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

Optimal. Leaf size=69 \[ -\frac {\sqrt {x^4-x^2-1} x}{16 \left (2 x^4-x^2-2\right )}-\frac {1}{16} \sqrt {\frac {3}{2}} \tan ^{-1}\left (\frac {\sqrt {\frac {3}{2}} x}{\sqrt {x^4-x^2-1}}\right ) \]

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Rubi [C]  time = 13.50, antiderivative size = 6713, normalized size of antiderivative = 97.29, number of steps used = 156, number of rules used = 14, integrand size = 51, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.274, Rules used = {6728, 6742, 1226, 1187, 1098, 1184, 1214, 1456, 540, 421, 419, 538, 537, 1208}

result too large to display

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(x*(1 - Sqrt[5] - 2*x^2))/(17*Sqrt[-1 - x^2 + x^4]) - ((17 - 2*Sqrt[17])*x*(1 - Sqrt[5] - 2*x^2))/(544*Sqrt[-1
- x^2 + x^4]) - (x*(1 - Sqrt[5] - 2*x^2))/(34*(1 - Sqrt[17])*Sqrt[-1 - x^2 + x^4]) - (x*(1 - Sqrt[5] - 2*x^2)
)/(34*(1 + Sqrt[17])*Sqrt[-1 - x^2 + x^4]) - ((17 + 2*Sqrt[17])*x*(1 - Sqrt[5] - 2*x^2))/(544*Sqrt[-1 - x^2 +
x^4]) + (x*Sqrt[-1 - x^2 + x^4])/(68*(1 - Sqrt[17] - 4*x^2)) + (4*x*Sqrt[-1 - x^2 + x^4])/(17*(1 - Sqrt[17])*(
1 - Sqrt[17] - 4*x^2)) + (x*Sqrt[-1 - x^2 + x^4])/(68*(1 + Sqrt[17] - 4*x^2)) + (4*x*Sqrt[-1 - x^2 + x^4])/(17
*(1 + Sqrt[17])*(1 + Sqrt[17] - 4*x^2)) + (5^(1/4)*Sqrt[-2 - (1 - Sqrt[5])*x^2]*Sqrt[(2 + (1 + Sqrt[5])*x^2)/(
2 + (1 - Sqrt[5])*x^2)]*EllipticE[ArcSin[(Sqrt[2]*5^(1/4)*x)/Sqrt[-2 - (1 - Sqrt[5])*x^2]], (5 - Sqrt[5])/10])
/(17*Sqrt[(2 + (1 - Sqrt[5])*x^2)^(-1)]*Sqrt[-1 - x^2 + x^4]) - (5^(1/4)*(17 - 2*Sqrt[17])*Sqrt[-2 - (1 - Sqrt
[5])*x^2]*Sqrt[(2 + (1 + Sqrt[5])*x^2)/(2 + (1 - Sqrt[5])*x^2)]*EllipticE[ArcSin[(Sqrt[2]*5^(1/4)*x)/Sqrt[-2 -
(1 - Sqrt[5])*x^2]], (5 - Sqrt[5])/10])/(544*Sqrt[(2 + (1 - Sqrt[5])*x^2)^(-1)]*Sqrt[-1 - x^2 + x^4]) - (5^(1
/4)*Sqrt[-2 - (1 - Sqrt[5])*x^2]*Sqrt[(2 + (1 + Sqrt[5])*x^2)/(2 + (1 - Sqrt[5])*x^2)]*EllipticE[ArcSin[(Sqrt[
2]*5^(1/4)*x)/Sqrt[-2 - (1 - Sqrt[5])*x^2]], (5 - Sqrt[5])/10])/(34*(1 - Sqrt[17])*Sqrt[(2 + (1 - Sqrt[5])*x^2
)^(-1)]*Sqrt[-1 - x^2 + x^4]) - (5^(1/4)*Sqrt[-2 - (1 - Sqrt[5])*x^2]*Sqrt[(2 + (1 + Sqrt[5])*x^2)/(2 + (1 - S
qrt[5])*x^2)]*EllipticE[ArcSin[(Sqrt[2]*5^(1/4)*x)/Sqrt[-2 - (1 - Sqrt[5])*x^2]], (5 - Sqrt[5])/10])/(34*(1 +
Sqrt[17])*Sqrt[(2 + (1 - Sqrt[5])*x^2)^(-1)]*Sqrt[-1 - x^2 + x^4]) - (5^(1/4)*(17 + 2*Sqrt[17])*Sqrt[-2 - (1 -
Sqrt[5])*x^2]*Sqrt[(2 + (1 + Sqrt[5])*x^2)/(2 + (1 - Sqrt[5])*x^2)]*EllipticE[ArcSin[(Sqrt[2]*5^(1/4)*x)/Sqrt
[-2 - (1 - Sqrt[5])*x^2]], (5 - Sqrt[5])/10])/(544*Sqrt[(2 + (1 - Sqrt[5])*x^2)^(-1)]*Sqrt[-1 - x^2 + x^4]) +
(3*(1 + Sqrt[5])*(1 - Sqrt[17])*Sqrt[-1 + Sqrt[5] + 2*x^2]*Sqrt[1 - (2*x^2)/(1 + Sqrt[5])]*EllipticF[ArcSin[Sq
rt[2/(1 + Sqrt[5])]*x], (-3 - Sqrt[5])/2])/(64*Sqrt[2]*(3 + 2*Sqrt[5] - Sqrt[17])*Sqrt[-1 - x^2 + x^4]) - ((1
+ Sqrt[5])*(1 + Sqrt[17])*Sqrt[-1 + Sqrt[5] + 2*x^2]*Sqrt[1 - (2*x^2)/(1 + Sqrt[5])]*EllipticF[ArcSin[Sqrt[2/(
1 + Sqrt[5])]*x], (-3 - Sqrt[5])/2])/(32*Sqrt[34]*(1 + 2*Sqrt[5] - Sqrt[17])*Sqrt[-1 - x^2 + x^4]) + ((1 + Sqr
t[5])*(1 + Sqrt[17])*(2 + Sqrt[17])*Sqrt[-1 + Sqrt[5] + 2*x^2]*Sqrt[1 - (2*x^2)/(1 + Sqrt[5])]*EllipticF[ArcSi
n[Sqrt[2/(1 + Sqrt[5])]*x], (-3 - Sqrt[5])/2])/(64*Sqrt[34]*(1 + 2*Sqrt[5] - Sqrt[17])*Sqrt[-1 - x^2 + x^4]) +
((1 + Sqrt[5])*(1 - Sqrt[17])*Sqrt[-1 + Sqrt[5] + 2*x^2]*Sqrt[1 - (2*x^2)/(1 + Sqrt[5])]*EllipticF[ArcSin[Sqr
t[2/(1 + Sqrt[5])]*x], (-3 - Sqrt[5])/2])/(32*Sqrt[34]*(1 + 2*Sqrt[5] + Sqrt[17])*Sqrt[-1 - x^2 + x^4]) - ((1
+ Sqrt[5])*(1 - Sqrt[17])*(2 - Sqrt[17])*Sqrt[-1 + Sqrt[5] + 2*x^2]*Sqrt[1 - (2*x^2)/(1 + Sqrt[5])]*EllipticF[
ArcSin[Sqrt[2/(1 + Sqrt[5])]*x], (-3 - Sqrt[5])/2])/(64*Sqrt[34]*(1 + 2*Sqrt[5] + Sqrt[17])*Sqrt[-1 - x^2 + x^
4]) + (3*(1 + Sqrt[5])*(1 + Sqrt[17])*Sqrt[-1 + Sqrt[5] + 2*x^2]*Sqrt[1 - (2*x^2)/(1 + Sqrt[5])]*EllipticF[Arc
Sin[Sqrt[2/(1 + Sqrt[5])]*x], (-3 - Sqrt[5])/2])/(64*Sqrt[2]*(3 + 2*Sqrt[5] + Sqrt[17])*Sqrt[-1 - x^2 + x^4])
+ ((1 + Sqrt[5])*(1 + Sqrt[17])*Sqrt[-1 + Sqrt[5] + 2*x^2]*Sqrt[1 - (2*x^2)/(1 + Sqrt[5])]*EllipticF[ArcSin[Sq
rt[2/(1 + Sqrt[5])]*x], (-3 - Sqrt[5])/2])/(8*Sqrt[34]*(8 - Sqrt[5] - Sqrt[85])*Sqrt[-1 - x^2 + x^4]) + ((1 +
Sqrt[5])*(9 + Sqrt[17])*Sqrt[-1 + Sqrt[5] + 2*x^2]*Sqrt[1 - (2*x^2)/(1 + Sqrt[5])]*EllipticF[ArcSin[Sqrt[2/(1
+ Sqrt[5])]*x], (-3 - Sqrt[5])/2])/(64*Sqrt[34]*(8 - Sqrt[5] - Sqrt[85])*Sqrt[-1 - x^2 + x^4]) - ((1 + Sqrt[5]
)*(1 - Sqrt[17])*Sqrt[-1 + Sqrt[5] + 2*x^2]*Sqrt[1 - (2*x^2)/(1 + Sqrt[5])]*EllipticF[ArcSin[Sqrt[2/(1 + Sqrt[
5])]*x], (-3 - Sqrt[5])/2])/(8*Sqrt[34]*(8 - Sqrt[5] + Sqrt[85])*Sqrt[-1 - x^2 + x^4]) - ((1 + Sqrt[5])*(9 - S
qrt[17])*Sqrt[-1 + Sqrt[5] + 2*x^2]*Sqrt[1 - (2*x^2)/(1 + Sqrt[5])]*EllipticF[ArcSin[Sqrt[2/(1 + Sqrt[5])]*x],
(-3 - Sqrt[5])/2])/(64*Sqrt[34]*(8 - Sqrt[5] + Sqrt[85])*Sqrt[-1 - x^2 + x^4]) + ((3 - 2*Sqrt[5] - Sqrt[17])*
Sqrt[-2 - (1 - Sqrt[5])*x^2]*Sqrt[(2 + (1 + Sqrt[5])*x^2)/(2 + (1 - Sqrt[5])*x^2)]*EllipticF[ArcSin[(Sqrt[2]*5
^(1/4)*x)/Sqrt[-2 - (1 - Sqrt[5])*x^2]], (5 - Sqrt[5])/10])/(128*5^(1/4)*Sqrt[(2 + (1 - Sqrt[5])*x^2)^(-1)]*Sq
rt[-1 - x^2 + x^4]) - (3*(1 - Sqrt[17])*Sqrt[-2 - (1 - Sqrt[5])*x^2]*Sqrt[(2 + (1 + Sqrt[5])*x^2)/(2 + (1 - Sq
rt[5])*x^2)]*EllipticF[ArcSin[(Sqrt[2]*5^(1/4)*x)/Sqrt[-2 - (1 - Sqrt[5])*x^2]], (5 - Sqrt[5])/10])/(64*5^(1/4
)*(3 + 2*Sqrt[5] - Sqrt[17])*Sqrt[(2 + (1 - Sqrt[5])*x^2)^(-1)]*Sqrt[-1 - x^2 + x^4]) + ((3 + 2*Sqrt[5] - Sqrt
[17])*Sqrt[-2 - (1 - Sqrt[5])*x^2]*Sqrt[(2 + (1 + Sqrt[5])*x^2)/(2 + (1 - Sqrt[5])*x^2)]*EllipticF[ArcSin[(Sqr
t[2]*5^(1/4)*x)/Sqrt[-2 - (1 - Sqrt[5])*x^2]], (5 - Sqrt[5])/10])/(2176*5^(1/4)*Sqrt[(2 + (1 - Sqrt[5])*x^2)^(
-1)]*Sqrt[-1 - x^2 + x^4]) + ((3 + 2*Sqrt[5] - Sqrt[17])*Sqrt[-2 - (1 - Sqrt[5])*x^2]*Sqrt[(2 + (1 + Sqrt[5])*
x^2)/(2 + (1 - Sqrt[5])*x^2)]*EllipticF[ArcSin[(Sqrt[2]*5^(1/4)*x)/Sqrt[-2 - (1 - Sqrt[5])*x^2]], (5 - Sqrt[5]
)/10])/(136*5^(1/4)*(1 - Sqrt[17])*Sqrt[(2 + (1 - Sqrt[5])*x^2)^(-1)]*Sqrt[-1 - x^2 + x^4]) + ((17 + Sqrt[17])
*Sqrt[-2 - (1 - Sqrt[5])*x^2]*Sqrt[(2 + (1 + Sqrt[5])*x^2)/(2 + (1 - Sqrt[5])*x^2)]*EllipticF[ArcSin[(Sqrt[2]*
5^(1/4)*x)/Sqrt[-2 - (1 - Sqrt[5])*x^2]], (5 - Sqrt[5])/10])/(544*5^(1/4)*(1 + 2*Sqrt[5] - Sqrt[17])*Sqrt[(2 +
(1 - Sqrt[5])*x^2)^(-1)]*Sqrt[-1 - x^2 + x^4]) - ((17 - 2*Sqrt[17])*(1 - 2*Sqrt[5] + Sqrt[17])*Sqrt[-2 - (1 -
Sqrt[5])*x^2]*Sqrt[(2 + (1 + Sqrt[5])*x^2)/(2 + (1 - Sqrt[5])*x^2)]*EllipticF[ArcSin[(Sqrt[2]*5^(1/4)*x)/Sqrt
[-2 - (1 - Sqrt[5])*x^2]], (5 - Sqrt[5])/10])/(2176*5^(1/4)*Sqrt[(2 + (1 - Sqrt[5])*x^2)^(-1)]*Sqrt[-1 - x^2 +
x^4]) + ((3 - 2*Sqrt[5] + Sqrt[17])*Sqrt[-2 - (1 - Sqrt[5])*x^2]*Sqrt[(2 + (1 + Sqrt[5])*x^2)/(2 + (1 - Sqrt[
5])*x^2)]*EllipticF[ArcSin[(Sqrt[2]*5^(1/4)*x)/Sqrt[-2 - (1 - Sqrt[5])*x^2]], (5 - Sqrt[5])/10])/(128*5^(1/4)*
Sqrt[(2 + (1 - Sqrt[5])*x^2)^(-1)]*Sqrt[-1 - x^2 + x^4]) - ((51 - 19*Sqrt[17])*Sqrt[-2 - (1 - Sqrt[5])*x^2]*Sq
rt[(2 + (1 + Sqrt[5])*x^2)/(2 + (1 - Sqrt[5])*x^2)]*EllipticF[ArcSin[(Sqrt[2]*5^(1/4)*x)/Sqrt[-2 - (1 - Sqrt[5
])*x^2]], (5 - Sqrt[5])/10])/(1088*5^(1/4)*(1 + 2*Sqrt[5] + Sqrt[17])*Sqrt[(2 + (1 - Sqrt[5])*x^2)^(-1)]*Sqrt[
-1 - x^2 + x^4]) + ((17 - Sqrt[17])*Sqrt[-2 - (1 - Sqrt[5])*x^2]*Sqrt[(2 + (1 + Sqrt[5])*x^2)/(2 + (1 - Sqrt[5
])*x^2)]*EllipticF[ArcSin[(Sqrt[2]*5^(1/4)*x)/Sqrt[-2 - (1 - Sqrt[5])*x^2]], (5 - Sqrt[5])/10])/(544*5^(1/4)*(
1 + 2*Sqrt[5] + Sqrt[17])*Sqrt[(2 + (1 - Sqrt[5])*x^2)^(-1)]*Sqrt[-1 - x^2 + x^4]) - (3*(1 + Sqrt[17])*Sqrt[-2
- (1 - Sqrt[5])*x^2]*Sqrt[(2 + (1 + Sqrt[5])*x^2)/(2 + (1 - Sqrt[5])*x^2)]*EllipticF[ArcSin[(Sqrt[2]*5^(1/4)*
x)/Sqrt[-2 - (1 - Sqrt[5])*x^2]], (5 - Sqrt[5])/10])/(64*5^(1/4)*(3 + 2*Sqrt[5] + Sqrt[17])*Sqrt[(2 + (1 - Sqr
t[5])*x^2)^(-1)]*Sqrt[-1 - x^2 + x^4]) + ((3 + 2*Sqrt[5] + Sqrt[17])*Sqrt[-2 - (1 - Sqrt[5])*x^2]*Sqrt[(2 + (1
+ Sqrt[5])*x^2)/(2 + (1 - Sqrt[5])*x^2)]*EllipticF[ArcSin[(Sqrt[2]*5^(1/4)*x)/Sqrt[-2 - (1 - Sqrt[5])*x^2]],
(5 - Sqrt[5])/10])/(2176*5^(1/4)*Sqrt[(2 + (1 - Sqrt[5])*x^2)^(-1)]*Sqrt[-1 - x^2 + x^4]) + ((3 + 2*Sqrt[5] +
Sqrt[17])*Sqrt[-2 - (1 - Sqrt[5])*x^2]*Sqrt[(2 + (1 + Sqrt[5])*x^2)/(2 + (1 - Sqrt[5])*x^2)]*EllipticF[ArcSin[
(Sqrt[2]*5^(1/4)*x)/Sqrt[-2 - (1 - Sqrt[5])*x^2]], (5 - Sqrt[5])/10])/(136*5^(1/4)*(1 + Sqrt[17])*Sqrt[(2 + (1
- Sqrt[5])*x^2)^(-1)]*Sqrt[-1 - x^2 + x^4]) - ((1 - 2*Sqrt[5] - Sqrt[17])*(17 + 2*Sqrt[17])*Sqrt[-2 - (1 - Sq
rt[5])*x^2]*Sqrt[(2 + (1 + Sqrt[5])*x^2)/(2 + (1 - Sqrt[5])*x^2)]*EllipticF[ArcSin[(Sqrt[2]*5^(1/4)*x)/Sqrt[-2
- (1 - Sqrt[5])*x^2]], (5 - Sqrt[5])/10])/(2176*5^(1/4)*Sqrt[(2 + (1 - Sqrt[5])*x^2)^(-1)]*Sqrt[-1 - x^2 + x^
4]) - ((51 + 19*Sqrt[17])*Sqrt[-2 - (1 - Sqrt[5])*x^2]*Sqrt[(2 + (1 + Sqrt[5])*x^2)/(2 + (1 - Sqrt[5])*x^2)]*E
llipticF[ArcSin[(Sqrt[2]*5^(1/4)*x)/Sqrt[-2 - (1 - Sqrt[5])*x^2]], (5 - Sqrt[5])/10])/(1088*5^(1/4)*(1 + 2*Sqr
t[5] - Sqrt[17])*Sqrt[(2 + (1 - Sqrt[5])*x^2)^(-1)]*Sqrt[-1 - x^2 + x^4]) - ((17 + Sqrt[17])*Sqrt[-2 - (1 - Sq
rt[5])*x^2]*Sqrt[(2 + (1 + Sqrt[5])*x^2)/(2 + (1 - Sqrt[5])*x^2)]*EllipticF[ArcSin[(Sqrt[2]*5^(1/4)*x)/Sqrt[-2
- (1 - Sqrt[5])*x^2]], (5 - Sqrt[5])/10])/(136*5^(1/4)*(8 - Sqrt[5] - Sqrt[85])*Sqrt[(2 + (1 - Sqrt[5])*x^2)^
(-1)]*Sqrt[-1 - x^2 + x^4]) - ((17 + 9*Sqrt[17])*Sqrt[-2 - (1 - Sqrt[5])*x^2]*Sqrt[(2 + (1 + Sqrt[5])*x^2)/(2
+ (1 - Sqrt[5])*x^2)]*EllipticF[ArcSin[(Sqrt[2]*5^(1/4)*x)/Sqrt[-2 - (1 - Sqrt[5])*x^2]], (5 - Sqrt[5])/10])/(
1088*5^(1/4)*(8 - Sqrt[5] - Sqrt[85])*Sqrt[(2 + (1 - Sqrt[5])*x^2)^(-1)]*Sqrt[-1 - x^2 + x^4]) - ((17 - 9*Sqrt
[17])*Sqrt[-2 - (1 - Sqrt[5])*x^2]*Sqrt[(2 + (1 + Sqrt[5])*x^2)/(2 + (1 - Sqrt[5])*x^2)]*EllipticF[ArcSin[(Sqr
t[2]*5^(1/4)*x)/Sqrt[-2 - (1 - Sqrt[5])*x^2]], (5 - Sqrt[5])/10])/(1088*5^(1/4)*(8 - Sqrt[5] + Sqrt[85])*Sqrt[
(2 + (1 - Sqrt[5])*x^2)^(-1)]*Sqrt[-1 - x^2 + x^4]) - ((17 - Sqrt[17])*Sqrt[-2 - (1 - Sqrt[5])*x^2]*Sqrt[(2 +
(1 + Sqrt[5])*x^2)/(2 + (1 - Sqrt[5])*x^2)]*EllipticF[ArcSin[(Sqrt[2]*5^(1/4)*x)/Sqrt[-2 - (1 - Sqrt[5])*x^2]]
, (5 - Sqrt[5])/10])/(136*5^(1/4)*(8 - Sqrt[5] + Sqrt[85])*Sqrt[(2 + (1 - Sqrt[5])*x^2)^(-1)]*Sqrt[-1 - x^2 +
x^4]) - ((17 - Sqrt[17]*(1 - 2*Sqrt[5]))*Sqrt[-2 - (1 - Sqrt[5])*x^2]*Sqrt[(2 + (1 + Sqrt[5])*x^2)/(2 + (1 - S
qrt[5])*x^2)]*EllipticF[ArcSin[(Sqrt[2]*5^(1/4)*x)/Sqrt[-2 - (1 - Sqrt[5])*x^2]], (5 - Sqrt[5])/10])/(1088*5^(
1/4)*Sqrt[(2 + (1 - Sqrt[5])*x^2)^(-1)]*Sqrt[-1 - x^2 + x^4]) - ((17 + Sqrt[17]*(1 - 2*Sqrt[5]))*Sqrt[-2 - (1
- Sqrt[5])*x^2]*Sqrt[(2 + (1 + Sqrt[5])*x^2)/(2 + (1 - Sqrt[5])*x^2)]*EllipticF[ArcSin[(Sqrt[2]*5^(1/4)*x)/Sqr
t[-2 - (1 - Sqrt[5])*x^2]], (5 - Sqrt[5])/10])/(1088*5^(1/4)*Sqrt[(2 + (1 - Sqrt[5])*x^2)^(-1)]*Sqrt[-1 - x^2
+ x^4]) - (3*(1 + Sqrt[5])*Sqrt[-1 + Sqrt[5] + 2*x^2]*Sqrt[1 - (2*x^2)/(1 + Sqrt[5])]*EllipticPi[(-2*(1 + Sqrt
[5]))/(1 - Sqrt[17]), ArcSin[Sqrt[2/(1 + Sqrt[5])]*x], (-3 - Sqrt[5])/2])/(64*Sqrt[2]*Sqrt[-1 - x^2 + x^4]) +
(3*(1 + Sqrt[5])*Sqrt[-1 + Sqrt[5] + 2*x^2]*Sqrt[1 - (2*x^2)/(1 + Sqrt[5])]*EllipticPi[(2*(1 + Sqrt[5]))/(1 -
Sqrt[17]), ArcSin[Sqrt[2/(1 + Sqrt[5])]*x], (-3 - Sqrt[5])/2])/(64*Sqrt[34]*Sqrt[-1 - x^2 + x^4]) + ((1 + Sqrt
[5])*Sqrt[-1 + Sqrt[5] + 2*x^2]*Sqrt[1 - (2*x^2)/(1 + Sqrt[5])]*EllipticPi[(2*(1 + Sqrt[5]))/(1 - Sqrt[17]), A
rcSin[Sqrt[2/(1 + Sqrt[5])]*x], (-3 - Sqrt[5])/2])/(4*Sqrt[34]*(1 - Sqrt[17])*Sqrt[-1 - x^2 + x^4]) - ((1 + Sq
rt[5])*(2 - Sqrt[17])*Sqrt[-1 + Sqrt[5] + 2*x^2]*Sqrt[1 - (2*x^2)/(1 + Sqrt[5])]*EllipticPi[(2*(1 + Sqrt[5]))/
(1 - Sqrt[17]), ArcSin[Sqrt[2/(1 + Sqrt[5])]*x], (-3 - Sqrt[5])/2])/(64*Sqrt[34]*Sqrt[-1 - x^2 + x^4]) - (3*(1
+ Sqrt[5])*Sqrt[-1 + Sqrt[5] + 2*x^2]*Sqrt[1 - (2*x^2)/(1 + Sqrt[5])]*EllipticPi[(-2*(1 + Sqrt[5]))/(1 + Sqrt
[17]), ArcSin[Sqrt[2/(1 + Sqrt[5])]*x], (-3 - Sqrt[5])/2])/(64*Sqrt[2]*Sqrt[-1 - x^2 + x^4]) - (3*(1 + Sqrt[5]
)*Sqrt[-1 + Sqrt[5] + 2*x^2]*Sqrt[1 - (2*x^2)/(1 + Sqrt[5])]*EllipticPi[(2*(1 + Sqrt[5]))/(1 + Sqrt[17]), ArcS
in[Sqrt[2/(1 + Sqrt[5])]*x], (-3 - Sqrt[5])/2])/(64*Sqrt[34]*Sqrt[-1 - x^2 + x^4]) - ((1 + Sqrt[5])*Sqrt[-1 +
Sqrt[5] + 2*x^2]*Sqrt[1 - (2*x^2)/(1 + Sqrt[5])]*EllipticPi[(2*(1 + Sqrt[5]))/(1 + Sqrt[17]), ArcSin[Sqrt[2/(1
+ Sqrt[5])]*x], (-3 - Sqrt[5])/2])/(4*Sqrt[34]*(1 + Sqrt[17])*Sqrt[-1 - x^2 + x^4]) + ((1 + Sqrt[5])*(2 + Sqr
t[17])*Sqrt[-1 + Sqrt[5] + 2*x^2]*Sqrt[1 - (2*x^2)/(1 + Sqrt[5])]*EllipticPi[(2*(1 + Sqrt[5]))/(1 + Sqrt[17]),
ArcSin[Sqrt[2/(1 + Sqrt[5])]*x], (-3 - Sqrt[5])/2])/(64*Sqrt[34]*Sqrt[-1 - x^2 + x^4])

Rule 419

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Simp[(1*EllipticF[ArcSin[Rt[-(d/c),
2]*x], (b*c)/(a*d)])/(Sqrt[a]*Sqrt[c]*Rt[-(d/c), 2]), x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] &
& GtQ[a, 0] &&  !(NegQ[b/a] && SimplerSqrtQ[-(b/a), -(d/c)])

Rule 421

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

Rule 537

Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x_)^2]), x_Symbol] :> Simp[(1*Ellipt
icPi[(b*c)/(a*d), ArcSin[Rt[-(d/c), 2]*x], (c*f)/(d*e)])/(a*Sqrt[c]*Sqrt[e]*Rt[-(d/c), 2]), x] /; FreeQ[{a, b,
c, d, e, f}, x] &&  !GtQ[d/c, 0] && GtQ[c, 0] && GtQ[e, 0] &&  !( !GtQ[f/e, 0] && SimplerSqrtQ[-(f/e), -(d/c)
])

Rule 538

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

Rule 540

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

Rule 1098

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

Rule 1184

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

Rule 1187

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[(2*c*d - e*(b - q))/(2*c), Int[1/Sqrt[a + b*x^2 + c*x^4], x], x] + Dist[e/(2*c), Int[(b - q + 2*c*x^2)/
Sqrt[a + b*x^2 + c*x^4], x], x] /; NeQ[2*c*d - e*(b - q), 0]] /; FreeQ[{a, b, c, d, e}, x] && GtQ[b^2 - 4*a*c,
0] && LtQ[a, 0] && GtQ[c, 0]

Rule 1208

Int[((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_)/((d_) + (e_.)*(x_)^2), x_Symbol] :> -Dist[(e^2)^(-1), Int[(c*d -
b*e - c*e*x^2)*(a + b*x^2 + c*x^4)^(p - 1), x], x] + Dist[(c*d^2 - b*d*e + a*e^2)/e^2, Int[(a + b*x^2 + c*x^4
)^(p - 1)/(d + e*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2
, 0] && IGtQ[p + 1/2, 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 1226

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

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 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^4\right ) \left (1+x^4\right ) \sqrt {-1-x^2+x^4}}{\left (-2-x^2+2 x^4\right )^2 \left (-2+x^2+2 x^4\right )} \, dx &=\int \left (\frac {\left (4+x^2\right ) \sqrt {-1-x^2+x^4}}{8 \left (-2-x^2+2 x^4\right )^2}+\frac {\left (1+4 x^2\right ) \sqrt {-1-x^2+x^4}}{16 \left (-2-x^2+2 x^4\right )}+\frac {\left (-1-4 x^2\right ) \sqrt {-1-x^2+x^4}}{16 \left (-2+x^2+2 x^4\right )}\right ) \, dx\\ &=\frac {1}{16} \int \frac {\left (1+4 x^2\right ) \sqrt {-1-x^2+x^4}}{-2-x^2+2 x^4} \, dx+\frac {1}{16} \int \frac {\left (-1-4 x^2\right ) \sqrt {-1-x^2+x^4}}{-2+x^2+2 x^4} \, dx+\frac {1}{8} \int \frac {\left (4+x^2\right ) \sqrt {-1-x^2+x^4}}{\left (-2-x^2+2 x^4\right )^2} \, dx\\ &=\frac {1}{16} \int \left (\frac {\left (4+\frac {8}{\sqrt {17}}\right ) \sqrt {-1-x^2+x^4}}{-1-\sqrt {17}+4 x^2}+\frac {\left (4-\frac {8}{\sqrt {17}}\right ) \sqrt {-1-x^2+x^4}}{-1+\sqrt {17}+4 x^2}\right ) \, dx+\frac {1}{16} \int \left (-\frac {4 \sqrt {-1-x^2+x^4}}{1-\sqrt {17}+4 x^2}-\frac {4 \sqrt {-1-x^2+x^4}}{1+\sqrt {17}+4 x^2}\right ) \, dx+\frac {1}{8} \int \left (\frac {4 \sqrt {-1-x^2+x^4}}{\left (-2-x^2+2 x^4\right )^2}+\frac {x^2 \sqrt {-1-x^2+x^4}}{\left (-2-x^2+2 x^4\right )^2}\right ) \, dx\\ &=\frac {1}{8} \int \frac {x^2 \sqrt {-1-x^2+x^4}}{\left (-2-x^2+2 x^4\right )^2} \, dx-\frac {1}{4} \int \frac {\sqrt {-1-x^2+x^4}}{1-\sqrt {17}+4 x^2} \, dx-\frac {1}{4} \int \frac {\sqrt {-1-x^2+x^4}}{1+\sqrt {17}+4 x^2} \, dx+\frac {1}{2} \int \frac {\sqrt {-1-x^2+x^4}}{\left (-2-x^2+2 x^4\right )^2} \, dx+\frac {1}{68} \left (17-2 \sqrt {17}\right ) \int \frac {\sqrt {-1-x^2+x^4}}{-1+\sqrt {17}+4 x^2} \, dx+\frac {1}{68} \left (17+2 \sqrt {17}\right ) \int \frac {\sqrt {-1-x^2+x^4}}{-1-\sqrt {17}+4 x^2} \, dx \end {align*}

rest of steps removed due to Latex formating problem.

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Mathematica [C]  time = 1.82, size = 477, normalized size = 6.91 \begin {gather*} \frac {-2 \sqrt {1+\sqrt {5}} x^5+2 \sqrt {1+\sqrt {5}} x^3-3 i \sqrt {2} \sqrt {-x^4+x^2+1} \left (2 x^4-x^2-2\right ) F\left (i \sinh ^{-1}\left (\sqrt {\frac {2}{-1+\sqrt {5}}} x\right )|\frac {1}{2} \left (-3+\sqrt {5}\right )\right )+6 i \sqrt {2} \sqrt {-x^4+x^2+1} x^4 \Pi \left (\frac {2 \left (-1+\sqrt {5}\right )}{1+\sqrt {17}};i \sinh ^{-1}\left (\sqrt {\frac {2}{-1+\sqrt {5}}} x\right )|\frac {1}{2} \left (-3+\sqrt {5}\right )\right )-3 i \sqrt {2} \sqrt {-x^4+x^2+1} x^2 \Pi \left (\frac {2 \left (-1+\sqrt {5}\right )}{1+\sqrt {17}};i \sinh ^{-1}\left (\sqrt {\frac {2}{-1+\sqrt {5}}} x\right )|\frac {1}{2} \left (-3+\sqrt {5}\right )\right )+3 i \sqrt {2} \sqrt {-x^4+x^2+1} \left (2 x^4-x^2-2\right ) \Pi \left (-\frac {2 \left (-1+\sqrt {5}\right )}{-1+\sqrt {17}};i \sinh ^{-1}\left (\sqrt {\frac {2}{-1+\sqrt {5}}} x\right )|\frac {1}{2} \left (-3+\sqrt {5}\right )\right )-6 i \sqrt {2} \sqrt {-x^4+x^2+1} \Pi \left (\frac {2 \left (-1+\sqrt {5}\right )}{1+\sqrt {17}};i \sinh ^{-1}\left (\sqrt {\frac {2}{-1+\sqrt {5}}} x\right )|\frac {1}{2} \left (-3+\sqrt {5}\right )\right )+2 \sqrt {1+\sqrt {5}} x}{32 \sqrt {1+\sqrt {5}} \sqrt {x^4-x^2-1} \left (2 x^4-x^2-2\right )} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(2*Sqrt[1 + Sqrt[5]]*x + 2*Sqrt[1 + Sqrt[5]]*x^3 - 2*Sqrt[1 + Sqrt[5]]*x^5 - (3*I)*Sqrt[2]*Sqrt[1 + x^2 - x^4]
*(-2 - x^2 + 2*x^4)*EllipticF[I*ArcSinh[Sqrt[2/(-1 + Sqrt[5])]*x], (-3 + Sqrt[5])/2] + (3*I)*Sqrt[2]*Sqrt[1 +
x^2 - x^4]*(-2 - x^2 + 2*x^4)*EllipticPi[(-2*(-1 + Sqrt[5]))/(-1 + Sqrt[17]), I*ArcSinh[Sqrt[2/(-1 + Sqrt[5])]
*x], (-3 + Sqrt[5])/2] - (6*I)*Sqrt[2]*Sqrt[1 + x^2 - x^4]*EllipticPi[(2*(-1 + Sqrt[5]))/(1 + Sqrt[17]), I*Arc
Sinh[Sqrt[2/(-1 + Sqrt[5])]*x], (-3 + Sqrt[5])/2] - (3*I)*Sqrt[2]*x^2*Sqrt[1 + x^2 - x^4]*EllipticPi[(2*(-1 +
Sqrt[5]))/(1 + Sqrt[17]), I*ArcSinh[Sqrt[2/(-1 + Sqrt[5])]*x], (-3 + Sqrt[5])/2] + (6*I)*Sqrt[2]*x^4*Sqrt[1 +
x^2 - x^4]*EllipticPi[(2*(-1 + Sqrt[5]))/(1 + Sqrt[17]), I*ArcSinh[Sqrt[2/(-1 + Sqrt[5])]*x], (-3 + Sqrt[5])/2
])/(32*Sqrt[1 + Sqrt[5]]*Sqrt[-1 - x^2 + x^4]*(-2 - x^2 + 2*x^4))

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IntegrateAlgebraic [A]  time = 0.89, size = 69, normalized size = 1.00 \begin {gather*} -\frac {x \sqrt {-1-x^2+x^4}}{16 \left (-2-x^2+2 x^4\right )}-\frac {1}{16} \sqrt {\frac {3}{2}} \tan ^{-1}\left (\frac {\sqrt {\frac {3}{2}} x}{\sqrt {-1-x^2+x^4}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/16*(x*Sqrt[-1 - x^2 + x^4])/(-2 - x^2 + 2*x^4) - (Sqrt[3/2]*ArcTan[(Sqrt[3/2]*x)/Sqrt[-1 - x^2 + x^4]])/16

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fricas [A]  time = 0.53, size = 87, normalized size = 1.26 \begin {gather*} -\frac {\sqrt {3} \sqrt {2} {\left (2 \, x^{4} - x^{2} - 2\right )} \arctan \left (\frac {2 \, \sqrt {3} \sqrt {2} \sqrt {x^{4} - x^{2} - 1} x}{2 \, x^{4} - 5 \, x^{2} - 2}\right ) + 4 \, \sqrt {x^{4} - x^{2} - 1} x}{64 \, {\left (2 \, x^{4} - x^{2} - 2\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/64*(sqrt(3)*sqrt(2)*(2*x^4 - x^2 - 2)*arctan(2*sqrt(3)*sqrt(2)*sqrt(x^4 - x^2 - 1)*x/(2*x^4 - 5*x^2 - 2)) +
4*sqrt(x^4 - x^2 - 1)*x)/(2*x^4 - x^2 - 2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.53, size = 75, normalized size = 1.09

method	result	size
elliptic	\(\frac {\left (\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {x^{4}-x^{2}-1}\, \sqrt {2}\, \sqrt {3}}{3 x}\right )}{16}-\frac {\sqrt {x^{4}-x^{2}-1}\, \sqrt {2}}{64 x \left (\frac {x^{4}-x^{2}-1}{2 x^{2}}+\frac {1}{4}\right )}\right ) \sqrt {2}}{2}\)	\(75\)
trager	\(-\frac {x \sqrt {x^{4}-x^{2}-1}}{16 \left (2 x^{4}-x^{2}-2\right )}-\frac {\RootOf \left (\textit {\_Z}^{2}+6\right ) \ln \left (-\frac {2 \RootOf \left (\textit {\_Z}^{2}+6\right ) x^{4}-5 \RootOf \left (\textit {\_Z}^{2}+6\right ) x^{2}+12 x \sqrt {x^{4}-x^{2}-1}-2 \RootOf \left (\textit {\_Z}^{2}+6\right )}{2 x^{4}+x^{2}-2}\right )}{64}\)	\(100\)
default	\(-\frac {x \sqrt {x^{4}-x^{2}-1}}{16 \left (2 x^{4}-x^{2}-2\right )}+\frac {3 \sqrt {1-\left (-\frac {1}{2}-\frac {\sqrt {5}}{2}\right ) x^{2}}\, \sqrt {1-\left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right ) x^{2}}\, \EllipticF \left (\frac {x \sqrt {-2-2 \sqrt {5}}}{2}, \frac {i \sqrt {5}}{2}-\frac {i}{2}\right )}{16 \sqrt {-2-2 \sqrt {5}}\, \sqrt {x^{4}-x^{2}-1}}+\frac {\sqrt {2}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (2 \textit {\_Z}^{4}+\textit {\_Z}^{2}-2\right )}{\sum }\underline {\hspace {1.25 ex}}\alpha \left (-\frac {2 \sqrt {3}\, \arctanh \left (\frac {\left (2 \underline {\hspace {1.25 ex}}\alpha ^{2}-1\right ) \left (-5 \underline {\hspace {1.25 ex}}\alpha ^{2}+2 x^{2}-6\right )}{4 \sqrt {-\frac {3 \underline {\hspace {1.25 ex}}\alpha ^{2}}{2}}\, \sqrt {x^{4}-x^{2}-1}}\right )}{\sqrt {-\underline {\hspace {1.25 ex}}\alpha ^{2}}}-\frac {3 \left (2 \underline {\hspace {1.25 ex}}\alpha ^{3}+\underline {\hspace {1.25 ex}}\alpha \right ) \sqrt {x^{2}+2+\sqrt {5}\, x^{2}}\, \sqrt {x^{2}+2-\sqrt {5}\, x^{2}}\, \EllipticPi \left (\sqrt {-\frac {1}{2}-\frac {\sqrt {5}}{2}}\, x , \frac {\underline {\hspace {1.25 ex}}\alpha ^{2}}{2}-\frac {\underline {\hspace {1.25 ex}}\alpha ^{2} \sqrt {5}}{2}+\frac {1}{4}-\frac {\sqrt {5}}{4}, \frac {\sqrt {\frac {\sqrt {5}}{2}-\frac {1}{2}}}{\sqrt {-\frac {1}{2}-\frac {\sqrt {5}}{2}}}\right )}{\sqrt {-\sqrt {5}-1}\, \sqrt {x^{4}-x^{2}-1}}\right )\right )}{256}\)	\(293\)
risch	\(-\frac {x \sqrt {x^{4}-x^{2}-1}}{16 \left (2 x^{4}-x^{2}-2\right )}+\frac {3 \sqrt {1-\left (-\frac {1}{2}-\frac {\sqrt {5}}{2}\right ) x^{2}}\, \sqrt {1-\left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right ) x^{2}}\, \EllipticF \left (\frac {x \sqrt {-2-2 \sqrt {5}}}{2}, \frac {i \sqrt {5}}{2}-\frac {i}{2}\right )}{16 \sqrt {-2-2 \sqrt {5}}\, \sqrt {x^{4}-x^{2}-1}}+\frac {\sqrt {2}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (2 \textit {\_Z}^{4}+\textit {\_Z}^{2}-2\right )}{\sum }\underline {\hspace {1.25 ex}}\alpha \left (-\frac {2 \sqrt {3}\, \arctanh \left (\frac {\left (2 \underline {\hspace {1.25 ex}}\alpha ^{2}-1\right ) \left (-5 \underline {\hspace {1.25 ex}}\alpha ^{2}+2 x^{2}-6\right )}{4 \sqrt {-\frac {3 \underline {\hspace {1.25 ex}}\alpha ^{2}}{2}}\, \sqrt {x^{4}-x^{2}-1}}\right )}{\sqrt {-\underline {\hspace {1.25 ex}}\alpha ^{2}}}-\frac {3 \left (2 \underline {\hspace {1.25 ex}}\alpha ^{3}+\underline {\hspace {1.25 ex}}\alpha \right ) \sqrt {x^{2}+2+\sqrt {5}\, x^{2}}\, \sqrt {x^{2}+2-\sqrt {5}\, x^{2}}\, \EllipticPi \left (\sqrt {-\frac {1}{2}-\frac {\sqrt {5}}{2}}\, x , \frac {\underline {\hspace {1.25 ex}}\alpha ^{2}}{2}-\frac {\underline {\hspace {1.25 ex}}\alpha ^{2} \sqrt {5}}{2}+\frac {1}{4}-\frac {\sqrt {5}}{4}, \frac {\sqrt {\frac {\sqrt {5}}{2}-\frac {1}{2}}}{\sqrt {-\frac {1}{2}-\frac {\sqrt {5}}{2}}}\right )}{\sqrt {-\sqrt {5}-1}\, \sqrt {x^{4}-x^{2}-1}}\right )\right )}{256}\)	\(293\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(1/16*3^(1/2)*arctan(1/3*(x^4-x^2-1)^(1/2)*2^(1/2)/x*3^(1/2))-1/64*(x^4-x^2-1)^(1/2)*2^(1/2)/x/(1/2*(x^4-x
^2-1)/x^2+1/4))*2^(1/2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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