# Maple integration test file: "5 Inverse trig functions\5.1 Inverse sine\5.1.5 Inverse sine functions.txt"

lst:=[

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

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

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

# n>0
[(d+e*x)^3*(a+b*arcsin(c*x)),x,5,-1/32*b*(8*c^4*d^4+24*c^2*d^2*e^2+3*e^4)*arcsin(c*x)/(c^4*e)+1/4*(d+e*x)^4*(a+b*arcsin(c*x))/e+7/48*b*d*(d+e*x)^2*sqrt(1-c^2*x^2)/c+1/16*b*(d+e*x)^3*sqrt(1-c^2*x^2)/c+1/96*b*(4*d*(19*c^2*d^2+16*e^2)+e*(26*c^2*d^2+9*e^2)*x)*sqrt(1-c^2*x^2)/c^3],
[(d+e*x)^2*(a+b*arcsin(c*x)),x,4,-1/6*b*d*(2*d^2+3*e^2/c^2)*arcsin(c*x)/e+1/3*(d+e*x)^3*(a+b*arcsin(c*x))/e+1/9*b*(d+e*x)^2*sqrt(1-c^2*x^2)/c+1/18*b*(4*(4*c^2*d^2+e^2)+5*c^2*d*e*x)*sqrt(1-c^2*x^2)/c^3],
[(d+e*x)*(a+b*arcsin(c*x)),x,4,-1/4*b*(2*d^2+e^2/c^2)*arcsin(c*x)/e+1/2*(d+e*x)^2*(a+b*arcsin(c*x))/e+3/4*b*d*sqrt(1-c^2*x^2)/c+1/4*b*(d+e*x)*sqrt(1-c^2*x^2)/c],
[a+b*arcsin(c*x),x,3,a*x+b*x*arcsin(c*x)+b*sqrt(1-c^2*x^2)/c],
[(a+b*arcsin(c*x))/(d+e*x),x,8,-1/2*I*(a+b*arcsin(c*x))^2/(b*e)+(a+b*arcsin(c*x))*log(1-I*e*exp(I*arcsin(c*x))/(c*d-sqrt(c^2*d^2-e^2)))/e+(a+b*arcsin(c*x))*log(1-I*e*exp(I*arcsin(c*x))/(c*d+sqrt(c^2*d^2-e^2)))/e-I*b*polylog(2,I*e*exp(I*arcsin(c*x))/(c*d-sqrt(c^2*d^2-e^2)))/e-I*b*polylog(2,I*e*exp(I*arcsin(c*x))/(c*d+sqrt(c^2*d^2-e^2)))/e],
[(a+b*arcsin(c*x))/(d+e*x)^2,x,3,(-a-b*arcsin(c*x))/(e*(d+e*x))+b*c*arctan((e+c^2*d*x)/(sqrt(c^2*d^2-e^2)*sqrt(1-c^2*x^2)))/(e*sqrt(c^2*d^2-e^2))],
[(a+b*arcsin(c*x))/(d+e*x)^3,x,4,1/2*(-a-b*arcsin(c*x))/(e*(d+e*x)^2)+1/2*b*c^3*d*arctan((e+c^2*d*x)/(sqrt(c^2*d^2-e^2)*sqrt(1-c^2*x^2)))/(e*(c^2*d^2-e^2)^(3/2))+1/2*b*c*sqrt(1-c^2*x^2)/((c^2*d^2-e^2)*(d+e*x))],
[(a+b*arcsin(c*x))/(d+e*x)^4,x,5,1/3*(-a-b*arcsin(c*x))/(e*(d+e*x)^3)+1/6*b*c^3*(2*c^2*d^2+e^2)*arctan((e+c^2*d*x)/(sqrt(c^2*d^2-e^2)*sqrt(1-c^2*x^2)))/(e*(c^2*d^2-e^2)^(5/2))+1/6*b*c*sqrt(1-c^2*x^2)/((c^2*d^2-e^2)*(d+e*x)^2)+1/2*b*c^3*d*sqrt(1-c^2*x^2)/((c^2*d^2-e^2)^2*(d+e*x))],
[(d+e*x)^3*(a+b*arcsin(c*x))^2,x,18,-2*b^2*d^3*x-4/3*b^2*d*e^2*x/c^2-3/4*b^2*d^2*e*x^2-3/32*b^2*e^3*x^2/c^2-2/9*b^2*d*e^2*x^3-1/32*b^2*e^3*x^4-1/4*d^4*(a+b*arcsin(c*x))^2/e-3/4*d^2*e*(a+b*arcsin(c*x))^2/c^2-3/32*e^3*(a+b*arcsin(c*x))^2/c^4+1/4*(d+e*x)^4*(a+b*arcsin(c*x))^2/e+2*b*d^3*(a+b*arcsin(c*x))*sqrt(1-c^2*x^2)/c+4/3*b*d*e^2*(a+b*arcsin(c*x))*sqrt(1-c^2*x^2)/c^3+3/2*b*d^2*e*x*(a+b*arcsin(c*x))*sqrt(1-c^2*x^2)/c+3/16*b*e^3*x*(a+b*arcsin(c*x))*sqrt(1-c^2*x^2)/c^3+2/3*b*d*e^2*x^2*(a+b*arcsin(c*x))*sqrt(1-c^2*x^2)/c+1/8*b*e^3*x^3*(a+b*arcsin(c*x))*sqrt(1-c^2*x^2)/c],
[(d+e*x)^2*(a+b*arcsin(c*x))^2,x,13,-2*b^2*d^2*x-4/9*b^2*e^2*x/c^2-1/2*b^2*d*e*x^2-2/27*b^2*e^2*x^3-1/3*d^3*(a+b*arcsin(c*x))^2/e-1/2*d*e*(a+b*arcsin(c*x))^2/c^2+1/3*(d+e*x)^3*(a+b*arcsin(c*x))^2/e+2*b*d^2*(a+b*arcsin(c*x))*sqrt(1-c^2*x^2)/c+4/9*b*e^2*(a+b*arcsin(c*x))*sqrt(1-c^2*x^2)/c^3+b*d*e*x*(a+b*arcsin(c*x))*sqrt(1-c^2*x^2)/c+2/9*b*e^2*x^2*(a+b*arcsin(c*x))*sqrt(1-c^2*x^2)/c],
[(d+e*x)*(a+b*arcsin(c*x))^2,x,9,-2*b^2*d*x-1/4*b^2*e*x^2-1/2*d^2*(a+b*arcsin(c*x))^2/e-1/4*e*(a+b*arcsin(c*x))^2/c^2+1/2*(d+e*x)^2*(a+b*arcsin(c*x))^2/e+2*b*d*(a+b*arcsin(c*x))*sqrt(1-c^2*x^2)/c+1/2*b*e*x*(a+b*arcsin(c*x))*sqrt(1-c^2*x^2)/c],
[(a+b*arcsin(c*x))^2,x,3,-2*b^2*x+x*(a+b*arcsin(c*x))^2+2*b*(a+b*arcsin(c*x))*sqrt(1-c^2*x^2)/c],
[(a+b*arcsin(c*x))^2/(d+e*x),x,10,-1/3*I*(a+b*arcsin(c*x))^3/(b*e)+(a+b*arcsin(c*x))^2*log(1-I*e*exp(I*arcsin(c*x))/(c*d-sqrt(c^2*d^2-e^2)))/e+(a+b*arcsin(c*x))^2*log(1-I*e*exp(I*arcsin(c*x))/(c*d+sqrt(c^2*d^2-e^2)))/e-2*I*b*(a+b*arcsin(c*x))*polylog(2,I*e*exp(I*arcsin(c*x))/(c*d-sqrt(c^2*d^2-e^2)))/e-2*I*b*(a+b*arcsin(c*x))*polylog(2,I*e*exp(I*arcsin(c*x))/(c*d+sqrt(c^2*d^2-e^2)))/e+2*b^2*polylog(3,I*e*exp(I*arcsin(c*x))/(c*d-sqrt(c^2*d^2-e^2)))/e+2*b^2*polylog(3,I*e*exp(I*arcsin(c*x))/(c*d+sqrt(c^2*d^2-e^2)))/e],
[(a+b*arcsin(c*x))^2/(d+e*x)^2,x,10,-(a+b*arcsin(c*x))^2/(e*(d+e*x))-2*I*b*c*(a+b*arcsin(c*x))*log(1-I*e*exp(I*arcsin(c*x))/(c*d-sqrt(c^2*d^2-e^2)))/(e*sqrt(c^2*d^2-e^2))+2*I*b*c*(a+b*arcsin(c*x))*log(1-I*e*exp(I*arcsin(c*x))/(c*d+sqrt(c^2*d^2-e^2)))/(e*sqrt(c^2*d^2-e^2))-2*b^2*c*polylog(2,I*e*exp(I*arcsin(c*x))/(c*d-sqrt(c^2*d^2-e^2)))/(e*sqrt(c^2*d^2-e^2))+2*b^2*c*polylog(2,I*e*exp(I*arcsin(c*x))/(c*d+sqrt(c^2*d^2-e^2)))/(e*sqrt(c^2*d^2-e^2))],
[(a+b*arcsin(c*x))^2/(d+e*x)^3,x,13,-1/2*(a+b*arcsin(c*x))^2/(e*(d+e*x)^2)-b^2*c^2*log(d+e*x)/(e*(c^2*d^2-e^2))-I*b*c^3*d*(a+b*arcsin(c*x))*log(1-I*e*exp(I*arcsin(c*x))/(c*d-sqrt(c^2*d^2-e^2)))/(e*(c^2*d^2-e^2)^(3/2))+I*b*c^3*d*(a+b*arcsin(c*x))*log(1-I*e*exp(I*arcsin(c*x))/(c*d+sqrt(c^2*d^2-e^2)))/(e*(c^2*d^2-e^2)^(3/2))-b^2*c^3*d*polylog(2,I*e*exp(I*arcsin(c*x))/(c*d-sqrt(c^2*d^2-e^2)))/(e*(c^2*d^2-e^2)^(3/2))+b^2*c^3*d*polylog(2,I*e*exp(I*arcsin(c*x))/(c*d+sqrt(c^2*d^2-e^2)))/(e*(c^2*d^2-e^2)^(3/2))+b*c*(a+b*arcsin(c*x))*sqrt(1-c^2*x^2)/((c^2*d^2-e^2)*(d+e*x))],

# n<0
[(d+e*x)^3/(a+b*arcsin(c*x)),x,27,d^3*Ci(a/b+arcsin(c*x))*cos(a/b)/(b*c)+3/4*d*e^2*Ci(a/b+arcsin(c*x))*cos(a/b)/(b*c^3)-3/4*d*e^2*Ci(3*a/b+3*arcsin(c*x))*cos(3*a/b)/(b*c^3)+3/2*d^2*e*cos(2*a/b)*Si(2*a/b+2*arcsin(c*x))/(b*c^2)+1/4*e^3*cos(2*a/b)*Si(2*a/b+2*arcsin(c*x))/(b*c^4)-1/8*e^3*cos(4*a/b)*Si(4*a/b+4*arcsin(c*x))/(b*c^4)+d^3*Si(a/b+arcsin(c*x))*sin(a/b)/(b*c)+3/4*d*e^2*Si(a/b+arcsin(c*x))*sin(a/b)/(b*c^3)-3/2*d^2*e*Ci(2*a/b+2*arcsin(c*x))*sin(2*a/b)/(b*c^2)-1/4*e^3*Ci(2*a/b+2*arcsin(c*x))*sin(2*a/b)/(b*c^4)-3/4*d*e^2*Si(3*a/b+3*arcsin(c*x))*sin(3*a/b)/(b*c^3)+1/8*e^3*Ci(4*a/b+4*arcsin(c*x))*sin(4*a/b)/(b*c^4)],
[(d+e*x)^2/(a+b*arcsin(c*x)),x,17,d^2*Ci(a/b+arcsin(c*x))*cos(a/b)/(b*c)+1/4*e^2*Ci(a/b+arcsin(c*x))*cos(a/b)/(b*c^3)-1/4*e^2*Ci(3*a/b+3*arcsin(c*x))*cos(3*a/b)/(b*c^3)+d*e*cos(2*a/b)*Si(2*a/b+2*arcsin(c*x))/(b*c^2)+d^2*Si(a/b+arcsin(c*x))*sin(a/b)/(b*c)+1/4*e^2*Si(a/b+arcsin(c*x))*sin(a/b)/(b*c^3)-d*e*Ci(2*a/b+2*arcsin(c*x))*sin(2*a/b)/(b*c^2)-1/4*e^2*Si(3*a/b+3*arcsin(c*x))*sin(3*a/b)/(b*c^3)],
[(d+e*x)/(a+b*arcsin(c*x)),x,11,d*Ci(a/b+arcsin(c*x))*cos(a/b)/(b*c)+1/2*e*cos(2*a/b)*Si(2*a/b+2*arcsin(c*x))/(b*c^2)+d*Si(a/b+arcsin(c*x))*sin(a/b)/(b*c)-1/2*e*Ci(2*a/b+2*arcsin(c*x))*sin(2*a/b)/(b*c^2)],
[1/(a+b*arcsin(c*x)),x,4,Ci((a+b*arcsin(c*x))/b)*cos(a/b)/(b*c)+Si((a+b*arcsin(c*x))/b)*sin(a/b)/(b*c)],
[1/((d+e*x)*(a+b*arcsin(c*x))),x,0,Unintegrable(1/((d+e*x)*(a+b*arcsin(c*x))),x)],
[1/((d+e*x)^2*(a+b*arcsin(c*x))),x,0,Unintegrable(1/((d+e*x)^2*(a+b*arcsin(c*x))),x)],
[(d+e*x)^2/(a+b*arcsin(c*x))^2,x,19,2*d*e*Ci(2*(a+b*arcsin(c*x))/b)*cos(2*a/b)/(b^2*c^2)-d^2*cos(a/b)*Si((a+b*arcsin(c*x))/b)/(b^2*c)-1/4*e^2*cos(a/b)*Si((a+b*arcsin(c*x))/b)/(b^2*c^3)+3/4*e^2*cos(3*a/b)*Si(3*(a+b*arcsin(c*x))/b)/(b^2*c^3)+d^2*Ci((a+b*arcsin(c*x))/b)*sin(a/b)/(b^2*c)+1/4*e^2*Ci((a+b*arcsin(c*x))/b)*sin(a/b)/(b^2*c^3)+2*d*e*Si(2*(a+b*arcsin(c*x))/b)*sin(2*a/b)/(b^2*c^2)-3/4*e^2*Ci(3*(a+b*arcsin(c*x))/b)*sin(3*a/b)/(b^2*c^3)-d^2*sqrt(1-c^2*x^2)/(b*c*(a+b*arcsin(c*x)))-2*d*e*x*sqrt(1-c^2*x^2)/(b*c*(a+b*arcsin(c*x)))-e^2*x^2*sqrt(1-c^2*x^2)/(b*c*(a+b*arcsin(c*x)))],
[(d+e*x)/(a+b*arcsin(c*x))^2,x,11,e*Ci(2*(a+b*arcsin(c*x))/b)*cos(2*a/b)/(b^2*c^2)-d*cos(a/b)*Si((a+b*arcsin(c*x))/b)/(b^2*c)+d*Ci((a+b*arcsin(c*x))/b)*sin(a/b)/(b^2*c)+e*Si(2*(a+b*arcsin(c*x))/b)*sin(2*a/b)/(b^2*c^2)-d*sqrt(1-c^2*x^2)/(b*c*(a+b*arcsin(c*x)))-e*x*sqrt(1-c^2*x^2)/(b*c*(a+b*arcsin(c*x)))],
[1/(a+b*arcsin(c*x))^2,x,5,-cos(a/b)*Si((a+b*arcsin(c*x))/b)/(b^2*c)+Ci((a+b*arcsin(c*x))/b)*sin(a/b)/(b^2*c)-sqrt(1-c^2*x^2)/(b*c*(a+b*arcsin(c*x)))],
[1/((d+e*x)*(a+b*arcsin(c*x))^2),x,0,Unintegrable(1/((d+e*x)*(a+b*arcsin(c*x))^2),x)],
[1/((d+e*x)^2*(a+b*arcsin(c*x))^2),x,0,Unintegrable(1/((d+e*x)^2*(a+b*arcsin(c*x))^2),x)],

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

# Integrands of the form (d+e x)^m (a+b ArcSin[c x])^n with m symbolic
[(d+e*x)^m*(a+b*arcsin(c*x))^2,x,1,(d+e*x)^(1+m)*(a+b*arcsin(c*x))^2/(e*(1+m))-2*b*c*Unintegrable((d+e*x)^(1+m)*(a+b*arcsin(c*x))/sqrt(1-c^2*x^2),x)/(e*(1+m))],
[(d+e*x)^m*(a+b*arcsin(c*x)),x,3,(d+e*x)^(1+m)*(a+b*arcsin(c*x))/(e*(1+m))-b*c*(d+e*x)^(2+m)*AppellF1(2+m,1/2,1/2,3+m,c*(d+e*x)/(c*d-e),c*(d+e*x)/(c*d+e))*sqrt(1-c*(d+e*x)/(c*d-e))*sqrt(1-c*(d+e*x)/(c*d+e))/(e^2*(1+m)*(2+m)*sqrt(1-c^2*x^2))],
[(d+e*x)^m/(a+b*arcsin(c*x)),x,0,Unintegrable((d+e*x)^m/(a+b*arcsin(c*x)),x)],
[(d+e*x)^m/(a+b*arcsin(c*x))^2,x,0,Unintegrable((d+e*x)^m/(a+b*arcsin(c*x))^2,x)],

# Integrands of the form (f+g x)^m (d+e x^2)^(p/2) (a+b ArcSin[c x])^n where c^2 d+e=0

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

# p>0
[(f+g*x)^3*(a+b*arcsin(c*x))*sqrt(d-c^2*d*x^2),x,16,1/2*f^3*x*(a+b*arcsin(c*x))*sqrt(d-c^2*d*x^2)-3/8*f*g^2*x*(a+b*arcsin(c*x))*sqrt(d-c^2*d*x^2)/c^2+3/4*f*g^2*x^3*(a+b*arcsin(c*x))*sqrt(d-c^2*d*x^2)-f^2*g*(1-c^2*x^2)*(a+b*arcsin(c*x))*sqrt(d-c^2*d*x^2)/c^2-1/3*g^3*(1-c^2*x^2)*(a+b*arcsin(c*x))*sqrt(d-c^2*d*x^2)/c^4+1/5*g^3*(1-c^2*x^2)^2*(a+b*arcsin(c*x))*sqrt(d-c^2*d*x^2)/c^4+b*f^2*g*x*sqrt(d-c^2*d*x^2)/(c*sqrt(1-c^2*x^2))+2/15*b*g^3*x*sqrt(d-c^2*d*x^2)/(c^3*sqrt(1-c^2*x^2))-1/4*b*c*f^3*x^2*sqrt(d-c^2*d*x^2)/sqrt(1-c^2*x^2)+3/16*b*f*g^2*x^2*sqrt(d-c^2*d*x^2)/(c*sqrt(1-c^2*x^2))-1/3*b*c*f^2*g*x^3*sqrt(d-c^2*d*x^2)/sqrt(1-c^2*x^2)+1/45*b*g^3*x^3*sqrt(d-c^2*d*x^2)/(c*sqrt(1-c^2*x^2))-3/16*b*c*f*g^2*x^4*sqrt(d-c^2*d*x^2)/sqrt(1-c^2*x^2)-1/25*b*c*g^3*x^5*sqrt(d-c^2*d*x^2)/sqrt(1-c^2*x^2)+1/4*f^3*(a+b*arcsin(c*x))^2*sqrt(d-c^2*d*x^2)/(b*c*sqrt(1-c^2*x^2))+3/16*f*g^2*(a+b*arcsin(c*x))^2*sqrt(d-c^2*d*x^2)/(b*c^3*sqrt(1-c^2*x^2))],
[(f+g*x)^2*(a+b*arcsin(c*x))*sqrt(d-c^2*d*x^2),x,13,1/2*f^2*x*(a+b*arcsin(c*x))*sqrt(d-c^2*d*x^2)-1/8*g^2*x*(a+b*arcsin(c*x))*sqrt(d-c^2*d*x^2)/c^2+1/4*g^2*x^3*(a+b*arcsin(c*x))*sqrt(d-c^2*d*x^2)-2/3*f*g*(1-c^2*x^2)*(a+b*arcsin(c*x))*sqrt(d-c^2*d*x^2)/c^2+2/3*b*f*g*x*sqrt(d-c^2*d*x^2)/(c*sqrt(1-c^2*x^2))-1/4*b*c*f^2*x^2*sqrt(d-c^2*d*x^2)/sqrt(1-c^2*x^2)+1/16*b*g^2*x^2*sqrt(d-c^2*d*x^2)/(c*sqrt(1-c^2*x^2))-2/9*b*c*f*g*x^3*sqrt(d-c^2*d*x^2)/sqrt(1-c^2*x^2)-1/16*b*c*g^2*x^4*sqrt(d-c^2*d*x^2)/sqrt(1-c^2*x^2)+1/4*f^2*(a+b*arcsin(c*x))^2*sqrt(d-c^2*d*x^2)/(b*c*sqrt(1-c^2*x^2))+1/16*g^2*(a+b*arcsin(c*x))^2*sqrt(d-c^2*d*x^2)/(b*c^3*sqrt(1-c^2*x^2))],
[(f+g*x)*(a+b*arcsin(c*x))*sqrt(d-c^2*d*x^2),x,8,1/2*f*x*(a+b*arcsin(c*x))*sqrt(d-c^2*d*x^2)-1/3*g*(1-c^2*x^2)*(a+b*arcsin(c*x))*sqrt(d-c^2*d*x^2)/c^2+1/3*b*g*x*sqrt(d-c^2*d*x^2)/(c*sqrt(1-c^2*x^2))-1/4*b*c*f*x^2*sqrt(d-c^2*d*x^2)/sqrt(1-c^2*x^2)-1/9*b*c*g*x^3*sqrt(d-c^2*d*x^2)/sqrt(1-c^2*x^2)+1/4*f*(a+b*arcsin(c*x))^2*sqrt(d-c^2*d*x^2)/(b*c*sqrt(1-c^2*x^2))],
[(a+b*arcsin(c*x))*sqrt(d-c^2*d*x^2)/(f+g*x),x,22,a*sqrt(d-c^2*d*x^2)/g+b*arcsin(c*x)*sqrt(d-c^2*d*x^2)/g-b*c*x*sqrt(d-c^2*d*x^2)/(g*sqrt(1-c^2*x^2))+1/2*c*x*(a+b*arcsin(c*x))^2*sqrt(d-c^2*d*x^2)/(b*g*sqrt(1-c^2*x^2))-1/2*(1-c^2*f^2/g^2)*(a+b*arcsin(c*x))^2*sqrt(d-c^2*d*x^2)/(b*c*(f+g*x)*sqrt(1-c^2*x^2))-a*arctan((g+c^2*f*x)/(sqrt(c^2*f^2-g^2)*sqrt(1-c^2*x^2)))*sqrt(c^2*f^2-g^2)*sqrt(d-c^2*d*x^2)/(g^2*sqrt(1-c^2*x^2))+I*b*arcsin(c*x)*log(1-I*exp(I*arcsin(c*x))*g/(c*f-sqrt(c^2*f^2-g^2)))*sqrt(c^2*f^2-g^2)*sqrt(d-c^2*d*x^2)/(g^2*sqrt(1-c^2*x^2))-I*b*arcsin(c*x)*log(1-I*exp(I*arcsin(c*x))*g/(c*f+sqrt(c^2*f^2-g^2)))*sqrt(c^2*f^2-g^2)*sqrt(d-c^2*d*x^2)/(g^2*sqrt(1-c^2*x^2))+b*polylog(2,I*exp(I*arcsin(c*x))*g/(c*f-sqrt(c^2*f^2-g^2)))*sqrt(c^2*f^2-g^2)*sqrt(d-c^2*d*x^2)/(g^2*sqrt(1-c^2*x^2))-b*polylog(2,I*exp(I*arcsin(c*x))*g/(c*f+sqrt(c^2*f^2-g^2)))*sqrt(c^2*f^2-g^2)*sqrt(d-c^2*d*x^2)/(g^2*sqrt(1-c^2*x^2))+1/2*(a+b*arcsin(c*x))^2*sqrt(1-c^2*x^2)*sqrt(d-c^2*d*x^2)/(b*c*(f+g*x))],
[(a+b*arcsin(c*x))*sqrt(d-c^2*d*x^2)/(f+g*x)^2,x,35,-a*sqrt(d-c^2*d*x^2)/(g*(f+g*x))-b*arcsin(c*x)*sqrt(d-c^2*d*x^2)/(g*(f+g*x))-a*c^3*f^2*arcsin(c*x)*sqrt(d-c^2*d*x^2)/(g^2*(c^2*f^2-g^2)*sqrt(1-c^2*x^2))-1/2*b*c^3*f^2*arcsin(c*x)^2*sqrt(d-c^2*d*x^2)/(g^2*(c^2*f^2-g^2)*sqrt(1-c^2*x^2))+1/2*(g+c^2*f*x)^2*(a+b*arcsin(c*x))^2*sqrt(d-c^2*d*x^2)/(b*c*(c^2*f^2-g^2)*(f+g*x)^2*sqrt(1-c^2*x^2))+b*c*log(f+g*x)*sqrt(d-c^2*d*x^2)/(g^2*sqrt(1-c^2*x^2))+a*c^2*f*arctan((g+c^2*f*x)/(sqrt(c^2*f^2-g^2)*sqrt(1-c^2*x^2)))*sqrt(d-c^2*d*x^2)/(g^2*sqrt(c^2*f^2-g^2)*sqrt(1-c^2*x^2))-I*b*c^2*f*arcsin(c*x)*log(1-I*exp(I*arcsin(c*x))*g/(c*f-sqrt(c^2*f^2-g^2)))*sqrt(d-c^2*d*x^2)/(g^2*sqrt(c^2*f^2-g^2)*sqrt(1-c^2*x^2))+I*b*c^2*f*arcsin(c*x)*log(1-I*exp(I*arcsin(c*x))*g/(c*f+sqrt(c^2*f^2-g^2)))*sqrt(d-c^2*d*x^2)/(g^2*sqrt(c^2*f^2-g^2)*sqrt(1-c^2*x^2))-b*c^2*f*polylog(2,I*exp(I*arcsin(c*x))*g/(c*f-sqrt(c^2*f^2-g^2)))*sqrt(d-c^2*d*x^2)/(g^2*sqrt(c^2*f^2-g^2)*sqrt(1-c^2*x^2))+b*c^2*f*polylog(2,I*exp(I*arcsin(c*x))*g/(c*f+sqrt(c^2*f^2-g^2)))*sqrt(d-c^2*d*x^2)/(g^2*sqrt(c^2*f^2-g^2)*sqrt(1-c^2*x^2))+1/2*(a+b*arcsin(c*x))^2*sqrt(1-c^2*x^2)*sqrt(d-c^2*d*x^2)/(b*c*(f+g*x)^2)],
[(f+g*x)^3*(d-c^2*d*x^2)^(3/2)*(a+b*arcsin(c*x)),x,24,3/8*d*f^3*x*(a+b*arcsin(c*x))*sqrt(d-c^2*d*x^2)-3/16*d*f*g^2*x*(a+b*arcsin(c*x))*sqrt(d-c^2*d*x^2)/c^2+3/8*d*f*g^2*x^3*(a+b*arcsin(c*x))*sqrt(d-c^2*d*x^2)+1/4*d*f^3*x*(1-c^2*x^2)*(a+b*arcsin(c*x))*sqrt(d-c^2*d*x^2)+1/2*d*f*g^2*x^3*(1-c^2*x^2)*(a+b*arcsin(c*x))*sqrt(d-c^2*d*x^2)-3/5*d*f^2*g*(1-c^2*x^2)^2*(a+b*arcsin(c*x))*sqrt(d-c^2*d*x^2)/c^2-1/5*d*g^3*(1-c^2*x^2)^2*(a+b*arcsin(c*x))*sqrt(d-c^2*d*x^2)/c^4+1/7*d*g^3*(1-c^2*x^2)^3*(a+b*arcsin(c*x))*sqrt(d-c^2*d*x^2)/c^4+3/5*b*d*f^2*g*x*sqrt(d-c^2*d*x^2)/(c*sqrt(1-c^2*x^2))+2/35*b*d*g^3*x*sqrt(d-c^2*d*x^2)/(c^3*sqrt(1-c^2*x^2))-5/16*b*c*d*f^3*x^2*sqrt(d-c^2*d*x^2)/sqrt(1-c^2*x^2)+3/32*b*d*f*g^2*x^2*sqrt(d-c^2*d*x^2)/(c*sqrt(1-c^2*x^2))-2/5*b*c*d*f^2*g*x^3*sqrt(d-c^2*d*x^2)/sqrt(1-c^2*x^2)+1/105*b*d*g^3*x^3*sqrt(d-c^2*d*x^2)/(c*sqrt(1-c^2*x^2))+1/16*b*c^3*d*f^3*x^4*sqrt(d-c^2*d*x^2)/sqrt(1-c^2*x^2)-7/32*b*c*d*f*g^2*x^4*sqrt(d-c^2*d*x^2)/sqrt(1-c^2*x^2)+3/25*b*c^3*d*f^2*g*x^5*sqrt(d-c^2*d*x^2)/sqrt(1-c^2*x^2)-8/175*b*c*d*g^3*x^5*sqrt(d-c^2*d*x^2)/sqrt(1-c^2*x^2)+1/12*b*c^3*d*f*g^2*x^6*sqrt(d-c^2*d*x^2)/sqrt(1-c^2*x^2)+1/49*b*c^3*d*g^3*x^7*sqrt(d-c^2*d*x^2)/sqrt(1-c^2*x^2)+3/16*d*f^3*(a+b*arcsin(c*x))^2*sqrt(d-c^2*d*x^2)/(b*c*sqrt(1-c^2*x^2))+3/32*d*f*g^2*(a+b*arcsin(c*x))^2*sqrt(d-c^2*d*x^2)/(b*c^3*sqrt(1-c^2*x^2))],
[(f+g*x)^2*(d-c^2*d*x^2)^(3/2)*(a+b*arcsin(c*x)),x,20,3/8*d*f^2*x*(a+b*arcsin(c*x))*sqrt(d-c^2*d*x^2)-1/16*d*g^2*x*(a+b*arcsin(c*x))*sqrt(d-c^2*d*x^2)/c^2+1/8*d*g^2*x^3*(a+b*arcsin(c*x))*sqrt(d-c^2*d*x^2)+1/4*d*f^2*x*(1-c^2*x^2)*(a+b*arcsin(c*x))*sqrt(d-c^2*d*x^2)+1/6*d*g^2*x^3*(1-c^2*x^2)*(a+b*arcsin(c*x))*sqrt(d-c^2*d*x^2)-2/5*d*f*g*(1-c^2*x^2)^2*(a+b*arcsin(c*x))*sqrt(d-c^2*d*x^2)/c^2+2/5*b*d*f*g*x*sqrt(d-c^2*d*x^2)/(c*sqrt(1-c^2*x^2))-5/16*b*c*d*f^2*x^2*sqrt(d-c^2*d*x^2)/sqrt(1-c^2*x^2)+1/32*b*d*g^2*x^2*sqrt(d-c^2*d*x^2)/(c*sqrt(1-c^2*x^2))-4/15*b*c*d*f*g*x^3*sqrt(d-c^2*d*x^2)/sqrt(1-c^2*x^2)+1/16*b*c^3*d*f^2*x^4*sqrt(d-c^2*d*x^2)/sqrt(1-c^2*x^2)-7/96*b*c*d*g^2*x^4*sqrt(d-c^2*d*x^2)/sqrt(1-c^2*x^2)+2/25*b*c^3*d*f*g*x^5*sqrt(d-c^2*d*x^2)/sqrt(1-c^2*x^2)+1/36*b*c^3*d*g^2*x^6*sqrt(d-c^2*d*x^2)/sqrt(1-c^2*x^2)+3/16*d*f^2*(a+b*arcsin(c*x))^2*sqrt(d-c^2*d*x^2)/(b*c*sqrt(1-c^2*x^2))+1/32*d*g^2*(a+b*arcsin(c*x))^2*sqrt(d-c^2*d*x^2)/(b*c^3*sqrt(1-c^2*x^2))],
[(f+g*x)*(d-c^2*d*x^2)^(3/2)*(a+b*arcsin(c*x)),x,12,3/8*d*f*x*(a+b*arcsin(c*x))*sqrt(d-c^2*d*x^2)+1/4*d*f*x*(1-c^2*x^2)*(a+b*arcsin(c*x))*sqrt(d-c^2*d*x^2)-1/5*d*g*(1-c^2*x^2)^2*(a+b*arcsin(c*x))*sqrt(d-c^2*d*x^2)/c^2+1/5*b*d*g*x*sqrt(d-c^2*d*x^2)/(c*sqrt(1-c^2*x^2))-5/16*b*c*d*f*x^2*sqrt(d-c^2*d*x^2)/sqrt(1-c^2*x^2)-2/15*b*c*d*g*x^3*sqrt(d-c^2*d*x^2)/sqrt(1-c^2*x^2)+1/16*b*c^3*d*f*x^4*sqrt(d-c^2*d*x^2)/sqrt(1-c^2*x^2)+1/25*b*c^3*d*g*x^5*sqrt(d-c^2*d*x^2)/sqrt(1-c^2*x^2)+3/16*d*f*(a+b*arcsin(c*x))^2*sqrt(d-c^2*d*x^2)/(b*c*sqrt(1-c^2*x^2))],
[(d-c^2*d*x^2)^(3/2)*(a+b*arcsin(c*x))/(f+g*x),x,29,-a*d*(c*f-g)*(c*f+g)*sqrt(d-c^2*d*x^2)/g^3-b*d*(c*f-g)*(c*f+g)*arcsin(c*x)*sqrt(d-c^2*d*x^2)/g^3+1/2*c^2*d*f*x*(a+b*arcsin(c*x))*sqrt(d-c^2*d*x^2)/g^2+1/3*d*(1-c^2*x^2)*(a+b*arcsin(c*x))*sqrt(d-c^2*d*x^2)/g-1/3*b*c*d*x*sqrt(d-c^2*d*x^2)/(g*sqrt(1-c^2*x^2))+b*c*d*(c*f-g)*(c*f+g)*x*sqrt(d-c^2*d*x^2)/(g^3*sqrt(1-c^2*x^2))-1/4*b*c^3*d*f*x^2*sqrt(d-c^2*d*x^2)/(g^2*sqrt(1-c^2*x^2))+1/9*b*c^3*d*x^3*sqrt(d-c^2*d*x^2)/(g*sqrt(1-c^2*x^2))+1/4*c*d*f*(a+b*arcsin(c*x))^2*sqrt(d-c^2*d*x^2)/(b*g^2*sqrt(1-c^2*x^2))-1/2*c*d*(c*f-g)*(c*f+g)*x*(a+b*arcsin(c*x))^2*sqrt(d-c^2*d*x^2)/(b*g^3*sqrt(1-c^2*x^2))-1/2*d*(c^2*f^2-g^2)^2*(a+b*arcsin(c*x))^2*sqrt(d-c^2*d*x^2)/(b*c*g^4*(f+g*x)*sqrt(1-c^2*x^2))+a*d*(c^2*f^2-g^2)^(3/2)*arctan((g+c^2*f*x)/(sqrt(c^2*f^2-g^2)*sqrt(1-c^2*x^2)))*sqrt(d-c^2*d*x^2)/(g^4*sqrt(1-c^2*x^2))-I*b*d*(c^2*f^2-g^2)^(3/2)*arcsin(c*x)*log(1-I*exp(I*arcsin(c*x))*g/(c*f-sqrt(c^2*f^2-g^2)))*sqrt(d-c^2*d*x^2)/(g^4*sqrt(1-c^2*x^2))+I*b*d*(c^2*f^2-g^2)^(3/2)*arcsin(c*x)*log(1-I*exp(I*arcsin(c*x))*g/(c*f+sqrt(c^2*f^2-g^2)))*sqrt(d-c^2*d*x^2)/(g^4*sqrt(1-c^2*x^2))-b*d*(c^2*f^2-g^2)^(3/2)*polylog(2,I*exp(I*arcsin(c*x))*g/(c*f-sqrt(c^2*f^2-g^2)))*sqrt(d-c^2*d*x^2)/(g^4*sqrt(1-c^2*x^2))+b*d*(c^2*f^2-g^2)^(3/2)*polylog(2,I*exp(I*arcsin(c*x))*g/(c*f+sqrt(c^2*f^2-g^2)))*sqrt(d-c^2*d*x^2)/(g^4*sqrt(1-c^2*x^2))-1/2*d*(c*f-g)*(c*f+g)*(a+b*arcsin(c*x))^2*sqrt(1-c^2*x^2)*sqrt(d-c^2*d*x^2)/(b*c*g^2*(f+g*x))],

#  {(d - c^2*d*x^2)^(3/2)*(a + b*ArcSin[c*x])/(f + g*x)^2, x, 71, (2*a*c^2*d*f*Sqrt[d - c^2*d*x^2])/g^3 + (a*d*(c*f - g)*(c*f + g)*Sqrt[d - c^2*d*x^2])/(g^3*(f + g*x)) - (2*b*c^3*d*f*x*Sqrt[d - c^2*d*x^2])/(g^3*Sqrt[1 - c^2*x^2]) + (b*c^3*d*x^2*Sqrt[d - c^2*d*x^2])/(4*g^2*Sqrt[1 - c^2*x^2]) + (2*b*c^2*d*f*Sqrt[d - c^2*d*x^2]*ArcSin[c*x])/g^3 + (b*d*(c*f - g)*(c*f + g)*Sqrt[d - c^2*d*x^2]*ArcSin[c*x])/(g^3*(f + g*x)) + (a*c^3*d*f^2*Sqrt[d - c^2*d*x^2]*ArcSin[c*x])/(g^4*Sqrt[1 - c^2*x^2]) + (b*c^3*d*f^2*Sqrt[d - c^2*d*x^2]*ArcSin[c*x]^2)/(2*g^4*Sqrt[1 - c^2*x^2]) - (c^2*d*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(2*g^2) - (c*d*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/(4*b*g^2*Sqrt[1 - c^2*x^2]) + (c^3*d*f*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/(b*g^3*Sqrt[1 - c^2*x^2]) - (d*(g + c^2*f*x)^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/(2*b*c*g^2*(f + g*x)^2*Sqrt[1 - c^2*x^2]) - (c*d*f*(1 - (c^2*f^2)/g^2)*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/(b*g^2*(f + g*x)*Sqrt[1 - c^2*x^2]) - (d*(c*f - g)*(c*f + g)*Sqrt[1 - c^2*x^2]*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/(2*b*c*g^2*(f + g*x)^2) + (c*d*f*Sqrt[1 - c^2*x^2]*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/(b*g^2*(f + g*x)) - (3*a*c^2*d*f*Sqrt[c^2*f^2 - g^2]*Sqrt[d - c^2*d*x^2]*ArcTan[(g + c^2*f*x)/(Sqrt[c^2*f^2 - g^2]*Sqrt[1 - c^2*x^2])])/(g^4*Sqrt[1 - c^2*x^2]) + (3*I*b*c^2*d*f*Sqrt[c^2*f^2 - g^2]*Sqrt[d - c^2*d*x^2]*ArcSin[c*x]*Log[1 - (I*E^(I*ArcSin[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g^2])])/(g^4*Sqrt[1 - c^2*x^2]) - (3*I*b*c^2*d*f*Sqrt[c^2*f^2 - g^2]*Sqrt[d - c^2*d*x^2]*ArcSin[c*x]*Log[1 - (I*E^(I*ArcSin[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2])])/(g^4*Sqrt[1 - c^2*x^2]) - (b*c*d*(c*f - g)*(c*f + g)*Sqrt[d - c^2*d*x^2]*Log[f + g*x])/(g^4*Sqrt[1 - c^2*x^2]) + (3*b*c^2*d*f*Sqrt[c^2*f^2 - g^2]*Sqrt[d - c^2*d*x^2]*PolyLog[2, (I*E^(I*ArcSin[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g^2])])/(g^4*Sqrt[1 - c^2*x^2]) - (3*b*c^2*d*f*Sqrt[c^2*f^2 - g^2]*Sqrt[d - c^2*d*x^2]*PolyLog[2, (I*E^(I*ArcSin[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2])])/(g^4*Sqrt[1 - c^2*x^2])} 
[(f+g*x)^3*(d-c^2*d*x^2)^(5/2)*(a+b*arcsin(c*x)),x,30,1/36*b*d^2*f^3*(1-c^2*x^2)^(5/2)*sqrt(d-c^2*d*x^2)/c+5/16*d^2*f^3*x*(a+b*arcsin(c*x))*sqrt(d-c^2*d*x^2)-15/128*d^2*f*g^2*x*(a+b*arcsin(c*x))*sqrt(d-c^2*d*x^2)/c^2+15/64*d^2*f*g^2*x^3*(a+b*arcsin(c*x))*sqrt(d-c^2*d*x^2)+5/24*d^2*f^3*x*(1-c^2*x^2)*(a+b*arcsin(c*x))*sqrt(d-c^2*d*x^2)+5/16*d^2*f*g^2*x^3*(1-c^2*x^2)*(a+b*arcsin(c*x))*sqrt(d-c^2*d*x^2)+1/6*d^2*f^3*x*(1-c^2*x^2)^2*(a+b*arcsin(c*x))*sqrt(d-c^2*d*x^2)+3/8*d^2*f*g^2*x^3*(1-c^2*x^2)^2*(a+b*arcsin(c*x))*sqrt(d-c^2*d*x^2)-3/7*d^2*f^2*g*(1-c^2*x^2)^3*(a+b*arcsin(c*x))*sqrt(d-c^2*d*x^2)/c^2-1/7*d^2*g^3*(1-c^2*x^2)^3*(a+b*arcsin(c*x))*sqrt(d-c^2*d*x^2)/c^4+1/9*d^2*g^3*(1-c^2*x^2)^4*(a+b*arcsin(c*x))*sqrt(d-c^2*d*x^2)/c^4+3/7*b*d^2*f^2*g*x*sqrt(d-c^2*d*x^2)/(c*sqrt(1-c^2*x^2))+2/63*b*d^2*g^3*x*sqrt(d-c^2*d*x^2)/(c^3*sqrt(1-c^2*x^2))-25/96*b*c*d^2*f^3*x^2*sqrt(d-c^2*d*x^2)/sqrt(1-c^2*x^2)+15/256*b*d^2*f*g^2*x^2*sqrt(d-c^2*d*x^2)/(c*sqrt(1-c^2*x^2))-3/7*b*c*d^2*f^2*g*x^3*sqrt(d-c^2*d*x^2)/sqrt(1-c^2*x^2)+1/189*b*d^2*g^3*x^3*sqrt(d-c^2*d*x^2)/(c*sqrt(1-c^2*x^2))+5/96*b*c^3*d^2*f^3*x^4*sqrt(d-c^2*d*x^2)/sqrt(1-c^2*x^2)-59/256*b*c*d^2*f*g^2*x^4*sqrt(d-c^2*d*x^2)/sqrt(1-c^2*x^2)+9/35*b*c^3*d^2*f^2*g*x^5*sqrt(d-c^2*d*x^2)/sqrt(1-c^2*x^2)-1/21*b*c*d^2*g^3*x^5*sqrt(d-c^2*d*x^2)/sqrt(1-c^2*x^2)+17/96*b*c^3*d^2*f*g^2*x^6*sqrt(d-c^2*d*x^2)/sqrt(1-c^2*x^2)-3/49*b*c^5*d^2*f^2*g*x^7*sqrt(d-c^2*d*x^2)/sqrt(1-c^2*x^2)+19/441*b*c^3*d^2*g^3*x^7*sqrt(d-c^2*d*x^2)/sqrt(1-c^2*x^2)-3/64*b*c^5*d^2*f*g^2*x^8*sqrt(d-c^2*d*x^2)/sqrt(1-c^2*x^2)-1/81*b*c^5*d^2*g^3*x^9*sqrt(d-c^2*d*x^2)/sqrt(1-c^2*x^2)+5/32*d^2*f^3*(a+b*arcsin(c*x))^2*sqrt(d-c^2*d*x^2)/(b*c*sqrt(1-c^2*x^2))+15/256*d^2*f*g^2*(a+b*arcsin(c*x))^2*sqrt(d-c^2*d*x^2)/(b*c^3*sqrt(1-c^2*x^2))],
[(f+g*x)^2*(d-c^2*d*x^2)^(5/2)*(a+b*arcsin(c*x)),x,26,1/36*b*d^2*f^2*(1-c^2*x^2)^(5/2)*sqrt(d-c^2*d*x^2)/c+5/16*d^2*f^2*x*(a+b*arcsin(c*x))*sqrt(d-c^2*d*x^2)-5/128*d^2*g^2*x*(a+b*arcsin(c*x))*sqrt(d-c^2*d*x^2)/c^2+5/64*d^2*g^2*x^3*(a+b*arcsin(c*x))*sqrt(d-c^2*d*x^2)+5/24*d^2*f^2*x*(1-c^2*x^2)*(a+b*arcsin(c*x))*sqrt(d-c^2*d*x^2)+5/48*d^2*g^2*x^3*(1-c^2*x^2)*(a+b*arcsin(c*x))*sqrt(d-c^2*d*x^2)+1/6*d^2*f^2*x*(1-c^2*x^2)^2*(a+b*arcsin(c*x))*sqrt(d-c^2*d*x^2)+1/8*d^2*g^2*x^3*(1-c^2*x^2)^2*(a+b*arcsin(c*x))*sqrt(d-c^2*d*x^2)-2/7*d^2*f*g*(1-c^2*x^2)^3*(a+b*arcsin(c*x))*sqrt(d-c^2*d*x^2)/c^2+2/7*b*d^2*f*g*x*sqrt(d-c^2*d*x^2)/(c*sqrt(1-c^2*x^2))-25/96*b*c*d^2*f^2*x^2*sqrt(d-c^2*d*x^2)/sqrt(1-c^2*x^2)+5/256*b*d^2*g^2*x^2*sqrt(d-c^2*d*x^2)/(c*sqrt(1-c^2*x^2))-2/7*b*c*d^2*f*g*x^3*sqrt(d-c^2*d*x^2)/sqrt(1-c^2*x^2)+5/96*b*c^3*d^2*f^2*x^4*sqrt(d-c^2*d*x^2)/sqrt(1-c^2*x^2)-59/768*b*c*d^2*g^2*x^4*sqrt(d-c^2*d*x^2)/sqrt(1-c^2*x^2)+6/35*b*c^3*d^2*f*g*x^5*sqrt(d-c^2*d*x^2)/sqrt(1-c^2*x^2)+17/288*b*c^3*d^2*g^2*x^6*sqrt(d-c^2*d*x^2)/sqrt(1-c^2*x^2)-2/49*b*c^5*d^2*f*g*x^7*sqrt(d-c^2*d*x^2)/sqrt(1-c^2*x^2)-1/64*b*c^5*d^2*g^2*x^8*sqrt(d-c^2*d*x^2)/sqrt(1-c^2*x^2)+5/32*d^2*f^2*(a+b*arcsin(c*x))^2*sqrt(d-c^2*d*x^2)/(b*c*sqrt(1-c^2*x^2))+5/256*d^2*g^2*(a+b*arcsin(c*x))^2*sqrt(d-c^2*d*x^2)/(b*c^3*sqrt(1-c^2*x^2))],
[(f+g*x)*(d-c^2*d*x^2)^(5/2)*(a+b*arcsin(c*x)),x,14,1/36*b*d^2*f*(1-c^2*x^2)^(5/2)*sqrt(d-c^2*d*x^2)/c+5/16*d^2*f*x*(a+b*arcsin(c*x))*sqrt(d-c^2*d*x^2)+5/24*d^2*f*x*(1-c^2*x^2)*(a+b*arcsin(c*x))*sqrt(d-c^2*d*x^2)+1/6*d^2*f*x*(1-c^2*x^2)^2*(a+b*arcsin(c*x))*sqrt(d-c^2*d*x^2)-1/7*d^2*g*(1-c^2*x^2)^3*(a+b*arcsin(c*x))*sqrt(d-c^2*d*x^2)/c^2+1/7*b*d^2*g*x*sqrt(d-c^2*d*x^2)/(c*sqrt(1-c^2*x^2))-25/96*b*c*d^2*f*x^2*sqrt(d-c^2*d*x^2)/sqrt(1-c^2*x^2)-1/7*b*c*d^2*g*x^3*sqrt(d-c^2*d*x^2)/sqrt(1-c^2*x^2)+5/96*b*c^3*d^2*f*x^4*sqrt(d-c^2*d*x^2)/sqrt(1-c^2*x^2)+3/35*b*c^3*d^2*g*x^5*sqrt(d-c^2*d*x^2)/sqrt(1-c^2*x^2)-1/49*b*c^5*d^2*g*x^7*sqrt(d-c^2*d*x^2)/sqrt(1-c^2*x^2)+5/32*d^2*f*(a+b*arcsin(c*x))^2*sqrt(d-c^2*d*x^2)/(b*c*sqrt(1-c^2*x^2))],
[(d-c^2*d*x^2)^(5/2)*(a+b*arcsin(c*x))/(f+g*x),x,37,a*d^2*(c^2*f^2-g^2)^2*sqrt(d-c^2*d*x^2)/g^5+b*d^2*(c^2*f^2-g^2)^2*arcsin(c*x)*sqrt(d-c^2*d*x^2)/g^5+1/8*c^2*d^2*f*x*(a+b*arcsin(c*x))*sqrt(d-c^2*d*x^2)/g^2-1/2*c^2*d^2*f*(c^2*f^2-2*g^2)*x*(a+b*arcsin(c*x))*sqrt(d-c^2*d*x^2)/g^4-1/4*c^4*d^2*f*x^3*(a+b*arcsin(c*x))*sqrt(d-c^2*d*x^2)/g^2-1/3*d^2*(1-c^2*x^2)*(a+b*arcsin(c*x))*sqrt(d-c^2*d*x^2)/g-1/3*d^2*(c^2*f^2-2*g^2)*(1-c^2*x^2)*(a+b*arcsin(c*x))*sqrt(d-c^2*d*x^2)/g^3+1/5*d^2*(1-c^2*x^2)^2*(a+b*arcsin(c*x))*sqrt(d-c^2*d*x^2)/g+2/15*b*c*d^2*x*sqrt(d-c^2*d*x^2)/(g*sqrt(1-c^2*x^2))+1/3*b*c*d^2*(c^2*f^2-2*g^2)*x*sqrt(d-c^2*d*x^2)/(g^3*sqrt(1-c^2*x^2))-b*c*d^2*(c^2*f^2-g^2)^2*x*sqrt(d-c^2*d*x^2)/(g^5*sqrt(1-c^2*x^2))-1/16*b*c^3*d^2*f*x^2*sqrt(d-c^2*d*x^2)/(g^2*sqrt(1-c^2*x^2))+1/4*b*c^3*d^2*f*(c^2*f^2-2*g^2)*x^2*sqrt(d-c^2*d*x^2)/(g^4*sqrt(1-c^2*x^2))+1/45*b*c^3*d^2*x^3*sqrt(d-c^2*d*x^2)/(g*sqrt(1-c^2*x^2))-1/9*b*c^3*d^2*(c^2*f^2-2*g^2)*x^3*sqrt(d-c^2*d*x^2)/(g^3*sqrt(1-c^2*x^2))+1/16*b*c^5*d^2*f*x^4*sqrt(d-c^2*d*x^2)/(g^2*sqrt(1-c^2*x^2))-1/25*b*c^5*d^2*x^5*sqrt(d-c^2*d*x^2)/(g*sqrt(1-c^2*x^2))-1/16*c*d^2*f*(a+b*arcsin(c*x))^2*sqrt(d-c^2*d*x^2)/(b*g^2*sqrt(1-c^2*x^2))-1/4*c*d^2*f*(c^2*f^2-2*g^2)*(a+b*arcsin(c*x))^2*sqrt(d-c^2*d*x^2)/(b*g^4*sqrt(1-c^2*x^2))+1/2*c*d^2*(c^2*f^2-g^2)^2*x*(a+b*arcsin(c*x))^2*sqrt(d-c^2*d*x^2)/(b*g^5*sqrt(1-c^2*x^2))+1/2*d^2*(c^2*f^2-g^2)^3*(a+b*arcsin(c*x))^2*sqrt(d-c^2*d*x^2)/(b*c*g^6*(f+g*x)*sqrt(1-c^2*x^2))-a*d^2*(c^2*f^2-g^2)^(5/2)*arctan((g+c^2*f*x)/(sqrt(c^2*f^2-g^2)*sqrt(1-c^2*x^2)))*sqrt(d-c^2*d*x^2)/(g^6*sqrt(1-c^2*x^2))+I*b*d^2*(c^2*f^2-g^2)^(5/2)*arcsin(c*x)*log(1-I*exp(I*arcsin(c*x))*g/(c*f-sqrt(c^2*f^2-g^2)))*sqrt(d-c^2*d*x^2)/(g^6*sqrt(1-c^2*x^2))-I*b*d^2*(c^2*f^2-g^2)^(5/2)*arcsin(c*x)*log(1-I*exp(I*arcsin(c*x))*g/(c*f+sqrt(c^2*f^2-g^2)))*sqrt(d-c^2*d*x^2)/(g^6*sqrt(1-c^2*x^2))+b*d^2*(c^2*f^2-g^2)^(5/2)*polylog(2,I*exp(I*arcsin(c*x))*g/(c*f-sqrt(c^2*f^2-g^2)))*sqrt(d-c^2*d*x^2)/(g^6*sqrt(1-c^2*x^2))-b*d^2*(c^2*f^2-g^2)^(5/2)*polylog(2,I*exp(I*arcsin(c*x))*g/(c*f+sqrt(c^2*f^2-g^2)))*sqrt(d-c^2*d*x^2)/(g^6*sqrt(1-c^2*x^2))+1/2*d^2*(c^2*f^2-g^2)^2*(a+b*arcsin(c*x))^2*sqrt(1-c^2*x^2)*sqrt(d-c^2*d*x^2)/(b*c*g^4*(f+g*x))],

#  {(d - c^2*d*x^2)^(5/2)*(a + b*ArcSin[c*x])/(f + g*x)^2, x, 78, -((4*a*c^2*d^2*f*(c*f - g)*(c*f + g)*Sqrt[d - c^2*d*x^2])/g^5) - (a*d^2*(c^2*f^2 - g^2)^2*Sqrt[d - c^2*d*x^2])/(g^5*(f + g*x)) - (2*b*c^3*d^2*f*x*Sqrt[d - c^2*d*x^2])/(3*g^3*Sqrt[1 - c^2*x^2]) + (4*b*c^3*d^2*f*(c*f - g)*(c*f + g)*x*Sqrt[d - c^2*d*x^2])/(g^5*Sqrt[1 - c^2*x^2]) + (b*c^3*d^2*x^2*Sqrt[d - c^2*d*x^2])/(16*g^2*Sqrt[1 - c^2*x^2]) - (b*c^3*d^2*(3*c^2*f^2 - 2*g^2)*x^2*Sqrt[d - c^2*d*x^2])/(4*g^4*Sqrt[1 - c^2*x^2]) + (2*b*c^5*d^2*f*x^3*Sqrt[d - c^2*d*x^2])/(9*g^3*Sqrt[1 - c^2*x^2]) - (b*c^5*d^2*x^4*Sqrt[d - c^2*d*x^2])/(16*g^2*Sqrt[1 - c^2*x^2]) - (4*b*c^2*d^2*f*(c*f - g)*(c*f + g)*Sqrt[d - c^2*d*x^2]*ArcSin[c*x])/g^5 - (b*d^2*(c^2*f^2 - g^2)^2*Sqrt[d - c^2*d*x^2]*ArcSin[c*x])/(g^5*(f + g*x)) - (a*c^3*d^2*f^2*(c*f - g)*(c*f + g)*Sqrt[d - c^2*d*x^2]*ArcSin[c*x])/(g^6*Sqrt[1 - c^2*x^2]) - (b*c^3*d^2*f^2*(c*f - g)*(c*f + g)*Sqrt[d - c^2*d*x^2]*ArcSin[c*x]^2)/(2*g^6*Sqrt[1 - c^2*x^2]) - (c^2*d^2*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(8*g^2) + (c^2*d^2*(3*c^2*f^2 - 2*g^2)*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(2*g^4) + (c^4*d^2*x^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(4*g^2) + (2*c^2*d^2*f*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(3*g^3) + (c*d^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/(16*b*g^2*Sqrt[1 - c^2*x^2]) + (c*d^2*(3*c^2*f^2 - 2*g^2)*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/(4*b*g^4*Sqrt[1 - c^2*x^2]) - (2*c^3*d^2*f*(c*f - g)*(c*f + g)*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/(b*g^5*Sqrt[1 - c^2*x^2]) + (d^2*(c*f - g)*(c*f + g)*(g + c^2*f*x)^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/(2*b*c*g^4*(f + g*x)^2*Sqrt[1 - c^2*x^2]) - (2*c*d^2*f*(c^2*f^2 - g^2)^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/(b*g^6*(f + g*x)*Sqrt[1 - c^2*x^2]) + (d^2*(c^2*f^2 - g^2)^2*Sqrt[1 - c^2*x^2]*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/(2*b*c*g^4*(f + g*x)^2) - (2*c*d^2*f*(c*f - g)*(c*f + g)*Sqrt[1 - c^2*x^2]*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/(b*g^4*(f + g*x)) + (5*a*c^2*d^2*f*(c^2*f^2 - g^2)^(3/2)*Sqrt[d - c^2*d*x^2]*ArcTan[(g + c^2*f*x)/(Sqrt[c^2*f^2 - g^2]*Sqrt[1 - c^2*x^2])])/(g^6*Sqrt[1 - c^2*x^2]) - (5*I*b*c^2*d^2*f*(c^2*f^2 - g^2)^(3/2)*Sqrt[d - c^2*d*x^2]*ArcSin[c*x]*Log[1 - (I*E^(I*ArcSin[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g^2])])/(g^6*Sqrt[1 - c^2*x^2]) + (5*I*b*c^2*d^2*f*(c^2*f^2 - g^2)^(3/2)*Sqrt[d - c^2*d*x^2]*ArcSin[c*x]*Log[1 - (I*E^(I*ArcSin[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2])])/(g^6*Sqrt[1 - c^2*x^2]) + (b*c*d^2*(c^2*f^2 - g^2)^2*Sqrt[d - c^2*d*x^2]*Log[f + g*x])/(g^6*Sqrt[1 - c^2*x^2]) - (5*b*c^2*d^2*f*(c^2*f^2 - g^2)^(3/2)*Sqrt[d - c^2*d*x^2]*PolyLog[2, (I*E^(I*ArcSin[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g^2])])/(g^6*Sqrt[1 - c^2*x^2]) + (5*b*c^2*d^2*f*(c^2*f^2 - g^2)^(3/2)*Sqrt[d - c^2*d*x^2]*PolyLog[2, (I*E^(I*ArcSin[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2])])/(g^6*Sqrt[1 - c^2*x^2])} 

# p<0
[(f+g*x)^3*(a+b*arcsin(c*x))/sqrt(d-c^2*d*x^2),x,13,-3*f^2*g*(1-c^2*x^2)*(a+b*arcsin(c*x))/(c^2*sqrt(d-c^2*d*x^2))-2/3*g^3*(1-c^2*x^2)*(a+b*arcsin(c*x))/(c^4*sqrt(d-c^2*d*x^2))-3/2*f*g^2*x*(1-c^2*x^2)*(a+b*arcsin(c*x))/(c^2*sqrt(d-c^2*d*x^2))-1/3*g^3*x^2*(1-c^2*x^2)*(a+b*arcsin(c*x))/(c^2*sqrt(d-c^2*d*x^2))+3*b*f^2*g*x*sqrt(1-c^2*x^2)/(c*sqrt(d-c^2*d*x^2))+2/3*b*g^3*x*sqrt(1-c^2*x^2)/(c^3*sqrt(d-c^2*d*x^2))+3/4*b*f*g^2*x^2*sqrt(1-c^2*x^2)/(c*sqrt(d-c^2*d*x^2))+1/9*b*g^3*x^3*sqrt(1-c^2*x^2)/(c*sqrt(d-c^2*d*x^2))+1/2*f^3*(a+b*arcsin(c*x))^2*sqrt(1-c^2*x^2)/(b*c*sqrt(d-c^2*d*x^2))+3/4*f*g^2*(a+b*arcsin(c*x))^2*sqrt(1-c^2*x^2)/(b*c^3*sqrt(d-c^2*d*x^2))],
[(f+g*x)^2*(a+b*arcsin(c*x))/sqrt(d-c^2*d*x^2),x,9,-2*f*g*(1-c^2*x^2)*(a+b*arcsin(c*x))/(c^2*sqrt(d-c^2*d*x^2))-1/2*g^2*x*(1-c^2*x^2)*(a+b*arcsin(c*x))/(c^2*sqrt(d-c^2*d*x^2))+2*b*f*g*x*sqrt(1-c^2*x^2)/(c*sqrt(d-c^2*d*x^2))+1/4*b*g^2*x^2*sqrt(1-c^2*x^2)/(c*sqrt(d-c^2*d*x^2))+1/2*f^2*(a+b*arcsin(c*x))^2*sqrt(1-c^2*x^2)/(b*c*sqrt(d-c^2*d*x^2))+1/4*g^2*(a+b*arcsin(c*x))^2*sqrt(1-c^2*x^2)/(b*c^3*sqrt(d-c^2*d*x^2))],
[(f+g*x)*(a+b*arcsin(c*x))/sqrt(d-c^2*d*x^2),x,6,-g*(1-c^2*x^2)*(a+b*arcsin(c*x))/(c^2*sqrt(d-c^2*d*x^2))+b*g*x*sqrt(1-c^2*x^2)/(c*sqrt(d-c^2*d*x^2))+1/2*f*(a+b*arcsin(c*x))^2*sqrt(1-c^2*x^2)/(b*c*sqrt(d-c^2*d*x^2))],
[(a+b*arcsin(c*x))/((f+g*x)*sqrt(d-c^2*d*x^2)),x,10,-I*(a+b*arcsin(c*x))*log(1-I*exp(I*arcsin(c*x))*g/(c*f-sqrt(c^2*f^2-g^2)))*sqrt(1-c^2*x^2)/(sqrt(c^2*f^2-g^2)*sqrt(d-c^2*d*x^2))+I*(a+b*arcsin(c*x))*log(1-I*exp(I*arcsin(c*x))*g/(c*f+sqrt(c^2*f^2-g^2)))*sqrt(1-c^2*x^2)/(sqrt(c^2*f^2-g^2)*sqrt(d-c^2*d*x^2))-b*polylog(2,I*exp(I*arcsin(c*x))*g/(c*f-sqrt(c^2*f^2-g^2)))*sqrt(1-c^2*x^2)/(sqrt(c^2*f^2-g^2)*sqrt(d-c^2*d*x^2))+b*polylog(2,I*exp(I*arcsin(c*x))*g/(c*f+sqrt(c^2*f^2-g^2)))*sqrt(1-c^2*x^2)/(sqrt(c^2*f^2-g^2)*sqrt(d-c^2*d*x^2))],
[(a+b*arcsin(c*x))/((f+g*x)^2*sqrt(d-c^2*d*x^2)),x,13,g*(1-c^2*x^2)*(a+b*arcsin(c*x))/((c^2*f^2-g^2)*(f+g*x)*sqrt(d-c^2*d*x^2))-b*c*log(f+g*x)*sqrt(1-c^2*x^2)/((c^2*f^2-g^2)*sqrt(d-c^2*d*x^2))-I*c^2*f*(a+b*arcsin(c*x))*log(1-I*exp(I*arcsin(c*x))*g/(c*f-sqrt(c^2*f^2-g^2)))*sqrt(1-c^2*x^2)/((c^2*f^2-g^2)^(3/2)*sqrt(d-c^2*d*x^2))+I*c^2*f*(a+b*arcsin(c*x))*log(1-I*exp(I*arcsin(c*x))*g/(c*f+sqrt(c^2*f^2-g^2)))*sqrt(1-c^2*x^2)/((c^2*f^2-g^2)^(3/2)*sqrt(d-c^2*d*x^2))-b*c^2*f*polylog(2,I*exp(I*arcsin(c*x))*g/(c*f-sqrt(c^2*f^2-g^2)))*sqrt(1-c^2*x^2)/((c^2*f^2-g^2)^(3/2)*sqrt(d-c^2*d*x^2))+b*c^2*f*polylog(2,I*exp(I*arcsin(c*x))*g/(c*f+sqrt(c^2*f^2-g^2)))*sqrt(1-c^2*x^2)/((c^2*f^2-g^2)^(3/2)*sqrt(d-c^2*d*x^2))],
[(f+g*x)^3*(a+b*arcsin(c*x))/(d-c^2*d*x^2)^(3/2),x,11,(g*(3*c^2*f^2+g^2)+c^2*f*(c^2*f^2+3*g^2)*x)*(a+b*arcsin(c*x))/(c^4*d*sqrt(d-c^2*d*x^2))+g^3*(1-c^2*x^2)*(a+b*arcsin(c*x))/(c^4*d*sqrt(d-c^2*d*x^2))-b*g^3*x*sqrt(1-c^2*x^2)/(c^3*d*sqrt(d-c^2*d*x^2))-3/2*f*g^2*(a+b*arcsin(c*x))^2*sqrt(1-c^2*x^2)/(b*c^3*d*sqrt(d-c^2*d*x^2))+1/2*b*(c*f+g)^3*log(1-c*x)*sqrt(1-c^2*x^2)/(c^4*d*sqrt(d-c^2*d*x^2))+1/2*b*(c*f-g)^3*log(1+c*x)*sqrt(1-c^2*x^2)/(c^4*d*sqrt(d-c^2*d*x^2))],
[(f+g*x)^2*(a+b*arcsin(c*x))/(d-c^2*d*x^2)^(3/2),x,8,(2*f*g+(c^2*f^2+g^2)*x)*(a+b*arcsin(c*x))/(c^2*d*sqrt(d-c^2*d*x^2))-1/2*g^2*(a+b*arcsin(c*x))^2*sqrt(1-c^2*x^2)/(b*c^3*d*sqrt(d-c^2*d*x^2))+1/2*b*(c*f+g)^2*log(1-c*x)*sqrt(1-c^2*x^2)/(c^3*d*sqrt(d-c^2*d*x^2))+1/2*b*(c*f-g)^2*log(1+c*x)*sqrt(1-c^2*x^2)/(c^3*d*sqrt(d-c^2*d*x^2))],
[(f+g*x)*(a+b*arcsin(c*x))/(d-c^2*d*x^2)^(3/2),x,6,(g+c^2*f*x)*(a+b*arcsin(c*x))/(c^2*d*sqrt(d-c^2*d*x^2))+1/2*b*(c*f+g)*log(1-c*x)*sqrt(1-c^2*x^2)/(c^2*d*sqrt(d-c^2*d*x^2))+1/2*b*(c*f-g)*log(1+c*x)*sqrt(1-c^2*x^2)/(c^2*d*sqrt(d-c^2*d*x^2))],
[(a+b*arcsin(c*x))/((f+g*x)*(d-c^2*d*x^2)^(3/2)),x,20,-1/2*(a+b*arcsin(c*x))*cot(1/4*Pi+1/2*arcsin(c*x))*sqrt(1-c^2*x^2)/(d*(c*f-g)*sqrt(d-c^2*d*x^2))+b*log(cos(1/4*Pi+1/2*arcsin(c*x)))*sqrt(1-c^2*x^2)/(d*(c*f+g)*sqrt(d-c^2*d*x^2))+b*log(sin(1/4*Pi+1/2*arcsin(c*x)))*sqrt(1-c^2*x^2)/(d*(c*f-g)*sqrt(d-c^2*d*x^2))+I*g^2*(a+b*arcsin(c*x))*log(1-I*exp(I*arcsin(c*x))*g/(c*f-sqrt(c^2*f^2-g^2)))*sqrt(1-c^2*x^2)/(d*(c^2*f^2-g^2)^(3/2)*sqrt(d-c^2*d*x^2))-I*g^2*(a+b*arcsin(c*x))*log(1-I*exp(I*arcsin(c*x))*g/(c*f+sqrt(c^2*f^2-g^2)))*sqrt(1-c^2*x^2)/(d*(c^2*f^2-g^2)^(3/2)*sqrt(d-c^2*d*x^2))+b*g^2*polylog(2,I*exp(I*arcsin(c*x))*g/(c*f-sqrt(c^2*f^2-g^2)))*sqrt(1-c^2*x^2)/(d*(c^2*f^2-g^2)^(3/2)*sqrt(d-c^2*d*x^2))-b*g^2*polylog(2,I*exp(I*arcsin(c*x))*g/(c*f+sqrt(c^2*f^2-g^2)))*sqrt(1-c^2*x^2)/(d*(c^2*f^2-g^2)^(3/2)*sqrt(d-c^2*d*x^2))+1/2*(a+b*arcsin(c*x))*sqrt(1-c^2*x^2)*tan(1/4*Pi+1/2*arcsin(c*x))/(d*(c*f+g)*sqrt(d-c^2*d*x^2))],

#  {1/(d - c^2*d*x^2)^(3/2)*(a + b*ArcSin[c*x])/(f + g*x)^2, x, 47, -((a*g^3*(1 - c^2*x^2))/(d*(c*f - g)^2*(c*f + g)^2*(f + g*x)*Sqrt[d - c^2*d*x^2])) - (b*g^3*(1 - c^2*x^2)*ArcSin[c*x])/(d*(c*f - g)^2*(c*f + g)^2*(f + g*x)*Sqrt[d - c^2*d*x^2]) - (6*a*c^2*f*g^2*Sqrt[1 - c^2*x^2]*ArcTan[(g + c*f*Tan[(1/2)*ArcSin[c*x]])/Sqrt[c^2*f^2 - g^2]])/(d*(c*f - g)^2*(c*f + g)^2*Sqrt[c^2*f^2 - g^2]*Sqrt[d - c^2*d*x^2]) + (3*I*b*c^2*f*g^2*Sqrt[1 - c^2*x^2]*ArcSin[c*x]*Log[1 - (I*E^(I*ArcSin[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g^2])])/(d*(c*f - g)^2*(c*f + g)^2*Sqrt[c^2*f^2 - g^2]*Sqrt[d - c^2*d*x^2]) - (3*I*b*c^2*f*g^2*Sqrt[1 - c^2*x^2]*ArcSin[c*x]*Log[1 - (I*E^(I*ArcSin[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2])])/(d*(c*f - g)^2*(c*f + g)^2*Sqrt[c^2*f^2 - g^2]*Sqrt[d - c^2*d*x^2]) + (b*c*g^2*Sqrt[1 - c^2*x^2]*Log[f + g*x])/(d*(c*f - g)^2*(c*f + g)^2*Sqrt[d - c^2*d*x^2]) + (b*c*Sqrt[1 - c^2*x^2]*Log[Cos[Pi/4 - (1/2)*ArcSin[c*x]]])/(d*(c*f - g)^2*Sqrt[d - c^2*d*x^2]) + (b*c*Sqrt[1 - c^2*x^2]*Log[Cos[Pi/4 + (1/2)*ArcSin[c*x]]])/(d*(c*f + g)^2*Sqrt[d - c^2*d*x^2]) + (3*b*c^2*f*g^2*Sqrt[1 - c^2*x^2]*PolyLog[2, (I*E^(I*ArcSin[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g^2])])/(d*(c*f - g)^2*(c*f + g)^2*Sqrt[c^2*f^2 - g^2]*Sqrt[d - c^2*d*x^2]) - (3*b*c^2*f*g^2*Sqrt[1 - c^2*x^2]*PolyLog[2, (I*E^(I*ArcSin[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2])])/(d*(c*f - g)^2*(c*f + g)^2*Sqrt[c^2*f^2 - g^2]*Sqrt[d - c^2*d*x^2]) - (c*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*Tan[Pi/4 - (1/2)*ArcSin[c*x]])/(2*d*(c*f - g)^2*Sqrt[d - c^2*d*x^2]) + (c*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*Tan[Pi/4 + (1/2)*ArcSin[c*x]])/(2*d*(c*f + g)^2*Sqrt[d - c^2*d*x^2])} 
[(f+g*x)^4*(a+b*arcsin(c*x))/(d-c^2*d*x^2)^(5/2),x,13,1/3*(f+g*x)*(g*(c^2*f^2-3*g^2)+2*c^2*f*(c^2*f^2-2*g^2)*x)*(a+b*arcsin(c*x))/(c^4*d^2*sqrt(d-c^2*d*x^2))+1/3*(g+c^2*f*x)*(f+g*x)^3*(a+b*arcsin(c*x))/(c^2*d^2*(1-c^2*x^2)*sqrt(d-c^2*d*x^2))+1/3*f*g*(2*c^2*f^2-5*g^2)*(1-c^2*x^2)*(a+b*arcsin(c*x))/(c^4*d^2*sqrt(d-c^2*d*x^2))-1/6*b*(f+g*x)^2*(c^2*f^2+g^2+2*c^2*f*g*x)/(c^3*d^2*sqrt(1-c^2*x^2)*sqrt(d-c^2*d*x^2))-1/3*b*f*g^3*x*sqrt(1-c^2*x^2)/(c^3*d^2*sqrt(d-c^2*d*x^2))-1/2*b*g^4*arcsin(c*x)^2*sqrt(1-c^2*x^2)/(c^5*d^2*sqrt(d-c^2*d*x^2))+g^4*arcsin(c*x)*(a+b*arcsin(c*x))*sqrt(1-c^2*x^2)/(c^5*d^2*sqrt(d-c^2*d*x^2))+1/3*b*(c*f-2*g)*(c*f+g)^3*log(1-c*x)*sqrt(1-c^2*x^2)/(c^5*d^2*sqrt(d-c^2*d*x^2))+1/3*b*(c*f-g)^3*(c*f+2*g)*log(1+c*x)*sqrt(1-c^2*x^2)/(c^5*d^2*sqrt(d-c^2*d*x^2)),1/3*(f+g*x)*(g*(c^2*f^2-3*g^2)+2*c^2*f*(c^2*f^2-2*g^2)*x)*(a+b*arcsin(c*x))/(c^4*d^2*sqrt(d-c^2*d*x^2))+1/3*(g+c^2*f*x)*(f+g*x)^3*(a+b*arcsin(c*x))/(c^2*d^2*(1-c^2*x^2)*sqrt(d-c^2*d*x^2))+1/3*f*g*(2*c^2*f^2-5*g^2)*(1-c^2*x^2)*(a+b*arcsin(c*x))/(c^4*d^2*sqrt(d-c^2*d*x^2))-1/6*b*(f+g*x)^2*(c^2*f^2+g^2+2*c^2*f*g*x)/(c^3*d^2*sqrt(1-c^2*x^2)*sqrt(d-c^2*d*x^2))-2/3*b*f*g^3*x*sqrt(1-c^2*x^2)/(c^3*d^2*sqrt(d-c^2*d*x^2))-1/3*b*f*g*(2*c^2*f^2-5*g^2)*x*sqrt(1-c^2*x^2)/(c^3*d^2*sqrt(d-c^2*d*x^2))+2/3*b*f*g*(c^2*f^2-2*g^2)*x*sqrt(1-c^2*x^2)/(c^3*d^2*sqrt(d-c^2*d*x^2))-1/2*b*g^4*arcsin(c*x)^2*sqrt(1-c^2*x^2)/(c^5*d^2*sqrt(d-c^2*d*x^2))+g^4*arcsin(c*x)*(a+b*arcsin(c*x))*sqrt(1-c^2*x^2)/(c^5*d^2*sqrt(d-c^2*d*x^2))+1/6*b*(2*c*f-3*g)*(c*f+g)^3*log(1-c*x)*sqrt(1-c^2*x^2)/(c^5*d^2*sqrt(d-c^2*d*x^2))-1/6*b*g*(c*f+g)^3*log(1-c*x)*sqrt(1-c^2*x^2)/(c^5*d^2*sqrt(d-c^2*d*x^2))+1/6*b*(c*f-g)^3*g*log(1+c*x)*sqrt(1-c^2*x^2)/(c^5*d^2*sqrt(d-c^2*d*x^2))+1/6*b*(c*f-g)^3*(2*c*f+3*g)*log(1+c*x)*sqrt(1-c^2*x^2)/(c^5*d^2*sqrt(d-c^2*d*x^2))],
[(f+g*x)^3*(a+b*arcsin(c*x))/(d-c^2*d*x^2)^(5/2),x,10,2/3*(c*f-g)*(c*f+g)*(g+c^2*f*x)*(a+b*arcsin(c*x))/(c^4*d^2*sqrt(d-c^2*d*x^2))+1/3*(g+c^2*f*x)*(f+g*x)^2*(a+b*arcsin(c*x))/(c^2*d^2*(1-c^2*x^2)*sqrt(d-c^2*d*x^2))-1/6*b*(f+g*x)*(c^2*f^2+g^2+2*c^2*f*g*x)/(c^3*d^2*sqrt(1-c^2*x^2)*sqrt(d-c^2*d*x^2))+1/3*b*(c*f-g)*(c*f+g)^2*log(1-c*x)*sqrt(1-c^2*x^2)/(c^4*d^2*sqrt(d-c^2*d*x^2))-1/12*b*g*(c*f+g)^2*log(1-c*x)*sqrt(1-c^2*x^2)/(c^4*d^2*sqrt(d-c^2*d*x^2))+1/12*b*(c*f-g)^2*g*log(1+c*x)*sqrt(1-c^2*x^2)/(c^4*d^2*sqrt(d-c^2*d*x^2))+1/3*b*(c*f-g)^2*(c*f+g)*log(1+c*x)*sqrt(1-c^2*x^2)/(c^4*d^2*sqrt(d-c^2*d*x^2))],
[(f+g*x)^2*(a+b*arcsin(c*x))/(d-c^2*d*x^2)^(5/2),x,10,2/3*f*(g+c^2*f*x)*(a+b*arcsin(c*x))/(c^2*d^2*sqrt(d-c^2*d*x^2))+1/3*x*(f+g*x)^2*(a+b*arcsin(c*x))/(d^2*(1-c^2*x^2)*sqrt(d-c^2*d*x^2))-1/6*b*x*(2*f*g+(c^2*f^2+g^2)*x)/(c*d^2*sqrt(1-c^2*x^2)*sqrt(d-c^2*d*x^2))+1/6*b*(2*c*f-g)*(c*f+g)*log(1-c*x)*sqrt(1-c^2*x^2)/(c^3*d^2*sqrt(d-c^2*d*x^2))+1/6*b*(c*f-g)*(2*c*f+g)*log(1+c*x)*sqrt(1-c^2*x^2)/(c^3*d^2*sqrt(d-c^2*d*x^2)),2/3*f*(g+c^2*f*x)*(a+b*arcsin(c*x))/(c^2*d^2*sqrt(d-c^2*d*x^2))+1/3*x*(f+g*x)^2*(a+b*arcsin(c*x))/(d^2*(1-c^2*x^2)*sqrt(d-c^2*d*x^2))-1/6*b*(f+g*x)^2/(c*d^2*sqrt(1-c^2*x^2)*sqrt(d-c^2*d*x^2))+1/3*b*f*(c*f+g)*log(1-c*x)*sqrt(1-c^2*x^2)/(c^2*d^2*sqrt(d-c^2*d*x^2))-1/6*b*g*(c*f+g)*log(1-c*x)*sqrt(1-c^2*x^2)/(c^3*d^2*sqrt(d-c^2*d*x^2))+1/3*b*f*(c*f-g)*log(1+c*x)*sqrt(1-c^2*x^2)/(c^2*d^2*sqrt(d-c^2*d*x^2))+1/6*b*(c*f-g)*g*log(1+c*x)*sqrt(1-c^2*x^2)/(c^3*d^2*sqrt(d-c^2*d*x^2))],
[(f+g*x)*(a+b*arcsin(c*x))/(d-c^2*d*x^2)^(5/2),x,6,2/3*f*x*(a+b*arcsin(c*x))/(d^2*sqrt(d-c^2*d*x^2))+1/3*(g+c^2*f*x)*(a+b*arcsin(c*x))/(c^2*d^2*(1-c^2*x^2)*sqrt(d-c^2*d*x^2))-1/6*b*(f+g*x)/(c*d^2*sqrt(1-c^2*x^2)*sqrt(d-c^2*d*x^2))-1/6*b*g*arctanh(c*x)*sqrt(1-c^2*x^2)/(c^2*d^2*sqrt(d-c^2*d*x^2))+1/3*b*f*log(1-c^2*x^2)*sqrt(1-c^2*x^2)/(c*d^2*sqrt(d-c^2*d*x^2))],
[(a+b*arcsin(c*x))/((f+g*x)*(d-c^2*d*x^2)^(5/2)),x,30,-1/4*(c*f-2*g)*(a+b*arcsin(c*x))*cot(1/4*Pi+1/2*arcsin(c*x))*sqrt(1-c^2*x^2)/(d^2*(c*f-g)^2*sqrt(d-c^2*d*x^2))-1/12*(a+b*arcsin(c*x))*cot(1/4*Pi+1/2*arcsin(c*x))*sqrt(1-c^2*x^2)/(d^2*(c*f-g)*sqrt(d-c^2*d*x^2))-1/24*b*csc(1/4*Pi+1/2*arcsin(c*x))^2*sqrt(1-c^2*x^2)/(d^2*(c*f-g)*sqrt(d-c^2*d*x^2))-1/24*(a+b*arcsin(c*x))*cot(1/4*Pi+1/2*arcsin(c*x))*csc(1/4*Pi+1/2*arcsin(c*x))^2*sqrt(1-c^2*x^2)/(d^2*(c*f-g)*sqrt(d-c^2*d*x^2))+1/6*b*log(cos(1/4*Pi+1/2*arcsin(c*x)))*sqrt(1-c^2*x^2)/(d^2*(c*f+g)*sqrt(d-c^2*d*x^2))+1/2*b*(c*f+2*g)*log(cos(1/4*Pi+1/2*arcsin(c*x)))*sqrt(1-c^2*x^2)/(d^2*(c*f+g)^2*sqrt(d-c^2*d*x^2))+1/2*b*(c*f-2*g)*log(sin(1/4*Pi+1/2*arcsin(c*x)))*sqrt(1-c^2*x^2)/(d^2*(c*f-g)^2*sqrt(d-c^2*d*x^2))+1/6*b*log(sin(1/4*Pi+1/2*arcsin(c*x)))*sqrt(1-c^2*x^2)/(d^2*(c*f-g)*sqrt(d-c^2*d*x^2))-I*g^4*(a+b*arcsin(c*x))*log(1-I*exp(I*arcsin(c*x))*g/(c*f-sqrt(c^2*f^2-g^2)))*sqrt(1-c^2*x^2)/(d^2*(c^2*f^2-g^2)^(5/2)*sqrt(d-c^2*d*x^2))+I*g^4*(a+b*arcsin(c*x))*log(1-I*exp(I*arcsin(c*x))*g/(c*f+sqrt(c^2*f^2-g^2)))*sqrt(1-c^2*x^2)/(d^2*(c^2*f^2-g^2)^(5/2)*sqrt(d-c^2*d*x^2))-b*g^4*polylog(2,I*exp(I*arcsin(c*x))*g/(c*f-sqrt(c^2*f^2-g^2)))*sqrt(1-c^2*x^2)/(d^2*(c^2*f^2-g^2)^(5/2)*sqrt(d-c^2*d*x^2))+b*g^4*polylog(2,I*exp(I*arcsin(c*x))*g/(c*f+sqrt(c^2*f^2-g^2)))*sqrt(1-c^2*x^2)/(d^2*(c^2*f^2-g^2)^(5/2)*sqrt(d-c^2*d*x^2))-1/24*b*sec(1/4*Pi+1/2*arcsin(c*x))^2*sqrt(1-c^2*x^2)/(d^2*(c*f+g)*sqrt(d-c^2*d*x^2))+1/12*(a+b*arcsin(c*x))*sqrt(1-c^2*x^2)*tan(1/4*Pi+1/2*arcsin(c*x))/(d^2*(c*f+g)*sqrt(d-c^2*d*x^2))+1/4*(c*f+2*g)*(a+b*arcsin(c*x))*sqrt(1-c^2*x^2)*tan(1/4*Pi+1/2*arcsin(c*x))/(d^2*(c*f+g)^2*sqrt(d-c^2*d*x^2))+1/24*(a+b*arcsin(c*x))*sec(1/4*Pi+1/2*arcsin(c*x))^2*sqrt(1-c^2*x^2)*tan(1/4*Pi+1/2*arcsin(c*x))/(d^2*(c*f+g)*sqrt(d-c^2*d*x^2))],

#  {1/(d - c^2*d*x^2)^(5/2)*(a + b*ArcSin[c*x])/(f + g*x)^2, x, 57, (a*g^5*(1 - c^2*x^2))/(d^2*(c*f - g)^3*(c*f + g)^3*(f + g*x)*Sqrt[d - c^2*d*x^2]) + (b*g^5*(1 - c^2*x^2)*ArcSin[c*x])/(d^2*(c*f - g)^3*(c*f + g)^3*(f + g*x)*Sqrt[d - c^2*d*x^2]) + (10*a*c^2*f*g^4*Sqrt[1 - c^2*x^2]*ArcTan[(g + c*f*Tan[(1/2)*ArcSin[c*x]])/Sqrt[c^2*f^2 - g^2]])/(d^2*(c*f - g)^3*(c*f + g)^3*Sqrt[c^2*f^2 - g^2]*Sqrt[d - c^2*d*x^2]) - (5*I*b*c^2*f*g^4*Sqrt[1 - c^2*x^2]*ArcSin[c*x]*Log[1 - (I*E^(I*ArcSin[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g^2])])/(d^2*(c*f - g)^3*(c*f + g)^3*Sqrt[c^2*f^2 - g^2]*Sqrt[d - c^2*d*x^2]) + (5*I*b*c^2*f*g^4*Sqrt[1 - c^2*x^2]*ArcSin[c*x]*Log[1 - (I*E^(I*ArcSin[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2])])/(d^2*(c*f - g)^3*(c*f + g)^3*Sqrt[c^2*f^2 - g^2]*Sqrt[d - c^2*d*x^2]) - (b*c*g^4*Sqrt[1 - c^2*x^2]*Log[f + g*x])/(d^2*(c*f - g)^3*(c*f + g)^3*Sqrt[d - c^2*d*x^2]) + (b*c*(c*f - 3*g)*Sqrt[1 - c^2*x^2]*Log[Cos[Pi/4 - (1/2)*ArcSin[c*x]]])/(2*d^2*(c*f - g)^3*Sqrt[d - c^2*d*x^2]) + (b*c*Sqrt[1 - c^2*x^2]*Log[Cos[Pi/4 - (1/2)*ArcSin[c*x]]])/(6*d^2*(c*f - g)^2*Sqrt[d - c^2*d*x^2]) + (b*c*Sqrt[1 - c^2*x^2]*Log[Cos[Pi/4 + (1/2)*ArcSin[c*x]]])/(6*d^2*(c*f + g)^2*Sqrt[d - c^2*d*x^2]) + (b*c*(c*f + 3*g)*Sqrt[1 - c^2*x^2]*Log[Cos[Pi/4 + (1/2)*ArcSin[c*x]]])/(2*d^2*(c*f + g)^3*Sqrt[d - c^2*d*x^2]) - (5*b*c^2*f*g^4*Sqrt[1 - c^2*x^2]*PolyLog[2, (I*E^(I*ArcSin[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g^2])])/(d^2*(c*f - g)^3*(c*f + g)^3*Sqrt[c^2*f^2 - g^2]*Sqrt[d - c^2*d*x^2]) + (5*b*c^2*f*g^4*Sqrt[1 - c^2*x^2]*PolyLog[2, (I*E^(I*ArcSin[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2])])/(d^2*(c*f - g)^3*(c*f + g)^3*Sqrt[c^2*f^2 - g^2]*Sqrt[d - c^2*d*x^2]) - (b*c*Sqrt[1 - c^2*x^2]*Sec[Pi/4 - (1/2)*ArcSin[c*x]]^2)/(24*d^2*(c*f - g)^2*Sqrt[d - c^2*d*x^2]) - (b*c*Sqrt[1 - c^2*x^2]*Sec[Pi/4 + (1/2)*ArcSin[c*x]]^2)/(24*d^2*(c*f + g)^2*Sqrt[d - c^2*d*x^2]) - (c*(c*f - 3*g)*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*Tan[Pi/4 - (1/2)*ArcSin[c*x]])/(4*d^2*(c*f - g)^3*Sqrt[d - c^2*d*x^2]) - (c*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*Tan[Pi/4 - (1/2)*ArcSin[c*x]])/(12*d^2*(c*f - g)^2*Sqrt[d - c^2*d*x^2]) - (c*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*Sec[Pi/4 - (1/2)*ArcSin[c*x]]^2*Tan[Pi/4 - (1/2)*ArcSin[c*x]])/(24*d^2*(c*f - g)^2*Sqrt[d - c^2*d*x^2]) + (c*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*Tan[Pi/4 + (1/2)*ArcSin[c*x]])/(12*d^2*(c*f + g)^2*Sqrt[d - c^2*d*x^2]) + (c*(c*f + 3*g)*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*Tan[Pi/4 + (1/2)*ArcSin[c*x]])/(4*d^2*(c*f + g)^3*Sqrt[d - c^2*d*x^2]) + (c*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*Sec[Pi/4 + (1/2)*ArcSin[c*x]]^2*Tan[Pi/4 + (1/2)*ArcSin[c*x]])/(24*d^2*(c*f + g)^2*Sqrt[d - c^2*d*x^2])} 

# Integrands of the form (f+g x)^m (d-c^2 d x^2)^(p/2) (a+b ArcSin[c x])^2

# p>0
[(f+g*x)^3*(a+b*arcsin(c*x))^2*sqrt(d-c^2*d*x^2),x,37,4/3*b^2*f^2*g*sqrt(d-c^2*d*x^2)/c^2+52/225*b^2*g^3*sqrt(d-c^2*d*x^2)/c^4-1/4*b^2*f^3*x*sqrt(d-c^2*d*x^2)+3/64*b^2*f*g^2*x*sqrt(d-c^2*d*x^2)/c^2-3/32*b^2*f*g^2*x^3*sqrt(d-c^2*d*x^2)+2/9*b^2*f^2*g*(1-c^2*x^2)*sqrt(d-c^2*d*x^2)/c^2+26/675*b^2*g^3*(1-c^2*x^2)*sqrt(d-c^2*d*x^2)/c^4-2/125*b^2*g^3*(1-c^2*x^2)^2*sqrt(d-c^2*d*x^2)/c^4-2/15*g^3*(a+b*arcsin(c*x))^2*sqrt(d-c^2*d*x^2)/c^4+1/2*f^3*x*(a+b*arcsin(c*x))^2*sqrt(d-c^2*d*x^2)-3/8*f*g^2*x*(a+b*arcsin(c*x))^2*sqrt(d-c^2*d*x^2)/c^2-1/15*g^3*x^2*(a+b*arcsin(c*x))^2*sqrt(d-c^2*d*x^2)/c^2+3/4*f*g^2*x^3*(a+b*arcsin(c*x))^2*sqrt(d-c^2*d*x^2)+1/5*g^3*x^4*(a+b*arcsin(c*x))^2*sqrt(d-c^2*d*x^2)-f^2*g*(1-c^2*x^2)*(a+b*arcsin(c*x))^2*sqrt(d-c^2*d*x^2)/c^2+4/15*a*b*g^3*x*sqrt(d-c^2*d*x^2)/(c^3*sqrt(1-c^2*x^2))+1/4*b^2*f^3*arcsin(c*x)*sqrt(d-c^2*d*x^2)/(c*sqrt(1-c^2*x^2))-3/64*b^2*f*g^2*arcsin(c*x)*sqrt(d-c^2*d*x^2)/(c^3*sqrt(1-c^2*x^2))+4/15*b^2*g^3*x*arcsin(c*x)*sqrt(d-c^2*d*x^2)/(c^3*sqrt(1-c^2*x^2))+2*b*f^2*g*x*(a+b*arcsin(c*x))*sqrt(d-c^2*d*x^2)/(c*sqrt(1-c^2*x^2))-1/2*b*c*f^3*x^2*(a+b*arcsin(c*x))*sqrt(d-c^2*d*x^2)/sqrt(1-c^2*x^2)+3/8*b*f*g^2*x^2*(a+b*arcsin(c*x))*sqrt(d-c^2*d*x^2)/(c*sqrt(1-c^2*x^2))-2/3*b*c*f^2*g*x^3*(a+b*arcsin(c*x))*sqrt(d-c^2*d*x^2)/sqrt(1-c^2*x^2)+2/45*b*g^3*x^3*(a+b*arcsin(c*x))*sqrt(d-c^2*d*x^2)/(c*sqrt(1-c^2*x^2))-3/8*b*c*f*g^2*x^4*(a+b*arcsin(c*x))*sqrt(d-c^2*d*x^2)/sqrt(1-c^2*x^2)-2/25*b*c*g^3*x^5*(a+b*arcsin(c*x))*sqrt(d-c^2*d*x^2)/sqrt(1-c^2*x^2)+1/6*f^3*(a+b*arcsin(c*x))^3*sqrt(d-c^2*d*x^2)/(b*c*sqrt(1-c^2*x^2))+1/8*f*g^2*(a+b*arcsin(c*x))^3*sqrt(d-c^2*d*x^2)/(b*c^3*sqrt(1-c^2*x^2))],
[(f+g*x)^2*(a+b*arcsin(c*x))^2*sqrt(d-c^2*d*x^2),x,23,8/9*b^2*f*g*sqrt(d-c^2*d*x^2)/c^2-1/4*b^2*f^2*x*sqrt(d-c^2*d*x^2)+1/64*b^2*g^2*x*sqrt(d-c^2*d*x^2)/c^2-1/32*b^2*g^2*x^3*sqrt(d-c^2*d*x^2)+4/27*b^2*f*g*(1-c^2*x^2)*sqrt(d-c^2*d*x^2)/c^2+1/2*f^2*x*(a+b*arcsin(c*x))^2*sqrt(d-c^2*d*x^2)-1/8*g^2*x*(a+b*arcsin(c*x))^2*sqrt(d-c^2*d*x^2)/c^2+1/4*g^2*x^3*(a+b*arcsin(c*x))^2*sqrt(d-c^2*d*x^2)-2/3*f*g*(1-c^2*x^2)*(a+b*arcsin(c*x))^2*sqrt(d-c^2*d*x^2)/c^2+1/4*b^2*f^2*arcsin(c*x)*sqrt(d-c^2*d*x^2)/(c*sqrt(1-c^2*x^2))-1/64*b^2*g^2*arcsin(c*x)*sqrt(d-c^2*d*x^2)/(c^3*sqrt(1-c^2*x^2))+4/3*b*f*g*x*(a+b*arcsin(c*x))*sqrt(d-c^2*d*x^2)/(c*sqrt(1-c^2*x^2))-1/2*b*c*f^2*x^2*(a+b*arcsin(c*x))*sqrt(d-c^2*d*x^2)/sqrt(1-c^2*x^2)+1/8*b*g^2*x^2*(a+b*arcsin(c*x))*sqrt(d-c^2*d*x^2)/(c*sqrt(1-c^2*x^2))-4/9*b*c*f*g*x^3*(a+b*arcsin(c*x))*sqrt(d-c^2*d*x^2)/sqrt(1-c^2*x^2)-1/8*b*c*g^2*x^4*(a+b*arcsin(c*x))*sqrt(d-c^2*d*x^2)/sqrt(1-c^2*x^2)+1/6*f^2*(a+b*arcsin(c*x))^3*sqrt(d-c^2*d*x^2)/(b*c*sqrt(1-c^2*x^2))+1/24*g^2*(a+b*arcsin(c*x))^3*sqrt(d-c^2*d*x^2)/(b*c^3*sqrt(1-c^2*x^2))],
[(f+g*x)*(a+b*arcsin(c*x))^2*sqrt(d-c^2*d*x^2),x,13,4/9*b^2*g*sqrt(d-c^2*d*x^2)/c^2-1/4*b^2*f*x*sqrt(d-c^2*d*x^2)+2/27*b^2*g*(1-c^2*x^2)*sqrt(d-c^2*d*x^2)/c^2+1/2*f*x*(a+b*arcsin(c*x))^2*sqrt(d-c^2*d*x^2)-1/3*g*(1-c^2*x^2)*(a+b*arcsin(c*x))^2*sqrt(d-c^2*d*x^2)/c^2+1/4*b^2*f*arcsin(c*x)*sqrt(d-c^2*d*x^2)/(c*sqrt(1-c^2*x^2))+2/3*b*g*x*(a+b*arcsin(c*x))*sqrt(d-c^2*d*x^2)/(c*sqrt(1-c^2*x^2))-1/2*b*c*f*x^2*(a+b*arcsin(c*x))*sqrt(d-c^2*d*x^2)/sqrt(1-c^2*x^2)-2/9*b*c*g*x^3*(a+b*arcsin(c*x))*sqrt(d-c^2*d*x^2)/sqrt(1-c^2*x^2)+1/6*f*(a+b*arcsin(c*x))^3*sqrt(d-c^2*d*x^2)/(b*c*sqrt(1-c^2*x^2))],
[(a+b*arcsin(c*x))^2*sqrt(d-c^2*d*x^2)/(f+g*x),x,38,a^2*sqrt(d-c^2*d*x^2)/g-2*b^2*sqrt(d-c^2*d*x^2)/g+2*a*b*arcsin(c*x)*sqrt(d-c^2*d*x^2)/g+b^2*arcsin(c*x)^2*sqrt(d-c^2*d*x^2)/g-2*a*b*c*x*sqrt(d-c^2*d*x^2)/(g*sqrt(1-c^2*x^2))-2*b^2*c*x*arcsin(c*x)*sqrt(d-c^2*d*x^2)/(g*sqrt(1-c^2*x^2))+1/3*c*x*(a+b*arcsin(c*x))^3*sqrt(d-c^2*d*x^2)/(b*g*sqrt(1-c^2*x^2))-1/3*(1-c^2*f^2/g^2)*(a+b*arcsin(c*x))^3*sqrt(d-c^2*d*x^2)/(b*c*(f+g*x)*sqrt(1-c^2*x^2))-a^2*arctan((g+c^2*f*x)/(sqrt(c^2*f^2-g^2)*sqrt(1-c^2*x^2)))*sqrt(c^2*f^2-g^2)*sqrt(d-c^2*d*x^2)/(g^2*sqrt(1-c^2*x^2))+2*I*a*b*arcsin(c*x)*log(1-I*exp(I*arcsin(c*x))*g/(c*f-sqrt(c^2*f^2-g^2)))*sqrt(c^2*f^2-g^2)*sqrt(d-c^2*d*x^2)/(g^2*sqrt(1-c^2*x^2))+I*b^2*arcsin(c*x)^2*log(1-I*exp(I*arcsin(c*x))*g/(c*f-sqrt(c^2*f^2-g^2)))*sqrt(c^2*f^2-g^2)*sqrt(d-c^2*d*x^2)/(g^2*sqrt(1-c^2*x^2))-2*I*a*b*arcsin(c*x)*log(1-I*exp(I*arcsin(c*x))*g/(c*f+sqrt(c^2*f^2-g^2)))*sqrt(c^2*f^2-g^2)*sqrt(d-c^2*d*x^2)/(g^2*sqrt(1-c^2*x^2))-I*b^2*arcsin(c*x)^2*log(1-I*exp(I*arcsin(c*x))*g/(c*f+sqrt(c^2*f^2-g^2)))*sqrt(c^2*f^2-g^2)*sqrt(d-c^2*d*x^2)/(g^2*sqrt(1-c^2*x^2))+2*a*b*polylog(2,I*exp(I*arcsin(c*x))*g/(c*f-sqrt(c^2*f^2-g^2)))*sqrt(c^2*f^2-g^2)*sqrt(d-c^2*d*x^2)/(g^2*sqrt(1-c^2*x^2))+2*b^2*arcsin(c*x)*polylog(2,I*exp(I*arcsin(c*x))*g/(c*f-sqrt(c^2*f^2-g^2)))*sqrt(c^2*f^2-g^2)*sqrt(d-c^2*d*x^2)/(g^2*sqrt(1-c^2*x^2))-2*a*b*polylog(2,I*exp(I*arcsin(c*x))*g/(c*f+sqrt(c^2*f^2-g^2)))*sqrt(c^2*f^2-g^2)*sqrt(d-c^2*d*x^2)/(g^2*sqrt(1-c^2*x^2))-2*b^2*arcsin(c*x)*polylog(2,I*exp(I*arcsin(c*x))*g/(c*f+sqrt(c^2*f^2-g^2)))*sqrt(c^2*f^2-g^2)*sqrt(d-c^2*d*x^2)/(g^2*sqrt(1-c^2*x^2))+2*I*b^2*polylog(3,I*exp(I*arcsin(c*x))*g/(c*f-sqrt(c^2*f^2-g^2)))*sqrt(c^2*f^2-g^2)*sqrt(d-c^2*d*x^2)/(g^2*sqrt(1-c^2*x^2))-2*I*b^2*polylog(3,I*exp(I*arcsin(c*x))*g/(c*f+sqrt(c^2*f^2-g^2)))*sqrt(c^2*f^2-g^2)*sqrt(d-c^2*d*x^2)/(g^2*sqrt(1-c^2*x^2))+1/3*(a+b*arcsin(c*x))^3*sqrt(1-c^2*x^2)*sqrt(d-c^2*d*x^2)/(b*c*(f+g*x))],

#  {Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2/(f + g*x)^2, x, 0, 0} 
[(f+g*x)^3*(d-c^2*d*x^2)^(3/2)*(a+b*arcsin(c*x))^2,x,56,16/25*b^2*d*f^2*g*sqrt(d-c^2*d*x^2)/c^2+304/3675*b^2*d*g^3*sqrt(d-c^2*d*x^2)/c^4-15/64*b^2*d*f^3*x*sqrt(d-c^2*d*x^2)-7/384*b^2*d*f*g^2*x*sqrt(d-c^2*d*x^2)/c^2-43/576*b^2*d*f*g^2*x^3*sqrt(d-c^2*d*x^2)+1/36*b^2*c^2*d*f*g^2*x^5*sqrt(d-c^2*d*x^2)+8/75*b^2*d*f^2*g*(1-c^2*x^2)*sqrt(d-c^2*d*x^2)/c^2+152/11025*b^2*d*g^3*(1-c^2*x^2)*sqrt(d-c^2*d*x^2)/c^4-1/32*b^2*d*f^3*x*(1-c^2*x^2)*sqrt(d-c^2*d*x^2)+6/125*b^2*d*f^2*g*(1-c^2*x^2)^2*sqrt(d-c^2*d*x^2)/c^2+38/6125*b^2*d*g^3*(1-c^2*x^2)^2*sqrt(d-c^2*d*x^2)/c^4-2/343*b^2*d*g^3*(1-c^2*x^2)^3*sqrt(d-c^2*d*x^2)/c^4+1/8*b*d*f^3*(1-c^2*x^2)^(3/2)*(a+b*arcsin(c*x))*sqrt(d-c^2*d*x^2)/c-2/35*d*g^3*(a+b*arcsin(c*x))^2*sqrt(d-c^2*d*x^2)/c^4+3/8*d*f^3*x*(a+b*arcsin(c*x))^2*sqrt(d-c^2*d*x^2)-3/16*d*f*g^2*x*(a+b*arcsin(c*x))^2*sqrt(d-c^2*d*x^2)/c^2-1/35*d*g^3*x^2*(a+b*arcsin(c*x))^2*sqrt(d-c^2*d*x^2)/c^2+3/8*d*f*g^2*x^3*(a+b*arcsin(c*x))^2*sqrt(d-c^2*d*x^2)+3/35*d*g^3*x^4*(a+b*arcsin(c*x))^2*sqrt(d-c^2*d*x^2)+1/4*d*f^3*x*(1-c^2*x^2)*(a+b*arcsin(c*x))^2*sqrt(d-c^2*d*x^2)+1/2*d*f*g^2*x^3*(1-c^2*x^2)*(a+b*arcsin(c*x))^2*sqrt(d-c^2*d*x^2)+1/7*d*g^3*x^4*(1-c^2*x^2)*(a+b*arcsin(c*x))^2*sqrt(d-c^2*d*x^2)-3/5*d*f^2*g*(1-c^2*x^2)^2*(a+b*arcsin(c*x))^2*sqrt(d-c^2*d*x^2)/c^2+4/35*a*b*d*g^3*x*sqrt(d-c^2*d*x^2)/(c^3*sqrt(1-c^2*x^2))+9/64*b^2*d*f^3*arcsin(c*x)*sqrt(d-c^2*d*x^2)/(c*sqrt(1-c^2*x^2))+7/384*b^2*d*f*g^2*arcsin(c*x)*sqrt(d-c^2*d*x^2)/(c^3*sqrt(1-c^2*x^2))+4/35*b^2*d*g^3*x*arcsin(c*x)*sqrt(d-c^2*d*x^2)/(c^3*sqrt(1-c^2*x^2))+6/5*b*d*f^2*g*x*(a+b*arcsin(c*x))*sqrt(d-c^2*d*x^2)/(c*sqrt(1-c^2*x^2))-3/8*b*c*d*f^3*x^2*(a+b*arcsin(c*x))*sqrt(d-c^2*d*x^2)/sqrt(1-c^2*x^2)+3/16*b*d*f*g^2*x^2*(a+b*arcsin(c*x))*sqrt(d-c^2*d*x^2)/(c*sqrt(1-c^2*x^2))-4/5*b*c*d*f^2*g*x^3*(a+b*arcsin(c*x))*sqrt(d-c^2*d*x^2)/sqrt(1-c^2*x^2)+2/105*b*d*g^3*x^3*(a+b*arcsin(c*x))*sqrt(d-c^2*d*x^2)/(c*sqrt(1-c^2*x^2))-7/16*b*c*d*f*g^2*x^4*(a+b*arcsin(c*x))*sqrt(d-c^2*d*x^2)/sqrt(1-c^2*x^2)+6/25*b*c^3*d*f^2*g*x^5*(a+b*arcsin(c*x))*sqrt(d-c^2*d*x^2)/sqrt(1-c^2*x^2)-16/175*b*c*d*g^3*x^5*(a+b*arcsin(c*x))*sqrt(d-c^2*d*x^2)/sqrt(1-c^2*x^2)+1/6*b*c^3*d*f*g^2*x^6*(a+b*arcsin(c*x))*sqrt(d-c^2*d*x^2)/sqrt(1-c^2*x^2)+2/49*b*c^3*d*g^3*x^7*(a+b*arcsin(c*x))*sqrt(d-c^2*d*x^2)/sqrt(1-c^2*x^2)+1/8*d*f^3*(a+b*arcsin(c*x))^3*sqrt(d-c^2*d*x^2)/(b*c*sqrt(1-c^2*x^2))+1/16*d*f*g^2*(a+b*arcsin(c*x))^3*sqrt(d-c^2*d*x^2)/(b*c^3*sqrt(1-c^2*x^2))],
[(f+g*x)^2*(d-c^2*d*x^2)^(3/2)*(a+b*arcsin(c*x))^2,x,36,32/75*b^2*d*f*g*sqrt(d-c^2*d*x^2)/c^2-15/64*b^2*d*f^2*x*sqrt(d-c^2*d*x^2)-7/1152*b^2*d*g^2*x*sqrt(d-c^2*d*x^2)/c^2-43/1728*b^2*d*g^2*x^3*sqrt(d-c^2*d*x^2)+1/108*b^2*c^2*d*g^2*x^5*sqrt(d-c^2*d*x^2)+16/225*b^2*d*f*g*(1-c^2*x^2)*sqrt(d-c^2*d*x^2)/c^2-1/32*b^2*d*f^2*x*(1-c^2*x^2)*sqrt(d-c^2*d*x^2)+4/125*b^2*d*f*g*(1-c^2*x^2)^2*sqrt(d-c^2*d*x^2)/c^2+1/8*b*d*f^2*(1-c^2*x^2)^(3/2)*(a+b*arcsin(c*x))*sqrt(d-c^2*d*x^2)/c+3/8*d*f^2*x*(a+b*arcsin(c*x))^2*sqrt(d-c^2*d*x^2)-1/16*d*g^2*x*(a+b*arcsin(c*x))^2*sqrt(d-c^2*d*x^2)/c^2+1/8*d*g^2*x^3*(a+b*arcsin(c*x))^2*sqrt(d-c^2*d*x^2)+1/4*d*f^2*x*(1-c^2*x^2)*(a+b*arcsin(c*x))^2*sqrt(d-c^2*d*x^2)+1/6*d*g^2*x^3*(1-c^2*x^2)*(a+b*arcsin(c*x))^2*sqrt(d-c^2*d*x^2)-2/5*d*f*g*(1-c^2*x^2)^2*(a+b*arcsin(c*x))^2*sqrt(d-c^2*d*x^2)/c^2+9/64*b^2*d*f^2*arcsin(c*x)*sqrt(d-c^2*d*x^2)/(c*sqrt(1-c^2*x^2))+7/1152*b^2*d*g^2*arcsin(c*x)*sqrt(d-c^2*d*x^2)/(c^3*sqrt(1-c^2*x^2))+4/5*b*d*f*g*x*(a+b*arcsin(c*x))*sqrt(d-c^2*d*x^2)/(c*sqrt(1-c^2*x^2))-3/8*b*c*d*f^2*x^2*(a+b*arcsin(c*x))*sqrt(d-c^2*d*x^2)/sqrt(1-c^2*x^2)+1/16*b*d*g^2*x^2*(a+b*arcsin(c*x))*sqrt(d-c^2*d*x^2)/(c*sqrt(1-c^2*x^2))-8/15*b*c*d*f*g*x^3*(a+b*arcsin(c*x))*sqrt(d-c^2*d*x^2)/sqrt(1-c^2*x^2)-7/48*b*c*d*g^2*x^4*(a+b*arcsin(c*x))*sqrt(d-c^2*d*x^2)/sqrt(1-c^2*x^2)+4/25*b*c^3*d*f*g*x^5*(a+b*arcsin(c*x))*sqrt(d-c^2*d*x^2)/sqrt(1-c^2*x^2)+1/18*b*c^3*d*g^2*x^6*(a+b*arcsin(c*x))*sqrt(d-c^2*d*x^2)/sqrt(1-c^2*x^2)+1/8*d*f^2*(a+b*arcsin(c*x))^3*sqrt(d-c^2*d*x^2)/(b*c*sqrt(1-c^2*x^2))+1/48*d*g^2*(a+b*arcsin(c*x))^3*sqrt(d-c^2*d*x^2)/(b*c^3*sqrt(1-c^2*x^2))],
[(f+g*x)*(d-c^2*d*x^2)^(3/2)*(a+b*arcsin(c*x))^2,x,19,16/75*b^2*d*g*sqrt(d-c^2*d*x^2)/c^2-15/64*b^2*d*f*x*sqrt(d-c^2*d*x^2)+8/225*b^2*d*g*(1-c^2*x^2)*sqrt(d-c^2*d*x^2)/c^2-1/32*b^2*d*f*x*(1-c^2*x^2)*sqrt(d-c^2*d*x^2)+2/125*b^2*d*g*(1-c^2*x^2)^2*sqrt(d-c^2*d*x^2)/c^2+1/8*b*d*f*(1-c^2*x^2)^(3/2)*(a+b*arcsin(c*x))*sqrt(d-c^2*d*x^2)/c+3/8*d*f*x*(a+b*arcsin(c*x))^2*sqrt(d-c^2*d*x^2)+1/4*d*f*x*(1-c^2*x^2)*(a+b*arcsin(c*x))^2*sqrt(d-c^2*d*x^2)-1/5*d*g*(1-c^2*x^2)^2*(a+b*arcsin(c*x))^2*sqrt(d-c^2*d*x^2)/c^2+9/64*b^2*d*f*arcsin(c*x)*sqrt(d-c^2*d*x^2)/(c*sqrt(1-c^2*x^2))+2/5*b*d*g*x*(a+b*arcsin(c*x))*sqrt(d-c^2*d*x^2)/(c*sqrt(1-c^2*x^2))-3/8*b*c*d*f*x^2*(a+b*arcsin(c*x))*sqrt(d-c^2*d*x^2)/sqrt(1-c^2*x^2)-4/15*b*c*d*g*x^3*(a+b*arcsin(c*x))*sqrt(d-c^2*d*x^2)/sqrt(1-c^2*x^2)+2/25*b*c^3*d*g*x^5*(a+b*arcsin(c*x))*sqrt(d-c^2*d*x^2)/sqrt(1-c^2*x^2)+1/8*d*f*(a+b*arcsin(c*x))^3*sqrt(d-c^2*d*x^2)/(b*c*sqrt(1-c^2*x^2))],
[(d-c^2*d*x^2)^(3/2)*(a+b*arcsin(c*x))^2/(f+g*x),x,50,-4/9*b^2*d*sqrt(d-c^2*d*x^2)/g-a^2*d*(c*f-g)*(c*f+g)*sqrt(d-c^2*d*x^2)/g^3+2*b^2*d*(c*f-g)*(c*f+g)*sqrt(d-c^2*d*x^2)/g^3-1/4*b^2*c^2*d*f*x*sqrt(d-c^2*d*x^2)/g^2-2/27*b^2*d*(1-c^2*x^2)*sqrt(d-c^2*d*x^2)/g-2*a*b*d*(c*f-g)*(c*f+g)*arcsin(c*x)*sqrt(d-c^2*d*x^2)/g^3-b^2*d*(c*f-g)*(c*f+g)*arcsin(c*x)^2*sqrt(d-c^2*d*x^2)/g^3+1/2*c^2*d*f*x*(a+b*arcsin(c*x))^2*sqrt(d-c^2*d*x^2)/g^2+1/3*d*(1-c^2*x^2)*(a+b*arcsin(c*x))^2*sqrt(d-c^2*d*x^2)/g+2*a*b*c*d*(c*f-g)*(c*f+g)*x*sqrt(d-c^2*d*x^2)/(g^3*sqrt(1-c^2*x^2))+1/4*b^2*c*d*f*arcsin(c*x)*sqrt(d-c^2*d*x^2)/(g^2*sqrt(1-c^2*x^2))+2*b^2*c*d*(c*f-g)*(c*f+g)*x*arcsin(c*x)*sqrt(d-c^2*d*x^2)/(g^3*sqrt(1-c^2*x^2))-2/3*b*c*d*x*(a+b*arcsin(c*x))*sqrt(d-c^2*d*x^2)/(g*sqrt(1-c^2*x^2))-1/2*b*c^3*d*f*x^2*(a+b*arcsin(c*x))*sqrt(d-c^2*d*x^2)/(g^2*sqrt(1-c^2*x^2))+2/9*b*c^3*d*x^3*(a+b*arcsin(c*x))*sqrt(d-c^2*d*x^2)/(g*sqrt(1-c^2*x^2))+1/6*c*d*f*(a+b*arcsin(c*x))^3*sqrt(d-c^2*d*x^2)/(b*g^2*sqrt(1-c^2*x^2))-1/3*c*d*(c*f-g)*(c*f+g)*x*(a+b*arcsin(c*x))^3*sqrt(d-c^2*d*x^2)/(b*g^3*sqrt(1-c^2*x^2))-1/3*d*(c^2*f^2-g^2)^2*(a+b*arcsin(c*x))^3*sqrt(d-c^2*d*x^2)/(b*c*g^4*(f+g*x)*sqrt(1-c^2*x^2))+a^2*d*(c^2*f^2-g^2)^(3/2)*arctan((g+c^2*f*x)/(sqrt(c^2*f^2-g^2)*sqrt(1-c^2*x^2)))*sqrt(d-c^2*d*x^2)/(g^4*sqrt(1-c^2*x^2))-2*I*a*b*d*(c^2*f^2-g^2)^(3/2)*arcsin(c*x)*log(1-I*exp(I*arcsin(c*x))*g/(c*f-sqrt(c^2*f^2-g^2)))*sqrt(d-c^2*d*x^2)/(g^4*sqrt(1-c^2*x^2))-I*b^2*d*(c^2*f^2-g^2)^(3/2)*arcsin(c*x)^2*log(1-I*exp(I*arcsin(c*x))*g/(c*f-sqrt(c^2*f^2-g^2)))*sqrt(d-c^2*d*x^2)/(g^4*sqrt(1-c^2*x^2))+2*I*a*b*d*(c^2*f^2-g^2)^(3/2)*arcsin(c*x)*log(1-I*exp(I*arcsin(c*x))*g/(c*f+sqrt(c^2*f^2-g^2)))*sqrt(d-c^2*d*x^2)/(g^4*sqrt(1-c^2*x^2))+I*b^2*d*(c^2*f^2-g^2)^(3/2)*arcsin(c*x)^2*log(1-I*exp(I*arcsin(c*x))*g/(c*f+sqrt(c^2*f^2-g^2)))*sqrt(d-c^2*d*x^2)/(g^4*sqrt(1-c^2*x^2))-2*a*b*d*(c^2*f^2-g^2)^(3/2)*polylog(2,I*exp(I*arcsin(c*x))*g/(c*f-sqrt(c^2*f^2-g^2)))*sqrt(d-c^2*d*x^2)/(g^4*sqrt(1-c^2*x^2))-2*b^2*d*(c^2*f^2-g^2)^(3/2)*arcsin(c*x)*polylog(2,I*exp(I*arcsin(c*x))*g/(c*f-sqrt(c^2*f^2-g^2)))*sqrt(d-c^2*d*x^2)/(g^4*sqrt(1-c^2*x^2))+2*a*b*d*(c^2*f^2-g^2)^(3/2)*polylog(2,I*exp(I*arcsin(c*x))*g/(c*f+sqrt(c^2*f^2-g^2)))*sqrt(d-c^2*d*x^2)/(g^4*sqrt(1-c^2*x^2))+2*b^2*d*(c^2*f^2-g^2)^(3/2)*arcsin(c*x)*polylog(2,I*exp(I*arcsin(c*x))*g/(c*f+sqrt(c^2*f^2-g^2)))*sqrt(d-c^2*d*x^2)/(g^4*sqrt(1-c^2*x^2))-2*I*b^2*d*(c^2*f^2-g^2)^(3/2)*polylog(3,I*exp(I*arcsin(c*x))*g/(c*f-sqrt(c^2*f^2-g^2)))*sqrt(d-c^2*d*x^2)/(g^4*sqrt(1-c^2*x^2))+2*I*b^2*d*(c^2*f^2-g^2)^(3/2)*polylog(3,I*exp(I*arcsin(c*x))*g/(c*f+sqrt(c^2*f^2-g^2)))*sqrt(d-c^2*d*x^2)/(g^4*sqrt(1-c^2*x^2))-1/3*d*(c*f-g)*(c*f+g)*(a+b*arcsin(c*x))^3*sqrt(1-c^2*x^2)*sqrt(d-c^2*d*x^2)/(b*c*g^2*(f+g*x))],

#  {(d - c^2*d*x^2)^(3/2)*(a + b*ArcSin[c*x])^2/(f + g*x)^2, x, 0, 0} 
[(f+g*x)^3*(d-c^2*d*x^2)^(5/2)*(a+b*arcsin(c*x))^2,x,77,96/245*b^2*d^2*f^2*g*sqrt(d-c^2*d*x^2)/c^2+160/3969*b^2*d^2*g^3*sqrt(d-c^2*d*x^2)/c^4-245/1152*b^2*d^2*f^3*x*sqrt(d-c^2*d*x^2)-359/12288*b^2*d^2*f*g^2*x*sqrt(d-c^2*d*x^2)/c^2-1079/18432*b^2*d^2*f*g^2*x^3*sqrt(d-c^2*d*x^2)+209/4608*b^2*c^2*d^2*f*g^2*x^5*sqrt(d-c^2*d*x^2)-3/256*b^2*c^4*d^2*f*g^2*x^7*sqrt(d-c^2*d*x^2)+16/245*b^2*d^2*f^2*g*(1-c^2*x^2)*sqrt(d-c^2*d*x^2)/c^2+80/11907*b^2*d^2*g^3*(1-c^2*x^2)*sqrt(d-c^2*d*x^2)/c^4-65/1728*b^2*d^2*f^3*x*(1-c^2*x^2)*sqrt(d-c^2*d*x^2)+36/1225*b^2*d^2*f^2*g*(1-c^2*x^2)^2*sqrt(d-c^2*d*x^2)/c^2+4/1323*b^2*d^2*g^3*(1-c^2*x^2)^2*sqrt(d-c^2*d*x^2)/c^4-1/108*b^2*d^2*f^3*x*(1-c^2*x^2)^2*sqrt(d-c^2*d*x^2)+6/343*b^2*d^2*f^2*g*(1-c^2*x^2)^3*sqrt(d-c^2*d*x^2)/c^2+50/27783*b^2*d^2*g^3*(1-c^2*x^2)^3*sqrt(d-c^2*d*x^2)/c^4-2/729*b^2*d^2*g^3*(1-c^2*x^2)^4*sqrt(d-c^2*d*x^2)/c^4+5/48*b*d^2*f^3*(1-c^2*x^2)^(3/2)*(a+b*arcsin(c*x))*sqrt(d-c^2*d*x^2)/c+1/18*b*d^2*f^3*(1-c^2*x^2)^(5/2)*(a+b*arcsin(c*x))*sqrt(d-c^2*d*x^2)/c-2/63*d^2*g^3*(a+b*arcsin(c*x))^2*sqrt(d-c^2*d*x^2)/c^4+5/16*d^2*f^3*x*(a+b*arcsin(c*x))^2*sqrt(d-c^2*d*x^2)-15/128*d^2*f*g^2*x*(a+b*arcsin(c*x))^2*sqrt(d-c^2*d*x^2)/c^2-1/63*d^2*g^3*x^2*(a+b*arcsin(c*x))^2*sqrt(d-c^2*d*x^2)/c^2+15/64*d^2*f*g^2*x^3*(a+b*arcsin(c*x))^2*sqrt(d-c^2*d*x^2)+1/21*d^2*g^3*x^4*(a+b*arcsin(c*x))^2*sqrt(d-c^2*d*x^2)+5/24*d^2*f^3*x*(1-c^2*x^2)*(a+b*arcsin(c*x))^2*sqrt(d-c^2*d*x^2)+5/16*d^2*f*g^2*x^3*(1-c^2*x^2)*(a+b*arcsin(c*x))^2*sqrt(d-c^2*d*x^2)+5/63*d^2*g^3*x^4*(1-c^2*x^2)*(a+b*arcsin(c*x))^2*sqrt(d-c^2*d*x^2)+1/6*d^2*f^3*x*(1-c^2*x^2)^2*(a+b*arcsin(c*x))^2*sqrt(d-c^2*d*x^2)+3/8*d^2*f*g^2*x^3*(1-c^2*x^2)^2*(a+b*arcsin(c*x))^2*sqrt(d-c^2*d*x^2)+1/9*d^2*g^3*x^4*(1-c^2*x^2)^2*(a+b*arcsin(c*x))^2*sqrt(d-c^2*d*x^2)-3/7*d^2*f^2*g*(1-c^2*x^2)^3*(a+b*arcsin(c*x))^2*sqrt(d-c^2*d*x^2)/c^2+4/63*a*b*d^2*g^3*x*sqrt(d-c^2*d*x^2)/(c^3*sqrt(1-c^2*x^2))+115/1152*b^2*d^2*f^3*arcsin(c*x)*sqrt(d-c^2*d*x^2)/(c*sqrt(1-c^2*x^2))+359/12288*b^2*d^2*f*g^2*arcsin(c*x)*sqrt(d-c^2*d*x^2)/(c^3*sqrt(1-c^2*x^2))+4/63*b^2*d^2*g^3*x*arcsin(c*x)*sqrt(d-c^2*d*x^2)/(c^3*sqrt(1-c^2*x^2))+6/7*b*d^2*f^2*g*x*(a+b*arcsin(c*x))*sqrt(d-c^2*d*x^2)/(c*sqrt(1-c^2*x^2))-5/16*b*c*d^2*f^3*x^2*(a+b*arcsin(c*x))*sqrt(d-c^2*d*x^2)/sqrt(1-c^2*x^2)+15/128*b*d^2*f*g^2*x^2*(a+b*arcsin(c*x))*sqrt(d-c^2*d*x^2)/(c*sqrt(1-c^2*x^2))-6/7*b*c*d^2*f^2*g*x^3*(a+b*arcsin(c*x))*sqrt(d-c^2*d*x^2)/sqrt(1-c^2*x^2)+2/189*b*d^2*g^3*x^3*(a+b*arcsin(c*x))*sqrt(d-c^2*d*x^2)/(c*sqrt(1-c^2*x^2))-59/128*b*c*d^2*f*g^2*x^4*(a+b*arcsin(c*x))*sqrt(d-c^2*d*x^2)/sqrt(1-c^2*x^2)+18/35*b*c^3*d^2*f^2*g*x^5*(a+b*arcsin(c*x))*sqrt(d-c^2*d*x^2)/sqrt(1-c^2*x^2)-2/21*b*c*d^2*g^3*x^5*(a+b*arcsin(c*x))*sqrt(d-c^2*d*x^2)/sqrt(1-c^2*x^2)+17/48*b*c^3*d^2*f*g^2*x^6*(a+b*arcsin(c*x))*sqrt(d-c^2*d*x^2)/sqrt(1-c^2*x^2)-6/49*b*c^5*d^2*f^2*g*x^7*(a+b*arcsin(c*x))*sqrt(d-c^2*d*x^2)/sqrt(1-c^2*x^2)+38/441*b*c^3*d^2*g^3*x^7*(a+b*arcsin(c*x))*sqrt(d-c^2*d*x^2)/sqrt(1-c^2*x^2)-3/32*b*c^5*d^2*f*g^2*x^8*(a+b*arcsin(c*x))*sqrt(d-c^2*d*x^2)/sqrt(1-c^2*x^2)-2/81*b*c^5*d^2*g^3*x^9*(a+b*arcsin(c*x))*sqrt(d-c^2*d*x^2)/sqrt(1-c^2*x^2)+5/48*d^2*f^3*(a+b*arcsin(c*x))^3*sqrt(d-c^2*d*x^2)/(b*c*sqrt(1-c^2*x^2))+5/128*d^2*f*g^2*(a+b*arcsin(c*x))^3*sqrt(d-c^2*d*x^2)/(b*c^3*sqrt(1-c^2*x^2))],
[(f+g*x)^2*(d-c^2*d*x^2)^(5/2)*(a+b*arcsin(c*x))^2,x,50,64/245*b^2*d^2*f*g*sqrt(d-c^2*d*x^2)/c^2-245/1152*b^2*d^2*f^2*x*sqrt(d-c^2*d*x^2)-359/36864*b^2*d^2*g^2*x*sqrt(d-c^2*d*x^2)/c^2-1079/55296*b^2*d^2*g^2*x^3*sqrt(d-c^2*d*x^2)+209/13824*b^2*c^2*d^2*g^2*x^5*sqrt(d-c^2*d*x^2)-1/256*b^2*c^4*d^2*g^2*x^7*sqrt(d-c^2*d*x^2)+32/735*b^2*d^2*f*g*(1-c^2*x^2)*sqrt(d-c^2*d*x^2)/c^2-65/1728*b^2*d^2*f^2*x*(1-c^2*x^2)*sqrt(d-c^2*d*x^2)+24/1225*b^2*d^2*f*g*(1-c^2*x^2)^2*sqrt(d-c^2*d*x^2)/c^2-1/108*b^2*d^2*f^2*x*(1-c^2*x^2)^2*sqrt(d-c^2*d*x^2)+4/343*b^2*d^2*f*g*(1-c^2*x^2)^3*sqrt(d-c^2*d*x^2)/c^2+5/48*b*d^2*f^2*(1-c^2*x^2)^(3/2)*(a+b*arcsin(c*x))*sqrt(d-c^2*d*x^2)/c+1/18*b*d^2*f^2*(1-c^2*x^2)^(5/2)*(a+b*arcsin(c*x))*sqrt(d-c^2*d*x^2)/c+5/16*d^2*f^2*x*(a+b*arcsin(c*x))^2*sqrt(d-c^2*d*x^2)-5/128*d^2*g^2*x*(a+b*arcsin(c*x))^2*sqrt(d-c^2*d*x^2)/c^2+5/64*d^2*g^2*x^3*(a+b*arcsin(c*x))^2*sqrt(d-c^2*d*x^2)+5/24*d^2*f^2*x*(1-c^2*x^2)*(a+b*arcsin(c*x))^2*sqrt(d-c^2*d*x^2)+5/48*d^2*g^2*x^3*(1-c^2*x^2)*(a+b*arcsin(c*x))^2*sqrt(d-c^2*d*x^2)+1/6*d^2*f^2*x*(1-c^2*x^2)^2*(a+b*arcsin(c*x))^2*sqrt(d-c^2*d*x^2)+1/8*d^2*g^2*x^3*(1-c^2*x^2)^2*(a+b*arcsin(c*x))^2*sqrt(d-c^2*d*x^2)-2/7*d^2*f*g*(1-c^2*x^2)^3*(a+b*arcsin(c*x))^2*sqrt(d-c^2*d*x^2)/c^2+115/1152*b^2*d^2*f^2*arcsin(c*x)*sqrt(d-c^2*d*x^2)/(c*sqrt(1-c^2*x^2))+359/36864*b^2*d^2*g^2*arcsin(c*x)*sqrt(d-c^2*d*x^2)/(c^3*sqrt(1-c^2*x^2))+4/7*b*d^2*f*g*x*(a+b*arcsin(c*x))*sqrt(d-c^2*d*x^2)/(c*sqrt(1-c^2*x^2))-5/16*b*c*d^2*f^2*x^2*(a+b*arcsin(c*x))*sqrt(d-c^2*d*x^2)/sqrt(1-c^2*x^2)+5/128*b*d^2*g^2*x^2*(a+b*arcsin(c*x))*sqrt(d-c^2*d*x^2)/(c*sqrt(1-c^2*x^2))-4/7*b*c*d^2*f*g*x^3*(a+b*arcsin(c*x))*sqrt(d-c^2*d*x^2)/sqrt(1-c^2*x^2)-59/384*b*c*d^2*g^2*x^4*(a+b*arcsin(c*x))*sqrt(d-c^2*d*x^2)/sqrt(1-c^2*x^2)+12/35*b*c^3*d^2*f*g*x^5*(a+b*arcsin(c*x))*sqrt(d-c^2*d*x^2)/sqrt(1-c^2*x^2)+17/144*b*c^3*d^2*g^2*x^6*(a+b*arcsin(c*x))*sqrt(d-c^2*d*x^2)/sqrt(1-c^2*x^2)-4/49*b*c^5*d^2*f*g*x^7*(a+b*arcsin(c*x))*sqrt(d-c^2*d*x^2)/sqrt(1-c^2*x^2)-1/32*b*c^5*d^2*g^2*x^8*(a+b*arcsin(c*x))*sqrt(d-c^2*d*x^2)/sqrt(1-c^2*x^2)+5/48*d^2*f^2*(a+b*arcsin(c*x))^3*sqrt(d-c^2*d*x^2)/(b*c*sqrt(1-c^2*x^2))+5/384*d^2*g^2*(a+b*arcsin(c*x))^3*sqrt(d-c^2*d*x^2)/(b*c^3*sqrt(1-c^2*x^2))],
[(f+g*x)*(d-c^2*d*x^2)^(5/2)*(a+b*arcsin(c*x))^2,x,25,32/245*b^2*d^2*g*sqrt(d-c^2*d*x^2)/c^2-245/1152*b^2*d^2*f*x*sqrt(d-c^2*d*x^2)+16/735*b^2*d^2*g*(1-c^2*x^2)*sqrt(d-c^2*d*x^2)/c^2-65/1728*b^2*d^2*f*x*(1-c^2*x^2)*sqrt(d-c^2*d*x^2)+12/1225*b^2*d^2*g*(1-c^2*x^2)^2*sqrt(d-c^2*d*x^2)/c^2-1/108*b^2*d^2*f*x*(1-c^2*x^2)^2*sqrt(d-c^2*d*x^2)+2/343*b^2*d^2*g*(1-c^2*x^2)^3*sqrt(d-c^2*d*x^2)/c^2+5/48*b*d^2*f*(1-c^2*x^2)^(3/2)*(a+b*arcsin(c*x))*sqrt(d-c^2*d*x^2)/c+1/18*b*d^2*f*(1-c^2*x^2)^(5/2)*(a+b*arcsin(c*x))*sqrt(d-c^2*d*x^2)/c+5/16*d^2*f*x*(a+b*arcsin(c*x))^2*sqrt(d-c^2*d*x^2)+5/24*d^2*f*x*(1-c^2*x^2)*(a+b*arcsin(c*x))^2*sqrt(d-c^2*d*x^2)+1/6*d^2*f*x*(1-c^2*x^2)^2*(a+b*arcsin(c*x))^2*sqrt(d-c^2*d*x^2)-1/7*d^2*g*(1-c^2*x^2)^3*(a+b*arcsin(c*x))^2*sqrt(d-c^2*d*x^2)/c^2+115/1152*b^2*d^2*f*arcsin(c*x)*sqrt(d-c^2*d*x^2)/(c*sqrt(1-c^2*x^2))+2/7*b*d^2*g*x*(a+b*arcsin(c*x))*sqrt(d-c^2*d*x^2)/(c*sqrt(1-c^2*x^2))-5/16*b*c*d^2*f*x^2*(a+b*arcsin(c*x))*sqrt(d-c^2*d*x^2)/sqrt(1-c^2*x^2)-2/7*b*c*d^2*g*x^3*(a+b*arcsin(c*x))*sqrt(d-c^2*d*x^2)/sqrt(1-c^2*x^2)+6/35*b*c^3*d^2*g*x^5*(a+b*arcsin(c*x))*sqrt(d-c^2*d*x^2)/sqrt(1-c^2*x^2)-2/49*b*c^5*d^2*g*x^7*(a+b*arcsin(c*x))*sqrt(d-c^2*d*x^2)/sqrt(1-c^2*x^2)+5/48*d^2*f*(a+b*arcsin(c*x))^3*sqrt(d-c^2*d*x^2)/(b*c*sqrt(1-c^2*x^2))],
[(d-c^2*d*x^2)^(5/2)*(a+b*arcsin(c*x))^2/(f+g*x),x,74,52/225*b^2*d^2*sqrt(d-c^2*d*x^2)/g+4/9*b^2*d^2*(c^2*f^2-2*g^2)*sqrt(d-c^2*d*x^2)/g^3+a^2*d^2*(c^2*f^2-g^2)^2*sqrt(d-c^2*d*x^2)/g^5-2*b^2*d^2*(c^2*f^2-g^2)^2*sqrt(d-c^2*d*x^2)/g^5-1/64*b^2*c^2*d^2*f*x*sqrt(d-c^2*d*x^2)/g^2+1/4*b^2*c^2*d^2*f*(c^2*f^2-2*g^2)*x*sqrt(d-c^2*d*x^2)/g^4+1/32*b^2*c^4*d^2*f*x^3*sqrt(d-c^2*d*x^2)/g^2+26/675*b^2*d^2*(1-c^2*x^2)*sqrt(d-c^2*d*x^2)/g+2/27*b^2*d^2*(c^2*f^2-2*g^2)*(1-c^2*x^2)*sqrt(d-c^2*d*x^2)/g^3-2/125*b^2*d^2*(1-c^2*x^2)^2*sqrt(d-c^2*d*x^2)/g+2*a*b*d^2*(c^2*f^2-g^2)^2*arcsin(c*x)*sqrt(d-c^2*d*x^2)/g^5+b^2*d^2*(c^2*f^2-g^2)^2*arcsin(c*x)^2*sqrt(d-c^2*d*x^2)/g^5-2/15*d^2*(a+b*arcsin(c*x))^2*sqrt(d-c^2*d*x^2)/g+1/8*c^2*d^2*f*x*(a+b*arcsin(c*x))^2*sqrt(d-c^2*d*x^2)/g^2-1/2*c^2*d^2*f*(c^2*f^2-2*g^2)*x*(a+b*arcsin(c*x))^2*sqrt(d-c^2*d*x^2)/g^4-1/15*c^2*d^2*x^2*(a+b*arcsin(c*x))^2*sqrt(d-c^2*d*x^2)/g-1/4*c^4*d^2*f*x^3*(a+b*arcsin(c*x))^2*sqrt(d-c^2*d*x^2)/g^2+1/5*c^4*d^2*x^4*(a+b*arcsin(c*x))^2*sqrt(d-c^2*d*x^2)/g-1/3*d^2*(c^2*f^2-2*g^2)*(1-c^2*x^2)*(a+b*arcsin(c*x))^2*sqrt(d-c^2*d*x^2)/g^3+4/15*a*b*c*d^2*x*sqrt(d-c^2*d*x^2)/(g*sqrt(1-c^2*x^2))-2*a*b*c*d^2*(c^2*f^2-g^2)^2*x*sqrt(d-c^2*d*x^2)/(g^5*sqrt(1-c^2*x^2))+1/64*b^2*c*d^2*f*arcsin(c*x)*sqrt(d-c^2*d*x^2)/(g^2*sqrt(1-c^2*x^2))-1/4*b^2*c*d^2*f*(c^2*f^2-2*g^2)*arcsin(c*x)*sqrt(d-c^2*d*x^2)/(g^4*sqrt(1-c^2*x^2))+4/15*b^2*c*d^2*x*arcsin(c*x)*sqrt(d-c^2*d*x^2)/(g*sqrt(1-c^2*x^2))-2*b^2*c*d^2*(c^2*f^2-g^2)^2*x*arcsin(c*x)*sqrt(d-c^2*d*x^2)/(g^5*sqrt(1-c^2*x^2))+2/3*b*c*d^2*(c^2*f^2-2*g^2)*x*(a+b*arcsin(c*x))*sqrt(d-c^2*d*x^2)/(g^3*sqrt(1-c^2*x^2))-1/8*b*c^3*d^2*f*x^2*(a+b*arcsin(c*x))*sqrt(d-c^2*d*x^2)/(g^2*sqrt(1-c^2*x^2))+1/2*b*c^3*d^2*f*(c^2*f^2-2*g^2)*x^2*(a+b*arcsin(c*x))*sqrt(d-c^2*d*x^2)/(g^4*sqrt(1-c^2*x^2))+2/45*b*c^3*d^2*x^3*(a+b*arcsin(c*x))*sqrt(d-c^2*d*x^2)/(g*sqrt(1-c^2*x^2))-2/9*b*c^3*d^2*(c^2*f^2-2*g^2)*x^3*(a+b*arcsin(c*x))*sqrt(d-c^2*d*x^2)/(g^3*sqrt(1-c^2*x^2))+1/8*b*c^5*d^2*f*x^4*(a+b*arcsin(c*x))*sqrt(d-c^2*d*x^2)/(g^2*sqrt(1-c^2*x^2))-2/25*b*c^5*d^2*x^5*(a+b*arcsin(c*x))*sqrt(d-c^2*d*x^2)/(g*sqrt(1-c^2*x^2))-1/24*c*d^2*f*(a+b*arcsin(c*x))^3*sqrt(d-c^2*d*x^2)/(b*g^2*sqrt(1-c^2*x^2))-1/6*c*d^2*f*(c^2*f^2-2*g^2)*(a+b*arcsin(c*x))^3*sqrt(d-c^2*d*x^2)/(b*g^4*sqrt(1-c^2*x^2))+1/3*c*d^2*(c^2*f^2-g^2)^2*x*(a+b*arcsin(c*x))^3*sqrt(d-c^2*d*x^2)/(b*g^5*sqrt(1-c^2*x^2))+1/3*d^2*(c^2*f^2-g^2)^3*(a+b*arcsin(c*x))^3*sqrt(d-c^2*d*x^2)/(b*c*g^6*(f+g*x)*sqrt(1-c^2*x^2))-a^2*d^2*(c^2*f^2-g^2)^(5/2)*arctan((g+c^2*f*x)/(sqrt(c^2*f^2-g^2)*sqrt(1-c^2*x^2)))*sqrt(d-c^2*d*x^2)/(g^6*sqrt(1-c^2*x^2))+2*I*a*b*d^2*(c^2*f^2-g^2)^(5/2)*arcsin(c*x)*log(1-I*exp(I*arcsin(c*x))*g/(c*f-sqrt(c^2*f^2-g^2)))*sqrt(d-c^2*d*x^2)/(g^6*sqrt(1-c^2*x^2))+I*b^2*d^2*(c^2*f^2-g^2)^(5/2)*arcsin(c*x)^2*log(1-I*exp(I*arcsin(c*x))*g/(c*f-sqrt(c^2*f^2-g^2)))*sqrt(d-c^2*d*x^2)/(g^6*sqrt(1-c^2*x^2))-2*I*a*b*d^2*(c^2*f^2-g^2)^(5/2)*arcsin(c*x)*log(1-I*exp(I*arcsin(c*x))*g/(c*f+sqrt(c^2*f^2-g^2)))*sqrt(d-c^2*d*x^2)/(g^6*sqrt(1-c^2*x^2))-I*b^2*d^2*(c^2*f^2-g^2)^(5/2)*arcsin(c*x)^2*log(1-I*exp(I*arcsin(c*x))*g/(c*f+sqrt(c^2*f^2-g^2)))*sqrt(d-c^2*d*x^2)/(g^6*sqrt(1-c^2*x^2))+2*a*b*d^2*(c^2*f^2-g^2)^(5/2)*polylog(2,I*exp(I*arcsin(c*x))*g/(c*f-sqrt(c^2*f^2-g^2)))*sqrt(d-c^2*d*x^2)/(g^6*sqrt(1-c^2*x^2))+2*b^2*d^2*(c^2*f^2-g^2)^(5/2)*arcsin(c*x)*polylog(2,I*exp(I*arcsin(c*x))*g/(c*f-sqrt(c^2*f^2-g^2)))*sqrt(d-c^2*d*x^2)/(g^6*sqrt(1-c^2*x^2))-2*a*b*d^2*(c^2*f^2-g^2)^(5/2)*polylog(2,I*exp(I*arcsin(c*x))*g/(c*f+sqrt(c^2*f^2-g^2)))*sqrt(d-c^2*d*x^2)/(g^6*sqrt(1-c^2*x^2))-2*b^2*d^2*(c^2*f^2-g^2)^(5/2)*arcsin(c*x)*polylog(2,I*exp(I*arcsin(c*x))*g/(c*f+sqrt(c^2*f^2-g^2)))*sqrt(d-c^2*d*x^2)/(g^6*sqrt(1-c^2*x^2))+2*I*b^2*d^2*(c^2*f^2-g^2)^(5/2)*polylog(3,I*exp(I*arcsin(c*x))*g/(c*f-sqrt(c^2*f^2-g^2)))*sqrt(d-c^2*d*x^2)/(g^6*sqrt(1-c^2*x^2))-2*I*b^2*d^2*(c^2*f^2-g^2)^(5/2)*polylog(3,I*exp(I*arcsin(c*x))*g/(c*f+sqrt(c^2*f^2-g^2)))*sqrt(d-c^2*d*x^2)/(g^6*sqrt(1-c^2*x^2))+1/3*d^2*(c^2*f^2-g^2)^2*(a+b*arcsin(c*x))^3*sqrt(1-c^2*x^2)*sqrt(d-c^2*d*x^2)/(b*c*g^4*(f+g*x))],

#  {(d - c^2*d*x^2)^(5/2)*(a + b*ArcSin[c*x])^2/(f + g*x)^2, x, 0, 0} 

# p<0
[(f+g*x)^3*(a+b*arcsin(c*x))^2/sqrt(d-c^2*d*x^2),x,17,6*b^2*f^2*g*(1-c^2*x^2)/(c^2*sqrt(d-c^2*d*x^2))+14/9*b^2*g^3*(1-c^2*x^2)/(c^4*sqrt(d-c^2*d*x^2))+3/4*b^2*f*g^2*x*(1-c^2*x^2)/(c^2*sqrt(d-c^2*d*x^2))-2/27*b^2*g^3*(1-c^2*x^2)^2/(c^4*sqrt(d-c^2*d*x^2))-3*f^2*g*(1-c^2*x^2)*(a+b*arcsin(c*x))^2/(c^2*sqrt(d-c^2*d*x^2))-2/3*g^3*(1-c^2*x^2)*(a+b*arcsin(c*x))^2/(c^4*sqrt(d-c^2*d*x^2))-3/2*f*g^2*x*(1-c^2*x^2)*(a+b*arcsin(c*x))^2/(c^2*sqrt(d-c^2*d*x^2))-1/3*g^3*x^2*(1-c^2*x^2)*(a+b*arcsin(c*x))^2/(c^2*sqrt(d-c^2*d*x^2))-3/4*b^2*f*g^2*arcsin(c*x)*sqrt(1-c^2*x^2)/(c^3*sqrt(d-c^2*d*x^2))+6*b*f^2*g*x*(a+b*arcsin(c*x))*sqrt(1-c^2*x^2)/(c*sqrt(d-c^2*d*x^2))+4/3*b*g^3*x*(a+b*arcsin(c*x))*sqrt(1-c^2*x^2)/(c^3*sqrt(d-c^2*d*x^2))+3/2*b*f*g^2*x^2*(a+b*arcsin(c*x))*sqrt(1-c^2*x^2)/(c*sqrt(d-c^2*d*x^2))+2/9*b*g^3*x^3*(a+b*arcsin(c*x))*sqrt(1-c^2*x^2)/(c*sqrt(d-c^2*d*x^2))+1/3*f^3*(a+b*arcsin(c*x))^3*sqrt(1-c^2*x^2)/(b*c*sqrt(d-c^2*d*x^2))+1/2*f*g^2*(a+b*arcsin(c*x))^3*sqrt(1-c^2*x^2)/(b*c^3*sqrt(d-c^2*d*x^2))],
[(f+g*x)^2*(a+b*arcsin(c*x))^2/sqrt(d-c^2*d*x^2),x,11,4*b^2*f*g*(1-c^2*x^2)/(c^2*sqrt(d-c^2*d*x^2))+1/4*b^2*g^2*x*(1-c^2*x^2)/(c^2*sqrt(d-c^2*d*x^2))-2*f*g*(1-c^2*x^2)*(a+b*arcsin(c*x))^2/(c^2*sqrt(d-c^2*d*x^2))-1/2*g^2*x*(1-c^2*x^2)*(a+b*arcsin(c*x))^2/(c^2*sqrt(d-c^2*d*x^2))-1/4*b^2*g^2*arcsin(c*x)*sqrt(1-c^2*x^2)/(c^3*sqrt(d-c^2*d*x^2))+4*b*f*g*x*(a+b*arcsin(c*x))*sqrt(1-c^2*x^2)/(c*sqrt(d-c^2*d*x^2))+1/2*b*g^2*x^2*(a+b*arcsin(c*x))*sqrt(1-c^2*x^2)/(c*sqrt(d-c^2*d*x^2))+1/3*f^2*(a+b*arcsin(c*x))^3*sqrt(1-c^2*x^2)/(b*c*sqrt(d-c^2*d*x^2))+1/6*g^2*(a+b*arcsin(c*x))^3*sqrt(1-c^2*x^2)/(b*c^3*sqrt(d-c^2*d*x^2))],
[(f+g*x)*(a+b*arcsin(c*x))^2/sqrt(d-c^2*d*x^2),x,8,2*b^2*g*(1-c^2*x^2)/(c^2*sqrt(d-c^2*d*x^2))-g*(1-c^2*x^2)*(a+b*arcsin(c*x))^2/(c^2*sqrt(d-c^2*d*x^2))+2*b*g*x*(a+b*arcsin(c*x))*sqrt(1-c^2*x^2)/(c*sqrt(d-c^2*d*x^2))+1/3*f*(a+b*arcsin(c*x))^3*sqrt(1-c^2*x^2)/(b*c*sqrt(d-c^2*d*x^2)),2*b^2*g*(1-c^2*x^2)/(c^2*sqrt(d-c^2*d*x^2))-g*(1-c^2*x^2)*(a+b*arcsin(c*x))^2/(c^2*sqrt(d-c^2*d*x^2))+2*a*b*g*x*sqrt(1-c^2*x^2)/(c*sqrt(d-c^2*d*x^2))+2*b^2*g*x*arcsin(c*x)*sqrt(1-c^2*x^2)/(c*sqrt(d-c^2*d*x^2))+1/3*f*(a+b*arcsin(c*x))^3*sqrt(1-c^2*x^2)/(b*c*sqrt(d-c^2*d*x^2))],
[(a+b*arcsin(c*x))^2/((f+g*x)*sqrt(d-c^2*d*x^2)),x,12,-I*(a+b*arcsin(c*x))^2*log(1-I*exp(I*arcsin(c*x))*g/(c*f-sqrt(c^2*f^2-g^2)))*sqrt(1-c^2*x^2)/(sqrt(c^2*f^2-g^2)*sqrt(d-c^2*d*x^2))+I*(a+b*arcsin(c*x))^2*log(1-I*exp(I*arcsin(c*x))*g/(c*f+sqrt(c^2*f^2-g^2)))*sqrt(1-c^2*x^2)/(sqrt(c^2*f^2-g^2)*sqrt(d-c^2*d*x^2))-2*b*(a+b*arcsin(c*x))*polylog(2,I*exp(I*arcsin(c*x))*g/(c*f-sqrt(c^2*f^2-g^2)))*sqrt(1-c^2*x^2)/(sqrt(c^2*f^2-g^2)*sqrt(d-c^2*d*x^2))+2*b*(a+b*arcsin(c*x))*polylog(2,I*exp(I*arcsin(c*x))*g/(c*f+sqrt(c^2*f^2-g^2)))*sqrt(1-c^2*x^2)/(sqrt(c^2*f^2-g^2)*sqrt(d-c^2*d*x^2))-2*I*b^2*polylog(3,I*exp(I*arcsin(c*x))*g/(c*f-sqrt(c^2*f^2-g^2)))*sqrt(1-c^2*x^2)/(sqrt(c^2*f^2-g^2)*sqrt(d-c^2*d*x^2))+2*I*b^2*polylog(3,I*exp(I*arcsin(c*x))*g/(c*f+sqrt(c^2*f^2-g^2)))*sqrt(1-c^2*x^2)/(sqrt(c^2*f^2-g^2)*sqrt(d-c^2*d*x^2))],
[(a+b*arcsin(c*x))^2/((f+g*x)^2*sqrt(d-c^2*d*x^2)),x,20,g*(1-c^2*x^2)*(a+b*arcsin(c*x))^2/((c^2*f^2-g^2)*(f+g*x)*sqrt(d-c^2*d*x^2))+I*c*(a+b*arcsin(c*x))^2*sqrt(1-c^2*x^2)/((c^2*f^2-g^2)*sqrt(d-c^2*d*x^2))-2*b*c*(a+b*arcsin(c*x))*log(1-I*exp(I*arcsin(c*x))*g/(c*f-sqrt(c^2*f^2-g^2)))*sqrt(1-c^2*x^2)/((c^2*f^2-g^2)*sqrt(d-c^2*d*x^2))-I*c^2*f*(a+b*arcsin(c*x))^2*log(1-I*exp(I*arcsin(c*x))*g/(c*f-sqrt(c^2*f^2-g^2)))*sqrt(1-c^2*x^2)/((c^2*f^2-g^2)^(3/2)*sqrt(d-c^2*d*x^2))-2*b*c*(a+b*arcsin(c*x))*log(1-I*exp(I*arcsin(c*x))*g/(c*f+sqrt(c^2*f^2-g^2)))*sqrt(1-c^2*x^2)/((c^2*f^2-g^2)*sqrt(d-c^2*d*x^2))+I*c^2*f*(a+b*arcsin(c*x))^2*log(1-I*exp(I*arcsin(c*x))*g/(c*f+sqrt(c^2*f^2-g^2)))*sqrt(1-c^2*x^2)/((c^2*f^2-g^2)^(3/2)*sqrt(d-c^2*d*x^2))+2*I*b^2*c*polylog(2,I*exp(I*arcsin(c*x))*g/(c*f-sqrt(c^2*f^2-g^2)))*sqrt(1-c^2*x^2)/((c^2*f^2-g^2)*sqrt(d-c^2*d*x^2))-2*b*c^2*f*(a+b*arcsin(c*x))*polylog(2,I*exp(I*arcsin(c*x))*g/(c*f-sqrt(c^2*f^2-g^2)))*sqrt(1-c^2*x^2)/((c^2*f^2-g^2)^(3/2)*sqrt(d-c^2*d*x^2))+2*I*b^2*c*polylog(2,I*exp(I*arcsin(c*x))*g/(c*f+sqrt(c^2*f^2-g^2)))*sqrt(1-c^2*x^2)/((c^2*f^2-g^2)*sqrt(d-c^2*d*x^2))+2*b*c^2*f*(a+b*arcsin(c*x))*polylog(2,I*exp(I*arcsin(c*x))*g/(c*f+sqrt(c^2*f^2-g^2)))*sqrt(1-c^2*x^2)/((c^2*f^2-g^2)^(3/2)*sqrt(d-c^2*d*x^2))-2*I*b^2*c^2*f*polylog(3,I*exp(I*arcsin(c*x))*g/(c*f-sqrt(c^2*f^2-g^2)))*sqrt(1-c^2*x^2)/((c^2*f^2-g^2)^(3/2)*sqrt(d-c^2*d*x^2))+2*I*b^2*c^2*f*polylog(3,I*exp(I*arcsin(c*x))*g/(c*f+sqrt(c^2*f^2-g^2)))*sqrt(1-c^2*x^2)/((c^2*f^2-g^2)^(3/2)*sqrt(d-c^2*d*x^2))],
[(f+g*x)^3*(a+b*arcsin(c*x))^2/(d-c^2*d*x^2)^(3/2),x,23,-2*b^2*g^3*(1-c^2*x^2)/(c^4*d*sqrt(d-c^2*d*x^2))+g*(3*c^2*f^2+g^2)*(a+b*arcsin(c*x))^2/(c^4*d*sqrt(d-c^2*d*x^2))+f*(f^2+3*g^2/c^2)*x*(a+b*arcsin(c*x))^2/(d*sqrt(d-c^2*d*x^2))+g^3*(1-c^2*x^2)*(a+b*arcsin(c*x))^2/(c^4*d*sqrt(d-c^2*d*x^2))-2*a*b*g^3*x*sqrt(1-c^2*x^2)/(c^3*d*sqrt(d-c^2*d*x^2))-2*b^2*g^3*x*arcsin(c*x)*sqrt(1-c^2*x^2)/(c^3*d*sqrt(d-c^2*d*x^2))-I*f*(c^2*f^2+3*g^2)*(a+b*arcsin(c*x))^2*sqrt(1-c^2*x^2)/(c^3*d*sqrt(d-c^2*d*x^2))-f*g^2*(a+b*arcsin(c*x))^3*sqrt(1-c^2*x^2)/(b*c^3*d*sqrt(d-c^2*d*x^2))+4*I*b*g*(3*c^2*f^2+g^2)*(a+b*arcsin(c*x))*arctan(exp(I*arcsin(c*x)))*sqrt(1-c^2*x^2)/(c^4*d*sqrt(d-c^2*d*x^2))+2*b*f*(c^2*f^2+3*g^2)*(a+b*arcsin(c*x))*log(1+exp(2*I*arcsin(c*x)))*sqrt(1-c^2*x^2)/(c^3*d*sqrt(d-c^2*d*x^2))-2*I*b^2*g*(3*c^2*f^2+g^2)*polylog(2,-I*exp(I*arcsin(c*x)))*sqrt(1-c^2*x^2)/(c^4*d*sqrt(d-c^2*d*x^2))+2*I*b^2*g*(3*c^2*f^2+g^2)*polylog(2,I*exp(I*arcsin(c*x)))*sqrt(1-c^2*x^2)/(c^4*d*sqrt(d-c^2*d*x^2))-I*b^2*f*(c^2*f^2+3*g^2)*polylog(2,-exp(2*I*arcsin(c*x)))*sqrt(1-c^2*x^2)/(c^3*d*sqrt(d-c^2*d*x^2))],
[(f+g*x)^2*(a+b*arcsin(c*x))^2/(d-c^2*d*x^2)^(3/2),x,19,2*f*g*(a+b*arcsin(c*x))^2/(c^2*d*sqrt(d-c^2*d*x^2))+(c^2*f^2+g^2)*x*(a+b*arcsin(c*x))^2/(c^2*d*sqrt(d-c^2*d*x^2))-I*(c^2*f^2+g^2)*(a+b*arcsin(c*x))^2*sqrt(1-c^2*x^2)/(c^3*d*sqrt(d-c^2*d*x^2))-1/3*g^2*(a+b*arcsin(c*x))^3*sqrt(1-c^2*x^2)/(b*c^3*d*sqrt(d-c^2*d*x^2))+8*I*b*f*g*(a+b*arcsin(c*x))*arctan(exp(I*arcsin(c*x)))*sqrt(1-c^2*x^2)/(c^2*d*sqrt(d-c^2*d*x^2))+2*b*(c^2*f^2+g^2)*(a+b*arcsin(c*x))*log(1+exp(2*I*arcsin(c*x)))*sqrt(1-c^2*x^2)/(c^3*d*sqrt(d-c^2*d*x^2))-4*I*b^2*f*g*polylog(2,-I*exp(I*arcsin(c*x)))*sqrt(1-c^2*x^2)/(c^2*d*sqrt(d-c^2*d*x^2))+4*I*b^2*f*g*polylog(2,I*exp(I*arcsin(c*x)))*sqrt(1-c^2*x^2)/(c^2*d*sqrt(d-c^2*d*x^2))-I*b^2*(c^2*f^2+g^2)*polylog(2,-exp(2*I*arcsin(c*x)))*sqrt(1-c^2*x^2)/(c^3*d*sqrt(d-c^2*d*x^2))],
[(f+g*x)*(a+b*arcsin(c*x))^2/(d-c^2*d*x^2)^(3/2),x,16,g*(a+b*arcsin(c*x))^2/(c^2*d*sqrt(d-c^2*d*x^2))+f*x*(a+b*arcsin(c*x))^2/(d*sqrt(d-c^2*d*x^2))-I*f*(a+b*arcsin(c*x))^2*sqrt(1-c^2*x^2)/(c*d*sqrt(d-c^2*d*x^2))+4*I*b*g*(a+b*arcsin(c*x))*arctan(exp(I*arcsin(c*x)))*sqrt(1-c^2*x^2)/(c^2*d*sqrt(d-c^2*d*x^2))+2*b*f*(a+b*arcsin(c*x))*log(1+exp(2*I*arcsin(c*x)))*sqrt(1-c^2*x^2)/(c*d*sqrt(d-c^2*d*x^2))-2*I*b^2*g*polylog(2,-I*exp(I*arcsin(c*x)))*sqrt(1-c^2*x^2)/(c^2*d*sqrt(d-c^2*d*x^2))+2*I*b^2*g*polylog(2,I*exp(I*arcsin(c*x)))*sqrt(1-c^2*x^2)/(c^2*d*sqrt(d-c^2*d*x^2))-I*b^2*f*polylog(2,-exp(2*I*arcsin(c*x)))*sqrt(1-c^2*x^2)/(c*d*sqrt(d-c^2*d*x^2))],
[(a+b*arcsin(c*x))^2/((f+g*x)*(d-c^2*d*x^2)^(3/2)),x,28,-1/2*I*(a+b*arcsin(c*x))^2*sqrt(1-c^2*x^2)/(d*(c*f-g)*sqrt(d-c^2*d*x^2))+1/2*I*(a+b*arcsin(c*x))^2*sqrt(1-c^2*x^2)/(d*(c*f+g)*sqrt(d-c^2*d*x^2))-1/2*(a+b*arcsin(c*x))^2*cot(1/4*Pi+1/2*arcsin(c*x))*sqrt(1-c^2*x^2)/(d*(c*f-g)*sqrt(d-c^2*d*x^2))+2*b*(a+b*arcsin(c*x))*log(1+(-I)/exp(I*arcsin(c*x)))*sqrt(1-c^2*x^2)/(d*(c*f+g)*sqrt(d-c^2*d*x^2))+2*b*(a+b*arcsin(c*x))*log(1-I*exp(I*arcsin(c*x)))*sqrt(1-c^2*x^2)/(d*(c*f-g)*sqrt(d-c^2*d*x^2))+I*g^2*(a+b*arcsin(c*x))^2*log(1-I*exp(I*arcsin(c*x))*g/(c*f-sqrt(c^2*f^2-g^2)))*sqrt(1-c^2*x^2)/(d*(c^2*f^2-g^2)^(3/2)*sqrt(d-c^2*d*x^2))-I*g^2*(a+b*arcsin(c*x))^2*log(1-I*exp(I*arcsin(c*x))*g/(c*f+sqrt(c^2*f^2-g^2)))*sqrt(1-c^2*x^2)/(d*(c^2*f^2-g^2)^(3/2)*sqrt(d-c^2*d*x^2))+2*I*b^2*polylog(2,I/exp(I*arcsin(c*x)))*sqrt(1-c^2*x^2)/(d*(c*f+g)*sqrt(d-c^2*d*x^2))-2*I*b^2*polylog(2,I*exp(I*arcsin(c*x)))*sqrt(1-c^2*x^2)/(d*(c*f-g)*sqrt(d-c^2*d*x^2))+2*b*g^2*(a+b*arcsin(c*x))*polylog(2,I*exp(I*arcsin(c*x))*g/(c*f-sqrt(c^2*f^2-g^2)))*sqrt(1-c^2*x^2)/(d*(c^2*f^2-g^2)^(3/2)*sqrt(d-c^2*d*x^2))-2*b*g^2*(a+b*arcsin(c*x))*polylog(2,I*exp(I*arcsin(c*x))*g/(c*f+sqrt(c^2*f^2-g^2)))*sqrt(1-c^2*x^2)/(d*(c^2*f^2-g^2)^(3/2)*sqrt(d-c^2*d*x^2))+2*I*b^2*g^2*polylog(3,I*exp(I*arcsin(c*x))*g/(c*f-sqrt(c^2*f^2-g^2)))*sqrt(1-c^2*x^2)/(d*(c^2*f^2-g^2)^(3/2)*sqrt(d-c^2*d*x^2))-2*I*b^2*g^2*polylog(3,I*exp(I*arcsin(c*x))*g/(c*f+sqrt(c^2*f^2-g^2)))*sqrt(1-c^2*x^2)/(d*(c^2*f^2-g^2)^(3/2)*sqrt(d-c^2*d*x^2))+1/2*(a+b*arcsin(c*x))^2*sqrt(1-c^2*x^2)*tan(1/4*Pi+1/2*arcsin(c*x))/(d*(c*f+g)*sqrt(d-c^2*d*x^2))],

#  {1/(d - c^2*d*x^2)^(3/2)*(a + b*ArcSin[c*x])^2/(f + g*x)^2, x, 7, 0} 
[(f+g*x)^3*(a+b*arcsin(c*x))^2/(d-c^2*d*x^2)^(5/2),x,37,-1/12*I*(c*f-g)^3*(a+b*arcsin(c*x))^2*sqrt(1-c^2*x^2)/(c^4*d^2*sqrt(d-c^2*d*x^2))+1/4*I*(c*f-2*g)*(c*f+g)^2*(a+b*arcsin(c*x))^2*sqrt(1-c^2*x^2)/(c^4*d^2*sqrt(d-c^2*d*x^2))+1/12*I*(c*f+g)^3*(a+b*arcsin(c*x))^2*sqrt(1-c^2*x^2)/(c^4*d^2*sqrt(d-c^2*d*x^2))-1/4*I*(c*f-g)^2*(c*f+2*g)*(a+b*arcsin(c*x))^2*sqrt(1-c^2*x^2)/(c^4*d^2*sqrt(d-c^2*d*x^2))-1/6*b^2*(c*f-g)^3*cot(1/4*Pi+1/2*arcsin(c*x))*sqrt(1-c^2*x^2)/(c^4*d^2*sqrt(d-c^2*d*x^2))-1/12*(c*f-g)^3*(a+b*arcsin(c*x))^2*cot(1/4*Pi+1/2*arcsin(c*x))*sqrt(1-c^2*x^2)/(c^4*d^2*sqrt(d-c^2*d*x^2))-1/4*(c*f-g)^2*(c*f+2*g)*(a+b*arcsin(c*x))^2*cot(1/4*Pi+1/2*arcsin(c*x))*sqrt(1-c^2*x^2)/(c^4*d^2*sqrt(d-c^2*d*x^2))-1/12*b*(c*f-g)^3*(a+b*arcsin(c*x))*csc(1/4*Pi+1/2*arcsin(c*x))^2*sqrt(1-c^2*x^2)/(c^4*d^2*sqrt(d-c^2*d*x^2))-1/24*(c*f-g)^3*(a+b*arcsin(c*x))^2*cot(1/4*Pi+1/2*arcsin(c*x))*csc(1/4*Pi+1/2*arcsin(c*x))^2*sqrt(1-c^2*x^2)/(c^4*d^2*sqrt(d-c^2*d*x^2))+b*(c*f-2*g)*(c*f+g)^2*(a+b*arcsin(c*x))*log(1+(-I)/exp(I*arcsin(c*x)))*sqrt(1-c^2*x^2)/(c^4*d^2*sqrt(d-c^2*d*x^2))+1/3*b*(c*f+g)^3*(a+b*arcsin(c*x))*log(1+(-I)/exp(I*arcsin(c*x)))*sqrt(1-c^2*x^2)/(c^4*d^2*sqrt(d-c^2*d*x^2))+1/3*b*(c*f-g)^3*(a+b*arcsin(c*x))*log(1-I*exp(I*arcsin(c*x)))*sqrt(1-c^2*x^2)/(c^4*d^2*sqrt(d-c^2*d*x^2))+b*(c*f-g)^2*(c*f+2*g)*(a+b*arcsin(c*x))*log(1-I*exp(I*arcsin(c*x)))*sqrt(1-c^2*x^2)/(c^4*d^2*sqrt(d-c^2*d*x^2))+I*b^2*(c*f-2*g)*(c*f+g)^2*polylog(2,I/exp(I*arcsin(c*x)))*sqrt(1-c^2*x^2)/(c^4*d^2*sqrt(d-c^2*d*x^2))+1/3*I*b^2*(c*f+g)^3*polylog(2,I/exp(I*arcsin(c*x)))*sqrt(1-c^2*x^2)/(c^4*d^2*sqrt(d-c^2*d*x^2))-1/3*I*b^2*(c*f-g)^3*polylog(2,I*exp(I*arcsin(c*x)))*sqrt(1-c^2*x^2)/(c^4*d^2*sqrt(d-c^2*d*x^2))-I*b^2*(c*f-g)^2*(c*f+2*g)*polylog(2,I*exp(I*arcsin(c*x)))*sqrt(1-c^2*x^2)/(c^4*d^2*sqrt(d-c^2*d*x^2))-1/12*b*(c*f+g)^3*(a+b*arcsin(c*x))*sec(1/4*Pi+1/2*arcsin(c*x))^2*sqrt(1-c^2*x^2)/(c^4*d^2*sqrt(d-c^2*d*x^2))+1/6*b^2*(c*f+g)^3*sqrt(1-c^2*x^2)*tan(1/4*Pi+1/2*arcsin(c*x))/(c^4*d^2*sqrt(d-c^2*d*x^2))+1/4*(c*f-2*g)*(c*f+g)^2*(a+b*arcsin(c*x))^2*sqrt(1-c^2*x^2)*tan(1/4*Pi+1/2*arcsin(c*x))/(c^4*d^2*sqrt(d-c^2*d*x^2))+1/12*(c*f+g)^3*(a+b*arcsin(c*x))^2*sqrt(1-c^2*x^2)*tan(1/4*Pi+1/2*arcsin(c*x))/(c^4*d^2*sqrt(d-c^2*d*x^2))+1/24*(c*f+g)^3*(a+b*arcsin(c*x))^2*sec(1/4*Pi+1/2*arcsin(c*x))^2*sqrt(1-c^2*x^2)*tan(1/4*Pi+1/2*arcsin(c*x))/(c^4*d^2*sqrt(d-c^2*d*x^2))],
[(f+g*x)^2*(a+b*arcsin(c*x))^2/(d-c^2*d*x^2)^(5/2),x,30,2/3*b^2*f*g/(c^2*d^2*sqrt(d-c^2*d*x^2))+1/3*b^2*f^2*x/(d^2*sqrt(d-c^2*d*x^2))+1/3*b^2*g^2*x/(c^2*d^2*sqrt(d-c^2*d*x^2))+2/3*f^2*x*(a+b*arcsin(c*x))^2/(d^2*sqrt(d-c^2*d*x^2))+2/3*f*g*(a+b*arcsin(c*x))^2/(c^2*d^2*(1-c^2*x^2)*sqrt(d-c^2*d*x^2))+1/3*f^2*x*(a+b*arcsin(c*x))^2/(d^2*(1-c^2*x^2)*sqrt(d-c^2*d*x^2))+1/3*g^2*x^3*(a+b*arcsin(c*x))^2/(d^2*(1-c^2*x^2)*sqrt(d-c^2*d*x^2))-1/3*b*f^2*(a+b*arcsin(c*x))/(c*d^2*sqrt(1-c^2*x^2)*sqrt(d-c^2*d*x^2))-2/3*b*f*g*x*(a+b*arcsin(c*x))/(c*d^2*sqrt(1-c^2*x^2)*sqrt(d-c^2*d*x^2))-1/3*b*g^2*x^2*(a+b*arcsin(c*x))/(c*d^2*sqrt(1-c^2*x^2)*sqrt(d-c^2*d*x^2))-1/3*b^2*g^2*arcsin(c*x)*sqrt(1-c^2*x^2)/(c^3*d^2*sqrt(d-c^2*d*x^2))-2/3*I*f^2*(a+b*arcsin(c*x))^2*sqrt(1-c^2*x^2)/(c*d^2*sqrt(d-c^2*d*x^2))+1/3*I*g^2*(a+b*arcsin(c*x))^2*sqrt(1-c^2*x^2)/(c^3*d^2*sqrt(d-c^2*d*x^2))+4/3*I*b*f*g*(a+b*arcsin(c*x))*arctan(exp(I*arcsin(c*x)))*sqrt(1-c^2*x^2)/(c^2*d^2*sqrt(d-c^2*d*x^2))+4/3*b*f^2*(a+b*arcsin(c*x))*log(1+exp(2*I*arcsin(c*x)))*sqrt(1-c^2*x^2)/(c*d^2*sqrt(d-c^2*d*x^2))-2/3*b*g^2*(a+b*arcsin(c*x))*log(1+exp(2*I*arcsin(c*x)))*sqrt(1-c^2*x^2)/(c^3*d^2*sqrt(d-c^2*d*x^2))-2/3*I*b^2*f*g*polylog(2,-I*exp(I*arcsin(c*x)))*sqrt(1-c^2*x^2)/(c^2*d^2*sqrt(d-c^2*d*x^2))+2/3*I*b^2*f*g*polylog(2,I*exp(I*arcsin(c*x)))*sqrt(1-c^2*x^2)/(c^2*d^2*sqrt(d-c^2*d*x^2))-2/3*I*b^2*f^2*polylog(2,-exp(2*I*arcsin(c*x)))*sqrt(1-c^2*x^2)/(c*d^2*sqrt(d-c^2*d*x^2))+1/3*I*b^2*g^2*polylog(2,-exp(2*I*arcsin(c*x)))*sqrt(1-c^2*x^2)/(c^3*d^2*sqrt(d-c^2*d*x^2))],
[(f+g*x)*(a+b*arcsin(c*x))^2/(d-c^2*d*x^2)^(5/2),x,21,1/3*b^2*g/(c^2*d^2*sqrt(d-c^2*d*x^2))+1/3*b^2*f*x/(d^2*sqrt(d-c^2*d*x^2))+2/3*f*x*(a+b*arcsin(c*x))^2/(d^2*sqrt(d-c^2*d*x^2))+1/3*g*(a+b*arcsin(c*x))^2/(c^2*d^2*(1-c^2*x^2)*sqrt(d-c^2*d*x^2))+1/3*f*x*(a+b*arcsin(c*x))^2/(d^2*(1-c^2*x^2)*sqrt(d-c^2*d*x^2))-1/3*b*f*(a+b*arcsin(c*x))/(c*d^2*sqrt(1-c^2*x^2)*sqrt(d-c^2*d*x^2))-1/3*b*g*x*(a+b*arcsin(c*x))/(c*d^2*sqrt(1-c^2*x^2)*sqrt(d-c^2*d*x^2))-2/3*I*f*(a+b*arcsin(c*x))^2*sqrt(1-c^2*x^2)/(c*d^2*sqrt(d-c^2*d*x^2))+2/3*I*b*g*(a+b*arcsin(c*x))*arctan(exp(I*arcsin(c*x)))*sqrt(1-c^2*x^2)/(c^2*d^2*sqrt(d-c^2*d*x^2))+4/3*b*f*(a+b*arcsin(c*x))*log(1+exp(2*I*arcsin(c*x)))*sqrt(1-c^2*x^2)/(c*d^2*sqrt(d-c^2*d*x^2))-1/3*I*b^2*g*polylog(2,-I*exp(I*arcsin(c*x)))*sqrt(1-c^2*x^2)/(c^2*d^2*sqrt(d-c^2*d*x^2))+1/3*I*b^2*g*polylog(2,I*exp(I*arcsin(c*x)))*sqrt(1-c^2*x^2)/(c^2*d^2*sqrt(d-c^2*d*x^2))-2/3*I*b^2*f*polylog(2,-exp(2*I*arcsin(c*x)))*sqrt(1-c^2*x^2)/(c*d^2*sqrt(d-c^2*d*x^2))],

#  {1/(d - c^2*d*x^2)^(5/2)*(a + b*ArcSin[c*x])^2/(f + g*x)^1, x, 48, If[$VersionNumber>=8, (I*(c*f - 2*g)*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2)/(4*d^2*(c*f - g)^2*Sqrt[d - c^2*d*x^2]) + (I*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2)/(12*d^2*(c*f - g)*Sqrt[d - c^2*d*x^2]) - (I*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2)/(12*d^2*(c*f + g)*Sqrt[d - c^2*d*x^2]) - (I*(c*f + 2*g)*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2)/(4*d^2*(c*f + g)^2*Sqrt[d - c^2*d*x^2]) + (b*(c*f - 2*g)*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*Log[1 + E^((I*Pi)/2 - I*ArcSin[c*x])])/(d^2*(c*f - g)^2*Sqrt[d - c^2*d*x^2]) + (b*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*Log[1 + E^((I*Pi)/2 - I*ArcSin[c*x])])/(3*d^2*(c*f - g)*Sqrt[d - c^2*d*x^2]) + (b*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*Log[1 + E^((I*Pi)/2 + I*ArcSin[c*x])])/(3*d^2*(c*f + g)*Sqrt[d - c^2*d*x^2]) + (b*(c*f + 2*g)*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*Log[1 + E^((I*Pi)/2 + I*ArcSin[c*x])])/(d^2*(c*f + g)^2*Sqrt[d - c^2*d*x^2]) - (I*g^4*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2*Log[1 - (I*E^(I*ArcSin[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g^2])])/(d^2*(c*f - g)^2*(c*f + g)^2*Sqrt[c^2*f^2 - g^2]*Sqrt[d - c^2*d*x^2]) + (I*g^4*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2*Log[1 - (I*E^(I*ArcSin[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2])])/(d^2*(c*f - g)^2*(c*f + g)^2*Sqrt[c^2*f^2 - g^2]*Sqrt[d - c^2*d*x^2]) + (I*b^2*(c*f - 2*g)*Sqrt[1 - c^2*x^2]*PolyLog[2, -E^((I*Pi)/2 - I*ArcSin[c*x])])/(d^2*(c*f - g)^2*Sqrt[d - c^2*d*x^2]) + (I*b^2*Sqrt[1 - c^2*x^2]*PolyLog[2, -E^((I*Pi)/2 - I*ArcSin[c*x])])/(3*d^2*(c*f - g)*Sqrt[d - c^2*d*x^2]) - (I*b^2*Sqrt[1 - c^2*x^2]*PolyLog[2, -E^((I*Pi)/2 + I*ArcSin[c*x])])/(3*d^2*(c*f + g)*Sqrt[d - c^2*d*x^2]) - (I*b^2*(c*f + 2*g)*Sqrt[1 - c^2*x^2]*PolyLog[2, -E^((I*Pi)/2 + I*ArcSin[c*x])])/(d^2*(c*f + g)^2*Sqrt[d - c^2*d*x^2]) - (2*b*g^4*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*PolyLog[2, (I*E^(I*ArcSin[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g^2])])/(d^2*(c*f - g)^2*(c*f + g)^2*Sqrt[c^2*f^2 - g^2]*Sqrt[d - c^2*d*x^2]) + (2*b*g^4*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*PolyLog[2, (I*E^(I*ArcSin[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2])])/(d^2*(c*f - g)^2*(c*f + g)^2*Sqrt[c^2*f^2 - g^2]*Sqrt[d - c^2*d*x^2]) - (2*I*b^2*g^4*Sqrt[1 - c^2*x^2]*PolyLog[3, (I*E^(I*ArcSin[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g^2])])/(d^2*(c*f - g)^2*(c*f + g)^2*Sqrt[c^2*f^2 - g^2]*Sqrt[d - c^2*d*x^2]) + (2*I*b^2*g^4*Sqrt[1 - c^2*x^2]*PolyLog[3, (I*E^(I*ArcSin[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2])])/(d^2*(c*f - g)^2*(c*f + g)^2*Sqrt[c^2*f^2 - g^2]*Sqrt[d - c^2*d*x^2]) - (b*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*Sec[Pi/4 - (1/2)*ArcSin[c*x]]^2)/(12*d^2*(c*f - g)*Sqrt[d - c^2*d*x^2]) - (b*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*Sec[Pi/4 + (1/2)*ArcSin[c*x]]^2)/(12*d^2*(c*f + g)*Sqrt[d - c^2*d*x^2]) - (b^2*Sqrt[1 - c^2*x^2]*Tan[Pi/4 - (1/2)*ArcSin[c*x]])/(6*d^2*(c*f - g)*Sqrt[d - c^2*d*x^2]) - ((c*f - 2*g)*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2*Tan[Pi/4 - (1/2)*ArcSin[c*x]])/(4*d^2*(c*f - g)^2*Sqrt[d - c^2*d*x^2]) - (Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2*Tan[Pi/4 - (1/2)*ArcSin[c*x]])/(12*d^2*(c*f - g)*Sqrt[d - c^2*d*x^2]) - (Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2*Sec[Pi/4 - (1/2)*ArcSin[c*x]]^2*Tan[Pi/4 - (1/2)*ArcSin[c*x]])/(24*d^2*(c*f - g)*Sqrt[d - c^2*d*x^2]) + (b^2*Sqrt[1 - c^2*x^2]*Tan[Pi/4 + (1/2)*ArcSin[c*x]])/(6*d^2*(c*f + g)*Sqrt[d - c^2*d*x^2]) + (Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2*Tan[Pi/4 + (1/2)*ArcSin[c*x]])/(12*d^2*(c*f + g)*Sqrt[d - c^2*d*x^2]) + ((c*f + 2*g)*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2*Tan[Pi/4 + (1/2)*ArcSin[c*x]])/(4*d^2*(c*f + g)^2*Sqrt[d - c^2*d*x^2]) + (Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2*Sec[Pi/4 + (1/2)*ArcSin[c*x]]^2*Tan[Pi/4 + (1/2)*ArcSin[c*x]])/(24*d^2*(c*f + g)*Sqrt[d - c^2*d*x^2]), (I*(c*f - 2*g)*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2)/(4*d^2*(c*f - g)^2*Sqrt[d - c^2*d*x^2]) + (I*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2)/(12*d^2*(c*f - g)*Sqrt[d - c^2*d*x^2]) - (I*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2)/(12*d^2*(c*f + g)*Sqrt[d - c^2*d*x^2]) - (I*(c*f + 2*g)*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2)/(4*d^2*(c*f + g)^2*Sqrt[d - c^2*d*x^2]) + (b*(c*f - 2*g)*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*Log[1 + E^((1/2)*I*(Pi - 2*ArcSin[c*x]))])/(d^2*(c*f - g)^2*Sqrt[d - c^2*d*x^2]) + (b*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*Log[1 + E^((1/2)*I*(Pi - 2*ArcSin[c*x]))])/(3*d^2*(c*f - g)*Sqrt[d - c^2*d*x^2]) + (b*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*Log[1 + E^((1/2)*I*(Pi + 2*ArcSin[c*x]))])/(3*d^2*(c*f + g)*Sqrt[d - c^2*d*x^2]) + (b*(c*f + 2*g)*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*Log[1 + E^((1/2)*I*(Pi + 2*ArcSin[c*x]))])/(d^2*(c*f + g)^2*Sqrt[d - c^2*d*x^2]) - (I*g^4*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2*Log[1 - (I*E^(I*ArcSin[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g^2])])/(d^2*(c^2*f^2 - g^2)^(5/2)*Sqrt[d - c^2*d*x^2]) + (I*g^4*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2*Log[1 - (I*E^(I*ArcSin[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2])])/(d^2*(c^2*f^2 - g^2)^(5/2)*Sqrt[d - c^2*d*x^2]) + (I*b^2*(c*f - 2*g)*Sqrt[1 - c^2*x^2]*PolyLog[2, -E^((1/2)*I*(Pi - 2*ArcSin[c*x]))])/(d^2*(c*f - g)^2*Sqrt[d - c^2*d*x^2]) + (I*b^2*Sqrt[1 - c^2*x^2]*PolyLog[2, -E^((1/2)*I*(Pi - 2*ArcSin[c*x]))])/(3*d^2*(c*f - g)*Sqrt[d - c^2*d*x^2]) - (I*b^2*Sqrt[1 - c^2*x^2]*PolyLog[2, -E^((1/2)*I*(Pi + 2*ArcSin[c*x]))])/(3*d^2*(c*f + g)*Sqrt[d - c^2*d*x^2]) - (I*b^2*(c*f + 2*g)*Sqrt[1 - c^2*x^2]*PolyLog[2, -E^((1/2)*I*(Pi + 2*ArcSin[c*x]))])/(d^2*(c*f + g)^2*Sqrt[d - c^2*d*x^2]) - (2*b*g^4*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*PolyLog[2, (I*E^(I*ArcSin[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g^2])])/(d^2*(c^2*f^2 - g^2)^(5/2)*Sqrt[d - c^2*d*x^2]) + (2*b*g^4*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*PolyLog[2, (I*E^(I*ArcSin[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2])])/(d^2*(c^2*f^2 - g^2)^(5/2)*Sqrt[d - c^2*d*x^2]) - (2*I*b^2*g^4*Sqrt[1 - c^2*x^2]*PolyLog[3, (I*E^(I*ArcSin[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g^2])])/(d^2*(c^2*f^2 - g^2)^(5/2)*Sqrt[d - c^2*d*x^2]) + (2*I*b^2*g^4*Sqrt[1 - c^2*x^2]*PolyLog[3, (I*E^(I*ArcSin[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2])])/(d^2*(c^2*f^2 - g^2)^(5/2)*Sqrt[d - c^2*d*x^2]) - (b*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*Sec[Pi/4 - (1/2)*ArcSin[c*x]]^2)/(12*d^2*(c*f - g)*Sqrt[d - c^2*d*x^2]) - (b*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*Sec[Pi/4 + (1/2)*ArcSin[c*x]]^2)/(12*d^2*(c*f + g)*Sqrt[d - c^2*d*x^2]) - (b^2*Sqrt[1 - c^2*x^2]*Tan[Pi/4 - (1/2)*ArcSin[c*x]])/(6*d^2*(c*f - g)*Sqrt[d - c^2*d*x^2]) - ((c*f - 2*g)*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2*Tan[Pi/4 - (1/2)*ArcSin[c*x]])/(4*d^2*(c*f - g)^2*Sqrt[d - c^2*d*x^2]) - (Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2*Tan[Pi/4 - (1/2)*ArcSin[c*x]])/(12*d^2*(c*f - g)*Sqrt[d - c^2*d*x^2]) - (Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2*Sec[Pi/4 - (1/2)*ArcSin[c*x]]^2*Tan[Pi/4 - (1/2)*ArcSin[c*x]])/(24*d^2*(c*f - g)*Sqrt[d - c^2*d*x^2]) + (b^2*Sqrt[1 - c^2*x^2]*Tan[Pi/4 + (1/2)*ArcSin[c*x]])/(6*d^2*(c*f + g)*Sqrt[d - c^2*d*x^2]) + (Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2*Tan[Pi/4 + (1/2)*ArcSin[c*x]])/(12*d^2*(c*f + g)*Sqrt[d - c^2*d*x^2]) + ((c*f + 2*g)*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2*Tan[Pi/4 + (1/2)*ArcSin[c*x]])/(4*d^2*(c*f + g)^2*Sqrt[d - c^2*d*x^2]) + (Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2*Sec[Pi/4 + (1/2)*ArcSin[c*x]]^2*Tan[Pi/4 + (1/2)*ArcSin[c*x]])/(24*d^2*(c*f + g)*Sqrt[d - c^2*d*x^2])]} 

#  {1/(d - c^2*d*x^2)^(3/2)*(a + b*ArcSin[c*x])^2/(f + g*x)^2, x, 7, 0} 

# Integrands of the form Log[h (f + g x)^m] (d+e x^2)^(p/2) (a+b ArcSin[c x])^n where c^2 d+e=0

# Integrands of the form Log[h (f + g x)^m] (d-c^2 d x^2)^(p/2) (a+b ArcSin[c x])^n

# p>0

# p<0
[(a+b*arcsin(c*x))^n*log(h*(f+g*x)^m)/sqrt(1-c^2*x^2),x,0,Unintegrable((a+b*arcsin(c*x))^n*log(h*(f+g*x)^m)/sqrt(1-c^2*x^2),x)],
[(a+b*arcsin(c*x))^3*log(h*(f+g*x)^m)/sqrt(1-c^2*x^2),x,15,1/20*I*m*(a+b*arcsin(c*x))^5/(b^2*c)+1/4*(a+b*arcsin(c*x))^4*log(h*(f+g*x)^m)/(b*c)-1/4*m*(a+b*arcsin(c*x))^4*log(1-I*exp(I*arcsin(c*x))*g/(c*f-sqrt(c^2*f^2-g^2)))/(b*c)-1/4*m*(a+b*arcsin(c*x))^4*log(1-I*exp(I*arcsin(c*x))*g/(c*f+sqrt(c^2*f^2-g^2)))/(b*c)+I*m*(a+b*arcsin(c*x))^3*polylog(2,I*exp(I*arcsin(c*x))*g/(c*f-sqrt(c^2*f^2-g^2)))/c+I*m*(a+b*arcsin(c*x))^3*polylog(2,I*exp(I*arcsin(c*x))*g/(c*f+sqrt(c^2*f^2-g^2)))/c-3*b*m*(a+b*arcsin(c*x))^2*polylog(3,I*exp(I*arcsin(c*x))*g/(c*f-sqrt(c^2*f^2-g^2)))/c-3*b*m*(a+b*arcsin(c*x))^2*polylog(3,I*exp(I*arcsin(c*x))*g/(c*f+sqrt(c^2*f^2-g^2)))/c-6*I*b^2*m*(a+b*arcsin(c*x))*polylog(4,I*exp(I*arcsin(c*x))*g/(c*f-sqrt(c^2*f^2-g^2)))/c-6*I*b^2*m*(a+b*arcsin(c*x))*polylog(4,I*exp(I*arcsin(c*x))*g/(c*f+sqrt(c^2*f^2-g^2)))/c+6*b^3*m*polylog(5,I*exp(I*arcsin(c*x))*g/(c*f-sqrt(c^2*f^2-g^2)))/c+6*b^3*m*polylog(5,I*exp(I*arcsin(c*x))*g/(c*f+sqrt(c^2*f^2-g^2)))/c],
[(a+b*arcsin(c*x))^2*log(h*(f+g*x)^m)/sqrt(1-c^2*x^2),x,13,1/12*I*m*(a+b*arcsin(c*x))^4/(b^2*c)+1/3*(a+b*arcsin(c*x))^3*log(h*(f+g*x)^m)/(b*c)-1/3*m*(a+b*arcsin(c*x))^3*log(1-I*exp(I*arcsin(c*x))*g/(c*f-sqrt(c^2*f^2-g^2)))/(b*c)-1/3*m*(a+b*arcsin(c*x))^3*log(1-I*exp(I*arcsin(c*x))*g/(c*f+sqrt(c^2*f^2-g^2)))/(b*c)+I*m*(a+b*arcsin(c*x))^2*polylog(2,I*exp(I*arcsin(c*x))*g/(c*f-sqrt(c^2*f^2-g^2)))/c+I*m*(a+b*arcsin(c*x))^2*polylog(2,I*exp(I*arcsin(c*x))*g/(c*f+sqrt(c^2*f^2-g^2)))/c-2*b*m*(a+b*arcsin(c*x))*polylog(3,I*exp(I*arcsin(c*x))*g/(c*f-sqrt(c^2*f^2-g^2)))/c-2*b*m*(a+b*arcsin(c*x))*polylog(3,I*exp(I*arcsin(c*x))*g/(c*f+sqrt(c^2*f^2-g^2)))/c-2*I*b^2*m*polylog(4,I*exp(I*arcsin(c*x))*g/(c*f-sqrt(c^2*f^2-g^2)))/c-2*I*b^2*m*polylog(4,I*exp(I*arcsin(c*x))*g/(c*f+sqrt(c^2*f^2-g^2)))/c],
[(a+b*arcsin(c*x))*log(h*(f+g*x)^m)/sqrt(1-c^2*x^2),x,11,1/6*I*m*(a+b*arcsin(c*x))^3/(b^2*c)+1/2*(a+b*arcsin(c*x))^2*log(h*(f+g*x)^m)/(b*c)-1/2*m*(a+b*arcsin(c*x))^2*log(1-I*exp(I*arcsin(c*x))*g/(c*f-sqrt(c^2*f^2-g^2)))/(b*c)-1/2*m*(a+b*arcsin(c*x))^2*log(1-I*exp(I*arcsin(c*x))*g/(c*f+sqrt(c^2*f^2-g^2)))/(b*c)+I*m*(a+b*arcsin(c*x))*polylog(2,I*exp(I*arcsin(c*x))*g/(c*f-sqrt(c^2*f^2-g^2)))/c+I*m*(a+b*arcsin(c*x))*polylog(2,I*exp(I*arcsin(c*x))*g/(c*f+sqrt(c^2*f^2-g^2)))/c-b*m*polylog(3,I*exp(I*arcsin(c*x))*g/(c*f-sqrt(c^2*f^2-g^2)))/c-b*m*polylog(3,I*exp(I*arcsin(c*x))*g/(c*f+sqrt(c^2*f^2-g^2)))/c],
[log(h*(f+g*x)^m)/sqrt(1-c^2*x^2),x,9,1/2*I*m*arcsin(c*x)^2/c+arcsin(c*x)*log(h*(f+g*x)^m)/c-m*arcsin(c*x)*log(1-I*exp(I*arcsin(c*x))*g/(c*f-sqrt(c^2*f^2-g^2)))/c-m*arcsin(c*x)*log(1-I*exp(I*arcsin(c*x))*g/(c*f+sqrt(c^2*f^2-g^2)))/c+I*m*polylog(2,I*exp(I*arcsin(c*x))*g/(c*f-sqrt(c^2*f^2-g^2)))/c+I*m*polylog(2,I*exp(I*arcsin(c*x))*g/(c*f+sqrt(c^2*f^2-g^2)))/c],
[log(h*(f+g*x)^m)/((a+b*arcsin(c*x))*sqrt(1-c^2*x^2)),x,0,Unintegrable(log(h*(f+g*x)^m)/((a+b*arcsin(c*x))*sqrt(1-c^2*x^2)),x)],

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

# Integrands of the form Px (d+e x)^m (a+b ArcSin[c x])
[(d+e*x)^3*(f+g*x)*(a+b*arcsin(c*x)),x,6,-1/32*b*(8*c^2*d^2*(3*e*f+d*g)+3*e^2*(e*f+3*d*g))*arcsin(c*x)/c^4+d^3*f*x*(a+b*arcsin(c*x))+1/2*d^2*(3*e*f+d*g)*x^2*(a+b*arcsin(c*x))+d*e*(e*f+d*g)*x^3*(a+b*arcsin(c*x))+1/4*e^2*(e*f+3*d*g)*x^4*(a+b*arcsin(c*x))+1/5*e^3*g*x^5*(a+b*arcsin(c*x))+1/75*b*e*(4*e^2*g+25*c^2*d*(e*f+d*g))*x^2*sqrt(1-c^2*x^2)/c^3+1/16*b*e^2*(e*f+3*d*g)*x^3*sqrt(1-c^2*x^2)/c+1/25*b*e^3*g*x^4*sqrt(1-c^2*x^2)/c+1/2400*b*(32*(75*c^4*d^3*f+8*e^3*g+50*c^2*d*e*(e*f+d*g))+75*c^2*(8*c^2*d^2*(3*e*f+d*g)+3*e^2*(e*f+3*d*g))*x)*sqrt(1-c^2*x^2)/c^5],
[(d+e*x)^2*(f+g*x)*(a+b*arcsin(c*x)),x,6,-1/32*b*(3*e^2*g+8*c^2*d*(2*e*f+d*g))*arcsin(c*x)/c^4+d^2*f*x*(a+b*arcsin(c*x))+1/2*d*(2*e*f+d*g)*x^2*(a+b*arcsin(c*x))+1/3*e*(e*f+2*d*g)*x^3*(a+b*arcsin(c*x))+1/4*e^2*g*x^4*(a+b*arcsin(c*x))+1/9*b*e*(e*f+2*d*g)*x^2*sqrt(1-c^2*x^2)/c+1/16*b*e^2*g*x^3*sqrt(1-c^2*x^2)/c+1/288*b*(32*(9*c^2*d^2*f+2*e*(e*f+2*d*g))+9*(3*e^2*g+8*c^2*d*(2*e*f+d*g))*x)*sqrt(1-c^2*x^2)/c^3],
[(d+e*x)*(f+g*x)*(a+b*arcsin(c*x)),x,5,-1/4*b*(e*f+d*g)*arcsin(c*x)/c^2+d*f*x*(a+b*arcsin(c*x))+1/2*(e*f+d*g)*x^2*(a+b*arcsin(c*x))+1/3*e*g*x^3*(a+b*arcsin(c*x))+1/9*b*e*g*x^2*sqrt(1-c^2*x^2)/c+1/36*b*(4*(9*c^2*d*f+2*e*g)+9*c^2*(e*f+d*g)*x)*sqrt(1-c^2*x^2)/c^3],
[(f+g*x)*(a+b*arcsin(c*x))/(d+e*x),x,14,-1/2*I*b*(e*f-d*g)*arcsin(c*x)^2/e^2+g*x*(a+b*arcsin(c*x))/e-b*(e*f-d*g)*arcsin(c*x)*log(d+e*x)/e^2+(e*f-d*g)*(a+b*arcsin(c*x))*log(d+e*x)/e^2+b*(e*f-d*g)*arcsin(c*x)*log(1-I*e*exp(I*arcsin(c*x))/(c*d-sqrt(c^2*d^2-e^2)))/e^2+b*(e*f-d*g)*arcsin(c*x)*log(1-I*e*exp(I*arcsin(c*x))/(c*d+sqrt(c^2*d^2-e^2)))/e^2-I*b*(e*f-d*g)*polylog(2,I*e*exp(I*arcsin(c*x))/(c*d-sqrt(c^2*d^2-e^2)))/e^2-I*b*(e*f-d*g)*polylog(2,I*e*exp(I*arcsin(c*x))/(c*d+sqrt(c^2*d^2-e^2)))/e^2+b*g*sqrt(1-c^2*x^2)/(c*e)],
[(f+g*x)*(a+b*arcsin(c*x))/(d+e*x)^2,x,15,-1/2*I*b*g*arcsin(c*x)^2/e^2-(e*f-d*g)*(a+b*arcsin(c*x))/(e^2*(d+e*x))-b*g*arcsin(c*x)*log(d+e*x)/e^2+g*(a+b*arcsin(c*x))*log(d+e*x)/e^2+b*g*arcsin(c*x)*log(1-I*e*exp(I*arcsin(c*x))/(c*d-sqrt(c^2*d^2-e^2)))/e^2+b*g*arcsin(c*x)*log(1-I*e*exp(I*arcsin(c*x))/(c*d+sqrt(c^2*d^2-e^2)))/e^2-I*b*g*polylog(2,I*e*exp(I*arcsin(c*x))/(c*d-sqrt(c^2*d^2-e^2)))/e^2-I*b*g*polylog(2,I*e*exp(I*arcsin(c*x))/(c*d+sqrt(c^2*d^2-e^2)))/e^2+b*c*(e*f-d*g)*arctan((e+c^2*d*x)/(sqrt(c^2*d^2-e^2)*sqrt(1-c^2*x^2)))/(e^2*sqrt(c^2*d^2-e^2))],
[(f+g*x)*(a+b*arcsin(c*x))/(d+e*x)^3,x,7,1/2*b*g^2*arcsin(c*x)/(e^2*(e*f-d*g))-1/2*(f+g*x)^2*(a+b*arcsin(c*x))/((e*f-d*g)*(d+e*x)^2)-1/2*b*c*(2*e^2*g-c^2*d*(e*f+d*g))*arctan((e+c^2*d*x)/(sqrt(c^2*d^2-e^2)*sqrt(1-c^2*x^2)))/(e^2*(c^2*d^2-e^2)^(3/2))+1/2*b*c*(e*f-d*g)*sqrt(1-c^2*x^2)/(e*(c^2*d^2-e^2)*(d+e*x))],
[(f+g*x)*(a+b*arcsin(c*x))/(d+e*x)^4,x,6,-1/3*(e*f-d*g)*(a+b*arcsin(c*x))/(e^2*(d+e*x)^3)-1/2*g*(a+b*arcsin(c*x))/(e^2*(d+e*x)^2)+1/6*b*c^3*(e^2*(e*f-4*d*g)+c^2*d^2*(2*e*f+d*g))*arctan((e+c^2*d*x)/(sqrt(c^2*d^2-e^2)*sqrt(1-c^2*x^2)))/(e^2*(c^2*d^2-e^2)^(5/2))+1/6*b*c*(e*f-d*g)*sqrt(1-c^2*x^2)/(e*(c^2*d^2-e^2)*(d+e*x)^2)+1/2*b*c*(c^2*d*f-e*g)*sqrt(1-c^2*x^2)/((c^2*d^2-e^2)^2*(d+e*x))],
[(f+g*x)*(a+b*arcsin(c*x))/(d+e*x)^5,x,7,-1/4*(e*f-d*g)*(a+b*arcsin(c*x))/(e^2*(d+e*x)^4)-1/3*g*(a+b*arcsin(c*x))/(e^2*(d+e*x)^3)-1/24*b*c^3*(4*e^4*g-c^2*d*e^2*(9*e*f-13*d*g)-2*c^4*d^3*(3*e*f+d*g))*arctan((e+c^2*d*x)/(sqrt(c^2*d^2-e^2)*sqrt(1-c^2*x^2)))/(e^2*(c^2*d^2-e^2)^(7/2))+1/12*b*c*(e*f-d*g)*sqrt(1-c^2*x^2)/(e*(c^2*d^2-e^2)*(d+e*x)^3)-1/24*b*c*(4*e^2*g-c^2*d*(5*e*f-d*g))*sqrt(1-c^2*x^2)/(e*(c^2*d^2-e^2)^2*(d+e*x)^2)+1/24*b*c^3*(4*e^2*(e*f-4*d*g)+c^2*d^2*(11*e*f+d*g))*sqrt(1-c^2*x^2)/(e*(c^2*d^2-e^2)^3*(d+e*x))],
[(f+g*x)*(a+b*arcsin(c*x))/(d+e*x)^6,x,8,-1/5*(e*f-d*g)*(a+b*arcsin(c*x))/(e^2*(d+e*x)^5)-1/4*g*(a+b*arcsin(c*x))/(e^2*(d+e*x)^4)+1/40*b*c^5*(c^2*d^2*e^2*(24*e*f-19*d*g)+3*e^4*(e*f-6*d*g)+2*c^4*d^4*(4*e*f+d*g))*arctan((e+c^2*d*x)/(sqrt(c^2*d^2-e^2)*sqrt(1-c^2*x^2)))/(e^2*(c^2*d^2-e^2)^(9/2))+1/20*b*c*(e*f-d*g)*sqrt(1-c^2*x^2)/(e*(c^2*d^2-e^2)*(d+e*x)^4)-1/60*b*c*(5*e^2*g-c^2*d*(7*e*f-2*d*g))*sqrt(1-c^2*x^2)/(e*(c^2*d^2-e^2)^2*(d+e*x)^3)+1/120*b*c^3*(e^2*(9*e*f-34*d*g)+c^2*d^2*(26*e*f-d*g))*sqrt(1-c^2*x^2)/(e*(c^2*d^2-e^2)^3*(d+e*x)^2)-1/24*b*c^3*(4*e^4*g-c^2*d*e^2*(11*e*f-18*d*g)-c^4*d^3*(10*e*f+d*g))*sqrt(1-c^2*x^2)/(e*(c^2*d^2-e^2)^4*(d+e*x))],
[(d+e*x)^3*(f+g*x+h*x^2)*(a+b*arcsin(c*x)),x,8,-1/96*b*(24*c^4*d^2*(3*e*f+d*g)+5*e^3*h+9*c^2*e*(e^2*f+3*d*e*g+3*d^2*h))*arcsin(c*x)/c^6+d^3*f*x*(a+b*arcsin(c*x))+1/2*d^2*(3*e*f+d*g)*x^2*(a+b*arcsin(c*x))+1/3*d*(3*e^2*f+3*d*e*g+d^2*h)*x^3*(a+b*arcsin(c*x))+1/4*e*(e^2*f+3*d*e*g+3*d^2*h)*x^4*(a+b*arcsin(c*x))+1/5*e^2*(e*g+3*d*h)*x^5*(a+b*arcsin(c*x))+1/6*e^3*h*x^6*(a+b*arcsin(c*x))+1/225*b*(12*e^2*(e*g+3*d*h)+25*c^2*d*(3*e^2*f+3*d*e*g+d^2*h))*x^2*sqrt(1-c^2*x^2)/c^3+1/144*b*e*(5*e^2*h+9*c^2*(e^2*f+3*d*e*g+3*d^2*h))*x^3*sqrt(1-c^2*x^2)/c^3+1/25*b*e^2*(e*g+3*d*h)*x^4*sqrt(1-c^2*x^2)/c+1/36*b*e^3*h*x^5*sqrt(1-c^2*x^2)/c+1/7200*b*(32*(225*c^4*d^3*f+24*e^2*(e*g+3*d*h)+50*c^2*d*(3*e^2*f+3*d*e*g+d^2*h))+75*(24*c^4*d^2*(3*e*f+d*g)+5*e^3*h+9*c^2*e*(e^2*f+3*d*e*g+3*d^2*h))*x)*sqrt(1-c^2*x^2)/c^5],
[(d+e*x)^2*(f+g*x+h*x^2)*(a+b*arcsin(c*x)),x,7,-1/32*b*(8*c^2*d*(2*e*f+d*g)+3*e*(e*g+2*d*h))*arcsin(c*x)/c^4+d^2*f*x*(a+b*arcsin(c*x))+1/2*d*(2*e*f+d*g)*x^2*(a+b*arcsin(c*x))+1/3*(e^2*f+2*d*e*g+d^2*h)*x^3*(a+b*arcsin(c*x))+1/4*e*(e*g+2*d*h)*x^4*(a+b*arcsin(c*x))+1/5*e^2*h*x^5*(a+b*arcsin(c*x))+1/225*b*(12*e^2*h+25*c^2*(e^2*f+2*d*e*g+d^2*h))*x^2*sqrt(1-c^2*x^2)/c^3+1/16*b*e*(e*g+2*d*h)*x^3*sqrt(1-c^2*x^2)/c+1/25*b*e^2*h*x^4*sqrt(1-c^2*x^2)/c+1/7200*b*(32*(225*c^4*d^2*f+24*e^2*h+50*c^2*(e^2*f+2*d*e*g+d^2*h))+225*c^2*(8*c^2*d*(2*e*f+d*g)+3*e*(e*g+2*d*h))*x)*sqrt(1-c^2*x^2)/c^5],
[(d+e*x)*(f+g*x+h*x^2)*(a+b*arcsin(c*x)),x,6,-1/32*b*(8*c^2*(e*f+d*g)+3*e*h)*arcsin(c*x)/c^4+d*f*x*(a+b*arcsin(c*x))+1/2*(e*f+d*g)*x^2*(a+b*arcsin(c*x))+1/3*(e*g+d*h)*x^3*(a+b*arcsin(c*x))+1/4*e*h*x^4*(a+b*arcsin(c*x))+1/9*b*(e*g+d*h)*x^2*sqrt(1-c^2*x^2)/c+1/16*b*e*h*x^3*sqrt(1-c^2*x^2)/c+1/288*b*(32*(9*c^2*d*f+2*e*g+2*d*h)+9*(8*c^2*(e*f+d*g)+3*e*h)*x)*sqrt(1-c^2*x^2)/c^3],
[(f+g*x+h*x^2)*(a+b*arcsin(c*x))/(d+e*x),x,15,-1/4*b*h*arcsin(c*x)/(c^2*e)-1/2*I*b*(e^2*f-d*e*g+d^2*h)*arcsin(c*x)^2/e^3+(e*g-d*h)*x*(a+b*arcsin(c*x))/e^2+1/2*h*x^2*(a+b*arcsin(c*x))/e-b*(e^2*f-d*e*g+d^2*h)*arcsin(c*x)*log(d+e*x)/e^3+(e^2*f-d*e*g+d^2*h)*(a+b*arcsin(c*x))*log(d+e*x)/e^3+b*(e^2*f-d*e*g+d^2*h)*arcsin(c*x)*log(1-I*e*exp(I*arcsin(c*x))/(c*d-sqrt(c^2*d^2-e^2)))/e^3+b*(e^2*f-d*e*g+d^2*h)*arcsin(c*x)*log(1-I*e*exp(I*arcsin(c*x))/(c*d+sqrt(c^2*d^2-e^2)))/e^3-I*b*(e^2*f-d*e*g+d^2*h)*polylog(2,I*e*exp(I*arcsin(c*x))/(c*d-sqrt(c^2*d^2-e^2)))/e^3-I*b*(e^2*f-d*e*g+d^2*h)*polylog(2,I*e*exp(I*arcsin(c*x))/(c*d+sqrt(c^2*d^2-e^2)))/e^3+1/4*b*(4*(e*g-d*h)+e*h*x)*sqrt(1-c^2*x^2)/(c*e^2)],
[(f+g*x+h*x^2)*(a+b*arcsin(c*x))/(d+e*x)^2,x,16,-1/2*I*b*(e*g-2*d*h)*arcsin(c*x)^2/e^3+h*x*(a+b*arcsin(c*x))/e^2-(e^2*f-d*e*g+d^2*h)*(a+b*arcsin(c*x))/(e^3*(d+e*x))-b*(e*g-2*d*h)*arcsin(c*x)*log(d+e*x)/e^3+(e*g-2*d*h)*(a+b*arcsin(c*x))*log(d+e*x)/e^3+b*(e*g-2*d*h)*arcsin(c*x)*log(1-I*e*exp(I*arcsin(c*x))/(c*d-sqrt(c^2*d^2-e^2)))/e^3+b*(e*g-2*d*h)*arcsin(c*x)*log(1-I*e*exp(I*arcsin(c*x))/(c*d+sqrt(c^2*d^2-e^2)))/e^3-I*b*(e*g-2*d*h)*polylog(2,I*e*exp(I*arcsin(c*x))/(c*d-sqrt(c^2*d^2-e^2)))/e^3-I*b*(e*g-2*d*h)*polylog(2,I*e*exp(I*arcsin(c*x))/(c*d+sqrt(c^2*d^2-e^2)))/e^3+b*c*(e^2*f-d*e*g+d^2*h)*arctan((e+c^2*d*x)/(sqrt(c^2*d^2-e^2)*sqrt(1-c^2*x^2)))/(e^3*sqrt(c^2*d^2-e^2))+b*h*sqrt(1-c^2*x^2)/(c*e^2)],
[(f+g*x+h*x^2)*(a+b*arcsin(c*x))/(d+e*x)^3,x,16,-1/2*I*b*h*arcsin(c*x)^2/e^3-1/2*(e^2*f-d*e*g+d^2*h)*(a+b*arcsin(c*x))/(e^3*(d+e*x)^2)-(e*g-2*d*h)*(a+b*arcsin(c*x))/(e^3*(d+e*x))-1/2*b*c*(2*e^2*(e*g-2*d*h)-c^2*d*(e^2*f+d*e*g-3*d^2*h))*arctan((e+c^2*d*x)/(sqrt(c^2*d^2-e^2)*sqrt(1-c^2*x^2)))/(e^3*(c^2*d^2-e^2)^(3/2))-b*h*arcsin(c*x)*log(d+e*x)/e^3+h*(a+b*arcsin(c*x))*log(d+e*x)/e^3+b*h*arcsin(c*x)*log(1-I*e*exp(I*arcsin(c*x))/(c*d-sqrt(c^2*d^2-e^2)))/e^3+b*h*arcsin(c*x)*log(1-I*e*exp(I*arcsin(c*x))/(c*d+sqrt(c^2*d^2-e^2)))/e^3-I*b*h*polylog(2,I*e*exp(I*arcsin(c*x))/(c*d-sqrt(c^2*d^2-e^2)))/e^3-I*b*h*polylog(2,I*e*exp(I*arcsin(c*x))/(c*d+sqrt(c^2*d^2-e^2)))/e^3+1/2*b*c*(e^2*f-d*e*g+d^2*h)*sqrt(1-c^2*x^2)/(e^2*(c^2*d^2-e^2)*(d+e*x))],
[(f+g*x+h*x^2)*(a+b*arcsin(c*x))/(d+e*x)^4,x,6,-1/3*(e^2*f-d*e*g+d^2*h)*(a+b*arcsin(c*x))/(e^3*(d+e*x)^3)-1/2*(e*g-2*d*h)*(a+b*arcsin(c*x))/(e^3*(d+e*x)^2)-h*(a+b*arcsin(c*x))/(e^3*(d+e*x))+1/6*b*c*(6*e^4*h+c^2*e^2*(e^2*f-4*d*e*g-5*d^2*h)+c^4*d^2*(2*e^2*f+d*e*g+2*d^2*h))*arctan((e+c^2*d*x)/(sqrt(c^2*d^2-e^2)*sqrt(1-c^2*x^2)))/(e^3*(c^2*d^2-e^2)^(5/2))+1/6*b*c*(e^2*f-d*e*g+d^2*h)*sqrt(1-c^2*x^2)/(e^2*(c^2*d^2-e^2)*(d+e*x)^2)-1/2*b*c*(e^2*(e*g-2*d*h)-c^2*(d*e^2*f-d^3*h))*sqrt(1-c^2*x^2)/(e^2*(c^2*d^2-e^2)^2*(d+e*x))],
[(f+g*x+h*x^2)*(a+b*arcsin(c*x))/(d+e*x)^5,x,7,-1/4*(e^2*f-d*e*g+d^2*h)*(a+b*arcsin(c*x))/(e^3*(d+e*x)^4)-1/3*(e*g-2*d*h)*(a+b*arcsin(c*x))/(e^3*(d+e*x)^3)-1/2*h*(a+b*arcsin(c*x))/(e^3*(d+e*x)^2)-1/24*b*c^3*(4*e^4*(e*g-5*d*h)-c^2*d*e^2*(9*e^2*f-13*d*e*g-7*d^2*h)-2*c^4*d^3*(3*e^2*f+d*e*g+d^2*h))*arctan((e+c^2*d*x)/(sqrt(c^2*d^2-e^2)*sqrt(1-c^2*x^2)))/(e^3*(c^2*d^2-e^2)^(7/2))+1/12*b*c*(e^2*f-d*e*g+d^2*h)*sqrt(1-c^2*x^2)/(e^2*(c^2*d^2-e^2)*(d+e*x)^3)-1/24*b*c*(4*e^2*(e*g-2*d*h)-c^2*d*(5*e^2*f-d*e*g-3*d^2*h))*sqrt(1-c^2*x^2)/(e^2*(c^2*d^2-e^2)^2*(d+e*x)^2)+1/24*b*c*(12*e^4*h+c^4*d^2*(11*e^2*f+d*e*g-d^2*h)+4*c^2*e^2*(e^2*f-4*d*e*g+d^2*h))*sqrt(1-c^2*x^2)/(e^2*(c^2*d^2-e^2)^3*(d+e*x))],
[(f+g*x+h*x^2)*(a+b*arcsin(c*x))/(d+e*x)^6,x,8,-1/5*(e^2*f-d*e*g+d^2*h)*(a+b*arcsin(c*x))/(e^3*(d+e*x)^5)-1/4*(e*g-2*d*h)*(a+b*arcsin(c*x))/(e^3*(d+e*x)^4)-1/3*h*(a+b*arcsin(c*x))/(e^3*(d+e*x)^3)+1/120*b*c^3*(20*e^6*h+3*c^4*d^2*e^2*(24*e^2*f-19*d*e*g-6*d^2*h)+2*c^6*d^4*(12*e^2*f+3*d*e*g+2*d^2*h)+9*c^2*e^4*(e^2*f-6*d*e*g+11*d^2*h))*arctan((e+c^2*d*x)/(sqrt(c^2*d^2-e^2)*sqrt(1-c^2*x^2)))/(e^3*(c^2*d^2-e^2)^(9/2))+1/20*b*c*(e^2*f-d*e*g+d^2*h)*sqrt(1-c^2*x^2)/(e^2*(c^2*d^2-e^2)*(d+e*x)^4)-1/60*b*c*(5*e^2*(e*g-2*d*h)-c^2*d*(7*e^2*f-2*d*e*g-3*d^2*h))*sqrt(1-c^2*x^2)/(e^2*(c^2*d^2-e^2)^2*(d+e*x)^3)+1/120*b*c*(20*e^4*h+c^4*d^2*(26*e^2*f-d*e*g-4*d^2*h)+c^2*e^2*(9*e^2*f-34*d*e*g+19*d^2*h))*sqrt(1-c^2*x^2)/(e^2*(c^2*d^2-e^2)^3*(d+e*x)^2)+1/24*b*c^3*(c^4*d^3*(10*e*f+d*g)-4*e^3*(e*g-5*d*h)+c^2*d*e*(11*e^2*f-18*d*e*g+d^2*h))*sqrt(1-c^2*x^2)/(e*(c^2*d^2-e^2)^4*(d+e*x))],
[(d+e*x)^3*(f+g*x+h*x^2+i*x^3)*(a+b*arcsin(c*x)),x,9,-1/96*b*(24*c^4*d^2*(3*e*f+d*g)+5*e^2*(e*h+3*d*i)+9*c^2*(e^3*f+3*d*e^2*g+3*d^2*e*h+d^3*i))*arcsin(c*x)/c^6+d^3*f*x*(a+b*arcsin(c*x))+1/2*d^2*(3*e*f+d*g)*x^2*(a+b*arcsin(c*x))+1/3*d*(3*e^2*f+3*d*e*g+d^2*h)*x^3*(a+b*arcsin(c*x))+1/4*(e^3*f+3*d*e^2*g+3*d^2*e*h+d^3*i)*x^4*(a+b*arcsin(c*x))+1/5*e*(e^2*g+3*d*e*h+3*d^2*i)*x^5*(a+b*arcsin(c*x))+1/6*e^2*(e*h+3*d*i)*x^6*(a+b*arcsin(c*x))+1/7*e^3*i*x^7*(a+b*arcsin(c*x))+1/11025*b*(1225*c^4*d*(3*e^2*f+3*d*e*g+d^2*h)+360*e^3*i+588*c^2*e*(e^2*g+3*d*e*h+3*d^2*i))*x^2*sqrt(1-c^2*x^2)/c^5+1/144*b*(5*e^2*(e*h+3*d*i)+9*c^2*(e^3*f+3*d*e^2*g+3*d^2*e*h+d^3*i))*x^3*sqrt(1-c^2*x^2)/c^3+1/1225*b*e*(30*e^2*i+49*c^2*(e^2*g+3*d*e*h+3*d^2*i))*x^4*sqrt(1-c^2*x^2)/c^3+1/36*b*e^2*(e*h+3*d*i)*x^5*sqrt(1-c^2*x^2)/c+1/49*b*e^3*i*x^6*sqrt(1-c^2*x^2)/c+1/352800*b*(32*(11025*c^6*d^3*f+2450*c^4*d*(3*e^2*f+3*d*e*g+d^2*h)+720*e^3*i+1176*c^2*e*(e^2*g+3*d*e*h+3*d^2*i))+3675*c^2*(24*c^4*d^2*(3*e*f+d*g)+5*e^2*(e*h+3*d*i)+9*c^2*(e^3*f+3*d*e^2*g+3*d^2*e*h+d^3*i))*x)*sqrt(1-c^2*x^2)/c^7],
[(d+e*x)^2*(f+g*x+h*x^2+i*x^3)*(a+b*arcsin(c*x)),x,8,-1/96*b*(24*c^4*d*(2*e*f+d*g)+5*e^2*i+9*c^2*(e^2*g+2*d*e*h+d^2*i))*arcsin(c*x)/c^6+d^2*f*x*(a+b*arcsin(c*x))+1/2*d*(2*e*f+d*g)*x^2*(a+b*arcsin(c*x))+1/3*(e^2*f+2*d*e*g+d^2*h)*x^3*(a+b*arcsin(c*x))+1/4*(e^2*g+2*d*e*h+d^2*i)*x^4*(a+b*arcsin(c*x))+1/5*e*(e*h+2*d*i)*x^5*(a+b*arcsin(c*x))+1/6*e^2*i*x^6*(a+b*arcsin(c*x))+1/225*b*(25*c^2*(e^2*f+2*d*e*g+d^2*h)+12*e*(e*h+2*d*i))*x^2*sqrt(1-c^2*x^2)/c^3+1/144*b*(5*e^2*i+9*c^2*(e^2*g+2*d*e*h+d^2*i))*x^3*sqrt(1-c^2*x^2)/c^3+1/25*b*e*(e*h+2*d*i)*x^4*sqrt(1-c^2*x^2)/c+1/36*b*e^2*i*x^5*sqrt(1-c^2*x^2)/c+1/7200*b*(32*(225*c^4*d^2*f+50*c^2*(e^2*f+2*d*e*g+d^2*h)+24*e*(e*h+2*d*i))+75*(24*c^4*d*(2*e*f+d*g)+5*e^2*i+9*c^2*(e^2*g+2*d*e*h+d^2*i))*x)*sqrt(1-c^2*x^2)/c^5],
[(d+e*x)*(f+g*x+h*x^2+i*x^3)*(a+b*arcsin(c*x)),x,7,-1/32*b*(8*c^2*(e*f+d*g)+3*(e*h+d*i))*arcsin(c*x)/c^4+d*f*x*(a+b*arcsin(c*x))+1/2*(e*f+d*g)*x^2*(a+b*arcsin(c*x))+1/3*(e*g+d*h)*x^3*(a+b*arcsin(c*x))+1/4*(e*h+d*i)*x^4*(a+b*arcsin(c*x))+1/5*e*i*x^5*(a+b*arcsin(c*x))+1/225*b*(25*c^2*(e*g+d*h)+12*e*i)*x^2*sqrt(1-c^2*x^2)/c^3+1/16*b*(e*h+d*i)*x^3*sqrt(1-c^2*x^2)/c+1/25*b*e*i*x^4*sqrt(1-c^2*x^2)/c+1/7200*b*(32*(225*c^4*d*f+50*c^2*(e*g+d*h)+24*e*i)+225*c^2*(8*c^2*(e*f+d*g)+3*(e*h+d*i))*x)*sqrt(1-c^2*x^2)/c^5],
[(f+g*x+h*x^2+i*x^3)*(a+b*arcsin(c*x))/(d+e*x),x,16,-1/4*b*(e*h-d*i)*arcsin(c*x)/(c^2*e^2)-1/2*I*b*(e^3*f-d*e^2*g+d^2*e*h-d^3*i)*arcsin(c*x)^2/e^4+(e^2*g-d*e*h+d^2*i)*x*(a+b*arcsin(c*x))/e^3+1/2*(e*h-d*i)*x^2*(a+b*arcsin(c*x))/e^2+1/3*i*x^3*(a+b*arcsin(c*x))/e-b*(e^3*f-d*e^2*g+d^2*e*h-d^3*i)*arcsin(c*x)*log(d+e*x)/e^4+(e^3*f-d*e^2*g+d^2*e*h-d^3*i)*(a+b*arcsin(c*x))*log(d+e*x)/e^4+b*(e^3*f-d*e^2*g+d^2*e*h-d^3*i)*arcsin(c*x)*log(1-I*e*exp(I*arcsin(c*x))/(c*d-sqrt(c^2*d^2-e^2)))/e^4+b*(e^3*f-d*e^2*g+d^2*e*h-d^3*i)*arcsin(c*x)*log(1-I*e*exp(I*arcsin(c*x))/(c*d+sqrt(c^2*d^2-e^2)))/e^4-I*b*(e^3*f-d*e^2*g+d^2*e*h-d^3*i)*polylog(2,I*e*exp(I*arcsin(c*x))/(c*d-sqrt(c^2*d^2-e^2)))/e^4-I*b*(e^3*f-d*e^2*g+d^2*e*h-d^3*i)*polylog(2,I*e*exp(I*arcsin(c*x))/(c*d+sqrt(c^2*d^2-e^2)))/e^4+1/9*b*i*x^2*sqrt(1-c^2*x^2)/(c*e)+1/36*b*(4*(2*e^2*i+9*c^2*(e^2*g-d*e*h+d^2*i))+9*c^2*e*(e*h-d*i)*x)*sqrt(1-c^2*x^2)/(c^3*e^3)],
[(f+g*x+h*x^2+i*x^3)*(a+b*arcsin(c*x))/(d+e*x)^2,x,18,-1/4*b*i*arcsin(c*x)/(c^2*e^2)-1/2*I*b*(e^2*g-2*d*e*h+3*d^2*i)*arcsin(c*x)^2/e^4+(e*h-2*d*i)*x*(a+b*arcsin(c*x))/e^3+1/2*i*x^2*(a+b*arcsin(c*x))/e^2-(e^3*f-d*e^2*g+d^2*e*h-d^3*i)*(a+b*arcsin(c*x))/(e^4*(d+e*x))-b*(e^2*g-2*d*e*h+3*d^2*i)*arcsin(c*x)*log(d+e*x)/e^4+(e^2*g-2*d*e*h+3*d^2*i)*(a+b*arcsin(c*x))*log(d+e*x)/e^4+b*(e^2*g-2*d*e*h+3*d^2*i)*arcsin(c*x)*log(1-I*e*exp(I*arcsin(c*x))/(c*d-sqrt(c^2*d^2-e^2)))/e^4+b*(e^2*g-2*d*e*h+3*d^2*i)*arcsin(c*x)*log(1-I*e*exp(I*arcsin(c*x))/(c*d+sqrt(c^2*d^2-e^2)))/e^4-I*b*(e^2*g-2*d*e*h+3*d^2*i)*polylog(2,I*e*exp(I*arcsin(c*x))/(c*d-sqrt(c^2*d^2-e^2)))/e^4-I*b*(e^2*g-2*d*e*h+3*d^2*i)*polylog(2,I*e*exp(I*arcsin(c*x))/(c*d+sqrt(c^2*d^2-e^2)))/e^4+b*c*(e^3*f-d*e^2*g+d^2*e*h-d^3*i)*arctan((e+c^2*d*x)/(sqrt(c^2*d^2-e^2)*sqrt(1-c^2*x^2)))/(e^4*sqrt(c^2*d^2-e^2))+b*(e*h-2*d*i)*sqrt(1-c^2*x^2)/(c*e^3)+1/4*b*i*x*sqrt(1-c^2*x^2)/(c*e^2)],
[(f+g*x+h*x^2+i*x^3)*(a+b*arcsin(c*x))/(d+e*x)^3,x,30,-1/2*I*b*(e*h-3*d*i)*arcsin(c*x)^2/e^4+i*x*(a+b*arcsin(c*x))/e^3-1/2*(e^3*f-d*e^2*g+d^2*e*h-d^3*i)*(a+b*arcsin(c*x))/(e^4*(d+e*x)^2)-(e^2*g-2*d*e*h+3*d^2*i)*(a+b*arcsin(c*x))/(e^4*(d+e*x))+5/2*b*c^3*d^4*i*arctan((e+c^2*d*x)/(sqrt(c^2*d^2-e^2)*sqrt(1-c^2*x^2)))/(e^4*(c^2*d^2-e^2)^(3/2))-1/2*b*c*d^2*(3*c^2*d*h+4*e*i)*arctan((e+c^2*d*x)/(sqrt(c^2*d^2-e^2)*sqrt(1-c^2*x^2)))/(e^3*(c^2*d^2-e^2)^(3/2))+1/2*b*c*d*(4*e^2*(e*h-2*d*i)+c^2*(d*e^2*g+4*d^3*i))*arctan((e+c^2*d*x)/(sqrt(c^2*d^2-e^2)*sqrt(1-c^2*x^2)))/(e^4*(c^2*d^2-e^2)^(3/2))-1/2*b*c*(2*e^4*g-6*d^2*e^2*i-c^2*(d*e^3*f-4*d^4*i))*arctan((e+c^2*d*x)/(sqrt(c^2*d^2-e^2)*sqrt(1-c^2*x^2)))/(e^4*(c^2*d^2-e^2)^(3/2))-b*(e*h-3*d*i)*arcsin(c*x)*log(d+e*x)/e^4+(e*h-3*d*i)*(a+b*arcsin(c*x))*log(d+e*x)/e^4+b*(e*h-3*d*i)*arcsin(c*x)*log(1-I*e*exp(I*arcsin(c*x))/(c*d-sqrt(c^2*d^2-e^2)))/e^4+b*(e*h-3*d*i)*arcsin(c*x)*log(1-I*e*exp(I*arcsin(c*x))/(c*d+sqrt(c^2*d^2-e^2)))/e^4-I*b*(e*h-3*d*i)*polylog(2,I*e*exp(I*arcsin(c*x))/(c*d-sqrt(c^2*d^2-e^2)))/e^4-I*b*(e*h-3*d*i)*polylog(2,I*e*exp(I*arcsin(c*x))/(c*d+sqrt(c^2*d^2-e^2)))/e^4+b*i*sqrt(1-c^2*x^2)/(c*e^3)+5/2*b*c*d^3*i*sqrt(1-c^2*x^2)/(e^3*(c^2*d^2-e^2)*(d+e*x))-1/2*b*c*d^2*(3*e*h+4*d*i)*sqrt(1-c^2*x^2)/(e^3*(c^2*d^2-e^2)*(d+e*x))+1/2*b*c*d*(e^2*g+4*d*e*h-4*d^2*i)*sqrt(1-c^2*x^2)/(e^3*(c^2*d^2-e^2)*(d+e*x))+1/2*b*c*(e^3*f-2*d*e^2*g+2*d^3*i)*sqrt(1-c^2*x^2)/(e^3*(c^2*d^2-e^2)*(d+e*x))],
[(f+g*x+h*x^2+i*x^3)*(a+b*arcsin(c*x))/(d+e*x)^4,x,29,-1/2*I*b*i*arcsin(c*x)^2/e^4-1/3*(e^3*f-d*e^2*g+d^2*e*h-d^3*i)*(a+b*arcsin(c*x))/(e^4*(d+e*x)^3)-1/2*(e^2*g-2*d*e*h+3*d^2*i)*(a+b*arcsin(c*x))/(e^4*(d+e*x)^2)-(e*h-3*d*i)*(a+b*arcsin(c*x))/(e^4*(d+e*x))+1/12*b*c*(4*c^4*d^2*f+12*e^2*h+c^2*(2*e^2*f-9*d*e*g+6*d^2*h))*arctan((e+c^2*d*x)/(sqrt(c^2*d^2-e^2)*sqrt(1-c^2*x^2)))/(e*(c^2*d^2-e^2)^(5/2))-11/12*b*c^3*d^3*(2*c^2*d^2+e^2)*i*arctan((e+c^2*d*x)/(sqrt(c^2*d^2-e^2)*sqrt(1-c^2*x^2)))/(e^4*(c^2*d^2-e^2)^(5/2))+1/12*b*c^3*d^2*(4*c^2*d^2*h+e*(2*e*h+81*d*i))*arctan((e+c^2*d*x)/(sqrt(c^2*d^2-e^2)*sqrt(1-c^2*x^2)))/(e^3*(c^2*d^2-e^2)^(5/2))+1/12*b*c*d*(2*c^4*d^2*g-36*e^2*i+c^2*(e^2*g-18*d*e*h-18*d^2*i))*arctan((e+c^2*d*x)/(sqrt(c^2*d^2-e^2)*sqrt(1-c^2*x^2)))/(e^2*(c^2*d^2-e^2)^(5/2))-b*i*arcsin(c*x)*log(d+e*x)/e^4+i*(a+b*arcsin(c*x))*log(d+e*x)/e^4+b*i*arcsin(c*x)*log(1-I*e*exp(I*arcsin(c*x))/(c*d-sqrt(c^2*d^2-e^2)))/e^4+b*i*arcsin(c*x)*log(1-I*e*exp(I*arcsin(c*x))/(c*d+sqrt(c^2*d^2-e^2)))/e^4-I*b*i*polylog(2,I*e*exp(I*arcsin(c*x))/(c*d-sqrt(c^2*d^2-e^2)))/e^4-I*b*i*polylog(2,I*e*exp(I*arcsin(c*x))/(c*d+sqrt(c^2*d^2-e^2)))/e^4+1/12*b*c*(2*e^2*f-3*d*e*g+6*d^2*h)*sqrt(1-c^2*x^2)/(e^2*(c^2*d^2-e^2)*(d+e*x)^2)-11/12*b*c*d^3*i*sqrt(1-c^2*x^2)/(e^3*(c^2*d^2-e^2)*(d+e*x)^2)+1/12*b*c*d^2*(2*e*h+27*d*i)*sqrt(1-c^2*x^2)/(e^3*(c^2*d^2-e^2)*(d+e*x)^2)+1/12*b*c*d*(e^2*g-6*d*e*h-18*d^2*i)*sqrt(1-c^2*x^2)/(e^3*(c^2*d^2-e^2)*(d+e*x)^2)-1/4*b*c*(2*e^2*(e*g-4*d*h)-c^2*d*(2*e^2*f-d*e*g-2*d^2*h))*sqrt(1-c^2*x^2)/(e^2*(c^2*d^2-e^2)^2*(d+e*x))-11/4*b*c^3*d^4*i*sqrt(1-c^2*x^2)/(e^3*(c^2*d^2-e^2)^2*(d+e*x))+1/4*b*c*d^2*(18*e^2*i+c^2*d*(2*e*h+9*d*i))*sqrt(1-c^2*x^2)/(e^3*(c^2*d^2-e^2)^2*(d+e*x))-1/4*b*c*d*(4*e^2*(e*h+6*d*i)-c^2*d*(e^2*g-2*d*e*h+6*d^2*i))*sqrt(1-c^2*x^2)/(e^3*(c^2*d^2-e^2)^2*(d+e*x))],

# Integrands of the form Px (d+e x)^m (a+b ArcSin[c x])^2
[(f+g*x)*(a+b*arcsin(c*x))^2/(d+e*x)^3,x,33,a*b*g^2*arcsin(c*x)/(e^2*(e*f-d*g))+1/2*b^2*g^2*arcsin(c*x)^2/(e^2*(e*f-d*g))-1/2*(f+g*x)^2*(a+b*arcsin(c*x))^2/((e*f-d*g)*(d+e*x)^2)-a*b*c*(2*e^2*g-c^2*d*(e*f+d*g))*arctan((e+c^2*d*x)/(sqrt(c^2*d^2-e^2)*sqrt(1-c^2*x^2)))/(e^2*(c^2*d^2-e^2)^(3/2))-b^2*c^2*(e*f-d*g)*log(d+e*x)/(e^2*(c^2*d^2-e^2))-I*b^2*c^3*d*(e*f-d*g)*arcsin(c*x)*log(1-I*e*exp(I*arcsin(c*x))/(c*d-sqrt(c^2*d^2-e^2)))/(e^2*(c^2*d^2-e^2)^(3/2))+I*b^2*c^3*d*(e*f-d*g)*arcsin(c*x)*log(1-I*e*exp(I*arcsin(c*x))/(c*d+sqrt(c^2*d^2-e^2)))/(e^2*(c^2*d^2-e^2)^(3/2))-b^2*c^3*d*(e*f-d*g)*polylog(2,I*e*exp(I*arcsin(c*x))/(c*d-sqrt(c^2*d^2-e^2)))/(e^2*(c^2*d^2-e^2)^(3/2))+b^2*c^3*d*(e*f-d*g)*polylog(2,I*e*exp(I*arcsin(c*x))/(c*d+sqrt(c^2*d^2-e^2)))/(e^2*(c^2*d^2-e^2)^(3/2))-2*I*b^2*c*g*arcsin(c*x)*log(1-I*e*exp(I*arcsin(c*x))/(c*d-sqrt(c^2*d^2-e^2)))/(e^2*sqrt(c^2*d^2-e^2))+2*I*b^2*c*g*arcsin(c*x)*log(1-I*e*exp(I*arcsin(c*x))/(c*d+sqrt(c^2*d^2-e^2)))/(e^2*sqrt(c^2*d^2-e^2))-2*b^2*c*g*polylog(2,I*e*exp(I*arcsin(c*x))/(c*d-sqrt(c^2*d^2-e^2)))/(e^2*sqrt(c^2*d^2-e^2))+2*b^2*c*g*polylog(2,I*e*exp(I*arcsin(c*x))/(c*d+sqrt(c^2*d^2-e^2)))/(e^2*sqrt(c^2*d^2-e^2))+a*b*c*(e*f-d*g)*sqrt(1-c^2*x^2)/(e*(c^2*d^2-e^2)*(d+e*x))+b^2*c*(e*f-d*g)*arcsin(c*x)*sqrt(1-c^2*x^2)/(e*(c^2*d^2-e^2)*(d+e*x))],
[(f+g*x)^2*(a+b*arcsin(c*x))^2/(d+e*x)^3,x,55,-1/2*a^2*(e*f-d*g)^2/(e^3*(d+e*x)^2)-2*a^2*g*(e*f-d*g)/(e^3*(d+e*x))-a*b*(e*f-d*g)^2*arcsin(c*x)/(e^3*(d+e*x)^2)-4*a*b*g*(e*f-d*g)*arcsin(c*x)/(e^3*(d+e*x))-I*a*b*g^2*arcsin(c*x)^2/e^3-1/2*b^2*(e*f-d*g)^2*arcsin(c*x)^2/(e^3*(d+e*x)^2)-2*b^2*g*(e*f-d*g)*arcsin(c*x)^2/(e^3*(d+e*x))-1/3*I*b^2*g^2*arcsin(c*x)^3/e^3-a*b*c*(e*f-d*g)*(4*e^2*g-c^2*d*(e*f+3*d*g))*arctan((e+c^2*d*x)/(sqrt(c^2*d^2-e^2)*sqrt(1-c^2*x^2)))/(e^3*(c^2*d^2-e^2)^(3/2))+a^2*g^2*log(d+e*x)/e^3-b^2*c^2*(e*f-d*g)^2*log(d+e*x)/(e^3*(c^2*d^2-e^2))+2*a*b*g^2*arcsin(c*x)*log(1-I*e*exp(I*arcsin(c*x))/(c*d-sqrt(c^2*d^2-e^2)))/e^3-I*b^2*c^3*d*(e*f-d*g)^2*arcsin(c*x)*log(1-I*e*exp(I*arcsin(c*x))/(c*d-sqrt(c^2*d^2-e^2)))/(e^3*(c^2*d^2-e^2)^(3/2))+b^2*g^2*arcsin(c*x)^2*log(1-I*e*exp(I*arcsin(c*x))/(c*d-sqrt(c^2*d^2-e^2)))/e^3+2*a*b*g^2*arcsin(c*x)*log(1-I*e*exp(I*arcsin(c*x))/(c*d+sqrt(c^2*d^2-e^2)))/e^3+I*b^2*c^3*d*(e*f-d*g)^2*arcsin(c*x)*log(1-I*e*exp(I*arcsin(c*x))/(c*d+sqrt(c^2*d^2-e^2)))/(e^3*(c^2*d^2-e^2)^(3/2))+b^2*g^2*arcsin(c*x)^2*log(1-I*e*exp(I*arcsin(c*x))/(c*d+sqrt(c^2*d^2-e^2)))/e^3-2*I*a*b*g^2*polylog(2,I*e*exp(I*arcsin(c*x))/(c*d-sqrt(c^2*d^2-e^2)))/e^3-b^2*c^3*d*(e*f-d*g)^2*polylog(2,I*e*exp(I*arcsin(c*x))/(c*d-sqrt(c^2*d^2-e^2)))/(e^3*(c^2*d^2-e^2)^(3/2))-2*I*b^2*g^2*arcsin(c*x)*polylog(2,I*e*exp(I*arcsin(c*x))/(c*d-sqrt(c^2*d^2-e^2)))/e^3-2*I*a*b*g^2*polylog(2,I*e*exp(I*arcsin(c*x))/(c*d+sqrt(c^2*d^2-e^2)))/e^3+b^2*c^3*d*(e*f-d*g)^2*polylog(2,I*e*exp(I*arcsin(c*x))/(c*d+sqrt(c^2*d^2-e^2)))/(e^3*(c^2*d^2-e^2)^(3/2))-2*I*b^2*g^2*arcsin(c*x)*polylog(2,I*e*exp(I*arcsin(c*x))/(c*d+sqrt(c^2*d^2-e^2)))/e^3+2*b^2*g^2*polylog(3,I*e*exp(I*arcsin(c*x))/(c*d-sqrt(c^2*d^2-e^2)))/e^3+2*b^2*g^2*polylog(3,I*e*exp(I*arcsin(c*x))/(c*d+sqrt(c^2*d^2-e^2)))/e^3-4*I*b^2*c*g*(e*f-d*g)*arcsin(c*x)*log(1-I*e*exp(I*arcsin(c*x))/(c*d-sqrt(c^2*d^2-e^2)))/(e^3*sqrt(c^2*d^2-e^2))+4*I*b^2*c*g*(e*f-d*g)*arcsin(c*x)*log(1-I*e*exp(I*arcsin(c*x))/(c*d+sqrt(c^2*d^2-e^2)))/(e^3*sqrt(c^2*d^2-e^2))-4*b^2*c*g*(e*f-d*g)*polylog(2,I*e*exp(I*arcsin(c*x))/(c*d-sqrt(c^2*d^2-e^2)))/(e^3*sqrt(c^2*d^2-e^2))+4*b^2*c*g*(e*f-d*g)*polylog(2,I*e*exp(I*arcsin(c*x))/(c*d+sqrt(c^2*d^2-e^2)))/(e^3*sqrt(c^2*d^2-e^2))+a*b*c*(e*f-d*g)^2*sqrt(1-c^2*x^2)/(e^2*(c^2*d^2-e^2)*(d+e*x))+b^2*c*(e*f-d*g)^2*arcsin(c*x)*sqrt(1-c^2*x^2)/(e^2*(c^2*d^2-e^2)*(d+e*x))],
[(g+h*x)^3*(d+e*x+f*x^2)*(a+b*arcsin(c*x))^2,x,35,-2*b^2*d*g^3*x-16/75*b^2*h^2*(3*f*g+e*h)*x/c^4-4/9*b^2*g*(f*g^2+3*h*(e*g+d*h))*x/c^2-5/96*b^2*f*h^3*x^2/c^4-1/4*b^2*g^2*(e*g+3*d*h)*x^2-3/32*b^2*h*(3*f*g^2+h*(3*e*g+d*h))*x^2/c^2-8/225*b^2*h^2*(3*f*g+e*h)*x^3/c^2-2/27*b^2*g*(f*g^2+3*h*(e*g+d*h))*x^3-5/288*b^2*f*h^3*x^4/c^2-1/32*b^2*h*(3*f*g^2+h*(3*e*g+d*h))*x^4-2/125*b^2*h^2*(3*f*g+e*h)*x^5-1/108*b^2*f*h^3*x^6-5/96*f*h^3*(a+b*arcsin(c*x))^2/c^6-1/4*g^2*(e*g+3*d*h)*(a+b*arcsin(c*x))^2/c^2-3/32*h*(3*f*g^2+h*(3*e*g+d*h))*(a+b*arcsin(c*x))^2/c^4+d*g^3*x*(a+b*arcsin(c*x))^2+1/2*g^2*(e*g+3*d*h)*x^2*(a+b*arcsin(c*x))^2+1/3*g*(f*g^2+3*h*(e*g+d*h))*x^3*(a+b*arcsin(c*x))^2+1/4*h*(3*f*g^2+h*(3*e*g+d*h))*x^4*(a+b*arcsin(c*x))^2+1/5*h^2*(3*f*g+e*h)*x^5*(a+b*arcsin(c*x))^2+1/6*f*h^3*x^6*(a+b*arcsin(c*x))^2+2*b*d*g^3*(a+b*arcsin(c*x))*sqrt(1-c^2*x^2)/c+16/75*b*h^2*(3*f*g+e*h)*(a+b*arcsin(c*x))*sqrt(1-c^2*x^2)/c^5+4/9*b*g*(f*g^2+3*h*(e*g+d*h))*(a+b*arcsin(c*x))*sqrt(1-c^2*x^2)/c^3+5/48*b*f*h^3*x*(a+b*arcsin(c*x))*sqrt(1-c^2*x^2)/c^5+1/2*b*g^2*(e*g+3*d*h)*x*(a+b*arcsin(c*x))*sqrt(1-c^2*x^2)/c+3/16*b*h*(3*f*g^2+h*(3*e*g+d*h))*x*(a+b*arcsin(c*x))*sqrt(1-c^2*x^2)/c^3+8/75*b*h^2*(3*f*g+e*h)*x^2*(a+b*arcsin(c*x))*sqrt(1-c^2*x^2)/c^3+2/9*b*g*(f*g^2+3*h*(e*g+d*h))*x^2*(a+b*arcsin(c*x))*sqrt(1-c^2*x^2)/c+5/72*b*f*h^3*x^3*(a+b*arcsin(c*x))*sqrt(1-c^2*x^2)/c^3+1/8*b*h*(3*f*g^2+h*(3*e*g+d*h))*x^3*(a+b*arcsin(c*x))*sqrt(1-c^2*x^2)/c+2/25*b*h^2*(3*f*g+e*h)*x^4*(a+b*arcsin(c*x))*sqrt(1-c^2*x^2)/c+1/18*b*f*h^3*x^5*(a+b*arcsin(c*x))*sqrt(1-c^2*x^2)/c],
[(g+h*x)^2*(d+e*x+f*x^2)*(a+b*arcsin(c*x))^2,x,27,-2*b^2*d*g^2*x-16/75*b^2*f*h^2*x/c^4-4/9*b^2*(f*g^2+h*(2*e*g+d*h))*x/c^2-1/4*b^2*g*(e*g+2*d*h)*x^2-3/32*b^2*h*(2*f*g+e*h)*x^2/c^2-8/225*b^2*f*h^2*x^3/c^2-2/27*b^2*(f*g^2+h*(2*e*g+d*h))*x^3-1/32*b^2*h*(2*f*g+e*h)*x^4-2/125*b^2*f*h^2*x^5-1/4*g*(e*g+2*d*h)*(a+b*arcsin(c*x))^2/c^2-3/32*h*(2*f*g+e*h)*(a+b*arcsin(c*x))^2/c^4+d*g^2*x*(a+b*arcsin(c*x))^2+1/2*g*(e*g+2*d*h)*x^2*(a+b*arcsin(c*x))^2+1/3*(f*g^2+h*(2*e*g+d*h))*x^3*(a+b*arcsin(c*x))^2+1/4*h*(2*f*g+e*h)*x^4*(a+b*arcsin(c*x))^2+1/5*f*h^2*x^5*(a+b*arcsin(c*x))^2+2*b*d*g^2*(a+b*arcsin(c*x))*sqrt(1-c^2*x^2)/c+16/75*b*f*h^2*(a+b*arcsin(c*x))*sqrt(1-c^2*x^2)/c^5+4/9*b*(f*g^2+h*(2*e*g+d*h))*(a+b*arcsin(c*x))*sqrt(1-c^2*x^2)/c^3+1/2*b*g*(e*g+2*d*h)*x*(a+b*arcsin(c*x))*sqrt(1-c^2*x^2)/c+3/16*b*h*(2*f*g+e*h)*x*(a+b*arcsin(c*x))*sqrt(1-c^2*x^2)/c^3+8/75*b*f*h^2*x^2*(a+b*arcsin(c*x))*sqrt(1-c^2*x^2)/c^3+2/9*b*(f*g^2+h*(2*e*g+d*h))*x^2*(a+b*arcsin(c*x))*sqrt(1-c^2*x^2)/c+1/8*b*h*(2*f*g+e*h)*x^3*(a+b*arcsin(c*x))*sqrt(1-c^2*x^2)/c+2/25*b*f*h^2*x^4*(a+b*arcsin(c*x))*sqrt(1-c^2*x^2)/c],
[(g+h*x)*(d+e*x+f*x^2)*(a+b*arcsin(c*x))^2,x,20,-2*b^2*d*g*x-4/9*b^2*(f*g+e*h)*x/c^2-3/32*b^2*f*h*x^2/c^2-1/4*b^2*(e*g+d*h)*x^2-2/27*b^2*(f*g+e*h)*x^3-1/32*b^2*f*h*x^4-3/32*f*h*(a+b*arcsin(c*x))^2/c^4-1/4*(e*g+d*h)*(a+b*arcsin(c*x))^2/c^2+d*g*x*(a+b*arcsin(c*x))^2+1/2*(e*g+d*h)*x^2*(a+b*arcsin(c*x))^2+1/3*(f*g+e*h)*x^3*(a+b*arcsin(c*x))^2+1/4*f*h*x^4*(a+b*arcsin(c*x))^2+2*b*d*g*(a+b*arcsin(c*x))*sqrt(1-c^2*x^2)/c+4/9*b*(f*g+e*h)*(a+b*arcsin(c*x))*sqrt(1-c^2*x^2)/c^3+3/16*b*f*h*x*(a+b*arcsin(c*x))*sqrt(1-c^2*x^2)/c^3+1/2*b*(e*g+d*h)*x*(a+b*arcsin(c*x))*sqrt(1-c^2*x^2)/c+2/9*b*(f*g+e*h)*x^2*(a+b*arcsin(c*x))*sqrt(1-c^2*x^2)/c+1/8*b*f*h*x^3*(a+b*arcsin(c*x))*sqrt(1-c^2*x^2)/c],
[(d+e*x+f*x^2)*(a+b*arcsin(c*x))^2/(g+h*x),x,38,-a^2*(f*g-e*h)*x/h^2+2*b^2*(f*g-e*h)*x/h^2+1/2*a^2*f*x^2/h-1/4*b^2*f*x^2/h-1/2*a*b*f*arcsin(c*x)/(c^2*h)-2*a*b*(f*g-e*h)*x*arcsin(c*x)/h^2+a*b*f*x^2*arcsin(c*x)/h-1/4*b^2*f*arcsin(c*x)^2/(c^2*h)-I*a*b*(f*g^2-e*g*h+d*h^2)*arcsin(c*x)^2/h^3-b^2*(f*g-e*h)*x*arcsin(c*x)^2/h^2+1/2*b^2*f*x^2*arcsin(c*x)^2/h-1/3*I*b^2*(f*g^2-e*g*h+d*h^2)*arcsin(c*x)^3/h^3+a^2*(f*g^2-e*g*h+d*h^2)*log(g+h*x)/h^3+2*a*b*(f*g^2-e*g*h+d*h^2)*arcsin(c*x)*log(1-I*exp(I*arcsin(c*x))*h/(c*g-sqrt(c^2*g^2-h^2)))/h^3+b^2*(f*g^2-e*g*h+d*h^2)*arcsin(c*x)^2*log(1-I*exp(I*arcsin(c*x))*h/(c*g-sqrt(c^2*g^2-h^2)))/h^3+2*a*b*(f*g^2-e*g*h+d*h^2)*arcsin(c*x)*log(1-I*exp(I*arcsin(c*x))*h/(c*g+sqrt(c^2*g^2-h^2)))/h^3+b^2*(f*g^2-e*g*h+d*h^2)*arcsin(c*x)^2*log(1-I*exp(I*arcsin(c*x))*h/(c*g+sqrt(c^2*g^2-h^2)))/h^3-2*I*a*b*(f*g^2-e*g*h+d*h^2)*polylog(2,I*exp(I*arcsin(c*x))*h/(c*g-sqrt(c^2*g^2-h^2)))/h^3-2*I*b^2*(f*g^2-e*g*h+d*h^2)*arcsin(c*x)*polylog(2,I*exp(I*arcsin(c*x))*h/(c*g-sqrt(c^2*g^2-h^2)))/h^3-2*I*a*b*(f*g^2-e*g*h+d*h^2)*polylog(2,I*exp(I*arcsin(c*x))*h/(c*g+sqrt(c^2*g^2-h^2)))/h^3-2*I*b^2*(f*g^2-e*g*h+d*h^2)*arcsin(c*x)*polylog(2,I*exp(I*arcsin(c*x))*h/(c*g+sqrt(c^2*g^2-h^2)))/h^3+2*b^2*(f*g^2-e*g*h+d*h^2)*polylog(3,I*exp(I*arcsin(c*x))*h/(c*g-sqrt(c^2*g^2-h^2)))/h^3+2*b^2*(f*g^2-e*g*h+d*h^2)*polylog(3,I*exp(I*arcsin(c*x))*h/(c*g+sqrt(c^2*g^2-h^2)))/h^3-1/2*a*b*(4*(f*g-e*h)-f*h*x)*sqrt(1-c^2*x^2)/(c*h^2)-2*b^2*(f*g-e*h)*arcsin(c*x)*sqrt(1-c^2*x^2)/(c*h^2)+1/2*b^2*f*x*arcsin(c*x)*sqrt(1-c^2*x^2)/(c*h)],
[(d+e*x+f*x^2)*(a+b*arcsin(c*x))^2/(g+h*x)^2,x,45,a^2*f*x/h^2-2*b^2*f*x/h^2-a^2*(f*g^2-e*g*h+d*h^2)/(h^3*(g+h*x))+2*a*b*f*x*arcsin(c*x)/h^2-2*a*b*(f*g^2-e*g*h+d*h^2)*arcsin(c*x)/(h^3*(g+h*x))+I*a*b*(2*f*g-e*h)*arcsin(c*x)^2/h^3+b^2*f*x*arcsin(c*x)^2/h^2-b^2*(f*g^2-e*g*h+d*h^2)*arcsin(c*x)^2/(h^3*(g+h*x))+1/3*I*b^2*(2*f*g-e*h)*arcsin(c*x)^3/h^3-a^2*(2*f*g-e*h)*log(g+h*x)/h^3-2*a*b*(2*f*g-e*h)*arcsin(c*x)*log(1-I*exp(I*arcsin(c*x))*h/(c*g-sqrt(c^2*g^2-h^2)))/h^3-b^2*(2*f*g-e*h)*arcsin(c*x)^2*log(1-I*exp(I*arcsin(c*x))*h/(c*g-sqrt(c^2*g^2-h^2)))/h^3-2*a*b*(2*f*g-e*h)*arcsin(c*x)*log(1-I*exp(I*arcsin(c*x))*h/(c*g+sqrt(c^2*g^2-h^2)))/h^3-b^2*(2*f*g-e*h)*arcsin(c*x)^2*log(1-I*exp(I*arcsin(c*x))*h/(c*g+sqrt(c^2*g^2-h^2)))/h^3+2*I*a*b*(2*f*g-e*h)*polylog(2,I*exp(I*arcsin(c*x))*h/(c*g-sqrt(c^2*g^2-h^2)))/h^3+2*I*b^2*(2*f*g-e*h)*arcsin(c*x)*polylog(2,I*exp(I*arcsin(c*x))*h/(c*g-sqrt(c^2*g^2-h^2)))/h^3+2*I*a*b*(2*f*g-e*h)*polylog(2,I*exp(I*arcsin(c*x))*h/(c*g+sqrt(c^2*g^2-h^2)))/h^3+2*I*b^2*(2*f*g-e*h)*arcsin(c*x)*polylog(2,I*exp(I*arcsin(c*x))*h/(c*g+sqrt(c^2*g^2-h^2)))/h^3-2*b^2*(2*f*g-e*h)*polylog(3,I*exp(I*arcsin(c*x))*h/(c*g-sqrt(c^2*g^2-h^2)))/h^3-2*b^2*(2*f*g-e*h)*polylog(3,I*exp(I*arcsin(c*x))*h/(c*g+sqrt(c^2*g^2-h^2)))/h^3+2*a*b*c*(f*g^2-e*g*h+d*h^2)*arctan((h+c^2*g*x)/(sqrt(c^2*g^2-h^2)*sqrt(1-c^2*x^2)))/(h^3*sqrt(c^2*g^2-h^2))-2*I*b^2*c*(f*g^2-e*g*h+d*h^2)*arcsin(c*x)*log(1-I*exp(I*arcsin(c*x))*h/(c*g-sqrt(c^2*g^2-h^2)))/(h^3*sqrt(c^2*g^2-h^2))+2*I*b^2*c*(f*g^2-e*g*h+d*h^2)*arcsin(c*x)*log(1-I*exp(I*arcsin(c*x))*h/(c*g+sqrt(c^2*g^2-h^2)))/(h^3*sqrt(c^2*g^2-h^2))-2*b^2*c*(f*g^2-e*g*h+d*h^2)*polylog(2,I*exp(I*arcsin(c*x))*h/(c*g-sqrt(c^2*g^2-h^2)))/(h^3*sqrt(c^2*g^2-h^2))+2*b^2*c*(f*g^2-e*g*h+d*h^2)*polylog(2,I*exp(I*arcsin(c*x))*h/(c*g+sqrt(c^2*g^2-h^2)))/(h^3*sqrt(c^2*g^2-h^2))+2*a*b*f*sqrt(1-c^2*x^2)/(c*h^2)+2*b^2*f*arcsin(c*x)*sqrt(1-c^2*x^2)/(c*h^2)],
[(e*f+2*d*h*x+e*h*x^2)*(a+b*arcsin(c*x))^2/(d+e*x)^2,x,20,-2*b^2*h*x/e+h*x*(a+b*arcsin(c*x))^2/e-(f-d^2*h/e^2)*(a+b*arcsin(c*x))^2/(d+e*x)+2*a*b*c*(e^2*f-d^2*h)*arctan((e+c^2*d*x)/(sqrt(c^2*d^2-e^2)*sqrt(1-c^2*x^2)))/(e^2*sqrt(c^2*d^2-e^2))-2*I*b^2*c*(e^2*f-d^2*h)*arcsin(c*x)*log(1-I*e*exp(I*arcsin(c*x))/(c*d-sqrt(c^2*d^2-e^2)))/(e^2*sqrt(c^2*d^2-e^2))+2*I*b^2*c*(e^2*f-d^2*h)*arcsin(c*x)*log(1-I*e*exp(I*arcsin(c*x))/(c*d+sqrt(c^2*d^2-e^2)))/(e^2*sqrt(c^2*d^2-e^2))-2*b^2*c*(e^2*f-d^2*h)*polylog(2,I*e*exp(I*arcsin(c*x))/(c*d-sqrt(c^2*d^2-e^2)))/(e^2*sqrt(c^2*d^2-e^2))+2*b^2*c*(e^2*f-d^2*h)*polylog(2,I*e*exp(I*arcsin(c*x))/(c*d+sqrt(c^2*d^2-e^2)))/(e^2*sqrt(c^2*d^2-e^2))+2*a*b*h*sqrt(1-c^2*x^2)/(c*e)+2*b^2*h*arcsin(c*x)*sqrt(1-c^2*x^2)/(c*e)],
[(e*f+2*d*h*x+e*h*x^2)^2*(a+b*arcsin(c*x))^2/(d+e*x)^2,x,32,-4/9*b^2*h^2*x/c^2-2*b^2*h*(2*e^2*f-d^2*h)*x/e^2-1/2*b^2*d*h^2*x^2/e-2/27*b^2*h^2*x^3-1/3*a*b*d*(2*c^2*d^2+3*e^2)*h^2*arcsin(c*x)/(c^2*e^3)-1/3*b^2*d^3*h^2*arcsin(c*x)^2/e^3-1/2*b^2*d*h^2*arcsin(c*x)^2/(c^2*e)+2*h*(e^2*f-d^2*h)*x*(a+b*arcsin(c*x))^2/e^2-(e^2*f-d^2*h)^2*(a+b*arcsin(c*x))^2/(e^3*(d+e*x))+1/3*h^2*(d+e*x)^3*(a+b*arcsin(c*x))^2/e^3+2*a*b*c*(e^2*f-d^2*h)^2*arctan((e+c^2*d*x)/(sqrt(c^2*d^2-e^2)*sqrt(1-c^2*x^2)))/(e^3*sqrt(c^2*d^2-e^2))-2*I*b^2*c*(e^2*f-d^2*h)^2*arcsin(c*x)*log(1-I*e*exp(I*arcsin(c*x))/(c*d-sqrt(c^2*d^2-e^2)))/(e^3*sqrt(c^2*d^2-e^2))+2*I*b^2*c*(e^2*f-d^2*h)^2*arcsin(c*x)*log(1-I*e*exp(I*arcsin(c*x))/(c*d+sqrt(c^2*d^2-e^2)))/(e^3*sqrt(c^2*d^2-e^2))-2*b^2*c*(e^2*f-d^2*h)^2*polylog(2,I*e*exp(I*arcsin(c*x))/(c*d-sqrt(c^2*d^2-e^2)))/(e^3*sqrt(c^2*d^2-e^2))+2*b^2*c*(e^2*f-d^2*h)^2*polylog(2,I*e*exp(I*arcsin(c*x))/(c*d+sqrt(c^2*d^2-e^2)))/(e^3*sqrt(c^2*d^2-e^2))+1/9*a*b*h*(4*e^2*h+c^2*(36*e^2*f-25*d^2*h))*sqrt(1-c^2*x^2)/(c^3*e^2)+5/9*a*b*d*h^2*(d+e*x)*sqrt(1-c^2*x^2)/(c*e^2)+2/9*a*b*h^2*(d+e*x)^2*sqrt(1-c^2*x^2)/(c*e^2)+4/9*b^2*h^2*arcsin(c*x)*sqrt(1-c^2*x^2)/c^3+2*b^2*h*(2*e^2*f-d^2*h)*arcsin(c*x)*sqrt(1-c^2*x^2)/(c*e^2)+b^2*d*h^2*x*arcsin(c*x)*sqrt(1-c^2*x^2)/(c*e)+2/9*b^2*h^2*x^2*arcsin(c*x)*sqrt(1-c^2*x^2)/c],

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

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

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

# n>0
[x^3*arcsin(a+b*x),x,6,-1/32*(3+24*a^2+8*a^4)*arcsin(a+b*x)/b^4+1/4*x^4*arcsin(a+b*x)-7/48*a*x^2*sqrt(1-(a+b*x)^2)/b^2+1/16*x^3*sqrt(1-(a+b*x)^2)/b-1/96*(4*a*(16+19*a^2)-(9+26*a^2)*(a+b*x))*sqrt(1-(a+b*x)^2)/b^4],
[x^2*arcsin(a+b*x),x,5,1/6*a*(3+2*a^2)*arcsin(a+b*x)/b^3+1/3*x^3*arcsin(a+b*x)+1/9*x^2*sqrt(1-(a+b*x)^2)/b+1/18*(4+11*a^2-5*a*b*x)*sqrt(1-(a+b*x)^2)/b^3],
[x*arcsin(a+b*x),x,5,-1/4*(1+2*a^2)*arcsin(a+b*x)/b^2+1/2*x^2*arcsin(a+b*x)-3/4*a*sqrt(1-(a+b*x)^2)/b^2+1/4*x*sqrt(1-(a+b*x)^2)/b],
[arcsin(a+b*x),x,3,(a+b*x)*arcsin(a+b*x)/b+sqrt(1-(a+b*x)^2)/b],
[arcsin(a+b*x)/x,x,9,-1/2*I*arcsin(a+b*x)^2+arcsin(a+b*x)*log(1-exp(I*arcsin(a+b*x))/(I*a-sqrt(1-a^2)))+arcsin(a+b*x)*log(1-exp(I*arcsin(a+b*x))/(I*a+sqrt(1-a^2)))-I*polylog(2,exp(I*arcsin(a+b*x))/(I*a-sqrt(1-a^2)))-I*polylog(2,exp(I*arcsin(a+b*x))/(I*a+sqrt(1-a^2)))],
[arcsin(a+b*x)/x^2,x,4,-arcsin(a+b*x)/x-b*arctanh((1-a*(a+b*x))/(sqrt(1-a^2)*sqrt(1-(a+b*x)^2)))/sqrt(1-a^2)],
[arcsin(a+b*x)/x^3,x,5,-1/2*arcsin(a+b*x)/x^2-1/2*a*b^2*arctanh((1-a*(a+b*x))/(sqrt(1-a^2)*sqrt(1-(a+b*x)^2)))/(1-a^2)^(3/2)-1/2*b*sqrt(1-(a+b*x)^2)/((1-a^2)*x)],
[arcsin(a+b*x)/x^4,x,6,-1/3*arcsin(a+b*x)/x^3-1/6*(1+2*a^2)*b^3*arctanh((1-a*(a+b*x))/(sqrt(1-a^2)*sqrt(1-(a+b*x)^2)))/(1-a^2)^(5/2)-1/6*b*sqrt(1-(a+b*x)^2)/((1-a^2)*x^2)-1/2*a*b^2*sqrt(1-(a+b*x)^2)/((1-a^2)^2*x)],
[arcsin(a+b*x)/x^5,x,7,-1/4*arcsin(a+b*x)/x^4-1/8*a*(3+2*a^2)*b^4*arctanh((1-a*(a+b*x))/(sqrt(1-a^2)*sqrt(1-(a+b*x)^2)))/(1-a^2)^(7/2)-1/12*b*sqrt(1-(a+b*x)^2)/((1-a^2)*x^3)-5/24*a*b^2*sqrt(1-(a+b*x)^2)/((1-a^2)^2*x^2)-1/24*(4+11*a^2)*b^3*sqrt(1-(a+b*x)^2)/((1-a^2)^3*x)],
[x^3*arcsin(a+b*x)^2,x,19,4/3*a*x/b^3+2*a^3*x/b^3-3/32*(a+b*x)^2/b^4-3/4*a^2*(a+b*x)^2/b^4+2/9*a*(a+b*x)^3/b^4-1/32*(a+b*x)^4/b^4-3/32*arcsin(a+b*x)^2/b^4-3/4*a^2*arcsin(a+b*x)^2/b^4-1/4*a^4*arcsin(a+b*x)^2/b^4+1/4*x^4*arcsin(a+b*x)^2-4/3*a*arcsin(a+b*x)*sqrt(1-(a+b*x)^2)/b^4-2*a^3*arcsin(a+b*x)*sqrt(1-(a+b*x)^2)/b^4+3/16*(a+b*x)*arcsin(a+b*x)*sqrt(1-(a+b*x)^2)/b^4+3/2*a^2*(a+b*x)*arcsin(a+b*x)*sqrt(1-(a+b*x)^2)/b^4-2/3*a*(a+b*x)^2*arcsin(a+b*x)*sqrt(1-(a+b*x)^2)/b^4+1/8*(a+b*x)^3*arcsin(a+b*x)*sqrt(1-(a+b*x)^2)/b^4],
[x^2*arcsin(a+b*x)^2,x,14,-4/9*x/b^2-2*a^2*x/b^2+1/2*a*(a+b*x)^2/b^3-2/27*(a+b*x)^3/b^3+1/2*a*arcsin(a+b*x)^2/b^3+1/3*a^3*arcsin(a+b*x)^2/b^3+1/3*x^3*arcsin(a+b*x)^2+4/9*arcsin(a+b*x)*sqrt(1-(a+b*x)^2)/b^3+2*a^2*arcsin(a+b*x)*sqrt(1-(a+b*x)^2)/b^3-a*(a+b*x)*arcsin(a+b*x)*sqrt(1-(a+b*x)^2)/b^3+2/9*(a+b*x)^2*arcsin(a+b*x)*sqrt(1-(a+b*x)^2)/b^3],
[x*arcsin(a+b*x)^2,x,10,2*a*x/b-1/4*(a+b*x)^2/b^2-1/4*arcsin(a+b*x)^2/b^2-1/2*a^2*arcsin(a+b*x)^2/b^2+1/2*x^2*arcsin(a+b*x)^2-2*a*arcsin(a+b*x)*sqrt(1-(a+b*x)^2)/b^2+1/2*(a+b*x)*arcsin(a+b*x)*sqrt(1-(a+b*x)^2)/b^2],
[arcsin(a+b*x)^2,x,4,-2*x+(a+b*x)*arcsin(a+b*x)^2/b+2*arcsin(a+b*x)*sqrt(1-(a+b*x)^2)/b],
[arcsin(a+b*x)^2/x,x,11,-1/3*I*arcsin(a+b*x)^3+arcsin(a+b*x)^2*log(1-exp(I*arcsin(a+b*x))/(I*a-sqrt(1-a^2)))+arcsin(a+b*x)^2*log(1-exp(I*arcsin(a+b*x))/(I*a+sqrt(1-a^2)))-2*I*arcsin(a+b*x)*polylog(2,exp(I*arcsin(a+b*x))/(I*a-sqrt(1-a^2)))-2*I*arcsin(a+b*x)*polylog(2,exp(I*arcsin(a+b*x))/(I*a+sqrt(1-a^2)))+2*polylog(3,exp(I*arcsin(a+b*x))/(I*a-sqrt(1-a^2)))+2*polylog(3,exp(I*arcsin(a+b*x))/(I*a+sqrt(1-a^2)))],
[arcsin(a+b*x)^2/x^2,x,11,-arcsin(a+b*x)^2/x-2*b*arcsin(a+b*x)*log(1-exp(I*arcsin(a+b*x))/(I*a-sqrt(1-a^2)))/sqrt(1-a^2)+2*b*arcsin(a+b*x)*log(1-exp(I*arcsin(a+b*x))/(I*a+sqrt(1-a^2)))/sqrt(1-a^2)+2*I*b*polylog(2,exp(I*arcsin(a+b*x))/(I*a-sqrt(1-a^2)))/sqrt(1-a^2)-2*I*b*polylog(2,exp(I*arcsin(a+b*x))/(I*a+sqrt(1-a^2)))/sqrt(1-a^2),-arcsin(a+b*x)^2/x+2*I*b*arcsin(a+b*x)*log(1+I*exp(I*arcsin(a+b*x))/(a-sqrt(-1+a^2)))/sqrt(-1+a^2)-2*I*b*arcsin(a+b*x)*log(1+I*exp(I*arcsin(a+b*x))/(a+sqrt(-1+a^2)))/sqrt(-1+a^2)+2*b*polylog(2,-I*exp(I*arcsin(a+b*x))/(a-sqrt(-1+a^2)))/sqrt(-1+a^2)-2*b*polylog(2,-I*exp(I*arcsin(a+b*x))/(a+sqrt(-1+a^2)))/sqrt(-1+a^2)],
[arcsin(a+b*x)^2/x^3,x,14,-1/2*arcsin(a+b*x)^2/x^2+b^2*log(x)/(1-a^2)-I*a*b^2*arcsin(a+b*x)*log(1+I*exp(I*arcsin(a+b*x))/(a-sqrt(-1+a^2)))/(-1+a^2)^(3/2)+I*a*b^2*arcsin(a+b*x)*log(1+I*exp(I*arcsin(a+b*x))/(a+sqrt(-1+a^2)))/(-1+a^2)^(3/2)-a*b^2*polylog(2,-I*exp(I*arcsin(a+b*x))/(a-sqrt(-1+a^2)))/(-1+a^2)^(3/2)+a*b^2*polylog(2,-I*exp(I*arcsin(a+b*x))/(a+sqrt(-1+a^2)))/(-1+a^2)^(3/2)-b*arcsin(a+b*x)*sqrt(1-(a+b*x)^2)/((1-a^2)*x)],

#  {ArcSin[a + b*x]^2/x^4, x, 40, -(b^2/(3*(1 - a^2)*x)) - (b*Sqrt[1 - (a + b*x)^2]*ArcSin[a + b*x])/(3*(1 - a^2)*x^2) - (a*b^2*Sqrt[1 - (a + b*x)^2]*ArcSin[a + b*x])/((1 - a^2)^2*x) - ArcSin[a + b*x]^2/(3*x^3) + (I*a^2*b^3*ArcSin[a + b*x]*Log[1 + (I*E^(I*ArcSin[a + b*x]))/(a - Sqrt[-1 + a^2])])/(-1 + a^2)^(5/2) - (I*b^3*ArcSin[a + b*x]*Log[1 + (I*E^(I*ArcSin[a + b*x]))/(a - Sqrt[-1 + a^2])])/(3*(-1 + a^2)^(3/2)) - (I*a^2*b^3*ArcSin[a + b*x]*Log[1 + (I*E^(I*ArcSin[a + b*x]))/(a + Sqrt[-1 + a^2])])/(-1 + a^2)^(5/2) + (I*b^3*ArcSin[a + b*x]*Log[1 + (I*E^(I*ArcSin[a + b*x]))/(a + Sqrt[-1 + a^2])])/(3*(-1 + a^2)^(3/2)) + (a*b^3*Log[x])/(1 - a^2)^2 + (a^2*b^3*PolyLog[2, -((I*E^(I*ArcSin[a + b*x]))/(a - Sqrt[-1 + a^2]))])/(-1 + a^2)^(5/2) - (b^3*PolyLog[2, -((I*E^(I*ArcSin[a + b*x]))/(a - Sqrt[-1 + a^2]))])/(3*(-1 + a^2)^(3/2)) - (a^2*b^3*PolyLog[2, -((I*E^(I*ArcSin[a + b*x]))/(a + Sqrt[-1 + a^2]))])/(-1 + a^2)^(5/2) + (b^3*PolyLog[2, -((I*E^(I*ArcSin[a + b*x]))/(a + Sqrt[-1 + a^2]))])/(3*(-1 + a^2)^(3/2))} 
[x^2*arcsin(a+b*x)^3,x,18,2/27*(1-(a+b*x)^2)^(3/2)/b^3-3/4*a*arcsin(a+b*x)/b^3-4/3*(a+b*x)*arcsin(a+b*x)/b^3-6*a^2*(a+b*x)*arcsin(a+b*x)/b^3+3/2*a*(a+b*x)^2*arcsin(a+b*x)/b^3-2/9*(a+b*x)^3*arcsin(a+b*x)/b^3+1/2*a*arcsin(a+b*x)^3/b^3+1/3*a^3*arcsin(a+b*x)^3/b^3+1/3*x^3*arcsin(a+b*x)^3-14/9*sqrt(1-(a+b*x)^2)/b^3-6*a^2*sqrt(1-(a+b*x)^2)/b^3+3/4*a*(a+b*x)*sqrt(1-(a+b*x)^2)/b^3+2/3*arcsin(a+b*x)^2*sqrt(1-(a+b*x)^2)/b^3+3*a^2*arcsin(a+b*x)^2*sqrt(1-(a+b*x)^2)/b^3-3/2*a*(a+b*x)*arcsin(a+b*x)^2*sqrt(1-(a+b*x)^2)/b^3+1/3*(a+b*x)^2*arcsin(a+b*x)^2*sqrt(1-(a+b*x)^2)/b^3],
[x*arcsin(a+b*x)^3,x,12,3/8*arcsin(a+b*x)/b^2+6*a*(a+b*x)*arcsin(a+b*x)/b^2-3/4*(a+b*x)^2*arcsin(a+b*x)/b^2-1/4*arcsin(a+b*x)^3/b^2-1/2*a^2*arcsin(a+b*x)^3/b^2+1/2*x^2*arcsin(a+b*x)^3+6*a*sqrt(1-(a+b*x)^2)/b^2-3/8*(a+b*x)*sqrt(1-(a+b*x)^2)/b^2-3*a*arcsin(a+b*x)^2*sqrt(1-(a+b*x)^2)/b^2+3/4*(a+b*x)*arcsin(a+b*x)^2*sqrt(1-(a+b*x)^2)/b^2],
[arcsin(a+b*x)^3,x,5,-6*(a+b*x)*arcsin(a+b*x)/b+(a+b*x)*arcsin(a+b*x)^3/b-6*sqrt(1-(a+b*x)^2)/b+3*arcsin(a+b*x)^2*sqrt(1-(a+b*x)^2)/b],
[arcsin(a+b*x)^3/x,x,13,-1/4*I*arcsin(a+b*x)^4+arcsin(a+b*x)^3*log(1-exp(I*arcsin(a+b*x))/(I*a-sqrt(1-a^2)))+arcsin(a+b*x)^3*log(1-exp(I*arcsin(a+b*x))/(I*a+sqrt(1-a^2)))-3*I*arcsin(a+b*x)^2*polylog(2,exp(I*arcsin(a+b*x))/(I*a-sqrt(1-a^2)))-3*I*arcsin(a+b*x)^2*polylog(2,exp(I*arcsin(a+b*x))/(I*a+sqrt(1-a^2)))+6*arcsin(a+b*x)*polylog(3,exp(I*arcsin(a+b*x))/(I*a-sqrt(1-a^2)))+6*arcsin(a+b*x)*polylog(3,exp(I*arcsin(a+b*x))/(I*a+sqrt(1-a^2)))+6*I*polylog(4,exp(I*arcsin(a+b*x))/(I*a-sqrt(1-a^2)))+6*I*polylog(4,exp(I*arcsin(a+b*x))/(I*a+sqrt(1-a^2)))],
[arcsin(a+b*x)^3/x^2,x,13,-arcsin(a+b*x)^3/x+3*I*b*arcsin(a+b*x)^2*log(1+I*exp(I*arcsin(a+b*x))/(a-sqrt(-1+a^2)))/sqrt(-1+a^2)-3*I*b*arcsin(a+b*x)^2*log(1+I*exp(I*arcsin(a+b*x))/(a+sqrt(-1+a^2)))/sqrt(-1+a^2)+6*b*arcsin(a+b*x)*polylog(2,-I*exp(I*arcsin(a+b*x))/(a-sqrt(-1+a^2)))/sqrt(-1+a^2)-6*b*arcsin(a+b*x)*polylog(2,-I*exp(I*arcsin(a+b*x))/(a+sqrt(-1+a^2)))/sqrt(-1+a^2)+6*I*b*polylog(3,-I*exp(I*arcsin(a+b*x))/(a-sqrt(-1+a^2)))/sqrt(-1+a^2)-6*I*b*polylog(3,-I*exp(I*arcsin(a+b*x))/(a+sqrt(-1+a^2)))/sqrt(-1+a^2)],

#  {ArcSin[a + b*x]^3/x^3, x, 21, -((3*I*b^2*ArcSin[a + b*x]^2)/(2*(1 - a^2))) - (3*b*Sqrt[1 - (a + b*x)^2]*ArcSin[a + b*x]^2)/(2*(1 - a^2)*x) - ArcSin[a + b*x]^3/(2*x^2) + (3*b^2*ArcSin[a + b*x]*Log[1 - E^(I*ArcSin[a + b*x])/(I*a - Sqrt[1 - a^2])])/(1 - a^2) + (3*b^2*ArcSin[a + b*x]*Log[1 - E^(I*ArcSin[a + b*x])/(I*a + Sqrt[1 - a^2])])/(1 - a^2) - (3*I*a*b^2*ArcSin[a + b*x]^2*Log[1 + (I*E^(I*ArcSin[a + b*x]))/(a - Sqrt[-1 + a^2])])/(2*(-1 + a^2)^(3/2)) + (3*I*a*b^2*ArcSin[a + b*x]^2*Log[1 + (I*E^(I*ArcSin[a + b*x]))/(a + Sqrt[-1 + a^2])])/(2*(-1 + a^2)^(3/2)) - (3*I*b^2*PolyLog[2, E^(I*ArcSin[a + b*x])/(I*a - Sqrt[1 - a^2])])/(1 - a^2) - (3*I*b^2*PolyLog[2, E^(I*ArcSin[a + b*x])/(I*a + Sqrt[1 - a^2])])/(1 - a^2) - (3*a*b^2*ArcSin[a + b*x]*PolyLog[2, -((I*E^(I*ArcSin[a + b*x]))/(a - Sqrt[-1 + a^2]))])/(-1 + a^2)^(3/2) + (3*a*b^2*ArcSin[a + b*x]*PolyLog[2, -((I*E^(I*ArcSin[a + b*x]))/(a + Sqrt[-1 + a^2]))])/(-1 + a^2)^(3/2) - (3*I*a*b^2*PolyLog[3, -((I*E^(I*ArcSin[a + b*x]))/(a - Sqrt[-1 + a^2]))])/(-1 + a^2)^(3/2) + (3*I*a*b^2*PolyLog[3, -((I*E^(I*ArcSin[a + b*x]))/(a + Sqrt[-1 + a^2]))])/(-1 + a^2)^(3/2)} 

# n<0
[x^2/arcsin(a+b*x),x,14,1/4*Ci(arcsin(a+b*x))/b^3+a^2*Ci(arcsin(a+b*x))/b^3-1/4*Ci(3*arcsin(a+b*x))/b^3-a*Si(2*arcsin(a+b*x))/b^3],
[x/arcsin(a+b*x),x,10,-a*Ci(arcsin(a+b*x))/b^2+1/2*Si(2*arcsin(a+b*x))/b^2],
[1/arcsin(a+b*x),x,3,Ci(arcsin(a+b*x))/b],
[1/(x*arcsin(a+b*x)),x,1,Unintegrable(1/(x*arcsin(a+b*x)),x)],
[x^2/arcsin(a+b*x)^2,x,12,-2*a*Ci(2*arcsin(a+b*x))/b^3-1/4*(1+4*a^2)*Si(arcsin(a+b*x))/b^3+3/4*Si(3*arcsin(a+b*x))/b^3-x^2*sqrt(1-(a+b*x)^2)/(b*arcsin(a+b*x)),-2*a*Ci(2*arcsin(a+b*x))/b^3-1/4*Si(arcsin(a+b*x))/b^3-a^2*Si(arcsin(a+b*x))/b^3+3/4*Si(3*arcsin(a+b*x))/b^3-a^2*sqrt(1-(a+b*x)^2)/(b^3*arcsin(a+b*x))+2*a*(a+b*x)*sqrt(1-(a+b*x)^2)/(b^3*arcsin(a+b*x))-(a+b*x)^2*sqrt(1-(a+b*x)^2)/(b^3*arcsin(a+b*x))],
[x/arcsin(a+b*x)^2,x,8,Ci(2*arcsin(a+b*x))/b^2+a*Si(arcsin(a+b*x))/b^2-x*sqrt(1-(a+b*x)^2)/(b*arcsin(a+b*x)),Ci(2*arcsin(a+b*x))/b^2+a*Si(arcsin(a+b*x))/b^2+a*sqrt(1-(a+b*x)^2)/(b^2*arcsin(a+b*x))-(a+b*x)*sqrt(1-(a+b*x)^2)/(b^2*arcsin(a+b*x))],
[1/arcsin(a+b*x)^2,x,4,-Si(arcsin(a+b*x))/b-sqrt(1-(a+b*x)^2)/(b*arcsin(a+b*x))],
[1/(x*arcsin(a+b*x)^2),x,1,Unintegrable(1/(x*arcsin(a+b*x)^2),x)],
[x^2/arcsin(a+b*x)^3,x,24,1/2*a^2*(a+b*x)/(b^3*arcsin(a+b*x))-2*a*(a+b*x)^2/(b^3*arcsin(a+b*x))+1/8*(9*a+b*x)/(b^3*arcsin(a+b*x))-1/8*(1+4*a^2)*Ci(arcsin(a+b*x))/b^3+9/8*Ci(3*arcsin(a+b*x))/b^3+2*a*Si(2*arcsin(a+b*x))/b^3-3/8*sin(3*arcsin(a+b*x))/(b^3*arcsin(a+b*x))-1/2*x^2*sqrt(1-(a+b*x)^2)/(b*arcsin(a+b*x)^2),a/(b^3*arcsin(a+b*x))+(-a-b*x)/(b^3*arcsin(a+b*x))+1/2*a^2*(a+b*x)/(b^3*arcsin(a+b*x))-2*a*(a+b*x)^2/(b^3*arcsin(a+b*x))+3/2*(a+b*x)^3/(b^3*arcsin(a+b*x))-1/8*Ci(arcsin(a+b*x))/b^3-1/2*a^2*Ci(arcsin(a+b*x))/b^3+9/8*Ci(3*arcsin(a+b*x))/b^3+2*a*Si(2*arcsin(a+b*x))/b^3-1/2*a^2*sqrt(1-(a+b*x)^2)/(b^3*arcsin(a+b*x)^2)+a*(a+b*x)*sqrt(1-(a+b*x)^2)/(b^3*arcsin(a+b*x)^2)-1/2*(a+b*x)^2*sqrt(1-(a+b*x)^2)/(b^3*arcsin(a+b*x)^2)],
[x/arcsin(a+b*x)^3,x,14,-1/2*a*(a+b*x)/(b^2*arcsin(a+b*x))+1/2*(-1+2*(a+b*x)^2)/(b^2*arcsin(a+b*x))+1/2*a*Ci(arcsin(a+b*x))/b^2-Si(2*arcsin(a+b*x))/b^2-1/2*x*sqrt(1-(a+b*x)^2)/(b*arcsin(a+b*x)^2),(-1/2)/(b^2*arcsin(a+b*x))-1/2*a*(a+b*x)/(b^2*arcsin(a+b*x))+(a+b*x)^2/(b^2*arcsin(a+b*x))+1/2*a*Ci(arcsin(a+b*x))/b^2-Si(2*arcsin(a+b*x))/b^2+1/2*a*sqrt(1-(a+b*x)^2)/(b^2*arcsin(a+b*x)^2)-1/2*(a+b*x)*sqrt(1-(a+b*x)^2)/(b^2*arcsin(a+b*x)^2)],
[1/arcsin(a+b*x)^3,x,5,1/2*(a+b*x)/(b*arcsin(a+b*x))-1/2*Ci(arcsin(a+b*x))/b-1/2*sqrt(1-(a+b*x)^2)/(b*arcsin(a+b*x)^2)],
[1/(x*arcsin(a+b*x)^3),x,1,Unintegrable(1/(x*arcsin(a+b*x)^3),x)],

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

# n>0
[x^2*(a+b*arcsin(c+d*x))^(1/2),x,23,1/12*cos(3*a/b)*FresnelS(sqrt(6/Pi)*sqrt(a+b*arcsin(c+d*x))/sqrt(b))*sqrt(b)*sqrt(1/6*Pi)/d^3-1/12*FresnelC(sqrt(6/Pi)*sqrt(a+b*arcsin(c+d*x))/sqrt(b))*sin(3*a/b)*sqrt(b)*sqrt(1/6*Pi)/d^3-1/4*cos(a/b)*FresnelS(sqrt(2/Pi)*sqrt(a+b*arcsin(c+d*x))/sqrt(b))*sqrt(b)*sqrt(1/2*Pi)/d^3-c^2*cos(a/b)*FresnelS(sqrt(2/Pi)*sqrt(a+b*arcsin(c+d*x))/sqrt(b))*sqrt(b)*sqrt(1/2*Pi)/d^3+1/4*FresnelC(sqrt(2/Pi)*sqrt(a+b*arcsin(c+d*x))/sqrt(b))*sin(a/b)*sqrt(b)*sqrt(1/2*Pi)/d^3+c^2*FresnelC(sqrt(2/Pi)*sqrt(a+b*arcsin(c+d*x))/sqrt(b))*sin(a/b)*sqrt(b)*sqrt(1/2*Pi)/d^3-1/4*c*cos(2*a/b)*FresnelC(2*sqrt(a+b*arcsin(c+d*x))/(sqrt(b)*sqrt(Pi)))*sqrt(b)*sqrt(Pi)/d^3-1/4*c*FresnelS(2*sqrt(a+b*arcsin(c+d*x))/(sqrt(b)*sqrt(Pi)))*sin(2*a/b)*sqrt(b)*sqrt(Pi)/d^3+c^2*(c+d*x)*sqrt(a+b*arcsin(c+d*x))/d^3+1/3*(c+d*x)^3*sqrt(a+b*arcsin(c+d*x))/d^3+1/2*c*cos(2*arcsin(c+d*x))*sqrt(a+b*arcsin(c+d*x))/d^3],
[x*(a+b*arcsin(c+d*x))^(1/2),x,14,c*cos(a/b)*FresnelS(sqrt(2/Pi)*sqrt(a+b*arcsin(c+d*x))/sqrt(b))*sqrt(b)*sqrt(1/2*Pi)/d^2-c*FresnelC(sqrt(2/Pi)*sqrt(a+b*arcsin(c+d*x))/sqrt(b))*sin(a/b)*sqrt(b)*sqrt(1/2*Pi)/d^2+1/8*cos(2*a/b)*FresnelC(2*sqrt(a+b*arcsin(c+d*x))/(sqrt(b)*sqrt(Pi)))*sqrt(b)*sqrt(Pi)/d^2+1/8*FresnelS(2*sqrt(a+b*arcsin(c+d*x))/(sqrt(b)*sqrt(Pi)))*sin(2*a/b)*sqrt(b)*sqrt(Pi)/d^2-c*(c+d*x)*sqrt(a+b*arcsin(c+d*x))/d^2-1/4*cos(2*arcsin(c+d*x))*sqrt(a+b*arcsin(c+d*x))/d^2],
[(a+b*arcsin(c+d*x))^(1/2),x,8,-cos(a/b)*FresnelS(sqrt(2/Pi)*sqrt(a+b*arcsin(c+d*x))/sqrt(b))*sqrt(b)*sqrt(1/2*Pi)/d+FresnelC(sqrt(2/Pi)*sqrt(a+b*arcsin(c+d*x))/sqrt(b))*sin(a/b)*sqrt(b)*sqrt(1/2*Pi)/d+(c+d*x)*sqrt(a+b*arcsin(c+d*x))/d],
[x*(a+b*arcsin(c+d*x))^(3/2),x,16,-c*(c+d*x)*(a+b*arcsin(c+d*x))^(3/2)/d^2-1/4*(a+b*arcsin(c+d*x))^(3/2)*cos(2*arcsin(c+d*x))/d^2+3/2*b^(3/2)*c*cos(a/b)*FresnelC(sqrt(2/Pi)*sqrt(a+b*arcsin(c+d*x))/sqrt(b))*sqrt(1/2*Pi)/d^2+3/2*b^(3/2)*c*FresnelS(sqrt(2/Pi)*sqrt(a+b*arcsin(c+d*x))/sqrt(b))*sin(a/b)*sqrt(1/2*Pi)/d^2-3/32*b^(3/2)*cos(2*a/b)*FresnelS(2*sqrt(a+b*arcsin(c+d*x))/(sqrt(b)*sqrt(Pi)))*sqrt(Pi)/d^2+3/32*b^(3/2)*FresnelC(2*sqrt(a+b*arcsin(c+d*x))/(sqrt(b)*sqrt(Pi)))*sin(2*a/b)*sqrt(Pi)/d^2+3/16*b*sin(2*arcsin(c+d*x))*sqrt(a+b*arcsin(c+d*x))/d^2-3/2*b*c*sqrt(1-(c+d*x)^2)*sqrt(a+b*arcsin(c+d*x))/d^2],
[(a+b*arcsin(c+d*x))^(3/2),x,9,(c+d*x)*(a+b*arcsin(c+d*x))^(3/2)/d-3/2*b^(3/2)*cos(a/b)*FresnelC(sqrt(2/Pi)*sqrt(a+b*arcsin(c+d*x))/sqrt(b))*sqrt(1/2*Pi)/d-3/2*b^(3/2)*FresnelS(sqrt(2/Pi)*sqrt(a+b*arcsin(c+d*x))/sqrt(b))*sin(a/b)*sqrt(1/2*Pi)/d+3/2*b*sqrt(1-(c+d*x)^2)*sqrt(a+b*arcsin(c+d*x))/d],
[x*(a+b*arcsin(c+d*x))^(5/2),x,18,-c*(c+d*x)*(a+b*arcsin(c+d*x))^(5/2)/d^2-1/4*(a+b*arcsin(c+d*x))^(5/2)*cos(2*arcsin(c+d*x))/d^2+5/16*b*(a+b*arcsin(c+d*x))^(3/2)*sin(2*arcsin(c+d*x))/d^2-15/4*b^(5/2)*c*cos(a/b)*FresnelS(sqrt(2/Pi)*sqrt(a+b*arcsin(c+d*x))/sqrt(b))*sqrt(1/2*Pi)/d^2+15/4*b^(5/2)*c*FresnelC(sqrt(2/Pi)*sqrt(a+b*arcsin(c+d*x))/sqrt(b))*sin(a/b)*sqrt(1/2*Pi)/d^2-15/128*b^(5/2)*cos(2*a/b)*FresnelC(2*sqrt(a+b*arcsin(c+d*x))/(sqrt(b)*sqrt(Pi)))*sqrt(Pi)/d^2-15/128*b^(5/2)*FresnelS(2*sqrt(a+b*arcsin(c+d*x))/(sqrt(b)*sqrt(Pi)))*sin(2*a/b)*sqrt(Pi)/d^2-5/2*b*c*(a+b*arcsin(c+d*x))^(3/2)*sqrt(1-(c+d*x)^2)/d^2+15/4*b^2*c*(c+d*x)*sqrt(a+b*arcsin(c+d*x))/d^2+15/64*b^2*cos(2*arcsin(c+d*x))*sqrt(a+b*arcsin(c+d*x))/d^2],
[(a+b*arcsin(c+d*x))^(5/2),x,10,(c+d*x)*(a+b*arcsin(c+d*x))^(5/2)/d+15/4*b^(5/2)*cos(a/b)*FresnelS(sqrt(2/Pi)*sqrt(a+b*arcsin(c+d*x))/sqrt(b))*sqrt(1/2*Pi)/d-15/4*b^(5/2)*FresnelC(sqrt(2/Pi)*sqrt(a+b*arcsin(c+d*x))/sqrt(b))*sin(a/b)*sqrt(1/2*Pi)/d+5/2*b*(a+b*arcsin(c+d*x))^(3/2)*sqrt(1-(c+d*x)^2)/d-15/4*b^2*(c+d*x)*sqrt(a+b*arcsin(c+d*x))/d],
[(a+b*arcsin(c+d*x))^(7/2),x,11,-35/4*b^2*(c+d*x)*(a+b*arcsin(c+d*x))^(3/2)/d+(c+d*x)*(a+b*arcsin(c+d*x))^(7/2)/d+105/8*b^(7/2)*cos(a/b)*FresnelC(sqrt(2/Pi)*sqrt(a+b*arcsin(c+d*x))/sqrt(b))*sqrt(1/2*Pi)/d+105/8*b^(7/2)*FresnelS(sqrt(2/Pi)*sqrt(a+b*arcsin(c+d*x))/sqrt(b))*sin(a/b)*sqrt(1/2*Pi)/d+7/2*b*(a+b*arcsin(c+d*x))^(5/2)*sqrt(1-(c+d*x)^2)/d-105/8*b^3*sqrt(1-(c+d*x)^2)*sqrt(a+b*arcsin(c+d*x))/d],

# n<0
[x^2/(a+b*arcsin(c+d*x))^(1/2),x,20,-1/2*cos(3*a/b)*FresnelC(sqrt(6/Pi)*sqrt(a+b*arcsin(c+d*x))/sqrt(b))*sqrt(1/6*Pi)/(d^3*sqrt(b))-1/2*FresnelS(sqrt(6/Pi)*sqrt(a+b*arcsin(c+d*x))/sqrt(b))*sin(3*a/b)*sqrt(1/6*Pi)/(d^3*sqrt(b))+1/2*cos(a/b)*FresnelC(sqrt(2/Pi)*sqrt(a+b*arcsin(c+d*x))/sqrt(b))*sqrt(1/2*Pi)/(d^3*sqrt(b))+1/2*FresnelS(sqrt(2/Pi)*sqrt(a+b*arcsin(c+d*x))/sqrt(b))*sin(a/b)*sqrt(1/2*Pi)/(d^3*sqrt(b))-c*cos(2*a/b)*FresnelS(2*sqrt(a+b*arcsin(c+d*x))/(sqrt(b)*sqrt(Pi)))*sqrt(Pi)/(d^3*sqrt(b))+c*FresnelC(2*sqrt(a+b*arcsin(c+d*x))/(sqrt(b)*sqrt(Pi)))*sin(2*a/b)*sqrt(Pi)/(d^3*sqrt(b))+c^2*cos(a/b)*FresnelC(sqrt(2/Pi)*sqrt(a+b*arcsin(c+d*x))/sqrt(b))*sqrt(2*Pi)/(d^3*sqrt(b))+c^2*FresnelS(sqrt(2/Pi)*sqrt(a+b*arcsin(c+d*x))/sqrt(b))*sin(a/b)*sqrt(2*Pi)/(d^3*sqrt(b))],
[x/(a+b*arcsin(c+d*x))^(1/2),x,12,1/2*cos(2*a/b)*FresnelS(2*sqrt(a+b*arcsin(c+d*x))/(sqrt(b)*sqrt(Pi)))*sqrt(Pi)/(d^2*sqrt(b))-1/2*FresnelC(2*sqrt(a+b*arcsin(c+d*x))/(sqrt(b)*sqrt(Pi)))*sin(2*a/b)*sqrt(Pi)/(d^2*sqrt(b))-c*cos(a/b)*FresnelC(sqrt(2/Pi)*sqrt(a+b*arcsin(c+d*x))/sqrt(b))*sqrt(2*Pi)/(d^2*sqrt(b))-c*FresnelS(sqrt(2/Pi)*sqrt(a+b*arcsin(c+d*x))/sqrt(b))*sin(a/b)*sqrt(2*Pi)/(d^2*sqrt(b))],
[1/(a+b*arcsin(c+d*x))^(1/2),x,7,cos(a/b)*FresnelC(sqrt(2/Pi)*sqrt(a+b*arcsin(c+d*x))/sqrt(b))*sqrt(2*Pi)/(d*sqrt(b))+FresnelS(sqrt(2/Pi)*sqrt(a+b*arcsin(c+d*x))/sqrt(b))*sin(a/b)*sqrt(2*Pi)/(d*sqrt(b))],
[x/(a+b*arcsin(c+d*x))^(3/2),x,16,2*cos(2*a/b)*FresnelC(2*sqrt(a+b*arcsin(c+d*x))/(sqrt(b)*sqrt(Pi)))*sqrt(Pi)/(b^(3/2)*d^2)+2*FresnelS(2*sqrt(a+b*arcsin(c+d*x))/(sqrt(b)*sqrt(Pi)))*sin(2*a/b)*sqrt(Pi)/(b^(3/2)*d^2)+2*c*cos(a/b)*FresnelS(sqrt(2/Pi)*sqrt(a+b*arcsin(c+d*x))/sqrt(b))*sqrt(2*Pi)/(b^(3/2)*d^2)-2*c*FresnelC(sqrt(2/Pi)*sqrt(a+b*arcsin(c+d*x))/sqrt(b))*sin(a/b)*sqrt(2*Pi)/(b^(3/2)*d^2)+2*c*sqrt(1-(c+d*x)^2)/(b*d^2*sqrt(a+b*arcsin(c+d*x)))-2*(c+d*x)*sqrt(1-(c+d*x)^2)/(b*d^2*sqrt(a+b*arcsin(c+d*x)))],
[1/(a+b*arcsin(c+d*x))^(3/2),x,8,-2*cos(a/b)*FresnelS(sqrt(2/Pi)*sqrt(a+b*arcsin(c+d*x))/sqrt(b))*sqrt(2*Pi)/(b^(3/2)*d)+2*FresnelC(sqrt(2/Pi)*sqrt(a+b*arcsin(c+d*x))/sqrt(b))*sin(a/b)*sqrt(2*Pi)/(b^(3/2)*d)-2*sqrt(1-(c+d*x)^2)/(b*d*sqrt(a+b*arcsin(c+d*x)))],
[x/(a+b*arcsin(c+d*x))^(5/2),x,22,-8/3*cos(2*a/b)*FresnelS(2*sqrt(a+b*arcsin(c+d*x))/(sqrt(b)*sqrt(Pi)))*sqrt(Pi)/(b^(5/2)*d^2)+8/3*FresnelC(2*sqrt(a+b*arcsin(c+d*x))/(sqrt(b)*sqrt(Pi)))*sin(2*a/b)*sqrt(Pi)/(b^(5/2)*d^2)+4/3*c*cos(a/b)*FresnelC(sqrt(2/Pi)*sqrt(a+b*arcsin(c+d*x))/sqrt(b))*sqrt(2*Pi)/(b^(5/2)*d^2)+4/3*c*FresnelS(sqrt(2/Pi)*sqrt(a+b*arcsin(c+d*x))/sqrt(b))*sin(a/b)*sqrt(2*Pi)/(b^(5/2)*d^2)+2/3*c*sqrt(1-(c+d*x)^2)/(b*d^2*(a+b*arcsin(c+d*x))^(3/2))-2/3*(c+d*x)*sqrt(1-(c+d*x)^2)/(b*d^2*(a+b*arcsin(c+d*x))^(3/2))+(-4/3)/(b^2*d^2*sqrt(a+b*arcsin(c+d*x)))-4/3*c*(c+d*x)/(b^2*d^2*sqrt(a+b*arcsin(c+d*x)))+8/3*(c+d*x)^2/(b^2*d^2*sqrt(a+b*arcsin(c+d*x)))],
[1/(a+b*arcsin(c+d*x))^(5/2),x,9,-4/3*cos(a/b)*FresnelC(sqrt(2/Pi)*sqrt(a+b*arcsin(c+d*x))/sqrt(b))*sqrt(2*Pi)/(b^(5/2)*d)-4/3*FresnelS(sqrt(2/Pi)*sqrt(a+b*arcsin(c+d*x))/sqrt(b))*sin(a/b)*sqrt(2*Pi)/(b^(5/2)*d)-2/3*sqrt(1-(c+d*x)^2)/(b*d*(a+b*arcsin(c+d*x))^(3/2))+4/3*(c+d*x)/(b^2*d*sqrt(a+b*arcsin(c+d*x)))],
[x/(a+b*arcsin(c+d*x))^(7/2),x,21,(-4/15)/(b^2*d^2*(a+b*arcsin(c+d*x))^(3/2))-4/15*c*(c+d*x)/(b^2*d^2*(a+b*arcsin(c+d*x))^(3/2))+8/15*(c+d*x)^2/(b^2*d^2*(a+b*arcsin(c+d*x))^(3/2))-32/15*cos(2*a/b)*FresnelC(2*sqrt(a+b*arcsin(c+d*x))/(sqrt(b)*sqrt(Pi)))*sqrt(Pi)/(b^(7/2)*d^2)-32/15*FresnelS(2*sqrt(a+b*arcsin(c+d*x))/(sqrt(b)*sqrt(Pi)))*sin(2*a/b)*sqrt(Pi)/(b^(7/2)*d^2)-8/15*c*cos(a/b)*FresnelS(sqrt(2/Pi)*sqrt(a+b*arcsin(c+d*x))/sqrt(b))*sqrt(2*Pi)/(b^(7/2)*d^2)+8/15*c*FresnelC(sqrt(2/Pi)*sqrt(a+b*arcsin(c+d*x))/sqrt(b))*sin(a/b)*sqrt(2*Pi)/(b^(7/2)*d^2)+2/5*c*sqrt(1-(c+d*x)^2)/(b*d^2*(a+b*arcsin(c+d*x))^(5/2))-2/5*(c+d*x)*sqrt(1-(c+d*x)^2)/(b*d^2*(a+b*arcsin(c+d*x))^(5/2))-8/15*c*sqrt(1-(c+d*x)^2)/(b^3*d^2*sqrt(a+b*arcsin(c+d*x)))+32/15*(c+d*x)*sqrt(1-(c+d*x)^2)/(b^3*d^2*sqrt(a+b*arcsin(c+d*x)))],
[1/(a+b*arcsin(c+d*x))^(7/2),x,10,4/15*(c+d*x)/(b^2*d*(a+b*arcsin(c+d*x))^(3/2))+8/15*cos(a/b)*FresnelS(sqrt(2/Pi)*sqrt(a+b*arcsin(c+d*x))/sqrt(b))*sqrt(2*Pi)/(b^(7/2)*d)-8/15*FresnelC(sqrt(2/Pi)*sqrt(a+b*arcsin(c+d*x))/sqrt(b))*sin(a/b)*sqrt(2*Pi)/(b^(7/2)*d)-2/5*sqrt(1-(c+d*x)^2)/(b*d*(a+b*arcsin(c+d*x))^(5/2))+8/15*sqrt(1-(c+d*x)^2)/(b^3*d*sqrt(a+b*arcsin(c+d*x)))],

# Integrands of the form x^m (a+b ArcSin[c+d x])^n with n symbolic
[x^m*(a+b*arcsin(c+d*x))^n,x,1,Unintegrable(x^m*(a+b*arcsin(c+d*x))^n,x)],
[x^2*(a+b*arcsin(c+d*x))^n,x,22,-1/8*I*(a+b*arcsin(c+d*x))^n*GAMMA(1+n,-I*(a+b*arcsin(c+d*x))/b)/(d^3*exp(I*a/b)*(-I*(a+b*arcsin(c+d*x))/b)^n)-1/2*I*c^2*(a+b*arcsin(c+d*x))^n*GAMMA(1+n,-I*(a+b*arcsin(c+d*x))/b)/(d^3*exp(I*a/b)*(-I*(a+b*arcsin(c+d*x))/b)^n)+1/8*I*exp(I*a/b)*(a+b*arcsin(c+d*x))^n*GAMMA(1+n,I*(a+b*arcsin(c+d*x))/b)/(d^3*(I*(a+b*arcsin(c+d*x))/b)^n)+1/2*I*c^2*exp(I*a/b)*(a+b*arcsin(c+d*x))^n*GAMMA(1+n,I*(a+b*arcsin(c+d*x))/b)/(d^3*(I*(a+b*arcsin(c+d*x))/b)^n)+2^(-2-n)*c*(a+b*arcsin(c+d*x))^n*GAMMA(1+n,-2*I*(a+b*arcsin(c+d*x))/b)/(d^3*exp(2*I*a/b)*(-I*(a+b*arcsin(c+d*x))/b)^n)+2^(-2-n)*c*exp(2*I*a/b)*(a+b*arcsin(c+d*x))^n*GAMMA(1+n,2*I*(a+b*arcsin(c+d*x))/b)/(d^3*(I*(a+b*arcsin(c+d*x))/b)^n)+1/8*I*3^(-1-n)*(a+b*arcsin(c+d*x))^n*GAMMA(1+n,-3*I*(a+b*arcsin(c+d*x))/b)/(d^3*exp(3*I*a/b)*(-I*(a+b*arcsin(c+d*x))/b)^n)-1/8*I*3^(-1-n)*exp(3*I*a/b)*(a+b*arcsin(c+d*x))^n*GAMMA(1+n,3*I*(a+b*arcsin(c+d*x))/b)/(d^3*(I*(a+b*arcsin(c+d*x))/b)^n)],
[x*(a+b*arcsin(c+d*x))^n,x,14,1/2*I*c*(a+b*arcsin(c+d*x))^n*GAMMA(1+n,-I*(a+b*arcsin(c+d*x))/b)/(d^2*exp(I*a/b)*(-I*(a+b*arcsin(c+d*x))/b)^n)-1/2*I*c*exp(I*a/b)*(a+b*arcsin(c+d*x))^n*GAMMA(1+n,I*(a+b*arcsin(c+d*x))/b)/(d^2*(I*(a+b*arcsin(c+d*x))/b)^n)-2^(-3-n)*(a+b*arcsin(c+d*x))^n*GAMMA(1+n,-2*I*(a+b*arcsin(c+d*x))/b)/(d^2*exp(2*I*a/b)*(-I*(a+b*arcsin(c+d*x))/b)^n)-2^(-3-n)*exp(2*I*a/b)*(a+b*arcsin(c+d*x))^n*GAMMA(1+n,2*I*(a+b*arcsin(c+d*x))/b)/(d^2*(I*(a+b*arcsin(c+d*x))/b)^n)],
[(a+b*arcsin(c+d*x))^n,x,5,-1/2*I*(a+b*arcsin(c+d*x))^n*GAMMA(1+n,-I*(a+b*arcsin(c+d*x))/b)/(d*exp(I*a/b)*(-I*(a+b*arcsin(c+d*x))/b)^n)+1/2*I*exp(I*a/b)*(a+b*arcsin(c+d*x))^n*GAMMA(1+n,I*(a+b*arcsin(c+d*x))/b)/(d*(I*(a+b*arcsin(c+d*x))/b)^n)],
[(a+b*arcsin(c+d*x))^n/x,x,1,Unintegrable((a+b*arcsin(c+d*x))^n/x,x)],

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

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

# n>0
[(c*e+d*e*x)^4*(a+b*arcsin(c+d*x)),x,6,-2/15*b*e^4*(1-(c+d*x)^2)^(3/2)/d+1/25*b*e^4*(1-(c+d*x)^2)^(5/2)/d+1/5*e^4*(c+d*x)^5*(a+b*arcsin(c+d*x))/d+1/5*b*e^4*sqrt(1-(c+d*x)^2)/d],
[(c*e+d*e*x)^3*(a+b*arcsin(c+d*x)),x,6,-3/32*b*e^3*arcsin(c+d*x)/d+1/4*e^3*(c+d*x)^4*(a+b*arcsin(c+d*x))/d+3/32*b*e^3*(c+d*x)*sqrt(1-(c+d*x)^2)/d+1/16*b*e^3*(c+d*x)^3*sqrt(1-(c+d*x)^2)/d],
[(c*e+d*e*x)^2*(a+b*arcsin(c+d*x)),x,6,-1/9*b*e^2*(1-(c+d*x)^2)^(3/2)/d+1/3*e^2*(c+d*x)^3*(a+b*arcsin(c+d*x))/d+1/3*b*e^2*sqrt(1-(c+d*x)^2)/d],
[(c*e+d*e*x)*(a+b*arcsin(c+d*x)),x,5,-1/4*b*e*arcsin(c+d*x)/d+1/2*e*(c+d*x)^2*(a+b*arcsin(c+d*x))/d+1/4*b*e*(c+d*x)*sqrt(1-(c+d*x)^2)/d],
[a+b*arcsin(c+d*x),x,4,a*x+b*(c+d*x)*arcsin(c+d*x)/d+b*sqrt(1-(c+d*x)^2)/d],
[(a+b*arcsin(c+d*x))/(c*e+d*e*x),x,7,-1/2*I*(a+b*arcsin(c+d*x))^2/(b*d*e)+(a+b*arcsin(c+d*x))*log(1-exp(2*I*arcsin(c+d*x)))/(d*e)-1/2*I*b*polylog(2,exp(2*I*arcsin(c+d*x)))/(d*e)],
[(a+b*arcsin(c+d*x))/(c*e+d*e*x)^2,x,6,(-a-b*arcsin(c+d*x))/(d*e^2*(c+d*x))-b*arctanh(sqrt(1-(c+d*x)^2))/(d*e^2)],
[(a+b*arcsin(c+d*x))/(c*e+d*e*x)^3,x,4,1/2*(-a-b*arcsin(c+d*x))/(d*e^3*(c+d*x)^2)-1/2*b*sqrt(1-(c+d*x)^2)/(d*e^3*(c+d*x))],
[(a+b*arcsin(c+d*x))/(c*e+d*e*x)^4,x,7,1/3*(-a-b*arcsin(c+d*x))/(d*e^4*(c+d*x)^3)-1/6*b*arctanh(sqrt(1-(c+d*x)^2))/(d*e^4)-1/6*b*sqrt(1-(c+d*x)^2)/(d*e^4*(c+d*x)^2)],
[(a+b*arcsin(c+d*x))/(c*e+d*e*x)^5,x,5,1/4*(-a-b*arcsin(c+d*x))/(d*e^5*(c+d*x)^4)-1/12*b*sqrt(1-(c+d*x)^2)/(d*e^5*(c+d*x)^3)-1/6*b*sqrt(1-(c+d*x)^2)/(d*e^5*(c+d*x))],
[(a+b*arcsin(c+d*x))/(c*e+d*e*x)^6,x,8,1/5*(-a-b*arcsin(c+d*x))/(d*e^6*(c+d*x)^5)-3/40*b*arctanh(sqrt(1-(c+d*x)^2))/(d*e^6)-1/20*b*sqrt(1-(c+d*x)^2)/(d*e^6*(c+d*x)^4)-3/40*b*sqrt(1-(c+d*x)^2)/(d*e^6*(c+d*x)^2)],
[(c*e+d*e*x)^4*(a+b*arcsin(c+d*x))^2,x,9,-16/75*b^2*e^4*x-8/225*b^2*e^4*(c+d*x)^3/d-2/125*b^2*e^4*(c+d*x)^5/d+1/5*e^4*(c+d*x)^5*(a+b*arcsin(c+d*x))^2/d+16/75*b*e^4*(a+b*arcsin(c+d*x))*sqrt(1-(c+d*x)^2)/d+8/75*b*e^4*(c+d*x)^2*(a+b*arcsin(c+d*x))*sqrt(1-(c+d*x)^2)/d+2/25*b*e^4*(c+d*x)^4*(a+b*arcsin(c+d*x))*sqrt(1-(c+d*x)^2)/d],
[(c*e+d*e*x)^3*(a+b*arcsin(c+d*x))^2,x,8,-3/32*b^2*e^3*(c+d*x)^2/d-1/32*b^2*e^3*(c+d*x)^4/d-3/32*e^3*(a+b*arcsin(c+d*x))^2/d+1/4*e^3*(c+d*x)^4*(a+b*arcsin(c+d*x))^2/d+3/16*b*e^3*(c+d*x)*(a+b*arcsin(c+d*x))*sqrt(1-(c+d*x)^2)/d+1/8*b*e^3*(c+d*x)^3*(a+b*arcsin(c+d*x))*sqrt(1-(c+d*x)^2)/d],
[(c*e+d*e*x)^2*(a+b*arcsin(c+d*x))^2,x,7,-4/9*b^2*e^2*x-2/27*b^2*e^2*(c+d*x)^3/d+1/3*e^2*(c+d*x)^3*(a+b*arcsin(c+d*x))^2/d+4/9*b*e^2*(a+b*arcsin(c+d*x))*sqrt(1-(c+d*x)^2)/d+2/9*b*e^2*(c+d*x)^2*(a+b*arcsin(c+d*x))*sqrt(1-(c+d*x)^2)/d],
[(c*e+d*e*x)*(a+b*arcsin(c+d*x))^2,x,6,-1/4*b^2*e*(c+d*x)^2/d-1/4*e*(a+b*arcsin(c+d*x))^2/d+1/2*e*(c+d*x)^2*(a+b*arcsin(c+d*x))^2/d+1/2*b*e*(c+d*x)*(a+b*arcsin(c+d*x))*sqrt(1-(c+d*x)^2)/d],
[(a+b*arcsin(c+d*x))^2,x,4,-2*b^2*x+(c+d*x)*(a+b*arcsin(c+d*x))^2/d+2*b*(a+b*arcsin(c+d*x))*sqrt(1-(c+d*x)^2)/d],
[(a+b*arcsin(c+d*x))^2/(c*e+d*e*x),x,8,-1/3*I*(a+b*arcsin(c+d*x))^3/(b*d*e)+(a+b*arcsin(c+d*x))^2*log(1-exp(2*I*arcsin(c+d*x)))/(d*e)-I*b*(a+b*arcsin(c+d*x))*polylog(2,exp(2*I*arcsin(c+d*x)))/(d*e)+1/2*b^2*polylog(3,exp(2*I*arcsin(c+d*x)))/(d*e)],
[(a+b*arcsin(c+d*x))^2/(c*e+d*e*x)^2,x,9,-(a+b*arcsin(c+d*x))^2/(d*e^2*(c+d*x))-4*b*(a+b*arcsin(c+d*x))*arctanh(exp(I*arcsin(c+d*x)))/(d*e^2)+2*I*b^2*polylog(2,-exp(I*arcsin(c+d*x)))/(d*e^2)-2*I*b^2*polylog(2,exp(I*arcsin(c+d*x)))/(d*e^2)],
[(a+b*arcsin(c+d*x))^2/(c*e+d*e*x)^3,x,5,-1/2*(a+b*arcsin(c+d*x))^2/(d*e^3*(c+d*x)^2)+b^2*log(c+d*x)/(d*e^3)-b*(a+b*arcsin(c+d*x))*sqrt(1-(c+d*x)^2)/(d*e^3*(c+d*x))],
[(a+b*arcsin(c+d*x))^2/(c*e+d*e*x)^4,x,11,-1/3*b^2/(d*e^4*(c+d*x))-1/3*(a+b*arcsin(c+d*x))^2/(d*e^4*(c+d*x)^3)-2/3*b*(a+b*arcsin(c+d*x))*arctanh(exp(I*arcsin(c+d*x)))/(d*e^4)+1/3*I*b^2*polylog(2,-exp(I*arcsin(c+d*x)))/(d*e^4)-1/3*I*b^2*polylog(2,exp(I*arcsin(c+d*x)))/(d*e^4)-1/3*b*(a+b*arcsin(c+d*x))*sqrt(1-(c+d*x)^2)/(d*e^4*(c+d*x)^2)],
[(c*e+d*e*x)^4*(a+b*arcsin(c+d*x))^3,x,17,-16/25*a*b^2*e^4*x+76/1125*b^3*e^4*(1-(c+d*x)^2)^(3/2)/d-6/625*b^3*e^4*(1-(c+d*x)^2)^(5/2)/d-16/25*b^3*e^4*(c+d*x)*arcsin(c+d*x)/d-8/75*b^2*e^4*(c+d*x)^3*(a+b*arcsin(c+d*x))/d-6/125*b^2*e^4*(c+d*x)^5*(a+b*arcsin(c+d*x))/d+1/5*e^4*(c+d*x)^5*(a+b*arcsin(c+d*x))^3/d-298/375*b^3*e^4*sqrt(1-(c+d*x)^2)/d+8/25*b*e^4*(a+b*arcsin(c+d*x))^2*sqrt(1-(c+d*x)^2)/d+4/25*b*e^4*(c+d*x)^2*(a+b*arcsin(c+d*x))^2*sqrt(1-(c+d*x)^2)/d+3/25*b*e^4*(c+d*x)^4*(a+b*arcsin(c+d*x))^2*sqrt(1-(c+d*x)^2)/d],
[(c*e+d*e*x)^3*(a+b*arcsin(c+d*x))^3,x,13,45/256*b^3*e^3*arcsin(c+d*x)/d-9/32*b^2*e^3*(c+d*x)^2*(a+b*arcsin(c+d*x))/d-3/32*b^2*e^3*(c+d*x)^4*(a+b*arcsin(c+d*x))/d-3/32*e^3*(a+b*arcsin(c+d*x))^3/d+1/4*e^3*(c+d*x)^4*(a+b*arcsin(c+d*x))^3/d-45/256*b^3*e^3*(c+d*x)*sqrt(1-(c+d*x)^2)/d-3/128*b^3*e^3*(c+d*x)^3*sqrt(1-(c+d*x)^2)/d+9/32*b*e^3*(c+d*x)*(a+b*arcsin(c+d*x))^2*sqrt(1-(c+d*x)^2)/d+3/16*b*e^3*(c+d*x)^3*(a+b*arcsin(c+d*x))^2*sqrt(1-(c+d*x)^2)/d],
[(c*e+d*e*x)^2*(a+b*arcsin(c+d*x))^3,x,12,-4/3*a*b^2*e^2*x+2/27*b^3*e^2*(1-(c+d*x)^2)^(3/2)/d-4/3*b^3*e^2*(c+d*x)*arcsin(c+d*x)/d-2/9*b^2*e^2*(c+d*x)^3*(a+b*arcsin(c+d*x))/d+1/3*e^2*(c+d*x)^3*(a+b*arcsin(c+d*x))^3/d-14/9*b^3*e^2*sqrt(1-(c+d*x)^2)/d+2/3*b*e^2*(a+b*arcsin(c+d*x))^2*sqrt(1-(c+d*x)^2)/d+1/3*b*e^2*(c+d*x)^2*(a+b*arcsin(c+d*x))^2*sqrt(1-(c+d*x)^2)/d],
[(c*e+d*e*x)*(a+b*arcsin(c+d*x))^3,x,8,3/8*b^3*e*arcsin(c+d*x)/d-3/4*b^2*e*(c+d*x)^2*(a+b*arcsin(c+d*x))/d-1/4*e*(a+b*arcsin(c+d*x))^3/d+1/2*e*(c+d*x)^2*(a+b*arcsin(c+d*x))^3/d-3/8*b^3*e*(c+d*x)*sqrt(1-(c+d*x)^2)/d+3/4*b*e*(c+d*x)*(a+b*arcsin(c+d*x))^2*sqrt(1-(c+d*x)^2)/d],
[(a+b*arcsin(c+d*x))^3,x,6,-6*a*b^2*x-6*b^3*(c+d*x)*arcsin(c+d*x)/d+(c+d*x)*(a+b*arcsin(c+d*x))^3/d-6*b^3*sqrt(1-(c+d*x)^2)/d+3*b*(a+b*arcsin(c+d*x))^2*sqrt(1-(c+d*x)^2)/d],
[(a+b*arcsin(c+d*x))^3/(c*e+d*e*x),x,9,-1/4*I*(a+b*arcsin(c+d*x))^4/(b*d*e)+(a+b*arcsin(c+d*x))^3*log(1-exp(2*I*arcsin(c+d*x)))/(d*e)-3/2*I*b*(a+b*arcsin(c+d*x))^2*polylog(2,exp(2*I*arcsin(c+d*x)))/(d*e)+3/2*b^2*(a+b*arcsin(c+d*x))*polylog(3,exp(2*I*arcsin(c+d*x)))/(d*e)+3/4*I*b^3*polylog(4,exp(2*I*arcsin(c+d*x)))/(d*e)],
[(a+b*arcsin(c+d*x))^3/(c*e+d*e*x)^2,x,11,-(a+b*arcsin(c+d*x))^3/(d*e^2*(c+d*x))-6*b*(a+b*arcsin(c+d*x))^2*arctanh(exp(I*arcsin(c+d*x)))/(d*e^2)+6*I*b^2*(a+b*arcsin(c+d*x))*polylog(2,-exp(I*arcsin(c+d*x)))/(d*e^2)-6*I*b^2*(a+b*arcsin(c+d*x))*polylog(2,exp(I*arcsin(c+d*x)))/(d*e^2)-6*b^3*polylog(3,-exp(I*arcsin(c+d*x)))/(d*e^2)+6*b^3*polylog(3,exp(I*arcsin(c+d*x)))/(d*e^2)],
[(a+b*arcsin(c+d*x))^3/(c*e+d*e*x)^3,x,9,-3/2*I*b*(a+b*arcsin(c+d*x))^2/(d*e^3)-1/2*(a+b*arcsin(c+d*x))^3/(d*e^3*(c+d*x)^2)+3*b^2*(a+b*arcsin(c+d*x))*log(1-exp(2*I*arcsin(c+d*x)))/(d*e^3)-3/2*I*b^3*polylog(2,exp(2*I*arcsin(c+d*x)))/(d*e^3)-3/2*b*(a+b*arcsin(c+d*x))^2*sqrt(1-(c+d*x)^2)/(d*e^3*(c+d*x))],
[(a+b*arcsin(c+d*x))^3/(c*e+d*e*x)^4,x,16,-b^2*(a+b*arcsin(c+d*x))/(d*e^4*(c+d*x))-1/3*(a+b*arcsin(c+d*x))^3/(d*e^4*(c+d*x)^3)-b*(a+b*arcsin(c+d*x))^2*arctanh(exp(I*arcsin(c+d*x)))/(d*e^4)-b^3*arctanh(sqrt(1-(c+d*x)^2))/(d*e^4)+I*b^2*(a+b*arcsin(c+d*x))*polylog(2,-exp(I*arcsin(c+d*x)))/(d*e^4)-I*b^2*(a+b*arcsin(c+d*x))*polylog(2,exp(I*arcsin(c+d*x)))/(d*e^4)-b^3*polylog(3,-exp(I*arcsin(c+d*x)))/(d*e^4)+b^3*polylog(3,exp(I*arcsin(c+d*x)))/(d*e^4)-1/2*b*(a+b*arcsin(c+d*x))^2*sqrt(1-(c+d*x)^2)/(d*e^4*(c+d*x)^2)],
[(c*e+d*e*x)^3*(a+b*arcsin(c+d*x))^4,x,16,45/128*b^4*e^3*(c+d*x)^2/d+3/128*b^4*e^3*(c+d*x)^4/d+45/128*b^2*e^3*(a+b*arcsin(c+d*x))^2/d-9/16*b^2*e^3*(c+d*x)^2*(a+b*arcsin(c+d*x))^2/d-3/16*b^2*e^3*(c+d*x)^4*(a+b*arcsin(c+d*x))^2/d-3/32*e^3*(a+b*arcsin(c+d*x))^4/d+1/4*e^3*(c+d*x)^4*(a+b*arcsin(c+d*x))^4/d-45/64*b^3*e^3*(c+d*x)*(a+b*arcsin(c+d*x))*sqrt(1-(c+d*x)^2)/d-3/32*b^3*e^3*(c+d*x)^3*(a+b*arcsin(c+d*x))*sqrt(1-(c+d*x)^2)/d+3/8*b*e^3*(c+d*x)*(a+b*arcsin(c+d*x))^3*sqrt(1-(c+d*x)^2)/d+1/4*b*e^3*(c+d*x)^3*(a+b*arcsin(c+d*x))^3*sqrt(1-(c+d*x)^2)/d],
[(c*e+d*e*x)^2*(a+b*arcsin(c+d*x))^4,x,13,160/27*b^4*e^2*x+8/81*b^4*e^2*(c+d*x)^3/d-8/3*b^2*e^2*(c+d*x)*(a+b*arcsin(c+d*x))^2/d-4/9*b^2*e^2*(c+d*x)^3*(a+b*arcsin(c+d*x))^2/d+1/3*e^2*(c+d*x)^3*(a+b*arcsin(c+d*x))^4/d-160/27*b^3*e^2*(a+b*arcsin(c+d*x))*sqrt(1-(c+d*x)^2)/d-8/27*b^3*e^2*(c+d*x)^2*(a+b*arcsin(c+d*x))*sqrt(1-(c+d*x)^2)/d+8/9*b*e^2*(a+b*arcsin(c+d*x))^3*sqrt(1-(c+d*x)^2)/d+4/9*b*e^2*(c+d*x)^2*(a+b*arcsin(c+d*x))^3*sqrt(1-(c+d*x)^2)/d],
[(c*e+d*e*x)*(a+b*arcsin(c+d*x))^4,x,9,3/4*b^4*e*(c+d*x)^2/d+3/4*b^2*e*(a+b*arcsin(c+d*x))^2/d-3/2*b^2*e*(c+d*x)^2*(a+b*arcsin(c+d*x))^2/d-1/4*e*(a+b*arcsin(c+d*x))^4/d+1/2*e*(c+d*x)^2*(a+b*arcsin(c+d*x))^4/d-3/2*b^3*e*(c+d*x)*(a+b*arcsin(c+d*x))*sqrt(1-(c+d*x)^2)/d+b*e*(c+d*x)*(a+b*arcsin(c+d*x))^3*sqrt(1-(c+d*x)^2)/d],
[(a+b*arcsin(c+d*x))^4,x,6,24*b^4*x-12*b^2*(c+d*x)*(a+b*arcsin(c+d*x))^2/d+(c+d*x)*(a+b*arcsin(c+d*x))^4/d-24*b^3*(a+b*arcsin(c+d*x))*sqrt(1-(c+d*x)^2)/d+4*b*(a+b*arcsin(c+d*x))^3*sqrt(1-(c+d*x)^2)/d],
[(a+b*arcsin(c+d*x))^4/(c*e+d*e*x),x,10,-1/5*I*(a+b*arcsin(c+d*x))^5/(b*d*e)+(a+b*arcsin(c+d*x))^4*log(1-exp(2*I*arcsin(c+d*x)))/(d*e)-2*I*b*(a+b*arcsin(c+d*x))^3*polylog(2,exp(2*I*arcsin(c+d*x)))/(d*e)+3*b^2*(a+b*arcsin(c+d*x))^2*polylog(3,exp(2*I*arcsin(c+d*x)))/(d*e)+3*I*b^3*(a+b*arcsin(c+d*x))*polylog(4,exp(2*I*arcsin(c+d*x)))/(d*e)-3/2*b^4*polylog(5,exp(2*I*arcsin(c+d*x)))/(d*e)],
[(a+b*arcsin(c+d*x))^4/(c*e+d*e*x)^2,x,13,-(a+b*arcsin(c+d*x))^4/(d*e^2*(c+d*x))-8*b*(a+b*arcsin(c+d*x))^3*arctanh(exp(I*arcsin(c+d*x)))/(d*e^2)+12*I*b^2*(a+b*arcsin(c+d*x))^2*polylog(2,-exp(I*arcsin(c+d*x)))/(d*e^2)-12*I*b^2*(a+b*arcsin(c+d*x))^2*polylog(2,exp(I*arcsin(c+d*x)))/(d*e^2)-24*b^3*(a+b*arcsin(c+d*x))*polylog(3,-exp(I*arcsin(c+d*x)))/(d*e^2)+24*b^3*(a+b*arcsin(c+d*x))*polylog(3,exp(I*arcsin(c+d*x)))/(d*e^2)-24*I*b^4*polylog(4,-exp(I*arcsin(c+d*x)))/(d*e^2)+24*I*b^4*polylog(4,exp(I*arcsin(c+d*x)))/(d*e^2)],
[(a+b*arcsin(c+d*x))^4/(c*e+d*e*x)^3,x,10,-2*I*b*(a+b*arcsin(c+d*x))^3/(d*e^3)-1/2*(a+b*arcsin(c+d*x))^4/(d*e^3*(c+d*x)^2)+6*b^2*(a+b*arcsin(c+d*x))^2*log(1-exp(2*I*arcsin(c+d*x)))/(d*e^3)-6*I*b^3*(a+b*arcsin(c+d*x))*polylog(2,exp(2*I*arcsin(c+d*x)))/(d*e^3)+3*b^4*polylog(3,exp(2*I*arcsin(c+d*x)))/(d*e^3)-2*b*(a+b*arcsin(c+d*x))^3*sqrt(1-(c+d*x)^2)/(d*e^3*(c+d*x))],
[(a+b*arcsin(c+d*x))^4/(c*e+d*e*x)^4,x,21,-2*b^2*(a+b*arcsin(c+d*x))^2/(d*e^4*(c+d*x))-1/3*(a+b*arcsin(c+d*x))^4/(d*e^4*(c+d*x)^3)-8*b^3*(a+b*arcsin(c+d*x))*arctanh(exp(I*arcsin(c+d*x)))/(d*e^4)-4/3*b*(a+b*arcsin(c+d*x))^3*arctanh(exp(I*arcsin(c+d*x)))/(d*e^4)+4*I*b^4*polylog(2,-exp(I*arcsin(c+d*x)))/(d*e^4)+2*I*b^2*(a+b*arcsin(c+d*x))^2*polylog(2,-exp(I*arcsin(c+d*x)))/(d*e^4)-4*I*b^4*polylog(2,exp(I*arcsin(c+d*x)))/(d*e^4)-2*I*b^2*(a+b*arcsin(c+d*x))^2*polylog(2,exp(I*arcsin(c+d*x)))/(d*e^4)-4*b^3*(a+b*arcsin(c+d*x))*polylog(3,-exp(I*arcsin(c+d*x)))/(d*e^4)+4*b^3*(a+b*arcsin(c+d*x))*polylog(3,exp(I*arcsin(c+d*x)))/(d*e^4)-4*I*b^4*polylog(4,-exp(I*arcsin(c+d*x)))/(d*e^4)+4*I*b^4*polylog(4,exp(I*arcsin(c+d*x)))/(d*e^4)-2/3*b*(a+b*arcsin(c+d*x))^3*sqrt(1-(c+d*x)^2)/(d*e^4*(c+d*x)^2)],
[(a+b*arcsin(c+d*x))^5,x,8,120*a*b^4*x+120*b^5*(c+d*x)*arcsin(c+d*x)/d-20*b^2*(c+d*x)*(a+b*arcsin(c+d*x))^3/d+(c+d*x)*(a+b*arcsin(c+d*x))^5/d+120*b^5*sqrt(1-(c+d*x)^2)/d-60*b^3*(a+b*arcsin(c+d*x))^2*sqrt(1-(c+d*x)^2)/d+5*b*(a+b*arcsin(c+d*x))^4*sqrt(1-(c+d*x)^2)/d],

# n<0
[(c*e+d*e*x)^4/(a+b*arcsin(c+d*x)),x,14,1/8*e^4*Ci((a+b*arcsin(c+d*x))/b)*cos(a/b)/(b*d)-3/16*e^4*Ci(3*(a+b*arcsin(c+d*x))/b)*cos(3*a/b)/(b*d)+1/16*e^4*Ci(5*(a+b*arcsin(c+d*x))/b)*cos(5*a/b)/(b*d)+1/8*e^4*Si((a+b*arcsin(c+d*x))/b)*sin(a/b)/(b*d)-3/16*e^4*Si(3*(a+b*arcsin(c+d*x))/b)*sin(3*a/b)/(b*d)+1/16*e^4*Si(5*(a+b*arcsin(c+d*x))/b)*sin(5*a/b)/(b*d)],
[(c*e+d*e*x)^3/(a+b*arcsin(c+d*x)),x,11,1/4*e^3*cos(2*a/b)*Si(2*(a+b*arcsin(c+d*x))/b)/(b*d)-1/8*e^3*cos(4*a/b)*Si(4*(a+b*arcsin(c+d*x))/b)/(b*d)-1/4*e^3*Ci(2*(a+b*arcsin(c+d*x))/b)*sin(2*a/b)/(b*d)+1/8*e^3*Ci(4*(a+b*arcsin(c+d*x))/b)*sin(4*a/b)/(b*d)],
[(c*e+d*e*x)^2/(a+b*arcsin(c+d*x)),x,11,1/4*e^2*Ci((a+b*arcsin(c+d*x))/b)*cos(a/b)/(b*d)-1/4*e^2*Ci(3*(a+b*arcsin(c+d*x))/b)*cos(3*a/b)/(b*d)+1/4*e^2*Si((a+b*arcsin(c+d*x))/b)*sin(a/b)/(b*d)-1/4*e^2*Si(3*(a+b*arcsin(c+d*x))/b)*sin(3*a/b)/(b*d)],
[(c*e+d*e*x)/(a+b*arcsin(c+d*x)),x,8,1/2*e*cos(2*a/b)*Si(2*(a+b*arcsin(c+d*x))/b)/(b*d)-1/2*e*Ci(2*(a+b*arcsin(c+d*x))/b)*sin(2*a/b)/(b*d)],
[1/(a+b*arcsin(c+d*x)),x,5,Ci((a+b*arcsin(c+d*x))/b)*cos(a/b)/(b*d)+Si((a+b*arcsin(c+d*x))/b)*sin(a/b)/(b*d)],
[1/((c*e+d*e*x)*(a+b*arcsin(c+d*x))),x,2,Unintegrable(1/((c+d*x)*(a+b*arcsin(c+d*x))),x)/e],
[(c*e+d*e*x)^4/(a+b*arcsin(c+d*x))^2,x,13,-1/8*e^4*cos(a/b)*Si((a+b*arcsin(c+d*x))/b)/(b^2*d)+9/16*e^4*cos(3*a/b)*Si(3*(a+b*arcsin(c+d*x))/b)/(b^2*d)-5/16*e^4*cos(5*a/b)*Si(5*(a+b*arcsin(c+d*x))/b)/(b^2*d)+1/8*e^4*Ci((a+b*arcsin(c+d*x))/b)*sin(a/b)/(b^2*d)-9/16*e^4*Ci(3*(a+b*arcsin(c+d*x))/b)*sin(3*a/b)/(b^2*d)+5/16*e^4*Ci(5*(a+b*arcsin(c+d*x))/b)*sin(5*a/b)/(b^2*d)-e^4*(c+d*x)^4*sqrt(1-(c+d*x)^2)/(b*d*(a+b*arcsin(c+d*x)))],
[(c*e+d*e*x)^3/(a+b*arcsin(c+d*x))^2,x,10,1/2*e^3*Ci(2*(a+b*arcsin(c+d*x))/b)*cos(2*a/b)/(b^2*d)-1/2*e^3*Ci(4*(a+b*arcsin(c+d*x))/b)*cos(4*a/b)/(b^2*d)+1/2*e^3*Si(2*(a+b*arcsin(c+d*x))/b)*sin(2*a/b)/(b^2*d)-1/2*e^3*Si(4*(a+b*arcsin(c+d*x))/b)*sin(4*a/b)/(b^2*d)-e^3*(c+d*x)^3*sqrt(1-(c+d*x)^2)/(b*d*(a+b*arcsin(c+d*x)))],
[(c*e+d*e*x)^2/(a+b*arcsin(c+d*x))^2,x,10,-1/4*e^2*cos(a/b)*Si((a+b*arcsin(c+d*x))/b)/(b^2*d)+3/4*e^2*cos(3*a/b)*Si(3*(a+b*arcsin(c+d*x))/b)/(b^2*d)+1/4*e^2*Ci((a+b*arcsin(c+d*x))/b)*sin(a/b)/(b^2*d)-3/4*e^2*Ci(3*(a+b*arcsin(c+d*x))/b)*sin(3*a/b)/(b^2*d)-e^2*(c+d*x)^2*sqrt(1-(c+d*x)^2)/(b*d*(a+b*arcsin(c+d*x)))],
[(c*e+d*e*x)/(a+b*arcsin(c+d*x))^2,x,6,e*Ci(2*(a+b*arcsin(c+d*x))/b)*cos(2*a/b)/(b^2*d)+e*Si(2*(a+b*arcsin(c+d*x))/b)*sin(2*a/b)/(b^2*d)-e*(c+d*x)*sqrt(1-(c+d*x)^2)/(b*d*(a+b*arcsin(c+d*x)))],
[1/(a+b*arcsin(c+d*x))^2,x,6,-cos(a/b)*Si((a+b*arcsin(c+d*x))/b)/(b^2*d)+Ci((a+b*arcsin(c+d*x))/b)*sin(a/b)/(b^2*d)-sqrt(1-(c+d*x)^2)/(b*d*(a+b*arcsin(c+d*x)))],
[1/((c*e+d*e*x)*(a+b*arcsin(c+d*x))^2),x,2,Unintegrable(1/((c+d*x)*(a+b*arcsin(c+d*x))^2),x)/e],
[(c*e+d*e*x)^4/(a+b*arcsin(c+d*x))^3,x,26,-2*e^4*(c+d*x)^3/(b^2*d*(a+b*arcsin(c+d*x)))+5/2*e^4*(c+d*x)^5/(b^2*d*(a+b*arcsin(c+d*x)))-1/16*e^4*Ci((a+b*arcsin(c+d*x))/b)*cos(a/b)/(b^3*d)+27/32*e^4*Ci(3*(a+b*arcsin(c+d*x))/b)*cos(3*a/b)/(b^3*d)-25/32*e^4*Ci(5*(a+b*arcsin(c+d*x))/b)*cos(5*a/b)/(b^3*d)-1/16*e^4*Si((a+b*arcsin(c+d*x))/b)*sin(a/b)/(b^3*d)+27/32*e^4*Si(3*(a+b*arcsin(c+d*x))/b)*sin(3*a/b)/(b^3*d)-25/32*e^4*Si(5*(a+b*arcsin(c+d*x))/b)*sin(5*a/b)/(b^3*d)-1/2*e^4*(c+d*x)^4*sqrt(1-(c+d*x)^2)/(b*d*(a+b*arcsin(c+d*x))^2)],
[(c*e+d*e*x)^3/(a+b*arcsin(c+d*x))^3,x,20,-3/2*e^3*(c+d*x)^2/(b^2*d*(a+b*arcsin(c+d*x)))+2*e^3*(c+d*x)^4/(b^2*d*(a+b*arcsin(c+d*x)))-1/2*e^3*cos(2*a/b)*Si(2*(a+b*arcsin(c+d*x))/b)/(b^3*d)+e^3*cos(4*a/b)*Si(4*(a+b*arcsin(c+d*x))/b)/(b^3*d)+1/2*e^3*Ci(2*(a+b*arcsin(c+d*x))/b)*sin(2*a/b)/(b^3*d)-e^3*Ci(4*(a+b*arcsin(c+d*x))/b)*sin(4*a/b)/(b^3*d)-1/2*e^3*(c+d*x)^3*sqrt(1-(c+d*x)^2)/(b*d*(a+b*arcsin(c+d*x))^2)],
[(c*e+d*e*x)^2/(a+b*arcsin(c+d*x))^3,x,18,-e^2*(c+d*x)/(b^2*d*(a+b*arcsin(c+d*x)))+3/2*e^2*(c+d*x)^3/(b^2*d*(a+b*arcsin(c+d*x)))-1/8*e^2*Ci((a+b*arcsin(c+d*x))/b)*cos(a/b)/(b^3*d)+9/8*e^2*Ci(3*(a+b*arcsin(c+d*x))/b)*cos(3*a/b)/(b^3*d)-1/8*e^2*Si((a+b*arcsin(c+d*x))/b)*sin(a/b)/(b^3*d)+9/8*e^2*Si(3*(a+b*arcsin(c+d*x))/b)*sin(3*a/b)/(b^3*d)-1/2*e^2*(c+d*x)^2*sqrt(1-(c+d*x)^2)/(b*d*(a+b*arcsin(c+d*x))^2)],
[(c*e+d*e*x)/(a+b*arcsin(c+d*x))^3,x,11,-1/2*e/(b^2*d*(a+b*arcsin(c+d*x)))+e*(c+d*x)^2/(b^2*d*(a+b*arcsin(c+d*x)))-e*cos(2*a/b)*Si(2*(a+b*arcsin(c+d*x))/b)/(b^3*d)+e*Ci(2*(a+b*arcsin(c+d*x))/b)*sin(2*a/b)/(b^3*d)-1/2*e*(c+d*x)*sqrt(1-(c+d*x)^2)/(b*d*(a+b*arcsin(c+d*x))^2)],
[1/(a+b*arcsin(c+d*x))^3,x,7,1/2*(c+d*x)/(b^2*d*(a+b*arcsin(c+d*x)))-1/2*Ci((a+b*arcsin(c+d*x))/b)*cos(a/b)/(b^3*d)-1/2*Si((a+b*arcsin(c+d*x))/b)*sin(a/b)/(b^3*d)-1/2*sqrt(1-(c+d*x)^2)/(b*d*(a+b*arcsin(c+d*x))^2)],
[1/((c*e+d*e*x)*(a+b*arcsin(c+d*x))^3),x,2,Unintegrable(1/((c+d*x)*(a+b*arcsin(c+d*x))^3),x)/e],
[(c*e+d*e*x)^4/(a+b*arcsin(c+d*x))^4,x,24,-2/3*e^4*(c+d*x)^3/(b^2*d*(a+b*arcsin(c+d*x))^2)+5/6*e^4*(c+d*x)^5/(b^2*d*(a+b*arcsin(c+d*x))^2)+1/48*e^4*cos(a/b)*Si((a+b*arcsin(c+d*x))/b)/(b^4*d)-27/32*e^4*cos(3*a/b)*Si(3*(a+b*arcsin(c+d*x))/b)/(b^4*d)+125/96*e^4*cos(5*a/b)*Si(5*(a+b*arcsin(c+d*x))/b)/(b^4*d)-1/48*e^4*Ci((a+b*arcsin(c+d*x))/b)*sin(a/b)/(b^4*d)+27/32*e^4*Ci(3*(a+b*arcsin(c+d*x))/b)*sin(3*a/b)/(b^4*d)-125/96*e^4*Ci(5*(a+b*arcsin(c+d*x))/b)*sin(5*a/b)/(b^4*d)-1/3*e^4*(c+d*x)^4*sqrt(1-(c+d*x)^2)/(b*d*(a+b*arcsin(c+d*x))^3)-2*e^4*(c+d*x)^2*sqrt(1-(c+d*x)^2)/(b^3*d*(a+b*arcsin(c+d*x)))+25/6*e^4*(c+d*x)^4*sqrt(1-(c+d*x)^2)/(b^3*d*(a+b*arcsin(c+d*x)))],
[(c*e+d*e*x)^3/(a+b*arcsin(c+d*x))^4,x,17,-1/2*e^3*(c+d*x)^2/(b^2*d*(a+b*arcsin(c+d*x))^2)+2/3*e^3*(c+d*x)^4/(b^2*d*(a+b*arcsin(c+d*x))^2)-1/3*e^3*Ci(2*(a+b*arcsin(c+d*x))/b)*cos(2*a/b)/(b^4*d)+4/3*e^3*Ci(4*(a+b*arcsin(c+d*x))/b)*cos(4*a/b)/(b^4*d)-1/3*e^3*Si(2*(a+b*arcsin(c+d*x))/b)*sin(2*a/b)/(b^4*d)+4/3*e^3*Si(4*(a+b*arcsin(c+d*x))/b)*sin(4*a/b)/(b^4*d)-1/3*e^3*(c+d*x)^3*sqrt(1-(c+d*x)^2)/(b*d*(a+b*arcsin(c+d*x))^3)-e^3*(c+d*x)*sqrt(1-(c+d*x)^2)/(b^3*d*(a+b*arcsin(c+d*x)))+8/3*e^3*(c+d*x)^3*sqrt(1-(c+d*x)^2)/(b^3*d*(a+b*arcsin(c+d*x)))],
[(c*e+d*e*x)^2/(a+b*arcsin(c+d*x))^4,x,18,-1/3*e^2*(c+d*x)/(b^2*d*(a+b*arcsin(c+d*x))^2)+1/2*e^2*(c+d*x)^3/(b^2*d*(a+b*arcsin(c+d*x))^2)+1/24*e^2*cos(a/b)*Si((a+b*arcsin(c+d*x))/b)/(b^4*d)-9/8*e^2*cos(3*a/b)*Si(3*(a+b*arcsin(c+d*x))/b)/(b^4*d)-1/24*e^2*Ci((a+b*arcsin(c+d*x))/b)*sin(a/b)/(b^4*d)+9/8*e^2*Ci(3*(a+b*arcsin(c+d*x))/b)*sin(3*a/b)/(b^4*d)-1/3*e^2*(c+d*x)^2*sqrt(1-(c+d*x)^2)/(b*d*(a+b*arcsin(c+d*x))^3)-1/3*e^2*sqrt(1-(c+d*x)^2)/(b^3*d*(a+b*arcsin(c+d*x)))+3/2*e^2*(c+d*x)^2*sqrt(1-(c+d*x)^2)/(b^3*d*(a+b*arcsin(c+d*x)))],
[(c*e+d*e*x)/(a+b*arcsin(c+d*x))^4,x,9,-1/6*e/(b^2*d*(a+b*arcsin(c+d*x))^2)+1/3*e*(c+d*x)^2/(b^2*d*(a+b*arcsin(c+d*x))^2)-2/3*e*Ci(2*(a+b*arcsin(c+d*x))/b)*cos(2*a/b)/(b^4*d)-2/3*e*Si(2*(a+b*arcsin(c+d*x))/b)*sin(2*a/b)/(b^4*d)-1/3*e*(c+d*x)*sqrt(1-(c+d*x)^2)/(b*d*(a+b*arcsin(c+d*x))^3)+2/3*e*(c+d*x)*sqrt(1-(c+d*x)^2)/(b^3*d*(a+b*arcsin(c+d*x)))],
[1/(a+b*arcsin(c+d*x))^4,x,8,1/6*(c+d*x)/(b^2*d*(a+b*arcsin(c+d*x))^2)+1/6*cos(a/b)*Si((a+b*arcsin(c+d*x))/b)/(b^4*d)-1/6*Ci((a+b*arcsin(c+d*x))/b)*sin(a/b)/(b^4*d)-1/3*sqrt(1-(c+d*x)^2)/(b*d*(a+b*arcsin(c+d*x))^3)+1/6*sqrt(1-(c+d*x)^2)/(b^3*d*(a+b*arcsin(c+d*x)))],
[1/((c*e+d*e*x)*(a+b*arcsin(c+d*x))^4),x,2,Unintegrable(1/((c+d*x)*(a+b*arcsin(c+d*x))^4),x)/e],
[1/(a+b*arcsin(c+d*x))^5,x,9,1/12*(c+d*x)/(b^2*d*(a+b*arcsin(c+d*x))^3)+1/24*(-c-d*x)/(b^4*d*(a+b*arcsin(c+d*x)))+1/24*Ci((a+b*arcsin(c+d*x))/b)*cos(a/b)/(b^5*d)+1/24*Si((a+b*arcsin(c+d*x))/b)*sin(a/b)/(b^5*d)-1/4*sqrt(1-(c+d*x)^2)/(b*d*(a+b*arcsin(c+d*x))^4)+1/24*sqrt(1-(c+d*x)^2)/(b^3*d*(a+b*arcsin(c+d*x))^2)],

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

# n>0
[(c*e+d*e*x)^3*sqrt(a+b*arcsin(c+d*x)),x,16,-1/64*e^3*cos(4*a/b)*FresnelC(2*sqrt(2/Pi)*sqrt(a+b*arcsin(c+d*x))/sqrt(b))*sqrt(b)*sqrt(1/2*Pi)/d-1/64*e^3*FresnelS(2*sqrt(2/Pi)*sqrt(a+b*arcsin(c+d*x))/sqrt(b))*sin(4*a/b)*sqrt(b)*sqrt(1/2*Pi)/d+1/16*e^3*cos(2*a/b)*FresnelC(2*sqrt(a+b*arcsin(c+d*x))/(sqrt(b)*sqrt(Pi)))*sqrt(b)*sqrt(Pi)/d+1/16*e^3*FresnelS(2*sqrt(a+b*arcsin(c+d*x))/(sqrt(b)*sqrt(Pi)))*sin(2*a/b)*sqrt(b)*sqrt(Pi)/d-3/32*e^3*sqrt(a+b*arcsin(c+d*x))/d+1/4*e^3*(c+d*x)^4*sqrt(a+b*arcsin(c+d*x))/d],
[(c*e+d*e*x)^2*sqrt(a+b*arcsin(c+d*x)),x,16,1/12*e^2*cos(3*a/b)*FresnelS(sqrt(6/Pi)*sqrt(a+b*arcsin(c+d*x))/sqrt(b))*sqrt(b)*sqrt(1/6*Pi)/d-1/12*e^2*FresnelC(sqrt(6/Pi)*sqrt(a+b*arcsin(c+d*x))/sqrt(b))*sin(3*a/b)*sqrt(b)*sqrt(1/6*Pi)/d-1/4*e^2*cos(a/b)*FresnelS(sqrt(2/Pi)*sqrt(a+b*arcsin(c+d*x))/sqrt(b))*sqrt(b)*sqrt(1/2*Pi)/d+1/4*e^2*FresnelC(sqrt(2/Pi)*sqrt(a+b*arcsin(c+d*x))/sqrt(b))*sin(a/b)*sqrt(b)*sqrt(1/2*Pi)/d+1/3*e^2*(c+d*x)^3*sqrt(a+b*arcsin(c+d*x))/d],
[(c*e+d*e*x)*sqrt(a+b*arcsin(c+d*x)),x,11,1/8*e*cos(2*a/b)*FresnelC(2*sqrt(a+b*arcsin(c+d*x))/(sqrt(b)*sqrt(Pi)))*sqrt(b)*sqrt(Pi)/d+1/8*e*FresnelS(2*sqrt(a+b*arcsin(c+d*x))/(sqrt(b)*sqrt(Pi)))*sin(2*a/b)*sqrt(b)*sqrt(Pi)/d-1/4*e*sqrt(a+b*arcsin(c+d*x))/d+1/2*e*(c+d*x)^2*sqrt(a+b*arcsin(c+d*x))/d],
[sqrt(a+b*arcsin(c+d*x)),x,8,-cos(a/b)*FresnelS(sqrt(2/Pi)*sqrt(a+b*arcsin(c+d*x))/sqrt(b))*sqrt(b)*sqrt(1/2*Pi)/d+FresnelC(sqrt(2/Pi)*sqrt(a+b*arcsin(c+d*x))/sqrt(b))*sin(a/b)*sqrt(b)*sqrt(1/2*Pi)/d+(c+d*x)*sqrt(a+b*arcsin(c+d*x))/d],
[sqrt(a+b*arcsin(c+d*x))/(c*e+d*e*x),x,2,Unintegrable(sqrt(a+b*arcsin(c+d*x))/(c+d*x),x)/e],
[(c*e+d*e*x)^3*(a+b*arcsin(c+d*x))^(3/2),x,27,-3/32*e^3*(a+b*arcsin(c+d*x))^(3/2)/d+1/4*e^3*(c+d*x)^4*(a+b*arcsin(c+d*x))^(3/2)/d+3/512*b^(3/2)*e^3*cos(4*a/b)*FresnelS(2*sqrt(2/Pi)*sqrt(a+b*arcsin(c+d*x))/sqrt(b))*sqrt(1/2*Pi)/d-3/512*b^(3/2)*e^3*FresnelC(2*sqrt(2/Pi)*sqrt(a+b*arcsin(c+d*x))/sqrt(b))*sin(4*a/b)*sqrt(1/2*Pi)/d-3/64*b^(3/2)*e^3*cos(2*a/b)*FresnelS(2*sqrt(a+b*arcsin(c+d*x))/(sqrt(b)*sqrt(Pi)))*sqrt(Pi)/d+3/64*b^(3/2)*e^3*FresnelC(2*sqrt(a+b*arcsin(c+d*x))/(sqrt(b)*sqrt(Pi)))*sin(2*a/b)*sqrt(Pi)/d+9/64*b*e^3*(c+d*x)*sqrt(1-(c+d*x)^2)*sqrt(a+b*arcsin(c+d*x))/d+3/32*b*e^3*(c+d*x)^3*sqrt(1-(c+d*x)^2)*sqrt(a+b*arcsin(c+d*x))/d],
[(c*e+d*e*x)^2*(a+b*arcsin(c+d*x))^(3/2),x,24,1/3*e^2*(c+d*x)^3*(a+b*arcsin(c+d*x))^(3/2)/d+1/24*b^(3/2)*e^2*cos(3*a/b)*FresnelC(sqrt(6/Pi)*sqrt(a+b*arcsin(c+d*x))/sqrt(b))*sqrt(1/6*Pi)/d+1/24*b^(3/2)*e^2*FresnelS(sqrt(6/Pi)*sqrt(a+b*arcsin(c+d*x))/sqrt(b))*sin(3*a/b)*sqrt(1/6*Pi)/d-3/8*b^(3/2)*e^2*cos(a/b)*FresnelC(sqrt(2/Pi)*sqrt(a+b*arcsin(c+d*x))/sqrt(b))*sqrt(1/2*Pi)/d-3/8*b^(3/2)*e^2*FresnelS(sqrt(2/Pi)*sqrt(a+b*arcsin(c+d*x))/sqrt(b))*sin(a/b)*sqrt(1/2*Pi)/d+1/3*b*e^2*sqrt(1-(c+d*x)^2)*sqrt(a+b*arcsin(c+d*x))/d+1/6*b*e^2*(c+d*x)^2*sqrt(1-(c+d*x)^2)*sqrt(a+b*arcsin(c+d*x))/d],
[(c*e+d*e*x)*(a+b*arcsin(c+d*x))^(3/2),x,13,-1/4*e*(a+b*arcsin(c+d*x))^(3/2)/d+1/2*e*(c+d*x)^2*(a+b*arcsin(c+d*x))^(3/2)/d-3/32*b^(3/2)*e*cos(2*a/b)*FresnelS(2*sqrt(a+b*arcsin(c+d*x))/(sqrt(b)*sqrt(Pi)))*sqrt(Pi)/d+3/32*b^(3/2)*e*FresnelC(2*sqrt(a+b*arcsin(c+d*x))/(sqrt(b)*sqrt(Pi)))*sin(2*a/b)*sqrt(Pi)/d+3/8*b*e*(c+d*x)*sqrt(1-(c+d*x)^2)*sqrt(a+b*arcsin(c+d*x))/d],
[(a+b*arcsin(c+d*x))^(3/2),x,9,(c+d*x)*(a+b*arcsin(c+d*x))^(3/2)/d-3/2*b^(3/2)*cos(a/b)*FresnelC(sqrt(2/Pi)*sqrt(a+b*arcsin(c+d*x))/sqrt(b))*sqrt(1/2*Pi)/d-3/2*b^(3/2)*FresnelS(sqrt(2/Pi)*sqrt(a+b*arcsin(c+d*x))/sqrt(b))*sin(a/b)*sqrt(1/2*Pi)/d+3/2*b*sqrt(1-(c+d*x)^2)*sqrt(a+b*arcsin(c+d*x))/d],
[(a+b*arcsin(c+d*x))^(3/2)/(c*e+d*e*x),x,2,Unintegrable((a+b*arcsin(c+d*x))^(3/2)/(c+d*x),x)/e],
[(c*e+d*e*x)^3*(a+b*arcsin(c+d*x))^(5/2),x,29,-3/32*e^3*(a+b*arcsin(c+d*x))^(5/2)/d+1/4*e^3*(c+d*x)^4*(a+b*arcsin(c+d*x))^(5/2)/d+15/4096*b^(5/2)*e^3*cos(4*a/b)*FresnelC(2*sqrt(2/Pi)*sqrt(a+b*arcsin(c+d*x))/sqrt(b))*sqrt(1/2*Pi)/d+15/4096*b^(5/2)*e^3*FresnelS(2*sqrt(2/Pi)*sqrt(a+b*arcsin(c+d*x))/sqrt(b))*sin(4*a/b)*sqrt(1/2*Pi)/d-15/256*b^(5/2)*e^3*cos(2*a/b)*FresnelC(2*sqrt(a+b*arcsin(c+d*x))/(sqrt(b)*sqrt(Pi)))*sqrt(Pi)/d-15/256*b^(5/2)*e^3*FresnelS(2*sqrt(a+b*arcsin(c+d*x))/(sqrt(b)*sqrt(Pi)))*sin(2*a/b)*sqrt(Pi)/d+15/64*b*e^3*(c+d*x)*(a+b*arcsin(c+d*x))^(3/2)*sqrt(1-(c+d*x)^2)/d+5/32*b*e^3*(c+d*x)^3*(a+b*arcsin(c+d*x))^(3/2)*sqrt(1-(c+d*x)^2)/d+225/2048*b^2*e^3*sqrt(a+b*arcsin(c+d*x))/d-45/256*b^2*e^3*(c+d*x)^2*sqrt(a+b*arcsin(c+d*x))/d-15/256*b^2*e^3*(c+d*x)^4*sqrt(a+b*arcsin(c+d*x))/d],
[(c*e+d*e*x)^2*(a+b*arcsin(c+d*x))^(5/2),x,26,1/3*e^2*(c+d*x)^3*(a+b*arcsin(c+d*x))^(5/2)/d-5/144*b^(5/2)*e^2*cos(3*a/b)*FresnelS(sqrt(6/Pi)*sqrt(a+b*arcsin(c+d*x))/sqrt(b))*sqrt(1/6*Pi)/d+5/144*b^(5/2)*e^2*FresnelC(sqrt(6/Pi)*sqrt(a+b*arcsin(c+d*x))/sqrt(b))*sin(3*a/b)*sqrt(1/6*Pi)/d+15/16*b^(5/2)*e^2*cos(a/b)*FresnelS(sqrt(2/Pi)*sqrt(a+b*arcsin(c+d*x))/sqrt(b))*sqrt(1/2*Pi)/d-15/16*b^(5/2)*e^2*FresnelC(sqrt(2/Pi)*sqrt(a+b*arcsin(c+d*x))/sqrt(b))*sin(a/b)*sqrt(1/2*Pi)/d+5/9*b*e^2*(a+b*arcsin(c+d*x))^(3/2)*sqrt(1-(c+d*x)^2)/d+5/18*b*e^2*(c+d*x)^2*(a+b*arcsin(c+d*x))^(3/2)*sqrt(1-(c+d*x)^2)/d-5/6*b^2*e^2*(c+d*x)*sqrt(a+b*arcsin(c+d*x))/d-5/36*b^2*e^2*(c+d*x)^3*sqrt(a+b*arcsin(c+d*x))/d],
[(c*e+d*e*x)*(a+b*arcsin(c+d*x))^(5/2),x,14,-1/4*e*(a+b*arcsin(c+d*x))^(5/2)/d+1/2*e*(c+d*x)^2*(a+b*arcsin(c+d*x))^(5/2)/d-15/128*b^(5/2)*e*cos(2*a/b)*FresnelC(2*sqrt(a+b*arcsin(c+d*x))/(sqrt(b)*sqrt(Pi)))*sqrt(Pi)/d-15/128*b^(5/2)*e*FresnelS(2*sqrt(a+b*arcsin(c+d*x))/(sqrt(b)*sqrt(Pi)))*sin(2*a/b)*sqrt(Pi)/d+5/8*b*e*(c+d*x)*(a+b*arcsin(c+d*x))^(3/2)*sqrt(1-(c+d*x)^2)/d+15/64*b^2*e*sqrt(a+b*arcsin(c+d*x))/d-15/32*b^2*e*(c+d*x)^2*sqrt(a+b*arcsin(c+d*x))/d],
[(a+b*arcsin(c+d*x))^(5/2),x,10,(c+d*x)*(a+b*arcsin(c+d*x))^(5/2)/d+15/4*b^(5/2)*cos(a/b)*FresnelS(sqrt(2/Pi)*sqrt(a+b*arcsin(c+d*x))/sqrt(b))*sqrt(1/2*Pi)/d-15/4*b^(5/2)*FresnelC(sqrt(2/Pi)*sqrt(a+b*arcsin(c+d*x))/sqrt(b))*sin(a/b)*sqrt(1/2*Pi)/d+5/2*b*(a+b*arcsin(c+d*x))^(3/2)*sqrt(1-(c+d*x)^2)/d-15/4*b^2*(c+d*x)*sqrt(a+b*arcsin(c+d*x))/d],
[(a+b*arcsin(c+d*x))^(5/2)/(c*e+d*e*x),x,2,Unintegrable((a+b*arcsin(c+d*x))^(5/2)/(c+d*x),x)/e],
[(c*e+d*e*x)^2*(a+b*arcsin(c+d*x))^(7/2),x,35,-35/18*b^2*e^2*(c+d*x)*(a+b*arcsin(c+d*x))^(3/2)/d-35/108*b^2*e^2*(c+d*x)^3*(a+b*arcsin(c+d*x))^(3/2)/d+1/3*e^2*(c+d*x)^3*(a+b*arcsin(c+d*x))^(7/2)/d-35/864*b^(7/2)*e^2*cos(3*a/b)*FresnelC(sqrt(6/Pi)*sqrt(a+b*arcsin(c+d*x))/sqrt(b))*sqrt(1/6*Pi)/d-35/864*b^(7/2)*e^2*FresnelS(sqrt(6/Pi)*sqrt(a+b*arcsin(c+d*x))/sqrt(b))*sin(3*a/b)*sqrt(1/6*Pi)/d+105/32*b^(7/2)*e^2*cos(a/b)*FresnelC(sqrt(2/Pi)*sqrt(a+b*arcsin(c+d*x))/sqrt(b))*sqrt(1/2*Pi)/d+105/32*b^(7/2)*e^2*FresnelS(sqrt(2/Pi)*sqrt(a+b*arcsin(c+d*x))/sqrt(b))*sin(a/b)*sqrt(1/2*Pi)/d+7/9*b*e^2*(a+b*arcsin(c+d*x))^(5/2)*sqrt(1-(c+d*x)^2)/d+7/18*b*e^2*(c+d*x)^2*(a+b*arcsin(c+d*x))^(5/2)*sqrt(1-(c+d*x)^2)/d-175/54*b^3*e^2*sqrt(1-(c+d*x)^2)*sqrt(a+b*arcsin(c+d*x))/d-35/216*b^3*e^2*(c+d*x)^2*sqrt(1-(c+d*x)^2)*sqrt(a+b*arcsin(c+d*x))/d],
[(c*e+d*e*x)*(a+b*arcsin(c+d*x))^(7/2),x,16,35/64*b^2*e*(a+b*arcsin(c+d*x))^(3/2)/d-35/32*b^2*e*(c+d*x)^2*(a+b*arcsin(c+d*x))^(3/2)/d-1/4*e*(a+b*arcsin(c+d*x))^(7/2)/d+1/2*e*(c+d*x)^2*(a+b*arcsin(c+d*x))^(7/2)/d+105/512*b^(7/2)*e*cos(2*a/b)*FresnelS(2*sqrt(a+b*arcsin(c+d*x))/(sqrt(b)*sqrt(Pi)))*sqrt(Pi)/d-105/512*b^(7/2)*e*FresnelC(2*sqrt(a+b*arcsin(c+d*x))/(sqrt(b)*sqrt(Pi)))*sin(2*a/b)*sqrt(Pi)/d+7/8*b*e*(c+d*x)*(a+b*arcsin(c+d*x))^(5/2)*sqrt(1-(c+d*x)^2)/d-105/128*b^3*e*(c+d*x)*sqrt(1-(c+d*x)^2)*sqrt(a+b*arcsin(c+d*x))/d],
[(a+b*arcsin(c+d*x))^(7/2),x,11,-35/4*b^2*(c+d*x)*(a+b*arcsin(c+d*x))^(3/2)/d+(c+d*x)*(a+b*arcsin(c+d*x))^(7/2)/d+105/8*b^(7/2)*cos(a/b)*FresnelC(sqrt(2/Pi)*sqrt(a+b*arcsin(c+d*x))/sqrt(b))*sqrt(1/2*Pi)/d+105/8*b^(7/2)*FresnelS(sqrt(2/Pi)*sqrt(a+b*arcsin(c+d*x))/sqrt(b))*sin(a/b)*sqrt(1/2*Pi)/d+7/2*b*(a+b*arcsin(c+d*x))^(5/2)*sqrt(1-(c+d*x)^2)/d-105/8*b^3*sqrt(1-(c+d*x)^2)*sqrt(a+b*arcsin(c+d*x))/d],
[(a+b*arcsin(c+d*x))^(7/2)/(c*e+d*e*x),x,2,Unintegrable((a+b*arcsin(c+d*x))^(7/2)/(c+d*x),x)/e],

# n<0
[(c*e+d*e*x)^4/sqrt(a+b*arcsin(c+d*x)),x,20,1/8*e^4*cos(5*a/b)*FresnelC(sqrt(10/Pi)*sqrt(a+b*arcsin(c+d*x))/sqrt(b))*sqrt(1/10*Pi)/(d*sqrt(b))+1/8*e^4*FresnelS(sqrt(10/Pi)*sqrt(a+b*arcsin(c+d*x))/sqrt(b))*sin(5*a/b)*sqrt(1/10*Pi)/(d*sqrt(b))+1/4*e^4*cos(a/b)*FresnelC(sqrt(2/Pi)*sqrt(a+b*arcsin(c+d*x))/sqrt(b))*sqrt(1/2*Pi)/(d*sqrt(b))+1/4*e^4*FresnelS(sqrt(2/Pi)*sqrt(a+b*arcsin(c+d*x))/sqrt(b))*sin(a/b)*sqrt(1/2*Pi)/(d*sqrt(b))-1/8*e^4*cos(3*a/b)*FresnelC(sqrt(6/Pi)*sqrt(a+b*arcsin(c+d*x))/sqrt(b))*sqrt(3/2*Pi)/(d*sqrt(b))-1/8*e^4*FresnelS(sqrt(6/Pi)*sqrt(a+b*arcsin(c+d*x))/sqrt(b))*sin(3*a/b)*sqrt(3/2*Pi)/(d*sqrt(b))],
[(c*e+d*e*x)^3/sqrt(a+b*arcsin(c+d*x)),x,15,-1/8*e^3*cos(4*a/b)*FresnelS(2*sqrt(2/Pi)*sqrt(a+b*arcsin(c+d*x))/sqrt(b))*sqrt(1/2*Pi)/(d*sqrt(b))+1/8*e^3*FresnelC(2*sqrt(2/Pi)*sqrt(a+b*arcsin(c+d*x))/sqrt(b))*sin(4*a/b)*sqrt(1/2*Pi)/(d*sqrt(b))+1/4*e^3*cos(2*a/b)*FresnelS(2*sqrt(a+b*arcsin(c+d*x))/(sqrt(b)*sqrt(Pi)))*sqrt(Pi)/(d*sqrt(b))-1/4*e^3*FresnelC(2*sqrt(a+b*arcsin(c+d*x))/(sqrt(b)*sqrt(Pi)))*sin(2*a/b)*sqrt(Pi)/(d*sqrt(b))],
[(c*e+d*e*x)^2/sqrt(a+b*arcsin(c+d*x)),x,15,-1/2*e^2*cos(3*a/b)*FresnelC(sqrt(6/Pi)*sqrt(a+b*arcsin(c+d*x))/sqrt(b))*sqrt(1/6*Pi)/(d*sqrt(b))-1/2*e^2*FresnelS(sqrt(6/Pi)*sqrt(a+b*arcsin(c+d*x))/sqrt(b))*sin(3*a/b)*sqrt(1/6*Pi)/(d*sqrt(b))+1/2*e^2*cos(a/b)*FresnelC(sqrt(2/Pi)*sqrt(a+b*arcsin(c+d*x))/sqrt(b))*sqrt(1/2*Pi)/(d*sqrt(b))+1/2*e^2*FresnelS(sqrt(2/Pi)*sqrt(a+b*arcsin(c+d*x))/sqrt(b))*sin(a/b)*sqrt(1/2*Pi)/(d*sqrt(b))],
[(c*e+d*e*x)/sqrt(a+b*arcsin(c+d*x)),x,10,1/2*e*cos(2*a/b)*FresnelS(2*sqrt(a+b*arcsin(c+d*x))/(sqrt(b)*sqrt(Pi)))*sqrt(Pi)/(d*sqrt(b))-1/2*e*FresnelC(2*sqrt(a+b*arcsin(c+d*x))/(sqrt(b)*sqrt(Pi)))*sin(2*a/b)*sqrt(Pi)/(d*sqrt(b))],
[1/sqrt(a+b*arcsin(c+d*x)),x,7,cos(a/b)*FresnelC(sqrt(2/Pi)*sqrt(a+b*arcsin(c+d*x))/sqrt(b))*sqrt(2*Pi)/(d*sqrt(b))+FresnelS(sqrt(2/Pi)*sqrt(a+b*arcsin(c+d*x))/sqrt(b))*sin(a/b)*sqrt(2*Pi)/(d*sqrt(b))],
[1/((c*e+d*e*x)*sqrt(a+b*arcsin(c+d*x))),x,2,Unintegrable(1/((c+d*x)*sqrt(a+b*arcsin(c+d*x))),x)/e],
[(c*e+d*e*x)^4/(a+b*arcsin(c+d*x))^(3/2),x,19,-1/2*e^4*cos(a/b)*FresnelS(sqrt(2/Pi)*sqrt(a+b*arcsin(c+d*x))/sqrt(b))*sqrt(1/2*Pi)/(b^(3/2)*d)+1/2*e^4*FresnelC(sqrt(2/Pi)*sqrt(a+b*arcsin(c+d*x))/sqrt(b))*sin(a/b)*sqrt(1/2*Pi)/(b^(3/2)*d)+3/4*e^4*cos(3*a/b)*FresnelS(sqrt(6/Pi)*sqrt(a+b*arcsin(c+d*x))/sqrt(b))*sqrt(3/2*Pi)/(b^(3/2)*d)-3/4*e^4*FresnelC(sqrt(6/Pi)*sqrt(a+b*arcsin(c+d*x))/sqrt(b))*sin(3*a/b)*sqrt(3/2*Pi)/(b^(3/2)*d)-1/4*e^4*cos(5*a/b)*FresnelS(sqrt(10/Pi)*sqrt(a+b*arcsin(c+d*x))/sqrt(b))*sqrt(5/2*Pi)/(b^(3/2)*d)+1/4*e^4*FresnelC(sqrt(10/Pi)*sqrt(a+b*arcsin(c+d*x))/sqrt(b))*sin(5*a/b)*sqrt(5/2*Pi)/(b^(3/2)*d)-2*e^4*(c+d*x)^4*sqrt(1-(c+d*x)^2)/(b*d*sqrt(a+b*arcsin(c+d*x)))],
[(c*e+d*e*x)^3/(a+b*arcsin(c+d*x))^(3/2),x,14,-e^3*cos(4*a/b)*FresnelC(2*sqrt(2/Pi)*sqrt(a+b*arcsin(c+d*x))/sqrt(b))*sqrt(1/2*Pi)/(b^(3/2)*d)-e^3*FresnelS(2*sqrt(2/Pi)*sqrt(a+b*arcsin(c+d*x))/sqrt(b))*sin(4*a/b)*sqrt(1/2*Pi)/(b^(3/2)*d)+e^3*cos(2*a/b)*FresnelC(2*sqrt(a+b*arcsin(c+d*x))/(sqrt(b)*sqrt(Pi)))*sqrt(Pi)/(b^(3/2)*d)+e^3*FresnelS(2*sqrt(a+b*arcsin(c+d*x))/(sqrt(b)*sqrt(Pi)))*sin(2*a/b)*sqrt(Pi)/(b^(3/2)*d)-2*e^3*(c+d*x)^3*sqrt(1-(c+d*x)^2)/(b*d*sqrt(a+b*arcsin(c+d*x)))],
[(c*e+d*e*x)^2/(a+b*arcsin(c+d*x))^(3/2),x,14,-e^2*cos(a/b)*FresnelS(sqrt(2/Pi)*sqrt(a+b*arcsin(c+d*x))/sqrt(b))*sqrt(1/2*Pi)/(b^(3/2)*d)+e^2*FresnelC(sqrt(2/Pi)*sqrt(a+b*arcsin(c+d*x))/sqrt(b))*sin(a/b)*sqrt(1/2*Pi)/(b^(3/2)*d)+e^2*cos(3*a/b)*FresnelS(sqrt(6/Pi)*sqrt(a+b*arcsin(c+d*x))/sqrt(b))*sqrt(3/2*Pi)/(b^(3/2)*d)-e^2*FresnelC(sqrt(6/Pi)*sqrt(a+b*arcsin(c+d*x))/sqrt(b))*sin(3*a/b)*sqrt(3/2*Pi)/(b^(3/2)*d)-2*e^2*(c+d*x)^2*sqrt(1-(c+d*x)^2)/(b*d*sqrt(a+b*arcsin(c+d*x)))],
[(c*e+d*e*x)/(a+b*arcsin(c+d*x))^(3/2),x,8,2*e*cos(2*a/b)*FresnelC(2*sqrt(a+b*arcsin(c+d*x))/(sqrt(b)*sqrt(Pi)))*sqrt(Pi)/(b^(3/2)*d)+2*e*FresnelS(2*sqrt(a+b*arcsin(c+d*x))/(sqrt(b)*sqrt(Pi)))*sin(2*a/b)*sqrt(Pi)/(b^(3/2)*d)-2*e*(c+d*x)*sqrt(1-(c+d*x)^2)/(b*d*sqrt(a+b*arcsin(c+d*x)))],
[1/(a+b*arcsin(c+d*x))^(3/2),x,8,-2*cos(a/b)*FresnelS(sqrt(2/Pi)*sqrt(a+b*arcsin(c+d*x))/sqrt(b))*sqrt(2*Pi)/(b^(3/2)*d)+2*FresnelC(sqrt(2/Pi)*sqrt(a+b*arcsin(c+d*x))/sqrt(b))*sin(a/b)*sqrt(2*Pi)/(b^(3/2)*d)-2*sqrt(1-(c+d*x)^2)/(b*d*sqrt(a+b*arcsin(c+d*x)))],
[1/((c*e+d*e*x)*(a+b*arcsin(c+d*x))^(3/2)),x,2,Unintegrable(1/((c+d*x)*(a+b*arcsin(c+d*x))^(3/2)),x)/e],
[(c*e+d*e*x)^3/(a+b*arcsin(c+d*x))^(5/2),x,26,-4/3*e^3*cos(2*a/b)*FresnelS(2*sqrt(a+b*arcsin(c+d*x))/(sqrt(b)*sqrt(Pi)))*sqrt(Pi)/(b^(5/2)*d)+4/3*e^3*FresnelC(2*sqrt(a+b*arcsin(c+d*x))/(sqrt(b)*sqrt(Pi)))*sin(2*a/b)*sqrt(Pi)/(b^(5/2)*d)+4/3*e^3*cos(4*a/b)*FresnelS(2*sqrt(2/Pi)*sqrt(a+b*arcsin(c+d*x))/sqrt(b))*sqrt(2*Pi)/(b^(5/2)*d)-4/3*e^3*FresnelC(2*sqrt(2/Pi)*sqrt(a+b*arcsin(c+d*x))/sqrt(b))*sin(4*a/b)*sqrt(2*Pi)/(b^(5/2)*d)-2/3*e^3*(c+d*x)^3*sqrt(1-(c+d*x)^2)/(b*d*(a+b*arcsin(c+d*x))^(3/2))-4*e^3*(c+d*x)^2/(b^2*d*sqrt(a+b*arcsin(c+d*x)))+16/3*e^3*(c+d*x)^4/(b^2*d*sqrt(a+b*arcsin(c+d*x)))],
[(c*e+d*e*x)^2/(a+b*arcsin(c+d*x))^(5/2),x,24,-1/3*e^2*cos(a/b)*FresnelC(sqrt(2/Pi)*sqrt(a+b*arcsin(c+d*x))/sqrt(b))*sqrt(2*Pi)/(b^(5/2)*d)-1/3*e^2*FresnelS(sqrt(2/Pi)*sqrt(a+b*arcsin(c+d*x))/sqrt(b))*sin(a/b)*sqrt(2*Pi)/(b^(5/2)*d)+e^2*cos(3*a/b)*FresnelC(sqrt(6/Pi)*sqrt(a+b*arcsin(c+d*x))/sqrt(b))*sqrt(6*Pi)/(b^(5/2)*d)+e^2*FresnelS(sqrt(6/Pi)*sqrt(a+b*arcsin(c+d*x))/sqrt(b))*sin(3*a/b)*sqrt(6*Pi)/(b^(5/2)*d)-2/3*e^2*(c+d*x)^2*sqrt(1-(c+d*x)^2)/(b*d*(a+b*arcsin(c+d*x))^(3/2))-8/3*e^2*(c+d*x)/(b^2*d*sqrt(a+b*arcsin(c+d*x)))+4*e^2*(c+d*x)^3/(b^2*d*sqrt(a+b*arcsin(c+d*x)))],
[(c*e+d*e*x)/(a+b*arcsin(c+d*x))^(5/2),x,13,-8/3*e*cos(2*a/b)*FresnelS(2*sqrt(a+b*arcsin(c+d*x))/(sqrt(b)*sqrt(Pi)))*sqrt(Pi)/(b^(5/2)*d)+8/3*e*FresnelC(2*sqrt(a+b*arcsin(c+d*x))/(sqrt(b)*sqrt(Pi)))*sin(2*a/b)*sqrt(Pi)/(b^(5/2)*d)-2/3*e*(c+d*x)*sqrt(1-(c+d*x)^2)/(b*d*(a+b*arcsin(c+d*x))^(3/2))-4/3*e/(b^2*d*sqrt(a+b*arcsin(c+d*x)))+8/3*e*(c+d*x)^2/(b^2*d*sqrt(a+b*arcsin(c+d*x)))],
[1/(a+b*arcsin(c+d*x))^(5/2),x,9,-4/3*cos(a/b)*FresnelC(sqrt(2/Pi)*sqrt(a+b*arcsin(c+d*x))/sqrt(b))*sqrt(2*Pi)/(b^(5/2)*d)-4/3*FresnelS(sqrt(2/Pi)*sqrt(a+b*arcsin(c+d*x))/sqrt(b))*sin(a/b)*sqrt(2*Pi)/(b^(5/2)*d)-2/3*sqrt(1-(c+d*x)^2)/(b*d*(a+b*arcsin(c+d*x))^(3/2))+4/3*(c+d*x)/(b^2*d*sqrt(a+b*arcsin(c+d*x)))],
[1/((c*e+d*e*x)*(a+b*arcsin(c+d*x))^(5/2)),x,2,Unintegrable(1/((c+d*x)*(a+b*arcsin(c+d*x))^(5/2)),x)/e],
[(c*e+d*e*x)^3/(a+b*arcsin(c+d*x))^(7/2),x,23,-4/5*e^3*(c+d*x)^2/(b^2*d*(a+b*arcsin(c+d*x))^(3/2))+16/15*e^3*(c+d*x)^4/(b^2*d*(a+b*arcsin(c+d*x))^(3/2))-16/15*e^3*cos(2*a/b)*FresnelC(2*sqrt(a+b*arcsin(c+d*x))/(sqrt(b)*sqrt(Pi)))*sqrt(Pi)/(b^(7/2)*d)-16/15*e^3*FresnelS(2*sqrt(a+b*arcsin(c+d*x))/(sqrt(b)*sqrt(Pi)))*sin(2*a/b)*sqrt(Pi)/(b^(7/2)*d)+32/15*e^3*cos(4*a/b)*FresnelC(2*sqrt(2/Pi)*sqrt(a+b*arcsin(c+d*x))/sqrt(b))*sqrt(2*Pi)/(b^(7/2)*d)+32/15*e^3*FresnelS(2*sqrt(2/Pi)*sqrt(a+b*arcsin(c+d*x))/sqrt(b))*sin(4*a/b)*sqrt(2*Pi)/(b^(7/2)*d)-2/5*e^3*(c+d*x)^3*sqrt(1-(c+d*x)^2)/(b*d*(a+b*arcsin(c+d*x))^(5/2))-16/5*e^3*(c+d*x)*sqrt(1-(c+d*x)^2)/(b^3*d*sqrt(a+b*arcsin(c+d*x)))+128/15*e^3*(c+d*x)^3*sqrt(1-(c+d*x)^2)/(b^3*d*sqrt(a+b*arcsin(c+d*x)))],
[(c*e+d*e*x)^2/(a+b*arcsin(c+d*x))^(7/2),x,24,-8/15*e^2*(c+d*x)/(b^2*d*(a+b*arcsin(c+d*x))^(3/2))+4/5*e^2*(c+d*x)^3/(b^2*d*(a+b*arcsin(c+d*x))^(3/2))+2/15*e^2*cos(a/b)*FresnelS(sqrt(2/Pi)*sqrt(a+b*arcsin(c+d*x))/sqrt(b))*sqrt(2*Pi)/(b^(7/2)*d)-2/15*e^2*FresnelC(sqrt(2/Pi)*sqrt(a+b*arcsin(c+d*x))/sqrt(b))*sin(a/b)*sqrt(2*Pi)/(b^(7/2)*d)-6/5*e^2*cos(3*a/b)*FresnelS(sqrt(6/Pi)*sqrt(a+b*arcsin(c+d*x))/sqrt(b))*sqrt(6*Pi)/(b^(7/2)*d)+6/5*e^2*FresnelC(sqrt(6/Pi)*sqrt(a+b*arcsin(c+d*x))/sqrt(b))*sin(3*a/b)*sqrt(6*Pi)/(b^(7/2)*d)-2/5*e^2*(c+d*x)^2*sqrt(1-(c+d*x)^2)/(b*d*(a+b*arcsin(c+d*x))^(5/2))-16/15*e^2*sqrt(1-(c+d*x)^2)/(b^3*d*sqrt(a+b*arcsin(c+d*x)))+24/5*e^2*(c+d*x)^2*sqrt(1-(c+d*x)^2)/(b^3*d*sqrt(a+b*arcsin(c+d*x)))],
[(c*e+d*e*x)/(a+b*arcsin(c+d*x))^(7/2),x,11,-4/15*e/(b^2*d*(a+b*arcsin(c+d*x))^(3/2))+8/15*e*(c+d*x)^2/(b^2*d*(a+b*arcsin(c+d*x))^(3/2))-32/15*e*cos(2*a/b)*FresnelC(2*sqrt(a+b*arcsin(c+d*x))/(sqrt(b)*sqrt(Pi)))*sqrt(Pi)/(b^(7/2)*d)-32/15*e*FresnelS(2*sqrt(a+b*arcsin(c+d*x))/(sqrt(b)*sqrt(Pi)))*sin(2*a/b)*sqrt(Pi)/(b^(7/2)*d)-2/5*e*(c+d*x)*sqrt(1-(c+d*x)^2)/(b*d*(a+b*arcsin(c+d*x))^(5/2))+32/15*e*(c+d*x)*sqrt(1-(c+d*x)^2)/(b^3*d*sqrt(a+b*arcsin(c+d*x)))],
[1/(a+b*arcsin(c+d*x))^(7/2),x,10,4/15*(c+d*x)/(b^2*d*(a+b*arcsin(c+d*x))^(3/2))+8/15*cos(a/b)*FresnelS(sqrt(2/Pi)*sqrt(a+b*arcsin(c+d*x))/sqrt(b))*sqrt(2*Pi)/(b^(7/2)*d)-8/15*FresnelC(sqrt(2/Pi)*sqrt(a+b*arcsin(c+d*x))/sqrt(b))*sin(a/b)*sqrt(2*Pi)/(b^(7/2)*d)-2/5*sqrt(1-(c+d*x)^2)/(b*d*(a+b*arcsin(c+d*x))^(5/2))+8/15*sqrt(1-(c+d*x)^2)/(b^3*d*sqrt(a+b*arcsin(c+d*x)))],
[1/((c*e+d*e*x)*(a+b*arcsin(c+d*x))^(7/2)),x,2,Unintegrable(1/((c+d*x)*(a+b*arcsin(c+d*x))^(7/2)),x)/e],

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

# n>0
[(c*e+d*e*x)^(7/2)*(a+b*arcsin(c+d*x)),x,7,2/9*(e*(c+d*x))^(9/2)*(a+b*arcsin(c+d*x))/(d*e)+28/135*b*e^3*EllipticE(sqrt(1-c-d*x)/sqrt(2),sqrt(2))*sqrt(e*(c+d*x))/(d*sqrt(c+d*x))+28/405*b*e^2*(e*(c+d*x))^(3/2)*sqrt(1-(c+d*x)^2)/d+4/81*b*(e*(c+d*x))^(7/2)*sqrt(1-(c+d*x)^2)/d],
[(c*e+d*e*x)^(5/2)*(a+b*arcsin(c+d*x)),x,6,2/7*(e*(c+d*x))^(7/2)*(a+b*arcsin(c+d*x))/(d*e)-20/147*b*e^(5/2)*EllipticF(sqrt(e*(c+d*x))/sqrt(e),I)/d+4/49*b*(e*(c+d*x))^(5/2)*sqrt(1-(c+d*x)^2)/d+20/147*b*e^2*sqrt(e*(c+d*x))*sqrt(1-(c+d*x)^2)/d],
[(c*e+d*e*x)^(3/2)*(a+b*arcsin(c+d*x)),x,6,2/5*(e*(c+d*x))^(5/2)*(a+b*arcsin(c+d*x))/(d*e)+12/25*b*e*EllipticE(sqrt(1-c-d*x)/sqrt(2),sqrt(2))*sqrt(e*(c+d*x))/(d*sqrt(c+d*x))+4/25*b*(e*(c+d*x))^(3/2)*sqrt(1-(c+d*x)^2)/d],
[(c*e+d*e*x)^(1/2)*(a+b*arcsin(c+d*x)),x,5,2/3*(e*(c+d*x))^(3/2)*(a+b*arcsin(c+d*x))/(d*e)-4/9*b*EllipticF(sqrt(e*(c+d*x))/sqrt(e),I)*sqrt(e)/d+4/9*b*sqrt(e*(c+d*x))*sqrt(1-(c+d*x)^2)/d],
[(a+b*arcsin(c+d*x))/(c*e+d*e*x)^(1/2),x,5,2*(a+b*arcsin(c+d*x))*sqrt(e*(c+d*x))/(d*e)+4*b*EllipticE(sqrt(1-c-d*x)/sqrt(2),sqrt(2))*sqrt(e*(c+d*x))/(d*e*sqrt(c+d*x))],
[(a+b*arcsin(c+d*x))/(c*e+d*e*x)^(3/2),x,4,4*b*EllipticF(sqrt(e*(c+d*x))/sqrt(e),I)/(d*e^(3/2))-2*(a+b*arcsin(c+d*x))/(d*e*sqrt(e*(c+d*x)))],
[(a+b*arcsin(c+d*x))/(c*e+d*e*x)^(5/2),x,6,-2/3*(a+b*arcsin(c+d*x))/(d*e*(e*(c+d*x))^(3/2))+4/3*b*EllipticE(sqrt(1-c-d*x)/sqrt(2),sqrt(2))*sqrt(e*(c+d*x))/(d*e^3*sqrt(c+d*x))-4/3*b*sqrt(1-(c+d*x)^2)/(d*e^2*sqrt(e*(c+d*x)))],
[(a+b*arcsin(c+d*x))/(c*e+d*e*x)^(7/2),x,5,-2/5*(a+b*arcsin(c+d*x))/(d*e*(e*(c+d*x))^(5/2))+4/15*b*EllipticF(sqrt(e*(c+d*x))/sqrt(e),I)/(d*e^(7/2))-4/15*b*sqrt(1-(c+d*x)^2)/(d*e^2*(e*(c+d*x))^(3/2))],
[(a+b*arcsin(c+d*x))/(c*e+d*e*x)^(9/2),x,7,-2/7*(a+b*arcsin(c+d*x))/(d*e*(e*(c+d*x))^(7/2))+12/35*b*EllipticE(sqrt(1-c-d*x)/sqrt(2),sqrt(2))*sqrt(e*(c+d*x))/(d*e^5*sqrt(c+d*x))-4/35*b*sqrt(1-(c+d*x)^2)/(d*e^2*(e*(c+d*x))^(5/2))-12/35*b*sqrt(1-(c+d*x)^2)/(d*e^4*sqrt(e*(c+d*x)))],
[(a+b*arcsin(c+d*x))/(c*e+d*e*x)^(11/2),x,6,-2/9*(a+b*arcsin(c+d*x))/(d*e*(e*(c+d*x))^(9/2))+20/189*b*EllipticF(sqrt(e*(c+d*x))/sqrt(e),I)/(d*e^(11/2))-4/63*b*sqrt(1-(c+d*x)^2)/(d*e^2*(e*(c+d*x))^(7/2))-20/189*b*sqrt(1-(c+d*x)^2)/(d*e^4*(e*(c+d*x))^(3/2))],
[(c*e+d*e*x)^(7/2)*(a+b*arcsin(c+d*x))^2,x,3,2/9*(e*(c+d*x))^(9/2)*(a+b*arcsin(c+d*x))^2/(d*e)-8/99*b*(e*(c+d*x))^(11/2)*(a+b*arcsin(c+d*x))*hypergeom([1/2,11/4],[15/4],(c+d*x)^2)/(d*e^2)+16/1287*b^2*(e*(c+d*x))^(13/2)*HypergeometricPFQ([1,13/4,13/4],[15/4,17/4],(c+d*x)^2)/(d*e^3)],
[(c*e+d*e*x)^(5/2)*(a+b*arcsin(c+d*x))^2,x,3,2/7*(e*(c+d*x))^(7/2)*(a+b*arcsin(c+d*x))^2/(d*e)-8/63*b*(e*(c+d*x))^(9/2)*(a+b*arcsin(c+d*x))*hypergeom([1/2,9/4],[13/4],(c+d*x)^2)/(d*e^2)+16/693*b^2*(e*(c+d*x))^(11/2)*HypergeometricPFQ([1,11/4,11/4],[13/4,15/4],(c+d*x)^2)/(d*e^3)],
[(c*e+d*e*x)^(3/2)*(a+b*arcsin(c+d*x))^2,x,3,2/5*(e*(c+d*x))^(5/2)*(a+b*arcsin(c+d*x))^2/(d*e)-8/35*b*(e*(c+d*x))^(7/2)*(a+b*arcsin(c+d*x))*hypergeom([1/2,7/4],[11/4],(c+d*x)^2)/(d*e^2)+16/315*b^2*(e*(c+d*x))^(9/2)*HypergeometricPFQ([1,9/4,9/4],[11/4,13/4],(c+d*x)^2)/(d*e^3)],
[(c*e+d*e*x)^(1/2)*(a+b*arcsin(c+d*x))^2,x,3,2/3*(e*(c+d*x))^(3/2)*(a+b*arcsin(c+d*x))^2/(d*e)-8/15*b*(e*(c+d*x))^(5/2)*(a+b*arcsin(c+d*x))*hypergeom([1/2,5/4],[9/4],(c+d*x)^2)/(d*e^2)+16/105*b^2*(e*(c+d*x))^(7/2)*HypergeometricPFQ([1,7/4,7/4],[9/4,11/4],(c+d*x)^2)/(d*e^3)],
[(a+b*arcsin(c+d*x))^2/(c*e+d*e*x)^(1/2),x,3,-8/3*b*(e*(c+d*x))^(3/2)*(a+b*arcsin(c+d*x))*hypergeom([1/2,3/4],[7/4],(c+d*x)^2)/(d*e^2)+16/15*b^2*(e*(c+d*x))^(5/2)*HypergeometricPFQ([1,5/4,5/4],[7/4,9/4],(c+d*x)^2)/(d*e^3)+2*(a+b*arcsin(c+d*x))^2*sqrt(e*(c+d*x))/(d*e)],
[(a+b*arcsin(c+d*x))^2/(c*e+d*e*x)^(3/2),x,3,-16/3*b^2*(e*(c+d*x))^(3/2)*HypergeometricPFQ([3/4,3/4,1],[5/4,7/4],(c+d*x)^2)/(d*e^3)-2*(a+b*arcsin(c+d*x))^2/(d*e*sqrt(e*(c+d*x)))+8*b*(a+b*arcsin(c+d*x))*hypergeom([1/4,1/2],[5/4],(c+d*x)^2)*sqrt(e*(c+d*x))/(d*e^2)],
[(a+b*arcsin(c+d*x))^2/(c*e+d*e*x)^(5/2),x,3,-2/3*(a+b*arcsin(c+d*x))^2/(d*e*(e*(c+d*x))^(3/2))-8/3*b*(a+b*arcsin(c+d*x))*hypergeom([-1/4,1/2],[3/4],(c+d*x)^2)/(d*e^2*sqrt(e*(c+d*x)))+16/3*b^2*HypergeometricPFQ([1/4,1/4,1],[3/4,5/4],(c+d*x)^2)*sqrt(e*(c+d*x))/(d*e^3)],
[(a+b*arcsin(c+d*x))^2/(c*e+d*e*x)^(7/2),x,3,-2/5*(a+b*arcsin(c+d*x))^2/(d*e*(e*(c+d*x))^(5/2))-8/15*b*(a+b*arcsin(c+d*x))*hypergeom([-3/4,1/2],[1/4],(c+d*x)^2)/(d*e^2*(e*(c+d*x))^(3/2))-16/15*b^2*HypergeometricPFQ([-1/4,-1/4,1],[1/4,3/4],(c+d*x)^2)/(d*e^3*sqrt(e*(c+d*x)))],
[(a+b*arcsin(c+d*x))^2/(c*e+d*e*x)^(9/2),x,3,-2/7*(a+b*arcsin(c+d*x))^2/(d*e*(e*(c+d*x))^(7/2))-8/35*b*(a+b*arcsin(c+d*x))*hypergeom([-5/4,1/2],[-1/4],(c+d*x)^2)/(d*e^2*(e*(c+d*x))^(5/2))-16/105*b^2*HypergeometricPFQ([-3/4,-3/4,1],[-1/4,1/4],(c+d*x)^2)/(d*e^3*(e*(c+d*x))^(3/2))],
[(c*e+d*e*x)^(1/2)*(a+b*arcsin(c+d*x))^3,x,2,2/3*(e*(c+d*x))^(3/2)*(a+b*arcsin(c+d*x))^3/(d*e)-2*b*Unintegrable((e*(c+d*x))^(3/2)*(a+b*arcsin(c+d*x))^2/sqrt(1-(c+d*x)^2),x)/e],
[(a+b*arcsin(c+d*x))^3/(c*e+d*e*x)^(1/2),x,2,2*(a+b*arcsin(c+d*x))^3*sqrt(e*(c+d*x))/(d*e)-6*b*Unintegrable((a+b*arcsin(c+d*x))^2*sqrt(e*(c+d*x))/sqrt(1-(c+d*x)^2),x)/e],
[(a+b*arcsin(c+d*x))^3/(c*e+d*e*x)^(3/2),x,2,-2*(a+b*arcsin(c+d*x))^3/(d*e*sqrt(e*(c+d*x)))+6*b*Unintegrable((a+b*arcsin(c+d*x))^2/(sqrt(e*(c+d*x))*sqrt(1-(c+d*x)^2)),x)/e],
[(a+b*arcsin(c+d*x))^3/(c*e+d*e*x)^(5/2),x,2,-2/3*(a+b*arcsin(c+d*x))^3/(d*e*(e*(c+d*x))^(3/2))+2*b*Unintegrable((a+b*arcsin(c+d*x))^2/((e*(c+d*x))^(3/2)*sqrt(1-(c+d*x)^2)),x)/e],
[(c*e+d*e*x)^(1/2)*(a+b*arcsin(c+d*x))^4,x,2,2/3*(e*(c+d*x))^(3/2)*(a+b*arcsin(c+d*x))^4/(d*e)-8/3*b*Unintegrable((e*(c+d*x))^(3/2)*(a+b*arcsin(c+d*x))^3/sqrt(1-(c+d*x)^2),x)/e],
[(a+b*arcsin(c+d*x))^4/(c*e+d*e*x)^(1/2),x,2,2*(a+b*arcsin(c+d*x))^4*sqrt(e*(c+d*x))/(d*e)-8*b*Unintegrable((a+b*arcsin(c+d*x))^3*sqrt(e*(c+d*x))/sqrt(1-(c+d*x)^2),x)/e],
[(a+b*arcsin(c+d*x))^4/(c*e+d*e*x)^(3/2),x,2,-2*(a+b*arcsin(c+d*x))^4/(d*e*sqrt(e*(c+d*x)))+8*b*Unintegrable((a+b*arcsin(c+d*x))^3/(sqrt(e*(c+d*x))*sqrt(1-(c+d*x)^2)),x)/e],
[(a+b*arcsin(c+d*x))^4/(c*e+d*e*x)^(5/2),x,2,-2/3*(a+b*arcsin(c+d*x))^4/(d*e*(e*(c+d*x))^(3/2))+8/3*b*Unintegrable((a+b*arcsin(c+d*x))^3/((e*(c+d*x))^(3/2)*sqrt(1-(c+d*x)^2)),x)/e],

# n<0

# Integrands of the form (c e+d e x)^m (a+b ArcSin[c+d x])^n with m symbolic
[(c*e+d*e*x)^m*(a+b*arcsin(c+d*x))^4,x,2,(e*(c+d*x))^(1+m)*(a+b*arcsin(c+d*x))^4/(d*e*(1+m))-4*b*Unintegrable((e*(c+d*x))^(1+m)*(a+b*arcsin(c+d*x))^3/sqrt(1-(c+d*x)^2),x)/(e*(1+m))],
[(c*e+d*e*x)^m*(a+b*arcsin(c+d*x))^3,x,2,(e*(c+d*x))^(1+m)*(a+b*arcsin(c+d*x))^3/(d*e*(1+m))-3*b*Unintegrable((e*(c+d*x))^(1+m)*(a+b*arcsin(c+d*x))^2/sqrt(1-(c+d*x)^2),x)/(e*(1+m))],
[(c*e+d*e*x)^m*(a+b*arcsin(c+d*x))^2,x,3,(e*(c+d*x))^(1+m)*(a+b*arcsin(c+d*x))^2/(d*e*(1+m))-2*b*(e*(c+d*x))^(2+m)*(a+b*arcsin(c+d*x))*hypergeom([1/2,1/2*(2+m)],[1/2*(4+m)],(c+d*x)^2)/(d*e^2*(1+m)*(2+m))+2*b^2*(e*(c+d*x))^(3+m)*HypergeometricPFQ([1,3/2+1/2*m,3/2+1/2*m],[2+1/2*m,5/2+1/2*m],(c+d*x)^2)/(d*e^3*(3+m)*(2+3*m+m^2))],
[(c*e+d*e*x)^m*(a+b*arcsin(c+d*x)),x,3,(e*(c+d*x))^(1+m)*(a+b*arcsin(c+d*x))/(d*e*(1+m))-b*(e*(c+d*x))^(2+m)*hypergeom([1/2,1/2*(2+m)],[1/2*(4+m)],(c+d*x)^2)/(d*e^2*(1+m)*(2+m))],
[(c*e+d*e*x)^m/(a+b*arcsin(c+d*x)),x,1,Unintegrable((e*(c+d*x))^m/(a+b*arcsin(c+d*x)),x)],

# Integrands of the form (1-(a+b x)^2)^(m/2) ArcSin[a+b x]^n

# m>0
[arcsin(a+b*x)^3*sqrt(1-a^2-2*a*b*x-b^2*x^2),x,7,3/8*(a+b*x)^2/b+3/8*arcsin(a+b*x)^2/b-3/4*(a+b*x)^2*arcsin(a+b*x)^2/b+1/8*arcsin(a+b*x)^4/b-3/4*(a+b*x)*arcsin(a+b*x)*sqrt(1-(a+b*x)^2)/b+1/2*(a+b*x)*arcsin(a+b*x)^3*sqrt(1-(a+b*x)^2)/b],
[arcsin(a+b*x)^2*sqrt(1-a^2-2*a*b*x-b^2*x^2),x,6,1/4*arcsin(a+b*x)/b-1/2*(a+b*x)^2*arcsin(a+b*x)/b+1/6*arcsin(a+b*x)^3/b-1/4*(a+b*x)*sqrt(1-(a+b*x)^2)/b+1/2*(a+b*x)*arcsin(a+b*x)^2*sqrt(1-(a+b*x)^2)/b],
[arcsin(a+b*x)*sqrt(1-a^2-2*a*b*x-b^2*x^2),x,4,-1/4*(a+b*x)^2/b+1/4*arcsin(a+b*x)^2/b+1/2*(a+b*x)*arcsin(a+b*x)*sqrt(1-(a+b*x)^2)/b],
[sqrt(1-a^2-2*a*b*x-b^2*x^2)/arcsin(a+b*x),x,5,1/2*Ci(2*arcsin(a+b*x))/b+1/2*log(arcsin(a+b*x))/b],
[sqrt(1-a^2-2*a*b*x-b^2*x^2)/arcsin(a+b*x)^2,x,6,(-1+(a+b*x)^2)/(b*arcsin(a+b*x))-Si(2*arcsin(a+b*x))/b],
[sqrt(1-a^2-2*a*b*x-b^2*x^2)/arcsin(a+b*x)^3,x,4,1/2*(-1+(a+b*x)^2)/(b*arcsin(a+b*x)^2)-Ci(2*arcsin(a+b*x))/b+(a+b*x)*sqrt(1-(a+b*x)^2)/(b*arcsin(a+b*x))],
[sqrt(1-a^2-2*a*b*x-b^2*x^2)/arcsin(a+b*x)^4,x,9,1/3*(-1+(a+b*x)^2)/(b*arcsin(a+b*x)^3)+1/3/(b*arcsin(a+b*x))-2/3*(a+b*x)^2/(b*arcsin(a+b*x))+2/3*Si(2*arcsin(a+b*x))/b+1/3*(a+b*x)*sqrt(1-(a+b*x)^2)/(b*arcsin(a+b*x)^2)],
[(1-a^2-2*a*b*x-b^2*x^2)^(3/2)*arcsin(a+b*x)^3,x,15,51/128*(a+b*x)^2/b-3/128*(a+b*x)^4/b-3/32*(a+b*x)*(1-(a+b*x)^2)^(3/2)*arcsin(a+b*x)/b+27/128*arcsin(a+b*x)^2/b-9/16*(a+b*x)^2*arcsin(a+b*x)^2/b+3/16*(1-(a+b*x)^2)^2*arcsin(a+b*x)^2/b+1/4*(a+b*x)*(1-(a+b*x)^2)^(3/2)*arcsin(a+b*x)^3/b+3/32*arcsin(a+b*x)^4/b-45/64*(a+b*x)*arcsin(a+b*x)*sqrt(1-(a+b*x)^2)/b+3/8*(a+b*x)*arcsin(a+b*x)^3*sqrt(1-(a+b*x)^2)/b],
[(1-a^2-2*a*b*x-b^2*x^2)^(3/2)*arcsin(a+b*x)^2,x,11,-1/32*(a+b*x)*(1-(a+b*x)^2)^(3/2)/b+9/64*arcsin(a+b*x)/b-3/8*(a+b*x)^2*arcsin(a+b*x)/b+1/8*(1-(a+b*x)^2)^2*arcsin(a+b*x)/b+1/4*(a+b*x)*(1-(a+b*x)^2)^(3/2)*arcsin(a+b*x)^2/b+1/8*arcsin(a+b*x)^3/b-15/64*(a+b*x)*sqrt(1-(a+b*x)^2)/b+3/8*(a+b*x)*arcsin(a+b*x)^2*sqrt(1-(a+b*x)^2)/b],
[(1-a^2-2*a*b*x-b^2*x^2)^(3/2)*arcsin(a+b*x),x,7,-5/16*(a+b*x)^2/b+1/16*(a+b*x)^4/b+1/4*(a+b*x)*(1-(a+b*x)^2)^(3/2)*arcsin(a+b*x)/b+3/16*arcsin(a+b*x)^2/b+3/8*(a+b*x)*arcsin(a+b*x)*sqrt(1-(a+b*x)^2)/b],
[(1-a^2-2*a*b*x-b^2*x^2)^(3/2)/arcsin(a+b*x),x,6,1/2*Ci(2*arcsin(a+b*x))/b+1/8*Ci(4*arcsin(a+b*x))/b+3/8*log(arcsin(a+b*x))/b],
[(1-a^2-2*a*b*x-b^2*x^2)^(3/2)/arcsin(a+b*x)^2,x,7,-(1-(a+b*x)^2)^2/(b*arcsin(a+b*x))-Si(2*arcsin(a+b*x))/b-1/2*Si(4*arcsin(a+b*x))/b],
[(1-a^2-2*a*b*x-b^2*x^2)^(3/2)/arcsin(a+b*x)^3,x,11,-1/2*(1-(a+b*x)^2)^2/(b*arcsin(a+b*x)^2)+2*(a+b*x)*(1-(a+b*x)^2)^(3/2)/(b*arcsin(a+b*x))-Ci(2*arcsin(a+b*x))/b-Ci(4*arcsin(a+b*x))/b],
[(1-a^2-2*a*b*x-b^2*x^2)^(3/2)/arcsin(a+b*x)^4,x,18,-1/3*(1-(a+b*x)^2)^2/(b*arcsin(a+b*x)^3)+2/3*(a+b*x)*(1-(a+b*x)^2)^(3/2)/(b*arcsin(a+b*x)^2)+2/3*(1-(a+b*x)^2)/(b*arcsin(a+b*x))-8/3*(a+b*x)^2*(1-(a+b*x)^2)/(b*arcsin(a+b*x))+2/3*Si(2*arcsin(a+b*x))/b+4/3*Si(4*arcsin(a+b*x))/b],

# m<0
[arcsin(a+b*x)^n/sqrt(1-a^2-2*a*b*x-b^2*x^2),x,2,arcsin(a+b*x)^(1+n)/(b*(1+n))],
[arcsin(a+b*x)^2/sqrt(1-a^2-2*a*b*x-b^2*x^2),x,2,1/3*arcsin(a+b*x)^3/b],
[arcsin(a+b*x)/sqrt(1-a^2-2*a*b*x-b^2*x^2),x,2,1/2*arcsin(a+b*x)^2/b],
[1/(arcsin(a+b*x)*sqrt(1-a^2-2*a*b*x-b^2*x^2)),x,2,log(arcsin(a+b*x))/b],
[1/(arcsin(a+b*x)^2*sqrt(1-a^2-2*a*b*x-b^2*x^2)),x,2,(-1)/(b*arcsin(a+b*x))],
[1/(arcsin(a+b*x)^3*sqrt(1-a^2-2*a*b*x-b^2*x^2)),x,2,(-1/2)/(b*arcsin(a+b*x)^2)],
[arcsin(a+b*x)^3/(1-a^2-2*a*b*x-b^2*x^2)^(3/2),x,8,-I*arcsin(a+b*x)^3/b+3*arcsin(a+b*x)^2*log(1+exp(2*I*arcsin(a+b*x)))/b-3*I*arcsin(a+b*x)*polylog(2,-exp(2*I*arcsin(a+b*x)))/b+3/2*polylog(3,-exp(2*I*arcsin(a+b*x)))/b+(a+b*x)*arcsin(a+b*x)^3/(b*sqrt(1-(a+b*x)^2))],
[arcsin(a+b*x)^2/(1-a^2-2*a*b*x-b^2*x^2)^(3/2),x,7,-I*arcsin(a+b*x)^2/b+2*arcsin(a+b*x)*log(1+exp(2*I*arcsin(a+b*x)))/b-I*polylog(2,-exp(2*I*arcsin(a+b*x)))/b+(a+b*x)*arcsin(a+b*x)^2/(b*sqrt(1-(a+b*x)^2))],
[arcsin(a+b*x)/(1-a^2-2*a*b*x-b^2*x^2)^(3/2),x,3,1/2*log(1-(a+b*x)^2)/b+(a+b*x)*arcsin(a+b*x)/(b*sqrt(1-(a+b*x)^2))],
[1/((1-a^2-2*a*b*x-b^2*x^2)^(3/2)*arcsin(a+b*x)),x,1,Unintegrable(1/((1-(a+b*x)^2)^(3/2)*arcsin(a+b*x)),x)],
[1/((1-a^2-2*a*b*x-b^2*x^2)^(3/2)*arcsin(a+b*x)^2),x,2,(-1)/(b*(1-(a+b*x)^2)*arcsin(a+b*x))+2*Unintegrable((a+b*x)/((1-(a+b*x)^2)^2*arcsin(a+b*x)),x)],
[arcsin(a+b*x)/sqrt(c-c*(a+b*x)^2),x,2,1/2*arcsin(a+b*x)^2*sqrt(1-(a+b*x)^2)/(b*sqrt(c-c*(a+b*x)^2))],
[arcsin(a+b*x)/sqrt((1-a^2)*c-2*a*b*c*x-b^2*c*x^2),x,2,1/2*arcsin(a+b*x)^2*sqrt(1-(a+b*x)^2)/(b*sqrt(c-c*(a+b*x)^2))],

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

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

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

# n>0
[x^9*(a+b*arcsin(c*x^2)),x,5,-1/15*b*(1-c^2*x^4)^(3/2)/c^5+1/50*b*(1-c^2*x^4)^(5/2)/c^5+1/10*x^10*(a+b*arcsin(c*x^2))+1/10*b*sqrt(1-c^2*x^4)/c^5],
[x^7*(a+b*arcsin(c*x^2)),x,6,-3/64*b*arcsin(c*x^2)/c^4+1/8*x^8*(a+b*arcsin(c*x^2))+3/64*b*x^2*sqrt(1-c^2*x^4)/c^3+1/32*b*x^6*sqrt(1-c^2*x^4)/c],
[x^5*(a+b*arcsin(c*x^2)),x,5,-1/18*b*(1-c^2*x^4)^(3/2)/c^3+1/6*x^6*(a+b*arcsin(c*x^2))+1/6*b*sqrt(1-c^2*x^4)/c^3],
[x^3*(a+b*arcsin(c*x^2)),x,5,-1/8*b*arcsin(c*x^2)/c^2+1/4*x^4*(a+b*arcsin(c*x^2))+1/8*b*x^2*sqrt(1-c^2*x^4)/c],
[x*(a+b*arcsin(c*x^2)),x,4,1/2*a*x^2+1/2*b*x^2*arcsin(c*x^2)+1/2*b*sqrt(1-c^2*x^4)/c],
[(a+b*arcsin(c*x^2))/x,x,7,-1/4*I*b*arcsin(c*x^2)^2+1/2*b*arcsin(c*x^2)*log(1-exp(2*I*arcsin(c*x^2)))+a*log(x)-1/4*I*b*polylog(2,exp(2*I*arcsin(c*x^2)))],
[(a+b*arcsin(c*x^2))/x^3,x,5,1/2*(-a-b*arcsin(c*x^2))/x^2-1/2*b*c*arctanh(sqrt(1-c^2*x^4))],
[(a+b*arcsin(c*x^2))/x^5,x,3,1/4*(-a-b*arcsin(c*x^2))/x^4-1/4*b*c*sqrt(1-c^2*x^4)/x^2],
[(a+b*arcsin(c*x^2))/x^7,x,6,1/6*(-a-b*arcsin(c*x^2))/x^6-1/12*b*c^3*arctanh(sqrt(1-c^2*x^4))-1/12*b*c*sqrt(1-c^2*x^4)/x^4],
[(a+b*arcsin(c*x^2))/x^9,x,4,1/8*(-a-b*arcsin(c*x^2))/x^8-1/24*b*c*sqrt(1-c^2*x^4)/x^6-1/12*b*c^3*sqrt(1-c^2*x^4)/x^2],
[(a+b*arcsin(c*x^2))/x^11,x,7,1/10*(-a-b*arcsin(c*x^2))/x^10-3/80*b*c^5*arctanh(sqrt(1-c^2*x^4))-1/40*b*c*sqrt(1-c^2*x^4)/x^8-3/80*b*c^3*sqrt(1-c^2*x^4)/x^4],
[(a+b*arcsin(c*x^2))/x^13,x,5,1/12*(-a-b*arcsin(c*x^2))/x^12-1/60*b*c*sqrt(1-c^2*x^4)/x^10-1/45*b*c^3*sqrt(1-c^2*x^4)/x^6-2/45*b*c^5*sqrt(1-c^2*x^4)/x^2],
[x^6*(a+b*arcsin(c*x^2)),x,5,1/7*x^7*(a+b*arcsin(c*x^2))-10/147*b*EllipticF(x*sqrt(c),I)/c^(7/2)+10/147*b*x*sqrt(1-c^2*x^4)/c^3+2/49*b*x^5*sqrt(1-c^2*x^4)/c],
[x^4*(a+b*arcsin(c*x^2)),x,7,1/5*x^5*(a+b*arcsin(c*x^2))-6/25*b*EllipticE(x*sqrt(c),I)/c^(5/2)+6/25*b*EllipticF(x*sqrt(c),I)/c^(5/2)+2/25*b*x^3*sqrt(1-c^2*x^4)/c],
[x^2*(a+b*arcsin(c*x^2)),x,4,1/3*x^3*(a+b*arcsin(c*x^2))-2/9*b*EllipticF(x*sqrt(c),I)/c^(3/2)+2/9*b*x*sqrt(1-c^2*x^4)/c],
[a+b*arcsin(c*x^2),x,7,a*x+b*x*arcsin(c*x^2)-2*b*EllipticE(x*sqrt(c),I)/sqrt(c)+2*b*EllipticF(x*sqrt(c),I)/sqrt(c)],
[(a+b*arcsin(c*x^2))/x^2,x,3,(-a-b*arcsin(c*x^2))/x+2*b*EllipticF(x*sqrt(c),I)*sqrt(c)],
[(a+b*arcsin(c*x^2))/x^4,x,7,1/3*(-a-b*arcsin(c*x^2))/x^3-2/3*b*c^(3/2)*EllipticE(x*sqrt(c),I)+2/3*b*c^(3/2)*EllipticF(x*sqrt(c),I)-2/3*b*c*sqrt(1-c^2*x^4)/x],
[(a+b*arcsin(c*x^2))/x^6,x,4,1/5*(-a-b*arcsin(c*x^2))/x^5+2/15*b*c^(5/2)*EllipticF(x*sqrt(c),I)-2/15*b*c*sqrt(1-c^2*x^4)/x^3],
[(a+b*arcsin(c*x^2))/x^8,x,8,1/7*(-a-b*arcsin(c*x^2))/x^7-6/35*b*c^(7/2)*EllipticE(x*sqrt(c),I)+6/35*b*c^(7/2)*EllipticF(x*sqrt(c),I)-2/35*b*c*sqrt(1-c^2*x^4)/x^5-6/35*b*c^3*sqrt(1-c^2*x^4)/x],
[arcsin(a*x^5)/x,x,5,-1/10*I*arcsin(a*x^5)^2+1/5*arcsin(a*x^5)*log(1-exp(2*I*arcsin(a*x^5)))-1/10*I*polylog(2,exp(2*I*arcsin(a*x^5)))],
[x^2*arcsin(sqrt(x)),x,8,5/96*arcsin(1-2*x)+1/3*x^3*arcsin(sqrt(x))+5/72*x^(3/2)*sqrt(1-x)+1/18*x^(5/2)*sqrt(1-x)+5/48*sqrt(1-x)*sqrt(x)],
[x*arcsin(sqrt(x)),x,7,3/32*arcsin(1-2*x)+1/2*x^2*arcsin(sqrt(x))+1/8*x^(3/2)*sqrt(1-x)+3/16*sqrt(1-x)*sqrt(x)],
[arcsin(sqrt(x)),x,6,1/4*arcsin(1-2*x)+x*arcsin(sqrt(x))+1/2*sqrt(1-x)*sqrt(x)],
[arcsin(sqrt(x))/x,x,5,-I*arcsin(sqrt(x))^2+2*arcsin(sqrt(x))*log(1-exp(2*I*arcsin(sqrt(x))))-I*polylog(2,exp(2*I*arcsin(sqrt(x))))],
[arcsin(sqrt(x))/x^2,x,3,-arcsin(sqrt(x))/x-sqrt(1-x)/sqrt(x)],
[arcsin(sqrt(x))/x^3,x,4,-1/2*arcsin(sqrt(x))/x^2-1/6*sqrt(1-x)/x^(3/2)-1/3*sqrt(1-x)/sqrt(x)],
[arcsin(sqrt(x))/x^4,x,5,-1/3*arcsin(sqrt(x))/x^3-1/15*sqrt(1-x)/x^(5/2)-4/45*sqrt(1-x)/x^(3/2)-8/45*sqrt(1-x)/sqrt(x)],
[arcsin(sqrt(x))/x^5,x,6,-1/4*arcsin(sqrt(x))/x^4-1/28*sqrt(1-x)/x^(7/2)-3/70*sqrt(1-x)/x^(5/2)-2/35*sqrt(1-x)/x^(3/2)-4/35*sqrt(1-x)/sqrt(x)],

# n<0
[x^4*(a+b*arcsin(c/x)),x,7,1/5*x^5*(a+b*arcsin(c/x))+3/40*b*c^5*arctanh(sqrt(1-c^2/x^2))+3/40*b*c^3*x^2*sqrt(1-c^2/x^2)+1/20*b*c*x^4*sqrt(1-c^2/x^2)],
[x^3*(a+b*arcsin(c/x)),x,4,1/4*x^4*(a+b*arcsin(c/x))+1/6*b*c^3*x*sqrt(1-c^2/x^2)+1/12*b*c*x^3*sqrt(1-c^2/x^2)],
[x^2*(a+b*arcsin(c/x)),x,6,1/3*x^3*(a+b*arcsin(c/x))+1/6*b*c^3*arctanh(sqrt(1-c^2/x^2))+1/6*b*c*x^2*sqrt(1-c^2/x^2)],
[x*(a+b*arcsin(c/x)),x,3,1/2*x^2*(a+b*arcsin(c/x))+1/2*b*c*x*sqrt(1-c^2/x^2)],
[a+b*arcsin(c/x),x,6,a*x+b*x*arccsc(x/c)+b*c*arctanh(sqrt(1-c^2/x^2))],
[(a+b*arcsin(c/x))/x,x,7,1/2*I*b*arcsin(c/x)^2-b*arcsin(c/x)*log(1-exp(2*I*arcsin(c/x)))+a*log(x)+1/2*I*b*polylog(2,exp(2*I*arcsin(c/x)))],
[(a+b*arcsin(c/x))/x^2,x,4,-a/x-b*arccsc(x/c)/x-b*sqrt(1-c^2/x^2)/c],
[(a+b*arcsin(c/x))/x^3,x,5,1/4*b*arccsc(x/c)/c^2+1/2*(-a-b*arcsin(c/x))/x^2-1/4*b*sqrt(1-c^2/x^2)/(c*x)],
[(a+b*arcsin(c/x))/x^4,x,5,1/9*b*(1-c^2/x^2)^(3/2)/c^3+1/3*(-a-b*arcsin(c/x))/x^3-1/3*b*sqrt(1-c^2/x^2)/c^3],
[(a+b*arcsin(c/x))/x^5,x,6,3/32*b*arccsc(x/c)/c^4+1/4*(-a-b*arcsin(c/x))/x^4-1/16*b*sqrt(1-c^2/x^2)/(c*x^3)-3/32*b*sqrt(1-c^2/x^2)/(c^3*x)],

# Integrands of the form x^m (a+b ArcSin[c x^n]) with n symbolic
[x^m*(a+b*arcsin(c*x^n)),x,3,x^(1+m)*(a+b*arcsin(c*x^n))/(1+m)-b*c*n*x^(1+m+n)*hypergeom([1/2,1/2*(1+m+n)/n],[1/2*(1+m+3*n)/n],c^2*x^(2*n))/((1+m)*(1+m+n))],
[x^2*(a+b*arcsin(c*x^n)),x,3,1/3*x^3*(a+b*arcsin(c*x^n))-1/3*b*c*n*x^(3+n)*hypergeom([1/2,1/2*(3+n)/n],[3/2*(1+n)/n],c^2*x^(2*n))/(3+n)],
[x*(a+b*arcsin(c*x^n)),x,3,1/2*x^2*(a+b*arcsin(c*x^n))-1/2*b*c*n*x^(2+n)*hypergeom([1/2,1/2*(2+n)/n],[1/2*(3+2/n)],c^2*x^(2*n))/(2+n)],
[a+b*arcsin(c*x^n),x,4,a*x+b*x*arcsin(c*x^n)-b*c*n*x^(1+n)*hypergeom([1/2,1/2*(1+n)/n],[1/2*(3+1/n)],c^2*x^(2*n))/(1+n)],
[(a+b*arcsin(c*x^n))/x,x,7,-1/2*I*b*arcsin(c*x^n)^2/n+b*arcsin(c*x^n)*log(1-exp(2*I*arcsin(c*x^n)))/n+a*log(x)-1/2*I*b*polylog(2,exp(2*I*arcsin(c*x^n)))/n],
[(a+b*arcsin(c*x^n))/x^2,x,3,(-a-b*arcsin(c*x^n))/x-b*c*n*x^(-1+n)*hypergeom([1/2,1/2*(-1+n)/n],[1/2*(3+(-1)/n)],c^2*x^(2*n))/(1-n)],
[(a+b*arcsin(c*x^n))/x^3,x,3,1/2*(-a-b*arcsin(c*x^n))/x^2-1/2*b*c*n*x^(-2+n)*hypergeom([1/2,1/2*(1+(-2)/n)],[1/2*(3+(-2)/n)],c^2*x^(2*n))/(2-n)],

# Integrands of the form x^m (a+b ArcSin[c+d x^n])
[x^5*(a+b*arcsin(c+d*x^2)),x,7,1/12*b*c*(3+2*c^2)*arcsin(c+d*x^2)/d^3+1/6*x^6*(a+b*arcsin(c+d*x^2))+1/18*b*x^4*sqrt(1-c^2-2*c*d*x^2-d^2*x^4)/d+1/36*b*(4+11*c^2-5*c*d*x^2)*sqrt(1-c^2-2*c*d*x^2-d^2*x^4)/d^3],
[x^3*(a+b*arcsin(c+d*x^2)),x,7,-1/8*b*(1+2*c^2)*arcsin(c+d*x^2)/d^2+1/4*x^4*(a+b*arcsin(c+d*x^2))-3/8*b*c*sqrt(1-c^2-2*c*d*x^2-d^2*x^4)/d^2+1/8*b*x^2*sqrt(1-c^2-2*c*d*x^2-d^2*x^4)/d],
[x*(a+b*arcsin(c+d*x^2)),x,5,1/2*a*x^2+1/2*b*(c+d*x^2)*arcsin(c+d*x^2)/d+1/2*b*sqrt(1-(c+d*x^2)^2)/d],
[(a+b*arcsin(c+d*x^2))/x,x,12,-1/4*I*b*arcsin(c+d*x^2)^2+a*log(x)+1/2*b*arcsin(c+d*x^2)*log(1-exp(I*arcsin(c+d*x^2))/(I*c-sqrt(1-c^2)))+1/2*b*arcsin(c+d*x^2)*log(1-exp(I*arcsin(c+d*x^2))/(I*c+sqrt(1-c^2)))-1/2*I*b*polylog(2,exp(I*arcsin(c+d*x^2))/(I*c-sqrt(1-c^2)))-1/2*I*b*polylog(2,exp(I*arcsin(c+d*x^2))/(I*c+sqrt(1-c^2)))],
[(a+b*arcsin(c+d*x^2))/x^3,x,5,1/2*(-a-b*arcsin(c+d*x^2))/x^2-1/2*b*d*arctanh((1-c^2-c*d*x^2)/(sqrt(1-c^2)*sqrt(1-c^2-2*c*d*x^2-d^2*x^4)))/sqrt(1-c^2)],
[(a+b*arcsin(c+d*x^2))/x^5,x,6,1/4*(-a-b*arcsin(c+d*x^2))/x^4-1/4*b*c*d^2*arctanh((1-c^2-c*d*x^2)/(sqrt(1-c^2)*sqrt(1-c^2-2*c*d*x^2-d^2*x^4)))/(1-c^2)^(3/2)-1/4*b*d*sqrt(1-c^2-2*c*d*x^2-d^2*x^4)/((1-c^2)*x^2)],
[(a+b*arcsin(c+d*x^2))/x^7,x,7,1/6*(-a-b*arcsin(c+d*x^2))/x^6-1/12*b*(1+2*c^2)*d^3*arctanh((1-c^2-c*d*x^2)/(sqrt(1-c^2)*sqrt(1-c^2-2*c*d*x^2-d^2*x^4)))/(1-c^2)^(5/2)-1/12*b*d*sqrt(1-c^2-2*c*d*x^2-d^2*x^4)/((1-c^2)*x^4)-1/4*b*c*d^2*sqrt(1-c^2-2*c*d*x^2-d^2*x^4)/((1-c^2)^2*x^2)],
[x^4*(a+b*arcsin(c+d*x^2)),x,8,1/5*x^5*(a+b*arcsin(c+d*x^2))-2/75*b*(1+c)*(9+23*c^2)*EllipticE(x*sqrt(d)/sqrt(1-c),sqrt((-1+c)/(1+c)))*sqrt(1-c)*sqrt(1-d*x^2/(1-c))*sqrt(1+d*x^2/(1+c))/(d^(5/2)*sqrt(1-c^2-2*c*d*x^2-d^2*x^4))+2/75*b*(1+c)*(9+8*c+15*c^2)*EllipticF(x*sqrt(d)/sqrt(1-c),sqrt((-1+c)/(1+c)))*sqrt(1-c)*sqrt(1-d*x^2/(1-c))*sqrt(1+d*x^2/(1+c))/(d^(5/2)*sqrt(1-c^2-2*c*d*x^2-d^2*x^4))-16/75*b*c*x*sqrt(1-c^2-2*c*d*x^2-d^2*x^4)/d^2+2/25*b*x^3*sqrt(1-c^2-2*c*d*x^2-d^2*x^4)/d],
[x^2*(a+b*arcsin(c+d*x^2)),x,7,1/3*x^3*(a+b*arcsin(c+d*x^2))+8/9*b*c*(1+c)*EllipticE(x*sqrt(d)/sqrt(1-c),sqrt((-1+c)/(1+c)))*sqrt(1-c)*sqrt(1-d*x^2/(1-c))*sqrt(1+d*x^2/(1+c))/(d^(3/2)*sqrt(1-c^2-2*c*d*x^2-d^2*x^4))-2/9*b*(1+c)*(1+3*c)*EllipticF(x*sqrt(d)/sqrt(1-c),sqrt((-1+c)/(1+c)))*sqrt(1-c)*sqrt(1-d*x^2/(1-c))*sqrt(1+d*x^2/(1+c))/(d^(3/2)*sqrt(1-c^2-2*c*d*x^2-d^2*x^4))+2/9*b*x*sqrt(1-c^2-2*c*d*x^2-d^2*x^4)/d],
[a+b*arcsin(c+d*x^2),x,7,a*x+b*x*arcsin(c+d*x^2)-2*b*(1+c)*EllipticE(x*sqrt(d)/sqrt(1-c),sqrt((-1+c)/(1+c)))*sqrt(1-c)*sqrt(1-d*x^2/(1-c))*sqrt(1+d*x^2/(1+c))/(sqrt(d)*sqrt(1-c^2-2*c*d*x^2-d^2*x^4))+2*b*(1+c)*EllipticF(x*sqrt(d)/sqrt(1-c),sqrt((-1+c)/(1+c)))*sqrt(1-c)*sqrt(1-d*x^2/(1-c))*sqrt(1+d*x^2/(1+c))/(sqrt(d)*sqrt(1-c^2-2*c*d*x^2-d^2*x^4))],
[(a+b*arcsin(c+d*x^2))/x^2,x,4,(-a-b*arcsin(c+d*x^2))/x+2*b*EllipticF(x*sqrt(d)/sqrt(1-c),sqrt((-1+c)/(1+c)))*sqrt(1-c)*sqrt(d)*sqrt(1-d*x^2/(1-c))*sqrt(1+d*x^2/(1+c))/sqrt(1-c^2-2*c*d*x^2-d^2*x^4)],
[(a+b*arcsin(c+d*x^2))/x^4,x,8,1/3*(-a-b*arcsin(c+d*x^2))/x^3-2/3*b*d^(3/2)*EllipticE(x*sqrt(d)/sqrt(1-c),sqrt((-1+c)/(1+c)))*sqrt(1-d*x^2/(1-c))*sqrt(1+d*x^2/(1+c))/(sqrt(1-c)*sqrt(1-c^2-2*c*d*x^2-d^2*x^4))+2/3*b*d^(3/2)*EllipticF(x*sqrt(d)/sqrt(1-c),sqrt((-1+c)/(1+c)))*sqrt(1-d*x^2/(1-c))*sqrt(1+d*x^2/(1+c))/(sqrt(1-c)*sqrt(1-c^2-2*c*d*x^2-d^2*x^4))-2/3*b*d*sqrt(1-c^2-2*c*d*x^2-d^2*x^4)/((1-c^2)*x)],
[(a+b*arcsin(c+d*x^2))/x^6,x,8,1/5*(-a-b*arcsin(c+d*x^2))/x^5-8/15*b*c*d^(5/2)*EllipticE(x*sqrt(d)/sqrt(1-c),sqrt((-1+c)/(1+c)))*sqrt(1-d*x^2/(1-c))*sqrt(1+d*x^2/(1+c))/((1-c^2)*sqrt(1-c)*sqrt(1-c^2-2*c*d*x^2-d^2*x^4))+2/15*b*(1+3*c)*d^(5/2)*EllipticF(x*sqrt(d)/sqrt(1-c),sqrt((-1+c)/(1+c)))*sqrt(1-d*x^2/(1-c))*sqrt(1+d*x^2/(1+c))/((1-c^2)*sqrt(1-c)*sqrt(1-c^2-2*c*d*x^2-d^2*x^4))-2/15*b*d*sqrt(1-c^2-2*c*d*x^2-d^2*x^4)/((1-c^2)*x^3)-8/15*b*c*d^2*sqrt(1-c^2-2*c*d*x^2-d^2*x^4)/((1-c^2)^2*x)],
[x^3*arcsin(a+b*x^4),x,4,1/4*(a+b*x^4)*arcsin(a+b*x^4)/b+1/4*sqrt(1-(a+b*x^4)^2)/b],
[x^(-1+n)*arcsin(a+b*x^n),x,4,(a+b*x^n)*arcsin(a+b*x^n)/(b*n)+sqrt(1-(a+b*x^n)^2)/(b*n)],

# Integrands of the form (a+b ArcSin[c+d x^2])^n

# Integrands of the form (a+b ArcSin[c+d x^2])^n when c^2=1
[(a+b*arcsin(1+d*x^2))^4,x,3,384*b^4*x-48*b^2*x*(a+b*arcsin(1+d*x^2))^2+x*(a+b*arcsin(1+d*x^2))^4-192*b^3*(a+b*arcsin(1+d*x^2))*sqrt(-2*d*x^2-d^2*x^4)/(d*x)+8*b*(a+b*arcsin(1+d*x^2))^3*sqrt(-2*d*x^2-d^2*x^4)/(d*x)],
[(a+b*arcsin(1+d*x^2))^3,x,5,-24*a*b^2*x-24*b^3*x*arcsin(1+d*x^2)+x*(a+b*arcsin(1+d*x^2))^3-48*b^3*sqrt(-2*d*x^2-d^2*x^4)/(d*x)+6*b*(a+b*arcsin(1+d*x^2))^2*sqrt(-2*d*x^2-d^2*x^4)/(d*x)],
[(a+b*arcsin(1+d*x^2))^2,x,2,-8*b^2*x+x*(a+b*arcsin(1+d*x^2))^2+4*b*(a+b*arcsin(1+d*x^2))*sqrt(-2*d*x^2-d^2*x^4)/(d*x)],
[a+b*arcsin(1+d*x^2),x,4,a*x+b*x*arcsin(1+d*x^2)+2*b*sqrt(-2*d*x^2-d^2*x^4)/(d*x)],
[1/(a+b*arcsin(1+d*x^2)),x,1,-1/2*x*Ci(1/2*(a+b*arcsin(1+d*x^2))/b)*(cos(1/2*a/b)-sin(1/2*a/b))/(b*(cos(1/2*arcsin(1+d*x^2))-sin(1/2*arcsin(1+d*x^2))))-1/2*x*Si(1/2*(a+b*arcsin(1+d*x^2))/b)*(cos(1/2*a/b)+sin(1/2*a/b))/(b*(cos(1/2*arcsin(1+d*x^2))-sin(1/2*arcsin(1+d*x^2))))],
[1/(a+b*arcsin(1+d*x^2))^2,x,1,1/4*x*Si(1/2*(a+b*arcsin(1+d*x^2))/b)*(cos(1/2*a/b)-sin(1/2*a/b))/(b^2*(cos(1/2*arcsin(1+d*x^2))-sin(1/2*arcsin(1+d*x^2))))-1/4*x*Ci(1/2*(a+b*arcsin(1+d*x^2))/b)*(cos(1/2*a/b)+sin(1/2*a/b))/(b^2*(cos(1/2*arcsin(1+d*x^2))-sin(1/2*arcsin(1+d*x^2))))-1/2*sqrt(-2*d*x^2-d^2*x^4)/(b*d*x*(a+b*arcsin(1+d*x^2)))],
[1/(a+b*arcsin(1+d*x^2))^3,x,2,1/8*x/(b^2*(a+b*arcsin(1+d*x^2)))+1/16*x*Ci(1/2*(a+b*arcsin(1+d*x^2))/b)*(cos(1/2*a/b)-sin(1/2*a/b))/(b^3*(cos(1/2*arcsin(1+d*x^2))-sin(1/2*arcsin(1+d*x^2))))+1/16*x*Si(1/2*(a+b*arcsin(1+d*x^2))/b)*(cos(1/2*a/b)+sin(1/2*a/b))/(b^3*(cos(1/2*arcsin(1+d*x^2))-sin(1/2*arcsin(1+d*x^2))))-1/4*sqrt(-2*d*x^2-d^2*x^4)/(b*d*x*(a+b*arcsin(1+d*x^2))^2)],
[(a+b*arcsin(-1+d*x^2))^4,x,3,384*b^4*x-48*b^2*x*(a-b*arcsin(1-d*x^2))^2+x*(a-b*arcsin(1-d*x^2))^4-192*b^3*(a-b*arcsin(1-d*x^2))*sqrt(2*d*x^2-d^2*x^4)/(d*x)+8*b*(a-b*arcsin(1-d*x^2))^3*sqrt(2*d*x^2-d^2*x^4)/(d*x)],
[(a+b*arcsin(-1+d*x^2))^3,x,5,-24*a*b^2*x+24*b^3*x*arcsin(1-d*x^2)+x*(a-b*arcsin(1-d*x^2))^3-48*b^3*sqrt(2*d*x^2-d^2*x^4)/(d*x)+6*b*(a-b*arcsin(1-d*x^2))^2*sqrt(2*d*x^2-d^2*x^4)/(d*x)],
[(a+b*arcsin(-1+d*x^2))^2,x,2,-8*b^2*x+x*(a-b*arcsin(1-d*x^2))^2+4*b*(a-b*arcsin(1-d*x^2))*sqrt(2*d*x^2-d^2*x^4)/(d*x)],
[a+b*arcsin(-1+d*x^2),x,4,a*x-b*x*arcsin(1-d*x^2)+2*b*sqrt(2*d*x^2-d^2*x^4)/(d*x)],
[1/(a+b*arcsin(-1+d*x^2)),x,1,-1/2*x*Si(1/2*a/b-1/2*arcsin(1-d*x^2))*(cos(1/2*a/b)-sin(1/2*a/b))/(b*(cos(1/2*arcsin(1-d*x^2))-sin(1/2*arcsin(1-d*x^2))))+1/2*x*Ci(1/2*(-a+b*arcsin(1-d*x^2))/b)*(cos(1/2*a/b)+sin(1/2*a/b))/(b*(cos(1/2*arcsin(1-d*x^2))-sin(1/2*arcsin(1-d*x^2))))],
[1/(a+b*arcsin(-1+d*x^2))^2,x,1,-1/4*x*Ci(1/2*(-a+b*arcsin(1-d*x^2))/b)*(cos(1/2*a/b)-sin(1/2*a/b))/(b^2*(cos(1/2*arcsin(1-d*x^2))-sin(1/2*arcsin(1-d*x^2))))-1/4*x*Si(1/2*a/b-1/2*arcsin(1-d*x^2))*(cos(1/2*a/b)+sin(1/2*a/b))/(b^2*(cos(1/2*arcsin(1-d*x^2))-sin(1/2*arcsin(1-d*x^2))))-1/2*sqrt(2*d*x^2-d^2*x^4)/(b*d*x*(a-b*arcsin(1-d*x^2)))],
[1/(a+b*arcsin(-1+d*x^2))^3,x,2,1/8*x/(b^2*(a-b*arcsin(1-d*x^2)))+1/16*x*Si(1/2*a/b-1/2*arcsin(1-d*x^2))*(cos(1/2*a/b)-sin(1/2*a/b))/(b^3*(cos(1/2*arcsin(1-d*x^2))-sin(1/2*arcsin(1-d*x^2))))-1/16*x*Ci(1/2*(-a+b*arcsin(1-d*x^2))/b)*(cos(1/2*a/b)+sin(1/2*a/b))/(b^3*(cos(1/2*arcsin(1-d*x^2))-sin(1/2*arcsin(1-d*x^2))))-1/4*sqrt(2*d*x^2-d^2*x^4)/(b*d*x*(a-b*arcsin(1-d*x^2))^2)],
[arcsin(1+x^2)^2,x,2,-8*x+x*arcsin(1+x^2)^2+4*arcsin(1+x^2)*sqrt(-2*x^2-x^4)/x],
[arcsin(1-x^2)^2,x,2,-8*x+x*arcsin(1-x^2)^2-4*arcsin(1-x^2)*sqrt(2*x^2-x^4)/x],

# Integrands of the form (a+b ArcSin[c+d x^2])^(n/2) when c^2=1
[(a+b*arcsin(1+d*x^2))^(5/2),x,2,x*(a+b*arcsin(1+d*x^2))^(5/2)-15*x*FresnelS(sqrt(1/b)*sqrt(a+b*arcsin(1+d*x^2))/sqrt(Pi))*(cos(1/2*a/b)-sin(1/2*a/b))*sqrt(Pi)/((1/b)^(5/2)*(cos(1/2*arcsin(1+d*x^2))-sin(1/2*arcsin(1+d*x^2))))+15*x*FresnelC(sqrt(1/b)*sqrt(a+b*arcsin(1+d*x^2))/sqrt(Pi))*(cos(1/2*a/b)+sin(1/2*a/b))*sqrt(Pi)/((1/b)^(5/2)*(cos(1/2*arcsin(1+d*x^2))-sin(1/2*arcsin(1+d*x^2))))+5*b*(a+b*arcsin(1+d*x^2))^(3/2)*sqrt(-2*d*x^2-d^2*x^4)/(d*x)-15*b^2*x*sqrt(a+b*arcsin(1+d*x^2))],
[(a+b*arcsin(1+d*x^2))^(3/2),x,2,x*(a+b*arcsin(1+d*x^2))^(3/2)+3*b^(3/2)*x*FresnelC(sqrt(a+b*arcsin(1+d*x^2))/(sqrt(b)*sqrt(Pi)))*(cos(1/2*a/b)-sin(1/2*a/b))*sqrt(Pi)/(cos(1/2*arcsin(1+d*x^2))-sin(1/2*arcsin(1+d*x^2)))+3*b^(3/2)*x*FresnelS(sqrt(a+b*arcsin(1+d*x^2))/(sqrt(b)*sqrt(Pi)))*(cos(1/2*a/b)+sin(1/2*a/b))*sqrt(Pi)/(cos(1/2*arcsin(1+d*x^2))-sin(1/2*arcsin(1+d*x^2)))+3*b*sqrt(-2*d*x^2-d^2*x^4)*sqrt(a+b*arcsin(1+d*x^2))/(d*x)],
[(a+b*arcsin(1+d*x^2))^(1/2),x,1,x*FresnelS(sqrt(1/b)*sqrt(a+b*arcsin(1+d*x^2))/sqrt(Pi))*(cos(1/2*a/b)-sin(1/2*a/b))*sqrt(Pi)/((cos(1/2*arcsin(1+d*x^2))-sin(1/2*arcsin(1+d*x^2)))*sqrt(1/b))-x*FresnelC(sqrt(1/b)*sqrt(a+b*arcsin(1+d*x^2))/sqrt(Pi))*(cos(1/2*a/b)+sin(1/2*a/b))*sqrt(Pi)/((cos(1/2*arcsin(1+d*x^2))-sin(1/2*arcsin(1+d*x^2)))*sqrt(1/b))+x*sqrt(a+b*arcsin(1+d*x^2))],
[1/(a+b*arcsin(1+d*x^2))^(1/2),x,1,-x*FresnelC(sqrt(a+b*arcsin(1+d*x^2))/(sqrt(b)*sqrt(Pi)))*(cos(1/2*a/b)-sin(1/2*a/b))*sqrt(Pi)/((cos(1/2*arcsin(1+d*x^2))-sin(1/2*arcsin(1+d*x^2)))*sqrt(b))-x*FresnelS(sqrt(a+b*arcsin(1+d*x^2))/(sqrt(b)*sqrt(Pi)))*(cos(1/2*a/b)+sin(1/2*a/b))*sqrt(Pi)/((cos(1/2*arcsin(1+d*x^2))-sin(1/2*arcsin(1+d*x^2)))*sqrt(b))],
[1/(a+b*arcsin(1+d*x^2))^(3/2),x,1,(1/b)^(3/2)*x*FresnelS(sqrt(1/b)*sqrt(a+b*arcsin(1+d*x^2))/sqrt(Pi))*(cos(1/2*a/b)-sin(1/2*a/b))*sqrt(Pi)/(cos(1/2*arcsin(1+d*x^2))-sin(1/2*arcsin(1+d*x^2)))-(1/b)^(3/2)*x*FresnelC(sqrt(1/b)*sqrt(a+b*arcsin(1+d*x^2))/sqrt(Pi))*(cos(1/2*a/b)+sin(1/2*a/b))*sqrt(Pi)/(cos(1/2*arcsin(1+d*x^2))-sin(1/2*arcsin(1+d*x^2)))-sqrt(-2*d*x^2-d^2*x^4)/(b*d*x*sqrt(a+b*arcsin(1+d*x^2)))],
[1/(a+b*arcsin(1+d*x^2))^(5/2),x,2,1/3*x*FresnelC(sqrt(a+b*arcsin(1+d*x^2))/(sqrt(b)*sqrt(Pi)))*(cos(1/2*a/b)-sin(1/2*a/b))*sqrt(Pi)/(b^(5/2)*(cos(1/2*arcsin(1+d*x^2))-sin(1/2*arcsin(1+d*x^2))))+1/3*x*FresnelS(sqrt(a+b*arcsin(1+d*x^2))/(sqrt(b)*sqrt(Pi)))*(cos(1/2*a/b)+sin(1/2*a/b))*sqrt(Pi)/(b^(5/2)*(cos(1/2*arcsin(1+d*x^2))-sin(1/2*arcsin(1+d*x^2))))-1/3*sqrt(-2*d*x^2-d^2*x^4)/(b*d*x*(a+b*arcsin(1+d*x^2))^(3/2))+1/3*x/(b^2*sqrt(a+b*arcsin(1+d*x^2)))],
[1/(a+b*arcsin(1+d*x^2))^(7/2),x,2,1/15*x/(b^2*(a+b*arcsin(1+d*x^2))^(3/2))-1/15*(1/b)^(7/2)*x*FresnelS(sqrt(1/b)*sqrt(a+b*arcsin(1+d*x^2))/sqrt(Pi))*(cos(1/2*a/b)-sin(1/2*a/b))*sqrt(Pi)/(cos(1/2*arcsin(1+d*x^2))-sin(1/2*arcsin(1+d*x^2)))+1/15*(1/b)^(7/2)*x*FresnelC(sqrt(1/b)*sqrt(a+b*arcsin(1+d*x^2))/sqrt(Pi))*(cos(1/2*a/b)+sin(1/2*a/b))*sqrt(Pi)/(cos(1/2*arcsin(1+d*x^2))-sin(1/2*arcsin(1+d*x^2)))-1/5*sqrt(-2*d*x^2-d^2*x^4)/(b*d*x*(a+b*arcsin(1+d*x^2))^(5/2))+1/15*sqrt(-2*d*x^2-d^2*x^4)/(b^3*d*x*sqrt(a+b*arcsin(1+d*x^2)))],
[(a+b*arcsin(-1+d*x^2))^(5/2),x,2,x*(a-b*arcsin(1-d*x^2))^(5/2)+15*x*FresnelC(sqrt((-1)/b)*sqrt(a-b*arcsin(1-d*x^2))/sqrt(Pi))*(cos(1/2*a/b)-sin(1/2*a/b))*sqrt(Pi)/(((-1)/b)^(5/2)*(cos(1/2*arcsin(1-d*x^2))-sin(1/2*arcsin(1-d*x^2))))-15*x*FresnelS(sqrt((-1)/b)*sqrt(a-b*arcsin(1-d*x^2))/sqrt(Pi))*(cos(1/2*a/b)+sin(1/2*a/b))*sqrt(Pi)/(((-1)/b)^(5/2)*(cos(1/2*arcsin(1-d*x^2))-sin(1/2*arcsin(1-d*x^2))))+5*b*(a-b*arcsin(1-d*x^2))^(3/2)*sqrt(2*d*x^2-d^2*x^4)/(d*x)-15*b^2*x*sqrt(a-b*arcsin(1-d*x^2))],
[(a+b*arcsin(-1+d*x^2))^(3/2),x,2,x*(a-b*arcsin(1-d*x^2))^(3/2)+3*(-b)^(3/2)*x*FresnelS(sqrt(a-b*arcsin(1-d*x^2))/(sqrt(-b)*sqrt(Pi)))*(cos(1/2*a/b)-sin(1/2*a/b))*sqrt(Pi)/(cos(1/2*arcsin(1-d*x^2))-sin(1/2*arcsin(1-d*x^2)))+3*(-b)^(3/2)*x*FresnelC(sqrt(a-b*arcsin(1-d*x^2))/(sqrt(-b)*sqrt(Pi)))*(cos(1/2*a/b)+sin(1/2*a/b))*sqrt(Pi)/(cos(1/2*arcsin(1-d*x^2))-sin(1/2*arcsin(1-d*x^2)))+3*b*sqrt(2*d*x^2-d^2*x^4)*sqrt(a-b*arcsin(1-d*x^2))/(d*x)],
[(a+b*arcsin(-1+d*x^2))^(1/2),x,1,-x*FresnelC(sqrt((-1)/b)*sqrt(a-b*arcsin(1-d*x^2))/sqrt(Pi))*(cos(1/2*a/b)-sin(1/2*a/b))*sqrt(Pi)/((cos(1/2*arcsin(1-d*x^2))-sin(1/2*arcsin(1-d*x^2)))*sqrt((-1)/b))+x*FresnelS(sqrt((-1)/b)*sqrt(a-b*arcsin(1-d*x^2))/sqrt(Pi))*(cos(1/2*a/b)+sin(1/2*a/b))*sqrt(Pi)/((cos(1/2*arcsin(1-d*x^2))-sin(1/2*arcsin(1-d*x^2)))*sqrt((-1)/b))+x*sqrt(a-b*arcsin(1-d*x^2))],
[1/(a+b*arcsin(-1+d*x^2))^(1/2),x,1,-x*FresnelS(sqrt(a-b*arcsin(1-d*x^2))/(sqrt(-b)*sqrt(Pi)))*(cos(1/2*a/b)-sin(1/2*a/b))*sqrt(Pi)/((cos(1/2*arcsin(1-d*x^2))-sin(1/2*arcsin(1-d*x^2)))*sqrt(-b))-x*FresnelC(sqrt(a-b*arcsin(1-d*x^2))/(sqrt(-b)*sqrt(Pi)))*(cos(1/2*a/b)+sin(1/2*a/b))*sqrt(Pi)/((cos(1/2*arcsin(1-d*x^2))-sin(1/2*arcsin(1-d*x^2)))*sqrt(-b))],
[1/(a+b*arcsin(-1+d*x^2))^(3/2),x,1,-((-1)/b)^(3/2)*x*FresnelC(sqrt((-1)/b)*sqrt(a-b*arcsin(1-d*x^2))/sqrt(Pi))*(cos(1/2*a/b)-sin(1/2*a/b))*sqrt(Pi)/(cos(1/2*arcsin(1-d*x^2))-sin(1/2*arcsin(1-d*x^2)))+((-1)/b)^(3/2)*x*FresnelS(sqrt((-1)/b)*sqrt(a-b*arcsin(1-d*x^2))/sqrt(Pi))*(cos(1/2*a/b)+sin(1/2*a/b))*sqrt(Pi)/(cos(1/2*arcsin(1-d*x^2))-sin(1/2*arcsin(1-d*x^2)))-sqrt(2*d*x^2-d^2*x^4)/(b*d*x*sqrt(a-b*arcsin(1-d*x^2)))],
[1/(a+b*arcsin(-1+d*x^2))^(5/2),x,2,1/3*x*FresnelS(sqrt(a-b*arcsin(1-d*x^2))/(sqrt(-b)*sqrt(Pi)))*(cos(1/2*a/b)-sin(1/2*a/b))*sqrt(Pi)/((-b)^(5/2)*(cos(1/2*arcsin(1-d*x^2))-sin(1/2*arcsin(1-d*x^2))))+1/3*x*FresnelC(sqrt(a-b*arcsin(1-d*x^2))/(sqrt(-b)*sqrt(Pi)))*(cos(1/2*a/b)+sin(1/2*a/b))*sqrt(Pi)/((-b)^(5/2)*(cos(1/2*arcsin(1-d*x^2))-sin(1/2*arcsin(1-d*x^2))))-1/3*sqrt(2*d*x^2-d^2*x^4)/(b*d*x*(a-b*arcsin(1-d*x^2))^(3/2))+1/3*x/(b^2*sqrt(a-b*arcsin(1-d*x^2)))],
[1/(a+b*arcsin(-1+d*x^2))^(7/2),x,2,1/15*x/(b^2*(a-b*arcsin(1-d*x^2))^(3/2))+1/15*((-1)/b)^(7/2)*x*FresnelC(sqrt((-1)/b)*sqrt(a-b*arcsin(1-d*x^2))/sqrt(Pi))*(cos(1/2*a/b)-sin(1/2*a/b))*sqrt(Pi)/(cos(1/2*arcsin(1-d*x^2))-sin(1/2*arcsin(1-d*x^2)))-1/15*((-1)/b)^(7/2)*x*FresnelS(sqrt((-1)/b)*sqrt(a-b*arcsin(1-d*x^2))/sqrt(Pi))*(cos(1/2*a/b)+sin(1/2*a/b))*sqrt(Pi)/(cos(1/2*arcsin(1-d*x^2))-sin(1/2*arcsin(1-d*x^2)))-1/5*sqrt(2*d*x^2-d^2*x^4)/(b*d*x*(a-b*arcsin(1-d*x^2))^(5/2))+1/15*sqrt(2*d*x^2-d^2*x^4)/(b^3*d*x*sqrt(a-b*arcsin(1-d*x^2)))],

# Integrands of the form u (a+b ArcSin[Sqrt[1-c x]/Sqrt[1+c x]])^n
[(a+b*arcsin(sqrt(1-c*x)/sqrt(1+c*x)))^n/(1-c^2*x^2),x,0,Unintegrable((a+b*arcsin(sqrt(1-c*x)/sqrt(1+c*x)))^n/(1-c^2*x^2),x)],
[(a+b*arcsin(sqrt(1-c*x)/sqrt(1+c*x)))^3/(1-c^2*x^2),x,8,1/4*I*(a+b*arcsin(sqrt(1-c*x)/sqrt(1+c*x)))^4/(b*c)-(a+b*arcsin(sqrt(1-c*x)/sqrt(1+c*x)))^3*log(1-exp(2*I*arcsin(sqrt(1-c*x)/sqrt(1+c*x))))/c+3/2*I*b*(a+b*arcsin(sqrt(1-c*x)/sqrt(1+c*x)))^2*polylog(2,exp(2*I*arcsin(sqrt(1-c*x)/sqrt(1+c*x))))/c-3/2*b^2*(a+b*arcsin(sqrt(1-c*x)/sqrt(1+c*x)))*polylog(3,exp(2*I*arcsin(sqrt(1-c*x)/sqrt(1+c*x))))/c-3/4*I*b^3*polylog(4,exp(2*I*arcsin(sqrt(1-c*x)/sqrt(1+c*x))))/c],
[(a+b*arcsin(sqrt(1-c*x)/sqrt(1+c*x)))^2/(1-c^2*x^2),x,7,1/3*I*(a+b*arcsin(sqrt(1-c*x)/sqrt(1+c*x)))^3/(b*c)-(a+b*arcsin(sqrt(1-c*x)/sqrt(1+c*x)))^2*log(1-exp(2*I*arcsin(sqrt(1-c*x)/sqrt(1+c*x))))/c+I*b*(a+b*arcsin(sqrt(1-c*x)/sqrt(1+c*x)))*polylog(2,exp(2*I*arcsin(sqrt(1-c*x)/sqrt(1+c*x))))/c-1/2*b^2*polylog(3,exp(2*I*arcsin(sqrt(1-c*x)/sqrt(1+c*x))))/c],
[(a+b*arcsin(sqrt(1-c*x)/sqrt(1+c*x)))/(1-c^2*x^2),x,6,1/2*I*(a+b*arcsin(sqrt(1-c*x)/sqrt(1+c*x)))^2/(b*c)-(a+b*arcsin(sqrt(1-c*x)/sqrt(1+c*x)))*log(1-exp(2*I*arcsin(sqrt(1-c*x)/sqrt(1+c*x))))/c+1/2*I*b*polylog(2,exp(2*I*arcsin(sqrt(1-c*x)/sqrt(1+c*x))))/c],
[1/((1-c^2*x^2)*(a+b*arcsin(sqrt(1-c*x)/sqrt(1+c*x)))),x,0,Unintegrable(1/((1-c^2*x^2)*(a+b*arcsin(sqrt(1-c*x)/sqrt(1+c*x)))),x)],
[1/((1-c^2*x^2)*(a+b*arcsin(sqrt(1-c*x)/sqrt(1+c*x)))^2),x,0,Unintegrable(1/((1-c^2*x^2)*(a+b*arcsin(sqrt(1-c*x)/sqrt(1+c*x)))^2),x)],

# Integrands involving inverse sines of exponentials
[exp(x)*arcsin(exp(x)),x,3,exp(x)*arcsin(exp(x))+sqrt(1-exp(2*x))],
[arcsin(c*exp(a+b*x)),x,6,-1/2*I*arcsin(c*exp(a+b*x))^2/b+arcsin(c*exp(a+b*x))*log(1-exp(2*I*arcsin(c*exp(a+b*x))))/b-1/2*I*polylog(2,exp(2*I*arcsin(c*exp(a+b*x))))/b],

# Integrands involving exponentials of inverse sines

# Integrands of the form x^m E^(ArcSin[a x]^n)
[exp(arcsin(a*x))*x^3,x,6,-1/10*exp(arcsin(a*x))*cos(2*arcsin(a*x))/a^4+1/34*exp(arcsin(a*x))*cos(4*arcsin(a*x))/a^4+1/20*exp(arcsin(a*x))*sin(2*arcsin(a*x))/a^4-1/136*exp(arcsin(a*x))*sin(4*arcsin(a*x))/a^4],
[exp(arcsin(a*x))*x^2,x,6,1/8*exp(arcsin(a*x))*x/a^2-1/40*exp(arcsin(a*x))*cos(3*arcsin(a*x))/a^3-3/40*exp(arcsin(a*x))*sin(3*arcsin(a*x))/a^3+1/8*exp(arcsin(a*x))*sqrt(1-a^2*x^2)/a^3],
[exp(arcsin(a*x))*x,x,5,-1/5*exp(arcsin(a*x))*cos(2*arcsin(a*x))/a^2+1/10*exp(arcsin(a*x))*sin(2*arcsin(a*x))/a^2],
[exp(arcsin(a*x)),x,2,1/2*exp(arcsin(a*x))*x+1/2*exp(arcsin(a*x))*sqrt(1-a^2*x^2)/a],
[exp(arcsin(a*x))/x,x,6,I*exp(arcsin(a*x))-2*I*exp(arcsin(a*x))*hypergeom([-1/2*I,1],[1-1/2*I],exp(2*I*arcsin(a*x)))],
[exp(arcsin(a*x))/x^2,x,6,(1-I)*a*exp((1+I)*arcsin(a*x))*hypergeom([1/2-1/2*I,1],[3/2-1/2*I],exp(2*I*arcsin(a*x)))+(-2+2*I)*a*exp((1+I)*arcsin(a*x))*hypergeom([1/2-1/2*I,2],[3/2-1/2*I],exp(2*I*arcsin(a*x)))],
[exp(arcsin(a*x)^2)*x^3,x,12,1/16*I*E*erfi(-I+arcsin(a*x))*sqrt(Pi)/a^4-1/16*I*E*erfi(I+arcsin(a*x))*sqrt(Pi)/a^4-1/32*I*exp(4)*erfi(-2*I+arcsin(a*x))*sqrt(Pi)/a^4+1/32*I*exp(4)*erfi(2*I+arcsin(a*x))*sqrt(Pi)/a^4],
[exp(arcsin(a*x)^2)*x^2,x,12,1/16*exp(1/4)*erfi(1/2*(-I+2*arcsin(a*x)))*sqrt(Pi)/a^3+1/16*exp(1/4)*erfi(1/2*(I+2*arcsin(a*x)))*sqrt(Pi)/a^3-1/16*exp(9/4)*erfi(1/2*(-3*I+2*arcsin(a*x)))*sqrt(Pi)/a^3-1/16*exp(9/4)*erfi(1/2*(3*I+2*arcsin(a*x)))*sqrt(Pi)/a^3],
[exp(arcsin(a*x)^2)*x,x,8,1/8*I*E*erfi(-I+arcsin(a*x))*sqrt(Pi)/a^2-1/8*I*E*erfi(I+arcsin(a*x))*sqrt(Pi)/a^2],
[exp(arcsin(a*x)^2),x,7,1/4*exp(1/4)*erfi(1/2*(-I+2*arcsin(a*x)))*sqrt(Pi)/a+1/4*exp(1/4)*erfi(1/2*(I+2*arcsin(a*x)))*sqrt(Pi)/a],
[exp(arcsin(a*x)^2)/x,x,2,a*CannotIntegrate(exp(arcsin(a*x)^2)/(a*x),x)],
[exp(arcsin(a*x)^2)/x^2,x,2,a^2*CannotIntegrate(exp(arcsin(a*x)^2)/(a^2*x^2),x)],

# Integrands of the form x^m E^(ArcSin[a+b x]^n)
[exp(arcsin(a+b*x))*x^3,x,17,-3/8*a*exp(arcsin(a+b*x))*(a+b*x)/b^4-1/2*a^3*exp(arcsin(a+b*x))*(a+b*x)/b^4-1/10*exp(arcsin(a+b*x))*cos(2*arcsin(a+b*x))/b^4-3/5*a^2*exp(arcsin(a+b*x))*cos(2*arcsin(a+b*x))/b^4+3/40*a*exp(arcsin(a+b*x))*cos(3*arcsin(a+b*x))/b^4+1/34*exp(arcsin(a+b*x))*cos(4*arcsin(a+b*x))/b^4+1/20*exp(arcsin(a+b*x))*sin(2*arcsin(a+b*x))/b^4+3/10*a^2*exp(arcsin(a+b*x))*sin(2*arcsin(a+b*x))/b^4+9/40*a*exp(arcsin(a+b*x))*sin(3*arcsin(a+b*x))/b^4-1/136*exp(arcsin(a+b*x))*sin(4*arcsin(a+b*x))/b^4-3/8*a*exp(arcsin(a+b*x))*sqrt(1-(a+b*x)^2)/b^4-1/2*a^3*exp(arcsin(a+b*x))*sqrt(1-(a+b*x)^2)/b^4],
[exp(arcsin(a+b*x))*x^2,x,13,1/8*exp(arcsin(a+b*x))*(a+b*x)/b^3+1/2*a^2*exp(arcsin(a+b*x))*(a+b*x)/b^3+2/5*a*exp(arcsin(a+b*x))*cos(2*arcsin(a+b*x))/b^3-1/40*exp(arcsin(a+b*x))*cos(3*arcsin(a+b*x))/b^3-1/5*a*exp(arcsin(a+b*x))*sin(2*arcsin(a+b*x))/b^3-3/40*exp(arcsin(a+b*x))*sin(3*arcsin(a+b*x))/b^3+1/8*exp(arcsin(a+b*x))*sqrt(1-(a+b*x)^2)/b^3+1/2*a^2*exp(arcsin(a+b*x))*sqrt(1-(a+b*x)^2)/b^3],
[exp(arcsin(a+b*x))*x,x,9,-1/2*a*exp(arcsin(a+b*x))*(a+b*x)/b^2-1/5*exp(arcsin(a+b*x))*cos(2*arcsin(a+b*x))/b^2+1/10*exp(arcsin(a+b*x))*sin(2*arcsin(a+b*x))/b^2-1/2*a*exp(arcsin(a+b*x))*sqrt(1-(a+b*x)^2)/b^2],
[exp(arcsin(a+b*x)),x,2,1/2*exp(arcsin(a+b*x))*(a+b*x)/b+1/2*exp(arcsin(a+b*x))*sqrt(1-(a+b*x)^2)/b],
[exp(arcsin(a+b*x))/x,x,3,b*CannotIntegrate(exp(arcsin(a+b*x))/(b*x),x)],
[exp(arcsin(a+b*x))/x^2,x,3,b^2*CannotIntegrate(exp(arcsin(a+b*x))/(b^2*x^2),x)],
[exp(arcsin(a+b*x)^2)*x^3,x,37,1/16*I*E*erfi(-I+arcsin(a+b*x))*sqrt(Pi)/b^4+3/8*I*a^2*E*erfi(-I+arcsin(a+b*x))*sqrt(Pi)/b^4-1/16*I*E*erfi(I+arcsin(a+b*x))*sqrt(Pi)/b^4-3/8*I*a^2*E*erfi(I+arcsin(a+b*x))*sqrt(Pi)/b^4-1/32*I*exp(4)*erfi(-2*I+arcsin(a+b*x))*sqrt(Pi)/b^4+1/32*I*exp(4)*erfi(2*I+arcsin(a+b*x))*sqrt(Pi)/b^4-3/16*a*exp(1/4)*erfi(1/2*(-I+2*arcsin(a+b*x)))*sqrt(Pi)/b^4-1/4*a^3*exp(1/4)*erfi(1/2*(-I+2*arcsin(a+b*x)))*sqrt(Pi)/b^4-3/16*a*exp(1/4)*erfi(1/2*(I+2*arcsin(a+b*x)))*sqrt(Pi)/b^4-1/4*a^3*exp(1/4)*erfi(1/2*(I+2*arcsin(a+b*x)))*sqrt(Pi)/b^4+3/16*a*exp(9/4)*erfi(1/2*(-3*I+2*arcsin(a+b*x)))*sqrt(Pi)/b^4+3/16*a*exp(9/4)*erfi(1/2*(3*I+2*arcsin(a+b*x)))*sqrt(Pi)/b^4],
[exp(arcsin(a+b*x)^2)*x^2,x,27,-1/4*I*a*E*erfi(-I+arcsin(a+b*x))*sqrt(Pi)/b^3+1/4*I*a*E*erfi(I+arcsin(a+b*x))*sqrt(Pi)/b^3+1/16*exp(1/4)*erfi(1/2*(-I+2*arcsin(a+b*x)))*sqrt(Pi)/b^3+1/4*a^2*exp(1/4)*erfi(1/2*(-I+2*arcsin(a+b*x)))*sqrt(Pi)/b^3+1/16*exp(1/4)*erfi(1/2*(I+2*arcsin(a+b*x)))*sqrt(Pi)/b^3+1/4*a^2*exp(1/4)*erfi(1/2*(I+2*arcsin(a+b*x)))*sqrt(Pi)/b^3-1/16*exp(9/4)*erfi(1/2*(-3*I+2*arcsin(a+b*x)))*sqrt(Pi)/b^3-1/16*exp(9/4)*erfi(1/2*(3*I+2*arcsin(a+b*x)))*sqrt(Pi)/b^3],
[exp(arcsin(a+b*x)^2)*x,x,17,1/8*I*E*erfi(-I+arcsin(a+b*x))*sqrt(Pi)/b^2-1/8*I*E*erfi(I+arcsin(a+b*x))*sqrt(Pi)/b^2-1/4*a*exp(1/4)*erfi(1/2*(-I+2*arcsin(a+b*x)))*sqrt(Pi)/b^2-1/4*a*exp(1/4)*erfi(1/2*(I+2*arcsin(a+b*x)))*sqrt(Pi)/b^2],
[exp(arcsin(a+b*x)^2),x,7,1/4*exp(1/4)*erfi(1/2*(-I+2*arcsin(a+b*x)))*sqrt(Pi)/b+1/4*exp(1/4)*erfi(1/2*(I+2*arcsin(a+b*x)))*sqrt(Pi)/b],
[exp(arcsin(a+b*x)^2)/x,x,3,b*CannotIntegrate(exp(arcsin(a+b*x)^2)/(b*x),x)],
[exp(arcsin(a+b*x)^2)/x^2,x,3,b^2*CannotIntegrate(exp(arcsin(a+b*x)^2)/(b^2*x^2),x)],

# Integrands of the form (1-a^2 x^2)^(m/2) E^(ArcSin[a x]^n)
[exp(arcsin(a*x))*(1-a^2*x^2)^(5/2),x,7,144/629*exp(arcsin(a*x))/a+72/629*exp(arcsin(a*x))*(1-a^2*x^2)/a+120/629*exp(arcsin(a*x))*x*(1-a^2*x^2)^(3/2)+30/629*exp(arcsin(a*x))*(1-a^2*x^2)^2/a+6/37*exp(arcsin(a*x))*x*(1-a^2*x^2)^(5/2)+1/37*exp(arcsin(a*x))*(1-a^2*x^2)^3/a+144/629*exp(arcsin(a*x))*x*sqrt(1-a^2*x^2)],
[exp(arcsin(a*x))*(1-a^2*x^2)^(3/2),x,6,24/85*exp(arcsin(a*x))/a+12/85*exp(arcsin(a*x))*(1-a^2*x^2)/a+4/17*exp(arcsin(a*x))*x*(1-a^2*x^2)^(3/2)+1/17*exp(arcsin(a*x))*(1-a^2*x^2)^2/a+24/85*exp(arcsin(a*x))*x*sqrt(1-a^2*x^2)],
[exp(arcsin(a*x))*(1-a^2*x^2)^(1/2),x,5,2/5*exp(arcsin(a*x))/a+1/5*exp(arcsin(a*x))*(1-a^2*x^2)/a+2/5*exp(arcsin(a*x))*x*sqrt(1-a^2*x^2)],
[exp(arcsin(a*x))/(1-a^2*x^2)^(1/2),x,4,exp(arcsin(a*x))/a],
[exp(arcsin(a*x))/(1-a^2*x^2)^(3/2),x,4,(4/5-8/5*I)*exp((1+2*I)*arcsin(a*x))*hypergeom([1-1/2*I,2],[2-1/2*I],-exp(2*I*arcsin(a*x)))/a],
[exp(arcsin(a*x))/(1-a^2*x^2)^(5/2),x,5,1/3*exp(arcsin(a*x))*x/(1-a^2*x^2)^(3/2)-1/6*exp(arcsin(a*x))/(a*(1-a^2*x^2))+(2/3-4/3*I)*exp((1+2*I)*arcsin(a*x))*hypergeom([1-1/2*I,2],[2-1/2*I],-exp(2*I*arcsin(a*x)))/a],

# Miscellaneous integrands involving inverse sines
[arcsin(c/(a+b*x)),x,6,(a+b*x)*arccsc(a/c+b*x/c)/b+c*arctanh(sqrt(1-c^2/(a+b*x)^2))/b],
[x/arcsin(sin(x)),x,-1,arcsin(sin(x))+log(arcsin(sin(x)))*(-arcsin(sin(x))+x*sec(x)*sqrt(cos(x)^2))],
[arcsin(sqrt(1+b*x^2))^n/sqrt(1+b*x^2),x,2,arcsin(sqrt(1+b*x^2))^(1+n)*sqrt(-b*x^2)/(b*(1+n)*x)],
[1/(arcsin(sqrt(1+b*x^2))*sqrt(1+b*x^2)),x,2,log(arcsin(sqrt(1+b*x^2)))*sqrt(-b*x^2)/(b*x)],

#  Following integrands are equal. 
[x/(1-x^2)+1/(arcsin(x)*sqrt(1-x^2)),x,3,-1/2*log(1-x^2)+log(arcsin(x))],
[(x*arcsin(x)+sqrt(1-x^2))/(arcsin(x)-x^2*arcsin(x)),x,-1,-1/2*log(1-x^2)+log(arcsin(x))]]:
