No closed form solution

Robert Israel — Mon, 15 Mar 2010 18:54:00 Z

It is very likely that the antiderivative can't be expressed in "closed form", and there doesn't appear to be anything special about the endpoint 0.2 that might lead one to think that special techniques (such as residues) could be applied to the definite integral. Perhaps you could use power-series solutions, e.g. (with J your integral)

> sort(mtaylor(J, [a,b,c,d]), [a,b,c,d], ascending);

.2000000000+.1000000000*d^2+.4000000000e-1*c*d+.5333333333e-2*c^2+.8000000000e-2*b*d+.2400000000e-2*b*c+.2880000000e-3*b^2+.1600000000e-2*a*d+.5120000000e-3*a*c+.1280000000e-3*a*b+.1462857143e-4*a^2-.2500000000e-1*d^4-.2000000000e-1*c*d^3-.7999999999e-2*c^2*d^2-.1600000000e-2*c^3*d-.1280000000e-3*c^4-.4000000000e-2*b*d^3-.3600000000e-2*b*c*d^2-.1152000000e-2*b*c^2*d-.1280000000e-3*b*c^3-.4320000000e-3*b^2*d^2-.2880000000e-3*b^2*c*d-.4937142857e-4*b^2*c^2-.2468571428e-4*b^3*d-.8640000000e-5*b^3*c-.5759999999e-6*b^4-.8000000000e-3*a*d^3-.7680000000e-3*a*c*d^2-.2560000001e-3*a*c^2*d-.2925714286e-4*a*c^3-.1920000000e-3*a*b*d^2-.1316571428e-3*a*b*c*d-.2304000000e-4*a*b*c^2-.1728000000e-4*a*b^2*d-.6144000000e-5*a*b^2*c-.5529600000e-6*a*b^3-.2194285714e-4*a^2*d^2-.1536000000e-4*a^2*c*d-.2730666666e-5*a^2*c^2-.4096000001e-5*a^2*b*d-.1474560000e-5*a^2*b*c-.2010763636e-6*a^2*b^2-.3276800000e-6*a^3*d-.1191563636e-6*a^3*c-.3276800000e-7*a^3*b-.2016492308e-8*a^4

Hmm, interesting: the <maple> tag doesn't preserve the order of terms in the sum. That should have been

Int ( sqrt( product ) )

Axel Vogt — Mon, 15 Mar 2010 19:49:23 Z

I played a bit in an old Maple, which reports 6 explicite solutions for the term under the sqrt (have not checked them),

may be you check Int ( sqrt( product ) )

MaplePrimes - answers and comments on Question, How to solve an integration with unknown constants inside of it ???

No closed form solution

Int ( sqrt( product ) )