Suppose that $t f(t) \to 0$ when $t \to \infty$ and $t f(t)\to 0$ when $t \to 0$.
For the following integral,
$$I(z)=\int_0^{\infty} f(t) \cos (z t) \mathrm{d}t,\qquad z>0 \tag{1}$$
We can apply the integration by parts and obtain:
$$I(z)=(1/z)\int_0^{\infty} f(t) \mathrm{d}\sin (z t)=-(1/z)\int_0^{\infty}\sin (z t)f'(t)\mathrm{d}t\tag{2}$$
Suppose that we have another integral ($\phi(z) \in \mathbb{R}$):
$$J(z,\phi(z))=\int_0^{\infty} f(t) \cos \left(\mathrm{e}^{i \phi(z)} z t \right) \mathrm{d}t,\qquad z>0 \tag{3}$$
Can we also apply the integration by parts and obtain the following result?
$$J(z,\phi(z))=(1/z)\mathrm{e}^{-i \phi(z)}\int_0^{\infty} f(t) \mathrm{d}\sin \left(\mathrm{e}^{i \phi(z)} z t \right)=$$ $$-(1/z)\mathrm{e}^{-i \phi(z)}\int_0^{\infty}\sin \left(\mathrm{e}^{i \phi(z)} z t \right)f'(t)\mathrm{d}t\tag{4}$$