Evaluate the indefinite integral of (tan^3x+tanx)/(tan x+1)?

We'll factorize by tan x at numerator and we'll
get:

f(x) = (tan x)*[(tan x)^2 + 1]/(tan x +
1)

Now, we'll calculate the indefinite
integral:

Int f(x)dx = Int (tan x)*[(tan x)^2 + 1]dx/(tan x +
1)

We'll substitute tan x = t

x =
arctan t => dx = dt/(1 + t^2)

We'll re-write the
integral:

Int ydx = Int (t)*[(t)^2 + 1]dt/(1 + t^2)*(t+
1)

We'll simplify by (1 + t^2) and we'll
get:

Int tdt/(t+ 1) = Int (t+1)dt/(t+1) - Int
dt/(t+1)

Int tdt/(t+ 1) = Int dt - Int
dt/(t+1)

Int tdt/(t+ 1) = t - ln |t+1| +
C

We'll substitute t by tan x and we'll
get:

The requested indefinite integral is: Int tanx
dx/cosx(sinx+cosx) = tan x - ln |tan x + 1| + C.