Evaluate: `intsin^(-1)((2x+2)/(sqrt(4x^2+8x+13)))dx`

Explanation

`I=int sin^(-1)((2x+2)/(sqrt(4x^(2)+8x+13)))dx`
`=int sin^(-1)((2x+2)/(sqrt((2x+2)^(2)+3^(2))))dx`
`[ "Put " 2x+2=3 tan theta " or " 2 dx =3 sec^(2)theta d theta]`
`="sin"^(-1)((3 tan theta)/(3 sec theta))(3)/(2)sec^(2) theta d theta`
`=(3)/(2)int theta sec^(2) theta d theta`
`=(3)/(2){theta tan theta-int tan theta d theta }+C`
`=(3)/(2){theta tan theta-log|sec theta| }+C`
`=(3)/(2){(2x+2)/(3)"tan"^(-1)((2x+2)/(3))-log(sqrt(1+((2x+2)/(3))^(2)))}+C`
`=(3)/(2){(2)/(3)(x+1)"tan"^(-1)((2)/(3)(x+1))-log sqrt(4x^(2)+8x+13)}+C`