Integrand size = 139, antiderivative size = 23 \[ \int e^{81+e^{2 x} \left (4+8 e^3+4 e^6\right )+18 x+x^2+4 e^6 x^2+e^3 \left (36 x+4 x^2\right )+e^x \left (36+4 x+8 e^6 x+e^3 (36+12 x)\right )} \left (18+e^{2 x} \left (8+16 e^3+8 e^6\right )+2 x+8 e^6 x+e^3 (36+8 x)+e^x \left (40+4 x+e^6 (8+8 x)+e^3 (48+12 x)\right )\right ) \, dx=1+e^{\left (9-x+2 \left (1+e^3\right ) \left (e^x+x\right )\right )^2} \]
Time = 2.05 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int e^{81+e^{2 x} \left (4+8 e^3+4 e^6\right )+18 x+x^2+4 e^6 x^2+e^3 \left (36 x+4 x^2\right )+e^x \left (36+4 x+8 e^6 x+e^3 (36+12 x)\right )} \left (18+e^{2 x} \left (8+16 e^3+8 e^6\right )+2 x+8 e^6 x+e^3 (36+8 x)+e^x \left (40+4 x+e^6 (8+8 x)+e^3 (48+12 x)\right )\right ) \, dx=e^{\left (9+2 e^x+2 e^{3+x}+x+2 e^3 x\right )^2} \]
Integrate[E^(81 + E^(2*x)*(4 + 8*E^3 + 4*E^6) + 18*x + x^2 + 4*E^6*x^2 + E ^3*(36*x + 4*x^2) + E^x*(36 + 4*x + 8*E^6*x + E^3*(36 + 12*x)))*(18 + E^(2 *x)*(8 + 16*E^3 + 8*E^6) + 2*x + 8*E^6*x + E^3*(36 + 8*x) + E^x*(40 + 4*x + E^6*(8 + 8*x) + E^3*(48 + 12*x))),x]
Time = 1.86 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.029, Rules used = {6, 7239, 27, 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.
\(\displaystyle \int \left (8 e^6 x+2 x+e^3 (8 x+36)+e^x \left (4 x+e^6 (8 x+8)+e^3 (12 x+48)+40\right )+\left (8+16 e^3+8 e^6\right ) e^{2 x}+18\right ) \exp \left (4 e^6 x^2+x^2+e^3 \left (4 x^2+36 x\right )+18 x+e^x \left (8 e^6 x+4 x+e^3 (12 x+36)+36\right )+\left (4+8 e^3+4 e^6\right ) e^{2 x}+81\right ) \, dx\) |
\(\Big \downarrow \) 6 |
\(\displaystyle \int \left (\left (2+8 e^6\right ) x+e^3 (8 x+36)+e^x \left (4 x+e^6 (8 x+8)+e^3 (12 x+48)+40\right )+\left (8+16 e^3+8 e^6\right ) e^{2 x}+18\right ) \exp \left (4 e^6 x^2+x^2+e^3 \left (4 x^2+36 x\right )+18 x+e^x \left (8 e^6 x+4 x+e^3 (12 x+36)+36\right )+\left (4+8 e^3+4 e^6\right ) e^{2 x}+81\right )dx\) |
\(\Big \downarrow \) 7239 |
\(\displaystyle \int 2 e^{\left (\left (1+2 e^3\right ) x+2 \left (1+e^3\right ) e^x+9\right )^2} \left (2 \left (1+e^3\right ) e^x+1+2 e^3\right ) \left (\left (1+2 e^3\right ) x+2 \left (1+e^3\right ) e^x+9\right )dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle 2 \int e^{\left (\left (1+2 e^3\right ) x+2 e^x \left (1+e^3\right )+9\right )^2} \left (1+2 e^3+2 e^x \left (1+e^3\right )\right ) \left (\left (1+2 e^3\right ) x+2 e^x \left (1+e^3\right )+9\right )dx\) |
\(\Big \downarrow \) 7257 |
\(\displaystyle e^{\left (\left (1+2 e^3\right ) x+2 \left (1+e^3\right ) e^x+9\right )^2}\) |
Int[E^(81 + E^(2*x)*(4 + 8*E^3 + 4*E^6) + 18*x + x^2 + 4*E^6*x^2 + E^3*(36 *x + 4*x^2) + E^x*(36 + 4*x + 8*E^6*x + E^3*(36 + 12*x)))*(18 + E^(2*x)*(8 + 16*E^3 + 8*E^6) + 2*x + 8*E^6*x + E^3*(36 + 8*x) + E^x*(40 + 4*x + E^6* (8 + 8*x) + E^3*(48 + 12*x))),x]
3.30.63.3.1 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[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, 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]
Timed out.
\[\int \left (\left (8 \,{\mathrm e}^{6}+16 \,{\mathrm e}^{3}+8\right ) {\mathrm e}^{2 x}+\left (\left (8 x +8\right ) {\mathrm e}^{6}+\left (12 x +48\right ) {\mathrm e}^{3}+4 x +40\right ) {\mathrm e}^{x}+8 x \,{\mathrm e}^{6}+\left (8 x +36\right ) {\mathrm e}^{3}+2 x +18\right ) {\mathrm e}^{\left (8 \,{\mathrm e}^{3}+4 \,{\mathrm e}^{6}+4\right ) {\mathrm e}^{2 x}+\left (8 x \,{\mathrm e}^{6}+\left (12 x +36\right ) {\mathrm e}^{3}+4 x +36\right ) {\mathrm e}^{x}+4 x^{2} {\mathrm e}^{6}+\left (4 x^{2}+36 x \right ) {\mathrm e}^{3}+x^{2}+18 x +81}d x\]
int(((8*exp(3)^2+16*exp(3)+8)*exp(x)^2+((8*x+8)*exp(3)^2+(12*x+48)*exp(3)+ 4*x+40)*exp(x)+8*x*exp(3)^2+(8*x+36)*exp(3)+2*x+18)*exp((4*exp(3)^2+8*exp( 3)+4)*exp(x)^2+(8*x*exp(3)^2+(12*x+36)*exp(3)+4*x+36)*exp(x)+4*x^2*exp(3)^ 2+(4*x^2+36*x)*exp(3)+x^2+18*x+81),x)
int(((8*exp(3)^2+16*exp(3)+8)*exp(x)^2+((8*x+8)*exp(3)^2+(12*x+48)*exp(3)+ 4*x+40)*exp(x)+8*x*exp(3)^2+(8*x+36)*exp(3)+2*x+18)*exp((4*exp(3)^2+8*exp( 3)+4)*exp(x)^2+(8*x*exp(3)^2+(12*x+36)*exp(3)+4*x+36)*exp(x)+4*x^2*exp(3)^ 2+(4*x^2+36*x)*exp(3)+x^2+18*x+81),x)
Leaf count of result is larger than twice the leaf count of optimal. 60 vs. \(2 (20) = 40\).
Time = 0.25 (sec) , antiderivative size = 60, normalized size of antiderivative = 2.61 \[ \int e^{81+e^{2 x} \left (4+8 e^3+4 e^6\right )+18 x+x^2+4 e^6 x^2+e^3 \left (36 x+4 x^2\right )+e^x \left (36+4 x+8 e^6 x+e^3 (36+12 x)\right )} \left (18+e^{2 x} \left (8+16 e^3+8 e^6\right )+2 x+8 e^6 x+e^3 (36+8 x)+e^x \left (40+4 x+e^6 (8+8 x)+e^3 (48+12 x)\right )\right ) \, dx=e^{\left (4 \, x^{2} e^{6} + x^{2} + 4 \, {\left (x^{2} + 9 \, x\right )} e^{3} + 4 \, {\left (e^{6} + 2 \, e^{3} + 1\right )} e^{\left (2 \, x\right )} + 4 \, {\left (2 \, x e^{6} + 3 \, {\left (x + 3\right )} e^{3} + x + 9\right )} e^{x} + 18 \, x + 81\right )} \]
integrate(((8*exp(3)^2+16*exp(3)+8)*exp(x)^2+((8*x+8)*exp(3)^2+(12*x+48)*e xp(3)+4*x+40)*exp(x)+8*x*exp(3)^2+(8*x+36)*exp(3)+2*x+18)*exp((4*exp(3)^2+ 8*exp(3)+4)*exp(x)^2+(8*x*exp(3)^2+(12*x+36)*exp(3)+4*x+36)*exp(x)+4*x^2*e xp(3)^2+(4*x^2+36*x)*exp(3)+x^2+18*x+81),x, algorithm=\
e^(4*x^2*e^6 + x^2 + 4*(x^2 + 9*x)*e^3 + 4*(e^6 + 2*e^3 + 1)*e^(2*x) + 4*( 2*x*e^6 + 3*(x + 3)*e^3 + x + 9)*e^x + 18*x + 81)
Leaf count of result is larger than twice the leaf count of optimal. 70 vs. \(2 (20) = 40\).
Time = 0.23 (sec) , antiderivative size = 70, normalized size of antiderivative = 3.04 \[ \int e^{81+e^{2 x} \left (4+8 e^3+4 e^6\right )+18 x+x^2+4 e^6 x^2+e^3 \left (36 x+4 x^2\right )+e^x \left (36+4 x+8 e^6 x+e^3 (36+12 x)\right )} \left (18+e^{2 x} \left (8+16 e^3+8 e^6\right )+2 x+8 e^6 x+e^3 (36+8 x)+e^x \left (40+4 x+e^6 (8+8 x)+e^3 (48+12 x)\right )\right ) \, dx=e^{x^{2} + 4 x^{2} e^{6} + 18 x + \left (4 x^{2} + 36 x\right ) e^{3} + \left (4 x + 8 x e^{6} + \left (12 x + 36\right ) e^{3} + 36\right ) e^{x} + \left (4 + 8 e^{3} + 4 e^{6}\right ) e^{2 x} + 81} \]
integrate(((8*exp(3)**2+16*exp(3)+8)*exp(x)**2+((8*x+8)*exp(3)**2+(12*x+48 )*exp(3)+4*x+40)*exp(x)+8*x*exp(3)**2+(8*x+36)*exp(3)+2*x+18)*exp((4*exp(3 )**2+8*exp(3)+4)*exp(x)**2+(8*x*exp(3)**2+(12*x+36)*exp(3)+4*x+36)*exp(x)+ 4*x**2*exp(3)**2+(4*x**2+36*x)*exp(3)+x**2+18*x+81),x)
exp(x**2 + 4*x**2*exp(6) + 18*x + (4*x**2 + 36*x)*exp(3) + (4*x + 8*x*exp( 6) + (12*x + 36)*exp(3) + 36)*exp(x) + (4 + 8*exp(3) + 4*exp(6))*exp(2*x) + 81)
Leaf count of result is larger than twice the leaf count of optimal. 79 vs. \(2 (20) = 40\).
Time = 0.72 (sec) , antiderivative size = 79, normalized size of antiderivative = 3.43 \[ \int e^{81+e^{2 x} \left (4+8 e^3+4 e^6\right )+18 x+x^2+4 e^6 x^2+e^3 \left (36 x+4 x^2\right )+e^x \left (36+4 x+8 e^6 x+e^3 (36+12 x)\right )} \left (18+e^{2 x} \left (8+16 e^3+8 e^6\right )+2 x+8 e^6 x+e^3 (36+8 x)+e^x \left (40+4 x+e^6 (8+8 x)+e^3 (48+12 x)\right )\right ) \, dx=e^{\left (4 \, x^{2} e^{6} + 4 \, x^{2} e^{3} + x^{2} + 36 \, x e^{3} + 8 \, x e^{\left (x + 6\right )} + 12 \, x e^{\left (x + 3\right )} + 4 \, x e^{x} + 18 \, x + 4 \, e^{\left (2 \, x\right )} + 4 \, e^{\left (2 \, x + 6\right )} + 8 \, e^{\left (2 \, x + 3\right )} + 36 \, e^{\left (x + 3\right )} + 36 \, e^{x} + 81\right )} \]
integrate(((8*exp(3)^2+16*exp(3)+8)*exp(x)^2+((8*x+8)*exp(3)^2+(12*x+48)*e xp(3)+4*x+40)*exp(x)+8*x*exp(3)^2+(8*x+36)*exp(3)+2*x+18)*exp((4*exp(3)^2+ 8*exp(3)+4)*exp(x)^2+(8*x*exp(3)^2+(12*x+36)*exp(3)+4*x+36)*exp(x)+4*x^2*e xp(3)^2+(4*x^2+36*x)*exp(3)+x^2+18*x+81),x, algorithm=\
e^(4*x^2*e^6 + 4*x^2*e^3 + x^2 + 36*x*e^3 + 8*x*e^(x + 6) + 12*x*e^(x + 3) + 4*x*e^x + 18*x + 4*e^(2*x) + 4*e^(2*x + 6) + 8*e^(2*x + 3) + 36*e^(x + 3) + 36*e^x + 81)
Leaf count of result is larger than twice the leaf count of optimal. 79 vs. \(2 (20) = 40\).
Time = 0.60 (sec) , antiderivative size = 79, normalized size of antiderivative = 3.43 \[ \int e^{81+e^{2 x} \left (4+8 e^3+4 e^6\right )+18 x+x^2+4 e^6 x^2+e^3 \left (36 x+4 x^2\right )+e^x \left (36+4 x+8 e^6 x+e^3 (36+12 x)\right )} \left (18+e^{2 x} \left (8+16 e^3+8 e^6\right )+2 x+8 e^6 x+e^3 (36+8 x)+e^x \left (40+4 x+e^6 (8+8 x)+e^3 (48+12 x)\right )\right ) \, dx=e^{\left (4 \, x^{2} e^{6} + 4 \, x^{2} e^{3} + x^{2} + 36 \, x e^{3} + 8 \, x e^{\left (x + 6\right )} + 12 \, x e^{\left (x + 3\right )} + 4 \, x e^{x} + 18 \, x + 4 \, e^{\left (2 \, x\right )} + 4 \, e^{\left (2 \, x + 6\right )} + 8 \, e^{\left (2 \, x + 3\right )} + 36 \, e^{\left (x + 3\right )} + 36 \, e^{x} + 81\right )} \]
integrate(((8*exp(3)^2+16*exp(3)+8)*exp(x)^2+((8*x+8)*exp(3)^2+(12*x+48)*e xp(3)+4*x+40)*exp(x)+8*x*exp(3)^2+(8*x+36)*exp(3)+2*x+18)*exp((4*exp(3)^2+ 8*exp(3)+4)*exp(x)^2+(8*x*exp(3)^2+(12*x+36)*exp(3)+4*x+36)*exp(x)+4*x^2*e xp(3)^2+(4*x^2+36*x)*exp(3)+x^2+18*x+81),x, algorithm=\
e^(4*x^2*e^6 + 4*x^2*e^3 + x^2 + 36*x*e^3 + 8*x*e^(x + 6) + 12*x*e^(x + 3) + 4*x*e^x + 18*x + 4*e^(2*x) + 4*e^(2*x + 6) + 8*e^(2*x + 3) + 36*e^(x + 3) + 36*e^x + 81)
Time = 11.65 (sec) , antiderivative size = 92, normalized size of antiderivative = 4.00 \[ \int e^{81+e^{2 x} \left (4+8 e^3+4 e^6\right )+18 x+x^2+4 e^6 x^2+e^3 \left (36 x+4 x^2\right )+e^x \left (36+4 x+8 e^6 x+e^3 (36+12 x)\right )} \left (18+e^{2 x} \left (8+16 e^3+8 e^6\right )+2 x+8 e^6 x+e^3 (36+8 x)+e^x \left (40+4 x+e^6 (8+8 x)+e^3 (48+12 x)\right )\right ) \, dx={\mathrm {e}}^{4\,{\mathrm {e}}^{2\,x}}\,{\mathrm {e}}^{4\,x^2\,{\mathrm {e}}^3}\,{\mathrm {e}}^{4\,x^2\,{\mathrm {e}}^6}\,{\mathrm {e}}^{36\,{\mathrm {e}}^3\,{\mathrm {e}}^x}\,{\mathrm {e}}^{4\,x\,{\mathrm {e}}^x}\,{\mathrm {e}}^{18\,x}\,{\mathrm {e}}^{x^2}\,{\mathrm {e}}^{81}\,{\mathrm {e}}^{4\,{\mathrm {e}}^{2\,x}\,{\mathrm {e}}^6}\,{\mathrm {e}}^{8\,{\mathrm {e}}^{2\,x}\,{\mathrm {e}}^3}\,{\mathrm {e}}^{8\,x\,{\mathrm {e}}^6\,{\mathrm {e}}^x}\,{\mathrm {e}}^{12\,x\,{\mathrm {e}}^3\,{\mathrm {e}}^x}\,{\mathrm {e}}^{36\,{\mathrm {e}}^x}\,{\mathrm {e}}^{36\,x\,{\mathrm {e}}^3} \]
int(exp(18*x + exp(3)*(36*x + 4*x^2) + exp(2*x)*(8*exp(3) + 4*exp(6) + 4) + 4*x^2*exp(6) + exp(x)*(4*x + 8*x*exp(6) + exp(3)*(12*x + 36) + 36) + x^2 + 81)*(2*x + 8*x*exp(6) + exp(x)*(4*x + exp(6)*(8*x + 8) + exp(3)*(12*x + 48) + 40) + exp(2*x)*(16*exp(3) + 8*exp(6) + 8) + exp(3)*(8*x + 36) + 18) ,x)