Thanks "@Charles Matthews". I did email Prof. Berndt (after seeing your comment) and he suggested me to look at this paper:

• Some definite integrals connected with Gauss's sums, Mess. Math. 44 (1915), 75-85.

The result appears in Page $60$ of the above link.

Thanks "@Charles Matthews". I did email Prof. Berndt (after seeing your comment) and he suggested me to look at this paper:

• Some definite integrals connected with Gauss's sums, Mess. Math. 44 (1915), 75-85.

The result appears here in Page 60. with proof.

Thanks "@Charles Matthews". I did email Prof. Berndt (after seeing your comment) and he suggested me to look at this paper:

• Some definite integrals connected with Gauss's sums, Mess. Math. 44 (1915), 75-85;

The result appears in Page $60$ of the above link.

Thanks "@Charles Matthews". I did email Prof. Berndt (after seeing your comment) and he suggested me to look at this paper:

• Some definite integrals connected with Gauss's sums, Mess. Math. 44 (1915), 75-85.

The result appears in Page $60$ of the above link.