3.96.37 \(\int \frac {-13+e^4 (52-32 x)+8 x+e^{2 e^x} (-1+4 e^4+e^x (6-2 x+e^4 (-24+8 x)))+e^{e^x} (10-4 x+e^4 (-40+16 x)+e^x (-12+10 x-2 x^2+e^4 (48-40 x+8 x^2)))}{e^4} \, dx\)

Optimal. Leaf size=32 \[ \left (4-\frac {1}{e^4}\right ) (-3+x) \left (-3+\left (2-e^{e^x}-x\right )^2-x^2\right ) \]

________________________________________________________________________________________

Rubi [B]  time = 0.09, antiderivative size = 81, normalized size of antiderivative = 2.53, number of steps used = 4, number of rules used = 2, integrand size = 99, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.020, Rules used = {12, 2288} \begin {gather*} \frac {4 x^2}{e^4}-2 e^{e^x-4} \left (x^2-4 e^4 \left (x^2-5 x+6\right )-5 x+6\right )-\frac {1}{4} (13-8 x)^2+e^{2 e^x-4} \left (-4 e^4 (3-x)-x+3\right )-\frac {13 x}{e^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-13 + E^4*(52 - 32*x) + 8*x + E^(2*E^x)*(-1 + 4*E^4 + E^x*(6 - 2*x + E^4*(-24 + 8*x))) + E^E^x*(10 - 4*x
+ E^4*(-40 + 16*x) + E^x*(-12 + 10*x - 2*x^2 + E^4*(48 - 40*x + 8*x^2))))/E^4,x]

[Out]

-1/4*(13 - 8*x)^2 + E^(-4 + 2*E^x)*(3 - 4*E^4*(3 - x) - x) - (13*x)/E^4 + (4*x^2)/E^4 - 2*E^(-4 + E^x)*(6 - 5*
x + x^2 - 4*E^4*(6 - 5*x + x^2))

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 2288

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 {gather*} \begin {aligned} \text {integral} &=\frac {\int \left (-13+e^4 (52-32 x)+8 x+e^{2 e^x} \left (-1+4 e^4+e^x \left (6-2 x+e^4 (-24+8 x)\right )\right )+e^{e^x} \left (10-4 x+e^4 (-40+16 x)+e^x \left (-12+10 x-2 x^2+e^4 \left (48-40 x+8 x^2\right )\right )\right )\right ) \, dx}{e^4}\\ &=-\frac {1}{4} (13-8 x)^2-\frac {13 x}{e^4}+\frac {4 x^2}{e^4}+\frac {\int e^{2 e^x} \left (-1+4 e^4+e^x \left (6-2 x+e^4 (-24+8 x)\right )\right ) \, dx}{e^4}+\frac {\int e^{e^x} \left (10-4 x+e^4 (-40+16 x)+e^x \left (-12+10 x-2 x^2+e^4 \left (48-40 x+8 x^2\right )\right )\right ) \, dx}{e^4}\\ &=-\frac {1}{4} (13-8 x)^2+e^{-4+2 e^x} \left (3-4 e^4 (3-x)-x\right )-\frac {13 x}{e^4}+\frac {4 x^2}{e^4}-2 e^{-4+e^x} \left (6-5 x+x^2-4 e^4 \left (6-5 x+x^2\right )\right )\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.10, size = 47, normalized size = 1.47 \begin {gather*} \frac {\left (-1+4 e^4\right ) \left (e^{2 e^x} (-3+x)+13 x-4 x^2+e^{e^x} \left (12-10 x+2 x^2\right )\right )}{e^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-13 + E^4*(52 - 32*x) + 8*x + E^(2*E^x)*(-1 + 4*E^4 + E^x*(6 - 2*x + E^4*(-24 + 8*x))) + E^E^x*(10
- 4*x + E^4*(-40 + 16*x) + E^x*(-12 + 10*x - 2*x^2 + E^4*(48 - 40*x + 8*x^2))))/E^4,x]

[Out]

((-1 + 4*E^4)*(E^(2*E^x)*(-3 + x) + 13*x - 4*x^2 + E^E^x*(12 - 10*x + 2*x^2)))/E^4

________________________________________________________________________________________

fricas [B]  time = 0.46, size = 68, normalized size = 2.12 \begin {gather*} {\left (4 \, x^{2} - 4 \, {\left (4 \, x^{2} - 13 \, x\right )} e^{4} + {\left (4 \, {\left (x - 3\right )} e^{4} - x + 3\right )} e^{\left (2 \, e^{x}\right )} - 2 \, {\left (x^{2} - 4 \, {\left (x^{2} - 5 \, x + 6\right )} e^{4} - 5 \, x + 6\right )} e^{\left (e^{x}\right )} - 13 \, x\right )} e^{\left (-4\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((((8*x-24)*exp(4)+6-2*x)*exp(x)+4*exp(4)-1)*exp(exp(x))^2+(((8*x^2-40*x+48)*exp(4)-2*x^2+10*x-12)*e
xp(x)+(16*x-40)*exp(4)+10-4*x)*exp(exp(x))+(-32*x+52)*exp(4)+8*x-13)/exp(4),x, algorithm="fricas")

[Out]

(4*x^2 - 4*(4*x^2 - 13*x)*e^4 + (4*(x - 3)*e^4 - x + 3)*e^(2*e^x) - 2*(x^2 - 4*(x^2 - 5*x + 6)*e^4 - 5*x + 6)*
e^(e^x) - 13*x)*e^(-4)

________________________________________________________________________________________

giac [B]  time = 0.22, size = 119, normalized size = 3.72 \begin {gather*} {\left (4 \, x^{2} - 4 \, {\left (4 \, x^{2} - 13 \, x\right )} e^{4} + 2 \, {\left (4 \, x^{2} e^{\left (x + e^{x} + 4\right )} - x^{2} e^{\left (x + e^{x}\right )} - 20 \, x e^{\left (x + e^{x} + 4\right )} + 5 \, x e^{\left (x + e^{x}\right )} + 24 \, e^{\left (x + e^{x} + 4\right )} - 6 \, e^{\left (x + e^{x}\right )}\right )} e^{\left (-x\right )} - x e^{\left (2 \, e^{x}\right )} + 4 \, x e^{\left (2 \, e^{x} + 4\right )} - 13 \, x + 3 \, e^{\left (2 \, e^{x}\right )} - 12 \, e^{\left (2 \, e^{x} + 4\right )}\right )} e^{\left (-4\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((((8*x-24)*exp(4)+6-2*x)*exp(x)+4*exp(4)-1)*exp(exp(x))^2+(((8*x^2-40*x+48)*exp(4)-2*x^2+10*x-12)*e
xp(x)+(16*x-40)*exp(4)+10-4*x)*exp(exp(x))+(-32*x+52)*exp(4)+8*x-13)/exp(4),x, algorithm="giac")

[Out]

(4*x^2 - 4*(4*x^2 - 13*x)*e^4 + 2*(4*x^2*e^(x + e^x + 4) - x^2*e^(x + e^x) - 20*x*e^(x + e^x + 4) + 5*x*e^(x +
e^x) + 24*e^(x + e^x + 4) - 6*e^(x + e^x))*e^(-x) - x*e^(2*e^x) + 4*x*e^(2*e^x + 4) - 13*x + 3*e^(2*e^x) - 12
*e^(2*e^x + 4))*e^(-4)

________________________________________________________________________________________

maple [B]  time = 0.12, size = 76, normalized size = 2.38

method	result	size
risch	\(-16 x^{2}+52 x +4 x^{2} {\mathrm e}^{-4}-13 x \,{\mathrm e}^{-4}+\left (4 x \,{\mathrm e}^{4}-12 \,{\mathrm e}^{4}-x +3\right ) {\mathrm e}^{-4+2 \,{\mathrm e}^{x}}+\left (8 x^{2} {\mathrm e}^{4}-40 x \,{\mathrm e}^{4}-2 x^{2}+48 \,{\mathrm e}^{4}+10 x -12\right ) {\mathrm e}^{{\mathrm e}^{x}-4}\)	\(76\)
default	\({\mathrm e}^{-4} \left (-13 x +{\mathrm e}^{4} \left (-16 x^{2}+52 x \right )+\left (-12 \,{\mathrm e}^{4}+3\right ) {\mathrm e}^{2 \,{\mathrm e}^{x}}+\left (4 \,{\mathrm e}^{4}-1\right ) x \,{\mathrm e}^{2 \,{\mathrm e}^{x}}+\left (48 \,{\mathrm e}^{4}-12\right ) {\mathrm e}^{{\mathrm e}^{x}}+\left (-40 \,{\mathrm e}^{4}+10\right ) x \,{\mathrm e}^{{\mathrm e}^{x}}+\left (8 \,{\mathrm e}^{4}-2\right ) x^{2} {\mathrm e}^{{\mathrm e}^{x}}+4 x^{2}\right )\)	\(86\)
norman	\(\left (4 \,{\mathrm e}^{4}-1\right ) {\mathrm e}^{-4} x \,{\mathrm e}^{2 \,{\mathrm e}^{x}}+13 \left (4 \,{\mathrm e}^{4}-1\right ) {\mathrm e}^{-4} x -4 \left (4 \,{\mathrm e}^{4}-1\right ) {\mathrm e}^{-4} x^{2}+12 \left (4 \,{\mathrm e}^{4}-1\right ) {\mathrm e}^{-4} {\mathrm e}^{{\mathrm e}^{x}}-3 \left (4 \,{\mathrm e}^{4}-1\right ) {\mathrm e}^{-4} {\mathrm e}^{2 \,{\mathrm e}^{x}}-10 \left (4 \,{\mathrm e}^{4}-1\right ) {\mathrm e}^{-4} x \,{\mathrm e}^{{\mathrm e}^{x}}+2 \left (4 \,{\mathrm e}^{4}-1\right ) {\mathrm e}^{-4} x^{2} {\mathrm e}^{{\mathrm e}^{x}}\)	\(113\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((((8*x-24)*exp(4)+6-2*x)*exp(x)+4*exp(4)-1)*exp(exp(x))^2+(((8*x^2-40*x+48)*exp(4)-2*x^2+10*x-12)*exp(x)+
(16*x-40)*exp(4)+10-4*x)*exp(exp(x))+(-32*x+52)*exp(4)+8*x-13)/exp(4),x,method=_RETURNVERBOSE)

[Out]

-16*x^2+52*x+4*x^2*exp(-4)-13*x*exp(-4)+(4*x*exp(4)-12*exp(4)-x+3)*exp(-4+2*exp(x))+(8*x^2*exp(4)-40*x*exp(4)-
2*x^2+48*exp(4)+10*x-12)*exp(exp(x)-4)

________________________________________________________________________________________

maxima [B]  time = 0.36, size = 75, normalized size = 2.34 \begin {gather*} {\left (4 \, x^{2} - 4 \, {\left (4 \, x^{2} - 13 \, x\right )} e^{4} + {\left (x {\left (4 \, e^{4} - 1\right )} - 12 \, e^{4} + 3\right )} e^{\left (2 \, e^{x}\right )} + 2 \, {\left (x^{2} {\left (4 \, e^{4} - 1\right )} - 5 \, x {\left (4 \, e^{4} - 1\right )} + 24 \, e^{4} - 6\right )} e^{\left (e^{x}\right )} - 13 \, x\right )} e^{\left (-4\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((((8*x-24)*exp(4)+6-2*x)*exp(x)+4*exp(4)-1)*exp(exp(x))^2+(((8*x^2-40*x+48)*exp(4)-2*x^2+10*x-12)*e
xp(x)+(16*x-40)*exp(4)+10-4*x)*exp(exp(x))+(-32*x+52)*exp(4)+8*x-13)/exp(4),x, algorithm="maxima")

[Out]

(4*x^2 - 4*(4*x^2 - 13*x)*e^4 + (x*(4*e^4 - 1) - 12*e^4 + 3)*e^(2*e^x) + 2*(x^2*(4*e^4 - 1) - 5*x*(4*e^4 - 1)
+ 24*e^4 - 6)*e^(e^x) - 13*x)*e^(-4)

________________________________________________________________________________________

mupad [B]  time = 7.32, size = 89, normalized size = 2.78 \begin {gather*} {\mathrm {e}}^{{\mathrm {e}}^x}\,\left ({\mathrm {e}}^{-4}\,\left (8\,{\mathrm {e}}^4-2\right )\,x^2-{\mathrm {e}}^{-4}\,\left (40\,{\mathrm {e}}^4-10\right )\,x+{\mathrm {e}}^{-4}\,\left (48\,{\mathrm {e}}^4-12\right )\right )-{\mathrm {e}}^{2\,{\mathrm {e}}^x}\,\left ({\mathrm {e}}^{-4}\,\left (12\,{\mathrm {e}}^4-3\right )-x\,{\mathrm {e}}^{-4}\,\left (4\,{\mathrm {e}}^4-1\right )\right )+x\,{\mathrm {e}}^{-4}\,\left (52\,{\mathrm {e}}^4-13\right )-\frac {x^2\,{\mathrm {e}}^{-4}\,\left (32\,{\mathrm {e}}^4-8\right )}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(-4)*(8*x + exp(2*exp(x))*(4*exp(4) + exp(x)*(exp(4)*(8*x - 24) - 2*x + 6) - 1) + exp(exp(x))*(exp(x)*(
10*x + exp(4)*(8*x^2 - 40*x + 48) - 2*x^2 - 12) - 4*x + exp(4)*(16*x - 40) + 10) - exp(4)*(32*x - 52) - 13),x)

[Out]

exp(exp(x))*(exp(-4)*(48*exp(4) - 12) - x*exp(-4)*(40*exp(4) - 10) + x^2*exp(-4)*(8*exp(4) - 2)) - exp(2*exp(x
))*(exp(-4)*(12*exp(4) - 3) - x*exp(-4)*(4*exp(4) - 1)) + x*exp(-4)*(52*exp(4) - 13) - (x^2*exp(-4)*(32*exp(4)
- 8))/2

________________________________________________________________________________________

sympy [B]  time = 0.35, size = 102, normalized size = 3.19 \begin {gather*} \frac {x^{2} \left (4 - 16 e^{4}\right )}{e^{4}} + \frac {x \left (-13 + 52 e^{4}\right )}{e^{4}} + \frac {\left (- x e^{4} + 4 x e^{8} - 12 e^{8} + 3 e^{4}\right ) e^{2 e^{x}} + \left (- 2 x^{2} e^{4} + 8 x^{2} e^{8} - 40 x e^{8} + 10 x e^{4} - 12 e^{4} + 48 e^{8}\right ) e^{e^{x}}}{e^{8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((((8*x-24)*exp(4)+6-2*x)*exp(x)+4*exp(4)-1)*exp(exp(x))**2+(((8*x**2-40*x+48)*exp(4)-2*x**2+10*x-12
)*exp(x)+(16*x-40)*exp(4)+10-4*x)*exp(exp(x))+(-32*x+52)*exp(4)+8*x-13)/exp(4),x)

[Out]

x**2*(4 - 16*exp(4))*exp(-4) + x*(-13 + 52*exp(4))*exp(-4) + ((-x*exp(4) + 4*x*exp(8) - 12*exp(8) + 3*exp(4))*
exp(2*exp(x)) + (-2*x**2*exp(4) + 8*x**2*exp(8) - 40*x*exp(8) + 10*x*exp(4) - 12*exp(4) + 48*exp(8))*exp(exp(x
)))*exp(-8)

________________________________________________________________________________________