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Maxima is the symbolic calculation software that Sage uses for integrals. It seems to be unable to evaluate this integral:

sage: maxima('integrate(sqrt(cos(x)^2+25), x)') 'integrate(sqrt(cos(x)^2+25),x)

On the other hand, now that we're talking about interfacing with Maxima directly, you could use Maxima's dblint function for this integral:

sage: maxima('r(x):= 0')
sage: maxima('s(x):= 2*pi')
sage: maxima('g(x,y):= sqrt(x^2*cos(y)^2 + 25)')
sage: BI = maxima('dblint(g, r, s, 3,5)')
sage: BI.n()

Unfortunately, I don't know how to access this directly from Sage.