A1=Sqrt[(d*I3-1)/(I1*(I3-I1))];
A2=Sqrt[(d*I3-1)/(I2*(I3-I2))];
A3=Sqrt[(1-d*I1)/(I3*(I3-I1))];
B=Sqrt[(1-d*I1)*(I3-I2)/(I1*I2*I3)];
m=((d*I3-1)*(I2-I1))/((1-d*I1)*(I3-I2));
ω1=A1*JacobiCN[B*t,m];
ω2=A2*JacobiSN[B*t,m];
ω3=A3*JacobiDN[B*t,m];

eq1=(I1*D[ω1,t]-(I2-I3)*ω2*ω3)/(JacobiDN[B*t,m]*JacobiSN[B*t,m]);
eq1a=Simplify[eq1,Assumptions->{I1<I3,I2<I3,d*I1<1,d*I3>1,I3>0,I1>0}];
Simplify[Expand[(eq1a) ^2],Assumptions->{I1<I3,I2<I3,d*I1<1,d*I3>1,I3>0,I1>0}]

eq2=(I2*D[ω2,t]-(I3-I1)*ω3*ω1)/(JacobiDN[B*t,m]*JacobiCN[B*t,m]);
eq2a=Simplify[eq2,Assumptions->{I1<I3,I2<I3,d*I1<1,d*I3>1,I3>0,I1>0}];
Simplify[Expand[(eq2a) ^2],Assumptions->{I1<I3,I2<I3,d*I1<1,d*I3>1,I3>0,I1>0}]

eq3=(I3*D[ω3,t]-(I1-I2)*ω1*ω2)/(JacobiSN[B*t,m]*JacobiCN[B*t,m]);
eq3a=Simplify[eq3,Assumptions->{I1<I3,I2<I3,d*I1<1,d*I3>1,I3>0,I1>0}];
Simplify[Expand[(eq3a) ^2],Assumptions->{I1<I3,I2<I3,d*I1<1,d*I3>1,I3>0,I1>0}]