3.57.46 \(\int \frac {-48 e^{2 x}-3 e^{2 e^x+2 x}-27 x^4+e^x (72 x-108 x^2)+e^{e^x} (24 e^{2 x}-12 e^{3 x}+18 e^x x^2)}{81 x^4+54 x^5+9 x^6+e^{2 e^x+2 x} (9+6 x+x^2)+e^{2 x} (256+128 x+16 x^2)+e^x (288 x^2+168 x^3+24 x^4)+e^{e^x} (e^{2 x} (-96-56 x-8 x^2)+e^x (-54 x^2-36 x^3-6 x^4))} \, dx\)
Optimal. Leaf size=30 \[ \frac {3}{3+x+\frac {4}{4-e^{e^x}+3 e^{-x} x^2}} \]
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Rubi [F] time = 86.93, antiderivative size = 0, normalized size of antiderivative = 0.00,
number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used =
{} \begin {gather*} \int \frac {-48 e^{2 x}-3 e^{2 e^x+2 x}-27 x^4+e^x \left (72 x-108 x^2\right )+e^{e^x} \left (24 e^{2 x}-12 e^{3 x}+18 e^x x^2\right )}{81 x^4+54 x^5+9 x^6+e^{2 e^x+2 x} \left (9+6 x+x^2\right )+e^{2 x} \left (256+128 x+16 x^2\right )+e^x \left (288 x^2+168 x^3+24 x^4\right )+e^{e^x} \left (e^{2 x} \left (-96-56 x-8 x^2\right )+e^x \left (-54 x^2-36 x^3-6 x^4\right )\right )} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
Int[(-48*E^(2*x) - 3*E^(2*E^x + 2*x) - 27*x^4 + E^x*(72*x - 108*x^2) + E^E^x*(24*E^(2*x) - 12*E^(3*x) + 18*E^x
*x^2))/(81*x^4 + 54*x^5 + 9*x^6 + E^(2*E^x + 2*x)*(9 + 6*x + x^2) + E^(2*x)*(256 + 128*x + 16*x^2) + E^x*(288*
x^2 + 168*x^3 + 24*x^4) + E^E^x*(E^(2*x)*(-96 - 56*x - 8*x^2) + E^x*(-54*x^2 - 36*x^3 - 6*x^4))),x]
[Out]
3/(3 + x) - 1152*Defer[Int][x^2/(-16 + 3*E^E^x - 4*x + E^E^x*x)^3, x] - 288*Defer[Int][x^3/(-16 + 3*E^E^x - 4*
x + E^E^x*x)^3, x] - 12*Defer[Int][E^(E^x + x)/(-16 + 3*E^E^x - 4*x + E^E^x*x)^2, x] - 72*Defer[Int][x^2/(-16
+ 3*E^E^x - 4*x + E^E^x*x)^2, x] - 48*Defer[Int][1/((3 + x)^2*(-16 + 3*E^E^x - 4*x + E^E^x*x)^2), x] - 24*Defe
r[Int][1/((3 + x)^2*(-16 + 3*E^E^x - 4*x + E^E^x*x)), x] - 62208*Defer[Int][(E^E^x*x^3)/((-16 + 3*E^E^x - 4*x
+ E^E^x*x)^3*(-16*E^x + 3*E^(E^x + x) - 4*E^x*x + E^(E^x + x)*x - 9*x^2 - 3*x^3)^2), x] + 5832*Defer[Int][(E^(
2*E^x)*x^3)/((-16 + 3*E^E^x - 4*x + E^E^x*x)^3*(-16*E^x + 3*E^(E^x + x) - 4*E^x*x + E^(E^x + x)*x - 9*x^2 - 3*
x^3)^2), x] - 27216*Defer[Int][(E^E^x*x^4)/((-16 + 3*E^E^x - 4*x + E^E^x*x)^3*(-16*E^x + 3*E^(E^x + x) - 4*E^x
*x + E^(E^x + x)*x - 9*x^2 - 3*x^3)^2), x] + 2916*Defer[Int][(E^(2*E^x)*x^4)/((-16 + 3*E^E^x - 4*x + E^E^x*x)^
3*(-16*E^x + 3*E^(E^x + x) - 4*E^x*x + E^(E^x + x)*x - 9*x^2 - 3*x^3)^2), x] + 10800*Defer[Int][(E^E^x*x^5)/((
-16 + 3*E^E^x - 4*x + E^E^x*x)^3*(-16*E^x + 3*E^(E^x + x) - 4*E^x*x + E^(E^x + x)*x - 9*x^2 - 3*x^3)^2), x] -
972*Defer[Int][(E^(2*E^x)*x^5)/((-16 + 3*E^E^x - 4*x + E^E^x*x)^3*(-16*E^x + 3*E^(E^x + x) - 4*E^x*x + E^(E^x
+ x)*x - 9*x^2 - 3*x^3)^2), x] - 1836*Defer[Int][(E^E^x*x^6)/((-16 + 3*E^E^x - 4*x + E^E^x*x)^3*(-16*E^x + 3*E
^(E^x + x) - 4*E^x*x + E^(E^x + x)*x - 9*x^2 - 3*x^3)^2), x] - 756*Defer[Int][(E^(2*E^x)*x^6)/((-16 + 3*E^E^x
- 4*x + E^E^x*x)^3*(-16*E^x + 3*E^(E^x + x) - 4*E^x*x + E^(E^x + x)*x - 9*x^2 - 3*x^3)^2), x] - 7884*Defer[Int
][(E^E^x*x^7)/((-16 + 3*E^E^x - 4*x + E^E^x*x)^3*(-16*E^x + 3*E^(E^x + x) - 4*E^x*x + E^(E^x + x)*x - 9*x^2 -
3*x^3)^2), x] - 108*Defer[Int][(E^(2*E^x)*x^7)/((-16 + 3*E^E^x - 4*x + E^E^x*x)^3*(-16*E^x + 3*E^(E^x + x) - 4
*E^x*x + E^(E^x + x)*x - 9*x^2 - 3*x^3)^2), x] - 2916*Defer[Int][(E^E^x*x^8)/((-16 + 3*E^E^x - 4*x + E^E^x*x)^
3*(-16*E^x + 3*E^(E^x + x) - 4*E^x*x + E^(E^x + x)*x - 9*x^2 - 3*x^3)^2), x] - 324*Defer[Int][(E^E^x*x^9)/((-1
6 + 3*E^E^x - 4*x + E^E^x*x)^3*(-16*E^x + 3*E^(E^x + x) - 4*E^x*x + E^(E^x + x)*x - 9*x^2 - 3*x^3)^2), x] - 69
12*Defer[Int][(E^E^x*x)/((-16 + 3*E^E^x - 4*x + E^E^x*x)^3*(-16*E^x + 3*E^(E^x + x) - 4*E^x*x + E^(E^x + x)*x
- 9*x^2 - 3*x^3)), x] + 648*Defer[Int][(E^(2*E^x)*x)/((-16 + 3*E^E^x - 4*x + E^E^x*x)^3*(-16*E^x + 3*E^(E^x +
x) - 4*E^x*x + E^(E^x + x)*x - 9*x^2 - 3*x^3)), x] + 1440*Defer[Int][(E^E^x*x^2)/((-16 + 3*E^E^x - 4*x + E^E^x
*x)^3*(-16*E^x + 3*E^(E^x + x) - 4*E^x*x + E^(E^x + x)*x - 9*x^2 - 3*x^3)), x] - 108*Defer[Int][(E^(2*E^x)*x^2
)/((-16 + 3*E^E^x - 4*x + E^E^x*x)^3*(-16*E^x + 3*E^(E^x + x) - 4*E^x*x + E^(E^x + x)*x - 9*x^2 - 3*x^3)), x]
+ 2016*Defer[Int][(E^E^x*x^3)/((-16 + 3*E^E^x - 4*x + E^E^x*x)^3*(-16*E^x + 3*E^(E^x + x) - 4*E^x*x + E^(E^x +
x)*x - 9*x^2 - 3*x^3)), x] - 216*Defer[Int][(E^(2*E^x)*x^3)/((-16 + 3*E^E^x - 4*x + E^E^x*x)^3*(-16*E^x + 3*E
^(E^x + x) - 4*E^x*x + E^(E^x + x)*x - 9*x^2 - 3*x^3)), x] - 2628*Defer[Int][(E^E^x*x^4)/((-16 + 3*E^E^x - 4*x
+ E^E^x*x)^3*(-16*E^x + 3*E^(E^x + x) - 4*E^x*x + E^(E^x + x)*x - 9*x^2 - 3*x^3)), x] - 36*Defer[Int][(E^(2*E
^x)*x^4)/((-16 + 3*E^E^x - 4*x + E^E^x*x)^3*(-16*E^x + 3*E^(E^x + x) - 4*E^x*x + E^(E^x + x)*x - 9*x^2 - 3*x^3
)), x] - 1944*Defer[Int][(E^E^x*x^5)/((-16 + 3*E^E^x - 4*x + E^E^x*x)^3*(-16*E^x + 3*E^(E^x + x) - 4*E^x*x + E
^(E^x + x)*x - 9*x^2 - 3*x^3)), x] - 324*Defer[Int][(E^E^x*x^6)/((-16 + 3*E^E^x - 4*x + E^E^x*x)^3*(-16*E^x +
3*E^(E^x + x) - 4*E^x*x + E^(E^x + x)*x - 9*x^2 - 3*x^3)), x] - 18432*Defer[Int][x/((-16 + 3*E^E^x - 4*x + E^E
^x*x)^3*(16*E^x - 3*E^(E^x + x) + 4*E^x*x - E^(E^x + x)*x + 9*x^2 + 3*x^3)), x] + 4608*Defer[Int][x^2/((-16 +
3*E^E^x - 4*x + E^E^x*x)^3*(16*E^x - 3*E^(E^x + x) + 4*E^x*x - E^(E^x + x)*x + 9*x^2 + 3*x^3)), x] + 4608*Defe
r[Int][x^3/((-16 + 3*E^E^x - 4*x + E^E^x*x)^3*(16*E^x - 3*E^(E^x + x) + 4*E^x*x - E^(E^x + x)*x + 9*x^2 + 3*x^
3)), x] + 576*Defer[Int][x^4/((-16 + 3*E^E^x - 4*x + E^E^x*x)^3*(16*E^x - 3*E^(E^x + x) + 4*E^x*x - E^(E^x + x
)*x + 9*x^2 + 3*x^3)), x] + 165888*Defer[Int][x^3/((E^E^x*(3 + x) - 4*(4 + x))^3*(E^(E^x + x)*(3 + x) - 3*x^2*
(3 + x) - 4*E^x*(4 + x))^2), x] + 62208*Defer[Int][x^4/((E^E^x*(3 + x) - 4*(4 + x))^3*(E^(E^x + x)*(3 + x) - 3
*x^2*(3 + x) - 4*E^x*(4 + x))^2), x] - 29376*Defer[Int][x^5/((E^E^x*(3 + x) - 4*(4 + x))^3*(E^(E^x + x)*(3 + x
) - 3*x^2*(3 + x) - 4*E^x*(4 + x))^2), x] - 15552*Defer[Int][x^6/((E^E^x*(3 + x) - 4*(4 + x))^3*(E^(E^x + x)*(
3 + x) - 3*x^2*(3 + x) - 4*E^x*(4 + x))^2), x] - 1728*Defer[Int][x^7/((E^E^x*(3 + x) - 4*(4 + x))^3*(E^(E^x +
x)*(3 + x) - 3*x^2*(3 + x) - 4*E^x*(4 + x))^2), x]
Rubi steps
Aborted
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Mathematica [A] time = 0.09, size = 52, normalized size = 1.73 \begin {gather*} -\frac {3 \left (4 e^x-e^{e^x+x}+3 x^2\right )}{e^{e^x+x} (3+x)-3 x^2 (3+x)-4 e^x (4+x)} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[(-48*E^(2*x) - 3*E^(2*E^x + 2*x) - 27*x^4 + E^x*(72*x - 108*x^2) + E^E^x*(24*E^(2*x) - 12*E^(3*x) +
18*E^x*x^2))/(81*x^4 + 54*x^5 + 9*x^6 + E^(2*E^x + 2*x)*(9 + 6*x + x^2) + E^(2*x)*(256 + 128*x + 16*x^2) + E^x
*(288*x^2 + 168*x^3 + 24*x^4) + E^E^x*(E^(2*x)*(-96 - 56*x - 8*x^2) + E^x*(-54*x^2 - 36*x^3 - 6*x^4))),x]
[Out]
(-3*(4*E^x - E^(E^x + x) + 3*x^2))/(E^(E^x + x)*(3 + x) - 3*x^2*(3 + x) - 4*E^x*(4 + x))
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fricas [A] time = 0.55, size = 49, normalized size = 1.63 \begin {gather*} \frac {3 \, {\left (3 \, x^{2} - e^{\left (x + e^{x}\right )} + 4 \, e^{x}\right )}}{3 \, x^{3} + 9 \, x^{2} - {\left (x + 3\right )} e^{\left (x + e^{x}\right )} + 4 \, {\left (x + 4\right )} e^{x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((-3*exp(x)^2*exp(exp(x))^2+(-12*exp(x)^3+24*exp(x)^2+18*exp(x)*x^2)*exp(exp(x))-48*exp(x)^2+(-108*x^
2+72*x)*exp(x)-27*x^4)/((x^2+6*x+9)*exp(x)^2*exp(exp(x))^2+((-8*x^2-56*x-96)*exp(x)^2+(-6*x^4-36*x^3-54*x^2)*e
xp(x))*exp(exp(x))+(16*x^2+128*x+256)*exp(x)^2+(24*x^4+168*x^3+288*x^2)*exp(x)+9*x^6+54*x^5+81*x^4),x, algorit
hm="fricas")
[Out]
3*(3*x^2 - e^(x + e^x) + 4*e^x)/(3*x^3 + 9*x^2 - (x + 3)*e^(x + e^x) + 4*(x + 4)*e^x)
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giac [B] time = 0.40, size = 56, normalized size = 1.87 \begin {gather*} \frac {3 \, {\left (3 \, x^{2} - e^{\left (x + e^{x}\right )} + 4 \, e^{x}\right )}}{3 \, x^{3} + 9 \, x^{2} - x e^{\left (x + e^{x}\right )} + 4 \, x e^{x} - 3 \, e^{\left (x + e^{x}\right )} + 16 \, e^{x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((-3*exp(x)^2*exp(exp(x))^2+(-12*exp(x)^3+24*exp(x)^2+18*exp(x)*x^2)*exp(exp(x))-48*exp(x)^2+(-108*x^
2+72*x)*exp(x)-27*x^4)/((x^2+6*x+9)*exp(x)^2*exp(exp(x))^2+((-8*x^2-56*x-96)*exp(x)^2+(-6*x^4-36*x^3-54*x^2)*e
xp(x))*exp(exp(x))+(16*x^2+128*x+256)*exp(x)^2+(24*x^4+168*x^3+288*x^2)*exp(x)+9*x^6+54*x^5+81*x^4),x, algorit
hm="giac")
[Out]
3*(3*x^2 - e^(x + e^x) + 4*e^x)/(3*x^3 + 9*x^2 - x*e^(x + e^x) + 4*x*e^x - 3*e^(x + e^x) + 16*e^x)
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maple [A] time = 0.07, size = 55, normalized size = 1.83
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\(\frac {3}{3+x}-\frac {12 \,{\mathrm e}^{x}}{\left (3+x \right ) \left (3 x^{3}-x \,{\mathrm e}^{{\mathrm e}^{x}+x}+9 x^{2}+4 \,{\mathrm e}^{x} x -3 \,{\mathrm e}^{{\mathrm e}^{x}+x}+16 \,{\mathrm e}^{x}\right )}\) |
\(55\) |
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Verification of antiderivative is not currently implemented for this CAS.
[In]
int((-3*exp(x)^2*exp(exp(x))^2+(-12*exp(x)^3+24*exp(x)^2+18*exp(x)*x^2)*exp(exp(x))-48*exp(x)^2+(-108*x^2+72*x
)*exp(x)-27*x^4)/((x^2+6*x+9)*exp(x)^2*exp(exp(x))^2+((-8*x^2-56*x-96)*exp(x)^2+(-6*x^4-36*x^3-54*x^2)*exp(x))
*exp(exp(x))+(16*x^2+128*x+256)*exp(x)^2+(24*x^4+168*x^3+288*x^2)*exp(x)+9*x^6+54*x^5+81*x^4),x,method=_RETURN
VERBOSE)
[Out]
3/(3+x)-12*exp(x)/(3+x)/(3*x^3-x*exp(exp(x)+x)+9*x^2+4*exp(x)*x-3*exp(exp(x)+x)+16*exp(x))
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maxima [A] time = 0.53, size = 49, normalized size = 1.63 \begin {gather*} \frac {3 \, {\left (3 \, x^{2} - e^{\left (x + e^{x}\right )} + 4 \, e^{x}\right )}}{3 \, x^{3} + 9 \, x^{2} - {\left (x + 3\right )} e^{\left (x + e^{x}\right )} + 4 \, {\left (x + 4\right )} e^{x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((-3*exp(x)^2*exp(exp(x))^2+(-12*exp(x)^3+24*exp(x)^2+18*exp(x)*x^2)*exp(exp(x))-48*exp(x)^2+(-108*x^
2+72*x)*exp(x)-27*x^4)/((x^2+6*x+9)*exp(x)^2*exp(exp(x))^2+((-8*x^2-56*x-96)*exp(x)^2+(-6*x^4-36*x^3-54*x^2)*e
xp(x))*exp(exp(x))+(16*x^2+128*x+256)*exp(x)^2+(24*x^4+168*x^3+288*x^2)*exp(x)+9*x^6+54*x^5+81*x^4),x, algorit
hm="maxima")
[Out]
3*(3*x^2 - e^(x + e^x) + 4*e^x)/(3*x^3 + 9*x^2 - (x + 3)*e^(x + e^x) + 4*(x + 4)*e^x)
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mupad [F] time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int -\frac {48\,{\mathrm {e}}^{2\,x}+3\,{\mathrm {e}}^{2\,x}\,{\mathrm {e}}^{2\,{\mathrm {e}}^x}-{\mathrm {e}}^{{\mathrm {e}}^x}\,\left (24\,{\mathrm {e}}^{2\,x}-12\,{\mathrm {e}}^{3\,x}+18\,x^2\,{\mathrm {e}}^x\right )-{\mathrm {e}}^x\,\left (72\,x-108\,x^2\right )+27\,x^4}{{\mathrm {e}}^{2\,x}\,\left (16\,x^2+128\,x+256\right )+{\mathrm {e}}^x\,\left (24\,x^4+168\,x^3+288\,x^2\right )-{\mathrm {e}}^{{\mathrm {e}}^x}\,\left ({\mathrm {e}}^{2\,x}\,\left (8\,x^2+56\,x+96\right )+{\mathrm {e}}^x\,\left (6\,x^4+36\,x^3+54\,x^2\right )\right )+81\,x^4+54\,x^5+9\,x^6+{\mathrm {e}}^{2\,x}\,{\mathrm {e}}^{2\,{\mathrm {e}}^x}\,\left (x^2+6\,x+9\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(-(48*exp(2*x) + 3*exp(2*x)*exp(2*exp(x)) - exp(exp(x))*(24*exp(2*x) - 12*exp(3*x) + 18*x^2*exp(x)) - exp(x
)*(72*x - 108*x^2) + 27*x^4)/(exp(2*x)*(128*x + 16*x^2 + 256) + exp(x)*(288*x^2 + 168*x^3 + 24*x^4) - exp(exp(
x))*(exp(2*x)*(56*x + 8*x^2 + 96) + exp(x)*(54*x^2 + 36*x^3 + 6*x^4)) + 81*x^4 + 54*x^5 + 9*x^6 + exp(2*x)*exp
(2*exp(x))*(6*x + x^2 + 9)),x)
[Out]
int(-(48*exp(2*x) + 3*exp(2*x)*exp(2*exp(x)) - exp(exp(x))*(24*exp(2*x) - 12*exp(3*x) + 18*x^2*exp(x)) - exp(x
)*(72*x - 108*x^2) + 27*x^4)/(exp(2*x)*(128*x + 16*x^2 + 256) + exp(x)*(288*x^2 + 168*x^3 + 24*x^4) - exp(exp(
x))*(exp(2*x)*(56*x + 8*x^2 + 96) + exp(x)*(54*x^2 + 36*x^3 + 6*x^4)) + 81*x^4 + 54*x^5 + 9*x^6 + exp(2*x)*exp
(2*exp(x))*(6*x + x^2 + 9)), x)
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sympy [B] time = 0.71, size = 68, normalized size = 2.27 \begin {gather*} \frac {12 e^{x}}{- 3 x^{4} - 18 x^{3} - 4 x^{2} e^{x} - 27 x^{2} - 28 x e^{x} + \left (x^{2} e^{x} + 6 x e^{x} + 9 e^{x}\right ) e^{e^{x}} - 48 e^{x}} + \frac {3}{x + 3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((-3*exp(x)**2*exp(exp(x))**2+(-12*exp(x)**3+24*exp(x)**2+18*exp(x)*x**2)*exp(exp(x))-48*exp(x)**2+(-
108*x**2+72*x)*exp(x)-27*x**4)/((x**2+6*x+9)*exp(x)**2*exp(exp(x))**2+((-8*x**2-56*x-96)*exp(x)**2+(-6*x**4-36
*x**3-54*x**2)*exp(x))*exp(exp(x))+(16*x**2+128*x+256)*exp(x)**2+(24*x**4+168*x**3+288*x**2)*exp(x)+9*x**6+54*
x**5+81*x**4),x)
[Out]
12*exp(x)/(-3*x**4 - 18*x**3 - 4*x**2*exp(x) - 27*x**2 - 28*x*exp(x) + (x**2*exp(x) + 6*x*exp(x) + 9*exp(x))*e
xp(exp(x)) - 48*exp(x)) + 3/(x + 3)
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