The documentation for the option AllSolutions for int says that the results are always valid for all real parameter values (in the endpoints). That seems like a pretty major claim. Each of these three is already wrong for a=-1/2, b=1/2:

int(1/ln(t), t = a .. b, AllSolutions);
piecewise(ln(a) < ln(b), piecewise(And(1 < b, a < 1), undefined, piecewise(a = 1, infinity,
Ei(1, -ln(a)))+piecewise(b = 1, -infinity, -Ei(1, -ln(b)))), ln(b) = ln(a), 0, ln(b) < ln(a),
-piecewise(And(1 < a, b < 1), undefined, piecewise(b = 1, infinity, Ei(1, -ln(b)))+
piecewise(a = 1, -infinity, -Ei(1, -ln(a)))))
int(sqrt(t^2-1+I*t), t = a .. b, AllSolutions);
piecewise(a < b, (1/2)*sqrt(b^2-1+I*b)*b+I*sqrt(b^2-1+I*b)*(1/4)-3*ln(-2*signum(0, -b, 1)^2*
b^2+2*sqrt(b^4-b^2+1)*signum(0, -b, 1)^2+4*b*sqrt(2*sqrt(b^4-b^2+1)+2*b^2-2)+2*
signum(0, -b, 1)^2-2*signum(0, -b, 1)*sqrt(2*sqrt(b^4-b^2+1)-2*b^2+2)+6*b^2+2*
sqrt(b^4-b^2+1)-1)*(1/16)-3*ln((-I*(signum(0, -b, 1)*sqrt(2*sqrt(b^4-b^2+1)-2*b^2+2)-1+I*
sqrt(2*sqrt(b^4-b^2+1)+2*b^2-2)+(2*I)*b))*(1/sqrt(-2*signum(0, -b, 1)^2*b^2+2*sqrt(b^4-b^2+1)*
signum(0, -b, 1)^2+4*b*sqrt(2*sqrt(b^4-b^2+1)+2*b^2-2)+2*signum(0, -b, 1)^2-2*signum(0, -b, 1)*
sqrt(2*sqrt(b^4-b^2+1)-2*b^2+2)+6*b^2+2*sqrt(b^4-b^2+1)-1)))*(1/8)-(1/2)*sqrt(a^2-1+I*a)*a-I*
sqrt(a^2-1+I*a)*(1/4)+3*ln(-2*signum(0, -a, -1)^2*a^2+2*sqrt(a^4-a^2+1)*signum(0, -a, -1)^2+
4*a*sqrt(2*sqrt(a^4-a^2+1)+2*a^2-2)+2*signum(0, -a, -1)^2-2*signum(0, -a, -1)*sqrt(2*
sqrt(a^4-a^2+1)-2*a^2+2)+6*a^2+2*sqrt(a^4-a^2+1)-1)*(1/16)+3*ln((-I*(signum(0, -a, -1)*sqrt(2*
sqrt(a^4-a^2+1)-2*a^2+2)+I*sqrt(2*sqrt(a^4-a^2+1)+2*a^2-2)-1+(2*I)*a))*(1/sqrt(-2*
signum(0, -a, -1)^2*a^2+2*sqrt(a^4-a^2+1)*signum(0, -a, -1)^2+4*a*sqrt(2*sqrt(a^4-a^2+1)+2*
a^2-2)+2*signum(0, -a, -1)^2-2*signum(0, -a, -1)*sqrt(2*sqrt(a^4-a^2+1)-2*a^2+2)+6*a^2+2*
sqrt(a^4-a^2+1)-1)))*(1/8), b = a, 0, b < a, -(1/2)*sqrt(a^2-1+I*a)*a-I*sqrt(a^2-1+I*a)*(1/4)+
3*ln(-2*signum(0, -a, 1)^2*a^2+2*signum(0, -a, 1)^2*sqrt(a^4-a^2+1)+4*a*sqrt(2*sqrt(a^4-a^2+1)+
2*a^2-2)+2*signum(0, -a, 1)^2-2*signum(0, -a, 1)*sqrt(2*sqrt(a^4-a^2+1)-2*a^2+2)+6*a^2+
2*sqrt(a^4-a^2+1)-1)*(1/16)+3*ln((-I*(signum(0, -a, 1)*sqrt(2*sqrt(a^4-a^2+1)-2*a^2+2)+
I*sqrt(2*sqrt(a^4-a^2+1)+2*a^2-2)-1+(2*I)*a))*(1/sqrt(-2*signum(0, -a, 1)^2*a^2+
2*signum(0, -a, 1)^2*sqrt(a^4-a^2+1)+4*a*sqrt(2*sqrt(a^4-a^2+1)+2*a^2-2)+2*signum(0, -a, 1)^2-
2*signum(0, -a, 1)*sqrt(2*sqrt(a^4-a^2+1)-2*a^2+2)+6*a^2+2*sqrt(a^4-a^2+1)-1)))*(1/8)+(1/2)*
sqrt(b^2-1+I*b)*b+I*sqrt(b^2-1+I*b)*(1/4)-3*ln(-2*signum(0, -b, -1)^2*b^2+2*sqrt(b^4-b^2+1)*
signum(0, -b, -1)^2+4*b*sqrt(2*sqrt(b^4-b^2+1)+2*b^2-2)+2*signum(0, -b, -1)^2-
2*signum(0, -b, -1)*sqrt(2*sqrt(b^4-b^2+1)-2*b^2+2)+6*b^2+2*sqrt(b^4-b^2+1)-1)*(1/16)-
3*ln(-(I*signum(0, -b, -1)*sqrt(2*sqrt(b^4-b^2+1)-2*b^2+2)-I-sqrt(2*sqrt(b^4-b^2+1)+2*b^2-2)-
2*b)/sqrt(-2*signum(0, -b, -1)^2*b^2+2*sqrt(b^4-b^2+1)*signum(0, -b, -1)^2+4*b*sqrt(2*
sqrt(b^4-b^2+1)+2*b^2-2)+2*signum(0, -b, -1)^2-2*signum(0, -b, -1)*sqrt(2*sqrt(b^4-b^2+1)-
2*b^2+2)+6*b^2+2*sqrt(b^4-b^2+1)-1))*(1/8))
int(arctan(t+2*I), t = a .. b, AllSolutions);
piecewise(a < b, piecewise(a < 0, I*arctan(4*a/(a^2-3))*(1/2)+(1/4)*ln(a^2+1)+(1/4)*ln(a^2+9)-
(2*I)*arctan(2*I+a)-arctan(2*I+a)*a+I*Pi*(1/2), a = 0, -I*Pi+3*ln(3)*(1/2), 0 < a, I*arctan(4*a/
(a^2-3))*(1/2)+(1/4)*ln(a^2+1)+(1/4)*ln(a^2+9)-(2*I)*arctan(2*I+a)-arctan(2*I+a)*a-I*Pi*(1/2))+
piecewise(b < 0, -I*arctan(4*b/(b^2-3))*(1/2)-(1/4)*ln(b^2+1)-(1/4)*ln(b^2+9)+(2*I)*
arctan(2*I+b)+arctan(2*I+b)*b-I*Pi*(1/2), b = 0, -I*Pi-3*ln(3)*(1/2), 0 < b, -I*arctan(4*b/
(b^2-3))*(1/2)-(1/4)*ln(b^2+1)-(1/4)*ln(b^2+9)+(2*I)*arctan(2*I+b)+arctan(2*I+b)*b+I*Pi*(1/2))+
piecewise(And(0 < b, a < 0), -(2*I)*Pi, 0), b = a, 0, b < a, piecewise(b < 0, -I*arctan(4*b/
(b^2-3))*(1/2)-(1/4)*ln(b^2+1)-(1/4)*ln(b^2+9)+(2*I)*arctan(2*I+b)+arctan(2*I+b)*b-I*Pi*(1/2),
b = 0, I*Pi-3*ln(3)*(1/2), 0 < b, -I*arctan(4*b/(b^2-3))*(1/2)-(1/4)*ln(b^2+1)-(1/4)*ln(b^2+9)+
(2*I)*arctan(2*I+b)+arctan(2*I+b)*b+I*Pi*(1/2))+piecewise(a < 0, I*arctan(4*a/(a^2-3))*(1/2)+
(1/4)*ln(a^2+1)+(1/4)*ln(a^2+9)-(2*I)*arctan(2*I+a)-arctan(2*I+a)*a+I*Pi*(1/2), a = 0,
I*Pi+3*ln(3)*(1/2), 0 < a, I*arctan(4*a/(a^2-3))*(1/2)+(1/4)*ln(a^2+1)+(1/4)*ln(a^2+9)-(2*I)*
arctan(2*I+a)-arctan(2*I+a)*a-I*Pi*(1/2))-piecewise(And(0 < a, b < 0), -(2*I)*Pi, 0))

The first one probably has the correct answer inside, but it has conditions like ln(a)<ln(b), so that case never gets selected when the values are complex.