> 
ig:=((3+x^2)*(16*x^2+x^4)^(1/4))/(1+x^2);

> 
S:=simplify([allvalues(int(convert(ig,RootOf),x))],size):

Stage1: firststage indefinite integration
Stage2: secondstage indefinite integration
Norman: enter RischNorman integrator
Norman: exit RischNorman integrator
int/algrisch/int: Risch/Trager's algorithm for algebraic function
int/algrisch/int: entering integrator at time 9.029
int/algrisch/int: function field has degree 4
int/algrisch/int: computation of an integral basis: start time 9.031
int/algrisch/int: computation of an integral basis: end time 9.038
int/algrisch/int: normalization at infinity: start time 9.039
int/algrisch/int: normalization at infinity: end time 9.048
int/algrisch/int: genus of the function field 3
int/algrisch/int: computation of the algebraic part: start time 9.059
int/algrisch/int: computation of the algebraic part: end time 9.060
int/algrisch/int: computation of the transcendental part: start time 9.063
int/algrisch/transcpar: computing a basis for the residues at time 9.068
int/algrisch/residues: computing a splitting field at time 9.068
int/algrisch/transcpar: basis for the residues computed at time 9.103
int/algrisch/transcpar: dimension is 2
int/algrisch/transcpar: building divisors at time 9.300
int/algrisch/transcpar: testing divisors for principality at time 9.605
int/algrisch/goodprime: searching for a good prime at time 9.606
int/algrisch/goodprime: good prime found at time 9.704
int/algrisch/goodprime: searching for a good prime at time 9.704
int/algrisch/goodprime: good prime found at time 9.762
int/algrisch/areprinc: the divisor is principal: time 10.084
int/algrisch/areprinc: the divisor is principal: time 11.833
int/algrisch/transcpar: divisors proven pincipal at time at time 11.833
int/algrisch/transcpar: generators computed at time 11.834
int/algrisch/transcpar: orders are [1 1]
int/algrisch/transcpar: check that the candidate is an actual antiderivative
int/algrisch/transcpar: the antiderivative is elementary
int/algrisch/transcpar: antiderivative is (1/2)*ln((RootOf(_Z^4*_z^4_z^4+6*_z^21 index = 1)^3*_z^3RootOf(_Z^4*_z^4_z^4+6*_z^21 index = 1)^2*_z^4+RootOf(_Z^4*_z^4_z^4+6*_z^21 index = 1)^2*_z^23*_z^3*RootOf(_Z^4*_z^4_z^4+6*_z^21 index = 1)+RootOf(_Z^4*_z^4_z^4+6*_z^21 index = 1)*_z5*_z^2+1)/((_z+1)*(_z1)*_z^2))+(1/2)*RootOf(_Z^2+1)*ln((RootOf(_Z^4*_z^4_z^4+6*_z^21 index = 1)^2*RootOf(_Z^2+1)*_z^4+RootOf(_Z^4*_z^4_z^4+6*_z^21 index = 1)^3*_z^3RootOf(_Z^2+1)*RootOf(_Z^4*_z^4_z^4+6*_z^21 index = 1)^2*_z^2+3*_z^3*RootOf(_Z^4*_z^4_z^4+6*_z^21 index = 1)5*RootOf(_Z^2+1)*_z^2RootOf(_Z^4*_z^4_z^4+6*_z^21 index = 1)*_z+RootOf(_Z^2+1))/((_z+1)*(_z1)*_z^2))
int/algrisch/int: computation of the transcendental part: end time 12.040
int/algrisch/int: exiting integrator for algebraic functions at time 12.041
> 
simplify( diff(S[1],x)  ig ), simplify( diff(S[2],x)  ig );

