Integral using Euler's substitution Question:
Solve this:
$$\displaystyle \int \frac {2x-\sqrt {4x^2-x+1} } {x-1} \,dx$$
Our solution: 
$$16 \ln|t+4|-\ln|t+0.25|+C$$ when $t=\sqrt {4x^2-x+1}-2x$ (using Euler's formula)
But wolfram's solution was totally different. We were wondering if these two are equal.
We found this  question very helpful
Integral( using Euler substitution)
but we were wondering how could one check that hyperbolic solution and Euler solution are equal?

Tools that might help for a direct check if the two solutions are equal up to a constant ... and perhaps useful in the future.


*

*We have that the expression $\sqrt{4x^2-x+1}-2x$ can be re-written as $\frac{(\sqrt{4x^2-x+1}-2x)(\sqrt{4x^2-x+1}+2x)}{\sqrt{4x^2-x+1}-2x}=\frac{(4x^2-x+1)-4x^2}{\sqrt{4x^2-x+1}+2x}$. 

*Moreover, if $\frac{(4x^2-x+1)-4x^2}{\sqrt{4x^2-x+1}+2x}$ is inside a logarithm it can be re-written as $\log(1-x)-\log(\sqrt{4x^2-x+1}+2x)$.

*The inverse of $\sinh$ can be computed in terms of logarithms: $y:=\sinh(x)=\frac{e^x-e^{-x}}{2}$, from where $e^{2x}-2ye^x-1=0$, and then $e^x=\frac{2y+\sqrt{4y^2+4}}{2}$. Then we get $\sinh^{-1}(x)=\log(y+\sqrt{y^2+1})$.
