\left (e^{4}\right )}\right )} - x - e^{3}}\right )} \]
integrate((3*x^2*exp(exp(exp(4)))^8+(-exp(3)^2-2*x*exp(3)-x^2)*exp(exp(exp (4)))^4)*exp((x*exp(3)+x^2)*exp(exp(exp(4)))^4/(3*x*exp(exp(exp(4)))^4-exp (3)-x))/(9*x^2*exp(exp(exp(4)))^8+(-6*x*exp(3)-6*x^2)*exp(exp(exp(4)))^4+e xp(3)^2+2*x*exp(3)+x^2),x, algorithm=\
Time = 0.36 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.21 \[ \int \frac {e^{\frac {e^{4 e^{e^4}} \left (e^3 x+x^2\right )}{-e^3-x+3 e^{4 e^{e^4}} x}} \left (3 e^{8 e^{e^4}} x^2+e^{4 e^{e^4}} \left (-e^6-2 e^3 x-x^2\right )\right )}{e^6+2 e^3 x+x^2+9 e^{8 e^{e^4}} x^2+e^{4 e^{e^4}} \left (-6 e^3 x-6 x^2\right )} \, dx=e^{\frac {\left (x^{2} + x e^{3}\right ) e^{4 e^{e^{4}}}}{- x + 3 x e^{4 e^{e^{4}}} - e^{3}}} \]
integrate((3*x**2*exp(exp(exp(4)))**8+(-exp(3)**2-2*x*exp(3)-x**2)*exp(exp (exp(4)))**4)*exp((x*exp(3)+x**2)*exp(exp(exp(4)))**4/(3*x*exp(exp(exp(4)) )**4-exp(3)-x))/(9*x**2*exp(exp(exp(4)))**8+(-6*x*exp(3)-6*x**2)*exp(exp(e xp(4)))**4+exp(3)**2+2*x*exp(3)+x**2),x)
Leaf count of result is larger than twice the leaf count of optimal. 218 vs. \(2 (23) = 46\).
Time = 0.43 (sec) , antiderivative size = 218, normalized size of antiderivative = 7.79 \[ \int \frac {e^{\frac {e^{4 e^{e^4}} \left (e^3 x+x^2\right )}{-e^3-x+3 e^{4 e^{e^4}} x}} \left (3 e^{8 e^{e^4}} x^2+e^{4 e^{e^4}} \left (-e^6-2 e^3 x-x^2\right )\right )}{e^6+2 e^3 x+x^2+9 e^{8 e^{e^4}} x^2+e^{4 e^{e^4}} \left (-6 e^3 x-6 x^2\right )} \, dx=e^{\left (\frac {3 \, x e^{\left (8 \, e^{\left (e^{4}\right )}\right )}}{9 \, e^{\left (8 \, e^{\left (e^{4}\right )}\right )} - 6 \, e^{\left (4 \, e^{\left (e^{4}\right )}\right )} + 1} - \frac {x e^{\left (4 \, e^{\left (e^{4}\right )}\right )}}{9 \, e^{\left (8 \, e^{\left (e^{4}\right )}\right )} - 6 \, e^{\left (4 \, e^{\left (e^{4}\right )}\right )} + 1} + \frac {e^{\left (4 \, e^{\left (e^{4}\right )} + 6\right )}}{x {\left (27 \, e^{\left (12 \, e^{\left (e^{4}\right )}\right )} - 27 \, e^{\left (8 \, e^{\left (e^{4}\right )}\right )} + 9 \, e^{\left (4 \, e^{\left (e^{4}\right )}\right )} - 1\right )} - e^{3} - 9 \, e^{\left (8 \, e^{\left (e^{4}\right )} + 3\right )} + 6 \, e^{\left (4 \, e^{\left (e^{4}\right )} + 3\right )}} + \frac {e^{\left (4 \, e^{\left (e^{4}\right )} + 6\right )}}{x {\left (9 \, e^{\left (8 \, e^{\left (e^{4}\right )}\right )} - 6 \, e^{\left (4 \, e^{\left (e^{4}\right )}\right )} + 1\right )} + e^{3} - 3 \, e^{\left (4 \, e^{\left (e^{4}\right )} + 3\right )}} + \frac {e^{\left (4 \, e^{\left (e^{4}\right )} + 3\right )}}{9 \, e^{\left (8 \, e^{\left (e^{4}\right )}\right )} - 6 \, e^{\left (4 \, e^{\left (e^{4}\right )}\right )} + 1} + \frac {e^{\left (4 \, e^{\left (e^{4}\right )} + 3\right )}}{3 \, e^{\left (4 \, e^{\left (e^{4}\right )}\right )} - 1}\right )} \]
integrate((3*x^2*exp(exp(exp(4)))^8+(-exp(3)^2-2*x*exp(3)-x^2)*exp(exp(exp (4)))^4)*exp((x*exp(3)+x^2)*exp(exp(exp(4)))^4/(3*x*exp(exp(exp(4)))^4-exp (3)-x))/(9*x^2*exp(exp(exp(4)))^8+(-6*x*exp(3)-6*x^2)*exp(exp(exp(4)))^4+e xp(3)^2+2*x*exp(3)+x^2),x, algorithm=\
e^(3*x*e^(8*e^(e^4))/(9*e^(8*e^(e^4)) - 6*e^(4*e^(e^4)) + 1) - x*e^(4*e^(e ^4))/(9*e^(8*e^(e^4)) - 6*e^(4*e^(e^4)) + 1) + e^(4*e^(e^4) + 6)/(x*(27*e^ (12*e^(e^4)) - 27*e^(8*e^(e^4)) + 9*e^(4*e^(e^4)) - 1) - e^3 - 9*e^(8*e^(e ^4) + 3) + 6*e^(4*e^(e^4) + 3)) + e^(4*e^(e^4) + 6)/(x*(9*e^(8*e^(e^4)) - 6*e^(4*e^(e^4)) + 1) + e^3 - 3*e^(4*e^(e^4) + 3)) + e^(4*e^(e^4) + 3)/(9*e ^(8*e^(e^4)) - 6*e^(4*e^(e^4)) + 1) + e^(4*e^(e^4) + 3)/(3*e^(4*e^(e^4)) - 1))
Time = 0.54 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.50 \[ \int \frac {e^{\frac {e^{4 e^{e^4}} \left (e^3 x+x^2\right )}{-e^3-x+3 e^{4 e^{e^4}} x}} \left (3 e^{8 e^{e^4}} x^2+e^{4 e^{e^4}} \left (-e^6-2 e^3 x-x^2\right )\right )}{e^6+2 e^3 x+x^2+9 e^{8 e^{e^4}} x^2+e^{4 e^{e^4}} \left (-6 e^3 x-6 x^2\right )} \, dx=e^{\left (\frac {x^{2} e^{\left (4 \, e^{\left (e^{4}\right )}\right )} + x e^{\left (4 \, e^{\left (e^{4}\right )} + 3\right )}}{3 \, x e^{\left (4 \, e^{\left (e^{4}\right )}\right )} - x - e^{3}}\right )} \]
integrate((3*x^2*exp(exp(exp(4)))^8+(-exp(3)^2-2*x*exp(3)-x^2)*exp(exp(exp (4)))^4)*exp((x*exp(3)+x^2)*exp(exp(exp(4)))^4/(3*x*exp(exp(exp(4)))^4-exp (3)-x))/(9*x^2*exp(exp(exp(4)))^8+(-6*x*exp(3)-6*x^2)*exp(exp(exp(4)))^4+e xp(3)^2+2*x*exp(3)+x^2),x, algorithm=\
Time = 8.72 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.39 \[ \int \frac {e^{\frac {e^{4 e^{e^4}} \left (e^3 x+x^2\right )}{-e^3-x+3 e^{4 e^{e^4}} x}} \left (3 e^{8 e^{e^4}} x^2+e^{4 e^{e^4}} \left (-e^6-2 e^3 x-x^2\right )\right )}{e^6+2 e^3 x+x^2+9 e^{8 e^{e^4}} x^2+e^{4 e^{e^4}} \left (-6 e^3 x-6 x^2\right )} \, dx={\mathrm {e}}^{-\frac {{\mathrm {e}}^{4\,{\mathrm {e}}^{{\mathrm {e}}^4}}\,x^2+{\mathrm {e}}^3\,{\mathrm {e}}^{4\,{\mathrm {e}}^{{\mathrm {e}}^4}}\,x}{x+{\mathrm {e}}^3-3\,x\,{\mathrm {e}}^{4\,{\mathrm {e}}^{{\mathrm {e}}^4}}}} \]
int((exp(-(exp(4*exp(exp(4)))*(x*exp(3) + x^2))/(x + exp(3) - 3*x*exp(4*ex p(exp(4)))))*(3*x^2*exp(8*exp(exp(4))) - exp(4*exp(exp(4)))*(exp(6) + 2*x* exp(3) + x^2)))/(exp(6) + 2*x*exp(3) - exp(4*exp(exp(4)))*(6*x*exp(3) + 6* x^2) + 9*x^2*exp(8*exp(exp(4))) + x^2),x)