# Maple integration test file: "4 Trig functions\4.2 Cosine\4.2.2.1 (a+b cos)^m (c+d cos)^n.txt"

lst:=[

# Integrands of the form (d Cos[e+f x])^n (a+b Cos[e+f x])^m

# Integrands of the form (d Cos[e+f x])^n (a+a Cos[e+f x])^m

# Integrands of the form Cos[e+f x]^m (a+a Cos[e+f x])^n

# n>0
[cos(c+d*x)^5*(a+a*cos(c+d*x)),x,7,5/16*a*x+a*sin(c+d*x)/d+5/16*a*cos(c+d*x)*sin(c+d*x)/d+5/24*a*cos(c+d*x)^3*sin(c+d*x)/d+1/6*a*cos(c+d*x)^5*sin(c+d*x)/d-2/3*a*sin(c+d*x)^3/d+1/5*a*sin(c+d*x)^5/d],
[cos(c+d*x)^4*(a+a*cos(c+d*x)),x,6,3/8*a*x+a*sin(c+d*x)/d+3/8*a*cos(c+d*x)*sin(c+d*x)/d+1/4*a*cos(c+d*x)^3*sin(c+d*x)/d-2/3*a*sin(c+d*x)^3/d+1/5*a*sin(c+d*x)^5/d],
[cos(c+d*x)^3*(a+a*cos(c+d*x)),x,6,3/8*a*x+a*sin(c+d*x)/d+3/8*a*cos(c+d*x)*sin(c+d*x)/d+1/4*a*cos(c+d*x)^3*sin(c+d*x)/d-1/3*a*sin(c+d*x)^3/d],
[cos(c+d*x)^2*(a+a*cos(c+d*x)),x,5,1/2*a*x+a*sin(c+d*x)/d+1/2*a*cos(c+d*x)*sin(c+d*x)/d-1/3*a*sin(c+d*x)^3/d],
[cos(c+d*x)*(a+a*cos(c+d*x)),x,1,1/2*a*x+a*sin(c+d*x)/d+1/2*a*cos(c+d*x)*sin(c+d*x)/d],
[a+a*cos(c+d*x),x,2,a*x+a*sin(c+d*x)/d],
[(a+a*cos(c+d*x))*sec(c+d*x),x,2,a*x+a*arctanh(sin(c+d*x))/d],
[(a+a*cos(c+d*x))*sec(c+d*x)^2,x,4,a*arctanh(sin(c+d*x))/d+a*tan(c+d*x)/d],
[(a+a*cos(c+d*x))*sec(c+d*x)^3,x,5,1/2*a*arctanh(sin(c+d*x))/d+a*tan(c+d*x)/d+1/2*a*sec(c+d*x)*tan(c+d*x)/d],
[(a+a*cos(c+d*x))*sec(c+d*x)^4,x,5,1/2*a*arctanh(sin(c+d*x))/d+a*tan(c+d*x)/d+1/2*a*sec(c+d*x)*tan(c+d*x)/d+1/3*a*tan(c+d*x)^3/d],
[(a+a*cos(c+d*x))*sec(c+d*x)^5,x,6,3/8*a*arctanh(sin(c+d*x))/d+a*tan(c+d*x)/d+3/8*a*sec(c+d*x)*tan(c+d*x)/d+1/4*a*sec(c+d*x)^3*tan(c+d*x)/d+1/3*a*tan(c+d*x)^3/d],
[(a+a*cos(c+d*x))*sec(c+d*x)^6,x,6,3/8*a*arctanh(sin(c+d*x))/d+a*tan(c+d*x)/d+3/8*a*sec(c+d*x)*tan(c+d*x)/d+1/4*a*sec(c+d*x)^3*tan(c+d*x)/d+2/3*a*tan(c+d*x)^3/d+1/5*a*tan(c+d*x)^5/d],
[cos(c+d*x)^4*(a+a*cos(c+d*x))^2,x,11,11/16*a^2*x+2*a^2*sin(c+d*x)/d+11/16*a^2*cos(c+d*x)*sin(c+d*x)/d+11/24*a^2*cos(c+d*x)^3*sin(c+d*x)/d+1/6*a^2*cos(c+d*x)^5*sin(c+d*x)/d-4/3*a^2*sin(c+d*x)^3/d+2/5*a^2*sin(c+d*x)^5/d],
[cos(c+d*x)^3*(a+a*cos(c+d*x))^2,x,9,3/4*a^2*x+2*a^2*sin(c+d*x)/d+3/4*a^2*cos(c+d*x)*sin(c+d*x)/d+1/2*a^2*cos(c+d*x)^3*sin(c+d*x)/d-a^2*sin(c+d*x)^3/d+1/5*a^2*sin(c+d*x)^5/d],
[cos(c+d*x)^2*(a+a*cos(c+d*x))^2,x,9,7/8*a^2*x+2*a^2*sin(c+d*x)/d+7/8*a^2*cos(c+d*x)*sin(c+d*x)/d+1/4*a^2*cos(c+d*x)^3*sin(c+d*x)/d-2/3*a^2*sin(c+d*x)^3/d],
[cos(c+d*x)*(a+a*cos(c+d*x))^2,x,2,a^2*x+2*a^2*sin(c+d*x)/d+a^2*cos(c+d*x)*sin(c+d*x)/d-1/3*a^2*sin(c+d*x)^3/d,a^2*x+4/3*a^2*sin(c+d*x)/d+1/3*a^2*cos(c+d*x)*sin(c+d*x)/d+1/3*(a+a*cos(c+d*x))^2*sin(c+d*x)/d],
[(a+a*cos(c+d*x))^2,x,1,3/2*a^2*x+2*a^2*sin(c+d*x)/d+1/2*a^2*cos(c+d*x)*sin(c+d*x)/d],
[(a+a*cos(c+d*x))^2*sec(c+d*x),x,3,2*a^2*x+a^2*arctanh(sin(c+d*x))/d+a^2*sin(c+d*x)/d],
[(a+a*cos(c+d*x))^2*sec(c+d*x)^2,x,5,a^2*x+2*a^2*arctanh(sin(c+d*x))/d+a^2*tan(c+d*x)/d],
[(a+a*cos(c+d*x))^2*sec(c+d*x)^3,x,7,3/2*a^2*arctanh(sin(c+d*x))/d+2*a^2*tan(c+d*x)/d+1/2*a^2*sec(c+d*x)*tan(c+d*x)/d],
[(a+a*cos(c+d*x))^2*sec(c+d*x)^4,x,8,a^2*arctanh(sin(c+d*x))/d+2*a^2*tan(c+d*x)/d+a^2*sec(c+d*x)*tan(c+d*x)/d+1/3*a^2*tan(c+d*x)^3/d],
[(a+a*cos(c+d*x))^2*sec(c+d*x)^5,x,9,7/8*a^2*arctanh(sin(c+d*x))/d+2*a^2*tan(c+d*x)/d+7/8*a^2*sec(c+d*x)*tan(c+d*x)/d+1/4*a^2*sec(c+d*x)^3*tan(c+d*x)/d+2/3*a^2*tan(c+d*x)^3/d],
[cos(c+d*x)^3*(a+a*cos(c+d*x))^3,x,13,23/16*a^3*x+4*a^3*sin(c+d*x)/d+23/16*a^3*cos(c+d*x)*sin(c+d*x)/d+23/24*a^3*cos(c+d*x)^3*sin(c+d*x)/d+1/6*a^3*cos(c+d*x)^5*sin(c+d*x)/d-7/3*a^3*sin(c+d*x)^3/d+3/5*a^3*sin(c+d*x)^5/d],
[cos(c+d*x)^2*(a+a*cos(c+d*x))^3,x,11,13/8*a^3*x+4*a^3*sin(c+d*x)/d+13/8*a^3*cos(c+d*x)*sin(c+d*x)/d+3/4*a^3*cos(c+d*x)^3*sin(c+d*x)/d-5/3*a^3*sin(c+d*x)^3/d+1/5*a^3*sin(c+d*x)^5/d],
[cos(c+d*x)*(a+a*cos(c+d*x))^3,x,8,15/8*a^3*x+4*a^3*sin(c+d*x)/d+15/8*a^3*cos(c+d*x)*sin(c+d*x)/d+1/4*a^3*cos(c+d*x)^3*sin(c+d*x)/d-a^3*sin(c+d*x)^3/d,15/8*a^3*x+3*a^3*sin(c+d*x)/d+9/8*a^3*cos(c+d*x)*sin(c+d*x)/d+1/4*(a+a*cos(c+d*x))^3*sin(c+d*x)/d-1/4*a^3*sin(c+d*x)^3/d],
[(a+a*cos(c+d*x))^3,x,7,5/2*a^3*x+4*a^3*sin(c+d*x)/d+3/2*a^3*cos(c+d*x)*sin(c+d*x)/d-1/3*a^3*sin(c+d*x)^3/d],
[(a+a*cos(c+d*x))^3*sec(c+d*x),x,6,7/2*a^3*x+a^3*arctanh(sin(c+d*x))/d+3*a^3*sin(c+d*x)/d+1/2*a^3*cos(c+d*x)*sin(c+d*x)/d],
[(a+a*cos(c+d*x))^3*sec(c+d*x)^2,x,6,3*a^3*x+3*a^3*arctanh(sin(c+d*x))/d+a^3*sin(c+d*x)/d+a^3*tan(c+d*x)/d],
[(a+a*cos(c+d*x))^3*sec(c+d*x)^3,x,7,a^3*x+7/2*a^3*arctanh(sin(c+d*x))/d+3*a^3*tan(c+d*x)/d+1/2*a^3*sec(c+d*x)*tan(c+d*x)/d],
[(a+a*cos(c+d*x))^3*sec(c+d*x)^4,x,9,5/2*a^3*arctanh(sin(c+d*x))/d+4*a^3*tan(c+d*x)/d+3/2*a^3*sec(c+d*x)*tan(c+d*x)/d+1/3*a^3*tan(c+d*x)^3/d],
[(a+a*cos(c+d*x))^3*sec(c+d*x)^5,x,11,15/8*a^3*arctanh(sin(c+d*x))/d+4*a^3*tan(c+d*x)/d+15/8*a^3*sec(c+d*x)*tan(c+d*x)/d+1/4*a^3*sec(c+d*x)^3*tan(c+d*x)/d+a^3*tan(c+d*x)^3/d],
[(a+a*cos(c+d*x))^3*sec(c+d*x)^6,x,11,13/8*a^3*arctanh(sin(c+d*x))/d+4*a^3*tan(c+d*x)/d+13/8*a^3*sec(c+d*x)*tan(c+d*x)/d+3/4*a^3*sec(c+d*x)^3*tan(c+d*x)/d+5/3*a^3*tan(c+d*x)^3/d+1/5*a^3*tan(c+d*x)^5/d],
[cos(c+d*x)^2*(a+a*cos(c+d*x))^4,x,15,49/16*a^4*x+8*a^4*sin(c+d*x)/d+49/16*a^4*cos(c+d*x)*sin(c+d*x)/d+41/24*a^4*cos(c+d*x)^3*sin(c+d*x)/d+1/6*a^4*cos(c+d*x)^5*sin(c+d*x)/d-4*a^4*sin(c+d*x)^3/d+4/5*a^4*sin(c+d*x)^5/d],
[cos(c+d*x)*(a+a*cos(c+d*x))^4,x,11,7/2*a^4*x+8*a^4*sin(c+d*x)/d+7/2*a^4*cos(c+d*x)*sin(c+d*x)/d+a^4*cos(c+d*x)^3*sin(c+d*x)/d-8/3*a^4*sin(c+d*x)^3/d+1/5*a^4*sin(c+d*x)^5/d,7/2*a^4*x+32/5*a^4*sin(c+d*x)/d+27/10*a^4*cos(c+d*x)*sin(c+d*x)/d+1/5*a^4*cos(c+d*x)^3*sin(c+d*x)/d+1/5*(a+a*cos(c+d*x))^4*sin(c+d*x)/d-16/15*a^4*sin(c+d*x)^3/d],
[(a+a*cos(c+d*x))^4,x,10,35/8*a^4*x+8*a^4*sin(c+d*x)/d+27/8*a^4*cos(c+d*x)*sin(c+d*x)/d+1/4*a^4*cos(c+d*x)^3*sin(c+d*x)/d-4/3*a^4*sin(c+d*x)^3/d],
[(a+a*cos(c+d*x))^4*sec(c+d*x),x,8,6*a^4*x+a^4*arctanh(sin(c+d*x))/d+7*a^4*sin(c+d*x)/d+2*a^4*cos(c+d*x)*sin(c+d*x)/d-1/3*a^4*sin(c+d*x)^3/d],
[(a+a*cos(c+d*x))^4*sec(c+d*x)^2,x,8,13/2*a^4*x+4*a^4*arctanh(sin(c+d*x))/d+4*a^4*sin(c+d*x)/d+1/2*a^4*cos(c+d*x)*sin(c+d*x)/d+a^4*tan(c+d*x)/d],
[(a+a*cos(c+d*x))^4*sec(c+d*x)^3,x,8,4*a^4*x+13/2*a^4*arctanh(sin(c+d*x))/d+a^4*sin(c+d*x)/d+4*a^4*tan(c+d*x)/d+1/2*a^4*sec(c+d*x)*tan(c+d*x)/d],
[(a+a*cos(c+d*x))^4*sec(c+d*x)^4,x,9,a^4*x+6*a^4*arctanh(sin(c+d*x))/d+7*a^4*tan(c+d*x)/d+2*a^4*sec(c+d*x)*tan(c+d*x)/d+1/3*a^4*tan(c+d*x)^3/d],
[(a+a*cos(c+d*x))^4*sec(c+d*x)^5,x,12,35/8*a^4*arctanh(sin(c+d*x))/d+8*a^4*tan(c+d*x)/d+27/8*a^4*sec(c+d*x)*tan(c+d*x)/d+1/4*a^4*sec(c+d*x)^3*tan(c+d*x)/d+4/3*a^4*tan(c+d*x)^3/d],
[(a+a*cos(c+d*x))^4*sec(c+d*x)^6,x,13,7/2*a^4*arctanh(sin(c+d*x))/d+8*a^4*tan(c+d*x)/d+7/2*a^4*sec(c+d*x)*tan(c+d*x)/d+a^4*sec(c+d*x)^3*tan(c+d*x)/d+8/3*a^4*tan(c+d*x)^3/d+1/5*a^4*tan(c+d*x)^5/d],
[(a+a*cos(c+d*x))^4*sec(c+d*x)^7,x,15,49/16*a^4*arctanh(sin(c+d*x))/d+8*a^4*tan(c+d*x)/d+49/16*a^4*sec(c+d*x)*tan(c+d*x)/d+41/24*a^4*sec(c+d*x)^3*tan(c+d*x)/d+1/6*a^4*sec(c+d*x)^5*tan(c+d*x)/d+4*a^4*tan(c+d*x)^3/d+4/5*a^4*tan(c+d*x)^5/d],

# n<0
[cos(c+d*x)^5/(a+a*cos(c+d*x)),x,7,15/8*x/a-4*sin(c+d*x)/(a*d)+15/8*cos(c+d*x)*sin(c+d*x)/(a*d)+5/4*cos(c+d*x)^3*sin(c+d*x)/(a*d)-cos(c+d*x)^4*sin(c+d*x)/(d*(a+a*cos(c+d*x)))+4/3*sin(c+d*x)^3/(a*d)],
[cos(c+d*x)^4/(a+a*cos(c+d*x)),x,6,-3/2*x/a+4*sin(c+d*x)/(a*d)-3/2*cos(c+d*x)*sin(c+d*x)/(a*d)-cos(c+d*x)^3*sin(c+d*x)/(d*(a+a*cos(c+d*x)))-4/3*sin(c+d*x)^3/(a*d)],
[cos(c+d*x)^3/(a+a*cos(c+d*x)),x,2,3/2*x/a-2*sin(c+d*x)/(a*d)+3/2*cos(c+d*x)*sin(c+d*x)/(a*d)-cos(c+d*x)^2*sin(c+d*x)/(d*(a+a*cos(c+d*x)))],
[cos(c+d*x)^2/(a+a*cos(c+d*x)),x,4,-x/a+sin(c+d*x)/(a*d)+sin(c+d*x)/(a*d*(1+cos(c+d*x)))],
[cos(c+d*x)/(a+a*cos(c+d*x)),x,2,x/a-sin(c+d*x)/(d*(a+a*cos(c+d*x)))],
[1/(a+a*cos(c+d*x)),x,1,sin(c+d*x)/(d*(a+a*cos(c+d*x)))],
[sec(c+d*x)/(a+a*cos(c+d*x)),x,3,arctanh(sin(c+d*x))/(a*d)-sin(c+d*x)/(d*(a+a*cos(c+d*x)))],
[sec(c+d*x)^2/(a+a*cos(c+d*x)),x,5,-arctanh(sin(c+d*x))/(a*d)+2*tan(c+d*x)/(a*d)-tan(c+d*x)/(d*(a+a*cos(c+d*x)))],
[sec(c+d*x)^3/(a+a*cos(c+d*x)),x,6,3/2*arctanh(sin(c+d*x))/(a*d)-2*tan(c+d*x)/(a*d)+3/2*sec(c+d*x)*tan(c+d*x)/(a*d)-sec(c+d*x)*tan(c+d*x)/(d*(a+a*cos(c+d*x)))],
[sec(c+d*x)^4/(a+a*cos(c+d*x)),x,6,-3/2*arctanh(sin(c+d*x))/(a*d)+4*tan(c+d*x)/(a*d)-3/2*sec(c+d*x)*tan(c+d*x)/(a*d)-sec(c+d*x)^2*tan(c+d*x)/(d*(a+a*cos(c+d*x)))+4/3*tan(c+d*x)^3/(a*d)],
[cos(c+d*x)^5/(a+a*cos(c+d*x))^2,x,7,-5*x/a^2+12*sin(c+d*x)/(a^2*d)-5*cos(c+d*x)*sin(c+d*x)/(a^2*d)-10/3*cos(c+d*x)^3*sin(c+d*x)/(a^2*d*(1+cos(c+d*x)))-1/3*cos(c+d*x)^4*sin(c+d*x)/(d*(a+a*cos(c+d*x))^2)-4*sin(c+d*x)^3/(a^2*d)],
[cos(c+d*x)^4/(a+a*cos(c+d*x))^2,x,3,7/2*x/a^2-16/3*sin(c+d*x)/(a^2*d)+7/2*cos(c+d*x)*sin(c+d*x)/(a^2*d)-8/3*cos(c+d*x)^2*sin(c+d*x)/(a^2*d*(1+cos(c+d*x)))-1/3*cos(c+d*x)^3*sin(c+d*x)/(d*(a+a*cos(c+d*x))^2)],
[cos(c+d*x)^3/(a+a*cos(c+d*x))^2,x,6,-2*x/a^2+4/3*sin(c+d*x)/(a^2*d)+2*sin(c+d*x)/(a^2*d*(1+cos(c+d*x)))-1/3*cos(c+d*x)^2*sin(c+d*x)/(d*(a+a*cos(c+d*x))^2)],
[cos(c+d*x)^2/(a+a*cos(c+d*x))^2,x,3,x/a^2-5/3*sin(c+d*x)/(a^2*d*(1+cos(c+d*x)))+1/3*sin(c+d*x)/(d*(a+a*cos(c+d*x))^2)],
[cos(c+d*x)/(a+a*cos(c+d*x))^2,x,2,-1/3*sin(c+d*x)/(d*(a+a*cos(c+d*x))^2)+2/3*sin(c+d*x)/(d*(a^2+a^2*cos(c+d*x)))],
[1/(a+a*cos(c+d*x))^2,x,2,1/3*sin(c+d*x)/(d*(a+a*cos(c+d*x))^2)+1/3*sin(c+d*x)/(d*(a^2+a^2*cos(c+d*x)))],
[sec(c+d*x)/(a+a*cos(c+d*x))^2,x,4,arctanh(sin(c+d*x))/(a^2*d)-4/3*sin(c+d*x)/(a^2*d*(1+cos(c+d*x)))-1/3*sin(c+d*x)/(d*(a+a*cos(c+d*x))^2)],
[sec(c+d*x)^2/(a+a*cos(c+d*x))^2,x,6,-2*arctanh(sin(c+d*x))/(a^2*d)+10/3*tan(c+d*x)/(a^2*d)-2*tan(c+d*x)/(a^2*d*(1+cos(c+d*x)))-1/3*tan(c+d*x)/(d*(a+a*cos(c+d*x))^2)],
[sec(c+d*x)^3/(a+a*cos(c+d*x))^2,x,7,7/2*arctanh(sin(c+d*x))/(a^2*d)-16/3*tan(c+d*x)/(a^2*d)+7/2*sec(c+d*x)*tan(c+d*x)/(a^2*d)-8/3*sec(c+d*x)*tan(c+d*x)/(a^2*d*(1+cos(c+d*x)))-1/3*sec(c+d*x)*tan(c+d*x)/(d*(a+a*cos(c+d*x))^2)],
[sec(c+d*x)^4/(a+a*cos(c+d*x))^2,x,7,-5*arctanh(sin(c+d*x))/(a^2*d)+12*tan(c+d*x)/(a^2*d)-5*sec(c+d*x)*tan(c+d*x)/(a^2*d)-10/3*sec(c+d*x)^2*tan(c+d*x)/(a^2*d*(1+cos(c+d*x)))-1/3*sec(c+d*x)^2*tan(c+d*x)/(d*(a+a*cos(c+d*x))^2)+4*tan(c+d*x)^3/(a^2*d)],
[cos(c+d*x)^5/(a+a*cos(c+d*x))^3,x,4,13/2*x/a^3-152/15*sin(c+d*x)/(a^3*d)+13/2*cos(c+d*x)*sin(c+d*x)/(a^3*d)-1/5*cos(c+d*x)^4*sin(c+d*x)/(d*(a+a*cos(c+d*x))^3)-11/15*cos(c+d*x)^3*sin(c+d*x)/(a*d*(a+a*cos(c+d*x))^2)-76/15*cos(c+d*x)^2*sin(c+d*x)/(d*(a^3+a^3*cos(c+d*x)))],
[cos(c+d*x)^4/(a+a*cos(c+d*x))^3,x,7,-3*x/a^3+9/5*sin(c+d*x)/(a^3*d)-1/5*cos(c+d*x)^3*sin(c+d*x)/(d*(a+a*cos(c+d*x))^3)-3/5*cos(c+d*x)^2*sin(c+d*x)/(a*d*(a+a*cos(c+d*x))^2)+3*sin(c+d*x)/(d*(a^3+a^3*cos(c+d*x)))],
[cos(c+d*x)^3/(a+a*cos(c+d*x))^3,x,5,x/a^3-1/5*cos(c+d*x)^2*sin(c+d*x)/(d*(a+a*cos(c+d*x))^3)+7/15*sin(c+d*x)/(a*d*(a+a*cos(c+d*x))^2)-29/15*sin(c+d*x)/(d*(a^3+a^3*cos(c+d*x)))],
[cos(c+d*x)^2/(a+a*cos(c+d*x))^3,x,3,1/5*sin(c+d*x)/(d*(a+a*cos(c+d*x))^3)-8/15*sin(c+d*x)/(a*d*(a+a*cos(c+d*x))^2)+7/15*sin(c+d*x)/(d*(a^3+a^3*cos(c+d*x)))],
[cos(c+d*x)/(a+a*cos(c+d*x))^3,x,3,-1/5*sin(c+d*x)/(d*(a+a*cos(c+d*x))^3)+1/5*sin(c+d*x)/(a*d*(a+a*cos(c+d*x))^2)+1/5*sin(c+d*x)/(d*(a^3+a^3*cos(c+d*x)))],
[1/(a+a*cos(c+d*x))^3,x,3,1/5*sin(c+d*x)/(d*(a+a*cos(c+d*x))^3)+2/15*sin(c+d*x)/(a*d*(a+a*cos(c+d*x))^2)+2/15*sin(c+d*x)/(d*(a^3+a^3*cos(c+d*x)))],
[sec(c+d*x)/(a+a*cos(c+d*x))^3,x,5,arctanh(sin(c+d*x))/(a^3*d)-1/5*sin(c+d*x)/(d*(a+a*cos(c+d*x))^3)-7/15*sin(c+d*x)/(a*d*(a+a*cos(c+d*x))^2)-22/15*sin(c+d*x)/(d*(a^3+a^3*cos(c+d*x)))],
[sec(c+d*x)^2/(a+a*cos(c+d*x))^3,x,7,-3*arctanh(sin(c+d*x))/(a^3*d)+24/5*tan(c+d*x)/(a^3*d)-1/5*tan(c+d*x)/(d*(a+a*cos(c+d*x))^3)-3/5*tan(c+d*x)/(a*d*(a+a*cos(c+d*x))^2)-3*tan(c+d*x)/(d*(a^3+a^3*cos(c+d*x)))],
[sec(c+d*x)^3/(a+a*cos(c+d*x))^3,x,8,13/2*arctanh(sin(c+d*x))/(a^3*d)-152/15*tan(c+d*x)/(a^3*d)+13/2*sec(c+d*x)*tan(c+d*x)/(a^3*d)-1/5*sec(c+d*x)*tan(c+d*x)/(d*(a+a*cos(c+d*x))^3)-11/15*sec(c+d*x)*tan(c+d*x)/(a*d*(a+a*cos(c+d*x))^2)-76/15*sec(c+d*x)*tan(c+d*x)/(d*(a^3+a^3*cos(c+d*x)))],
[cos(c+d*x)^6/(a+a*cos(c+d*x))^4,x,5,21/2*x/a^4-576/35*sin(c+d*x)/(a^4*d)+21/2*cos(c+d*x)*sin(c+d*x)/(a^4*d)-43/35*cos(c+d*x)^3*sin(c+d*x)/(a^4*d*(1+cos(c+d*x))^2)-288/35*cos(c+d*x)^2*sin(c+d*x)/(a^4*d*(1+cos(c+d*x)))-1/7*cos(c+d*x)^5*sin(c+d*x)/(d*(a+a*cos(c+d*x))^4)-2/5*cos(c+d*x)^4*sin(c+d*x)/(a*d*(a+a*cos(c+d*x))^3)],
[cos(c+d*x)^5/(a+a*cos(c+d*x))^4,x,8,-4*x/a^4+244/105*sin(c+d*x)/(a^4*d)-88/105*cos(c+d*x)^2*sin(c+d*x)/(a^4*d*(1+cos(c+d*x))^2)+4*sin(c+d*x)/(a^4*d*(1+cos(c+d*x)))-1/7*cos(c+d*x)^4*sin(c+d*x)/(d*(a+a*cos(c+d*x))^4)-12/35*cos(c+d*x)^3*sin(c+d*x)/(a*d*(a+a*cos(c+d*x))^3)],
[cos(c+d*x)^4/(a+a*cos(c+d*x))^4,x,6,x/a^4+11/21*sin(c+d*x)/(a^4*d*(1+cos(c+d*x))^2)-43/21*sin(c+d*x)/(a^4*d*(1+cos(c+d*x)))-1/7*cos(c+d*x)^3*sin(c+d*x)/(d*(a+a*cos(c+d*x))^4)-2/7*cos(c+d*x)^2*sin(c+d*x)/(a*d*(a+a*cos(c+d*x))^3)],
[cos(c+d*x)^3/(a+a*cos(c+d*x))^4,x,5,-18/35*sin(c+d*x)/(a^4*d*(1+cos(c+d*x))^2)+12/35*sin(c+d*x)/(a^4*d*(1+cos(c+d*x)))-1/7*cos(c+d*x)^2*sin(c+d*x)/(d*(a+a*cos(c+d*x))^4)+8/35*sin(c+d*x)/(a*d*(a+a*cos(c+d*x))^3)],
[cos(c+d*x)^2/(a+a*cos(c+d*x))^4,x,4,1/7*sin(c+d*x)/(d*(a+a*cos(c+d*x))^4)-11/35*sin(c+d*x)/(a*d*(a+a*cos(c+d*x))^3)+13/105*sin(c+d*x)/(d*(a^2+a^2*cos(c+d*x))^2)+13/105*sin(c+d*x)/(d*(a^4+a^4*cos(c+d*x)))],
[cos(c+d*x)/(a+a*cos(c+d*x))^4,x,4,-1/7*sin(c+d*x)/(d*(a+a*cos(c+d*x))^4)+4/35*sin(c+d*x)/(a*d*(a+a*cos(c+d*x))^3)+8/105*sin(c+d*x)/(d*(a^2+a^2*cos(c+d*x))^2)+8/105*sin(c+d*x)/(d*(a^4+a^4*cos(c+d*x)))],
[1/(a+a*cos(c+d*x))^4,x,4,1/7*sin(c+d*x)/(d*(a+a*cos(c+d*x))^4)+3/35*sin(c+d*x)/(a*d*(a+a*cos(c+d*x))^3)+2/35*sin(c+d*x)/(d*(a^2+a^2*cos(c+d*x))^2)+2/35*sin(c+d*x)/(d*(a^4+a^4*cos(c+d*x)))],
[sec(c+d*x)/(a+a*cos(c+d*x))^4,x,6,arctanh(sin(c+d*x))/(a^4*d)-11/21*sin(c+d*x)/(a^4*d*(1+cos(c+d*x))^2)-32/21*sin(c+d*x)/(a^4*d*(1+cos(c+d*x)))-1/7*sin(c+d*x)/(d*(a+a*cos(c+d*x))^4)-2/7*sin(c+d*x)/(a*d*(a+a*cos(c+d*x))^3)],
[sec(c+d*x)^2/(a+a*cos(c+d*x))^4,x,8,-4*arctanh(sin(c+d*x))/(a^4*d)+664/105*tan(c+d*x)/(a^4*d)-88/105*tan(c+d*x)/(a^4*d*(1+cos(c+d*x))^2)-4*tan(c+d*x)/(a^4*d*(1+cos(c+d*x)))-1/7*tan(c+d*x)/(d*(a+a*cos(c+d*x))^4)-12/35*tan(c+d*x)/(a*d*(a+a*cos(c+d*x))^3)],
[sec(c+d*x)^3/(a+a*cos(c+d*x))^4,x,9,21/2*arctanh(sin(c+d*x))/(a^4*d)-576/35*tan(c+d*x)/(a^4*d)+21/2*sec(c+d*x)*tan(c+d*x)/(a^4*d)-43/35*sec(c+d*x)*tan(c+d*x)/(a^4*d*(1+cos(c+d*x))^2)-288/35*sec(c+d*x)*tan(c+d*x)/(a^4*d*(1+cos(c+d*x)))-1/7*sec(c+d*x)*tan(c+d*x)/(d*(a+a*cos(c+d*x))^4)-2/5*sec(c+d*x)*tan(c+d*x)/(a*d*(a+a*cos(c+d*x))^3)],
[cos(c+d*x)^7/(a+a*cos(c+d*x))^5,x,6,31/2*x/a^5-7664/315*sin(c+d*x)/(a^5*d)+31/2*cos(c+d*x)*sin(c+d*x)/(a^5*d)-1/9*cos(c+d*x)^6*sin(c+d*x)/(d*(a+a*cos(c+d*x))^5)-17/63*cos(c+d*x)^5*sin(c+d*x)/(a*d*(a+a*cos(c+d*x))^4)-28/45*cos(c+d*x)^4*sin(c+d*x)/(a^2*d*(a+a*cos(c+d*x))^3)-577/315*cos(c+d*x)^3*sin(c+d*x)/(a^3*d*(a+a*cos(c+d*x))^2)-3832/315*cos(c+d*x)^2*sin(c+d*x)/(d*(a^5+a^5*cos(c+d*x)))],
[cos(c+d*x)^6/(a+a*cos(c+d*x))^5,x,9,-5*x/a^5+181/63*sin(c+d*x)/(a^5*d)-1/9*cos(c+d*x)^5*sin(c+d*x)/(d*(a+a*cos(c+d*x))^5)-5/21*cos(c+d*x)^4*sin(c+d*x)/(a*d*(a+a*cos(c+d*x))^4)-29/63*cos(c+d*x)^3*sin(c+d*x)/(a^2*d*(a+a*cos(c+d*x))^3)-67/63*cos(c+d*x)^2*sin(c+d*x)/(a^3*d*(a+a*cos(c+d*x))^2)+5*sin(c+d*x)/(d*(a^5+a^5*cos(c+d*x)))],
[cos(c+d*x)^5/(a+a*cos(c+d*x))^5,x,7,x/a^5-1/9*cos(c+d*x)^4*sin(c+d*x)/(d*(a+a*cos(c+d*x))^5)-13/63*cos(c+d*x)^3*sin(c+d*x)/(a*d*(a+a*cos(c+d*x))^4)-34/105*cos(c+d*x)^2*sin(c+d*x)/(a^2*d*(a+a*cos(c+d*x))^3)+173/315*sin(c+d*x)/(a^3*d*(a+a*cos(c+d*x))^2)-661/315*sin(c+d*x)/(d*(a^5+a^5*cos(c+d*x)))],
[cos(c+d*x)^4/(a+a*cos(c+d*x))^5,x,6,-1/9*cos(c+d*x)^3*sin(c+d*x)/(d*(a+a*cos(c+d*x))^5)-11/63*cos(c+d*x)^2*sin(c+d*x)/(a*d*(a+a*cos(c+d*x))^4)+67/315*sin(c+d*x)/(a^2*d*(a+a*cos(c+d*x))^3)-142/315*sin(c+d*x)/(a^3*d*(a+a*cos(c+d*x))^2)+83/315*sin(c+d*x)/(d*(a^5+a^5*cos(c+d*x)))],
[cos(c+d*x)^3/(a+a*cos(c+d*x))^5,x,6,-1/9*cos(c+d*x)^2*sin(c+d*x)/(d*(a+a*cos(c+d*x))^5)+1/7*sin(c+d*x)/(a*d*(a+a*cos(c+d*x))^4)-17/63*sin(c+d*x)/(a^2*d*(a+a*cos(c+d*x))^3)+5/63*sin(c+d*x)/(a^3*d*(a+a*cos(c+d*x))^2)+5/63*sin(c+d*x)/(d*(a^5+a^5*cos(c+d*x)))],
[cos(c+d*x)^2/(a+a*cos(c+d*x))^5,x,5,1/9*sin(c+d*x)/(d*(a+a*cos(c+d*x))^5)-2/9*sin(c+d*x)/(a*d*(a+a*cos(c+d*x))^4)+1/15*sin(c+d*x)/(a^2*d*(a+a*cos(c+d*x))^3)+2/45*sin(c+d*x)/(a^3*d*(a+a*cos(c+d*x))^2)+2/45*sin(c+d*x)/(d*(a^5+a^5*cos(c+d*x)))],
[cos(c+d*x)/(a+a*cos(c+d*x))^5,x,5,-1/9*sin(c+d*x)/(d*(a+a*cos(c+d*x))^5)+5/63*sin(c+d*x)/(a*d*(a+a*cos(c+d*x))^4)+1/21*sin(c+d*x)/(a^2*d*(a+a*cos(c+d*x))^3)+2/63*sin(c+d*x)/(a*d*(a^2+a^2*cos(c+d*x))^2)+2/63*sin(c+d*x)/(d*(a^5+a^5*cos(c+d*x)))],
[1/(a+a*cos(c+d*x))^5,x,5,1/9*sin(c+d*x)/(d*(a+a*cos(c+d*x))^5)+4/63*sin(c+d*x)/(a*d*(a+a*cos(c+d*x))^4)+4/105*sin(c+d*x)/(a^2*d*(a+a*cos(c+d*x))^3)+8/315*sin(c+d*x)/(a*d*(a^2+a^2*cos(c+d*x))^2)+8/315*sin(c+d*x)/(d*(a^5+a^5*cos(c+d*x)))],
[sec(c+d*x)/(a+a*cos(c+d*x))^5,x,7,arctanh(sin(c+d*x))/(a^5*d)-1/9*sin(c+d*x)/(d*(a+a*cos(c+d*x))^5)-13/63*sin(c+d*x)/(a*d*(a+a*cos(c+d*x))^4)-34/105*sin(c+d*x)/(a^2*d*(a+a*cos(c+d*x))^3)-173/315*sin(c+d*x)/(a^3*d*(a+a*cos(c+d*x))^2)-488/315*sin(c+d*x)/(d*(a^5+a^5*cos(c+d*x)))],
[sec(c+d*x)^2/(a+a*cos(c+d*x))^5,x,9,-5*arctanh(sin(c+d*x))/(a^5*d)+496/63*tan(c+d*x)/(a^5*d)-1/9*tan(c+d*x)/(d*(a+a*cos(c+d*x))^5)-5/21*tan(c+d*x)/(a*d*(a+a*cos(c+d*x))^4)-29/63*tan(c+d*x)/(a^2*d*(a+a*cos(c+d*x))^3)-67/63*tan(c+d*x)/(a^3*d*(a+a*cos(c+d*x))^2)-5*tan(c+d*x)/(d*(a^5+a^5*cos(c+d*x)))],
[sec(c+d*x)^3/(a+a*cos(c+d*x))^5,x,10,31/2*arctanh(sin(c+d*x))/(a^5*d)-7664/315*tan(c+d*x)/(a^5*d)+31/2*sec(c+d*x)*tan(c+d*x)/(a^5*d)-1/9*sec(c+d*x)*tan(c+d*x)/(d*(a+a*cos(c+d*x))^5)-17/63*sec(c+d*x)*tan(c+d*x)/(a*d*(a+a*cos(c+d*x))^4)-28/45*sec(c+d*x)*tan(c+d*x)/(a^2*d*(a+a*cos(c+d*x))^3)-577/315*sec(c+d*x)*tan(c+d*x)/(a^3*d*(a+a*cos(c+d*x))^2)-3832/315*sec(c+d*x)*tan(c+d*x)/(d*(a^5+a^5*cos(c+d*x)))],
[cos(c+d*x)^5/(a+a*cos(c+d*x))^6,x,7,130/693*sin(c+d*x)/(a^6*d*(1+cos(c+d*x))^3)-268/693*sin(c+d*x)/(a^6*d*(1+cos(c+d*x))^2)+146/693*sin(c+d*x)/(a^6*d*(1+cos(c+d*x)))-1/11*cos(c+d*x)^4*sin(c+d*x)/(d*(a+a*cos(c+d*x))^6)-14/99*cos(c+d*x)^3*sin(c+d*x)/(a*d*(a+a*cos(c+d*x))^5)-118/693*cos(c+d*x)^2*sin(c+d*x)/(a^2*d*(a+a*cos(c+d*x))^4)],
[cos(c+d*x)^4/(a+a*cos(c+d*x))^6,x,7,-241/1155*sin(c+d*x)/(a^6*d*(1+cos(c+d*x))^3)+61/1155*sin(c+d*x)/(a^6*d*(1+cos(c+d*x))^2)+61/1155*sin(c+d*x)/(a^6*d*(1+cos(c+d*x)))-1/11*cos(c+d*x)^3*sin(c+d*x)/(d*(a+a*cos(c+d*x))^6)-4/33*cos(c+d*x)^2*sin(c+d*x)/(a*d*(a+a*cos(c+d*x))^5)+9/77*sin(c+d*x)/(a^2*d*(a+a*cos(c+d*x))^4)],

# Integrands of the form Cos[e+f x]^m (a+a Cos[e+f x])^(n/2)

# n>0
[cos(c+d*x)^4*(a+a*cos(c+d*x))^(1/2),x,5,32/105*(a+a*cos(c+d*x))^(3/2)*sin(c+d*x)/(a*d)+32/45*a*sin(c+d*x)/(d*sqrt(a+a*cos(c+d*x)))+16/63*a*cos(c+d*x)^3*sin(c+d*x)/(d*sqrt(a+a*cos(c+d*x)))+2/9*a*cos(c+d*x)^4*sin(c+d*x)/(d*sqrt(a+a*cos(c+d*x)))-64/315*sin(c+d*x)*sqrt(a+a*cos(c+d*x))/d],
[cos(c+d*x)^3*(a+a*cos(c+d*x))^(1/2),x,4,12/35*(a+a*cos(c+d*x))^(3/2)*sin(c+d*x)/(a*d)+4/5*a*sin(c+d*x)/(d*sqrt(a+a*cos(c+d*x)))+2/7*a*cos(c+d*x)^3*sin(c+d*x)/(d*sqrt(a+a*cos(c+d*x)))-8/35*sin(c+d*x)*sqrt(a+a*cos(c+d*x))/d],
[cos(c+d*x)^2*(a+a*cos(c+d*x))^(1/2),x,3,2/5*(a+a*cos(c+d*x))^(3/2)*sin(c+d*x)/(a*d)+14/15*a*sin(c+d*x)/(d*sqrt(a+a*cos(c+d*x)))-4/15*sin(c+d*x)*sqrt(a+a*cos(c+d*x))/d],
[cos(c+d*x)*(a+a*cos(c+d*x))^(1/2),x,2,2/3*a*sin(c+d*x)/(d*sqrt(a+a*cos(c+d*x)))+2/3*sin(c+d*x)*sqrt(a+a*cos(c+d*x))/d],
[(a+a*cos(c+d*x))^(1/2),x,1,2*a*sin(c+d*x)/(d*sqrt(a+a*cos(c+d*x)))],
[(a+a*cos(c+d*x))^(1/2)*sec(c+d*x),x,2,2*arctanh(sin(c+d*x)*sqrt(a)/sqrt(a+a*cos(c+d*x)))*sqrt(a)/d],
[(a+a*cos(c+d*x))^(1/2)*sec(c+d*x)^2,x,3,arctanh(sin(c+d*x)*sqrt(a)/sqrt(a+a*cos(c+d*x)))*sqrt(a)/d+a*tan(c+d*x)/(d*sqrt(a+a*cos(c+d*x)))],
[(a+a*cos(c+d*x))^(1/2)*sec(c+d*x)^3,x,4,3/4*arctanh(sin(c+d*x)*sqrt(a)/sqrt(a+a*cos(c+d*x)))*sqrt(a)/d+3/4*a*tan(c+d*x)/(d*sqrt(a+a*cos(c+d*x)))+1/2*a*sec(c+d*x)*tan(c+d*x)/(d*sqrt(a+a*cos(c+d*x)))],
[(a+a*cos(c+d*x))^(1/2)*sec(c+d*x)^4,x,5,5/8*arctanh(sin(c+d*x)*sqrt(a)/sqrt(a+a*cos(c+d*x)))*sqrt(a)/d+5/8*a*tan(c+d*x)/(d*sqrt(a+a*cos(c+d*x)))+5/12*a*sec(c+d*x)*tan(c+d*x)/(d*sqrt(a+a*cos(c+d*x)))+1/3*a*sec(c+d*x)^2*tan(c+d*x)/(d*sqrt(a+a*cos(c+d*x)))],
[cos(c+d*x)^3*(a+a*cos(c+d*x))^(3/2),x,6,68/105*(a+a*cos(c+d*x))^(3/2)*sin(c+d*x)/d+68/45*a^2*sin(c+d*x)/(d*sqrt(a+a*cos(c+d*x)))+34/63*a^2*cos(c+d*x)^3*sin(c+d*x)/(d*sqrt(a+a*cos(c+d*x)))+2/9*a^2*cos(c+d*x)^4*sin(c+d*x)/(d*sqrt(a+a*cos(c+d*x)))-136/315*a*sin(c+d*x)*sqrt(a+a*cos(c+d*x))/d],
[cos(c+d*x)^2*(a+a*cos(c+d*x))^(3/2),x,4,-4/35*(a+a*cos(c+d*x))^(3/2)*sin(c+d*x)/d+2/7*(a+a*cos(c+d*x))^(5/2)*sin(c+d*x)/(a*d)+152/105*a^2*sin(c+d*x)/(d*sqrt(a+a*cos(c+d*x)))+38/105*a*sin(c+d*x)*sqrt(a+a*cos(c+d*x))/d],
[cos(c+d*x)*(a+a*cos(c+d*x))^(3/2),x,3,2/5*(a+a*cos(c+d*x))^(3/2)*sin(c+d*x)/d+8/5*a^2*sin(c+d*x)/(d*sqrt(a+a*cos(c+d*x)))+2/5*a*sin(c+d*x)*sqrt(a+a*cos(c+d*x))/d],
[(a+a*cos(c+d*x))^(3/2),x,2,8/3*a^2*sin(c+d*x)/(d*sqrt(a+a*cos(c+d*x)))+2/3*a*sin(c+d*x)*sqrt(a+a*cos(c+d*x))/d],
[(a+a*cos(c+d*x))^(3/2)*sec(c+d*x),x,4,2*a^(3/2)*arctanh(sin(c+d*x)*sqrt(a)/sqrt(a+a*cos(c+d*x)))/d+2*a^2*sin(c+d*x)/(d*sqrt(a+a*cos(c+d*x)))],
[(a+a*cos(c+d*x))^(3/2)*sec(c+d*x)^2,x,4,3*a^(3/2)*arctanh(sin(c+d*x)*sqrt(a)/sqrt(a+a*cos(c+d*x)))/d+a^2*tan(c+d*x)/(d*sqrt(a+a*cos(c+d*x)))],
[(a+a*cos(c+d*x))^(3/2)*sec(c+d*x)^3,x,5,7/4*a^(3/2)*arctanh(sin(c+d*x)*sqrt(a)/sqrt(a+a*cos(c+d*x)))/d+7/4*a^2*tan(c+d*x)/(d*sqrt(a+a*cos(c+d*x)))+1/2*a^2*sec(c+d*x)*tan(c+d*x)/(d*sqrt(a+a*cos(c+d*x)))],
[(a+a*cos(c+d*x))^(3/2)*sec(c+d*x)^4,x,6,11/8*a^(3/2)*arctanh(sin(c+d*x)*sqrt(a)/sqrt(a+a*cos(c+d*x)))/d+11/8*a^2*tan(c+d*x)/(d*sqrt(a+a*cos(c+d*x)))+11/12*a^2*sec(c+d*x)*tan(c+d*x)/(d*sqrt(a+a*cos(c+d*x)))+1/3*a^2*sec(c+d*x)^2*tan(c+d*x)/(d*sqrt(a+a*cos(c+d*x)))],
[cos(c+d*x)^3*(a+a*cos(c+d*x))^(5/2),x,6,284/231*a*(a+a*cos(c+d*x))^(3/2)*sin(c+d*x)/d+284/99*a^3*sin(c+d*x)/(d*sqrt(a+a*cos(c+d*x)))+710/693*a^3*cos(c+d*x)^3*sin(c+d*x)/(d*sqrt(a+a*cos(c+d*x)))+46/99*a^3*cos(c+d*x)^4*sin(c+d*x)/(d*sqrt(a+a*cos(c+d*x)))-568/693*a^2*sin(c+d*x)*sqrt(a+a*cos(c+d*x))/d+2/11*a^2*cos(c+d*x)^4*sin(c+d*x)*sqrt(a+a*cos(c+d*x))/d],
[cos(c+d*x)^2*(a+a*cos(c+d*x))^(5/2),x,5,26/105*a*(a+a*cos(c+d*x))^(3/2)*sin(c+d*x)/d-4/63*(a+a*cos(c+d*x))^(5/2)*sin(c+d*x)/d+2/9*(a+a*cos(c+d*x))^(7/2)*sin(c+d*x)/(a*d)+832/315*a^3*sin(c+d*x)/(d*sqrt(a+a*cos(c+d*x)))+208/315*a^2*sin(c+d*x)*sqrt(a+a*cos(c+d*x))/d],
[cos(c+d*x)*(a+a*cos(c+d*x))^(5/2),x,4,2/7*a*(a+a*cos(c+d*x))^(3/2)*sin(c+d*x)/d+2/7*(a+a*cos(c+d*x))^(5/2)*sin(c+d*x)/d+64/21*a^3*sin(c+d*x)/(d*sqrt(a+a*cos(c+d*x)))+16/21*a^2*sin(c+d*x)*sqrt(a+a*cos(c+d*x))/d],
[(a+a*cos(c+d*x))^(5/2),x,3,2/5*a*(a+a*cos(c+d*x))^(3/2)*sin(c+d*x)/d+64/15*a^3*sin(c+d*x)/(d*sqrt(a+a*cos(c+d*x)))+16/15*a^2*sin(c+d*x)*sqrt(a+a*cos(c+d*x))/d],
[(a+a*cos(c+d*x))^(5/2)*sec(c+d*x),x,4,2*a^(5/2)*arctanh(sin(c+d*x)*sqrt(a)/sqrt(a+a*cos(c+d*x)))/d+14/3*a^3*sin(c+d*x)/(d*sqrt(a+a*cos(c+d*x)))+2/3*a^2*sin(c+d*x)*sqrt(a+a*cos(c+d*x))/d],
[(a+a*cos(c+d*x))^(5/2)*sec(c+d*x)^2,x,4,5*a^(5/2)*arctanh(sin(c+d*x)*sqrt(a)/sqrt(a+a*cos(c+d*x)))/d+a^3*sin(c+d*x)/(d*sqrt(a+a*cos(c+d*x)))+a^2*sqrt(a+a*cos(c+d*x))*tan(c+d*x)/d],
[(a+a*cos(c+d*x))^(5/2)*sec(c+d*x)^3,x,4,19/4*a^(5/2)*arctanh(sin(c+d*x)*sqrt(a)/sqrt(a+a*cos(c+d*x)))/d+9/4*a^3*tan(c+d*x)/(d*sqrt(a+a*cos(c+d*x)))+1/2*a^2*sec(c+d*x)*sqrt(a+a*cos(c+d*x))*tan(c+d*x)/d],
[(a+a*cos(c+d*x))^(5/2)*sec(c+d*x)^4,x,5,25/8*a^(5/2)*arctanh(sin(c+d*x)*sqrt(a)/sqrt(a+a*cos(c+d*x)))/d+25/8*a^3*tan(c+d*x)/(d*sqrt(a+a*cos(c+d*x)))+13/12*a^3*sec(c+d*x)*tan(c+d*x)/(d*sqrt(a+a*cos(c+d*x)))+1/3*a^2*sec(c+d*x)^2*sqrt(a+a*cos(c+d*x))*tan(c+d*x)/d],
[(a+a*cos(c+d*x))^(5/2)*sec(c+d*x)^5,x,6,163/64*a^(5/2)*arctanh(sin(c+d*x)*sqrt(a)/sqrt(a+a*cos(c+d*x)))/d+163/64*a^3*tan(c+d*x)/(d*sqrt(a+a*cos(c+d*x)))+163/96*a^3*sec(c+d*x)*tan(c+d*x)/(d*sqrt(a+a*cos(c+d*x)))+17/24*a^3*sec(c+d*x)^2*tan(c+d*x)/(d*sqrt(a+a*cos(c+d*x)))+1/4*a^2*sec(c+d*x)^3*sqrt(a+a*cos(c+d*x))*tan(c+d*x)/d],
[(a+a*cos(c+d*x))^(7/2),x,4,24/35*a^2*(a+a*cos(c+d*x))^(3/2)*sin(c+d*x)/d+2/7*a*(a+a*cos(c+d*x))^(5/2)*sin(c+d*x)/d+256/35*a^4*sin(c+d*x)/(d*sqrt(a+a*cos(c+d*x)))+64/35*a^3*sin(c+d*x)*sqrt(a+a*cos(c+d*x))/d],

# n<0
[cos(c+d*x)^4/(a+a*cos(c+d*x))^(1/2),x,7,arctanh(sin(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(a+a*cos(c+d*x))))*sqrt(2)/(d*sqrt(a))-148/105*sin(c+d*x)/(d*sqrt(a+a*cos(c+d*x)))-2/35*cos(c+d*x)^2*sin(c+d*x)/(d*sqrt(a+a*cos(c+d*x)))+2/7*cos(c+d*x)^3*sin(c+d*x)/(d*sqrt(a+a*cos(c+d*x)))+62/105*sin(c+d*x)*sqrt(a+a*cos(c+d*x))/(a*d)],
[cos(c+d*x)^3/(a+a*cos(c+d*x))^(1/2),x,6,-arctanh(sin(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(a+a*cos(c+d*x))))*sqrt(2)/(d*sqrt(a))+28/15*sin(c+d*x)/(d*sqrt(a+a*cos(c+d*x)))+2/5*cos(c+d*x)^2*sin(c+d*x)/(d*sqrt(a+a*cos(c+d*x)))-2/15*sin(c+d*x)*sqrt(a+a*cos(c+d*x))/(a*d)],
[cos(c+d*x)^2/(a+a*cos(c+d*x))^(1/2),x,4,arctanh(sin(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(a+a*cos(c+d*x))))*sqrt(2)/(d*sqrt(a))-4/3*sin(c+d*x)/(d*sqrt(a+a*cos(c+d*x)))+2/3*sin(c+d*x)*sqrt(a+a*cos(c+d*x))/(a*d)],
[cos(c+d*x)/(a+a*cos(c+d*x))^(1/2),x,3,-arctanh(sin(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(a+a*cos(c+d*x))))*sqrt(2)/(d*sqrt(a))+2*sin(c+d*x)/(d*sqrt(a+a*cos(c+d*x)))],
[1/(a+a*cos(c+d*x))^(1/2),x,2,arctanh(sin(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(a+a*cos(c+d*x))))*sqrt(2)/(d*sqrt(a))],
[sec(c+d*x)/sqrt(a+a*cos(c+d*x)),x,5,2*arctanh(sin(c+d*x)*sqrt(a)/sqrt(a+a*cos(c+d*x)))/(d*sqrt(a))-arctanh(sin(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(a+a*cos(c+d*x))))*sqrt(2)/(d*sqrt(a))],
[sec(c+d*x)^2/sqrt(a+a*cos(c+d*x)),x,6,-arctanh(sin(c+d*x)*sqrt(a)/sqrt(a+a*cos(c+d*x)))/(d*sqrt(a))+arctanh(sin(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(a+a*cos(c+d*x))))*sqrt(2)/(d*sqrt(a))+tan(c+d*x)/(d*sqrt(a+a*cos(c+d*x)))],
[sec(c+d*x)^3/sqrt(a+a*cos(c+d*x)),x,7,7/4*arctanh(sin(c+d*x)*sqrt(a)/sqrt(a+a*cos(c+d*x)))/(d*sqrt(a))-arctanh(sin(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(a+a*cos(c+d*x))))*sqrt(2)/(d*sqrt(a))-1/4*tan(c+d*x)/(d*sqrt(a+a*cos(c+d*x)))+1/2*sec(c+d*x)*tan(c+d*x)/(d*sqrt(a+a*cos(c+d*x)))],
[sec(c+d*x)^4/sqrt(a+a*cos(c+d*x)),x,8,-9/8*arctanh(sin(c+d*x)*sqrt(a)/sqrt(a+a*cos(c+d*x)))/(d*sqrt(a))+arctanh(sin(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(a+a*cos(c+d*x))))*sqrt(2)/(d*sqrt(a))+7/8*tan(c+d*x)/(d*sqrt(a+a*cos(c+d*x)))-1/12*sec(c+d*x)*tan(c+d*x)/(d*sqrt(a+a*cos(c+d*x)))+1/3*sec(c+d*x)^2*tan(c+d*x)/(d*sqrt(a+a*cos(c+d*x)))],
[cos(c+d*x)^4/(a+a*cos(c+d*x))^(3/2),x,7,-1/2*cos(c+d*x)^3*sin(c+d*x)/(d*(a+a*cos(c+d*x))^(3/2))-15/2*arctanh(sin(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(a+a*cos(c+d*x))))/(a^(3/2)*d*sqrt(2))+31/5*sin(c+d*x)/(a*d*sqrt(a+a*cos(c+d*x)))+9/10*cos(c+d*x)^2*sin(c+d*x)/(a*d*sqrt(a+a*cos(c+d*x)))-13/10*sin(c+d*x)*sqrt(a+a*cos(c+d*x))/(a^2*d)],
[cos(c+d*x)^3/(a+a*cos(c+d*x))^(3/2),x,6,-1/2*cos(c+d*x)^2*sin(c+d*x)/(d*(a+a*cos(c+d*x))^(3/2))+11/2*arctanh(sin(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(a+a*cos(c+d*x))))/(a^(3/2)*d*sqrt(2))-13/3*sin(c+d*x)/(a*d*sqrt(a+a*cos(c+d*x)))+7/6*sin(c+d*x)*sqrt(a+a*cos(c+d*x))/(a^2*d)],
[cos(c+d*x)^2/(a+a*cos(c+d*x))^(3/2),x,4,1/2*sin(c+d*x)/(d*(a+a*cos(c+d*x))^(3/2))-7/2*arctanh(sin(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(a+a*cos(c+d*x))))/(a^(3/2)*d*sqrt(2))+2*sin(c+d*x)/(a*d*sqrt(a+a*cos(c+d*x)))],
[cos(c+d*x)/(a+a*cos(c+d*x))^(3/2),x,3,-1/2*sin(c+d*x)/(d*(a+a*cos(c+d*x))^(3/2))+3/2*arctanh(sin(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(a+a*cos(c+d*x))))/(a^(3/2)*d*sqrt(2))],
[1/(a+a*cos(c+d*x))^(3/2),x,3,1/2*sin(c+d*x)/(d*(a+a*cos(c+d*x))^(3/2))+1/2*arctanh(sin(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(a+a*cos(c+d*x))))/(a^(3/2)*d*sqrt(2))],
[sec(c+d*x)/(a+a*cos(c+d*x))^(3/2),x,6,2*arctanh(sin(c+d*x)*sqrt(a)/sqrt(a+a*cos(c+d*x)))/(a^(3/2)*d)-1/2*sin(c+d*x)/(d*(a+a*cos(c+d*x))^(3/2))-5/2*arctanh(sin(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(a+a*cos(c+d*x))))/(a^(3/2)*d*sqrt(2))],
[sec(c+d*x)^2/(a+a*cos(c+d*x))^(3/2),x,7,-3*arctanh(sin(c+d*x)*sqrt(a)/sqrt(a+a*cos(c+d*x)))/(a^(3/2)*d)+9/2*arctanh(sin(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(a+a*cos(c+d*x))))/(a^(3/2)*d*sqrt(2))-1/2*tan(c+d*x)/(d*(a+a*cos(c+d*x))^(3/2))+3/2*tan(c+d*x)/(a*d*sqrt(a+a*cos(c+d*x)))],
[sec(c+d*x)^3/(a+a*cos(c+d*x))^(3/2),x,8,19/4*arctanh(sin(c+d*x)*sqrt(a)/sqrt(a+a*cos(c+d*x)))/(a^(3/2)*d)-13/2*arctanh(sin(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(a+a*cos(c+d*x))))/(a^(3/2)*d*sqrt(2))-1/2*sec(c+d*x)*tan(c+d*x)/(d*(a+a*cos(c+d*x))^(3/2))-7/4*tan(c+d*x)/(a*d*sqrt(a+a*cos(c+d*x)))+sec(c+d*x)*tan(c+d*x)/(a*d*sqrt(a+a*cos(c+d*x)))],
[cos(c+d*x)^4/(a+a*cos(c+d*x))^(5/2),x,7,-1/4*cos(c+d*x)^3*sin(c+d*x)/(d*(a+a*cos(c+d*x))^(5/2))-17/16*cos(c+d*x)^2*sin(c+d*x)/(a*d*(a+a*cos(c+d*x))^(3/2))+163/16*arctanh(sin(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(a+a*cos(c+d*x))))/(a^(5/2)*d*sqrt(2))-197/24*sin(c+d*x)/(a^2*d*sqrt(a+a*cos(c+d*x)))+95/48*sin(c+d*x)*sqrt(a+a*cos(c+d*x))/(a^3*d)],
[cos(c+d*x)^3/(a+a*cos(c+d*x))^(5/2),x,6,-1/4*cos(c+d*x)^2*sin(c+d*x)/(d*(a+a*cos(c+d*x))^(5/2))+13/16*sin(c+d*x)/(a*d*(a+a*cos(c+d*x))^(3/2))-75/16*arctanh(sin(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(a+a*cos(c+d*x))))/(a^(5/2)*d*sqrt(2))+9/4*sin(c+d*x)/(a^2*d*sqrt(a+a*cos(c+d*x)))],
[cos(c+d*x)^2/(a+a*cos(c+d*x))^(5/2),x,4,1/4*sin(c+d*x)/(d*(a+a*cos(c+d*x))^(5/2))-13/16*sin(c+d*x)/(a*d*(a+a*cos(c+d*x))^(3/2))+19/16*arctanh(sin(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(a+a*cos(c+d*x))))/(a^(5/2)*d*sqrt(2))],
[cos(c+d*x)/(a+a*cos(c+d*x))^(5/2),x,4,-1/4*sin(c+d*x)/(d*(a+a*cos(c+d*x))^(5/2))+5/16*sin(c+d*x)/(a*d*(a+a*cos(c+d*x))^(3/2))+5/16*arctanh(sin(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(a+a*cos(c+d*x))))/(a^(5/2)*d*sqrt(2))],
[1/(a+a*cos(c+d*x))^(5/2),x,4,1/4*sin(c+d*x)/(d*(a+a*cos(c+d*x))^(5/2))+3/16*sin(c+d*x)/(a*d*(a+a*cos(c+d*x))^(3/2))+3/16*arctanh(sin(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(a+a*cos(c+d*x))))/(a^(5/2)*d*sqrt(2))],
[sec(c+d*x)/(a+a*cos(c+d*x))^(5/2),x,7,2*arctanh(sin(c+d*x)*sqrt(a)/sqrt(a+a*cos(c+d*x)))/(a^(5/2)*d)-1/4*sin(c+d*x)/(d*(a+a*cos(c+d*x))^(5/2))-11/16*sin(c+d*x)/(a*d*(a+a*cos(c+d*x))^(3/2))-43/16*arctanh(sin(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(a+a*cos(c+d*x))))/(a^(5/2)*d*sqrt(2))],
[sec(c+d*x)^2/(a+a*cos(c+d*x))^(5/2),x,8,-5*arctanh(sin(c+d*x)*sqrt(a)/sqrt(a+a*cos(c+d*x)))/(a^(5/2)*d)+115/16*arctanh(sin(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(a+a*cos(c+d*x))))/(a^(5/2)*d*sqrt(2))-1/4*tan(c+d*x)/(d*(a+a*cos(c+d*x))^(5/2))-15/16*tan(c+d*x)/(a*d*(a+a*cos(c+d*x))^(3/2))+35/16*tan(c+d*x)/(a^2*d*sqrt(a+a*cos(c+d*x)))],

# Integrands of the form Cos[e+f x]^(m/2) (a+a Cos[e+f x])^n

# n>0
[cos(c+d*x)^(5/2)*(a+a*cos(c+d*x)),x,6,6/5*a*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/d+10/21*a*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))/d+2/5*a*cos(c+d*x)^(3/2)*sin(c+d*x)/d+2/7*a*cos(c+d*x)^(5/2)*sin(c+d*x)/d+10/21*a*sin(c+d*x)*sqrt(cos(c+d*x))/d],
[cos(c+d*x)^(3/2)*(a+a*cos(c+d*x)),x,5,6/5*a*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/d+2/3*a*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))/d+2/5*a*cos(c+d*x)^(3/2)*sin(c+d*x)/d+2/3*a*sin(c+d*x)*sqrt(cos(c+d*x))/d],
[cos(c+d*x)^(1/2)*(a+a*cos(c+d*x)),x,4,2*a*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/d+2/3*a*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))/d+2/3*a*sin(c+d*x)*sqrt(cos(c+d*x))/d],
[(a+a*cos(c+d*x))/cos(c+d*x)^(1/2),x,3,2*a*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/d+2*a*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))/d],
[(a+a*cos(c+d*x))/cos(c+d*x)^(3/2),x,4,-2*a*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/d+2*a*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))/d+2*a*sin(c+d*x)/(d*sqrt(cos(c+d*x)))],
[(a+a*cos(c+d*x))/cos(c+d*x)^(5/2),x,5,-2*a*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/d+2/3*a*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))/d+2/3*a*sin(c+d*x)/(d*cos(c+d*x)^(3/2))+2*a*sin(c+d*x)/(d*sqrt(cos(c+d*x)))],
[(a+a*cos(c+d*x))/cos(c+d*x)^(7/2),x,6,-6/5*a*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/d+2/3*a*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))/d+2/5*a*sin(c+d*x)/(d*cos(c+d*x)^(5/2))+2/3*a*sin(c+d*x)/(d*cos(c+d*x)^(3/2))+6/5*a*sin(c+d*x)/(d*sqrt(cos(c+d*x)))],
[cos(c+d*x)^(5/2)*(a+a*cos(c+d*x))^2,x,10,32/15*a^2*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/d+20/21*a^2*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))/d+32/45*a^2*cos(c+d*x)^(3/2)*sin(c+d*x)/d+4/7*a^2*cos(c+d*x)^(5/2)*sin(c+d*x)/d+2/9*a^2*cos(c+d*x)^(7/2)*sin(c+d*x)/d+20/21*a^2*sin(c+d*x)*sqrt(cos(c+d*x))/d],
[cos(c+d*x)^(3/2)*(a+a*cos(c+d*x))^2,x,9,12/5*a^2*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/d+8/7*a^2*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))/d+4/5*a^2*cos(c+d*x)^(3/2)*sin(c+d*x)/d+2/7*a^2*cos(c+d*x)^(5/2)*sin(c+d*x)/d+8/7*a^2*sin(c+d*x)*sqrt(cos(c+d*x))/d],
[cos(c+d*x)^(1/2)*(a+a*cos(c+d*x))^2,x,7,16/5*a^2*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/d+4/3*a^2*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))/d+2/5*a^2*cos(c+d*x)^(3/2)*sin(c+d*x)/d+4/3*a^2*sin(c+d*x)*sqrt(cos(c+d*x))/d],
[(a+a*cos(c+d*x))^2/cos(c+d*x)^(1/2),x,6,4*a^2*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/d+8/3*a^2*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))/d+2/3*a^2*sin(c+d*x)*sqrt(cos(c+d*x))/d],
[(a+a*cos(c+d*x))^2/cos(c+d*x)^(3/2),x,6,4*a^2*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))/d+2*a^2*sin(c+d*x)/(d*sqrt(cos(c+d*x)))],
[(a+a*cos(c+d*x))^2/cos(c+d*x)^(5/2),x,7,-4*a^2*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/d+8/3*a^2*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))/d+2/3*a^2*sin(c+d*x)/(d*cos(c+d*x)^(3/2))+4*a^2*sin(c+d*x)/(d*sqrt(cos(c+d*x)))],
[(a+a*cos(c+d*x))^2/cos(c+d*x)^(7/2),x,9,-16/5*a^2*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/d+4/3*a^2*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))/d+2/5*a^2*sin(c+d*x)/(d*cos(c+d*x)^(5/2))+4/3*a^2*sin(c+d*x)/(d*cos(c+d*x)^(3/2))+16/5*a^2*sin(c+d*x)/(d*sqrt(cos(c+d*x)))],
[cos(c+d*x)^(3/2)*(a+a*cos(c+d*x))^3,x,12,68/15*a^3*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/d+44/21*a^3*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))/d+68/45*a^3*cos(c+d*x)^(3/2)*sin(c+d*x)/d+6/7*a^3*cos(c+d*x)^(5/2)*sin(c+d*x)/d+2/9*a^3*cos(c+d*x)^(7/2)*sin(c+d*x)/d+44/21*a^3*sin(c+d*x)*sqrt(cos(c+d*x))/d],
[cos(c+d*x)^(1/2)*(a+a*cos(c+d*x))^3,x,10,28/5*a^3*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/d+52/21*a^3*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))/d+6/5*a^3*cos(c+d*x)^(3/2)*sin(c+d*x)/d+2/7*a^3*cos(c+d*x)^(5/2)*sin(c+d*x)/d+52/21*a^3*sin(c+d*x)*sqrt(cos(c+d*x))/d],
[(a+a*cos(c+d*x))^3/cos(c+d*x)^(1/2),x,8,36/5*a^3*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/d+4*a^3*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))/d+2/5*a^3*cos(c+d*x)^(3/2)*sin(c+d*x)/d+2*a^3*sin(c+d*x)*sqrt(cos(c+d*x))/d],
[(a+a*cos(c+d*x))^3/cos(c+d*x)^(3/2),x,8,4*a^3*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/d+20/3*a^3*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))/d+2*a^3*sin(c+d*x)/(d*sqrt(cos(c+d*x)))+2/3*a^3*sin(c+d*x)*sqrt(cos(c+d*x))/d],
[(a+a*cos(c+d*x))^3/cos(c+d*x)^(5/2),x,8,-4*a^3*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/d+20/3*a^3*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))/d+2/3*a^3*sin(c+d*x)/(d*cos(c+d*x)^(3/2))+6*a^3*sin(c+d*x)/(d*sqrt(cos(c+d*x)))],
[(a+a*cos(c+d*x))^3/cos(c+d*x)^(7/2),x,10,-36/5*a^3*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/d+4*a^3*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))/d+2/5*a^3*sin(c+d*x)/(d*cos(c+d*x)^(5/2))+2*a^3*sin(c+d*x)/(d*cos(c+d*x)^(3/2))+36/5*a^3*sin(c+d*x)/(d*sqrt(cos(c+d*x)))],
[(a+a*cos(c+d*x))^3/cos(c+d*x)^(9/2),x,12,-28/5*a^3*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/d+52/21*a^3*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))/d+2/7*a^3*sin(c+d*x)/(d*cos(c+d*x)^(7/2))+6/5*a^3*sin(c+d*x)/(d*cos(c+d*x)^(5/2))+52/21*a^3*sin(c+d*x)/(d*cos(c+d*x)^(3/2))+28/5*a^3*sin(c+d*x)/(d*sqrt(cos(c+d*x)))],
[cos(c+d*x)^(3/2)*(a+a*cos(c+d*x))^4,x,16,128/15*a^4*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/d+904/231*a^4*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))/d+128/45*a^4*cos(c+d*x)^(3/2)*sin(c+d*x)/d+150/77*a^4*cos(c+d*x)^(5/2)*sin(c+d*x)/d+8/9*a^4*cos(c+d*x)^(7/2)*sin(c+d*x)/d+2/11*a^4*cos(c+d*x)^(9/2)*sin(c+d*x)/d+904/231*a^4*sin(c+d*x)*sqrt(cos(c+d*x))/d],
[cos(c+d*x)^(1/2)*(a+a*cos(c+d*x))^4,x,13,152/15*a^4*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/d+32/7*a^4*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))/d+122/45*a^4*cos(c+d*x)^(3/2)*sin(c+d*x)/d+8/7*a^4*cos(c+d*x)^(5/2)*sin(c+d*x)/d+2/9*a^4*cos(c+d*x)^(7/2)*sin(c+d*x)/d+32/7*a^4*sin(c+d*x)*sqrt(cos(c+d*x))/d],
[(a+a*cos(c+d*x))^4/cos(c+d*x)^(1/2),x,11,64/5*a^4*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/d+136/21*a^4*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))/d+8/5*a^4*cos(c+d*x)^(3/2)*sin(c+d*x)/d+2/7*a^4*cos(c+d*x)^(5/2)*sin(c+d*x)/d+94/21*a^4*sin(c+d*x)*sqrt(cos(c+d*x))/d],
[(a+a*cos(c+d*x))^4/cos(c+d*x)^(3/2),x,10,56/5*a^4*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/d+32/3*a^4*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))/d+2/5*a^4*cos(c+d*x)^(3/2)*sin(c+d*x)/d+2*a^4*sin(c+d*x)/(d*sqrt(cos(c+d*x)))+8/3*a^4*sin(c+d*x)*sqrt(cos(c+d*x))/d],
[(a+a*cos(c+d*x))^4/cos(c+d*x)^(5/2),x,10,40/3*a^4*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))/d+2/3*a^4*sin(c+d*x)/(d*cos(c+d*x)^(3/2))+8*a^4*sin(c+d*x)/(d*sqrt(cos(c+d*x)))+2/3*a^4*sin(c+d*x)*sqrt(cos(c+d*x))/d],
[(a+a*cos(c+d*x))^4/cos(c+d*x)^(7/2),x,11,-56/5*a^4*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/d+32/3*a^4*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))/d+2/5*a^4*sin(c+d*x)/(d*cos(c+d*x)^(5/2))+8/3*a^4*sin(c+d*x)/(d*cos(c+d*x)^(3/2))+66/5*a^4*sin(c+d*x)/(d*sqrt(cos(c+d*x)))],
[(a+a*cos(c+d*x))^4/cos(c+d*x)^(9/2),x,13,-64/5*a^4*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/d+136/21*a^4*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))/d+2/7*a^4*sin(c+d*x)/(d*cos(c+d*x)^(7/2))+8/5*a^4*sin(c+d*x)/(d*cos(c+d*x)^(5/2))+94/21*a^4*sin(c+d*x)/(d*cos(c+d*x)^(3/2))+64/5*a^4*sin(c+d*x)/(d*sqrt(cos(c+d*x)))],

# n<0
[cos(c+d*x)^(7/2)/(a+a*cos(c+d*x)),x,6,21/5*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/(a*d)-5/3*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))/(a*d)+7/5*cos(c+d*x)^(3/2)*sin(c+d*x)/(a*d)-cos(c+d*x)^(5/2)*sin(c+d*x)/(d*(a+a*cos(c+d*x)))-5/3*sin(c+d*x)*sqrt(cos(c+d*x))/(a*d)],
[cos(c+d*x)^(5/2)/(a+a*cos(c+d*x)),x,5,-3*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/(a*d)+5/3*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))/(a*d)-cos(c+d*x)^(3/2)*sin(c+d*x)/(d*(a+a*cos(c+d*x)))+5/3*sin(c+d*x)*sqrt(cos(c+d*x))/(a*d)],
[cos(c+d*x)^(3/2)/(a+a*cos(c+d*x)),x,4,3*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/(a*d)-sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))/(a*d)-sin(c+d*x)*sqrt(cos(c+d*x))/(d*(a+a*cos(c+d*x)))],
[cos(c+d*x)^(1/2)/(a+a*cos(c+d*x)),x,4,-sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/(a*d)+sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))/(a*d)+sin(c+d*x)*sqrt(cos(c+d*x))/(d*(a+a*cos(c+d*x)))],
[1/(cos(c+d*x)^(1/2)*(a+a*cos(c+d*x))),x,4,sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/(a*d)+sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))/(a*d)-sin(c+d*x)*sqrt(cos(c+d*x))/(d*(a+a*cos(c+d*x)))],
[1/(cos(c+d*x)^(3/2)*(a+a*cos(c+d*x))),x,5,-3*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/(a*d)-sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))/(a*d)+3*sin(c+d*x)/(a*d*sqrt(cos(c+d*x)))-sin(c+d*x)/(d*(a+a*cos(c+d*x))*sqrt(cos(c+d*x)))],
[1/(cos(c+d*x)^(5/2)*(a+a*cos(c+d*x))),x,6,3*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/(a*d)+5/3*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))/(a*d)+5/3*sin(c+d*x)/(a*d*cos(c+d*x)^(3/2))-sin(c+d*x)/(d*cos(c+d*x)^(3/2)*(a+a*cos(c+d*x)))-3*sin(c+d*x)/(a*d*sqrt(cos(c+d*x)))],
[cos(c+d*x)^(9/2)/(a+a*cos(c+d*x))^2,x,7,56/5*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/(a^2*d)-5*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))/(a^2*d)+56/15*cos(c+d*x)^(3/2)*sin(c+d*x)/(a^2*d)-3*cos(c+d*x)^(5/2)*sin(c+d*x)/(a^2*d*(1+cos(c+d*x)))-1/3*cos(c+d*x)^(7/2)*sin(c+d*x)/(d*(a+a*cos(c+d*x))^2)-5*sin(c+d*x)*sqrt(cos(c+d*x))/(a^2*d)],
[cos(c+d*x)^(7/2)/(a+a*cos(c+d*x))^2,x,6,-7*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/(a^2*d)+10/3*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))/(a^2*d)-7/3*cos(c+d*x)^(3/2)*sin(c+d*x)/(a^2*d*(1+cos(c+d*x)))-1/3*cos(c+d*x)^(5/2)*sin(c+d*x)/(d*(a+a*cos(c+d*x))^2)+10/3*sin(c+d*x)*sqrt(cos(c+d*x))/(a^2*d)],
[cos(c+d*x)^(5/2)/(a+a*cos(c+d*x))^2,x,5,4*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/(a^2*d)-5/3*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))/(a^2*d)-1/3*cos(c+d*x)^(3/2)*sin(c+d*x)/(d*(a+a*cos(c+d*x))^2)-5/3*sin(c+d*x)*sqrt(cos(c+d*x))/(a^2*d*(1+cos(c+d*x)))],
[cos(c+d*x)^(3/2)/(a+a*cos(c+d*x))^2,x,5,-sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/(a^2*d)+2/3*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))/(a^2*d)+sin(c+d*x)*sqrt(cos(c+d*x))/(a^2*d*(1+cos(c+d*x)))-1/3*sin(c+d*x)*sqrt(cos(c+d*x))/(d*(a+a*cos(c+d*x))^2)],
[cos(c+d*x)^(1/2)/(a+a*cos(c+d*x))^2,x,3,1/3*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))/(a^2*d)+1/3*sin(c+d*x)*sqrt(cos(c+d*x))/(d*(a+a*cos(c+d*x))^2)],
[1/(cos(c+d*x)^(1/2)*(a+a*cos(c+d*x))^2),x,5,sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/(a^2*d)+2/3*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))/(a^2*d)-sin(c+d*x)*sqrt(cos(c+d*x))/(a^2*d*(1+cos(c+d*x)))-1/3*sin(c+d*x)*sqrt(cos(c+d*x))/(d*(a+a*cos(c+d*x))^2)],
[1/(cos(c+d*x)^(3/2)*(a+a*cos(c+d*x))^2),x,6,-4*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/(a^2*d)-5/3*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))/(a^2*d)+4*sin(c+d*x)/(a^2*d*sqrt(cos(c+d*x)))-5/3*sin(c+d*x)/(a^2*d*(1+cos(c+d*x))*sqrt(cos(c+d*x)))-1/3*sin(c+d*x)/(d*(a+a*cos(c+d*x))^2*sqrt(cos(c+d*x)))],
[1/(cos(c+d*x)^(5/2)*(a+a*cos(c+d*x))^2),x,7,7*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/(a^2*d)+10/3*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))/(a^2*d)+10/3*sin(c+d*x)/(a^2*d*cos(c+d*x)^(3/2))-7/3*sin(c+d*x)/(a^2*d*cos(c+d*x)^(3/2)*(1+cos(c+d*x)))-1/3*sin(c+d*x)/(d*cos(c+d*x)^(3/2)*(a+a*cos(c+d*x))^2)-7*sin(c+d*x)/(a^2*d*sqrt(cos(c+d*x)))],
[cos(c+d*x)^(11/2)/(a+a*cos(c+d*x))^3,x,8,231/10*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/(a^3*d)-21/2*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))/(a^3*d)+77/10*cos(c+d*x)^(3/2)*sin(c+d*x)/(a^3*d)-1/5*cos(c+d*x)^(9/2)*sin(c+d*x)/(d*(a+a*cos(c+d*x))^3)-4/5*cos(c+d*x)^(7/2)*sin(c+d*x)/(a*d*(a+a*cos(c+d*x))^2)-63/10*cos(c+d*x)^(5/2)*sin(c+d*x)/(d*(a^3+a^3*cos(c+d*x)))-21/2*sin(c+d*x)*sqrt(cos(c+d*x))/(a^3*d)],
[cos(c+d*x)^(9/2)/(a+a*cos(c+d*x))^3,x,7,-119/10*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/(a^3*d)+11/2*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))/(a^3*d)-1/5*cos(c+d*x)^(7/2)*sin(c+d*x)/(d*(a+a*cos(c+d*x))^3)-2/3*cos(c+d*x)^(5/2)*sin(c+d*x)/(a*d*(a+a*cos(c+d*x))^2)-119/30*cos(c+d*x)^(3/2)*sin(c+d*x)/(d*(a^3+a^3*cos(c+d*x)))+11/2*sin(c+d*x)*sqrt(cos(c+d*x))/(a^3*d)],
[cos(c+d*x)^(7/2)/(a+a*cos(c+d*x))^3,x,6,49/10*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/(a^3*d)-13/6*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))/(a^3*d)-1/5*cos(c+d*x)^(5/2)*sin(c+d*x)/(d*(a+a*cos(c+d*x))^3)-8/15*cos(c+d*x)^(3/2)*sin(c+d*x)/(a*d*(a+a*cos(c+d*x))^2)-13/6*sin(c+d*x)*sqrt(cos(c+d*x))/(d*(a^3+a^3*cos(c+d*x)))],
[cos(c+d*x)^(5/2)/(a+a*cos(c+d*x))^3,x,6,-9/10*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/(a^3*d)+1/2*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))/(a^3*d)-1/5*cos(c+d*x)^(3/2)*sin(c+d*x)/(d*(a+a*cos(c+d*x))^3)-2/5*sin(c+d*x)*sqrt(cos(c+d*x))/(a*d*(a+a*cos(c+d*x))^2)+9/10*sin(c+d*x)*sqrt(cos(c+d*x))/(d*(a^3+a^3*cos(c+d*x)))],
[cos(c+d*x)^(3/2)/(a+a*cos(c+d*x))^3,x,6,-1/10*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/(a^3*d)+1/6*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))/(a^3*d)-1/5*sin(c+d*x)*sqrt(cos(c+d*x))/(d*(a+a*cos(c+d*x))^3)+4/15*sin(c+d*x)*sqrt(cos(c+d*x))/(a*d*(a+a*cos(c+d*x))^2)+1/10*sin(c+d*x)*sqrt(cos(c+d*x))/(d*(a^3+a^3*cos(c+d*x)))],
[cos(c+d*x)^(1/2)/(a+a*cos(c+d*x))^3,x,6,1/10*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/(a^3*d)+1/6*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))/(a^3*d)+1/5*sin(c+d*x)*sqrt(cos(c+d*x))/(d*(a+a*cos(c+d*x))^3)+1/15*sin(c+d*x)*sqrt(cos(c+d*x))/(a*d*(a+a*cos(c+d*x))^2)-1/10*sin(c+d*x)*sqrt(cos(c+d*x))/(d*(a^3+a^3*cos(c+d*x)))],
[1/(cos(c+d*x)^(1/2)*(a+a*cos(c+d*x))^3),x,6,9/10*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/(a^3*d)+1/2*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))/(a^3*d)-1/5*sin(c+d*x)*sqrt(cos(c+d*x))/(d*(a+a*cos(c+d*x))^3)-2/5*sin(c+d*x)*sqrt(cos(c+d*x))/(a*d*(a+a*cos(c+d*x))^2)-9/10*sin(c+d*x)*sqrt(cos(c+d*x))/(d*(a^3+a^3*cos(c+d*x)))],
[1/(cos(c+d*x)^(3/2)*(a+a*cos(c+d*x))^3),x,7,-49/10*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/(a^3*d)-13/6*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))/(a^3*d)+49/10*sin(c+d*x)/(a^3*d*sqrt(cos(c+d*x)))-1/5*sin(c+d*x)/(d*(a+a*cos(c+d*x))^3*sqrt(cos(c+d*x)))-8/15*sin(c+d*x)/(a*d*(a+a*cos(c+d*x))^2*sqrt(cos(c+d*x)))-13/6*sin(c+d*x)/(d*(a^3+a^3*cos(c+d*x))*sqrt(cos(c+d*x)))],
[1/(cos(c+d*x)^(5/2)*(a+a*cos(c+d*x))^3),x,8,119/10*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/(a^3*d)+11/2*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))/(a^3*d)+11/2*sin(c+d*x)/(a^3*d*cos(c+d*x)^(3/2))-1/5*sin(c+d*x)/(d*cos(c+d*x)^(3/2)*(a+a*cos(c+d*x))^3)-2/3*sin(c+d*x)/(a*d*cos(c+d*x)^(3/2)*(a+a*cos(c+d*x))^2)-119/30*sin(c+d*x)/(d*cos(c+d*x)^(3/2)*(a^3+a^3*cos(c+d*x)))-119/10*sin(c+d*x)/(a^3*d*sqrt(cos(c+d*x)))],

# Integrands of the form Cos[e+f x]^(m/2) (a+a Cos[e+f x])^(n/2)

# n>0
[cos(c+d*x)^(5/2)*(a+a*cos(c+d*x))^(1/2),x,5,5/8*arcsin(sin(c+d*x)*sqrt(a)/sqrt(a+a*cos(c+d*x)))*sqrt(a)/d+5/12*a*cos(c+d*x)^(3/2)*sin(c+d*x)/(d*sqrt(a+a*cos(c+d*x)))+1/3*a*cos(c+d*x)^(5/2)*sin(c+d*x)/(d*sqrt(a+a*cos(c+d*x)))+5/8*a*sin(c+d*x)*sqrt(cos(c+d*x))/(d*sqrt(a+a*cos(c+d*x)))],
[cos(c+d*x)^(3/2)*(a+a*cos(c+d*x))^(1/2),x,4,3/4*arcsin(sin(c+d*x)*sqrt(a)/sqrt(a+a*cos(c+d*x)))*sqrt(a)/d+1/2*a*cos(c+d*x)^(3/2)*sin(c+d*x)/(d*sqrt(a+a*cos(c+d*x)))+3/4*a*sin(c+d*x)*sqrt(cos(c+d*x))/(d*sqrt(a+a*cos(c+d*x)))],
[cos(c+d*x)^(1/2)*(a+a*cos(c+d*x))^(1/2),x,3,arcsin(sin(c+d*x)*sqrt(a)/sqrt(a+a*cos(c+d*x)))*sqrt(a)/d+a*sin(c+d*x)*sqrt(cos(c+d*x))/(d*sqrt(a+a*cos(c+d*x)))],
[(a+a*cos(c+d*x))^(1/2)/cos(c+d*x)^(1/2),x,2,2*arcsin(sin(c+d*x)*sqrt(a)/sqrt(a+a*cos(c+d*x)))*sqrt(a)/d],
[(a+a*cos(c+d*x))^(1/2)/cos(c+d*x)^(3/2),x,1,2*a*sin(c+d*x)/(d*sqrt(cos(c+d*x))*sqrt(a+a*cos(c+d*x)))],
[(a+a*cos(c+d*x))^(1/2)/cos(c+d*x)^(5/2),x,2,2/3*a*sin(c+d*x)/(d*cos(c+d*x)^(3/2)*sqrt(a+a*cos(c+d*x)))+4/3*a*sin(c+d*x)/(d*sqrt(cos(c+d*x))*sqrt(a+a*cos(c+d*x)))],
[(a+a*cos(c+d*x))^(1/2)/cos(c+d*x)^(7/2),x,3,2/5*a*sin(c+d*x)/(d*cos(c+d*x)^(5/2)*sqrt(a+a*cos(c+d*x)))+8/15*a*sin(c+d*x)/(d*cos(c+d*x)^(3/2)*sqrt(a+a*cos(c+d*x)))+16/15*a*sin(c+d*x)/(d*sqrt(cos(c+d*x))*sqrt(a+a*cos(c+d*x)))],
[(a+a*cos(c+d*x))^(1/2)/cos(c+d*x)^(9/2),x,4,2/7*a*sin(c+d*x)/(d*cos(c+d*x)^(7/2)*sqrt(a+a*cos(c+d*x)))+12/35*a*sin(c+d*x)/(d*cos(c+d*x)^(5/2)*sqrt(a+a*cos(c+d*x)))+16/35*a*sin(c+d*x)/(d*cos(c+d*x)^(3/2)*sqrt(a+a*cos(c+d*x)))+32/35*a*sin(c+d*x)/(d*sqrt(cos(c+d*x))*sqrt(a+a*cos(c+d*x)))],
[cos(c+d*x)^(3/2)*(a+a*cos(c+d*x))^(3/2),x,6,11/8*a^(3/2)*arcsin(sin(c+d*x)*sqrt(a)/sqrt(a+a*cos(c+d*x)))/d+11/12*a^2*cos(c+d*x)^(3/2)*sin(c+d*x)/(d*sqrt(a+a*cos(c+d*x)))+1/3*a^2*cos(c+d*x)^(5/2)*sin(c+d*x)/(d*sqrt(a+a*cos(c+d*x)))+11/8*a^2*sin(c+d*x)*sqrt(cos(c+d*x))/(d*sqrt(a+a*cos(c+d*x)))],
[cos(c+d*x)^(1/2)*(a+a*cos(c+d*x))^(3/2),x,5,7/4*a^(3/2)*arcsin(sin(c+d*x)*sqrt(a)/sqrt(a+a*cos(c+d*x)))/d+1/2*a^2*cos(c+d*x)^(3/2)*sin(c+d*x)/(d*sqrt(a+a*cos(c+d*x)))+7/4*a^2*sin(c+d*x)*sqrt(cos(c+d*x))/(d*sqrt(a+a*cos(c+d*x)))],
[(a+a*cos(c+d*x))^(3/2)/cos(c+d*x)^(1/2),x,4,3*a^(3/2)*arcsin(sin(c+d*x)*sqrt(a)/sqrt(a+a*cos(c+d*x)))/d+a^2*sin(c+d*x)*sqrt(cos(c+d*x))/(d*sqrt(a+a*cos(c+d*x)))],
[(a+a*cos(c+d*x))^(3/2)/cos(c+d*x)^(3/2),x,4,2*a^(3/2)*arcsin(sin(c+d*x)*sqrt(a)/sqrt(a+a*cos(c+d*x)))/d+2*a^2*sin(c+d*x)/(d*sqrt(cos(c+d*x))*sqrt(a+a*cos(c+d*x)))],
[(a+a*cos(c+d*x))^(3/2)/cos(c+d*x)^(5/2),x,3,2/3*a^2*sin(c+d*x)/(d*cos(c+d*x)^(3/2)*sqrt(a+a*cos(c+d*x)))+10/3*a^2*sin(c+d*x)/(d*sqrt(cos(c+d*x))*sqrt(a+a*cos(c+d*x)))],
[(a+a*cos(c+d*x))^(3/2)/cos(c+d*x)^(7/2),x,4,2/5*a^2*sin(c+d*x)/(d*cos(c+d*x)^(5/2)*sqrt(a+a*cos(c+d*x)))+6/5*a^2*sin(c+d*x)/(d*cos(c+d*x)^(3/2)*sqrt(a+a*cos(c+d*x)))+12/5*a^2*sin(c+d*x)/(d*sqrt(cos(c+d*x))*sqrt(a+a*cos(c+d*x)))],
[(a+a*cos(c+d*x))^(3/2)/cos(c+d*x)^(9/2),x,5,2/7*a^2*sin(c+d*x)/(d*cos(c+d*x)^(7/2)*sqrt(a+a*cos(c+d*x)))+26/35*a^2*sin(c+d*x)/(d*cos(c+d*x)^(5/2)*sqrt(a+a*cos(c+d*x)))+104/105*a^2*sin(c+d*x)/(d*cos(c+d*x)^(3/2)*sqrt(a+a*cos(c+d*x)))+208/105*a^2*sin(c+d*x)/(d*sqrt(cos(c+d*x))*sqrt(a+a*cos(c+d*x)))],
[cos(c+d*x)^(3/2)*(a+a*cos(c+d*x))^(5/2),x,6,163/64*a^(5/2)*arcsin(sin(c+d*x)*sqrt(a)/sqrt(a+a*cos(c+d*x)))/d+163/96*a^3*cos(c+d*x)^(3/2)*sin(c+d*x)/(d*sqrt(a+a*cos(c+d*x)))+17/24*a^3*cos(c+d*x)^(5/2)*sin(c+d*x)/(d*sqrt(a+a*cos(c+d*x)))+163/64*a^3*sin(c+d*x)*sqrt(cos(c+d*x))/(d*sqrt(a+a*cos(c+d*x)))+1/4*a^2*cos(c+d*x)^(5/2)*sin(c+d*x)*sqrt(a+a*cos(c+d*x))/d],
[cos(c+d*x)^(1/2)*(a+a*cos(c+d*x))^(5/2),x,5,25/8*a^(5/2)*arcsin(sin(c+d*x)*sqrt(a)/sqrt(a+a*cos(c+d*x)))/d+13/12*a^3*cos(c+d*x)^(3/2)*sin(c+d*x)/(d*sqrt(a+a*cos(c+d*x)))+25/8*a^3*sin(c+d*x)*sqrt(cos(c+d*x))/(d*sqrt(a+a*cos(c+d*x)))+1/3*a^2*cos(c+d*x)^(3/2)*sin(c+d*x)*sqrt(a+a*cos(c+d*x))/d],
[(a+a*cos(c+d*x))^(5/2)/cos(c+d*x)^(1/2),x,4,19/4*a^(5/2)*arcsin(sin(c+d*x)*sqrt(a)/sqrt(a+a*cos(c+d*x)))/d+9/4*a^3*sin(c+d*x)*sqrt(cos(c+d*x))/(d*sqrt(a+a*cos(c+d*x)))+1/2*a^2*sin(c+d*x)*sqrt(cos(c+d*x))*sqrt(a+a*cos(c+d*x))/d],
[(a+a*cos(c+d*x))^(5/2)/cos(c+d*x)^(3/2),x,4,5*a^(5/2)*arcsin(sin(c+d*x)*sqrt(a)/sqrt(a+a*cos(c+d*x)))/d-a^3*sin(c+d*x)*sqrt(cos(c+d*x))/(d*sqrt(a+a*cos(c+d*x)))+2*a^2*sin(c+d*x)*sqrt(a+a*cos(c+d*x))/(d*sqrt(cos(c+d*x)))],
[(a+a*cos(c+d*x))^(5/2)/cos(c+d*x)^(5/2),x,4,2*a^(5/2)*arcsin(sin(c+d*x)*sqrt(a)/sqrt(a+a*cos(c+d*x)))/d+14/3*a^3*sin(c+d*x)/(d*sqrt(cos(c+d*x))*sqrt(a+a*cos(c+d*x)))+2/3*a^2*sin(c+d*x)*sqrt(a+a*cos(c+d*x))/(d*cos(c+d*x)^(3/2))],
[(a+a*cos(c+d*x))^(5/2)/cos(c+d*x)^(7/2),x,3,22/15*a^3*sin(c+d*x)/(d*cos(c+d*x)^(3/2)*sqrt(a+a*cos(c+d*x)))+86/15*a^3*sin(c+d*x)/(d*sqrt(cos(c+d*x))*sqrt(a+a*cos(c+d*x)))+2/5*a^2*sin(c+d*x)*sqrt(a+a*cos(c+d*x))/(d*cos(c+d*x)^(5/2))],
[(a+a*cos(c+d*x))^(5/2)/cos(c+d*x)^(9/2),x,4,6/7*a^3*sin(c+d*x)/(d*cos(c+d*x)^(5/2)*sqrt(a+a*cos(c+d*x)))+46/21*a^3*sin(c+d*x)/(d*cos(c+d*x)^(3/2)*sqrt(a+a*cos(c+d*x)))+92/21*a^3*sin(c+d*x)/(d*sqrt(cos(c+d*x))*sqrt(a+a*cos(c+d*x)))+2/7*a^2*sin(c+d*x)*sqrt(a+a*cos(c+d*x))/(d*cos(c+d*x)^(7/2))],
[(a+a*cos(c+d*x))^(5/2)/cos(c+d*x)^(11/2),x,5,38/63*a^3*sin(c+d*x)/(d*cos(c+d*x)^(7/2)*sqrt(a+a*cos(c+d*x)))+146/105*a^3*sin(c+d*x)/(d*cos(c+d*x)^(5/2)*sqrt(a+a*cos(c+d*x)))+584/315*a^3*sin(c+d*x)/(d*cos(c+d*x)^(3/2)*sqrt(a+a*cos(c+d*x)))+1168/315*a^3*sin(c+d*x)/(d*sqrt(cos(c+d*x))*sqrt(a+a*cos(c+d*x)))+2/9*a^2*sin(c+d*x)*sqrt(a+a*cos(c+d*x))/(d*cos(c+d*x)^(9/2))],
[(a+a*cos(c+d*x))^(3/2)/cos(c+d*x)^(5/4),x,2,4*a^2*sin(c+d*x)/(d*cos(c+d*x)^(1/4)*sqrt(a+a*cos(c+d*x)))],
[sqrt(a+a*cos(e+f*x))/sqrt(cos(e+f*x)),x,2,2*arcsin(sin(e+f*x)*sqrt(a)/sqrt(a+a*cos(e+f*x)))*sqrt(a)/f],
[sqrt(a-a*cos(e+f*x))/sqrt(-cos(e+f*x)),x,2,-2*arcsin(sin(e+f*x)*sqrt(a)/sqrt(a-a*cos(e+f*x)))*sqrt(a)/f],

# n<0
[cos(c+d*x)^(5/2)/(a+a*cos(c+d*x))^(1/2),x,7,7/4*arcsin(sin(c+d*x)*sqrt(a)/sqrt(a+a*cos(c+d*x)))/(d*sqrt(a))-arctan(sin(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(cos(c+d*x))*sqrt(a+a*cos(c+d*x))))*sqrt(2)/(d*sqrt(a))+1/2*cos(c+d*x)^(3/2)*sin(c+d*x)/(d*sqrt(a+a*cos(c+d*x)))-1/4*sin(c+d*x)*sqrt(cos(c+d*x))/(d*sqrt(a+a*cos(c+d*x)))],
[cos(c+d*x)^(3/2)/(a+a*cos(c+d*x))^(1/2),x,6,-arcsin(sin(c+d*x)*sqrt(a)/sqrt(a+a*cos(c+d*x)))/(d*sqrt(a))+arctan(sin(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(cos(c+d*x))*sqrt(a+a*cos(c+d*x))))*sqrt(2)/(d*sqrt(a))+sin(c+d*x)*sqrt(cos(c+d*x))/(d*sqrt(a+a*cos(c+d*x)))],
[cos(c+d*x)^(1/2)/(a+a*cos(c+d*x))^(1/2),x,5,2*arcsin(sin(c+d*x)*sqrt(a)/sqrt(a+a*cos(c+d*x)))/(d*sqrt(a))-arctan(sin(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(cos(c+d*x))*sqrt(a+a*cos(c+d*x))))*sqrt(2)/(d*sqrt(a))],
[1/(cos(c+d*x)^(1/2)*(a+a*cos(c+d*x))^(1/2)),x,2,arctan(sin(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(cos(c+d*x))*sqrt(a+a*cos(c+d*x))))*sqrt(2)/(d*sqrt(a))],
[1/(cos(c+d*x)^(3/2)*(a+a*cos(c+d*x))^(1/2)),x,4,-arctan(sin(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(cos(c+d*x))*sqrt(a+a*cos(c+d*x))))*sqrt(2)/(d*sqrt(a))+2*sin(c+d*x)/(d*sqrt(cos(c+d*x))*sqrt(a+a*cos(c+d*x)))],
[1/(cos(c+d*x)^(5/2)*(a+a*cos(c+d*x))^(1/2)),x,5,arctan(sin(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(cos(c+d*x))*sqrt(a+a*cos(c+d*x))))*sqrt(2)/(d*sqrt(a))+2/3*sin(c+d*x)/(d*cos(c+d*x)^(3/2)*sqrt(a+a*cos(c+d*x)))-2/3*sin(c+d*x)/(d*sqrt(cos(c+d*x))*sqrt(a+a*cos(c+d*x)))],
[1/(cos(c+d*x)^(7/2)*(a+a*cos(c+d*x))^(1/2)),x,6,-arctan(sin(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(cos(c+d*x))*sqrt(a+a*cos(c+d*x))))*sqrt(2)/(d*sqrt(a))+2/5*sin(c+d*x)/(d*cos(c+d*x)^(5/2)*sqrt(a+a*cos(c+d*x)))-2/15*sin(c+d*x)/(d*cos(c+d*x)^(3/2)*sqrt(a+a*cos(c+d*x)))+26/15*sin(c+d*x)/(d*sqrt(cos(c+d*x))*sqrt(a+a*cos(c+d*x)))],
[cos(c+d*x)^(5/2)/(1+cos(c+d*x))^(1/2),x,7,7/4*arcsin(sin(c+d*x)/sqrt(1+cos(c+d*x)))/d-arcsin(sin(c+d*x)/(1+cos(c+d*x)))*sqrt(2)/d+1/2*cos(c+d*x)^(3/2)*sin(c+d*x)/(d*sqrt(1+cos(c+d*x)))-1/4*sin(c+d*x)*sqrt(cos(c+d*x))/(d*sqrt(1+cos(c+d*x)))],
[cos(c+d*x)^(3/2)/(1+cos(c+d*x))^(1/2),x,6,-arcsin(sin(c+d*x)/sqrt(1+cos(c+d*x)))/d+arcsin(sin(c+d*x)/(1+cos(c+d*x)))*sqrt(2)/d+sin(c+d*x)*sqrt(cos(c+d*x))/(d*sqrt(1+cos(c+d*x)))],
[cos(c+d*x)^(1/2)/(1+cos(c+d*x))^(1/2),x,5,2*arcsin(sin(c+d*x)/sqrt(1+cos(c+d*x)))/d-arcsin(sin(c+d*x)/(1+cos(c+d*x)))*sqrt(2)/d],
[1/(cos(c+d*x)^(1/2)*(1+cos(c+d*x))^(1/2)),x,2,arcsin(sin(c+d*x)/(1+cos(c+d*x)))*sqrt(2)/d],
[1/(cos(c+d*x)^(3/2)*(1+cos(c+d*x))^(1/2)),x,3,-arcsin(sin(c+d*x)/(1+cos(c+d*x)))*sqrt(2)/d+2*sin(c+d*x)/(d*sqrt(cos(c+d*x))*sqrt(1+cos(c+d*x)))],
[1/(cos(c+d*x)^(5/2)*(1+cos(c+d*x))^(1/2)),x,5,arcsin(sin(c+d*x)/(1+cos(c+d*x)))*sqrt(2)/d+2/3*sin(c+d*x)/(d*cos(c+d*x)^(3/2)*sqrt(1+cos(c+d*x)))-2/3*sin(c+d*x)/(d*sqrt(cos(c+d*x))*sqrt(1+cos(c+d*x)))],
[1/(cos(c+d*x)^(7/2)*(1+cos(c+d*x))^(1/2)),x,6,-arcsin(sin(c+d*x)/(1+cos(c+d*x)))*sqrt(2)/d+2/5*sin(c+d*x)/(d*cos(c+d*x)^(5/2)*sqrt(1+cos(c+d*x)))-2/15*sin(c+d*x)/(d*cos(c+d*x)^(3/2)*sqrt(1+cos(c+d*x)))+26/15*sin(c+d*x)/(d*sqrt(cos(c+d*x))*sqrt(1+cos(c+d*x)))],
[cos(c+d*x)^(5/2)/(a+a*cos(c+d*x))^(3/2),x,7,-3*arcsin(sin(c+d*x)*sqrt(a)/sqrt(a+a*cos(c+d*x)))/(a^(3/2)*d)-1/2*cos(c+d*x)^(3/2)*sin(c+d*x)/(d*(a+a*cos(c+d*x))^(3/2))+9/2*arctan(sin(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(cos(c+d*x))*sqrt(a+a*cos(c+d*x))))/(a^(3/2)*d*sqrt(2))+3/2*sin(c+d*x)*sqrt(cos(c+d*x))/(a*d*sqrt(a+a*cos(c+d*x)))],
[cos(c+d*x)^(3/2)/(a+a*cos(c+d*x))^(3/2),x,6,2*arcsin(sin(c+d*x)*sqrt(a)/sqrt(a+a*cos(c+d*x)))/(a^(3/2)*d)-5/2*arctan(sin(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(cos(c+d*x))*sqrt(a+a*cos(c+d*x))))/(a^(3/2)*d*sqrt(2))-1/2*sin(c+d*x)*sqrt(cos(c+d*x))/(d*(a+a*cos(c+d*x))^(3/2))],
[cos(c+d*x)^(1/2)/(a+a*cos(c+d*x))^(3/2),x,4,1/2*arctan(sin(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(cos(c+d*x))*sqrt(a+a*cos(c+d*x))))/(a^(3/2)*d*sqrt(2))+1/2*sin(c+d*x)*sqrt(cos(c+d*x))/(d*(a+a*cos(c+d*x))^(3/2))],
[1/(cos(c+d*x)^(1/2)*(a+a*cos(c+d*x))^(3/2)),x,4,3/2*arctan(sin(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(cos(c+d*x))*sqrt(a+a*cos(c+d*x))))/(a^(3/2)*d*sqrt(2))-1/2*sin(c+d*x)*sqrt(cos(c+d*x))/(d*(a+a*cos(c+d*x))^(3/2))],
[1/(cos(c+d*x)^(3/2)*(a+a*cos(c+d*x))^(3/2)),x,5,-7/2*arctan(sin(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(cos(c+d*x))*sqrt(a+a*cos(c+d*x))))/(a^(3/2)*d*sqrt(2))-1/2*sin(c+d*x)/(d*(a+a*cos(c+d*x))^(3/2)*sqrt(cos(c+d*x)))+5/2*sin(c+d*x)/(a*d*sqrt(cos(c+d*x))*sqrt(a+a*cos(c+d*x)))],
[1/(cos(c+d*x)^(5/2)*(a+a*cos(c+d*x))^(3/2)),x,6,-1/2*sin(c+d*x)/(d*cos(c+d*x)^(3/2)*(a+a*cos(c+d*x))^(3/2))+11/2*arctan(sin(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(cos(c+d*x))*sqrt(a+a*cos(c+d*x))))/(a^(3/2)*d*sqrt(2))+7/6*sin(c+d*x)/(a*d*cos(c+d*x)^(3/2)*sqrt(a+a*cos(c+d*x)))-19/6*sin(c+d*x)/(a*d*sqrt(cos(c+d*x))*sqrt(a+a*cos(c+d*x)))],
[cos(c+d*x)^(7/2)/(a+a*cos(c+d*x))^(5/2),x,8,-5*arcsin(sin(c+d*x)*sqrt(a)/sqrt(a+a*cos(c+d*x)))/(a^(5/2)*d)-1/4*cos(c+d*x)^(5/2)*sin(c+d*x)/(d*(a+a*cos(c+d*x))^(5/2))-15/16*cos(c+d*x)^(3/2)*sin(c+d*x)/(a*d*(a+a*cos(c+d*x))^(3/2))+115/16*arctan(sin(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(cos(c+d*x))*sqrt(a+a*cos(c+d*x))))/(a^(5/2)*d*sqrt(2))+35/16*sin(c+d*x)*sqrt(cos(c+d*x))/(a^2*d*sqrt(a+a*cos(c+d*x)))],
[cos(c+d*x)^(5/2)/(a+a*cos(c+d*x))^(5/2),x,7,2*arcsin(sin(c+d*x)*sqrt(a)/sqrt(a+a*cos(c+d*x)))/(a^(5/2)*d)-1/4*cos(c+d*x)^(3/2)*sin(c+d*x)/(d*(a+a*cos(c+d*x))^(5/2))-43/16*arctan(sin(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(cos(c+d*x))*sqrt(a+a*cos(c+d*x))))/(a^(5/2)*d*sqrt(2))-11/16*sin(c+d*x)*sqrt(cos(c+d*x))/(a*d*(a+a*cos(c+d*x))^(3/2))],
[cos(c+d*x)^(3/2)/(a+a*cos(c+d*x))^(5/2),x,5,3/16*arctan(sin(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(cos(c+d*x))*sqrt(a+a*cos(c+d*x))))/(a^(5/2)*d*sqrt(2))-1/4*sin(c+d*x)*sqrt(cos(c+d*x))/(d*(a+a*cos(c+d*x))^(5/2))+7/16*sin(c+d*x)*sqrt(cos(c+d*x))/(a*d*(a+a*cos(c+d*x))^(3/2))],
[cos(c+d*x)^(1/2)/(a+a*cos(c+d*x))^(5/2),x,5,5/16*arctan(sin(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(cos(c+d*x))*sqrt(a+a*cos(c+d*x))))/(a^(5/2)*d*sqrt(2))+1/4*sin(c+d*x)*sqrt(cos(c+d*x))/(d*(a+a*cos(c+d*x))^(5/2))+1/16*sin(c+d*x)*sqrt(cos(c+d*x))/(a*d*(a+a*cos(c+d*x))^(3/2))],
[1/(cos(c+d*x)^(1/2)*(a+a*cos(c+d*x))^(5/2)),x,5,19/16*arctan(sin(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(cos(c+d*x))*sqrt(a+a*cos(c+d*x))))/(a^(5/2)*d*sqrt(2))-1/4*sin(c+d*x)*sqrt(cos(c+d*x))/(d*(a+a*cos(c+d*x))^(5/2))-9/16*sin(c+d*x)*sqrt(cos(c+d*x))/(a*d*(a+a*cos(c+d*x))^(3/2))],
[1/(cos(c+d*x)^(3/2)*(a+a*cos(c+d*x))^(5/2)),x,6,-75/16*arctan(sin(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(cos(c+d*x))*sqrt(a+a*cos(c+d*x))))/(a^(5/2)*d*sqrt(2))-1/4*sin(c+d*x)/(d*(a+a*cos(c+d*x))^(5/2)*sqrt(cos(c+d*x)))-13/16*sin(c+d*x)/(a*d*(a+a*cos(c+d*x))^(3/2)*sqrt(cos(c+d*x)))+49/16*sin(c+d*x)/(a^2*d*sqrt(cos(c+d*x))*sqrt(a+a*cos(c+d*x)))],
[1/(cos(c+d*x)^(5/2)*(a+a*cos(c+d*x))^(5/2)),x,7,-1/4*sin(c+d*x)/(d*cos(c+d*x)^(3/2)*(a+a*cos(c+d*x))^(5/2))-17/16*sin(c+d*x)/(a*d*cos(c+d*x)^(3/2)*(a+a*cos(c+d*x))^(3/2))+163/16*arctan(sin(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(cos(c+d*x))*sqrt(a+a*cos(c+d*x))))/(a^(5/2)*d*sqrt(2))+95/48*sin(c+d*x)/(a^2*d*cos(c+d*x)^(3/2)*sqrt(a+a*cos(c+d*x)))-299/48*sin(c+d*x)/(a^2*d*sqrt(cos(c+d*x))*sqrt(a+a*cos(c+d*x)))],
[cos(c+d*x)^(9/2)/(a+a*cos(c+d*x))^(7/2),x,9,-7*arcsin(sin(c+d*x)*sqrt(a)/sqrt(a+a*cos(c+d*x)))/(a^(7/2)*d)-1/6*cos(c+d*x)^(7/2)*sin(c+d*x)/(d*(a+a*cos(c+d*x))^(7/2))-7/16*cos(c+d*x)^(5/2)*sin(c+d*x)/(a*d*(a+a*cos(c+d*x))^(5/2))-259/192*cos(c+d*x)^(3/2)*sin(c+d*x)/(a^2*d*(a+a*cos(c+d*x))^(3/2))+637/64*arctan(sin(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(cos(c+d*x))*sqrt(a+a*cos(c+d*x))))/(a^(7/2)*d*sqrt(2))+189/64*sin(c+d*x)*sqrt(cos(c+d*x))/(a^3*d*sqrt(a+a*cos(c+d*x)))],
[cos(c+d*x)^(7/2)/(a+a*cos(c+d*x))^(7/2),x,8,2*arcsin(sin(c+d*x)*sqrt(a)/sqrt(a+a*cos(c+d*x)))/(a^(7/2)*d)-1/6*cos(c+d*x)^(5/2)*sin(c+d*x)/(d*(a+a*cos(c+d*x))^(7/2))-17/48*cos(c+d*x)^(3/2)*sin(c+d*x)/(a*d*(a+a*cos(c+d*x))^(5/2))-177/64*arctan(sin(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(cos(c+d*x))*sqrt(a+a*cos(c+d*x))))/(a^(7/2)*d*sqrt(2))-49/64*sin(c+d*x)*sqrt(cos(c+d*x))/(a^2*d*(a+a*cos(c+d*x))^(3/2))],
[cos(c+d*x)^(5/2)/(a+a*cos(c+d*x))^(7/2),x,6,-1/6*cos(c+d*x)^(3/2)*sin(c+d*x)/(d*(a+a*cos(c+d*x))^(7/2))+5/64*arctan(sin(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(cos(c+d*x))*sqrt(a+a*cos(c+d*x))))/(a^(7/2)*d*sqrt(2))-13/48*sin(c+d*x)*sqrt(cos(c+d*x))/(a*d*(a+a*cos(c+d*x))^(5/2))+67/192*sin(c+d*x)*sqrt(cos(c+d*x))/(a^2*d*(a+a*cos(c+d*x))^(3/2))],
[cos(c+d*x)^(3/2)/(a+a*cos(c+d*x))^(7/2),x,6,7/64*arctan(sin(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(cos(c+d*x))*sqrt(a+a*cos(c+d*x))))/(a^(7/2)*d*sqrt(2))-1/6*sin(c+d*x)*sqrt(cos(c+d*x))/(d*(a+a*cos(c+d*x))^(7/2))+3/16*sin(c+d*x)*sqrt(cos(c+d*x))/(a*d*(a+a*cos(c+d*x))^(5/2))+17/192*sin(c+d*x)*sqrt(cos(c+d*x))/(a^2*d*(a+a*cos(c+d*x))^(3/2))],
[cos(c+d*x)^(1/2)/(a+a*cos(c+d*x))^(7/2),x,6,13/64*arctan(sin(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(cos(c+d*x))*sqrt(a+a*cos(c+d*x))))/(a^(7/2)*d*sqrt(2))+1/6*sin(c+d*x)*sqrt(cos(c+d*x))/(d*(a+a*cos(c+d*x))^(7/2))+1/16*sin(c+d*x)*sqrt(cos(c+d*x))/(a*d*(a+a*cos(c+d*x))^(5/2))-5/192*sin(c+d*x)*sqrt(cos(c+d*x))/(a^2*d*(a+a*cos(c+d*x))^(3/2))],
[1/(cos(c+d*x)^(1/2)*(a+a*cos(c+d*x))^(7/2)),x,6,63/64*arctan(sin(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(cos(c+d*x))*sqrt(a+a*cos(c+d*x))))/(a^(7/2)*d*sqrt(2))-1/6*sin(c+d*x)*sqrt(cos(c+d*x))/(d*(a+a*cos(c+d*x))^(7/2))-5/16*sin(c+d*x)*sqrt(cos(c+d*x))/(a*d*(a+a*cos(c+d*x))^(5/2))-103/192*sin(c+d*x)*sqrt(cos(c+d*x))/(a^2*d*(a+a*cos(c+d*x))^(3/2))],
[1/(cos(c+d*x)^(3/2)*(a+a*cos(c+d*x))^(7/2)),x,7,-363/64*arctan(sin(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(cos(c+d*x))*sqrt(a+a*cos(c+d*x))))/(a^(7/2)*d*sqrt(2))-1/6*sin(c+d*x)/(d*(a+a*cos(c+d*x))^(7/2)*sqrt(cos(c+d*x)))-19/48*sin(c+d*x)/(a*d*(a+a*cos(c+d*x))^(5/2)*sqrt(cos(c+d*x)))-199/192*sin(c+d*x)/(a^2*d*(a+a*cos(c+d*x))^(3/2)*sqrt(cos(c+d*x)))+691/192*sin(c+d*x)/(a^3*d*sqrt(cos(c+d*x))*sqrt(a+a*cos(c+d*x)))],
[1/(cos(c+d*x)^(5/2)*(a+a*cos(c+d*x))^(7/2)),x,8,-1/6*sin(c+d*x)/(d*cos(c+d*x)^(3/2)*(a+a*cos(c+d*x))^(7/2))-23/48*sin(c+d*x)/(a*d*cos(c+d*x)^(3/2)*(a+a*cos(c+d*x))^(5/2))-109/64*sin(c+d*x)/(a^2*d*cos(c+d*x)^(3/2)*(a+a*cos(c+d*x))^(3/2))+1015/64*arctan(sin(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(cos(c+d*x))*sqrt(a+a*cos(c+d*x))))/(a^(7/2)*d*sqrt(2))+193/64*sin(c+d*x)/(a^3*d*cos(c+d*x)^(3/2)*sqrt(a+a*cos(c+d*x)))-629/64*sin(c+d*x)/(a^3*d*sqrt(cos(c+d*x))*sqrt(a+a*cos(c+d*x)))],
[cos(c+d*x)^(7/2)/(a+a*cos(c+d*x))^(9/2),x,7,-1/8*cos(c+d*x)^(5/2)*sin(c+d*x)/(d*(a+a*cos(c+d*x))^(9/2))-19/96*cos(c+d*x)^(3/2)*sin(c+d*x)/(a*d*(a+a*cos(c+d*x))^(7/2))+35/1024*arctan(sin(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(cos(c+d*x))*sqrt(a+a*cos(c+d*x))))/(a^(9/2)*d*sqrt(2))-187/768*sin(c+d*x)*sqrt(cos(c+d*x))/(a^2*d*(a+a*cos(c+d*x))^(5/2))+853/3072*sin(c+d*x)*sqrt(cos(c+d*x))/(a^3*d*(a+a*cos(c+d*x))^(3/2))],
[cos(c+d*x)^(5/2)/(a+a*cos(c+d*x))^(9/2),x,7,-1/8*cos(c+d*x)^(3/2)*sin(c+d*x)/(d*(a+a*cos(c+d*x))^(9/2))+45/1024*arctan(sin(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(cos(c+d*x))*sqrt(a+a*cos(c+d*x))))/(a^(9/2)*d*sqrt(2))-5/32*sin(c+d*x)*sqrt(cos(c+d*x))/(a*d*(a+a*cos(c+d*x))^(7/2))+33/256*sin(c+d*x)*sqrt(cos(c+d*x))/(a^2*d*(a+a*cos(c+d*x))^(5/2))+73/1024*sin(c+d*x)*sqrt(cos(c+d*x))/(a^3*d*(a+a*cos(c+d*x))^(3/2))],
[1/(sqrt(cos(x))*sqrt(1+cos(x))),x,2,arcsin(sin(x)/(1+cos(x)))*sqrt(2)],
[1/(sqrt(cos(x))*sqrt(a+a*cos(x))),x,2,arctan(sin(x)*sqrt(a)/(sqrt(2)*sqrt(cos(x))*sqrt(a+a*cos(x))))*sqrt(2)/sqrt(a)],

# Integrands of the form Cos[e+f x]^(m/2) (a - a Cos[e+f x])^(n/2)

# n>0
[cos(c+d*x)^(3/2)*(a-a*cos(c+d*x))^(1/2),x,4,-3/4*arctanh(sin(c+d*x)*sqrt(a)/(sqrt(cos(c+d*x))*sqrt(a-a*cos(c+d*x))))*sqrt(a)/d-1/2*a*cos(c+d*x)^(3/2)*sin(c+d*x)/(d*sqrt(a-a*cos(c+d*x)))+3/4*a*sin(c+d*x)*sqrt(cos(c+d*x))/(d*sqrt(a-a*cos(c+d*x)))],
[cos(c+d*x)^(1/2)*(a-a*cos(c+d*x))^(1/2),x,3,arctanh(sin(c+d*x)*sqrt(a)/(sqrt(cos(c+d*x))*sqrt(a-a*cos(c+d*x))))*sqrt(a)/d-a*sin(c+d*x)*sqrt(cos(c+d*x))/(d*sqrt(a-a*cos(c+d*x)))],
[(a-a*cos(c+d*x))^(1/2)/cos(c+d*x)^(1/2),x,2,-2*arctanh(sin(c+d*x)*sqrt(a)/(sqrt(cos(c+d*x))*sqrt(a-a*cos(c+d*x))))*sqrt(a)/d],
[(a-a*cos(c+d*x))^(1/2)/cos(c+d*x)^(3/2),x,1,2*a*sin(c+d*x)/(d*sqrt(cos(c+d*x))*sqrt(a-a*cos(c+d*x)))],
[(a-a*cos(c+d*x))^(1/2)/cos(c+d*x)^(5/2),x,2,2/3*a*sin(c+d*x)/(d*cos(c+d*x)^(3/2)*sqrt(a-a*cos(c+d*x)))-4/3*a*sin(c+d*x)/(d*sqrt(cos(c+d*x))*sqrt(a-a*cos(c+d*x)))],
[(a-a*cos(c+d*x))^(1/2)/cos(c+d*x)^(7/2),x,3,2/5*a*sin(c+d*x)/(d*cos(c+d*x)^(5/2)*sqrt(a-a*cos(c+d*x)))-8/15*a*sin(c+d*x)/(d*cos(c+d*x)^(3/2)*sqrt(a-a*cos(c+d*x)))+16/15*a*sin(c+d*x)/(d*sqrt(cos(c+d*x))*sqrt(a-a*cos(c+d*x)))],
[(1-cos(c+d*x))^(1/2)*cos(c+d*x)^(3/2),x,4,-3/4*arctanh(sin(c+d*x)/(sqrt(1-cos(c+d*x))*sqrt(cos(c+d*x))))/d-1/2*cos(c+d*x)^(3/2)*sin(c+d*x)/(d*sqrt(1-cos(c+d*x)))+3/4*sin(c+d*x)*sqrt(cos(c+d*x))/(d*sqrt(1-cos(c+d*x)))],
[(1-cos(c+d*x))^(1/2)*cos(c+d*x)^(1/2),x,3,arctanh(sin(c+d*x)/(sqrt(1-cos(c+d*x))*sqrt(cos(c+d*x))))/d-sin(c+d*x)*sqrt(cos(c+d*x))/(d*sqrt(1-cos(c+d*x)))],
[(1-cos(c+d*x))^(1/2)/cos(c+d*x)^(1/2),x,2,-2*arctanh(sin(c+d*x)/(sqrt(1-cos(c+d*x))*sqrt(cos(c+d*x))))/d],
[(1-cos(c+d*x))^(1/2)/cos(c+d*x)^(3/2),x,1,2*sin(c+d*x)/(d*sqrt(1-cos(c+d*x))*sqrt(cos(c+d*x)))],
[(1-cos(c+d*x))^(1/2)/cos(c+d*x)^(5/2),x,2,2/3*sin(c+d*x)/(d*cos(c+d*x)^(3/2)*sqrt(1-cos(c+d*x)))-4/3*sin(c+d*x)/(d*sqrt(1-cos(c+d*x))*sqrt(cos(c+d*x)))],
[(1-cos(c+d*x))^(1/2)/cos(c+d*x)^(7/2),x,3,2/5*sin(c+d*x)/(d*cos(c+d*x)^(5/2)*sqrt(1-cos(c+d*x)))-8/15*sin(c+d*x)/(d*cos(c+d*x)^(3/2)*sqrt(1-cos(c+d*x)))+16/15*sin(c+d*x)/(d*sqrt(1-cos(c+d*x))*sqrt(cos(c+d*x)))],

# n<0
[cos(c+d*x)^(5/2)/(a-a*cos(c+d*x))^(1/2),x,7,7/4*arctanh(sin(c+d*x)*sqrt(a)/(sqrt(cos(c+d*x))*sqrt(a-a*cos(c+d*x))))/(d*sqrt(a))-arctanh(sin(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(cos(c+d*x))*sqrt(a-a*cos(c+d*x))))*sqrt(2)/(d*sqrt(a))+1/2*cos(c+d*x)^(3/2)*sin(c+d*x)/(d*sqrt(a-a*cos(c+d*x)))+1/4*sin(c+d*x)*sqrt(cos(c+d*x))/(d*sqrt(a-a*cos(c+d*x)))],
[cos(c+d*x)^(3/2)/(a-a*cos(c+d*x))^(1/2),x,6,arctanh(sin(c+d*x)*sqrt(a)/(sqrt(cos(c+d*x))*sqrt(a-a*cos(c+d*x))))/(d*sqrt(a))-arctanh(sin(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(cos(c+d*x))*sqrt(a-a*cos(c+d*x))))*sqrt(2)/(d*sqrt(a))+sin(c+d*x)*sqrt(cos(c+d*x))/(d*sqrt(a-a*cos(c+d*x)))],
[cos(c+d*x)^(1/2)/(a-a*cos(c+d*x))^(1/2),x,5,2*arctanh(sin(c+d*x)*sqrt(a)/(sqrt(cos(c+d*x))*sqrt(a-a*cos(c+d*x))))/(d*sqrt(a))-arctanh(sin(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(cos(c+d*x))*sqrt(a-a*cos(c+d*x))))*sqrt(2)/(d*sqrt(a))],
[1/(cos(c+d*x)^(1/2)*(a-a*cos(c+d*x))^(1/2)),x,2,-arctanh(sin(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(cos(c+d*x))*sqrt(a-a*cos(c+d*x))))*sqrt(2)/(d*sqrt(a))],
[1/(cos(c+d*x)^(3/2)*(a-a*cos(c+d*x))^(1/2)),x,4,-arctanh(sin(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(cos(c+d*x))*sqrt(a-a*cos(c+d*x))))*sqrt(2)/(d*sqrt(a))+2*sin(c+d*x)/(d*sqrt(cos(c+d*x))*sqrt(a-a*cos(c+d*x)))],
[1/(cos(c+d*x)^(5/2)*(a-a*cos(c+d*x))^(1/2)),x,5,-arctanh(sin(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(cos(c+d*x))*sqrt(a-a*cos(c+d*x))))*sqrt(2)/(d*sqrt(a))+2/3*sin(c+d*x)/(d*cos(c+d*x)^(3/2)*sqrt(a-a*cos(c+d*x)))+2/3*sin(c+d*x)/(d*sqrt(cos(c+d*x))*sqrt(a-a*cos(c+d*x)))],
[1/(cos(c+d*x)^(7/2)*(a-a*cos(c+d*x))^(1/2)),x,6,-arctanh(sin(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(cos(c+d*x))*sqrt(a-a*cos(c+d*x))))*sqrt(2)/(d*sqrt(a))+2/5*sin(c+d*x)/(d*cos(c+d*x)^(5/2)*sqrt(a-a*cos(c+d*x)))+2/15*sin(c+d*x)/(d*cos(c+d*x)^(3/2)*sqrt(a-a*cos(c+d*x)))+26/15*sin(c+d*x)/(d*sqrt(cos(c+d*x))*sqrt(a-a*cos(c+d*x)))],
[cos(c+d*x)^(5/2)/(1-cos(c+d*x))^(1/2),x,7,7/4*arctanh(sin(c+d*x)/(sqrt(1-cos(c+d*x))*sqrt(cos(c+d*x))))/d-arctanh(sin(c+d*x)/(sqrt(2)*sqrt(1-cos(c+d*x))*sqrt(cos(c+d*x))))*sqrt(2)/d+1/2*cos(c+d*x)^(3/2)*sin(c+d*x)/(d*sqrt(1-cos(c+d*x)))+1/4*sin(c+d*x)*sqrt(cos(c+d*x))/(d*sqrt(1-cos(c+d*x)))],
[cos(c+d*x)^(3/2)/(1-cos(c+d*x))^(1/2),x,6,arctanh(sin(c+d*x)/(sqrt(1-cos(c+d*x))*sqrt(cos(c+d*x))))/d-arctanh(sin(c+d*x)/(sqrt(2)*sqrt(1-cos(c+d*x))*sqrt(cos(c+d*x))))*sqrt(2)/d+sin(c+d*x)*sqrt(cos(c+d*x))/(d*sqrt(1-cos(c+d*x)))],
[cos(c+d*x)^(1/2)/(1-cos(c+d*x))^(1/2),x,5,2*arctanh(sin(c+d*x)/(sqrt(1-cos(c+d*x))*sqrt(cos(c+d*x))))/d-arctanh(sin(c+d*x)/(sqrt(2)*sqrt(1-cos(c+d*x))*sqrt(cos(c+d*x))))*sqrt(2)/d],
[1/((1-cos(c+d*x))^(1/2)*cos(c+d*x)^(1/2)),x,2,-arctanh(sin(c+d*x)/(sqrt(2)*sqrt(1-cos(c+d*x))*sqrt(cos(c+d*x))))*sqrt(2)/d],
[1/((1-cos(c+d*x))^(1/2)*cos(c+d*x)^(3/2)),x,3,-arctanh(sin(c+d*x)/(sqrt(2)*sqrt(1-cos(c+d*x))*sqrt(cos(c+d*x))))*sqrt(2)/d+2*sin(c+d*x)/(d*sqrt(1-cos(c+d*x))*sqrt(cos(c+d*x)))],
[1/((1-cos(c+d*x))^(1/2)*cos(c+d*x)^(5/2)),x,5,-arctanh(sin(c+d*x)/(sqrt(2)*sqrt(1-cos(c+d*x))*sqrt(cos(c+d*x))))*sqrt(2)/d+2/3*sin(c+d*x)/(d*cos(c+d*x)^(3/2)*sqrt(1-cos(c+d*x)))+2/3*sin(c+d*x)/(d*sqrt(1-cos(c+d*x))*sqrt(cos(c+d*x)))],

# Integrands of the form Cos[e+f x]^(m/3) (a+a Cos[e+f x])^(n/3)

# n>0
[cos(c+d*x)^(4/3)*(a+a*cos(c+d*x))^(1/3),x,3,2^(5/6)*AppellF1(1/2,-4/3,1/6,3/2,1-cos(c+d*x),1/2*(1-cos(c+d*x)))*(a+a*cos(c+d*x))^(1/3)*sin(c+d*x)/(d*(1+cos(c+d*x))^(5/6))],
[cos(c+d*x)^(4/3)*(a+a*cos(c+d*x))^(2/3),x,3,2*2^(1/6)*AppellF1(1/2,-4/3,-1/6,3/2,1-cos(c+d*x),1/2*(1-cos(c+d*x)))*(a+a*cos(c+d*x))^(2/3)*sin(c+d*x)/(d*(1+cos(c+d*x))^(7/6))],
[cos(c+d*x)^(5/3)*(a+a*cos(c+d*x))^(2/3),x,3,2*2^(1/6)*AppellF1(1/2,-5/3,-1/6,3/2,1-cos(c+d*x),1/2*(1-cos(c+d*x)))*(a+a*cos(c+d*x))^(2/3)*sin(c+d*x)/(d*(1+cos(c+d*x))^(7/6))],

# n<0

# Integrands of the form Sec[e+f x]^(m/2) (a+a Cos[e+f x])^n

# n>0
[(a+a*cos(c+d*x))*sec(c+d*x)^(7/2),x,9,2/3*a*sec(c+d*x)^(3/2)*sin(c+d*x)/d+2/5*a*sec(c+d*x)^(5/2)*sin(c+d*x)/d+6/5*a*sin(c+d*x)*sqrt(sec(c+d*x))/d-6/5*a*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/d+2/3*a*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/d],
[(a+a*cos(c+d*x))*sec(c+d*x)^(5/2),x,8,2/3*a*sec(c+d*x)^(3/2)*sin(c+d*x)/d+2*a*sin(c+d*x)*sqrt(sec(c+d*x))/d-2*a*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/d+2/3*a*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/d],
[(a+a*cos(c+d*x))*sec(c+d*x)^(3/2),x,7,2*a*sin(c+d*x)*sqrt(sec(c+d*x))/d-2*a*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/d+2*a*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/d],
[(a+a*cos(c+d*x))*sec(c+d*x)^(1/2),x,6,2*a*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/d+2*a*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/d],
[(a+a*cos(c+d*x))/sec(c+d*x)^(1/2),x,7,2/3*a*sin(c+d*x)/(d*sqrt(sec(c+d*x)))+2*a*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/d+2/3*a*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/d],
[(a+a*cos(c+d*x))/sec(c+d*x)^(3/2),x,8,2/5*a*sin(c+d*x)/(d*sec(c+d*x)^(3/2))+2/3*a*sin(c+d*x)/(d*sqrt(sec(c+d*x)))+6/5*a*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/d+2/3*a*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/d],
[(a+a*cos(c+d*x))/sec(c+d*x)^(5/2),x,9,2/7*a*sin(c+d*x)/(d*sec(c+d*x)^(5/2))+2/5*a*sin(c+d*x)/(d*sec(c+d*x)^(3/2))+10/21*a*sin(c+d*x)/(d*sqrt(sec(c+d*x)))+6/5*a*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/d+10/21*a*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/d],
[(a+a*cos(c+d*x))^2*sec(c+d*x)^(7/2),x,9,4/3*a^2*sec(c+d*x)^(3/2)*sin(c+d*x)/d+2/5*a^2*sec(c+d*x)^(5/2)*sin(c+d*x)/d+16/5*a^2*sin(c+d*x)*sqrt(sec(c+d*x))/d-16/5*a^2*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/d+4/3*a^2*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/d],
[(a+a*cos(c+d*x))^2*sec(c+d*x)^(5/2),x,8,2/3*a^2*sec(c+d*x)^(3/2)*sin(c+d*x)/d+4*a^2*sin(c+d*x)*sqrt(sec(c+d*x))/d-4*a^2*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/d+8/3*a^2*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/d],
[(a+a*cos(c+d*x))^2*sec(c+d*x)^(3/2),x,5,2*a^2*sin(c+d*x)*sqrt(sec(c+d*x))/d+4*a^2*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/d],
[(a+a*cos(c+d*x))^2*sec(c+d*x)^(1/2),x,7,2/3*a^2*sin(c+d*x)/(d*sqrt(sec(c+d*x)))+4*a^2*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/d+8/3*a^2*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/d],
[(a+a*cos(c+d*x))^2/sec(c+d*x)^(1/2),x,8,2/5*a^2*sin(c+d*x)/(d*sec(c+d*x)^(3/2))+4/3*a^2*sin(c+d*x)/(d*sqrt(sec(c+d*x)))+16/5*a^2*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/d+4/3*a^2*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/d],
[(a+a*cos(c+d*x))^2/sec(c+d*x)^(3/2),x,9,2/7*a^2*sin(c+d*x)/(d*sec(c+d*x)^(5/2))+4/5*a^2*sin(c+d*x)/(d*sec(c+d*x)^(3/2))+8/7*a^2*sin(c+d*x)/(d*sqrt(sec(c+d*x)))+12/5*a^2*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/d+8/7*a^2*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/d],
[(a+a*cos(c+d*x))^3*sec(c+d*x)^(9/2),x,17,52/21*a^3*sec(c+d*x)^(3/2)*sin(c+d*x)/d+6/5*a^3*sec(c+d*x)^(5/2)*sin(c+d*x)/d+2/7*a^3*sec(c+d*x)^(7/2)*sin(c+d*x)/d+28/5*a^3*sin(c+d*x)*sqrt(sec(c+d*x))/d-28/5*a^3*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/d+52/21*a^3*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/d],
[(a+a*cos(c+d*x))^3*sec(c+d*x)^(7/2),x,15,2*a^3*sec(c+d*x)^(3/2)*sin(c+d*x)/d+2/5*a^3*sec(c+d*x)^(5/2)*sin(c+d*x)/d+36/5*a^3*sin(c+d*x)*sqrt(sec(c+d*x))/d-36/5*a^3*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/d+4*a^3*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/d],
[(a+a*cos(c+d*x))^3*sec(c+d*x)^(5/2),x,13,2/3*a^3*sec(c+d*x)^(3/2)*sin(c+d*x)/d+6*a^3*sin(c+d*x)*sqrt(sec(c+d*x))/d-4*a^3*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/d+20/3*a^3*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/d],
[(a+a*cos(c+d*x))^3*sec(c+d*x)^(3/2),x,13,2/3*a^3*sin(c+d*x)/(d*sqrt(sec(c+d*x)))+2*a^3*sin(c+d*x)*sqrt(sec(c+d*x))/d+4*a^3*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/d+20/3*a^3*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/d],
[(a+a*cos(c+d*x))^3*sec(c+d*x)^(1/2),x,13,2/5*a^3*sin(c+d*x)/(d*sec(c+d*x)^(3/2))+2*a^3*sin(c+d*x)/(d*sqrt(sec(c+d*x)))+36/5*a^3*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/d+4*a^3*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/d],
[(a+a*cos(c+d*x))^3/sec(c+d*x)^(1/2),x,15,2/7*a^3*sin(c+d*x)/(d*sec(c+d*x)^(5/2))+6/5*a^3*sin(c+d*x)/(d*sec(c+d*x)^(3/2))+52/21*a^3*sin(c+d*x)/(d*sqrt(sec(c+d*x)))+28/5*a^3*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/d+52/21*a^3*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/d],
[(a+a*cos(c+d*x))^3/sec(c+d*x)^(3/2),x,17,2/9*a^3*sin(c+d*x)/(d*sec(c+d*x)^(7/2))+6/7*a^3*sin(c+d*x)/(d*sec(c+d*x)^(5/2))+68/45*a^3*sin(c+d*x)/(d*sec(c+d*x)^(3/2))+44/21*a^3*sin(c+d*x)/(d*sqrt(sec(c+d*x)))+68/15*a^3*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/d+44/21*a^3*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/d],
[(a+a*cos(c+d*x))^4*sec(c+d*x)^(9/2),x,19,94/21*a^4*sec(c+d*x)^(3/2)*sin(c+d*x)/d+8/5*a^4*sec(c+d*x)^(5/2)*sin(c+d*x)/d+2/7*a^4*sec(c+d*x)^(7/2)*sin(c+d*x)/d+64/5*a^4*sin(c+d*x)*sqrt(sec(c+d*x))/d-64/5*a^4*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/d+136/21*a^4*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/d],
[(a+a*cos(c+d*x))^4*sec(c+d*x)^(7/2),x,17,8/3*a^4*sec(c+d*x)^(3/2)*sin(c+d*x)/d+2/5*a^4*sec(c+d*x)^(5/2)*sin(c+d*x)/d+66/5*a^4*sin(c+d*x)*sqrt(sec(c+d*x))/d-56/5*a^4*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/d+32/3*a^4*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/d],
[(a+a*cos(c+d*x))^4*sec(c+d*x)^(5/2),x,16,2/3*a^4*sec(c+d*x)^(3/2)*sin(c+d*x)/d+2/3*a^4*sin(c+d*x)/(d*sqrt(sec(c+d*x)))+8*a^4*sin(c+d*x)*sqrt(sec(c+d*x))/d+40/3*a^4*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/d],
[(a+a*cos(c+d*x))^4*sec(c+d*x)^(3/2),x,16,2/5*a^4*sin(c+d*x)/(d*sec(c+d*x)^(3/2))+8/3*a^4*sin(c+d*x)/(d*sqrt(sec(c+d*x)))+2*a^4*sin(c+d*x)*sqrt(sec(c+d*x))/d+56/5*a^4*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/d+32/3*a^4*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/d],
[(a+a*cos(c+d*x))^4*sec(c+d*x)^(1/2),x,17,2/7*a^4*sin(c+d*x)/(d*sec(c+d*x)^(5/2))+8/5*a^4*sin(c+d*x)/(d*sec(c+d*x)^(3/2))+94/21*a^4*sin(c+d*x)/(d*sqrt(sec(c+d*x)))+64/5*a^4*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/d+136/21*a^4*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/d],
[(a+a*cos(c+d*x))^4/sec(c+d*x)^(1/2),x,19,2/9*a^4*sin(c+d*x)/(d*sec(c+d*x)^(7/2))+8/7*a^4*sin(c+d*x)/(d*sec(c+d*x)^(5/2))+122/45*a^4*sin(c+d*x)/(d*sec(c+d*x)^(3/2))+32/7*a^4*sin(c+d*x)/(d*sqrt(sec(c+d*x)))+152/15*a^4*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/d+32/7*a^4*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/d],

# n<0
[sec(c+d*x)^(5/2)/(a+a*cos(c+d*x)),x,9,5/3*sec(c+d*x)^(3/2)*sin(c+d*x)/(a*d)-sec(c+d*x)^(5/2)*sin(c+d*x)/(d*(a+a*sec(c+d*x)))-3*sin(c+d*x)*sqrt(sec(c+d*x))/(a*d)+3*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(a*d)+5/3*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(a*d)],
[sec(c+d*x)^(3/2)/(a+a*cos(c+d*x)),x,8,-sec(c+d*x)^(3/2)*sin(c+d*x)/(d*(a+a*sec(c+d*x)))+3*sin(c+d*x)*sqrt(sec(c+d*x))/(a*d)-3*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(a*d)-sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(a*d)],
[sec(c+d*x)^(1/2)/(a+a*cos(c+d*x)),x,7,-sin(c+d*x)*sqrt(sec(c+d*x))/(d*(a+a*sec(c+d*x)))+sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(a*d)+sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(a*d)],
[1/((a+a*cos(c+d*x))*sec(c+d*x)^(1/2)),x,7,sin(c+d*x)*sqrt(sec(c+d*x))/(d*(a+a*sec(c+d*x)))-sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(a*d)+sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(a*d)],
[1/((a+a*cos(c+d*x))*sec(c+d*x)^(3/2)),x,7,-sin(c+d*x)*sqrt(sec(c+d*x))/(d*(a+a*sec(c+d*x)))+3*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(a*d)-sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(a*d)],
[1/((a+a*cos(c+d*x))*sec(c+d*x)^(5/2)),x,8,5/3*sin(c+d*x)/(a*d*sqrt(sec(c+d*x)))-sin(c+d*x)/(d*(a+a*sec(c+d*x))*sqrt(sec(c+d*x)))-3*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(a*d)+5/3*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(a*d)],
[1/((a+a*cos(c+d*x))*sec(c+d*x)^(7/2)),x,9,7/5*sin(c+d*x)/(a*d*sec(c+d*x)^(3/2))-sin(c+d*x)/(d*sec(c+d*x)^(3/2)*(a+a*sec(c+d*x)))-5/3*sin(c+d*x)/(a*d*sqrt(sec(c+d*x)))+21/5*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(a*d)-5/3*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(a*d)],
[sec(c+d*x)^(5/2)/(a+a*cos(c+d*x))^2,x,10,10/3*sec(c+d*x)^(3/2)*sin(c+d*x)/(a^2*d)-7/3*sec(c+d*x)^(5/2)*sin(c+d*x)/(a^2*d*(1+sec(c+d*x)))-1/3*sec(c+d*x)^(7/2)*sin(c+d*x)/(d*(a+a*sec(c+d*x))^2)-7*sin(c+d*x)*sqrt(sec(c+d*x))/(a^2*d)+7*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(a^2*d)+10/3*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(a^2*d)],
[sec(c+d*x)^(3/2)/(a+a*cos(c+d*x))^2,x,9,-5/3*sec(c+d*x)^(3/2)*sin(c+d*x)/(a^2*d*(1+sec(c+d*x)))-1/3*sec(c+d*x)^(5/2)*sin(c+d*x)/(d*(a+a*sec(c+d*x))^2)+4*sin(c+d*x)*sqrt(sec(c+d*x))/(a^2*d)-4*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(a^2*d)-5/3*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(a^2*d)],
[sec(c+d*x)^(1/2)/(a+a*cos(c+d*x))^2,x,8,-1/3*sec(c+d*x)^(3/2)*sin(c+d*x)/(d*(a+a*sec(c+d*x))^2)-sin(c+d*x)*sqrt(sec(c+d*x))/(a^2*d*(1+sec(c+d*x)))+sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(a^2*d)+2/3*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(a^2*d)],
[1/((a+a*cos(c+d*x))^2*sec(c+d*x)^(1/2)),x,5,1/3*sec(c+d*x)^(3/2)*sin(c+d*x)/(d*(a+a*sec(c+d*x))^2)+1/3*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(a^2*d)],
[1/((a+a*cos(c+d*x))^2*sec(c+d*x)^(3/2)),x,8,-1/3*sec(c+d*x)^(3/2)*sin(c+d*x)/(d*(a+a*sec(c+d*x))^2)+sin(c+d*x)*sqrt(sec(c+d*x))/(a^2*d*(1+sec(c+d*x)))-sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(a^2*d)+2/3*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(a^2*d)],
[1/((a+a*cos(c+d*x))^2*sec(c+d*x)^(5/2)),x,8,-5/3*sin(c+d*x)*sqrt(sec(c+d*x))/(a^2*d*(1+sec(c+d*x)))-1/3*sin(c+d*x)*sqrt(sec(c+d*x))/(d*(a+a*sec(c+d*x))^2)+4*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(a^2*d)-5/3*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(a^2*d)],
[1/((a+a*cos(c+d*x))^2*sec(c+d*x)^(7/2)),x,9,10/3*sin(c+d*x)/(a^2*d*sqrt(sec(c+d*x)))-7/3*sin(c+d*x)/(a^2*d*(1+sec(c+d*x))*sqrt(sec(c+d*x)))-1/3*sin(c+d*x)/(d*(a+a*sec(c+d*x))^2*sqrt(sec(c+d*x)))-7*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(a^2*d)+10/3*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(a^2*d)],
[1/((a+a*cos(c+d*x))^2*sec(c+d*x)^(9/2)),x,10,56/15*sin(c+d*x)/(a^2*d*sec(c+d*x)^(3/2))-3*sin(c+d*x)/(a^2*d*sec(c+d*x)^(3/2)*(1+sec(c+d*x)))-1/3*sin(c+d*x)/(d*sec(c+d*x)^(3/2)*(a+a*sec(c+d*x))^2)-5*sin(c+d*x)/(a^2*d*sqrt(sec(c+d*x)))+56/5*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(a^2*d)-5*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(a^2*d)],
[sec(c+d*x)^(3/2)/(a+a*cos(c+d*x))^3,x,10,-1/5*sec(c+d*x)^(7/2)*sin(c+d*x)/(d*(a+a*sec(c+d*x))^3)-8/15*sec(c+d*x)^(5/2)*sin(c+d*x)/(a*d*(a+a*sec(c+d*x))^2)-13/6*sec(c+d*x)^(3/2)*sin(c+d*x)/(d*(a^3+a^3*sec(c+d*x)))+49/10*sin(c+d*x)*sqrt(sec(c+d*x))/(a^3*d)-49/10*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(a^3*d)-13/6*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(a^3*d)],
[sec(c+d*x)^(1/2)/(a+a*cos(c+d*x))^3,x,9,-1/5*sec(c+d*x)^(5/2)*sin(c+d*x)/(d*(a+a*sec(c+d*x))^3)-2/5*sec(c+d*x)^(3/2)*sin(c+d*x)/(a*d*(a+a*sec(c+d*x))^2)-9/10*sin(c+d*x)*sqrt(sec(c+d*x))/(d*(a^3+a^3*sec(c+d*x)))+9/10*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(a^3*d)+1/2*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(a^3*d)],
[1/((a+a*cos(c+d*x))^3*sec(c+d*x)^(1/2)),x,9,-1/5*sec(c+d*x)^(3/2)*sin(c+d*x)/(d*(a+a*sec(c+d*x))^3)-4/15*sin(c+d*x)*sqrt(sec(c+d*x))/(a*d*(a+a*sec(c+d*x))^2)+1/6*sin(c+d*x)*sqrt(sec(c+d*x))/(d*(a^3+a^3*sec(c+d*x)))+1/10*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(a^3*d)+1/6*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(a^3*d)],
[1/((a+a*cos(c+d*x))^3*sec(c+d*x)^(3/2)),x,9,1/5*sec(c+d*x)^(3/2)*sin(c+d*x)/(d*(a+a*sec(c+d*x))^3)-1/15*sin(c+d*x)*sqrt(sec(c+d*x))/(a*d*(a+a*sec(c+d*x))^2)+1/6*sin(c+d*x)*sqrt(sec(c+d*x))/(d*(a^3+a^3*sec(c+d*x)))-1/10*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(a^3*d)+1/6*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(a^3*d)],
[1/((a+a*cos(c+d*x))^3*sec(c+d*x)^(5/2)),x,9,-1/5*sec(c+d*x)^(3/2)*sin(c+d*x)/(d*(a+a*sec(c+d*x))^3)+2/5*sin(c+d*x)*sqrt(sec(c+d*x))/(a*d*(a+a*sec(c+d*x))^2)+1/2*sin(c+d*x)*sqrt(sec(c+d*x))/(d*(a^3+a^3*sec(c+d*x)))-9/10*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(a^3*d)+1/2*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(a^3*d)],
[1/((a+a*cos(c+d*x))^3*sec(c+d*x)^(7/2)),x,9,-1/5*sin(c+d*x)*sqrt(sec(c+d*x))/(d*(a+a*sec(c+d*x))^3)-8/15*sin(c+d*x)*sqrt(sec(c+d*x))/(a*d*(a+a*sec(c+d*x))^2)-13/6*sin(c+d*x)*sqrt(sec(c+d*x))/(d*(a^3+a^3*sec(c+d*x)))+49/10*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(a^3*d)-13/6*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(a^3*d)],
[1/((a+a*cos(c+d*x))^3*sec(c+d*x)^(9/2)),x,10,11/2*sin(c+d*x)/(a^3*d*sqrt(sec(c+d*x)))-1/5*sin(c+d*x)/(d*(a+a*sec(c+d*x))^3*sqrt(sec(c+d*x)))-2/3*sin(c+d*x)/(a*d*(a+a*sec(c+d*x))^2*sqrt(sec(c+d*x)))-119/30*sin(c+d*x)/(d*(a^3+a^3*sec(c+d*x))*sqrt(sec(c+d*x)))-119/10*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(a^3*d)+11/2*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(a^3*d)],

# Integrands of the form Sec[e+f x]^(m/2) (a+a Cos[e+f x])^(n/2)

# n>0
[sec(c+d*x)^(9/2)*sqrt(a+a*cos(c+d*x)),x,5,16/35*a*sec(c+d*x)^(3/2)*sin(c+d*x)/(d*sqrt(a+a*cos(c+d*x)))+12/35*a*sec(c+d*x)^(5/2)*sin(c+d*x)/(d*sqrt(a+a*cos(c+d*x)))+2/7*a*sec(c+d*x)^(7/2)*sin(c+d*x)/(d*sqrt(a+a*cos(c+d*x)))+32/35*a*sin(c+d*x)*sqrt(sec(c+d*x))/(d*sqrt(a+a*cos(c+d*x)))],
[sec(c+d*x)^(7/2)*sqrt(a+a*cos(c+d*x)),x,4,8/15*a*sec(c+d*x)^(3/2)*sin(c+d*x)/(d*sqrt(a+a*cos(c+d*x)))+2/5*a*sec(c+d*x)^(5/2)*sin(c+d*x)/(d*sqrt(a+a*cos(c+d*x)))+16/15*a*sin(c+d*x)*sqrt(sec(c+d*x))/(d*sqrt(a+a*cos(c+d*x)))],
[sec(c+d*x)^(5/2)*sqrt(a+a*cos(c+d*x)),x,3,2/3*a*sec(c+d*x)^(3/2)*sin(c+d*x)/(d*sqrt(a+a*cos(c+d*x)))+4/3*a*sin(c+d*x)*sqrt(sec(c+d*x))/(d*sqrt(a+a*cos(c+d*x)))],
[sec(c+d*x)^(3/2)*sqrt(a+a*cos(c+d*x)),x,2,2*a*sin(c+d*x)*sqrt(sec(c+d*x))/(d*sqrt(a+a*cos(c+d*x)))],
[sec(c+d*x)^(1/2)*sqrt(a+a*cos(c+d*x)),x,3,2*arcsin(sin(c+d*x)*sqrt(a)/sqrt(a+a*cos(c+d*x)))*sqrt(a)*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/d],
[sqrt(a+a*cos(c+d*x))/sec(c+d*x)^(1/2),x,4,a*sin(c+d*x)/(d*sqrt(a+a*cos(c+d*x))*sqrt(sec(c+d*x)))+arcsin(sin(c+d*x)*sqrt(a)/sqrt(a+a*cos(c+d*x)))*sqrt(a)*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/d],
[sqrt(a+a*cos(c+d*x))/sec(c+d*x)^(3/2),x,5,1/2*a*sin(c+d*x)/(d*sec(c+d*x)^(3/2)*sqrt(a+a*cos(c+d*x)))+3/4*a*sin(c+d*x)/(d*sqrt(a+a*cos(c+d*x))*sqrt(sec(c+d*x)))+3/4*arcsin(sin(c+d*x)*sqrt(a)/sqrt(a+a*cos(c+d*x)))*sqrt(a)*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/d],
[(a+a*cos(c+d*x))^(3/2)*sec(c+d*x)^(9/2),x,6,104/105*a^2*sec(c+d*x)^(3/2)*sin(c+d*x)/(d*sqrt(a+a*cos(c+d*x)))+26/35*a^2*sec(c+d*x)^(5/2)*sin(c+d*x)/(d*sqrt(a+a*cos(c+d*x)))+2/7*a^2*sec(c+d*x)^(7/2)*sin(c+d*x)/(d*sqrt(a+a*cos(c+d*x)))+208/105*a^2*sin(c+d*x)*sqrt(sec(c+d*x))/(d*sqrt(a+a*cos(c+d*x)))],
[(a+a*cos(c+d*x))^(3/2)*sec(c+d*x)^(7/2),x,5,6/5*a^2*sec(c+d*x)^(3/2)*sin(c+d*x)/(d*sqrt(a+a*cos(c+d*x)))+2/5*a^2*sec(c+d*x)^(5/2)*sin(c+d*x)/(d*sqrt(a+a*cos(c+d*x)))+12/5*a^2*sin(c+d*x)*sqrt(sec(c+d*x))/(d*sqrt(a+a*cos(c+d*x)))],
[(a+a*cos(c+d*x))^(3/2)*sec(c+d*x)^(5/2),x,4,2/3*a^2*sec(c+d*x)^(3/2)*sin(c+d*x)/(d*sqrt(a+a*cos(c+d*x)))+10/3*a^2*sin(c+d*x)*sqrt(sec(c+d*x))/(d*sqrt(a+a*cos(c+d*x)))],
[(a+a*cos(c+d*x))^(3/2)*sec(c+d*x)^(3/2),x,5,2*a^(3/2)*arcsin(sin(c+d*x)*sqrt(a)/sqrt(a+a*cos(c+d*x)))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/d+2*a^2*sin(c+d*x)*sqrt(sec(c+d*x))/(d*sqrt(a+a*cos(c+d*x)))],
[(a+a*cos(c+d*x))^(3/2)*sec(c+d*x)^(1/2),x,5,a^2*sin(c+d*x)/(d*sqrt(a+a*cos(c+d*x))*sqrt(sec(c+d*x)))+3*a^(3/2)*arcsin(sin(c+d*x)*sqrt(a)/sqrt(a+a*cos(c+d*x)))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/d],
[(a+a*cos(c+d*x))^(3/2)/sec(c+d*x)^(1/2),x,6,1/2*a^2*sin(c+d*x)/(d*sec(c+d*x)^(3/2)*sqrt(a+a*cos(c+d*x)))+7/4*a^2*sin(c+d*x)/(d*sqrt(a+a*cos(c+d*x))*sqrt(sec(c+d*x)))+7/4*a^(3/2)*arcsin(sin(c+d*x)*sqrt(a)/sqrt(a+a*cos(c+d*x)))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/d],
[(a+a*cos(c+d*x))^(3/2)/sec(c+d*x)^(3/2),x,7,1/3*a^2*sin(c+d*x)/(d*sec(c+d*x)^(5/2)*sqrt(a+a*cos(c+d*x)))+11/12*a^2*sin(c+d*x)/(d*sec(c+d*x)^(3/2)*sqrt(a+a*cos(c+d*x)))+11/8*a^2*sin(c+d*x)/(d*sqrt(a+a*cos(c+d*x))*sqrt(sec(c+d*x)))+11/8*a^(3/2)*arcsin(sin(c+d*x)*sqrt(a)/sqrt(a+a*cos(c+d*x)))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/d],
[(a+a*cos(c+d*x))^(5/2)*sec(c+d*x)^(11/2),x,6,584/315*a^3*sec(c+d*x)^(3/2)*sin(c+d*x)/(d*sqrt(a+a*cos(c+d*x)))+146/105*a^3*sec(c+d*x)^(5/2)*sin(c+d*x)/(d*sqrt(a+a*cos(c+d*x)))+38/63*a^3*sec(c+d*x)^(7/2)*sin(c+d*x)/(d*sqrt(a+a*cos(c+d*x)))+2/9*a^2*sec(c+d*x)^(9/2)*sin(c+d*x)*sqrt(a+a*cos(c+d*x))/d+1168/315*a^3*sin(c+d*x)*sqrt(sec(c+d*x))/(d*sqrt(a+a*cos(c+d*x)))],
[(a+a*cos(c+d*x))^(5/2)*sec(c+d*x)^(9/2),x,5,46/21*a^3*sec(c+d*x)^(3/2)*sin(c+d*x)/(d*sqrt(a+a*cos(c+d*x)))+6/7*a^3*sec(c+d*x)^(5/2)*sin(c+d*x)/(d*sqrt(a+a*cos(c+d*x)))+2/7*a^2*sec(c+d*x)^(7/2)*sin(c+d*x)*sqrt(a+a*cos(c+d*x))/d+92/21*a^3*sin(c+d*x)*sqrt(sec(c+d*x))/(d*sqrt(a+a*cos(c+d*x)))],
[(a+a*cos(c+d*x))^(5/2)*sec(c+d*x)^(7/2),x,4,22/15*a^3*sec(c+d*x)^(3/2)*sin(c+d*x)/(d*sqrt(a+a*cos(c+d*x)))+2/5*a^2*sec(c+d*x)^(5/2)*sin(c+d*x)*sqrt(a+a*cos(c+d*x))/d+86/15*a^3*sin(c+d*x)*sqrt(sec(c+d*x))/(d*sqrt(a+a*cos(c+d*x)))],
[(a+a*cos(c+d*x))^(5/2)*sec(c+d*x)^(5/2),x,5,2/3*a^2*sec(c+d*x)^(3/2)*sin(c+d*x)*sqrt(a+a*cos(c+d*x))/d+2*a^(5/2)*arcsin(sin(c+d*x)*sqrt(a)/sqrt(a+a*cos(c+d*x)))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/d+14/3*a^3*sin(c+d*x)*sqrt(sec(c+d*x))/(d*sqrt(a+a*cos(c+d*x)))],
[(a+a*cos(c+d*x))^(5/2)*sec(c+d*x)^(3/2),x,5,-a^3*sin(c+d*x)/(d*sqrt(a+a*cos(c+d*x))*sqrt(sec(c+d*x)))+5*a^(5/2)*arcsin(sin(c+d*x)*sqrt(a)/sqrt(a+a*cos(c+d*x)))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/d+2*a^2*sin(c+d*x)*sqrt(a+a*cos(c+d*x))*sqrt(sec(c+d*x))/d],
[(a+a*cos(c+d*x))^(5/2)*sec(c+d*x)^(1/2),x,5,9/4*a^3*sin(c+d*x)/(d*sqrt(a+a*cos(c+d*x))*sqrt(sec(c+d*x)))+1/2*a^2*sin(c+d*x)*sqrt(a+a*cos(c+d*x))/(d*sqrt(sec(c+d*x)))+19/4*a^(5/2)*arcsin(sin(c+d*x)*sqrt(a)/sqrt(a+a*cos(c+d*x)))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/d],
[(a+a*cos(c+d*x))^(5/2)/sec(c+d*x)^(1/2),x,6,13/12*a^3*sin(c+d*x)/(d*sec(c+d*x)^(3/2)*sqrt(a+a*cos(c+d*x)))+1/3*a^2*sin(c+d*x)*sqrt(a+a*cos(c+d*x))/(d*sec(c+d*x)^(3/2))+25/8*a^3*sin(c+d*x)/(d*sqrt(a+a*cos(c+d*x))*sqrt(sec(c+d*x)))+25/8*a^(5/2)*arcsin(sin(c+d*x)*sqrt(a)/sqrt(a+a*cos(c+d*x)))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/d],
[(a+a*cos(c+d*x))^(5/2)/sec(c+d*x)^(3/2),x,7,17/24*a^3*sin(c+d*x)/(d*sec(c+d*x)^(5/2)*sqrt(a+a*cos(c+d*x)))+163/96*a^3*sin(c+d*x)/(d*sec(c+d*x)^(3/2)*sqrt(a+a*cos(c+d*x)))+1/4*a^2*sin(c+d*x)*sqrt(a+a*cos(c+d*x))/(d*sec(c+d*x)^(5/2))+163/64*a^3*sin(c+d*x)/(d*sqrt(a+a*cos(c+d*x))*sqrt(sec(c+d*x)))+163/64*a^(5/2)*arcsin(sin(c+d*x)*sqrt(a)/sqrt(a+a*cos(c+d*x)))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/d],

# n<0
[sec(c+d*x)^(7/2)/sqrt(1+cos(c+d*x)),x,7,-2/15*sec(c+d*x)^(3/2)*sin(c+d*x)/(d*sqrt(1+cos(c+d*x)))+2/5*sec(c+d*x)^(5/2)*sin(c+d*x)/(d*sqrt(1+cos(c+d*x)))-arcsin(sin(c+d*x)/(1+cos(c+d*x)))*sqrt(2)*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/d+26/15*sin(c+d*x)*sqrt(sec(c+d*x))/(d*sqrt(1+cos(c+d*x)))],
[sec(c+d*x)^(5/2)/sqrt(1+cos(c+d*x)),x,6,2/3*sec(c+d*x)^(3/2)*sin(c+d*x)/(d*sqrt(1+cos(c+d*x)))+arcsin(sin(c+d*x)/(1+cos(c+d*x)))*sqrt(2)*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/d-2/3*sin(c+d*x)*sqrt(sec(c+d*x))/(d*sqrt(1+cos(c+d*x)))],
[sec(c+d*x)^(3/2)/sqrt(1+cos(c+d*x)),x,4,-arcsin(sin(c+d*x)/(1+cos(c+d*x)))*sqrt(2)*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/d+2*sin(c+d*x)*sqrt(sec(c+d*x))/(d*sqrt(1+cos(c+d*x)))],
[sec(c+d*x)^(1/2)/sqrt(1+cos(c+d*x)),x,3,arcsin(sin(c+d*x)/(1+cos(c+d*x)))*sqrt(2)*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/d],
[1/(sec(c+d*x)^(1/2)*sqrt(1+cos(c+d*x))),x,6,2*arcsin(sin(c+d*x)/sqrt(1+cos(c+d*x)))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/d-arcsin(sin(c+d*x)/(1+cos(c+d*x)))*sqrt(2)*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/d],
[1/(sec(c+d*x)^(3/2)*sqrt(1+cos(c+d*x))),x,7,sin(c+d*x)/(d*sqrt(1+cos(c+d*x))*sqrt(sec(c+d*x)))-arcsin(sin(c+d*x)/sqrt(1+cos(c+d*x)))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/d+arcsin(sin(c+d*x)/(1+cos(c+d*x)))*sqrt(2)*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/d],
[sec(c+d*x)^(7/2)/sqrt(a+a*cos(c+d*x)),x,7,-2/15*sec(c+d*x)^(3/2)*sin(c+d*x)/(d*sqrt(a+a*cos(c+d*x)))+2/5*sec(c+d*x)^(5/2)*sin(c+d*x)/(d*sqrt(a+a*cos(c+d*x)))-arctan(sin(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(cos(c+d*x))*sqrt(a+a*cos(c+d*x))))*sqrt(2)*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(d*sqrt(a))+26/15*sin(c+d*x)*sqrt(sec(c+d*x))/(d*sqrt(a+a*cos(c+d*x)))],
[sec(c+d*x)^(5/2)/sqrt(a+a*cos(c+d*x)),x,6,2/3*sec(c+d*x)^(3/2)*sin(c+d*x)/(d*sqrt(a+a*cos(c+d*x)))+arctan(sin(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(cos(c+d*x))*sqrt(a+a*cos(c+d*x))))*sqrt(2)*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(d*sqrt(a))-2/3*sin(c+d*x)*sqrt(sec(c+d*x))/(d*sqrt(a+a*cos(c+d*x)))],
[sec(c+d*x)^(3/2)/sqrt(a+a*cos(c+d*x)),x,5,-arctan(sin(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(cos(c+d*x))*sqrt(a+a*cos(c+d*x))))*sqrt(2)*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(d*sqrt(a))+2*sin(c+d*x)*sqrt(sec(c+d*x))/(d*sqrt(a+a*cos(c+d*x)))],
[sec(c+d*x)^(1/2)/sqrt(a+a*cos(c+d*x)),x,3,arctan(sin(c+d*x)*sqrt(a)*sqrt(sec(c+d*x))/(sqrt(2)*sqrt(a+a*cos(c+d*x))))*sqrt(2)/(d*sqrt(a)),arctan(sin(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(cos(c+d*x))*sqrt(a+a*cos(c+d*x))))*sqrt(2)*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(d*sqrt(a))],
[1/(sec(c+d*x)^(1/2)*sqrt(a+a*cos(c+d*x))),x,6,2*arctan(sin(c+d*x)*sqrt(a)*sqrt(sec(c+d*x))/sqrt(a+a*cos(c+d*x)))/(d*sqrt(a))-arctan(sin(c+d*x)*sqrt(a)*sqrt(sec(c+d*x))/(sqrt(2)*sqrt(a+a*cos(c+d*x))))*sqrt(2)/(d*sqrt(a)),2*arcsin(sin(c+d*x)*sqrt(a)/sqrt(a+a*cos(c+d*x)))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(d*sqrt(a))-arctan(sin(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(cos(c+d*x))*sqrt(a+a*cos(c+d*x))))*sqrt(2)*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(d*sqrt(a))],
[1/(sec(c+d*x)^(3/2)*sqrt(a+a*cos(c+d*x))),x,7,sin(c+d*x)/(d*sqrt(a+a*cos(c+d*x))*sqrt(sec(c+d*x)))-arcsin(sin(c+d*x)*sqrt(a)/sqrt(a+a*cos(c+d*x)))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(d*sqrt(a))+arctan(sin(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(cos(c+d*x))*sqrt(a+a*cos(c+d*x))))*sqrt(2)*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(d*sqrt(a))],
[sec(c+d*x)^(5/2)/(a+a*cos(c+d*x))^(3/2),x,7,-1/2*sec(c+d*x)^(3/2)*sin(c+d*x)/(d*(a+a*cos(c+d*x))^(3/2))+7/6*sec(c+d*x)^(3/2)*sin(c+d*x)/(a*d*sqrt(a+a*cos(c+d*x)))+11/2*arctan(sin(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(cos(c+d*x))*sqrt(a+a*cos(c+d*x))))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(a^(3/2)*d*sqrt(2))-19/6*sin(c+d*x)*sqrt(sec(c+d*x))/(a*d*sqrt(a+a*cos(c+d*x)))],
[sec(c+d*x)^(3/2)/(a+a*cos(c+d*x))^(3/2),x,6,-1/2*sin(c+d*x)*sqrt(sec(c+d*x))/(d*(a+a*cos(c+d*x))^(3/2))-7/2*arctan(sin(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(cos(c+d*x))*sqrt(a+a*cos(c+d*x))))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(a^(3/2)*d*sqrt(2))+5/2*sin(c+d*x)*sqrt(sec(c+d*x))/(a*d*sqrt(a+a*cos(c+d*x)))],
[sec(c+d*x)^(1/2)/(a+a*cos(c+d*x))^(3/2),x,5,-1/2*sin(c+d*x)/(d*(a+a*cos(c+d*x))^(3/2)*sqrt(sec(c+d*x)))+3/2*arctan(sin(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(cos(c+d*x))*sqrt(a+a*cos(c+d*x))))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(a^(3/2)*d*sqrt(2))],
[1/((a+a*cos(c+d*x))^(3/2)*sec(c+d*x)^(1/2)),x,5,1/2*sin(c+d*x)/(d*(a+a*cos(c+d*x))^(3/2)*sqrt(sec(c+d*x)))+1/2*arctan(sin(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(cos(c+d*x))*sqrt(a+a*cos(c+d*x))))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(a^(3/2)*d*sqrt(2))],
[1/((a+a*cos(c+d*x))^(3/2)*sec(c+d*x)^(3/2)),x,7,-1/2*sin(c+d*x)/(d*(a+a*cos(c+d*x))^(3/2)*sqrt(sec(c+d*x)))+2*arcsin(sin(c+d*x)*sqrt(a)/sqrt(a+a*cos(c+d*x)))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(a^(3/2)*d)-5/2*arctan(sin(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(cos(c+d*x))*sqrt(a+a*cos(c+d*x))))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(a^(3/2)*d*sqrt(2))],
[1/((a+a*cos(c+d*x))^(3/2)*sec(c+d*x)^(5/2)),x,8,-1/2*sin(c+d*x)/(d*(a+a*cos(c+d*x))^(3/2)*sec(c+d*x)^(3/2))+3/2*sin(c+d*x)/(a*d*sqrt(a+a*cos(c+d*x))*sqrt(sec(c+d*x)))-3*arcsin(sin(c+d*x)*sqrt(a)/sqrt(a+a*cos(c+d*x)))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(a^(3/2)*d)+9/2*arctan(sin(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(cos(c+d*x))*sqrt(a+a*cos(c+d*x))))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(a^(3/2)*d*sqrt(2))],
[sec(c+d*x)^(5/2)/(a+a*cos(c+d*x))^(5/2),x,8,-1/4*sec(c+d*x)^(3/2)*sin(c+d*x)/(d*(a+a*cos(c+d*x))^(5/2))-17/16*sec(c+d*x)^(3/2)*sin(c+d*x)/(a*d*(a+a*cos(c+d*x))^(3/2))+95/48*sec(c+d*x)^(3/2)*sin(c+d*x)/(a^2*d*sqrt(a+a*cos(c+d*x)))+163/16*arctan(sin(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(cos(c+d*x))*sqrt(a+a*cos(c+d*x))))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(a^(5/2)*d*sqrt(2))-299/48*sin(c+d*x)*sqrt(sec(c+d*x))/(a^2*d*sqrt(a+a*cos(c+d*x)))],
[sec(c+d*x)^(3/2)/(a+a*cos(c+d*x))^(5/2),x,7,-1/4*sin(c+d*x)*sqrt(sec(c+d*x))/(d*(a+a*cos(c+d*x))^(5/2))-13/16*sin(c+d*x)*sqrt(sec(c+d*x))/(a*d*(a+a*cos(c+d*x))^(3/2))-75/16*arctan(sin(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(cos(c+d*x))*sqrt(a+a*cos(c+d*x))))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(a^(5/2)*d*sqrt(2))+49/16*sin(c+d*x)*sqrt(sec(c+d*x))/(a^2*d*sqrt(a+a*cos(c+d*x)))],
[sec(c+d*x)^(1/2)/(a+a*cos(c+d*x))^(5/2),x,6,-1/4*sin(c+d*x)/(d*(a+a*cos(c+d*x))^(5/2)*sqrt(sec(c+d*x)))-9/16*sin(c+d*x)/(a*d*(a+a*cos(c+d*x))^(3/2)*sqrt(sec(c+d*x)))+19/16*arctan(sin(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(cos(c+d*x))*sqrt(a+a*cos(c+d*x))))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(a^(5/2)*d*sqrt(2))],
[1/((a+a*cos(c+d*x))^(5/2)*sec(c+d*x)^(1/2)),x,6,1/4*sin(c+d*x)/(d*(a+a*cos(c+d*x))^(5/2)*sqrt(sec(c+d*x)))+1/16*sin(c+d*x)/(a*d*(a+a*cos(c+d*x))^(3/2)*sqrt(sec(c+d*x)))+5/16*arctan(sin(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(cos(c+d*x))*sqrt(a+a*cos(c+d*x))))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(a^(5/2)*d*sqrt(2))],
[1/((a+a*cos(c+d*x))^(5/2)*sec(c+d*x)^(3/2)),x,6,-1/4*sin(c+d*x)/(d*(a+a*cos(c+d*x))^(5/2)*sqrt(sec(c+d*x)))+7/16*sin(c+d*x)/(a*d*(a+a*cos(c+d*x))^(3/2)*sqrt(sec(c+d*x)))+3/16*arctan(sin(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(cos(c+d*x))*sqrt(a+a*cos(c+d*x))))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(a^(5/2)*d*sqrt(2))],
[1/((a+a*cos(c+d*x))^(5/2)*sec(c+d*x)^(5/2)),x,8,-1/4*sin(c+d*x)/(d*(a+a*cos(c+d*x))^(5/2)*sec(c+d*x)^(3/2))-11/16*sin(c+d*x)/(a*d*(a+a*cos(c+d*x))^(3/2)*sqrt(sec(c+d*x)))+2*arcsin(sin(c+d*x)*sqrt(a)/sqrt(a+a*cos(c+d*x)))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(a^(5/2)*d)-43/16*arctan(sin(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(cos(c+d*x))*sqrt(a+a*cos(c+d*x))))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(a^(5/2)*d*sqrt(2))],
[1/((a+a*cos(c+d*x))^(5/2)*sec(c+d*x)^(7/2)),x,9,-1/4*sin(c+d*x)/(d*(a+a*cos(c+d*x))^(5/2)*sec(c+d*x)^(5/2))-15/16*sin(c+d*x)/(a*d*(a+a*cos(c+d*x))^(3/2)*sec(c+d*x)^(3/2))+35/16*sin(c+d*x)/(a^2*d*sqrt(a+a*cos(c+d*x))*sqrt(sec(c+d*x)))-5*arcsin(sin(c+d*x)*sqrt(a)/sqrt(a+a*cos(c+d*x)))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(a^(5/2)*d)+115/16*arctan(sin(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(cos(c+d*x))*sqrt(a+a*cos(c+d*x))))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(a^(5/2)*d*sqrt(2))],
[sec(c+d*x)^(5/2)/(a+a*cos(c+d*x))^(7/2),x,9,-1/6*sec(c+d*x)^(3/2)*sin(c+d*x)/(d*(a+a*cos(c+d*x))^(7/2))-23/48*sec(c+d*x)^(3/2)*sin(c+d*x)/(a*d*(a+a*cos(c+d*x))^(5/2))-109/64*sec(c+d*x)^(3/2)*sin(c+d*x)/(a^2*d*(a+a*cos(c+d*x))^(3/2))+193/64*sec(c+d*x)^(3/2)*sin(c+d*x)/(a^3*d*sqrt(a+a*cos(c+d*x)))+1015/64*arctan(sin(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(cos(c+d*x))*sqrt(a+a*cos(c+d*x))))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(a^(7/2)*d*sqrt(2))-629/64*sin(c+d*x)*sqrt(sec(c+d*x))/(a^3*d*sqrt(a+a*cos(c+d*x)))],
[sec(c+d*x)^(3/2)/(a+a*cos(c+d*x))^(7/2),x,8,-1/6*sin(c+d*x)*sqrt(sec(c+d*x))/(d*(a+a*cos(c+d*x))^(7/2))-19/48*sin(c+d*x)*sqrt(sec(c+d*x))/(a*d*(a+a*cos(c+d*x))^(5/2))-199/192*sin(c+d*x)*sqrt(sec(c+d*x))/(a^2*d*(a+a*cos(c+d*x))^(3/2))-363/64*arctan(sin(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(cos(c+d*x))*sqrt(a+a*cos(c+d*x))))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(a^(7/2)*d*sqrt(2))+691/192*sin(c+d*x)*sqrt(sec(c+d*x))/(a^3*d*sqrt(a+a*cos(c+d*x)))],
[sec(c+d*x)^(1/2)/(a+a*cos(c+d*x))^(7/2),x,7,-1/6*sin(c+d*x)/(d*(a+a*cos(c+d*x))^(7/2)*sqrt(sec(c+d*x)))-5/16*sin(c+d*x)/(a*d*(a+a*cos(c+d*x))^(5/2)*sqrt(sec(c+d*x)))-103/192*sin(c+d*x)/(a^2*d*(a+a*cos(c+d*x))^(3/2)*sqrt(sec(c+d*x)))+63/64*arctan(sin(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(cos(c+d*x))*sqrt(a+a*cos(c+d*x))))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(a^(7/2)*d*sqrt(2))],
[1/((a+a*cos(c+d*x))^(7/2)*sec(c+d*x)^(1/2)),x,7,1/6*sin(c+d*x)/(d*(a+a*cos(c+d*x))^(7/2)*sqrt(sec(c+d*x)))+1/16*sin(c+d*x)/(a*d*(a+a*cos(c+d*x))^(5/2)*sqrt(sec(c+d*x)))-5/192*sin(c+d*x)/(a^2*d*(a+a*cos(c+d*x))^(3/2)*sqrt(sec(c+d*x)))+13/64*arctan(sin(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(cos(c+d*x))*sqrt(a+a*cos(c+d*x))))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(a^(7/2)*d*sqrt(2))],
[1/((a+a*cos(c+d*x))^(7/2)*sec(c+d*x)^(3/2)),x,7,-1/6*sin(c+d*x)/(d*(a+a*cos(c+d*x))^(7/2)*sqrt(sec(c+d*x)))+3/16*sin(c+d*x)/(a*d*(a+a*cos(c+d*x))^(5/2)*sqrt(sec(c+d*x)))+17/192*sin(c+d*x)/(a^2*d*(a+a*cos(c+d*x))^(3/2)*sqrt(sec(c+d*x)))+7/64*arctan(sin(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(cos(c+d*x))*sqrt(a+a*cos(c+d*x))))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(a^(7/2)*d*sqrt(2))],
[1/((a+a*cos(c+d*x))^(7/2)*sec(c+d*x)^(5/2)),x,7,-1/6*sin(c+d*x)/(d*(a+a*cos(c+d*x))^(7/2)*sec(c+d*x)^(3/2))-13/48*sin(c+d*x)/(a*d*(a+a*cos(c+d*x))^(5/2)*sqrt(sec(c+d*x)))+67/192*sin(c+d*x)/(a^2*d*(a+a*cos(c+d*x))^(3/2)*sqrt(sec(c+d*x)))+5/64*arctan(sin(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(cos(c+d*x))*sqrt(a+a*cos(c+d*x))))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(a^(7/2)*d*sqrt(2))],
[1/((a+a*cos(c+d*x))^(7/2)*sec(c+d*x)^(7/2)),x,9,-1/6*sin(c+d*x)/(d*(a+a*cos(c+d*x))^(7/2)*sec(c+d*x)^(5/2))-17/48*sin(c+d*x)/(a*d*(a+a*cos(c+d*x))^(5/2)*sec(c+d*x)^(3/2))-49/64*sin(c+d*x)/(a^2*d*(a+a*cos(c+d*x))^(3/2)*sqrt(sec(c+d*x)))+2*arcsin(sin(c+d*x)*sqrt(a)/sqrt(a+a*cos(c+d*x)))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(a^(7/2)*d)-177/64*arctan(sin(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(cos(c+d*x))*sqrt(a+a*cos(c+d*x))))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(a^(7/2)*d*sqrt(2))],
[1/((a+a*cos(c+d*x))^(7/2)*sec(c+d*x)^(9/2)),x,10,-1/6*sin(c+d*x)/(d*(a+a*cos(c+d*x))^(7/2)*sec(c+d*x)^(7/2))-7/16*sin(c+d*x)/(a*d*(a+a*cos(c+d*x))^(5/2)*sec(c+d*x)^(5/2))-259/192*sin(c+d*x)/(a^2*d*(a+a*cos(c+d*x))^(3/2)*sec(c+d*x)^(3/2))+189/64*sin(c+d*x)/(a^3*d*sqrt(a+a*cos(c+d*x))*sqrt(sec(c+d*x)))-7*arcsin(sin(c+d*x)*sqrt(a)/sqrt(a+a*cos(c+d*x)))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(a^(7/2)*d)+637/64*arctan(sin(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(cos(c+d*x))*sqrt(a+a*cos(c+d*x))))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(a^(7/2)*d*sqrt(2))],
[1/((a+a*cos(c+d*x))^(9/2)*sec(c+d*x)^(5/2)),x,8,-1/8*sin(c+d*x)/(d*(a+a*cos(c+d*x))^(9/2)*sec(c+d*x)^(3/2))-5/32*sin(c+d*x)/(a*d*(a+a*cos(c+d*x))^(7/2)*sqrt(sec(c+d*x)))+33/256*sin(c+d*x)/(a^2*d*(a+a*cos(c+d*x))^(5/2)*sqrt(sec(c+d*x)))+73/1024*sin(c+d*x)/(a^3*d*(a+a*cos(c+d*x))^(3/2)*sqrt(sec(c+d*x)))+45/1024*arctan(sin(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(cos(c+d*x))*sqrt(a+a*cos(c+d*x))))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(a^(9/2)*d*sqrt(2))],
[1/((a+a*cos(c+d*x))^(9/2)*sec(c+d*x)^(7/2)),x,8,-1/8*sin(c+d*x)/(d*(a+a*cos(c+d*x))^(9/2)*sec(c+d*x)^(5/2))-19/96*sin(c+d*x)/(a*d*(a+a*cos(c+d*x))^(7/2)*sec(c+d*x)^(3/2))-187/768*sin(c+d*x)/(a^2*d*(a+a*cos(c+d*x))^(5/2)*sqrt(sec(c+d*x)))+853/3072*sin(c+d*x)/(a^3*d*(a+a*cos(c+d*x))^(3/2)*sqrt(sec(c+d*x)))+35/1024*arctan(sin(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(cos(c+d*x))*sqrt(a+a*cos(c+d*x))))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(a^(9/2)*d*sqrt(2))],

# Integrands of the form Sec[e+f x]^(m/4) / (a+a Cos[e+f x])^(n/2)
[(a+a*cos(c+d*x))^(3/2)*sec(c+d*x)^(5/4),x,3,4*a^2*sec(c+d*x)^(1/4)*sin(c+d*x)/(d*sqrt(a+a*cos(c+d*x)))],

# Integrands of the form Cos[e+f x]^m (a+a Cos[e+f x])^n with m and/or n symbolic
[cos(c+d*x)^m*(a+a*cos(c+d*x))^4,x,7,a^4*(55+29*m+4*m^2)*cos(c+d*x)^(1+m)*sin(c+d*x)/(d*(4+m)*(6+5*m+m^2))+cos(c+d*x)^(1+m)*(a^2+a^2*cos(c+d*x))^2*sin(c+d*x)/(d*(4+m))+2*(5+m)*cos(c+d*x)^(1+m)*(a^4+a^4*cos(c+d*x))*sin(c+d*x)/(d*(3+m)*(4+m))-a^4*(35+40*m+8*m^2)*cos(c+d*x)^(1+m)*hypergeom([1/2,1/2*(1+m)],[1/2*(3+m)],cos(c+d*x)^2)*sin(c+d*x)/(d*(8+14*m+7*m^2+m^3)*sqrt(sin(c+d*x)^2))-4*a^4*(5+2*m)*cos(c+d*x)^(2+m)*hypergeom([1/2,1/2*(2+m)],[1/2*(4+m)],cos(c+d*x)^2)*sin(c+d*x)/(d*(2+m)*(3+m)*sqrt(sin(c+d*x)^2))],
[cos(c+d*x)^m*(a+a*cos(c+d*x))^3,x,6,a^3*(7+2*m)*cos(c+d*x)^(1+m)*sin(c+d*x)/(d*(2+m)*(3+m))+cos(c+d*x)^(1+m)*(a^3+a^3*cos(c+d*x))*sin(c+d*x)/(d*(3+m))-a^3*(5+4*m)*cos(c+d*x)^(1+m)*hypergeom([1/2,1/2*(1+m)],[1/2*(3+m)],cos(c+d*x)^2)*sin(c+d*x)/(d*(1+m)*(2+m)*sqrt(sin(c+d*x)^2))-a^3*(11+4*m)*cos(c+d*x)^(2+m)*hypergeom([1/2,1/2*(2+m)],[1/2*(4+m)],cos(c+d*x)^2)*sin(c+d*x)/(d*(2+m)*(3+m)*sqrt(sin(c+d*x)^2))],
[cos(c+d*x)^m*(a+a*cos(c+d*x))^2,x,4,a^2*cos(c+d*x)^(1+m)*sin(c+d*x)/(d*(2+m))-a^2*(3+2*m)*cos(c+d*x)^(1+m)*hypergeom([1/2,1/2*(1+m)],[1/2*(3+m)],cos(c+d*x)^2)*sin(c+d*x)/(d*(1+m)*(2+m)*sqrt(sin(c+d*x)^2))-2*a^2*cos(c+d*x)^(2+m)*hypergeom([1/2,1/2*(2+m)],[1/2*(4+m)],cos(c+d*x)^2)*sin(c+d*x)/(d*(2+m)*sqrt(sin(c+d*x)^2))],
[cos(c+d*x)^m*(a+a*cos(c+d*x)),x,3,-a*cos(c+d*x)^(1+m)*hypergeom([1/2,1/2*(1+m)],[1/2*(3+m)],cos(c+d*x)^2)*sin(c+d*x)/(d*(1+m)*sqrt(sin(c+d*x)^2))-a*cos(c+d*x)^(2+m)*hypergeom([1/2,1/2*(2+m)],[1/2*(4+m)],cos(c+d*x)^2)*sin(c+d*x)/(d*(2+m)*sqrt(sin(c+d*x)^2))],
[cos(c+d*x)^m/(a+a*cos(c+d*x)),x,4,cos(c+d*x)^m*sin(c+d*x)/(d*(a+a*cos(c+d*x)))-cos(c+d*x)^m*hypergeom([1/2,1/2*m],[1/2*(2+m)],cos(c+d*x)^2)*sin(c+d*x)/(a*d*sqrt(sin(c+d*x)^2))+m*cos(c+d*x)^(1+m)*hypergeom([1/2,1/2*(1+m)],[1/2*(3+m)],cos(c+d*x)^2)*sin(c+d*x)/(a*d*(1+m)*sqrt(sin(c+d*x)^2))],
[cos(c+d*x)^m/(a+a*cos(c+d*x))^2,x,5,-2/3*(1-m)*cos(c+d*x)^(1+m)*sin(c+d*x)/(a^2*d*(1+cos(c+d*x)))-1/3*cos(c+d*x)^(1+m)*sin(c+d*x)/(d*(a+a*cos(c+d*x))^2)+1/3*(1-2*m)*m*cos(c+d*x)^(1+m)*hypergeom([1/2,1/2*(1+m)],[1/2*(3+m)],cos(c+d*x)^2)*sin(c+d*x)/(a^2*d*(1+m)*sqrt(sin(c+d*x)^2))-2/3*(1-m)*(1+m)*cos(c+d*x)^(2+m)*hypergeom([1/2,1/2*(2+m)],[1/2*(4+m)],cos(c+d*x)^2)*sin(c+d*x)/(a^2*d*(2+m)*sqrt(sin(c+d*x)^2))],

# Integrands of the form (d Cos[e+f x])^n (a+b Cos[e+f x])^m

# Integrands of the form Cos[e+f x]^m (a+b Cos[e+f x])^n

# n>0
[cos(c+d*x)^7*(a+b*cos(c+d*x)),x,8,35/128*b*x+a*sin(c+d*x)/d+35/128*b*cos(c+d*x)*sin(c+d*x)/d+35/192*b*cos(c+d*x)^3*sin(c+d*x)/d+7/48*b*cos(c+d*x)^5*sin(c+d*x)/d+1/8*b*cos(c+d*x)^7*sin(c+d*x)/d-a*sin(c+d*x)^3/d+3/5*a*sin(c+d*x)^5/d-1/7*a*sin(c+d*x)^7/d],
[cos(c+d*x)^6*(a+b*cos(c+d*x)),x,7,5/16*a*x+b*sin(c+d*x)/d+5/16*a*cos(c+d*x)*sin(c+d*x)/d+5/24*a*cos(c+d*x)^3*sin(c+d*x)/d+1/6*a*cos(c+d*x)^5*sin(c+d*x)/d-b*sin(c+d*x)^3/d+3/5*b*sin(c+d*x)^5/d-1/7*b*sin(c+d*x)^7/d],
[cos(c+d*x)^5*(a+b*cos(c+d*x)),x,7,5/16*b*x+a*sin(c+d*x)/d+5/16*b*cos(c+d*x)*sin(c+d*x)/d+5/24*b*cos(c+d*x)^3*sin(c+d*x)/d+1/6*b*cos(c+d*x)^5*sin(c+d*x)/d-2/3*a*sin(c+d*x)^3/d+1/5*a*sin(c+d*x)^5/d],
[cos(c+d*x)^4*(a+b*cos(c+d*x)),x,6,3/8*a*x+b*sin(c+d*x)/d+3/8*a*cos(c+d*x)*sin(c+d*x)/d+1/4*a*cos(c+d*x)^3*sin(c+d*x)/d-2/3*b*sin(c+d*x)^3/d+1/5*b*sin(c+d*x)^5/d],
[cos(c+d*x)^3*(a+b*cos(c+d*x)),x,6,3/8*b*x+a*sin(c+d*x)/d+3/8*b*cos(c+d*x)*sin(c+d*x)/d+1/4*b*cos(c+d*x)^3*sin(c+d*x)/d-1/3*a*sin(c+d*x)^3/d],
[cos(c+d*x)^2*(a+b*cos(c+d*x)),x,5,1/2*a*x+b*sin(c+d*x)/d+1/2*a*cos(c+d*x)*sin(c+d*x)/d-1/3*b*sin(c+d*x)^3/d],
[cos(c+d*x)*(a+b*cos(c+d*x)),x,1,1/2*b*x+a*sin(c+d*x)/d+1/2*b*cos(c+d*x)*sin(c+d*x)/d],
[a+b*cos(c+d*x),x,2,a*x+b*sin(c+d*x)/d],
[(a+b*cos(c+d*x))*sec(c+d*x),x,2,b*x+a*arctanh(sin(c+d*x))/d],
[(a+b*cos(c+d*x))*sec(c+d*x)^2,x,4,b*arctanh(sin(c+d*x))/d+a*tan(c+d*x)/d],
[(a+b*cos(c+d*x))*sec(c+d*x)^3,x,5,1/2*a*arctanh(sin(c+d*x))/d+b*tan(c+d*x)/d+1/2*a*sec(c+d*x)*tan(c+d*x)/d],
[(a+b*cos(c+d*x))*sec(c+d*x)^4,x,5,1/2*b*arctanh(sin(c+d*x))/d+a*tan(c+d*x)/d+1/2*b*sec(c+d*x)*tan(c+d*x)/d+1/3*a*tan(c+d*x)^3/d],
[(a+b*cos(c+d*x))*sec(c+d*x)^5,x,6,3/8*a*arctanh(sin(c+d*x))/d+b*tan(c+d*x)/d+3/8*a*sec(c+d*x)*tan(c+d*x)/d+1/4*a*sec(c+d*x)^3*tan(c+d*x)/d+1/3*b*tan(c+d*x)^3/d],
[(a+b*cos(c+d*x))*sec(c+d*x)^6,x,6,3/8*b*arctanh(sin(c+d*x))/d+a*tan(c+d*x)/d+3/8*b*sec(c+d*x)*tan(c+d*x)/d+1/4*b*sec(c+d*x)^3*tan(c+d*x)/d+2/3*a*tan(c+d*x)^3/d+1/5*a*tan(c+d*x)^5/d],
[cos(c+d*x)^4*(a+b*cos(c+d*x))^2,x,7,1/16*(6*a^2+5*b^2)*x+2*a*b*sin(c+d*x)/d+1/16*(6*a^2+5*b^2)*cos(c+d*x)*sin(c+d*x)/d+1/24*(6*a^2+5*b^2)*cos(c+d*x)^3*sin(c+d*x)/d+1/6*b^2*cos(c+d*x)^5*sin(c+d*x)/d-4/3*a*b*sin(c+d*x)^3/d+2/5*a*b*sin(c+d*x)^5/d],
[cos(c+d*x)^3*(a+b*cos(c+d*x))^2,x,7,3/4*a*b*x+(a^2+b^2)*sin(c+d*x)/d+3/4*a*b*cos(c+d*x)*sin(c+d*x)/d+1/2*a*b*cos(c+d*x)^3*sin(c+d*x)/d-1/3*(a^2+2*b^2)*sin(c+d*x)^3/d+1/5*b^2*sin(c+d*x)^5/d],
[cos(c+d*x)^2*(a+b*cos(c+d*x))^2,x,6,1/8*(4*a^2+3*b^2)*x+2*a*b*sin(c+d*x)/d+1/8*(4*a^2+3*b^2)*cos(c+d*x)*sin(c+d*x)/d+1/4*b^2*cos(c+d*x)^3*sin(c+d*x)/d-2/3*a*b*sin(c+d*x)^3/d],
[cos(c+d*x)*(a+b*cos(c+d*x))^2,x,2,a*b*x+2/3*(a^2+b^2)*sin(c+d*x)/d+1/3*a*b*cos(c+d*x)*sin(c+d*x)/d+1/3*(a+b*cos(c+d*x))^2*sin(c+d*x)/d],
[(a+b*cos(c+d*x))^2,x,1,1/2*(2*a^2+b^2)*x+2*a*b*sin(c+d*x)/d+1/2*b^2*cos(c+d*x)*sin(c+d*x)/d],
[(a+b*cos(c+d*x))^2*sec(c+d*x),x,3,2*a*b*x+a^2*arctanh(sin(c+d*x))/d+b^2*sin(c+d*x)/d],
[(a+b*cos(c+d*x))^2*sec(c+d*x)^2,x,4,b^2*x+2*a*b*arctanh(sin(c+d*x))/d+a^2*tan(c+d*x)/d],
[(a+b*cos(c+d*x))^2*sec(c+d*x)^3,x,5,1/2*(a^2+2*b^2)*arctanh(sin(c+d*x))/d+2*a*b*tan(c+d*x)/d+1/2*a^2*sec(c+d*x)*tan(c+d*x)/d],
[(a+b*cos(c+d*x))^2*sec(c+d*x)^4,x,6,a*b*arctanh(sin(c+d*x))/d+1/3*(2*a^2+3*b^2)*tan(c+d*x)/d+a*b*sec(c+d*x)*tan(c+d*x)/d+1/3*a^2*sec(c+d*x)^2*tan(c+d*x)/d],
[(a+b*cos(c+d*x))^2*sec(c+d*x)^5,x,6,1/8*(3*a^2+4*b^2)*arctanh(sin(c+d*x))/d+2*a*b*tan(c+d*x)/d+1/8*(3*a^2+4*b^2)*sec(c+d*x)*tan(c+d*x)/d+1/4*a^2*sec(c+d*x)^3*tan(c+d*x)/d+2/3*a*b*tan(c+d*x)^3/d],
[(a+b*cos(c+d*x))^2*sec(c+d*x)^6,x,7,3/4*a*b*arctanh(sin(c+d*x))/d+1/5*(4*a^2+5*b^2)*tan(c+d*x)/d+3/4*a*b*sec(c+d*x)*tan(c+d*x)/d+1/2*a*b*sec(c+d*x)^3*tan(c+d*x)/d+1/5*a^2*sec(c+d*x)^4*tan(c+d*x)/d+1/15*(4*a^2+5*b^2)*tan(c+d*x)^3/d],
[cos(c+d*x)^3*(a+b*cos(c+d*x))^3,x,8,9/8*a^2*b*x+5/16*b^3*x+a*(a^2+3*b^2)*sin(c+d*x)/d+1/16*b*(18*a^2+5*b^2)*cos(c+d*x)*sin(c+d*x)/d+1/24*b*(18*a^2+5*b^2)*cos(c+d*x)^3*sin(c+d*x)/d+1/6*b^3*cos(c+d*x)^5*sin(c+d*x)/d-1/3*a*(a^2+6*b^2)*sin(c+d*x)^3/d+3/5*a*b^2*sin(c+d*x)^5/d,1/16*b*(18*a^2+5*b^2)*x+1/5*a*(5*a^2+12*b^2)*sin(c+d*x)/d+1/16*b*(18*a^2+5*b^2)*cos(c+d*x)*sin(c+d*x)/d+1/24*b*(18*a^2+5*b^2)*cos(c+d*x)^3*sin(c+d*x)/d+13/30*a*b^2*cos(c+d*x)^4*sin(c+d*x)/d+1/6*b^2*cos(c+d*x)^4*(a+b*cos(c+d*x))*sin(c+d*x)/d-1/15*a*(5*a^2+12*b^2)*sin(c+d*x)^3/d],
[cos(c+d*x)^2*(a+b*cos(c+d*x))^3,x,4,1/8*a*(4*a^2+9*b^2)*x-1/30*(3*a^4-52*a^2*b^2-16*b^4)*sin(c+d*x)/(b*d)-1/120*a*(6*a^2-71*b^2)*cos(c+d*x)*sin(c+d*x)/d-1/60*(3*a^2-16*b^2)*(a+b*cos(c+d*x))^2*sin(c+d*x)/(b*d)-1/20*a*(a+b*cos(c+d*x))^3*sin(c+d*x)/(b*d)+1/5*(a+b*cos(c+d*x))^4*sin(c+d*x)/(b*d)],
[cos(c+d*x)*(a+b*cos(c+d*x))^3,x,3,3/8*b*(4*a^2+b^2)*x+1/2*a*(a^2+4*b^2)*sin(c+d*x)/d+1/8*b*(2*a^2+3*b^2)*cos(c+d*x)*sin(c+d*x)/d+1/4*a*(a+b*cos(c+d*x))^2*sin(c+d*x)/d+1/4*(a+b*cos(c+d*x))^3*sin(c+d*x)/d],
[(a+b*cos(c+d*x))^3,x,2,a^3*x+3/2*a*b^2*x+b*(3*a^2+b^2)*sin(c+d*x)/d+3/2*a*b^2*cos(c+d*x)*sin(c+d*x)/d-1/3*b^3*sin(c+d*x)^3/d,1/2*a*(2*a^2+3*b^2)*x+2/3*b*(4*a^2+b^2)*sin(c+d*x)/d+5/6*a*b^2*cos(c+d*x)*sin(c+d*x)/d+1/3*b*(a+b*cos(c+d*x))^2*sin(c+d*x)/d],
[(a+b*cos(c+d*x))^3*sec(c+d*x),x,4,1/2*b*(6*a^2+b^2)*x+a^3*arctanh(sin(c+d*x))/d+5/2*a*b^2*sin(c+d*x)/d+1/2*b^2*(a+b*cos(c+d*x))*sin(c+d*x)/d],
[(a+b*cos(c+d*x))^3*sec(c+d*x)^2,x,4,3*a*b^2*x+3*a^2*b*arctanh(sin(c+d*x))/d-b*(a^2-b^2)*sin(c+d*x)/d+a^2*(a+b*cos(c+d*x))*tan(c+d*x)/d],
[(a+b*cos(c+d*x))^3*sec(c+d*x)^3,x,4,b^3*x+1/2*a*(a^2+6*b^2)*arctanh(sin(c+d*x))/d+5/2*a^2*b*tan(c+d*x)/d+1/2*a^2*(a+b*cos(c+d*x))*sec(c+d*x)*tan(c+d*x)/d],
[(a+b*cos(c+d*x))^3*sec(c+d*x)^4,x,6,1/2*b*(3*a^2+2*b^2)*arctanh(sin(c+d*x))/d+1/3*a*(2*a^2+9*b^2)*tan(c+d*x)/d+7/6*a^2*b*sec(c+d*x)*tan(c+d*x)/d+1/3*a^2*(a+b*cos(c+d*x))*sec(c+d*x)^2*tan(c+d*x)/d],
[(a+b*cos(c+d*x))^3*sec(c+d*x)^5,x,7,3/8*a*(a^2+4*b^2)*arctanh(sin(c+d*x))/d+b*(2*a^2+b^2)*tan(c+d*x)/d+3/8*a*(a^2+4*b^2)*sec(c+d*x)*tan(c+d*x)/d+3/4*a^2*b*sec(c+d*x)^2*tan(c+d*x)/d+1/4*a^2*(a+b*cos(c+d*x))*sec(c+d*x)^3*tan(c+d*x)/d],
[(a+b*cos(c+d*x))^3*sec(c+d*x)^6,x,7,1/8*b*(9*a^2+4*b^2)*arctanh(sin(c+d*x))/d+1/5*a*(4*a^2+15*b^2)*tan(c+d*x)/d+1/8*b*(9*a^2+4*b^2)*sec(c+d*x)*tan(c+d*x)/d+11/20*a^2*b*sec(c+d*x)^3*tan(c+d*x)/d+1/5*a^2*(a+b*cos(c+d*x))*sec(c+d*x)^4*tan(c+d*x)/d+1/15*a*(4*a^2+15*b^2)*tan(c+d*x)^3/d],
[cos(c+d*x)^3*(a+b*cos(c+d*x))^4,x,9,1/4*a*b*(6*a^2+5*b^2)*x+1/35*(35*a^4+168*a^2*b^2+24*b^4)*sin(c+d*x)/d+1/4*a*b*(6*a^2+5*b^2)*cos(c+d*x)*sin(c+d*x)/d+1/6*a*b*(6*a^2+5*b^2)*cos(c+d*x)^3*sin(c+d*x)/d+1/35*b^2*(37*a^2+6*b^2)*cos(c+d*x)^4*sin(c+d*x)/d+8/21*a*b^3*cos(c+d*x)^5*sin(c+d*x)/d+1/7*b^2*cos(c+d*x)^4*(a+b*cos(c+d*x))^2*sin(c+d*x)/d-1/105*(35*a^4+168*a^2*b^2+24*b^4)*sin(c+d*x)^3/d],
[cos(c+d*x)^2*(a+b*cos(c+d*x))^4,x,5,1/16*(8*a^4+36*a^2*b^2+5*b^4)*x-1/60*a*(4*a^4-121*a^2*b^2-128*b^4)*sin(c+d*x)/(b*d)-1/240*(8*a^4-178*a^2*b^2-75*b^4)*cos(c+d*x)*sin(c+d*x)/d-1/120*a*(4*a^2-53*b^2)*(a+b*cos(c+d*x))^2*sin(c+d*x)/(b*d)-1/120*(4*a^2-25*b^2)*(a+b*cos(c+d*x))^3*sin(c+d*x)/(b*d)-1/30*a*(a+b*cos(c+d*x))^4*sin(c+d*x)/(b*d)+1/6*(a+b*cos(c+d*x))^5*sin(c+d*x)/(b*d)],
[cos(c+d*x)*(a+b*cos(c+d*x))^4,x,4,1/2*a*b*(4*a^2+3*b^2)*x+2/15*(3*a^4+28*a^2*b^2+4*b^4)*sin(c+d*x)/d+1/30*a*b*(6*a^2+29*b^2)*cos(c+d*x)*sin(c+d*x)/d+1/15*(3*a^2+4*b^2)*(a+b*cos(c+d*x))^2*sin(c+d*x)/d+1/5*a*(a+b*cos(c+d*x))^3*sin(c+d*x)/d+1/5*(a+b*cos(c+d*x))^4*sin(c+d*x)/d],
[(a+b*cos(c+d*x))^4,x,3,1/8*(8*a^4+24*a^2*b^2+3*b^4)*x+1/6*a*b*(19*a^2+16*b^2)*sin(c+d*x)/d+1/24*b^2*(26*a^2+9*b^2)*cos(c+d*x)*sin(c+d*x)/d+7/12*a*b*(a+b*cos(c+d*x))^2*sin(c+d*x)/d+1/4*b*(a+b*cos(c+d*x))^3*sin(c+d*x)/d],
[(a+b*cos(c+d*x))^4*sec(c+d*x),x,5,2*a*b*(2*a^2+b^2)*x+a^4*arctanh(sin(c+d*x))/d+1/3*b^2*(17*a^2+2*b^2)*sin(c+d*x)/d+4/3*a*b^3*cos(c+d*x)*sin(c+d*x)/d+1/3*b^2*(a+b*cos(c+d*x))^2*sin(c+d*x)/d],
[(a+b*cos(c+d*x))^4*sec(c+d*x)^2,x,5,1/2*b^2*(12*a^2+b^2)*x+4*a^3*b*arctanh(sin(c+d*x))/d-2*a*b*(a^2-2*b^2)*sin(c+d*x)/d-1/2*b^2*(2*a^2-b^2)*cos(c+d*x)*sin(c+d*x)/d+a^2*(a+b*cos(c+d*x))^2*tan(c+d*x)/d],
[(a+b*cos(c+d*x))^4*sec(c+d*x)^3,x,5,4*a*b^3*x+1/2*a^2*(a^2+12*b^2)*arctanh(sin(c+d*x))/d-1/2*b^2*(a^2-2*b^2)*sin(c+d*x)/d+3*a^3*b*tan(c+d*x)/d+1/2*a^2*(a+b*cos(c+d*x))^2*sec(c+d*x)*tan(c+d*x)/d],
[(a+b*cos(c+d*x))^4*sec(c+d*x)^4,x,5,b^4*x+2*a*b*(a^2+2*b^2)*arctanh(sin(c+d*x))/d+1/3*a^2*(2*a^2+17*b^2)*tan(c+d*x)/d+4/3*a^3*b*sec(c+d*x)*tan(c+d*x)/d+1/3*a^2*(a+b*cos(c+d*x))^2*sec(c+d*x)^2*tan(c+d*x)/d],
[(a+b*cos(c+d*x))^4*sec(c+d*x)^5,x,7,1/8*(3*a^4+24*a^2*b^2+8*b^4)*arctanh(sin(c+d*x))/d+4/3*a*b*(2*a^2+3*b^2)*tan(c+d*x)/d+1/8*a^2*(3*a^2+22*b^2)*sec(c+d*x)*tan(c+d*x)/d+5/6*a^3*b*sec(c+d*x)^2*tan(c+d*x)/d+1/4*a^2*(a+b*cos(c+d*x))^2*sec(c+d*x)^3*tan(c+d*x)/d],
[(a+b*cos(c+d*x))^4*sec(c+d*x)^6,x,8,1/2*a*b*(3*a^2+4*b^2)*arctanh(sin(c+d*x))/d+1/15*(8*a^4+60*a^2*b^2+15*b^4)*tan(c+d*x)/d+1/2*a*b*(3*a^2+4*b^2)*sec(c+d*x)*tan(c+d*x)/d+1/15*a^2*(4*a^2+27*b^2)*sec(c+d*x)^2*tan(c+d*x)/d+3/5*a^3*b*sec(c+d*x)^3*tan(c+d*x)/d+1/5*a^2*(a+b*cos(c+d*x))^2*sec(c+d*x)^4*tan(c+d*x)/d],
[(a+b*cos(c+d*x))^4*sec(c+d*x)^7,x,8,1/16*(5*a^4+36*a^2*b^2+8*b^4)*arctanh(sin(c+d*x))/d+4/5*a*b*(4*a^2+5*b^2)*tan(c+d*x)/d+1/16*(5*a^4+36*a^2*b^2+8*b^4)*sec(c+d*x)*tan(c+d*x)/d+1/24*a^2*(5*a^2+32*b^2)*sec(c+d*x)^3*tan(c+d*x)/d+7/15*a^3*b*sec(c+d*x)^4*tan(c+d*x)/d+1/6*a^2*(a+b*cos(c+d*x))^2*sec(c+d*x)^5*tan(c+d*x)/d+4/15*a*b*(4*a^2+5*b^2)*tan(c+d*x)^3/d],

# n<0
[cos(c+d*x)^5/(a+b*cos(c+d*x)),x,7,1/8*(8*a^4+4*a^2*b^2+3*b^4)*x/b^5-1/3*a*(3*a^2+2*b^2)*sin(c+d*x)/(b^4*d)+1/8*(4*a^2+3*b^2)*cos(c+d*x)*sin(c+d*x)/(b^3*d)-1/3*a*cos(c+d*x)^2*sin(c+d*x)/(b^2*d)+1/4*cos(c+d*x)^3*sin(c+d*x)/(b*d)-2*a^5*arctan(sqrt(a-b)*tan(1/2*(c+d*x))/sqrt(a+b))/(b^5*d*sqrt(a-b)*sqrt(a+b))],
[cos(c+d*x)^4/(a+b*cos(c+d*x)),x,6,-1/2*a*(2*a^2+b^2)*x/b^4+1/3*(3*a^2+2*b^2)*sin(c+d*x)/(b^3*d)-1/2*a*cos(c+d*x)*sin(c+d*x)/(b^2*d)+1/3*cos(c+d*x)^2*sin(c+d*x)/(b*d)+2*a^4*arctan(sqrt(a-b)*tan(1/2*(c+d*x))/sqrt(a+b))/(b^4*d*sqrt(a-b)*sqrt(a+b))],
[cos(c+d*x)^3/(a+b*cos(c+d*x)),x,5,1/2*(2*a^2+b^2)*x/b^3-a*sin(c+d*x)/(b^2*d)+1/2*cos(c+d*x)*sin(c+d*x)/(b*d)-2*a^3*arctan(sqrt(a-b)*tan(1/2*(c+d*x))/sqrt(a+b))/(b^3*d*sqrt(a-b)*sqrt(a+b))],
[cos(c+d*x)^2/(a+b*cos(c+d*x)),x,5,-a*x/b^2+sin(c+d*x)/(b*d)+2*a^2*arctan(sqrt(a-b)*tan(1/2*(c+d*x))/sqrt(a+b))/(b^2*d*sqrt(a-b)*sqrt(a+b))],
[cos(c+d*x)/(a+b*cos(c+d*x)),x,3,x/b-2*a*arctan(sqrt(a-b)*tan(1/2*(c+d*x))/sqrt(a+b))/(b*d*sqrt(a-b)*sqrt(a+b))],
[1/(a+b*cos(c+d*x)),x,2,2*arctan(sqrt(a-b)*tan(1/2*(c+d*x))/sqrt(a+b))/(d*sqrt(a-b)*sqrt(a+b))],
[sec(c+d*x)/(a+b*cos(c+d*x)),x,4,arctanh(sin(c+d*x))/(a*d)-2*b*arctan(sqrt(a-b)*tan(1/2*(c+d*x))/sqrt(a+b))/(a*d*sqrt(a-b)*sqrt(a+b))],
[sec(c+d*x)^2/(a+b*cos(c+d*x)),x,6,-b*arctanh(sin(c+d*x))/(a^2*d)+2*b^2*arctan(sqrt(a-b)*tan(1/2*(c+d*x))/sqrt(a+b))/(a^2*d*sqrt(a-b)*sqrt(a+b))+tan(c+d*x)/(a*d)],
[sec(c+d*x)^3/(a+b*cos(c+d*x)),x,6,1/2*(a^2+2*b^2)*arctanh(sin(c+d*x))/(a^3*d)-2*b^3*arctan(sqrt(a-b)*tan(1/2*(c+d*x))/sqrt(a+b))/(a^3*d*sqrt(a-b)*sqrt(a+b))-b*tan(c+d*x)/(a^2*d)+1/2*sec(c+d*x)*tan(c+d*x)/(a*d)],
[sec(c+d*x)^4/(a+b*cos(c+d*x)),x,7,-1/2*b*(a^2+2*b^2)*arctanh(sin(c+d*x))/(a^4*d)+2*b^4*arctan(sqrt(a-b)*tan(1/2*(c+d*x))/sqrt(a+b))/(a^4*d*sqrt(a-b)*sqrt(a+b))+1/3*(2*a^2+3*b^2)*tan(c+d*x)/(a^3*d)-1/2*b*sec(c+d*x)*tan(c+d*x)/(a^2*d)+1/3*sec(c+d*x)^2*tan(c+d*x)/(a*d)],
[cos(c+d*x)^5/(a+b*cos(c+d*x))^2,x,7,-a*(4*a^2+b^2)*x/b^5+2*a^4*(4*a^2-5*b^2)*arctan(sqrt(a-b)*tan(1/2*(c+d*x))/sqrt(a+b))/((a-b)^(3/2)*b^5*(a+b)^(3/2)*d)+1/3*(12*a^4-7*a^2*b^2-2*b^4)*sin(c+d*x)/(b^4*(a^2-b^2)*d)-a*(2*a^2-b^2)*cos(c+d*x)*sin(c+d*x)/(b^3*(a^2-b^2)*d)+1/3*(4*a^2-b^2)*cos(c+d*x)^2*sin(c+d*x)/(b^2*(a^2-b^2)*d)-a^2*cos(c+d*x)^3*sin(c+d*x)/(b*(a^2-b^2)*d*(a+b*cos(c+d*x)))],
[cos(c+d*x)^4/(a+b*cos(c+d*x))^2,x,6,1/2*(6*a^2+b^2)*x/b^4-2*a^3*(3*a^2-4*b^2)*arctanh((a-b)*sin(c+d*x)/((1+cos(c+d*x))*sqrt(-a^2+b^2)))/(b^4*(-a^2+b^2)^(3/2)*d)-2*a*sin(c+d*x)/(b^3*d)+1/2*cos(c+d*x)*sin(c+d*x)/(b^2*d)-a^4*sin(c+d*x)/(b^3*(a^2-b^2)*d*(a+b*cos(c+d*x))),1/2*(6*a^2+b^2)*x/b^4-2*a^3*(3*a^2-4*b^2)*arctan(sqrt(a-b)*tan(1/2*(c+d*x))/sqrt(a+b))/((a-b)^(3/2)*b^4*(a+b)^(3/2)*d)-a*(3*a^2-2*b^2)*sin(c+d*x)/(b^3*(a^2-b^2)*d)+1/2*(3*a^2-b^2)*cos(c+d*x)*sin(c+d*x)/(b^2*(a^2-b^2)*d)-a^2*cos(c+d*x)^2*sin(c+d*x)/(b*(a^2-b^2)*d*(a+b*cos(c+d*x)))],
[cos(c+d*x)^3/(a+b*cos(c+d*x))^2,x,5,-2*a*x/b^3+2*a^2*(2*a^2-3*b^2)*arctan(sqrt(a-b)*tan(1/2*(c+d*x))/sqrt(a+b))/((a-b)^(3/2)*b^3*(a+b)^(3/2)*d)+(2*a^2-b^2)*sin(c+d*x)/(b^2*(a^2-b^2)*d)-a^2*cos(c+d*x)*sin(c+d*x)/(b*(a^2-b^2)*d*(a+b*cos(c+d*x)))],
[cos(c+d*x)^2/(a+b*cos(c+d*x))^2,x,4,x/b^2-2*a*(a^2-2*b^2)*arctan(sqrt(a-b)*tan(1/2*(c+d*x))/sqrt(a+b))/((a-b)^(3/2)*b^2*(a+b)^(3/2)*d)-a^2*sin(c+d*x)/(b*(a^2-b^2)*d*(a+b*cos(c+d*x)))],
[cos(c+d*x)/(a+b*cos(c+d*x))^2,x,4,-2*b*arctan(sqrt(a-b)*tan(1/2*(c+d*x))/sqrt(a+b))/((a-b)^(3/2)*(a+b)^(3/2)*d)+a*sin(c+d*x)/((a^2-b^2)*d*(a+b*cos(c+d*x)))],
[1/(a+b*cos(c+d*x))^2,x,4,2*a*arctan(sqrt(a-b)*tan(1/2*(c+d*x))/sqrt(a+b))/((a-b)^(3/2)*(a+b)^(3/2)*d)-b*sin(c+d*x)/((a^2-b^2)*d*(a+b*cos(c+d*x)))],
[sec(c+d*x)/(a+b*cos(c+d*x))^2,x,5,-2*b*(2*a^2-b^2)*arctan(sqrt(a-b)*tan(1/2*(c+d*x))/sqrt(a+b))/(a^2*(a-b)^(3/2)*(a+b)^(3/2)*d)+arctanh(sin(c+d*x))/(a^2*d)+b^2*sin(c+d*x)/(a*(a^2-b^2)*d*(a+b*cos(c+d*x)))],
[sec(c+d*x)^2/(a+b*cos(c+d*x))^2,x,6,2*b^2*(3*a^2-2*b^2)*arctan(sqrt(a-b)*tan(1/2*(c+d*x))/sqrt(a+b))/(a^3*(a-b)^(3/2)*(a+b)^(3/2)*d)-2*b*arctanh(sin(c+d*x))/(a^3*d)+(a^2-2*b^2)*tan(c+d*x)/(a^2*(a^2-b^2)*d)+b^2*tan(c+d*x)/(a*(a^2-b^2)*d*(a+b*cos(c+d*x)))],
[sec(c+d*x)^3/(a+b*cos(c+d*x))^2,x,7,-2*b^3*(4*a^2-3*b^2)*arctan(sqrt(a-b)*tan(1/2*(c+d*x))/sqrt(a+b))/(a^4*(a-b)^(3/2)*(a+b)^(3/2)*d)+1/2*(a^2+6*b^2)*arctanh(sin(c+d*x))/(a^4*d)-b*(2*a^2-3*b^2)*tan(c+d*x)/(a^3*(a^2-b^2)*d)+1/2*(a^2-3*b^2)*sec(c+d*x)*tan(c+d*x)/(a^2*(a^2-b^2)*d)+b^2*sec(c+d*x)*tan(c+d*x)/(a*(a^2-b^2)*d*(a+b*cos(c+d*x)))],
[sec(c+d*x)^4/(a+b*cos(c+d*x))^2,x,8,2*b^4*(5*a^2-4*b^2)*arctan(sqrt(a-b)*tan(1/2*(c+d*x))/sqrt(a+b))/(a^5*(a-b)^(3/2)*(a+b)^(3/2)*d)-b*(a^2+4*b^2)*arctanh(sin(c+d*x))/(a^5*d)+1/3*(2*a^4+7*a^2*b^2-12*b^4)*tan(c+d*x)/(a^4*(a^2-b^2)*d)-b*(a^2-2*b^2)*sec(c+d*x)*tan(c+d*x)/(a^3*(a^2-b^2)*d)+1/3*(a^2-4*b^2)*sec(c+d*x)^2*tan(c+d*x)/(a^2*(a^2-b^2)*d)+b^2*sec(c+d*x)^2*tan(c+d*x)/(a*(a^2-b^2)*d*(a+b*cos(c+d*x)))],
[cos(c+d*x)^5/(a+b*cos(c+d*x))^3,x,7,1/2*(12*a^2+b^2)*x/b^5-a^3*(12*a^4-29*a^2*b^2+20*b^4)*arctan(sqrt(a-b)*tan(1/2*(c+d*x))/sqrt(a+b))/((a-b)^(5/2)*b^5*(a+b)^(5/2)*d)-3/2*a*(4*a^4-7*a^2*b^2+2*b^4)*sin(c+d*x)/(b^4*(a^2-b^2)^2*d)+1/2*(6*a^4-10*a^2*b^2+b^4)*cos(c+d*x)*sin(c+d*x)/(b^3*(a^2-b^2)^2*d)-1/2*a^2*cos(c+d*x)^3*sin(c+d*x)/(b*(a^2-b^2)*d*(a+b*cos(c+d*x))^2)-1/2*a^2*(4*a^2-7*b^2)*cos(c+d*x)^2*sin(c+d*x)/(b^2*(a^2-b^2)^2*d*(a+b*cos(c+d*x)))],
[cos(c+d*x)^4/(a+b*cos(c+d*x))^3,x,6,-3*a*x/b^4+3*a^2*(2*a^4-5*a^2*b^2+4*b^4)*arctan(sqrt(a-b)*tan(1/2*(c+d*x))/sqrt(a+b))/((a-b)^(5/2)*b^4*(a+b)^(5/2)*d)+1/2*(3*a^2-2*b^2)*sin(c+d*x)/(b^3*(a^2-b^2)*d)-1/2*a^2*cos(c+d*x)^2*sin(c+d*x)/(b*(a^2-b^2)*d*(a+b*cos(c+d*x))^2)+3/2*a^3*(a^2-2*b^2)*sin(c+d*x)/(b^3*(a^2-b^2)^2*d*(a+b*cos(c+d*x)))],
[cos(c+d*x)^3/(a+b*cos(c+d*x))^3,x,5,x/b^3-a*(2*a^4-5*a^2*b^2+6*b^4)*arctan(sqrt(a-b)*tan(1/2*(c+d*x))/sqrt(a+b))/((a-b)^(5/2)*b^3*(a+b)^(5/2)*d)-1/2*a^2*cos(c+d*x)*sin(c+d*x)/(b*(a^2-b^2)*d*(a+b*cos(c+d*x))^2)-1/2*a^2*(2*a^2-5*b^2)*sin(c+d*x)/(b^2*(a^2-b^2)^2*d*(a+b*cos(c+d*x)))],
[cos(c+d*x)^2/(a+b*cos(c+d*x))^3,x,5,(a^2+2*b^2)*arctan(sqrt(a-b)*tan(1/2*(c+d*x))/sqrt(a+b))/((a-b)^(5/2)*(a+b)^(5/2)*d)-1/2*a^2*sin(c+d*x)/(b*(a^2-b^2)*d*(a+b*cos(c+d*x))^2)+1/2*a*(a^2-4*b^2)*sin(c+d*x)/(b*(a^2-b^2)^2*d*(a+b*cos(c+d*x)))],
[cos(c+d*x)/(a+b*cos(c+d*x))^3,x,5,-3*a*b*arctan(sqrt(a-b)*tan(1/2*(c+d*x))/sqrt(a+b))/((a-b)^(5/2)*(a+b)^(5/2)*d)+1/2*a*sin(c+d*x)/((a^2-b^2)*d*(a+b*cos(c+d*x))^2)+1/2*(a^2+2*b^2)*sin(c+d*x)/((a^2-b^2)^2*d*(a+b*cos(c+d*x)))],
[1/(a+b*cos(c+d*x))^3,x,5,(2*a^2+b^2)*arctan(sqrt(a-b)*tan(1/2*(c+d*x))/sqrt(a+b))/((a-b)^(5/2)*(a+b)^(5/2)*d)-1/2*b*sin(c+d*x)/((a^2-b^2)*d*(a+b*cos(c+d*x))^2)-3/2*a*b*sin(c+d*x)/((a^2-b^2)^2*d*(a+b*cos(c+d*x)))],
[sec(c+d*x)/(a+b*cos(c+d*x))^3,x,6,-b*(6*a^4-5*a^2*b^2+2*b^4)*arctan(sqrt(a-b)*tan(1/2*(c+d*x))/sqrt(a+b))/(a^3*(a-b)^(5/2)*(a+b)^(5/2)*d)+arctanh(sin(c+d*x))/(a^3*d)+1/2*b^2*sin(c+d*x)/(a*(a^2-b^2)*d*(a+b*cos(c+d*x))^2)+1/2*b^2*(5*a^2-2*b^2)*sin(c+d*x)/(a^2*(a^2-b^2)^2*d*(a+b*cos(c+d*x)))],
[sec(c+d*x)^2/(a+b*cos(c+d*x))^3,x,7,3*b^2*(4*a^4-5*a^2*b^2+2*b^4)*arctan(sqrt(a-b)*tan(1/2*(c+d*x))/sqrt(a+b))/(a^4*(a-b)^(5/2)*(a+b)^(5/2)*d)-3*b*arctanh(sin(c+d*x))/(a^4*d)+1/2*(2*a^4-11*a^2*b^2+6*b^4)*tan(c+d*x)/(a^3*(a^2-b^2)^2*d)+1/2*b^2*tan(c+d*x)/(a*(a^2-b^2)*d*(a+b*cos(c+d*x))^2)+3/2*b^2*(2*a^2-b^2)*tan(c+d*x)/(a^2*(a^2-b^2)^2*d*(a+b*cos(c+d*x)))],
[sec(c+d*x)^3/(a+b*cos(c+d*x))^3,x,8,-b^3*(20*a^4-29*a^2*b^2+12*b^4)*arctan(sqrt(a-b)*tan(1/2*(c+d*x))/sqrt(a+b))/(a^5*(a-b)^(5/2)*(a+b)^(5/2)*d)+1/2*(a^2+12*b^2)*arctanh(sin(c+d*x))/(a^5*d)-3/2*b*(2*a^4-7*a^2*b^2+4*b^4)*tan(c+d*x)/(a^4*(a^2-b^2)^2*d)+1/2*(a^4-10*a^2*b^2+6*b^4)*sec(c+d*x)*tan(c+d*x)/(a^3*(a^2-b^2)^2*d)+1/2*b^2*sec(c+d*x)*tan(c+d*x)/(a*(a^2-b^2)*d*(a+b*cos(c+d*x))^2)+1/2*b^2*(7*a^2-4*b^2)*sec(c+d*x)*tan(c+d*x)/(a^2*(a^2-b^2)^2*d*(a+b*cos(c+d*x)))],
[cos(c+d*x)^5/(a+b*cos(c+d*x))^4,x,7,-4*a*x/b^5+a^2*(8*a^6-28*a^4*b^2+35*a^2*b^4-20*b^6)*arctan(sqrt(a-b)*tan(1/2*(c+d*x))/sqrt(a+b))/((a-b)^(7/2)*b^5*(a+b)^(7/2)*d)+1/6*(12*a^4-23*a^2*b^2+6*b^4)*sin(c+d*x)/(b^4*(a^2-b^2)^2*d)-1/3*a^2*cos(c+d*x)^3*sin(c+d*x)/(b*(a^2-b^2)*d*(a+b*cos(c+d*x))^3)-1/6*a^2*(4*a^2-9*b^2)*cos(c+d*x)^2*sin(c+d*x)/(b^2*(a^2-b^2)^2*d*(a+b*cos(c+d*x))^2)+1/2*a^3*(4*a^4-11*a^2*b^2+12*b^4)*sin(c+d*x)/(b^4*(a^2-b^2)^3*d*(a+b*cos(c+d*x)))],
[cos(c+d*x)^4/(a+b*cos(c+d*x))^4,x,6,x/b^4-a*(2*a^6-7*a^4*b^2+8*a^2*b^4-8*b^6)*arctan(sqrt(a-b)*tan(1/2*(c+d*x))/sqrt(a+b))/((a-b)^(7/2)*b^4*(a+b)^(7/2)*d)-1/3*a^2*cos(c+d*x)^2*sin(c+d*x)/(b*(a^2-b^2)*d*(a+b*cos(c+d*x))^3)+1/6*a^3*(3*a^2-8*b^2)*sin(c+d*x)/(b^3*(a^2-b^2)^2*d*(a+b*cos(c+d*x))^2)-1/6*a^2*(9*a^4-28*a^2*b^2+34*b^4)*sin(c+d*x)/(b^3*(a^2-b^2)^3*d*(a+b*cos(c+d*x)))],
[cos(c+d*x)^3/(a+b*cos(c+d*x))^4,x,6,-b*(3*a^2+2*b^2)*arctan(sqrt(a-b)*tan(1/2*(c+d*x))/sqrt(a+b))/((a-b)^(7/2)*(a+b)^(7/2)*d)-1/3*a^2*cos(c+d*x)*sin(c+d*x)/(b*(a^2-b^2)*d*(a+b*cos(c+d*x))^3)-1/6*a^2*(2*a^2-7*b^2)*sin(c+d*x)/(b^2*(a^2-b^2)^2*d*(a+b*cos(c+d*x))^2)+1/6*a*(2*a^4-5*a^2*b^2+18*b^4)*sin(c+d*x)/(b^2*(a^2-b^2)^3*d*(a+b*cos(c+d*x)))],
[cos(c+d*x)^2/(a+b*cos(c+d*x))^4,x,6,a*(a^2+4*b^2)*arctan(sqrt(a-b)*tan(1/2*(c+d*x))/sqrt(a+b))/((a-b)^(7/2)*(a+b)^(7/2)*d)-1/3*a^2*sin(c+d*x)/(b*(a^2-b^2)*d*(a+b*cos(c+d*x))^3)+1/6*a*(a^2-6*b^2)*sin(c+d*x)/(b*(a^2-b^2)^2*d*(a+b*cos(c+d*x))^2)+1/6*(a^4-10*a^2*b^2-6*b^4)*sin(c+d*x)/(b*(a^2-b^2)^3*d*(a+b*cos(c+d*x)))],
[cos(c+d*x)/(a+b*cos(c+d*x))^4,x,6,-b*(4*a^2+b^2)*arctan(sqrt(a-b)*tan(1/2*(c+d*x))/sqrt(a+b))/((a-b)^(7/2)*(a+b)^(7/2)*d)+1/3*a*sin(c+d*x)/((a^2-b^2)*d*(a+b*cos(c+d*x))^3)+1/6*(2*a^2+3*b^2)*sin(c+d*x)/((a^2-b^2)^2*d*(a+b*cos(c+d*x))^2)+1/6*a*(2*a^2+13*b^2)*sin(c+d*x)/((a^2-b^2)^3*d*(a+b*cos(c+d*x)))],
[1/(a+b*cos(c+d*x))^4,x,6,a*(2*a^2+3*b^2)*arctan(sqrt(a-b)*tan(1/2*(c+d*x))/sqrt(a+b))/((a-b)^(7/2)*(a+b)^(7/2)*d)-1/3*b*sin(c+d*x)/((a^2-b^2)*d*(a+b*cos(c+d*x))^3)-5/6*a*b*sin(c+d*x)/((a^2-b^2)^2*d*(a+b*cos(c+d*x))^2)-1/6*b*(11*a^2+4*b^2)*sin(c+d*x)/((a^2-b^2)^3*d*(a+b*cos(c+d*x)))],
[sec(c+d*x)/(a+b*cos(c+d*x))^4,x,7,-b*(8*a^6-8*a^4*b^2+7*a^2*b^4-2*b^6)*arctan(sqrt(a-b)*tan(1/2*(c+d*x))/sqrt(a+b))/(a^4*(a-b)^(7/2)*(a+b)^(7/2)*d)+arctanh(sin(c+d*x))/(a^4*d)+1/3*b^2*sin(c+d*x)/(a*(a^2-b^2)*d*(a+b*cos(c+d*x))^3)+1/6*b^2*(8*a^2-3*b^2)*sin(c+d*x)/(a^2*(a^2-b^2)^2*d*(a+b*cos(c+d*x))^2)+1/6*b^2*(26*a^4-17*a^2*b^2+6*b^4)*sin(c+d*x)/(a^3*(a^2-b^2)^3*d*(a+b*cos(c+d*x)))],
[sec(c+d*x)^2/(a+b*cos(c+d*x))^4,x,8,b^2*(20*a^6-35*a^4*b^2+28*a^2*b^4-8*b^6)*arctan(sqrt(a-b)*tan(1/2*(c+d*x))/sqrt(a+b))/(a^5*(a-b)^(7/2)*(a+b)^(7/2)*d)-4*b*arctanh(sin(c+d*x))/(a^5*d)+1/6*(6*a^6-65*a^4*b^2+68*a^2*b^4-24*b^6)*tan(c+d*x)/(a^4*(a^2-b^2)^3*d)+1/3*b^2*tan(c+d*x)/(a*(a^2-b^2)*d*(a+b*cos(c+d*x))^3)+1/6*b^2*(9*a^2-4*b^2)*tan(c+d*x)/(a^2*(a^2-b^2)^2*d*(a+b*cos(c+d*x))^2)+1/2*b^2*(12*a^4-11*a^2*b^2+4*b^4)*tan(c+d*x)/(a^3*(a^2-b^2)^3*d*(a+b*cos(c+d*x)))],

# Integrands of the form Cos[e+f x]^m (a+b Cos[e+f x])^(n/2)

# n>0
[cos(c+d*x)^3*sqrt(a+b*cos(c+d*x)),x,8,-8/35*a*(a+b*cos(c+d*x))^(3/2)*sin(c+d*x)/(b^2*d)+2/7*cos(c+d*x)*(a+b*cos(c+d*x))^(3/2)*sin(c+d*x)/(b*d)+2/105*(8*a^2+25*b^2)*sin(c+d*x)*sqrt(a+b*cos(c+d*x))/(b^2*d)+2/105*a*(8*a^2+19*b^2)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2*b/(a+b)))*sqrt(a+b*cos(c+d*x))/(b^3*d*sqrt((a+b*cos(c+d*x))/(a+b)))-2/105*(8*a^4+17*a^2*b^2-25*b^4)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2*b/(a+b)))*sqrt((a+b*cos(c+d*x))/(a+b))/(b^3*d*sqrt(a+b*cos(c+d*x)))],
[cos(c+d*x)^2*sqrt(a+b*cos(c+d*x)),x,7,2/5*(a+b*cos(c+d*x))^(3/2)*sin(c+d*x)/(b*d)-4/15*a*sin(c+d*x)*sqrt(a+b*cos(c+d*x))/(b*d)-2/15*(2*a^2-9*b^2)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2*b/(a+b)))*sqrt(a+b*cos(c+d*x))/(b^2*d*sqrt((a+b*cos(c+d*x))/(a+b)))+4/15*a*(a^2-b^2)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2*b/(a+b)))*sqrt((a+b*cos(c+d*x))/(a+b))/(b^2*d*sqrt(a+b*cos(c+d*x)))],
[cos(c+d*x)*sqrt(a+b*cos(c+d*x)),x,6,2/3*sin(c+d*x)*sqrt(a+b*cos(c+d*x))/d+2/3*a*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2*b/(a+b)))*sqrt(a+b*cos(c+d*x))/(b*d*sqrt((a+b*cos(c+d*x))/(a+b)))-2/3*(a^2-b^2)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2*b/(a+b)))*sqrt((a+b*cos(c+d*x))/(a+b))/(b*d*sqrt(a+b*cos(c+d*x)))],
[sqrt(a+b*cos(c+d*x)),x,2,2*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2*b/(a+b)))*sqrt(a+b*cos(c+d*x))/(d*sqrt((a+b*cos(c+d*x))/(a+b)))],
[sec(c+d*x)*sqrt(a+b*cos(c+d*x)),x,5,2*b*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2*b/(a+b)))*sqrt((a+b*cos(c+d*x))/(a+b))/(d*sqrt(a+b*cos(c+d*x)))+2*a*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2,sqrt(2*b/(a+b)))*sqrt((a+b*cos(c+d*x))/(a+b))/(d*sqrt(a+b*cos(c+d*x)))],
[sec(c+d*x)^2*sqrt(a+b*cos(c+d*x)),x,9,-sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2*b/(a+b)))*sqrt(a+b*cos(c+d*x))/(d*sqrt((a+b*cos(c+d*x))/(a+b)))+a*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2*b/(a+b)))*sqrt((a+b*cos(c+d*x))/(a+b))/(d*sqrt(a+b*cos(c+d*x)))+b*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2,sqrt(2*b/(a+b)))*sqrt((a+b*cos(c+d*x))/(a+b))/(d*sqrt(a+b*cos(c+d*x)))+sqrt(a+b*cos(c+d*x))*tan(c+d*x)/d],
[sec(c+d*x)^3*sqrt(a+b*cos(c+d*x)),x,10,-1/4*b*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2*b/(a+b)))*sqrt(a+b*cos(c+d*x))/(a*d*sqrt((a+b*cos(c+d*x))/(a+b)))+3/4*b*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2*b/(a+b)))*sqrt((a+b*cos(c+d*x))/(a+b))/(d*sqrt(a+b*cos(c+d*x)))+1/4*(4*a^2-b^2)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2,sqrt(2*b/(a+b)))*sqrt((a+b*cos(c+d*x))/(a+b))/(a*d*sqrt(a+b*cos(c+d*x)))+1/4*b*sqrt(a+b*cos(c+d*x))*tan(c+d*x)/(a*d)+1/2*sec(c+d*x)*sqrt(a+b*cos(c+d*x))*tan(c+d*x)/d],
[cos(c+d*x)^3*(a+b*cos(c+d*x))^(3/2),x,9,2/315*(8*a^2+49*b^2)*(a+b*cos(c+d*x))^(3/2)*sin(c+d*x)/(b^2*d)-8/63*a*(a+b*cos(c+d*x))^(5/2)*sin(c+d*x)/(b^2*d)+2/9*cos(c+d*x)*(a+b*cos(c+d*x))^(5/2)*sin(c+d*x)/(b*d)+2/315*a*(8*a^2+39*b^2)*sin(c+d*x)*sqrt(a+b*cos(c+d*x))/(b^2*d)+2/315*(8*a^4+33*a^2*b^2+147*b^4)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2*b/(a+b)))*sqrt(a+b*cos(c+d*x))/(b^3*d*sqrt((a+b*cos(c+d*x))/(a+b)))-2/315*a*(8*a^4+31*a^2*b^2-39*b^4)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2*b/(a+b)))*sqrt((a+b*cos(c+d*x))/(a+b))/(b^3*d*sqrt(a+b*cos(c+d*x)))],
[cos(c+d*x)^2*(a+b*cos(c+d*x))^(3/2),x,8,-4/35*a*(a+b*cos(c+d*x))^(3/2)*sin(c+d*x)/(b*d)+2/7*(a+b*cos(c+d*x))^(5/2)*sin(c+d*x)/(b*d)-2/105*(6*a^2-25*b^2)*sin(c+d*x)*sqrt(a+b*cos(c+d*x))/(b*d)-4/105*a*(3*a^2-41*b^2)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2*b/(a+b)))*sqrt(a+b*cos(c+d*x))/(b^2*d*sqrt((a+b*cos(c+d*x))/(a+b)))+2/105*(6*a^4-31*a^2*b^2+25*b^4)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2*b/(a+b)))*sqrt((a+b*cos(c+d*x))/(a+b))/(b^2*d*sqrt(a+b*cos(c+d*x)))],
[cos(c+d*x)*(a+b*cos(c+d*x))^(3/2),x,7,2/5*(a+b*cos(c+d*x))^(3/2)*sin(c+d*x)/d+2/5*a*sin(c+d*x)*sqrt(a+b*cos(c+d*x))/d+2/5*(a^2+3*b^2)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2*b/(a+b)))*sqrt(a+b*cos(c+d*x))/(b*d*sqrt((a+b*cos(c+d*x))/(a+b)))-2/5*a*(a^2-b^2)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2*b/(a+b)))*sqrt((a+b*cos(c+d*x))/(a+b))/(b*d*sqrt(a+b*cos(c+d*x)))],
[(a+b*cos(c+d*x))^(3/2),x,6,2/3*b*sin(c+d*x)*sqrt(a+b*cos(c+d*x))/d+8/3*a*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2*b/(a+b)))*sqrt(a+b*cos(c+d*x))/(d*sqrt((a+b*cos(c+d*x))/(a+b)))-2/3*(a^2-b^2)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2*b/(a+b)))*sqrt((a+b*cos(c+d*x))/(a+b))/(d*sqrt(a+b*cos(c+d*x)))],
[(a+b*cos(c+d*x))^(3/2)*sec(c+d*x),x,8,2*b*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2*b/(a+b)))*sqrt(a+b*cos(c+d*x))/(d*sqrt((a+b*cos(c+d*x))/(a+b)))+2*a*b*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2*b/(a+b)))*sqrt((a+b*cos(c+d*x))/(a+b))/(d*sqrt(a+b*cos(c+d*x)))+2*a^2*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2,sqrt(2*b/(a+b)))*sqrt((a+b*cos(c+d*x))/(a+b))/(d*sqrt(a+b*cos(c+d*x)))],
[(a+b*cos(c+d*x))^(3/2)*sec(c+d*x)^2,x,9,-a*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2*b/(a+b)))*sqrt(a+b*cos(c+d*x))/(d*sqrt((a+b*cos(c+d*x))/(a+b)))+(a^2+2*b^2)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2*b/(a+b)))*sqrt((a+b*cos(c+d*x))/(a+b))/(d*sqrt(a+b*cos(c+d*x)))+3*a*b*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2,sqrt(2*b/(a+b)))*sqrt((a+b*cos(c+d*x))/(a+b))/(d*sqrt(a+b*cos(c+d*x)))+a*sqrt(a+b*cos(c+d*x))*tan(c+d*x)/d],
[(a+b*cos(c+d*x))^(3/2)*sec(c+d*x)^3,x,10,-5/4*b*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2*b/(a+b)))*sqrt(a+b*cos(c+d*x))/(d*sqrt((a+b*cos(c+d*x))/(a+b)))+7/4*a*b*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2*b/(a+b)))*sqrt((a+b*cos(c+d*x))/(a+b))/(d*sqrt(a+b*cos(c+d*x)))+1/4*(4*a^2+3*b^2)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2,sqrt(2*b/(a+b)))*sqrt((a+b*cos(c+d*x))/(a+b))/(d*sqrt(a+b*cos(c+d*x)))+5/4*b*sqrt(a+b*cos(c+d*x))*tan(c+d*x)/d+1/2*a*sec(c+d*x)*sqrt(a+b*cos(c+d*x))*tan(c+d*x)/d],
[cos(c+d*x)^3*(a+b*cos(c+d*x))^(5/2),x,10,2/693*a*(8*a^2+67*b^2)*(a+b*cos(c+d*x))^(3/2)*sin(c+d*x)/(b^2*d)+2/693*(8*a^2+81*b^2)*(a+b*cos(c+d*x))^(5/2)*sin(c+d*x)/(b^2*d)-8/99*a*(a+b*cos(c+d*x))^(7/2)*sin(c+d*x)/(b^2*d)+2/11*cos(c+d*x)*(a+b*cos(c+d*x))^(7/2)*sin(c+d*x)/(b*d)+2/693*(8*a^4+57*a^2*b^2+135*b^4)*sin(c+d*x)*sqrt(a+b*cos(c+d*x))/(b^2*d)+2/693*a*(8*a^4+51*a^2*b^2+741*b^4)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2*b/(a+b)))*sqrt(a+b*cos(c+d*x))/(b^3*d*sqrt((a+b*cos(c+d*x))/(a+b)))-2/693*(8*a^6+49*a^4*b^2+78*a^2*b^4-135*b^6)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2*b/(a+b)))*sqrt((a+b*cos(c+d*x))/(a+b))/(b^3*d*sqrt(a+b*cos(c+d*x)))],
[cos(c+d*x)^2*(a+b*cos(c+d*x))^(5/2),x,9,-2/315*(10*a^2-49*b^2)*(a+b*cos(c+d*x))^(3/2)*sin(c+d*x)/(b*d)-4/63*a*(a+b*cos(c+d*x))^(5/2)*sin(c+d*x)/(b*d)+2/9*(a+b*cos(c+d*x))^(7/2)*sin(c+d*x)/(b*d)-4/315*a*(5*a^2-57*b^2)*sin(c+d*x)*sqrt(a+b*cos(c+d*x))/(b*d)-2/315*(10*a^4-279*a^2*b^2-147*b^4)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2*b/(a+b)))*sqrt(a+b*cos(c+d*x))/(b^2*d*sqrt((a+b*cos(c+d*x))/(a+b)))+4/315*a*(5*a^4-62*a^2*b^2+57*b^4)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2*b/(a+b)))*sqrt((a+b*cos(c+d*x))/(a+b))/(b^2*d*sqrt(a+b*cos(c+d*x)))],
[cos(c+d*x)*(a+b*cos(c+d*x))^(5/2),x,8,2/7*a*(a+b*cos(c+d*x))^(3/2)*sin(c+d*x)/d+2/7*(a+b*cos(c+d*x))^(5/2)*sin(c+d*x)/d+2/21*(3*a^2+5*b^2)*sin(c+d*x)*sqrt(a+b*cos(c+d*x))/d+2/21*a*(3*a^2+29*b^2)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2*b/(a+b)))*sqrt(a+b*cos(c+d*x))/(b*d*sqrt((a+b*cos(c+d*x))/(a+b)))-2/21*(3*a^4+2*a^2*b^2-5*b^4)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2*b/(a+b)))*sqrt((a+b*cos(c+d*x))/(a+b))/(b*d*sqrt(a+b*cos(c+d*x)))],
[(a+b*cos(c+d*x))^(5/2),x,7,2/5*b*(a+b*cos(c+d*x))^(3/2)*sin(c+d*x)/d+16/15*a*b*sin(c+d*x)*sqrt(a+b*cos(c+d*x))/d+2/15*(23*a^2+9*b^2)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2*b/(a+b)))*sqrt(a+b*cos(c+d*x))/(d*sqrt((a+b*cos(c+d*x))/(a+b)))-16/15*a*(a^2-b^2)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2*b/(a+b)))*sqrt((a+b*cos(c+d*x))/(a+b))/(d*sqrt(a+b*cos(c+d*x)))],
[(a+b*cos(c+d*x))^(5/2)*sec(c+d*x),x,9,2/3*b^2*sin(c+d*x)*sqrt(a+b*cos(c+d*x))/d+14/3*a*b*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2*b/(a+b)))*sqrt(a+b*cos(c+d*x))/(d*sqrt((a+b*cos(c+d*x))/(a+b)))+2/3*b*(2*a^2+b^2)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2*b/(a+b)))*sqrt((a+b*cos(c+d*x))/(a+b))/(d*sqrt(a+b*cos(c+d*x)))+2*a^3*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2,sqrt(2*b/(a+b)))*sqrt((a+b*cos(c+d*x))/(a+b))/(d*sqrt(a+b*cos(c+d*x)))],
[(a+b*cos(c+d*x))^(5/2)*sec(c+d*x)^2,x,9,-(a^2-2*b^2)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2*b/(a+b)))*sqrt(a+b*cos(c+d*x))/(d*sqrt((a+b*cos(c+d*x))/(a+b)))+a*(a^2+4*b^2)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2*b/(a+b)))*sqrt((a+b*cos(c+d*x))/(a+b))/(d*sqrt(a+b*cos(c+d*x)))+5*a^2*b*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2,sqrt(2*b/(a+b)))*sqrt((a+b*cos(c+d*x))/(a+b))/(d*sqrt(a+b*cos(c+d*x)))+a^2*sqrt(a+b*cos(c+d*x))*tan(c+d*x)/d],
[(a+b*cos(c+d*x))^(5/2)*sec(c+d*x)^3,x,10,-9/4*a*b*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2*b/(a+b)))*sqrt(a+b*cos(c+d*x))/(d*sqrt((a+b*cos(c+d*x))/(a+b)))+1/4*b*(11*a^2+8*b^2)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2*b/(a+b)))*sqrt((a+b*cos(c+d*x))/(a+b))/(d*sqrt(a+b*cos(c+d*x)))+1/4*a*(4*a^2+15*b^2)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2,sqrt(2*b/(a+b)))*sqrt((a+b*cos(c+d*x))/(a+b))/(d*sqrt(a+b*cos(c+d*x)))+9/4*a*b*sqrt(a+b*cos(c+d*x))*tan(c+d*x)/d+1/2*a^2*sec(c+d*x)*sqrt(a+b*cos(c+d*x))*tan(c+d*x)/d],
[(a+b*cos(c+d*x))^(5/2)*sec(c+d*x)^4,x,11,-1/24*(16*a^2+33*b^2)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2*b/(a+b)))*sqrt(a+b*cos(c+d*x))/(d*sqrt((a+b*cos(c+d*x))/(a+b)))+1/24*a*(16*a^2+59*b^2)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2*b/(a+b)))*sqrt((a+b*cos(c+d*x))/(a+b))/(d*sqrt(a+b*cos(c+d*x)))+5/8*b*(4*a^2+b^2)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2,sqrt(2*b/(a+b)))*sqrt((a+b*cos(c+d*x))/(a+b))/(d*sqrt(a+b*cos(c+d*x)))+1/24*(16*a^2+33*b^2)*sqrt(a+b*cos(c+d*x))*tan(c+d*x)/d+13/12*a*b*sec(c+d*x)*sqrt(a+b*cos(c+d*x))*tan(c+d*x)/d+1/3*a^2*sec(c+d*x)^2*sqrt(a+b*cos(c+d*x))*tan(c+d*x)/d],
[(a+b*cos(c+d*x))^(7/2),x,8,24/35*a*b*(a+b*cos(c+d*x))^(3/2)*sin(c+d*x)/d+2/7*b*(a+b*cos(c+d*x))^(5/2)*sin(c+d*x)/d+2/105*b*(71*a^2+25*b^2)*sin(c+d*x)*sqrt(a+b*cos(c+d*x))/d+32/105*a*(11*a^2+13*b^2)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2*b/(a+b)))*sqrt(a+b*cos(c+d*x))/(d*sqrt((a+b*cos(c+d*x))/(a+b)))-2/105*(71*a^4-46*a^2*b^2-25*b^4)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2*b/(a+b)))*sqrt((a+b*cos(c+d*x))/(a+b))/(d*sqrt(a+b*cos(c+d*x)))],
[cos(c+d*x)^3*sqrt(3+4*cos(c+d*x)),x,6,-3/70*(3+4*cos(c+d*x))^(3/2)*sin(c+d*x)/d+1/14*cos(c+d*x)*(3+4*cos(c+d*x))^(3/2)*sin(c+d*x)/d+47/20*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(8/7))/(d*sqrt(7))+59/60*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(8/7))/(d*sqrt(7))+59/105*sin(c+d*x)*sqrt(3+4*cos(c+d*x))/d],
[cos(c+d*x)^2*sqrt(3+4*cos(c+d*x)),x,5,1/10*(3+4*cos(c+d*x))^(3/2)*sin(c+d*x)/d+21/20*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(8/7))*sqrt(7)/d-1/20*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(8/7))*sqrt(7)/d-1/5*sin(c+d*x)*sqrt(3+4*cos(c+d*x))/d],
[cos(c+d*x)*sqrt(3+4*cos(c+d*x)),x,4,1/2*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(8/7))*sqrt(7)/d+1/6*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(8/7))*sqrt(7)/d+2/3*sin(c+d*x)*sqrt(3+4*cos(c+d*x))/d],
[sqrt(3+4*cos(c+d*x)),x,1,2*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(8/7))*sqrt(7)/d],
[sec(c+d*x)*sqrt(3+4*cos(c+d*x)),x,3,8*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(8/7))/(d*sqrt(7))+6*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2,sqrt(8/7))/(d*sqrt(7))],
[sec(c+d*x)^2*sqrt(3+4*cos(c+d*x)),x,6,3*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(8/7))/(d*sqrt(7))+4*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2,sqrt(8/7))/(d*sqrt(7))-sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(8/7))*sqrt(7)/d+sqrt(3+4*cos(c+d*x))*tan(c+d*x)/d],
[sec(c+d*x)^3*sqrt(3+4*cos(c+d*x)),x,7,3*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(8/7))/(d*sqrt(7))+5/3*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2,sqrt(8/7))/(d*sqrt(7))-1/3*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(8/7))*sqrt(7)/d+1/3*sqrt(3+4*cos(c+d*x))*tan(c+d*x)/d+1/2*sec(c+d*x)*sqrt(3+4*cos(c+d*x))*tan(c+d*x)/d],
[cos(c+d*x)^3*sqrt(3-4*cos(c+d*x)),x,6,-3/70*(3-4*cos(c+d*x))^(3/2)*sin(c+d*x)/d-1/14*(3-4*cos(c+d*x))^(3/2)*cos(c+d*x)*sin(c+d*x)/d-47/20*sqrt(cos(1/2*(c+Pi+d*x))^2)/cos(1/2*(c+Pi+d*x))*EllipticE(sin(1/2*(c+Pi+d*x)),sqrt(8/7))/(d*sqrt(7))-59/60*sqrt(cos(1/2*(c+Pi+d*x))^2)/cos(1/2*(c+Pi+d*x))*EllipticF(sin(1/2*(c+Pi+d*x)),sqrt(8/7))/(d*sqrt(7))+59/105*sin(c+d*x)*sqrt(3-4*cos(c+d*x))/d],
[cos(c+d*x)^2*sqrt(3-4*cos(c+d*x)),x,5,-1/10*(3-4*cos(c+d*x))^(3/2)*sin(c+d*x)/d+21/20*sqrt(cos(1/2*(c+Pi+d*x))^2)/cos(1/2*(c+Pi+d*x))*EllipticE(sin(1/2*(c+Pi+d*x)),sqrt(8/7))*sqrt(7)/d-1/20*sqrt(cos(1/2*(c+Pi+d*x))^2)/cos(1/2*(c+Pi+d*x))*EllipticF(sin(1/2*(c+Pi+d*x)),sqrt(8/7))*sqrt(7)/d+1/5*sin(c+d*x)*sqrt(3-4*cos(c+d*x))/d],
[cos(c+d*x)*sqrt(3-4*cos(c+d*x)),x,4,-1/2*sqrt(cos(1/2*(c+Pi+d*x))^2)/cos(1/2*(c+Pi+d*x))*EllipticE(sin(1/2*(c+Pi+d*x)),sqrt(8/7))*sqrt(7)/d-1/6*sqrt(cos(1/2*(c+Pi+d*x))^2)/cos(1/2*(c+Pi+d*x))*EllipticF(sin(1/2*(c+Pi+d*x)),sqrt(8/7))*sqrt(7)/d+2/3*sin(c+d*x)*sqrt(3-4*cos(c+d*x))/d],
[sqrt(3-4*cos(c+d*x)),x,1,2*sqrt(cos(1/2*(c+Pi+d*x))^2)/cos(1/2*(c+Pi+d*x))*EllipticE(sin(1/2*(c+Pi+d*x)),sqrt(8/7))*sqrt(7)/d],
[sec(c+d*x)*sqrt(3-4*cos(c+d*x)),x,3,-8*sqrt(cos(1/2*(c+Pi+d*x))^2)/cos(1/2*(c+Pi+d*x))*EllipticF(sin(1/2*(c+Pi+d*x)),sqrt(8/7))/(d*sqrt(7))-6*sqrt(cos(1/2*(c+Pi+d*x))^2)/cos(1/2*(c+Pi+d*x))*EllipticPi(sin(1/2*(c+Pi+d*x)),2,sqrt(8/7))/(d*sqrt(7))],
[sec(c+d*x)^2*sqrt(3-4*cos(c+d*x)),x,6,3*sqrt(cos(1/2*(c+Pi+d*x))^2)/cos(1/2*(c+Pi+d*x))*EllipticF(sin(1/2*(c+Pi+d*x)),sqrt(8/7))/(d*sqrt(7))+4*sqrt(cos(1/2*(c+Pi+d*x))^2)/cos(1/2*(c+Pi+d*x))*EllipticPi(sin(1/2*(c+Pi+d*x)),2,sqrt(8/7))/(d*sqrt(7))-sqrt(cos(1/2*(c+Pi+d*x))^2)/cos(1/2*(c+Pi+d*x))*EllipticE(sin(1/2*(c+Pi+d*x)),sqrt(8/7))*sqrt(7)/d+sqrt(3-4*cos(c+d*x))*tan(c+d*x)/d],
[sec(c+d*x)^3*sqrt(3-4*cos(c+d*x)),x,7,-3*sqrt(cos(1/2*(c+Pi+d*x))^2)/cos(1/2*(c+Pi+d*x))*EllipticF(sin(1/2*(c+Pi+d*x)),sqrt(8/7))/(d*sqrt(7))-5/3*sqrt(cos(1/2*(c+Pi+d*x))^2)/cos(1/2*(c+Pi+d*x))*EllipticPi(sin(1/2*(c+Pi+d*x)),2,sqrt(8/7))/(d*sqrt(7))+1/3*sqrt(cos(1/2*(c+Pi+d*x))^2)/cos(1/2*(c+Pi+d*x))*EllipticE(sin(1/2*(c+Pi+d*x)),sqrt(8/7))*sqrt(7)/d-1/3*sqrt(3-4*cos(c+d*x))*tan(c+d*x)/d+1/2*sec(c+d*x)*sqrt(3-4*cos(c+d*x))*tan(c+d*x)/d],

# n<0
[cos(c+d*x)^3/sqrt(a+b*cos(c+d*x)),x,7,-8/15*a*sin(c+d*x)*sqrt(a+b*cos(c+d*x))/(b^2*d)+2/5*cos(c+d*x)*sin(c+d*x)*sqrt(a+b*cos(c+d*x))/(b*d)+2/15*(8*a^2+9*b^2)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2*b/(a+b)))*sqrt(a+b*cos(c+d*x))/(b^3*d*sqrt((a+b*cos(c+d*x))/(a+b)))-2/15*a*(8*a^2+7*b^2)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2*b/(a+b)))*sqrt((a+b*cos(c+d*x))/(a+b))/(b^3*d*sqrt(a+b*cos(c+d*x)))],
[cos(c+d*x)^2/sqrt(a+b*cos(c+d*x)),x,6,2/3*sin(c+d*x)*sqrt(a+b*cos(c+d*x))/(b*d)-4/3*a*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2*b/(a+b)))*sqrt(a+b*cos(c+d*x))/(b^2*d*sqrt((a+b*cos(c+d*x))/(a+b)))+2/3*(2*a^2+b^2)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2*b/(a+b)))*sqrt((a+b*cos(c+d*x))/(a+b))/(b^2*d*sqrt(a+b*cos(c+d*x)))],
[cos(c+d*x)/sqrt(a+b*cos(c+d*x)),x,5,2*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2*b/(a+b)))*sqrt(a+b*cos(c+d*x))/(b*d*sqrt((a+b*cos(c+d*x))/(a+b)))-2*a*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2*b/(a+b)))*sqrt((a+b*cos(c+d*x))/(a+b))/(b*d*sqrt(a+b*cos(c+d*x)))],
[1/sqrt(a+b*cos(c+d*x)),x,2,2*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2*b/(a+b)))*sqrt((a+b*cos(c+d*x))/(a+b))/(d*sqrt(a+b*cos(c+d*x)))],
[sec(c+d*x)/sqrt(a+b*cos(c+d*x)),x,2,2*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2,sqrt(2*b/(a+b)))*sqrt((a+b*cos(c+d*x))/(a+b))/(d*sqrt(a+b*cos(c+d*x)))],
[sec(c+d*x)^2/sqrt(a+b*cos(c+d*x)),x,9,-sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2*b/(a+b)))*sqrt(a+b*cos(c+d*x))/(a*d*sqrt((a+b*cos(c+d*x))/(a+b)))+sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2*b/(a+b)))*sqrt((a+b*cos(c+d*x))/(a+b))/(d*sqrt(a+b*cos(c+d*x)))-b*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2,sqrt(2*b/(a+b)))*sqrt((a+b*cos(c+d*x))/(a+b))/(a*d*sqrt(a+b*cos(c+d*x)))+sqrt(a+b*cos(c+d*x))*tan(c+d*x)/(a*d)],
[sec(c+d*x)^3/sqrt(a+b*cos(c+d*x)),x,10,3/4*b*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2*b/(a+b)))*sqrt(a+b*cos(c+d*x))/(a^2*d*sqrt((a+b*cos(c+d*x))/(a+b)))-1/4*b*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2*b/(a+b)))*sqrt((a+b*cos(c+d*x))/(a+b))/(a*d*sqrt(a+b*cos(c+d*x)))+1/4*(4*a^2+3*b^2)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2,sqrt(2*b/(a+b)))*sqrt((a+b*cos(c+d*x))/(a+b))/(a^2*d*sqrt(a+b*cos(c+d*x)))-3/4*b*sqrt(a+b*cos(c+d*x))*tan(c+d*x)/(a^2*d)+1/2*sec(c+d*x)*sqrt(a+b*cos(c+d*x))*tan(c+d*x)/(a*d)],
[cos(c+d*x)^4/(a+b*cos(c+d*x))^(3/2),x,8,-2*a^2*cos(c+d*x)^2*sin(c+d*x)/(b*(a^2-b^2)*d*sqrt(a+b*cos(c+d*x)))-2/5*a*(8*a^2-3*b^2)*sin(c+d*x)*sqrt(a+b*cos(c+d*x))/(b^3*(a^2-b^2)*d)+2/5*(6*a^2-b^2)*cos(c+d*x)*sin(c+d*x)*sqrt(a+b*cos(c+d*x))/(b^2*(a^2-b^2)*d)+2/5*(16*a^4-8*a^2*b^2-3*b^4)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2*b/(a+b)))*sqrt(a+b*cos(c+d*x))/(b^4*(a^2-b^2)*d*sqrt((a+b*cos(c+d*x))/(a+b)))-8/5*a*(4*a^2+b^2)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2*b/(a+b)))*sqrt((a+b*cos(c+d*x))/(a+b))/(b^4*d*sqrt(a+b*cos(c+d*x)))],
[cos(c+d*x)^3/(a+b*cos(c+d*x))^(3/2),x,7,-2*a^2*cos(c+d*x)*sin(c+d*x)/(b*(a^2-b^2)*d*sqrt(a+b*cos(c+d*x)))+2/3*(4*a^2-b^2)*sin(c+d*x)*sqrt(a+b*cos(c+d*x))/(b^2*(a^2-b^2)*d)-2/3*a*(8*a^2-5*b^2)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2*b/(a+b)))*sqrt(a+b*cos(c+d*x))/(b^3*(a^2-b^2)*d*sqrt((a+b*cos(c+d*x))/(a+b)))+2/3*(8*a^2+b^2)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2*b/(a+b)))*sqrt((a+b*cos(c+d*x))/(a+b))/(b^3*d*sqrt(a+b*cos(c+d*x)))],
[cos(c+d*x)^2/(a+b*cos(c+d*x))^(3/2),x,6,-2*a^2*sin(c+d*x)/(b*(a^2-b^2)*d*sqrt(a+b*cos(c+d*x)))+2*(2*a^2-b^2)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2*b/(a+b)))*sqrt(a+b*cos(c+d*x))/(b^2*(a^2-b^2)*d*sqrt((a+b*cos(c+d*x))/(a+b)))-4*a*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2*b/(a+b)))*sqrt((a+b*cos(c+d*x))/(a+b))/(b^2*d*sqrt(a+b*cos(c+d*x)))],
[cos(c+d*x)/(a+b*cos(c+d*x))^(3/2),x,6,2*a*sin(c+d*x)/((a^2-b^2)*d*sqrt(a+b*cos(c+d*x)))-2*a*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2*b/(a+b)))*sqrt(a+b*cos(c+d*x))/(b*(a^2-b^2)*d*sqrt((a+b*cos(c+d*x))/(a+b)))+2*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2*b/(a+b)))*sqrt((a+b*cos(c+d*x))/(a+b))/(b*d*sqrt(a+b*cos(c+d*x)))],
[1/(a+b*cos(c+d*x))^(3/2),x,4,-2*b*sin(c+d*x)/((a^2-b^2)*d*sqrt(a+b*cos(c+d*x)))+2*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2*b/(a+b)))*sqrt(a+b*cos(c+d*x))/((a^2-b^2)*d*sqrt((a+b*cos(c+d*x))/(a+b)))],
[sec(c+d*x)/(a+b*cos(c+d*x))^(3/2),x,7,2*b^2*sin(c+d*x)/(a*(a^2-b^2)*d*sqrt(a+b*cos(c+d*x)))-2*b*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2*b/(a+b)))*sqrt(a+b*cos(c+d*x))/(a*(a^2-b^2)*d*sqrt((a+b*cos(c+d*x))/(a+b)))+2*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2,sqrt(2*b/(a+b)))*sqrt((a+b*cos(c+d*x))/(a+b))/(a*d*sqrt(a+b*cos(c+d*x)))],
[sec(c+d*x)^2/(a+b*cos(c+d*x))^(3/2),x,10,b*(a^2-3*b^2)*sin(c+d*x)/(a^2*(a^2-b^2)*d*sqrt(a+b*cos(c+d*x)))-(a^2-3*b^2)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2*b/(a+b)))*sqrt(a+b*cos(c+d*x))/(a^2*(a^2-b^2)*d*sqrt((a+b*cos(c+d*x))/(a+b)))+sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2*b/(a+b)))*sqrt((a+b*cos(c+d*x))/(a+b))/(a*d*sqrt(a+b*cos(c+d*x)))-3*b*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2,sqrt(2*b/(a+b)))*sqrt((a+b*cos(c+d*x))/(a+b))/(a^2*d*sqrt(a+b*cos(c+d*x)))+tan(c+d*x)/(a*d*sqrt(a+b*cos(c+d*x)))],
[sec(c+d*x)^3/(a+b*cos(c+d*x))^(3/2),x,11,-1/4*b^2*(7*a^2-15*b^2)*sin(c+d*x)/(a^3*(a^2-b^2)*d*sqrt(a+b*cos(c+d*x)))+1/4*b*(7*a^2-15*b^2)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2*b/(a+b)))*sqrt(a+b*cos(c+d*x))/(a^3*(a^2-b^2)*d*sqrt((a+b*cos(c+d*x))/(a+b)))-5/4*b*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2*b/(a+b)))*sqrt((a+b*cos(c+d*x))/(a+b))/(a^2*d*sqrt(a+b*cos(c+d*x)))+1/4*(4*a^2+15*b^2)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2,sqrt(2*b/(a+b)))*sqrt((a+b*cos(c+d*x))/(a+b))/(a^3*d*sqrt(a+b*cos(c+d*x)))-5/4*b*tan(c+d*x)/(a^2*d*sqrt(a+b*cos(c+d*x)))+1/2*sec(c+d*x)*tan(c+d*x)/(a*d*sqrt(a+b*cos(c+d*x)))],
[cos(c+d*x)^5/(a+b*cos(c+d*x))^(5/2),x,9,-2/3*a^2*cos(c+d*x)^3*sin(c+d*x)/(b*(a^2-b^2)*d*(a+b*cos(c+d*x))^(3/2))-8/3*a^2*(2*a^2-3*b^2)*cos(c+d*x)^2*sin(c+d*x)/(b^2*(a^2-b^2)^2*d*sqrt(a+b*cos(c+d*x)))-4/15*a*(32*a^4-49*a^2*b^2+7*b^4)*sin(c+d*x)*sqrt(a+b*cos(c+d*x))/(b^4*(a^2-b^2)^2*d)+2/15*(48*a^4-71*a^2*b^2+3*b^4)*cos(c+d*x)*sin(c+d*x)*sqrt(a+b*cos(c+d*x))/(b^3*(a^2-b^2)^2*d)+2/15*(128*a^6-212*a^4*b^2+55*a^2*b^4+9*b^6)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2*b/(a+b)))*sqrt(a+b*cos(c+d*x))/(b^5*(a^2-b^2)^2*d*sqrt((a+b*cos(c+d*x))/(a+b)))-2/15*a*(128*a^4-116*a^2*b^2-17*b^4)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2*b/(a+b)))*sqrt((a+b*cos(c+d*x))/(a+b))/(b^5*(a^2-b^2)*d*sqrt(a+b*cos(c+d*x)))],
[cos(c+d*x)^4/(a+b*cos(c+d*x))^(5/2),x,8,-2/3*a^2*cos(c+d*x)^2*sin(c+d*x)/(b*(a^2-b^2)*d*(a+b*cos(c+d*x))^(3/2))+4/3*a^3*(3*a^2-5*b^2)*sin(c+d*x)/(b^3*(a^2-b^2)^2*d*sqrt(a+b*cos(c+d*x)))+2/3*(2*a^2-b^2)*sin(c+d*x)*sqrt(a+b*cos(c+d*x))/(b^3*(a^2-b^2)*d)-8/3*a*(4*a^4-7*a^2*b^2+2*b^4)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2*b/(a+b)))*sqrt(a+b*cos(c+d*x))/(b^4*(a^2-b^2)^2*d*sqrt((a+b*cos(c+d*x))/(a+b)))+2/3*(16*a^4-16*a^2*b^2-b^4)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2*b/(a+b)))*sqrt((a+b*cos(c+d*x))/(a+b))/(b^4*(a^2-b^2)*d*sqrt(a+b*cos(c+d*x)))],
[cos(c+d*x)^3/(a+b*cos(c+d*x))^(5/2),x,7,-2/3*a^2*cos(c+d*x)*sin(c+d*x)/(b*(a^2-b^2)*d*(a+b*cos(c+d*x))^(3/2))-8/3*a^2*(a^2-2*b^2)*sin(c+d*x)/(b^2*(a^2-b^2)^2*d*sqrt(a+b*cos(c+d*x)))+2/3*(8*a^4-15*a^2*b^2+3*b^4)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2*b/(a+b)))*sqrt(a+b*cos(c+d*x))/(b^3*(a^2-b^2)^2*d*sqrt((a+b*cos(c+d*x))/(a+b)))-2/3*a*(8*a^2-9*b^2)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2*b/(a+b)))*sqrt((a+b*cos(c+d*x))/(a+b))/(b^3*(a^2-b^2)*d*sqrt(a+b*cos(c+d*x)))],
[cos(c+d*x)^2/(a+b*cos(c+d*x))^(5/2),x,7,-2/3*a^2*sin(c+d*x)/(b*(a^2-b^2)*d*(a+b*cos(c+d*x))^(3/2))+4/3*a*(a^2-3*b^2)*sin(c+d*x)/(b*(a^2-b^2)^2*d*sqrt(a+b*cos(c+d*x)))-4/3*a*(a^2-3*b^2)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2*b/(a+b)))*sqrt(a+b*cos(c+d*x))/(b^2*(a^2-b^2)^2*d*sqrt((a+b*cos(c+d*x))/(a+b)))+2/3*(2*a^2-3*b^2)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2*b/(a+b)))*sqrt((a+b*cos(c+d*x))/(a+b))/(b^2*(a^2-b^2)*d*sqrt(a+b*cos(c+d*x)))],
[cos(c+d*x)/(a+b*cos(c+d*x))^(5/2),x,7,2/3*a*sin(c+d*x)/((a^2-b^2)*d*(a+b*cos(c+d*x))^(3/2))+2/3*(a^2+3*b^2)*sin(c+d*x)/((a^2-b^2)^2*d*sqrt(a+b*cos(c+d*x)))-2/3*(a^2+3*b^2)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2*b/(a+b)))*sqrt(a+b*cos(c+d*x))/(b*(a^2-b^2)^2*d*sqrt((a+b*cos(c+d*x))/(a+b)))+2/3*a*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2*b/(a+b)))*sqrt((a+b*cos(c+d*x))/(a+b))/(b*(a^2-b^2)*d*sqrt(a+b*cos(c+d*x)))],
[1/(a+b*cos(c+d*x))^(5/2),x,7,-2/3*b*sin(c+d*x)/((a^2-b^2)*d*(a+b*cos(c+d*x))^(3/2))-8/3*a*b*sin(c+d*x)/((a^2-b^2)^2*d*sqrt(a+b*cos(c+d*x)))+8/3*a*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2*b/(a+b)))*sqrt(a+b*cos(c+d*x))/((a^2-b^2)^2*d*sqrt((a+b*cos(c+d*x))/(a+b)))-2/3*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2*b/(a+b)))*sqrt((a+b*cos(c+d*x))/(a+b))/((a^2-b^2)*d*sqrt(a+b*cos(c+d*x)))],
[sec(c+d*x)/(a+b*cos(c+d*x))^(5/2),x,10,2/3*b^2*sin(c+d*x)/(a*(a^2-b^2)*d*(a+b*cos(c+d*x))^(3/2))+2/3*b^2*(7*a^2-3*b^2)*sin(c+d*x)/(a^2*(a^2-b^2)^2*d*sqrt(a+b*cos(c+d*x)))-2/3*b*(7*a^2-3*b^2)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2*b/(a+b)))*sqrt(a+b*cos(c+d*x))/(a^2*(a^2-b^2)^2*d*sqrt((a+b*cos(c+d*x))/(a+b)))+2/3*b*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2*b/(a+b)))*sqrt((a+b*cos(c+d*x))/(a+b))/(a*(a^2-b^2)*d*sqrt(a+b*cos(c+d*x)))+2*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2,sqrt(2*b/(a+b)))*sqrt((a+b*cos(c+d*x))/(a+b))/(a^2*d*sqrt(a+b*cos(c+d*x)))],
[sec(c+d*x)^2/(a+b*cos(c+d*x))^(5/2),x,11,1/3*b*(3*a^2-5*b^2)*sin(c+d*x)/(a^2*(a^2-b^2)*d*(a+b*cos(c+d*x))^(3/2))+1/3*b*(3*a^4-26*a^2*b^2+15*b^4)*sin(c+d*x)/(a^3*(a^2-b^2)^2*d*sqrt(a+b*cos(c+d*x)))-1/3*(3*a^4-26*a^2*b^2+15*b^4)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2*b/(a+b)))*sqrt(a+b*cos(c+d*x))/(a^3*(a^2-b^2)^2*d*sqrt((a+b*cos(c+d*x))/(a+b)))+1/3*(3*a^2-5*b^2)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2*b/(a+b)))*sqrt((a+b*cos(c+d*x))/(a+b))/(a^2*(a^2-b^2)*d*sqrt(a+b*cos(c+d*x)))-5*b*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2,sqrt(2*b/(a+b)))*sqrt((a+b*cos(c+d*x))/(a+b))/(a^3*d*sqrt(a+b*cos(c+d*x)))+tan(c+d*x)/(a*d*(a+b*cos(c+d*x))^(3/2))],
[1/(a+b*cos(c+d*x))^(7/2),x,8,-2/5*b*sin(c+d*x)/((a^2-b^2)*d*(a+b*cos(c+d*x))^(5/2))-16/15*a*b*sin(c+d*x)/((a^2-b^2)^2*d*(a+b*cos(c+d*x))^(3/2))-2/15*b*(23*a^2+9*b^2)*sin(c+d*x)/((a^2-b^2)^3*d*sqrt(a+b*cos(c+d*x)))+2/15*(23*a^2+9*b^2)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2*b/(a+b)))*sqrt(a+b*cos(c+d*x))/((a^2-b^2)^3*d*sqrt((a+b*cos(c+d*x))/(a+b)))-16/15*a*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2*b/(a+b)))*sqrt((a+b*cos(c+d*x))/(a+b))/((a^2-b^2)^2*d*sqrt(a+b*cos(c+d*x)))],
[cos(c+d*x)^3/sqrt(3+4*cos(c+d*x)),x,5,-23/20*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(8/7))/(d*sqrt(7))+9/20*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(8/7))*sqrt(7)/d-1/10*sin(c+d*x)*sqrt(3+4*cos(c+d*x))/d+1/10*cos(c+d*x)*sin(c+d*x)*sqrt(3+4*cos(c+d*x))/d],
[cos(c+d*x)^2/sqrt(3+4*cos(c+d*x)),x,4,17/12*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(8/7))/(d*sqrt(7))-1/4*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(8/7))*sqrt(7)/d+1/6*sin(c+d*x)*sqrt(3+4*cos(c+d*x))/d],
[cos(c+d*x)/sqrt(3+4*cos(c+d*x)),x,3,-3/2*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(8/7))/(d*sqrt(7))+1/2*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(8/7))*sqrt(7)/d],
[1/sqrt(3+4*cos(c+d*x)),x,1,2*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(8/7))/(d*sqrt(7))],
[sec(c+d*x)/sqrt(3+4*cos(c+d*x)),x,1,2*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2,sqrt(8/7))/(d*sqrt(7))],
[sec(c+d*x)^2/sqrt(3+4*cos(c+d*x)),x,6,sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(8/7))/(d*sqrt(7))-4/3*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2,sqrt(8/7))/(d*sqrt(7))-1/3*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(8/7))*sqrt(7)/d+1/3*sqrt(3+4*cos(c+d*x))*tan(c+d*x)/d],
[sec(c+d*x)^3/sqrt(3+4*cos(c+d*x)),x,7,-1/3*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(8/7))/(d*sqrt(7))+1/3*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(8/7))*sqrt(7)/d+1/3*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2,sqrt(8/7))*sqrt(7)/d-1/3*sqrt(3+4*cos(c+d*x))*tan(c+d*x)/d+1/6*sec(c+d*x)*sqrt(3+4*cos(c+d*x))*tan(c+d*x)/d],
[cos(c+d*x)^3/sqrt(3-4*cos(c+d*x)),x,5,23/20*sqrt(cos(1/2*(c+Pi+d*x))^2)/cos(1/2*(c+Pi+d*x))*EllipticF(sin(1/2*(c+Pi+d*x)),sqrt(8/7))/(d*sqrt(7))-9/20*sqrt(cos(1/2*(c+Pi+d*x))^2)/cos(1/2*(c+Pi+d*x))*EllipticE(sin(1/2*(c+Pi+d*x)),sqrt(8/7))*sqrt(7)/d-1/10*sin(c+d*x)*sqrt(3-4*cos(c+d*x))/d-1/10*cos(c+d*x)*sin(c+d*x)*sqrt(3-4*cos(c+d*x))/d],
[cos(c+d*x)^2/sqrt(3-4*cos(c+d*x)),x,4,17/12*sqrt(cos(1/2*(c+Pi+d*x))^2)/cos(1/2*(c+Pi+d*x))*EllipticF(sin(1/2*(c+Pi+d*x)),sqrt(8/7))/(d*sqrt(7))-1/4*sqrt(cos(1/2*(c+Pi+d*x))^2)/cos(1/2*(c+Pi+d*x))*EllipticE(sin(1/2*(c+Pi+d*x)),sqrt(8/7))*sqrt(7)/d-1/6*sin(c+d*x)*sqrt(3-4*cos(c+d*x))/d],
[cos(c+d*x)/sqrt(3-4*cos(c+d*x)),x,3,3/2*sqrt(cos(1/2*(c+Pi+d*x))^2)/cos(1/2*(c+Pi+d*x))*EllipticF(sin(1/2*(c+Pi+d*x)),sqrt(8/7))/(d*sqrt(7))-1/2*sqrt(cos(1/2*(c+Pi+d*x))^2)/cos(1/2*(c+Pi+d*x))*EllipticE(sin(1/2*(c+Pi+d*x)),sqrt(8/7))*sqrt(7)/d],
[1/sqrt(3-4*cos(c+d*x)),x,1,2*sqrt(cos(1/2*(c+Pi+d*x))^2)/cos(1/2*(c+Pi+d*x))*EllipticF(sin(1/2*(c+Pi+d*x)),sqrt(8/7))/(d*sqrt(7))],
[sec(c+d*x)/sqrt(3-4*cos(c+d*x)),x,1,-2*sqrt(cos(1/2*(c+Pi+d*x))^2)/cos(1/2*(c+Pi+d*x))*EllipticPi(sin(1/2*(c+Pi+d*x)),2,sqrt(8/7))/(d*sqrt(7))],
[sec(c+d*x)^2/sqrt(3-4*cos(c+d*x)),x,6,sqrt(cos(1/2*(c+Pi+d*x))^2)/cos(1/2*(c+Pi+d*x))*EllipticF(sin(1/2*(c+Pi+d*x)),sqrt(8/7))/(d*sqrt(7))-4/3*sqrt(cos(1/2*(c+Pi+d*x))^2)/cos(1/2*(c+Pi+d*x))*EllipticPi(sin(1/2*(c+Pi+d*x)),2,sqrt(8/7))/(d*sqrt(7))-1/3*sqrt(cos(1/2*(c+Pi+d*x))^2)/cos(1/2*(c+Pi+d*x))*EllipticE(sin(1/2*(c+Pi+d*x)),sqrt(8/7))*sqrt(7)/d+1/3*sqrt(3-4*cos(c+d*x))*tan(c+d*x)/d],
[sec(c+d*x)^3/sqrt(3-4*cos(c+d*x)),x,7,1/3*sqrt(cos(1/2*(c+Pi+d*x))^2)/cos(1/2*(c+Pi+d*x))*EllipticF(sin(1/2*(c+Pi+d*x)),sqrt(8/7))/(d*sqrt(7))-1/3*sqrt(cos(1/2*(c+Pi+d*x))^2)/cos(1/2*(c+Pi+d*x))*EllipticE(sin(1/2*(c+Pi+d*x)),sqrt(8/7))*sqrt(7)/d-1/3*sqrt(cos(1/2*(c+Pi+d*x))^2)/cos(1/2*(c+Pi+d*x))*EllipticPi(sin(1/2*(c+Pi+d*x)),2,sqrt(8/7))*sqrt(7)/d+1/3*sqrt(3-4*cos(c+d*x))*tan(c+d*x)/d+1/6*sec(c+d*x)*sqrt(3-4*cos(c+d*x))*tan(c+d*x)/d],

# Integrands of the form Cos[e+f x]^(m/2) (a+b Cos[e+f x])^n

# n>0
[cos(c+d*x)^(5/2)*(A+B*cos(c+d*x)),x,6,6/5*A*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/d+10/21*B*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))/d+2/5*A*cos(c+d*x)^(3/2)*sin(c+d*x)/d+2/7*B*cos(c+d*x)^(5/2)*sin(c+d*x)/d+10/21*B*sin(c+d*x)*sqrt(cos(c+d*x))/d],
[cos(c+d*x)^(3/2)*(A+B*cos(c+d*x)),x,5,6/5*B*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/d+2/3*A*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))/d+2/5*B*cos(c+d*x)^(3/2)*sin(c+d*x)/d+2/3*A*sin(c+d*x)*sqrt(cos(c+d*x))/d],
[(A+B*cos(c+d*x))*sqrt(cos(c+d*x)),x,4,2*A*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/d+2/3*B*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))/d+2/3*B*sin(c+d*x)*sqrt(cos(c+d*x))/d],
[(A+B*cos(c+d*x))/sqrt(cos(c+d*x)),x,3,2*B*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/d+2*A*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))/d],
[(A+B*cos(c+d*x))/cos(c+d*x)^(3/2),x,4,-2*A*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/d+2*B*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))/d+2*A*sin(c+d*x)/(d*sqrt(cos(c+d*x)))],
[(A+B*cos(c+d*x))/cos(c+d*x)^(5/2),x,5,-2*B*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/d+2/3*A*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))/d+2/3*A*sin(c+d*x)/(d*cos(c+d*x)^(3/2))+2*B*sin(c+d*x)/(d*sqrt(cos(c+d*x)))],
[(A+B*cos(c+d*x))/cos(c+d*x)^(7/2),x,6,-6/5*A*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/d+2/3*B*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))/d+2/5*A*sin(c+d*x)/(d*cos(c+d*x)^(5/2))+2/3*B*sin(c+d*x)/(d*cos(c+d*x)^(3/2))+6/5*A*sin(c+d*x)/(d*sqrt(cos(c+d*x)))],
[cos(c+d*x)^(5/2)*(a+b*cos(c+d*x))^2,x,7,2/15*(9*a^2+7*b^2)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/d+20/21*a*b*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))/d+2/45*(9*a^2+7*b^2)*cos(c+d*x)^(3/2)*sin(c+d*x)/d+4/7*a*b*cos(c+d*x)^(5/2)*sin(c+d*x)/d+2/9*b^2*cos(c+d*x)^(7/2)*sin(c+d*x)/d+20/21*a*b*sin(c+d*x)*sqrt(cos(c+d*x))/d],
[cos(c+d*x)^(3/2)*(a+b*cos(c+d*x))^2,x,6,12/5*a*b*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/d+2/21*(7*a^2+5*b^2)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))/d+4/5*a*b*cos(c+d*x)^(3/2)*sin(c+d*x)/d+2/7*b^2*cos(c+d*x)^(5/2)*sin(c+d*x)/d+2/21*(7*a^2+5*b^2)*sin(c+d*x)*sqrt(cos(c+d*x))/d],
[cos(c+d*x)^(1/2)*(a+b*cos(c+d*x))^2,x,5,2/5*(5*a^2+3*b^2)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/d+4/3*a*b*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))/d+2/5*b^2*cos(c+d*x)^(3/2)*sin(c+d*x)/d+4/3*a*b*sin(c+d*x)*sqrt(cos(c+d*x))/d],
[(a+b*cos(c+d*x))^2/cos(c+d*x)^(1/2),x,4,4*a*b*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/d+2/3*(3*a^2+b^2)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))/d+2/3*b^2*sin(c+d*x)*sqrt(cos(c+d*x))/d],
[(a+b*cos(c+d*x))^2/cos(c+d*x)^(3/2),x,4,-2*(a^2-b^2)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/d+4*a*b*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))/d+2*a^2*sin(c+d*x)/(d*sqrt(cos(c+d*x)))],
[(a+b*cos(c+d*x))^2/cos(c+d*x)^(5/2),x,5,-4*a*b*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/d+2/3*(a^2+3*b^2)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))/d+2/3*a^2*sin(c+d*x)/(d*cos(c+d*x)^(3/2))+4*a*b*sin(c+d*x)/(d*sqrt(cos(c+d*x)))],
[(a+b*cos(c+d*x))^2/cos(c+d*x)^(7/2),x,6,-2/5*(3*a^2+5*b^2)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/d+4/3*a*b*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))/d+2/5*a^2*sin(c+d*x)/(d*cos(c+d*x)^(5/2))+4/3*a*b*sin(c+d*x)/(d*cos(c+d*x)^(3/2))+2/5*(3*a^2+5*b^2)*sin(c+d*x)/(d*sqrt(cos(c+d*x)))],
[cos(c+d*x)^(3/2)*(a+b*cos(c+d*x))^3,x,7,2/15*b*(27*a^2+7*b^2)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/d+2/21*a*(7*a^2+15*b^2)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))/d+2/45*b*(27*a^2+7*b^2)*cos(c+d*x)^(3/2)*sin(c+d*x)/d+40/63*a*b^2*cos(c+d*x)^(5/2)*sin(c+d*x)/d+2/9*b^2*cos(c+d*x)^(5/2)*(a+b*cos(c+d*x))*sin(c+d*x)/d+2/21*a*(7*a^2+15*b^2)*sin(c+d*x)*sqrt(cos(c+d*x))/d],
[cos(c+d*x)^(1/2)*(a+b*cos(c+d*x))^3,x,6,2/5*a*(5*a^2+9*b^2)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/d+2/21*b*(21*a^2+5*b^2)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))/d+32/35*a*b^2*cos(c+d*x)^(3/2)*sin(c+d*x)/d+2/7*b^2*cos(c+d*x)^(3/2)*(a+b*cos(c+d*x))*sin(c+d*x)/d+2/21*b*(21*a^2+5*b^2)*sin(c+d*x)*sqrt(cos(c+d*x))/d],
[(a+b*cos(c+d*x))^3/cos(c+d*x)^(1/2),x,5,6/5*b*(5*a^2+b^2)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/d+2*a*(a^2+b^2)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))/d+8/5*a*b^2*sin(c+d*x)*sqrt(cos(c+d*x))/d+2/5*b^2*(a+b*cos(c+d*x))*sin(c+d*x)*sqrt(cos(c+d*x))/d],
[(a+b*cos(c+d*x))^3/cos(c+d*x)^(3/2),x,5,-2*a*(a^2-3*b^2)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/d+2/3*b*(9*a^2+b^2)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))/d+2*a^2*(a+b*cos(c+d*x))*sin(c+d*x)/(d*sqrt(cos(c+d*x)))-2/3*b*(3*a^2-b^2)*sin(c+d*x)*sqrt(cos(c+d*x))/d],
[(a+b*cos(c+d*x))^3/cos(c+d*x)^(5/2),x,5,-2*b*(3*a^2-b^2)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/d+2/3*a*(a^2+9*b^2)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))/d+2/3*a^2*(a+b*cos(c+d*x))*sin(c+d*x)/(d*cos(c+d*x)^(3/2))+16/3*a^2*b*sin(c+d*x)/(d*sqrt(cos(c+d*x)))],
[(a+b*cos(c+d*x))^3/cos(c+d*x)^(7/2),x,6,-6/5*a*(a^2+5*b^2)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/d+2*b*(a^2+b^2)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))/d+8/5*a^2*b*sin(c+d*x)/(d*cos(c+d*x)^(3/2))+2/5*a^2*(a+b*cos(c+d*x))*sin(c+d*x)/(d*cos(c+d*x)^(5/2))+6/5*a*(a^2+5*b^2)*sin(c+d*x)/(d*sqrt(cos(c+d*x)))],
[(a+b*cos(c+d*x))^3/cos(c+d*x)^(9/2),x,7,-2/5*b*(9*a^2+5*b^2)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/d+2/21*a*(5*a^2+21*b^2)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))/d+32/35*a^2*b*sin(c+d*x)/(d*cos(c+d*x)^(5/2))+2/21*a*(5*a^2+21*b^2)*sin(c+d*x)/(d*cos(c+d*x)^(3/2))+2/7*a^2*(a+b*cos(c+d*x))*sin(c+d*x)/(d*cos(c+d*x)^(7/2))+2/5*b*(9*a^2+5*b^2)*sin(c+d*x)/(d*sqrt(cos(c+d*x)))],

# n<0
[cos(c+d*x)^(5/2)/(a+b*cos(c+d*x)),x,6,-2*a*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/(b^2*d)+2/3*(3*a^2+b^2)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))/(b^3*d)-2*a^3*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2*b/(a+b),sqrt(2))/(b^3*(a+b)*d)+2/3*sin(c+d*x)*sqrt(cos(c+d*x))/(b*d)],
[cos(c+d*x)^(3/2)/(a+b*cos(c+d*x)),x,5,2*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/(b*d)-2*a*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))/(b^2*d)+2*a^2*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2*b/(a+b),sqrt(2))/(b^2*(a+b)*d)],
[cos(c+d*x)^(1/2)/(a+b*cos(c+d*x)),x,3,2*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))/(b*d)-2*a*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2*b/(a+b),sqrt(2))/(b*(a+b)*d)],
[1/(cos(c+d*x)^(1/2)*(a+b*cos(c+d*x))),x,1,2*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2*b/(a+b),sqrt(2))/((a+b)*d)],
[1/(cos(c+d*x)^(3/2)*(a+b*cos(c+d*x))),x,5,-2*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/(a*d)-2*b*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2*b/(a+b),sqrt(2))/(a*(a+b)*d)+2*sin(c+d*x)/(a*d*sqrt(cos(c+d*x)))],
[1/(cos(c+d*x)^(5/2)*(a+b*cos(c+d*x))),x,7,2*b*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/(a^2*d)+2/3*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))/(a*d)+2*b^2*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2*b/(a+b),sqrt(2))/(a^2*(a+b)*d)+2/3*sin(c+d*x)/(a*d*cos(c+d*x)^(3/2))-2*b*sin(c+d*x)/(a^2*d*sqrt(cos(c+d*x)))],
[cos(c+d*x)^(7/2)/(a+b*cos(c+d*x))^2,x,7,-a*(5*a^2-4*b^2)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/(b^3*(a^2-b^2)*d)+1/3*(15*a^4-16*a^2*b^2-2*b^4)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))/(b^4*(a^2-b^2)*d)-a^3*(5*a^2-7*b^2)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2*b/(a+b),sqrt(2))/((a-b)*b^4*(a+b)^2*d)-a^2*cos(c+d*x)^(3/2)*sin(c+d*x)/(b*(a^2-b^2)*d*(a+b*cos(c+d*x)))+1/3*(5*a^2-2*b^2)*sin(c+d*x)*sqrt(cos(c+d*x))/(b^2*(a^2-b^2)*d)],
[cos(c+d*x)^(5/2)/(a+b*cos(c+d*x))^2,x,6,(3*a^2-2*b^2)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/(b^2*(a^2-b^2)*d)-a*(3*a^2-4*b^2)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))/(b^3*(a^2-b^2)*d)+a^2*(3*a^2-5*b^2)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2*b/(a+b),sqrt(2))/((a-b)*b^3*(a+b)^2*d)-a^2*sin(c+d*x)*sqrt(cos(c+d*x))/(b*(a^2-b^2)*d*(a+b*cos(c+d*x)))],
[cos(c+d*x)^(3/2)/(a+b*cos(c+d*x))^2,x,6,-a*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/(b*(a^2-b^2)*d)+(a^2-2*b^2)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))/(b^2*(a^2-b^2)*d)-a*(a^2-3*b^2)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2*b/(a+b),sqrt(2))/((a-b)*b^2*(a+b)^2*d)+a*sin(c+d*x)*sqrt(cos(c+d*x))/((a^2-b^2)*d*(a+b*cos(c+d*x)))],
[cos(c+d*x)^(1/2)/(a+b*cos(c+d*x))^2,x,6,sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/((a^2-b^2)*d)+a*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))/(b*(a^2-b^2)*d)-(a^2+b^2)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2*b/(a+b),sqrt(2))/((a-b)*b*(a+b)^2*d)-b*sin(c+d*x)*sqrt(cos(c+d*x))/((a^2-b^2)*d*(a+b*cos(c+d*x)))],
[1/(cos(c+d*x)^(1/2)*(a+b*cos(c+d*x))^2),x,6,-b*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/(a*(a^2-b^2)*d)-sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))/((a^2-b^2)*d)+(3*a^2-b^2)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2*b/(a+b),sqrt(2))/(a*(a-b)*(a+b)^2*d)+b^2*sin(c+d*x)*sqrt(cos(c+d*x))/(a*(a^2-b^2)*d*(a+b*cos(c+d*x)))],
[1/(cos(c+d*x)^(3/2)*(a+b*cos(c+d*x))^2),x,7,-(2*a^2-3*b^2)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/(a^2*(a^2-b^2)*d)+b*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))/(a*(a^2-b^2)*d)-b*(5*a^2-3*b^2)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2*b/(a+b),sqrt(2))/(a^2*(a-b)*(a+b)^2*d)+(2*a^2-3*b^2)*sin(c+d*x)/(a^2*(a^2-b^2)*d*sqrt(cos(c+d*x)))+b^2*sin(c+d*x)/(a*(a^2-b^2)*d*(a+b*cos(c+d*x))*sqrt(cos(c+d*x)))],
[1/(cos(c+d*x)^(5/2)*(a+b*cos(c+d*x))^2),x,8,b*(4*a^2-5*b^2)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/(a^3*(a^2-b^2)*d)+1/3*(2*a^2-5*b^2)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))/(a^2*(a^2-b^2)*d)+b^2*(7*a^2-5*b^2)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2*b/(a+b),sqrt(2))/(a^3*(a-b)*(a+b)^2*d)+1/3*(2*a^2-5*b^2)*sin(c+d*x)/(a^2*(a^2-b^2)*d*cos(c+d*x)^(3/2))+b^2*sin(c+d*x)/(a*(a^2-b^2)*d*cos(c+d*x)^(3/2)*(a+b*cos(c+d*x)))-b*(4*a^2-5*b^2)*sin(c+d*x)/(a^3*(a^2-b^2)*d*sqrt(cos(c+d*x)))],
[cos(c+d*x)^(9/2)/(a+b*cos(c+d*x))^3,x,8,-1/4*a*(35*a^4-65*a^2*b^2+24*b^4)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/(b^4*(a^2-b^2)^2*d)+1/12*(105*a^6-223*a^4*b^2+128*a^2*b^4+8*b^6)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))/(b^5*(a^2-b^2)^2*d)-1/4*a^3*(35*a^4-86*a^2*b^2+63*b^4)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2*b/(a+b),sqrt(2))/((a-b)^2*b^5*(a+b)^3*d)-1/2*a^2*cos(c+d*x)^(5/2)*sin(c+d*x)/(b*(a^2-b^2)*d*(a+b*cos(c+d*x))^2)-1/4*a^2*(7*a^2-13*b^2)*cos(c+d*x)^(3/2)*sin(c+d*x)/(b^2*(a^2-b^2)^2*d*(a+b*cos(c+d*x)))+1/12*(35*a^4-61*a^2*b^2+8*b^4)*sin(c+d*x)*sqrt(cos(c+d*x))/(b^3*(a^2-b^2)^2*d)],
[cos(c+d*x)^(7/2)/(a+b*cos(c+d*x))^3,x,7,1/4*(15*a^4-29*a^2*b^2+8*b^4)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/(b^3*(a^2-b^2)^2*d)-3/4*a*(5*a^4-11*a^2*b^2+8*b^4)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))/(b^4*(a^2-b^2)^2*d)+1/4*a^2*(15*a^4-38*a^2*b^2+35*b^4)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2*b/(a+b),sqrt(2))/((a-b)^2*b^4*(a+b)^3*d)-1/2*a^2*cos(c+d*x)^(3/2)*sin(c+d*x)/(b*(a^2-b^2)*d*(a+b*cos(c+d*x))^2)-1/4*a^2*(5*a^2-11*b^2)*sin(c+d*x)*sqrt(cos(c+d*x))/(b^2*(a^2-b^2)^2*d*(a+b*cos(c+d*x)))],
[cos(c+d*x)^(5/2)/(a+b*cos(c+d*x))^3,x,7,-3/4*a*(a^2-3*b^2)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/(b^2*(a^2-b^2)^2*d)+1/4*(3*a^4-5*a^2*b^2+8*b^4)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))/(b^3*(a^2-b^2)^2*d)-3/4*a*(a^4-2*a^2*b^2+5*b^4)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2*b/(a+b),sqrt(2))/((a-b)^2*b^3*(a+b)^3*d)-1/2*a^2*sin(c+d*x)*sqrt(cos(c+d*x))/(b*(a^2-b^2)*d*(a+b*cos(c+d*x))^2)+3/4*a*(a^2-3*b^2)*sin(c+d*x)*sqrt(cos(c+d*x))/(b*(a^2-b^2)^2*d*(a+b*cos(c+d*x)))],
[cos(c+d*x)^(3/2)/(a+b*cos(c+d*x))^3,x,7,-1/4*(a^2+5*b^2)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/(b*(a^2-b^2)^2*d)+1/4*a*(a^2-7*b^2)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))/(b^2*(a^2-b^2)^2*d)-1/4*(a^4-10*a^2*b^2-3*b^4)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2*b/(a+b),sqrt(2))/((a-b)^2*b^2*(a+b)^3*d)+1/2*a*sin(c+d*x)*sqrt(cos(c+d*x))/((a^2-b^2)*d*(a+b*cos(c+d*x))^2)+1/4*(a^2+5*b^2)*sin(c+d*x)*sqrt(cos(c+d*x))/((a^2-b^2)^2*d*(a+b*cos(c+d*x)))],
[cos(c+d*x)^(1/2)/(a+b*cos(c+d*x))^3,x,7,1/4*(5*a^2+b^2)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/(a*(a^2-b^2)^2*d)+3/4*(a^2+b^2)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))/(b*(a^2-b^2)^2*d)-1/4*(3*a^4+10*a^2*b^2-b^4)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2*b/(a+b),sqrt(2))/(a*(a-b)^2*b*(a+b)^3*d)-1/2*b*sin(c+d*x)*sqrt(cos(c+d*x))/((a^2-b^2)*d*(a+b*cos(c+d*x))^2)-1/4*b*(5*a^2+b^2)*sin(c+d*x)*sqrt(cos(c+d*x))/(a*(a^2-b^2)^2*d*(a+b*cos(c+d*x)))],
[1/(cos(c+d*x)^(1/2)*(a+b*cos(c+d*x))^3),x,7,-3/4*b*(3*a^2-b^2)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/(a^2*(a^2-b^2)^2*d)-1/4*(7*a^2-b^2)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))/(a*(a^2-b^2)^2*d)+3/4*(5*a^4-2*a^2*b^2+b^4)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2*b/(a+b),sqrt(2))/(a^2*(a-b)^2*(a+b)^3*d)+1/2*b^2*sin(c+d*x)*sqrt(cos(c+d*x))/(a*(a^2-b^2)*d*(a+b*cos(c+d*x))^2)+3/4*b^2*(3*a^2-b^2)*sin(c+d*x)*sqrt(cos(c+d*x))/(a^2*(a^2-b^2)^2*d*(a+b*cos(c+d*x)))],
[1/(cos(c+d*x)^(3/2)*(a+b*cos(c+d*x))^3),x,8,-1/4*(8*a^4-29*a^2*b^2+15*b^4)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/(a^3*(a^2-b^2)^2*d)+1/4*b*(11*a^2-5*b^2)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))/(a^2*(a^2-b^2)^2*d)-1/4*b*(35*a^4-38*a^2*b^2+15*b^4)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2*b/(a+b),sqrt(2))/(a^3*(a-b)^2*(a+b)^3*d)+1/4*(8*a^4-29*a^2*b^2+15*b^4)*sin(c+d*x)/(a^3*(a^2-b^2)^2*d*sqrt(cos(c+d*x)))+1/2*b^2*sin(c+d*x)/(a*(a^2-b^2)*d*(a+b*cos(c+d*x))^2*sqrt(cos(c+d*x)))+1/4*b^2*(11*a^2-5*b^2)*sin(c+d*x)/(a^2*(a^2-b^2)^2*d*(a+b*cos(c+d*x))*sqrt(cos(c+d*x)))],
[1/(cos(c+d*x)^(5/2)*(a+b*cos(c+d*x))^3),x,9,1/4*b*(24*a^4-65*a^2*b^2+35*b^4)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))/(a^4*(a^2-b^2)^2*d)+1/12*(8*a^4-61*a^2*b^2+35*b^4)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))/(a^3*(a^2-b^2)^2*d)+1/4*b^2*(63*a^4-86*a^2*b^2+35*b^4)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2*b/(a+b),sqrt(2))/(a^4*(a-b)^2*(a+b)^3*d)+1/12*(8*a^4-61*a^2*b^2+35*b^4)*sin(c+d*x)/(a^3*(a^2-b^2)^2*d*cos(c+d*x)^(3/2))+1/2*b^2*sin(c+d*x)/(a*(a^2-b^2)*d*cos(c+d*x)^(3/2)*(a+b*cos(c+d*x))^2)+1/4*b^2*(13*a^2-7*b^2)*sin(c+d*x)/(a^2*(a^2-b^2)^2*d*cos(c+d*x)^(3/2)*(a+b*cos(c+d*x)))-1/4*b*(24*a^4-65*a^2*b^2+35*b^4)*sin(c+d*x)/(a^4*(a^2-b^2)^2*d*sqrt(cos(c+d*x)))],

# Integrands of the form Cos[e+f x]^(m/2) (a+b Cos[e+f x])^(n/2)

# n>0
[cos(c+d*x)^(3/2)*(a+b*cos(c+d*x))^(1/2),x,7,1/4*a*sin(c+d*x)*sqrt(a+b*cos(c+d*x))/(b*d*sqrt(cos(c+d*x)))+1/2*sin(c+d*x)*sqrt(cos(c+d*x))*sqrt(a+b*cos(c+d*x))/d-1/4*(a-b)*cot(c+d*x)*EllipticE(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(b*d)+1/4*(a+2*b)*cot(c+d*x)*EllipticF(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(b*d)+1/4*(a^2-4*b^2)*cot(c+d*x)*EllipticPi(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),(a+b)/b,sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(b^2*d)],
[cos(c+d*x)^(1/2)*(a+b*cos(c+d*x))^(1/2),x,7,sin(c+d*x)*sqrt(a+b*cos(c+d*x))/(d*sqrt(cos(c+d*x)))-(a-b)*cot(c+d*x)*EllipticE(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a*d)+cot(c+d*x)*EllipticF(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/d-a*cot(c+d*x)*EllipticPi(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),(a+b)/b,sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(b*d)],
[(a+b*cos(c+d*x))^(1/2)/cos(c+d*x)^(1/2),x,1,-2*(a+b*cos(c+d*x))*csc(c+d*x)*EllipticPi(sqrt(a+b)*sqrt(cos(c+d*x))/sqrt(a+b*cos(c+d*x)),b/(a+b),sqrt((-a+b)/(a+b)))*sqrt(a*(1-cos(c+d*x))/(a+b*cos(c+d*x)))*sqrt(a*(1+cos(c+d*x))/(a+b*cos(c+d*x)))/(d*sqrt(a+b))],
[(a+b*cos(c+d*x))^(1/2)/cos(c+d*x)^(3/2),x,3,2*(a-b)*cot(c+d*x)*EllipticE(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a*d)-2*(a-b)*cot(c+d*x)*EllipticF(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a*d)],
[(a+b*cos(c+d*x))^(1/2)/cos(c+d*x)^(5/2),x,4,2/3*sin(c+d*x)*sqrt(a+b*cos(c+d*x))/(d*cos(c+d*x)^(3/2))+2/3*(a-b)*b*cot(c+d*x)*EllipticE(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a^2*d)+2/3*(a-b)*cot(c+d*x)*EllipticF(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a*d)],
[(a+b*cos(c+d*x))^(1/2)/cos(c+d*x)^(7/2),x,5,2/5*sin(c+d*x)*sqrt(a+b*cos(c+d*x))/(d*cos(c+d*x)^(5/2))+2/15*b*sin(c+d*x)*sqrt(a+b*cos(c+d*x))/(a*d*cos(c+d*x)^(3/2))+2/15*(a-b)*(9*a^2-2*b^2)*cot(c+d*x)*EllipticE(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a^3*d)-2/15*(a-b)*(9*a+2*b)*cot(c+d*x)*EllipticF(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a^2*d)],
[(a+b*cos(c+d*x))^(1/2)/cos(c+d*x)^(9/2),x,6,2/7*sin(c+d*x)*sqrt(a+b*cos(c+d*x))/(d*cos(c+d*x)^(7/2))+2/35*b*sin(c+d*x)*sqrt(a+b*cos(c+d*x))/(a*d*cos(c+d*x)^(5/2))+2/105*(25*a^2-4*b^2)*sin(c+d*x)*sqrt(a+b*cos(c+d*x))/(a^2*d*cos(c+d*x)^(3/2))+2/105*(a-b)*b*(19*a^2+8*b^2)*cot(c+d*x)*EllipticE(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a^4*d)+2/105*(a-b)*(25*a^2+6*a*b+8*b^2)*cot(c+d*x)*EllipticF(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a^3*d)],
[cos(c+d*x)^(3/2)*(a+b*cos(c+d*x))^(3/2),x,8,1/3*(a+b*cos(c+d*x))^(3/2)*sin(c+d*x)*sqrt(cos(c+d*x))/d+1/24*(3*a^2+16*b^2)*sin(c+d*x)*sqrt(a+b*cos(c+d*x))/(b*d*sqrt(cos(c+d*x)))+1/4*a*sin(c+d*x)*sqrt(cos(c+d*x))*sqrt(a+b*cos(c+d*x))/d-1/24*(a-b)*(3*a^2+16*b^2)*cot(c+d*x)*EllipticE(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a*b*d)+1/24*(a+2*b)*(3*a+8*b)*cot(c+d*x)*EllipticF(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(b*d)+1/8*a*(a^2-12*b^2)*cot(c+d*x)*EllipticPi(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),(a+b)/b,sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(b^2*d)],
[cos(c+d*x)^(1/2)*(a+b*cos(c+d*x))^(3/2),x,8,1/2*(a+b*cos(c+d*x))^(3/2)*sin(c+d*x)/(d*sqrt(cos(c+d*x)))+3/4*a*sin(c+d*x)*sqrt(a+b*cos(c+d*x))/(d*sqrt(cos(c+d*x)))-5/4*(a-b)*cot(c+d*x)*EllipticE(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/d+1/4*(5*a+2*b)*cot(c+d*x)*EllipticF(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/d-1/4*(3*a^2+4*b^2)*cot(c+d*x)*EllipticPi(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),(a+b)/b,sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(b*d)],
[(a+b*cos(c+d*x))^(3/2)/cos(c+d*x)^(1/2),x,6,b*sin(c+d*x)*sqrt(a+b*cos(c+d*x))/(d*sqrt(cos(c+d*x)))-(a-b)*b*cot(c+d*x)*EllipticE(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a*d)+(2*a+b)*cot(c+d*x)*EllipticF(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/d-3*a*cot(c+d*x)*EllipticPi(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),(a+b)/b,sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/d],
[(a+b*cos(c+d*x))^(3/2)/cos(c+d*x)^(3/2),x,5,2*(a-b)*cot(c+d*x)*EllipticE(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/d-2*(a-2*b)*cot(c+d*x)*EllipticF(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/d-2*b*cot(c+d*x)*EllipticPi(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),(a+b)/b,sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/d],
[(a+b*cos(c+d*x))^(3/2)/cos(c+d*x)^(5/2),x,4,2/3*a*sin(c+d*x)*sqrt(a+b*cos(c+d*x))/(d*cos(c+d*x)^(3/2))+8/3*(a-b)*b*cot(c+d*x)*EllipticE(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a*d)+2/3*(a-3*b)*(a-b)*cot(c+d*x)*EllipticF(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a*d)],
[(a+b*cos(c+d*x))^(3/2)/cos(c+d*x)^(7/2),x,5,2/5*a*sin(c+d*x)*sqrt(a+b*cos(c+d*x))/(d*cos(c+d*x)^(5/2))+4/5*b*sin(c+d*x)*sqrt(a+b*cos(c+d*x))/(d*cos(c+d*x)^(3/2))+2/5*(a-b)*(3*a^2+b^2)*cot(c+d*x)*EllipticE(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a^2*d)-2/5*(a-b)*(3*a-b)*cot(c+d*x)*EllipticF(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a*d)],
[(a+b*cos(c+d*x))^(3/2)/cos(c+d*x)^(9/2),x,6,2/7*a*sin(c+d*x)*sqrt(a+b*cos(c+d*x))/(d*cos(c+d*x)^(7/2))+16/35*b*sin(c+d*x)*sqrt(a+b*cos(c+d*x))/(d*cos(c+d*x)^(5/2))+2/105*(25*a^2+3*b^2)*sin(c+d*x)*sqrt(a+b*cos(c+d*x))/(a*d*cos(c+d*x)^(3/2))+4/105*(a-b)*b*(41*a^2-3*b^2)*cot(c+d*x)*EllipticE(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a^3*d)+2/105*(a-b)*(25*a^2-57*a*b-6*b^2)*cot(c+d*x)*EllipticF(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a^2*d)],
[(a+b*cos(c+d*x))^(3/2)/cos(c+d*x)^(11/2),x,7,2/9*a*sin(c+d*x)*sqrt(a+b*cos(c+d*x))/(d*cos(c+d*x)^(9/2))+20/63*b*sin(c+d*x)*sqrt(a+b*cos(c+d*x))/(d*cos(c+d*x)^(7/2))+2/315*(49*a^2+3*b^2)*sin(c+d*x)*sqrt(a+b*cos(c+d*x))/(a*d*cos(c+d*x)^(5/2))+8/315*b*(22*a^2-b^2)*sin(c+d*x)*sqrt(a+b*cos(c+d*x))/(a^2*d*cos(c+d*x)^(3/2))+2/315*(a-b)*(147*a^4+33*a^2*b^2+8*b^4)*cot(c+d*x)*EllipticE(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a^4*d)-2/315*(a-b)*(147*a^3-39*a^2*b-6*a*b^2-8*b^3)*cot(c+d*x)*EllipticF(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a^3*d)],
[cos(c+d*x)^(1/2)*(a+b*cos(c+d*x))^(5/2),x,8,1/3*b^2*cos(c+d*x)^(3/2)*sin(c+d*x)*sqrt(a+b*cos(c+d*x))/d+1/24*(33*a^2+16*b^2)*sin(c+d*x)*sqrt(a+b*cos(c+d*x))/(d*sqrt(cos(c+d*x)))+13/12*a*b*sin(c+d*x)*sqrt(cos(c+d*x))*sqrt(a+b*cos(c+d*x))/d-1/24*(a-b)*(33*a^2+16*b^2)*cot(c+d*x)*EllipticE(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a*d)+1/24*(33*a^2+26*a*b+16*b^2)*cot(c+d*x)*EllipticF(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/d-5/8*a*(a^2+4*b^2)*cot(c+d*x)*EllipticPi(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),(a+b)/b,sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(b*d)],
[(a+b*cos(c+d*x))^(5/2)/cos(c+d*x)^(1/2),x,7,9/4*a*b*sin(c+d*x)*sqrt(a+b*cos(c+d*x))/(d*sqrt(cos(c+d*x)))+1/2*b^2*sin(c+d*x)*sqrt(cos(c+d*x))*sqrt(a+b*cos(c+d*x))/d-9/4*(a-b)*b*cot(c+d*x)*EllipticE(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/d+1/4*(8*a^2+9*a*b+2*b^2)*cot(c+d*x)*EllipticF(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/d-1/4*(15*a^2+4*b^2)*cot(c+d*x)*EllipticPi(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),(a+b)/b,sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/d],
[(a+b*cos(c+d*x))^(5/2)/cos(c+d*x)^(3/2),x,7,2*a^2*sin(c+d*x)*sqrt(a+b*cos(c+d*x))/(d*sqrt(cos(c+d*x)))-(2*a^2-b^2)*sin(c+d*x)*sqrt(a+b*cos(c+d*x))/(d*sqrt(cos(c+d*x)))+(a-b)*(2*a^2-b^2)*cot(c+d*x)*EllipticE(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a*d)-(2*a^2-6*a*b-b^2)*cot(c+d*x)*EllipticF(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/d-5*a*b*cot(c+d*x)*EllipticPi(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),(a+b)/b,sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/d],
[(a+b*cos(c+d*x))^(5/2)/cos(c+d*x)^(5/2),x,6,2/3*a^2*sin(c+d*x)*sqrt(a+b*cos(c+d*x))/(d*cos(c+d*x)^(3/2))+14/3*(a-b)*b*cot(c+d*x)*EllipticE(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/d+2/3*(a^2-7*a*b+9*b^2)*cot(c+d*x)*EllipticF(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/d-2*b^2*cot(c+d*x)*EllipticPi(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),(a+b)/b,sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/d],
[(a+b*cos(c+d*x))^(5/2)/cos(c+d*x)^(7/2),x,5,2/5*a^2*sin(c+d*x)*sqrt(a+b*cos(c+d*x))/(d*cos(c+d*x)^(5/2))+22/15*a*b*sin(c+d*x)*sqrt(a+b*cos(c+d*x))/(d*cos(c+d*x)^(3/2))+2/15*(a-b)*(9*a^2+23*b^2)*cot(c+d*x)*EllipticE(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a*d)-2/15*(a-b)*(9*a^2-8*a*b+15*b^2)*cot(c+d*x)*EllipticF(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a*d)],
[(a+b*cos(c+d*x))^(5/2)/cos(c+d*x)^(9/2),x,6,2/7*a^2*sin(c+d*x)*sqrt(a+b*cos(c+d*x))/(d*cos(c+d*x)^(7/2))+6/7*a*b*sin(c+d*x)*sqrt(a+b*cos(c+d*x))/(d*cos(c+d*x)^(5/2))+2/21*(5*a^2+9*b^2)*sin(c+d*x)*sqrt(a+b*cos(c+d*x))/(d*cos(c+d*x)^(3/2))+2/21*(a-b)*b*(29*a^2+3*b^2)*cot(c+d*x)*EllipticE(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a^2*d)+2/21*(a-b)*(5*a^2-24*a*b+3*b^2)*cot(c+d*x)*EllipticF(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a*d)],
[(a+b*cos(c+d*x))^(5/2)/cos(c+d*x)^(11/2),x,7,2/9*a^2*sin(c+d*x)*sqrt(a+b*cos(c+d*x))/(d*cos(c+d*x)^(9/2))+38/63*a*b*sin(c+d*x)*sqrt(a+b*cos(c+d*x))/(d*cos(c+d*x)^(7/2))+2/315*(49*a^2+75*b^2)*sin(c+d*x)*sqrt(a+b*cos(c+d*x))/(d*cos(c+d*x)^(5/2))+2/315*b*(163*a^2+5*b^2)*sin(c+d*x)*sqrt(a+b*cos(c+d*x))/(a*d*cos(c+d*x)^(3/2))+2/315*(a-b)*(147*a^4+279*a^2*b^2-10*b^4)*cot(c+d*x)*EllipticE(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a^3*d)-2/315*(a-b)*(147*a^3-114*a^2*b+165*a*b^2+10*b^3)*cot(c+d*x)*EllipticF(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a^2*d)],
[(a+b*cos(c+d*x))^(5/2)/cos(c+d*x)^(13/2),x,8,2/11*a^2*sin(c+d*x)*sqrt(a+b*cos(c+d*x))/(d*cos(c+d*x)^(11/2))+46/99*a*b*sin(c+d*x)*sqrt(a+b*cos(c+d*x))/(d*cos(c+d*x)^(9/2))+2/693*(81*a^2+113*b^2)*sin(c+d*x)*sqrt(a+b*cos(c+d*x))/(d*cos(c+d*x)^(7/2))+2/693*b*(229*a^2+3*b^2)*sin(c+d*x)*sqrt(a+b*cos(c+d*x))/(a*d*cos(c+d*x)^(5/2))+2/693*(135*a^4+205*a^2*b^2-4*b^4)*sin(c+d*x)*sqrt(a+b*cos(c+d*x))/(a^2*d*cos(c+d*x)^(3/2))+2/693*(a-b)*b*(741*a^4+51*a^2*b^2+8*b^4)*cot(c+d*x)*EllipticE(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a^4*d)+2/693*(a-b)*(135*a^4-606*a^3*b+57*a^2*b^2+6*a*b^3+8*b^4)*cot(c+d*x)*EllipticF(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a^3*d)],

# n<0
[cos(c+d*x)^(3/2)/(a+b*cos(c+d*x))^(1/2),x,8,sin(c+d*x)*sqrt(a+b*cos(c+d*x))/(b*d*sqrt(cos(c+d*x)))-(a-b)*cot(c+d*x)*EllipticE(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a*b*d)+cot(c+d*x)*EllipticF(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(b*d)+a*cot(c+d*x)*EllipticPi(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),(a+b)/b,sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(b^2*d),a*sin(c+d*x)/(b*d*sqrt(cos(c+d*x))*sqrt(a+b*cos(c+d*x)))+sin(c+d*x)*sqrt(cos(c+d*x))/(d*sqrt(a+b*cos(c+d*x)))-(a-b)*cot(c+d*x)*EllipticE(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a*b*d)+cot(c+d*x)*EllipticF(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(b*d)+a*cot(c+d*x)*EllipticPi(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),(a+b)/b,sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(b^2*d)],
[cos(c+d*x)^(1/2)/(a+b*cos(c+d*x))^(1/2),x,1,-2*cot(c+d*x)*EllipticPi(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),(a+b)/b,sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(b*d)],
[1/(cos(c+d*x)^(1/2)*(a+b*cos(c+d*x))^(1/2)),x,1,2*cot(c+d*x)*EllipticF(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a*d)],
[1/(cos(c+d*x)^(3/2)*(a+b*cos(c+d*x))^(1/2)),x,3,2*(a-b)*cot(c+d*x)*EllipticE(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a^2*d)-2*cot(c+d*x)*EllipticF(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a*d)],
[1/(cos(c+d*x)^(5/2)*(a+b*cos(c+d*x))^(1/2)),x,4,2/3*sin(c+d*x)*sqrt(a+b*cos(c+d*x))/(a*d*cos(c+d*x)^(3/2))-4/3*(a-b)*b*cot(c+d*x)*EllipticE(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a^3*d)+2/3*(a+2*b)*cot(c+d*x)*EllipticF(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a^2*d)],
[cos(c+d*x)^(5/2)/(a+b*cos(c+d*x))^(3/2),x,7,-2*a^2*sin(c+d*x)*sqrt(cos(c+d*x))/(b*(a^2-b^2)*d*sqrt(a+b*cos(c+d*x)))+(3*a^2-b^2)*sin(c+d*x)*sqrt(a+b*cos(c+d*x))/(b^2*(a^2-b^2)*d*sqrt(cos(c+d*x)))-(3*a^2-b^2)*cot(c+d*x)*EllipticE(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a*b^2*d*sqrt(a+b))+(3*a+b)*cot(c+d*x)*EllipticF(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(b^2*d*sqrt(a+b))+3*a*cot(c+d*x)*EllipticPi(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),(a+b)/b,sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(b^3*d)],
[cos(c+d*x)^(3/2)/(a+b*cos(c+d*x))^(3/2),x,6,-2*a^2*sin(c+d*x)/(b*(a^2-b^2)*d*sqrt(cos(c+d*x))*sqrt(a+b*cos(c+d*x)))+2*cot(c+d*x)*EllipticE(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(b*d*sqrt(a+b))-2*cot(c+d*x)*EllipticF(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(b*d*sqrt(a+b))-2*cot(c+d*x)*EllipticPi(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),(a+b)/b,sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(b^2*d)],
[cos(c+d*x)^(1/2)/(a+b*cos(c+d*x))^(3/2),x,4,2*a*sin(c+d*x)/((a^2-b^2)*d*sqrt(cos(c+d*x))*sqrt(a+b*cos(c+d*x)))-2*cot(c+d*x)*EllipticE(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a*d*sqrt(a+b))+2*cot(c+d*x)*EllipticF(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a*d*sqrt(a+b))],
[1/(cos(c+d*x)^(1/2)*(a+b*cos(c+d*x))^(3/2)),x,4,-2*b*sin(c+d*x)/((a^2-b^2)*d*sqrt(cos(c+d*x))*sqrt(a+b*cos(c+d*x)))+2*b*cot(c+d*x)*EllipticE(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a^2*d*sqrt(a+b))+2*cot(c+d*x)*EllipticF(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a*d*sqrt(a+b))],
[1/(cos(c+d*x)^(3/2)*(a+b*cos(c+d*x))^(3/2)),x,4,2*b^2*sin(c+d*x)/(a*(a^2-b^2)*d*sqrt(cos(c+d*x))*sqrt(a+b*cos(c+d*x)))+2*(a^2-2*b^2)*cot(c+d*x)*EllipticE(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a^3*d*sqrt(a+b))-2*(a+2*b)*cot(c+d*x)*EllipticF(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a^2*d*sqrt(a+b))],
[1/(cos(c+d*x)^(5/2)*(a+b*cos(c+d*x))^(3/2)),x,5,2*b^2*sin(c+d*x)/(a*(a^2-b^2)*d*cos(c+d*x)^(3/2)*sqrt(a+b*cos(c+d*x)))+2/3*(a^2-4*b^2)*sin(c+d*x)*sqrt(a+b*cos(c+d*x))/(a^2*(a^2-b^2)*d*cos(c+d*x)^(3/2))-2/3*b*(5*a^2-8*b^2)*cot(c+d*x)*EllipticE(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a^4*d*sqrt(a+b))+2/3*(a+2*b)*(a+4*b)*cot(c+d*x)*EllipticF(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a^3*d*sqrt(a+b))],
[1/(cos(c+d*x)^(7/2)*(a+b*cos(c+d*x))^(3/2)),x,6,2*b^2*sin(c+d*x)/(a*(a^2-b^2)*d*cos(c+d*x)^(5/2)*sqrt(a+b*cos(c+d*x)))+2/5*(a^2-6*b^2)*sin(c+d*x)*sqrt(a+b*cos(c+d*x))/(a^2*(a^2-b^2)*d*cos(c+d*x)^(5/2))-2/5*b*(3*a^2-8*b^2)*sin(c+d*x)*sqrt(a+b*cos(c+d*x))/(a^3*(a^2-b^2)*d*cos(c+d*x)^(3/2))+2/5*(3*a^4+8*a^2*b^2-16*b^4)*cot(c+d*x)*EllipticE(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a^5*d*sqrt(a+b))-2/5*(3*a+4*b)*(a^2+4*b^2)*cot(c+d*x)*EllipticF(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a^4*d*sqrt(a+b))],
[cos(c+d*x)^(5/2)/(a+b*cos(c+d*x))^(5/2),x,7,-2/3*a^2*sin(c+d*x)*sqrt(cos(c+d*x))/(b*(a^2-b^2)*d*(a+b*cos(c+d*x))^(3/2))-2/3*a^2*(3*a^2-7*b^2)*sin(c+d*x)/(b^2*(a^2-b^2)^2*d*sqrt(cos(c+d*x))*sqrt(a+b*cos(c+d*x)))+2/3*(3*a^2-7*b^2)*cot(c+d*x)*EllipticE(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/((a-b)*b^2*(a+b)^(3/2)*d)-2/3*(3*a^2+a*b-6*b^2)*cot(c+d*x)*EllipticF(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/((a-b)*b^2*(a+b)^(3/2)*d)-2*cot(c+d*x)*EllipticPi(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),(a+b)/b,sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(b^3*d)],
[cos(c+d*x)^(3/2)/(a+b*cos(c+d*x))^(5/2),x,5,2/3*a*sin(c+d*x)*sqrt(cos(c+d*x))/((a^2-b^2)*d*(a+b*cos(c+d*x))^(3/2))-8/3*a*b*sin(c+d*x)/((a^2-b^2)^2*d*sqrt(cos(c+d*x))*sqrt(a+b*cos(c+d*x)))+8/3*b*cot(c+d*x)*EllipticE(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a*(a-b)*(a+b)^(3/2)*d)+2/3*(a-3*b)*cot(c+d*x)*EllipticF(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a*(a-b)*(a+b)^(3/2)*d)],
[cos(c+d*x)^(1/2)/(a+b*cos(c+d*x))^(5/2),x,5,-2/3*b*sin(c+d*x)*sqrt(cos(c+d*x))/((a^2-b^2)*d*(a+b*cos(c+d*x))^(3/2))+2/3*(3*a^2+b^2)*sin(c+d*x)/((a^2-b^2)^2*d*sqrt(cos(c+d*x))*sqrt(a+b*cos(c+d*x)))-2/3*(3*a^2+b^2)*cot(c+d*x)*EllipticE(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a^2*(a-b)*(a+b)^(3/2)*d)+2/3*(3*a-b)*cot(c+d*x)*EllipticF(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a*(a-b)*(a+b)^(3/2)*d)],
[1/(cos(c+d*x)^(1/2)*(a+b*cos(c+d*x))^(5/2)),x,5,2/3*b^2*sin(c+d*x)*sqrt(cos(c+d*x))/(a*(a^2-b^2)*d*(a+b*cos(c+d*x))^(3/2))-4/3*b*(3*a^2-b^2)*sin(c+d*x)/(a*(a^2-b^2)^2*d*sqrt(cos(c+d*x))*sqrt(a+b*cos(c+d*x)))+4/3*b*(3*a^2-b^2)*cot(c+d*x)*EllipticE(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a^3*(a-b)*(a+b)^(3/2)*d)+2/3*(3*a^2-3*a*b-2*b^2)*cot(c+d*x)*EllipticF(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a^2*(a-b)*(a+b)^(3/2)*d)],
[1/(cos(c+d*x)^(3/2)*(a+b*cos(c+d*x))^(5/2)),x,5,2/3*b^2*sin(c+d*x)/(a*(a^2-b^2)*d*(a+b*cos(c+d*x))^(3/2)*sqrt(cos(c+d*x)))+8/3*b^2*(2*a^2-b^2)*sin(c+d*x)/(a^2*(a^2-b^2)^2*d*sqrt(cos(c+d*x))*sqrt(a+b*cos(c+d*x)))+2/3*(3*a^4-15*a^2*b^2+8*b^4)*cot(c+d*x)*EllipticE(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a^4*(a-b)*(a+b)^(3/2)*d)-2/3*(3*a^3+9*a^2*b-6*a*b^2-8*b^3)*cot(c+d*x)*EllipticF(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a^3*(a-b)*(a+b)^(3/2)*d)],
[1/(cos(c+d*x)^(5/2)*(a+b*cos(c+d*x))^(5/2)),x,6,2/3*b^2*sin(c+d*x)/(a*(a^2-b^2)*d*cos(c+d*x)^(3/2)*(a+b*cos(c+d*x))^(3/2))+4/3*b^2*(5*a^2-3*b^2)*sin(c+d*x)/(a^2*(a^2-b^2)^2*d*cos(c+d*x)^(3/2)*sqrt(a+b*cos(c+d*x)))+2/3*(a^4-13*a^2*b^2+8*b^4)*sin(c+d*x)*sqrt(a+b*cos(c+d*x))/(a^3*(a^2-b^2)^2*d*cos(c+d*x)^(3/2))-8/3*b*(2*a^4-7*a^2*b^2+4*b^4)*cot(c+d*x)*EllipticE(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a^5*(a-b)*(a+b)^(3/2)*d)+2/3*(a^4+9*a^3*b+16*a^2*b^2-12*a*b^3-16*b^4)*cot(c+d*x)*EllipticF(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a^4*(a-b)*(a+b)^(3/2)*d)],
[1/(sqrt(cos(c+d*x))*sqrt(2+3*cos(c+d*x))),x,1,2*EllipticF(sin(c+d*x)/(1+cos(c+d*x)),sqrt(1/5))/(d*sqrt(5))],
[1/(sqrt(cos(c+d*x))*sqrt(-2+3*cos(c+d*x))),x,1,2*EllipticF(sin(c+d*x)/(1+cos(c+d*x)),sqrt(5))/d],
[1/(sqrt(2-3*cos(c+d*x))*sqrt(cos(c+d*x))),x,2,-2*EllipticF(sin(c+d*x)/(1-cos(c+d*x)),sqrt(1/5))*sqrt(-cos(c+d*x))/(d*sqrt(5)*sqrt(cos(c+d*x)))],
[1/(sqrt(-2-3*cos(c+d*x))*sqrt(cos(c+d*x))),x,2,-2*EllipticF(sin(c+d*x)/(1-cos(c+d*x)),sqrt(5))*sqrt(-cos(c+d*x))/(d*sqrt(cos(c+d*x)))],
[1/(sqrt(cos(c+d*x))*sqrt(3+2*cos(c+d*x))),x,1,2*cot(c+d*x)*EllipticF(sqrt(3+2*cos(c+d*x))/(sqrt(5)*sqrt(cos(c+d*x))),sqrt(-5))*sqrt(-tan(c+d*x)^2)/d],
[1/(sqrt(3-2*cos(c+d*x))*sqrt(cos(c+d*x))),x,1,2*cot(c+d*x)*EllipticF(sqrt(3-2*cos(c+d*x))/sqrt(cos(c+d*x)),sqrt(-1/5))*sqrt(-tan(c+d*x)^2)/(d*sqrt(5))],
[1/(sqrt(cos(c+d*x))*sqrt(-3+2*cos(c+d*x))),x,2,-2*csc(c+d*x)*EllipticF(sqrt(-3+2*cos(c+d*x))/sqrt(-cos(c+d*x)),sqrt(-1/5))*sqrt(-cos(c+d*x))*sqrt(cos(c+d*x))*sqrt(-tan(c+d*x)^2)/(d*sqrt(5))],
[1/(sqrt(-3-2*cos(c+d*x))*sqrt(cos(c+d*x))),x,2,-2*csc(c+d*x)*EllipticF(sqrt(-3-2*cos(c+d*x))/(sqrt(5)*sqrt(-cos(c+d*x))),sqrt(-5))*sqrt(-cos(c+d*x))*sqrt(cos(c+d*x))*sqrt(-tan(c+d*x)^2)/d],
[1/(sqrt(-cos(c+d*x))*sqrt(2+3*cos(c+d*x))),x,2,2*EllipticF(sin(c+d*x)/(1+cos(c+d*x)),sqrt(1/5))*sqrt(cos(c+d*x))/(d*sqrt(5)*sqrt(-cos(c+d*x)))],
[1/(sqrt(-cos(c+d*x))*sqrt(-2+3*cos(c+d*x))),x,2,2*EllipticF(sin(c+d*x)/(1+cos(c+d*x)),sqrt(5))*sqrt(cos(c+d*x))/(d*sqrt(-cos(c+d*x)))],
[1/(sqrt(2-3*cos(c+d*x))*sqrt(-cos(c+d*x))),x,1,-2*EllipticF(sin(c+d*x)/(1-cos(c+d*x)),sqrt(1/5))/(d*sqrt(5))],
[1/(sqrt(-2-3*cos(c+d*x))*sqrt(-cos(c+d*x))),x,1,-2*EllipticF(sin(c+d*x)/(1-cos(c+d*x)),sqrt(5))/d],
[1/(sqrt(-cos(c+d*x))*sqrt(3+2*cos(c+d*x))),x,2,2*cos(c+d*x)^(3/2)*csc(c+d*x)*EllipticF(sqrt(3+2*cos(c+d*x))/(sqrt(5)*sqrt(cos(c+d*x))),sqrt(-5))*sqrt(-tan(c+d*x)^2)/(d*sqrt(-cos(c+d*x)))],
[1/(sqrt(3-2*cos(c+d*x))*sqrt(-cos(c+d*x))),x,2,2*cos(c+d*x)^(3/2)*csc(c+d*x)*EllipticF(sqrt(3-2*cos(c+d*x))/sqrt(cos(c+d*x)),sqrt(-1/5))*sqrt(-tan(c+d*x)^2)/(d*sqrt(5)*sqrt(-cos(c+d*x)))],
[1/(sqrt(-cos(c+d*x))*sqrt(-3+2*cos(c+d*x))),x,1,-2*cot(c+d*x)*EllipticF(sqrt(-3+2*cos(c+d*x))/sqrt(-cos(c+d*x)),sqrt(-1/5))*sqrt(-tan(c+d*x)^2)/(d*sqrt(5))],
[1/(sqrt(-3-2*cos(c+d*x))*sqrt(-cos(c+d*x))),x,1,-2*cot(c+d*x)*EllipticF(sqrt(-3-2*cos(c+d*x))/(sqrt(5)*sqrt(-cos(c+d*x))),sqrt(-5))*sqrt(-tan(c+d*x)^2)/d],
[sqrt(cos(c+d*x))/sqrt(2+3*cos(c+d*x)),x,1,-4/3*cot(c+d*x)*EllipticPi(sqrt(2+3*cos(c+d*x))/(sqrt(5)*sqrt(cos(c+d*x))),5/3,sqrt(5))*sqrt(-1-sec(c+d*x))*sqrt(1-sec(c+d*x))/d],
[sqrt(cos(c+d*x))/sqrt(-2+3*cos(c+d*x)),x,1,-4/3*cot(c+d*x)*EllipticPi(sqrt(-2+3*cos(c+d*x))/sqrt(cos(c+d*x)),1/3,sqrt(1/5))*sqrt(-1+sec(c+d*x))*sqrt(1+sec(c+d*x))/(d*sqrt(5))],
[sqrt(cos(c+d*x))/sqrt(2-3*cos(c+d*x)),x,2,-4/3*cos(c+d*x)^(3/2)*csc(c+d*x)*EllipticPi(sqrt(2-3*cos(c+d*x))/sqrt(-cos(c+d*x)),1/3,sqrt(1/5))*sqrt(-1+sec(c+d*x))*sqrt(1+sec(c+d*x))/(d*sqrt(5)*sqrt(-cos(c+d*x)))],
[sqrt(cos(c+d*x))/sqrt(-2-3*cos(c+d*x)),x,2,-4/3*cos(c+d*x)^(3/2)*csc(c+d*x)*EllipticPi(sqrt(-2-3*cos(c+d*x))/(sqrt(5)*sqrt(-cos(c+d*x))),5/3,sqrt(5))*sqrt(-1-sec(c+d*x))*sqrt(1-sec(c+d*x))/(d*sqrt(-cos(c+d*x)))],
[sqrt(cos(c+d*x))/sqrt(3+2*cos(c+d*x)),x,1,-3*cot(c+d*x)*EllipticPi(sqrt(3+2*cos(c+d*x))/(sqrt(5)*sqrt(cos(c+d*x))),5/2,sqrt(-5))*sqrt(1-sec(c+d*x))*sqrt(1+sec(c+d*x))/d],
[sqrt(cos(c+d*x))/sqrt(3-2*cos(c+d*x)),x,1,3*cot(c+d*x)*EllipticPi(sqrt(3-2*cos(c+d*x))/sqrt(cos(c+d*x)),-1/2,sqrt(-1/5))*sqrt(1-sec(c+d*x))*sqrt(1+sec(c+d*x))/(d*sqrt(5))],
[sqrt(cos(c+d*x))/sqrt(-3+2*cos(c+d*x)),x,2,3*cos(c+d*x)^(3/2)*csc(c+d*x)*EllipticPi(sqrt(-3+2*cos(c+d*x))/sqrt(-cos(c+d*x)),-1/2,sqrt(-1/5))*sqrt(1-sec(c+d*x))*sqrt(1+sec(c+d*x))/(d*sqrt(5)*sqrt(-cos(c+d*x)))],
[sqrt(cos(c+d*x))/sqrt(-3-2*cos(c+d*x)),x,2,-3*cos(c+d*x)^(3/2)*csc(c+d*x)*EllipticPi(sqrt(-3-2*cos(c+d*x))/(sqrt(5)*sqrt(-cos(c+d*x))),5/2,sqrt(-5))*sqrt(1-sec(c+d*x))*sqrt(1+sec(c+d*x))/(d*sqrt(-cos(c+d*x)))],
[sqrt(-cos(c+d*x))/sqrt(2+3*cos(c+d*x)),x,2,-4/3*csc(c+d*x)*EllipticPi(sqrt(2+3*cos(c+d*x))/(sqrt(5)*sqrt(cos(c+d*x))),5/3,sqrt(5))*sqrt(-cos(c+d*x))*sqrt(cos(c+d*x))*sqrt(-1-sec(c+d*x))*sqrt(1-sec(c+d*x))/d],
[sqrt(-cos(c+d*x))/sqrt(-2+3*cos(c+d*x)),x,2,-4/3*csc(c+d*x)*EllipticPi(sqrt(-2+3*cos(c+d*x))/sqrt(cos(c+d*x)),1/3,sqrt(1/5))*sqrt(-cos(c+d*x))*sqrt(cos(c+d*x))*sqrt(-1+sec(c+d*x))*sqrt(1+sec(c+d*x))/(d*sqrt(5))],
[sqrt(-cos(c+d*x))/sqrt(2-3*cos(c+d*x)),x,1,-4/3*cot(c+d*x)*EllipticPi(sqrt(2-3*cos(c+d*x))/sqrt(-cos(c+d*x)),1/3,sqrt(1/5))*sqrt(-1+sec(c+d*x))*sqrt(1+sec(c+d*x))/(d*sqrt(5))],
[sqrt(-cos(c+d*x))/sqrt(-2-3*cos(c+d*x)),x,1,-4/3*cot(c+d*x)*EllipticPi(sqrt(-2-3*cos(c+d*x))/(sqrt(5)*sqrt(-cos(c+d*x))),5/3,sqrt(5))*sqrt(-1-sec(c+d*x))*sqrt(1-sec(c+d*x))/d],

#  For some reason, Mathematica cannot verify the following two antiderivatives correct. 
[sqrt(-cos(c+d*x))/sqrt(3+2*cos(c+d*x)),x,2,-3*csc(c+d*x)*EllipticPi(sqrt(3+2*cos(c+d*x))/(sqrt(5)*sqrt(cos(c+d*x))),5/2,sqrt(-5))*sqrt(-cos(c+d*x))*sqrt(cos(c+d*x))*sqrt(1-sec(c+d*x))*sqrt(1+sec(c+d*x))/d],
[sqrt(-cos(c+d*x))/sqrt(3-2*cos(c+d*x)),x,2,3*csc(c+d*x)*EllipticPi(sqrt(3-2*cos(c+d*x))/sqrt(cos(c+d*x)),-1/2,sqrt(-1/5))*sqrt(-cos(c+d*x))*sqrt(cos(c+d*x))*sqrt(1-sec(c+d*x))*sqrt(1+sec(c+d*x))/(d*sqrt(5))],
[sqrt(-cos(c+d*x))/sqrt(-3+2*cos(c+d*x)),x,1,3*cot(c+d*x)*EllipticPi(sqrt(-3+2*cos(c+d*x))/sqrt(-cos(c+d*x)),-1/2,sqrt(-1/5))*sqrt(1-sec(c+d*x))*sqrt(1+sec(c+d*x))/(d*sqrt(5))],
[sqrt(-cos(c+d*x))/sqrt(-3-2*cos(c+d*x)),x,1,-3*cot(c+d*x)*EllipticPi(sqrt(-3-2*cos(c+d*x))/(sqrt(5)*sqrt(-cos(c+d*x))),5/2,sqrt(-5))*sqrt(1-sec(c+d*x))*sqrt(1+sec(c+d*x))/d],

# Integrands of the form Cos[e+f x]^(m/3) (a+b Cos[e+f x])^n

# n>0

# n<0
[cos(c+d*x)^(2/3)/(a+b*cos(c+d*x)),x,5,-b*AppellF1(1/2,-1/3,1,3/2,sin(c+d*x)^2,-b^2*sin(c+d*x)^2/(a^2-b^2))*cos(c+d*x)^(2/3)*sin(c+d*x)/((a^2-b^2)*d*(cos(c+d*x)^2)^(1/3))+a*AppellF1(1/2,1/6,1,3/2,sin(c+d*x)^2,-b^2*sin(c+d*x)^2/(a^2-b^2))*(cos(c+d*x)^2)^(1/6)*sin(c+d*x)/((a^2-b^2)*d*cos(c+d*x)^(1/3))],
[cos(c+d*x)^(1/3)/(a+b*cos(c+d*x)),x,5,-b*AppellF1(1/2,-1/6,1,3/2,sin(c+d*x)^2,-b^2*sin(c+d*x)^2/(a^2-b^2))*cos(c+d*x)^(1/3)*sin(c+d*x)/((a^2-b^2)*d*(cos(c+d*x)^2)^(1/6))+a*AppellF1(1/2,1/3,1,3/2,sin(c+d*x)^2,-b^2*sin(c+d*x)^2/(a^2-b^2))*(cos(c+d*x)^2)^(1/3)*sin(c+d*x)/((a^2-b^2)*d*cos(c+d*x)^(2/3))],
[1/(cos(c+d*x)^(1/3)*(a+b*cos(c+d*x))),x,5,-b*AppellF1(1/2,1/6,1,3/2,sin(c+d*x)^2,-b^2*sin(c+d*x)^2/(a^2-b^2))*(cos(c+d*x)^2)^(1/6)*sin(c+d*x)/((a^2-b^2)*d*cos(c+d*x)^(1/3))+a*AppellF1(1/2,2/3,1,3/2,sin(c+d*x)^2,-b^2*sin(c+d*x)^2/(a^2-b^2))*(cos(c+d*x)^2)^(2/3)*sin(c+d*x)/((a^2-b^2)*d*cos(c+d*x)^(4/3))],
[1/(cos(c+d*x)^(2/3)*(a+b*cos(c+d*x))),x,5,-b*AppellF1(1/2,1/3,1,3/2,sin(c+d*x)^2,-b^2*sin(c+d*x)^2/(a^2-b^2))*(cos(c+d*x)^2)^(1/3)*sin(c+d*x)/((a^2-b^2)*d*cos(c+d*x)^(2/3))+a*AppellF1(1/2,5/6,1,3/2,sin(c+d*x)^2,-b^2*sin(c+d*x)^2/(a^2-b^2))*(cos(c+d*x)^2)^(5/6)*sin(c+d*x)/((a^2-b^2)*d*cos(c+d*x)^(5/3))],

# Integrands of the form Cos[e+f x]^(m/3) (a+b Cos[e+f x])^(n/2)

# n>0

# n<0
[cos(c+d*x)^(7/3)/sqrt(a+b*cos(c+d*x)),x,0,Unintegrable(cos(c+d*x)^(7/3)/sqrt(a+b*cos(c+d*x)),x)],
[cos(c+d*x)^(5/3)/sqrt(a+b*cos(c+d*x)),x,0,Unintegrable(cos(c+d*x)^(5/3)/sqrt(a+b*cos(c+d*x)),x)],
[cos(c+d*x)^(4/3)/sqrt(a+b*cos(c+d*x)),x,0,Unintegrable(cos(c+d*x)^(4/3)/sqrt(a+b*cos(c+d*x)),x)],
[cos(c+d*x)^(2/3)/sqrt(a+b*cos(c+d*x)),x,0,Unintegrable(cos(c+d*x)^(2/3)/sqrt(a+b*cos(c+d*x)),x)],
[cos(c+d*x)^(1/3)/sqrt(a+b*cos(c+d*x)),x,0,Unintegrable(cos(c+d*x)^(1/3)/sqrt(a+b*cos(c+d*x)),x)],
[1/(cos(c+d*x)^(1/3)*sqrt(a+b*cos(c+d*x))),x,0,Unintegrable(1/(cos(c+d*x)^(1/3)*sqrt(a+b*cos(c+d*x))),x)],
[1/(cos(c+d*x)^(2/3)*sqrt(a+b*cos(c+d*x))),x,0,Unintegrable(1/(cos(c+d*x)^(2/3)*sqrt(a+b*cos(c+d*x))),x)],
[1/(cos(c+d*x)^(4/3)*sqrt(a+b*cos(c+d*x))),x,0,Unintegrable(1/(cos(c+d*x)^(4/3)*sqrt(a+b*cos(c+d*x))),x)],
[1/(cos(c+d*x)^(5/3)*sqrt(a+b*cos(c+d*x))),x,0,Unintegrable(1/(cos(c+d*x)^(5/3)*sqrt(a+b*cos(c+d*x))),x)],
[1/(cos(c+d*x)^(7/3)*sqrt(a+b*cos(c+d*x))),x,0,Unintegrable(1/(cos(c+d*x)^(7/3)*sqrt(a+b*cos(c+d*x))),x)],

# Integrands of the form Sec[e+f x]^(m/2) (a+b Cos[e+f x])^n

# n>0
[(A+B*cos(c+d*x))*sec(c+d*x)^(7/2),x,9,2/3*B*sec(c+d*x)^(3/2)*sin(c+d*x)/d+2/5*A*sec(c+d*x)^(5/2)*sin(c+d*x)/d+6/5*A*sin(c+d*x)*sqrt(sec(c+d*x))/d-6/5*A*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/d+2/3*B*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/d],
[(A+B*cos(c+d*x))*sec(c+d*x)^(5/2),x,8,2/3*A*sec(c+d*x)^(3/2)*sin(c+d*x)/d+2*B*sin(c+d*x)*sqrt(sec(c+d*x))/d-2*B*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/d+2/3*A*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/d],
[(A+B*cos(c+d*x))*sec(c+d*x)^(3/2),x,7,2*A*sin(c+d*x)*sqrt(sec(c+d*x))/d-2*A*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/d+2*B*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/d],
[(A+B*cos(c+d*x))*sqrt(sec(c+d*x)),x,6,2*B*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/d+2*A*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/d],
[(A+B*cos(c+d*x))/sqrt(sec(c+d*x)),x,7,2/3*B*sin(c+d*x)/(d*sqrt(sec(c+d*x)))+2*A*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/d+2/3*B*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/d],
[(A+B*cos(c+d*x))/sec(c+d*x)^(3/2),x,8,2/5*B*sin(c+d*x)/(d*sec(c+d*x)^(3/2))+2/3*A*sin(c+d*x)/(d*sqrt(sec(c+d*x)))+6/5*B*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/d+2/3*A*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/d],
[(A+B*cos(c+d*x))/sec(c+d*x)^(5/2),x,9,2/7*B*sin(c+d*x)/(d*sec(c+d*x)^(5/2))+2/5*A*sin(c+d*x)/(d*sec(c+d*x)^(3/2))+10/21*B*sin(c+d*x)/(d*sqrt(sec(c+d*x)))+6/5*A*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/d+10/21*B*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/d],
[(a+b*cos(c+d*x))^2*sec(c+d*x)^(9/2),x,10,2/21*(5*a^2+7*b^2)*sec(c+d*x)^(3/2)*sin(c+d*x)/d+4/5*a*b*sec(c+d*x)^(5/2)*sin(c+d*x)/d+2/7*a^2*sec(c+d*x)^(7/2)*sin(c+d*x)/d+12/5*a*b*sin(c+d*x)*sqrt(sec(c+d*x))/d-12/5*a*b*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/d+2/21*(5*a^2+7*b^2)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/d],
[(a+b*cos(c+d*x))^2*sec(c+d*x)^(7/2),x,9,4/3*a*b*sec(c+d*x)^(3/2)*sin(c+d*x)/d+2/5*a^2*sec(c+d*x)^(5/2)*sin(c+d*x)/d+2/5*(3*a^2+5*b^2)*sin(c+d*x)*sqrt(sec(c+d*x))/d-2/5*(3*a^2+5*b^2)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/d+4/3*a*b*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/d],
[(a+b*cos(c+d*x))^2*sec(c+d*x)^(5/2),x,8,2/3*a^2*sec(c+d*x)^(3/2)*sin(c+d*x)/d+4*a*b*sin(c+d*x)*sqrt(sec(c+d*x))/d-4*a*b*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/d+2/3*(a^2+3*b^2)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/d],
[(a+b*cos(c+d*x))^2*sec(c+d*x)^(3/2),x,7,2*a^2*sin(c+d*x)*sqrt(sec(c+d*x))/d-2*(a^2-b^2)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/d+4*a*b*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/d],
[(a+b*cos(c+d*x))^2*sec(c+d*x)^(1/2),x,7,2/3*b^2*sin(c+d*x)/(d*sqrt(sec(c+d*x)))+4*a*b*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/d+2/3*(3*a^2+b^2)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/d],
[(a+b*cos(c+d*x))^2/sec(c+d*x)^(1/2),x,8,2/5*b^2*sin(c+d*x)/(d*sec(c+d*x)^(3/2))+4/3*a*b*sin(c+d*x)/(d*sqrt(sec(c+d*x)))+2/5*(5*a^2+3*b^2)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/d+4/3*a*b*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/d],
[(a+b*cos(c+d*x))^2/sec(c+d*x)^(3/2),x,9,2/7*b^2*sin(c+d*x)/(d*sec(c+d*x)^(5/2))+4/5*a*b*sin(c+d*x)/(d*sec(c+d*x)^(3/2))+2/21*(7*a^2+5*b^2)*sin(c+d*x)/(d*sqrt(sec(c+d*x)))+12/5*a*b*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/d+2/21*(7*a^2+5*b^2)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/d],
[(a+b*cos(c+d*x))^2/sec(c+d*x)^(5/2),x,10,2/9*b^2*sin(c+d*x)/(d*sec(c+d*x)^(7/2))+4/7*a*b*sin(c+d*x)/(d*sec(c+d*x)^(5/2))+2/45*(9*a^2+7*b^2)*sin(c+d*x)/(d*sec(c+d*x)^(3/2))+20/21*a*b*sin(c+d*x)/(d*sqrt(sec(c+d*x)))+2/15*(9*a^2+7*b^2)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/d+20/21*a*b*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/d],
[(a+b*cos(c+d*x))^3*sec(c+d*x)^(9/2),x,10,2/21*a*(5*a^2+21*b^2)*sec(c+d*x)^(3/2)*sin(c+d*x)/d+32/35*a^2*b*sec(c+d*x)^(5/2)*sin(c+d*x)/d+2/7*a^2*sec(c+d*x)^(5/2)*(b+a*sec(c+d*x))*sin(c+d*x)/d+2/5*b*(9*a^2+5*b^2)*sin(c+d*x)*sqrt(sec(c+d*x))/d-2/5*b*(9*a^2+5*b^2)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/d+2/21*a*(5*a^2+21*b^2)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/d],
[(a+b*cos(c+d*x))^3*sec(c+d*x)^(7/2),x,9,8/5*a^2*b*sec(c+d*x)^(3/2)*sin(c+d*x)/d+2/5*a^2*sec(c+d*x)^(3/2)*(b+a*sec(c+d*x))*sin(c+d*x)/d+6/5*a*(a^2+5*b^2)*sin(c+d*x)*sqrt(sec(c+d*x))/d-6/5*a*(a^2+5*b^2)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/d+2*b*(a^2+b^2)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/d],
[(a+b*cos(c+d*x))^3*sec(c+d*x)^(5/2),x,8,16/3*a^2*b*sin(c+d*x)*sqrt(sec(c+d*x))/d+2/3*a^2*(b+a*sec(c+d*x))*sin(c+d*x)*sqrt(sec(c+d*x))/d-2*b*(3*a^2-b^2)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/d+2/3*a*(a^2+9*b^2)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/d],
[(a+b*cos(c+d*x))^3*sec(c+d*x)^(3/2),x,8,2/3*b^2*(b+a*sec(c+d*x))*sin(c+d*x)/(d*sqrt(sec(c+d*x)))+2/3*a*(3*a^2-b^2)*sin(c+d*x)*sqrt(sec(c+d*x))/d-2*a*(a^2-3*b^2)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/d+2/3*b*(9*a^2+b^2)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/d],
[(a+b*cos(c+d*x))^3*sec(c+d*x)^(1/2),x,8,2/5*b^2*(b+a*sec(c+d*x))*sin(c+d*x)/(d*sec(c+d*x)^(3/2))+8/5*a*b^2*sin(c+d*x)/(d*sqrt(sec(c+d*x)))+6/5*b*(5*a^2+b^2)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/d+2*a*(a^2+b^2)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/d],
[(a+b*cos(c+d*x))^3/sec(c+d*x)^(1/2),x,9,32/35*a*b^2*sin(c+d*x)/(d*sec(c+d*x)^(3/2))+2/7*b^2*(b+a*sec(c+d*x))*sin(c+d*x)/(d*sec(c+d*x)^(5/2))+2/21*b*(21*a^2+5*b^2)*sin(c+d*x)/(d*sqrt(sec(c+d*x)))+2/5*a*(5*a^2+9*b^2)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/d+2/21*b*(21*a^2+5*b^2)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/d],
[(a+b*cos(c+d*x))^3/sec(c+d*x)^(3/2),x,10,40/63*a*b^2*sin(c+d*x)/(d*sec(c+d*x)^(5/2))+2/45*b*(27*a^2+7*b^2)*sin(c+d*x)/(d*sec(c+d*x)^(3/2))+2/9*b^2*(b+a*sec(c+d*x))*sin(c+d*x)/(d*sec(c+d*x)^(7/2))+2/21*a*(7*a^2+15*b^2)*sin(c+d*x)/(d*sqrt(sec(c+d*x)))+2/15*b*(27*a^2+7*b^2)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/d+2/21*a*(7*a^2+15*b^2)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/d],

# n<0
[sec(c+d*x)^(5/2)/(a+b*cos(c+d*x)),x,11,2/3*sec(c+d*x)^(3/2)*sin(c+d*x)/(a*d)-2*b*sin(c+d*x)*sqrt(sec(c+d*x))/(a^2*d)+2*b*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(a^2*d)+2/3*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(a*d)+2*b^2*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2*b/(a+b),sqrt(2))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(a^2*(a+b)*d)],
[sec(c+d*x)^(3/2)/(a+b*cos(c+d*x)),x,7,2*sin(c+d*x)*sqrt(sec(c+d*x))/(a*d)-2*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(a*d)-2*b*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2*b/(a+b),sqrt(2))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(a*(a+b)*d)],
[sec(c+d*x)^(1/2)/(a+b*cos(c+d*x)),x,3,2*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2*b/(a+b),sqrt(2))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/((a+b)*d)],
[1/((a+b*cos(c+d*x))*sec(c+d*x)^(1/2)),x,5,2*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(b*d)-2*a*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2*b/(a+b),sqrt(2))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(b*(a+b)*d)],
[1/((a+b*cos(c+d*x))*sec(c+d*x)^(3/2)),x,9,2*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(b*d)-2*a*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(b^2*d)+2*a^2*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2*b/(a+b),sqrt(2))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(b^2*(a+b)*d)],
[1/((a+b*cos(c+d*x))*sec(c+d*x)^(5/2)),x,10,2/3*sin(c+d*x)/(b*d*sqrt(sec(c+d*x)))-2*a*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(b^2*d)+2/3*(3*a^2+b^2)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(b^3*d)-2*a^3*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2*b/(a+b),sqrt(2))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(b^3*(a+b)*d)],
[sec(c+d*x)^(5/2)/(a+b*cos(c+d*x))^2,x,12,1/3*(2*a^2-5*b^2)*sec(c+d*x)^(3/2)*sin(c+d*x)/(a^2*(a^2-b^2)*d)+b^2*sec(c+d*x)^(5/2)*sin(c+d*x)/(a*(a^2-b^2)*d*(b+a*sec(c+d*x)))-b*(4*a^2-5*b^2)*sin(c+d*x)*sqrt(sec(c+d*x))/(a^3*(a^2-b^2)*d)+b*(4*a^2-5*b^2)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(a^3*(a^2-b^2)*d)+1/3*(2*a^2-5*b^2)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(a^2*(a^2-b^2)*d)+b^2*(7*a^2-5*b^2)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2*b/(a+b),sqrt(2))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(a^3*(a-b)*(a+b)^2*d)],
[sec(c+d*x)^(3/2)/(a+b*cos(c+d*x))^2,x,11,b^2*sec(c+d*x)^(3/2)*sin(c+d*x)/(a*(a^2-b^2)*d*(b+a*sec(c+d*x)))+(2*a^2-3*b^2)*sin(c+d*x)*sqrt(sec(c+d*x))/(a^2*(a^2-b^2)*d)-(2*a^2-3*b^2)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(a^2*(a^2-b^2)*d)+b*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(a*(a^2-b^2)*d)-b*(5*a^2-3*b^2)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2*b/(a+b),sqrt(2))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(a^2*(a-b)*(a+b)^2*d)],
[sec(c+d*x)^(1/2)/(a+b*cos(c+d*x))^2,x,10,b^2*sin(c+d*x)*sqrt(sec(c+d*x))/(a*(a^2-b^2)*d*(b+a*sec(c+d*x)))-b*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(a*(a^2-b^2)*d)-sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/((a^2-b^2)*d)+(3*a^2-b^2)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2*b/(a+b),sqrt(2))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(a*(a-b)*(a+b)^2*d)],
[1/((a+b*cos(c+d*x))^2*sec(c+d*x)^(1/2)),x,10,-b*sin(c+d*x)*sqrt(sec(c+d*x))/((a^2-b^2)*d*(b+a*sec(c+d*x)))+sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/((a^2-b^2)*d)+a*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(b*(a^2-b^2)*d)-(a^2+b^2)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2*b/(a+b),sqrt(2))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/((a-b)*b*(a+b)^2*d)],
[1/((a+b*cos(c+d*x))^2*sec(c+d*x)^(3/2)),x,10,a*sin(c+d*x)*sqrt(sec(c+d*x))/((a^2-b^2)*d*(b+a*sec(c+d*x)))-a*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(b*(a^2-b^2)*d)+(a^2-2*b^2)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(b^2*(a^2-b^2)*d)-a*(a^2-3*b^2)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2*b/(a+b),sqrt(2))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/((a-b)*b^2*(a+b)^2*d)],
[1/((a+b*cos(c+d*x))^2*sec(c+d*x)^(5/2)),x,10,-a^2*sin(c+d*x)*sqrt(sec(c+d*x))/(b*(a^2-b^2)*d*(b+a*sec(c+d*x)))+(3*a^2-2*b^2)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(b^2*(a^2-b^2)*d)-a*(3*a^2-4*b^2)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(b^3*(a^2-b^2)*d)+a^2*(3*a^2-5*b^2)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2*b/(a+b),sqrt(2))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/((a-b)*b^3*(a+b)^2*d)],
[sec(c+d*x)^(5/2)/(a+b*cos(c+d*x))^3,x,13,1/12*(8*a^4-61*a^2*b^2+35*b^4)*sec(c+d*x)^(3/2)*sin(c+d*x)/(a^3*(a^2-b^2)^2*d)+1/2*b^2*sec(c+d*x)^(7/2)*sin(c+d*x)/(a*(a^2-b^2)*d*(b+a*sec(c+d*x))^2)+1/4*b^2*(13*a^2-7*b^2)*sec(c+d*x)^(5/2)*sin(c+d*x)/(a^2*(a^2-b^2)^2*d*(b+a*sec(c+d*x)))-1/4*b*(24*a^4-65*a^2*b^2+35*b^4)*sin(c+d*x)*sqrt(sec(c+d*x))/(a^4*(a^2-b^2)^2*d)+1/4*b*(24*a^4-65*a^2*b^2+35*b^4)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(a^4*(a^2-b^2)^2*d)+1/12*(8*a^4-61*a^2*b^2+35*b^4)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(a^3*(a^2-b^2)^2*d)+1/4*b^2*(63*a^4-86*a^2*b^2+35*b^4)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2*b/(a+b),sqrt(2))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(a^4*(a-b)^2*(a+b)^3*d)],
[sec(c+d*x)^(3/2)/(a+b*cos(c+d*x))^3,x,12,1/2*b^2*sec(c+d*x)^(5/2)*sin(c+d*x)/(a*(a^2-b^2)*d*(b+a*sec(c+d*x))^2)+1/4*b^2*(11*a^2-5*b^2)*sec(c+d*x)^(3/2)*sin(c+d*x)/(a^2*(a^2-b^2)^2*d*(b+a*sec(c+d*x)))+1/4*(8*a^4-29*a^2*b^2+15*b^4)*sin(c+d*x)*sqrt(sec(c+d*x))/(a^3*(a^2-b^2)^2*d)-1/4*(8*a^4-29*a^2*b^2+15*b^4)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(a^3*(a^2-b^2)^2*d)+1/4*b*(11*a^2-5*b^2)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(a^2*(a^2-b^2)^2*d)-1/4*b*(35*a^4-38*a^2*b^2+15*b^4)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2*b/(a+b),sqrt(2))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(a^3*(a-b)^2*(a+b)^3*d)],
[sec(c+d*x)^(1/2)/(a+b*cos(c+d*x))^3,x,11,1/2*b^2*sec(c+d*x)^(3/2)*sin(c+d*x)/(a*(a^2-b^2)*d*(b+a*sec(c+d*x))^2)+3/4*b^2*(3*a^2-b^2)*sin(c+d*x)*sqrt(sec(c+d*x))/(a^2*(a^2-b^2)^2*d*(b+a*sec(c+d*x)))-3/4*b*(3*a^2-b^2)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(a^2*(a^2-b^2)^2*d)-1/4*(7*a^2-b^2)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(a*(a^2-b^2)^2*d)+3/4*(5*a^4-2*a^2*b^2+b^4)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2*b/(a+b),sqrt(2))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(a^2*(a-b)^2*(a+b)^3*d)],
[1/((a+b*cos(c+d*x))^3*sec(c+d*x)^(1/2)),x,11,1/2*b^2*sin(c+d*x)*sqrt(sec(c+d*x))/(a*(a^2-b^2)*d*(b+a*sec(c+d*x))^2)-1/4*b*(7*a^2-b^2)*sin(c+d*x)*sqrt(sec(c+d*x))/(a*(a^2-b^2)^2*d*(b+a*sec(c+d*x)))+1/4*(5*a^2+b^2)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(a*(a^2-b^2)^2*d)+3/4*(a^2+b^2)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(b*(a^2-b^2)^2*d)-1/4*(3*a^4+10*a^2*b^2-b^4)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2*b/(a+b),sqrt(2))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(a*(a-b)^2*b*(a+b)^3*d)],
[1/((a+b*cos(c+d*x))^3*sec(c+d*x)^(3/2)),x,11,-1/2*b*sin(c+d*x)*sqrt(sec(c+d*x))/((a^2-b^2)*d*(b+a*sec(c+d*x))^2)+3/4*(a^2+b^2)*sin(c+d*x)*sqrt(sec(c+d*x))/((a^2-b^2)^2*d*(b+a*sec(c+d*x)))-1/4*(a^2+5*b^2)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(b*(a^2-b^2)^2*d)+1/4*a*(a^2-7*b^2)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(b^2*(a^2-b^2)^2*d)-1/4*(a^4-10*a^2*b^2-3*b^4)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2*b/(a+b),sqrt(2))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/((a-b)^2*b^2*(a+b)^3*d)],
[1/((a+b*cos(c+d*x))^3*sec(c+d*x)^(5/2)),x,11,1/2*a*sin(c+d*x)*sqrt(sec(c+d*x))/((a^2-b^2)*d*(b+a*sec(c+d*x))^2)+1/4*a*(a^2-7*b^2)*sin(c+d*x)*sqrt(sec(c+d*x))/(b*(a^2-b^2)^2*d*(b+a*sec(c+d*x)))-3/4*a*(a^2-3*b^2)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(b^2*(a^2-b^2)^2*d)+1/4*(3*a^4-5*a^2*b^2+8*b^4)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(b^3*(a^2-b^2)^2*d)-3/4*a*(a^4-2*a^2*b^2+5*b^4)*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2*b/(a+b),sqrt(2))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/((a-b)^2*b^3*(a+b)^3*d)],

# Integrands of the form Sec[e+f x]^(m/2) (a+b Cos[e+f x])^(n/2)

# n>0
[sec(c+d*x)^(7/2)*sqrt(a+b*cos(c+d*x)),x,6,2/15*b*sec(c+d*x)^(3/2)*sin(c+d*x)*sqrt(a+b*cos(c+d*x))/(a*d)+2/5*sec(c+d*x)^(5/2)*sin(c+d*x)*sqrt(a+b*cos(c+d*x))/d+2/15*(a-b)*(9*a^2-2*b^2)*csc(c+d*x)*EllipticE(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(cos(c+d*x))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a^3*d*sqrt(sec(c+d*x)))-2/15*(a-b)*(9*a+2*b)*csc(c+d*x)*EllipticF(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(cos(c+d*x))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a^2*d*sqrt(sec(c+d*x)))],
[sec(c+d*x)^(5/2)*sqrt(a+b*cos(c+d*x)),x,5,2/3*sec(c+d*x)^(3/2)*sin(c+d*x)*sqrt(a+b*cos(c+d*x))/d+2/3*(a-b)*b*csc(c+d*x)*EllipticE(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(cos(c+d*x))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a^2*d*sqrt(sec(c+d*x)))+2/3*(a-b)*csc(c+d*x)*EllipticF(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(cos(c+d*x))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a*d*sqrt(sec(c+d*x)))],
[sec(c+d*x)^(3/2)*sqrt(a+b*cos(c+d*x)),x,4,2*(a-b)*csc(c+d*x)*EllipticE(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(cos(c+d*x))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a*d*sqrt(sec(c+d*x)))-2*(a-b)*csc(c+d*x)*EllipticF(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(cos(c+d*x))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a*d*sqrt(sec(c+d*x)))],
[sqrt(a+b*cos(c+d*x))*sqrt(sec(c+d*x)),x,2,-2*(a+b*cos(c+d*x))*csc(c+d*x)*EllipticPi(sqrt(a+b)*sqrt(cos(c+d*x))/sqrt(a+b*cos(c+d*x)),b/(a+b),sqrt((-a+b)/(a+b)))*sqrt(cos(c+d*x))*sqrt(a*(1-cos(c+d*x))/(a+b*cos(c+d*x)))*sqrt(a*(1+cos(c+d*x))/(a+b*cos(c+d*x)))*sqrt(sec(c+d*x))/(d*sqrt(a+b))],
[sqrt(a+b*cos(c+d*x))/sqrt(sec(c+d*x)),x,8,sin(c+d*x)*sqrt(a+b*cos(c+d*x))*sqrt(sec(c+d*x))/d-(a-b)*csc(c+d*x)*EllipticE(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(cos(c+d*x))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a*d*sqrt(sec(c+d*x)))+csc(c+d*x)*EllipticF(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(cos(c+d*x))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(d*sqrt(sec(c+d*x)))-a*csc(c+d*x)*EllipticPi(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),(a+b)/b,sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(cos(c+d*x))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(b*d*sqrt(sec(c+d*x)))],
[sqrt(a+b*cos(c+d*x))/sec(c+d*x)^(3/2),x,8,1/2*sin(c+d*x)*sqrt(a+b*cos(c+d*x))/(d*sqrt(sec(c+d*x)))+1/4*a*sin(c+d*x)*sqrt(a+b*cos(c+d*x))*sqrt(sec(c+d*x))/(b*d)-1/4*(a-b)*csc(c+d*x)*EllipticE(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(cos(c+d*x))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(b*d*sqrt(sec(c+d*x)))+1/4*(a+2*b)*csc(c+d*x)*EllipticF(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(cos(c+d*x))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(b*d*sqrt(sec(c+d*x)))+1/4*(a^2-4*b^2)*csc(c+d*x)*EllipticPi(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),(a+b)/b,sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(cos(c+d*x))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(b^2*d*sqrt(sec(c+d*x)))],
[(a+b*cos(c+d*x))^(3/2)*sec(c+d*x)^(9/2),x,7,2/105*(25*a^2+3*b^2)*sec(c+d*x)^(3/2)*sin(c+d*x)*sqrt(a+b*cos(c+d*x))/(a*d)+16/35*b*sec(c+d*x)^(5/2)*sin(c+d*x)*sqrt(a+b*cos(c+d*x))/d+2/7*a*sec(c+d*x)^(7/2)*sin(c+d*x)*sqrt(a+b*cos(c+d*x))/d+4/105*(a-b)*b*(41*a^2-3*b^2)*csc(c+d*x)*EllipticE(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(cos(c+d*x))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a^3*d*sqrt(sec(c+d*x)))+2/105*(a-b)*(25*a^2-57*a*b-6*b^2)*csc(c+d*x)*EllipticF(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(cos(c+d*x))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a^2*d*sqrt(sec(c+d*x)))],
[(a+b*cos(c+d*x))^(3/2)*sec(c+d*x)^(7/2),x,6,4/5*b*sec(c+d*x)^(3/2)*sin(c+d*x)*sqrt(a+b*cos(c+d*x))/d+2/5*a*sec(c+d*x)^(5/2)*sin(c+d*x)*sqrt(a+b*cos(c+d*x))/d+2/5*(a-b)*(3*a^2+b^2)*csc(c+d*x)*EllipticE(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(cos(c+d*x))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a^2*d*sqrt(sec(c+d*x)))-2/5*(a-b)*(3*a-b)*csc(c+d*x)*EllipticF(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(cos(c+d*x))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a*d*sqrt(sec(c+d*x)))],
[(a+b*cos(c+d*x))^(3/2)*sec(c+d*x)^(5/2),x,5,2/3*a*sec(c+d*x)^(3/2)*sin(c+d*x)*sqrt(a+b*cos(c+d*x))/d+8/3*(a-b)*b*csc(c+d*x)*EllipticE(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(cos(c+d*x))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a*d*sqrt(sec(c+d*x)))+2/3*(a-3*b)*(a-b)*csc(c+d*x)*EllipticF(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(cos(c+d*x))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a*d*sqrt(sec(c+d*x)))],
[(a+b*cos(c+d*x))^(3/2)*sec(c+d*x)^(3/2),x,6,2*(a-b)*csc(c+d*x)*EllipticE(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(cos(c+d*x))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(d*sqrt(sec(c+d*x)))-2*(a-2*b)*csc(c+d*x)*EllipticF(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(cos(c+d*x))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(d*sqrt(sec(c+d*x)))-2*b*csc(c+d*x)*EllipticPi(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),(a+b)/b,sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(cos(c+d*x))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(d*sqrt(sec(c+d*x)))],
[(a+b*cos(c+d*x))^(3/2)*sqrt(sec(c+d*x)),x,7,b*sin(c+d*x)*sqrt(a+b*cos(c+d*x))*sqrt(sec(c+d*x))/d-(a-b)*b*csc(c+d*x)*EllipticE(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(cos(c+d*x))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a*d*sqrt(sec(c+d*x)))+(2*a+b)*csc(c+d*x)*EllipticF(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(cos(c+d*x))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(d*sqrt(sec(c+d*x)))-3*a*csc(c+d*x)*EllipticPi(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),(a+b)/b,sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(cos(c+d*x))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(d*sqrt(sec(c+d*x)))],
[(a+b*cos(c+d*x))^(3/2)/sqrt(sec(c+d*x)),x,9,1/2*(a+b*cos(c+d*x))^(3/2)*sin(c+d*x)*sqrt(sec(c+d*x))/d+3/4*a*sin(c+d*x)*sqrt(a+b*cos(c+d*x))*sqrt(sec(c+d*x))/d-5/4*(a-b)*csc(c+d*x)*EllipticE(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(cos(c+d*x))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(d*sqrt(sec(c+d*x)))+1/4*(5*a+2*b)*csc(c+d*x)*EllipticF(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(cos(c+d*x))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(d*sqrt(sec(c+d*x)))-1/4*(3*a^2+4*b^2)*csc(c+d*x)*EllipticPi(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),(a+b)/b,sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(cos(c+d*x))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(b*d*sqrt(sec(c+d*x)))],
[(a+b*cos(c+d*x))^(3/2)/sec(c+d*x)^(3/2),x,9,1/3*(a+b*cos(c+d*x))^(3/2)*sin(c+d*x)/(d*sqrt(sec(c+d*x)))+1/4*a*sin(c+d*x)*sqrt(a+b*cos(c+d*x))/(d*sqrt(sec(c+d*x)))+1/24*(3*a^2+16*b^2)*sin(c+d*x)*sqrt(a+b*cos(c+d*x))*sqrt(sec(c+d*x))/(b*d)-1/24*(a-b)*(3*a^2+16*b^2)*csc(c+d*x)*EllipticE(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(cos(c+d*x))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a*b*d*sqrt(sec(c+d*x)))+1/24*(a+2*b)*(3*a+8*b)*csc(c+d*x)*EllipticF(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(cos(c+d*x))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(b*d*sqrt(sec(c+d*x)))+1/8*a*(a^2-12*b^2)*csc(c+d*x)*EllipticPi(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),(a+b)/b,sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(cos(c+d*x))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(b^2*d*sqrt(sec(c+d*x)))],
[(a+b*cos(c+d*x))^(5/2)*sec(c+d*x)^(11/2),x,8,2/315*b*(163*a^2+5*b^2)*sec(c+d*x)^(3/2)*sin(c+d*x)*sqrt(a+b*cos(c+d*x))/(a*d)+2/315*(49*a^2+75*b^2)*sec(c+d*x)^(5/2)*sin(c+d*x)*sqrt(a+b*cos(c+d*x))/d+38/63*a*b*sec(c+d*x)^(7/2)*sin(c+d*x)*sqrt(a+b*cos(c+d*x))/d+2/9*a^2*sec(c+d*x)^(9/2)*sin(c+d*x)*sqrt(a+b*cos(c+d*x))/d+2/315*(a-b)*(147*a^4+279*a^2*b^2-10*b^4)*csc(c+d*x)*EllipticE(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(cos(c+d*x))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a^3*d*sqrt(sec(c+d*x)))-2/315*(a-b)*(147*a^3-114*a^2*b+165*a*b^2+10*b^3)*csc(c+d*x)*EllipticF(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(cos(c+d*x))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a^2*d*sqrt(sec(c+d*x)))],
[(a+b*cos(c+d*x))^(5/2)*sec(c+d*x)^(9/2),x,7,2/21*(5*a^2+9*b^2)*sec(c+d*x)^(3/2)*sin(c+d*x)*sqrt(a+b*cos(c+d*x))/d+6/7*a*b*sec(c+d*x)^(5/2)*sin(c+d*x)*sqrt(a+b*cos(c+d*x))/d+2/7*a^2*sec(c+d*x)^(7/2)*sin(c+d*x)*sqrt(a+b*cos(c+d*x))/d+2/21*(a-b)*b*(29*a^2+3*b^2)*csc(c+d*x)*EllipticE(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(cos(c+d*x))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a^2*d*sqrt(sec(c+d*x)))+2/21*(a-b)*(5*a^2-24*a*b+3*b^2)*csc(c+d*x)*EllipticF(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(cos(c+d*x))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a*d*sqrt(sec(c+d*x)))],
[(a+b*cos(c+d*x))^(5/2)*sec(c+d*x)^(7/2),x,6,22/15*a*b*sec(c+d*x)^(3/2)*sin(c+d*x)*sqrt(a+b*cos(c+d*x))/d+2/5*a^2*sec(c+d*x)^(5/2)*sin(c+d*x)*sqrt(a+b*cos(c+d*x))/d+2/15*(a-b)*(9*a^2+23*b^2)*csc(c+d*x)*EllipticE(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(cos(c+d*x))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a*d*sqrt(sec(c+d*x)))-2/15*(a-b)*(9*a^2-8*a*b+15*b^2)*csc(c+d*x)*EllipticF(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(cos(c+d*x))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a*d*sqrt(sec(c+d*x)))],
[(a+b*cos(c+d*x))^(5/2)*sec(c+d*x)^(5/2),x,7,2/3*a^2*sec(c+d*x)^(3/2)*sin(c+d*x)*sqrt(a+b*cos(c+d*x))/d+14/3*(a-b)*b*csc(c+d*x)*EllipticE(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(cos(c+d*x))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(d*sqrt(sec(c+d*x)))+2/3*(a^2-7*a*b+9*b^2)*csc(c+d*x)*EllipticF(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(cos(c+d*x))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(d*sqrt(sec(c+d*x)))-2*b^2*csc(c+d*x)*EllipticPi(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),(a+b)/b,sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(cos(c+d*x))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(d*sqrt(sec(c+d*x)))],
[(a+b*cos(c+d*x))^(5/2)*sec(c+d*x)^(3/2),x,8,2*a^2*sin(c+d*x)*sqrt(a+b*cos(c+d*x))*sqrt(sec(c+d*x))/d-(2*a^2-b^2)*sin(c+d*x)*sqrt(a+b*cos(c+d*x))*sqrt(sec(c+d*x))/d+(a-b)*(2*a^2-b^2)*csc(c+d*x)*EllipticE(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(cos(c+d*x))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a*d*sqrt(sec(c+d*x)))-(2*a^2-6*a*b-b^2)*csc(c+d*x)*EllipticF(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(cos(c+d*x))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(d*sqrt(sec(c+d*x)))-5*a*b*csc(c+d*x)*EllipticPi(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),(a+b)/b,sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(cos(c+d*x))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(d*sqrt(sec(c+d*x)))],
[(a+b*cos(c+d*x))^(5/2)*sqrt(sec(c+d*x)),x,8,1/2*b^2*sin(c+d*x)*sqrt(a+b*cos(c+d*x))/(d*sqrt(sec(c+d*x)))+9/4*a*b*sin(c+d*x)*sqrt(a+b*cos(c+d*x))*sqrt(sec(c+d*x))/d-9/4*(a-b)*b*csc(c+d*x)*EllipticE(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(cos(c+d*x))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(d*sqrt(sec(c+d*x)))+1/4*(8*a^2+9*a*b+2*b^2)*csc(c+d*x)*EllipticF(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(cos(c+d*x))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(d*sqrt(sec(c+d*x)))-1/4*(15*a^2+4*b^2)*csc(c+d*x)*EllipticPi(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),(a+b)/b,sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(cos(c+d*x))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(d*sqrt(sec(c+d*x)))],
[(a+b*cos(c+d*x))^(5/2)/sqrt(sec(c+d*x)),x,9,1/3*b^2*sin(c+d*x)*sqrt(a+b*cos(c+d*x))/(d*sec(c+d*x)^(3/2))+13/12*a*b*sin(c+d*x)*sqrt(a+b*cos(c+d*x))/(d*sqrt(sec(c+d*x)))+1/24*(33*a^2+16*b^2)*sin(c+d*x)*sqrt(a+b*cos(c+d*x))*sqrt(sec(c+d*x))/d-1/24*(a-b)*(33*a^2+16*b^2)*csc(c+d*x)*EllipticE(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(cos(c+d*x))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a*d*sqrt(sec(c+d*x)))+1/24*(33*a^2+26*a*b+16*b^2)*csc(c+d*x)*EllipticF(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(cos(c+d*x))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(d*sqrt(sec(c+d*x)))-5/8*a*(a^2+4*b^2)*csc(c+d*x)*EllipticPi(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),(a+b)/b,sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(cos(c+d*x))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(b*d*sqrt(sec(c+d*x)))],
[(a+b*cos(c+d*x))^(5/2)/sec(c+d*x)^(3/2),x,10,1/4*b^2*sin(c+d*x)*sqrt(a+b*cos(c+d*x))/(d*sec(c+d*x)^(5/2))+17/24*a*b*sin(c+d*x)*sqrt(a+b*cos(c+d*x))/(d*sec(c+d*x)^(3/2))+1/96*(59*a^2+36*b^2)*sin(c+d*x)*sqrt(a+b*cos(c+d*x))/(d*sqrt(sec(c+d*x)))+1/192*a*(15*a^2+284*b^2)*sin(c+d*x)*sqrt(a+b*cos(c+d*x))*sqrt(sec(c+d*x))/(b*d)-1/192*(a-b)*(15*a^2+284*b^2)*csc(c+d*x)*EllipticE(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(cos(c+d*x))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(b*d*sqrt(sec(c+d*x)))+1/192*(15*a^3+118*a^2*b+284*a*b^2+72*b^3)*csc(c+d*x)*EllipticF(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(cos(c+d*x))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(b*d*sqrt(sec(c+d*x)))+1/64*(5*a^4-120*a^2*b^2-48*b^4)*csc(c+d*x)*EllipticPi(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),(a+b)/b,sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(cos(c+d*x))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(b^2*d*sqrt(sec(c+d*x)))],

# n<0
[sec(c+d*x)^(5/2)/sqrt(a+b*cos(c+d*x)),x,5,2/3*sec(c+d*x)^(3/2)*sin(c+d*x)*sqrt(a+b*cos(c+d*x))/(a*d)-4/3*(a-b)*b*csc(c+d*x)*EllipticE(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(cos(c+d*x))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a^3*d*sqrt(sec(c+d*x)))+2/3*(a+2*b)*csc(c+d*x)*EllipticF(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(cos(c+d*x))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a^2*d*sqrt(sec(c+d*x)))],
[sec(c+d*x)^(3/2)/sqrt(a+b*cos(c+d*x)),x,4,2*(a-b)*csc(c+d*x)*EllipticE(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(cos(c+d*x))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a^2*d*sqrt(sec(c+d*x)))-2*csc(c+d*x)*EllipticF(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(cos(c+d*x))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a*d*sqrt(sec(c+d*x)))],
[sqrt(sec(c+d*x))/sqrt(a+b*cos(c+d*x)),x,2,2*csc(c+d*x)*EllipticF(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(cos(c+d*x))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a*d*sqrt(sec(c+d*x)))],
[1/(sqrt(a+b*cos(c+d*x))*sqrt(sec(c+d*x))),x,2,-2*csc(c+d*x)*EllipticPi(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),(a+b)/b,sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(cos(c+d*x))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(b*d*sqrt(sec(c+d*x)))],
[1/(sec(c+d*x)^(3/2)*sqrt(a+b*cos(c+d*x))),x,9,sin(c+d*x)/(d*sqrt(a+b*cos(c+d*x))*sqrt(sec(c+d*x)))+a*sin(c+d*x)*sqrt(sec(c+d*x))/(b*d*sqrt(a+b*cos(c+d*x)))-(a-b)*csc(c+d*x)*EllipticE(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(cos(c+d*x))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a*b*d*sqrt(sec(c+d*x)))+csc(c+d*x)*EllipticF(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(cos(c+d*x))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(b*d*sqrt(sec(c+d*x)))+a*csc(c+d*x)*EllipticPi(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),(a+b)/b,sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(cos(c+d*x))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(b^2*d*sqrt(sec(c+d*x)))],
[1/(sec(c+d*x)^(5/2)*sqrt(a+b*cos(c+d*x))),x,8,1/2*sin(c+d*x)*sqrt(a+b*cos(c+d*x))/(b*d*sqrt(sec(c+d*x)))-3/4*a*sin(c+d*x)*sqrt(a+b*cos(c+d*x))*sqrt(sec(c+d*x))/(b^2*d)+3/4*(a-b)*csc(c+d*x)*EllipticE(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(cos(c+d*x))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(b^2*d*sqrt(sec(c+d*x)))-1/4*(3*a-2*b)*csc(c+d*x)*EllipticF(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(cos(c+d*x))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(b^2*d*sqrt(sec(c+d*x)))-1/4*(3*a^2+4*b^2)*csc(c+d*x)*EllipticPi(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),(a+b)/b,sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(cos(c+d*x))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(b^3*d*sqrt(sec(c+d*x)))],
[sec(c+d*x)^(5/2)/(a+b*cos(c+d*x))^(3/2),x,6,2*b^2*sec(c+d*x)^(3/2)*sin(c+d*x)/(a*(a^2-b^2)*d*sqrt(a+b*cos(c+d*x)))+2/3*(a^2-4*b^2)*sec(c+d*x)^(3/2)*sin(c+d*x)*sqrt(a+b*cos(c+d*x))/(a^2*(a^2-b^2)*d)-2/3*b*(5*a^2-8*b^2)*csc(c+d*x)*EllipticE(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(cos(c+d*x))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a^4*d*sqrt(a+b)*sqrt(sec(c+d*x)))+2/3*(a+2*b)*(a+4*b)*csc(c+d*x)*EllipticF(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(cos(c+d*x))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a^3*d*sqrt(a+b)*sqrt(sec(c+d*x)))],
[sec(c+d*x)^(3/2)/(a+b*cos(c+d*x))^(3/2),x,5,2*b^2*sin(c+d*x)*sqrt(sec(c+d*x))/(a*(a^2-b^2)*d*sqrt(a+b*cos(c+d*x)))+2*(a^2-2*b^2)*csc(c+d*x)*EllipticE(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(cos(c+d*x))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a^3*d*sqrt(a+b)*sqrt(sec(c+d*x)))-2*(a+2*b)*csc(c+d*x)*EllipticF(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(cos(c+d*x))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a^2*d*sqrt(a+b)*sqrt(sec(c+d*x)))],
[sqrt(sec(c+d*x))/(a+b*cos(c+d*x))^(3/2),x,5,-2*b*sin(c+d*x)*sqrt(sec(c+d*x))/((a^2-b^2)*d*sqrt(a+b*cos(c+d*x)))+2*b*csc(c+d*x)*EllipticE(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(cos(c+d*x))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a^2*d*sqrt(a+b)*sqrt(sec(c+d*x)))+2*csc(c+d*x)*EllipticF(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(cos(c+d*x))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a*d*sqrt(a+b)*sqrt(sec(c+d*x)))],
[1/((a+b*cos(c+d*x))^(3/2)*sqrt(sec(c+d*x))),x,5,2*a*sin(c+d*x)*sqrt(sec(c+d*x))/((a^2-b^2)*d*sqrt(a+b*cos(c+d*x)))-2*csc(c+d*x)*EllipticE(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(cos(c+d*x))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a*d*sqrt(a+b)*sqrt(sec(c+d*x)))+2*csc(c+d*x)*EllipticF(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(cos(c+d*x))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a*d*sqrt(a+b)*sqrt(sec(c+d*x)))],
[1/((a+b*cos(c+d*x))^(3/2)*sec(c+d*x)^(3/2)),x,7,-2*a^2*sin(c+d*x)*sqrt(sec(c+d*x))/(b*(a^2-b^2)*d*sqrt(a+b*cos(c+d*x)))+2*csc(c+d*x)*EllipticE(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(cos(c+d*x))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(b*d*sqrt(a+b)*sqrt(sec(c+d*x)))-2*csc(c+d*x)*EllipticF(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(cos(c+d*x))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(b*d*sqrt(a+b)*sqrt(sec(c+d*x)))-2*csc(c+d*x)*EllipticPi(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),(a+b)/b,sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(cos(c+d*x))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(b^2*d*sqrt(sec(c+d*x)))],
[1/((a+b*cos(c+d*x))^(3/2)*sec(c+d*x)^(5/2)),x,8,-2*a^2*sin(c+d*x)/(b*(a^2-b^2)*d*sqrt(a+b*cos(c+d*x))*sqrt(sec(c+d*x)))+(3*a^2-b^2)*sin(c+d*x)*sqrt(a+b*cos(c+d*x))*sqrt(sec(c+d*x))/(b^2*(a^2-b^2)*d)-(3*a^2-b^2)*csc(c+d*x)*EllipticE(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(cos(c+d*x))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a*b^2*d*sqrt(a+b)*sqrt(sec(c+d*x)))+(3*a+b)*csc(c+d*x)*EllipticF(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(cos(c+d*x))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(b^2*d*sqrt(a+b)*sqrt(sec(c+d*x)))+3*a*csc(c+d*x)*EllipticPi(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),(a+b)/b,sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(cos(c+d*x))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(b^3*d*sqrt(sec(c+d*x)))],
[sec(c+d*x)^(5/2)/(a+b*cos(c+d*x))^(5/2),x,7,2/3*b^2*sec(c+d*x)^(3/2)*sin(c+d*x)/(a*(a^2-b^2)*d*(a+b*cos(c+d*x))^(3/2))+4/3*b^2*(5*a^2-3*b^2)*sec(c+d*x)^(3/2)*sin(c+d*x)/(a^2*(a^2-b^2)^2*d*sqrt(a+b*cos(c+d*x)))+2/3*(a^4-13*a^2*b^2+8*b^4)*sec(c+d*x)^(3/2)*sin(c+d*x)*sqrt(a+b*cos(c+d*x))/(a^3*(a^2-b^2)^2*d)-8/3*b*(2*a^4-7*a^2*b^2+4*b^4)*csc(c+d*x)*EllipticE(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(cos(c+d*x))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a^5*(a-b)*(a+b)^(3/2)*d*sqrt(sec(c+d*x)))+2/3*(a^4+9*a^3*b+16*a^2*b^2-12*a*b^3-16*b^4)*csc(c+d*x)*EllipticF(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(cos(c+d*x))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a^4*(a-b)*(a+b)^(3/2)*d*sqrt(sec(c+d*x)))],
[sec(c+d*x)^(3/2)/(a+b*cos(c+d*x))^(5/2),x,6,2/3*b^2*sin(c+d*x)*sqrt(sec(c+d*x))/(a*(a^2-b^2)*d*(a+b*cos(c+d*x))^(3/2))+8/3*b^2*(2*a^2-b^2)*sin(c+d*x)*sqrt(sec(c+d*x))/(a^2*(a^2-b^2)^2*d*sqrt(a+b*cos(c+d*x)))+2/3*(3*a^4-15*a^2*b^2+8*b^4)*csc(c+d*x)*EllipticE(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(cos(c+d*x))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a^4*(a-b)*(a+b)^(3/2)*d*sqrt(sec(c+d*x)))-2/3*(3*a^3+9*a^2*b-6*a*b^2-8*b^3)*csc(c+d*x)*EllipticF(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(cos(c+d*x))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a^3*(a-b)*(a+b)^(3/2)*d*sqrt(sec(c+d*x)))],
[sqrt(sec(c+d*x))/(a+b*cos(c+d*x))^(5/2),x,6,2/3*b^2*sin(c+d*x)/(a*(a^2-b^2)*d*(a+b*cos(c+d*x))^(3/2)*sqrt(sec(c+d*x)))-4/3*b*(3*a^2-b^2)*sin(c+d*x)*sqrt(sec(c+d*x))/(a*(a^2-b^2)^2*d*sqrt(a+b*cos(c+d*x)))+4/3*b*(3*a^2-b^2)*csc(c+d*x)*EllipticE(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(cos(c+d*x))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a^3*(a-b)*(a+b)^(3/2)*d*sqrt(sec(c+d*x)))+2/3*(3*a^2-3*a*b-2*b^2)*csc(c+d*x)*EllipticF(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(cos(c+d*x))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a^2*(a-b)*(a+b)^(3/2)*d*sqrt(sec(c+d*x)))],
[1/((a+b*cos(c+d*x))^(5/2)*sqrt(sec(c+d*x))),x,6,-2/3*b*sin(c+d*x)/((a^2-b^2)*d*(a+b*cos(c+d*x))^(3/2)*sqrt(sec(c+d*x)))+2/3*(3*a^2+b^2)*sin(c+d*x)*sqrt(sec(c+d*x))/((a^2-b^2)^2*d*sqrt(a+b*cos(c+d*x)))-2/3*(3*a^2+b^2)*csc(c+d*x)*EllipticE(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(cos(c+d*x))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a^2*(a-b)*(a+b)^(3/2)*d*sqrt(sec(c+d*x)))+2/3*(3*a-b)*csc(c+d*x)*EllipticF(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(cos(c+d*x))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a*(a-b)*(a+b)^(3/2)*d*sqrt(sec(c+d*x)))],
[1/((a+b*cos(c+d*x))^(5/2)*sec(c+d*x)^(3/2)),x,6,2/3*a*sin(c+d*x)/((a^2-b^2)*d*(a+b*cos(c+d*x))^(3/2)*sqrt(sec(c+d*x)))-8/3*a*b*sin(c+d*x)*sqrt(sec(c+d*x))/((a^2-b^2)^2*d*sqrt(a+b*cos(c+d*x)))+8/3*b*csc(c+d*x)*EllipticE(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(cos(c+d*x))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a*(a-b)*(a+b)^(3/2)*d*sqrt(sec(c+d*x)))+2/3*(a-3*b)*csc(c+d*x)*EllipticF(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(cos(c+d*x))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(a*(a-b)*(a+b)^(3/2)*d*sqrt(sec(c+d*x)))],
[1/((a+b*cos(c+d*x))^(5/2)*sec(c+d*x)^(5/2)),x,8,-2/3*a^2*sin(c+d*x)/(b*(a^2-b^2)*d*(a+b*cos(c+d*x))^(3/2)*sqrt(sec(c+d*x)))-2/3*a^2*(3*a^2-7*b^2)*sin(c+d*x)*sqrt(sec(c+d*x))/(b^2*(a^2-b^2)^2*d*sqrt(a+b*cos(c+d*x)))+2/3*(3*a^2-7*b^2)*csc(c+d*x)*EllipticE(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(cos(c+d*x))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/((a-b)*b^2*(a+b)^(3/2)*d*sqrt(sec(c+d*x)))-2/3*(3*a^2+a*b-6*b^2)*csc(c+d*x)*EllipticF(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),sqrt((-a-b)/(a-b)))*sqrt(cos(c+d*x))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/((a-b)*b^2*(a+b)^(3/2)*d*sqrt(sec(c+d*x)))-2*csc(c+d*x)*EllipticPi(sqrt(a+b*cos(c+d*x))/(sqrt(a+b)*sqrt(cos(c+d*x))),(a+b)/b,sqrt((-a-b)/(a-b)))*sqrt(a+b)*sqrt(cos(c+d*x))*sqrt(a*(1-sec(c+d*x))/(a+b))*sqrt(a*(1+sec(c+d*x))/(a-b))/(b^3*d*sqrt(sec(c+d*x)))],

# Integrands of the form Cos[e+f x]^m (a+b Cos[e+f x])^n with m and/or n symbolic
[cos(c+d*x)^m*(a+b*cos(c+d*x))^4,x,6,b^2*(b^2*(3+m)+a^2*(22+5*m))*cos(c+d*x)^(1+m)*sin(c+d*x)/(d*(2+m)*(4+m))+2*a*b^3*(5+m)*cos(c+d*x)^(2+m)*sin(c+d*x)/(d*(3+m)*(4+m))+b^2*cos(c+d*x)^(1+m)*(a+b*cos(c+d*x))^2*sin(c+d*x)/(d*(4+m))-(b^4*(3+4*m+m^2)+6*a^2*b^2*(4+5*m+m^2)+a^4*(8+6*m+m^2))*cos(c+d*x)^(1+m)*hypergeom([1/2,1/2*(1+m)],[1/2*(3+m)],cos(c+d*x)^2)*sin(c+d*x)/(d*(4+m)*(2+3*m+m^2)*sqrt(sin(c+d*x)^2))-4*a*b*(b^2*(2+m)+a^2*(3+m))*cos(c+d*x)^(2+m)*hypergeom([1/2,1/2*(2+m)],[1/2*(4+m)],cos(c+d*x)^2)*sin(c+d*x)/(d*(2+m)*(3+m)*sqrt(sin(c+d*x)^2))],
[cos(c+d*x)^m*(a+b*cos(c+d*x))^3,x,5,a*b^2*(7+2*m)*cos(c+d*x)^(1+m)*sin(c+d*x)/(d*(2+m)*(3+m))+b^2*cos(c+d*x)^(1+m)*(a+b*cos(c+d*x))*sin(c+d*x)/(d*(3+m))-a*(3*b^2*(1+m)+a^2*(2+m))*cos(c+d*x)^(1+m)*hypergeom([1/2,1/2*(1+m)],[1/2*(3+m)],cos(c+d*x)^2)*sin(c+d*x)/(d*(1+m)*(2+m)*sqrt(sin(c+d*x)^2))-b*(b^2*(2+m)+3*a^2*(3+m))*cos(c+d*x)^(2+m)*hypergeom([1/2,1/2*(2+m)],[1/2*(4+m)],cos(c+d*x)^2)*sin(c+d*x)/(d*(2+m)*(3+m)*sqrt(sin(c+d*x)^2))],
[cos(c+d*x)^m*(a+b*cos(c+d*x))^2,x,4,b^2*cos(c+d*x)^(1+m)*sin(c+d*x)/(d*(2+m))-(b^2*(1+m)+a^2*(2+m))*cos(c+d*x)^(1+m)*hypergeom([1/2,1/2*(1+m)],[1/2*(3+m)],cos(c+d*x)^2)*sin(c+d*x)/(d*(1+m)*(2+m)*sqrt(sin(c+d*x)^2))-2*a*b*cos(c+d*x)^(2+m)*hypergeom([1/2,1/2*(2+m)],[1/2*(4+m)],cos(c+d*x)^2)*sin(c+d*x)/(d*(2+m)*sqrt(sin(c+d*x)^2))],
[cos(c+d*x)^m*(a+b*cos(c+d*x)),x,3,-a*cos(c+d*x)^(1+m)*hypergeom([1/2,1/2*(1+m)],[1/2*(3+m)],cos(c+d*x)^2)*sin(c+d*x)/(d*(1+m)*sqrt(sin(c+d*x)^2))-b*cos(c+d*x)^(2+m)*hypergeom([1/2,1/2*(2+m)],[1/2*(4+m)],cos(c+d*x)^2)*sin(c+d*x)/(d*(2+m)*sqrt(sin(c+d*x)^2))],
[cos(c+d*x)^m/(a+b*cos(c+d*x)),x,5,a*AppellF1(1/2,1/2*(1-m),1,3/2,sin(c+d*x)^2,-b^2*sin(c+d*x)^2/(a^2-b^2))*cos(c+d*x)^(-1+m)*(cos(c+d*x)^2)^(1/2*(1-m))*sin(c+d*x)/((a^2-b^2)*d)-b*AppellF1(1/2,-1/2*m,1,3/2,sin(c+d*x)^2,-b^2*sin(c+d*x)^2/(a^2-b^2))*cos(c+d*x)^m*sin(c+d*x)/((a^2-b^2)*d*(cos(c+d*x)^2)^(1/2*m))],
[cos(c+d*x)^m/(a+b*cos(c+d*x))^2,x,8,b^2*AppellF1(1/2,1/2*(-1-m),2,3/2,sin(c+d*x)^2,-b^2*sin(c+d*x)^2/(a^2-b^2))*cos(c+d*x)^(1+m)*(cos(c+d*x)^2)^(1/2*(-1-m))*sin(c+d*x)/((a^2-b^2)^2*d)+a^2*AppellF1(1/2,1/2*(1-m),2,3/2,sin(c+d*x)^2,-b^2*sin(c+d*x)^2/(a^2-b^2))*cos(c+d*x)^(-1+m)*(cos(c+d*x)^2)^(1/2*(1-m))*sin(c+d*x)/((a^2-b^2)^2*d)-2*a*b*AppellF1(1/2,-1/2*m,2,3/2,sin(c+d*x)^2,-b^2*sin(c+d*x)^2/(a^2-b^2))*cos(c+d*x)^m*sin(c+d*x)/((a^2-b^2)^2*d*(cos(c+d*x)^2)^(1/2*m))],
[(a+b*cos(c+d*x))^3*sec(c+d*x)^m,x,8,-a^2*b*(1-2*m)*sec(c+d*x)^(-2+m)*sin(c+d*x)/(d*(2-3*m+m^2))-a^2*sec(c+d*x)^(-2+m)*(b+a*sec(c+d*x))*sin(c+d*x)/(d*(1-m))-b*(b^2*(2-m)+3*a^2*(3-m))*hypergeom([1/2,1/2*(4-m)],[1/2*(6-m)],cos(c+d*x)^2)*sec(c+d*x)^(-4+m)*sin(c+d*x)/(d*(8-6*m+m^2)*sqrt(sin(c+d*x)^2))-a*(3*b^2*(1-m)+a^2*(2-m))*hypergeom([1/2,1/2*(3-m)],[1/2*(5-m)],cos(c+d*x)^2)*sec(c+d*x)^(-3+m)*sin(c+d*x)/(d*(3-4*m+m^2)*sqrt(sin(c+d*x)^2))],
[(a+b*cos(c+d*x))^2*sec(c+d*x)^m,x,7,-a^2*sec(c+d*x)^(-1+m)*sin(c+d*x)/(d*(1-m))-(b^2*(1-m)+a^2*(2-m))*hypergeom([1/2,1/2*(3-m)],[1/2*(5-m)],cos(c+d*x)^2)*sec(c+d*x)^(-3+m)*sin(c+d*x)/(d*(3-4*m+m^2)*sqrt(sin(c+d*x)^2))-2*a*b*hypergeom([1/2,1/2*(2-m)],[1/2*(4-m)],cos(c+d*x)^2)*sec(c+d*x)^(-2+m)*sin(c+d*x)/(d*(2-m)*sqrt(sin(c+d*x)^2))],
[(a+b*cos(c+d*x))*sec(c+d*x)^m,x,6,-b*hypergeom([1/2,1/2*(2-m)],[1/2*(4-m)],cos(c+d*x)^2)*sec(c+d*x)^(-2+m)*sin(c+d*x)/(d*(2-m)*sqrt(sin(c+d*x)^2))-a*hypergeom([1/2,1/2*(1-m)],[1/2*(3-m)],cos(c+d*x)^2)*sec(c+d*x)^(-1+m)*sin(c+d*x)/(d*(1-m)*sqrt(sin(c+d*x)^2))],

# Integrands of the form (a+b Cos[e+f x])^m (c+d Cos[e+f x])^n

# Integrands of the form (a+a Cos[e+f x])^m (c-c Cos[e+f x])^n

# Integrands of the form (a+a Cos[e+f x])^m (c+d Cos[e+f x])^n
[sqrt(1-cos(x))/sqrt(a-cos(x)),x,2,-2*arctan(sin(x)/(sqrt(1-cos(x))*sqrt(a-cos(x))))],
[sqrt((1-cos(x))/(a-cos(x))),x,3,-2*arctan(sin(x)/(sqrt(1-cos(x))*sqrt(a-cos(x))))*sqrt((1-cos(x))/(a-cos(x)))*sqrt(a-cos(x))/sqrt(1-cos(x))],
[(a+a*cos(c+d*x))*(-1/2*B+B*cos(c+d*x)),x,1,1/2*a*B*sin(c+d*x)/d+1/2*a*B*cos(c+d*x)*sin(c+d*x)/d],
[(a+a*cos(c+d*x))^4*(-4/5*B+B*cos(c+d*x)),x,1,1/5*B*(a+a*cos(c+d*x))^4*sin(c+d*x)/d],
[(a+a*cos(c+d*x))^n*(-B*n/(1+n)+B*cos(c+d*x)),x,1,B*(a+a*cos(c+d*x))^n*sin(c+d*x)/(d*(1+n))],
[(-3/2*B+B*cos(c+d*x))/(a+a*cos(c+d*x))^3,x,1,-1/2*B*sin(c+d*x)/(d*(a+a*cos(c+d*x))^3)],
[(a+a*cos(c+d*x))^(3/2)*(-3/5*B+B*cos(c+d*x)),x,1,2/5*B*(a+a*cos(c+d*x))^(3/2)*sin(c+d*x)/d],
[(B+B*cos(c+d*x))/(a+a*cos(c+d*x))^(1/2),x,2,2*B*sin(c+d*x)/(d*sqrt(a+a*cos(c+d*x)))],
[(-5/3*B+B*cos(c+d*x))/(a+a*cos(c+d*x))^(5/2),x,1,-2/3*B*sin(c+d*x)/(d*(a+a*cos(c+d*x))^(5/2))],
[(a+a*cos(c+d*x))^(2/3)*(A+B*cos(c+d*x)),x,3,3/5*B*(a+a*cos(c+d*x))^(2/3)*sin(c+d*x)/d+2/5*2^(1/6)*(5*A+2*B)*(a+a*cos(c+d*x))^(2/3)*hypergeom([-1/6,1/2],[3/2],1/2*(1-cos(c+d*x)))*sin(c+d*x)/(d*(1+cos(c+d*x))^(7/6))],
[(a+a*cos(c+d*x))^(1/3)*(A+B*cos(c+d*x)),x,3,3/4*B*(a+a*cos(c+d*x))^(1/3)*sin(c+d*x)/d+1/2*(4*A+B)*(a+a*cos(c+d*x))^(1/3)*hypergeom([1/6,1/2],[3/2],1/2*(1-cos(c+d*x)))*sin(c+d*x)/(2^(1/6)*d*(1+cos(c+d*x))^(5/6))],
[(A+B*cos(c+d*x))/(a+a*cos(c+d*x))^(1/3),x,3,3/2*B*sin(c+d*x)/(d*(a+a*cos(c+d*x))^(1/3))+(2*A-B)*hypergeom([1/2,5/6],[3/2],1/2*(1-cos(c+d*x)))*sin(c+d*x)/(2^(5/6)*d*(1+cos(c+d*x))^(1/6)*(a+a*cos(c+d*x))^(1/3))],
[(A+B*cos(c+d*x))/(a+a*cos(c+d*x))^(2/3),x,3,3*(A-B)*sin(c+d*x)/(d*(a+a*cos(c+d*x))^(2/3))-2^(5/6)*(A-2*B)*(a+a*cos(c+d*x))^(1/3)*hypergeom([1/6,1/2],[3/2],1/2*(1-cos(c+d*x)))*sin(c+d*x)/(a*d*(1+cos(c+d*x))^(5/6))],

# Integrands of the form (a+b Cos[e+f x])^m (c+d Cos[e+f x])^n
[(b*B/a+B*cos(c+d*x))/(a+b*cos(c+d*x)),x,3,B*x/b-2*B*arctan(sqrt(a-b)*tan(1/2*(c+d*x))/sqrt(a+b))*sqrt(a-b)*sqrt(a+b)/(a*b*d)],
[(a+b*cos(c+d*x))/(b+a*cos(c+d*x))^2,x,2,sin(c+d*x)/(d*(b+a*cos(c+d*x)))],
[(3+cos(c+d*x))/(2-cos(c+d*x)),x,2,-x+5*x/sqrt(3)+10*arctan(sin(c+d*x)/(2-cos(c+d*x)+sqrt(3)))/(d*sqrt(3))],
[(a*B+b*B*cos(c+d*x))/(a+b*cos(c+d*x))^(1/2),x,3,2*B*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2*b/(a+b)))*sqrt(a+b*cos(c+d*x))/(d*sqrt((a+b*cos(c+d*x))/(a+b)))],

# Integrands of the form (a+b Cos[e+f x])^(n/3) (c+d Cos[e+f x])^n

# n>0
[(a+b*cos(c+d*x))^(2/3)*(A+B*cos(c+d*x)),x,7,(a+b)*B*AppellF1(1/2,1/2,-5/3,3/2,1/2*(1-cos(c+d*x)),b*(1-cos(c+d*x))/(a+b))*(a+b*cos(c+d*x))^(2/3)*sin(c+d*x)*sqrt(2)/(b*d*((a+b*cos(c+d*x))/(a+b))^(2/3)*sqrt(1+cos(c+d*x)))+(A*b-a*B)*AppellF1(1/2,1/2,-2/3,3/2,1/2*(1-cos(c+d*x)),b*(1-cos(c+d*x))/(a+b))*(a+b*cos(c+d*x))^(2/3)*sin(c+d*x)*sqrt(2)/(b*d*((a+b*cos(c+d*x))/(a+b))^(2/3)*sqrt(1+cos(c+d*x)))],
[(a+b*cos(c+d*x))^(1/3)*(A+B*cos(c+d*x)),x,7,(a+b)*B*AppellF1(1/2,1/2,-4/3,3/2,1/2*(1-cos(c+d*x)),b*(1-cos(c+d*x))/(a+b))*(a+b*cos(c+d*x))^(1/3)*sin(c+d*x)*sqrt(2)/(b*d*((a+b*cos(c+d*x))/(a+b))^(1/3)*sqrt(1+cos(c+d*x)))+(A*b-a*B)*AppellF1(1/2,1/2,-1/3,3/2,1/2*(1-cos(c+d*x)),b*(1-cos(c+d*x))/(a+b))*(a+b*cos(c+d*x))^(1/3)*sin(c+d*x)*sqrt(2)/(b*d*((a+b*cos(c+d*x))/(a+b))^(1/3)*sqrt(1+cos(c+d*x)))],

# n<0
[(A+B*cos(c+d*x))/(a+b*cos(c+d*x))^(1/3),x,7,B*AppellF1(1/2,1/2,-2/3,3/2,1/2*(1-cos(c+d*x)),b*(1-cos(c+d*x))/(a+b))*(a+b*cos(c+d*x))^(2/3)*sin(c+d*x)*sqrt(2)/(b*d*((a+b*cos(c+d*x))/(a+b))^(2/3)*sqrt(1+cos(c+d*x)))+(A*b-a*B)*AppellF1(1/2,1/2,1/3,3/2,1/2*(1-cos(c+d*x)),b*(1-cos(c+d*x))/(a+b))*((a+b*cos(c+d*x))/(a+b))^(1/3)*sin(c+d*x)*sqrt(2)/(b*d*(a+b*cos(c+d*x))^(1/3)*sqrt(1+cos(c+d*x)))],
[(A+B*cos(c+d*x))/(a+b*cos(c+d*x))^(2/3),x,7,B*AppellF1(1/2,1/2,-1/3,3/2,1/2*(1-cos(c+d*x)),b*(1-cos(c+d*x))/(a+b))*(a+b*cos(c+d*x))^(1/3)*sin(c+d*x)*sqrt(2)/(b*d*((a+b*cos(c+d*x))/(a+b))^(1/3)*sqrt(1+cos(c+d*x)))+(A*b-a*B)*AppellF1(1/2,1/2,2/3,3/2,1/2*(1-cos(c+d*x)),b*(1-cos(c+d*x))/(a+b))*((a+b*cos(c+d*x))/(a+b))^(2/3)*sin(c+d*x)*sqrt(2)/(b*d*(a+b*cos(c+d*x))^(2/3)*sqrt(1+cos(c+d*x)))],

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

# Integrands of the form (a+b Cos[e+f x]) (c Cos[e+f x])^m (d Cos[e+f x])^n

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

# n>0
[cos(c+d*x)^2*(b*cos(c+d*x))^(1/2)*(A+B*cos(c+d*x)),x,9,2/5*A*(b*cos(c+d*x))^(3/2)*sin(c+d*x)/(b*d)+2/7*B*(b*cos(c+d*x))^(5/2)*sin(c+d*x)/(b^2*d)+10/21*b*B*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))/(d*sqrt(b*cos(c+d*x)))+10/21*B*sin(c+d*x)*sqrt(b*cos(c+d*x))/d+6/5*A*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(b*cos(c+d*x))/(d*sqrt(cos(c+d*x)))],
[cos(c+d*x)*(b*cos(c+d*x))^(1/2)*(A+B*cos(c+d*x)),x,8,2/5*B*(b*cos(c+d*x))^(3/2)*sin(c+d*x)/(b*d)+2/3*A*b*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))/(d*sqrt(b*cos(c+d*x)))+2/3*A*sin(c+d*x)*sqrt(b*cos(c+d*x))/d+6/5*B*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(b*cos(c+d*x))/(d*sqrt(cos(c+d*x)))],
[(b*cos(c+d*x))^(1/2)*(A+B*cos(c+d*x)),x,6,2/3*b*B*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))/(d*sqrt(b*cos(c+d*x)))+2/3*B*sin(c+d*x)*sqrt(b*cos(c+d*x))/d+2*A*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(b*cos(c+d*x))/(d*sqrt(cos(c+d*x)))],
[(b*cos(c+d*x))^(1/2)*(A+B*cos(c+d*x))*sec(c+d*x),x,6,2*A*b*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))/(d*sqrt(b*cos(c+d*x)))+2*B*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(b*cos(c+d*x))/(d*sqrt(cos(c+d*x)))],
[(b*cos(c+d*x))^(1/2)*(A+B*cos(c+d*x))*sec(c+d*x)^2,x,7,2*A*b*sin(c+d*x)/(d*sqrt(b*cos(c+d*x)))+2*b*B*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))/(d*sqrt(b*cos(c+d*x)))-2*A*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(b*cos(c+d*x))/(d*sqrt(cos(c+d*x)))],
[(b*cos(c+d*x))^(1/2)*(A+B*cos(c+d*x))*sec(c+d*x)^3,x,8,2/3*A*b^2*sin(c+d*x)/(d*(b*cos(c+d*x))^(3/2))+2*b*B*sin(c+d*x)/(d*sqrt(b*cos(c+d*x)))+2/3*A*b*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))/(d*sqrt(b*cos(c+d*x)))-2*B*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(b*cos(c+d*x))/(d*sqrt(cos(c+d*x)))],
[(b*cos(c+d*x))^(1/2)*(A+B*cos(c+d*x))*sec(c+d*x)^4,x,9,2/5*A*b^3*sin(c+d*x)/(d*(b*cos(c+d*x))^(5/2))+2/3*b^2*B*sin(c+d*x)/(d*(b*cos(c+d*x))^(3/2))+6/5*A*b*sin(c+d*x)/(d*sqrt(b*cos(c+d*x)))+2/3*b*B*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))/(d*sqrt(b*cos(c+d*x)))-6/5*A*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(b*cos(c+d*x))/(d*sqrt(cos(c+d*x)))],
[cos(c+d*x)*(b*cos(c+d*x))^(3/2)*(A+B*cos(c+d*x)),x,9,2/5*A*(b*cos(c+d*x))^(3/2)*sin(c+d*x)/d+2/7*B*(b*cos(c+d*x))^(5/2)*sin(c+d*x)/(b*d)+10/21*b^2*B*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))/(d*sqrt(b*cos(c+d*x)))+10/21*b*B*sin(c+d*x)*sqrt(b*cos(c+d*x))/d+6/5*A*b*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(b*cos(c+d*x))/(d*sqrt(cos(c+d*x)))],
[(b*cos(c+d*x))^(3/2)*(A+B*cos(c+d*x)),x,7,2/5*B*(b*cos(c+d*x))^(3/2)*sin(c+d*x)/d+2/3*A*b^2*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))/(d*sqrt(b*cos(c+d*x)))+2/3*A*b*sin(c+d*x)*sqrt(b*cos(c+d*x))/d+6/5*b*B*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(b*cos(c+d*x))/(d*sqrt(cos(c+d*x)))],
[(b*cos(c+d*x))^(3/2)*(A+B*cos(c+d*x))*sec(c+d*x),x,7,2/3*b^2*B*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))/(d*sqrt(b*cos(c+d*x)))+2/3*b*B*sin(c+d*x)*sqrt(b*cos(c+d*x))/d+2*A*b*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(b*cos(c+d*x))/(d*sqrt(cos(c+d*x)))],
[(b*cos(c+d*x))^(3/2)*(A+B*cos(c+d*x))*sec(c+d*x)^2,x,6,2*A*b^2*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))/(d*sqrt(b*cos(c+d*x)))+2*b*B*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(b*cos(c+d*x))/(d*sqrt(cos(c+d*x)))],
[(b*cos(c+d*x))^(3/2)*(A+B*cos(c+d*x))*sec(c+d*x)^3,x,7,2*A*b^2*sin(c+d*x)/(d*sqrt(b*cos(c+d*x)))+2*b^2*B*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))/(d*sqrt(b*cos(c+d*x)))-2*A*b*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(b*cos(c+d*x))/(d*sqrt(cos(c+d*x)))],
[(b*cos(c+d*x))^(3/2)*(A+B*cos(c+d*x))*sec(c+d*x)^4,x,8,2/3*A*b^3*sin(c+d*x)/(d*(b*cos(c+d*x))^(3/2))+2*b^2*B*sin(c+d*x)/(d*sqrt(b*cos(c+d*x)))+2/3*A*b^2*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))/(d*sqrt(b*cos(c+d*x)))-2*b*B*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(b*cos(c+d*x))/(d*sqrt(cos(c+d*x)))],
[(b*cos(c+d*x))^(3/2)*(A+B*cos(c+d*x))*sec(c+d*x)^5,x,9,2/5*A*b^4*sin(c+d*x)/(d*(b*cos(c+d*x))^(5/2))+2/3*b^3*B*sin(c+d*x)/(d*(b*cos(c+d*x))^(3/2))+6/5*A*b^2*sin(c+d*x)/(d*sqrt(b*cos(c+d*x)))+2/3*b^2*B*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))/(d*sqrt(b*cos(c+d*x)))-6/5*A*b*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(b*cos(c+d*x))/(d*sqrt(cos(c+d*x)))],
[(b*cos(c+d*x))^(5/2)*(A+B*cos(c+d*x)),x,8,2/5*A*b*(b*cos(c+d*x))^(3/2)*sin(c+d*x)/d+2/7*B*(b*cos(c+d*x))^(5/2)*sin(c+d*x)/d+10/21*b^3*B*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))/(d*sqrt(b*cos(c+d*x)))+10/21*b^2*B*sin(c+d*x)*sqrt(b*cos(c+d*x))/d+6/5*A*b^2*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(b*cos(c+d*x))/(d*sqrt(cos(c+d*x)))],
[(b*cos(c+d*x))^(5/2)*(A+B*cos(c+d*x))*sec(c+d*x),x,8,2/5*b*B*(b*cos(c+d*x))^(3/2)*sin(c+d*x)/d+2/3*A*b^3*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))/(d*sqrt(b*cos(c+d*x)))+2/3*A*b^2*sin(c+d*x)*sqrt(b*cos(c+d*x))/d+6/5*b^2*B*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(b*cos(c+d*x))/(d*sqrt(cos(c+d*x)))],
[(b*cos(c+d*x))^(5/2)*(A+B*cos(c+d*x))*sec(c+d*x)^2,x,7,2/3*b^3*B*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))/(d*sqrt(b*cos(c+d*x)))+2/3*b^2*B*sin(c+d*x)*sqrt(b*cos(c+d*x))/d+2*A*b^2*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(b*cos(c+d*x))/(d*sqrt(cos(c+d*x)))],
[(b*cos(c+d*x))^(5/2)*(A+B*cos(c+d*x))*sec(c+d*x)^3,x,6,2*A*b^3*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))/(d*sqrt(b*cos(c+d*x)))+2*b^2*B*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(b*cos(c+d*x))/(d*sqrt(cos(c+d*x)))],
[(b*cos(c+d*x))^(5/2)*(A+B*cos(c+d*x))*sec(c+d*x)^4,x,7,2*A*b^3*sin(c+d*x)/(d*sqrt(b*cos(c+d*x)))+2*b^3*B*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))/(d*sqrt(b*cos(c+d*x)))-2*A*b^2*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(b*cos(c+d*x))/(d*sqrt(cos(c+d*x)))],
[(b*cos(c+d*x))^(5/2)*(A+B*cos(c+d*x))*sec(c+d*x)^5,x,8,2/3*A*b^4*sin(c+d*x)/(d*(b*cos(c+d*x))^(3/2))+2*b^3*B*sin(c+d*x)/(d*sqrt(b*cos(c+d*x)))+2/3*A*b^3*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))/(d*sqrt(b*cos(c+d*x)))-2*b^2*B*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(b*cos(c+d*x))/(d*sqrt(cos(c+d*x)))],
[(b*cos(c+d*x))^(5/2)*(A+B*cos(c+d*x))*sec(c+d*x)^6,x,9,2/5*A*b^5*sin(c+d*x)/(d*(b*cos(c+d*x))^(5/2))+2/3*b^4*B*sin(c+d*x)/(d*(b*cos(c+d*x))^(3/2))+6/5*A*b^3*sin(c+d*x)/(d*sqrt(b*cos(c+d*x)))+2/3*b^3*B*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))/(d*sqrt(b*cos(c+d*x)))-6/5*A*b^2*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(b*cos(c+d*x))/(d*sqrt(cos(c+d*x)))],

# n<0
[cos(c+d*x)^3*(A+B*cos(c+d*x))/(b*cos(c+d*x))^(1/2),x,9,2/5*A*(b*cos(c+d*x))^(3/2)*sin(c+d*x)/(b^2*d)+2/7*B*(b*cos(c+d*x))^(5/2)*sin(c+d*x)/(b^3*d)+10/21*B*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))/(d*sqrt(b*cos(c+d*x)))+10/21*B*sin(c+d*x)*sqrt(b*cos(c+d*x))/(b*d)+6/5*A*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(b*cos(c+d*x))/(b*d*sqrt(cos(c+d*x)))],
[cos(c+d*x)^2*(A+B*cos(c+d*x))/(b*cos(c+d*x))^(1/2),x,8,2/5*B*(b*cos(c+d*x))^(3/2)*sin(c+d*x)/(b^2*d)+2/3*A*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))/(d*sqrt(b*cos(c+d*x)))+2/3*A*sin(c+d*x)*sqrt(b*cos(c+d*x))/(b*d)+6/5*B*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(b*cos(c+d*x))/(b*d*sqrt(cos(c+d*x)))],
[cos(c+d*x)*(A+B*cos(c+d*x))/(b*cos(c+d*x))^(1/2),x,7,2/3*B*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))/(d*sqrt(b*cos(c+d*x)))+2/3*B*sin(c+d*x)*sqrt(b*cos(c+d*x))/(b*d)+2*A*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(b*cos(c+d*x))/(b*d*sqrt(cos(c+d*x)))],
[(A+B*cos(c+d*x))/(b*cos(c+d*x))^(1/2),x,5,2*A*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))/(d*sqrt(b*cos(c+d*x)))+2*B*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(b*cos(c+d*x))/(b*d*sqrt(cos(c+d*x)))],
[(A+B*cos(c+d*x))*sec(c+d*x)/(b*cos(c+d*x))^(1/2),x,7,2*A*sin(c+d*x)/(d*sqrt(b*cos(c+d*x)))+2*B*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))/(d*sqrt(b*cos(c+d*x)))-2*A*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(b*cos(c+d*x))/(b*d*sqrt(cos(c+d*x)))],
[(A+B*cos(c+d*x))*sec(c+d*x)^2/(b*cos(c+d*x))^(1/2),x,8,2/3*A*b*sin(c+d*x)/(d*(b*cos(c+d*x))^(3/2))+2*B*sin(c+d*x)/(d*sqrt(b*cos(c+d*x)))+2/3*A*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))/(d*sqrt(b*cos(c+d*x)))-2*B*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(b*cos(c+d*x))/(b*d*sqrt(cos(c+d*x)))],
[(A+B*cos(c+d*x))*sec(c+d*x)^3/(b*cos(c+d*x))^(1/2),x,9,2/5*A*b^2*sin(c+d*x)/(d*(b*cos(c+d*x))^(5/2))+2/3*b*B*sin(c+d*x)/(d*(b*cos(c+d*x))^(3/2))+6/5*A*sin(c+d*x)/(d*sqrt(b*cos(c+d*x)))+2/3*B*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))/(d*sqrt(b*cos(c+d*x)))-6/5*A*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(b*cos(c+d*x))/(b*d*sqrt(cos(c+d*x)))],
[cos(c+d*x)^4*(A+B*cos(c+d*x))/(b*cos(c+d*x))^(3/2),x,9,2/5*A*(b*cos(c+d*x))^(3/2)*sin(c+d*x)/(b^3*d)+2/7*B*(b*cos(c+d*x))^(5/2)*sin(c+d*x)/(b^4*d)+10/21*B*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))/(b*d*sqrt(b*cos(c+d*x)))+10/21*B*sin(c+d*x)*sqrt(b*cos(c+d*x))/(b^2*d)+6/5*A*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(b*cos(c+d*x))/(b^2*d*sqrt(cos(c+d*x)))],
[cos(c+d*x)^3*(A+B*cos(c+d*x))/(b*cos(c+d*x))^(3/2),x,8,2/5*B*(b*cos(c+d*x))^(3/2)*sin(c+d*x)/(b^3*d)+2/3*A*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))/(b*d*sqrt(b*cos(c+d*x)))+2/3*A*sin(c+d*x)*sqrt(b*cos(c+d*x))/(b^2*d)+6/5*B*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(b*cos(c+d*x))/(b^2*d*sqrt(cos(c+d*x)))],
[cos(c+d*x)^2*(A+B*cos(c+d*x))/(b*cos(c+d*x))^(3/2),x,7,2/3*B*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))/(b*d*sqrt(b*cos(c+d*x)))+2/3*B*sin(c+d*x)*sqrt(b*cos(c+d*x))/(b^2*d)+2*A*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(b*cos(c+d*x))/(b^2*d*sqrt(cos(c+d*x)))],
[cos(c+d*x)*(A+B*cos(c+d*x))/(b*cos(c+d*x))^(3/2),x,6,2*A*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))/(b*d*sqrt(b*cos(c+d*x)))+2*B*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(b*cos(c+d*x))/(b^2*d*sqrt(cos(c+d*x)))],
[(A+B*cos(c+d*x))/(b*cos(c+d*x))^(3/2),x,6,2*A*sin(c+d*x)/(b*d*sqrt(b*cos(c+d*x)))+2*B*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))/(b*d*sqrt(b*cos(c+d*x)))-2*A*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(b*cos(c+d*x))/(b^2*d*sqrt(cos(c+d*x)))],
[(A+B*cos(c+d*x))*sec(c+d*x)/(b*cos(c+d*x))^(3/2),x,8,2/3*A*sin(c+d*x)/(d*(b*cos(c+d*x))^(3/2))+2*B*sin(c+d*x)/(b*d*sqrt(b*cos(c+d*x)))+2/3*A*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))/(b*d*sqrt(b*cos(c+d*x)))-2*B*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(b*cos(c+d*x))/(b^2*d*sqrt(cos(c+d*x)))],
[(A+B*cos(c+d*x))*sec(c+d*x)^2/(b*cos(c+d*x))^(3/2),x,9,2/5*A*b*sin(c+d*x)/(d*(b*cos(c+d*x))^(5/2))+2/3*B*sin(c+d*x)/(d*(b*cos(c+d*x))^(3/2))+6/5*A*sin(c+d*x)/(b*d*sqrt(b*cos(c+d*x)))+2/3*B*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))/(b*d*sqrt(b*cos(c+d*x)))-6/5*A*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(b*cos(c+d*x))/(b^2*d*sqrt(cos(c+d*x)))],
[cos(c+d*x)^5*(A+B*cos(c+d*x))/(b*cos(c+d*x))^(5/2),x,9,2/5*A*(b*cos(c+d*x))^(3/2)*sin(c+d*x)/(b^4*d)+2/7*B*(b*cos(c+d*x))^(5/2)*sin(c+d*x)/(b^5*d)+10/21*B*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))/(b^2*d*sqrt(b*cos(c+d*x)))+10/21*B*sin(c+d*x)*sqrt(b*cos(c+d*x))/(b^3*d)+6/5*A*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(b*cos(c+d*x))/(b^3*d*sqrt(cos(c+d*x)))],
[cos(c+d*x)^4*(A+B*cos(c+d*x))/(b*cos(c+d*x))^(5/2),x,8,2/5*B*(b*cos(c+d*x))^(3/2)*sin(c+d*x)/(b^4*d)+2/3*A*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))/(b^2*d*sqrt(b*cos(c+d*x)))+2/3*A*sin(c+d*x)*sqrt(b*cos(c+d*x))/(b^3*d)+6/5*B*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(b*cos(c+d*x))/(b^3*d*sqrt(cos(c+d*x)))],
[cos(c+d*x)^3*(A+B*cos(c+d*x))/(b*cos(c+d*x))^(5/2),x,7,2/3*B*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))/(b^2*d*sqrt(b*cos(c+d*x)))+2/3*B*sin(c+d*x)*sqrt(b*cos(c+d*x))/(b^3*d)+2*A*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(b*cos(c+d*x))/(b^3*d*sqrt(cos(c+d*x)))],
[cos(c+d*x)^2*(A+B*cos(c+d*x))/(b*cos(c+d*x))^(5/2),x,6,2*A*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))/(b^2*d*sqrt(b*cos(c+d*x)))+2*B*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(b*cos(c+d*x))/(b^3*d*sqrt(cos(c+d*x)))],
[cos(c+d*x)*(A+B*cos(c+d*x))/(b*cos(c+d*x))^(5/2),x,7,2*A*sin(c+d*x)/(b^2*d*sqrt(b*cos(c+d*x)))+2*B*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))/(b^2*d*sqrt(b*cos(c+d*x)))-2*A*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(b*cos(c+d*x))/(b^3*d*sqrt(cos(c+d*x)))],
[(A+B*cos(c+d*x))/(b*cos(c+d*x))^(5/2),x,7,2/3*A*sin(c+d*x)/(b*d*(b*cos(c+d*x))^(3/2))+2*B*sin(c+d*x)/(b^2*d*sqrt(b*cos(c+d*x)))+2/3*A*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))/(b^2*d*sqrt(b*cos(c+d*x)))-2*B*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(b*cos(c+d*x))/(b^3*d*sqrt(cos(c+d*x)))],
[(A+B*cos(c+d*x))*sec(c+d*x)/(b*cos(c+d*x))^(5/2),x,9,2/5*A*sin(c+d*x)/(d*(b*cos(c+d*x))^(5/2))+2/3*B*sin(c+d*x)/(b*d*(b*cos(c+d*x))^(3/2))+6/5*A*sin(c+d*x)/(b^2*d*sqrt(b*cos(c+d*x)))+2/3*B*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))/(b^2*d*sqrt(b*cos(c+d*x)))-6/5*A*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(b*cos(c+d*x))/(b^3*d*sqrt(cos(c+d*x)))],
[(A+B*cos(c+d*x))/(b*cos(c+d*x))^(7/2),x,8,2/5*A*sin(c+d*x)/(b*d*(b*cos(c+d*x))^(5/2))+2/3*B*sin(c+d*x)/(b^2*d*(b*cos(c+d*x))^(3/2))+6/5*A*sin(c+d*x)/(b^3*d*sqrt(b*cos(c+d*x)))+2/3*B*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2))*sqrt(cos(c+d*x))/(b^3*d*sqrt(b*cos(c+d*x)))-6/5*A*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(2))*sqrt(b*cos(c+d*x))/(b^4*d*sqrt(cos(c+d*x)))],

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

# n>0
[cos(c+d*x)^(5/2)*(b*cos(c+d*x))^(1/2)*(A+B*cos(c+d*x)),x,7,1/4*B*cos(c+d*x)^(5/2)*sin(c+d*x)*sqrt(b*cos(c+d*x))/d+3/8*B*x*sqrt(b*cos(c+d*x))/sqrt(cos(c+d*x))+A*sin(c+d*x)*sqrt(b*cos(c+d*x))/(d*sqrt(cos(c+d*x)))-1/3*A*sin(c+d*x)^3*sqrt(b*cos(c+d*x))/(d*sqrt(cos(c+d*x)))+3/8*B*sin(c+d*x)*sqrt(cos(c+d*x))*sqrt(b*cos(c+d*x))/d],
[cos(c+d*x)^(3/2)*(b*cos(c+d*x))^(1/2)*(A+B*cos(c+d*x)),x,6,1/2*A*x*sqrt(b*cos(c+d*x))/sqrt(cos(c+d*x))+B*sin(c+d*x)*sqrt(b*cos(c+d*x))/(d*sqrt(cos(c+d*x)))-1/3*B*sin(c+d*x)^3*sqrt(b*cos(c+d*x))/(d*sqrt(cos(c+d*x)))+1/2*A*sin(c+d*x)*sqrt(cos(c+d*x))*sqrt(b*cos(c+d*x))/d],
[cos(c+d*x)^(1/2)*(b*cos(c+d*x))^(1/2)*(A+B*cos(c+d*x)),x,2,1/2*B*x*sqrt(b*cos(c+d*x))/sqrt(cos(c+d*x))+A*sin(c+d*x)*sqrt(b*cos(c+d*x))/(d*sqrt(cos(c+d*x)))+1/2*B*sin(c+d*x)*sqrt(cos(c+d*x))*sqrt(b*cos(c+d*x))/d],
[(b*cos(c+d*x))^(1/2)*(A+B*cos(c+d*x))/cos(c+d*x)^(1/2),x,3,A*x*sqrt(b*cos(c+d*x))/sqrt(cos(c+d*x))+B*sin(c+d*x)*sqrt(b*cos(c+d*x))/(d*sqrt(cos(c+d*x)))],
[(b*cos(c+d*x))^(1/2)*(A+B*cos(c+d*x))/cos(c+d*x)^(3/2),x,3,B*x*sqrt(b*cos(c+d*x))/sqrt(cos(c+d*x))+A*arctanh(sin(c+d*x))*sqrt(b*cos(c+d*x))/(d*sqrt(cos(c+d*x)))],
[(b*cos(c+d*x))^(1/2)*(A+B*cos(c+d*x))/cos(c+d*x)^(5/2),x,5,A*sin(c+d*x)*sqrt(b*cos(c+d*x))/(d*cos(c+d*x)^(3/2))+B*arctanh(sin(c+d*x))*sqrt(b*cos(c+d*x))/(d*sqrt(cos(c+d*x)))],
[(b*cos(c+d*x))^(1/2)*(A+B*cos(c+d*x))/cos(c+d*x)^(7/2),x,6,1/2*A*sin(c+d*x)*sqrt(b*cos(c+d*x))/(d*cos(c+d*x)^(5/2))+B*sin(c+d*x)*sqrt(b*cos(c+d*x))/(d*cos(c+d*x)^(3/2))+1/2*A*arctanh(sin(c+d*x))*sqrt(b*cos(c+d*x))/(d*sqrt(cos(c+d*x)))],
[(b*cos(c+d*x))^(1/2)*(A+B*cos(c+d*x))/cos(c+d*x)^(9/2),x,6,1/2*B*sin(c+d*x)*sqrt(b*cos(c+d*x))/(d*cos(c+d*x)^(5/2))+A*sin(c+d*x)*sqrt(b*cos(c+d*x))/(d*cos(c+d*x)^(3/2))+1/3*A*sin(c+d*x)^3*sqrt(b*cos(c+d*x))/(d*cos(c+d*x)^(7/2))+1/2*B*arctanh(sin(c+d*x))*sqrt(b*cos(c+d*x))/(d*sqrt(cos(c+d*x)))],
[cos(c+d*x)^(3/2)*(b*cos(c+d*x))^(3/2)*(A+B*cos(c+d*x)),x,7,1/4*b*B*cos(c+d*x)^(5/2)*sin(c+d*x)*sqrt(b*cos(c+d*x))/d+3/8*b*B*x*sqrt(b*cos(c+d*x))/sqrt(cos(c+d*x))+A*b*sin(c+d*x)*sqrt(b*cos(c+d*x))/(d*sqrt(cos(c+d*x)))-1/3*A*b*sin(c+d*x)^3*sqrt(b*cos(c+d*x))/(d*sqrt(cos(c+d*x)))+3/8*b*B*sin(c+d*x)*sqrt(cos(c+d*x))*sqrt(b*cos(c+d*x))/d],
[cos(c+d*x)^(1/2)*(b*cos(c+d*x))^(3/2)*(A+B*cos(c+d*x)),x,6,1/2*A*b*x*sqrt(b*cos(c+d*x))/sqrt(cos(c+d*x))+b*B*sin(c+d*x)*sqrt(b*cos(c+d*x))/(d*sqrt(cos(c+d*x)))-1/3*b*B*sin(c+d*x)^3*sqrt(b*cos(c+d*x))/(d*sqrt(cos(c+d*x)))+1/2*A*b*sin(c+d*x)*sqrt(cos(c+d*x))*sqrt(b*cos(c+d*x))/d],
[(b*cos(c+d*x))^(3/2)*(A+B*cos(c+d*x))/cos(c+d*x)^(1/2),x,2,1/2*b*B*x*sqrt(b*cos(c+d*x))/sqrt(cos(c+d*x))+A*b*sin(c+d*x)*sqrt(b*cos(c+d*x))/(d*sqrt(cos(c+d*x)))+1/2*b*B*sin(c+d*x)*sqrt(cos(c+d*x))*sqrt(b*cos(c+d*x))/d],
[(b*cos(c+d*x))^(3/2)*(A+B*cos(c+d*x))/cos(c+d*x)^(3/2),x,3,A*b*x*sqrt(b*cos(c+d*x))/sqrt(cos(c+d*x))+b*B*sin(c+d*x)*sqrt(b*cos(c+d*x))/(d*sqrt(cos(c+d*x)))],
[(b*cos(c+d*x))^(3/2)*(A+B*cos(c+d*x))/cos(c+d*x)^(5/2),x,3,b*B*x*sqrt(b*cos(c+d*x))/sqrt(cos(c+d*x))+A*b*arctanh(sin(c+d*x))*sqrt(b*cos(c+d*x))/(d*sqrt(cos(c+d*x)))],
[(b*cos(c+d*x))^(3/2)*(A+B*cos(c+d*x))/cos(c+d*x)^(7/2),x,5,A*b*sin(c+d*x)*sqrt(b*cos(c+d*x))/(d*cos(c+d*x)^(3/2))+b*B*arctanh(sin(c+d*x))*sqrt(b*cos(c+d*x))/(d*sqrt(cos(c+d*x)))],
[(b*cos(c+d*x))^(3/2)*(A+B*cos(c+d*x))/cos(c+d*x)^(9/2),x,6,1/2*A*b*sin(c+d*x)*sqrt(b*cos(c+d*x))/(d*cos(c+d*x)^(5/2))+b*B*sin(c+d*x)*sqrt(b*cos(c+d*x))/(d*cos(c+d*x)^(3/2))+1/2*A*b*arctanh(sin(c+d*x))*sqrt(b*cos(c+d*x))/(d*sqrt(cos(c+d*x)))],
[(b*cos(c+d*x))^(3/2)*(A+B*cos(c+d*x))/cos(c+d*x)^(11/2),x,6,1/2*b*B*sin(c+d*x)*sqrt(b*cos(c+d*x))/(d*cos(c+d*x)^(5/2))+A*b*sin(c+d*x)*sqrt(b*cos(c+d*x))/(d*cos(c+d*x)^(3/2))+1/3*A*b*sin(c+d*x)^3*sqrt(b*cos(c+d*x))/(d*cos(c+d*x)^(7/2))+1/2*b*B*arctanh(sin(c+d*x))*sqrt(b*cos(c+d*x))/(d*sqrt(cos(c+d*x)))],
[cos(c+d*x)^(1/2)*(b*cos(c+d*x))^(5/2)*(A+B*cos(c+d*x)),x,7,1/4*b^2*B*cos(c+d*x)^(5/2)*sin(c+d*x)*sqrt(b*cos(c+d*x))/d+3/8*b^2*B*x*sqrt(b*cos(c+d*x))/sqrt(cos(c+d*x))+A*b^2*sin(c+d*x)*sqrt(b*cos(c+d*x))/(d*sqrt(cos(c+d*x)))-1/3*A*b^2*sin(c+d*x)^3*sqrt(b*cos(c+d*x))/(d*sqrt(cos(c+d*x)))+3/8*b^2*B*sin(c+d*x)*sqrt(cos(c+d*x))*sqrt(b*cos(c+d*x))/d],
[(b*cos(c+d*x))^(5/2)*(A+B*cos(c+d*x))/cos(c+d*x)^(1/2),x,6,1/2*A*b^2*x*sqrt(b*cos(c+d*x))/sqrt(cos(c+d*x))+b^2*B*sin(c+d*x)*sqrt(b*cos(c+d*x))/(d*sqrt(cos(c+d*x)))-1/3*b^2*B*sin(c+d*x)^3*sqrt(b*cos(c+d*x))/(d*sqrt(cos(c+d*x)))+1/2*A*b^2*sin(c+d*x)*sqrt(cos(c+d*x))*sqrt(b*cos(c+d*x))/d],
[(b*cos(c+d*x))^(5/2)*(A+B*cos(c+d*x))/cos(c+d*x)^(3/2),x,2,1/2*b^2*B*x*sqrt(b*cos(c+d*x))/sqrt(cos(c+d*x))+A*b^2*sin(c+d*x)*sqrt(b*cos(c+d*x))/(d*sqrt(cos(c+d*x)))+1/2*b^2*B*sin(c+d*x)*sqrt(cos(c+d*x))*sqrt(b*cos(c+d*x))/d],
[(b*cos(c+d*x))^(5/2)*(A+B*cos(c+d*x))/cos(c+d*x)^(5/2),x,3,A*b^2*x*sqrt(b*cos(c+d*x))/sqrt(cos(c+d*x))+b^2*B*sin(c+d*x)*sqrt(b*cos(c+d*x))/(d*sqrt(cos(c+d*x)))],
[(b*cos(c+d*x))^(5/2)*(A+B*cos(c+d*x))/cos(c+d*x)^(7/2),x,3,b^2*B*x*sqrt(b*cos(c+d*x))/sqrt(cos(c+d*x))+A*b^2*arctanh(sin(c+d*x))*sqrt(b*cos(c+d*x))/(d*sqrt(cos(c+d*x)))],
[(b*cos(c+d*x))^(5/2)*(A+B*cos(c+d*x))/cos(c+d*x)^(9/2),x,5,A*b^2*sin(c+d*x)*sqrt(b*cos(c+d*x))/(d*cos(c+d*x)^(3/2))+b^2*B*arctanh(sin(c+d*x))*sqrt(b*cos(c+d*x))/(d*sqrt(cos(c+d*x)))],
[(b*cos(c+d*x))^(5/2)*(A+B*cos(c+d*x))/cos(c+d*x)^(11/2),x,6,1/2*A*b^2*sin(c+d*x)*sqrt(b*cos(c+d*x))/(d*cos(c+d*x)^(5/2))+b^2*B*sin(c+d*x)*sqrt(b*cos(c+d*x))/(d*cos(c+d*x)^(3/2))+1/2*A*b^2*arctanh(sin(c+d*x))*sqrt(b*cos(c+d*x))/(d*sqrt(cos(c+d*x)))],
[(b*cos(c+d*x))^(5/2)*(A+B*cos(c+d*x))/cos(c+d*x)^(13/2),x,6,1/2*b^2*B*sin(c+d*x)*sqrt(b*cos(c+d*x))/(d*cos(c+d*x)^(5/2))+A*b^2*sin(c+d*x)*sqrt(b*cos(c+d*x))/(d*cos(c+d*x)^(3/2))+1/3*A*b^2*sin(c+d*x)^3*sqrt(b*cos(c+d*x))/(d*cos(c+d*x)^(7/2))+1/2*b^2*B*arctanh(sin(c+d*x))*sqrt(b*cos(c+d*x))/(d*sqrt(cos(c+d*x)))],

# n<0
[cos(c+d*x)^(5/2)*(A+B*cos(c+d*x))/(b*cos(c+d*x))^(1/2),x,6,1/2*A*cos(c+d*x)^(3/2)*sin(c+d*x)/(d*sqrt(b*cos(c+d*x)))+1/2*A*x*sqrt(cos(c+d*x))/sqrt(b*cos(c+d*x))+B*sin(c+d*x)*sqrt(cos(c+d*x))/(d*sqrt(b*cos(c+d*x)))-1/3*B*sin(c+d*x)^3*sqrt(cos(c+d*x))/(d*sqrt(b*cos(c+d*x)))],
[cos(c+d*x)^(3/2)*(A+B*cos(c+d*x))/(b*cos(c+d*x))^(1/2),x,2,1/2*B*cos(c+d*x)^(3/2)*sin(c+d*x)/(d*sqrt(b*cos(c+d*x)))+1/2*B*x*sqrt(cos(c+d*x))/sqrt(b*cos(c+d*x))+A*sin(c+d*x)*sqrt(cos(c+d*x))/(d*sqrt(b*cos(c+d*x)))],
[cos(c+d*x)^(1/2)*(A+B*cos(c+d*x))/(b*cos(c+d*x))^(1/2),x,3,A*x*sqrt(cos(c+d*x))/sqrt(b*cos(c+d*x))+B*sin(c+d*x)*sqrt(cos(c+d*x))/(d*sqrt(b*cos(c+d*x)))],
[(A+B*cos(c+d*x))/(cos(c+d*x)^(1/2)*(b*cos(c+d*x))^(1/2)),x,3,B*x*sqrt(cos(c+d*x))/sqrt(b*cos(c+d*x))+A*arctanh(sin(c+d*x))*sqrt(cos(c+d*x))/(d*sqrt(b*cos(c+d*x)))],
[(A+B*cos(c+d*x))/(cos(c+d*x)^(3/2)*(b*cos(c+d*x))^(1/2)),x,5,A*sin(c+d*x)/(d*sqrt(cos(c+d*x))*sqrt(b*cos(c+d*x)))+B*arctanh(sin(c+d*x))*sqrt(cos(c+d*x))/(d*sqrt(b*cos(c+d*x)))],
[(A+B*cos(c+d*x))/(cos(c+d*x)^(5/2)*(b*cos(c+d*x))^(1/2)),x,6,1/2*A*sin(c+d*x)/(d*cos(c+d*x)^(3/2)*sqrt(b*cos(c+d*x)))+B*sin(c+d*x)/(d*sqrt(cos(c+d*x))*sqrt(b*cos(c+d*x)))+1/2*A*arctanh(sin(c+d*x))*sqrt(cos(c+d*x))/(d*sqrt(b*cos(c+d*x)))],
[(A+B*cos(c+d*x))/(cos(c+d*x)^(7/2)*(b*cos(c+d*x))^(1/2)),x,6,1/2*B*sin(c+d*x)/(d*cos(c+d*x)^(3/2)*sqrt(b*cos(c+d*x)))+1/3*A*sin(c+d*x)^3/(d*cos(c+d*x)^(5/2)*sqrt(b*cos(c+d*x)))+A*sin(c+d*x)/(d*sqrt(cos(c+d*x))*sqrt(b*cos(c+d*x)))+1/2*B*arctanh(sin(c+d*x))*sqrt(cos(c+d*x))/(d*sqrt(b*cos(c+d*x)))],
[cos(c+d*x)^(7/2)*(A+B*cos(c+d*x))/(b*cos(c+d*x))^(3/2),x,6,1/2*A*cos(c+d*x)^(3/2)*sin(c+d*x)/(b*d*sqrt(b*cos(c+d*x)))+1/2*A*x*sqrt(cos(c+d*x))/(b*sqrt(b*cos(c+d*x)))+B*sin(c+d*x)*sqrt(cos(c+d*x))/(b*d*sqrt(b*cos(c+d*x)))-1/3*B*sin(c+d*x)^3*sqrt(cos(c+d*x))/(b*d*sqrt(b*cos(c+d*x)))],
[cos(c+d*x)^(5/2)*(A+B*cos(c+d*x))/(b*cos(c+d*x))^(3/2),x,2,1/2*B*cos(c+d*x)^(3/2)*sin(c+d*x)/(b*d*sqrt(b*cos(c+d*x)))+1/2*B*x*sqrt(cos(c+d*x))/(b*sqrt(b*cos(c+d*x)))+A*sin(c+d*x)*sqrt(cos(c+d*x))/(b*d*sqrt(b*cos(c+d*x)))],
[cos(c+d*x)^(3/2)*(A+B*cos(c+d*x))/(b*cos(c+d*x))^(3/2),x,3,A*x*sqrt(cos(c+d*x))/(b*sqrt(b*cos(c+d*x)))+B*sin(c+d*x)*sqrt(cos(c+d*x))/(b*d*sqrt(b*cos(c+d*x)))],
[cos(c+d*x)^(1/2)*(A+B*cos(c+d*x))/(b*cos(c+d*x))^(3/2),x,3,B*x*sqrt(cos(c+d*x))/(b*sqrt(b*cos(c+d*x)))+A*arctanh(sin(c+d*x))*sqrt(cos(c+d*x))/(b*d*sqrt(b*cos(c+d*x)))],
[(A+B*cos(c+d*x))/(cos(c+d*x)^(1/2)*(b*cos(c+d*x))^(3/2)),x,5,A*sin(c+d*x)/(b*d*sqrt(cos(c+d*x))*sqrt(b*cos(c+d*x)))+B*arctanh(sin(c+d*x))*sqrt(cos(c+d*x))/(b*d*sqrt(b*cos(c+d*x)))],
[(A+B*cos(c+d*x))/(cos(c+d*x)^(3/2)*(b*cos(c+d*x))^(3/2)),x,6,1/2*A*sin(c+d*x)/(b*d*cos(c+d*x)^(3/2)*sqrt(b*cos(c+d*x)))+B*sin(c+d*x)/(b*d*sqrt(cos(c+d*x))*sqrt(b*cos(c+d*x)))+1/2*A*arctanh(sin(c+d*x))*sqrt(cos(c+d*x))/(b*d*sqrt(b*cos(c+d*x)))],
[(A+B*cos(c+d*x))/(cos(c+d*x)^(5/2)*(b*cos(c+d*x))^(3/2)),x,6,1/2*B*sin(c+d*x)/(b*d*cos(c+d*x)^(3/2)*sqrt(b*cos(c+d*x)))+1/3*A*sin(c+d*x)^3/(b*d*cos(c+d*x)^(5/2)*sqrt(b*cos(c+d*x)))+A*sin(c+d*x)/(b*d*sqrt(cos(c+d*x))*sqrt(b*cos(c+d*x)))+1/2*B*arctanh(sin(c+d*x))*sqrt(cos(c+d*x))/(b*d*sqrt(b*cos(c+d*x)))],
[cos(c+d*x)^(9/2)*(A+B*cos(c+d*x))/(b*cos(c+d*x))^(5/2),x,6,1/2*A*cos(c+d*x)^(3/2)*sin(c+d*x)/(b^2*d*sqrt(b*cos(c+d*x)))+1/2*A*x*sqrt(cos(c+d*x))/(b^2*sqrt(b*cos(c+d*x)))+B*sin(c+d*x)*sqrt(cos(c+d*x))/(b^2*d*sqrt(b*cos(c+d*x)))-1/3*B*sin(c+d*x)^3*sqrt(cos(c+d*x))/(b^2*d*sqrt(b*cos(c+d*x)))],
[cos(c+d*x)^(7/2)*(A+B*cos(c+d*x))/(b*cos(c+d*x))^(5/2),x,2,1/2*B*cos(c+d*x)^(3/2)*sin(c+d*x)/(b^2*d*sqrt(b*cos(c+d*x)))+1/2*B*x*sqrt(cos(c+d*x))/(b^2*sqrt(b*cos(c+d*x)))+A*sin(c+d*x)*sqrt(cos(c+d*x))/(b^2*d*sqrt(b*cos(c+d*x)))],
[cos(c+d*x)^(5/2)*(A+B*cos(c+d*x))/(b*cos(c+d*x))^(5/2),x,3,A*x*sqrt(cos(c+d*x))/(b^2*sqrt(b*cos(c+d*x)))+B*sin(c+d*x)*sqrt(cos(c+d*x))/(b^2*d*sqrt(b*cos(c+d*x)))],
[cos(c+d*x)^(3/2)*(A+B*cos(c+d*x))/(b*cos(c+d*x))^(5/2),x,3,B*x*sqrt(cos(c+d*x))/(b^2*sqrt(b*cos(c+d*x)))+A*arctanh(sin(c+d*x))*sqrt(cos(c+d*x))/(b^2*d*sqrt(b*cos(c+d*x)))],
[cos(c+d*x)^(1/2)*(A+B*cos(c+d*x))/(b*cos(c+d*x))^(5/2),x,5,A*sin(c+d*x)/(b^2*d*sqrt(cos(c+d*x))*sqrt(b*cos(c+d*x)))+B*arctanh(sin(c+d*x))*sqrt(cos(c+d*x))/(b^2*d*sqrt(b*cos(c+d*x)))],
[(A+B*cos(c+d*x))/(cos(c+d*x)^(1/2)*(b*cos(c+d*x))^(5/2)),x,6,1/2*A*sin(c+d*x)/(b^2*d*cos(c+d*x)^(3/2)*sqrt(b*cos(c+d*x)))+B*sin(c+d*x)/(b^2*d*sqrt(cos(c+d*x))*sqrt(b*cos(c+d*x)))+1/2*A*arctanh(sin(c+d*x))*sqrt(cos(c+d*x))/(b^2*d*sqrt(b*cos(c+d*x)))],
[(A+B*cos(c+d*x))/(cos(c+d*x)^(3/2)*(b*cos(c+d*x))^(5/2)),x,6,1/2*B*sin(c+d*x)/(b^2*d*cos(c+d*x)^(3/2)*sqrt(b*cos(c+d*x)))+1/3*A*sin(c+d*x)^3/(b^2*d*cos(c+d*x)^(5/2)*sqrt(b*cos(c+d*x)))+A*sin(c+d*x)/(b^2*d*sqrt(cos(c+d*x))*sqrt(b*cos(c+d*x)))+1/2*B*arctanh(sin(c+d*x))*sqrt(cos(c+d*x))/(b^2*d*sqrt(b*cos(c+d*x)))],

# Integrands of the  form (a+b Cos[e+f x]) (c Cos[e+f x])^m (d Cos[e+f x])^(n/3)

# n>0
[cos(c+d*x)^2*(b*cos(c+d*x))^(1/3)*(A+B*cos(c+d*x)),x,4,-3/10*A*(b*cos(c+d*x))^(10/3)*hypergeom([1/2,5/3],[8/3],cos(c+d*x)^2)*sin(c+d*x)/(b^3*d*sqrt(sin(c+d*x)^2))-3/13*B*(b*cos(c+d*x))^(13/3)*hypergeom([1/2,13/6],[19/6],cos(c+d*x)^2)*sin(c+d*x)/(b^4*d*sqrt(sin(c+d*x)^2))],
[cos(c+d*x)*(b*cos(c+d*x))^(1/3)*(A+B*cos(c+d*x)),x,4,-3/7*A*(b*cos(c+d*x))^(7/3)*hypergeom([1/2,7/6],[13/6],cos(c+d*x)^2)*sin(c+d*x)/(b^2*d*sqrt(sin(c+d*x)^2))-3/10*B*(b*cos(c+d*x))^(10/3)*hypergeom([1/2,5/3],[8/3],cos(c+d*x)^2)*sin(c+d*x)/(b^3*d*sqrt(sin(c+d*x)^2))],
[(b*cos(c+d*x))^(1/3)*(A+B*cos(c+d*x)),x,3,-3/4*A*(b*cos(c+d*x))^(4/3)*hypergeom([1/2,2/3],[5/3],cos(c+d*x)^2)*sin(c+d*x)/(b*d*sqrt(sin(c+d*x)^2))-3/7*B*(b*cos(c+d*x))^(7/3)*hypergeom([1/2,7/6],[13/6],cos(c+d*x)^2)*sin(c+d*x)/(b^2*d*sqrt(sin(c+d*x)^2))],
[(b*cos(c+d*x))^(1/3)*(A+B*cos(c+d*x))*sec(c+d*x),x,4,-3*A*(b*cos(c+d*x))^(1/3)*hypergeom([1/6,1/2],[7/6],cos(c+d*x)^2)*sin(c+d*x)/(d*sqrt(sin(c+d*x)^2))-3/4*B*(b*cos(c+d*x))^(4/3)*hypergeom([1/2,2/3],[5/3],cos(c+d*x)^2)*sin(c+d*x)/(b*d*sqrt(sin(c+d*x)^2))],
[(b*cos(c+d*x))^(1/3)*(A+B*cos(c+d*x))*sec(c+d*x)^2,x,4,3/2*A*b*hypergeom([-1/3,1/2],[2/3],cos(c+d*x)^2)*sin(c+d*x)/(d*(b*cos(c+d*x))^(2/3)*sqrt(sin(c+d*x)^2))-3*B*(b*cos(c+d*x))^(1/3)*hypergeom([1/6,1/2],[7/6],cos(c+d*x)^2)*sin(c+d*x)/(d*sqrt(sin(c+d*x)^2))],
[(b*cos(c+d*x))^(1/3)*(A+B*cos(c+d*x))*sec(c+d*x)^3,x,4,3/5*A*b^2*hypergeom([-5/6,1/2],[1/6],cos(c+d*x)^2)*sin(c+d*x)/(d*(b*cos(c+d*x))^(5/3)*sqrt(sin(c+d*x)^2))+3/2*b*B*hypergeom([-1/3,1/2],[2/3],cos(c+d*x)^2)*sin(c+d*x)/(d*(b*cos(c+d*x))^(2/3)*sqrt(sin(c+d*x)^2))],
[cos(c+d*x)^2*(b*cos(c+d*x))^(4/3)*(A+B*cos(c+d*x)),x,4,-3/13*A*(b*cos(c+d*x))^(13/3)*hypergeom([1/2,13/6],[19/6],cos(c+d*x)^2)*sin(c+d*x)/(b^3*d*sqrt(sin(c+d*x)^2))-3/16*B*(b*cos(c+d*x))^(16/3)*hypergeom([1/2,8/3],[11/3],cos(c+d*x)^2)*sin(c+d*x)/(b^4*d*sqrt(sin(c+d*x)^2))],
[cos(c+d*x)*(b*cos(c+d*x))^(4/3)*(A+B*cos(c+d*x)),x,4,-3/10*A*(b*cos(c+d*x))^(10/3)*hypergeom([1/2,5/3],[8/3],cos(c+d*x)^2)*sin(c+d*x)/(b^2*d*sqrt(sin(c+d*x)^2))-3/13*B*(b*cos(c+d*x))^(13/3)*hypergeom([1/2,13/6],[19/6],cos(c+d*x)^2)*sin(c+d*x)/(b^3*d*sqrt(sin(c+d*x)^2))],
[(b*cos(c+d*x))^(4/3)*(A+B*cos(c+d*x)),x,3,-3/7*A*(b*cos(c+d*x))^(7/3)*hypergeom([1/2,7/6],[13/6],cos(c+d*x)^2)*sin(c+d*x)/(b*d*sqrt(sin(c+d*x)^2))-3/10*B*(b*cos(c+d*x))^(10/3)*hypergeom([1/2,5/3],[8/3],cos(c+d*x)^2)*sin(c+d*x)/(b^2*d*sqrt(sin(c+d*x)^2))],
[(b*cos(c+d*x))^(4/3)*(A+B*cos(c+d*x))*sec(c+d*x),x,4,-3/4*A*(b*cos(c+d*x))^(4/3)*hypergeom([1/2,2/3],[5/3],cos(c+d*x)^2)*sin(c+d*x)/(d*sqrt(sin(c+d*x)^2))-3/7*B*(b*cos(c+d*x))^(7/3)*hypergeom([1/2,7/6],[13/6],cos(c+d*x)^2)*sin(c+d*x)/(b*d*sqrt(sin(c+d*x)^2))],
[(b*cos(c+d*x))^(4/3)*(A+B*cos(c+d*x))*sec(c+d*x)^2,x,4,-3*A*b*(b*cos(c+d*x))^(1/3)*hypergeom([1/6,1/2],[7/6],cos(c+d*x)^2)*sin(c+d*x)/(d*sqrt(sin(c+d*x)^2))-3/4*B*(b*cos(c+d*x))^(4/3)*hypergeom([1/2,2/3],[5/3],cos(c+d*x)^2)*sin(c+d*x)/(d*sqrt(sin(c+d*x)^2))],
[(b*cos(c+d*x))^(4/3)*(A+B*cos(c+d*x))*sec(c+d*x)^3,x,4,3/2*A*b^2*hypergeom([-1/3,1/2],[2/3],cos(c+d*x)^2)*sin(c+d*x)/(d*(b*cos(c+d*x))^(2/3)*sqrt(sin(c+d*x)^2))-3*b*B*(b*cos(c+d*x))^(1/3)*hypergeom([1/6,1/2],[7/6],cos(c+d*x)^2)*sin(c+d*x)/(d*sqrt(sin(c+d*x)^2))],

# n<0
[cos(c+d*x)^2*(A+B*cos(c+d*x))/(b*cos(c+d*x))^(2/3),x,4,-3/7*A*(b*cos(c+d*x))^(7/3)*hypergeom([1/2,7/6],[13/6],cos(c+d*x)^2)*sin(c+d*x)/(b^3*d*sqrt(sin(c+d*x)^2))-3/10*B*(b*cos(c+d*x))^(10/3)*hypergeom([1/2,5/3],[8/3],cos(c+d*x)^2)*sin(c+d*x)/(b^4*d*sqrt(sin(c+d*x)^2))],
[cos(c+d*x)*(A+B*cos(c+d*x))/(b*cos(c+d*x))^(2/3),x,4,-3/4*A*(b*cos(c+d*x))^(4/3)*hypergeom([1/2,2/3],[5/3],cos(c+d*x)^2)*sin(c+d*x)/(b^2*d*sqrt(sin(c+d*x)^2))-3/7*B*(b*cos(c+d*x))^(7/3)*hypergeom([1/2,7/6],[13/6],cos(c+d*x)^2)*sin(c+d*x)/(b^3*d*sqrt(sin(c+d*x)^2))],
[(A+B*cos(c+d*x))/(b*cos(c+d*x))^(2/3),x,3,-3*A*(b*cos(c+d*x))^(1/3)*hypergeom([1/6,1/2],[7/6],cos(c+d*x)^2)*sin(c+d*x)/(b*d*sqrt(sin(c+d*x)^2))-3/4*B*(b*cos(c+d*x))^(4/3)*hypergeom([1/2,2/3],[5/3],cos(c+d*x)^2)*sin(c+d*x)/(b^2*d*sqrt(sin(c+d*x)^2))],
[(A+B*cos(c+d*x))*sec(c+d*x)/(b*cos(c+d*x))^(2/3),x,4,3/2*A*hypergeom([-1/3,1/2],[2/3],cos(c+d*x)^2)*sin(c+d*x)/(d*(b*cos(c+d*x))^(2/3)*sqrt(sin(c+d*x)^2))-3*B*(b*cos(c+d*x))^(1/3)*hypergeom([1/6,1/2],[7/6],cos(c+d*x)^2)*sin(c+d*x)/(b*d*sqrt(sin(c+d*x)^2))],
[(A+B*cos(c+d*x))*sec(c+d*x)^2/(b*cos(c+d*x))^(2/3),x,4,3/5*A*b*hypergeom([-5/6,1/2],[1/6],cos(c+d*x)^2)*sin(c+d*x)/(d*(b*cos(c+d*x))^(5/3)*sqrt(sin(c+d*x)^2))+3/2*B*hypergeom([-1/3,1/2],[2/3],cos(c+d*x)^2)*sin(c+d*x)/(d*(b*cos(c+d*x))^(2/3)*sqrt(sin(c+d*x)^2))],
[(A+B*cos(c+d*x))*sec(c+d*x)^3/(b*cos(c+d*x))^(2/3),x,4,3/8*A*b^2*hypergeom([-4/3,1/2],[-1/3],cos(c+d*x)^2)*sin(c+d*x)/(d*(b*cos(c+d*x))^(8/3)*sqrt(sin(c+d*x)^2))+3/5*b*B*hypergeom([-5/6,1/2],[1/6],cos(c+d*x)^2)*sin(c+d*x)/(d*(b*cos(c+d*x))^(5/3)*sqrt(sin(c+d*x)^2))],
[cos(c+d*x)^2*(A+B*cos(c+d*x))/(b*cos(c+d*x))^(4/3),x,4,-3/5*A*(b*cos(c+d*x))^(5/3)*hypergeom([1/2,5/6],[11/6],cos(c+d*x)^2)*sin(c+d*x)/(b^3*d*sqrt(sin(c+d*x)^2))-3/8*B*(b*cos(c+d*x))^(8/3)*hypergeom([1/2,4/3],[7/3],cos(c+d*x)^2)*sin(c+d*x)/(b^4*d*sqrt(sin(c+d*x)^2))],
[cos(c+d*x)*(A+B*cos(c+d*x))/(b*cos(c+d*x))^(4/3),x,4,-3/2*A*(b*cos(c+d*x))^(2/3)*hypergeom([1/3,1/2],[4/3],cos(c+d*x)^2)*sin(c+d*x)/(b^2*d*sqrt(sin(c+d*x)^2))-3/5*B*(b*cos(c+d*x))^(5/3)*hypergeom([1/2,5/6],[11/6],cos(c+d*x)^2)*sin(c+d*x)/(b^3*d*sqrt(sin(c+d*x)^2))],
[(A+B*cos(c+d*x))/(b*cos(c+d*x))^(4/3),x,3,3*A*hypergeom([-1/6,1/2],[5/6],cos(c+d*x)^2)*sin(c+d*x)/(b*d*(b*cos(c+d*x))^(1/3)*sqrt(sin(c+d*x)^2))-3/2*B*(b*cos(c+d*x))^(2/3)*hypergeom([1/3,1/2],[4/3],cos(c+d*x)^2)*sin(c+d*x)/(b^2*d*sqrt(sin(c+d*x)^2))],
[(A+B*cos(c+d*x))*sec(c+d*x)/(b*cos(c+d*x))^(4/3),x,4,3/4*A*hypergeom([-2/3,1/2],[1/3],cos(c+d*x)^2)*sin(c+d*x)/(d*(b*cos(c+d*x))^(4/3)*sqrt(sin(c+d*x)^2))+3*B*hypergeom([-1/6,1/2],[5/6],cos(c+d*x)^2)*sin(c+d*x)/(b*d*(b*cos(c+d*x))^(1/3)*sqrt(sin(c+d*x)^2))],
[(A+B*cos(c+d*x))*sec(c+d*x)^2/(b*cos(c+d*x))^(4/3),x,4,3/7*A*b*hypergeom([-7/6,1/2],[-1/6],cos(c+d*x)^2)*sin(c+d*x)/(d*(b*cos(c+d*x))^(7/3)*sqrt(sin(c+d*x)^2))+3/4*B*hypergeom([-2/3,1/2],[1/3],cos(c+d*x)^2)*sin(c+d*x)/(d*(b*cos(c+d*x))^(4/3)*sqrt(sin(c+d*x)^2))],
[(A+B*cos(c+d*x))*sec(c+d*x)^3/(b*cos(c+d*x))^(4/3),x,4,3/10*A*b^2*hypergeom([-5/3,1/2],[-2/3],cos(c+d*x)^2)*sin(c+d*x)/(d*(b*cos(c+d*x))^(10/3)*sqrt(sin(c+d*x)^2))+3/7*b*B*hypergeom([-7/6,1/2],[-1/6],cos(c+d*x)^2)*sin(c+d*x)/(d*(b*cos(c+d*x))^(7/3)*sqrt(sin(c+d*x)^2))],

# Integrands of the  form (a+b Cos[e+f x]) (c Cos[e+f x])^m (d Cos[e+f x])^n) with m and/or n symbolic
[cos(c+d*x)^m*(b*cos(c+d*x))^n*(A+B*cos(c+d*x)),x,4,-A*cos(c+d*x)^(1+m)*(b*cos(c+d*x))^n*hypergeom([1/2,1/2*(1+m+n)],[1/2*(3+m+n)],cos(c+d*x)^2)*sin(c+d*x)/(d*(1+m+n)*sqrt(sin(c+d*x)^2))-B*cos(c+d*x)^(2+m)*(b*cos(c+d*x))^n*hypergeom([1/2,1/2*(2+m+n)],[1/2*(4+m+n)],cos(c+d*x)^2)*sin(c+d*x)/(d*(2+m+n)*sqrt(sin(c+d*x)^2))],
[cos(c+d*x)^2*(b*cos(c+d*x))^n*(A+B*cos(c+d*x)),x,4,-A*(b*cos(c+d*x))^(3+n)*hypergeom([1/2,1/2*(3+n)],[1/2*(5+n)],cos(c+d*x)^2)*sin(c+d*x)/(b^3*d*(3+n)*sqrt(sin(c+d*x)^2))-B*(b*cos(c+d*x))^(4+n)*hypergeom([1/2,1/2*(4+n)],[1/2*(6+n)],cos(c+d*x)^2)*sin(c+d*x)/(b^4*d*(4+n)*sqrt(sin(c+d*x)^2))],
[cos(c+d*x)*(b*cos(c+d*x))^n*(A+B*cos(c+d*x)),x,4,-A*(b*cos(c+d*x))^(2+n)*hypergeom([1/2,1/2*(2+n)],[1/2*(4+n)],cos(c+d*x)^2)*sin(c+d*x)/(b^2*d*(2+n)*sqrt(sin(c+d*x)^2))-B*(b*cos(c+d*x))^(3+n)*hypergeom([1/2,1/2*(3+n)],[1/2*(5+n)],cos(c+d*x)^2)*sin(c+d*x)/(b^3*d*(3+n)*sqrt(sin(c+d*x)^2))],
[(b*cos(c+d*x))^n*(A+B*cos(c+d*x)),x,3,-A*(b*cos(c+d*x))^(1+n)*hypergeom([1/2,1/2*(1+n)],[1/2*(3+n)],cos(c+d*x)^2)*sin(c+d*x)/(b*d*(1+n)*sqrt(sin(c+d*x)^2))-B*(b*cos(c+d*x))^(2+n)*hypergeom([1/2,1/2*(2+n)],[1/2*(4+n)],cos(c+d*x)^2)*sin(c+d*x)/(b^2*d*(2+n)*sqrt(sin(c+d*x)^2))],
[(b*cos(c+d*x))^n*(A+B*cos(c+d*x))*sec(c+d*x),x,4,-A*(b*cos(c+d*x))^n*hypergeom([1/2,1/2*n],[1/2*(2+n)],cos(c+d*x)^2)*sin(c+d*x)/(d*n*sqrt(sin(c+d*x)^2))-B*(b*cos(c+d*x))^(1+n)*hypergeom([1/2,1/2*(1+n)],[1/2*(3+n)],cos(c+d*x)^2)*sin(c+d*x)/(b*d*(1+n)*sqrt(sin(c+d*x)^2))],
[(b*cos(c+d*x))^n*(A+B*cos(c+d*x))*sec(c+d*x)^2,x,4,A*b*(b*cos(c+d*x))^(-1+n)*hypergeom([1/2,1/2*(-1+n)],[1/2*(1+n)],cos(c+d*x)^2)*sin(c+d*x)/(d*(1-n)*sqrt(sin(c+d*x)^2))-B*(b*cos(c+d*x))^n*hypergeom([1/2,1/2*n],[1/2*(2+n)],cos(c+d*x)^2)*sin(c+d*x)/(d*n*sqrt(sin(c+d*x)^2))],
[(b*cos(c+d*x))^n*(A+B*cos(c+d*x))*sec(c+d*x)^3,x,4,A*b^2*(b*cos(c+d*x))^(-2+n)*hypergeom([1/2,1/2*(-2+n)],[1/2*n],cos(c+d*x)^2)*sin(c+d*x)/(d*(2-n)*sqrt(sin(c+d*x)^2))+b*B*(b*cos(c+d*x))^(-1+n)*hypergeom([1/2,1/2*(-1+n)],[1/2*(1+n)],cos(c+d*x)^2)*sin(c+d*x)/(d*(1-n)*sqrt(sin(c+d*x)^2))],
[(b*cos(c+d*x))^n*(A+B*cos(c+d*x))*sec(c+d*x)^4,x,4,A*b^3*(b*cos(c+d*x))^(-3+n)*hypergeom([1/2,1/2*(-3+n)],[1/2*(-1+n)],cos(c+d*x)^2)*sin(c+d*x)/(d*(3-n)*sqrt(sin(c+d*x)^2))+b^2*B*(b*cos(c+d*x))^(-2+n)*hypergeom([1/2,1/2*(-2+n)],[1/2*n],cos(c+d*x)^2)*sin(c+d*x)/(d*(2-n)*sqrt(sin(c+d*x)^2))],
[cos(c+d*x)^(5/2)*(b*cos(c+d*x))^n*(A+B*cos(c+d*x)),x,4,-2*A*cos(c+d*x)^(7/2)*(b*cos(c+d*x))^n*hypergeom([1/2,1/4*(7+2*n)],[1/4*(11+2*n)],cos(c+d*x)^2)*sin(c+d*x)/(d*(7+2*n)*sqrt(sin(c+d*x)^2))-2*B*cos(c+d*x)^(9/2)*(b*cos(c+d*x))^n*hypergeom([1/2,1/4*(9+2*n)],[1/4*(13+2*n)],cos(c+d*x)^2)*sin(c+d*x)/(d*(9+2*n)*sqrt(sin(c+d*x)^2))],
[cos(c+d*x)^(3/2)*(b*cos(c+d*x))^n*(A+B*cos(c+d*x)),x,4,-2*A*cos(c+d*x)^(5/2)*(b*cos(c+d*x))^n*hypergeom([1/2,1/4*(5+2*n)],[1/4*(9+2*n)],cos(c+d*x)^2)*sin(c+d*x)/(d*(5+2*n)*sqrt(sin(c+d*x)^2))-2*B*cos(c+d*x)^(7/2)*(b*cos(c+d*x))^n*hypergeom([1/2,1/4*(7+2*n)],[1/4*(11+2*n)],cos(c+d*x)^2)*sin(c+d*x)/(d*(7+2*n)*sqrt(sin(c+d*x)^2))],
[(b*cos(c+d*x))^n*(A+B*cos(c+d*x))*sqrt(cos(c+d*x)),x,4,-2*A*cos(c+d*x)^(3/2)*(b*cos(c+d*x))^n*hypergeom([1/2,1/4*(3+2*n)],[1/4*(7+2*n)],cos(c+d*x)^2)*sin(c+d*x)/(d*(3+2*n)*sqrt(sin(c+d*x)^2))-2*B*cos(c+d*x)^(5/2)*(b*cos(c+d*x))^n*hypergeom([1/2,1/4*(5+2*n)],[1/4*(9+2*n)],cos(c+d*x)^2)*sin(c+d*x)/(d*(5+2*n)*sqrt(sin(c+d*x)^2))],
[(b*cos(c+d*x))^n*(A+B*cos(c+d*x))/sqrt(cos(c+d*x)),x,4,-2*B*cos(c+d*x)^(3/2)*(b*cos(c+d*x))^n*hypergeom([1/2,1/4*(3+2*n)],[1/4*(7+2*n)],cos(c+d*x)^2)*sin(c+d*x)/(d*(3+2*n)*sqrt(sin(c+d*x)^2))-2*A*(b*cos(c+d*x))^n*hypergeom([1/2,1/4*(1+2*n)],[1/4*(5+2*n)],cos(c+d*x)^2)*sin(c+d*x)*sqrt(cos(c+d*x))/(d*(1+2*n)*sqrt(sin(c+d*x)^2))],
[(b*cos(c+d*x))^n*(A+B*cos(c+d*x))/cos(c+d*x)^(3/2),x,4,2*A*(b*cos(c+d*x))^n*hypergeom([1/2,1/4*(-1+2*n)],[1/4*(3+2*n)],cos(c+d*x)^2)*sin(c+d*x)/(d*(1-2*n)*sqrt(cos(c+d*x))*sqrt(sin(c+d*x)^2))-2*B*(b*cos(c+d*x))^n*hypergeom([1/2,1/4*(1+2*n)],[1/4*(5+2*n)],cos(c+d*x)^2)*sin(c+d*x)*sqrt(cos(c+d*x))/(d*(1+2*n)*sqrt(sin(c+d*x)^2))],
[(b*cos(c+d*x))^n*(A+B*cos(c+d*x))/cos(c+d*x)^(5/2),x,4,2*A*(b*cos(c+d*x))^n*hypergeom([1/2,1/4*(-3+2*n)],[1/4*(1+2*n)],cos(c+d*x)^2)*sin(c+d*x)/(d*(3-2*n)*cos(c+d*x)^(3/2)*sqrt(sin(c+d*x)^2))+2*B*(b*cos(c+d*x))^n*hypergeom([1/2,1/4*(-1+2*n)],[1/4*(3+2*n)],cos(c+d*x)^2)*sin(c+d*x)/(d*(1-2*n)*sqrt(cos(c+d*x))*sqrt(sin(c+d*x)^2))],
[(b*cos(c+d*x))^n*(A+B*cos(c+d*x))/cos(c+d*x)^(7/2),x,4,2*A*(b*cos(c+d*x))^n*hypergeom([1/2,1/4*(-5+2*n)],[1/4*(-1+2*n)],cos(c+d*x)^2)*sin(c+d*x)/(d*(5-2*n)*cos(c+d*x)^(5/2)*sqrt(sin(c+d*x)^2))+2*B*(b*cos(c+d*x))^n*hypergeom([1/2,1/4*(-3+2*n)],[1/4*(1+2*n)],cos(c+d*x)^2)*sin(c+d*x)/(d*(3-2*n)*cos(c+d*x)^(3/2)*sqrt(sin(c+d*x)^2))],
[(b*cos(c+d*x))^n*(A+B*cos(c+d*x))/cos(c+d*x)^(9/2),x,4,2*A*(b*cos(c+d*x))^n*hypergeom([1/2,1/4*(-7+2*n)],[1/4*(-3+2*n)],cos(c+d*x)^2)*sin(c+d*x)/(d*(7-2*n)*cos(c+d*x)^(7/2)*sqrt(sin(c+d*x)^2))+2*B*(b*cos(c+d*x))^n*hypergeom([1/2,1/4*(-5+2*n)],[1/4*(-1+2*n)],cos(c+d*x)^2)*sin(c+d*x)/(d*(5-2*n)*cos(c+d*x)^(5/2)*sqrt(sin(c+d*x)^2))],
[cos(c+d*x)^m*(b*cos(c+d*x))^(4/3)*(A+B*cos(c+d*x)),x,4,-3*A*b*cos(c+d*x)^(2+m)*(b*cos(c+d*x))^(1/3)*hypergeom([1/2,1/6*(7+3*m)],[1/6*(13+3*m)],cos(c+d*x)^2)*sin(c+d*x)/(d*(7+3*m)*sqrt(sin(c+d*x)^2))-3*b*B*cos(c+d*x)^(3+m)*(b*cos(c+d*x))^(1/3)*hypergeom([1/2,1/6*(10+3*m)],[1/6*(16+3*m)],cos(c+d*x)^2)*sin(c+d*x)/(d*(10+3*m)*sqrt(sin(c+d*x)^2))],
[cos(c+d*x)^m*(b*cos(c+d*x))^(2/3)*(A+B*cos(c+d*x)),x,4,-3*A*cos(c+d*x)^(1+m)*(b*cos(c+d*x))^(2/3)*hypergeom([1/2,1/6*(5+3*m)],[1/6*(11+3*m)],cos(c+d*x)^2)*sin(c+d*x)/(d*(5+3*m)*sqrt(sin(c+d*x)^2))-3*B*cos(c+d*x)^(2+m)*(b*cos(c+d*x))^(2/3)*hypergeom([1/2,1/6*(8+3*m)],[1/6*(14+3*m)],cos(c+d*x)^2)*sin(c+d*x)/(d*(8+3*m)*sqrt(sin(c+d*x)^2))],
[cos(c+d*x)^m*(b*cos(c+d*x))^(1/3)*(A+B*cos(c+d*x)),x,4,-3*A*cos(c+d*x)^(1+m)*(b*cos(c+d*x))^(1/3)*hypergeom([1/2,1/6*(4+3*m)],[1/6*(10+3*m)],cos(c+d*x)^2)*sin(c+d*x)/(d*(4+3*m)*sqrt(sin(c+d*x)^2))-3*B*cos(c+d*x)^(2+m)*(b*cos(c+d*x))^(1/3)*hypergeom([1/2,1/6*(7+3*m)],[1/6*(13+3*m)],cos(c+d*x)^2)*sin(c+d*x)/(d*(7+3*m)*sqrt(sin(c+d*x)^2))],
[cos(c+d*x)^m*(A+B*cos(c+d*x))/(b*cos(c+d*x))^(1/3),x,4,-3*A*cos(c+d*x)^(1+m)*hypergeom([1/2,1/6*(2+3*m)],[1/6*(8+3*m)],cos(c+d*x)^2)*sin(c+d*x)/(d*(2+3*m)*(b*cos(c+d*x))^(1/3)*sqrt(sin(c+d*x)^2))-3*B*cos(c+d*x)^(2+m)*hypergeom([1/2,1/6*(5+3*m)],[1/6*(11+3*m)],cos(c+d*x)^2)*sin(c+d*x)/(d*(5+3*m)*(b*cos(c+d*x))^(1/3)*sqrt(sin(c+d*x)^2))],
[cos(c+d*x)^m*(A+B*cos(c+d*x))/(b*cos(c+d*x))^(2/3),x,4,-3*A*cos(c+d*x)^(1+m)*hypergeom([1/2,1/6*(1+3*m)],[1/6*(7+3*m)],cos(c+d*x)^2)*sin(c+d*x)/(d*(1+3*m)*(b*cos(c+d*x))^(2/3)*sqrt(sin(c+d*x)^2))-3*B*cos(c+d*x)^(2+m)*hypergeom([1/2,1/6*(4+3*m)],[1/6*(10+3*m)],cos(c+d*x)^2)*sin(c+d*x)/(d*(4+3*m)*(b*cos(c+d*x))^(2/3)*sqrt(sin(c+d*x)^2))],
[cos(c+d*x)^m*(A+B*cos(c+d*x))/(b*cos(c+d*x))^(4/3),x,4,3*A*cos(c+d*x)^m*hypergeom([1/2,1/6*(-1+3*m)],[1/6*(5+3*m)],cos(c+d*x)^2)*sin(c+d*x)/(b*d*(1-3*m)*(b*cos(c+d*x))^(1/3)*sqrt(sin(c+d*x)^2))-3*B*cos(c+d*x)^(1+m)*hypergeom([1/2,1/6*(2+3*m)],[1/6*(8+3*m)],cos(c+d*x)^2)*sin(c+d*x)/(b*d*(2+3*m)*(b*cos(c+d*x))^(1/3)*sqrt(sin(c+d*x)^2))]]:
