## Symbolic Integration I: Transcendental Functions, Second by Manuel Bronstein

By Manuel Bronstein

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In, e i , . . , en) (* Complete partial fraction decomposition *) (* Given a Euclidean domain D, positive integ ers Ti, e i , . . ,dn G D\ {0} with gcd{di^dj) = 1 for i j^ j , return ao, ai,i,. , ai,ei, • • • , an,i, •. n. ^ -C^ such that aij d^^ • • • dn and either aij = 0 or i^{aij) < u(di). *) ( a o , a i , . . ,an) <— PartialPractioii(a,

Here is H e r m i t e R e d u c e on x^ - 24x^ - 4x^ + 8x - 8 ^ ~~ x^ + 6x6 + 12x4 + 8^2 ^ ^V^) • A squarefree factorization of the denominator of / is D =: x^ + 6x^ + 12x^ + 8x^ = x2(x2 + 2)^ = DIDI and the partial fraction decomposition of / is: / = X- 1 -4X - 6x3 __ 18^2 __ i 2 x + 8 (x2 + 2)3 Here is the rest of the Hermite reduction for / : i V 3 A. 2 The Hermite Reduction 41 Thus, 24x^ - 4^2 + 8x - 8 , 1 6x dx = -\, ^ ^ (x2 +, ^,n 2)2 x» + 6J:6 + 12^4 + 8x2 x-3 f dx 77——: + x^ + 2 We also mention t h e following variant of Hermite's algorithm t h a t does not require a partial fraction decomposition of / : let D = D1D2 • • • D'^ be a squarefree factorization of D and suppose t h a t m > 2 (otherwise D is already squarefree).

Dn G D\ {0} with gcd{di^dj) = 1 for i j^ j , return ao, ai,i,. , ai,ei, • • • , an,i, •. n. ^ -C^ such that aij d^^ • • • dn and either aij = 0 or i^{aij) < u(di). *) ( a o , a i , . . ,an) <— PartialPractioii(a,