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The integral is studied in On Certain Indefinite Integrals Involving Bessel Functions (1958).
_{(The $i=0$ integral is $g(a,0,x)$ in the notation of that paper, and as Robert Israel points out, the $i=1$ integral is simply related.)}
The paper is behind a paywall, so I have not studied it.

And then there is Tables of some indefinite integral of Bessel functions of integer order (2017), which examines the $i=0$ integral in great detail and gives several series expansions, see pages 87 and following.

I reproduce one such formula:

The integral is studied in On Certain Indefinite Integrals Involving Bessel Functions (1958).
_{(The $i=0$ integral is $g(a,0,x)$ in the notation of that paper, and as Robert Israel points out, the $i=1$ integral is simply related.)}
The paper is behind a paywall, so I have not read it.

And then there is Tables of some indefinite integral of Bessel functions of integer order (2017), which examines the $i=0$ integral in great detail and gives several series expansions, see pages 87 and following.

I reproduce one such formula:

The integral is studied in On Certain Indefinite Integrals Involving Bessel Functions (1958).
_{(The $i=0$ integral is $g(a,0,x)$ in the notation of that paper, and as Robert Israel points out, the $i=1$ integral is simply related.)}
The paper is behind a paywall, so I have not studied it.

And then there is Tables of some indefinite integral of bessel functions of integer order, (2017) which examines the $i=0$ integral and gives both a small-$x$ and a large-$x$ series expansion, see pages 87 and following.

The integral is studied in On Certain Indefinite Integrals Involving Bessel Functions (1958).
_{(The $i=0$ integral is $g(a,0,x)$ in the notation of that paper, and as Robert Israel points out, the $i=1$ integral is simply related.)}
The paper is behind a paywall, so I have not studied it.

And then there is Tables of some indefinite integral of Bessel functions of integer order (2017), which examines the $i=0$ integral in great detail and gives several series expansions, see pages 87 and following.

I reproduce one such formula:

The integral is studied in On Certain Indefinite Integrals Involving Bessel Functions (1958).
_{(The $i=0$ integral is $g(a,0,x)$ in the notation of that paper, and as Robert Israel points out, the $i=1$ integral is simply related.)}
The paper is behind a paywall, so I have not studied it.

And then there is Tables of some indefinite integral of bessel functions of integer order, (2017) which examines the $i=0$ integral and gives both a small-$x$ and a large-$x$ series expansion.

For small $x$ we can use the power series from page 87,

For large $x$ the power series in $1/x$ is given on page 89:

There are also small-$a$ and large-$a$ expansions, which I won't reproduce here.

The integral is studied in On Certain Indefinite Integrals Involving Bessel Functions (1958).
_{(The $i=0$ integral is $g(a,0,x)$ in the notation of that paper, and as Robert Israel points out, the $i=1$ integral is simply related.)}
The paper is behind a paywall, so I have not studied it.

And then there is Tables of some indefinite integral of bessel functions of integer order, (2017) which examines the $i=0$ integral and gives both a small-$x$ and a large-$x$ series expansion, see pages 87 and following.