Old does not mean easy, like this integral.

Calculus Level 5

\[\int_0^{\frac{\sqrt{2} }{2} } \frac{\arcsin x}{x} dx = \frac{a}{b}G +\frac{c}{d}\pi^k \ln(m),\]

where \(a,b,c,d,k,m\) are positive integers and \(\gcd(a,b)=\gcd(c,d)=1\), and \(G\) is the Catalan constant.

Find \(a+b+c+d+k+m\).


Note: The Catalan constant \(G\) is defined by \[G=\sum_{n=0}^{\infty} \frac{(-1)^n}{(2n+1)^2}. \]

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