# Integral of Sqrt[1/x-1/a] Cosh[Sqrt[x]]

MMA delivers: $$\int_0^a\sqrt{\frac{1}{x}-\frac{1}{a}}\cosh\left(\sqrt{x}\right)\textrm{d}x=\pi I_1(\sqrt{a})$$ where $$a>0$$ and $$I_1$$ is the modified Bessel function of $$1^\textrm{st}$$ kind and order $$1$$.

1. How was this equation by MMA integrated? Rubi cannot integrate it, so it can be assumed that it is not a simple integration.

Edit: Question 1 was also posted in a more suitable forum and also answered there.

1. Why MMA does give a solution only if it is a definite integral and if upper border is $$a$$? (For indefinite integral or if instead of $$a$$ we set upper border to $$1$$ MMA delivers no solution.)

Edit: Question 2 was answered below.

$Assumptions=a>0; Integrate[Sqrt[(1/x-1/a)] Cosh[Sqrt[x]],{x,0,a}] (*\[Pi] BesselI[1,Sqrt[a]*)  MMA 12.3 ## 1 Answer $Version

(* "12.3.1 for Mac OS X x86 (64-bit) (June 19, 2021)" *)

\$Assumptions = a > 0;

Integrate[Sqrt[(1/x - 1/a)] Cosh[Sqrt[x]], {x, 0, a}]

(* π BesselI[1, Sqrt[a]] *)

% /. a -> 1

(* π BesselI[1, 1] *)


For a == 1 you need to replace both instances of a: in the argument as well as the integration bounds.

Integrate[Sqrt[(1/x - 1/1)] Cosh[Sqrt[x]], {x, 0, 1}]

(* π BesselI[1, 1] *)