If I have a given function that is defined as an integral, like of something that's clearly hard to work with like the gamma function or the exponential integral or the cosine integral or the error function or etc, is there a way to define an inverse of that function in a similar form that's NOT a cop-out dy integral? Not some dy integral that forces me to switch to integrating along the y-axis for no reason that solves absolutely nothing and never will, but something that starts as f(x) = integral(g(x))dx and hopefully takes the form of f-1(x) = integral g^-1(x)dx or something similar, something that is still defined on the domain of x.

In fact, you know what, let's take an example: arctangent. The arctangent function can be described as the integral of 1/sqrt(1+x^2) which after complex analysis or trig substitution we know actually takes the form of a complicated complex-valued logarithm containing a polynomial. But now, let's say I wanted to define the actual regular tangent function just starting out with the integral definition of the arctangent function (which for verification purposes we know takes the form of a sum/quotient of real/imaginary exponents)...

**Edited by SFNQuestions, 17 April 2017 - 01:43 AM.**