# Maple integration test file: "4 Trig functions\4.5 Secant\4.5.1.2 (d sec)^n (a+b sec)^m.txt"

lst:=[

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

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

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

# m>0
[sec(c+d*x)^4*(a+a*sec(c+d*x)),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],
[sec(c+d*x)^3*(a+a*sec(c+d*x)),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],
[sec(c+d*x)^2*(a+a*sec(c+d*x)),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],
[sec(c+d*x)*(a+a*sec(c+d*x)),x,4,a*arctanh(sin(c+d*x))/d+a*tan(c+d*x)/d],
[a+a*sec(c+d*x),x,2,a*x+a*arctanh(sin(c+d*x))/d],
[cos(c+d*x)*(a+a*sec(c+d*x)),x,3,a*x+a*sin(c+d*x)/d],
[cos(c+d*x)^2*(a+a*sec(c+d*x)),x,4,1/2*a*x+a*sin(c+d*x)/d+1/2*a*cos(c+d*x)*sin(c+d*x)/d],
[cos(c+d*x)^3*(a+a*sec(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)^4*(a+a*sec(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],
[sec(c+d*x)^4*(a+a*sec(c+d*x))^2,x,7,3/4*a^2*arctanh(sin(c+d*x))/d+9/5*a^2*tan(c+d*x)/d+3/4*a^2*sec(c+d*x)*tan(c+d*x)/d+1/2*a^2*sec(c+d*x)^3*tan(c+d*x)/d+1/5*a^2*sec(c+d*x)^4*tan(c+d*x)/d+3/5*a^2*tan(c+d*x)^3/d],
[sec(c+d*x)^3*(a+a*sec(c+d*x))^2,x,6,7/8*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],
[sec(c+d*x)^2*(a+a*sec(c+d*x))^2,x,6,a^2*arctanh(sin(c+d*x))/d+5/3*a^2*tan(c+d*x)/d+a^2*sec(c+d*x)*tan(c+d*x)/d+1/3*a^2*sec(c+d*x)^2*tan(c+d*x)/d],
[sec(c+d*x)*(a+a*sec(c+d*x))^2,x,5,3/2*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*sec(c+d*x))^2,x,4,a^2*x+2*a^2*arctanh(sin(c+d*x))/d+a^2*tan(c+d*x)/d],
[cos(c+d*x)*(a+a*sec(c+d*x))^2,x,4,2*a^2*x+a^2*arctanh(sin(c+d*x))/d+a^2*sin(c+d*x)/d],
[cos(c+d*x)^2*(a+a*sec(c+d*x))^2,x,4,3/2*a^2*x+2*a^2*sin(c+d*x)/d+1/2*a^2*cos(c+d*x)*sin(c+d*x)/d],
[cos(c+d*x)^3*(a+a*sec(c+d*x))^2,x,6,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],
[cos(c+d*x)^4*(a+a*sec(c+d*x))^2,x,6,7/8*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)^5*(a+a*sec(c+d*x))^2,x,8,3/4*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],
[sec(c+d*x)^3*(a+a*sec(c+d*x))^3,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],
[sec(c+d*x)^2*(a+a*sec(c+d*x))^3,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],
[sec(c+d*x)*(a+a*sec(c+d*x))^3,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*sec(c+d*x))^3,x,5,a^3*x+7/2*a^3*arctanh(sin(c+d*x))/d+5/2*a^3*tan(c+d*x)/d+1/2*(a^3+a^3*sec(c+d*x))*tan(c+d*x)/d],
[cos(c+d*x)*(a+a*sec(c+d*x))^3,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],
[cos(c+d*x)^2*(a+a*sec(c+d*x))^3,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],
[cos(c+d*x)^3*(a+a*sec(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],
[cos(c+d*x)^4*(a+a*sec(c+d*x))^3,x,10,15/8*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],
[cos(c+d*x)^5*(a+a*sec(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)^6*(a+a*sec(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],
[sec(c+d*x)^3*(a+a*sec(c+d*x))^4,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],
[sec(c+d*x)^2*(a+a*sec(c+d*x))^4,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],
[sec(c+d*x)*(a+a*sec(c+d*x))^4,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*sec(c+d*x))^4,x,6,a^4*x+6*a^4*arctanh(sin(c+d*x))/d+5*a^4*tan(c+d*x)/d+1/3*(a^2+a^2*sec(c+d*x))^2*tan(c+d*x)/d+4/3*(a^4+a^4*sec(c+d*x))*tan(c+d*x)/d],
[cos(c+d*x)*(a+a*sec(c+d*x))^4,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],
[cos(c+d*x)^2*(a+a*sec(c+d*x))^4,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],
[cos(c+d*x)^3*(a+a*sec(c+d*x))^4,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],
[cos(c+d*x)^4*(a+a*sec(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],
[cos(c+d*x)^5*(a+a*sec(c+d*x))^4,x,12,7/2*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],
[cos(c+d*x)^6*(a+a*sec(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)^7*(a+a*sec(c+d*x))^4,x,15,11/4*a^4*x+8*a^4*sin(c+d*x)/d+11/4*a^4*cos(c+d*x)*sin(c+d*x)/d+11/6*a^4*cos(c+d*x)^3*sin(c+d*x)/d+2/3*a^4*cos(c+d*x)^5*sin(c+d*x)/d-16/3*a^4*sin(c+d*x)^3/d+9/5*a^4*sin(c+d*x)^5/d-1/7*a^4*sin(c+d*x)^7/d],
[sec(c+d*x)^3*(a+a*sec(c+d*x))^5,x,17,93/16*a^5*arctanh(sin(c+d*x))/d+16*a^5*tan(c+d*x)/d+93/16*a^5*sec(c+d*x)*tan(c+d*x)/d+85/24*a^5*sec(c+d*x)^3*tan(c+d*x)/d+5/6*a^5*sec(c+d*x)^5*tan(c+d*x)/d+28/3*a^5*tan(c+d*x)^3/d+13/5*a^5*tan(c+d*x)^5/d+1/7*a^5*tan(c+d*x)^7/d],

# m<0
[sec(c+d*x)^5/(a+a*sec(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)^3*tan(c+d*x)/(d*(a+a*sec(c+d*x)))+4/3*tan(c+d*x)^3/(a*d)],
[sec(c+d*x)^4/(a+a*sec(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)^2*tan(c+d*x)/(d*(a+a*sec(c+d*x)))],
[sec(c+d*x)^3/(a+a*sec(c+d*x)),x,4,-arctanh(sin(c+d*x))/(a*d)+tan(c+d*x)/(a*d)+tan(c+d*x)/(d*(a+a*sec(c+d*x)))],
[sec(c+d*x)^2/(a+a*sec(c+d*x)),x,3,arctanh(sin(c+d*x))/(a*d)-tan(c+d*x)/(d*(a+a*sec(c+d*x)))],
[sec(c+d*x)/(a+a*sec(c+d*x)),x,1,tan(c+d*x)/(d*(a+a*sec(c+d*x)))],
[1/(a+a*sec(c+d*x)),x,2,x/a-tan(c+d*x)/(d*(a+a*sec(c+d*x)))],
[cos(c+d*x)/(a+a*sec(c+d*x)),x,4,-x/a+2*sin(c+d*x)/(a*d)-sin(c+d*x)/(d*(a+a*sec(c+d*x)))],
[cos(c+d*x)^2/(a+a*sec(c+d*x)),x,5,3/2*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)*sin(c+d*x)/(d*(a+a*sec(c+d*x)))],
[cos(c+d*x)^3/(a+a*sec(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)^2*sin(c+d*x)/(d*(a+a*sec(c+d*x)))-4/3*sin(c+d*x)^3/(a*d)],
[cos(c+d*x)^4/(a+a*sec(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)^3*sin(c+d*x)/(d*(a+a*sec(c+d*x)))+4/3*sin(c+d*x)^3/(a*d)],
[sec(c+d*x)^5/(a+a*sec(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)^2*tan(c+d*x)/(a^2*d*(1+sec(c+d*x)))-1/3*sec(c+d*x)^3*tan(c+d*x)/(d*(a+a*sec(c+d*x))^2)],
[sec(c+d*x)^4/(a+a*sec(c+d*x))^2,x,6,-2*arctanh(sin(c+d*x))/(a^2*d)+4/3*tan(c+d*x)/(a^2*d)+2*tan(c+d*x)/(a^2*d*(1+sec(c+d*x)))-1/3*sec(c+d*x)^2*tan(c+d*x)/(d*(a+a*sec(c+d*x))^2)],
[sec(c+d*x)^3/(a+a*sec(c+d*x))^2,x,4,arctanh(sin(c+d*x))/(a^2*d)-5/3*tan(c+d*x)/(a^2*d*(1+sec(c+d*x)))+1/3*tan(c+d*x)/(d*(a+a*sec(c+d*x))^2)],
[sec(c+d*x)^2/(a+a*sec(c+d*x))^2,x,2,-1/3*tan(c+d*x)/(d*(a+a*sec(c+d*x))^2)+2/3*tan(c+d*x)/(d*(a^2+a^2*sec(c+d*x)))],
[sec(c+d*x)/(a+a*sec(c+d*x))^2,x,2,1/3*tan(c+d*x)/(d*(a+a*sec(c+d*x))^2)+1/3*tan(c+d*x)/(d*(a^2+a^2*sec(c+d*x)))],
[1/(a+a*sec(c+d*x))^2,x,3,x/a^2-4/3*tan(c+d*x)/(a^2*d*(1+sec(c+d*x)))-1/3*tan(c+d*x)/(d*(a+a*sec(c+d*x))^2)],
[cos(c+d*x)/(a+a*sec(c+d*x))^2,x,5,-2*x/a^2+10/3*sin(c+d*x)/(a^2*d)-2*sin(c+d*x)/(a^2*d*(1+sec(c+d*x)))-1/3*sin(c+d*x)/(d*(a+a*sec(c+d*x))^2)],
[cos(c+d*x)^2/(a+a*sec(c+d*x))^2,x,6,7/2*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)*sin(c+d*x)/(a^2*d*(1+sec(c+d*x)))-1/3*cos(c+d*x)*sin(c+d*x)/(d*(a+a*sec(c+d*x))^2)],
[cos(c+d*x)^3/(a+a*sec(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)^2*sin(c+d*x)/(a^2*d*(1+sec(c+d*x)))-1/3*cos(c+d*x)^2*sin(c+d*x)/(d*(a+a*sec(c+d*x))^2)-4*sin(c+d*x)^3/(a^2*d)],
[sec(c+d*x)^6/(a+a*sec(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)^4*tan(c+d*x)/(d*(a+a*sec(c+d*x))^3)-11/15*sec(c+d*x)^3*tan(c+d*x)/(a*d*(a+a*sec(c+d*x))^2)-76/15*sec(c+d*x)^2*tan(c+d*x)/(d*(a^3+a^3*sec(c+d*x)))],
[sec(c+d*x)^5/(a+a*sec(c+d*x))^3,x,7,-3*arctanh(sin(c+d*x))/(a^3*d)+9/5*tan(c+d*x)/(a^3*d)-1/5*sec(c+d*x)^3*tan(c+d*x)/(d*(a+a*sec(c+d*x))^3)-3/5*sec(c+d*x)^2*tan(c+d*x)/(a*d*(a+a*sec(c+d*x))^2)+3*tan(c+d*x)/(d*(a^3+a^3*sec(c+d*x)))],
[sec(c+d*x)^4/(a+a*sec(c+d*x))^3,x,5,arctanh(sin(c+d*x))/(a^3*d)-1/5*sec(c+d*x)^2*tan(c+d*x)/(d*(a+a*sec(c+d*x))^3)+7/15*tan(c+d*x)/(a*d*(a+a*sec(c+d*x))^2)-29/15*tan(c+d*x)/(d*(a^3+a^3*sec(c+d*x)))],
[sec(c+d*x)^3/(a+a*sec(c+d*x))^3,x,3,1/5*tan(c+d*x)/(d*(a+a*sec(c+d*x))^3)-8/15*tan(c+d*x)/(a*d*(a+a*sec(c+d*x))^2)+7/15*tan(c+d*x)/(d*(a^3+a^3*sec(c+d*x)))],
[sec(c+d*x)^2/(a+a*sec(c+d*x))^3,x,3,-1/5*tan(c+d*x)/(d*(a+a*sec(c+d*x))^3)+1/5*tan(c+d*x)/(a*d*(a+a*sec(c+d*x))^2)+1/5*tan(c+d*x)/(d*(a^3+a^3*sec(c+d*x)))],
[sec(c+d*x)/(a+a*sec(c+d*x))^3,x,3,1/5*tan(c+d*x)/(d*(a+a*sec(c+d*x))^3)+2/15*tan(c+d*x)/(a*d*(a+a*sec(c+d*x))^2)+2/15*tan(c+d*x)/(d*(a^3+a^3*sec(c+d*x)))],
[1/(a+a*sec(c+d*x))^3,x,4,x/a^3-1/5*tan(c+d*x)/(d*(a+a*sec(c+d*x))^3)-7/15*tan(c+d*x)/(a*d*(a+a*sec(c+d*x))^2)-22/15*tan(c+d*x)/(d*(a^3+a^3*sec(c+d*x)))],
[cos(c+d*x)/(a+a*sec(c+d*x))^3,x,6,-3*x/a^3+24/5*sin(c+d*x)/(a^3*d)-1/5*sin(c+d*x)/(d*(a+a*sec(c+d*x))^3)-3/5*sin(c+d*x)/(a*d*(a+a*sec(c+d*x))^2)-3*sin(c+d*x)/(d*(a^3+a^3*sec(c+d*x)))],
[cos(c+d*x)^2/(a+a*sec(c+d*x))^3,x,7,13/2*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)*sin(c+d*x)/(d*(a+a*sec(c+d*x))^3)-11/15*cos(c+d*x)*sin(c+d*x)/(a*d*(a+a*sec(c+d*x))^2)-76/15*cos(c+d*x)*sin(c+d*x)/(d*(a^3+a^3*sec(c+d*x)))],
[sec(c+d*x)^7/(a+a*sec(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)^3*tan(c+d*x)/(a^4*d*(1+sec(c+d*x))^2)-288/35*sec(c+d*x)^2*tan(c+d*x)/(a^4*d*(1+sec(c+d*x)))-1/7*sec(c+d*x)^5*tan(c+d*x)/(d*(a+a*sec(c+d*x))^4)-2/5*sec(c+d*x)^4*tan(c+d*x)/(a*d*(a+a*sec(c+d*x))^3)],
[sec(c+d*x)^6/(a+a*sec(c+d*x))^4,x,8,-4*arctanh(sin(c+d*x))/(a^4*d)+244/105*tan(c+d*x)/(a^4*d)-88/105*sec(c+d*x)^2*tan(c+d*x)/(a^4*d*(1+sec(c+d*x))^2)+4*tan(c+d*x)/(a^4*d*(1+sec(c+d*x)))-1/7*sec(c+d*x)^4*tan(c+d*x)/(d*(a+a*sec(c+d*x))^4)-12/35*sec(c+d*x)^3*tan(c+d*x)/(a*d*(a+a*sec(c+d*x))^3)],
[sec(c+d*x)^5/(a+a*sec(c+d*x))^4,x,6,arctanh(sin(c+d*x))/(a^4*d)+11/21*tan(c+d*x)/(a^4*d*(1+sec(c+d*x))^2)-43/21*tan(c+d*x)/(a^4*d*(1+sec(c+d*x)))-1/7*sec(c+d*x)^3*tan(c+d*x)/(d*(a+a*sec(c+d*x))^4)-2/7*sec(c+d*x)^2*tan(c+d*x)/(a*d*(a+a*sec(c+d*x))^3)],
[sec(c+d*x)^4/(a+a*sec(c+d*x))^4,x,4,1/7*sec(c+d*x)^3*tan(c+d*x)/(d*(a+a*sec(c+d*x))^4)+3/35*tan(c+d*x)/(a*d*(a+a*sec(c+d*x))^3)-8/35*tan(c+d*x)/(d*(a^2+a^2*sec(c+d*x))^2)+1/5*tan(c+d*x)/(d*(a^4+a^4*sec(c+d*x)))],
[sec(c+d*x)^3/(a+a*sec(c+d*x))^4,x,4,1/7*tan(c+d*x)/(d*(a+a*sec(c+d*x))^4)-11/35*tan(c+d*x)/(a*d*(a+a*sec(c+d*x))^3)+13/105*tan(c+d*x)/(d*(a^2+a^2*sec(c+d*x))^2)+13/105*tan(c+d*x)/(d*(a^4+a^4*sec(c+d*x)))],
[sec(c+d*x)^2/(a+a*sec(c+d*x))^4,x,4,-1/7*tan(c+d*x)/(d*(a+a*sec(c+d*x))^4)+4/35*tan(c+d*x)/(a*d*(a+a*sec(c+d*x))^3)+8/105*tan(c+d*x)/(d*(a^2+a^2*sec(c+d*x))^2)+8/105*tan(c+d*x)/(d*(a^4+a^4*sec(c+d*x)))],
[sec(c+d*x)/(a+a*sec(c+d*x))^4,x,4,1/7*tan(c+d*x)/(d*(a+a*sec(c+d*x))^4)+3/35*tan(c+d*x)/(a*d*(a+a*sec(c+d*x))^3)+2/35*tan(c+d*x)/(d*(a^2+a^2*sec(c+d*x))^2)+2/35*tan(c+d*x)/(d*(a^4+a^4*sec(c+d*x)))],
[1/(a+a*sec(c+d*x))^4,x,5,x/a^4-11/21*tan(c+d*x)/(a^4*d*(1+sec(c+d*x))^2)-32/21*tan(c+d*x)/(a^4*d*(1+sec(c+d*x)))-1/7*tan(c+d*x)/(d*(a+a*sec(c+d*x))^4)-2/7*tan(c+d*x)/(a*d*(a+a*sec(c+d*x))^3)],
[cos(c+d*x)/(a+a*sec(c+d*x))^4,x,7,-4*x/a^4+664/105*sin(c+d*x)/(a^4*d)-88/105*sin(c+d*x)/(a^4*d*(1+sec(c+d*x))^2)-4*sin(c+d*x)/(a^4*d*(1+sec(c+d*x)))-1/7*sin(c+d*x)/(d*(a+a*sec(c+d*x))^4)-12/35*sin(c+d*x)/(a*d*(a+a*sec(c+d*x))^3)],
[cos(c+d*x)^2/(a+a*sec(c+d*x))^4,x,8,21/2*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)*sin(c+d*x)/(a^4*d*(1+sec(c+d*x))^2)-288/35*cos(c+d*x)*sin(c+d*x)/(a^4*d*(1+sec(c+d*x)))-1/7*cos(c+d*x)*sin(c+d*x)/(d*(a+a*sec(c+d*x))^4)-2/5*cos(c+d*x)*sin(c+d*x)/(a*d*(a+a*sec(c+d*x))^3)],
[sec(c+d*x)^7/(a+a*sec(c+d*x))^5,x,9,-5*arctanh(sin(c+d*x))/(a^5*d)+181/63*tan(c+d*x)/(a^5*d)-1/9*sec(c+d*x)^5*tan(c+d*x)/(d*(a+a*sec(c+d*x))^5)-5/21*sec(c+d*x)^4*tan(c+d*x)/(a*d*(a+a*sec(c+d*x))^4)-29/63*sec(c+d*x)^3*tan(c+d*x)/(a^2*d*(a+a*sec(c+d*x))^3)-67/63*sec(c+d*x)^2*tan(c+d*x)/(a^3*d*(a+a*sec(c+d*x))^2)+5*tan(c+d*x)/(d*(a^5+a^5*sec(c+d*x)))],
[sec(c+d*x)^6/(a+a*sec(c+d*x))^5,x,7,arctanh(sin(c+d*x))/(a^5*d)-1/9*sec(c+d*x)^4*tan(c+d*x)/(d*(a+a*sec(c+d*x))^5)-13/63*sec(c+d*x)^3*tan(c+d*x)/(a*d*(a+a*sec(c+d*x))^4)-34/105*sec(c+d*x)^2*tan(c+d*x)/(a^2*d*(a+a*sec(c+d*x))^3)+173/315*tan(c+d*x)/(a^3*d*(a+a*sec(c+d*x))^2)-661/315*tan(c+d*x)/(d*(a^5+a^5*sec(c+d*x)))],
[sec(c+d*x)^5/(a+a*sec(c+d*x))^5,x,5,1/9*sec(c+d*x)^4*tan(c+d*x)/(d*(a+a*sec(c+d*x))^5)+4/63*sec(c+d*x)^3*tan(c+d*x)/(a*d*(a+a*sec(c+d*x))^4)+4/105*tan(c+d*x)/(a^2*d*(a+a*sec(c+d*x))^3)-32/315*tan(c+d*x)/(a*d*(a^2+a^2*sec(c+d*x))^2)+4/45*tan(c+d*x)/(d*(a^5+a^5*sec(c+d*x)))],
[sec(c+d*x)^4/(a+a*sec(c+d*x))^5,x,5,-1/9*sec(c+d*x)^4*tan(c+d*x)/(d*(a+a*sec(c+d*x))^5)+5/63*sec(c+d*x)^3*tan(c+d*x)/(a*d*(a+a*sec(c+d*x))^4)+1/21*tan(c+d*x)/(a^2*d*(a+a*sec(c+d*x))^3)-8/63*tan(c+d*x)/(a*d*(a^2+a^2*sec(c+d*x))^2)+1/9*tan(c+d*x)/(d*(a^5+a^5*sec(c+d*x)))],
[sec(c+d*x)^3/(a+a*sec(c+d*x))^5,x,5,1/9*tan(c+d*x)/(d*(a+a*sec(c+d*x))^5)-2/9*tan(c+d*x)/(a*d*(a+a*sec(c+d*x))^4)+1/15*tan(c+d*x)/(a^2*d*(a+a*sec(c+d*x))^3)+2/45*tan(c+d*x)/(a^3*d*(a+a*sec(c+d*x))^2)+2/45*tan(c+d*x)/(d*(a^5+a^5*sec(c+d*x)))],
[sec(c+d*x)^2/(a+a*sec(c+d*x))^5,x,5,-1/9*tan(c+d*x)/(d*(a+a*sec(c+d*x))^5)+5/63*tan(c+d*x)/(a*d*(a+a*sec(c+d*x))^4)+1/21*tan(c+d*x)/(a^2*d*(a+a*sec(c+d*x))^3)+2/63*tan(c+d*x)/(a*d*(a^2+a^2*sec(c+d*x))^2)+2/63*tan(c+d*x)/(d*(a^5+a^5*sec(c+d*x)))],
[sec(c+d*x)/(a+a*sec(c+d*x))^5,x,5,1/9*tan(c+d*x)/(d*(a+a*sec(c+d*x))^5)+4/63*tan(c+d*x)/(a*d*(a+a*sec(c+d*x))^4)+4/105*tan(c+d*x)/(a^2*d*(a+a*sec(c+d*x))^3)+8/315*tan(c+d*x)/(a*d*(a^2+a^2*sec(c+d*x))^2)+8/315*tan(c+d*x)/(d*(a^5+a^5*sec(c+d*x)))],
[1/(a+a*sec(c+d*x))^5,x,6,x/a^5-1/9*tan(c+d*x)/(d*(a+a*sec(c+d*x))^5)-13/63*tan(c+d*x)/(a*d*(a+a*sec(c+d*x))^4)-34/105*tan(c+d*x)/(a^2*d*(a+a*sec(c+d*x))^3)-173/315*tan(c+d*x)/(a^3*d*(a+a*sec(c+d*x))^2)-488/315*tan(c+d*x)/(d*(a^5+a^5*sec(c+d*x)))],
[cos(c+d*x)/(a+a*sec(c+d*x))^5,x,8,-5*x/a^5+496/63*sin(c+d*x)/(a^5*d)-1/9*sin(c+d*x)/(d*(a+a*sec(c+d*x))^5)-5/21*sin(c+d*x)/(a*d*(a+a*sec(c+d*x))^4)-29/63*sin(c+d*x)/(a^2*d*(a+a*sec(c+d*x))^3)-67/63*sin(c+d*x)/(a^3*d*(a+a*sec(c+d*x))^2)-5*sin(c+d*x)/(d*(a^5+a^5*sec(c+d*x)))],
[cos(c+d*x)^2/(a+a*sec(c+d*x))^5,x,9,31/2*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)*sin(c+d*x)/(d*(a+a*sec(c+d*x))^5)-17/63*cos(c+d*x)*sin(c+d*x)/(a*d*(a+a*sec(c+d*x))^4)-28/45*cos(c+d*x)*sin(c+d*x)/(a^2*d*(a+a*sec(c+d*x))^3)-577/315*cos(c+d*x)*sin(c+d*x)/(a^3*d*(a+a*sec(c+d*x))^2)-3832/315*cos(c+d*x)*sin(c+d*x)/(d*(a^5+a^5*sec(c+d*x)))],

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

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

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

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

# m>0
[sec(c+d*x)^3*(a+a*sec(c+d*x))^(2/3),x,7,-9/40*(a+a*sec(c+d*x))^(2/3)*tan(c+d*x)/d+57/80*(a+a*sec(c+d*x))^(2/3)*tan(c+d*x)/(d*(1+sec(c+d*x)))+3/8*(a+a*sec(c+d*x))^(5/3)*tan(c+d*x)/(a*d)-19/80*3^(3/4)*sqrt(cos(arccos((2^(1/3)-(1+sec(c+d*x))^(1/3)*(1-sqrt(3)))/(2^(1/3)-(1+sec(c+d*x))^(1/3)*(1+sqrt(3)))))^2)/cos(arccos((2^(1/3)-(1+sec(c+d*x))^(1/3)*(1-sqrt(3)))/(2^(1/3)-(1+sec(c+d*x))^(1/3)*(1+sqrt(3)))))*EllipticF(sin(arccos((2^(1/3)-(1+sec(c+d*x))^(1/3)*(1-sqrt(3)))/(2^(1/3)-(1+sec(c+d*x))^(1/3)*(1+sqrt(3))))),sqrt(1/4*(2+sqrt(3))))*(a+a*sec(c+d*x))^(2/3)*(2^(1/3)-(1+sec(c+d*x))^(1/3))*sqrt((2^(2/3)+2^(1/3)*(1+sec(c+d*x))^(1/3)+(1+sec(c+d*x))^(2/3))/(2^(1/3)-(1+sec(c+d*x))^(1/3)*(1+sqrt(3)))^2)*tan(c+d*x)/(2^(1/3)*d*(1-sec(c+d*x))*(1+sec(c+d*x))*sqrt(-(1+sec(c+d*x))^(1/3)*(2^(1/3)-(1+sec(c+d*x))^(1/3))/(2^(1/3)-(1+sec(c+d*x))^(1/3)*(1+sqrt(3)))^2))],
[sec(c+d*x)^2*(a+a*sec(c+d*x))^(2/3),x,6,3/5*(a+a*sec(c+d*x))^(2/3)*tan(c+d*x)/d+3/5*(a+a*sec(c+d*x))^(2/3)*tan(c+d*x)/(d*(1+sec(c+d*x)))-1/5*3^(3/4)*sqrt(cos(arccos((2^(1/3)-(1+sec(c+d*x))^(1/3)*(1-sqrt(3)))/(2^(1/3)-(1+sec(c+d*x))^(1/3)*(1+sqrt(3)))))^2)/cos(arccos((2^(1/3)-(1+sec(c+d*x))^(1/3)*(1-sqrt(3)))/(2^(1/3)-(1+sec(c+d*x))^(1/3)*(1+sqrt(3)))))*EllipticF(sin(arccos((2^(1/3)-(1+sec(c+d*x))^(1/3)*(1-sqrt(3)))/(2^(1/3)-(1+sec(c+d*x))^(1/3)*(1+sqrt(3))))),sqrt(1/4*(2+sqrt(3))))*(a+a*sec(c+d*x))^(2/3)*(2^(1/3)-(1+sec(c+d*x))^(1/3))*sqrt((2^(2/3)+2^(1/3)*(1+sec(c+d*x))^(1/3)+(1+sec(c+d*x))^(2/3))/(2^(1/3)-(1+sec(c+d*x))^(1/3)*(1+sqrt(3)))^2)*tan(c+d*x)/(2^(1/3)*d*(1-sec(c+d*x))*(1+sec(c+d*x))*sqrt(-(1+sec(c+d*x))^(1/3)*(2^(1/3)-(1+sec(c+d*x))^(1/3))/(2^(1/3)-(1+sec(c+d*x))^(1/3)*(1+sqrt(3)))^2))],
[sec(c+d*x)*(a+a*sec(c+d*x))^(2/3),x,5,3/2*(a+a*sec(c+d*x))^(2/3)*tan(c+d*x)/(d*(1+sec(c+d*x)))-1/2*3^(3/4)*sqrt(cos(arccos((2^(1/3)-(1+sec(c+d*x))^(1/3)*(1-sqrt(3)))/(2^(1/3)-(1+sec(c+d*x))^(1/3)*(1+sqrt(3)))))^2)/cos(arccos((2^(1/3)-(1+sec(c+d*x))^(1/3)*(1-sqrt(3)))/(2^(1/3)-(1+sec(c+d*x))^(1/3)*(1+sqrt(3)))))*EllipticF(sin(arccos((2^(1/3)-(1+sec(c+d*x))^(1/3)*(1-sqrt(3)))/(2^(1/3)-(1+sec(c+d*x))^(1/3)*(1+sqrt(3))))),sqrt(1/4*(2+sqrt(3))))*(a+a*sec(c+d*x))^(2/3)*(2^(1/3)-(1+sec(c+d*x))^(1/3))*sqrt((2^(2/3)+2^(1/3)*(1+sec(c+d*x))^(1/3)+(1+sec(c+d*x))^(2/3))/(2^(1/3)-(1+sec(c+d*x))^(1/3)*(1+sqrt(3)))^2)*tan(c+d*x)/(2^(1/3)*d*(1-sec(c+d*x))*(1+sec(c+d*x))*sqrt(-(1+sec(c+d*x))^(1/3)*(2^(1/3)-(1+sec(c+d*x))^(1/3))/(2^(1/3)-(1+sec(c+d*x))^(1/3)*(1+sqrt(3)))^2))],
[(a+a*sec(c+d*x))^(2/3),x,3,3/7*AppellF1(7/6,1/2,1,13/6,1/2*(1+sec(c+d*x)),1+sec(c+d*x))*(a+a*sec(c+d*x))^(2/3)*sqrt(2)*tan(c+d*x)/(d*sqrt(1-sec(c+d*x)))],
[cos(c+d*x)*(a+a*sec(c+d*x))^(2/3),x,3,-3/7*AppellF1(7/6,1/2,2,13/6,1/2*(1+sec(c+d*x)),1+sec(c+d*x))*(a+a*sec(c+d*x))^(2/3)*sqrt(2)*tan(c+d*x)/(d*sqrt(1-sec(c+d*x)))],
[sec(c+d*x)^3*(a+a*sec(c+d*x))^(5/3),x,8,147/440*a*(a+a*sec(c+d*x))^(2/3)*tan(c+d*x)/d+1029/880*a*(a+a*sec(c+d*x))^(2/3)*tan(c+d*x)/(d*(1+sec(c+d*x)))-9/88*(a+a*sec(c+d*x))^(5/3)*tan(c+d*x)/d+3/11*(a+a*sec(c+d*x))^(8/3)*tan(c+d*x)/(a*d)-343/880*3^(3/4)*a*sqrt(cos(arccos((2^(1/3)-(1+sec(c+d*x))^(1/3)*(1-sqrt(3)))/(2^(1/3)-(1+sec(c+d*x))^(1/3)*(1+sqrt(3)))))^2)/cos(arccos((2^(1/3)-(1+sec(c+d*x))^(1/3)*(1-sqrt(3)))/(2^(1/3)-(1+sec(c+d*x))^(1/3)*(1+sqrt(3)))))*EllipticF(sin(arccos((2^(1/3)-(1+sec(c+d*x))^(1/3)*(1-sqrt(3)))/(2^(1/3)-(1+sec(c+d*x))^(1/3)*(1+sqrt(3))))),sqrt(1/4*(2+sqrt(3))))*(a+a*sec(c+d*x))^(2/3)*(2^(1/3)-(1+sec(c+d*x))^(1/3))*sqrt((2^(2/3)+2^(1/3)*(1+sec(c+d*x))^(1/3)+(1+sec(c+d*x))^(2/3))/(2^(1/3)-(1+sec(c+d*x))^(1/3)*(1+sqrt(3)))^2)*tan(c+d*x)/(2^(1/3)*d*(1-sec(c+d*x))*(1+sec(c+d*x))*sqrt(-(1+sec(c+d*x))^(1/3)*(2^(1/3)-(1+sec(c+d*x))^(1/3))/(2^(1/3)-(1+sec(c+d*x))^(1/3)*(1+sqrt(3)))^2))],
[sec(c+d*x)^2*(a+a*sec(c+d*x))^(5/3),x,7,3/8*a*(a+a*sec(c+d*x))^(2/3)*tan(c+d*x)/d+21/16*a*(a+a*sec(c+d*x))^(2/3)*tan(c+d*x)/(d*(1+sec(c+d*x)))+3/8*(a+a*sec(c+d*x))^(5/3)*tan(c+d*x)/d-7/16*3^(3/4)*a*sqrt(cos(arccos((2^(1/3)-(1+sec(c+d*x))^(1/3)*(1-sqrt(3)))/(2^(1/3)-(1+sec(c+d*x))^(1/3)*(1+sqrt(3)))))^2)/cos(arccos((2^(1/3)-(1+sec(c+d*x))^(1/3)*(1-sqrt(3)))/(2^(1/3)-(1+sec(c+d*x))^(1/3)*(1+sqrt(3)))))*EllipticF(sin(arccos((2^(1/3)-(1+sec(c+d*x))^(1/3)*(1-sqrt(3)))/(2^(1/3)-(1+sec(c+d*x))^(1/3)*(1+sqrt(3))))),sqrt(1/4*(2+sqrt(3))))*(a+a*sec(c+d*x))^(2/3)*(2^(1/3)-(1+sec(c+d*x))^(1/3))*sqrt((2^(2/3)+2^(1/3)*(1+sec(c+d*x))^(1/3)+(1+sec(c+d*x))^(2/3))/(2^(1/3)-(1+sec(c+d*x))^(1/3)*(1+sqrt(3)))^2)*tan(c+d*x)/(2^(1/3)*d*(1-sec(c+d*x))*(1+sec(c+d*x))*sqrt(-(1+sec(c+d*x))^(1/3)*(2^(1/3)-(1+sec(c+d*x))^(1/3))/(2^(1/3)-(1+sec(c+d*x))^(1/3)*(1+sqrt(3)))^2))],
[sec(c+d*x)*(a+a*sec(c+d*x))^(5/3),x,6,3/5*a*(a+a*sec(c+d*x))^(2/3)*tan(c+d*x)/d+21/10*a*(a+a*sec(c+d*x))^(2/3)*tan(c+d*x)/(d*(1+sec(c+d*x)))-7/10*3^(3/4)*a*sqrt(cos(arccos((2^(1/3)-(1+sec(c+d*x))^(1/3)*(1-sqrt(3)))/(2^(1/3)-(1+sec(c+d*x))^(1/3)*(1+sqrt(3)))))^2)/cos(arccos((2^(1/3)-(1+sec(c+d*x))^(1/3)*(1-sqrt(3)))/(2^(1/3)-(1+sec(c+d*x))^(1/3)*(1+sqrt(3)))))*EllipticF(sin(arccos((2^(1/3)-(1+sec(c+d*x))^(1/3)*(1-sqrt(3)))/(2^(1/3)-(1+sec(c+d*x))^(1/3)*(1+sqrt(3))))),sqrt(1/4*(2+sqrt(3))))*(a+a*sec(c+d*x))^(2/3)*(2^(1/3)-(1+sec(c+d*x))^(1/3))*sqrt((2^(2/3)+2^(1/3)*(1+sec(c+d*x))^(1/3)+(1+sec(c+d*x))^(2/3))/(2^(1/3)-(1+sec(c+d*x))^(1/3)*(1+sqrt(3)))^2)*tan(c+d*x)/(2^(1/3)*d*(1-sec(c+d*x))*(1+sec(c+d*x))*sqrt(-(1+sec(c+d*x))^(1/3)*(2^(1/3)-(1+sec(c+d*x))^(1/3))/(2^(1/3)-(1+sec(c+d*x))^(1/3)*(1+sqrt(3)))^2))],
[(a+a*sec(c+d*x))^(5/3),x,3,3/13*a*AppellF1(13/6,1/2,1,19/6,1/2*(1+sec(c+d*x)),1+sec(c+d*x))*(1+sec(c+d*x))*(a+a*sec(c+d*x))^(2/3)*sqrt(2)*tan(c+d*x)/(d*sqrt(1-sec(c+d*x)))],
[cos(c+d*x)*(a+a*sec(c+d*x))^(5/3),x,3,-3/13*a*AppellF1(13/6,1/2,2,19/6,1/2*(1+sec(c+d*x)),1+sec(c+d*x))*(1+sec(c+d*x))*(a+a*sec(c+d*x))^(2/3)*sqrt(2)*tan(c+d*x)/(d*sqrt(1-sec(c+d*x)))],

# m<0
[sec(c+d*x)^4/(a+a*sec(c+d*x))^(1/3),x,7,99/80*tan(c+d*x)/(d*(a+a*sec(c+d*x))^(1/3))+3/8*sec(c+d*x)^2*tan(c+d*x)/(d*(a+a*sec(c+d*x))^(1/3))-3/40*(a+a*sec(c+d*x))^(2/3)*tan(c+d*x)/(a*d)+37/80*3^(3/4)*sqrt(cos(arccos((2^(1/3)-(1+sec(c+d*x))^(1/3)*(1-sqrt(3)))/(2^(1/3)-(1+sec(c+d*x))^(1/3)*(1+sqrt(3)))))^2)/cos(arccos((2^(1/3)-(1+sec(c+d*x))^(1/3)*(1-sqrt(3)))/(2^(1/3)-(1+sec(c+d*x))^(1/3)*(1+sqrt(3)))))*EllipticF(sin(arccos((2^(1/3)-(1+sec(c+d*x))^(1/3)*(1-sqrt(3)))/(2^(1/3)-(1+sec(c+d*x))^(1/3)*(1+sqrt(3))))),sqrt(1/4*(2+sqrt(3))))*(2^(1/3)-(1+sec(c+d*x))^(1/3))*sqrt((2^(2/3)+2^(1/3)*(1+sec(c+d*x))^(1/3)+(1+sec(c+d*x))^(2/3))/(2^(1/3)-(1+sec(c+d*x))^(1/3)*(1+sqrt(3)))^2)*tan(c+d*x)/(2^(1/3)*d*(1-sec(c+d*x))*(a+a*sec(c+d*x))^(1/3)*sqrt(-(1+sec(c+d*x))^(1/3)*(2^(1/3)-(1+sec(c+d*x))^(1/3))/(2^(1/3)-(1+sec(c+d*x))^(1/3)*(1+sqrt(3)))^2))],
[sec(c+d*x)^3/(a+a*sec(c+d*x))^(1/3),x,6,-9/10*tan(c+d*x)/(d*(a+a*sec(c+d*x))^(1/3))+3/5*(a+a*sec(c+d*x))^(2/3)*tan(c+d*x)/(a*d)-7/10*3^(3/4)*sqrt(cos(arccos((2^(1/3)-(1+sec(c+d*x))^(1/3)*(1-sqrt(3)))/(2^(1/3)-(1+sec(c+d*x))^(1/3)*(1+sqrt(3)))))^2)/cos(arccos((2^(1/3)-(1+sec(c+d*x))^(1/3)*(1-sqrt(3)))/(2^(1/3)-(1+sec(c+d*x))^(1/3)*(1+sqrt(3)))))*EllipticF(sin(arccos((2^(1/3)-(1+sec(c+d*x))^(1/3)*(1-sqrt(3)))/(2^(1/3)-(1+sec(c+d*x))^(1/3)*(1+sqrt(3))))),sqrt(1/4*(2+sqrt(3))))*(2^(1/3)-(1+sec(c+d*x))^(1/3))*sqrt((2^(2/3)+2^(1/3)*(1+sec(c+d*x))^(1/3)+(1+sec(c+d*x))^(2/3))/(2^(1/3)-(1+sec(c+d*x))^(1/3)*(1+sqrt(3)))^2)*tan(c+d*x)/(2^(1/3)*d*(1-sec(c+d*x))*(a+a*sec(c+d*x))^(1/3)*sqrt(-(1+sec(c+d*x))^(1/3)*(2^(1/3)-(1+sec(c+d*x))^(1/3))/(2^(1/3)-(1+sec(c+d*x))^(1/3)*(1+sqrt(3)))^2))],
[sec(c+d*x)^2/(a+a*sec(c+d*x))^(1/3),x,5,3/2*tan(c+d*x)/(d*(a+a*sec(c+d*x))^(1/3))+1/2*3^(3/4)*sqrt(cos(arccos((2^(1/3)-(1+sec(c+d*x))^(1/3)*(1-sqrt(3)))/(2^(1/3)-(1+sec(c+d*x))^(1/3)*(1+sqrt(3)))))^2)/cos(arccos((2^(1/3)-(1+sec(c+d*x))^(1/3)*(1-sqrt(3)))/(2^(1/3)-(1+sec(c+d*x))^(1/3)*(1+sqrt(3)))))*EllipticF(sin(arccos((2^(1/3)-(1+sec(c+d*x))^(1/3)*(1-sqrt(3)))/(2^(1/3)-(1+sec(c+d*x))^(1/3)*(1+sqrt(3))))),sqrt(1/4*(2+sqrt(3))))*(2^(1/3)-(1+sec(c+d*x))^(1/3))*sqrt((2^(2/3)+2^(1/3)*(1+sec(c+d*x))^(1/3)+(1+sec(c+d*x))^(2/3))/(2^(1/3)-(1+sec(c+d*x))^(1/3)*(1+sqrt(3)))^2)*tan(c+d*x)/(2^(1/3)*d*(1-sec(c+d*x))*(a+a*sec(c+d*x))^(1/3)*sqrt(-(1+sec(c+d*x))^(1/3)*(2^(1/3)-(1+sec(c+d*x))^(1/3))/(2^(1/3)-(1+sec(c+d*x))^(1/3)*(1+sqrt(3)))^2))],
[sec(c+d*x)/(a+a*sec(c+d*x))^(1/3),x,4,-3^(3/4)*sqrt(cos(arccos((2^(1/3)-(1+sec(c+d*x))^(1/3)*(1-sqrt(3)))/(2^(1/3)-(1+sec(c+d*x))^(1/3)*(1+sqrt(3)))))^2)/cos(arccos((2^(1/3)-(1+sec(c+d*x))^(1/3)*(1-sqrt(3)))/(2^(1/3)-(1+sec(c+d*x))^(1/3)*(1+sqrt(3)))))*EllipticF(sin(arccos((2^(1/3)-(1+sec(c+d*x))^(1/3)*(1-sqrt(3)))/(2^(1/3)-(1+sec(c+d*x))^(1/3)*(1+sqrt(3))))),sqrt(1/4*(2+sqrt(3))))*(2^(1/3)-(1+sec(c+d*x))^(1/3))*sqrt((2^(2/3)+2^(1/3)*(1+sec(c+d*x))^(1/3)+(1+sec(c+d*x))^(2/3))/(2^(1/3)-(1+sec(c+d*x))^(1/3)*(1+sqrt(3)))^2)*tan(c+d*x)/(2^(1/3)*d*(1-sec(c+d*x))*(a+a*sec(c+d*x))^(1/3)*sqrt(-(1+sec(c+d*x))^(1/3)*(2^(1/3)-(1+sec(c+d*x))^(1/3))/(2^(1/3)-(1+sec(c+d*x))^(1/3)*(1+sqrt(3)))^2))],
[1/(a+a*sec(c+d*x))^(1/3),x,3,3*AppellF1(1/6,1/2,1,7/6,1/2*(1+sec(c+d*x)),1+sec(c+d*x))*sqrt(2)*tan(c+d*x)/(d*(a+a*sec(c+d*x))^(1/3)*sqrt(1-sec(c+d*x)))],
[cos(c+d*x)/(a+a*sec(c+d*x))^(1/3),x,3,-3*AppellF1(1/6,1/2,2,7/6,1/2*(1+sec(c+d*x)),1+sec(c+d*x))*sqrt(2)*tan(c+d*x)/(d*(a+a*sec(c+d*x))^(1/3)*sqrt(1-sec(c+d*x)))],
[sec(c+d*x)^4/(a+a*sec(c+d*x))^(5/3),x,9,-33/28*tan(c+d*x)/(d*(a+a*sec(c+d*x))^(5/3))+3/4*sec(c+d*x)^2*tan(c+d*x)/(d*(a+a*sec(c+d*x))^(5/3))+135/14*tan(c+d*x)/(a*d*(a+a*sec(c+d*x))^(2/3))+375/28*(a+a*sec(c+d*x))^(1/3)*(1+sqrt(3))*tan(c+d*x)/(a^2*d*(1+sec(c+d*x))^(2/3)*(2^(1/3)-(1+sec(c+d*x))^(1/3)*(1+sqrt(3))))-375/14*3^(1/4)*sqrt(cos(arccos((2^(1/3)-(1+sec(c+d*x))^(1/3)*(1-sqrt(3)))/(2^(1/3)-(1+sec(c+d*x))^(1/3)*(1+sqrt(3)))))^2)/cos(arccos((2^(1/3)-(1+sec(c+d*x))^(1/3)*(1-sqrt(3)))/(2^(1/3)-(1+sec(c+d*x))^(1/3)*(1+sqrt(3)))))*EllipticE(sin(arccos((2^(1/3)-(1+sec(c+d*x))^(1/3)*(1-sqrt(3)))/(2^(1/3)-(1+sec(c+d*x))^(1/3)*(1+sqrt(3))))),sqrt(1/4*(2+sqrt(3))))*(a+a*sec(c+d*x))^(1/3)*(2^(1/3)-(1+sec(c+d*x))^(1/3))*sqrt((2^(2/3)+2^(1/3)*(1+sec(c+d*x))^(1/3)+(1+sec(c+d*x))^(2/3))/(2^(1/3)-(1+sec(c+d*x))^(1/3)*(1+sqrt(3)))^2)*tan(c+d*x)/(2^(2/3)*a^2*d*(1-sec(c+d*x))*(1+sec(c+d*x))^(2/3)*sqrt(-(1+sec(c+d*x))^(1/3)*(2^(1/3)-(1+sec(c+d*x))^(1/3))/(2^(1/3)-(1+sec(c+d*x))^(1/3)*(1+sqrt(3)))^2))-125/28*3^(3/4)*sqrt(cos(arccos((2^(1/3)-(1+sec(c+d*x))^(1/3)*(1-sqrt(3)))/(2^(1/3)-(1+sec(c+d*x))^(1/3)*(1+sqrt(3)))))^2)/cos(arccos((2^(1/3)-(1+sec(c+d*x))^(1/3)*(1-sqrt(3)))/(2^(1/3)-(1+sec(c+d*x))^(1/3)*(1+sqrt(3)))))*EllipticF(sin(arccos((2^(1/3)-(1+sec(c+d*x))^(1/3)*(1-sqrt(3)))/(2^(1/3)-(1+sec(c+d*x))^(1/3)*(1+sqrt(3))))),sqrt(1/4*(2+sqrt(3))))*(a+a*sec(c+d*x))^(1/3)*(2^(1/3)-(1+sec(c+d*x))^(1/3))*(1-sqrt(3))*sqrt((2^(2/3)+2^(1/3)*(1+sec(c+d*x))^(1/3)+(1+sec(c+d*x))^(2/3))/(2^(1/3)-(1+sec(c+d*x))^(1/3)*(1+sqrt(3)))^2)*tan(c+d*x)/(2^(2/3)*a^2*d*(1-sec(c+d*x))*(1+sec(c+d*x))^(2/3)*sqrt(-(1+sec(c+d*x))^(1/3)*(2^(1/3)-(1+sec(c+d*x))^(1/3))/(2^(1/3)-(1+sec(c+d*x))^(1/3)*(1+sqrt(3)))^2))],
[sec(c+d*x)^3/(a+a*sec(c+d*x))^(5/3),x,8,3/7*tan(c+d*x)/(d*(a+a*sec(c+d*x))^(5/3))-36/7*tan(c+d*x)/(a*d*(a+a*sec(c+d*x))^(2/3))-57/7*(a+a*sec(c+d*x))^(1/3)*(1+sqrt(3))*tan(c+d*x)/(a^2*d*(1+sec(c+d*x))^(2/3)*(2^(1/3)-(1+sec(c+d*x))^(1/3)*(1+sqrt(3))))+57/7*2^(1/3)*3^(1/4)*sqrt(cos(arccos((2^(1/3)-(1+sec(c+d*x))^(1/3)*(1-sqrt(3)))/(2^(1/3)-(1+sec(c+d*x))^(1/3)*(1+sqrt(3)))))^2)/cos(arccos((2^(1/3)-(1+sec(c+d*x))^(1/3)*(1-sqrt(3)))/(2^(1/3)-(1+sec(c+d*x))^(1/3)*(1+sqrt(3)))))*EllipticE(sin(arccos((2^(1/3)-(1+sec(c+d*x))^(1/3)*(1-sqrt(3)))/(2^(1/3)-(1+sec(c+d*x))^(1/3)*(1+sqrt(3))))),sqrt(1/4*(2+sqrt(3))))*(a+a*sec(c+d*x))^(1/3)*(2^(1/3)-(1+sec(c+d*x))^(1/3))*sqrt((2^(2/3)+2^(1/3)*(1+sec(c+d*x))^(1/3)+(1+sec(c+d*x))^(2/3))/(2^(1/3)-(1+sec(c+d*x))^(1/3)*(1+sqrt(3)))^2)*tan(c+d*x)/(a^2*d*(1-sec(c+d*x))*(1+sec(c+d*x))^(2/3)*sqrt(-(1+sec(c+d*x))^(1/3)*(2^(1/3)-(1+sec(c+d*x))^(1/3))/(2^(1/3)-(1+sec(c+d*x))^(1/3)*(1+sqrt(3)))^2))+19/7*3^(3/4)*sqrt(cos(arccos((2^(1/3)-(1+sec(c+d*x))^(1/3)*(1-sqrt(3)))/(2^(1/3)-(1+sec(c+d*x))^(1/3)*(1+sqrt(3)))))^2)/cos(arccos((2^(1/3)-(1+sec(c+d*x))^(1/3)*(1-sqrt(3)))/(2^(1/3)-(1+sec(c+d*x))^(1/3)*(1+sqrt(3)))))*EllipticF(sin(arccos((2^(1/3)-(1+sec(c+d*x))^(1/3)*(1-sqrt(3)))/(2^(1/3)-(1+sec(c+d*x))^(1/3)*(1+sqrt(3))))),sqrt(1/4*(2+sqrt(3))))*(a+a*sec(c+d*x))^(1/3)*(2^(1/3)-(1+sec(c+d*x))^(1/3))*(1-sqrt(3))*sqrt((2^(2/3)+2^(1/3)*(1+sec(c+d*x))^(1/3)+(1+sec(c+d*x))^(2/3))/(2^(1/3)-(1+sec(c+d*x))^(1/3)*(1+sqrt(3)))^2)*tan(c+d*x)/(2^(2/3)*a^2*d*(1-sec(c+d*x))*(1+sec(c+d*x))^(2/3)*sqrt(-(1+sec(c+d*x))^(1/3)*(2^(1/3)-(1+sec(c+d*x))^(1/3))/(2^(1/3)-(1+sec(c+d*x))^(1/3)*(1+sqrt(3)))^2))],
[sec(c+d*x)^2/(a+a*sec(c+d*x))^(5/3),x,8,-3/7*tan(c+d*x)/(d*(a+a*sec(c+d*x))^(5/3))+15/7*tan(c+d*x)/(a*d*(a+a*sec(c+d*x))^(2/3))+15/7*(1+sec(c+d*x))^(1/3)*(1+sqrt(3))*tan(c+d*x)/(a*d*(a+a*sec(c+d*x))^(2/3)*(2^(1/3)-(1+sec(c+d*x))^(1/3)*(1+sqrt(3))))-15/7*2^(1/3)*3^(1/4)*sqrt(cos(arccos((2^(1/3)-(1+sec(c+d*x))^(1/3)*(1-sqrt(3)))/(2^(1/3)-(1+sec(c+d*x))^(1/3)*(1+sqrt(3)))))^2)/cos(arccos((2^(1/3)-(1+sec(c+d*x))^(1/3)*(1-sqrt(3)))/(2^(1/3)-(1+sec(c+d*x))^(1/3)*(1+sqrt(3)))))*EllipticE(sin(arccos((2^(1/3)-(1+sec(c+d*x))^(1/3)*(1-sqrt(3)))/(2^(1/3)-(1+sec(c+d*x))^(1/3)*(1+sqrt(3))))),sqrt(1/4*(2+sqrt(3))))*(1+sec(c+d*x))^(1/3)*(2^(1/3)-(1+sec(c+d*x))^(1/3))*sqrt((2^(2/3)+2^(1/3)*(1+sec(c+d*x))^(1/3)+(1+sec(c+d*x))^(2/3))/(2^(1/3)-(1+sec(c+d*x))^(1/3)*(1+sqrt(3)))^2)*tan(c+d*x)/(a*d*(1-sec(c+d*x))*(a+a*sec(c+d*x))^(2/3)*sqrt(-(1+sec(c+d*x))^(1/3)*(2^(1/3)-(1+sec(c+d*x))^(1/3))/(2^(1/3)-(1+sec(c+d*x))^(1/3)*(1+sqrt(3)))^2))-5/7*3^(3/4)*sqrt(cos(arccos((2^(1/3)-(1+sec(c+d*x))^(1/3)*(1-sqrt(3)))/(2^(1/3)-(1+sec(c+d*x))^(1/3)*(1+sqrt(3)))))^2)/cos(arccos((2^(1/3)-(1+sec(c+d*x))^(1/3)*(1-sqrt(3)))/(2^(1/3)-(1+sec(c+d*x))^(1/3)*(1+sqrt(3)))))*EllipticF(sin(arccos((2^(1/3)-(1+sec(c+d*x))^(1/3)*(1-sqrt(3)))/(2^(1/3)-(1+sec(c+d*x))^(1/3)*(1+sqrt(3))))),sqrt(1/4*(2+sqrt(3))))*(1+sec(c+d*x))^(1/3)*(2^(1/3)-(1+sec(c+d*x))^(1/3))*(1-sqrt(3))*sqrt((2^(2/3)+2^(1/3)*(1+sec(c+d*x))^(1/3)+(1+sec(c+d*x))^(2/3))/(2^(1/3)-(1+sec(c+d*x))^(1/3)*(1+sqrt(3)))^2)*tan(c+d*x)/(2^(2/3)*a*d*(1-sec(c+d*x))*(a+a*sec(c+d*x))^(2/3)*sqrt(-(1+sec(c+d*x))^(1/3)*(2^(1/3)-(1+sec(c+d*x))^(1/3))/(2^(1/3)-(1+sec(c+d*x))^(1/3)*(1+sqrt(3)))^2))],
[sec(c+d*x)/(a+a*sec(c+d*x))^(5/3),x,8,6/7*tan(c+d*x)/(a*d*(a+a*sec(c+d*x))^(2/3))+3/7*tan(c+d*x)/(a*d*(1+sec(c+d*x))*(a+a*sec(c+d*x))^(2/3))+6/7*(1+sec(c+d*x))^(1/3)*(1+sqrt(3))*tan(c+d*x)/(a*d*(a+a*sec(c+d*x))^(2/3)*(2^(1/3)-(1+sec(c+d*x))^(1/3)*(1+sqrt(3))))-6/7*2^(1/3)*3^(1/4)*sqrt(cos(arccos((2^(1/3)-(1+sec(c+d*x))^(1/3)*(1-sqrt(3)))/(2^(1/3)-(1+sec(c+d*x))^(1/3)*(1+sqrt(3)))))^2)/cos(arccos((2^(1/3)-(1+sec(c+d*x))^(1/3)*(1-sqrt(3)))/(2^(1/3)-(1+sec(c+d*x))^(1/3)*(1+sqrt(3)))))*EllipticE(sin(arccos((2^(1/3)-(1+sec(c+d*x))^(1/3)*(1-sqrt(3)))/(2^(1/3)-(1+sec(c+d*x))^(1/3)*(1+sqrt(3))))),sqrt(1/4*(2+sqrt(3))))*(1+sec(c+d*x))^(1/3)*(2^(1/3)-(1+sec(c+d*x))^(1/3))*sqrt((2^(2/3)+2^(1/3)*(1+sec(c+d*x))^(1/3)+(1+sec(c+d*x))^(2/3))/(2^(1/3)-(1+sec(c+d*x))^(1/3)*(1+sqrt(3)))^2)*tan(c+d*x)/(a*d*(1-sec(c+d*x))*(a+a*sec(c+d*x))^(2/3)*sqrt(-(1+sec(c+d*x))^(1/3)*(2^(1/3)-(1+sec(c+d*x))^(1/3))/(2^(1/3)-(1+sec(c+d*x))^(1/3)*(1+sqrt(3)))^2))-1/7*2^(1/3)*3^(3/4)*sqrt(cos(arccos((2^(1/3)-(1+sec(c+d*x))^(1/3)*(1-sqrt(3)))/(2^(1/3)-(1+sec(c+d*x))^(1/3)*(1+sqrt(3)))))^2)/cos(arccos((2^(1/3)-(1+sec(c+d*x))^(1/3)*(1-sqrt(3)))/(2^(1/3)-(1+sec(c+d*x))^(1/3)*(1+sqrt(3)))))*EllipticF(sin(arccos((2^(1/3)-(1+sec(c+d*x))^(1/3)*(1-sqrt(3)))/(2^(1/3)-(1+sec(c+d*x))^(1/3)*(1+sqrt(3))))),sqrt(1/4*(2+sqrt(3))))*(1+sec(c+d*x))^(1/3)*(2^(1/3)-(1+sec(c+d*x))^(1/3))*(1-sqrt(3))*sqrt((2^(2/3)+2^(1/3)*(1+sec(c+d*x))^(1/3)+(1+sec(c+d*x))^(2/3))/(2^(1/3)-(1+sec(c+d*x))^(1/3)*(1+sqrt(3)))^2)*tan(c+d*x)/(a*d*(1-sec(c+d*x))*(a+a*sec(c+d*x))^(2/3)*sqrt(-(1+sec(c+d*x))^(1/3)*(2^(1/3)-(1+sec(c+d*x))^(1/3))/(2^(1/3)-(1+sec(c+d*x))^(1/3)*(1+sqrt(3)))^2))],
[1/(a+a*sec(c+d*x))^(5/3),x,3,-3/7*AppellF1(-7/6,1/2,1,-1/6,1/2*(1+sec(c+d*x)),1+sec(c+d*x))*sqrt(2)*tan(c+d*x)/(a*d*(1+sec(c+d*x))*(a+a*sec(c+d*x))^(2/3)*sqrt(1-sec(c+d*x)))],
[cos(c+d*x)/(a+a*sec(c+d*x))^(5/3),x,3,3/7*AppellF1(-7/6,1/2,2,-1/6,1/2*(1+sec(c+d*x)),1+sec(c+d*x))*sqrt(2)*tan(c+d*x)/(a*d*(1+sec(c+d*x))*(a+a*sec(c+d*x))^(2/3)*sqrt(1-sec(c+d*x)))],

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

# n>0
[sec(c+d*x)^(5/2)*(a+a*sec(c+d*x)),x,8,2/3*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],
[sec(c+d*x)^(3/2)*(a+a*sec(c+d*x)),x,7,2/3*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],
[sec(c+d*x)^(1/2)*(a+a*sec(c+d*x)),x,6,2*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*sec(c+d*x))/sec(c+d*x)^(1/2),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))*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*sec(c+d*x))/sec(c+d*x)^(3/2),x,6,2/3*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*sec(c+d*x))/sec(c+d*x)^(5/2),x,7,2/5*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*sec(c+d*x))/sec(c+d*x)^(7/2),x,8,2/7*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],
[sec(c+d*x)^(5/2)*(a+a*sec(c+d*x))^2,x,9,8/7*a^2*sec(c+d*x)^(3/2)*sin(c+d*x)/d+4/5*a^2*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^2*sin(c+d*x)*sqrt(sec(c+d*x))/d-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],
[sec(c+d*x)^(3/2)*(a+a*sec(c+d*x))^2,x,8,4/3*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],
[sec(c+d*x)^(1/2)*(a+a*sec(c+d*x))^2,x,7,2/3*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*sec(c+d*x))^2/sec(c+d*x)^(1/2),x,4,2*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*sec(c+d*x))^2/sec(c+d*x)^(3/2),x,6,2/3*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*sec(c+d*x))^2/sec(c+d*x)^(5/2),x,7,2/5*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*sec(c+d*x))^2/sec(c+d*x)^(7/2),x,8,2/7*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],
[sec(c+d*x)^(3/2)*(a+a*sec(c+d*x))^3,x,16,52/21*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],
[sec(c+d*x)^(1/2)*(a+a*sec(c+d*x))^3,x,14,2*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*sec(c+d*x))^3/sec(c+d*x)^(1/2),x,12,2/3*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*sec(c+d*x))^3/sec(c+d*x)^(3/2),x,12,2/3*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*sec(c+d*x))^3/sec(c+d*x)^(5/2),x,12,2/5*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*sec(c+d*x))^3/sec(c+d*x)^(7/2),x,14,2/7*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*sec(c+d*x))^3/sec(c+d*x)^(9/2),x,16,2/9*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],
[sec(c+d*x)^(3/2)*(a+a*sec(c+d*x))^4,x,21,32/7*a^4*sec(c+d*x)^(3/2)*sin(c+d*x)/d+122/45*a^4*sec(c+d*x)^(5/2)*sin(c+d*x)/d+8/7*a^4*sec(c+d*x)^(7/2)*sin(c+d*x)/d+2/9*a^4*sec(c+d*x)^(9/2)*sin(c+d*x)/d+152/15*a^4*sin(c+d*x)*sqrt(sec(c+d*x))/d-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],
[sec(c+d*x)^(1/2)*(a+a*sec(c+d*x))^4,x,18,94/21*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*sec(c+d*x))^4/sec(c+d*x)^(1/2),x,16,8/3*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*sec(c+d*x))^4/sec(c+d*x)^(3/2),x,15,2/3*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*sec(c+d*x))^4/sec(c+d*x)^(5/2),x,15,2/5*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*sec(c+d*x))^4/sec(c+d*x)^(7/2),x,16,2/7*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*sec(c+d*x))^4/sec(c+d*x)^(9/2),x,18,2/9*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],
[(a+a*sec(c+d*x))^4/sec(c+d*x)^(11/2),x,21,2/11*a^4*sin(c+d*x)/(d*sec(c+d*x)^(9/2))+8/9*a^4*sin(c+d*x)/(d*sec(c+d*x)^(7/2))+150/77*a^4*sin(c+d*x)/(d*sec(c+d*x)^(5/2))+128/45*a^4*sin(c+d*x)/(d*sec(c+d*x)^(3/2))+904/231*a^4*sin(c+d*x)/(d*sqrt(sec(c+d*x)))+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))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/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))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/d],

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

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

# n>0
[sec(c+d*x)^(5/2)*sqrt(a+a*sec(c+d*x)),x,4,3/4*arcsinh(sqrt(a)*tan(c+d*x)/sqrt(a+a*sec(c+d*x)))*sqrt(a)/d+3/4*a*sec(c+d*x)^(3/2)*sin(c+d*x)/(d*sqrt(a+a*sec(c+d*x)))+1/2*a*sec(c+d*x)^(5/2)*sin(c+d*x)/(d*sqrt(a+a*sec(c+d*x)))],
[sec(c+d*x)^(3/2)*sqrt(a+a*sec(c+d*x)),x,3,arcsinh(sqrt(a)*tan(c+d*x)/sqrt(a+a*sec(c+d*x)))*sqrt(a)/d+a*sec(c+d*x)^(3/2)*sin(c+d*x)/(d*sqrt(a+a*sec(c+d*x)))],
[sec(c+d*x)^(1/2)*sqrt(a+a*sec(c+d*x)),x,2,2*arcsinh(sqrt(a)*tan(c+d*x)/sqrt(a+a*sec(c+d*x)))*sqrt(a)/d],
[sqrt(a+a*sec(c+d*x))/sec(c+d*x)^(1/2),x,1,2*a*sin(c+d*x)*sqrt(sec(c+d*x))/(d*sqrt(a+a*sec(c+d*x)))],
[sqrt(a+a*sec(c+d*x))/sec(c+d*x)^(3/2),x,2,2/3*a*sin(c+d*x)/(d*sqrt(sec(c+d*x))*sqrt(a+a*sec(c+d*x)))+4/3*a*sin(c+d*x)*sqrt(sec(c+d*x))/(d*sqrt(a+a*sec(c+d*x)))],
[sqrt(a+a*sec(c+d*x))/sec(c+d*x)^(5/2),x,3,2/5*a*sin(c+d*x)/(d*sec(c+d*x)^(3/2)*sqrt(a+a*sec(c+d*x)))+8/15*a*sin(c+d*x)/(d*sqrt(sec(c+d*x))*sqrt(a+a*sec(c+d*x)))+16/15*a*sin(c+d*x)*sqrt(sec(c+d*x))/(d*sqrt(a+a*sec(c+d*x)))],
[sqrt(a+a*sec(c+d*x))/sec(c+d*x)^(7/2),x,4,2/7*a*sin(c+d*x)/(d*sec(c+d*x)^(5/2)*sqrt(a+a*sec(c+d*x)))+12/35*a*sin(c+d*x)/(d*sec(c+d*x)^(3/2)*sqrt(a+a*sec(c+d*x)))+16/35*a*sin(c+d*x)/(d*sqrt(sec(c+d*x))*sqrt(a+a*sec(c+d*x)))+32/35*a*sin(c+d*x)*sqrt(sec(c+d*x))/(d*sqrt(a+a*sec(c+d*x)))],
[sec(c+d*x)^(5/2)*(a+a*sec(c+d*x))^(3/2),x,6,11/8*a^(3/2)*arcsinh(sqrt(a)*tan(c+d*x)/sqrt(a+a*sec(c+d*x)))/d+11/8*a^2*sec(c+d*x)^(3/2)*sin(c+d*x)/(d*sqrt(a+a*sec(c+d*x)))+11/12*a^2*sec(c+d*x)^(5/2)*sin(c+d*x)/(d*sqrt(a+a*sec(c+d*x)))+1/3*a^2*sec(c+d*x)^(7/2)*sin(c+d*x)/(d*sqrt(a+a*sec(c+d*x)))],
[sec(c+d*x)^(3/2)*(a+a*sec(c+d*x))^(3/2),x,5,7/4*a^(3/2)*arcsinh(sqrt(a)*tan(c+d*x)/sqrt(a+a*sec(c+d*x)))/d+7/4*a^2*sec(c+d*x)^(3/2)*sin(c+d*x)/(d*sqrt(a+a*sec(c+d*x)))+1/2*a^2*sec(c+d*x)^(5/2)*sin(c+d*x)/(d*sqrt(a+a*sec(c+d*x)))],
[sec(c+d*x)^(1/2)*(a+a*sec(c+d*x))^(3/2),x,4,3*a^(3/2)*arcsinh(sqrt(a)*tan(c+d*x)/sqrt(a+a*sec(c+d*x)))/d+a^2*sec(c+d*x)^(3/2)*sin(c+d*x)/(d*sqrt(a+a*sec(c+d*x)))],
[(a+a*sec(c+d*x))^(3/2)/sec(c+d*x)^(1/2),x,4,2*a^(3/2)*arcsinh(sqrt(a)*tan(c+d*x)/sqrt(a+a*sec(c+d*x)))/d+2*a^2*sin(c+d*x)*sqrt(sec(c+d*x))/(d*sqrt(a+a*sec(c+d*x)))],
[(a+a*sec(c+d*x))^(3/2)/sec(c+d*x)^(3/2),x,2,8/3*a^2*sin(c+d*x)*sqrt(sec(c+d*x))/(d*sqrt(a+a*sec(c+d*x)))+2/3*a*sin(c+d*x)*sqrt(a+a*sec(c+d*x))/(d*sqrt(sec(c+d*x)))],
[(a+a*sec(c+d*x))^(3/2)/sec(c+d*x)^(5/2),x,3,2/5*(a+a*sec(c+d*x))^(3/2)*sin(c+d*x)/(d*sec(c+d*x)^(3/2))+8/5*a^2*sin(c+d*x)*sqrt(sec(c+d*x))/(d*sqrt(a+a*sec(c+d*x)))+2/5*a*sin(c+d*x)*sqrt(a+a*sec(c+d*x))/(d*sqrt(sec(c+d*x)))],
[(a+a*sec(c+d*x))^(3/2)/sec(c+d*x)^(7/2),x,5,2/7*a^2*sin(c+d*x)/(d*sec(c+d*x)^(5/2)*sqrt(a+a*sec(c+d*x)))+26/35*a^2*sin(c+d*x)/(d*sec(c+d*x)^(3/2)*sqrt(a+a*sec(c+d*x)))+104/105*a^2*sin(c+d*x)/(d*sqrt(sec(c+d*x))*sqrt(a+a*sec(c+d*x)))+208/105*a^2*sin(c+d*x)*sqrt(sec(c+d*x))/(d*sqrt(a+a*sec(c+d*x)))],
[(a+a*sec(c+d*x))^(3/2)/sec(c+d*x)^(9/2),x,6,2/9*a^2*sin(c+d*x)/(d*sec(c+d*x)^(7/2)*sqrt(a+a*sec(c+d*x)))+34/63*a^2*sin(c+d*x)/(d*sec(c+d*x)^(5/2)*sqrt(a+a*sec(c+d*x)))+68/105*a^2*sin(c+d*x)/(d*sec(c+d*x)^(3/2)*sqrt(a+a*sec(c+d*x)))+272/315*a^2*sin(c+d*x)/(d*sqrt(sec(c+d*x))*sqrt(a+a*sec(c+d*x)))+544/315*a^2*sin(c+d*x)*sqrt(sec(c+d*x))/(d*sqrt(a+a*sec(c+d*x)))],
[sec(c+d*x)^(5/2)*(a+a*sec(c+d*x))^(5/2),x,6,163/64*a^(5/2)*arcsinh(sqrt(a)*tan(c+d*x)/sqrt(a+a*sec(c+d*x)))/d+163/64*a^3*sec(c+d*x)^(3/2)*sin(c+d*x)/(d*sqrt(a+a*sec(c+d*x)))+163/96*a^3*sec(c+d*x)^(5/2)*sin(c+d*x)/(d*sqrt(a+a*sec(c+d*x)))+17/24*a^3*sec(c+d*x)^(7/2)*sin(c+d*x)/(d*sqrt(a+a*sec(c+d*x)))+1/4*a^2*sec(c+d*x)^(7/2)*sin(c+d*x)*sqrt(a+a*sec(c+d*x))/d],
[sec(c+d*x)^(3/2)*(a+a*sec(c+d*x))^(5/2),x,5,25/8*a^(5/2)*arcsinh(sqrt(a)*tan(c+d*x)/sqrt(a+a*sec(c+d*x)))/d+25/8*a^3*sec(c+d*x)^(3/2)*sin(c+d*x)/(d*sqrt(a+a*sec(c+d*x)))+13/12*a^3*sec(c+d*x)^(5/2)*sin(c+d*x)/(d*sqrt(a+a*sec(c+d*x)))+1/3*a^2*sec(c+d*x)^(5/2)*sin(c+d*x)*sqrt(a+a*sec(c+d*x))/d],
[sec(c+d*x)^(1/2)*(a+a*sec(c+d*x))^(5/2),x,4,19/4*a^(5/2)*arcsinh(sqrt(a)*tan(c+d*x)/sqrt(a+a*sec(c+d*x)))/d+9/4*a^3*sec(c+d*x)^(3/2)*sin(c+d*x)/(d*sqrt(a+a*sec(c+d*x)))+1/2*a^2*sec(c+d*x)^(3/2)*sin(c+d*x)*sqrt(a+a*sec(c+d*x))/d],
[(a+a*sec(c+d*x))^(5/2)/sec(c+d*x)^(1/2),x,4,5*a^(5/2)*arcsinh(sqrt(a)*tan(c+d*x)/sqrt(a+a*sec(c+d*x)))/d+a^3*sin(c+d*x)*sqrt(sec(c+d*x))/(d*sqrt(a+a*sec(c+d*x)))+a^2*sin(c+d*x)*sqrt(sec(c+d*x))*sqrt(a+a*sec(c+d*x))/d],
[(a+a*sec(c+d*x))^(5/2)/sec(c+d*x)^(3/2),x,4,2*a^(5/2)*arcsinh(sqrt(a)*tan(c+d*x)/sqrt(a+a*sec(c+d*x)))/d+14/3*a^3*sin(c+d*x)*sqrt(sec(c+d*x))/(d*sqrt(a+a*sec(c+d*x)))+2/3*a^2*sin(c+d*x)*sqrt(a+a*sec(c+d*x))/(d*sqrt(sec(c+d*x)))],
[(a+a*sec(c+d*x))^(5/2)/sec(c+d*x)^(5/2),x,3,2/5*a*(a+a*sec(c+d*x))^(3/2)*sin(c+d*x)/(d*sec(c+d*x)^(3/2))+64/15*a^3*sin(c+d*x)*sqrt(sec(c+d*x))/(d*sqrt(a+a*sec(c+d*x)))+16/15*a^2*sin(c+d*x)*sqrt(a+a*sec(c+d*x))/(d*sqrt(sec(c+d*x)))],
[(a+a*sec(c+d*x))^(5/2)/sec(c+d*x)^(7/2),x,4,2/7*a*(a+a*sec(c+d*x))^(3/2)*sin(c+d*x)/(d*sec(c+d*x)^(3/2))+2/7*(a+a*sec(c+d*x))^(5/2)*sin(c+d*x)/(d*sec(c+d*x)^(5/2))+64/21*a^3*sin(c+d*x)*sqrt(sec(c+d*x))/(d*sqrt(a+a*sec(c+d*x)))+16/21*a^2*sin(c+d*x)*sqrt(a+a*sec(c+d*x))/(d*sqrt(sec(c+d*x)))],
[(a+a*sec(c+d*x))^(5/2)/sec(c+d*x)^(9/2),x,5,38/63*a^3*sin(c+d*x)/(d*sec(c+d*x)^(5/2)*sqrt(a+a*sec(c+d*x)))+146/105*a^3*sin(c+d*x)/(d*sec(c+d*x)^(3/2)*sqrt(a+a*sec(c+d*x)))+584/315*a^3*sin(c+d*x)/(d*sqrt(sec(c+d*x))*sqrt(a+a*sec(c+d*x)))+1168/315*a^3*sin(c+d*x)*sqrt(sec(c+d*x))/(d*sqrt(a+a*sec(c+d*x)))+2/9*a^2*sin(c+d*x)*sqrt(a+a*sec(c+d*x))/(d*sec(c+d*x)^(7/2))],
[(a+a*sec(c+d*x))^(5/2)/sec(c+d*x)^(11/2),x,6,46/99*a^3*sin(c+d*x)/(d*sec(c+d*x)^(7/2)*sqrt(a+a*sec(c+d*x)))+710/693*a^3*sin(c+d*x)/(d*sec(c+d*x)^(5/2)*sqrt(a+a*sec(c+d*x)))+284/231*a^3*sin(c+d*x)/(d*sec(c+d*x)^(3/2)*sqrt(a+a*sec(c+d*x)))+1136/693*a^3*sin(c+d*x)/(d*sqrt(sec(c+d*x))*sqrt(a+a*sec(c+d*x)))+2272/693*a^3*sin(c+d*x)*sqrt(sec(c+d*x))/(d*sqrt(a+a*sec(c+d*x)))+2/11*a^2*sin(c+d*x)*sqrt(a+a*sec(c+d*x))/(d*sec(c+d*x)^(9/2))],
[(a+a*sec(c+d*x))^(3/2)/sec(c+d*x)^(1/4),x,2,4*a^2*sec(c+d*x)^(3/4)*sin(c+d*x)/(d*sqrt(a+a*sec(c+d*x)))],
[sqrt(sec(e+f*x))*sqrt(a+a*sec(e+f*x)),x,2,2*arcsinh(sqrt(a)*tan(e+f*x)/sqrt(a+a*sec(e+f*x)))*sqrt(a)/f],
[sqrt(-sec(e+f*x))*sqrt(a-a*sec(e+f*x)),x,2,2*arcsinh(sqrt(a)*tan(e+f*x)/sqrt(a-a*sec(e+f*x)))*sqrt(a)/f],

# n<0
[sec(c+d*x)^(5/2)/sqrt(a+a*sec(c+d*x)),x,6,-arcsinh(sqrt(a)*tan(c+d*x)/sqrt(a+a*sec(c+d*x)))/(d*sqrt(a))+arctanh(sin(c+d*x)*sqrt(a)*sqrt(sec(c+d*x))/(sqrt(2)*sqrt(a+a*sec(c+d*x))))*sqrt(2)/(d*sqrt(a))+sec(c+d*x)^(3/2)*sin(c+d*x)/(d*sqrt(a+a*sec(c+d*x)))],
[sec(c+d*x)^(3/2)/sqrt(a+a*sec(c+d*x)),x,5,2*arcsinh(sqrt(a)*tan(c+d*x)/sqrt(a+a*sec(c+d*x)))/(d*sqrt(a))-arctanh(sin(c+d*x)*sqrt(a)*sqrt(sec(c+d*x))/(sqrt(2)*sqrt(a+a*sec(c+d*x))))*sqrt(2)/(d*sqrt(a))],
[sqrt(sec(c+d*x))/sqrt(a+a*sec(c+d*x)),x,2,arctanh(sin(c+d*x)*sqrt(a)*sqrt(sec(c+d*x))/(sqrt(2)*sqrt(a+a*sec(c+d*x))))*sqrt(2)/(d*sqrt(a))],
[1/(sqrt(sec(c+d*x))*sqrt(a+a*sec(c+d*x))),x,3,-arctanh(sin(c+d*x)*sqrt(a)*sqrt(sec(c+d*x))/(sqrt(2)*sqrt(a+a*sec(c+d*x))))*sqrt(2)/(d*sqrt(a))+2*sin(c+d*x)*sqrt(sec(c+d*x))/(d*sqrt(a+a*sec(c+d*x)))],
[1/(sec(c+d*x)^(3/2)*sqrt(a+a*sec(c+d*x))),x,4,arctanh(sin(c+d*x)*sqrt(a)*sqrt(sec(c+d*x))/(sqrt(2)*sqrt(a+a*sec(c+d*x))))*sqrt(2)/(d*sqrt(a))+2/3*sin(c+d*x)/(d*sqrt(sec(c+d*x))*sqrt(a+a*sec(c+d*x)))-2/3*sin(c+d*x)*sqrt(sec(c+d*x))/(d*sqrt(a+a*sec(c+d*x)))],
[1/(sec(c+d*x)^(5/2)*sqrt(a+a*sec(c+d*x))),x,5,-arctanh(sin(c+d*x)*sqrt(a)*sqrt(sec(c+d*x))/(sqrt(2)*sqrt(a+a*sec(c+d*x))))*sqrt(2)/(d*sqrt(a))+2/5*sin(c+d*x)/(d*sec(c+d*x)^(3/2)*sqrt(a+a*sec(c+d*x)))-2/15*sin(c+d*x)/(d*sqrt(sec(c+d*x))*sqrt(a+a*sec(c+d*x)))+26/15*sin(c+d*x)*sqrt(sec(c+d*x))/(d*sqrt(a+a*sec(c+d*x)))],
[sec(c+d*x)^(7/2)/(a+a*sec(c+d*x))^(3/2),x,7,-3*arcsinh(sqrt(a)*tan(c+d*x)/sqrt(a+a*sec(c+d*x)))/(a^(3/2)*d)-1/2*sec(c+d*x)^(5/2)*sin(c+d*x)/(d*(a+a*sec(c+d*x))^(3/2))+9/2*arctanh(sin(c+d*x)*sqrt(a)*sqrt(sec(c+d*x))/(sqrt(2)*sqrt(a+a*sec(c+d*x))))/(a^(3/2)*d*sqrt(2))+3/2*sec(c+d*x)^(3/2)*sin(c+d*x)/(a*d*sqrt(a+a*sec(c+d*x)))],
[sec(c+d*x)^(5/2)/(a+a*sec(c+d*x))^(3/2),x,6,2*arcsinh(sqrt(a)*tan(c+d*x)/sqrt(a+a*sec(c+d*x)))/(a^(3/2)*d)-1/2*sec(c+d*x)^(3/2)*sin(c+d*x)/(d*(a+a*sec(c+d*x))^(3/2))-5/2*arctanh(sin(c+d*x)*sqrt(a)*sqrt(sec(c+d*x))/(sqrt(2)*sqrt(a+a*sec(c+d*x))))/(a^(3/2)*d*sqrt(2))],
[sec(c+d*x)^(3/2)/(a+a*sec(c+d*x))^(3/2),x,3,1/2*sec(c+d*x)^(3/2)*sin(c+d*x)/(d*(a+a*sec(c+d*x))^(3/2))+1/2*arctanh(sin(c+d*x)*sqrt(a)*sqrt(sec(c+d*x))/(sqrt(2)*sqrt(a+a*sec(c+d*x))))/(a^(3/2)*d*sqrt(2))],
[sqrt(sec(c+d*x))/(a+a*sec(c+d*x))^(3/2),x,3,-1/2*sec(c+d*x)^(3/2)*sin(c+d*x)/(d*(a+a*sec(c+d*x))^(3/2))+3/2*arctanh(sin(c+d*x)*sqrt(a)*sqrt(sec(c+d*x))/(sqrt(2)*sqrt(a+a*sec(c+d*x))))/(a^(3/2)*d*sqrt(2))],
[1/((a+a*sec(c+d*x))^(3/2)*sqrt(sec(c+d*x))),x,4,-7/2*arctanh(sin(c+d*x)*sqrt(a)*sqrt(sec(c+d*x))/(sqrt(2)*sqrt(a+a*sec(c+d*x))))/(a^(3/2)*d*sqrt(2))-1/2*sin(c+d*x)*sqrt(sec(c+d*x))/(d*(a+a*sec(c+d*x))^(3/2))+5/2*sin(c+d*x)*sqrt(sec(c+d*x))/(a*d*sqrt(a+a*sec(c+d*x)))],
[1/(sec(c+d*x)^(3/2)*(a+a*sec(c+d*x))^(3/2)),x,5,11/2*arctanh(sin(c+d*x)*sqrt(a)*sqrt(sec(c+d*x))/(sqrt(2)*sqrt(a+a*sec(c+d*x))))/(a^(3/2)*d*sqrt(2))-1/2*sin(c+d*x)/(d*(a+a*sec(c+d*x))^(3/2)*sqrt(sec(c+d*x)))+7/6*sin(c+d*x)/(a*d*sqrt(sec(c+d*x))*sqrt(a+a*sec(c+d*x)))-19/6*sin(c+d*x)*sqrt(sec(c+d*x))/(a*d*sqrt(a+a*sec(c+d*x)))],
[1/(sec(c+d*x)^(5/2)*(a+a*sec(c+d*x))^(3/2)),x,6,-1/2*sin(c+d*x)/(d*sec(c+d*x)^(3/2)*(a+a*sec(c+d*x))^(3/2))-15/2*arctanh(sin(c+d*x)*sqrt(a)*sqrt(sec(c+d*x))/(sqrt(2)*sqrt(a+a*sec(c+d*x))))/(a^(3/2)*d*sqrt(2))+9/10*sin(c+d*x)/(a*d*sec(c+d*x)^(3/2)*sqrt(a+a*sec(c+d*x)))-13/10*sin(c+d*x)/(a*d*sqrt(sec(c+d*x))*sqrt(a+a*sec(c+d*x)))+49/10*sin(c+d*x)*sqrt(sec(c+d*x))/(a*d*sqrt(a+a*sec(c+d*x)))],
[sec(c+d*x)^(9/2)/(a+a*sec(c+d*x))^(5/2),x,8,-5*arcsinh(sqrt(a)*tan(c+d*x)/sqrt(a+a*sec(c+d*x)))/(a^(5/2)*d)-1/4*sec(c+d*x)^(7/2)*sin(c+d*x)/(d*(a+a*sec(c+d*x))^(5/2))-15/16*sec(c+d*x)^(5/2)*sin(c+d*x)/(a*d*(a+a*sec(c+d*x))^(3/2))+115/16*arctanh(sin(c+d*x)*sqrt(a)*sqrt(sec(c+d*x))/(sqrt(2)*sqrt(a+a*sec(c+d*x))))/(a^(5/2)*d*sqrt(2))+35/16*sec(c+d*x)^(3/2)*sin(c+d*x)/(a^2*d*sqrt(a+a*sec(c+d*x)))],
[sec(c+d*x)^(7/2)/(a+a*sec(c+d*x))^(5/2),x,7,2*arcsinh(sqrt(a)*tan(c+d*x)/sqrt(a+a*sec(c+d*x)))/(a^(5/2)*d)-1/4*sec(c+d*x)^(5/2)*sin(c+d*x)/(d*(a+a*sec(c+d*x))^(5/2))-11/16*sec(c+d*x)^(3/2)*sin(c+d*x)/(a*d*(a+a*sec(c+d*x))^(3/2))-43/16*arctanh(sin(c+d*x)*sqrt(a)*sqrt(sec(c+d*x))/(sqrt(2)*sqrt(a+a*sec(c+d*x))))/(a^(5/2)*d*sqrt(2))],
[sec(c+d*x)^(5/2)/(a+a*sec(c+d*x))^(5/2),x,4,1/4*sec(c+d*x)^(5/2)*sin(c+d*x)/(d*(a+a*sec(c+d*x))^(5/2))+3/16*sec(c+d*x)^(3/2)*sin(c+d*x)/(a*d*(a+a*sec(c+d*x))^(3/2))+3/16*arctanh(sin(c+d*x)*sqrt(a)*sqrt(sec(c+d*x))/(sqrt(2)*sqrt(a+a*sec(c+d*x))))/(a^(5/2)*d*sqrt(2))],
[sec(c+d*x)^(3/2)/(a+a*sec(c+d*x))^(5/2),x,4,-1/4*sec(c+d*x)^(5/2)*sin(c+d*x)/(d*(a+a*sec(c+d*x))^(5/2))+5/16*sec(c+d*x)^(3/2)*sin(c+d*x)/(a*d*(a+a*sec(c+d*x))^(3/2))+5/16*arctanh(sin(c+d*x)*sqrt(a)*sqrt(sec(c+d*x))/(sqrt(2)*sqrt(a+a*sec(c+d*x))))/(a^(5/2)*d*sqrt(2))],
[sqrt(sec(c+d*x))/(a+a*sec(c+d*x))^(5/2),x,4,-1/4*sec(c+d*x)^(3/2)*sin(c+d*x)/(d*(a+a*sec(c+d*x))^(5/2))-9/16*sec(c+d*x)^(3/2)*sin(c+d*x)/(a*d*(a+a*sec(c+d*x))^(3/2))+19/16*arctanh(sin(c+d*x)*sqrt(a)*sqrt(sec(c+d*x))/(sqrt(2)*sqrt(a+a*sec(c+d*x))))/(a^(5/2)*d*sqrt(2))],
[1/((a+a*sec(c+d*x))^(5/2)*sqrt(sec(c+d*x))),x,5,-75/16*arctanh(sin(c+d*x)*sqrt(a)*sqrt(sec(c+d*x))/(sqrt(2)*sqrt(a+a*sec(c+d*x))))/(a^(5/2)*d*sqrt(2))-1/4*sin(c+d*x)*sqrt(sec(c+d*x))/(d*(a+a*sec(c+d*x))^(5/2))-13/16*sin(c+d*x)*sqrt(sec(c+d*x))/(a*d*(a+a*sec(c+d*x))^(3/2))+49/16*sin(c+d*x)*sqrt(sec(c+d*x))/(a^2*d*sqrt(a+a*sec(c+d*x)))],
[1/(sec(c+d*x)^(3/2)*(a+a*sec(c+d*x))^(5/2)),x,6,163/16*arctanh(sin(c+d*x)*sqrt(a)*sqrt(sec(c+d*x))/(sqrt(2)*sqrt(a+a*sec(c+d*x))))/(a^(5/2)*d*sqrt(2))-1/4*sin(c+d*x)/(d*(a+a*sec(c+d*x))^(5/2)*sqrt(sec(c+d*x)))-17/16*sin(c+d*x)/(a*d*(a+a*sec(c+d*x))^(3/2)*sqrt(sec(c+d*x)))+95/48*sin(c+d*x)/(a^2*d*sqrt(sec(c+d*x))*sqrt(a+a*sec(c+d*x)))-299/48*sin(c+d*x)*sqrt(sec(c+d*x))/(a^2*d*sqrt(a+a*sec(c+d*x)))],
[sec(c+d*x)^(7/2)/sqrt(1+sec(c+d*x)),x,7,7/4*arcsinh(tan(c+d*x)/sqrt(1+sec(c+d*x)))/d-arcsinh(tan(c+d*x)/(1+sec(c+d*x)))*sqrt(2)/d-1/4*sec(c+d*x)^(3/2)*sin(c+d*x)/(d*sqrt(1+sec(c+d*x)))+1/2*sec(c+d*x)^(5/2)*sin(c+d*x)/(d*sqrt(1+sec(c+d*x)))],
[sec(c+d*x)^(5/2)/sqrt(1+sec(c+d*x)),x,6,-arcsinh(tan(c+d*x)/sqrt(1+sec(c+d*x)))/d+arcsinh(tan(c+d*x)/(1+sec(c+d*x)))*sqrt(2)/d+sec(c+d*x)^(3/2)*sin(c+d*x)/(d*sqrt(1+sec(c+d*x)))],
[sec(c+d*x)^(3/2)/sqrt(1+sec(c+d*x)),x,5,2*arcsinh(tan(c+d*x)/sqrt(1+sec(c+d*x)))/d-arcsinh(tan(c+d*x)/(1+sec(c+d*x)))*sqrt(2)/d],
[sqrt(sec(c+d*x))/sqrt(1+sec(c+d*x)),x,2,arcsinh(tan(c+d*x)/(1+sec(c+d*x)))*sqrt(2)/d],
[1/(sqrt(sec(c+d*x))*sqrt(1+sec(c+d*x))),x,3,-arcsinh(tan(c+d*x)/(1+sec(c+d*x)))*sqrt(2)/d+2*sin(c+d*x)*sqrt(sec(c+d*x))/(d*sqrt(1+sec(c+d*x)))],
[1/(sec(c+d*x)^(3/2)*sqrt(1+sec(c+d*x))),x,4,arcsinh(tan(c+d*x)/(1+sec(c+d*x)))*sqrt(2)/d+2/3*sin(c+d*x)/(d*sqrt(sec(c+d*x))*sqrt(1+sec(c+d*x)))-2/3*sin(c+d*x)*sqrt(sec(c+d*x))/(d*sqrt(1+sec(c+d*x)))],
[1/(sec(c+d*x)^(5/2)*sqrt(1+sec(c+d*x))),x,5,-arcsinh(tan(c+d*x)/(1+sec(c+d*x)))*sqrt(2)/d+2/5*sin(c+d*x)/(d*sec(c+d*x)^(3/2)*sqrt(1+sec(c+d*x)))-2/15*sin(c+d*x)/(d*sqrt(sec(c+d*x))*sqrt(1+sec(c+d*x)))+26/15*sin(c+d*x)*sqrt(sec(c+d*x))/(d*sqrt(1+sec(c+d*x)))],

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

# n>0
[(e*sec(c+d*x))^(4/3)*sqrt(a+a*sec(c+d*x)),x,4,6/5*a*e*(e*sec(c+d*x))^(1/3)*tan(c+d*x)/(d*sqrt(a+a*sec(c+d*x)))+4/5*3^(3/4)*a^2*e*EllipticF((-(e*sec(c+d*x))^(1/3)+e^(1/3)*(1-sqrt(3)))/(-(e*sec(c+d*x))^(1/3)+e^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*(e^(1/3)-(e*sec(c+d*x))^(1/3))*sqrt(2+sqrt(3))*sqrt((e^(2/3)+e^(1/3)*(e*sec(c+d*x))^(1/3)+(e*sec(c+d*x))^(2/3))/(-(e*sec(c+d*x))^(1/3)+e^(1/3)*(1+sqrt(3)))^2)*tan(c+d*x)/(d*(a-a*sec(c+d*x))*sqrt(a+a*sec(c+d*x))*sqrt(e^(1/3)*(e^(1/3)-(e*sec(c+d*x))^(1/3))/(-(e*sec(c+d*x))^(1/3)+e^(1/3)*(1+sqrt(3)))^2))],
[(e*sec(c+d*x))^(1/3)*sqrt(a+a*sec(c+d*x)),x,3,2*3^(3/4)*a^2*EllipticF((-(e*sec(c+d*x))^(1/3)+e^(1/3)*(1-sqrt(3)))/(-(e*sec(c+d*x))^(1/3)+e^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*(e^(1/3)-(e*sec(c+d*x))^(1/3))*sqrt(2+sqrt(3))*sqrt((e^(2/3)+e^(1/3)*(e*sec(c+d*x))^(1/3)+(e*sec(c+d*x))^(2/3))/(-(e*sec(c+d*x))^(1/3)+e^(1/3)*(1+sqrt(3)))^2)*tan(c+d*x)/(d*(a-a*sec(c+d*x))*sqrt(a+a*sec(c+d*x))*sqrt(e^(1/3)*(e^(1/3)-(e*sec(c+d*x))^(1/3))/(-(e*sec(c+d*x))^(1/3)+e^(1/3)*(1+sqrt(3)))^2))],
[sqrt(a+a*sec(c+d*x))/(e*sec(c+d*x))^(2/3),x,4,3/2*a*tan(c+d*x)/(d*(e*sec(c+d*x))^(2/3)*sqrt(a+a*sec(c+d*x)))+1/2*3^(3/4)*a^2*EllipticF((-(e*sec(c+d*x))^(1/3)+e^(1/3)*(1-sqrt(3)))/(-(e*sec(c+d*x))^(1/3)+e^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*(e^(1/3)-(e*sec(c+d*x))^(1/3))*sqrt(2+sqrt(3))*sqrt((e^(2/3)+e^(1/3)*(e*sec(c+d*x))^(1/3)+(e*sec(c+d*x))^(2/3))/(-(e*sec(c+d*x))^(1/3)+e^(1/3)*(1+sqrt(3)))^2)*tan(c+d*x)/(d*e*(a-a*sec(c+d*x))*sqrt(a+a*sec(c+d*x))*sqrt(e^(1/3)*(e^(1/3)-(e*sec(c+d*x))^(1/3))/(-(e*sec(c+d*x))^(1/3)+e^(1/3)*(1+sqrt(3)))^2))],
[(e*sec(c+d*x))^(8/3)*sqrt(a+a*sec(c+d*x)),x,7,60/91*a*e^2*(e*sec(c+d*x))^(2/3)*tan(c+d*x)/(d*sqrt(a+a*sec(c+d*x)))+6/13*a*e*(e*sec(c+d*x))^(5/3)*tan(c+d*x)/(d*sqrt(a+a*sec(c+d*x)))-240/91*a*e^3*tan(c+d*x)/(d*(-(e*sec(c+d*x))^(1/3)+e^(1/3)*(1+sqrt(3)))*sqrt(a+a*sec(c+d*x)))-80/91*3^(3/4)*a^2*e^(7/3)*EllipticF((-(e*sec(c+d*x))^(1/3)+e^(1/3)*(1-sqrt(3)))/(-(e*sec(c+d*x))^(1/3)+e^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*(e^(1/3)-(e*sec(c+d*x))^(1/3))*sqrt(2)*sqrt((e^(2/3)+e^(1/3)*(e*sec(c+d*x))^(1/3)+(e*sec(c+d*x))^(2/3))/(-(e*sec(c+d*x))^(1/3)+e^(1/3)*(1+sqrt(3)))^2)*tan(c+d*x)/(d*(a-a*sec(c+d*x))*sqrt(a+a*sec(c+d*x))*sqrt(e^(1/3)*(e^(1/3)-(e*sec(c+d*x))^(1/3))/(-(e*sec(c+d*x))^(1/3)+e^(1/3)*(1+sqrt(3)))^2))+120/91*3^(1/4)*a^2*e^(7/3)*EllipticE((-(e*sec(c+d*x))^(1/3)+e^(1/3)*(1-sqrt(3)))/(-(e*sec(c+d*x))^(1/3)+e^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*(e^(1/3)-(e*sec(c+d*x))^(1/3))*sqrt(2-sqrt(3))*sqrt((e^(2/3)+e^(1/3)*(e*sec(c+d*x))^(1/3)+(e*sec(c+d*x))^(2/3))/(-(e*sec(c+d*x))^(1/3)+e^(1/3)*(1+sqrt(3)))^2)*tan(c+d*x)/(d*(a-a*sec(c+d*x))*sqrt(a+a*sec(c+d*x))*sqrt(e^(1/3)*(e^(1/3)-(e*sec(c+d*x))^(1/3))/(-(e*sec(c+d*x))^(1/3)+e^(1/3)*(1+sqrt(3)))^2))],
[(e*sec(c+d*x))^(5/3)*sqrt(a+a*sec(c+d*x)),x,6,6/7*a*e*(e*sec(c+d*x))^(2/3)*tan(c+d*x)/(d*sqrt(a+a*sec(c+d*x)))-24/7*a*e^2*tan(c+d*x)/(d*(-(e*sec(c+d*x))^(1/3)+e^(1/3)*(1+sqrt(3)))*sqrt(a+a*sec(c+d*x)))-8/7*3^(3/4)*a^2*e^(4/3)*EllipticF((-(e*sec(c+d*x))^(1/3)+e^(1/3)*(1-sqrt(3)))/(-(e*sec(c+d*x))^(1/3)+e^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*(e^(1/3)-(e*sec(c+d*x))^(1/3))*sqrt(2)*sqrt((e^(2/3)+e^(1/3)*(e*sec(c+d*x))^(1/3)+(e*sec(c+d*x))^(2/3))/(-(e*sec(c+d*x))^(1/3)+e^(1/3)*(1+sqrt(3)))^2)*tan(c+d*x)/(d*(a-a*sec(c+d*x))*sqrt(a+a*sec(c+d*x))*sqrt(e^(1/3)*(e^(1/3)-(e*sec(c+d*x))^(1/3))/(-(e*sec(c+d*x))^(1/3)+e^(1/3)*(1+sqrt(3)))^2))+12/7*3^(1/4)*a^2*e^(4/3)*EllipticE((-(e*sec(c+d*x))^(1/3)+e^(1/3)*(1-sqrt(3)))/(-(e*sec(c+d*x))^(1/3)+e^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*(e^(1/3)-(e*sec(c+d*x))^(1/3))*sqrt(2-sqrt(3))*sqrt((e^(2/3)+e^(1/3)*(e*sec(c+d*x))^(1/3)+(e*sec(c+d*x))^(2/3))/(-(e*sec(c+d*x))^(1/3)+e^(1/3)*(1+sqrt(3)))^2)*tan(c+d*x)/(d*(a-a*sec(c+d*x))*sqrt(a+a*sec(c+d*x))*sqrt(e^(1/3)*(e^(1/3)-(e*sec(c+d*x))^(1/3))/(-(e*sec(c+d*x))^(1/3)+e^(1/3)*(1+sqrt(3)))^2))],
[(e*sec(c+d*x))^(2/3)*sqrt(a+a*sec(c+d*x)),x,5,-6*a*e*tan(c+d*x)/(d*(-(e*sec(c+d*x))^(1/3)+e^(1/3)*(1+sqrt(3)))*sqrt(a+a*sec(c+d*x)))-2*3^(3/4)*a^2*e^(1/3)*EllipticF((-(e*sec(c+d*x))^(1/3)+e^(1/3)*(1-sqrt(3)))/(-(e*sec(c+d*x))^(1/3)+e^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*(e^(1/3)-(e*sec(c+d*x))^(1/3))*sqrt(2)*sqrt((e^(2/3)+e^(1/3)*(e*sec(c+d*x))^(1/3)+(e*sec(c+d*x))^(2/3))/(-(e*sec(c+d*x))^(1/3)+e^(1/3)*(1+sqrt(3)))^2)*tan(c+d*x)/(d*(a-a*sec(c+d*x))*sqrt(a+a*sec(c+d*x))*sqrt(e^(1/3)*(e^(1/3)-(e*sec(c+d*x))^(1/3))/(-(e*sec(c+d*x))^(1/3)+e^(1/3)*(1+sqrt(3)))^2))+3*3^(1/4)*a^2*e^(1/3)*EllipticE((-(e*sec(c+d*x))^(1/3)+e^(1/3)*(1-sqrt(3)))/(-(e*sec(c+d*x))^(1/3)+e^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*(e^(1/3)-(e*sec(c+d*x))^(1/3))*sqrt(2-sqrt(3))*sqrt((e^(2/3)+e^(1/3)*(e*sec(c+d*x))^(1/3)+(e*sec(c+d*x))^(2/3))/(-(e*sec(c+d*x))^(1/3)+e^(1/3)*(1+sqrt(3)))^2)*tan(c+d*x)/(d*(a-a*sec(c+d*x))*sqrt(a+a*sec(c+d*x))*sqrt(e^(1/3)*(e^(1/3)-(e*sec(c+d*x))^(1/3))/(-(e*sec(c+d*x))^(1/3)+e^(1/3)*(1+sqrt(3)))^2))],
[sqrt(a+a*sec(c+d*x))/(e*sec(c+d*x))^(1/3),x,6,3*a*tan(c+d*x)/(d*(e*sec(c+d*x))^(1/3)*sqrt(a+a*sec(c+d*x)))+3*a*tan(c+d*x)/(d*(-(e*sec(c+d*x))^(1/3)+e^(1/3)*(1+sqrt(3)))*sqrt(a+a*sec(c+d*x)))+3^(3/4)*a^2*EllipticF((-(e*sec(c+d*x))^(1/3)+e^(1/3)*(1-sqrt(3)))/(-(e*sec(c+d*x))^(1/3)+e^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*(e^(1/3)-(e*sec(c+d*x))^(1/3))*sqrt(2)*sqrt((e^(2/3)+e^(1/3)*(e*sec(c+d*x))^(1/3)+(e*sec(c+d*x))^(2/3))/(-(e*sec(c+d*x))^(1/3)+e^(1/3)*(1+sqrt(3)))^2)*tan(c+d*x)/(d*e^(2/3)*(a-a*sec(c+d*x))*sqrt(a+a*sec(c+d*x))*sqrt(e^(1/3)*(e^(1/3)-(e*sec(c+d*x))^(1/3))/(-(e*sec(c+d*x))^(1/3)+e^(1/3)*(1+sqrt(3)))^2))-3/2*3^(1/4)*a^2*EllipticE((-(e*sec(c+d*x))^(1/3)+e^(1/3)*(1-sqrt(3)))/(-(e*sec(c+d*x))^(1/3)+e^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*(e^(1/3)-(e*sec(c+d*x))^(1/3))*sqrt(2-sqrt(3))*sqrt((e^(2/3)+e^(1/3)*(e*sec(c+d*x))^(1/3)+(e*sec(c+d*x))^(2/3))/(-(e*sec(c+d*x))^(1/3)+e^(1/3)*(1+sqrt(3)))^2)*tan(c+d*x)/(d*e^(2/3)*(a-a*sec(c+d*x))*sqrt(a+a*sec(c+d*x))*sqrt(e^(1/3)*(e^(1/3)-(e*sec(c+d*x))^(1/3))/(-(e*sec(c+d*x))^(1/3)+e^(1/3)*(1+sqrt(3)))^2))],
[sqrt(a+a*sec(c+d*x))/(e*sec(c+d*x))^(4/3),x,7,3/4*a*tan(c+d*x)/(d*(e*sec(c+d*x))^(4/3)*sqrt(a+a*sec(c+d*x)))+15/8*a*tan(c+d*x)/(d*e*(e*sec(c+d*x))^(1/3)*sqrt(a+a*sec(c+d*x)))+15/8*a*tan(c+d*x)/(d*e*(-(e*sec(c+d*x))^(1/3)+e^(1/3)*(1+sqrt(3)))*sqrt(a+a*sec(c+d*x)))+5/4*3^(3/4)*a^2*EllipticF((-(e*sec(c+d*x))^(1/3)+e^(1/3)*(1-sqrt(3)))/(-(e*sec(c+d*x))^(1/3)+e^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*(e^(1/3)-(e*sec(c+d*x))^(1/3))*sqrt((e^(2/3)+e^(1/3)*(e*sec(c+d*x))^(1/3)+(e*sec(c+d*x))^(2/3))/(-(e*sec(c+d*x))^(1/3)+e^(1/3)*(1+sqrt(3)))^2)*tan(c+d*x)/(d*e^(5/3)*(a-a*sec(c+d*x))*sqrt(2)*sqrt(a+a*sec(c+d*x))*sqrt(e^(1/3)*(e^(1/3)-(e*sec(c+d*x))^(1/3))/(-(e*sec(c+d*x))^(1/3)+e^(1/3)*(1+sqrt(3)))^2))-15/16*3^(1/4)*a^2*EllipticE((-(e*sec(c+d*x))^(1/3)+e^(1/3)*(1-sqrt(3)))/(-(e*sec(c+d*x))^(1/3)+e^(1/3)*(1+sqrt(3))),sqrt(-7-4*sqrt(3)))*(e^(1/3)-(e*sec(c+d*x))^(1/3))*sqrt(2-sqrt(3))*sqrt((e^(2/3)+e^(1/3)*(e*sec(c+d*x))^(1/3)+(e*sec(c+d*x))^(2/3))/(-(e*sec(c+d*x))^(1/3)+e^(1/3)*(1+sqrt(3)))^2)*tan(c+d*x)/(d*e^(5/3)*(a-a*sec(c+d*x))*sqrt(a+a*sec(c+d*x))*sqrt(e^(1/3)*(e^(1/3)-(e*sec(c+d*x))^(1/3))/(-(e*sec(c+d*x))^(1/3)+e^(1/3)*(1+sqrt(3)))^2))],

# n<0
[(e*sec(c+d*x))^(2/3)/sqrt(a+a*sec(c+d*x)),x,4,-3/2*AppellF1(2/3,1/2,1,5/3,sec(c+d*x),-sec(c+d*x))*(e*sec(c+d*x))^(2/3)*tan(c+d*x)/(d*sqrt(1-sec(c+d*x))*sqrt(a+a*sec(c+d*x)))],
[(e*sec(c+d*x))^(1/3)/sqrt(a+a*sec(c+d*x)),x,4,-3*AppellF1(1/3,1/2,1,4/3,sec(c+d*x),-sec(c+d*x))*(e*sec(c+d*x))^(1/3)*tan(c+d*x)/(d*sqrt(1-sec(c+d*x))*sqrt(a+a*sec(c+d*x)))],
[1/((e*sec(c+d*x))^(1/3)*sqrt(a+a*sec(c+d*x))),x,4,3*AppellF1(-1/3,1/2,1,2/3,sec(c+d*x),-sec(c+d*x))*tan(c+d*x)/(d*(e*sec(c+d*x))^(1/3)*sqrt(1-sec(c+d*x))*sqrt(a+a*sec(c+d*x)))],
[1/((e*sec(c+d*x))^(2/3)*sqrt(a+a*sec(c+d*x))),x,4,3/2*AppellF1(-2/3,1/2,1,1/3,sec(c+d*x),-sec(c+d*x))*tan(c+d*x)/(d*(e*sec(c+d*x))^(2/3)*sqrt(1-sec(c+d*x))*sqrt(a+a*sec(c+d*x)))],

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

# n>0
[sec(c+d*x)^(4/3)*(a+a*sec(c+d*x))^(1/3),x,3,2^(5/6)*AppellF1(1/2,-1/3,1/6,3/2,1-sec(c+d*x),1/2*(1-sec(c+d*x)))*(a+a*sec(c+d*x))^(1/3)*tan(c+d*x)/(d*(1+sec(c+d*x))^(5/6))],
[sec(c+d*x)^(4/3)*(a+a*sec(c+d*x))^(2/3),x,3,2*2^(1/6)*AppellF1(1/2,-1/3,-1/6,3/2,1-sec(c+d*x),1/2*(1-sec(c+d*x)))*(a+a*sec(c+d*x))^(2/3)*tan(c+d*x)/(d*(1+sec(c+d*x))^(7/6))],
[sec(c+d*x)^(5/3)*(a+a*sec(c+d*x))^(2/3),x,-3,-3/2*a*sec(c+d*x)^(5/3)*sin(c+d*x)/(d*(a*(1+sec(c+d*x)))^(1/3))+9/4*sec(c+d*x)^(2/3)*(a*(1+sec(c+d*x)))^(2/3)*sin(c+d*x)/d-9/4*(a*(1+sec(c+d*x)))^(2/3)*tan(c+d*x)/(d*(1/(1+cos(c+d*x)))^(1/3)*(1+sec(c+d*x))^(7/3))+1/8*hypergeom([1/4,1/3],[5/4],tan(1/2*(c+d*x))^4)*(cos(c+d*x)*sec(1/2*(c+d*x))^4)^(1/3)*(a*(1+sec(c+d*x)))^(2/3)*tan(c+d*x)/(d*(1/(1+cos(c+d*x)))^(1/3)*(1+sec(c+d*x))^(4/3))-5/8*hypergeom([1/3,3/4],[7/4],tan(1/2*(c+d*x))^4)*(cos(c+d*x)*sec(1/2*(c+d*x))^4)^(1/3)*(a*(1+sec(c+d*x)))^(2/3)*tan(c+d*x)^3/(d*(1/(1+cos(c+d*x)))^(1/3)*(1+sec(c+d*x))^(10/3))],
[(a+a*sec(c+d*x))^(4/3)/sec(c+d*x)^(1/3),x,3,2*2^(5/6)*a*AppellF1(1/2,4/3,-5/6,3/2,1-sec(c+d*x),1/2*(1-sec(c+d*x)))*(a+a*sec(c+d*x))^(1/3)*tan(c+d*x)/(d*(1+sec(c+d*x))^(5/6))],

# n<0

# Integrands of the form (d Sec[e+f x])^n (a+a Sec[e+f x])^m with n symbolic
[sec(e+f*x)^n*(a+a*sec(e+f*x))^4,x,8,a^4*(30+21*n+4*n^2)*sec(e+f*x)^(1+n)*sin(e+f*x)/(f*(6+11*n+6*n^2+n^3))+sec(e+f*x)^(1+n)*(a^2+a^2*sec(e+f*x))^2*sin(e+f*x)/(f*(3+n))+2*(4+n)*sec(e+f*x)^(1+n)*(a^4+a^4*sec(e+f*x))*sin(e+f*x)/(f*(2+n)*(3+n))-a^4*(3+24*n+8*n^2)*hypergeom([1/2,1/2*(1-n)],[1/2*(3-n)],cos(e+f*x)^2)*sec(e+f*x)^(-1+n)*sin(e+f*x)/(f*(3+n)*(1-n^2)*sqrt(sin(e+f*x)^2))+4*a^4*(3+2*n)*hypergeom([1/2,-1/2*n],[1/2*(2-n)],cos(e+f*x)^2)*sec(e+f*x)^n*sin(e+f*x)/(f*n*(2+n)*sqrt(sin(e+f*x)^2))],
[sec(e+f*x)^n*(a+a*sec(e+f*x))^3,x,7,a^3*(5+2*n)*sec(e+f*x)^(1+n)*sin(e+f*x)/(f*(1+n)*(2+n))+sec(e+f*x)^(1+n)*(a^3+a^3*sec(e+f*x))*sin(e+f*x)/(f*(2+n))-a^3*(1+4*n)*hypergeom([1/2,1/2*(1-n)],[1/2*(3-n)],cos(e+f*x)^2)*sec(e+f*x)^(-1+n)*sin(e+f*x)/(f*(1-n^2)*sqrt(sin(e+f*x)^2))+a^3*(7+4*n)*hypergeom([1/2,-1/2*n],[1/2*(2-n)],cos(e+f*x)^2)*sec(e+f*x)^n*sin(e+f*x)/(f*n*(2+n)*sqrt(sin(e+f*x)^2))],
[sec(e+f*x)^n*(a+a*sec(e+f*x))^2,x,6,a^2*sec(e+f*x)^(1+n)*sin(e+f*x)/(f*(1+n))-a^2*(1+2*n)*hypergeom([1/2,1/2*(1-n)],[1/2*(3-n)],cos(e+f*x)^2)*sec(e+f*x)^(-1+n)*sin(e+f*x)/(f*(1-n^2)*sqrt(sin(e+f*x)^2))+2*a^2*hypergeom([1/2,-1/2*n],[1/2*(2-n)],cos(e+f*x)^2)*sec(e+f*x)^n*sin(e+f*x)/(f*n*sqrt(sin(e+f*x)^2))],
[sec(e+f*x)^n*(a+a*sec(e+f*x)),x,5,-a*hypergeom([1/2,1/2*(1-n)],[1/2*(3-n)],cos(e+f*x)^2)*sec(e+f*x)^(-1+n)*sin(e+f*x)/(f*(1-n)*sqrt(sin(e+f*x)^2))+a*hypergeom([1/2,-1/2*n],[1/2*(2-n)],cos(e+f*x)^2)*sec(e+f*x)^n*sin(e+f*x)/(f*n*sqrt(sin(e+f*x)^2))],
[sec(e+f*x)^n/(a+a*sec(e+f*x)),x,6,sec(e+f*x)^n*sin(e+f*x)/(f*(a+a*sec(e+f*x)))+(1-n)*hypergeom([1/2,1/2*(2-n)],[1/2*(4-n)],cos(e+f*x)^2)*sec(e+f*x)^(-2+n)*sin(e+f*x)/(a*f*(2-n)*sqrt(sin(e+f*x)^2))-hypergeom([1/2,1/2*(1-n)],[1/2*(3-n)],cos(e+f*x)^2)*sec(e+f*x)^(-1+n)*sin(e+f*x)/(a*f*sqrt(sin(e+f*x)^2))],
[sec(e+f*x)^n/(a+a*sec(e+f*x))^2,x,7,-2/3*(2-n)*sec(e+f*x)^(1+n)*sin(e+f*x)/(a^2*f*(1+sec(e+f*x)))-1/3*sec(e+f*x)^(1+n)*sin(e+f*x)/(f*(a+a*sec(e+f*x))^2)-1/3*(3-2*n)*hypergeom([1/2,1/2*(1-n)],[1/2*(3-n)],cos(e+f*x)^2)*sec(e+f*x)^(-1+n)*sin(e+f*x)/(a^2*f*sqrt(sin(e+f*x)^2))+2/3*(2-n)*hypergeom([1/2,-1/2*n],[1/2*(2-n)],cos(e+f*x)^2)*sec(e+f*x)^n*sin(e+f*x)/(a^2*f*sqrt(sin(e+f*x)^2))],
[sec(e+f*x)^n*(1+sec(e+f*x))^(5/2),x,4,2*(7+4*n)*sec(e+f*x)^(1+n)*sin(e+f*x)/(f*(3+8*n+4*n^2)*sqrt(1+sec(e+f*x)))+2*sec(e+f*x)^(1+n)*sin(e+f*x)*sqrt(1+sec(e+f*x))/(f*(3+2*n))+2*(3+24*n+16*n^2)*hypergeom([1/2,1-n],[3/2],1-sec(e+f*x))*tan(e+f*x)/(f*(3+8*n+4*n^2)*sqrt(1+sec(e+f*x)))],
[sec(e+f*x)^n*(1+sec(e+f*x))^(3/2),x,4,2*sec(e+f*x)^(1+n)*sin(e+f*x)/(f*(1+2*n)*sqrt(1+sec(e+f*x)))+2*(1+4*n)*hypergeom([1/2,1-n],[3/2],1-sec(e+f*x))*tan(e+f*x)/(f*(1+2*n)*sqrt(1+sec(e+f*x)))],
[sec(e+f*x)^n*(1+sec(e+f*x))^(1/2),x,2,2*hypergeom([1/2,1-n],[3/2],1-sec(e+f*x))*tan(e+f*x)/(f*sqrt(1+sec(e+f*x)))],
[sec(e+f*x)^n/(1+sec(e+f*x))^(1/2),x,3,AppellF1(1/2,1-n,1,3/2,1-sec(e+f*x),1/2*(1-sec(e+f*x)))*tan(e+f*x)/(f*sqrt(1+sec(e+f*x)))],
[sec(e+f*x)^n/(1+sec(e+f*x))^(3/2),x,3,1/2*AppellF1(1/2,1-n,2,3/2,1-sec(e+f*x),1/2*(1-sec(e+f*x)))*tan(e+f*x)/(f*sqrt(1+sec(e+f*x)))],
[(-sec(e+f*x))^n*(1+sec(e+f*x))^(3/2),x,4,2*(-sec(e+f*x))^n*tan(e+f*x)/(f*(1+2*n)*sqrt(1+sec(e+f*x)))-(1+4*n)*hypergeom([1/2,n],[1+n],sec(e+f*x))*(-sec(e+f*x))^n*tan(e+f*x)/(f*n*(1+2*n)*sqrt(1-sec(e+f*x))*sqrt(1+sec(e+f*x)))],
[(-sec(e+f*x))^n*(1+sec(e+f*x))^(1/2),x,2,-hypergeom([1/2,n],[1+n],sec(e+f*x))*(-sec(e+f*x))^n*tan(e+f*x)/(f*n*sqrt(1-sec(e+f*x))*sqrt(1+sec(e+f*x)))],
[(-sec(e+f*x))^n/(1+sec(e+f*x))^(1/2),x,2,-AppellF1(n,1/2,1,1+n,sec(e+f*x),-sec(e+f*x))*(-sec(e+f*x))^n*tan(e+f*x)/(f*n*sqrt(1-sec(e+f*x))*sqrt(1+sec(e+f*x)))],
[(-sec(e+f*x))^n/(1+sec(e+f*x))^(3/2),x,2,-AppellF1(n,1/2,2,1+n,sec(e+f*x),-sec(e+f*x))*(-sec(e+f*x))^n*tan(e+f*x)/(f*n*sqrt(1-sec(e+f*x))*sqrt(1+sec(e+f*x)))],
[(d*sec(e+f*x))^n*(1+sec(e+f*x))^(3/2),x,4,2*(d*sec(e+f*x))^n*tan(e+f*x)/(f*(1+2*n)*sqrt(1+sec(e+f*x)))-(1+4*n)*hypergeom([1/2,n],[1+n],sec(e+f*x))*(d*sec(e+f*x))^n*tan(e+f*x)/(f*n*(1+2*n)*sqrt(1-sec(e+f*x))*sqrt(1+sec(e+f*x)))],
[(d*sec(e+f*x))^n*(1+sec(e+f*x))^(1/2),x,2,-hypergeom([1/2,n],[1+n],sec(e+f*x))*(d*sec(e+f*x))^n*tan(e+f*x)/(f*n*sqrt(1-sec(e+f*x))*sqrt(1+sec(e+f*x)))],
[(d*sec(e+f*x))^n/(1+sec(e+f*x))^(1/2),x,2,-AppellF1(n,1/2,1,1+n,sec(e+f*x),-sec(e+f*x))*(d*sec(e+f*x))^n*tan(e+f*x)/(f*n*sqrt(1-sec(e+f*x))*sqrt(1+sec(e+f*x)))],
[(d*sec(e+f*x))^n/(1+sec(e+f*x))^(3/2),x,2,-AppellF1(n,1/2,2,1+n,sec(e+f*x),-sec(e+f*x))*(d*sec(e+f*x))^n*tan(e+f*x)/(f*n*sqrt(1-sec(e+f*x))*sqrt(1+sec(e+f*x)))],
[sec(e+f*x)^n*(a+a*sec(e+f*x))^(5/2),x,4,2*a^3*(7+4*n)*sec(e+f*x)^(1+n)*sin(e+f*x)/(f*(3+8*n+4*n^2)*sqrt(a+a*sec(e+f*x)))+2*a^2*sec(e+f*x)^(1+n)*sin(e+f*x)*sqrt(a+a*sec(e+f*x))/(f*(3+2*n))+2*a^3*(3+24*n+16*n^2)*hypergeom([1/2,1-n],[3/2],1-sec(e+f*x))*tan(e+f*x)/(f*(3+8*n+4*n^2)*sqrt(a+a*sec(e+f*x)))],
[sec(e+f*x)^n*(a+a*sec(e+f*x))^(3/2),x,4,2*a^2*sec(e+f*x)^(1+n)*sin(e+f*x)/(f*(1+2*n)*sqrt(a+a*sec(e+f*x)))+2*a^2*(1+4*n)*hypergeom([1/2,1-n],[3/2],1-sec(e+f*x))*tan(e+f*x)/(f*(1+2*n)*sqrt(a+a*sec(e+f*x)))],
[sec(e+f*x)^n*(a+a*sec(e+f*x))^(1/2),x,2,2*a*hypergeom([1/2,1-n],[3/2],1-sec(e+f*x))*tan(e+f*x)/(f*sqrt(a+a*sec(e+f*x)))],
[sec(e+f*x)^n/(a+a*sec(e+f*x))^(1/2),x,4,AppellF1(1/2,1-n,1,3/2,1-sec(e+f*x),1/2*(1-sec(e+f*x)))*tan(e+f*x)/(f*sqrt(a+a*sec(e+f*x)))],
[sec(e+f*x)^n/(a+a*sec(e+f*x))^(3/2),x,4,1/2*AppellF1(1/2,1-n,2,3/2,1-sec(e+f*x),1/2*(1-sec(e+f*x)))*tan(e+f*x)/(a*f*sqrt(a+a*sec(e+f*x)))],
[(-sec(e+f*x))^n*(a+a*sec(e+f*x))^(3/2),x,5,2*a^2*(1+4*n)*hypergeom([1/2,1-n],[3/2],1-sec(e+f*x))*(-sec(e+f*x))^n*sec(e+f*x)^(1-n)*sin(e+f*x)/(f*(1+2*n)*sqrt(a+a*sec(e+f*x)))+2*a^2*(-sec(e+f*x))^n*tan(e+f*x)/(f*(1+2*n)*sqrt(a+a*sec(e+f*x)))],
[(-sec(e+f*x))^n*(a+a*sec(e+f*x))^(1/2),x,3,2*a*hypergeom([1/2,1-n],[3/2],1-sec(e+f*x))*(-sec(e+f*x))^n*sec(e+f*x)^(1-n)*sin(e+f*x)/(f*sqrt(a+a*sec(e+f*x)))],
[(-sec(e+f*x))^n/(a+a*sec(e+f*x))^(1/2),x,3,-AppellF1(n,1/2,1,1+n,sec(e+f*x),-sec(e+f*x))*(-sec(e+f*x))^n*tan(e+f*x)/(f*n*sqrt(1-sec(e+f*x))*sqrt(a+a*sec(e+f*x)))],
[(-sec(e+f*x))^n/(a+a*sec(e+f*x))^(3/2),x,3,-AppellF1(n,1/2,2,1+n,sec(e+f*x),-sec(e+f*x))*(-sec(e+f*x))^n*tan(e+f*x)/(a*f*n*sqrt(1-sec(e+f*x))*sqrt(a+a*sec(e+f*x)))],
[(d*sec(e+f*x))^n*(a+a*sec(e+f*x))^(3/2),x,5,2*a^2*(1+4*n)*hypergeom([1/2,1-n],[3/2],1-sec(e+f*x))*sec(e+f*x)^(1-n)*(d*sec(e+f*x))^n*sin(e+f*x)/(f*(1+2*n)*sqrt(a+a*sec(e+f*x)))+2*a^2*(d*sec(e+f*x))^n*tan(e+f*x)/(f*(1+2*n)*sqrt(a+a*sec(e+f*x)))],
[(d*sec(e+f*x))^n*(a+a*sec(e+f*x))^(1/2),x,3,2*a*hypergeom([1/2,1-n],[3/2],1-sec(e+f*x))*sec(e+f*x)^(1-n)*(d*sec(e+f*x))^n*sin(e+f*x)/(f*sqrt(a+a*sec(e+f*x)))],
[(d*sec(e+f*x))^n/(a+a*sec(e+f*x))^(1/2),x,3,-AppellF1(n,1/2,1,1+n,sec(e+f*x),-sec(e+f*x))*(d*sec(e+f*x))^n*tan(e+f*x)/(f*n*sqrt(1-sec(e+f*x))*sqrt(a+a*sec(e+f*x)))],
[(d*sec(e+f*x))^n/(a+a*sec(e+f*x))^(3/2),x,3,-AppellF1(n,1/2,2,1+n,sec(e+f*x),-sec(e+f*x))*(d*sec(e+f*x))^n*tan(e+f*x)/(a*f*n*sqrt(1-sec(e+f*x))*sqrt(a+a*sec(e+f*x)))],
[(-sec(e+f*x))^n*(a-a*sec(e+f*x))^(5/2),x,4,2*a^3*(3+24*n+16*n^2)*hypergeom([1/2,1-n],[3/2],1+sec(e+f*x))*tan(e+f*x)/(f*(3+8*n+4*n^2)*sqrt(a-a*sec(e+f*x)))+2*a^3*(7+4*n)*(-sec(e+f*x))^n*tan(e+f*x)/(f*(3+8*n+4*n^2)*sqrt(a-a*sec(e+f*x)))+2*a^2*(-sec(e+f*x))^n*sqrt(a-a*sec(e+f*x))*tan(e+f*x)/(f*(3+2*n))],
[(-sec(e+f*x))^n*(a-a*sec(e+f*x))^(3/2),x,4,2*a^2*(1+4*n)*hypergeom([1/2,1-n],[3/2],1+sec(e+f*x))*tan(e+f*x)/(f*(1+2*n)*sqrt(a-a*sec(e+f*x)))+2*a^2*(-sec(e+f*x))^n*tan(e+f*x)/(f*(1+2*n)*sqrt(a-a*sec(e+f*x)))],
[(-sec(e+f*x))^n*(a-a*sec(e+f*x))^(1/2),x,2,2*a*hypergeom([1/2,1-n],[3/2],1+sec(e+f*x))*tan(e+f*x)/(f*sqrt(a-a*sec(e+f*x)))],
[(-sec(e+f*x))^n/(a-a*sec(e+f*x))^(1/2),x,4,AppellF1(1/2,1-n,1,3/2,1+sec(e+f*x),1/2*(1+sec(e+f*x)))*tan(e+f*x)/(f*sqrt(a-a*sec(e+f*x)))],
[(-sec(e+f*x))^n/(a-a*sec(e+f*x))^(3/2),x,4,1/2*AppellF1(1/2,1-n,2,3/2,1+sec(e+f*x),1/2*(1+sec(e+f*x)))*tan(e+f*x)/(a*f*sqrt(a-a*sec(e+f*x)))],
[sec(e+f*x)^n*(a-a*sec(e+f*x))^(3/2),x,5,2*a^2*sec(e+f*x)^(1+n)*sin(e+f*x)/(f*(1+2*n)*sqrt(a-a*sec(e+f*x)))+2*a^2*(1+4*n)*hypergeom([1/2,1-n],[3/2],1+sec(e+f*x))*sec(e+f*x)^(1+n)*sin(e+f*x)/(f*(1+2*n)*(-sec(e+f*x))^n*sqrt(a-a*sec(e+f*x)))],
[sec(e+f*x)^n*(a-a*sec(e+f*x))^(1/2),x,3,2*a*hypergeom([1/2,1-n],[3/2],1+sec(e+f*x))*sec(e+f*x)^(1+n)*sin(e+f*x)/(f*(-sec(e+f*x))^n*sqrt(a-a*sec(e+f*x)))],
[(d*sec(e+f*x))^n*(a-a*sec(e+f*x))^(3/2),x,5,2*a^2*(d*sec(e+f*x))^n*tan(e+f*x)/(f*(1+2*n)*sqrt(a-a*sec(e+f*x)))+2*a^2*(1+4*n)*hypergeom([1/2,1-n],[3/2],1+sec(e+f*x))*(d*sec(e+f*x))^n*tan(e+f*x)/(f*(1+2*n)*(-sec(e+f*x))^n*sqrt(a-a*sec(e+f*x)))],
[(d*sec(e+f*x))^n*(a-a*sec(e+f*x))^(1/2),x,3,2*a*hypergeom([1/2,1-n],[3/2],1+sec(e+f*x))*(d*sec(e+f*x))^n*tan(e+f*x)/(f*(-sec(e+f*x))^n*sqrt(a-a*sec(e+f*x)))],

# Integrands of the form (d Sec[e+f x])^n (a+a Sec[e+f x])^m with m symbolic
[sec(e+f*x)^n*(1+sec(e+f*x))^m,x,2,2^(1/2+m)*AppellF1(1/2,1-n,1/2-m,3/2,1-sec(e+f*x),1/2*(1-sec(e+f*x)))*tan(e+f*x)/(f*sqrt(1+sec(e+f*x)))],
[(1-sec(e+f*x))^m*sec(e+f*x)^n,x,2,AppellF1(1/2+m,1-n,1/2,3/2+m,1-sec(e+f*x),1/2*(1-sec(e+f*x)))*(1-sec(e+f*x))^m*sqrt(2)*tan(e+f*x)/(f*(1+2*m)*sqrt(1+sec(e+f*x)))],
[sec(e+f*x)^n*(a+a*sec(e+f*x))^m,x,3,2^(1/2+m)*AppellF1(1/2,1-n,1/2-m,3/2,1-sec(e+f*x),1/2*(1-sec(e+f*x)))*(1+sec(e+f*x))^(-1/2-m)*(a+a*sec(e+f*x))^m*tan(e+f*x)/f],
[sec(e+f*x)^n*(a-a*sec(e+f*x))^m,x,3,AppellF1(1/2+m,1-n,1/2,3/2+m,1-sec(e+f*x),1/2*(1-sec(e+f*x)))*(a-a*sec(e+f*x))^m*sqrt(2)*tan(e+f*x)/(f*(1+2*m)*sqrt(1+sec(e+f*x)))],
[(-sec(e+f*x))^n*(1+sec(e+f*x))^m,x,2,AppellF1(1/2+m,1-n,1/2,3/2+m,1+sec(e+f*x),1/2*(1+sec(e+f*x)))*(1+sec(e+f*x))^m*sqrt(2)*tan(e+f*x)/(f*(1+2*m)*sqrt(1-sec(e+f*x)))],
[(1-sec(e+f*x))^m*(-sec(e+f*x))^n,x,2,2^(1/2+m)*AppellF1(1/2,1-n,1/2-m,3/2,1+sec(e+f*x),1/2*(1+sec(e+f*x)))*tan(e+f*x)/(f*sqrt(1-sec(e+f*x)))],
[(-sec(e+f*x))^n*(a+a*sec(e+f*x))^m,x,3,AppellF1(1/2+m,1-n,1/2,3/2+m,1+sec(e+f*x),1/2*(1+sec(e+f*x)))*(a+a*sec(e+f*x))^m*sqrt(2)*tan(e+f*x)/(f*(1+2*m)*sqrt(1-sec(e+f*x)))],
[(-sec(e+f*x))^n*(a-a*sec(e+f*x))^m,x,3,2^(1/2+m)*AppellF1(1/2,1-n,1/2-m,3/2,1+sec(e+f*x),1/2*(1+sec(e+f*x)))*(1-sec(e+f*x))^(-1/2-m)*(a-a*sec(e+f*x))^m*tan(e+f*x)/f],
[(d*sec(e+f*x))^n*(1+sec(e+f*x))^m,x,2,-AppellF1(n,1/2,1/2-m,1+n,sec(e+f*x),-sec(e+f*x))*(d*sec(e+f*x))^n*tan(e+f*x)/(f*n*sqrt(1-sec(e+f*x))*sqrt(1+sec(e+f*x)))],
[(1-sec(e+f*x))^m*(d*sec(e+f*x))^n,x,2,-AppellF1(n,1/2-m,1/2,1+n,sec(e+f*x),-sec(e+f*x))*(d*sec(e+f*x))^n*tan(e+f*x)/(f*n*sqrt(1-sec(e+f*x))*sqrt(1+sec(e+f*x)))],
[(d*sec(e+f*x))^n*(a+a*sec(e+f*x))^m,x,3,-AppellF1(n,1/2,1/2-m,1+n,sec(e+f*x),-sec(e+f*x))*(d*sec(e+f*x))^n*(1+sec(e+f*x))^(-1/2-m)*(a+a*sec(e+f*x))^m*tan(e+f*x)/(f*n*sqrt(1-sec(e+f*x)))],
[(d*sec(e+f*x))^n*(a-a*sec(e+f*x))^m,x,3,-AppellF1(n,1/2-m,1/2,1+n,sec(e+f*x),-sec(e+f*x))*(1-sec(e+f*x))^(-1/2-m)*(d*sec(e+f*x))^n*(a-a*sec(e+f*x))^m*tan(e+f*x)/(f*n*sqrt(1+sec(e+f*x)))],
[sec(e+f*x)^4*(a+a*sec(e+f*x))^m,x,6,(4+m)*(a+a*sec(e+f*x))^m*tan(e+f*x)/(f*(6+11*m+6*m^2+m^3))+sec(e+f*x)^2*(a+a*sec(e+f*x))^m*tan(e+f*x)/(f*(3+m))+2^(1/2+m)*m*(5+3*m+m^2)*hypergeom([1/2,1/2-m],[3/2],1/2*(1-sec(e+f*x)))*(1+sec(e+f*x))^(-1/2-m)*(a+a*sec(e+f*x))^m*tan(e+f*x)/(f*(6+11*m+6*m^2+m^3))+m*(a+a*sec(e+f*x))^(1+m)*tan(e+f*x)/(a*f*(6+5*m+m^2))],
[sec(e+f*x)^3*(a+a*sec(e+f*x))^m,x,5,-(a+a*sec(e+f*x))^m*tan(e+f*x)/(f*(2+3*m+m^2))+2^(1/2+m)*(1+m+m^2)*hypergeom([1/2,1/2-m],[3/2],1/2*(1-sec(e+f*x)))*(1+sec(e+f*x))^(-1/2-m)*(a+a*sec(e+f*x))^m*tan(e+f*x)/(f*(2+3*m+m^2))+(a+a*sec(e+f*x))^(1+m)*tan(e+f*x)/(a*f*(2+m))],
[sec(e+f*x)^2*(a+a*sec(e+f*x))^m,x,4,(a+a*sec(e+f*x))^m*tan(e+f*x)/(f*(1+m))+2^(1/2+m)*m*hypergeom([1/2,1/2-m],[3/2],1/2*(1-sec(e+f*x)))*(1+sec(e+f*x))^(-1/2-m)*(a+a*sec(e+f*x))^m*tan(e+f*x)/(f*(1+m))],
[sec(e+f*x)*(a+a*sec(e+f*x))^m,x,3,2^(1/2+m)*hypergeom([1/2,1/2-m],[3/2],1/2*(1-sec(e+f*x)))*(1+sec(e+f*x))^(-1/2-m)*(a+a*sec(e+f*x))^m*tan(e+f*x)/f],
[(a+a*sec(e+f*x))^m,x,3,AppellF1(1/2+m,1/2,1,3/2+m,1/2*(1+sec(e+f*x)),1+sec(e+f*x))*(a+a*sec(e+f*x))^m*sqrt(2)*tan(e+f*x)/(f*(1+2*m)*sqrt(1-sec(e+f*x)))],
[cos(e+f*x)*(a+a*sec(e+f*x))^m,x,3,-AppellF1(1/2+m,1/2,2,3/2+m,1/2*(1+sec(e+f*x)),1+sec(e+f*x))*(a+a*sec(e+f*x))^m*sqrt(2)*tan(e+f*x)/(f*(1+2*m)*sqrt(1-sec(e+f*x)))],
[(d*sec(e+f*x))^(3/2)*(a+a*sec(e+f*x))^m,x,3,-2/3*AppellF1(3/2,1/2,1/2-m,5/2,sec(e+f*x),-sec(e+f*x))*(d*sec(e+f*x))^(3/2)*(1+sec(e+f*x))^(-1/2-m)*(a+a*sec(e+f*x))^m*tan(e+f*x)/(f*sqrt(1-sec(e+f*x)))],
[(d*sec(e+f*x))^(1/2)*(a+a*sec(e+f*x))^m,x,3,-2*AppellF1(1/2,1/2,1/2-m,3/2,sec(e+f*x),-sec(e+f*x))*(1+sec(e+f*x))^(-1/2-m)*(a+a*sec(e+f*x))^m*sqrt(d*sec(e+f*x))*tan(e+f*x)/(f*sqrt(1-sec(e+f*x)))],
[(a+a*sec(e+f*x))^m/(d*sec(e+f*x))^(1/2),x,3,2*AppellF1(-1/2,1/2,1/2-m,1/2,sec(e+f*x),-sec(e+f*x))*(1+sec(e+f*x))^(-1/2-m)*(a+a*sec(e+f*x))^m*tan(e+f*x)/(f*sqrt(1-sec(e+f*x))*sqrt(d*sec(e+f*x)))],
[(a+a*sec(e+f*x))^m/(d*sec(e+f*x))^(3/2),x,3,2/3*AppellF1(-3/2,1/2,1/2-m,-1/2,sec(e+f*x),-sec(e+f*x))*(1+sec(e+f*x))^(-1/2-m)*(a+a*sec(e+f*x))^m*tan(e+f*x)/(f*(d*sec(e+f*x))^(3/2)*sqrt(1-sec(e+f*x)))],

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

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

# n>0
[cos(c+d*x)^(7/2)*(a+a*sec(c+d*x)),x,7,6/5*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)^(5/2)*(a+a*sec(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+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)^(3/2)*(a+a*sec(c+d*x)),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)*sqrt(cos(c+d*x))/d],
[cos(c+d*x)^(1/2)*(a+a*sec(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*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*sec(c+d*x))/cos(c+d*x)^(1/2),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*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*sec(c+d*x))/cos(c+d*x)^(3/2),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))/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*sec(c+d*x))/cos(c+d*x)^(5/2),x,7,-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)))],
[(a+a*sec(c+d*x))/cos(c+d*x)^(7/2),x,8,-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+2/7*a*sin(c+d*x)/(d*cos(c+d*x)^(7/2))+2/5*a*sin(c+d*x)/(d*cos(c+d*x)^(5/2))+10/21*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)^(9/2)*(a+a*sec(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)^(7/2)*(a+a*sec(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)^(5/2)*(a+a*sec(c+d*x))^2,x,8,16/5*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],
[cos(c+d*x)^(3/2)*(a+a*sec(c+d*x))^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)*sqrt(cos(c+d*x))/d],
[cos(c+d*x)^(1/2)*(a+a*sec(c+d*x))^2,x,5,4*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*sec(c+d*x))^2/cos(c+d*x)^(1/2),x,8,-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*sec(c+d*x))^2/cos(c+d*x)^(3/2),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)))],
[(a+a*sec(c+d*x))^2/cos(c+d*x)^(5/2),x,10,-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+2/7*a^2*sin(c+d*x)/(d*cos(c+d*x)^(7/2))+4/5*a^2*sin(c+d*x)/(d*cos(c+d*x)^(5/2))+8/7*a^2*sin(c+d*x)/(d*cos(c+d*x)^(3/2))+12/5*a^2*sin(c+d*x)/(d*sqrt(cos(c+d*x)))],
[cos(c+d*x)^(9/2)*(a+a*sec(c+d*x))^3,x,17,68/15*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)^(7/2)*(a+a*sec(c+d*x))^3,x,15,28/5*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],
[cos(c+d*x)^(5/2)*(a+a*sec(c+d*x))^3,x,13,36/5*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],
[cos(c+d*x)^(3/2)*(a+a*sec(c+d*x))^3,x,13,4*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],
[cos(c+d*x)^(1/2)*(a+a*sec(c+d*x))^3,x,13,-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*sec(c+d*x))^3/cos(c+d*x)^(1/2),x,15,-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*sec(c+d*x))^3/cos(c+d*x)^(3/2),x,17,-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)))],

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

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

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

# n<0
[cos(c+d*x)^(5/2)/sqrt(a+a*sec(c+d*x)),x,6,-arctanh(sin(c+d*x)*sqrt(a)*sqrt(sec(c+d*x))/(sqrt(2)*sqrt(a+a*sec(c+d*x))))*sqrt(2)*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(d*sqrt(a))+2/5*cos(c+d*x)^(3/2)*sin(c+d*x)/(d*sqrt(a+a*sec(c+d*x)))+26/15*sin(c+d*x)/(d*sqrt(cos(c+d*x))*sqrt(a+a*sec(c+d*x)))-2/15*sin(c+d*x)*sqrt(cos(c+d*x))/(d*sqrt(a+a*sec(c+d*x)))],
[cos(c+d*x)^(3/2)/sqrt(a+a*sec(c+d*x)),x,5,arctanh(sin(c+d*x)*sqrt(a)*sqrt(sec(c+d*x))/(sqrt(2)*sqrt(a+a*sec(c+d*x))))*sqrt(2)*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(d*sqrt(a))-2/3*sin(c+d*x)/(d*sqrt(cos(c+d*x))*sqrt(a+a*sec(c+d*x)))+2/3*sin(c+d*x)*sqrt(cos(c+d*x))/(d*sqrt(a+a*sec(c+d*x)))],
[sqrt(cos(c+d*x))/sqrt(a+a*sec(c+d*x)),x,4,-arctanh(sin(c+d*x)*sqrt(a)*sqrt(sec(c+d*x))/(sqrt(2)*sqrt(a+a*sec(c+d*x))))*sqrt(2)*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(d*sqrt(a))+2*sin(c+d*x)/(d*sqrt(cos(c+d*x))*sqrt(a+a*sec(c+d*x)))],
[1/(sqrt(cos(c+d*x))*sqrt(a+a*sec(c+d*x))),x,3,arctanh(sin(c+d*x)*sqrt(a)/(sqrt(2)*sqrt(cos(c+d*x))*sqrt(a+a*sec(c+d*x))))*sqrt(2)/(d*sqrt(a)),arctanh(sin(c+d*x)*sqrt(a)*sqrt(sec(c+d*x))/(sqrt(2)*sqrt(a+a*sec(c+d*x))))*sqrt(2)*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(d*sqrt(a))],
[1/(cos(c+d*x)^(3/2)*sqrt(a+a*sec(c+d*x))),x,6,2*arcsinh(sqrt(a)*tan(c+d*x)/sqrt(a+a*sec(c+d*x)))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(d*sqrt(a))-arctanh(sin(c+d*x)*sqrt(a)*sqrt(sec(c+d*x))/(sqrt(2)*sqrt(a+a*sec(c+d*x))))*sqrt(2)*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(d*sqrt(a))],
[1/(cos(c+d*x)^(5/2)*sqrt(a+a*sec(c+d*x))),x,7,-arcsinh(sqrt(a)*tan(c+d*x)/sqrt(a+a*sec(c+d*x)))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(d*sqrt(a))+arctanh(sin(c+d*x)*sqrt(a)*sqrt(sec(c+d*x))/(sqrt(2)*sqrt(a+a*sec(c+d*x))))*sqrt(2)*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(d*sqrt(a))+sin(c+d*x)/(d*cos(c+d*x)^(3/2)*sqrt(a+a*sec(c+d*x)))],
[1/(cos(c+d*x)^(7/2)*sqrt(a+a*sec(c+d*x))),x,8,7/4*arcsinh(sqrt(a)*tan(c+d*x)/sqrt(a+a*sec(c+d*x)))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(d*sqrt(a))-arctanh(sin(c+d*x)*sqrt(a)*sqrt(sec(c+d*x))/(sqrt(2)*sqrt(a+a*sec(c+d*x))))*sqrt(2)*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(d*sqrt(a))+1/2*sin(c+d*x)/(d*cos(c+d*x)^(5/2)*sqrt(a+a*sec(c+d*x)))-1/4*sin(c+d*x)/(d*cos(c+d*x)^(3/2)*sqrt(a+a*sec(c+d*x)))],
[cos(c+d*x)^(5/2)/(a+a*sec(c+d*x))^(3/2),x,7,-1/2*cos(c+d*x)^(3/2)*sin(c+d*x)/(d*(a+a*sec(c+d*x))^(3/2))-15/2*arctanh(sin(c+d*x)*sqrt(a)*sqrt(sec(c+d*x))/(sqrt(2)*sqrt(a+a*sec(c+d*x))))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(a^(3/2)*d*sqrt(2))+9/10*cos(c+d*x)^(3/2)*sin(c+d*x)/(a*d*sqrt(a+a*sec(c+d*x)))+49/10*sin(c+d*x)/(a*d*sqrt(cos(c+d*x))*sqrt(a+a*sec(c+d*x)))-13/10*sin(c+d*x)*sqrt(cos(c+d*x))/(a*d*sqrt(a+a*sec(c+d*x)))],
[cos(c+d*x)^(3/2)/(a+a*sec(c+d*x))^(3/2),x,6,-1/2*sin(c+d*x)*sqrt(cos(c+d*x))/(d*(a+a*sec(c+d*x))^(3/2))+11/2*arctanh(sin(c+d*x)*sqrt(a)*sqrt(sec(c+d*x))/(sqrt(2)*sqrt(a+a*sec(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)/(a*d*sqrt(cos(c+d*x))*sqrt(a+a*sec(c+d*x)))+7/6*sin(c+d*x)*sqrt(cos(c+d*x))/(a*d*sqrt(a+a*sec(c+d*x)))],
[sqrt(cos(c+d*x))/(a+a*sec(c+d*x))^(3/2),x,5,-1/2*sin(c+d*x)/(d*(a+a*sec(c+d*x))^(3/2)*sqrt(cos(c+d*x)))-7/2*arctanh(sin(c+d*x)*sqrt(a)*sqrt(sec(c+d*x))/(sqrt(2)*sqrt(a+a*sec(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)/(a*d*sqrt(cos(c+d*x))*sqrt(a+a*sec(c+d*x)))],
[1/((a+a*sec(c+d*x))^(3/2)*sqrt(cos(c+d*x))),x,4,-1/2*sin(c+d*x)/(d*cos(c+d*x)^(3/2)*(a+a*sec(c+d*x))^(3/2))+3/2*arctanh(sin(c+d*x)*sqrt(a)*sqrt(sec(c+d*x))/(sqrt(2)*sqrt(a+a*sec(c+d*x))))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(a^(3/2)*d*sqrt(2))],
[1/(cos(c+d*x)^(3/2)*(a+a*sec(c+d*x))^(3/2)),x,4,1/2*sin(c+d*x)/(d*cos(c+d*x)^(3/2)*(a+a*sec(c+d*x))^(3/2))+1/2*arctanh(sin(c+d*x)*sqrt(a)*sqrt(sec(c+d*x))/(sqrt(2)*sqrt(a+a*sec(c+d*x))))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(a^(3/2)*d*sqrt(2))],
[1/(cos(c+d*x)^(5/2)*(a+a*sec(c+d*x))^(3/2)),x,7,-1/2*sin(c+d*x)/(d*cos(c+d*x)^(3/2)*(a+a*sec(c+d*x))^(3/2))+2*arcsinh(sqrt(a)*tan(c+d*x)/sqrt(a+a*sec(c+d*x)))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(a^(3/2)*d)-5/2*arctanh(sin(c+d*x)*sqrt(a)*sqrt(sec(c+d*x))/(sqrt(2)*sqrt(a+a*sec(c+d*x))))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(a^(3/2)*d*sqrt(2))],
[1/(cos(c+d*x)^(7/2)*(a+a*sec(c+d*x))^(3/2)),x,8,-1/2*sin(c+d*x)/(d*cos(c+d*x)^(5/2)*(a+a*sec(c+d*x))^(3/2))-3*arcsinh(sqrt(a)*tan(c+d*x)/sqrt(a+a*sec(c+d*x)))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(a^(3/2)*d)+9/2*arctanh(sin(c+d*x)*sqrt(a)*sqrt(sec(c+d*x))/(sqrt(2)*sqrt(a+a*sec(c+d*x))))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(a^(3/2)*d*sqrt(2))+3/2*sin(c+d*x)/(a*d*cos(c+d*x)^(3/2)*sqrt(a+a*sec(c+d*x)))],
[cos(c+d*x)^(3/2)/(a+a*sec(c+d*x))^(5/2),x,7,-1/4*sin(c+d*x)*sqrt(cos(c+d*x))/(d*(a+a*sec(c+d*x))^(5/2))-17/16*sin(c+d*x)*sqrt(cos(c+d*x))/(a*d*(a+a*sec(c+d*x))^(3/2))+163/16*arctanh(sin(c+d*x)*sqrt(a)*sqrt(sec(c+d*x))/(sqrt(2)*sqrt(a+a*sec(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)/(a^2*d*sqrt(cos(c+d*x))*sqrt(a+a*sec(c+d*x)))+95/48*sin(c+d*x)*sqrt(cos(c+d*x))/(a^2*d*sqrt(a+a*sec(c+d*x)))],
[sqrt(cos(c+d*x))/(a+a*sec(c+d*x))^(5/2),x,6,-1/4*sin(c+d*x)/(d*(a+a*sec(c+d*x))^(5/2)*sqrt(cos(c+d*x)))-13/16*sin(c+d*x)/(a*d*(a+a*sec(c+d*x))^(3/2)*sqrt(cos(c+d*x)))-75/16*arctanh(sin(c+d*x)*sqrt(a)*sqrt(sec(c+d*x))/(sqrt(2)*sqrt(a+a*sec(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)/(a^2*d*sqrt(cos(c+d*x))*sqrt(a+a*sec(c+d*x)))],
[1/((a+a*sec(c+d*x))^(5/2)*sqrt(cos(c+d*x))),x,5,-1/4*sin(c+d*x)/(d*cos(c+d*x)^(3/2)*(a+a*sec(c+d*x))^(5/2))-9/16*sin(c+d*x)/(a*d*cos(c+d*x)^(3/2)*(a+a*sec(c+d*x))^(3/2))+19/16*arctanh(sin(c+d*x)*sqrt(a)*sqrt(sec(c+d*x))/(sqrt(2)*sqrt(a+a*sec(c+d*x))))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(a^(5/2)*d*sqrt(2))],
[1/(cos(c+d*x)^(3/2)*(a+a*sec(c+d*x))^(5/2)),x,5,-1/4*sin(c+d*x)/(d*cos(c+d*x)^(5/2)*(a+a*sec(c+d*x))^(5/2))+5/16*sin(c+d*x)/(a*d*cos(c+d*x)^(3/2)*(a+a*sec(c+d*x))^(3/2))+5/16*arctanh(sin(c+d*x)*sqrt(a)*sqrt(sec(c+d*x))/(sqrt(2)*sqrt(a+a*sec(c+d*x))))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(a^(5/2)*d*sqrt(2))],
[1/(cos(c+d*x)^(5/2)*(a+a*sec(c+d*x))^(5/2)),x,5,1/4*sin(c+d*x)/(d*cos(c+d*x)^(5/2)*(a+a*sec(c+d*x))^(5/2))+3/16*sin(c+d*x)/(a*d*cos(c+d*x)^(3/2)*(a+a*sec(c+d*x))^(3/2))+3/16*arctanh(sin(c+d*x)*sqrt(a)*sqrt(sec(c+d*x))/(sqrt(2)*sqrt(a+a*sec(c+d*x))))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(a^(5/2)*d*sqrt(2))],
[1/(cos(c+d*x)^(7/2)*(a+a*sec(c+d*x))^(5/2)),x,8,-1/4*sin(c+d*x)/(d*cos(c+d*x)^(5/2)*(a+a*sec(c+d*x))^(5/2))-11/16*sin(c+d*x)/(a*d*cos(c+d*x)^(3/2)*(a+a*sec(c+d*x))^(3/2))+2*arcsinh(sqrt(a)*tan(c+d*x)/sqrt(a+a*sec(c+d*x)))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(a^(5/2)*d)-43/16*arctanh(sin(c+d*x)*sqrt(a)*sqrt(sec(c+d*x))/(sqrt(2)*sqrt(a+a*sec(c+d*x))))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(a^(5/2)*d*sqrt(2))],
[1/(cos(c+d*x)^(9/2)*(a+a*sec(c+d*x))^(5/2)),x,9,-1/4*sin(c+d*x)/(d*cos(c+d*x)^(7/2)*(a+a*sec(c+d*x))^(5/2))-15/16*sin(c+d*x)/(a*d*cos(c+d*x)^(5/2)*(a+a*sec(c+d*x))^(3/2))-5*arcsinh(sqrt(a)*tan(c+d*x)/sqrt(a+a*sec(c+d*x)))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(a^(5/2)*d)+115/16*arctanh(sin(c+d*x)*sqrt(a)*sqrt(sec(c+d*x))/(sqrt(2)*sqrt(a+a*sec(c+d*x))))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/(a^(5/2)*d*sqrt(2))+35/16*sin(c+d*x)/(a^2*d*cos(c+d*x)^(3/2)*sqrt(a+a*sec(c+d*x)))],

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

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

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

# n>0
[sec(c+d*x)^4*(a+b*sec(c+d*x)),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+1/3*a*tan(c+d*x)^3/d],
[sec(c+d*x)^3*(a+b*sec(c+d*x)),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+1/3*b*tan(c+d*x)^3/d],
[sec(c+d*x)^2*(a+b*sec(c+d*x)),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],
[sec(c+d*x)*(a+b*sec(c+d*x)),x,4,a*arctanh(sin(c+d*x))/d+b*tan(c+d*x)/d],
[a+b*sec(c+d*x),x,2,a*x+b*arctanh(sin(c+d*x))/d],
[cos(c+d*x)*(a+b*sec(c+d*x)),x,3,b*x+a*sin(c+d*x)/d],
[cos(c+d*x)^2*(a+b*sec(c+d*x)),x,4,1/2*a*x+b*sin(c+d*x)/d+1/2*a*cos(c+d*x)*sin(c+d*x)/d],
[cos(c+d*x)^3*(a+b*sec(c+d*x)),x,5,1/2*b*x+a*sin(c+d*x)/d+1/2*b*cos(c+d*x)*sin(c+d*x)/d-1/3*a*sin(c+d*x)^3/d],
[cos(c+d*x)^4*(a+b*sec(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-1/3*b*sin(c+d*x)^3/d],
[cos(c+d*x)^5*(a+b*sec(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-2/3*a*sin(c+d*x)^3/d+1/5*a*sin(c+d*x)^5/d],
[sec(c+d*x)^4*(a+b*sec(c+d*x))^2,x,7,3/4*a*b*arctanh(sin(c+d*x))/d+1/5*(5*a^2+4*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*b^2*sec(c+d*x)^4*tan(c+d*x)/d+1/15*(5*a^2+4*b^2)*tan(c+d*x)^3/d],
[sec(c+d*x)^3*(a+b*sec(c+d*x))^2,x,6,1/8*(4*a^2+3*b^2)*arctanh(sin(c+d*x))/d+2*a*b*tan(c+d*x)/d+1/8*(4*a^2+3*b^2)*sec(c+d*x)*tan(c+d*x)/d+1/4*b^2*sec(c+d*x)^3*tan(c+d*x)/d+2/3*a*b*tan(c+d*x)^3/d],
[sec(c+d*x)^2*(a+b*sec(c+d*x))^2,x,6,a*b*arctanh(sin(c+d*x))/d+1/3*(3*a^2+2*b^2)*tan(c+d*x)/d+a*b*sec(c+d*x)*tan(c+d*x)/d+1/3*b^2*sec(c+d*x)^2*tan(c+d*x)/d],
[sec(c+d*x)*(a+b*sec(c+d*x))^2,x,5,1/2*(2*a^2+b^2)*arctanh(sin(c+d*x))/d+2*a*b*tan(c+d*x)/d+1/2*b^2*sec(c+d*x)*tan(c+d*x)/d],
[(a+b*sec(c+d*x))^2,x,4,a^2*x+2*a*b*arctanh(sin(c+d*x))/d+b^2*tan(c+d*x)/d],
[cos(c+d*x)*(a+b*sec(c+d*x))^2,x,4,2*a*b*x+b^2*arctanh(sin(c+d*x))/d+a^2*sin(c+d*x)/d],
[cos(c+d*x)^2*(a+b*sec(c+d*x))^2,x,4,1/2*(a^2+2*b^2)*x+2*a*b*sin(c+d*x)/d+1/2*a^2*cos(c+d*x)*sin(c+d*x)/d],
[cos(c+d*x)^3*(a+b*sec(c+d*x))^2,x,6,a*b*x+(a^2+b^2)*sin(c+d*x)/d+a*b*cos(c+d*x)*sin(c+d*x)/d-1/3*a^2*sin(c+d*x)^3/d],
[cos(c+d*x)^4*(a+b*sec(c+d*x))^2,x,6,1/8*(3*a^2+4*b^2)*x+2*a*b*sin(c+d*x)/d+1/8*(3*a^2+4*b^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*b*sin(c+d*x)^3/d],
[cos(c+d*x)^5*(a+b*sec(c+d*x))^2,x,8,3/4*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*(2*a^2+b^2)*sin(c+d*x)^3/d+1/5*a^2*sin(c+d*x)^5/d],
[sec(c+d*x)^3*(a+b*sec(c+d*x))^3,x,8,1/8*a*(4*a^2+9*b^2)*arctanh(sin(c+d*x))/d-1/30*(3*a^4-52*a^2*b^2-16*b^4)*tan(c+d*x)/(b*d)-1/120*a*(6*a^2-71*b^2)*sec(c+d*x)*tan(c+d*x)/d-1/60*(3*a^2-16*b^2)*(a+b*sec(c+d*x))^2*tan(c+d*x)/(b*d)-1/20*a*(a+b*sec(c+d*x))^3*tan(c+d*x)/(b*d)+1/5*(a+b*sec(c+d*x))^4*tan(c+d*x)/(b*d)],
[sec(c+d*x)^2*(a+b*sec(c+d*x))^3,x,7,3/8*b*(4*a^2+b^2)*arctanh(sin(c+d*x))/d+1/2*a*(a^2+4*b^2)*tan(c+d*x)/d+1/8*b*(2*a^2+3*b^2)*sec(c+d*x)*tan(c+d*x)/d+1/4*a*(a+b*sec(c+d*x))^2*tan(c+d*x)/d+1/4*(a+b*sec(c+d*x))^3*tan(c+d*x)/d],
[sec(c+d*x)*(a+b*sec(c+d*x))^3,x,6,1/2*a*(2*a^2+3*b^2)*arctanh(sin(c+d*x))/d+2/3*b*(4*a^2+b^2)*tan(c+d*x)/d+5/6*a*b^2*sec(c+d*x)*tan(c+d*x)/d+1/3*b*(a+b*sec(c+d*x))^2*tan(c+d*x)/d],
[(a+b*sec(c+d*x))^3,x,5,a^3*x+1/2*b*(6*a^2+b^2)*arctanh(sin(c+d*x))/d+5/2*a*b^2*tan(c+d*x)/d+1/2*b^2*(a+b*sec(c+d*x))*tan(c+d*x)/d],
[cos(c+d*x)*(a+b*sec(c+d*x))^3,x,5,3*a^2*b*x+3*a*b^2*arctanh(sin(c+d*x))/d+a*(a^2-b^2)*sin(c+d*x)/d+b^2*(a+b*sec(c+d*x))*sin(c+d*x)/d],
[cos(c+d*x)^2*(a+b*sec(c+d*x))^3,x,5,1/2*a*(a^2+6*b^2)*x+b^3*arctanh(sin(c+d*x))/d+5/2*a^2*b*sin(c+d*x)/d+1/2*a^2*cos(c+d*x)*(a+b*sec(c+d*x))*sin(c+d*x)/d],
[cos(c+d*x)^3*(a+b*sec(c+d*x))^3,x,5,1/2*b*(3*a^2+2*b^2)*x+1/3*a*(2*a^2+9*b^2)*sin(c+d*x)/d+7/6*a^2*b*cos(c+d*x)*sin(c+d*x)/d+1/3*a^2*cos(c+d*x)^2*(a+b*sec(c+d*x))*sin(c+d*x)/d],
[cos(c+d*x)^4*(a+b*sec(c+d*x))^3,x,7,3/8*a*(a^2+4*b^2)*x+1/4*b*(11*a^2+4*b^2)*sin(c+d*x)/d+3/8*a*(a^2+4*b^2)*cos(c+d*x)*sin(c+d*x)/d+1/4*a^2*cos(c+d*x)^3*(a+b*sec(c+d*x))*sin(c+d*x)/d-3/4*a^2*b*sin(c+d*x)^3/d],
[cos(c+d*x)^5*(a+b*sec(c+d*x))^3,x,7,1/8*b*(9*a^2+4*b^2)*x+1/5*a*(4*a^2+15*b^2)*sin(c+d*x)/d+1/8*b*(9*a^2+4*b^2)*cos(c+d*x)*sin(c+d*x)/d+11/20*a^2*b*cos(c+d*x)^3*sin(c+d*x)/d+1/5*a^2*cos(c+d*x)^4*(a+b*sec(c+d*x))*sin(c+d*x)/d-1/15*a*(4*a^2+15*b^2)*sin(c+d*x)^3/d],
[cos(c+d*x)^6*(a+b*sec(c+d*x))^3,x,9,1/16*a*(5*a^2+18*b^2)*x+1/6*b*(17*a^2+6*b^2)*sin(c+d*x)/d+1/16*a*(5*a^2+18*b^2)*cos(c+d*x)*sin(c+d*x)/d+1/24*a*(5*a^2+18*b^2)*cos(c+d*x)^3*sin(c+d*x)/d+1/6*a^2*cos(c+d*x)^5*(a+b*sec(c+d*x))*sin(c+d*x)/d-1/3*b*(5*a^2+b^2)*sin(c+d*x)^3/d+13/30*a^2*b*sin(c+d*x)^5/d],
[sec(c+d*x)^3*(a+b*sec(c+d*x))^4,x,9,1/16*(8*a^4+36*a^2*b^2+5*b^4)*arctanh(sin(c+d*x))/d-1/60*a*(4*a^4-121*a^2*b^2-128*b^4)*tan(c+d*x)/(b*d)-1/240*(8*a^4-178*a^2*b^2-75*b^4)*sec(c+d*x)*tan(c+d*x)/d-1/120*a*(4*a^2-53*b^2)*(a+b*sec(c+d*x))^2*tan(c+d*x)/(b*d)-1/120*(4*a^2-25*b^2)*(a+b*sec(c+d*x))^3*tan(c+d*x)/(b*d)-1/30*a*(a+b*sec(c+d*x))^4*tan(c+d*x)/(b*d)+1/6*(a+b*sec(c+d*x))^5*tan(c+d*x)/(b*d)],
[sec(c+d*x)^2*(a+b*sec(c+d*x))^4,x,8,1/2*a*b*(4*a^2+3*b^2)*arctanh(sin(c+d*x))/d+2/15*(3*a^4+28*a^2*b^2+4*b^4)*tan(c+d*x)/d+1/30*a*b*(6*a^2+29*b^2)*sec(c+d*x)*tan(c+d*x)/d+1/15*(3*a^2+4*b^2)*(a+b*sec(c+d*x))^2*tan(c+d*x)/d+1/5*a*(a+b*sec(c+d*x))^3*tan(c+d*x)/d+1/5*(a+b*sec(c+d*x))^4*tan(c+d*x)/d],
[sec(c+d*x)*(a+b*sec(c+d*x))^4,x,7,1/8*(8*a^4+24*a^2*b^2+3*b^4)*arctanh(sin(c+d*x))/d+1/6*a*b*(19*a^2+16*b^2)*tan(c+d*x)/d+1/24*b^2*(26*a^2+9*b^2)*sec(c+d*x)*tan(c+d*x)/d+7/12*a*b*(a+b*sec(c+d*x))^2*tan(c+d*x)/d+1/4*b*(a+b*sec(c+d*x))^3*tan(c+d*x)/d],
[(a+b*sec(c+d*x))^4,x,6,a^4*x+2*a*b*(2*a^2+b^2)*arctanh(sin(c+d*x))/d+1/3*b^2*(17*a^2+2*b^2)*tan(c+d*x)/d+4/3*a*b^3*sec(c+d*x)*tan(c+d*x)/d+1/3*b^2*(a+b*sec(c+d*x))^2*tan(c+d*x)/d],
[cos(c+d*x)*(a+b*sec(c+d*x))^4,x,6,4*a^3*b*x+1/2*b^2*(12*a^2+b^2)*arctanh(sin(c+d*x))/d+1/2*a^2*(2*a^2-b^2)*sin(c+d*x)/d+1/2*b^2*(a+b*sec(c+d*x))^2*sin(c+d*x)/d+3*a*b^3*tan(c+d*x)/d],
[cos(c+d*x)^2*(a+b*sec(c+d*x))^4,x,6,1/2*a^2*(a^2+12*b^2)*x+4*a*b^3*arctanh(sin(c+d*x))/d+3*a^3*b*sin(c+d*x)/d+1/2*a^2*cos(c+d*x)*(a+b*sec(c+d*x))^2*sin(c+d*x)/d-1/2*b^2*(a^2-2*b^2)*tan(c+d*x)/d],
[cos(c+d*x)^3*(a+b*sec(c+d*x))^4,x,6,2*a*b*(a^2+2*b^2)*x+b^4*arctanh(sin(c+d*x))/d+1/3*a^2*(2*a^2+17*b^2)*sin(c+d*x)/d+4/3*a^3*b*cos(c+d*x)*sin(c+d*x)/d+1/3*a^2*cos(c+d*x)^2*(a+b*sec(c+d*x))^2*sin(c+d*x)/d],
[cos(c+d*x)^4*(a+b*sec(c+d*x))^4,x,6,1/8*(3*a^4+24*a^2*b^2+8*b^4)*x+4/3*a*b*(2*a^2+3*b^2)*sin(c+d*x)/d+1/8*a^2*(3*a^2+22*b^2)*cos(c+d*x)*sin(c+d*x)/d+5/6*a^3*b*cos(c+d*x)^2*sin(c+d*x)/d+1/4*a^2*cos(c+d*x)^3*(a+b*sec(c+d*x))^2*sin(c+d*x)/d],
[cos(c+d*x)^5*(a+b*sec(c+d*x))^4,x,8,1/2*a*b*(3*a^2+4*b^2)*x+1/5*(4*a^4+29*a^2*b^2+5*b^4)*sin(c+d*x)/d+1/2*a*b*(3*a^2+4*b^2)*cos(c+d*x)*sin(c+d*x)/d+3/5*a^3*b*cos(c+d*x)^3*sin(c+d*x)/d+1/5*a^2*cos(c+d*x)^4*(a+b*sec(c+d*x))^2*sin(c+d*x)/d-1/15*a^2*(4*a^2+27*b^2)*sin(c+d*x)^3/d],
[cos(c+d*x)^6*(a+b*sec(c+d*x))^4,x,8,1/16*(5*a^4+36*a^2*b^2+8*b^4)*x+4/5*a*b*(4*a^2+5*b^2)*sin(c+d*x)/d+1/16*(5*a^4+36*a^2*b^2+8*b^4)*cos(c+d*x)*sin(c+d*x)/d+1/24*a^2*(5*a^2+32*b^2)*cos(c+d*x)^3*sin(c+d*x)/d+7/15*a^3*b*cos(c+d*x)^4*sin(c+d*x)/d+1/6*a^2*cos(c+d*x)^5*(a+b*sec(c+d*x))^2*sin(c+d*x)/d-4/15*a*b*(4*a^2+5*b^2)*sin(c+d*x)^3/d],
[(a+b*sec(c+d*x))^5,x,7,a^5*x+1/8*b*(40*a^4+40*a^2*b^2+3*b^4)*arctanh(sin(c+d*x))/d+1/6*a*b^2*(53*a^2+20*b^2)*tan(c+d*x)/d+1/24*b^3*(58*a^2+9*b^2)*sec(c+d*x)*tan(c+d*x)/d+11/12*a*b^2*(a+b*sec(c+d*x))^2*tan(c+d*x)/d+1/4*b^2*(a+b*sec(c+d*x))^3*tan(c+d*x)/d],

# n<0
[sec(c+d*x)^5/(a+b*sec(c+d*x)),x,8,-1/2*a*(2*a^2+b^2)*arctanh(sin(c+d*x))/(b^4*d)+2*a^4*arctanh(sqrt(a-b)*tan(1/2*(c+d*x))/sqrt(a+b))/(b^4*d*sqrt(a-b)*sqrt(a+b))+1/3*(3*a^2+2*b^2)*tan(c+d*x)/(b^3*d)-1/2*a*sec(c+d*x)*tan(c+d*x)/(b^2*d)+1/3*sec(c+d*x)^2*tan(c+d*x)/(b*d)],
[sec(c+d*x)^4/(a+b*sec(c+d*x)),x,7,1/2*(2*a^2+b^2)*arctanh(sin(c+d*x))/(b^3*d)-2*a^3*arctanh(sqrt(a-b)*tan(1/2*(c+d*x))/sqrt(a+b))/(b^3*d*sqrt(a-b)*sqrt(a+b))-a*tan(c+d*x)/(b^2*d)+1/2*sec(c+d*x)*tan(c+d*x)/(b*d)],
[sec(c+d*x)^3/(a+b*sec(c+d*x)),x,6,-a*arctanh(sin(c+d*x))/(b^2*d)+2*a^2*arctanh(sqrt(a-b)*tan(1/2*(c+d*x))/sqrt(a+b))/(b^2*d*sqrt(a-b)*sqrt(a+b))+tan(c+d*x)/(b*d)],
[sec(c+d*x)^2/(a+b*sec(c+d*x)),x,5,arctanh(sin(c+d*x))/(b*d)-2*a*arctanh(sqrt(a-b)*tan(1/2*(c+d*x))/sqrt(a+b))/(b*d*sqrt(a-b)*sqrt(a+b))],
[sec(c+d*x)/(a+b*sec(c+d*x)),x,3,2*arctanh(sqrt(a-b)*tan(1/2*(c+d*x))/sqrt(a+b))/(d*sqrt(a-b)*sqrt(a+b))],
[1/(a+b*sec(c+d*x)),x,3,x/a-2*b*arctanh(sqrt(a-b)*tan(1/2*(c+d*x))/sqrt(a+b))/(a*d*sqrt(a-b)*sqrt(a+b))],
[cos(c+d*x)/(a+b*sec(c+d*x)),x,5,-b*x/a^2+sin(c+d*x)/(a*d)+2*b^2*arctanh(sqrt(a-b)*tan(1/2*(c+d*x))/sqrt(a+b))/(a^2*d*sqrt(a-b)*sqrt(a+b))],
[cos(c+d*x)^2/(a+b*sec(c+d*x)),x,6,1/2*(a^2+2*b^2)*x/a^3-b*sin(c+d*x)/(a^2*d)+1/2*cos(c+d*x)*sin(c+d*x)/(a*d)-2*b^3*arctanh(sqrt(a-b)*tan(1/2*(c+d*x))/sqrt(a+b))/(a^3*d*sqrt(a-b)*sqrt(a+b))],
[cos(c+d*x)^3/(a+b*sec(c+d*x)),x,7,-1/2*b*(a^2+2*b^2)*x/a^4+1/3*(2*a^2+3*b^2)*sin(c+d*x)/(a^3*d)-1/2*b*cos(c+d*x)*sin(c+d*x)/(a^2*d)+1/3*cos(c+d*x)^2*sin(c+d*x)/(a*d)+2*b^4*arctanh(sqrt(a-b)*tan(1/2*(c+d*x))/sqrt(a+b))/(a^4*d*sqrt(a-b)*sqrt(a+b))],
[cos(c+d*x)^4/(a+b*sec(c+d*x)),x,8,1/8*(3*a^4+4*a^2*b^2+8*b^4)*x/a^5-1/3*b*(2*a^2+3*b^2)*sin(c+d*x)/(a^4*d)+1/8*(3*a^2+4*b^2)*cos(c+d*x)*sin(c+d*x)/(a^3*d)-1/3*b*cos(c+d*x)^2*sin(c+d*x)/(a^2*d)+1/4*cos(c+d*x)^3*sin(c+d*x)/(a*d)-2*b^5*arctanh(sqrt(a-b)*tan(1/2*(c+d*x))/sqrt(a+b))/(a^5*d*sqrt(a-b)*sqrt(a+b))],
[sec(c+d*x)^5/(a+b*sec(c+d*x))^2,x,8,1/2*(6*a^2+b^2)*arctanh(sin(c+d*x))/(b^4*d)-2*a^3*(3*a^2-4*b^2)*arctanh(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)*tan(c+d*x)/(b^3*(a^2-b^2)*d)+1/2*(3*a^2-b^2)*sec(c+d*x)*tan(c+d*x)/(b^2*(a^2-b^2)*d)-a^2*sec(c+d*x)^2*tan(c+d*x)/(b*(a^2-b^2)*d*(a+b*sec(c+d*x)))],
[sec(c+d*x)^4/(a+b*sec(c+d*x))^2,x,7,-2*a*arctanh(sin(c+d*x))/(b^3*d)+2*a^2*(2*a^2-3*b^2)*arctanh(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)*tan(c+d*x)/(b^2*(a^2-b^2)*d)-a^2*sec(c+d*x)*tan(c+d*x)/(b*(a^2-b^2)*d*(a+b*sec(c+d*x)))],
[sec(c+d*x)^3/(a+b*sec(c+d*x))^2,x,6,arctanh(sin(c+d*x))/(b^2*d)-2*a*(a^2-2*b^2)*arctanh(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*tan(c+d*x)/(b*(a^2-b^2)*d*(a+b*sec(c+d*x)))],
[sec(c+d*x)^2/(a+b*sec(c+d*x))^2,x,5,-2*b*arctanh(sqrt(a-b)*tan(1/2*(c+d*x))/sqrt(a+b))/((a-b)^(3/2)*(a+b)^(3/2)*d)+a*tan(c+d*x)/((a^2-b^2)*d*(a+b*sec(c+d*x)))],
[sec(c+d*x)/(a+b*sec(c+d*x))^2,x,5,2*a*arctanh(sqrt(a-b)*tan(1/2*(c+d*x))/sqrt(a+b))/((a-b)^(3/2)*(a+b)^(3/2)*d)-b*tan(c+d*x)/((a^2-b^2)*d*(a+b*sec(c+d*x)))],
[1/(a+b*sec(c+d*x))^2,x,5,x/a^2-2*b*(2*a^2-b^2)*arctanh(sqrt(a-b)*tan(1/2*(c+d*x))/sqrt(a+b))/(a^2*(a-b)^(3/2)*(a+b)^(3/2)*d)+b^2*tan(c+d*x)/(a*(a^2-b^2)*d*(a+b*sec(c+d*x)))],
[cos(c+d*x)/(a+b*sec(c+d*x))^2,x,6,-2*b*x/a^3+2*b^2*(3*a^2-2*b^2)*arctanh(sqrt(a-b)*tan(1/2*(c+d*x))/sqrt(a+b))/(a^3*(a-b)^(3/2)*(a+b)^(3/2)*d)+(a^2-2*b^2)*sin(c+d*x)/(a^2*(a^2-b^2)*d)+b^2*sin(c+d*x)/(a*(a^2-b^2)*d*(a+b*sec(c+d*x)))],
[cos(c+d*x)^2/(a+b*sec(c+d*x))^2,x,7,1/2*(a^2+6*b^2)*x/a^4-2*b^3*(4*a^2-3*b^2)*arctanh(sqrt(a-b)*tan(1/2*(c+d*x))/sqrt(a+b))/(a^4*(a-b)^(3/2)*(a+b)^(3/2)*d)-b*(2*a^2-3*b^2)*sin(c+d*x)/(a^3*(a^2-b^2)*d)+1/2*(a^2-3*b^2)*cos(c+d*x)*sin(c+d*x)/(a^2*(a^2-b^2)*d)+b^2*cos(c+d*x)*sin(c+d*x)/(a*(a^2-b^2)*d*(a+b*sec(c+d*x)))],
[cos(c+d*x)^3/(a+b*sec(c+d*x))^2,x,8,-b*(a^2+4*b^2)*x/a^5+2*b^4*(5*a^2-4*b^2)*arctanh(sqrt(a-b)*tan(1/2*(c+d*x))/sqrt(a+b))/(a^5*(a-b)^(3/2)*(a+b)^(3/2)*d)+1/3*(2*a^4+7*a^2*b^2-12*b^4)*sin(c+d*x)/(a^4*(a^2-b^2)*d)-b*(a^2-2*b^2)*cos(c+d*x)*sin(c+d*x)/(a^3*(a^2-b^2)*d)+1/3*(a^2-4*b^2)*cos(c+d*x)^2*sin(c+d*x)/(a^2*(a^2-b^2)*d)+b^2*cos(c+d*x)^2*sin(c+d*x)/(a*(a^2-b^2)*d*(a+b*sec(c+d*x)))],
[sec(c+d*x)^5/(a+b*sec(c+d*x))^3,x,8,-3*a*arctanh(sin(c+d*x))/(b^4*d)+3*a^2*(2*a^4-5*a^2*b^2+4*b^4)*arctanh(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)*tan(c+d*x)/(b^3*(a^2-b^2)*d)-1/2*a^2*sec(c+d*x)^2*tan(c+d*x)/(b*(a^2-b^2)*d*(a+b*sec(c+d*x))^2)+3/2*a^3*(a^2-2*b^2)*tan(c+d*x)/(b^3*(a^2-b^2)^2*d*(a+b*sec(c+d*x)))],
[sec(c+d*x)^4/(a+b*sec(c+d*x))^3,x,7,arctanh(sin(c+d*x))/(b^3*d)-a*(2*a^4-5*a^2*b^2+6*b^4)*arctanh(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*sec(c+d*x)*tan(c+d*x)/(b*(a^2-b^2)*d*(a+b*sec(c+d*x))^2)-1/2*a^2*(2*a^2-5*b^2)*tan(c+d*x)/(b^2*(a^2-b^2)^2*d*(a+b*sec(c+d*x)))],
[sec(c+d*x)^3/(a+b*sec(c+d*x))^3,x,6,(a^2+2*b^2)*arctanh(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*tan(c+d*x)/(b*(a^2-b^2)*d*(a+b*sec(c+d*x))^2)+1/2*a*(a^2-4*b^2)*tan(c+d*x)/(b*(a^2-b^2)^2*d*(a+b*sec(c+d*x)))],
[sec(c+d*x)^2/(a+b*sec(c+d*x))^3,x,6,-3*a*b*arctanh(sqrt(a-b)*tan(1/2*(c+d*x))/sqrt(a+b))/((a-b)^(5/2)*(a+b)^(5/2)*d)+1/2*a*tan(c+d*x)/((a^2-b^2)*d*(a+b*sec(c+d*x))^2)+1/2*(a^2+2*b^2)*tan(c+d*x)/((a^2-b^2)^2*d*(a+b*sec(c+d*x)))],
[sec(c+d*x)/(a+b*sec(c+d*x))^3,x,6,(2*a^2+b^2)*arctanh(sqrt(a-b)*tan(1/2*(c+d*x))/sqrt(a+b))/((a-b)^(5/2)*(a+b)^(5/2)*d)-1/2*b*tan(c+d*x)/((a^2-b^2)*d*(a+b*sec(c+d*x))^2)-3/2*a*b*tan(c+d*x)/((a^2-b^2)^2*d*(a+b*sec(c+d*x)))],
[1/(a+b*sec(c+d*x))^3,x,6,x/a^3-b*(6*a^4-5*a^2*b^2+2*b^4)*arctanh(sqrt(a-b)*tan(1/2*(c+d*x))/sqrt(a+b))/(a^3*(a-b)^(5/2)*(a+b)^(5/2)*d)+1/2*b^2*tan(c+d*x)/(a*(a^2-b^2)*d*(a+b*sec(c+d*x))^2)+1/2*b^2*(5*a^2-2*b^2)*tan(c+d*x)/(a^2*(a^2-b^2)^2*d*(a+b*sec(c+d*x)))],
[cos(c+d*x)/(a+b*sec(c+d*x))^3,x,7,-3*b*x/a^4+3*b^2*(4*a^4-5*a^2*b^2+2*b^4)*arctanh(sqrt(a-b)*tan(1/2*(c+d*x))/sqrt(a+b))/(a^4*(a-b)^(5/2)*(a+b)^(5/2)*d)+1/2*(2*a^4-11*a^2*b^2+6*b^4)*sin(c+d*x)/(a^3*(a^2-b^2)^2*d)+1/2*b^2*sin(c+d*x)/(a*(a^2-b^2)*d*(a+b*sec(c+d*x))^2)+3/2*b^2*(2*a^2-b^2)*sin(c+d*x)/(a^2*(a^2-b^2)^2*d*(a+b*sec(c+d*x)))],
[cos(c+d*x)^2/(a+b*sec(c+d*x))^3,x,8,1/2*(a^2+12*b^2)*x/a^5-b^3*(20*a^4-29*a^2*b^2+12*b^4)*arctanh(sqrt(a-b)*tan(1/2*(c+d*x))/sqrt(a+b))/(a^5*(a-b)^(5/2)*(a+b)^(5/2)*d)-3/2*b*(2*a^4-7*a^2*b^2+4*b^4)*sin(c+d*x)/(a^4*(a^2-b^2)^2*d)+1/2*(a^4-10*a^2*b^2+6*b^4)*cos(c+d*x)*sin(c+d*x)/(a^3*(a^2-b^2)^2*d)+1/2*b^2*cos(c+d*x)*sin(c+d*x)/(a*(a^2-b^2)*d*(a+b*sec(c+d*x))^2)+1/2*b^2*(7*a^2-4*b^2)*cos(c+d*x)*sin(c+d*x)/(a^2*(a^2-b^2)^2*d*(a+b*sec(c+d*x)))],
[sec(c+d*x)^6/(a+b*sec(c+d*x))^4,x,9,-4*a*arctanh(sin(c+d*x))/(b^5*d)+a^2*(8*a^6-28*a^4*b^2+35*a^2*b^4-20*b^6)*arctanh(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)*tan(c+d*x)/(b^4*(a^2-b^2)^2*d)-1/3*a^2*sec(c+d*x)^3*tan(c+d*x)/(b*(a^2-b^2)*d*(a+b*sec(c+d*x))^3)-1/6*a^2*(4*a^2-9*b^2)*sec(c+d*x)^2*tan(c+d*x)/(b^2*(a^2-b^2)^2*d*(a+b*sec(c+d*x))^2)+1/2*a^3*(4*a^4-11*a^2*b^2+12*b^4)*tan(c+d*x)/(b^4*(a^2-b^2)^3*d*(a+b*sec(c+d*x)))],
[sec(c+d*x)^5/(a+b*sec(c+d*x))^4,x,8,arctanh(sin(c+d*x))/(b^4*d)-a*(2*a^6-7*a^4*b^2+8*a^2*b^4-8*b^6)*arctanh(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*sec(c+d*x)^2*tan(c+d*x)/(b*(a^2-b^2)*d*(a+b*sec(c+d*x))^3)+1/6*a^3*(3*a^2-8*b^2)*tan(c+d*x)/(b^3*(a^2-b^2)^2*d*(a+b*sec(c+d*x))^2)-1/6*a^2*(9*a^4-28*a^2*b^2+34*b^4)*tan(c+d*x)/(b^3*(a^2-b^2)^3*d*(a+b*sec(c+d*x)))],
[sec(c+d*x)^4/(a+b*sec(c+d*x))^4,x,7,-b*(3*a^2+2*b^2)*arctanh(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*sec(c+d*x)*tan(c+d*x)/(b*(a^2-b^2)*d*(a+b*sec(c+d*x))^3)-1/6*a^2*(2*a^2-7*b^2)*tan(c+d*x)/(b^2*(a^2-b^2)^2*d*(a+b*sec(c+d*x))^2)+1/6*a*(2*a^4-5*a^2*b^2+18*b^4)*tan(c+d*x)/(b^2*(a^2-b^2)^3*d*(a+b*sec(c+d*x)))],
[sec(c+d*x)^3/(a+b*sec(c+d*x))^4,x,7,a*(a^2+4*b^2)*arctanh(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*tan(c+d*x)/(b*(a^2-b^2)*d*(a+b*sec(c+d*x))^3)+1/6*a*(a^2-6*b^2)*tan(c+d*x)/(b*(a^2-b^2)^2*d*(a+b*sec(c+d*x))^2)+1/6*(a^4-10*a^2*b^2-6*b^4)*tan(c+d*x)/(b*(a^2-b^2)^3*d*(a+b*sec(c+d*x)))],
[sec(c+d*x)^2/(a+b*sec(c+d*x))^4,x,7,-b*(4*a^2+b^2)*arctanh(sqrt(a-b)*tan(1/2*(c+d*x))/sqrt(a+b))/((a-b)^(7/2)*(a+b)^(7/2)*d)+1/3*a*tan(c+d*x)/((a^2-b^2)*d*(a+b*sec(c+d*x))^3)+1/6*(2*a^2+3*b^2)*tan(c+d*x)/((a^2-b^2)^2*d*(a+b*sec(c+d*x))^2)+1/6*a*(2*a^2+13*b^2)*tan(c+d*x)/((a^2-b^2)^3*d*(a+b*sec(c+d*x)))],
[sec(c+d*x)/(a+b*sec(c+d*x))^4,x,7,a*(2*a^2+3*b^2)*arctanh(sqrt(a-b)*tan(1/2*(c+d*x))/sqrt(a+b))/((a-b)^(7/2)*(a+b)^(7/2)*d)-1/3*b*tan(c+d*x)/((a^2-b^2)*d*(a+b*sec(c+d*x))^3)-5/6*a*b*tan(c+d*x)/((a^2-b^2)^2*d*(a+b*sec(c+d*x))^2)-1/6*b*(11*a^2+4*b^2)*tan(c+d*x)/((a^2-b^2)^3*d*(a+b*sec(c+d*x)))],
[1/(a+b*sec(c+d*x))^4,x,7,x/a^4-b*(8*a^6-8*a^4*b^2+7*a^2*b^4-2*b^6)*arctanh(sqrt(a-b)*tan(1/2*(c+d*x))/sqrt(a+b))/(a^4*(a-b)^(7/2)*(a+b)^(7/2)*d)+1/3*b^2*tan(c+d*x)/(a*(a^2-b^2)*d*(a+b*sec(c+d*x))^3)+1/6*b^2*(8*a^2-3*b^2)*tan(c+d*x)/(a^2*(a^2-b^2)^2*d*(a+b*sec(c+d*x))^2)+1/6*b^2*(26*a^4-17*a^2*b^2+6*b^4)*tan(c+d*x)/(a^3*(a^2-b^2)^3*d*(a+b*sec(c+d*x)))],
[cos(c+d*x)/(a+b*sec(c+d*x))^4,x,8,-4*b*x/a^5+b^2*(20*a^6-35*a^4*b^2+28*a^2*b^4-8*b^6)*arctanh(sqrt(a-b)*tan(1/2*(c+d*x))/sqrt(a+b))/(a^5*(a-b)^(7/2)*(a+b)^(7/2)*d)+1/6*(6*a^6-65*a^4*b^2+68*a^2*b^4-24*b^6)*sin(c+d*x)/(a^4*(a^2-b^2)^3*d)+1/3*b^2*sin(c+d*x)/(a*(a^2-b^2)*d*(a+b*sec(c+d*x))^3)+1/6*b^2*(9*a^2-4*b^2)*sin(c+d*x)/(a^2*(a^2-b^2)^2*d*(a+b*sec(c+d*x))^2)+1/2*b^2*(12*a^4-11*a^2*b^2+4*b^4)*sin(c+d*x)/(a^3*(a^2-b^2)^3*d*(a+b*sec(c+d*x)))],
[cos(c+d*x)^2/(a+b*sec(c+d*x))^4,x,9,1/2*(a^2+20*b^2)*x/a^6-b^3*(40*a^6-84*a^4*b^2+69*a^2*b^4-20*b^6)*arctanh(sqrt(a-b)*tan(1/2*(c+d*x))/sqrt(a+b))/(a^6*(a-b)^(7/2)*(a+b)^(7/2)*d)-1/6*b*(24*a^6-146*a^4*b^2+167*a^2*b^4-60*b^6)*sin(c+d*x)/(a^5*(a^2-b^2)^3*d)+1/2*(a^6-23*a^4*b^2+27*a^2*b^4-10*b^6)*cos(c+d*x)*sin(c+d*x)/(a^4*(a^2-b^2)^3*d)+1/3*b^2*cos(c+d*x)*sin(c+d*x)/(a*(a^2-b^2)*d*(a+b*sec(c+d*x))^3)+5/6*b^2*(2*a^2-b^2)*cos(c+d*x)*sin(c+d*x)/(a^2*(a^2-b^2)^2*d*(a+b*sec(c+d*x))^2)+1/6*b^2*(48*a^4-53*a^2*b^2+20*b^4)*cos(c+d*x)*sin(c+d*x)/(a^3*(a^2-b^2)^3*d*(a+b*sec(c+d*x)))],
[1/(3+5*sec(c+d*x)),x,2,-1/12*x+5/6*arctan(sin(c+d*x)/(3+cos(c+d*x)))/d],
[1/(3+5*sec(c+d*x))^2,x,4,29/576*x+35/288*arctan(sin(c+d*x)/(3+cos(c+d*x)))/d-25/48*tan(c+d*x)/(d*(3+5*sec(c+d*x)))],
[1/(3+5*sec(c+d*x))^3,x,5,-1007/55296*x+3055/27648*arctan(sin(c+d*x)/(3+cos(c+d*x)))/d-25/96*tan(c+d*x)/(d*(3+5*sec(c+d*x))^2)-125/4608*tan(c+d*x)/(d*(3+5*sec(c+d*x)))],
[1/(3+5*sec(c+d*x))^4,x,6,21553/2654208*x+11215/1327104*arctan(sin(c+d*x)/(3+cos(c+d*x)))/d-25/144*tan(c+d*x)/(d*(3+5*sec(c+d*x))^3)-25/4608*tan(c+d*x)/(d*(3+5*sec(c+d*x))^2)-16925/221184*tan(c+d*x)/(d*(3+5*sec(c+d*x)))],
[1/(5+3*sec(c+d*x)),x,3,1/5*x+3/20*log(2*cos(1/2*(c+d*x))-sin(1/2*(c+d*x)))/d-3/20*log(2*cos(1/2*(c+d*x))+sin(1/2*(c+d*x)))/d],
[1/(5+3*sec(c+d*x))^2,x,5,1/25*x+123/1600*log(2*cos(1/2*(c+d*x))-sin(1/2*(c+d*x)))/d-123/1600*log(2*cos(1/2*(c+d*x))+sin(1/2*(c+d*x)))/d+9/80*tan(c+d*x)/(d*(5+3*sec(c+d*x)))],
[1/(5+3*sec(c+d*x))^3,x,6,1/125*x+8361/256000*log(2*cos(1/2*(c+d*x))-sin(1/2*(c+d*x)))/d-8361/256000*log(2*cos(1/2*(c+d*x))+sin(1/2*(c+d*x)))/d+9/160*tan(c+d*x)/(d*(5+3*sec(c+d*x))^2)+963/12800*tan(c+d*x)/(d*(5+3*sec(c+d*x)))],
[1/(5+3*sec(c+d*x))^4,x,7,1/625*x+278151/20480000*log(2*cos(1/2*(c+d*x))-sin(1/2*(c+d*x)))/d-278151/20480000*log(2*cos(1/2*(c+d*x))+sin(1/2*(c+d*x)))/d+3/80*tan(c+d*x)/(d*(5+3*sec(c+d*x))^3)+519/12800*tan(c+d*x)/(d*(5+3*sec(c+d*x))^2)+38733/1024000*tan(c+d*x)/(d*(5+3*sec(c+d*x)))],

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

# n>0
[sec(c+d*x)^3*sqrt(a+b*sec(c+d*x)),x,5,2/15*(a-b)*(2*a^2-9*b^2)*cot(c+d*x)*EllipticE(sqrt(a+b*sec(c+d*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(a+b)*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/(b^3*d)+2/15*(a-b)*(2*a+9*b)*cot(c+d*x)*EllipticF(sqrt(a+b*sec(c+d*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(a+b)*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/(b^2*d)+2/5*(a+b*sec(c+d*x))^(3/2)*tan(c+d*x)/(b*d)-4/15*a*sqrt(a+b*sec(c+d*x))*tan(c+d*x)/(b*d)],
[sec(c+d*x)^2*sqrt(a+b*sec(c+d*x)),x,4,-2/3*a*(a-b)*cot(c+d*x)*EllipticE(sqrt(a+b*sec(c+d*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(a+b)*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/(b^2*d)-2/3*(a-b)*cot(c+d*x)*EllipticF(sqrt(a+b*sec(c+d*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(a+b)*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/(b*d)+2/3*sqrt(a+b*sec(c+d*x))*tan(c+d*x)/d],
[sec(c+d*x)*sqrt(a+b*sec(c+d*x)),x,3,-2*(a-b)*cot(c+d*x)*EllipticE(sqrt(a+b*sec(c+d*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(a+b)*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/(b*d)+2*(a-b)*cot(c+d*x)*EllipticF(sqrt(a+b*sec(c+d*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(a+b)*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/(b*d)],
[sqrt(a+b*sec(c+d*x)),x,1,-2*cot(c+d*x)*EllipticPi(sqrt(a+b)/sqrt(a+b*sec(c+d*x)),a/(a+b),sqrt((a-b)/(a+b)))*(a+b*sec(c+d*x))*sqrt(-b*(1-sec(c+d*x))/(a+b*sec(c+d*x)))*sqrt(b*(1+sec(c+d*x))/(a+b*sec(c+d*x)))/(d*sqrt(a+b))],
[cos(c+d*x)*sqrt(a+b*sec(c+d*x)),x,6,(a-b)*cot(c+d*x)*EllipticE(sqrt(a+b*sec(c+d*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(a+b)*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/(b*d)+cot(c+d*x)*EllipticF(sqrt(a+b*sec(c+d*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(a+b)*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/d-b*cot(c+d*x)*EllipticPi(sqrt(a+b*sec(c+d*x))/sqrt(a+b),(a+b)/a,sqrt((a+b)/(a-b)))*sqrt(a+b)*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/(a*d)+sin(c+d*x)*sqrt(a+b*sec(c+d*x))/d],
[cos(c+d*x)^2*sqrt(a+b*sec(c+d*x)),x,7,1/4*(a-b)*cot(c+d*x)*EllipticE(sqrt(a+b*sec(c+d*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(a+b)*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/(a*d)+1/4*(2*a+b)*cot(c+d*x)*EllipticF(sqrt(a+b*sec(c+d*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(a+b)*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/(a*d)-1/4*(4*a^2-b^2)*cot(c+d*x)*EllipticPi(sqrt(a+b*sec(c+d*x))/sqrt(a+b),(a+b)/a,sqrt((a+b)/(a-b)))*sqrt(a+b)*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/(a^2*d)+1/4*b*sin(c+d*x)*sqrt(a+b*sec(c+d*x))/(a*d)+1/2*cos(c+d*x)*sin(c+d*x)*sqrt(a+b*sec(c+d*x))/d],
[sec(c+d*x)^4*(a+b*sec(c+d*x))^(3/2),x,7,-2/315*(a-b)*(8*a^4+33*a^2*b^2+147*b^4)*cot(c+d*x)*EllipticE(sqrt(a+b*sec(c+d*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(a+b)*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/(b^4*d)-2/315*(a-b)*(8*a^3+6*a^2*b+39*a*b^2-147*b^3)*cot(c+d*x)*EllipticF(sqrt(a+b*sec(c+d*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(a+b)*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/(b^3*d)+2/315*(8*a^2+49*b^2)*(a+b*sec(c+d*x))^(3/2)*tan(c+d*x)/(b^2*d)-8/63*a*(a+b*sec(c+d*x))^(5/2)*tan(c+d*x)/(b^2*d)+2/9*sec(c+d*x)*(a+b*sec(c+d*x))^(5/2)*tan(c+d*x)/(b*d)+2/315*a*(8*a^2+39*b^2)*sqrt(a+b*sec(c+d*x))*tan(c+d*x)/(b^2*d)],
[sec(c+d*x)^3*(a+b*sec(c+d*x))^(3/2),x,6,4/105*a*(a-b)*(3*a^2-41*b^2)*cot(c+d*x)*EllipticE(sqrt(a+b*sec(c+d*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(a+b)*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/(b^3*d)+2/105*(a-b)*(6*a^2+57*a*b-25*b^2)*cot(c+d*x)*EllipticF(sqrt(a+b*sec(c+d*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(a+b)*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/(b^2*d)-4/35*a*(a+b*sec(c+d*x))^(3/2)*tan(c+d*x)/(b*d)+2/7*(a+b*sec(c+d*x))^(5/2)*tan(c+d*x)/(b*d)-2/105*(6*a^2-25*b^2)*sqrt(a+b*sec(c+d*x))*tan(c+d*x)/(b*d)],
[sec(c+d*x)^2*(a+b*sec(c+d*x))^(3/2),x,5,-2/5*(a-b)*(a^2+3*b^2)*cot(c+d*x)*EllipticE(sqrt(a+b*sec(c+d*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(a+b)*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/(b^2*d)-2/5*(a-3*b)*(a-b)*cot(c+d*x)*EllipticF(sqrt(a+b*sec(c+d*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(a+b)*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/(b*d)+2/5*(a+b*sec(c+d*x))^(3/2)*tan(c+d*x)/d+2/5*a*sqrt(a+b*sec(c+d*x))*tan(c+d*x)/d],
[sec(c+d*x)*(a+b*sec(c+d*x))^(3/2),x,4,-8/3*a*(a-b)*cot(c+d*x)*EllipticE(sqrt(a+b*sec(c+d*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(a+b)*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/(b*d)+2/3*(a-b)*(3*a-b)*cot(c+d*x)*EllipticF(sqrt(a+b*sec(c+d*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(a+b)*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/(b*d)+2/3*b*sqrt(a+b*sec(c+d*x))*tan(c+d*x)/d],
[(a+b*sec(c+d*x))^(3/2),x,5,-2*(a-b)*cot(c+d*x)*EllipticE(sqrt(a+b*sec(c+d*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(a+b)*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/d+2*(2*a-b)*cot(c+d*x)*EllipticF(sqrt(a+b*sec(c+d*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(a+b)*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/d-2*a*cot(c+d*x)*EllipticPi(sqrt(a+b*sec(c+d*x))/sqrt(a+b),(a+b)/a,sqrt((a+b)/(a-b)))*sqrt(a+b)*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/d],
[cos(c+d*x)*(a+b*sec(c+d*x))^(3/2),x,6,a*(a-b)*cot(c+d*x)*EllipticE(sqrt(a+b*sec(c+d*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(a+b)*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/(b*d)+(a+2*b)*cot(c+d*x)*EllipticF(sqrt(a+b*sec(c+d*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(a+b)*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/d-3*b*cot(c+d*x)*EllipticPi(sqrt(a+b*sec(c+d*x))/sqrt(a+b),(a+b)/a,sqrt((a+b)/(a-b)))*sqrt(a+b)*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/d+a*sin(c+d*x)*sqrt(a+b*sec(c+d*x))/d],
[cos(c+d*x)^2*(a+b*sec(c+d*x))^(3/2),x,7,5/4*(a-b)*cot(c+d*x)*EllipticE(sqrt(a+b*sec(c+d*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(a+b)*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/d+1/4*(2*a+5*b)*cot(c+d*x)*EllipticF(sqrt(a+b*sec(c+d*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(a+b)*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/d-1/4*(4*a^2+3*b^2)*cot(c+d*x)*EllipticPi(sqrt(a+b*sec(c+d*x))/sqrt(a+b),(a+b)/a,sqrt((a+b)/(a-b)))*sqrt(a+b)*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/(a*d)+5/4*b*sin(c+d*x)*sqrt(a+b*sec(c+d*x))/d+1/2*a*cos(c+d*x)*sin(c+d*x)*sqrt(a+b*sec(c+d*x))/d],
[sec(c+d*x)^4*(a+b*sec(c+d*x))^(5/2),x,8,-2/693*a*(a-b)*(8*a^4+51*a^2*b^2+741*b^4)*cot(c+d*x)*EllipticE(sqrt(a+b*sec(c+d*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(a+b)*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/(b^4*d)-2/693*(a-b)*(8*a^4+6*a^3*b+57*a^2*b^2-606*a*b^3+135*b^4)*cot(c+d*x)*EllipticF(sqrt(a+b*sec(c+d*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(a+b)*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/(b^3*d)+2/693*a*(8*a^2+67*b^2)*(a+b*sec(c+d*x))^(3/2)*tan(c+d*x)/(b^2*d)+2/693*(8*a^2+81*b^2)*(a+b*sec(c+d*x))^(5/2)*tan(c+d*x)/(b^2*d)-8/99*a*(a+b*sec(c+d*x))^(7/2)*tan(c+d*x)/(b^2*d)+2/11*sec(c+d*x)*(a+b*sec(c+d*x))^(7/2)*tan(c+d*x)/(b*d)+2/693*(8*a^4+57*a^2*b^2+135*b^4)*sqrt(a+b*sec(c+d*x))*tan(c+d*x)/(b^2*d)],
[sec(c+d*x)^3*(a+b*sec(c+d*x))^(5/2),x,7,2/315*(a-b)*(10*a^4-279*a^2*b^2-147*b^4)*cot(c+d*x)*EllipticE(sqrt(a+b*sec(c+d*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(a+b)*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/(b^3*d)+2/315*(a-b)*(10*a^3+165*a^2*b-114*a*b^2+147*b^3)*cot(c+d*x)*EllipticF(sqrt(a+b*sec(c+d*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(a+b)*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/(b^2*d)-2/315*(10*a^2-49*b^2)*(a+b*sec(c+d*x))^(3/2)*tan(c+d*x)/(b*d)-4/63*a*(a+b*sec(c+d*x))^(5/2)*tan(c+d*x)/(b*d)+2/9*(a+b*sec(c+d*x))^(7/2)*tan(c+d*x)/(b*d)-4/315*a*(5*a^2-57*b^2)*sqrt(a+b*sec(c+d*x))*tan(c+d*x)/(b*d)],
[sec(c+d*x)^2*(a+b*sec(c+d*x))^(5/2),x,6,-2/21*a*(a-b)*(3*a^2+29*b^2)*cot(c+d*x)*EllipticE(sqrt(a+b*sec(c+d*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(a+b)*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/(b^2*d)-2/21*(a-b)*(3*a^2-24*a*b+5*b^2)*cot(c+d*x)*EllipticF(sqrt(a+b*sec(c+d*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(a+b)*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/(b*d)+2/7*a*(a+b*sec(c+d*x))^(3/2)*tan(c+d*x)/d+2/7*(a+b*sec(c+d*x))^(5/2)*tan(c+d*x)/d+2/21*(3*a^2+5*b^2)*sqrt(a+b*sec(c+d*x))*tan(c+d*x)/d],
[sec(c+d*x)*(a+b*sec(c+d*x))^(5/2),x,5,-2/15*(a-b)*(23*a^2+9*b^2)*cot(c+d*x)*EllipticE(sqrt(a+b*sec(c+d*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(a+b)*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/(b*d)+2/15*(a-b)*(15*a^2-8*a*b+9*b^2)*cot(c+d*x)*EllipticF(sqrt(a+b*sec(c+d*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(a+b)*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/(b*d)+2/5*b*(a+b*sec(c+d*x))^(3/2)*tan(c+d*x)/d+16/15*a*b*sqrt(a+b*sec(c+d*x))*tan(c+d*x)/d],
[(a+b*sec(c+d*x))^(5/2),x,6,-14/3*a*(a-b)*cot(c+d*x)*EllipticE(sqrt(a+b*sec(c+d*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(a+b)*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/d+2/3*(9*a^2-7*a*b+b^2)*cot(c+d*x)*EllipticF(sqrt(a+b*sec(c+d*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(a+b)*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/d-2*a^2*cot(c+d*x)*EllipticPi(sqrt(a+b*sec(c+d*x))/sqrt(a+b),(a+b)/a,sqrt((a+b)/(a-b)))*sqrt(a+b)*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/d+2/3*b^2*sqrt(a+b*sec(c+d*x))*tan(c+d*x)/d],
[cos(c+d*x)*(a+b*sec(c+d*x))^(5/2),x,6,(a-b)*(a^2-2*b^2)*cot(c+d*x)*EllipticE(sqrt(a+b*sec(c+d*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(a+b)*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/(b*d)+(a^2+6*a*b-2*b^2)*cot(c+d*x)*EllipticF(sqrt(a+b*sec(c+d*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(a+b)*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/d-5*a*b*cot(c+d*x)*EllipticPi(sqrt(a+b*sec(c+d*x))/sqrt(a+b),(a+b)/a,sqrt((a+b)/(a-b)))*sqrt(a+b)*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/d+a^2*sin(c+d*x)*sqrt(a+b*sec(c+d*x))/d],
[cos(c+d*x)^2*(a+b*sec(c+d*x))^(5/2),x,7,9/4*a*(a-b)*cot(c+d*x)*EllipticE(sqrt(a+b*sec(c+d*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(a+b)*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/d+1/4*(2*a^2+9*a*b+8*b^2)*cot(c+d*x)*EllipticF(sqrt(a+b*sec(c+d*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(a+b)*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/d-1/4*(4*a^2+15*b^2)*cot(c+d*x)*EllipticPi(sqrt(a+b*sec(c+d*x))/sqrt(a+b),(a+b)/a,sqrt((a+b)/(a-b)))*sqrt(a+b)*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/d+9/4*a*b*sin(c+d*x)*sqrt(a+b*sec(c+d*x))/d+1/2*a^2*cos(c+d*x)*sin(c+d*x)*sqrt(a+b*sec(c+d*x))/d],
[cos(c+d*x)^3*(a+b*sec(c+d*x))^(5/2),x,8,1/24*(a-b)*(16*a^2+33*b^2)*cot(c+d*x)*EllipticE(sqrt(a+b*sec(c+d*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(a+b)*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/(b*d)+1/24*(16*a^2+26*a*b+33*b^2)*cot(c+d*x)*EllipticF(sqrt(a+b*sec(c+d*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(a+b)*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/d-5/8*b*(4*a^2+b^2)*cot(c+d*x)*EllipticPi(sqrt(a+b*sec(c+d*x))/sqrt(a+b),(a+b)/a,sqrt((a+b)/(a-b)))*sqrt(a+b)*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/(a*d)+1/24*(16*a^2+33*b^2)*sin(c+d*x)*sqrt(a+b*sec(c+d*x))/d+13/12*a*b*cos(c+d*x)*sin(c+d*x)*sqrt(a+b*sec(c+d*x))/d+1/3*a^2*cos(c+d*x)^2*sin(c+d*x)*sqrt(a+b*sec(c+d*x))/d],
[cos(c+d*x)^4*(a+b*sec(c+d*x))^(5/2),x,9,1/192*(a-b)*(284*a^2+15*b^2)*cot(c+d*x)*EllipticE(sqrt(a+b*sec(c+d*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(a+b)*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/(a*d)+1/192*(72*a^3+284*a^2*b+118*a*b^2+15*b^3)*cot(c+d*x)*EllipticF(sqrt(a+b*sec(c+d*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(a+b)*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/(a*d)-1/64*(48*a^4+120*a^2*b^2-5*b^4)*cot(c+d*x)*EllipticPi(sqrt(a+b*sec(c+d*x))/sqrt(a+b),(a+b)/a,sqrt((a+b)/(a-b)))*sqrt(a+b)*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/(a^2*d)+1/192*b*(284*a^2+15*b^2)*sin(c+d*x)*sqrt(a+b*sec(c+d*x))/(a*d)+1/96*(36*a^2+59*b^2)*cos(c+d*x)*sin(c+d*x)*sqrt(a+b*sec(c+d*x))/d+17/24*a*b*cos(c+d*x)^2*sin(c+d*x)*sqrt(a+b*sec(c+d*x))/d+1/4*a^2*cos(c+d*x)^3*sin(c+d*x)*sqrt(a+b*sec(c+d*x))/d],
[(a+b*sec(c+d*x))^(7/2),x,7,-2/15*(a-b)*(58*a^2+9*b^2)*cot(c+d*x)*EllipticE(sqrt(a+b*sec(c+d*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(a+b)*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/d+2/15*(60*a^3-58*a^2*b+22*a*b^2-9*b^3)*cot(c+d*x)*EllipticF(sqrt(a+b*sec(c+d*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(a+b)*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/d-2*a^3*cot(c+d*x)*EllipticPi(sqrt(a+b*sec(c+d*x))/sqrt(a+b),(a+b)/a,sqrt((a+b)/(a-b)))*sqrt(a+b)*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/d+2/5*b^2*(a+b*sec(c+d*x))^(3/2)*tan(c+d*x)/d+26/15*a*b^2*sqrt(a+b*sec(c+d*x))*tan(c+d*x)/d],

# n<0
[sec(c+d*x)^5/sqrt(a+b*sec(c+d*x)),x,6,8/105*a*(a-b)*(12*a^2+11*b^2)*cot(c+d*x)*EllipticE(sqrt(a+b*sec(c+d*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(a+b)*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/(b^5*d)+2/105*(48*a^3-12*a^2*b+44*a*b^2+25*b^3)*cot(c+d*x)*EllipticF(sqrt(a+b*sec(c+d*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(a+b)*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/(b^4*d)+2/105*(24*a^2+25*b^2)*sqrt(a+b*sec(c+d*x))*tan(c+d*x)/(b^3*d)-12/35*a*sec(c+d*x)*sqrt(a+b*sec(c+d*x))*tan(c+d*x)/(b^2*d)+2/7*sec(c+d*x)^2*sqrt(a+b*sec(c+d*x))*tan(c+d*x)/(b*d)],
[sec(c+d*x)^4/sqrt(a+b*sec(c+d*x)),x,5,-2/15*(a-b)*(8*a^2+9*b^2)*cot(c+d*x)*EllipticE(sqrt(a+b*sec(c+d*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(a+b)*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/(b^4*d)-2/15*(8*a^2-2*a*b+9*b^2)*cot(c+d*x)*EllipticF(sqrt(a+b*sec(c+d*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(a+b)*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/(b^3*d)-8/15*a*sqrt(a+b*sec(c+d*x))*tan(c+d*x)/(b^2*d)+2/5*sec(c+d*x)*sqrt(a+b*sec(c+d*x))*tan(c+d*x)/(b*d)],
[sec(c+d*x)^3/sqrt(a+b*sec(c+d*x)),x,4,4/3*a*(a-b)*cot(c+d*x)*EllipticE(sqrt(a+b*sec(c+d*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(a+b)*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/(b^3*d)+2/3*(2*a+b)*cot(c+d*x)*EllipticF(sqrt(a+b*sec(c+d*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(a+b)*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/(b^2*d)+2/3*sqrt(a+b*sec(c+d*x))*tan(c+d*x)/(b*d)],
[sec(c+d*x)^2/sqrt(a+b*sec(c+d*x)),x,3,-2*(a-b)*cot(c+d*x)*EllipticE(sqrt(a+b*sec(c+d*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(a+b)*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/(b^2*d)-2*cot(c+d*x)*EllipticF(sqrt(a+b*sec(c+d*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(a+b)*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/(b*d)],
[sec(c+d*x)/sqrt(a+b*sec(c+d*x)),x,1,2*cot(c+d*x)*EllipticF(sqrt(a+b*sec(c+d*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(a+b)*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/(b*d)],
[1/sqrt(a+b*sec(c+d*x)),x,1,-2*cot(c+d*x)*EllipticPi(sqrt(a+b*sec(c+d*x))/sqrt(a+b),(a+b)/a,sqrt((a+b)/(a-b)))*sqrt(a+b)*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/(a*d)],
[cos(c+d*x)/sqrt(a+b*sec(c+d*x)),x,6,(a-b)*cot(c+d*x)*EllipticE(sqrt(a+b*sec(c+d*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(a+b)*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/(a*b*d)+cot(c+d*x)*EllipticF(sqrt(a+b*sec(c+d*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(a+b)*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/(a*d)+b*cot(c+d*x)*EllipticPi(sqrt(a+b*sec(c+d*x))/sqrt(a+b),(a+b)/a,sqrt((a+b)/(a-b)))*sqrt(a+b)*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/(a^2*d)+sin(c+d*x)*sqrt(a+b*sec(c+d*x))/(a*d)],
[cos(c+d*x)^2/sqrt(a+b*sec(c+d*x)),x,7,-3/4*(a-b)*cot(c+d*x)*EllipticE(sqrt(a+b*sec(c+d*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(a+b)*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/(a^2*d)+1/4*(2*a-3*b)*cot(c+d*x)*EllipticF(sqrt(a+b*sec(c+d*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(a+b)*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/(a^2*d)-1/4*(4*a^2+3*b^2)*cot(c+d*x)*EllipticPi(sqrt(a+b*sec(c+d*x))/sqrt(a+b),(a+b)/a,sqrt((a+b)/(a-b)))*sqrt(a+b)*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/(a^3*d)-3/4*b*sin(c+d*x)*sqrt(a+b*sec(c+d*x))/(a^2*d)+1/2*cos(c+d*x)*sin(c+d*x)*sqrt(a+b*sec(c+d*x))/(a*d)],
[sec(c+d*x)^5/(a+b*sec(c+d*x))^(3/2),x,6,-2/5*(16*a^4-8*a^2*b^2-3*b^4)*cot(c+d*x)*EllipticE(sqrt(a+b*sec(c+d*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/(b^5*d*sqrt(a+b))-2/5*(4*a+3*b)*(4*a^2+b^2)*cot(c+d*x)*EllipticF(sqrt(a+b*sec(c+d*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/(b^4*d*sqrt(a+b))-2*a^2*sec(c+d*x)^2*tan(c+d*x)/(b*(a^2-b^2)*d*sqrt(a+b*sec(c+d*x)))-2/5*a*(8*a^2-3*b^2)*sqrt(a+b*sec(c+d*x))*tan(c+d*x)/(b^3*(a^2-b^2)*d)+2/5*(6*a^2-b^2)*sec(c+d*x)*sqrt(a+b*sec(c+d*x))*tan(c+d*x)/(b^2*(a^2-b^2)*d)],
[sec(c+d*x)^4/(a+b*sec(c+d*x))^(3/2),x,5,2/3*a*(8*a^2-5*b^2)*cot(c+d*x)*EllipticE(sqrt(a+b*sec(c+d*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/(b^4*d*sqrt(a+b))+2/3*(2*a+b)*(4*a+b)*cot(c+d*x)*EllipticF(sqrt(a+b*sec(c+d*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/(b^3*d*sqrt(a+b))-2*a^2*sec(c+d*x)*tan(c+d*x)/(b*(a^2-b^2)*d*sqrt(a+b*sec(c+d*x)))+2/3*(4*a^2-b^2)*sqrt(a+b*sec(c+d*x))*tan(c+d*x)/(b^2*(a^2-b^2)*d)],
[sec(c+d*x)^3/(a+b*sec(c+d*x))^(3/2),x,4,-2*(2*a^2-b^2)*cot(c+d*x)*EllipticE(sqrt(a+b*sec(c+d*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/(b^3*d*sqrt(a+b))-2*(2*a+b)*cot(c+d*x)*EllipticF(sqrt(a+b*sec(c+d*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/(b^2*d*sqrt(a+b))-2*a^2*tan(c+d*x)/(b*(a^2-b^2)*d*sqrt(a+b*sec(c+d*x)))],
[sec(c+d*x)^2/(a+b*sec(c+d*x))^(3/2),x,4,2*a*cot(c+d*x)*EllipticE(sqrt(a+b*sec(c+d*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/(b^2*d*sqrt(a+b))+2*cot(c+d*x)*EllipticF(sqrt(a+b*sec(c+d*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/(b*d*sqrt(a+b))+2*a*tan(c+d*x)/((a^2-b^2)*d*sqrt(a+b*sec(c+d*x)))],
[sec(c+d*x)/(a+b*sec(c+d*x))^(3/2),x,5,-2*cot(c+d*x)*EllipticE(sqrt(a+b*sec(c+d*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/(b*d*sqrt(a+b))+2*cot(c+d*x)*EllipticF(sqrt(a+b*sec(c+d*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/(b*d*sqrt(a+b))-2*b*tan(c+d*x)/((a^2-b^2)*d*sqrt(a+b*sec(c+d*x)))],
[1/(a+b*sec(c+d*x))^(3/2),x,6,2*cot(c+d*x)*EllipticE(sqrt(a+b*sec(c+d*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/(a*d*sqrt(a+b))-2*cot(c+d*x)*EllipticF(sqrt(a+b*sec(c+d*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/(a*d*sqrt(a+b))-2*cot(c+d*x)*EllipticPi(sqrt(a+b*sec(c+d*x))/sqrt(a+b),(a+b)/a,sqrt((a+b)/(a-b)))*sqrt(a+b)*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/(a^2*d)+2*b^2*tan(c+d*x)/(a*(a^2-b^2)*d*sqrt(a+b*sec(c+d*x)))],
[cos(c+d*x)/(a+b*sec(c+d*x))^(3/2),x,7,(a^2-3*b^2)*cot(c+d*x)*EllipticE(sqrt(a+b*sec(c+d*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/(a^2*b*d*sqrt(a+b))+(a+3*b)*cot(c+d*x)*EllipticF(sqrt(a+b*sec(c+d*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/(a^2*d*sqrt(a+b))+3*b*cot(c+d*x)*EllipticPi(sqrt(a+b*sec(c+d*x))/sqrt(a+b),(a+b)/a,sqrt((a+b)/(a-b)))*sqrt(a+b)*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/(a^3*d)+sin(c+d*x)/(a*d*sqrt(a+b*sec(c+d*x)))+b*(a^2-3*b^2)*tan(c+d*x)/(a^2*(a^2-b^2)*d*sqrt(a+b*sec(c+d*x)))],
[cos(c+d*x)^2/(a+b*sec(c+d*x))^(3/2),x,8,-1/4*(7*a^2-15*b^2)*cot(c+d*x)*EllipticE(sqrt(a+b*sec(c+d*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/(a^3*d*sqrt(a+b))+1/4*(2*a^2-5*a*b-15*b^2)*cot(c+d*x)*EllipticF(sqrt(a+b*sec(c+d*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/(a^3*d*sqrt(a+b))-1/4*(4*a^2+15*b^2)*cot(c+d*x)*EllipticPi(sqrt(a+b*sec(c+d*x))/sqrt(a+b),(a+b)/a,sqrt((a+b)/(a-b)))*sqrt(a+b)*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/(a^4*d)-5/4*b*sin(c+d*x)/(a^2*d*sqrt(a+b*sec(c+d*x)))+1/2*cos(c+d*x)*sin(c+d*x)/(a*d*sqrt(a+b*sec(c+d*x)))-1/4*b^2*(7*a^2-15*b^2)*tan(c+d*x)/(a^3*(a^2-b^2)*d*sqrt(a+b*sec(c+d*x)))],
[sec(c+d*x)^5/(a+b*sec(c+d*x))^(5/2),x,6,8/3*a*(4*a^4-7*a^2*b^2+2*b^4)*cot(c+d*x)*EllipticE(sqrt(a+b*sec(c+d*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/((a-b)*b^5*(a+b)^(3/2)*d)+2/3*(16*a^4+12*a^3*b-16*a^2*b^2-9*a*b^3-b^4)*cot(c+d*x)*EllipticF(sqrt(a+b*sec(c+d*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/((a-b)*b^4*(a+b)^(3/2)*d)-2/3*a^2*sec(c+d*x)^2*tan(c+d*x)/(b*(a^2-b^2)*d*(a+b*sec(c+d*x))^(3/2))+4/3*a^3*(3*a^2-5*b^2)*tan(c+d*x)/(b^3*(a^2-b^2)^2*d*sqrt(a+b*sec(c+d*x)))+2/3*(2*a^2-b^2)*sqrt(a+b*sec(c+d*x))*tan(c+d*x)/(b^3*(a^2-b^2)*d)],
[sec(c+d*x)^4/(a+b*sec(c+d*x))^(5/2),x,5,-2/3*(8*a^4-15*a^2*b^2+3*b^4)*cot(c+d*x)*EllipticE(sqrt(a+b*sec(c+d*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/((a-b)*b^4*(a+b)^(3/2)*d)-2/3*(8*a^3+6*a^2*b-9*a*b^2-3*b^3)*cot(c+d*x)*EllipticF(sqrt(a+b*sec(c+d*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/((a-b)*b^3*(a+b)^(3/2)*d)-2/3*a^2*sec(c+d*x)*tan(c+d*x)/(b*(a^2-b^2)*d*(a+b*sec(c+d*x))^(3/2))-8/3*a^2*(a^2-2*b^2)*tan(c+d*x)/(b^2*(a^2-b^2)^2*d*sqrt(a+b*sec(c+d*x)))],
[sec(c+d*x)^3/(a+b*sec(c+d*x))^(5/2),x,5,4/3*a*(a^2-3*b^2)*cot(c+d*x)*EllipticE(sqrt(a+b*sec(c+d*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/((a-b)*b^3*(a+b)^(3/2)*d)+2/3*(2*a^2+3*a*b-3*b^2)*cot(c+d*x)*EllipticF(sqrt(a+b*sec(c+d*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/((a-b)*b^2*(a+b)^(3/2)*d)-2/3*a^2*tan(c+d*x)/(b*(a^2-b^2)*d*(a+b*sec(c+d*x))^(3/2))+4/3*a*(a^2-3*b^2)*tan(c+d*x)/(b*(a^2-b^2)^2*d*sqrt(a+b*sec(c+d*x)))],
[sec(c+d*x)^2/(a+b*sec(c+d*x))^(5/2),x,5,2/3*(a^2+3*b^2)*cot(c+d*x)*EllipticE(sqrt(a+b*sec(c+d*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/((a-b)*b^2*(a+b)^(3/2)*d)+2/3*(a-3*b)*cot(c+d*x)*EllipticF(sqrt(a+b*sec(c+d*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/((a-b)*b*(a+b)^(3/2)*d)+2/3*a*tan(c+d*x)/((a^2-b^2)*d*(a+b*sec(c+d*x))^(3/2))+2/3*(a^2+3*b^2)*tan(c+d*x)/((a^2-b^2)^2*d*sqrt(a+b*sec(c+d*x)))],
[sec(c+d*x)/(a+b*sec(c+d*x))^(5/2),x,5,-8/3*a*cot(c+d*x)*EllipticE(sqrt(a+b*sec(c+d*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/((a-b)*b*(a+b)^(3/2)*d)+2/3*(3*a-b)*cot(c+d*x)*EllipticF(sqrt(a+b*sec(c+d*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/((a-b)*b*(a+b)^(3/2)*d)-2/3*b*tan(c+d*x)/((a^2-b^2)*d*(a+b*sec(c+d*x))^(3/2))-8/3*a*b*tan(c+d*x)/((a^2-b^2)^2*d*sqrt(a+b*sec(c+d*x)))],
[1/(a+b*sec(c+d*x))^(5/2),x,7,2/3*(7*a^2-3*b^2)*cot(c+d*x)*EllipticE(sqrt(a+b*sec(c+d*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/(a^2*(a-b)*(a+b)^(3/2)*d)-2/3*(6*a^2-a*b-3*b^2)*cot(c+d*x)*EllipticF(sqrt(a+b*sec(c+d*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/(a^2*(a-b)*(a+b)^(3/2)*d)-2*cot(c+d*x)*EllipticPi(sqrt(a+b*sec(c+d*x))/sqrt(a+b),(a+b)/a,sqrt((a+b)/(a-b)))*sqrt(a+b)*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/(a^3*d)+2/3*b^2*tan(c+d*x)/(a*(a^2-b^2)*d*(a+b*sec(c+d*x))^(3/2))+2/3*b^2*(7*a^2-3*b^2)*tan(c+d*x)/(a^2*(a^2-b^2)^2*d*sqrt(a+b*sec(c+d*x)))],
[cos(c+d*x)/(a+b*sec(c+d*x))^(5/2),x,8,sin(c+d*x)/(a*d*(a+b*sec(c+d*x))^(3/2))+1/3*(3*a^4-26*a^2*b^2+15*b^4)*cot(c+d*x)*EllipticE(sqrt(a+b*sec(c+d*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/(a^3*(a-b)*b*(a+b)^(3/2)*d)+1/3*(3*a^3+21*a^2*b-5*a*b^2-15*b^3)*cot(c+d*x)*EllipticF(sqrt(a+b*sec(c+d*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/(a^3*(a-b)*(a+b)^(3/2)*d)+5*b*cot(c+d*x)*EllipticPi(sqrt(a+b*sec(c+d*x))/sqrt(a+b),(a+b)/a,sqrt((a+b)/(a-b)))*sqrt(a+b)*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/(a^4*d)+1/3*b*(3*a^2-5*b^2)*tan(c+d*x)/(a^2*(a^2-b^2)*d*(a+b*sec(c+d*x))^(3/2))+1/3*b*(3*a^4-26*a^2*b^2+15*b^4)*tan(c+d*x)/(a^3*(a^2-b^2)^2*d*sqrt(a+b*sec(c+d*x)))],
[cos(c+d*x)^2/(a+b*sec(c+d*x))^(5/2),x,9,-7/4*b*sin(c+d*x)/(a^2*d*(a+b*sec(c+d*x))^(3/2))+1/2*cos(c+d*x)*sin(c+d*x)/(a*d*(a+b*sec(c+d*x))^(3/2))-1/12*(33*a^4-170*a^2*b^2+105*b^4)*cot(c+d*x)*EllipticE(sqrt(a+b*sec(c+d*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/(a^4*(a-b)*(a+b)^(3/2)*d)+1/12*(a+3*b)*(6*a^3-45*a^2*b+35*b^3)*cot(c+d*x)*EllipticF(sqrt(a+b*sec(c+d*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/(a^4*(a-b)*(a+b)^(3/2)*d)-1/4*(4*a^2+35*b^2)*cot(c+d*x)*EllipticPi(sqrt(a+b*sec(c+d*x))/sqrt(a+b),(a+b)/a,sqrt((a+b)/(a-b)))*sqrt(a+b)*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/(a^5*d)-1/12*b^2*(27*a^2-35*b^2)*tan(c+d*x)/(a^3*(a^2-b^2)*d*(a+b*sec(c+d*x))^(3/2))-1/12*b^2*(33*a^4-170*a^2*b^2+105*b^4)*tan(c+d*x)/(a^4*(a^2-b^2)^2*d*sqrt(a+b*sec(c+d*x)))],
[1/(a+b*sec(c+d*x))^(7/2),x,8,2/15*(58*a^4-41*a^2*b^2+15*b^4)*cot(c+d*x)*EllipticE(sqrt(a+b*sec(c+d*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/(a^3*(a-b)^2*(a+b)^(5/2)*d)-2/15*(45*a^4-13*a^3*b-36*a^2*b^2+5*a*b^3+15*b^4)*cot(c+d*x)*EllipticF(sqrt(a+b*sec(c+d*x))/sqrt(a+b),sqrt((a+b)/(a-b)))*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/(a^3*(a-b)^2*(a+b)^(5/2)*d)-2*cot(c+d*x)*EllipticPi(sqrt(a+b*sec(c+d*x))/sqrt(a+b),(a+b)/a,sqrt((a+b)/(a-b)))*sqrt(a+b)*sqrt(b*(1-sec(c+d*x))/(a+b))*sqrt(-b*(1+sec(c+d*x))/(a-b))/(a^4*d)+2/5*b^2*tan(c+d*x)/(a*(a^2-b^2)*d*(a+b*sec(c+d*x))^(5/2))+2/15*b^2*(13*a^2-5*b^2)*tan(c+d*x)/(a^2*(a^2-b^2)^2*d*(a+b*sec(c+d*x))^(3/2))+2/15*b^2*(58*a^4-41*a^2*b^2+15*b^4)*tan(c+d*x)/(a^3*(a^2-b^2)^3*d*sqrt(a+b*sec(c+d*x)))],

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

# n>0
[sec(c+d*x)^(5/2)*(a+b*sec(c+d*x)),x,8,2/3*a*sec(c+d*x)^(3/2)*sin(c+d*x)/d+2/5*b*sec(c+d*x)^(5/2)*sin(c+d*x)/d+6/5*b*sin(c+d*x)*sqrt(sec(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(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],
[sec(c+d*x)^(3/2)*(a+b*sec(c+d*x)),x,7,2/3*b*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*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],
[sec(c+d*x)^(1/2)*(a+b*sec(c+d*x)),x,6,2*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*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*sec(c+d*x))/sec(c+d*x)^(1/2),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))*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*sec(c+d*x))/sec(c+d*x)^(3/2),x,6,2/3*a*sin(c+d*x)/(d*sqrt(sec(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(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*sec(c+d*x))/sec(c+d*x)^(5/2),x,7,2/5*a*sin(c+d*x)/(d*sec(c+d*x)^(3/2))+2/3*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+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*sec(c+d*x))/sec(c+d*x)^(7/2),x,8,2/7*a*sin(c+d*x)/(d*sec(c+d*x)^(5/2))+2/5*b*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*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+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],
[sec(c+d*x)^(5/2)*(a+b*sec(c+d*x))^2,x,9,2/21*(7*a^2+5*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*b^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*(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],
[sec(c+d*x)^(3/2)*(a+b*sec(c+d*x))^2,x,8,4/3*a*b*sec(c+d*x)^(3/2)*sin(c+d*x)/d+2/5*b^2*sec(c+d*x)^(5/2)*sin(c+d*x)/d+2/5*(5*a^2+3*b^2)*sin(c+d*x)*sqrt(sec(c+d*x))/d-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],
[sec(c+d*x)^(1/2)*(a+b*sec(c+d*x))^2,x,7,2/3*b^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*(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*sec(c+d*x))^2/sec(c+d*x)^(1/2),x,6,2*b^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*sec(c+d*x))^2/sec(c+d*x)^(3/2),x,6,2/3*a^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*(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*sec(c+d*x))^2/sec(c+d*x)^(5/2),x,7,2/5*a^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*(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*sec(c+d*x))^2/sec(c+d*x)^(7/2),x,8,2/7*a^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*(5*a^2+7*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*(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],
[sec(c+d*x)^(3/2)*(a+b*sec(c+d*x))^3,x,9,2/21*b*(21*a^2+5*b^2)*sec(c+d*x)^(3/2)*sin(c+d*x)/d+32/35*a*b^2*sec(c+d*x)^(5/2)*sin(c+d*x)/d+2/7*b^2*sec(c+d*x)^(5/2)*(a+b*sec(c+d*x))*sin(c+d*x)/d+2/5*a*(5*a^2+9*b^2)*sin(c+d*x)*sqrt(sec(c+d*x))/d-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],
[sec(c+d*x)^(1/2)*(a+b*sec(c+d*x))^3,x,8,8/5*a*b^2*sec(c+d*x)^(3/2)*sin(c+d*x)/d+2/5*b^2*sec(c+d*x)^(3/2)*(a+b*sec(c+d*x))*sin(c+d*x)/d+6/5*b*(5*a^2+b^2)*sin(c+d*x)*sqrt(sec(c+d*x))/d-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*sec(c+d*x))^3/sec(c+d*x)^(1/2),x,7,16/3*a*b^2*sin(c+d*x)*sqrt(sec(c+d*x))/d+2/3*b^2*(a+b*sec(c+d*x))*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*sec(c+d*x))^3/sec(c+d*x)^(3/2),x,7,2/3*a^2*(a+b*sec(c+d*x))*sin(c+d*x)/(d*sqrt(sec(c+d*x)))-2/3*b*(a^2-3*b^2)*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*sec(c+d*x))^3/sec(c+d*x)^(5/2),x,7,2/5*a^2*(a+b*sec(c+d*x))*sin(c+d*x)/(d*sec(c+d*x)^(3/2))+8/5*a^2*b*sin(c+d*x)/(d*sqrt(sec(c+d*x)))+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*sec(c+d*x))^3/sec(c+d*x)^(7/2),x,8,32/35*a^2*b*sin(c+d*x)/(d*sec(c+d*x)^(3/2))+2/7*a^2*(a+b*sec(c+d*x))*sin(c+d*x)/(d*sec(c+d*x)^(5/2))+2/21*a*(5*a^2+21*b^2)*sin(c+d*x)/(d*sqrt(sec(c+d*x)))+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*sec(c+d*x))^3/sec(c+d*x)^(9/2),x,9,40/63*a^2*b*sin(c+d*x)/(d*sec(c+d*x)^(5/2))+2/45*a*(7*a^2+27*b^2)*sin(c+d*x)/(d*sec(c+d*x)^(3/2))+2/9*a^2*(a+b*sec(c+d*x))*sin(c+d*x)/(d*sec(c+d*x)^(7/2))+2/21*b*(15*a^2+7*b^2)*sin(c+d*x)/(d*sqrt(sec(c+d*x)))+2/15*a*(7*a^2+27*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*(15*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],
[sec(c+d*x)^(3/2)*(a+b*sec(c+d*x))^4,x,10,8/21*a*b*(7*a^2+5*b^2)*sec(c+d*x)^(3/2)*sin(c+d*x)/d+14/45*b^2*(7*a^2+b^2)*sec(c+d*x)^(5/2)*sin(c+d*x)/d+44/63*a*b^3*sec(c+d*x)^(7/2)*sin(c+d*x)/d+2/9*b^2*sec(c+d*x)^(5/2)*(a+b*sec(c+d*x))^2*sin(c+d*x)/d+2/15*(15*a^4+54*a^2*b^2+7*b^4)*sin(c+d*x)*sqrt(sec(c+d*x))/d-2/15*(15*a^4+54*a^2*b^2+7*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))/d+8/21*a*b*(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],
[sec(c+d*x)^(1/2)*(a+b*sec(c+d*x))^4,x,9,2/21*b^2*(39*a^2+5*b^2)*sec(c+d*x)^(3/2)*sin(c+d*x)/d+36/35*a*b^3*sec(c+d*x)^(5/2)*sin(c+d*x)/d+2/7*b^2*sec(c+d*x)^(3/2)*(a+b*sec(c+d*x))^2*sin(c+d*x)/d+8/5*a*b*(5*a^2+3*b^2)*sin(c+d*x)*sqrt(sec(c+d*x))/d-8/5*a*b*(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+2/21*(21*a^4+42*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))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/d],
[(a+b*sec(c+d*x))^4/sec(c+d*x)^(1/2),x,8,28/15*a*b^3*sec(c+d*x)^(3/2)*sin(c+d*x)/d+2/5*b^2*(29*a^2+3*b^2)*sin(c+d*x)*sqrt(sec(c+d*x))/d+2/5*b^2*(a+b*sec(c+d*x))^2*sin(c+d*x)*sqrt(sec(c+d*x))/d+2/5*(5*a^4-30*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))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/d+8/3*a*b*(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*sec(c+d*x))^4/sec(c+d*x)^(3/2),x,8,-2/3*b^2*(a^2-b^2)*sec(c+d*x)^(3/2)*sin(c+d*x)/d+2/3*a^2*(a+b*sec(c+d*x))^2*sin(c+d*x)/(d*sqrt(sec(c+d*x)))-4/3*a*b*(a^2-6*b^2)*sin(c+d*x)*sqrt(sec(c+d*x))/d+8*a*b*(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^4+18*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))*sqrt(cos(c+d*x))*sqrt(sec(c+d*x))/d],
[(a+b*sec(c+d*x))^4/sec(c+d*x)^(5/2),x,8,2/5*a^2*(a+b*sec(c+d*x))^2*sin(c+d*x)/(d*sec(c+d*x)^(3/2))+28/15*a^3*b*sin(c+d*x)/(d*sqrt(sec(c+d*x)))-2/5*b^2*(a^2-5*b^2)*sin(c+d*x)*sqrt(sec(c+d*x))/d+2/5*(3*a^4+30*a^2*b^2-5*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))/d+8/3*a*b*(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*sec(c+d*x))^4/sec(c+d*x)^(7/2),x,8,36/35*a^3*b*sin(c+d*x)/(d*sec(c+d*x)^(3/2))+2/7*a^2*(a+b*sec(c+d*x))^2*sin(c+d*x)/(d*sec(c+d*x)^(5/2))+2/21*a^2*(5*a^2+39*b^2)*sin(c+d*x)/(d*sqrt(sec(c+d*x)))+8/5*a*b*(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+2/21*(5*a^4+42*a^2*b^2+21*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))/d],
[(a+b*sec(c+d*x))^4/sec(c+d*x)^(9/2),x,9,44/63*a^3*b*sin(c+d*x)/(d*sec(c+d*x)^(5/2))+14/45*a^2*(a^2+7*b^2)*sin(c+d*x)/(d*sec(c+d*x)^(3/2))+2/9*a^2*(a+b*sec(c+d*x))^2*sin(c+d*x)/(d*sec(c+d*x)^(7/2))+8/21*a*b*(5*a^2+7*b^2)*sin(c+d*x)/(d*sqrt(sec(c+d*x)))+2/15*(7*a^4+54*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))/d+8/21*a*b*(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*sec(c+d*x))^4/sec(c+d*x)^(11/2),x,10,52/99*a^3*b*sin(c+d*x)/(d*sec(c+d*x)^(7/2))+2/77*a^2*(9*a^2+59*b^2)*sin(c+d*x)/(d*sec(c+d*x)^(5/2))+8/45*a*b*(7*a^2+9*b^2)*sin(c+d*x)/(d*sec(c+d*x)^(3/2))+2/11*a^2*(a+b*sec(c+d*x))^2*sin(c+d*x)/(d*sec(c+d*x)^(9/2))+2/231*(45*a^4+330*a^2*b^2+77*b^4)*sin(c+d*x)/(d*sqrt(sec(c+d*x)))+8/15*a*b*(7*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+2/231*(45*a^4+330*a^2*b^2+77*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))/d],

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

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

# n>0
[sec(c+d*x)^(3/2)*(a+b*sec(c+d*x))^(1/2),x,12,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+b)))*sqrt((b+a*cos(c+d*x))/(a+b))*sqrt(sec(c+d*x))/(d*sqrt(a+b*sec(c+d*x)))+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*a/(a+b)))*sqrt((b+a*cos(c+d*x))/(a+b))*sqrt(sec(c+d*x))/(d*sqrt(a+b*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*a/(a+b)))*sqrt(a+b*sec(c+d*x))/(d*sqrt((b+a*cos(c+d*x))/(a+b))*sqrt(sec(c+d*x)))+sin(c+d*x)*sqrt(sec(c+d*x))*sqrt(a+b*sec(c+d*x))/d],
[sec(c+d*x)^(1/2)*(a+b*sec(c+d*x))^(1/2),x,7,2*a*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2*a/(a+b)))*sqrt((b+a*cos(c+d*x))/(a+b))*sqrt(sec(c+d*x))/(d*sqrt(a+b*sec(c+d*x)))+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*a/(a+b)))*sqrt((b+a*cos(c+d*x))/(a+b))*sqrt(sec(c+d*x))/(d*sqrt(a+b*sec(c+d*x)))],
[(a+b*sec(c+d*x))^(1/2)/sec(c+d*x)^(1/2),x,3,2*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+b)))*sqrt(a+b*sec(c+d*x))/(d*sqrt((b+a*cos(c+d*x))/(a+b))*sqrt(sec(c+d*x)))],
[(a+b*sec(c+d*x))^(1/2)/sec(c+d*x)^(3/2),x,8,2/3*(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*a/(a+b)))*sqrt((b+a*cos(c+d*x))/(a+b))*sqrt(sec(c+d*x))/(a*d*sqrt(a+b*sec(c+d*x)))+2/3*sin(c+d*x)*sqrt(a+b*sec(c+d*x))/(d*sqrt(sec(c+d*x)))+2/3*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+b)))*sqrt(a+b*sec(c+d*x))/(a*d*sqrt((b+a*cos(c+d*x))/(a+b))*sqrt(sec(c+d*x)))],
[(a+b*sec(c+d*x))^(1/2)/sec(c+d*x)^(5/2),x,9,-4/15*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*a/(a+b)))*sqrt((b+a*cos(c+d*x))/(a+b))*sqrt(sec(c+d*x))/(a^2*d*sqrt(a+b*sec(c+d*x)))+2/5*sin(c+d*x)*sqrt(a+b*sec(c+d*x))/(d*sec(c+d*x)^(3/2))+2/15*b*sin(c+d*x)*sqrt(a+b*sec(c+d*x))/(a*d*sqrt(sec(c+d*x)))+2/15*(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*a/(a+b)))*sqrt(a+b*sec(c+d*x))/(a^2*d*sqrt((b+a*cos(c+d*x))/(a+b))*sqrt(sec(c+d*x)))],
[(a+b*sec(c+d*x))^(1/2)/sec(c+d*x)^(7/2),x,10,2/105*(25*a^4-17*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*a/(a+b)))*sqrt((b+a*cos(c+d*x))/(a+b))*sqrt(sec(c+d*x))/(a^3*d*sqrt(a+b*sec(c+d*x)))+2/7*sin(c+d*x)*sqrt(a+b*sec(c+d*x))/(d*sec(c+d*x)^(5/2))+2/35*b*sin(c+d*x)*sqrt(a+b*sec(c+d*x))/(a*d*sec(c+d*x)^(3/2))+2/105*(25*a^2-4*b^2)*sin(c+d*x)*sqrt(a+b*sec(c+d*x))/(a^2*d*sqrt(sec(c+d*x)))+2/105*b*(19*a^2+8*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+b)))*sqrt(a+b*sec(c+d*x))/(a^3*d*sqrt((b+a*cos(c+d*x))/(a+b))*sqrt(sec(c+d*x)))],
[sec(c+d*x)^(3/2)*(a+b*sec(c+d*x))^(3/2),x,13,7/4*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*a/(a+b)))*sqrt((b+a*cos(c+d*x))/(a+b))*sqrt(sec(c+d*x))/(d*sqrt(a+b*sec(c+d*x)))+1/4*(3*a^2+4*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*a/(a+b)))*sqrt((b+a*cos(c+d*x))/(a+b))*sqrt(sec(c+d*x))/(d*sqrt(a+b*sec(c+d*x)))+1/2*b*sec(c+d*x)^(3/2)*sin(c+d*x)*sqrt(a+b*sec(c+d*x))/d-5/4*a*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+b)))*sqrt(a+b*sec(c+d*x))/(d*sqrt((b+a*cos(c+d*x))/(a+b))*sqrt(sec(c+d*x)))+5/4*a*sin(c+d*x)*sqrt(sec(c+d*x))*sqrt(a+b*sec(c+d*x))/d],
[sec(c+d*x)^(1/2)*(a+b*sec(c+d*x))^(3/2),x,12,(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*a/(a+b)))*sqrt((b+a*cos(c+d*x))/(a+b))*sqrt(sec(c+d*x))/(d*sqrt(a+b*sec(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*a/(a+b)))*sqrt((b+a*cos(c+d*x))/(a+b))*sqrt(sec(c+d*x))/(d*sqrt(a+b*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*a/(a+b)))*sqrt(a+b*sec(c+d*x))/(d*sqrt((b+a*cos(c+d*x))/(a+b))*sqrt(sec(c+d*x)))+b*sin(c+d*x)*sqrt(sec(c+d*x))*sqrt(a+b*sec(c+d*x))/d],
[(a+b*sec(c+d*x))^(3/2)/sec(c+d*x)^(1/2),x,11,2*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*a/(a+b)))*sqrt((b+a*cos(c+d*x))/(a+b))*sqrt(sec(c+d*x))/(d*sqrt(a+b*sec(c+d*x)))+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*a/(a+b)))*sqrt((b+a*cos(c+d*x))/(a+b))*sqrt(sec(c+d*x))/(d*sqrt(a+b*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*a/(a+b)))*sqrt(a+b*sec(c+d*x))/(d*sqrt((b+a*cos(c+d*x))/(a+b))*sqrt(sec(c+d*x)))],
[(a+b*sec(c+d*x))^(3/2)/sec(c+d*x)^(3/2),x,8,2/3*(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*a/(a+b)))*sqrt((b+a*cos(c+d*x))/(a+b))*sqrt(sec(c+d*x))/(d*sqrt(a+b*sec(c+d*x)))+2/3*a*sin(c+d*x)*sqrt(a+b*sec(c+d*x))/(d*sqrt(sec(c+d*x)))+8/3*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+b)))*sqrt(a+b*sec(c+d*x))/(d*sqrt((b+a*cos(c+d*x))/(a+b))*sqrt(sec(c+d*x)))],
[(a+b*sec(c+d*x))^(3/2)/sec(c+d*x)^(5/2),x,9,2/5*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*a/(a+b)))*sqrt((b+a*cos(c+d*x))/(a+b))*sqrt(sec(c+d*x))/(a*d*sqrt(a+b*sec(c+d*x)))+2/5*a*sin(c+d*x)*sqrt(a+b*sec(c+d*x))/(d*sec(c+d*x)^(3/2))+4/5*b*sin(c+d*x)*sqrt(a+b*sec(c+d*x))/(d*sqrt(sec(c+d*x)))+2/5*(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/(a+b)))*sqrt(a+b*sec(c+d*x))/(a*d*sqrt((b+a*cos(c+d*x))/(a+b))*sqrt(sec(c+d*x)))],
[(a+b*sec(c+d*x))^(3/2)/sec(c+d*x)^(7/2),x,10,2/105*(25*a^4-31*a^2*b^2+6*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/(a+b)))*sqrt((b+a*cos(c+d*x))/(a+b))*sqrt(sec(c+d*x))/(a^2*d*sqrt(a+b*sec(c+d*x)))+2/7*a*sin(c+d*x)*sqrt(a+b*sec(c+d*x))/(d*sec(c+d*x)^(5/2))+16/35*b*sin(c+d*x)*sqrt(a+b*sec(c+d*x))/(d*sec(c+d*x)^(3/2))+2/105*(25*a^2+3*b^2)*sin(c+d*x)*sqrt(a+b*sec(c+d*x))/(a*d*sqrt(sec(c+d*x)))+4/105*b*(41*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/(a+b)))*sqrt(a+b*sec(c+d*x))/(a^2*d*sqrt((b+a*cos(c+d*x))/(a+b))*sqrt(sec(c+d*x)))],
[sec(c+d*x)^(3/2)*(a+b*sec(c+d*x))^(5/2),x,14,1/24*b*(59*a^2+16*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+b)))*sqrt((b+a*cos(c+d*x))/(a+b))*sqrt(sec(c+d*x))/(d*sqrt(a+b*sec(c+d*x)))+5/8*a*(a^2+4*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*a/(a+b)))*sqrt((b+a*cos(c+d*x))/(a+b))*sqrt(sec(c+d*x))/(d*sqrt(a+b*sec(c+d*x)))+13/12*a*b*sec(c+d*x)^(3/2)*sin(c+d*x)*sqrt(a+b*sec(c+d*x))/d+1/3*b^2*sec(c+d*x)^(5/2)*sin(c+d*x)*sqrt(a+b*sec(c+d*x))/d-1/24*(33*a^2+16*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+b)))*sqrt(a+b*sec(c+d*x))/(d*sqrt((b+a*cos(c+d*x))/(a+b))*sqrt(sec(c+d*x)))+1/24*(33*a^2+16*b^2)*sin(c+d*x)*sqrt(sec(c+d*x))*sqrt(a+b*sec(c+d*x))/d],
[sec(c+d*x)^(1/2)*(a+b*sec(c+d*x))^(5/2),x,13,1/4*a*(8*a^2+11*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+b)))*sqrt((b+a*cos(c+d*x))/(a+b))*sqrt(sec(c+d*x))/(d*sqrt(a+b*sec(c+d*x)))+1/4*b*(15*a^2+4*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*a/(a+b)))*sqrt((b+a*cos(c+d*x))/(a+b))*sqrt(sec(c+d*x))/(d*sqrt(a+b*sec(c+d*x)))+1/2*b^2*sec(c+d*x)^(3/2)*sin(c+d*x)*sqrt(a+b*sec(c+d*x))/d-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*a/(a+b)))*sqrt(a+b*sec(c+d*x))/(d*sqrt((b+a*cos(c+d*x))/(a+b))*sqrt(sec(c+d*x)))+9/4*a*b*sin(c+d*x)*sqrt(sec(c+d*x))*sqrt(a+b*sec(c+d*x))/d],
[(a+b*sec(c+d*x))^(5/2)/sec(c+d*x)^(1/2),x,12,b*(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*a/(a+b)))*sqrt((b+a*cos(c+d*x))/(a+b))*sqrt(sec(c+d*x))/(d*sqrt(a+b*sec(c+d*x)))+5*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*a/(a+b)))*sqrt((b+a*cos(c+d*x))/(a+b))*sqrt(sec(c+d*x))/(d*sqrt(a+b*sec(c+d*x)))+(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*a/(a+b)))*sqrt(a+b*sec(c+d*x))/(d*sqrt((b+a*cos(c+d*x))/(a+b))*sqrt(sec(c+d*x)))+b^2*sin(c+d*x)*sqrt(sec(c+d*x))*sqrt(a+b*sec(c+d*x))/d],
[(a+b*sec(c+d*x))^(5/2)/sec(c+d*x)^(3/2),x,12,2/3*a*(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*a/(a+b)))*sqrt((b+a*cos(c+d*x))/(a+b))*sqrt(sec(c+d*x))/(d*sqrt(a+b*sec(c+d*x)))+2*b^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*a/(a+b)))*sqrt((b+a*cos(c+d*x))/(a+b))*sqrt(sec(c+d*x))/(d*sqrt(a+b*sec(c+d*x)))+2/3*a^2*sin(c+d*x)*sqrt(a+b*sec(c+d*x))/(d*sqrt(sec(c+d*x)))+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*a/(a+b)))*sqrt(a+b*sec(c+d*x))/(d*sqrt((b+a*cos(c+d*x))/(a+b))*sqrt(sec(c+d*x)))],
[(a+b*sec(c+d*x))^(5/2)/sec(c+d*x)^(5/2),x,9,16/15*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*a/(a+b)))*sqrt((b+a*cos(c+d*x))/(a+b))*sqrt(sec(c+d*x))/(d*sqrt(a+b*sec(c+d*x)))+2/5*a^2*sin(c+d*x)*sqrt(a+b*sec(c+d*x))/(d*sec(c+d*x)^(3/2))+22/15*a*b*sin(c+d*x)*sqrt(a+b*sec(c+d*x))/(d*sqrt(sec(c+d*x)))+2/15*(9*a^2+23*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+b)))*sqrt(a+b*sec(c+d*x))/(d*sqrt((b+a*cos(c+d*x))/(a+b))*sqrt(sec(c+d*x)))],
[(a+b*sec(c+d*x))^(5/2)/sec(c+d*x)^(7/2),x,10,2/21*(5*a^4-2*a^2*b^2-3*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/(a+b)))*sqrt((b+a*cos(c+d*x))/(a+b))*sqrt(sec(c+d*x))/(a*d*sqrt(a+b*sec(c+d*x)))+2/7*a^2*sin(c+d*x)*sqrt(a+b*sec(c+d*x))/(d*sec(c+d*x)^(5/2))+6/7*a*b*sin(c+d*x)*sqrt(a+b*sec(c+d*x))/(d*sec(c+d*x)^(3/2))+2/21*(5*a^2+9*b^2)*sin(c+d*x)*sqrt(a+b*sec(c+d*x))/(d*sqrt(sec(c+d*x)))+2/21*b*(29*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/(a+b)))*sqrt(a+b*sec(c+d*x))/(a*d*sqrt((b+a*cos(c+d*x))/(a+b))*sqrt(sec(c+d*x)))],
[(a+b*sec(c+d*x))^(5/2)/sec(c+d*x)^(9/2),x,11,4/315*b*(57*a^4-62*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*a/(a+b)))*sqrt((b+a*cos(c+d*x))/(a+b))*sqrt(sec(c+d*x))/(a^2*d*sqrt(a+b*sec(c+d*x)))+2/9*a^2*sin(c+d*x)*sqrt(a+b*sec(c+d*x))/(d*sec(c+d*x)^(7/2))+38/63*a*b*sin(c+d*x)*sqrt(a+b*sec(c+d*x))/(d*sec(c+d*x)^(5/2))+2/315*(49*a^2+75*b^2)*sin(c+d*x)*sqrt(a+b*sec(c+d*x))/(d*sec(c+d*x)^(3/2))+2/315*b*(163*a^2+5*b^2)*sin(c+d*x)*sqrt(a+b*sec(c+d*x))/(a*d*sqrt(sec(c+d*x)))+2/315*(147*a^4+279*a^2*b^2-10*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/(a+b)))*sqrt(a+b*sec(c+d*x))/(a^2*d*sqrt((b+a*cos(c+d*x))/(a+b))*sqrt(sec(c+d*x)))],

# n<0
[sec(c+d*x)^(7/2)/(a+b*sec(c+d*x))^(1/2),x,13,-1/4*a*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2*a/(a+b)))*sqrt((b+a*cos(c+d*x))/(a+b))*sqrt(sec(c+d*x))/(b*d*sqrt(a+b*sec(c+d*x)))+1/4*(3*a^2+4*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*a/(a+b)))*sqrt((b+a*cos(c+d*x))/(a+b))*sqrt(sec(c+d*x))/(b^2*d*sqrt(a+b*sec(c+d*x)))+1/2*sec(c+d*x)^(3/2)*sin(c+d*x)*sqrt(a+b*sec(c+d*x))/(b*d)+3/4*a*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+b)))*sqrt(a+b*sec(c+d*x))/(b^2*d*sqrt((b+a*cos(c+d*x))/(a+b))*sqrt(sec(c+d*x)))-3/4*a*sin(c+d*x)*sqrt(sec(c+d*x))*sqrt(a+b*sec(c+d*x))/(b^2*d)],
[sec(c+d*x)^(5/2)/(a+b*sec(c+d*x))^(1/2),x,12,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+b)))*sqrt((b+a*cos(c+d*x))/(a+b))*sqrt(sec(c+d*x))/(d*sqrt(a+b*sec(c+d*x)))-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*a/(a+b)))*sqrt((b+a*cos(c+d*x))/(a+b))*sqrt(sec(c+d*x))/(b*d*sqrt(a+b*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*a/(a+b)))*sqrt(a+b*sec(c+d*x))/(b*d*sqrt((b+a*cos(c+d*x))/(a+b))*sqrt(sec(c+d*x)))+sin(c+d*x)*sqrt(sec(c+d*x))*sqrt(a+b*sec(c+d*x))/(b*d)],
[sec(c+d*x)^(3/2)/(a+b*sec(c+d*x))^(1/2),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,sqrt(2*a/(a+b)))*sqrt((b+a*cos(c+d*x))/(a+b))*sqrt(sec(c+d*x))/(d*sqrt(a+b*sec(c+d*x)))],
[sec(c+d*x)^(1/2)/(a+b*sec(c+d*x))^(1/2),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*a/(a+b)))*sqrt((b+a*cos(c+d*x))/(a+b))*sqrt(sec(c+d*x))/(d*sqrt(a+b*sec(c+d*x)))],
[1/(sec(c+d*x)^(1/2)*(a+b*sec(c+d*x))^(1/2)),x,7,-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*a/(a+b)))*sqrt((b+a*cos(c+d*x))/(a+b))*sqrt(sec(c+d*x))/(a*d*sqrt(a+b*sec(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*a/(a+b)))*sqrt(a+b*sec(c+d*x))/(a*d*sqrt((b+a*cos(c+d*x))/(a+b))*sqrt(sec(c+d*x)))],
[1/(sec(c+d*x)^(3/2)*(a+b*sec(c+d*x))^(1/2)),x,8,2/3*(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*a/(a+b)))*sqrt((b+a*cos(c+d*x))/(a+b))*sqrt(sec(c+d*x))/(a^2*d*sqrt(a+b*sec(c+d*x)))+2/3*sin(c+d*x)*sqrt(a+b*sec(c+d*x))/(a*d*sqrt(sec(c+d*x)))-4/3*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+b)))*sqrt(a+b*sec(c+d*x))/(a^2*d*sqrt((b+a*cos(c+d*x))/(a+b))*sqrt(sec(c+d*x)))],
[1/(sec(c+d*x)^(5/2)*(a+b*sec(c+d*x))^(1/2)),x,9,-2/15*b*(7*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*a/(a+b)))*sqrt((b+a*cos(c+d*x))/(a+b))*sqrt(sec(c+d*x))/(a^3*d*sqrt(a+b*sec(c+d*x)))+2/5*sin(c+d*x)*sqrt(a+b*sec(c+d*x))/(a*d*sec(c+d*x)^(3/2))-8/15*b*sin(c+d*x)*sqrt(a+b*sec(c+d*x))/(a^2*d*sqrt(sec(c+d*x)))+2/15*(9*a^2+8*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+b)))*sqrt(a+b*sec(c+d*x))/(a^3*d*sqrt((b+a*cos(c+d*x))/(a+b))*sqrt(sec(c+d*x)))],
[sec(c+d*x)^(7/2)/(a+b*sec(c+d*x))^(3/2),x,13,-2*a^2*sec(c+d*x)^(3/2)*sin(c+d*x)/(b*(a^2-b^2)*d*sqrt(a+b*sec(c+d*x)))+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+b)))*sqrt((b+a*cos(c+d*x))/(a+b))*sqrt(sec(c+d*x))/(b*d*sqrt(a+b*sec(c+d*x)))-3*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*a/(a+b)))*sqrt((b+a*cos(c+d*x))/(a+b))*sqrt(sec(c+d*x))/(b^2*d*sqrt(a+b*sec(c+d*x)))-(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/(a+b)))*sqrt(a+b*sec(c+d*x))/(b^2*(a^2-b^2)*d*sqrt((b+a*cos(c+d*x))/(a+b))*sqrt(sec(c+d*x)))+(3*a^2-b^2)*sin(c+d*x)*sqrt(sec(c+d*x))*sqrt(a+b*sec(c+d*x))/(b^2*(a^2-b^2)*d)],
[sec(c+d*x)^(5/2)/(a+b*sec(c+d*x))^(3/2),x,9,-2*a^2*sin(c+d*x)*sqrt(sec(c+d*x))/(b*(a^2-b^2)*d*sqrt(a+b*sec(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*a/(a+b)))*sqrt((b+a*cos(c+d*x))/(a+b))*sqrt(sec(c+d*x))/(b*d*sqrt(a+b*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*a/(a+b)))*sqrt(a+b*sec(c+d*x))/(b*(a^2-b^2)*d*sqrt((b+a*cos(c+d*x))/(a+b))*sqrt(sec(c+d*x)))],
[sec(c+d*x)^(3/2)/(a+b*sec(c+d*x))^(3/2),x,5,2*a*sin(c+d*x)*sqrt(sec(c+d*x))/((a^2-b^2)*d*sqrt(a+b*sec(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*a/(a+b)))*sqrt(a+b*sec(c+d*x))/((a^2-b^2)*d*sqrt((b+a*cos(c+d*x))/(a+b))*sqrt(sec(c+d*x)))],
[sec(c+d*x)^(1/2)/(a+b*sec(c+d*x))^(3/2),x,8,-2*b*sin(c+d*x)*sqrt(sec(c+d*x))/((a^2-b^2)*d*sqrt(a+b*sec(c+d*x)))+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+b)))*sqrt((b+a*cos(c+d*x))/(a+b))*sqrt(sec(c+d*x))/(a*d*sqrt(a+b*sec(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*a/(a+b)))*sqrt(a+b*sec(c+d*x))/(a*(a^2-b^2)*d*sqrt((b+a*cos(c+d*x))/(a+b))*sqrt(sec(c+d*x)))],
[1/(sec(c+d*x)^(1/2)*(a+b*sec(c+d*x))^(3/2)),x,8,2*b^2*sin(c+d*x)*sqrt(sec(c+d*x))/(a*(a^2-b^2)*d*sqrt(a+b*sec(c+d*x)))-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*a/(a+b)))*sqrt((b+a*cos(c+d*x))/(a+b))*sqrt(sec(c+d*x))/(a^2*d*sqrt(a+b*sec(c+d*x)))+2*(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*a/(a+b)))*sqrt(a+b*sec(c+d*x))/(a^2*(a^2-b^2)*d*sqrt((b+a*cos(c+d*x))/(a+b))*sqrt(sec(c+d*x)))],
[1/(sec(c+d*x)^(3/2)*(a+b*sec(c+d*x))^(3/2)),x,9,2*b^2*sin(c+d*x)/(a*(a^2-b^2)*d*sqrt(sec(c+d*x))*sqrt(a+b*sec(c+d*x)))+2/3*(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*a/(a+b)))*sqrt((b+a*cos(c+d*x))/(a+b))*sqrt(sec(c+d*x))/(a^3*d*sqrt(a+b*sec(c+d*x)))+2/3*(a^2-4*b^2)*sin(c+d*x)*sqrt(a+b*sec(c+d*x))/(a^2*(a^2-b^2)*d*sqrt(sec(c+d*x)))-2/3*b*(5*a^2-8*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+b)))*sqrt(a+b*sec(c+d*x))/(a^3*(a^2-b^2)*d*sqrt((b+a*cos(c+d*x))/(a+b))*sqrt(sec(c+d*x)))],
[1/(sec(c+d*x)^(5/2)*(a+b*sec(c+d*x))^(3/2)),x,10,2*b^2*sin(c+d*x)/(a*(a^2-b^2)*d*sec(c+d*x)^(3/2)*sqrt(a+b*sec(c+d*x)))-8/5*b*(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*a/(a+b)))*sqrt((b+a*cos(c+d*x))/(a+b))*sqrt(sec(c+d*x))/(a^4*d*sqrt(a+b*sec(c+d*x)))+2/5*(a^2-6*b^2)*sin(c+d*x)*sqrt(a+b*sec(c+d*x))/(a^2*(a^2-b^2)*d*sec(c+d*x)^(3/2))-2/5*b*(3*a^2-8*b^2)*sin(c+d*x)*sqrt(a+b*sec(c+d*x))/(a^3*(a^2-b^2)*d*sqrt(sec(c+d*x)))+2/5*(3*a^4+8*a^2*b^2-16*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/(a+b)))*sqrt(a+b*sec(c+d*x))/(a^4*(a^2-b^2)*d*sqrt((b+a*cos(c+d*x))/(a+b))*sqrt(sec(c+d*x)))],
[sec(c+d*x)^(9/2)/(a+b*sec(c+d*x))^(5/2),x,14,-2/3*a^2*sec(c+d*x)^(5/2)*sin(c+d*x)/(b*(a^2-b^2)*d*(a+b*sec(c+d*x))^(3/2))-2/3*a^2*(5*a^2-9*b^2)*sec(c+d*x)^(3/2)*sin(c+d*x)/(b^2*(a^2-b^2)^2*d*sqrt(a+b*sec(c+d*x)))+1/3*(5*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*a/(a+b)))*sqrt((b+a*cos(c+d*x))/(a+b))*sqrt(sec(c+d*x))/(b^2*(a^2-b^2)*d*sqrt(a+b*sec(c+d*x)))-5*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*a/(a+b)))*sqrt((b+a*cos(c+d*x))/(a+b))*sqrt(sec(c+d*x))/(b^3*d*sqrt(a+b*sec(c+d*x)))-1/3*(15*a^4-26*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*a/(a+b)))*sqrt(a+b*sec(c+d*x))/(b^3*(a^2-b^2)^2*d*sqrt((b+a*cos(c+d*x))/(a+b))*sqrt(sec(c+d*x)))+1/3*(15*a^4-26*a^2*b^2+3*b^4)*sin(c+d*x)*sqrt(sec(c+d*x))*sqrt(a+b*sec(c+d*x))/(b^3*(a^2-b^2)^2*d)],
[sec(c+d*x)^(7/2)/(a+b*sec(c+d*x))^(5/2),x,13,-2/3*a^2*sec(c+d*x)^(3/2)*sin(c+d*x)/(b*(a^2-b^2)*d*(a+b*sec(c+d*x))^(3/2))-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*sec(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*a/(a+b)))*sqrt((b+a*cos(c+d*x))/(a+b))*sqrt(sec(c+d*x))/(b*(a^2-b^2)*d*sqrt(a+b*sec(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*a/(a+b)))*sqrt((b+a*cos(c+d*x))/(a+b))*sqrt(sec(c+d*x))/(b^2*d*sqrt(a+b*sec(c+d*x)))+2/3*a*(3*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*a/(a+b)))*sqrt(a+b*sec(c+d*x))/(b^2*(a^2-b^2)^2*d*sqrt((b+a*cos(c+d*x))/(a+b))*sqrt(sec(c+d*x)))],
[sec(c+d*x)^(5/2)/(a+b*sec(c+d*x))^(5/2),x,9,-2/3*a^2*sin(c+d*x)*sqrt(sec(c+d*x))/(b*(a^2-b^2)*d*(a+b*sec(c+d*x))^(3/2))+2/3*a*(a^2-5*b^2)*sin(c+d*x)*sqrt(sec(c+d*x))/(b*(a^2-b^2)^2*d*sqrt(a+b*sec(c+d*x)))+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/(a+b)))*sqrt((b+a*cos(c+d*x))/(a+b))*sqrt(sec(c+d*x))/((a^2-b^2)*d*sqrt(a+b*sec(c+d*x)))+8/3*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+b)))*sqrt(a+b*sec(c+d*x))/((a^2-b^2)^2*d*sqrt((b+a*cos(c+d*x))/(a+b))*sqrt(sec(c+d*x)))],
[sec(c+d*x)^(3/2)/(a+b*sec(c+d*x))^(5/2),x,9,2/3*a*sin(c+d*x)*sqrt(sec(c+d*x))/((a^2-b^2)*d*(a+b*sec(c+d*x))^(3/2))+4/3*(a^2+b^2)*sin(c+d*x)*sqrt(sec(c+d*x))/((a^2-b^2)^2*d*sqrt(a+b*sec(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*a/(a+b)))*sqrt((b+a*cos(c+d*x))/(a+b))*sqrt(sec(c+d*x))/(a*(a^2-b^2)*d*sqrt(a+b*sec(c+d*x)))-2/3*(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/(a+b)))*sqrt(a+b*sec(c+d*x))/(a*(a^2-b^2)^2*d*sqrt((b+a*cos(c+d*x))/(a+b))*sqrt(sec(c+d*x)))],
[sec(c+d*x)^(1/2)/(a+b*sec(c+d*x))^(5/2),x,9,-2/3*b*sin(c+d*x)*sqrt(sec(c+d*x))/((a^2-b^2)*d*(a+b*sec(c+d*x))^(3/2))-2/3*b*(5*a^2-b^2)*sin(c+d*x)*sqrt(sec(c+d*x))/(a*(a^2-b^2)^2*d*sqrt(a+b*sec(c+d*x)))+2/3*(3*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*a/(a+b)))*sqrt((b+a*cos(c+d*x))/(a+b))*sqrt(sec(c+d*x))/(a^2*(a^2-b^2)*d*sqrt(a+b*sec(c+d*x)))+4/3*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/(a+b)))*sqrt(a+b*sec(c+d*x))/(a^2*(a^2-b^2)^2*d*sqrt((b+a*cos(c+d*x))/(a+b))*sqrt(sec(c+d*x)))],
[1/(sec(c+d*x)^(1/2)*(a+b*sec(c+d*x))^(5/2)),x,9,2/3*b^2*sin(c+d*x)*sqrt(sec(c+d*x))/(a*(a^2-b^2)*d*(a+b*sec(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*sec(c+d*x)))-2/3*b*(9*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*a/(a+b)))*sqrt((b+a*cos(c+d*x))/(a+b))*sqrt(sec(c+d*x))/(a^3*(a^2-b^2)*d*sqrt(a+b*sec(c+d*x)))+2/3*(3*a^4-15*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*a/(a+b)))*sqrt(a+b*sec(c+d*x))/(a^3*(a^2-b^2)^2*d*sqrt((b+a*cos(c+d*x))/(a+b))*sqrt(sec(c+d*x)))],
[1/(sec(c+d*x)^(3/2)*(a+b*sec(c+d*x))^(5/2)),x,10,2/3*b^2*sin(c+d*x)/(a*(a^2-b^2)*d*(a+b*sec(c+d*x))^(3/2)*sqrt(sec(c+d*x)))+4/3*b^2*(5*a^2-3*b^2)*sin(c+d*x)/(a^2*(a^2-b^2)^2*d*sqrt(sec(c+d*x))*sqrt(a+b*sec(c+d*x)))+2/3*(a^4+16*a^2*b^2-16*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/(a+b)))*sqrt((b+a*cos(c+d*x))/(a+b))*sqrt(sec(c+d*x))/(a^4*(a^2-b^2)*d*sqrt(a+b*sec(c+d*x)))+2/3*(a^4-13*a^2*b^2+8*b^4)*sin(c+d*x)*sqrt(a+b*sec(c+d*x))/(a^3*(a^2-b^2)^2*d*sqrt(sec(c+d*x)))-8/3*b*(2*a^4-7*a^2*b^2+4*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/(a+b)))*sqrt(a+b*sec(c+d*x))/(a^4*(a^2-b^2)^2*d*sqrt((b+a*cos(c+d*x))/(a+b))*sqrt(sec(c+d*x)))],
[1/(sec(c+d*x)^(5/2)*(a+b*sec(c+d*x))^(5/2)),x,11,2/3*b^2*sin(c+d*x)/(a*(a^2-b^2)*d*sec(c+d*x)^(3/2)*(a+b*sec(c+d*x))^(3/2))+8/3*b^2*(3*a^2-2*b^2)*sin(c+d*x)/(a^2*(a^2-b^2)^2*d*sec(c+d*x)^(3/2)*sqrt(a+b*sec(c+d*x)))-2/15*b*(17*a^4+116*a^2*b^2-128*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/(a+b)))*sqrt((b+a*cos(c+d*x))/(a+b))*sqrt(sec(c+d*x))/(a^5*(a^2-b^2)*d*sqrt(a+b*sec(c+d*x)))+2/15*(3*a^4-71*a^2*b^2+48*b^4)*sin(c+d*x)*sqrt(a+b*sec(c+d*x))/(a^3*(a^2-b^2)^2*d*sec(c+d*x)^(3/2))-4/15*b*(7*a^4-49*a^2*b^2+32*b^4)*sin(c+d*x)*sqrt(a+b*sec(c+d*x))/(a^4*(a^2-b^2)^2*d*sqrt(sec(c+d*x)))+2/15*(9*a^6+55*a^4*b^2-212*a^2*b^4+128*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*a/(a+b)))*sqrt(a+b*sec(c+d*x))/(a^5*(a^2-b^2)^2*d*sqrt((b+a*cos(c+d*x))/(a+b))*sqrt(sec(c+d*x)))],
[1/(sqrt(sec(c+d*x))*sqrt(2+3*sec(c+d*x))),x,5,-3*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(4/5))*sqrt(3+2*cos(c+d*x))*sqrt(sec(c+d*x))/(d*sqrt(5)*sqrt(2+3*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(4/5))*sqrt(5)*sqrt(2+3*sec(c+d*x))/(d*sqrt(3+2*cos(c+d*x))*sqrt(sec(c+d*x)))],
[1/(sqrt(sec(c+d*x))*sqrt(-2+3*sec(c+d*x))),x,5,3*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(-4))*sqrt(3-2*cos(c+d*x))*sqrt(sec(c+d*x))/(d*sqrt(-2+3*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(-4))*sqrt(-2+3*sec(c+d*x))/(d*sqrt(3-2*cos(c+d*x))*sqrt(sec(c+d*x)))],
[1/(sqrt(2-3*sec(c+d*x))*sqrt(sec(c+d*x))),x,7,sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(-4))*sqrt(2-3*sec(c+d*x))/(d*sqrt(3-2*cos(c+d*x))*sqrt(sec(c+d*x)))+3*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(-4))*sqrt(3-2*cos(c+d*x))*sqrt(sec(c+d*x))/(d*sqrt(2-3*sec(c+d*x)))],
[1/(sqrt(-2-3*sec(c+d*x))*sqrt(sec(c+d*x))),x,7,-sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(4/5))*sqrt(5)*sqrt(-2-3*sec(c+d*x))/(d*sqrt(3+2*cos(c+d*x))*sqrt(sec(c+d*x)))-3*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(4/5))*sqrt(3+2*cos(c+d*x))*sqrt(sec(c+d*x))/(d*sqrt(5)*sqrt(-2-3*sec(c+d*x)))],
[1/(sqrt(sec(c+d*x))*sqrt(3+2*sec(c+d*x))),x,5,-4/3*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(6/5))*sqrt(2+3*cos(c+d*x))*sqrt(sec(c+d*x))/(d*sqrt(5)*sqrt(3+2*sec(c+d*x)))+2/3*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(6/5))*sqrt(5)*sqrt(3+2*sec(c+d*x))/(d*sqrt(2+3*cos(c+d*x))*sqrt(sec(c+d*x)))],
[1/(sqrt(3-2*sec(c+d*x))*sqrt(sec(c+d*x))),x,5,2/3*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticE(sin(1/2*(c+d*x)),sqrt(6))*sqrt(3-2*sec(c+d*x))/(d*sqrt(-2+3*cos(c+d*x))*sqrt(sec(c+d*x)))+4/3*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(6))*sqrt(-2+3*cos(c+d*x))*sqrt(sec(c+d*x))/(d*sqrt(3-2*sec(c+d*x)))],
[1/(sqrt(sec(c+d*x))*sqrt(-3+2*sec(c+d*x))),x,5,4/3*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(6/5))*sqrt(2-3*cos(c+d*x))*sqrt(sec(c+d*x))/(d*sqrt(5)*sqrt(-3+2*sec(c+d*x)))-2/3*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(6/5))*sqrt(5)*sqrt(-3+2*sec(c+d*x))/(d*sqrt(2-3*cos(c+d*x))*sqrt(sec(c+d*x)))],
[1/(sqrt(-3-2*sec(c+d*x))*sqrt(sec(c+d*x))),x,5,-2/3*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(6))*sqrt(-3-2*sec(c+d*x))/(d*sqrt(-2-3*cos(c+d*x))*sqrt(sec(c+d*x)))-4/3*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(6))*sqrt(-2-3*cos(c+d*x))*sqrt(sec(c+d*x))/(d*sqrt(-3-2*sec(c+d*x)))],
[sqrt(sec(c+d*x))/sqrt(2+3*sec(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(4/5))*sqrt(3+2*cos(c+d*x))*sqrt(sec(c+d*x))/(d*sqrt(5)*sqrt(2+3*sec(c+d*x)))],
[sqrt(sec(c+d*x))/sqrt(-2+3*sec(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(-4))*sqrt(3-2*cos(c+d*x))*sqrt(sec(c+d*x))/(d*sqrt(-2+3*sec(c+d*x)))],
[sqrt(sec(c+d*x))/sqrt(2-3*sec(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(-4))*sqrt(3-2*cos(c+d*x))*sqrt(sec(c+d*x))/(d*sqrt(2-3*sec(c+d*x)))],
[sqrt(sec(c+d*x))/sqrt(-2-3*sec(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(4/5))*sqrt(3+2*cos(c+d*x))*sqrt(sec(c+d*x))/(d*sqrt(5)*sqrt(-2-3*sec(c+d*x)))],
[sqrt(sec(c+d*x))/sqrt(3+2*sec(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(6/5))*sqrt(2+3*cos(c+d*x))*sqrt(sec(c+d*x))/(d*sqrt(5)*sqrt(3+2*sec(c+d*x)))],
[sqrt(sec(c+d*x))/sqrt(3-2*sec(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(6))*sqrt(-2+3*cos(c+d*x))*sqrt(sec(c+d*x))/(d*sqrt(3-2*sec(c+d*x)))],
[sqrt(sec(c+d*x))/sqrt(-3+2*sec(c+d*x)),x,2,2*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(6/5))*sqrt(2-3*cos(c+d*x))*sqrt(sec(c+d*x))/(d*sqrt(5)*sqrt(-3+2*sec(c+d*x)))],
[sqrt(sec(c+d*x))/sqrt(-3-2*sec(c+d*x)),x,2,2*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(6))*sqrt(-2-3*cos(c+d*x))*sqrt(sec(c+d*x))/(d*sqrt(-3-2*sec(c+d*x)))],

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

# n>0
[sec(c+d*x)*(a+b*sec(c+d*x))^(1/3),x,3,AppellF1(1/2,1/2,-1/3,3/2,1/2*(1-sec(c+d*x)),b*(1-sec(c+d*x))/(a+b))*(a+b*sec(c+d*x))^(1/3)*sqrt(2)*tan(c+d*x)/(d*((a+b*sec(c+d*x))/(a+b))^(1/3)*sqrt(1+sec(c+d*x)))],
[(a+b*sec(c+d*x))^(1/3),x,0,Unintegrable((a+b*sec(c+d*x))^(1/3),x)],
[sec(c+d*x)^4*(a+b*sec(c+d*x))^(2/3),x,10,3/220*(9*a^2+32*b^2)*(a+b*sec(c+d*x))^(2/3)*tan(c+d*x)/(b^2*d)-9/44*a*(a+b*sec(c+d*x))^(5/3)*tan(c+d*x)/(b^2*d)+3/11*sec(c+d*x)*(a+b*sec(c+d*x))^(5/3)*tan(c+d*x)/(b*d)+1/110*a*(18*a^2+49*b^2)*AppellF1(1/2,1/2,-2/3,3/2,1/2*(1-sec(c+d*x)),b*(1-sec(c+d*x))/(a+b))*(a+b*sec(c+d*x))^(2/3)*tan(c+d*x)/(b^3*d*((a+b*sec(c+d*x))/(a+b))^(2/3)*sqrt(2)*sqrt(1+sec(c+d*x)))-1/55*(9*a^4+23*a^2*b^2-32*b^4)*AppellF1(1/2,1/2,1/3,3/2,1/2*(1-sec(c+d*x)),b*(1-sec(c+d*x))/(a+b))*((a+b*sec(c+d*x))/(a+b))^(1/3)*tan(c+d*x)/(b^3*d*(a+b*sec(c+d*x))^(1/3)*sqrt(2)*sqrt(1+sec(c+d*x)))],
[sec(c+d*x)^3*(a+b*sec(c+d*x))^(2/3),x,9,-9/40*a*(a+b*sec(c+d*x))^(2/3)*tan(c+d*x)/(b*d)+3/8*(a+b*sec(c+d*x))^(5/3)*tan(c+d*x)/(b*d)-1/20*(6*a^2-25*b^2)*AppellF1(1/2,1/2,-2/3,3/2,1/2*(1-sec(c+d*x)),b*(1-sec(c+d*x))/(a+b))*(a+b*sec(c+d*x))^(2/3)*tan(c+d*x)/(b^2*d*((a+b*sec(c+d*x))/(a+b))^(2/3)*sqrt(2)*sqrt(1+sec(c+d*x)))+3/10*a*(a^2-b^2)*AppellF1(1/2,1/2,1/3,3/2,1/2*(1-sec(c+d*x)),b*(1-sec(c+d*x))/(a+b))*((a+b*sec(c+d*x))/(a+b))^(1/3)*tan(c+d*x)/(b^2*d*(a+b*sec(c+d*x))^(1/3)*sqrt(2)*sqrt(1+sec(c+d*x)))],
[sec(c+d*x)^2*(a+b*sec(c+d*x))^(2/3),x,8,3/5*(a+b*sec(c+d*x))^(2/3)*tan(c+d*x)/d+2/5*a*AppellF1(1/2,1/2,-2/3,3/2,1/2*(1-sec(c+d*x)),b*(1-sec(c+d*x))/(a+b))*(a+b*sec(c+d*x))^(2/3)*sqrt(2)*tan(c+d*x)/(b*d*((a+b*sec(c+d*x))/(a+b))^(2/3)*sqrt(1+sec(c+d*x)))-2/5*(a^2-b^2)*AppellF1(1/2,1/2,1/3,3/2,1/2*(1-sec(c+d*x)),b*(1-sec(c+d*x))/(a+b))*((a+b*sec(c+d*x))/(a+b))^(1/3)*sqrt(2)*tan(c+d*x)/(b*d*(a+b*sec(c+d*x))^(1/3)*sqrt(1+sec(c+d*x)))],
[sec(c+d*x)*(a+b*sec(c+d*x))^(2/3),x,3,AppellF1(1/2,1/2,-2/3,3/2,1/2*(1-sec(c+d*x)),b*(1-sec(c+d*x))/(a+b))*(a+b*sec(c+d*x))^(2/3)*sqrt(2)*tan(c+d*x)/(d*((a+b*sec(c+d*x))/(a+b))^(2/3)*sqrt(1+sec(c+d*x)))],
[(a+b*sec(c+d*x))^(2/3),x,0,Unintegrable((a+b*sec(c+d*x))^(2/3),x)],
[sec(c+d*x)*(a+b*sec(c+d*x))^(4/3),x,3,(a+b)*AppellF1(1/2,1/2,-4/3,3/2,1/2*(1-sec(c+d*x)),b*(1-sec(c+d*x))/(a+b))*(a+b*sec(c+d*x))^(1/3)*sqrt(2)*tan(c+d*x)/(d*((a+b*sec(c+d*x))/(a+b))^(1/3)*sqrt(1+sec(c+d*x)))],
[(a+b*sec(c+d*x))^(4/3),x,0,Unintegrable((a+b*sec(c+d*x))^(4/3),x)],
[sec(c+d*x)^4*(a+b*sec(c+d*x))^(5/3),x,11,3/1232*a*(18*a^2+97*b^2)*(a+b*sec(c+d*x))^(2/3)*tan(c+d*x)/(b^2*d)+3/1232*(18*a^2+121*b^2)*(a+b*sec(c+d*x))^(5/3)*tan(c+d*x)/(b^2*d)-9/77*a*(a+b*sec(c+d*x))^(8/3)*tan(c+d*x)/(b^2*d)+3/14*sec(c+d*x)*(a+b*sec(c+d*x))^(8/3)*tan(c+d*x)/(b*d)+1/616*(36*a^4+164*a^2*b^2+605*b^4)*AppellF1(1/2,1/2,-2/3,3/2,1/2*(1-sec(c+d*x)),b*(1-sec(c+d*x))/(a+b))*(a+b*sec(c+d*x))^(2/3)*tan(c+d*x)/(b^3*d*((a+b*sec(c+d*x))/(a+b))^(2/3)*sqrt(2)*sqrt(1+sec(c+d*x)))-1/308*a*(18*a^4+79*a^2*b^2-97*b^4)*AppellF1(1/2,1/2,1/3,3/2,1/2*(1-sec(c+d*x)),b*(1-sec(c+d*x))/(a+b))*((a+b*sec(c+d*x))/(a+b))^(1/3)*tan(c+d*x)/(b^3*d*(a+b*sec(c+d*x))^(1/3)*sqrt(2)*sqrt(1+sec(c+d*x)))],
[sec(c+d*x)^3*(a+b*sec(c+d*x))^(5/3),x,10,-3/440*(15*a^2-64*b^2)*(a+b*sec(c+d*x))^(2/3)*tan(c+d*x)/(b*d)-9/88*a*(a+b*sec(c+d*x))^(5/3)*tan(c+d*x)/(b*d)+3/11*(a+b*sec(c+d*x))^(8/3)*tan(c+d*x)/(b*d)-1/220*a*(30*a^2-373*b^2)*AppellF1(1/2,1/2,-2/3,3/2,1/2*(1-sec(c+d*x)),b*(1-sec(c+d*x))/(a+b))*(a+b*sec(c+d*x))^(2/3)*tan(c+d*x)/(b^2*d*((a+b*sec(c+d*x))/(a+b))^(2/3)*sqrt(2)*sqrt(1+sec(c+d*x)))+1/110*(15*a^4-79*a^2*b^2+64*b^4)*AppellF1(1/2,1/2,1/3,3/2,1/2*(1-sec(c+d*x)),b*(1-sec(c+d*x))/(a+b))*((a+b*sec(c+d*x))/(a+b))^(1/3)*tan(c+d*x)/(b^2*d*(a+b*sec(c+d*x))^(1/3)*sqrt(2)*sqrt(1+sec(c+d*x)))],
[sec(c+d*x)^2*(a+b*sec(c+d*x))^(5/3),x,9,3/8*a*(a+b*sec(c+d*x))^(2/3)*tan(c+d*x)/d+3/8*(a+b*sec(c+d*x))^(5/3)*tan(c+d*x)/d+1/4*(2*a^2+5*b^2)*AppellF1(1/2,1/2,-2/3,3/2,1/2*(1-sec(c+d*x)),b*(1-sec(c+d*x))/(a+b))*(a+b*sec(c+d*x))^(2/3)*tan(c+d*x)/(b*d*((a+b*sec(c+d*x))/(a+b))^(2/3)*sqrt(2)*sqrt(1+sec(c+d*x)))-1/2*a*(a^2-b^2)*AppellF1(1/2,1/2,1/3,3/2,1/2*(1-sec(c+d*x)),b*(1-sec(c+d*x))/(a+b))*((a+b*sec(c+d*x))/(a+b))^(1/3)*tan(c+d*x)/(b*d*(a+b*sec(c+d*x))^(1/3)*sqrt(2)*sqrt(1+sec(c+d*x)))],
[sec(c+d*x)*(a+b*sec(c+d*x))^(5/3),x,3,(a+b)*AppellF1(1/2,1/2,-5/3,3/2,1/2*(1-sec(c+d*x)),b*(1-sec(c+d*x))/(a+b))*(a+b*sec(c+d*x))^(2/3)*sqrt(2)*tan(c+d*x)/(d*((a+b*sec(c+d*x))/(a+b))^(2/3)*sqrt(1+sec(c+d*x)))],
[(a+b*sec(c+d*x))^(5/3),x,0,Unintegrable((a+b*sec(c+d*x))^(5/3),x)],

# n<0
[sec(c+d*x)^4/(a+b*sec(c+d*x))^(1/3),x,9,-9/20*a*(a+b*sec(c+d*x))^(2/3)*tan(c+d*x)/(b^2*d)+3/8*sec(c+d*x)*(a+b*sec(c+d*x))^(2/3)*tan(c+d*x)/(b*d)+1/20*(18*a^2+25*b^2)*AppellF1(1/2,1/2,-2/3,3/2,1/2*(1-sec(c+d*x)),b*(1-sec(c+d*x))/(a+b))*(a+b*sec(c+d*x))^(2/3)*tan(c+d*x)/(b^3*d*((a+b*sec(c+d*x))/(a+b))^(2/3)*sqrt(2)*sqrt(1+sec(c+d*x)))-1/10*a*(9*a^2+11*b^2)*AppellF1(1/2,1/2,1/3,3/2,1/2*(1-sec(c+d*x)),b*(1-sec(c+d*x))/(a+b))*((a+b*sec(c+d*x))/(a+b))^(1/3)*tan(c+d*x)/(b^3*d*(a+b*sec(c+d*x))^(1/3)*sqrt(2)*sqrt(1+sec(c+d*x)))],
[sec(c+d*x)^3/(a+b*sec(c+d*x))^(1/3),x,8,3/5*(a+b*sec(c+d*x))^(2/3)*tan(c+d*x)/(b*d)-3/5*a*AppellF1(1/2,1/2,-2/3,3/2,1/2*(1-sec(c+d*x)),b*(1-sec(c+d*x))/(a+b))*(a+b*sec(c+d*x))^(2/3)*sqrt(2)*tan(c+d*x)/(b^2*d*((a+b*sec(c+d*x))/(a+b))^(2/3)*sqrt(1+sec(c+d*x)))+1/5*(3*a^2+2*b^2)*AppellF1(1/2,1/2,1/3,3/2,1/2*(1-sec(c+d*x)),b*(1-sec(c+d*x))/(a+b))*((a+b*sec(c+d*x))/(a+b))^(1/3)*sqrt(2)*tan(c+d*x)/(b^2*d*(a+b*sec(c+d*x))^(1/3)*sqrt(1+sec(c+d*x)))],
[sec(c+d*x)^2/(a+b*sec(c+d*x))^(1/3),x,7,AppellF1(1/2,1/2,-2/3,3/2,1/2*(1-sec(c+d*x)),b*(1-sec(c+d*x))/(a+b))*(a+b*sec(c+d*x))^(2/3)*sqrt(2)*tan(c+d*x)/(b*d*((a+b*sec(c+d*x))/(a+b))^(2/3)*sqrt(1+sec(c+d*x)))-a*AppellF1(1/2,1/2,1/3,3/2,1/2*(1-sec(c+d*x)),b*(1-sec(c+d*x))/(a+b))*((a+b*sec(c+d*x))/(a+b))^(1/3)*sqrt(2)*tan(c+d*x)/(b*d*(a+b*sec(c+d*x))^(1/3)*sqrt(1+sec(c+d*x)))],
[sec(c+d*x)/(a+b*sec(c+d*x))^(1/3),x,3,AppellF1(1/2,1/2,1/3,3/2,1/2*(1-sec(c+d*x)),b*(1-sec(c+d*x))/(a+b))*((a+b*sec(c+d*x))/(a+b))^(1/3)*sqrt(2)*tan(c+d*x)/(d*(a+b*sec(c+d*x))^(1/3)*sqrt(1+sec(c+d*x)))],
[1/(a+b*sec(c+d*x))^(1/3),x,0,Unintegrable(1/(a+b*sec(c+d*x))^(1/3),x)],
[sec(c+d*x)/(a+b*sec(c+d*x))^(2/3),x,3,AppellF1(1/2,1/2,2/3,3/2,1/2*(1-sec(c+d*x)),b*(1-sec(c+d*x))/(a+b))*((a+b*sec(c+d*x))/(a+b))^(2/3)*sqrt(2)*tan(c+d*x)/(d*(a+b*sec(c+d*x))^(2/3)*sqrt(1+sec(c+d*x)))],
[1/(a+b*sec(c+d*x))^(2/3),x,0,Unintegrable(1/(a+b*sec(c+d*x))^(2/3),x)],
[sec(c+d*x)/(a+b*sec(c+d*x))^(4/3),x,3,AppellF1(1/2,1/2,4/3,3/2,1/2*(1-sec(c+d*x)),b*(1-sec(c+d*x))/(a+b))*((a+b*sec(c+d*x))/(a+b))^(1/3)*sqrt(2)*tan(c+d*x)/((a+b)*d*(a+b*sec(c+d*x))^(1/3)*sqrt(1+sec(c+d*x)))],
[1/(a+b*sec(c+d*x))^(4/3),x,0,Unintegrable(1/(a+b*sec(c+d*x))^(4/3),x)],
[sec(c+d*x)^4/(a+b*sec(c+d*x))^(5/3),x,9,-3/2*a^2*sec(c+d*x)*tan(c+d*x)/(b*(a^2-b^2)*d*(a+b*sec(c+d*x))^(2/3))+3/4*(3*a^2-b^2)*(a+b*sec(c+d*x))^(1/3)*tan(c+d*x)/(b^2*(a^2-b^2)*d)-1/2*a*(9*a^2-7*b^2)*AppellF1(1/2,1/2,-1/3,3/2,1/2*(1-sec(c+d*x)),b*(1-sec(c+d*x))/(a+b))*(a+b*sec(c+d*x))^(1/3)*tan(c+d*x)/(b^3*(a^2-b^2)*d*((a+b*sec(c+d*x))/(a+b))^(1/3)*sqrt(2)*sqrt(1+sec(c+d*x)))+1/2*(9*a^4-10*a^2*b^2-b^4)*AppellF1(1/2,1/2,2/3,3/2,1/2*(1-sec(c+d*x)),b*(1-sec(c+d*x))/(a+b))*((a+b*sec(c+d*x))/(a+b))^(2/3)*tan(c+d*x)/(b^3*(a^2-b^2)*d*(a+b*sec(c+d*x))^(2/3)*sqrt(2)*sqrt(1+sec(c+d*x)))],
[sec(c+d*x)^3/(a+b*sec(c+d*x))^(5/3),x,8,-3/2*a^2*tan(c+d*x)/(b*(a^2-b^2)*d*(a+b*sec(c+d*x))^(2/3))+(3*a^2-2*b^2)*AppellF1(1/2,1/2,-1/3,3/2,1/2*(1-sec(c+d*x)),b*(1-sec(c+d*x))/(a+b))*(a+b*sec(c+d*x))^(1/3)*tan(c+d*x)/(b^2*(a^2-b^2)*d*((a+b*sec(c+d*x))/(a+b))^(1/3)*sqrt(2)*sqrt(1+sec(c+d*x)))-a*(3*a^2-4*b^2)*AppellF1(1/2,1/2,2/3,3/2,1/2*(1-sec(c+d*x)),b*(1-sec(c+d*x))/(a+b))*((a+b*sec(c+d*x))/(a+b))^(2/3)*tan(c+d*x)/(b^2*(a^2-b^2)*d*(a+b*sec(c+d*x))^(2/3)*sqrt(2)*sqrt(1+sec(c+d*x)))],
[sec(c+d*x)^2/(a+b*sec(c+d*x))^(5/3),x,8,3/2*a*tan(c+d*x)/((a^2-b^2)*d*(a+b*sec(c+d*x))^(2/3))-a*AppellF1(1/2,1/2,-1/3,3/2,1/2*(1-sec(c+d*x)),b*(1-sec(c+d*x))/(a+b))*(a+b*sec(c+d*x))^(1/3)*tan(c+d*x)/(b*(a^2-b^2)*d*((a+b*sec(c+d*x))/(a+b))^(1/3)*sqrt(2)*sqrt(1+sec(c+d*x)))+(a^2-2*b^2)*AppellF1(1/2,1/2,2/3,3/2,1/2*(1-sec(c+d*x)),b*(1-sec(c+d*x))/(a+b))*((a+b*sec(c+d*x))/(a+b))^(2/3)*tan(c+d*x)/(b*(a^2-b^2)*d*(a+b*sec(c+d*x))^(2/3)*sqrt(2)*sqrt(1+sec(c+d*x)))],
[sec(c+d*x)/(a+b*sec(c+d*x))^(5/3),x,3,AppellF1(1/2,1/2,5/3,3/2,1/2*(1-sec(c+d*x)),b*(1-sec(c+d*x))/(a+b))*((a+b*sec(c+d*x))/(a+b))^(2/3)*sqrt(2)*tan(c+d*x)/((a+b)*d*(a+b*sec(c+d*x))^(2/3)*sqrt(1+sec(c+d*x)))],
[1/(a+b*sec(c+d*x))^(5/3),x,0,Unintegrable(1/(a+b*sec(c+d*x))^(5/3),x)],

# Integrands of the form (d Sec[e+f x])^(n/3) (a+b Sec[e+f x])^m
[sec(c+d*x)^(2/3)/(a+b*sec(c+d*x)),x,6,a*AppellF1(1/2,-1/6,1,3/2,sin(c+d*x)^2,a^2*sin(c+d*x)^2/(a^2-b^2))*sin(c+d*x)/((a^2-b^2)*d*(cos(c+d*x)^2)^(1/6)*sec(c+d*x)^(1/3))-b*AppellF1(1/2,1/3,1,3/2,sin(c+d*x)^2,a^2*sin(c+d*x)^2/(a^2-b^2))*(cos(c+d*x)^2)^(1/3)*sec(c+d*x)^(2/3)*sin(c+d*x)/((a^2-b^2)*d)],
[sec(c+d*x)^(1/3)/(a+b*sec(c+d*x)),x,6,a*AppellF1(1/2,-1/3,1,3/2,sin(c+d*x)^2,a^2*sin(c+d*x)^2/(a^2-b^2))*sin(c+d*x)/((a^2-b^2)*d*(cos(c+d*x)^2)^(1/3)*sec(c+d*x)^(2/3))-b*AppellF1(1/2,1/6,1,3/2,sin(c+d*x)^2,a^2*sin(c+d*x)^2/(a^2-b^2))*(cos(c+d*x)^2)^(1/6)*sec(c+d*x)^(1/3)*sin(c+d*x)/((a^2-b^2)*d)],
[1/(sec(c+d*x)^(1/3)*(a+b*sec(c+d*x))),x,6,-b*AppellF1(1/2,-1/6,1,3/2,sin(c+d*x)^2,a^2*sin(c+d*x)^2/(a^2-b^2))*sin(c+d*x)/((a^2-b^2)*d*(cos(c+d*x)^2)^(1/6)*sec(c+d*x)^(1/3))+a*AppellF1(1/2,-2/3,1,3/2,sin(c+d*x)^2,a^2*sin(c+d*x)^2/(a^2-b^2))*(cos(c+d*x)^2)^(1/3)*sec(c+d*x)^(2/3)*sin(c+d*x)/((a^2-b^2)*d)],
[1/(sec(c+d*x)^(2/3)*(a+b*sec(c+d*x))),x,6,-b*AppellF1(1/2,-1/3,1,3/2,sin(c+d*x)^2,a^2*sin(c+d*x)^2/(a^2-b^2))*sin(c+d*x)/((a^2-b^2)*d*(cos(c+d*x)^2)^(1/3)*sec(c+d*x)^(2/3))+a*AppellF1(1/2,-5/6,1,3/2,sin(c+d*x)^2,a^2*sin(c+d*x)^2/(a^2-b^2))*(cos(c+d*x)^2)^(1/6)*sec(c+d*x)^(1/3)*sin(c+d*x)/((a^2-b^2)*d)],

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

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

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

# Integrands of the form (d Sec[e+f x])^n (a+b Sec[e+f x])^m with n symbolic
[(d*sec(e+f*x))^n*(a+b*sec(e+f*x))^3,x,7,-a*d*(3*b^2*n+a^2*(1+n))*hypergeom([1/2,1/2*(1-n)],[1/2*(3-n)],cos(e+f*x)^2)*(d*sec(e+f*x))^(-1+n)*sin(e+f*x)/(f*(1-n^2)*sqrt(sin(e+f*x)^2))+b*(b^2*(1+n)+3*a^2*(2+n))*hypergeom([1/2,-1/2*n],[1/2*(2-n)],cos(e+f*x)^2)*(d*sec(e+f*x))^n*sin(e+f*x)/(f*n*(2+n)*sqrt(sin(e+f*x)^2))+a*b^2*(5+2*n)*(d*sec(e+f*x))^n*tan(e+f*x)/(f*(1+n)*(2+n))+b^2*(d*sec(e+f*x))^n*(a+b*sec(e+f*x))*tan(e+f*x)/(f*(2+n))],
[(d*sec(e+f*x))^n*(a+b*sec(e+f*x))^2,x,6,-d*(b^2*n+a^2*(1+n))*hypergeom([1/2,1/2*(1-n)],[1/2*(3-n)],cos(e+f*x)^2)*(d*sec(e+f*x))^(-1+n)*sin(e+f*x)/(f*(1-n^2)*sqrt(sin(e+f*x)^2))+2*a*b*hypergeom([1/2,-1/2*n],[1/2*(2-n)],cos(e+f*x)^2)*(d*sec(e+f*x))^n*sin(e+f*x)/(f*n*sqrt(sin(e+f*x)^2))+b^2*(d*sec(e+f*x))^n*tan(e+f*x)/(f*(1+n))],
[(d*sec(e+f*x))^n*(a+b*sec(e+f*x)),x,5,-a*d*hypergeom([1/2,1/2*(1-n)],[1/2*(3-n)],cos(e+f*x)^2)*(d*sec(e+f*x))^(-1+n)*sin(e+f*x)/(f*(1-n)*sqrt(sin(e+f*x)^2))+b*hypergeom([1/2,-1/2*n],[1/2*(2-n)],cos(e+f*x)^2)*(d*sec(e+f*x))^n*sin(e+f*x)/(f*n*sqrt(sin(e+f*x)^2))],
[(d*sec(e+f*x))^n/(a+b*sec(e+f*x)),x,6,a*AppellF1(1/2,1/2*(-1+n),1,3/2,sin(e+f*x)^2,a^2*sin(e+f*x)^2/(a^2-b^2))*cos(e+f*x)*(cos(e+f*x)^2)^(1/2*(-1+n))*(d*sec(e+f*x))^n*sin(e+f*x)/((a^2-b^2)*f)-b*AppellF1(1/2,1/2*n,1,3/2,sin(e+f*x)^2,a^2*sin(e+f*x)^2/(a^2-b^2))*(cos(e+f*x)^2)^(1/2*n)*(d*sec(e+f*x))^n*sin(e+f*x)/((a^2-b^2)*f)],
[(d*sec(e+f*x))^n/(a+b*sec(e+f*x))^2,x,9,a^2*AppellF1(1/2,1/2*(-3+n),2,3/2,sin(e+f*x)^2,a^2*sin(e+f*x)^2/(a^2-b^2))*cos(e+f*x)*(cos(e+f*x)^2)^(1/2*(-1+n))*(d*sec(e+f*x))^n*sin(e+f*x)/((a^2-b^2)^2*f)+b^2*AppellF1(1/2,1/2*(-1+n),2,3/2,sin(e+f*x)^2,a^2*sin(e+f*x)^2/(a^2-b^2))*cos(e+f*x)*(cos(e+f*x)^2)^(1/2*(-1+n))*(d*sec(e+f*x))^n*sin(e+f*x)/((a^2-b^2)^2*f)-2*a*b*AppellF1(1/2,1/2*(-2+n),2,3/2,sin(e+f*x)^2,a^2*sin(e+f*x)^2/(a^2-b^2))*(cos(e+f*x)^2)^(1/2*n)*(d*sec(e+f*x))^n*sin(e+f*x)/((a^2-b^2)^2*f)],
[(d*sec(e+f*x))^n*(a+b*sec(e+f*x))^(3/2),x,0,Unintegrable((d*sec(e+f*x))^n*(a+b*sec(e+f*x))^(3/2),x)],
[(d*sec(e+f*x))^n*(a+b*sec(e+f*x))^(1/2),x,0,Unintegrable((d*sec(e+f*x))^n*sqrt(a+b*sec(e+f*x)),x)],
[(d*sec(e+f*x))^n/(a+b*sec(e+f*x))^(1/2),x,0,Unintegrable((d*sec(e+f*x))^n/sqrt(a+b*sec(e+f*x)),x)],
[(d*sec(e+f*x))^n/(a+b*sec(e+f*x))^(3/2),x,0,Unintegrable((d*sec(e+f*x))^n/(a+b*sec(e+f*x))^(3/2),x)],

# Integrands of the form (d Sec[e+f x])^n (a+b Sec[e+f x])^m with m symbolic
[sec(e+f*x)^n*(a+b*sec(e+f*x))^m,x,0,Unintegrable(sec(e+f*x)^n*(a+b*sec(e+f*x))^m,x)],
[(d*sec(e+f*x))^n*(a+b*sec(e+f*x))^m,x,0,Unintegrable((d*sec(e+f*x))^n*(a+b*sec(e+f*x))^m,x)],
[sec(e+f*x)^3*(a+b*sec(e+f*x))^m,x,8,(a+b*sec(e+f*x))^(1+m)*tan(e+f*x)/(b*f*(2+m))-a*(a+b)*AppellF1(1/2,1/2,-1-m,3/2,1/2*(1-sec(e+f*x)),b*(1-sec(e+f*x))/(a+b))*(a+b*sec(e+f*x))^m*sqrt(2)*tan(e+f*x)/(b^2*f*(2+m)*((a+b*sec(e+f*x))/(a+b))^m*sqrt(1+sec(e+f*x)))+(a^2+b^2*(1+m))*AppellF1(1/2,1/2,-m,3/2,1/2*(1-sec(e+f*x)),b*(1-sec(e+f*x))/(a+b))*(a+b*sec(e+f*x))^m*sqrt(2)*tan(e+f*x)/(b^2*f*(2+m)*((a+b*sec(e+f*x))/(a+b))^m*sqrt(1+sec(e+f*x)))],
[sec(e+f*x)^2*(a+b*sec(e+f*x))^m,x,7,(a+b)*AppellF1(1/2,1/2,-1-m,3/2,1/2*(1-sec(e+f*x)),b*(1-sec(e+f*x))/(a+b))*(a+b*sec(e+f*x))^m*sqrt(2)*tan(e+f*x)/(b*f*((a+b*sec(e+f*x))/(a+b))^m*sqrt(1+sec(e+f*x)))-a*AppellF1(1/2,1/2,-m,3/2,1/2*(1-sec(e+f*x)),b*(1-sec(e+f*x))/(a+b))*(a+b*sec(e+f*x))^m*sqrt(2)*tan(e+f*x)/(b*f*((a+b*sec(e+f*x))/(a+b))^m*sqrt(1+sec(e+f*x)))],
[sec(e+f*x)*(a+b*sec(e+f*x))^m,x,3,AppellF1(1/2,1/2,-m,3/2,1/2*(1-sec(e+f*x)),b*(1-sec(e+f*x))/(a+b))*(a+b*sec(e+f*x))^m*sqrt(2)*tan(e+f*x)/(f*((a+b*sec(e+f*x))/(a+b))^m*sqrt(1+sec(e+f*x)))],
[(a+b*sec(e+f*x))^m,x,0,Unintegrable((a+b*sec(e+f*x))^m,x)],
[cos(e+f*x)*(a+b*sec(e+f*x))^m,x,0,Unintegrable(cos(e+f*x)*(a+b*sec(e+f*x))^m,x)],
[cos(e+f*x)^2*(a+b*sec(e+f*x))^m,x,0,Unintegrable(cos(e+f*x)^2*(a+b*sec(e+f*x))^m,x)],

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

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

# n>0
[cos(c+d*x)^(9/2)*(a+b*sec(c+d*x)),x,8,14/15*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+14/45*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+2/9*a*cos(c+d*x)^(7/2)*sin(c+d*x)/d+10/21*b*sin(c+d*x)*sqrt(cos(c+d*x))/d],
[cos(c+d*x)^(7/2)*(a+b*sec(c+d*x)),x,7,6/5*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+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*b*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)^(5/2)*(a+b*sec(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+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*cos(c+d*x)^(3/2)*sin(c+d*x)/d+2/3*b*sin(c+d*x)*sqrt(cos(c+d*x))/d],
[cos(c+d*x)^(3/2)*(a+b*sec(c+d*x)),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)*sqrt(cos(c+d*x))/d],
[cos(c+d*x)^(1/2)*(a+b*sec(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*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],
[(a+b*sec(c+d*x))/cos(c+d*x)^(1/2),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*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*b*sin(c+d*x)/(d*sqrt(cos(c+d*x)))],
[(a+b*sec(c+d*x))/cos(c+d*x)^(3/2),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))/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)/(d*cos(c+d*x)^(3/2))+2*a*sin(c+d*x)/(d*sqrt(cos(c+d*x)))],
[(a+b*sec(c+d*x))/cos(c+d*x)^(5/2),x,7,-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*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*b*sin(c+d*x)/(d*sqrt(cos(c+d*x)))],
[cos(c+d*x)^(9/2)*(a+b*sec(c+d*x))^2,x,10,2/15*(7*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+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*(7*a^2+9*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*a^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)^(7/2)*(a+b*sec(c+d*x))^2,x,9,12/5*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*(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))/d+4/5*a*b*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+2/21*(5*a^2+7*b^2)*sin(c+d*x)*sqrt(cos(c+d*x))/d],
[cos(c+d*x)^(5/2)*(a+b*sec(c+d*x))^2,x,8,2/5*(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*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],
[cos(c+d*x)^(3/2)*(a+b*sec(c+d*x))^2,x,7,4*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)*sqrt(cos(c+d*x))/d],
[cos(c+d*x)^(1/2)*(a+b*sec(c+d*x))^2,x,7,2*(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*b^2*sin(c+d*x)/(d*sqrt(cos(c+d*x)))],
[(a+b*sec(c+d*x))^2/cos(c+d*x)^(1/2),x,8,-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)/(d*cos(c+d*x)^(3/2))+4*a*b*sin(c+d*x)/(d*sqrt(cos(c+d*x)))],
[(a+b*sec(c+d*x))^2/cos(c+d*x)^(3/2),x,9,-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*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*(5*a^2+3*b^2)*sin(c+d*x)/(d*sqrt(cos(c+d*x)))],
[(a+b*sec(c+d*x))^2/cos(c+d*x)^(5/2),x,10,-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+2/7*b^2*sin(c+d*x)/(d*cos(c+d*x)^(7/2))+4/5*a*b*sin(c+d*x)/(d*cos(c+d*x)^(5/2))+2/21*(7*a^2+5*b^2)*sin(c+d*x)/(d*cos(c+d*x)^(3/2))+12/5*a*b*sin(c+d*x)/(d*sqrt(cos(c+d*x)))],
[cos(c+d*x)^(9/2)*(a+b*sec(c+d*x))^3,x,10,2/15*a*(7*a^2+27*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*(15*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))/d+2/45*a*(7*a^2+27*b^2)*cos(c+d*x)^(3/2)*sin(c+d*x)/d+40/63*a^2*b*cos(c+d*x)^(5/2)*sin(c+d*x)/d+2/9*a^2*cos(c+d*x)^(7/2)*(a+b*sec(c+d*x))*sin(c+d*x)/d+2/21*b*(15*a^2+7*b^2)*sin(c+d*x)*sqrt(cos(c+d*x))/d],
[cos(c+d*x)^(7/2)*(a+b*sec(c+d*x))^3,x,9,2/5*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*cos(c+d*x)^(3/2)*sin(c+d*x)/d+2/7*a^2*cos(c+d*x)^(5/2)*(a+b*sec(c+d*x))*sin(c+d*x)/d+2/21*a*(5*a^2+21*b^2)*sin(c+d*x)*sqrt(cos(c+d*x))/d],
[cos(c+d*x)^(5/2)*(a+b*sec(c+d*x))^3,x,8,6/5*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+2/5*a^2*cos(c+d*x)^(3/2)*(a+b*sec(c+d*x))*sin(c+d*x)/d+8/5*a^2*b*sin(c+d*x)*sqrt(cos(c+d*x))/d],
[cos(c+d*x)^(3/2)*(a+b*sec(c+d*x))^3,x,8,2*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*b*(a^2-3*b^2)*sin(c+d*x)/(d*sqrt(cos(c+d*x)))+2/3*a^2*(a+b*sec(c+d*x))*sin(c+d*x)*sqrt(cos(c+d*x))/d],
[cos(c+d*x)^(1/2)*(a+b*sec(c+d*x))^3,x,8,2*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+16/3*a*b^2*sin(c+d*x)/(d*sqrt(cos(c+d*x)))+2/3*b^2*(a+b*sec(c+d*x))*sin(c+d*x)/(d*sqrt(cos(c+d*x)))],
[(a+b*sec(c+d*x))^3/cos(c+d*x)^(1/2),x,9,-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)/(d*cos(c+d*x)^(3/2))+2/5*b^2*(a+b*sec(c+d*x))*sin(c+d*x)/(d*cos(c+d*x)^(3/2))+6/5*b*(5*a^2+b^2)*sin(c+d*x)/(d*sqrt(cos(c+d*x)))],
[(a+b*sec(c+d*x))^3/cos(c+d*x)^(3/2),x,10,-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*sin(c+d*x)/(d*cos(c+d*x)^(5/2))+2/21*b*(21*a^2+5*b^2)*sin(c+d*x)/(d*cos(c+d*x)^(3/2))+2/7*b^2*(a+b*sec(c+d*x))*sin(c+d*x)/(d*cos(c+d*x)^(5/2))+2/5*a*(5*a^2+9*b^2)*sin(c+d*x)/(d*sqrt(cos(c+d*x)))],

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

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

# n>0
[cos(c+d*x)^(5/2)*sqrt(a+b*sec(c+d*x)),x,10,-4/15*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*a/(a+b)))*sqrt((b+a*cos(c+d*x))/(a+b))/(a^2*d*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x)))+2/5*cos(c+d*x)^(3/2)*sin(c+d*x)*sqrt(a+b*sec(c+d*x))/d+2/15*b*sin(c+d*x)*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x))/(a*d)+2/15*(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*a/(a+b)))*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x))/(a^2*d*sqrt((b+a*cos(c+d*x))/(a+b)))],
[cos(c+d*x)^(3/2)*sqrt(a+b*sec(c+d*x)),x,9,2/3*(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*a/(a+b)))*sqrt((b+a*cos(c+d*x))/(a+b))/(a*d*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x)))+2/3*sin(c+d*x)*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x))/d+2/3*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+b)))*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x))/(a*d*sqrt((b+a*cos(c+d*x))/(a+b)))],
[cos(c+d*x)^(1/2)*sqrt(a+b*sec(c+d*x)),x,4,2*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+b)))*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x))/(d*sqrt((b+a*cos(c+d*x))/(a+b)))],
[sqrt(a+b*sec(c+d*x))/cos(c+d*x)^(1/2),x,8,2*a*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2*a/(a+b)))*sqrt((b+a*cos(c+d*x))/(a+b))/(d*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x)))+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*a/(a+b)))*sqrt((b+a*cos(c+d*x))/(a+b))/(d*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x)))],
[sqrt(a+b*sec(c+d*x))/cos(c+d*x)^(3/2),x,13,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+b)))*sqrt((b+a*cos(c+d*x))/(a+b))/(d*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x)))+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*a/(a+b)))*sqrt((b+a*cos(c+d*x))/(a+b))/(d*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x)))+sin(c+d*x)*sqrt(a+b*sec(c+d*x))/(d*sqrt(cos(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*a/(a+b)))*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x))/(d*sqrt((b+a*cos(c+d*x))/(a+b)))],
[cos(c+d*x)^(7/2)*(a+b*sec(c+d*x))^(3/2),x,11,2/105*(25*a^4-31*a^2*b^2+6*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/(a+b)))*sqrt((b+a*cos(c+d*x))/(a+b))/(a^2*d*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x)))+16/35*b*cos(c+d*x)^(3/2)*sin(c+d*x)*sqrt(a+b*sec(c+d*x))/d+2/7*a*cos(c+d*x)^(5/2)*sin(c+d*x)*sqrt(a+b*sec(c+d*x))/d+2/105*(25*a^2+3*b^2)*sin(c+d*x)*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x))/(a*d)+4/105*b*(41*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/(a+b)))*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x))/(a^2*d*sqrt((b+a*cos(c+d*x))/(a+b)))],
[cos(c+d*x)^(5/2)*(a+b*sec(c+d*x))^(3/2),x,10,2/5*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*a/(a+b)))*sqrt((b+a*cos(c+d*x))/(a+b))/(a*d*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x)))+2/5*a*cos(c+d*x)^(3/2)*sin(c+d*x)*sqrt(a+b*sec(c+d*x))/d+4/5*b*sin(c+d*x)*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x))/d+2/5*(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/(a+b)))*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x))/(a*d*sqrt((b+a*cos(c+d*x))/(a+b)))],
[cos(c+d*x)^(3/2)*(a+b*sec(c+d*x))^(3/2),x,9,2/3*(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*a/(a+b)))*sqrt((b+a*cos(c+d*x))/(a+b))/(d*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x)))+2/3*a*sin(c+d*x)*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x))/d+8/3*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+b)))*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x))/(d*sqrt((b+a*cos(c+d*x))/(a+b)))],
[cos(c+d*x)^(1/2)*(a+b*sec(c+d*x))^(3/2),x,12,2*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*a/(a+b)))*sqrt((b+a*cos(c+d*x))/(a+b))/(d*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x)))+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*a/(a+b)))*sqrt((b+a*cos(c+d*x))/(a+b))/(d*sqrt(cos(c+d*x))*sqrt(a+b*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*a/(a+b)))*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x))/(d*sqrt((b+a*cos(c+d*x))/(a+b)))],
[(a+b*sec(c+d*x))^(3/2)/cos(c+d*x)^(1/2),x,13,(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*a/(a+b)))*sqrt((b+a*cos(c+d*x))/(a+b))/(d*sqrt(cos(c+d*x))*sqrt(a+b*sec(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*a/(a+b)))*sqrt((b+a*cos(c+d*x))/(a+b))/(d*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x)))+b*sin(c+d*x)*sqrt(a+b*sec(c+d*x))/(d*sqrt(cos(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*a/(a+b)))*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x))/(d*sqrt((b+a*cos(c+d*x))/(a+b)))],
[(a+b*sec(c+d*x))^(3/2)/cos(c+d*x)^(3/2),x,14,7/4*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*a/(a+b)))*sqrt((b+a*cos(c+d*x))/(a+b))/(d*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x)))+1/4*(3*a^2+4*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*a/(a+b)))*sqrt((b+a*cos(c+d*x))/(a+b))/(d*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x)))+1/2*b*sin(c+d*x)*sqrt(a+b*sec(c+d*x))/(d*cos(c+d*x)^(3/2))+5/4*a*sin(c+d*x)*sqrt(a+b*sec(c+d*x))/(d*sqrt(cos(c+d*x)))-5/4*a*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+b)))*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x))/(d*sqrt((b+a*cos(c+d*x))/(a+b)))],
[cos(c+d*x)^(9/2)*(a+b*sec(c+d*x))^(5/2),x,12,4/315*b*(57*a^4-62*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*a/(a+b)))*sqrt((b+a*cos(c+d*x))/(a+b))/(a^2*d*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x)))+2/315*(49*a^2+75*b^2)*cos(c+d*x)^(3/2)*sin(c+d*x)*sqrt(a+b*sec(c+d*x))/d+38/63*a*b*cos(c+d*x)^(5/2)*sin(c+d*x)*sqrt(a+b*sec(c+d*x))/d+2/9*a^2*cos(c+d*x)^(7/2)*sin(c+d*x)*sqrt(a+b*sec(c+d*x))/d+2/315*b*(163*a^2+5*b^2)*sin(c+d*x)*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x))/(a*d)+2/315*(147*a^4+279*a^2*b^2-10*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/(a+b)))*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x))/(a^2*d*sqrt((b+a*cos(c+d*x))/(a+b)))],
[cos(c+d*x)^(7/2)*(a+b*sec(c+d*x))^(5/2),x,11,2/21*(5*a^4-2*a^2*b^2-3*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/(a+b)))*sqrt((b+a*cos(c+d*x))/(a+b))/(a*d*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x)))+6/7*a*b*cos(c+d*x)^(3/2)*sin(c+d*x)*sqrt(a+b*sec(c+d*x))/d+2/7*a^2*cos(c+d*x)^(5/2)*sin(c+d*x)*sqrt(a+b*sec(c+d*x))/d+2/21*(5*a^2+9*b^2)*sin(c+d*x)*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x))/d+2/21*b*(29*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/(a+b)))*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x))/(a*d*sqrt((b+a*cos(c+d*x))/(a+b)))],
[cos(c+d*x)^(5/2)*(a+b*sec(c+d*x))^(5/2),x,10,16/15*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*a/(a+b)))*sqrt((b+a*cos(c+d*x))/(a+b))/(d*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x)))+2/5*a^2*cos(c+d*x)^(3/2)*sin(c+d*x)*sqrt(a+b*sec(c+d*x))/d+22/15*a*b*sin(c+d*x)*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x))/d+2/15*(9*a^2+23*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+b)))*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x))/(d*sqrt((b+a*cos(c+d*x))/(a+b)))],
[cos(c+d*x)^(3/2)*(a+b*sec(c+d*x))^(5/2),x,13,2/3*a*(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*a/(a+b)))*sqrt((b+a*cos(c+d*x))/(a+b))/(d*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x)))+2*b^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*a/(a+b)))*sqrt((b+a*cos(c+d*x))/(a+b))/(d*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x)))+2/3*a^2*sin(c+d*x)*sqrt(cos(c+d*x))*sqrt(a+b*sec(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*a/(a+b)))*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x))/(d*sqrt((b+a*cos(c+d*x))/(a+b)))],
[cos(c+d*x)^(1/2)*(a+b*sec(c+d*x))^(5/2),x,13,b*(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*a/(a+b)))*sqrt((b+a*cos(c+d*x))/(a+b))/(d*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x)))+5*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*a/(a+b)))*sqrt((b+a*cos(c+d*x))/(a+b))/(d*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x)))+b^2*sin(c+d*x)*sqrt(a+b*sec(c+d*x))/(d*sqrt(cos(c+d*x)))+(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*a/(a+b)))*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x))/(d*sqrt((b+a*cos(c+d*x))/(a+b)))],
[(a+b*sec(c+d*x))^(5/2)/cos(c+d*x)^(1/2),x,14,1/4*a*(8*a^2+11*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+b)))*sqrt((b+a*cos(c+d*x))/(a+b))/(d*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x)))+1/4*b*(15*a^2+4*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*a/(a+b)))*sqrt((b+a*cos(c+d*x))/(a+b))/(d*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x)))+1/2*b^2*sin(c+d*x)*sqrt(a+b*sec(c+d*x))/(d*cos(c+d*x)^(3/2))+9/4*a*b*sin(c+d*x)*sqrt(a+b*sec(c+d*x))/(d*sqrt(cos(c+d*x)))-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*a/(a+b)))*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x))/(d*sqrt((b+a*cos(c+d*x))/(a+b)))],
[(a+b*sec(c+d*x))^(5/2)/cos(c+d*x)^(3/2),x,15,1/24*b*(59*a^2+16*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+b)))*sqrt((b+a*cos(c+d*x))/(a+b))/(d*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x)))+5/8*a*(a^2+4*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*a/(a+b)))*sqrt((b+a*cos(c+d*x))/(a+b))/(d*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x)))+1/3*b^2*sin(c+d*x)*sqrt(a+b*sec(c+d*x))/(d*cos(c+d*x)^(5/2))+13/12*a*b*sin(c+d*x)*sqrt(a+b*sec(c+d*x))/(d*cos(c+d*x)^(3/2))+1/24*(33*a^2+16*b^2)*sin(c+d*x)*sqrt(a+b*sec(c+d*x))/(d*sqrt(cos(c+d*x)))-1/24*(33*a^2+16*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+b)))*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x))/(d*sqrt((b+a*cos(c+d*x))/(a+b)))],

# n<0
[cos(c+d*x)^(5/2)/sqrt(a+b*sec(c+d*x)),x,10,-2/15*b*(7*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*a/(a+b)))*sqrt((b+a*cos(c+d*x))/(a+b))/(a^3*d*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x)))+2/5*cos(c+d*x)^(3/2)*sin(c+d*x)*sqrt(a+b*sec(c+d*x))/(a*d)-8/15*b*sin(c+d*x)*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x))/(a^2*d)+2/15*(9*a^2+8*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+b)))*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x))/(a^3*d*sqrt((b+a*cos(c+d*x))/(a+b)))],
[cos(c+d*x)^(3/2)/sqrt(a+b*sec(c+d*x)),x,9,2/3*(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*a/(a+b)))*sqrt((b+a*cos(c+d*x))/(a+b))/(a^2*d*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x)))+2/3*sin(c+d*x)*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x))/(a*d)-4/3*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+b)))*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x))/(a^2*d*sqrt((b+a*cos(c+d*x))/(a+b)))],
[cos(c+d*x)^(1/2)/sqrt(a+b*sec(c+d*x)),x,8,-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*a/(a+b)))*sqrt((b+a*cos(c+d*x))/(a+b))/(a*d*sqrt(cos(c+d*x))*sqrt(a+b*sec(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*a/(a+b)))*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x))/(a*d*sqrt((b+a*cos(c+d*x))/(a+b)))],
[1/(cos(c+d*x)^(1/2)*sqrt(a+b*sec(c+d*x))),x,4,2*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+b)))*sqrt((b+a*cos(c+d*x))/(a+b))/(d*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x)))],
[1/(cos(c+d*x)^(3/2)*sqrt(a+b*sec(c+d*x))),x,4,2*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticPi(sin(1/2*(c+d*x)),2,sqrt(2*a/(a+b)))*sqrt((b+a*cos(c+d*x))/(a+b))/(d*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x)))],
[1/(cos(c+d*x)^(5/2)*sqrt(a+b*sec(c+d*x))),x,13,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+b)))*sqrt((b+a*cos(c+d*x))/(a+b))/(d*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x)))-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*a/(a+b)))*sqrt((b+a*cos(c+d*x))/(a+b))/(b*d*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x)))+sin(c+d*x)*sqrt(a+b*sec(c+d*x))/(b*d*sqrt(cos(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*a/(a+b)))*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x))/(b*d*sqrt((b+a*cos(c+d*x))/(a+b)))],
[1/(cos(c+d*x)^(7/2)*sqrt(a+b*sec(c+d*x))),x,14,-1/4*a*sqrt(cos(1/2*(c+d*x))^2)/cos(1/2*(c+d*x))*EllipticF(sin(1/2*(c+d*x)),sqrt(2*a/(a+b)))*sqrt((b+a*cos(c+d*x))/(a+b))/(b*d*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x)))+1/4*(3*a^2+4*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*a/(a+b)))*sqrt((b+a*cos(c+d*x))/(a+b))/(b^2*d*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x)))+1/2*sin(c+d*x)*sqrt(a+b*sec(c+d*x))/(b*d*cos(c+d*x)^(3/2))-3/4*a*sin(c+d*x)*sqrt(a+b*sec(c+d*x))/(b^2*d*sqrt(cos(c+d*x)))+3/4*a*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+b)))*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x))/(b^2*d*sqrt((b+a*cos(c+d*x))/(a+b)))],
[cos(c+d*x)^(5/2)/(a+b*sec(c+d*x))^(3/2),x,11,2*b^2*cos(c+d*x)^(3/2)*sin(c+d*x)/(a*(a^2-b^2)*d*sqrt(a+b*sec(c+d*x)))-8/5*b*(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*a/(a+b)))*sqrt((b+a*cos(c+d*x))/(a+b))/(a^4*d*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x)))+2/5*(a^2-6*b^2)*cos(c+d*x)^(3/2)*sin(c+d*x)*sqrt(a+b*sec(c+d*x))/(a^2*(a^2-b^2)*d)-2/5*b*(3*a^2-8*b^2)*sin(c+d*x)*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x))/(a^3*(a^2-b^2)*d)+2/5*(3*a^4+8*a^2*b^2-16*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/(a+b)))*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x))/(a^4*(a^2-b^2)*d*sqrt((b+a*cos(c+d*x))/(a+b)))],
[cos(c+d*x)^(3/2)/(a+b*sec(c+d*x))^(3/2),x,10,2*b^2*sin(c+d*x)*sqrt(cos(c+d*x))/(a*(a^2-b^2)*d*sqrt(a+b*sec(c+d*x)))+2/3*(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*a/(a+b)))*sqrt((b+a*cos(c+d*x))/(a+b))/(a^3*d*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x)))+2/3*(a^2-4*b^2)*sin(c+d*x)*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x))/(a^2*(a^2-b^2)*d)-2/3*b*(5*a^2-8*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+b)))*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x))/(a^3*(a^2-b^2)*d*sqrt((b+a*cos(c+d*x))/(a+b)))],
[cos(c+d*x)^(1/2)/(a+b*sec(c+d*x))^(3/2),x,9,2*b^2*sin(c+d*x)/(a*(a^2-b^2)*d*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x)))-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*a/(a+b)))*sqrt((b+a*cos(c+d*x))/(a+b))/(a^2*d*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x)))+2*(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*a/(a+b)))*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x))/(a^2*(a^2-b^2)*d*sqrt((b+a*cos(c+d*x))/(a+b)))],
[1/(cos(c+d*x)^(1/2)*(a+b*sec(c+d*x))^(3/2)),x,9,-2*b*sin(c+d*x)/((a^2-b^2)*d*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x)))+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+b)))*sqrt((b+a*cos(c+d*x))/(a+b))/(a*d*sqrt(cos(c+d*x))*sqrt(a+b*sec(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*a/(a+b)))*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x))/(a*(a^2-b^2)*d*sqrt((b+a*cos(c+d*x))/(a+b)))],
[1/(cos(c+d*x)^(3/2)*(a+b*sec(c+d*x))^(3/2)),x,6,2*a*sin(c+d*x)/((a^2-b^2)*d*sqrt(cos(c+d*x))*sqrt(a+b*sec(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*a/(a+b)))*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x))/((a^2-b^2)*d*sqrt((b+a*cos(c+d*x))/(a+b)))],
[1/(cos(c+d*x)^(5/2)*(a+b*sec(c+d*x))^(3/2)),x,10,-2*a^2*sin(c+d*x)/(b*(a^2-b^2)*d*sqrt(cos(c+d*x))*sqrt(a+b*sec(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*a/(a+b)))*sqrt((b+a*cos(c+d*x))/(a+b))/(b*d*sqrt(cos(c+d*x))*sqrt(a+b*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*a/(a+b)))*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x))/(b*(a^2-b^2)*d*sqrt((b+a*cos(c+d*x))/(a+b)))],
[1/(cos(c+d*x)^(7/2)*(a+b*sec(c+d*x))^(3/2)),x,14,-2*a^2*sin(c+d*x)/(b*(a^2-b^2)*d*cos(c+d*x)^(3/2)*sqrt(a+b*sec(c+d*x)))+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+b)))*sqrt((b+a*cos(c+d*x))/(a+b))/(b*d*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x)))-3*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*a/(a+b)))*sqrt((b+a*cos(c+d*x))/(a+b))/(b^2*d*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x)))+(3*a^2-b^2)*sin(c+d*x)*sqrt(a+b*sec(c+d*x))/(b^2*(a^2-b^2)*d*sqrt(cos(c+d*x)))-(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/(a+b)))*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x))/(b^2*(a^2-b^2)*d*sqrt((b+a*cos(c+d*x))/(a+b)))],
[cos(c+d*x)^(3/2)/(a+b*sec(c+d*x))^(5/2),x,11,2/3*b^2*sin(c+d*x)*sqrt(cos(c+d*x))/(a*(a^2-b^2)*d*(a+b*sec(c+d*x))^(3/2))+4/3*b^2*(5*a^2-3*b^2)*sin(c+d*x)*sqrt(cos(c+d*x))/(a^2*(a^2-b^2)^2*d*sqrt(a+b*sec(c+d*x)))+2/3*(a^4+16*a^2*b^2-16*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/(a+b)))*sqrt((b+a*cos(c+d*x))/(a+b))/(a^4*(a^2-b^2)*d*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x)))+2/3*(a^4-13*a^2*b^2+8*b^4)*sin(c+d*x)*sqrt(cos(c+d*x))*sqrt(a+b*sec(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)*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+b)))*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x))/(a^4*(a^2-b^2)^2*d*sqrt((b+a*cos(c+d*x))/(a+b)))],
[cos(c+d*x)^(1/2)/(a+b*sec(c+d*x))^(5/2),x,10,2/3*b^2*sin(c+d*x)/(a*(a^2-b^2)*d*(a+b*sec(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*sec(c+d*x)))-2/3*b*(9*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*a/(a+b)))*sqrt((b+a*cos(c+d*x))/(a+b))/(a^3*(a^2-b^2)*d*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x)))+2/3*(3*a^4-15*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*a/(a+b)))*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x))/(a^3*(a^2-b^2)^2*d*sqrt((b+a*cos(c+d*x))/(a+b)))],
[1/(cos(c+d*x)^(1/2)*(a+b*sec(c+d*x))^(5/2)),x,10,-2/3*b*sin(c+d*x)/((a^2-b^2)*d*(a+b*sec(c+d*x))^(3/2)*sqrt(cos(c+d*x)))-2/3*b*(5*a^2-b^2)*sin(c+d*x)/(a*(a^2-b^2)^2*d*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x)))+2/3*(3*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*a/(a+b)))*sqrt((b+a*cos(c+d*x))/(a+b))/(a^2*(a^2-b^2)*d*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x)))+4/3*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/(a+b)))*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x))/(a^2*(a^2-b^2)^2*d*sqrt((b+a*cos(c+d*x))/(a+b)))],
[1/(cos(c+d*x)^(3/2)*(a+b*sec(c+d*x))^(5/2)),x,10,2/3*a*sin(c+d*x)/((a^2-b^2)*d*(a+b*sec(c+d*x))^(3/2)*sqrt(cos(c+d*x)))+4/3*(a^2+b^2)*sin(c+d*x)/((a^2-b^2)^2*d*sqrt(cos(c+d*x))*sqrt(a+b*sec(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*a/(a+b)))*sqrt((b+a*cos(c+d*x))/(a+b))/(a*(a^2-b^2)*d*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x)))-2/3*(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/(a+b)))*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x))/(a*(a^2-b^2)^2*d*sqrt((b+a*cos(c+d*x))/(a+b)))],
[1/(cos(c+d*x)^(5/2)*(a+b*sec(c+d*x))^(5/2)),x,10,-2/3*a^2*sin(c+d*x)/(b*(a^2-b^2)*d*(a+b*sec(c+d*x))^(3/2)*sqrt(cos(c+d*x)))+2/3*a*(a^2-5*b^2)*sin(c+d*x)/(b*(a^2-b^2)^2*d*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x)))+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/(a+b)))*sqrt((b+a*cos(c+d*x))/(a+b))/((a^2-b^2)*d*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x)))+8/3*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+b)))*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x))/((a^2-b^2)^2*d*sqrt((b+a*cos(c+d*x))/(a+b)))],
[1/(cos(c+d*x)^(7/2)*(a+b*sec(c+d*x))^(5/2)),x,14,-2/3*a^2*sin(c+d*x)/(b*(a^2-b^2)*d*cos(c+d*x)^(3/2)*(a+b*sec(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*sec(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*a/(a+b)))*sqrt((b+a*cos(c+d*x))/(a+b))/(b*(a^2-b^2)*d*sqrt(cos(c+d*x))*sqrt(a+b*sec(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*a/(a+b)))*sqrt((b+a*cos(c+d*x))/(a+b))/(b^2*d*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x)))+2/3*a*(3*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*a/(a+b)))*sqrt(cos(c+d*x))*sqrt(a+b*sec(c+d*x))/(b^2*(a^2-b^2)^2*d*sqrt((b+a*cos(c+d*x))/(a+b)))],

# Integrands of the form (d Cos[e+f x])^n (a+b Sec[e+f x])^m with n symbolic
[(d*cos(e+f*x))^n*(a+b*sec(e+f*x))^3,x,8,-b*(b^2*(1-n)+3*a^2*(2-n))*(d*cos(e+f*x))^n*hypergeom([1/2,1/2*n],[1/2*(2+n)],cos(e+f*x)^2)*sin(e+f*x)/(f*(2-n)*n*sqrt(sin(e+f*x)^2))-a*(a^2*(1-n)-3*b^2*n)*cos(e+f*x)*(d*cos(e+f*x))^n*hypergeom([1/2,1/2*(1+n)],[1/2*(3+n)],cos(e+f*x)^2)*sin(e+f*x)/(f*(1-n^2)*sqrt(sin(e+f*x)^2))+a*b^2*(5-2*n)*(d*cos(e+f*x))^n*tan(e+f*x)/(f*(2-3*n+n^2))+b^2*(d*cos(e+f*x))^n*(a+b*sec(e+f*x))*tan(e+f*x)/(f*(2-n))],
[(d*cos(e+f*x))^n*(a+b*sec(e+f*x))^2,x,7,-2*a*b*(d*cos(e+f*x))^n*hypergeom([1/2,1/2*n],[1/2*(2+n)],cos(e+f*x)^2)*sin(e+f*x)/(f*n*sqrt(sin(e+f*x)^2))-(a^2*(1-n)-b^2*n)*cos(e+f*x)*(d*cos(e+f*x))^n*hypergeom([1/2,1/2*(1+n)],[1/2*(3+n)],cos(e+f*x)^2)*sin(e+f*x)/(f*(1-n^2)*sqrt(sin(e+f*x)^2))+b^2*(d*cos(e+f*x))^n*tan(e+f*x)/(f*(1-n))],
[(d*cos(e+f*x))^n*(a+b*sec(e+f*x)),x,5,-b*(d*cos(e+f*x))^n*hypergeom([1/2,1/2*n],[1/2*(2+n)],cos(e+f*x)^2)*sin(e+f*x)/(f*n*sqrt(sin(e+f*x)^2))-a*(d*cos(e+f*x))^(1+n)*hypergeom([1/2,1/2*(1+n)],[1/2*(3+n)],cos(e+f*x)^2)*sin(e+f*x)/(d*f*(1+n)*sqrt(sin(e+f*x)^2))],
[(d*cos(e+f*x))^n/(a+b*sec(e+f*x)),x,7,a*AppellF1(1/2,1/2*(-1-n),1,3/2,sin(e+f*x)^2,a^2*sin(e+f*x)^2/(a^2-b^2))*cos(e+f*x)*(d*cos(e+f*x))^n*(cos(e+f*x)^2)^(1/2*(-1-n))*sin(e+f*x)/((a^2-b^2)*f)-b*AppellF1(1/2,-1/2*n,1,3/2,sin(e+f*x)^2,a^2*sin(e+f*x)^2/(a^2-b^2))*(d*cos(e+f*x))^n*sin(e+f*x)/((a^2-b^2)*f*(cos(e+f*x)^2)^(1/2*n))],
[(d*cos(e+f*x))^n/(a+b*sec(e+f*x))^2,x,10,a^2*AppellF1(1/2,1/2*(-3-n),2,3/2,sin(e+f*x)^2,a^2*sin(e+f*x)^2/(a^2-b^2))*cos(e+f*x)*(d*cos(e+f*x))^n*(cos(e+f*x)^2)^(1/2*(-1-n))*sin(e+f*x)/((a^2-b^2)^2*f)+b^2*AppellF1(1/2,1/2*(-1-n),2,3/2,sin(e+f*x)^2,a^2*sin(e+f*x)^2/(a^2-b^2))*cos(e+f*x)*(d*cos(e+f*x))^n*(cos(e+f*x)^2)^(1/2*(-1-n))*sin(e+f*x)/((a^2-b^2)^2*f)-2*a*b*AppellF1(1/2,1/2*(-2-n),2,3/2,sin(e+f*x)^2,a^2*sin(e+f*x)^2/(a^2-b^2))*(d*cos(e+f*x))^n*sin(e+f*x)/((a^2-b^2)^2*f*(cos(e+f*x)^2)^(1/2*n))]]:
