# Maple integration test file: "5 Inverse trig functions\5.5 Inverse secant\5.5.1 u (a+b arcsec(c x))^n.txt"

lst:=[

# Integrands of the form u (a+b ArcSec[c x])^n

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

# Integrands of the form x^m (a+b ArcSec[c x])^n

# n>0
[x^6*(a+b*arcsec(c*x)),x,7,1/7*x^7*(a+b*arcsec(c*x))-5/112*b*arctanh(sqrt(1+(-1)/(c^2*x^2)))/c^7-5/112*b*x^2*sqrt(1+(-1)/(c^2*x^2))/c^5-5/168*b*x^4*sqrt(1+(-1)/(c^2*x^2))/c^3-1/42*b*x^6*sqrt(1+(-1)/(c^2*x^2))/c],
[x^5*(a+b*arcsec(c*x)),x,4,1/6*x^6*(a+b*arcsec(c*x))-4/45*b*x*sqrt(1+(-1)/(c^2*x^2))/c^5-2/45*b*x^3*sqrt(1+(-1)/(c^2*x^2))/c^3-1/30*b*x^5*sqrt(1+(-1)/(c^2*x^2))/c],
[x^4*(a+b*arcsec(c*x)),x,6,1/5*x^5*(a+b*arcsec(c*x))-3/40*b*arctanh(sqrt(1+(-1)/(c^2*x^2)))/c^5-3/40*b*x^2*sqrt(1+(-1)/(c^2*x^2))/c^3-1/20*b*x^4*sqrt(1+(-1)/(c^2*x^2))/c],
[x^3*(a+b*arcsec(c*x)),x,3,1/4*x^4*(a+b*arcsec(c*x))-1/6*b*x*sqrt(1+(-1)/(c^2*x^2))/c^3-1/12*b*x^3*sqrt(1+(-1)/(c^2*x^2))/c],
[x^2*(a+b*arcsec(c*x)),x,5,1/3*x^3*(a+b*arcsec(c*x))-1/6*b*arctanh(sqrt(1+(-1)/(c^2*x^2)))/c^3-1/6*b*x^2*sqrt(1+(-1)/(c^2*x^2))/c],
[x*(a+b*arcsec(c*x)),x,2,1/2*x^2*(a+b*arcsec(c*x))-1/2*b*x*sqrt(1+(-1)/(c^2*x^2))/c],
[a+b*arcsec(c*x),x,5,a*x+b*x*arcsec(c*x)-b*arctanh(sqrt(1+(-1)/(c^2*x^2)))/c],
[(a+b*arcsec(c*x))/x,x,6,1/2*I*(a+b*arcsec(c*x))^2/b-(a+b*arcsec(c*x))*log(1+exp(2*I*arcsec(c*x)))+1/2*I*b*polylog(2,-exp(2*I*arcsec(c*x)))],
[(a+b*arcsec(c*x))/x^2,x,2,(-a-b*arcsec(c*x))/x+b*c*sqrt(1+(-1)/(c^2*x^2))],
[(a+b*arcsec(c*x))/x^3,x,4,-1/4*b*c^2*arccsc(c*x)+1/2*(-a-b*arcsec(c*x))/x^2+1/4*b*c*sqrt(1+(-1)/(c^2*x^2))/x],
[(a+b*arcsec(c*x))/x^4,x,4,-1/9*b*c^3*(1+(-1)/(c^2*x^2))^(3/2)+1/3*(-a-b*arcsec(c*x))/x^3+1/3*b*c^3*sqrt(1+(-1)/(c^2*x^2))],
[(a+b*arcsec(c*x))/x^5,x,5,-3/32*b*c^4*arccsc(c*x)+1/4*(-a-b*arcsec(c*x))/x^4+1/16*b*c*sqrt(1+(-1)/(c^2*x^2))/x^3+3/32*b*c^3*sqrt(1+(-1)/(c^2*x^2))/x],
[(a+b*arcsec(c*x))/x^6,x,4,-2/15*b*c^5*(1+(-1)/(c^2*x^2))^(3/2)+1/25*b*c^5*(1+(-1)/(c^2*x^2))^(5/2)+1/5*(-a-b*arcsec(c*x))/x^5+1/5*b*c^5*sqrt(1+(-1)/(c^2*x^2))],
[(a+b*arcsec(c*x))/x^7,x,6,-5/96*b*c^6*arccsc(c*x)+1/6*(-a-b*arcsec(c*x))/x^6+1/36*b*c*sqrt(1+(-1)/(c^2*x^2))/x^5+5/144*b*c^3*sqrt(1+(-1)/(c^2*x^2))/x^3+5/96*b*c^5*sqrt(1+(-1)/(c^2*x^2))/x],
[x^3*(a+b*arcsec(c*x))^2,x,5,1/12*b^2*x^2/c^2+1/4*x^4*(a+b*arcsec(c*x))^2+1/3*b^2*log(x)/c^4-1/3*b*x*(a+b*arcsec(c*x))*sqrt(1+(-1)/(c^2*x^2))/c^3-1/6*b*x^3*(a+b*arcsec(c*x))*sqrt(1+(-1)/(c^2*x^2))/c],
[x^2*(a+b*arcsec(c*x))^2,x,8,1/3*b^2*x/c^2+1/3*x^3*(a+b*arcsec(c*x))^2+2/3*I*b*(a+b*arcsec(c*x))*arctan(exp(I*arcsec(c*x)))/c^3-1/3*I*b^2*polylog(2,-I*exp(I*arcsec(c*x)))/c^3+1/3*I*b^2*polylog(2,I*exp(I*arcsec(c*x)))/c^3-1/3*b*x^2*(a+b*arcsec(c*x))*sqrt(1+(-1)/(c^2*x^2))/c],
[x*(a+b*arcsec(c*x))^2,x,4,1/2*x^2*(a+b*arcsec(c*x))^2+b^2*log(x)/c^2-b*x*(a+b*arcsec(c*x))*sqrt(1+(-1)/(c^2*x^2))/c],
[(a+b*arcsec(c*x))^2,x,7,x*(a+b*arcsec(c*x))^2+4*I*b*(a+b*arcsec(c*x))*arctan(exp(I*arcsec(c*x)))/c-2*I*b^2*polylog(2,-I*exp(I*arcsec(c*x)))/c+2*I*b^2*polylog(2,I*exp(I*arcsec(c*x)))/c],
[(a+b*arcsec(c*x))^2/x,x,6,1/3*I*(a+b*arcsec(c*x))^3/b-(a+b*arcsec(c*x))^2*log(1+exp(2*I*arcsec(c*x)))+I*b*(a+b*arcsec(c*x))*polylog(2,-exp(2*I*arcsec(c*x)))-1/2*b^2*polylog(3,-exp(2*I*arcsec(c*x)))],
[(a+b*arcsec(c*x))^2/x^2,x,4,2*b^2/x-(a+b*arcsec(c*x))^2/x+2*b*c*(a+b*arcsec(c*x))*sqrt(1+(-1)/(c^2*x^2))],
[(a+b*arcsec(c*x))^2/x^3,x,4,1/4*b^2/x^2-1/2*a*b*c^2*arcsec(c*x)-1/4*b^2*c^2*arcsec(c*x)^2+1/2*(c^2+(-1)/x^2)*(a+b*arcsec(c*x))^2+1/2*b*c*(a+b*arcsec(c*x))*sqrt(1+(-1)/(c^2*x^2))/x],
[(a+b*arcsec(c*x))^2/x^4,x,5,2/27*b^2/x^3+4/9*b^2*c^2/x-1/3*(a+b*arcsec(c*x))^2/x^3+4/9*b*c^3*(a+b*arcsec(c*x))*sqrt(1+(-1)/(c^2*x^2))+2/9*b*c*(a+b*arcsec(c*x))*sqrt(1+(-1)/(c^2*x^2))/x^2],
[(a+b*arcsec(c*x))^2/x^5,x,5,1/32*b^2/x^4+3/32*b^2*c^2/x^2+3/16*a*b*c^4*arcsec(c*x)+3/32*b^2*c^4*arcsec(c*x)^2-1/4*(a+b*arcsec(c*x))^2/x^4+1/8*b*c*(a+b*arcsec(c*x))*sqrt(1+(-1)/(c^2*x^2))/x^3+3/16*b*c^3*(a+b*arcsec(c*x))*sqrt(1+(-1)/(c^2*x^2))/x],
[x^3*(a+b*arcsec(c*x))^3,x,10,1/4*b^2*x^2*(a+b*arcsec(c*x))/c^2+1/2*I*b*(a+b*arcsec(c*x))^2/c^4+1/4*x^4*(a+b*arcsec(c*x))^3-b^2*(a+b*arcsec(c*x))*log(1+exp(2*I*arcsec(c*x)))/c^4+1/2*I*b^3*polylog(2,-exp(2*I*arcsec(c*x)))/c^4-1/4*b^3*x*sqrt(1+(-1)/(c^2*x^2))/c^3-1/2*b*x*(a+b*arcsec(c*x))^2*sqrt(1+(-1)/(c^2*x^2))/c^3-1/4*b*x^3*(a+b*arcsec(c*x))^2*sqrt(1+(-1)/(c^2*x^2))/c],
[x^2*(a+b*arcsec(c*x))^3,x,11,b^2*x*(a+b*arcsec(c*x))/c^2+1/3*x^3*(a+b*arcsec(c*x))^3+I*b*(a+b*arcsec(c*x))^2*arctan(exp(I*arcsec(c*x)))/c^3-b^3*arctanh(sqrt(1+(-1)/(c^2*x^2)))/c^3-I*b^2*(a+b*arcsec(c*x))*polylog(2,-I*exp(I*arcsec(c*x)))/c^3+I*b^2*(a+b*arcsec(c*x))*polylog(2,I*exp(I*arcsec(c*x)))/c^3+b^3*polylog(3,-I*exp(I*arcsec(c*x)))/c^3-b^3*polylog(3,I*exp(I*arcsec(c*x)))/c^3-1/2*b*x^2*(a+b*arcsec(c*x))^2*sqrt(1+(-1)/(c^2*x^2))/c],
[x*(a+b*arcsec(c*x))^3,x,7,3/2*I*b*(a+b*arcsec(c*x))^2/c^2+1/2*x^2*(a+b*arcsec(c*x))^3-3*b^2*(a+b*arcsec(c*x))*log(1+exp(2*I*arcsec(c*x)))/c^2+3/2*I*b^3*polylog(2,-exp(2*I*arcsec(c*x)))/c^2-3/2*b*x*(a+b*arcsec(c*x))^2*sqrt(1+(-1)/(c^2*x^2))/c],
[(a+b*arcsec(c*x))^3,x,9,x*(a+b*arcsec(c*x))^3+6*I*b*(a+b*arcsec(c*x))^2*arctan(exp(I*arcsec(c*x)))/c-6*I*b^2*(a+b*arcsec(c*x))*polylog(2,-I*exp(I*arcsec(c*x)))/c+6*I*b^2*(a+b*arcsec(c*x))*polylog(2,I*exp(I*arcsec(c*x)))/c+6*b^3*polylog(3,-I*exp(I*arcsec(c*x)))/c-6*b^3*polylog(3,I*exp(I*arcsec(c*x)))/c],
[(a+b*arcsec(c*x))^3/x,x,7,1/4*I*(a+b*arcsec(c*x))^4/b-(a+b*arcsec(c*x))^3*log(1+exp(2*I*arcsec(c*x)))+3/2*I*b*(a+b*arcsec(c*x))^2*polylog(2,-exp(2*I*arcsec(c*x)))-3/2*b^2*(a+b*arcsec(c*x))*polylog(3,-exp(2*I*arcsec(c*x)))-3/4*I*b^3*polylog(4,-exp(2*I*arcsec(c*x)))],
[(a+b*arcsec(c*x))^3/x^2,x,5,6*b^2*(a+b*arcsec(c*x))/x-(a+b*arcsec(c*x))^3/x-6*b^3*c*sqrt(1+(-1)/(c^2*x^2))+3*b*c*(a+b*arcsec(c*x))^2*sqrt(1+(-1)/(c^2*x^2))],
[(a+b*arcsec(c*x))^3/x^3,x,6,3/8*b^3*c^2*arcsec(c*x)-3/4*b^2*(c^2+(-1)/x^2)*(a+b*arcsec(c*x))-1/4*c^2*(a+b*arcsec(c*x))^3+1/2*(c^2+(-1)/x^2)*(a+b*arcsec(c*x))^3-3/8*b^3*c*sqrt(1+(-1)/(c^2*x^2))/x+3/4*b*c*(a+b*arcsec(c*x))^2*sqrt(1+(-1)/(c^2*x^2))/x],
[(a+b*arcsec(c*x))^3/x^4,x,8,2/27*b^3*c^3*(1+(-1)/(c^2*x^2))^(3/2)+2/9*b^2*(a+b*arcsec(c*x))/x^3+4/3*b^2*c^2*(a+b*arcsec(c*x))/x-1/3*(a+b*arcsec(c*x))^3/x^3-14/9*b^3*c^3*sqrt(1+(-1)/(c^2*x^2))+2/3*b*c^3*(a+b*arcsec(c*x))^2*sqrt(1+(-1)/(c^2*x^2))+1/3*b*c*(a+b*arcsec(c*x))^2*sqrt(1+(-1)/(c^2*x^2))/x^2],
[(a+b*arcsec(c*x))^3/x^5,x,10,-45/256*b^3*c^4*arcsec(c*x)+3/32*b^2*(a+b*arcsec(c*x))/x^4+9/32*b^2*c^2*(a+b*arcsec(c*x))/x^2+3/32*c^4*(a+b*arcsec(c*x))^3-1/4*(a+b*arcsec(c*x))^3/x^4-3/128*b^3*c*sqrt(1+(-1)/(c^2*x^2))/x^3-45/256*b^3*c^3*sqrt(1+(-1)/(c^2*x^2))/x+3/16*b*c*(a+b*arcsec(c*x))^2*sqrt(1+(-1)/(c^2*x^2))/x^3+9/32*b*c^3*(a+b*arcsec(c*x))^2*sqrt(1+(-1)/(c^2*x^2))/x],

# n<0
[x/(a+b*arcsec(c*x)),x,0,Unintegrable(x/(a+b*arcsec(c*x)),x)],
[1/(a+b*arcsec(c*x)),x,0,Unintegrable(1/(a+b*arcsec(c*x)),x)],
[1/(x*(a+b*arcsec(c*x))),x,0,Unintegrable(1/(x*(a+b*arcsec(c*x))),x)],
[1/(x^2*(a+b*arcsec(c*x))),x,4,c*cos(a/b)*Si(a/b+arcsec(c*x))/b-c*Ci(a/b+arcsec(c*x))*sin(a/b)/b],
[1/(x^3*(a+b*arcsec(c*x))),x,6,1/2*c^2*cos(2*a/b)*Si(2*a/b+2*arcsec(c*x))/b-1/2*c^2*Ci(2*a/b+2*arcsec(c*x))*sin(2*a/b)/b],
[1/(x^4*(a+b*arcsec(c*x))),x,9,1/4*c^3*cos(a/b)*Si(a/b+arcsec(c*x))/b+1/4*c^3*cos(3*a/b)*Si(3*a/b+3*arcsec(c*x))/b-1/4*c^3*Ci(a/b+arcsec(c*x))*sin(a/b)/b-1/4*c^3*Ci(3*a/b+3*arcsec(c*x))*sin(3*a/b)/b],
[x/(a+b*arcsec(c*x))^2,x,0,Unintegrable(x/(a+b*arcsec(c*x))^2,x)],
[1/(a+b*arcsec(c*x))^2,x,0,Unintegrable(1/(a+b*arcsec(c*x))^2,x)],
[1/(x*(a+b*arcsec(c*x))^2),x,0,Unintegrable(1/(x*(a+b*arcsec(c*x))^2),x)],
[1/(x^2*(a+b*arcsec(c*x))^2),x,5,c*Ci(a/b+arcsec(c*x))*cos(a/b)/b^2+c*Si(a/b+arcsec(c*x))*sin(a/b)/b^2-c*sqrt(1+(-1)/(c^2*x^2))/(b*(a+b*arcsec(c*x)))],
[1/(x^3*(a+b*arcsec(c*x))^2),x,7,c^2*Ci(2*a/b+2*arcsec(c*x))*cos(2*a/b)/b^2+c^2*Si(2*a/b+2*arcsec(c*x))*sin(2*a/b)/b^2-1/2*c^2*sin(2*arcsec(c*x))/(b*(a+b*arcsec(c*x)))],
[1/(x^4*(a+b*arcsec(c*x))^2),x,11,1/4*c^3*Ci(a/b+arcsec(c*x))*cos(a/b)/b^2+3/4*c^3*Ci(3*a/b+3*arcsec(c*x))*cos(3*a/b)/b^2+1/4*c^3*Si(a/b+arcsec(c*x))*sin(a/b)/b^2+3/4*c^3*Si(3*a/b+3*arcsec(c*x))*sin(3*a/b)/b^2-1/4*c^3*sin(3*arcsec(c*x))/(b*(a+b*arcsec(c*x)))-1/4*c^3*sqrt(1+(-1)/(c^2*x^2))/(b*(a+b*arcsec(c*x)))],
[x/(a+b*arcsec(c*x))^3,x,0,Unintegrable(x/(a+b*arcsec(c*x))^3,x)],
[1/(a+b*arcsec(c*x))^3,x,0,Unintegrable(1/(a+b*arcsec(c*x))^3,x)],
[1/(x*(a+b*arcsec(c*x))^3),x,0,Unintegrable(1/(x*(a+b*arcsec(c*x))^3),x)],
[1/(x^2*(a+b*arcsec(c*x))^3),x,6,(-1/2)/(b^2*x*(a+b*arcsec(c*x)))-1/2*c*cos(a/b)*Si(a/b+arcsec(c*x))/b^3+1/2*c*Ci(a/b+arcsec(c*x))*sin(a/b)/b^3-1/2*c*sqrt(1+(-1)/(c^2*x^2))/(b*(a+b*arcsec(c*x))^2)],
[1/(x^3*(a+b*arcsec(c*x))^3),x,8,-1/2*c^2*cos(2*arcsec(c*x))/(b^2*(a+b*arcsec(c*x)))-c^2*cos(2*a/b)*Si(2*a/b+2*arcsec(c*x))/b^3+c^2*Ci(2*a/b+2*arcsec(c*x))*sin(2*a/b)/b^3-1/4*c^2*sin(2*arcsec(c*x))/(b*(a+b*arcsec(c*x))^2)],
[1/(x^4*(a+b*arcsec(c*x))^3),x,13,-1/8*c^2/(b^2*x*(a+b*arcsec(c*x)))-3/8*c^3*cos(3*arcsec(c*x))/(b^2*(a+b*arcsec(c*x)))-1/8*c^3*cos(a/b)*Si(a/b+arcsec(c*x))/b^3-9/8*c^3*cos(3*a/b)*Si(3*a/b+3*arcsec(c*x))/b^3+1/8*c^3*Ci(a/b+arcsec(c*x))*sin(a/b)/b^3+9/8*c^3*Ci(3*a/b+3*arcsec(c*x))*sin(3*a/b)/b^3-1/8*c^3*sin(3*arcsec(c*x))/(b*(a+b*arcsec(c*x))^2)-1/8*c^3*sqrt(1+(-1)/(c^2*x^2))/(b*(a+b*arcsec(c*x))^2)],

# Integrands of the form (d x)^m (a+b ArcSec[c x])^n with m symbolic
[(d*x)^m*(a+b*arcsec(c*x))^3,x,0,Unintegrable((d*x)^m*(a+b*arcsec(c*x))^3,x)],
[(d*x)^m*(a+b*arcsec(c*x))^2,x,0,Unintegrable((d*x)^m*(a+b*arcsec(c*x))^2,x)],
[(d*x)^m*(a+b*arcsec(c*x)),x,3,(d*x)^(1+m)*(a+b*arcsec(c*x))/(d*(1+m))-b*(d*x)^m*hypergeom([1/2,-1/2*m],[1-1/2*m],1/(c^2*x^2))/(c*m*(1+m))],
[(d*x)^m/(a+b*arcsec(c*x)),x,0,Unintegrable((d*x)^m/(a+b*arcsec(c*x)),x)],
[(d*x)^m/(a+b*arcsec(c*x))^2,x,0,Unintegrable((d*x)^m/(a+b*arcsec(c*x))^2,x)],

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

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

# n>0
[(d+e*x)^3*(a+b*arcsec(c*x)),x,11,1/4*b*d^4*arccsc(c*x)/e+1/4*(d+e*x)^4*(a+b*arcsec(c*x))/e-1/2*b*d*(2*c^2*d^2+e^2)*arctanh(sqrt(1+(-1)/(c^2*x^2)))/c^3-1/6*b*e*(9*c^2*d^2+e^2)*x*sqrt(1+(-1)/(c^2*x^2))/c^3-1/2*b*d*e^2*x^2*sqrt(1+(-1)/(c^2*x^2))/c-1/12*b*e^3*x^3*sqrt(1+(-1)/(c^2*x^2))/c],
[(d+e*x)^2*(a+b*arcsec(c*x)),x,10,1/3*b*d^3*arccsc(c*x)/e+1/3*(d+e*x)^3*(a+b*arcsec(c*x))/e-1/6*b*(6*c^2*d^2+e^2)*arctanh(sqrt(1+(-1)/(c^2*x^2)))/c^3-b*d*e*x*sqrt(1+(-1)/(c^2*x^2))/c-1/6*b*e^2*x^2*sqrt(1+(-1)/(c^2*x^2))/c],
[(d+e*x)*(a+b*arcsec(c*x)),x,9,1/2*b*d^2*arccsc(c*x)/e+1/2*(d+e*x)^2*(a+b*arcsec(c*x))/e-b*d*arctanh(sqrt(1+(-1)/(c^2*x^2)))/c-1/2*b*e*x*sqrt(1+(-1)/(c^2*x^2))/c],
[a+b*arcsec(c*x),x,5,a*x+b*x*arcsec(c*x)-b*arctanh(sqrt(1+(-1)/(c^2*x^2)))/c],
[(a+b*arcsec(c*x))/(d+e*x),x,4,-(a+b*arcsec(c*x))*log(1+exp(2*I*arcsec(c*x)))/e+(a+b*arcsec(c*x))*log(1+exp(I*arcsec(c*x))*(e-sqrt(-c^2*d^2+e^2))/(c*d))/e+(a+b*arcsec(c*x))*log(1+exp(I*arcsec(c*x))*(e+sqrt(-c^2*d^2+e^2))/(c*d))/e+1/2*I*b*polylog(2,-exp(2*I*arcsec(c*x)))/e-I*b*polylog(2,-exp(I*arcsec(c*x))*(e-sqrt(-c^2*d^2+e^2))/(c*d))/e-I*b*polylog(2,-exp(I*arcsec(c*x))*(e+sqrt(-c^2*d^2+e^2))/(c*d))/e],
[(a+b*arcsec(c*x))/(d+e*x)^2,x,7,-b*arccsc(c*x)/(d*e)+(-a-b*arcsec(c*x))/(e*(d+e*x))-b*arctanh((c^2*d+e/x)/(c*sqrt(c^2*d^2-e^2)*sqrt(1+(-1)/(c^2*x^2))))/(d*sqrt(c^2*d^2-e^2))],
[(a+b*arcsec(c*x))/(d+e*x)^3,x,8,-1/2*b*arccsc(c*x)/(d^2*e)+1/2*(-a-b*arcsec(c*x))/(e*(d+e*x)^2)-1/2*b*(2*c^2*d^2-e^2)*arctanh((c^2*d+e/x)/(c*sqrt(c^2*d^2-e^2)*sqrt(1+(-1)/(c^2*x^2))))/(d^2*(c^2*d^2-e^2)^(3/2))+1/2*b*c*e*sqrt(1+(-1)/(c^2*x^2))/(d*(c^2*d^2-e^2)*(e+d/x))],

# n<0

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

# n>0
[(d+e*x)^(3/2)*(a+b*arcsec(c*x)),x,22,2/5*(d+e*x)^(5/2)*(a+b*arcsec(c*x))/e+4/15*b*e*(1-c^2*x^2)*sqrt(d+e*x)/(c^3*x*sqrt(1+(-1)/(c^2*x^2)))+28/15*b*d*EllipticE(sqrt(1-c*x)/sqrt(2),sqrt(2*e/(c*d+e)))*sqrt(d+e*x)*sqrt(1-c^2*x^2)/(c^2*x*sqrt(1+(-1)/(c^2*x^2))*sqrt(c*(d+e*x)/(c*d+e)))+4/15*b*(2*c^2*d^2+e^2)*EllipticF(sqrt(1-c*x)/sqrt(2),sqrt(2*e/(c*d+e)))*sqrt(c*(d+e*x)/(c*d+e))*sqrt(1-c^2*x^2)/(c^4*x*sqrt(1+(-1)/(c^2*x^2))*sqrt(d+e*x))+4/5*b*d^3*EllipticPi(sqrt(1-c*x)/sqrt(2),2,sqrt(2*e/(c*d+e)))*sqrt(c*(d+e*x)/(c*d+e))*sqrt(1-c^2*x^2)/(c*e*x*sqrt(1+(-1)/(c^2*x^2))*sqrt(d+e*x))],
[(d+e*x)^(1/2)*(a+b*arcsec(c*x)),x,15,2/3*(d+e*x)^(3/2)*(a+b*arcsec(c*x))/e+4/3*b*EllipticE(sqrt(1-c*x)/sqrt(2),sqrt(2*e/(c*d+e)))*sqrt(d+e*x)*sqrt(1-c^2*x^2)/(c^2*x*sqrt(1+(-1)/(c^2*x^2))*sqrt(c*(d+e*x)/(c*d+e)))+4/3*b*d*EllipticF(sqrt(1-c*x)/sqrt(2),sqrt(2*e/(c*d+e)))*sqrt(c*(d+e*x)/(c*d+e))*sqrt(1-c^2*x^2)/(c^2*x*sqrt(1+(-1)/(c^2*x^2))*sqrt(d+e*x))+4/3*b*d^2*EllipticPi(sqrt(1-c*x)/sqrt(2),2,sqrt(2*e/(c*d+e)))*sqrt(c*(d+e*x)/(c*d+e))*sqrt(1-c^2*x^2)/(c*e*x*sqrt(1+(-1)/(c^2*x^2))*sqrt(d+e*x))],
[(a+b*arcsec(c*x))/(d+e*x)^(1/2),x,9,2*(a+b*arcsec(c*x))*sqrt(d+e*x)/e+4*b*EllipticF(sqrt(1-c*x)/sqrt(2),sqrt(2*e/(c*d+e)))*sqrt(c*(d+e*x)/(c*d+e))*sqrt(1-c^2*x^2)/(c^2*x*sqrt(1+(-1)/(c^2*x^2))*sqrt(d+e*x))+4*b*d*EllipticPi(sqrt(1-c*x)/sqrt(2),2,sqrt(2*e/(c*d+e)))*sqrt(c*(d+e*x)/(c*d+e))*sqrt(1-c^2*x^2)/(c*e*x*sqrt(1+(-1)/(c^2*x^2))*sqrt(d+e*x))],
[(a+b*arcsec(c*x))/(d+e*x)^(3/2),x,6,-2*(a+b*arcsec(c*x))/(e*sqrt(d+e*x))-4*b*EllipticPi(sqrt(1-c*x)/sqrt(2),2,sqrt(2*e/(c*d+e)))*sqrt(c*(d+e*x)/(c*d+e))*sqrt(1-c^2*x^2)/(c*e*x*sqrt(1+(-1)/(c^2*x^2))*sqrt(d+e*x))],
[(a+b*arcsec(c*x))/(d+e*x)^(5/2),x,12,-2/3*(a+b*arcsec(c*x))/(e*(d+e*x)^(3/2))-4/3*b*e*(1-c^2*x^2)/(c*d*(c^2*d^2-e^2)*x*sqrt(1+(-1)/(c^2*x^2))*sqrt(d+e*x))+4/3*b*EllipticE(sqrt(1-c*x)/sqrt(2),sqrt(2*e/(c*d+e)))*sqrt(d+e*x)*sqrt(1-c^2*x^2)/(d*(c^2*d^2-e^2)*x*sqrt(1+(-1)/(c^2*x^2))*sqrt(c*(d+e*x)/(c*d+e)))-4/3*b*EllipticPi(sqrt(1-c*x)/sqrt(2),2,sqrt(2*e/(c*d+e)))*sqrt(c*(d+e*x)/(c*d+e))*sqrt(1-c^2*x^2)/(c*d*e*x*sqrt(1+(-1)/(c^2*x^2))*sqrt(d+e*x))],
[(a+b*arcsec(c*x))/(d+e*x)^(7/2),x,19,-2/5*(a+b*arcsec(c*x))/(e*(d+e*x)^(5/2))-4/15*b*e*(1-c^2*x^2)/(c*d*(c^2*d^2-e^2)*x*(d+e*x)^(3/2)*sqrt(1+(-1)/(c^2*x^2)))-16/15*b*c*e*(1-c^2*x^2)/((c^2*d^2-e^2)^2*x*sqrt(1+(-1)/(c^2*x^2))*sqrt(d+e*x))-4/5*b*e*(1-c^2*x^2)/(c*d^2*(c^2*d^2-e^2)*x*sqrt(1+(-1)/(c^2*x^2))*sqrt(d+e*x))+4/15*b*(7*c^2*d^2-3*e^2)*EllipticE(sqrt(1-c*x)/sqrt(2),sqrt(2*e/(c*d+e)))*sqrt(d+e*x)*sqrt(1-c^2*x^2)/((c^2*d^3-d*e^2)^2*x*sqrt(1+(-1)/(c^2*x^2))*sqrt(c*(d+e*x)/(c*d+e)))-4/15*b*EllipticF(sqrt(1-c*x)/sqrt(2),sqrt(2*e/(c*d+e)))*sqrt(c*(d+e*x)/(c*d+e))*sqrt(1-c^2*x^2)/(d*(c^2*d^2-e^2)*x*sqrt(1+(-1)/(c^2*x^2))*sqrt(d+e*x))-4/5*b*EllipticPi(sqrt(1-c*x)/sqrt(2),2,sqrt(2*e/(c*d+e)))*sqrt(c*(d+e*x)/(c*d+e))*sqrt(1-c^2*x^2)/(c*d^2*e*x*sqrt(1+(-1)/(c^2*x^2))*sqrt(d+e*x)),-2/5*(a+b*arcsec(c*x))/(e*(d+e*x)^(5/2))-4/15*b*e*(1-c^2*x^2)/(c*d*(c^2*d^2-e^2)*x*(d+e*x)^(3/2)*sqrt(1+(-1)/(c^2*x^2)))-16/15*b*c*e*(1-c^2*x^2)/((c^2*d^2-e^2)^2*x*sqrt(1+(-1)/(c^2*x^2))*sqrt(d+e*x))-4/5*b*e*(1-c^2*x^2)/(c*d^2*(c^2*d^2-e^2)*x*sqrt(1+(-1)/(c^2*x^2))*sqrt(d+e*x))+16/15*b*c^2*EllipticE(sqrt(1-c*x)/sqrt(2),sqrt(2*e/(c*d+e)))*sqrt(d+e*x)*sqrt(1-c^2*x^2)/((c^2*d^2-e^2)^2*x*sqrt(1+(-1)/(c^2*x^2))*sqrt(c*(d+e*x)/(c*d+e)))+4/5*b*EllipticE(sqrt(1-c*x)/sqrt(2),sqrt(2*e/(c*d+e)))*sqrt(d+e*x)*sqrt(1-c^2*x^2)/(d^2*(c^2*d^2-e^2)*x*sqrt(1+(-1)/(c^2*x^2))*sqrt(c*(d+e*x)/(c*d+e)))-4/15*b*EllipticF(sqrt(1-c*x)/sqrt(2),sqrt(2*e/(c*d+e)))*sqrt(c*(d+e*x)/(c*d+e))*sqrt(1-c^2*x^2)/(d*(c^2*d^2-e^2)*x*sqrt(1+(-1)/(c^2*x^2))*sqrt(d+e*x))-4/5*b*EllipticPi(sqrt(1-c*x)/sqrt(2),2,sqrt(2*e/(c*d+e)))*sqrt(c*(d+e*x)/(c*d+e))*sqrt(1-c^2*x^2)/(c*d^2*e*x*sqrt(1+(-1)/(c^2*x^2))*sqrt(d+e*x))],

# n<0

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

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

# p>0
[x^4*(d+e*x^2)*(a+b*arcsec(c*x)),x,7,1/5*d*x^5*(a+b*arcsec(c*x))+1/7*e*x^7*(a+b*arcsec(c*x))-1/560*b*(42*c^2*d+25*e)*x*arctanh(c*x/sqrt(-1+c^2*x^2))/(c^6*sqrt(c^2*x^2))-1/560*b*(42*c^2*d+25*e)*x^2*sqrt(-1+c^2*x^2)/(c^5*sqrt(c^2*x^2))-1/840*b*(42*c^2*d+25*e)*x^4*sqrt(-1+c^2*x^2)/(c^3*sqrt(c^2*x^2))-1/42*b*e*x^6*sqrt(-1+c^2*x^2)/(c*sqrt(c^2*x^2))],
[x^2*(d+e*x^2)*(a+b*arcsec(c*x)),x,6,1/3*d*x^3*(a+b*arcsec(c*x))+1/5*e*x^5*(a+b*arcsec(c*x))-1/120*b*(20*c^2*d+9*e)*x*arctanh(c*x/sqrt(-1+c^2*x^2))/(c^4*sqrt(c^2*x^2))-1/120*b*(20*c^2*d+9*e)*x^2*sqrt(-1+c^2*x^2)/(c^3*sqrt(c^2*x^2))-1/20*b*e*x^4*sqrt(-1+c^2*x^2)/(c*sqrt(c^2*x^2))],
[(d+e*x^2)*(a+b*arcsec(c*x)),x,5,d*x*(a+b*arcsec(c*x))+1/3*e*x^3*(a+b*arcsec(c*x))-1/6*b*(6*c^2*d+e)*x*arctanh(c*x/sqrt(-1+c^2*x^2))/(c^2*sqrt(c^2*x^2))-1/6*b*e*x^2*sqrt(-1+c^2*x^2)/(c*sqrt(c^2*x^2))],
[(d+e*x^2)*(a+b*arcsec(c*x))/x^2,x,4,-d*(a+b*arcsec(c*x))/x+e*x*(a+b*arcsec(c*x))-b*e*x*arctanh(c*x/sqrt(-1+c^2*x^2))/sqrt(c^2*x^2)+b*c*d*sqrt(-1+c^2*x^2)/sqrt(c^2*x^2)],
[(d+e*x^2)*(a+b*arcsec(c*x))/x^4,x,4,-1/3*d*(a+b*arcsec(c*x))/x^3-e*(a+b*arcsec(c*x))/x+1/9*b*c*(2*c^2*d+9*e)*sqrt(-1+c^2*x^2)/sqrt(c^2*x^2)+1/9*b*c*d*sqrt(-1+c^2*x^2)/(x^2*sqrt(c^2*x^2))],
[(d+e*x^2)*(a+b*arcsec(c*x))/x^6,x,5,-1/5*d*(a+b*arcsec(c*x))/x^5-1/3*e*(a+b*arcsec(c*x))/x^3+2/225*b*c^3*(12*c^2*d+25*e)*sqrt(-1+c^2*x^2)/sqrt(c^2*x^2)+1/25*b*c*d*sqrt(-1+c^2*x^2)/(x^4*sqrt(c^2*x^2))+1/225*b*c*(12*c^2*d+25*e)*sqrt(-1+c^2*x^2)/(x^2*sqrt(c^2*x^2))],
[(d+e*x^2)*(a+b*arcsec(c*x))/x^8,x,6,-1/7*d*(a+b*arcsec(c*x))/x^7-1/5*e*(a+b*arcsec(c*x))/x^5+8/3675*b*c^5*(30*c^2*d+49*e)*sqrt(-1+c^2*x^2)/sqrt(c^2*x^2)+1/49*b*c*d*sqrt(-1+c^2*x^2)/(x^6*sqrt(c^2*x^2))+1/1225*b*c*(30*c^2*d+49*e)*sqrt(-1+c^2*x^2)/(x^4*sqrt(c^2*x^2))+4/3675*b*c^3*(30*c^2*d+49*e)*sqrt(-1+c^2*x^2)/(x^2*sqrt(c^2*x^2))],
[x^5*(d+e*x^2)*(a+b*arcsec(c*x)),x,5,1/6*d*x^6*(a+b*arcsec(c*x))+1/8*e*x^8*(a+b*arcsec(c*x))-1/72*b*(8*c^2*d+9*e)*x*(-1+c^2*x^2)^(3/2)/(c^7*sqrt(c^2*x^2))-1/120*b*(4*c^2*d+9*e)*x*(-1+c^2*x^2)^(5/2)/(c^7*sqrt(c^2*x^2))-1/56*b*e*x*(-1+c^2*x^2)^(7/2)/(c^7*sqrt(c^2*x^2))-1/24*b*(4*c^2*d+3*e)*x*sqrt(-1+c^2*x^2)/(c^7*sqrt(c^2*x^2))],
[x^3*(d+e*x^2)*(a+b*arcsec(c*x)),x,5,1/4*d*x^4*(a+b*arcsec(c*x))+1/6*e*x^6*(a+b*arcsec(c*x))-1/36*b*(3*c^2*d+4*e)*x*(-1+c^2*x^2)^(3/2)/(c^5*sqrt(c^2*x^2))-1/30*b*e*x*(-1+c^2*x^2)^(5/2)/(c^5*sqrt(c^2*x^2))-1/12*b*(3*c^2*d+2*e)*x*sqrt(-1+c^2*x^2)/(c^5*sqrt(c^2*x^2))],
[x*(d+e*x^2)*(a+b*arcsec(c*x)),x,6,1/4*(d+e*x^2)^2*(a+b*arcsec(c*x))/e-1/12*b*e*x*(-1+c^2*x^2)^(3/2)/(c^3*sqrt(c^2*x^2))-1/4*b*c*d^2*x*arctan(sqrt(-1+c^2*x^2))/(e*sqrt(c^2*x^2))-1/4*b*(2*c^2*d+e)*x*sqrt(-1+c^2*x^2)/(c^3*sqrt(c^2*x^2))],
[(d+e*x^2)*(a+b*arcsec(c*x))/x,x,11,-1/2*I*b*d*arccsc(c*x)^2+1/2*e*x^2*(a+b*arcsec(c*x))+b*d*arccsc(c*x)*log(1-exp(2*I*arccsc(c*x)))-b*d*arccsc(c*x)*log(1/x)-d*(a+b*arcsec(c*x))*log(1/x)-1/2*I*b*d*polylog(2,exp(2*I*arccsc(c*x)))-1/2*b*e*x*sqrt(1+(-1)/(c^2*x^2))/c],
[(d+e*x^2)*(a+b*arcsec(c*x))/x^3,x,13,-1/4*b*c^2*d*arccsc(c*x)-1/2*I*b*e*arccsc(c*x)^2-1/2*d*(a+b*arcsec(c*x))/x^2+b*e*arccsc(c*x)*log(1-exp(2*I*arccsc(c*x)))-b*e*arccsc(c*x)*log(1/x)-e*(a+b*arcsec(c*x))*log(1/x)-1/2*I*b*e*polylog(2,exp(2*I*arccsc(c*x)))+1/4*b*c*d*sqrt(1+(-1)/(c^2*x^2))/x],
[x^2*(d+e*x^2)^2*(a+b*arcsec(c*x)),x,7,1/3*d^2*x^3*(a+b*arcsec(c*x))+2/5*d*e*x^5*(a+b*arcsec(c*x))+1/7*e^2*x^7*(a+b*arcsec(c*x))-1/1680*b*(280*c^4*d^2+252*c^2*d*e+75*e^2)*x*arctanh(c*x/sqrt(-1+c^2*x^2))/(c^6*sqrt(c^2*x^2))-1/1680*b*(280*c^4*d^2+252*c^2*d*e+75*e^2)*x^2*sqrt(-1+c^2*x^2)/(c^5*sqrt(c^2*x^2))-1/840*b*e*(84*c^2*d+25*e)*x^4*sqrt(-1+c^2*x^2)/(c^3*sqrt(c^2*x^2))-1/42*b*e^2*x^6*sqrt(-1+c^2*x^2)/(c*sqrt(c^2*x^2))],
[(d+e*x^2)^2*(a+b*arcsec(c*x)),x,6,d^2*x*(a+b*arcsec(c*x))+2/3*d*e*x^3*(a+b*arcsec(c*x))+1/5*e^2*x^5*(a+b*arcsec(c*x))-1/120*b*(120*c^4*d^2+40*c^2*d*e+9*e^2)*x*arctanh(c*x/sqrt(-1+c^2*x^2))/(c^4*sqrt(c^2*x^2))-1/120*b*e*(40*c^2*d+9*e)*x^2*sqrt(-1+c^2*x^2)/(c^3*sqrt(c^2*x^2))-1/20*b*e^2*x^4*sqrt(-1+c^2*x^2)/(c*sqrt(c^2*x^2))],
[(d+e*x^2)^2*(a+b*arcsec(c*x))/x^2,x,6,-d^2*(a+b*arcsec(c*x))/x+2*d*e*x*(a+b*arcsec(c*x))+1/3*e^2*x^3*(a+b*arcsec(c*x))-1/6*b*e*(12*c^2*d+e)*x*arctanh(c*x/sqrt(-1+c^2*x^2))/(c^2*sqrt(c^2*x^2))+b*c*d^2*sqrt(-1+c^2*x^2)/sqrt(c^2*x^2)-1/6*b*e^2*x^2*sqrt(-1+c^2*x^2)/(c*sqrt(c^2*x^2))],
[(d+e*x^2)^2*(a+b*arcsec(c*x))/x^4,x,6,-1/3*d^2*(a+b*arcsec(c*x))/x^3-2*d*e*(a+b*arcsec(c*x))/x+e^2*x*(a+b*arcsec(c*x))-b*e^2*x*arctanh(c*x/sqrt(-1+c^2*x^2))/sqrt(c^2*x^2)+2/9*b*c*d*(c^2*d+9*e)*sqrt(-1+c^2*x^2)/sqrt(c^2*x^2)+1/9*b*c*d^2*sqrt(-1+c^2*x^2)/(x^2*sqrt(c^2*x^2))],
[(d+e*x^2)^2*(a+b*arcsec(c*x))/x^6,x,5,-1/5*d^2*(a+b*arcsec(c*x))/x^5-2/3*d*e*(a+b*arcsec(c*x))/x^3-e^2*(a+b*arcsec(c*x))/x+1/225*b*c*(24*c^4*d^2+100*c^2*d*e+225*e^2)*sqrt(-1+c^2*x^2)/sqrt(c^2*x^2)+1/25*b*c*d^2*sqrt(-1+c^2*x^2)/(x^4*sqrt(c^2*x^2))+2/225*b*c*d*(6*c^2*d+25*e)*sqrt(-1+c^2*x^2)/(x^2*sqrt(c^2*x^2))],
[(d+e*x^2)^2*(a+b*arcsec(c*x))/x^8,x,6,-1/7*d^2*(a+b*arcsec(c*x))/x^7-2/5*d*e*(a+b*arcsec(c*x))/x^5-1/3*e^2*(a+b*arcsec(c*x))/x^3+2/11025*b*c^3*(360*c^4*d^2+1176*c^2*d*e+1225*e^2)*sqrt(-1+c^2*x^2)/sqrt(c^2*x^2)+1/49*b*c*d^2*sqrt(-1+c^2*x^2)/(x^6*sqrt(c^2*x^2))+2/1225*b*c*d*(15*c^2*d+49*e)*sqrt(-1+c^2*x^2)/(x^4*sqrt(c^2*x^2))+1/11025*b*c*(360*c^4*d^2+1176*c^2*d*e+1225*e^2)*sqrt(-1+c^2*x^2)/(x^2*sqrt(c^2*x^2))],
[x^3*(d+e*x^2)^2*(a+b*arcsec(c*x)),x,5,1/4*d^2*x^4*(a+b*arcsec(c*x))+1/3*d*e*x^6*(a+b*arcsec(c*x))+1/8*e^2*x^8*(a+b*arcsec(c*x))-1/72*b*(6*c^4*d^2+16*c^2*d*e+9*e^2)*x*(-1+c^2*x^2)^(3/2)/(c^7*sqrt(c^2*x^2))-1/120*b*e*(8*c^2*d+9*e)*x*(-1+c^2*x^2)^(5/2)/(c^7*sqrt(c^2*x^2))-1/56*b*e^2*x*(-1+c^2*x^2)^(7/2)/(c^7*sqrt(c^2*x^2))-1/24*b*(6*c^4*d^2+8*c^2*d*e+3*e^2)*x*sqrt(-1+c^2*x^2)/(c^7*sqrt(c^2*x^2))],
[x*(d+e*x^2)^2*(a+b*arcsec(c*x)),x,6,1/6*(d+e*x^2)^3*(a+b*arcsec(c*x))/e-1/18*b*e*(3*c^2*d+2*e)*x*(-1+c^2*x^2)^(3/2)/(c^5*sqrt(c^2*x^2))-1/30*b*e^2*x*(-1+c^2*x^2)^(5/2)/(c^5*sqrt(c^2*x^2))-1/6*b*c*d^3*x*arctan(sqrt(-1+c^2*x^2))/(e*sqrt(c^2*x^2))-1/6*b*(3*c^4*d^2+3*c^2*d*e+e^2)*x*sqrt(-1+c^2*x^2)/(c^5*sqrt(c^2*x^2))],
[(d+e*x^2)^2*(a+b*arcsec(c*x))/x,x,12,-1/2*I*b*d^2*arccsc(c*x)^2+d*e*x^2*(a+b*arcsec(c*x))+1/4*e^2*x^4*(a+b*arcsec(c*x))+b*d^2*arccsc(c*x)*log(1-exp(2*I*arccsc(c*x)))-b*d^2*arccsc(c*x)*log(1/x)-d^2*(a+b*arcsec(c*x))*log(1/x)-1/2*I*b*d^2*polylog(2,exp(2*I*arccsc(c*x)))-1/6*b*e*(6*c^2*d+e)*x*sqrt(1+(-1)/(c^2*x^2))/c^3-1/12*b*e^2*x^3*sqrt(1+(-1)/(c^2*x^2))/c],
[(d+e*x^2)^2*(a+b*arcsec(c*x))/x^3,x,14,-1/4*b*c^2*d^2*arccsc(c*x)-I*b*d*e*arccsc(c*x)^2-1/2*d^2*(a+b*arcsec(c*x))/x^2+1/2*e^2*x^2*(a+b*arcsec(c*x))+2*b*d*e*arccsc(c*x)*log(1-exp(2*I*arccsc(c*x)))-2*b*d*e*arccsc(c*x)*log(1/x)-2*d*e*(a+b*arcsec(c*x))*log(1/x)-I*b*d*e*polylog(2,exp(2*I*arccsc(c*x)))+1/4*b*c*d^2*sqrt(1+(-1)/(c^2*x^2))/x-1/2*b*e^2*x*sqrt(1+(-1)/(c^2*x^2))/c],

# p<0

#  {x^3*(a + b*ArcSec[c*x])/(d + e*x^2), x, 47, -((b*Sqrt[1 - 1/(c^2*x^2)]*x)/(2*c*e)) + (I*b*d*ArcSec[c*x]^2)/(2*e^2) + (x^2*(a + b*ArcSec[c*x]))/(2*e) - (I*d*(a + b*ArcSec[c*x])^2)/(2*b*e^2) - (b*d*ArcSec[c*x]*Log[1 - (c*Sqrt[-d]*E^(I*ArcSec[c*x]))/(Sqrt[e] - Sqrt[c^2*d + e])])/(2*e^2) - (b*d*ArcSec[c*x]*Log[1 + (c*Sqrt[-d]*E^(I*ArcSec[c*x]))/(Sqrt[e] - Sqrt[c^2*d + e])])/(2*e^2) - (b*d*ArcSec[c*x]*Log[1 - (c*Sqrt[-d]*E^(I*ArcSec[c*x]))/(Sqrt[e] + Sqrt[c^2*d + e])])/(2*e^2) - (b*d*ArcSec[c*x]*Log[1 + (c*Sqrt[-d]*E^(I*ArcSec[c*x]))/(Sqrt[e] + Sqrt[c^2*d + e])])/(2*e^2) + (d*(a + b*ArcSec[c*x])*Log[1 + E^(2*I*ArcSec[c*x])])/e^2 - (a*d*Log[Sqrt[e] - Sqrt[-d]/x])/(2*e^2) - (a*d*Log[Sqrt[e] + Sqrt[-d]/x])/(2*e^2) + (I*b*d*PolyLog[2, -((c*Sqrt[-d]*E^(I*ArcSec[c*x]))/(Sqrt[e] - Sqrt[c^2*d + e]))])/(2*e^2) + (I*b*d*PolyLog[2, (c*Sqrt[-d]*E^(I*ArcSec[c*x]))/(Sqrt[e] - Sqrt[c^2*d + e])])/(2*e^2) + (I*b*d*PolyLog[2, -((c*Sqrt[-d]*E^(I*ArcSec[c*x]))/(Sqrt[e] + Sqrt[c^2*d + e]))])/(2*e^2) + (I*b*d*PolyLog[2, (c*Sqrt[-d]*E^(I*ArcSec[c*x]))/(Sqrt[e] + Sqrt[c^2*d + e])])/(2*e^2) - (I*b*d*PolyLog[2, -E^(2*I*ArcSec[c*x])])/(2*e^2)} 
[x^2*(a+b*arcsec(c*x))/(d+e*x^2),x,25,x*(a+b*arcsec(c*x))/e-b*arctanh(sqrt(1+(-1)/(c^2*x^2)))/(c*e)+1/2*(a+b*arcsec(c*x))*log(1-c*exp(I*arcsec(c*x))*sqrt(-d)/(sqrt(e)-sqrt(c^2*d+e)))*sqrt(-d)/e^(3/2)-1/2*(a+b*arcsec(c*x))*log(1+c*exp(I*arcsec(c*x))*sqrt(-d)/(sqrt(e)-sqrt(c^2*d+e)))*sqrt(-d)/e^(3/2)+1/2*(a+b*arcsec(c*x))*log(1-c*exp(I*arcsec(c*x))*sqrt(-d)/(sqrt(e)+sqrt(c^2*d+e)))*sqrt(-d)/e^(3/2)-1/2*(a+b*arcsec(c*x))*log(1+c*exp(I*arcsec(c*x))*sqrt(-d)/(sqrt(e)+sqrt(c^2*d+e)))*sqrt(-d)/e^(3/2)+1/2*I*b*polylog(2,-c*exp(I*arcsec(c*x))*sqrt(-d)/(sqrt(e)-sqrt(c^2*d+e)))*sqrt(-d)/e^(3/2)-1/2*I*b*polylog(2,c*exp(I*arcsec(c*x))*sqrt(-d)/(sqrt(e)-sqrt(c^2*d+e)))*sqrt(-d)/e^(3/2)+1/2*I*b*polylog(2,-c*exp(I*arcsec(c*x))*sqrt(-d)/(sqrt(e)+sqrt(c^2*d+e)))*sqrt(-d)/e^(3/2)-1/2*I*b*polylog(2,c*exp(I*arcsec(c*x))*sqrt(-d)/(sqrt(e)+sqrt(c^2*d+e)))*sqrt(-d)/e^(3/2)],
[x*(a+b*arcsec(c*x))/(d+e*x^2),x,26,-(a+b*arcsec(c*x))*log(1+exp(2*I*arcsec(c*x)))/e+1/2*(a+b*arcsec(c*x))*log(1-c*exp(I*arcsec(c*x))*sqrt(-d)/(sqrt(e)-sqrt(c^2*d+e)))/e+1/2*(a+b*arcsec(c*x))*log(1+c*exp(I*arcsec(c*x))*sqrt(-d)/(sqrt(e)-sqrt(c^2*d+e)))/e+1/2*(a+b*arcsec(c*x))*log(1-c*exp(I*arcsec(c*x))*sqrt(-d)/(sqrt(e)+sqrt(c^2*d+e)))/e+1/2*(a+b*arcsec(c*x))*log(1+c*exp(I*arcsec(c*x))*sqrt(-d)/(sqrt(e)+sqrt(c^2*d+e)))/e+1/2*I*b*polylog(2,-exp(2*I*arcsec(c*x)))/e-1/2*I*b*polylog(2,-c*exp(I*arcsec(c*x))*sqrt(-d)/(sqrt(e)-sqrt(c^2*d+e)))/e-1/2*I*b*polylog(2,c*exp(I*arcsec(c*x))*sqrt(-d)/(sqrt(e)-sqrt(c^2*d+e)))/e-1/2*I*b*polylog(2,-c*exp(I*arcsec(c*x))*sqrt(-d)/(sqrt(e)+sqrt(c^2*d+e)))/e-1/2*I*b*polylog(2,c*exp(I*arcsec(c*x))*sqrt(-d)/(sqrt(e)+sqrt(c^2*d+e)))/e],
[(a+b*arcsec(c*x))/(d+e*x^2),x,19,1/2*(a+b*arcsec(c*x))*log(1-c*exp(I*arcsec(c*x))*sqrt(-d)/(sqrt(e)-sqrt(c^2*d+e)))/(sqrt(-d)*sqrt(e))-1/2*(a+b*arcsec(c*x))*log(1+c*exp(I*arcsec(c*x))*sqrt(-d)/(sqrt(e)-sqrt(c^2*d+e)))/(sqrt(-d)*sqrt(e))+1/2*(a+b*arcsec(c*x))*log(1-c*exp(I*arcsec(c*x))*sqrt(-d)/(sqrt(e)+sqrt(c^2*d+e)))/(sqrt(-d)*sqrt(e))-1/2*(a+b*arcsec(c*x))*log(1+c*exp(I*arcsec(c*x))*sqrt(-d)/(sqrt(e)+sqrt(c^2*d+e)))/(sqrt(-d)*sqrt(e))+1/2*I*b*polylog(2,-c*exp(I*arcsec(c*x))*sqrt(-d)/(sqrt(e)-sqrt(c^2*d+e)))/(sqrt(-d)*sqrt(e))-1/2*I*b*polylog(2,c*exp(I*arcsec(c*x))*sqrt(-d)/(sqrt(e)-sqrt(c^2*d+e)))/(sqrt(-d)*sqrt(e))+1/2*I*b*polylog(2,-c*exp(I*arcsec(c*x))*sqrt(-d)/(sqrt(e)+sqrt(c^2*d+e)))/(sqrt(-d)*sqrt(e))-1/2*I*b*polylog(2,c*exp(I*arcsec(c*x))*sqrt(-d)/(sqrt(e)+sqrt(c^2*d+e)))/(sqrt(-d)*sqrt(e))],
[(a+b*arcsec(c*x))/(x*(d+e*x^2)),x,19,1/2*I*(a+b*arcsec(c*x))^2/(b*d)-1/2*(a+b*arcsec(c*x))*log(1-c*exp(I*arcsec(c*x))*sqrt(-d)/(sqrt(e)-sqrt(c^2*d+e)))/d-1/2*(a+b*arcsec(c*x))*log(1+c*exp(I*arcsec(c*x))*sqrt(-d)/(sqrt(e)-sqrt(c^2*d+e)))/d-1/2*(a+b*arcsec(c*x))*log(1-c*exp(I*arcsec(c*x))*sqrt(-d)/(sqrt(e)+sqrt(c^2*d+e)))/d-1/2*(a+b*arcsec(c*x))*log(1+c*exp(I*arcsec(c*x))*sqrt(-d)/(sqrt(e)+sqrt(c^2*d+e)))/d+1/2*I*b*polylog(2,-c*exp(I*arcsec(c*x))*sqrt(-d)/(sqrt(e)-sqrt(c^2*d+e)))/d+1/2*I*b*polylog(2,c*exp(I*arcsec(c*x))*sqrt(-d)/(sqrt(e)-sqrt(c^2*d+e)))/d+1/2*I*b*polylog(2,-c*exp(I*arcsec(c*x))*sqrt(-d)/(sqrt(e)+sqrt(c^2*d+e)))/d+1/2*I*b*polylog(2,c*exp(I*arcsec(c*x))*sqrt(-d)/(sqrt(e)+sqrt(c^2*d+e)))/d],
[(a+b*arcsec(c*x))/(x^2*(d+e*x^2)),x,24,-a/(d*x)-b*arcsec(c*x)/(d*x)+1/2*(a+b*arcsec(c*x))*log(1-c*exp(I*arcsec(c*x))*sqrt(-d)/(sqrt(e)-sqrt(c^2*d+e)))*sqrt(e)/(-d)^(3/2)-1/2*(a+b*arcsec(c*x))*log(1+c*exp(I*arcsec(c*x))*sqrt(-d)/(sqrt(e)-sqrt(c^2*d+e)))*sqrt(e)/(-d)^(3/2)+1/2*(a+b*arcsec(c*x))*log(1-c*exp(I*arcsec(c*x))*sqrt(-d)/(sqrt(e)+sqrt(c^2*d+e)))*sqrt(e)/(-d)^(3/2)-1/2*(a+b*arcsec(c*x))*log(1+c*exp(I*arcsec(c*x))*sqrt(-d)/(sqrt(e)+sqrt(c^2*d+e)))*sqrt(e)/(-d)^(3/2)+1/2*I*b*polylog(2,-c*exp(I*arcsec(c*x))*sqrt(-d)/(sqrt(e)-sqrt(c^2*d+e)))*sqrt(e)/(-d)^(3/2)-1/2*I*b*polylog(2,c*exp(I*arcsec(c*x))*sqrt(-d)/(sqrt(e)-sqrt(c^2*d+e)))*sqrt(e)/(-d)^(3/2)+1/2*I*b*polylog(2,-c*exp(I*arcsec(c*x))*sqrt(-d)/(sqrt(e)+sqrt(c^2*d+e)))*sqrt(e)/(-d)^(3/2)-1/2*I*b*polylog(2,c*exp(I*arcsec(c*x))*sqrt(-d)/(sqrt(e)+sqrt(c^2*d+e)))*sqrt(e)/(-d)^(3/2)+b*c*sqrt(1+(-1)/(c^2*x^2))/d],
[x^5*(a+b*arcsec(c*x))/(d+e*x^2)^2,x,31,1/2*d*(a+b*arcsec(c*x))/(e^2*(e+d/x^2))+1/2*x^2*(a+b*arcsec(c*x))/e^2+2*d*(a+b*arcsec(c*x))*log(1+exp(2*I*arcsec(c*x)))/e^3-d*(a+b*arcsec(c*x))*log(1-c*exp(I*arcsec(c*x))*sqrt(-d)/(sqrt(e)-sqrt(c^2*d+e)))/e^3-d*(a+b*arcsec(c*x))*log(1+c*exp(I*arcsec(c*x))*sqrt(-d)/(sqrt(e)-sqrt(c^2*d+e)))/e^3-d*(a+b*arcsec(c*x))*log(1-c*exp(I*arcsec(c*x))*sqrt(-d)/(sqrt(e)+sqrt(c^2*d+e)))/e^3-d*(a+b*arcsec(c*x))*log(1+c*exp(I*arcsec(c*x))*sqrt(-d)/(sqrt(e)+sqrt(c^2*d+e)))/e^3-I*b*d*polylog(2,-exp(2*I*arcsec(c*x)))/e^3+I*b*d*polylog(2,-c*exp(I*arcsec(c*x))*sqrt(-d)/(sqrt(e)-sqrt(c^2*d+e)))/e^3+I*b*d*polylog(2,c*exp(I*arcsec(c*x))*sqrt(-d)/(sqrt(e)-sqrt(c^2*d+e)))/e^3+I*b*d*polylog(2,-c*exp(I*arcsec(c*x))*sqrt(-d)/(sqrt(e)+sqrt(c^2*d+e)))/e^3+I*b*d*polylog(2,c*exp(I*arcsec(c*x))*sqrt(-d)/(sqrt(e)+sqrt(c^2*d+e)))/e^3+1/2*b*d*arctan(sqrt(c^2*d+e)/(c*x*sqrt(e)*sqrt(1+(-1)/(c^2*x^2))))/(e^(5/2)*sqrt(c^2*d+e))-1/2*b*x*sqrt(1+(-1)/(c^2*x^2))/(c*e^2)],
[x^3*(a+b*arcsec(c*x))/(d+e*x^2)^2,x,29,1/2*(-a-b*arcsec(c*x))/(e*(e+d/x^2))-(a+b*arcsec(c*x))*log(1+exp(2*I*arcsec(c*x)))/e^2+1/2*(a+b*arcsec(c*x))*log(1-c*exp(I*arcsec(c*x))*sqrt(-d)/(sqrt(e)-sqrt(c^2*d+e)))/e^2+1/2*(a+b*arcsec(c*x))*log(1+c*exp(I*arcsec(c*x))*sqrt(-d)/(sqrt(e)-sqrt(c^2*d+e)))/e^2+1/2*(a+b*arcsec(c*x))*log(1-c*exp(I*arcsec(c*x))*sqrt(-d)/(sqrt(e)+sqrt(c^2*d+e)))/e^2+1/2*(a+b*arcsec(c*x))*log(1+c*exp(I*arcsec(c*x))*sqrt(-d)/(sqrt(e)+sqrt(c^2*d+e)))/e^2+1/2*I*b*polylog(2,-exp(2*I*arcsec(c*x)))/e^2-1/2*I*b*polylog(2,-c*exp(I*arcsec(c*x))*sqrt(-d)/(sqrt(e)-sqrt(c^2*d+e)))/e^2-1/2*I*b*polylog(2,c*exp(I*arcsec(c*x))*sqrt(-d)/(sqrt(e)-sqrt(c^2*d+e)))/e^2-1/2*I*b*polylog(2,-c*exp(I*arcsec(c*x))*sqrt(-d)/(sqrt(e)+sqrt(c^2*d+e)))/e^2-1/2*I*b*polylog(2,c*exp(I*arcsec(c*x))*sqrt(-d)/(sqrt(e)+sqrt(c^2*d+e)))/e^2-1/2*b*arctan(sqrt(c^2*d+e)/(c*x*sqrt(e)*sqrt(1+(-1)/(c^2*x^2))))/(e^(3/2)*sqrt(c^2*d+e))],
[x*(a+b*arcsec(c*x))/(d+e*x^2)^2,x,7,1/2*(-a-b*arcsec(c*x))/(e*(d+e*x^2))+1/2*b*c*x*arctan(sqrt(-1+c^2*x^2))/(d*e*sqrt(c^2*x^2))-1/2*b*c*x*arctan(sqrt(e)*sqrt(-1+c^2*x^2)/sqrt(c^2*d+e))/(d*sqrt(e)*sqrt(c^2*d+e)*sqrt(c^2*x^2))],
[(a+b*arcsec(c*x))/(x*(d+e*x^2)^2),x,24,-1/2*e*(a+b*arcsec(c*x))/(d^2*(e+d/x^2))+1/2*I*(a+b*arcsec(c*x))^2/(b*d^2)-1/2*(a+b*arcsec(c*x))*log(1-c*exp(I*arcsec(c*x))*sqrt(-d)/(sqrt(e)-sqrt(c^2*d+e)))/d^2-1/2*(a+b*arcsec(c*x))*log(1+c*exp(I*arcsec(c*x))*sqrt(-d)/(sqrt(e)-sqrt(c^2*d+e)))/d^2-1/2*(a+b*arcsec(c*x))*log(1-c*exp(I*arcsec(c*x))*sqrt(-d)/(sqrt(e)+sqrt(c^2*d+e)))/d^2-1/2*(a+b*arcsec(c*x))*log(1+c*exp(I*arcsec(c*x))*sqrt(-d)/(sqrt(e)+sqrt(c^2*d+e)))/d^2+1/2*I*b*polylog(2,-c*exp(I*arcsec(c*x))*sqrt(-d)/(sqrt(e)-sqrt(c^2*d+e)))/d^2+1/2*I*b*polylog(2,c*exp(I*arcsec(c*x))*sqrt(-d)/(sqrt(e)-sqrt(c^2*d+e)))/d^2+1/2*I*b*polylog(2,-c*exp(I*arcsec(c*x))*sqrt(-d)/(sqrt(e)+sqrt(c^2*d+e)))/d^2+1/2*I*b*polylog(2,c*exp(I*arcsec(c*x))*sqrt(-d)/(sqrt(e)+sqrt(c^2*d+e)))/d^2-1/2*b*arctan(sqrt(c^2*d+e)/(c*x*sqrt(e)*sqrt(1+(-1)/(c^2*x^2))))*sqrt(e)/(d^2*sqrt(c^2*d+e))],
[x^4*(a+b*arcsec(c*x))/(d+e*x^2)^2,x,51,x*(a+b*arcsec(c*x))/e^2-b*arctanh(sqrt(1+(-1)/(c^2*x^2)))/(c*e^2)+3/4*(a+b*arcsec(c*x))*log(1-c*exp(I*arcsec(c*x))*sqrt(-d)/(sqrt(e)-sqrt(c^2*d+e)))*sqrt(-d)/e^(5/2)-3/4*(a+b*arcsec(c*x))*log(1+c*exp(I*arcsec(c*x))*sqrt(-d)/(sqrt(e)-sqrt(c^2*d+e)))*sqrt(-d)/e^(5/2)+3/4*(a+b*arcsec(c*x))*log(1-c*exp(I*arcsec(c*x))*sqrt(-d)/(sqrt(e)+sqrt(c^2*d+e)))*sqrt(-d)/e^(5/2)-3/4*(a+b*arcsec(c*x))*log(1+c*exp(I*arcsec(c*x))*sqrt(-d)/(sqrt(e)+sqrt(c^2*d+e)))*sqrt(-d)/e^(5/2)+3/4*I*b*polylog(2,-c*exp(I*arcsec(c*x))*sqrt(-d)/(sqrt(e)-sqrt(c^2*d+e)))*sqrt(-d)/e^(5/2)-3/4*I*b*polylog(2,c*exp(I*arcsec(c*x))*sqrt(-d)/(sqrt(e)-sqrt(c^2*d+e)))*sqrt(-d)/e^(5/2)+3/4*I*b*polylog(2,-c*exp(I*arcsec(c*x))*sqrt(-d)/(sqrt(e)+sqrt(c^2*d+e)))*sqrt(-d)/e^(5/2)-3/4*I*b*polylog(2,c*exp(I*arcsec(c*x))*sqrt(-d)/(sqrt(e)+sqrt(c^2*d+e)))*sqrt(-d)/e^(5/2)-1/4*d*(a+b*arcsec(c*x))/(e^2*(-d/x+sqrt(-d)*sqrt(e)))+1/4*d*(a+b*arcsec(c*x))/(e^2*(d/x+sqrt(-d)*sqrt(e)))-1/4*b*arctanh((c^2*d-sqrt(-d)*sqrt(e)/x)/(c*sqrt(d)*sqrt(c^2*d+e)*sqrt(1+(-1)/(c^2*x^2))))*sqrt(d)/(e^2*sqrt(c^2*d+e))-1/4*b*arctanh((c^2*d+sqrt(-d)*sqrt(e)/x)/(c*sqrt(d)*sqrt(c^2*d+e)*sqrt(1+(-1)/(c^2*x^2))))*sqrt(d)/(e^2*sqrt(c^2*d+e))],
[x^2*(a+b*arcsec(c*x))/(d+e*x^2)^2,x,27,1/4*(a+b*arcsec(c*x))*log(1-c*exp(I*arcsec(c*x))*sqrt(-d)/(sqrt(e)-sqrt(c^2*d+e)))/(e^(3/2)*sqrt(-d))-1/4*(a+b*arcsec(c*x))*log(1+c*exp(I*arcsec(c*x))*sqrt(-d)/(sqrt(e)-sqrt(c^2*d+e)))/(e^(3/2)*sqrt(-d))+1/4*(a+b*arcsec(c*x))*log(1-c*exp(I*arcsec(c*x))*sqrt(-d)/(sqrt(e)+sqrt(c^2*d+e)))/(e^(3/2)*sqrt(-d))-1/4*(a+b*arcsec(c*x))*log(1+c*exp(I*arcsec(c*x))*sqrt(-d)/(sqrt(e)+sqrt(c^2*d+e)))/(e^(3/2)*sqrt(-d))+1/4*I*b*polylog(2,-c*exp(I*arcsec(c*x))*sqrt(-d)/(sqrt(e)-sqrt(c^2*d+e)))/(e^(3/2)*sqrt(-d))-1/4*I*b*polylog(2,c*exp(I*arcsec(c*x))*sqrt(-d)/(sqrt(e)-sqrt(c^2*d+e)))/(e^(3/2)*sqrt(-d))+1/4*I*b*polylog(2,-c*exp(I*arcsec(c*x))*sqrt(-d)/(sqrt(e)+sqrt(c^2*d+e)))/(e^(3/2)*sqrt(-d))-1/4*I*b*polylog(2,c*exp(I*arcsec(c*x))*sqrt(-d)/(sqrt(e)+sqrt(c^2*d+e)))/(e^(3/2)*sqrt(-d))+1/4*(a+b*arcsec(c*x))/(e*(-d/x+sqrt(-d)*sqrt(e)))+1/4*(-a-b*arcsec(c*x))/(e*(d/x+sqrt(-d)*sqrt(e)))+1/4*b*arctanh((c^2*d-sqrt(-d)*sqrt(e)/x)/(c*sqrt(d)*sqrt(c^2*d+e)*sqrt(1+(-1)/(c^2*x^2))))/(e*sqrt(d)*sqrt(c^2*d+e))+1/4*b*arctanh((c^2*d+sqrt(-d)*sqrt(e)/x)/(c*sqrt(d)*sqrt(c^2*d+e)*sqrt(1+(-1)/(c^2*x^2))))/(e*sqrt(d)*sqrt(c^2*d+e))],
[(a+b*arcsec(c*x))/(d+e*x^2)^2,x,47,-1/4*(a+b*arcsec(c*x))*log(1-c*exp(I*arcsec(c*x))*sqrt(-d)/(sqrt(e)-sqrt(c^2*d+e)))/((-d)^(3/2)*sqrt(e))+1/4*(a+b*arcsec(c*x))*log(1+c*exp(I*arcsec(c*x))*sqrt(-d)/(sqrt(e)-sqrt(c^2*d+e)))/((-d)^(3/2)*sqrt(e))-1/4*(a+b*arcsec(c*x))*log(1-c*exp(I*arcsec(c*x))*sqrt(-d)/(sqrt(e)+sqrt(c^2*d+e)))/((-d)^(3/2)*sqrt(e))+1/4*(a+b*arcsec(c*x))*log(1+c*exp(I*arcsec(c*x))*sqrt(-d)/(sqrt(e)+sqrt(c^2*d+e)))/((-d)^(3/2)*sqrt(e))-1/4*I*b*polylog(2,-c*exp(I*arcsec(c*x))*sqrt(-d)/(sqrt(e)-sqrt(c^2*d+e)))/((-d)^(3/2)*sqrt(e))+1/4*I*b*polylog(2,c*exp(I*arcsec(c*x))*sqrt(-d)/(sqrt(e)-sqrt(c^2*d+e)))/((-d)^(3/2)*sqrt(e))-1/4*I*b*polylog(2,-c*exp(I*arcsec(c*x))*sqrt(-d)/(sqrt(e)+sqrt(c^2*d+e)))/((-d)^(3/2)*sqrt(e))+1/4*I*b*polylog(2,c*exp(I*arcsec(c*x))*sqrt(-d)/(sqrt(e)+sqrt(c^2*d+e)))/((-d)^(3/2)*sqrt(e))+1/4*(-a-b*arcsec(c*x))/(d*(-d/x+sqrt(-d)*sqrt(e)))+1/4*(a+b*arcsec(c*x))/(d*(d/x+sqrt(-d)*sqrt(e)))-1/4*b*arctanh((c^2*d-sqrt(-d)*sqrt(e)/x)/(c*sqrt(d)*sqrt(c^2*d+e)*sqrt(1+(-1)/(c^2*x^2))))/(d^(3/2)*sqrt(c^2*d+e))-1/4*b*arctanh((c^2*d+sqrt(-d)*sqrt(e)/x)/(c*sqrt(d)*sqrt(c^2*d+e)*sqrt(1+(-1)/(c^2*x^2))))/(d^(3/2)*sqrt(c^2*d+e))],
[(a+b*arcsec(c*x))/(x^2*(d+e*x^2)^2),x,50,-a/(d^2*x)-b*arcsec(c*x)/(d^2*x)-3/4*(a+b*arcsec(c*x))*log(1-c*exp(I*arcsec(c*x))*sqrt(-d)/(sqrt(e)-sqrt(c^2*d+e)))*sqrt(e)/(-d)^(5/2)+3/4*(a+b*arcsec(c*x))*log(1+c*exp(I*arcsec(c*x))*sqrt(-d)/(sqrt(e)-sqrt(c^2*d+e)))*sqrt(e)/(-d)^(5/2)-3/4*(a+b*arcsec(c*x))*log(1-c*exp(I*arcsec(c*x))*sqrt(-d)/(sqrt(e)+sqrt(c^2*d+e)))*sqrt(e)/(-d)^(5/2)+3/4*(a+b*arcsec(c*x))*log(1+c*exp(I*arcsec(c*x))*sqrt(-d)/(sqrt(e)+sqrt(c^2*d+e)))*sqrt(e)/(-d)^(5/2)-3/4*I*b*polylog(2,-c*exp(I*arcsec(c*x))*sqrt(-d)/(sqrt(e)-sqrt(c^2*d+e)))*sqrt(e)/(-d)^(5/2)+3/4*I*b*polylog(2,c*exp(I*arcsec(c*x))*sqrt(-d)/(sqrt(e)-sqrt(c^2*d+e)))*sqrt(e)/(-d)^(5/2)-3/4*I*b*polylog(2,-c*exp(I*arcsec(c*x))*sqrt(-d)/(sqrt(e)+sqrt(c^2*d+e)))*sqrt(e)/(-d)^(5/2)+3/4*I*b*polylog(2,c*exp(I*arcsec(c*x))*sqrt(-d)/(sqrt(e)+sqrt(c^2*d+e)))*sqrt(e)/(-d)^(5/2)+1/4*e*(a+b*arcsec(c*x))/(d^2*(-d/x+sqrt(-d)*sqrt(e)))-1/4*e*(a+b*arcsec(c*x))/(d^2*(d/x+sqrt(-d)*sqrt(e)))+1/4*b*e*arctanh((c^2*d-sqrt(-d)*sqrt(e)/x)/(c*sqrt(d)*sqrt(c^2*d+e)*sqrt(1+(-1)/(c^2*x^2))))/(d^(5/2)*sqrt(c^2*d+e))+1/4*b*e*arctanh((c^2*d+sqrt(-d)*sqrt(e)/x)/(c*sqrt(d)*sqrt(c^2*d+e)*sqrt(1+(-1)/(c^2*x^2))))/(d^(5/2)*sqrt(c^2*d+e))+b*c*sqrt(1+(-1)/(c^2*x^2))/d^2],
[x^5*(a+b*arcsec(c*x))/(d+e*x^2)^3,x,33,1/4*(-a-b*arcsec(c*x))/(e*(e+d/x^2)^2)+1/2*(-a-b*arcsec(c*x))/(e^2*(e+d/x^2))-1/8*b*(c^2*d+2*e)*arctan(sqrt(c^2*d+e)/(c*x*sqrt(e)*sqrt(1+(-1)/(c^2*x^2))))/(e^(5/2)*(c^2*d+e)^(3/2))-(a+b*arcsec(c*x))*log(1+exp(2*I*arcsec(c*x)))/e^3+1/2*(a+b*arcsec(c*x))*log(1-c*exp(I*arcsec(c*x))*sqrt(-d)/(sqrt(e)-sqrt(c^2*d+e)))/e^3+1/2*(a+b*arcsec(c*x))*log(1+c*exp(I*arcsec(c*x))*sqrt(-d)/(sqrt(e)-sqrt(c^2*d+e)))/e^3+1/2*(a+b*arcsec(c*x))*log(1-c*exp(I*arcsec(c*x))*sqrt(-d)/(sqrt(e)+sqrt(c^2*d+e)))/e^3+1/2*(a+b*arcsec(c*x))*log(1+c*exp(I*arcsec(c*x))*sqrt(-d)/(sqrt(e)+sqrt(c^2*d+e)))/e^3+1/2*I*b*polylog(2,-exp(2*I*arcsec(c*x)))/e^3-1/2*I*b*polylog(2,-c*exp(I*arcsec(c*x))*sqrt(-d)/(sqrt(e)-sqrt(c^2*d+e)))/e^3-1/2*I*b*polylog(2,c*exp(I*arcsec(c*x))*sqrt(-d)/(sqrt(e)-sqrt(c^2*d+e)))/e^3-1/2*I*b*polylog(2,-c*exp(I*arcsec(c*x))*sqrt(-d)/(sqrt(e)+sqrt(c^2*d+e)))/e^3-1/2*I*b*polylog(2,c*exp(I*arcsec(c*x))*sqrt(-d)/(sqrt(e)+sqrt(c^2*d+e)))/e^3-1/2*b*arctan(sqrt(c^2*d+e)/(c*x*sqrt(e)*sqrt(1+(-1)/(c^2*x^2))))/(e^(5/2)*sqrt(c^2*d+e))-1/8*b*c*d*sqrt(1+(-1)/(c^2*x^2))/(e^2*(c^2*d+e)*(e+d/x^2)*x)],
[x^3*(a+b*arcsec(c*x))/(d+e*x^2)^3,x,6,1/4*x^4*(a+b*arcsec(c*x))/(d*(d+e*x^2)^2)-1/8*b*c*(c^2*d+2*e)*x*arctan(sqrt(e)*sqrt(-1+c^2*x^2)/sqrt(c^2*d+e))/(d*e^(3/2)*(c^2*d+e)^(3/2)*sqrt(c^2*x^2))+1/8*b*c*x*sqrt(-1+c^2*x^2)/(e*(c^2*d+e)*(d+e*x^2)*sqrt(c^2*x^2))],
[x*(a+b*arcsec(c*x))/(d+e*x^2)^3,x,8,1/4*(-a-b*arcsec(c*x))/(e*(d+e*x^2)^2)+1/4*b*c*x*arctan(sqrt(-1+c^2*x^2))/(d^2*e*sqrt(c^2*x^2))-1/8*b*c*(3*c^2*d+2*e)*x*arctan(sqrt(e)*sqrt(-1+c^2*x^2)/sqrt(c^2*d+e))/(d^2*(c^2*d+e)^(3/2)*sqrt(e)*sqrt(c^2*x^2))-1/8*b*c*x*sqrt(-1+c^2*x^2)/(d*(c^2*d+e)*(d+e*x^2)*sqrt(c^2*x^2))],
[(a+b*arcsec(c*x))/(x*(d+e*x^2)^3),x,28,1/4*e^2*(a+b*arcsec(c*x))/(d^3*(e+d/x^2)^2)-e*(a+b*arcsec(c*x))/(d^3*(e+d/x^2))+1/2*I*(a+b*arcsec(c*x))^2/(b*d^3)-1/2*(a+b*arcsec(c*x))*log(1-c*exp(I*arcsec(c*x))*sqrt(-d)/(sqrt(e)-sqrt(c^2*d+e)))/d^3-1/2*(a+b*arcsec(c*x))*log(1+c*exp(I*arcsec(c*x))*sqrt(-d)/(sqrt(e)-sqrt(c^2*d+e)))/d^3-1/2*(a+b*arcsec(c*x))*log(1-c*exp(I*arcsec(c*x))*sqrt(-d)/(sqrt(e)+sqrt(c^2*d+e)))/d^3-1/2*(a+b*arcsec(c*x))*log(1+c*exp(I*arcsec(c*x))*sqrt(-d)/(sqrt(e)+sqrt(c^2*d+e)))/d^3+1/2*I*b*polylog(2,-c*exp(I*arcsec(c*x))*sqrt(-d)/(sqrt(e)-sqrt(c^2*d+e)))/d^3+1/2*I*b*polylog(2,c*exp(I*arcsec(c*x))*sqrt(-d)/(sqrt(e)-sqrt(c^2*d+e)))/d^3+1/2*I*b*polylog(2,-c*exp(I*arcsec(c*x))*sqrt(-d)/(sqrt(e)+sqrt(c^2*d+e)))/d^3+1/2*I*b*polylog(2,c*exp(I*arcsec(c*x))*sqrt(-d)/(sqrt(e)+sqrt(c^2*d+e)))/d^3+1/8*b*(c^2*d+2*e)*arctan(sqrt(c^2*d+e)/(c*x*sqrt(e)*sqrt(1+(-1)/(c^2*x^2))))*sqrt(e)/(d^3*(c^2*d+e)^(3/2))-b*arctan(sqrt(c^2*d+e)/(c*x*sqrt(e)*sqrt(1+(-1)/(c^2*x^2))))*sqrt(e)/(d^3*sqrt(c^2*d+e))+1/8*b*c*e*sqrt(1+(-1)/(c^2*x^2))/(d^2*(c^2*d+e)*(e+d/x^2)*x)],
[x^4*(a+b*arcsec(c*x))/(d+e*x^2)^3,x,35,3/16*(a+b*arcsec(c*x))*log(1-c*exp(I*arcsec(c*x))*sqrt(-d)/(sqrt(e)-sqrt(c^2*d+e)))/(e^(5/2)*sqrt(-d))-3/16*(a+b*arcsec(c*x))*log(1+c*exp(I*arcsec(c*x))*sqrt(-d)/(sqrt(e)-sqrt(c^2*d+e)))/(e^(5/2)*sqrt(-d))+3/16*(a+b*arcsec(c*x))*log(1-c*exp(I*arcsec(c*x))*sqrt(-d)/(sqrt(e)+sqrt(c^2*d+e)))/(e^(5/2)*sqrt(-d))-3/16*(a+b*arcsec(c*x))*log(1+c*exp(I*arcsec(c*x))*sqrt(-d)/(sqrt(e)+sqrt(c^2*d+e)))/(e^(5/2)*sqrt(-d))+3/16*I*b*polylog(2,-c*exp(I*arcsec(c*x))*sqrt(-d)/(sqrt(e)-sqrt(c^2*d+e)))/(e^(5/2)*sqrt(-d))-3/16*I*b*polylog(2,c*exp(I*arcsec(c*x))*sqrt(-d)/(sqrt(e)-sqrt(c^2*d+e)))/(e^(5/2)*sqrt(-d))+3/16*I*b*polylog(2,-c*exp(I*arcsec(c*x))*sqrt(-d)/(sqrt(e)+sqrt(c^2*d+e)))/(e^(5/2)*sqrt(-d))-3/16*I*b*polylog(2,c*exp(I*arcsec(c*x))*sqrt(-d)/(sqrt(e)+sqrt(c^2*d+e)))/(e^(5/2)*sqrt(-d))+1/16*b*arctanh((c^2*d-sqrt(-d)*sqrt(e)/x)/(c*sqrt(d)*sqrt(c^2*d+e)*sqrt(1+(-1)/(c^2*x^2))))/(e*(c^2*d+e)^(3/2)*sqrt(d))+1/16*b*arctanh((c^2*d+sqrt(-d)*sqrt(e)/x)/(c*sqrt(d)*sqrt(c^2*d+e)*sqrt(1+(-1)/(c^2*x^2))))/(e*(c^2*d+e)^(3/2)*sqrt(d))+1/16*(a+b*arcsec(c*x))*sqrt(-d)/(e^(3/2)*(-d/x+sqrt(-d)*sqrt(e))^2)+3/16*(a+b*arcsec(c*x))/(e^2*(-d/x+sqrt(-d)*sqrt(e)))-1/16*(a+b*arcsec(c*x))*sqrt(-d)/(e^(3/2)*(d/x+sqrt(-d)*sqrt(e))^2)-3/16*(a+b*arcsec(c*x))/(e^2*(d/x+sqrt(-d)*sqrt(e)))+3/16*b*arctanh((c^2*d-sqrt(-d)*sqrt(e)/x)/(c*sqrt(d)*sqrt(c^2*d+e)*sqrt(1+(-1)/(c^2*x^2))))/(e^2*sqrt(d)*sqrt(c^2*d+e))+3/16*b*arctanh((c^2*d+sqrt(-d)*sqrt(e)/x)/(c*sqrt(d)*sqrt(c^2*d+e)*sqrt(1+(-1)/(c^2*x^2))))/(e^2*sqrt(d)*sqrt(c^2*d+e))+1/16*b*c*sqrt(-d)*sqrt(1+(-1)/(c^2*x^2))/(e^(3/2)*(c^2*d+e)*(-d/x+sqrt(-d)*sqrt(e)))+1/16*b*c*sqrt(-d)*sqrt(1+(-1)/(c^2*x^2))/(e^(3/2)*(c^2*d+e)*(d/x+sqrt(-d)*sqrt(e)))],
[x^2*(a+b*arcsec(c*x))/(d+e*x^2)^3,x,63,-1/16*b*arctanh((c^2*d-sqrt(-d)*sqrt(e)/x)/(c*sqrt(d)*sqrt(c^2*d+e)*sqrt(1+(-1)/(c^2*x^2))))/(d^(3/2)*(c^2*d+e)^(3/2))-1/16*b*arctanh((c^2*d+sqrt(-d)*sqrt(e)/x)/(c*sqrt(d)*sqrt(c^2*d+e)*sqrt(1+(-1)/(c^2*x^2))))/(d^(3/2)*(c^2*d+e)^(3/2))-1/16*(a+b*arcsec(c*x))*log(1-c*exp(I*arcsec(c*x))*sqrt(-d)/(sqrt(e)-sqrt(c^2*d+e)))/((-d)^(3/2)*e^(3/2))+1/16*(a+b*arcsec(c*x))*log(1+c*exp(I*arcsec(c*x))*sqrt(-d)/(sqrt(e)-sqrt(c^2*d+e)))/((-d)^(3/2)*e^(3/2))-1/16*(a+b*arcsec(c*x))*log(1-c*exp(I*arcsec(c*x))*sqrt(-d)/(sqrt(e)+sqrt(c^2*d+e)))/((-d)^(3/2)*e^(3/2))+1/16*(a+b*arcsec(c*x))*log(1+c*exp(I*arcsec(c*x))*sqrt(-d)/(sqrt(e)+sqrt(c^2*d+e)))/((-d)^(3/2)*e^(3/2))-1/16*I*b*polylog(2,-c*exp(I*arcsec(c*x))*sqrt(-d)/(sqrt(e)-sqrt(c^2*d+e)))/((-d)^(3/2)*e^(3/2))+1/16*I*b*polylog(2,c*exp(I*arcsec(c*x))*sqrt(-d)/(sqrt(e)-sqrt(c^2*d+e)))/((-d)^(3/2)*e^(3/2))-1/16*I*b*polylog(2,-c*exp(I*arcsec(c*x))*sqrt(-d)/(sqrt(e)+sqrt(c^2*d+e)))/((-d)^(3/2)*e^(3/2))+1/16*I*b*polylog(2,c*exp(I*arcsec(c*x))*sqrt(-d)/(sqrt(e)+sqrt(c^2*d+e)))/((-d)^(3/2)*e^(3/2))+1/16*(a+b*arcsec(c*x))/(sqrt(-d)*sqrt(e)*(-d/x+sqrt(-d)*sqrt(e))^2)+1/16*(a+b*arcsec(c*x))/(d*e*(-d/x+sqrt(-d)*sqrt(e)))+1/16*(-a-b*arcsec(c*x))/(sqrt(-d)*sqrt(e)*(d/x+sqrt(-d)*sqrt(e))^2)+1/16*(-a-b*arcsec(c*x))/(d*e*(d/x+sqrt(-d)*sqrt(e)))+1/16*b*arctanh((c^2*d-sqrt(-d)*sqrt(e)/x)/(c*sqrt(d)*sqrt(c^2*d+e)*sqrt(1+(-1)/(c^2*x^2))))/(d^(3/2)*e*sqrt(c^2*d+e))+1/16*b*arctanh((c^2*d+sqrt(-d)*sqrt(e)/x)/(c*sqrt(d)*sqrt(c^2*d+e)*sqrt(1+(-1)/(c^2*x^2))))/(d^(3/2)*e*sqrt(c^2*d+e))+1/16*b*c*sqrt(1+(-1)/(c^2*x^2))/((c^2*d+e)*sqrt(-d)*sqrt(e)*(-d/x+sqrt(-d)*sqrt(e)))+1/16*b*c*sqrt(1+(-1)/(c^2*x^2))/((c^2*d+e)*sqrt(-d)*sqrt(e)*(d/x+sqrt(-d)*sqrt(e)))],
[(a+b*arcsec(c*x))/(d+e*x^2)^3,x,81,1/16*b*e*arctanh((c^2*d-sqrt(-d)*sqrt(e)/x)/(c*sqrt(d)*sqrt(c^2*d+e)*sqrt(1+(-1)/(c^2*x^2))))/(d^(5/2)*(c^2*d+e)^(3/2))+1/16*b*e*arctanh((c^2*d+sqrt(-d)*sqrt(e)/x)/(c*sqrt(d)*sqrt(c^2*d+e)*sqrt(1+(-1)/(c^2*x^2))))/(d^(5/2)*(c^2*d+e)^(3/2))+3/16*(a+b*arcsec(c*x))*log(1-c*exp(I*arcsec(c*x))*sqrt(-d)/(sqrt(e)-sqrt(c^2*d+e)))/((-d)^(5/2)*sqrt(e))-3/16*(a+b*arcsec(c*x))*log(1+c*exp(I*arcsec(c*x))*sqrt(-d)/(sqrt(e)-sqrt(c^2*d+e)))/((-d)^(5/2)*sqrt(e))+3/16*(a+b*arcsec(c*x))*log(1-c*exp(I*arcsec(c*x))*sqrt(-d)/(sqrt(e)+sqrt(c^2*d+e)))/((-d)^(5/2)*sqrt(e))-3/16*(a+b*arcsec(c*x))*log(1+c*exp(I*arcsec(c*x))*sqrt(-d)/(sqrt(e)+sqrt(c^2*d+e)))/((-d)^(5/2)*sqrt(e))+3/16*I*b*polylog(2,-c*exp(I*arcsec(c*x))*sqrt(-d)/(sqrt(e)-sqrt(c^2*d+e)))/((-d)^(5/2)*sqrt(e))-3/16*I*b*polylog(2,c*exp(I*arcsec(c*x))*sqrt(-d)/(sqrt(e)-sqrt(c^2*d+e)))/((-d)^(5/2)*sqrt(e))+3/16*I*b*polylog(2,-c*exp(I*arcsec(c*x))*sqrt(-d)/(sqrt(e)+sqrt(c^2*d+e)))/((-d)^(5/2)*sqrt(e))-3/16*I*b*polylog(2,c*exp(I*arcsec(c*x))*sqrt(-d)/(sqrt(e)+sqrt(c^2*d+e)))/((-d)^(5/2)*sqrt(e))+1/16*(a+b*arcsec(c*x))*sqrt(e)/((-d)^(3/2)*(-d/x+sqrt(-d)*sqrt(e))^2)-5/16*(a+b*arcsec(c*x))/(d^2*(-d/x+sqrt(-d)*sqrt(e)))-1/16*(a+b*arcsec(c*x))*sqrt(e)/((-d)^(3/2)*(d/x+sqrt(-d)*sqrt(e))^2)+5/16*(a+b*arcsec(c*x))/(d^2*(d/x+sqrt(-d)*sqrt(e)))-5/16*b*arctanh((c^2*d-sqrt(-d)*sqrt(e)/x)/(c*sqrt(d)*sqrt(c^2*d+e)*sqrt(1+(-1)/(c^2*x^2))))/(d^(5/2)*sqrt(c^2*d+e))-5/16*b*arctanh((c^2*d+sqrt(-d)*sqrt(e)/x)/(c*sqrt(d)*sqrt(c^2*d+e)*sqrt(1+(-1)/(c^2*x^2))))/(d^(5/2)*sqrt(c^2*d+e))+1/16*b*c*sqrt(e)*sqrt(1+(-1)/(c^2*x^2))/((-d)^(3/2)*(c^2*d+e)*(-d/x+sqrt(-d)*sqrt(e)))+1/16*b*c*sqrt(e)*sqrt(1+(-1)/(c^2*x^2))/((-d)^(3/2)*(c^2*d+e)*(d/x+sqrt(-d)*sqrt(e)))],

#  {(a + b*ArcSec[c*x])/(x^2*(d + e*x^2)^3), x, 96, (b*c*Sqrt[1 - 1/(c^2*x^2)])/d^3 + (b*c*e^(3/2)*Sqrt[1 - 1/(c^2*x^2)])/(16*(-d)^(5/2)*(c^2*d + e)*(Sqrt[-d]*Sqrt[e] - d/x)) + (b*c*e^(3/2)*Sqrt[1 - 1/(c^2*x^2)])/(16*(-d)^(5/2)*(c^2*d + e)*(Sqrt[-d]*Sqrt[e] + d/x)) - a/(d^3*x) + (a*e^2)/(4*d^3*(e + d/x^2)^2*x) - (9*a*e)/(8*d^3*(e + d/x^2)*x) + (15*a*Sqrt[e]*ArcCot[(Sqrt[e]*x)/Sqrt[d]])/(8*d^(7/2)) + (b*e^(3/2)*ArcSec[c*x])/(16*(-d)^(5/2)*(Sqrt[-d]*Sqrt[e] - d/x)^2) + (9*b*e*ArcSec[c*x])/(16*d^3*(Sqrt[-d]*Sqrt[e] - d/x)) - (b*e^(3/2)*ArcSec[c*x])/(16*(-d)^(5/2)*(Sqrt[-d]*Sqrt[e] + d/x)^2) - (9*b*e*ArcSec[c*x])/(16*d^3*(Sqrt[-d]*Sqrt[e] + d/x)) - (b*ArcSec[c*x])/(d^3*x) - (b*e^2*ArcTanh[(c^2*d - (Sqrt[-d]*Sqrt[e])/x)/(c*Sqrt[d]*Sqrt[c^2*d + e]*Sqrt[1 - 1/(c^2*x^2)])])/(16*d^(7/2)*(c^2*d + e)^(3/2)) + (9*b*e*ArcTanh[(c^2*d - (Sqrt[-d]*Sqrt[e])/x)/(c*Sqrt[d]*Sqrt[c^2*d + e]*Sqrt[1 - 1/(c^2*x^2)])])/(16*d^(7/2)*Sqrt[c^2*d + e]) - (b*e^2*ArcTanh[(c^2*d + (Sqrt[-d]*Sqrt[e])/x)/(c*Sqrt[d]*Sqrt[c^2*d + e]*Sqrt[1 - 1/(c^2*x^2)])])/(16*d^(7/2)*(c^2*d + e)^(3/2)) + (9*b*e*ArcTanh[(c^2*d + (Sqrt[-d]*Sqrt[e])/x)/(c*Sqrt[d]*Sqrt[c^2*d + e]*Sqrt[1 - 1/(c^2*x^2)])])/(16*d^(7/2)*Sqrt[c^2*d + e]) + (15*b*Sqrt[e]*ArcSec[c*x]*Log[1 - (c*Sqrt[-d]*E^(I*ArcSec[c*x]))/(Sqrt[e] - c*Sqrt[d + e/c^2])])/(16*(-d)^(7/2)) - (15*b*Sqrt[e]*ArcSec[c*x]*Log[1 + (c*Sqrt[-d]*E^(I*ArcSec[c*x]))/(Sqrt[e] - c*Sqrt[d + e/c^2])])/(16*(-d)^(7/2)) + (15*b*Sqrt[e]*ArcSec[c*x]*Log[1 - (c*Sqrt[-d]*E^(I*ArcSec[c*x]))/(Sqrt[e] + c*Sqrt[d + e/c^2])])/(16*(-d)^(7/2)) - (15*b*Sqrt[e]*ArcSec[c*x]*Log[1 + (c*Sqrt[-d]*E^(I*ArcSec[c*x]))/(Sqrt[e] + c*Sqrt[d + e/c^2])])/(16*(-d)^(7/2)) + (15*I*b*Sqrt[e]*PolyLog[2, -((c*Sqrt[-d]*E^(I*ArcSec[c*x]))/(Sqrt[e] - c*Sqrt[d + e/c^2]))])/(16*(-d)^(7/2)) - (15*I*b*Sqrt[e]*PolyLog[2, (c*Sqrt[-d]*E^(I*ArcSec[c*x]))/(Sqrt[e] - c*Sqrt[d + e/c^2])])/(16*(-d)^(7/2)) + (15*I*b*Sqrt[e]*PolyLog[2, -((c*Sqrt[-d]*E^(I*ArcSec[c*x]))/(Sqrt[e] + c*Sqrt[d + e/c^2]))])/(16*(-d)^(7/2)) - (15*I*b*Sqrt[e]*PolyLog[2, (c*Sqrt[-d]*E^(I*ArcSec[c*x]))/(Sqrt[e] + c*Sqrt[d + e/c^2])])/(16*(-d)^(7/2))} 

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

# p>0
[x^5*(a+b*arcsec(c*x))*sqrt(d+e*x^2),x,13,1/3*d^2*(d+e*x^2)^(3/2)*(a+b*arcsec(c*x))/e^3-2/5*d*(d+e*x^2)^(5/2)*(a+b*arcsec(c*x))/e^3+1/7*(d+e*x^2)^(7/2)*(a+b*arcsec(c*x))/e^3+8/105*b*c*d^(7/2)*x*arctan(sqrt(d+e*x^2)/(sqrt(d)*sqrt(-1+c^2*x^2)))/(e^3*sqrt(c^2*x^2))-1/1680*b*(105*c^6*d^3-35*c^4*d^2*e+63*c^2*d*e^2+75*e^3)*x*arctanh(sqrt(e)*sqrt(-1+c^2*x^2)/(c*sqrt(d+e*x^2)))/(c^6*e^(5/2)*sqrt(c^2*x^2))+1/840*b*(29*c^2*d-25*e)*x*(d+e*x^2)^(3/2)*sqrt(-1+c^2*x^2)/(c^3*e^2*sqrt(c^2*x^2))-1/42*b*x*(d+e*x^2)^(5/2)*sqrt(-1+c^2*x^2)/(c*e^2*sqrt(c^2*x^2))+1/1680*b*(23*c^4*d^2+12*c^2*d*e-75*e^2)*x*sqrt(-1+c^2*x^2)*sqrt(d+e*x^2)/(c^5*e^2*sqrt(c^2*x^2))],
[x^3*(a+b*arcsec(c*x))*sqrt(d+e*x^2),x,12,-1/3*d*(d+e*x^2)^(3/2)*(a+b*arcsec(c*x))/e^2+1/5*(d+e*x^2)^(5/2)*(a+b*arcsec(c*x))/e^2-2/15*b*c*d^(5/2)*x*arctan(sqrt(d+e*x^2)/(sqrt(d)*sqrt(-1+c^2*x^2)))/(e^2*sqrt(c^2*x^2))+1/120*b*(15*c^4*d^2-10*c^2*d*e-9*e^2)*x*arctanh(sqrt(e)*sqrt(-1+c^2*x^2)/(c*sqrt(d+e*x^2)))/(c^4*e^(3/2)*sqrt(c^2*x^2))-1/20*b*x*(d+e*x^2)^(3/2)*sqrt(-1+c^2*x^2)/(c*e*sqrt(c^2*x^2))-1/120*b*(c^2*d+9*e)*x*sqrt(-1+c^2*x^2)*sqrt(d+e*x^2)/(c^3*e*sqrt(c^2*x^2))],
[x*(a+b*arcsec(c*x))*sqrt(d+e*x^2),x,9,1/3*(d+e*x^2)^(3/2)*(a+b*arcsec(c*x))/e+1/3*b*c*d^(3/2)*x*arctan(sqrt(d+e*x^2)/(sqrt(d)*sqrt(-1+c^2*x^2)))/(e*sqrt(c^2*x^2))-1/6*b*(3*c^2*d+e)*x*arctanh(sqrt(e)*sqrt(-1+c^2*x^2)/(c*sqrt(d+e*x^2)))/(c^2*sqrt(e)*sqrt(c^2*x^2))-1/6*b*x*sqrt(-1+c^2*x^2)*sqrt(d+e*x^2)/(c*sqrt(c^2*x^2))],
[(a+b*arcsec(c*x))*sqrt(d+e*x^2)/x,x,0,Unintegrable((a+b*arcsec(c*x))*sqrt(d+e*x^2)/x,x)],
[(a+b*arcsec(c*x))*sqrt(d+e*x^2)/x^3,x,0,Unintegrable((a+b*arcsec(c*x))*sqrt(d+e*x^2)/x^3,x)],
[x^2*(a+b*arcsec(c*x))*sqrt(d+e*x^2),x,0,Unintegrable(x^2*(a+b*arcsec(c*x))*sqrt(d+e*x^2),x)],
[(a+b*arcsec(c*x))*sqrt(d+e*x^2),x,0,Unintegrable((a+b*arcsec(c*x))*sqrt(d+e*x^2),x)],
[(a+b*arcsec(c*x))*sqrt(d+e*x^2)/x^2,x,0,Unintegrable((a+b*arcsec(c*x))*sqrt(d+e*x^2)/x^2,x)],
[(a+b*arcsec(c*x))*sqrt(d+e*x^2)/x^4,x,11,-1/3*(d+e*x^2)^(3/2)*(a+b*arcsec(c*x))/(d*x^3)+2/9*b*c*(c^2*d+2*e)*sqrt(-1+c^2*x^2)*sqrt(d+e*x^2)/(d*sqrt(c^2*x^2))+1/9*b*c*sqrt(-1+c^2*x^2)*sqrt(d+e*x^2)/(x^2*sqrt(c^2*x^2))-2/9*b*c^2*(c^2*d+2*e)*x*EllipticE(c*x,sqrt(-e/(c^2*d)))*sqrt(1-c^2*x^2)*sqrt(d+e*x^2)/(d*sqrt(c^2*x^2)*sqrt(-1+c^2*x^2)*sqrt(1+e*x^2/d))+1/9*b*(c^2*d+e)*(2*c^2*d+3*e)*x*EllipticF(c*x,sqrt(-e/(c^2*d)))*sqrt(1-c^2*x^2)*sqrt(1+e*x^2/d)/(d*sqrt(c^2*x^2)*sqrt(-1+c^2*x^2)*sqrt(d+e*x^2))],
[(a+b*arcsec(c*x))*sqrt(d+e*x^2)/x^6,x,32,-1/5*(d+e*x^2)^(3/2)*(a+b*arcsec(c*x))/(d*x^5)+2/15*e*(d+e*x^2)^(3/2)*(a+b*arcsec(c*x))/(d^2*x^3)-2/15*b*c*e^2*sqrt(-1+c^2*x^2)*sqrt(d+e*x^2)/(d^2*sqrt(c^2*x^2))+1/45*b*c*e*(2*c^2*d+e)*sqrt(-1+c^2*x^2)*sqrt(d+e*x^2)/(d^2*sqrt(c^2*x^2))+1/75*b*c*(8*c^4*d^2+3*c^2*d*e-2*e^2)*sqrt(-1+c^2*x^2)*sqrt(d+e*x^2)/(d^2*sqrt(c^2*x^2))+1/25*b*c*sqrt(-1+c^2*x^2)*sqrt(d+e*x^2)/(x^4*sqrt(c^2*x^2))+1/45*b*c*e*sqrt(-1+c^2*x^2)*sqrt(d+e*x^2)/(d*x^2*sqrt(c^2*x^2))+1/75*b*c*(4*c^2*d+e)*sqrt(-1+c^2*x^2)*sqrt(d+e*x^2)/(d*x^2*sqrt(c^2*x^2))+2/15*b*c^2*e^2*x*EllipticE(c*x,sqrt(-e/(c^2*d)))*sqrt(1-c^2*x^2)*sqrt(d+e*x^2)/(d^2*sqrt(c^2*x^2)*sqrt(-1+c^2*x^2)*sqrt(1+e*x^2/d))-1/45*b*c^2*e*(2*c^2*d+e)*x*EllipticE(c*x,sqrt(-e/(c^2*d)))*sqrt(1-c^2*x^2)*sqrt(d+e*x^2)/(d^2*sqrt(c^2*x^2)*sqrt(-1+c^2*x^2)*sqrt(1+e*x^2/d))-1/75*b*c^2*(8*c^4*d^2+3*c^2*d*e-2*e^2)*x*EllipticE(c*x,sqrt(-e/(c^2*d)))*sqrt(1-c^2*x^2)*sqrt(d+e*x^2)/(d^2*sqrt(c^2*x^2)*sqrt(-1+c^2*x^2)*sqrt(1+e*x^2/d))+1/75*b*c^2*(8*c^2*d-e)*(c^2*d+e)*x*EllipticF(c*x,sqrt(-e/(c^2*d)))*sqrt(1-c^2*x^2)*sqrt(1+e*x^2/d)/(d*sqrt(c^2*x^2)*sqrt(-1+c^2*x^2)*sqrt(d+e*x^2))+2/45*b*c^2*e*(c^2*d+e)*x*EllipticF(c*x,sqrt(-e/(c^2*d)))*sqrt(1-c^2*x^2)*sqrt(1+e*x^2/d)/(d*sqrt(c^2*x^2)*sqrt(-1+c^2*x^2)*sqrt(d+e*x^2))-2/15*b*e^2*(c^2*d+e)*x*EllipticF(c*x,sqrt(-e/(c^2*d)))*sqrt(1-c^2*x^2)*sqrt(1+e*x^2/d)/(d^2*sqrt(c^2*x^2)*sqrt(-1+c^2*x^2)*sqrt(d+e*x^2))],
[x^3*(d+e*x^2)^(3/2)*(a+b*arcsec(c*x)),x,12,-1/5*d*(d+e*x^2)^(5/2)*(a+b*arcsec(c*x))/e^2+1/7*(d+e*x^2)^(7/2)*(a+b*arcsec(c*x))/e^2-2/35*b*c*d^(7/2)*x*arctan(sqrt(d+e*x^2)/(sqrt(d)*sqrt(-1+c^2*x^2)))/(e^2*sqrt(c^2*x^2))+1/560*b*(35*c^6*d^3-35*c^4*d^2*e-63*c^2*d*e^2-25*e^3)*x*arctanh(sqrt(e)*sqrt(-1+c^2*x^2)/(c*sqrt(d+e*x^2)))/(c^6*e^(3/2)*sqrt(c^2*x^2))-1/840*b*(13*c^2*d+25*e)*x*(d+e*x^2)^(3/2)*sqrt(-1+c^2*x^2)/(c^3*e*sqrt(c^2*x^2))-1/42*b*x*(d+e*x^2)^(5/2)*sqrt(-1+c^2*x^2)/(c*e*sqrt(c^2*x^2))+1/560*b*(3*c^4*d^2-38*c^2*d*e-25*e^2)*x*sqrt(-1+c^2*x^2)*sqrt(d+e*x^2)/(c^5*e*sqrt(c^2*x^2))],
[x*(d+e*x^2)^(3/2)*(a+b*arcsec(c*x)),x,10,1/5*(d+e*x^2)^(5/2)*(a+b*arcsec(c*x))/e+1/5*b*c*d^(5/2)*x*arctan(sqrt(d+e*x^2)/(sqrt(d)*sqrt(-1+c^2*x^2)))/(e*sqrt(c^2*x^2))-1/40*b*(15*c^4*d^2+10*c^2*d*e+3*e^2)*x*arctanh(sqrt(e)*sqrt(-1+c^2*x^2)/(c*sqrt(d+e*x^2)))/(c^4*sqrt(e)*sqrt(c^2*x^2))-1/20*b*x*(d+e*x^2)^(3/2)*sqrt(-1+c^2*x^2)/(c*sqrt(c^2*x^2))-1/40*b*(7*c^2*d+3*e)*x*sqrt(-1+c^2*x^2)*sqrt(d+e*x^2)/(c^3*sqrt(c^2*x^2))],
[(d+e*x^2)^(3/2)*(a+b*arcsec(c*x))/x,x,0,Unintegrable((d+e*x^2)^(3/2)*(a+b*arcsec(c*x))/x,x)],
[(d+e*x^2)^(3/2)*(a+b*arcsec(c*x))/x^3,x,0,Unintegrable((d+e*x^2)^(3/2)*(a+b*arcsec(c*x))/x^3,x)],
[x^2*(d+e*x^2)^(3/2)*(a+b*arcsec(c*x)),x,0,Unintegrable(x^2*(d+e*x^2)^(3/2)*(a+b*arcsec(c*x)),x)],
[(d+e*x^2)^(3/2)*(a+b*arcsec(c*x)),x,0,Unintegrable((d+e*x^2)^(3/2)*(a+b*arcsec(c*x)),x)],
[(d+e*x^2)^(3/2)*(a+b*arcsec(c*x))/x^2,x,0,Unintegrable((d+e*x^2)^(3/2)*(a+b*arcsec(c*x))/x^2,x)],
[(d+e*x^2)^(3/2)*(a+b*arcsec(c*x))/x^4,x,0,Unintegrable((d+e*x^2)^(3/2)*(a+b*arcsec(c*x))/x^4,x)],
[(d+e*x^2)^(3/2)*(a+b*arcsec(c*x))/x^6,x,12,-1/5*(d+e*x^2)^(5/2)*(a+b*arcsec(c*x))/(d*x^5)+1/25*b*c*(d+e*x^2)^(3/2)*sqrt(-1+c^2*x^2)/(x^4*sqrt(c^2*x^2))+1/75*b*c*(8*c^4*d^2+23*c^2*d*e+23*e^2)*sqrt(-1+c^2*x^2)*sqrt(d+e*x^2)/(d*sqrt(c^2*x^2))+4/75*b*c*(c^2*d+2*e)*sqrt(-1+c^2*x^2)*sqrt(d+e*x^2)/(x^2*sqrt(c^2*x^2))-1/75*b*c^2*(8*c^4*d^2+23*c^2*d*e+23*e^2)*x*EllipticE(c*x,sqrt(-e/(c^2*d)))*sqrt(1-c^2*x^2)*sqrt(d+e*x^2)/(d*sqrt(c^2*x^2)*sqrt(-1+c^2*x^2)*sqrt(1+e*x^2/d))+1/75*b*(c^2*d+e)*(8*c^4*d^2+19*c^2*d*e+15*e^2)*x*EllipticF(c*x,sqrt(-e/(c^2*d)))*sqrt(1-c^2*x^2)*sqrt(1+e*x^2/d)/(d*sqrt(c^2*x^2)*sqrt(-1+c^2*x^2)*sqrt(d+e*x^2))],
[(d+e*x^2)^(3/2)*(a+b*arcsec(c*x))/x^8,x,13,-1/7*(d+e*x^2)^(5/2)*(a+b*arcsec(c*x))/(d*x^7)+2/35*e*(d+e*x^2)^(5/2)*(a+b*arcsec(c*x))/(d^2*x^5)+1/1225*b*c*(30*c^2*d+11*e)*(d+e*x^2)^(3/2)*sqrt(-1+c^2*x^2)/(d*x^4*sqrt(c^2*x^2))+1/49*b*c*(d+e*x^2)^(5/2)*sqrt(-1+c^2*x^2)/(d*x^6*sqrt(c^2*x^2))+1/3675*b*c*(240*c^6*d^3+528*c^4*d^2*e+193*c^2*d*e^2-247*e^3)*sqrt(-1+c^2*x^2)*sqrt(d+e*x^2)/(d^2*sqrt(c^2*x^2))+1/3675*b*c*(120*c^4*d^2+159*c^2*d*e-37*e^2)*sqrt(-1+c^2*x^2)*sqrt(d+e*x^2)/(d*x^2*sqrt(c^2*x^2))-1/3675*b*c^2*(240*c^6*d^3+528*c^4*d^2*e+193*c^2*d*e^2-247*e^3)*x*EllipticE(c*x,sqrt(-e/(c^2*d)))*sqrt(1-c^2*x^2)*sqrt(d+e*x^2)/(d^2*sqrt(c^2*x^2)*sqrt(-1+c^2*x^2)*sqrt(1+e*x^2/d))+2/3675*b*(c^2*d+e)*(120*c^6*d^3+204*c^4*d^2*e+17*c^2*d*e^2-105*e^3)*x*EllipticF(c*x,sqrt(-e/(c^2*d)))*sqrt(1-c^2*x^2)*sqrt(1+e*x^2/d)/(d^2*sqrt(c^2*x^2)*sqrt(-1+c^2*x^2)*sqrt(d+e*x^2))],

# p<0
[x^5*(a+b*arcsec(c*x))/sqrt(d+e*x^2),x,11,-2/3*d*(d+e*x^2)^(3/2)*(a+b*arcsec(c*x))/e^3+1/5*(d+e*x^2)^(5/2)*(a+b*arcsec(c*x))/e^3+8/15*b*c*d^(5/2)*x*arctan(sqrt(d+e*x^2)/(sqrt(d)*sqrt(-1+c^2*x^2)))/(e^3*sqrt(c^2*x^2))-1/120*b*(45*c^4*d^2-10*c^2*d*e+9*e^2)*x*arctanh(sqrt(e)*sqrt(-1+c^2*x^2)/(c*sqrt(d+e*x^2)))/(c^4*e^(5/2)*sqrt(c^2*x^2))-1/20*b*x*(d+e*x^2)^(3/2)*sqrt(-1+c^2*x^2)/(c*e^2*sqrt(c^2*x^2))+d^2*(a+b*arcsec(c*x))*sqrt(d+e*x^2)/e^3+1/120*b*(19*c^2*d-9*e)*x*sqrt(-1+c^2*x^2)*sqrt(d+e*x^2)/(c^3*e^2*sqrt(c^2*x^2))],
[x^3*(a+b*arcsec(c*x))/sqrt(d+e*x^2),x,10,1/3*(d+e*x^2)^(3/2)*(a+b*arcsec(c*x))/e^2-2/3*b*c*d^(3/2)*x*arctan(sqrt(d+e*x^2)/(sqrt(d)*sqrt(-1+c^2*x^2)))/(e^2*sqrt(c^2*x^2))+1/6*b*(3*c^2*d-e)*x*arctanh(sqrt(e)*sqrt(-1+c^2*x^2)/(c*sqrt(d+e*x^2)))/(c^2*e^(3/2)*sqrt(c^2*x^2))-d*(a+b*arcsec(c*x))*sqrt(d+e*x^2)/e^2-1/6*b*x*sqrt(-1+c^2*x^2)*sqrt(d+e*x^2)/(c*e*sqrt(c^2*x^2))],
[x*(a+b*arcsec(c*x))/sqrt(d+e*x^2),x,9,b*c*x*arctan(sqrt(d+e*x^2)/(sqrt(d)*sqrt(-1+c^2*x^2)))*sqrt(d)/(e*sqrt(c^2*x^2))-b*x*arctanh(sqrt(e)*sqrt(-1+c^2*x^2)/(c*sqrt(d+e*x^2)))/(sqrt(e)*sqrt(c^2*x^2))+(a+b*arcsec(c*x))*sqrt(d+e*x^2)/e],
[(a+b*arcsec(c*x))/(x*sqrt(d+e*x^2)),x,0,Unintegrable((a+b*arcsec(c*x))/(x*sqrt(d+e*x^2)),x)],
[(a+b*arcsec(c*x))/(x^3*sqrt(d+e*x^2)),x,0,Unintegrable((a+b*arcsec(c*x))/(x^3*sqrt(d+e*x^2)),x)],
[x^2*(a+b*arcsec(c*x))/sqrt(d+e*x^2),x,0,Unintegrable(x^2*(a+b*arcsec(c*x))/sqrt(d+e*x^2),x)],
[(a+b*arcsec(c*x))/sqrt(d+e*x^2),x,0,Unintegrable((a+b*arcsec(c*x))/sqrt(d+e*x^2),x)],
[(a+b*arcsec(c*x))/(x^2*sqrt(d+e*x^2)),x,11,-(a+b*arcsec(c*x))*sqrt(d+e*x^2)/(d*x)+b*c*sqrt(-1+c^2*x^2)*sqrt(d+e*x^2)/(d*sqrt(c^2*x^2))-b*c^2*x*EllipticE(c*x,sqrt(-e/(c^2*d)))*sqrt(1-c^2*x^2)*sqrt(d+e*x^2)/(d*sqrt(c^2*x^2)*sqrt(-1+c^2*x^2)*sqrt(1+e*x^2/d))+b*(c^2*d+e)*x*EllipticF(c*x,sqrt(-e/(c^2*d)))*sqrt(1-c^2*x^2)*sqrt(1+e*x^2/d)/(d*sqrt(c^2*x^2)*sqrt(-1+c^2*x^2)*sqrt(d+e*x^2))],
[(a+b*arcsec(c*x))/(x^4*sqrt(d+e*x^2)),x,11,-1/3*(a+b*arcsec(c*x))*sqrt(d+e*x^2)/(d*x^3)+2/3*e*(a+b*arcsec(c*x))*sqrt(d+e*x^2)/(d^2*x)+1/9*b*c*(2*c^2*d-5*e)*sqrt(-1+c^2*x^2)*sqrt(d+e*x^2)/(d^2*sqrt(c^2*x^2))+1/9*b*c*sqrt(-1+c^2*x^2)*sqrt(d+e*x^2)/(d*x^2*sqrt(c^2*x^2))-1/9*b*c^2*(2*c^2*d-5*e)*x*EllipticE(c*x,sqrt(-e/(c^2*d)))*sqrt(1-c^2*x^2)*sqrt(d+e*x^2)/(d^2*sqrt(c^2*x^2)*sqrt(-1+c^2*x^2)*sqrt(1+e*x^2/d))+2/9*b*(c^2*d-3*e)*(c^2*d+e)*x*EllipticF(c*x,sqrt(-e/(c^2*d)))*sqrt(1-c^2*x^2)*sqrt(1+e*x^2/d)/(d^2*sqrt(c^2*x^2)*sqrt(-1+c^2*x^2)*sqrt(d+e*x^2))],
[(a+b*arcsec(c*x))/(x^6*sqrt(d+e*x^2)),x,32,-1/5*(a+b*arcsec(c*x))*sqrt(d+e*x^2)/(d*x^5)+4/15*e*(a+b*arcsec(c*x))*sqrt(d+e*x^2)/(d^2*x^3)-8/15*e^2*(a+b*arcsec(c*x))*sqrt(d+e*x^2)/(d^3*x)+8/15*b*c*e^2*sqrt(-1+c^2*x^2)*sqrt(d+e*x^2)/(d^3*sqrt(c^2*x^2))-4/45*b*c*e*(2*c^2*d+e)*sqrt(-1+c^2*x^2)*sqrt(d+e*x^2)/(d^3*sqrt(c^2*x^2))+1/75*b*c*(8*c^4*d^2+3*c^2*d*e-2*e^2)*sqrt(-1+c^2*x^2)*sqrt(d+e*x^2)/(d^3*sqrt(c^2*x^2))+1/25*b*c*sqrt(-1+c^2*x^2)*sqrt(d+e*x^2)/(d*x^4*sqrt(c^2*x^2))-4/45*b*c*e*sqrt(-1+c^2*x^2)*sqrt(d+e*x^2)/(d^2*x^2*sqrt(c^2*x^2))+1/75*b*c*(4*c^2*d+e)*sqrt(-1+c^2*x^2)*sqrt(d+e*x^2)/(d^2*x^2*sqrt(c^2*x^2))-8/15*b*c^2*e^2*x*EllipticE(c*x,sqrt(-e/(c^2*d)))*sqrt(1-c^2*x^2)*sqrt(d+e*x^2)/(d^3*sqrt(c^2*x^2)*sqrt(-1+c^2*x^2)*sqrt(1+e*x^2/d))+4/45*b*c^2*e*(2*c^2*d+e)*x*EllipticE(c*x,sqrt(-e/(c^2*d)))*sqrt(1-c^2*x^2)*sqrt(d+e*x^2)/(d^3*sqrt(c^2*x^2)*sqrt(-1+c^2*x^2)*sqrt(1+e*x^2/d))-1/75*b*c^2*(8*c^4*d^2+3*c^2*d*e-2*e^2)*x*EllipticE(c*x,sqrt(-e/(c^2*d)))*sqrt(1-c^2*x^2)*sqrt(d+e*x^2)/(d^3*sqrt(c^2*x^2)*sqrt(-1+c^2*x^2)*sqrt(1+e*x^2/d))+1/75*b*c^2*(8*c^2*d-e)*(c^2*d+e)*x*EllipticF(c*x,sqrt(-e/(c^2*d)))*sqrt(1-c^2*x^2)*sqrt(1+e*x^2/d)/(d^2*sqrt(c^2*x^2)*sqrt(-1+c^2*x^2)*sqrt(d+e*x^2))-8/45*b*c^2*e*(c^2*d+e)*x*EllipticF(c*x,sqrt(-e/(c^2*d)))*sqrt(1-c^2*x^2)*sqrt(1+e*x^2/d)/(d^2*sqrt(c^2*x^2)*sqrt(-1+c^2*x^2)*sqrt(d+e*x^2))+8/15*b*e^2*(c^2*d+e)*x*EllipticF(c*x,sqrt(-e/(c^2*d)))*sqrt(1-c^2*x^2)*sqrt(1+e*x^2/d)/(d^3*sqrt(c^2*x^2)*sqrt(-1+c^2*x^2)*sqrt(d+e*x^2))],
[x^5*(a+b*arcsec(c*x))/(d+e*x^2)^(3/2),x,10,1/3*(d+e*x^2)^(3/2)*(a+b*arcsec(c*x))/e^3-8/3*b*c*d^(3/2)*x*arctan(sqrt(d+e*x^2)/(sqrt(d)*sqrt(-1+c^2*x^2)))/(e^3*sqrt(c^2*x^2))+1/6*b*(9*c^2*d-e)*x*arctanh(sqrt(e)*sqrt(-1+c^2*x^2)/(c*sqrt(d+e*x^2)))/(c^2*e^(5/2)*sqrt(c^2*x^2))-d^2*(a+b*arcsec(c*x))/(e^3*sqrt(d+e*x^2))-2*d*(a+b*arcsec(c*x))*sqrt(d+e*x^2)/e^3-1/6*b*x*sqrt(-1+c^2*x^2)*sqrt(d+e*x^2)/(c*e^2*sqrt(c^2*x^2))],
[x^3*(a+b*arcsec(c*x))/(d+e*x^2)^(3/2),x,9,-b*x*arctanh(sqrt(e)*sqrt(-1+c^2*x^2)/(c*sqrt(d+e*x^2)))/(e^(3/2)*sqrt(c^2*x^2))+2*b*c*x*arctan(sqrt(d+e*x^2)/(sqrt(d)*sqrt(-1+c^2*x^2)))*sqrt(d)/(e^2*sqrt(c^2*x^2))+d*(a+b*arcsec(c*x))/(e^2*sqrt(d+e*x^2))+(a+b*arcsec(c*x))*sqrt(d+e*x^2)/e^2],
[x*(a+b*arcsec(c*x))/(d+e*x^2)^(3/2),x,4,-b*c*x*arctan(sqrt(d+e*x^2)/(sqrt(d)*sqrt(-1+c^2*x^2)))/(e*sqrt(d)*sqrt(c^2*x^2))+(-a-b*arcsec(c*x))/(e*sqrt(d+e*x^2))],
[(a+b*arcsec(c*x))/(x*(d+e*x^2)^(3/2)),x,0,Unintegrable((a+b*arcsec(c*x))/(x*(d+e*x^2)^(3/2)),x)],
[(a+b*arcsec(c*x))/(x^3*(d+e*x^2)^(3/2)),x,0,Unintegrable((a+b*arcsec(c*x))/(x^3*(d+e*x^2)^(3/2)),x)],
[x^4*(a+b*arcsec(c*x))/(d+e*x^2)^(3/2),x,0,Unintegrable(x^4*(a+b*arcsec(c*x))/(d+e*x^2)^(3/2),x)],
[x^2*(a+b*arcsec(c*x))/(d+e*x^2)^(3/2),x,0,Unintegrable(x^2*(a+b*arcsec(c*x))/(d+e*x^2)^(3/2),x)],
[(a+b*arcsec(c*x))/(d+e*x^2)^(3/2),x,5,x*(a+b*arcsec(c*x))/(d*sqrt(d+e*x^2))-b*x*EllipticF(c*x,sqrt(-e/(c^2*d)))*sqrt(1-c^2*x^2)*sqrt(1+e*x^2/d)/(d*sqrt(c^2*x^2)*sqrt(-1+c^2*x^2)*sqrt(d+e*x^2))],
[(a+b*arcsec(c*x))/(x^2*(d+e*x^2)^(3/2)),x,10,(-a-b*arcsec(c*x))/(d*x*sqrt(d+e*x^2))-2*e*x*(a+b*arcsec(c*x))/(d^2*sqrt(d+e*x^2))+b*c*sqrt(-1+c^2*x^2)*sqrt(d+e*x^2)/(d^2*sqrt(c^2*x^2))-b*c^2*x*EllipticE(c*x,sqrt(-e/(c^2*d)))*sqrt(1-c^2*x^2)*sqrt(d+e*x^2)/(d^2*sqrt(c^2*x^2)*sqrt(-1+c^2*x^2)*sqrt(1+e*x^2/d))+b*(c^2*d+2*e)*x*EllipticF(c*x,sqrt(-e/(c^2*d)))*sqrt(1-c^2*x^2)*sqrt(1+e*x^2/d)/(d^2*sqrt(c^2*x^2)*sqrt(-1+c^2*x^2)*sqrt(d+e*x^2))],
[(a+b*arcsec(c*x))/(x^4*(d+e*x^2)^(3/2)),x,25,1/3*(-a-b*arcsec(c*x))/(d*x^3*sqrt(d+e*x^2))+4/3*e*(a+b*arcsec(c*x))/(d^2*x*sqrt(d+e*x^2))+8/3*e^2*x*(a+b*arcsec(c*x))/(d^3*sqrt(d+e*x^2))+2/9*b*c*(c^2*d-e)*sqrt(-1+c^2*x^2)*sqrt(d+e*x^2)/(d^3*sqrt(c^2*x^2))-4/3*b*c*e*sqrt(-1+c^2*x^2)*sqrt(d+e*x^2)/(d^3*sqrt(c^2*x^2))+1/9*b*c*sqrt(-1+c^2*x^2)*sqrt(d+e*x^2)/(d^2*x^2*sqrt(c^2*x^2))-2/9*b*c^2*(c^2*d-e)*x*EllipticE(c*x,sqrt(-e/(c^2*d)))*sqrt(1-c^2*x^2)*sqrt(d+e*x^2)/(d^3*sqrt(c^2*x^2)*sqrt(-1+c^2*x^2)*sqrt(1+e*x^2/d))+4/3*b*c^2*e*x*EllipticE(c*x,sqrt(-e/(c^2*d)))*sqrt(1-c^2*x^2)*sqrt(d+e*x^2)/(d^3*sqrt(c^2*x^2)*sqrt(-1+c^2*x^2)*sqrt(1+e*x^2/d))+1/9*b*c^2*(2*c^2*d-e)*x*EllipticF(c*x,sqrt(-e/(c^2*d)))*sqrt(1-c^2*x^2)*sqrt(1+e*x^2/d)/(d^2*sqrt(c^2*x^2)*sqrt(-1+c^2*x^2)*sqrt(d+e*x^2))-4/3*b*c^2*e*x*EllipticF(c*x,sqrt(-e/(c^2*d)))*sqrt(1-c^2*x^2)*sqrt(1+e*x^2/d)/(d^2*sqrt(c^2*x^2)*sqrt(-1+c^2*x^2)*sqrt(d+e*x^2))-8/3*b*e^2*x*EllipticF(c*x,sqrt(-e/(c^2*d)))*sqrt(1-c^2*x^2)*sqrt(1+e*x^2/d)/(d^3*sqrt(c^2*x^2)*sqrt(-1+c^2*x^2)*sqrt(d+e*x^2))],
[x^5*(a+b*arcsec(c*x))/(d+e*x^2)^(5/2),x,10,-1/3*d^2*(a+b*arcsec(c*x))/(e^3*(d+e*x^2)^(3/2))-b*x*arctanh(sqrt(e)*sqrt(-1+c^2*x^2)/(c*sqrt(d+e*x^2)))/(e^(5/2)*sqrt(c^2*x^2))+8/3*b*c*x*arctan(sqrt(d+e*x^2)/(sqrt(d)*sqrt(-1+c^2*x^2)))*sqrt(d)/(e^3*sqrt(c^2*x^2))+2*d*(a+b*arcsec(c*x))/(e^3*sqrt(d+e*x^2))-1/3*b*c*d*x*sqrt(-1+c^2*x^2)/(e^2*(c^2*d+e)*sqrt(c^2*x^2)*sqrt(d+e*x^2))+(a+b*arcsec(c*x))*sqrt(d+e*x^2)/e^3],
[x^3*(a+b*arcsec(c*x))/(d+e*x^2)^(5/2),x,7,1/3*d*(a+b*arcsec(c*x))/(e^2*(d+e*x^2)^(3/2))-2/3*b*c*x*arctan(sqrt(d+e*x^2)/(sqrt(d)*sqrt(-1+c^2*x^2)))/(e^2*sqrt(d)*sqrt(c^2*x^2))+(-a-b*arcsec(c*x))/(e^2*sqrt(d+e*x^2))+1/3*b*c*x*sqrt(-1+c^2*x^2)/(e*(c^2*d+e)*sqrt(c^2*x^2)*sqrt(d+e*x^2))],
[x*(a+b*arcsec(c*x))/(d+e*x^2)^(5/2),x,5,1/3*(-a-b*arcsec(c*x))/(e*(d+e*x^2)^(3/2))-1/3*b*c*x*arctan(sqrt(d+e*x^2)/(sqrt(d)*sqrt(-1+c^2*x^2)))/(d^(3/2)*e*sqrt(c^2*x^2))-1/3*b*c*x*sqrt(-1+c^2*x^2)/(d*(c^2*d+e)*sqrt(c^2*x^2)*sqrt(d+e*x^2))],
[(a+b*arcsec(c*x))/(x*(d+e*x^2)^(5/2)),x,0,Unintegrable((a+b*arcsec(c*x))/(x*(d+e*x^2)^(5/2)),x)],
[(a+b*arcsec(c*x))/(x^3*(d+e*x^2)^(5/2)),x,0,Unintegrable((a+b*arcsec(c*x))/(x^3*(d+e*x^2)^(5/2)),x)],
[x^6*(a+b*arcsec(c*x))/(d+e*x^2)^(5/2),x,0,Unintegrable(x^6*(a+b*arcsec(c*x))/(d+e*x^2)^(5/2),x)],
[x^4*(a+b*arcsec(c*x))/(d+e*x^2)^(5/2),x,0,Unintegrable(x^4*(a+b*arcsec(c*x))/(d+e*x^2)^(5/2),x)],
[x^2*(a+b*arcsec(c*x))/(d+e*x^2)^(5/2),x,10,1/3*x^3*(a+b*arcsec(c*x))/(d*(d+e*x^2)^(3/2))-1/3*b*c*x^2*sqrt(-1+c^2*x^2)/(d*(c^2*d+e)*sqrt(c^2*x^2)*sqrt(d+e*x^2))+1/3*b*c^2*x*EllipticE(c*x,sqrt(-e/(c^2*d)))*sqrt(1-c^2*x^2)*sqrt(d+e*x^2)/(d*e*(c^2*d+e)*sqrt(c^2*x^2)*sqrt(-1+c^2*x^2)*sqrt(1+e*x^2/d))-1/3*b*x*EllipticF(c*x,sqrt(-e/(c^2*d)))*sqrt(1-c^2*x^2)*sqrt(1+e*x^2/d)/(d*e*sqrt(c^2*x^2)*sqrt(-1+c^2*x^2)*sqrt(d+e*x^2))],
[(a+b*arcsec(c*x))/(d+e*x^2)^(5/2),x,10,1/3*x*(a+b*arcsec(c*x))/(d*(d+e*x^2)^(3/2))+2/3*x*(a+b*arcsec(c*x))/(d^2*sqrt(d+e*x^2))+1/3*b*c*e*x^2*sqrt(-1+c^2*x^2)/(d^2*(c^2*d+e)*sqrt(c^2*x^2)*sqrt(d+e*x^2))-1/3*b*c^2*x*EllipticE(c*x,sqrt(-e/(c^2*d)))*sqrt(1-c^2*x^2)*sqrt(d+e*x^2)/(d^2*(c^2*d+e)*sqrt(c^2*x^2)*sqrt(-1+c^2*x^2)*sqrt(1+e*x^2/d))-2/3*b*x*EllipticF(c*x,sqrt(-e/(c^2*d)))*sqrt(1-c^2*x^2)*sqrt(1+e*x^2/d)/(d^2*sqrt(c^2*x^2)*sqrt(-1+c^2*x^2)*sqrt(d+e*x^2))],
[(a+b*arcsec(c*x))/(x^2*(d+e*x^2)^(5/2)),x,26,(-a-b*arcsec(c*x))/(d*x*(d+e*x^2)^(3/2))-4/3*e*x*(a+b*arcsec(c*x))/(d^2*(d+e*x^2)^(3/2))-8/3*e*x*(a+b*arcsec(c*x))/(d^3*sqrt(d+e*x^2))-b*c*e*sqrt(-1+c^2*x^2)/(d^2*(c^2*d+e)*sqrt(c^2*x^2)*sqrt(d+e*x^2))-4/3*b*c*e^2*x^2*sqrt(-1+c^2*x^2)/(d^3*(c^2*d+e)*sqrt(c^2*x^2)*sqrt(d+e*x^2))+b*c*(c^2*d+2*e)*sqrt(-1+c^2*x^2)*sqrt(d+e*x^2)/(d^3*(c^2*d+e)*sqrt(c^2*x^2))+4/3*b*c^2*e*x*EllipticE(c*x,sqrt(-e/(c^2*d)))*sqrt(1-c^2*x^2)*sqrt(d+e*x^2)/(d^3*(c^2*d+e)*sqrt(c^2*x^2)*sqrt(-1+c^2*x^2)*sqrt(1+e*x^2/d))-b*c^2*(c^2*d+2*e)*x*EllipticE(c*x,sqrt(-e/(c^2*d)))*sqrt(1-c^2*x^2)*sqrt(d+e*x^2)/(d^3*(c^2*d+e)*sqrt(c^2*x^2)*sqrt(-1+c^2*x^2)*sqrt(1+e*x^2/d))+b*c^2*x*EllipticF(c*x,sqrt(-e/(c^2*d)))*sqrt(1-c^2*x^2)*sqrt(1+e*x^2/d)/(d^2*sqrt(c^2*x^2)*sqrt(-1+c^2*x^2)*sqrt(d+e*x^2))+8/3*b*e*x*EllipticF(c*x,sqrt(-e/(c^2*d)))*sqrt(1-c^2*x^2)*sqrt(1+e*x^2/d)/(d^3*sqrt(c^2*x^2)*sqrt(-1+c^2*x^2)*sqrt(d+e*x^2))],

# Integrands of the form (f x)^m (d+e x^2)^p (a+b ArcSec[c x]) when m symbolic
[(f*x)^m*(d+e*x^2)^3*(a+b*arcsec(c*x)),x,6,d^3*(f*x)^(1+m)*(a+b*arcsec(c*x))/(f*(1+m))+3*d^2*e*(f*x)^(3+m)*(a+b*arcsec(c*x))/(f^3*(3+m))+3*d*e^2*(f*x)^(5+m)*(a+b*arcsec(c*x))/(f^5*(5+m))+e^3*(f*x)^(7+m)*(a+b*arcsec(c*x))/(f^7*(7+m))-b*(c^6*d^3*(2+m)*(4+m)*(6+m)/(1+m)+e*(1+m)*(e^2*(15+8*m+m^2)^2+3*c^2*d*e*(3+m)^2*(42+13*m+m^2)+3*c^4*d^2*(840+638*m+179*m^2+22*m^3+m^4))/(105+71*m+15*m^2+m^3))*x*(f*x)^(1+m)*hypergeom([1/2,1/2*(1+m)],[1/2*(3+m)],c^2*x^2)*sqrt(1-c^2*x^2)/(c^5*f*(1+m)*(2+m)*(4+m)*(6+m)*sqrt(c^2*x^2)*sqrt(-1+c^2*x^2))-b*e*(e^2*(15+8*m+m^2)^2+3*c^2*d*e*(3+m)^2*(42+13*m+m^2)+3*c^4*d^2*(840+638*m+179*m^2+22*m^3+m^4))*x*(f*x)^(1+m)*sqrt(-1+c^2*x^2)/(c^5*f*(6+m)*(8+6*m+m^2)*(105+71*m+15*m^2+m^3)*sqrt(c^2*x^2))-b*e^2*(e*(5+m)^2+3*c^2*d*(42+13*m+m^2))*x*(f*x)^(3+m)*sqrt(-1+c^2*x^2)/(c^3*f^3*(4+m)*(5+m)*(6+m)*(7+m)*sqrt(c^2*x^2))-b*e^3*x*(f*x)^(5+m)*sqrt(-1+c^2*x^2)/(c*f^5*(6+m)*(7+m)*sqrt(c^2*x^2))],
[(f*x)^m*(d+e*x^2)^2*(a+b*arcsec(c*x)),x,6,d^2*(f*x)^(1+m)*(a+b*arcsec(c*x))/(f*(1+m))+2*d*e*(f*x)^(3+m)*(a+b*arcsec(c*x))/(f^3*(3+m))+e^2*(f*x)^(5+m)*(a+b*arcsec(c*x))/(f^5*(5+m))-b*(c^4*d^2*(2+m)*(3+m)*(4+m)*(5+m)+e*(1+m)^2*(e*(3+m)^2+2*c^2*d*(20+9*m+m^2)))*x*(f*x)^(1+m)*hypergeom([1/2,1/2*(1+m)],[1/2*(3+m)],c^2*x^2)*sqrt(1-c^2*x^2)/(c^3*f*(1+m)^2*(2+m)*(3+m)*(4+m)*(5+m)*sqrt(c^2*x^2)*sqrt(-1+c^2*x^2))-b*e*(e*(3+m)^2+2*c^2*d*(20+9*m+m^2))*x*(f*x)^(1+m)*sqrt(-1+c^2*x^2)/(c^3*f*(4+m)*(5+m)*(6+5*m+m^2)*sqrt(c^2*x^2))-b*e^2*x*(f*x)^(3+m)*sqrt(-1+c^2*x^2)/(c*f^3*(4+m)*(5+m)*sqrt(c^2*x^2))],
[(f*x)^m*(d+e*x^2)*(a+b*arcsec(c*x)),x,5,d*x^(1+m)*(a+b*arcsec(c*x))/(1+m)+e*x^(3+m)*(a+b*arcsec(c*x))/(3+m)-b*e*x^(2+m)*sqrt(-1+c^2*x^2)/(c*(6+5*m+m^2)*sqrt(c^2*x^2))+b*(e*(1+m)^2+c^2*d*(2+m)*(3+m))*x^(2+m)*hypergeom([1,1/2*(2+m)],[1/2*(3+m)],c^2*x^2)*sqrt(-1+c^2*x^2)/(c*(1+m)^2*(2+m)*(3+m)*sqrt(c^2*x^2)),d*(f*x)^(1+m)*(a+b*arcsec(c*x))/(f*(1+m))+e*(f*x)^(3+m)*(a+b*arcsec(c*x))/(f^3*(3+m))-b*(e*(1+m)^2+c^2*d*(2+m)*(3+m))*x*(f*x)^(1+m)*hypergeom([1/2,1/2*(1+m)],[1/2*(3+m)],c^2*x^2)*sqrt(1-c^2*x^2)/(c*f*(1+m)^2*(2+m)*(3+m)*sqrt(c^2*x^2)*sqrt(-1+c^2*x^2))-b*e*x*(f*x)^(1+m)*sqrt(-1+c^2*x^2)/(c*f*(6+5*m+m^2)*sqrt(c^2*x^2))],
[(f*x)^m*(a+b*arcsec(c*x))/(d+e*x^2),x,0,Unintegrable((f*x)^m*(a+b*arcsec(c*x))/(d+e*x^2),x)],
[(f*x)^m*(a+b*arcsec(c*x))/(d+e*x^2)^2,x,0,Unintegrable((f*x)^m*(a+b*arcsec(c*x))/(d+e*x^2)^2,x)],
[(f*x)^m*(d+e*x^2)^(3/2)*(a+b*arcsec(c*x)),x,0,Unintegrable((f*x)^m*(d+e*x^2)^(3/2)*(a+b*arcsec(c*x)),x)],
[(f*x)^m*(d+e*x^2)^(1/2)*(a+b*arcsec(c*x)),x,0,Unintegrable((f*x)^m*(a+b*arcsec(c*x))*sqrt(d+e*x^2),x)],
[(f*x)^m*(a+b*arcsec(c*x))/(d+e*x^2)^(1/2),x,0,Unintegrable((f*x)^m*(a+b*arcsec(c*x))/sqrt(d+e*x^2),x)],
[(f*x)^m*(a+b*arcsec(c*x))/(d+e*x^2)^(3/2),x,0,Unintegrable((f*x)^m*(a+b*arcsec(c*x))/(d+e*x^2)^(3/2),x)],

# Integrands of the form (f x)^m (d+e x^4)^p (a+b ArcSec[c x])^n

# Integrands of the form x^m (d+e x^4)^(p/2) (a+b ArcSec[c x])

# p>0

# p<0
[x^11*(a+b*arcsec(c*x))/sqrt(1-c^4*x^4),x,16,1/3*(1-c^4*x^4)^(3/2)*(a+b*arcsec(c*x))/c^12-1/10*(1-c^4*x^4)^(5/2)*(a+b*arcsec(c*x))/c^12-7/90*b*(1+c^2*x^2)^(3/2)*sqrt(1-c^2*x^2)/(c^13*x*sqrt(1+(-1)/(c^2*x^2)))+13/150*b*(1+c^2*x^2)^(5/2)*sqrt(1-c^2*x^2)/(c^13*x*sqrt(1+(-1)/(c^2*x^2)))-3/70*b*(1+c^2*x^2)^(7/2)*sqrt(1-c^2*x^2)/(c^13*x*sqrt(1+(-1)/(c^2*x^2)))+1/90*b*(1+c^2*x^2)^(9/2)*sqrt(1-c^2*x^2)/(c^13*x*sqrt(1+(-1)/(c^2*x^2)))-4/15*b*arctanh(sqrt(1+c^2*x^2))*sqrt(1-c^2*x^2)/(c^13*x*sqrt(1+(-1)/(c^2*x^2)))+4/15*b*sqrt(1-c^2*x^2)*sqrt(1+c^2*x^2)/(c^13*x*sqrt(1+(-1)/(c^2*x^2)))-1/2*(a+b*arcsec(c*x))*sqrt(1-c^4*x^4)/c^12],
[x^7*(a+b*arcsec(c*x))/sqrt(1-c^4*x^4),x,13,1/6*(1-c^4*x^4)^(3/2)*(a+b*arcsec(c*x))/c^8-1/18*b*(1+c^2*x^2)^(3/2)*sqrt(1-c^2*x^2)/(c^9*x*sqrt(1+(-1)/(c^2*x^2)))+1/30*b*(1+c^2*x^2)^(5/2)*sqrt(1-c^2*x^2)/(c^9*x*sqrt(1+(-1)/(c^2*x^2)))-1/3*b*arctanh(sqrt(1+c^2*x^2))*sqrt(1-c^2*x^2)/(c^9*x*sqrt(1+(-1)/(c^2*x^2)))+1/3*b*sqrt(1-c^2*x^2)*sqrt(1+c^2*x^2)/(c^9*x*sqrt(1+(-1)/(c^2*x^2)))-1/2*(a+b*arcsec(c*x))*sqrt(1-c^4*x^4)/c^8],
[x^3*(a+b*arcsec(c*x))/sqrt(1-c^4*x^4),x,8,-1/2*b*x*arctan(sqrt(1-c^4*x^4)/sqrt(-1+c^2*x^2))/(c^3*sqrt(c^2*x^2))-1/2*(a+b*arcsec(c*x))*sqrt(1-c^4*x^4)/c^4+1/2*b*x*sqrt(1-c^4*x^4)/(c^3*sqrt(c^2*x^2)*sqrt(-1+c^2*x^2)),-1/2*b*arctanh(sqrt(1+c^2*x^2))*sqrt(1-c^2*x^2)/(c^5*x*sqrt(1+(-1)/(c^2*x^2)))+1/2*b*sqrt(1-c^2*x^2)*sqrt(1+c^2*x^2)/(c^5*x*sqrt(1+(-1)/(c^2*x^2)))-1/2*(a+b*arcsec(c*x))*sqrt(1-c^4*x^4)/c^4],
[(a+b*arcsec(c*x))/(x*sqrt(1-c^4*x^4)),x,0,Unintegrable((a+b*arcsec(c*x))/(x*sqrt(1-c^4*x^4)),x)],
[(a+b*arcsec(c*x))/(x^5*sqrt(1-c^4*x^4)),x,0,Unintegrable((a+b*arcsec(c*x))/(x^5*sqrt(1-c^4*x^4)),x)]]:

# Integrands of the form u (a+b ArcSec[c x])^n
