\(\int e^{8+2 e^{4+2 x^2}+2 x+12 x^3+2 x^6+2 e^{2+x^2} (-6-2 x^3)} (2 x+2 x^2+8 e^{4+2 x^2} x^3+36 x^4+12 x^7+e^{2+x^2} (-24 x^3-12 x^4-8 x^6)) \, dx\) [7462]

Optimal result

Integrand size = 102, antiderivative size = 29 \[ \int e^{8+2 e^{4+2 x^2}+2 x+12 x^3+2 x^6+2 e^{2+x^2} \left (-6-2 x^3\right )} \left (2 x+2 x^2+8 e^{4+2 x^2} x^3+36 x^4+12 x^7+e^{2+x^2} \left (-24 x^3-12 x^4-8 x^6\right )\right ) \, dx=e^{-10+2 x+2 \left (-3+e^{2+x^2}-x^3\right )^2} x^2 \]

[Out]

x^2*exp(x-5+(exp(x^2+2)-3-x^3)^2)^2

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(149\) vs. \(2(29)=58\).

Time = 1.47 (sec) , antiderivative size = 149, normalized size of antiderivative = 5.14, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.010, Rules used = {2326} \[ \int e^{8+2 e^{4+2 x^2}+2 x+12 x^3+2 x^6+2 e^{2+x^2} \left (-6-2 x^3\right )} \left (2 x+2 x^2+8 e^{4+2 x^2} x^3+36 x^4+12 x^7+e^{2+x^2} \left (-24 x^3-12 x^4-8 x^6\right )\right ) \, dx=\frac {\left (6 x^7+18 x^4+x^2+4 e^{2 x^2+4} x^3-2 e^{x^2+2} \left (2 x^6+3 x^4+6 x^3\right )\right ) \exp \left (2 x^6+12 x^3+2 e^{2 x^2+4}-4 e^{x^2+2} \left (x^3+3\right )+2 x+8\right )}{6 x^5-6 e^{x^2+2} x^2+18 x^2+4 e^{2 x^2+4} x-4 e^{x^2+2} \left (x^3+3\right ) x+1} \]

[In]

Int[E^(8 + 2*E^(4 + 2*x^2) + 2*x + 12*x^3 + 2*x^6 + 2*E^(2 + x^2)*(-6 - 2*x^3))*(2*x + 2*x^2 + 8*E^(4 + 2*x^2)
*x^3 + 36*x^4 + 12*x^7 + E^(2 + x^2)*(-24*x^3 - 12*x^4 - 8*x^6)),x]

[Out]

(E^(8 + 2*E^(4 + 2*x^2) + 2*x + 12*x^3 + 2*x^6 - 4*E^(2 + x^2)*(3 + x^3))*(x^2 + 4*E^(4 + 2*x^2)*x^3 + 18*x^4
+ 6*x^7 - 2*E^(2 + x^2)*(6*x^3 + 3*x^4 + 2*x^6)))/(1 + 4*E^(4 + 2*x^2)*x + 18*x^2 - 6*E^(2 + x^2)*x^2 + 6*x^5
- 4*E^(2 + x^2)*x*(3 + x^3))

Rule 2326

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]

Rubi steps \begin{align*} \text {integral}& = \frac {\exp \left (8+2 e^{4+2 x^2}+2 x+12 x^3+2 x^6-4 e^{2+x^2} \left (3+x^3\right )\right ) \left (x^2+4 e^{4+2 x^2} x^3+18 x^4+6 x^7-2 e^{2+x^2} \left (6 x^3+3 x^4+2 x^6\right )\right )}{1+4 e^{4+2 x^2} x+18 x^2-6 e^{2+x^2} x^2+6 x^5-4 e^{2+x^2} x \left (3+x^3\right )} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.45 \[ \int e^{8+2 e^{4+2 x^2}+2 x+12 x^3+2 x^6+2 e^{2+x^2} \left (-6-2 x^3\right )} \left (2 x+2 x^2+8 e^{4+2 x^2} x^3+36 x^4+12 x^7+e^{2+x^2} \left (-24 x^3-12 x^4-8 x^6\right )\right ) \, dx=e^{2 \left (4+e^{4+2 x^2}+x+6 x^3+x^6-2 e^{2+x^2} \left (3+x^3\right )\right )} x^2 \]

[In]

Integrate[E^(8 + 2*E^(4 + 2*x^2) + 2*x + 12*x^3 + 2*x^6 + 2*E^(2 + x^2)*(-6 - 2*x^3))*(2*x + 2*x^2 + 8*E^(4 +
2*x^2)*x^3 + 36*x^4 + 12*x^7 + E^(2 + x^2)*(-24*x^3 - 12*x^4 - 8*x^6)),x]

[Out]

E^(2*(4 + E^(4 + 2*x^2) + x + 6*x^3 + x^6 - 2*E^(2 + x^2)*(3 + x^3)))*x^2

Maple [A] (verified)

Time = 0.17 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.41

method	result	size
parallelrisch	\(x^{2} {\mathrm e}^{2 \,{\mathrm e}^{2 x^{2}+4}+2 \left (-2 x^{3}-6\right ) {\mathrm e}^{x^{2}+2}+2 x^{6}+12 x^{3}+2 x +8}\)	\(41\)
risch	\(x^{2} {\mathrm e}^{2 x^{6}-4 \,{\mathrm e}^{x^{2}+2} x^{3}+12 x^{3}-12 \,{\mathrm e}^{x^{2}+2}+2 \,{\mathrm e}^{2 x^{2}+4}+2 x +8}\)	\(50\)

[In]

int((8*x^3*exp(x^2+2)^2+(-8*x^6-12*x^4-24*x^3)*exp(x^2+2)+12*x^7+36*x^4+2*x^2+2*x)*exp(exp(x^2+2)^2+(-2*x^3-6)
*exp(x^2+2)+x^6+6*x^3+x+4)^2,x,method=_RETURNVERBOSE)

[Out]

x^2*exp(exp(x^2+2)^2+(-2*x^3-6)*exp(x^2+2)+x^6+6*x^3+x+4)^2

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.48 \[ \int e^{8+2 e^{4+2 x^2}+2 x+12 x^3+2 x^6+2 e^{2+x^2} \left (-6-2 x^3\right )} \left (2 x+2 x^2+8 e^{4+2 x^2} x^3+36 x^4+12 x^7+e^{2+x^2} \left (-24 x^3-12 x^4-8 x^6\right )\right ) \, dx=x^{2} e^{\left (2 \, x^{6} + 12 \, x^{3} - 4 \, {\left (x^{3} + 3\right )} e^{\left (x^{2} + 2\right )} + 2 \, x + 2 \, e^{\left (2 \, x^{2} + 4\right )} + 8\right )} \]

[In]

integrate((8*x^3*exp(x^2+2)^2+(-8*x^6-12*x^4-24*x^3)*exp(x^2+2)+12*x^7+36*x^4+2*x^2+2*x)*exp(exp(x^2+2)^2+(-2*
x^3-6)*exp(x^2+2)+x^6+6*x^3+x+4)^2,x, algorithm="fricas")

[Out]

x^2*e^(2*x^6 + 12*x^3 - 4*(x^3 + 3)*e^(x^2 + 2) + 2*x + 2*e^(2*x^2 + 4) + 8)

Sympy [A] (verification not implemented)

Time = 2.84 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.59 \[ \int e^{8+2 e^{4+2 x^2}+2 x+12 x^3+2 x^6+2 e^{2+x^2} \left (-6-2 x^3\right )} \left (2 x+2 x^2+8 e^{4+2 x^2} x^3+36 x^4+12 x^7+e^{2+x^2} \left (-24 x^3-12 x^4-8 x^6\right )\right ) \, dx=x^{2} e^{2 x^{6} + 12 x^{3} + 2 x + 2 \left (- 2 x^{3} - 6\right ) e^{x^{2} + 2} + 2 e^{2 x^{2} + 4} + 8} \]

[In]

integrate((8*x**3*exp(x**2+2)**2+(-8*x**6-12*x**4-24*x**3)*exp(x**2+2)+12*x**7+36*x**4+2*x**2+2*x)*exp(exp(x**
2+2)**2+(-2*x**3-6)*exp(x**2+2)+x**6+6*x**3+x+4)**2,x)

[Out]

x**2*exp(2*x**6 + 12*x**3 + 2*x + 2*(-2*x**3 - 6)*exp(x**2 + 2) + 2*exp(2*x**2 + 4) + 8)

Maxima [A] (verification not implemented)

none

Time = 0.50 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.69 \[ \int e^{8+2 e^{4+2 x^2}+2 x+12 x^3+2 x^6+2 e^{2+x^2} \left (-6-2 x^3\right )} \left (2 x+2 x^2+8 e^{4+2 x^2} x^3+36 x^4+12 x^7+e^{2+x^2} \left (-24 x^3-12 x^4-8 x^6\right )\right ) \, dx=x^{2} e^{\left (2 \, x^{6} - 4 \, x^{3} e^{\left (x^{2} + 2\right )} + 12 \, x^{3} + 2 \, x + 2 \, e^{\left (2 \, x^{2} + 4\right )} - 12 \, e^{\left (x^{2} + 2\right )} + 8\right )} \]

[In]

integrate((8*x^3*exp(x^2+2)^2+(-8*x^6-12*x^4-24*x^3)*exp(x^2+2)+12*x^7+36*x^4+2*x^2+2*x)*exp(exp(x^2+2)^2+(-2*
x^3-6)*exp(x^2+2)+x^6+6*x^3+x+4)^2,x, algorithm="maxima")

[Out]

x^2*e^(2*x^6 - 4*x^3*e^(x^2 + 2) + 12*x^3 + 2*x + 2*e^(2*x^2 + 4) - 12*e^(x^2 + 2) + 8)

Giac [F]

\[ \int e^{8+2 e^{4+2 x^2}+2 x+12 x^3+2 x^6+2 e^{2+x^2} \left (-6-2 x^3\right )} \left (2 x+2 x^2+8 e^{4+2 x^2} x^3+36 x^4+12 x^7+e^{2+x^2} \left (-24 x^3-12 x^4-8 x^6\right )\right ) \, dx=\int { 2 \, {\left (6 \, x^{7} + 18 \, x^{4} + 4 \, x^{3} e^{\left (2 \, x^{2} + 4\right )} + x^{2} - 2 \, {\left (2 \, x^{6} + 3 \, x^{4} + 6 \, x^{3}\right )} e^{\left (x^{2} + 2\right )} + x\right )} e^{\left (2 \, x^{6} + 12 \, x^{3} - 4 \, {\left (x^{3} + 3\right )} e^{\left (x^{2} + 2\right )} + 2 \, x + 2 \, e^{\left (2 \, x^{2} + 4\right )} + 8\right )} \,d x } \]

[In]

integrate((8*x^3*exp(x^2+2)^2+(-8*x^6-12*x^4-24*x^3)*exp(x^2+2)+12*x^7+36*x^4+2*x^2+2*x)*exp(exp(x^2+2)^2+(-2*
x^3-6)*exp(x^2+2)+x^6+6*x^3+x+4)^2,x, algorithm="giac")

[Out]

integrate(2*(6*x^7 + 18*x^4 + 4*x^3*e^(2*x^2 + 4) + x^2 - 2*(2*x^6 + 3*x^4 + 6*x^3)*e^(x^2 + 2) + x)*e^(2*x^6
+ 12*x^3 - 4*(x^3 + 3)*e^(x^2 + 2) + 2*x + 2*e^(2*x^2 + 4) + 8), x)

Mupad [B] (verification not implemented)

Time = 13.01 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.86 \[ \int e^{8+2 e^{4+2 x^2}+2 x+12 x^3+2 x^6+2 e^{2+x^2} \left (-6-2 x^3\right )} \left (2 x+2 x^2+8 e^{4+2 x^2} x^3+36 x^4+12 x^7+e^{2+x^2} \left (-24 x^3-12 x^4-8 x^6\right )\right ) \, dx=x^2\,{\mathrm {e}}^{2\,x}\,{\mathrm {e}}^{-4\,x^3\,{\mathrm {e}}^{x^2}\,{\mathrm {e}}^2}\,{\mathrm {e}}^8\,{\mathrm {e}}^{-12\,{\mathrm {e}}^{x^2}\,{\mathrm {e}}^2}\,{\mathrm {e}}^{2\,x^6}\,{\mathrm {e}}^{12\,x^3}\,{\mathrm {e}}^{2\,{\mathrm {e}}^4\,{\mathrm {e}}^{2\,x^2}} \]

[In]

int(exp(2*x + 2*exp(2*x^2 + 4) - 2*exp(x^2 + 2)*(2*x^3 + 6) + 12*x^3 + 2*x^6 + 8)*(2*x + 8*x^3*exp(2*x^2 + 4)
+ 2*x^2 + 36*x^4 + 12*x^7 - exp(x^2 + 2)*(24*x^3 + 12*x^4 + 8*x^6)),x)

[Out]

x^2*exp(2*x)*exp(-4*x^3*exp(x^2)*exp(2))*exp(8)*exp(-12*exp(x^2)*exp(2))*exp(2*x^6)*exp(12*x^3)*exp(2*exp(4)*e
xp(2*x^2))