Evaluation of this definite complex integral

Question

I want to obtain the symbolic expression for this definite complex integral:

$\int_0^z \frac{1}{b^2+z^2} \, dz$

Then doing a subtition z in the Integral's upper boundary

z = I*((b*(t - 1))/(t + 1))

I did the following in MMA..

ClearAll[j, j1, t, j2]

z = I*((b*(t - 1))/(t + 1))

j = Integrate[1/(b^2 + z^2), {z, 0, z}];

j = j /. z -> (I*b*(-1 + t))/(1 + t)

Got a error message ?

Some background information

Answer 1

Make the substitution inside the integral:

integrand=1/(b^2 + z^2) /. z -> (I b (t - 1)/(t + 1));

calculate dt:

dt=D[(I b (t - 1)/(t + 1)), t]

do the integral:

res=Integrate[integrand dt, t]

To write this as a function of z we need to inverse the replacement:

res /. Solve[z == (I b (t - 1)/(t + 1)), t][[1]]