Integration solution needed

Question

I need to solve the following integration $$\int_0^{\infty}\exp \left(-ax^4-bx^2[\pi/2-\arctan(\frac{c}{x^2d})]\right)dx$$ where $a,b,c,d$ are all constants and $\geq 0$. I know the following formula $$\int_0^{\infty}\exp(-ax-bx^2)dx=\sqrt{\frac{\pi}{b}}\exp(\frac{a^2}{4b})Q(\frac{a}{\sqrt{2b}})$$ where $Q(x)$ is the famous Q function. But I can not apply this formula because the form in my first equation does not matches exactly with the form of my second equation (because of the presence of $\arctan(\frac{c}{x^2d})$). How can I solve this problem? Is there any approximation of $\arctan()$ that can be used to make the form of my first equation look like the second equation. Many thanks in advance for your help.