Old does not mean easy, like this integral.

Calculus Level 5

022arcsinxxdx=abG+cdπkln(m),\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,ma,b,c,d,k,m are positive integers and gcd(a,b)=gcd(c,d)=1\gcd(a,b)=\gcd(c,d)=1, and GG is the Catalan constant.

Find a+b+c+d+k+ma+b+c+d+k+m.


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

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