3.6.3 \(\int \frac {(-2+x^3) \sqrt {2+x^2+2 x^3}}{(1+x^3) (1+x^2+x^3)} \, dx\)
Optimal. Leaf size=39 \[ -2 \tan ^{-1}\left (\frac {x}{\sqrt {2 x^3+x^2+2}}\right )-2 \tanh ^{-1}\left (\frac {x}{\sqrt {2 x^3+x^2+2}}\right ) \]
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Rubi [F] time = 74.09, 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 {\left (-2+x^3\right ) \sqrt {2+x^2+2 x^3}}{\left (1+x^3\right ) \left (1+x^2+x^3\right )} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
Int[((-2 + x^3)*Sqrt[2 + x^2 + 2*x^3])/((1 + x^3)*(1 + x^2 + x^3)),x]
[Out]
(-2*Sqrt[2 + x^2 + 2*x^3])/3 + ((1 - I*Sqrt[3])*Sqrt[2 + x^2 + 2*x^3])/3 + ((1 + I*Sqrt[3])*Sqrt[2 + x^2 + 2*x
^3])/3 - ((5*I)*Sqrt[2]*Sqrt[2 + x^2 + 2*x^3]*EllipticE[ArcSin[((109 - 6*Sqrt[330])^(1/6)*Sqrt[I*(2 - (1 + I*S
qrt[3])/(109 - 6*Sqrt[330])^(1/3) - (1 - I*Sqrt[3])*(109 - 6*Sqrt[330])^(1/3) + 12*x)])/(3^(1/4)*Sqrt[2*(1 - (
109 - 6*Sqrt[330])^(2/3))])], (-2*Sqrt[3]*(1 - (109 - 6*Sqrt[330])^(2/3)))/(3*I - Sqrt[3] + (3*I + Sqrt[3])*(1
09 - 6*Sqrt[330])^(2/3))])/((109 - 6*Sqrt[330])^(1/6)*Sqrt[(1 + (109 - 6*Sqrt[330])^(-1/3) + (109 - 6*Sqrt[330
])^(1/3) + 6*x)/(3 + I*Sqrt[3] + (3 - I*Sqrt[3])*(109 - 6*Sqrt[330])^(2/3))]*Sqrt[-1 + (109 - 6*Sqrt[330])^(-2
/3) + (109 - 6*Sqrt[330])^(2/3) - ((109 - 6*Sqrt[330])^(-1/3) + (109 - 6*Sqrt[330])^(1/3))*(1 + 6*x) + (1 + 6*
x)^2]) - ((13*I - Sqrt[3])*Sqrt[2 + x^2 + 2*x^3]*EllipticE[ArcSin[((109 - 6*Sqrt[330])^(1/6)*Sqrt[I*(2 - (1 +
I*Sqrt[3])/(109 - 6*Sqrt[330])^(1/3) + I*(I + Sqrt[3])*(109 - 6*Sqrt[330])^(1/3) + 12*x)])/(3^(1/4)*Sqrt[2*(1
- (109 - 6*Sqrt[330])^(2/3))])], (-2*Sqrt[3]*(1 - (109 - 6*Sqrt[330])^(2/3)))/(3*I - Sqrt[3] + (3*I + Sqrt[3])
*(109 - 6*Sqrt[330])^(2/3))])/(Sqrt[2]*(109 - 6*Sqrt[330])^(1/6)*Sqrt[(1 + (109 - 6*Sqrt[330])^(-1/3) + (109 -
6*Sqrt[330])^(1/3) + 6*x)/(3 + I*Sqrt[3] + (3 - I*Sqrt[3])*(109 - 6*Sqrt[330])^(2/3))]*Sqrt[-1 + (109 - 6*Sqr
t[330])^(-2/3) + (109 - 6*Sqrt[330])^(2/3) - ((1 + (109 - 6*Sqrt[330])^(2/3))*(1 + 6*x))/(109 - 6*Sqrt[330])^(
1/3) + (1 + 6*x)^2]) - ((13*I + Sqrt[3])*Sqrt[2 + x^2 + 2*x^3]*EllipticE[ArcSin[((109 - 6*Sqrt[330])^(1/6)*Sqr
t[I*(2 - (1 + I*Sqrt[3])/(109 - 6*Sqrt[330])^(1/3) + I*(I + Sqrt[3])*(109 - 6*Sqrt[330])^(1/3) + 12*x)])/(3^(1
/4)*Sqrt[2*(1 - (109 - 6*Sqrt[330])^(2/3))])], (-2*Sqrt[3]*(1 - (109 - 6*Sqrt[330])^(2/3)))/(3*I - Sqrt[3] + (
3*I + Sqrt[3])*(109 - 6*Sqrt[330])^(2/3))])/(Sqrt[2]*(109 - 6*Sqrt[330])^(1/6)*Sqrt[(1 + (109 - 6*Sqrt[330])^(
-1/3) + (109 - 6*Sqrt[330])^(1/3) + 6*x)/(3 + I*Sqrt[3] + (3 - I*Sqrt[3])*(109 - 6*Sqrt[330])^(2/3))]*Sqrt[-1
+ (109 - 6*Sqrt[330])^(-2/3) + (109 - 6*Sqrt[330])^(2/3) - ((1 + (109 - 6*Sqrt[330])^(2/3))*(1 + 6*x))/(109 -
6*Sqrt[330])^(1/3) + (1 + 6*x)^2]) + ((46*I)*Sqrt[2]*(109 - 6*Sqrt[330])^(1/6)*Sqrt[(1 + (109 - 6*Sqrt[330])^(
-1/3) + (109 - 6*Sqrt[330])^(1/3) + 6*x)/(3 + I*Sqrt[3] + (3 - I*Sqrt[3])*(109 - 6*Sqrt[330])^(2/3))]*Sqrt[2 +
x^2 + 2*x^3]*EllipticF[ArcSin[((109 - 6*Sqrt[330])^(1/6)*Sqrt[I*(2 - (1 + I*Sqrt[3])/(109 - 6*Sqrt[330])^(1/3
) - (1 - I*Sqrt[3])*(109 - 6*Sqrt[330])^(1/3) + 12*x)])/(3^(1/4)*Sqrt[2*(1 - (109 - 6*Sqrt[330])^(2/3))])], (-
2*Sqrt[3]*(1 - (109 - 6*Sqrt[330])^(2/3)))/(3*I - Sqrt[3] + (3*I + Sqrt[3])*(109 - 6*Sqrt[330])^(2/3))])/((1 +
(109 - 6*Sqrt[330])^(-1/3) + (109 - 6*Sqrt[330])^(1/3) + 6*x)*Sqrt[-1 + (109 - 6*Sqrt[330])^(-2/3) + (109 - 6
*Sqrt[330])^(2/3) - ((109 - 6*Sqrt[330])^(-1/3) + (109 - 6*Sqrt[330])^(1/3))*(1 + 6*x) + (1 + 6*x)^2]) + ((10*
I)*Sqrt[2]*(1 + (109 - 6*Sqrt[330])^(2/3))*Sqrt[(1 + (109 - 6*Sqrt[330])^(-1/3) + (109 - 6*Sqrt[330])^(1/3) +
6*x)/(3 + I*Sqrt[3] + (3 - I*Sqrt[3])*(109 - 6*Sqrt[330])^(2/3))]*Sqrt[2 + x^2 + 2*x^3]*EllipticF[ArcSin[((109
- 6*Sqrt[330])^(1/6)*Sqrt[I*(2 - (1 + I*Sqrt[3])/(109 - 6*Sqrt[330])^(1/3) - (1 - I*Sqrt[3])*(109 - 6*Sqrt[33
0])^(1/3) + 12*x)])/(3^(1/4)*Sqrt[2*(1 - (109 - 6*Sqrt[330])^(2/3))])], (-2*Sqrt[3]*(1 - (109 - 6*Sqrt[330])^(
2/3)))/(3*I - Sqrt[3] + (3*I + Sqrt[3])*(109 - 6*Sqrt[330])^(2/3))])/((109 - 6*Sqrt[330])^(1/6)*(1 + (109 - 6*
Sqrt[330])^(-1/3) + (109 - 6*Sqrt[330])^(1/3) + 6*x)*Sqrt[-1 + (109 - 6*Sqrt[330])^(-2/3) + (109 - 6*Sqrt[330]
)^(2/3) - ((109 - 6*Sqrt[330])^(-1/3) + (109 - 6*Sqrt[330])^(1/3))*(1 + 6*x) + (1 + 6*x)^2]) + (Sqrt[2]*(13*I
- 24*Sqrt[3])*(1 + I*Sqrt[3])*(109 - 6*Sqrt[330])^(1/6)*Sqrt[(1 + (109 - 6*Sqrt[330])^(-1/3) + (109 - 6*Sqrt[3
30])^(1/3) + 6*x)/(3 + I*Sqrt[3] + (3 - I*Sqrt[3])*(109 - 6*Sqrt[330])^(2/3))]*Sqrt[2 + x^2 + 2*x^3]*EllipticF
[ArcSin[((109 - 6*Sqrt[330])^(1/6)*Sqrt[I*(2 - (1 + I*Sqrt[3])/(109 - 6*Sqrt[330])^(1/3) + I*(I + Sqrt[3])*(10
9 - 6*Sqrt[330])^(1/3) + 12*x)])/(3^(1/4)*Sqrt[2*(1 - (109 - 6*Sqrt[330])^(2/3))])], (-2*Sqrt[3]*(1 - (109 - 6
*Sqrt[330])^(2/3)))/(3*I - Sqrt[3] + (3*I + Sqrt[3])*(109 - 6*Sqrt[330])^(2/3))])/((1 + (109 - 6*Sqrt[330])^(-
1/3) + (109 - 6*Sqrt[330])^(1/3) + 6*x)*Sqrt[-1 + (109 - 6*Sqrt[330])^(-2/3) + (109 - 6*Sqrt[330])^(2/3) - ((1
+ (109 - 6*Sqrt[330])^(2/3))*(1 + 6*x))/(109 - 6*Sqrt[330])^(1/3) + (1 + 6*x)^2]) + (Sqrt[2]*(13 - (24*I)*Sqr
t[3])*(I + Sqrt[3])*(109 - 6*Sqrt[330])^(1/6)*Sqrt[(1 + (109 - 6*Sqrt[330])^(-1/3) + (109 - 6*Sqrt[330])^(1/3)
+ 6*x)/(3 + I*Sqrt[3] + (3 - I*Sqrt[3])*(109 - 6*Sqrt[330])^(2/3))]*Sqrt[2 + x^2 + 2*x^3]*EllipticF[ArcSin[((
109 - 6*Sqrt[330])^(1/6)*Sqrt[I*(2 - (1 + I*Sqrt[3])/(109 - 6*Sqrt[330])^(1/3) + I*(I + Sqrt[3])*(109 - 6*Sqrt
[330])^(1/3) + 12*x)])/(3^(1/4)*Sqrt[2*(1 - (109 - 6*Sqrt[330])^(2/3))])], (-2*Sqrt[3]*(1 - (109 - 6*Sqrt[330]
)^(2/3)))/(3*I - Sqrt[3] + (3*I + Sqrt[3])*(109 - 6*Sqrt[330])^(2/3))])/((1 + (109 - 6*Sqrt[330])^(-1/3) + (10
9 - 6*Sqrt[330])^(1/3) + 6*x)*Sqrt[-1 + (109 - 6*Sqrt[330])^(-2/3) + (109 - 6*Sqrt[330])^(2/3) - ((1 + (109 -
6*Sqrt[330])^(2/3))*(1 + 6*x))/(109 - 6*Sqrt[330])^(1/3) + (1 + 6*x)^2]) + (Sqrt[2]*(4 + (3*I)*Sqrt[3])*(I + S
qrt[3])*(1 + (109 - 6*Sqrt[330])^(2/3))*Sqrt[(1 + (109 - 6*Sqrt[330])^(-1/3) + (109 - 6*Sqrt[330])^(1/3) + 6*x
)/(3 + I*Sqrt[3] + (3 - I*Sqrt[3])*(109 - 6*Sqrt[330])^(2/3))]*Sqrt[2 + x^2 + 2*x^3]*EllipticF[ArcSin[((109 -
6*Sqrt[330])^(1/6)*Sqrt[I*(2 - (1 + I*Sqrt[3])/(109 - 6*Sqrt[330])^(1/3) + I*(I + Sqrt[3])*(109 - 6*Sqrt[330])
^(1/3) + 12*x)])/(3^(1/4)*Sqrt[2*(1 - (109 - 6*Sqrt[330])^(2/3))])], (-2*Sqrt[3]*(1 - (109 - 6*Sqrt[330])^(2/3
)))/(3*I - Sqrt[3] + (3*I + Sqrt[3])*(109 - 6*Sqrt[330])^(2/3))])/((109 - 6*Sqrt[330])^(1/6)*(1 + (109 - 6*Sqr
t[330])^(-1/3) + (109 - 6*Sqrt[330])^(1/3) + 6*x)*Sqrt[-1 + (109 - 6*Sqrt[330])^(-2/3) + (109 - 6*Sqrt[330])^(
2/3) - ((1 + (109 - 6*Sqrt[330])^(2/3))*(1 + 6*x))/(109 - 6*Sqrt[330])^(1/3) + (1 + 6*x)^2]) + (Sqrt[2]*(1 + I
*Sqrt[3])*(4*I + 3*Sqrt[3])*(1 + (109 - 6*Sqrt[330])^(2/3))*Sqrt[(1 + (109 - 6*Sqrt[330])^(-1/3) + (109 - 6*Sq
rt[330])^(1/3) + 6*x)/(3 + I*Sqrt[3] + (3 - I*Sqrt[3])*(109 - 6*Sqrt[330])^(2/3))]*Sqrt[2 + x^2 + 2*x^3]*Ellip
ticF[ArcSin[((109 - 6*Sqrt[330])^(1/6)*Sqrt[I*(2 - (1 + I*Sqrt[3])/(109 - 6*Sqrt[330])^(1/3) + I*(I + Sqrt[3])
*(109 - 6*Sqrt[330])^(1/3) + 12*x)])/(3^(1/4)*Sqrt[2*(1 - (109 - 6*Sqrt[330])^(2/3))])], (-2*Sqrt[3]*(1 - (109
- 6*Sqrt[330])^(2/3)))/(3*I - Sqrt[3] + (3*I + Sqrt[3])*(109 - 6*Sqrt[330])^(2/3))])/((109 - 6*Sqrt[330])^(1/
6)*(1 + (109 - 6*Sqrt[330])^(-1/3) + (109 - 6*Sqrt[330])^(1/3) + 6*x)*Sqrt[-1 + (109 - 6*Sqrt[330])^(-2/3) + (
109 - 6*Sqrt[330])^(2/3) - ((1 + (109 - 6*Sqrt[330])^(2/3))*(1 + 6*x))/(109 - 6*Sqrt[330])^(1/3) + (1 + 6*x)^2
]) - (27*(1 - I*Sqrt[3])^2*Sqrt[2*(109 - 6*Sqrt[330])*(3 - I*Sqrt[3] + (3 + I*Sqrt[3])*(109 - 6*Sqrt[330])^(2/
3))]*Sqrt[2 + (I*(I + Sqrt[3]))/(109 - 6*Sqrt[330])^(1/3) - (1 + I*Sqrt[3])*(109 - 6*Sqrt[330])^(1/3) + 12*x]*
Sqrt[2 - (1 + I*Sqrt[3])/(109 - 6*Sqrt[330])^(1/3) + I*(I + Sqrt[3])*(109 - 6*Sqrt[330])^(1/3) + 12*x]*Sqrt[2
+ x^2 + 2*x^3]*Sqrt[1 - (2*(1 + (109 - 6*Sqrt[330])^(2/3) + (109 - 6*Sqrt[330])^(1/3)*(1 + 6*x)))/(3 + I*Sqrt[
3] + (3 - I*Sqrt[3])*(109 - 6*Sqrt[330])^(2/3))]*Sqrt[1 - (2*(1 + (109 - 6*Sqrt[330])^(2/3) + (109 - 6*Sqrt[33
0])^(1/3)*(1 + 6*x)))/(3 - I*Sqrt[3] + (3 + I*Sqrt[3])*(109 - 6*Sqrt[330])^(2/3))]*EllipticPi[(3*I + Sqrt[3] +
(3*I - Sqrt[3])*(109 - 6*Sqrt[330])^(2/3))/(2*(I + ((8*I - 6*Sqrt[3])*(109 - 6*Sqrt[330])^(1/3))/2 + I*(109 -
6*Sqrt[330])^(2/3))), ArcSin[(Sqrt[2]*(109 - 6*Sqrt[330])^(1/6)*Sqrt[1 + (109 - 6*Sqrt[330])^(-1/3) + (109 -
6*Sqrt[330])^(1/3) + 6*x])/Sqrt[3 - I*Sqrt[3] + (3 + I*Sqrt[3])*(109 - 6*Sqrt[330])^(2/3)]], (3*I + Sqrt[3] +
(3*I - Sqrt[3])*(109 - 6*Sqrt[330])^(2/3))/(3*I - Sqrt[3] + (3*I + Sqrt[3])*(109 - 6*Sqrt[330])^(2/3))])/((1 +
(4 + (3*I)*Sqrt[3])*(109 - 6*Sqrt[330])^(1/3) + (109 - 6*Sqrt[330])^(2/3))*Sqrt[1 + (109 - 6*Sqrt[330])^(-1/3
) + (109 - 6*Sqrt[330])^(1/3) + 6*x]*Sqrt[-1 + I*Sqrt[3] - (1 + I*Sqrt[3])*(109 - 6*Sqrt[330])^(2/3) + 2*(109
- 6*Sqrt[330])^(1/3)*(1 + 6*x)]*Sqrt[-1 - I*Sqrt[3] + I*(I + Sqrt[3])*(109 - 6*Sqrt[330])^(2/3) + 2*(109 - 6*S
qrt[330])^(1/3)*(1 + 6*x)]*(1 - (109 - 6*Sqrt[330])^(-2/3) - (109 - 6*Sqrt[330])^(2/3) + ((109 - 6*Sqrt[330])^
(-1/3) + (109 - 6*Sqrt[330])^(1/3))*(1 + 6*x) - (1 + 6*x)^2)) - (108*Sqrt[2*(109 - 6*Sqrt[330])*(3 - I*Sqrt[3]
+ (3 + I*Sqrt[3])*(109 - 6*Sqrt[330])^(2/3))]*Sqrt[2 + (I*(I + Sqrt[3]))/(109 - 6*Sqrt[330])^(1/3) - (1 + I*S
qrt[3])*(109 - 6*Sqrt[330])^(1/3) + 12*x]*Sqrt[2 - (1 + I*Sqrt[3])/(109 - 6*Sqrt[330])^(1/3) + I*(I + Sqrt[3])
*(109 - 6*Sqrt[330])^(1/3) + 12*x]*Sqrt[2 + x^2 + 2*x^3]*Sqrt[1 - (2*(1 + (109 - 6*Sqrt[330])^(2/3) + (109 - 6
*Sqrt[330])^(1/3)*(1 + 6*x)))/(3 + I*Sqrt[3] + (3 - I*Sqrt[3])*(109 - 6*Sqrt[330])^(2/3))]*Sqrt[1 - (2*(1 + (1
09 - 6*Sqrt[330])^(2/3) + (109 - 6*Sqrt[330])^(1/3)*(1 + 6*x)))/(3 - I*Sqrt[3] + (3 + I*Sqrt[3])*(109 - 6*Sqrt
[330])^(2/3))]*EllipticPi[(3 - I*Sqrt[3] + (3 + I*Sqrt[3])*(109 - 6*Sqrt[330])^(2/3))/(2*(1 - 5*(109 - 6*Sqrt[
330])^(1/3) + (109 - 6*Sqrt[330])^(2/3))), ArcSin[(Sqrt[2]*(109 - 6*Sqrt[330])^(1/6)*Sqrt[1 + (109 - 6*Sqrt[33
0])^(-1/3) + (109 - 6*Sqrt[330])^(1/3) + 6*x])/Sqrt[3 - I*Sqrt[3] + (3 + I*Sqrt[3])*(109 - 6*Sqrt[330])^(2/3)]
], (3*I + Sqrt[3] + (3*I - Sqrt[3])*(109 - 6*Sqrt[330])^(2/3))/(3*I - Sqrt[3] + (3*I + Sqrt[3])*(109 - 6*Sqrt[
330])^(2/3))])/((1 - 5*(109 - 6*Sqrt[330])^(1/3) + (109 - 6*Sqrt[330])^(2/3))*Sqrt[1 + (109 - 6*Sqrt[330])^(-1
/3) + (109 - 6*Sqrt[330])^(1/3) + 6*x]*Sqrt[-1 + I*Sqrt[3] - (1 + I*Sqrt[3])*(109 - 6*Sqrt[330])^(2/3) + 2*(10
9 - 6*Sqrt[330])^(1/3)*(1 + 6*x)]*Sqrt[-1 - I*Sqrt[3] + I*(I + Sqrt[3])*(109 - 6*Sqrt[330])^(2/3) + 2*(109 - 6
*Sqrt[330])^(1/3)*(1 + 6*x)]*(1 - (109 - 6*Sqrt[330])^(-2/3) - (109 - 6*Sqrt[330])^(2/3) + ((109 - 6*Sqrt[330]
)^(-1/3) + (109 - 6*Sqrt[330])^(1/3))*(1 + 6*x) - (1 + 6*x)^2)) - (27*(1 + I*Sqrt[3])^2*Sqrt[2*(109 - 6*Sqrt[3
30])*(3 - I*Sqrt[3] + (3 + I*Sqrt[3])*(109 - 6*Sqrt[330])^(2/3))]*Sqrt[2 + (I*(I + Sqrt[3]))/(109 - 6*Sqrt[330
])^(1/3) - (1 + I*Sqrt[3])*(109 - 6*Sqrt[330])^(1/3) + 12*x]*Sqrt[2 - (1 + I*Sqrt[3])/(109 - 6*Sqrt[330])^(1/3
) + I*(I + Sqrt[3])*(109 - 6*Sqrt[330])^(1/3) + 12*x]*Sqrt[2 + x^2 + 2*x^3]*Sqrt[1 - (2*(1 + (109 - 6*Sqrt[330
])^(2/3) + (109 - 6*Sqrt[330])^(1/3)*(1 + 6*x)))/(3 + I*Sqrt[3] + (3 - I*Sqrt[3])*(109 - 6*Sqrt[330])^(2/3))]*
Sqrt[1 - (2*(1 + (109 - 6*Sqrt[330])^(2/3) + (109 - 6*Sqrt[330])^(1/3)*(1 + 6*x)))/(3 - I*Sqrt[3] + (3 + I*Sqr
t[3])*(109 - 6*Sqrt[330])^(2/3))]*EllipticPi[(3 - I*Sqrt[3] + (3 + I*Sqrt[3])*(109 - 6*Sqrt[330])^(2/3))/(2*(1
+ (4 - (3*I)*Sqrt[3])*(109 - 6*Sqrt[330])^(1/3) + (109 - 6*Sqrt[330])^(2/3))), ArcSin[(Sqrt[2]*(109 - 6*Sqrt[
330])^(1/6)*Sqrt[1 + (109 - 6*Sqrt[330])^(-1/3) + (109 - 6*Sqrt[330])^(1/3) + 6*x])/Sqrt[3 - I*Sqrt[3] + (3 +
I*Sqrt[3])*(109 - 6*Sqrt[330])^(2/3)]], (3*I + Sqrt[3] + (3*I - Sqrt[3])*(109 - 6*Sqrt[330])^(2/3))/(3*I - Sqr
t[3] + (3*I + Sqrt[3])*(109 - 6*Sqrt[330])^(2/3))])/((1 + (4 - (3*I)*Sqrt[3])*(109 - 6*Sqrt[330])^(1/3) + (109
- 6*Sqrt[330])^(2/3))*Sqrt[1 + (109 - 6*Sqrt[330])^(-1/3) + (109 - 6*Sqrt[330])^(1/3) + 6*x]*Sqrt[-1 + I*Sqrt
[3] - (1 + I*Sqrt[3])*(109 - 6*Sqrt[330])^(2/3) + 2*(109 - 6*Sqrt[330])^(1/3)*(1 + 6*x)]*Sqrt[-1 - I*Sqrt[3] +
I*(I + Sqrt[3])*(109 - 6*Sqrt[330])^(2/3) + 2*(109 - 6*Sqrt[330])^(1/3)*(1 + 6*x)]*(1 - (109 - 6*Sqrt[330])^(
-2/3) - (109 - 6*Sqrt[330])^(2/3) + ((109 - 6*Sqrt[330])^(-1/3) + (109 - 6*Sqrt[330])^(1/3))*(1 + 6*x) - (1 +
6*x)^2)) - 2*Defer[Int][Sqrt[2 + x^2 + 2*x^3]/(1 + x^2 + x^3), x] - 3*Defer[Int][(x*Sqrt[2 + x^2 + 2*x^3])/(1
+ x^2 + x^3), x]
Rubi steps
\begin {align*} \int \frac {\left (-2+x^3\right ) \sqrt {2+x^2+2 x^3}}{\left (1+x^3\right ) \left (1+x^2+x^3\right )} \, dx &=\int \left (\frac {\sqrt {2+x^2+2 x^3}}{-1-x}+\frac {(1+x) \sqrt {2+x^2+2 x^3}}{1-x+x^2}+\frac {(-2-3 x) \sqrt {2+x^2+2 x^3}}{1+x^2+x^3}\right ) \, dx\\ &=\int \frac {\sqrt {2+x^2+2 x^3}}{-1-x} \, dx+\int \frac {(1+x) \sqrt {2+x^2+2 x^3}}{1-x+x^2} \, dx+\int \frac {(-2-3 x) \sqrt {2+x^2+2 x^3}}{1+x^2+x^3} \, dx\\ &=\int \left (\frac {\left (1-i \sqrt {3}\right ) \sqrt {2+x^2+2 x^3}}{-1-i \sqrt {3}+2 x}+\frac {\left (1+i \sqrt {3}\right ) \sqrt {2+x^2+2 x^3}}{-1+i \sqrt {3}+2 x}\right ) \, dx+\int \left (-\frac {2 \sqrt {2+x^2+2 x^3}}{1+x^2+x^3}-\frac {3 x \sqrt {2+x^2+2 x^3}}{1+x^2+x^3}\right ) \, dx+\operatorname {Subst}\left (\int \frac {\sqrt {\frac {109}{54}-\frac {x}{6}+2 x^3}}{-\frac {5}{6}-x} \, dx,x,\frac {1}{6}+x\right ) \end {align*}
rest of steps removed due to Latex formating problem.
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Mathematica [C] time = 6.34, size = 10734, normalized size = 275.23 \begin {gather*} \text {Result too large to show} \end {gather*}
Warning: Unable to verify antiderivative.
[In]
Integrate[((-2 + x^3)*Sqrt[2 + x^2 + 2*x^3])/((1 + x^3)*(1 + x^2 + x^3)),x]
[Out]
Result too large to show
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IntegrateAlgebraic [A] time = 0.35, size = 39, normalized size = 1.00 \begin {gather*} -2 \tan ^{-1}\left (\frac {x}{\sqrt {2+x^2+2 x^3}}\right )-2 \tanh ^{-1}\left (\frac {x}{\sqrt {2+x^2+2 x^3}}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
IntegrateAlgebraic[((-2 + x^3)*Sqrt[2 + x^2 + 2*x^3])/((1 + x^3)*(1 + x^2 + x^3)),x]
[Out]
-2*ArcTan[x/Sqrt[2 + x^2 + 2*x^3]] - 2*ArcTanh[x/Sqrt[2 + x^2 + 2*x^3]]
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fricas [A] time = 0.48, size = 66, normalized size = 1.69 \begin {gather*} \arctan \left (\frac {\sqrt {2 \, x^{3} + x^{2} + 2} {\left (x^{3} + 1\right )}}{2 \, x^{4} + x^{3} + 2 \, x}\right ) + \log \left (\frac {x^{3} + x^{2} - \sqrt {2 \, x^{3} + x^{2} + 2} x + 1}{x^{3} + 1}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((x^3-2)*(2*x^3+x^2+2)^(1/2)/(x^3+1)/(x^3+x^2+1),x, algorithm="fricas")
[Out]
arctan(sqrt(2*x^3 + x^2 + 2)*(x^3 + 1)/(2*x^4 + x^3 + 2*x)) + log((x^3 + x^2 - sqrt(2*x^3 + x^2 + 2)*x + 1)/(x
^3 + 1))
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {2 \, x^{3} + x^{2} + 2} {\left (x^{3} - 2\right )}}{{\left (x^{3} + x^{2} + 1\right )} {\left (x^{3} + 1\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((x^3-2)*(2*x^3+x^2+2)^(1/2)/(x^3+1)/(x^3+x^2+1),x, algorithm="giac")
[Out]
integrate(sqrt(2*x^3 + x^2 + 2)*(x^3 - 2)/((x^3 + x^2 + 1)*(x^3 + 1)), x)
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maple [C] time = 4.17, size = 98, normalized size = 2.51
| | |
| method |
result |
size |
| | |
| trager |
\(-\ln \left (-\frac {x^{3}+\sqrt {2 x^{3}+x^{2}+2}\, x +x^{2}+1}{\left (1+x \right ) \left (x^{2}-x +1\right )}\right )+\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (-\frac {-\RootOf \left (\textit {\_Z}^{2}+1\right ) x^{3}+\sqrt {2 x^{3}+x^{2}+2}\, x -\RootOf \left (\textit {\_Z}^{2}+1\right )}{x^{3}+x^{2}+1}\right )\) |
\(98\) |
| default |
\(\text {Expression too large to display}\) |
\(4717\) |
| elliptic |
\(\text {Expression too large to display}\) |
\(73157\) |
| | |
|
|
|
|
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((x^3-2)*(2*x^3+x^2+2)^(1/2)/(x^3+1)/(x^3+x^2+1),x,method=_RETURNVERBOSE)
[Out]
-ln(-(x^3+(2*x^3+x^2+2)^(1/2)*x+x^2+1)/(1+x)/(x^2-x+1))+RootOf(_Z^2+1)*ln(-(-RootOf(_Z^2+1)*x^3+(2*x^3+x^2+2)^
(1/2)*x-RootOf(_Z^2+1))/(x^3+x^2+1))
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {2 \, x^{3} + x^{2} + 2} {\left (x^{3} - 2\right )}}{{\left (x^{3} + x^{2} + 1\right )} {\left (x^{3} + 1\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((x^3-2)*(2*x^3+x^2+2)^(1/2)/(x^3+1)/(x^3+x^2+1),x, algorithm="maxima")
[Out]
integrate(sqrt(2*x^3 + x^2 + 2)*(x^3 - 2)/((x^3 + x^2 + 1)*(x^3 + 1)), x)
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mupad [B] time = 1.60, size = 2611, normalized size = 66.95
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(((x^3 - 2)*(x^2 + 2*x^3 + 2)^(1/2))/((x^3 + 1)*(x^2 + x^3 + 1)),x)
[Out]
symsum(-(2*(-(x + (3^(1/2)*(1/(36*(109/216 - (55^(1/2)*216^(1/2))/216)^(1/3)) - (109/216 - (55^(1/2)*216^(1/2)
)/216)^(1/3))*1i)/2 - 1/(72*(109/216 - (55^(1/2)*216^(1/2))/216)^(1/3)) - (109/216 - (55^(1/2)*216^(1/2))/216)
^(1/3)/2 + 1/6)/(1/(24*(109/216 - (55^(1/2)*216^(1/2))/216)^(1/3)) - (3^(1/2)*(1/(36*(109/216 - (55^(1/2)*216^
(1/2))/216)^(1/3)) - (109/216 - (55^(1/2)*216^(1/2))/216)^(1/3))*1i)/2 + (3*(109/216 - (55^(1/2)*216^(1/2))/21
6)^(1/3))/2))^(1/2)*((x + 1/(36*(109/216 - (55^(1/2)*216^(1/2))/216)^(1/3)) + (109/216 - (55^(1/2)*216^(1/2))/
216)^(1/3) + 1/6)/(1/(24*(109/216 - (55^(1/2)*216^(1/2))/216)^(1/3)) - (3^(1/2)*(1/(36*(109/216 - (55^(1/2)*21
6^(1/2))/216)^(1/3)) - (109/216 - (55^(1/2)*216^(1/2))/216)^(1/3))*1i)/2 + (3*(109/216 - (55^(1/2)*216^(1/2))/
216)^(1/3))/2))^(1/2)*(x^2/2 + x^3 + 1)^(1/2)*(1/(24*(109/216 - (55^(1/2)*216^(1/2))/216)^(1/3)) - (3^(1/2)*(1
/(36*(109/216 - (55^(1/2)*216^(1/2))/216)^(1/3)) - (109/216 - (55^(1/2)*216^(1/2))/216)^(1/3))*1i)/2 + (3*(109
/216 - (55^(1/2)*216^(1/2))/216)^(1/3))/2)*ellipticPi(-(1/(24*(109/216 - (55^(1/2)*216^(1/2))/216)^(1/3)) - (3
^(1/2)*(1/(36*(109/216 - (55^(1/2)*216^(1/2))/216)^(1/3)) - (109/216 - (55^(1/2)*216^(1/2))/216)^(1/3))*1i)/2
+ (3*(109/216 - (55^(1/2)*216^(1/2))/216)^(1/3))/2)/(_X187 + (3^(1/2)*(1/(36*(109/216 - (55^(1/2)*216^(1/2))/2
16)^(1/3)) - (109/216 - (55^(1/2)*216^(1/2))/216)^(1/3))*1i)/2 - 1/(72*(109/216 - (55^(1/2)*216^(1/2))/216)^(1
/3)) - (109/216 - (55^(1/2)*216^(1/2))/216)^(1/3)/2 + 1/6), asin((-(x + (3^(1/2)*(1/(36*(109/216 - (55^(1/2)*2
16^(1/2))/216)^(1/3)) - (109/216 - (55^(1/2)*216^(1/2))/216)^(1/3))*1i)/2 - 1/(72*(109/216 - (55^(1/2)*216^(1/
2))/216)^(1/3)) - (109/216 - (55^(1/2)*216^(1/2))/216)^(1/3)/2 + 1/6)/(1/(24*(109/216 - (55^(1/2)*216^(1/2))/2
16)^(1/3)) - (3^(1/2)*(1/(36*(109/216 - (55^(1/2)*216^(1/2))/216)^(1/3)) - (109/216 - (55^(1/2)*216^(1/2))/216
)^(1/3))*1i)/2 + (3*(109/216 - (55^(1/2)*216^(1/2))/216)^(1/3))/2))^(1/2)), (3^(1/2)*(1/(24*(109/216 - (55^(1/
2)*216^(1/2))/216)^(1/3)) - (3^(1/2)*(1/(36*(109/216 - (55^(1/2)*216^(1/2))/216)^(1/3)) - (109/216 - (55^(1/2)
*216^(1/2))/216)^(1/3))*1i)/2 + (3*(109/216 - (55^(1/2)*216^(1/2))/216)^(1/3))/2)*1i)/(3*(1/(36*(109/216 - (55
^(1/2)*216^(1/2))/216)^(1/3)) - (109/216 - (55^(1/2)*216^(1/2))/216)^(1/3))))*(-(3^(1/2)*((3^(1/2)*(1/(36*(109
/216 - (55^(1/2)*216^(1/2))/216)^(1/3)) - (109/216 - (55^(1/2)*216^(1/2))/216)^(1/3))*1i)/2 - x + 1/(72*(109/2
16 - (55^(1/2)*216^(1/2))/216)^(1/3)) + (109/216 - (55^(1/2)*216^(1/2))/216)^(1/3)/2 - 1/6)*1i)/(3*(1/(36*(109
/216 - (55^(1/2)*216^(1/2))/216)^(1/3)) - (109/216 - (55^(1/2)*216^(1/2))/216)^(1/3))))^(1/2)*(4*_X187^2 + 6*_
X187^3 + _X187^5 + 6))/(((_X187^3 + 1)*(2*_X187 + 3*_X187^2) + 3*_X187^2*(_X187^2 + _X187^3 + 1))*(x^2 + 2*x^3
+ 2)^(1/2)*(x^2/2 - x*(((3^(1/2)*(1/(36*(109/216 - (55^(1/2)*216^(1/2))/216)^(1/3)) - (109/216 - (55^(1/2)*21
6^(1/2))/216)^(1/3))*1i)/2 - 1/(72*(109/216 - (55^(1/2)*216^(1/2))/216)^(1/3)) - (109/216 - (55^(1/2)*216^(1/2
))/216)^(1/3)/2 + 1/6)*((3^(1/2)*(1/(36*(109/216 - (55^(1/2)*216^(1/2))/216)^(1/3)) - (109/216 - (55^(1/2)*216
^(1/2))/216)^(1/3))*1i)/2 + 1/(72*(109/216 - (55^(1/2)*216^(1/2))/216)^(1/3)) + (109/216 - (55^(1/2)*216^(1/2)
)/216)^(1/3)/2 - 1/6) - (1/(36*(109/216 - (55^(1/2)*216^(1/2))/216)^(1/3)) + (109/216 - (55^(1/2)*216^(1/2))/2
16)^(1/3) + 1/6)*((3^(1/2)*(1/(36*(109/216 - (55^(1/2)*216^(1/2))/216)^(1/3)) - (109/216 - (55^(1/2)*216^(1/2)
)/216)^(1/3))*1i)/2 - 1/(72*(109/216 - (55^(1/2)*216^(1/2))/216)^(1/3)) - (109/216 - (55^(1/2)*216^(1/2))/216)
^(1/3)/2 + 1/6) + (1/(36*(109/216 - (55^(1/2)*216^(1/2))/216)^(1/3)) + (109/216 - (55^(1/2)*216^(1/2))/216)^(1
/3) + 1/6)*((3^(1/2)*(1/(36*(109/216 - (55^(1/2)*216^(1/2))/216)^(1/3)) - (109/216 - (55^(1/2)*216^(1/2))/216)
^(1/3))*1i)/2 + 1/(72*(109/216 - (55^(1/2)*216^(1/2))/216)^(1/3)) + (109/216 - (55^(1/2)*216^(1/2))/216)^(1/3)
/2 - 1/6)) + x^3 - (1/(36*(109/216 - (55^(1/2)*216^(1/2))/216)^(1/3)) + (109/216 - (55^(1/2)*216^(1/2))/216)^(
1/3) + 1/6)*((3^(1/2)*(1/(36*(109/216 - (55^(1/2)*216^(1/2))/216)^(1/3)) - (109/216 - (55^(1/2)*216^(1/2))/216
)^(1/3))*1i)/2 - 1/(72*(109/216 - (55^(1/2)*216^(1/2))/216)^(1/3)) - (109/216 - (55^(1/2)*216^(1/2))/216)^(1/3
)/2 + 1/6)*((3^(1/2)*(1/(36*(109/216 - (55^(1/2)*216^(1/2))/216)^(1/3)) - (109/216 - (55^(1/2)*216^(1/2))/216)
^(1/3))*1i)/2 + 1/(72*(109/216 - (55^(1/2)*216^(1/2))/216)^(1/3)) + (109/216 - (55^(1/2)*216^(1/2))/216)^(1/3)
/2 - 1/6))^(1/2)*(_X187 + (3^(1/2)*(1/(36*(109/216 - (55^(1/2)*216^(1/2))/216)^(1/3)) - (109/216 - (55^(1/2)*2
16^(1/2))/216)^(1/3))*1i)/2 - 1/(72*(109/216 - (55^(1/2)*216^(1/2))/216)^(1/3)) - (109/216 - (55^(1/2)*216^(1/
2))/216)^(1/3)/2 + 1/6)), _X187 in {-1, 1/2 - (3^(1/2)*1i)/2, (3^(1/2)*1i)/2 + 1/2} union root(z^3 + z^2 + 1,
z)) - (4*(-(x + (3^(1/2)*(1/(36*(109/216 - (55^(1/2)*216^(1/2))/216)^(1/3)) - (109/216 - (55^(1/2)*216^(1/2))/
216)^(1/3))*1i)/2 - 1/(72*(109/216 - (55^(1/2)*216^(1/2))/216)^(1/3)) - (109/216 - (55^(1/2)*216^(1/2))/216)^(
1/3)/2 + 1/6)/(1/(24*(109/216 - (55^(1/2)*216^(1/2))/216)^(1/3)) - (3^(1/2)*(1/(36*(109/216 - (55^(1/2)*216^(1
/2))/216)^(1/3)) - (109/216 - (55^(1/2)*216^(1/2))/216)^(1/3))*1i)/2 + (3*(109/216 - (55^(1/2)*216^(1/2))/216)
^(1/3))/2))^(1/2)*((x + 1/(36*(109/216 - (55^(1/2)*216^(1/2))/216)^(1/3)) + (109/216 - (55^(1/2)*216^(1/2))/21
6)^(1/3) + 1/6)/(1/(24*(109/216 - (55^(1/2)*216^(1/2))/216)^(1/3)) - (3^(1/2)*(1/(36*(109/216 - (55^(1/2)*216^
(1/2))/216)^(1/3)) - (109/216 - (55^(1/2)*216^(1/2))/216)^(1/3))*1i)/2 + (3*(109/216 - (55^(1/2)*216^(1/2))/21
6)^(1/3))/2))^(1/2)*ellipticF(asin((-(x + (3^(1/2)*(1/(36*(109/216 - (55^(1/2)*216^(1/2))/216)^(1/3)) - (109/2
16 - (55^(1/2)*216^(1/2))/216)^(1/3))*1i)/2 - 1/(72*(109/216 - (55^(1/2)*216^(1/2))/216)^(1/3)) - (109/216 - (
55^(1/2)*216^(1/2))/216)^(1/3)/2 + 1/6)/(1/(24*(109/216 - (55^(1/2)*216^(1/2))/216)^(1/3)) - (3^(1/2)*(1/(36*(
109/216 - (55^(1/2)*216^(1/2))/216)^(1/3)) - (109/216 - (55^(1/2)*216^(1/2))/216)^(1/3))*1i)/2 + (3*(109/216 -
(55^(1/2)*216^(1/2))/216)^(1/3))/2))^(1/2)), (3^(1/2)*(1/(24*(109/216 - (55^(1/2)*216^(1/2))/216)^(1/3)) - (3
^(1/2)*(1/(36*(109/216 - (55^(1/2)*216^(1/2))/216)^(1/3)) - (109/216 - (55^(1/2)*216^(1/2))/216)^(1/3))*1i)/2
+ (3*(109/216 - (55^(1/2)*216^(1/2))/216)^(1/3))/2)*1i)/(3*(1/(36*(109/216 - (55^(1/2)*216^(1/2))/216)^(1/3))
- (109/216 - (55^(1/2)*216^(1/2))/216)^(1/3))))*(x^2/2 + x^3 + 1)^(1/2)*(1/(24*(109/216 - (55^(1/2)*216^(1/2))
/216)^(1/3)) - (3^(1/2)*(1/(36*(109/216 - (55^(1/2)*216^(1/2))/216)^(1/3)) - (109/216 - (55^(1/2)*216^(1/2))/2
16)^(1/3))*1i)/2 + (3*(109/216 - (55^(1/2)*216^(1/2))/216)^(1/3))/2)*(-(3^(1/2)*((3^(1/2)*(1/(36*(109/216 - (5
5^(1/2)*216^(1/2))/216)^(1/3)) - (109/216 - (55^(1/2)*216^(1/2))/216)^(1/3))*1i)/2 - x + 1/(72*(109/216 - (55^
(1/2)*216^(1/2))/216)^(1/3)) + (109/216 - (55^(1/2)*216^(1/2))/216)^(1/3)/2 - 1/6)*1i)/(3*(1/(36*(109/216 - (5
5^(1/2)*216^(1/2))/216)^(1/3)) - (109/216 - (55^(1/2)*216^(1/2))/216)^(1/3))))^(1/2))/((x^2 + 2*x^3 + 2)^(1/2)
*(x^2/2 - x*(((3^(1/2)*(1/(36*(109/216 - (55^(1/2)*216^(1/2))/216)^(1/3)) - (109/216 - (55^(1/2)*216^(1/2))/21
6)^(1/3))*1i)/2 - 1/(72*(109/216 - (55^(1/2)*216^(1/2))/216)^(1/3)) - (109/216 - (55^(1/2)*216^(1/2))/216)^(1/
3)/2 + 1/6)*((3^(1/2)*(1/(36*(109/216 - (55^(1/2)*216^(1/2))/216)^(1/3)) - (109/216 - (55^(1/2)*216^(1/2))/216
)^(1/3))*1i)/2 + 1/(72*(109/216 - (55^(1/2)*216^(1/2))/216)^(1/3)) + (109/216 - (55^(1/2)*216^(1/2))/216)^(1/3
)/2 - 1/6) - (1/(36*(109/216 - (55^(1/2)*216^(1/2))/216)^(1/3)) + (109/216 - (55^(1/2)*216^(1/2))/216)^(1/3) +
1/6)*((3^(1/2)*(1/(36*(109/216 - (55^(1/2)*216^(1/2))/216)^(1/3)) - (109/216 - (55^(1/2)*216^(1/2))/216)^(1/3
))*1i)/2 - 1/(72*(109/216 - (55^(1/2)*216^(1/2))/216)^(1/3)) - (109/216 - (55^(1/2)*216^(1/2))/216)^(1/3)/2 +
1/6) + (1/(36*(109/216 - (55^(1/2)*216^(1/2))/216)^(1/3)) + (109/216 - (55^(1/2)*216^(1/2))/216)^(1/3) + 1/6)*
((3^(1/2)*(1/(36*(109/216 - (55^(1/2)*216^(1/2))/216)^(1/3)) - (109/216 - (55^(1/2)*216^(1/2))/216)^(1/3))*1i)
/2 + 1/(72*(109/216 - (55^(1/2)*216^(1/2))/216)^(1/3)) + (109/216 - (55^(1/2)*216^(1/2))/216)^(1/3)/2 - 1/6))
+ x^3 - (1/(36*(109/216 - (55^(1/2)*216^(1/2))/216)^(1/3)) + (109/216 - (55^(1/2)*216^(1/2))/216)^(1/3) + 1/6)
*((3^(1/2)*(1/(36*(109/216 - (55^(1/2)*216^(1/2))/216)^(1/3)) - (109/216 - (55^(1/2)*216^(1/2))/216)^(1/3))*1i
)/2 - 1/(72*(109/216 - (55^(1/2)*216^(1/2))/216)^(1/3)) - (109/216 - (55^(1/2)*216^(1/2))/216)^(1/3)/2 + 1/6)*
((3^(1/2)*(1/(36*(109/216 - (55^(1/2)*216^(1/2))/216)^(1/3)) - (109/216 - (55^(1/2)*216^(1/2))/216)^(1/3))*1i)
/2 + 1/(72*(109/216 - (55^(1/2)*216^(1/2))/216)^(1/3)) + (109/216 - (55^(1/2)*216^(1/2))/216)^(1/3)/2 - 1/6))^
(1/2))
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (x^{3} - 2\right ) \sqrt {2 x^{3} + x^{2} + 2}}{\left (x + 1\right ) \left (x^{2} - x + 1\right ) \left (x^{3} + x^{2} + 1\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((x**3-2)*(2*x**3+x**2+2)**(1/2)/(x**3+1)/(x**3+x**2+1),x)
[Out]
Integral((x**3 - 2)*sqrt(2*x**3 + x**2 + 2)/((x + 1)*(x**2 - x + 1)*(x**3 + x**2 + 1)), x)
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