Let y=x²+1, then dy=2xdx, so xdx=dy/2.
The integrand simplifies:
[(2sin(y)-sin(2y))/(sin(2y)+2sin(y))](dy/2)=
[(2sin(y)-2sin(y)cos(y))/(2sin(y)cos(y)+2sin(y)](dy/2)=
Cancel common factor sin(y):
(1-cos(y))/(2(1+cos(y)))dy=
Multiply by (1-cos(y))/(1-cos(y)):
(1-cos(y))²/(2(1-cos²(y)))dy=
(1-cos(y))²/(2sin²(y))dy=
(1-2cos(y)+cos²(y))/(2sin²(y))dy=
(csc²(y)/2-cot(y)csc(y)+cot²(y)/2)dy.
Derivative of cot(y) is -csc²(y).
Derivative of csc(y) is -cot(y)csc(y).
cot²(y)=csc²(y)-1.
∫(csc²(y)/2-cot(y)csc(y)+cot²(y)/2)dy=
∫(csc²(y)/2-cot(y)csc(y)+(csc²(y)-1)/2)dy=
∫(csc²(y)-cot(y)csc(y)-½)dy=
-cot(y)+csc(y)-y/2+c, where c is the constant of integration.
Replacing y with x²+1:
-cot(x²+1)+csc(x²+1)-(x²+1)/2+c or -cot(x²+1)+csc(x²+1)-x²+C (combining constants).
