Rational functions of trigonometric functions

QUESTION:

Rational functions of trigonometric functions An integrand with trigonometric functions in the numerator and denominator can often be converted to a rational integrand using the substitution u=tan(x/2) or \(x=2 \tan ^{-1} u\). The following relations are used in making this change of variables.

A: \(d x=\frac{2}{1+u^{2}} d u\) 

B: \(\sin x=\frac{2 u}{1+u^{2}}\) 

C: \(\cos x=\frac{1-u^{2}}{1+u^{2}}\)

Evaluate \(\int \frac{d x}{1+\sin x+\cos x}\).