This is an arc length integral. Apparently you wish to compute the length of that curve in the plane, with x going from 0 to pi/2. As a numerical integral, this is easy.
f=log(45.*sin(x.^2) + cos(x));
gfun = matlabFunction(g);
First, it alway makes sense to plot everything, before you just throw it into any general tool.
I don't epect that arc length integral has an exact solution. I might also be slightly worried about what seems to be a singularity at 0, though an integration tool should survive that. You can use vpaintegral, if you wish to stay in the symbolic domain.
Or you can use integral.
Which seems to agree.
And, since I have a tool on the file exchange that can compute the arclength of a curve, I might try this:
ffun = matlabFunction(f);
which also seems to agree pretty well.