\(\int (-1+e^{36+e^{-16+24 x}+36 x+21 x^2+6 x^3+x^4+e^{-8+12 x} (-12-6 x-2 x^2)} (36+24 e^{-16+24 x}+42 x+18 x^2+4 x^3+e^{-8+12 x} (-150-76 x-24 x^2))) \, dx\) [3359]

Optimal result

Integrand size = 90, antiderivative size = 30 \[ \int \left (-1+e^{36+e^{-16+24 x}+36 x+21 x^2+6 x^3+x^4+e^{-8+12 x} \left (-12-6 x-2 x^2\right )} \left (36+24 e^{-16+24 x}+42 x+18 x^2+4 x^3+e^{-8+12 x} \left (-150-76 x-24 x^2\right )\right )\right ) \, dx=1+e^{\left (-e^{2 (-4+6 x)}+x+x^2+2 (3+x)\right )^2}-x \]

[Out]

exp((3*x+6-exp(12*x-8)+x^2)^2)+1-x

Rubi [A] (verified)

Time = 2.48 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.27, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.022, Rules used = {6873, 6838} \[ \int \left (-1+e^{36+e^{-16+24 x}+36 x+21 x^2+6 x^3+x^4+e^{-8+12 x} \left (-12-6 x-2 x^2\right )} \left (36+24 e^{-16+24 x}+42 x+18 x^2+4 x^3+e^{-8+12 x} \left (-150-76 x-24 x^2\right )\right )\right ) \, dx=e^{\frac {\left (e^8 x^2+3 e^8 x-e^{12 x}+6 e^8\right )^2}{e^{16}}}-x \]

[In]

Int[-1 + E^(36 + E^(-16 + 24*x) + 36*x + 21*x^2 + 6*x^3 + x^4 + E^(-8 + 12*x)*(-12 - 6*x - 2*x^2))*(36 + 24*E^
(-16 + 24*x) + 42*x + 18*x^2 + 4*x^3 + E^(-8 + 12*x)*(-150 - 76*x - 24*x^2)),x]

[Out]

E^((6*E^8 - E^(12*x) + 3*E^8*x + E^8*x^2)^2/E^16) - x

Rule 6838

Int[(F_)^(v_)*(u_), x_Symbol] :> With[{q = DerivativeDivides[v, u, x]}, Simp[q*(F^v/Log[F]), x] /;  !FalseQ[q]
] /; FreeQ[F, x]

Rule 6873

Int[u_, x_Symbol] :> With[{v = NormalizeIntegrand[u, x]}, Int[v, x] /; v =!= u]

Rubi steps \begin{align*} \text {integral}& = -x+\int \exp \left (36+e^{-16+24 x}+36 x+21 x^2+6 x^3+x^4+e^{-8+12 x} \left (-12-6 x-2 x^2\right )\right ) \left (36+24 e^{-16+24 x}+42 x+18 x^2+4 x^3+e^{-8+12 x} \left (-150-76 x-24 x^2\right )\right ) \, dx \\ & = -x+\int e^{\frac {\left (6 e^8-e^{12 x}+3 e^8 x+e^8 x^2\right )^2}{e^{16}}} \left (36+24 e^{-16+24 x}+42 x+18 x^2+4 x^3+e^{-8+12 x} \left (-150-76 x-24 x^2\right )\right ) \, dx \\ & = e^{\frac {\left (6 e^8-e^{12 x}+3 e^8 x+e^8 x^2\right )^2}{e^{16}}}-x \\ \end{align*}

Mathematica [A] (verified)

Time = 5.20 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.03 \[ \int \left (-1+e^{36+e^{-16+24 x}+36 x+21 x^2+6 x^3+x^4+e^{-8+12 x} \left (-12-6 x-2 x^2\right )} \left (36+24 e^{-16+24 x}+42 x+18 x^2+4 x^3+e^{-8+12 x} \left (-150-76 x-24 x^2\right )\right )\right ) \, dx=e^{\frac {\left (e^{12 x}-e^8 \left (6+3 x+x^2\right )\right )^2}{e^{16}}}-x \]

[In]

Integrate[-1 + E^(36 + E^(-16 + 24*x) + 36*x + 21*x^2 + 6*x^3 + x^4 + E^(-8 + 12*x)*(-12 - 6*x - 2*x^2))*(36 +
24*E^(-16 + 24*x) + 42*x + 18*x^2 + 4*x^3 + E^(-8 + 12*x)*(-150 - 76*x - 24*x^2)),x]

[Out]

E^((E^(12*x) - E^8*(6 + 3*x + x^2))^2/E^16) - x

Maple [A] (verified)

Time = 1.55 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.63

method	result	size
default	\(-x +{\mathrm e}^{{\mathrm e}^{24 x -16}+\left (-2 x^{2}-6 x -12\right ) {\mathrm e}^{12 x -8}+x^{4}+6 x^{3}+21 x^{2}+36 x +36}\)	\(49\)
norman	\(-x +{\mathrm e}^{{\mathrm e}^{24 x -16}+\left (-2 x^{2}-6 x -12\right ) {\mathrm e}^{12 x -8}+x^{4}+6 x^{3}+21 x^{2}+36 x +36}\)	\(49\)
parallelrisch	\(-x +{\mathrm e}^{{\mathrm e}^{24 x -16}+\left (-2 x^{2}-6 x -12\right ) {\mathrm e}^{12 x -8}+x^{4}+6 x^{3}+21 x^{2}+36 x +36}\)	\(49\)
risch	\(-x +{\mathrm e}^{x^{4}-2 \,{\mathrm e}^{12 x -8} x^{2}+6 x^{3}-6 \,{\mathrm e}^{12 x -8} x +21 x^{2}-12 \,{\mathrm e}^{12 x -8}+{\mathrm e}^{24 x -16}+36 x +36}\)	\(58\)

[In]

int((24*exp(12*x-8)^2+(-24*x^2-76*x-150)*exp(12*x-8)+4*x^3+18*x^2+42*x+36)*exp(exp(12*x-8)^2+(-2*x^2-6*x-12)*e
xp(12*x-8)+x^4+6*x^3+21*x^2+36*x+36)-1,x,method=_RETURNVERBOSE)

[Out]

-x+exp(exp(12*x-8)^2+(-2*x^2-6*x-12)*exp(12*x-8)+x^4+6*x^3+21*x^2+36*x+36)

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.50 \[ \int \left (-1+e^{36+e^{-16+24 x}+36 x+21 x^2+6 x^3+x^4+e^{-8+12 x} \left (-12-6 x-2 x^2\right )} \left (36+24 e^{-16+24 x}+42 x+18 x^2+4 x^3+e^{-8+12 x} \left (-150-76 x-24 x^2\right )\right )\right ) \, dx=-x + e^{\left (x^{4} + 6 \, x^{3} + 21 \, x^{2} - 2 \, {\left (x^{2} + 3 \, x + 6\right )} e^{\left (12 \, x - 8\right )} + 36 \, x + e^{\left (24 \, x - 16\right )} + 36\right )} \]

[In]

integrate((24*exp(12*x-8)^2+(-24*x^2-76*x-150)*exp(12*x-8)+4*x^3+18*x^2+42*x+36)*exp(exp(12*x-8)^2+(-2*x^2-6*x
-12)*exp(12*x-8)+x^4+6*x^3+21*x^2+36*x+36)-1,x, algorithm="fricas")

[Out]

-x + e^(x^4 + 6*x^3 + 21*x^2 - 2*(x^2 + 3*x + 6)*e^(12*x - 8) + 36*x + e^(24*x - 16) + 36)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 46 vs. \(2 (20) = 40\).

Time = 0.16 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.53 \[ \int \left (-1+e^{36+e^{-16+24 x}+36 x+21 x^2+6 x^3+x^4+e^{-8+12 x} \left (-12-6 x-2 x^2\right )} \left (36+24 e^{-16+24 x}+42 x+18 x^2+4 x^3+e^{-8+12 x} \left (-150-76 x-24 x^2\right )\right )\right ) \, dx=- x + e^{x^{4} + 6 x^{3} + 21 x^{2} + 36 x + \left (- 2 x^{2} - 6 x - 12\right ) e^{12 x - 8} + e^{24 x - 16} + 36} \]

[In]

integrate((24*exp(12*x-8)**2+(-24*x**2-76*x-150)*exp(12*x-8)+4*x**3+18*x**2+42*x+36)*exp(exp(12*x-8)**2+(-2*x*
*2-6*x-12)*exp(12*x-8)+x**4+6*x**3+21*x**2+36*x+36)-1,x)

[Out]

-x + exp(x**4 + 6*x**3 + 21*x**2 + 36*x + (-2*x**2 - 6*x - 12)*exp(12*x - 8) + exp(24*x - 16) + 36)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 57 vs. \(2 (24) = 48\).

Time = 0.59 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.90 \[ \int \left (-1+e^{36+e^{-16+24 x}+36 x+21 x^2+6 x^3+x^4+e^{-8+12 x} \left (-12-6 x-2 x^2\right )} \left (36+24 e^{-16+24 x}+42 x+18 x^2+4 x^3+e^{-8+12 x} \left (-150-76 x-24 x^2\right )\right )\right ) \, dx=-x + e^{\left (x^{4} + 6 \, x^{3} - 2 \, x^{2} e^{\left (12 \, x - 8\right )} + 21 \, x^{2} - 6 \, x e^{\left (12 \, x - 8\right )} + 36 \, x + e^{\left (24 \, x - 16\right )} - 12 \, e^{\left (12 \, x - 8\right )} + 36\right )} \]

[In]

integrate((24*exp(12*x-8)^2+(-24*x^2-76*x-150)*exp(12*x-8)+4*x^3+18*x^2+42*x+36)*exp(exp(12*x-8)^2+(-2*x^2-6*x
-12)*exp(12*x-8)+x^4+6*x^3+21*x^2+36*x+36)-1,x, algorithm="maxima")

[Out]

-x + e^(x^4 + 6*x^3 - 2*x^2*e^(12*x - 8) + 21*x^2 - 6*x*e^(12*x - 8) + 36*x + e^(24*x - 16) - 12*e^(12*x - 8)
+ 36)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 57 vs. \(2 (24) = 48\).

Time = 0.39 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.90 \[ \int \left (-1+e^{36+e^{-16+24 x}+36 x+21 x^2+6 x^3+x^4+e^{-8+12 x} \left (-12-6 x-2 x^2\right )} \left (36+24 e^{-16+24 x}+42 x+18 x^2+4 x^3+e^{-8+12 x} \left (-150-76 x-24 x^2\right )\right )\right ) \, dx=-x + e^{\left (x^{4} + 6 \, x^{3} - 2 \, x^{2} e^{\left (12 \, x - 8\right )} + 21 \, x^{2} - 6 \, x e^{\left (12 \, x - 8\right )} + 36 \, x + e^{\left (24 \, x - 16\right )} - 12 \, e^{\left (12 \, x - 8\right )} + 36\right )} \]

[In]

integrate((24*exp(12*x-8)^2+(-24*x^2-76*x-150)*exp(12*x-8)+4*x^3+18*x^2+42*x+36)*exp(exp(12*x-8)^2+(-2*x^2-6*x
-12)*exp(12*x-8)+x^4+6*x^3+21*x^2+36*x+36)-1,x, algorithm="giac")

[Out]

-x + e^(x^4 + 6*x^3 - 2*x^2*e^(12*x - 8) + 21*x^2 - 6*x*e^(12*x - 8) + 36*x + e^(24*x - 16) - 12*e^(12*x - 8)
+ 36)

Mupad [B] (verification not implemented)

Time = 0.28 (sec) , antiderivative size = 66, normalized size of antiderivative = 2.20 \[ \int \left (-1+e^{36+e^{-16+24 x}+36 x+21 x^2+6 x^3+x^4+e^{-8+12 x} \left (-12-6 x-2 x^2\right )} \left (36+24 e^{-16+24 x}+42 x+18 x^2+4 x^3+e^{-8+12 x} \left (-150-76 x-24 x^2\right )\right )\right ) \, dx={\mathrm {e}}^{36\,x}\,{\mathrm {e}}^{x^4}\,{\mathrm {e}}^{-2\,x^2\,{\mathrm {e}}^{12\,x}\,{\mathrm {e}}^{-8}}\,{\mathrm {e}}^{36}\,{\mathrm {e}}^{-12\,{\mathrm {e}}^{12\,x}\,{\mathrm {e}}^{-8}}\,{\mathrm {e}}^{{\mathrm {e}}^{24\,x}\,{\mathrm {e}}^{-16}}\,{\mathrm {e}}^{6\,x^3}\,{\mathrm {e}}^{21\,x^2}\,{\mathrm {e}}^{-6\,x\,{\mathrm {e}}^{12\,x}\,{\mathrm {e}}^{-8}}-x \]

[In]

int(exp(36*x + exp(24*x - 16) - exp(12*x - 8)*(6*x + 2*x^2 + 12) + 21*x^2 + 6*x^3 + x^4 + 36)*(42*x + 24*exp(2
4*x - 16) - exp(12*x - 8)*(76*x + 24*x^2 + 150) + 18*x^2 + 4*x^3 + 36) - 1,x)

[Out]

exp(36*x)*exp(x^4)*exp(-2*x^2*exp(12*x)*exp(-8))*exp(36)*exp(-12*exp(12*x)*exp(-8))*exp(exp(24*x)*exp(-16))*ex
p(6*x^3)*exp(21*x^2)*exp(-6*x*exp(12*x)*exp(-8)) - x