3.45.28 \(\int \frac {200+150 x-75 e x-100 x^2+e^{18-12 e^x+2 e^{2 x}} (8+6 x-3 e x-4 x^2+e^x (48 x+24 x^2-12 e x^2-12 x^3)+e^{2 x} (-16 x-8 x^2+4 e x^2+4 x^3))+e^{9-6 e^x+e^{2 x}} (-80-60 x+30 e x+40 x^2+e^{2 x} (80 x+40 x^2-20 e x^2-20 x^3)+e^x (-240 x-120 x^2+60 e x^2+60 x^3))}{16 x^3+16 x^4-4 x^5+e^2 x^5-4 x^6+x^7+e (-8 x^4-4 x^5+2 x^6)} \, dx\)
Optimal. Leaf size=28 \[ \frac {\left (-5+e^{\left (-3+e^x\right )^2}\right )^2}{x^3 \left (-2+e-\frac {4}{x}+x\right )} \]
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Rubi [F] time = 32.81, 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 {200+150 x-75 e x-100 x^2+e^{18-12 e^x+2 e^{2 x}} \left (8+6 x-3 e x-4 x^2+e^x \left (48 x+24 x^2-12 e x^2-12 x^3\right )+e^{2 x} \left (-16 x-8 x^2+4 e x^2+4 x^3\right )\right )+e^{9-6 e^x+e^{2 x}} \left (-80-60 x+30 e x+40 x^2+e^{2 x} \left (80 x+40 x^2-20 e x^2-20 x^3\right )+e^x \left (-240 x-120 x^2+60 e x^2+60 x^3\right )\right )}{16 x^3+16 x^4-4 x^5+e^2 x^5-4 x^6+x^7+e \left (-8 x^4-4 x^5+2 x^6\right )} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
Int[(200 + 150*x - 75*E*x - 100*x^2 + E^(18 - 12*E^x + 2*E^(2*x))*(8 + 6*x - 3*E*x - 4*x^2 + E^x*(48*x + 24*x^
2 - 12*E*x^2 - 12*x^3) + E^(2*x)*(-16*x - 8*x^2 + 4*E*x^2 + 4*x^3)) + E^(9 - 6*E^x + E^(2*x))*(-80 - 60*x + 30
*E*x + 40*x^2 + E^(2*x)*(80*x + 40*x^2 - 20*E*x^2 - 20*x^3) + E^x*(-240*x - 120*x^2 + 60*E*x^2 + 60*x^3)))/(16
*x^3 + 16*x^4 - 4*x^5 + E^2*x^5 - 4*x^6 + x^7 + E*(-8*x^4 - 4*x^5 + 2*x^6)),x]
[Out]
(-25*(44 - 12*E + 3*E^2))/(4*(20 - 4*E + E^2)*x^2) - (75*(2 - E)*(16 - 4*E + E^2))/(8*(20 - 4*E + E^2)*x) + (2
5*(2 - E)*(56 - 12*E + 3*E^2))/(8*(20 - 4*E + E^2)*x) - (25*(12 - 4*E + E^2 - (2 - E)*x))/((20 - 4*E + E^2)*(4
+ (2 - E)*x - x^2)) + (50*(12 - 4*E + E^2 - (2 - E)*x))/((20 - 4*E + E^2)*x^2*(4 + (2 - E)*x - x^2)) + (75*(2
- E)*(12 - 4*E + E^2 - (2 - E)*x))/(4*(20 - 4*E + E^2)*x*(4 + (2 - E)*x - x^2)) - (25*(2 - E)*(28 - 4*E + E^2
)*ArcTanh[(2 - E - 2*x)/Sqrt[20 - 4*E + E^2]])/(4*(20 - 4*E + E^2)^(3/2)) - (75*(2 - E)*(208 - 128*E + 48*E^2
- 8*E^3 + E^4)*ArcTanh[(2 - E - 2*x)/Sqrt[20 - 4*E + E^2]])/(32*(20 - 4*E + E^2)^(3/2)) + (25*(2 - E)*(848 - 4
16*E + 152*E^2 - 24*E^3 + 3*E^4)*ArcTanh[(2 - E - 2*x)/Sqrt[20 - 4*E + E^2]])/(32*(20 - 4*E + E^2)^(3/2)) - (2
5*Log[x])/4 - (75*(2 - E)^2*Log[x])/32 + (25*(20 - 12*E + 3*E^2)*Log[x])/32 + (25*Log[4 + (2 - E)*x - x^2])/8
+ (75*(2 - E)^2*Log[4 + (2 - E)*x - x^2])/64 - (25*(20 - 12*E + 3*E^2)*Log[4 + (2 - E)*x - x^2])/64 + (5*(1 +
(2 - E)/Sqrt[20 - 4*E + E^2])*Defer[Int][E^(-3 + E^x)^2/(2 - E - Sqrt[20 - 4*E + E^2] - 2*x), x])/2 + (15*(2 -
E)*(2 - E + (8 - 4*E + E^2)/Sqrt[20 - 4*E + E^2])*Defer[Int][E^(-3 + E^x)^2/(2 - E - Sqrt[20 - 4*E + E^2] - 2
*x), x])/16 - (5*(20 - 12*E + 3*E^2 + (3*(24 - 20*E + 6*E^2 - E^3))/Sqrt[20 - 4*E + E^2])*Defer[Int][E^(-3 + E
^x)^2/(2 - E - Sqrt[20 - 4*E + E^2] - 2*x), x])/16 - ((1 + (2 - E)/Sqrt[20 - 4*E + E^2])*Defer[Int][E^(2*(-3 +
E^x)^2)/(2 - E - Sqrt[20 - 4*E + E^2] - 2*x), x])/4 - (3*(2 - E)*(2 - E + (8 - 4*E + E^2)/Sqrt[20 - 4*E + E^2
])*Defer[Int][E^(2*(-3 + E^x)^2)/(2 - E - Sqrt[20 - 4*E + E^2] - 2*x), x])/32 + ((20 - 12*E + 3*E^2 + (3*(24 -
20*E + 6*E^2 - E^3))/Sqrt[20 - 4*E + E^2])*Defer[Int][E^(2*(-3 + E^x)^2)/(2 - E - Sqrt[20 - 4*E + E^2] - 2*x)
, x])/32 + ((2 - E + (12 - 4*E + E^2)/Sqrt[20 - 4*E + E^2])*Defer[Int][E^(2*(9 - 6*E^x + E^(2*x) + x))/(2 - E
- Sqrt[20 - 4*E + E^2] - 2*x), x])/4 + (15*(2 - E + (12 - 4*E + E^2)/Sqrt[20 - 4*E + E^2])*Defer[Int][E^((-3 +
E^x)^2 + x)/(2 - E - Sqrt[20 - 4*E + E^2] - 2*x), x])/4 - (3*(2 - E + (12 - 4*E + E^2)/Sqrt[20 - 4*E + E^2])*
Defer[Int][E^(2*(-3 + E^x)^2 + x)/(2 - E - Sqrt[20 - 4*E + E^2] - 2*x), x])/4 - (5*(2 - E + (12 - 4*E + E^2)/S
qrt[20 - 4*E + E^2])*Defer[Int][E^((-3 + E^x)^2 + 2*x)/(2 - E - Sqrt[20 - 4*E + E^2] - 2*x), x])/4 - (40*(2 -
E)*Defer[Int][E^(-3 + E^x)^2/(2 - E + Sqrt[20 - 4*E + E^2] - 2*x)^2, x])/(20 - 4*E + E^2) - (15*(2 - E)*(8 - 4
*E + E^2)*Defer[Int][E^(-3 + E^x)^2/(2 - E + Sqrt[20 - 4*E + E^2] - 2*x)^2, x])/(2*(20 - 4*E + E^2)) + (5*(2 -
E)*(12 - 4*E + E^2)*Defer[Int][E^(-3 + E^x)^2/(2 - E + Sqrt[20 - 4*E + E^2] - 2*x)^2, x])/(20 - 4*E + E^2) +
(20*(2 - E + Sqrt[20 - 4*E + E^2])*Defer[Int][E^(-3 + E^x)^2/(2 - E + Sqrt[20 - 4*E + E^2] - 2*x)^2, x])/(20 -
4*E + E^2) + (15*(2 - E)^2*(2 - E + Sqrt[20 - 4*E + E^2])*Defer[Int][E^(-3 + E^x)^2/(2 - E + Sqrt[20 - 4*E +
E^2] - 2*x)^2, x])/(4*(20 - 4*E + E^2)) - (5*(8 - 4*E + E^2)*(2 - E + Sqrt[20 - 4*E + E^2])*Defer[Int][E^(-3 +
E^x)^2/(2 - E + Sqrt[20 - 4*E + E^2] - 2*x)^2, x])/(2*(20 - 4*E + E^2)) + (4*(2 - E)*Defer[Int][E^(2*(-3 + E^
x)^2)/(2 - E + Sqrt[20 - 4*E + E^2] - 2*x)^2, x])/(20 - 4*E + E^2) + (3*(2 - E)*(8 - 4*E + E^2)*Defer[Int][E^(
2*(-3 + E^x)^2)/(2 - E + Sqrt[20 - 4*E + E^2] - 2*x)^2, x])/(4*(20 - 4*E + E^2)) - ((2 - E)*(12 - 4*E + E^2)*D
efer[Int][E^(2*(-3 + E^x)^2)/(2 - E + Sqrt[20 - 4*E + E^2] - 2*x)^2, x])/(2*(20 - 4*E + E^2)) - (2*(2 - E + Sq
rt[20 - 4*E + E^2])*Defer[Int][E^(2*(-3 + E^x)^2)/(2 - E + Sqrt[20 - 4*E + E^2] - 2*x)^2, x])/(20 - 4*E + E^2)
- (3*(2 - E)^2*(2 - E + Sqrt[20 - 4*E + E^2])*Defer[Int][E^(2*(-3 + E^x)^2)/(2 - E + Sqrt[20 - 4*E + E^2] - 2
*x)^2, x])/(8*(20 - 4*E + E^2)) + ((8 - 4*E + E^2)*(2 - E + Sqrt[20 - 4*E + E^2])*Defer[Int][E^(2*(-3 + E^x)^2
)/(2 - E + Sqrt[20 - 4*E + E^2] - 2*x)^2, x])/(4*(20 - 4*E + E^2)) - (20*(2 - E)*Defer[Int][E^(-3 + E^x)^2/(2
- E + Sqrt[20 - 4*E + E^2] - 2*x), x])/(20 - 4*E + E^2)^(3/2) + (15*(2 - E)^3*Defer[Int][E^(-3 + E^x)^2/(2 - E
+ Sqrt[20 - 4*E + E^2] - 2*x), x])/(4*(20 - 4*E + E^2)^(3/2)) - (10*(2 - E)*(8 - 4*E + E^2)*Defer[Int][E^(-3
+ E^x)^2/(2 - E + Sqrt[20 - 4*E + E^2] - 2*x), x])/(20 - 4*E + E^2)^(3/2) + (5*(2 - E)*(12 - 4*E + E^2)*Defer[
Int][E^(-3 + E^x)^2/(2 - E + Sqrt[20 - 4*E + E^2] - 2*x), x])/(20 - 4*E + E^2)^(3/2) + (5*(1 - (2 - E)/Sqrt[20
- 4*E + E^2])*Defer[Int][E^(-3 + E^x)^2/(2 - E + Sqrt[20 - 4*E + E^2] - 2*x), x])/2 + (15*(2 - E)*(2 - E - (8
- 4*E + E^2)/Sqrt[20 - 4*E + E^2])*Defer[Int][E^(-3 + E^x)^2/(2 - E + Sqrt[20 - 4*E + E^2] - 2*x), x])/16 - (
5*(20 - 12*E + 3*E^2 - (3*(24 - 20*E + 6*E^2 - E^3))/Sqrt[20 - 4*E + E^2])*Defer[Int][E^(-3 + E^x)^2/(2 - E +
Sqrt[20 - 4*E + E^2] - 2*x), x])/16 + (2*(2 - E)*Defer[Int][E^(2*(-3 + E^x)^2)/(2 - E + Sqrt[20 - 4*E + E^2] -
2*x), x])/(20 - 4*E + E^2)^(3/2) - (3*(2 - E)^3*Defer[Int][E^(2*(-3 + E^x)^2)/(2 - E + Sqrt[20 - 4*E + E^2] -
2*x), x])/(8*(20 - 4*E + E^2)^(3/2)) + ((2 - E)*(8 - 4*E + E^2)*Defer[Int][E^(2*(-3 + E^x)^2)/(2 - E + Sqrt[2
0 - 4*E + E^2] - 2*x), x])/(20 - 4*E + E^2)^(3/2) - ((2 - E)*(12 - 4*E + E^2)*Defer[Int][E^(2*(-3 + E^x)^2)/(2
- E + Sqrt[20 - 4*E + E^2] - 2*x), x])/(2*(20 - 4*E + E^2)^(3/2)) - ((1 - (2 - E)/Sqrt[20 - 4*E + E^2])*Defer
[Int][E^(2*(-3 + E^x)^2)/(2 - E + Sqrt[20 - 4*E + E^2] - 2*x), x])/4 - (3*(2 - E)*(2 - E - (8 - 4*E + E^2)/Sqr
t[20 - 4*E + E^2])*Defer[Int][E^(2*(-3 + E^x)^2)/(2 - E + Sqrt[20 - 4*E + E^2] - 2*x), x])/32 + ((20 - 12*E +
3*E^2 - (3*(24 - 20*E + 6*E^2 - E^3))/Sqrt[20 - 4*E + E^2])*Defer[Int][E^(2*(-3 + E^x)^2)/(2 - E + Sqrt[20 - 4
*E + E^2] - 2*x), x])/32 + ((2 - E - (12 - 4*E + E^2)/Sqrt[20 - 4*E + E^2])*Defer[Int][E^(2*(9 - 6*E^x + E^(2*
x) + x))/(2 - E + Sqrt[20 - 4*E + E^2] - 2*x), x])/4 + (15*(2 - E - (12 - 4*E + E^2)/Sqrt[20 - 4*E + E^2])*Def
er[Int][E^((-3 + E^x)^2 + x)/(2 - E + Sqrt[20 - 4*E + E^2] - 2*x), x])/4 - (3*(2 - E - (12 - 4*E + E^2)/Sqrt[2
0 - 4*E + E^2])*Defer[Int][E^(2*(-3 + E^x)^2 + x)/(2 - E + Sqrt[20 - 4*E + E^2] - 2*x), x])/4 - (5*(2 - E - (1
2 - 4*E + E^2)/Sqrt[20 - 4*E + E^2])*Defer[Int][E^((-3 + E^x)^2 + 2*x)/(2 - E + Sqrt[20 - 4*E + E^2] - 2*x), x
])/4 - 5*Defer[Int][E^(-3 + E^x)^2/x^3, x] + Defer[Int][E^(2*(-3 + E^x)^2)/x^3, x]/2 + (5*(2 - E)*Defer[Int][E
^(-3 + E^x)^2/x^2, x])/8 - ((2 - E)*Defer[Int][E^(2*(-3 + E^x)^2)/x^2, x])/16 - Defer[Int][E^(2*(9 - 6*E^x + E
^(2*x) + x))/x^2, x] - 15*Defer[Int][E^((-3 + E^x)^2 + x)/x^2, x] + 3*Defer[Int][E^(2*(-3 + E^x)^2 + x)/x^2, x
] + 5*Defer[Int][E^((-3 + E^x)^2 + 2*x)/x^2, x] + (5*Defer[Int][E^(-3 + E^x)^2/x, x])/2 + (15*(2 - E)^2*Defer[
Int][E^(-3 + E^x)^2/x, x])/16 - (5*(20 - 12*E + 3*E^2)*Defer[Int][E^(-3 + E^x)^2/x, x])/16 - Defer[Int][E^(2*(
-3 + E^x)^2)/x, x]/4 - (3*(2 - E)^2*Defer[Int][E^(2*(-3 + E^x)^2)/x, x])/32 + ((20 - 12*E + 3*E^2)*Defer[Int][
E^(2*(-3 + E^x)^2)/x, x])/32 + ((2 - E)*Defer[Int][E^(2*(9 - 6*E^x + E^(2*x) + x))/x, x])/4 + (15*(2 - E)*Defe
r[Int][E^((-3 + E^x)^2 + x)/x, x])/4 - (3*(2 - E)*Defer[Int][E^(2*(-3 + E^x)^2 + x)/x, x])/4 - (5*(2 - E)*Defe
r[Int][E^((-3 + E^x)^2 + 2*x)/x, x])/4 - (40*(2 - E)*Defer[Int][E^(-3 + E^x)^2/(-2 + E + Sqrt[20 - 4*E + E^2]
+ 2*x)^2, x])/(20 - 4*E + E^2) - (15*(2 - E)*(8 - 4*E + E^2)*Defer[Int][E^(-3 + E^x)^2/(-2 + E + Sqrt[20 - 4*E
+ E^2] + 2*x)^2, x])/(2*(20 - 4*E + E^2)) + (5*(2 - E)*(12 - 4*E + E^2)*Defer[Int][E^(-3 + E^x)^2/(-2 + E + S
qrt[20 - 4*E + E^2] + 2*x)^2, x])/(20 - 4*E + E^2) + (20*(2 - E - Sqrt[20 - 4*E + E^2])*Defer[Int][E^(-3 + E^x
)^2/(-2 + E + Sqrt[20 - 4*E + E^2] + 2*x)^2, x])/(20 - 4*E + E^2) + (15*(2 - E)^2*(2 - E - Sqrt[20 - 4*E + E^2
])*Defer[Int][E^(-3 + E^x)^2/(-2 + E + Sqrt[20 - 4*E + E^2] + 2*x)^2, x])/(4*(20 - 4*E + E^2)) - (5*(8 - 4*E +
E^2)*(2 - E - Sqrt[20 - 4*E + E^2])*Defer[Int][E^(-3 + E^x)^2/(-2 + E + Sqrt[20 - 4*E + E^2] + 2*x)^2, x])/(2
*(20 - 4*E + E^2)) + (4*(2 - E)*Defer[Int][E^(2*(-3 + E^x)^2)/(-2 + E + Sqrt[20 - 4*E + E^2] + 2*x)^2, x])/(20
- 4*E + E^2) + (3*(2 - E)*(8 - 4*E + E^2)*Defer[Int][E^(2*(-3 + E^x)^2)/(-2 + E + Sqrt[20 - 4*E + E^2] + 2*x)
^2, x])/(4*(20 - 4*E + E^2)) - ((2 - E)*(12 - 4*E + E^2)*Defer[Int][E^(2*(-3 + E^x)^2)/(-2 + E + Sqrt[20 - 4*E
+ E^2] + 2*x)^2, x])/(2*(20 - 4*E + E^2)) - (2*(2 - E - Sqrt[20 - 4*E + E^2])*Defer[Int][E^(2*(-3 + E^x)^2)/(
-2 + E + Sqrt[20 - 4*E + E^2] + 2*x)^2, x])/(20 - 4*E + E^2) - (3*(2 - E)^2*(2 - E - Sqrt[20 - 4*E + E^2])*Def
er[Int][E^(2*(-3 + E^x)^2)/(-2 + E + Sqrt[20 - 4*E + E^2] + 2*x)^2, x])/(8*(20 - 4*E + E^2)) + ((8 - 4*E + E^2
)*(2 - E - Sqrt[20 - 4*E + E^2])*Defer[Int][E^(2*(-3 + E^x)^2)/(-2 + E + Sqrt[20 - 4*E + E^2] + 2*x)^2, x])/(4
*(20 - 4*E + E^2)) - (20*(2 - E)*Defer[Int][E^(-3 + E^x)^2/(-2 + E + Sqrt[20 - 4*E + E^2] + 2*x), x])/(20 - 4*
E + E^2)^(3/2) + (15*(2 - E)^3*Defer[Int][E^(-3 + E^x)^2/(-2 + E + Sqrt[20 - 4*E + E^2] + 2*x), x])/(4*(20 - 4
*E + E^2)^(3/2)) - (10*(2 - E)*(8 - 4*E + E^2)*Defer[Int][E^(-3 + E^x)^2/(-2 + E + Sqrt[20 - 4*E + E^2] + 2*x)
, x])/(20 - 4*E + E^2)^(3/2) + (5*(2 - E)*(12 - 4*E + E^2)*Defer[Int][E^(-3 + E^x)^2/(-2 + E + Sqrt[20 - 4*E +
E^2] + 2*x), x])/(20 - 4*E + E^2)^(3/2) + (2*(2 - E)*Defer[Int][E^(2*(-3 + E^x)^2)/(-2 + E + Sqrt[20 - 4*E +
E^2] + 2*x), x])/(20 - 4*E + E^2)^(3/2) - (3*(2 - E)^3*Defer[Int][E^(2*(-3 + E^x)^2)/(-2 + E + Sqrt[20 - 4*E +
E^2] + 2*x), x])/(8*(20 - 4*E + E^2)^(3/2)) + ((2 - E)*(8 - 4*E + E^2)*Defer[Int][E^(2*(-3 + E^x)^2)/(-2 + E
+ Sqrt[20 - 4*E + E^2] + 2*x), x])/(20 - 4*E + E^2)^(3/2) - ((2 - E)*(12 - 4*E + E^2)*Defer[Int][E^(2*(-3 + E^
x)^2)/(-2 + E + Sqrt[20 - 4*E + E^2] + 2*x), x])/(2*(20 - 4*E + E^2)^(3/2))
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {200+150 x-75 e x-100 x^2+e^{18-12 e^x+2 e^{2 x}} \left (8+6 x-3 e x-4 x^2+e^x \left (48 x+24 x^2-12 e x^2-12 x^3\right )+e^{2 x} \left (-16 x-8 x^2+4 e x^2+4 x^3\right )\right )+e^{9-6 e^x+e^{2 x}} \left (-80-60 x+30 e x+40 x^2+e^{2 x} \left (80 x+40 x^2-20 e x^2-20 x^3\right )+e^x \left (-240 x-120 x^2+60 e x^2+60 x^3\right )\right )}{16 x^3+16 x^4+\left (-4+e^2\right ) x^5-4 x^6+x^7+e \left (-8 x^4-4 x^5+2 x^6\right )} \, dx\\ &=\int \frac {200+(150-75 e) x-100 x^2+e^{18-12 e^x+2 e^{2 x}} \left (8+6 x-3 e x-4 x^2+e^x \left (48 x+24 x^2-12 e x^2-12 x^3\right )+e^{2 x} \left (-16 x-8 x^2+4 e x^2+4 x^3\right )\right )+e^{9-6 e^x+e^{2 x}} \left (-80-60 x+30 e x+40 x^2+e^{2 x} \left (80 x+40 x^2-20 e x^2-20 x^3\right )+e^x \left (-240 x-120 x^2+60 e x^2+60 x^3\right )\right )}{16 x^3+16 x^4+\left (-4+e^2\right ) x^5-4 x^6+x^7+e \left (-8 x^4-4 x^5+2 x^6\right )} \, dx\\ &=\int \frac {200-75 (-2+e) x-100 x^2+10 e^{\left (-3+e^x\right )^2} \left (-8-6 x+3 e x+4 x^2+6 e^x x \left (-4+(-2+e) x+x^2\right )-2 e^{2 x} x \left (-4+(-2+e) x+x^2\right )\right )+e^{2 \left (-3+e^x\right )^2} \left (8+6 x-3 e x-4 x^2-12 e^x x \left (-4+(-2+e) x+x^2\right )+4 e^{2 x} x \left (-4+(-2+e) x+x^2\right )\right )}{x^3 \left (4+(2-e) x-x^2\right )^2} \, dx\\ &=\int \left (\frac {200}{x^3 \left (4+(2-e) x-x^2\right )^2}-\frac {80 e^{\left (-3+e^x\right )^2}}{x^3 \left (4+(2-e) x-x^2\right )^2}+\frac {8 e^{2 \left (-3+e^x\right )^2}}{x^3 \left (4+(2-e) x-x^2\right )^2}+\frac {75 (2-e)}{x^2 \left (4+(2-e) x-x^2\right )^2}-\frac {60 \left (1-\frac {e}{2}\right ) e^{\left (-3+e^x\right )^2}}{x^2 \left (4+(2-e) x-x^2\right )^2}+\frac {6 \left (1-\frac {e}{2}\right ) e^{2 \left (-3+e^x\right )^2}}{x^2 \left (4+(2-e) x-x^2\right )^2}-\frac {100}{x \left (4+(2-e) x-x^2\right )^2}+\frac {40 e^{\left (-3+e^x\right )^2}}{x \left (4+(2-e) x-x^2\right )^2}-\frac {4 e^{2 \left (-3+e^x\right )^2}}{x \left (4+(2-e) x-x^2\right )^2}+\frac {4 e^{\left (-3+e^x\right )^2+2 x} \left (5-e^{\left (-3+e^x\right )^2}\right )}{x^2 \left (4+(2-e) x-x^2\right )}+\frac {12 e^{\left (-3+e^x\right )^2+x} \left (-5+e^{\left (-3+e^x\right )^2}\right )}{x^2 \left (4+(2-e) x-x^2\right )}\right ) \, dx\\ &=-\left (4 \int \frac {e^{2 \left (-3+e^x\right )^2}}{x \left (4+(2-e) x-x^2\right )^2} \, dx\right )+4 \int \frac {e^{\left (-3+e^x\right )^2+2 x} \left (5-e^{\left (-3+e^x\right )^2}\right )}{x^2 \left (4+(2-e) x-x^2\right )} \, dx+8 \int \frac {e^{2 \left (-3+e^x\right )^2}}{x^3 \left (4+(2-e) x-x^2\right )^2} \, dx+12 \int \frac {e^{\left (-3+e^x\right )^2+x} \left (-5+e^{\left (-3+e^x\right )^2}\right )}{x^2 \left (4+(2-e) x-x^2\right )} \, dx+40 \int \frac {e^{\left (-3+e^x\right )^2}}{x \left (4+(2-e) x-x^2\right )^2} \, dx-80 \int \frac {e^{\left (-3+e^x\right )^2}}{x^3 \left (4+(2-e) x-x^2\right )^2} \, dx-100 \int \frac {1}{x \left (4+(2-e) x-x^2\right )^2} \, dx+200 \int \frac {1}{x^3 \left (4+(2-e) x-x^2\right )^2} \, dx+(3 (2-e)) \int \frac {e^{2 \left (-3+e^x\right )^2}}{x^2 \left (4+(2-e) x-x^2\right )^2} \, dx-(30 (2-e)) \int \frac {e^{\left (-3+e^x\right )^2}}{x^2 \left (4+(2-e) x-x^2\right )^2} \, dx+(75 (2-e)) \int \frac {1}{x^2 \left (4+(2-e) x-x^2\right )^2} \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}
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Mathematica [A] time = 0.36, size = 44, normalized size = 1.57 \begin {gather*} \frac {e^{-12 e^x} \left (-5 e^{6 e^x}+e^{9+e^{2 x}}\right )^2}{x^2 \left (-4+(-2+e) x+x^2\right )} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[(200 + 150*x - 75*E*x - 100*x^2 + E^(18 - 12*E^x + 2*E^(2*x))*(8 + 6*x - 3*E*x - 4*x^2 + E^x*(48*x +
24*x^2 - 12*E*x^2 - 12*x^3) + E^(2*x)*(-16*x - 8*x^2 + 4*E*x^2 + 4*x^3)) + E^(9 - 6*E^x + E^(2*x))*(-80 - 60*
x + 30*E*x + 40*x^2 + E^(2*x)*(80*x + 40*x^2 - 20*E*x^2 - 20*x^3) + E^x*(-240*x - 120*x^2 + 60*E*x^2 + 60*x^3)
))/(16*x^3 + 16*x^4 - 4*x^5 + E^2*x^5 - 4*x^6 + x^7 + E*(-8*x^4 - 4*x^5 + 2*x^6)),x]
[Out]
(-5*E^(6*E^x) + E^(9 + E^(2*x)))^2/(E^(12*E^x)*x^2*(-4 + (-2 + E)*x + x^2))
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fricas [A] time = 0.62, size = 51, normalized size = 1.82 \begin {gather*} \frac {e^{\left (2 \, e^{\left (2 \, x\right )} - 12 \, e^{x} + 18\right )} - 10 \, e^{\left (e^{\left (2 \, x\right )} - 6 \, e^{x} + 9\right )} + 25}{x^{4} + x^{3} e - 2 \, x^{3} - 4 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((((4*x^2*exp(1)+4*x^3-8*x^2-16*x)*exp(x)^2+(-12*x^2*exp(1)-12*x^3+24*x^2+48*x)*exp(x)-3*x*exp(1)-4*x
^2+6*x+8)*exp(exp(x)^2-6*exp(x)+9)^2+((-20*x^2*exp(1)-20*x^3+40*x^2+80*x)*exp(x)^2+(60*x^2*exp(1)+60*x^3-120*x
^2-240*x)*exp(x)+30*x*exp(1)+40*x^2-60*x-80)*exp(exp(x)^2-6*exp(x)+9)-75*x*exp(1)-100*x^2+150*x+200)/(x^5*exp(
1)^2+(2*x^6-4*x^5-8*x^4)*exp(1)+x^7-4*x^6-4*x^5+16*x^4+16*x^3),x, algorithm="fricas")
[Out]
(e^(2*e^(2*x) - 12*e^x + 18) - 10*e^(e^(2*x) - 6*e^x + 9) + 25)/(x^4 + x^3*e - 2*x^3 - 4*x^2)
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int -\frac {100 \, x^{2} + 75 \, x e + {\left (4 \, x^{2} + 3 \, x e - 4 \, {\left (x^{3} + x^{2} e - 2 \, x^{2} - 4 \, x\right )} e^{\left (2 \, x\right )} + 12 \, {\left (x^{3} + x^{2} e - 2 \, x^{2} - 4 \, x\right )} e^{x} - 6 \, x - 8\right )} e^{\left (2 \, e^{\left (2 \, x\right )} - 12 \, e^{x} + 18\right )} - 10 \, {\left (4 \, x^{2} + 3 \, x e - 2 \, {\left (x^{3} + x^{2} e - 2 \, x^{2} - 4 \, x\right )} e^{\left (2 \, x\right )} + 6 \, {\left (x^{3} + x^{2} e - 2 \, x^{2} - 4 \, x\right )} e^{x} - 6 \, x - 8\right )} e^{\left (e^{\left (2 \, x\right )} - 6 \, e^{x} + 9\right )} - 150 \, x - 200}{x^{7} - 4 \, x^{6} + x^{5} e^{2} - 4 \, x^{5} + 16 \, x^{4} + 16 \, x^{3} + 2 \, {\left (x^{6} - 2 \, x^{5} - 4 \, x^{4}\right )} e}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((((4*x^2*exp(1)+4*x^3-8*x^2-16*x)*exp(x)^2+(-12*x^2*exp(1)-12*x^3+24*x^2+48*x)*exp(x)-3*x*exp(1)-4*x
^2+6*x+8)*exp(exp(x)^2-6*exp(x)+9)^2+((-20*x^2*exp(1)-20*x^3+40*x^2+80*x)*exp(x)^2+(60*x^2*exp(1)+60*x^3-120*x
^2-240*x)*exp(x)+30*x*exp(1)+40*x^2-60*x-80)*exp(exp(x)^2-6*exp(x)+9)-75*x*exp(1)-100*x^2+150*x+200)/(x^5*exp(
1)^2+(2*x^6-4*x^5-8*x^4)*exp(1)+x^7-4*x^6-4*x^5+16*x^4+16*x^3),x, algorithm="giac")
[Out]
integrate(-(100*x^2 + 75*x*e + (4*x^2 + 3*x*e - 4*(x^3 + x^2*e - 2*x^2 - 4*x)*e^(2*x) + 12*(x^3 + x^2*e - 2*x^
2 - 4*x)*e^x - 6*x - 8)*e^(2*e^(2*x) - 12*e^x + 18) - 10*(4*x^2 + 3*x*e - 2*(x^3 + x^2*e - 2*x^2 - 4*x)*e^(2*x
) + 6*(x^3 + x^2*e - 2*x^2 - 4*x)*e^x - 6*x - 8)*e^(e^(2*x) - 6*e^x + 9) - 150*x - 200)/(x^7 - 4*x^6 + x^5*e^2
- 4*x^5 + 16*x^4 + 16*x^3 + 2*(x^6 - 2*x^5 - 4*x^4)*e), x)
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maple [B] time = 0.30, size = 82, normalized size = 2.93
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risch |
\(\frac {25}{x^{2} \left (x \,{\mathrm e}+x^{2}-2 x -4\right )}+\frac {{\mathrm e}^{2 \,{\mathrm e}^{2 x}-12 \,{\mathrm e}^{x}+18}}{x^{2} \left (x \,{\mathrm e}+x^{2}-2 x -4\right )}-\frac {10 \,{\mathrm e}^{{\mathrm e}^{2 x}-6 \,{\mathrm e}^{x}+9}}{x^{2} \left (x \,{\mathrm e}+x^{2}-2 x -4\right )}\) |
\(82\) |
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Verification of antiderivative is not currently implemented for this CAS.
[In]
int((((4*x^2*exp(1)+4*x^3-8*x^2-16*x)*exp(x)^2+(-12*x^2*exp(1)-12*x^3+24*x^2+48*x)*exp(x)-3*x*exp(1)-4*x^2+6*x
+8)*exp(exp(x)^2-6*exp(x)+9)^2+((-20*x^2*exp(1)-20*x^3+40*x^2+80*x)*exp(x)^2+(60*x^2*exp(1)+60*x^3-120*x^2-240
*x)*exp(x)+30*x*exp(1)+40*x^2-60*x-80)*exp(exp(x)^2-6*exp(x)+9)-75*x*exp(1)-100*x^2+150*x+200)/(x^5*exp(1)^2+(
2*x^6-4*x^5-8*x^4)*exp(1)+x^7-4*x^6-4*x^5+16*x^4+16*x^3),x,method=_RETURNVERBOSE)
[Out]
25/x^2/(x*exp(1)+x^2-2*x-4)+1/x^2/(x*exp(1)+x^2-2*x-4)*exp(2*exp(2*x)-12*exp(x)+18)-10/x^2/(x*exp(1)+x^2-2*x-4
)*exp(exp(2*x)-6*exp(x)+9)
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maxima [B] time = 0.51, size = 759, normalized size = 27.11 result too large to
display
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((((4*x^2*exp(1)+4*x^3-8*x^2-16*x)*exp(x)^2+(-12*x^2*exp(1)-12*x^3+24*x^2+48*x)*exp(x)-3*x*exp(1)-4*x
^2+6*x+8)*exp(exp(x)^2-6*exp(x)+9)^2+((-20*x^2*exp(1)-20*x^3+40*x^2+80*x)*exp(x)^2+(60*x^2*exp(1)+60*x^3-120*x
^2-240*x)*exp(x)+30*x*exp(1)+40*x^2-60*x-80)*exp(exp(x)^2-6*exp(x)+9)-75*x*exp(1)-100*x^2+150*x+200)/(x^5*exp(
1)^2+(2*x^6-4*x^5-8*x^4)*exp(1)+x^7-4*x^6-4*x^5+16*x^4+16*x^3),x, algorithm="maxima")
[Out]
75/64*((e - 2)*log(x^2 + x*(e - 2) - 4) - 2*(e - 2)*log(x) + (e^4 - 8*e^3 + 48*e^2 - 128*e + 208)*log((2*x - s
qrt(e^2 - 4*e + 20) + e - 2)/(2*x + sqrt(e^2 - 4*e + 20) + e - 2))/(e^2 - 4*e + 20)^(3/2) + 8*(x^2*(e^2 - 4*e
+ 16) + x*(e^3 - 6*e^2 + 26*e - 36) - 2*e^2 + 8*e - 40)/(x^3*(e^2 - 4*e + 20) + x^2*(e^3 - 6*e^2 + 28*e - 40)
- 4*x*(e^2 - 4*e + 20)))*e - 25/64*(3*e^2 - 12*e + 20)*log(x^2 + x*(e - 2) - 4) - 75/32*(e - 2)*log(x^2 + x*(e
- 2) - 4) + 25/32*(3*e^2 - 12*e + 20)*log(x) + 75/16*(e - 2)*log(x) + (e^(2*e^(2*x) + 18) - 10*e^(e^(2*x) + 6
*e^x + 9))*e^(-12*e^x)/(x^4 + x^3*(e - 2) - 4*x^2) - 25/64*(3*e^5 - 30*e^4 + 200*e^3 - 720*e^2 + 1680*e - 1696
)*log((2*x - sqrt(e^2 - 4*e + 20) + e - 2)/(2*x + sqrt(e^2 - 4*e + 20) + e - 2))/(e^2 - 4*e + 20)^(3/2) - 75/3
2*(e^4 - 8*e^3 + 48*e^2 - 128*e + 208)*log((2*x - sqrt(e^2 - 4*e + 20) + e - 2)/(2*x + sqrt(e^2 - 4*e + 20) +
e - 2))/(e^2 - 4*e + 20)^(3/2) + 25/8*(e^3 - 6*e^2 + 36*e - 56)*log((2*x - sqrt(e^2 - 4*e + 20) + e - 2)/(2*x
+ sqrt(e^2 - 4*e + 20) + e - 2))/(e^2 - 4*e + 20)^(3/2) - 25/8*(x^3*(3*e^3 - 18*e^2 + 80*e - 112) + x^2*(3*e^4
- 24*e^3 + 122*e^2 - 296*e + 312) - 6*x*(e^3 - 6*e^2 + 28*e - 40) - 8*e^2 + 32*e - 160)/(x^4*(e^2 - 4*e + 20)
+ x^3*(e^3 - 6*e^2 + 28*e - 40) - 4*x^2*(e^2 - 4*e + 20)) - 75/4*(x^2*(e^2 - 4*e + 16) + x*(e^3 - 6*e^2 + 26*
e - 36) - 2*e^2 + 8*e - 40)/(x^3*(e^2 - 4*e + 20) + x^2*(e^3 - 6*e^2 + 28*e - 40) - 4*x*(e^2 - 4*e + 20)) + 25
*(x*(e - 2) + e^2 - 4*e + 12)/(x^2*(e^2 - 4*e + 20) + x*(e^3 - 6*e^2 + 28*e - 40) - 4*e^2 + 16*e - 80) + 25/8*
log(x^2 + x*(e - 2) - 4) - 25/4*log(x)
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mupad [B] time = 4.14, size = 54, normalized size = 1.93 \begin {gather*} -\frac {{\mathrm {e}}^{2\,{\mathrm {e}}^{2\,x}-12\,{\mathrm {e}}^x+18}\,{\left (5\,{\mathrm {e}}^{6\,{\mathrm {e}}^x-{\mathrm {e}}^{2\,x}-9}-1\right )}^2}{x^2\,\left (2\,x-x\,\mathrm {e}-x^2+4\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((150*x + exp(2*exp(2*x) - 12*exp(x) + 18)*(6*x - 3*x*exp(1) + exp(x)*(48*x - 12*x^2*exp(1) + 24*x^2 - 12*x
^3) - 4*x^2 - exp(2*x)*(16*x - 4*x^2*exp(1) + 8*x^2 - 4*x^3) + 8) - 75*x*exp(1) - exp(exp(2*x) - 6*exp(x) + 9)
*(60*x - 30*x*exp(1) + exp(x)*(240*x - 60*x^2*exp(1) + 120*x^2 - 60*x^3) - 40*x^2 - exp(2*x)*(80*x - 20*x^2*ex
p(1) + 40*x^2 - 20*x^3) + 80) - 100*x^2 + 200)/(x^5*exp(2) - exp(1)*(8*x^4 + 4*x^5 - 2*x^6) + 16*x^3 + 16*x^4
- 4*x^5 - 4*x^6 + x^7),x)
[Out]
-(exp(2*exp(2*x) - 12*exp(x) + 18)*(5*exp(6*exp(x) - exp(2*x) - 9) - 1)^2)/(x^2*(2*x - x*exp(1) - x^2 + 4))
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sympy [B] time = 1.20, size = 146, normalized size = 5.21 \begin {gather*} \frac {\left (- 10 x^{4} - 10 e x^{3} + 20 x^{3} + 40 x^{2}\right ) e^{e^{2 x} - 6 e^{x} + 9} + \left (x^{4} - 2 x^{3} + e x^{3} - 4 x^{2}\right ) e^{2 e^{2 x} - 12 e^{x} + 18}}{x^{8} - 4 x^{7} + 2 e x^{7} - 4 e x^{6} - 4 x^{6} + x^{6} e^{2} - 8 e x^{5} + 16 x^{5} + 16 x^{4}} + \frac {25}{x^{4} + x^{3} \left (-2 + e\right ) - 4 x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((((4*x**2*exp(1)+4*x**3-8*x**2-16*x)*exp(x)**2+(-12*x**2*exp(1)-12*x**3+24*x**2+48*x)*exp(x)-3*x*exp
(1)-4*x**2+6*x+8)*exp(exp(x)**2-6*exp(x)+9)**2+((-20*x**2*exp(1)-20*x**3+40*x**2+80*x)*exp(x)**2+(60*x**2*exp(
1)+60*x**3-120*x**2-240*x)*exp(x)+30*x*exp(1)+40*x**2-60*x-80)*exp(exp(x)**2-6*exp(x)+9)-75*x*exp(1)-100*x**2+
150*x+200)/(x**5*exp(1)**2+(2*x**6-4*x**5-8*x**4)*exp(1)+x**7-4*x**6-4*x**5+16*x**4+16*x**3),x)
[Out]
((-10*x**4 - 10*E*x**3 + 20*x**3 + 40*x**2)*exp(exp(2*x) - 6*exp(x) + 9) + (x**4 - 2*x**3 + E*x**3 - 4*x**2)*e
xp(2*exp(2*x) - 12*exp(x) + 18))/(x**8 - 4*x**7 + 2*E*x**7 - 4*E*x**6 - 4*x**6 + x**6*exp(2) - 8*E*x**5 + 16*x
**5 + 16*x**4) + 25/(x**4 + x**3*(-2 + E) - 4*x**2)
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