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

Optimal. Leaf size=30 \[ 1+e^{\left (-e^{2 (-4+6 x)}+x+x^2+2 (3+x)\right )^2}-x \]

________________________________________________________________________________________

Rubi [A]  time = 3.72, antiderivative size = 38, normalized size of antiderivative = 1.27, number of steps used = 3, number of rules used = 2, integrand size = 90, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.022, Rules used = {6741, 6706} \begin {gather*} e^{\frac {\left (e^8 x^2+3 e^8 x-e^{12 x}+6 e^8\right )^2}{e^{16}}}-x \end {gather*}

Antiderivative was successfully verified.

[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 6706

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 6741

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

Rubi steps

\begin {gather*} \begin {aligned} \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 {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 2.43, size = 31, normalized size = 1.03 \begin {gather*} e^{\frac {\left (e^{12 x}-e^8 \left (6+3 x+x^2\right )\right )^2}{e^{16}}}-x \end {gather*}

Antiderivative was successfully verified.

[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

________________________________________________________________________________________

fricas [A]  time = 0.64, size = 45, normalized size = 1.50 \begin {gather*} -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 )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

giac [B]  time = 0.43, size = 57, normalized size = 1.90 \begin {gather*} -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 )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

maple [A]  time = 0.12, size = 49, normalized size = 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	\(-x +{\mathrm e}^{{\mathrm e}^{24 x -16}+\left (-2 x^{2}-6 x -12\right ) {\mathrm e}^{12 x -8}+x^{4}+6 x^{3}+21 x^{2}+36 x +36}\)	\(49\)
risch	\(-x +{\mathrm e}^{x^{4}-2 \,{\mathrm e}^{12 x -8} x^{2}+6 x^{3}-6 \,{\mathrm e}^{12 x -8} x +21 x^{2}+{\mathrm e}^{24 x -16}-12 \,{\mathrm e}^{12 x -8}+36 x +36}\)	\(58\)

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

maxima [B]  time = 1.15, size = 57, normalized size = 1.90 \begin {gather*} -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 )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

mupad [B]  time = 0.26, size = 66, normalized size = 2.20 \begin {gather*} {\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 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

________________________________________________________________________________________

sympy [B]  time = 0.32, size = 46, normalized size = 1.53 \begin {gather*} - 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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________