The integral of the given expression

(x^m +m^x )÷(m-x)

is-

The above function cannot be integrated because there are important restriction on antiderivatives that can be expressed as elementary functions

According to the Liouville's theorem for differential algebra:-

The antiderivatives of certain elementary functions cannot themselves be expressed as elementary functions. A standard example of such a function is e^{{-x^{2}}}, whose antiderivative is (with a multiplier of a constant) the error function, familiar from statistics. Other examples include the functions {\frac  {\sin(x)}{x}} and x^{x}.

Liouville's theorem states that elementary antiderivatives if they exist, must be in the same differential field as the function, plus possibly a finite number of logarithms.

Hope it helps.