Evaluate the definite integral: `int_(-1/(sqrt(3)))^(1/(sqrt(3)))((x^4)/(1-x^4))cos^(01)((2x)/(1+x^2))dxdot`

Explanation

Let `I=int_(-1//sqrt(3))^(1//sqrt(3))(x^(4))/(1-x^(4))cos^(-1)((2x)/(1+x^(2)))dx`
`=int_(1//sqrt(3))^(1//sqrt(3))((-x)^(4))/(1-(-x)^(4))cos^(-1)((2(-x))/(1+(-x)^(2)))dx`
`=int_(-1//sqrt(3))^(1//sqrt(3))(x^(4))/(1-x^(4))[pi-cos^(-1)((2x)/(1+x^(2)))]dx`
`=pi int_(-1//sqrt(3))^(1/sqrt(3))(x^(4))/(1-x^(4))dx-I`
`:.2I=2pi int_(0)^(1//sqrt(3))(x^(4))/(1-x^(4))dx`
`:.I=pi int_(0)^(1//sqrt(3))[-1+1/(1-x^(4))]dx`
`=-pi int_(0)^(1//sqrt(3))dx+(pi)/2 int_(0)^(1//sqrt(3))[1/(1-x^(2))+1/(1+x^(2))]dx`
`=-(pi)/(sqrt(3))+(pi)/2[(-1/2 log_(e)|(1-x)/(1+x)|+tan^(-1)x)]_(0)^(1//sqrt(3))`
`=-(pi)/(sqrt(3))+(pi)/2(1/2log_(e)|(sqrt(3)+1)/(sqrt(3)-1)|+(pi)/6)`