To solve the integral $\int f\left( g(x) \right) dx$ for $f,g:\mathbb{R}\rightarrow\mathbb{R}$ we can use a straightforward substitution (1, 2).
Is there any general way to solve the following integral, for $f:\mathbb{R}^n\rightarrow\mathbb{R}$, $g_i:\mathbb{R}\rightarrow\mathbb{R}$? $$ \int f\left( g_1(x), ..., g_n(x) \right) dx $$
I will respond to your specific question that we've been discussing in the comments. I want your integrand to be of the form $\nabla\phi(\vec g(t))\cdot \vec g'(t)$. We will get close, but will not get what you want. You have $\vec g(t) = t\vec a + \vec b$, so $\vec g' = \vec a$. With $F(x_1,\dots,x_n) = \dfrac{e^{x_k}}{e^{x_1} + \dots + e^{x_n}}$, we try $\phi(x_1,\dots,x_n) = \ln(e^{x_1} + \dots +e^{x_n})$. Then $$\nabla\phi = \frac1{e^{x_1} + \dots + e^{x_n}}(e^{x_1},e^{x_2},\dots,e^{x_n}).$$ The only way to isolate to a single $e^{x_k}$ is to have a function $\vec g(t)$ whose only non-constant component is the $k$th. The best I can do for you is to get $$\int \frac{\sum a_ie^{a_it+b_i}}{\sum e^{a_it+b_i}}dt = \ln\left(\sum e^{a_it+b_i}\right).$$ Of course, we didn't need any multivariable calculus to get this.