Identity between two possible outputs in an integral process

Identity between two possible outputs in an integral process

I have this:
$$\int{\frac{45}{\sqrt{x^2-1}}}dx$$
This can be solved in at least 2 different ways:
1) With trigonometric functions:
$$45sin^{-1}x+C$$
or,
2) By logarithms
$$45log\sqrt{x^2-1}+x+C$$
If I am right and by:
law: if a=b and b=c then a=c
Question
If this means that:
$$45log\sqrt{x^2-1}+x+C=45sin^{-1}x+C$$
Is this an equality? What is the right relation of the two terms? What
does this means?