3.16.56 \(\int \frac {(-1+x^2) (1-x+x^2-x^3+x^4)}{(1-x+x^2)^2 (1+x+x^2) \sqrt {1+3 x^2+x^4}} \, dx\)

Optimal. Leaf size=107 \[ \frac {\sqrt {x^4+3 x^2+1}}{4 \left (x^2-x+1\right )}-\sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {2} x}{x^2+\sqrt {x^4+3 x^2+1}-x+1}\right )-\frac {\tanh ^{-1}\left (\frac {\sqrt {2} x}{x^2+\sqrt {x^4+3 x^2+1}+x+1}\right )}{2 \sqrt {2}} \]

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Rubi [C]  time = 10.76, antiderivative size = 5419, normalized size of antiderivative = 50.64, number of steps used = 136, number of rules used = 18, integrand size = 54, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6688, 6728, 1099, 6742, 1726, 1741, 12, 1247, 724, 204, 1716, 1189, 1135, 1214, 1456, 539, 1724, 2}

result too large to display

Warning: Unable to verify antiderivative.

[In]

Int[((-1 + x^2)*(1 - x + x^2 - x^3 + x^4))/((1 - x + x^2)^2*(1 + x + x^2)*Sqrt[1 + 3*x^2 + x^4]),x]

[Out]

(x*(3 + Sqrt[5] + 2*x^2))/(12*(1 - I*Sqrt[3])*Sqrt[1 + 3*x^2 + x^4]) + (x*(3 + Sqrt[5] + 2*x^2))/(12*(1 + I*Sq
rt[3])*Sqrt[1 + 3*x^2 + x^4]) + ((I - Sqrt[3])*x*(3 + Sqrt[5] + 2*x^2))/(24*(I + Sqrt[3])*Sqrt[1 + 3*x^2 + x^4
]) + ((I + Sqrt[3])*x*(3 + Sqrt[5] + 2*x^2))/(24*(I - Sqrt[3])*Sqrt[1 + 3*x^2 + x^4]) + Sqrt[1 + 3*x^2 + x^4]/
(3*(1 + I*Sqrt[3])*(1 - I*Sqrt[3] - 2*x)) + ((I + Sqrt[3])*Sqrt[1 + 3*x^2 + x^4])/(6*(I - Sqrt[3])*(1 - I*Sqrt
[3] - 2*x)) + Sqrt[1 + 3*x^2 + x^4]/(3*(1 - I*Sqrt[3])*(1 + I*Sqrt[3] - 2*x)) + ((I - Sqrt[3])*Sqrt[1 + 3*x^2
+ x^4])/(6*(I + Sqrt[3])*(1 + I*Sqrt[3] - 2*x)) - ((1 - I*Sqrt[3])*ArcTan[(1 - (3*I)*Sqrt[3] + 2*(2 - I*Sqrt[3
])*x^2)/(4*Sqrt[1 + I*Sqrt[3]]*Sqrt[1 + 3*x^2 + x^4])])/(16*Sqrt[1 + I*Sqrt[3]]) + ((I/4)*ArcTan[(1 - (3*I)*Sq
rt[3] + 2*(2 - I*Sqrt[3])*x^2)/(4*Sqrt[1 + I*Sqrt[3]]*Sqrt[1 + 3*x^2 + x^4])])/Sqrt[3*(1 + I*Sqrt[3])] + ((1 +
(3*I)*Sqrt[3])*ArcTan[(1 - (3*I)*Sqrt[3] + 2*(2 - I*Sqrt[3])*x^2)/(4*Sqrt[1 + I*Sqrt[3]]*Sqrt[1 + 3*x^2 + x^4
])])/(24*(1 + I*Sqrt[3])^(3/2)) + ((1 - (5*I)*Sqrt[3])*ArcTan[(1 - (3*I)*Sqrt[3] + 2*(2 - I*Sqrt[3])*x^2)/(4*S
qrt[1 + I*Sqrt[3]]*Sqrt[1 + 3*x^2 + x^4])])/(16*Sqrt[1 + I*Sqrt[3]]) - ((I - 3*Sqrt[3])*(I + Sqrt[3])*ArcTan[(
1 - (3*I)*Sqrt[3] + 2*(2 - I*Sqrt[3])*x^2)/(4*Sqrt[1 + I*Sqrt[3]]*Sqrt[1 + 3*x^2 + x^4])])/(48*(1 + I*Sqrt[3])
^(3/2)) - ((I/4)*ArcTan[(1 + (3*I)*Sqrt[3] + 2*(2 + I*Sqrt[3])*x^2)/(4*Sqrt[1 - I*Sqrt[3]]*Sqrt[1 + 3*x^2 + x^
4])])/Sqrt[3*(1 - I*Sqrt[3])] + ((5 - I*Sqrt[3])*ArcTan[(1 + (3*I)*Sqrt[3] + 2*(2 + I*Sqrt[3])*x^2)/(4*Sqrt[1
- I*Sqrt[3]]*Sqrt[1 + 3*x^2 + x^4])])/(24*(1 - I*Sqrt[3])^(3/2)) - ((1 + I*Sqrt[3])*ArcTan[(1 + (3*I)*Sqrt[3]
+ 2*(2 + I*Sqrt[3])*x^2)/(4*Sqrt[1 - I*Sqrt[3]]*Sqrt[1 + 3*x^2 + x^4])])/(16*Sqrt[1 - I*Sqrt[3]]) + ((1 - (3*I
)*Sqrt[3])*ArcTan[(1 + (3*I)*Sqrt[3] + 2*(2 + I*Sqrt[3])*x^2)/(4*Sqrt[1 - I*Sqrt[3]]*Sqrt[1 + 3*x^2 + x^4])])/
(24*(1 - I*Sqrt[3])^(3/2)) + ((1 + (5*I)*Sqrt[3])*ArcTan[(1 + (3*I)*Sqrt[3] + 2*(2 + I*Sqrt[3])*x^2)/(4*Sqrt[1
- I*Sqrt[3]]*Sqrt[1 + 3*x^2 + x^4])])/(16*Sqrt[1 - I*Sqrt[3]]) - (Sqrt[(3 + Sqrt[5])/2]*Sqrt[(2 + (3 - Sqrt[5
])*x^2)/(2 + (3 + Sqrt[5])*x^2)]*(2 + (3 + Sqrt[5])*x^2)*EllipticE[ArcTan[Sqrt[(3 + Sqrt[5])/2]*x], (-5 + 3*Sq
rt[5])/2])/(12*(1 - I*Sqrt[3])*Sqrt[1 + 3*x^2 + x^4]) - (Sqrt[(3 + Sqrt[5])/2]*Sqrt[(2 + (3 - Sqrt[5])*x^2)/(2
+ (3 + Sqrt[5])*x^2)]*(2 + (3 + Sqrt[5])*x^2)*EllipticE[ArcTan[Sqrt[(3 + Sqrt[5])/2]*x], (-5 + 3*Sqrt[5])/2])
/(12*(1 + I*Sqrt[3])*Sqrt[1 + 3*x^2 + x^4]) - ((I - Sqrt[3])*Sqrt[(3 + Sqrt[5])/2]*Sqrt[(2 + (3 - Sqrt[5])*x^2
)/(2 + (3 + Sqrt[5])*x^2)]*(2 + (3 + Sqrt[5])*x^2)*EllipticE[ArcTan[Sqrt[(3 + Sqrt[5])/2]*x], (-5 + 3*Sqrt[5])
/2])/(24*(I + Sqrt[3])*Sqrt[1 + 3*x^2 + x^4]) - ((I + Sqrt[3])*Sqrt[(3 + Sqrt[5])/2]*Sqrt[(2 + (3 - Sqrt[5])*x
^2)/(2 + (3 + Sqrt[5])*x^2)]*(2 + (3 + Sqrt[5])*x^2)*EllipticE[ArcTan[Sqrt[(3 + Sqrt[5])/2]*x], (-5 + 3*Sqrt[5
])/2])/(24*(I - Sqrt[3])*Sqrt[1 + 3*x^2 + x^4]) + (7*Sqrt[(2 + (3 - Sqrt[5])*x^2)/(2 + (3 + Sqrt[5])*x^2)]*(2
+ (3 + Sqrt[5])*x^2)*EllipticF[ArcTan[Sqrt[(3 + Sqrt[5])/2]*x], (-5 + 3*Sqrt[5])/2])/(6*Sqrt[2*(3 + Sqrt[5])]*
Sqrt[1 + 3*x^2 + x^4]) + ((1 - I*Sqrt[3])*Sqrt[(2 + (3 - Sqrt[5])*x^2)/(2 + (3 + Sqrt[5])*x^2)]*(2 + (3 + Sqrt
[5])*x^2)*EllipticF[ArcTan[Sqrt[(3 + Sqrt[5])/2]*x], (-5 + 3*Sqrt[5])/2])/(24*Sqrt[2*(3 + Sqrt[5])]*Sqrt[1 + 3
*x^2 + x^4]) + ((1 + I*Sqrt[3])*Sqrt[(2 + (3 - Sqrt[5])*x^2)/(2 + (3 + Sqrt[5])*x^2)]*(2 + (3 + Sqrt[5])*x^2)*
EllipticF[ArcTan[Sqrt[(3 + Sqrt[5])/2]*x], (-5 + 3*Sqrt[5])/2])/(24*Sqrt[2*(3 + Sqrt[5])]*Sqrt[1 + 3*x^2 + x^4
]) - ((5*I - Sqrt[3])*Sqrt[(2 + (3 - Sqrt[5])*x^2)/(2 + (3 + Sqrt[5])*x^2)]*(2 + (3 + Sqrt[5])*x^2)*EllipticF[
ArcTan[Sqrt[(3 + Sqrt[5])/2]*x], (-5 + 3*Sqrt[5])/2])/(6*(I - Sqrt[3])*(2 - I*Sqrt[3] - Sqrt[5])*Sqrt[2*(3 + S
qrt[5])]*Sqrt[1 + 3*x^2 + x^4]) - ((2*I + Sqrt[3])*Sqrt[(2 + (3 - Sqrt[5])*x^2)/(2 + (3 + Sqrt[5])*x^2)]*(2 +
(3 + Sqrt[5])*x^2)*EllipticF[ArcTan[Sqrt[(3 + Sqrt[5])/2]*x], (-5 + 3*Sqrt[5])/2])/(3*(I - Sqrt[3])*(2 - I*Sqr
t[3] - Sqrt[5])*Sqrt[2*(3 + Sqrt[5])]*Sqrt[1 + 3*x^2 + x^4]) - ((2*I - Sqrt[3])*Sqrt[(2 + (3 - Sqrt[5])*x^2)/(
2 + (3 + Sqrt[5])*x^2)]*(2 + (3 + Sqrt[5])*x^2)*EllipticF[ArcTan[Sqrt[(3 + Sqrt[5])/2]*x], (-5 + 3*Sqrt[5])/2]
)/(3*(I + Sqrt[3])*(2 + I*Sqrt[3] - Sqrt[5])*Sqrt[2*(3 + Sqrt[5])]*Sqrt[1 + 3*x^2 + x^4]) + ((I + Sqrt[3])*Sqr
t[(2 + (3 - Sqrt[5])*x^2)/(2 + (3 + Sqrt[5])*x^2)]*(2 + (3 + Sqrt[5])*x^2)*EllipticF[ArcTan[Sqrt[(3 + Sqrt[5])
/2]*x], (-5 + 3*Sqrt[5])/2])/(4*(2*I - Sqrt[3] - I*Sqrt[5])*Sqrt[2*(3 + Sqrt[5])]*Sqrt[1 + 3*x^2 + x^4]) + ((7
*I + 3*Sqrt[3])*Sqrt[(2 + (3 - Sqrt[5])*x^2)/(2 + (3 + Sqrt[5])*x^2)]*(2 + (3 + Sqrt[5])*x^2)*EllipticF[ArcTan
[Sqrt[(3 + Sqrt[5])/2]*x], (-5 + 3*Sqrt[5])/2])/(4*(2*I - Sqrt[3] - I*Sqrt[5])*Sqrt[2*(3 + Sqrt[5])]*Sqrt[1 +
3*x^2 + x^4]) + ((7*I - 3*Sqrt[3])*Sqrt[(2 + (3 - Sqrt[5])*x^2)/(2 + (3 + Sqrt[5])*x^2)]*(2 + (3 + Sqrt[5])*x^
2)*EllipticF[ArcTan[Sqrt[(3 + Sqrt[5])/2]*x], (-5 + 3*Sqrt[5])/2])/(4*(2*I + Sqrt[3] - I*Sqrt[5])*Sqrt[2*(3 +
Sqrt[5])]*Sqrt[1 + 3*x^2 + x^4]) + ((I - Sqrt[3])*Sqrt[(2 + (3 - Sqrt[5])*x^2)/(2 + (3 + Sqrt[5])*x^2)]*(2 + (
3 + Sqrt[5])*x^2)*EllipticF[ArcTan[Sqrt[(3 + Sqrt[5])/2]*x], (-5 + 3*Sqrt[5])/2])/(4*(2*I + Sqrt[3] - I*Sqrt[5
])*Sqrt[2*(3 + Sqrt[5])]*Sqrt[1 + 3*x^2 + x^4]) - ((I + Sqrt[3])*Sqrt[(2 + (3 - Sqrt[5])*x^2)/(2 + (3 + Sqrt[5
])*x^2)]*(2 + (3 + Sqrt[5])*x^2)*EllipticF[ArcTan[Sqrt[(3 + Sqrt[5])/2]*x], (-5 + 3*Sqrt[5])/2])/(2*(2 - I*Sqr
t[3] - Sqrt[5])*Sqrt[6*(3 + Sqrt[5])]*Sqrt[1 + 3*x^2 + x^4]) + ((I - Sqrt[3])*Sqrt[(2 + (3 - Sqrt[5])*x^2)/(2
+ (3 + Sqrt[5])*x^2)]*(2 + (3 + Sqrt[5])*x^2)*EllipticF[ArcTan[Sqrt[(3 + Sqrt[5])/2]*x], (-5 + 3*Sqrt[5])/2])/
(2*(2 + I*Sqrt[3] - Sqrt[5])*Sqrt[6*(3 + Sqrt[5])]*Sqrt[1 + 3*x^2 + x^4]) - ((5*I + Sqrt[3])*Sqrt[(2 + (3 - Sq
rt[5])*x^2)/(2 + (3 + Sqrt[5])*x^2)]*(2 + (3 + Sqrt[5])*x^2)*EllipticF[ArcTan[Sqrt[(3 + Sqrt[5])/2]*x], (-5 +
3*Sqrt[5])/2])/(6*Sqrt[2*(3 + Sqrt[5])]*(5*I + Sqrt[3] - I*Sqrt[5] - Sqrt[15])*Sqrt[1 + 3*x^2 + x^4]) + ((1 +
I*Sqrt[3])*Sqrt[(9 - 4*Sqrt[5])/3]*(3 + Sqrt[5] + 2*x^2)*EllipticPi[1 - (2*(3 - Sqrt[5]))/(I - Sqrt[3])^2, Arc
Tan[Sqrt[(3 + Sqrt[5])/2]*x], (-5 + 3*Sqrt[5])/2])/(2*(5*I + Sqrt[3] - I*Sqrt[5] - Sqrt[15])*Sqrt[(3 + Sqrt[5]
+ 2*x^2)/(3 - Sqrt[5] + 2*x^2)]*Sqrt[1 + 3*x^2 + x^4]) + ((I - Sqrt[3])*(1 + (5*I)*Sqrt[3])*Sqrt[9 - 4*Sqrt[5
]]*(3 + Sqrt[5] + 2*x^2)*EllipticPi[1 - (2*(3 - Sqrt[5]))/(I - Sqrt[3])^2, ArcTan[Sqrt[(3 + Sqrt[5])/2]*x], (-
5 + 3*Sqrt[5])/2])/(8*(5*I + Sqrt[3] - I*Sqrt[5] - Sqrt[15])*Sqrt[(3 + Sqrt[5] + 2*x^2)/(3 - Sqrt[5] + 2*x^2)]
*Sqrt[1 + 3*x^2 + x^4]) - ((I + Sqrt[3])*Sqrt[9 - 4*Sqrt[5]]*(3 + Sqrt[5] + 2*x^2)*EllipticPi[1 - (2*(3 - Sqrt
[5]))/(I - Sqrt[3])^2, ArcTan[Sqrt[(3 + Sqrt[5])/2]*x], (-5 + 3*Sqrt[5])/2])/(4*(5*I + Sqrt[3] - I*Sqrt[5] - S
qrt[15])*Sqrt[(3 + Sqrt[5] + 2*x^2)/(3 - Sqrt[5] + 2*x^2)]*Sqrt[1 + 3*x^2 + x^4]) - ((I - Sqrt[3])*(I + 3*Sqrt
[3])*Sqrt[9 - 4*Sqrt[5]]*(3 + Sqrt[5] + 2*x^2)*EllipticPi[1 - (2*(3 - Sqrt[5]))/(I - Sqrt[3])^2, ArcTan[Sqrt[(
3 + Sqrt[5])/2]*x], (-5 + 3*Sqrt[5])/2])/(24*(1 - (3*I)*Sqrt[3] + Sqrt[5] + I*Sqrt[15])*Sqrt[(3 + Sqrt[5] + 2*
x^2)/(3 - Sqrt[5] + 2*x^2)]*Sqrt[1 + 3*x^2 + x^4]) + ((I + Sqrt[3])*(I + 3*Sqrt[3])*Sqrt[9 - 4*Sqrt[5]]*(3 + S
qrt[5] + 2*x^2)*EllipticPi[1 - (2*(3 - Sqrt[5]))/(I - Sqrt[3])^2, ArcTan[Sqrt[(3 + Sqrt[5])/2]*x], (-5 + 3*Sqr
t[5])/2])/(24*(1 - (3*I)*Sqrt[3] + Sqrt[5] + I*Sqrt[15])*Sqrt[(3 + Sqrt[5] + 2*x^2)/(3 - Sqrt[5] + 2*x^2)]*Sqr
t[1 + 3*x^2 + x^4]) - ((1 - I*Sqrt[3])*Sqrt[(9 - 4*Sqrt[5])/3]*(3 + Sqrt[5] + 2*x^2)*EllipticPi[1 - (2*(3 - Sq
rt[5]))/(I + Sqrt[3])^2, ArcTan[Sqrt[(3 + Sqrt[5])/2]*x], (-5 + 3*Sqrt[5])/2])/(2*(5*I - Sqrt[3] - I*Sqrt[5] +
Sqrt[15])*Sqrt[(3 + Sqrt[5] + 2*x^2)/(3 - Sqrt[5] + 2*x^2)]*Sqrt[1 + 3*x^2 + x^4]) - ((I - Sqrt[3])*Sqrt[9 -
4*Sqrt[5]]*(3 + Sqrt[5] + 2*x^2)*EllipticPi[1 - (2*(3 - Sqrt[5]))/(I + Sqrt[3])^2, ArcTan[Sqrt[(3 + Sqrt[5])/2
]*x], (-5 + 3*Sqrt[5])/2])/(4*(5*I - Sqrt[3] - I*Sqrt[5] + Sqrt[15])*Sqrt[(3 + Sqrt[5] + 2*x^2)/(3 - Sqrt[5] +
2*x^2)]*Sqrt[1 + 3*x^2 + x^4]) + ((1 - (5*I)*Sqrt[3])*(I + Sqrt[3])*Sqrt[9 - 4*Sqrt[5]]*(3 + Sqrt[5] + 2*x^2)
*EllipticPi[1 - (2*(3 - Sqrt[5]))/(I + Sqrt[3])^2, ArcTan[Sqrt[(3 + Sqrt[5])/2]*x], (-5 + 3*Sqrt[5])/2])/(8*(5
*I - Sqrt[3] - I*Sqrt[5] + Sqrt[15])*Sqrt[(3 + Sqrt[5] + 2*x^2)/(3 - Sqrt[5] + 2*x^2)]*Sqrt[1 + 3*x^2 + x^4])
- ((I - Sqrt[3])*(1 + (3*I)*Sqrt[3])*Sqrt[9 - 4*Sqrt[5]]*(3 + Sqrt[5] + 2*x^2)*EllipticPi[1 - (2*(3 - Sqrt[5])
)/(I + Sqrt[3])^2, ArcTan[Sqrt[(3 + Sqrt[5])/2]*x], (-5 + 3*Sqrt[5])/2])/(24*(I - 3*Sqrt[3] + I*Sqrt[5] + Sqrt
[15])*Sqrt[(3 + Sqrt[5] + 2*x^2)/(3 - Sqrt[5] + 2*x^2)]*Sqrt[1 + 3*x^2 + x^4]) + ((1 + (3*I)*Sqrt[3])*(I + Sqr
t[3])*Sqrt[9 - 4*Sqrt[5]]*(3 + Sqrt[5] + 2*x^2)*EllipticPi[1 - (2*(3 - Sqrt[5]))/(I + Sqrt[3])^2, ArcTan[Sqrt[
(3 + Sqrt[5])/2]*x], (-5 + 3*Sqrt[5])/2])/(24*(I - 3*Sqrt[3] + I*Sqrt[5] + Sqrt[15])*Sqrt[(3 + Sqrt[5] + 2*x^2
)/(3 - Sqrt[5] + 2*x^2)]*Sqrt[1 + 3*x^2 + x^4])

Rule 2

Int[(u_.)*((a_.) + (b_.)*(x_)^(n_.))^(p_.), x_Symbol] :> Int[u*a^p, x] /; FreeQ[{a, b, n, p}, x] && EqQ[b, 0]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 539

Int[Sqrt[(c_) + (d_.)*(x_)^2]/(((a_) + (b_.)*(x_)^2)*Sqrt[(e_) + (f_.)*(x_)^2]), x_Symbol] :> Simp[(c*Sqrt[e +
f*x^2]*EllipticPi[1 - (b*c)/(a*d), ArcTan[Rt[d/c, 2]*x], 1 - (c*f)/(d*e)])/(a*e*Rt[d/c, 2]*Sqrt[c + d*x^2]*Sq
rt[(c*(e + f*x^2))/(e*(c + d*x^2))]), x] /; FreeQ[{a, b, c, d, e, f}, x] && PosQ[d/c]

Rule 724

Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symbol] :> Dist[-2, Subst[Int[1/(4*c*d
^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, (2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a,
b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[2*c*d - b*e, 0]

Rule 1099

Int[1/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[((2*a + (b +
q)*x^2)*Sqrt[(2*a + (b - q)*x^2)/(2*a + (b + q)*x^2)]*EllipticF[ArcTan[Rt[(b + q)/(2*a), 2]*x], (2*q)/(b + q)]
)/(2*a*Rt[(b + q)/(2*a), 2]*Sqrt[a + b*x^2 + c*x^4]), x] /; PosQ[(b + q)/a] &&  !(PosQ[(b - q)/a] && SimplerSq
rtQ[(b - q)/(2*a), (b + q)/(2*a)])] /; FreeQ[{a, b, c}, x] && GtQ[b^2 - 4*a*c, 0]

Rule 1135

Int[(x_)^2/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[(x*(b +
q + 2*c*x^2))/(2*c*Sqrt[a + b*x^2 + c*x^4]), x] - Simp[(Rt[(b + q)/(2*a), 2]*(2*a + (b + q)*x^2)*Sqrt[(2*a + (
b - q)*x^2)/(2*a + (b + q)*x^2)]*EllipticE[ArcTan[Rt[(b + q)/(2*a), 2]*x], (2*q)/(b + q)])/(2*c*Sqrt[a + b*x^2
+ c*x^4]), x] /; PosQ[(b + q)/a] &&  !(PosQ[(b - q)/a] && SimplerSqrtQ[(b - q)/(2*a), (b + q)/(2*a)])] /; Fre
eQ[{a, b, c}, x] && GtQ[b^2 - 4*a*c, 0]

Rule 1189

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}
, Dist[d, Int[1/Sqrt[a + b*x^2 + c*x^4], x], x] + Dist[e, Int[x^2/Sqrt[a + b*x^2 + c*x^4], x], x] /; PosQ[(b +
q)/a] || PosQ[(b - q)/a]] /; FreeQ[{a, b, c, d, e}, x] && GtQ[b^2 - 4*a*c, 0]

Rule 1214

Int[1/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4]), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c,
2]}, Dist[(2*c)/(2*c*d - e*(b - q)), Int[1/Sqrt[a + b*x^2 + c*x^4], x], x] - Dist[e/(2*c*d - e*(b - q)), Int[
(b - q + 2*c*x^2)/((d + e*x^2)*Sqrt[a + b*x^2 + c*x^4]), x], x]] /; FreeQ[{a, b, c, d, e}, x] && GtQ[b^2 - 4*a
*c, 0] &&  !LtQ[c, 0]

Rule 1247

Int[(x_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2, Subst[
Int[(d + e*x)^q*(a + b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, d, e, p, q}, x]

Rule 1456

Int[((d_) + (e_.)*(x_)^(n_))^(q_.)*((f_) + (g_.)*(x_)^(n_))^(r_.)*((a_) + (b_.)*(x_)^(n_) + (c_.)*(x_)^(n2_))^
(p_), x_Symbol] :> Dist[(a + b*x^n + c*x^(2*n))^FracPart[p]/((d + e*x^n)^FracPart[p]*(a/d + (c*x^n)/e)^FracPar
t[p]), Int[(d + e*x^n)^(p + q)*(f + g*x^n)^r*(a/d + (c*x^n)/e)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, n, p,
q, r}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] &&  !IntegerQ[p]

Rule 1716

Int[(P4x_)/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4]), x_Symbol] :> With[{A = Coeff[P4x,
x, 0], B = Coeff[P4x, x, 2], C = Coeff[P4x, x, 4]}, -Dist[(e^2)^(-1), Int[(C*d - B*e - C*e*x^2)/Sqrt[a + b*x^
2 + c*x^4], x], x] + Dist[(C*d^2 - B*d*e + A*e^2)/e^2, Int[1/((d + e*x^2)*Sqrt[a + b*x^2 + c*x^4]), x], x]] /;
FreeQ[{a, b, c, d, e}, x] && PolyQ[P4x, x^2, 2] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && Ne
Q[c*d^2 - a*e^2, 0]

Rule 1724

Int[1/(((d_) + (e_.)*(x_))*Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4]), x_Symbol] :> Dist[d, Int[1/((d^2 - e^2*x
^2)*Sqrt[a + b*x^2 + c*x^4]), x], x] - Dist[e, Int[x/((d^2 - e^2*x^2)*Sqrt[a + b*x^2 + c*x^4]), x], x] /; Free
Q[{a, b, c, d, e}, x]

Rule 1726

Int[((d_) + (e_.)*(x_))^(q_)/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> Simp[(e^3*(d + e*x)^(q + 1
)*Sqrt[a + b*x^2 + c*x^4])/((q + 1)*(c*d^4 + b*d^2*e^2 + a*e^4)), x] + Dist[1/((q + 1)*(c*d^4 + b*d^2*e^2 + a*
e^4)), Int[((d + e*x)^(q + 1)*Simp[d*(q + 1)*(c*d^2 + b*e^2) - e*(c*d^2*(q + 1) + b*e^2*(q + 2))*x + c*d*e^2*(
q + 1)*x^2 - c*e^3*(q + 3)*x^3, x])/Sqrt[a + b*x^2 + c*x^4], x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[c*d^4
+ b*d^2*e^2 + a*e^4, 0] && ILtQ[q, -1]

Rule 1741

Int[(Px_)/(((d_) + (e_.)*(x_))*Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4]), x_Symbol] :> With[{A = Coeff[Px, x,
0], B = Coeff[Px, x, 1], C = Coeff[Px, x, 2], D = Coeff[Px, x, 3]}, Int[(x*(B*d - A*e + (d*D - C*e)*x^2))/((d^
2 - e^2*x^2)*Sqrt[a + b*x^2 + c*x^4]), x] + Int[(A*d + (C*d - B*e)*x^2 - D*e*x^4)/((d^2 - e^2*x^2)*Sqrt[a + b*
x^2 + c*x^4]), x]] /; FreeQ[{a, b, c, d, e}, x] && PolyQ[Px, x] && LeQ[Expon[Px, x], 3] && NeQ[c*d^4 + b*d^2*e
^2 + a*e^4, 0]

Rule 6688

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, x]]

Rule 6728

Int[(u_)/((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a +
b*x^n + c*x^(2*n)), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps

\begin {align*} \int \frac {\left (-1+x^2\right ) \left (1-x+x^2-x^3+x^4\right )}{\left (1-x+x^2\right )^2 \left (1+x+x^2\right ) \sqrt {1+3 x^2+x^4}} \, dx &=\int \frac {-1+x-x^5+x^6}{\left (1-x+x^2\right )^2 \left (1+x+x^2\right ) \sqrt {1+3 x^2+x^4}} \, dx\\ &=\int \left (\frac {1}{\sqrt {1+3 x^2+x^4}}+\frac {1+x}{2 \left (1-x+x^2\right )^2 \sqrt {1+3 x^2+x^4}}+\frac {-8+x}{4 \left (1-x+x^2\right ) \sqrt {1+3 x^2+x^4}}+\frac {-2-x}{4 \left (1+x+x^2\right ) \sqrt {1+3 x^2+x^4}}\right ) \, dx\\ &=\frac {1}{4} \int \frac {-8+x}{\left (1-x+x^2\right ) \sqrt {1+3 x^2+x^4}} \, dx+\frac {1}{4} \int \frac {-2-x}{\left (1+x+x^2\right ) \sqrt {1+3 x^2+x^4}} \, dx+\frac {1}{2} \int \frac {1+x}{\left (1-x+x^2\right )^2 \sqrt {1+3 x^2+x^4}} \, dx+\int \frac {1}{\sqrt {1+3 x^2+x^4}} \, dx\\ &=\text {rest of steps removed due to Latex formating problem} \end {align*}

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Mathematica [C]  time = 1.71, size = 767, normalized size = 7.17 \begin {gather*} \frac {-40 i \left (1+\sqrt [3]{-1}\right ) \left (x^2-x+1\right ) \sqrt {2 x^2-\sqrt {5}+3} \sqrt {2 x^2+\sqrt {5}+3} F\left (i \sinh ^{-1}\left (\sqrt {\frac {2}{3+\sqrt {5}}} x\right )|\frac {7}{2}+\frac {3 \sqrt {5}}{2}\right )+40 (-1)^{5/6} \left (x^2-x+1\right ) \sqrt {2 x^2-\sqrt {5}+3} \sqrt {2 x^2+\sqrt {5}+3} \Pi \left (\frac {1}{2} \sqrt [3]{-1} \left (3+\sqrt {5}\right );i \sinh ^{-1}\left (\sqrt {\frac {2}{3+\sqrt {5}}} x\right )|\frac {1}{2} \left (7+3 \sqrt {5}\right )\right )+40 i \left (x^2-x+1\right ) \sqrt {2 x^2-\sqrt {5}+3} \sqrt {2 x^2+\sqrt {5}+3} \Pi \left (\frac {1}{2} \sqrt [3]{-1} \left (3+\sqrt {5}\right );i \sinh ^{-1}\left (\sqrt {\frac {2}{3+\sqrt {5}}} x\right )|\frac {1}{2} \left (7+3 \sqrt {5}\right )\right )+40 (-1)^{5/6} \left (x^2-x+1\right ) \sqrt {2 x^2-\sqrt {5}+3} \sqrt {2 x^2+\sqrt {5}+3} \Pi \left (-\frac {1}{2} (-1)^{2/3} \left (3+\sqrt {5}\right );i \sinh ^{-1}\left (\sqrt {\frac {2}{3+\sqrt {5}}} x\right )|\frac {1}{2} \left (7+3 \sqrt {5}\right )\right )+40 i \left (x^2-x+1\right ) \sqrt {2 x^2-\sqrt {5}+3} \sqrt {2 x^2+\sqrt {5}+3} \Pi \left (-\frac {1}{2} (-1)^{2/3} \left (3+\sqrt {5}\right );i \sinh ^{-1}\left (\sqrt {\frac {2}{3+\sqrt {5}}} x\right )|\frac {1}{2} \left (7+3 \sqrt {5}\right )\right )+8 \left (1+\sqrt [3]{-1}\right ) \sqrt {6-2 \sqrt {5}} \left (x^4+3 x^2+1\right )+3 \left (1+\sqrt [3]{-1}\right ) \sqrt {6-2 \sqrt {5}} \left (x^2-x+1\right ) \sqrt {x^4+3 x^2+1} \left (\left (1-i \sqrt {3}\right )^{3/2} \tan ^{-1}\left (\frac {\left (4-2 i \sqrt {3}\right ) x^2-3 i \sqrt {3}+1}{4 \sqrt {1+i \sqrt {3}} \sqrt {x^4+3 x^2+1}}\right )+\left (1+i \sqrt {3}\right )^{3/2} \tan ^{-1}\left (\frac {\left (4+2 i \sqrt {3}\right ) x^2+3 i \sqrt {3}+1}{4 \sqrt {1-i \sqrt {3}} \sqrt {x^4+3 x^2+1}}\right )\right )}{32 \left (1+\sqrt [3]{-1}\right ) \sqrt {6-2 \sqrt {5}} \left (x^2-x+1\right ) \sqrt {x^4+3 x^2+1}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[((-1 + x^2)*(1 - x + x^2 - x^3 + x^4))/((1 - x + x^2)^2*(1 + x + x^2)*Sqrt[1 + 3*x^2 + x^4]),x]

[Out]

(8*(1 + (-1)^(1/3))*Sqrt[6 - 2*Sqrt[5]]*(1 + 3*x^2 + x^4) + 3*(1 + (-1)^(1/3))*Sqrt[6 - 2*Sqrt[5]]*(1 - x + x^
2)*Sqrt[1 + 3*x^2 + x^4]*((1 - I*Sqrt[3])^(3/2)*ArcTan[(1 - (3*I)*Sqrt[3] + (4 - (2*I)*Sqrt[3])*x^2)/(4*Sqrt[1
+ I*Sqrt[3]]*Sqrt[1 + 3*x^2 + x^4])] + (1 + I*Sqrt[3])^(3/2)*ArcTan[(1 + (3*I)*Sqrt[3] + (4 + (2*I)*Sqrt[3])*
x^2)/(4*Sqrt[1 - I*Sqrt[3]]*Sqrt[1 + 3*x^2 + x^4])]) - (40*I)*(1 + (-1)^(1/3))*(1 - x + x^2)*Sqrt[3 - Sqrt[5]
+ 2*x^2]*Sqrt[3 + Sqrt[5] + 2*x^2]*EllipticF[I*ArcSinh[Sqrt[2/(3 + Sqrt[5])]*x], 7/2 + (3*Sqrt[5])/2] + (40*I)
*(1 - x + x^2)*Sqrt[3 - Sqrt[5] + 2*x^2]*Sqrt[3 + Sqrt[5] + 2*x^2]*EllipticPi[((-1)^(1/3)*(3 + Sqrt[5]))/2, I*
ArcSinh[Sqrt[2/(3 + Sqrt[5])]*x], (7 + 3*Sqrt[5])/2] + 40*(-1)^(5/6)*(1 - x + x^2)*Sqrt[3 - Sqrt[5] + 2*x^2]*S
qrt[3 + Sqrt[5] + 2*x^2]*EllipticPi[((-1)^(1/3)*(3 + Sqrt[5]))/2, I*ArcSinh[Sqrt[2/(3 + Sqrt[5])]*x], (7 + 3*S
qrt[5])/2] + (40*I)*(1 - x + x^2)*Sqrt[3 - Sqrt[5] + 2*x^2]*Sqrt[3 + Sqrt[5] + 2*x^2]*EllipticPi[-1/2*((-1)^(2
/3)*(3 + Sqrt[5])), I*ArcSinh[Sqrt[2/(3 + Sqrt[5])]*x], (7 + 3*Sqrt[5])/2] + 40*(-1)^(5/6)*(1 - x + x^2)*Sqrt[
3 - Sqrt[5] + 2*x^2]*Sqrt[3 + Sqrt[5] + 2*x^2]*EllipticPi[-1/2*((-1)^(2/3)*(3 + Sqrt[5])), I*ArcSinh[Sqrt[2/(3
+ Sqrt[5])]*x], (7 + 3*Sqrt[5])/2])/(32*(1 + (-1)^(1/3))*Sqrt[6 - 2*Sqrt[5]]*(1 - x + x^2)*Sqrt[1 + 3*x^2 + x
^4])

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IntegrateAlgebraic [A]  time = 2.10, size = 107, normalized size = 1.00 \begin {gather*} \frac {\sqrt {1+3 x^2+x^4}}{4 \left (1-x+x^2\right )}-\sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {2} x}{1-x+x^2+\sqrt {1+3 x^2+x^4}}\right )-\frac {\tanh ^{-1}\left (\frac {\sqrt {2} x}{1+x+x^2+\sqrt {1+3 x^2+x^4}}\right )}{2 \sqrt {2}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((-1 + x^2)*(1 - x + x^2 - x^3 + x^4))/((1 - x + x^2)^2*(1 + x + x^2)*Sqrt[1 + 3*x^2 + x^4]
),x]

[Out]

Sqrt[1 + 3*x^2 + x^4]/(4*(1 - x + x^2)) - Sqrt[2]*ArcTanh[(Sqrt[2]*x)/(1 - x + x^2 + Sqrt[1 + 3*x^2 + x^4])] -
ArcTanh[(Sqrt[2]*x)/(1 + x + x^2 + Sqrt[1 + 3*x^2 + x^4])]/(2*Sqrt[2])

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fricas [B]  time = 0.54, size = 184, normalized size = 1.72 \begin {gather*} \frac {\sqrt {2} {\left (x^{2} - x + 1\right )} \log \left (\frac {3 \, x^{4} - 2 \, x^{3} + 2 \, \sqrt {2} \sqrt {x^{4} + 3 \, x^{2} + 1} {\left (x^{2} - x + 1\right )} + 9 \, x^{2} - 2 \, x + 3}{x^{4} + 2 \, x^{3} + 3 \, x^{2} + 2 \, x + 1}\right ) + 4 \, \sqrt {2} {\left (x^{2} - x + 1\right )} \log \left (\frac {3 \, x^{4} + 2 \, x^{3} - 2 \, \sqrt {2} \sqrt {x^{4} + 3 \, x^{2} + 1} {\left (x^{2} + x + 1\right )} + 9 \, x^{2} + 2 \, x + 3}{x^{4} - 2 \, x^{3} + 3 \, x^{2} - 2 \, x + 1}\right ) + 4 \, \sqrt {x^{4} + 3 \, x^{2} + 1}}{16 \, {\left (x^{2} - x + 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2-1)*(x^4-x^3+x^2-x+1)/(x^2-x+1)^2/(x^2+x+1)/(x^4+3*x^2+1)^(1/2),x, algorithm="fricas")

[Out]

1/16*(sqrt(2)*(x^2 - x + 1)*log((3*x^4 - 2*x^3 + 2*sqrt(2)*sqrt(x^4 + 3*x^2 + 1)*(x^2 - x + 1) + 9*x^2 - 2*x +
3)/(x^4 + 2*x^3 + 3*x^2 + 2*x + 1)) + 4*sqrt(2)*(x^2 - x + 1)*log((3*x^4 + 2*x^3 - 2*sqrt(2)*sqrt(x^4 + 3*x^2
+ 1)*(x^2 + x + 1) + 9*x^2 + 2*x + 3)/(x^4 - 2*x^3 + 3*x^2 - 2*x + 1)) + 4*sqrt(x^4 + 3*x^2 + 1))/(x^2 - x +
1)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (x^{4} - x^{3} + x^{2} - x + 1\right )} {\left (x^{2} - 1\right )}}{\sqrt {x^{4} + 3 \, x^{2} + 1} {\left (x^{2} + x + 1\right )} {\left (x^{2} - x + 1\right )}^{2}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2-1)*(x^4-x^3+x^2-x+1)/(x^2-x+1)^2/(x^2+x+1)/(x^4+3*x^2+1)^(1/2),x, algorithm="giac")

[Out]

integrate((x^4 - x^3 + x^2 - x + 1)*(x^2 - 1)/(sqrt(x^4 + 3*x^2 + 1)*(x^2 + x + 1)*(x^2 - x + 1)^2), x)

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maple [C]  time = 1.65, size = 320, normalized size = 2.99

method	result	size
trager	\(\frac {\sqrt {x^{4}+3 x^{2}+1}}{4 x^{2}-4 x +4}+\frac {\RootOf \left (\textit {\_Z}^{2}-2\right ) \ln \left (-\frac {-7 \RootOf \left (\textit {\_Z}^{2}-2\right ) x^{10}-29 \RootOf \left (\textit {\_Z}^{2}-2\right ) x^{9}+10 \sqrt {x^{4}+3 x^{2}+1}\, x^{8}-89 \RootOf \left (\textit {\_Z}^{2}-2\right ) x^{8}+40 \sqrt {x^{4}+3 x^{2}+1}\, x^{7}-190 \RootOf \left (\textit {\_Z}^{2}-2\right ) x^{7}+116 \sqrt {x^{4}+3 x^{2}+1}\, x^{6}-283 \RootOf \left (\textit {\_Z}^{2}-2\right ) x^{6}+192 \sqrt {x^{4}+3 x^{2}+1}\, x^{5}-363 \RootOf \left (\textit {\_Z}^{2}-2\right ) x^{5}+270 \sqrt {x^{4}+3 x^{2}+1}\, x^{4}-283 \RootOf \left (\textit {\_Z}^{2}-2\right ) x^{4}+192 \sqrt {x^{4}+3 x^{2}+1}\, x^{3}-190 \RootOf \left (\textit {\_Z}^{2}-2\right ) x^{3}+116 \sqrt {x^{4}+3 x^{2}+1}\, x^{2}-89 \RootOf \left (\textit {\_Z}^{2}-2\right ) x^{2}+40 \sqrt {x^{4}+3 x^{2}+1}\, x -29 \RootOf \left (\textit {\_Z}^{2}-2\right ) x +10 \sqrt {x^{4}+3 x^{2}+1}-7 \RootOf \left (\textit {\_Z}^{2}-2\right )}{\left (x^{2}+x +1\right ) \left (x^{2}-x +1\right )^{4}}\right )}{8}\)	\(320\)
elliptic	\(-\frac {\sqrt {-\frac {\left (x^{2}-1\right )^{2}}{\left (-x^{2}-1\right )^{2}}+5}\, \left (3 \sqrt {2}\, \arctanh \left (\frac {\sqrt {-\frac {\left (x^{2}-1\right )^{2}}{\left (-x^{2}-1\right )^{2}}+5}\, \sqrt {2}}{4}\right )+\sqrt {6}\, \arctan \left (\frac {2 \sqrt {6}\, \sqrt {-\frac {\left (x^{2}-1\right )^{2}}{\left (-x^{2}-1\right )^{2}}+5}\, \left (x^{2}-1\right )}{3 \left (\frac {\left (x^{2}-1\right )^{2}}{\left (-x^{2}-1\right )^{2}}-5\right ) \left (-x^{2}-1\right )}\right )\right )}{12 \sqrt {-\frac {\frac {\left (x^{2}-1\right )^{2}}{\left (-x^{2}-1\right )^{2}}-5}{\left (\frac {x^{2}-1}{-x^{2}-1}+1\right )^{2}}}\, \left (\frac {x^{2}-1}{-x^{2}-1}+1\right )}-\frac {\sqrt {-\frac {\left (x^{2}-1\right )^{2}}{\left (-x^{2}-1\right )^{2}}+5}\, \left (\frac {3 \sqrt {2}\, \arctanh \left (\frac {\sqrt {-\frac {\left (x^{2}-1\right )^{2}}{\left (-x^{2}-1\right )^{2}}+5}\, \sqrt {2}}{4}\right ) \left (x^{2}-1\right )^{2}}{\left (-x^{2}-1\right )^{2}}-\frac {2 \sqrt {6}\, \arctan \left (\frac {2 \sqrt {6}\, \sqrt {-\frac {\left (x^{2}-1\right )^{2}}{\left (-x^{2}-1\right )^{2}}+5}\, \left (x^{2}-1\right )}{3 \left (\frac {\left (x^{2}-1\right )^{2}}{\left (-x^{2}-1\right )^{2}}-5\right ) \left (-x^{2}-1\right )}\right ) \left (x^{2}-1\right )^{2}}{\left (-x^{2}-1\right )^{2}}+9 \sqrt {2}\, \arctanh \left (\frac {\sqrt {-\frac {\left (x^{2}-1\right )^{2}}{\left (-x^{2}-1\right )^{2}}+5}\, \sqrt {2}}{4}\right )-6 \sqrt {6}\, \arctan \left (\frac {2 \sqrt {6}\, \sqrt {-\frac {\left (x^{2}-1\right )^{2}}{\left (-x^{2}-1\right )^{2}}+5}\, \left (x^{2}-1\right )}{3 \left (\frac {\left (x^{2}-1\right )^{2}}{\left (-x^{2}-1\right )^{2}}-5\right ) \left (-x^{2}-1\right )}\right )-12 \sqrt {-\frac {\left (x^{2}-1\right )^{2}}{\left (-x^{2}-1\right )^{2}}+5}\right )}{24 \sqrt {-\frac {\frac {\left (x^{2}-1\right )^{2}}{\left (-x^{2}-1\right )^{2}}-5}{\left (\frac {x^{2}-1}{-x^{2}-1}+1\right )^{2}}}\, \left (\frac {x^{2}-1}{-x^{2}-1}+1\right ) \left (\frac {\left (x^{2}-1\right )^{2}}{\left (-x^{2}-1\right )^{2}}+3\right )}+\frac {\left (\frac {1}{-8+\frac {4 \sqrt {x^{4}+3 x^{2}+1}\, \sqrt {2}}{x}}+\frac {5 \ln \left (-1+\frac {\sqrt {x^{4}+3 x^{2}+1}\, \sqrt {2}}{2 x}\right )}{8}+\frac {1}{8+\frac {4 \sqrt {x^{4}+3 x^{2}+1}\, \sqrt {2}}{x}}-\frac {5 \ln \left (1+\frac {\sqrt {x^{4}+3 x^{2}+1}\, \sqrt {2}}{2 x}\right )}{8}\right ) \sqrt {2}}{2}\)	\(661\)
risch	\(\frac {\sqrt {x^{4}+3 x^{2}+1}}{4 x^{2}-4 x +4}+\frac {5 \sqrt {1-\left (\frac {\sqrt {5}}{2}-\frac {3}{2}\right ) x^{2}}\, \sqrt {1-\left (-\frac {3}{2}-\frac {\sqrt {5}}{2}\right ) x^{2}}\, \EllipticF \left (x \left (\frac {i \sqrt {5}}{2}-\frac {i}{2}\right ), \frac {3}{2}+\frac {\sqrt {5}}{2}\right )}{4 \left (\frac {i \sqrt {5}}{2}-\frac {i}{2}\right ) \sqrt {x^{4}+3 x^{2}+1}}+\frac {\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (\frac {\arctanh \left (\frac {\left (-2+i \sqrt {3}\right ) \left (7 x^{2}+\frac {11}{2}-\frac {5 i \sqrt {3}}{2}\right )}{14 \sqrt {-1-i \sqrt {3}}\, \sqrt {x^{4}+3 x^{2}+1}}\right )}{2 \sqrt {-1-i \sqrt {3}}}-\frac {\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {1-\left (\frac {\sqrt {5}}{2}-\frac {3}{2}\right ) x^{2}}\, \sqrt {1-\left (-\frac {3}{2}-\frac {\sqrt {5}}{2}\right ) x^{2}}\, \EllipticPi \left (\sqrt {\frac {\sqrt {5}}{2}-\frac {3}{2}}\, x , -\frac {\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {5}}{2}+\frac {3}{4}-\frac {3 i \sqrt {3}}{4}, \frac {\sqrt {-\frac {3}{2}-\frac {\sqrt {5}}{2}}}{\sqrt {\frac {\sqrt {5}}{2}-\frac {3}{2}}}\right )}{\sqrt {\frac {\sqrt {5}}{2}-\frac {3}{2}}\, \sqrt {x^{4}+3 x^{2}+1}}\right )}{4}+\frac {\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \left (\frac {\arctanh \left (\frac {\left (-2-i \sqrt {3}\right ) \left (7 x^{2}+\frac {11}{2}+\frac {5 i \sqrt {3}}{2}\right )}{14 \sqrt {-1+i \sqrt {3}}\, \sqrt {x^{4}+3 x^{2}+1}}\right )}{2 \sqrt {-1+i \sqrt {3}}}-\frac {\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {1-\left (\frac {\sqrt {5}}{2}-\frac {3}{2}\right ) x^{2}}\, \sqrt {1-\left (-\frac {3}{2}-\frac {\sqrt {5}}{2}\right ) x^{2}}\, \EllipticPi \left (\sqrt {\frac {\sqrt {5}}{2}-\frac {3}{2}}\, x , -\frac {\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {5}}{2}+\frac {3}{4}+\frac {3 i \sqrt {3}}{4}, \frac {\sqrt {-\frac {3}{2}-\frac {\sqrt {5}}{2}}}{\sqrt {\frac {\sqrt {5}}{2}-\frac {3}{2}}}\right )}{\sqrt {\frac {\sqrt {5}}{2}-\frac {3}{2}}\, \sqrt {x^{4}+3 x^{2}+1}}\right )}{4}+\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {\arctanh \left (\frac {\left (2+i \sqrt {3}\right ) \left (7 x^{2}+\frac {11}{2}+\frac {5 i \sqrt {3}}{2}\right )}{14 \sqrt {-1+i \sqrt {3}}\, \sqrt {x^{4}+3 x^{2}+1}}\right )}{2 \sqrt {-1+i \sqrt {3}}}-\frac {\left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {1-\left (\frac {\sqrt {5}}{2}-\frac {3}{2}\right ) x^{2}}\, \sqrt {1-\left (-\frac {3}{2}-\frac {\sqrt {5}}{2}\right ) x^{2}}\, \EllipticPi \left (\sqrt {\frac {\sqrt {5}}{2}-\frac {3}{2}}\, x , \frac {\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {5}}{2}+\frac {3}{4}+\frac {3 i \sqrt {3}}{4}, \frac {\sqrt {-\frac {3}{2}-\frac {\sqrt {5}}{2}}}{\sqrt {\frac {\sqrt {5}}{2}-\frac {3}{2}}}\right )}{\sqrt {\frac {\sqrt {5}}{2}-\frac {3}{2}}\, \sqrt {x^{4}+3 x^{2}+1}}\right )+\left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \left (-\frac {\arctanh \left (\frac {\left (2-i \sqrt {3}\right ) \left (7 x^{2}+\frac {11}{2}-\frac {5 i \sqrt {3}}{2}\right )}{14 \sqrt {-1-i \sqrt {3}}\, \sqrt {x^{4}+3 x^{2}+1}}\right )}{2 \sqrt {-1-i \sqrt {3}}}-\frac {\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {1-\left (\frac {\sqrt {5}}{2}-\frac {3}{2}\right ) x^{2}}\, \sqrt {1-\left (-\frac {3}{2}-\frac {\sqrt {5}}{2}\right ) x^{2}}\, \EllipticPi \left (\sqrt {\frac {\sqrt {5}}{2}-\frac {3}{2}}\, x , \frac {\left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {5}}{2}+\frac {3}{4}-\frac {3 i \sqrt {3}}{4}, \frac {\sqrt {-\frac {3}{2}-\frac {\sqrt {5}}{2}}}{\sqrt {\frac {\sqrt {5}}{2}-\frac {3}{2}}}\right )}{\sqrt {\frac {\sqrt {5}}{2}-\frac {3}{2}}\, \sqrt {x^{4}+3 x^{2}+1}}\right )\)	\(836\)
default	\(\frac {5 \sqrt {1-\left (\frac {\sqrt {5}}{2}-\frac {3}{2}\right ) x^{2}}\, \sqrt {1-\left (-\frac {3}{2}-\frac {\sqrt {5}}{2}\right ) x^{2}}\, \EllipticF \left (x \left (\frac {i \sqrt {5}}{2}-\frac {i}{2}\right ), \frac {3}{2}+\frac {\sqrt {5}}{2}\right )}{4 \left (\frac {i \sqrt {5}}{2}-\frac {i}{2}\right ) \sqrt {x^{4}+3 x^{2}+1}}+\frac {\left (\frac {1}{2}+\frac {5 i \sqrt {3}}{2}\right ) \left (-\frac {\arctanh \left (\frac {\left (2+i \sqrt {3}\right ) \left (7 x^{2}+\frac {11}{2}+\frac {5 i \sqrt {3}}{2}\right )}{14 \sqrt {-1+i \sqrt {3}}\, \sqrt {x^{4}+3 x^{2}+1}}\right )}{2 \sqrt {-1+i \sqrt {3}}}-\frac {\left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {1-\left (\frac {\sqrt {5}}{2}-\frac {3}{2}\right ) x^{2}}\, \sqrt {1-\left (-\frac {3}{2}-\frac {\sqrt {5}}{2}\right ) x^{2}}\, \EllipticPi \left (\sqrt {\frac {\sqrt {5}}{2}-\frac {3}{2}}\, x , \frac {\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {5}}{2}+\frac {3}{4}+\frac {3 i \sqrt {3}}{4}, \frac {\sqrt {-\frac {3}{2}-\frac {\sqrt {5}}{2}}}{\sqrt {\frac {\sqrt {5}}{2}-\frac {3}{2}}}\right )}{\sqrt {\frac {\sqrt {5}}{2}-\frac {3}{2}}\, \sqrt {x^{4}+3 x^{2}+1}}\right )}{4}+\frac {\left (\frac {1}{2}-\frac {5 i \sqrt {3}}{2}\right ) \left (-\frac {\arctanh \left (\frac {\left (2-i \sqrt {3}\right ) \left (7 x^{2}+\frac {11}{2}-\frac {5 i \sqrt {3}}{2}\right )}{14 \sqrt {-1-i \sqrt {3}}\, \sqrt {x^{4}+3 x^{2}+1}}\right )}{2 \sqrt {-1-i \sqrt {3}}}-\frac {\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {1-\left (\frac {\sqrt {5}}{2}-\frac {3}{2}\right ) x^{2}}\, \sqrt {1-\left (-\frac {3}{2}-\frac {\sqrt {5}}{2}\right ) x^{2}}\, \EllipticPi \left (\sqrt {\frac {\sqrt {5}}{2}-\frac {3}{2}}\, x , \frac {\left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {5}}{2}+\frac {3}{4}-\frac {3 i \sqrt {3}}{4}, \frac {\sqrt {-\frac {3}{2}-\frac {\sqrt {5}}{2}}}{\sqrt {\frac {\sqrt {5}}{2}-\frac {3}{2}}}\right )}{\sqrt {\frac {\sqrt {5}}{2}-\frac {3}{2}}\, \sqrt {x^{4}+3 x^{2}+1}}\right )}{4}+\frac {\sqrt {x^{4}+3 x^{2}+1}}{4 x^{2}-4 x +4}+\frac {\left (\frac {3}{4}-\frac {i \sqrt {3}}{4}\right ) \left (-\frac {\arctanh \left (\frac {\left (2+i \sqrt {3}\right ) \left (7 x^{2}+\frac {11}{2}+\frac {5 i \sqrt {3}}{2}\right )}{14 \sqrt {-1+i \sqrt {3}}\, \sqrt {x^{4}+3 x^{2}+1}}\right )}{2 \sqrt {-1+i \sqrt {3}}}-\frac {\left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {1-\left (\frac {\sqrt {5}}{2}-\frac {3}{2}\right ) x^{2}}\, \sqrt {1-\left (-\frac {3}{2}-\frac {\sqrt {5}}{2}\right ) x^{2}}\, \EllipticPi \left (\sqrt {\frac {\sqrt {5}}{2}-\frac {3}{2}}\, x , \frac {\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {5}}{2}+\frac {3}{4}+\frac {3 i \sqrt {3}}{4}, \frac {\sqrt {-\frac {3}{2}-\frac {\sqrt {5}}{2}}}{\sqrt {\frac {\sqrt {5}}{2}-\frac {3}{2}}}\right )}{\sqrt {\frac {\sqrt {5}}{2}-\frac {3}{2}}\, \sqrt {x^{4}+3 x^{2}+1}}\right )}{2}+\frac {\left (\frac {3}{4}+\frac {i \sqrt {3}}{4}\right ) \left (-\frac {\arctanh \left (\frac {\left (2-i \sqrt {3}\right ) \left (7 x^{2}+\frac {11}{2}-\frac {5 i \sqrt {3}}{2}\right )}{14 \sqrt {-1-i \sqrt {3}}\, \sqrt {x^{4}+3 x^{2}+1}}\right )}{2 \sqrt {-1-i \sqrt {3}}}-\frac {\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {1-\left (\frac {\sqrt {5}}{2}-\frac {3}{2}\right ) x^{2}}\, \sqrt {1-\left (-\frac {3}{2}-\frac {\sqrt {5}}{2}\right ) x^{2}}\, \EllipticPi \left (\sqrt {\frac {\sqrt {5}}{2}-\frac {3}{2}}\, x , \frac {\left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {5}}{2}+\frac {3}{4}-\frac {3 i \sqrt {3}}{4}, \frac {\sqrt {-\frac {3}{2}-\frac {\sqrt {5}}{2}}}{\sqrt {\frac {\sqrt {5}}{2}-\frac {3}{2}}}\right )}{\sqrt {\frac {\sqrt {5}}{2}-\frac {3}{2}}\, \sqrt {x^{4}+3 x^{2}+1}}\right )}{2}+\frac {\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (\frac {\arctanh \left (\frac {\left (-2+i \sqrt {3}\right ) \left (7 x^{2}+\frac {11}{2}-\frac {5 i \sqrt {3}}{2}\right )}{14 \sqrt {-1-i \sqrt {3}}\, \sqrt {x^{4}+3 x^{2}+1}}\right )}{2 \sqrt {-1-i \sqrt {3}}}-\frac {\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {1-\left (\frac {\sqrt {5}}{2}-\frac {3}{2}\right ) x^{2}}\, \sqrt {1-\left (-\frac {3}{2}-\frac {\sqrt {5}}{2}\right ) x^{2}}\, \EllipticPi \left (\sqrt {\frac {\sqrt {5}}{2}-\frac {3}{2}}\, x , -\frac {\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {5}}{2}+\frac {3}{4}-\frac {3 i \sqrt {3}}{4}, \frac {\sqrt {-\frac {3}{2}-\frac {\sqrt {5}}{2}}}{\sqrt {\frac {\sqrt {5}}{2}-\frac {3}{2}}}\right )}{\sqrt {\frac {\sqrt {5}}{2}-\frac {3}{2}}\, \sqrt {x^{4}+3 x^{2}+1}}\right )}{4}+\frac {\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \left (\frac {\arctanh \left (\frac {\left (-2-i \sqrt {3}\right ) \left (7 x^{2}+\frac {11}{2}+\frac {5 i \sqrt {3}}{2}\right )}{14 \sqrt {-1+i \sqrt {3}}\, \sqrt {x^{4}+3 x^{2}+1}}\right )}{2 \sqrt {-1+i \sqrt {3}}}-\frac {\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {1-\left (\frac {\sqrt {5}}{2}-\frac {3}{2}\right ) x^{2}}\, \sqrt {1-\left (-\frac {3}{2}-\frac {\sqrt {5}}{2}\right ) x^{2}}\, \EllipticPi \left (\sqrt {\frac {\sqrt {5}}{2}-\frac {3}{2}}\, x , -\frac {\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {5}}{2}+\frac {3}{4}+\frac {3 i \sqrt {3}}{4}, \frac {\sqrt {-\frac {3}{2}-\frac {\sqrt {5}}{2}}}{\sqrt {\frac {\sqrt {5}}{2}-\frac {3}{2}}}\right )}{\sqrt {\frac {\sqrt {5}}{2}-\frac {3}{2}}\, \sqrt {x^{4}+3 x^{2}+1}}\right )}{4}\)	\(1206\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^2-1)*(x^4-x^3+x^2-x+1)/(x^2-x+1)^2/(x^2+x+1)/(x^4+3*x^2+1)^(1/2),x,method=_RETURNVERBOSE)

[Out]

1/4*(x^4+3*x^2+1)^(1/2)/(x^2-x+1)+1/8*RootOf(_Z^2-2)*ln(-(-7*RootOf(_Z^2-2)*x^10-29*RootOf(_Z^2-2)*x^9+10*(x^4
+3*x^2+1)^(1/2)*x^8-89*RootOf(_Z^2-2)*x^8+40*(x^4+3*x^2+1)^(1/2)*x^7-190*RootOf(_Z^2-2)*x^7+116*(x^4+3*x^2+1)^
(1/2)*x^6-283*RootOf(_Z^2-2)*x^6+192*(x^4+3*x^2+1)^(1/2)*x^5-363*RootOf(_Z^2-2)*x^5+270*(x^4+3*x^2+1)^(1/2)*x^
4-283*RootOf(_Z^2-2)*x^4+192*(x^4+3*x^2+1)^(1/2)*x^3-190*RootOf(_Z^2-2)*x^3+116*(x^4+3*x^2+1)^(1/2)*x^2-89*Roo
tOf(_Z^2-2)*x^2+40*(x^4+3*x^2+1)^(1/2)*x-29*RootOf(_Z^2-2)*x+10*(x^4+3*x^2+1)^(1/2)-7*RootOf(_Z^2-2))/(x^2+x+1
)/(x^2-x+1)^4)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (x^{4} - x^{3} + x^{2} - x + 1\right )} {\left (x^{2} - 1\right )}}{\sqrt {x^{4} + 3 \, x^{2} + 1} {\left (x^{2} + x + 1\right )} {\left (x^{2} - x + 1\right )}^{2}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2-1)*(x^4-x^3+x^2-x+1)/(x^2-x+1)^2/(x^2+x+1)/(x^4+3*x^2+1)^(1/2),x, algorithm="maxima")

[Out]

integrate((x^4 - x^3 + x^2 - x + 1)*(x^2 - 1)/(sqrt(x^4 + 3*x^2 + 1)*(x^2 + x + 1)*(x^2 - x + 1)^2), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\left (x^2-1\right )\,\left (x^4-x^3+x^2-x+1\right )}{{\left (x^2-x+1\right )}^2\,\sqrt {x^4+3\,x^2+1}\,\left (x^2+x+1\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((x^2 - 1)*(x^2 - x - x^3 + x^4 + 1))/((x^2 - x + 1)^2*(3*x^2 + x^4 + 1)^(1/2)*(x + x^2 + 1)),x)

[Out]

int(((x^2 - 1)*(x^2 - x - x^3 + x^4 + 1))/((x^2 - x + 1)^2*(3*x^2 + x^4 + 1)^(1/2)*(x + x^2 + 1)), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (x - 1\right ) \left (x + 1\right ) \left (x^{4} - x^{3} + x^{2} - x + 1\right )}{\left (x^{2} - x + 1\right )^{2} \left (x^{2} + x + 1\right ) \sqrt {x^{4} + 3 x^{2} + 1}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**2-1)*(x**4-x**3+x**2-x+1)/(x**2-x+1)**2/(x**2+x+1)/(x**4+3*x**2+1)**(1/2),x)

[Out]

Integral((x - 1)*(x + 1)*(x**4 - x**3 + x**2 - x + 1)/((x**2 - x + 1)**2*(x**2 + x + 1)*sqrt(x**4 + 3*x**2 + 1
)), x)

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