I have the following principal value integral: $$\mathcal{P}\int_0^\infty\frac{x^4}{\left(1+\frac{x^2}{B^2}\right)^{4}\sqrt{C^2+x^2}\left[\sqrt{C^2+a^2}-\sqrt{C^2+x^2}\right]}dx$$ where $a,B,C\in\mathbb{R}^+$, (and of course the singularity is at $x=a$). If it's analytically solvable (which I highly doubt), I'd like to know how to solve it. If not, how can I manipulate it to equate it to an expression in terms of "ordinary" (non-principal-value) integrals?

It has been suggested by someone I know that I might be able to make use of the result: $$\mathcal{P}\int_0^\infty\frac{dx}{C^2-x^2}=0$$ for $C\in\mathbb{R}$, but I am unsure how to make this result useful.

I have the following principal value integral: $$\mathcal{P}\int_0^\infty\frac{x^4}{\left(1+\frac{x^2}{B^2}\right)^{4}\sqrt{C^2+x^2}\left[\sqrt{C^2+a^2}-\sqrt{C^2+x^2}\right]}dx$$ where $a,B,C\in\mathbb{R}^+$, (and of course the singularity is at $x=a$). If it's analytically solvable (which I highly doubt), I'd like to know how to solve it. (As a commenter has pointed out, there is actually an elementary antiderivative for the indefinite integral, but the form is rather complicated so I'd rather just reduce it to a non-principal form and evaluate it numerically.) How can I manipulate it to equate it to an expression in terms of "ordinary" (non-principal-value) integrals?

It has been suggested by someone I know that I might be able to make use of the result: $$\mathcal{P}\int_0^\infty\frac{dx}{C^2-x^2}=0$$ for $C\in\mathbb{R}$, but I am unsure how to make this result useful.

I have the following principal value integral: $$\mathcal{P}\int_0^\infty\frac{x^4}{\left(1+\frac{x^2}{B^2}\right)^{4}\sqrt{C^2+x^2}\left[\sqrt{C^2+a^2}-\sqrt{C^2+x^2}\right]}dx$$ where $a,B,C\in\mathbb{R}^+$, (and of course the singularity is at $x=a$). If it's analytically solvable (which I highly doubt), I'd like to know how to solve it. If not, how can I manipulate it to equate it to an expression in terms of "ordinary" (non-principal-value) integrals?

I have the following principal value integral: $$\mathcal{P}\int_0^\infty\frac{x^4}{\left(1+\frac{x^2}{B^2}\right)^{4}\sqrt{C^2+x^2}\left[\sqrt{C^2+a^2}-\sqrt{C^2+x^2}\right]}dx$$ where $a,B,C\in\mathbb{R}^+$, (and of course the singularity is at $x=a$). If it's analytically solvable (which I highly doubt), I'd like to know how to solve it. If not, how can I manipulate it to equate it to an expression in terms of "ordinary" (non-principal-value) integrals?

It has been suggested by someone I know that I might be able to make use of the result: $$\mathcal{P}\int_0^\infty\frac{dx}{C^2-x^2}=0$$ for $C\in\mathbb{R}$, but I am unsure how to make this result useful.