Finding $\int\frac{\sin x+\tan x}{\cos x+\csc x}dx$ Finding $\displaystyle \int\frac{\sin x+\tan x}{\cos x+\csc x}dx$
what i try
$\displaystyle \Lambda =\int\frac{\sin^2 x(1+\cos x)}{\cos x(\sin x\cos x+1)}dx$
$\displaystyle \Lambda=\int\frac{\sin^4 x}{\cos x(1-\cos x)(\sin x\cos x+1)}dx$
How do i solve it help me please 

Hint:
$$\text{Define }I=\underbrace{\int\dfrac{\tan x}{\cos x+\csc x}\mathrm dx}_{I_1}+\underbrace{\int\dfrac{\sin x}{\cos x+\csc x}\mathrm dx}_{I_2}$$

For $I_1$, substitute $u=\tan(x/2)$ (Weierstrass substitution) and perform Partial Fraction Decomposition twice.

For $I_2$, rewrite everything in terms of $\tan x$ and $\sec x$, let $v=\tan x$ and perform Partial Fraction Decomposition.

The antiderivative is not very pleasing, have a look: $$I=\ln\left(\left|\sec\left(x\right)\right|\right)-\dfrac{\ln\left(\tan^2\left(x\right)+\tan\left(x\right)+1\right)}{2}+\dfrac{1}{\sqrt{3}}\left[\arctan\left(\frac{2\tan\left(x\right)+1}{\sqrt{3}}\right)-\ln\left(\left(\tan\left(\frac{x}{2}\right)-1\right)\left(\tan\left(\frac{x}{2}\right)+\sqrt{3}\right)+2\right)\\ +\ln\left(\left(\tan\left(\frac{x}{2}\right)-1\right)\left(\tan\left(\frac{x}{2}\right)-\sqrt{3}\right)+2\right)\right]+\ln\left(\left|\tan\left(\dfrac{x}{2}\right)+1\right|\right)-\ln\left(\left|\tan\left(\dfrac{x}{2}\right)-1\right|\right)$$
