Integrand size = 344, antiderivative size = 34 \[ \int e^{-e^{4 x} x^2+4 e^{2 e^{4-e^{x^2}}+5 x} x^2-6 e^{4 e^{4-e^{x^2}}+6 x} x^2+4 e^{6 e^{4-e^{x^2}}+7 x} x^2-e^{8 e^{4-e^{x^2}}+8 x} x^2} \left (e^{4 x} \left (-6 x-12 x^2\right )+e^{8 e^{4-e^{x^2}}+8 x} \left (-6 x-24 x^2+48 e^{4-e^{x^2}+x^2} x^3\right )+e^{4 e^{4-e^{x^2}}+4 x} \left (144 e^{4-e^{x^2}+2 x+x^2} x^3+e^{2 x} \left (-36 x-108 x^2\right )\right )+e^{2 e^{4-e^{x^2}}+2 x} \left (-48 e^{4-e^{x^2}+3 x+x^2} x^3+e^{3 x} \left (24 x+60 x^2\right )\right )+e^{6 e^{4-e^{x^2}}+6 x} \left (-144 e^{4-e^{x^2}+x+x^2} x^3+e^x \left (24 x+84 x^2\right )\right )\right ) \, dx=3 e^{-\left (e^x-e^{2 \left (e^{4-e^{x^2}}+x\right )}\right )^4 x^2} \] Output:
3/exp(x^2*(exp(x)-exp(2*exp(4-exp(x^2))+2*x))^4)
Time = 0.33 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.03 \[ \int e^{-e^{4 x} x^2+4 e^{2 e^{4-e^{x^2}}+5 x} x^2-6 e^{4 e^{4-e^{x^2}}+6 x} x^2+4 e^{6 e^{4-e^{x^2}}+7 x} x^2-e^{8 e^{4-e^{x^2}}+8 x} x^2} \left (e^{4 x} \left (-6 x-12 x^2\right )+e^{8 e^{4-e^{x^2}}+8 x} \left (-6 x-24 x^2+48 e^{4-e^{x^2}+x^2} x^3\right )+e^{4 e^{4-e^{x^2}}+4 x} \left (144 e^{4-e^{x^2}+2 x+x^2} x^3+e^{2 x} \left (-36 x-108 x^2\right )\right )+e^{2 e^{4-e^{x^2}}+2 x} \left (-48 e^{4-e^{x^2}+3 x+x^2} x^3+e^{3 x} \left (24 x+60 x^2\right )\right )+e^{6 e^{4-e^{x^2}}+6 x} \left (-144 e^{4-e^{x^2}+x+x^2} x^3+e^x \left (24 x+84 x^2\right )\right )\right ) \, dx=3 e^{-e^{4 x} \left (-1+e^{2 e^{4-e^{x^2}}+x}\right )^4 x^2} \] Input:
Integrate[E^(-(E^(4*x)*x^2) + 4*E^(2*E^(4 - E^x^2) + 5*x)*x^2 - 6*E^(4*E^( 4 - E^x^2) + 6*x)*x^2 + 4*E^(6*E^(4 - E^x^2) + 7*x)*x^2 - E^(8*E^(4 - E^x^ 2) + 8*x)*x^2)*(E^(4*x)*(-6*x - 12*x^2) + E^(8*E^(4 - E^x^2) + 8*x)*(-6*x - 24*x^2 + 48*E^(4 - E^x^2 + x^2)*x^3) + E^(4*E^(4 - E^x^2) + 4*x)*(144*E^ (4 - E^x^2 + 2*x + x^2)*x^3 + E^(2*x)*(-36*x - 108*x^2)) + E^(2*E^(4 - E^x ^2) + 2*x)*(-48*E^(4 - E^x^2 + 3*x + x^2)*x^3 + E^(3*x)*(24*x + 60*x^2)) + E^(6*E^(4 - E^x^2) + 6*x)*(-144*E^(4 - E^x^2 + x + x^2)*x^3 + E^x*(24*x + 84*x^2))),x]
Output:
3/E^(E^(4*x)*(-1 + E^(2*E^(4 - E^x^2) + x))^4*x^2)
Leaf count is larger than twice the leaf count of optimal. \(111\) vs. \(2(34)=68\).
Time = 7.24 (sec) , antiderivative size = 111, normalized size of antiderivative = 3.26, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.003, Rules used = {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.
| \(\displaystyle \int \left (e^{4 x} \left (-12 x^2-6 x\right )+e^{8 e^{4-e^{x^2}}+8 x} \left (-24 x^2+48 e^{x^2-e^{x^2}+4} x^3-6 x\right )+e^{4 e^{4-e^{x^2}}+4 x} \left (e^{2 x} \left (-108 x^2-36 x\right )+144 e^{x^2-e^{x^2}+2 x+4} x^3\right )+e^{2 e^{4-e^{x^2}}+2 x} \left (e^{3 x} \left (60 x^2+24 x\right )-48 e^{x^2-e^{x^2}+3 x+4} x^3\right )+e^{6 e^{4-e^{x^2}}+6 x} \left (e^x \left (84 x^2+24 x\right )-144 e^{x^2-e^{x^2}+x+4} x^3\right )\right ) \exp \left (-e^{4 x} x^2+4 e^{2 e^{4-e^{x^2}}+5 x} x^2-6 e^{4 e^{4-e^{x^2}}+6 x} x^2+4 e^{6 e^{4-e^{x^2}}+7 x} x^2-e^{8 e^{4-e^{x^2}}+8 x} x^2\right ) \, dx\) |
| \(\Big \downarrow \) 7257 |
| \(\displaystyle 3 \exp \left (-e^{4 x} x^2+4 e^{2 e^{4-e^{x^2}}+5 x} x^2-6 e^{4 e^{4-e^{x^2}}+6 x} x^2+4 e^{6 e^{4-e^{x^2}}+7 x} x^2-e^{8 e^{4-e^{x^2}}+8 x} x^2\right )\) |
Input:
Int[E^(-(E^(4*x)*x^2) + 4*E^(2*E^(4 - E^x^2) + 5*x)*x^2 - 6*E^(4*E^(4 - E^ x^2) + 6*x)*x^2 + 4*E^(6*E^(4 - E^x^2) + 7*x)*x^2 - E^(8*E^(4 - E^x^2) + 8 *x)*x^2)*(E^(4*x)*(-6*x - 12*x^2) + E^(8*E^(4 - E^x^2) + 8*x)*(-6*x - 24*x ^2 + 48*E^(4 - E^x^2 + x^2)*x^3) + E^(4*E^(4 - E^x^2) + 4*x)*(144*E^(4 - E ^x^2 + 2*x + x^2)*x^3 + E^(2*x)*(-36*x - 108*x^2)) + E^(2*E^(4 - E^x^2) + 2*x)*(-48*E^(4 - E^x^2 + 3*x + x^2)*x^3 + E^(3*x)*(24*x + 60*x^2)) + E^(6* E^(4 - E^x^2) + 6*x)*(-144*E^(4 - E^x^2 + x + x^2)*x^3 + E^x*(24*x + 84*x^ 2))),x]
Output:
3*E^(-(E^(4*x)*x^2) + 4*E^(2*E^(4 - E^x^2) + 5*x)*x^2 - 6*E^(4*E^(4 - E^x^ 2) + 6*x)*x^2 + 4*E^(6*E^(4 - E^x^2) + 7*x)*x^2 - E^(8*E^(4 - E^x^2) + 8*x )*x^2)
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]
Leaf count of result is larger than twice the leaf count of optimal. \(83\) vs. \(2(32)=64\).
Time = 37.20 (sec) , antiderivative size = 84, normalized size of antiderivative = 2.47
| method | result | size |
| risch | \(3 \,{\mathrm e}^{-x^{2} \left ({\mathrm e}^{8 \,{\mathrm e}^{4-{\mathrm e}^{x^{2}}}+8 x}-4 \,{\mathrm e}^{7 x +6 \,{\mathrm e}^{4-{\mathrm e}^{x^{2}}}}+6 \,{\mathrm e}^{6 x +4 \,{\mathrm e}^{4-{\mathrm e}^{x^{2}}}}-4 \,{\mathrm e}^{5 x +2 \,{\mathrm e}^{4-{\mathrm e}^{x^{2}}}}+{\mathrm e}^{4 x}\right )}\) | \(84\) |
| parallelrisch | \(3 \,{\mathrm e}^{-x^{2} \left ({\mathrm e}^{8 \,{\mathrm e}^{4-{\mathrm e}^{x^{2}}}+8 x}-4 \,{\mathrm e}^{7 x +6 \,{\mathrm e}^{4-{\mathrm e}^{x^{2}}}}+6 \,{\mathrm e}^{6 x +4 \,{\mathrm e}^{4-{\mathrm e}^{x^{2}}}}-4 \,{\mathrm e}^{5 x +2 \,{\mathrm e}^{4-{\mathrm e}^{x^{2}}}}+{\mathrm e}^{4 x}\right )}\) | \(101\) |
Input:
int(((48*x^3*exp(x^2)*exp(4-exp(x^2))-24*x^2-6*x)*exp(2*exp(4-exp(x^2))+2* x)^4+(-144*x^3*exp(x)*exp(x^2)*exp(4-exp(x^2))+(84*x^2+24*x)*exp(x))*exp(2 *exp(4-exp(x^2))+2*x)^3+(144*x^3*exp(x)^2*exp(x^2)*exp(4-exp(x^2))+(-108*x ^2-36*x)*exp(x)^2)*exp(2*exp(4-exp(x^2))+2*x)^2+(-48*x^3*exp(x)^3*exp(x^2) *exp(4-exp(x^2))+(60*x^2+24*x)*exp(x)^3)*exp(2*exp(4-exp(x^2))+2*x)+(-12*x ^2-6*x)*exp(x)^4)/exp(x^2*exp(2*exp(4-exp(x^2))+2*x)^4-4*x^2*exp(x)*exp(2* exp(4-exp(x^2))+2*x)^3+6*x^2*exp(x)^2*exp(2*exp(4-exp(x^2))+2*x)^2-4*x^2*e xp(x)^3*exp(2*exp(4-exp(x^2))+2*x)+x^2*exp(x)^4),x,method=_RETURNVERBOSE)
Output:
3*exp(-x^2*(exp(8*exp(4-exp(x^2))+8*x)-4*exp(7*x+6*exp(4-exp(x^2)))+6*exp( 6*x+4*exp(4-exp(x^2)))-4*exp(5*x+2*exp(4-exp(x^2)))+exp(4*x)))
Leaf count of result is larger than twice the leaf count of optimal. 216 vs. \(2 (31) = 62\).
Time = 0.11 (sec) , antiderivative size = 216, normalized size of antiderivative = 6.35 \[ \int e^{-e^{4 x} x^2+4 e^{2 e^{4-e^{x^2}}+5 x} x^2-6 e^{4 e^{4-e^{x^2}}+6 x} x^2+4 e^{6 e^{4-e^{x^2}}+7 x} x^2-e^{8 e^{4-e^{x^2}}+8 x} x^2} \left (e^{4 x} \left (-6 x-12 x^2\right )+e^{8 e^{4-e^{x^2}}+8 x} \left (-6 x-24 x^2+48 e^{4-e^{x^2}+x^2} x^3\right )+e^{4 e^{4-e^{x^2}}+4 x} \left (144 e^{4-e^{x^2}+2 x+x^2} x^3+e^{2 x} \left (-36 x-108 x^2\right )\right )+e^{2 e^{4-e^{x^2}}+2 x} \left (-48 e^{4-e^{x^2}+3 x+x^2} x^3+e^{3 x} \left (24 x+60 x^2\right )\right )+e^{6 e^{4-e^{x^2}}+6 x} \left (-144 e^{4-e^{x^2}+x+x^2} x^3+e^x \left (24 x+84 x^2\right )\right )\right ) \, dx=3 \, e^{\left (-{\left (x^{2} e^{\left (4 \, {\left (5 \, x e^{\left (x^{2} + 3 \, x\right )} + 2 \, e^{\left (x^{2} + 3 \, x - e^{\left (x^{2}\right )} + 4\right )}\right )} e^{\left (-x^{2} - 3 \, x\right )}\right )} - 4 \, x^{2} e^{\left (3 \, {\left (5 \, x e^{\left (x^{2} + 3 \, x\right )} + 2 \, e^{\left (x^{2} + 3 \, x - e^{\left (x^{2}\right )} + 4\right )}\right )} e^{\left (-x^{2} - 3 \, x\right )} + 4 \, x\right )} + 6 \, x^{2} e^{\left (2 \, {\left (5 \, x e^{\left (x^{2} + 3 \, x\right )} + 2 \, e^{\left (x^{2} + 3 \, x - e^{\left (x^{2}\right )} + 4\right )}\right )} e^{\left (-x^{2} - 3 \, x\right )} + 8 \, x\right )} - 4 \, x^{2} e^{\left ({\left (5 \, x e^{\left (x^{2} + 3 \, x\right )} + 2 \, e^{\left (x^{2} + 3 \, x - e^{\left (x^{2}\right )} + 4\right )}\right )} e^{\left (-x^{2} - 3 \, x\right )} + 12 \, x\right )} + x^{2} e^{\left (16 \, x\right )}\right )} e^{\left (-12 \, x\right )}\right )} \] Input:
integrate(((48*x^3*exp(x^2)*exp(4-exp(x^2))-24*x^2-6*x)*exp(2*exp(4-exp(x^ 2))+2*x)^4+(-144*x^3*exp(x)*exp(x^2)*exp(4-exp(x^2))+(84*x^2+24*x)*exp(x)) *exp(2*exp(4-exp(x^2))+2*x)^3+(144*x^3*exp(x)^2*exp(x^2)*exp(4-exp(x^2))+( -108*x^2-36*x)*exp(x)^2)*exp(2*exp(4-exp(x^2))+2*x)^2+(-48*x^3*exp(x)^3*ex p(x^2)*exp(4-exp(x^2))+(60*x^2+24*x)*exp(x)^3)*exp(2*exp(4-exp(x^2))+2*x)+ (-12*x^2-6*x)*exp(x)^4)/exp(x^2*exp(2*exp(4-exp(x^2))+2*x)^4-4*x^2*exp(x)* exp(2*exp(4-exp(x^2))+2*x)^3+6*x^2*exp(x)^2*exp(2*exp(4-exp(x^2))+2*x)^2-4 *x^2*exp(x)^3*exp(2*exp(4-exp(x^2))+2*x)+x^2*exp(x)^4),x, algorithm="frica s")
Output:
3*e^(-(x^2*e^(4*(5*x*e^(x^2 + 3*x) + 2*e^(x^2 + 3*x - e^(x^2) + 4))*e^(-x^ 2 - 3*x)) - 4*x^2*e^(3*(5*x*e^(x^2 + 3*x) + 2*e^(x^2 + 3*x - e^(x^2) + 4)) *e^(-x^2 - 3*x) + 4*x) + 6*x^2*e^(2*(5*x*e^(x^2 + 3*x) + 2*e^(x^2 + 3*x - e^(x^2) + 4))*e^(-x^2 - 3*x) + 8*x) - 4*x^2*e^((5*x*e^(x^2 + 3*x) + 2*e^(x ^2 + 3*x - e^(x^2) + 4))*e^(-x^2 - 3*x) + 12*x) + x^2*e^(16*x))*e^(-12*x))
Timed out. \[ \int e^{-e^{4 x} x^2+4 e^{2 e^{4-e^{x^2}}+5 x} x^2-6 e^{4 e^{4-e^{x^2}}+6 x} x^2+4 e^{6 e^{4-e^{x^2}}+7 x} x^2-e^{8 e^{4-e^{x^2}}+8 x} x^2} \left (e^{4 x} \left (-6 x-12 x^2\right )+e^{8 e^{4-e^{x^2}}+8 x} \left (-6 x-24 x^2+48 e^{4-e^{x^2}+x^2} x^3\right )+e^{4 e^{4-e^{x^2}}+4 x} \left (144 e^{4-e^{x^2}+2 x+x^2} x^3+e^{2 x} \left (-36 x-108 x^2\right )\right )+e^{2 e^{4-e^{x^2}}+2 x} \left (-48 e^{4-e^{x^2}+3 x+x^2} x^3+e^{3 x} \left (24 x+60 x^2\right )\right )+e^{6 e^{4-e^{x^2}}+6 x} \left (-144 e^{4-e^{x^2}+x+x^2} x^3+e^x \left (24 x+84 x^2\right )\right )\right ) \, dx=\text {Timed out} \] Input:
integrate(((48*x**3*exp(x**2)*exp(4-exp(x**2))-24*x**2-6*x)*exp(2*exp(4-ex p(x**2))+2*x)**4+(-144*x**3*exp(x)*exp(x**2)*exp(4-exp(x**2))+(84*x**2+24* x)*exp(x))*exp(2*exp(4-exp(x**2))+2*x)**3+(144*x**3*exp(x)**2*exp(x**2)*ex p(4-exp(x**2))+(-108*x**2-36*x)*exp(x)**2)*exp(2*exp(4-exp(x**2))+2*x)**2+ (-48*x**3*exp(x)**3*exp(x**2)*exp(4-exp(x**2))+(60*x**2+24*x)*exp(x)**3)*e xp(2*exp(4-exp(x**2))+2*x)+(-12*x**2-6*x)*exp(x)**4)/exp(x**2*exp(2*exp(4- exp(x**2))+2*x)**4-4*x**2*exp(x)*exp(2*exp(4-exp(x**2))+2*x)**3+6*x**2*exp (x)**2*exp(2*exp(4-exp(x**2))+2*x)**2-4*x**2*exp(x)**3*exp(2*exp(4-exp(x** 2))+2*x)+x**2*exp(x)**4),x)
Output:
Timed out
Leaf count of result is larger than twice the leaf count of optimal. 97 vs. \(2 (31) = 62\).
Time = 0.34 (sec) , antiderivative size = 97, normalized size of antiderivative = 2.85 \[ \int e^{-e^{4 x} x^2+4 e^{2 e^{4-e^{x^2}}+5 x} x^2-6 e^{4 e^{4-e^{x^2}}+6 x} x^2+4 e^{6 e^{4-e^{x^2}}+7 x} x^2-e^{8 e^{4-e^{x^2}}+8 x} x^2} \left (e^{4 x} \left (-6 x-12 x^2\right )+e^{8 e^{4-e^{x^2}}+8 x} \left (-6 x-24 x^2+48 e^{4-e^{x^2}+x^2} x^3\right )+e^{4 e^{4-e^{x^2}}+4 x} \left (144 e^{4-e^{x^2}+2 x+x^2} x^3+e^{2 x} \left (-36 x-108 x^2\right )\right )+e^{2 e^{4-e^{x^2}}+2 x} \left (-48 e^{4-e^{x^2}+3 x+x^2} x^3+e^{3 x} \left (24 x+60 x^2\right )\right )+e^{6 e^{4-e^{x^2}}+6 x} \left (-144 e^{4-e^{x^2}+x+x^2} x^3+e^x \left (24 x+84 x^2\right )\right )\right ) \, dx=3 \, e^{\left (-x^{2} e^{\left (4 \, x\right )} - x^{2} e^{\left (8 \, x + 8 \, e^{\left (-e^{\left (x^{2}\right )} + 4\right )}\right )} + 4 \, x^{2} e^{\left (7 \, x + 6 \, e^{\left (-e^{\left (x^{2}\right )} + 4\right )}\right )} - 6 \, x^{2} e^{\left (6 \, x + 4 \, e^{\left (-e^{\left (x^{2}\right )} + 4\right )}\right )} + 4 \, x^{2} e^{\left (5 \, x + 2 \, e^{\left (-e^{\left (x^{2}\right )} + 4\right )}\right )}\right )} \] Input:
integrate(((48*x^3*exp(x^2)*exp(4-exp(x^2))-24*x^2-6*x)*exp(2*exp(4-exp(x^ 2))+2*x)^4+(-144*x^3*exp(x)*exp(x^2)*exp(4-exp(x^2))+(84*x^2+24*x)*exp(x)) *exp(2*exp(4-exp(x^2))+2*x)^3+(144*x^3*exp(x)^2*exp(x^2)*exp(4-exp(x^2))+( -108*x^2-36*x)*exp(x)^2)*exp(2*exp(4-exp(x^2))+2*x)^2+(-48*x^3*exp(x)^3*ex p(x^2)*exp(4-exp(x^2))+(60*x^2+24*x)*exp(x)^3)*exp(2*exp(4-exp(x^2))+2*x)+ (-12*x^2-6*x)*exp(x)^4)/exp(x^2*exp(2*exp(4-exp(x^2))+2*x)^4-4*x^2*exp(x)* exp(2*exp(4-exp(x^2))+2*x)^3+6*x^2*exp(x)^2*exp(2*exp(4-exp(x^2))+2*x)^2-4 *x^2*exp(x)^3*exp(2*exp(4-exp(x^2))+2*x)+x^2*exp(x)^4),x, algorithm="maxim a")
Output:
3*e^(-x^2*e^(4*x) - x^2*e^(8*x + 8*e^(-e^(x^2) + 4)) + 4*x^2*e^(7*x + 6*e^ (-e^(x^2) + 4)) - 6*x^2*e^(6*x + 4*e^(-e^(x^2) + 4)) + 4*x^2*e^(5*x + 2*e^ (-e^(x^2) + 4)))
\[ \int e^{-e^{4 x} x^2+4 e^{2 e^{4-e^{x^2}}+5 x} x^2-6 e^{4 e^{4-e^{x^2}}+6 x} x^2+4 e^{6 e^{4-e^{x^2}}+7 x} x^2-e^{8 e^{4-e^{x^2}}+8 x} x^2} \left (e^{4 x} \left (-6 x-12 x^2\right )+e^{8 e^{4-e^{x^2}}+8 x} \left (-6 x-24 x^2+48 e^{4-e^{x^2}+x^2} x^3\right )+e^{4 e^{4-e^{x^2}}+4 x} \left (144 e^{4-e^{x^2}+2 x+x^2} x^3+e^{2 x} \left (-36 x-108 x^2\right )\right )+e^{2 e^{4-e^{x^2}}+2 x} \left (-48 e^{4-e^{x^2}+3 x+x^2} x^3+e^{3 x} \left (24 x+60 x^2\right )\right )+e^{6 e^{4-e^{x^2}}+6 x} \left (-144 e^{4-e^{x^2}+x+x^2} x^3+e^x \left (24 x+84 x^2\right )\right )\right ) \, dx=\int { -6 \, {\left ({\left (2 \, x^{2} + x\right )} e^{\left (4 \, x\right )} - {\left (8 \, x^{3} e^{\left (x^{2} - e^{\left (x^{2}\right )} + 4\right )} - 4 \, x^{2} - x\right )} e^{\left (8 \, x + 8 \, e^{\left (-e^{\left (x^{2}\right )} + 4\right )}\right )} + 2 \, {\left (12 \, x^{3} e^{\left (x^{2} + x - e^{\left (x^{2}\right )} + 4\right )} - {\left (7 \, x^{2} + 2 \, x\right )} e^{x}\right )} e^{\left (6 \, x + 6 \, e^{\left (-e^{\left (x^{2}\right )} + 4\right )}\right )} - 6 \, {\left (4 \, x^{3} e^{\left (x^{2} + 2 \, x - e^{\left (x^{2}\right )} + 4\right )} - {\left (3 \, x^{2} + x\right )} e^{\left (2 \, x\right )}\right )} e^{\left (4 \, x + 4 \, e^{\left (-e^{\left (x^{2}\right )} + 4\right )}\right )} + 2 \, {\left (4 \, x^{3} e^{\left (x^{2} + 3 \, x - e^{\left (x^{2}\right )} + 4\right )} - {\left (5 \, x^{2} + 2 \, x\right )} e^{\left (3 \, x\right )}\right )} e^{\left (2 \, x + 2 \, e^{\left (-e^{\left (x^{2}\right )} + 4\right )}\right )}\right )} e^{\left (-x^{2} e^{\left (4 \, x\right )} - x^{2} e^{\left (8 \, x + 8 \, e^{\left (-e^{\left (x^{2}\right )} + 4\right )}\right )} + 4 \, x^{2} e^{\left (7 \, x + 6 \, e^{\left (-e^{\left (x^{2}\right )} + 4\right )}\right )} - 6 \, x^{2} e^{\left (6 \, x + 4 \, e^{\left (-e^{\left (x^{2}\right )} + 4\right )}\right )} + 4 \, x^{2} e^{\left (5 \, x + 2 \, e^{\left (-e^{\left (x^{2}\right )} + 4\right )}\right )}\right )} \,d x } \] Input:
integrate(((48*x^3*exp(x^2)*exp(4-exp(x^2))-24*x^2-6*x)*exp(2*exp(4-exp(x^ 2))+2*x)^4+(-144*x^3*exp(x)*exp(x^2)*exp(4-exp(x^2))+(84*x^2+24*x)*exp(x)) *exp(2*exp(4-exp(x^2))+2*x)^3+(144*x^3*exp(x)^2*exp(x^2)*exp(4-exp(x^2))+( -108*x^2-36*x)*exp(x)^2)*exp(2*exp(4-exp(x^2))+2*x)^2+(-48*x^3*exp(x)^3*ex p(x^2)*exp(4-exp(x^2))+(60*x^2+24*x)*exp(x)^3)*exp(2*exp(4-exp(x^2))+2*x)+ (-12*x^2-6*x)*exp(x)^4)/exp(x^2*exp(2*exp(4-exp(x^2))+2*x)^4-4*x^2*exp(x)* exp(2*exp(4-exp(x^2))+2*x)^3+6*x^2*exp(x)^2*exp(2*exp(4-exp(x^2))+2*x)^2-4 *x^2*exp(x)^3*exp(2*exp(4-exp(x^2))+2*x)+x^2*exp(x)^4),x, algorithm="giac" )
Output:
integrate(-6*((2*x^2 + x)*e^(4*x) - (8*x^3*e^(x^2 - e^(x^2) + 4) - 4*x^2 - x)*e^(8*x + 8*e^(-e^(x^2) + 4)) + 2*(12*x^3*e^(x^2 + x - e^(x^2) + 4) - ( 7*x^2 + 2*x)*e^x)*e^(6*x + 6*e^(-e^(x^2) + 4)) - 6*(4*x^3*e^(x^2 + 2*x - e ^(x^2) + 4) - (3*x^2 + x)*e^(2*x))*e^(4*x + 4*e^(-e^(x^2) + 4)) + 2*(4*x^3 *e^(x^2 + 3*x - e^(x^2) + 4) - (5*x^2 + 2*x)*e^(3*x))*e^(2*x + 2*e^(-e^(x^ 2) + 4)))*e^(-x^2*e^(4*x) - x^2*e^(8*x + 8*e^(-e^(x^2) + 4)) + 4*x^2*e^(7* x + 6*e^(-e^(x^2) + 4)) - 6*x^2*e^(6*x + 4*e^(-e^(x^2) + 4)) + 4*x^2*e^(5* x + 2*e^(-e^(x^2) + 4))), x)
Time = 2.31 (sec) , antiderivative size = 100, normalized size of antiderivative = 2.94 \[ \int e^{-e^{4 x} x^2+4 e^{2 e^{4-e^{x^2}}+5 x} x^2-6 e^{4 e^{4-e^{x^2}}+6 x} x^2+4 e^{6 e^{4-e^{x^2}}+7 x} x^2-e^{8 e^{4-e^{x^2}}+8 x} x^2} \left (e^{4 x} \left (-6 x-12 x^2\right )+e^{8 e^{4-e^{x^2}}+8 x} \left (-6 x-24 x^2+48 e^{4-e^{x^2}+x^2} x^3\right )+e^{4 e^{4-e^{x^2}}+4 x} \left (144 e^{4-e^{x^2}+2 x+x^2} x^3+e^{2 x} \left (-36 x-108 x^2\right )\right )+e^{2 e^{4-e^{x^2}}+2 x} \left (-48 e^{4-e^{x^2}+3 x+x^2} x^3+e^{3 x} \left (24 x+60 x^2\right )\right )+e^{6 e^{4-e^{x^2}}+6 x} \left (-144 e^{4-e^{x^2}+x+x^2} x^3+e^x \left (24 x+84 x^2\right )\right )\right ) \, dx=3\,{\mathrm {e}}^{4\,x^2\,{\mathrm {e}}^{2\,{\mathrm {e}}^{-{\mathrm {e}}^{x^2}}\,{\mathrm {e}}^4}\,{\mathrm {e}}^{5\,x}}\,{\mathrm {e}}^{-6\,x^2\,{\mathrm {e}}^{4\,{\mathrm {e}}^{-{\mathrm {e}}^{x^2}}\,{\mathrm {e}}^4}\,{\mathrm {e}}^{6\,x}}\,{\mathrm {e}}^{4\,x^2\,{\mathrm {e}}^{6\,{\mathrm {e}}^{-{\mathrm {e}}^{x^2}}\,{\mathrm {e}}^4}\,{\mathrm {e}}^{7\,x}}\,{\mathrm {e}}^{-x^2\,{\mathrm {e}}^{8\,{\mathrm {e}}^{-{\mathrm {e}}^{x^2}}\,{\mathrm {e}}^4}\,{\mathrm {e}}^{8\,x}}\,{\mathrm {e}}^{-x^2\,{\mathrm {e}}^{4\,x}} \] Input:
int(-exp(4*x^2*exp(6*x + 6*exp(4 - exp(x^2)))*exp(x) - x^2*exp(8*x + 8*exp (4 - exp(x^2))) - x^2*exp(4*x) + 4*x^2*exp(3*x)*exp(2*x + 2*exp(4 - exp(x^ 2))) - 6*x^2*exp(2*x)*exp(4*x + 4*exp(4 - exp(x^2))))*(exp(4*x)*(6*x + 12* x^2) - exp(2*x + 2*exp(4 - exp(x^2)))*(exp(3*x)*(24*x + 60*x^2) - 48*x^3*e xp(3*x)*exp(x^2)*exp(4 - exp(x^2))) + exp(4*x + 4*exp(4 - exp(x^2)))*(exp( 2*x)*(36*x + 108*x^2) - 144*x^3*exp(2*x)*exp(x^2)*exp(4 - exp(x^2))) - exp (6*x + 6*exp(4 - exp(x^2)))*(exp(x)*(24*x + 84*x^2) - 144*x^3*exp(x^2)*exp (4 - exp(x^2))*exp(x)) + exp(8*x + 8*exp(4 - exp(x^2)))*(6*x + 24*x^2 - 48 *x^3*exp(x^2)*exp(4 - exp(x^2)))),x)
Output:
3*exp(4*x^2*exp(2*exp(-exp(x^2))*exp(4))*exp(5*x))*exp(-6*x^2*exp(4*exp(-e xp(x^2))*exp(4))*exp(6*x))*exp(4*x^2*exp(6*exp(-exp(x^2))*exp(4))*exp(7*x) )*exp(-x^2*exp(8*exp(-exp(x^2))*exp(4))*exp(8*x))*exp(-x^2*exp(4*x))
Time = 2.32 (sec) , antiderivative size = 150, normalized size of antiderivative = 4.41 \[ \int e^{-e^{4 x} x^2+4 e^{2 e^{4-e^{x^2}}+5 x} x^2-6 e^{4 e^{4-e^{x^2}}+6 x} x^2+4 e^{6 e^{4-e^{x^2}}+7 x} x^2-e^{8 e^{4-e^{x^2}}+8 x} x^2} \left (e^{4 x} \left (-6 x-12 x^2\right )+e^{8 e^{4-e^{x^2}}+8 x} \left (-6 x-24 x^2+48 e^{4-e^{x^2}+x^2} x^3\right )+e^{4 e^{4-e^{x^2}}+4 x} \left (144 e^{4-e^{x^2}+2 x+x^2} x^3+e^{2 x} \left (-36 x-108 x^2\right )\right )+e^{2 e^{4-e^{x^2}}+2 x} \left (-48 e^{4-e^{x^2}+3 x+x^2} x^3+e^{3 x} \left (24 x+60 x^2\right )\right )+e^{6 e^{4-e^{x^2}}+6 x} \left (-144 e^{4-e^{x^2}+x+x^2} x^3+e^x \left (24 x+84 x^2\right )\right )\right ) \, dx=\frac {3 e^{4 e^{\frac {7 e^{e^{x^{2}}} x +6 e^{4}}{e^{e^{x^{2}}}}} x^{2}+4 e^{\frac {5 e^{e^{x^{2}}} x +2 e^{4}}{e^{e^{x^{2}}}}} x^{2}}}{e^{e^{\frac {8 e^{e^{x^{2}}} x +8 e^{4}}{e^{e^{x^{2}}}}} x^{2}+6 e^{\frac {6 e^{e^{x^{2}}} x +4 e^{4}}{e^{e^{x^{2}}}}} x^{2}+e^{4 x} x^{2}}} \] Input:
int(((48*x^3*exp(x^2)*exp(4-exp(x^2))-24*x^2-6*x)*exp(2*exp(4-exp(x^2))+2* x)^4+(-144*x^3*exp(x)*exp(x^2)*exp(4-exp(x^2))+(84*x^2+24*x)*exp(x))*exp(2 *exp(4-exp(x^2))+2*x)^3+(144*x^3*exp(x)^2*exp(x^2)*exp(4-exp(x^2))+(-108*x ^2-36*x)*exp(x)^2)*exp(2*exp(4-exp(x^2))+2*x)^2+(-48*x^3*exp(x)^3*exp(x^2) *exp(4-exp(x^2))+(60*x^2+24*x)*exp(x)^3)*exp(2*exp(4-exp(x^2))+2*x)+(-12*x ^2-6*x)*exp(x)^4)/exp(x^2*exp(2*exp(4-exp(x^2))+2*x)^4-4*x^2*exp(x)*exp(2* exp(4-exp(x^2))+2*x)^3+6*x^2*exp(x)^2*exp(2*exp(4-exp(x^2))+2*x)^2-4*x^2*e xp(x)^3*exp(2*exp(4-exp(x^2))+2*x)+x^2*exp(x)^4),x)
Output:
(3*e**(4*e**((7*e**(e**(x**2))*x + 6*e**4)/e**(e**(x**2)))*x**2 + 4*e**((5 *e**(e**(x**2))*x + 2*e**4)/e**(e**(x**2)))*x**2))/e**(e**((8*e**(e**(x**2 ))*x + 8*e**4)/e**(e**(x**2)))*x**2 + 6*e**((6*e**(e**(x**2))*x + 4*e**4)/ e**(e**(x**2)))*x**2 + e**(4*x)*x**2)