# Maple integration test file: "5 Inverse trig functions\5.3 Inverse tangent\5.3.4 u (a+b arctan(c x))^p.txt"

lst:=[

# Integrands of the form (f x)^m (d+e x)^q (a+b ArcTan[c x])^p

# Integrands of the form x^m (d+e x)^q (a+b ArcTan[c x])^p with c^2 d^2+e^2=0

# Integrands of the form x^m (d+e x)^q (a+b ArcTan[c x])^1 with c^2 d^2+e^2=0

# q>0
[x^3*(d+I*c*d*x)*(a+b*arctan(c*x)),x,7,1/4*b*d*x/c^3+1/10*I*b*d*x^2/c^2-1/12*b*d*x^3/c-1/20*I*b*d*x^4-1/4*b*d*arctan(c*x)/c^4+1/4*d*x^4*(a+b*arctan(c*x))+1/5*I*c*d*x^5*(a+b*arctan(c*x))-1/10*I*b*d*log(1+c^2*x^2)/c^4],
[x^2*(d+I*c*d*x)*(a+b*arctan(c*x)),x,7,1/4*I*b*d*x/c^2-1/6*b*d*x^2/c-1/12*I*b*d*x^3-1/4*I*b*d*arctan(c*x)/c^3+1/3*d*x^3*(a+b*arctan(c*x))+1/4*I*c*d*x^4*(a+b*arctan(c*x))+1/6*b*d*log(1+c^2*x^2)/c^3],
[x*(d+I*c*d*x)*(a+b*arctan(c*x)),x,7,-1/2*b*d*x/c-1/6*I*b*d*x^2+1/2*b*d*arctan(c*x)/c^2+1/2*d*x^2*(a+b*arctan(c*x))+1/3*I*c*d*x^3*(a+b*arctan(c*x))+1/6*I*b*d*log(1+c^2*x^2)/c^2],
[(d+I*c*d*x)*(a+b*arctan(c*x)),x,4,-1/2*I*b*d*x-1/2*I*d*(1+I*c*x)^2*(a+b*arctan(c*x))/c-b*d*log(I+c*x)/c],
[(d+I*c*d*x)*(a+b*arctan(c*x))/x,x,8,I*a*c*d*x+I*b*c*d*x*arctan(c*x)+a*d*log(x)-1/2*I*b*d*log(1+c^2*x^2)+1/2*I*b*d*polylog(2,-I*c*x)-1/2*I*b*d*polylog(2,I*c*x)],
[(d+I*c*d*x)*(a+b*arctan(c*x))/x^2,x,10,-d*(a+b*arctan(c*x))/x+I*a*c*d*log(x)+b*c*d*log(x)-1/2*b*c*d*log(1+c^2*x^2)-1/2*b*c*d*polylog(2,-I*c*x)+1/2*b*c*d*polylog(2,I*c*x)],
[(d+I*c*d*x)*(a+b*arctan(c*x))/x^3,x,4,-1/2*b*c*d/x-1/2*d*(1+I*c*x)^2*(a+b*arctan(c*x))/x^2+I*b*c^2*d*log(x)-I*b*c^2*d*log(I+c*x)],
[(d+I*c*d*x)*(a+b*arctan(c*x))/x^4,x,4,-1/6*b*c*d/x^2-1/2*I*b*c^2*d/x-1/3*d*(a+b*arctan(c*x))/x^3-1/2*I*c*d*(a+b*arctan(c*x))/x^2-1/3*b*c^3*d*log(x)-1/12*b*c^3*d*log(I-c*x)+5/12*b*c^3*d*log(I+c*x)],
[(d+I*c*d*x)*(a+b*arctan(c*x))/x^5,x,4,-1/12*b*c*d/x^3-1/6*I*b*c^2*d/x^2+1/4*b*c^3*d/x-1/4*d*(a+b*arctan(c*x))/x^4-1/3*I*c*d*(a+b*arctan(c*x))/x^3-1/3*I*b*c^4*d*log(x)+1/24*I*b*c^4*d*log(I-c*x)+7/24*I*b*c^4*d*log(I+c*x)],
[x^3*(d+I*c*d*x)^2*(a+b*arctan(c*x)),x,7,5/12*b*d^2*x/c^3+1/5*I*b*d^2*x^2/c^2-5/36*b*d^2*x^3/c-1/10*I*b*d^2*x^4+1/30*b*c*d^2*x^5-5/12*b*d^2*arctan(c*x)/c^4+1/4*d^2*x^4*(a+b*arctan(c*x))+2/5*I*c*d^2*x^5*(a+b*arctan(c*x))-1/6*c^2*d^2*x^6*(a+b*arctan(c*x))-1/5*I*b*d^2*log(1+c^2*x^2)/c^4],
[x^2*(d+I*c*d*x)^2*(a+b*arctan(c*x)),x,7,1/2*I*b*d^2*x/c^2-4/15*b*d^2*x^2/c-1/6*I*b*d^2*x^3+1/20*b*c*d^2*x^4-1/2*I*b*d^2*arctan(c*x)/c^3+1/3*d^2*x^3*(a+b*arctan(c*x))+1/2*I*c*d^2*x^4*(a+b*arctan(c*x))-1/5*c^2*d^2*x^5*(a+b*arctan(c*x))+4/15*b*d^2*log(1+c^2*x^2)/c^3],
[x*(d+I*c*d*x)^2*(a+b*arctan(c*x)),x,7,-3/4*b*d^2*x/c-1/3*I*b*d^2*x^2+1/12*b*c*d^2*x^3+3/4*b*d^2*arctan(c*x)/c^2+1/2*d^2*x^2*(a+b*arctan(c*x))+2/3*I*c*d^2*x^3*(a+b*arctan(c*x))-1/4*c^2*d^2*x^4*(a+b*arctan(c*x))+1/3*I*b*d^2*log(1+c^2*x^2)/c^2],
[(d+I*c*d*x)^2*(a+b*arctan(c*x)),x,4,-2/3*I*b*d^2*x-1/6*b*d^2*(1+I*c*x)^2/c-1/3*I*d^2*(1+I*c*x)^3*(a+b*arctan(c*x))/c-4/3*b*d^2*log(1-I*c*x)/c],
[(d+I*c*d*x)^2*(a+b*arctan(c*x))/x,x,11,2*I*a*c*d^2*x+1/2*b*c*d^2*x-1/2*b*d^2*arctan(c*x)+2*I*b*c*d^2*x*arctan(c*x)-1/2*c^2*d^2*x^2*(a+b*arctan(c*x))+a*d^2*log(x)-I*b*d^2*log(1+c^2*x^2)+1/2*I*b*d^2*polylog(2,-I*c*x)-1/2*I*b*d^2*polylog(2,I*c*x)],
[(d+I*c*d*x)^2*(a+b*arctan(c*x))/x^2,x,13,-a*c^2*d^2*x-b*c^2*d^2*x*arctan(c*x)-d^2*(a+b*arctan(c*x))/x+2*I*a*c*d^2*log(x)+b*c*d^2*log(x)-b*c*d^2*polylog(2,-I*c*x)+b*c*d^2*polylog(2,I*c*x)],
[(d+I*c*d*x)^2*(a+b*arctan(c*x))/x^3,x,13,-1/2*b*c*d^2/x-1/2*b*c^2*d^2*arctan(c*x)-1/2*d^2*(a+b*arctan(c*x))/x^2-2*I*c*d^2*(a+b*arctan(c*x))/x-a*c^2*d^2*log(x)+2*I*b*c^2*d^2*log(x)-I*b*c^2*d^2*log(1+c^2*x^2)-1/2*I*b*c^2*d^2*polylog(2,-I*c*x)+1/2*I*b*c^2*d^2*polylog(2,I*c*x)],
[(d+I*c*d*x)^2*(a+b*arctan(c*x))/x^4,x,4,-1/6*b*c*d^2/x^2-I*b*c^2*d^2/x-1/3*d^2*(1+I*c*x)^3*(a+b*arctan(c*x))/x^3-4/3*b*c^3*d^2*log(x)+4/3*b*c^3*d^2*log(I+c*x)],
[(d+I*c*d*x)^2*(a+b*arctan(c*x))/x^5,x,4,-1/12*b*c*d^2/x^3-1/3*I*b*c^2*d^2/x^2+3/4*b*c^3*d^2/x-1/4*d^2*(a+b*arctan(c*x))/x^4-2/3*I*c*d^2*(a+b*arctan(c*x))/x^3+1/2*c^2*d^2*(a+b*arctan(c*x))/x^2-2/3*I*b*c^4*d^2*log(x)-1/24*I*b*c^4*d^2*log(I-c*x)+17/24*I*b*c^4*d^2*log(I+c*x)],
[(d+I*c*d*x)^2*(a+b*arctan(c*x))/x^6,x,4,-1/20*b*c*d^2/x^4-1/6*I*b*c^2*d^2/x^3+4/15*b*c^3*d^2/x^2+1/2*I*b*c^4*d^2/x-1/5*d^2*(a+b*arctan(c*x))/x^5-1/2*I*c*d^2*(a+b*arctan(c*x))/x^4+1/3*c^2*d^2*(a+b*arctan(c*x))/x^3+8/15*b*c^5*d^2*log(x)-1/60*b*c^5*d^2*log(I-c*x)-31/60*b*c^5*d^2*log(I+c*x)],
[x^3*(d+I*c*d*x)^3*(a+b*arctan(c*x)),x,7,3/4*b*d^3*x/c^3+13/35*I*b*d^3*x^2/c^2-1/4*b*d^3*x^3/c-13/70*I*b*d^3*x^4+1/10*b*c*d^3*x^5+1/42*I*b*c^2*d^3*x^6-3/4*b*d^3*arctan(c*x)/c^4+1/4*d^3*x^4*(a+b*arctan(c*x))+3/5*I*c*d^3*x^5*(a+b*arctan(c*x))-1/2*c^2*d^3*x^6*(a+b*arctan(c*x))-1/7*I*c^3*d^3*x^7*(a+b*arctan(c*x))-13/35*I*b*d^3*log(1+c^2*x^2)/c^4],
[x^2*(d+I*c*d*x)^3*(a+b*arctan(c*x)),x,7,11/12*I*b*d^3*x/c^2-7/15*b*d^3*x^2/c-11/36*I*b*d^3*x^3+3/20*b*c*d^3*x^4+1/30*I*b*c^2*d^3*x^5-11/12*I*b*d^3*arctan(c*x)/c^3+1/3*d^3*x^3*(a+b*arctan(c*x))+3/4*I*c*d^3*x^4*(a+b*arctan(c*x))-3/5*c^2*d^3*x^5*(a+b*arctan(c*x))-1/6*I*c^3*d^3*x^6*(a+b*arctan(c*x))+7/15*b*d^3*log(1+c^2*x^2)/c^3],
[x*(d+I*c*d*x)^3*(a+b*arctan(c*x)),x,4,-3/5*b*d^3*x/c-3/20*I*b*d^3*(I-c*x)^2/c^2-1/20*b*d^3*(I-c*x)^3/c^2+1/20*I*b*d^3*(I-c*x)^4/c^2+1/4*d^3*(1+I*c*x)^4*(a+b*arctan(c*x))/c^2-1/5*d^3*(1+I*c*x)^5*(a+b*arctan(c*x))/c^2+6/5*I*b*d^3*log(I+c*x)/c^2],
[(d+I*c*d*x)^3*(a+b*arctan(c*x)),x,4,-I*b*d^3*x-1/4*b*d^3*(1+I*c*x)^2/c-1/12*b*d^3*(1+I*c*x)^3/c-1/4*I*d^3*(1+I*c*x)^4*(a+b*arctan(c*x))/c-2*b*d^3*log(1-I*c*x)/c],
[(d+I*c*d*x)^3*(a+b*arctan(c*x))/x,x,15,3*I*a*c*d^3*x+3/2*b*c*d^3*x+1/6*I*b*c^2*d^3*x^2-3/2*b*d^3*arctan(c*x)+3*I*b*c*d^3*x*arctan(c*x)-3/2*c^2*d^3*x^2*(a+b*arctan(c*x))-1/3*I*c^3*d^3*x^3*(a+b*arctan(c*x))+a*d^3*log(x)-5/3*I*b*d^3*log(1+c^2*x^2)+1/2*I*b*d^3*polylog(2,-I*c*x)-1/2*I*b*d^3*polylog(2,I*c*x)],
[(d+I*c*d*x)^3*(a+b*arctan(c*x))/x^2,x,16,-3*a*c^2*d^3*x+1/2*I*b*c^2*d^3*x-1/2*I*b*c*d^3*arctan(c*x)-3*b*c^2*d^3*x*arctan(c*x)-d^3*(a+b*arctan(c*x))/x-1/2*I*c^3*d^3*x^2*(a+b*arctan(c*x))+3*I*a*c*d^3*log(x)+b*c*d^3*log(x)+b*c*d^3*log(1+c^2*x^2)-3/2*b*c*d^3*polylog(2,-I*c*x)+3/2*b*c*d^3*polylog(2,I*c*x)],
[(d+I*c*d*x)^3*(a+b*arctan(c*x))/x^3,x,16,-1/2*b*c*d^3/x-I*a*c^3*d^3*x-1/2*b*c^2*d^3*arctan(c*x)-I*b*c^3*d^3*x*arctan(c*x)-1/2*d^3*(a+b*arctan(c*x))/x^2-3*I*c*d^3*(a+b*arctan(c*x))/x-3*a*c^2*d^3*log(x)+3*I*b*c^2*d^3*log(x)-I*b*c^2*d^3*log(1+c^2*x^2)-3/2*I*b*c^2*d^3*polylog(2,-I*c*x)+3/2*I*b*c^2*d^3*polylog(2,I*c*x)],
[(d+I*c*d*x)^3*(a+b*arctan(c*x))/x^4,x,17,-1/6*b*c*d^3/x^2-3/2*I*b*c^2*d^3/x-3/2*I*b*c^3*d^3*arctan(c*x)-1/3*d^3*(a+b*arctan(c*x))/x^3-3/2*I*c*d^3*(a+b*arctan(c*x))/x^2+3*c^2*d^3*(a+b*arctan(c*x))/x-I*a*c^3*d^3*log(x)-10/3*b*c^3*d^3*log(x)+5/3*b*c^3*d^3*log(1+c^2*x^2)+1/2*b*c^3*d^3*polylog(2,-I*c*x)-1/2*b*c^3*d^3*polylog(2,I*c*x)],
[(d+I*c*d*x)^3*(a+b*arctan(c*x))/x^5,x,4,-1/12*b*c*d^3/x^3-1/2*I*b*c^2*d^3/x^2+7/4*b*c^3*d^3/x-1/4*d^3*(1+I*c*x)^4*(a+b*arctan(c*x))/x^4-2*I*b*c^4*d^3*log(x)+2*I*b*c^4*d^3*log(I+c*x)],
[(d+I*c*d*x)^3*(a+b*arctan(c*x))/x^6,x,4,-1/20*b*c*d^3/x^4-1/4*I*b*c^2*d^3/x^3+3/5*b*c^3*d^3/x^2+5/4*I*b*c^4*d^3/x-1/5*d^3*(1+I*c*x)^4*(a+b*arctan(c*x))/x^5+1/20*I*c*d^3*(1+I*c*x)^4*(a+b*arctan(c*x))/x^4+6/5*b*c^5*d^3*log(x)-6/5*b*c^5*d^3*log(I+c*x)],
[(d+I*c*d*x)^3*(a+b*arctan(c*x))/x^7,x,4,-1/30*b*c*d^3/x^5-3/20*I*b*c^2*d^3/x^4+11/36*b*c^3*d^3/x^3+7/15*I*b*c^4*d^3/x^2-11/12*b*c^5*d^3/x-1/6*d^3*(a+b*arctan(c*x))/x^6-3/5*I*c*d^3*(a+b*arctan(c*x))/x^5+3/4*c^2*d^3*(a+b*arctan(c*x))/x^4+1/3*I*c^3*d^3*(a+b*arctan(c*x))/x^3+14/15*I*b*c^6*d^3*log(x)-1/120*I*b*c^6*d^3*log(I-c*x)-37/40*I*b*c^6*d^3*log(I+c*x)],
[x^3*(d+I*c*d*x)^4*(a+b*arctan(c*x)),x,7,11/8*b*d^4*x/c^3+24/35*I*b*d^4*x^2/c^2-11/24*b*d^4*x^3/c-12/35*I*b*d^4*x^4+9/40*b*c*d^4*x^5+2/21*I*b*c^2*d^4*x^6-1/56*b*c^3*d^4*x^7-11/8*b*d^4*arctan(c*x)/c^4+1/4*d^4*x^4*(a+b*arctan(c*x))+4/5*I*c*d^4*x^5*(a+b*arctan(c*x))-c^2*d^4*x^6*(a+b*arctan(c*x))-4/7*I*c^3*d^4*x^7*(a+b*arctan(c*x))+1/8*c^4*d^4*x^8*(a+b*arctan(c*x))-24/35*I*b*d^4*log(1+c^2*x^2)/c^4],
[x^2*(d+I*c*d*x)^4*(a+b*arctan(c*x)),x,4,5/3*I*b*d^4*x/c^2-88/105*b*d^4*x^2/c-5/9*I*b*d^4*x^3+47/140*b*c*d^4*x^4+2/15*I*b*c^2*d^4*x^5-1/42*b*c^3*d^4*x^6+1/5*I*d^4*(1+I*c*x)^5*(a+b*arctan(c*x))/c^3-1/3*I*d^4*(1+I*c*x)^6*(a+b*arctan(c*x))/c^3+1/7*I*d^4*(1+I*c*x)^7*(a+b*arctan(c*x))/c^3+176/105*b*d^4*log(I+c*x)/c^3],
[x*(d+I*c*d*x)^4*(a+b*arctan(c*x)),x,4,-16/15*b*d^4*x/c-4/15*I*b*d^4*(I-c*x)^2/c^2-4/45*b*d^4*(I-c*x)^3/c^2+1/30*I*b*d^4*(I-c*x)^4/c^2+1/30*b*d^4*(I-c*x)^5/c^2+1/5*d^4*(1+I*c*x)^5*(a+b*arctan(c*x))/c^2-1/6*d^4*(1+I*c*x)^6*(a+b*arctan(c*x))/c^2+32/15*I*b*d^4*log(I+c*x)/c^2],
[(d+I*c*d*x)^4*(a+b*arctan(c*x)),x,4,-8/5*I*b*d^4*x-2/5*b*d^4*(1+I*c*x)^2/c-2/15*b*d^4*(1+I*c*x)^3/c-1/20*b*d^4*(1+I*c*x)^4/c-1/5*I*d^4*(1+I*c*x)^5*(a+b*arctan(c*x))/c-16/5*b*d^4*log(1-I*c*x)/c],
[(d+I*c*d*x)^4*(a+b*arctan(c*x))/x,x,19,4*I*a*c*d^4*x+13/4*b*c*d^4*x+2/3*I*b*c^2*d^4*x^2-1/12*b*c^3*d^4*x^3-13/4*b*d^4*arctan(c*x)+4*I*b*c*d^4*x*arctan(c*x)-3*c^2*d^4*x^2*(a+b*arctan(c*x))-4/3*I*c^3*d^4*x^3*(a+b*arctan(c*x))+1/4*c^4*d^4*x^4*(a+b*arctan(c*x))+a*d^4*log(x)-8/3*I*b*d^4*log(1+c^2*x^2)+1/2*I*b*d^4*polylog(2,-I*c*x)-1/2*I*b*d^4*polylog(2,I*c*x)],
[(d+I*c*d*x)^4*(a+b*arctan(c*x))/x^2,x,20,-6*a*c^2*d^4*x+2*I*b*c^2*d^4*x-1/6*b*c^3*d^4*x^2-2*I*b*c*d^4*arctan(c*x)-6*b*c^2*d^4*x*arctan(c*x)-d^4*(a+b*arctan(c*x))/x-2*I*c^3*d^4*x^2*(a+b*arctan(c*x))+1/3*c^4*d^4*x^3*(a+b*arctan(c*x))+4*I*a*c*d^4*log(x)+b*c*d^4*log(x)+8/3*b*c*d^4*log(1+c^2*x^2)-2*b*c*d^4*polylog(2,-I*c*x)+2*b*c*d^4*polylog(2,I*c*x)],
[(d+I*c*d*x)^4*(a+b*arctan(c*x))/x^3,x,19,-1/2*b*c*d^4/x-4*I*a*c^3*d^4*x-1/2*b*c^3*d^4*x-4*I*b*c^3*d^4*x*arctan(c*x)-1/2*d^4*(a+b*arctan(c*x))/x^2-4*I*c*d^4*(a+b*arctan(c*x))/x+1/2*c^4*d^4*x^2*(a+b*arctan(c*x))-6*a*c^2*d^4*log(x)+4*I*b*c^2*d^4*log(x)-3*I*b*c^2*d^4*polylog(2,-I*c*x)+3*I*b*c^2*d^4*polylog(2,I*c*x)],
[(d+I*c*d*x)^4*(a+b*arctan(c*x))/x^4,x,20,-1/6*b*c*d^4/x^2-2*I*b*c^2*d^4/x+a*c^4*d^4*x-2*I*b*c^3*d^4*arctan(c*x)+b*c^4*d^4*x*arctan(c*x)-1/3*d^4*(a+b*arctan(c*x))/x^3-2*I*c*d^4*(a+b*arctan(c*x))/x^2+6*c^2*d^4*(a+b*arctan(c*x))/x-4*I*a*c^3*d^4*log(x)-19/3*b*c^3*d^4*log(x)+8/3*b*c^3*d^4*log(1+c^2*x^2)+2*b*c^3*d^4*polylog(2,-I*c*x)-2*b*c^3*d^4*polylog(2,I*c*x)],
[(d+I*c*d*x)^4*(a+b*arctan(c*x))/x^5,x,21,-1/12*b*c*d^4/x^3-2/3*I*b*c^2*d^4/x^2+13/4*b*c^3*d^4/x+13/4*b*c^4*d^4*arctan(c*x)-1/4*d^4*(a+b*arctan(c*x))/x^4-4/3*I*c*d^4*(a+b*arctan(c*x))/x^3+3*c^2*d^4*(a+b*arctan(c*x))/x^2+4*I*c^3*d^4*(a+b*arctan(c*x))/x+a*c^4*d^4*log(x)-16/3*I*b*c^4*d^4*log(x)+8/3*I*b*c^4*d^4*log(1+c^2*x^2)+1/2*I*b*c^4*d^4*polylog(2,-I*c*x)-1/2*I*b*c^4*d^4*polylog(2,I*c*x)],
[(d+I*c*d*x)^4*(a+b*arctan(c*x))/x^6,x,4,-1/20*b*c*d^4/x^4-1/3*I*b*c^2*d^4/x^3+11/10*b*c^3*d^4/x^2+3*I*b*c^4*d^4/x-1/5*d^4*(1+I*c*x)^5*(a+b*arctan(c*x))/x^5+16/5*b*c^5*d^4*log(x)-16/5*b*c^5*d^4*log(I+c*x)],
[(d+I*c*d*x)^4*(a+b*arctan(c*x))/x^7,x,4,-1/30*b*c*d^4/x^5-1/5*I*b*c^2*d^4/x^4+5/9*b*c^3*d^4/x^3+16/15*I*b*c^4*d^4/x^2-13/6*b*c^5*d^4/x-1/6*d^4*(1+I*c*x)^5*(a+b*arctan(c*x))/x^6+1/30*I*c*d^4*(1+I*c*x)^5*(a+b*arctan(c*x))/x^5+32/15*I*b*c^6*d^4*log(x)-32/15*I*b*c^6*d^4*log(I+c*x)],
[(d+I*c*d*x)^4*(a+b*arctan(c*x))/x^8,x,4,-1/42*b*c*d^4/x^6-2/15*I*b*c^2*d^4/x^5+47/140*b*c^3*d^4/x^4+5/9*I*b*c^4*d^4/x^3-88/105*b*c^5*d^4/x^2-5/3*I*b*c^6*d^4/x-1/7*d^4*(a+b*arctan(c*x))/x^7-2/3*I*c*d^4*(a+b*arctan(c*x))/x^6+6/5*c^2*d^4*(a+b*arctan(c*x))/x^5+I*c^3*d^4*(a+b*arctan(c*x))/x^4-1/3*c^4*d^4*(a+b*arctan(c*x))/x^3-176/105*b*c^7*d^4*log(x)+1/210*b*c^7*d^4*log(I-c*x)+117/70*b*c^7*d^4*log(I+c*x)],

# q<0
[x^3*(a+b*arctan(c*x))/(d+I*c*d*x),x,16,I*a*x/(c^3*d)-1/2*b*x/(c^3*d)+1/6*I*b*x^2/(c^2*d)+1/2*b*arctan(c*x)/(c^4*d)+I*b*x*arctan(c*x)/(c^3*d)+1/2*x^2*(a+b*arctan(c*x))/(c^2*d)-1/3*I*x^3*(a+b*arctan(c*x))/(c*d)+(a+b*arctan(c*x))*log(2/(1+I*c*x))/(c^4*d)-2/3*I*b*log(1+c^2*x^2)/(c^4*d)+1/2*I*b*polylog(2,1+(-2)/(1+I*c*x))/(c^4*d)],
[x^2*(a+b*arctan(c*x))/(d+I*c*d*x),x,11,a*x/(c^2*d)+1/2*I*b*x/(c^2*d)-1/2*I*b*arctan(c*x)/(c^3*d)+b*x*arctan(c*x)/(c^2*d)-1/2*I*x^2*(a+b*arctan(c*x))/(c*d)-I*(a+b*arctan(c*x))*log(2/(1+I*c*x))/(c^3*d)-1/2*b*log(1+c^2*x^2)/(c^3*d)+1/2*b*polylog(2,1+(-2)/(1+I*c*x))/(c^3*d)],
[x*(a+b*arctan(c*x))/(d+I*c*d*x),x,7,-I*a*x/(c*d)-I*b*x*arctan(c*x)/(c*d)-(a+b*arctan(c*x))*log(2/(1+I*c*x))/(c^2*d)+1/2*I*b*log(1+c^2*x^2)/(c^2*d)-1/2*I*b*polylog(2,1+(-2)/(1+I*c*x))/(c^2*d)],
[(a+b*arctan(c*x))/(d+I*c*d*x),x,3,I*(a+b*arctan(c*x))*log(2/(1+I*c*x))/(c*d)-1/2*b*polylog(2,1+(-2)/(1+I*c*x))/(c*d)],
[(a+b*arctan(c*x))/(x*(d+I*c*d*x)),x,2,(a+b*arctan(c*x))*log(2+(-2)/(1+I*c*x))/d+1/2*I*b*polylog(2,-1+2/(1+I*c*x))/d],
[(a+b*arctan(c*x))/(x^2*(d+I*c*d*x)),x,8,(-a-b*arctan(c*x))/(d*x)+b*c*log(x)/d-1/2*b*c*log(1+c^2*x^2)/d-I*c*(a+b*arctan(c*x))*log(2+(-2)/(1+I*c*x))/d+1/2*b*c*polylog(2,-1+2/(1+I*c*x))/d],
[(a+b*arctan(c*x))/(x^3*(d+I*c*d*x)),x,12,-1/2*b*c/(d*x)-1/2*b*c^2*arctan(c*x)/d+1/2*(-a-b*arctan(c*x))/(d*x^2)+I*c*(a+b*arctan(c*x))/(d*x)-I*b*c^2*log(x)/d+1/2*I*b*c^2*log(1+c^2*x^2)/d-c^2*(a+b*arctan(c*x))*log(2+(-2)/(1+I*c*x))/d-1/2*I*b*c^2*polylog(2,-1+2/(1+I*c*x))/d],
[(a+b*arctan(c*x))/(x^4*(d+I*c*d*x)),x,17,-1/6*b*c/(d*x^2)+1/2*I*b*c^2/(d*x)+1/2*I*b*c^3*arctan(c*x)/d+1/3*(-a-b*arctan(c*x))/(d*x^3)+1/2*I*c*(a+b*arctan(c*x))/(d*x^2)+c^2*(a+b*arctan(c*x))/(d*x)-4/3*b*c^3*log(x)/d+2/3*b*c^3*log(1+c^2*x^2)/d+I*c^3*(a+b*arctan(c*x))*log(2+(-2)/(1+I*c*x))/d-1/2*b*c^3*polylog(2,-1+2/(1+I*c*x))/d],
[x^3*(a+b*arctan(c*x))/(d+I*c*d*x)^2,x,16,-2*I*a*x/(c^3*d^2)+1/2*b*x/(c^3*d^2)+1/2*b/(c^4*d^2*(I-c*x))-b*arctan(c*x)/(c^4*d^2)-2*I*b*x*arctan(c*x)/(c^3*d^2)-1/2*x^2*(a+b*arctan(c*x))/(c^2*d^2)+I*(a+b*arctan(c*x))/(c^4*d^2*(I-c*x))-3*(a+b*arctan(c*x))*log(2/(1+I*c*x))/(c^4*d^2)+I*b*log(1+c^2*x^2)/(c^4*d^2)-3/2*I*b*polylog(2,1+(-2)/(1+I*c*x))/(c^4*d^2)],
[x^2*(a+b*arctan(c*x))/(d+I*c*d*x)^2,x,13,-a*x/(c^2*d^2)-1/2*I*b/(c^3*d^2*(I-c*x))+1/2*I*b*arctan(c*x)/(c^3*d^2)-b*x*arctan(c*x)/(c^2*d^2)+(a+b*arctan(c*x))/(c^3*d^2*(I-c*x))+2*I*(a+b*arctan(c*x))*log(2/(1+I*c*x))/(c^3*d^2)+1/2*b*log(1+c^2*x^2)/(c^3*d^2)-b*polylog(2,1+(-2)/(1+I*c*x))/(c^3*d^2)],
[x*(a+b*arctan(c*x))/(d+I*c*d*x)^2,x,10,-1/2*b/(c^2*d^2*(I-c*x))+1/2*b*arctan(c*x)/(c^2*d^2)-I*(a+b*arctan(c*x))/(c^2*d^2*(I-c*x))+(a+b*arctan(c*x))*log(2/(1+I*c*x))/(c^2*d^2)+1/2*I*b*polylog(2,1+(-2)/(1+I*c*x))/(c^2*d^2)],
[(a+b*arctan(c*x))/(d+I*c*d*x)^2,x,5,1/2*I*b/(c*d^2*(I-c*x))-1/2*I*b*arctan(c*x)/(c*d^2)+I*(a+b*arctan(c*x))/(c*d^2*(1+I*c*x))],
[(a+b*arctan(c*x))/(x*(d+I*c*d*x)^2),x,13,1/2*b/(d^2*(I-c*x))-1/2*b*arctan(c*x)/d^2+I*(a+b*arctan(c*x))/(d^2*(I-c*x))+a*log(x)/d^2+(a+b*arctan(c*x))*log(2/(1+I*c*x))/d^2+1/2*I*b*polylog(2,-I*c*x)/d^2-1/2*I*b*polylog(2,I*c*x)/d^2+1/2*I*b*polylog(2,1+(-2)/(1+I*c*x))/d^2],
[(a+b*arctan(c*x))/(x^2*(d+I*c*d*x)^2),x,18,-1/2*I*b*c/(d^2*(I-c*x))+1/2*I*b*c*arctan(c*x)/d^2+(-a-b*arctan(c*x))/(d^2*x)+c*(a+b*arctan(c*x))/(d^2*(I-c*x))-2*I*a*c*log(x)/d^2+b*c*log(x)/d^2-2*I*c*(a+b*arctan(c*x))*log(2/(1+I*c*x))/d^2-1/2*b*c*log(1+c^2*x^2)/d^2+b*c*polylog(2,-I*c*x)/d^2-b*c*polylog(2,I*c*x)/d^2+b*c*polylog(2,1+(-2)/(1+I*c*x))/d^2],
[(a+b*arctan(c*x))/(x^3*(d+I*c*d*x)^2),x,21,-1/2*b*c/(d^2*x)-1/2*b*c^2/(d^2*(I-c*x))+1/2*(-a-b*arctan(c*x))/(d^2*x^2)+2*I*c*(a+b*arctan(c*x))/(d^2*x)-I*c^2*(a+b*arctan(c*x))/(d^2*(I-c*x))-3*a*c^2*log(x)/d^2-2*I*b*c^2*log(x)/d^2-3*c^2*(a+b*arctan(c*x))*log(2/(1+I*c*x))/d^2+I*b*c^2*log(1+c^2*x^2)/d^2-3/2*I*b*c^2*polylog(2,-I*c*x)/d^2+3/2*I*b*c^2*polylog(2,I*c*x)/d^2-3/2*I*b*c^2*polylog(2,1+(-2)/(1+I*c*x))/d^2],
[x^4*(a+b*arctan(c*x))/(d+I*c*d*x)^3,x,21,-3*a*x/(c^4*d^3)-1/2*I*b*x/(c^4*d^3)-1/8*b/(c^5*d^3*(I-c*x)^2)-15/8*I*b/(c^5*d^3*(I-c*x))+19/8*I*b*arctan(c*x)/(c^5*d^3)-3*b*x*arctan(c*x)/(c^4*d^3)+1/2*I*x^2*(a+b*arctan(c*x))/(c^3*d^3)-1/2*I*(a+b*arctan(c*x))/(c^5*d^3*(I-c*x)^2)+4*(a+b*arctan(c*x))/(c^5*d^3*(I-c*x))+6*I*(a+b*arctan(c*x))*log(2/(1+I*c*x))/(c^5*d^3)+3/2*b*log(1+c^2*x^2)/(c^5*d^3)-3*b*polylog(2,1+(-2)/(1+I*c*x))/(c^5*d^3)],
[x^3*(a+b*arctan(c*x))/(d+I*c*d*x)^3,x,18,I*a*x/(c^3*d^3)+1/8*I*b/(c^4*d^3*(I-c*x)^2)-11/8*b/(c^4*d^3*(I-c*x))+11/8*b*arctan(c*x)/(c^4*d^3)+I*b*x*arctan(c*x)/(c^3*d^3)+1/2*(-a-b*arctan(c*x))/(c^4*d^3*(I-c*x)^2)-3*I*(a+b*arctan(c*x))/(c^4*d^3*(I-c*x))+3*(a+b*arctan(c*x))*log(2/(1+I*c*x))/(c^4*d^3)-1/2*I*b*log(1+c^2*x^2)/(c^4*d^3)+3/2*I*b*polylog(2,1+(-2)/(1+I*c*x))/(c^4*d^3)],
[x^2*(a+b*arctan(c*x))/(d+I*c*d*x)^3,x,15,1/8*b/(c^3*d^3*(I-c*x)^2)+7/8*I*b/(c^3*d^3*(I-c*x))-7/8*I*b*arctan(c*x)/(c^3*d^3)+1/2*I*(a+b*arctan(c*x))/(c^3*d^3*(I-c*x)^2)-2*(a+b*arctan(c*x))/(c^3*d^3*(I-c*x))-I*(a+b*arctan(c*x))*log(2/(1+I*c*x))/(c^3*d^3)+1/2*b*polylog(2,1+(-2)/(1+I*c*x))/(c^3*d^3)],
[x*(a+b*arctan(c*x))/(d+I*c*d*x)^3,x,5,-1/8*I*b/(c^2*d^3*(I-c*x)^2)+3/8*b/(c^2*d^3*(I-c*x))+1/8*b*arctan(c*x)/(c^2*d^3)+1/2*x^2*(a+b*arctan(c*x))/(d^3*(1+I*c*x)^2)],
[(a+b*arctan(c*x))/(d+I*c*d*x)^3,x,5,-1/8*b/(c*d^3*(I-c*x)^2)+1/8*I*b/(c*d^3*(I-c*x))-1/8*I*b*arctan(c*x)/(c*d^3)+1/2*I*(a+b*arctan(c*x))/(c*d^3*(1+I*c*x)^2)],
[(a+b*arctan(c*x))/(x*(d+I*c*d*x)^3),x,18,1/8*I*b/(d^3*(I-c*x)^2)+5/8*b/(d^3*(I-c*x))-5/8*b*arctan(c*x)/d^3+1/2*(-a-b*arctan(c*x))/(d^3*(I-c*x)^2)+I*(a+b*arctan(c*x))/(d^3*(I-c*x))+a*log(x)/d^3+(a+b*arctan(c*x))*log(2/(1+I*c*x))/d^3+1/2*I*b*polylog(2,-I*c*x)/d^3-1/2*I*b*polylog(2,I*c*x)/d^3+1/2*I*b*polylog(2,1+(-2)/(1+I*c*x))/d^3],
[(a+b*arctan(c*x))/(x^2*(d+I*c*d*x)^3),x,23,1/8*b*c/(d^3*(I-c*x)^2)-9/8*I*b*c/(d^3*(I-c*x))+9/8*I*b*c*arctan(c*x)/d^3+(-a-b*arctan(c*x))/(d^3*x)+1/2*I*c*(a+b*arctan(c*x))/(d^3*(I-c*x)^2)+2*c*(a+b*arctan(c*x))/(d^3*(I-c*x))-3*I*a*c*log(x)/d^3+b*c*log(x)/d^3-3*I*c*(a+b*arctan(c*x))*log(2/(1+I*c*x))/d^3-1/2*b*c*log(1+c^2*x^2)/d^3+3/2*b*c*polylog(2,-I*c*x)/d^3-3/2*b*c*polylog(2,I*c*x)/d^3+3/2*b*c*polylog(2,1+(-2)/(1+I*c*x))/d^3],
[(a+b*arctan(c*x))/(x^3*(d+I*c*d*x)^3),x,26,-1/2*b*c/(d^3*x)-1/8*I*b*c^2/(d^3*(I-c*x)^2)-13/8*b*c^2/(d^3*(I-c*x))+9/8*b*c^2*arctan(c*x)/d^3+1/2*(-a-b*arctan(c*x))/(d^3*x^2)+3*I*c*(a+b*arctan(c*x))/(d^3*x)+1/2*c^2*(a+b*arctan(c*x))/(d^3*(I-c*x)^2)-3*I*c^2*(a+b*arctan(c*x))/(d^3*(I-c*x))-6*a*c^2*log(x)/d^3-3*I*b*c^2*log(x)/d^3-6*c^2*(a+b*arctan(c*x))*log(2/(1+I*c*x))/d^3+3/2*I*b*c^2*log(1+c^2*x^2)/d^3-3*I*b*c^2*polylog(2,-I*c*x)/d^3+3*I*b*c^2*polylog(2,I*c*x)/d^3-3*I*b*c^2*polylog(2,1+(-2)/(1+I*c*x))/d^3],
[(a+b*arctan(c*x))/(1+I*c*x)^4,x,5,-1/18*I*b/(c*(I-c*x)^3)-1/24*b/(c*(I-c*x)^2)+1/24*I*b/(c*(I-c*x))-1/24*I*b*arctan(c*x)/c+1/3*I*(a+b*arctan(c*x))/(c*(1+I*c*x)^3)],
[arctan(a*x)/(c*x+I*a*c*x^2),x,3,arctan(a*x)*log(2+(-2)/(1+I*a*x))/c+1/2*I*polylog(2,-1+2/(1+I*a*x))/c],

# Integrands of the form x^m (d+e x)^q (a+b ArcTan[c x])^2 with c^2 d^2+e^2=0

# q>0
[x^3*(d+I*c*d*x)*(a+b*arctan(c*x))^2,x,27,1/2*a*b*d*x/c^3-3/10*I*b^2*d*x/c^3+1/12*b^2*d*x^2/c^2+1/30*I*b^2*d*x^3/c+3/10*I*b^2*d*arctan(c*x)/c^4+1/2*b^2*d*x*arctan(c*x)/c^3+1/5*I*b*d*x^2*(a+b*arctan(c*x))/c^2-1/6*b*d*x^3*(a+b*arctan(c*x))/c-1/10*I*b*d*x^4*(a+b*arctan(c*x))-9/20*d*(a+b*arctan(c*x))^2/c^4+1/4*d*x^4*(a+b*arctan(c*x))^2+1/5*I*c*d*x^5*(a+b*arctan(c*x))^2+2/5*I*b*d*(a+b*arctan(c*x))*log(2/(1+I*c*x))/c^4-1/3*b^2*d*log(1+c^2*x^2)/c^4-1/5*b^2*d*polylog(2,1+(-2)/(1+I*c*x))/c^4],
[x^2*(d+I*c*d*x)*(a+b*arctan(c*x))^2,x,22,1/2*I*a*b*d*x/c^2+1/3*b^2*d*x/c^2+1/12*I*b^2*d*x^2/c-1/3*b^2*d*arctan(c*x)/c^3+1/2*I*b^2*d*x*arctan(c*x)/c^2-1/3*b*d*x^2*(a+b*arctan(c*x))/c-1/6*I*b*d*x^3*(a+b*arctan(c*x))-7/12*I*d*(a+b*arctan(c*x))^2/c^3+1/3*d*x^3*(a+b*arctan(c*x))^2+1/4*I*c*d*x^4*(a+b*arctan(c*x))^2-2/3*b*d*(a+b*arctan(c*x))*log(2/(1+I*c*x))/c^3-1/3*I*b^2*d*log(1+c^2*x^2)/c^3-1/3*I*b^2*d*polylog(2,1+(-2)/(1+I*c*x))/c^3],
[x*(d+I*c*d*x)*(a+b*arctan(c*x))^2,x,17,-a*b*d*x/c+1/3*I*b^2*d*x/c-1/3*I*b^2*d*arctan(c*x)/c^2-b^2*d*x*arctan(c*x)/c-1/3*I*b*d*x^2*(a+b*arctan(c*x))+5/6*d*(a+b*arctan(c*x))^2/c^2+1/2*d*x^2*(a+b*arctan(c*x))^2+1/3*I*c*d*x^3*(a+b*arctan(c*x))^2-2/3*I*b*d*(a+b*arctan(c*x))*log(2/(1+I*c*x))/c^2+1/2*b^2*d*log(1+c^2*x^2)/c^2+1/3*b^2*d*polylog(2,1+(-2)/(1+I*c*x))/c^2],
[(d+I*c*d*x)*(a+b*arctan(c*x))^2,x,9,-I*a*b*d*x-I*b^2*d*x*arctan(c*x)-1/2*I*d*(1+I*c*x)^2*(a+b*arctan(c*x))^2/c+2*b*d*(a+b*arctan(c*x))*log(2/(1-I*c*x))/c+1/2*I*b^2*d*log(1+c^2*x^2)/c-I*b^2*d*polylog(2,1+(-2)/(1-I*c*x))/c],
[(d+I*c*d*x)*(a+b*arctan(c*x))^2/x,x,13,-d*(a+b*arctan(c*x))^2+I*c*d*x*(a+b*arctan(c*x))^2+2*d*(a+b*arctan(c*x))^2*arctanh(1+(-2)/(1+I*c*x))+2*I*b*d*(a+b*arctan(c*x))*log(2/(1+I*c*x))-b^2*d*polylog(2,1+(-2)/(1+I*c*x))-I*b*d*(a+b*arctan(c*x))*polylog(2,1+(-2)/(1+I*c*x))+I*b*d*(a+b*arctan(c*x))*polylog(2,-1+2/(1+I*c*x))-1/2*b^2*d*polylog(3,1+(-2)/(1+I*c*x))+1/2*b^2*d*polylog(3,-1+2/(1+I*c*x))],
[(d+I*c*d*x)*(a+b*arctan(c*x))^2/x^2,x,12,-I*c*d*(a+b*arctan(c*x))^2-d*(a+b*arctan(c*x))^2/x+2*I*c*d*(a+b*arctan(c*x))^2*arctanh(1+(-2)/(1+I*c*x))+2*b*c*d*(a+b*arctan(c*x))*log(2+(-2)/(1-I*c*x))-I*b^2*c*d*polylog(2,-1+2/(1-I*c*x))+b*c*d*(a+b*arctan(c*x))*polylog(2,1+(-2)/(1+I*c*x))-b*c*d*(a+b*arctan(c*x))*polylog(2,-1+2/(1+I*c*x))-1/2*I*b^2*c*d*polylog(3,1+(-2)/(1+I*c*x))+1/2*I*b^2*c*d*polylog(3,-1+2/(1+I*c*x))],
[(d+I*c*d*x)*(a+b*arctan(c*x))^2/x^3,x,14,-b*c*d*(a+b*arctan(c*x))/x+1/2*c^2*d*(a+b*arctan(c*x))^2-1/2*d*(a+b*arctan(c*x))^2/x^2-I*c*d*(a+b*arctan(c*x))^2/x+b^2*c^2*d*log(x)-1/2*b^2*c^2*d*log(1+c^2*x^2)+2*I*b*c^2*d*(a+b*arctan(c*x))*log(2+(-2)/(1-I*c*x))+b^2*c^2*d*polylog(2,-1+2/(1-I*c*x))],
[(d+I*c*d*x)*(a+b*arctan(c*x))^2/x^4,x,18,-1/3*b^2*c^2*d/x-1/3*b^2*c^3*d*arctan(c*x)-1/3*b*c*d*(a+b*arctan(c*x))/x^2-I*b*c^2*d*(a+b*arctan(c*x))/x-1/6*I*c^3*d*(a+b*arctan(c*x))^2-1/3*d*(a+b*arctan(c*x))^2/x^3-1/2*I*c*d*(a+b*arctan(c*x))^2/x^2+I*b^2*c^3*d*log(x)-1/2*I*b^2*c^3*d*log(1+c^2*x^2)-2/3*b*c^3*d*(a+b*arctan(c*x))*log(2+(-2)/(1-I*c*x))+1/3*I*b^2*c^3*d*polylog(2,-1+2/(1-I*c*x))],
[x^3*(d+I*c*d*x)^2*(a+b*arctan(c*x))^2,x,43,5/6*a*b*d^2*x/c^3-3/5*I*b^2*d^2*x/c^3+31/180*b^2*d^2*x^2/c^2+1/15*I*b^2*d^2*x^3/c-1/60*b^2*d^2*x^4+3/5*I*b^2*d^2*arctan(c*x)/c^4+5/6*b^2*d^2*x*arctan(c*x)/c^3+2/5*I*b*d^2*x^2*(a+b*arctan(c*x))/c^2-5/18*b*d^2*x^3*(a+b*arctan(c*x))/c-1/5*I*b*d^2*x^4*(a+b*arctan(c*x))+1/15*b*c*d^2*x^5*(a+b*arctan(c*x))-49/60*d^2*(a+b*arctan(c*x))^2/c^4+1/4*d^2*x^4*(a+b*arctan(c*x))^2+2/5*I*c*d^2*x^5*(a+b*arctan(c*x))^2-1/6*c^2*d^2*x^6*(a+b*arctan(c*x))^2+4/5*I*b*d^2*(a+b*arctan(c*x))*log(2/(1+I*c*x))/c^4-53/90*b^2*d^2*log(1+c^2*x^2)/c^4-2/5*b^2*d^2*polylog(2,1+(-2)/(1+I*c*x))/c^4],
[x^2*(d+I*c*d*x)^2*(a+b*arctan(c*x))^2,x,36,I*a*b*d^2*x/c^2+19/30*b^2*d^2*x/c^2+1/6*I*b^2*d^2*x^2/c-1/30*b^2*d^2*x^3-19/30*b^2*d^2*arctan(c*x)/c^3+I*b^2*d^2*x*arctan(c*x)/c^2-8/15*b*d^2*x^2*(a+b*arctan(c*x))/c-1/3*I*b*d^2*x^3*(a+b*arctan(c*x))+1/10*b*c*d^2*x^4*(a+b*arctan(c*x))-31/30*I*d^2*(a+b*arctan(c*x))^2/c^3+1/3*d^2*x^3*(a+b*arctan(c*x))^2+1/2*I*c*d^2*x^4*(a+b*arctan(c*x))^2-1/5*c^2*d^2*x^5*(a+b*arctan(c*x))^2-16/15*b*d^2*(a+b*arctan(c*x))*log(2/(1+I*c*x))/c^3-2/3*I*b^2*d^2*log(1+c^2*x^2)/c^3-8/15*I*b^2*d^2*polylog(2,1+(-2)/(1+I*c*x))/c^3],
[x*(d+I*c*d*x)^2*(a+b*arctan(c*x))^2,x,28,-3/2*a*b*d^2*x/c+2/3*I*b^2*d^2*x/c-1/12*b^2*d^2*x^2-2/3*I*b^2*d^2*arctan(c*x)/c^2-3/2*b^2*d^2*x*arctan(c*x)/c-2/3*I*b*d^2*x^2*(a+b*arctan(c*x))+1/6*b*c*d^2*x^3*(a+b*arctan(c*x))+17/12*d^2*(a+b*arctan(c*x))^2/c^2+1/2*d^2*x^2*(a+b*arctan(c*x))^2+2/3*I*c*d^2*x^3*(a+b*arctan(c*x))^2-1/4*c^2*d^2*x^4*(a+b*arctan(c*x))^2-4/3*I*b*d^2*(a+b*arctan(c*x))*log(2/(1+I*c*x))/c^2+5/6*b^2*d^2*log(1+c^2*x^2)/c^2+2/3*b^2*d^2*polylog(2,1+(-2)/(1+I*c*x))/c^2],
[(d+I*c*d*x)^2*(a+b*arctan(c*x))^2,x,12,-2*I*a*b*d^2*x-1/3*b^2*d^2*x+1/3*b^2*d^2*arctan(c*x)/c-2*I*b^2*d^2*x*arctan(c*x)+1/3*b*c*d^2*x^2*(a+b*arctan(c*x))-1/3*I*d^2*(1+I*c*x)^3*(a+b*arctan(c*x))^2/c+8/3*b*d^2*(a+b*arctan(c*x))*log(2/(1-I*c*x))/c+I*b^2*d^2*log(1+c^2*x^2)/c-4/3*I*b^2*d^2*polylog(2,1+(-2)/(1-I*c*x))/c],
[(d+I*c*d*x)^2*(a+b*arctan(c*x))^2/x,x,19,a*b*c*d^2*x+b^2*c*d^2*x*arctan(c*x)-5/2*d^2*(a+b*arctan(c*x))^2+2*I*c*d^2*x*(a+b*arctan(c*x))^2-1/2*c^2*d^2*x^2*(a+b*arctan(c*x))^2+2*d^2*(a+b*arctan(c*x))^2*arctanh(1+(-2)/(1+I*c*x))+4*I*b*d^2*(a+b*arctan(c*x))*log(2/(1+I*c*x))-1/2*b^2*d^2*log(1+c^2*x^2)-2*b^2*d^2*polylog(2,1+(-2)/(1+I*c*x))-I*b*d^2*(a+b*arctan(c*x))*polylog(2,1+(-2)/(1+I*c*x))+I*b*d^2*(a+b*arctan(c*x))*polylog(2,-1+2/(1+I*c*x))-1/2*b^2*d^2*polylog(3,1+(-2)/(1+I*c*x))+1/2*b^2*d^2*polylog(3,-1+2/(1+I*c*x))],
[(d+I*c*d*x)^2*(a+b*arctan(c*x))^2/x^2,x,17,-2*I*c*d^2*(a+b*arctan(c*x))^2-d^2*(a+b*arctan(c*x))^2/x-c^2*d^2*x*(a+b*arctan(c*x))^2+4*I*c*d^2*(a+b*arctan(c*x))^2*arctanh(1+(-2)/(1+I*c*x))-2*b*c*d^2*(a+b*arctan(c*x))*log(2/(1+I*c*x))+2*b*c*d^2*(a+b*arctan(c*x))*log(2+(-2)/(1-I*c*x))-I*b^2*c*d^2*polylog(2,-1+2/(1-I*c*x))-I*b^2*c*d^2*polylog(2,1+(-2)/(1+I*c*x))+2*b*c*d^2*(a+b*arctan(c*x))*polylog(2,1+(-2)/(1+I*c*x))-2*b*c*d^2*(a+b*arctan(c*x))*polylog(2,-1+2/(1+I*c*x))-I*b^2*c*d^2*polylog(3,1+(-2)/(1+I*c*x))+I*b^2*c*d^2*polylog(3,-1+2/(1+I*c*x))],
[(d+I*c*d*x)^2*(a+b*arctan(c*x))^2/x^3,x,20,-b*c*d^2*(a+b*arctan(c*x))/x+3/2*c^2*d^2*(a+b*arctan(c*x))^2-1/2*d^2*(a+b*arctan(c*x))^2/x^2-2*I*c*d^2*(a+b*arctan(c*x))^2/x-2*c^2*d^2*(a+b*arctan(c*x))^2*arctanh(1+(-2)/(1+I*c*x))+b^2*c^2*d^2*log(x)-1/2*b^2*c^2*d^2*log(1+c^2*x^2)+4*I*b*c^2*d^2*(a+b*arctan(c*x))*log(2+(-2)/(1-I*c*x))+2*b^2*c^2*d^2*polylog(2,-1+2/(1-I*c*x))+I*b*c^2*d^2*(a+b*arctan(c*x))*polylog(2,1+(-2)/(1+I*c*x))-I*b*c^2*d^2*(a+b*arctan(c*x))*polylog(2,-1+2/(1+I*c*x))+1/2*b^2*c^2*d^2*polylog(3,1+(-2)/(1+I*c*x))-1/2*b^2*c^2*d^2*polylog(3,-1+2/(1+I*c*x))],
[(d+I*c*d*x)^2*(a+b*arctan(c*x))^2/x^4,x,16,-1/3*b^2*c^2*d^2/x-1/3*b^2*c^3*d^2*arctan(c*x)-1/3*b*c*d^2*(a+b*arctan(c*x))/x^2-2*I*b*c^2*d^2*(a+b*arctan(c*x))/x-1/3*d^2*(1+I*c*x)^3*(a+b*arctan(c*x))^2/x^3-8/3*a*b*c^3*d^2*log(x)+2*I*b^2*c^3*d^2*log(x)-8/3*b*c^3*d^2*(a+b*arctan(c*x))*log(2/(1-I*c*x))-I*b^2*c^3*d^2*log(1+c^2*x^2)-4/3*I*b^2*c^3*d^2*polylog(2,-I*c*x)+4/3*I*b^2*c^3*d^2*polylog(2,I*c*x)+4/3*I*b^2*c^3*d^2*polylog(2,1+(-2)/(1-I*c*x))],
[x^3*(d+I*c*d*x)^3*(a+b*arctan(c*x))^2,x,62,3/2*a*b*d^3*x/c^3-122/105*I*b^2*d^3*x/c^3+7/20*b^2*d^3*x^2/c^2+44/315*I*b^2*d^3*x^3/c-1/20*b^2*d^3*x^4-1/105*I*b^2*c*d^3*x^5+122/105*I*b^2*d^3*arctan(c*x)/c^4+3/2*b^2*d^3*x*arctan(c*x)/c^3+26/35*I*b*d^3*x^2*(a+b*arctan(c*x))/c^2-1/2*b*d^3*x^3*(a+b*arctan(c*x))/c-13/35*I*b*d^3*x^4*(a+b*arctan(c*x))+1/5*b*c*d^3*x^5*(a+b*arctan(c*x))+1/21*I*b*c^2*d^3*x^6*(a+b*arctan(c*x))-209/140*d^3*(a+b*arctan(c*x))^2/c^4+1/4*d^3*x^4*(a+b*arctan(c*x))^2+3/5*I*c*d^3*x^5*(a+b*arctan(c*x))^2-1/2*c^2*d^3*x^6*(a+b*arctan(c*x))^2-1/7*I*c^3*d^3*x^7*(a+b*arctan(c*x))^2+52/35*I*b*d^3*(a+b*arctan(c*x))*log(2/(1+I*c*x))/c^4-11/10*b^2*d^3*log(1+c^2*x^2)/c^4-26/35*b^2*d^3*polylog(2,1+(-2)/(1+I*c*x))/c^4],
[x^2*(d+I*c*d*x)^3*(a+b*arctan(c*x))^2,x,52,11/6*I*a*b*d^3*x/c^2+37/30*b^2*d^3*x/c^2+61/180*I*b^2*d^3*x^2/c-1/10*b^2*d^3*x^3-1/60*I*b^2*c*d^3*x^4-37/30*b^2*d^3*arctan(c*x)/c^3+11/6*I*b^2*d^3*x*arctan(c*x)/c^2-14/15*b*d^3*x^2*(a+b*arctan(c*x))/c-11/18*I*b*d^3*x^3*(a+b*arctan(c*x))+3/10*b*c*d^3*x^4*(a+b*arctan(c*x))+1/15*I*b*c^2*d^3*x^5*(a+b*arctan(c*x))-37/20*I*d^3*(a+b*arctan(c*x))^2/c^3+1/3*d^3*x^3*(a+b*arctan(c*x))^2+3/4*I*c*d^3*x^4*(a+b*arctan(c*x))^2-3/5*c^2*d^3*x^5*(a+b*arctan(c*x))^2-1/6*I*c^3*d^3*x^6*(a+b*arctan(c*x))^2-28/15*b*d^3*(a+b*arctan(c*x))*log(2/(1+I*c*x))/c^3-113/90*I*b^2*d^3*log(1+c^2*x^2)/c^3-14/15*I*b^2*d^3*polylog(2,1+(-2)/(1+I*c*x))/c^3],
[x*(d+I*c*d*x)^3*(a+b*arctan(c*x))^2,x,38,-5/2*a*b*d^3*x/c+13/10*I*b^2*d^3*x/c-1/4*b^2*d^3*x^2-1/30*I*b^2*c*d^3*x^3-13/10*I*b^2*d^3*arctan(c*x)/c^2-5/2*b^2*d^3*x*arctan(c*x)/c-6/5*I*b*d^3*x^2*(a+b*arctan(c*x))+1/2*b*c*d^3*x^3*(a+b*arctan(c*x))+1/10*I*b*c^2*d^3*x^4*(a+b*arctan(c*x))+1/4*d^3*(1+I*c*x)^4*(a+b*arctan(c*x))^2/c^2-1/5*d^3*(1+I*c*x)^5*(a+b*arctan(c*x))^2/c^2-12/5*I*b*d^3*(a+b*arctan(c*x))*log(2/(1-I*c*x))/c^2+3/2*b^2*d^3*log(1+c^2*x^2)/c^2-6/5*b^2*d^3*polylog(2,1+(-2)/(1-I*c*x))/c^2],
[(d+I*c*d*x)^3*(a+b*arctan(c*x))^2,x,16,-7/2*I*a*b*d^3*x-b^2*d^3*x-1/12*I*b^2*c*d^3*x^2+b^2*d^3*arctan(c*x)/c-7/2*I*b^2*d^3*x*arctan(c*x)+b*c*d^3*x^2*(a+b*arctan(c*x))+1/6*I*b*c^2*d^3*x^3*(a+b*arctan(c*x))-1/4*I*d^3*(1+I*c*x)^4*(a+b*arctan(c*x))^2/c+4*b*d^3*(a+b*arctan(c*x))*log(2/(1-I*c*x))/c+11/6*I*b^2*d^3*log(1+c^2*x^2)/c-2*I*b^2*d^3*polylog(2,1+(-2)/(1-I*c*x))/c],
[(d+I*c*d*x)^3*(a+b*arctan(c*x))^2/x,x,28,3*a*b*c*d^3*x-1/3*I*b^2*c*d^3*x+1/3*I*b^2*d^3*arctan(c*x)+3*b^2*c*d^3*x*arctan(c*x)+1/3*I*b*c^2*d^3*x^2*(a+b*arctan(c*x))-29/6*d^3*(a+b*arctan(c*x))^2+3*I*c*d^3*x*(a+b*arctan(c*x))^2-3/2*c^2*d^3*x^2*(a+b*arctan(c*x))^2-1/3*I*c^3*d^3*x^3*(a+b*arctan(c*x))^2+2*d^3*(a+b*arctan(c*x))^2*arctanh(1+(-2)/(1+I*c*x))+20/3*I*b*d^3*(a+b*arctan(c*x))*log(2/(1+I*c*x))-3/2*b^2*d^3*log(1+c^2*x^2)-10/3*b^2*d^3*polylog(2,1+(-2)/(1+I*c*x))-I*b*d^3*(a+b*arctan(c*x))*polylog(2,1+(-2)/(1+I*c*x))+I*b*d^3*(a+b*arctan(c*x))*polylog(2,-1+2/(1+I*c*x))-1/2*b^2*d^3*polylog(3,1+(-2)/(1+I*c*x))+1/2*b^2*d^3*polylog(3,-1+2/(1+I*c*x))],
[(d+I*c*d*x)^3*(a+b*arctan(c*x))^2/x^2,x,23,I*a*b*c^2*d^3*x+I*b^2*c^2*d^3*x*arctan(c*x)-9/2*I*c*d^3*(a+b*arctan(c*x))^2-d^3*(a+b*arctan(c*x))^2/x-3*c^2*d^3*x*(a+b*arctan(c*x))^2-1/2*I*c^3*d^3*x^2*(a+b*arctan(c*x))^2+6*I*c*d^3*(a+b*arctan(c*x))^2*arctanh(1+(-2)/(1+I*c*x))-6*b*c*d^3*(a+b*arctan(c*x))*log(2/(1+I*c*x))-1/2*I*b^2*c*d^3*log(1+c^2*x^2)+2*b*c*d^3*(a+b*arctan(c*x))*log(2+(-2)/(1-I*c*x))-I*b^2*c*d^3*polylog(2,-1+2/(1-I*c*x))-3*I*b^2*c*d^3*polylog(2,1+(-2)/(1+I*c*x))+3*b*c*d^3*(a+b*arctan(c*x))*polylog(2,1+(-2)/(1+I*c*x))-3*b*c*d^3*(a+b*arctan(c*x))*polylog(2,-1+2/(1+I*c*x))-3/2*I*b^2*c*d^3*polylog(3,1+(-2)/(1+I*c*x))+3/2*I*b^2*c*d^3*polylog(3,-1+2/(1+I*c*x))],
[(d+I*c*d*x)^3*(a+b*arctan(c*x))^2/x^3,x,25,-b*c*d^3*(a+b*arctan(c*x))/x+7/2*c^2*d^3*(a+b*arctan(c*x))^2-1/2*d^3*(a+b*arctan(c*x))^2/x^2-3*I*c*d^3*(a+b*arctan(c*x))^2/x-I*c^3*d^3*x*(a+b*arctan(c*x))^2-6*c^2*d^3*(a+b*arctan(c*x))^2*arctanh(1+(-2)/(1+I*c*x))+b^2*c^2*d^3*log(x)-2*I*b*c^2*d^3*(a+b*arctan(c*x))*log(2/(1+I*c*x))-1/2*b^2*c^2*d^3*log(1+c^2*x^2)+6*I*b*c^2*d^3*(a+b*arctan(c*x))*log(2+(-2)/(1-I*c*x))+3*b^2*c^2*d^3*polylog(2,-1+2/(1-I*c*x))+b^2*c^2*d^3*polylog(2,1+(-2)/(1+I*c*x))+3*I*b*c^2*d^3*(a+b*arctan(c*x))*polylog(2,1+(-2)/(1+I*c*x))-3*I*b*c^2*d^3*(a+b*arctan(c*x))*polylog(2,-1+2/(1+I*c*x))+3/2*b^2*c^2*d^3*polylog(3,1+(-2)/(1+I*c*x))-3/2*b^2*c^2*d^3*polylog(3,-1+2/(1+I*c*x))],
[(d+I*c*d*x)^3*(a+b*arctan(c*x))^2/x^4,x,28,-1/3*b^2*c^2*d^3/x-1/3*b^2*c^3*d^3*arctan(c*x)-1/3*b*c*d^3*(a+b*arctan(c*x))/x^2-3*I*b*c^2*d^3*(a+b*arctan(c*x))/x+11/6*I*c^3*d^3*(a+b*arctan(c*x))^2-1/3*d^3*(a+b*arctan(c*x))^2/x^3-3/2*I*c*d^3*(a+b*arctan(c*x))^2/x^2+3*c^2*d^3*(a+b*arctan(c*x))^2/x-2*I*c^3*d^3*(a+b*arctan(c*x))^2*arctanh(1+(-2)/(1+I*c*x))+3*I*b^2*c^3*d^3*log(x)-3/2*I*b^2*c^3*d^3*log(1+c^2*x^2)-20/3*b*c^3*d^3*(a+b*arctan(c*x))*log(2+(-2)/(1-I*c*x))+10/3*I*b^2*c^3*d^3*polylog(2,-1+2/(1-I*c*x))-b*c^3*d^3*(a+b*arctan(c*x))*polylog(2,1+(-2)/(1+I*c*x))+b*c^3*d^3*(a+b*arctan(c*x))*polylog(2,-1+2/(1+I*c*x))+1/2*I*b^2*c^3*d^3*polylog(3,1+(-2)/(1+I*c*x))-1/2*I*b^2*c^3*d^3*polylog(3,-1+2/(1+I*c*x))],
[(d+I*c*d*x)^3*(a+b*arctan(c*x))^2/x^5,x,20,-1/12*b^2*c^2*d^3/x^2-I*b^2*c^3*d^3/x-I*b^2*c^4*d^3*arctan(c*x)-1/6*b*c*d^3*(a+b*arctan(c*x))/x^3-I*b*c^2*d^3*(a+b*arctan(c*x))/x^2+7/2*b*c^3*d^3*(a+b*arctan(c*x))/x-1/4*d^3*(1+I*c*x)^4*(a+b*arctan(c*x))^2/x^4-4*I*a*b*c^4*d^3*log(x)-11/3*b^2*c^4*d^3*log(x)-4*I*b*c^4*d^3*(a+b*arctan(c*x))*log(2/(1-I*c*x))+11/6*b^2*c^4*d^3*log(1+c^2*x^2)+2*b^2*c^4*d^3*polylog(2,-I*c*x)-2*b^2*c^4*d^3*polylog(2,I*c*x)-2*b^2*c^4*d^3*polylog(2,1+(-2)/(1-I*c*x))],
[(d+I*c*d*x)^3*(a+b*arctan(c*x))^2/x^6,x,24,-1/30*b^2*c^2*d^3/x^3-1/4*I*b^2*c^3*d^3/x^2+13/10*b^2*c^4*d^3/x+13/10*b^2*c^5*d^3*arctan(c*x)-1/10*b*c*d^3*(a+b*arctan(c*x))/x^4-1/2*I*b*c^2*d^3*(a+b*arctan(c*x))/x^3+6/5*b*c^3*d^3*(a+b*arctan(c*x))/x^2+5/2*I*b*c^4*d^3*(a+b*arctan(c*x))/x-1/5*d^3*(1+I*c*x)^4*(a+b*arctan(c*x))^2/x^5+1/20*I*c*d^3*(1+I*c*x)^4*(a+b*arctan(c*x))^2/x^4+12/5*a*b*c^5*d^3*log(x)-3*I*b^2*c^5*d^3*log(x)+12/5*b*c^5*d^3*(a+b*arctan(c*x))*log(2/(1-I*c*x))+3/2*I*b^2*c^5*d^3*log(1+c^2*x^2)+6/5*I*b^2*c^5*d^3*polylog(2,-I*c*x)-6/5*I*b^2*c^5*d^3*polylog(2,I*c*x)-6/5*I*b^2*c^5*d^3*polylog(2,1+(-2)/(1-I*c*x))],
[(d+I*c*d*x)^3*(a+b*arctan(c*x))^2/x^7,x,31,-1/60*b^2*c^2*d^3/x^4-1/10*I*b^2*c^3*d^3/x^3+61/180*b^2*c^4*d^3/x^2+37/30*I*b^2*c^5*d^3/x+37/30*I*b^2*c^6*d^3*arctan(c*x)-1/15*b*c*d^3*(a+b*arctan(c*x))/x^5-3/10*I*b*c^2*d^3*(a+b*arctan(c*x))/x^4+11/18*b*c^3*d^3*(a+b*arctan(c*x))/x^3+14/15*I*b*c^4*d^3*(a+b*arctan(c*x))/x^2-11/6*b*c^5*d^3*(a+b*arctan(c*x))/x-1/6*d^3*(a+b*arctan(c*x))^2/x^6-3/5*I*c*d^3*(a+b*arctan(c*x))^2/x^5+3/4*c^2*d^3*(a+b*arctan(c*x))^2/x^4+1/3*I*c^3*d^3*(a+b*arctan(c*x))^2/x^3+28/15*I*a*b*c^6*d^3*log(x)+113/45*b^2*c^6*d^3*log(x)+37/20*I*b*c^6*d^3*(a+b*arctan(c*x))*log(2/(1-I*c*x))+1/60*I*b*c^6*d^3*(a+b*arctan(c*x))*log(2/(1+I*c*x))-113/90*b^2*c^6*d^3*log(1+c^2*x^2)-14/15*b^2*c^6*d^3*polylog(2,-I*c*x)+14/15*b^2*c^6*d^3*polylog(2,I*c*x)+37/40*b^2*c^6*d^3*polylog(2,1+(-2)/(1-I*c*x))-1/120*b^2*c^6*d^3*polylog(2,1+(-2)/(1+I*c*x))],

# q<0
[x^3*(a+b*arctan(c*x))^2/(d+I*c*d*x),x,26,-a*b*x/(c^3*d)-1/3*I*b^2*x/(c^3*d)+1/3*I*b^2*arctan(c*x)/(c^4*d)-b^2*x*arctan(c*x)/(c^3*d)+1/3*I*b*x^2*(a+b*arctan(c*x))/(c^2*d)-5/6*(a+b*arctan(c*x))^2/(c^4*d)+I*x*(a+b*arctan(c*x))^2/(c^3*d)+1/2*x^2*(a+b*arctan(c*x))^2/(c^2*d)-1/3*I*x^3*(a+b*arctan(c*x))^2/(c*d)+8/3*I*b*(a+b*arctan(c*x))*log(2/(1+I*c*x))/(c^4*d)+(a+b*arctan(c*x))^2*log(2/(1+I*c*x))/(c^4*d)+1/2*b^2*log(1+c^2*x^2)/(c^4*d)-4/3*b^2*polylog(2,1+(-2)/(1+I*c*x))/(c^4*d)+I*b*(a+b*arctan(c*x))*polylog(2,1+(-2)/(1+I*c*x))/(c^4*d)+1/2*b^2*polylog(3,1+(-2)/(1+I*c*x))/(c^4*d)],
[x^2*(a+b*arctan(c*x))^2/(d+I*c*d*x),x,16,I*a*b*x/(c^2*d)+I*b^2*x*arctan(c*x)/(c^2*d)+1/2*I*(a+b*arctan(c*x))^2/(c^3*d)+x*(a+b*arctan(c*x))^2/(c^2*d)-1/2*I*x^2*(a+b*arctan(c*x))^2/(c*d)+2*b*(a+b*arctan(c*x))*log(2/(1+I*c*x))/(c^3*d)-I*(a+b*arctan(c*x))^2*log(2/(1+I*c*x))/(c^3*d)-1/2*I*b^2*log(1+c^2*x^2)/(c^3*d)+I*b^2*polylog(2,1+(-2)/(1+I*c*x))/(c^3*d)+b*(a+b*arctan(c*x))*polylog(2,1+(-2)/(1+I*c*x))/(c^3*d)-1/2*I*b^2*polylog(3,1+(-2)/(1+I*c*x))/(c^3*d)],
[x*(a+b*arctan(c*x))^2/(d+I*c*d*x),x,9,(a+b*arctan(c*x))^2/(c^2*d)-I*x*(a+b*arctan(c*x))^2/(c*d)-2*I*b*(a+b*arctan(c*x))*log(2/(1+I*c*x))/(c^2*d)-(a+b*arctan(c*x))^2*log(2/(1+I*c*x))/(c^2*d)+b^2*polylog(2,1+(-2)/(1+I*c*x))/(c^2*d)-I*b*(a+b*arctan(c*x))*polylog(2,1+(-2)/(1+I*c*x))/(c^2*d)-1/2*b^2*polylog(3,1+(-2)/(1+I*c*x))/(c^2*d)],
[(a+b*arctan(c*x))^2/(d+I*c*d*x),x,3,I*(a+b*arctan(c*x))^2*log(2/(1+I*c*x))/(c*d)-b*(a+b*arctan(c*x))*polylog(2,1+(-2)/(1+I*c*x))/(c*d)+1/2*I*b^2*polylog(3,1+(-2)/(1+I*c*x))/(c*d)],
[(a+b*arctan(c*x))^2/(x*(d+I*c*d*x)),x,3,(a+b*arctan(c*x))^2*log(2+(-2)/(1+I*c*x))/d+I*b*(a+b*arctan(c*x))*polylog(2,-1+2/(1+I*c*x))/d+1/2*b^2*polylog(3,-1+2/(1+I*c*x))/d],
[(a+b*arctan(c*x))^2/(x^2*(d+I*c*d*x)),x,8,-I*c*(a+b*arctan(c*x))^2/d-(a+b*arctan(c*x))^2/(d*x)+2*b*c*(a+b*arctan(c*x))*log(2+(-2)/(1-I*c*x))/d-I*c*(a+b*arctan(c*x))^2*log(2+(-2)/(1+I*c*x))/d-I*b^2*c*polylog(2,-1+2/(1-I*c*x))/d+b*c*(a+b*arctan(c*x))*polylog(2,-1+2/(1+I*c*x))/d-1/2*I*b^2*c*polylog(3,-1+2/(1+I*c*x))/d],
[(a+b*arctan(c*x))^2/(x^3*(d+I*c*d*x)),x,17,-b*c*(a+b*arctan(c*x))/(d*x)-3/2*c^2*(a+b*arctan(c*x))^2/d-1/2*(a+b*arctan(c*x))^2/(d*x^2)+I*c*(a+b*arctan(c*x))^2/(d*x)+b^2*c^2*log(x)/d-1/2*b^2*c^2*log(1+c^2*x^2)/d-2*I*b*c^2*(a+b*arctan(c*x))*log(2+(-2)/(1-I*c*x))/d-c^2*(a+b*arctan(c*x))^2*log(2+(-2)/(1+I*c*x))/d-b^2*c^2*polylog(2,-1+2/(1-I*c*x))/d-I*b*c^2*(a+b*arctan(c*x))*polylog(2,-1+2/(1+I*c*x))/d-1/2*b^2*c^2*polylog(3,-1+2/(1+I*c*x))/d],
[(a+b*arctan(c*x))^2/(x^4*(d+I*c*d*x)),x,26,-1/3*b^2*c^2/(d*x)-1/3*b^2*c^3*arctan(c*x)/d-1/3*b*c*(a+b*arctan(c*x))/(d*x^2)+I*b*c^2*(a+b*arctan(c*x))/(d*x)+11/6*I*c^3*(a+b*arctan(c*x))^2/d-1/3*(a+b*arctan(c*x))^2/(d*x^3)+1/2*I*c*(a+b*arctan(c*x))^2/(d*x^2)+c^2*(a+b*arctan(c*x))^2/(d*x)-I*b^2*c^3*log(x)/d+1/2*I*b^2*c^3*log(1+c^2*x^2)/d-8/3*b*c^3*(a+b*arctan(c*x))*log(2+(-2)/(1-I*c*x))/d+I*c^3*(a+b*arctan(c*x))^2*log(2+(-2)/(1+I*c*x))/d+4/3*I*b^2*c^3*polylog(2,-1+2/(1-I*c*x))/d-b*c^3*(a+b*arctan(c*x))*polylog(2,-1+2/(1+I*c*x))/d+1/2*I*b^2*c^3*polylog(3,-1+2/(1+I*c*x))/d],
[x^4*(a+b*arctan(c*x))^2/(d+I*c*d*x)^2,x,33,2*I*a*b*x/(c^4*d^2)-1/3*b^2*x/(c^4*d^2)+1/2*b^2/(c^5*d^2*(I-c*x))-1/6*b^2*arctan(c*x)/(c^5*d^2)+2*I*b^2*x*arctan(c*x)/(c^4*d^2)+1/3*b*x^2*(a+b*arctan(c*x))/(c^3*d^2)+I*b*(a+b*arctan(c*x))/(c^5*d^2*(I-c*x))+11/6*I*(a+b*arctan(c*x))^2/(c^5*d^2)+3*x*(a+b*arctan(c*x))^2/(c^4*d^2)-I*x^2*(a+b*arctan(c*x))^2/(c^3*d^2)-1/3*x^3*(a+b*arctan(c*x))^2/(c^2*d^2)-(a+b*arctan(c*x))^2/(c^5*d^2*(I-c*x))+20/3*b*(a+b*arctan(c*x))*log(2/(1+I*c*x))/(c^5*d^2)-4*I*(a+b*arctan(c*x))^2*log(2/(1+I*c*x))/(c^5*d^2)-I*b^2*log(1+c^2*x^2)/(c^5*d^2)+10/3*I*b^2*polylog(2,1+(-2)/(1+I*c*x))/(c^5*d^2)+4*b*(a+b*arctan(c*x))*polylog(2,1+(-2)/(1+I*c*x))/(c^5*d^2)-2*I*b^2*polylog(3,1+(-2)/(1+I*c*x))/(c^5*d^2)],
[x^3*(a+b*arctan(c*x))^2/(d+I*c*d*x)^2,x,24,a*b*x/(c^3*d^2)-1/2*I*b^2/(c^4*d^2*(I-c*x))+1/2*I*b^2*arctan(c*x)/(c^4*d^2)+b^2*x*arctan(c*x)/(c^3*d^2)+b*(a+b*arctan(c*x))/(c^4*d^2*(I-c*x))+(a+b*arctan(c*x))^2/(c^4*d^2)-2*I*x*(a+b*arctan(c*x))^2/(c^3*d^2)-1/2*x^2*(a+b*arctan(c*x))^2/(c^2*d^2)+I*(a+b*arctan(c*x))^2/(c^4*d^2*(I-c*x))-4*I*b*(a+b*arctan(c*x))*log(2/(1+I*c*x))/(c^4*d^2)-3*(a+b*arctan(c*x))^2*log(2/(1+I*c*x))/(c^4*d^2)-1/2*b^2*log(1+c^2*x^2)/(c^4*d^2)+2*b^2*polylog(2,1+(-2)/(1+I*c*x))/(c^4*d^2)-3*I*b*(a+b*arctan(c*x))*polylog(2,1+(-2)/(1+I*c*x))/(c^4*d^2)-3/2*b^2*polylog(3,1+(-2)/(1+I*c*x))/(c^4*d^2)],
[x^2*(a+b*arctan(c*x))^2/(d+I*c*d*x)^2,x,18,-1/2*b^2/(c^3*d^2*(I-c*x))+1/2*b^2*arctan(c*x)/(c^3*d^2)-I*b*(a+b*arctan(c*x))/(c^3*d^2*(I-c*x))-1/2*I*(a+b*arctan(c*x))^2/(c^3*d^2)-x*(a+b*arctan(c*x))^2/(c^2*d^2)+(a+b*arctan(c*x))^2/(c^3*d^2*(I-c*x))-2*b*(a+b*arctan(c*x))*log(2/(1+I*c*x))/(c^3*d^2)+2*I*(a+b*arctan(c*x))^2*log(2/(1+I*c*x))/(c^3*d^2)-I*b^2*polylog(2,1+(-2)/(1+I*c*x))/(c^3*d^2)-2*b*(a+b*arctan(c*x))*polylog(2,1+(-2)/(1+I*c*x))/(c^3*d^2)+I*b^2*polylog(3,1+(-2)/(1+I*c*x))/(c^3*d^2)],
[x*(a+b*arctan(c*x))^2/(d+I*c*d*x)^2,x,13,1/2*I*b^2/(c^2*d^2*(I-c*x))-1/2*I*b^2*arctan(c*x)/(c^2*d^2)-b*(a+b*arctan(c*x))/(c^2*d^2*(I-c*x))+1/2*(a+b*arctan(c*x))^2/(c^2*d^2)-I*(a+b*arctan(c*x))^2/(c^2*d^2*(I-c*x))+(a+b*arctan(c*x))^2*log(2/(1+I*c*x))/(c^2*d^2)+I*b*(a+b*arctan(c*x))*polylog(2,1+(-2)/(1+I*c*x))/(c^2*d^2)+1/2*b^2*polylog(3,1+(-2)/(1+I*c*x))/(c^2*d^2)],
[(a+b*arctan(c*x))^2/(d+I*c*d*x)^2,x,8,1/2*b^2/(c*d^2*(I-c*x))-1/2*b^2*arctan(c*x)/(c*d^2)+I*b*(a+b*arctan(c*x))/(c*d^2*(I-c*x))-1/2*I*(a+b*arctan(c*x))^2/(c*d^2)+I*(a+b*arctan(c*x))^2/(c*d^2*(1+I*c*x))],
[(a+b*arctan(c*x))^2/(x*(d+I*c*d*x)^2),x,19,-1/2*I*b^2/(d^2*(I-c*x))+1/2*I*b^2*arctan(c*x)/d^2+b*(a+b*arctan(c*x))/(d^2*(I-c*x))-1/2*(a+b*arctan(c*x))^2/d^2+I*(a+b*arctan(c*x))^2/(d^2*(I-c*x))+2*(a+b*arctan(c*x))^2*arctanh(1+(-2)/(1+I*c*x))/d^2+(a+b*arctan(c*x))^2*log(2/(1+I*c*x))/d^2+I*b*(a+b*arctan(c*x))*polylog(2,-1+2/(1+I*c*x))/d^2+1/2*b^2*polylog(3,-1+2/(1+I*c*x))/d^2],
[(a+b*arctan(c*x))^2/(x^2*(d+I*c*d*x)^2),x,23,-1/2*b^2*c/(d^2*(I-c*x))+1/2*b^2*c*arctan(c*x)/d^2-I*b*c*(a+b*arctan(c*x))/(d^2*(I-c*x))-1/2*I*c*(a+b*arctan(c*x))^2/d^2-(a+b*arctan(c*x))^2/(d^2*x)+c*(a+b*arctan(c*x))^2/(d^2*(I-c*x))-4*I*c*(a+b*arctan(c*x))^2*arctanh(1+(-2)/(1+I*c*x))/d^2-2*I*c*(a+b*arctan(c*x))^2*log(2/(1+I*c*x))/d^2+2*b*c*(a+b*arctan(c*x))*log(2+(-2)/(1-I*c*x))/d^2-I*b^2*c*polylog(2,-1+2/(1-I*c*x))/d^2+2*b*c*(a+b*arctan(c*x))*polylog(2,-1+2/(1+I*c*x))/d^2-I*b^2*c*polylog(3,-1+2/(1+I*c*x))/d^2],
[(a+b*arctan(c*x))^2/(x^3*(d+I*c*d*x)^2),x,31,1/2*I*b^2*c^2/(d^2*(I-c*x))-1/2*I*b^2*c^2*arctan(c*x)/d^2-b*c*(a+b*arctan(c*x))/(d^2*x)-b*c^2*(a+b*arctan(c*x))/(d^2*(I-c*x))-2*c^2*(a+b*arctan(c*x))^2/d^2-1/2*(a+b*arctan(c*x))^2/(d^2*x^2)+2*I*c*(a+b*arctan(c*x))^2/(d^2*x)-I*c^2*(a+b*arctan(c*x))^2/(d^2*(I-c*x))-6*c^2*(a+b*arctan(c*x))^2*arctanh(1+(-2)/(1+I*c*x))/d^2+b^2*c^2*log(x)/d^2-3*c^2*(a+b*arctan(c*x))^2*log(2/(1+I*c*x))/d^2-1/2*b^2*c^2*log(1+c^2*x^2)/d^2-4*I*b*c^2*(a+b*arctan(c*x))*log(2+(-2)/(1-I*c*x))/d^2-2*b^2*c^2*polylog(2,-1+2/(1-I*c*x))/d^2-3*I*b*c^2*(a+b*arctan(c*x))*polylog(2,-1+2/(1+I*c*x))/d^2-3/2*b^2*c^2*polylog(3,-1+2/(1+I*c*x))/d^2],
[x^4*(a+b*arctan(c*x))^2/(d+I*c*d*x)^3,x,37,-I*a*b*x/(c^4*d^3)+1/16*I*b^2/(c^5*d^3*(I-c*x)^2)-29/16*b^2/(c^5*d^3*(I-c*x))+29/16*b^2*arctan(c*x)/(c^5*d^3)-I*b^2*x*arctan(c*x)/(c^4*d^3)-1/4*b*(a+b*arctan(c*x))/(c^5*d^3*(I-c*x)^2)-15/4*I*b*(a+b*arctan(c*x))/(c^5*d^3*(I-c*x))-5/8*I*(a+b*arctan(c*x))^2/(c^5*d^3)-3*x*(a+b*arctan(c*x))^2/(c^4*d^3)+1/2*I*x^2*(a+b*arctan(c*x))^2/(c^3*d^3)-1/2*I*(a+b*arctan(c*x))^2/(c^5*d^3*(I-c*x)^2)+4*(a+b*arctan(c*x))^2/(c^5*d^3*(I-c*x))-6*b*(a+b*arctan(c*x))*log(2/(1+I*c*x))/(c^5*d^3)+6*I*(a+b*arctan(c*x))^2*log(2/(1+I*c*x))/(c^5*d^3)+1/2*I*b^2*log(1+c^2*x^2)/(c^5*d^3)-3*I*b^2*polylog(2,1+(-2)/(1+I*c*x))/(c^5*d^3)-6*b*(a+b*arctan(c*x))*polylog(2,1+(-2)/(1+I*c*x))/(c^5*d^3)+3*I*b^2*polylog(3,1+(-2)/(1+I*c*x))/(c^5*d^3)],
[x^3*(a+b*arctan(c*x))^2/(d+I*c*d*x)^3,x,31,1/16*b^2/(c^4*d^3*(I-c*x)^2)+21/16*I*b^2/(c^4*d^3*(I-c*x))-21/16*I*b^2*arctan(c*x)/(c^4*d^3)+1/4*I*b*(a+b*arctan(c*x))/(c^4*d^3*(I-c*x)^2)-11/4*b*(a+b*arctan(c*x))/(c^4*d^3*(I-c*x))+3/8*(a+b*arctan(c*x))^2/(c^4*d^3)+I*x*(a+b*arctan(c*x))^2/(c^3*d^3)-1/2*(a+b*arctan(c*x))^2/(c^4*d^3*(I-c*x)^2)-3*I*(a+b*arctan(c*x))^2/(c^4*d^3*(I-c*x))+2*I*b*(a+b*arctan(c*x))*log(2/(1+I*c*x))/(c^4*d^3)+3*(a+b*arctan(c*x))^2*log(2/(1+I*c*x))/(c^4*d^3)-b^2*polylog(2,1+(-2)/(1+I*c*x))/(c^4*d^3)+3*I*b*(a+b*arctan(c*x))*polylog(2,1+(-2)/(1+I*c*x))/(c^4*d^3)+3/2*b^2*polylog(3,1+(-2)/(1+I*c*x))/(c^4*d^3)],
[x^2*(a+b*arctan(c*x))^2/(d+I*c*d*x)^3,x,26,-1/16*I*b^2/(c^3*d^3*(I-c*x)^2)+13/16*b^2/(c^3*d^3*(I-c*x))-13/16*b^2*arctan(c*x)/(c^3*d^3)+1/4*b*(a+b*arctan(c*x))/(c^3*d^3*(I-c*x)^2)+7/4*I*b*(a+b*arctan(c*x))/(c^3*d^3*(I-c*x))-7/8*I*(a+b*arctan(c*x))^2/(c^3*d^3)+1/2*I*(a+b*arctan(c*x))^2/(c^3*d^3*(I-c*x)^2)-2*(a+b*arctan(c*x))^2/(c^3*d^3*(I-c*x))-I*(a+b*arctan(c*x))^2*log(2/(1+I*c*x))/(c^3*d^3)+b*(a+b*arctan(c*x))*polylog(2,1+(-2)/(1+I*c*x))/(c^3*d^3)-1/2*I*b^2*polylog(3,1+(-2)/(1+I*c*x))/(c^3*d^3)],
[x*(a+b*arctan(c*x))^2/(d+I*c*d*x)^3,x,13,-1/16*b^2/(c^2*d^3*(I-c*x)^2)-5/16*I*b^2/(c^2*d^3*(I-c*x))+5/16*I*b^2*arctan(c*x)/(c^2*d^3)-1/4*I*b*(a+b*arctan(c*x))/(c^2*d^3*(I-c*x)^2)+3/4*b*(a+b*arctan(c*x))/(c^2*d^3*(I-c*x))+1/8*(a+b*arctan(c*x))^2/(c^2*d^3)+1/2*x^2*(a+b*arctan(c*x))^2/(d^3*(1+I*c*x)^2)],
[(a+b*arctan(c*x))^2/(d+I*c*d*x)^3,x,13,1/16*I*b^2/(c*d^3*(I-c*x)^2)+3/16*b^2/(c*d^3*(I-c*x))-3/16*b^2*arctan(c*x)/(c*d^3)-1/4*b*(a+b*arctan(c*x))/(c*d^3*(I-c*x)^2)+1/4*I*b*(a+b*arctan(c*x))/(c*d^3*(I-c*x))-1/8*I*(a+b*arctan(c*x))^2/(c*d^3)+1/2*I*(a+b*arctan(c*x))^2/(c*d^3*(1+I*c*x)^2)],
[(a+b*arctan(c*x))^2/(x*(d+I*c*d*x)^3),x,32,1/16*b^2/(d^3*(I-c*x)^2)-11/16*I*b^2/(d^3*(I-c*x))+11/16*I*b^2*arctan(c*x)/d^3+1/4*I*b*(a+b*arctan(c*x))/(d^3*(I-c*x)^2)+5/4*b*(a+b*arctan(c*x))/(d^3*(I-c*x))-5/8*(a+b*arctan(c*x))^2/d^3-1/2*(a+b*arctan(c*x))^2/(d^3*(I-c*x)^2)+I*(a+b*arctan(c*x))^2/(d^3*(I-c*x))+2*(a+b*arctan(c*x))^2*arctanh(1+(-2)/(1+I*c*x))/d^3+(a+b*arctan(c*x))^2*log(2/(1+I*c*x))/d^3+I*b*(a+b*arctan(c*x))*polylog(2,-1+2/(1+I*c*x))/d^3+1/2*b^2*polylog(3,-1+2/(1+I*c*x))/d^3],
[(a+b*arctan(c*x))^2/(x^2*(d+I*c*d*x)^3),x,36,-1/16*I*b^2*c/(d^3*(I-c*x)^2)-19/16*b^2*c/(d^3*(I-c*x))+19/16*b^2*c*arctan(c*x)/d^3+1/4*b*c*(a+b*arctan(c*x))/(d^3*(I-c*x)^2)-9/4*I*b*c*(a+b*arctan(c*x))/(d^3*(I-c*x))+1/8*I*c*(a+b*arctan(c*x))^2/d^3-(a+b*arctan(c*x))^2/(d^3*x)+1/2*I*c*(a+b*arctan(c*x))^2/(d^3*(I-c*x)^2)+2*c*(a+b*arctan(c*x))^2/(d^3*(I-c*x))-6*I*c*(a+b*arctan(c*x))^2*arctanh(1+(-2)/(1+I*c*x))/d^3-3*I*c*(a+b*arctan(c*x))^2*log(2/(1+I*c*x))/d^3+2*b*c*(a+b*arctan(c*x))*log(2+(-2)/(1-I*c*x))/d^3-I*b^2*c*polylog(2,-1+2/(1-I*c*x))/d^3+3*b*c*(a+b*arctan(c*x))*polylog(2,-1+2/(1+I*c*x))/d^3-3/2*I*b^2*c*polylog(3,-1+2/(1+I*c*x))/d^3],
[(a+b*arctan(c*x))^2/(1+I*c*x)^4,x,18,-1/54*b^2/(c*(I-c*x)^3)+5/144*I*b^2/(c*(I-c*x)^2)+11/144*b^2/(c*(I-c*x))-11/144*b^2*arctan(c*x)/c-1/9*I*b*(a+b*arctan(c*x))/(c*(I-c*x)^3)-1/12*b*(a+b*arctan(c*x))/(c*(I-c*x)^2)+1/12*I*b*(a+b*arctan(c*x))/(c*(I-c*x))-1/24*I*(a+b*arctan(c*x))^2/c+1/3*I*(a+b*arctan(c*x))^2/(c*(1+I*c*x)^3)],
[arctan(a*x)^2/(c*x-I*a*c*x^2),x,4,arctan(a*x)^2*log(2+(-2)/(1-I*a*x))/c-I*arctan(a*x)*polylog(2,-1+2/(1-I*a*x))/c+1/2*polylog(3,-1+2/(1-I*a*x))/c],

# Integrands of the form x^m (d+e x)^q (a+b ArcTan[c x])^3 with c^2 d^2+e^2=0
[(d+I*c*d*x)^3*(a+b*arctan(c*x))^3,x,26,-3*a*b^2*d^3*x+1/4*I*b^3*d^3*x-1/4*I*b^3*d^3*arctan(c*x)/c-3*b^3*d^3*x*arctan(c*x)-1/4*I*b^2*c*d^3*x^2*(a+b*arctan(c*x))+7*b*d^3*(a+b*arctan(c*x))^2/c-21/4*I*b*d^3*x*(a+b*arctan(c*x))^2+3/2*b*c*d^3*x^2*(a+b*arctan(c*x))^2+1/4*I*b*c^2*d^3*x^3*(a+b*arctan(c*x))^2-1/4*I*d^3*(1+I*c*x)^4*(a+b*arctan(c*x))^3/c+6*b*d^3*(a+b*arctan(c*x))^2*log(2/(1-I*c*x))/c-11*I*b^2*d^3*(a+b*arctan(c*x))*log(2/(1+I*c*x))/c+3/2*b^3*d^3*log(1+c^2*x^2)/c-6*I*b^2*d^3*(a+b*arctan(c*x))*polylog(2,1+(-2)/(1-I*c*x))/c+11/2*b^3*d^3*polylog(2,1+(-2)/(1+I*c*x))/c+3*b^3*d^3*polylog(3,1+(-2)/(1-I*c*x))/c],
[(d+I*c*d*x)^2*(a+b*arctan(c*x))^3,x,17,-a*b^2*d^2*x-b^3*d^2*x*arctan(c*x)+7/2*b*d^2*(a+b*arctan(c*x))^2/c-3*I*b*d^2*x*(a+b*arctan(c*x))^2+1/2*b*c*d^2*x^2*(a+b*arctan(c*x))^2-1/3*I*d^2*(1+I*c*x)^3*(a+b*arctan(c*x))^3/c+4*b*d^2*(a+b*arctan(c*x))^2*log(2/(1-I*c*x))/c-6*I*b^2*d^2*(a+b*arctan(c*x))*log(2/(1+I*c*x))/c+1/2*b^3*d^2*log(1+c^2*x^2)/c-4*I*b^2*d^2*(a+b*arctan(c*x))*polylog(2,1+(-2)/(1-I*c*x))/c+3*b^3*d^2*polylog(2,1+(-2)/(1+I*c*x))/c+2*b^3*d^2*polylog(3,1+(-2)/(1-I*c*x))/c],
[(d+I*c*d*x)*(a+b*arctan(c*x))^3,x,11,3/2*b*d*(a+b*arctan(c*x))^2/c-3/2*I*b*d*x*(a+b*arctan(c*x))^2-1/2*I*d*(1+I*c*x)^2*(a+b*arctan(c*x))^3/c+3*b*d*(a+b*arctan(c*x))^2*log(2/(1-I*c*x))/c-3*I*b^2*d*(a+b*arctan(c*x))*log(2/(1+I*c*x))/c-3*I*b^2*d*(a+b*arctan(c*x))*polylog(2,1+(-2)/(1-I*c*x))/c+3/2*b^3*d*polylog(2,1+(-2)/(1+I*c*x))/c+3/2*b^3*d*polylog(3,1+(-2)/(1-I*c*x))/c],
[(a+b*arctan(c*x))^3/(d+I*c*d*x),x,4,I*(a+b*arctan(c*x))^3*log(2/(1+I*c*x))/(c*d)-3/2*b*(a+b*arctan(c*x))^2*polylog(2,1+(-2)/(1+I*c*x))/(c*d)+3/2*I*b^2*(a+b*arctan(c*x))*polylog(3,1+(-2)/(1+I*c*x))/(c*d)+3/4*b^3*polylog(4,1+(-2)/(1+I*c*x))/(c*d)],
[(a+b*arctan(c*x))^3/(d+I*c*d*x)^2,x,11,-3/4*I*b^3/(c*d^2*(I-c*x))+3/4*I*b^3*arctan(c*x)/(c*d^2)+3/2*b^2*(a+b*arctan(c*x))/(c*d^2*(I-c*x))-3/4*b*(a+b*arctan(c*x))^2/(c*d^2)+3/2*I*b*(a+b*arctan(c*x))^2/(c*d^2*(I-c*x))-1/2*I*(a+b*arctan(c*x))^3/(c*d^2)+I*(a+b*arctan(c*x))^3/(c*d^2*(1+I*c*x))],
[(a+b*arctan(c*x))^3/(d+I*c*d*x)^3,x,24,3/64*b^3/(c*d^3*(I-c*x)^2)-21/64*I*b^3/(c*d^3*(I-c*x))+21/64*I*b^3*arctan(c*x)/(c*d^3)+3/16*I*b^2*(a+b*arctan(c*x))/(c*d^3*(I-c*x)^2)+9/16*b^2*(a+b*arctan(c*x))/(c*d^3*(I-c*x))-9/32*b*(a+b*arctan(c*x))^2/(c*d^3)-3/8*b*(a+b*arctan(c*x))^2/(c*d^3*(I-c*x)^2)+3/8*I*b*(a+b*arctan(c*x))^2/(c*d^3*(I-c*x))-1/8*I*(a+b*arctan(c*x))^3/(c*d^3)+1/2*I*(a+b*arctan(c*x))^3/(c*d^3*(1+I*c*x)^2)],
[(a+b*arctan(c*x))^3/(d+I*c*d*x)^4,x,42,1/108*I*b^3/(c*d^4*(I-c*x)^3)+19/576*b^3/(c*d^4*(I-c*x)^2)-85/576*I*b^3/(c*d^4*(I-c*x))+85/576*I*b^3*arctan(c*x)/(c*d^4)-1/18*b^2*(a+b*arctan(c*x))/(c*d^4*(I-c*x)^3)+5/48*I*b^2*(a+b*arctan(c*x))/(c*d^4*(I-c*x)^2)+11/48*b^2*(a+b*arctan(c*x))/(c*d^4*(I-c*x))-11/96*b*(a+b*arctan(c*x))^2/(c*d^4)-1/6*I*b*(a+b*arctan(c*x))^2/(c*d^4*(I-c*x)^3)-1/8*b*(a+b*arctan(c*x))^2/(c*d^4*(I-c*x)^2)+1/8*I*b*(a+b*arctan(c*x))^2/(c*d^4*(I-c*x))-1/24*I*(a+b*arctan(c*x))^3/(c*d^4)+1/3*I*(a+b*arctan(c*x))^3/(c*d^4*(1+I*c*x)^3)],
[x^2*(a+b*arctan(c*x))^3/(d+I*c*d*x),x,19,-3/2*b*(a+b*arctan(c*x))^2/(c^3*d)+3/2*I*b*x*(a+b*arctan(c*x))^2/(c^2*d)+1/2*I*(a+b*arctan(c*x))^3/(c^3*d)+x*(a+b*arctan(c*x))^3/(c^2*d)-1/2*I*x^2*(a+b*arctan(c*x))^3/(c*d)+3*I*b^2*(a+b*arctan(c*x))*log(2/(1+I*c*x))/(c^3*d)+3*b*(a+b*arctan(c*x))^2*log(2/(1+I*c*x))/(c^3*d)-I*(a+b*arctan(c*x))^3*log(2/(1+I*c*x))/(c^3*d)-3/2*b^3*polylog(2,1+(-2)/(1+I*c*x))/(c^3*d)+3*I*b^2*(a+b*arctan(c*x))*polylog(2,1+(-2)/(1+I*c*x))/(c^3*d)+3/2*b*(a+b*arctan(c*x))^2*polylog(2,1+(-2)/(1+I*c*x))/(c^3*d)+3/2*b^3*polylog(3,1+(-2)/(1+I*c*x))/(c^3*d)-3/2*I*b^2*(a+b*arctan(c*x))*polylog(3,1+(-2)/(1+I*c*x))/(c^3*d)-3/4*b^3*polylog(4,1+(-2)/(1+I*c*x))/(c^3*d)],
[x*(a+b*arctan(c*x))^3/(d+I*c*d*x),x,10,(a+b*arctan(c*x))^3/(c^2*d)-I*x*(a+b*arctan(c*x))^3/(c*d)-3*I*b*(a+b*arctan(c*x))^2*log(2/(1+I*c*x))/(c^2*d)-(a+b*arctan(c*x))^3*log(2/(1+I*c*x))/(c^2*d)+3*b^2*(a+b*arctan(c*x))*polylog(2,1+(-2)/(1+I*c*x))/(c^2*d)-3/2*I*b*(a+b*arctan(c*x))^2*polylog(2,1+(-2)/(1+I*c*x))/(c^2*d)-3/2*I*b^3*polylog(3,1+(-2)/(1+I*c*x))/(c^2*d)-3/2*b^2*(a+b*arctan(c*x))*polylog(3,1+(-2)/(1+I*c*x))/(c^2*d)+3/4*I*b^3*polylog(4,1+(-2)/(1+I*c*x))/(c^2*d)],
[(a+b*arctan(c*x))^3/(d+I*c*d*x),x,4,I*(a+b*arctan(c*x))^3*log(2/(1+I*c*x))/(c*d)-3/2*b*(a+b*arctan(c*x))^2*polylog(2,1+(-2)/(1+I*c*x))/(c*d)+3/2*I*b^2*(a+b*arctan(c*x))*polylog(3,1+(-2)/(1+I*c*x))/(c*d)+3/4*b^3*polylog(4,1+(-2)/(1+I*c*x))/(c*d)],
[(a+b*arctan(c*x))^3/(x*(d+I*c*d*x)),x,4,(a+b*arctan(c*x))^3*log(2+(-2)/(1+I*c*x))/d+3/2*I*b*(a+b*arctan(c*x))^2*polylog(2,-1+2/(1+I*c*x))/d+3/2*b^2*(a+b*arctan(c*x))*polylog(3,-1+2/(1+I*c*x))/d-3/4*I*b^3*polylog(4,-1+2/(1+I*c*x))/d],
[(a+b*arctan(c*x))^3/(x^2*(d+I*c*d*x)),x,10,-I*c*(a+b*arctan(c*x))^3/d-(a+b*arctan(c*x))^3/(d*x)+3*b*c*(a+b*arctan(c*x))^2*log(2+(-2)/(1-I*c*x))/d-I*c*(a+b*arctan(c*x))^3*log(2+(-2)/(1+I*c*x))/d-3*I*b^2*c*(a+b*arctan(c*x))*polylog(2,-1+2/(1-I*c*x))/d+3/2*b*c*(a+b*arctan(c*x))^2*polylog(2,-1+2/(1+I*c*x))/d+3/2*b^3*c*polylog(3,-1+2/(1-I*c*x))/d-3/2*I*b^2*c*(a+b*arctan(c*x))*polylog(3,-1+2/(1+I*c*x))/d-3/4*b^3*c*polylog(4,-1+2/(1+I*c*x))/d],
[(a+b*arctan(c*x))^3/(x^3*(d+I*c*d*x)),x,18,-3/2*I*b*c^2*(a+b*arctan(c*x))^2/d-3/2*b*c*(a+b*arctan(c*x))^2/(d*x)-3/2*c^2*(a+b*arctan(c*x))^3/d-1/2*(a+b*arctan(c*x))^3/(d*x^2)+I*c*(a+b*arctan(c*x))^3/(d*x)+3*b^2*c^2*(a+b*arctan(c*x))*log(2+(-2)/(1-I*c*x))/d-3*I*b*c^2*(a+b*arctan(c*x))^2*log(2+(-2)/(1-I*c*x))/d-c^2*(a+b*arctan(c*x))^3*log(2+(-2)/(1+I*c*x))/d-3/2*I*b^3*c^2*polylog(2,-1+2/(1-I*c*x))/d-3*b^2*c^2*(a+b*arctan(c*x))*polylog(2,-1+2/(1-I*c*x))/d-3/2*I*b*c^2*(a+b*arctan(c*x))^2*polylog(2,-1+2/(1+I*c*x))/d-3/2*I*b^3*c^2*polylog(3,-1+2/(1-I*c*x))/d-3/2*b^2*c^2*(a+b*arctan(c*x))*polylog(3,-1+2/(1+I*c*x))/d+3/4*I*b^3*c^2*polylog(4,-1+2/(1+I*c*x))/d],

# Integrands of the form x^m (d+e x)^q (a+b ArcTan[c x])^4 with c^2 d^2+e^2=0

# Integrands of the form x^m (d+e x)^q (a+b ArcTan[c x])^-1 with c^2 d^2+e^2=0
[1/((d+I*c*d*x)*(a+b*arctan(c*x))),x,0,Unintegrable(1/((d+I*c*d*x)*(a+b*arctan(c*x))),x)],

# Integrands of the form (f x)^m (d+e x)^q (a+b ArcTan[c x])^p

# Integrands of the form x^m / (d+e x) (a+b ArcTan[c x])^p

# p>0
[x^3*(a+b*arctan(c*x))/(d+e*x),x,16,a*d^2*x/e^3+1/2*b*d*x/(c*e^2)-1/6*b*x^2/(c*e)-1/2*b*d*arctan(c*x)/(c^2*e^2)+b*d^2*x*arctan(c*x)/e^3-1/2*d*x^2*(a+b*arctan(c*x))/e^2+1/3*x^3*(a+b*arctan(c*x))/e+d^3*(a+b*arctan(c*x))*log(2/(1-I*c*x))/e^4-d^3*(a+b*arctan(c*x))*log(2*c*(d+e*x)/((c*d+I*e)*(1-I*c*x)))/e^4-1/2*b*d^2*log(1+c^2*x^2)/(c*e^3)+1/6*b*log(1+c^2*x^2)/(c^3*e)-1/2*I*b*d^3*polylog(2,1+(-2)/(1-I*c*x))/e^4+1/2*I*b*d^3*polylog(2,1-2*c*(d+e*x)/((c*d+I*e)*(1-I*c*x)))/e^4],
[x^2*(a+b*arctan(c*x))/(d+e*x),x,12,-a*d*x/e^2-1/2*b*x/(c*e)+1/2*b*arctan(c*x)/(c^2*e)-b*d*x*arctan(c*x)/e^2+1/2*x^2*(a+b*arctan(c*x))/e-d^2*(a+b*arctan(c*x))*log(2/(1-I*c*x))/e^3+d^2*(a+b*arctan(c*x))*log(2*c*(d+e*x)/((c*d+I*e)*(1-I*c*x)))/e^3+1/2*b*d*log(1+c^2*x^2)/(c*e^2)+1/2*I*b*d^2*polylog(2,1+(-2)/(1-I*c*x))/e^3-1/2*I*b*d^2*polylog(2,1-2*c*(d+e*x)/((c*d+I*e)*(1-I*c*x)))/e^3],
[x*(a+b*arctan(c*x))/(d+e*x),x,9,a*x/e+b*x*arctan(c*x)/e+d*(a+b*arctan(c*x))*log(2/(1-I*c*x))/e^2-d*(a+b*arctan(c*x))*log(2*c*(d+e*x)/((c*d+I*e)*(1-I*c*x)))/e^2-1/2*b*log(1+c^2*x^2)/(c*e)-1/2*I*b*d*polylog(2,1+(-2)/(1-I*c*x))/e^2+1/2*I*b*d*polylog(2,1-2*c*(d+e*x)/((c*d+I*e)*(1-I*c*x)))/e^2],
[(a+b*arctan(c*x))/(d+e*x),x,4,-(a+b*arctan(c*x))*log(2/(1-I*c*x))/e+(a+b*arctan(c*x))*log(2*c*(d+e*x)/((c*d+I*e)*(1-I*c*x)))/e+1/2*I*b*polylog(2,1+(-2)/(1-I*c*x))/e-1/2*I*b*polylog(2,1-2*c*(d+e*x)/((c*d+I*e)*(1-I*c*x)))/e],
[(a+b*arctan(c*x))/(x*(d+e*x)),x,9,a*log(x)/d+(a+b*arctan(c*x))*log(2/(1-I*c*x))/d-(a+b*arctan(c*x))*log(2*c*(d+e*x)/((c*d+I*e)*(1-I*c*x)))/d+1/2*I*b*polylog(2,-I*c*x)/d-1/2*I*b*polylog(2,I*c*x)/d-1/2*I*b*polylog(2,1+(-2)/(1-I*c*x))/d+1/2*I*b*polylog(2,1-2*c*(d+e*x)/((c*d+I*e)*(1-I*c*x)))/d],
[(a+b*arctan(c*x))/(x^2*(d+e*x)),x,14,(-a-b*arctan(c*x))/(d*x)+b*c*log(x)/d-a*e*log(x)/d^2-e*(a+b*arctan(c*x))*log(2/(1-I*c*x))/d^2+e*(a+b*arctan(c*x))*log(2*c*(d+e*x)/((c*d+I*e)*(1-I*c*x)))/d^2-1/2*b*c*log(1+c^2*x^2)/d-1/2*I*b*e*polylog(2,-I*c*x)/d^2+1/2*I*b*e*polylog(2,I*c*x)/d^2+1/2*I*b*e*polylog(2,1+(-2)/(1-I*c*x))/d^2-1/2*I*b*e*polylog(2,1-2*c*(d+e*x)/((c*d+I*e)*(1-I*c*x)))/d^2],
[(a+b*arctan(c*x))/(x^3*(d+e*x)),x,17,-1/2*b*c/(d*x)-1/2*b*c^2*arctan(c*x)/d+1/2*(-a-b*arctan(c*x))/(d*x^2)+e*(a+b*arctan(c*x))/(d^2*x)-b*c*e*log(x)/d^2+a*e^2*log(x)/d^3+e^2*(a+b*arctan(c*x))*log(2/(1-I*c*x))/d^3-e^2*(a+b*arctan(c*x))*log(2*c*(d+e*x)/((c*d+I*e)*(1-I*c*x)))/d^3+1/2*b*c*e*log(1+c^2*x^2)/d^2+1/2*I*b*e^2*polylog(2,-I*c*x)/d^3-1/2*I*b*e^2*polylog(2,I*c*x)/d^3-1/2*I*b*e^2*polylog(2,1+(-2)/(1-I*c*x))/d^3+1/2*I*b*e^2*polylog(2,1-2*c*(d+e*x)/((c*d+I*e)*(1-I*c*x)))/d^3],
[x^3*(a+b*arctan(c*x))^2/(d+e*x),x,23,a*b*d*x/(c*e^2)+1/3*b^2*x/(c^2*e)-1/3*b^2*arctan(c*x)/(c^3*e)+b^2*d*x*arctan(c*x)/(c*e^2)-1/3*b*x^2*(a+b*arctan(c*x))/(c*e)+I*d^2*(a+b*arctan(c*x))^2/(c*e^3)-1/2*d*(a+b*arctan(c*x))^2/(c^2*e^2)-1/3*I*(a+b*arctan(c*x))^2/(c^3*e)+d^2*x*(a+b*arctan(c*x))^2/e^3-1/2*d*x^2*(a+b*arctan(c*x))^2/e^2+1/3*x^3*(a+b*arctan(c*x))^2/e+d^3*(a+b*arctan(c*x))^2*log(2/(1-I*c*x))/e^4+2*b*d^2*(a+b*arctan(c*x))*log(2/(1+I*c*x))/(c*e^3)-2/3*b*(a+b*arctan(c*x))*log(2/(1+I*c*x))/(c^3*e)-d^3*(a+b*arctan(c*x))^2*log(2*c*(d+e*x)/((c*d+I*e)*(1-I*c*x)))/e^4-1/2*b^2*d*log(1+c^2*x^2)/(c^2*e^2)-I*b*d^3*(a+b*arctan(c*x))*polylog(2,1+(-2)/(1-I*c*x))/e^4+I*b^2*d^2*polylog(2,1+(-2)/(1+I*c*x))/(c*e^3)-1/3*I*b^2*polylog(2,1+(-2)/(1+I*c*x))/(c^3*e)+I*b*d^3*(a+b*arctan(c*x))*polylog(2,1-2*c*(d+e*x)/((c*d+I*e)*(1-I*c*x)))/e^4+1/2*b^2*d^3*polylog(3,1+(-2)/(1-I*c*x))/e^4-1/2*b^2*d^3*polylog(3,1-2*c*(d+e*x)/((c*d+I*e)*(1-I*c*x)))/e^4],
[x^2*(a+b*arctan(c*x))^2/(d+e*x),x,14,-a*b*x/(c*e)-b^2*x*arctan(c*x)/(c*e)-I*d*(a+b*arctan(c*x))^2/(c*e^2)+1/2*(a+b*arctan(c*x))^2/(c^2*e)-d*x*(a+b*arctan(c*x))^2/e^2+1/2*x^2*(a+b*arctan(c*x))^2/e-d^2*(a+b*arctan(c*x))^2*log(2/(1-I*c*x))/e^3-2*b*d*(a+b*arctan(c*x))*log(2/(1+I*c*x))/(c*e^2)+d^2*(a+b*arctan(c*x))^2*log(2*c*(d+e*x)/((c*d+I*e)*(1-I*c*x)))/e^3+1/2*b^2*log(1+c^2*x^2)/(c^2*e)+I*b*d^2*(a+b*arctan(c*x))*polylog(2,1+(-2)/(1-I*c*x))/e^3-I*b^2*d*polylog(2,1+(-2)/(1+I*c*x))/(c*e^2)-I*b*d^2*(a+b*arctan(c*x))*polylog(2,1-2*c*(d+e*x)/((c*d+I*e)*(1-I*c*x)))/e^3-1/2*b^2*d^2*polylog(3,1+(-2)/(1-I*c*x))/e^3+1/2*b^2*d^2*polylog(3,1-2*c*(d+e*x)/((c*d+I*e)*(1-I*c*x)))/e^3],
[x*(a+b*arctan(c*x))^2/(d+e*x),x,8,I*(a+b*arctan(c*x))^2/(c*e)+x*(a+b*arctan(c*x))^2/e+d*(a+b*arctan(c*x))^2*log(2/(1-I*c*x))/e^2+2*b*(a+b*arctan(c*x))*log(2/(1+I*c*x))/(c*e)-d*(a+b*arctan(c*x))^2*log(2*c*(d+e*x)/((c*d+I*e)*(1-I*c*x)))/e^2-I*b*d*(a+b*arctan(c*x))*polylog(2,1+(-2)/(1-I*c*x))/e^2+I*b^2*polylog(2,1+(-2)/(1+I*c*x))/(c*e)+I*b*d*(a+b*arctan(c*x))*polylog(2,1-2*c*(d+e*x)/((c*d+I*e)*(1-I*c*x)))/e^2+1/2*b^2*d*polylog(3,1+(-2)/(1-I*c*x))/e^2-1/2*b^2*d*polylog(3,1-2*c*(d+e*x)/((c*d+I*e)*(1-I*c*x)))/e^2],
[(a+b*arctan(c*x))^2/(d+e*x),x,1,-(a+b*arctan(c*x))^2*log(2/(1-I*c*x))/e+(a+b*arctan(c*x))^2*log(2*c*(d+e*x)/((c*d+I*e)*(1-I*c*x)))/e+I*b*(a+b*arctan(c*x))*polylog(2,1+(-2)/(1-I*c*x))/e-I*b*(a+b*arctan(c*x))*polylog(2,1-2*c*(d+e*x)/((c*d+I*e)*(1-I*c*x)))/e-1/2*b^2*polylog(3,1+(-2)/(1-I*c*x))/e+1/2*b^2*polylog(3,1-2*c*(d+e*x)/((c*d+I*e)*(1-I*c*x)))/e],
[(a+b*arctan(c*x))^2/(x*(d+e*x)),x,9,2*(a+b*arctan(c*x))^2*arctanh(1+(-2)/(1+I*c*x))/d+(a+b*arctan(c*x))^2*log(2/(1-I*c*x))/d-(a+b*arctan(c*x))^2*log(2*c*(d+e*x)/((c*d+I*e)*(1-I*c*x)))/d-I*b*(a+b*arctan(c*x))*polylog(2,1+(-2)/(1-I*c*x))/d-I*b*(a+b*arctan(c*x))*polylog(2,1+(-2)/(1+I*c*x))/d+I*b*(a+b*arctan(c*x))*polylog(2,-1+2/(1+I*c*x))/d+I*b*(a+b*arctan(c*x))*polylog(2,1-2*c*(d+e*x)/((c*d+I*e)*(1-I*c*x)))/d+1/2*b^2*polylog(3,1+(-2)/(1-I*c*x))/d-1/2*b^2*polylog(3,1+(-2)/(1+I*c*x))/d+1/2*b^2*polylog(3,-1+2/(1+I*c*x))/d-1/2*b^2*polylog(3,1-2*c*(d+e*x)/((c*d+I*e)*(1-I*c*x)))/d],
[(a+b*arctan(c*x))^2/(x^2*(d+e*x)),x,13,-I*c*(a+b*arctan(c*x))^2/d-(a+b*arctan(c*x))^2/(d*x)-2*e*(a+b*arctan(c*x))^2*arctanh(1+(-2)/(1+I*c*x))/d^2-e*(a+b*arctan(c*x))^2*log(2/(1-I*c*x))/d^2+e*(a+b*arctan(c*x))^2*log(2*c*(d+e*x)/((c*d+I*e)*(1-I*c*x)))/d^2+2*b*c*(a+b*arctan(c*x))*log(2+(-2)/(1-I*c*x))/d+I*b*e*(a+b*arctan(c*x))*polylog(2,1+(-2)/(1-I*c*x))/d^2-I*b^2*c*polylog(2,-1+2/(1-I*c*x))/d+I*b*e*(a+b*arctan(c*x))*polylog(2,1+(-2)/(1+I*c*x))/d^2-I*b*e*(a+b*arctan(c*x))*polylog(2,-1+2/(1+I*c*x))/d^2-I*b*e*(a+b*arctan(c*x))*polylog(2,1-2*c*(d+e*x)/((c*d+I*e)*(1-I*c*x)))/d^2-1/2*b^2*e*polylog(3,1+(-2)/(1-I*c*x))/d^2+1/2*b^2*e*polylog(3,1+(-2)/(1+I*c*x))/d^2-1/2*b^2*e*polylog(3,-1+2/(1+I*c*x))/d^2+1/2*b^2*e*polylog(3,1-2*c*(d+e*x)/((c*d+I*e)*(1-I*c*x)))/d^2],
[(a+b*arctan(c*x))^2/(x^3*(d+e*x)),x,21,-b*c*(a+b*arctan(c*x))/(d*x)-1/2*c^2*(a+b*arctan(c*x))^2/d+I*c*e*(a+b*arctan(c*x))^2/d^2-1/2*(a+b*arctan(c*x))^2/(d*x^2)+e*(a+b*arctan(c*x))^2/(d^2*x)+2*e^2*(a+b*arctan(c*x))^2*arctanh(1+(-2)/(1+I*c*x))/d^3+b^2*c^2*log(x)/d+e^2*(a+b*arctan(c*x))^2*log(2/(1-I*c*x))/d^3-e^2*(a+b*arctan(c*x))^2*log(2*c*(d+e*x)/((c*d+I*e)*(1-I*c*x)))/d^3-1/2*b^2*c^2*log(1+c^2*x^2)/d-2*b*c*e*(a+b*arctan(c*x))*log(2+(-2)/(1-I*c*x))/d^2-I*b*e^2*(a+b*arctan(c*x))*polylog(2,1+(-2)/(1-I*c*x))/d^3+I*b^2*c*e*polylog(2,-1+2/(1-I*c*x))/d^2-I*b*e^2*(a+b*arctan(c*x))*polylog(2,1+(-2)/(1+I*c*x))/d^3+I*b*e^2*(a+b*arctan(c*x))*polylog(2,-1+2/(1+I*c*x))/d^3+I*b*e^2*(a+b*arctan(c*x))*polylog(2,1-2*c*(d+e*x)/((c*d+I*e)*(1-I*c*x)))/d^3+1/2*b^2*e^2*polylog(3,1+(-2)/(1-I*c*x))/d^3-1/2*b^2*e^2*polylog(3,1+(-2)/(1+I*c*x))/d^3+1/2*b^2*e^2*polylog(3,-1+2/(1+I*c*x))/d^3-1/2*b^2*e^2*polylog(3,1-2*c*(d+e*x)/((c*d+I*e)*(1-I*c*x)))/d^3],

# p<0
[1/((d+e*x)*(a+b*arctan(c*x))),x,0,Unintegrable(1/((d+e*x)*(a+b*arctan(c*x))),x)],

# Integrands of the form (f x)^m (d+e x^2)^q (a+b ArcTan[c x])^p with e=c^2 d

# Integrands of the form (f x)^m (d+c^2 d x^2)^q (a+b ArcTan[c x])^1

# Integrands of the form x^m (c+d x^2)^p ArcTan[a x]^1 with d=a^2 c

# p>0
[x^3*(c+a^2*c*x^2)*arctan(a*x),x,9,1/12*c*x/a^3-1/36*c*x^3/a-1/30*a*c*x^5-1/12*c*arctan(a*x)/a^4+1/4*c*x^4*arctan(a*x)+1/6*a^2*c*x^6*arctan(a*x)],
[x^2*(c+a^2*c*x^2)*arctan(a*x),x,9,-1/15*c*x^2/a-1/20*a*c*x^4+1/3*c*x^3*arctan(a*x)+1/5*a^2*c*x^5*arctan(a*x)+1/15*c*log(1+a^2*x^2)/a^3],
[x*(c+a^2*c*x^2)*arctan(a*x),x,2,-1/4*c*x/a-1/12*a*c*x^3+1/4*c*(1+a^2*x^2)^2*arctan(a*x)/a^2],
[(c+a^2*c*x^2)*arctan(a*x),x,3,-1/6*a*c*x^2+c*x*arctan(a*x)+1/3*a^2*c*x^3*arctan(a*x)-1/3*c*log(1+a^2*x^2)/a,-1/6*c*(1+a^2*x^2)/a+2/3*c*x*arctan(a*x)+1/3*c*x*(1+a^2*x^2)*arctan(a*x)-1/3*c*log(1+a^2*x^2)/a],
[(c+a^2*c*x^2)*arctan(a*x)/x,x,7,-1/2*a*c*x+1/2*c*arctan(a*x)+1/2*a^2*c*x^2*arctan(a*x)+1/2*I*c*polylog(2,-I*a*x)-1/2*I*c*polylog(2,I*a*x)],
[(c+a^2*c*x^2)*arctan(a*x)/x^2,x,8,-c*arctan(a*x)/x+a^2*c*x*arctan(a*x)+a*c*log(x)-a*c*log(1+a^2*x^2)],
[(c+a^2*c*x^2)*arctan(a*x)/x^3,x,7,-1/2*a*c/x-1/2*a^2*c*arctan(a*x)-1/2*c*arctan(a*x)/x^2+1/2*I*a^2*c*polylog(2,-I*a*x)-1/2*I*a^2*c*polylog(2,I*a*x)],
[(c+a^2*c*x^2)*arctan(a*x)/x^4,x,10,-1/6*a*c/x^2-1/3*c*arctan(a*x)/x^3-a^2*c*arctan(a*x)/x+2/3*a^3*c*log(x)-1/3*a^3*c*log(1+a^2*x^2)],
[x^3*(c+a^2*c*x^2)^2*arctan(a*x),x,14,1/24*c^2*x/a^3-1/72*c^2*x^3/a-1/24*a*c^2*x^5-1/56*a^3*c^2*x^7-1/24*c^2*arctan(a*x)/a^4+1/4*c^2*x^4*arctan(a*x)+1/3*a^2*c^2*x^6*arctan(a*x)+1/8*a^4*c^2*x^8*arctan(a*x)],
[x^2*(c+a^2*c*x^2)^2*arctan(a*x),x,14,-4/105*c^2*x^2/a-9/140*a*c^2*x^4-1/42*a^3*c^2*x^6+1/3*c^2*x^3*arctan(a*x)+2/5*a^2*c^2*x^5*arctan(a*x)+1/7*a^4*c^2*x^7*arctan(a*x)+4/105*c^2*log(1+a^2*x^2)/a^3],
[x*(c+a^2*c*x^2)^2*arctan(a*x),x,3,-1/6*c^2*x/a-1/9*a*c^2*x^3-1/30*a^3*c^2*x^5+1/6*c^2*(1+a^2*x^2)^3*arctan(a*x)/a^2],
[(c+a^2*c*x^2)^2*arctan(a*x),x,4,-2/15*c^2*(1+a^2*x^2)/a-1/20*c^2*(1+a^2*x^2)^2/a+8/15*c^2*x*arctan(a*x)+4/15*c^2*x*(1+a^2*x^2)*arctan(a*x)+1/5*c^2*x*(1+a^2*x^2)^2*arctan(a*x)-4/15*c^2*log(1+a^2*x^2)/a],
[(c+a^2*c*x^2)^2*arctan(a*x)/x,x,12,-3/4*a*c^2*x-1/12*a^3*c^2*x^3+3/4*c^2*arctan(a*x)+a^2*c^2*x^2*arctan(a*x)+1/4*a^4*c^2*x^4*arctan(a*x)+1/2*I*c^2*polylog(2,-I*a*x)-1/2*I*c^2*polylog(2,I*a*x)],
[(c+a^2*c*x^2)^2*arctan(a*x)/x^2,x,13,-1/6*a^3*c^2*x^2-c^2*arctan(a*x)/x+2*a^2*c^2*x*arctan(a*x)+1/3*a^4*c^2*x^3*arctan(a*x)+a*c^2*log(x)-4/3*a*c^2*log(1+a^2*x^2)],
[(c+a^2*c*x^2)^2*arctan(a*x)/x^3,x,11,-1/2*a*c^2/x-1/2*a^3*c^2*x-1/2*c^2*arctan(a*x)/x^2+1/2*a^4*c^2*x^2*arctan(a*x)+I*a^2*c^2*polylog(2,-I*a*x)-I*a^2*c^2*polylog(2,I*a*x)],
[(c+a^2*c*x^2)^2*arctan(a*x)/x^4,x,13,-1/6*a*c^2/x^2-1/3*c^2*arctan(a*x)/x^3-2*a^2*c^2*arctan(a*x)/x+a^4*c^2*x*arctan(a*x)+5/3*a^3*c^2*log(x)-4/3*a^3*c^2*log(1+a^2*x^2)],
[x^3*(c+a^2*c*x^2)^3*arctan(a*x),x,18,1/40*c^3*x/a^3-1/120*c^3*x^3/a-9/200*a*c^3*x^5-11/280*a^3*c^3*x^7-1/90*a^5*c^3*x^9-1/40*c^3*arctan(a*x)/a^4+1/4*c^3*x^4*arctan(a*x)+1/2*a^2*c^3*x^6*arctan(a*x)+3/8*a^4*c^3*x^8*arctan(a*x)+1/10*a^6*c^3*x^10*arctan(a*x)],
[x^2*(c+a^2*c*x^2)^3*arctan(a*x),x,18,-8/315*c^3*x^2/a-89/1260*a*c^3*x^4-10/189*a^3*c^3*x^6-1/72*a^5*c^3*x^8+1/3*c^3*x^3*arctan(a*x)+3/5*a^2*c^3*x^5*arctan(a*x)+3/7*a^4*c^3*x^7*arctan(a*x)+1/9*a^6*c^3*x^9*arctan(a*x)+8/315*c^3*log(1+a^2*x^2)/a^3],
[x*(c+a^2*c*x^2)^3*arctan(a*x),x,3,-1/8*c^3*x/a-1/8*a*c^3*x^3-3/40*a^3*c^3*x^5-1/56*a^5*c^3*x^7+1/8*c^3*(1+a^2*x^2)^4*arctan(a*x)/a^2],
[(c+a^2*c*x^2)^3*arctan(a*x),x,5,-4/35*c^3*(1+a^2*x^2)/a-3/70*c^3*(1+a^2*x^2)^2/a-1/42*c^3*(1+a^2*x^2)^3/a+16/35*c^3*x*arctan(a*x)+8/35*c^3*x*(1+a^2*x^2)*arctan(a*x)+6/35*c^3*x*(1+a^2*x^2)^2*arctan(a*x)+1/7*c^3*x*(1+a^2*x^2)^3*arctan(a*x)-8/35*c^3*log(1+a^2*x^2)/a],
[(c+a^2*c*x^2)^3*arctan(a*x)/x,x,16,-11/12*a*c^3*x-7/36*a^3*c^3*x^3-1/30*a^5*c^3*x^5+11/12*c^3*arctan(a*x)+3/2*a^2*c^3*x^2*arctan(a*x)+3/4*a^4*c^3*x^4*arctan(a*x)+1/6*a^6*c^3*x^6*arctan(a*x)+1/2*I*c^3*polylog(2,-I*a*x)-1/2*I*c^3*polylog(2,I*a*x)],
[(c+a^2*c*x^2)^3*arctan(a*x)/x^2,x,17,-2/5*a^3*c^3*x^2-1/20*a^5*c^3*x^4-c^3*arctan(a*x)/x+3*a^2*c^3*x*arctan(a*x)+a^4*c^3*x^3*arctan(a*x)+1/5*a^6*c^3*x^5*arctan(a*x)+a*c^3*log(x)-8/5*a*c^3*log(1+a^2*x^2)],
[(c+a^2*c*x^2)^3*arctan(a*x)/x^3,x,15,-1/2*a*c^3/x-5/4*a^3*c^3*x-1/12*a^5*c^3*x^3+3/4*a^2*c^3*arctan(a*x)-1/2*c^3*arctan(a*x)/x^2+3/2*a^4*c^3*x^2*arctan(a*x)+1/4*a^6*c^3*x^4*arctan(a*x)+3/2*I*a^2*c^3*polylog(2,-I*a*x)-3/2*I*a^2*c^3*polylog(2,I*a*x)],
[(c+a^2*c*x^2)^3*arctan(a*x)/x^4,x,17,-1/6*a*c^3/x^2-1/6*a^5*c^3*x^2-1/3*c^3*arctan(a*x)/x^3-3*a^2*c^3*arctan(a*x)/x+3*a^4*c^3*x*arctan(a*x)+1/3*a^6*c^3*x^3*arctan(a*x)+8/3*a^3*c^3*log(x)-8/3*a^3*c^3*log(1+a^2*x^2)],

# p<0
[x^4*arctan(a*x)/(c+a^2*c*x^2),x,9,-1/6*x^2/(a^3*c)-x*arctan(a*x)/(a^4*c)+1/3*x^3*arctan(a*x)/(a^2*c)+1/2*arctan(a*x)^2/(a^5*c)+2/3*log(1+a^2*x^2)/(a^5*c)],
[x^3*arctan(a*x)/(c+a^2*c*x^2),x,8,-1/2*x/(a^3*c)+1/2*arctan(a*x)/(a^4*c)+1/2*x^2*arctan(a*x)/(a^2*c)+1/2*I*arctan(a*x)^2/(a^4*c)+arctan(a*x)*log(2/(1+I*a*x))/(a^4*c)+1/2*I*polylog(2,1+(-2)/(1+I*a*x))/(a^4*c)],
[x^2*arctan(a*x)/(c+a^2*c*x^2),x,4,x*arctan(a*x)/(a^2*c)-1/2*arctan(a*x)^2/(a^3*c)-1/2*log(1+a^2*x^2)/(a^3*c)],
[x*arctan(a*x)/(c+a^2*c*x^2),x,4,-1/2*I*arctan(a*x)^2/(a^2*c)-arctan(a*x)*log(2/(1+I*a*x))/(a^2*c)-1/2*I*polylog(2,1+(-2)/(1+I*a*x))/(a^2*c)],
[arctan(a*x)/(c+a^2*c*x^2),x,1,1/2*arctan(a*x)^2/(a*c)],
[arctan(a*x)/(x*(c+a^2*c*x^2)),x,3,-1/2*I*arctan(a*x)^2/c+arctan(a*x)*log(2+(-2)/(1-I*a*x))/c-1/2*I*polylog(2,-1+2/(1-I*a*x))/c],
[arctan(a*x)/(x^2*(c+a^2*c*x^2)),x,7,-arctan(a*x)/(c*x)-1/2*a*arctan(a*x)^2/c+a*log(x)/c-1/2*a*log(1+a^2*x^2)/c],
[arctan(a*x)/(x^3*(c+a^2*c*x^2)),x,7,-1/2*a/(c*x)-1/2*a^2*arctan(a*x)/c-1/2*arctan(a*x)/(c*x^2)+1/2*I*a^2*arctan(a*x)^2/c-a^2*arctan(a*x)*log(2+(-2)/(1-I*a*x))/c+1/2*I*a^2*polylog(2,-1+2/(1-I*a*x))/c],
[arctan(a*x)/(x^4*(c+a^2*c*x^2)),x,12,-1/6*a/(c*x^2)-1/3*arctan(a*x)/(c*x^3)+a^2*arctan(a*x)/(c*x)+1/2*a^3*arctan(a*x)^2/c-4/3*a^3*log(x)/c+2/3*a^3*log(1+a^2*x^2)/c],
[x^5*arctan(a*x)/(c+a^2*c*x^2)^2,x,17,-1/2*x/(a^5*c^2)+1/4*x/(a^5*c^2*(1+a^2*x^2))+3/4*arctan(a*x)/(a^6*c^2)+1/2*x^2*arctan(a*x)/(a^4*c^2)-1/2*arctan(a*x)/(a^6*c^2*(1+a^2*x^2))+I*arctan(a*x)^2/(a^6*c^2)+2*arctan(a*x)*log(2/(1+I*a*x))/(a^6*c^2)+I*polylog(2,1+(-2)/(1+I*a*x))/(a^6*c^2)],
[x^4*arctan(a*x)/(c+a^2*c*x^2)^2,x,7,1/4/(a^5*c^2*(1+a^2*x^2))+x*arctan(a*x)/(a^4*c^2)+1/2*x*arctan(a*x)/(a^4*c^2*(1+a^2*x^2))-3/4*arctan(a*x)^2/(a^5*c^2)-1/2*log(1+a^2*x^2)/(a^5*c^2)],
[x^3*arctan(a*x)/(c+a^2*c*x^2)^2,x,8,-1/4*x/(a^3*c^2*(1+a^2*x^2))-1/4*arctan(a*x)/(a^4*c^2)+1/2*arctan(a*x)/(a^4*c^2*(1+a^2*x^2))-1/2*I*arctan(a*x)^2/(a^4*c^2)-arctan(a*x)*log(2/(1+I*a*x))/(a^4*c^2)-1/2*I*polylog(2,1+(-2)/(1+I*a*x))/(a^4*c^2)],
[x^2*arctan(a*x)/(c+a^2*c*x^2)^2,x,2,(-1/4)/(a^3*c^2*(1+a^2*x^2))-1/2*x*arctan(a*x)/(a^2*c^2*(1+a^2*x^2))+1/4*arctan(a*x)^2/(a^3*c^2)],
[x*arctan(a*x)/(c+a^2*c*x^2)^2,x,3,1/4*x/(a*c^2*(1+a^2*x^2))+1/4*arctan(a*x)/(a^2*c^2)-1/2*arctan(a*x)/(a^2*c^2*(1+a^2*x^2))],
[arctan(a*x)/(c+a^2*c*x^2)^2,x,2,1/4/(a*c^2*(1+a^2*x^2))+1/2*x*arctan(a*x)/(c^2*(1+a^2*x^2))+1/4*arctan(a*x)^2/(a*c^2)],
[arctan(a*x)/(x*(c+a^2*c*x^2)^2),x,7,-1/4*a*x/(c^2*(1+a^2*x^2))-1/4*arctan(a*x)/c^2+1/2*arctan(a*x)/(c^2*(1+a^2*x^2))-1/2*I*arctan(a*x)^2/c^2+arctan(a*x)*log(2+(-2)/(1-I*a*x))/c^2-1/2*I*polylog(2,-1+2/(1-I*a*x))/c^2],
[arctan(a*x)/(x^2*(c+a^2*c*x^2)^2),x,10,-1/4*a/(c^2*(1+a^2*x^2))-arctan(a*x)/(c^2*x)-1/2*a^2*x*arctan(a*x)/(c^2*(1+a^2*x^2))-3/4*a*arctan(a*x)^2/c^2+a*log(x)/c^2-1/2*a*log(1+a^2*x^2)/c^2],
[arctan(a*x)/(x^3*(c+a^2*c*x^2)^2),x,15,-1/2*a/(c^2*x)+1/4*a^3*x/(c^2*(1+a^2*x^2))-1/4*a^2*arctan(a*x)/c^2-1/2*arctan(a*x)/(c^2*x^2)-1/2*a^2*arctan(a*x)/(c^2*(1+a^2*x^2))+I*a^2*arctan(a*x)^2/c^2-2*a^2*arctan(a*x)*log(2+(-2)/(1-I*a*x))/c^2+I*a^2*polylog(2,-1+2/(1-I*a*x))/c^2],
[arctan(a*x)/(x^4*(c+a^2*c*x^2)^2),x,23,-1/6*a/(c^2*x^2)+1/4*a^3/(c^2*(1+a^2*x^2))-1/3*arctan(a*x)/(c^2*x^3)+2*a^2*arctan(a*x)/(c^2*x)+1/2*a^4*x*arctan(a*x)/(c^2*(1+a^2*x^2))+5/4*a^3*arctan(a*x)^2/c^2-7/3*a^3*log(x)/c^2+7/6*a^3*log(1+a^2*x^2)/c^2],
[x^3*arctan(a*x)/(c+a^2*c*x^2)^3,x,4,1/16*x^3/(a*c^3*(1+a^2*x^2)^2)+3/32*x/(a^3*c^3*(1+a^2*x^2))-3/32*arctan(a*x)/(a^4*c^3)+1/4*x^4*arctan(a*x)/(c^3*(1+a^2*x^2)^2)],
[x^2*arctan(a*x)/(c+a^2*c*x^2)^3,x,3,(-1/16)/(a^3*c^3*(1+a^2*x^2)^2)+1/16/(a^3*c^3*(1+a^2*x^2))-1/4*x*arctan(a*x)/(a^2*c^3*(1+a^2*x^2)^2)+1/8*x*arctan(a*x)/(a^2*c^3*(1+a^2*x^2))+1/16*arctan(a*x)^2/(a^3*c^3)],
[x*arctan(a*x)/(c+a^2*c*x^2)^3,x,4,1/16*x/(a*c^3*(1+a^2*x^2)^2)+3/32*x/(a*c^3*(1+a^2*x^2))+3/32*arctan(a*x)/(a^2*c^3)-1/4*arctan(a*x)/(a^2*c^3*(1+a^2*x^2)^2)],
[arctan(a*x)/(c+a^2*c*x^2)^3,x,3,1/16/(a*c^3*(1+a^2*x^2)^2)+3/16/(a*c^3*(1+a^2*x^2))+1/4*x*arctan(a*x)/(c^3*(1+a^2*x^2)^2)+3/8*x*arctan(a*x)/(c^3*(1+a^2*x^2))+3/16*arctan(a*x)^2/(a*c^3)],
[arctan(a*x)/(x*(c+a^2*c*x^2)^3),x,12,-1/16*a*x/(c^3*(1+a^2*x^2)^2)-11/32*a*x/(c^3*(1+a^2*x^2))-11/32*arctan(a*x)/c^3+1/4*arctan(a*x)/(c^3*(1+a^2*x^2)^2)+1/2*arctan(a*x)/(c^3*(1+a^2*x^2))-1/2*I*arctan(a*x)^2/c^3+arctan(a*x)*log(2+(-2)/(1-I*a*x))/c^3-1/2*I*polylog(2,-1+2/(1-I*a*x))/c^3],
[arctan(a*x)/(x^2*(c+a^2*c*x^2)^3),x,14,-1/16*a/(c^3*(1+a^2*x^2)^2)-7/16*a/(c^3*(1+a^2*x^2))-arctan(a*x)/(c^3*x)-1/4*a^2*x*arctan(a*x)/(c^3*(1+a^2*x^2)^2)-7/8*a^2*x*arctan(a*x)/(c^3*(1+a^2*x^2))-15/16*a*arctan(a*x)^2/c^3+a*log(x)/c^3-1/2*a*log(1+a^2*x^2)/c^3],
[arctan(a*x)/(x^3*(c+a^2*c*x^2)^3),x,28,-1/2*a/(c^3*x)+1/16*a^3*x/(c^3*(1+a^2*x^2)^2)+19/32*a^3*x/(c^3*(1+a^2*x^2))+3/32*a^2*arctan(a*x)/c^3-1/2*arctan(a*x)/(c^3*x^2)-1/4*a^2*arctan(a*x)/(c^3*(1+a^2*x^2)^2)-a^2*arctan(a*x)/(c^3*(1+a^2*x^2))+3/2*I*a^2*arctan(a*x)^2/c^3-3*a^2*arctan(a*x)*log(2+(-2)/(1-I*a*x))/c^3+3/2*I*a^2*polylog(2,-1+2/(1-I*a*x))/c^3],
[arctan(a*x)/(x^4*(c+a^2*c*x^2)^3),x,38,-1/6*a/(c^3*x^2)+1/16*a^3/(c^3*(1+a^2*x^2)^2)+11/16*a^3/(c^3*(1+a^2*x^2))-1/3*arctan(a*x)/(c^3*x^3)+3*a^2*arctan(a*x)/(c^3*x)+1/4*a^4*x*arctan(a*x)/(c^3*(1+a^2*x^2)^2)+11/8*a^4*x*arctan(a*x)/(c^3*(1+a^2*x^2))+35/16*a^3*arctan(a*x)^2/c^3-10/3*a^3*log(x)/c^3+5/3*a^3*log(1+a^2*x^2)/c^3],

# Integrands of the form x^m (c+d x^2)^(p/2) ArcTan[a x]^1 with d=a^2 c

# p>0
[x^3*arctan(a*x)*sqrt(c+a^2*c*x^2),x,12,11/120*arctanh(a*x*sqrt(c)/sqrt(c+a^2*c*x^2))*sqrt(c)/a^4+1/24*x*sqrt(c+a^2*c*x^2)/a^3-1/20*x^3*sqrt(c+a^2*c*x^2)/a-2/15*arctan(a*x)*sqrt(c+a^2*c*x^2)/a^4+1/15*x^2*arctan(a*x)*sqrt(c+a^2*c*x^2)/a^2+1/5*x^4*arctan(a*x)*sqrt(c+a^2*c*x^2)],
[x^2*arctan(a*x)*sqrt(c+a^2*c*x^2),x,8,-1/12*(c+a^2*c*x^2)^(3/2)/(a^3*c)+1/4*I*c*arctan(a*x)*arctan(sqrt(1+I*a*x)/sqrt(1-I*a*x))*sqrt(1+a^2*x^2)/(a^3*sqrt(c+a^2*c*x^2))-1/8*I*c*polylog(2,-I*sqrt(1+I*a*x)/sqrt(1-I*a*x))*sqrt(1+a^2*x^2)/(a^3*sqrt(c+a^2*c*x^2))+1/8*I*c*polylog(2,I*sqrt(1+I*a*x)/sqrt(1-I*a*x))*sqrt(1+a^2*x^2)/(a^3*sqrt(c+a^2*c*x^2))+1/8*sqrt(c+a^2*c*x^2)/a^3+1/8*x*arctan(a*x)*sqrt(c+a^2*c*x^2)/a^2+1/4*x^3*arctan(a*x)*sqrt(c+a^2*c*x^2)],
[x*arctan(a*x)*sqrt(c+a^2*c*x^2),x,4,1/3*(c+a^2*c*x^2)^(3/2)*arctan(a*x)/(a^2*c)-1/6*arctanh(a*x*sqrt(c)/sqrt(c+a^2*c*x^2))*sqrt(c)/a^2-1/6*x*sqrt(c+a^2*c*x^2)/a],
[arctan(a*x)*sqrt(c+a^2*c*x^2),x,3,-I*c*arctan(a*x)*arctan(sqrt(1+I*a*x)/sqrt(1-I*a*x))*sqrt(1+a^2*x^2)/(a*sqrt(c+a^2*c*x^2))+1/2*I*c*polylog(2,-I*sqrt(1+I*a*x)/sqrt(1-I*a*x))*sqrt(1+a^2*x^2)/(a*sqrt(c+a^2*c*x^2))-1/2*I*c*polylog(2,I*sqrt(1+I*a*x)/sqrt(1-I*a*x))*sqrt(1+a^2*x^2)/(a*sqrt(c+a^2*c*x^2))-1/2*sqrt(c+a^2*c*x^2)/a+1/2*x*arctan(a*x)*sqrt(c+a^2*c*x^2)],
[arctan(a*x)*sqrt(c+a^2*c*x^2)/x,x,5,-arctanh(a*x*sqrt(c)/sqrt(c+a^2*c*x^2))*sqrt(c)-2*c*arctan(a*x)*arctanh(sqrt(1+I*a*x)/sqrt(1-I*a*x))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)+I*c*polylog(2,-sqrt(1+I*a*x)/sqrt(1-I*a*x))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)-I*c*polylog(2,sqrt(1+I*a*x)/sqrt(1-I*a*x))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)+arctan(a*x)*sqrt(c+a^2*c*x^2)],
[arctan(a*x)*sqrt(c+a^2*c*x^2)/x^2,x,7,-a*arctanh(sqrt(c+a^2*c*x^2)/sqrt(c))*sqrt(c)-2*I*a*c*arctan(a*x)*arctan(sqrt(1+I*a*x)/sqrt(1-I*a*x))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)+I*a*c*polylog(2,-I*sqrt(1+I*a*x)/sqrt(1-I*a*x))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)-I*a*c*polylog(2,I*sqrt(1+I*a*x)/sqrt(1-I*a*x))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)-arctan(a*x)*sqrt(c+a^2*c*x^2)/x],
[arctan(a*x)*sqrt(c+a^2*c*x^2)/x^3,x,6,-a^2*c*arctan(a*x)*arctanh(sqrt(1+I*a*x)/sqrt(1-I*a*x))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)+1/2*I*a^2*c*polylog(2,-sqrt(1+I*a*x)/sqrt(1-I*a*x))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)-1/2*I*a^2*c*polylog(2,sqrt(1+I*a*x)/sqrt(1-I*a*x))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)-1/2*a*sqrt(c+a^2*c*x^2)/x-1/2*arctan(a*x)*sqrt(c+a^2*c*x^2)/x^2],
[arctan(a*x)*sqrt(c+a^2*c*x^2)/x^4,x,5,-1/3*(c+a^2*c*x^2)^(3/2)*arctan(a*x)/(c*x^3)-1/6*a^3*arctanh(sqrt(c+a^2*c*x^2)/sqrt(c))*sqrt(c)-1/6*a*sqrt(c+a^2*c*x^2)/x^2],
[x^3*(c+a^2*c*x^2)^(3/2)*arctan(a*x),x,31,17/560*c^(3/2)*arctanh(a*x*sqrt(c)/sqrt(c+a^2*c*x^2))/a^4+3/112*c*x*sqrt(c+a^2*c*x^2)/a^3-23/840*c*x^3*sqrt(c+a^2*c*x^2)/a-1/42*a*c*x^5*sqrt(c+a^2*c*x^2)-2/35*c*arctan(a*x)*sqrt(c+a^2*c*x^2)/a^4+1/35*c*x^2*arctan(a*x)*sqrt(c+a^2*c*x^2)/a^2+8/35*c*x^4*arctan(a*x)*sqrt(c+a^2*c*x^2)+1/7*a^2*c*x^6*arctan(a*x)*sqrt(c+a^2*c*x^2)],
[x^2*(c+a^2*c*x^2)^(3/2)*arctan(a*x),x,21,1/72*(c+a^2*c*x^2)^(3/2)/a^3-1/30*(c+a^2*c*x^2)^(5/2)/(a^3*c)+1/8*I*c^2*arctan(a*x)*arctan(sqrt(1+I*a*x)/sqrt(1-I*a*x))*sqrt(1+a^2*x^2)/(a^3*sqrt(c+a^2*c*x^2))-1/16*I*c^2*polylog(2,-I*sqrt(1+I*a*x)/sqrt(1-I*a*x))*sqrt(1+a^2*x^2)/(a^3*sqrt(c+a^2*c*x^2))+1/16*I*c^2*polylog(2,I*sqrt(1+I*a*x)/sqrt(1-I*a*x))*sqrt(1+a^2*x^2)/(a^3*sqrt(c+a^2*c*x^2))+1/16*c*sqrt(c+a^2*c*x^2)/a^3+1/16*c*x*arctan(a*x)*sqrt(c+a^2*c*x^2)/a^2+7/24*c*x^3*arctan(a*x)*sqrt(c+a^2*c*x^2)+1/6*a^2*c*x^5*arctan(a*x)*sqrt(c+a^2*c*x^2)],
[x*(c+a^2*c*x^2)^(3/2)*arctan(a*x),x,5,-1/20*x*(c+a^2*c*x^2)^(3/2)/a+1/5*(c+a^2*c*x^2)^(5/2)*arctan(a*x)/(a^2*c)-3/40*c^(3/2)*arctanh(a*x*sqrt(c)/sqrt(c+a^2*c*x^2))/a^2-3/40*c*x*sqrt(c+a^2*c*x^2)/a],
[(c+a^2*c*x^2)^(3/2)*arctan(a*x),x,4,-1/12*(c+a^2*c*x^2)^(3/2)/a+1/4*x*(c+a^2*c*x^2)^(3/2)*arctan(a*x)-3/4*I*c^2*arctan(a*x)*arctan(sqrt(1+I*a*x)/sqrt(1-I*a*x))*sqrt(1+a^2*x^2)/(a*sqrt(c+a^2*c*x^2))+3/8*I*c^2*polylog(2,-I*sqrt(1+I*a*x)/sqrt(1-I*a*x))*sqrt(1+a^2*x^2)/(a*sqrt(c+a^2*c*x^2))-3/8*I*c^2*polylog(2,I*sqrt(1+I*a*x)/sqrt(1-I*a*x))*sqrt(1+a^2*x^2)/(a*sqrt(c+a^2*c*x^2))-3/8*c*sqrt(c+a^2*c*x^2)/a+3/8*c*x*arctan(a*x)*sqrt(c+a^2*c*x^2)],
[(c+a^2*c*x^2)^(3/2)*arctan(a*x)/x,x,10,1/3*(c+a^2*c*x^2)^(3/2)*arctan(a*x)-7/6*c^(3/2)*arctanh(a*x*sqrt(c)/sqrt(c+a^2*c*x^2))-2*c^2*arctan(a*x)*arctanh(sqrt(1+I*a*x)/sqrt(1-I*a*x))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)+I*c^2*polylog(2,-sqrt(1+I*a*x)/sqrt(1-I*a*x))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)-I*c^2*polylog(2,sqrt(1+I*a*x)/sqrt(1-I*a*x))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)-1/6*a*c*x*sqrt(c+a^2*c*x^2)+c*arctan(a*x)*sqrt(c+a^2*c*x^2)],
[(c+a^2*c*x^2)^(3/2)*arctan(a*x)/x^2,x,11,-a*c^(3/2)*arctanh(sqrt(c+a^2*c*x^2)/sqrt(c))-3*I*a*c^2*arctan(a*x)*arctan(sqrt(1+I*a*x)/sqrt(1-I*a*x))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)+3/2*I*a*c^2*polylog(2,-I*sqrt(1+I*a*x)/sqrt(1-I*a*x))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)-3/2*I*a*c^2*polylog(2,I*sqrt(1+I*a*x)/sqrt(1-I*a*x))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)-1/2*a*c*sqrt(c+a^2*c*x^2)-c*arctan(a*x)*sqrt(c+a^2*c*x^2)/x+1/2*a^2*c*x*arctan(a*x)*sqrt(c+a^2*c*x^2)],
[(c+a^2*c*x^2)^(3/2)*arctan(a*x)/x^3,x,12,-a^2*c^(3/2)*arctanh(a*x*sqrt(c)/sqrt(c+a^2*c*x^2))-3*a^2*c^2*arctan(a*x)*arctanh(sqrt(1+I*a*x)/sqrt(1-I*a*x))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)+3/2*I*a^2*c^2*polylog(2,-sqrt(1+I*a*x)/sqrt(1-I*a*x))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)-3/2*I*a^2*c^2*polylog(2,sqrt(1+I*a*x)/sqrt(1-I*a*x))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)-1/2*a*c*sqrt(c+a^2*c*x^2)/x+a^2*c*arctan(a*x)*sqrt(c+a^2*c*x^2)-1/2*c*arctan(a*x)*sqrt(c+a^2*c*x^2)/x^2],
[(c+a^2*c*x^2)^(3/2)*arctan(a*x)/x^4,x,13,-1/3*(c+a^2*c*x^2)^(3/2)*arctan(a*x)/x^3-7/6*a^3*c^(3/2)*arctanh(sqrt(c+a^2*c*x^2)/sqrt(c))-2*I*a^3*c^2*arctan(a*x)*arctan(sqrt(1+I*a*x)/sqrt(1-I*a*x))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)+I*a^3*c^2*polylog(2,-I*sqrt(1+I*a*x)/sqrt(1-I*a*x))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)-I*a^3*c^2*polylog(2,I*sqrt(1+I*a*x)/sqrt(1-I*a*x))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)-1/6*a*c*sqrt(c+a^2*c*x^2)/x^2-a^2*c*arctan(a*x)*sqrt(c+a^2*c*x^2)/x],
[x^3*(c+a^2*c*x^2)^(5/2)*arctan(a*x),x,76,115/8064*c^(5/2)*arctanh(a*x*sqrt(c)/sqrt(c+a^2*c*x^2))/a^4+47/2688*c^2*x*sqrt(c+a^2*c*x^2)/a^3-205/12096*c^2*x^3*sqrt(c+a^2*c*x^2)/a-103/3024*a*c^2*x^5*sqrt(c+a^2*c*x^2)-1/72*a^3*c^2*x^7*sqrt(c+a^2*c*x^2)-2/63*c^2*arctan(a*x)*sqrt(c+a^2*c*x^2)/a^4+1/63*c^2*x^2*arctan(a*x)*sqrt(c+a^2*c*x^2)/a^2+5/21*c^2*x^4*arctan(a*x)*sqrt(c+a^2*c*x^2)+19/63*a^2*c^2*x^6*arctan(a*x)*sqrt(c+a^2*c*x^2)+1/9*a^4*c^2*x^8*arctan(a*x)*sqrt(c+a^2*c*x^2)],
[x^2*(c+a^2*c*x^2)^(5/2)*arctan(a*x),x,51,5/576*c*(c+a^2*c*x^2)^(3/2)/a^3+1/240*(c+a^2*c*x^2)^(5/2)/a^3-1/56*(c+a^2*c*x^2)^(7/2)/(a^3*c)+5/64*I*c^3*arctan(a*x)*arctan(sqrt(1+I*a*x)/sqrt(1-I*a*x))*sqrt(1+a^2*x^2)/(a^3*sqrt(c+a^2*c*x^2))-5/128*I*c^3*polylog(2,-I*sqrt(1+I*a*x)/sqrt(1-I*a*x))*sqrt(1+a^2*x^2)/(a^3*sqrt(c+a^2*c*x^2))+5/128*I*c^3*polylog(2,I*sqrt(1+I*a*x)/sqrt(1-I*a*x))*sqrt(1+a^2*x^2)/(a^3*sqrt(c+a^2*c*x^2))+5/128*c^2*sqrt(c+a^2*c*x^2)/a^3+5/128*c^2*x*arctan(a*x)*sqrt(c+a^2*c*x^2)/a^2+59/192*c^2*x^3*arctan(a*x)*sqrt(c+a^2*c*x^2)+17/48*a^2*c^2*x^5*arctan(a*x)*sqrt(c+a^2*c*x^2)+1/8*a^4*c^2*x^7*arctan(a*x)*sqrt(c+a^2*c*x^2)],
[x*(c+a^2*c*x^2)^(5/2)*arctan(a*x),x,6,-5/168*c*x*(c+a^2*c*x^2)^(3/2)/a-1/42*x*(c+a^2*c*x^2)^(5/2)/a+1/7*(c+a^2*c*x^2)^(7/2)*arctan(a*x)/(a^2*c)-5/112*c^(5/2)*arctanh(a*x*sqrt(c)/sqrt(c+a^2*c*x^2))/a^2-5/112*c^2*x*sqrt(c+a^2*c*x^2)/a],
[(c+a^2*c*x^2)^(5/2)*arctan(a*x),x,5,-5/72*c*(c+a^2*c*x^2)^(3/2)/a-1/30*(c+a^2*c*x^2)^(5/2)/a+5/24*c*x*(c+a^2*c*x^2)^(3/2)*arctan(a*x)+1/6*x*(c+a^2*c*x^2)^(5/2)*arctan(a*x)-5/8*I*c^3*arctan(a*x)*arctan(sqrt(1+I*a*x)/sqrt(1-I*a*x))*sqrt(1+a^2*x^2)/(a*sqrt(c+a^2*c*x^2))+5/16*I*c^3*polylog(2,-I*sqrt(1+I*a*x)/sqrt(1-I*a*x))*sqrt(1+a^2*x^2)/(a*sqrt(c+a^2*c*x^2))-5/16*I*c^3*polylog(2,I*sqrt(1+I*a*x)/sqrt(1-I*a*x))*sqrt(1+a^2*x^2)/(a*sqrt(c+a^2*c*x^2))-5/16*c^2*sqrt(c+a^2*c*x^2)/a+5/16*c^2*x*arctan(a*x)*sqrt(c+a^2*c*x^2)],
[(c+a^2*c*x^2)^(5/2)*arctan(a*x)/x,x,16,-1/20*a*c*x*(c+a^2*c*x^2)^(3/2)+1/3*c*(c+a^2*c*x^2)^(3/2)*arctan(a*x)+1/5*(c+a^2*c*x^2)^(5/2)*arctan(a*x)-149/120*c^(5/2)*arctanh(a*x*sqrt(c)/sqrt(c+a^2*c*x^2))-2*c^3*arctan(a*x)*arctanh(sqrt(1+I*a*x)/sqrt(1-I*a*x))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)+I*c^3*polylog(2,-sqrt(1+I*a*x)/sqrt(1-I*a*x))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)-I*c^3*polylog(2,sqrt(1+I*a*x)/sqrt(1-I*a*x))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)-29/120*a*c^2*x*sqrt(c+a^2*c*x^2)+c^2*arctan(a*x)*sqrt(c+a^2*c*x^2)],
[(c+a^2*c*x^2)^(5/2)*arctan(a*x)/x^2,x,16,-1/12*a*c*(c+a^2*c*x^2)^(3/2)+1/4*a^2*c*x*(c+a^2*c*x^2)^(3/2)*arctan(a*x)-a*c^(5/2)*arctanh(sqrt(c+a^2*c*x^2)/sqrt(c))-15/4*I*a*c^3*arctan(a*x)*arctan(sqrt(1+I*a*x)/sqrt(1-I*a*x))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)+15/8*I*a*c^3*polylog(2,-I*sqrt(1+I*a*x)/sqrt(1-I*a*x))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)-15/8*I*a*c^3*polylog(2,I*sqrt(1+I*a*x)/sqrt(1-I*a*x))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)-7/8*a*c^2*sqrt(c+a^2*c*x^2)-c^2*arctan(a*x)*sqrt(c+a^2*c*x^2)/x+7/8*a^2*c^2*x*arctan(a*x)*sqrt(c+a^2*c*x^2)],
[(c+a^2*c*x^2)^(5/2)*arctan(a*x)/x^3,x,23,1/3*a^2*c*(c+a^2*c*x^2)^(3/2)*arctan(a*x)-13/6*a^2*c^(5/2)*arctanh(a*x*sqrt(c)/sqrt(c+a^2*c*x^2))-5*a^2*c^3*arctan(a*x)*arctanh(sqrt(1+I*a*x)/sqrt(1-I*a*x))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)+5/2*I*a^2*c^3*polylog(2,-sqrt(1+I*a*x)/sqrt(1-I*a*x))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)-5/2*I*a^2*c^3*polylog(2,sqrt(1+I*a*x)/sqrt(1-I*a*x))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)-1/2*a*c^2*sqrt(c+a^2*c*x^2)/x-1/6*a^3*c^2*x*sqrt(c+a^2*c*x^2)+2*a^2*c^2*arctan(a*x)*sqrt(c+a^2*c*x^2)-1/2*c^2*arctan(a*x)*sqrt(c+a^2*c*x^2)/x^2],
[(c+a^2*c*x^2)^(5/2)*arctan(a*x)/x^4,x,25,-1/3*c*(c+a^2*c*x^2)^(3/2)*arctan(a*x)/x^3-13/6*a^3*c^(5/2)*arctanh(sqrt(c+a^2*c*x^2)/sqrt(c))-5*I*a^3*c^3*arctan(a*x)*arctan(sqrt(1+I*a*x)/sqrt(1-I*a*x))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)+5/2*I*a^3*c^3*polylog(2,-I*sqrt(1+I*a*x)/sqrt(1-I*a*x))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)-5/2*I*a^3*c^3*polylog(2,I*sqrt(1+I*a*x)/sqrt(1-I*a*x))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)-1/2*a^3*c^2*sqrt(c+a^2*c*x^2)-1/6*a*c^2*sqrt(c+a^2*c*x^2)/x^2-2*a^2*c^2*arctan(a*x)*sqrt(c+a^2*c*x^2)/x+1/2*a^4*c^2*x*arctan(a*x)*sqrt(c+a^2*c*x^2)],

# p<0
[x^3*arctan(a*x)/sqrt(c+a^2*c*x^2),x,7,5/6*arctanh(a*x*sqrt(c)/sqrt(c+a^2*c*x^2))/(a^4*sqrt(c))-1/6*x*sqrt(c+a^2*c*x^2)/(a^3*c)-2/3*arctan(a*x)*sqrt(c+a^2*c*x^2)/(a^4*c)+1/3*x^2*arctan(a*x)*sqrt(c+a^2*c*x^2)/(a^2*c)],
[x^2*arctan(a*x)/sqrt(c+a^2*c*x^2),x,4,I*arctan(a*x)*arctan(sqrt(1+I*a*x)/sqrt(1-I*a*x))*sqrt(1+a^2*x^2)/(a^3*sqrt(c+a^2*c*x^2))-1/2*I*polylog(2,-I*sqrt(1+I*a*x)/sqrt(1-I*a*x))*sqrt(1+a^2*x^2)/(a^3*sqrt(c+a^2*c*x^2))+1/2*I*polylog(2,I*sqrt(1+I*a*x)/sqrt(1-I*a*x))*sqrt(1+a^2*x^2)/(a^3*sqrt(c+a^2*c*x^2))-1/2*sqrt(c+a^2*c*x^2)/(a^3*c)+1/2*x*arctan(a*x)*sqrt(c+a^2*c*x^2)/(a^2*c)],
[x*arctan(a*x)/sqrt(c+a^2*c*x^2),x,3,-arctanh(a*x*sqrt(c)/sqrt(c+a^2*c*x^2))/(a^2*sqrt(c))+arctan(a*x)*sqrt(c+a^2*c*x^2)/(a^2*c)],
[arctan(a*x)/sqrt(c+a^2*c*x^2),x,2,-2*I*arctan(a*x)*arctan(sqrt(1+I*a*x)/sqrt(1-I*a*x))*sqrt(1+a^2*x^2)/(a*sqrt(c+a^2*c*x^2))+I*polylog(2,-I*sqrt(1+I*a*x)/sqrt(1-I*a*x))*sqrt(1+a^2*x^2)/(a*sqrt(c+a^2*c*x^2))-I*polylog(2,I*sqrt(1+I*a*x)/sqrt(1-I*a*x))*sqrt(1+a^2*x^2)/(a*sqrt(c+a^2*c*x^2))],
[arctan(a*x)/(x*sqrt(c+a^2*c*x^2)),x,2,-2*arctan(a*x)*arctanh(sqrt(1+I*a*x)/sqrt(1-I*a*x))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)+I*polylog(2,-sqrt(1+I*a*x)/sqrt(1-I*a*x))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)-I*polylog(2,sqrt(1+I*a*x)/sqrt(1-I*a*x))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)],
[arctan(a*x)/(x^2*sqrt(c+a^2*c*x^2)),x,4,-a*arctanh(sqrt(c+a^2*c*x^2)/sqrt(c))/sqrt(c)-arctan(a*x)*sqrt(c+a^2*c*x^2)/(c*x)],
[arctan(a*x)/(x^3*sqrt(c+a^2*c*x^2)),x,4,a^2*arctan(a*x)*arctanh(sqrt(1+I*a*x)/sqrt(1-I*a*x))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)-1/2*I*a^2*polylog(2,-sqrt(1+I*a*x)/sqrt(1-I*a*x))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)+1/2*I*a^2*polylog(2,sqrt(1+I*a*x)/sqrt(1-I*a*x))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)-1/2*a*sqrt(c+a^2*c*x^2)/(c*x)-1/2*arctan(a*x)*sqrt(c+a^2*c*x^2)/(c*x^2)],
[arctan(a*x)/(x^4*sqrt(c+a^2*c*x^2)),x,9,5/6*a^3*arctanh(sqrt(c+a^2*c*x^2)/sqrt(c))/sqrt(c)-1/6*a*sqrt(c+a^2*c*x^2)/(c*x^2)-1/3*arctan(a*x)*sqrt(c+a^2*c*x^2)/(c*x^3)+2/3*a^2*arctan(a*x)*sqrt(c+a^2*c*x^2)/(c*x)],
[x^3*arctan(a*x)/(c+a^2*c*x^2)^(3/2),x,6,-arctanh(a*x*sqrt(c)/sqrt(c+a^2*c*x^2))/(a^4*c^(3/2))-x/(a^3*c*sqrt(c+a^2*c*x^2))+arctan(a*x)/(a^4*c*sqrt(c+a^2*c*x^2))+arctan(a*x)*sqrt(c+a^2*c*x^2)/(a^4*c^2)],
[x^2*arctan(a*x)/(c+a^2*c*x^2)^(3/2),x,3,(-1)/(a^3*c*sqrt(c+a^2*c*x^2))-x*arctan(a*x)/(a^2*c*sqrt(c+a^2*c*x^2))-2*I*arctan(a*x)*arctan(sqrt(1+I*a*x)/sqrt(1-I*a*x))*sqrt(1+a^2*x^2)/(a^3*c*sqrt(c+a^2*c*x^2))+I*polylog(2,-I*sqrt(1+I*a*x)/sqrt(1-I*a*x))*sqrt(1+a^2*x^2)/(a^3*c*sqrt(c+a^2*c*x^2))-I*polylog(2,I*sqrt(1+I*a*x)/sqrt(1-I*a*x))*sqrt(1+a^2*x^2)/(a^3*c*sqrt(c+a^2*c*x^2))],
[x*arctan(a*x)/(c+a^2*c*x^2)^(3/2),x,2,x/(a*c*sqrt(c+a^2*c*x^2))-arctan(a*x)/(a^2*c*sqrt(c+a^2*c*x^2))],
[arctan(a*x)/(c+a^2*c*x^2)^(3/2),x,1,1/(a*c*sqrt(c+a^2*c*x^2))+x*arctan(a*x)/(c*sqrt(c+a^2*c*x^2))],
[arctan(a*x)/(x*(c+a^2*c*x^2)^(3/2)),x,5,-a*x/(c*sqrt(c+a^2*c*x^2))+arctan(a*x)/(c*sqrt(c+a^2*c*x^2))-2*arctan(a*x)*arctanh(sqrt(1+I*a*x)/sqrt(1-I*a*x))*sqrt(1+a^2*x^2)/(c*sqrt(c+a^2*c*x^2))+I*polylog(2,-sqrt(1+I*a*x)/sqrt(1-I*a*x))*sqrt(1+a^2*x^2)/(c*sqrt(c+a^2*c*x^2))-I*polylog(2,sqrt(1+I*a*x)/sqrt(1-I*a*x))*sqrt(1+a^2*x^2)/(c*sqrt(c+a^2*c*x^2))],
[arctan(a*x)/(x^2*(c+a^2*c*x^2)^(3/2)),x,6,-a*arctanh(sqrt(c+a^2*c*x^2)/sqrt(c))/c^(3/2)-a/(c*sqrt(c+a^2*c*x^2))-a^2*x*arctan(a*x)/(c*sqrt(c+a^2*c*x^2))-arctan(a*x)*sqrt(c+a^2*c*x^2)/(c^2*x)],
[arctan(a*x)/(x^3*(c+a^2*c*x^2)^(3/2)),x,10,a^3*x/(c*sqrt(c+a^2*c*x^2))-a^2*arctan(a*x)/(c*sqrt(c+a^2*c*x^2))+3*a^2*arctan(a*x)*arctanh(sqrt(1+I*a*x)/sqrt(1-I*a*x))*sqrt(1+a^2*x^2)/(c*sqrt(c+a^2*c*x^2))-3/2*I*a^2*polylog(2,-sqrt(1+I*a*x)/sqrt(1-I*a*x))*sqrt(1+a^2*x^2)/(c*sqrt(c+a^2*c*x^2))+3/2*I*a^2*polylog(2,sqrt(1+I*a*x)/sqrt(1-I*a*x))*sqrt(1+a^2*x^2)/(c*sqrt(c+a^2*c*x^2))-1/2*a*sqrt(c+a^2*c*x^2)/(c^2*x)-1/2*arctan(a*x)*sqrt(c+a^2*c*x^2)/(c^2*x^2)],
[arctan(a*x)/(x^4*(c+a^2*c*x^2)^(3/2)),x,16,11/6*a^3*arctanh(sqrt(c+a^2*c*x^2)/sqrt(c))/c^(3/2)+a^3/(c*sqrt(c+a^2*c*x^2))+a^4*x*arctan(a*x)/(c*sqrt(c+a^2*c*x^2))-1/6*a*sqrt(c+a^2*c*x^2)/(c^2*x^2)-1/3*arctan(a*x)*sqrt(c+a^2*c*x^2)/(c^2*x^3)+5/3*a^2*arctan(a*x)*sqrt(c+a^2*c*x^2)/(c^2*x)],
[x^5*arctan(a*x)/(c+a^2*c*x^2)^(5/2),x,10,-1/9*x^3/(a^3*c*(c+a^2*c*x^2)^(3/2))+1/3*x^2*arctan(a*x)/(a^4*c*(c+a^2*c*x^2)^(3/2))-arctanh(a*x*sqrt(c)/sqrt(c+a^2*c*x^2))/(a^6*c^(5/2))-5/3*x/(a^5*c^2*sqrt(c+a^2*c*x^2))+5/3*arctan(a*x)/(a^6*c^2*sqrt(c+a^2*c*x^2))+arctan(a*x)*sqrt(c+a^2*c*x^2)/(a^6*c^3)],
[x^4*arctan(a*x)/(c+a^2*c*x^2)^(5/2),x,8,1/9/(a^5*c*(c+a^2*c*x^2)^(3/2))-1/3*x^3*arctan(a*x)/(a^2*c*(c+a^2*c*x^2)^(3/2))+(-4/3)/(a^5*c^2*sqrt(c+a^2*c*x^2))-x*arctan(a*x)/(a^4*c^2*sqrt(c+a^2*c*x^2))-2*I*arctan(a*x)*arctan(sqrt(1+I*a*x)/sqrt(1-I*a*x))*sqrt(1+a^2*x^2)/(a^5*c^2*sqrt(c+a^2*c*x^2))+I*polylog(2,-I*sqrt(1+I*a*x)/sqrt(1-I*a*x))*sqrt(1+a^2*x^2)/(a^5*c^2*sqrt(c+a^2*c*x^2))-I*polylog(2,I*sqrt(1+I*a*x)/sqrt(1-I*a*x))*sqrt(1+a^2*x^2)/(a^5*c^2*sqrt(c+a^2*c*x^2))],
[x^3*arctan(a*x)/(c+a^2*c*x^2)^(5/2),x,3,1/9*x^3/(a*c*(c+a^2*c*x^2)^(3/2))-1/3*x^2*arctan(a*x)/(a^2*c*(c+a^2*c*x^2)^(3/2))+2/3*x/(a^3*c^2*sqrt(c+a^2*c*x^2))-2/3*arctan(a*x)/(a^4*c^2*sqrt(c+a^2*c*x^2))],
[x^2*arctan(a*x)/(c+a^2*c*x^2)^(5/2),x,4,(-1/9)/(a^3*c*(c+a^2*c*x^2)^(3/2))+1/3*x^3*arctan(a*x)/(c*(c+a^2*c*x^2)^(3/2))+1/3/(a^3*c^2*sqrt(c+a^2*c*x^2))],
[x*arctan(a*x)/(c+a^2*c*x^2)^(5/2),x,3,1/9*x/(a*c*(c+a^2*c*x^2)^(3/2))-1/3*arctan(a*x)/(a^2*c*(c+a^2*c*x^2)^(3/2))+2/9*x/(a*c^2*sqrt(c+a^2*c*x^2))],
[arctan(a*x)/(c+a^2*c*x^2)^(5/2),x,2,1/9/(a*c*(c+a^2*c*x^2)^(3/2))+1/3*x*arctan(a*x)/(c*(c+a^2*c*x^2)^(3/2))+2/3/(a*c^2*sqrt(c+a^2*c*x^2))+2/3*x*arctan(a*x)/(c^2*sqrt(c+a^2*c*x^2))],
[arctan(a*x)/(x*(c+a^2*c*x^2)^(5/2)),x,9,-1/9*a*x/(c*(c+a^2*c*x^2)^(3/2))+1/3*arctan(a*x)/(c*(c+a^2*c*x^2)^(3/2))-11/9*a*x/(c^2*sqrt(c+a^2*c*x^2))+arctan(a*x)/(c^2*sqrt(c+a^2*c*x^2))-2*arctan(a*x)*arctanh(sqrt(1+I*a*x)/sqrt(1-I*a*x))*sqrt(1+a^2*x^2)/(c^2*sqrt(c+a^2*c*x^2))+I*polylog(2,-sqrt(1+I*a*x)/sqrt(1-I*a*x))*sqrt(1+a^2*x^2)/(c^2*sqrt(c+a^2*c*x^2))-I*polylog(2,sqrt(1+I*a*x)/sqrt(1-I*a*x))*sqrt(1+a^2*x^2)/(c^2*sqrt(c+a^2*c*x^2))],
[arctan(a*x)/(x^2*(c+a^2*c*x^2)^(5/2)),x,9,-1/9*a/(c*(c+a^2*c*x^2)^(3/2))-1/3*a^2*x*arctan(a*x)/(c*(c+a^2*c*x^2)^(3/2))-a*arctanh(sqrt(c+a^2*c*x^2)/sqrt(c))/c^(5/2)-5/3*a/(c^2*sqrt(c+a^2*c*x^2))-5/3*a^2*x*arctan(a*x)/(c^2*sqrt(c+a^2*c*x^2))-arctan(a*x)*sqrt(c+a^2*c*x^2)/(c^3*x)],

# Integrands of the form x^m (c+d x^2)^p ArcTan[a x]^1 with d=a^2 c and m symbolic
[x^m*(c+a^2*c*x^2)^3*arctan(a*x),x,10,c^3*x^(1+m)*arctan(a*x)/(1+m)+3*a^2*c^3*x^(3+m)*arctan(a*x)/(3+m)+3*a^4*c^3*x^(5+m)*arctan(a*x)/(5+m)+a^6*c^3*x^(7+m)*arctan(a*x)/(7+m)-a*c^3*x^(2+m)*hypergeom([1,1/2*(2+m)],[1/2*(4+m)],-a^2*x^2)/(2+3*m+m^2)-3*a^3*c^3*x^(4+m)*hypergeom([1,1/2*(4+m)],[1/2*(6+m)],-a^2*x^2)/(12+7*m+m^2)-3*a^5*c^3*x^(6+m)*hypergeom([1,1/2*(6+m)],[1/2*(8+m)],-a^2*x^2)/((5+m)*(6+m))-a^7*c^3*x^(8+m)*hypergeom([1,1/2*(8+m)],[1/2*(10+m)],-a^2*x^2)/((7+m)*(8+m))],
[x^m*(c+a^2*c*x^2)^2*arctan(a*x),x,8,c^2*x^(1+m)*arctan(a*x)/(1+m)+2*a^2*c^2*x^(3+m)*arctan(a*x)/(3+m)+a^4*c^2*x^(5+m)*arctan(a*x)/(5+m)-a*c^2*x^(2+m)*hypergeom([1,1/2*(2+m)],[1/2*(4+m)],-a^2*x^2)/(2+3*m+m^2)-2*a^3*c^2*x^(4+m)*hypergeom([1,1/2*(4+m)],[1/2*(6+m)],-a^2*x^2)/(12+7*m+m^2)-a^5*c^2*x^(6+m)*hypergeom([1,1/2*(6+m)],[1/2*(8+m)],-a^2*x^2)/((5+m)*(6+m))],
[x^m*(c+a^2*c*x^2)*arctan(a*x),x,5,c*x^(1+m)*arctan(a*x)/(1+m)+a^2*c*x^(3+m)*arctan(a*x)/(3+m)-a*c*x^(2+m)*hypergeom([1,1/2*(2+m)],[1/2*(4+m)],-a^2*x^2)/(2+3*m+m^2)-a^3*c*x^(4+m)*hypergeom([1,1/2*(4+m)],[1/2*(6+m)],-a^2*x^2)/(12+7*m+m^2)],
[x^m*arctan(a*x)/(c+a^2*c*x^2),x,0,Unintegrable(x^m*arctan(a*x)/(c+a^2*c*x^2),x)],
[x^m*arctan(a*x)/(c+a^2*c*x^2)^2,x,0,Unintegrable(x^m*arctan(a*x)/(c+a^2*c*x^2)^2,x)],
[x^m*(c+a^2*c*x^2)^(5/2)*arctan(a*x),x,0,Unintegrable(x^m*(c+a^2*c*x^2)^(5/2)*arctan(a*x),x)],
[x^m*(c+a^2*c*x^2)^(3/2)*arctan(a*x),x,0,Unintegrable(x^m*(c+a^2*c*x^2)^(3/2)*arctan(a*x),x)],
[x^m*(c+a^2*c*x^2)^(1/2)*arctan(a*x),x,3,x^(1+m)*arctan(a*x)*sqrt(c+a^2*c*x^2)/(2+m)-a*x^(2+m)*hypergeom([1,1/2*(3+m)],[1/2*(4+m)],-a^2*x^2)*sqrt(c+a^2*c*x^2)/(2+m)^2+c*Unintegrable(x^m*arctan(a*x)/sqrt(c+a^2*c*x^2),x)/(2+m),-a*c*x^(2+m)*hypergeom([1/2,1/2*(2+m)],[1/2*(4+m)],-a^2*x^2)*sqrt(1+a^2*x^2)/((2+m)^2*sqrt(c+a^2*c*x^2))+x^(1+m)*arctan(a*x)*sqrt(c+a^2*c*x^2)/(2+m)+c*Unintegrable(x^m*arctan(a*x)/sqrt(c+a^2*c*x^2),x)/(2+m)],
[x^m*arctan(a*x)/(c+a^2*c*x^2)^(1/2),x,0,Unintegrable(x^m*arctan(a*x)/sqrt(c+a^2*c*x^2),x)],
[x^m*arctan(a*x)/(c+a^2*c*x^2)^(3/2),x,0,Unintegrable(x^m*arctan(a*x)/(c+a^2*c*x^2)^(3/2),x)],

# Integrands of the form (f x)^m (d+c^2 d x^2)^q (a+b ArcTan[c x])^2

# Integrands of the form x^m (c+d x^2)^p ArcTan[a x]^2 with d=a^2 c

# p>0
[x^3*(c+a^2*c*x^2)*arctan(a*x)^2,x,26,-1/180*c*x^2/a^2+1/60*c*x^4+1/6*c*x*arctan(a*x)/a^3-1/18*c*x^3*arctan(a*x)/a-1/15*a*c*x^5*arctan(a*x)-1/12*c*arctan(a*x)^2/a^4+1/4*c*x^4*arctan(a*x)^2+1/6*a^2*c*x^6*arctan(a*x)^2-7/90*c*log(1+a^2*x^2)/a^4],
[x^2*(c+a^2*c*x^2)*arctan(a*x)^2,x,24,1/30*c*x/a^2+1/30*c*x^3-1/30*c*arctan(a*x)/a^3-2/15*c*x^2*arctan(a*x)/a-1/10*a*c*x^4*arctan(a*x)-2/15*I*c*arctan(a*x)^2/a^3+1/3*c*x^3*arctan(a*x)^2+1/5*a^2*c*x^5*arctan(a*x)^2-4/15*c*arctan(a*x)*log(2/(1+I*a*x))/a^3-2/15*I*c*polylog(2,1+(-2)/(1+I*a*x))/a^3],
[x*(c+a^2*c*x^2)*arctan(a*x)^2,x,4,1/12*c*(1+a^2*x^2)/a^2-1/3*c*x*arctan(a*x)/a-1/6*c*x*(1+a^2*x^2)*arctan(a*x)/a+1/4*c*(1+a^2*x^2)^2*arctan(a*x)^2/a^2+1/6*c*log(1+a^2*x^2)/a^2],
[(c+a^2*c*x^2)*arctan(a*x)^2,x,7,1/3*c*x-1/3*c*(1+a^2*x^2)*arctan(a*x)/a+2/3*I*c*arctan(a*x)^2/a+2/3*c*x*arctan(a*x)^2+1/3*c*x*(1+a^2*x^2)*arctan(a*x)^2+4/3*c*arctan(a*x)*log(2/(1+I*a*x))/a+2/3*I*c*polylog(2,1+(-2)/(1+I*a*x))/a],
[(c+a^2*c*x^2)*arctan(a*x)^2/x,x,12,-a*c*x*arctan(a*x)+1/2*c*arctan(a*x)^2+1/2*a^2*c*x^2*arctan(a*x)^2+2*c*arctan(a*x)^2*arctanh(1+(-2)/(1+I*a*x))+1/2*c*log(1+a^2*x^2)-I*c*arctan(a*x)*polylog(2,1+(-2)/(1+I*a*x))+I*c*arctan(a*x)*polylog(2,-1+2/(1+I*a*x))-1/2*c*polylog(3,1+(-2)/(1+I*a*x))+1/2*c*polylog(3,-1+2/(1+I*a*x))],
[(c+a^2*c*x^2)*arctan(a*x)^2/x^2,x,10,-c*arctan(a*x)^2/x+a^2*c*x*arctan(a*x)^2+2*a*c*arctan(a*x)*log(2/(1+I*a*x))+2*a*c*arctan(a*x)*log(2+(-2)/(1-I*a*x))-I*a*c*polylog(2,-1+2/(1-I*a*x))+I*a*c*polylog(2,1+(-2)/(1+I*a*x))],
[(c+a^2*c*x^2)*arctan(a*x)^2/x^3,x,15,-a*c*arctan(a*x)/x-1/2*a^2*c*arctan(a*x)^2-1/2*c*arctan(a*x)^2/x^2+2*a^2*c*arctan(a*x)^2*arctanh(1+(-2)/(1+I*a*x))+a^2*c*log(x)-1/2*a^2*c*log(1+a^2*x^2)-I*a^2*c*arctan(a*x)*polylog(2,1+(-2)/(1+I*a*x))+I*a^2*c*arctan(a*x)*polylog(2,-1+2/(1+I*a*x))-1/2*a^2*c*polylog(3,1+(-2)/(1+I*a*x))+1/2*a^2*c*polylog(3,-1+2/(1+I*a*x))],
[(c+a^2*c*x^2)*arctan(a*x)^2/x^4,x,13,-1/3*a^2*c/x-1/3*a^3*c*arctan(a*x)-1/3*a*c*arctan(a*x)/x^2-2/3*I*a^3*c*arctan(a*x)^2-1/3*c*arctan(a*x)^2/x^3-a^2*c*arctan(a*x)^2/x+4/3*a^3*c*arctan(a*x)*log(2+(-2)/(1-I*a*x))-2/3*I*a^3*c*polylog(2,-1+2/(1-I*a*x))],
[x^3*(c+a^2*c*x^2)^2*arctan(a*x)^2,x,47,-5/504*c^2*x^2/a^2+1/84*c^2*x^4+1/168*a^2*c^2*x^6+1/12*c^2*x*arctan(a*x)/a^3-1/36*c^2*x^3*arctan(a*x)/a-1/12*a*c^2*x^5*arctan(a*x)-1/28*a^3*c^2*x^7*arctan(a*x)-1/24*c^2*arctan(a*x)^2/a^4+1/4*c^2*x^4*arctan(a*x)^2+1/3*a^2*c^2*x^6*arctan(a*x)^2+1/8*a^4*c^2*x^8*arctan(a*x)^2-2/63*c^2*log(1+a^2*x^2)/a^4],
[x^2*(c+a^2*c*x^2)^2*arctan(a*x)^2,x,44,-1/210*c^2*x/a^2+17/630*c^2*x^3+1/105*a^2*c^2*x^5+1/210*c^2*arctan(a*x)/a^3-8/105*c^2*x^2*arctan(a*x)/a-9/70*a*c^2*x^4*arctan(a*x)-1/21*a^3*c^2*x^6*arctan(a*x)-8/105*I*c^2*arctan(a*x)^2/a^3+1/3*c^2*x^3*arctan(a*x)^2+2/5*a^2*c^2*x^5*arctan(a*x)^2+1/7*a^4*c^2*x^7*arctan(a*x)^2-16/105*c^2*arctan(a*x)*log(2/(1+I*a*x))/a^3-8/105*I*c^2*polylog(2,1+(-2)/(1+I*a*x))/a^3],
[x*(c+a^2*c*x^2)^2*arctan(a*x)^2,x,5,2/45*c^2*(1+a^2*x^2)/a^2+1/60*c^2*(1+a^2*x^2)^2/a^2-8/45*c^2*x*arctan(a*x)/a-4/45*c^2*x*(1+a^2*x^2)*arctan(a*x)/a-1/15*c^2*x*(1+a^2*x^2)^2*arctan(a*x)/a+1/6*c^2*(1+a^2*x^2)^3*arctan(a*x)^2/a^2+4/45*c^2*log(1+a^2*x^2)/a^2],
[(c+a^2*c*x^2)^2*arctan(a*x)^2,x,9,11/30*c^2*x+1/30*a^2*c^2*x^3-4/15*c^2*(1+a^2*x^2)*arctan(a*x)/a-1/10*c^2*(1+a^2*x^2)^2*arctan(a*x)/a+8/15*I*c^2*arctan(a*x)^2/a+8/15*c^2*x*arctan(a*x)^2+4/15*c^2*x*(1+a^2*x^2)*arctan(a*x)^2+1/5*c^2*x*(1+a^2*x^2)^2*arctan(a*x)^2+16/15*c^2*arctan(a*x)*log(2/(1+I*a*x))/a+8/15*I*c^2*polylog(2,1+(-2)/(1+I*a*x))/a],
[(c+a^2*c*x^2)^2*arctan(a*x)^2/x,x,23,1/12*a^2*c^2*x^2-3/2*a*c^2*x*arctan(a*x)-1/6*a^3*c^2*x^3*arctan(a*x)+3/4*c^2*arctan(a*x)^2+a^2*c^2*x^2*arctan(a*x)^2+1/4*a^4*c^2*x^4*arctan(a*x)^2+2*c^2*arctan(a*x)^2*arctanh(1+(-2)/(1+I*a*x))+2/3*c^2*log(1+a^2*x^2)-I*c^2*arctan(a*x)*polylog(2,1+(-2)/(1+I*a*x))+I*c^2*arctan(a*x)*polylog(2,-1+2/(1+I*a*x))-1/2*c^2*polylog(3,1+(-2)/(1+I*a*x))+1/2*c^2*polylog(3,-1+2/(1+I*a*x))],
[(c+a^2*c*x^2)^2*arctan(a*x)^2/x^2,x,20,1/3*a^2*c^2*x-1/3*a*c^2*arctan(a*x)-1/3*a^3*c^2*x^2*arctan(a*x)+2/3*I*a*c^2*arctan(a*x)^2-c^2*arctan(a*x)^2/x+2*a^2*c^2*x*arctan(a*x)^2+1/3*a^4*c^2*x^3*arctan(a*x)^2+10/3*a*c^2*arctan(a*x)*log(2/(1+I*a*x))+2*a*c^2*arctan(a*x)*log(2+(-2)/(1-I*a*x))-I*a*c^2*polylog(2,-1+2/(1-I*a*x))+5/3*I*a*c^2*polylog(2,1+(-2)/(1+I*a*x))],
[(c+a^2*c*x^2)^2*arctan(a*x)^2/x^3,x,21,-a*c^2*arctan(a*x)/x-a^3*c^2*x*arctan(a*x)-1/2*c^2*arctan(a*x)^2/x^2+1/2*a^4*c^2*x^2*arctan(a*x)^2+4*a^2*c^2*arctan(a*x)^2*arctanh(1+(-2)/(1+I*a*x))+a^2*c^2*log(x)-2*I*a^2*c^2*arctan(a*x)*polylog(2,1+(-2)/(1+I*a*x))+2*I*a^2*c^2*arctan(a*x)*polylog(2,-1+2/(1+I*a*x))-a^2*c^2*polylog(3,1+(-2)/(1+I*a*x))+a^2*c^2*polylog(3,-1+2/(1+I*a*x))],
[(c+a^2*c*x^2)^2*arctan(a*x)^2/x^4,x,19,-1/3*a^2*c^2/x-1/3*a^3*c^2*arctan(a*x)-1/3*a*c^2*arctan(a*x)/x^2-2/3*I*a^3*c^2*arctan(a*x)^2-1/3*c^2*arctan(a*x)^2/x^3-2*a^2*c^2*arctan(a*x)^2/x+a^4*c^2*x*arctan(a*x)^2+2*a^3*c^2*arctan(a*x)*log(2/(1+I*a*x))+10/3*a^3*c^2*arctan(a*x)*log(2+(-2)/(1-I*a*x))-5/3*I*a^3*c^2*polylog(2,-1+2/(1-I*a*x))+I*a^3*c^2*polylog(2,1+(-2)/(1+I*a*x))],
[x^3*(c+a^2*c*x^2)^3*arctan(a*x)^2,x,72,-107/12600*c^3*x^2/a^2+53/6300*c^3*x^4+71/7560*a^2*c^3*x^6+1/360*a^4*c^3*x^8+1/20*c^3*x*arctan(a*x)/a^3-1/60*c^3*x^3*arctan(a*x)/a-9/100*a*c^3*x^5*arctan(a*x)-11/140*a^3*c^3*x^7*arctan(a*x)-1/45*a^5*c^3*x^9*arctan(a*x)-1/40*c^3*arctan(a*x)^2/a^4+1/4*c^3*x^4*arctan(a*x)^2+1/2*a^2*c^3*x^6*arctan(a*x)^2+3/8*a^4*c^3*x^8*arctan(a*x)^2+1/10*a^6*c^3*x^10*arctan(a*x)^2-26/1575*c^3*log(1+a^2*x^2)/a^4],
[x^2*(c+a^2*c*x^2)^3*arctan(a*x)^2,x,68,-47/3780*c^3*x/a^2+239/11340*c^3*x^3+59/3780*a^2*c^3*x^5+1/252*a^4*c^3*x^7+47/3780*c^3*arctan(a*x)/a^3-16/315*c^3*x^2*arctan(a*x)/a-89/630*a*c^3*x^4*arctan(a*x)-20/189*a^3*c^3*x^6*arctan(a*x)-1/36*a^5*c^3*x^8*arctan(a*x)-16/315*I*c^3*arctan(a*x)^2/a^3+1/3*c^3*x^3*arctan(a*x)^2+3/5*a^2*c^3*x^5*arctan(a*x)^2+3/7*a^4*c^3*x^7*arctan(a*x)^2+1/9*a^6*c^3*x^9*arctan(a*x)^2-32/315*c^3*arctan(a*x)*log(2/(1+I*a*x))/a^3-16/315*I*c^3*polylog(2,1+(-2)/(1+I*a*x))/a^3],
[x*(c+a^2*c*x^2)^3*arctan(a*x)^2,x,6,1/35*c^3*(1+a^2*x^2)/a^2+3/280*c^3*(1+a^2*x^2)^2/a^2+1/168*c^3*(1+a^2*x^2)^3/a^2-4/35*c^3*x*arctan(a*x)/a-2/35*c^3*x*(1+a^2*x^2)*arctan(a*x)/a-3/70*c^3*x*(1+a^2*x^2)^2*arctan(a*x)/a-1/28*c^3*x*(1+a^2*x^2)^3*arctan(a*x)/a+1/8*c^3*(1+a^2*x^2)^4*arctan(a*x)^2/a^2+2/35*c^3*log(1+a^2*x^2)/a^2],
[(c+a^2*c*x^2)^3*arctan(a*x)^2,x,12,38/105*c^3*x+19/315*a^2*c^3*x^3+1/105*a^4*c^3*x^5-8/35*c^3*(1+a^2*x^2)*arctan(a*x)/a-3/35*c^3*(1+a^2*x^2)^2*arctan(a*x)/a-1/21*c^3*(1+a^2*x^2)^3*arctan(a*x)/a+16/35*I*c^3*arctan(a*x)^2/a+16/35*c^3*x*arctan(a*x)^2+8/35*c^3*x*(1+a^2*x^2)*arctan(a*x)^2+6/35*c^3*x*(1+a^2*x^2)^2*arctan(a*x)^2+1/7*c^3*x*(1+a^2*x^2)^3*arctan(a*x)^2+32/35*c^3*arctan(a*x)*log(2/(1+I*a*x))/a+16/35*I*c^3*polylog(2,1+(-2)/(1+I*a*x))/a],
[(c+a^2*c*x^2)^3*arctan(a*x)^2/x,x,38,29/180*a^2*c^3*x^2+1/60*a^4*c^3*x^4-11/6*a*c^3*x*arctan(a*x)-7/18*a^3*c^3*x^3*arctan(a*x)-1/15*a^5*c^3*x^5*arctan(a*x)+11/12*c^3*arctan(a*x)^2+3/2*a^2*c^3*x^2*arctan(a*x)^2+3/4*a^4*c^3*x^4*arctan(a*x)^2+1/6*a^6*c^3*x^6*arctan(a*x)^2+2*c^3*arctan(a*x)^2*arctanh(1+(-2)/(1+I*a*x))+34/45*c^3*log(1+a^2*x^2)-I*c^3*arctan(a*x)*polylog(2,1+(-2)/(1+I*a*x))+I*c^3*arctan(a*x)*polylog(2,-1+2/(1+I*a*x))-1/2*c^3*polylog(3,1+(-2)/(1+I*a*x))+1/2*c^3*polylog(3,-1+2/(1+I*a*x))],
[(c+a^2*c*x^2)^3*arctan(a*x)^2/x^2,x,34,7/10*a^2*c^3*x+1/30*a^4*c^3*x^3-7/10*a*c^3*arctan(a*x)-4/5*a^3*c^3*x^2*arctan(a*x)-1/10*a^5*c^3*x^4*arctan(a*x)+6/5*I*a*c^3*arctan(a*x)^2-c^3*arctan(a*x)^2/x+3*a^2*c^3*x*arctan(a*x)^2+a^4*c^3*x^3*arctan(a*x)^2+1/5*a^6*c^3*x^5*arctan(a*x)^2+22/5*a*c^3*arctan(a*x)*log(2/(1+I*a*x))+2*a*c^3*arctan(a*x)*log(2+(-2)/(1-I*a*x))-I*a*c^3*polylog(2,-1+2/(1-I*a*x))+11/5*I*a*c^3*polylog(2,1+(-2)/(1+I*a*x))],
[(c+a^2*c*x^2)^3*arctan(a*x)^2/x^3,x,31,1/12*a^4*c^3*x^2-a*c^3*arctan(a*x)/x-5/2*a^3*c^3*x*arctan(a*x)-1/6*a^5*c^3*x^3*arctan(a*x)+3/4*a^2*c^3*arctan(a*x)^2-1/2*c^3*arctan(a*x)^2/x^2+3/2*a^4*c^3*x^2*arctan(a*x)^2+1/4*a^6*c^3*x^4*arctan(a*x)^2+6*a^2*c^3*arctan(a*x)^2*arctanh(1+(-2)/(1+I*a*x))+a^2*c^3*log(x)+2/3*a^2*c^3*log(1+a^2*x^2)-3*I*a^2*c^3*arctan(a*x)*polylog(2,1+(-2)/(1+I*a*x))+3*I*a^2*c^3*arctan(a*x)*polylog(2,-1+2/(1+I*a*x))-3/2*a^2*c^3*polylog(3,1+(-2)/(1+I*a*x))+3/2*a^2*c^3*polylog(3,-1+2/(1+I*a*x))],
[(c+a^2*c*x^2)^3*arctan(a*x)^2/x^4,x,28,-1/3*a^2*c^3/x+1/3*a^4*c^3*x-2/3*a^3*c^3*arctan(a*x)-1/3*a*c^3*arctan(a*x)/x^2-1/3*a^5*c^3*x^2*arctan(a*x)-1/3*c^3*arctan(a*x)^2/x^3-3*a^2*c^3*arctan(a*x)^2/x+3*a^4*c^3*x*arctan(a*x)^2+1/3*a^6*c^3*x^3*arctan(a*x)^2+16/3*a^3*c^3*arctan(a*x)*log(2/(1+I*a*x))+16/3*a^3*c^3*arctan(a*x)*log(2+(-2)/(1-I*a*x))-8/3*I*a^3*c^3*polylog(2,-1+2/(1-I*a*x))+8/3*I*a^3*c^3*polylog(2,1+(-2)/(1+I*a*x))],

# p<0
[x^4*arctan(a*x)^2/(c+a^2*c*x^2),x,17,1/3*x/(a^4*c)-1/3*arctan(a*x)/(a^5*c)-1/3*x^2*arctan(a*x)/(a^3*c)-4/3*I*arctan(a*x)^2/(a^5*c)-x*arctan(a*x)^2/(a^4*c)+1/3*x^3*arctan(a*x)^2/(a^2*c)+1/3*arctan(a*x)^3/(a^5*c)-8/3*arctan(a*x)*log(2/(1+I*a*x))/(a^5*c)-4/3*I*polylog(2,1+(-2)/(1+I*a*x))/(a^5*c)],
[x^3*arctan(a*x)^2/(c+a^2*c*x^2),x,10,-x*arctan(a*x)/(a^3*c)+1/2*arctan(a*x)^2/(a^4*c)+1/2*x^2*arctan(a*x)^2/(a^2*c)+1/3*I*arctan(a*x)^3/(a^4*c)+arctan(a*x)^2*log(2/(1+I*a*x))/(a^4*c)+1/2*log(1+a^2*x^2)/(a^4*c)+I*arctan(a*x)*polylog(2,1+(-2)/(1+I*a*x))/(a^4*c)+1/2*polylog(3,1+(-2)/(1+I*a*x))/(a^4*c)],
[x^2*arctan(a*x)^2/(c+a^2*c*x^2),x,7,I*arctan(a*x)^2/(a^3*c)+x*arctan(a*x)^2/(a^2*c)-1/3*arctan(a*x)^3/(a^3*c)+2*arctan(a*x)*log(2/(1+I*a*x))/(a^3*c)+I*polylog(2,1+(-2)/(1+I*a*x))/(a^3*c)],
[x*arctan(a*x)^2/(c+a^2*c*x^2),x,4,-1/3*I*arctan(a*x)^3/(a^2*c)-arctan(a*x)^2*log(2/(1+I*a*x))/(a^2*c)-I*arctan(a*x)*polylog(2,1+(-2)/(1+I*a*x))/(a^2*c)-1/2*polylog(3,1+(-2)/(1+I*a*x))/(a^2*c)],
[arctan(a*x)^2/(c+a^2*c*x^2),x,1,1/3*arctan(a*x)^3/(a*c)],
[arctan(a*x)^2/(x*(c+a^2*c*x^2)),x,4,-1/3*I*arctan(a*x)^3/c+arctan(a*x)^2*log(2+(-2)/(1-I*a*x))/c-I*arctan(a*x)*polylog(2,-1+2/(1-I*a*x))/c+1/2*polylog(3,-1+2/(1-I*a*x))/c],
[arctan(a*x)^2/(x^2*(c+a^2*c*x^2)),x,6,-I*a*arctan(a*x)^2/c-arctan(a*x)^2/(c*x)-1/3*a*arctan(a*x)^3/c+2*a*arctan(a*x)*log(2+(-2)/(1-I*a*x))/c-I*a*polylog(2,-1+2/(1-I*a*x))/c],
[arctan(a*x)^2/(x^3*(c+a^2*c*x^2)),x,13,-a*arctan(a*x)/(c*x)-1/2*a^2*arctan(a*x)^2/c-1/2*arctan(a*x)^2/(c*x^2)+1/3*I*a^2*arctan(a*x)^3/c+a^2*log(x)/c-1/2*a^2*log(1+a^2*x^2)/c-a^2*arctan(a*x)^2*log(2+(-2)/(1-I*a*x))/c+I*a^2*arctan(a*x)*polylog(2,-1+2/(1-I*a*x))/c-1/2*a^2*polylog(3,-1+2/(1-I*a*x))/c],
[arctan(a*x)^2/(x^4*(c+a^2*c*x^2)),x,15,-1/3*a^2/(c*x)-1/3*a^3*arctan(a*x)/c-1/3*a*arctan(a*x)/(c*x^2)+4/3*I*a^3*arctan(a*x)^2/c-1/3*arctan(a*x)^2/(c*x^3)+a^2*arctan(a*x)^2/(c*x)+1/3*a^3*arctan(a*x)^3/c-8/3*a^3*arctan(a*x)*log(2+(-2)/(1-I*a*x))/c+4/3*I*a^3*polylog(2,-1+2/(1-I*a*x))/c],
[x^3*arctan(a*x)^2/(c+a^2*c*x^2)^2,x,8,(-1/4)/(a^4*c^2*(1+a^2*x^2))-1/2*x*arctan(a*x)/(a^3*c^2*(1+a^2*x^2))-1/4*arctan(a*x)^2/(a^4*c^2)+1/2*arctan(a*x)^2/(a^4*c^2*(1+a^2*x^2))-1/3*I*arctan(a*x)^3/(a^4*c^2)-arctan(a*x)^2*log(2/(1+I*a*x))/(a^4*c^2)-I*arctan(a*x)*polylog(2,1+(-2)/(1+I*a*x))/(a^4*c^2)-1/2*polylog(3,1+(-2)/(1+I*a*x))/(a^4*c^2)],
[x^2*arctan(a*x)^2/(c+a^2*c*x^2)^2,x,4,1/4*x/(a^2*c^2*(1+a^2*x^2))+1/4*arctan(a*x)/(a^3*c^2)-1/2*arctan(a*x)/(a^3*c^2*(1+a^2*x^2))-1/2*x*arctan(a*x)^2/(a^2*c^2*(1+a^2*x^2))+1/6*arctan(a*x)^3/(a^3*c^2)],
[x*arctan(a*x)^2/(c+a^2*c*x^2)^2,x,3,1/4/(a^2*c^2*(1+a^2*x^2))+1/2*x*arctan(a*x)/(a*c^2*(1+a^2*x^2))+1/4*arctan(a*x)^2/(a^2*c^2)-1/2*arctan(a*x)^2/(a^2*c^2*(1+a^2*x^2))],
[arctan(a*x)^2/(c+a^2*c*x^2)^2,x,4,-1/4*x/(c^2*(1+a^2*x^2))-1/4*arctan(a*x)/(a*c^2)+1/2*arctan(a*x)/(a*c^2*(1+a^2*x^2))+1/2*x*arctan(a*x)^2/(c^2*(1+a^2*x^2))+1/6*arctan(a*x)^3/(a*c^2)],
[arctan(a*x)^2/(x*(c+a^2*c*x^2)^2),x,8,(-1/4)/(c^2*(1+a^2*x^2))-1/2*a*x*arctan(a*x)/(c^2*(1+a^2*x^2))-1/4*arctan(a*x)^2/c^2+1/2*arctan(a*x)^2/(c^2*(1+a^2*x^2))-1/3*I*arctan(a*x)^3/c^2+arctan(a*x)^2*log(2+(-2)/(1-I*a*x))/c^2-I*arctan(a*x)*polylog(2,-1+2/(1-I*a*x))/c^2+1/2*polylog(3,-1+2/(1-I*a*x))/c^2],
[arctan(a*x)^2/(x^2*(c+a^2*c*x^2)^2),x,11,1/4*a^2*x/(c^2*(1+a^2*x^2))+1/4*a*arctan(a*x)/c^2-1/2*a*arctan(a*x)/(c^2*(1+a^2*x^2))-I*a*arctan(a*x)^2/c^2-arctan(a*x)^2/(c^2*x)-1/2*a^2*x*arctan(a*x)^2/(c^2*(1+a^2*x^2))-1/2*a*arctan(a*x)^3/c^2+2*a*arctan(a*x)*log(2+(-2)/(1-I*a*x))/c^2-I*a*polylog(2,-1+2/(1-I*a*x))/c^2],
[arctan(a*x)^2/(x^3*(c+a^2*c*x^2)^2),x,22,1/4*a^2/(c^2*(1+a^2*x^2))-a*arctan(a*x)/(c^2*x)+1/2*a^3*x*arctan(a*x)/(c^2*(1+a^2*x^2))-1/4*a^2*arctan(a*x)^2/c^2-1/2*arctan(a*x)^2/(c^2*x^2)-1/2*a^2*arctan(a*x)^2/(c^2*(1+a^2*x^2))+2/3*I*a^2*arctan(a*x)^3/c^2+a^2*log(x)/c^2-1/2*a^2*log(1+a^2*x^2)/c^2-2*a^2*arctan(a*x)^2*log(2+(-2)/(1-I*a*x))/c^2+2*I*a^2*arctan(a*x)*polylog(2,-1+2/(1-I*a*x))/c^2-a^2*polylog(3,-1+2/(1-I*a*x))/c^2],
[arctan(a*x)^2/(x^4*(c+a^2*c*x^2)^2),x,27,-1/3*a^2/(c^2*x)-1/4*a^4*x/(c^2*(1+a^2*x^2))-7/12*a^3*arctan(a*x)/c^2-1/3*a*arctan(a*x)/(c^2*x^2)+1/2*a^3*arctan(a*x)/(c^2*(1+a^2*x^2))+7/3*I*a^3*arctan(a*x)^2/c^2-1/3*arctan(a*x)^2/(c^2*x^3)+2*a^2*arctan(a*x)^2/(c^2*x)+1/2*a^4*x*arctan(a*x)^2/(c^2*(1+a^2*x^2))+5/6*a^3*arctan(a*x)^3/c^2-14/3*a^3*arctan(a*x)*log(2+(-2)/(1-I*a*x))/c^2+7/3*I*a^3*polylog(2,-1+2/(1-I*a*x))/c^2],
[x^3*arctan(a*x)^2/(c+a^2*c*x^2)^3,x,4,-1/32*x^4/(c^3*(1+a^2*x^2)^2)+3/32/(a^4*c^3*(1+a^2*x^2))+1/8*x^3*arctan(a*x)/(a*c^3*(1+a^2*x^2)^2)+3/16*x*arctan(a*x)/(a^3*c^3*(1+a^2*x^2))-3/32*arctan(a*x)^2/(a^4*c^3)+1/4*x^4*arctan(a*x)^2/(c^3*(1+a^2*x^2)^2)],
[x^2*arctan(a*x)^2/(c+a^2*c*x^2)^3,x,13,1/32*x/(a^2*c^3*(1+a^2*x^2)^2)-1/64*x/(a^2*c^3*(1+a^2*x^2))-1/64*arctan(a*x)/(a^3*c^3)-1/8*arctan(a*x)/(a^3*c^3*(1+a^2*x^2)^2)+1/8*arctan(a*x)/(a^3*c^3*(1+a^2*x^2))-1/4*x*arctan(a*x)^2/(a^2*c^3*(1+a^2*x^2)^2)+1/8*x*arctan(a*x)^2/(a^2*c^3*(1+a^2*x^2))+1/24*arctan(a*x)^3/(a^3*c^3)],
[x*arctan(a*x)^2/(c+a^2*c*x^2)^3,x,4,1/32/(a^2*c^3*(1+a^2*x^2)^2)+3/32/(a^2*c^3*(1+a^2*x^2))+1/8*x*arctan(a*x)/(a*c^3*(1+a^2*x^2)^2)+3/16*x*arctan(a*x)/(a*c^3*(1+a^2*x^2))+3/32*arctan(a*x)^2/(a^2*c^3)-1/4*arctan(a*x)^2/(a^2*c^3*(1+a^2*x^2)^2)],
[arctan(a*x)^2/(c+a^2*c*x^2)^3,x,8,-1/32*x/(c^3*(1+a^2*x^2)^2)-15/64*x/(c^3*(1+a^2*x^2))-15/64*arctan(a*x)/(a*c^3)+1/8*arctan(a*x)/(a*c^3*(1+a^2*x^2)^2)+3/8*arctan(a*x)/(a*c^3*(1+a^2*x^2))+1/4*x*arctan(a*x)^2/(c^3*(1+a^2*x^2)^2)+3/8*x*arctan(a*x)^2/(c^3*(1+a^2*x^2))+1/8*arctan(a*x)^3/(a*c^3)],
[arctan(a*x)^2/(x*(c+a^2*c*x^2)^3),x,13,(-1/32)/(c^3*(1+a^2*x^2)^2)+(-11/32)/(c^3*(1+a^2*x^2))-1/8*a*x*arctan(a*x)/(c^3*(1+a^2*x^2)^2)-11/16*a*x*arctan(a*x)/(c^3*(1+a^2*x^2))-11/32*arctan(a*x)^2/c^3+1/4*arctan(a*x)^2/(c^3*(1+a^2*x^2)^2)+1/2*arctan(a*x)^2/(c^3*(1+a^2*x^2))-1/3*I*arctan(a*x)^3/c^3+arctan(a*x)^2*log(2+(-2)/(1-I*a*x))/c^3-I*arctan(a*x)*polylog(2,-1+2/(1-I*a*x))/c^3+1/2*polylog(3,-1+2/(1-I*a*x))/c^3],
[arctan(a*x)^2/(x^2*(c+a^2*c*x^2)^3),x,20,1/32*a^2*x/(c^3*(1+a^2*x^2)^2)+31/64*a^2*x/(c^3*(1+a^2*x^2))+31/64*a*arctan(a*x)/c^3-1/8*a*arctan(a*x)/(c^3*(1+a^2*x^2)^2)-7/8*a*arctan(a*x)/(c^3*(1+a^2*x^2))-I*a*arctan(a*x)^2/c^3-arctan(a*x)^2/(c^3*x)-1/4*a^2*x*arctan(a*x)^2/(c^3*(1+a^2*x^2)^2)-7/8*a^2*x*arctan(a*x)^2/(c^3*(1+a^2*x^2))-5/8*a*arctan(a*x)^3/c^3+2*a*arctan(a*x)*log(2+(-2)/(1-I*a*x))/c^3-I*a*polylog(2,-1+2/(1-I*a*x))/c^3],
[arctan(a*x)^2/(x^3*(c+a^2*c*x^2)^3),x,36,1/32*a^2/(c^3*(1+a^2*x^2)^2)+19/32*a^2/(c^3*(1+a^2*x^2))-a*arctan(a*x)/(c^3*x)+1/8*a^3*x*arctan(a*x)/(c^3*(1+a^2*x^2)^2)+19/16*a^3*x*arctan(a*x)/(c^3*(1+a^2*x^2))+3/32*a^2*arctan(a*x)^2/c^3-1/2*arctan(a*x)^2/(c^3*x^2)-1/4*a^2*arctan(a*x)^2/(c^3*(1+a^2*x^2)^2)-a^2*arctan(a*x)^2/(c^3*(1+a^2*x^2))+I*a^2*arctan(a*x)^3/c^3+a^2*log(x)/c^3-1/2*a^2*log(1+a^2*x^2)/c^3-3*a^2*arctan(a*x)^2*log(2+(-2)/(1-I*a*x))/c^3+3*I*a^2*arctan(a*x)*polylog(2,-1+2/(1-I*a*x))/c^3-3/2*a^2*polylog(3,-1+2/(1-I*a*x))/c^3],
[arctan(a*x)^2/(x^4*(c+a^2*c*x^2)^3),x,48,-1/3*a^2/(c^3*x)-1/32*a^4*x/(c^3*(1+a^2*x^2)^2)-47/64*a^4*x/(c^3*(1+a^2*x^2))-205/192*a^3*arctan(a*x)/c^3-1/3*a*arctan(a*x)/(c^3*x^2)+1/8*a^3*arctan(a*x)/(c^3*(1+a^2*x^2)^2)+11/8*a^3*arctan(a*x)/(c^3*(1+a^2*x^2))+10/3*I*a^3*arctan(a*x)^2/c^3-1/3*arctan(a*x)^2/(c^3*x^3)+3*a^2*arctan(a*x)^2/(c^3*x)+1/4*a^4*x*arctan(a*x)^2/(c^3*(1+a^2*x^2)^2)+11/8*a^4*x*arctan(a*x)^2/(c^3*(1+a^2*x^2))+35/24*a^3*arctan(a*x)^3/c^3-20/3*a^3*arctan(a*x)*log(2+(-2)/(1-I*a*x))/c^3+10/3*I*a^3*polylog(2,-1+2/(1-I*a*x))/c^3],

# Integrands of the form x^m (c+d x^2)^(p/2) ArcTan[a x]^2 with d=a^2 c

# p>0
[x^3*arctan(a*x)^2*sqrt(c+a^2*c*x^2),x,26,1/30*(c+a^2*c*x^2)^(3/2)/(a^4*c)-11/30*I*c*arctan(a*x)*arctan(sqrt(1+I*a*x)/sqrt(1-I*a*x))*sqrt(1+a^2*x^2)/(a^4*sqrt(c+a^2*c*x^2))+11/60*I*c*polylog(2,-I*sqrt(1+I*a*x)/sqrt(1-I*a*x))*sqrt(1+a^2*x^2)/(a^4*sqrt(c+a^2*c*x^2))-11/60*I*c*polylog(2,I*sqrt(1+I*a*x)/sqrt(1-I*a*x))*sqrt(1+a^2*x^2)/(a^4*sqrt(c+a^2*c*x^2))-11/60*sqrt(c+a^2*c*x^2)/a^4+1/12*x*arctan(a*x)*sqrt(c+a^2*c*x^2)/a^3-1/10*x^3*arctan(a*x)*sqrt(c+a^2*c*x^2)/a-2/15*arctan(a*x)^2*sqrt(c+a^2*c*x^2)/a^4+1/15*x^2*arctan(a*x)^2*sqrt(c+a^2*c*x^2)/a^2+1/5*x^4*arctan(a*x)^2*sqrt(c+a^2*c*x^2)],
[x^2*arctan(a*x)^2*sqrt(c+a^2*c*x^2),x,35,-1/6*arctanh(a*x*sqrt(c)/sqrt(c+a^2*c*x^2))*sqrt(c)/a^3+1/4*I*c*arctan(exp(I*arctan(a*x)))*arctan(a*x)^2*sqrt(1+a^2*x^2)/(a^3*sqrt(c+a^2*c*x^2))-1/4*I*c*arctan(a*x)*polylog(2,-I*exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/(a^3*sqrt(c+a^2*c*x^2))+1/4*I*c*arctan(a*x)*polylog(2,I*exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/(a^3*sqrt(c+a^2*c*x^2))+1/4*c*polylog(3,-I*exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/(a^3*sqrt(c+a^2*c*x^2))-1/4*c*polylog(3,I*exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/(a^3*sqrt(c+a^2*c*x^2))+1/12*x*sqrt(c+a^2*c*x^2)/a^2+1/12*arctan(a*x)*sqrt(c+a^2*c*x^2)/a^3-1/6*x^2*arctan(a*x)*sqrt(c+a^2*c*x^2)/a+1/8*x*arctan(a*x)^2*sqrt(c+a^2*c*x^2)/a^2+1/4*x^3*arctan(a*x)^2*sqrt(c+a^2*c*x^2)],
[x*arctan(a*x)^2*sqrt(c+a^2*c*x^2),x,4,1/3*(c+a^2*c*x^2)^(3/2)*arctan(a*x)^2/(a^2*c)+2/3*I*c*arctan(a*x)*arctan(sqrt(1+I*a*x)/sqrt(1-I*a*x))*sqrt(1+a^2*x^2)/(a^2*sqrt(c+a^2*c*x^2))-1/3*I*c*polylog(2,-I*sqrt(1+I*a*x)/sqrt(1-I*a*x))*sqrt(1+a^2*x^2)/(a^2*sqrt(c+a^2*c*x^2))+1/3*I*c*polylog(2,I*sqrt(1+I*a*x)/sqrt(1-I*a*x))*sqrt(1+a^2*x^2)/(a^2*sqrt(c+a^2*c*x^2))+1/3*sqrt(c+a^2*c*x^2)/a^2-1/3*x*arctan(a*x)*sqrt(c+a^2*c*x^2)/a],
[arctan(a*x)^2*sqrt(c+a^2*c*x^2),x,12,arctanh(a*x*sqrt(c)/sqrt(c+a^2*c*x^2))*sqrt(c)/a-I*c*arctan(exp(I*arctan(a*x)))*arctan(a*x)^2*sqrt(1+a^2*x^2)/(a*sqrt(c+a^2*c*x^2))+I*c*arctan(a*x)*polylog(2,-I*exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/(a*sqrt(c+a^2*c*x^2))-I*c*arctan(a*x)*polylog(2,I*exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/(a*sqrt(c+a^2*c*x^2))-c*polylog(3,-I*exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/(a*sqrt(c+a^2*c*x^2))+c*polylog(3,I*exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/(a*sqrt(c+a^2*c*x^2))-arctan(a*x)*sqrt(c+a^2*c*x^2)/a+1/2*x*arctan(a*x)^2*sqrt(c+a^2*c*x^2)],
[arctan(a*x)^2*sqrt(c+a^2*c*x^2)/x,x,13,4*I*c*arctan(a*x)*arctan(sqrt(1+I*a*x)/sqrt(1-I*a*x))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)-2*c*arctan(a*x)^2*arctanh(exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)+2*I*c*arctan(a*x)*polylog(2,-exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)-2*I*c*arctan(a*x)*polylog(2,exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)-2*I*c*polylog(2,-I*sqrt(1+I*a*x)/sqrt(1-I*a*x))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)+2*I*c*polylog(2,I*sqrt(1+I*a*x)/sqrt(1-I*a*x))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)-2*c*polylog(3,-exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)+2*c*polylog(3,exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)+arctan(a*x)^2*sqrt(c+a^2*c*x^2)],
[arctan(a*x)^2*sqrt(c+a^2*c*x^2)/x^2,x,13,-2*I*a*c*arctan(exp(I*arctan(a*x)))*arctan(a*x)^2*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)-4*a*c*arctan(a*x)*arctanh(sqrt(1+I*a*x)/sqrt(1-I*a*x))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)+2*I*a*c*arctan(a*x)*polylog(2,-I*exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)-2*I*a*c*arctan(a*x)*polylog(2,I*exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)+2*I*a*c*polylog(2,-sqrt(1+I*a*x)/sqrt(1-I*a*x))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)-2*I*a*c*polylog(2,sqrt(1+I*a*x)/sqrt(1-I*a*x))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)-2*a*c*polylog(3,-I*exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)+2*a*c*polylog(3,I*exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)-arctan(a*x)^2*sqrt(c+a^2*c*x^2)/x],
[arctan(a*x)^2*sqrt(c+a^2*c*x^2)/x^3,x,24,-a^2*arctanh(sqrt(c+a^2*c*x^2)/sqrt(c))*sqrt(c)-a^2*c*arctan(a*x)^2*arctanh(exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)+I*a^2*c*arctan(a*x)*polylog(2,-exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)-I*a^2*c*arctan(a*x)*polylog(2,exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)-a^2*c*polylog(3,-exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)+a^2*c*polylog(3,exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)-a*arctan(a*x)*sqrt(c+a^2*c*x^2)/x-1/2*arctan(a*x)^2*sqrt(c+a^2*c*x^2)/x^2],
[arctan(a*x)^2*sqrt(c+a^2*c*x^2)/x^4,x,7,-1/3*(c+a^2*c*x^2)^(3/2)*arctan(a*x)^2/(c*x^3)-2/3*a^3*c*arctan(a*x)*arctanh(sqrt(1+I*a*x)/sqrt(1-I*a*x))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)+1/3*I*a^3*c*polylog(2,-sqrt(1+I*a*x)/sqrt(1-I*a*x))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)-1/3*I*a^3*c*polylog(2,sqrt(1+I*a*x)/sqrt(1-I*a*x))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)-1/3*a^2*sqrt(c+a^2*c*x^2)/x-1/3*a*arctan(a*x)*sqrt(c+a^2*c*x^2)/x^2],
[x^3*(c+a^2*c*x^2)^(3/2)*arctan(a*x)^2,x,75,-17/1260*(c+a^2*c*x^2)^(3/2)/a^4+1/105*(c+a^2*c*x^2)^(5/2)/(a^4*c)-17/140*I*c^2*arctan(a*x)*arctan(sqrt(1+I*a*x)/sqrt(1-I*a*x))*sqrt(1+a^2*x^2)/(a^4*sqrt(c+a^2*c*x^2))+17/280*I*c^2*polylog(2,-I*sqrt(1+I*a*x)/sqrt(1-I*a*x))*sqrt(1+a^2*x^2)/(a^4*sqrt(c+a^2*c*x^2))-17/280*I*c^2*polylog(2,I*sqrt(1+I*a*x)/sqrt(1-I*a*x))*sqrt(1+a^2*x^2)/(a^4*sqrt(c+a^2*c*x^2))-17/280*c*sqrt(c+a^2*c*x^2)/a^4+3/56*c*x*arctan(a*x)*sqrt(c+a^2*c*x^2)/a^3-23/420*c*x^3*arctan(a*x)*sqrt(c+a^2*c*x^2)/a-1/21*a*c*x^5*arctan(a*x)*sqrt(c+a^2*c*x^2)-2/35*c*arctan(a*x)^2*sqrt(c+a^2*c*x^2)/a^4+1/35*c*x^2*arctan(a*x)^2*sqrt(c+a^2*c*x^2)/a^2+8/35*c*x^4*arctan(a*x)^2*sqrt(c+a^2*c*x^2)+1/7*a^2*c*x^6*arctan(a*x)^2*sqrt(c+a^2*c*x^2)],
[x^2*(c+a^2*c*x^2)^(3/2)*arctan(a*x)^2,x,92,-41/360*c^(3/2)*arctanh(a*x*sqrt(c)/sqrt(c+a^2*c*x^2))/a^3+1/8*I*c^2*arctan(exp(I*arctan(a*x)))*arctan(a*x)^2*sqrt(1+a^2*x^2)/(a^3*sqrt(c+a^2*c*x^2))-1/8*I*c^2*arctan(a*x)*polylog(2,-I*exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/(a^3*sqrt(c+a^2*c*x^2))+1/8*I*c^2*arctan(a*x)*polylog(2,I*exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/(a^3*sqrt(c+a^2*c*x^2))+1/8*c^2*polylog(3,-I*exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/(a^3*sqrt(c+a^2*c*x^2))-1/8*c^2*polylog(3,I*exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/(a^3*sqrt(c+a^2*c*x^2))+1/36*c*x*sqrt(c+a^2*c*x^2)/a^2+1/60*c*x^3*sqrt(c+a^2*c*x^2)+31/360*c*arctan(a*x)*sqrt(c+a^2*c*x^2)/a^3-19/180*c*x^2*arctan(a*x)*sqrt(c+a^2*c*x^2)/a-1/15*a*c*x^4*arctan(a*x)*sqrt(c+a^2*c*x^2)+1/16*c*x*arctan(a*x)^2*sqrt(c+a^2*c*x^2)/a^2+7/24*c*x^3*arctan(a*x)^2*sqrt(c+a^2*c*x^2)+1/6*a^2*c*x^5*arctan(a*x)^2*sqrt(c+a^2*c*x^2)],
[x*(c+a^2*c*x^2)^(3/2)*arctan(a*x)^2,x,5,1/30*(c+a^2*c*x^2)^(3/2)/a^2-1/10*x*(c+a^2*c*x^2)^(3/2)*arctan(a*x)/a+1/5*(c+a^2*c*x^2)^(5/2)*arctan(a*x)^2/(a^2*c)+3/10*I*c^2*arctan(a*x)*arctan(sqrt(1+I*a*x)/sqrt(1-I*a*x))*sqrt(1+a^2*x^2)/(a^2*sqrt(c+a^2*c*x^2))-3/20*I*c^2*polylog(2,-I*sqrt(1+I*a*x)/sqrt(1-I*a*x))*sqrt(1+a^2*x^2)/(a^2*sqrt(c+a^2*c*x^2))+3/20*I*c^2*polylog(2,I*sqrt(1+I*a*x)/sqrt(1-I*a*x))*sqrt(1+a^2*x^2)/(a^2*sqrt(c+a^2*c*x^2))+3/20*c*sqrt(c+a^2*c*x^2)/a^2-3/20*c*x*arctan(a*x)*sqrt(c+a^2*c*x^2)/a],
[(c+a^2*c*x^2)^(3/2)*arctan(a*x)^2,x,16,-1/6*(c+a^2*c*x^2)^(3/2)*arctan(a*x)/a+1/4*x*(c+a^2*c*x^2)^(3/2)*arctan(a*x)^2+5/6*c^(3/2)*arctanh(a*x*sqrt(c)/sqrt(c+a^2*c*x^2))/a-3/4*I*c^2*arctan(exp(I*arctan(a*x)))*arctan(a*x)^2*sqrt(1+a^2*x^2)/(a*sqrt(c+a^2*c*x^2))+3/4*I*c^2*arctan(a*x)*polylog(2,-I*exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/(a*sqrt(c+a^2*c*x^2))-3/4*I*c^2*arctan(a*x)*polylog(2,I*exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/(a*sqrt(c+a^2*c*x^2))-3/4*c^2*polylog(3,-I*exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/(a*sqrt(c+a^2*c*x^2))+3/4*c^2*polylog(3,I*exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/(a*sqrt(c+a^2*c*x^2))+1/12*c*x*sqrt(c+a^2*c*x^2)-3/4*c*arctan(a*x)*sqrt(c+a^2*c*x^2)/a+3/8*c*x*arctan(a*x)^2*sqrt(c+a^2*c*x^2)],
[(c+a^2*c*x^2)^(3/2)*arctan(a*x)^2/x,x,18,1/3*(c+a^2*c*x^2)^(3/2)*arctan(a*x)^2+14/3*I*c^2*arctan(a*x)*arctan(sqrt(1+I*a*x)/sqrt(1-I*a*x))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)-2*c^2*arctan(a*x)^2*arctanh(exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)+2*I*c^2*arctan(a*x)*polylog(2,-exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)-2*I*c^2*arctan(a*x)*polylog(2,exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)-7/3*I*c^2*polylog(2,-I*sqrt(1+I*a*x)/sqrt(1-I*a*x))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)+7/3*I*c^2*polylog(2,I*sqrt(1+I*a*x)/sqrt(1-I*a*x))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)-2*c^2*polylog(3,-exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)+2*c^2*polylog(3,exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)+1/3*c*sqrt(c+a^2*c*x^2)-1/3*a*c*x*arctan(a*x)*sqrt(c+a^2*c*x^2)+c*arctan(a*x)^2*sqrt(c+a^2*c*x^2)],
[(c+a^2*c*x^2)^(3/2)*arctan(a*x)^2/x^2,x,26,a*c^(3/2)*arctanh(a*x*sqrt(c)/sqrt(c+a^2*c*x^2))-3*I*a*c^2*arctan(exp(I*arctan(a*x)))*arctan(a*x)^2*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)-4*a*c^2*arctan(a*x)*arctanh(sqrt(1+I*a*x)/sqrt(1-I*a*x))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)+3*I*a*c^2*arctan(a*x)*polylog(2,-I*exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)-3*I*a*c^2*arctan(a*x)*polylog(2,I*exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)+2*I*a*c^2*polylog(2,-sqrt(1+I*a*x)/sqrt(1-I*a*x))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)-2*I*a*c^2*polylog(2,sqrt(1+I*a*x)/sqrt(1-I*a*x))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)-3*a*c^2*polylog(3,-I*exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)+3*a*c^2*polylog(3,I*exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)-a*c*arctan(a*x)*sqrt(c+a^2*c*x^2)-c*arctan(a*x)^2*sqrt(c+a^2*c*x^2)/x+1/2*a^2*c*x*arctan(a*x)^2*sqrt(c+a^2*c*x^2)],
[(c+a^2*c*x^2)^(3/2)*arctan(a*x)^2/x^3,x,38,-a^2*c^(3/2)*arctanh(sqrt(c+a^2*c*x^2)/sqrt(c))+4*I*a^2*c^2*arctan(a*x)*arctan(sqrt(1+I*a*x)/sqrt(1-I*a*x))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)-3*a^2*c^2*arctan(a*x)^2*arctanh(exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)+3*I*a^2*c^2*arctan(a*x)*polylog(2,-exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)-3*I*a^2*c^2*arctan(a*x)*polylog(2,exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)-2*I*a^2*c^2*polylog(2,-I*sqrt(1+I*a*x)/sqrt(1-I*a*x))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)+2*I*a^2*c^2*polylog(2,I*sqrt(1+I*a*x)/sqrt(1-I*a*x))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)-3*a^2*c^2*polylog(3,-exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)+3*a^2*c^2*polylog(3,exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)-a*c*arctan(a*x)*sqrt(c+a^2*c*x^2)/x+a^2*c*arctan(a*x)^2*sqrt(c+a^2*c*x^2)-1/2*c*arctan(a*x)^2*sqrt(c+a^2*c*x^2)/x^2],
[(c+a^2*c*x^2)^(3/2)*arctan(a*x)^2/x^4,x,21,-1/3*(c+a^2*c*x^2)^(3/2)*arctan(a*x)^2/x^3-2*I*a^3*c^2*arctan(exp(I*arctan(a*x)))*arctan(a*x)^2*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)-14/3*a^3*c^2*arctan(a*x)*arctanh(sqrt(1+I*a*x)/sqrt(1-I*a*x))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)+2*I*a^3*c^2*arctan(a*x)*polylog(2,-I*exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)-2*I*a^3*c^2*arctan(a*x)*polylog(2,I*exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)+7/3*I*a^3*c^2*polylog(2,-sqrt(1+I*a*x)/sqrt(1-I*a*x))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)-7/3*I*a^3*c^2*polylog(2,sqrt(1+I*a*x)/sqrt(1-I*a*x))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)-2*a^3*c^2*polylog(3,-I*exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)+2*a^3*c^2*polylog(3,I*exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)-1/3*a^2*c*sqrt(c+a^2*c*x^2)/x-1/3*a*c*arctan(a*x)*sqrt(c+a^2*c*x^2)/x^2-a^2*c*arctan(a*x)^2*sqrt(c+a^2*c*x^2)/x],
[x^3*(c+a^2*c*x^2)^(5/2)*arctan(a*x)^2,x,203,-115/18144*c*(c+a^2*c*x^2)^(3/2)/a^4-23/7560*(c+a^2*c*x^2)^(5/2)/a^4+1/252*(c+a^2*c*x^2)^(7/2)/(a^4*c)-115/2016*I*c^3*arctan(a*x)*arctan(sqrt(1+I*a*x)/sqrt(1-I*a*x))*sqrt(1+a^2*x^2)/(a^4*sqrt(c+a^2*c*x^2))+115/4032*I*c^3*polylog(2,-I*sqrt(1+I*a*x)/sqrt(1-I*a*x))*sqrt(1+a^2*x^2)/(a^4*sqrt(c+a^2*c*x^2))-115/4032*I*c^3*polylog(2,I*sqrt(1+I*a*x)/sqrt(1-I*a*x))*sqrt(1+a^2*x^2)/(a^4*sqrt(c+a^2*c*x^2))-115/4032*c^2*sqrt(c+a^2*c*x^2)/a^4+47/1344*c^2*x*arctan(a*x)*sqrt(c+a^2*c*x^2)/a^3-205/6048*c^2*x^3*arctan(a*x)*sqrt(c+a^2*c*x^2)/a-103/1512*a*c^2*x^5*arctan(a*x)*sqrt(c+a^2*c*x^2)-1/36*a^3*c^2*x^7*arctan(a*x)*sqrt(c+a^2*c*x^2)-2/63*c^2*arctan(a*x)^2*sqrt(c+a^2*c*x^2)/a^4+1/63*c^2*x^2*arctan(a*x)^2*sqrt(c+a^2*c*x^2)/a^2+5/21*c^2*x^4*arctan(a*x)^2*sqrt(c+a^2*c*x^2)+19/63*a^2*c^2*x^6*arctan(a*x)^2*sqrt(c+a^2*c*x^2)+1/9*a^4*c^2*x^8*arctan(a*x)^2*sqrt(c+a^2*c*x^2)],
[x^2*(c+a^2*c*x^2)^(5/2)*arctan(a*x)^2,x,238,-397/5040*c^(5/2)*arctanh(a*x*sqrt(c)/sqrt(c+a^2*c*x^2))/a^3+5/64*I*c^3*arctan(exp(I*arctan(a*x)))*arctan(a*x)^2*sqrt(1+a^2*x^2)/(a^3*sqrt(c+a^2*c*x^2))-5/64*I*c^3*arctan(a*x)*polylog(2,-I*exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/(a^3*sqrt(c+a^2*c*x^2))+5/64*I*c^3*arctan(a*x)*polylog(2,I*exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/(a^3*sqrt(c+a^2*c*x^2))+5/64*c^3*polylog(3,-I*exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/(a^3*sqrt(c+a^2*c*x^2))-5/64*c^3*polylog(3,I*exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/(a^3*sqrt(c+a^2*c*x^2))+43/4032*c^2*x*sqrt(c+a^2*c*x^2)/a^2+29/1680*c^2*x^3*sqrt(c+a^2*c*x^2)+1/168*a^2*c^2*x^5*sqrt(c+a^2*c*x^2)+1373/20160*c^2*arctan(a*x)*sqrt(c+a^2*c*x^2)/a^3-737/10080*c^2*x^2*arctan(a*x)*sqrt(c+a^2*c*x^2)/a-83/840*a*c^2*x^4*arctan(a*x)*sqrt(c+a^2*c*x^2)-1/28*a^3*c^2*x^6*arctan(a*x)*sqrt(c+a^2*c*x^2)+5/128*c^2*x*arctan(a*x)^2*sqrt(c+a^2*c*x^2)/a^2+59/192*c^2*x^3*arctan(a*x)^2*sqrt(c+a^2*c*x^2)+17/48*a^2*c^2*x^5*arctan(a*x)^2*sqrt(c+a^2*c*x^2)+1/8*a^4*c^2*x^7*arctan(a*x)^2*sqrt(c+a^2*c*x^2)],
[x*(c+a^2*c*x^2)^(5/2)*arctan(a*x)^2,x,6,5/252*c*(c+a^2*c*x^2)^(3/2)/a^2+1/105*(c+a^2*c*x^2)^(5/2)/a^2-5/84*c*x*(c+a^2*c*x^2)^(3/2)*arctan(a*x)/a-1/21*x*(c+a^2*c*x^2)^(5/2)*arctan(a*x)/a+1/7*(c+a^2*c*x^2)^(7/2)*arctan(a*x)^2/(a^2*c)+5/28*I*c^3*arctan(a*x)*arctan(sqrt(1+I*a*x)/sqrt(1-I*a*x))*sqrt(1+a^2*x^2)/(a^2*sqrt(c+a^2*c*x^2))-5/56*I*c^3*polylog(2,-I*sqrt(1+I*a*x)/sqrt(1-I*a*x))*sqrt(1+a^2*x^2)/(a^2*sqrt(c+a^2*c*x^2))+5/56*I*c^3*polylog(2,I*sqrt(1+I*a*x)/sqrt(1-I*a*x))*sqrt(1+a^2*x^2)/(a^2*sqrt(c+a^2*c*x^2))+5/56*c^2*sqrt(c+a^2*c*x^2)/a^2-5/56*c^2*x*arctan(a*x)*sqrt(c+a^2*c*x^2)/a],
[(c+a^2*c*x^2)^(5/2)*arctan(a*x)^2,x,21,1/60*c*x*(c+a^2*c*x^2)^(3/2)-5/36*c*(c+a^2*c*x^2)^(3/2)*arctan(a*x)/a-1/15*(c+a^2*c*x^2)^(5/2)*arctan(a*x)/a+5/24*c*x*(c+a^2*c*x^2)^(3/2)*arctan(a*x)^2+1/6*x*(c+a^2*c*x^2)^(5/2)*arctan(a*x)^2+259/360*c^(5/2)*arctanh(a*x*sqrt(c)/sqrt(c+a^2*c*x^2))/a-5/8*I*c^3*arctan(exp(I*arctan(a*x)))*arctan(a*x)^2*sqrt(1+a^2*x^2)/(a*sqrt(c+a^2*c*x^2))+5/8*I*c^3*arctan(a*x)*polylog(2,-I*exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/(a*sqrt(c+a^2*c*x^2))-5/8*I*c^3*arctan(a*x)*polylog(2,I*exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/(a*sqrt(c+a^2*c*x^2))-5/8*c^3*polylog(3,-I*exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/(a*sqrt(c+a^2*c*x^2))+5/8*c^3*polylog(3,I*exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/(a*sqrt(c+a^2*c*x^2))+17/180*c^2*x*sqrt(c+a^2*c*x^2)-5/8*c^2*arctan(a*x)*sqrt(c+a^2*c*x^2)/a+5/16*c^2*x*arctan(a*x)^2*sqrt(c+a^2*c*x^2)],
[(c+a^2*c*x^2)^(5/2)*arctan(a*x)^2/x,x,24,1/30*c*(c+a^2*c*x^2)^(3/2)-1/10*a*c*x*(c+a^2*c*x^2)^(3/2)*arctan(a*x)+1/3*c*(c+a^2*c*x^2)^(3/2)*arctan(a*x)^2+1/5*(c+a^2*c*x^2)^(5/2)*arctan(a*x)^2+149/30*I*c^3*arctan(a*x)*arctan(sqrt(1+I*a*x)/sqrt(1-I*a*x))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)-2*c^3*arctan(a*x)^2*arctanh(exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)+2*I*c^3*arctan(a*x)*polylog(2,-exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)-2*I*c^3*arctan(a*x)*polylog(2,exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)-149/60*I*c^3*polylog(2,-I*sqrt(1+I*a*x)/sqrt(1-I*a*x))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)+149/60*I*c^3*polylog(2,I*sqrt(1+I*a*x)/sqrt(1-I*a*x))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)-2*c^3*polylog(3,-exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)+2*c^3*polylog(3,exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)+29/60*c^2*sqrt(c+a^2*c*x^2)-29/60*a*c^2*x*arctan(a*x)*sqrt(c+a^2*c*x^2)+c^2*arctan(a*x)^2*sqrt(c+a^2*c*x^2)],
[(c+a^2*c*x^2)^(5/2)*arctan(a*x)^2/x^2,x,43,-1/6*a*c*(c+a^2*c*x^2)^(3/2)*arctan(a*x)+1/4*a^2*c*x*(c+a^2*c*x^2)^(3/2)*arctan(a*x)^2+11/6*a*c^(5/2)*arctanh(a*x*sqrt(c)/sqrt(c+a^2*c*x^2))-15/4*I*a*c^3*arctan(exp(I*arctan(a*x)))*arctan(a*x)^2*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)-4*a*c^3*arctan(a*x)*arctanh(sqrt(1+I*a*x)/sqrt(1-I*a*x))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)+15/4*I*a*c^3*arctan(a*x)*polylog(2,-I*exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)-15/4*I*a*c^3*arctan(a*x)*polylog(2,I*exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)+2*I*a*c^3*polylog(2,-sqrt(1+I*a*x)/sqrt(1-I*a*x))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)-2*I*a*c^3*polylog(2,sqrt(1+I*a*x)/sqrt(1-I*a*x))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)-15/4*a*c^3*polylog(3,-I*exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)+15/4*a*c^3*polylog(3,I*exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)+1/12*a^2*c^2*x*sqrt(c+a^2*c*x^2)-7/4*a*c^2*arctan(a*x)*sqrt(c+a^2*c*x^2)-c^2*arctan(a*x)^2*sqrt(c+a^2*c*x^2)/x+7/8*a^2*c^2*x*arctan(a*x)^2*sqrt(c+a^2*c*x^2)],
[(c+a^2*c*x^2)^(5/2)*arctan(a*x)^2/x^3,x,57,1/3*a^2*c*(c+a^2*c*x^2)^(3/2)*arctan(a*x)^2-a^2*c^(5/2)*arctanh(sqrt(c+a^2*c*x^2)/sqrt(c))+26/3*I*a^2*c^3*arctan(a*x)*arctan(sqrt(1+I*a*x)/sqrt(1-I*a*x))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)-5*a^2*c^3*arctan(a*x)^2*arctanh(exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)+5*I*a^2*c^3*arctan(a*x)*polylog(2,-exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)-5*I*a^2*c^3*arctan(a*x)*polylog(2,exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)-13/3*I*a^2*c^3*polylog(2,-I*sqrt(1+I*a*x)/sqrt(1-I*a*x))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)+13/3*I*a^2*c^3*polylog(2,I*sqrt(1+I*a*x)/sqrt(1-I*a*x))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)-5*a^2*c^3*polylog(3,-exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)+5*a^2*c^3*polylog(3,exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)+1/3*a^2*c^2*sqrt(c+a^2*c*x^2)-a*c^2*arctan(a*x)*sqrt(c+a^2*c*x^2)/x-1/3*a^3*c^2*x*arctan(a*x)*sqrt(c+a^2*c*x^2)+2*a^2*c^2*arctan(a*x)^2*sqrt(c+a^2*c*x^2)-1/2*c^2*arctan(a*x)^2*sqrt(c+a^2*c*x^2)/x^2],
[(c+a^2*c*x^2)^(5/2)*arctan(a*x)^2/x^4,x,48,-1/3*c*(c+a^2*c*x^2)^(3/2)*arctan(a*x)^2/x^3+a^3*c^(5/2)*arctanh(a*x*sqrt(c)/sqrt(c+a^2*c*x^2))-5*I*a^3*c^3*arctan(exp(I*arctan(a*x)))*arctan(a*x)^2*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)-26/3*a^3*c^3*arctan(a*x)*arctanh(sqrt(1+I*a*x)/sqrt(1-I*a*x))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)+5*I*a^3*c^3*arctan(a*x)*polylog(2,-I*exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)-5*I*a^3*c^3*arctan(a*x)*polylog(2,I*exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)+13/3*I*a^3*c^3*polylog(2,-sqrt(1+I*a*x)/sqrt(1-I*a*x))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)-13/3*I*a^3*c^3*polylog(2,sqrt(1+I*a*x)/sqrt(1-I*a*x))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)-5*a^3*c^3*polylog(3,-I*exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)+5*a^3*c^3*polylog(3,I*exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)-1/3*a^2*c^2*sqrt(c+a^2*c*x^2)/x-a^3*c^2*arctan(a*x)*sqrt(c+a^2*c*x^2)-1/3*a*c^2*arctan(a*x)*sqrt(c+a^2*c*x^2)/x^2-2*a^2*c^2*arctan(a*x)^2*sqrt(c+a^2*c*x^2)/x+1/2*a^4*c^2*x*arctan(a*x)^2*sqrt(c+a^2*c*x^2)],

# p<0
[x^3*arctan(a*x)^2/sqrt(c+a^2*c*x^2),x,8,-10/3*I*arctan(a*x)*arctan(sqrt(1+I*a*x)/sqrt(1-I*a*x))*sqrt(1+a^2*x^2)/(a^4*sqrt(c+a^2*c*x^2))+5/3*I*polylog(2,-I*sqrt(1+I*a*x)/sqrt(1-I*a*x))*sqrt(1+a^2*x^2)/(a^4*sqrt(c+a^2*c*x^2))-5/3*I*polylog(2,I*sqrt(1+I*a*x)/sqrt(1-I*a*x))*sqrt(1+a^2*x^2)/(a^4*sqrt(c+a^2*c*x^2))+1/3*sqrt(c+a^2*c*x^2)/(a^4*c)-1/3*x*arctan(a*x)*sqrt(c+a^2*c*x^2)/(a^3*c)-2/3*arctan(a*x)^2*sqrt(c+a^2*c*x^2)/(a^4*c)+1/3*x^2*arctan(a*x)^2*sqrt(c+a^2*c*x^2)/(a^2*c)],
[x^2*arctan(a*x)^2/sqrt(c+a^2*c*x^2),x,13,arctanh(a*x*sqrt(c)/sqrt(c+a^2*c*x^2))/(a^3*sqrt(c))+I*arctan(exp(I*arctan(a*x)))*arctan(a*x)^2*sqrt(1+a^2*x^2)/(a^3*sqrt(c+a^2*c*x^2))-I*arctan(a*x)*polylog(2,-I*exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/(a^3*sqrt(c+a^2*c*x^2))+I*arctan(a*x)*polylog(2,I*exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/(a^3*sqrt(c+a^2*c*x^2))+polylog(3,-I*exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/(a^3*sqrt(c+a^2*c*x^2))-polylog(3,I*exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/(a^3*sqrt(c+a^2*c*x^2))-arctan(a*x)*sqrt(c+a^2*c*x^2)/(a^3*c)+1/2*x*arctan(a*x)^2*sqrt(c+a^2*c*x^2)/(a^2*c)],
[x*arctan(a*x)^2/sqrt(c+a^2*c*x^2),x,3,4*I*arctan(a*x)*arctan(sqrt(1+I*a*x)/sqrt(1-I*a*x))*sqrt(1+a^2*x^2)/(a^2*sqrt(c+a^2*c*x^2))-2*I*polylog(2,-I*sqrt(1+I*a*x)/sqrt(1-I*a*x))*sqrt(1+a^2*x^2)/(a^2*sqrt(c+a^2*c*x^2))+2*I*polylog(2,I*sqrt(1+I*a*x)/sqrt(1-I*a*x))*sqrt(1+a^2*x^2)/(a^2*sqrt(c+a^2*c*x^2))+arctan(a*x)^2*sqrt(c+a^2*c*x^2)/(a^2*c)],
[arctan(a*x)^2/sqrt(c+a^2*c*x^2),x,9,-2*I*arctan(exp(I*arctan(a*x)))*arctan(a*x)^2*sqrt(1+a^2*x^2)/(a*sqrt(c+a^2*c*x^2))+2*I*arctan(a*x)*polylog(2,-I*exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/(a*sqrt(c+a^2*c*x^2))-2*I*arctan(a*x)*polylog(2,I*exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/(a*sqrt(c+a^2*c*x^2))-2*polylog(3,-I*exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/(a*sqrt(c+a^2*c*x^2))+2*polylog(3,I*exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/(a*sqrt(c+a^2*c*x^2))],
[arctan(a*x)^2/(x*sqrt(c+a^2*c*x^2)),x,9,-2*arctan(a*x)^2*arctanh(exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)+2*I*arctan(a*x)*polylog(2,-exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)-2*I*arctan(a*x)*polylog(2,exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)-2*polylog(3,-exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)+2*polylog(3,exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)],
[arctan(a*x)^2/(x^2*sqrt(c+a^2*c*x^2)),x,3,-4*a*arctan(a*x)*arctanh(sqrt(1+I*a*x)/sqrt(1-I*a*x))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)+2*I*a*polylog(2,-sqrt(1+I*a*x)/sqrt(1-I*a*x))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)-2*I*a*polylog(2,sqrt(1+I*a*x)/sqrt(1-I*a*x))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)-arctan(a*x)^2*sqrt(c+a^2*c*x^2)/(c*x)],
[arctan(a*x)^2/(x^3*sqrt(c+a^2*c*x^2)),x,14,-a^2*arctanh(sqrt(c+a^2*c*x^2)/sqrt(c))/sqrt(c)+a^2*arctan(a*x)^2*arctanh(exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)-I*a^2*arctan(a*x)*polylog(2,-exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)+I*a^2*arctan(a*x)*polylog(2,exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)+a^2*polylog(3,-exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)-a^2*polylog(3,exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)-a*arctan(a*x)*sqrt(c+a^2*c*x^2)/(c*x)-1/2*arctan(a*x)^2*sqrt(c+a^2*c*x^2)/(c*x^2)],
[arctan(a*x)^2/(x^4*sqrt(c+a^2*c*x^2)),x,8,10/3*a^3*arctan(a*x)*arctanh(sqrt(1+I*a*x)/sqrt(1-I*a*x))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)-5/3*I*a^3*polylog(2,-sqrt(1+I*a*x)/sqrt(1-I*a*x))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)+5/3*I*a^3*polylog(2,sqrt(1+I*a*x)/sqrt(1-I*a*x))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)-1/3*a^2*sqrt(c+a^2*c*x^2)/(c*x)-1/3*a*arctan(a*x)*sqrt(c+a^2*c*x^2)/(c*x^2)-1/3*arctan(a*x)^2*sqrt(c+a^2*c*x^2)/(c*x^3)+2/3*a^2*arctan(a*x)^2*sqrt(c+a^2*c*x^2)/(c*x)],
[x^3*arctan(a*x)^2/(c+a^2*c*x^2)^(3/2),x,6,(-2)/(a^4*c*sqrt(c+a^2*c*x^2))-2*x*arctan(a*x)/(a^3*c*sqrt(c+a^2*c*x^2))+arctan(a*x)^2/(a^4*c*sqrt(c+a^2*c*x^2))+4*I*arctan(a*x)*arctan(sqrt(1+I*a*x)/sqrt(1-I*a*x))*sqrt(1+a^2*x^2)/(a^4*c*sqrt(c+a^2*c*x^2))-2*I*polylog(2,-I*sqrt(1+I*a*x)/sqrt(1-I*a*x))*sqrt(1+a^2*x^2)/(a^4*c*sqrt(c+a^2*c*x^2))+2*I*polylog(2,I*sqrt(1+I*a*x)/sqrt(1-I*a*x))*sqrt(1+a^2*x^2)/(a^4*c*sqrt(c+a^2*c*x^2))+arctan(a*x)^2*sqrt(c+a^2*c*x^2)/(a^4*c^2)],
[x^2*arctan(a*x)^2/(c+a^2*c*x^2)^(3/2),x,12,2*x/(a^2*c*sqrt(c+a^2*c*x^2))-2*arctan(a*x)/(a^3*c*sqrt(c+a^2*c*x^2))-x*arctan(a*x)^2/(a^2*c*sqrt(c+a^2*c*x^2))-2*I*arctan(exp(I*arctan(a*x)))*arctan(a*x)^2*sqrt(1+a^2*x^2)/(a^3*c*sqrt(c+a^2*c*x^2))+2*I*arctan(a*x)*polylog(2,-I*exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/(a^3*c*sqrt(c+a^2*c*x^2))-2*I*arctan(a*x)*polylog(2,I*exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/(a^3*c*sqrt(c+a^2*c*x^2))-2*polylog(3,-I*exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/(a^3*c*sqrt(c+a^2*c*x^2))+2*polylog(3,I*exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/(a^3*c*sqrt(c+a^2*c*x^2))],
[x*arctan(a*x)^2/(c+a^2*c*x^2)^(3/2),x,2,2/(a^2*c*sqrt(c+a^2*c*x^2))+2*x*arctan(a*x)/(a*c*sqrt(c+a^2*c*x^2))-arctan(a*x)^2/(a^2*c*sqrt(c+a^2*c*x^2))],
[arctan(a*x)^2/(c+a^2*c*x^2)^(3/2),x,2,-2*x/(c*sqrt(c+a^2*c*x^2))+2*arctan(a*x)/(a*c*sqrt(c+a^2*c*x^2))+x*arctan(a*x)^2/(c*sqrt(c+a^2*c*x^2))],
[arctan(a*x)^2/(x*(c+a^2*c*x^2)^(3/2)),x,12,(-2)/(c*sqrt(c+a^2*c*x^2))-2*a*x*arctan(a*x)/(c*sqrt(c+a^2*c*x^2))+arctan(a*x)^2/(c*sqrt(c+a^2*c*x^2))-2*arctan(a*x)^2*arctanh(exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/(c*sqrt(c+a^2*c*x^2))+2*I*arctan(a*x)*polylog(2,-exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/(c*sqrt(c+a^2*c*x^2))-2*I*arctan(a*x)*polylog(2,exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/(c*sqrt(c+a^2*c*x^2))-2*polylog(3,-exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/(c*sqrt(c+a^2*c*x^2))+2*polylog(3,exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/(c*sqrt(c+a^2*c*x^2))],
[arctan(a*x)^2/(x^2*(c+a^2*c*x^2)^(3/2)),x,6,2*a^2*x/(c*sqrt(c+a^2*c*x^2))-2*a*arctan(a*x)/(c*sqrt(c+a^2*c*x^2))-a^2*x*arctan(a*x)^2/(c*sqrt(c+a^2*c*x^2))-4*a*arctan(a*x)*arctanh(sqrt(1+I*a*x)/sqrt(1-I*a*x))*sqrt(1+a^2*x^2)/(c*sqrt(c+a^2*c*x^2))+2*I*a*polylog(2,-sqrt(1+I*a*x)/sqrt(1-I*a*x))*sqrt(1+a^2*x^2)/(c*sqrt(c+a^2*c*x^2))-2*I*a*polylog(2,sqrt(1+I*a*x)/sqrt(1-I*a*x))*sqrt(1+a^2*x^2)/(c*sqrt(c+a^2*c*x^2))-arctan(a*x)^2*sqrt(c+a^2*c*x^2)/(c^2*x)],
[arctan(a*x)^2/(x^3*(c+a^2*c*x^2)^(3/2)),x,27,-a^2*arctanh(sqrt(c+a^2*c*x^2)/sqrt(c))/c^(3/2)+2*a^2/(c*sqrt(c+a^2*c*x^2))+2*a^3*x*arctan(a*x)/(c*sqrt(c+a^2*c*x^2))-a^2*arctan(a*x)^2/(c*sqrt(c+a^2*c*x^2))+3*a^2*arctan(a*x)^2*arctanh(exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/(c*sqrt(c+a^2*c*x^2))-3*I*a^2*arctan(a*x)*polylog(2,-exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/(c*sqrt(c+a^2*c*x^2))+3*I*a^2*arctan(a*x)*polylog(2,exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/(c*sqrt(c+a^2*c*x^2))+3*a^2*polylog(3,-exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/(c*sqrt(c+a^2*c*x^2))-3*a^2*polylog(3,exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/(c*sqrt(c+a^2*c*x^2))-a*arctan(a*x)*sqrt(c+a^2*c*x^2)/(c^2*x)-1/2*arctan(a*x)^2*sqrt(c+a^2*c*x^2)/(c^2*x^2)],
[arctan(a*x)^2/(x^4*(c+a^2*c*x^2)^(3/2)),x,15,-2*a^4*x/(c*sqrt(c+a^2*c*x^2))+2*a^3*arctan(a*x)/(c*sqrt(c+a^2*c*x^2))+a^4*x*arctan(a*x)^2/(c*sqrt(c+a^2*c*x^2))+22/3*a^3*arctan(a*x)*arctanh(sqrt(1+I*a*x)/sqrt(1-I*a*x))*sqrt(1+a^2*x^2)/(c*sqrt(c+a^2*c*x^2))-11/3*I*a^3*polylog(2,-sqrt(1+I*a*x)/sqrt(1-I*a*x))*sqrt(1+a^2*x^2)/(c*sqrt(c+a^2*c*x^2))+11/3*I*a^3*polylog(2,sqrt(1+I*a*x)/sqrt(1-I*a*x))*sqrt(1+a^2*x^2)/(c*sqrt(c+a^2*c*x^2))-1/3*a^2*sqrt(c+a^2*c*x^2)/(c^2*x)-1/3*a*arctan(a*x)*sqrt(c+a^2*c*x^2)/(c^2*x^2)-1/3*arctan(a*x)^2*sqrt(c+a^2*c*x^2)/(c^2*x^3)+5/3*a^2*arctan(a*x)^2*sqrt(c+a^2*c*x^2)/(c^2*x)],
[x^5*arctan(a*x)^2/(c+a^2*c*x^2)^(5/2),x,13,2/27/(a^6*c*(c+a^2*c*x^2)^(3/2))-2/9*x^3*arctan(a*x)/(a^3*c*(c+a^2*c*x^2)^(3/2))+1/3*x^2*arctan(a*x)^2/(a^4*c*(c+a^2*c*x^2)^(3/2))+(-32/9)/(a^6*c^2*sqrt(c+a^2*c*x^2))-10/3*x*arctan(a*x)/(a^5*c^2*sqrt(c+a^2*c*x^2))+5/3*arctan(a*x)^2/(a^6*c^2*sqrt(c+a^2*c*x^2))+4*I*arctan(a*x)*arctan(sqrt(1+I*a*x)/sqrt(1-I*a*x))*sqrt(1+a^2*x^2)/(a^6*c^2*sqrt(c+a^2*c*x^2))-2*I*polylog(2,-I*sqrt(1+I*a*x)/sqrt(1-I*a*x))*sqrt(1+a^2*x^2)/(a^6*c^2*sqrt(c+a^2*c*x^2))+2*I*polylog(2,I*sqrt(1+I*a*x)/sqrt(1-I*a*x))*sqrt(1+a^2*x^2)/(a^6*c^2*sqrt(c+a^2*c*x^2))+arctan(a*x)^2*sqrt(c+a^2*c*x^2)/(a^6*c^3)],
[x^4*arctan(a*x)^2/(c+a^2*c*x^2)^(5/2),x,17,2/27*x^3/(a^2*c*(c+a^2*c*x^2)^(3/2))-2/9*x^2*arctan(a*x)/(a^3*c*(c+a^2*c*x^2)^(3/2))-1/3*x^3*arctan(a*x)^2/(a^2*c*(c+a^2*c*x^2)^(3/2))+22/9*x/(a^4*c^2*sqrt(c+a^2*c*x^2))-22/9*arctan(a*x)/(a^5*c^2*sqrt(c+a^2*c*x^2))-x*arctan(a*x)^2/(a^4*c^2*sqrt(c+a^2*c*x^2))-2*I*arctan(exp(I*arctan(a*x)))*arctan(a*x)^2*sqrt(1+a^2*x^2)/(a^5*c^2*sqrt(c+a^2*c*x^2))+2*I*arctan(a*x)*polylog(2,-I*exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/(a^5*c^2*sqrt(c+a^2*c*x^2))-2*I*arctan(a*x)*polylog(2,I*exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/(a^5*c^2*sqrt(c+a^2*c*x^2))-2*polylog(3,-I*exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/(a^5*c^2*sqrt(c+a^2*c*x^2))+2*polylog(3,I*exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/(a^5*c^2*sqrt(c+a^2*c*x^2))],
[x^3*arctan(a*x)^2/(c+a^2*c*x^2)^(5/2),x,6,(-2/27)/(a^4*c*(c+a^2*c*x^2)^(3/2))+2/9*x^3*arctan(a*x)/(a*c*(c+a^2*c*x^2)^(3/2))-1/3*x^2*arctan(a*x)^2/(a^2*c*(c+a^2*c*x^2)^(3/2))+14/9/(a^4*c^2*sqrt(c+a^2*c*x^2))+4/3*x*arctan(a*x)/(a^3*c^2*sqrt(c+a^2*c*x^2))-2/3*arctan(a*x)^2/(a^4*c^2*sqrt(c+a^2*c*x^2))],
[x^2*arctan(a*x)^2/(c+a^2*c*x^2)^(5/2),x,4,-2/27*x^3/(c*(c+a^2*c*x^2)^(3/2))+2/9*x^2*arctan(a*x)/(a*c*(c+a^2*c*x^2)^(3/2))+1/3*x^3*arctan(a*x)^2/(c*(c+a^2*c*x^2)^(3/2))-4/9*x/(a^2*c^2*sqrt(c+a^2*c*x^2))+4/9*arctan(a*x)/(a^3*c^2*sqrt(c+a^2*c*x^2))],
[x*arctan(a*x)^2/(c+a^2*c*x^2)^(5/2),x,3,2/27/(a^2*c*(c+a^2*c*x^2)^(3/2))+2/9*x*arctan(a*x)/(a*c*(c+a^2*c*x^2)^(3/2))-1/3*arctan(a*x)^2/(a^2*c*(c+a^2*c*x^2)^(3/2))+4/9/(a^2*c^2*sqrt(c+a^2*c*x^2))+4/9*x*arctan(a*x)/(a*c^2*sqrt(c+a^2*c*x^2))],
[arctan(a*x)^2/(c+a^2*c*x^2)^(5/2),x,5,-2/27*x/(c*(c+a^2*c*x^2)^(3/2))+2/9*arctan(a*x)/(a*c*(c+a^2*c*x^2)^(3/2))+1/3*x*arctan(a*x)^2/(c*(c+a^2*c*x^2)^(3/2))-40/27*x/(c^2*sqrt(c+a^2*c*x^2))+4/3*arctan(a*x)/(a*c^2*sqrt(c+a^2*c*x^2))+2/3*x*arctan(a*x)^2/(c^2*sqrt(c+a^2*c*x^2))],
[arctan(a*x)^2/(x*(c+a^2*c*x^2)^(5/2)),x,16,(-2/27)/(c*(c+a^2*c*x^2)^(3/2))-2/9*a*x*arctan(a*x)/(c*(c+a^2*c*x^2)^(3/2))+1/3*arctan(a*x)^2/(c*(c+a^2*c*x^2)^(3/2))+(-22/9)/(c^2*sqrt(c+a^2*c*x^2))-22/9*a*x*arctan(a*x)/(c^2*sqrt(c+a^2*c*x^2))+arctan(a*x)^2/(c^2*sqrt(c+a^2*c*x^2))-2*arctan(a*x)^2*arctanh(exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/(c^2*sqrt(c+a^2*c*x^2))+2*I*arctan(a*x)*polylog(2,-exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/(c^2*sqrt(c+a^2*c*x^2))-2*I*arctan(a*x)*polylog(2,exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/(c^2*sqrt(c+a^2*c*x^2))-2*polylog(3,-exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/(c^2*sqrt(c+a^2*c*x^2))+2*polylog(3,exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/(c^2*sqrt(c+a^2*c*x^2))],
[arctan(a*x)^2/(x^2*(c+a^2*c*x^2)^(5/2)),x,12,2/27*a^2*x/(c*(c+a^2*c*x^2)^(3/2))-2/9*a*arctan(a*x)/(c*(c+a^2*c*x^2)^(3/2))-1/3*a^2*x*arctan(a*x)^2/(c*(c+a^2*c*x^2)^(3/2))+94/27*a^2*x/(c^2*sqrt(c+a^2*c*x^2))-10/3*a*arctan(a*x)/(c^2*sqrt(c+a^2*c*x^2))-5/3*a^2*x*arctan(a*x)^2/(c^2*sqrt(c+a^2*c*x^2))-4*a*arctan(a*x)*arctanh(sqrt(1+I*a*x)/sqrt(1-I*a*x))*sqrt(1+a^2*x^2)/(c^2*sqrt(c+a^2*c*x^2))+2*I*a*polylog(2,-sqrt(1+I*a*x)/sqrt(1-I*a*x))*sqrt(1+a^2*x^2)/(c^2*sqrt(c+a^2*c*x^2))-2*I*a*polylog(2,sqrt(1+I*a*x)/sqrt(1-I*a*x))*sqrt(1+a^2*x^2)/(c^2*sqrt(c+a^2*c*x^2))-arctan(a*x)^2*sqrt(c+a^2*c*x^2)/(c^3*x)],

# Integrands of the form x^m (c+d x^2)^p ArcTan[a x]^2 with d=a^2 c and m symbolic
[x^m*(c+a^2*c*x^2)^2*arctan(a*x)^2,x,0,Unintegrable(x^m*(c+a^2*c*x^2)^2*arctan(a*x)^2,x)],
[x^m*(c+a^2*c*x^2)*arctan(a*x)^2,x,0,Unintegrable(x^m*(c+a^2*c*x^2)*arctan(a*x)^2,x)],
[x^m*arctan(a*x)^2/(c+a^2*c*x^2),x,0,Unintegrable(x^m*arctan(a*x)^2/(c+a^2*c*x^2),x)],
[x^m*arctan(a*x)^2/(c+a^2*c*x^2)^2,x,0,Unintegrable(x^m*arctan(a*x)^2/(c+a^2*c*x^2)^2,x)],
[x^m*(c+a^2*c*x^2)^(3/2)*arctan(a*x)^2,x,0,Unintegrable(x^m*(c+a^2*c*x^2)^(3/2)*arctan(a*x)^2,x)],
[x^m*(c+a^2*c*x^2)^(1/2)*arctan(a*x)^2,x,0,Unintegrable(x^m*arctan(a*x)^2*sqrt(c+a^2*c*x^2),x)],
[x^m*arctan(a*x)^2/(c+a^2*c*x^2)^(1/2),x,0,Unintegrable(x^m*arctan(a*x)^2/sqrt(c+a^2*c*x^2),x)],
[x^m*arctan(a*x)^2/(c+a^2*c*x^2)^(3/2),x,0,Unintegrable(x^m*arctan(a*x)^2/(c+a^2*c*x^2)^(3/2),x)],

# Integrands of the form (f x)^m (d+c^2 d x^2)^q (a+b ArcTan[c x])^3

# Integrands of the form x^m (c+d x^2)^p ArcTan[a x]^3 with d=a^2 c

# p>0
[x^3*(c+a^2*c*x^2)*arctan(a*x)^3,x,52,1/15*c*x/a^3-1/60*c*x^3/a-1/15*c*arctan(a*x)/a^4-1/60*c*x^2*arctan(a*x)/a^2+1/20*c*x^4*arctan(a*x)+7/30*I*c*arctan(a*x)^2/a^4+1/4*c*x*arctan(a*x)^2/a^3-1/12*c*x^3*arctan(a*x)^2/a-1/10*a*c*x^5*arctan(a*x)^2-1/12*c*arctan(a*x)^3/a^4+1/4*c*x^4*arctan(a*x)^3+1/6*a^2*c*x^6*arctan(a*x)^3+7/15*c*arctan(a*x)*log(2/(1+I*a*x))/a^4+7/30*I*c*polylog(2,1+(-2)/(1+I*a*x))/a^4],
[x^2*(c+a^2*c*x^2)*arctan(a*x)^3,x,34,-1/20*c*x^2/a+1/10*c*x*arctan(a*x)/a^2+1/10*c*x^3*arctan(a*x)-1/20*c*arctan(a*x)^2/a^3-1/5*c*x^2*arctan(a*x)^2/a-3/20*a*c*x^4*arctan(a*x)^2-2/15*I*c*arctan(a*x)^3/a^3+1/3*c*x^3*arctan(a*x)^3+1/5*a^2*c*x^5*arctan(a*x)^3-2/5*c*arctan(a*x)^2*log(2/(1+I*a*x))/a^3-2/5*I*c*arctan(a*x)*polylog(2,1+(-2)/(1+I*a*x))/a^3-1/5*c*polylog(3,1+(-2)/(1+I*a*x))/a^3],
[x*(c+a^2*c*x^2)*arctan(a*x)^3,x,8,-1/4*c*x/a+1/4*c*(1+a^2*x^2)*arctan(a*x)/a^2-1/2*I*c*arctan(a*x)^2/a^2-1/2*c*x*arctan(a*x)^2/a-1/4*c*x*(1+a^2*x^2)*arctan(a*x)^2/a+1/4*c*(1+a^2*x^2)^2*arctan(a*x)^3/a^2-c*arctan(a*x)*log(2/(1+I*a*x))/a^2-1/2*I*c*polylog(2,1+(-2)/(1+I*a*x))/a^2],
[(c+a^2*c*x^2)*arctan(a*x)^3,x,8,c*x*arctan(a*x)-1/2*c*(1+a^2*x^2)*arctan(a*x)^2/a+2/3*I*c*arctan(a*x)^3/a+2/3*c*x*arctan(a*x)^3+1/3*c*x*(1+a^2*x^2)*arctan(a*x)^3+2*c*arctan(a*x)^2*log(2/(1+I*a*x))/a-1/2*c*log(1+a^2*x^2)/a+2*I*c*arctan(a*x)*polylog(2,1+(-2)/(1+I*a*x))/a+c*polylog(3,1+(-2)/(1+I*a*x))/a],
[(c+a^2*c*x^2)*arctan(a*x)^3/x,x,17,-3/2*I*c*arctan(a*x)^2-3/2*a*c*x*arctan(a*x)^2+1/2*c*arctan(a*x)^3+1/2*a^2*c*x^2*arctan(a*x)^3+2*c*arctan(a*x)^3*arctanh(1+(-2)/(1+I*a*x))-3*c*arctan(a*x)*log(2/(1+I*a*x))-3/2*I*c*polylog(2,1+(-2)/(1+I*a*x))-3/2*I*c*arctan(a*x)^2*polylog(2,1+(-2)/(1+I*a*x))+3/2*I*c*arctan(a*x)^2*polylog(2,-1+2/(1+I*a*x))-3/2*c*arctan(a*x)*polylog(3,1+(-2)/(1+I*a*x))+3/2*c*arctan(a*x)*polylog(3,-1+2/(1+I*a*x))+3/4*I*c*polylog(4,1+(-2)/(1+I*a*x))-3/4*I*c*polylog(4,-1+2/(1+I*a*x))],
[(c+a^2*c*x^2)*arctan(a*x)^3/x^2,x,11,-c*arctan(a*x)^3/x+a^2*c*x*arctan(a*x)^3+3*a*c*arctan(a*x)^2*log(2/(1+I*a*x))+3*a*c*arctan(a*x)^2*log(2+(-2)/(1-I*a*x))-3*I*a*c*arctan(a*x)*polylog(2,-1+2/(1-I*a*x))+3*I*a*c*arctan(a*x)*polylog(2,1+(-2)/(1+I*a*x))+3/2*a*c*polylog(3,-1+2/(1-I*a*x))+3/2*a*c*polylog(3,1+(-2)/(1+I*a*x))],
[(c+a^2*c*x^2)*arctan(a*x)^3/x^3,x,16,-3/2*I*a^2*c*arctan(a*x)^2-3/2*a*c*arctan(a*x)^2/x-1/2*a^2*c*arctan(a*x)^3-1/2*c*arctan(a*x)^3/x^2+2*a^2*c*arctan(a*x)^3*arctanh(1+(-2)/(1+I*a*x))+3*a^2*c*arctan(a*x)*log(2+(-2)/(1-I*a*x))-3/2*I*a^2*c*polylog(2,-1+2/(1-I*a*x))-3/2*I*a^2*c*arctan(a*x)^2*polylog(2,1+(-2)/(1+I*a*x))+3/2*I*a^2*c*arctan(a*x)^2*polylog(2,-1+2/(1+I*a*x))-3/2*a^2*c*arctan(a*x)*polylog(3,1+(-2)/(1+I*a*x))+3/2*a^2*c*arctan(a*x)*polylog(3,-1+2/(1+I*a*x))+3/4*I*a^2*c*polylog(4,1+(-2)/(1+I*a*x))-3/4*I*a^2*c*polylog(4,-1+2/(1+I*a*x))],
[(c+a^2*c*x^2)*arctan(a*x)^3/x^4,x,20,-a^2*c*arctan(a*x)/x-1/2*a^3*c*arctan(a*x)^2-1/2*a*c*arctan(a*x)^2/x^2-2/3*I*a^3*c*arctan(a*x)^3-1/3*c*arctan(a*x)^3/x^3-a^2*c*arctan(a*x)^3/x+a^3*c*log(x)-1/2*a^3*c*log(1+a^2*x^2)+2*a^3*c*arctan(a*x)^2*log(2+(-2)/(1-I*a*x))-2*I*a^3*c*arctan(a*x)*polylog(2,-1+2/(1-I*a*x))+a^3*c*polylog(3,-1+2/(1-I*a*x))],
[x^3*(c+a^2*c*x^2)^2*arctan(a*x)^3,x,106,1/21*c^2*x/a^3-1/168*c^2*x^3/a-1/280*a*c^2*x^5-1/21*c^2*arctan(a*x)/a^4-5/168*c^2*x^2*arctan(a*x)/a^2+1/28*c^2*x^4*arctan(a*x)+1/56*a^2*c^2*x^6*arctan(a*x)+2/21*I*c^2*arctan(a*x)^2/a^4+1/8*c^2*x*arctan(a*x)^2/a^3-1/24*c^2*x^3*arctan(a*x)^2/a-1/8*a*c^2*x^5*arctan(a*x)^2-3/56*a^3*c^2*x^7*arctan(a*x)^2-1/24*c^2*arctan(a*x)^3/a^4+1/4*c^2*x^4*arctan(a*x)^3+1/3*a^2*c^2*x^6*arctan(a*x)^3+1/8*a^4*c^2*x^8*arctan(a*x)^3+4/21*c^2*arctan(a*x)*log(2/(1+I*a*x))/a^4+2/21*I*c^2*polylog(2,1+(-2)/(1+I*a*x))/a^4],
[x^2*(c+a^2*c*x^2)^2*arctan(a*x)^3,x,73,-11/420*c^2*x^2/a-1/140*a*c^2*x^4-1/70*c^2*x*arctan(a*x)/a^2+17/210*c^2*x^3*arctan(a*x)+1/35*a^2*c^2*x^5*arctan(a*x)+1/140*c^2*arctan(a*x)^2/a^3-4/35*c^2*x^2*arctan(a*x)^2/a-27/140*a*c^2*x^4*arctan(a*x)^2-1/14*a^3*c^2*x^6*arctan(a*x)^2-8/105*I*c^2*arctan(a*x)^3/a^3+1/3*c^2*x^3*arctan(a*x)^3+2/5*a^2*c^2*x^5*arctan(a*x)^3+1/7*a^4*c^2*x^7*arctan(a*x)^3-8/35*c^2*arctan(a*x)^2*log(2/(1+I*a*x))/a^3+1/30*c^2*log(1+a^2*x^2)/a^3-8/35*I*c^2*arctan(a*x)*polylog(2,1+(-2)/(1+I*a*x))/a^3-4/35*c^2*polylog(3,1+(-2)/(1+I*a*x))/a^3],
[x*(c+a^2*c*x^2)^2*arctan(a*x)^3,x,10,-11/60*c^2*x/a-1/60*a*c^2*x^3+2/15*c^2*(1+a^2*x^2)*arctan(a*x)/a^2+1/20*c^2*(1+a^2*x^2)^2*arctan(a*x)/a^2-4/15*I*c^2*arctan(a*x)^2/a^2-4/15*c^2*x*arctan(a*x)^2/a-2/15*c^2*x*(1+a^2*x^2)*arctan(a*x)^2/a-1/10*c^2*x*(1+a^2*x^2)^2*arctan(a*x)^2/a+1/6*c^2*(1+a^2*x^2)^3*arctan(a*x)^3/a^2-8/15*c^2*arctan(a*x)*log(2/(1+I*a*x))/a^2-4/15*I*c^2*polylog(2,1+(-2)/(1+I*a*x))/a^2],
[(c+a^2*c*x^2)^2*arctan(a*x)^3,x,12,-1/20*c^2*(1+a^2*x^2)/a+c^2*x*arctan(a*x)+1/10*c^2*x*(1+a^2*x^2)*arctan(a*x)-2/5*c^2*(1+a^2*x^2)*arctan(a*x)^2/a-3/20*c^2*(1+a^2*x^2)^2*arctan(a*x)^2/a+8/15*I*c^2*arctan(a*x)^3/a+8/15*c^2*x*arctan(a*x)^3+4/15*c^2*x*(1+a^2*x^2)*arctan(a*x)^3+1/5*c^2*x*(1+a^2*x^2)^2*arctan(a*x)^3+8/5*c^2*arctan(a*x)^2*log(2/(1+I*a*x))/a-1/2*c^2*log(1+a^2*x^2)/a+8/5*I*c^2*arctan(a*x)*polylog(2,1+(-2)/(1+I*a*x))/a+4/5*c^2*polylog(3,1+(-2)/(1+I*a*x))/a],
[(c+a^2*c*x^2)^2*arctan(a*x)^3/x,x,36,-1/4*a*c^2*x+1/4*c^2*arctan(a*x)+1/4*a^2*c^2*x^2*arctan(a*x)-2*I*c^2*arctan(a*x)^2-9/4*a*c^2*x*arctan(a*x)^2-1/4*a^3*c^2*x^3*arctan(a*x)^2+3/4*c^2*arctan(a*x)^3+a^2*c^2*x^2*arctan(a*x)^3+1/4*a^4*c^2*x^4*arctan(a*x)^3+2*c^2*arctan(a*x)^3*arctanh(1+(-2)/(1+I*a*x))-4*c^2*arctan(a*x)*log(2/(1+I*a*x))-2*I*c^2*polylog(2,1+(-2)/(1+I*a*x))-3/2*I*c^2*arctan(a*x)^2*polylog(2,1+(-2)/(1+I*a*x))+3/2*I*c^2*arctan(a*x)^2*polylog(2,-1+2/(1+I*a*x))-3/2*c^2*arctan(a*x)*polylog(3,1+(-2)/(1+I*a*x))+3/2*c^2*arctan(a*x)*polylog(3,-1+2/(1+I*a*x))+3/4*I*c^2*polylog(4,1+(-2)/(1+I*a*x))-3/4*I*c^2*polylog(4,-1+2/(1+I*a*x))],
[(c+a^2*c*x^2)^2*arctan(a*x)^3/x^2,x,23,a^2*c^2*x*arctan(a*x)-1/2*a*c^2*arctan(a*x)^2-1/2*a^3*c^2*x^2*arctan(a*x)^2+2/3*I*a*c^2*arctan(a*x)^3-c^2*arctan(a*x)^3/x+2*a^2*c^2*x*arctan(a*x)^3+1/3*a^4*c^2*x^3*arctan(a*x)^3+5*a*c^2*arctan(a*x)^2*log(2/(1+I*a*x))-1/2*a*c^2*log(1+a^2*x^2)+3*a*c^2*arctan(a*x)^2*log(2+(-2)/(1-I*a*x))-3*I*a*c^2*arctan(a*x)*polylog(2,-1+2/(1-I*a*x))+5*I*a*c^2*arctan(a*x)*polylog(2,1+(-2)/(1+I*a*x))+3/2*a*c^2*polylog(3,-1+2/(1-I*a*x))+5/2*a*c^2*polylog(3,1+(-2)/(1+I*a*x))],
[(c+a^2*c*x^2)^2*arctan(a*x)^3/x^3,x,25,-3*I*a^2*c^2*arctan(a*x)^2-3/2*a*c^2*arctan(a*x)^2/x-3/2*a^3*c^2*x*arctan(a*x)^2-1/2*c^2*arctan(a*x)^3/x^2+1/2*a^4*c^2*x^2*arctan(a*x)^3+4*a^2*c^2*arctan(a*x)^3*arctanh(1+(-2)/(1+I*a*x))-3*a^2*c^2*arctan(a*x)*log(2/(1+I*a*x))+3*a^2*c^2*arctan(a*x)*log(2+(-2)/(1-I*a*x))-3/2*I*a^2*c^2*polylog(2,-1+2/(1-I*a*x))-3/2*I*a^2*c^2*polylog(2,1+(-2)/(1+I*a*x))-3*I*a^2*c^2*arctan(a*x)^2*polylog(2,1+(-2)/(1+I*a*x))+3*I*a^2*c^2*arctan(a*x)^2*polylog(2,-1+2/(1+I*a*x))-3*a^2*c^2*arctan(a*x)*polylog(3,1+(-2)/(1+I*a*x))+3*a^2*c^2*arctan(a*x)*polylog(3,-1+2/(1+I*a*x))+3/2*I*a^2*c^2*polylog(4,1+(-2)/(1+I*a*x))-3/2*I*a^2*c^2*polylog(4,-1+2/(1+I*a*x))],
[(c+a^2*c*x^2)^2*arctan(a*x)^3/x^4,x,26,-a^2*c^2*arctan(a*x)/x-1/2*a^3*c^2*arctan(a*x)^2-1/2*a*c^2*arctan(a*x)^2/x^2-2/3*I*a^3*c^2*arctan(a*x)^3-1/3*c^2*arctan(a*x)^3/x^3-2*a^2*c^2*arctan(a*x)^3/x+a^4*c^2*x*arctan(a*x)^3+a^3*c^2*log(x)+3*a^3*c^2*arctan(a*x)^2*log(2/(1+I*a*x))-1/2*a^3*c^2*log(1+a^2*x^2)+5*a^3*c^2*arctan(a*x)^2*log(2+(-2)/(1-I*a*x))-5*I*a^3*c^2*arctan(a*x)*polylog(2,-1+2/(1-I*a*x))+3*I*a^3*c^2*arctan(a*x)*polylog(2,1+(-2)/(1+I*a*x))+5/2*a^3*c^2*polylog(3,-1+2/(1-I*a*x))+3/2*a^3*c^2*polylog(3,1+(-2)/(1+I*a*x))],
[x^3*(c+a^2*c*x^2)^3*arctan(a*x)^3,x,184,389/12600*c^3*x/a^3-17/9450*c^3*x^3/a-1/252*a*c^3*x^5-1/840*a^3*c^3*x^7-389/12600*c^3*arctan(a*x)/a^4-107/4200*c^3*x^2*arctan(a*x)/a^2+53/2100*c^3*x^4*arctan(a*x)+71/2520*a^2*c^3*x^6*arctan(a*x)+1/120*a^4*c^3*x^8*arctan(a*x)+26/525*I*c^3*arctan(a*x)^2/a^4+3/40*c^3*x*arctan(a*x)^2/a^3-1/40*c^3*x^3*arctan(a*x)^2/a-27/200*a*c^3*x^5*arctan(a*x)^2-33/280*a^3*c^3*x^7*arctan(a*x)^2-1/30*a^5*c^3*x^9*arctan(a*x)^2-1/40*c^3*arctan(a*x)^3/a^4+1/4*c^3*x^4*arctan(a*x)^3+1/2*a^2*c^3*x^6*arctan(a*x)^3+3/8*a^4*c^3*x^8*arctan(a*x)^3+1/10*a^6*c^3*x^10*arctan(a*x)^3+52/525*c^3*arctan(a*x)*log(2/(1+I*a*x))/a^4+26/525*I*c^3*polylog(2,1+(-2)/(1+I*a*x))/a^4],
[x^2*(c+a^2*c*x^2)^3*arctan(a*x)^3,x,132,-107/7560*c^3*x^2/a-11/1260*a*c^3*x^4-1/504*a^3*c^3*x^6-47/1260*c^3*x*arctan(a*x)/a^2+239/3780*c^3*x^3*arctan(a*x)+59/1260*a^2*c^3*x^5*arctan(a*x)+1/84*a^4*c^3*x^7*arctan(a*x)+47/2520*c^3*arctan(a*x)^2/a^3-8/105*c^3*x^2*arctan(a*x)^2/a-89/420*a*c^3*x^4*arctan(a*x)^2-10/63*a^3*c^3*x^6*arctan(a*x)^2-1/24*a^5*c^3*x^8*arctan(a*x)^2-16/315*I*c^3*arctan(a*x)^3/a^3+1/3*c^3*x^3*arctan(a*x)^3+3/5*a^2*c^3*x^5*arctan(a*x)^3+3/7*a^4*c^3*x^7*arctan(a*x)^3+1/9*a^6*c^3*x^9*arctan(a*x)^3-16/105*c^3*arctan(a*x)^2*log(2/(1+I*a*x))/a^3+31/945*c^3*log(1+a^2*x^2)/a^3-16/105*I*c^3*arctan(a*x)*polylog(2,1+(-2)/(1+I*a*x))/a^3-8/105*c^3*polylog(3,1+(-2)/(1+I*a*x))/a^3],
[x*(c+a^2*c*x^2)^3*arctan(a*x)^3,x,13,-19/140*c^3*x/a-19/840*a*c^3*x^3-1/280*a^3*c^3*x^5+3/35*c^3*(1+a^2*x^2)*arctan(a*x)/a^2+9/280*c^3*(1+a^2*x^2)^2*arctan(a*x)/a^2+1/56*c^3*(1+a^2*x^2)^3*arctan(a*x)/a^2-6/35*I*c^3*arctan(a*x)^2/a^2-6/35*c^3*x*arctan(a*x)^2/a-3/35*c^3*x*(1+a^2*x^2)*arctan(a*x)^2/a-9/140*c^3*x*(1+a^2*x^2)^2*arctan(a*x)^2/a-3/56*c^3*x*(1+a^2*x^2)^3*arctan(a*x)^2/a+1/8*c^3*(1+a^2*x^2)^4*arctan(a*x)^3/a^2-12/35*c^3*arctan(a*x)*log(2/(1+I*a*x))/a^2-6/35*I*c^3*polylog(2,1+(-2)/(1+I*a*x))/a^2],
[(c+a^2*c*x^2)^3*arctan(a*x)^3,x,17,-13/210*c^3*(1+a^2*x^2)/a-1/140*c^3*(1+a^2*x^2)^2/a+14/15*c^3*x*arctan(a*x)+13/105*c^3*x*(1+a^2*x^2)*arctan(a*x)+1/35*c^3*x*(1+a^2*x^2)^2*arctan(a*x)-12/35*c^3*(1+a^2*x^2)*arctan(a*x)^2/a-9/70*c^3*(1+a^2*x^2)^2*arctan(a*x)^2/a-1/14*c^3*(1+a^2*x^2)^3*arctan(a*x)^2/a+16/35*I*c^3*arctan(a*x)^3/a+16/35*c^3*x*arctan(a*x)^3+8/35*c^3*x*(1+a^2*x^2)*arctan(a*x)^3+6/35*c^3*x*(1+a^2*x^2)^2*arctan(a*x)^3+1/7*c^3*x*(1+a^2*x^2)^3*arctan(a*x)^3+48/35*c^3*arctan(a*x)^2*log(2/(1+I*a*x))/a-7/15*c^3*log(1+a^2*x^2)/a+48/35*I*c^3*arctan(a*x)*polylog(2,1+(-2)/(1+I*a*x))/a+24/35*c^3*polylog(3,1+(-2)/(1+I*a*x))/a],
[(c+a^2*c*x^2)^3*arctan(a*x)^3/x,x,69,-13/30*a*c^3*x-1/60*a^3*c^3*x^3+13/30*c^3*arctan(a*x)+29/60*a^2*c^3*x^2*arctan(a*x)+1/20*a^4*c^3*x^4*arctan(a*x)-34/15*I*c^3*arctan(a*x)^2-11/4*a*c^3*x*arctan(a*x)^2-7/12*a^3*c^3*x^3*arctan(a*x)^2-1/10*a^5*c^3*x^5*arctan(a*x)^2+11/12*c^3*arctan(a*x)^3+3/2*a^2*c^3*x^2*arctan(a*x)^3+3/4*a^4*c^3*x^4*arctan(a*x)^3+1/6*a^6*c^3*x^6*arctan(a*x)^3+2*c^3*arctan(a*x)^3*arctanh(1+(-2)/(1+I*a*x))-68/15*c^3*arctan(a*x)*log(2/(1+I*a*x))-34/15*I*c^3*polylog(2,1+(-2)/(1+I*a*x))-3/2*I*c^3*arctan(a*x)^2*polylog(2,1+(-2)/(1+I*a*x))+3/2*I*c^3*arctan(a*x)^2*polylog(2,-1+2/(1+I*a*x))-3/2*c^3*arctan(a*x)*polylog(3,1+(-2)/(1+I*a*x))+3/2*c^3*arctan(a*x)*polylog(3,-1+2/(1+I*a*x))+3/4*I*c^3*polylog(4,1+(-2)/(1+I*a*x))-3/4*I*c^3*polylog(4,-1+2/(1+I*a*x))],
[(c+a^2*c*x^2)^3*arctan(a*x)^3/x^2,x,45,-1/20*a^3*c^3*x^2+21/10*a^2*c^3*x*arctan(a*x)+1/10*a^4*c^3*x^3*arctan(a*x)-21/20*a*c^3*arctan(a*x)^2-6/5*a^3*c^3*x^2*arctan(a*x)^2-3/20*a^5*c^3*x^4*arctan(a*x)^2+6/5*I*a*c^3*arctan(a*x)^3-c^3*arctan(a*x)^3/x+3*a^2*c^3*x*arctan(a*x)^3+a^4*c^3*x^3*arctan(a*x)^3+1/5*a^6*c^3*x^5*arctan(a*x)^3+33/5*a*c^3*arctan(a*x)^2*log(2/(1+I*a*x))-a*c^3*log(1+a^2*x^2)+3*a*c^3*arctan(a*x)^2*log(2+(-2)/(1-I*a*x))-3*I*a*c^3*arctan(a*x)*polylog(2,-1+2/(1-I*a*x))+33/5*I*a*c^3*arctan(a*x)*polylog(2,1+(-2)/(1+I*a*x))+3/2*a*c^3*polylog(3,-1+2/(1-I*a*x))+33/10*a*c^3*polylog(3,1+(-2)/(1+I*a*x))],
[(c+a^2*c*x^2)^3*arctan(a*x)^3/x^3,x,43,-1/4*a^3*c^3*x+1/4*a^2*c^3*arctan(a*x)+1/4*a^4*c^3*x^2*arctan(a*x)-5*I*a^2*c^3*arctan(a*x)^2-3/2*a*c^3*arctan(a*x)^2/x-15/4*a^3*c^3*x*arctan(a*x)^2-1/4*a^5*c^3*x^3*arctan(a*x)^2+3/4*a^2*c^3*arctan(a*x)^3-1/2*c^3*arctan(a*x)^3/x^2+3/2*a^4*c^3*x^2*arctan(a*x)^3+1/4*a^6*c^3*x^4*arctan(a*x)^3+6*a^2*c^3*arctan(a*x)^3*arctanh(1+(-2)/(1+I*a*x))-7*a^2*c^3*arctan(a*x)*log(2/(1+I*a*x))+3*a^2*c^3*arctan(a*x)*log(2+(-2)/(1-I*a*x))-3/2*I*a^2*c^3*polylog(2,-1+2/(1-I*a*x))-7/2*I*a^2*c^3*polylog(2,1+(-2)/(1+I*a*x))-9/2*I*a^2*c^3*arctan(a*x)^2*polylog(2,1+(-2)/(1+I*a*x))+9/2*I*a^2*c^3*arctan(a*x)^2*polylog(2,-1+2/(1+I*a*x))-9/2*a^2*c^3*arctan(a*x)*polylog(3,1+(-2)/(1+I*a*x))+9/2*a^2*c^3*arctan(a*x)*polylog(3,-1+2/(1+I*a*x))+9/4*I*a^2*c^3*polylog(4,1+(-2)/(1+I*a*x))-9/4*I*a^2*c^3*polylog(4,-1+2/(1+I*a*x))],
[(c+a^2*c*x^2)^3*arctan(a*x)^3/x^4,x,37,-a^2*c^3*arctan(a*x)/x+a^4*c^3*x*arctan(a*x)-a^3*c^3*arctan(a*x)^2-1/2*a*c^3*arctan(a*x)^2/x^2-1/2*a^5*c^3*x^2*arctan(a*x)^2-1/3*c^3*arctan(a*x)^3/x^3-3*a^2*c^3*arctan(a*x)^3/x+3*a^4*c^3*x*arctan(a*x)^3+1/3*a^6*c^3*x^3*arctan(a*x)^3+a^3*c^3*log(x)+8*a^3*c^3*arctan(a*x)^2*log(2/(1+I*a*x))-a^3*c^3*log(1+a^2*x^2)+8*a^3*c^3*arctan(a*x)^2*log(2+(-2)/(1-I*a*x))-8*I*a^3*c^3*arctan(a*x)*polylog(2,-1+2/(1-I*a*x))+8*I*a^3*c^3*arctan(a*x)*polylog(2,1+(-2)/(1+I*a*x))+4*a^3*c^3*polylog(3,-1+2/(1-I*a*x))+4*a^3*c^3*polylog(3,1+(-2)/(1+I*a*x))],

# p<0
[x^4*arctan(a*x)^3/(c+a^2*c*x^2),x,19,x*arctan(a*x)/(a^4*c)-1/2*arctan(a*x)^2/(a^5*c)-1/2*x^2*arctan(a*x)^2/(a^3*c)-4/3*I*arctan(a*x)^3/(a^5*c)-x*arctan(a*x)^3/(a^4*c)+1/3*x^3*arctan(a*x)^3/(a^2*c)+1/4*arctan(a*x)^4/(a^5*c)-4*arctan(a*x)^2*log(2/(1+I*a*x))/(a^5*c)-1/2*log(1+a^2*x^2)/(a^5*c)-4*I*arctan(a*x)*polylog(2,1+(-2)/(1+I*a*x))/(a^5*c)-2*polylog(3,1+(-2)/(1+I*a*x))/(a^5*c)],
[x^3*arctan(a*x)^3/(c+a^2*c*x^2),x,14,-3/2*I*arctan(a*x)^2/(a^4*c)-3/2*x*arctan(a*x)^2/(a^3*c)+1/2*arctan(a*x)^3/(a^4*c)+1/2*x^2*arctan(a*x)^3/(a^2*c)+1/4*I*arctan(a*x)^4/(a^4*c)-3*arctan(a*x)*log(2/(1+I*a*x))/(a^4*c)+arctan(a*x)^3*log(2/(1+I*a*x))/(a^4*c)-3/2*I*polylog(2,1+(-2)/(1+I*a*x))/(a^4*c)+3/2*I*arctan(a*x)^2*polylog(2,1+(-2)/(1+I*a*x))/(a^4*c)+3/2*arctan(a*x)*polylog(3,1+(-2)/(1+I*a*x))/(a^4*c)-3/4*I*polylog(4,1+(-2)/(1+I*a*x))/(a^4*c)],
[x^2*arctan(a*x)^3/(c+a^2*c*x^2),x,7,I*arctan(a*x)^3/(a^3*c)+x*arctan(a*x)^3/(a^2*c)-1/4*arctan(a*x)^4/(a^3*c)+3*arctan(a*x)^2*log(2/(1+I*a*x))/(a^3*c)+3*I*arctan(a*x)*polylog(2,1+(-2)/(1+I*a*x))/(a^3*c)+3/2*polylog(3,1+(-2)/(1+I*a*x))/(a^3*c)],
[x*arctan(a*x)^3/(c+a^2*c*x^2),x,5,-1/4*I*arctan(a*x)^4/(a^2*c)-arctan(a*x)^3*log(2/(1+I*a*x))/(a^2*c)-3/2*I*arctan(a*x)^2*polylog(2,1+(-2)/(1+I*a*x))/(a^2*c)-3/2*arctan(a*x)*polylog(3,1+(-2)/(1+I*a*x))/(a^2*c)+3/4*I*polylog(4,1+(-2)/(1+I*a*x))/(a^2*c)],
[arctan(a*x)^3/(c+a^2*c*x^2),x,1,1/4*arctan(a*x)^4/(a*c)],
[arctan(a*x)^3/(x*(c+a^2*c*x^2)),x,5,-1/4*I*arctan(a*x)^4/c+arctan(a*x)^3*log(2+(-2)/(1-I*a*x))/c-3/2*I*arctan(a*x)^2*polylog(2,-1+2/(1-I*a*x))/c+3/2*arctan(a*x)*polylog(3,-1+2/(1-I*a*x))/c+3/4*I*polylog(4,-1+2/(1-I*a*x))/c],
[arctan(a*x)^3/(x^2*(c+a^2*c*x^2)),x,7,-I*a*arctan(a*x)^3/c-arctan(a*x)^3/(c*x)-1/4*a*arctan(a*x)^4/c+3*a*arctan(a*x)^2*log(2+(-2)/(1-I*a*x))/c-3*I*a*arctan(a*x)*polylog(2,-1+2/(1-I*a*x))/c+3/2*a*polylog(3,-1+2/(1-I*a*x))/c],
[arctan(a*x)^3/(x^3*(c+a^2*c*x^2)),x,13,-3/2*I*a^2*arctan(a*x)^2/c-3/2*a*arctan(a*x)^2/(c*x)-1/2*a^2*arctan(a*x)^3/c-1/2*arctan(a*x)^3/(c*x^2)+1/4*I*a^2*arctan(a*x)^4/c+3*a^2*arctan(a*x)*log(2+(-2)/(1-I*a*x))/c-a^2*arctan(a*x)^3*log(2+(-2)/(1-I*a*x))/c-3/2*I*a^2*polylog(2,-1+2/(1-I*a*x))/c+3/2*I*a^2*arctan(a*x)^2*polylog(2,-1+2/(1-I*a*x))/c-3/2*a^2*arctan(a*x)*polylog(3,-1+2/(1-I*a*x))/c-3/4*I*a^2*polylog(4,-1+2/(1-I*a*x))/c],
[arctan(a*x)^3/(x^4*(c+a^2*c*x^2)),x,22,-a^2*arctan(a*x)/(c*x)-1/2*a^3*arctan(a*x)^2/c-1/2*a*arctan(a*x)^2/(c*x^2)+4/3*I*a^3*arctan(a*x)^3/c-1/3*arctan(a*x)^3/(c*x^3)+a^2*arctan(a*x)^3/(c*x)+1/4*a^3*arctan(a*x)^4/c+a^3*log(x)/c-1/2*a^3*log(1+a^2*x^2)/c-4*a^3*arctan(a*x)^2*log(2+(-2)/(1-I*a*x))/c+4*I*a^3*arctan(a*x)*polylog(2,-1+2/(1-I*a*x))/c-2*a^3*polylog(3,-1+2/(1-I*a*x))/c],
[x^3*arctan(a*x)^3/(c+a^2*c*x^2)^2,x,11,3/8*x/(a^3*c^2*(1+a^2*x^2))+3/8*arctan(a*x)/(a^4*c^2)-3/4*arctan(a*x)/(a^4*c^2*(1+a^2*x^2))-3/4*x*arctan(a*x)^2/(a^3*c^2*(1+a^2*x^2))-1/4*arctan(a*x)^3/(a^4*c^2)+1/2*arctan(a*x)^3/(a^4*c^2*(1+a^2*x^2))-1/4*I*arctan(a*x)^4/(a^4*c^2)-arctan(a*x)^3*log(2/(1+I*a*x))/(a^4*c^2)-3/2*I*arctan(a*x)^2*polylog(2,1+(-2)/(1+I*a*x))/(a^4*c^2)-3/2*arctan(a*x)*polylog(3,1+(-2)/(1+I*a*x))/(a^4*c^2)+3/4*I*polylog(4,1+(-2)/(1+I*a*x))/(a^4*c^2)],
[x^2*arctan(a*x)^3/(c+a^2*c*x^2)^2,x,4,3/8/(a^3*c^2*(1+a^2*x^2))+3/4*x*arctan(a*x)/(a^2*c^2*(1+a^2*x^2))+3/8*arctan(a*x)^2/(a^3*c^2)-3/4*arctan(a*x)^2/(a^3*c^2*(1+a^2*x^2))-1/2*x*arctan(a*x)^3/(a^2*c^2*(1+a^2*x^2))+1/8*arctan(a*x)^4/(a^3*c^2)],
[x*arctan(a*x)^3/(c+a^2*c*x^2)^2,x,5,-3/8*x/(a*c^2*(1+a^2*x^2))-3/8*arctan(a*x)/(a^2*c^2)+3/4*arctan(a*x)/(a^2*c^2*(1+a^2*x^2))+3/4*x*arctan(a*x)^2/(a*c^2*(1+a^2*x^2))+1/4*arctan(a*x)^3/(a^2*c^2)-1/2*arctan(a*x)^3/(a^2*c^2*(1+a^2*x^2))],
[arctan(a*x)^3/(c+a^2*c*x^2)^2,x,4,(-3/8)/(a*c^2*(1+a^2*x^2))-3/4*x*arctan(a*x)/(c^2*(1+a^2*x^2))-3/8*arctan(a*x)^2/(a*c^2)+3/4*arctan(a*x)^2/(a*c^2*(1+a^2*x^2))+1/2*x*arctan(a*x)^3/(c^2*(1+a^2*x^2))+1/8*arctan(a*x)^4/(a*c^2)],
[arctan(a*x)^3/(x*(c+a^2*c*x^2)^2),x,11,3/8*a*x/(c^2*(1+a^2*x^2))+3/8*arctan(a*x)/c^2-3/4*arctan(a*x)/(c^2*(1+a^2*x^2))-3/4*a*x*arctan(a*x)^2/(c^2*(1+a^2*x^2))-1/4*arctan(a*x)^3/c^2+1/2*arctan(a*x)^3/(c^2*(1+a^2*x^2))-1/4*I*arctan(a*x)^4/c^2+arctan(a*x)^3*log(2+(-2)/(1-I*a*x))/c^2-3/2*I*arctan(a*x)^2*polylog(2,-1+2/(1-I*a*x))/c^2+3/2*arctan(a*x)*polylog(3,-1+2/(1-I*a*x))/c^2+3/4*I*polylog(4,-1+2/(1-I*a*x))/c^2],
[arctan(a*x)^3/(x^2*(c+a^2*c*x^2)^2),x,12,3/8*a/(c^2*(1+a^2*x^2))+3/4*a^2*x*arctan(a*x)/(c^2*(1+a^2*x^2))+3/8*a*arctan(a*x)^2/c^2-3/4*a*arctan(a*x)^2/(c^2*(1+a^2*x^2))-I*a*arctan(a*x)^3/c^2-arctan(a*x)^3/(c^2*x)-1/2*a^2*x*arctan(a*x)^3/(c^2*(1+a^2*x^2))-3/8*a*arctan(a*x)^4/c^2+3*a*arctan(a*x)^2*log(2+(-2)/(1-I*a*x))/c^2-3*I*a*arctan(a*x)*polylog(2,-1+2/(1-I*a*x))/c^2+3/2*a*polylog(3,-1+2/(1-I*a*x))/c^2],
[arctan(a*x)^3/(x^3*(c+a^2*c*x^2)^2),x,25,-3/8*a^3*x/(c^2*(1+a^2*x^2))-3/8*a^2*arctan(a*x)/c^2+3/4*a^2*arctan(a*x)/(c^2*(1+a^2*x^2))-3/2*I*a^2*arctan(a*x)^2/c^2-3/2*a*arctan(a*x)^2/(c^2*x)+3/4*a^3*x*arctan(a*x)^2/(c^2*(1+a^2*x^2))-1/4*a^2*arctan(a*x)^3/c^2-1/2*arctan(a*x)^3/(c^2*x^2)-1/2*a^2*arctan(a*x)^3/(c^2*(1+a^2*x^2))+1/2*I*a^2*arctan(a*x)^4/c^2+3*a^2*arctan(a*x)*log(2+(-2)/(1-I*a*x))/c^2-2*a^2*arctan(a*x)^3*log(2+(-2)/(1-I*a*x))/c^2-3/2*I*a^2*polylog(2,-1+2/(1-I*a*x))/c^2+3*I*a^2*arctan(a*x)^2*polylog(2,-1+2/(1-I*a*x))/c^2-3*a^2*arctan(a*x)*polylog(3,-1+2/(1-I*a*x))/c^2-3/2*I*a^2*polylog(4,-1+2/(1-I*a*x))/c^2],
[arctan(a*x)^3/(x^4*(c+a^2*c*x^2)^2),x,35,-3/8*a^3/(c^2*(1+a^2*x^2))-a^2*arctan(a*x)/(c^2*x)-3/4*a^4*x*arctan(a*x)/(c^2*(1+a^2*x^2))-7/8*a^3*arctan(a*x)^2/c^2-1/2*a*arctan(a*x)^2/(c^2*x^2)+3/4*a^3*arctan(a*x)^2/(c^2*(1+a^2*x^2))+7/3*I*a^3*arctan(a*x)^3/c^2-1/3*arctan(a*x)^3/(c^2*x^3)+2*a^2*arctan(a*x)^3/(c^2*x)+1/2*a^4*x*arctan(a*x)^3/(c^2*(1+a^2*x^2))+5/8*a^3*arctan(a*x)^4/c^2+a^3*log(x)/c^2-1/2*a^3*log(1+a^2*x^2)/c^2-7*a^3*arctan(a*x)^2*log(2+(-2)/(1-I*a*x))/c^2+7*I*a^3*arctan(a*x)*polylog(2,-1+2/(1-I*a*x))/c^2-7/2*a^3*polylog(3,-1+2/(1-I*a*x))/c^2],
[x^3*arctan(a*x)^3/(c+a^2*c*x^2)^3,x,9,-3/128*x^3/(a*c^3*(1+a^2*x^2)^2)-45/256*x/(a^3*c^3*(1+a^2*x^2))-27/256*arctan(a*x)/(a^4*c^3)-3/32*x^4*arctan(a*x)/(c^3*(1+a^2*x^2)^2)+9/32*arctan(a*x)/(a^4*c^3*(1+a^2*x^2))+3/16*x^3*arctan(a*x)^2/(a*c^3*(1+a^2*x^2)^2)+9/32*x*arctan(a*x)^2/(a^3*c^3*(1+a^2*x^2))-3/32*arctan(a*x)^3/(a^4*c^3)+1/4*x^4*arctan(a*x)^3/(c^3*(1+a^2*x^2)^2)],
[x^2*arctan(a*x)^3/(c+a^2*c*x^2)^3,x,13,3/128/(a^3*c^3*(1+a^2*x^2)^2)+(-3/128)/(a^3*c^3*(1+a^2*x^2))+3/32*x*arctan(a*x)/(a^2*c^3*(1+a^2*x^2)^2)-3/64*x*arctan(a*x)/(a^2*c^3*(1+a^2*x^2))-3/128*arctan(a*x)^2/(a^3*c^3)-3/16*arctan(a*x)^2/(a^3*c^3*(1+a^2*x^2)^2)+3/16*arctan(a*x)^2/(a^3*c^3*(1+a^2*x^2))-1/4*x*arctan(a*x)^3/(a^2*c^3*(1+a^2*x^2)^2)+1/8*x*arctan(a*x)^3/(a^2*c^3*(1+a^2*x^2))+1/32*arctan(a*x)^4/(a^3*c^3)],
[x*arctan(a*x)^3/(c+a^2*c*x^2)^3,x,9,-3/128*x/(a*c^3*(1+a^2*x^2)^2)-45/256*x/(a*c^3*(1+a^2*x^2))-45/256*arctan(a*x)/(a^2*c^3)+3/32*arctan(a*x)/(a^2*c^3*(1+a^2*x^2)^2)+9/32*arctan(a*x)/(a^2*c^3*(1+a^2*x^2))+3/16*x*arctan(a*x)^2/(a*c^3*(1+a^2*x^2)^2)+9/32*x*arctan(a*x)^2/(a*c^3*(1+a^2*x^2))+3/32*arctan(a*x)^3/(a^2*c^3)-1/4*arctan(a*x)^3/(a^2*c^3*(1+a^2*x^2)^2)],
[arctan(a*x)^3/(c+a^2*c*x^2)^3,x,8,(-3/128)/(a*c^3*(1+a^2*x^2)^2)+(-45/128)/(a*c^3*(1+a^2*x^2))-3/32*x*arctan(a*x)/(c^3*(1+a^2*x^2)^2)-45/64*x*arctan(a*x)/(c^3*(1+a^2*x^2))-45/128*arctan(a*x)^2/(a*c^3)+3/16*arctan(a*x)^2/(a*c^3*(1+a^2*x^2)^2)+9/16*arctan(a*x)^2/(a*c^3*(1+a^2*x^2))+1/4*x*arctan(a*x)^3/(c^3*(1+a^2*x^2)^2)+3/8*x*arctan(a*x)^3/(c^3*(1+a^2*x^2))+3/32*arctan(a*x)^4/(a*c^3)],
[arctan(a*x)^3/(x*(c+a^2*c*x^2)^3),x,21,3/128*a*x/(c^3*(1+a^2*x^2)^2)+141/256*a*x/(c^3*(1+a^2*x^2))+141/256*arctan(a*x)/c^3-3/32*arctan(a*x)/(c^3*(1+a^2*x^2)^2)-33/32*arctan(a*x)/(c^3*(1+a^2*x^2))-3/16*a*x*arctan(a*x)^2/(c^3*(1+a^2*x^2)^2)-33/32*a*x*arctan(a*x)^2/(c^3*(1+a^2*x^2))-11/32*arctan(a*x)^3/c^3+1/4*arctan(a*x)^3/(c^3*(1+a^2*x^2)^2)+1/2*arctan(a*x)^3/(c^3*(1+a^2*x^2))-1/4*I*arctan(a*x)^4/c^3+arctan(a*x)^3*log(2+(-2)/(1-I*a*x))/c^3-3/2*I*arctan(a*x)^2*polylog(2,-1+2/(1-I*a*x))/c^3+3/2*arctan(a*x)*polylog(3,-1+2/(1-I*a*x))/c^3+3/4*I*polylog(4,-1+2/(1-I*a*x))/c^3],
[arctan(a*x)^3/(x^2*(c+a^2*c*x^2)^3),x,21,3/128*a/(c^3*(1+a^2*x^2)^2)+93/128*a/(c^3*(1+a^2*x^2))+3/32*a^2*x*arctan(a*x)/(c^3*(1+a^2*x^2)^2)+93/64*a^2*x*arctan(a*x)/(c^3*(1+a^2*x^2))+93/128*a*arctan(a*x)^2/c^3-3/16*a*arctan(a*x)^2/(c^3*(1+a^2*x^2)^2)-21/16*a*arctan(a*x)^2/(c^3*(1+a^2*x^2))-I*a*arctan(a*x)^3/c^3-arctan(a*x)^3/(c^3*x)-1/4*a^2*x*arctan(a*x)^3/(c^3*(1+a^2*x^2)^2)-7/8*a^2*x*arctan(a*x)^3/(c^3*(1+a^2*x^2))-15/32*a*arctan(a*x)^4/c^3+3*a*arctan(a*x)^2*log(2+(-2)/(1-I*a*x))/c^3-3*I*a*arctan(a*x)*polylog(2,-1+2/(1-I*a*x))/c^3+3/2*a*polylog(3,-1+2/(1-I*a*x))/c^3],
[arctan(a*x)^3/(x^3*(c+a^2*c*x^2)^3),x,47,-3/128*a^3*x/(c^3*(1+a^2*x^2)^2)-237/256*a^3*x/(c^3*(1+a^2*x^2))-237/256*a^2*arctan(a*x)/c^3+3/32*a^2*arctan(a*x)/(c^3*(1+a^2*x^2)^2)+57/32*a^2*arctan(a*x)/(c^3*(1+a^2*x^2))-3/2*I*a^2*arctan(a*x)^2/c^3-3/2*a*arctan(a*x)^2/(c^3*x)+3/16*a^3*x*arctan(a*x)^2/(c^3*(1+a^2*x^2)^2)+57/32*a^3*x*arctan(a*x)^2/(c^3*(1+a^2*x^2))+3/32*a^2*arctan(a*x)^3/c^3-1/2*arctan(a*x)^3/(c^3*x^2)-1/4*a^2*arctan(a*x)^3/(c^3*(1+a^2*x^2)^2)-a^2*arctan(a*x)^3/(c^3*(1+a^2*x^2))+3/4*I*a^2*arctan(a*x)^4/c^3+3*a^2*arctan(a*x)*log(2+(-2)/(1-I*a*x))/c^3-3*a^2*arctan(a*x)^3*log(2+(-2)/(1-I*a*x))/c^3-3/2*I*a^2*polylog(2,-1+2/(1-I*a*x))/c^3+9/2*I*a^2*arctan(a*x)^2*polylog(2,-1+2/(1-I*a*x))/c^3-9/2*a^2*arctan(a*x)*polylog(3,-1+2/(1-I*a*x))/c^3-9/4*I*a^2*polylog(4,-1+2/(1-I*a*x))/c^3],
[arctan(a*x)^3/(x^4*(c+a^2*c*x^2)^3),x,57,-3/128*a^3/(c^3*(1+a^2*x^2)^2)-141/128*a^3/(c^3*(1+a^2*x^2))-a^2*arctan(a*x)/(c^3*x)-3/32*a^4*x*arctan(a*x)/(c^3*(1+a^2*x^2)^2)-141/64*a^4*x*arctan(a*x)/(c^3*(1+a^2*x^2))-205/128*a^3*arctan(a*x)^2/c^3-1/2*a*arctan(a*x)^2/(c^3*x^2)+3/16*a^3*arctan(a*x)^2/(c^3*(1+a^2*x^2)^2)+33/16*a^3*arctan(a*x)^2/(c^3*(1+a^2*x^2))+10/3*I*a^3*arctan(a*x)^3/c^3-1/3*arctan(a*x)^3/(c^3*x^3)+3*a^2*arctan(a*x)^3/(c^3*x)+1/4*a^4*x*arctan(a*x)^3/(c^3*(1+a^2*x^2)^2)+11/8*a^4*x*arctan(a*x)^3/(c^3*(1+a^2*x^2))+35/32*a^3*arctan(a*x)^4/c^3+a^3*log(x)/c^3-1/2*a^3*log(1+a^2*x^2)/c^3-10*a^3*arctan(a*x)^2*log(2+(-2)/(1-I*a*x))/c^3+10*I*a^3*arctan(a*x)*polylog(2,-1+2/(1-I*a*x))/c^3-5*a^3*polylog(3,-1+2/(1-I*a*x))/c^3],

# Integrands of the form x^m (c+d x^2)^(p/2) ArcTan[a x]^3 with d=a^2 c

# p>0
[x^3*arctan(a*x)^3*sqrt(c+a^2*c*x^2),x,71,1/2*arctanh(a*x*sqrt(c)/sqrt(c+a^2*c*x^2))*sqrt(c)/a^4-11/20*I*c*arctan(exp(I*arctan(a*x)))*arctan(a*x)^2*sqrt(1+a^2*x^2)/(a^4*sqrt(c+a^2*c*x^2))+11/20*I*c*arctan(a*x)*polylog(2,-I*exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/(a^4*sqrt(c+a^2*c*x^2))-11/20*I*c*arctan(a*x)*polylog(2,I*exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/(a^4*sqrt(c+a^2*c*x^2))-11/20*c*polylog(3,-I*exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/(a^4*sqrt(c+a^2*c*x^2))+11/20*c*polylog(3,I*exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/(a^4*sqrt(c+a^2*c*x^2))-1/20*x*sqrt(c+a^2*c*x^2)/a^3-9/20*arctan(a*x)*sqrt(c+a^2*c*x^2)/a^4+1/10*x^2*arctan(a*x)*sqrt(c+a^2*c*x^2)/a^2+1/8*x*arctan(a*x)^2*sqrt(c+a^2*c*x^2)/a^3-3/20*x^3*arctan(a*x)^2*sqrt(c+a^2*c*x^2)/a-2/15*arctan(a*x)^3*sqrt(c+a^2*c*x^2)/a^4+1/15*x^2*arctan(a*x)^3*sqrt(c+a^2*c*x^2)/a^2+1/5*x^4*arctan(a*x)^3*sqrt(c+a^2*c*x^2)],
[x^2*arctan(a*x)^3*sqrt(c+a^2*c*x^2),x,40,1/4*I*c*arctan(exp(I*arctan(a*x)))*arctan(a*x)^3*sqrt(1+a^2*x^2)/(a^3*sqrt(c+a^2*c*x^2))+I*c*arctan(a*x)*arctan(sqrt(1+I*a*x)/sqrt(1-I*a*x))*sqrt(1+a^2*x^2)/(a^3*sqrt(c+a^2*c*x^2))-3/8*I*c*arctan(a*x)^2*polylog(2,-I*exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/(a^3*sqrt(c+a^2*c*x^2))+3/8*I*c*arctan(a*x)^2*polylog(2,I*exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/(a^3*sqrt(c+a^2*c*x^2))-1/2*I*c*polylog(2,-I*sqrt(1+I*a*x)/sqrt(1-I*a*x))*sqrt(1+a^2*x^2)/(a^3*sqrt(c+a^2*c*x^2))+1/2*I*c*polylog(2,I*sqrt(1+I*a*x)/sqrt(1-I*a*x))*sqrt(1+a^2*x^2)/(a^3*sqrt(c+a^2*c*x^2))+3/4*c*arctan(a*x)*polylog(3,-I*exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/(a^3*sqrt(c+a^2*c*x^2))-3/4*c*arctan(a*x)*polylog(3,I*exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/(a^3*sqrt(c+a^2*c*x^2))+3/4*I*c*polylog(4,-I*exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/(a^3*sqrt(c+a^2*c*x^2))-3/4*I*c*polylog(4,I*exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/(a^3*sqrt(c+a^2*c*x^2))-1/4*sqrt(c+a^2*c*x^2)/a^3+1/4*x*arctan(a*x)*sqrt(c+a^2*c*x^2)/a^2+1/8*arctan(a*x)^2*sqrt(c+a^2*c*x^2)/a^3-1/4*x^2*arctan(a*x)^2*sqrt(c+a^2*c*x^2)/a+1/8*x*arctan(a*x)^3*sqrt(c+a^2*c*x^2)/a^2+1/4*x^3*arctan(a*x)^3*sqrt(c+a^2*c*x^2)],
[x*arctan(a*x)^3*sqrt(c+a^2*c*x^2),x,13,1/3*(c+a^2*c*x^2)^(3/2)*arctan(a*x)^3/(a^2*c)-arctanh(a*x*sqrt(c)/sqrt(c+a^2*c*x^2))*sqrt(c)/a^2+I*c*arctan(exp(I*arctan(a*x)))*arctan(a*x)^2*sqrt(1+a^2*x^2)/(a^2*sqrt(c+a^2*c*x^2))-I*c*arctan(a*x)*polylog(2,-I*exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/(a^2*sqrt(c+a^2*c*x^2))+I*c*arctan(a*x)*polylog(2,I*exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/(a^2*sqrt(c+a^2*c*x^2))+c*polylog(3,-I*exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/(a^2*sqrt(c+a^2*c*x^2))-c*polylog(3,I*exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/(a^2*sqrt(c+a^2*c*x^2))+arctan(a*x)*sqrt(c+a^2*c*x^2)/a^2-1/2*x*arctan(a*x)^2*sqrt(c+a^2*c*x^2)/a],
[arctan(a*x)^3*sqrt(c+a^2*c*x^2),x,14,-I*c*arctan(exp(I*arctan(a*x)))*arctan(a*x)^3*sqrt(1+a^2*x^2)/(a*sqrt(c+a^2*c*x^2))-6*I*c*arctan(a*x)*arctan(sqrt(1+I*a*x)/sqrt(1-I*a*x))*sqrt(1+a^2*x^2)/(a*sqrt(c+a^2*c*x^2))+3/2*I*c*arctan(a*x)^2*polylog(2,-I*exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/(a*sqrt(c+a^2*c*x^2))-3/2*I*c*arctan(a*x)^2*polylog(2,I*exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/(a*sqrt(c+a^2*c*x^2))+3*I*c*polylog(2,-I*sqrt(1+I*a*x)/sqrt(1-I*a*x))*sqrt(1+a^2*x^2)/(a*sqrt(c+a^2*c*x^2))-3*I*c*polylog(2,I*sqrt(1+I*a*x)/sqrt(1-I*a*x))*sqrt(1+a^2*x^2)/(a*sqrt(c+a^2*c*x^2))-3*c*arctan(a*x)*polylog(3,-I*exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/(a*sqrt(c+a^2*c*x^2))+3*c*arctan(a*x)*polylog(3,I*exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/(a*sqrt(c+a^2*c*x^2))-3*I*c*polylog(4,-I*exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/(a*sqrt(c+a^2*c*x^2))+3*I*c*polylog(4,I*exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/(a*sqrt(c+a^2*c*x^2))-3/2*arctan(a*x)^2*sqrt(c+a^2*c*x^2)/a+1/2*x*arctan(a*x)^3*sqrt(c+a^2*c*x^2)],
[arctan(a*x)^3*sqrt(c+a^2*c*x^2)/x,x,22,6*I*c*arctan(exp(I*arctan(a*x)))*arctan(a*x)^2*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)-2*c*arctan(a*x)^3*arctanh(exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)+3*I*c*arctan(a*x)^2*polylog(2,-exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)-6*I*c*arctan(a*x)*polylog(2,-I*exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)+6*I*c*arctan(a*x)*polylog(2,I*exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)-3*I*c*arctan(a*x)^2*polylog(2,exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)-6*c*arctan(a*x)*polylog(3,-exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)+6*c*polylog(3,-I*exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)-6*c*polylog(3,I*exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)+6*c*arctan(a*x)*polylog(3,exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)-6*I*c*polylog(4,-exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)+6*I*c*polylog(4,exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)+arctan(a*x)^3*sqrt(c+a^2*c*x^2)],
[arctan(a*x)^3*sqrt(c+a^2*c*x^2)/x^2,x,22,-2*I*a*c*arctan(exp(I*arctan(a*x)))*arctan(a*x)^3*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)-6*a*c*arctan(a*x)^2*arctanh(exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)+6*I*a*c*arctan(a*x)*polylog(2,-exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)+3*I*a*c*arctan(a*x)^2*polylog(2,-I*exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)-3*I*a*c*arctan(a*x)^2*polylog(2,I*exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)-6*I*a*c*arctan(a*x)*polylog(2,exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)-6*a*c*polylog(3,-exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)-6*a*c*arctan(a*x)*polylog(3,-I*exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)+6*a*c*arctan(a*x)*polylog(3,I*exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)+6*a*c*polylog(3,exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)-6*I*a*c*polylog(4,-I*exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)+6*I*a*c*polylog(4,I*exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)-arctan(a*x)^3*sqrt(c+a^2*c*x^2)/x],
[arctan(a*x)^3*sqrt(c+a^2*c*x^2)/x^3,x,27,-a^2*c*arctan(a*x)^3*arctanh(exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)-6*a^2*c*arctan(a*x)*arctanh(sqrt(1+I*a*x)/sqrt(1-I*a*x))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)+3/2*I*a^2*c*arctan(a*x)^2*polylog(2,-exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)-3/2*I*a^2*c*arctan(a*x)^2*polylog(2,exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)+3*I*a^2*c*polylog(2,-sqrt(1+I*a*x)/sqrt(1-I*a*x))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)-3*I*a^2*c*polylog(2,sqrt(1+I*a*x)/sqrt(1-I*a*x))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)-3*a^2*c*arctan(a*x)*polylog(3,-exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)+3*a^2*c*arctan(a*x)*polylog(3,exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)-3*I*a^2*c*polylog(4,-exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)+3*I*a^2*c*polylog(4,exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)-3/2*a*arctan(a*x)^2*sqrt(c+a^2*c*x^2)/x-1/2*arctan(a*x)^3*sqrt(c+a^2*c*x^2)/x^2],
[arctan(a*x)^3*sqrt(c+a^2*c*x^2)/x^4,x,25,-1/3*(c+a^2*c*x^2)^(3/2)*arctan(a*x)^3/(c*x^3)-a^3*arctanh(sqrt(c+a^2*c*x^2)/sqrt(c))*sqrt(c)-a^3*c*arctan(a*x)^2*arctanh(exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)+I*a^3*c*arctan(a*x)*polylog(2,-exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)-I*a^3*c*arctan(a*x)*polylog(2,exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)-a^3*c*polylog(3,-exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)+a^3*c*polylog(3,exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)-a^2*arctan(a*x)*sqrt(c+a^2*c*x^2)/x-1/2*a*arctan(a*x)^2*sqrt(c+a^2*c*x^2)/x^2],
[x^3*(c+a^2*c*x^2)^(3/2)*arctan(a*x)^3,x,200,23/120*c^(3/2)*arctanh(a*x*sqrt(c)/sqrt(c+a^2*c*x^2))/a^4-51/280*I*c^2*arctan(exp(I*arctan(a*x)))*arctan(a*x)^2*sqrt(1+a^2*x^2)/(a^4*sqrt(c+a^2*c*x^2))+51/280*I*c^2*arctan(a*x)*polylog(2,-I*exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/(a^4*sqrt(c+a^2*c*x^2))-51/280*I*c^2*arctan(a*x)*polylog(2,I*exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/(a^4*sqrt(c+a^2*c*x^2))-51/280*c^2*polylog(3,-I*exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/(a^4*sqrt(c+a^2*c*x^2))+51/280*c^2*polylog(3,I*exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/(a^4*sqrt(c+a^2*c*x^2))+1/420*c*x*sqrt(c+a^2*c*x^2)/a^3-1/140*c*x^3*sqrt(c+a^2*c*x^2)/a-163/840*c*arctan(a*x)*sqrt(c+a^2*c*x^2)/a^4+1/60*c*x^2*arctan(a*x)*sqrt(c+a^2*c*x^2)/a^2+1/35*c*x^4*arctan(a*x)*sqrt(c+a^2*c*x^2)+9/112*c*x*arctan(a*x)^2*sqrt(c+a^2*c*x^2)/a^3-23/280*c*x^3*arctan(a*x)^2*sqrt(c+a^2*c*x^2)/a-1/14*a*c*x^5*arctan(a*x)^2*sqrt(c+a^2*c*x^2)-2/35*c*arctan(a*x)^3*sqrt(c+a^2*c*x^2)/a^4+1/35*c*x^2*arctan(a*x)^3*sqrt(c+a^2*c*x^2)/a^2+8/35*c*x^4*arctan(a*x)^3*sqrt(c+a^2*c*x^2)+1/7*a^2*c*x^6*arctan(a*x)^3*sqrt(c+a^2*c*x^2)],
[x^2*(c+a^2*c*x^2)^(3/2)*arctan(a*x)^3,x,108,-1/60*(c+a^2*c*x^2)^(3/2)/a^3+1/8*I*c^2*arctan(exp(I*arctan(a*x)))*arctan(a*x)^3*sqrt(1+a^2*x^2)/(a^3*sqrt(c+a^2*c*x^2))+41/60*I*c^2*arctan(a*x)*arctan(sqrt(1+I*a*x)/sqrt(1-I*a*x))*sqrt(1+a^2*x^2)/(a^3*sqrt(c+a^2*c*x^2))-3/16*I*c^2*arctan(a*x)^2*polylog(2,-I*exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/(a^3*sqrt(c+a^2*c*x^2))+3/16*I*c^2*arctan(a*x)^2*polylog(2,I*exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/(a^3*sqrt(c+a^2*c*x^2))-41/120*I*c^2*polylog(2,-I*sqrt(1+I*a*x)/sqrt(1-I*a*x))*sqrt(1+a^2*x^2)/(a^3*sqrt(c+a^2*c*x^2))+41/120*I*c^2*polylog(2,I*sqrt(1+I*a*x)/sqrt(1-I*a*x))*sqrt(1+a^2*x^2)/(a^3*sqrt(c+a^2*c*x^2))+3/8*c^2*arctan(a*x)*polylog(3,-I*exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/(a^3*sqrt(c+a^2*c*x^2))-3/8*c^2*arctan(a*x)*polylog(3,I*exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/(a^3*sqrt(c+a^2*c*x^2))+3/8*I*c^2*polylog(4,-I*exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/(a^3*sqrt(c+a^2*c*x^2))-3/8*I*c^2*polylog(4,I*exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/(a^3*sqrt(c+a^2*c*x^2))-1/30*c*sqrt(c+a^2*c*x^2)/a^3+1/12*c*x*arctan(a*x)*sqrt(c+a^2*c*x^2)/a^2+1/20*c*x^3*arctan(a*x)*sqrt(c+a^2*c*x^2)+31/240*c*arctan(a*x)^2*sqrt(c+a^2*c*x^2)/a^3-19/120*c*x^2*arctan(a*x)^2*sqrt(c+a^2*c*x^2)/a-1/10*a*c*x^4*arctan(a*x)^2*sqrt(c+a^2*c*x^2)+1/16*c*x*arctan(a*x)^3*sqrt(c+a^2*c*x^2)/a^2+7/24*c*x^3*arctan(a*x)^3*sqrt(c+a^2*c*x^2)+1/6*a^2*c*x^5*arctan(a*x)^3*sqrt(c+a^2*c*x^2)],
[x*(c+a^2*c*x^2)^(3/2)*arctan(a*x)^3,x,17,1/10*(c+a^2*c*x^2)^(3/2)*arctan(a*x)/a^2-3/20*x*(c+a^2*c*x^2)^(3/2)*arctan(a*x)^2/a+1/5*(c+a^2*c*x^2)^(5/2)*arctan(a*x)^3/(a^2*c)-1/2*c^(3/2)*arctanh(a*x*sqrt(c)/sqrt(c+a^2*c*x^2))/a^2+9/20*I*c^2*arctan(exp(I*arctan(a*x)))*arctan(a*x)^2*sqrt(1+a^2*x^2)/(a^2*sqrt(c+a^2*c*x^2))-9/20*I*c^2*arctan(a*x)*polylog(2,-I*exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/(a^2*sqrt(c+a^2*c*x^2))+9/20*I*c^2*arctan(a*x)*polylog(2,I*exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/(a^2*sqrt(c+a^2*c*x^2))+9/20*c^2*polylog(3,-I*exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/(a^2*sqrt(c+a^2*c*x^2))-9/20*c^2*polylog(3,I*exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/(a^2*sqrt(c+a^2*c*x^2))-1/20*c*x*sqrt(c+a^2*c*x^2)/a+9/20*c*arctan(a*x)*sqrt(c+a^2*c*x^2)/a^2-9/40*c*x*arctan(a*x)^2*sqrt(c+a^2*c*x^2)/a],
[(c+a^2*c*x^2)^(3/2)*arctan(a*x)^3,x,18,-1/4*(c+a^2*c*x^2)^(3/2)*arctan(a*x)^2/a+1/4*x*(c+a^2*c*x^2)^(3/2)*arctan(a*x)^3-3/4*I*c^2*arctan(exp(I*arctan(a*x)))*arctan(a*x)^3*sqrt(1+a^2*x^2)/(a*sqrt(c+a^2*c*x^2))-5*I*c^2*arctan(a*x)*arctan(sqrt(1+I*a*x)/sqrt(1-I*a*x))*sqrt(1+a^2*x^2)/(a*sqrt(c+a^2*c*x^2))+9/8*I*c^2*arctan(a*x)^2*polylog(2,-I*exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/(a*sqrt(c+a^2*c*x^2))-9/8*I*c^2*arctan(a*x)^2*polylog(2,I*exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/(a*sqrt(c+a^2*c*x^2))+5/2*I*c^2*polylog(2,-I*sqrt(1+I*a*x)/sqrt(1-I*a*x))*sqrt(1+a^2*x^2)/(a*sqrt(c+a^2*c*x^2))-5/2*I*c^2*polylog(2,I*sqrt(1+I*a*x)/sqrt(1-I*a*x))*sqrt(1+a^2*x^2)/(a*sqrt(c+a^2*c*x^2))-9/4*c^2*arctan(a*x)*polylog(3,-I*exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/(a*sqrt(c+a^2*c*x^2))+9/4*c^2*arctan(a*x)*polylog(3,I*exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/(a*sqrt(c+a^2*c*x^2))-9/4*I*c^2*polylog(4,-I*exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/(a*sqrt(c+a^2*c*x^2))+9/4*I*c^2*polylog(4,I*exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/(a*sqrt(c+a^2*c*x^2))-1/4*c*sqrt(c+a^2*c*x^2)/a+1/4*c*x*arctan(a*x)*sqrt(c+a^2*c*x^2)-9/8*c*arctan(a*x)^2*sqrt(c+a^2*c*x^2)/a+3/8*c*x*arctan(a*x)^3*sqrt(c+a^2*c*x^2)],
[(c+a^2*c*x^2)^(3/2)*arctan(a*x)^3/x,x,36,1/3*(c+a^2*c*x^2)^(3/2)*arctan(a*x)^3-c^(3/2)*arctanh(a*x*sqrt(c)/sqrt(c+a^2*c*x^2))+7*I*c^2*arctan(exp(I*arctan(a*x)))*arctan(a*x)^2*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)-2*c^2*arctan(a*x)^3*arctanh(exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)+3*I*c^2*arctan(a*x)^2*polylog(2,-exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)-7*I*c^2*arctan(a*x)*polylog(2,-I*exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)+7*I*c^2*arctan(a*x)*polylog(2,I*exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)-3*I*c^2*arctan(a*x)^2*polylog(2,exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)-6*c^2*arctan(a*x)*polylog(3,-exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)+7*c^2*polylog(3,-I*exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)-7*c^2*polylog(3,I*exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)+6*c^2*arctan(a*x)*polylog(3,exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)-6*I*c^2*polylog(4,-exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)+6*I*c^2*polylog(4,exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)+c*arctan(a*x)*sqrt(c+a^2*c*x^2)-1/2*a*c*x*arctan(a*x)^2*sqrt(c+a^2*c*x^2)+c*arctan(a*x)^3*sqrt(c+a^2*c*x^2)],
[(c+a^2*c*x^2)^(3/2)*arctan(a*x)^3/x^2,x,37,-3*I*a*c^2*arctan(exp(I*arctan(a*x)))*arctan(a*x)^3*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)-6*I*a*c^2*arctan(a*x)*arctan(sqrt(1+I*a*x)/sqrt(1-I*a*x))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)-6*a*c^2*arctan(a*x)^2*arctanh(exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)+6*I*a*c^2*arctan(a*x)*polylog(2,-exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)+9/2*I*a*c^2*arctan(a*x)^2*polylog(2,-I*exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)-9/2*I*a*c^2*arctan(a*x)^2*polylog(2,I*exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)-6*I*a*c^2*arctan(a*x)*polylog(2,exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)+3*I*a*c^2*polylog(2,-I*sqrt(1+I*a*x)/sqrt(1-I*a*x))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)-3*I*a*c^2*polylog(2,I*sqrt(1+I*a*x)/sqrt(1-I*a*x))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)-6*a*c^2*polylog(3,-exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)-9*a*c^2*arctan(a*x)*polylog(3,-I*exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)+9*a*c^2*arctan(a*x)*polylog(3,I*exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)+6*a*c^2*polylog(3,exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)-9*I*a*c^2*polylog(4,-I*exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)+9*I*a*c^2*polylog(4,I*exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)-3/2*a*c*arctan(a*x)^2*sqrt(c+a^2*c*x^2)-c*arctan(a*x)^3*sqrt(c+a^2*c*x^2)/x+1/2*a^2*c*x*arctan(a*x)^3*sqrt(c+a^2*c*x^2)],
[(c+a^2*c*x^2)^(3/2)*arctan(a*x)^3/x^3,x,50,6*I*a^2*c^2*arctan(exp(I*arctan(a*x)))*arctan(a*x)^2*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)-3*a^2*c^2*arctan(a*x)^3*arctanh(exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)-6*a^2*c^2*arctan(a*x)*arctanh(sqrt(1+I*a*x)/sqrt(1-I*a*x))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)+9/2*I*a^2*c^2*arctan(a*x)^2*polylog(2,-exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)-6*I*a^2*c^2*arctan(a*x)*polylog(2,-I*exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)+6*I*a^2*c^2*arctan(a*x)*polylog(2,I*exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)-9/2*I*a^2*c^2*arctan(a*x)^2*polylog(2,exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)+3*I*a^2*c^2*polylog(2,-sqrt(1+I*a*x)/sqrt(1-I*a*x))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)-3*I*a^2*c^2*polylog(2,sqrt(1+I*a*x)/sqrt(1-I*a*x))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)-9*a^2*c^2*arctan(a*x)*polylog(3,-exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)+6*a^2*c^2*polylog(3,-I*exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)-6*a^2*c^2*polylog(3,I*exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)+9*a^2*c^2*arctan(a*x)*polylog(3,exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)-9*I*a^2*c^2*polylog(4,-exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)+9*I*a^2*c^2*polylog(4,exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)-3/2*a*c*arctan(a*x)^2*sqrt(c+a^2*c*x^2)/x+a^2*c*arctan(a*x)^3*sqrt(c+a^2*c*x^2)-1/2*c*arctan(a*x)^3*sqrt(c+a^2*c*x^2)/x^2],
[(c+a^2*c*x^2)^(3/2)*arctan(a*x)^3/x^4,x,48,-1/3*(c+a^2*c*x^2)^(3/2)*arctan(a*x)^3/x^3-a^3*c^(3/2)*arctanh(sqrt(c+a^2*c*x^2)/sqrt(c))-2*I*a^3*c^2*arctan(exp(I*arctan(a*x)))*arctan(a*x)^3*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)-7*a^3*c^2*arctan(a*x)^2*arctanh(exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)+7*I*a^3*c^2*arctan(a*x)*polylog(2,-exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)+3*I*a^3*c^2*arctan(a*x)^2*polylog(2,-I*exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)-3*I*a^3*c^2*arctan(a*x)^2*polylog(2,I*exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)-7*I*a^3*c^2*arctan(a*x)*polylog(2,exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)-7*a^3*c^2*polylog(3,-exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)-6*a^3*c^2*arctan(a*x)*polylog(3,-I*exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)+6*a^3*c^2*arctan(a*x)*polylog(3,I*exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)+7*a^3*c^2*polylog(3,exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)-6*I*a^3*c^2*polylog(4,-I*exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)+6*I*a^3*c^2*polylog(4,I*exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)-a^2*c*arctan(a*x)*sqrt(c+a^2*c*x^2)/x-1/2*a*c*arctan(a*x)^2*sqrt(c+a^2*c*x^2)/x^2-a^2*c*arctan(a*x)^3*sqrt(c+a^2*c*x^2)/x],
[x^3*(c+a^2*c*x^2)^(5/2)*arctan(a*x)^3,x,547,1433/15120*c^(5/2)*arctanh(a*x*sqrt(c)/sqrt(c+a^2*c*x^2))/a^4-115/1344*I*c^3*arctan(exp(I*arctan(a*x)))*arctan(a*x)^2*sqrt(1+a^2*x^2)/(a^4*sqrt(c+a^2*c*x^2))+115/1344*I*c^3*arctan(a*x)*polylog(2,-I*exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/(a^4*sqrt(c+a^2*c*x^2))-115/1344*I*c^3*arctan(a*x)*polylog(2,I*exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/(a^4*sqrt(c+a^2*c*x^2))-115/1344*c^3*polylog(3,-I*exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/(a^4*sqrt(c+a^2*c*x^2))+115/1344*c^3*polylog(3,I*exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/(a^4*sqrt(c+a^2*c*x^2))+85/12096*c^2*x*sqrt(c+a^2*c*x^2)/a^3-1/240*c^2*x^3*sqrt(c+a^2*c*x^2)/a-1/504*a*c^2*x^5*sqrt(c+a^2*c*x^2)-6157/60480*c^2*arctan(a*x)*sqrt(c+a^2*c*x^2)/a^4-47/30240*c^2*x^2*arctan(a*x)*sqrt(c+a^2*c*x^2)/a^2+67/2520*c^2*x^4*arctan(a*x)*sqrt(c+a^2*c*x^2)+1/84*a^2*c^2*x^6*arctan(a*x)*sqrt(c+a^2*c*x^2)+47/896*c^2*x*arctan(a*x)^2*sqrt(c+a^2*c*x^2)/a^3-205/4032*c^2*x^3*arctan(a*x)^2*sqrt(c+a^2*c*x^2)/a-103/1008*a*c^2*x^5*arctan(a*x)^2*sqrt(c+a^2*c*x^2)-1/24*a^3*c^2*x^7*arctan(a*x)^2*sqrt(c+a^2*c*x^2)-2/63*c^2*arctan(a*x)^3*sqrt(c+a^2*c*x^2)/a^4+1/63*c^2*x^2*arctan(a*x)^3*sqrt(c+a^2*c*x^2)/a^2+5/21*c^2*x^4*arctan(a*x)^3*sqrt(c+a^2*c*x^2)+19/63*a^2*c^2*x^6*arctan(a*x)^3*sqrt(c+a^2*c*x^2)+1/9*a^4*c^2*x^8*arctan(a*x)^3*sqrt(c+a^2*c*x^2)],
[x^2*(c+a^2*c*x^2)^(5/2)*arctan(a*x)^3,x,293,-3/560*c*(c+a^2*c*x^2)^(3/2)/a^3-1/280*(c+a^2*c*x^2)^(5/2)/a^3+5/64*I*c^3*arctan(exp(I*arctan(a*x)))*arctan(a*x)^3*sqrt(1+a^2*x^2)/(a^3*sqrt(c+a^2*c*x^2))+397/840*I*c^3*arctan(a*x)*arctan(sqrt(1+I*a*x)/sqrt(1-I*a*x))*sqrt(1+a^2*x^2)/(a^3*sqrt(c+a^2*c*x^2))-15/128*I*c^3*arctan(a*x)^2*polylog(2,-I*exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/(a^3*sqrt(c+a^2*c*x^2))+15/128*I*c^3*arctan(a*x)^2*polylog(2,I*exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/(a^3*sqrt(c+a^2*c*x^2))-397/1680*I*c^3*polylog(2,-I*sqrt(1+I*a*x)/sqrt(1-I*a*x))*sqrt(1+a^2*x^2)/(a^3*sqrt(c+a^2*c*x^2))+397/1680*I*c^3*polylog(2,I*sqrt(1+I*a*x)/sqrt(1-I*a*x))*sqrt(1+a^2*x^2)/(a^3*sqrt(c+a^2*c*x^2))+15/64*c^3*arctan(a*x)*polylog(3,-I*exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/(a^3*sqrt(c+a^2*c*x^2))-15/64*c^3*arctan(a*x)*polylog(3,I*exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/(a^3*sqrt(c+a^2*c*x^2))+15/64*I*c^3*polylog(4,-I*exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/(a^3*sqrt(c+a^2*c*x^2))-15/64*I*c^3*polylog(4,I*exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/(a^3*sqrt(c+a^2*c*x^2))+13/6720*c^2*sqrt(c+a^2*c*x^2)/a^3+43/1344*c^2*x*arctan(a*x)*sqrt(c+a^2*c*x^2)/a^2+29/560*c^2*x^3*arctan(a*x)*sqrt(c+a^2*c*x^2)+1/56*a^2*c^2*x^5*arctan(a*x)*sqrt(c+a^2*c*x^2)+1373/13440*c^2*arctan(a*x)^2*sqrt(c+a^2*c*x^2)/a^3-737/6720*c^2*x^2*arctan(a*x)^2*sqrt(c+a^2*c*x^2)/a-83/560*a*c^2*x^4*arctan(a*x)^2*sqrt(c+a^2*c*x^2)-3/56*a^3*c^2*x^6*arctan(a*x)^2*sqrt(c+a^2*c*x^2)+5/128*c^2*x*arctan(a*x)^3*sqrt(c+a^2*c*x^2)/a^2+59/192*c^2*x^3*arctan(a*x)^3*sqrt(c+a^2*c*x^2)+17/48*a^2*c^2*x^5*arctan(a*x)^3*sqrt(c+a^2*c*x^2)+1/8*a^4*c^2*x^7*arctan(a*x)^3*sqrt(c+a^2*c*x^2)],
[x*(c+a^2*c*x^2)^(5/2)*arctan(a*x)^3,x,22,-1/140*c*x*(c+a^2*c*x^2)^(3/2)/a+5/84*c*(c+a^2*c*x^2)^(3/2)*arctan(a*x)/a^2+1/35*(c+a^2*c*x^2)^(5/2)*arctan(a*x)/a^2-5/56*c*x*(c+a^2*c*x^2)^(3/2)*arctan(a*x)^2/a-1/14*x*(c+a^2*c*x^2)^(5/2)*arctan(a*x)^2/a+1/7*(c+a^2*c*x^2)^(7/2)*arctan(a*x)^3/(a^2*c)-37/120*c^(5/2)*arctanh(a*x*sqrt(c)/sqrt(c+a^2*c*x^2))/a^2+15/56*I*c^3*arctan(exp(I*arctan(a*x)))*arctan(a*x)^2*sqrt(1+a^2*x^2)/(a^2*sqrt(c+a^2*c*x^2))-15/56*I*c^3*arctan(a*x)*polylog(2,-I*exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/(a^2*sqrt(c+a^2*c*x^2))+15/56*I*c^3*arctan(a*x)*polylog(2,I*exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/(a^2*sqrt(c+a^2*c*x^2))+15/56*c^3*polylog(3,-I*exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/(a^2*sqrt(c+a^2*c*x^2))-15/56*c^3*polylog(3,I*exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/(a^2*sqrt(c+a^2*c*x^2))-17/420*c^2*x*sqrt(c+a^2*c*x^2)/a+15/56*c^2*arctan(a*x)*sqrt(c+a^2*c*x^2)/a^2-15/112*c^2*x*arctan(a*x)^2*sqrt(c+a^2*c*x^2)/a],
[(c+a^2*c*x^2)^(5/2)*arctan(a*x)^3,x,23,-1/60*c*(c+a^2*c*x^2)^(3/2)/a+1/20*c*x*(c+a^2*c*x^2)^(3/2)*arctan(a*x)-5/24*c*(c+a^2*c*x^2)^(3/2)*arctan(a*x)^2/a-1/10*(c+a^2*c*x^2)^(5/2)*arctan(a*x)^2/a+5/24*c*x*(c+a^2*c*x^2)^(3/2)*arctan(a*x)^3+1/6*x*(c+a^2*c*x^2)^(5/2)*arctan(a*x)^3-5/8*I*c^3*arctan(exp(I*arctan(a*x)))*arctan(a*x)^3*sqrt(1+a^2*x^2)/(a*sqrt(c+a^2*c*x^2))-259/60*I*c^3*arctan(a*x)*arctan(sqrt(1+I*a*x)/sqrt(1-I*a*x))*sqrt(1+a^2*x^2)/(a*sqrt(c+a^2*c*x^2))+15/16*I*c^3*arctan(a*x)^2*polylog(2,-I*exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/(a*sqrt(c+a^2*c*x^2))-15/16*I*c^3*arctan(a*x)^2*polylog(2,I*exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/(a*sqrt(c+a^2*c*x^2))+259/120*I*c^3*polylog(2,-I*sqrt(1+I*a*x)/sqrt(1-I*a*x))*sqrt(1+a^2*x^2)/(a*sqrt(c+a^2*c*x^2))-259/120*I*c^3*polylog(2,I*sqrt(1+I*a*x)/sqrt(1-I*a*x))*sqrt(1+a^2*x^2)/(a*sqrt(c+a^2*c*x^2))-15/8*c^3*arctan(a*x)*polylog(3,-I*exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/(a*sqrt(c+a^2*c*x^2))+15/8*c^3*arctan(a*x)*polylog(3,I*exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/(a*sqrt(c+a^2*c*x^2))-15/8*I*c^3*polylog(4,-I*exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/(a*sqrt(c+a^2*c*x^2))+15/8*I*c^3*polylog(4,I*exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/(a*sqrt(c+a^2*c*x^2))-17/60*c^2*sqrt(c+a^2*c*x^2)/a+17/60*c^2*x*arctan(a*x)*sqrt(c+a^2*c*x^2)-15/16*c^2*arctan(a*x)^2*sqrt(c+a^2*c*x^2)/a+5/16*c^2*x*arctan(a*x)^3*sqrt(c+a^2*c*x^2)],
[(c+a^2*c*x^2)^(5/2)*arctan(a*x)^3/x,x,54,1/10*c*(c+a^2*c*x^2)^(3/2)*arctan(a*x)-3/20*a*c*x*(c+a^2*c*x^2)^(3/2)*arctan(a*x)^2+1/3*c*(c+a^2*c*x^2)^(3/2)*arctan(a*x)^3+1/5*(c+a^2*c*x^2)^(5/2)*arctan(a*x)^3-3/2*c^(5/2)*arctanh(a*x*sqrt(c)/sqrt(c+a^2*c*x^2))+149/20*I*c^3*arctan(exp(I*arctan(a*x)))*arctan(a*x)^2*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)-2*c^3*arctan(a*x)^3*arctanh(exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)+3*I*c^3*arctan(a*x)^2*polylog(2,-exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)-149/20*I*c^3*arctan(a*x)*polylog(2,-I*exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)+149/20*I*c^3*arctan(a*x)*polylog(2,I*exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)-3*I*c^3*arctan(a*x)^2*polylog(2,exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)-6*c^3*arctan(a*x)*polylog(3,-exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)+149/20*c^3*polylog(3,-I*exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)-149/20*c^3*polylog(3,I*exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)+6*c^3*arctan(a*x)*polylog(3,exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)-6*I*c^3*polylog(4,-exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)+6*I*c^3*polylog(4,exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)-1/20*a*c^2*x*sqrt(c+a^2*c*x^2)+29/20*c^2*arctan(a*x)*sqrt(c+a^2*c*x^2)-29/40*a*c^2*x*arctan(a*x)^2*sqrt(c+a^2*c*x^2)+c^2*arctan(a*x)^3*sqrt(c+a^2*c*x^2)],
[(c+a^2*c*x^2)^(5/2)*arctan(a*x)^3/x^2,x,56,-1/4*a*c*(c+a^2*c*x^2)^(3/2)*arctan(a*x)^2+1/4*a^2*c*x*(c+a^2*c*x^2)^(3/2)*arctan(a*x)^3-15/4*I*a*c^3*arctan(exp(I*arctan(a*x)))*arctan(a*x)^3*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)-11*I*a*c^3*arctan(a*x)*arctan(sqrt(1+I*a*x)/sqrt(1-I*a*x))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)-6*a*c^3*arctan(a*x)^2*arctanh(exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)+6*I*a*c^3*arctan(a*x)*polylog(2,-exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)+45/8*I*a*c^3*arctan(a*x)^2*polylog(2,-I*exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)-45/8*I*a*c^3*arctan(a*x)^2*polylog(2,I*exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)-6*I*a*c^3*arctan(a*x)*polylog(2,exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)+11/2*I*a*c^3*polylog(2,-I*sqrt(1+I*a*x)/sqrt(1-I*a*x))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)-11/2*I*a*c^3*polylog(2,I*sqrt(1+I*a*x)/sqrt(1-I*a*x))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)-6*a*c^3*polylog(3,-exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)-45/4*a*c^3*arctan(a*x)*polylog(3,-I*exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)+45/4*a*c^3*arctan(a*x)*polylog(3,I*exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)+6*a*c^3*polylog(3,exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)-45/4*I*a*c^3*polylog(4,-I*exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)+45/4*I*a*c^3*polylog(4,I*exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)-1/4*a*c^2*sqrt(c+a^2*c*x^2)+1/4*a^2*c^2*x*arctan(a*x)*sqrt(c+a^2*c*x^2)-21/8*a*c^2*arctan(a*x)^2*sqrt(c+a^2*c*x^2)-c^2*arctan(a*x)^3*sqrt(c+a^2*c*x^2)/x+7/8*a^2*c^2*x*arctan(a*x)^3*sqrt(c+a^2*c*x^2)],
[(c+a^2*c*x^2)^(5/2)*arctan(a*x)^3/x^3,x,87,1/3*a^2*c*(c+a^2*c*x^2)^(3/2)*arctan(a*x)^3-a^2*c^(5/2)*arctanh(a*x*sqrt(c)/sqrt(c+a^2*c*x^2))+13*I*a^2*c^3*arctan(exp(I*arctan(a*x)))*arctan(a*x)^2*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)-5*a^2*c^3*arctan(a*x)^3*arctanh(exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)-6*a^2*c^3*arctan(a*x)*arctanh(sqrt(1+I*a*x)/sqrt(1-I*a*x))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)+15/2*I*a^2*c^3*arctan(a*x)^2*polylog(2,-exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)-13*I*a^2*c^3*arctan(a*x)*polylog(2,-I*exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)+13*I*a^2*c^3*arctan(a*x)*polylog(2,I*exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)-15/2*I*a^2*c^3*arctan(a*x)^2*polylog(2,exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)+3*I*a^2*c^3*polylog(2,-sqrt(1+I*a*x)/sqrt(1-I*a*x))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)-3*I*a^2*c^3*polylog(2,sqrt(1+I*a*x)/sqrt(1-I*a*x))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)-15*a^2*c^3*arctan(a*x)*polylog(3,-exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)+13*a^2*c^3*polylog(3,-I*exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)-13*a^2*c^3*polylog(3,I*exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)+15*a^2*c^3*arctan(a*x)*polylog(3,exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)-15*I*a^2*c^3*polylog(4,-exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)+15*I*a^2*c^3*polylog(4,exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)+a^2*c^2*arctan(a*x)*sqrt(c+a^2*c*x^2)-3/2*a*c^2*arctan(a*x)^2*sqrt(c+a^2*c*x^2)/x-1/2*a^3*c^2*x*arctan(a*x)^2*sqrt(c+a^2*c*x^2)+2*a^2*c^2*arctan(a*x)^3*sqrt(c+a^2*c*x^2)-1/2*c^2*arctan(a*x)^3*sqrt(c+a^2*c*x^2)/x^2],
[(c+a^2*c*x^2)^(5/2)*arctan(a*x)^3/x^4,x,86,-1/3*c*(c+a^2*c*x^2)^(3/2)*arctan(a*x)^3/x^3-a^3*c^(5/2)*arctanh(sqrt(c+a^2*c*x^2)/sqrt(c))-5*I*a^3*c^3*arctan(exp(I*arctan(a*x)))*arctan(a*x)^3*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)-6*I*a^3*c^3*arctan(a*x)*arctan(sqrt(1+I*a*x)/sqrt(1-I*a*x))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)-13*a^3*c^3*arctan(a*x)^2*arctanh(exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)+13*I*a^3*c^3*arctan(a*x)*polylog(2,-exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)+15/2*I*a^3*c^3*arctan(a*x)^2*polylog(2,-I*exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)-15/2*I*a^3*c^3*arctan(a*x)^2*polylog(2,I*exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)-13*I*a^3*c^3*arctan(a*x)*polylog(2,exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)+3*I*a^3*c^3*polylog(2,-I*sqrt(1+I*a*x)/sqrt(1-I*a*x))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)-3*I*a^3*c^3*polylog(2,I*sqrt(1+I*a*x)/sqrt(1-I*a*x))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)-13*a^3*c^3*polylog(3,-exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)-15*a^3*c^3*arctan(a*x)*polylog(3,-I*exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)+15*a^3*c^3*arctan(a*x)*polylog(3,I*exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)+13*a^3*c^3*polylog(3,exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)-15*I*a^3*c^3*polylog(4,-I*exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)+15*I*a^3*c^3*polylog(4,I*exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)-a^2*c^2*arctan(a*x)*sqrt(c+a^2*c*x^2)/x-3/2*a^3*c^2*arctan(a*x)^2*sqrt(c+a^2*c*x^2)-1/2*a*c^2*arctan(a*x)^2*sqrt(c+a^2*c*x^2)/x^2-2*a^2*c^2*arctan(a*x)^3*sqrt(c+a^2*c*x^2)/x+1/2*a^4*c^2*x*arctan(a*x)^3*sqrt(c+a^2*c*x^2)],

# p<0
[x^3*arctan(a*x)^3/sqrt(c+a^2*c*x^2),x,24,-arctanh(a*x*sqrt(c)/sqrt(c+a^2*c*x^2))/(a^4*sqrt(c))-5*I*arctan(exp(I*arctan(a*x)))*arctan(a*x)^2*sqrt(1+a^2*x^2)/(a^4*sqrt(c+a^2*c*x^2))+5*I*arctan(a*x)*polylog(2,-I*exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/(a^4*sqrt(c+a^2*c*x^2))-5*I*arctan(a*x)*polylog(2,I*exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/(a^4*sqrt(c+a^2*c*x^2))-5*polylog(3,-I*exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/(a^4*sqrt(c+a^2*c*x^2))+5*polylog(3,I*exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/(a^4*sqrt(c+a^2*c*x^2))+arctan(a*x)*sqrt(c+a^2*c*x^2)/(a^4*c)-1/2*x*arctan(a*x)^2*sqrt(c+a^2*c*x^2)/(a^3*c)-2/3*arctan(a*x)^3*sqrt(c+a^2*c*x^2)/(a^4*c)+1/3*x^2*arctan(a*x)^3*sqrt(c+a^2*c*x^2)/(a^2*c)],
[x^2*arctan(a*x)^3/sqrt(c+a^2*c*x^2),x,15,I*arctan(exp(I*arctan(a*x)))*arctan(a*x)^3*sqrt(1+a^2*x^2)/(a^3*sqrt(c+a^2*c*x^2))-6*I*arctan(a*x)*arctan(sqrt(1+I*a*x)/sqrt(1-I*a*x))*sqrt(1+a^2*x^2)/(a^3*sqrt(c+a^2*c*x^2))-3/2*I*arctan(a*x)^2*polylog(2,-I*exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/(a^3*sqrt(c+a^2*c*x^2))+3/2*I*arctan(a*x)^2*polylog(2,I*exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/(a^3*sqrt(c+a^2*c*x^2))+3*I*polylog(2,-I*sqrt(1+I*a*x)/sqrt(1-I*a*x))*sqrt(1+a^2*x^2)/(a^3*sqrt(c+a^2*c*x^2))-3*I*polylog(2,I*sqrt(1+I*a*x)/sqrt(1-I*a*x))*sqrt(1+a^2*x^2)/(a^3*sqrt(c+a^2*c*x^2))+3*arctan(a*x)*polylog(3,-I*exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/(a^3*sqrt(c+a^2*c*x^2))-3*arctan(a*x)*polylog(3,I*exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/(a^3*sqrt(c+a^2*c*x^2))+3*I*polylog(4,-I*exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/(a^3*sqrt(c+a^2*c*x^2))-3*I*polylog(4,I*exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/(a^3*sqrt(c+a^2*c*x^2))-3/2*arctan(a*x)^2*sqrt(c+a^2*c*x^2)/(a^3*c)+1/2*x*arctan(a*x)^3*sqrt(c+a^2*c*x^2)/(a^2*c)],
[x*arctan(a*x)^3/sqrt(c+a^2*c*x^2),x,10,6*I*arctan(exp(I*arctan(a*x)))*arctan(a*x)^2*sqrt(1+a^2*x^2)/(a^2*sqrt(c+a^2*c*x^2))-6*I*arctan(a*x)*polylog(2,-I*exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/(a^2*sqrt(c+a^2*c*x^2))+6*I*arctan(a*x)*polylog(2,I*exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/(a^2*sqrt(c+a^2*c*x^2))+6*polylog(3,-I*exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/(a^2*sqrt(c+a^2*c*x^2))-6*polylog(3,I*exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/(a^2*sqrt(c+a^2*c*x^2))+arctan(a*x)^3*sqrt(c+a^2*c*x^2)/(a^2*c)],
[arctan(a*x)^3/sqrt(c+a^2*c*x^2),x,11,-2*I*arctan(exp(I*arctan(a*x)))*arctan(a*x)^3*sqrt(1+a^2*x^2)/(a*sqrt(c+a^2*c*x^2))+3*I*arctan(a*x)^2*polylog(2,-I*exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/(a*sqrt(c+a^2*c*x^2))-3*I*arctan(a*x)^2*polylog(2,I*exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/(a*sqrt(c+a^2*c*x^2))-6*arctan(a*x)*polylog(3,-I*exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/(a*sqrt(c+a^2*c*x^2))+6*arctan(a*x)*polylog(3,I*exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/(a*sqrt(c+a^2*c*x^2))-6*I*polylog(4,-I*exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/(a*sqrt(c+a^2*c*x^2))+6*I*polylog(4,I*exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/(a*sqrt(c+a^2*c*x^2))],
[arctan(a*x)^3/(x*sqrt(c+a^2*c*x^2)),x,11,-2*arctan(a*x)^3*arctanh(exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)+3*I*arctan(a*x)^2*polylog(2,-exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)-3*I*arctan(a*x)^2*polylog(2,exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)-6*arctan(a*x)*polylog(3,-exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)+6*arctan(a*x)*polylog(3,exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)-6*I*polylog(4,-exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)+6*I*polylog(4,exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)],
[arctan(a*x)^3/(x^2*sqrt(c+a^2*c*x^2)),x,10,-6*a*arctan(a*x)^2*arctanh(exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)+6*I*a*arctan(a*x)*polylog(2,-exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)-6*I*a*arctan(a*x)*polylog(2,exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)-6*a*polylog(3,-exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)+6*a*polylog(3,exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)-arctan(a*x)^3*sqrt(c+a^2*c*x^2)/(c*x)],
[arctan(a*x)^3/(x^3*sqrt(c+a^2*c*x^2)),x,15,a^2*arctan(a*x)^3*arctanh(exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)-6*a^2*arctan(a*x)*arctanh(sqrt(1+I*a*x)/sqrt(1-I*a*x))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)-3/2*I*a^2*arctan(a*x)^2*polylog(2,-exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)+3/2*I*a^2*arctan(a*x)^2*polylog(2,exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)+3*I*a^2*polylog(2,-sqrt(1+I*a*x)/sqrt(1-I*a*x))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)-3*I*a^2*polylog(2,sqrt(1+I*a*x)/sqrt(1-I*a*x))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)+3*a^2*arctan(a*x)*polylog(3,-exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)-3*a^2*arctan(a*x)*polylog(3,exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)+3*I*a^2*polylog(4,-exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)-3*I*a^2*polylog(4,exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)-3/2*a*arctan(a*x)^2*sqrt(c+a^2*c*x^2)/(c*x)-1/2*arctan(a*x)^3*sqrt(c+a^2*c*x^2)/(c*x^2)],
[arctan(a*x)^3/(x^4*sqrt(c+a^2*c*x^2)),x,25,-a^3*arctanh(sqrt(c+a^2*c*x^2)/sqrt(c))/sqrt(c)+5*a^3*arctan(a*x)^2*arctanh(exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)-5*I*a^3*arctan(a*x)*polylog(2,-exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)+5*I*a^3*arctan(a*x)*polylog(2,exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)+5*a^3*polylog(3,-exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)-5*a^3*polylog(3,exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/sqrt(c+a^2*c*x^2)-a^2*arctan(a*x)*sqrt(c+a^2*c*x^2)/(c*x)-1/2*a*arctan(a*x)^2*sqrt(c+a^2*c*x^2)/(c*x^2)-1/3*arctan(a*x)^3*sqrt(c+a^2*c*x^2)/(c*x^3)+2/3*a^2*arctan(a*x)^3*sqrt(c+a^2*c*x^2)/(c*x)],
[x^3*arctan(a*x)^3/(c+a^2*c*x^2)^(3/2),x,14,6*x/(a^3*c*sqrt(c+a^2*c*x^2))-6*arctan(a*x)/(a^4*c*sqrt(c+a^2*c*x^2))-3*x*arctan(a*x)^2/(a^3*c*sqrt(c+a^2*c*x^2))+arctan(a*x)^3/(a^4*c*sqrt(c+a^2*c*x^2))+6*I*arctan(exp(I*arctan(a*x)))*arctan(a*x)^2*sqrt(1+a^2*x^2)/(a^4*c*sqrt(c+a^2*c*x^2))-6*I*arctan(a*x)*polylog(2,-I*exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/(a^4*c*sqrt(c+a^2*c*x^2))+6*I*arctan(a*x)*polylog(2,I*exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/(a^4*c*sqrt(c+a^2*c*x^2))+6*polylog(3,-I*exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/(a^4*c*sqrt(c+a^2*c*x^2))-6*polylog(3,I*exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/(a^4*c*sqrt(c+a^2*c*x^2))+arctan(a*x)^3*sqrt(c+a^2*c*x^2)/(a^4*c^2)],
[x^2*arctan(a*x)^3/(c+a^2*c*x^2)^(3/2),x,14,6/(a^3*c*sqrt(c+a^2*c*x^2))+6*x*arctan(a*x)/(a^2*c*sqrt(c+a^2*c*x^2))-3*arctan(a*x)^2/(a^3*c*sqrt(c+a^2*c*x^2))-x*arctan(a*x)^3/(a^2*c*sqrt(c+a^2*c*x^2))-2*I*arctan(exp(I*arctan(a*x)))*arctan(a*x)^3*sqrt(1+a^2*x^2)/(a^3*c*sqrt(c+a^2*c*x^2))+3*I*arctan(a*x)^2*polylog(2,-I*exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/(a^3*c*sqrt(c+a^2*c*x^2))-3*I*arctan(a*x)^2*polylog(2,I*exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/(a^3*c*sqrt(c+a^2*c*x^2))-6*arctan(a*x)*polylog(3,-I*exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/(a^3*c*sqrt(c+a^2*c*x^2))+6*arctan(a*x)*polylog(3,I*exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/(a^3*c*sqrt(c+a^2*c*x^2))-6*I*polylog(4,-I*exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/(a^3*c*sqrt(c+a^2*c*x^2))+6*I*polylog(4,I*exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/(a^3*c*sqrt(c+a^2*c*x^2))],
[x*arctan(a*x)^3/(c+a^2*c*x^2)^(3/2),x,3,-6*x/(a*c*sqrt(c+a^2*c*x^2))+6*arctan(a*x)/(a^2*c*sqrt(c+a^2*c*x^2))+3*x*arctan(a*x)^2/(a*c*sqrt(c+a^2*c*x^2))-arctan(a*x)^3/(a^2*c*sqrt(c+a^2*c*x^2))],
[arctan(a*x)^3/(c+a^2*c*x^2)^(3/2),x,2,(-6)/(a*c*sqrt(c+a^2*c*x^2))-6*x*arctan(a*x)/(c*sqrt(c+a^2*c*x^2))+3*arctan(a*x)^2/(a*c*sqrt(c+a^2*c*x^2))+x*arctan(a*x)^3/(c*sqrt(c+a^2*c*x^2))],
[arctan(a*x)^3/(x*(c+a^2*c*x^2)^(3/2)),x,15,6*a*x/(c*sqrt(c+a^2*c*x^2))-6*arctan(a*x)/(c*sqrt(c+a^2*c*x^2))-3*a*x*arctan(a*x)^2/(c*sqrt(c+a^2*c*x^2))+arctan(a*x)^3/(c*sqrt(c+a^2*c*x^2))-2*arctan(a*x)^3*arctanh(exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/(c*sqrt(c+a^2*c*x^2))+3*I*arctan(a*x)^2*polylog(2,-exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/(c*sqrt(c+a^2*c*x^2))-3*I*arctan(a*x)^2*polylog(2,exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/(c*sqrt(c+a^2*c*x^2))-6*arctan(a*x)*polylog(3,-exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/(c*sqrt(c+a^2*c*x^2))+6*arctan(a*x)*polylog(3,exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/(c*sqrt(c+a^2*c*x^2))-6*I*polylog(4,-exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/(c*sqrt(c+a^2*c*x^2))+6*I*polylog(4,exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/(c*sqrt(c+a^2*c*x^2))],
[arctan(a*x)^3/(x^2*(c+a^2*c*x^2)^(3/2)),x,13,6*a/(c*sqrt(c+a^2*c*x^2))+6*a^2*x*arctan(a*x)/(c*sqrt(c+a^2*c*x^2))-3*a*arctan(a*x)^2/(c*sqrt(c+a^2*c*x^2))-a^2*x*arctan(a*x)^3/(c*sqrt(c+a^2*c*x^2))-6*a*arctan(a*x)^2*arctanh(exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/(c*sqrt(c+a^2*c*x^2))+6*I*a*arctan(a*x)*polylog(2,-exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/(c*sqrt(c+a^2*c*x^2))-6*I*a*arctan(a*x)*polylog(2,exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/(c*sqrt(c+a^2*c*x^2))-6*a*polylog(3,-exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/(c*sqrt(c+a^2*c*x^2))+6*a*polylog(3,exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/(c*sqrt(c+a^2*c*x^2))-arctan(a*x)^3*sqrt(c+a^2*c*x^2)/(c^2*x)],
[x^5*arctan(a*x)^3/(c+a^2*c*x^2)^(5/2),x,22,2/27*x^3/(a^3*c*(c+a^2*c*x^2)^(3/2))-2/9*x^2*arctan(a*x)/(a^4*c*(c+a^2*c*x^2)^(3/2))-1/3*x^3*arctan(a*x)^2/(a^3*c*(c+a^2*c*x^2)^(3/2))+1/3*x^2*arctan(a*x)^3/(a^4*c*(c+a^2*c*x^2)^(3/2))+94/9*x/(a^5*c^2*sqrt(c+a^2*c*x^2))-94/9*arctan(a*x)/(a^6*c^2*sqrt(c+a^2*c*x^2))-5*x*arctan(a*x)^2/(a^5*c^2*sqrt(c+a^2*c*x^2))+5/3*arctan(a*x)^3/(a^6*c^2*sqrt(c+a^2*c*x^2))+6*I*arctan(exp(I*arctan(a*x)))*arctan(a*x)^2*sqrt(1+a^2*x^2)/(a^6*c^2*sqrt(c+a^2*c*x^2))-6*I*arctan(a*x)*polylog(2,-I*exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/(a^6*c^2*sqrt(c+a^2*c*x^2))+6*I*arctan(a*x)*polylog(2,I*exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/(a^6*c^2*sqrt(c+a^2*c*x^2))+6*polylog(3,-I*exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/(a^6*c^2*sqrt(c+a^2*c*x^2))-6*polylog(3,I*exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/(a^6*c^2*sqrt(c+a^2*c*x^2))+arctan(a*x)^3*sqrt(c+a^2*c*x^2)/(a^6*c^3)],
[x^4*arctan(a*x)^3/(c+a^2*c*x^2)^(5/2),x,22,(-2/27)/(a^5*c*(c+a^2*c*x^2)^(3/2))+2/9*x^3*arctan(a*x)/(a^2*c*(c+a^2*c*x^2)^(3/2))-1/3*x^2*arctan(a*x)^2/(a^3*c*(c+a^2*c*x^2)^(3/2))-1/3*x^3*arctan(a*x)^3/(a^2*c*(c+a^2*c*x^2)^(3/2))+68/9/(a^5*c^2*sqrt(c+a^2*c*x^2))+22/3*x*arctan(a*x)/(a^4*c^2*sqrt(c+a^2*c*x^2))-11/3*arctan(a*x)^2/(a^5*c^2*sqrt(c+a^2*c*x^2))-x*arctan(a*x)^3/(a^4*c^2*sqrt(c+a^2*c*x^2))-2*I*arctan(exp(I*arctan(a*x)))*arctan(a*x)^3*sqrt(1+a^2*x^2)/(a^5*c^2*sqrt(c+a^2*c*x^2))+3*I*arctan(a*x)^2*polylog(2,-I*exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/(a^5*c^2*sqrt(c+a^2*c*x^2))-3*I*arctan(a*x)^2*polylog(2,I*exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/(a^5*c^2*sqrt(c+a^2*c*x^2))-6*arctan(a*x)*polylog(3,-I*exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/(a^5*c^2*sqrt(c+a^2*c*x^2))+6*arctan(a*x)*polylog(3,I*exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/(a^5*c^2*sqrt(c+a^2*c*x^2))-6*I*polylog(4,-I*exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/(a^5*c^2*sqrt(c+a^2*c*x^2))+6*I*polylog(4,I*exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/(a^5*c^2*sqrt(c+a^2*c*x^2))],
[x^3*arctan(a*x)^3/(c+a^2*c*x^2)^(5/2),x,7,-2/27*x^3/(a*c*(c+a^2*c*x^2)^(3/2))+2/9*x^2*arctan(a*x)/(a^2*c*(c+a^2*c*x^2)^(3/2))+1/3*x^3*arctan(a*x)^2/(a*c*(c+a^2*c*x^2)^(3/2))-1/3*x^2*arctan(a*x)^3/(a^2*c*(c+a^2*c*x^2)^(3/2))-40/9*x/(a^3*c^2*sqrt(c+a^2*c*x^2))+40/9*arctan(a*x)/(a^4*c^2*sqrt(c+a^2*c*x^2))+2*x*arctan(a*x)^2/(a^3*c^2*sqrt(c+a^2*c*x^2))-2/3*arctan(a*x)^3/(a^4*c^2*sqrt(c+a^2*c*x^2))],
[x^2*arctan(a*x)^3/(c+a^2*c*x^2)^(5/2),x,7,2/27/(a^3*c*(c+a^2*c*x^2)^(3/2))-2/9*x^3*arctan(a*x)/(c*(c+a^2*c*x^2)^(3/2))+1/3*x^2*arctan(a*x)^2/(a*c*(c+a^2*c*x^2)^(3/2))+1/3*x^3*arctan(a*x)^3/(c*(c+a^2*c*x^2)^(3/2))+(-14/9)/(a^3*c^2*sqrt(c+a^2*c*x^2))-4/3*x*arctan(a*x)/(a^2*c^2*sqrt(c+a^2*c*x^2))+2/3*arctan(a*x)^2/(a^3*c^2*sqrt(c+a^2*c*x^2))],
[x*arctan(a*x)^3/(c+a^2*c*x^2)^(5/2),x,6,-2/27*x/(a*c*(c+a^2*c*x^2)^(3/2))+2/9*arctan(a*x)/(a^2*c*(c+a^2*c*x^2)^(3/2))+1/3*x*arctan(a*x)^2/(a*c*(c+a^2*c*x^2)^(3/2))-1/3*arctan(a*x)^3/(a^2*c*(c+a^2*c*x^2)^(3/2))-40/27*x/(a*c^2*sqrt(c+a^2*c*x^2))+4/3*arctan(a*x)/(a^2*c^2*sqrt(c+a^2*c*x^2))+2/3*x*arctan(a*x)^2/(a*c^2*sqrt(c+a^2*c*x^2))],
[arctan(a*x)^3/(c+a^2*c*x^2)^(5/2),x,5,(-2/27)/(a*c*(c+a^2*c*x^2)^(3/2))-2/9*x*arctan(a*x)/(c*(c+a^2*c*x^2)^(3/2))+1/3*arctan(a*x)^2/(a*c*(c+a^2*c*x^2)^(3/2))+1/3*x*arctan(a*x)^3/(c*(c+a^2*c*x^2)^(3/2))+(-40/9)/(a*c^2*sqrt(c+a^2*c*x^2))-40/9*x*arctan(a*x)/(c^2*sqrt(c+a^2*c*x^2))+2*arctan(a*x)^2/(a*c^2*sqrt(c+a^2*c*x^2))+2/3*x*arctan(a*x)^3/(c^2*sqrt(c+a^2*c*x^2))],
[arctan(a*x)^3/(x*(c+a^2*c*x^2)^(5/2)),x,22,2/27*a*x/(c*(c+a^2*c*x^2)^(3/2))-2/9*arctan(a*x)/(c*(c+a^2*c*x^2)^(3/2))-1/3*a*x*arctan(a*x)^2/(c*(c+a^2*c*x^2)^(3/2))+1/3*arctan(a*x)^3/(c*(c+a^2*c*x^2)^(3/2))+202/27*a*x/(c^2*sqrt(c+a^2*c*x^2))-22/3*arctan(a*x)/(c^2*sqrt(c+a^2*c*x^2))-11/3*a*x*arctan(a*x)^2/(c^2*sqrt(c+a^2*c*x^2))+arctan(a*x)^3/(c^2*sqrt(c+a^2*c*x^2))-2*arctan(a*x)^3*arctanh(exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/(c^2*sqrt(c+a^2*c*x^2))+3*I*arctan(a*x)^2*polylog(2,-exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/(c^2*sqrt(c+a^2*c*x^2))-3*I*arctan(a*x)^2*polylog(2,exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/(c^2*sqrt(c+a^2*c*x^2))-6*arctan(a*x)*polylog(3,-exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/(c^2*sqrt(c+a^2*c*x^2))+6*arctan(a*x)*polylog(3,exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/(c^2*sqrt(c+a^2*c*x^2))-6*I*polylog(4,-exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/(c^2*sqrt(c+a^2*c*x^2))+6*I*polylog(4,exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/(c^2*sqrt(c+a^2*c*x^2))],
[arctan(a*x)^3/(x^2*(c+a^2*c*x^2)^(5/2)),x,19,2/27*a/(c*(c+a^2*c*x^2)^(3/2))+2/9*a^2*x*arctan(a*x)/(c*(c+a^2*c*x^2)^(3/2))-1/3*a*arctan(a*x)^2/(c*(c+a^2*c*x^2)^(3/2))-1/3*a^2*x*arctan(a*x)^3/(c*(c+a^2*c*x^2)^(3/2))+94/9*a/(c^2*sqrt(c+a^2*c*x^2))+94/9*a^2*x*arctan(a*x)/(c^2*sqrt(c+a^2*c*x^2))-5*a*arctan(a*x)^2/(c^2*sqrt(c+a^2*c*x^2))-5/3*a^2*x*arctan(a*x)^3/(c^2*sqrt(c+a^2*c*x^2))-6*a*arctan(a*x)^2*arctanh(exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/(c^2*sqrt(c+a^2*c*x^2))+6*I*a*arctan(a*x)*polylog(2,-exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/(c^2*sqrt(c+a^2*c*x^2))-6*I*a*arctan(a*x)*polylog(2,exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/(c^2*sqrt(c+a^2*c*x^2))-6*a*polylog(3,-exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/(c^2*sqrt(c+a^2*c*x^2))+6*a*polylog(3,exp(I*arctan(a*x)))*sqrt(1+a^2*x^2)/(c^2*sqrt(c+a^2*c*x^2))-arctan(a*x)^3*sqrt(c+a^2*c*x^2)/(c^3*x)],

# Integrands of the form x^m (c+d x^2)^p ArcTan[a x]^3 with d=a^2 c and m symbolic
[x^m*(c+a^2*c*x^2)^2*arctan(a*x)^3,x,0,Unintegrable(x^m*(c+a^2*c*x^2)^2*arctan(a*x)^3,x)],
[x^m*(c+a^2*c*x^2)*arctan(a*x)^3,x,0,Unintegrable(x^m*(c+a^2*c*x^2)*arctan(a*x)^3,x)],
[x^m*arctan(a*x)^3/(c+a^2*c*x^2),x,0,Unintegrable(x^m*arctan(a*x)^3/(c+a^2*c*x^2),x)],
[x^m*arctan(a*x)^3/(c+a^2*c*x^2)^2,x,0,Unintegrable(x^m*arctan(a*x)^3/(c+a^2*c*x^2)^2,x)],
[x^m*(c+a^2*c*x^2)^(3/2)*arctan(a*x)^3,x,0,Unintegrable(x^m*(c+a^2*c*x^2)^(3/2)*arctan(a*x)^3,x)],
[x^m*arctan(a*x)^3*sqrt(c+a^2*c*x^2),x,0,Unintegrable(x^m*arctan(a*x)^3*sqrt(c+a^2*c*x^2),x)],
[x^m*arctan(a*x)^3/sqrt(c+a^2*c*x^2),x,0,Unintegrable(x^m*arctan(a*x)^3/sqrt(c+a^2*c*x^2),x)],
[x^m*arctan(a*x)^3/(c+a^2*c*x^2)^(3/2),x,0,Unintegrable(x^m*arctan(a*x)^3/(c+a^2*c*x^2)^(3/2),x)],

# Integrands of the form (f x)^m (d+c^2 d x^2)^q (a+b ArcTan[c x])^-1

# Integrands of the form x^m (c+d x^2)^p / ArcTan[a x]^1 with d=a^2 c

# p>0
[x*(c+a^2*c*x^2)/arctan(a*x),x,0,Unintegrable(x*(c+a^2*c*x^2)/arctan(a*x),x)],
[(c+a^2*c*x^2)/arctan(a*x),x,0,Unintegrable((c+a^2*c*x^2)/arctan(a*x),x)],
[(c+a^2*c*x^2)/(x*arctan(a*x)),x,0,Unintegrable((c+a^2*c*x^2)/(x*arctan(a*x)),x)],
[x*(c+a^2*c*x^2)^2/arctan(a*x),x,0,Unintegrable(x*(c+a^2*c*x^2)^2/arctan(a*x),x)],
[(c+a^2*c*x^2)^2/arctan(a*x),x,0,Unintegrable((c+a^2*c*x^2)^2/arctan(a*x),x)],
[(c+a^2*c*x^2)^2/(x*arctan(a*x)),x,0,Unintegrable((c+a^2*c*x^2)^2/(x*arctan(a*x)),x)],
[x*(c+a^2*c*x^2)^3/arctan(a*x),x,0,Unintegrable(x*(c+a^2*c*x^2)^3/arctan(a*x),x)],
[(c+a^2*c*x^2)^3/arctan(a*x),x,0,Unintegrable((c+a^2*c*x^2)^3/arctan(a*x),x)],
[(c+a^2*c*x^2)^3/(x*arctan(a*x)),x,0,Unintegrable((c+a^2*c*x^2)^3/(x*arctan(a*x)),x)],

# p<0
[x^2/((c+a^2*c*x^2)*arctan(a*x)),x,0,Unintegrable(x^2/((c+a^2*c*x^2)*arctan(a*x)),x)],
[x/((c+a^2*c*x^2)*arctan(a*x)),x,0,Unintegrable(x/((c+a^2*c*x^2)*arctan(a*x)),x)],
[1/((c+a^2*c*x^2)*arctan(a*x)),x,1,log(arctan(a*x))/(a*c)],
[1/(x*(c+a^2*c*x^2)*arctan(a*x)),x,0,Unintegrable(1/(x*(c+a^2*c*x^2)*arctan(a*x)),x)],
[1/(x^2*(c+a^2*c*x^2)*arctan(a*x)),x,0,Unintegrable(1/(x^2*(c+a^2*c*x^2)*arctan(a*x)),x)],
[x^4/((c+a^2*c*x^2)^2*arctan(a*x)),x,0,Unintegrable(x^4/((c+a^2*c*x^2)^2*arctan(a*x)),x)],
[x^3/((c+a^2*c*x^2)^2*arctan(a*x)),x,0,Unintegrable(x^3/((c+a^2*c*x^2)^2*arctan(a*x)),x)],
[x^2/((c+a^2*c*x^2)^2*arctan(a*x)),x,4,-1/2*Ci(2*arctan(a*x))/(a^3*c^2)+1/2*log(arctan(a*x))/(a^3*c^2)],
[x/((c+a^2*c*x^2)^2*arctan(a*x)),x,4,1/2*Si(2*arctan(a*x))/(a^2*c^2)],
[1/((c+a^2*c*x^2)^2*arctan(a*x)),x,4,1/2*Ci(2*arctan(a*x))/(a*c^2)+1/2*log(arctan(a*x))/(a*c^2)],
[1/(x*(c+a^2*c*x^2)^2*arctan(a*x)),x,0,Unintegrable(1/(x*(c+a^2*c*x^2)^2*arctan(a*x)),x)],
[1/(x^2*(c+a^2*c*x^2)^2*arctan(a*x)),x,0,Unintegrable(1/(x^2*(c+a^2*c*x^2)^2*arctan(a*x)),x)],
[x^6/((c+a^2*c*x^2)^3*arctan(a*x)),x,0,Unintegrable(x^6/((c+a^2*c*x^2)^3*arctan(a*x)),x)],
[x^5/((c+a^2*c*x^2)^3*arctan(a*x)),x,0,Unintegrable(x^5/((c+a^2*c*x^2)^3*arctan(a*x)),x)],
[x^4/((c+a^2*c*x^2)^3*arctan(a*x)),x,5,-1/2*Ci(2*arctan(a*x))/(a^5*c^3)+1/8*Ci(4*arctan(a*x))/(a^5*c^3)+3/8*log(arctan(a*x))/(a^5*c^3)],
[x^3/((c+a^2*c*x^2)^3*arctan(a*x)),x,5,1/4*Si(2*arctan(a*x))/(a^4*c^3)-1/8*Si(4*arctan(a*x))/(a^4*c^3)],
[x^2/((c+a^2*c*x^2)^3*arctan(a*x)),x,4,-1/8*Ci(4*arctan(a*x))/(a^3*c^3)+1/8*log(arctan(a*x))/(a^3*c^3)],
[x/((c+a^2*c*x^2)^3*arctan(a*x)),x,5,1/4*Si(2*arctan(a*x))/(a^2*c^3)+1/8*Si(4*arctan(a*x))/(a^2*c^3)],
[1/((c+a^2*c*x^2)^3*arctan(a*x)),x,5,1/2*Ci(2*arctan(a*x))/(a*c^3)+1/8*Ci(4*arctan(a*x))/(a*c^3)+3/8*log(arctan(a*x))/(a*c^3)],
[1/(x*(c+a^2*c*x^2)^3*arctan(a*x)),x,0,Unintegrable(1/(x*(c+a^2*c*x^2)^3*arctan(a*x)),x)],
[1/(x^2*(c+a^2*c*x^2)^3*arctan(a*x)),x,0,Unintegrable(1/(x^2*(c+a^2*c*x^2)^3*arctan(a*x)),x)],

# Integrands of the form x^m (c+d x^2)^(p/2) / ArcTan[a x]^1 with d=a^2 c

# p>0
[x*sqrt(c+a^2*c*x^2)/arctan(a*x),x,0,Unintegrable(x*sqrt(c+a^2*c*x^2)/arctan(a*x),x)],
[sqrt(c+a^2*c*x^2)/arctan(a*x),x,0,Unintegrable(sqrt(c+a^2*c*x^2)/arctan(a*x),x)],
[sqrt(c+a^2*c*x^2)/(x*arctan(a*x)),x,0,Unintegrable(sqrt(c+a^2*c*x^2)/(x*arctan(a*x)),x)],
[x*(c+a^2*c*x^2)^(3/2)/arctan(a*x),x,0,Unintegrable(x*(c+a^2*c*x^2)^(3/2)/arctan(a*x),x)],
[(c+a^2*c*x^2)^(3/2)/arctan(a*x),x,0,Unintegrable((c+a^2*c*x^2)^(3/2)/arctan(a*x),x)],
[(c+a^2*c*x^2)^(3/2)/(x*arctan(a*x)),x,0,Unintegrable((c+a^2*c*x^2)^(3/2)/(x*arctan(a*x)),x)],
[x*(c+a^2*c*x^2)^(5/2)/arctan(a*x),x,0,Unintegrable(x*(c+a^2*c*x^2)^(5/2)/arctan(a*x),x)],
[(c+a^2*c*x^2)^(5/2)/arctan(a*x),x,0,Unintegrable((c+a^2*c*x^2)^(5/2)/arctan(a*x),x)],
[(c+a^2*c*x^2)^(5/2)/(x*arctan(a*x)),x,0,Unintegrable((c+a^2*c*x^2)^(5/2)/(x*arctan(a*x)),x)],

# p<0
[x/(arctan(a*x)*sqrt(c+a^2*c*x^2)),x,0,Unintegrable(x/(arctan(a*x)*sqrt(c+a^2*c*x^2)),x)],
[1/(arctan(a*x)*sqrt(c+a^2*c*x^2)),x,0,Unintegrable(1/(arctan(a*x)*sqrt(c+a^2*c*x^2)),x)],
[1/(x*arctan(a*x)*sqrt(c+a^2*c*x^2)),x,0,Unintegrable(1/(x*arctan(a*x)*sqrt(c+a^2*c*x^2)),x)],
[x^3/((c+a^2*c*x^2)^(3/2)*arctan(a*x)),x,0,Unintegrable(x^3/((c+a^2*c*x^2)^(3/2)*arctan(a*x)),x)],
[x^2/((c+a^2*c*x^2)^(3/2)*arctan(a*x)),x,0,Unintegrable(x^2/((c+a^2*c*x^2)^(3/2)*arctan(a*x)),x)],
[x/((c+a^2*c*x^2)^(3/2)*arctan(a*x)),x,3,Si(arctan(a*x))*sqrt(1+a^2*x^2)/(a^2*c*sqrt(c+a^2*c*x^2))],
[1/((c+a^2*c*x^2)^(3/2)*arctan(a*x)),x,3,Ci(arctan(a*x))*sqrt(1+a^2*x^2)/(a*c*sqrt(c+a^2*c*x^2))],
[1/(x*(c+a^2*c*x^2)^(3/2)*arctan(a*x)),x,0,Unintegrable(1/(x*(c+a^2*c*x^2)^(3/2)*arctan(a*x)),x)],
[1/(x^2*(c+a^2*c*x^2)^(3/2)*arctan(a*x)),x,0,Unintegrable(1/(x^2*(c+a^2*c*x^2)^(3/2)*arctan(a*x)),x)],
[x^5/((c+a^2*c*x^2)^(5/2)*arctan(a*x)),x,0,Unintegrable(x^5/((c+a^2*c*x^2)^(5/2)*arctan(a*x)),x)],
[x^4/((c+a^2*c*x^2)^(5/2)*arctan(a*x)),x,0,Unintegrable(x^4/((c+a^2*c*x^2)^(5/2)*arctan(a*x)),x)],
[x^3/((c+a^2*c*x^2)^(5/2)*arctan(a*x)),x,6,3/4*Si(arctan(a*x))*sqrt(1+a^2*x^2)/(a^4*c^2*sqrt(c+a^2*c*x^2))-1/4*Si(3*arctan(a*x))*sqrt(1+a^2*x^2)/(a^4*c^2*sqrt(c+a^2*c*x^2))],
[x^2/((c+a^2*c*x^2)^(5/2)*arctan(a*x)),x,6,1/4*Ci(arctan(a*x))*sqrt(1+a^2*x^2)/(a^3*c^2*sqrt(c+a^2*c*x^2))-1/4*Ci(3*arctan(a*x))*sqrt(1+a^2*x^2)/(a^3*c^2*sqrt(c+a^2*c*x^2))],
[x/((c+a^2*c*x^2)^(5/2)*arctan(a*x)),x,6,1/4*Si(arctan(a*x))*sqrt(1+a^2*x^2)/(a^2*c^2*sqrt(c+a^2*c*x^2))+1/4*Si(3*arctan(a*x))*sqrt(1+a^2*x^2)/(a^2*c^2*sqrt(c+a^2*c*x^2))],
[1/((c+a^2*c*x^2)^(5/2)*arctan(a*x)),x,6,3/4*Ci(arctan(a*x))*sqrt(1+a^2*x^2)/(a*c^2*sqrt(c+a^2*c*x^2))+1/4*Ci(3*arctan(a*x))*sqrt(1+a^2*x^2)/(a*c^2*sqrt(c+a^2*c*x^2))],
[1/(x*(c+a^2*c*x^2)^(5/2)*arctan(a*x)),x,0,Unintegrable(1/(x*(c+a^2*c*x^2)^(5/2)*arctan(a*x)),x)],
[1/(x^2*(c+a^2*c*x^2)^(5/2)*arctan(a*x)),x,0,Unintegrable(1/(x^2*(c+a^2*c*x^2)^(5/2)*arctan(a*x)),x)],

# Integrands of the form x^m (c+d x^2)^p / ArcTan[a x]^1 with d=a^2 c and m symbolic
[x^m*(c+a^2*c*x^2)^3/arctan(a*x),x,0,Unintegrable(x^m*(c+a^2*c*x^2)^3/arctan(a*x),x)],
[x^m*(c+a^2*c*x^2)^2/arctan(a*x),x,0,Unintegrable(x^m*(c+a^2*c*x^2)^2/arctan(a*x),x)],
[x^m*(c+a^2*c*x^2)/arctan(a*x),x,0,Unintegrable(x^m*(c+a^2*c*x^2)/arctan(a*x),x)],
[x^m/((c+a^2*c*x^2)*arctan(a*x)),x,0,Unintegrable(x^m/((c+a^2*c*x^2)*arctan(a*x)),x)],
[x^m/((c+a^2*c*x^2)^2*arctan(a*x)),x,0,Unintegrable(x^m/((c+a^2*c*x^2)^2*arctan(a*x)),x)],
[x^m/((c+a^2*c*x^2)^3*arctan(a*x)),x,0,Unintegrable(x^m/((c+a^2*c*x^2)^3*arctan(a*x)),x)],
[x^m*(c+a^2*c*x^2)^(5/2)/arctan(a*x),x,0,Unintegrable(x^m*(c+a^2*c*x^2)^(5/2)/arctan(a*x),x)],
[x^m*(c+a^2*c*x^2)^(3/2)/arctan(a*x),x,0,Unintegrable(x^m*(c+a^2*c*x^2)^(3/2)/arctan(a*x),x)],
[x^m*sqrt(c+a^2*c*x^2)/arctan(a*x),x,0,Unintegrable(x^m*sqrt(c+a^2*c*x^2)/arctan(a*x),x)],
[x^m/(arctan(a*x)*sqrt(c+a^2*c*x^2)),x,0,Unintegrable(x^m/(arctan(a*x)*sqrt(c+a^2*c*x^2)),x)],
[x^m/((c+a^2*c*x^2)^(3/2)*arctan(a*x)),x,0,Unintegrable(x^m/((c+a^2*c*x^2)^(3/2)*arctan(a*x)),x)],
[x^m/((c+a^2*c*x^2)^(5/2)*arctan(a*x)),x,0,Unintegrable(x^m/((c+a^2*c*x^2)^(5/2)*arctan(a*x)),x)],

# Integrands of the form (f x)^m (d+c^2 d x^2)^q (a+b ArcTan[c x])^-2

# Integrands of the form x^m (d+c^2 d x^2)^q / (a+b ArcTan[c x])^2

# q>0
[x*(c+a^2*c*x^2)/arctan(a*x)^2,x,0,Unintegrable(x*(c+a^2*c*x^2)/arctan(a*x)^2,x)],
[(c+a^2*c*x^2)/arctan(a*x)^2,x,0,Unintegrable((c+a^2*c*x^2)/arctan(a*x)^2,x)],
[(c+a^2*c*x^2)/(x*arctan(a*x)^2),x,0,Unintegrable((c+a^2*c*x^2)/(x*arctan(a*x)^2),x)],
[x*(c+a^2*c*x^2)^2/arctan(a*x)^2,x,0,Unintegrable(x*(c+a^2*c*x^2)^2/arctan(a*x)^2,x)],
[(c+a^2*c*x^2)^2/arctan(a*x)^2,x,0,Unintegrable((c+a^2*c*x^2)^2/arctan(a*x)^2,x)],
[(c+a^2*c*x^2)^2/(x*arctan(a*x)^2),x,0,Unintegrable((c+a^2*c*x^2)^2/(x*arctan(a*x)^2),x)],
[x*(c+a^2*c*x^2)^3/arctan(a*x)^2,x,0,Unintegrable(x*(c+a^2*c*x^2)^3/arctan(a*x)^2,x)],
[(c+a^2*c*x^2)^3/arctan(a*x)^2,x,0,Unintegrable((c+a^2*c*x^2)^3/arctan(a*x)^2,x)],
[(c+a^2*c*x^2)^3/(x*arctan(a*x)^2),x,0,Unintegrable((c+a^2*c*x^2)^3/(x*arctan(a*x)^2),x)],

# q<0
[x^3/((c+a^2*c*x^2)*arctan(a*x)^2),x,1,-x^3/(a*c*arctan(a*x))+3*Unintegrable(x^2/arctan(a*x),x)/(a*c)],
[x^2/((c+a^2*c*x^2)*arctan(a*x)^2),x,1,-x^2/(a*c*arctan(a*x))+2*Unintegrable(x/arctan(a*x),x)/(a*c)],
[x/((c+a^2*c*x^2)*arctan(a*x)^2),x,1,-x/(a*c*arctan(a*x))+Unintegrable(1/arctan(a*x),x)/(a*c)],
[1/((c+a^2*c*x^2)*arctan(a*x)^2),x,1,(-1)/(a*c*arctan(a*x))],
[1/(x*(c+a^2*c*x^2)*arctan(a*x)^2),x,1,(-1)/(a*c*x*arctan(a*x))-Unintegrable(1/(x^2*arctan(a*x)),x)/(a*c)],
[1/(x^2*(c+a^2*c*x^2)*arctan(a*x)^2),x,1,(-1)/(a*c*x^2*arctan(a*x))-2*Unintegrable(1/(x^3*arctan(a*x)),x)/(a*c)],
[1/(x^3*(c+a^2*c*x^2)*arctan(a*x)^2),x,1,(-1)/(a*c*x^3*arctan(a*x))-3*Unintegrable(1/(x^4*arctan(a*x)),x)/(a*c)],
[1/(x^4*(c+a^2*c*x^2)*arctan(a*x)^2),x,1,(-1)/(a*c*x^4*arctan(a*x))-4*Unintegrable(1/(x^5*arctan(a*x)),x)/(a*c)],
[x^3/((c+a^2*c*x^2)^2*arctan(a*x)^2),x,11,-x/(a^3*c^2*arctan(a*x))+x/(a^3*c^2*(1+a^2*x^2)*arctan(a*x))-Ci(2*arctan(a*x))/(a^4*c^2)+Unintegrable(1/arctan(a*x),x)/(a^3*c^2)],
[x^2/((c+a^2*c*x^2)^2*arctan(a*x)^2),x,5,-x^2/(a*c^2*(1+a^2*x^2)*arctan(a*x))+Si(2*arctan(a*x))/(a^3*c^2)],
[x/((c+a^2*c*x^2)^2*arctan(a*x)^2),x,9,-x/(a*c^2*(1+a^2*x^2)*arctan(a*x))+Ci(2*arctan(a*x))/(a^2*c^2)],
[1/((c+a^2*c*x^2)^2*arctan(a*x)^2),x,5,(-1)/(a*c^2*(1+a^2*x^2)*arctan(a*x))-Si(2*arctan(a*x))/(a*c^2)],
[1/(x*(c+a^2*c*x^2)^2*arctan(a*x)^2),x,11,(-1)/(a*c^2*x*arctan(a*x))+a*x/(c^2*(1+a^2*x^2)*arctan(a*x))-Ci(2*arctan(a*x))/c^2-Unintegrable(1/(x^2*arctan(a*x)),x)/(a*c^2)],
[1/(x^2*(c+a^2*c*x^2)^2*arctan(a*x)^2),x,7,(-1)/(a*c^2*x^2*arctan(a*x))+a/(c^2*(1+a^2*x^2)*arctan(a*x))+a*Si(2*arctan(a*x))/c^2-2*Unintegrable(1/(x^3*arctan(a*x)),x)/(a*c^2)],
[1/(x^3*(c+a^2*c*x^2)^2*arctan(a*x)^2),x,13,(-1)/(a*c^2*x^3*arctan(a*x))+a/(c^2*x*arctan(a*x))-a^3*x/(c^2*(1+a^2*x^2)*arctan(a*x))+a^2*Ci(2*arctan(a*x))/c^2-3*Unintegrable(1/(x^4*arctan(a*x)),x)/(a*c^2)+a*Unintegrable(1/(x^2*arctan(a*x)),x)/c^2],
[1/(x^4*(c+a^2*c*x^2)^2*arctan(a*x)^2),x,9,(-1)/(a*c^2*x^4*arctan(a*x))+a/(c^2*x^2*arctan(a*x))-a^3/(c^2*(1+a^2*x^2)*arctan(a*x))-a^3*Si(2*arctan(a*x))/c^2-4*Unintegrable(1/(x^5*arctan(a*x)),x)/(a*c^2)+2*a*Unintegrable(1/(x^3*arctan(a*x)),x)/c^2],
[x^3/((c+a^2*c*x^2)^3*arctan(a*x)^2),x,20,x/(a^3*c^3*(1+a^2*x^2)^2*arctan(a*x))-x/(a^3*c^3*(1+a^2*x^2)*arctan(a*x))+1/2*Ci(2*arctan(a*x))/(a^4*c^3)-1/2*Ci(4*arctan(a*x))/(a^4*c^3)],
[x^2/((c+a^2*c*x^2)^3*arctan(a*x)^2),x,12,1/(a^3*c^3*(1+a^2*x^2)^2*arctan(a*x))+(-1)/(a^3*c^3*(1+a^2*x^2)*arctan(a*x))+1/2*Si(4*arctan(a*x))/(a^3*c^3)],
[x/((c+a^2*c*x^2)^3*arctan(a*x)^2),x,10,-x/(a*c^3*(1+a^2*x^2)^2*arctan(a*x))+1/2*Ci(2*arctan(a*x))/(a^2*c^3)+1/2*Ci(4*arctan(a*x))/(a^2*c^3)],
[1/((c+a^2*c*x^2)^3*arctan(a*x)^2),x,6,(-1)/(a*c^3*(1+a^2*x^2)^2*arctan(a*x))-Si(2*arctan(a*x))/(a*c^3)-1/2*Si(4*arctan(a*x))/(a*c^3)],
[1/(x*(c+a^2*c*x^2)^3*arctan(a*x)^2),x,22,(-1)/(a*c^3*x*arctan(a*x))+a*x/(c^3*(1+a^2*x^2)^2*arctan(a*x))+a*x/(c^3*(1+a^2*x^2)*arctan(a*x))-3/2*Ci(2*arctan(a*x))/c^3-1/2*Ci(4*arctan(a*x))/c^3-Unintegrable(1/(x^2*arctan(a*x)),x)/(a*c^3)],
[1/(x^2*(c+a^2*c*x^2)^3*arctan(a*x)^2),x,14,(-1)/(a*c^3*x^2*arctan(a*x))+a/(c^3*(1+a^2*x^2)^2*arctan(a*x))+a/(c^3*(1+a^2*x^2)*arctan(a*x))+2*a*Si(2*arctan(a*x))/c^3+1/2*a*Si(4*arctan(a*x))/c^3-2*Unintegrable(1/(x^3*arctan(a*x)),x)/(a*c^3)],
[1/(x^3*(c+a^2*c*x^2)^3*arctan(a*x)^2),x,36,(-1)/(a*c^3*x^3*arctan(a*x))+2*a/(c^3*x*arctan(a*x))-a^3*x/(c^3*(1+a^2*x^2)^2*arctan(a*x))-2*a^3*x/(c^3*(1+a^2*x^2)*arctan(a*x))+5/2*a^2*Ci(2*arctan(a*x))/c^3+1/2*a^2*Ci(4*arctan(a*x))/c^3-3*Unintegrable(1/(x^4*arctan(a*x)),x)/(a*c^3)+2*a*Unintegrable(1/(x^2*arctan(a*x)),x)/c^3],
[1/(x^4*(c+a^2*c*x^2)^3*arctan(a*x)^2),x,24,(-1)/(a*c^3*x^4*arctan(a*x))+2*a/(c^3*x^2*arctan(a*x))-a^3/(c^3*(1+a^2*x^2)^2*arctan(a*x))-2*a^3/(c^3*(1+a^2*x^2)*arctan(a*x))-3*a^3*Si(2*arctan(a*x))/c^3-1/2*a^3*Si(4*arctan(a*x))/c^3-4*Unintegrable(1/(x^5*arctan(a*x)),x)/(a*c^3)+4*a*Unintegrable(1/(x^3*arctan(a*x)),x)/c^3],

# Integrands of the form x^m (d+c^2 d x^2)^(q/2) / (a+b ArcTan[c x])^2

# q>0
[x*sqrt(c+a^2*c*x^2)/arctan(a*x)^2,x,0,Unintegrable(x*sqrt(c+a^2*c*x^2)/arctan(a*x)^2,x)],
[sqrt(c+a^2*c*x^2)/arctan(a*x)^2,x,0,Unintegrable(sqrt(c+a^2*c*x^2)/arctan(a*x)^2,x)],
[sqrt(c+a^2*c*x^2)/(x*arctan(a*x)^2),x,0,Unintegrable(sqrt(c+a^2*c*x^2)/(x*arctan(a*x)^2),x)],
[x*(c+a^2*c*x^2)^(3/2)/arctan(a*x)^2,x,0,Unintegrable(x*(c+a^2*c*x^2)^(3/2)/arctan(a*x)^2,x)],
[(c+a^2*c*x^2)^(3/2)/arctan(a*x)^2,x,0,Unintegrable((c+a^2*c*x^2)^(3/2)/arctan(a*x)^2,x)],
[(c+a^2*c*x^2)^(3/2)/(x*arctan(a*x)^2),x,0,Unintegrable((c+a^2*c*x^2)^(3/2)/(x*arctan(a*x)^2),x)],
[x*(c+a^2*c*x^2)^(5/2)/arctan(a*x)^2,x,0,Unintegrable(x*(c+a^2*c*x^2)^(5/2)/arctan(a*x)^2,x)],
[(c+a^2*c*x^2)^(5/2)/arctan(a*x)^2,x,0,Unintegrable((c+a^2*c*x^2)^(5/2)/arctan(a*x)^2,x)],
[(c+a^2*c*x^2)^(5/2)/(x*arctan(a*x)^2),x,0,Unintegrable((c+a^2*c*x^2)^(5/2)/(x*arctan(a*x)^2),x)],

# q<0
[x/(arctan(a*x)^2*sqrt(c+a^2*c*x^2)),x,0,Unintegrable(x/(arctan(a*x)^2*sqrt(c+a^2*c*x^2)),x)],
[1/(arctan(a*x)^2*sqrt(c+a^2*c*x^2)),x,0,Unintegrable(1/(arctan(a*x)^2*sqrt(c+a^2*c*x^2)),x)],
[1/(x*arctan(a*x)^2*sqrt(c+a^2*c*x^2)),x,1,-sqrt(c+a^2*c*x^2)/(a*c*x*arctan(a*x))-Unintegrable(1/(x^2*arctan(a*x)*sqrt(c+a^2*c*x^2)),x)/a],
[x^3/((c+a^2*c*x^2)^(3/2)*arctan(a*x)^2),x,5,x/(a^3*c*arctan(a*x)*sqrt(c+a^2*c*x^2))-Ci(arctan(a*x))*sqrt(1+a^2*x^2)/(a^4*c*sqrt(c+a^2*c*x^2))+Unintegrable(x/(arctan(a*x)^2*sqrt(c+a^2*c*x^2)),x)/(a^2*c)],
[x^2/((c+a^2*c*x^2)^(3/2)*arctan(a*x)^2),x,5,1/(a^3*c*arctan(a*x)*sqrt(c+a^2*c*x^2))+Si(arctan(a*x))*sqrt(1+a^2*x^2)/(a^3*c*sqrt(c+a^2*c*x^2))+Unintegrable(1/(arctan(a*x)^2*sqrt(c+a^2*c*x^2)),x)/(a^2*c)],
[x/((c+a^2*c*x^2)^(3/2)*arctan(a*x)^2),x,4,-x/(a*c*arctan(a*x)*sqrt(c+a^2*c*x^2))+Ci(arctan(a*x))*sqrt(1+a^2*x^2)/(a^2*c*sqrt(c+a^2*c*x^2))],
[1/((c+a^2*c*x^2)^(3/2)*arctan(a*x)^2),x,4,(-1)/(a*c*arctan(a*x)*sqrt(c+a^2*c*x^2))-Si(arctan(a*x))*sqrt(1+a^2*x^2)/(a*c*sqrt(c+a^2*c*x^2))],
[1/(x*(c+a^2*c*x^2)^(3/2)*arctan(a*x)^2),x,6,a*x/(c*arctan(a*x)*sqrt(c+a^2*c*x^2))-Ci(arctan(a*x))*sqrt(1+a^2*x^2)/(c*sqrt(c+a^2*c*x^2))-sqrt(c+a^2*c*x^2)/(a*c^2*x*arctan(a*x))-Unintegrable(1/(x^2*arctan(a*x)*sqrt(c+a^2*c*x^2)),x)/(a*c)],
[1/(x^2*(c+a^2*c*x^2)^(3/2)*arctan(a*x)^2),x,5,a/(c*arctan(a*x)*sqrt(c+a^2*c*x^2))+a*Si(arctan(a*x))*sqrt(1+a^2*x^2)/(c*sqrt(c+a^2*c*x^2))+Unintegrable(1/(x^2*arctan(a*x)^2*sqrt(c+a^2*c*x^2)),x)/c],
[1/(x^3*(c+a^2*c*x^2)^(3/2)*arctan(a*x)^2),x,7,-a^3*x/(c*arctan(a*x)*sqrt(c+a^2*c*x^2))+a^2*Ci(arctan(a*x))*sqrt(1+a^2*x^2)/(c*sqrt(c+a^2*c*x^2))+a*sqrt(c+a^2*c*x^2)/(c^2*x*arctan(a*x))+Unintegrable(1/(x^3*arctan(a*x)^2*sqrt(c+a^2*c*x^2)),x)/c+a*Unintegrable(1/(x^2*arctan(a*x)*sqrt(c+a^2*c*x^2)),x)/c],
[1/(x^4*(c+a^2*c*x^2)^(3/2)*arctan(a*x)^2),x,6,-a^3/(c*arctan(a*x)*sqrt(c+a^2*c*x^2))-a^3*Si(arctan(a*x))*sqrt(1+a^2*x^2)/(c*sqrt(c+a^2*c*x^2))+Unintegrable(1/(x^4*arctan(a*x)^2*sqrt(c+a^2*c*x^2)),x)/c-a^2*Unintegrable(1/(x^2*arctan(a*x)^2*sqrt(c+a^2*c*x^2)),x)/c],
[x^5/((c+a^2*c*x^2)^(5/2)*arctan(a*x)^2),x,13,x^3/(a^3*c*(c+a^2*c*x^2)^(3/2)*arctan(a*x))+x/(a^5*c^2*arctan(a*x)*sqrt(c+a^2*c*x^2))-7/4*Ci(arctan(a*x))*sqrt(1+a^2*x^2)/(a^6*c^2*sqrt(c+a^2*c*x^2))+3/4*Ci(3*arctan(a*x))*sqrt(1+a^2*x^2)/(a^6*c^2*sqrt(c+a^2*c*x^2))+Unintegrable(x/(arctan(a*x)^2*sqrt(c+a^2*c*x^2)),x)/(a^4*c^2)],
[x^4/((c+a^2*c*x^2)^(5/2)*arctan(a*x)^2),x,18,(-1)/(a^5*c*(c+a^2*c*x^2)^(3/2)*arctan(a*x))+2/(a^5*c^2*arctan(a*x)*sqrt(c+a^2*c*x^2))+5/4*Si(arctan(a*x))*sqrt(1+a^2*x^2)/(a^5*c^2*sqrt(c+a^2*c*x^2))-3/4*Si(3*arctan(a*x))*sqrt(1+a^2*x^2)/(a^5*c^2*sqrt(c+a^2*c*x^2))+Unintegrable(1/(arctan(a*x)^2*sqrt(c+a^2*c*x^2)),x)/(a^4*c^2)],
[x^3/((c+a^2*c*x^2)^(5/2)*arctan(a*x)^2),x,7,-x^3/(a*c*(c+a^2*c*x^2)^(3/2)*arctan(a*x))+3/4*Ci(arctan(a*x))*sqrt(1+a^2*x^2)/(a^4*c^2*sqrt(c+a^2*c*x^2))-3/4*Ci(3*arctan(a*x))*sqrt(1+a^2*x^2)/(a^4*c^2*sqrt(c+a^2*c*x^2))],
[x^2/((c+a^2*c*x^2)^(5/2)*arctan(a*x)^2),x,12,1/(a^3*c*(c+a^2*c*x^2)^(3/2)*arctan(a*x))+(-1)/(a^3*c^2*arctan(a*x)*sqrt(c+a^2*c*x^2))-1/4*Si(arctan(a*x))*sqrt(1+a^2*x^2)/(a^3*c^2*sqrt(c+a^2*c*x^2))+3/4*Si(3*arctan(a*x))*sqrt(1+a^2*x^2)/(a^3*c^2*sqrt(c+a^2*c*x^2))],
[x/((c+a^2*c*x^2)^(5/2)*arctan(a*x)^2),x,13,-x/(a*c*(c+a^2*c*x^2)^(3/2)*arctan(a*x))+1/4*Ci(arctan(a*x))*sqrt(1+a^2*x^2)/(a^2*c^2*sqrt(c+a^2*c*x^2))+3/4*Ci(3*arctan(a*x))*sqrt(1+a^2*x^2)/(a^2*c^2*sqrt(c+a^2*c*x^2))],
[1/((c+a^2*c*x^2)^(5/2)*arctan(a*x)^2),x,7,(-1)/(a*c*(c+a^2*c*x^2)^(3/2)*arctan(a*x))-3/4*Si(arctan(a*x))*sqrt(1+a^2*x^2)/(a*c^2*sqrt(c+a^2*c*x^2))-3/4*Si(3*arctan(a*x))*sqrt(1+a^2*x^2)/(a*c^2*sqrt(c+a^2*c*x^2))],
[1/(x*(c+a^2*c*x^2)^(5/2)*arctan(a*x)^2),x,20,a*x/(c*(c+a^2*c*x^2)^(3/2)*arctan(a*x))+a*x/(c^2*arctan(a*x)*sqrt(c+a^2*c*x^2))-5/4*Ci(arctan(a*x))*sqrt(1+a^2*x^2)/(c^2*sqrt(c+a^2*c*x^2))-3/4*Ci(3*arctan(a*x))*sqrt(1+a^2*x^2)/(c^2*sqrt(c+a^2*c*x^2))-sqrt(c+a^2*c*x^2)/(a*c^3*x*arctan(a*x))-Unintegrable(1/(x^2*arctan(a*x)*sqrt(c+a^2*c*x^2)),x)/(a*c^2)],
[1/(x^2*(c+a^2*c*x^2)^(5/2)*arctan(a*x)^2),x,13,a/(c*(c+a^2*c*x^2)^(3/2)*arctan(a*x))+a/(c^2*arctan(a*x)*sqrt(c+a^2*c*x^2))+7/4*a*Si(arctan(a*x))*sqrt(1+a^2*x^2)/(c^2*sqrt(c+a^2*c*x^2))+3/4*a*Si(3*arctan(a*x))*sqrt(1+a^2*x^2)/(c^2*sqrt(c+a^2*c*x^2))+Unintegrable(1/(x^2*arctan(a*x)^2*sqrt(c+a^2*c*x^2)),x)/c^2],
[1/(x^3*(c+a^2*c*x^2)^(5/2)*arctan(a*x)^2),x,28,-a^3*x/(c*(c+a^2*c*x^2)^(3/2)*arctan(a*x))-2*a^3*x/(c^2*arctan(a*x)*sqrt(c+a^2*c*x^2))+9/4*a^2*Ci(arctan(a*x))*sqrt(1+a^2*x^2)/(c^2*sqrt(c+a^2*c*x^2))+3/4*a^2*Ci(3*arctan(a*x))*sqrt(1+a^2*x^2)/(c^2*sqrt(c+a^2*c*x^2))+2*a*sqrt(c+a^2*c*x^2)/(c^3*x*arctan(a*x))+Unintegrable(1/(x^3*arctan(a*x)^2*sqrt(c+a^2*c*x^2)),x)/c^2+2*a*Unintegrable(1/(x^2*arctan(a*x)*sqrt(c+a^2*c*x^2)),x)/c^2],
[1/(x^4*(c+a^2*c*x^2)^(5/2)*arctan(a*x)^2),x,20,-a^3/(c*(c+a^2*c*x^2)^(3/2)*arctan(a*x))-2*a^3/(c^2*arctan(a*x)*sqrt(c+a^2*c*x^2))-11/4*a^3*Si(arctan(a*x))*sqrt(1+a^2*x^2)/(c^2*sqrt(c+a^2*c*x^2))-3/4*a^3*Si(3*arctan(a*x))*sqrt(1+a^2*x^2)/(c^2*sqrt(c+a^2*c*x^2))+Unintegrable(1/(x^4*arctan(a*x)^2*sqrt(c+a^2*c*x^2)),x)/c^2-2*a^2*Unintegrable(1/(x^2*arctan(a*x)^2*sqrt(c+a^2*c*x^2)),x)/c^2],

# Integrands of the form (f x)^(m/2) (d+c^2 d x^2)^q / (a+b ArcTan[c x])^2
[sqrt(f*x)/((d+c^2*d*x^2)^2*(a+b*arctan(c*x))^2),x,0,Unintegrable(sqrt(f*x)/((d+c^2*d*x^2)^2*(a+b*arctan(c*x))^2),x)],

# Integrands of the form (f x)^m (d+c^2 d x^2)^q / (a+b ArcTan[c x])^2 with m symbolic
[x^m*(c+a^2*c*x^2)^3/arctan(a*x)^2,x,0,Unintegrable(x^m*(c+a^2*c*x^2)^3/arctan(a*x)^2,x)],
[x^m*(c+a^2*c*x^2)^2/arctan(a*x)^2,x,0,Unintegrable(x^m*(c+a^2*c*x^2)^2/arctan(a*x)^2,x)],
[x^m*(c+a^2*c*x^2)/arctan(a*x)^2,x,0,Unintegrable(x^m*(c+a^2*c*x^2)/arctan(a*x)^2,x)],
[x^m/((c+a^2*c*x^2)*arctan(a*x)^2),x,1,-x^m/(a*c*arctan(a*x))+m*Unintegrable(x^(-1+m)/arctan(a*x),x)/(a*c)],
[x^m/((c+a^2*c*x^2)^2*arctan(a*x)^2),x,0,Unintegrable(x^m/((c+a^2*c*x^2)^2*arctan(a*x)^2),x)],
[x^m/((c+a^2*c*x^2)^3*arctan(a*x)^2),x,0,Unintegrable(x^m/((c+a^2*c*x^2)^3*arctan(a*x)^2),x)],
[x^m*(c+a^2*c*x^2)^(5/2)/arctan(a*x)^2,x,0,Unintegrable(x^m*(c+a^2*c*x^2)^(5/2)/arctan(a*x)^2,x)],
[x^m*(c+a^2*c*x^2)^(3/2)/arctan(a*x)^2,x,0,Unintegrable(x^m*(c+a^2*c*x^2)^(3/2)/arctan(a*x)^2,x)],
[x^m*sqrt(c+a^2*c*x^2)/arctan(a*x)^2,x,0,Unintegrable(x^m*sqrt(c+a^2*c*x^2)/arctan(a*x)^2,x)],
[x^m/(arctan(a*x)^2*sqrt(c+a^2*c*x^2)),x,0,Unintegrable(x^m/(arctan(a*x)^2*sqrt(c+a^2*c*x^2)),x)],
[x^m/((c+a^2*c*x^2)^(3/2)*arctan(a*x)^2),x,0,Unintegrable(x^m/((c+a^2*c*x^2)^(3/2)*arctan(a*x)^2),x)],
[x^m/((c+a^2*c*x^2)^(5/2)*arctan(a*x)^2),x,0,Unintegrable(x^m/((c+a^2*c*x^2)^(5/2)*arctan(a*x)^2),x)],

# Integrands of the form (f x)^m (d+c^2 d x^2)^q (a+b ArcTan[c x])^-3

# Integrands of the form x^m (c+d x^2)^p / ArcTan[a x]^3 with d=a^2 c

# p>0
[x*(c+a^2*c*x^2)/arctan(a*x)^3,x,0,Unintegrable(x*(c+a^2*c*x^2)/arctan(a*x)^3,x)],
[(c+a^2*c*x^2)/arctan(a*x)^3,x,0,Unintegrable((c+a^2*c*x^2)/arctan(a*x)^3,x)],
[(c+a^2*c*x^2)/(x*arctan(a*x)^3),x,0,Unintegrable((c+a^2*c*x^2)/(x*arctan(a*x)^3),x)],
[x*(c+a^2*c*x^2)^2/arctan(a*x)^3,x,0,Unintegrable(x*(c+a^2*c*x^2)^2/arctan(a*x)^3,x)],
[(c+a^2*c*x^2)^2/arctan(a*x)^3,x,0,Unintegrable((c+a^2*c*x^2)^2/arctan(a*x)^3,x)],
[(c+a^2*c*x^2)^2/(x*arctan(a*x)^3),x,0,Unintegrable((c+a^2*c*x^2)^2/(x*arctan(a*x)^3),x)],
[x*(c+a^2*c*x^2)^3/arctan(a*x)^3,x,0,Unintegrable(x*(c+a^2*c*x^2)^3/arctan(a*x)^3,x)],
[(c+a^2*c*x^2)^3/arctan(a*x)^3,x,0,Unintegrable((c+a^2*c*x^2)^3/arctan(a*x)^3,x)],
[(c+a^2*c*x^2)^3/(x*arctan(a*x)^3),x,0,Unintegrable((c+a^2*c*x^2)^3/(x*arctan(a*x)^3),x)],

# p<0
[x^3/((c+a^2*c*x^2)*arctan(a*x)^3),x,1,-1/2*x^3/(a*c*arctan(a*x)^2)+3/2*Unintegrable(x^2/arctan(a*x)^2,x)/(a*c)],
[x^2/((c+a^2*c*x^2)*arctan(a*x)^3),x,1,-1/2*x^2/(a*c*arctan(a*x)^2)+Unintegrable(x/arctan(a*x)^2,x)/(a*c)],
[x/((c+a^2*c*x^2)*arctan(a*x)^3),x,1,-1/2*x/(a*c*arctan(a*x)^2)+1/2*Unintegrable(1/arctan(a*x)^2,x)/(a*c)],
[1/((c+a^2*c*x^2)*arctan(a*x)^3),x,1,(-1/2)/(a*c*arctan(a*x)^2)],
[1/(x*(c+a^2*c*x^2)*arctan(a*x)^3),x,1,(-1/2)/(a*c*x*arctan(a*x)^2)-1/2*Unintegrable(1/(x^2*arctan(a*x)^2),x)/(a*c)],
[1/(x^2*(c+a^2*c*x^2)*arctan(a*x)^3),x,1,(-1/2)/(a*c*x^2*arctan(a*x)^2)-Unintegrable(1/(x^3*arctan(a*x)^2),x)/(a*c)],
[1/(x^3*(c+a^2*c*x^2)*arctan(a*x)^3),x,1,(-1/2)/(a*c*x^3*arctan(a*x)^2)-3/2*Unintegrable(1/(x^4*arctan(a*x)^2),x)/(a*c)],
[1/(x^4*(c+a^2*c*x^2)*arctan(a*x)^3),x,1,(-1/2)/(a*c*x^4*arctan(a*x)^2)-2*Unintegrable(1/(x^5*arctan(a*x)^2),x)/(a*c)],
[x^3/((c+a^2*c*x^2)^2*arctan(a*x)^3),x,7,-1/2*x/(a^3*c^2*arctan(a*x)^2)+1/2*x/(a^3*c^2*(1+a^2*x^2)*arctan(a*x)^2)+1/2*(1-a^2*x^2)/(a^4*c^2*(1+a^2*x^2)*arctan(a*x))+Si(2*arctan(a*x))/(a^4*c^2)+1/2*Unintegrable(1/arctan(a*x)^2,x)/(a^3*c^2)],
[x^2/((c+a^2*c*x^2)^2*arctan(a*x)^3),x,10,-1/2*x^2/(a*c^2*(1+a^2*x^2)*arctan(a*x)^2)-x/(a^2*c^2*(1+a^2*x^2)*arctan(a*x))+Ci(2*arctan(a*x))/(a^3*c^2)],
[x/((c+a^2*c*x^2)^2*arctan(a*x)^3),x,5,-1/2*x/(a*c^2*(1+a^2*x^2)*arctan(a*x)^2)+1/2*(-1+a^2*x^2)/(a^2*c^2*(1+a^2*x^2)*arctan(a*x))-Si(2*arctan(a*x))/(a^2*c^2)],
[1/((c+a^2*c*x^2)^2*arctan(a*x)^3),x,10,(-1/2)/(a*c^2*(1+a^2*x^2)*arctan(a*x)^2)+x/(c^2*(1+a^2*x^2)*arctan(a*x))-Ci(2*arctan(a*x))/(a*c^2)],
[1/(x*(c+a^2*c*x^2)^2*arctan(a*x)^3),x,7,(-1/2)/(a*c^2*x*arctan(a*x)^2)+1/2*a*x/(c^2*(1+a^2*x^2)*arctan(a*x)^2)+1/2*(1-a^2*x^2)/(c^2*(1+a^2*x^2)*arctan(a*x))+Si(2*arctan(a*x))/c^2-1/2*Unintegrable(1/(x^2*arctan(a*x)^2),x)/(a*c^2)],
[1/(x^2*(c+a^2*c*x^2)^2*arctan(a*x)^3),x,12,(-1/2)/(a*c^2*x^2*arctan(a*x)^2)+1/2*a/(c^2*(1+a^2*x^2)*arctan(a*x)^2)-a^2*x/(c^2*(1+a^2*x^2)*arctan(a*x))+a*Ci(2*arctan(a*x))/c^2-Unintegrable(1/(x^3*arctan(a*x)^2),x)/(a*c^2)],
[1/(x^3*(c+a^2*c*x^2)^2*arctan(a*x)^3),x,9,(-1/2)/(a*c^2*x^3*arctan(a*x)^2)+1/2*a/(c^2*x*arctan(a*x)^2)-1/2*a^3*x/(c^2*(1+a^2*x^2)*arctan(a*x)^2)-1/2*a^2*(1-a^2*x^2)/(c^2*(1+a^2*x^2)*arctan(a*x))-a^2*Si(2*arctan(a*x))/c^2-3/2*Unintegrable(1/(x^4*arctan(a*x)^2),x)/(a*c^2)+1/2*a*Unintegrable(1/(x^2*arctan(a*x)^2),x)/c^2],
[1/(x^4*(c+a^2*c*x^2)^2*arctan(a*x)^3),x,14,(-1/2)/(a*c^2*x^4*arctan(a*x)^2)+1/2*a/(c^2*x^2*arctan(a*x)^2)-1/2*a^3/(c^2*(1+a^2*x^2)*arctan(a*x)^2)+a^4*x/(c^2*(1+a^2*x^2)*arctan(a*x))-a^3*Ci(2*arctan(a*x))/c^2-2*Unintegrable(1/(x^5*arctan(a*x)^2),x)/(a*c^2)+a*Unintegrable(1/(x^3*arctan(a*x)^2),x)/c^2],
[x^3/((c+a^2*c*x^2)^3*arctan(a*x)^3),x,25,1/2*x/(a^3*c^3*(1+a^2*x^2)^2*arctan(a*x)^2)-1/2*x/(a^3*c^3*(1+a^2*x^2)*arctan(a*x)^2)+2/(a^4*c^3*(1+a^2*x^2)^2*arctan(a*x))+(-3/2)/(a^4*c^3*(1+a^2*x^2)*arctan(a*x))+1/2*(-1+a^2*x^2)/(a^4*c^3*(1+a^2*x^2)*arctan(a*x))-1/2*Si(2*arctan(a*x))/(a^4*c^3)+Si(4*arctan(a*x))/(a^4*c^3)],
[x^2/((c+a^2*c*x^2)^3*arctan(a*x)^3),x,22,1/2/(a^3*c^3*(1+a^2*x^2)^2*arctan(a*x)^2)+(-1/2)/(a^3*c^3*(1+a^2*x^2)*arctan(a*x)^2)-2*x/(a^2*c^3*(1+a^2*x^2)^2*arctan(a*x))+x/(a^2*c^3*(1+a^2*x^2)*arctan(a*x))+Ci(4*arctan(a*x))/(a^3*c^3)],
[x/((c+a^2*c*x^2)^3*arctan(a*x)^3),x,19,-1/2*x/(a*c^3*(1+a^2*x^2)^2*arctan(a*x)^2)+(-2)/(a^2*c^3*(1+a^2*x^2)^2*arctan(a*x))+3/2/(a^2*c^3*(1+a^2*x^2)*arctan(a*x))-1/2*Si(2*arctan(a*x))/(a^2*c^3)-Si(4*arctan(a*x))/(a^2*c^3)],
[1/((c+a^2*c*x^2)^3*arctan(a*x)^3),x,11,(-1/2)/(a*c^3*(1+a^2*x^2)^2*arctan(a*x)^2)+2*x/(c^3*(1+a^2*x^2)^2*arctan(a*x))-Ci(2*arctan(a*x))/(a*c^3)-Ci(4*arctan(a*x))/(a*c^3)],
[1/(x*(c+a^2*c*x^2)^3*arctan(a*x)^3),x,27,(-1/2)/(a*c^3*x*arctan(a*x)^2)+1/2*a*x/(c^3*(1+a^2*x^2)^2*arctan(a*x)^2)+1/2*a*x/(c^3*(1+a^2*x^2)*arctan(a*x)^2)+2/(c^3*(1+a^2*x^2)^2*arctan(a*x))+(-3/2)/(c^3*(1+a^2*x^2)*arctan(a*x))+1/2*(1-a^2*x^2)/(c^3*(1+a^2*x^2)*arctan(a*x))+3/2*Si(2*arctan(a*x))/c^3+Si(4*arctan(a*x))/c^3-1/2*Unintegrable(1/(x^2*arctan(a*x)^2),x)/(a*c^3)],
[1/(x^2*(c+a^2*c*x^2)^3*arctan(a*x)^3),x,24,(-1/2)/(a*c^3*x^2*arctan(a*x)^2)+1/2*a/(c^3*(1+a^2*x^2)^2*arctan(a*x)^2)+1/2*a/(c^3*(1+a^2*x^2)*arctan(a*x)^2)-2*a^2*x/(c^3*(1+a^2*x^2)^2*arctan(a*x))-a^2*x/(c^3*(1+a^2*x^2)*arctan(a*x))+2*a*Ci(2*arctan(a*x))/c^3+a*Ci(4*arctan(a*x))/c^3-Unintegrable(1/(x^3*arctan(a*x)^2),x)/(a*c^3)],
[1/(x^3*(c+a^2*c*x^2)^3*arctan(a*x)^3),x,37,(-1/2)/(a*c^3*x^3*arctan(a*x)^2)+a/(c^3*x*arctan(a*x)^2)-1/2*a^3*x/(c^3*(1+a^2*x^2)^2*arctan(a*x)^2)-a^3*x/(c^3*(1+a^2*x^2)*arctan(a*x)^2)-2*a^2/(c^3*(1+a^2*x^2)^2*arctan(a*x))+3/2*a^2/(c^3*(1+a^2*x^2)*arctan(a*x))-a^2*(1-a^2*x^2)/(c^3*(1+a^2*x^2)*arctan(a*x))-5/2*a^2*Si(2*arctan(a*x))/c^3-a^2*Si(4*arctan(a*x))/c^3-3/2*Unintegrable(1/(x^4*arctan(a*x)^2),x)/(a*c^3)+a*Unintegrable(1/(x^2*arctan(a*x)^2),x)/c^3],
[1/(x^4*(c+a^2*c*x^2)^3*arctan(a*x)^3),x,39,(-1/2)/(a*c^3*x^4*arctan(a*x)^2)+a/(c^3*x^2*arctan(a*x)^2)-1/2*a^3/(c^3*(1+a^2*x^2)^2*arctan(a*x)^2)-a^3/(c^3*(1+a^2*x^2)*arctan(a*x)^2)+2*a^4*x/(c^3*(1+a^2*x^2)^2*arctan(a*x))+2*a^4*x/(c^3*(1+a^2*x^2)*arctan(a*x))-3*a^3*Ci(2*arctan(a*x))/c^3-a^3*Ci(4*arctan(a*x))/c^3-2*Unintegrable(1/(x^5*arctan(a*x)^2),x)/(a*c^3)+2*a*Unintegrable(1/(x^3*arctan(a*x)^2),x)/c^3],
[x^3/((1+a^2*x^2)*arctan(a*x)^3)-3/2*x^2/(a*arctan(a*x)^2),x,2,-1/2*x^3/(a*arctan(a*x)^2)],

# Integrands of the form x^m (c+d x^2)^(p/2) / ArcTan[a x]^3 with d=a^2 c

# p>0
[x*sqrt(c+a^2*c*x^2)/arctan(a*x)^3,x,0,Unintegrable(x*sqrt(c+a^2*c*x^2)/arctan(a*x)^3,x)],
[sqrt(c+a^2*c*x^2)/arctan(a*x)^3,x,0,Unintegrable(sqrt(c+a^2*c*x^2)/arctan(a*x)^3,x)],
[sqrt(c+a^2*c*x^2)/(x*arctan(a*x)^3),x,0,Unintegrable(sqrt(c+a^2*c*x^2)/(x*arctan(a*x)^3),x)],
[x*(c+a^2*c*x^2)^(3/2)/arctan(a*x)^3,x,0,Unintegrable(x*(c+a^2*c*x^2)^(3/2)/arctan(a*x)^3,x)],
[(c+a^2*c*x^2)^(3/2)/arctan(a*x)^3,x,0,Unintegrable((c+a^2*c*x^2)^(3/2)/arctan(a*x)^3,x)],
[(c+a^2*c*x^2)^(3/2)/(x*arctan(a*x)^3),x,0,Unintegrable((c+a^2*c*x^2)^(3/2)/(x*arctan(a*x)^3),x)],
[x*(c+a^2*c*x^2)^(5/2)/arctan(a*x)^3,x,0,Unintegrable(x*(c+a^2*c*x^2)^(5/2)/arctan(a*x)^3,x)],
[(c+a^2*c*x^2)^(5/2)/arctan(a*x)^3,x,0,Unintegrable((c+a^2*c*x^2)^(5/2)/arctan(a*x)^3,x)],
[(c+a^2*c*x^2)^(5/2)/(x*arctan(a*x)^3),x,0,Unintegrable((c+a^2*c*x^2)^(5/2)/(x*arctan(a*x)^3),x)],

# p<0
[x/(arctan(a*x)^3*sqrt(c+a^2*c*x^2)),x,0,Unintegrable(x/(arctan(a*x)^3*sqrt(c+a^2*c*x^2)),x)],
[1/(arctan(a*x)^3*sqrt(c+a^2*c*x^2)),x,0,Unintegrable(1/(arctan(a*x)^3*sqrt(c+a^2*c*x^2)),x)],
[1/(x*arctan(a*x)^3*sqrt(c+a^2*c*x^2)),x,1,-1/2*sqrt(c+a^2*c*x^2)/(a*c*x*arctan(a*x)^2)-1/2*Unintegrable(1/(x^2*arctan(a*x)^2*sqrt(c+a^2*c*x^2)),x)/a],
[1/(x^2*arctan(a*x)^3*sqrt(c+a^2*c*x^2)),x,0,Unintegrable(1/(x^2*arctan(a*x)^3*sqrt(c+a^2*c*x^2)),x)],
[1/(x^3*arctan(a*x)^3*sqrt(c+a^2*c*x^2)),x,0,Unintegrable(1/(x^3*arctan(a*x)^3*sqrt(c+a^2*c*x^2)),x)],
[x^3/((c+a^2*c*x^2)^(3/2)*arctan(a*x)^3),x,6,1/2*x/(a^3*c*arctan(a*x)^2*sqrt(c+a^2*c*x^2))+1/2/(a^4*c*arctan(a*x)*sqrt(c+a^2*c*x^2))+1/2*Si(arctan(a*x))*sqrt(1+a^2*x^2)/(a^4*c*sqrt(c+a^2*c*x^2))+Unintegrable(x/(arctan(a*x)^3*sqrt(c+a^2*c*x^2)),x)/(a^2*c)],
[x^2/((c+a^2*c*x^2)^(3/2)*arctan(a*x)^3),x,6,1/2/(a^3*c*arctan(a*x)^2*sqrt(c+a^2*c*x^2))-1/2*x/(a^2*c*arctan(a*x)*sqrt(c+a^2*c*x^2))+1/2*Ci(arctan(a*x))*sqrt(1+a^2*x^2)/(a^3*c*sqrt(c+a^2*c*x^2))+Unintegrable(1/(arctan(a*x)^3*sqrt(c+a^2*c*x^2)),x)/(a^2*c)],
[x/((c+a^2*c*x^2)^(3/2)*arctan(a*x)^3),x,5,-1/2*x/(a*c*arctan(a*x)^2*sqrt(c+a^2*c*x^2))+(-1/2)/(a^2*c*arctan(a*x)*sqrt(c+a^2*c*x^2))-1/2*Si(arctan(a*x))*sqrt(1+a^2*x^2)/(a^2*c*sqrt(c+a^2*c*x^2))],
[1/((c+a^2*c*x^2)^(3/2)*arctan(a*x)^3),x,5,(-1/2)/(a*c*arctan(a*x)^2*sqrt(c+a^2*c*x^2))+1/2*x/(c*arctan(a*x)*sqrt(c+a^2*c*x^2))-1/2*Ci(arctan(a*x))*sqrt(1+a^2*x^2)/(a*c*sqrt(c+a^2*c*x^2))],
[1/(x*(c+a^2*c*x^2)^(3/2)*arctan(a*x)^3),x,7,1/2*a*x/(c*arctan(a*x)^2*sqrt(c+a^2*c*x^2))+1/2/(c*arctan(a*x)*sqrt(c+a^2*c*x^2))+1/2*Si(arctan(a*x))*sqrt(1+a^2*x^2)/(c*sqrt(c+a^2*c*x^2))-1/2*sqrt(c+a^2*c*x^2)/(a*c^2*x*arctan(a*x)^2)-1/2*Unintegrable(1/(x^2*arctan(a*x)^2*sqrt(c+a^2*c*x^2)),x)/(a*c)],
[1/(x^2*(c+a^2*c*x^2)^(3/2)*arctan(a*x)^3),x,6,1/2*a/(c*arctan(a*x)^2*sqrt(c+a^2*c*x^2))-1/2*a^2*x/(c*arctan(a*x)*sqrt(c+a^2*c*x^2))+1/2*a*Ci(arctan(a*x))*sqrt(1+a^2*x^2)/(c*sqrt(c+a^2*c*x^2))+Unintegrable(1/(x^2*arctan(a*x)^3*sqrt(c+a^2*c*x^2)),x)/c],
[1/(x^3*(c+a^2*c*x^2)^(3/2)*arctan(a*x)^3),x,8,-1/2*a^3*x/(c*arctan(a*x)^2*sqrt(c+a^2*c*x^2))-1/2*a^2/(c*arctan(a*x)*sqrt(c+a^2*c*x^2))-1/2*a^2*Si(arctan(a*x))*sqrt(1+a^2*x^2)/(c*sqrt(c+a^2*c*x^2))+1/2*a*sqrt(c+a^2*c*x^2)/(c^2*x*arctan(a*x)^2)+Unintegrable(1/(x^3*arctan(a*x)^3*sqrt(c+a^2*c*x^2)),x)/c+1/2*a*Unintegrable(1/(x^2*arctan(a*x)^2*sqrt(c+a^2*c*x^2)),x)/c],
[1/(x^4*(c+a^2*c*x^2)^(3/2)*arctan(a*x)^3),x,7,-1/2*a^3/(c*arctan(a*x)^2*sqrt(c+a^2*c*x^2))+1/2*a^4*x/(c*arctan(a*x)*sqrt(c+a^2*c*x^2))-1/2*a^3*Ci(arctan(a*x))*sqrt(1+a^2*x^2)/(c*sqrt(c+a^2*c*x^2))+Unintegrable(1/(x^4*arctan(a*x)^3*sqrt(c+a^2*c*x^2)),x)/c-a^2*Unintegrable(1/(x^2*arctan(a*x)^3*sqrt(c+a^2*c*x^2)),x)/c],
[x^5/((c+a^2*c*x^2)^(5/2)*arctan(a*x)^3),x,20,1/2*x^3/(a^3*c*(c+a^2*c*x^2)^(3/2)*arctan(a*x)^2)+(-3/2)/(a^6*c*(c+a^2*c*x^2)^(3/2)*arctan(a*x))+1/2*x/(a^5*c^2*arctan(a*x)^2*sqrt(c+a^2*c*x^2))+2/(a^6*c^2*arctan(a*x)*sqrt(c+a^2*c*x^2))+7/8*Si(arctan(a*x))*sqrt(1+a^2*x^2)/(a^6*c^2*sqrt(c+a^2*c*x^2))-9/8*Si(3*arctan(a*x))*sqrt(1+a^2*x^2)/(a^6*c^2*sqrt(c+a^2*c*x^2))+Unintegrable(x/(arctan(a*x)^3*sqrt(c+a^2*c*x^2)),x)/(a^4*c^2)],
[x^4/((c+a^2*c*x^2)^(5/2)*arctan(a*x)^3),x,27,(-1/2)/(a^5*c*(c+a^2*c*x^2)^(3/2)*arctan(a*x)^2)+3/2*x/(a^4*c*(c+a^2*c*x^2)^(3/2)*arctan(a*x))+1/(a^5*c^2*arctan(a*x)^2*sqrt(c+a^2*c*x^2))-x/(a^4*c^2*arctan(a*x)*sqrt(c+a^2*c*x^2))+5/8*Ci(arctan(a*x))*sqrt(1+a^2*x^2)/(a^5*c^2*sqrt(c+a^2*c*x^2))-9/8*Ci(3*arctan(a*x))*sqrt(1+a^2*x^2)/(a^5*c^2*sqrt(c+a^2*c*x^2))+Unintegrable(1/(arctan(a*x)^3*sqrt(c+a^2*c*x^2)),x)/(a^4*c^2)],
[x^3/((c+a^2*c*x^2)^(5/2)*arctan(a*x)^3),x,13,-1/2*x^3/(a*c*(c+a^2*c*x^2)^(3/2)*arctan(a*x)^2)+3/2/(a^4*c*(c+a^2*c*x^2)^(3/2)*arctan(a*x))+(-3/2)/(a^4*c^2*arctan(a*x)*sqrt(c+a^2*c*x^2))-3/8*Si(arctan(a*x))*sqrt(1+a^2*x^2)/(a^4*c^2*sqrt(c+a^2*c*x^2))+9/8*Si(3*arctan(a*x))*sqrt(1+a^2*x^2)/(a^4*c^2*sqrt(c+a^2*c*x^2))],
[x^2/((c+a^2*c*x^2)^(5/2)*arctan(a*x)^3),x,20,1/2/(a^3*c*(c+a^2*c*x^2)^(3/2)*arctan(a*x)^2)-3/2*x/(a^2*c*(c+a^2*c*x^2)^(3/2)*arctan(a*x))+(-1/2)/(a^3*c^2*arctan(a*x)^2*sqrt(c+a^2*c*x^2))+1/2*x/(a^2*c^2*arctan(a*x)*sqrt(c+a^2*c*x^2))-1/8*Ci(arctan(a*x))*sqrt(1+a^2*x^2)/(a^3*c^2*sqrt(c+a^2*c*x^2))+9/8*Ci(3*arctan(a*x))*sqrt(1+a^2*x^2)/(a^3*c^2*sqrt(c+a^2*c*x^2))],
[x/((c+a^2*c*x^2)^(5/2)*arctan(a*x)^3),x,20,-1/2*x/(a*c*(c+a^2*c*x^2)^(3/2)*arctan(a*x)^2)+(-3/2)/(a^2*c*(c+a^2*c*x^2)^(3/2)*arctan(a*x))+1/(a^2*c^2*arctan(a*x)*sqrt(c+a^2*c*x^2))-1/8*Si(arctan(a*x))*sqrt(1+a^2*x^2)/(a^2*c^2*sqrt(c+a^2*c*x^2))-9/8*Si(3*arctan(a*x))*sqrt(1+a^2*x^2)/(a^2*c^2*sqrt(c+a^2*c*x^2))],
[1/((c+a^2*c*x^2)^(5/2)*arctan(a*x)^3),x,14,(-1/2)/(a*c*(c+a^2*c*x^2)^(3/2)*arctan(a*x)^2)+3/2*x/(c*(c+a^2*c*x^2)^(3/2)*arctan(a*x))-3/8*Ci(arctan(a*x))*sqrt(1+a^2*x^2)/(a*c^2*sqrt(c+a^2*c*x^2))-9/8*Ci(3*arctan(a*x))*sqrt(1+a^2*x^2)/(a*c^2*sqrt(c+a^2*c*x^2))],
[1/(x*(c+a^2*c*x^2)^(5/2)*arctan(a*x)^3),x,28,1/2*a*x/(c*(c+a^2*c*x^2)^(3/2)*arctan(a*x)^2)+3/2/(c*(c+a^2*c*x^2)^(3/2)*arctan(a*x))+1/2*a*x/(c^2*arctan(a*x)^2*sqrt(c+a^2*c*x^2))+(-1/2)/(c^2*arctan(a*x)*sqrt(c+a^2*c*x^2))+5/8*Si(arctan(a*x))*sqrt(1+a^2*x^2)/(c^2*sqrt(c+a^2*c*x^2))+9/8*Si(3*arctan(a*x))*sqrt(1+a^2*x^2)/(c^2*sqrt(c+a^2*c*x^2))-1/2*sqrt(c+a^2*c*x^2)/(a*c^3*x*arctan(a*x)^2)-1/2*Unintegrable(1/(x^2*arctan(a*x)^2*sqrt(c+a^2*c*x^2)),x)/(a*c^2)],
[1/(x^2*(c+a^2*c*x^2)^(5/2)*arctan(a*x)^3),x,21,1/2*a/(c*(c+a^2*c*x^2)^(3/2)*arctan(a*x)^2)-3/2*a^2*x/(c*(c+a^2*c*x^2)^(3/2)*arctan(a*x))+1/2*a/(c^2*arctan(a*x)^2*sqrt(c+a^2*c*x^2))-1/2*a^2*x/(c^2*arctan(a*x)*sqrt(c+a^2*c*x^2))+7/8*a*Ci(arctan(a*x))*sqrt(1+a^2*x^2)/(c^2*sqrt(c+a^2*c*x^2))+9/8*a*Ci(3*arctan(a*x))*sqrt(1+a^2*x^2)/(c^2*sqrt(c+a^2*c*x^2))+Unintegrable(1/(x^2*arctan(a*x)^3*sqrt(c+a^2*c*x^2)),x)/c^2],

# Integrands of the form x^m (c+d x^2)^p / ArcTan[a x]^3 with d=a^2 c and m symbolic
[x^m*(c+a^2*c*x^2)^3/arctan(a*x)^3,x,0,Unintegrable(x^m*(c+a^2*c*x^2)^3/arctan(a*x)^3,x)],
[x^m*(c+a^2*c*x^2)^2/arctan(a*x)^3,x,0,Unintegrable(x^m*(c+a^2*c*x^2)^2/arctan(a*x)^3,x)],
[x^m*(c+a^2*c*x^2)/arctan(a*x)^3,x,0,Unintegrable(x^m*(c+a^2*c*x^2)/arctan(a*x)^3,x)],
[x^m/((c+a^2*c*x^2)*arctan(a*x)^3),x,1,-1/2*x^m/(a*c*arctan(a*x)^2)+1/2*m*Unintegrable(x^(-1+m)/arctan(a*x)^2,x)/(a*c)],
[x^m/((c+a^2*c*x^2)^2*arctan(a*x)^3),x,0,Unintegrable(x^m/((c+a^2*c*x^2)^2*arctan(a*x)^3),x)],
[x^m/((c+a^2*c*x^2)^3*arctan(a*x)^3),x,0,Unintegrable(x^m/((c+a^2*c*x^2)^3*arctan(a*x)^3),x)],
[x^m*(c+a^2*c*x^2)^(5/2)/arctan(a*x)^3,x,0,Unintegrable(x^m*(c+a^2*c*x^2)^(5/2)/arctan(a*x)^3,x)],
[x^m*(c+a^2*c*x^2)^(3/2)/arctan(a*x)^3,x,0,Unintegrable(x^m*(c+a^2*c*x^2)^(3/2)/arctan(a*x)^3,x)],
[x^m*sqrt(c+a^2*c*x^2)/arctan(a*x)^3,x,0,Unintegrable(x^m*sqrt(c+a^2*c*x^2)/arctan(a*x)^3,x)],
[x^m/(arctan(a*x)^3*sqrt(c+a^2*c*x^2)),x,0,Unintegrable(x^m/(arctan(a*x)^3*sqrt(c+a^2*c*x^2)),x)],
[x^m/((c+a^2*c*x^2)^(3/2)*arctan(a*x)^3),x,0,Unintegrable(x^m/((c+a^2*c*x^2)^(3/2)*arctan(a*x)^3),x)],
[x^m/((c+a^2*c*x^2)^(5/2)*arctan(a*x)^3),x,0,Unintegrable(x^m/((c+a^2*c*x^2)^(5/2)*arctan(a*x)^3),x)],

# Integrands of the form (f x)^m (d+c^2 d x^2)^q (a+b ArcTan[c x])^(1/2)

# Integrands of the form x^m (c+d x^2)^p ArcTan[a x]^(1/2) with d=a^2 c

# p>0
[x^m*(c+a^2*c*x^2)*sqrt(arctan(a*x)),x,0,Unintegrable(x^m*(c+a^2*c*x^2)*sqrt(arctan(a*x)),x)],
[x*(c+a^2*c*x^2)*sqrt(arctan(a*x)),x,1,1/4*c*(1+a^2*x^2)^2*sqrt(arctan(a*x))/a^2-1/8*Unintegrable((c+a^2*c*x^2)/sqrt(arctan(a*x)),x)/a],
[(c+a^2*c*x^2)*sqrt(arctan(a*x)),x,0,Unintegrable((c+a^2*c*x^2)*sqrt(arctan(a*x)),x)],
[(c+a^2*c*x^2)*sqrt(arctan(a*x))/x,x,0,Unintegrable((c+a^2*c*x^2)*sqrt(arctan(a*x))/x,x)],
[x^m*(c+a^2*c*x^2)^2*sqrt(arctan(a*x)),x,0,Unintegrable(x^m*(c+a^2*c*x^2)^2*sqrt(arctan(a*x)),x)],
[x*(c+a^2*c*x^2)^2*sqrt(arctan(a*x)),x,1,1/6*c^2*(1+a^2*x^2)^3*sqrt(arctan(a*x))/a^2-1/12*Unintegrable((c+a^2*c*x^2)^2/sqrt(arctan(a*x)),x)/a],
[(c+a^2*c*x^2)^2*sqrt(arctan(a*x)),x,0,Unintegrable((c+a^2*c*x^2)^2*sqrt(arctan(a*x)),x)],
[(c+a^2*c*x^2)^2*sqrt(arctan(a*x))/x,x,0,Unintegrable((c+a^2*c*x^2)^2*sqrt(arctan(a*x))/x,x)],
[x^m*(c+a^2*c*x^2)^3*sqrt(arctan(a*x)),x,0,Unintegrable(x^m*(c+a^2*c*x^2)^3*sqrt(arctan(a*x)),x)],
[x*(c+a^2*c*x^2)^3*sqrt(arctan(a*x)),x,1,1/8*c^3*(1+a^2*x^2)^4*sqrt(arctan(a*x))/a^2-1/16*Unintegrable((c+a^2*c*x^2)^3/sqrt(arctan(a*x)),x)/a],
[(c+a^2*c*x^2)^3*sqrt(arctan(a*x)),x,0,Unintegrable((c+a^2*c*x^2)^3*sqrt(arctan(a*x)),x)],
[(c+a^2*c*x^2)^3*sqrt(arctan(a*x))/x,x,0,Unintegrable((c+a^2*c*x^2)^3*sqrt(arctan(a*x))/x,x)],

# p<0
[x^m*sqrt(arctan(a*x))/(c+a^2*c*x^2),x,0,Unintegrable(x^m*sqrt(arctan(a*x))/(c+a^2*c*x^2),x)],
[x^3*sqrt(arctan(a*x))/(c+a^2*c*x^2),x,2,-2/3*x*arctan(a*x)^(3/2)/(a^3*c)+2/3*Unintegrable(arctan(a*x)^(3/2),x)/(a^3*c)+Unintegrable(x*sqrt(arctan(a*x)),x)/(a^2*c)],
[x^2*sqrt(arctan(a*x))/(c+a^2*c*x^2),x,2,-2/3*arctan(a*x)^(3/2)/(a^3*c)+Unintegrable(sqrt(arctan(a*x)),x)/(a^2*c)],
[x*sqrt(arctan(a*x))/(c+a^2*c*x^2),x,1,2/3*x*arctan(a*x)^(3/2)/(a*c)-2/3*Unintegrable(arctan(a*x)^(3/2),x)/(a*c)],
[sqrt(arctan(a*x))/(c+a^2*c*x^2),x,1,2/3*arctan(a*x)^(3/2)/(a*c)],
[sqrt(arctan(a*x))/(x*(c+a^2*c*x^2)),x,1,-2/3*I*arctan(a*x)^(3/2)/c+I*Unintegrable(sqrt(arctan(a*x))/(x*(I+a*x)),x)/c],
[sqrt(arctan(a*x))/(x^2*(c+a^2*c*x^2)),x,2,-2/3*a*arctan(a*x)^(3/2)/c+Unintegrable(sqrt(arctan(a*x))/x^2,x)/c],
[sqrt(arctan(a*x))/(x^3*(c+a^2*c*x^2)),x,2,2/3*I*a^2*arctan(a*x)^(3/2)/c+Unintegrable(sqrt(arctan(a*x))/x^3,x)/c-I*a^2*Unintegrable(sqrt(arctan(a*x))/(x*(I+a*x)),x)/c],
[sqrt(arctan(a*x))/(x^4*(c+a^2*c*x^2)),x,3,2/3*a^3*arctan(a*x)^(3/2)/c+Unintegrable(sqrt(arctan(a*x))/x^4,x)/c-a^2*Unintegrable(sqrt(arctan(a*x))/x^2,x)/c],
[x^m*sqrt(arctan(a*x))/(c+a^2*c*x^2)^2,x,0,Unintegrable(x^m*sqrt(arctan(a*x))/(c+a^2*c*x^2)^2,x)],
[x^3*sqrt(arctan(a*x))/(c+a^2*c*x^2)^2,x,0,Unintegrable(x^3*sqrt(arctan(a*x))/(c+a^2*c*x^2)^2,x)],
[x^2*sqrt(arctan(a*x))/(c+a^2*c*x^2)^2,x,6,1/3*arctan(a*x)^(3/2)/(a^3*c^2)+1/8*FresnelS(2*sqrt(arctan(a*x))/sqrt(Pi))*sqrt(Pi)/(a^3*c^2)-1/2*x*sqrt(arctan(a*x))/(a^2*c^2*(1+a^2*x^2))],
[x*sqrt(arctan(a*x))/(c+a^2*c*x^2)^2,x,6,1/8*FresnelC(2*sqrt(arctan(a*x))/sqrt(Pi))*sqrt(Pi)/(a^2*c^2)+1/4*sqrt(arctan(a*x))/(a^2*c^2)-1/2*sqrt(arctan(a*x))/(a^2*c^2*(1+a^2*x^2))],
[sqrt(arctan(a*x))/(c+a^2*c*x^2)^2,x,6,1/3*arctan(a*x)^(3/2)/(a*c^2)-1/8*FresnelS(2*sqrt(arctan(a*x))/sqrt(Pi))*sqrt(Pi)/(a*c^2)+1/2*x*sqrt(arctan(a*x))/(c^2*(1+a^2*x^2))],
[sqrt(arctan(a*x))/(x*(c+a^2*c*x^2)^2),x,0,Unintegrable(sqrt(arctan(a*x))/(x*(c+a^2*c*x^2)^2),x)],
[x^m*sqrt(arctan(a*x))/(c+a^2*c*x^2)^3,x,0,Unintegrable(x^m*sqrt(arctan(a*x))/(c+a^2*c*x^2)^3,x)],
[x^5*sqrt(arctan(a*x))/(c+a^2*c*x^2)^3,x,0,Unintegrable(x^5*sqrt(arctan(a*x))/(c+a^2*c*x^2)^3,x)],
[x^4*sqrt(arctan(a*x))/(c+a^2*c*x^2)^3,x,9,1/4*arctan(a*x)^(3/2)/(a^5*c^3)-1/64*FresnelS(2*sqrt(2/Pi)*sqrt(arctan(a*x)))*sqrt(1/2*Pi)/(a^5*c^3)+1/8*FresnelS(2*sqrt(arctan(a*x))/sqrt(Pi))*sqrt(Pi)/(a^5*c^3)-1/4*sin(2*arctan(a*x))*sqrt(arctan(a*x))/(a^5*c^3)+1/32*sin(4*arctan(a*x))*sqrt(arctan(a*x))/(a^5*c^3)],
[x^3*sqrt(arctan(a*x))/(c+a^2*c*x^2)^3,x,8,-1/64*FresnelC(2*sqrt(2/Pi)*sqrt(arctan(a*x)))*sqrt(1/2*Pi)/(a^4*c^3)+1/16*FresnelC(2*sqrt(arctan(a*x))/sqrt(Pi))*sqrt(Pi)/(a^4*c^3)-3/32*sqrt(arctan(a*x))/(a^4*c^3)+1/4*x^4*sqrt(arctan(a*x))/(c^3*(1+a^2*x^2)^2)],
[x^2*sqrt(arctan(a*x))/(c+a^2*c*x^2)^3,x,6,1/12*arctan(a*x)^(3/2)/(a^3*c^3)+1/64*FresnelS(2*sqrt(2/Pi)*sqrt(arctan(a*x)))*sqrt(1/2*Pi)/(a^3*c^3)-1/32*sin(4*arctan(a*x))*sqrt(arctan(a*x))/(a^3*c^3)],
[x*sqrt(arctan(a*x))/(c+a^2*c*x^2)^3,x,8,1/64*FresnelC(2*sqrt(2/Pi)*sqrt(arctan(a*x)))*sqrt(1/2*Pi)/(a^2*c^3)+1/16*FresnelC(2*sqrt(arctan(a*x))/sqrt(Pi))*sqrt(Pi)/(a^2*c^3)+3/32*sqrt(arctan(a*x))/(a^2*c^3)-1/4*sqrt(arctan(a*x))/(a^2*c^3*(1+a^2*x^2)^2)],
[sqrt(arctan(a*x))/(c+a^2*c*x^2)^3,x,9,1/4*arctan(a*x)^(3/2)/(a*c^3)-1/64*FresnelS(2*sqrt(2/Pi)*sqrt(arctan(a*x)))*sqrt(1/2*Pi)/(a*c^3)-1/8*FresnelS(2*sqrt(arctan(a*x))/sqrt(Pi))*sqrt(Pi)/(a*c^3)+1/4*sin(2*arctan(a*x))*sqrt(arctan(a*x))/(a*c^3)+1/32*sin(4*arctan(a*x))*sqrt(arctan(a*x))/(a*c^3)],
[sqrt(arctan(a*x))/(x*(c+a^2*c*x^2)^3),x,0,Unintegrable(sqrt(arctan(a*x))/(x*(c+a^2*c*x^2)^3),x)],

# Integrands of the form x^m (c+d x^2)^(p/2) ArcTan[a x]^(1/2) with d=a^2 c

# p>0
[x^m*sqrt(c+a^2*c*x^2)*sqrt(arctan(a*x)),x,0,Unintegrable(x^m*sqrt(c+a^2*c*x^2)*sqrt(arctan(a*x)),x)],
[x^2*sqrt(c+a^2*c*x^2)*sqrt(arctan(a*x)),x,0,Unintegrable(x^2*sqrt(c+a^2*c*x^2)*sqrt(arctan(a*x)),x)],
[x*sqrt(c+a^2*c*x^2)*sqrt(arctan(a*x)),x,1,1/3*(c+a^2*c*x^2)^(3/2)*sqrt(arctan(a*x))/(a^2*c)-1/6*Unintegrable(sqrt(c+a^2*c*x^2)/sqrt(arctan(a*x)),x)/a],
[sqrt(c+a^2*c*x^2)*sqrt(arctan(a*x)),x,0,Unintegrable(sqrt(c+a^2*c*x^2)*sqrt(arctan(a*x)),x)],
[x^m*(c+a^2*c*x^2)^(3/2)*sqrt(arctan(a*x)),x,0,Unintegrable(x^m*(c+a^2*c*x^2)^(3/2)*sqrt(arctan(a*x)),x)],
[x^2*(c+a^2*c*x^2)^(3/2)*sqrt(arctan(a*x)),x,0,Unintegrable(x^2*(c+a^2*c*x^2)^(3/2)*sqrt(arctan(a*x)),x)],
[x*(c+a^2*c*x^2)^(3/2)*sqrt(arctan(a*x)),x,1,1/5*(c+a^2*c*x^2)^(5/2)*sqrt(arctan(a*x))/(a^2*c)-1/10*Unintegrable((c+a^2*c*x^2)^(3/2)/sqrt(arctan(a*x)),x)/a],
[(c+a^2*c*x^2)^(3/2)*sqrt(arctan(a*x)),x,0,Unintegrable((c+a^2*c*x^2)^(3/2)*sqrt(arctan(a*x)),x)],
[x^m*(c+a^2*c*x^2)^(5/2)*sqrt(arctan(a*x)),x,0,Unintegrable(x^m*(c+a^2*c*x^2)^(5/2)*sqrt(arctan(a*x)),x)],
[x^2*(c+a^2*c*x^2)^(5/2)*sqrt(arctan(a*x)),x,0,Unintegrable(x^2*(c+a^2*c*x^2)^(5/2)*sqrt(arctan(a*x)),x)],
[x*(c+a^2*c*x^2)^(5/2)*sqrt(arctan(a*x)),x,1,1/7*(c+a^2*c*x^2)^(7/2)*sqrt(arctan(a*x))/(a^2*c)-1/14*Unintegrable((c+a^2*c*x^2)^(5/2)/sqrt(arctan(a*x)),x)/a],
[(c+a^2*c*x^2)^(5/2)*sqrt(arctan(a*x)),x,0,Unintegrable((c+a^2*c*x^2)^(5/2)*sqrt(arctan(a*x)),x)],

# p<0
[x^m*sqrt(arctan(a*x))/sqrt(c+a^2*c*x^2),x,0,Unintegrable(x^m*sqrt(arctan(a*x))/sqrt(c+a^2*c*x^2),x)],
[x^3*sqrt(arctan(a*x))/sqrt(c+a^2*c*x^2),x,2,-2/3*sqrt(c+a^2*c*x^2)*sqrt(arctan(a*x))/(a^4*c)+1/3*x^2*sqrt(c+a^2*c*x^2)*sqrt(arctan(a*x))/(a^2*c)+1/3*Unintegrable(1/(sqrt(c+a^2*c*x^2)*sqrt(arctan(a*x))),x)/a^3-1/6*Unintegrable(x^2/(sqrt(c+a^2*c*x^2)*sqrt(arctan(a*x))),x)/a],
[x^2*sqrt(arctan(a*x))/sqrt(c+a^2*c*x^2),x,1,1/2*x*sqrt(c+a^2*c*x^2)*sqrt(arctan(a*x))/(a^2*c)-1/4*Unintegrable(x/(sqrt(c+a^2*c*x^2)*sqrt(arctan(a*x))),x)/a-1/2*Unintegrable(sqrt(arctan(a*x))/sqrt(c+a^2*c*x^2),x)/a^2],
[x*sqrt(arctan(a*x))/sqrt(c+a^2*c*x^2),x,1,sqrt(c+a^2*c*x^2)*sqrt(arctan(a*x))/(a^2*c)-1/2*Unintegrable(1/(sqrt(c+a^2*c*x^2)*sqrt(arctan(a*x))),x)/a],
[sqrt(arctan(a*x))/sqrt(c+a^2*c*x^2),x,0,Unintegrable(sqrt(arctan(a*x))/sqrt(c+a^2*c*x^2),x)],
[sqrt(arctan(a*x))/(x*sqrt(c+a^2*c*x^2)),x,0,Unintegrable(sqrt(arctan(a*x))/(x*sqrt(c+a^2*c*x^2)),x)],
[sqrt(arctan(a*x))/(x^2*sqrt(c+a^2*c*x^2)),x,1,-sqrt(c+a^2*c*x^2)*sqrt(arctan(a*x))/(c*x)+1/2*a*Unintegrable(1/(x*sqrt(c+a^2*c*x^2)*sqrt(arctan(a*x))),x)],
[sqrt(arctan(a*x))/(x^3*sqrt(c+a^2*c*x^2)),x,1,-1/2*sqrt(c+a^2*c*x^2)*sqrt(arctan(a*x))/(c*x^2)+1/4*a*Unintegrable(1/(x^2*sqrt(c+a^2*c*x^2)*sqrt(arctan(a*x))),x)-1/2*a^2*Unintegrable(sqrt(arctan(a*x))/(x*sqrt(c+a^2*c*x^2)),x)],
[sqrt(arctan(a*x))/(x^4*sqrt(c+a^2*c*x^2)),x,2,-1/3*sqrt(c+a^2*c*x^2)*sqrt(arctan(a*x))/(c*x^3)+2/3*a^2*sqrt(c+a^2*c*x^2)*sqrt(arctan(a*x))/(c*x)+1/6*a*Unintegrable(1/(x^3*sqrt(c+a^2*c*x^2)*sqrt(arctan(a*x))),x)-1/3*a^3*Unintegrable(1/(x*sqrt(c+a^2*c*x^2)*sqrt(arctan(a*x))),x)],
[x^m*sqrt(arctan(a*x))/(c+a^2*c*x^2)^(3/2),x,0,Unintegrable(x^m*sqrt(arctan(a*x))/(c+a^2*c*x^2)^(3/2),x)],
[x^3*sqrt(arctan(a*x))/(c+a^2*c*x^2)^(3/2),x,0,Unintegrable(x^3*sqrt(arctan(a*x))/(c+a^2*c*x^2)^(3/2),x)],
[x^2*sqrt(arctan(a*x))/(c+a^2*c*x^2)^(3/2),x,0,Unintegrable(x^2*sqrt(arctan(a*x))/(c+a^2*c*x^2)^(3/2),x)],
[x*sqrt(arctan(a*x))/(c+a^2*c*x^2)^(3/2),x,5,FresnelC(sqrt(2/Pi)*sqrt(arctan(a*x)))*sqrt(1/2*Pi)*sqrt(1+a^2*x^2)/(a^2*c*sqrt(c+a^2*c*x^2))-sqrt(arctan(a*x))/(a^2*c*sqrt(c+a^2*c*x^2))],
[sqrt(arctan(a*x))/(c+a^2*c*x^2)^(3/2),x,5,-FresnelS(sqrt(2/Pi)*sqrt(arctan(a*x)))*sqrt(1/2*Pi)*sqrt(1+a^2*x^2)/(a*c*sqrt(c+a^2*c*x^2))+x*sqrt(arctan(a*x))/(c*sqrt(c+a^2*c*x^2))],
[sqrt(arctan(a*x))/(x*(c+a^2*c*x^2)^(3/2)),x,0,Unintegrable(sqrt(arctan(a*x))/(x*(c+a^2*c*x^2)^(3/2)),x)],
[sqrt(arctan(a*x))/(x^2*(c+a^2*c*x^2)^(3/2)),x,0,Unintegrable(sqrt(arctan(a*x))/(x^2*(c+a^2*c*x^2)^(3/2)),x)],
[x^m*sqrt(arctan(a*x))/(c+a^2*c*x^2)^(5/2),x,0,Unintegrable(x^m*sqrt(arctan(a*x))/(c+a^2*c*x^2)^(5/2),x)],
[x^4*sqrt(arctan(a*x))/(c+a^2*c*x^2)^(5/2),x,0,Unintegrable(x^4*sqrt(arctan(a*x))/(c+a^2*c*x^2)^(5/2),x)],
[x^3*sqrt(arctan(a*x))/(c+a^2*c*x^2)^(5/2),x,10,-1/12*FresnelC(sqrt(6/Pi)*sqrt(arctan(a*x)))*sqrt(1/6*Pi)*sqrt(1+a^2*x^2)/(a^4*c^2*sqrt(c+a^2*c*x^2))+3/4*FresnelC(sqrt(2/Pi)*sqrt(arctan(a*x)))*sqrt(1/2*Pi)*sqrt(1+a^2*x^2)/(a^4*c^2*sqrt(c+a^2*c*x^2))-3/4*sqrt(arctan(a*x))/(a^4*c^2*sqrt(c+a^2*c*x^2))+1/12*cos(3*arctan(a*x))*sqrt(1+a^2*x^2)*sqrt(arctan(a*x))/(a^4*c^2*sqrt(c+a^2*c*x^2))],
[x^2*sqrt(arctan(a*x))/(c+a^2*c*x^2)^(5/2),x,9,1/12*FresnelS(sqrt(6/Pi)*sqrt(arctan(a*x)))*sqrt(1/6*Pi)*sqrt(1+a^2*x^2)/(a^3*c^2*sqrt(c+a^2*c*x^2))-1/4*FresnelS(sqrt(2/Pi)*sqrt(arctan(a*x)))*sqrt(1/2*Pi)*sqrt(1+a^2*x^2)/(a^3*c^2*sqrt(c+a^2*c*x^2))+1/3*x^3*sqrt(arctan(a*x))/(c*(c+a^2*c*x^2)^(3/2))],
[x*sqrt(arctan(a*x))/(c+a^2*c*x^2)^(5/2),x,9,1/12*FresnelC(sqrt(6/Pi)*sqrt(arctan(a*x)))*sqrt(1/6*Pi)*sqrt(1+a^2*x^2)/(a^2*c^2*sqrt(c+a^2*c*x^2))+1/4*FresnelC(sqrt(2/Pi)*sqrt(arctan(a*x)))*sqrt(1/2*Pi)*sqrt(1+a^2*x^2)/(a^2*c^2*sqrt(c+a^2*c*x^2))-1/3*sqrt(arctan(a*x))/(a^2*c*(c+a^2*c*x^2)^(3/2))],
[sqrt(arctan(a*x))/(c+a^2*c*x^2)^(5/2),x,10,-1/12*FresnelS(sqrt(6/Pi)*sqrt(arctan(a*x)))*sqrt(1/6*Pi)*sqrt(1+a^2*x^2)/(a*c^2*sqrt(c+a^2*c*x^2))-3/4*FresnelS(sqrt(2/Pi)*sqrt(arctan(a*x)))*sqrt(1/2*Pi)*sqrt(1+a^2*x^2)/(a*c^2*sqrt(c+a^2*c*x^2))+3/4*x*sqrt(arctan(a*x))/(c^2*sqrt(c+a^2*c*x^2))+1/12*sin(3*arctan(a*x))*sqrt(1+a^2*x^2)*sqrt(arctan(a*x))/(a*c^2*sqrt(c+a^2*c*x^2))],
[sqrt(arctan(a*x))/(x*(c+a^2*c*x^2)^(5/2)),x,0,Unintegrable(sqrt(arctan(a*x))/(x*(c+a^2*c*x^2)^(5/2)),x)],

# Integrands of the form (f x)^m (d+c^2 d x^2)^q (a+b ArcTan[c x])^(3/2)

# Integrands of the form x^m (c+d x^2)^p ArcTan[a x]^(3/2) with d=a^2 c

# p>0
[x^m*(c+a^2*c*x^2)*arctan(a*x)^(3/2),x,0,Unintegrable(x^m*(c+a^2*c*x^2)*arctan(a*x)^(3/2),x)],
[x^2*(c+a^2*c*x^2)*arctan(a*x)^(3/2),x,0,Unintegrable(x^2*(c+a^2*c*x^2)*arctan(a*x)^(3/2),x)],
[x*(c+a^2*c*x^2)*arctan(a*x)^(3/2),x,1,1/4*c*(1+a^2*x^2)^2*arctan(a*x)^(3/2)/a^2-3/8*Unintegrable((c+a^2*c*x^2)*sqrt(arctan(a*x)),x)/a],
[(c+a^2*c*x^2)*arctan(a*x)^(3/2),x,1,1/3*c*x*(1+a^2*x^2)*arctan(a*x)^(3/2)-1/4*c*(1+a^2*x^2)*sqrt(arctan(a*x))/a+2/3*c*Unintegrable(arctan(a*x)^(3/2),x)+1/8*c*Unintegrable(1/sqrt(arctan(a*x)),x)],
[(c+a^2*c*x^2)*arctan(a*x)^(3/2)/x,x,0,Unintegrable((c+a^2*c*x^2)*arctan(a*x)^(3/2)/x,x)],
[(c+a^2*c*x^2)*arctan(a*x)^(3/2)/x^2,x,0,Unintegrable((c+a^2*c*x^2)*arctan(a*x)^(3/2)/x^2,x)],
[x^m*(c+a^2*c*x^2)^2*arctan(a*x)^(3/2),x,0,Unintegrable(x^m*(c+a^2*c*x^2)^2*arctan(a*x)^(3/2),x)],
[x^2*(c+a^2*c*x^2)^2*arctan(a*x)^(3/2),x,0,Unintegrable(x^2*(c+a^2*c*x^2)^2*arctan(a*x)^(3/2),x)],
[x*(c+a^2*c*x^2)^2*arctan(a*x)^(3/2),x,1,1/6*c^2*(1+a^2*x^2)^3*arctan(a*x)^(3/2)/a^2-1/4*Unintegrable((c+a^2*c*x^2)^2*sqrt(arctan(a*x)),x)/a],
[(c+a^2*c*x^2)^2*arctan(a*x)^(3/2),x,2,4/15*c^2*x*(1+a^2*x^2)*arctan(a*x)^(3/2)+1/5*c^2*x*(1+a^2*x^2)^2*arctan(a*x)^(3/2)-1/5*c^2*(1+a^2*x^2)*sqrt(arctan(a*x))/a-3/40*c^2*(1+a^2*x^2)^2*sqrt(arctan(a*x))/a+8/15*c^2*Unintegrable(arctan(a*x)^(3/2),x)+1/10*c^2*Unintegrable(1/sqrt(arctan(a*x)),x)+3/80*c*Unintegrable((c+a^2*c*x^2)/sqrt(arctan(a*x)),x)],
[(c+a^2*c*x^2)^2*arctan(a*x)^(3/2)/x,x,0,Unintegrable((c+a^2*c*x^2)^2*arctan(a*x)^(3/2)/x,x)],
[(c+a^2*c*x^2)^2*arctan(a*x)^(3/2)/x^2,x,0,Unintegrable((c+a^2*c*x^2)^2*arctan(a*x)^(3/2)/x^2,x)],
[x^m*(c+a^2*c*x^2)^3*arctan(a*x)^(3/2),x,0,Unintegrable(x^m*(c+a^2*c*x^2)^3*arctan(a*x)^(3/2),x)],
[x^2*(c+a^2*c*x^2)^3*arctan(a*x)^(3/2),x,0,Unintegrable(x^2*(c+a^2*c*x^2)^3*arctan(a*x)^(3/2),x)],
[x*(c+a^2*c*x^2)^3*arctan(a*x)^(3/2),x,1,1/8*c^3*(1+a^2*x^2)^4*arctan(a*x)^(3/2)/a^2-3/16*Unintegrable((c+a^2*c*x^2)^3*sqrt(arctan(a*x)),x)/a],
[(c+a^2*c*x^2)^3*arctan(a*x)^(3/2),x,3,8/35*c^3*x*(1+a^2*x^2)*arctan(a*x)^(3/2)+6/35*c^3*x*(1+a^2*x^2)^2*arctan(a*x)^(3/2)+1/7*c^3*x*(1+a^2*x^2)^3*arctan(a*x)^(3/2)-6/35*c^3*(1+a^2*x^2)*sqrt(arctan(a*x))/a-9/140*c^3*(1+a^2*x^2)^2*sqrt(arctan(a*x))/a-1/28*c^3*(1+a^2*x^2)^3*sqrt(arctan(a*x))/a+16/35*c^3*Unintegrable(arctan(a*x)^(3/2),x)+3/35*c^3*Unintegrable(1/sqrt(arctan(a*x)),x)+9/280*c^2*Unintegrable((c+a^2*c*x^2)/sqrt(arctan(a*x)),x)+1/56*c*Unintegrable((c+a^2*c*x^2)^2/sqrt(arctan(a*x)),x)],
[(c+a^2*c*x^2)^3*arctan(a*x)^(3/2)/x,x,0,Unintegrable((c+a^2*c*x^2)^3*arctan(a*x)^(3/2)/x,x)],
[(c+a^2*c*x^2)^3*arctan(a*x)^(3/2)/x^2,x,0,Unintegrable((c+a^2*c*x^2)^3*arctan(a*x)^(3/2)/x^2,x)],

# p<0
[x^m*arctan(a*x)^(3/2)/(c+a^2*c*x^2),x,0,Unintegrable(x^m*arctan(a*x)^(3/2)/(c+a^2*c*x^2),x)],
[x^3*arctan(a*x)^(3/2)/(c+a^2*c*x^2),x,2,-2/5*x*arctan(a*x)^(5/2)/(a^3*c)+Unintegrable(x*arctan(a*x)^(3/2),x)/(a^2*c)+2/5*Unintegrable(arctan(a*x)^(5/2),x)/(a^3*c)],
[x^2*arctan(a*x)^(3/2)/(c+a^2*c*x^2),x,2,-2/5*arctan(a*x)^(5/2)/(a^3*c)+Unintegrable(arctan(a*x)^(3/2),x)/(a^2*c)],
[x*arctan(a*x)^(3/2)/(c+a^2*c*x^2),x,1,2/5*x*arctan(a*x)^(5/2)/(a*c)-2/5*Unintegrable(arctan(a*x)^(5/2),x)/(a*c)],
[arctan(a*x)^(3/2)/(c+a^2*c*x^2),x,1,2/5*arctan(a*x)^(5/2)/(a*c)],
[arctan(a*x)^(3/2)/(x*(c+a^2*c*x^2)),x,1,-2/5*I*arctan(a*x)^(5/2)/c+I*Unintegrable(arctan(a*x)^(3/2)/(x*(I+a*x)),x)/c],
[arctan(a*x)^(3/2)/(x^2*(c+a^2*c*x^2)),x,2,-2/5*a*arctan(a*x)^(5/2)/c+Unintegrable(arctan(a*x)^(3/2)/x^2,x)/c],
[arctan(a*x)^(3/2)/(x^3*(c+a^2*c*x^2)),x,2,2/5*I*a^2*arctan(a*x)^(5/2)/c+Unintegrable(arctan(a*x)^(3/2)/x^3,x)/c-I*a^2*Unintegrable(arctan(a*x)^(3/2)/(x*(I+a*x)),x)/c],
[arctan(a*x)^(3/2)/(x^4*(c+a^2*c*x^2)),x,3,2/5*a^3*arctan(a*x)^(5/2)/c+Unintegrable(arctan(a*x)^(3/2)/x^4,x)/c-a^2*Unintegrable(arctan(a*x)^(3/2)/x^2,x)/c],
[x^m*arctan(a*x)^(3/2)/(c+a^2*c*x^2)^2,x,0,Unintegrable(x^m*arctan(a*x)^(3/2)/(c+a^2*c*x^2)^2,x)],
[x^3*arctan(a*x)^(3/2)/(c+a^2*c*x^2)^2,x,0,Unintegrable(x^3*arctan(a*x)^(3/2)/(c+a^2*c*x^2)^2,x)],
[x^2*arctan(a*x)^(3/2)/(c+a^2*c*x^2)^2,x,7,-1/2*x*arctan(a*x)^(3/2)/(a^2*c^2*(1+a^2*x^2))+1/5*arctan(a*x)^(5/2)/(a^3*c^2)+3/32*FresnelC(2*sqrt(arctan(a*x))/sqrt(Pi))*sqrt(Pi)/(a^3*c^2)+3/16*sqrt(arctan(a*x))/(a^3*c^2)-3/8*sqrt(arctan(a*x))/(a^3*c^2*(1+a^2*x^2))],
[x*arctan(a*x)^(3/2)/(c+a^2*c*x^2)^2,x,7,1/4*arctan(a*x)^(3/2)/(a^2*c^2)-1/2*arctan(a*x)^(3/2)/(a^2*c^2*(1+a^2*x^2))-3/32*FresnelS(2*sqrt(arctan(a*x))/sqrt(Pi))*sqrt(Pi)/(a^2*c^2)+3/8*x*sqrt(arctan(a*x))/(a*c^2*(1+a^2*x^2))],
[arctan(a*x)^(3/2)/(c+a^2*c*x^2)^2,x,7,1/2*x*arctan(a*x)^(3/2)/(c^2*(1+a^2*x^2))+1/5*arctan(a*x)^(5/2)/(a*c^2)-3/32*FresnelC(2*sqrt(arctan(a*x))/sqrt(Pi))*sqrt(Pi)/(a*c^2)-3/16*sqrt(arctan(a*x))/(a*c^2)+3/8*sqrt(arctan(a*x))/(a*c^2*(1+a^2*x^2))],
[arctan(a*x)^(3/2)/(x*(c+a^2*c*x^2)^2),x,0,Unintegrable(arctan(a*x)^(3/2)/(x*(c+a^2*c*x^2)^2),x)],
[x^m*arctan(a*x)^(3/2)/(c+a^2*c*x^2)^3,x,0,Unintegrable(x^m*arctan(a*x)^(3/2)/(c+a^2*c*x^2)^3,x)],
[x^5*arctan(a*x)^(3/2)/(c+a^2*c*x^2)^3,x,0,Unintegrable(x^5*arctan(a*x)^(3/2)/(c+a^2*c*x^2)^3,x)],
[x^4*arctan(a*x)^(3/2)/(c+a^2*c*x^2)^3,x,15,-1/4*x^3*arctan(a*x)^(3/2)/(a^2*c^3*(1+a^2*x^2)^2)-3/8*x*arctan(a*x)^(3/2)/(a^4*c^3*(1+a^2*x^2))+3/20*arctan(a*x)^(5/2)/(a^5*c^3)-3/512*FresnelC(2*sqrt(2/Pi)*sqrt(arctan(a*x)))*sqrt(1/2*Pi)/(a^5*c^3)+3/32*FresnelC(2*sqrt(arctan(a*x))/sqrt(Pi))*sqrt(Pi)/(a^5*c^3)+27/256*sqrt(arctan(a*x))/(a^5*c^3)+3/32*x^4*sqrt(arctan(a*x))/(a*c^3*(1+a^2*x^2)^2)-9/32*sqrt(arctan(a*x))/(a^5*c^3*(1+a^2*x^2))],
[x^3*arctan(a*x)^(3/2)/(c+a^2*c*x^2)^3,x,10,-3/32*arctan(a*x)^(3/2)/(a^4*c^3)+1/4*x^4*arctan(a*x)^(3/2)/(c^3*(1+a^2*x^2)^2)+3/512*FresnelS(2*sqrt(2/Pi)*sqrt(arctan(a*x)))*sqrt(1/2*Pi)/(a^4*c^3)-3/64*FresnelS(2*sqrt(arctan(a*x))/sqrt(Pi))*sqrt(Pi)/(a^4*c^3)+3/32*sin(2*arctan(a*x))*sqrt(arctan(a*x))/(a^4*c^3)-3/256*sin(4*arctan(a*x))*sqrt(arctan(a*x))/(a^4*c^3)],
[x^2*arctan(a*x)^(3/2)/(c+a^2*c*x^2)^3,x,7,1/20*arctan(a*x)^(5/2)/(a^3*c^3)-1/32*arctan(a*x)^(3/2)*sin(4*arctan(a*x))/(a^3*c^3)+3/512*FresnelC(2*sqrt(2/Pi)*sqrt(arctan(a*x)))*sqrt(1/2*Pi)/(a^3*c^3)-3/256*cos(4*arctan(a*x))*sqrt(arctan(a*x))/(a^3*c^3)],
[x*arctan(a*x)^(3/2)/(c+a^2*c*x^2)^3,x,10,3/32*arctan(a*x)^(3/2)/(a^2*c^3)-1/4*arctan(a*x)^(3/2)/(a^2*c^3*(1+a^2*x^2)^2)-3/512*FresnelS(2*sqrt(2/Pi)*sqrt(arctan(a*x)))*sqrt(1/2*Pi)/(a^2*c^3)-3/64*FresnelS(2*sqrt(arctan(a*x))/sqrt(Pi))*sqrt(Pi)/(a^2*c^3)+3/32*sin(2*arctan(a*x))*sqrt(arctan(a*x))/(a^2*c^3)+3/256*sin(4*arctan(a*x))*sqrt(arctan(a*x))/(a^2*c^3)],
[arctan(a*x)^(3/2)/(c+a^2*c*x^2)^3,x,15,1/4*x*arctan(a*x)^(3/2)/(c^3*(1+a^2*x^2)^2)+3/8*x*arctan(a*x)^(3/2)/(c^3*(1+a^2*x^2))+3/20*arctan(a*x)^(5/2)/(a*c^3)-3/512*FresnelC(2*sqrt(2/Pi)*sqrt(arctan(a*x)))*sqrt(1/2*Pi)/(a*c^3)-3/32*FresnelC(2*sqrt(arctan(a*x))/sqrt(Pi))*sqrt(Pi)/(a*c^3)-45/256*sqrt(arctan(a*x))/(a*c^3)+3/32*sqrt(arctan(a*x))/(a*c^3*(1+a^2*x^2)^2)+9/32*sqrt(arctan(a*x))/(a*c^3*(1+a^2*x^2))],
[arctan(a*x)^(3/2)/(x*(c+a^2*c*x^2)^3),x,0,Unintegrable(arctan(a*x)^(3/2)/(x*(c+a^2*c*x^2)^3),x)],

# Integrands of the form x^m (c+d x^2)^(p/2) ArcTan[a x]^(3/2) with d=a^2 c

# p>0
[x^m*arctan(a*x)^(3/2)*sqrt(c+a^2*c*x^2),x,0,Unintegrable(x^m*arctan(a*x)^(3/2)*sqrt(c+a^2*c*x^2),x)],
[x^2*arctan(a*x)^(3/2)*sqrt(c+a^2*c*x^2),x,0,Unintegrable(x^2*arctan(a*x)^(3/2)*sqrt(c+a^2*c*x^2),x)],
[x*arctan(a*x)^(3/2)*sqrt(c+a^2*c*x^2),x,1,1/3*(c+a^2*c*x^2)^(3/2)*arctan(a*x)^(3/2)/(a^2*c)-1/2*Unintegrable(sqrt(c+a^2*c*x^2)*sqrt(arctan(a*x)),x)/a],
[arctan(a*x)^(3/2)*sqrt(c+a^2*c*x^2),x,1,1/2*x*arctan(a*x)^(3/2)*sqrt(c+a^2*c*x^2)-3/4*sqrt(c+a^2*c*x^2)*sqrt(arctan(a*x))/a+1/2*c*Unintegrable(arctan(a*x)^(3/2)/sqrt(c+a^2*c*x^2),x)+3/8*c*Unintegrable(1/(sqrt(c+a^2*c*x^2)*sqrt(arctan(a*x))),x)],
[arctan(a*x)^(3/2)*sqrt(c+a^2*c*x^2)/x,x,0,Unintegrable(arctan(a*x)^(3/2)*sqrt(c+a^2*c*x^2)/x,x)],
[x^m*(c+a^2*c*x^2)^(3/2)*arctan(a*x)^(3/2),x,0,Unintegrable(x^m*(c+a^2*c*x^2)^(3/2)*arctan(a*x)^(3/2),x)],
[x^2*(c+a^2*c*x^2)^(3/2)*arctan(a*x)^(3/2),x,0,Unintegrable(x^2*(c+a^2*c*x^2)^(3/2)*arctan(a*x)^(3/2),x)],
[x*(c+a^2*c*x^2)^(3/2)*arctan(a*x)^(3/2),x,1,1/5*(c+a^2*c*x^2)^(5/2)*arctan(a*x)^(3/2)/(a^2*c)-3/10*Unintegrable((c+a^2*c*x^2)^(3/2)*sqrt(arctan(a*x)),x)/a],
[(c+a^2*c*x^2)^(3/2)*arctan(a*x)^(3/2),x,2,1/4*x*(c+a^2*c*x^2)^(3/2)*arctan(a*x)^(3/2)+3/8*c*x*arctan(a*x)^(3/2)*sqrt(c+a^2*c*x^2)-1/8*(c+a^2*c*x^2)^(3/2)*sqrt(arctan(a*x))/a-9/16*c*sqrt(c+a^2*c*x^2)*sqrt(arctan(a*x))/a+3/8*c^2*Unintegrable(arctan(a*x)^(3/2)/sqrt(c+a^2*c*x^2),x)+9/32*c^2*Unintegrable(1/(sqrt(c+a^2*c*x^2)*sqrt(arctan(a*x))),x)+1/16*c*Unintegrable(sqrt(c+a^2*c*x^2)/sqrt(arctan(a*x)),x)],
[(c+a^2*c*x^2)^(3/2)*arctan(a*x)^(3/2)/x,x,0,Unintegrable((c+a^2*c*x^2)^(3/2)*arctan(a*x)^(3/2)/x,x)],
[x^m*(c+a^2*c*x^2)^(5/2)*arctan(a*x)^(3/2),x,0,Unintegrable(x^m*(c+a^2*c*x^2)^(5/2)*arctan(a*x)^(3/2),x)],
[x^2*(c+a^2*c*x^2)^(5/2)*arctan(a*x)^(3/2),x,0,Unintegrable(x^2*(c+a^2*c*x^2)^(5/2)*arctan(a*x)^(3/2),x)],
[x*(c+a^2*c*x^2)^(5/2)*arctan(a*x)^(3/2),x,1,1/7*(c+a^2*c*x^2)^(7/2)*arctan(a*x)^(3/2)/(a^2*c)-3/14*Unintegrable((c+a^2*c*x^2)^(5/2)*sqrt(arctan(a*x)),x)/a],
[(c+a^2*c*x^2)^(5/2)*arctan(a*x)^(3/2),x,3,5/24*c*x*(c+a^2*c*x^2)^(3/2)*arctan(a*x)^(3/2)+1/6*x*(c+a^2*c*x^2)^(5/2)*arctan(a*x)^(3/2)+5/16*c^2*x*arctan(a*x)^(3/2)*sqrt(c+a^2*c*x^2)-5/48*c*(c+a^2*c*x^2)^(3/2)*sqrt(arctan(a*x))/a-1/20*(c+a^2*c*x^2)^(5/2)*sqrt(arctan(a*x))/a-15/32*c^2*sqrt(c+a^2*c*x^2)*sqrt(arctan(a*x))/a+5/16*c^3*Unintegrable(arctan(a*x)^(3/2)/sqrt(c+a^2*c*x^2),x)+1/40*c*Unintegrable((c+a^2*c*x^2)^(3/2)/sqrt(arctan(a*x)),x)+15/64*c^3*Unintegrable(1/(sqrt(c+a^2*c*x^2)*sqrt(arctan(a*x))),x)+5/96*c^2*Unintegrable(sqrt(c+a^2*c*x^2)/sqrt(arctan(a*x)),x)],
[(c+a^2*c*x^2)^(5/2)*arctan(a*x)^(3/2)/x,x,0,Unintegrable((c+a^2*c*x^2)^(5/2)*arctan(a*x)^(3/2)/x,x)],

# p<0
[x^m*arctan(a*x)^(3/2)/sqrt(c+a^2*c*x^2),x,0,Unintegrable(x^m*arctan(a*x)^(3/2)/sqrt(c+a^2*c*x^2),x)],
[x^3*arctan(a*x)^(3/2)/sqrt(c+a^2*c*x^2),x,3,-2/3*arctan(a*x)^(3/2)*sqrt(c+a^2*c*x^2)/(a^4*c)+1/3*x^2*arctan(a*x)^(3/2)*sqrt(c+a^2*c*x^2)/(a^2*c)-1/4*x*sqrt(c+a^2*c*x^2)*sqrt(arctan(a*x))/(a^3*c)+1/8*Unintegrable(x/(sqrt(c+a^2*c*x^2)*sqrt(arctan(a*x))),x)/a^2+5/4*Unintegrable(sqrt(arctan(a*x))/sqrt(c+a^2*c*x^2),x)/a^3],
[x^2*arctan(a*x)^(3/2)/sqrt(c+a^2*c*x^2),x,2,1/2*x*arctan(a*x)^(3/2)*sqrt(c+a^2*c*x^2)/(a^2*c)-3/4*sqrt(c+a^2*c*x^2)*sqrt(arctan(a*x))/(a^3*c)-1/2*Unintegrable(arctan(a*x)^(3/2)/sqrt(c+a^2*c*x^2),x)/a^2+3/8*Unintegrable(1/(sqrt(c+a^2*c*x^2)*sqrt(arctan(a*x))),x)/a^2],
[x*arctan(a*x)^(3/2)/sqrt(c+a^2*c*x^2),x,1,arctan(a*x)^(3/2)*sqrt(c+a^2*c*x^2)/(a^2*c)-3/2*Unintegrable(sqrt(arctan(a*x))/sqrt(c+a^2*c*x^2),x)/a],
[arctan(a*x)^(3/2)/sqrt(c+a^2*c*x^2),x,0,Unintegrable(arctan(a*x)^(3/2)/sqrt(c+a^2*c*x^2),x)],
[arctan(a*x)^(3/2)/(x*sqrt(c+a^2*c*x^2)),x,0,Unintegrable(arctan(a*x)^(3/2)/(x*sqrt(c+a^2*c*x^2)),x)],
[arctan(a*x)^(3/2)/(x^2*sqrt(c+a^2*c*x^2)),x,1,-arctan(a*x)^(3/2)*sqrt(c+a^2*c*x^2)/(c*x)+3/2*a*Unintegrable(sqrt(arctan(a*x))/(x*sqrt(c+a^2*c*x^2)),x)],
[arctan(a*x)^(3/2)/(x^3*sqrt(c+a^2*c*x^2)),x,2,-1/2*arctan(a*x)^(3/2)*sqrt(c+a^2*c*x^2)/(c*x^2)-3/4*a*sqrt(c+a^2*c*x^2)*sqrt(arctan(a*x))/(c*x)-1/2*a^2*Unintegrable(arctan(a*x)^(3/2)/(x*sqrt(c+a^2*c*x^2)),x)+3/8*a^2*Unintegrable(1/(x*sqrt(c+a^2*c*x^2)*sqrt(arctan(a*x))),x)],
[arctan(a*x)^(3/2)/(x^4*sqrt(c+a^2*c*x^2)),x,3,-1/3*arctan(a*x)^(3/2)*sqrt(c+a^2*c*x^2)/(c*x^3)+2/3*a^2*arctan(a*x)^(3/2)*sqrt(c+a^2*c*x^2)/(c*x)-1/4*a*sqrt(c+a^2*c*x^2)*sqrt(arctan(a*x))/(c*x^2)+1/8*a^2*Unintegrable(1/(x^2*sqrt(c+a^2*c*x^2)*sqrt(arctan(a*x))),x)-5/4*a^3*Unintegrable(sqrt(arctan(a*x))/(x*sqrt(c+a^2*c*x^2)),x)],
[x^m*arctan(a*x)^(3/2)/(c+a^2*c*x^2)^(3/2),x,0,Unintegrable(x^m*arctan(a*x)^(3/2)/(c+a^2*c*x^2)^(3/2),x)],
[x^3*arctan(a*x)^(3/2)/(c+a^2*c*x^2)^(3/2),x,0,Unintegrable(x^3*arctan(a*x)^(3/2)/(c+a^2*c*x^2)^(3/2),x)],
[x^2*arctan(a*x)^(3/2)/(c+a^2*c*x^2)^(3/2),x,0,Unintegrable(x^2*arctan(a*x)^(3/2)/(c+a^2*c*x^2)^(3/2),x)],
[x*arctan(a*x)^(3/2)/(c+a^2*c*x^2)^(3/2),x,6,-arctan(a*x)^(3/2)/(a^2*c*sqrt(c+a^2*c*x^2))-3/2*FresnelS(sqrt(2/Pi)*sqrt(arctan(a*x)))*sqrt(1/2*Pi)*sqrt(1+a^2*x^2)/(a^2*c*sqrt(c+a^2*c*x^2))+3/2*x*sqrt(arctan(a*x))/(a*c*sqrt(c+a^2*c*x^2))],
[arctan(a*x)^(3/2)/(c+a^2*c*x^2)^(3/2),x,5,x*arctan(a*x)^(3/2)/(c*sqrt(c+a^2*c*x^2))-3/2*FresnelC(sqrt(2/Pi)*sqrt(arctan(a*x)))*sqrt(1/2*Pi)*sqrt(1+a^2*x^2)/(a*c*sqrt(c+a^2*c*x^2))+3/2*sqrt(arctan(a*x))/(a*c*sqrt(c+a^2*c*x^2))],
[arctan(a*x)^(3/2)/(x*(c+a^2*c*x^2)^(3/2)),x,0,Unintegrable(arctan(a*x)^(3/2)/(x*(c+a^2*c*x^2)^(3/2)),x)],
[arctan(a*x)^(3/2)/(x^2*(c+a^2*c*x^2)^(3/2)),x,0,Unintegrable(arctan(a*x)^(3/2)/(x^2*(c+a^2*c*x^2)^(3/2)),x)],
[x^m*arctan(a*x)^(3/2)/(c+a^2*c*x^2)^(5/2),x,0,Unintegrable(x^m*arctan(a*x)^(3/2)/(c+a^2*c*x^2)^(5/2),x)],
[x^5*arctan(a*x)^(3/2)/(c+a^2*c*x^2)^(5/2),x,0,Unintegrable(x^5*arctan(a*x)^(3/2)/(c+a^2*c*x^2)^(5/2),x)],
[x^4*arctan(a*x)^(3/2)/(c+a^2*c*x^2)^(5/2),x,0,Unintegrable(x^4*arctan(a*x)^(3/2)/(c+a^2*c*x^2)^(5/2),x)],
[x^3*arctan(a*x)^(3/2)/(c+a^2*c*x^2)^(5/2),x,15,-1/3*x^2*arctan(a*x)^(3/2)/(a^2*c*(c+a^2*c*x^2)^(3/2))-2/3*arctan(a*x)^(3/2)/(a^4*c^2*sqrt(c+a^2*c*x^2))+1/24*FresnelS(sqrt(6/Pi)*sqrt(arctan(a*x)))*sqrt(1/6*Pi)*sqrt(1+a^2*x^2)/(a^4*c^2*sqrt(c+a^2*c*x^2))-9/8*FresnelS(sqrt(2/Pi)*sqrt(arctan(a*x)))*sqrt(1/2*Pi)*sqrt(1+a^2*x^2)/(a^4*c^2*sqrt(c+a^2*c*x^2))+1/6*x^3*sqrt(arctan(a*x))/(a*c*(c+a^2*c*x^2)^(3/2))+x*sqrt(arctan(a*x))/(a^3*c^2*sqrt(c+a^2*c*x^2))],
[x^2*arctan(a*x)^(3/2)/(c+a^2*c*x^2)^(5/2),x,11,1/3*x^3*arctan(a*x)^(3/2)/(c*(c+a^2*c*x^2)^(3/2))+1/24*FresnelC(sqrt(6/Pi)*sqrt(arctan(a*x)))*sqrt(1/6*Pi)*sqrt(1+a^2*x^2)/(a^3*c^2*sqrt(c+a^2*c*x^2))-3/8*FresnelC(sqrt(2/Pi)*sqrt(arctan(a*x)))*sqrt(1/2*Pi)*sqrt(1+a^2*x^2)/(a^3*c^2*sqrt(c+a^2*c*x^2))+3/8*sqrt(arctan(a*x))/(a^3*c^2*sqrt(c+a^2*c*x^2))-1/24*cos(3*arctan(a*x))*sqrt(1+a^2*x^2)*sqrt(arctan(a*x))/(a^3*c^2*sqrt(c+a^2*c*x^2))],
[x*arctan(a*x)^(3/2)/(c+a^2*c*x^2)^(5/2),x,11,-1/3*arctan(a*x)^(3/2)/(a^2*c*(c+a^2*c*x^2)^(3/2))-1/24*FresnelS(sqrt(6/Pi)*sqrt(arctan(a*x)))*sqrt(1/6*Pi)*sqrt(1+a^2*x^2)/(a^2*c^2*sqrt(c+a^2*c*x^2))-3/8*FresnelS(sqrt(2/Pi)*sqrt(arctan(a*x)))*sqrt(1/2*Pi)*sqrt(1+a^2*x^2)/(a^2*c^2*sqrt(c+a^2*c*x^2))+3/8*x*sqrt(arctan(a*x))/(a*c^2*sqrt(c+a^2*c*x^2))+1/24*sin(3*arctan(a*x))*sqrt(1+a^2*x^2)*sqrt(arctan(a*x))/(a^2*c^2*sqrt(c+a^2*c*x^2))],
[arctan(a*x)^(3/2)/(c+a^2*c*x^2)^(5/2),x,14,1/3*x*arctan(a*x)^(3/2)/(c*(c+a^2*c*x^2)^(3/2))+2/3*x*arctan(a*x)^(3/2)/(c^2*sqrt(c+a^2*c*x^2))-1/24*FresnelC(sqrt(6/Pi)*sqrt(arctan(a*x)))*sqrt(1/6*Pi)*sqrt(1+a^2*x^2)/(a*c^2*sqrt(c+a^2*c*x^2))-9/8*FresnelC(sqrt(2/Pi)*sqrt(arctan(a*x)))*sqrt(1/2*Pi)*sqrt(1+a^2*x^2)/(a*c^2*sqrt(c+a^2*c*x^2))+1/6*sqrt(arctan(a*x))/(a*c*(c+a^2*c*x^2)^(3/2))+sqrt(arctan(a*x))/(a*c^2*sqrt(c+a^2*c*x^2))],
[arctan(a*x)^(3/2)/(x*(c+a^2*c*x^2)^(5/2)),x,0,Unintegrable(arctan(a*x)^(3/2)/(x*(c+a^2*c*x^2)^(5/2)),x)],
[arctan(a*x)^(3/2)/(x^2*(c+a^2*c*x^2)^(5/2)),x,0,Unintegrable(arctan(a*x)^(3/2)/(x^2*(c+a^2*c*x^2)^(5/2)),x)],

# Integrands of the form (f x)^m (d+c^2 d x^2)^q (a+b ArcTan[c x])^(5/2)

# Integrands of the form x^m (c+d x^2)^p ArcTan[a x]^(5/2) with d=a^2 c

# p>0
[x^m*(c+a^2*c*x^2)*arctan(a*x)^(5/2),x,0,Unintegrable(x^m*(c+a^2*c*x^2)*arctan(a*x)^(5/2),x)],
[x^2*(c+a^2*c*x^2)*arctan(a*x)^(5/2),x,0,Unintegrable(x^2*(c+a^2*c*x^2)*arctan(a*x)^(5/2),x)],
[x*(c+a^2*c*x^2)*arctan(a*x)^(5/2),x,2,-5/24*c*x*(1+a^2*x^2)*arctan(a*x)^(3/2)/a+1/4*c*(1+a^2*x^2)^2*arctan(a*x)^(5/2)/a^2+5/32*c*(1+a^2*x^2)*sqrt(arctan(a*x))/a^2-5/12*c*Unintegrable(arctan(a*x)^(3/2),x)/a-5/64*c*Unintegrable(1/sqrt(arctan(a*x)),x)/a],
[(c+a^2*c*x^2)*arctan(a*x)^(5/2),x,1,-5/12*c*(1+a^2*x^2)*arctan(a*x)^(3/2)/a+1/3*c*x*(1+a^2*x^2)*arctan(a*x)^(5/2)+2/3*c*Unintegrable(arctan(a*x)^(5/2),x)+5/8*c*Unintegrable(sqrt(arctan(a*x)),x)],
[(c+a^2*c*x^2)*arctan(a*x)^(5/2)/x,x,0,Unintegrable((c+a^2*c*x^2)*arctan(a*x)^(5/2)/x,x)],
[(c+a^2*c*x^2)*arctan(a*x)^(5/2)/x^2,x,0,Unintegrable((c+a^2*c*x^2)*arctan(a*x)^(5/2)/x^2,x)],
[x^m*(c+a^2*c*x^2)^2*arctan(a*x)^(5/2),x,0,Unintegrable(x^m*(c+a^2*c*x^2)^2*arctan(a*x)^(5/2),x)],
[x^2*(c+a^2*c*x^2)^2*arctan(a*x)^(5/2),x,0,Unintegrable(x^2*(c+a^2*c*x^2)^2*arctan(a*x)^(5/2),x)],
[x*(c+a^2*c*x^2)^2*arctan(a*x)^(5/2),x,3,-1/9*c^2*x*(1+a^2*x^2)*arctan(a*x)^(3/2)/a-1/12*c^2*x*(1+a^2*x^2)^2*arctan(a*x)^(3/2)/a+1/6*c^2*(1+a^2*x^2)^3*arctan(a*x)^(5/2)/a^2+1/12*c^2*(1+a^2*x^2)*sqrt(arctan(a*x))/a^2+1/32*c^2*(1+a^2*x^2)^2*sqrt(arctan(a*x))/a^2-2/9*c^2*Unintegrable(arctan(a*x)^(3/2),x)/a-1/24*c^2*Unintegrable(1/sqrt(arctan(a*x)),x)/a-1/64*c*Unintegrable((c+a^2*c*x^2)/sqrt(arctan(a*x)),x)/a],
[(c+a^2*c*x^2)^2*arctan(a*x)^(5/2),x,2,-1/3*c^2*(1+a^2*x^2)*arctan(a*x)^(3/2)/a-1/8*c^2*(1+a^2*x^2)^2*arctan(a*x)^(3/2)/a+4/15*c^2*x*(1+a^2*x^2)*arctan(a*x)^(5/2)+1/5*c^2*x*(1+a^2*x^2)^2*arctan(a*x)^(5/2)+8/15*c^2*Unintegrable(arctan(a*x)^(5/2),x)+1/2*c^2*Unintegrable(sqrt(arctan(a*x)),x)+3/16*c*Unintegrable((c+a^2*c*x^2)*sqrt(arctan(a*x)),x)],
[(c+a^2*c*x^2)^2*arctan(a*x)^(5/2)/x,x,0,Unintegrable((c+a^2*c*x^2)^2*arctan(a*x)^(5/2)/x,x)],
[(c+a^2*c*x^2)^2*arctan(a*x)^(5/2)/x^2,x,0,Unintegrable((c+a^2*c*x^2)^2*arctan(a*x)^(5/2)/x^2,x)],
[x^m*(c+a^2*c*x^2)^3*arctan(a*x)^(5/2),x,0,Unintegrable(x^m*(c+a^2*c*x^2)^3*arctan(a*x)^(5/2),x)],
[x^2*(c+a^2*c*x^2)^3*arctan(a*x)^(5/2),x,0,Unintegrable(x^2*(c+a^2*c*x^2)^3*arctan(a*x)^(5/2),x)],
[x*(c+a^2*c*x^2)^3*arctan(a*x)^(5/2),x,4,-1/14*c^3*x*(1+a^2*x^2)*arctan(a*x)^(3/2)/a-3/56*c^3*x*(1+a^2*x^2)^2*arctan(a*x)^(3/2)/a-5/112*c^3*x*(1+a^2*x^2)^3*arctan(a*x)^(3/2)/a+1/8*c^3*(1+a^2*x^2)^4*arctan(a*x)^(5/2)/a^2+3/56*c^3*(1+a^2*x^2)*sqrt(arctan(a*x))/a^2+9/448*c^3*(1+a^2*x^2)^2*sqrt(arctan(a*x))/a^2+5/448*c^3*(1+a^2*x^2)^3*sqrt(arctan(a*x))/a^2-1/7*c^3*Unintegrable(arctan(a*x)^(3/2),x)/a-3/112*c^3*Unintegrable(1/sqrt(arctan(a*x)),x)/a-9/896*c^2*Unintegrable((c+a^2*c*x^2)/sqrt(arctan(a*x)),x)/a-5/896*c*Unintegrable((c+a^2*c*x^2)^2/sqrt(arctan(a*x)),x)/a],
[(c+a^2*c*x^2)^3*arctan(a*x)^(5/2),x,3,-2/7*c^3*(1+a^2*x^2)*arctan(a*x)^(3/2)/a-3/28*c^3*(1+a^2*x^2)^2*arctan(a*x)^(3/2)/a-5/84*c^3*(1+a^2*x^2)^3*arctan(a*x)^(3/2)/a+8/35*c^3*x*(1+a^2*x^2)*arctan(a*x)^(5/2)+6/35*c^3*x*(1+a^2*x^2)^2*arctan(a*x)^(5/2)+1/7*c^3*x*(1+a^2*x^2)^3*arctan(a*x)^(5/2)+16/35*c^3*Unintegrable(arctan(a*x)^(5/2),x)+3/7*c^3*Unintegrable(sqrt(arctan(a*x)),x)+9/56*c^2*Unintegrable((c+a^2*c*x^2)*sqrt(arctan(a*x)),x)+5/56*c*Unintegrable((c+a^2*c*x^2)^2*sqrt(arctan(a*x)),x)],
[(c+a^2*c*x^2)^3*arctan(a*x)^(5/2)/x,x,0,Unintegrable((c+a^2*c*x^2)^3*arctan(a*x)^(5/2)/x,x)],
[(c+a^2*c*x^2)^3*arctan(a*x)^(5/2)/x^2,x,0,Unintegrable((c+a^2*c*x^2)^3*arctan(a*x)^(5/2)/x^2,x)],

# p<0
[x^m*arctan(a*x)^(5/2)/(c+a^2*c*x^2),x,0,Unintegrable(x^m*arctan(a*x)^(5/2)/(c+a^2*c*x^2),x)],
[x^3*arctan(a*x)^(5/2)/(c+a^2*c*x^2),x,2,-2/7*x*arctan(a*x)^(7/2)/(a^3*c)+Unintegrable(x*arctan(a*x)^(5/2),x)/(a^2*c)+2/7*Unintegrable(arctan(a*x)^(7/2),x)/(a^3*c)],
[x^2*arctan(a*x)^(5/2)/(c+a^2*c*x^2),x,2,-2/7*arctan(a*x)^(7/2)/(a^3*c)+Unintegrable(arctan(a*x)^(5/2),x)/(a^2*c)],
[x*arctan(a*x)^(5/2)/(c+a^2*c*x^2),x,1,2/7*x*arctan(a*x)^(7/2)/(a*c)-2/7*Unintegrable(arctan(a*x)^(7/2),x)/(a*c)],
[arctan(a*x)^(5/2)/(c+a^2*c*x^2),x,1,2/7*arctan(a*x)^(7/2)/(a*c)],
[arctan(a*x)^(5/2)/(x*(c+a^2*c*x^2)),x,1,-2/7*I*arctan(a*x)^(7/2)/c+I*Unintegrable(arctan(a*x)^(5/2)/(x*(I+a*x)),x)/c],
[arctan(a*x)^(5/2)/(x^2*(c+a^2*c*x^2)),x,2,-2/7*a*arctan(a*x)^(7/2)/c+Unintegrable(arctan(a*x)^(5/2)/x^2,x)/c],
[arctan(a*x)^(5/2)/(x^3*(c+a^2*c*x^2)),x,2,2/7*I*a^2*arctan(a*x)^(7/2)/c+Unintegrable(arctan(a*x)^(5/2)/x^3,x)/c-I*a^2*Unintegrable(arctan(a*x)^(5/2)/(x*(I+a*x)),x)/c],
[arctan(a*x)^(5/2)/(x^4*(c+a^2*c*x^2)),x,3,2/7*a^3*arctan(a*x)^(7/2)/c+Unintegrable(arctan(a*x)^(5/2)/x^4,x)/c-a^2*Unintegrable(arctan(a*x)^(5/2)/x^2,x)/c],
[x^m*arctan(a*x)^(5/2)/(c+a^2*c*x^2)^2,x,0,Unintegrable(x^m*arctan(a*x)^(5/2)/(c+a^2*c*x^2)^2,x)],
[x^3*arctan(a*x)^(5/2)/(c+a^2*c*x^2)^2,x,0,Unintegrable(x^3*arctan(a*x)^(5/2)/(c+a^2*c*x^2)^2,x)],
[x^2*arctan(a*x)^(5/2)/(c+a^2*c*x^2)^2,x,8,5/16*arctan(a*x)^(3/2)/(a^3*c^2)-5/8*arctan(a*x)^(3/2)/(a^3*c^2*(1+a^2*x^2))-1/2*x*arctan(a*x)^(5/2)/(a^2*c^2*(1+a^2*x^2))+1/7*arctan(a*x)^(7/2)/(a^3*c^2)-15/128*FresnelS(2*sqrt(arctan(a*x))/sqrt(Pi))*sqrt(Pi)/(a^3*c^2)+15/32*x*sqrt(arctan(a*x))/(a^2*c^2*(1+a^2*x^2))],
[x*arctan(a*x)^(5/2)/(c+a^2*c*x^2)^2,x,8,5/8*x*arctan(a*x)^(3/2)/(a*c^2*(1+a^2*x^2))+1/4*arctan(a*x)^(5/2)/(a^2*c^2)-1/2*arctan(a*x)^(5/2)/(a^2*c^2*(1+a^2*x^2))-15/128*FresnelC(2*sqrt(arctan(a*x))/sqrt(Pi))*sqrt(Pi)/(a^2*c^2)-15/64*sqrt(arctan(a*x))/(a^2*c^2)+15/32*sqrt(arctan(a*x))/(a^2*c^2*(1+a^2*x^2))],
[arctan(a*x)^(5/2)/(c+a^2*c*x^2)^2,x,8,-5/16*arctan(a*x)^(3/2)/(a*c^2)+5/8*arctan(a*x)^(3/2)/(a*c^2*(1+a^2*x^2))+1/2*x*arctan(a*x)^(5/2)/(c^2*(1+a^2*x^2))+1/7*arctan(a*x)^(7/2)/(a*c^2)+15/128*FresnelS(2*sqrt(arctan(a*x))/sqrt(Pi))*sqrt(Pi)/(a*c^2)-15/32*x*sqrt(arctan(a*x))/(c^2*(1+a^2*x^2))],
[arctan(a*x)^(5/2)/(x*(c+a^2*c*x^2)^2),x,0,Unintegrable(arctan(a*x)^(5/2)/(x*(c+a^2*c*x^2)^2),x)],
[x^m*arctan(a*x)^(5/2)/(c+a^2*c*x^2)^3,x,0,Unintegrable(x^m*arctan(a*x)^(5/2)/(c+a^2*c*x^2)^3,x)],
[x^5*arctan(a*x)^(5/2)/(c+a^2*c*x^2)^3,x,0,Unintegrable(x^5*arctan(a*x)^(5/2)/(c+a^2*c*x^2)^3,x)],
[x^4*arctan(a*x)^(5/2)/(c+a^2*c*x^2)^3,x,18,45/256*arctan(a*x)^(3/2)/(a^5*c^3)+5/32*x^4*arctan(a*x)^(3/2)/(a*c^3*(1+a^2*x^2)^2)-15/32*arctan(a*x)^(3/2)/(a^5*c^3*(1+a^2*x^2))-1/4*x^3*arctan(a*x)^(5/2)/(a^2*c^3*(1+a^2*x^2)^2)-3/8*x*arctan(a*x)^(5/2)/(a^4*c^3*(1+a^2*x^2))+3/28*arctan(a*x)^(7/2)/(a^5*c^3)+15/4096*FresnelS(2*sqrt(2/Pi)*sqrt(arctan(a*x)))*sqrt(1/2*Pi)/(a^5*c^3)-15/128*FresnelS(2*sqrt(arctan(a*x))/sqrt(Pi))*sqrt(Pi)/(a^5*c^3)+45/128*x*sqrt(arctan(a*x))/(a^4*c^3*(1+a^2*x^2))+15/256*sin(2*arctan(a*x))*sqrt(arctan(a*x))/(a^5*c^3)-15/2048*sin(4*arctan(a*x))*sqrt(arctan(a*x))/(a^5*c^3)],
[x^3*arctan(a*x)^(5/2)/(c+a^2*c*x^2)^3,x,16,5/32*x^3*arctan(a*x)^(3/2)/(a*c^3*(1+a^2*x^2)^2)+15/64*x*arctan(a*x)^(3/2)/(a^3*c^3*(1+a^2*x^2))-3/32*arctan(a*x)^(5/2)/(a^4*c^3)+1/4*x^4*arctan(a*x)^(5/2)/(c^3*(1+a^2*x^2)^2)+15/4096*FresnelC(2*sqrt(2/Pi)*sqrt(arctan(a*x)))*sqrt(1/2*Pi)/(a^4*c^3)-15/256*FresnelC(2*sqrt(arctan(a*x))/sqrt(Pi))*sqrt(Pi)/(a^4*c^3)-135/2048*sqrt(arctan(a*x))/(a^4*c^3)-15/256*x^4*sqrt(arctan(a*x))/(c^3*(1+a^2*x^2)^2)+45/256*sqrt(arctan(a*x))/(a^4*c^3*(1+a^2*x^2))],
[x^2*arctan(a*x)^(5/2)/(c+a^2*c*x^2)^3,x,8,1/28*arctan(a*x)^(7/2)/(a^3*c^3)-5/256*arctan(a*x)^(3/2)*cos(4*arctan(a*x))/(a^3*c^3)-1/32*arctan(a*x)^(5/2)*sin(4*arctan(a*x))/(a^3*c^3)-15/4096*FresnelS(2*sqrt(2/Pi)*sqrt(arctan(a*x)))*sqrt(1/2*Pi)/(a^3*c^3)+15/2048*sin(4*arctan(a*x))*sqrt(arctan(a*x))/(a^3*c^3)],
[x*arctan(a*x)^(5/2)/(c+a^2*c*x^2)^3,x,16,5/32*x*arctan(a*x)^(3/2)/(a*c^3*(1+a^2*x^2)^2)+15/64*x*arctan(a*x)^(3/2)/(a*c^3*(1+a^2*x^2))+3/32*arctan(a*x)^(5/2)/(a^2*c^3)-1/4*arctan(a*x)^(5/2)/(a^2*c^3*(1+a^2*x^2)^2)-15/4096*FresnelC(2*sqrt(2/Pi)*sqrt(arctan(a*x)))*sqrt(1/2*Pi)/(a^2*c^3)-15/256*FresnelC(2*sqrt(arctan(a*x))/sqrt(Pi))*sqrt(Pi)/(a^2*c^3)-225/2048*sqrt(arctan(a*x))/(a^2*c^3)+15/256*sqrt(arctan(a*x))/(a^2*c^3*(1+a^2*x^2)^2)+45/256*sqrt(arctan(a*x))/(a^2*c^3*(1+a^2*x^2))],
[arctan(a*x)^(5/2)/(c+a^2*c*x^2)^3,x,18,-75/256*arctan(a*x)^(3/2)/(a*c^3)+5/32*arctan(a*x)^(3/2)/(a*c^3*(1+a^2*x^2)^2)+15/32*arctan(a*x)^(3/2)/(a*c^3*(1+a^2*x^2))+1/4*x*arctan(a*x)^(5/2)/(c^3*(1+a^2*x^2)^2)+3/8*x*arctan(a*x)^(5/2)/(c^3*(1+a^2*x^2))+3/28*arctan(a*x)^(7/2)/(a*c^3)+15/4096*FresnelS(2*sqrt(2/Pi)*sqrt(arctan(a*x)))*sqrt(1/2*Pi)/(a*c^3)+15/128*FresnelS(2*sqrt(arctan(a*x))/sqrt(Pi))*sqrt(Pi)/(a*c^3)-45/128*x*sqrt(arctan(a*x))/(c^3*(1+a^2*x^2))-15/256*sin(2*arctan(a*x))*sqrt(arctan(a*x))/(a*c^3)-15/2048*sin(4*arctan(a*x))*sqrt(arctan(a*x))/(a*c^3)],
[arctan(a*x)^(5/2)/(x*(c+a^2*c*x^2)^3),x,0,Unintegrable(arctan(a*x)^(5/2)/(x*(c+a^2*c*x^2)^3),x)],

# Integrands of the form x^m (c+d x^2)^(p/2) ArcTan[a x]^(5/2) with d=a^2 c

# p>0
[x^m*arctan(a*x)^(5/2)*sqrt(c+a^2*c*x^2),x,0,Unintegrable(x^m*arctan(a*x)^(5/2)*sqrt(c+a^2*c*x^2),x)],
[x^2*arctan(a*x)^(5/2)*sqrt(c+a^2*c*x^2),x,0,Unintegrable(x^2*arctan(a*x)^(5/2)*sqrt(c+a^2*c*x^2),x)],
[x*arctan(a*x)^(5/2)*sqrt(c+a^2*c*x^2),x,2,1/3*(c+a^2*c*x^2)^(3/2)*arctan(a*x)^(5/2)/(a^2*c)-5/12*x*arctan(a*x)^(3/2)*sqrt(c+a^2*c*x^2)/a+5/8*sqrt(c+a^2*c*x^2)*sqrt(arctan(a*x))/a^2-5/12*c*Unintegrable(arctan(a*x)^(3/2)/sqrt(c+a^2*c*x^2),x)/a-5/16*c*Unintegrable(1/(sqrt(c+a^2*c*x^2)*sqrt(arctan(a*x))),x)/a],
[arctan(a*x)^(5/2)*sqrt(c+a^2*c*x^2),x,1,-5/4*arctan(a*x)^(3/2)*sqrt(c+a^2*c*x^2)/a+1/2*x*arctan(a*x)^(5/2)*sqrt(c+a^2*c*x^2)+1/2*c*Unintegrable(arctan(a*x)^(5/2)/sqrt(c+a^2*c*x^2),x)+15/8*c*Unintegrable(sqrt(arctan(a*x))/sqrt(c+a^2*c*x^2),x)],
[arctan(a*x)^(5/2)*sqrt(c+a^2*c*x^2)/x,x,0,Unintegrable(arctan(a*x)^(5/2)*sqrt(c+a^2*c*x^2)/x,x)],
[x^m*(c+a^2*c*x^2)^(3/2)*arctan(a*x)^(5/2),x,0,Unintegrable(x^m*(c+a^2*c*x^2)^(3/2)*arctan(a*x)^(5/2),x)],
[x^2*(c+a^2*c*x^2)^(3/2)*arctan(a*x)^(5/2),x,0,Unintegrable(x^2*(c+a^2*c*x^2)^(3/2)*arctan(a*x)^(5/2),x)],
[x*(c+a^2*c*x^2)^(3/2)*arctan(a*x)^(5/2),x,3,-1/8*x*(c+a^2*c*x^2)^(3/2)*arctan(a*x)^(3/2)/a+1/5*(c+a^2*c*x^2)^(5/2)*arctan(a*x)^(5/2)/(a^2*c)-3/16*c*x*arctan(a*x)^(3/2)*sqrt(c+a^2*c*x^2)/a+1/16*(c+a^2*c*x^2)^(3/2)*sqrt(arctan(a*x))/a^2+9/32*c*sqrt(c+a^2*c*x^2)*sqrt(arctan(a*x))/a^2-3/16*c^2*Unintegrable(arctan(a*x)^(3/2)/sqrt(c+a^2*c*x^2),x)/a-9/64*c^2*Unintegrable(1/(sqrt(c+a^2*c*x^2)*sqrt(arctan(a*x))),x)/a-1/32*c*Unintegrable(sqrt(c+a^2*c*x^2)/sqrt(arctan(a*x)),x)/a],
[(c+a^2*c*x^2)^(3/2)*arctan(a*x)^(5/2),x,2,-5/24*(c+a^2*c*x^2)^(3/2)*arctan(a*x)^(3/2)/a+1/4*x*(c+a^2*c*x^2)^(3/2)*arctan(a*x)^(5/2)-15/16*c*arctan(a*x)^(3/2)*sqrt(c+a^2*c*x^2)/a+3/8*c*x*arctan(a*x)^(5/2)*sqrt(c+a^2*c*x^2)+3/8*c^2*Unintegrable(arctan(a*x)^(5/2)/sqrt(c+a^2*c*x^2),x)+45/32*c^2*Unintegrable(sqrt(arctan(a*x))/sqrt(c+a^2*c*x^2),x)+5/16*c*Unintegrable(sqrt(c+a^2*c*x^2)*sqrt(arctan(a*x)),x)],
[(c+a^2*c*x^2)^(3/2)*arctan(a*x)^(5/2)/x,x,0,Unintegrable((c+a^2*c*x^2)^(3/2)*arctan(a*x)^(5/2)/x,x)],
[x^m*(c+a^2*c*x^2)^(5/2)*arctan(a*x)^(5/2),x,0,Unintegrable(x^m*(c+a^2*c*x^2)^(5/2)*arctan(a*x)^(5/2),x)],
[x^2*(c+a^2*c*x^2)^(5/2)*arctan(a*x)^(5/2),x,0,Unintegrable(x^2*(c+a^2*c*x^2)^(5/2)*arctan(a*x)^(5/2),x)],
[x*(c+a^2*c*x^2)^(5/2)*arctan(a*x)^(5/2),x,4,-25/336*c*x*(c+a^2*c*x^2)^(3/2)*arctan(a*x)^(3/2)/a-5/84*x*(c+a^2*c*x^2)^(5/2)*arctan(a*x)^(3/2)/a+1/7*(c+a^2*c*x^2)^(7/2)*arctan(a*x)^(5/2)/(a^2*c)-25/224*c^2*x*arctan(a*x)^(3/2)*sqrt(c+a^2*c*x^2)/a+25/672*c*(c+a^2*c*x^2)^(3/2)*sqrt(arctan(a*x))/a^2+1/56*(c+a^2*c*x^2)^(5/2)*sqrt(arctan(a*x))/a^2+75/448*c^2*sqrt(c+a^2*c*x^2)*sqrt(arctan(a*x))/a^2-25/224*c^3*Unintegrable(arctan(a*x)^(3/2)/sqrt(c+a^2*c*x^2),x)/a-1/112*c*Unintegrable((c+a^2*c*x^2)^(3/2)/sqrt(arctan(a*x)),x)/a-75/896*c^3*Unintegrable(1/(sqrt(c+a^2*c*x^2)*sqrt(arctan(a*x))),x)/a-25/1344*c^2*Unintegrable(sqrt(c+a^2*c*x^2)/sqrt(arctan(a*x)),x)/a],
[(c+a^2*c*x^2)^(5/2)*arctan(a*x)^(5/2),x,3,-25/144*c*(c+a^2*c*x^2)^(3/2)*arctan(a*x)^(3/2)/a-1/12*(c+a^2*c*x^2)^(5/2)*arctan(a*x)^(3/2)/a+5/24*c*x*(c+a^2*c*x^2)^(3/2)*arctan(a*x)^(5/2)+1/6*x*(c+a^2*c*x^2)^(5/2)*arctan(a*x)^(5/2)-25/32*c^2*arctan(a*x)^(3/2)*sqrt(c+a^2*c*x^2)/a+5/16*c^2*x*arctan(a*x)^(5/2)*sqrt(c+a^2*c*x^2)+5/16*c^3*Unintegrable(arctan(a*x)^(5/2)/sqrt(c+a^2*c*x^2),x)+1/8*c*Unintegrable((c+a^2*c*x^2)^(3/2)*sqrt(arctan(a*x)),x)+75/64*c^3*Unintegrable(sqrt(arctan(a*x))/sqrt(c+a^2*c*x^2),x)+25/96*c^2*Unintegrable(sqrt(c+a^2*c*x^2)*sqrt(arctan(a*x)),x)],
[(c+a^2*c*x^2)^(5/2)*arctan(a*x)^(5/2)/x,x,0,Unintegrable((c+a^2*c*x^2)^(5/2)*arctan(a*x)^(5/2)/x,x)],

# p<0
[x^m*arctan(a*x)^(5/2)/sqrt(c+a^2*c*x^2),x,0,Unintegrable(x^m*arctan(a*x)^(5/2)/sqrt(c+a^2*c*x^2),x)],
[x^3*arctan(a*x)^(5/2)/sqrt(c+a^2*c*x^2),x,4,-5/12*x*arctan(a*x)^(3/2)*sqrt(c+a^2*c*x^2)/(a^3*c)-2/3*arctan(a*x)^(5/2)*sqrt(c+a^2*c*x^2)/(a^4*c)+1/3*x^2*arctan(a*x)^(5/2)*sqrt(c+a^2*c*x^2)/(a^2*c)+5/8*sqrt(c+a^2*c*x^2)*sqrt(arctan(a*x))/(a^4*c)+25/12*Unintegrable(arctan(a*x)^(3/2)/sqrt(c+a^2*c*x^2),x)/a^3-5/16*Unintegrable(1/(sqrt(c+a^2*c*x^2)*sqrt(arctan(a*x))),x)/a^3],
[x^2*arctan(a*x)^(5/2)/sqrt(c+a^2*c*x^2),x,2,-5/4*arctan(a*x)^(3/2)*sqrt(c+a^2*c*x^2)/(a^3*c)+1/2*x*arctan(a*x)^(5/2)*sqrt(c+a^2*c*x^2)/(a^2*c)-1/2*Unintegrable(arctan(a*x)^(5/2)/sqrt(c+a^2*c*x^2),x)/a^2+15/8*Unintegrable(sqrt(arctan(a*x))/sqrt(c+a^2*c*x^2),x)/a^2],
[x*arctan(a*x)^(5/2)/sqrt(c+a^2*c*x^2),x,1,arctan(a*x)^(5/2)*sqrt(c+a^2*c*x^2)/(a^2*c)-5/2*Unintegrable(arctan(a*x)^(3/2)/sqrt(c+a^2*c*x^2),x)/a],
[arctan(a*x)^(5/2)/sqrt(c+a^2*c*x^2),x,0,Unintegrable(arctan(a*x)^(5/2)/sqrt(c+a^2*c*x^2),x)],
[arctan(a*x)^(5/2)/(x*sqrt(c+a^2*c*x^2)),x,0,Unintegrable(arctan(a*x)^(5/2)/(x*sqrt(c+a^2*c*x^2)),x)],
[arctan(a*x)^(5/2)/(x^2*sqrt(c+a^2*c*x^2)),x,1,-arctan(a*x)^(5/2)*sqrt(c+a^2*c*x^2)/(c*x)+5/2*a*Unintegrable(arctan(a*x)^(3/2)/(x*sqrt(c+a^2*c*x^2)),x)],
[arctan(a*x)^(5/2)/(x^3*sqrt(c+a^2*c*x^2)),x,2,-5/4*a*arctan(a*x)^(3/2)*sqrt(c+a^2*c*x^2)/(c*x)-1/2*arctan(a*x)^(5/2)*sqrt(c+a^2*c*x^2)/(c*x^2)-1/2*a^2*Unintegrable(arctan(a*x)^(5/2)/(x*sqrt(c+a^2*c*x^2)),x)+15/8*a^2*Unintegrable(sqrt(arctan(a*x))/(x*sqrt(c+a^2*c*x^2)),x)],
[arctan(a*x)^(5/2)/(x^4*sqrt(c+a^2*c*x^2)),x,4,-5/12*a*arctan(a*x)^(3/2)*sqrt(c+a^2*c*x^2)/(c*x^2)-1/3*arctan(a*x)^(5/2)*sqrt(c+a^2*c*x^2)/(c*x^3)+2/3*a^2*arctan(a*x)^(5/2)*sqrt(c+a^2*c*x^2)/(c*x)-5/8*a^2*sqrt(c+a^2*c*x^2)*sqrt(arctan(a*x))/(c*x)-25/12*a^3*Unintegrable(arctan(a*x)^(3/2)/(x*sqrt(c+a^2*c*x^2)),x)+5/16*a^3*Unintegrable(1/(x*sqrt(c+a^2*c*x^2)*sqrt(arctan(a*x))),x)],
[x^m*arctan(a*x)^(5/2)/(c+a^2*c*x^2)^(3/2),x,0,Unintegrable(x^m*arctan(a*x)^(5/2)/(c+a^2*c*x^2)^(3/2),x)],
[x^2*arctan(a*x)^(5/2)/(c+a^2*c*x^2)^(3/2),x,0,Unintegrable(x^2*arctan(a*x)^(5/2)/(c+a^2*c*x^2)^(3/2),x)],
[x*arctan(a*x)^(5/2)/(c+a^2*c*x^2)^(3/2),x,6,5/2*x*arctan(a*x)^(3/2)/(a*c*sqrt(c+a^2*c*x^2))-arctan(a*x)^(5/2)/(a^2*c*sqrt(c+a^2*c*x^2))-15/4*FresnelC(sqrt(2/Pi)*sqrt(arctan(a*x)))*sqrt(1/2*Pi)*sqrt(1+a^2*x^2)/(a^2*c*sqrt(c+a^2*c*x^2))+15/4*sqrt(arctan(a*x))/(a^2*c*sqrt(c+a^2*c*x^2))],
[arctan(a*x)^(5/2)/(c+a^2*c*x^2)^(3/2),x,6,5/2*arctan(a*x)^(3/2)/(a*c*sqrt(c+a^2*c*x^2))+x*arctan(a*x)^(5/2)/(c*sqrt(c+a^2*c*x^2))+15/4*FresnelS(sqrt(2/Pi)*sqrt(arctan(a*x)))*sqrt(1/2*Pi)*sqrt(1+a^2*x^2)/(a*c*sqrt(c+a^2*c*x^2))-15/4*x*sqrt(arctan(a*x))/(c*sqrt(c+a^2*c*x^2))],
[arctan(a*x)^(5/2)/(x*(c+a^2*c*x^2)^(3/2)),x,0,Unintegrable(arctan(a*x)^(5/2)/(x*(c+a^2*c*x^2)^(3/2)),x)],
[x^m*arctan(a*x)^(5/2)/(c+a^2*c*x^2)^(5/2),x,0,Unintegrable(x^m*arctan(a*x)^(5/2)/(c+a^2*c*x^2)^(5/2),x)],
[x^4*arctan(a*x)^(5/2)/(c+a^2*c*x^2)^(5/2),x,0,Unintegrable(x^4*arctan(a*x)^(5/2)/(c+a^2*c*x^2)^(5/2),x)],
[x^3*arctan(a*x)^(5/2)/(c+a^2*c*x^2)^(5/2),x,17,5/18*x^3*arctan(a*x)^(3/2)/(a*c*(c+a^2*c*x^2)^(3/2))-1/3*x^2*arctan(a*x)^(5/2)/(a^2*c*(c+a^2*c*x^2)^(3/2))+5/3*x*arctan(a*x)^(3/2)/(a^3*c^2*sqrt(c+a^2*c*x^2))-2/3*arctan(a*x)^(5/2)/(a^4*c^2*sqrt(c+a^2*c*x^2))+5/144*FresnelC(sqrt(6/Pi)*sqrt(arctan(a*x)))*sqrt(1/6*Pi)*sqrt(1+a^2*x^2)/(a^4*c^2*sqrt(c+a^2*c*x^2))-45/16*FresnelC(sqrt(2/Pi)*sqrt(arctan(a*x)))*sqrt(1/2*Pi)*sqrt(1+a^2*x^2)/(a^4*c^2*sqrt(c+a^2*c*x^2))+45/16*sqrt(arctan(a*x))/(a^4*c^2*sqrt(c+a^2*c*x^2))-5/144*cos(3*arctan(a*x))*sqrt(1+a^2*x^2)*sqrt(arctan(a*x))/(a^4*c^2*sqrt(c+a^2*c*x^2))],
[x^2*arctan(a*x)^(5/2)/(c+a^2*c*x^2)^(5/2),x,16,5/18*x^2*arctan(a*x)^(3/2)/(a*c*(c+a^2*c*x^2)^(3/2))+1/3*x^3*arctan(a*x)^(5/2)/(c*(c+a^2*c*x^2)^(3/2))+5/9*arctan(a*x)^(3/2)/(a^3*c^2*sqrt(c+a^2*c*x^2))-5/144*FresnelS(sqrt(6/Pi)*sqrt(arctan(a*x)))*sqrt(1/6*Pi)*sqrt(1+a^2*x^2)/(a^3*c^2*sqrt(c+a^2*c*x^2))+15/16*FresnelS(sqrt(2/Pi)*sqrt(arctan(a*x)))*sqrt(1/2*Pi)*sqrt(1+a^2*x^2)/(a^3*c^2*sqrt(c+a^2*c*x^2))-5/36*x^3*sqrt(arctan(a*x))/(c*(c+a^2*c*x^2)^(3/2))-5/6*x*sqrt(arctan(a*x))/(a^2*c^2*sqrt(c+a^2*c*x^2))],
[x*arctan(a*x)^(5/2)/(c+a^2*c*x^2)^(5/2),x,15,5/18*x*arctan(a*x)^(3/2)/(a*c*(c+a^2*c*x^2)^(3/2))-1/3*arctan(a*x)^(5/2)/(a^2*c*(c+a^2*c*x^2)^(3/2))+5/9*x*arctan(a*x)^(3/2)/(a*c^2*sqrt(c+a^2*c*x^2))-5/144*FresnelC(sqrt(6/Pi)*sqrt(arctan(a*x)))*sqrt(1/6*Pi)*sqrt(1+a^2*x^2)/(a^2*c^2*sqrt(c+a^2*c*x^2))-15/16*FresnelC(sqrt(2/Pi)*sqrt(arctan(a*x)))*sqrt(1/2*Pi)*sqrt(1+a^2*x^2)/(a^2*c^2*sqrt(c+a^2*c*x^2))+5/36*sqrt(arctan(a*x))/(a^2*c*(c+a^2*c*x^2)^(3/2))+5/6*sqrt(arctan(a*x))/(a^2*c^2*sqrt(c+a^2*c*x^2))],
[arctan(a*x)^(5/2)/(c+a^2*c*x^2)^(5/2),x,17,5/18*arctan(a*x)^(3/2)/(a*c*(c+a^2*c*x^2)^(3/2))+1/3*x*arctan(a*x)^(5/2)/(c*(c+a^2*c*x^2)^(3/2))+5/3*arctan(a*x)^(3/2)/(a*c^2*sqrt(c+a^2*c*x^2))+2/3*x*arctan(a*x)^(5/2)/(c^2*sqrt(c+a^2*c*x^2))+5/144*FresnelS(sqrt(6/Pi)*sqrt(arctan(a*x)))*sqrt(1/6*Pi)*sqrt(1+a^2*x^2)/(a*c^2*sqrt(c+a^2*c*x^2))+45/16*FresnelS(sqrt(2/Pi)*sqrt(arctan(a*x)))*sqrt(1/2*Pi)*sqrt(1+a^2*x^2)/(a*c^2*sqrt(c+a^2*c*x^2))-45/16*x*sqrt(arctan(a*x))/(c^2*sqrt(c+a^2*c*x^2))-5/144*sin(3*arctan(a*x))*sqrt(1+a^2*x^2)*sqrt(arctan(a*x))/(a*c^2*sqrt(c+a^2*c*x^2))],
[arctan(a*x)^(5/2)/(x*(c+a^2*c*x^2)^(5/2)),x,0,Unintegrable(arctan(a*x)^(5/2)/(x*(c+a^2*c*x^2)^(5/2)),x)],

# Integrands of the form (f x)^m (d+c^2 d x^2)^q (a+b ArcTan[c x])^(-1/2)

# Integrands of the form x^m (c+d x^2)^p / ArcTan[a x]^(1/2) with d=a^2 c

# p>0
[x^m*(c+a^2*c*x^2)/sqrt(arctan(a*x)),x,0,Unintegrable(x^m*(c+a^2*c*x^2)/sqrt(arctan(a*x)),x)],
[x*(c+a^2*c*x^2)/sqrt(arctan(a*x)),x,0,Unintegrable(x*(c+a^2*c*x^2)/sqrt(arctan(a*x)),x)],
[(c+a^2*c*x^2)/sqrt(arctan(a*x)),x,0,Unintegrable((c+a^2*c*x^2)/sqrt(arctan(a*x)),x)],
[(c+a^2*c*x^2)/(x*sqrt(arctan(a*x))),x,0,Unintegrable((c+a^2*c*x^2)/(x*sqrt(arctan(a*x))),x)],
[x^m*(c+a^2*c*x^2)^2/sqrt(arctan(a*x)),x,0,Unintegrable(x^m*(c+a^2*c*x^2)^2/sqrt(arctan(a*x)),x)],
[x*(c+a^2*c*x^2)^2/sqrt(arctan(a*x)),x,0,Unintegrable(x*(c+a^2*c*x^2)^2/sqrt(arctan(a*x)),x)],
[(c+a^2*c*x^2)^2/sqrt(arctan(a*x)),x,0,Unintegrable((c+a^2*c*x^2)^2/sqrt(arctan(a*x)),x)],
[(c+a^2*c*x^2)^2/(x*sqrt(arctan(a*x))),x,0,Unintegrable((c+a^2*c*x^2)^2/(x*sqrt(arctan(a*x))),x)],
[x^m*(c+a^2*c*x^2)^3/sqrt(arctan(a*x)),x,0,Unintegrable(x^m*(c+a^2*c*x^2)^3/sqrt(arctan(a*x)),x)],
[x*(c+a^2*c*x^2)^3/sqrt(arctan(a*x)),x,0,Unintegrable(x*(c+a^2*c*x^2)^3/sqrt(arctan(a*x)),x)],
[(c+a^2*c*x^2)^3/sqrt(arctan(a*x)),x,0,Unintegrable((c+a^2*c*x^2)^3/sqrt(arctan(a*x)),x)],
[(c+a^2*c*x^2)^3/(x*sqrt(arctan(a*x))),x,0,Unintegrable((c+a^2*c*x^2)^3/(x*sqrt(arctan(a*x))),x)],

# p<0
[x^m/((c+a^2*c*x^2)*sqrt(arctan(a*x))),x,0,Unintegrable(x^m/((c+a^2*c*x^2)*sqrt(arctan(a*x))),x)],
[x/((c+a^2*c*x^2)*sqrt(arctan(a*x))),x,1,2*x*sqrt(arctan(a*x))/(a*c)-2*Unintegrable(sqrt(arctan(a*x)),x)/(a*c)],
[1/((c+a^2*c*x^2)*sqrt(arctan(a*x))),x,1,2*sqrt(arctan(a*x))/(a*c)],
[1/(x*(c+a^2*c*x^2)*sqrt(arctan(a*x))),x,0,Unintegrable(1/(x*(c+a^2*c*x^2)*sqrt(arctan(a*x))),x)],
[x^m/((c+a^2*c*x^2)^2*sqrt(arctan(a*x))),x,0,Unintegrable(x^m/((c+a^2*c*x^2)^2*sqrt(arctan(a*x))),x)],
[x^3/((c+a^2*c*x^2)^2*sqrt(arctan(a*x))),x,0,Unintegrable(x^3/((c+a^2*c*x^2)^2*sqrt(arctan(a*x))),x)],
[x^2/((c+a^2*c*x^2)^2*sqrt(arctan(a*x))),x,5,-1/2*FresnelC(2*sqrt(arctan(a*x))/sqrt(Pi))*sqrt(Pi)/(a^3*c^2)+sqrt(arctan(a*x))/(a^3*c^2)],
[x/((c+a^2*c*x^2)^2*sqrt(arctan(a*x))),x,5,1/2*FresnelS(2*sqrt(arctan(a*x))/sqrt(Pi))*sqrt(Pi)/(a^2*c^2)],
[1/((c+a^2*c*x^2)^2*sqrt(arctan(a*x))),x,5,1/2*FresnelC(2*sqrt(arctan(a*x))/sqrt(Pi))*sqrt(Pi)/(a*c^2)+sqrt(arctan(a*x))/(a*c^2)],
[1/(x*(c+a^2*c*x^2)^2*sqrt(arctan(a*x))),x,0,Unintegrable(1/(x*(c+a^2*c*x^2)^2*sqrt(arctan(a*x))),x)],
[x^m/((c+a^2*c*x^2)^3*sqrt(arctan(a*x))),x,0,Unintegrable(x^m/((c+a^2*c*x^2)^3*sqrt(arctan(a*x))),x)],
[x^5/((c+a^2*c*x^2)^3*sqrt(arctan(a*x))),x,0,Unintegrable(x^5/((c+a^2*c*x^2)^3*sqrt(arctan(a*x))),x)],
[x^4/((c+a^2*c*x^2)^3*sqrt(arctan(a*x))),x,7,1/8*FresnelC(2*sqrt(2/Pi)*sqrt(arctan(a*x)))*sqrt(1/2*Pi)/(a^5*c^3)-1/2*FresnelC(2*sqrt(arctan(a*x))/sqrt(Pi))*sqrt(Pi)/(a^5*c^3)+3/4*sqrt(arctan(a*x))/(a^5*c^3)],
[x^3/((c+a^2*c*x^2)^3*sqrt(arctan(a*x))),x,7,-1/8*FresnelS(2*sqrt(2/Pi)*sqrt(arctan(a*x)))*sqrt(1/2*Pi)/(a^4*c^3)+1/4*FresnelS(2*sqrt(arctan(a*x))/sqrt(Pi))*sqrt(Pi)/(a^4*c^3)],
[x^2/((c+a^2*c*x^2)^3*sqrt(arctan(a*x))),x,5,-1/8*FresnelC(2*sqrt(2/Pi)*sqrt(arctan(a*x)))*sqrt(1/2*Pi)/(a^3*c^3)+1/4*sqrt(arctan(a*x))/(a^3*c^3)],
[x/((c+a^2*c*x^2)^3*sqrt(arctan(a*x))),x,7,1/8*FresnelS(2*sqrt(2/Pi)*sqrt(arctan(a*x)))*sqrt(1/2*Pi)/(a^2*c^3)+1/4*FresnelS(2*sqrt(arctan(a*x))/sqrt(Pi))*sqrt(Pi)/(a^2*c^3)],
[1/((c+a^2*c*x^2)^3*sqrt(arctan(a*x))),x,7,1/8*FresnelC(2*sqrt(2/Pi)*sqrt(arctan(a*x)))*sqrt(1/2*Pi)/(a*c^3)+1/2*FresnelC(2*sqrt(arctan(a*x))/sqrt(Pi))*sqrt(Pi)/(a*c^3)+3/4*sqrt(arctan(a*x))/(a*c^3)],
[1/(x*(c+a^2*c*x^2)^3*sqrt(arctan(a*x))),x,0,Unintegrable(1/(x*(c+a^2*c*x^2)^3*sqrt(arctan(a*x))),x)],

# Integrands of the form x^m (c+d x^2)^(p/2) / ArcTan[a x]^(1/2) with d=a^2 c

# p>0
[x^m*sqrt(c+a^2*c*x^2)/sqrt(arctan(a*x)),x,0,Unintegrable(x^m*sqrt(c+a^2*c*x^2)/sqrt(arctan(a*x)),x)],
[x*sqrt(c+a^2*c*x^2)/sqrt(arctan(a*x)),x,0,Unintegrable(x*sqrt(c+a^2*c*x^2)/sqrt(arctan(a*x)),x)],
[sqrt(c+a^2*c*x^2)/sqrt(arctan(a*x)),x,0,Unintegrable(sqrt(c+a^2*c*x^2)/sqrt(arctan(a*x)),x)],
[sqrt(c+a^2*c*x^2)/(x*sqrt(arctan(a*x))),x,0,Unintegrable(sqrt(c+a^2*c*x^2)/(x*sqrt(arctan(a*x))),x)],
[x^m*(c+a^2*c*x^2)^(3/2)/sqrt(arctan(a*x)),x,0,Unintegrable(x^m*(c+a^2*c*x^2)^(3/2)/sqrt(arctan(a*x)),x)],
[x*(c+a^2*c*x^2)^(3/2)/sqrt(arctan(a*x)),x,0,Unintegrable(x*(c+a^2*c*x^2)^(3/2)/sqrt(arctan(a*x)),x)],
[(c+a^2*c*x^2)^(3/2)/sqrt(arctan(a*x)),x,0,Unintegrable((c+a^2*c*x^2)^(3/2)/sqrt(arctan(a*x)),x)],
[(c+a^2*c*x^2)^(3/2)/(x*sqrt(arctan(a*x))),x,0,Unintegrable((c+a^2*c*x^2)^(3/2)/(x*sqrt(arctan(a*x))),x)],
[x^m*(c+a^2*c*x^2)^(5/2)/sqrt(arctan(a*x)),x,0,Unintegrable(x^m*(c+a^2*c*x^2)^(5/2)/sqrt(arctan(a*x)),x)],
[x*(c+a^2*c*x^2)^(5/2)/sqrt(arctan(a*x)),x,0,Unintegrable(x*(c+a^2*c*x^2)^(5/2)/sqrt(arctan(a*x)),x)],
[(c+a^2*c*x^2)^(5/2)/sqrt(arctan(a*x)),x,0,Unintegrable((c+a^2*c*x^2)^(5/2)/sqrt(arctan(a*x)),x)],
[(c+a^2*c*x^2)^(5/2)/(x*sqrt(arctan(a*x))),x,0,Unintegrable((c+a^2*c*x^2)^(5/2)/(x*sqrt(arctan(a*x))),x)],

# p<0
[x^m/(sqrt(c+a^2*c*x^2)*sqrt(arctan(a*x))),x,0,Unintegrable(x^m/(sqrt(c+a^2*c*x^2)*sqrt(arctan(a*x))),x)],
[x/(sqrt(c+a^2*c*x^2)*sqrt(arctan(a*x))),x,0,Unintegrable(x/(sqrt(c+a^2*c*x^2)*sqrt(arctan(a*x))),x)],
[1/(sqrt(c+a^2*c*x^2)*sqrt(arctan(a*x))),x,0,Unintegrable(1/(sqrt(c+a^2*c*x^2)*sqrt(arctan(a*x))),x)],
[1/(x*sqrt(c+a^2*c*x^2)*sqrt(arctan(a*x))),x,0,Unintegrable(1/(x*sqrt(c+a^2*c*x^2)*sqrt(arctan(a*x))),x)],
[x^m/((c+a^2*c*x^2)^(3/2)*sqrt(arctan(a*x))),x,0,Unintegrable(x^m/((c+a^2*c*x^2)^(3/2)*sqrt(arctan(a*x))),x)],
[x^2/((c+a^2*c*x^2)^(3/2)*sqrt(arctan(a*x))),x,0,Unintegrable(x^2/((c+a^2*c*x^2)^(3/2)*sqrt(arctan(a*x))),x)],
[x/((c+a^2*c*x^2)^(3/2)*sqrt(arctan(a*x))),x,4,FresnelS(sqrt(2/Pi)*sqrt(arctan(a*x)))*sqrt(2*Pi)*sqrt(1+a^2*x^2)/(a^2*c*sqrt(c+a^2*c*x^2))],
[1/((c+a^2*c*x^2)^(3/2)*sqrt(arctan(a*x))),x,4,FresnelC(sqrt(2/Pi)*sqrt(arctan(a*x)))*sqrt(2*Pi)*sqrt(1+a^2*x^2)/(a*c*sqrt(c+a^2*c*x^2))],
[1/(x*(c+a^2*c*x^2)^(3/2)*sqrt(arctan(a*x))),x,0,Unintegrable(1/(x*(c+a^2*c*x^2)^(3/2)*sqrt(arctan(a*x))),x)],
[x^m/((c+a^2*c*x^2)^(5/2)*sqrt(arctan(a*x))),x,0,Unintegrable(x^m/((c+a^2*c*x^2)^(5/2)*sqrt(arctan(a*x))),x)],
[x^4/((c+a^2*c*x^2)^(5/2)*sqrt(arctan(a*x))),x,0,Unintegrable(x^4/((c+a^2*c*x^2)^(5/2)*sqrt(arctan(a*x))),x)],
[x^3/((c+a^2*c*x^2)^(5/2)*sqrt(arctan(a*x))),x,8,-1/2*FresnelS(sqrt(6/Pi)*sqrt(arctan(a*x)))*sqrt(1/6*Pi)*sqrt(1+a^2*x^2)/(a^4*c^2*sqrt(c+a^2*c*x^2))+3/2*FresnelS(sqrt(2/Pi)*sqrt(arctan(a*x)))*sqrt(1/2*Pi)*sqrt(1+a^2*x^2)/(a^4*c^2*sqrt(c+a^2*c*x^2))],
[x^2/((c+a^2*c*x^2)^(5/2)*sqrt(arctan(a*x))),x,8,-1/2*FresnelC(sqrt(6/Pi)*sqrt(arctan(a*x)))*sqrt(1/6*Pi)*sqrt(1+a^2*x^2)/(a^3*c^2*sqrt(c+a^2*c*x^2))+1/2*FresnelC(sqrt(2/Pi)*sqrt(arctan(a*x)))*sqrt(1/2*Pi)*sqrt(1+a^2*x^2)/(a^3*c^2*sqrt(c+a^2*c*x^2))],
[x/((c+a^2*c*x^2)^(5/2)*sqrt(arctan(a*x))),x,8,1/2*FresnelS(sqrt(6/Pi)*sqrt(arctan(a*x)))*sqrt(1/6*Pi)*sqrt(1+a^2*x^2)/(a^2*c^2*sqrt(c+a^2*c*x^2))+1/2*FresnelS(sqrt(2/Pi)*sqrt(arctan(a*x)))*sqrt(1/2*Pi)*sqrt(1+a^2*x^2)/(a^2*c^2*sqrt(c+a^2*c*x^2))],
[1/((c+a^2*c*x^2)^(5/2)*sqrt(arctan(a*x))),x,8,1/2*FresnelC(sqrt(6/Pi)*sqrt(arctan(a*x)))*sqrt(1/6*Pi)*sqrt(1+a^2*x^2)/(a*c^2*sqrt(c+a^2*c*x^2))+3/2*FresnelC(sqrt(2/Pi)*sqrt(arctan(a*x)))*sqrt(1/2*Pi)*sqrt(1+a^2*x^2)/(a*c^2*sqrt(c+a^2*c*x^2))],
[1/(x*(c+a^2*c*x^2)^(5/2)*sqrt(arctan(a*x))),x,0,Unintegrable(1/(x*(c+a^2*c*x^2)^(5/2)*sqrt(arctan(a*x))),x)],

# Integrands of the form (f x)^m (d+c^2 d x^2)^q (a+b ArcTan[c x])^(-3/2)

# Integrands of the form x^m (c+d x^2)^p / ArcTan[a x]^(3/2) with d=a^2 c

# p>0
[x^m*(c+a^2*c*x^2)/arctan(a*x)^(3/2),x,0,Unintegrable(x^m*(c+a^2*c*x^2)/arctan(a*x)^(3/2),x)],
[x*(c+a^2*c*x^2)/arctan(a*x)^(3/2),x,0,Unintegrable(x*(c+a^2*c*x^2)/arctan(a*x)^(3/2),x)],
[(c+a^2*c*x^2)/arctan(a*x)^(3/2),x,0,Unintegrable((c+a^2*c*x^2)/arctan(a*x)^(3/2),x)],
[(c+a^2*c*x^2)/(x*arctan(a*x)^(3/2)),x,0,Unintegrable((c+a^2*c*x^2)/(x*arctan(a*x)^(3/2)),x)],
[x^m*(c+a^2*c*x^2)^2/arctan(a*x)^(3/2),x,0,Unintegrable(x^m*(c+a^2*c*x^2)^2/arctan(a*x)^(3/2),x)],
[x*(c+a^2*c*x^2)^2/arctan(a*x)^(3/2),x,0,Unintegrable(x*(c+a^2*c*x^2)^2/arctan(a*x)^(3/2),x)],
[(c+a^2*c*x^2)^2/arctan(a*x)^(3/2),x,0,Unintegrable((c+a^2*c*x^2)^2/arctan(a*x)^(3/2),x)],
[(c+a^2*c*x^2)^2/(x*arctan(a*x)^(3/2)),x,0,Unintegrable((c+a^2*c*x^2)^2/(x*arctan(a*x)^(3/2)),x)],
[x^m*(c+a^2*c*x^2)^3/arctan(a*x)^(3/2),x,0,Unintegrable(x^m*(c+a^2*c*x^2)^3/arctan(a*x)^(3/2),x)],
[x*(c+a^2*c*x^2)^3/arctan(a*x)^(3/2),x,0,Unintegrable(x*(c+a^2*c*x^2)^3/arctan(a*x)^(3/2),x)],
[(c+a^2*c*x^2)^3/arctan(a*x)^(3/2),x,0,Unintegrable((c+a^2*c*x^2)^3/arctan(a*x)^(3/2),x)],
[(c+a^2*c*x^2)^3/(x*arctan(a*x)^(3/2)),x,0,Unintegrable((c+a^2*c*x^2)^3/(x*arctan(a*x)^(3/2)),x)],

# p<0
[x^m/((c+a^2*c*x^2)*arctan(a*x)^(3/2)),x,1,-2*x^m/(a*c*sqrt(arctan(a*x)))+2*m*Unintegrable(x^(-1+m)/sqrt(arctan(a*x)),x)/(a*c)],
[x/((c+a^2*c*x^2)*arctan(a*x)^(3/2)),x,1,-2*x/(a*c*sqrt(arctan(a*x)))+2*Unintegrable(1/sqrt(arctan(a*x)),x)/(a*c)],
[1/((c+a^2*c*x^2)*arctan(a*x)^(3/2)),x,1,(-2)/(a*c*sqrt(arctan(a*x)))],
[1/(x*(c+a^2*c*x^2)*arctan(a*x)^(3/2)),x,1,(-2)/(a*c*x*sqrt(arctan(a*x)))-2*Unintegrable(1/(x^2*sqrt(arctan(a*x))),x)/(a*c)],
[x^m/((c+a^2*c*x^2)^2*arctan(a*x)^(3/2)),x,0,Unintegrable(x^m/((c+a^2*c*x^2)^2*arctan(a*x)^(3/2)),x)],
[x^4/((c+a^2*c*x^2)^2*arctan(a*x)^(3/2)),x,1,-2*x^4/(a*c^2*(1+a^2*x^2)*sqrt(arctan(a*x)))+8*Unintegrable(x^3/((c+a^2*c*x^2)^2*sqrt(arctan(a*x))),x)/a+4*a*Unintegrable(x^5/((c+a^2*c*x^2)^2*sqrt(arctan(a*x))),x)],
[x^3/((c+a^2*c*x^2)^2*arctan(a*x)^(3/2)),x,6,-3*FresnelC(2*sqrt(arctan(a*x))/sqrt(Pi))*sqrt(Pi)/(a^4*c^2)-2*x^3/(a*c^2*(1+a^2*x^2)*sqrt(arctan(a*x)))+6*sqrt(arctan(a*x))/(a^4*c^2)+2*a*Unintegrable(x^4/((c+a^2*c*x^2)^2*sqrt(arctan(a*x))),x)],
[x^2/((c+a^2*c*x^2)^2*arctan(a*x)^(3/2)),x,6,2*FresnelS(2*sqrt(arctan(a*x))/sqrt(Pi))*sqrt(Pi)/(a^3*c^2)-2*x^2/(a*c^2*(1+a^2*x^2)*sqrt(arctan(a*x)))],
[x/((c+a^2*c*x^2)^2*arctan(a*x)^(3/2)),x,7,2*FresnelC(2*sqrt(arctan(a*x))/sqrt(Pi))*sqrt(Pi)/(a^2*c^2)-2*x/(a*c^2*(1+a^2*x^2)*sqrt(arctan(a*x)))+4*sqrt(arctan(a*x))/(a^2*c^2)-8*sqrt(arctan(a*x))/(a^2*c^2*(1+a^2*x^2))+4*(1-a^2*x^2)*sqrt(arctan(a*x))/(a^2*c^2*(1+a^2*x^2))],
[1/((c+a^2*c*x^2)^2*arctan(a*x)^(3/2)),x,6,-2*FresnelS(2*sqrt(arctan(a*x))/sqrt(Pi))*sqrt(Pi)/(a*c^2)+(-2)/(a*c^2*(1+a^2*x^2)*sqrt(arctan(a*x)))],
[1/(x*(c+a^2*c*x^2)^2*arctan(a*x)^(3/2)),x,6,-3*FresnelC(2*sqrt(arctan(a*x))/sqrt(Pi))*sqrt(Pi)/c^2+(-2)/(a*c^2*x*(1+a^2*x^2)*sqrt(arctan(a*x)))-6*sqrt(arctan(a*x))/c^2-2*Unintegrable(1/(x^2*(c+a^2*c*x^2)^2*sqrt(arctan(a*x))),x)/a],
[1/(x^2*(c+a^2*c*x^2)^2*arctan(a*x)^(3/2)),x,1,(-2)/(a*c^2*x^2*(1+a^2*x^2)*sqrt(arctan(a*x)))-4*Unintegrable(1/(x^3*(c+a^2*c*x^2)^2*sqrt(arctan(a*x))),x)/a-8*a*Unintegrable(1/(x*(c+a^2*c*x^2)^2*sqrt(arctan(a*x))),x)],
[1/(x^3*(c+a^2*c*x^2)^2*arctan(a*x)^(3/2)),x,1,(-2)/(a*c^2*x^3*(1+a^2*x^2)*sqrt(arctan(a*x)))-6*Unintegrable(1/(x^4*(c+a^2*c*x^2)^2*sqrt(arctan(a*x))),x)/a-10*a*Unintegrable(1/(x^2*(c+a^2*c*x^2)^2*sqrt(arctan(a*x))),x)],
[1/(x^4*(c+a^2*c*x^2)^2*arctan(a*x)^(3/2)),x,1,(-2)/(a*c^2*x^4*(1+a^2*x^2)*sqrt(arctan(a*x)))-8*Unintegrable(1/(x^5*(c+a^2*c*x^2)^2*sqrt(arctan(a*x))),x)/a-12*a*Unintegrable(1/(x^3*(c+a^2*c*x^2)^2*sqrt(arctan(a*x))),x)],
[x^m/((c+a^2*c*x^2)^3*arctan(a*x)^(3/2)),x,0,Unintegrable(x^m/((c+a^2*c*x^2)^3*arctan(a*x)^(3/2)),x)],
[x^3/((c+a^2*c*x^2)^3*arctan(a*x)^(3/2)),x,13,-FresnelC(2*sqrt(2/Pi)*sqrt(arctan(a*x)))*sqrt(1/2*Pi)/(a^4*c^3)+FresnelC(2*sqrt(arctan(a*x))/sqrt(Pi))*sqrt(Pi)/(a^4*c^3)-2*x^3/(a*c^3*(1+a^2*x^2)^2*sqrt(arctan(a*x)))],
[x^2/((c+a^2*c*x^2)^3*arctan(a*x)^(3/2)),x,15,FresnelS(2*sqrt(2/Pi)*sqrt(arctan(a*x)))*sqrt(1/2*Pi)/(a^3*c^3)-2*x^2/(a*c^3*(1+a^2*x^2)^2*sqrt(arctan(a*x)))],
[x/((c+a^2*c*x^2)^3*arctan(a*x)^(3/2)),x,13,FresnelC(2*sqrt(2/Pi)*sqrt(arctan(a*x)))*sqrt(1/2*Pi)/(a^2*c^3)+FresnelC(2*sqrt(arctan(a*x))/sqrt(Pi))*sqrt(Pi)/(a^2*c^3)-2*x/(a*c^3*(1+a^2*x^2)^2*sqrt(arctan(a*x)))],
[1/((c+a^2*c*x^2)^3*arctan(a*x)^(3/2)),x,8,-FresnelS(2*sqrt(2/Pi)*sqrt(arctan(a*x)))*sqrt(1/2*Pi)/(a*c^3)-2*FresnelS(2*sqrt(arctan(a*x))/sqrt(Pi))*sqrt(Pi)/(a*c^3)+(-2)/(a*c^3*(1+a^2*x^2)^2*sqrt(arctan(a*x)))],
[1/(x*(c+a^2*c*x^2)^3*arctan(a*x)^(3/2)),x,8,-5/4*FresnelC(2*sqrt(2/Pi)*sqrt(arctan(a*x)))*sqrt(1/2*Pi)/c^3-5*FresnelC(2*sqrt(arctan(a*x))/sqrt(Pi))*sqrt(Pi)/c^3+(-2)/(a*c^3*x*(1+a^2*x^2)^2*sqrt(arctan(a*x)))-15/2*sqrt(arctan(a*x))/c^3-2*Unintegrable(1/(x^2*(c+a^2*c*x^2)^3*sqrt(arctan(a*x))),x)/a],
[1/(x^2*(c+a^2*c*x^2)^3*arctan(a*x)^(3/2)),x,1,(-2)/(a*c^3*x^2*(1+a^2*x^2)^2*sqrt(arctan(a*x)))-4*Unintegrable(1/(x^3*(c+a^2*c*x^2)^3*sqrt(arctan(a*x))),x)/a-12*a*Unintegrable(1/(x*(c+a^2*c*x^2)^3*sqrt(arctan(a*x))),x)],
[1/(x^3*(c+a^2*c*x^2)^3*arctan(a*x)^(3/2)),x,1,(-2)/(a*c^3*x^3*(1+a^2*x^2)^2*sqrt(arctan(a*x)))-6*Unintegrable(1/(x^4*(c+a^2*c*x^2)^3*sqrt(arctan(a*x))),x)/a-14*a*Unintegrable(1/(x^2*(c+a^2*c*x^2)^3*sqrt(arctan(a*x))),x)],
[1/(x^4*(c+a^2*c*x^2)^3*arctan(a*x)^(3/2)),x,1,(-2)/(a*c^3*x^4*(1+a^2*x^2)^2*sqrt(arctan(a*x)))-8*Unintegrable(1/(x^5*(c+a^2*c*x^2)^3*sqrt(arctan(a*x))),x)/a-16*a*Unintegrable(1/(x^3*(c+a^2*c*x^2)^3*sqrt(arctan(a*x))),x)],

# Integrands of the form x^m (c+d x^2)^(p/2) / ArcTan[a x]^(3/2) with d=a^2 c

# p>0
[x^m*sqrt(c+a^2*c*x^2)/arctan(a*x)^(3/2),x,0,Unintegrable(x^m*sqrt(c+a^2*c*x^2)/arctan(a*x)^(3/2),x)],
[x*sqrt(c+a^2*c*x^2)/arctan(a*x)^(3/2),x,0,Unintegrable(x*sqrt(c+a^2*c*x^2)/arctan(a*x)^(3/2),x)],
[sqrt(c+a^2*c*x^2)/arctan(a*x)^(3/2),x,0,Unintegrable(sqrt(c+a^2*c*x^2)/arctan(a*x)^(3/2),x)],
[sqrt(c+a^2*c*x^2)/(x*arctan(a*x)^(3/2)),x,0,Unintegrable(sqrt(c+a^2*c*x^2)/(x*arctan(a*x)^(3/2)),x)],
[x^m*(c+a^2*c*x^2)^(3/2)/arctan(a*x)^(3/2),x,0,Unintegrable(x^m*(c+a^2*c*x^2)^(3/2)/arctan(a*x)^(3/2),x)],
[x*(c+a^2*c*x^2)^(3/2)/arctan(a*x)^(3/2),x,0,Unintegrable(x*(c+a^2*c*x^2)^(3/2)/arctan(a*x)^(3/2),x)],
[(c+a^2*c*x^2)^(3/2)/arctan(a*x)^(3/2),x,0,Unintegrable((c+a^2*c*x^2)^(3/2)/arctan(a*x)^(3/2),x)],
[(c+a^2*c*x^2)^(3/2)/(x*arctan(a*x)^(3/2)),x,0,Unintegrable((c+a^2*c*x^2)^(3/2)/(x*arctan(a*x)^(3/2)),x)],
[x^m*(c+a^2*c*x^2)^(5/2)/arctan(a*x)^(3/2),x,0,Unintegrable(x^m*(c+a^2*c*x^2)^(5/2)/arctan(a*x)^(3/2),x)],
[x*(c+a^2*c*x^2)^(5/2)/arctan(a*x)^(3/2),x,0,Unintegrable(x*(c+a^2*c*x^2)^(5/2)/arctan(a*x)^(3/2),x)],
[(c+a^2*c*x^2)^(5/2)/arctan(a*x)^(3/2),x,0,Unintegrable((c+a^2*c*x^2)^(5/2)/arctan(a*x)^(3/2),x)],
[(c+a^2*c*x^2)^(5/2)/(x*arctan(a*x)^(3/2)),x,0,Unintegrable((c+a^2*c*x^2)^(5/2)/(x*arctan(a*x)^(3/2)),x)],

# p<0
[x^m/(arctan(a*x)^(3/2)*sqrt(c+a^2*c*x^2)),x,0,Unintegrable(x^m/(arctan(a*x)^(3/2)*sqrt(c+a^2*c*x^2)),x)],
[x/(arctan(a*x)^(3/2)*sqrt(c+a^2*c*x^2)),x,0,Unintegrable(x/(arctan(a*x)^(3/2)*sqrt(c+a^2*c*x^2)),x)],
[1/(arctan(a*x)^(3/2)*sqrt(c+a^2*c*x^2)),x,0,Unintegrable(1/(arctan(a*x)^(3/2)*sqrt(c+a^2*c*x^2)),x)],
[1/(x*arctan(a*x)^(3/2)*sqrt(c+a^2*c*x^2)),x,1,-2*sqrt(c+a^2*c*x^2)/(a*c*x*sqrt(arctan(a*x)))-2*Unintegrable(1/(x^2*sqrt(c+a^2*c*x^2)*sqrt(arctan(a*x))),x)/a],
[1/(x^2*arctan(a*x)^(3/2)*sqrt(c+a^2*c*x^2)),x,0,Unintegrable(1/(x^2*arctan(a*x)^(3/2)*sqrt(c+a^2*c*x^2)),x)],
[x^m/((c+a^2*c*x^2)^(3/2)*arctan(a*x)^(3/2)),x,0,Unintegrable(x^m/((c+a^2*c*x^2)^(3/2)*arctan(a*x)^(3/2)),x)],
[x^3/((c+a^2*c*x^2)^(3/2)*arctan(a*x)^(3/2)),x,1,-2*x^3/(a*c*sqrt(c+a^2*c*x^2)*sqrt(arctan(a*x)))+6*Unintegrable(x^2/((c+a^2*c*x^2)^(3/2)*sqrt(arctan(a*x))),x)/a+4*a*Unintegrable(x^4/((c+a^2*c*x^2)^(3/2)*sqrt(arctan(a*x))),x)],
[x^2/((c+a^2*c*x^2)^(3/2)*arctan(a*x)^(3/2)),x,5,4*FresnelS(sqrt(2/Pi)*sqrt(arctan(a*x)))*sqrt(2*Pi)*sqrt(1+a^2*x^2)/(a^3*c*sqrt(c+a^2*c*x^2))-2*x^2/(a*c*sqrt(c+a^2*c*x^2)*sqrt(arctan(a*x)))+2*a*Unintegrable(x^3/((c+a^2*c*x^2)^(3/2)*sqrt(arctan(a*x))),x)],
[x/((c+a^2*c*x^2)^(3/2)*arctan(a*x)^(3/2)),x,5,2*FresnelC(sqrt(2/Pi)*sqrt(arctan(a*x)))*sqrt(2*Pi)*sqrt(1+a^2*x^2)/(a^2*c*sqrt(c+a^2*c*x^2))-2*x/(a*c*sqrt(c+a^2*c*x^2)*sqrt(arctan(a*x)))],
[1/((c+a^2*c*x^2)^(3/2)*arctan(a*x)^(3/2)),x,5,-2*FresnelS(sqrt(2/Pi)*sqrt(arctan(a*x)))*sqrt(2*Pi)*sqrt(1+a^2*x^2)/(a*c*sqrt(c+a^2*c*x^2))+(-2)/(a*c*sqrt(c+a^2*c*x^2)*sqrt(arctan(a*x)))],
[1/(x*(c+a^2*c*x^2)^(3/2)*arctan(a*x)^(3/2)),x,5,-4*FresnelC(sqrt(2/Pi)*sqrt(arctan(a*x)))*sqrt(2*Pi)*sqrt(1+a^2*x^2)/(c*sqrt(c+a^2*c*x^2))+(-2)/(a*c*x*sqrt(c+a^2*c*x^2)*sqrt(arctan(a*x)))-2*Unintegrable(1/(x^2*(c+a^2*c*x^2)^(3/2)*sqrt(arctan(a*x))),x)/a],
[1/(x^2*(c+a^2*c*x^2)^(3/2)*arctan(a*x)^(3/2)),x,1,(-2)/(a*c*x^2*sqrt(c+a^2*c*x^2)*sqrt(arctan(a*x)))-4*Unintegrable(1/(x^3*(c+a^2*c*x^2)^(3/2)*sqrt(arctan(a*x))),x)/a-6*a*Unintegrable(1/(x*(c+a^2*c*x^2)^(3/2)*sqrt(arctan(a*x))),x)],
[1/(x^3*(c+a^2*c*x^2)^(3/2)*arctan(a*x)^(3/2)),x,1,(-2)/(a*c*x^3*sqrt(c+a^2*c*x^2)*sqrt(arctan(a*x)))-6*Unintegrable(1/(x^4*(c+a^2*c*x^2)^(3/2)*sqrt(arctan(a*x))),x)/a-8*a*Unintegrable(1/(x^2*(c+a^2*c*x^2)^(3/2)*sqrt(arctan(a*x))),x)],
[1/(x^4*(c+a^2*c*x^2)^(3/2)*arctan(a*x)^(3/2)),x,1,(-2)/(a*c*x^4*sqrt(c+a^2*c*x^2)*sqrt(arctan(a*x)))-8*Unintegrable(1/(x^5*(c+a^2*c*x^2)^(3/2)*sqrt(arctan(a*x))),x)/a-10*a*Unintegrable(1/(x^3*(c+a^2*c*x^2)^(3/2)*sqrt(arctan(a*x))),x)],
[x^m/((c+a^2*c*x^2)^(5/2)*arctan(a*x)^(3/2)),x,0,Unintegrable(x^m/((c+a^2*c*x^2)^(5/2)*arctan(a*x)^(3/2)),x)],
[x^3/((c+a^2*c*x^2)^(5/2)*arctan(a*x)^(3/2)),x,9,3*FresnelC(sqrt(2/Pi)*sqrt(arctan(a*x)))*sqrt(1/2*Pi)*sqrt(1+a^2*x^2)/(a^4*c^2*sqrt(c+a^2*c*x^2))-FresnelC(sqrt(6/Pi)*sqrt(arctan(a*x)))*sqrt(3/2*Pi)*sqrt(1+a^2*x^2)/(a^4*c^2*sqrt(c+a^2*c*x^2))-2*x^3/(a*c*(c+a^2*c*x^2)^(3/2)*sqrt(arctan(a*x)))],
[x^2/((c+a^2*c*x^2)^(5/2)*arctan(a*x)^(3/2)),x,17,FresnelS(sqrt(6/Pi)*sqrt(arctan(a*x)))*sqrt(1/6*Pi)*sqrt(1+a^2*x^2)/(a^3*c^2*sqrt(c+a^2*c*x^2))-3*FresnelS(sqrt(2/Pi)*sqrt(arctan(a*x)))*sqrt(1/2*Pi)*sqrt(1+a^2*x^2)/(a^3*c^2*sqrt(c+a^2*c*x^2))+FresnelS(sqrt(6/Pi)*sqrt(arctan(a*x)))*sqrt(2/3*Pi)*sqrt(1+a^2*x^2)/(a^3*c^2*sqrt(c+a^2*c*x^2))+FresnelS(sqrt(2/Pi)*sqrt(arctan(a*x)))*sqrt(2*Pi)*sqrt(1+a^2*x^2)/(a^3*c^2*sqrt(c+a^2*c*x^2))-2*x^2/(a*c*(c+a^2*c*x^2)^(3/2)*sqrt(arctan(a*x)))],
[x/((c+a^2*c*x^2)^(5/2)*arctan(a*x)^(3/2)),x,17,FresnelC(sqrt(6/Pi)*sqrt(arctan(a*x)))*sqrt(1/6*Pi)*sqrt(1+a^2*x^2)/(a^2*c^2*sqrt(c+a^2*c*x^2))+3*FresnelC(sqrt(2/Pi)*sqrt(arctan(a*x)))*sqrt(1/2*Pi)*sqrt(1+a^2*x^2)/(a^2*c^2*sqrt(c+a^2*c*x^2))+FresnelC(sqrt(6/Pi)*sqrt(arctan(a*x)))*sqrt(2/3*Pi)*sqrt(1+a^2*x^2)/(a^2*c^2*sqrt(c+a^2*c*x^2))-FresnelC(sqrt(2/Pi)*sqrt(arctan(a*x)))*sqrt(2*Pi)*sqrt(1+a^2*x^2)/(a^2*c^2*sqrt(c+a^2*c*x^2))-2*x/(a*c*(c+a^2*c*x^2)^(3/2)*sqrt(arctan(a*x)))],
[1/((c+a^2*c*x^2)^(5/2)*arctan(a*x)^(3/2)),x,9,-3*FresnelS(sqrt(2/Pi)*sqrt(arctan(a*x)))*sqrt(1/2*Pi)*sqrt(1+a^2*x^2)/(a*c^2*sqrt(c+a^2*c*x^2))-FresnelS(sqrt(6/Pi)*sqrt(arctan(a*x)))*sqrt(3/2*Pi)*sqrt(1+a^2*x^2)/(a*c^2*sqrt(c+a^2*c*x^2))+(-2)/(a*c*(c+a^2*c*x^2)^(3/2)*sqrt(arctan(a*x)))],
[1/(x*(c+a^2*c*x^2)^(5/2)*arctan(a*x)^(3/2)),x,9,-2*FresnelC(sqrt(6/Pi)*sqrt(arctan(a*x)))*sqrt(2/3*Pi)*sqrt(1+a^2*x^2)/(c^2*sqrt(c+a^2*c*x^2))-6*FresnelC(sqrt(2/Pi)*sqrt(arctan(a*x)))*sqrt(2*Pi)*sqrt(1+a^2*x^2)/(c^2*sqrt(c+a^2*c*x^2))+(-2)/(a*c*x*(c+a^2*c*x^2)^(3/2)*sqrt(arctan(a*x)))-2*Unintegrable(1/(x^2*(c+a^2*c*x^2)^(5/2)*sqrt(arctan(a*x))),x)/a],
[1/(x^2*(c+a^2*c*x^2)^(5/2)*arctan(a*x)^(3/2)),x,1,(-2)/(a*c*x^2*(c+a^2*c*x^2)^(3/2)*sqrt(arctan(a*x)))-4*Unintegrable(1/(x^3*(c+a^2*c*x^2)^(5/2)*sqrt(arctan(a*x))),x)/a-10*a*Unintegrable(1/(x*(c+a^2*c*x^2)^(5/2)*sqrt(arctan(a*x))),x)],
[1/(x^3*(c+a^2*c*x^2)^(5/2)*arctan(a*x)^(3/2)),x,1,(-2)/(a*c*x^3*(c+a^2*c*x^2)^(3/2)*sqrt(arctan(a*x)))-6*Unintegrable(1/(x^4*(c+a^2*c*x^2)^(5/2)*sqrt(arctan(a*x))),x)/a-12*a*Unintegrable(1/(x^2*(c+a^2*c*x^2)^(5/2)*sqrt(arctan(a*x))),x)],
[1/(x^4*(c+a^2*c*x^2)^(5/2)*arctan(a*x)^(3/2)),x,1,(-2)/(a*c*x^4*(c+a^2*c*x^2)^(3/2)*sqrt(arctan(a*x)))-8*Unintegrable(1/(x^5*(c+a^2*c*x^2)^(5/2)*sqrt(arctan(a*x))),x)/a-14*a*Unintegrable(1/(x^3*(c+a^2*c*x^2)^(5/2)*sqrt(arctan(a*x))),x)],

# Integrands of the form (f x)^m (d+c^2 d x^2)^q (a+b ArcTan[c x])^(-5/2)

# Integrands of the form x^m (c+d x^2)^p / ArcTan[a x]^(5/2) with d=a^2 c

# p>0
[x^m*(c+a^2*c*x^2)/arctan(a*x)^(5/2),x,0,Unintegrable(x^m*(c+a^2*c*x^2)/arctan(a*x)^(5/2),x)],
[x*(c+a^2*c*x^2)/arctan(a*x)^(5/2),x,0,Unintegrable(x*(c+a^2*c*x^2)/arctan(a*x)^(5/2),x)],
[(c+a^2*c*x^2)/arctan(a*x)^(5/2),x,0,Unintegrable((c+a^2*c*x^2)/arctan(a*x)^(5/2),x)],
[(c+a^2*c*x^2)/(x*arctan(a*x)^(5/2)),x,0,Unintegrable((c+a^2*c*x^2)/(x*arctan(a*x)^(5/2)),x)],
[x^m*(c+a^2*c*x^2)^2/arctan(a*x)^(5/2),x,0,Unintegrable(x^m*(c+a^2*c*x^2)^2/arctan(a*x)^(5/2),x)],
[x*(c+a^2*c*x^2)^2/arctan(a*x)^(5/2),x,0,Unintegrable(x*(c+a^2*c*x^2)^2/arctan(a*x)^(5/2),x)],
[(c+a^2*c*x^2)^2/arctan(a*x)^(5/2),x,0,Unintegrable((c+a^2*c*x^2)^2/arctan(a*x)^(5/2),x)],
[(c+a^2*c*x^2)^2/(x*arctan(a*x)^(5/2)),x,0,Unintegrable((c+a^2*c*x^2)^2/(x*arctan(a*x)^(5/2)),x)],
[x^m*(c+a^2*c*x^2)^3/arctan(a*x)^(5/2),x,0,Unintegrable(x^m*(c+a^2*c*x^2)^3/arctan(a*x)^(5/2),x)],
[x*(c+a^2*c*x^2)^3/arctan(a*x)^(5/2),x,0,Unintegrable(x*(c+a^2*c*x^2)^3/arctan(a*x)^(5/2),x)],
[(c+a^2*c*x^2)^3/arctan(a*x)^(5/2),x,0,Unintegrable((c+a^2*c*x^2)^3/arctan(a*x)^(5/2),x)],
[(c+a^2*c*x^2)^3/(x*arctan(a*x)^(5/2)),x,0,Unintegrable((c+a^2*c*x^2)^3/(x*arctan(a*x)^(5/2)),x)],

# p<0
[x^m/((c+a^2*c*x^2)*arctan(a*x)^(5/2)),x,1,-2/3*x^m/(a*c*arctan(a*x)^(3/2))+2/3*m*Unintegrable(x^(-1+m)/arctan(a*x)^(3/2),x)/(a*c)],
[x/((c+a^2*c*x^2)*arctan(a*x)^(5/2)),x,1,-2/3*x/(a*c*arctan(a*x)^(3/2))+2/3*Unintegrable(1/arctan(a*x)^(3/2),x)/(a*c)],
[1/((c+a^2*c*x^2)*arctan(a*x)^(5/2)),x,1,(-2/3)/(a*c*arctan(a*x)^(3/2))],
[1/(x*(c+a^2*c*x^2)*arctan(a*x)^(5/2)),x,1,(-2/3)/(a*c*x*arctan(a*x)^(3/2))-2/3*Unintegrable(1/(x^2*arctan(a*x)^(3/2)),x)/(a*c)],
[x^m/((c+a^2*c*x^2)^2*arctan(a*x)^(5/2)),x,0,Unintegrable(x^m/((c+a^2*c*x^2)^2*arctan(a*x)^(5/2)),x)],
[x^3/((c+a^2*c*x^2)^2*arctan(a*x)^(5/2)),x,8,-2/3*x^3/(a*c^2*(1+a^2*x^2)*arctan(a*x)^(3/2))+4*FresnelS(2*sqrt(arctan(a*x))/sqrt(Pi))*sqrt(Pi)/(a^4*c^2)-4*x^2/(a^2*c^2*(1+a^2*x^2)*sqrt(arctan(a*x)))-4/3*x^4/(c^2*(1+a^2*x^2)*sqrt(arctan(a*x)))+16/3*Unintegrable(x^3/((c+a^2*c*x^2)^2*sqrt(arctan(a*x))),x)+8/3*a^2*Unintegrable(x^5/((c+a^2*c*x^2)^2*sqrt(arctan(a*x))),x)],
[x^2/((c+a^2*c*x^2)^2*arctan(a*x)^(5/2)),x,8,-2/3*x^2/(a*c^2*(1+a^2*x^2)*arctan(a*x)^(3/2))+8/3*FresnelC(2*sqrt(arctan(a*x))/sqrt(Pi))*sqrt(Pi)/(a^3*c^2)-8/3*x/(a^2*c^2*(1+a^2*x^2)*sqrt(arctan(a*x)))+16/3*sqrt(arctan(a*x))/(a^3*c^2)-32/3*sqrt(arctan(a*x))/(a^3*c^2*(1+a^2*x^2))+16/3*(1-a^2*x^2)*sqrt(arctan(a*x))/(a^3*c^2*(1+a^2*x^2))],
[x/((c+a^2*c*x^2)^2*arctan(a*x)^(5/2)),x,6,-2/3*x/(a*c^2*(1+a^2*x^2)*arctan(a*x)^(3/2))-8/3*FresnelS(2*sqrt(arctan(a*x))/sqrt(Pi))*sqrt(Pi)/(a^2*c^2)-4/3*(1-a^2*x^2)/(a^2*c^2*(1+a^2*x^2)*sqrt(arctan(a*x)))],
[1/((c+a^2*c*x^2)^2*arctan(a*x)^(5/2)),x,8,(-2/3)/(a*c^2*(1+a^2*x^2)*arctan(a*x)^(3/2))-8/3*FresnelC(2*sqrt(arctan(a*x))/sqrt(Pi))*sqrt(Pi)/(a*c^2)+8/3*x/(c^2*(1+a^2*x^2)*sqrt(arctan(a*x)))-16/3*sqrt(arctan(a*x))/(a*c^2)+32/3*sqrt(arctan(a*x))/(a*c^2*(1+a^2*x^2))-16/3*(1-a^2*x^2)*sqrt(arctan(a*x))/(a*c^2*(1+a^2*x^2))],
[1/(x*(c+a^2*c*x^2)^2*arctan(a*x)^(5/2)),x,8,(-2/3)/(a*c^2*x*(1+a^2*x^2)*arctan(a*x)^(3/2))+4*FresnelS(2*sqrt(arctan(a*x))/sqrt(Pi))*sqrt(Pi)/c^2+4/(c^2*(1+a^2*x^2)*sqrt(arctan(a*x)))+4/3/(a^2*c^2*x^2*(1+a^2*x^2)*sqrt(arctan(a*x)))+8/3*Unintegrable(1/(x^3*(c+a^2*c*x^2)^2*sqrt(arctan(a*x))),x)/a^2+16/3*Unintegrable(1/(x*(c+a^2*c*x^2)^2*sqrt(arctan(a*x))),x)],
[1/(x^2*(c+a^2*c*x^2)^2*arctan(a*x)^(5/2)),x,8,(-2/3)/(a*c^2*x^2*(1+a^2*x^2)*arctan(a*x)^(3/2))+8*a*FresnelC(2*sqrt(arctan(a*x))/sqrt(Pi))*sqrt(Pi)/c^2+8/3/(a^2*c^2*x^3*(1+a^2*x^2)*sqrt(arctan(a*x)))+16/3/(c^2*x*(1+a^2*x^2)*sqrt(arctan(a*x)))+16*a*sqrt(arctan(a*x))/c^2+8*Unintegrable(1/(x^4*(c+a^2*c*x^2)^2*sqrt(arctan(a*x))),x)/a^2+56/3*Unintegrable(1/(x^2*(c+a^2*c*x^2)^2*sqrt(arctan(a*x))),x)],
[1/(x^3*(c+a^2*c*x^2)^2*arctan(a*x)^(5/2)),x,3,(-2/3)/(a*c^2*x^3*(1+a^2*x^2)*arctan(a*x)^(3/2))+4/(a^2*c^2*x^4*(1+a^2*x^2)*sqrt(arctan(a*x)))+20/3/(c^2*x^2*(1+a^2*x^2)*sqrt(arctan(a*x)))+16*Unintegrable(1/(x^5*(c+a^2*c*x^2)^2*sqrt(arctan(a*x))),x)/a^2+112/3*Unintegrable(1/(x^3*(c+a^2*c*x^2)^2*sqrt(arctan(a*x))),x)+80/3*a^2*Unintegrable(1/(x*(c+a^2*c*x^2)^2*sqrt(arctan(a*x))),x)],
[1/(x^4*(c+a^2*c*x^2)^2*arctan(a*x)^(5/2)),x,3,(-2/3)/(a*c^2*x^4*(1+a^2*x^2)*arctan(a*x)^(3/2))+16/3/(a^2*c^2*x^5*(1+a^2*x^2)*sqrt(arctan(a*x)))+8/(c^2*x^3*(1+a^2*x^2)*sqrt(arctan(a*x)))+80/3*Unintegrable(1/(x^6*(c+a^2*c*x^2)^2*sqrt(arctan(a*x))),x)/a^2+184/3*Unintegrable(1/(x^4*(c+a^2*c*x^2)^2*sqrt(arctan(a*x))),x)+40*a^2*Unintegrable(1/(x^2*(c+a^2*c*x^2)^2*sqrt(arctan(a*x))),x)],
[x^m/((c+a^2*c*x^2)^3*arctan(a*x)^(5/2)),x,0,Unintegrable(x^m/((c+a^2*c*x^2)^3*arctan(a*x)^(5/2)),x)],
[x^3/((c+a^2*c*x^2)^3*arctan(a*x)^(5/2)),x,24,-2/3*x^3/(a*c^3*(1+a^2*x^2)^2*arctan(a*x)^(3/2))-4/3*FresnelS(2*sqrt(arctan(a*x))/sqrt(Pi))*sqrt(Pi)/(a^4*c^3)+4/3*FresnelS(2*sqrt(2/Pi)*sqrt(arctan(a*x)))*sqrt(2*Pi)/(a^4*c^3)-4*x^2/(a^2*c^3*(1+a^2*x^2)^2*sqrt(arctan(a*x)))+4/3*x^4/(c^3*(1+a^2*x^2)^2*sqrt(arctan(a*x)))],
[x^2/((c+a^2*c*x^2)^3*arctan(a*x)^(5/2)),x,27,-2/3*x^2/(a*c^3*(1+a^2*x^2)^2*arctan(a*x)^(3/2))+4/3*FresnelC(2*sqrt(2/Pi)*sqrt(arctan(a*x)))*sqrt(2*Pi)/(a^3*c^3)-8/3*x/(a^2*c^3*(1+a^2*x^2)^2*sqrt(arctan(a*x)))+8/3*x^3/(c^3*(1+a^2*x^2)^2*sqrt(arctan(a*x)))],
[x/((c+a^2*c*x^2)^3*arctan(a*x)^(5/2)),x,24,-2/3*x/(a*c^3*(1+a^2*x^2)^2*arctan(a*x)^(3/2))-4/3*FresnelS(2*sqrt(arctan(a*x))/sqrt(Pi))*sqrt(Pi)/(a^2*c^3)-4/3*FresnelS(2*sqrt(2/Pi)*sqrt(arctan(a*x)))*sqrt(2*Pi)/(a^2*c^3)+(-4/3)/(a^2*c^3*(1+a^2*x^2)^2*sqrt(arctan(a*x)))+4*x^2/(c^3*(1+a^2*x^2)^2*sqrt(arctan(a*x)))],
[1/((c+a^2*c*x^2)^3*arctan(a*x)^(5/2)),x,14,(-2/3)/(a*c^3*(1+a^2*x^2)^2*arctan(a*x)^(3/2))-8/3*FresnelC(2*sqrt(arctan(a*x))/sqrt(Pi))*sqrt(Pi)/(a*c^3)-4/3*FresnelC(2*sqrt(2/Pi)*sqrt(arctan(a*x)))*sqrt(2*Pi)/(a*c^3)+16/3*x/(c^3*(1+a^2*x^2)^2*sqrt(arctan(a*x)))],
[1/(x*(c+a^2*c*x^2)^3*arctan(a*x)^(5/2)),x,10,(-2/3)/(a*c^3*x*(1+a^2*x^2)^2*arctan(a*x)^(3/2))+20/3*FresnelS(2*sqrt(arctan(a*x))/sqrt(Pi))*sqrt(Pi)/c^3+5/3*FresnelS(2*sqrt(2/Pi)*sqrt(arctan(a*x)))*sqrt(2*Pi)/c^3+20/3/(c^3*(1+a^2*x^2)^2*sqrt(arctan(a*x)))+4/3/(a^2*c^3*x^2*(1+a^2*x^2)^2*sqrt(arctan(a*x)))+8/3*Unintegrable(1/(x^3*(c+a^2*c*x^2)^3*sqrt(arctan(a*x))),x)/a^2+8*Unintegrable(1/(x*(c+a^2*c*x^2)^3*sqrt(arctan(a*x))),x)],
[1/(x^2*(c+a^2*c*x^2)^3*arctan(a*x)^(5/2)),x,10,(-2/3)/(a*c^3*x^2*(1+a^2*x^2)^2*arctan(a*x)^(3/2))+5*a*FresnelC(2*sqrt(2/Pi)*sqrt(arctan(a*x)))*sqrt(1/2*Pi)/c^3+20*a*FresnelC(2*sqrt(arctan(a*x))/sqrt(Pi))*sqrt(Pi)/c^3+8/3/(a^2*c^3*x^3*(1+a^2*x^2)^2*sqrt(arctan(a*x)))+8/(c^3*x*(1+a^2*x^2)^2*sqrt(arctan(a*x)))+30*a*sqrt(arctan(a*x))/c^3+8*Unintegrable(1/(x^4*(c+a^2*c*x^2)^3*sqrt(arctan(a*x))),x)/a^2+80/3*Unintegrable(1/(x^2*(c+a^2*c*x^2)^3*sqrt(arctan(a*x))),x)],
[1/(x^3*(c+a^2*c*x^2)^3*arctan(a*x)^(5/2)),x,3,(-2/3)/(a*c^3*x^3*(1+a^2*x^2)^2*arctan(a*x)^(3/2))+4/(a^2*c^3*x^4*(1+a^2*x^2)^2*sqrt(arctan(a*x)))+28/3/(c^3*x^2*(1+a^2*x^2)^2*sqrt(arctan(a*x)))+16*Unintegrable(1/(x^5*(c+a^2*c*x^2)^3*sqrt(arctan(a*x))),x)/a^2+152/3*Unintegrable(1/(x^3*(c+a^2*c*x^2)^3*sqrt(arctan(a*x))),x)+56*a^2*Unintegrable(1/(x*(c+a^2*c*x^2)^3*sqrt(arctan(a*x))),x)],
[1/(x^4*(c+a^2*c*x^2)^3*arctan(a*x)^(5/2)),x,3,(-2/3)/(a*c^3*x^4*(1+a^2*x^2)^2*arctan(a*x)^(3/2))+16/3/(a^2*c^3*x^5*(1+a^2*x^2)^2*sqrt(arctan(a*x)))+32/3/(c^3*x^3*(1+a^2*x^2)^2*sqrt(arctan(a*x)))+80/3*Unintegrable(1/(x^6*(c+a^2*c*x^2)^3*sqrt(arctan(a*x))),x)/a^2+80*Unintegrable(1/(x^4*(c+a^2*c*x^2)^3*sqrt(arctan(a*x))),x)+224/3*a^2*Unintegrable(1/(x^2*(c+a^2*c*x^2)^3*sqrt(arctan(a*x))),x)],

# Integrands of the form x^m (c+d x^2)^(p/2) / ArcTan[a x]^(5/2) with d=a^2 c

# p>0
[x^m*sqrt(c+a^2*c*x^2)/arctan(a*x)^(5/2),x,0,Unintegrable(x^m*sqrt(c+a^2*c*x^2)/arctan(a*x)^(5/2),x)],
[x*sqrt(c+a^2*c*x^2)/arctan(a*x)^(5/2),x,0,Unintegrable(x*sqrt(c+a^2*c*x^2)/arctan(a*x)^(5/2),x)],
[sqrt(c+a^2*c*x^2)/arctan(a*x)^(5/2),x,0,Unintegrable(sqrt(c+a^2*c*x^2)/arctan(a*x)^(5/2),x)],
[sqrt(c+a^2*c*x^2)/(x*arctan(a*x)^(5/2)),x,0,Unintegrable(sqrt(c+a^2*c*x^2)/(x*arctan(a*x)^(5/2)),x)],
[x^m*(c+a^2*c*x^2)^(3/2)/arctan(a*x)^(5/2),x,0,Unintegrable(x^m*(c+a^2*c*x^2)^(3/2)/arctan(a*x)^(5/2),x)],
[x*(c+a^2*c*x^2)^(3/2)/arctan(a*x)^(5/2),x,0,Unintegrable(x*(c+a^2*c*x^2)^(3/2)/arctan(a*x)^(5/2),x)],
[(c+a^2*c*x^2)^(3/2)/arctan(a*x)^(5/2),x,0,Unintegrable((c+a^2*c*x^2)^(3/2)/arctan(a*x)^(5/2),x)],
[(c+a^2*c*x^2)^(3/2)/(x*arctan(a*x)^(5/2)),x,0,Unintegrable((c+a^2*c*x^2)^(3/2)/(x*arctan(a*x)^(5/2)),x)],
[x^m*(c+a^2*c*x^2)^(5/2)/arctan(a*x)^(5/2),x,0,Unintegrable(x^m*(c+a^2*c*x^2)^(5/2)/arctan(a*x)^(5/2),x)],
[x*(c+a^2*c*x^2)^(5/2)/arctan(a*x)^(5/2),x,0,Unintegrable(x*(c+a^2*c*x^2)^(5/2)/arctan(a*x)^(5/2),x)],
[(c+a^2*c*x^2)^(5/2)/arctan(a*x)^(5/2),x,0,Unintegrable((c+a^2*c*x^2)^(5/2)/arctan(a*x)^(5/2),x)],
[(c+a^2*c*x^2)^(5/2)/(x*arctan(a*x)^(5/2)),x,0,Unintegrable((c+a^2*c*x^2)^(5/2)/(x*arctan(a*x)^(5/2)),x)],

# p<0
[x^m/(arctan(a*x)^(5/2)*sqrt(c+a^2*c*x^2)),x,0,Unintegrable(x^m/(arctan(a*x)^(5/2)*sqrt(c+a^2*c*x^2)),x)],
[x/(arctan(a*x)^(5/2)*sqrt(c+a^2*c*x^2)),x,0,Unintegrable(x/(arctan(a*x)^(5/2)*sqrt(c+a^2*c*x^2)),x)],
[1/(arctan(a*x)^(5/2)*sqrt(c+a^2*c*x^2)),x,0,Unintegrable(1/(arctan(a*x)^(5/2)*sqrt(c+a^2*c*x^2)),x)],
[1/(x*arctan(a*x)^(5/2)*sqrt(c+a^2*c*x^2)),x,1,-2/3*sqrt(c+a^2*c*x^2)/(a*c*x*arctan(a*x)^(3/2))-2/3*Unintegrable(1/(x^2*arctan(a*x)^(3/2)*sqrt(c+a^2*c*x^2)),x)/a],
[1/(x^2*arctan(a*x)^(5/2)*sqrt(c+a^2*c*x^2)),x,0,Unintegrable(1/(x^2*arctan(a*x)^(5/2)*sqrt(c+a^2*c*x^2)),x)],
[x^m/((c+a^2*c*x^2)^(3/2)*arctan(a*x)^(5/2)),x,0,Unintegrable(x^m/((c+a^2*c*x^2)^(3/2)*arctan(a*x)^(5/2)),x)],
[x^3/((c+a^2*c*x^2)^(3/2)*arctan(a*x)^(5/2)),x,7,-2/3*x^3/(a*c*arctan(a*x)^(3/2)*sqrt(c+a^2*c*x^2))+8*FresnelS(sqrt(2/Pi)*sqrt(arctan(a*x)))*sqrt(2*Pi)*sqrt(1+a^2*x^2)/(a^4*c*sqrt(c+a^2*c*x^2))-4*x^2/(a^2*c*sqrt(c+a^2*c*x^2)*sqrt(arctan(a*x)))-8/3*x^4/(c*sqrt(c+a^2*c*x^2)*sqrt(arctan(a*x)))+44/3*Unintegrable(x^3/((c+a^2*c*x^2)^(3/2)*sqrt(arctan(a*x))),x)+8*a^2*Unintegrable(x^5/((c+a^2*c*x^2)^(3/2)*sqrt(arctan(a*x))),x)],
[x^2/((c+a^2*c*x^2)^(3/2)*arctan(a*x)^(5/2)),x,7,-2/3*x^2/(a*c*arctan(a*x)^(3/2)*sqrt(c+a^2*c*x^2))+8/3*FresnelC(sqrt(2/Pi)*sqrt(arctan(a*x)))*sqrt(2*Pi)*sqrt(1+a^2*x^2)/(a^3*c*sqrt(c+a^2*c*x^2))-8/3*x/(a^2*c*sqrt(c+a^2*c*x^2)*sqrt(arctan(a*x)))-4/3*x^3/(c*sqrt(c+a^2*c*x^2)*sqrt(arctan(a*x)))+4*Unintegrable(x^2/((c+a^2*c*x^2)^(3/2)*sqrt(arctan(a*x))),x)+8/3*a^2*Unintegrable(x^4/((c+a^2*c*x^2)^(3/2)*sqrt(arctan(a*x))),x)],
[x/((c+a^2*c*x^2)^(3/2)*arctan(a*x)^(5/2)),x,6,-2/3*x/(a*c*arctan(a*x)^(3/2)*sqrt(c+a^2*c*x^2))-4/3*FresnelS(sqrt(2/Pi)*sqrt(arctan(a*x)))*sqrt(2*Pi)*sqrt(1+a^2*x^2)/(a^2*c*sqrt(c+a^2*c*x^2))+(-4/3)/(a^2*c*sqrt(c+a^2*c*x^2)*sqrt(arctan(a*x)))],
[1/((c+a^2*c*x^2)^(3/2)*arctan(a*x)^(5/2)),x,6,(-2/3)/(a*c*arctan(a*x)^(3/2)*sqrt(c+a^2*c*x^2))-4/3*FresnelC(sqrt(2/Pi)*sqrt(arctan(a*x)))*sqrt(2*Pi)*sqrt(1+a^2*x^2)/(a*c*sqrt(c+a^2*c*x^2))+4/3*x/(c*sqrt(c+a^2*c*x^2)*sqrt(arctan(a*x)))],
[1/(x*(c+a^2*c*x^2)^(3/2)*arctan(a*x)^(5/2)),x,7,(-2/3)/(a*c*x*arctan(a*x)^(3/2)*sqrt(c+a^2*c*x^2))+8/3*FresnelS(sqrt(2/Pi)*sqrt(arctan(a*x)))*sqrt(2*Pi)*sqrt(1+a^2*x^2)/(c*sqrt(c+a^2*c*x^2))+8/3/(c*sqrt(c+a^2*c*x^2)*sqrt(arctan(a*x)))+4/3/(a^2*c*x^2*sqrt(c+a^2*c*x^2)*sqrt(arctan(a*x)))+8/3*Unintegrable(1/(x^3*(c+a^2*c*x^2)^(3/2)*sqrt(arctan(a*x))),x)/a^2+4*Unintegrable(1/(x*(c+a^2*c*x^2)^(3/2)*sqrt(arctan(a*x))),x)],
[1/(x^2*(c+a^2*c*x^2)^(3/2)*arctan(a*x)^(5/2)),x,7,(-2/3)/(a*c*x^2*arctan(a*x)^(3/2)*sqrt(c+a^2*c*x^2))+8*a*FresnelC(sqrt(2/Pi)*sqrt(arctan(a*x)))*sqrt(2*Pi)*sqrt(1+a^2*x^2)/(c*sqrt(c+a^2*c*x^2))+8/3/(a^2*c*x^3*sqrt(c+a^2*c*x^2)*sqrt(arctan(a*x)))+4/(c*x*sqrt(c+a^2*c*x^2)*sqrt(arctan(a*x)))+8*Unintegrable(1/(x^4*(c+a^2*c*x^2)^(3/2)*sqrt(arctan(a*x))),x)/a^2+44/3*Unintegrable(1/(x^2*(c+a^2*c*x^2)^(3/2)*sqrt(arctan(a*x))),x)],
[1/(x^3*(c+a^2*c*x^2)^(3/2)*arctan(a*x)^(5/2)),x,3,(-2/3)/(a*c*x^3*arctan(a*x)^(3/2)*sqrt(c+a^2*c*x^2))+4/(a^2*c*x^4*sqrt(c+a^2*c*x^2)*sqrt(arctan(a*x)))+16/3/(c*x^2*sqrt(c+a^2*c*x^2)*sqrt(arctan(a*x)))+16*Unintegrable(1/(x^5*(c+a^2*c*x^2)^(3/2)*sqrt(arctan(a*x))),x)/a^2+92/3*Unintegrable(1/(x^3*(c+a^2*c*x^2)^(3/2)*sqrt(arctan(a*x))),x)+16*a^2*Unintegrable(1/(x*(c+a^2*c*x^2)^(3/2)*sqrt(arctan(a*x))),x)],
[1/(x^4*(c+a^2*c*x^2)^(3/2)*arctan(a*x)^(5/2)),x,3,(-2/3)/(a*c*x^4*arctan(a*x)^(3/2)*sqrt(c+a^2*c*x^2))+16/3/(a^2*c*x^5*sqrt(c+a^2*c*x^2)*sqrt(arctan(a*x)))+20/3/(c*x^3*sqrt(c+a^2*c*x^2)*sqrt(arctan(a*x)))+80/3*Unintegrable(1/(x^6*(c+a^2*c*x^2)^(3/2)*sqrt(arctan(a*x))),x)/a^2+52*Unintegrable(1/(x^4*(c+a^2*c*x^2)^(3/2)*sqrt(arctan(a*x))),x)+80/3*a^2*Unintegrable(1/(x^2*(c+a^2*c*x^2)^(3/2)*sqrt(arctan(a*x))),x)],
[x^m/((c+a^2*c*x^2)^(5/2)*arctan(a*x)^(5/2)),x,0,Unintegrable(x^m/((c+a^2*c*x^2)^(5/2)*arctan(a*x)^(5/2)),x)],
[x^3/((c+a^2*c*x^2)^(5/2)*arctan(a*x)^(5/2)),x,18,-2/3*x^3/(a*c*(c+a^2*c*x^2)^(3/2)*arctan(a*x)^(3/2))-FresnelS(sqrt(2/Pi)*sqrt(arctan(a*x)))*sqrt(2*Pi)*sqrt(1+a^2*x^2)/(a^4*c^2*sqrt(c+a^2*c*x^2))+FresnelS(sqrt(6/Pi)*sqrt(arctan(a*x)))*sqrt(6*Pi)*sqrt(1+a^2*x^2)/(a^4*c^2*sqrt(c+a^2*c*x^2))-4*x^2/(a^2*c*(c+a^2*c*x^2)^(3/2)*sqrt(arctan(a*x)))],
[x^2/((c+a^2*c*x^2)^(5/2)*arctan(a*x)^(5/2)),x,27,-2/3*x^2/(a*c*(c+a^2*c*x^2)^(3/2)*arctan(a*x)^(3/2))-1/3*FresnelC(sqrt(2/Pi)*sqrt(arctan(a*x)))*sqrt(2*Pi)*sqrt(1+a^2*x^2)/(a^3*c^2*sqrt(c+a^2*c*x^2))+FresnelC(sqrt(6/Pi)*sqrt(arctan(a*x)))*sqrt(6*Pi)*sqrt(1+a^2*x^2)/(a^3*c^2*sqrt(c+a^2*c*x^2))-8/3*x/(a^2*c*(c+a^2*c*x^2)^(3/2)*sqrt(arctan(a*x)))+4/3*x^3/(c*(c+a^2*c*x^2)^(3/2)*sqrt(arctan(a*x)))],
[x/((c+a^2*c*x^2)^(5/2)*arctan(a*x)^(5/2)),x,27,-2/3*x/(a*c*(c+a^2*c*x^2)^(3/2)*arctan(a*x)^(3/2))-1/3*FresnelS(sqrt(2/Pi)*sqrt(arctan(a*x)))*sqrt(2*Pi)*sqrt(1+a^2*x^2)/(a^2*c^2*sqrt(c+a^2*c*x^2))-FresnelS(sqrt(6/Pi)*sqrt(arctan(a*x)))*sqrt(6*Pi)*sqrt(1+a^2*x^2)/(a^2*c^2*sqrt(c+a^2*c*x^2))+(-4/3)/(a^2*c*(c+a^2*c*x^2)^(3/2)*sqrt(arctan(a*x)))+8/3*x^2/(c*(c+a^2*c*x^2)^(3/2)*sqrt(arctan(a*x)))],
[1/((c+a^2*c*x^2)^(5/2)*arctan(a*x)^(5/2)),x,18,(-2/3)/(a*c*(c+a^2*c*x^2)^(3/2)*arctan(a*x)^(3/2))-FresnelC(sqrt(2/Pi)*sqrt(arctan(a*x)))*sqrt(2*Pi)*sqrt(1+a^2*x^2)/(a*c^2*sqrt(c+a^2*c*x^2))-FresnelC(sqrt(6/Pi)*sqrt(arctan(a*x)))*sqrt(6*Pi)*sqrt(1+a^2*x^2)/(a*c^2*sqrt(c+a^2*c*x^2))+4*x/(c*(c+a^2*c*x^2)^(3/2)*sqrt(arctan(a*x)))],
[1/(x*(c+a^2*c*x^2)^(5/2)*arctan(a*x)^(5/2)),x,11,(-2/3)/(a*c*x*(c+a^2*c*x^2)^(3/2)*arctan(a*x)^(3/2))+4*FresnelS(sqrt(6/Pi)*sqrt(arctan(a*x)))*sqrt(2/3*Pi)*sqrt(1+a^2*x^2)/(c^2*sqrt(c+a^2*c*x^2))+4*FresnelS(sqrt(2/Pi)*sqrt(arctan(a*x)))*sqrt(2*Pi)*sqrt(1+a^2*x^2)/(c^2*sqrt(c+a^2*c*x^2))+16/3/(c*(c+a^2*c*x^2)^(3/2)*sqrt(arctan(a*x)))+4/3/(a^2*c*x^2*(c+a^2*c*x^2)^(3/2)*sqrt(arctan(a*x)))+8/3*Unintegrable(1/(x^3*(c+a^2*c*x^2)^(5/2)*sqrt(arctan(a*x))),x)/a^2+20/3*Unintegrable(1/(x*(c+a^2*c*x^2)^(5/2)*sqrt(arctan(a*x))),x)],
[1/(x^2*(c+a^2*c*x^2)^(5/2)*arctan(a*x)^(5/2)),x,11,(-2/3)/(a*c*x^2*(c+a^2*c*x^2)^(3/2)*arctan(a*x)^(3/2))+20/3*a*FresnelC(sqrt(6/Pi)*sqrt(arctan(a*x)))*sqrt(2/3*Pi)*sqrt(1+a^2*x^2)/(c^2*sqrt(c+a^2*c*x^2))+20*a*FresnelC(sqrt(2/Pi)*sqrt(arctan(a*x)))*sqrt(2*Pi)*sqrt(1+a^2*x^2)/(c^2*sqrt(c+a^2*c*x^2))+8/3/(a^2*c*x^3*(c+a^2*c*x^2)^(3/2)*sqrt(arctan(a*x)))+20/3/(c*x*(c+a^2*c*x^2)^(3/2)*sqrt(arctan(a*x)))+8*Unintegrable(1/(x^4*(c+a^2*c*x^2)^(5/2)*sqrt(arctan(a*x))),x)/a^2+68/3*Unintegrable(1/(x^2*(c+a^2*c*x^2)^(5/2)*sqrt(arctan(a*x))),x)],
[1/(x^3*(c+a^2*c*x^2)^(5/2)*arctan(a*x)^(5/2)),x,3,(-2/3)/(a*c*x^3*(c+a^2*c*x^2)^(3/2)*arctan(a*x)^(3/2))+4/(a^2*c*x^4*(c+a^2*c*x^2)^(3/2)*sqrt(arctan(a*x)))+8/(c*x^2*(c+a^2*c*x^2)^(3/2)*sqrt(arctan(a*x)))+16*Unintegrable(1/(x^5*(c+a^2*c*x^2)^(5/2)*sqrt(arctan(a*x))),x)/a^2+44*Unintegrable(1/(x^3*(c+a^2*c*x^2)^(5/2)*sqrt(arctan(a*x))),x)+40*a^2*Unintegrable(1/(x*(c+a^2*c*x^2)^(5/2)*sqrt(arctan(a*x))),x)],
[1/(x^4*(c+a^2*c*x^2)^(5/2)*arctan(a*x)^(5/2)),x,3,(-2/3)/(a*c*x^4*(c+a^2*c*x^2)^(3/2)*arctan(a*x)^(3/2))+16/3/(a^2*c*x^5*(c+a^2*c*x^2)^(3/2)*sqrt(arctan(a*x)))+28/3/(c*x^3*(c+a^2*c*x^2)^(3/2)*sqrt(arctan(a*x)))+80/3*Unintegrable(1/(x^6*(c+a^2*c*x^2)^(5/2)*sqrt(arctan(a*x))),x)/a^2+212/3*Unintegrable(1/(x^4*(c+a^2*c*x^2)^(5/2)*sqrt(arctan(a*x))),x)+56*a^2*Unintegrable(1/(x^2*(c+a^2*c*x^2)^(5/2)*sqrt(arctan(a*x))),x)],

# Integrands of the form (f x)^m (d+c^2 d x^2)^q (a+b ArcTan[c x])^p with p symbolic

# Integrands of the form (f x)^m (d+c^2 d x^2)^q (a+b ArcTan[c x])^p with p symbolic
[x*arctan(a*x)^n/(c+a^2*c*x^2),x,1,x*arctan(a*x)^(1+n)/(a*c*(1+n))-Unintegrable(arctan(a*x)^(1+n),x)/(a*c*(1+n))],
[arctan(a*x)^n/(c+a^2*c*x^2),x,1,arctan(a*x)^(1+n)/(a*c*(1+n))],

# Integrands of the form (f x)^m (d+c^2 d x^2)^q (a+b ArcTan[c x])^p with p and q symbolic
[(f*x)^m*(d+c^2*d*x^2)^q*(a+b*arctan(c*x))^p,x,0,Unintegrable((f*x)^m*(d+c^2*d*x^2)^q*(a+b*arctan(c*x))^p,x)],

# Integrands of the form (f x)^m (d+e x^2)^q (a+b ArcTan[c x])^p

# Integrands of the form (f x)^m (d+e x^2)^p (a+b ArcTan[c x])^1

# Integrands of the form x^m (d+e x^2)^p (a+b ArcTan[c x])

# p>0
[x^3*(d+e*x^2)*(a+b*arctan(c*x)),x,5,1/12*b*(3*c^2*d-2*e)*x/c^5-1/36*b*(3*c^2*d-2*e)*x^3/c^3-1/30*b*e*x^5/c-1/12*b*(3*c^2*d-2*e)*arctan(c*x)/c^6+1/4*d*x^4*(a+b*arctan(c*x))+1/6*e*x^6*(a+b*arctan(c*x))],
[x^2*(d+e*x^2)*(a+b*arctan(c*x)),x,4,-1/30*b*(5*c^2*d-3*e)*x^2/c^3-1/20*b*e*x^4/c+1/3*d*x^3*(a+b*arctan(c*x))+1/5*e*x^5*(a+b*arctan(c*x))+1/30*b*(5*c^2*d-3*e)*log(1+c^2*x^2)/c^5],
[x*(d+e*x^2)*(a+b*arctan(c*x)),x,4,-1/4*b*(2*c^2*d-e)*x/c^3-1/12*b*e*x^3/c-1/4*b*(c^2*d-e)^2*arctan(c*x)/(c^4*e)+1/4*(d+e*x^2)^2*(a+b*arctan(c*x))/e],
[(d+e*x^2)*(a+b*arctan(c*x)),x,5,-1/6*b*e*x^2/c+d*x*(a+b*arctan(c*x))+1/3*e*x^3*(a+b*arctan(c*x))-1/6*b*(3*c^2*d-e)*log(1+c^2*x^2)/c^3],
[(d+e*x^2)*(a+b*arctan(c*x))/x,x,8,-1/2*b*e*x/c+1/2*b*e*arctan(c*x)/c^2+1/2*e*x^2*(a+b*arctan(c*x))+a*d*log(x)+1/2*I*b*d*polylog(2,-I*c*x)-1/2*I*b*d*polylog(2,I*c*x)],
[(d+e*x^2)*(a+b*arctan(c*x))/x^2,x,4,-d*(a+b*arctan(c*x))/x+e*x*(a+b*arctan(c*x))+b*c*d*log(x)-1/2*b*(c^2*d+e)*log(1+c^2*x^2)/c],
[(d+e*x^2)*(a+b*arctan(c*x))/x^3,x,8,-1/2*b*c*d/x-1/2*b*c^2*d*arctan(c*x)-1/2*d*(a+b*arctan(c*x))/x^2+a*e*log(x)+1/2*I*b*e*polylog(2,-I*c*x)-1/2*I*b*e*polylog(2,I*c*x)],
[(d+e*x^2)*(a+b*arctan(c*x))/x^4,x,5,-1/6*b*c*d/x^2-1/3*d*(a+b*arctan(c*x))/x^3-e*(a+b*arctan(c*x))/x-1/3*b*c*(c^2*d-3*e)*log(x)+1/6*b*c*(c^2*d-3*e)*log(1+c^2*x^2)],
[(d+e*x^2)*(a+b*arctan(c*x))/x^5,x,5,-1/12*b*c*d/x^3+1/4*b*c*(c^2*d-2*e)/x+1/4*b*c^2*(c^2*d-2*e)*arctan(c*x)-1/4*d*(a+b*arctan(c*x))/x^4-1/2*e*(a+b*arctan(c*x))/x^2],
[(d+e*x^2)*(a+b*arctan(c*x))/x^6,x,5,-1/20*b*c*d/x^4+1/30*b*c*(3*c^2*d-5*e)/x^2-1/5*d*(a+b*arctan(c*x))/x^5-1/3*e*(a+b*arctan(c*x))/x^3+1/15*b*c^3*(3*c^2*d-5*e)*log(x)-1/30*b*c^3*(3*c^2*d-5*e)*log(1+c^2*x^2)],
[(d+e*x^2)*(a+b*arctan(c*x))/x^7,x,6,-1/30*b*c*d/x^5+1/36*b*c*(2*c^2*d-3*e)/x^3-1/12*b*c^3*(2*c^2*d-3*e)/x-1/12*b*c^4*(2*c^2*d-3*e)*arctan(c*x)-1/6*d*(a+b*arctan(c*x))/x^6-1/4*e*(a+b*arctan(c*x))/x^4],
[x^3*(d+e*x^2)^2*(a+b*arctan(c*x)),x,4,1/24*b*(6*c^4*d^2-8*c^2*d*e+3*e^2)*x/c^7-1/72*b*(6*c^4*d^2-8*c^2*d*e+3*e^2)*x^3/c^5-1/120*b*(8*c^2*d-3*e)*e*x^5/c^3-1/56*b*e^2*x^7/c-1/24*b*(6*c^4*d^2-8*c^2*d*e+3*e^2)*arctan(c*x)/c^8+1/4*d^2*x^4*(a+b*arctan(c*x))+1/3*d*e*x^6*(a+b*arctan(c*x))+1/8*e^2*x^8*(a+b*arctan(c*x))],
[x^2*(d+e*x^2)^2*(a+b*arctan(c*x)),x,5,-1/210*b*(35*c^4*d^2-42*c^2*d*e+15*e^2)*x^2/c^5-1/140*b*(14*c^2*d-5*e)*e*x^4/c^3-1/42*b*e^2*x^6/c+1/3*d^2*x^3*(a+b*arctan(c*x))+2/5*d*e*x^5*(a+b*arctan(c*x))+1/7*e^2*x^7*(a+b*arctan(c*x))+1/210*b*(35*c^4*d^2-42*c^2*d*e+15*e^2)*log(1+c^2*x^2)/c^7],
[x*(d+e*x^2)^2*(a+b*arctan(c*x)),x,4,-1/6*b*(3*c^4*d^2-3*c^2*d*e+e^2)*x/c^5-1/18*b*(3*c^2*d-e)*e*x^3/c^3-1/30*b*e^2*x^5/c-1/6*b*(c^2*d-e)^3*arctan(c*x)/(c^6*e)+1/6*(d+e*x^2)^3*(a+b*arctan(c*x))/e],
[(d+e*x^2)^2*(a+b*arctan(c*x)),x,5,-1/30*b*(10*c^2*d-3*e)*e*x^2/c^3-1/20*b*e^2*x^4/c+d^2*x*(a+b*arctan(c*x))+2/3*d*e*x^3*(a+b*arctan(c*x))+1/5*e^2*x^5*(a+b*arctan(c*x))-1/30*b*(15*c^4*d^2-10*c^2*d*e+3*e^2)*log(1+c^2*x^2)/c^5],
[(d+e*x^2)^2*(a+b*arctan(c*x))/x,x,12,-b*d*e*x/c+1/4*b*e^2*x/c^3-1/12*b*e^2*x^3/c+b*d*e*arctan(c*x)/c^2-1/4*b*e^2*arctan(c*x)/c^4+d*e*x^2*(a+b*arctan(c*x))+1/4*e^2*x^4*(a+b*arctan(c*x))+a*d^2*log(x)+1/2*I*b*d^2*polylog(2,-I*c*x)-1/2*I*b*d^2*polylog(2,I*c*x)],
[(d+e*x^2)^2*(a+b*arctan(c*x))/x^2,x,4,-1/6*b*e^2*x^2/c-d^2*(a+b*arctan(c*x))/x+2*d*e*x*(a+b*arctan(c*x))+1/3*e^2*x^3*(a+b*arctan(c*x))+b*c*d^2*log(x)-1/6*b*(3*c^4*d^2+6*c^2*d*e-e^2)*log(1+c^2*x^2)/c^3],
[(d+e*x^2)^2*(a+b*arctan(c*x))/x^3,x,11,-1/2*b*c*d^2/x-1/2*b*e^2*x/c-1/2*b*c^2*d^2*arctan(c*x)+1/2*b*e^2*arctan(c*x)/c^2-1/2*d^2*(a+b*arctan(c*x))/x^2+1/2*e^2*x^2*(a+b*arctan(c*x))+2*a*d*e*log(x)+I*b*d*e*polylog(2,-I*c*x)-I*b*d*e*polylog(2,I*c*x)],
[(d+e*x^2)^2*(a+b*arctan(c*x))/x^4,x,5,-1/6*b*c*d^2/x^2-1/3*d^2*(a+b*arctan(c*x))/x^3-2*d*e*(a+b*arctan(c*x))/x+e^2*x*(a+b*arctan(c*x))-1/3*b*c*d*(c^2*d-6*e)*log(x)+1/6*b*(c^4*d^2-6*c^2*d*e-3*e^2)*log(1+c^2*x^2)/c],
[(d+e*x^2)^2*(a+b*arctan(c*x))/x^5,x,12,-1/12*b*c*d^2/x^3+1/4*b*c^3*d^2/x-b*c*d*e/x+1/4*b*c^4*d^2*arctan(c*x)-b*c^2*d*e*arctan(c*x)-1/4*d^2*(a+b*arctan(c*x))/x^4-d*e*(a+b*arctan(c*x))/x^2+a*e^2*log(x)+1/2*I*b*e^2*polylog(2,-I*c*x)-1/2*I*b*e^2*polylog(2,I*c*x)],
[(d+e*x^2)^2*(a+b*arctan(c*x))/x^6,x,5,-1/20*b*c*d^2/x^4+1/30*b*c*d*(3*c^2*d-10*e)/x^2-1/5*d^2*(a+b*arctan(c*x))/x^5-2/3*d*e*(a+b*arctan(c*x))/x^3-e^2*(a+b*arctan(c*x))/x+1/15*b*c*(3*c^4*d^2-10*c^2*d*e+15*e^2)*log(x)-1/30*b*c*(3*c^4*d^2-10*c^2*d*e+15*e^2)*log(1+c^2*x^2)],
[(d+e*x^2)^2*(a+b*arctan(c*x))/x^7,x,5,-1/30*b*c*d^2/x^5+1/18*b*c*d*(c^2*d-3*e)/x^3-1/6*b*c*(c^4*d^2-3*c^2*d*e+3*e^2)/x-1/6*b*(c^2*d-e)^3*arctan(c*x)/d-1/6*(d+e*x^2)^3*(a+b*arctan(c*x))/(d*x^6)],
[(d+e*x^2)^2*(a+b*arctan(c*x))/x^8,x,5,-1/42*b*c*d^2/x^6+1/140*b*c*d*(5*c^2*d-14*e)/x^4-1/210*b*c*(15*c^4*d^2-42*c^2*d*e+35*e^2)/x^2-1/7*d^2*(a+b*arctan(c*x))/x^7-2/5*d*e*(a+b*arctan(c*x))/x^5-1/3*e^2*(a+b*arctan(c*x))/x^3-1/105*b*c^3*(15*c^4*d^2-42*c^2*d*e+35*e^2)*log(x)+1/210*b*c^3*(15*c^4*d^2-42*c^2*d*e+35*e^2)*log(1+c^2*x^2)],
[x^3*(d+e*x^2)^3*(a+b*arctan(c*x)),x,-8,1/40*b*(10*c^6*d^3-20*c^4*d^2*e+15*c^2*d*e^2-4*e^3)*x/c^9-1/120*b*(10*c^6*d^3-20*c^4*d^2*e+15*c^2*d*e^2-4*e^3)*x^3/c^7-1/200*b*e*(20*c^4*d^2-15*c^2*d*e+4*e^2)*x^5/c^5-1/280*b*(15*c^2*d-4*e)*e^2*x^7/c^3-1/90*b*e^3*x^9/c+1/40*b*(c^2*d-e)^4*(c^2*d+4*e)*arctan(c*x)/(c^10*e^2)-1/8*d*(d+e*x^2)^4*(a+b*arctan(c*x))/e^2+1/10*(d+e*x^2)^5*(a+b*arctan(c*x))/e^2],
[x^2*(d+e*x^2)^3*(a+b*arctan(c*x)),x,5,-1/630*b*(105*c^6*d^3-189*c^4*d^2*e+135*c^2*d*e^2-35*e^3)*x^2/c^7-1/1260*b*e*(189*c^4*d^2-135*c^2*d*e+35*e^2)*x^4/c^5-1/378*b*(27*c^2*d-7*e)*e^2*x^6/c^3-1/72*b*e^3*x^8/c+1/3*d^3*x^3*(a+b*arctan(c*x))+3/5*d^2*e*x^5*(a+b*arctan(c*x))+3/7*d*e^2*x^7*(a+b*arctan(c*x))+1/9*e^3*x^9*(a+b*arctan(c*x))+1/630*b*(105*c^6*d^3-189*c^4*d^2*e+135*c^2*d*e^2-35*e^3)*log(1+c^2*x^2)/c^9],
[x*(d+e*x^2)^3*(a+b*arctan(c*x)),x,4,-1/8*b*(2*c^2*d-e)*(2*c^4*d^2-2*c^2*d*e+e^2)*x/c^7-1/24*b*e*(6*c^4*d^2-4*c^2*d*e+e^2)*x^3/c^5-1/40*b*(4*c^2*d-e)*e^2*x^5/c^3-1/56*b*e^3*x^7/c-1/8*b*(c^2*d-e)^4*arctan(c*x)/(c^8*e)+1/8*(d+e*x^2)^4*(a+b*arctan(c*x))/e],
[(d+e*x^2)^3*(a+b*arctan(c*x)),x,4,-1/70*b*e*(35*c^4*d^2-21*c^2*d*e+5*e^2)*x^2/c^5-1/140*b*(21*c^2*d-5*e)*e^2*x^4/c^3-1/42*b*e^3*x^6/c+d^3*x*(a+b*arctan(c*x))+d^2*e*x^3*(a+b*arctan(c*x))+3/5*d*e^2*x^5*(a+b*arctan(c*x))+1/7*e^3*x^7*(a+b*arctan(c*x))-1/70*b*(35*c^6*d^3-35*c^4*d^2*e+21*c^2*d*e^2-5*e^3)*log(1+c^2*x^2)/c^7],
[(d+e*x^2)^3*(a+b*arctan(c*x))/x,x,16,-3/2*b*d^2*e*x/c+3/4*b*d*e^2*x/c^3-1/6*b*e^3*x/c^5-1/4*b*d*e^2*x^3/c+1/18*b*e^3*x^3/c^3-1/30*b*e^3*x^5/c+3/2*b*d^2*e*arctan(c*x)/c^2-3/4*b*d*e^2*arctan(c*x)/c^4+1/6*b*e^3*arctan(c*x)/c^6+3/2*d^2*e*x^2*(a+b*arctan(c*x))+3/4*d*e^2*x^4*(a+b*arctan(c*x))+1/6*e^3*x^6*(a+b*arctan(c*x))+a*d^3*log(x)+1/2*I*b*d^3*polylog(2,-I*c*x)-1/2*I*b*d^3*polylog(2,I*c*x)],
[(d+e*x^2)^3*(a+b*arctan(c*x))/x^2,x,4,-1/10*b*(5*c^2*d-e)*e^2*x^2/c^3-1/20*b*e^3*x^4/c-d^3*(a+b*arctan(c*x))/x+3*d^2*e*x*(a+b*arctan(c*x))+d*e^2*x^3*(a+b*arctan(c*x))+1/5*e^3*x^5*(a+b*arctan(c*x))+b*c*d^3*log(x)-1/10*b*(5*c^6*d^3+15*c^4*d^2*e-5*c^2*d*e^2+e^3)*log(1+c^2*x^2)/c^5],
[(d+e*x^2)^3*(a+b*arctan(c*x))/x^3,x,15,-1/2*b*c*d^3/x-3/2*b*d*e^2*x/c+1/4*b*e^3*x/c^3-1/12*b*e^3*x^3/c-1/2*b*c^2*d^3*arctan(c*x)+3/2*b*d*e^2*arctan(c*x)/c^2-1/4*b*e^3*arctan(c*x)/c^4-1/2*d^3*(a+b*arctan(c*x))/x^2+3/2*d*e^2*x^2*(a+b*arctan(c*x))+1/4*e^3*x^4*(a+b*arctan(c*x))+3*a*d^2*e*log(x)+3/2*I*b*d^2*e*polylog(2,-I*c*x)-3/2*I*b*d^2*e*polylog(2,I*c*x)],
[(d+e*x^2)^3*(a+b*arctan(c*x))/x^4,x,5,-1/6*b*c*d^3/x^2-1/6*b*e^3*x^2/c-1/3*d^3*(a+b*arctan(c*x))/x^3-3*d^2*e*(a+b*arctan(c*x))/x+3*d*e^2*x*(a+b*arctan(c*x))+1/3*e^3*x^3*(a+b*arctan(c*x))-1/3*b*c*d^2*(c^2*d-9*e)*log(x)+1/6*b*(c^2*d+e)*(c^4*d^2-10*c^2*d*e+e^2)*log(1+c^2*x^2)/c^3],
[(d+e*x^2)^3*(a+b*arctan(c*x))/x^5,x,15,-1/12*b*c*d^3/x^3+1/4*b*c^3*d^3/x-3/2*b*c*d^2*e/x-1/2*b*e^3*x/c+1/4*b*c^4*d^3*arctan(c*x)-3/2*b*c^2*d^2*e*arctan(c*x)+1/2*b*e^3*arctan(c*x)/c^2-1/4*d^3*(a+b*arctan(c*x))/x^4-3/2*d^2*e*(a+b*arctan(c*x))/x^2+1/2*e^3*x^2*(a+b*arctan(c*x))+3*a*d*e^2*log(x)+3/2*I*b*d*e^2*polylog(2,-I*c*x)-3/2*I*b*d*e^2*polylog(2,I*c*x)],
[(d+e*x^2)^3*(a+b*arctan(c*x))/x^6,x,5,-1/20*b*c*d^3/x^4+1/10*b*c*d^2*(c^2*d-5*e)/x^2-1/5*d^3*(a+b*arctan(c*x))/x^5-d^2*e*(a+b*arctan(c*x))/x^3-3*d*e^2*(a+b*arctan(c*x))/x+e^3*x*(a+b*arctan(c*x))+1/5*b*c*d*(c^4*d^2-5*c^2*d*e+15*e^2)*log(x)-1/10*b*(c^6*d^3-5*c^4*d^2*e+15*c^2*d*e^2+5*e^3)*log(1+c^2*x^2)/c],
[(d+e*x^2)^3*(a+b*arctan(c*x))/x^7,x,17,-1/30*b*c*d^3/x^5+1/18*b*c^3*d^3/x^3-1/4*b*c*d^2*e/x^3-1/6*b*c^5*d^3/x+3/4*b*c^3*d^2*e/x-3/2*b*c*d*e^2/x-1/6*b*c^6*d^3*arctan(c*x)+3/4*b*c^4*d^2*e*arctan(c*x)-3/2*b*c^2*d*e^2*arctan(c*x)-1/6*d^3*(a+b*arctan(c*x))/x^6-3/4*d^2*e*(a+b*arctan(c*x))/x^4-3/2*d*e^2*(a+b*arctan(c*x))/x^2+a*e^3*log(x)+1/2*I*b*e^3*polylog(2,-I*c*x)-1/2*I*b*e^3*polylog(2,I*c*x)],
[(d+e*x^2)^3*(a+b*arctan(c*x))/x^8,x,5,-1/42*b*c*d^3/x^6+1/140*b*c*d^2*(5*c^2*d-21*e)/x^4-1/70*b*c*d*(5*c^4*d^2-21*c^2*d*e+35*e^2)/x^2-1/7*d^3*(a+b*arctan(c*x))/x^7-3/5*d^2*e*(a+b*arctan(c*x))/x^5-d*e^2*(a+b*arctan(c*x))/x^3-e^3*(a+b*arctan(c*x))/x-1/35*b*c*(5*c^6*d^3-21*c^4*d^2*e+35*c^2*d*e^2-35*e^3)*log(x)+1/70*b*c*(5*c^6*d^3-21*c^4*d^2*e+35*c^2*d*e^2-35*e^3)*log(1+c^2*x^2)],
[(d+e*x^2)^3*(a+b*arctan(c*x))/x^9,x,5,-1/56*b*c*d^3/x^7+1/40*b*c*d^2*(c^2*d-4*e)/x^5-1/24*b*c*d*(c^4*d^2-4*c^2*d*e+6*e^2)/x^3+1/8*b*c*(c^2*d-2*e)*(c^4*d^2-2*c^2*d*e+2*e^2)/x+1/8*b*(c^2*d-e)^4*arctan(c*x)/d-1/8*(d+e*x^2)^4*(a+b*arctan(c*x))/(d*x^8)],
[(c+d*x^2)^4*arctan(a*x),x,4,-1/630*d*(420*a^6*c^3-378*a^4*c^2*d+180*a^2*c*d^2-35*d^3)*x^2/a^7-1/1260*d^2*(378*a^4*c^2-180*a^2*c*d+35*d^2)*x^4/a^5-1/378*(36*a^2*c-7*d)*d^3*x^6/a^3-1/72*d^4*x^8/a+c^4*x*arctan(a*x)+4/3*c^3*d*x^3*arctan(a*x)+6/5*c^2*d^2*x^5*arctan(a*x)+4/7*c*d^3*x^7*arctan(a*x)+1/9*d^4*x^9*arctan(a*x)-1/630*(315*a^8*c^4-420*a^6*c^3*d+378*a^4*c^2*d^2-180*a^2*c*d^3+35*d^4)*log(1+a^2*x^2)/a^9],

# p<0
[x^3*(a+b*arctan(c*x))/(d+e*x^2),x,14,-1/2*b*x/(c*e)+1/2*b*arctan(c*x)/(c^2*e)+1/2*x^2*(a+b*arctan(c*x))/e+d*(a+b*arctan(c*x))*log(2/(1-I*c*x))/e^2-1/2*d*(a+b*arctan(c*x))*log(2*c*(sqrt(-d)-x*sqrt(e))/((1-I*c*x)*(c*sqrt(-d)-I*sqrt(e))))/e^2-1/2*d*(a+b*arctan(c*x))*log(2*c*(sqrt(-d)+x*sqrt(e))/((1-I*c*x)*(c*sqrt(-d)+I*sqrt(e))))/e^2-1/2*I*b*d*polylog(2,1+(-2)/(1-I*c*x))/e^2+1/4*I*b*d*polylog(2,1-2*c*(sqrt(-d)-x*sqrt(e))/((1-I*c*x)*(c*sqrt(-d)-I*sqrt(e))))/e^2+1/4*I*b*d*polylog(2,1-2*c*(sqrt(-d)+x*sqrt(e))/((1-I*c*x)*(c*sqrt(-d)+I*sqrt(e))))/e^2],
[x*(a+b*arctan(c*x))/(d+e*x^2),x,10,-(a+b*arctan(c*x))*log(2/(1-I*c*x))/e+1/2*(a+b*arctan(c*x))*log(2*c*(sqrt(-d)-x*sqrt(e))/((1-I*c*x)*(c*sqrt(-d)-I*sqrt(e))))/e+1/2*(a+b*arctan(c*x))*log(2*c*(sqrt(-d)+x*sqrt(e))/((1-I*c*x)*(c*sqrt(-d)+I*sqrt(e))))/e+1/2*I*b*polylog(2,1+(-2)/(1-I*c*x))/e-1/4*I*b*polylog(2,1-2*c*(sqrt(-d)-x*sqrt(e))/((1-I*c*x)*(c*sqrt(-d)-I*sqrt(e))))/e-1/4*I*b*polylog(2,1-2*c*(sqrt(-d)+x*sqrt(e))/((1-I*c*x)*(c*sqrt(-d)+I*sqrt(e))))/e],
[(a+b*arctan(c*x))/(x*(d+e*x^2)),x,15,a*log(x)/d+(a+b*arctan(c*x))*log(2/(1-I*c*x))/d-1/2*(a+b*arctan(c*x))*log(2*c*(sqrt(-d)-x*sqrt(e))/((1-I*c*x)*(c*sqrt(-d)-I*sqrt(e))))/d-1/2*(a+b*arctan(c*x))*log(2*c*(sqrt(-d)+x*sqrt(e))/((1-I*c*x)*(c*sqrt(-d)+I*sqrt(e))))/d+1/2*I*b*polylog(2,-I*c*x)/d-1/2*I*b*polylog(2,I*c*x)/d-1/2*I*b*polylog(2,1+(-2)/(1-I*c*x))/d+1/4*I*b*polylog(2,1-2*c*(sqrt(-d)-x*sqrt(e))/((1-I*c*x)*(c*sqrt(-d)-I*sqrt(e))))/d+1/4*I*b*polylog(2,1-2*c*(sqrt(-d)+x*sqrt(e))/((1-I*c*x)*(c*sqrt(-d)+I*sqrt(e))))/d],
[(a+b*arctan(c*x))/(x^3*(d+e*x^2)),x,19,-1/2*b*c/(d*x)-1/2*b*c^2*arctan(c*x)/d+1/2*(-a-b*arctan(c*x))/(d*x^2)-a*e*log(x)/d^2-e*(a+b*arctan(c*x))*log(2/(1-I*c*x))/d^2+1/2*e*(a+b*arctan(c*x))*log(2*c*(sqrt(-d)-x*sqrt(e))/((1-I*c*x)*(c*sqrt(-d)-I*sqrt(e))))/d^2+1/2*e*(a+b*arctan(c*x))*log(2*c*(sqrt(-d)+x*sqrt(e))/((1-I*c*x)*(c*sqrt(-d)+I*sqrt(e))))/d^2-1/2*I*b*e*polylog(2,-I*c*x)/d^2+1/2*I*b*e*polylog(2,I*c*x)/d^2+1/2*I*b*e*polylog(2,1+(-2)/(1-I*c*x))/d^2-1/4*I*b*e*polylog(2,1-2*c*(sqrt(-d)-x*sqrt(e))/((1-I*c*x)*(c*sqrt(-d)-I*sqrt(e))))/d^2-1/4*I*b*e*polylog(2,1-2*c*(sqrt(-d)+x*sqrt(e))/((1-I*c*x)*(c*sqrt(-d)+I*sqrt(e))))/d^2],
[x^2*(a+b*arctan(c*x))/(d+e*x^2),x,23,a*x/e+b*x*arctan(c*x)/e-1/2*b*log(1+c^2*x^2)/(c*e)-1/4*I*b*log(1+I*c*x)*log(c*(sqrt(-d)-x*sqrt(e))/(c*sqrt(-d)-I*sqrt(e)))*sqrt(-d)/e^(3/2)+1/4*I*b*log(1-I*c*x)*log(c*(sqrt(-d)-x*sqrt(e))/(c*sqrt(-d)+I*sqrt(e)))*sqrt(-d)/e^(3/2)-1/4*I*b*log(1-I*c*x)*log(c*(sqrt(-d)+x*sqrt(e))/(c*sqrt(-d)-I*sqrt(e)))*sqrt(-d)/e^(3/2)+1/4*I*b*log(1+I*c*x)*log(c*(sqrt(-d)+x*sqrt(e))/(c*sqrt(-d)+I*sqrt(e)))*sqrt(-d)/e^(3/2)+1/4*I*b*polylog(2,(I-c*x)*sqrt(e)/(c*sqrt(-d)+I*sqrt(e)))*sqrt(-d)/e^(3/2)+1/4*I*b*polylog(2,(I+c*x)*sqrt(e)/(c*sqrt(-d)+I*sqrt(e)))*sqrt(-d)/e^(3/2)-1/4*I*b*polylog(2,(1-I*c*x)*sqrt(e)/(I*c*sqrt(-d)+sqrt(e)))*sqrt(-d)/e^(3/2)-1/4*I*b*polylog(2,(1+I*c*x)*sqrt(e)/(I*c*sqrt(-d)+sqrt(e)))*sqrt(-d)/e^(3/2)-a*arctan(x*sqrt(e)/sqrt(d))*sqrt(d)/e^(3/2)],
[(a+b*arctan(c*x))/(d+e*x^2),x,19,-1/4*I*b*log(1+I*c*x)*log(c*(sqrt(-d)-x*sqrt(e))/(c*sqrt(-d)-I*sqrt(e)))/(sqrt(-d)*sqrt(e))+1/4*I*b*log(1-I*c*x)*log(c*(sqrt(-d)-x*sqrt(e))/(c*sqrt(-d)+I*sqrt(e)))/(sqrt(-d)*sqrt(e))-1/4*I*b*log(1-I*c*x)*log(c*(sqrt(-d)+x*sqrt(e))/(c*sqrt(-d)-I*sqrt(e)))/(sqrt(-d)*sqrt(e))+1/4*I*b*log(1+I*c*x)*log(c*(sqrt(-d)+x*sqrt(e))/(c*sqrt(-d)+I*sqrt(e)))/(sqrt(-d)*sqrt(e))+1/4*I*b*polylog(2,(I-c*x)*sqrt(e)/(c*sqrt(-d)+I*sqrt(e)))/(sqrt(-d)*sqrt(e))+1/4*I*b*polylog(2,(I+c*x)*sqrt(e)/(c*sqrt(-d)+I*sqrt(e)))/(sqrt(-d)*sqrt(e))-1/4*I*b*polylog(2,(1-I*c*x)*sqrt(e)/(I*c*sqrt(-d)+sqrt(e)))/(sqrt(-d)*sqrt(e))-1/4*I*b*polylog(2,(1+I*c*x)*sqrt(e)/(I*c*sqrt(-d)+sqrt(e)))/(sqrt(-d)*sqrt(e))+a*arctan(x*sqrt(e)/sqrt(d))/(sqrt(d)*sqrt(e))],
[(a+b*arctan(c*x))/(x^2*(d+e*x^2)),x,25,(-a-b*arctan(c*x))/(d*x)+b*c*log(x)/d-1/2*b*c*log(1+c^2*x^2)/d-a*arctan(x*sqrt(e)/sqrt(d))*sqrt(e)/d^(3/2)-1/4*I*b*log(1+I*c*x)*log(c*(sqrt(-d)-x*sqrt(e))/(c*sqrt(-d)-I*sqrt(e)))*sqrt(e)/(-d)^(3/2)+1/4*I*b*log(1-I*c*x)*log(c*(sqrt(-d)-x*sqrt(e))/(c*sqrt(-d)+I*sqrt(e)))*sqrt(e)/(-d)^(3/2)-1/4*I*b*log(1-I*c*x)*log(c*(sqrt(-d)+x*sqrt(e))/(c*sqrt(-d)-I*sqrt(e)))*sqrt(e)/(-d)^(3/2)+1/4*I*b*log(1+I*c*x)*log(c*(sqrt(-d)+x*sqrt(e))/(c*sqrt(-d)+I*sqrt(e)))*sqrt(e)/(-d)^(3/2)+1/4*I*b*polylog(2,(I-c*x)*sqrt(e)/(c*sqrt(-d)+I*sqrt(e)))*sqrt(e)/(-d)^(3/2)+1/4*I*b*polylog(2,(I+c*x)*sqrt(e)/(c*sqrt(-d)+I*sqrt(e)))*sqrt(e)/(-d)^(3/2)-1/4*I*b*polylog(2,(1-I*c*x)*sqrt(e)/(I*c*sqrt(-d)+sqrt(e)))*sqrt(e)/(-d)^(3/2)-1/4*I*b*polylog(2,(1+I*c*x)*sqrt(e)/(I*c*sqrt(-d)+sqrt(e)))*sqrt(e)/(-d)^(3/2)],
[x^3*(a+b*arctan(c*x))/(d+e*x^2)^2,x,16,-1/2*b*c^2*d*arctan(c*x)/((c^2*d-e)*e^2)+1/2*d*(a+b*arctan(c*x))/(e^2*(d+e*x^2))-(a+b*arctan(c*x))*log(2/(1-I*c*x))/e^2+1/2*(a+b*arctan(c*x))*log(2*c*(sqrt(-d)-x*sqrt(e))/((1-I*c*x)*(c*sqrt(-d)-I*sqrt(e))))/e^2+1/2*(a+b*arctan(c*x))*log(2*c*(sqrt(-d)+x*sqrt(e))/((1-I*c*x)*(c*sqrt(-d)+I*sqrt(e))))/e^2+1/2*I*b*polylog(2,1+(-2)/(1-I*c*x))/e^2-1/4*I*b*polylog(2,1-2*c*(sqrt(-d)-x*sqrt(e))/((1-I*c*x)*(c*sqrt(-d)-I*sqrt(e))))/e^2-1/4*I*b*polylog(2,1-2*c*(sqrt(-d)+x*sqrt(e))/((1-I*c*x)*(c*sqrt(-d)+I*sqrt(e))))/e^2+1/2*b*c*arctan(x*sqrt(e)/sqrt(d))*sqrt(d)/((c^2*d-e)*e^(3/2))],
[x*(a+b*arctan(c*x))/(d+e*x^2)^2,x,4,1/2*b*c^2*arctan(c*x)/((c^2*d-e)*e)+1/2*(-a-b*arctan(c*x))/(e*(d+e*x^2))-1/2*b*c*arctan(x*sqrt(e)/sqrt(d))/((c^2*d-e)*sqrt(d)*sqrt(e))],
[(a+b*arctan(c*x))/(x*(d+e*x^2)^2),x,19,-1/2*b*c^2*arctan(c*x)/(d*(c^2*d-e))+1/2*(a+b*arctan(c*x))/(d*(d+e*x^2))+a*log(x)/d^2+(a+b*arctan(c*x))*log(2/(1-I*c*x))/d^2-1/2*(a+b*arctan(c*x))*log(2*c*(sqrt(-d)-x*sqrt(e))/((1-I*c*x)*(c*sqrt(-d)-I*sqrt(e))))/d^2-1/2*(a+b*arctan(c*x))*log(2*c*(sqrt(-d)+x*sqrt(e))/((1-I*c*x)*(c*sqrt(-d)+I*sqrt(e))))/d^2+1/2*I*b*polylog(2,-I*c*x)/d^2-1/2*I*b*polylog(2,I*c*x)/d^2-1/2*I*b*polylog(2,1+(-2)/(1-I*c*x))/d^2+1/4*I*b*polylog(2,1-2*c*(sqrt(-d)-x*sqrt(e))/((1-I*c*x)*(c*sqrt(-d)-I*sqrt(e))))/d^2+1/4*I*b*polylog(2,1-2*c*(sqrt(-d)+x*sqrt(e))/((1-I*c*x)*(c*sqrt(-d)+I*sqrt(e))))/d^2+1/2*b*c*arctan(x*sqrt(e)/sqrt(d))*sqrt(e)/(d^(3/2)*(c^2*d-e))],
[(a+b*arctan(c*x))/(x^3*(d+e*x^2)^2),x,22,-1/2*b*c/(d^2*x)-1/2*b*c^2*arctan(c*x)/d^2+1/2*b*c^2*e*arctan(c*x)/(d^2*(c^2*d-e))+1/2*(-a-b*arctan(c*x))/(d^2*x^2)-1/2*e*(a+b*arctan(c*x))/(d^2*(d+e*x^2))-1/2*b*c*e^(3/2)*arctan(x*sqrt(e)/sqrt(d))/(d^(5/2)*(c^2*d-e))-2*a*e*log(x)/d^3-2*e*(a+b*arctan(c*x))*log(2/(1-I*c*x))/d^3+e*(a+b*arctan(c*x))*log(2*c*(sqrt(-d)-x*sqrt(e))/((1-I*c*x)*(c*sqrt(-d)-I*sqrt(e))))/d^3+e*(a+b*arctan(c*x))*log(2*c*(sqrt(-d)+x*sqrt(e))/((1-I*c*x)*(c*sqrt(-d)+I*sqrt(e))))/d^3-I*b*e*polylog(2,-I*c*x)/d^3+I*b*e*polylog(2,I*c*x)/d^3+I*b*e*polylog(2,1+(-2)/(1-I*c*x))/d^3-1/2*I*b*e*polylog(2,1-2*c*(sqrt(-d)-x*sqrt(e))/((1-I*c*x)*(c*sqrt(-d)-I*sqrt(e))))/d^3-1/2*I*b*e*polylog(2,1-2*c*(sqrt(-d)+x*sqrt(e))/((1-I*c*x)*(c*sqrt(-d)+I*sqrt(e))))/d^3],
[x^2*(a+b*arctan(c*x))/(d+e*x^2)^2,x,45,-1/2*x*(a+b*arctan(c*x))/(e*(d+e*x^2))+1/4*b*c*log(1+c^2*x^2)/((c^2*d-e)*e)-1/4*b*c*log(d+e*x^2)/((c^2*d-e)*e)-1/4*I*b*log(1+I*c*x)*log(c*(sqrt(-d)-x*sqrt(e))/(c*sqrt(-d)-I*sqrt(e)))/(e^(3/2)*sqrt(-d))+1/4*I*b*log(1-I*c*x)*log(c*(sqrt(-d)-x*sqrt(e))/(c*sqrt(-d)+I*sqrt(e)))/(e^(3/2)*sqrt(-d))-1/4*I*b*log(1-I*c*x)*log(c*(sqrt(-d)+x*sqrt(e))/(c*sqrt(-d)-I*sqrt(e)))/(e^(3/2)*sqrt(-d))+1/4*I*b*log(1+I*c*x)*log(c*(sqrt(-d)+x*sqrt(e))/(c*sqrt(-d)+I*sqrt(e)))/(e^(3/2)*sqrt(-d))+1/4*I*b*polylog(2,(I-c*x)*sqrt(e)/(c*sqrt(-d)+I*sqrt(e)))/(e^(3/2)*sqrt(-d))+1/4*I*b*polylog(2,(I+c*x)*sqrt(e)/(c*sqrt(-d)+I*sqrt(e)))/(e^(3/2)*sqrt(-d))-1/4*I*b*polylog(2,(1-I*c*x)*sqrt(e)/(I*c*sqrt(-d)+sqrt(e)))/(e^(3/2)*sqrt(-d))-1/4*I*b*polylog(2,(1+I*c*x)*sqrt(e)/(I*c*sqrt(-d)+sqrt(e)))/(e^(3/2)*sqrt(-d))+a*arctan(x*sqrt(e)/sqrt(d))/(e^(3/2)*sqrt(d))-1/2*(a+b*arctan(c*x))*arctan(x*sqrt(e)/sqrt(d))/(e^(3/2)*sqrt(d))+1/8*I*b*c*log(-(1+x*sqrt(-c^2))*sqrt(e)/(I*sqrt(-c^2)*sqrt(d)-sqrt(e)))*log(1-I*x*sqrt(e)/sqrt(d))/(e^(3/2)*sqrt(-c^2)*sqrt(d))-1/8*I*b*c*log((1-x*sqrt(-c^2))*sqrt(e)/(I*sqrt(-c^2)*sqrt(d)+sqrt(e)))*log(1-I*x*sqrt(e)/sqrt(d))/(e^(3/2)*sqrt(-c^2)*sqrt(d))+1/8*I*b*c*log(-(1-x*sqrt(-c^2))*sqrt(e)/(I*sqrt(-c^2)*sqrt(d)-sqrt(e)))*log(1+I*x*sqrt(e)/sqrt(d))/(e^(3/2)*sqrt(-c^2)*sqrt(d))-1/8*I*b*c*log((1+x*sqrt(-c^2))*sqrt(e)/(I*sqrt(-c^2)*sqrt(d)+sqrt(e)))*log(1+I*x*sqrt(e)/sqrt(d))/(e^(3/2)*sqrt(-c^2)*sqrt(d))-1/8*I*b*c*polylog(2,sqrt(-c^2)*(sqrt(d)-I*x*sqrt(e))/(sqrt(-c^2)*sqrt(d)-I*sqrt(e)))/(e^(3/2)*sqrt(-c^2)*sqrt(d))+1/8*I*b*c*polylog(2,sqrt(-c^2)*(sqrt(d)-I*x*sqrt(e))/(sqrt(-c^2)*sqrt(d)+I*sqrt(e)))/(e^(3/2)*sqrt(-c^2)*sqrt(d))-1/8*I*b*c*polylog(2,sqrt(-c^2)*(sqrt(d)+I*x*sqrt(e))/(sqrt(-c^2)*sqrt(d)-I*sqrt(e)))/(e^(3/2)*sqrt(-c^2)*sqrt(d))+1/8*I*b*c*polylog(2,sqrt(-c^2)*(sqrt(d)+I*x*sqrt(e))/(sqrt(-c^2)*sqrt(d)+I*sqrt(e)))/(e^(3/2)*sqrt(-c^2)*sqrt(d))],
[(a+b*arctan(c*x))/(d+e*x^2)^2,x,24,1/2*x*(a+b*arctan(c*x))/(d*(d+e*x^2))-1/4*b*c*log(1+c^2*x^2)/(d*(c^2*d-e))+1/4*b*c*log(d+e*x^2)/(d*(c^2*d-e))+1/2*(a+b*arctan(c*x))*arctan(x*sqrt(e)/sqrt(d))/(d^(3/2)*sqrt(e))-1/8*I*b*c*log(-(1+x*sqrt(-c^2))*sqrt(e)/(I*sqrt(-c^2)*sqrt(d)-sqrt(e)))*log(1-I*x*sqrt(e)/sqrt(d))/(d^(3/2)*sqrt(-c^2)*sqrt(e))+1/8*I*b*c*log((1-x*sqrt(-c^2))*sqrt(e)/(I*sqrt(-c^2)*sqrt(d)+sqrt(e)))*log(1-I*x*sqrt(e)/sqrt(d))/(d^(3/2)*sqrt(-c^2)*sqrt(e))-1/8*I*b*c*log(-(1-x*sqrt(-c^2))*sqrt(e)/(I*sqrt(-c^2)*sqrt(d)-sqrt(e)))*log(1+I*x*sqrt(e)/sqrt(d))/(d^(3/2)*sqrt(-c^2)*sqrt(e))+1/8*I*b*c*log((1+x*sqrt(-c^2))*sqrt(e)/(I*sqrt(-c^2)*sqrt(d)+sqrt(e)))*log(1+I*x*sqrt(e)/sqrt(d))/(d^(3/2)*sqrt(-c^2)*sqrt(e))+1/8*I*b*c*polylog(2,sqrt(-c^2)*(sqrt(d)-I*x*sqrt(e))/(sqrt(-c^2)*sqrt(d)-I*sqrt(e)))/(d^(3/2)*sqrt(-c^2)*sqrt(e))-1/8*I*b*c*polylog(2,sqrt(-c^2)*(sqrt(d)-I*x*sqrt(e))/(sqrt(-c^2)*sqrt(d)+I*sqrt(e)))/(d^(3/2)*sqrt(-c^2)*sqrt(e))+1/8*I*b*c*polylog(2,sqrt(-c^2)*(sqrt(d)+I*x*sqrt(e))/(sqrt(-c^2)*sqrt(d)-I*sqrt(e)))/(d^(3/2)*sqrt(-c^2)*sqrt(e))-1/8*I*b*c*polylog(2,sqrt(-c^2)*(sqrt(d)+I*x*sqrt(e))/(sqrt(-c^2)*sqrt(d)+I*sqrt(e)))/(d^(3/2)*sqrt(-c^2)*sqrt(e))],
[(a+b*arctan(c*x))/(x^2*(d+e*x^2)^2),x,50,(-a-b*arctan(c*x))/(d^2*x)-1/2*e*x*(a+b*arctan(c*x))/(d^2*(d+e*x^2))+b*c*log(x)/d^2-1/2*b*c*log(1+c^2*x^2)/d^2+1/4*b*c*e*log(1+c^2*x^2)/(d^2*(c^2*d-e))-1/4*b*c*e*log(d+e*x^2)/(d^2*(c^2*d-e))-a*arctan(x*sqrt(e)/sqrt(d))*sqrt(e)/d^(5/2)-1/2*(a+b*arctan(c*x))*arctan(x*sqrt(e)/sqrt(d))*sqrt(e)/d^(5/2)+1/4*I*b*log(1+I*c*x)*log(c*(sqrt(-d)-x*sqrt(e))/(c*sqrt(-d)-I*sqrt(e)))*sqrt(e)/(-d)^(5/2)-1/4*I*b*log(1-I*c*x)*log(c*(sqrt(-d)-x*sqrt(e))/(c*sqrt(-d)+I*sqrt(e)))*sqrt(e)/(-d)^(5/2)+1/4*I*b*log(1-I*c*x)*log(c*(sqrt(-d)+x*sqrt(e))/(c*sqrt(-d)-I*sqrt(e)))*sqrt(e)/(-d)^(5/2)-1/4*I*b*log(1+I*c*x)*log(c*(sqrt(-d)+x*sqrt(e))/(c*sqrt(-d)+I*sqrt(e)))*sqrt(e)/(-d)^(5/2)-1/4*I*b*polylog(2,(I-c*x)*sqrt(e)/(c*sqrt(-d)+I*sqrt(e)))*sqrt(e)/(-d)^(5/2)-1/4*I*b*polylog(2,(I+c*x)*sqrt(e)/(c*sqrt(-d)+I*sqrt(e)))*sqrt(e)/(-d)^(5/2)+1/4*I*b*polylog(2,(1-I*c*x)*sqrt(e)/(I*c*sqrt(-d)+sqrt(e)))*sqrt(e)/(-d)^(5/2)+1/4*I*b*polylog(2,(1+I*c*x)*sqrt(e)/(I*c*sqrt(-d)+sqrt(e)))*sqrt(e)/(-d)^(5/2)+1/8*I*b*c*log(-(1+x*sqrt(-c^2))*sqrt(e)/(I*sqrt(-c^2)*sqrt(d)-sqrt(e)))*log(1-I*x*sqrt(e)/sqrt(d))*sqrt(e)/(d^(5/2)*sqrt(-c^2))-1/8*I*b*c*log((1-x*sqrt(-c^2))*sqrt(e)/(I*sqrt(-c^2)*sqrt(d)+sqrt(e)))*log(1-I*x*sqrt(e)/sqrt(d))*sqrt(e)/(d^(5/2)*sqrt(-c^2))+1/8*I*b*c*log(-(1-x*sqrt(-c^2))*sqrt(e)/(I*sqrt(-c^2)*sqrt(d)-sqrt(e)))*log(1+I*x*sqrt(e)/sqrt(d))*sqrt(e)/(d^(5/2)*sqrt(-c^2))-1/8*I*b*c*log((1+x*sqrt(-c^2))*sqrt(e)/(I*sqrt(-c^2)*sqrt(d)+sqrt(e)))*log(1+I*x*sqrt(e)/sqrt(d))*sqrt(e)/(d^(5/2)*sqrt(-c^2))-1/8*I*b*c*polylog(2,sqrt(-c^2)*(sqrt(d)-I*x*sqrt(e))/(sqrt(-c^2)*sqrt(d)-I*sqrt(e)))*sqrt(e)/(d^(5/2)*sqrt(-c^2))+1/8*I*b*c*polylog(2,sqrt(-c^2)*(sqrt(d)-I*x*sqrt(e))/(sqrt(-c^2)*sqrt(d)+I*sqrt(e)))*sqrt(e)/(d^(5/2)*sqrt(-c^2))-1/8*I*b*c*polylog(2,sqrt(-c^2)*(sqrt(d)+I*x*sqrt(e))/(sqrt(-c^2)*sqrt(d)-I*sqrt(e)))*sqrt(e)/(d^(5/2)*sqrt(-c^2))+1/8*I*b*c*polylog(2,sqrt(-c^2)*(sqrt(d)+I*x*sqrt(e))/(sqrt(-c^2)*sqrt(d)+I*sqrt(e)))*sqrt(e)/(d^(5/2)*sqrt(-c^2))],
[x^5*(a+b*arctan(c*x))/(d+e*x^2)^3,x,21,-1/8*b*c*d*x/((c^2*d-e)*e^2*(d+e*x^2))+1/4*b*c^4*d^2*arctan(c*x)/((c^2*d-e)^2*e^3)-b*c^2*d*arctan(c*x)/((c^2*d-e)*e^3)-1/4*d^2*(a+b*arctan(c*x))/(e^3*(d+e*x^2)^2)+d*(a+b*arctan(c*x))/(e^3*(d+e*x^2))-(a+b*arctan(c*x))*log(2/(1-I*c*x))/e^3+1/2*(a+b*arctan(c*x))*log(2*c*(sqrt(-d)-x*sqrt(e))/((1-I*c*x)*(c*sqrt(-d)-I*sqrt(e))))/e^3+1/2*(a+b*arctan(c*x))*log(2*c*(sqrt(-d)+x*sqrt(e))/((1-I*c*x)*(c*sqrt(-d)+I*sqrt(e))))/e^3+1/2*I*b*polylog(2,1+(-2)/(1-I*c*x))/e^3-1/4*I*b*polylog(2,1-2*c*(sqrt(-d)-x*sqrt(e))/((1-I*c*x)*(c*sqrt(-d)-I*sqrt(e))))/e^3-1/4*I*b*polylog(2,1-2*c*(sqrt(-d)+x*sqrt(e))/((1-I*c*x)*(c*sqrt(-d)+I*sqrt(e))))/e^3+b*c*arctan(x*sqrt(e)/sqrt(d))*sqrt(d)/((c^2*d-e)*e^(5/2))-1/8*b*c*(3*c^2*d-e)*arctan(x*sqrt(e)/sqrt(d))*sqrt(d)/((c^2*d-e)^2*e^(5/2))],
[x^3*(a+b*arctan(c*x))/(d+e*x^2)^3,x,6,1/8*b*c*x/((c^2*d-e)*e*(d+e*x^2))-1/4*b*arctan(c*x)/(d*(c^2*d-e)^2)+1/4*x^4*(a+b*arctan(c*x))/(d*(d+e*x^2)^2)-1/8*b*c*(c^2*d-3*e)*arctan(x*sqrt(e)/sqrt(d))/((c^2*d-e)^2*e^(3/2)*sqrt(d))],
[x*(a+b*arctan(c*x))/(d+e*x^2)^3,x,5,-1/8*b*c*x/(d*(c^2*d-e)*(d+e*x^2))+1/4*b*c^4*arctan(c*x)/((c^2*d-e)^2*e)+1/4*(-a-b*arctan(c*x))/(e*(d+e*x^2)^2)-1/8*b*c*(3*c^2*d-e)*arctan(x*sqrt(e)/sqrt(d))/(d^(3/2)*(c^2*d-e)^2*sqrt(e))],
[(a+b*arctan(c*x))/(x*(d+e*x^2)^3),x,24,1/8*b*c*e*x/(d^2*(c^2*d-e)*(d+e*x^2))-1/4*b*c^4*arctan(c*x)/(d*(c^2*d-e)^2)-1/2*b*c^2*arctan(c*x)/(d^2*(c^2*d-e))+1/4*(a+b*arctan(c*x))/(d*(d+e*x^2)^2)+1/2*(a+b*arctan(c*x))/(d^2*(d+e*x^2))+a*log(x)/d^3+(a+b*arctan(c*x))*log(2/(1-I*c*x))/d^3-1/2*(a+b*arctan(c*x))*log(2*c*(sqrt(-d)-x*sqrt(e))/((1-I*c*x)*(c*sqrt(-d)-I*sqrt(e))))/d^3-1/2*(a+b*arctan(c*x))*log(2*c*(sqrt(-d)+x*sqrt(e))/((1-I*c*x)*(c*sqrt(-d)+I*sqrt(e))))/d^3+1/2*I*b*polylog(2,-I*c*x)/d^3-1/2*I*b*polylog(2,I*c*x)/d^3-1/2*I*b*polylog(2,1+(-2)/(1-I*c*x))/d^3+1/4*I*b*polylog(2,1-2*c*(sqrt(-d)-x*sqrt(e))/((1-I*c*x)*(c*sqrt(-d)-I*sqrt(e))))/d^3+1/4*I*b*polylog(2,1-2*c*(sqrt(-d)+x*sqrt(e))/((1-I*c*x)*(c*sqrt(-d)+I*sqrt(e))))/d^3+1/2*b*c*arctan(x*sqrt(e)/sqrt(d))*sqrt(e)/(d^(5/2)*(c^2*d-e))+1/8*b*c*(3*c^2*d-e)*arctan(x*sqrt(e)/sqrt(d))*sqrt(e)/(d^(5/2)*(c^2*d-e)^2)],
[(a+b*arctan(c*x))/(x^3*(d+e*x^2)^3),x,27,-1/2*b*c/(d^3*x)-1/8*b*c*e^2*x/(d^3*(c^2*d-e)*(d+e*x^2))-1/2*b*c^2*arctan(c*x)/d^3+1/4*b*c^4*e*arctan(c*x)/(d^2*(c^2*d-e)^2)+b*c^2*e*arctan(c*x)/(d^3*(c^2*d-e))+1/2*(-a-b*arctan(c*x))/(d^3*x^2)-1/4*e*(a+b*arctan(c*x))/(d^2*(d+e*x^2)^2)-e*(a+b*arctan(c*x))/(d^3*(d+e*x^2))-b*c*e^(3/2)*arctan(x*sqrt(e)/sqrt(d))/(d^(7/2)*(c^2*d-e))-1/8*b*c*(3*c^2*d-e)*e^(3/2)*arctan(x*sqrt(e)/sqrt(d))/(d^(7/2)*(c^2*d-e)^2)-3*a*e*log(x)/d^4-3*e*(a+b*arctan(c*x))*log(2/(1-I*c*x))/d^4+3/2*e*(a+b*arctan(c*x))*log(2*c*(sqrt(-d)-x*sqrt(e))/((1-I*c*x)*(c*sqrt(-d)-I*sqrt(e))))/d^4+3/2*e*(a+b*arctan(c*x))*log(2*c*(sqrt(-d)+x*sqrt(e))/((1-I*c*x)*(c*sqrt(-d)+I*sqrt(e))))/d^4-3/2*I*b*e*polylog(2,-I*c*x)/d^4+3/2*I*b*e*polylog(2,I*c*x)/d^4+3/2*I*b*e*polylog(2,1+(-2)/(1-I*c*x))/d^4-3/4*I*b*e*polylog(2,1-2*c*(sqrt(-d)-x*sqrt(e))/((1-I*c*x)*(c*sqrt(-d)-I*sqrt(e))))/d^4-3/4*I*b*e*polylog(2,1-2*c*(sqrt(-d)+x*sqrt(e))/((1-I*c*x)*(c*sqrt(-d)+I*sqrt(e))))/d^4],
[x^2*(a+b*arctan(c*x))/(d+e*x^2)^3,x,49,1/8*b*c/((c^2*d-e)*e*(d+e*x^2))-1/4*x*(a+b*arctan(c*x))/(e*(d+e*x^2)^2)+1/8*x*(a+b*arctan(c*x))/(d*e*(d+e*x^2))+1/8*(a+b*arctan(c*x))*arctan(x*sqrt(e)/sqrt(d))/(d^(3/2)*e^(3/2))+1/16*b*c*(5*c^2*d-3*e)*log(1+c^2*x^2)/(d*(c^2*d-e)^2*e)-1/4*b*c*log(1+c^2*x^2)/(d*(c^2*d-e)*e)-1/16*b*c*(5*c^2*d-3*e)*log(d+e*x^2)/(d*(c^2*d-e)^2*e)+1/4*b*c*log(d+e*x^2)/(d*(c^2*d-e)*e)-1/32*I*b*c*log(-(1+x*sqrt(-c^2))*sqrt(e)/(I*sqrt(-c^2)*sqrt(d)-sqrt(e)))*log(1-I*x*sqrt(e)/sqrt(d))/(d^(3/2)*e^(3/2)*sqrt(-c^2))+1/32*I*b*c*log((1-x*sqrt(-c^2))*sqrt(e)/(I*sqrt(-c^2)*sqrt(d)+sqrt(e)))*log(1-I*x*sqrt(e)/sqrt(d))/(d^(3/2)*e^(3/2)*sqrt(-c^2))-1/32*I*b*c*log(-(1-x*sqrt(-c^2))*sqrt(e)/(I*sqrt(-c^2)*sqrt(d)-sqrt(e)))*log(1+I*x*sqrt(e)/sqrt(d))/(d^(3/2)*e^(3/2)*sqrt(-c^2))+1/32*I*b*c*log((1+x*sqrt(-c^2))*sqrt(e)/(I*sqrt(-c^2)*sqrt(d)+sqrt(e)))*log(1+I*x*sqrt(e)/sqrt(d))/(d^(3/2)*e^(3/2)*sqrt(-c^2))+1/32*I*b*c*polylog(2,sqrt(-c^2)*(sqrt(d)-I*x*sqrt(e))/(sqrt(-c^2)*sqrt(d)-I*sqrt(e)))/(d^(3/2)*e^(3/2)*sqrt(-c^2))-1/32*I*b*c*polylog(2,sqrt(-c^2)*(sqrt(d)-I*x*sqrt(e))/(sqrt(-c^2)*sqrt(d)+I*sqrt(e)))/(d^(3/2)*e^(3/2)*sqrt(-c^2))+1/32*I*b*c*polylog(2,sqrt(-c^2)*(sqrt(d)+I*x*sqrt(e))/(sqrt(-c^2)*sqrt(d)-I*sqrt(e)))/(d^(3/2)*e^(3/2)*sqrt(-c^2))-1/32*I*b*c*polylog(2,sqrt(-c^2)*(sqrt(d)+I*x*sqrt(e))/(sqrt(-c^2)*sqrt(d)+I*sqrt(e)))/(d^(3/2)*e^(3/2)*sqrt(-c^2))],
[(a+b*arctan(c*x))/(d+e*x^2)^3,x,23,-1/8*b*c/(d*(c^2*d-e)*(d+e*x^2))+1/4*x*(a+b*arctan(c*x))/(d*(d+e*x^2)^2)+3/8*x*(a+b*arctan(c*x))/(d^2*(d+e*x^2))-1/16*b*c*(5*c^2*d-3*e)*log(1+c^2*x^2)/(d^2*(c^2*d-e)^2)+1/16*b*c*(5*c^2*d-3*e)*log(d+e*x^2)/(d^2*(c^2*d-e)^2)+3/8*(a+b*arctan(c*x))*arctan(x*sqrt(e)/sqrt(d))/(d^(5/2)*sqrt(e))-3/32*I*b*c*log(-(1+x*sqrt(-c^2))*sqrt(e)/(I*sqrt(-c^2)*sqrt(d)-sqrt(e)))*log(1-I*x*sqrt(e)/sqrt(d))/(d^(5/2)*sqrt(-c^2)*sqrt(e))+3/32*I*b*c*log((1-x*sqrt(-c^2))*sqrt(e)/(I*sqrt(-c^2)*sqrt(d)+sqrt(e)))*log(1-I*x*sqrt(e)/sqrt(d))/(d^(5/2)*sqrt(-c^2)*sqrt(e))-3/32*I*b*c*log(-(1-x*sqrt(-c^2))*sqrt(e)/(I*sqrt(-c^2)*sqrt(d)-sqrt(e)))*log(1+I*x*sqrt(e)/sqrt(d))/(d^(5/2)*sqrt(-c^2)*sqrt(e))+3/32*I*b*c*log((1+x*sqrt(-c^2))*sqrt(e)/(I*sqrt(-c^2)*sqrt(d)+sqrt(e)))*log(1+I*x*sqrt(e)/sqrt(d))/(d^(5/2)*sqrt(-c^2)*sqrt(e))+3/32*I*b*c*polylog(2,sqrt(-c^2)*(sqrt(d)-I*x*sqrt(e))/(sqrt(-c^2)*sqrt(d)-I*sqrt(e)))/(d^(5/2)*sqrt(-c^2)*sqrt(e))-3/32*I*b*c*polylog(2,sqrt(-c^2)*(sqrt(d)-I*x*sqrt(e))/(sqrt(-c^2)*sqrt(d)+I*sqrt(e)))/(d^(5/2)*sqrt(-c^2)*sqrt(e))+3/32*I*b*c*polylog(2,sqrt(-c^2)*(sqrt(d)+I*x*sqrt(e))/(sqrt(-c^2)*sqrt(d)-I*sqrt(e)))/(d^(5/2)*sqrt(-c^2)*sqrt(e))-3/32*I*b*c*polylog(2,sqrt(-c^2)*(sqrt(d)+I*x*sqrt(e))/(sqrt(-c^2)*sqrt(d)+I*sqrt(e)))/(d^(5/2)*sqrt(-c^2)*sqrt(e))],
[(a+b*arctan(c*x))/(x^2*(d+e*x^2)^3),x,73,1/8*b*c*e/(d^2*(c^2*d-e)*(d+e*x^2))+(-a-b*arctan(c*x))/(d^3*x)-1/4*e*x*(a+b*arctan(c*x))/(d^2*(d+e*x^2)^2)-7/8*e*x*(a+b*arctan(c*x))/(d^3*(d+e*x^2))+b*c*log(x)/d^3-1/2*b*c*log(1+c^2*x^2)/d^3+1/16*b*c*(5*c^2*d-3*e)*e*log(1+c^2*x^2)/(d^3*(c^2*d-e)^2)+1/4*b*c*e*log(1+c^2*x^2)/(d^3*(c^2*d-e))-1/16*b*c*(5*c^2*d-3*e)*e*log(d+e*x^2)/(d^3*(c^2*d-e)^2)-1/4*b*c*e*log(d+e*x^2)/(d^3*(c^2*d-e))-a*arctan(x*sqrt(e)/sqrt(d))*sqrt(e)/d^(7/2)-7/8*(a+b*arctan(c*x))*arctan(x*sqrt(e)/sqrt(d))*sqrt(e)/d^(7/2)-1/4*I*b*log(1+I*c*x)*log(c*(sqrt(-d)-x*sqrt(e))/(c*sqrt(-d)-I*sqrt(e)))*sqrt(e)/(-d)^(7/2)+1/4*I*b*log(1-I*c*x)*log(c*(sqrt(-d)-x*sqrt(e))/(c*sqrt(-d)+I*sqrt(e)))*sqrt(e)/(-d)^(7/2)-1/4*I*b*log(1-I*c*x)*log(c*(sqrt(-d)+x*sqrt(e))/(c*sqrt(-d)-I*sqrt(e)))*sqrt(e)/(-d)^(7/2)+1/4*I*b*log(1+I*c*x)*log(c*(sqrt(-d)+x*sqrt(e))/(c*sqrt(-d)+I*sqrt(e)))*sqrt(e)/(-d)^(7/2)+1/4*I*b*polylog(2,(I-c*x)*sqrt(e)/(c*sqrt(-d)+I*sqrt(e)))*sqrt(e)/(-d)^(7/2)+1/4*I*b*polylog(2,(I+c*x)*sqrt(e)/(c*sqrt(-d)+I*sqrt(e)))*sqrt(e)/(-d)^(7/2)-1/4*I*b*polylog(2,(1-I*c*x)*sqrt(e)/(I*c*sqrt(-d)+sqrt(e)))*sqrt(e)/(-d)^(7/2)-1/4*I*b*polylog(2,(1+I*c*x)*sqrt(e)/(I*c*sqrt(-d)+sqrt(e)))*sqrt(e)/(-d)^(7/2)+7/32*I*b*c*log(-(1+x*sqrt(-c^2))*sqrt(e)/(I*sqrt(-c^2)*sqrt(d)-sqrt(e)))*log(1-I*x*sqrt(e)/sqrt(d))*sqrt(e)/(d^(7/2)*sqrt(-c^2))-7/32*I*b*c*log((1-x*sqrt(-c^2))*sqrt(e)/(I*sqrt(-c^2)*sqrt(d)+sqrt(e)))*log(1-I*x*sqrt(e)/sqrt(d))*sqrt(e)/(d^(7/2)*sqrt(-c^2))+7/32*I*b*c*log(-(1-x*sqrt(-c^2))*sqrt(e)/(I*sqrt(-c^2)*sqrt(d)-sqrt(e)))*log(1+I*x*sqrt(e)/sqrt(d))*sqrt(e)/(d^(7/2)*sqrt(-c^2))-7/32*I*b*c*log((1+x*sqrt(-c^2))*sqrt(e)/(I*sqrt(-c^2)*sqrt(d)+sqrt(e)))*log(1+I*x*sqrt(e)/sqrt(d))*sqrt(e)/(d^(7/2)*sqrt(-c^2))-7/32*I*b*c*polylog(2,sqrt(-c^2)*(sqrt(d)-I*x*sqrt(e))/(sqrt(-c^2)*sqrt(d)-I*sqrt(e)))*sqrt(e)/(d^(7/2)*sqrt(-c^2))+7/32*I*b*c*polylog(2,sqrt(-c^2)*(sqrt(d)-I*x*sqrt(e))/(sqrt(-c^2)*sqrt(d)+I*sqrt(e)))*sqrt(e)/(d^(7/2)*sqrt(-c^2))-7/32*I*b*c*polylog(2,sqrt(-c^2)*(sqrt(d)+I*x*sqrt(e))/(sqrt(-c^2)*sqrt(d)-I*sqrt(e)))*sqrt(e)/(d^(7/2)*sqrt(-c^2))+7/32*I*b*c*polylog(2,sqrt(-c^2)*(sqrt(d)+I*x*sqrt(e))/(sqrt(-c^2)*sqrt(d)+I*sqrt(e)))*sqrt(e)/(d^(7/2)*sqrt(-c^2))],

# Integrands of the form x^m (d+e x^2)^(p/2) (a+b ArcTan[c x])

# p>0
[x^3*(d+e*x^2)^(1/2)*(a+b*arctan(c*x)),x,10,-1/20*b*x*(d+e*x^2)^(3/2)/(c*e)-1/3*d*(d+e*x^2)^(3/2)*(a+b*arctan(c*x))/e^2+1/5*(d+e*x^2)^(5/2)*(a+b*arctan(c*x))/e^2+1/15*b*(c^2*d-e)^(3/2)*(2*c^2*d+3*e)*arctan(x*sqrt(c^2*d-e)/sqrt(d+e*x^2))/(c^5*e^2)+1/120*b*(15*c^4*d^2+20*c^2*d*e-24*e^2)*arctanh(x*sqrt(e)/sqrt(d+e*x^2))/(c^5*e^(3/2))-1/120*b*(c^2*d-12*e)*x*sqrt(d+e*x^2)/(c^3*e)],
[x^2*(d+e*x^2)^(1/2)*(a+b*arctan(c*x)),x,5,-1/8*a*d^2*arctanh(x*sqrt(e)/sqrt(d+e*x^2))/e^(3/2)+1/8*a*d*x*sqrt(d+e*x^2)/e+1/4*a*x^3*sqrt(d+e*x^2)+b*Unintegrable(x^2*arctan(c*x)*sqrt(d+e*x^2),x)],
[x*(d+e*x^2)^(1/2)*(a+b*arctan(c*x)),x,7,1/3*(d+e*x^2)^(3/2)*(a+b*arctan(c*x))/e-1/3*b*(c^2*d-e)^(3/2)*arctan(x*sqrt(c^2*d-e)/sqrt(d+e*x^2))/(c^3*e)-1/6*b*(3*c^2*d-2*e)*arctanh(x*sqrt(e)/sqrt(d+e*x^2))/(c^3*sqrt(e))-1/6*b*x*sqrt(d+e*x^2)/c],
[(d+e*x^2)^(1/2)*(a+b*arctan(c*x)),x,0,Unintegrable((a+b*arctan(c*x))*sqrt(d+e*x^2),x)],
[(d+e*x^2)^(1/2)*(a+b*arctan(c*x))/x,x,5,-a*arctanh(sqrt(d+e*x^2)/sqrt(d))*sqrt(d)+a*sqrt(d+e*x^2)+b*Unintegrable(arctan(c*x)*sqrt(d+e*x^2)/x,x)],
[(d+e*x^2)^(1/2)*(a+b*arctan(c*x))/x^2,x,4,a*arctanh(x*sqrt(e)/sqrt(d+e*x^2))*sqrt(e)-a*sqrt(d+e*x^2)/x+b*Unintegrable(arctan(c*x)*sqrt(d+e*x^2)/x^2,x)],
[(d+e*x^2)^(1/2)*(a+b*arctan(c*x))/x^3,x,5,-1/2*a*e*arctanh(sqrt(d+e*x^2)/sqrt(d))/sqrt(d)-1/2*a*sqrt(d+e*x^2)/x^2+b*Unintegrable(arctan(c*x)*sqrt(d+e*x^2)/x^3,x)],
[(d+e*x^2)^(1/2)*(a+b*arctan(c*x))/x^4,x,9,-1/3*(d+e*x^2)^(3/2)*(a+b*arctan(c*x))/(d*x^3)-1/3*b*(c^2*d-e)^(3/2)*arctanh(c*sqrt(d+e*x^2)/sqrt(c^2*d-e))/d+1/6*b*c*(2*c^2*d-3*e)*arctanh(sqrt(d+e*x^2)/sqrt(d))/sqrt(d)-1/6*b*c*sqrt(d+e*x^2)/x^2],
[(d+e*x^2)^(1/2)*(a+b*arctan(c*x))/x^5,x,6,1/8*a*e^2*arctanh(sqrt(d+e*x^2)/sqrt(d))/d^(3/2)-1/4*a*sqrt(d+e*x^2)/x^4-1/8*a*e*sqrt(d+e*x^2)/(d*x^2)+b*Unintegrable(arctan(c*x)*sqrt(d+e*x^2)/x^5,x)],
[(d+e*x^2)^(1/2)*(a+b*arctan(c*x))/x^6,x,21,-1/5*(d+e*x^2)^(3/2)*(a+b*arctan(c*x))/(d*x^5)+2/15*e*(d+e*x^2)^(3/2)*(a+b*arctan(c*x))/(d^2*x^3)+1/30*b*c*(3*c^2*d-e)*e*arctanh(sqrt(d+e*x^2)/sqrt(d))/d^(3/2)+1/40*b*c*e^2*arctanh(sqrt(d+e*x^2)/sqrt(d))/d^(3/2)-1/15*b*c*(c^2*d-e)*(3*c^2*d+2*e)*arctanh(sqrt(d+e*x^2)/sqrt(d))/d^(3/2)+1/15*b*(c^2*d-e)^(3/2)*(3*c^2*d+2*e)*arctanh(c*sqrt(d+e*x^2)/sqrt(c^2*d-e))/d^2-1/20*b*c*sqrt(d+e*x^2)/x^4+1/30*b*c*(3*c^2*d-e)*sqrt(d+e*x^2)/(d*x^2)-1/40*b*c*e*sqrt(d+e*x^2)/(d*x^2)],
[x^3*(d+e*x^2)^(3/2)*(a+b*arctan(c*x)),x,10,-1/840*b*(13*c^2*d-30*e)*x*(d+e*x^2)^(3/2)/(c^3*e)-1/42*b*x*(d+e*x^2)^(5/2)/(c*e)-1/5*d*(d+e*x^2)^(5/2)*(a+b*arctan(c*x))/e^2+1/7*(d+e*x^2)^(7/2)*(a+b*arctan(c*x))/e^2+1/35*b*(c^2*d-e)^(5/2)*(2*c^2*d+5*e)*arctan(x*sqrt(c^2*d-e)/sqrt(d+e*x^2))/(c^7*e^2)+1/560*b*(35*c^6*d^3+70*c^4*d^2*e-168*c^2*d*e^2+80*e^3)*arctanh(x*sqrt(e)/sqrt(d+e*x^2))/(c^7*e^(3/2))+1/560*b*(3*c^4*d^2+54*c^2*d*e-40*e^2)*x*sqrt(d+e*x^2)/(c^5*e)],
[x^2*(d+e*x^2)^(3/2)*(a+b*arctan(c*x)),x,6,1/6*a*x^3*(d+e*x^2)^(3/2)-1/16*a*d^3*arctanh(x*sqrt(e)/sqrt(d+e*x^2))/e^(3/2)+1/16*a*d^2*x*sqrt(d+e*x^2)/e+1/8*a*d*x^3*sqrt(d+e*x^2)+b*Unintegrable(x^2*(d+e*x^2)^(3/2)*arctan(c*x),x)],
[x*(d+e*x^2)^(3/2)*(a+b*arctan(c*x)),x,8,-1/20*b*x*(d+e*x^2)^(3/2)/c+1/5*(d+e*x^2)^(5/2)*(a+b*arctan(c*x))/e-1/5*b*(c^2*d-e)^(5/2)*arctan(x*sqrt(c^2*d-e)/sqrt(d+e*x^2))/(c^5*e)-1/40*b*(15*c^4*d^2-20*c^2*d*e+8*e^2)*arctanh(x*sqrt(e)/sqrt(d+e*x^2))/(c^5*sqrt(e))-1/40*b*(7*c^2*d-4*e)*x*sqrt(d+e*x^2)/c^3],
[(d+e*x^2)^(3/2)*(a+b*arctan(c*x)),x,0,Unintegrable((d+e*x^2)^(3/2)*(a+b*arctan(c*x)),x)],
[(d+e*x^2)^(3/2)*(a+b*arctan(c*x))/x,x,6,1/3*a*(d+e*x^2)^(3/2)-a*d^(3/2)*arctanh(sqrt(d+e*x^2)/sqrt(d))+a*d*sqrt(d+e*x^2)+b*Unintegrable((d+e*x^2)^(3/2)*arctan(c*x)/x,x)],
[(d+e*x^2)^(3/2)*(a+b*arctan(c*x))/x^2,x,5,-a*(d+e*x^2)^(3/2)/x+3/2*a*d*arctanh(x*sqrt(e)/sqrt(d+e*x^2))*sqrt(e)+3/2*a*e*x*sqrt(d+e*x^2)+b*Unintegrable((d+e*x^2)^(3/2)*arctan(c*x)/x^2,x)],
[(d+e*x^2)^(3/2)*(a+b*arctan(c*x))/x^3,x,6,-1/2*a*(d+e*x^2)^(3/2)/x^2-3/2*a*e*arctanh(sqrt(d+e*x^2)/sqrt(d))*sqrt(d)+3/2*a*e*sqrt(d+e*x^2)+b*Unintegrable((d+e*x^2)^(3/2)*arctan(c*x)/x^3,x)],
[(d+e*x^2)^(3/2)*(a+b*arctan(c*x))/x^4,x,5,-1/3*a*(d+e*x^2)^(3/2)/x^3+a*e^(3/2)*arctanh(x*sqrt(e)/sqrt(d+e*x^2))-a*e*sqrt(d+e*x^2)/x+b*Unintegrable((d+e*x^2)^(3/2)*arctan(c*x)/x^4,x)],
[(d+e*x^2)^(3/2)*(a+b*arctan(c*x))/x^5,x,6,-1/4*a*(d+e*x^2)^(3/2)/x^4-3/8*a*e^2*arctanh(sqrt(d+e*x^2)/sqrt(d))/sqrt(d)-3/8*a*e*sqrt(d+e*x^2)/x^2+b*Unintegrable((d+e*x^2)^(3/2)*arctan(c*x)/x^5,x)],
[(d+e*x^2)^(3/2)*(a+b*arctan(c*x))/x^6,x,10,-1/20*b*c*(d+e*x^2)^(3/2)/x^4-1/5*(d+e*x^2)^(5/2)*(a+b*arctan(c*x))/(d*x^5)+1/5*b*(c^2*d-e)^(5/2)*arctanh(c*sqrt(d+e*x^2)/sqrt(c^2*d-e))/d-1/40*b*c*(8*c^4*d^2-20*c^2*d*e+15*e^2)*arctanh(sqrt(d+e*x^2)/sqrt(d))/sqrt(d)+1/40*b*c*(4*c^2*d-7*e)*sqrt(d+e*x^2)/x^2],
[x^3*(d+e*x^2)^(5/2)*(a+b*arctan(c*x)),x,11,-1/12096*b*(69*c^4*d^2-520*c^2*d*e+336*e^2)*x*(d+e*x^2)^(3/2)/(c^5*e)-1/3024*b*(33*c^2*d-56*e)*x*(d+e*x^2)^(5/2)/(c^3*e)-1/72*b*x*(d+e*x^2)^(7/2)/(c*e)-1/7*d*(d+e*x^2)^(7/2)*(a+b*arctan(c*x))/e^2+1/9*(d+e*x^2)^(9/2)*(a+b*arctan(c*x))/e^2+1/63*b*(c^2*d-e)^(7/2)*(2*c^2*d+7*e)*arctan(x*sqrt(c^2*d-e)/sqrt(d+e*x^2))/(c^9*e^2)+1/8064*b*(315*c^8*d^4+840*c^6*d^3*e-3024*c^4*d^2*e^2+2880*c^2*d*e^3-896*e^4)*arctanh(x*sqrt(e)/sqrt(d+e*x^2))/(c^9*e^(3/2))+1/8064*b*(59*c^6*d^3+712*c^4*d^2*e-1104*c^2*d*e^2+448*e^3)*x*sqrt(d+e*x^2)/(c^7*e)],
[x^2*(d+e*x^2)^(5/2)*(a+b*arctan(c*x)),x,7,5/48*a*d*x^3*(d+e*x^2)^(3/2)+1/8*a*x^3*(d+e*x^2)^(5/2)-5/128*a*d^4*arctanh(x*sqrt(e)/sqrt(d+e*x^2))/e^(3/2)+5/128*a*d^3*x*sqrt(d+e*x^2)/e+5/64*a*d^2*x^3*sqrt(d+e*x^2)+b*Unintegrable(x^2*(d+e*x^2)^(5/2)*arctan(c*x),x)],
[x*(d+e*x^2)^(5/2)*(a+b*arctan(c*x)),x,9,-1/168*b*(11*c^2*d-6*e)*x*(d+e*x^2)^(3/2)/c^3-1/42*b*x*(d+e*x^2)^(5/2)/c+1/7*(d+e*x^2)^(7/2)*(a+b*arctan(c*x))/e-1/7*b*(c^2*d-e)^(7/2)*arctan(x*sqrt(c^2*d-e)/sqrt(d+e*x^2))/(c^7*e)-1/112*b*(35*c^6*d^3-70*c^4*d^2*e+56*c^2*d*e^2-16*e^3)*arctanh(x*sqrt(e)/sqrt(d+e*x^2))/(c^7*sqrt(e))-1/112*b*(19*c^4*d^2-22*c^2*d*e+8*e^2)*x*sqrt(d+e*x^2)/c^5],
[(d+e*x^2)^(5/2)*(a+b*arctan(c*x)),x,0,Unintegrable((d+e*x^2)^(5/2)*(a+b*arctan(c*x)),x)],
[(d+e*x^2)^(5/2)*(a+b*arctan(c*x))/x,x,7,1/3*a*d*(d+e*x^2)^(3/2)+1/5*a*(d+e*x^2)^(5/2)-a*d^(5/2)*arctanh(sqrt(d+e*x^2)/sqrt(d))+a*d^2*sqrt(d+e*x^2)+b*Unintegrable((d+e*x^2)^(5/2)*arctan(c*x)/x,x)],
[(d+e*x^2)^(5/2)*(a+b*arctan(c*x))/x^2,x,6,5/4*a*e*x*(d+e*x^2)^(3/2)-a*(d+e*x^2)^(5/2)/x+15/8*a*d^2*arctanh(x*sqrt(e)/sqrt(d+e*x^2))*sqrt(e)+15/8*a*d*e*x*sqrt(d+e*x^2)+b*Unintegrable((d+e*x^2)^(5/2)*arctan(c*x)/x^2,x)],
[(d+e*x^2)^(5/2)*(a+b*arctan(c*x))/x^3,x,7,5/6*a*e*(d+e*x^2)^(3/2)-1/2*a*(d+e*x^2)^(5/2)/x^2-5/2*a*d^(3/2)*e*arctanh(sqrt(d+e*x^2)/sqrt(d))+5/2*a*d*e*sqrt(d+e*x^2)+b*Unintegrable((d+e*x^2)^(5/2)*arctan(c*x)/x^3,x)],
[(d+e*x^2)^(5/2)*(a+b*arctan(c*x))/x^4,x,6,-5/3*a*e*(d+e*x^2)^(3/2)/x-1/3*a*(d+e*x^2)^(5/2)/x^3+5/2*a*d*e^(3/2)*arctanh(x*sqrt(e)/sqrt(d+e*x^2))+5/2*a*e^2*x*sqrt(d+e*x^2)+b*Unintegrable((d+e*x^2)^(5/2)*arctan(c*x)/x^4,x)],

# p<0
[x^3*(a+b*arctan(c*x))/(d+e*x^2)^(1/2),x,8,1/3*(d+e*x^2)^(3/2)*(a+b*arctan(c*x))/e^2+1/6*b*(3*c^2*d+2*e)*arctanh(x*sqrt(e)/sqrt(d+e*x^2))/(c^3*e^(3/2))+1/3*b*(2*c^2*d+e)*arctan(x*sqrt(c^2*d-e)/sqrt(d+e*x^2))*sqrt(c^2*d-e)/(c^3*e^2)-1/6*b*x*sqrt(d+e*x^2)/(c*e)-d*(a+b*arctan(c*x))*sqrt(d+e*x^2)/e^2],
[x^2*(a+b*arctan(c*x))/(d+e*x^2)^(1/2),x,4,-1/2*a*d*arctanh(x*sqrt(e)/sqrt(d+e*x^2))/e^(3/2)+1/2*a*x*sqrt(d+e*x^2)/e+b*Unintegrable(x^2*arctan(c*x)/sqrt(d+e*x^2),x)],
[x*(a+b*arctan(c*x))/(d+e*x^2)^(1/2),x,6,-b*arctan(x*sqrt(c^2*d-e)/sqrt(d+e*x^2))*sqrt(c^2*d-e)/(c*e)-b*arctanh(x*sqrt(e)/sqrt(d+e*x^2))/(c*sqrt(e))+(a+b*arctan(c*x))*sqrt(d+e*x^2)/e],
[(a+b*arctan(c*x))/(d+e*x^2)^(1/2),x,0,Unintegrable((a+b*arctan(c*x))/sqrt(d+e*x^2),x)],
[(a+b*arctan(c*x))/(x*(d+e*x^2)^(1/2)),x,4,-a*arctanh(sqrt(d+e*x^2)/sqrt(d))/sqrt(d)+b*Unintegrable(arctan(c*x)/(x*sqrt(d+e*x^2)),x)],
[(a+b*arctan(c*x))/(x^2*(d+e*x^2)^(1/2)),x,7,-b*c*arctanh(sqrt(d+e*x^2)/sqrt(d))/sqrt(d)+b*arctanh(c*sqrt(d+e*x^2)/sqrt(c^2*d-e))*sqrt(c^2*d-e)/d-(a+b*arctan(c*x))*sqrt(d+e*x^2)/(d*x)],
[(a+b*arctan(c*x))/(x^3*(d+e*x^2)^(1/2)),x,5,1/2*a*e*arctanh(sqrt(d+e*x^2)/sqrt(d))/d^(3/2)-1/2*a*sqrt(d+e*x^2)/(d*x^2)+b*Unintegrable(arctan(c*x)/(x^3*sqrt(d+e*x^2)),x)],
[(a+b*arctan(c*x))/(x^4*(d+e*x^2)^(1/2)),x,9,1/6*b*c*(2*c^2*d+3*e)*arctanh(sqrt(d+e*x^2)/sqrt(d))/d^(3/2)-1/3*b*(c^2*d+2*e)*arctanh(c*sqrt(d+e*x^2)/sqrt(c^2*d-e))*sqrt(c^2*d-e)/d^2-1/6*b*c*sqrt(d+e*x^2)/(d*x^2)-1/3*(a+b*arctan(c*x))*sqrt(d+e*x^2)/(d*x^3)+2/3*e*(a+b*arctan(c*x))*sqrt(d+e*x^2)/(d^2*x)],
[x^3*(a+b*arctan(c*x))/(d+e*x^2)^(3/2),x,7,-b*arctanh(x*sqrt(e)/sqrt(d+e*x^2))/(c*e^(3/2))-b*(2*c^2*d-e)*arctan(x*sqrt(c^2*d-e)/sqrt(d+e*x^2))/(c*e^2*sqrt(c^2*d-e))+d*(a+b*arctan(c*x))/(e^2*sqrt(d+e*x^2))+(a+b*arctan(c*x))*sqrt(d+e*x^2)/e^2],
[x^2*(a+b*arctan(c*x))/(d+e*x^2)^(3/2),x,4,a*arctanh(x*sqrt(e)/sqrt(d+e*x^2))/e^(3/2)-a*x/(e*sqrt(d+e*x^2))+b*Unintegrable(x^2*arctan(c*x)/(d+e*x^2)^(3/2),x)],
[x*(a+b*arctan(c*x))/(d+e*x^2)^(3/2),x,3,b*c*arctan(x*sqrt(c^2*d-e)/sqrt(d+e*x^2))/(e*sqrt(c^2*d-e))+(-a-b*arctan(c*x))/(e*sqrt(d+e*x^2))],
[(a+b*arctan(c*x))/(d+e*x^2)^(3/2),x,5,b*arctanh(c*sqrt(d+e*x^2)/sqrt(c^2*d-e))/(d*sqrt(c^2*d-e))+x*(a+b*arctan(c*x))/(d*sqrt(d+e*x^2))],
[(a+b*arctan(c*x))/(x*(d+e*x^2)^(3/2)),x,5,-a*arctanh(sqrt(d+e*x^2)/sqrt(d))/d^(3/2)+a/(d*sqrt(d+e*x^2))+b*Unintegrable(arctan(c*x)/(x*(d+e*x^2)^(3/2)),x)],
[(a+b*arctan(c*x))/(x^2*(d+e*x^2)^(3/2)),x,8,-b*c*arctanh(sqrt(d+e*x^2)/sqrt(d))/d^(3/2)+b*(c^2*d-2*e)*arctanh(c*sqrt(d+e*x^2)/sqrt(c^2*d-e))/(d^2*sqrt(c^2*d-e))+(-a-b*arctan(c*x))/(d*x*sqrt(d+e*x^2))-2*e*x*(a+b*arctan(c*x))/(d^2*sqrt(d+e*x^2))],
[(a+b*arctan(c*x))/(x^3*(d+e*x^2)^(3/2)),x,6,3/2*a*e*arctanh(sqrt(d+e*x^2)/sqrt(d))/d^(5/2)-3/2*a*e/(d^2*sqrt(d+e*x^2))-1/2*a/(d*x^2*sqrt(d+e*x^2))+b*Unintegrable(arctan(c*x)/(x^3*(d+e*x^2)^(3/2)),x)],
[(a+b*arctan(c*x))/(x^4*(d+e*x^2)^(3/2)),x,14,1/6*b*c*e*arctanh(sqrt(d+e*x^2)/sqrt(d))/d^(5/2)+1/3*b*c*(c^2*d+4*e)*arctanh(sqrt(d+e*x^2)/sqrt(d))/d^(5/2)-1/3*b*(c^4*d^2+4*c^2*d*e-8*e^2)*arctanh(c*sqrt(d+e*x^2)/sqrt(c^2*d-e))/(d^3*sqrt(c^2*d-e))+1/3*(-a-b*arctan(c*x))/(d*x^3*sqrt(d+e*x^2))+4/3*e*(a+b*arctan(c*x))/(d^2*x*sqrt(d+e*x^2))+8/3*e^2*x*(a+b*arctan(c*x))/(d^3*sqrt(d+e*x^2))-1/6*b*c*sqrt(d+e*x^2)/(d^2*x^2)],
[x^4*(a+b*arctan(c*x))/(d+e*x^2)^(5/2),x,5,-1/3*a*x^3/(e*(d+e*x^2)^(3/2))+a*arctanh(x*sqrt(e)/sqrt(d+e*x^2))/e^(5/2)-a*x/(e^2*sqrt(d+e*x^2))+b*Unintegrable(x^4*arctan(c*x)/(d+e*x^2)^(5/2),x)],
[x^3*(a+b*arctan(c*x))/(d+e*x^2)^(5/2),x,6,1/3*d*(a+b*arctan(c*x))/(e^2*(d+e*x^2)^(3/2))+1/3*b*c*(2*c^2*d-3*e)*arctan(x*sqrt(c^2*d-e)/sqrt(d+e*x^2))/((c^2*d-e)^(3/2)*e^2)+1/3*b*c*x/((c^2*d-e)*e*sqrt(d+e*x^2))+(-a-b*arctan(c*x))/(e^2*sqrt(d+e*x^2))],
[x^2*(a+b*arctan(c*x))/(d+e*x^2)^(5/2),x,5,1/3*x^3*(a+b*arctan(c*x))/(d*(d+e*x^2)^(3/2))-1/3*b*arctanh(c*sqrt(d+e*x^2)/sqrt(c^2*d-e))/(d*(c^2*d-e)^(3/2))+1/3*b*c/((c^2*d-e)*e*sqrt(d+e*x^2))],
[x*(a+b*arctan(c*x))/(d+e*x^2)^(5/2),x,4,1/3*(-a-b*arctan(c*x))/(e*(d+e*x^2)^(3/2))+1/3*b*c^3*arctan(x*sqrt(c^2*d-e)/sqrt(d+e*x^2))/((c^2*d-e)^(3/2)*e)-1/3*b*c*x/(d*(c^2*d-e)*sqrt(d+e*x^2))],
[(a+b*arctan(c*x))/(d+e*x^2)^(5/2),x,7,1/3*x*(a+b*arctan(c*x))/(d*(d+e*x^2)^(3/2))+1/3*b*(3*c^2*d-2*e)*arctanh(c*sqrt(d+e*x^2)/sqrt(c^2*d-e))/(d^2*(c^2*d-e)^(3/2))-1/3*b*c/(d*(c^2*d-e)*sqrt(d+e*x^2))+2/3*x*(a+b*arctan(c*x))/(d^2*sqrt(d+e*x^2))],
[(a+b*arctan(c*x))/(x*(d+e*x^2)^(5/2)),x,6,1/3*a/(d*(d+e*x^2)^(3/2))-a*arctanh(sqrt(d+e*x^2)/sqrt(d))/d^(5/2)+a/(d^2*sqrt(d+e*x^2))+b*Unintegrable(arctan(c*x)/(x*(d+e*x^2)^(5/2)),x)],
[(a+b*arctan(c*x))/(x^2*(d+e*x^2)^(5/2)),x,13,(-a-b*arctan(c*x))/(d*x*(d+e*x^2)^(3/2))-4/3*e*x*(a+b*arctan(c*x))/(d^2*(d+e*x^2)^(3/2))-b*c*arctanh(sqrt(d+e*x^2)/sqrt(d))/d^(5/2)+1/3*b*(3*c^4*d^2-12*c^2*d*e+8*e^2)*arctanh(c*sqrt(d+e*x^2)/sqrt(c^2*d-e))/(d^3*(c^2*d-e)^(3/2))+b*c/(d^2*sqrt(d+e*x^2))-8/3*b*e/(c*d^3*sqrt(d+e*x^2))-1/3*b*(3*c^4*d^2-12*c^2*d*e+8*e^2)/(c*d^3*(c^2*d-e)*sqrt(d+e*x^2))-8/3*e*x*(a+b*arctan(c*x))/(d^3*sqrt(d+e*x^2))],
[(a+b*arctan(c*x))/(x^3*(d+e*x^2)^(5/2)),x,7,-5/6*a*e/(d^2*(d+e*x^2)^(3/2))-1/2*a/(d*x^2*(d+e*x^2)^(3/2))+5/2*a*e*arctanh(sqrt(d+e*x^2)/sqrt(d))/d^(7/2)-5/2*a*e/(d^3*sqrt(d+e*x^2))+b*Unintegrable(arctan(c*x)/(x^3*(d+e*x^2)^(5/2)),x)],
[(a+b*arctan(c*x))/(x^4*(d+e*x^2)^(5/2)),x,18,1/3*(-a-b*arctan(c*x))/(d*x^3*(d+e*x^2)^(3/2))+2*e*(a+b*arctan(c*x))/(d^2*x*(d+e*x^2)^(3/2))+8/3*e^2*x*(a+b*arctan(c*x))/(d^3*(d+e*x^2)^(3/2))+1/2*b*c*e*arctanh(sqrt(d+e*x^2)/sqrt(d))/d^(7/2)+1/3*b*c*(c^2*d+6*e)*arctanh(sqrt(d+e*x^2)/sqrt(d))/d^(7/2)-1/3*b*(c^2*d-2*e)*(c^4*d^2+8*c^2*d*e-8*e^2)*arctanh(c*sqrt(d+e*x^2)/sqrt(c^2*d-e))/(d^4*(c^2*d-e)^(3/2))-1/2*b*c*e/(d^3*sqrt(d+e*x^2))+16/3*b*e^2/(c*d^4*sqrt(d+e*x^2))-1/3*b*c*(c^2*d+6*e)/(d^3*sqrt(d+e*x^2))+1/3*b*(c^2*d-2*e)*(c^4*d^2+8*c^2*d*e-8*e^2)/(c*d^4*(c^2*d-e)*sqrt(d+e*x^2))-1/6*b*c/(d^2*x^2*sqrt(d+e*x^2))+16/3*e^2*x*(a+b*arctan(c*x))/(d^4*sqrt(d+e*x^2))],
[arctan(a*x)/(c+d*x^2)^(7/2),x,8,-1/15*a/(c*(a^2*c-d)*(c+d*x^2)^(3/2))+1/5*x*arctan(a*x)/(c*(c+d*x^2)^(5/2))+4/15*x*arctan(a*x)/(c^2*(c+d*x^2)^(3/2))+1/15*(15*a^4*c^2-20*a^2*c*d+8*d^2)*arctanh(a*sqrt(c+d*x^2)/sqrt(a^2*c-d))/(c^3*(a^2*c-d)^(5/2))-1/15*a*(7*a^2*c-4*d)/(c^2*(a^2*c-d)^2*sqrt(c+d*x^2))+8/15*x*arctan(a*x)/(c^3*sqrt(c+d*x^2))],
[arctan(a*x)/(c+d*x^2)^(9/2),x,8,-1/35*a/(c*(a^2*c-d)*(c+d*x^2)^(5/2))-1/105*a*(11*a^2*c-6*d)/(c^2*(a^2*c-d)^2*(c+d*x^2)^(3/2))+1/7*x*arctan(a*x)/(c*(c+d*x^2)^(7/2))+6/35*x*arctan(a*x)/(c^2*(c+d*x^2)^(5/2))+8/35*x*arctan(a*x)/(c^3*(c+d*x^2)^(3/2))+1/35*(35*a^6*c^3-70*a^4*c^2*d+56*a^2*c*d^2-16*d^3)*arctanh(a*sqrt(c+d*x^2)/sqrt(a^2*c-d))/(c^4*(a^2*c-d)^(7/2))-1/35*a*(19*a^4*c^2-22*a^2*c*d+8*d^2)/(c^3*(a^2*c-d)^3*sqrt(c+d*x^2))+16/35*x*arctan(a*x)/(c^4*sqrt(c+d*x^2))],

# Integrands of the form x^m (d+e x^2)^p (a+b ArcTan[c x]) with m symbolic
[x^m*(d+e*x^2)^3*(a+b*arctan(c*x)),x,4,-b*e*(e^2*(15+8*m+m^2)-3*c^2*d*e*(21+10*m+m^2)+3*c^4*d^2*(35+12*m+m^2))*x^(2+m)/(c^5*(2+m)*(7+m)*(15+8*m+m^2))+b*e^2*(e*(5+m)-3*c^2*d*(7+m))*x^(4+m)/(c^3*(4+m)*(5+m)*(7+m))-b*e^3*x^(6+m)/(c*(6+m)*(7+m))+d^3*x^(1+m)*(a+b*arctan(c*x))/(1+m)+3*d^2*e*x^(3+m)*(a+b*arctan(c*x))/(3+m)+3*d*e^2*x^(5+m)*(a+b*arctan(c*x))/(5+m)+e^3*x^(7+m)*(a+b*arctan(c*x))/(7+m)+b*(e^3*(15+23*m+9*m^2+m^3)-3*c^2*d*e^2*(21+31*m+11*m^2+m^3)+3*c^4*d^2*e*(35+47*m+13*m^2+m^3)-c^6*d^3*(105+71*m+15*m^2+m^3))*x^(2+m)*hypergeom([1,1/2*(2+m)],[1/2*(4+m)],-c^2*x^2)/(c^5*(35+12*m+m^2)*(6+11*m+6*m^2+m^3))],
[x^m*(d+e*x^2)^2*(a+b*arctan(c*x)),x,4,b*e*(e*(3+m)-2*c^2*d*(5+m))*x^(2+m)/(c^3*(5+m)*(6+5*m+m^2))-b*e^2*x^(4+m)/(c*(4+m)*(5+m))+d^2*x^(1+m)*(a+b*arctan(c*x))/(1+m)+2*d*e*x^(3+m)*(a+b*arctan(c*x))/(3+m)+e^2*x^(5+m)*(a+b*arctan(c*x))/(5+m)-b*(e^2*(3+4*m+m^2)-2*c^2*d*e*(5+6*m+m^2)+c^4*d^2*(15+8*m+m^2))*x^(2+m)*hypergeom([1,1/2*(2+m)],[1/2*(4+m)],-c^2*x^2)/(c^3*(2+3*m+m^2)*(15+8*m+m^2))],
[x^m*(d+e*x^2)*(a+b*arctan(c*x)),x,3,-b*e*x^(2+m)/(c*(6+5*m+m^2))+d*x^(1+m)*(a+b*arctan(c*x))/(1+m)+e*x^(3+m)*(a+b*arctan(c*x))/(3+m)-b*(c^2*d/(1+m)-e/(3+m))*x^(2+m)*hypergeom([1,1/2*(2+m)],[1/2*(4+m)],-c^2*x^2)/(c*(2+m))],
[x^m*(a+b*arctan(c*x))/(d+e*x^2),x,2,a*x^(1+m)*hypergeom([1,1/2*(1+m)],[1/2*(3+m)],-e*x^2/d)/(d*(1+m))+b*Unintegrable(x^m*arctan(c*x)/(d+e*x^2),x)],
[x^m*(a+b*arctan(c*x))/(d+e*x^2)^2,x,2,a*x^(1+m)*hypergeom([2,1/2*(1+m)],[1/2*(3+m)],-e*x^2/d)/(d^2*(1+m))+b*Unintegrable(x^m*arctan(c*x)/(d+e*x^2)^2,x)],
[x^m*(d+e*x^2)^(5/2)*(a+b*arctan(c*x)),x,3,a*x^(1+m)*(d+e*x^2)^(7/2)*hypergeom([1,1/2*(8+m)],[1/2*(3+m)],-e*x^2/d)/(d*(1+m))+b*Unintegrable(x^m*(d+e*x^2)^(5/2)*arctan(c*x),x),a*d^2*x^(1+m)*hypergeom([-5/2,1/2*(1+m)],[1/2*(3+m)],-e*x^2/d)*sqrt(d+e*x^2)/((1+m)*sqrt(1+e*x^2/d))+b*Unintegrable(x^m*(d+e*x^2)^(5/2)*arctan(c*x),x)],
[x^m*(d+e*x^2)^(3/2)*(a+b*arctan(c*x)),x,3,a*x^(1+m)*(d+e*x^2)^(5/2)*hypergeom([1,1/2*(6+m)],[1/2*(3+m)],-e*x^2/d)/(d*(1+m))+b*Unintegrable(x^m*(d+e*x^2)^(3/2)*arctan(c*x),x),a*d*x^(1+m)*hypergeom([-3/2,1/2*(1+m)],[1/2*(3+m)],-e*x^2/d)*sqrt(d+e*x^2)/((1+m)*sqrt(1+e*x^2/d))+b*Unintegrable(x^m*(d+e*x^2)^(3/2)*arctan(c*x),x)],
[x^m*(d+e*x^2)^(1/2)*(a+b*arctan(c*x)),x,3,a*x^(1+m)*(d+e*x^2)^(3/2)*hypergeom([1,1/2*(4+m)],[1/2*(3+m)],-e*x^2/d)/(d*(1+m))+b*Unintegrable(x^m*arctan(c*x)*sqrt(d+e*x^2),x),a*x^(1+m)*hypergeom([-1/2,1/2*(1+m)],[1/2*(3+m)],-e*x^2/d)*sqrt(d+e*x^2)/((1+m)*sqrt(1+e*x^2/d))+b*Unintegrable(x^m*arctan(c*x)*sqrt(d+e*x^2),x)],
[x^m*(a+b*arctan(c*x))/(d+e*x^2)^(1/2),x,3,a*x^(1+m)*hypergeom([1,1/2*(2+m)],[1/2*(3+m)],-e*x^2/d)*sqrt(d+e*x^2)/(d*(1+m))+b*Unintegrable(x^m*arctan(c*x)/sqrt(d+e*x^2),x),a*x^(1+m)*hypergeom([1/2,1/2*(1+m)],[1/2*(3+m)],-e*x^2/d)*sqrt(1+e*x^2/d)/((1+m)*sqrt(d+e*x^2))+b*Unintegrable(x^m*arctan(c*x)/sqrt(d+e*x^2),x)],
[x^m*(a+b*arctan(c*x))/(d+e*x^2)^(3/2),x,3,a*x^(1+m)*hypergeom([1,1/2*m],[1/2*(3+m)],-e*x^2/d)/(d*(1+m)*sqrt(d+e*x^2))+b*Unintegrable(x^m*arctan(c*x)/(d+e*x^2)^(3/2),x),a*x^(1+m)*hypergeom([3/2,1/2*(1+m)],[1/2*(3+m)],-e*x^2/d)*sqrt(1+e*x^2/d)/(d*(1+m)*sqrt(d+e*x^2))+b*Unintegrable(x^m*arctan(c*x)/(d+e*x^2)^(3/2),x)],
[x^m*(a+b*arctan(c*x))/(d+e*x^2)^(5/2),x,3,a*x^(1+m)*hypergeom([1,1/2*(-2+m)],[1/2*(3+m)],-e*x^2/d)/(d*(1+m)*(d+e*x^2)^(3/2))+b*Unintegrable(x^m*arctan(c*x)/(d+e*x^2)^(5/2),x),a*x^(1+m)*hypergeom([5/2,1/2*(1+m)],[1/2*(3+m)],-e*x^2/d)*sqrt(1+e*x^2/d)/(d^2*(1+m)*sqrt(d+e*x^2))+b*Unintegrable(x^m*arctan(c*x)/(d+e*x^2)^(5/2),x)],

# Integrands of the form x^m (d+e x^2)^p (a+b ArcTan[c x]) with p symbolic
[x^m*(d+e*x^2)^p*(a+b*arctan(c*x)),x,3,a*x^(1+m)*(d+e*x^2)^(1+p)*hypergeom([1,1/2*(3+m+2*p)],[1/2*(3+m)],-e*x^2/d)/(d*(1+m))+b*Unintegrable(x^m*(d+e*x^2)^p*arctan(c*x),x),a*x^(1+m)*(d+e*x^2)^p*hypergeom([1/2*(1+m),-p],[1/2*(3+m)],-e*x^2/d)/((1+m)*(1+e*x^2/d)^p)+b*Unintegrable(x^m*(d+e*x^2)^p*arctan(c*x),x)],
[x^(-2-2*p)*(d+e*x^2)^p*(a+b*arctan(c*x)),x,3,-a*x^(-1-2*p)*(d+e*x^2)^(1+p)*hypergeom([1/2,1],[1/2*(1-2*p)],-e*x^2/d)/(d*(1+2*p))+b*Unintegrable(x^(-2-2*p)*(d+e*x^2)^p*arctan(c*x),x),-a*x^(-1-2*p)*(d+e*x^2)^p*hypergeom([1/2*(-1-2*p),-p],[1/2*(1-2*p)],-e*x^2/d)/((1+2*p)*(1+e*x^2/d)^p)+b*Unintegrable(x^(-2-2*p)*(d+e*x^2)^p*arctan(c*x),x)],
[x^(-3-2*p)*(d+e*x^2)^p*(a+b*arctan(c*x)),x,4,-1/2*b*c*x^(-1-2*p)*(d+e*x^2)^p*AppellF1(1/2*(-1-2*p),1,-1-p,1/2*(1-2*p),-c^2*x^2,-e*x^2/d)/((1+3*p+2*p^2)*(1+e*x^2/d)^p)-1/2*(d+e*x^2)^(1+p)*(a+b*arctan(c*x))/(d*(1+p)*x^(2*(1+p)))],
[x^(-4-2*p)*(d+e*x^2)^p*(a+b*arctan(c*x)),x,3,-a*x^(-3-2*p)*(d+e*x^2)^(1+p)*hypergeom([-1/2,1],[1/2*(-1-2*p)],-e*x^2/d)/(d*(3+2*p))+b*Unintegrable(x^(-4-2*p)*(d+e*x^2)^p*arctan(c*x),x),-a*x^(-3-2*p)*(d+e*x^2)^p*hypergeom([1/2*(-3-2*p),-p],[1/2*(-1-2*p)],-e*x^2/d)/((3+2*p)*(1+e*x^2/d)^p)+b*Unintegrable(x^(-4-2*p)*(d+e*x^2)^p*arctan(c*x),x)],
[x^(-5-2*p)*(d+e*x^2)^p*(a+b*arctan(c*x)),x,8,-1/2*b*(e+c^2*d*(1+p))*x^(-3-2*p)*(d+e*x^2)^p*AppellF1(1/2*(-3-2*p),1,-1-p,1/2*(-1-2*p),-c^2*x^2,-e*x^2/d)/(c*d*(3+2*p)*(2+3*p+p^2)*(1+e*x^2/d)^p)+1/2*e*(d+e*x^2)^(1+p)*(a+b*arctan(c*x))/(d^2*(1+p)*(2+p)*x^(2*(1+p)))-1/2*(d+e*x^2)^(1+p)*(a+b*arctan(c*x))/(d*(2+p)*x^(2*(2+p)))+1/2*b*e*x^(-3-2*p)*(d+e*x^2)^p*hypergeom([1/2*(-3-2*p),-1-p],[1/2*(-1-2*p)],-e*x^2/d)/(c*d*(6+13*p+9*p^2+2*p^3)*(1+e*x^2/d)^p)],
[x^(-6-2*p)*(d+e*x^2)^p*(a+b*arctan(c*x)),x,3,-a*x^(-5-2*p)*(d+e*x^2)^(1+p)*hypergeom([-3/2,1],[1/2*(-3-2*p)],-e*x^2/d)/(d*(5+2*p))+b*Unintegrable(x^(-6-2*p)*(d+e*x^2)^p*arctan(c*x),x),-a*x^(-5-2*p)*(d+e*x^2)^p*hypergeom([1/2*(-5-2*p),-p],[1/2*(-3-2*p)],-e*x^2/d)/((5+2*p)*(1+e*x^2/d)^p)+b*Unintegrable(x^(-6-2*p)*(d+e*x^2)^p*arctan(c*x),x)],
[x^(-7-2*p)*(d+e*x^2)^p*(a+b*arctan(c*x)),x,10,-1/2*b*(2*e^2+2*c^2*d*e*(1+p)+c^4*d^2*(2+3*p+p^2))*x^(-5-2*p)*(d+e*x^2)^p*AppellF1(1/2*(-5-2*p),1,-1-p,1/2*(-3-2*p),-c^2*x^2,-e*x^2/d)/(c^3*d^2*(3+p)*(5+2*p)*(2+3*p+p^2)*(1+e*x^2/d)^p)-e^2*(d+e*x^2)^(1+p)*(a+b*arctan(c*x))/(d^3*(2+p)*(3+4*p+p^2)*x^(2*(1+p)))+e*(d+e*x^2)^(1+p)*(a+b*arctan(c*x))/(d^2*(2+p)*(3+p)*x^(2*(2+p)))-1/2*(d+e*x^2)^(1+p)*(a+b*arctan(c*x))/(d*(3+p)*x^(2*(3+p)))+b*e*(e+c^2*d*(1+p))*x^(-5-2*p)*(d+e*x^2)^p*hypergeom([1/2*(-5-2*p),-1-p],[1/2*(-3-2*p)],-e*x^2/d)/(c^3*d^2*(3+p)*(5+2*p)*(2+3*p+p^2)*(1+e*x^2/d)^p)-b*e^2*x^(-3-2*p)*(d+e*x^2)^p*hypergeom([1/2*(-3-2*p),-1-p],[1/2*(-1-2*p)],-e*x^2/d)/(c*d^2*(2+3*p+p^2)*(9+9*p+2*p^2)*(1+e*x^2/d)^p)],
[x^(-8-2*p)*(d+e*x^2)^p*(a+b*arctan(c*x)),x,3,-a*x^(-7-2*p)*(d+e*x^2)^(1+p)*hypergeom([-5/2,1],[1/2*(-5-2*p)],-e*x^2/d)/(d*(7+2*p))+b*Unintegrable(x^(-8-2*p)*(d+e*x^2)^p*arctan(c*x),x),-a*x^(-7-2*p)*(d+e*x^2)^p*hypergeom([1/2*(-7-2*p),-p],[1/2*(-5-2*p)],-e*x^2/d)/((7+2*p)*(1+e*x^2/d)^p)+b*Unintegrable(x^(-8-2*p)*(d+e*x^2)^p*arctan(c*x),x)],

# Integrands of the form (f x)^m (d+e x^2)^p (a+b ArcTan[c x])^2

# Integrands of the form x^m (d+e x^2)^p (a+b ArcTan[c x])^2

# p>0
[x^3*(d+e*x^2)*(a+b*arctan(c*x))^2,x,29,1/2*a*b*d*x/c^3-1/3*a*b*e*x/c^5+1/12*b^2*d*x^2/c^2-4/45*b^2*e*x^2/c^4+1/60*b^2*e*x^4/c^2+1/2*b^2*d*x*arctan(c*x)/c^3-1/3*b^2*e*x*arctan(c*x)/c^5-1/6*b*d*x^3*(a+b*arctan(c*x))/c+1/9*b*e*x^3*(a+b*arctan(c*x))/c^3-1/15*b*e*x^5*(a+b*arctan(c*x))/c-1/4*d*(a+b*arctan(c*x))^2/c^4+1/6*e*(a+b*arctan(c*x))^2/c^6+1/4*d*x^4*(a+b*arctan(c*x))^2+1/6*e*x^6*(a+b*arctan(c*x))^2-1/3*b^2*d*log(1+c^2*x^2)/c^4+23/90*b^2*e*log(1+c^2*x^2)/c^6],
[x^2*(d+e*x^2)*(a+b*arctan(c*x))^2,x,25,1/3*b^2*d*x/c^2-3/10*b^2*e*x/c^4+1/30*b^2*e*x^3/c^2-1/3*b^2*d*arctan(c*x)/c^3+3/10*b^2*e*arctan(c*x)/c^5-1/3*b*d*x^2*(a+b*arctan(c*x))/c+1/5*b*e*x^2*(a+b*arctan(c*x))/c^3-1/10*b*e*x^4*(a+b*arctan(c*x))/c-1/3*I*d*(a+b*arctan(c*x))^2/c^3+1/5*I*e*(a+b*arctan(c*x))^2/c^5+1/3*d*x^3*(a+b*arctan(c*x))^2+1/5*e*x^5*(a+b*arctan(c*x))^2-2/3*b*d*(a+b*arctan(c*x))*log(2/(1+I*c*x))/c^3+2/5*b*e*(a+b*arctan(c*x))*log(2/(1+I*c*x))/c^5-1/3*I*b^2*d*polylog(2,1+(-2)/(1+I*c*x))/c^3+1/5*I*b^2*e*polylog(2,1+(-2)/(1+I*c*x))/c^5],
[x*(d+e*x^2)*(a+b*arctan(c*x))^2,x,19,-a*b*d*x/c+1/2*a*b*e*x/c^3+1/12*b^2*e*x^2/c^2-b^2*d*x*arctan(c*x)/c+1/2*b^2*e*x*arctan(c*x)/c^3-1/6*b*e*x^3*(a+b*arctan(c*x))/c+1/2*d*(a+b*arctan(c*x))^2/c^2-1/4*e*(a+b*arctan(c*x))^2/c^4+1/2*d*x^2*(a+b*arctan(c*x))^2+1/4*e*x^4*(a+b*arctan(c*x))^2+1/2*b^2*d*log(1+c^2*x^2)/c^2-1/3*b^2*e*log(1+c^2*x^2)/c^4],
[(d+e*x^2)*(a+b*arctan(c*x))^2,x,16,1/3*b^2*e*x/c^2-1/3*b^2*e*arctan(c*x)/c^3-1/3*b*e*x^2*(a+b*arctan(c*x))/c+I*d*(a+b*arctan(c*x))^2/c-1/3*I*e*(a+b*arctan(c*x))^2/c^3+d*x*(a+b*arctan(c*x))^2+1/3*e*x^3*(a+b*arctan(c*x))^2+2*b*d*(a+b*arctan(c*x))*log(2/(1+I*c*x))/c-2/3*b*e*(a+b*arctan(c*x))*log(2/(1+I*c*x))/c^3+I*b^2*d*polylog(2,1+(-2)/(1+I*c*x))/c-1/3*I*b^2*e*polylog(2,1+(-2)/(1+I*c*x))/c^3],
[(d+e*x^2)*(a+b*arctan(c*x))^2/x,x,14,-a*b*e*x/c-b^2*e*x*arctan(c*x)/c+1/2*e*(a+b*arctan(c*x))^2/c^2+1/2*e*x^2*(a+b*arctan(c*x))^2+2*d*(a+b*arctan(c*x))^2*arctanh(1+(-2)/(1+I*c*x))+1/2*b^2*e*log(1+c^2*x^2)/c^2-I*b*d*(a+b*arctan(c*x))*polylog(2,1+(-2)/(1+I*c*x))+I*b*d*(a+b*arctan(c*x))*polylog(2,-1+2/(1+I*c*x))-1/2*b^2*d*polylog(3,1+(-2)/(1+I*c*x))+1/2*b^2*d*polylog(3,-1+2/(1+I*c*x))],
[(d+e*x^2)*(a+b*arctan(c*x))^2/x^2,x,11,-I*c*d*(a+b*arctan(c*x))^2+I*e*(a+b*arctan(c*x))^2/c-d*(a+b*arctan(c*x))^2/x+e*x*(a+b*arctan(c*x))^2+2*b*e*(a+b*arctan(c*x))*log(2/(1+I*c*x))/c+2*b*c*d*(a+b*arctan(c*x))*log(2+(-2)/(1-I*c*x))-I*b^2*c*d*polylog(2,-1+2/(1-I*c*x))+I*b^2*e*polylog(2,1+(-2)/(1+I*c*x))/c],
[(d+e*x^2)*(a+b*arctan(c*x))^2/x^3,x,16,-b*c*d*(a+b*arctan(c*x))/x-1/2*c^2*d*(a+b*arctan(c*x))^2-1/2*d*(a+b*arctan(c*x))^2/x^2+2*e*(a+b*arctan(c*x))^2*arctanh(1+(-2)/(1+I*c*x))+b^2*c^2*d*log(x)-1/2*b^2*c^2*d*log(1+c^2*x^2)-I*b*e*(a+b*arctan(c*x))*polylog(2,1+(-2)/(1+I*c*x))+I*b*e*(a+b*arctan(c*x))*polylog(2,-1+2/(1+I*c*x))-1/2*b^2*e*polylog(3,1+(-2)/(1+I*c*x))+1/2*b^2*e*polylog(3,-1+2/(1+I*c*x))],
[x^3*(d+e*x^2)^2*(a+b*arctan(c*x))^2,x,50,1/2*a*b*d^2*x/c^3-2/3*a*b*d*e*x/c^5+1/4*a*b*e^2*x/c^7+1/12*b^2*d^2*x^2/c^2-8/45*b^2*d*e*x^2/c^4+71/840*b^2*e^2*x^2/c^6+1/30*b^2*d*e*x^4/c^2-3/140*b^2*e^2*x^4/c^4+1/168*b^2*e^2*x^6/c^2+1/2*b^2*d^2*x*arctan(c*x)/c^3-2/3*b^2*d*e*x*arctan(c*x)/c^5+1/4*b^2*e^2*x*arctan(c*x)/c^7-1/6*b*d^2*x^3*(a+b*arctan(c*x))/c+2/9*b*d*e*x^3*(a+b*arctan(c*x))/c^3-1/12*b*e^2*x^3*(a+b*arctan(c*x))/c^5-2/15*b*d*e*x^5*(a+b*arctan(c*x))/c+1/20*b*e^2*x^5*(a+b*arctan(c*x))/c^3-1/28*b*e^2*x^7*(a+b*arctan(c*x))/c-1/4*d^2*(a+b*arctan(c*x))^2/c^4+1/3*d*e*(a+b*arctan(c*x))^2/c^6-1/8*e^2*(a+b*arctan(c*x))^2/c^8+1/4*d^2*x^4*(a+b*arctan(c*x))^2+1/3*d*e*x^6*(a+b*arctan(c*x))^2+1/8*e^2*x^8*(a+b*arctan(c*x))^2-1/3*b^2*d^2*log(1+c^2*x^2)/c^4+23/45*b^2*d*e*log(1+c^2*x^2)/c^6-22/105*b^2*e^2*log(1+c^2*x^2)/c^8],
[x^2*(d+e*x^2)^2*(a+b*arctan(c*x))^2,x,44,1/3*b^2*d^2*x/c^2-3/5*b^2*d*e*x/c^4+11/42*b^2*e^2*x/c^6+1/15*b^2*d*e*x^3/c^2-5/126*b^2*e^2*x^3/c^4+1/105*b^2*e^2*x^5/c^2-1/3*b^2*d^2*arctan(c*x)/c^3+3/5*b^2*d*e*arctan(c*x)/c^5-11/42*b^2*e^2*arctan(c*x)/c^7-1/3*b*d^2*x^2*(a+b*arctan(c*x))/c+2/5*b*d*e*x^2*(a+b*arctan(c*x))/c^3-1/7*b*e^2*x^2*(a+b*arctan(c*x))/c^5-1/5*b*d*e*x^4*(a+b*arctan(c*x))/c+1/14*b*e^2*x^4*(a+b*arctan(c*x))/c^3-1/21*b*e^2*x^6*(a+b*arctan(c*x))/c-1/3*I*d^2*(a+b*arctan(c*x))^2/c^3+2/5*I*d*e*(a+b*arctan(c*x))^2/c^5-1/7*I*e^2*(a+b*arctan(c*x))^2/c^7+1/3*d^2*x^3*(a+b*arctan(c*x))^2+2/5*d*e*x^5*(a+b*arctan(c*x))^2+1/7*e^2*x^7*(a+b*arctan(c*x))^2-2/3*b*d^2*(a+b*arctan(c*x))*log(2/(1+I*c*x))/c^3+4/5*b*d*e*(a+b*arctan(c*x))*log(2/(1+I*c*x))/c^5-2/7*b*e^2*(a+b*arctan(c*x))*log(2/(1+I*c*x))/c^7-1/3*I*b^2*d^2*polylog(2,1+(-2)/(1+I*c*x))/c^3+2/5*I*b^2*d*e*polylog(2,1+(-2)/(1+I*c*x))/c^5-1/7*I*b^2*e^2*polylog(2,1+(-2)/(1+I*c*x))/c^7],
[x*(d+e*x^2)^2*(a+b*arctan(c*x))^2,x,35,-a*b*d^2*x/c+a*b*d*e*x/c^3-1/3*a*b*e^2*x/c^5+1/6*b^2*d*e*x^2/c^2-4/45*b^2*e^2*x^2/c^4+1/60*b^2*e^2*x^4/c^2-b^2*d^2*x*arctan(c*x)/c+b^2*d*e*x*arctan(c*x)/c^3-1/3*b^2*e^2*x*arctan(c*x)/c^5-1/3*b*d*e*x^3*(a+b*arctan(c*x))/c+1/9*b*e^2*x^3*(a+b*arctan(c*x))/c^3-1/15*b*e^2*x^5*(a+b*arctan(c*x))/c+1/2*d^2*(a+b*arctan(c*x))^2/c^2-1/2*d*e*(a+b*arctan(c*x))^2/c^4+1/6*e^2*(a+b*arctan(c*x))^2/c^6+1/2*d^2*x^2*(a+b*arctan(c*x))^2+1/2*d*e*x^4*(a+b*arctan(c*x))^2+1/6*e^2*x^6*(a+b*arctan(c*x))^2+1/2*b^2*d^2*log(1+c^2*x^2)/c^2-2/3*b^2*d*e*log(1+c^2*x^2)/c^4+23/90*b^2*e^2*log(1+c^2*x^2)/c^6],
[(d+e*x^2)^2*(a+b*arctan(c*x))^2,x,30,2/3*b^2*d*e*x/c^2-3/10*b^2*e^2*x/c^4+1/30*b^2*e^2*x^3/c^2-2/3*b^2*d*e*arctan(c*x)/c^3+3/10*b^2*e^2*arctan(c*x)/c^5-2/3*b*d*e*x^2*(a+b*arctan(c*x))/c+1/5*b*e^2*x^2*(a+b*arctan(c*x))/c^3-1/10*b*e^2*x^4*(a+b*arctan(c*x))/c+I*d^2*(a+b*arctan(c*x))^2/c-2/3*I*d*e*(a+b*arctan(c*x))^2/c^3+1/5*I*e^2*(a+b*arctan(c*x))^2/c^5+d^2*x*(a+b*arctan(c*x))^2+2/3*d*e*x^3*(a+b*arctan(c*x))^2+1/5*e^2*x^5*(a+b*arctan(c*x))^2+2*b*d^2*(a+b*arctan(c*x))*log(2/(1+I*c*x))/c-4/3*b*d*e*(a+b*arctan(c*x))*log(2/(1+I*c*x))/c^3+2/5*b*e^2*(a+b*arctan(c*x))*log(2/(1+I*c*x))/c^5+I*b^2*d^2*polylog(2,1+(-2)/(1+I*c*x))/c-2/3*I*b^2*d*e*polylog(2,1+(-2)/(1+I*c*x))/c^3+1/5*I*b^2*e^2*polylog(2,1+(-2)/(1+I*c*x))/c^5],
[(d+e*x^2)^2*(a+b*arctan(c*x))^2/x,x,25,-2*a*b*d*e*x/c+1/2*a*b*e^2*x/c^3+1/12*b^2*e^2*x^2/c^2-2*b^2*d*e*x*arctan(c*x)/c+1/2*b^2*e^2*x*arctan(c*x)/c^3-1/6*b*e^2*x^3*(a+b*arctan(c*x))/c+d*e*(a+b*arctan(c*x))^2/c^2-1/4*e^2*(a+b*arctan(c*x))^2/c^4+d*e*x^2*(a+b*arctan(c*x))^2+1/4*e^2*x^4*(a+b*arctan(c*x))^2+2*d^2*(a+b*arctan(c*x))^2*arctanh(1+(-2)/(1+I*c*x))+b^2*d*e*log(1+c^2*x^2)/c^2-1/3*b^2*e^2*log(1+c^2*x^2)/c^4-I*b*d^2*(a+b*arctan(c*x))*polylog(2,1+(-2)/(1+I*c*x))+I*b*d^2*(a+b*arctan(c*x))*polylog(2,-1+2/(1+I*c*x))-1/2*b^2*d^2*polylog(3,1+(-2)/(1+I*c*x))+1/2*b^2*d^2*polylog(3,-1+2/(1+I*c*x))],
[(d+e*x^2)^2*(a+b*arctan(c*x))^2/x^2,x,20,1/3*b^2*e^2*x/c^2-1/3*b^2*e^2*arctan(c*x)/c^3-1/3*b*e^2*x^2*(a+b*arctan(c*x))/c-I*c*d^2*(a+b*arctan(c*x))^2+2*I*d*e*(a+b*arctan(c*x))^2/c-1/3*I*e^2*(a+b*arctan(c*x))^2/c^3-d^2*(a+b*arctan(c*x))^2/x+2*d*e*x*(a+b*arctan(c*x))^2+1/3*e^2*x^3*(a+b*arctan(c*x))^2+4*b*d*e*(a+b*arctan(c*x))*log(2/(1+I*c*x))/c-2/3*b*e^2*(a+b*arctan(c*x))*log(2/(1+I*c*x))/c^3+2*b*c*d^2*(a+b*arctan(c*x))*log(2+(-2)/(1-I*c*x))-I*b^2*c*d^2*polylog(2,-1+2/(1-I*c*x))+2*I*b^2*d*e*polylog(2,1+(-2)/(1+I*c*x))/c-1/3*I*b^2*e^2*polylog(2,1+(-2)/(1+I*c*x))/c^3],
[(d+e*x^2)^2*(a+b*arctan(c*x))^2/x^3,x,22,-a*b*e^2*x/c-b^2*e^2*x*arctan(c*x)/c-b*c*d^2*(a+b*arctan(c*x))/x-1/2*c^2*d^2*(a+b*arctan(c*x))^2+1/2*e^2*(a+b*arctan(c*x))^2/c^2-1/2*d^2*(a+b*arctan(c*x))^2/x^2+1/2*e^2*x^2*(a+b*arctan(c*x))^2+4*d*e*(a+b*arctan(c*x))^2*arctanh(1+(-2)/(1+I*c*x))+b^2*c^2*d^2*log(x)-1/2*b^2*c^2*d^2*log(1+c^2*x^2)+1/2*b^2*e^2*log(1+c^2*x^2)/c^2-2*I*b*d*e*(a+b*arctan(c*x))*polylog(2,1+(-2)/(1+I*c*x))+2*I*b*d*e*(a+b*arctan(c*x))*polylog(2,-1+2/(1+I*c*x))-b^2*d*e*polylog(3,1+(-2)/(1+I*c*x))+b^2*d*e*polylog(3,-1+2/(1+I*c*x))],

# p<0
[x^3*(a+b*arctan(c*x))^2/(d+e*x^2),x,11,-a*b*x/(c*e)-b^2*x*arctan(c*x)/(c*e)+1/2*(a+b*arctan(c*x))^2/(c^2*e)+1/2*x^2*(a+b*arctan(c*x))^2/e+d*(a+b*arctan(c*x))^2*log(2/(1-I*c*x))/e^2+1/2*b^2*log(1+c^2*x^2)/(c^2*e)-1/2*d*(a+b*arctan(c*x))^2*log(2*c*(sqrt(-d)-x*sqrt(e))/((1-I*c*x)*(c*sqrt(-d)-I*sqrt(e))))/e^2-1/2*d*(a+b*arctan(c*x))^2*log(2*c*(sqrt(-d)+x*sqrt(e))/((1-I*c*x)*(c*sqrt(-d)+I*sqrt(e))))/e^2-I*b*d*(a+b*arctan(c*x))*polylog(2,1+(-2)/(1-I*c*x))/e^2+1/2*I*b*d*(a+b*arctan(c*x))*polylog(2,1-2*c*(sqrt(-d)-x*sqrt(e))/((1-I*c*x)*(c*sqrt(-d)-I*sqrt(e))))/e^2+1/2*I*b*d*(a+b*arctan(c*x))*polylog(2,1-2*c*(sqrt(-d)+x*sqrt(e))/((1-I*c*x)*(c*sqrt(-d)+I*sqrt(e))))/e^2+1/2*b^2*d*polylog(3,1+(-2)/(1-I*c*x))/e^2-1/4*b^2*d*polylog(3,1-2*c*(sqrt(-d)-x*sqrt(e))/((1-I*c*x)*(c*sqrt(-d)-I*sqrt(e))))/e^2-1/4*b^2*d*polylog(3,1-2*c*(sqrt(-d)+x*sqrt(e))/((1-I*c*x)*(c*sqrt(-d)+I*sqrt(e))))/e^2],
[x^2*(a+b*arctan(c*x))^2/(d+e*x^2),x,10,I*(a+b*arctan(c*x))^2/(c*e)+x*(a+b*arctan(c*x))^2/e+2*b*(a+b*arctan(c*x))*log(2/(1+I*c*x))/(c*e)+I*b^2*polylog(2,1+(-2)/(1+I*c*x))/(c*e)+1/2*(a+b*arctan(c*x))^2*log(2*c*(sqrt(-d)-x*sqrt(e))/((1-I*c*x)*(c*sqrt(-d)-I*sqrt(e))))*sqrt(-d)/e^(3/2)-1/2*(a+b*arctan(c*x))^2*log(2*c*(sqrt(-d)+x*sqrt(e))/((1-I*c*x)*(c*sqrt(-d)+I*sqrt(e))))*sqrt(-d)/e^(3/2)-1/2*I*b*(a+b*arctan(c*x))*polylog(2,1-2*c*(sqrt(-d)-x*sqrt(e))/((1-I*c*x)*(c*sqrt(-d)-I*sqrt(e))))*sqrt(-d)/e^(3/2)+1/2*I*b*(a+b*arctan(c*x))*polylog(2,1-2*c*(sqrt(-d)+x*sqrt(e))/((1-I*c*x)*(c*sqrt(-d)+I*sqrt(e))))*sqrt(-d)/e^(3/2)+1/4*b^2*polylog(3,1-2*c*(sqrt(-d)-x*sqrt(e))/((1-I*c*x)*(c*sqrt(-d)-I*sqrt(e))))*sqrt(-d)/e^(3/2)-1/4*b^2*polylog(3,1-2*c*(sqrt(-d)+x*sqrt(e))/((1-I*c*x)*(c*sqrt(-d)+I*sqrt(e))))*sqrt(-d)/e^(3/2)],
[x*(a+b*arctan(c*x))^2/(d+e*x^2),x,4,-(a+b*arctan(c*x))^2*log(2/(1-I*c*x))/e+1/2*(a+b*arctan(c*x))^2*log(2*c*(sqrt(-d)-x*sqrt(e))/((1-I*c*x)*(c*sqrt(-d)-I*sqrt(e))))/e+1/2*(a+b*arctan(c*x))^2*log(2*c*(sqrt(-d)+x*sqrt(e))/((1-I*c*x)*(c*sqrt(-d)+I*sqrt(e))))/e+I*b*(a+b*arctan(c*x))*polylog(2,1+(-2)/(1-I*c*x))/e-1/2*I*b*(a+b*arctan(c*x))*polylog(2,1-2*c*(sqrt(-d)-x*sqrt(e))/((1-I*c*x)*(c*sqrt(-d)-I*sqrt(e))))/e-1/2*I*b*(a+b*arctan(c*x))*polylog(2,1-2*c*(sqrt(-d)+x*sqrt(e))/((1-I*c*x)*(c*sqrt(-d)+I*sqrt(e))))/e-1/2*b^2*polylog(3,1+(-2)/(1-I*c*x))/e+1/4*b^2*polylog(3,1-2*c*(sqrt(-d)-x*sqrt(e))/((1-I*c*x)*(c*sqrt(-d)-I*sqrt(e))))/e+1/4*b^2*polylog(3,1-2*c*(sqrt(-d)+x*sqrt(e))/((1-I*c*x)*(c*sqrt(-d)+I*sqrt(e))))/e],
[(a+b*arctan(c*x))^2/(d+e*x^2),x,4,1/2*(a+b*arctan(c*x))^2*log(2*c*(sqrt(-d)-x*sqrt(e))/((1-I*c*x)*(c*sqrt(-d)-I*sqrt(e))))/(sqrt(-d)*sqrt(e))-1/2*(a+b*arctan(c*x))^2*log(2*c*(sqrt(-d)+x*sqrt(e))/((1-I*c*x)*(c*sqrt(-d)+I*sqrt(e))))/(sqrt(-d)*sqrt(e))-1/2*I*b*(a+b*arctan(c*x))*polylog(2,1-2*c*(sqrt(-d)-x*sqrt(e))/((1-I*c*x)*(c*sqrt(-d)-I*sqrt(e))))/(sqrt(-d)*sqrt(e))+1/2*I*b*(a+b*arctan(c*x))*polylog(2,1-2*c*(sqrt(-d)+x*sqrt(e))/((1-I*c*x)*(c*sqrt(-d)+I*sqrt(e))))/(sqrt(-d)*sqrt(e))+1/4*b^2*polylog(3,1-2*c*(sqrt(-d)-x*sqrt(e))/((1-I*c*x)*(c*sqrt(-d)-I*sqrt(e))))/(sqrt(-d)*sqrt(e))-1/4*b^2*polylog(3,1-2*c*(sqrt(-d)+x*sqrt(e))/((1-I*c*x)*(c*sqrt(-d)+I*sqrt(e))))/(sqrt(-d)*sqrt(e))],
[(a+b*arctan(c*x))^2/(x*(d+e*x^2)),x,12,2*(a+b*arctan(c*x))^2*arctanh(1+(-2)/(1+I*c*x))/d+(a+b*arctan(c*x))^2*log(2/(1-I*c*x))/d-1/2*(a+b*arctan(c*x))^2*log(2*c*(sqrt(-d)-x*sqrt(e))/((1-I*c*x)*(c*sqrt(-d)-I*sqrt(e))))/d-1/2*(a+b*arctan(c*x))^2*log(2*c*(sqrt(-d)+x*sqrt(e))/((1-I*c*x)*(c*sqrt(-d)+I*sqrt(e))))/d-I*b*(a+b*arctan(c*x))*polylog(2,1+(-2)/(1-I*c*x))/d-I*b*(a+b*arctan(c*x))*polylog(2,1+(-2)/(1+I*c*x))/d+I*b*(a+b*arctan(c*x))*polylog(2,-1+2/(1+I*c*x))/d+1/2*I*b*(a+b*arctan(c*x))*polylog(2,1-2*c*(sqrt(-d)-x*sqrt(e))/((1-I*c*x)*(c*sqrt(-d)-I*sqrt(e))))/d+1/2*I*b*(a+b*arctan(c*x))*polylog(2,1-2*c*(sqrt(-d)+x*sqrt(e))/((1-I*c*x)*(c*sqrt(-d)+I*sqrt(e))))/d+1/2*b^2*polylog(3,1+(-2)/(1-I*c*x))/d-1/2*b^2*polylog(3,1+(-2)/(1+I*c*x))/d+1/2*b^2*polylog(3,-1+2/(1+I*c*x))/d-1/4*b^2*polylog(3,1-2*c*(sqrt(-d)-x*sqrt(e))/((1-I*c*x)*(c*sqrt(-d)-I*sqrt(e))))/d-1/4*b^2*polylog(3,1-2*c*(sqrt(-d)+x*sqrt(e))/((1-I*c*x)*(c*sqrt(-d)+I*sqrt(e))))/d],
[(a+b*arctan(c*x))^2/(x^2*(d+e*x^2)),x,9,-I*c*(a+b*arctan(c*x))^2/d-(a+b*arctan(c*x))^2/(d*x)+2*b*c*(a+b*arctan(c*x))*log(2+(-2)/(1-I*c*x))/d-I*b^2*c*polylog(2,-1+2/(1-I*c*x))/d+1/2*(a+b*arctan(c*x))^2*log(2*c*(sqrt(-d)-x*sqrt(e))/((1-I*c*x)*(c*sqrt(-d)-I*sqrt(e))))*sqrt(e)/(-d)^(3/2)-1/2*(a+b*arctan(c*x))^2*log(2*c*(sqrt(-d)+x*sqrt(e))/((1-I*c*x)*(c*sqrt(-d)+I*sqrt(e))))*sqrt(e)/(-d)^(3/2)-1/2*I*b*(a+b*arctan(c*x))*polylog(2,1-2*c*(sqrt(-d)-x*sqrt(e))/((1-I*c*x)*(c*sqrt(-d)-I*sqrt(e))))*sqrt(e)/(-d)^(3/2)+1/2*I*b*(a+b*arctan(c*x))*polylog(2,1-2*c*(sqrt(-d)+x*sqrt(e))/((1-I*c*x)*(c*sqrt(-d)+I*sqrt(e))))*sqrt(e)/(-d)^(3/2)+1/4*b^2*polylog(3,1-2*c*(sqrt(-d)-x*sqrt(e))/((1-I*c*x)*(c*sqrt(-d)-I*sqrt(e))))*sqrt(e)/(-d)^(3/2)-1/4*b^2*polylog(3,1-2*c*(sqrt(-d)+x*sqrt(e))/((1-I*c*x)*(c*sqrt(-d)+I*sqrt(e))))*sqrt(e)/(-d)^(3/2)],
[(a+b*arctan(c*x))^2/(x^3*(d+e*x^2)),x,21,-b*c*(a+b*arctan(c*x))/(d*x)-1/2*c^2*(a+b*arctan(c*x))^2/d-1/2*(a+b*arctan(c*x))^2/(d*x^2)-2*e*(a+b*arctan(c*x))^2*arctanh(1+(-2)/(1+I*c*x))/d^2+b^2*c^2*log(x)/d-e*(a+b*arctan(c*x))^2*log(2/(1-I*c*x))/d^2-1/2*b^2*c^2*log(1+c^2*x^2)/d+1/2*e*(a+b*arctan(c*x))^2*log(2*c*(sqrt(-d)-x*sqrt(e))/((1-I*c*x)*(c*sqrt(-d)-I*sqrt(e))))/d^2+1/2*e*(a+b*arctan(c*x))^2*log(2*c*(sqrt(-d)+x*sqrt(e))/((1-I*c*x)*(c*sqrt(-d)+I*sqrt(e))))/d^2+I*b*e*(a+b*arctan(c*x))*polylog(2,1+(-2)/(1-I*c*x))/d^2+I*b*e*(a+b*arctan(c*x))*polylog(2,1+(-2)/(1+I*c*x))/d^2-I*b*e*(a+b*arctan(c*x))*polylog(2,-1+2/(1+I*c*x))/d^2-1/2*I*b*e*(a+b*arctan(c*x))*polylog(2,1-2*c*(sqrt(-d)-x*sqrt(e))/((1-I*c*x)*(c*sqrt(-d)-I*sqrt(e))))/d^2-1/2*I*b*e*(a+b*arctan(c*x))*polylog(2,1-2*c*(sqrt(-d)+x*sqrt(e))/((1-I*c*x)*(c*sqrt(-d)+I*sqrt(e))))/d^2-1/2*b^2*e*polylog(3,1+(-2)/(1-I*c*x))/d^2+1/2*b^2*e*polylog(3,1+(-2)/(1+I*c*x))/d^2-1/2*b^2*e*polylog(3,-1+2/(1+I*c*x))/d^2+1/4*b^2*e*polylog(3,1-2*c*(sqrt(-d)-x*sqrt(e))/((1-I*c*x)*(c*sqrt(-d)-I*sqrt(e))))/d^2+1/4*b^2*e*polylog(3,1-2*c*(sqrt(-d)+x*sqrt(e))/((1-I*c*x)*(c*sqrt(-d)+I*sqrt(e))))/d^2],
[x^3*(a+b*arctan(c*x))^2/(d+e*x^2)^2,x,33,-1/2*c^2*d*(a+b*arctan(c*x))^2/((c^2*d-e)*e^2)-(a+b*arctan(c*x))^2*log(2/(1-I*c*x))/e^2+1/2*(a+b*arctan(c*x))^2*log(2*c*(sqrt(-d)-x*sqrt(e))/((1-I*c*x)*(c*sqrt(-d)-I*sqrt(e))))/e^2+1/2*(a+b*arctan(c*x))^2*log(2*c*(sqrt(-d)+x*sqrt(e))/((1-I*c*x)*(c*sqrt(-d)+I*sqrt(e))))/e^2+I*b*(a+b*arctan(c*x))*polylog(2,1+(-2)/(1-I*c*x))/e^2-1/2*I*b*(a+b*arctan(c*x))*polylog(2,1-2*c*(sqrt(-d)-x*sqrt(e))/((1-I*c*x)*(c*sqrt(-d)-I*sqrt(e))))/e^2-1/2*I*b*(a+b*arctan(c*x))*polylog(2,1-2*c*(sqrt(-d)+x*sqrt(e))/((1-I*c*x)*(c*sqrt(-d)+I*sqrt(e))))/e^2-1/2*b^2*polylog(3,1+(-2)/(1-I*c*x))/e^2+1/4*b^2*polylog(3,1-2*c*(sqrt(-d)-x*sqrt(e))/((1-I*c*x)*(c*sqrt(-d)-I*sqrt(e))))/e^2+1/4*b^2*polylog(3,1-2*c*(sqrt(-d)+x*sqrt(e))/((1-I*c*x)*(c*sqrt(-d)+I*sqrt(e))))/e^2-1/2*b*c*(a+b*arctan(c*x))*log(2*c*(sqrt(-d)-x*sqrt(e))/((1-I*c*x)*(c*sqrt(-d)-I*sqrt(e))))*sqrt(-d)/((c^2*d-e)*e^(3/2))+1/2*b*c*(a+b*arctan(c*x))*log(2*c*(sqrt(-d)+x*sqrt(e))/((1-I*c*x)*(c*sqrt(-d)+I*sqrt(e))))*sqrt(-d)/((c^2*d-e)*e^(3/2))+1/4*I*b^2*c*polylog(2,1-2*c*(sqrt(-d)-x*sqrt(e))/((1-I*c*x)*(c*sqrt(-d)-I*sqrt(e))))*sqrt(-d)/((c^2*d-e)*e^(3/2))-1/4*I*b^2*c*polylog(2,1-2*c*(sqrt(-d)+x*sqrt(e))/((1-I*c*x)*(c*sqrt(-d)+I*sqrt(e))))*sqrt(-d)/((c^2*d-e)*e^(3/2))+1/4*(a+b*arctan(c*x))^2/(e^2*(1-x*sqrt(e)/sqrt(-d)))+1/4*(a+b*arctan(c*x))^2/(e^2*(1+x*sqrt(e)/sqrt(-d)))],
[x^2*(a+b*arctan(c*x))^2/(d+e*x^2)^2,x,38,-1/2*I*c*(a+b*arctan(c*x))^2/((c^2*d-e)*e)+b*c*(a+b*arctan(c*x))*log(2/(1-I*c*x))/((c^2*d-e)*e)-b*c*(a+b*arctan(c*x))*log(2/(1+I*c*x))/((c^2*d-e)*e)-1/2*b*c*(a+b*arctan(c*x))*log(2*c*(sqrt(-d)-x*sqrt(e))/((1-I*c*x)*(c*sqrt(-d)-I*sqrt(e))))/((c^2*d-e)*e)-1/2*b*c*(a+b*arctan(c*x))*log(2*c*(sqrt(-d)+x*sqrt(e))/((1-I*c*x)*(c*sqrt(-d)+I*sqrt(e))))/((c^2*d-e)*e)-1/2*I*b^2*c*polylog(2,1+(-2)/(1-I*c*x))/((c^2*d-e)*e)-1/2*I*b^2*c*polylog(2,1+(-2)/(1+I*c*x))/((c^2*d-e)*e)+1/4*I*b^2*c*polylog(2,1-2*c*(sqrt(-d)-x*sqrt(e))/((1-I*c*x)*(c*sqrt(-d)-I*sqrt(e))))/((c^2*d-e)*e)+1/4*I*b^2*c*polylog(2,1-2*c*(sqrt(-d)+x*sqrt(e))/((1-I*c*x)*(c*sqrt(-d)+I*sqrt(e))))/((c^2*d-e)*e)+1/4*(a+b*arctan(c*x))^2*log(2*c*(sqrt(-d)-x*sqrt(e))/((1-I*c*x)*(c*sqrt(-d)-I*sqrt(e))))/(e^(3/2)*sqrt(-d))-1/4*(a+b*arctan(c*x))^2*log(2*c*(sqrt(-d)+x*sqrt(e))/((1-I*c*x)*(c*sqrt(-d)+I*sqrt(e))))/(e^(3/2)*sqrt(-d))-1/4*I*b*(a+b*arctan(c*x))*polylog(2,1-2*c*(sqrt(-d)-x*sqrt(e))/((1-I*c*x)*(c*sqrt(-d)-I*sqrt(e))))/(e^(3/2)*sqrt(-d))+1/4*I*b*(a+b*arctan(c*x))*polylog(2,1-2*c*(sqrt(-d)+x*sqrt(e))/((1-I*c*x)*(c*sqrt(-d)+I*sqrt(e))))/(e^(3/2)*sqrt(-d))+1/8*b^2*polylog(3,1-2*c*(sqrt(-d)-x*sqrt(e))/((1-I*c*x)*(c*sqrt(-d)-I*sqrt(e))))/(e^(3/2)*sqrt(-d))-1/8*b^2*polylog(3,1-2*c*(sqrt(-d)+x*sqrt(e))/((1-I*c*x)*(c*sqrt(-d)+I*sqrt(e))))/(e^(3/2)*sqrt(-d))+1/4*(a+b*arctan(c*x))^2/(e^(3/2)*(sqrt(-d)-x*sqrt(e)))-1/4*(a+b*arctan(c*x))^2/(e^(3/2)*(sqrt(-d)+x*sqrt(e)))],
[x*(a+b*arctan(c*x))^2/(d+e*x^2)^2,x,27,1/2*c^2*(a+b*arctan(c*x))^2/((c^2*d-e)*e)-1/2*b*c*(a+b*arctan(c*x))*log(2*c*(sqrt(-d)-x*sqrt(e))/((1-I*c*x)*(c*sqrt(-d)-I*sqrt(e))))/((c^2*d-e)*sqrt(-d)*sqrt(e))+1/2*b*c*(a+b*arctan(c*x))*log(2*c*(sqrt(-d)+x*sqrt(e))/((1-I*c*x)*(c*sqrt(-d)+I*sqrt(e))))/((c^2*d-e)*sqrt(-d)*sqrt(e))+1/4*I*b^2*c*polylog(2,1-2*c*(sqrt(-d)-x*sqrt(e))/((1-I*c*x)*(c*sqrt(-d)-I*sqrt(e))))/((c^2*d-e)*sqrt(-d)*sqrt(e))-1/4*I*b^2*c*polylog(2,1-2*c*(sqrt(-d)+x*sqrt(e))/((1-I*c*x)*(c*sqrt(-d)+I*sqrt(e))))/((c^2*d-e)*sqrt(-d)*sqrt(e))-1/4*(a+b*arctan(c*x))^2/(d*e*(1-x*sqrt(e)/sqrt(-d)))-1/4*(a+b*arctan(c*x))^2/(d*e*(1+x*sqrt(e)/sqrt(-d)))],
[(a+b*arctan(c*x))^2/(d+e*x^2)^2,x,32,1/2*I*c*(a+b*arctan(c*x))^2/(d*(c^2*d-e))-b*c*(a+b*arctan(c*x))*log(2/(1-I*c*x))/(d*(c^2*d-e))+b*c*(a+b*arctan(c*x))*log(2/(1+I*c*x))/(d*(c^2*d-e))+1/2*b*c*(a+b*arctan(c*x))*log(2*c*(sqrt(-d)-x*sqrt(e))/((1-I*c*x)*(c*sqrt(-d)-I*sqrt(e))))/(d*(c^2*d-e))+1/2*b*c*(a+b*arctan(c*x))*log(2*c*(sqrt(-d)+x*sqrt(e))/((1-I*c*x)*(c*sqrt(-d)+I*sqrt(e))))/(d*(c^2*d-e))+1/2*I*b^2*c*polylog(2,1+(-2)/(1-I*c*x))/(d*(c^2*d-e))+1/2*I*b^2*c*polylog(2,1+(-2)/(1+I*c*x))/(d*(c^2*d-e))-1/4*I*b^2*c*polylog(2,1-2*c*(sqrt(-d)-x*sqrt(e))/((1-I*c*x)*(c*sqrt(-d)-I*sqrt(e))))/(d*(c^2*d-e))-1/4*I*b^2*c*polylog(2,1-2*c*(sqrt(-d)+x*sqrt(e))/((1-I*c*x)*(c*sqrt(-d)+I*sqrt(e))))/(d*(c^2*d-e))-1/4*(a+b*arctan(c*x))^2*log(2*c*(sqrt(-d)-x*sqrt(e))/((1-I*c*x)*(c*sqrt(-d)-I*sqrt(e))))/((-d)^(3/2)*sqrt(e))+1/4*(a+b*arctan(c*x))^2*log(2*c*(sqrt(-d)+x*sqrt(e))/((1-I*c*x)*(c*sqrt(-d)+I*sqrt(e))))/((-d)^(3/2)*sqrt(e))+1/4*I*b*(a+b*arctan(c*x))*polylog(2,1-2*c*(sqrt(-d)-x*sqrt(e))/((1-I*c*x)*(c*sqrt(-d)-I*sqrt(e))))/((-d)^(3/2)*sqrt(e))-1/4*I*b*(a+b*arctan(c*x))*polylog(2,1-2*c*(sqrt(-d)+x*sqrt(e))/((1-I*c*x)*(c*sqrt(-d)+I*sqrt(e))))/((-d)^(3/2)*sqrt(e))-1/8*b^2*polylog(3,1-2*c*(sqrt(-d)-x*sqrt(e))/((1-I*c*x)*(c*sqrt(-d)-I*sqrt(e))))/((-d)^(3/2)*sqrt(e))+1/8*b^2*polylog(3,1-2*c*(sqrt(-d)+x*sqrt(e))/((1-I*c*x)*(c*sqrt(-d)+I*sqrt(e))))/((-d)^(3/2)*sqrt(e))-1/4*(a+b*arctan(c*x))^2/(d*sqrt(e)*(sqrt(-d)-x*sqrt(e)))+1/4*(a+b*arctan(c*x))^2/(d*sqrt(e)*(sqrt(-d)+x*sqrt(e)))],
[(a+b*arctan(c*x))^2/(x*(d+e*x^2)^2),x,39,-1/2*c^2*(a+b*arctan(c*x))^2/(d*(c^2*d-e))+2*(a+b*arctan(c*x))^2*arctanh(1+(-2)/(1+I*c*x))/d^2+(a+b*arctan(c*x))^2*log(2/(1-I*c*x))/d^2-1/2*(a+b*arctan(c*x))^2*log(2*c*(sqrt(-d)-x*sqrt(e))/((1-I*c*x)*(c*sqrt(-d)-I*sqrt(e))))/d^2-1/2*(a+b*arctan(c*x))^2*log(2*c*(sqrt(-d)+x*sqrt(e))/((1-I*c*x)*(c*sqrt(-d)+I*sqrt(e))))/d^2-I*b*(a+b*arctan(c*x))*polylog(2,1+(-2)/(1-I*c*x))/d^2-I*b*(a+b*arctan(c*x))*polylog(2,1+(-2)/(1+I*c*x))/d^2+I*b*(a+b*arctan(c*x))*polylog(2,-1+2/(1+I*c*x))/d^2+1/2*I*b*(a+b*arctan(c*x))*polylog(2,1-2*c*(sqrt(-d)-x*sqrt(e))/((1-I*c*x)*(c*sqrt(-d)-I*sqrt(e))))/d^2+1/2*I*b*(a+b*arctan(c*x))*polylog(2,1-2*c*(sqrt(-d)+x*sqrt(e))/((1-I*c*x)*(c*sqrt(-d)+I*sqrt(e))))/d^2+1/2*b^2*polylog(3,1+(-2)/(1-I*c*x))/d^2-1/2*b^2*polylog(3,1+(-2)/(1+I*c*x))/d^2+1/2*b^2*polylog(3,-1+2/(1+I*c*x))/d^2-1/4*b^2*polylog(3,1-2*c*(sqrt(-d)-x*sqrt(e))/((1-I*c*x)*(c*sqrt(-d)-I*sqrt(e))))/d^2-1/4*b^2*polylog(3,1-2*c*(sqrt(-d)+x*sqrt(e))/((1-I*c*x)*(c*sqrt(-d)+I*sqrt(e))))/d^2-1/2*b*c*(a+b*arctan(c*x))*log(2*c*(sqrt(-d)-x*sqrt(e))/((1-I*c*x)*(c*sqrt(-d)-I*sqrt(e))))*sqrt(e)/((-d)^(3/2)*(c^2*d-e))+1/2*b*c*(a+b*arctan(c*x))*log(2*c*(sqrt(-d)+x*sqrt(e))/((1-I*c*x)*(c*sqrt(-d)+I*sqrt(e))))*sqrt(e)/((-d)^(3/2)*(c^2*d-e))+1/4*I*b^2*c*polylog(2,1-2*c*(sqrt(-d)-x*sqrt(e))/((1-I*c*x)*(c*sqrt(-d)-I*sqrt(e))))*sqrt(e)/((-d)^(3/2)*(c^2*d-e))-1/4*I*b^2*c*polylog(2,1-2*c*(sqrt(-d)+x*sqrt(e))/((1-I*c*x)*(c*sqrt(-d)+I*sqrt(e))))*sqrt(e)/((-d)^(3/2)*(c^2*d-e))+1/4*(a+b*arctan(c*x))^2/(d^2*(1-x*sqrt(e)/sqrt(-d)))+1/4*(a+b*arctan(c*x))^2/(d^2*(1+x*sqrt(e)/sqrt(-d)))],
[(a+b*arctan(c*x))^2/(x^2*(d+e*x^2)^2),x,42,-I*c*(a+b*arctan(c*x))^2/d^2-1/2*I*c*e*(a+b*arctan(c*x))^2/(d^2*(c^2*d-e))-(a+b*arctan(c*x))^2/(d^2*x)+b*c*e*(a+b*arctan(c*x))*log(2/(1-I*c*x))/(d^2*(c^2*d-e))-b*c*e*(a+b*arctan(c*x))*log(2/(1+I*c*x))/(d^2*(c^2*d-e))+2*b*c*(a+b*arctan(c*x))*log(2+(-2)/(1-I*c*x))/d^2-1/2*b*c*e*(a+b*arctan(c*x))*log(2*c*(sqrt(-d)-x*sqrt(e))/((1-I*c*x)*(c*sqrt(-d)-I*sqrt(e))))/(d^2*(c^2*d-e))-1/2*b*c*e*(a+b*arctan(c*x))*log(2*c*(sqrt(-d)+x*sqrt(e))/((1-I*c*x)*(c*sqrt(-d)+I*sqrt(e))))/(d^2*(c^2*d-e))-1/2*I*b^2*c*e*polylog(2,1+(-2)/(1-I*c*x))/(d^2*(c^2*d-e))-I*b^2*c*polylog(2,-1+2/(1-I*c*x))/d^2-1/2*I*b^2*c*e*polylog(2,1+(-2)/(1+I*c*x))/(d^2*(c^2*d-e))+1/4*I*b^2*c*e*polylog(2,1-2*c*(sqrt(-d)-x*sqrt(e))/((1-I*c*x)*(c*sqrt(-d)-I*sqrt(e))))/(d^2*(c^2*d-e))+1/4*I*b^2*c*e*polylog(2,1-2*c*(sqrt(-d)+x*sqrt(e))/((1-I*c*x)*(c*sqrt(-d)+I*sqrt(e))))/(d^2*(c^2*d-e))-3/4*(a+b*arctan(c*x))^2*log(2*c*(sqrt(-d)-x*sqrt(e))/((1-I*c*x)*(c*sqrt(-d)-I*sqrt(e))))*sqrt(e)/(-d)^(5/2)+3/4*(a+b*arctan(c*x))^2*log(2*c*(sqrt(-d)+x*sqrt(e))/((1-I*c*x)*(c*sqrt(-d)+I*sqrt(e))))*sqrt(e)/(-d)^(5/2)+3/4*I*b*(a+b*arctan(c*x))*polylog(2,1-2*c*(sqrt(-d)-x*sqrt(e))/((1-I*c*x)*(c*sqrt(-d)-I*sqrt(e))))*sqrt(e)/(-d)^(5/2)-3/4*I*b*(a+b*arctan(c*x))*polylog(2,1-2*c*(sqrt(-d)+x*sqrt(e))/((1-I*c*x)*(c*sqrt(-d)+I*sqrt(e))))*sqrt(e)/(-d)^(5/2)-3/8*b^2*polylog(3,1-2*c*(sqrt(-d)-x*sqrt(e))/((1-I*c*x)*(c*sqrt(-d)-I*sqrt(e))))*sqrt(e)/(-d)^(5/2)+3/8*b^2*polylog(3,1-2*c*(sqrt(-d)+x*sqrt(e))/((1-I*c*x)*(c*sqrt(-d)+I*sqrt(e))))*sqrt(e)/(-d)^(5/2)+1/4*(a+b*arctan(c*x))^2*sqrt(e)/(d^2*(sqrt(-d)-x*sqrt(e)))-1/4*(a+b*arctan(c*x))^2*sqrt(e)/(d^2*(sqrt(-d)+x*sqrt(e)))],
[(a+b*arctan(c*x))^2/(x^3*(d+e*x^2)^2),x,47,-b*c*(a+b*arctan(c*x))/(d^2*x)-1/2*c^2*(a+b*arctan(c*x))^2/d^2+1/2*c^2*e*(a+b*arctan(c*x))^2/(d^2*(c^2*d-e))-1/2*(a+b*arctan(c*x))^2/(d^2*x^2)-4*e*(a+b*arctan(c*x))^2*arctanh(1+(-2)/(1+I*c*x))/d^3+b^2*c^2*log(x)/d^2-2*e*(a+b*arctan(c*x))^2*log(2/(1-I*c*x))/d^3-1/2*b^2*c^2*log(1+c^2*x^2)/d^2-1/2*b*c*e^(3/2)*(a+b*arctan(c*x))*log(2*c*(sqrt(-d)-x*sqrt(e))/((1-I*c*x)*(c*sqrt(-d)-I*sqrt(e))))/((-d)^(5/2)*(c^2*d-e))+e*(a+b*arctan(c*x))^2*log(2*c*(sqrt(-d)-x*sqrt(e))/((1-I*c*x)*(c*sqrt(-d)-I*sqrt(e))))/d^3+1/2*b*c*e^(3/2)*(a+b*arctan(c*x))*log(2*c*(sqrt(-d)+x*sqrt(e))/((1-I*c*x)*(c*sqrt(-d)+I*sqrt(e))))/((-d)^(5/2)*(c^2*d-e))+e*(a+b*arctan(c*x))^2*log(2*c*(sqrt(-d)+x*sqrt(e))/((1-I*c*x)*(c*sqrt(-d)+I*sqrt(e))))/d^3+2*I*b*e*(a+b*arctan(c*x))*polylog(2,1+(-2)/(1-I*c*x))/d^3+2*I*b*e*(a+b*arctan(c*x))*polylog(2,1+(-2)/(1+I*c*x))/d^3-2*I*b*e*(a+b*arctan(c*x))*polylog(2,-1+2/(1+I*c*x))/d^3+1/4*I*b^2*c*e^(3/2)*polylog(2,1-2*c*(sqrt(-d)-x*sqrt(e))/((1-I*c*x)*(c*sqrt(-d)-I*sqrt(e))))/((-d)^(5/2)*(c^2*d-e))-I*b*e*(a+b*arctan(c*x))*polylog(2,1-2*c*(sqrt(-d)-x*sqrt(e))/((1-I*c*x)*(c*sqrt(-d)-I*sqrt(e))))/d^3-1/4*I*b^2*c*e^(3/2)*polylog(2,1-2*c*(sqrt(-d)+x*sqrt(e))/((1-I*c*x)*(c*sqrt(-d)+I*sqrt(e))))/((-d)^(5/2)*(c^2*d-e))-I*b*e*(a+b*arctan(c*x))*polylog(2,1-2*c*(sqrt(-d)+x*sqrt(e))/((1-I*c*x)*(c*sqrt(-d)+I*sqrt(e))))/d^3-b^2*e*polylog(3,1+(-2)/(1-I*c*x))/d^3+b^2*e*polylog(3,1+(-2)/(1+I*c*x))/d^3-b^2*e*polylog(3,-1+2/(1+I*c*x))/d^3+1/2*b^2*e*polylog(3,1-2*c*(sqrt(-d)-x*sqrt(e))/((1-I*c*x)*(c*sqrt(-d)-I*sqrt(e))))/d^3+1/2*b^2*e*polylog(3,1-2*c*(sqrt(-d)+x*sqrt(e))/((1-I*c*x)*(c*sqrt(-d)+I*sqrt(e))))/d^3-1/4*e*(a+b*arctan(c*x))^2/(d^3*(1-x*sqrt(e)/sqrt(-d)))-1/4*e*(a+b*arctan(c*x))^2/(d^3*(1+x*sqrt(e)/sqrt(-d)))],

# Integrands of the form (h x)^m (d+e Log[f+g x^2]) (a+b ArcTan[c x])^p

# Integrands of the form x^m (d+e Log[f+g x^2]) (a+b ArcTan[c x])
[x^4*arctan(x)*log(1+x^2),x,24,-77/300*x^2+9/200*x^4-2/5*x*arctan(x)+2/15*x^3*arctan(x)-2/25*x^5*arctan(x)+1/5*arctan(x)^2+137/300*log(1+x^2)+1/10*x^2*log(1+x^2)-1/20*x^4*log(1+x^2)+1/5*x^5*arctan(x)*log(1+x^2)-1/20*log(1+x^2)^2],
[x^3*arctan(x)*log(1+x^2),x,14,-25/24*x+7/72*x^3+25/24*arctan(x)+1/4*x^2*arctan(x)-1/8*x^4*arctan(x)+1/4*x*log(1+x^2)-1/12*x^3*log(1+x^2)-1/4*arctan(x)*log(1+x^2)+1/4*x^4*arctan(x)*log(1+x^2)],
[x^2*arctan(x)*log(1+x^2),x,19,5/18*x^2+2/3*x*arctan(x)-2/9*x^3*arctan(x)-1/3*arctan(x)^2-11/18*log(1+x^2)-1/6*x^2*log(1+x^2)+1/3*x^3*arctan(x)*log(1+x^2)+1/12*log(1+x^2)^2],
[x*arctan(x)*log(1+x^2),x,7,3/2*x-3/2*arctan(x)-1/2*x^2*arctan(x)-1/2*x*log(1+x^2)+1/2*(1+x^2)*arctan(x)*log(1+x^2)],
[arctan(x)*log(1+x^2),x,8,-2*x*arctan(x)+arctan(x)^2+log(1+x^2)+x*arctan(x)*log(1+x^2)-1/4*log(1+x^2)^2],
[arctan(x)*log(1+x^2)/x,x,12,-1/2*I*log(1+I*x)^2*log(-I*x)+1/2*I*log(1-I*x)^2*log(I*x)+I*log(1-I*x)*polylog(2,1-I*x)-I*log(1+I*x)*polylog(2,1+I*x)-1/2*I*(log(1-I*x)+log(1+I*x)-log(1+x^2))*polylog(2,-I*x)+1/2*I*(log(1-I*x)+log(1+I*x)-log(1+x^2))*polylog(2,I*x)-I*polylog(3,1-I*x)+I*polylog(3,1+I*x)],
[arctan(x)*log(1+x^2)/x^2,x,8,arctan(x)^2-arctan(x)*log(1+x^2)/x-1/4*log(1+x^2)^2-1/2*polylog(2,-x^2)],
[arctan(x)*log(1+x^2)/x^3,x,6,arctan(x)-1/2*log(1+x^2)/x-1/2*arctan(x)*log(1+x^2)-1/2*arctan(x)*log(1+x^2)/x^2+1/2*I*polylog(2,-I*x)-1/2*I*polylog(2,I*x)],
[arctan(x)*log(1+x^2)/x^4,x,18,-2/3*arctan(x)/x-1/3*arctan(x)^2+log(x)-1/2*log(1+x^2)-1/6*log(1+x^2)/x^2-1/3*arctan(x)*log(1+x^2)/x^3+1/12*log(1+x^2)^2+1/6*polylog(2,-x^2)],
[arctan(x)*log(1+x^2)/x^5,x,12,(-5/12)/x-11/12*arctan(x)-1/4*arctan(x)/x^2-1/12*log(1+x^2)/x^3+1/4*log(1+x^2)/x+1/4*arctan(x)*log(1+x^2)-1/4*arctan(x)*log(1+x^2)/x^4-1/4*I*polylog(2,-I*x)+1/4*I*polylog(2,I*x)],
[arctan(x)*log(1+x^2)/x^6,x,26,(-7/60)/x^2-2/15*arctan(x)/x^3+2/5*arctan(x)/x+1/5*arctan(x)^2-5/6*log(x)+5/12*log(1+x^2)-1/20*log(1+x^2)/x^4+1/10*log(1+x^2)/x^2-1/5*arctan(x)*log(1+x^2)/x^5-1/20*log(1+x^2)^2-1/10*polylog(2,-x^2)],
[x^4*(a+b*arctan(c*x))*(d+e*log(1+c^2*x^2)),x,26,-2/5*a*e*x/c^4-77/300*b*e*x^2/c^3+2/15*a*e*x^3/c^2+9/200*b*e*x^4/c-2/25*a*e*x^5+2/5*a*e*arctan(c*x)/c^5-2/5*b*e*x*arctan(c*x)/c^4+2/15*b*e*x^3*arctan(c*x)/c^2-2/25*b*e*x^5*arctan(c*x)+1/5*b*e*arctan(c*x)^2/c^5+137/300*b*e*log(1+c^2*x^2)/c^5+1/20*b*e*log(1+c^2*x^2)^2/c^5+1/10*b*x^2*(d+e*log(1+c^2*x^2))/c^3-1/20*b*x^4*(d+e*log(1+c^2*x^2))/c+1/5*x^5*(a+b*arctan(c*x))*(d+e*log(1+c^2*x^2))-1/10*b*log(1+c^2*x^2)*(d+e*log(1+c^2*x^2))/c^5],
[x^3*(a+b*arctan(c*x))*(d+e*log(1+c^2*x^2)),x,14,1/8*b*(2*d-3*e)*x/c^3-2/3*b*e*x/c^3-1/24*b*(2*d-e)*x^3/c+1/18*b*e*x^3/c-1/8*b*(2*d-3*e)*arctan(c*x)/c^4+2/3*b*e*arctan(c*x)/c^4+1/4*e*x^2*(a+b*arctan(c*x))/c^2-1/8*e*x^4*(a+b*arctan(c*x))+1/4*b*e*x*log(1+c^2*x^2)/c^3-1/12*b*e*x^3*log(1+c^2*x^2)/c-1/4*e*(a+b*arctan(c*x))*log(1+c^2*x^2)/c^4+1/4*x^4*(a+b*arctan(c*x))*(d+e*log(1+c^2*x^2))],
[x^2*(a+b*arctan(c*x))*(d+e*log(1+c^2*x^2)),x,21,2/3*a*e*x/c^2+5/18*b*e*x^2/c-2/9*a*e*x^3-2/3*a*e*arctan(c*x)/c^3+2/3*b*e*x*arctan(c*x)/c^2-2/9*b*e*x^3*arctan(c*x)-1/3*b*e*arctan(c*x)^2/c^3-11/18*b*e*log(1+c^2*x^2)/c^3-1/12*b*e*log(1+c^2*x^2)^2/c^3-1/6*b*x^2*(d+e*log(1+c^2*x^2))/c+1/3*x^3*(a+b*arctan(c*x))*(d+e*log(1+c^2*x^2))+1/6*b*log(1+c^2*x^2)*(d+e*log(1+c^2*x^2))/c^3],
[x*(a+b*arctan(c*x))*(d+e*log(1+c^2*x^2)),x,7,-1/2*b*(d-e)*x/c+b*e*x/c+1/2*b*(d-e)*arctan(c*x)/c^2-b*e*arctan(c*x)/c^2+1/2*d*x^2*(a+b*arctan(c*x))-1/2*e*x^2*(a+b*arctan(c*x))-1/2*b*e*x*log(1+c^2*x^2)/c+1/2*e*(1+c^2*x^2)*(a+b*arctan(c*x))*log(1+c^2*x^2)/c^2],
[(a+b*arctan(c*x))*(d+e*log(1+c^2*x^2)),x,9,-2*a*e*x-2*b*e*x*arctan(c*x)+e*(a+b*arctan(c*x))^2/(b*c)+b*e*log(1+c^2*x^2)/c+x*(a+b*arctan(c*x))*(d+e*log(1+c^2*x^2))-1/4*b*(d+e*log(1+c^2*x^2))^2/(c*e)],
[(a+b*arctan(c*x))*(d+e*log(1+c^2*x^2))/x,x,18,a*d*log(x)+1/2*I*b*e*log(I*c*x)*log(1-I*c*x)^2-1/2*I*b*e*log(-I*c*x)*log(1+I*c*x)^2+1/2*I*b*d*polylog(2,-I*c*x)-1/2*I*b*e*(log(1-I*c*x)+log(1+I*c*x)-log(1+c^2*x^2))*polylog(2,-I*c*x)-1/2*I*b*d*polylog(2,I*c*x)+1/2*I*b*e*(log(1-I*c*x)+log(1+I*c*x)-log(1+c^2*x^2))*polylog(2,I*c*x)-1/2*a*e*polylog(2,-c^2*x^2)+I*b*e*log(1-I*c*x)*polylog(2,1-I*c*x)-I*b*e*log(1+I*c*x)*polylog(2,1+I*c*x)-I*b*e*polylog(3,1-I*c*x)+I*b*e*polylog(3,1+I*c*x)],
[(a+b*arctan(c*x))*(d+e*log(1+c^2*x^2))/x^2,x,6,c*e*(a+b*arctan(c*x))^2/b-(a+b*arctan(c*x))*(d+e*log(1+c^2*x^2))/x+1/2*b*c*(d+e*log(1+c^2*x^2))*log(1+(-1)/(1+c^2*x^2))-1/2*b*c*e*polylog(2,1/(1+c^2*x^2))],
[(a+b*arctan(c*x))*(d+e*log(1+c^2*x^2))/x^3,x,10,b*c^2*e*arctan(c*x)+a*c^2*e*log(x)-1/2*a*c^2*e*log(1+c^2*x^2)-1/2*b*c*(d+e*log(1+c^2*x^2))/x-1/2*b*c^2*arctan(c*x)*(d+e*log(1+c^2*x^2))-1/2*(a+b*arctan(c*x))*(d+e*log(1+c^2*x^2))/x^2+1/2*I*b*c^2*e*polylog(2,-I*c*x)-1/2*I*b*c^2*e*polylog(2,I*c*x)],
[(a+b*arctan(c*x))*(d+e*log(1+c^2*x^2))/x^4,x,15,-2/3*c^2*e*(a+b*arctan(c*x))/x-1/3*c^3*e*(a+b*arctan(c*x))^2/b+b*c^3*e*log(x)-1/3*b*c^3*e*log(1+c^2*x^2)-1/6*b*c*(1+c^2*x^2)*(d+e*log(1+c^2*x^2))/x^2-1/3*(a+b*arctan(c*x))*(d+e*log(1+c^2*x^2))/x^3-1/6*b*c^3*(d+e*log(1+c^2*x^2))*log(1+(-1)/(1+c^2*x^2))+1/6*b*c^3*e*polylog(2,1/(1+c^2*x^2))],
[(a+b*arctan(c*x))*(d+e*log(1+c^2*x^2))/x^5,x,15,-1/4*a*c^2*e/x^2-5/12*b*c^3*e/x-11/12*b*c^4*e*arctan(c*x)-1/4*b*c^2*e*arctan(c*x)/x^2-1/2*a*c^4*e*log(x)+1/4*a*c^4*e*log(1+c^2*x^2)-1/12*b*c*(d+e*log(1+c^2*x^2))/x^3+1/4*b*c^3*(d+e*log(1+c^2*x^2))/x+1/4*b*c^4*arctan(c*x)*(d+e*log(1+c^2*x^2))-1/4*(a+b*arctan(c*x))*(d+e*log(1+c^2*x^2))/x^4-1/4*I*b*c^4*e*polylog(2,-I*c*x)+1/4*I*b*c^4*e*polylog(2,I*c*x)],
[(a+b*arctan(c*x))*(d+e*log(1+c^2*x^2))/x^6,x,24,-7/60*b*c^3*e/x^2-2/15*c^2*e*(a+b*arctan(c*x))/x^3+2/5*c^4*e*(a+b*arctan(c*x))/x+1/5*c^5*e*(a+b*arctan(c*x))^2/b-5/6*b*c^5*e*log(x)+19/60*b*c^5*e*log(1+c^2*x^2)-1/20*b*c*(d+e*log(1+c^2*x^2))/x^4+1/10*b*c^3*(1+c^2*x^2)*(d+e*log(1+c^2*x^2))/x^2-1/5*(a+b*arctan(c*x))*(d+e*log(1+c^2*x^2))/x^5+1/10*b*c^5*(d+e*log(1+c^2*x^2))*log(1+(-1)/(1+c^2*x^2))-1/10*b*c^5*e*polylog(2,1/(1+c^2*x^2))],
[x*(a+b*arctan(c*x))*(d+e*log(f+g*x^2)),x,21,-1/2*b*(d-e)*x/c+b*e*x/c+1/2*b*(d-e)*arctan(c*x)/c^2+1/2*d*x^2*(a+b*arctan(c*x))-1/2*e*x^2*(a+b*arctan(c*x))-b*e*(c^2*f-g)*arctan(c*x)*log(2/(1-I*c*x))/(c^2*g)-1/2*b*e*x*log(f+g*x^2)/c-1/2*b*e*(c^2*f-g)*arctan(c*x)*log(f+g*x^2)/(c^2*g)+1/2*e*(f+g*x^2)*(a+b*arctan(c*x))*log(f+g*x^2)/g+1/2*b*e*(c^2*f-g)*arctan(c*x)*log(2*c*(sqrt(-f)-x*sqrt(g))/((1-I*c*x)*(c*sqrt(-f)-I*sqrt(g))))/(c^2*g)+1/2*b*e*(c^2*f-g)*arctan(c*x)*log(2*c*(sqrt(-f)+x*sqrt(g))/((1-I*c*x)*(c*sqrt(-f)+I*sqrt(g))))/(c^2*g)+1/2*I*b*e*(c^2*f-g)*polylog(2,1+(-2)/(1-I*c*x))/(c^2*g)-1/4*I*b*e*(c^2*f-g)*polylog(2,1-2*c*(sqrt(-f)-x*sqrt(g))/((1-I*c*x)*(c*sqrt(-f)-I*sqrt(g))))/(c^2*g)-1/4*I*b*e*(c^2*f-g)*polylog(2,1-2*c*(sqrt(-f)+x*sqrt(g))/((1-I*c*x)*(c*sqrt(-f)+I*sqrt(g))))/(c^2*g)-b*e*arctan(x*sqrt(g)/sqrt(f))*sqrt(f)/(c*sqrt(g))],
[(a+b*arctan(c*x))*(d+e*log(f+g*x^2)),x,28,-2*a*e*x-2*b*e*x*arctan(c*x)+b*e*log(1+c^2*x^2)/c+x*(a+b*arctan(c*x))*(d+e*log(f+g*x^2))-1/2*b*log(-g*(1+c^2*x^2)/(c^2*f-g))*(d+e*log(f+g*x^2))/c-1/2*b*e*polylog(2,c^2*(f+g*x^2)/(c^2*f-g))/c+1/2*I*b*e*log(1+I*c*x)*log(c*(sqrt(-f)-x*sqrt(g))/(c*sqrt(-f)-I*sqrt(g)))*sqrt(-f)/sqrt(g)-1/2*I*b*e*log(1-I*c*x)*log(c*(sqrt(-f)-x*sqrt(g))/(c*sqrt(-f)+I*sqrt(g)))*sqrt(-f)/sqrt(g)+1/2*I*b*e*log(1-I*c*x)*log(c*(sqrt(-f)+x*sqrt(g))/(c*sqrt(-f)-I*sqrt(g)))*sqrt(-f)/sqrt(g)-1/2*I*b*e*log(1+I*c*x)*log(c*(sqrt(-f)+x*sqrt(g))/(c*sqrt(-f)+I*sqrt(g)))*sqrt(-f)/sqrt(g)-1/2*I*b*e*polylog(2,(I-c*x)*sqrt(g)/(c*sqrt(-f)+I*sqrt(g)))*sqrt(-f)/sqrt(g)-1/2*I*b*e*polylog(2,(I+c*x)*sqrt(g)/(c*sqrt(-f)+I*sqrt(g)))*sqrt(-f)/sqrt(g)+1/2*I*b*e*polylog(2,(1-I*c*x)*sqrt(g)/(I*c*sqrt(-f)+sqrt(g)))*sqrt(-f)/sqrt(g)+1/2*I*b*e*polylog(2,(1+I*c*x)*sqrt(g)/(I*c*sqrt(-f)+sqrt(g)))*sqrt(-f)/sqrt(g)+2*a*e*arctan(x*sqrt(g)/sqrt(f))*sqrt(f)/sqrt(g)],
[(a+b*arctan(c*x))*(d+e*log(f+g*x^2))/x,x,8,b*e*CannotIntegrate(arctan(c*x)*log(f+g*x^2)/x,x)+a*d*log(x)+1/2*a*e*log(-g*x^2/f)*log(f+g*x^2)+1/2*I*b*d*polylog(2,-I*c*x)-1/2*I*b*d*polylog(2,I*c*x)+1/2*a*e*polylog(2,1+g*x^2/f)],
[(a+b*arctan(c*x))*(d+e*log(f+g*x^2))/x^2,x,28,-(a+b*arctan(c*x))*(d+e*log(f+g*x^2))/x+1/2*b*c*log(-g*x^2/f)*(d+e*log(f+g*x^2))-1/2*b*c*log(-g*(1+c^2*x^2)/(c^2*f-g))*(d+e*log(f+g*x^2))-1/2*b*c*e*polylog(2,c^2*(f+g*x^2)/(c^2*f-g))+1/2*b*c*e*polylog(2,1+g*x^2/f)-1/2*I*b*e*log(1+I*c*x)*log(c*(sqrt(-f)-x*sqrt(g))/(c*sqrt(-f)-I*sqrt(g)))*sqrt(g)/sqrt(-f)+1/2*I*b*e*log(1-I*c*x)*log(c*(sqrt(-f)-x*sqrt(g))/(c*sqrt(-f)+I*sqrt(g)))*sqrt(g)/sqrt(-f)-1/2*I*b*e*log(1-I*c*x)*log(c*(sqrt(-f)+x*sqrt(g))/(c*sqrt(-f)-I*sqrt(g)))*sqrt(g)/sqrt(-f)+1/2*I*b*e*log(1+I*c*x)*log(c*(sqrt(-f)+x*sqrt(g))/(c*sqrt(-f)+I*sqrt(g)))*sqrt(g)/sqrt(-f)+1/2*I*b*e*polylog(2,(I-c*x)*sqrt(g)/(c*sqrt(-f)+I*sqrt(g)))*sqrt(g)/sqrt(-f)+1/2*I*b*e*polylog(2,(I+c*x)*sqrt(g)/(c*sqrt(-f)+I*sqrt(g)))*sqrt(g)/sqrt(-f)-1/2*I*b*e*polylog(2,(1-I*c*x)*sqrt(g)/(I*c*sqrt(-f)+sqrt(g)))*sqrt(g)/sqrt(-f)-1/2*I*b*e*polylog(2,(1+I*c*x)*sqrt(g)/(I*c*sqrt(-f)+sqrt(g)))*sqrt(g)/sqrt(-f)+2*a*e*arctan(x*sqrt(g)/sqrt(f))*sqrt(g)/sqrt(f)],
[(a+b*arctan(c*x))*(d+e*log(f+g*x^2))/x^3,x,22,a*e*g*log(x)/f-b*e*(c^2*f-g)*arctan(c*x)*log(2/(1-I*c*x))/f-1/2*a*e*g*log(f+g*x^2)/f-1/2*b*c*(d+e*log(f+g*x^2))/x-1/2*b*c^2*arctan(c*x)*(d+e*log(f+g*x^2))-1/2*(a+b*arctan(c*x))*(d+e*log(f+g*x^2))/x^2+1/2*b*e*(c^2*f-g)*arctan(c*x)*log(2*c*(sqrt(-f)-x*sqrt(g))/((1-I*c*x)*(c*sqrt(-f)-I*sqrt(g))))/f+1/2*b*e*(c^2*f-g)*arctan(c*x)*log(2*c*(sqrt(-f)+x*sqrt(g))/((1-I*c*x)*(c*sqrt(-f)+I*sqrt(g))))/f+1/2*I*b*e*g*polylog(2,-I*c*x)/f-1/2*I*b*e*g*polylog(2,I*c*x)/f+1/2*I*b*e*(c^2*f-g)*polylog(2,1+(-2)/(1-I*c*x))/f-1/4*I*b*e*(c^2*f-g)*polylog(2,1-2*c*(sqrt(-f)-x*sqrt(g))/((1-I*c*x)*(c*sqrt(-f)-I*sqrt(g))))/f-1/4*I*b*e*(c^2*f-g)*polylog(2,1-2*c*(sqrt(-f)+x*sqrt(g))/((1-I*c*x)*(c*sqrt(-f)+I*sqrt(g))))/f+b*c*e*arctan(x*sqrt(g)/sqrt(f))*sqrt(g)/sqrt(f)]]:
