Integrate[cC1((((a x))^2)+1),x]

(cC1 x+(1/3) (a^2) cC1 (x^3))/.Solve[(cC1 x+(1/3) (a^2) cC1 (x^3)/.{x->1})==1,cC1][[1]]

Manipulate[Plot[((3 x)/(3+(a^2)))+(((a^2) (x^3))/(3+(a^2))),{x,0,1},PlotRange->{0,1.2}],{a,0.001,10}]