When $\int_{a}^{b} \big( f(x)+g(x) \big) \text{ d}x$ is easy to evaluate but $\int_{a}^{b} f(x) \text{ d}x$ and $\int_{a}^{b} g(x) \text{ d}x$ are not

Question

I would like to construct functions $f$ and $g$ such that $\int_{a}^{b} \big( f(x)+g(x) \big) \text{ d}x$ is possible and easy to evaluate, but $\int_{a}^{b} f(x) \text{ d}x$ and $\int_{a}^{b} g(x) \text{ d}x$ are both hard or impossible to evaluate. How can we construct such (nontrivial) functions?

For example, $$\int_{0}^{2} \big(\sqrt{1+x^3}+\sqrt[3]{x^2+2x} \big) \text{ d}x = 6.$$ In this case, we apply some results of integrals of inverse functions.

So is there any idea to construct such integrals, other than the idea of inverse functions?

Can we do the same for some indefinite integrals as well?

Answer 1

I'll add more in future.

$1$:

$$\int f = \int \log (\sqrt{1+x^2}+x )\mathrm dx$$ and $$\int g = \int\log (\sqrt{1+x^2}-x )\mathrm dx$$

$2:$

$$\int f = \int \sqrt{\tan x}\mathrm dx$$ $$\int g = \int \sqrt{\cot x}\mathrm dx$$