1,1,249,0,0.537678," ","integrate((a+b*csc(d*x+c)^2)^4,x, algorithm=""fricas"")","-\frac{4 \, {\left(105 \, a^{3} b + 105 \, a^{2} b^{2} + 56 \, a b^{3} + 12 \, b^{4}\right)} \cos\left(d x + c\right)^{7} - 14 \, {\left(90 \, a^{3} b + 105 \, a^{2} b^{2} + 56 \, a b^{3} + 12 \, b^{4}\right)} \cos\left(d x + c\right)^{5} + 70 \, {\left(18 \, a^{3} b + 24 \, a^{2} b^{2} + 14 \, a b^{3} + 3 \, b^{4}\right)} \cos\left(d x + c\right)^{3} - 105 \, {\left(4 \, a^{3} b + 6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4}\right)} \cos\left(d x + c\right) - 105 \, {\left(a^{4} d x \cos\left(d x + c\right)^{6} - 3 \, a^{4} d x \cos\left(d x + c\right)^{4} + 3 \, a^{4} d x \cos\left(d x + c\right)^{2} - a^{4} d x\right)} \sin\left(d x + c\right)}{105 \, {\left(d \cos\left(d x + c\right)^{6} - 3 \, d \cos\left(d x + c\right)^{4} + 3 \, d \cos\left(d x + c\right)^{2} - d\right)} \sin\left(d x + c\right)}"," ",0,"-1/105*(4*(105*a^3*b + 105*a^2*b^2 + 56*a*b^3 + 12*b^4)*cos(d*x + c)^7 - 14*(90*a^3*b + 105*a^2*b^2 + 56*a*b^3 + 12*b^4)*cos(d*x + c)^5 + 70*(18*a^3*b + 24*a^2*b^2 + 14*a*b^3 + 3*b^4)*cos(d*x + c)^3 - 105*(4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4)*cos(d*x + c) - 105*(a^4*d*x*cos(d*x + c)^6 - 3*a^4*d*x*cos(d*x + c)^4 + 3*a^4*d*x*cos(d*x + c)^2 - a^4*d*x)*sin(d*x + c))/((d*cos(d*x + c)^6 - 3*d*cos(d*x + c)^4 + 3*d*cos(d*x + c)^2 - d)*sin(d*x + c))","B",0
2,1,159,0,0.473101," ","integrate((a+b*csc(d*x+c)^2)^3,x, algorithm=""fricas"")","-\frac{{\left(45 \, a^{2} b + 30 \, a b^{2} + 8 \, b^{3}\right)} \cos\left(d x + c\right)^{5} - 5 \, {\left(18 \, a^{2} b + 15 \, a b^{2} + 4 \, b^{3}\right)} \cos\left(d x + c\right)^{3} + 15 \, {\left(3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right)} \cos\left(d x + c\right) - 15 \, {\left(a^{3} d x \cos\left(d x + c\right)^{4} - 2 \, a^{3} d x \cos\left(d x + c\right)^{2} + a^{3} d x\right)} \sin\left(d x + c\right)}{15 \, {\left(d \cos\left(d x + c\right)^{4} - 2 \, d \cos\left(d x + c\right)^{2} + d\right)} \sin\left(d x + c\right)}"," ",0,"-1/15*((45*a^2*b + 30*a*b^2 + 8*b^3)*cos(d*x + c)^5 - 5*(18*a^2*b + 15*a*b^2 + 4*b^3)*cos(d*x + c)^3 + 15*(3*a^2*b + 3*a*b^2 + b^3)*cos(d*x + c) - 15*(a^3*d*x*cos(d*x + c)^4 - 2*a^3*d*x*cos(d*x + c)^2 + a^3*d*x)*sin(d*x + c))/((d*cos(d*x + c)^4 - 2*d*cos(d*x + c)^2 + d)*sin(d*x + c))","B",0
3,1,91,0,0.402884," ","integrate((a+b*csc(d*x+c)^2)^2,x, algorithm=""fricas"")","-\frac{2 \, {\left(3 \, a b + b^{2}\right)} \cos\left(d x + c\right)^{3} - 3 \, {\left(2 \, a b + b^{2}\right)} \cos\left(d x + c\right) - 3 \, {\left(a^{2} d x \cos\left(d x + c\right)^{2} - a^{2} d x\right)} \sin\left(d x + c\right)}{3 \, {\left(d \cos\left(d x + c\right)^{2} - d\right)} \sin\left(d x + c\right)}"," ",0,"-1/3*(2*(3*a*b + b^2)*cos(d*x + c)^3 - 3*(2*a*b + b^2)*cos(d*x + c) - 3*(a^2*d*x*cos(d*x + c)^2 - a^2*d*x)*sin(d*x + c))/((d*cos(d*x + c)^2 - d)*sin(d*x + c))","B",0
4,1,32,0,0.430491," ","integrate(a+b*csc(d*x+c)^2,x, algorithm=""fricas"")","\frac{a d x \sin\left(d x + c\right) - b \cos\left(d x + c\right)}{d \sin\left(d x + c\right)}"," ",0,"(a*d*x*sin(d*x + c) - b*cos(d*x + c))/(d*sin(d*x + c))","A",0
5,1,260,0,0.451751," ","integrate(1/(a+b*csc(d*x+c)^2),x, algorithm=""fricas"")","\left[\frac{4 \, d x + \sqrt{-\frac{b}{a + b}} \log\left(\frac{{\left(a^{2} + 8 \, a b + 8 \, b^{2}\right)} \cos\left(d x + c\right)^{4} - 2 \, {\left(a^{2} + 5 \, a b + 4 \, b^{2}\right)} \cos\left(d x + c\right)^{2} + 4 \, {\left({\left(a^{2} + 3 \, a b + 2 \, b^{2}\right)} \cos\left(d x + c\right)^{3} - {\left(a^{2} + 2 \, a b + b^{2}\right)} \cos\left(d x + c\right)\right)} \sqrt{-\frac{b}{a + b}} \sin\left(d x + c\right) + a^{2} + 2 \, a b + b^{2}}{a^{2} \cos\left(d x + c\right)^{4} - 2 \, {\left(a^{2} + a b\right)} \cos\left(d x + c\right)^{2} + a^{2} + 2 \, a b + b^{2}}\right)}{4 \, a d}, \frac{2 \, d x + \sqrt{\frac{b}{a + b}} \arctan\left(\frac{{\left({\left(a + 2 \, b\right)} \cos\left(d x + c\right)^{2} - a - b\right)} \sqrt{\frac{b}{a + b}}}{2 \, b \cos\left(d x + c\right) \sin\left(d x + c\right)}\right)}{2 \, a d}\right]"," ",0,"[1/4*(4*d*x + sqrt(-b/(a + b))*log(((a^2 + 8*a*b + 8*b^2)*cos(d*x + c)^4 - 2*(a^2 + 5*a*b + 4*b^2)*cos(d*x + c)^2 + 4*((a^2 + 3*a*b + 2*b^2)*cos(d*x + c)^3 - (a^2 + 2*a*b + b^2)*cos(d*x + c))*sqrt(-b/(a + b))*sin(d*x + c) + a^2 + 2*a*b + b^2)/(a^2*cos(d*x + c)^4 - 2*(a^2 + a*b)*cos(d*x + c)^2 + a^2 + 2*a*b + b^2)))/(a*d), 1/2*(2*d*x + sqrt(b/(a + b))*arctan(1/2*((a + 2*b)*cos(d*x + c)^2 - a - b)*sqrt(b/(a + b))/(b*cos(d*x + c)*sin(d*x + c))))/(a*d)]","A",0
6,1,492,0,0.491064," ","integrate(1/(a+b*csc(d*x+c)^2)^2,x, algorithm=""fricas"")","\left[\frac{8 \, {\left(a^{2} + a b\right)} d x \cos\left(d x + c\right)^{2} - 4 \, a b \cos\left(d x + c\right) \sin\left(d x + c\right) - 8 \, {\left(a^{2} + 2 \, a b + b^{2}\right)} d x + {\left({\left(3 \, a^{2} + 2 \, a b\right)} \cos\left(d x + c\right)^{2} - 3 \, a^{2} - 5 \, a b - 2 \, b^{2}\right)} \sqrt{-\frac{b}{a + b}} \log\left(\frac{{\left(a^{2} + 8 \, a b + 8 \, b^{2}\right)} \cos\left(d x + c\right)^{4} - 2 \, {\left(a^{2} + 5 \, a b + 4 \, b^{2}\right)} \cos\left(d x + c\right)^{2} + 4 \, {\left({\left(a^{2} + 3 \, a b + 2 \, b^{2}\right)} \cos\left(d x + c\right)^{3} - {\left(a^{2} + 2 \, a b + b^{2}\right)} \cos\left(d x + c\right)\right)} \sqrt{-\frac{b}{a + b}} \sin\left(d x + c\right) + a^{2} + 2 \, a b + b^{2}}{a^{2} \cos\left(d x + c\right)^{4} - 2 \, {\left(a^{2} + a b\right)} \cos\left(d x + c\right)^{2} + a^{2} + 2 \, a b + b^{2}}\right)}{8 \, {\left({\left(a^{4} + a^{3} b\right)} d \cos\left(d x + c\right)^{2} - {\left(a^{4} + 2 \, a^{3} b + a^{2} b^{2}\right)} d\right)}}, \frac{4 \, {\left(a^{2} + a b\right)} d x \cos\left(d x + c\right)^{2} - 2 \, a b \cos\left(d x + c\right) \sin\left(d x + c\right) - 4 \, {\left(a^{2} + 2 \, a b + b^{2}\right)} d x + {\left({\left(3 \, a^{2} + 2 \, a b\right)} \cos\left(d x + c\right)^{2} - 3 \, a^{2} - 5 \, a b - 2 \, b^{2}\right)} \sqrt{\frac{b}{a + b}} \arctan\left(\frac{{\left({\left(a + 2 \, b\right)} \cos\left(d x + c\right)^{2} - a - b\right)} \sqrt{\frac{b}{a + b}}}{2 \, b \cos\left(d x + c\right) \sin\left(d x + c\right)}\right)}{4 \, {\left({\left(a^{4} + a^{3} b\right)} d \cos\left(d x + c\right)^{2} - {\left(a^{4} + 2 \, a^{3} b + a^{2} b^{2}\right)} d\right)}}\right]"," ",0,"[1/8*(8*(a^2 + a*b)*d*x*cos(d*x + c)^2 - 4*a*b*cos(d*x + c)*sin(d*x + c) - 8*(a^2 + 2*a*b + b^2)*d*x + ((3*a^2 + 2*a*b)*cos(d*x + c)^2 - 3*a^2 - 5*a*b - 2*b^2)*sqrt(-b/(a + b))*log(((a^2 + 8*a*b + 8*b^2)*cos(d*x + c)^4 - 2*(a^2 + 5*a*b + 4*b^2)*cos(d*x + c)^2 + 4*((a^2 + 3*a*b + 2*b^2)*cos(d*x + c)^3 - (a^2 + 2*a*b + b^2)*cos(d*x + c))*sqrt(-b/(a + b))*sin(d*x + c) + a^2 + 2*a*b + b^2)/(a^2*cos(d*x + c)^4 - 2*(a^2 + a*b)*cos(d*x + c)^2 + a^2 + 2*a*b + b^2)))/((a^4 + a^3*b)*d*cos(d*x + c)^2 - (a^4 + 2*a^3*b + a^2*b^2)*d), 1/4*(4*(a^2 + a*b)*d*x*cos(d*x + c)^2 - 2*a*b*cos(d*x + c)*sin(d*x + c) - 4*(a^2 + 2*a*b + b^2)*d*x + ((3*a^2 + 2*a*b)*cos(d*x + c)^2 - 3*a^2 - 5*a*b - 2*b^2)*sqrt(b/(a + b))*arctan(1/2*((a + 2*b)*cos(d*x + c)^2 - a - b)*sqrt(b/(a + b))/(b*cos(d*x + c)*sin(d*x + c))))/((a^4 + a^3*b)*d*cos(d*x + c)^2 - (a^4 + 2*a^3*b + a^2*b^2)*d)]","B",0
7,1,950,0,0.588479," ","integrate(1/(a+b*csc(d*x+c)^2)^3,x, algorithm=""fricas"")","\left[\frac{32 \, {\left(a^{4} + 2 \, a^{3} b + a^{2} b^{2}\right)} d x \cos\left(d x + c\right)^{4} - 64 \, {\left(a^{4} + 3 \, a^{3} b + 3 \, a^{2} b^{2} + a b^{3}\right)} d x \cos\left(d x + c\right)^{2} + 32 \, {\left(a^{4} + 4 \, a^{3} b + 6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4}\right)} d x + {\left({\left(15 \, a^{4} + 20 \, a^{3} b + 8 \, a^{2} b^{2}\right)} \cos\left(d x + c\right)^{4} + 15 \, a^{4} + 50 \, a^{3} b + 63 \, a^{2} b^{2} + 36 \, a b^{3} + 8 \, b^{4} - 2 \, {\left(15 \, a^{4} + 35 \, a^{3} b + 28 \, a^{2} b^{2} + 8 \, a b^{3}\right)} \cos\left(d x + c\right)^{2}\right)} \sqrt{-\frac{b}{a + b}} \log\left(\frac{{\left(a^{2} + 8 \, a b + 8 \, b^{2}\right)} \cos\left(d x + c\right)^{4} - 2 \, {\left(a^{2} + 5 \, a b + 4 \, b^{2}\right)} \cos\left(d x + c\right)^{2} + 4 \, {\left({\left(a^{2} + 3 \, a b + 2 \, b^{2}\right)} \cos\left(d x + c\right)^{3} - {\left(a^{2} + 2 \, a b + b^{2}\right)} \cos\left(d x + c\right)\right)} \sqrt{-\frac{b}{a + b}} \sin\left(d x + c\right) + a^{2} + 2 \, a b + b^{2}}{a^{2} \cos\left(d x + c\right)^{4} - 2 \, {\left(a^{2} + a b\right)} \cos\left(d x + c\right)^{2} + a^{2} + 2 \, a b + b^{2}}\right) - 4 \, {\left(3 \, {\left(3 \, a^{3} b + 2 \, a^{2} b^{2}\right)} \cos\left(d x + c\right)^{3} - {\left(9 \, a^{3} b + 13 \, a^{2} b^{2} + 4 \, a b^{3}\right)} \cos\left(d x + c\right)\right)} \sin\left(d x + c\right)}{32 \, {\left({\left(a^{7} + 2 \, a^{6} b + a^{5} b^{2}\right)} d \cos\left(d x + c\right)^{4} - 2 \, {\left(a^{7} + 3 \, a^{6} b + 3 \, a^{5} b^{2} + a^{4} b^{3}\right)} d \cos\left(d x + c\right)^{2} + {\left(a^{7} + 4 \, a^{6} b + 6 \, a^{5} b^{2} + 4 \, a^{4} b^{3} + a^{3} b^{4}\right)} d\right)}}, \frac{16 \, {\left(a^{4} + 2 \, a^{3} b + a^{2} b^{2}\right)} d x \cos\left(d x + c\right)^{4} - 32 \, {\left(a^{4} + 3 \, a^{3} b + 3 \, a^{2} b^{2} + a b^{3}\right)} d x \cos\left(d x + c\right)^{2} + 16 \, {\left(a^{4} + 4 \, a^{3} b + 6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4}\right)} d x + {\left({\left(15 \, a^{4} + 20 \, a^{3} b + 8 \, a^{2} b^{2}\right)} \cos\left(d x + c\right)^{4} + 15 \, a^{4} + 50 \, a^{3} b + 63 \, a^{2} b^{2} + 36 \, a b^{3} + 8 \, b^{4} - 2 \, {\left(15 \, a^{4} + 35 \, a^{3} b + 28 \, a^{2} b^{2} + 8 \, a b^{3}\right)} \cos\left(d x + c\right)^{2}\right)} \sqrt{\frac{b}{a + b}} \arctan\left(\frac{{\left({\left(a + 2 \, b\right)} \cos\left(d x + c\right)^{2} - a - b\right)} \sqrt{\frac{b}{a + b}}}{2 \, b \cos\left(d x + c\right) \sin\left(d x + c\right)}\right) - 2 \, {\left(3 \, {\left(3 \, a^{3} b + 2 \, a^{2} b^{2}\right)} \cos\left(d x + c\right)^{3} - {\left(9 \, a^{3} b + 13 \, a^{2} b^{2} + 4 \, a b^{3}\right)} \cos\left(d x + c\right)\right)} \sin\left(d x + c\right)}{16 \, {\left({\left(a^{7} + 2 \, a^{6} b + a^{5} b^{2}\right)} d \cos\left(d x + c\right)^{4} - 2 \, {\left(a^{7} + 3 \, a^{6} b + 3 \, a^{5} b^{2} + a^{4} b^{3}\right)} d \cos\left(d x + c\right)^{2} + {\left(a^{7} + 4 \, a^{6} b + 6 \, a^{5} b^{2} + 4 \, a^{4} b^{3} + a^{3} b^{4}\right)} d\right)}}\right]"," ",0,"[1/32*(32*(a^4 + 2*a^3*b + a^2*b^2)*d*x*cos(d*x + c)^4 - 64*(a^4 + 3*a^3*b + 3*a^2*b^2 + a*b^3)*d*x*cos(d*x + c)^2 + 32*(a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4)*d*x + ((15*a^4 + 20*a^3*b + 8*a^2*b^2)*cos(d*x + c)^4 + 15*a^4 + 50*a^3*b + 63*a^2*b^2 + 36*a*b^3 + 8*b^4 - 2*(15*a^4 + 35*a^3*b + 28*a^2*b^2 + 8*a*b^3)*cos(d*x + c)^2)*sqrt(-b/(a + b))*log(((a^2 + 8*a*b + 8*b^2)*cos(d*x + c)^4 - 2*(a^2 + 5*a*b + 4*b^2)*cos(d*x + c)^2 + 4*((a^2 + 3*a*b + 2*b^2)*cos(d*x + c)^3 - (a^2 + 2*a*b + b^2)*cos(d*x + c))*sqrt(-b/(a + b))*sin(d*x + c) + a^2 + 2*a*b + b^2)/(a^2*cos(d*x + c)^4 - 2*(a^2 + a*b)*cos(d*x + c)^2 + a^2 + 2*a*b + b^2)) - 4*(3*(3*a^3*b + 2*a^2*b^2)*cos(d*x + c)^3 - (9*a^3*b + 13*a^2*b^2 + 4*a*b^3)*cos(d*x + c))*sin(d*x + c))/((a^7 + 2*a^6*b + a^5*b^2)*d*cos(d*x + c)^4 - 2*(a^7 + 3*a^6*b + 3*a^5*b^2 + a^4*b^3)*d*cos(d*x + c)^2 + (a^7 + 4*a^6*b + 6*a^5*b^2 + 4*a^4*b^3 + a^3*b^4)*d), 1/16*(16*(a^4 + 2*a^3*b + a^2*b^2)*d*x*cos(d*x + c)^4 - 32*(a^4 + 3*a^3*b + 3*a^2*b^2 + a*b^3)*d*x*cos(d*x + c)^2 + 16*(a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4)*d*x + ((15*a^4 + 20*a^3*b + 8*a^2*b^2)*cos(d*x + c)^4 + 15*a^4 + 50*a^3*b + 63*a^2*b^2 + 36*a*b^3 + 8*b^4 - 2*(15*a^4 + 35*a^3*b + 28*a^2*b^2 + 8*a*b^3)*cos(d*x + c)^2)*sqrt(b/(a + b))*arctan(1/2*((a + 2*b)*cos(d*x + c)^2 - a - b)*sqrt(b/(a + b))/(b*cos(d*x + c)*sin(d*x + c))) - 2*(3*(3*a^3*b + 2*a^2*b^2)*cos(d*x + c)^3 - (9*a^3*b + 13*a^2*b^2 + 4*a*b^3)*cos(d*x + c))*sin(d*x + c))/((a^7 + 2*a^6*b + a^5*b^2)*d*cos(d*x + c)^4 - 2*(a^7 + 3*a^6*b + 3*a^5*b^2 + a^4*b^3)*d*cos(d*x + c)^2 + (a^7 + 4*a^6*b + 6*a^5*b^2 + 4*a^4*b^3 + a^3*b^4)*d)]","B",0
8,1,1592,0,0.791098," ","integrate(1/(a+b*csc(d*x+c)^2)^4,x, algorithm=""fricas"")","\left[\frac{192 \, {\left(a^{6} + 3 \, a^{5} b + 3 \, a^{4} b^{2} + a^{3} b^{3}\right)} d x \cos\left(d x + c\right)^{6} - 576 \, {\left(a^{6} + 4 \, a^{5} b + 6 \, a^{4} b^{2} + 4 \, a^{3} b^{3} + a^{2} b^{4}\right)} d x \cos\left(d x + c\right)^{4} + 576 \, {\left(a^{6} + 5 \, a^{5} b + 10 \, a^{4} b^{2} + 10 \, a^{3} b^{3} + 5 \, a^{2} b^{4} + a b^{5}\right)} d x \cos\left(d x + c\right)^{2} - 192 \, {\left(a^{6} + 6 \, a^{5} b + 15 \, a^{4} b^{2} + 20 \, a^{3} b^{3} + 15 \, a^{2} b^{4} + 6 \, a b^{5} + b^{6}\right)} d x + 3 \, {\left({\left(35 \, a^{6} + 70 \, a^{5} b + 56 \, a^{4} b^{2} + 16 \, a^{3} b^{3}\right)} \cos\left(d x + c\right)^{6} - 35 \, a^{6} - 175 \, a^{5} b - 371 \, a^{4} b^{2} - 429 \, a^{3} b^{3} - 286 \, a^{2} b^{4} - 104 \, a b^{5} - 16 \, b^{6} - 3 \, {\left(35 \, a^{6} + 105 \, a^{5} b + 126 \, a^{4} b^{2} + 72 \, a^{3} b^{3} + 16 \, a^{2} b^{4}\right)} \cos\left(d x + c\right)^{4} + 3 \, {\left(35 \, a^{6} + 140 \, a^{5} b + 231 \, a^{4} b^{2} + 198 \, a^{3} b^{3} + 88 \, a^{2} b^{4} + 16 \, a b^{5}\right)} \cos\left(d x + c\right)^{2}\right)} \sqrt{-\frac{b}{a + b}} \log\left(\frac{{\left(a^{2} + 8 \, a b + 8 \, b^{2}\right)} \cos\left(d x + c\right)^{4} - 2 \, {\left(a^{2} + 5 \, a b + 4 \, b^{2}\right)} \cos\left(d x + c\right)^{2} + 4 \, {\left({\left(a^{2} + 3 \, a b + 2 \, b^{2}\right)} \cos\left(d x + c\right)^{3} - {\left(a^{2} + 2 \, a b + b^{2}\right)} \cos\left(d x + c\right)\right)} \sqrt{-\frac{b}{a + b}} \sin\left(d x + c\right) + a^{2} + 2 \, a b + b^{2}}{a^{2} \cos\left(d x + c\right)^{4} - 2 \, {\left(a^{2} + a b\right)} \cos\left(d x + c\right)^{2} + a^{2} + 2 \, a b + b^{2}}\right) - 4 \, {\left({\left(87 \, a^{5} b + 116 \, a^{4} b^{2} + 44 \, a^{3} b^{3}\right)} \cos\left(d x + c\right)^{5} - 2 \, {\left(87 \, a^{5} b + 184 \, a^{4} b^{2} + 127 \, a^{3} b^{3} + 30 \, a^{2} b^{4}\right)} \cos\left(d x + c\right)^{3} + 3 \, {\left(29 \, a^{5} b + 84 \, a^{4} b^{2} + 89 \, a^{3} b^{3} + 42 \, a^{2} b^{4} + 8 \, a b^{5}\right)} \cos\left(d x + c\right)\right)} \sin\left(d x + c\right)}{192 \, {\left({\left(a^{10} + 3 \, a^{9} b + 3 \, a^{8} b^{2} + a^{7} b^{3}\right)} d \cos\left(d x + c\right)^{6} - 3 \, {\left(a^{10} + 4 \, a^{9} b + 6 \, a^{8} b^{2} + 4 \, a^{7} b^{3} + a^{6} b^{4}\right)} d \cos\left(d x + c\right)^{4} + 3 \, {\left(a^{10} + 5 \, a^{9} b + 10 \, a^{8} b^{2} + 10 \, a^{7} b^{3} + 5 \, a^{6} b^{4} + a^{5} b^{5}\right)} d \cos\left(d x + c\right)^{2} - {\left(a^{10} + 6 \, a^{9} b + 15 \, a^{8} b^{2} + 20 \, a^{7} b^{3} + 15 \, a^{6} b^{4} + 6 \, a^{5} b^{5} + a^{4} b^{6}\right)} d\right)}}, \frac{96 \, {\left(a^{6} + 3 \, a^{5} b + 3 \, a^{4} b^{2} + a^{3} b^{3}\right)} d x \cos\left(d x + c\right)^{6} - 288 \, {\left(a^{6} + 4 \, a^{5} b + 6 \, a^{4} b^{2} + 4 \, a^{3} b^{3} + a^{2} b^{4}\right)} d x \cos\left(d x + c\right)^{4} + 288 \, {\left(a^{6} + 5 \, a^{5} b + 10 \, a^{4} b^{2} + 10 \, a^{3} b^{3} + 5 \, a^{2} b^{4} + a b^{5}\right)} d x \cos\left(d x + c\right)^{2} - 96 \, {\left(a^{6} + 6 \, a^{5} b + 15 \, a^{4} b^{2} + 20 \, a^{3} b^{3} + 15 \, a^{2} b^{4} + 6 \, a b^{5} + b^{6}\right)} d x + 3 \, {\left({\left(35 \, a^{6} + 70 \, a^{5} b + 56 \, a^{4} b^{2} + 16 \, a^{3} b^{3}\right)} \cos\left(d x + c\right)^{6} - 35 \, a^{6} - 175 \, a^{5} b - 371 \, a^{4} b^{2} - 429 \, a^{3} b^{3} - 286 \, a^{2} b^{4} - 104 \, a b^{5} - 16 \, b^{6} - 3 \, {\left(35 \, a^{6} + 105 \, a^{5} b + 126 \, a^{4} b^{2} + 72 \, a^{3} b^{3} + 16 \, a^{2} b^{4}\right)} \cos\left(d x + c\right)^{4} + 3 \, {\left(35 \, a^{6} + 140 \, a^{5} b + 231 \, a^{4} b^{2} + 198 \, a^{3} b^{3} + 88 \, a^{2} b^{4} + 16 \, a b^{5}\right)} \cos\left(d x + c\right)^{2}\right)} \sqrt{\frac{b}{a + b}} \arctan\left(\frac{{\left({\left(a + 2 \, b\right)} \cos\left(d x + c\right)^{2} - a - b\right)} \sqrt{\frac{b}{a + b}}}{2 \, b \cos\left(d x + c\right) \sin\left(d x + c\right)}\right) - 2 \, {\left({\left(87 \, a^{5} b + 116 \, a^{4} b^{2} + 44 \, a^{3} b^{3}\right)} \cos\left(d x + c\right)^{5} - 2 \, {\left(87 \, a^{5} b + 184 \, a^{4} b^{2} + 127 \, a^{3} b^{3} + 30 \, a^{2} b^{4}\right)} \cos\left(d x + c\right)^{3} + 3 \, {\left(29 \, a^{5} b + 84 \, a^{4} b^{2} + 89 \, a^{3} b^{3} + 42 \, a^{2} b^{4} + 8 \, a b^{5}\right)} \cos\left(d x + c\right)\right)} \sin\left(d x + c\right)}{96 \, {\left({\left(a^{10} + 3 \, a^{9} b + 3 \, a^{8} b^{2} + a^{7} b^{3}\right)} d \cos\left(d x + c\right)^{6} - 3 \, {\left(a^{10} + 4 \, a^{9} b + 6 \, a^{8} b^{2} + 4 \, a^{7} b^{3} + a^{6} b^{4}\right)} d \cos\left(d x + c\right)^{4} + 3 \, {\left(a^{10} + 5 \, a^{9} b + 10 \, a^{8} b^{2} + 10 \, a^{7} b^{3} + 5 \, a^{6} b^{4} + a^{5} b^{5}\right)} d \cos\left(d x + c\right)^{2} - {\left(a^{10} + 6 \, a^{9} b + 15 \, a^{8} b^{2} + 20 \, a^{7} b^{3} + 15 \, a^{6} b^{4} + 6 \, a^{5} b^{5} + a^{4} b^{6}\right)} d\right)}}\right]"," ",0,"[1/192*(192*(a^6 + 3*a^5*b + 3*a^4*b^2 + a^3*b^3)*d*x*cos(d*x + c)^6 - 576*(a^6 + 4*a^5*b + 6*a^4*b^2 + 4*a^3*b^3 + a^2*b^4)*d*x*cos(d*x + c)^4 + 576*(a^6 + 5*a^5*b + 10*a^4*b^2 + 10*a^3*b^3 + 5*a^2*b^4 + a*b^5)*d*x*cos(d*x + c)^2 - 192*(a^6 + 6*a^5*b + 15*a^4*b^2 + 20*a^3*b^3 + 15*a^2*b^4 + 6*a*b^5 + b^6)*d*x + 3*((35*a^6 + 70*a^5*b + 56*a^4*b^2 + 16*a^3*b^3)*cos(d*x + c)^6 - 35*a^6 - 175*a^5*b - 371*a^4*b^2 - 429*a^3*b^3 - 286*a^2*b^4 - 104*a*b^5 - 16*b^6 - 3*(35*a^6 + 105*a^5*b + 126*a^4*b^2 + 72*a^3*b^3 + 16*a^2*b^4)*cos(d*x + c)^4 + 3*(35*a^6 + 140*a^5*b + 231*a^4*b^2 + 198*a^3*b^3 + 88*a^2*b^4 + 16*a*b^5)*cos(d*x + c)^2)*sqrt(-b/(a + b))*log(((a^2 + 8*a*b + 8*b^2)*cos(d*x + c)^4 - 2*(a^2 + 5*a*b + 4*b^2)*cos(d*x + c)^2 + 4*((a^2 + 3*a*b + 2*b^2)*cos(d*x + c)^3 - (a^2 + 2*a*b + b^2)*cos(d*x + c))*sqrt(-b/(a + b))*sin(d*x + c) + a^2 + 2*a*b + b^2)/(a^2*cos(d*x + c)^4 - 2*(a^2 + a*b)*cos(d*x + c)^2 + a^2 + 2*a*b + b^2)) - 4*((87*a^5*b + 116*a^4*b^2 + 44*a^3*b^3)*cos(d*x + c)^5 - 2*(87*a^5*b + 184*a^4*b^2 + 127*a^3*b^3 + 30*a^2*b^4)*cos(d*x + c)^3 + 3*(29*a^5*b + 84*a^4*b^2 + 89*a^3*b^3 + 42*a^2*b^4 + 8*a*b^5)*cos(d*x + c))*sin(d*x + c))/((a^10 + 3*a^9*b + 3*a^8*b^2 + a^7*b^3)*d*cos(d*x + c)^6 - 3*(a^10 + 4*a^9*b + 6*a^8*b^2 + 4*a^7*b^3 + a^6*b^4)*d*cos(d*x + c)^4 + 3*(a^10 + 5*a^9*b + 10*a^8*b^2 + 10*a^7*b^3 + 5*a^6*b^4 + a^5*b^5)*d*cos(d*x + c)^2 - (a^10 + 6*a^9*b + 15*a^8*b^2 + 20*a^7*b^3 + 15*a^6*b^4 + 6*a^5*b^5 + a^4*b^6)*d), 1/96*(96*(a^6 + 3*a^5*b + 3*a^4*b^2 + a^3*b^3)*d*x*cos(d*x + c)^6 - 288*(a^6 + 4*a^5*b + 6*a^4*b^2 + 4*a^3*b^3 + a^2*b^4)*d*x*cos(d*x + c)^4 + 288*(a^6 + 5*a^5*b + 10*a^4*b^2 + 10*a^3*b^3 + 5*a^2*b^4 + a*b^5)*d*x*cos(d*x + c)^2 - 96*(a^6 + 6*a^5*b + 15*a^4*b^2 + 20*a^3*b^3 + 15*a^2*b^4 + 6*a*b^5 + b^6)*d*x + 3*((35*a^6 + 70*a^5*b + 56*a^4*b^2 + 16*a^3*b^3)*cos(d*x + c)^6 - 35*a^6 - 175*a^5*b - 371*a^4*b^2 - 429*a^3*b^3 - 286*a^2*b^4 - 104*a*b^5 - 16*b^6 - 3*(35*a^6 + 105*a^5*b + 126*a^4*b^2 + 72*a^3*b^3 + 16*a^2*b^4)*cos(d*x + c)^4 + 3*(35*a^6 + 140*a^5*b + 231*a^4*b^2 + 198*a^3*b^3 + 88*a^2*b^4 + 16*a*b^5)*cos(d*x + c)^2)*sqrt(b/(a + b))*arctan(1/2*((a + 2*b)*cos(d*x + c)^2 - a - b)*sqrt(b/(a + b))/(b*cos(d*x + c)*sin(d*x + c))) - 2*((87*a^5*b + 116*a^4*b^2 + 44*a^3*b^3)*cos(d*x + c)^5 - 2*(87*a^5*b + 184*a^4*b^2 + 127*a^3*b^3 + 30*a^2*b^4)*cos(d*x + c)^3 + 3*(29*a^5*b + 84*a^4*b^2 + 89*a^3*b^3 + 42*a^2*b^4 + 8*a*b^5)*cos(d*x + c))*sin(d*x + c))/((a^10 + 3*a^9*b + 3*a^8*b^2 + a^7*b^3)*d*cos(d*x + c)^6 - 3*(a^10 + 4*a^9*b + 6*a^8*b^2 + 4*a^7*b^3 + a^6*b^4)*d*cos(d*x + c)^4 + 3*(a^10 + 5*a^9*b + 10*a^8*b^2 + 10*a^7*b^3 + 5*a^6*b^4 + a^5*b^5)*d*cos(d*x + c)^2 - (a^10 + 6*a^9*b + 15*a^8*b^2 + 20*a^7*b^3 + 15*a^6*b^4 + 6*a^5*b^5 + a^4*b^6)*d)]","B",0
9,1,1986,0,2.208404," ","integrate((a+b*csc(d*x+c)^2)^(5/2),x, algorithm=""fricas"")","\left[\frac{4 \, {\left(a^{2} \cos\left(d x + c\right)^{2} - a^{2}\right)} \sqrt{-a} \log\left(128 \, a^{4} \cos\left(d x + c\right)^{8} - 256 \, {\left(a^{4} + a^{3} b\right)} \cos\left(d x + c\right)^{6} + 160 \, {\left(a^{4} + 2 \, a^{3} b + a^{2} b^{2}\right)} \cos\left(d x + c\right)^{4} + a^{4} + 4 \, a^{3} b + 6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4} - 32 \, {\left(a^{4} + 3 \, a^{3} b + 3 \, a^{2} b^{2} + a b^{3}\right)} \cos\left(d x + c\right)^{2} - 8 \, {\left(16 \, a^{3} \cos\left(d x + c\right)^{7} - 24 \, {\left(a^{3} + a^{2} b\right)} \cos\left(d x + c\right)^{5} + 10 \, {\left(a^{3} + 2 \, a^{2} b + a b^{2}\right)} \cos\left(d x + c\right)^{3} - {\left(a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right)} \cos\left(d x + c\right)\right)} \sqrt{-a} \sqrt{\frac{a \cos\left(d x + c\right)^{2} - a - b}{\cos\left(d x + c\right)^{2} - 1}} \sin\left(d x + c\right)\right) \sin\left(d x + c\right) + {\left({\left(15 \, a^{2} + 10 \, a b + 3 \, b^{2}\right)} \cos\left(d x + c\right)^{2} - 15 \, a^{2} - 10 \, a b - 3 \, b^{2}\right)} \sqrt{b} \log\left(\frac{2 \, {\left({\left(a^{2} - 6 \, a b + b^{2}\right)} \cos\left(d x + c\right)^{4} - 2 \, {\left(a^{2} - 2 \, a b - 3 \, b^{2}\right)} \cos\left(d x + c\right)^{2} + 4 \, {\left({\left(a - b\right)} \cos\left(d x + c\right)^{3} - {\left(a + b\right)} \cos\left(d x + c\right)\right)} \sqrt{b} \sqrt{\frac{a \cos\left(d x + c\right)^{2} - a - b}{\cos\left(d x + c\right)^{2} - 1}} \sin\left(d x + c\right) + a^{2} + 2 \, a b + b^{2}\right)}}{\cos\left(d x + c\right)^{4} - 2 \, \cos\left(d x + c\right)^{2} + 1}\right) \sin\left(d x + c\right) - 4 \, {\left(3 \, {\left(3 \, a b + b^{2}\right)} \cos\left(d x + c\right)^{3} - {\left(9 \, a b + 5 \, b^{2}\right)} \cos\left(d x + c\right)\right)} \sqrt{\frac{a \cos\left(d x + c\right)^{2} - a - b}{\cos\left(d x + c\right)^{2} - 1}}}{32 \, {\left(d \cos\left(d x + c\right)^{2} - d\right)} \sin\left(d x + c\right)}, -\frac{{\left({\left(15 \, a^{2} + 10 \, a b + 3 \, b^{2}\right)} \cos\left(d x + c\right)^{2} - 15 \, a^{2} - 10 \, a b - 3 \, b^{2}\right)} \sqrt{-b} \arctan\left(-\frac{{\left({\left(a - b\right)} \cos\left(d x + c\right)^{2} - a - b\right)} \sqrt{-b} \sqrt{\frac{a \cos\left(d x + c\right)^{2} - a - b}{\cos\left(d x + c\right)^{2} - 1}} \sin\left(d x + c\right)}{2 \, {\left(a b \cos\left(d x + c\right)^{3} - {\left(a b + b^{2}\right)} \cos\left(d x + c\right)\right)}}\right) \sin\left(d x + c\right) - 2 \, {\left(a^{2} \cos\left(d x + c\right)^{2} - a^{2}\right)} \sqrt{-a} \log\left(128 \, a^{4} \cos\left(d x + c\right)^{8} - 256 \, {\left(a^{4} + a^{3} b\right)} \cos\left(d x + c\right)^{6} + 160 \, {\left(a^{4} + 2 \, a^{3} b + a^{2} b^{2}\right)} \cos\left(d x + c\right)^{4} + a^{4} + 4 \, a^{3} b + 6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4} - 32 \, {\left(a^{4} + 3 \, a^{3} b + 3 \, a^{2} b^{2} + a b^{3}\right)} \cos\left(d x + c\right)^{2} - 8 \, {\left(16 \, a^{3} \cos\left(d x + c\right)^{7} - 24 \, {\left(a^{3} + a^{2} b\right)} \cos\left(d x + c\right)^{5} + 10 \, {\left(a^{3} + 2 \, a^{2} b + a b^{2}\right)} \cos\left(d x + c\right)^{3} - {\left(a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right)} \cos\left(d x + c\right)\right)} \sqrt{-a} \sqrt{\frac{a \cos\left(d x + c\right)^{2} - a - b}{\cos\left(d x + c\right)^{2} - 1}} \sin\left(d x + c\right)\right) \sin\left(d x + c\right) + 2 \, {\left(3 \, {\left(3 \, a b + b^{2}\right)} \cos\left(d x + c\right)^{3} - {\left(9 \, a b + 5 \, b^{2}\right)} \cos\left(d x + c\right)\right)} \sqrt{\frac{a \cos\left(d x + c\right)^{2} - a - b}{\cos\left(d x + c\right)^{2} - 1}}}{16 \, {\left(d \cos\left(d x + c\right)^{2} - d\right)} \sin\left(d x + c\right)}, \frac{8 \, {\left(a^{2} \cos\left(d x + c\right)^{2} - a^{2}\right)} \sqrt{a} \arctan\left(\frac{{\left(8 \, a^{2} \cos\left(d x + c\right)^{4} - 8 \, {\left(a^{2} + a b\right)} \cos\left(d x + c\right)^{2} + a^{2} + 2 \, a b + b^{2}\right)} \sqrt{a} \sqrt{\frac{a \cos\left(d x + c\right)^{2} - a - b}{\cos\left(d x + c\right)^{2} - 1}} \sin\left(d x + c\right)}{4 \, {\left(2 \, a^{3} \cos\left(d x + c\right)^{5} - 3 \, {\left(a^{3} + a^{2} b\right)} \cos\left(d x + c\right)^{3} + {\left(a^{3} + 2 \, a^{2} b + a b^{2}\right)} \cos\left(d x + c\right)\right)}}\right) \sin\left(d x + c\right) + {\left({\left(15 \, a^{2} + 10 \, a b + 3 \, b^{2}\right)} \cos\left(d x + c\right)^{2} - 15 \, a^{2} - 10 \, a b - 3 \, b^{2}\right)} \sqrt{b} \log\left(\frac{2 \, {\left({\left(a^{2} - 6 \, a b + b^{2}\right)} \cos\left(d x + c\right)^{4} - 2 \, {\left(a^{2} - 2 \, a b - 3 \, b^{2}\right)} \cos\left(d x + c\right)^{2} + 4 \, {\left({\left(a - b\right)} \cos\left(d x + c\right)^{3} - {\left(a + b\right)} \cos\left(d x + c\right)\right)} \sqrt{b} \sqrt{\frac{a \cos\left(d x + c\right)^{2} - a - b}{\cos\left(d x + c\right)^{2} - 1}} \sin\left(d x + c\right) + a^{2} + 2 \, a b + b^{2}\right)}}{\cos\left(d x + c\right)^{4} - 2 \, \cos\left(d x + c\right)^{2} + 1}\right) \sin\left(d x + c\right) - 4 \, {\left(3 \, {\left(3 \, a b + b^{2}\right)} \cos\left(d x + c\right)^{3} - {\left(9 \, a b + 5 \, b^{2}\right)} \cos\left(d x + c\right)\right)} \sqrt{\frac{a \cos\left(d x + c\right)^{2} - a - b}{\cos\left(d x + c\right)^{2} - 1}}}{32 \, {\left(d \cos\left(d x + c\right)^{2} - d\right)} \sin\left(d x + c\right)}, \frac{4 \, {\left(a^{2} \cos\left(d x + c\right)^{2} - a^{2}\right)} \sqrt{a} \arctan\left(\frac{{\left(8 \, a^{2} \cos\left(d x + c\right)^{4} - 8 \, {\left(a^{2} + a b\right)} \cos\left(d x + c\right)^{2} + a^{2} + 2 \, a b + b^{2}\right)} \sqrt{a} \sqrt{\frac{a \cos\left(d x + c\right)^{2} - a - b}{\cos\left(d x + c\right)^{2} - 1}} \sin\left(d x + c\right)}{4 \, {\left(2 \, a^{3} \cos\left(d x + c\right)^{5} - 3 \, {\left(a^{3} + a^{2} b\right)} \cos\left(d x + c\right)^{3} + {\left(a^{3} + 2 \, a^{2} b + a b^{2}\right)} \cos\left(d x + c\right)\right)}}\right) \sin\left(d x + c\right) - {\left({\left(15 \, a^{2} + 10 \, a b + 3 \, b^{2}\right)} \cos\left(d x + c\right)^{2} - 15 \, a^{2} - 10 \, a b - 3 \, b^{2}\right)} \sqrt{-b} \arctan\left(-\frac{{\left({\left(a - b\right)} \cos\left(d x + c\right)^{2} - a - b\right)} \sqrt{-b} \sqrt{\frac{a \cos\left(d x + c\right)^{2} - a - b}{\cos\left(d x + c\right)^{2} - 1}} \sin\left(d x + c\right)}{2 \, {\left(a b \cos\left(d x + c\right)^{3} - {\left(a b + b^{2}\right)} \cos\left(d x + c\right)\right)}}\right) \sin\left(d x + c\right) - 2 \, {\left(3 \, {\left(3 \, a b + b^{2}\right)} \cos\left(d x + c\right)^{3} - {\left(9 \, a b + 5 \, b^{2}\right)} \cos\left(d x + c\right)\right)} \sqrt{\frac{a \cos\left(d x + c\right)^{2} - a - b}{\cos\left(d x + c\right)^{2} - 1}}}{16 \, {\left(d \cos\left(d x + c\right)^{2} - d\right)} \sin\left(d x + c\right)}\right]"," ",0,"[1/32*(4*(a^2*cos(d*x + c)^2 - a^2)*sqrt(-a)*log(128*a^4*cos(d*x + c)^8 - 256*(a^4 + a^3*b)*cos(d*x + c)^6 + 160*(a^4 + 2*a^3*b + a^2*b^2)*cos(d*x + c)^4 + a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4 - 32*(a^4 + 3*a^3*b + 3*a^2*b^2 + a*b^3)*cos(d*x + c)^2 - 8*(16*a^3*cos(d*x + c)^7 - 24*(a^3 + a^2*b)*cos(d*x + c)^5 + 10*(a^3 + 2*a^2*b + a*b^2)*cos(d*x + c)^3 - (a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cos(d*x + c))*sqrt(-a)*sqrt((a*cos(d*x + c)^2 - a - b)/(cos(d*x + c)^2 - 1))*sin(d*x + c))*sin(d*x + c) + ((15*a^2 + 10*a*b + 3*b^2)*cos(d*x + c)^2 - 15*a^2 - 10*a*b - 3*b^2)*sqrt(b)*log(2*((a^2 - 6*a*b + b^2)*cos(d*x + c)^4 - 2*(a^2 - 2*a*b - 3*b^2)*cos(d*x + c)^2 + 4*((a - b)*cos(d*x + c)^3 - (a + b)*cos(d*x + c))*sqrt(b)*sqrt((a*cos(d*x + c)^2 - a - b)/(cos(d*x + c)^2 - 1))*sin(d*x + c) + a^2 + 2*a*b + b^2)/(cos(d*x + c)^4 - 2*cos(d*x + c)^2 + 1))*sin(d*x + c) - 4*(3*(3*a*b + b^2)*cos(d*x + c)^3 - (9*a*b + 5*b^2)*cos(d*x + c))*sqrt((a*cos(d*x + c)^2 - a - b)/(cos(d*x + c)^2 - 1)))/((d*cos(d*x + c)^2 - d)*sin(d*x + c)), -1/16*(((15*a^2 + 10*a*b + 3*b^2)*cos(d*x + c)^2 - 15*a^2 - 10*a*b - 3*b^2)*sqrt(-b)*arctan(-1/2*((a - b)*cos(d*x + c)^2 - a - b)*sqrt(-b)*sqrt((a*cos(d*x + c)^2 - a - b)/(cos(d*x + c)^2 - 1))*sin(d*x + c)/(a*b*cos(d*x + c)^3 - (a*b + b^2)*cos(d*x + c)))*sin(d*x + c) - 2*(a^2*cos(d*x + c)^2 - a^2)*sqrt(-a)*log(128*a^4*cos(d*x + c)^8 - 256*(a^4 + a^3*b)*cos(d*x + c)^6 + 160*(a^4 + 2*a^3*b + a^2*b^2)*cos(d*x + c)^4 + a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4 - 32*(a^4 + 3*a^3*b + 3*a^2*b^2 + a*b^3)*cos(d*x + c)^2 - 8*(16*a^3*cos(d*x + c)^7 - 24*(a^3 + a^2*b)*cos(d*x + c)^5 + 10*(a^3 + 2*a^2*b + a*b^2)*cos(d*x + c)^3 - (a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cos(d*x + c))*sqrt(-a)*sqrt((a*cos(d*x + c)^2 - a - b)/(cos(d*x + c)^2 - 1))*sin(d*x + c))*sin(d*x + c) + 2*(3*(3*a*b + b^2)*cos(d*x + c)^3 - (9*a*b + 5*b^2)*cos(d*x + c))*sqrt((a*cos(d*x + c)^2 - a - b)/(cos(d*x + c)^2 - 1)))/((d*cos(d*x + c)^2 - d)*sin(d*x + c)), 1/32*(8*(a^2*cos(d*x + c)^2 - a^2)*sqrt(a)*arctan(1/4*(8*a^2*cos(d*x + c)^4 - 8*(a^2 + a*b)*cos(d*x + c)^2 + a^2 + 2*a*b + b^2)*sqrt(a)*sqrt((a*cos(d*x + c)^2 - a - b)/(cos(d*x + c)^2 - 1))*sin(d*x + c)/(2*a^3*cos(d*x + c)^5 - 3*(a^3 + a^2*b)*cos(d*x + c)^3 + (a^3 + 2*a^2*b + a*b^2)*cos(d*x + c)))*sin(d*x + c) + ((15*a^2 + 10*a*b + 3*b^2)*cos(d*x + c)^2 - 15*a^2 - 10*a*b - 3*b^2)*sqrt(b)*log(2*((a^2 - 6*a*b + b^2)*cos(d*x + c)^4 - 2*(a^2 - 2*a*b - 3*b^2)*cos(d*x + c)^2 + 4*((a - b)*cos(d*x + c)^3 - (a + b)*cos(d*x + c))*sqrt(b)*sqrt((a*cos(d*x + c)^2 - a - b)/(cos(d*x + c)^2 - 1))*sin(d*x + c) + a^2 + 2*a*b + b^2)/(cos(d*x + c)^4 - 2*cos(d*x + c)^2 + 1))*sin(d*x + c) - 4*(3*(3*a*b + b^2)*cos(d*x + c)^3 - (9*a*b + 5*b^2)*cos(d*x + c))*sqrt((a*cos(d*x + c)^2 - a - b)/(cos(d*x + c)^2 - 1)))/((d*cos(d*x + c)^2 - d)*sin(d*x + c)), 1/16*(4*(a^2*cos(d*x + c)^2 - a^2)*sqrt(a)*arctan(1/4*(8*a^2*cos(d*x + c)^4 - 8*(a^2 + a*b)*cos(d*x + c)^2 + a^2 + 2*a*b + b^2)*sqrt(a)*sqrt((a*cos(d*x + c)^2 - a - b)/(cos(d*x + c)^2 - 1))*sin(d*x + c)/(2*a^3*cos(d*x + c)^5 - 3*(a^3 + a^2*b)*cos(d*x + c)^3 + (a^3 + 2*a^2*b + a*b^2)*cos(d*x + c)))*sin(d*x + c) - ((15*a^2 + 10*a*b + 3*b^2)*cos(d*x + c)^2 - 15*a^2 - 10*a*b - 3*b^2)*sqrt(-b)*arctan(-1/2*((a - b)*cos(d*x + c)^2 - a - b)*sqrt(-b)*sqrt((a*cos(d*x + c)^2 - a - b)/(cos(d*x + c)^2 - 1))*sin(d*x + c)/(a*b*cos(d*x + c)^3 - (a*b + b^2)*cos(d*x + c)))*sin(d*x + c) - 2*(3*(3*a*b + b^2)*cos(d*x + c)^3 - (9*a*b + 5*b^2)*cos(d*x + c))*sqrt((a*cos(d*x + c)^2 - a - b)/(cos(d*x + c)^2 - 1)))/((d*cos(d*x + c)^2 - d)*sin(d*x + c))]","B",0
10,1,1607,0,1.021014," ","integrate((a+b*csc(d*x+c)^2)^(3/2),x, algorithm=""fricas"")","\left[\frac{\sqrt{-a} a \log\left(128 \, a^{4} \cos\left(d x + c\right)^{8} - 256 \, {\left(a^{4} + a^{3} b\right)} \cos\left(d x + c\right)^{6} + 160 \, {\left(a^{4} + 2 \, a^{3} b + a^{2} b^{2}\right)} \cos\left(d x + c\right)^{4} + a^{4} + 4 \, a^{3} b + 6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4} - 32 \, {\left(a^{4} + 3 \, a^{3} b + 3 \, a^{2} b^{2} + a b^{3}\right)} \cos\left(d x + c\right)^{2} - 8 \, {\left(16 \, a^{3} \cos\left(d x + c\right)^{7} - 24 \, {\left(a^{3} + a^{2} b\right)} \cos\left(d x + c\right)^{5} + 10 \, {\left(a^{3} + 2 \, a^{2} b + a b^{2}\right)} \cos\left(d x + c\right)^{3} - {\left(a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right)} \cos\left(d x + c\right)\right)} \sqrt{-a} \sqrt{\frac{a \cos\left(d x + c\right)^{2} - a - b}{\cos\left(d x + c\right)^{2} - 1}} \sin\left(d x + c\right)\right) \sin\left(d x + c\right) + {\left(3 \, a + b\right)} \sqrt{b} \log\left(\frac{2 \, {\left({\left(a^{2} - 6 \, a b + b^{2}\right)} \cos\left(d x + c\right)^{4} - 2 \, {\left(a^{2} - 2 \, a b - 3 \, b^{2}\right)} \cos\left(d x + c\right)^{2} + 4 \, {\left({\left(a - b\right)} \cos\left(d x + c\right)^{3} - {\left(a + b\right)} \cos\left(d x + c\right)\right)} \sqrt{b} \sqrt{\frac{a \cos\left(d x + c\right)^{2} - a - b}{\cos\left(d x + c\right)^{2} - 1}} \sin\left(d x + c\right) + a^{2} + 2 \, a b + b^{2}\right)}}{\cos\left(d x + c\right)^{4} - 2 \, \cos\left(d x + c\right)^{2} + 1}\right) \sin\left(d x + c\right) - 4 \, b \sqrt{\frac{a \cos\left(d x + c\right)^{2} - a - b}{\cos\left(d x + c\right)^{2} - 1}} \cos\left(d x + c\right)}{8 \, d \sin\left(d x + c\right)}, -\frac{2 \, {\left(3 \, a + b\right)} \sqrt{-b} \arctan\left(-\frac{{\left({\left(a - b\right)} \cos\left(d x + c\right)^{2} - a - b\right)} \sqrt{-b} \sqrt{\frac{a \cos\left(d x + c\right)^{2} - a - b}{\cos\left(d x + c\right)^{2} - 1}} \sin\left(d x + c\right)}{2 \, {\left(a b \cos\left(d x + c\right)^{3} - {\left(a b + b^{2}\right)} \cos\left(d x + c\right)\right)}}\right) \sin\left(d x + c\right) - \sqrt{-a} a \log\left(128 \, a^{4} \cos\left(d x + c\right)^{8} - 256 \, {\left(a^{4} + a^{3} b\right)} \cos\left(d x + c\right)^{6} + 160 \, {\left(a^{4} + 2 \, a^{3} b + a^{2} b^{2}\right)} \cos\left(d x + c\right)^{4} + a^{4} + 4 \, a^{3} b + 6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4} - 32 \, {\left(a^{4} + 3 \, a^{3} b + 3 \, a^{2} b^{2} + a b^{3}\right)} \cos\left(d x + c\right)^{2} - 8 \, {\left(16 \, a^{3} \cos\left(d x + c\right)^{7} - 24 \, {\left(a^{3} + a^{2} b\right)} \cos\left(d x + c\right)^{5} + 10 \, {\left(a^{3} + 2 \, a^{2} b + a b^{2}\right)} \cos\left(d x + c\right)^{3} - {\left(a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right)} \cos\left(d x + c\right)\right)} \sqrt{-a} \sqrt{\frac{a \cos\left(d x + c\right)^{2} - a - b}{\cos\left(d x + c\right)^{2} - 1}} \sin\left(d x + c\right)\right) \sin\left(d x + c\right) + 4 \, b \sqrt{\frac{a \cos\left(d x + c\right)^{2} - a - b}{\cos\left(d x + c\right)^{2} - 1}} \cos\left(d x + c\right)}{8 \, d \sin\left(d x + c\right)}, \frac{2 \, a^{\frac{3}{2}} \arctan\left(\frac{{\left(8 \, a^{2} \cos\left(d x + c\right)^{4} - 8 \, {\left(a^{2} + a b\right)} \cos\left(d x + c\right)^{2} + a^{2} + 2 \, a b + b^{2}\right)} \sqrt{a} \sqrt{\frac{a \cos\left(d x + c\right)^{2} - a - b}{\cos\left(d x + c\right)^{2} - 1}} \sin\left(d x + c\right)}{4 \, {\left(2 \, a^{3} \cos\left(d x + c\right)^{5} - 3 \, {\left(a^{3} + a^{2} b\right)} \cos\left(d x + c\right)^{3} + {\left(a^{3} + 2 \, a^{2} b + a b^{2}\right)} \cos\left(d x + c\right)\right)}}\right) \sin\left(d x + c\right) + {\left(3 \, a + b\right)} \sqrt{b} \log\left(\frac{2 \, {\left({\left(a^{2} - 6 \, a b + b^{2}\right)} \cos\left(d x + c\right)^{4} - 2 \, {\left(a^{2} - 2 \, a b - 3 \, b^{2}\right)} \cos\left(d x + c\right)^{2} + 4 \, {\left({\left(a - b\right)} \cos\left(d x + c\right)^{3} - {\left(a + b\right)} \cos\left(d x + c\right)\right)} \sqrt{b} \sqrt{\frac{a \cos\left(d x + c\right)^{2} - a - b}{\cos\left(d x + c\right)^{2} - 1}} \sin\left(d x + c\right) + a^{2} + 2 \, a b + b^{2}\right)}}{\cos\left(d x + c\right)^{4} - 2 \, \cos\left(d x + c\right)^{2} + 1}\right) \sin\left(d x + c\right) - 4 \, b \sqrt{\frac{a \cos\left(d x + c\right)^{2} - a - b}{\cos\left(d x + c\right)^{2} - 1}} \cos\left(d x + c\right)}{8 \, d \sin\left(d x + c\right)}, \frac{a^{\frac{3}{2}} \arctan\left(\frac{{\left(8 \, a^{2} \cos\left(d x + c\right)^{4} - 8 \, {\left(a^{2} + a b\right)} \cos\left(d x + c\right)^{2} + a^{2} + 2 \, a b + b^{2}\right)} \sqrt{a} \sqrt{\frac{a \cos\left(d x + c\right)^{2} - a - b}{\cos\left(d x + c\right)^{2} - 1}} \sin\left(d x + c\right)}{4 \, {\left(2 \, a^{3} \cos\left(d x + c\right)^{5} - 3 \, {\left(a^{3} + a^{2} b\right)} \cos\left(d x + c\right)^{3} + {\left(a^{3} + 2 \, a^{2} b + a b^{2}\right)} \cos\left(d x + c\right)\right)}}\right) \sin\left(d x + c\right) - {\left(3 \, a + b\right)} \sqrt{-b} \arctan\left(-\frac{{\left({\left(a - b\right)} \cos\left(d x + c\right)^{2} - a - b\right)} \sqrt{-b} \sqrt{\frac{a \cos\left(d x + c\right)^{2} - a - b}{\cos\left(d x + c\right)^{2} - 1}} \sin\left(d x + c\right)}{2 \, {\left(a b \cos\left(d x + c\right)^{3} - {\left(a b + b^{2}\right)} \cos\left(d x + c\right)\right)}}\right) \sin\left(d x + c\right) - 2 \, b \sqrt{\frac{a \cos\left(d x + c\right)^{2} - a - b}{\cos\left(d x + c\right)^{2} - 1}} \cos\left(d x + c\right)}{4 \, d \sin\left(d x + c\right)}\right]"," ",0,"[1/8*(sqrt(-a)*a*log(128*a^4*cos(d*x + c)^8 - 256*(a^4 + a^3*b)*cos(d*x + c)^6 + 160*(a^4 + 2*a^3*b + a^2*b^2)*cos(d*x + c)^4 + a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4 - 32*(a^4 + 3*a^3*b + 3*a^2*b^2 + a*b^3)*cos(d*x + c)^2 - 8*(16*a^3*cos(d*x + c)^7 - 24*(a^3 + a^2*b)*cos(d*x + c)^5 + 10*(a^3 + 2*a^2*b + a*b^2)*cos(d*x + c)^3 - (a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cos(d*x + c))*sqrt(-a)*sqrt((a*cos(d*x + c)^2 - a - b)/(cos(d*x + c)^2 - 1))*sin(d*x + c))*sin(d*x + c) + (3*a + b)*sqrt(b)*log(2*((a^2 - 6*a*b + b^2)*cos(d*x + c)^4 - 2*(a^2 - 2*a*b - 3*b^2)*cos(d*x + c)^2 + 4*((a - b)*cos(d*x + c)^3 - (a + b)*cos(d*x + c))*sqrt(b)*sqrt((a*cos(d*x + c)^2 - a - b)/(cos(d*x + c)^2 - 1))*sin(d*x + c) + a^2 + 2*a*b + b^2)/(cos(d*x + c)^4 - 2*cos(d*x + c)^2 + 1))*sin(d*x + c) - 4*b*sqrt((a*cos(d*x + c)^2 - a - b)/(cos(d*x + c)^2 - 1))*cos(d*x + c))/(d*sin(d*x + c)), -1/8*(2*(3*a + b)*sqrt(-b)*arctan(-1/2*((a - b)*cos(d*x + c)^2 - a - b)*sqrt(-b)*sqrt((a*cos(d*x + c)^2 - a - b)/(cos(d*x + c)^2 - 1))*sin(d*x + c)/(a*b*cos(d*x + c)^3 - (a*b + b^2)*cos(d*x + c)))*sin(d*x + c) - sqrt(-a)*a*log(128*a^4*cos(d*x + c)^8 - 256*(a^4 + a^3*b)*cos(d*x + c)^6 + 160*(a^4 + 2*a^3*b + a^2*b^2)*cos(d*x + c)^4 + a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4 - 32*(a^4 + 3*a^3*b + 3*a^2*b^2 + a*b^3)*cos(d*x + c)^2 - 8*(16*a^3*cos(d*x + c)^7 - 24*(a^3 + a^2*b)*cos(d*x + c)^5 + 10*(a^3 + 2*a^2*b + a*b^2)*cos(d*x + c)^3 - (a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cos(d*x + c))*sqrt(-a)*sqrt((a*cos(d*x + c)^2 - a - b)/(cos(d*x + c)^2 - 1))*sin(d*x + c))*sin(d*x + c) + 4*b*sqrt((a*cos(d*x + c)^2 - a - b)/(cos(d*x + c)^2 - 1))*cos(d*x + c))/(d*sin(d*x + c)), 1/8*(2*a^(3/2)*arctan(1/4*(8*a^2*cos(d*x + c)^4 - 8*(a^2 + a*b)*cos(d*x + c)^2 + a^2 + 2*a*b + b^2)*sqrt(a)*sqrt((a*cos(d*x + c)^2 - a - b)/(cos(d*x + c)^2 - 1))*sin(d*x + c)/(2*a^3*cos(d*x + c)^5 - 3*(a^3 + a^2*b)*cos(d*x + c)^3 + (a^3 + 2*a^2*b + a*b^2)*cos(d*x + c)))*sin(d*x + c) + (3*a + b)*sqrt(b)*log(2*((a^2 - 6*a*b + b^2)*cos(d*x + c)^4 - 2*(a^2 - 2*a*b - 3*b^2)*cos(d*x + c)^2 + 4*((a - b)*cos(d*x + c)^3 - (a + b)*cos(d*x + c))*sqrt(b)*sqrt((a*cos(d*x + c)^2 - a - b)/(cos(d*x + c)^2 - 1))*sin(d*x + c) + a^2 + 2*a*b + b^2)/(cos(d*x + c)^4 - 2*cos(d*x + c)^2 + 1))*sin(d*x + c) - 4*b*sqrt((a*cos(d*x + c)^2 - a - b)/(cos(d*x + c)^2 - 1))*cos(d*x + c))/(d*sin(d*x + c)), 1/4*(a^(3/2)*arctan(1/4*(8*a^2*cos(d*x + c)^4 - 8*(a^2 + a*b)*cos(d*x + c)^2 + a^2 + 2*a*b + b^2)*sqrt(a)*sqrt((a*cos(d*x + c)^2 - a - b)/(cos(d*x + c)^2 - 1))*sin(d*x + c)/(2*a^3*cos(d*x + c)^5 - 3*(a^3 + a^2*b)*cos(d*x + c)^3 + (a^3 + 2*a^2*b + a*b^2)*cos(d*x + c)))*sin(d*x + c) - (3*a + b)*sqrt(-b)*arctan(-1/2*((a - b)*cos(d*x + c)^2 - a - b)*sqrt(-b)*sqrt((a*cos(d*x + c)^2 - a - b)/(cos(d*x + c)^2 - 1))*sin(d*x + c)/(a*b*cos(d*x + c)^3 - (a*b + b^2)*cos(d*x + c)))*sin(d*x + c) - 2*b*sqrt((a*cos(d*x + c)^2 - a - b)/(cos(d*x + c)^2 - 1))*cos(d*x + c))/(d*sin(d*x + c))]","B",0
11,1,1341,0,0.665962," ","integrate((a+b*csc(d*x+c)^2)^(1/2),x, algorithm=""fricas"")","\left[\frac{\sqrt{-a} \log\left(128 \, a^{4} \cos\left(d x + c\right)^{8} - 256 \, {\left(a^{4} + a^{3} b\right)} \cos\left(d x + c\right)^{6} + 160 \, {\left(a^{4} + 2 \, a^{3} b + a^{2} b^{2}\right)} \cos\left(d x + c\right)^{4} + a^{4} + 4 \, a^{3} b + 6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4} - 32 \, {\left(a^{4} + 3 \, a^{3} b + 3 \, a^{2} b^{2} + a b^{3}\right)} \cos\left(d x + c\right)^{2} - 8 \, {\left(16 \, a^{3} \cos\left(d x + c\right)^{7} - 24 \, {\left(a^{3} + a^{2} b\right)} \cos\left(d x + c\right)^{5} + 10 \, {\left(a^{3} + 2 \, a^{2} b + a b^{2}\right)} \cos\left(d x + c\right)^{3} - {\left(a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right)} \cos\left(d x + c\right)\right)} \sqrt{-a} \sqrt{\frac{a \cos\left(d x + c\right)^{2} - a - b}{\cos\left(d x + c\right)^{2} - 1}} \sin\left(d x + c\right)\right) + 2 \, \sqrt{b} \log\left(\frac{2 \, {\left({\left(a^{2} - 6 \, a b + b^{2}\right)} \cos\left(d x + c\right)^{4} - 2 \, {\left(a^{2} - 2 \, a b - 3 \, b^{2}\right)} \cos\left(d x + c\right)^{2} + 4 \, {\left({\left(a - b\right)} \cos\left(d x + c\right)^{3} - {\left(a + b\right)} \cos\left(d x + c\right)\right)} \sqrt{b} \sqrt{\frac{a \cos\left(d x + c\right)^{2} - a - b}{\cos\left(d x + c\right)^{2} - 1}} \sin\left(d x + c\right) + a^{2} + 2 \, a b + b^{2}\right)}}{\cos\left(d x + c\right)^{4} - 2 \, \cos\left(d x + c\right)^{2} + 1}\right)}{8 \, d}, -\frac{4 \, \sqrt{-b} \arctan\left(-\frac{{\left({\left(a - b\right)} \cos\left(d x + c\right)^{2} - a - b\right)} \sqrt{-b} \sqrt{\frac{a \cos\left(d x + c\right)^{2} - a - b}{\cos\left(d x + c\right)^{2} - 1}} \sin\left(d x + c\right)}{2 \, {\left(a b \cos\left(d x + c\right)^{3} - {\left(a b + b^{2}\right)} \cos\left(d x + c\right)\right)}}\right) - \sqrt{-a} \log\left(128 \, a^{4} \cos\left(d x + c\right)^{8} - 256 \, {\left(a^{4} + a^{3} b\right)} \cos\left(d x + c\right)^{6} + 160 \, {\left(a^{4} + 2 \, a^{3} b + a^{2} b^{2}\right)} \cos\left(d x + c\right)^{4} + a^{4} + 4 \, a^{3} b + 6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4} - 32 \, {\left(a^{4} + 3 \, a^{3} b + 3 \, a^{2} b^{2} + a b^{3}\right)} \cos\left(d x + c\right)^{2} - 8 \, {\left(16 \, a^{3} \cos\left(d x + c\right)^{7} - 24 \, {\left(a^{3} + a^{2} b\right)} \cos\left(d x + c\right)^{5} + 10 \, {\left(a^{3} + 2 \, a^{2} b + a b^{2}\right)} \cos\left(d x + c\right)^{3} - {\left(a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right)} \cos\left(d x + c\right)\right)} \sqrt{-a} \sqrt{\frac{a \cos\left(d x + c\right)^{2} - a - b}{\cos\left(d x + c\right)^{2} - 1}} \sin\left(d x + c\right)\right)}{8 \, d}, \frac{\sqrt{a} \arctan\left(\frac{{\left(8 \, a^{2} \cos\left(d x + c\right)^{4} - 8 \, {\left(a^{2} + a b\right)} \cos\left(d x + c\right)^{2} + a^{2} + 2 \, a b + b^{2}\right)} \sqrt{a} \sqrt{\frac{a \cos\left(d x + c\right)^{2} - a - b}{\cos\left(d x + c\right)^{2} - 1}} \sin\left(d x + c\right)}{4 \, {\left(2 \, a^{3} \cos\left(d x + c\right)^{5} - 3 \, {\left(a^{3} + a^{2} b\right)} \cos\left(d x + c\right)^{3} + {\left(a^{3} + 2 \, a^{2} b + a b^{2}\right)} \cos\left(d x + c\right)\right)}}\right) + \sqrt{b} \log\left(\frac{2 \, {\left({\left(a^{2} - 6 \, a b + b^{2}\right)} \cos\left(d x + c\right)^{4} - 2 \, {\left(a^{2} - 2 \, a b - 3 \, b^{2}\right)} \cos\left(d x + c\right)^{2} + 4 \, {\left({\left(a - b\right)} \cos\left(d x + c\right)^{3} - {\left(a + b\right)} \cos\left(d x + c\right)\right)} \sqrt{b} \sqrt{\frac{a \cos\left(d x + c\right)^{2} - a - b}{\cos\left(d x + c\right)^{2} - 1}} \sin\left(d x + c\right) + a^{2} + 2 \, a b + b^{2}\right)}}{\cos\left(d x + c\right)^{4} - 2 \, \cos\left(d x + c\right)^{2} + 1}\right)}{4 \, d}, \frac{\sqrt{a} \arctan\left(\frac{{\left(8 \, a^{2} \cos\left(d x + c\right)^{4} - 8 \, {\left(a^{2} + a b\right)} \cos\left(d x + c\right)^{2} + a^{2} + 2 \, a b + b^{2}\right)} \sqrt{a} \sqrt{\frac{a \cos\left(d x + c\right)^{2} - a - b}{\cos\left(d x + c\right)^{2} - 1}} \sin\left(d x + c\right)}{4 \, {\left(2 \, a^{3} \cos\left(d x + c\right)^{5} - 3 \, {\left(a^{3} + a^{2} b\right)} \cos\left(d x + c\right)^{3} + {\left(a^{3} + 2 \, a^{2} b + a b^{2}\right)} \cos\left(d x + c\right)\right)}}\right) - 2 \, \sqrt{-b} \arctan\left(-\frac{{\left({\left(a - b\right)} \cos\left(d x + c\right)^{2} - a - b\right)} \sqrt{-b} \sqrt{\frac{a \cos\left(d x + c\right)^{2} - a - b}{\cos\left(d x + c\right)^{2} - 1}} \sin\left(d x + c\right)}{2 \, {\left(a b \cos\left(d x + c\right)^{3} - {\left(a b + b^{2}\right)} \cos\left(d x + c\right)\right)}}\right)}{4 \, d}\right]"," ",0,"[1/8*(sqrt(-a)*log(128*a^4*cos(d*x + c)^8 - 256*(a^4 + a^3*b)*cos(d*x + c)^6 + 160*(a^4 + 2*a^3*b + a^2*b^2)*cos(d*x + c)^4 + a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4 - 32*(a^4 + 3*a^3*b + 3*a^2*b^2 + a*b^3)*cos(d*x + c)^2 - 8*(16*a^3*cos(d*x + c)^7 - 24*(a^3 + a^2*b)*cos(d*x + c)^5 + 10*(a^3 + 2*a^2*b + a*b^2)*cos(d*x + c)^3 - (a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cos(d*x + c))*sqrt(-a)*sqrt((a*cos(d*x + c)^2 - a - b)/(cos(d*x + c)^2 - 1))*sin(d*x + c)) + 2*sqrt(b)*log(2*((a^2 - 6*a*b + b^2)*cos(d*x + c)^4 - 2*(a^2 - 2*a*b - 3*b^2)*cos(d*x + c)^2 + 4*((a - b)*cos(d*x + c)^3 - (a + b)*cos(d*x + c))*sqrt(b)*sqrt((a*cos(d*x + c)^2 - a - b)/(cos(d*x + c)^2 - 1))*sin(d*x + c) + a^2 + 2*a*b + b^2)/(cos(d*x + c)^4 - 2*cos(d*x + c)^2 + 1)))/d, -1/8*(4*sqrt(-b)*arctan(-1/2*((a - b)*cos(d*x + c)^2 - a - b)*sqrt(-b)*sqrt((a*cos(d*x + c)^2 - a - b)/(cos(d*x + c)^2 - 1))*sin(d*x + c)/(a*b*cos(d*x + c)^3 - (a*b + b^2)*cos(d*x + c))) - sqrt(-a)*log(128*a^4*cos(d*x + c)^8 - 256*(a^4 + a^3*b)*cos(d*x + c)^6 + 160*(a^4 + 2*a^3*b + a^2*b^2)*cos(d*x + c)^4 + a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4 - 32*(a^4 + 3*a^3*b + 3*a^2*b^2 + a*b^3)*cos(d*x + c)^2 - 8*(16*a^3*cos(d*x + c)^7 - 24*(a^3 + a^2*b)*cos(d*x + c)^5 + 10*(a^3 + 2*a^2*b + a*b^2)*cos(d*x + c)^3 - (a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cos(d*x + c))*sqrt(-a)*sqrt((a*cos(d*x + c)^2 - a - b)/(cos(d*x + c)^2 - 1))*sin(d*x + c)))/d, 1/4*(sqrt(a)*arctan(1/4*(8*a^2*cos(d*x + c)^4 - 8*(a^2 + a*b)*cos(d*x + c)^2 + a^2 + 2*a*b + b^2)*sqrt(a)*sqrt((a*cos(d*x + c)^2 - a - b)/(cos(d*x + c)^2 - 1))*sin(d*x + c)/(2*a^3*cos(d*x + c)^5 - 3*(a^3 + a^2*b)*cos(d*x + c)^3 + (a^3 + 2*a^2*b + a*b^2)*cos(d*x + c))) + sqrt(b)*log(2*((a^2 - 6*a*b + b^2)*cos(d*x + c)^4 - 2*(a^2 - 2*a*b - 3*b^2)*cos(d*x + c)^2 + 4*((a - b)*cos(d*x + c)^3 - (a + b)*cos(d*x + c))*sqrt(b)*sqrt((a*cos(d*x + c)^2 - a - b)/(cos(d*x + c)^2 - 1))*sin(d*x + c) + a^2 + 2*a*b + b^2)/(cos(d*x + c)^4 - 2*cos(d*x + c)^2 + 1)))/d, 1/4*(sqrt(a)*arctan(1/4*(8*a^2*cos(d*x + c)^4 - 8*(a^2 + a*b)*cos(d*x + c)^2 + a^2 + 2*a*b + b^2)*sqrt(a)*sqrt((a*cos(d*x + c)^2 - a - b)/(cos(d*x + c)^2 - 1))*sin(d*x + c)/(2*a^3*cos(d*x + c)^5 - 3*(a^3 + a^2*b)*cos(d*x + c)^3 + (a^3 + 2*a^2*b + a*b^2)*cos(d*x + c))) - 2*sqrt(-b)*arctan(-1/2*((a - b)*cos(d*x + c)^2 - a - b)*sqrt(-b)*sqrt((a*cos(d*x + c)^2 - a - b)/(cos(d*x + c)^2 - 1))*sin(d*x + c)/(a*b*cos(d*x + c)^3 - (a*b + b^2)*cos(d*x + c))))/d]","B",0
12,1,414,0,0.655293," ","integrate(1/(a+b*csc(d*x+c)^2)^(1/2),x, algorithm=""fricas"")","\left[-\frac{\sqrt{-a} \log\left(128 \, a^{4} \cos\left(d x + c\right)^{8} - 256 \, {\left(a^{4} + a^{3} b\right)} \cos\left(d x + c\right)^{6} + 160 \, {\left(a^{4} + 2 \, a^{3} b + a^{2} b^{2}\right)} \cos\left(d x + c\right)^{4} + a^{4} + 4 \, a^{3} b + 6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4} - 32 \, {\left(a^{4} + 3 \, a^{3} b + 3 \, a^{2} b^{2} + a b^{3}\right)} \cos\left(d x + c\right)^{2} + 8 \, {\left(16 \, a^{3} \cos\left(d x + c\right)^{7} - 24 \, {\left(a^{3} + a^{2} b\right)} \cos\left(d x + c\right)^{5} + 10 \, {\left(a^{3} + 2 \, a^{2} b + a b^{2}\right)} \cos\left(d x + c\right)^{3} - {\left(a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right)} \cos\left(d x + c\right)\right)} \sqrt{-a} \sqrt{\frac{a \cos\left(d x + c\right)^{2} - a - b}{\cos\left(d x + c\right)^{2} - 1}} \sin\left(d x + c\right)\right)}{8 \, a d}, \frac{\arctan\left(\frac{{\left(8 \, a^{2} \cos\left(d x + c\right)^{4} - 8 \, {\left(a^{2} + a b\right)} \cos\left(d x + c\right)^{2} + a^{2} + 2 \, a b + b^{2}\right)} \sqrt{a} \sqrt{\frac{a \cos\left(d x + c\right)^{2} - a - b}{\cos\left(d x + c\right)^{2} - 1}} \sin\left(d x + c\right)}{4 \, {\left(2 \, a^{3} \cos\left(d x + c\right)^{5} - 3 \, {\left(a^{3} + a^{2} b\right)} \cos\left(d x + c\right)^{3} + {\left(a^{3} + 2 \, a^{2} b + a b^{2}\right)} \cos\left(d x + c\right)\right)}}\right)}{4 \, \sqrt{a} d}\right]"," ",0,"[-1/8*sqrt(-a)*log(128*a^4*cos(d*x + c)^8 - 256*(a^4 + a^3*b)*cos(d*x + c)^6 + 160*(a^4 + 2*a^3*b + a^2*b^2)*cos(d*x + c)^4 + a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4 - 32*(a^4 + 3*a^3*b + 3*a^2*b^2 + a*b^3)*cos(d*x + c)^2 + 8*(16*a^3*cos(d*x + c)^7 - 24*(a^3 + a^2*b)*cos(d*x + c)^5 + 10*(a^3 + 2*a^2*b + a*b^2)*cos(d*x + c)^3 - (a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cos(d*x + c))*sqrt(-a)*sqrt((a*cos(d*x + c)^2 - a - b)/(cos(d*x + c)^2 - 1))*sin(d*x + c))/(a*d), 1/4*arctan(1/4*(8*a^2*cos(d*x + c)^4 - 8*(a^2 + a*b)*cos(d*x + c)^2 + a^2 + 2*a*b + b^2)*sqrt(a)*sqrt((a*cos(d*x + c)^2 - a - b)/(cos(d*x + c)^2 - 1))*sin(d*x + c)/(2*a^3*cos(d*x + c)^5 - 3*(a^3 + a^2*b)*cos(d*x + c)^3 + (a^3 + 2*a^2*b + a*b^2)*cos(d*x + c)))/(sqrt(a)*d)]","B",0
13,1,652,0,0.720889," ","integrate(1/(a+b*csc(d*x+c)^2)^(3/2),x, algorithm=""fricas"")","\left[-\frac{8 \, a b \sqrt{\frac{a \cos\left(d x + c\right)^{2} - a - b}{\cos\left(d x + c\right)^{2} - 1}} \cos\left(d x + c\right) \sin\left(d x + c\right) + {\left({\left(a^{2} + a b\right)} \cos\left(d x + c\right)^{2} - a^{2} - 2 \, a b - b^{2}\right)} \sqrt{-a} \log\left(128 \, a^{4} \cos\left(d x + c\right)^{8} - 256 \, {\left(a^{4} + a^{3} b\right)} \cos\left(d x + c\right)^{6} + 160 \, {\left(a^{4} + 2 \, a^{3} b + a^{2} b^{2}\right)} \cos\left(d x + c\right)^{4} + a^{4} + 4 \, a^{3} b + 6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4} - 32 \, {\left(a^{4} + 3 \, a^{3} b + 3 \, a^{2} b^{2} + a b^{3}\right)} \cos\left(d x + c\right)^{2} + 8 \, {\left(16 \, a^{3} \cos\left(d x + c\right)^{7} - 24 \, {\left(a^{3} + a^{2} b\right)} \cos\left(d x + c\right)^{5} + 10 \, {\left(a^{3} + 2 \, a^{2} b + a b^{2}\right)} \cos\left(d x + c\right)^{3} - {\left(a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right)} \cos\left(d x + c\right)\right)} \sqrt{-a} \sqrt{\frac{a \cos\left(d x + c\right)^{2} - a - b}{\cos\left(d x + c\right)^{2} - 1}} \sin\left(d x + c\right)\right)}{8 \, {\left({\left(a^{4} + a^{3} b\right)} d \cos\left(d x + c\right)^{2} - {\left(a^{4} + 2 \, a^{3} b + a^{2} b^{2}\right)} d\right)}}, -\frac{4 \, a b \sqrt{\frac{a \cos\left(d x + c\right)^{2} - a - b}{\cos\left(d x + c\right)^{2} - 1}} \cos\left(d x + c\right) \sin\left(d x + c\right) - {\left({\left(a^{2} + a b\right)} \cos\left(d x + c\right)^{2} - a^{2} - 2 \, a b - b^{2}\right)} \sqrt{a} \arctan\left(\frac{{\left(8 \, a^{2} \cos\left(d x + c\right)^{4} - 8 \, {\left(a^{2} + a b\right)} \cos\left(d x + c\right)^{2} + a^{2} + 2 \, a b + b^{2}\right)} \sqrt{a} \sqrt{\frac{a \cos\left(d x + c\right)^{2} - a - b}{\cos\left(d x + c\right)^{2} - 1}} \sin\left(d x + c\right)}{4 \, {\left(2 \, a^{3} \cos\left(d x + c\right)^{5} - 3 \, {\left(a^{3} + a^{2} b\right)} \cos\left(d x + c\right)^{3} + {\left(a^{3} + 2 \, a^{2} b + a b^{2}\right)} \cos\left(d x + c\right)\right)}}\right)}{4 \, {\left({\left(a^{4} + a^{3} b\right)} d \cos\left(d x + c\right)^{2} - {\left(a^{4} + 2 \, a^{3} b + a^{2} b^{2}\right)} d\right)}}\right]"," ",0,"[-1/8*(8*a*b*sqrt((a*cos(d*x + c)^2 - a - b)/(cos(d*x + c)^2 - 1))*cos(d*x + c)*sin(d*x + c) + ((a^2 + a*b)*cos(d*x + c)^2 - a^2 - 2*a*b - b^2)*sqrt(-a)*log(128*a^4*cos(d*x + c)^8 - 256*(a^4 + a^3*b)*cos(d*x + c)^6 + 160*(a^4 + 2*a^3*b + a^2*b^2)*cos(d*x + c)^4 + a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4 - 32*(a^4 + 3*a^3*b + 3*a^2*b^2 + a*b^3)*cos(d*x + c)^2 + 8*(16*a^3*cos(d*x + c)^7 - 24*(a^3 + a^2*b)*cos(d*x + c)^5 + 10*(a^3 + 2*a^2*b + a*b^2)*cos(d*x + c)^3 - (a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cos(d*x + c))*sqrt(-a)*sqrt((a*cos(d*x + c)^2 - a - b)/(cos(d*x + c)^2 - 1))*sin(d*x + c)))/((a^4 + a^3*b)*d*cos(d*x + c)^2 - (a^4 + 2*a^3*b + a^2*b^2)*d), -1/4*(4*a*b*sqrt((a*cos(d*x + c)^2 - a - b)/(cos(d*x + c)^2 - 1))*cos(d*x + c)*sin(d*x + c) - ((a^2 + a*b)*cos(d*x + c)^2 - a^2 - 2*a*b - b^2)*sqrt(a)*arctan(1/4*(8*a^2*cos(d*x + c)^4 - 8*(a^2 + a*b)*cos(d*x + c)^2 + a^2 + 2*a*b + b^2)*sqrt(a)*sqrt((a*cos(d*x + c)^2 - a - b)/(cos(d*x + c)^2 - 1))*sin(d*x + c)/(2*a^3*cos(d*x + c)^5 - 3*(a^3 + a^2*b)*cos(d*x + c)^3 + (a^3 + 2*a^2*b + a*b^2)*cos(d*x + c))))/((a^4 + a^3*b)*d*cos(d*x + c)^2 - (a^4 + 2*a^3*b + a^2*b^2)*d)]","B",0
14,1,973,0,1.359210," ","integrate(1/(a+b*csc(d*x+c)^2)^(5/2),x, algorithm=""fricas"")","\left[-\frac{3 \, {\left({\left(a^{4} + 2 \, a^{3} b + a^{2} b^{2}\right)} \cos\left(d x + c\right)^{4} + a^{4} + 4 \, a^{3} b + 6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4} - 2 \, {\left(a^{4} + 3 \, a^{3} b + 3 \, a^{2} b^{2} + a b^{3}\right)} \cos\left(d x + c\right)^{2}\right)} \sqrt{-a} \log\left(128 \, a^{4} \cos\left(d x + c\right)^{8} - 256 \, {\left(a^{4} + a^{3} b\right)} \cos\left(d x + c\right)^{6} + 160 \, {\left(a^{4} + 2 \, a^{3} b + a^{2} b^{2}\right)} \cos\left(d x + c\right)^{4} + a^{4} + 4 \, a^{3} b + 6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4} - 32 \, {\left(a^{4} + 3 \, a^{3} b + 3 \, a^{2} b^{2} + a b^{3}\right)} \cos\left(d x + c\right)^{2} + 8 \, {\left(16 \, a^{3} \cos\left(d x + c\right)^{7} - 24 \, {\left(a^{3} + a^{2} b\right)} \cos\left(d x + c\right)^{5} + 10 \, {\left(a^{3} + 2 \, a^{2} b + a b^{2}\right)} \cos\left(d x + c\right)^{3} - {\left(a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right)} \cos\left(d x + c\right)\right)} \sqrt{-a} \sqrt{\frac{a \cos\left(d x + c\right)^{2} - a - b}{\cos\left(d x + c\right)^{2} - 1}} \sin\left(d x + c\right)\right) + 8 \, {\left(2 \, {\left(3 \, a^{3} b + 2 \, a^{2} b^{2}\right)} \cos\left(d x + c\right)^{3} - 3 \, {\left(2 \, a^{3} b + 3 \, a^{2} b^{2} + a b^{3}\right)} \cos\left(d x + c\right)\right)} \sqrt{\frac{a \cos\left(d x + c\right)^{2} - a - b}{\cos\left(d x + c\right)^{2} - 1}} \sin\left(d x + c\right)}{24 \, {\left({\left(a^{7} + 2 \, a^{6} b + a^{5} b^{2}\right)} d \cos\left(d x + c\right)^{4} - 2 \, {\left(a^{7} + 3 \, a^{6} b + 3 \, a^{5} b^{2} + a^{4} b^{3}\right)} d \cos\left(d x + c\right)^{2} + {\left(a^{7} + 4 \, a^{6} b + 6 \, a^{5} b^{2} + 4 \, a^{4} b^{3} + a^{3} b^{4}\right)} d\right)}}, \frac{3 \, {\left({\left(a^{4} + 2 \, a^{3} b + a^{2} b^{2}\right)} \cos\left(d x + c\right)^{4} + a^{4} + 4 \, a^{3} b + 6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4} - 2 \, {\left(a^{4} + 3 \, a^{3} b + 3 \, a^{2} b^{2} + a b^{3}\right)} \cos\left(d x + c\right)^{2}\right)} \sqrt{a} \arctan\left(\frac{{\left(8 \, a^{2} \cos\left(d x + c\right)^{4} - 8 \, {\left(a^{2} + a b\right)} \cos\left(d x + c\right)^{2} + a^{2} + 2 \, a b + b^{2}\right)} \sqrt{a} \sqrt{\frac{a \cos\left(d x + c\right)^{2} - a - b}{\cos\left(d x + c\right)^{2} - 1}} \sin\left(d x + c\right)}{4 \, {\left(2 \, a^{3} \cos\left(d x + c\right)^{5} - 3 \, {\left(a^{3} + a^{2} b\right)} \cos\left(d x + c\right)^{3} + {\left(a^{3} + 2 \, a^{2} b + a b^{2}\right)} \cos\left(d x + c\right)\right)}}\right) - 4 \, {\left(2 \, {\left(3 \, a^{3} b + 2 \, a^{2} b^{2}\right)} \cos\left(d x + c\right)^{3} - 3 \, {\left(2 \, a^{3} b + 3 \, a^{2} b^{2} + a b^{3}\right)} \cos\left(d x + c\right)\right)} \sqrt{\frac{a \cos\left(d x + c\right)^{2} - a - b}{\cos\left(d x + c\right)^{2} - 1}} \sin\left(d x + c\right)}{12 \, {\left({\left(a^{7} + 2 \, a^{6} b + a^{5} b^{2}\right)} d \cos\left(d x + c\right)^{4} - 2 \, {\left(a^{7} + 3 \, a^{6} b + 3 \, a^{5} b^{2} + a^{4} b^{3}\right)} d \cos\left(d x + c\right)^{2} + {\left(a^{7} + 4 \, a^{6} b + 6 \, a^{5} b^{2} + 4 \, a^{4} b^{3} + a^{3} b^{4}\right)} d\right)}}\right]"," ",0,"[-1/24*(3*((a^4 + 2*a^3*b + a^2*b^2)*cos(d*x + c)^4 + a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4 - 2*(a^4 + 3*a^3*b + 3*a^2*b^2 + a*b^3)*cos(d*x + c)^2)*sqrt(-a)*log(128*a^4*cos(d*x + c)^8 - 256*(a^4 + a^3*b)*cos(d*x + c)^6 + 160*(a^4 + 2*a^3*b + a^2*b^2)*cos(d*x + c)^4 + a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4 - 32*(a^4 + 3*a^3*b + 3*a^2*b^2 + a*b^3)*cos(d*x + c)^2 + 8*(16*a^3*cos(d*x + c)^7 - 24*(a^3 + a^2*b)*cos(d*x + c)^5 + 10*(a^3 + 2*a^2*b + a*b^2)*cos(d*x + c)^3 - (a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cos(d*x + c))*sqrt(-a)*sqrt((a*cos(d*x + c)^2 - a - b)/(cos(d*x + c)^2 - 1))*sin(d*x + c)) + 8*(2*(3*a^3*b + 2*a^2*b^2)*cos(d*x + c)^3 - 3*(2*a^3*b + 3*a^2*b^2 + a*b^3)*cos(d*x + c))*sqrt((a*cos(d*x + c)^2 - a - b)/(cos(d*x + c)^2 - 1))*sin(d*x + c))/((a^7 + 2*a^6*b + a^5*b^2)*d*cos(d*x + c)^4 - 2*(a^7 + 3*a^6*b + 3*a^5*b^2 + a^4*b^3)*d*cos(d*x + c)^2 + (a^7 + 4*a^6*b + 6*a^5*b^2 + 4*a^4*b^3 + a^3*b^4)*d), 1/12*(3*((a^4 + 2*a^3*b + a^2*b^2)*cos(d*x + c)^4 + a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4 - 2*(a^4 + 3*a^3*b + 3*a^2*b^2 + a*b^3)*cos(d*x + c)^2)*sqrt(a)*arctan(1/4*(8*a^2*cos(d*x + c)^4 - 8*(a^2 + a*b)*cos(d*x + c)^2 + a^2 + 2*a*b + b^2)*sqrt(a)*sqrt((a*cos(d*x + c)^2 - a - b)/(cos(d*x + c)^2 - 1))*sin(d*x + c)/(2*a^3*cos(d*x + c)^5 - 3*(a^3 + a^2*b)*cos(d*x + c)^3 + (a^3 + 2*a^2*b + a*b^2)*cos(d*x + c))) - 4*(2*(3*a^3*b + 2*a^2*b^2)*cos(d*x + c)^3 - 3*(2*a^3*b + 3*a^2*b^2 + a*b^3)*cos(d*x + c))*sqrt((a*cos(d*x + c)^2 - a - b)/(cos(d*x + c)^2 - 1))*sin(d*x + c))/((a^7 + 2*a^6*b + a^5*b^2)*d*cos(d*x + c)^4 - 2*(a^7 + 3*a^6*b + 3*a^5*b^2 + a^4*b^3)*d*cos(d*x + c)^2 + (a^7 + 4*a^6*b + 6*a^5*b^2 + 4*a^4*b^3 + a^3*b^4)*d)]","B",0
15,1,1445,0,3.356925," ","integrate(1/(a+b*csc(d*x+c)^2)^(7/2),x, algorithm=""fricas"")","\left[-\frac{15 \, {\left({\left(a^{6} + 3 \, a^{5} b + 3 \, a^{4} b^{2} + a^{3} b^{3}\right)} \cos\left(d x + c\right)^{6} - a^{6} - 6 \, a^{5} b - 15 \, a^{4} b^{2} - 20 \, a^{3} b^{3} - 15 \, a^{2} b^{4} - 6 \, a b^{5} - b^{6} - 3 \, {\left(a^{6} + 4 \, a^{5} b + 6 \, a^{4} b^{2} + 4 \, a^{3} b^{3} + a^{2} b^{4}\right)} \cos\left(d x + c\right)^{4} + 3 \, {\left(a^{6} + 5 \, a^{5} b + 10 \, a^{4} b^{2} + 10 \, a^{3} b^{3} + 5 \, a^{2} b^{4} + a b^{5}\right)} \cos\left(d x + c\right)^{2}\right)} \sqrt{-a} \log\left(128 \, a^{4} \cos\left(d x + c\right)^{8} - 256 \, {\left(a^{4} + a^{3} b\right)} \cos\left(d x + c\right)^{6} + 160 \, {\left(a^{4} + 2 \, a^{3} b + a^{2} b^{2}\right)} \cos\left(d x + c\right)^{4} + a^{4} + 4 \, a^{3} b + 6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4} - 32 \, {\left(a^{4} + 3 \, a^{3} b + 3 \, a^{2} b^{2} + a b^{3}\right)} \cos\left(d x + c\right)^{2} + 8 \, {\left(16 \, a^{3} \cos\left(d x + c\right)^{7} - 24 \, {\left(a^{3} + a^{2} b\right)} \cos\left(d x + c\right)^{5} + 10 \, {\left(a^{3} + 2 \, a^{2} b + a b^{2}\right)} \cos\left(d x + c\right)^{3} - {\left(a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right)} \cos\left(d x + c\right)\right)} \sqrt{-a} \sqrt{\frac{a \cos\left(d x + c\right)^{2} - a - b}{\cos\left(d x + c\right)^{2} - 1}} \sin\left(d x + c\right)\right) + 8 \, {\left({\left(45 \, a^{5} b + 60 \, a^{4} b^{2} + 23 \, a^{3} b^{3}\right)} \cos\left(d x + c\right)^{5} - 5 \, {\left(18 \, a^{5} b + 39 \, a^{4} b^{2} + 28 \, a^{3} b^{3} + 7 \, a^{2} b^{4}\right)} \cos\left(d x + c\right)^{3} + 15 \, {\left(3 \, a^{5} b + 9 \, a^{4} b^{2} + 10 \, a^{3} b^{3} + 5 \, a^{2} b^{4} + a b^{5}\right)} \cos\left(d x + c\right)\right)} \sqrt{\frac{a \cos\left(d x + c\right)^{2} - a - b}{\cos\left(d x + c\right)^{2} - 1}} \sin\left(d x + c\right)}{120 \, {\left({\left(a^{10} + 3 \, a^{9} b + 3 \, a^{8} b^{2} + a^{7} b^{3}\right)} d \cos\left(d x + c\right)^{6} - 3 \, {\left(a^{10} + 4 \, a^{9} b + 6 \, a^{8} b^{2} + 4 \, a^{7} b^{3} + a^{6} b^{4}\right)} d \cos\left(d x + c\right)^{4} + 3 \, {\left(a^{10} + 5 \, a^{9} b + 10 \, a^{8} b^{2} + 10 \, a^{7} b^{3} + 5 \, a^{6} b^{4} + a^{5} b^{5}\right)} d \cos\left(d x + c\right)^{2} - {\left(a^{10} + 6 \, a^{9} b + 15 \, a^{8} b^{2} + 20 \, a^{7} b^{3} + 15 \, a^{6} b^{4} + 6 \, a^{5} b^{5} + a^{4} b^{6}\right)} d\right)}}, \frac{15 \, {\left({\left(a^{6} + 3 \, a^{5} b + 3 \, a^{4} b^{2} + a^{3} b^{3}\right)} \cos\left(d x + c\right)^{6} - a^{6} - 6 \, a^{5} b - 15 \, a^{4} b^{2} - 20 \, a^{3} b^{3} - 15 \, a^{2} b^{4} - 6 \, a b^{5} - b^{6} - 3 \, {\left(a^{6} + 4 \, a^{5} b + 6 \, a^{4} b^{2} + 4 \, a^{3} b^{3} + a^{2} b^{4}\right)} \cos\left(d x + c\right)^{4} + 3 \, {\left(a^{6} + 5 \, a^{5} b + 10 \, a^{4} b^{2} + 10 \, a^{3} b^{3} + 5 \, a^{2} b^{4} + a b^{5}\right)} \cos\left(d x + c\right)^{2}\right)} \sqrt{a} \arctan\left(\frac{{\left(8 \, a^{2} \cos\left(d x + c\right)^{4} - 8 \, {\left(a^{2} + a b\right)} \cos\left(d x + c\right)^{2} + a^{2} + 2 \, a b + b^{2}\right)} \sqrt{a} \sqrt{\frac{a \cos\left(d x + c\right)^{2} - a - b}{\cos\left(d x + c\right)^{2} - 1}} \sin\left(d x + c\right)}{4 \, {\left(2 \, a^{3} \cos\left(d x + c\right)^{5} - 3 \, {\left(a^{3} + a^{2} b\right)} \cos\left(d x + c\right)^{3} + {\left(a^{3} + 2 \, a^{2} b + a b^{2}\right)} \cos\left(d x + c\right)\right)}}\right) - 4 \, {\left({\left(45 \, a^{5} b + 60 \, a^{4} b^{2} + 23 \, a^{3} b^{3}\right)} \cos\left(d x + c\right)^{5} - 5 \, {\left(18 \, a^{5} b + 39 \, a^{4} b^{2} + 28 \, a^{3} b^{3} + 7 \, a^{2} b^{4}\right)} \cos\left(d x + c\right)^{3} + 15 \, {\left(3 \, a^{5} b + 9 \, a^{4} b^{2} + 10 \, a^{3} b^{3} + 5 \, a^{2} b^{4} + a b^{5}\right)} \cos\left(d x + c\right)\right)} \sqrt{\frac{a \cos\left(d x + c\right)^{2} - a - b}{\cos\left(d x + c\right)^{2} - 1}} \sin\left(d x + c\right)}{60 \, {\left({\left(a^{10} + 3 \, a^{9} b + 3 \, a^{8} b^{2} + a^{7} b^{3}\right)} d \cos\left(d x + c\right)^{6} - 3 \, {\left(a^{10} + 4 \, a^{9} b + 6 \, a^{8} b^{2} + 4 \, a^{7} b^{3} + a^{6} b^{4}\right)} d \cos\left(d x + c\right)^{4} + 3 \, {\left(a^{10} + 5 \, a^{9} b + 10 \, a^{8} b^{2} + 10 \, a^{7} b^{3} + 5 \, a^{6} b^{4} + a^{5} b^{5}\right)} d \cos\left(d x + c\right)^{2} - {\left(a^{10} + 6 \, a^{9} b + 15 \, a^{8} b^{2} + 20 \, a^{7} b^{3} + 15 \, a^{6} b^{4} + 6 \, a^{5} b^{5} + a^{4} b^{6}\right)} d\right)}}\right]"," ",0,"[-1/120*(15*((a^6 + 3*a^5*b + 3*a^4*b^2 + a^3*b^3)*cos(d*x + c)^6 - a^6 - 6*a^5*b - 15*a^4*b^2 - 20*a^3*b^3 - 15*a^2*b^4 - 6*a*b^5 - b^6 - 3*(a^6 + 4*a^5*b + 6*a^4*b^2 + 4*a^3*b^3 + a^2*b^4)*cos(d*x + c)^4 + 3*(a^6 + 5*a^5*b + 10*a^4*b^2 + 10*a^3*b^3 + 5*a^2*b^4 + a*b^5)*cos(d*x + c)^2)*sqrt(-a)*log(128*a^4*cos(d*x + c)^8 - 256*(a^4 + a^3*b)*cos(d*x + c)^6 + 160*(a^4 + 2*a^3*b + a^2*b^2)*cos(d*x + c)^4 + a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4 - 32*(a^4 + 3*a^3*b + 3*a^2*b^2 + a*b^3)*cos(d*x + c)^2 + 8*(16*a^3*cos(d*x + c)^7 - 24*(a^3 + a^2*b)*cos(d*x + c)^5 + 10*(a^3 + 2*a^2*b + a*b^2)*cos(d*x + c)^3 - (a^3 + 3*a^2*b + 3*a*b^2 + b^3)*cos(d*x + c))*sqrt(-a)*sqrt((a*cos(d*x + c)^2 - a - b)/(cos(d*x + c)^2 - 1))*sin(d*x + c)) + 8*((45*a^5*b + 60*a^4*b^2 + 23*a^3*b^3)*cos(d*x + c)^5 - 5*(18*a^5*b + 39*a^4*b^2 + 28*a^3*b^3 + 7*a^2*b^4)*cos(d*x + c)^3 + 15*(3*a^5*b + 9*a^4*b^2 + 10*a^3*b^3 + 5*a^2*b^4 + a*b^5)*cos(d*x + c))*sqrt((a*cos(d*x + c)^2 - a - b)/(cos(d*x + c)^2 - 1))*sin(d*x + c))/((a^10 + 3*a^9*b + 3*a^8*b^2 + a^7*b^3)*d*cos(d*x + c)^6 - 3*(a^10 + 4*a^9*b + 6*a^8*b^2 + 4*a^7*b^3 + a^6*b^4)*d*cos(d*x + c)^4 + 3*(a^10 + 5*a^9*b + 10*a^8*b^2 + 10*a^7*b^3 + 5*a^6*b^4 + a^5*b^5)*d*cos(d*x + c)^2 - (a^10 + 6*a^9*b + 15*a^8*b^2 + 20*a^7*b^3 + 15*a^6*b^4 + 6*a^5*b^5 + a^4*b^6)*d), 1/60*(15*((a^6 + 3*a^5*b + 3*a^4*b^2 + a^3*b^3)*cos(d*x + c)^6 - a^6 - 6*a^5*b - 15*a^4*b^2 - 20*a^3*b^3 - 15*a^2*b^4 - 6*a*b^5 - b^6 - 3*(a^6 + 4*a^5*b + 6*a^4*b^2 + 4*a^3*b^3 + a^2*b^4)*cos(d*x + c)^4 + 3*(a^6 + 5*a^5*b + 10*a^4*b^2 + 10*a^3*b^3 + 5*a^2*b^4 + a*b^5)*cos(d*x + c)^2)*sqrt(a)*arctan(1/4*(8*a^2*cos(d*x + c)^4 - 8*(a^2 + a*b)*cos(d*x + c)^2 + a^2 + 2*a*b + b^2)*sqrt(a)*sqrt((a*cos(d*x + c)^2 - a - b)/(cos(d*x + c)^2 - 1))*sin(d*x + c)/(2*a^3*cos(d*x + c)^5 - 3*(a^3 + a^2*b)*cos(d*x + c)^3 + (a^3 + 2*a^2*b + a*b^2)*cos(d*x + c))) - 4*((45*a^5*b + 60*a^4*b^2 + 23*a^3*b^3)*cos(d*x + c)^5 - 5*(18*a^5*b + 39*a^4*b^2 + 28*a^3*b^3 + 7*a^2*b^4)*cos(d*x + c)^3 + 15*(3*a^5*b + 9*a^4*b^2 + 10*a^3*b^3 + 5*a^2*b^4 + a*b^5)*cos(d*x + c))*sqrt((a*cos(d*x + c)^2 - a - b)/(cos(d*x + c)^2 - 1))*sin(d*x + c))/((a^10 + 3*a^9*b + 3*a^8*b^2 + a^7*b^3)*d*cos(d*x + c)^6 - 3*(a^10 + 4*a^9*b + 6*a^8*b^2 + 4*a^7*b^3 + a^6*b^4)*d*cos(d*x + c)^4 + 3*(a^10 + 5*a^9*b + 10*a^8*b^2 + 10*a^7*b^3 + 5*a^6*b^4 + a^5*b^5)*d*cos(d*x + c)^2 - (a^10 + 6*a^9*b + 15*a^8*b^2 + 20*a^7*b^3 + 15*a^6*b^4 + 6*a^5*b^5 + a^4*b^6)*d)]","B",0
16,1,193,0,0.462733," ","integrate((1+csc(x)^2)^(3/2),x, algorithm=""fricas"")","\frac{\arctan\left(\frac{{\left(\cos\left(x\right)^{3} - \cos\left(x\right)\right)} \sqrt{\frac{\cos\left(x\right)^{2} - 2}{\cos\left(x\right)^{2} - 1}} \sin\left(x\right) - \cos\left(x\right) \sin\left(x\right)}{\cos\left(x\right)^{4} - 3 \, \cos\left(x\right)^{2} + 1}\right) \sin\left(x\right) - \arctan\left(\frac{\sin\left(x\right)}{\cos\left(x\right)}\right) \sin\left(x\right) - 2 \, \log\left(-\cos\left(x\right)^{2} + \cos\left(x\right) \sin\left(x\right) - {\left(\cos\left(x\right)^{2} - \cos\left(x\right) \sin\left(x\right) - 1\right)} \sqrt{\frac{\cos\left(x\right)^{2} - 2}{\cos\left(x\right)^{2} - 1}} + 2\right) \sin\left(x\right) + 2 \, \log\left(-\cos\left(x\right)^{2} - \cos\left(x\right) \sin\left(x\right) - {\left(\cos\left(x\right)^{2} + \cos\left(x\right) \sin\left(x\right) - 1\right)} \sqrt{\frac{\cos\left(x\right)^{2} - 2}{\cos\left(x\right)^{2} - 1}} + 2\right) \sin\left(x\right) - \sqrt{\frac{\cos\left(x\right)^{2} - 2}{\cos\left(x\right)^{2} - 1}} \cos\left(x\right)}{2 \, \sin\left(x\right)}"," ",0,"1/2*(arctan(((cos(x)^3 - cos(x))*sqrt((cos(x)^2 - 2)/(cos(x)^2 - 1))*sin(x) - cos(x)*sin(x))/(cos(x)^4 - 3*cos(x)^2 + 1))*sin(x) - arctan(sin(x)/cos(x))*sin(x) - 2*log(-cos(x)^2 + cos(x)*sin(x) - (cos(x)^2 - cos(x)*sin(x) - 1)*sqrt((cos(x)^2 - 2)/(cos(x)^2 - 1)) + 2)*sin(x) + 2*log(-cos(x)^2 - cos(x)*sin(x) - (cos(x)^2 + cos(x)*sin(x) - 1)*sqrt((cos(x)^2 - 2)/(cos(x)^2 - 1)) + 2)*sin(x) - sqrt((cos(x)^2 - 2)/(cos(x)^2 - 1))*cos(x))/sin(x)","B",0
17,1,159,0,0.489799," ","integrate((1+csc(x)^2)^(1/2),x, algorithm=""fricas"")","\frac{1}{2} \, \arctan\left(\frac{{\left(\cos\left(x\right)^{3} - \cos\left(x\right)\right)} \sqrt{\frac{\cos\left(x\right)^{2} - 2}{\cos\left(x\right)^{2} - 1}} \sin\left(x\right) - \cos\left(x\right) \sin\left(x\right)}{\cos\left(x\right)^{4} - 3 \, \cos\left(x\right)^{2} + 1}\right) - \frac{1}{2} \, \arctan\left(\frac{\sin\left(x\right)}{\cos\left(x\right)}\right) - \frac{1}{2} \, \log\left(-\cos\left(x\right)^{2} + \cos\left(x\right) \sin\left(x\right) - {\left(\cos\left(x\right)^{2} - \cos\left(x\right) \sin\left(x\right) - 1\right)} \sqrt{\frac{\cos\left(x\right)^{2} - 2}{\cos\left(x\right)^{2} - 1}} + 2\right) + \frac{1}{2} \, \log\left(-\cos\left(x\right)^{2} - \cos\left(x\right) \sin\left(x\right) - {\left(\cos\left(x\right)^{2} + \cos\left(x\right) \sin\left(x\right) - 1\right)} \sqrt{\frac{\cos\left(x\right)^{2} - 2}{\cos\left(x\right)^{2} - 1}} + 2\right)"," ",0,"1/2*arctan(((cos(x)^3 - cos(x))*sqrt((cos(x)^2 - 2)/(cos(x)^2 - 1))*sin(x) - cos(x)*sin(x))/(cos(x)^4 - 3*cos(x)^2 + 1)) - 1/2*arctan(sin(x)/cos(x)) - 1/2*log(-cos(x)^2 + cos(x)*sin(x) - (cos(x)^2 - cos(x)*sin(x) - 1)*sqrt((cos(x)^2 - 2)/(cos(x)^2 - 1)) + 2) + 1/2*log(-cos(x)^2 - cos(x)*sin(x) - (cos(x)^2 + cos(x)*sin(x) - 1)*sqrt((cos(x)^2 - 2)/(cos(x)^2 - 1)) + 2)","B",0
18,1,65,0,0.462761," ","integrate(1/(1+csc(x)^2)^(1/2),x, algorithm=""fricas"")","\frac{1}{2} \, \arctan\left(\frac{{\left(\cos\left(x\right)^{3} - \cos\left(x\right)\right)} \sqrt{\frac{\cos\left(x\right)^{2} - 2}{\cos\left(x\right)^{2} - 1}} \sin\left(x\right) - \cos\left(x\right) \sin\left(x\right)}{\cos\left(x\right)^{4} - 3 \, \cos\left(x\right)^{2} + 1}\right) - \frac{1}{2} \, \arctan\left(\frac{\sin\left(x\right)}{\cos\left(x\right)}\right)"," ",0,"1/2*arctan(((cos(x)^3 - cos(x))*sqrt((cos(x)^2 - 2)/(cos(x)^2 - 1))*sin(x) - cos(x)*sin(x))/(cos(x)^4 - 3*cos(x)^2 + 1)) - 1/2*arctan(sin(x)/cos(x))","B",0
19,1,10,0,0.477928," ","integrate((1-csc(x)^2)^(3/2),x, algorithm=""fricas"")","x + \arctan\left(\frac{\cos\left(x\right)}{\sin\left(x\right)}\right)"," ",0,"x + arctan(cos(x)/sin(x))","A",0
20,1,10,0,0.445866," ","integrate((1-csc(x)^2)^(1/2),x, algorithm=""fricas"")","x + \arctan\left(\frac{\cos\left(x\right)}{\sin\left(x\right)}\right)"," ",0,"x + arctan(cos(x)/sin(x))","A",0
21,1,12,0,0.432792," ","integrate(1/(1-csc(x)^2)^(1/2),x, algorithm=""fricas"")","x - \arctan\left(\frac{\sin\left(x\right)}{\cos\left(x\right)}\right)"," ",0,"x - arctan(sin(x)/cos(x))","A",0
22,1,25,0,0.445182," ","integrate((-1+csc(x)^2)^(3/2),x, algorithm=""fricas"")","\frac{2 \, {\left(\cos\left(x\right)^{2} - 1\right)} \log\left(\frac{1}{2} \, \sin\left(x\right)\right) - 1}{2 \, {\left(\cos\left(x\right)^{2} - 1\right)}}"," ",0,"1/2*(2*(cos(x)^2 - 1)*log(1/2*sin(x)) - 1)/(cos(x)^2 - 1)","A",0
23,1,7,0,0.448981," ","integrate((-1+csc(x)^2)^(1/2),x, algorithm=""fricas"")","-\log\left(\frac{1}{2} \, \sin\left(x\right)\right)"," ",0,"-log(1/2*sin(x))","A",0
24,1,5,0,0.431926," ","integrate(1/(-1+csc(x)^2)^(1/2),x, algorithm=""fricas"")","\log\left(-\cos\left(x\right)\right)"," ",0,"log(-cos(x))","A",0
25,1,215,0,0.445587," ","integrate((-1-csc(x)^2)^(3/2),x, algorithm=""fricas"")","\frac{{\left(e^{\left(4 i \, x\right)} - 2 \, e^{\left(2 i \, x\right)} + 1\right)} \log\left(-2 \, \sqrt{e^{\left(4 i \, x\right)} - 6 \, e^{\left(2 i \, x\right)} + 1} {\left(e^{\left(2 i \, x\right)} - 1\right)} + 2 \, e^{\left(4 i \, x\right)} - 8 \, e^{\left(2 i \, x\right)} - 2\right) + {\left(4 i \, e^{\left(4 i \, x\right)} - 8 i \, e^{\left(2 i \, x\right)} + 4 i\right)} \log\left(\sqrt{e^{\left(4 i \, x\right)} - 6 \, e^{\left(2 i \, x\right)} + 1} - e^{\left(2 i \, x\right)} + 2 i + 1\right) - {\left(e^{\left(4 i \, x\right)} - 2 \, e^{\left(2 i \, x\right)} + 1\right)} \log\left(\sqrt{e^{\left(4 i \, x\right)} - 6 \, e^{\left(2 i \, x\right)} + 1} - e^{\left(2 i \, x\right)} + 1\right) + {\left(-4 i \, e^{\left(4 i \, x\right)} + 8 i \, e^{\left(2 i \, x\right)} - 4 i\right)} \log\left(\sqrt{e^{\left(4 i \, x\right)} - 6 \, e^{\left(2 i \, x\right)} + 1} - e^{\left(2 i \, x\right)} - 2 i + 1\right) - \sqrt{e^{\left(4 i \, x\right)} - 6 \, e^{\left(2 i \, x\right)} + 1} {\left(e^{\left(2 i \, x\right)} + 1\right)} - e^{\left(4 i \, x\right)} + 2 \, e^{\left(2 i \, x\right)} - 1}{2 \, {\left(e^{\left(4 i \, x\right)} - 2 \, e^{\left(2 i \, x\right)} + 1\right)}}"," ",0,"1/2*((e^(4*I*x) - 2*e^(2*I*x) + 1)*log(-2*sqrt(e^(4*I*x) - 6*e^(2*I*x) + 1)*(e^(2*I*x) - 1) + 2*e^(4*I*x) - 8*e^(2*I*x) - 2) + (4*I*e^(4*I*x) - 8*I*e^(2*I*x) + 4*I)*log(sqrt(e^(4*I*x) - 6*e^(2*I*x) + 1) - e^(2*I*x) + 2*I + 1) - (e^(4*I*x) - 2*e^(2*I*x) + 1)*log(sqrt(e^(4*I*x) - 6*e^(2*I*x) + 1) - e^(2*I*x) + 1) + (-4*I*e^(4*I*x) + 8*I*e^(2*I*x) - 4*I)*log(sqrt(e^(4*I*x) - 6*e^(2*I*x) + 1) - e^(2*I*x) - 2*I + 1) - sqrt(e^(4*I*x) - 6*e^(2*I*x) + 1)*(e^(2*I*x) + 1) - e^(4*I*x) + 2*e^(2*I*x) - 1)/(e^(4*I*x) - 2*e^(2*I*x) + 1)","C",0
26,1,115,0,0.430903," ","integrate((-1-csc(x)^2)^(1/2),x, algorithm=""fricas"")","-\frac{1}{2} \, \log\left(-2 \, \sqrt{e^{\left(4 i \, x\right)} - 6 \, e^{\left(2 i \, x\right)} + 1} {\left(e^{\left(2 i \, x\right)} - 1\right)} + 2 \, e^{\left(4 i \, x\right)} - 8 \, e^{\left(2 i \, x\right)} - 2\right) - i \, \log\left(\sqrt{e^{\left(4 i \, x\right)} - 6 \, e^{\left(2 i \, x\right)} + 1} - e^{\left(2 i \, x\right)} + 2 i + 1\right) + \frac{1}{2} \, \log\left(\sqrt{e^{\left(4 i \, x\right)} - 6 \, e^{\left(2 i \, x\right)} + 1} - e^{\left(2 i \, x\right)} + 1\right) + i \, \log\left(\sqrt{e^{\left(4 i \, x\right)} - 6 \, e^{\left(2 i \, x\right)} + 1} - e^{\left(2 i \, x\right)} - 2 i + 1\right)"," ",0,"-1/2*log(-2*sqrt(e^(4*I*x) - 6*e^(2*I*x) + 1)*(e^(2*I*x) - 1) + 2*e^(4*I*x) - 8*e^(2*I*x) - 2) - I*log(sqrt(e^(4*I*x) - 6*e^(2*I*x) + 1) - e^(2*I*x) + 2*I + 1) + 1/2*log(sqrt(e^(4*I*x) - 6*e^(2*I*x) + 1) - e^(2*I*x) + 1) + I*log(sqrt(e^(4*I*x) - 6*e^(2*I*x) + 1) - e^(2*I*x) - 2*I + 1)","C",0
27,1,65,0,0.409422," ","integrate(1/(-1-csc(x)^2)^(1/2),x, algorithm=""fricas"")","\frac{1}{2} \, \log\left(-2 \, \sqrt{e^{\left(4 i \, x\right)} - 6 \, e^{\left(2 i \, x\right)} + 1} {\left(e^{\left(2 i \, x\right)} - 1\right)} + 2 \, e^{\left(4 i \, x\right)} - 8 \, e^{\left(2 i \, x\right)} - 2\right) - \frac{1}{2} \, \log\left(\sqrt{e^{\left(4 i \, x\right)} - 6 \, e^{\left(2 i \, x\right)} + 1} - e^{\left(2 i \, x\right)} + 1\right)"," ",0,"1/2*log(-2*sqrt(e^(4*I*x) - 6*e^(2*I*x) + 1)*(e^(2*I*x) - 1) + 2*e^(4*I*x) - 8*e^(2*I*x) - 2) - 1/2*log(sqrt(e^(4*I*x) - 6*e^(2*I*x) + 1) - e^(2*I*x) + 1)","C",0
