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What makes me fear is the following:

sage: a = integrate(sqrt(1-1/4*cosh(x)^2),x)
sage: a.full_simplify()
cosh(x)

Which is definitely a wrong answer :(

Concerning your third question, the second compopent of the result of numerical_integral() is the error bound:

sage: numerical_integral(sqrt(1-1/4*cosh(x)^2),0,0.1)         
(0.08655430733928117, 9.609458488855213e-16)

What makes me fear is the following:

sage: a = integrate(sqrt(1-1/4*cosh(x)^2),x)
sage: a.full_simplify()
cosh(x)

Which is definitely a wrong answer :(

Concerning your third fourth question, the second compopent of the result of numerical_integral() is the error bound:

sage: numerical_integral(sqrt(1-1/4*cosh(x)^2),0,0.1)         
(0.08655430733928117, 9.609458488855213e-16)

What makes me fear is the following:

sage: a = integrate(sqrt(1-1/4*cosh(x)^2),x)
sage: a.full_simplify()
cosh(x)

Which is definitely a wrong answer :(

Concerning your fourth question, the second compopent component of the result of numerical_integral() is the error bound:

sage: numerical_integral(sqrt(1-1/4*cosh(x)^2),0,0.1)         
(0.08655430733928117, 9.609458488855213e-16)