3.50.93 \(\int \frac {40 x^3+42 x^4-2 x^6+e^{3 x} (2 x+6 x^2+2 x^3)+e^{2 x} (20 x+38 x^2-32 x^3-16 x^4)+e^x (50 x+40 x^2-182 x^3-24 x^4+16 x^5)+(-8 x^3-10 x^4-2 x^5+e^{2 x} (-4 x-10 x^2+2 x^3+2 x^4)+e^x (-20 x-28 x^2+60 x^3+20 x^4-2 x^5)) \log (9+27 x+9 x^2)+e^x (2 x+4 x^2-4 x^3-2 x^4) \log ^2(9+27 x+9 x^2)}{-x^3-3 x^4-x^5+e^{3 x} (1+3 x+x^2)+e^{2 x} (-3 x-9 x^2-3 x^3)+e^x (3 x^2+9 x^3+3 x^4)} \, dx\)
Optimal. Leaf size=32 \[ \left (x+x \left (-2+\frac {5-\log \left (9 \left (x+(1+x)^2\right )\right )}{-e^x+x}\right )\right )^2 \]
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Rubi [F] time = 51.15, 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 {40 x^3+42 x^4-2 x^6+e^{3 x} \left (2 x+6 x^2+2 x^3\right )+e^{2 x} \left (20 x+38 x^2-32 x^3-16 x^4\right )+e^x \left (50 x+40 x^2-182 x^3-24 x^4+16 x^5\right )+\left (-8 x^3-10 x^4-2 x^5+e^{2 x} \left (-4 x-10 x^2+2 x^3+2 x^4\right )+e^x \left (-20 x-28 x^2+60 x^3+20 x^4-2 x^5\right )\right ) \log \left (9+27 x+9 x^2\right )+e^x \left (2 x+4 x^2-4 x^3-2 x^4\right ) \log ^2\left (9+27 x+9 x^2\right )}{-x^3-3 x^4-x^5+e^{3 x} \left (1+3 x+x^2\right )+e^{2 x} \left (-3 x-9 x^2-3 x^3\right )+e^x \left (3 x^2+9 x^3+3 x^4\right )} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
Int[(40*x^3 + 42*x^4 - 2*x^6 + E^(3*x)*(2*x + 6*x^2 + 2*x^3) + E^(2*x)*(20*x + 38*x^2 - 32*x^3 - 16*x^4) + E^x
*(50*x + 40*x^2 - 182*x^3 - 24*x^4 + 16*x^5) + (-8*x^3 - 10*x^4 - 2*x^5 + E^(2*x)*(-4*x - 10*x^2 + 2*x^3 + 2*x
^4) + E^x*(-20*x - 28*x^2 + 60*x^3 + 20*x^4 - 2*x^5))*Log[9 + 27*x + 9*x^2] + E^x*(2*x + 4*x^2 - 4*x^3 - 2*x^4
)*Log[9 + 27*x + 9*x^2]^2)/(-x^3 - 3*x^4 - x^5 + E^(3*x)*(1 + 3*x + x^2) + E^(2*x)*(-3*x - 9*x^2 - 3*x^3) + E^
x*(3*x^2 + 9*x^3 + 3*x^4)),x]
[Out]
x^2 + 30*Defer[Int][(E^x - x)^(-2), x] - 6*Log[9*(1 + 3*x + x^2)]*Defer[Int][(E^x - x)^(-2), x] + 12*Sqrt[5]*D
efer[Int][1/((-3 + Sqrt[5] - 2*x)*(E^x - x)^2), x] + (228*Log[9*(1 + 3*x + x^2)]*Defer[Int][1/((-3 + Sqrt[5] -
2*x)*(E^x - x)^2), x])/Sqrt[5] - 48*Sqrt[5]*Log[9*(1 + 3*x + x^2)]*Defer[Int][1/((-3 + Sqrt[5] - 2*x)*(E^x -
x)^2), x] + 6*Defer[Int][(E^x - x)^(-1), x] + (172*Defer[Int][1/((-3 + Sqrt[5] - 2*x)*(E^x - x)), x])/Sqrt[5]
- 32*Sqrt[5]*Defer[Int][1/((-3 + Sqrt[5] - 2*x)*(E^x - x)), x] + 30*Defer[Int][x/(E^x - x)^2, x] - 16*Log[9*(1
+ 3*x + x^2)]*Defer[Int][x/(E^x - x)^2, x] + 16*Defer[Int][x/(E^x - x), x] - 4*Log[9*(1 + 3*x + x^2)]*Defer[I
nt][x/(E^x - x), x] + 50*Defer[Int][x^2/(E^x - x)^3, x] - 20*Log[9*(1 + 3*x + x^2)]*Defer[Int][x^2/(E^x - x)^3
, x] - 40*Defer[Int][x^2/(E^x - x)^2, x] + 18*Log[9*(1 + 3*x + x^2)]*Defer[Int][x^2/(E^x - x)^2, x] - 10*Defer
[Int][x^2/(E^x - x), x] + 2*Log[9*(1 + 3*x + x^2)]*Defer[Int][x^2/(E^x - x), x] - 50*Defer[Int][x^3/(E^x - x)^
3, x] + 20*Log[9*(1 + 3*x + x^2)]*Defer[Int][x^3/(E^x - x)^3, x] - 10*Defer[Int][x^3/(E^x - x)^2, x] + 2*Log[9
*(1 + 3*x + x^2)]*Defer[Int][x^3/(E^x - x)^2, x] - 14*(5 - 3*Sqrt[5])*Defer[Int][1/((E^x - x)^2*(3 - Sqrt[5] +
2*x)), x] + (14*(5 - 3*Sqrt[5])*Log[9*(1 + 3*x + x^2)]*Defer[Int][1/((E^x - x)^2*(3 - Sqrt[5] + 2*x)), x])/5
- (14*(5 - 3*Sqrt[5])*Defer[Int][1/((E^x - x)*(3 - Sqrt[5] + 2*x)), x])/5 + 12*Sqrt[5]*Defer[Int][1/((E^x - x)
^2*(3 + Sqrt[5] + 2*x)), x] - 14*(5 + 3*Sqrt[5])*Defer[Int][1/((E^x - x)^2*(3 + Sqrt[5] + 2*x)), x] + (228*Log
[9*(1 + 3*x + x^2)]*Defer[Int][1/((E^x - x)^2*(3 + Sqrt[5] + 2*x)), x])/Sqrt[5] - 48*Sqrt[5]*Log[9*(1 + 3*x +
x^2)]*Defer[Int][1/((E^x - x)^2*(3 + Sqrt[5] + 2*x)), x] + (14*(5 + 3*Sqrt[5])*Log[9*(1 + 3*x + x^2)]*Defer[In
t][1/((E^x - x)^2*(3 + Sqrt[5] + 2*x)), x])/5 + (172*Defer[Int][1/((E^x - x)*(3 + Sqrt[5] + 2*x)), x])/Sqrt[5]
- 32*Sqrt[5]*Defer[Int][1/((E^x - x)*(3 + Sqrt[5] + 2*x)), x] - (14*(5 + 3*Sqrt[5])*Defer[Int][1/((E^x - x)*(
3 + Sqrt[5] + 2*x)), x])/5 + 2*Defer[Int][(x*Log[9*(1 + 3*x + x^2)]^2)/(E^x - x)^2, x] + 2*Defer[Int][(x^2*Log
[9*(1 + 3*x + x^2)]^2)/(E^x - x)^3, x] - 2*Defer[Int][(x^2*Log[9*(1 + 3*x + x^2)]^2)/(E^x - x)^2, x] - 2*Defer
[Int][(x^3*Log[9*(1 + 3*x + x^2)]^2)/(E^x - x)^3, x] + (684*Defer[Int][Defer[Int][(E^x - x)^(-2), x]/(-3 + Sqr
t[5] - 2*x), x])/Sqrt[5] - 144*Sqrt[5]*Defer[Int][Defer[Int][(E^x - x)^(-2), x]/(-3 + Sqrt[5] - 2*x), x] + (12
*(5 - 3*Sqrt[5])*Defer[Int][Defer[Int][(E^x - x)^(-2), x]/(3 - Sqrt[5] + 2*x), x])/5 + (684*Defer[Int][Defer[I
nt][(E^x - x)^(-2), x]/(3 + Sqrt[5] + 2*x), x])/Sqrt[5] - 144*Sqrt[5]*Defer[Int][Defer[Int][(E^x - x)^(-2), x]
/(3 + Sqrt[5] + 2*x), x] + (12*(5 + 3*Sqrt[5])*Defer[Int][Defer[Int][(E^x - x)^(-2), x]/(3 + Sqrt[5] + 2*x), x
])/5 - (72*Defer[Int][Defer[Int][1/((-3 + Sqrt[5] - 2*x)*(E^x - x)^2), x]/(-3 + Sqrt[5] - 2*x), x])/5 - (24*(3
- Sqrt[5])*Defer[Int][Defer[Int][1/((-3 + Sqrt[5] - 2*x)*(E^x - x)^2), x]/(3 - Sqrt[5] + 2*x), x])/5 - (72*De
fer[Int][Defer[Int][1/((-3 + Sqrt[5] - 2*x)*(E^x - x)^2), x]/(3 + Sqrt[5] + 2*x), x])/5 + (24*(3 + Sqrt[5])*De
fer[Int][Defer[Int][1/((-3 + Sqrt[5] - 2*x)*(E^x - x)^2), x]/(3 + Sqrt[5] + 2*x), x])/5 - (336*Defer[Int][Defe
r[Int][x/(E^x - x)^2, x]/(-3 + Sqrt[5] - 2*x), x])/Sqrt[5] + 48*Sqrt[5]*Defer[Int][Defer[Int][x/(E^x - x)^2, x
]/(-3 + Sqrt[5] - 2*x), x] + (32*(5 - 3*Sqrt[5])*Defer[Int][Defer[Int][x/(E^x - x)^2, x]/(3 - Sqrt[5] + 2*x),
x])/5 - (336*Defer[Int][Defer[Int][x/(E^x - x)^2, x]/(3 + Sqrt[5] + 2*x), x])/Sqrt[5] + 48*Sqrt[5]*Defer[Int][
Defer[Int][x/(E^x - x)^2, x]/(3 + Sqrt[5] + 2*x), x] + (32*(5 + 3*Sqrt[5])*Defer[Int][Defer[Int][x/(E^x - x)^2
, x]/(3 + Sqrt[5] + 2*x), x])/5 - (24*Defer[Int][Defer[Int][x/(E^x - x), x]/(-3 + Sqrt[5] - 2*x), x])/Sqrt[5]
+ (8*(5 - 3*Sqrt[5])*Defer[Int][Defer[Int][x/(E^x - x), x]/(3 - Sqrt[5] + 2*x), x])/5 - (24*Defer[Int][Defer[I
nt][x/(E^x - x), x]/(3 + Sqrt[5] + 2*x), x])/Sqrt[5] + (8*(5 + 3*Sqrt[5])*Defer[Int][Defer[Int][x/(E^x - x), x
]/(3 + Sqrt[5] + 2*x), x])/5 - 24*Sqrt[5]*Defer[Int][Defer[Int][x^2/(E^x - x)^3, x]/(-3 + Sqrt[5] - 2*x), x] +
8*(5 - 3*Sqrt[5])*Defer[Int][Defer[Int][x^2/(E^x - x)^3, x]/(3 - Sqrt[5] + 2*x), x] - 24*Sqrt[5]*Defer[Int][D
efer[Int][x^2/(E^x - x)^3, x]/(3 + Sqrt[5] + 2*x), x] + 8*(5 + 3*Sqrt[5])*Defer[Int][Defer[Int][x^2/(E^x - x)^
3, x]/(3 + Sqrt[5] + 2*x), x] + (108*Defer[Int][Defer[Int][x^2/(E^x - x)^2, x]/(-3 + Sqrt[5] - 2*x), x])/Sqrt[
5] - (36*(5 - 3*Sqrt[5])*Defer[Int][Defer[Int][x^2/(E^x - x)^2, x]/(3 - Sqrt[5] + 2*x), x])/5 + (108*Defer[Int
][Defer[Int][x^2/(E^x - x)^2, x]/(3 + Sqrt[5] + 2*x), x])/Sqrt[5] - (36*(5 + 3*Sqrt[5])*Defer[Int][Defer[Int][
x^2/(E^x - x)^2, x]/(3 + Sqrt[5] + 2*x), x])/5 + (12*Defer[Int][Defer[Int][x^2/(E^x - x), x]/(-3 + Sqrt[5] - 2
*x), x])/Sqrt[5] - (4*(5 - 3*Sqrt[5])*Defer[Int][Defer[Int][x^2/(E^x - x), x]/(3 - Sqrt[5] + 2*x), x])/5 + (12
*Defer[Int][Defer[Int][x^2/(E^x - x), x]/(3 + Sqrt[5] + 2*x), x])/Sqrt[5] - (4*(5 + 3*Sqrt[5])*Defer[Int][Defe
r[Int][x^2/(E^x - x), x]/(3 + Sqrt[5] + 2*x), x])/5 + 24*Sqrt[5]*Defer[Int][Defer[Int][x^3/(E^x - x)^3, x]/(-3
+ Sqrt[5] - 2*x), x] - 8*(5 - 3*Sqrt[5])*Defer[Int][Defer[Int][x^3/(E^x - x)^3, x]/(3 - Sqrt[5] + 2*x), x] +
24*Sqrt[5]*Defer[Int][Defer[Int][x^3/(E^x - x)^3, x]/(3 + Sqrt[5] + 2*x), x] - 8*(5 + 3*Sqrt[5])*Defer[Int][De
fer[Int][x^3/(E^x - x)^3, x]/(3 + Sqrt[5] + 2*x), x] + (12*Defer[Int][Defer[Int][x^3/(E^x - x)^2, x]/(-3 + Sqr
t[5] - 2*x), x])/Sqrt[5] - (4*(5 - 3*Sqrt[5])*Defer[Int][Defer[Int][x^3/(E^x - x)^2, x]/(3 - Sqrt[5] + 2*x), x
])/5 + (12*Defer[Int][Defer[Int][x^3/(E^x - x)^2, x]/(3 + Sqrt[5] + 2*x), x])/Sqrt[5] - (4*(5 + 3*Sqrt[5])*Def
er[Int][Defer[Int][x^3/(E^x - x)^2, x]/(3 + Sqrt[5] + 2*x), x])/5 - (84*(3 - Sqrt[5])*Defer[Int][Defer[Int][1/
((E^x - x)^2*(3 - Sqrt[5] + 2*x)), x]/(-3 + Sqrt[5] - 2*x), x])/5 - (56*(7 - 3*Sqrt[5])*Defer[Int][Defer[Int][
1/((E^x - x)^2*(3 - Sqrt[5] + 2*x)), x]/(3 - Sqrt[5] + 2*x), x])/5 + (112*Defer[Int][Defer[Int][1/((E^x - x)^2
*(3 - Sqrt[5] + 2*x)), x]/(3 + Sqrt[5] + 2*x), x])/5 - (84*(3 - Sqrt[5])*Defer[Int][Defer[Int][1/((E^x - x)^2*
(3 - Sqrt[5] + 2*x)), x]/(3 + Sqrt[5] + 2*x), x])/5 - 24*(3 + Sqrt[5])*Defer[Int][Defer[Int][1/((E^x - x)^2*(3
+ Sqrt[5] + 2*x)), x]/(-3 + Sqrt[5] - 2*x), x] + (216*(7 + 3*Sqrt[5])*Defer[Int][Defer[Int][1/((E^x - x)^2*(3
+ Sqrt[5] + 2*x)), x]/(-3 + Sqrt[5] - 2*x), x])/5 + 96*(9 + 4*Sqrt[5])*Defer[Int][Defer[Int][1/((E^x - x)^2*(
3 + Sqrt[5] + 2*x)), x]/(-3 + Sqrt[5] - 2*x), x] - (144*(47 + 21*Sqrt[5])*Defer[Int][Defer[Int][1/((E^x - x)^2
*(3 + Sqrt[5] + 2*x)), x]/(-3 + Sqrt[5] - 2*x), x])/5 + (12*(123 + 55*Sqrt[5])*Defer[Int][Defer[Int][1/((E^x -
x)^2*(3 + Sqrt[5] + 2*x)), x]/(-3 + Sqrt[5] - 2*x), x])/5 - 32*Defer[Int][Defer[Int][1/((E^x - x)^2*(3 + Sqrt
[5] + 2*x)), x]/(3 - Sqrt[5] + 2*x), x] + (144*(3 + Sqrt[5])*Defer[Int][Defer[Int][1/((E^x - x)^2*(3 + Sqrt[5]
+ 2*x)), x]/(3 - Sqrt[5] + 2*x), x])/5 + 32*(7 + 3*Sqrt[5])*Defer[Int][Defer[Int][1/((E^x - x)^2*(3 + Sqrt[5]
+ 2*x)), x]/(3 - Sqrt[5] + 2*x), x] - (192*(9 + 4*Sqrt[5])*Defer[Int][Defer[Int][1/((E^x - x)^2*(3 + Sqrt[5]
+ 2*x)), x]/(3 - Sqrt[5] + 2*x), x])/5 + (8*(47 + 21*Sqrt[5])*Defer[Int][Defer[Int][1/((E^x - x)^2*(3 + Sqrt[5
] + 2*x)), x]/(3 - Sqrt[5] + 2*x), x])/5 - 24*(3 + Sqrt[5])*Defer[Int][Defer[Int][1/((E^x - x)^2*(3 + Sqrt[5]
+ 2*x)), x]/(3 + Sqrt[5] + 2*x), x] + (296*(7 + 3*Sqrt[5])*Defer[Int][Defer[Int][1/((E^x - x)^2*(3 + Sqrt[5] +
2*x)), x]/(3 + Sqrt[5] + 2*x), x])/5 + (192*(9 + 4*Sqrt[5])*Defer[Int][Defer[Int][1/((E^x - x)^2*(3 + Sqrt[5]
+ 2*x)), x]/(3 + Sqrt[5] + 2*x), x])/5 - (304*(47 + 21*Sqrt[5])*Defer[Int][Defer[Int][1/((E^x - x)^2*(3 + Sqr
t[5] + 2*x)), x]/(3 + Sqrt[5] + 2*x), x])/5 + (108*(123 + 55*Sqrt[5])*Defer[Int][Defer[Int][1/((E^x - x)^2*(3
+ Sqrt[5] + 2*x)), x]/(3 + Sqrt[5] + 2*x), x])/5 - (16*(161 + 72*Sqrt[5])*Defer[Int][Defer[Int][1/((E^x - x)^2
*(3 + Sqrt[5] + 2*x)), x]/(3 + Sqrt[5] + 2*x), x])/5 - (24*(3 + Sqrt[5])*Defer[Int][Defer[Int][1/((E^x - x)*(3
+ Sqrt[5] + 2*x)), x]/(-3 + Sqrt[5] - 2*x), x])/5 + 12*(7 + 3*Sqrt[5])*Defer[Int][Defer[Int][1/((E^x - x)*(3
+ Sqrt[5] + 2*x)), x]/(-3 + Sqrt[5] - 2*x), x] + (24*(9 + 4*Sqrt[5])*Defer[Int][Defer[Int][1/((E^x - x)*(3 + S
qrt[5] + 2*x)), x]/(-3 + Sqrt[5] - 2*x), x])/5 - (12*(47 + 21*Sqrt[5])*Defer[Int][Defer[Int][1/((E^x - x)*(3 +
Sqrt[5] + 2*x)), x]/(-3 + Sqrt[5] - 2*x), x])/5 - (32*Defer[Int][Defer[Int][1/((E^x - x)*(3 + Sqrt[5] + 2*x))
, x]/(3 - Sqrt[5] + 2*x), x])/5 + 8*(3 + Sqrt[5])*Defer[Int][Defer[Int][1/((E^x - x)*(3 + Sqrt[5] + 2*x)), x]/
(3 - Sqrt[5] + 2*x), x] + (8*(7 + 3*Sqrt[5])*Defer[Int][Defer[Int][1/((E^x - x)*(3 + Sqrt[5] + 2*x)), x]/(3 -
Sqrt[5] + 2*x), x])/5 - (16*(9 + 4*Sqrt[5])*Defer[Int][Defer[Int][1/((E^x - x)*(3 + Sqrt[5] + 2*x)), x]/(3 - S
qrt[5] + 2*x), x])/5 - (24*(3 + Sqrt[5])*Defer[Int][Defer[Int][1/((E^x - x)*(3 + Sqrt[5] + 2*x)), x]/(3 + Sqrt
[5] + 2*x), x])/5 + (76*(7 + 3*Sqrt[5])*Defer[Int][Defer[Int][1/((E^x - x)*(3 + Sqrt[5] + 2*x)), x]/(3 + Sqrt[
5] + 2*x), x])/5 - (56*(9 + 4*Sqrt[5])*Defer[Int][Defer[Int][1/((E^x - x)*(3 + Sqrt[5] + 2*x)), x]/(3 + Sqrt[5
] + 2*x), x])/5 - 4*(47 + 21*Sqrt[5])*Defer[Int][Defer[Int][1/((E^x - x)*(3 + Sqrt[5] + 2*x)), x]/(3 + Sqrt[5]
+ 2*x), x] + (8*(123 + 55*Sqrt[5])*Defer[Int][Defer[Int][1/((E^x - x)*(3 + Sqrt[5] + 2*x)), x]/(3 + Sqrt[5] +
2*x), x])/5
Rubi steps
Aborted
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Mathematica [A] time = 0.21, size = 34, normalized size = 1.06 \begin {gather*} \frac {x^2 \left (-5-e^x+x+\log \left (9 \left (1+3 x+x^2\right )\right )\right )^2}{\left (e^x-x\right )^2} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[(40*x^3 + 42*x^4 - 2*x^6 + E^(3*x)*(2*x + 6*x^2 + 2*x^3) + E^(2*x)*(20*x + 38*x^2 - 32*x^3 - 16*x^4)
+ E^x*(50*x + 40*x^2 - 182*x^3 - 24*x^4 + 16*x^5) + (-8*x^3 - 10*x^4 - 2*x^5 + E^(2*x)*(-4*x - 10*x^2 + 2*x^3
+ 2*x^4) + E^x*(-20*x - 28*x^2 + 60*x^3 + 20*x^4 - 2*x^5))*Log[9 + 27*x + 9*x^2] + E^x*(2*x + 4*x^2 - 4*x^3 -
2*x^4)*Log[9 + 27*x + 9*x^2]^2)/(-x^3 - 3*x^4 - x^5 + E^(3*x)*(1 + 3*x + x^2) + E^(2*x)*(-3*x - 9*x^2 - 3*x^3
) + E^x*(3*x^2 + 9*x^3 + 3*x^4)),x]
[Out]
(x^2*(-5 - E^x + x + Log[9*(1 + 3*x + x^2)])^2)/(E^x - x)^2
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fricas [B] time = 0.75, size = 97, normalized size = 3.03 \begin {gather*} \frac {x^{4} + x^{2} \log \left (9 \, x^{2} + 27 \, x + 9\right )^{2} - 10 \, x^{3} + x^{2} e^{\left (2 \, x\right )} + 25 \, x^{2} - 2 \, {\left (x^{3} - 5 \, x^{2}\right )} e^{x} + 2 \, {\left (x^{3} - x^{2} e^{x} - 5 \, x^{2}\right )} \log \left (9 \, x^{2} + 27 \, x + 9\right )}{x^{2} - 2 \, x e^{x} + e^{\left (2 \, x\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(((-2*x^4-4*x^3+4*x^2+2*x)*exp(x)*log(9*x^2+27*x+9)^2+((2*x^4+2*x^3-10*x^2-4*x)*exp(x)^2+(-2*x^5+20*x
^4+60*x^3-28*x^2-20*x)*exp(x)-2*x^5-10*x^4-8*x^3)*log(9*x^2+27*x+9)+(2*x^3+6*x^2+2*x)*exp(x)^3+(-16*x^4-32*x^3
+38*x^2+20*x)*exp(x)^2+(16*x^5-24*x^4-182*x^3+40*x^2+50*x)*exp(x)-2*x^6+42*x^4+40*x^3)/((x^2+3*x+1)*exp(x)^3+(
-3*x^3-9*x^2-3*x)*exp(x)^2+(3*x^4+9*x^3+3*x^2)*exp(x)-x^5-3*x^4-x^3),x, algorithm="fricas")
[Out]
(x^4 + x^2*log(9*x^2 + 27*x + 9)^2 - 10*x^3 + x^2*e^(2*x) + 25*x^2 - 2*(x^3 - 5*x^2)*e^x + 2*(x^3 - x^2*e^x -
5*x^2)*log(9*x^2 + 27*x + 9))/(x^2 - 2*x*e^x + e^(2*x))
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giac [B] time = 0.27, size = 159, normalized size = 4.97 \begin {gather*} \frac {x^{4} - 2 \, x^{3} e^{x} + 4 \, x^{3} \log \relax (3) - 4 \, x^{2} e^{x} \log \relax (3) + 4 \, x^{2} \log \relax (3)^{2} + 2 \, x^{3} \log \left (x^{2} + 3 \, x + 1\right ) - 2 \, x^{2} e^{x} \log \left (x^{2} + 3 \, x + 1\right ) + 4 \, x^{2} \log \relax (3) \log \left (x^{2} + 3 \, x + 1\right ) + x^{2} \log \left (x^{2} + 3 \, x + 1\right )^{2} - 10 \, x^{3} + x^{2} e^{\left (2 \, x\right )} + 10 \, x^{2} e^{x} - 20 \, x^{2} \log \relax (3) - 10 \, x^{2} \log \left (x^{2} + 3 \, x + 1\right ) + 25 \, x^{2}}{x^{2} - 2 \, x e^{x} + e^{\left (2 \, x\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(((-2*x^4-4*x^3+4*x^2+2*x)*exp(x)*log(9*x^2+27*x+9)^2+((2*x^4+2*x^3-10*x^2-4*x)*exp(x)^2+(-2*x^5+20*x
^4+60*x^3-28*x^2-20*x)*exp(x)-2*x^5-10*x^4-8*x^3)*log(9*x^2+27*x+9)+(2*x^3+6*x^2+2*x)*exp(x)^3+(-16*x^4-32*x^3
+38*x^2+20*x)*exp(x)^2+(16*x^5-24*x^4-182*x^3+40*x^2+50*x)*exp(x)-2*x^6+42*x^4+40*x^3)/((x^2+3*x+1)*exp(x)^3+(
-3*x^3-9*x^2-3*x)*exp(x)^2+(3*x^4+9*x^3+3*x^2)*exp(x)-x^5-3*x^4-x^3),x, algorithm="giac")
[Out]
(x^4 - 2*x^3*e^x + 4*x^3*log(3) - 4*x^2*e^x*log(3) + 4*x^2*log(3)^2 + 2*x^3*log(x^2 + 3*x + 1) - 2*x^2*e^x*log
(x^2 + 3*x + 1) + 4*x^2*log(3)*log(x^2 + 3*x + 1) + x^2*log(x^2 + 3*x + 1)^2 - 10*x^3 + x^2*e^(2*x) + 10*x^2*e
^x - 20*x^2*log(3) - 10*x^2*log(x^2 + 3*x + 1) + 25*x^2)/(x^2 - 2*x*e^x + e^(2*x))
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maple [B] time = 0.08, size = 91, normalized size = 2.84
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\(\frac {x^{2} \ln \left (9 x^{2}+27 x +9\right )^{2}}{\left (x -{\mathrm e}^{x}\right )^{2}}+\frac {2 x^{2} \left (-{\mathrm e}^{x}+x -5\right ) \ln \left (9 x^{2}+27 x +9\right )}{\left (x -{\mathrm e}^{x}\right )^{2}}+\frac {x^{2} \left (x^{2}-2 \,{\mathrm e}^{x} x +{\mathrm e}^{2 x}-10 x +10 \,{\mathrm e}^{x}+25\right )}{\left (x -{\mathrm e}^{x}\right )^{2}}\) |
\(91\) |
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Verification of antiderivative is not currently implemented for this CAS.
[In]
int(((-2*x^4-4*x^3+4*x^2+2*x)*exp(x)*ln(9*x^2+27*x+9)^2+((2*x^4+2*x^3-10*x^2-4*x)*exp(x)^2+(-2*x^5+20*x^4+60*x
^3-28*x^2-20*x)*exp(x)-2*x^5-10*x^4-8*x^3)*ln(9*x^2+27*x+9)+(2*x^3+6*x^2+2*x)*exp(x)^3+(-16*x^4-32*x^3+38*x^2+
20*x)*exp(x)^2+(16*x^5-24*x^4-182*x^3+40*x^2+50*x)*exp(x)-2*x^6+42*x^4+40*x^3)/((x^2+3*x+1)*exp(x)^3+(-3*x^3-9
*x^2-3*x)*exp(x)^2+(3*x^4+9*x^3+3*x^2)*exp(x)-x^5-3*x^4-x^3),x,method=_RETURNVERBOSE)
[Out]
x^2/(x-exp(x))^2*ln(9*x^2+27*x+9)^2+2*x^2*(-exp(x)+x-5)/(x-exp(x))^2*ln(9*x^2+27*x+9)+x^2*(x^2-2*exp(x)*x+exp(
2*x)-10*x+10*exp(x)+25)/(x-exp(x))^2
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maxima [B] time = 0.55, size = 120, normalized size = 3.75 \begin {gather*} \frac {x^{4} + 2 \, x^{3} {\left (2 \, \log \relax (3) - 5\right )} + x^{2} \log \left (x^{2} + 3 \, x + 1\right )^{2} + {\left (4 \, \log \relax (3)^{2} - 20 \, \log \relax (3) + 25\right )} x^{2} + x^{2} e^{\left (2 \, x\right )} - 2 \, {\left (x^{3} + x^{2} {\left (2 \, \log \relax (3) - 5\right )}\right )} e^{x} + 2 \, {\left (x^{3} + x^{2} {\left (2 \, \log \relax (3) - 5\right )} - x^{2} e^{x}\right )} \log \left (x^{2} + 3 \, x + 1\right )}{x^{2} - 2 \, x e^{x} + e^{\left (2 \, x\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(((-2*x^4-4*x^3+4*x^2+2*x)*exp(x)*log(9*x^2+27*x+9)^2+((2*x^4+2*x^3-10*x^2-4*x)*exp(x)^2+(-2*x^5+20*x
^4+60*x^3-28*x^2-20*x)*exp(x)-2*x^5-10*x^4-8*x^3)*log(9*x^2+27*x+9)+(2*x^3+6*x^2+2*x)*exp(x)^3+(-16*x^4-32*x^3
+38*x^2+20*x)*exp(x)^2+(16*x^5-24*x^4-182*x^3+40*x^2+50*x)*exp(x)-2*x^6+42*x^4+40*x^3)/((x^2+3*x+1)*exp(x)^3+(
-3*x^3-9*x^2-3*x)*exp(x)^2+(3*x^4+9*x^3+3*x^2)*exp(x)-x^5-3*x^4-x^3),x, algorithm="maxima")
[Out]
(x^4 + 2*x^3*(2*log(3) - 5) + x^2*log(x^2 + 3*x + 1)^2 + (4*log(3)^2 - 20*log(3) + 25)*x^2 + x^2*e^(2*x) - 2*(
x^3 + x^2*(2*log(3) - 5))*e^x + 2*(x^3 + x^2*(2*log(3) - 5) - x^2*e^x)*log(x^2 + 3*x + 1))/(x^2 - 2*x*e^x + e^
(2*x))
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mupad [B] time = 3.68, size = 137, normalized size = 4.28 \begin {gather*} x^2+\frac {10\,\left (x^2-x^3\right )}{\left (x-{\mathrm {e}}^x\right )\,\left (x-1\right )}-\frac {25\,\left (x^2-x^3\right )}{\left (x-1\right )\,\left ({\mathrm {e}}^{2\,x}-2\,x\,{\mathrm {e}}^x+x^2\right )}-\frac {\ln \left (9\,x^2+27\,x+9\right )\,\left (2\,x^2\,{\mathrm {e}}^x+10\,x^2-2\,x^3\right )}{{\mathrm {e}}^{2\,x}-2\,x\,{\mathrm {e}}^x+x^2}+\frac {x^2\,{\ln \left (9\,x^2+27\,x+9\right )}^2}{{\mathrm {e}}^{2\,x}-2\,x\,{\mathrm {e}}^x+x^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(-(exp(3*x)*(2*x + 6*x^2 + 2*x^3) - log(27*x + 9*x^2 + 9)*(exp(x)*(20*x + 28*x^2 - 60*x^3 - 20*x^4 + 2*x^5)
+ exp(2*x)*(4*x + 10*x^2 - 2*x^3 - 2*x^4) + 8*x^3 + 10*x^4 + 2*x^5) + exp(x)*(50*x + 40*x^2 - 182*x^3 - 24*x^
4 + 16*x^5) + exp(2*x)*(20*x + 38*x^2 - 32*x^3 - 16*x^4) + 40*x^3 + 42*x^4 - 2*x^6 + exp(x)*log(27*x + 9*x^2 +
9)^2*(2*x + 4*x^2 - 4*x^3 - 2*x^4))/(exp(2*x)*(3*x + 9*x^2 + 3*x^3) - exp(x)*(3*x^2 + 9*x^3 + 3*x^4) - exp(3*
x)*(3*x + x^2 + 1) + x^3 + 3*x^4 + x^5),x)
[Out]
x^2 + (10*(x^2 - x^3))/((x - exp(x))*(x - 1)) - (25*(x^2 - x^3))/((x - 1)*(exp(2*x) - 2*x*exp(x) + x^2)) - (lo
g(27*x + 9*x^2 + 9)*(2*x^2*exp(x) + 10*x^2 - 2*x^3))/(exp(2*x) - 2*x*exp(x) + x^2) + (x^2*log(27*x + 9*x^2 + 9
)^2)/(exp(2*x) - 2*x*exp(x) + x^2)
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sympy [B] time = 0.51, size = 104, normalized size = 3.25 \begin {gather*} x^{2} + \frac {2 x^{3} \log {\left (9 x^{2} + 27 x + 9 \right )} - 10 x^{3} + x^{2} \log {\left (9 x^{2} + 27 x + 9 \right )}^{2} - 10 x^{2} \log {\left (9 x^{2} + 27 x + 9 \right )} + 25 x^{2} + \left (- 2 x^{2} \log {\left (9 x^{2} + 27 x + 9 \right )} + 10 x^{2}\right ) e^{x}}{x^{2} - 2 x e^{x} + e^{2 x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(((-2*x**4-4*x**3+4*x**2+2*x)*exp(x)*ln(9*x**2+27*x+9)**2+((2*x**4+2*x**3-10*x**2-4*x)*exp(x)**2+(-2*
x**5+20*x**4+60*x**3-28*x**2-20*x)*exp(x)-2*x**5-10*x**4-8*x**3)*ln(9*x**2+27*x+9)+(2*x**3+6*x**2+2*x)*exp(x)*
*3+(-16*x**4-32*x**3+38*x**2+20*x)*exp(x)**2+(16*x**5-24*x**4-182*x**3+40*x**2+50*x)*exp(x)-2*x**6+42*x**4+40*
x**3)/((x**2+3*x+1)*exp(x)**3+(-3*x**3-9*x**2-3*x)*exp(x)**2+(3*x**4+9*x**3+3*x**2)*exp(x)-x**5-3*x**4-x**3),x
)
[Out]
x**2 + (2*x**3*log(9*x**2 + 27*x + 9) - 10*x**3 + x**2*log(9*x**2 + 27*x + 9)**2 - 10*x**2*log(9*x**2 + 27*x +
9) + 25*x**2 + (-2*x**2*log(9*x**2 + 27*x + 9) + 10*x**2)*exp(x))/(x**2 - 2*x*exp(x) + exp(2*x))
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