3.18.2 \(\int \frac {-16-32 x-16 x^2+e^{2 x+e^{2 x} (3 e^{\frac {2 x}{2+2 x}} x+6 e^{\frac {x}{2+2 x}} x^2+3 x^3)} (-108 x^2-288 x^3-252 x^4-72 x^5+e^{\frac {2 x}{2+2 x}} (-36-180 x-180 x^2-72 x^3)+e^{\frac {x}{2+2 x}} (-144 x-468 x^2-432 x^3-144 x^4))}{16 x^2+32 x^3+16 x^4+e^{2 e^{2 x} (3 e^{\frac {2 x}{2+2 x}} x+6 e^{\frac {x}{2+2 x}} x^2+3 x^3)} (9+18 x+9 x^2)+e^{e^{2 x} (3 e^{\frac {2 x}{2+2 x}} x+6 e^{\frac {x}{2+2 x}} x^2+3 x^3)} (24 x+48 x^2+24 x^3)} \, dx\)
Optimal. Leaf size=35 \[ \frac {4}{x+3 \left (e^{3 e^{2 x} x \left (e^{\frac {x}{2+2 x}}+x\right )^2}+x\right )} \]
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Rubi [F] time = 180.00, 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*} \text {\$Aborted} \end {gather*}
Verification is not applicable to the result.
[In]
Int[(-16 - 32*x - 16*x^2 + E^(2*x + E^(2*x)*(3*E^((2*x)/(2 + 2*x))*x + 6*E^(x/(2 + 2*x))*x^2 + 3*x^3))*(-108*x
^2 - 288*x^3 - 252*x^4 - 72*x^5 + E^((2*x)/(2 + 2*x))*(-36 - 180*x - 180*x^2 - 72*x^3) + E^(x/(2 + 2*x))*(-144
*x - 468*x^2 - 432*x^3 - 144*x^4)))/(16*x^2 + 32*x^3 + 16*x^4 + E^(2*E^(2*x)*(3*E^((2*x)/(2 + 2*x))*x + 6*E^(x
/(2 + 2*x))*x^2 + 3*x^3))*(9 + 18*x + 9*x^2) + E^(E^(2*x)*(3*E^((2*x)/(2 + 2*x))*x + 6*E^(x/(2 + 2*x))*x^2 + 3
*x^3))*(24*x + 48*x^2 + 24*x^3)),x]
[Out]
$Aborted
Rubi steps
Aborted
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Mathematica [A] time = 0.86, size = 35, normalized size = 1.00 \begin {gather*} \frac {4}{3 e^{3 e^{2 x} x \left (e^{\frac {x}{2+2 x}}+x\right )^2}+4 x} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[(-16 - 32*x - 16*x^2 + E^(2*x + E^(2*x)*(3*E^((2*x)/(2 + 2*x))*x + 6*E^(x/(2 + 2*x))*x^2 + 3*x^3))*(
-108*x^2 - 288*x^3 - 252*x^4 - 72*x^5 + E^((2*x)/(2 + 2*x))*(-36 - 180*x - 180*x^2 - 72*x^3) + E^(x/(2 + 2*x))
*(-144*x - 468*x^2 - 432*x^3 - 144*x^4)))/(16*x^2 + 32*x^3 + 16*x^4 + E^(2*E^(2*x)*(3*E^((2*x)/(2 + 2*x))*x +
6*E^(x/(2 + 2*x))*x^2 + 3*x^3))*(9 + 18*x + 9*x^2) + E^(E^(2*x)*(3*E^((2*x)/(2 + 2*x))*x + 6*E^(x/(2 + 2*x))*x
^2 + 3*x^3))*(24*x + 48*x^2 + 24*x^3)),x]
[Out]
4/(3*E^(3*E^(2*x)*x*(E^(x/(2 + 2*x)) + x)^2) + 4*x)
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fricas [A] time = 0.75, size = 57, normalized size = 1.63 \begin {gather*} \frac {4 \, e^{\left (2 \, x\right )}}{4 \, x e^{\left (2 \, x\right )} + 3 \, e^{\left (3 \, {\left (x^{3} + 2 \, x^{2} e^{\left (\frac {x}{2 \, {\left (x + 1\right )}}\right )} + x e^{\left (\frac {x}{x + 1}\right )}\right )} e^{\left (2 \, x\right )} + 2 \, x\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((((-72*x^3-180*x^2-180*x-36)*exp(x/(2*x+2))^2+(-144*x^4-432*x^3-468*x^2-144*x)*exp(x/(2*x+2))-72*x^5
-252*x^4-288*x^3-108*x^2)*exp(x)^2*exp((3*x*exp(x/(2*x+2))^2+6*x^2*exp(x/(2*x+2))+3*x^3)*exp(x)^2)-16*x^2-32*x
-16)/((9*x^2+18*x+9)*exp((3*x*exp(x/(2*x+2))^2+6*x^2*exp(x/(2*x+2))+3*x^3)*exp(x)^2)^2+(24*x^3+48*x^2+24*x)*ex
p((3*x*exp(x/(2*x+2))^2+6*x^2*exp(x/(2*x+2))+3*x^3)*exp(x)^2)+16*x^4+32*x^3+16*x^2),x, algorithm="fricas")
[Out]
4*e^(2*x)/(4*x*e^(2*x) + 3*e^(3*(x^3 + 2*x^2*e^(1/2*x/(x + 1)) + x*e^(x/(x + 1)))*e^(2*x) + 2*x))
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giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((((-72*x^3-180*x^2-180*x-36)*exp(x/(2*x+2))^2+(-144*x^4-432*x^3-468*x^2-144*x)*exp(x/(2*x+2))-72*x^5
-252*x^4-288*x^3-108*x^2)*exp(x)^2*exp((3*x*exp(x/(2*x+2))^2+6*x^2*exp(x/(2*x+2))+3*x^3)*exp(x)^2)-16*x^2-32*x
-16)/((9*x^2+18*x+9)*exp((3*x*exp(x/(2*x+2))^2+6*x^2*exp(x/(2*x+2))+3*x^3)*exp(x)^2)^2+(24*x^3+48*x^2+24*x)*ex
p((3*x*exp(x/(2*x+2))^2+6*x^2*exp(x/(2*x+2))+3*x^3)*exp(x)^2)+16*x^4+32*x^3+16*x^2),x, algorithm="giac")
[Out]
Timed out
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maple [A] time = 0.15, size = 43, normalized size = 1.23
|
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result |
size |
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| risch |
\(\frac {4}{4 x +3 \,{\mathrm e}^{3 x \left (2 x \,{\mathrm e}^{\frac {x}{2 x +2}}+x^{2}+{\mathrm e}^{\frac {x}{x +1}}\right ) {\mathrm e}^{2 x}}}\) |
\(43\) |
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Verification of antiderivative is not currently implemented for this CAS.
[In]
int((((-72*x^3-180*x^2-180*x-36)*exp(x/(2*x+2))^2+(-144*x^4-432*x^3-468*x^2-144*x)*exp(x/(2*x+2))-72*x^5-252*x
^4-288*x^3-108*x^2)*exp(x)^2*exp((3*x*exp(x/(2*x+2))^2+6*x^2*exp(x/(2*x+2))+3*x^3)*exp(x)^2)-16*x^2-32*x-16)/(
(9*x^2+18*x+9)*exp((3*x*exp(x/(2*x+2))^2+6*x^2*exp(x/(2*x+2))+3*x^3)*exp(x)^2)^2+(24*x^3+48*x^2+24*x)*exp((3*x
*exp(x/(2*x+2))^2+6*x^2*exp(x/(2*x+2))+3*x^3)*exp(x)^2)+16*x^4+32*x^3+16*x^2),x,method=_RETURNVERBOSE)
[Out]
4/(4*x+3*exp(3*x*(2*exp(1/2*x/(x+1))*x+x^2+exp(x/(x+1)))*exp(2*x)))
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 result too large to
display
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((((-72*x^3-180*x^2-180*x-36)*exp(x/(2*x+2))^2+(-144*x^4-432*x^3-468*x^2-144*x)*exp(x/(2*x+2))-72*x^5
-252*x^4-288*x^3-108*x^2)*exp(x)^2*exp((3*x*exp(x/(2*x+2))^2+6*x^2*exp(x/(2*x+2))+3*x^3)*exp(x)^2)-16*x^2-32*x
-16)/((9*x^2+18*x+9)*exp((3*x*exp(x/(2*x+2))^2+6*x^2*exp(x/(2*x+2))+3*x^3)*exp(x)^2)^2+(24*x^3+48*x^2+24*x)*ex
p((3*x*exp(x/(2*x+2))^2+6*x^2*exp(x/(2*x+2))+3*x^3)*exp(x)^2)+16*x^4+32*x^3+16*x^2),x, algorithm="maxima")
[Out]
4*(3*(4*x^5*e^(1/2) + 12*x^4*e^(1/2) + 13*x^3*e^(1/2) + 4*x^2*e^(1/2))*e^(2*x + 3/2/(x + 1)) + 3*(2*x^4*e + 5*
x^3*e + 5*x^2*e + x*e)*e^(2*x + 1/(x + 1)) - (x^2 - 3*(2*x^6 + 7*x^5 + 8*x^4 + 3*x^3)*e^(2*x) + 2*x + 1)*e^(2/
(x + 1)))/(3*(3*(4*x^5*e^(1/2) + 12*x^4*e^(1/2) + 13*x^3*e^(1/2) + 4*x^2*e^(1/2))*e^(2*x + 3/2/(x + 1)) + (3*(
2*x^4*e + 5*x^3*e + 5*x^2*e + x*e)*e^(2*x) - (x^2 - 3*(2*x^6 + 7*x^5 + 8*x^4 + 3*x^3)*e^(2*x) + 2*x + 1)*e^(1/
(x + 1)))*e^(1/(x + 1)))*e^(3*x^3*e^(2*x) + 6*x^2*e^(2*x - 1/2/(x + 1) + 1/2) + 3*x*e^(2*x - 1/(x + 1) + 1)) +
12*(4*x^6*e^(1/2) + 12*x^5*e^(1/2) + 13*x^4*e^(1/2) + 4*x^3*e^(1/2))*e^(2*x + 3/2/(x + 1)) + 4*(3*(2*x^5*e +
5*x^4*e + 5*x^3*e + x^2*e)*e^(2*x) - (x^3 + 2*x^2 - 3*(2*x^7 + 7*x^6 + 8*x^5 + 3*x^4)*e^(2*x) + x)*e^(1/(x + 1
)))*e^(1/(x + 1))) - 4*integrate(3*((3*(4*x^9*e^2 + 28*x^8*e^2 + 89*x^7*e^2 + 164*x^6*e^2 + 188*x^5*e^2 + 134*
x^4*e^2 + 56*x^3*e^2 + 12*x^2*e^2 + x*e^2)*e^(4*x) + (3*(12*x^11*e + 88*x^10*e + 295*x^9*e + 581*x^8*e + 722*x
^7*e + 572*x^6*e + 280*x^5*e + 77*x^4*e + 9*x^3*e)*e^(4*x) + (4*x^8*e + 28*x^7*e + 88*x^6*e + 156*x^5*e + 167*
x^4*e + 108*x^3*e + 39*x^2*e + 6*x*e)*e^(2*x))*e^(1/(x + 1)))*e^(3/2/(x + 1)) - (3*(4*x^9*e^2 + 28*x^8*e^2 + 8
9*x^7*e^2 + 164*x^6*e^2 + 188*x^5*e^2 + 134*x^4*e^2 + 56*x^3*e^2 + 12*x^2*e^2 + x*e^2)*e^(4*x + 1/(x + 1)) + (
3*(12*x^11*e + 88*x^10*e + 295*x^9*e + 581*x^8*e + 722*x^7*e + 572*x^6*e + 280*x^5*e + 77*x^4*e + 9*x^3*e)*e^(
4*x) + (4*x^8*e + 28*x^7*e + 88*x^6*e + 156*x^5*e + 167*x^4*e + 108*x^3*e + 39*x^2*e + 6*x*e)*e^(2*x))*e^(2/(x
+ 1)))*e^(1/2/(x + 1)))/(3*(9*(16*x^12*e + 128*x^11*e + 456*x^10*e + 936*x^9*e + 1201*x^8*e + 978*x^7*e + 489
*x^6*e + 136*x^5*e + 16*x^4*e)*e^(4*x + 5/2/(x + 1)) + (9*(4*x^10*e^2 + 28*x^9*e^2 + 89*x^8*e^2 + 164*x^7*e^2
+ 188*x^6*e^2 + 134*x^5*e^2 + 56*x^4*e^2 + 12*x^3*e^2 + x^2*e^2)*e^(4*x) + (x^6 + 6*x^5 + 15*x^4 + 20*x^3 + 15
*x^2 + 9*(4*x^14 + 36*x^13 + 141*x^12 + 314*x^11 + 435*x^10 + 384*x^9 + 211*x^8 + 66*x^7 + 9*x^6)*e^(4*x) - 6*
(2*x^10 + 15*x^9 + 48*x^8 + 85*x^7 + 90*x^6 + 57*x^5 + 20*x^4 + 3*x^3)*e^(2*x) + 6*x + 1)*e^(2/(x + 1)) + 6*(3
*(4*x^12*e + 32*x^11*e + 113*x^10*e + 229*x^9*e + 289*x^8*e + 230*x^7*e + 111*x^6*e + 29*x^5*e + 3*x^4*e)*e^(4
*x) - (2*x^8*e + 13*x^7*e + 37*x^6*e + 59*x^5*e + 56*x^4*e + 31*x^3*e + 9*x^2*e + x*e)*e^(2*x))*e^(1/(x + 1)))
*e^(3/2/(x + 1)) + 6*(3*(8*x^11*e^(3/2) + 60*x^10*e^(3/2) + 202*x^9*e^(3/2) + 393*x^8*e^(3/2) + 477*x^7*e^(3/2
) + 364*x^6*e^(3/2) + 167*x^5*e^(3/2) + 41*x^4*e^(3/2) + 4*x^3*e^(3/2))*e^(4*x + 1/(x + 1)) + (3*(8*x^13*e^(1/
2) + 68*x^12*e^(1/2) + 254*x^11*e^(1/2) + 543*x^10*e^(1/2) + 724*x^9*e^(1/2) + 614*x^8*e^(1/2) + 322*x^7*e^(1/
2) + 95*x^6*e^(1/2) + 12*x^5*e^(1/2))*e^(4*x) - (4*x^9*e^(1/2) + 28*x^8*e^(1/2) + 85*x^7*e^(1/2) + 144*x^6*e^(
1/2) + 146*x^5*e^(1/2) + 88*x^4*e^(1/2) + 29*x^3*e^(1/2) + 4*x^2*e^(1/2))*e^(2*x))*e^(2/(x + 1)))*e^(1/(x + 1)
))*e^(3*x^3*e^(2*x) + 6*x^2*e^(2*x - 1/2/(x + 1) + 1/2) + 3*x*e^(2*x - 1/(x + 1) + 1)) + 36*(16*x^13*e + 128*x
^12*e + 456*x^11*e + 936*x^10*e + 1201*x^9*e + 978*x^8*e + 489*x^7*e + 136*x^6*e + 16*x^5*e)*e^(4*x + 5/2/(x +
1)) + 4*(9*(4*x^11*e^2 + 28*x^10*e^2 + 89*x^9*e^2 + 164*x^8*e^2 + 188*x^7*e^2 + 134*x^6*e^2 + 56*x^5*e^2 + 12
*x^4*e^2 + x^3*e^2)*e^(4*x) + (x^7 + 6*x^6 + 15*x^5 + 20*x^4 + 15*x^3 + 6*x^2 + 9*(4*x^15 + 36*x^14 + 141*x^13
+ 314*x^12 + 435*x^11 + 384*x^10 + 211*x^9 + 66*x^8 + 9*x^7)*e^(4*x) - 6*(2*x^11 + 15*x^10 + 48*x^9 + 85*x^8
+ 90*x^7 + 57*x^6 + 20*x^5 + 3*x^4)*e^(2*x) + x)*e^(2/(x + 1)) + 6*(3*(4*x^13*e + 32*x^12*e + 113*x^11*e + 229
*x^10*e + 289*x^9*e + 230*x^8*e + 111*x^7*e + 29*x^6*e + 3*x^5*e)*e^(4*x) - (2*x^9*e + 13*x^8*e + 37*x^7*e + 5
9*x^6*e + 56*x^5*e + 31*x^4*e + 9*x^3*e + x^2*e)*e^(2*x))*e^(1/(x + 1)))*e^(3/2/(x + 1)) + 24*(3*(8*x^12*e^(3/
2) + 60*x^11*e^(3/2) + 202*x^10*e^(3/2) + 393*x^9*e^(3/2) + 477*x^8*e^(3/2) + 364*x^7*e^(3/2) + 167*x^6*e^(3/2
) + 41*x^5*e^(3/2) + 4*x^4*e^(3/2))*e^(4*x + 1/(x + 1)) + (3*(8*x^14*e^(1/2) + 68*x^13*e^(1/2) + 254*x^12*e^(1
/2) + 543*x^11*e^(1/2) + 724*x^10*e^(1/2) + 614*x^9*e^(1/2) + 322*x^8*e^(1/2) + 95*x^7*e^(1/2) + 12*x^6*e^(1/2
))*e^(4*x) - (4*x^10*e^(1/2) + 28*x^9*e^(1/2) + 85*x^8*e^(1/2) + 144*x^7*e^(1/2) + 146*x^6*e^(1/2) + 88*x^5*e^
(1/2) + 29*x^4*e^(1/2) + 4*x^3*e^(1/2))*e^(2*x))*e^(2/(x + 1)))*e^(1/(x + 1))), x)
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mupad [B] time = 3.43, size = 56, normalized size = 1.60 \begin {gather*} \frac {4}{4\,x+3\,{\mathrm {e}}^{6\,x^2\,{\mathrm {e}}^{2\,x}\,{\mathrm {e}}^{\frac {x}{2\,x+2}}}\,{\mathrm {e}}^{3\,x^3\,{\mathrm {e}}^{2\,x}}\,{\mathrm {e}}^{3\,x\,{\mathrm {e}}^{2\,x}\,{\mathrm {e}}^{\frac {x}{x+1}}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(-(32*x + 16*x^2 + exp(2*x)*exp(exp(2*x)*(6*x^2*exp(x/(2*x + 2)) + 3*x*exp((2*x)/(2*x + 2)) + 3*x^3))*(exp(
x/(2*x + 2))*(144*x + 468*x^2 + 432*x^3 + 144*x^4) + exp((2*x)/(2*x + 2))*(180*x + 180*x^2 + 72*x^3 + 36) + 10
8*x^2 + 288*x^3 + 252*x^4 + 72*x^5) + 16)/(exp(2*exp(2*x)*(6*x^2*exp(x/(2*x + 2)) + 3*x*exp((2*x)/(2*x + 2)) +
3*x^3))*(18*x + 9*x^2 + 9) + exp(exp(2*x)*(6*x^2*exp(x/(2*x + 2)) + 3*x*exp((2*x)/(2*x + 2)) + 3*x^3))*(24*x
+ 48*x^2 + 24*x^3) + 16*x^2 + 32*x^3 + 16*x^4),x)
[Out]
4/(4*x + 3*exp(6*x^2*exp(2*x)*exp(x/(2*x + 2)))*exp(3*x^3*exp(2*x))*exp(3*x*exp(2*x)*exp(x/(x + 1))))
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sympy [A] time = 1.45, size = 44, normalized size = 1.26 \begin {gather*} \frac {4}{4 x + 3 e^{\left (3 x^{3} + 6 x^{2} e^{\frac {x}{2 x + 2}} + 3 x e^{\frac {2 x}{2 x + 2}}\right ) e^{2 x}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((((-72*x**3-180*x**2-180*x-36)*exp(x/(2*x+2))**2+(-144*x**4-432*x**3-468*x**2-144*x)*exp(x/(2*x+2))-
72*x**5-252*x**4-288*x**3-108*x**2)*exp(x)**2*exp((3*x*exp(x/(2*x+2))**2+6*x**2*exp(x/(2*x+2))+3*x**3)*exp(x)*
*2)-16*x**2-32*x-16)/((9*x**2+18*x+9)*exp((3*x*exp(x/(2*x+2))**2+6*x**2*exp(x/(2*x+2))+3*x**3)*exp(x)**2)**2+(
24*x**3+48*x**2+24*x)*exp((3*x*exp(x/(2*x+2))**2+6*x**2*exp(x/(2*x+2))+3*x**3)*exp(x)**2)+16*x**4+32*x**3+16*x
**2),x)
[Out]
4/(4*x + 3*exp((3*x**3 + 6*x**2*exp(x/(2*x + 2)) + 3*x*exp(2*x/(2*x + 2)))*exp(2*x)))
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