1,1,168,0,0.541140," ","integrate(csc(x)^5/(a+a*csc(x)),x, algorithm=""fricas"")","\frac{32 \, \cos\left(x\right)^{4} + 14 \, \cos\left(x\right)^{3} - 48 \, \cos\left(x\right)^{2} + 9 \, {\left(\cos\left(x\right)^{4} - 2 \, \cos\left(x\right)^{2} - {\left(\cos\left(x\right)^{3} + \cos\left(x\right)^{2} - \cos\left(x\right) - 1\right)} \sin\left(x\right) + 1\right)} \log\left(\frac{1}{2} \, \cos\left(x\right) + \frac{1}{2}\right) - 9 \, {\left(\cos\left(x\right)^{4} - 2 \, \cos\left(x\right)^{2} - {\left(\cos\left(x\right)^{3} + \cos\left(x\right)^{2} - \cos\left(x\right) - 1\right)} \sin\left(x\right) + 1\right)} \log\left(-\frac{1}{2} \, \cos\left(x\right) + \frac{1}{2}\right) + 2 \, {\left(16 \, \cos\left(x\right)^{3} + 9 \, \cos\left(x\right)^{2} - 15 \, \cos\left(x\right) - 6\right)} \sin\left(x\right) - 18 \, \cos\left(x\right) + 12}{12 \, {\left(a \cos\left(x\right)^{4} - 2 \, a \cos\left(x\right)^{2} - {\left(a \cos\left(x\right)^{3} + a \cos\left(x\right)^{2} - a \cos\left(x\right) - a\right)} \sin\left(x\right) + a\right)}}"," ",0,"1/12*(32*cos(x)^4 + 14*cos(x)^3 - 48*cos(x)^2 + 9*(cos(x)^4 - 2*cos(x)^2 - (cos(x)^3 + cos(x)^2 - cos(x) - 1)*sin(x) + 1)*log(1/2*cos(x) + 1/2) - 9*(cos(x)^4 - 2*cos(x)^2 - (cos(x)^3 + cos(x)^2 - cos(x) - 1)*sin(x) + 1)*log(-1/2*cos(x) + 1/2) + 2*(16*cos(x)^3 + 9*cos(x)^2 - 15*cos(x) - 6)*sin(x) - 18*cos(x) + 12)/(a*cos(x)^4 - 2*a*cos(x)^2 - (a*cos(x)^3 + a*cos(x)^2 - a*cos(x) - a)*sin(x) + a)","B",0
2,1,134,0,0.474811," ","integrate(csc(x)^4/(a+a*csc(x)),x, algorithm=""fricas"")","\frac{8 \, \cos\left(x\right)^{3} + 6 \, \cos\left(x\right)^{2} - 3 \, {\left(\cos\left(x\right)^{3} + \cos\left(x\right)^{2} + {\left(\cos\left(x\right)^{2} - 1\right)} \sin\left(x\right) - \cos\left(x\right) - 1\right)} \log\left(\frac{1}{2} \, \cos\left(x\right) + \frac{1}{2}\right) + 3 \, {\left(\cos\left(x\right)^{3} + \cos\left(x\right)^{2} + {\left(\cos\left(x\right)^{2} - 1\right)} \sin\left(x\right) - \cos\left(x\right) - 1\right)} \log\left(-\frac{1}{2} \, \cos\left(x\right) + \frac{1}{2}\right) - 2 \, {\left(4 \, \cos\left(x\right)^{2} + \cos\left(x\right) - 2\right)} \sin\left(x\right) - 6 \, \cos\left(x\right) - 4}{4 \, {\left(a \cos\left(x\right)^{3} + a \cos\left(x\right)^{2} - a \cos\left(x\right) + {\left(a \cos\left(x\right)^{2} - a\right)} \sin\left(x\right) - a\right)}}"," ",0,"1/4*(8*cos(x)^3 + 6*cos(x)^2 - 3*(cos(x)^3 + cos(x)^2 + (cos(x)^2 - 1)*sin(x) - cos(x) - 1)*log(1/2*cos(x) + 1/2) + 3*(cos(x)^3 + cos(x)^2 + (cos(x)^2 - 1)*sin(x) - cos(x) - 1)*log(-1/2*cos(x) + 1/2) - 2*(4*cos(x)^2 + cos(x) - 2)*sin(x) - 6*cos(x) - 4)/(a*cos(x)^3 + a*cos(x)^2 - a*cos(x) + (a*cos(x)^2 - a)*sin(x) - a)","B",0
3,1,91,0,0.518373," ","integrate(csc(x)^3/(a+a*csc(x)),x, algorithm=""fricas"")","\frac{4 \, \cos\left(x\right)^{2} + {\left(\cos\left(x\right)^{2} - {\left(\cos\left(x\right) + 1\right)} \sin\left(x\right) - 1\right)} \log\left(\frac{1}{2} \, \cos\left(x\right) + \frac{1}{2}\right) - {\left(\cos\left(x\right)^{2} - {\left(\cos\left(x\right) + 1\right)} \sin\left(x\right) - 1\right)} \log\left(-\frac{1}{2} \, \cos\left(x\right) + \frac{1}{2}\right) + 2 \, {\left(2 \, \cos\left(x\right) + 1\right)} \sin\left(x\right) + 2 \, \cos\left(x\right) - 2}{2 \, {\left(a \cos\left(x\right)^{2} - {\left(a \cos\left(x\right) + a\right)} \sin\left(x\right) - a\right)}}"," ",0,"1/2*(4*cos(x)^2 + (cos(x)^2 - (cos(x) + 1)*sin(x) - 1)*log(1/2*cos(x) + 1/2) - (cos(x)^2 - (cos(x) + 1)*sin(x) - 1)*log(-1/2*cos(x) + 1/2) + 2*(2*cos(x) + 1)*sin(x) + 2*cos(x) - 2)/(a*cos(x)^2 - (a*cos(x) + a)*sin(x) - a)","B",0
4,1,53,0,0.488795," ","integrate(csc(x)^2/(a+a*csc(x)),x, algorithm=""fricas"")","-\frac{{\left(\cos\left(x\right) + \sin\left(x\right) + 1\right)} \log\left(\frac{1}{2} \, \cos\left(x\right) + \frac{1}{2}\right) - {\left(\cos\left(x\right) + \sin\left(x\right) + 1\right)} \log\left(-\frac{1}{2} \, \cos\left(x\right) + \frac{1}{2}\right) - 2 \, \cos\left(x\right) + 2 \, \sin\left(x\right) - 2}{2 \, {\left(a \cos\left(x\right) + a \sin\left(x\right) + a\right)}}"," ",0,"-1/2*((cos(x) + sin(x) + 1)*log(1/2*cos(x) + 1/2) - (cos(x) + sin(x) + 1)*log(-1/2*cos(x) + 1/2) - 2*cos(x) + 2*sin(x) - 2)/(a*cos(x) + a*sin(x) + a)","B",0
5,1,22,0,0.501165," ","integrate(csc(x)/(a+a*csc(x)),x, algorithm=""fricas"")","-\frac{\cos\left(x\right) - \sin\left(x\right) + 1}{a \cos\left(x\right) + a \sin\left(x\right) + a}"," ",0,"-(cos(x) - sin(x) + 1)/(a*cos(x) + a*sin(x) + a)","A",0
6,1,54,0,0.556388," ","integrate(1/(a+a*csc(d*x+c)),x, algorithm=""fricas"")","\frac{d x + {\left(d x + 1\right)} \cos\left(d x + c\right) + {\left(d x - 1\right)} \sin\left(d x + c\right) + 1}{a d \cos\left(d x + c\right) + a d \sin\left(d x + c\right) + a d}"," ",0,"(d*x + (d*x + 1)*cos(d*x + c) + (d*x - 1)*sin(d*x + c) + 1)/(a*d*cos(d*x + c) + a*d*sin(d*x + c) + a*d)","A",0
7,1,35,0,0.610241," ","integrate(sin(x)/(a+a*csc(x)),x, algorithm=""fricas"")","-\frac{{\left(x + 2\right)} \cos\left(x\right) + \cos\left(x\right)^{2} + {\left(x + \cos\left(x\right) - 1\right)} \sin\left(x\right) + x + 1}{a \cos\left(x\right) + a \sin\left(x\right) + a}"," ",0,"-((x + 2)*cos(x) + cos(x)^2 + (x + cos(x) - 1)*sin(x) + x + 1)/(a*cos(x) + a*sin(x) + a)","A",0
8,1,53,0,0.519304," ","integrate(sin(x)^2/(a+a*csc(x)),x, algorithm=""fricas"")","\frac{\cos\left(x\right)^{3} + 3 \, {\left(x + 1\right)} \cos\left(x\right) + 2 \, \cos\left(x\right)^{2} - {\left(\cos\left(x\right)^{2} - 3 \, x - \cos\left(x\right) + 2\right)} \sin\left(x\right) + 3 \, x + 2}{2 \, {\left(a \cos\left(x\right) + a \sin\left(x\right) + a\right)}}"," ",0,"1/2*(cos(x)^3 + 3*(x + 1)*cos(x) + 2*cos(x)^2 - (cos(x)^2 - 3*x - cos(x) + 2)*sin(x) + 3*x + 2)/(a*cos(x) + a*sin(x) + a)","A",0
9,1,70,0,0.449222," ","integrate(sin(x)^3/(a+a*csc(x)),x, algorithm=""fricas"")","\frac{2 \, \cos\left(x\right)^{4} - \cos\left(x\right)^{3} - 3 \, {\left(3 \, x + 5\right)} \cos\left(x\right) - 12 \, \cos\left(x\right)^{2} + {\left(2 \, \cos\left(x\right)^{3} + 3 \, \cos\left(x\right)^{2} - 9 \, x - 9 \, \cos\left(x\right) + 6\right)} \sin\left(x\right) - 9 \, x - 6}{6 \, {\left(a \cos\left(x\right) + a \sin\left(x\right) + a\right)}}"," ",0,"1/6*(2*cos(x)^4 - cos(x)^3 - 3*(3*x + 5)*cos(x) - 12*cos(x)^2 + (2*cos(x)^3 + 3*cos(x)^2 - 9*x - 9*cos(x) + 6)*sin(x) - 9*x - 6)/(a*cos(x) + a*sin(x) + a)","A",0
10,1,81,0,0.586516," ","integrate(sin(x)^4/(a+a*csc(x)),x, algorithm=""fricas"")","-\frac{6 \, \cos\left(x\right)^{5} + 8 \, \cos\left(x\right)^{4} - 25 \, \cos\left(x\right)^{3} - 45 \, {\left(x + 1\right)} \cos\left(x\right) - 48 \, \cos\left(x\right)^{2} - {\left(6 \, \cos\left(x\right)^{4} - 2 \, \cos\left(x\right)^{3} - 27 \, \cos\left(x\right)^{2} + 45 \, x + 21 \, \cos\left(x\right) - 24\right)} \sin\left(x\right) - 45 \, x - 24}{24 \, {\left(a \cos\left(x\right) + a \sin\left(x\right) + a\right)}}"," ",0,"-1/24*(6*cos(x)^5 + 8*cos(x)^4 - 25*cos(x)^3 - 45*(x + 1)*cos(x) - 48*cos(x)^2 - (6*cos(x)^4 - 2*cos(x)^3 - 27*cos(x)^2 + 45*x + 21*cos(x) - 24)*sin(x) - 45*x - 24)/(a*cos(x) + a*sin(x) + a)","A",0
11,1,124,0,0.473786," ","integrate(1/(a+a*csc(d*x+c))^2,x, algorithm=""fricas"")","\frac{{\left(3 \, d x - 5\right)} \cos\left(d x + c\right)^{2} - 6 \, d x - {\left(3 \, d x + 4\right)} \cos\left(d x + c\right) - {\left(6 \, d x + {\left(3 \, d x + 5\right)} \cos\left(d x + c\right) + 1\right)} \sin\left(d x + c\right) + 1}{3 \, {\left(a^{2} d \cos\left(d x + c\right)^{2} - a^{2} d \cos\left(d x + c\right) - 2 \, a^{2} d - {\left(a^{2} d \cos\left(d x + c\right) + 2 \, a^{2} d\right)} \sin\left(d x + c\right)\right)}}"," ",0,"1/3*((3*d*x - 5)*cos(d*x + c)^2 - 6*d*x - (3*d*x + 4)*cos(d*x + c) - (6*d*x + (3*d*x + 5)*cos(d*x + c) + 1)*sin(d*x + c) + 1)/(a^2*d*cos(d*x + c)^2 - a^2*d*cos(d*x + c) - 2*a^2*d - (a^2*d*cos(d*x + c) + 2*a^2*d)*sin(d*x + c))","B",0
12,1,181,0,0.519561," ","integrate(1/(a+a*csc(d*x+c))^3,x, algorithm=""fricas"")","\frac{{\left(15 \, d x + 32\right)} \cos\left(d x + c\right)^{3} + {\left(45 \, d x - 19\right)} \cos\left(d x + c\right)^{2} - 60 \, d x - 6 \, {\left(5 \, d x + 9\right)} \cos\left(d x + c\right) + {\left({\left(15 \, d x - 32\right)} \cos\left(d x + c\right)^{2} - 60 \, d x - 3 \, {\left(10 \, d x + 17\right)} \cos\left(d x + c\right) + 3\right)} \sin\left(d x + c\right) - 3}{15 \, {\left(a^{3} d \cos\left(d x + c\right)^{3} + 3 \, a^{3} d \cos\left(d x + c\right)^{2} - 2 \, a^{3} d \cos\left(d x + c\right) - 4 \, a^{3} d + {\left(a^{3} d \cos\left(d x + c\right)^{2} - 2 \, a^{3} d \cos\left(d x + c\right) - 4 \, a^{3} d\right)} \sin\left(d x + c\right)\right)}}"," ",0,"1/15*((15*d*x + 32)*cos(d*x + c)^3 + (45*d*x - 19)*cos(d*x + c)^2 - 60*d*x - 6*(5*d*x + 9)*cos(d*x + c) + ((15*d*x - 32)*cos(d*x + c)^2 - 60*d*x - 3*(10*d*x + 17)*cos(d*x + c) + 3)*sin(d*x + c) - 3)/(a^3*d*cos(d*x + c)^3 + 3*a^3*d*cos(d*x + c)^2 - 2*a^3*d*cos(d*x + c) - 4*a^3*d + (a^3*d*cos(d*x + c)^2 - 2*a^3*d*cos(d*x + c) - 4*a^3*d)*sin(d*x + c))","B",0
13,1,318,0,0.564747," ","integrate((a+a*csc(x))^(5/2),x, algorithm=""fricas"")","\left[\frac{3 \, {\left(a^{2} \cos\left(x\right)^{2} - a^{2} - {\left(a^{2} \cos\left(x\right) + a^{2}\right)} \sin\left(x\right)\right)} \sqrt{-a} \log\left(\frac{2 \, a \cos\left(x\right)^{2} - 2 \, {\left(\cos\left(x\right)^{2} + {\left(\cos\left(x\right) + 1\right)} \sin\left(x\right) - 1\right)} \sqrt{-a} \sqrt{\frac{a \sin\left(x\right) + a}{\sin\left(x\right)}} + a \cos\left(x\right) - {\left(2 \, a \cos\left(x\right) + a\right)} \sin\left(x\right) - a}{\cos\left(x\right) + \sin\left(x\right) + 1}\right) + 2 \, {\left(8 \, a^{2} \cos\left(x\right)^{2} + a^{2} \cos\left(x\right) - 7 \, a^{2} + {\left(8 \, a^{2} \cos\left(x\right) + 7 \, a^{2}\right)} \sin\left(x\right)\right)} \sqrt{\frac{a \sin\left(x\right) + a}{\sin\left(x\right)}}}{3 \, {\left(\cos\left(x\right)^{2} - {\left(\cos\left(x\right) + 1\right)} \sin\left(x\right) - 1\right)}}, \frac{2 \, {\left(3 \, {\left(a^{2} \cos\left(x\right)^{2} - a^{2} - {\left(a^{2} \cos\left(x\right) + a^{2}\right)} \sin\left(x\right)\right)} \sqrt{a} \arctan\left(-\frac{\sqrt{a} \sqrt{\frac{a \sin\left(x\right) + a}{\sin\left(x\right)}} {\left(\cos\left(x\right) - \sin\left(x\right) + 1\right)}}{a \cos\left(x\right) + a \sin\left(x\right) + a}\right) + {\left(8 \, a^{2} \cos\left(x\right)^{2} + a^{2} \cos\left(x\right) - 7 \, a^{2} + {\left(8 \, a^{2} \cos\left(x\right) + 7 \, a^{2}\right)} \sin\left(x\right)\right)} \sqrt{\frac{a \sin\left(x\right) + a}{\sin\left(x\right)}}\right)}}{3 \, {\left(\cos\left(x\right)^{2} - {\left(\cos\left(x\right) + 1\right)} \sin\left(x\right) - 1\right)}}\right]"," ",0,"[1/3*(3*(a^2*cos(x)^2 - a^2 - (a^2*cos(x) + a^2)*sin(x))*sqrt(-a)*log((2*a*cos(x)^2 - 2*(cos(x)^2 + (cos(x) + 1)*sin(x) - 1)*sqrt(-a)*sqrt((a*sin(x) + a)/sin(x)) + a*cos(x) - (2*a*cos(x) + a)*sin(x) - a)/(cos(x) + sin(x) + 1)) + 2*(8*a^2*cos(x)^2 + a^2*cos(x) - 7*a^2 + (8*a^2*cos(x) + 7*a^2)*sin(x))*sqrt((a*sin(x) + a)/sin(x)))/(cos(x)^2 - (cos(x) + 1)*sin(x) - 1), 2/3*(3*(a^2*cos(x)^2 - a^2 - (a^2*cos(x) + a^2)*sin(x))*sqrt(a)*arctan(-sqrt(a)*sqrt((a*sin(x) + a)/sin(x))*(cos(x) - sin(x) + 1)/(a*cos(x) + a*sin(x) + a)) + (8*a^2*cos(x)^2 + a^2*cos(x) - 7*a^2 + (8*a^2*cos(x) + 7*a^2)*sin(x))*sqrt((a*sin(x) + a)/sin(x)))/(cos(x)^2 - (cos(x) + 1)*sin(x) - 1)]","B",0
14,1,212,0,0.456392," ","integrate((a+a*csc(x))^(3/2),x, algorithm=""fricas"")","\left[\frac{{\left(a \cos\left(x\right) + a \sin\left(x\right) + a\right)} \sqrt{-a} \log\left(\frac{2 \, a \cos\left(x\right)^{2} - 2 \, {\left(\cos\left(x\right)^{2} + {\left(\cos\left(x\right) + 1\right)} \sin\left(x\right) - 1\right)} \sqrt{-a} \sqrt{\frac{a \sin\left(x\right) + a}{\sin\left(x\right)}} + a \cos\left(x\right) - {\left(2 \, a \cos\left(x\right) + a\right)} \sin\left(x\right) - a}{\cos\left(x\right) + \sin\left(x\right) + 1}\right) - 2 \, {\left(a \cos\left(x\right) - a \sin\left(x\right) + a\right)} \sqrt{\frac{a \sin\left(x\right) + a}{\sin\left(x\right)}}}{\cos\left(x\right) + \sin\left(x\right) + 1}, \frac{2 \, {\left({\left(a \cos\left(x\right) + a \sin\left(x\right) + a\right)} \sqrt{a} \arctan\left(-\frac{\sqrt{a} \sqrt{\frac{a \sin\left(x\right) + a}{\sin\left(x\right)}} {\left(\cos\left(x\right) - \sin\left(x\right) + 1\right)}}{a \cos\left(x\right) + a \sin\left(x\right) + a}\right) - {\left(a \cos\left(x\right) - a \sin\left(x\right) + a\right)} \sqrt{\frac{a \sin\left(x\right) + a}{\sin\left(x\right)}}\right)}}{\cos\left(x\right) + \sin\left(x\right) + 1}\right]"," ",0,"[((a*cos(x) + a*sin(x) + a)*sqrt(-a)*log((2*a*cos(x)^2 - 2*(cos(x)^2 + (cos(x) + 1)*sin(x) - 1)*sqrt(-a)*sqrt((a*sin(x) + a)/sin(x)) + a*cos(x) - (2*a*cos(x) + a)*sin(x) - a)/(cos(x) + sin(x) + 1)) - 2*(a*cos(x) - a*sin(x) + a)*sqrt((a*sin(x) + a)/sin(x)))/(cos(x) + sin(x) + 1), 2*((a*cos(x) + a*sin(x) + a)*sqrt(a)*arctan(-sqrt(a)*sqrt((a*sin(x) + a)/sin(x))*(cos(x) - sin(x) + 1)/(a*cos(x) + a*sin(x) + a)) - (a*cos(x) - a*sin(x) + a)*sqrt((a*sin(x) + a)/sin(x)))/(cos(x) + sin(x) + 1)]","B",0
15,1,120,0,0.545597," ","integrate((a+a*csc(x))^(1/2),x, algorithm=""fricas"")","\left[\sqrt{-a} \log\left(\frac{2 \, a \cos\left(x\right)^{2} - 2 \, {\left(\cos\left(x\right)^{2} + {\left(\cos\left(x\right) + 1\right)} \sin\left(x\right) - 1\right)} \sqrt{-a} \sqrt{\frac{a \sin\left(x\right) + a}{\sin\left(x\right)}} + a \cos\left(x\right) - {\left(2 \, a \cos\left(x\right) + a\right)} \sin\left(x\right) - a}{\cos\left(x\right) + \sin\left(x\right) + 1}\right), 2 \, \sqrt{a} \arctan\left(-\frac{\sqrt{a} \sqrt{\frac{a \sin\left(x\right) + a}{\sin\left(x\right)}} {\left(\cos\left(x\right) - \sin\left(x\right) + 1\right)}}{a \cos\left(x\right) + a \sin\left(x\right) + a}\right)\right]"," ",0,"[sqrt(-a)*log((2*a*cos(x)^2 - 2*(cos(x)^2 + (cos(x) + 1)*sin(x) - 1)*sqrt(-a)*sqrt((a*sin(x) + a)/sin(x)) + a*cos(x) - (2*a*cos(x) + a)*sin(x) - a)/(cos(x) + sin(x) + 1)), 2*sqrt(a)*arctan(-sqrt(a)*sqrt((a*sin(x) + a)/sin(x))*(cos(x) - sin(x) + 1)/(a*cos(x) + a*sin(x) + a))]","B",0
16,1,219,0,0.537265," ","integrate(1/(a+a*csc(x))^(1/2),x, algorithm=""fricas"")","\left[\frac{\sqrt{2} a \sqrt{-\frac{1}{a}} \log\left(\frac{\sqrt{2} \sqrt{\frac{a \sin\left(x\right) + a}{\sin\left(x\right)}} \sqrt{-\frac{1}{a}} \sin\left(x\right) + \cos\left(x\right)}{\sin\left(x\right) + 1}\right) - \sqrt{-a} \log\left(\frac{2 \, a \cos\left(x\right)^{2} + 2 \, {\left(\cos\left(x\right)^{2} + {\left(\cos\left(x\right) + 1\right)} \sin\left(x\right) - 1\right)} \sqrt{-a} \sqrt{\frac{a \sin\left(x\right) + a}{\sin\left(x\right)}} + a \cos\left(x\right) - {\left(2 \, a \cos\left(x\right) + a\right)} \sin\left(x\right) - a}{\cos\left(x\right) + \sin\left(x\right) + 1}\right)}{a}, -\frac{2 \, {\left(\sqrt{2} \sqrt{a} \arctan\left(\frac{\sqrt{2} \sqrt{\frac{a \sin\left(x\right) + a}{\sin\left(x\right)}} \sin\left(x\right)}{\sqrt{a} {\left(\cos\left(x\right) + \sin\left(x\right) + 1\right)}}\right) - \sqrt{a} \arctan\left(-\frac{\sqrt{a} \sqrt{\frac{a \sin\left(x\right) + a}{\sin\left(x\right)}} {\left(\cos\left(x\right) - \sin\left(x\right) + 1\right)}}{a \cos\left(x\right) + a \sin\left(x\right) + a}\right)\right)}}{a}\right]"," ",0,"[(sqrt(2)*a*sqrt(-1/a)*log((sqrt(2)*sqrt((a*sin(x) + a)/sin(x))*sqrt(-1/a)*sin(x) + cos(x))/(sin(x) + 1)) - sqrt(-a)*log((2*a*cos(x)^2 + 2*(cos(x)^2 + (cos(x) + 1)*sin(x) - 1)*sqrt(-a)*sqrt((a*sin(x) + a)/sin(x)) + a*cos(x) - (2*a*cos(x) + a)*sin(x) - a)/(cos(x) + sin(x) + 1)))/a, -2*(sqrt(2)*sqrt(a)*arctan(sqrt(2)*sqrt((a*sin(x) + a)/sin(x))*sin(x)/(sqrt(a)*(cos(x) + sin(x) + 1))) - sqrt(a)*arctan(-sqrt(a)*sqrt((a*sin(x) + a)/sin(x))*(cos(x) - sin(x) + 1)/(a*cos(x) + a*sin(x) + a)))/a]","A",0
17,1,427,0,0.605441," ","integrate(1/(a+a*csc(x))^(3/2),x, algorithm=""fricas"")","\left[-\frac{5 \, \sqrt{2} {\left(\cos\left(x\right)^{2} - {\left(\cos\left(x\right) + 2\right)} \sin\left(x\right) - \cos\left(x\right) - 2\right)} \sqrt{-a} \log\left(-\frac{\sqrt{2} \sqrt{-a} \sqrt{\frac{a \sin\left(x\right) + a}{\sin\left(x\right)}} \sin\left(x\right) - a \cos\left(x\right)}{\sin\left(x\right) + 1}\right) + 4 \, {\left(\cos\left(x\right)^{2} - {\left(\cos\left(x\right) + 2\right)} \sin\left(x\right) - \cos\left(x\right) - 2\right)} \sqrt{-a} \log\left(\frac{2 \, a \cos\left(x\right)^{2} + 2 \, {\left(\cos\left(x\right)^{2} + {\left(\cos\left(x\right) + 1\right)} \sin\left(x\right) - 1\right)} \sqrt{-a} \sqrt{\frac{a \sin\left(x\right) + a}{\sin\left(x\right)}} + a \cos\left(x\right) - {\left(2 \, a \cos\left(x\right) + a\right)} \sin\left(x\right) - a}{\cos\left(x\right) + \sin\left(x\right) + 1}\right) + 2 \, {\left(\cos\left(x\right)^{2} + {\left(\cos\left(x\right) + 1\right)} \sin\left(x\right) - 1\right)} \sqrt{\frac{a \sin\left(x\right) + a}{\sin\left(x\right)}}}{4 \, {\left(a^{2} \cos\left(x\right)^{2} - a^{2} \cos\left(x\right) - 2 \, a^{2} - {\left(a^{2} \cos\left(x\right) + 2 \, a^{2}\right)} \sin\left(x\right)\right)}}, \frac{5 \, \sqrt{2} {\left(\cos\left(x\right)^{2} - {\left(\cos\left(x\right) + 2\right)} \sin\left(x\right) - \cos\left(x\right) - 2\right)} \sqrt{a} \arctan\left(\frac{\sqrt{2} \sqrt{a} \sqrt{\frac{a \sin\left(x\right) + a}{\sin\left(x\right)}} {\left(\cos\left(x\right) + 1\right)}}{a \cos\left(x\right) + a \sin\left(x\right) + a}\right) + 4 \, {\left(\cos\left(x\right)^{2} - {\left(\cos\left(x\right) + 2\right)} \sin\left(x\right) - \cos\left(x\right) - 2\right)} \sqrt{a} \arctan\left(-\frac{\sqrt{a} \sqrt{\frac{a \sin\left(x\right) + a}{\sin\left(x\right)}} {\left(\cos\left(x\right) - \sin\left(x\right) + 1\right)}}{a \cos\left(x\right) + a \sin\left(x\right) + a}\right) - {\left(\cos\left(x\right)^{2} + {\left(\cos\left(x\right) + 1\right)} \sin\left(x\right) - 1\right)} \sqrt{\frac{a \sin\left(x\right) + a}{\sin\left(x\right)}}}{2 \, {\left(a^{2} \cos\left(x\right)^{2} - a^{2} \cos\left(x\right) - 2 \, a^{2} - {\left(a^{2} \cos\left(x\right) + 2 \, a^{2}\right)} \sin\left(x\right)\right)}}\right]"," ",0,"[-1/4*(5*sqrt(2)*(cos(x)^2 - (cos(x) + 2)*sin(x) - cos(x) - 2)*sqrt(-a)*log(-(sqrt(2)*sqrt(-a)*sqrt((a*sin(x) + a)/sin(x))*sin(x) - a*cos(x))/(sin(x) + 1)) + 4*(cos(x)^2 - (cos(x) + 2)*sin(x) - cos(x) - 2)*sqrt(-a)*log((2*a*cos(x)^2 + 2*(cos(x)^2 + (cos(x) + 1)*sin(x) - 1)*sqrt(-a)*sqrt((a*sin(x) + a)/sin(x)) + a*cos(x) - (2*a*cos(x) + a)*sin(x) - a)/(cos(x) + sin(x) + 1)) + 2*(cos(x)^2 + (cos(x) + 1)*sin(x) - 1)*sqrt((a*sin(x) + a)/sin(x)))/(a^2*cos(x)^2 - a^2*cos(x) - 2*a^2 - (a^2*cos(x) + 2*a^2)*sin(x)), 1/2*(5*sqrt(2)*(cos(x)^2 - (cos(x) + 2)*sin(x) - cos(x) - 2)*sqrt(a)*arctan(sqrt(2)*sqrt(a)*sqrt((a*sin(x) + a)/sin(x))*(cos(x) + 1)/(a*cos(x) + a*sin(x) + a)) + 4*(cos(x)^2 - (cos(x) + 2)*sin(x) - cos(x) - 2)*sqrt(a)*arctan(-sqrt(a)*sqrt((a*sin(x) + a)/sin(x))*(cos(x) - sin(x) + 1)/(a*cos(x) + a*sin(x) + a)) - (cos(x)^2 + (cos(x) + 1)*sin(x) - 1)*sqrt((a*sin(x) + a)/sin(x)))/(a^2*cos(x)^2 - a^2*cos(x) - 2*a^2 - (a^2*cos(x) + 2*a^2)*sin(x))]","B",0
18,1,546,0,0.483284," ","integrate(1/(a+a*csc(x))^(5/2),x, algorithm=""fricas"")","\left[-\frac{43 \, \sqrt{2} {\left(\cos\left(x\right)^{3} + 3 \, \cos\left(x\right)^{2} + {\left(\cos\left(x\right)^{2} - 2 \, \cos\left(x\right) - 4\right)} \sin\left(x\right) - 2 \, \cos\left(x\right) - 4\right)} \sqrt{-a} \log\left(-\frac{\sqrt{2} \sqrt{-a} \sqrt{\frac{a \sin\left(x\right) + a}{\sin\left(x\right)}} \sin\left(x\right) - a \cos\left(x\right)}{\sin\left(x\right) + 1}\right) + 32 \, {\left(\cos\left(x\right)^{3} + 3 \, \cos\left(x\right)^{2} + {\left(\cos\left(x\right)^{2} - 2 \, \cos\left(x\right) - 4\right)} \sin\left(x\right) - 2 \, \cos\left(x\right) - 4\right)} \sqrt{-a} \log\left(\frac{2 \, a \cos\left(x\right)^{2} + 2 \, {\left(\cos\left(x\right)^{2} + {\left(\cos\left(x\right) + 1\right)} \sin\left(x\right) - 1\right)} \sqrt{-a} \sqrt{\frac{a \sin\left(x\right) + a}{\sin\left(x\right)}} + a \cos\left(x\right) - {\left(2 \, a \cos\left(x\right) + a\right)} \sin\left(x\right) - a}{\cos\left(x\right) + \sin\left(x\right) + 1}\right) - 2 \, {\left(15 \, \cos\left(x\right)^{3} + 4 \, \cos\left(x\right)^{2} - {\left(15 \, \cos\left(x\right)^{2} + 11 \, \cos\left(x\right) - 4\right)} \sin\left(x\right) - 15 \, \cos\left(x\right) - 4\right)} \sqrt{\frac{a \sin\left(x\right) + a}{\sin\left(x\right)}}}{32 \, {\left(a^{3} \cos\left(x\right)^{3} + 3 \, a^{3} \cos\left(x\right)^{2} - 2 \, a^{3} \cos\left(x\right) - 4 \, a^{3} + {\left(a^{3} \cos\left(x\right)^{2} - 2 \, a^{3} \cos\left(x\right) - 4 \, a^{3}\right)} \sin\left(x\right)\right)}}, \frac{43 \, \sqrt{2} {\left(\cos\left(x\right)^{3} + 3 \, \cos\left(x\right)^{2} + {\left(\cos\left(x\right)^{2} - 2 \, \cos\left(x\right) - 4\right)} \sin\left(x\right) - 2 \, \cos\left(x\right) - 4\right)} \sqrt{a} \arctan\left(\frac{\sqrt{2} \sqrt{a} \sqrt{\frac{a \sin\left(x\right) + a}{\sin\left(x\right)}} {\left(\cos\left(x\right) + 1\right)}}{a \cos\left(x\right) + a \sin\left(x\right) + a}\right) + 32 \, {\left(\cos\left(x\right)^{3} + 3 \, \cos\left(x\right)^{2} + {\left(\cos\left(x\right)^{2} - 2 \, \cos\left(x\right) - 4\right)} \sin\left(x\right) - 2 \, \cos\left(x\right) - 4\right)} \sqrt{a} \arctan\left(-\frac{\sqrt{a} \sqrt{\frac{a \sin\left(x\right) + a}{\sin\left(x\right)}} {\left(\cos\left(x\right) - \sin\left(x\right) + 1\right)}}{a \cos\left(x\right) + a \sin\left(x\right) + a}\right) + {\left(15 \, \cos\left(x\right)^{3} + 4 \, \cos\left(x\right)^{2} - {\left(15 \, \cos\left(x\right)^{2} + 11 \, \cos\left(x\right) - 4\right)} \sin\left(x\right) - 15 \, \cos\left(x\right) - 4\right)} \sqrt{\frac{a \sin\left(x\right) + a}{\sin\left(x\right)}}}{16 \, {\left(a^{3} \cos\left(x\right)^{3} + 3 \, a^{3} \cos\left(x\right)^{2} - 2 \, a^{3} \cos\left(x\right) - 4 \, a^{3} + {\left(a^{3} \cos\left(x\right)^{2} - 2 \, a^{3} \cos\left(x\right) - 4 \, a^{3}\right)} \sin\left(x\right)\right)}}\right]"," ",0,"[-1/32*(43*sqrt(2)*(cos(x)^3 + 3*cos(x)^2 + (cos(x)^2 - 2*cos(x) - 4)*sin(x) - 2*cos(x) - 4)*sqrt(-a)*log(-(sqrt(2)*sqrt(-a)*sqrt((a*sin(x) + a)/sin(x))*sin(x) - a*cos(x))/(sin(x) + 1)) + 32*(cos(x)^3 + 3*cos(x)^2 + (cos(x)^2 - 2*cos(x) - 4)*sin(x) - 2*cos(x) - 4)*sqrt(-a)*log((2*a*cos(x)^2 + 2*(cos(x)^2 + (cos(x) + 1)*sin(x) - 1)*sqrt(-a)*sqrt((a*sin(x) + a)/sin(x)) + a*cos(x) - (2*a*cos(x) + a)*sin(x) - a)/(cos(x) + sin(x) + 1)) - 2*(15*cos(x)^3 + 4*cos(x)^2 - (15*cos(x)^2 + 11*cos(x) - 4)*sin(x) - 15*cos(x) - 4)*sqrt((a*sin(x) + a)/sin(x)))/(a^3*cos(x)^3 + 3*a^3*cos(x)^2 - 2*a^3*cos(x) - 4*a^3 + (a^3*cos(x)^2 - 2*a^3*cos(x) - 4*a^3)*sin(x)), 1/16*(43*sqrt(2)*(cos(x)^3 + 3*cos(x)^2 + (cos(x)^2 - 2*cos(x) - 4)*sin(x) - 2*cos(x) - 4)*sqrt(a)*arctan(sqrt(2)*sqrt(a)*sqrt((a*sin(x) + a)/sin(x))*(cos(x) + 1)/(a*cos(x) + a*sin(x) + a)) + 32*(cos(x)^3 + 3*cos(x)^2 + (cos(x)^2 - 2*cos(x) - 4)*sin(x) - 2*cos(x) - 4)*sqrt(a)*arctan(-sqrt(a)*sqrt((a*sin(x) + a)/sin(x))*(cos(x) - sin(x) + 1)/(a*cos(x) + a*sin(x) + a)) + (15*cos(x)^3 + 4*cos(x)^2 - (15*cos(x)^2 + 11*cos(x) - 4)*sin(x) - 15*cos(x) - 4)*sqrt((a*sin(x) + a)/sin(x)))/(a^3*cos(x)^3 + 3*a^3*cos(x)^2 - 2*a^3*cos(x) - 4*a^3 + (a^3*cos(x)^2 - 2*a^3*cos(x) - 4*a^3)*sin(x))]","B",0
19,1,283,0,0.803827," ","integrate(csc(f*x+e)^(1/2)*(a+a*csc(f*x+e))^(1/2),x, algorithm=""fricas"")","\left[\frac{\sqrt{a} \log\left(\frac{a \cos\left(f x + e\right)^{3} - 7 \, a \cos\left(f x + e\right)^{2} - 9 \, a \cos\left(f x + e\right) + {\left(a \cos\left(f x + e\right)^{2} + 8 \, a \cos\left(f x + e\right) - a\right)} \sin\left(f x + e\right) + \frac{4 \, {\left(\cos\left(f x + e\right)^{3} + 3 \, \cos\left(f x + e\right)^{2} - {\left(\cos\left(f x + e\right)^{2} - 2 \, \cos\left(f x + e\right) - 3\right)} \sin\left(f x + e\right) - \cos\left(f x + e\right) - 3\right)} \sqrt{a} \sqrt{\frac{a \sin\left(f x + e\right) + a}{\sin\left(f x + e\right)}}}{\sqrt{\sin\left(f x + e\right)}} - a}{\cos\left(f x + e\right)^{3} + \cos\left(f x + e\right)^{2} + {\left(\cos\left(f x + e\right)^{2} - 1\right)} \sin\left(f x + e\right) - \cos\left(f x + e\right) - 1}\right)}{2 \, f}, \frac{\sqrt{-a} \arctan\left(\frac{{\left(\cos\left(f x + e\right)^{2} + 2 \, \sin\left(f x + e\right) - 1\right)} \sqrt{-a} \sqrt{\frac{a \sin\left(f x + e\right) + a}{\sin\left(f x + e\right)}}}{2 \, a \cos\left(f x + e\right) \sqrt{\sin\left(f x + e\right)}}\right)}{f}\right]"," ",0,"[1/2*sqrt(a)*log((a*cos(f*x + e)^3 - 7*a*cos(f*x + e)^2 - 9*a*cos(f*x + e) + (a*cos(f*x + e)^2 + 8*a*cos(f*x + e) - a)*sin(f*x + e) + 4*(cos(f*x + e)^3 + 3*cos(f*x + e)^2 - (cos(f*x + e)^2 - 2*cos(f*x + e) - 3)*sin(f*x + e) - cos(f*x + e) - 3)*sqrt(a)*sqrt((a*sin(f*x + e) + a)/sin(f*x + e))/sqrt(sin(f*x + e)) - a)/(cos(f*x + e)^3 + cos(f*x + e)^2 + (cos(f*x + e)^2 - 1)*sin(f*x + e) - cos(f*x + e) - 1))/f, sqrt(-a)*arctan(1/2*(cos(f*x + e)^2 + 2*sin(f*x + e) - 1)*sqrt(-a)*sqrt((a*sin(f*x + e) + a)/sin(f*x + e))/(a*cos(f*x + e)*sqrt(sin(f*x + e))))/f]","B",0
20,1,296,0,1.084038," ","integrate((-csc(f*x+e))^(1/2)*(a-a*csc(f*x+e))^(1/2),x, algorithm=""fricas"")","\left[\frac{\sqrt{a} \log\left(\frac{a \cos\left(f x + e\right)^{3} - 7 \, a \cos\left(f x + e\right)^{2} - 4 \, {\left(\cos\left(f x + e\right)^{3} + 3 \, \cos\left(f x + e\right)^{2} + {\left(\cos\left(f x + e\right)^{2} - 2 \, \cos\left(f x + e\right) - 3\right)} \sin\left(f x + e\right) - \cos\left(f x + e\right) - 3\right)} \sqrt{a} \sqrt{\frac{a \sin\left(f x + e\right) - a}{\sin\left(f x + e\right)}} \sqrt{-\frac{1}{\sin\left(f x + e\right)}} - 9 \, a \cos\left(f x + e\right) - {\left(a \cos\left(f x + e\right)^{2} + 8 \, a \cos\left(f x + e\right) - a\right)} \sin\left(f x + e\right) - a}{\cos\left(f x + e\right)^{3} + \cos\left(f x + e\right)^{2} - {\left(\cos\left(f x + e\right)^{2} - 1\right)} \sin\left(f x + e\right) - \cos\left(f x + e\right) - 1}\right)}{2 \, f}, \frac{\sqrt{-a} \arctan\left(-\frac{{\left(\cos\left(f x + e\right)^{2} - 2 \, \sin\left(f x + e\right) - 1\right)} \sqrt{-a} \sqrt{\frac{a \sin\left(f x + e\right) - a}{\sin\left(f x + e\right)}} \sqrt{-\frac{1}{\sin\left(f x + e\right)}}}{2 \, a \cos\left(f x + e\right)}\right)}{f}\right]"," ",0,"[1/2*sqrt(a)*log((a*cos(f*x + e)^3 - 7*a*cos(f*x + e)^2 - 4*(cos(f*x + e)^3 + 3*cos(f*x + e)^2 + (cos(f*x + e)^2 - 2*cos(f*x + e) - 3)*sin(f*x + e) - cos(f*x + e) - 3)*sqrt(a)*sqrt((a*sin(f*x + e) - a)/sin(f*x + e))*sqrt(-1/sin(f*x + e)) - 9*a*cos(f*x + e) - (a*cos(f*x + e)^2 + 8*a*cos(f*x + e) - a)*sin(f*x + e) - a)/(cos(f*x + e)^3 + cos(f*x + e)^2 - (cos(f*x + e)^2 - 1)*sin(f*x + e) - cos(f*x + e) - 1))/f, sqrt(-a)*arctan(-1/2*(cos(f*x + e)^2 - 2*sin(f*x + e) - 1)*sqrt(-a)*sqrt((a*sin(f*x + e) - a)/sin(f*x + e))*sqrt(-1/sin(f*x + e))/(a*cos(f*x + e)))/f]","B",0
21,0,0,0,0.951952," ","integrate(csc(d*x+c)^(4/3)*(a+a*csc(d*x+c))^(1/2),x, algorithm=""fricas"")","{\rm integral}\left(\sqrt{a \csc\left(d x + c\right) + a} \csc\left(d x + c\right)^{\frac{4}{3}}, x\right)"," ",0,"integral(sqrt(a*csc(d*x + c) + a)*csc(d*x + c)^(4/3), x)","F",0
22,0,0,0,0.905999," ","integrate(csc(d*x+c)^(1/3)*(a+a*csc(d*x+c))^(1/2),x, algorithm=""fricas"")","{\rm integral}\left(\sqrt{a \csc\left(d x + c\right) + a} \csc\left(d x + c\right)^{\frac{1}{3}}, x\right)"," ",0,"integral(sqrt(a*csc(d*x + c) + a)*csc(d*x + c)^(1/3), x)","F",0
23,0,0,0,0.701493," ","integrate((a+a*csc(d*x+c))^(1/2)/csc(d*x+c)^(2/3),x, algorithm=""fricas"")","{\rm integral}\left(\frac{\sqrt{a \csc\left(d x + c\right) + a}}{\csc\left(d x + c\right)^{\frac{2}{3}}}, x\right)"," ",0,"integral(sqrt(a*csc(d*x + c) + a)/csc(d*x + c)^(2/3), x)","F",0
24,0,0,0,0.890663," ","integrate(csc(d*x+c)^(5/3)*(a+a*csc(d*x+c))^(1/2),x, algorithm=""fricas"")","{\rm integral}\left(\sqrt{a \csc\left(d x + c\right) + a} \csc\left(d x + c\right)^{\frac{5}{3}}, x\right)"," ",0,"integral(sqrt(a*csc(d*x + c) + a)*csc(d*x + c)^(5/3), x)","F",0
25,0,0,0,1.454099," ","integrate(csc(d*x+c)^(2/3)*(a+a*csc(d*x+c))^(1/2),x, algorithm=""fricas"")","{\rm integral}\left(\sqrt{a \csc\left(d x + c\right) + a} \csc\left(d x + c\right)^{\frac{2}{3}}, x\right)"," ",0,"integral(sqrt(a*csc(d*x + c) + a)*csc(d*x + c)^(2/3), x)","F",0
26,0,0,0,1.652821," ","integrate((a+a*csc(d*x+c))^(1/2)/csc(d*x+c)^(1/3),x, algorithm=""fricas"")","{\rm integral}\left(\frac{\sqrt{a \csc\left(d x + c\right) + a}}{\csc\left(d x + c\right)^{\frac{1}{3}}}, x\right)"," ",0,"integral(sqrt(a*csc(d*x + c) + a)/csc(d*x + c)^(1/3), x)","F",0
27,0,0,0,1.887867," ","integrate((a+a*csc(d*x+c))^(1/2)/csc(d*x+c)^(4/3),x, algorithm=""fricas"")","{\rm integral}\left(\frac{\sqrt{a \csc\left(d x + c\right) + a}}{\csc\left(d x + c\right)^{\frac{4}{3}}}, x\right)"," ",0,"integral(sqrt(a*csc(d*x + c) + a)/csc(d*x + c)^(4/3), x)","F",0
28,0,0,0,0.944007," ","integrate(csc(d*x+c)^n*(a+a*csc(d*x+c))^(1/2),x, algorithm=""fricas"")","{\rm integral}\left(\sqrt{a \csc\left(d x + c\right) + a} \csc\left(d x + c\right)^{n}, x\right)"," ",0,"integral(sqrt(a*csc(d*x + c) + a)*csc(d*x + c)^n, x)","F",0
29,0,0,0,0.646317," ","integrate(csc(d*x+c)^n*(a-a*csc(d*x+c))^(1/2),x, algorithm=""fricas"")","{\rm integral}\left(\sqrt{-a \csc\left(d x + c\right) + a} \csc\left(d x + c\right)^{n}, x\right)"," ",0,"integral(sqrt(-a*csc(d*x + c) + a)*csc(d*x + c)^n, x)","F",0
30,0,0,0,0.528021," ","integrate(csc(f*x+e)^3*(a+a*csc(f*x+e))^m,x, algorithm=""fricas"")","{\rm integral}\left({\left(a \csc\left(f x + e\right) + a\right)}^{m} \csc\left(f x + e\right)^{3}, x\right)"," ",0,"integral((a*csc(f*x + e) + a)^m*csc(f*x + e)^3, x)","F",0
31,0,0,0,0.581817," ","integrate(csc(f*x+e)^2*(a+a*csc(f*x+e))^m,x, algorithm=""fricas"")","{\rm integral}\left({\left(a \csc\left(f x + e\right) + a\right)}^{m} \csc\left(f x + e\right)^{2}, x\right)"," ",0,"integral((a*csc(f*x + e) + a)^m*csc(f*x + e)^2, x)","F",0
32,0,0,0,0.536077," ","integrate(csc(f*x+e)*(a+a*csc(f*x+e))^m,x, algorithm=""fricas"")","{\rm integral}\left({\left(a \csc\left(f x + e\right) + a\right)}^{m} \csc\left(f x + e\right), x\right)"," ",0,"integral((a*csc(f*x + e) + a)^m*csc(f*x + e), x)","F",0
33,0,0,0,0.832613," ","integrate((a+a*csc(f*x+e))^m,x, algorithm=""fricas"")","{\rm integral}\left({\left(a \csc\left(f x + e\right) + a\right)}^{m}, x\right)"," ",0,"integral((a*csc(f*x + e) + a)^m, x)","F",0
34,0,0,0,0.532345," ","integrate((a+a*csc(f*x+e))^m*sin(f*x+e),x, algorithm=""fricas"")","{\rm integral}\left({\left(a \csc\left(f x + e\right) + a\right)}^{m} \sin\left(f x + e\right), x\right)"," ",0,"integral((a*csc(f*x + e) + a)^m*sin(f*x + e), x)","F",0
35,0,0,0,0.606036," ","integrate((a+a*csc(f*x+e))^m*sin(f*x+e)^2,x, algorithm=""fricas"")","{\rm integral}\left(-{\left(\cos\left(f x + e\right)^{2} - 1\right)} {\left(a \csc\left(f x + e\right) + a\right)}^{m}, x\right)"," ",0,"integral(-(cos(f*x + e)^2 - 1)*(a*csc(f*x + e) + a)^m, x)","F",0
36,1,217,0,0.783677," ","integrate((a+b*csc(d*x+c))^4,x, algorithm=""fricas"")","-\frac{2 \, {\left(9 \, a^{2} b^{2} + b^{4}\right)} \cos\left(d x + c\right)^{3} - 3 \, {\left(2 \, a^{3} b + a b^{3} - {\left(2 \, a^{3} b + a b^{3}\right)} \cos\left(d x + c\right)^{2}\right)} \log\left(\frac{1}{2} \, \cos\left(d x + c\right) + \frac{1}{2}\right) \sin\left(d x + c\right) + 3 \, {\left(2 \, a^{3} b + a b^{3} - {\left(2 \, a^{3} b + a b^{3}\right)} \cos\left(d x + c\right)^{2}\right)} \log\left(-\frac{1}{2} \, \cos\left(d x + c\right) + \frac{1}{2}\right) \sin\left(d x + c\right) - 3 \, {\left(6 \, a^{2} b^{2} + b^{4}\right)} \cos\left(d x + c\right) - 3 \, {\left(a^{4} d x \cos\left(d x + c\right)^{2} - a^{4} d x + 2 \, a b^{3} \cos\left(d x + c\right)\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*(9*a^2*b^2 + b^4)*cos(d*x + c)^3 - 3*(2*a^3*b + a*b^3 - (2*a^3*b + a*b^3)*cos(d*x + c)^2)*log(1/2*cos(d*x + c) + 1/2)*sin(d*x + c) + 3*(2*a^3*b + a*b^3 - (2*a^3*b + a*b^3)*cos(d*x + c)^2)*log(-1/2*cos(d*x + c) + 1/2)*sin(d*x + c) - 3*(6*a^2*b^2 + b^4)*cos(d*x + c) - 3*(a^4*d*x*cos(d*x + c)^2 - a^4*d*x + 2*a*b^3*cos(d*x + c))*sin(d*x + c))/((d*cos(d*x + c)^2 - d)*sin(d*x + c))","B",0
37,1,155,0,0.550948," ","integrate((a+b*csc(d*x+c))^3,x, algorithm=""fricas"")","\frac{4 \, a^{3} d x \cos\left(d x + c\right)^{2} - 4 \, a^{3} d x + 12 \, a b^{2} \cos\left(d x + c\right) \sin\left(d x + c\right) + 2 \, b^{3} \cos\left(d x + c\right) + {\left(6 \, a^{2} b + b^{3} - {\left(6 \, a^{2} b + b^{3}\right)} \cos\left(d x + c\right)^{2}\right)} \log\left(\frac{1}{2} \, \cos\left(d x + c\right) + \frac{1}{2}\right) - {\left(6 \, a^{2} b + b^{3} - {\left(6 \, a^{2} b + b^{3}\right)} \cos\left(d x + c\right)^{2}\right)} \log\left(-\frac{1}{2} \, \cos\left(d x + c\right) + \frac{1}{2}\right)}{4 \, {\left(d \cos\left(d x + c\right)^{2} - d\right)}}"," ",0,"1/4*(4*a^3*d*x*cos(d*x + c)^2 - 4*a^3*d*x + 12*a*b^2*cos(d*x + c)*sin(d*x + c) + 2*b^3*cos(d*x + c) + (6*a^2*b + b^3 - (6*a^2*b + b^3)*cos(d*x + c)^2)*log(1/2*cos(d*x + c) + 1/2) - (6*a^2*b + b^3 - (6*a^2*b + b^3)*cos(d*x + c)^2)*log(-1/2*cos(d*x + c) + 1/2))/(d*cos(d*x + c)^2 - d)","B",0
38,1,77,0,0.679898," ","integrate((a+b*csc(d*x+c))^2,x, algorithm=""fricas"")","\frac{a^{2} d x \sin\left(d x + c\right) - a b \log\left(\frac{1}{2} \, \cos\left(d x + c\right) + \frac{1}{2}\right) \sin\left(d x + c\right) + a b \log\left(-\frac{1}{2} \, \cos\left(d x + c\right) + \frac{1}{2}\right) \sin\left(d x + c\right) - b^{2} \cos\left(d x + c\right)}{d \sin\left(d x + c\right)}"," ",0,"(a^2*d*x*sin(d*x + c) - a*b*log(1/2*cos(d*x + c) + 1/2)*sin(d*x + c) + a*b*log(-1/2*cos(d*x + c) + 1/2)*sin(d*x + c) - b^2*cos(d*x + c))/(d*sin(d*x + c))","B",0
39,1,607,0,0.735574," ","integrate(csc(x)^5/(a+b*csc(x)),x, algorithm=""fricas"")","\left[\frac{4 \, {\left(3 \, a^{4} b - a^{2} b^{3} - 2 \, b^{5}\right)} \cos\left(x\right)^{3} - 6 \, {\left(a^{4} \cos\left(x\right)^{2} - a^{4}\right)} \sqrt{a^{2} - b^{2}} \log\left(-\frac{{\left(a^{2} - 2 \, b^{2}\right)} \cos\left(x\right)^{2} + 2 \, a b \sin\left(x\right) + a^{2} + b^{2} - 2 \, {\left(b \cos\left(x\right) \sin\left(x\right) + a \cos\left(x\right)\right)} \sqrt{a^{2} - b^{2}}}{a^{2} \cos\left(x\right)^{2} - 2 \, a b \sin\left(x\right) - a^{2} - b^{2}}\right) \sin\left(x\right) + 6 \, {\left(a^{3} b^{2} - a b^{4}\right)} \cos\left(x\right) \sin\left(x\right) + 3 \, {\left(2 \, a^{5} - a^{3} b^{2} - a b^{4} - {\left(2 \, a^{5} - a^{3} b^{2} - a b^{4}\right)} \cos\left(x\right)^{2}\right)} \log\left(\frac{1}{2} \, \cos\left(x\right) + \frac{1}{2}\right) \sin\left(x\right) - 3 \, {\left(2 \, a^{5} - a^{3} b^{2} - a b^{4} - {\left(2 \, a^{5} - a^{3} b^{2} - a b^{4}\right)} \cos\left(x\right)^{2}\right)} \log\left(-\frac{1}{2} \, \cos\left(x\right) + \frac{1}{2}\right) \sin\left(x\right) - 12 \, {\left(a^{4} b - b^{5}\right)} \cos\left(x\right)}{12 \, {\left(a^{2} b^{4} - b^{6} - {\left(a^{2} b^{4} - b^{6}\right)} \cos\left(x\right)^{2}\right)} \sin\left(x\right)}, \frac{4 \, {\left(3 \, a^{4} b - a^{2} b^{3} - 2 \, b^{5}\right)} \cos\left(x\right)^{3} + 12 \, {\left(a^{4} \cos\left(x\right)^{2} - a^{4}\right)} \sqrt{-a^{2} + b^{2}} \arctan\left(-\frac{\sqrt{-a^{2} + b^{2}} {\left(b \sin\left(x\right) + a\right)}}{{\left(a^{2} - b^{2}\right)} \cos\left(x\right)}\right) \sin\left(x\right) + 6 \, {\left(a^{3} b^{2} - a b^{4}\right)} \cos\left(x\right) \sin\left(x\right) + 3 \, {\left(2 \, a^{5} - a^{3} b^{2} - a b^{4} - {\left(2 \, a^{5} - a^{3} b^{2} - a b^{4}\right)} \cos\left(x\right)^{2}\right)} \log\left(\frac{1}{2} \, \cos\left(x\right) + \frac{1}{2}\right) \sin\left(x\right) - 3 \, {\left(2 \, a^{5} - a^{3} b^{2} - a b^{4} - {\left(2 \, a^{5} - a^{3} b^{2} - a b^{4}\right)} \cos\left(x\right)^{2}\right)} \log\left(-\frac{1}{2} \, \cos\left(x\right) + \frac{1}{2}\right) \sin\left(x\right) - 12 \, {\left(a^{4} b - b^{5}\right)} \cos\left(x\right)}{12 \, {\left(a^{2} b^{4} - b^{6} - {\left(a^{2} b^{4} - b^{6}\right)} \cos\left(x\right)^{2}\right)} \sin\left(x\right)}\right]"," ",0,"[1/12*(4*(3*a^4*b - a^2*b^3 - 2*b^5)*cos(x)^3 - 6*(a^4*cos(x)^2 - a^4)*sqrt(a^2 - b^2)*log(-((a^2 - 2*b^2)*cos(x)^2 + 2*a*b*sin(x) + a^2 + b^2 - 2*(b*cos(x)*sin(x) + a*cos(x))*sqrt(a^2 - b^2))/(a^2*cos(x)^2 - 2*a*b*sin(x) - a^2 - b^2))*sin(x) + 6*(a^3*b^2 - a*b^4)*cos(x)*sin(x) + 3*(2*a^5 - a^3*b^2 - a*b^4 - (2*a^5 - a^3*b^2 - a*b^4)*cos(x)^2)*log(1/2*cos(x) + 1/2)*sin(x) - 3*(2*a^5 - a^3*b^2 - a*b^4 - (2*a^5 - a^3*b^2 - a*b^4)*cos(x)^2)*log(-1/2*cos(x) + 1/2)*sin(x) - 12*(a^4*b - b^5)*cos(x))/((a^2*b^4 - b^6 - (a^2*b^4 - b^6)*cos(x)^2)*sin(x)), 1/12*(4*(3*a^4*b - a^2*b^3 - 2*b^5)*cos(x)^3 + 12*(a^4*cos(x)^2 - a^4)*sqrt(-a^2 + b^2)*arctan(-sqrt(-a^2 + b^2)*(b*sin(x) + a)/((a^2 - b^2)*cos(x)))*sin(x) + 6*(a^3*b^2 - a*b^4)*cos(x)*sin(x) + 3*(2*a^5 - a^3*b^2 - a*b^4 - (2*a^5 - a^3*b^2 - a*b^4)*cos(x)^2)*log(1/2*cos(x) + 1/2)*sin(x) - 3*(2*a^5 - a^3*b^2 - a*b^4 - (2*a^5 - a^3*b^2 - a*b^4)*cos(x)^2)*log(-1/2*cos(x) + 1/2)*sin(x) - 12*(a^4*b - b^5)*cos(x))/((a^2*b^4 - b^6 - (a^2*b^4 - b^6)*cos(x)^2)*sin(x))]","B",0
40,1,524,0,0.867352," ","integrate(csc(x)^4/(a+b*csc(x)),x, algorithm=""fricas"")","\left[\frac{4 \, {\left(a^{3} b - a b^{3}\right)} \cos\left(x\right) \sin\left(x\right) - 2 \, {\left(a^{3} \cos\left(x\right)^{2} - a^{3}\right)} \sqrt{a^{2} - b^{2}} \log\left(\frac{{\left(a^{2} - 2 \, b^{2}\right)} \cos\left(x\right)^{2} + 2 \, a b \sin\left(x\right) + a^{2} + b^{2} + 2 \, {\left(b \cos\left(x\right) \sin\left(x\right) + a \cos\left(x\right)\right)} \sqrt{a^{2} - b^{2}}}{a^{2} \cos\left(x\right)^{2} - 2 \, a b \sin\left(x\right) - a^{2} - b^{2}}\right) - 2 \, {\left(a^{2} b^{2} - b^{4}\right)} \cos\left(x\right) - {\left(2 \, a^{4} - a^{2} b^{2} - b^{4} - {\left(2 \, a^{4} - a^{2} b^{2} - b^{4}\right)} \cos\left(x\right)^{2}\right)} \log\left(\frac{1}{2} \, \cos\left(x\right) + \frac{1}{2}\right) + {\left(2 \, a^{4} - a^{2} b^{2} - b^{4} - {\left(2 \, a^{4} - a^{2} b^{2} - b^{4}\right)} \cos\left(x\right)^{2}\right)} \log\left(-\frac{1}{2} \, \cos\left(x\right) + \frac{1}{2}\right)}{4 \, {\left(a^{2} b^{3} - b^{5} - {\left(a^{2} b^{3} - b^{5}\right)} \cos\left(x\right)^{2}\right)}}, \frac{4 \, {\left(a^{3} b - a b^{3}\right)} \cos\left(x\right) \sin\left(x\right) - 4 \, {\left(a^{3} \cos\left(x\right)^{2} - a^{3}\right)} \sqrt{-a^{2} + b^{2}} \arctan\left(-\frac{\sqrt{-a^{2} + b^{2}} {\left(b \sin\left(x\right) + a\right)}}{{\left(a^{2} - b^{2}\right)} \cos\left(x\right)}\right) - 2 \, {\left(a^{2} b^{2} - b^{4}\right)} \cos\left(x\right) - {\left(2 \, a^{4} - a^{2} b^{2} - b^{4} - {\left(2 \, a^{4} - a^{2} b^{2} - b^{4}\right)} \cos\left(x\right)^{2}\right)} \log\left(\frac{1}{2} \, \cos\left(x\right) + \frac{1}{2}\right) + {\left(2 \, a^{4} - a^{2} b^{2} - b^{4} - {\left(2 \, a^{4} - a^{2} b^{2} - b^{4}\right)} \cos\left(x\right)^{2}\right)} \log\left(-\frac{1}{2} \, \cos\left(x\right) + \frac{1}{2}\right)}{4 \, {\left(a^{2} b^{3} - b^{5} - {\left(a^{2} b^{3} - b^{5}\right)} \cos\left(x\right)^{2}\right)}}\right]"," ",0,"[1/4*(4*(a^3*b - a*b^3)*cos(x)*sin(x) - 2*(a^3*cos(x)^2 - a^3)*sqrt(a^2 - b^2)*log(((a^2 - 2*b^2)*cos(x)^2 + 2*a*b*sin(x) + a^2 + b^2 + 2*(b*cos(x)*sin(x) + a*cos(x))*sqrt(a^2 - b^2))/(a^2*cos(x)^2 - 2*a*b*sin(x) - a^2 - b^2)) - 2*(a^2*b^2 - b^4)*cos(x) - (2*a^4 - a^2*b^2 - b^4 - (2*a^4 - a^2*b^2 - b^4)*cos(x)^2)*log(1/2*cos(x) + 1/2) + (2*a^4 - a^2*b^2 - b^4 - (2*a^4 - a^2*b^2 - b^4)*cos(x)^2)*log(-1/2*cos(x) + 1/2))/(a^2*b^3 - b^5 - (a^2*b^3 - b^5)*cos(x)^2), 1/4*(4*(a^3*b - a*b^3)*cos(x)*sin(x) - 4*(a^3*cos(x)^2 - a^3)*sqrt(-a^2 + b^2)*arctan(-sqrt(-a^2 + b^2)*(b*sin(x) + a)/((a^2 - b^2)*cos(x))) - 2*(a^2*b^2 - b^4)*cos(x) - (2*a^4 - a^2*b^2 - b^4 - (2*a^4 - a^2*b^2 - b^4)*cos(x)^2)*log(1/2*cos(x) + 1/2) + (2*a^4 - a^2*b^2 - b^4 - (2*a^4 - a^2*b^2 - b^4)*cos(x)^2)*log(-1/2*cos(x) + 1/2))/(a^2*b^3 - b^5 - (a^2*b^3 - b^5)*cos(x)^2)]","B",0
41,1,308,0,0.672215," ","integrate(csc(x)^3/(a+b*csc(x)),x, algorithm=""fricas"")","\left[\frac{\sqrt{a^{2} - b^{2}} a^{2} \log\left(-\frac{{\left(a^{2} - 2 \, b^{2}\right)} \cos\left(x\right)^{2} + 2 \, a b \sin\left(x\right) + a^{2} + b^{2} - 2 \, {\left(b \cos\left(x\right) \sin\left(x\right) + a \cos\left(x\right)\right)} \sqrt{a^{2} - b^{2}}}{a^{2} \cos\left(x\right)^{2} - 2 \, a b \sin\left(x\right) - a^{2} - b^{2}}\right) \sin\left(x\right) + {\left(a^{3} - a b^{2}\right)} \log\left(\frac{1}{2} \, \cos\left(x\right) + \frac{1}{2}\right) \sin\left(x\right) - {\left(a^{3} - a b^{2}\right)} \log\left(-\frac{1}{2} \, \cos\left(x\right) + \frac{1}{2}\right) \sin\left(x\right) - 2 \, {\left(a^{2} b - b^{3}\right)} \cos\left(x\right)}{2 \, {\left(a^{2} b^{2} - b^{4}\right)} \sin\left(x\right)}, -\frac{2 \, \sqrt{-a^{2} + b^{2}} a^{2} \arctan\left(-\frac{\sqrt{-a^{2} + b^{2}} {\left(b \sin\left(x\right) + a\right)}}{{\left(a^{2} - b^{2}\right)} \cos\left(x\right)}\right) \sin\left(x\right) - {\left(a^{3} - a b^{2}\right)} \log\left(\frac{1}{2} \, \cos\left(x\right) + \frac{1}{2}\right) \sin\left(x\right) + {\left(a^{3} - a b^{2}\right)} \log\left(-\frac{1}{2} \, \cos\left(x\right) + \frac{1}{2}\right) \sin\left(x\right) + 2 \, {\left(a^{2} b - b^{3}\right)} \cos\left(x\right)}{2 \, {\left(a^{2} b^{2} - b^{4}\right)} \sin\left(x\right)}\right]"," ",0,"[1/2*(sqrt(a^2 - b^2)*a^2*log(-((a^2 - 2*b^2)*cos(x)^2 + 2*a*b*sin(x) + a^2 + b^2 - 2*(b*cos(x)*sin(x) + a*cos(x))*sqrt(a^2 - b^2))/(a^2*cos(x)^2 - 2*a*b*sin(x) - a^2 - b^2))*sin(x) + (a^3 - a*b^2)*log(1/2*cos(x) + 1/2)*sin(x) - (a^3 - a*b^2)*log(-1/2*cos(x) + 1/2)*sin(x) - 2*(a^2*b - b^3)*cos(x))/((a^2*b^2 - b^4)*sin(x)), -1/2*(2*sqrt(-a^2 + b^2)*a^2*arctan(-sqrt(-a^2 + b^2)*(b*sin(x) + a)/((a^2 - b^2)*cos(x)))*sin(x) - (a^3 - a*b^2)*log(1/2*cos(x) + 1/2)*sin(x) + (a^3 - a*b^2)*log(-1/2*cos(x) + 1/2)*sin(x) + 2*(a^2*b - b^3)*cos(x))/((a^2*b^2 - b^4)*sin(x))]","B",0
42,1,245,0,0.663432," ","integrate(csc(x)^2/(a+b*csc(x)),x, algorithm=""fricas"")","\left[\frac{\sqrt{a^{2} - b^{2}} a \log\left(\frac{{\left(a^{2} - 2 \, b^{2}\right)} \cos\left(x\right)^{2} + 2 \, a b \sin\left(x\right) + a^{2} + b^{2} + 2 \, {\left(b \cos\left(x\right) \sin\left(x\right) + a \cos\left(x\right)\right)} \sqrt{a^{2} - b^{2}}}{a^{2} \cos\left(x\right)^{2} - 2 \, a b \sin\left(x\right) - a^{2} - b^{2}}\right) - {\left(a^{2} - b^{2}\right)} \log\left(\frac{1}{2} \, \cos\left(x\right) + \frac{1}{2}\right) + {\left(a^{2} - b^{2}\right)} \log\left(-\frac{1}{2} \, \cos\left(x\right) + \frac{1}{2}\right)}{2 \, {\left(a^{2} b - b^{3}\right)}}, \frac{2 \, \sqrt{-a^{2} + b^{2}} a \arctan\left(-\frac{\sqrt{-a^{2} + b^{2}} {\left(b \sin\left(x\right) + a\right)}}{{\left(a^{2} - b^{2}\right)} \cos\left(x\right)}\right) - {\left(a^{2} - b^{2}\right)} \log\left(\frac{1}{2} \, \cos\left(x\right) + \frac{1}{2}\right) + {\left(a^{2} - b^{2}\right)} \log\left(-\frac{1}{2} \, \cos\left(x\right) + \frac{1}{2}\right)}{2 \, {\left(a^{2} b - b^{3}\right)}}\right]"," ",0,"[1/2*(sqrt(a^2 - b^2)*a*log(((a^2 - 2*b^2)*cos(x)^2 + 2*a*b*sin(x) + a^2 + b^2 + 2*(b*cos(x)*sin(x) + a*cos(x))*sqrt(a^2 - b^2))/(a^2*cos(x)^2 - 2*a*b*sin(x) - a^2 - b^2)) - (a^2 - b^2)*log(1/2*cos(x) + 1/2) + (a^2 - b^2)*log(-1/2*cos(x) + 1/2))/(a^2*b - b^3), 1/2*(2*sqrt(-a^2 + b^2)*a*arctan(-sqrt(-a^2 + b^2)*(b*sin(x) + a)/((a^2 - b^2)*cos(x))) - (a^2 - b^2)*log(1/2*cos(x) + 1/2) + (a^2 - b^2)*log(-1/2*cos(x) + 1/2))/(a^2*b - b^3)]","B",0
43,1,154,0,0.528553," ","integrate(csc(x)/(a+b*csc(x)),x, algorithm=""fricas"")","\left[\frac{\log\left(-\frac{{\left(a^{2} - 2 \, b^{2}\right)} \cos\left(x\right)^{2} + 2 \, a b \sin\left(x\right) + a^{2} + b^{2} - 2 \, {\left(b \cos\left(x\right) \sin\left(x\right) + a \cos\left(x\right)\right)} \sqrt{a^{2} - b^{2}}}{a^{2} \cos\left(x\right)^{2} - 2 \, a b \sin\left(x\right) - a^{2} - b^{2}}\right)}{2 \, \sqrt{a^{2} - b^{2}}}, -\frac{\sqrt{-a^{2} + b^{2}} \arctan\left(-\frac{\sqrt{-a^{2} + b^{2}} {\left(b \sin\left(x\right) + a\right)}}{{\left(a^{2} - b^{2}\right)} \cos\left(x\right)}\right)}{a^{2} - b^{2}}\right]"," ",0,"[1/2*log(-((a^2 - 2*b^2)*cos(x)^2 + 2*a*b*sin(x) + a^2 + b^2 - 2*(b*cos(x)*sin(x) + a*cos(x))*sqrt(a^2 - b^2))/(a^2*cos(x)^2 - 2*a*b*sin(x) - a^2 - b^2))/sqrt(a^2 - b^2), -sqrt(-a^2 + b^2)*arctan(-sqrt(-a^2 + b^2)*(b*sin(x) + a)/((a^2 - b^2)*cos(x)))/(a^2 - b^2)]","A",0
44,1,238,0,0.614055," ","integrate(1/(a+b*csc(d*x+c)),x, algorithm=""fricas"")","\left[\frac{2 \, {\left(a^{2} - b^{2}\right)} d x + \sqrt{a^{2} - b^{2}} b \log\left(\frac{{\left(a^{2} - 2 \, b^{2}\right)} \cos\left(d x + c\right)^{2} + 2 \, a b \sin\left(d x + c\right) + a^{2} + b^{2} + 2 \, {\left(b \cos\left(d x + c\right) \sin\left(d x + c\right) + a \cos\left(d x + c\right)\right)} \sqrt{a^{2} - b^{2}}}{a^{2} \cos\left(d x + c\right)^{2} - 2 \, a b \sin\left(d x + c\right) - a^{2} - b^{2}}\right)}{2 \, {\left(a^{3} - a b^{2}\right)} d}, \frac{{\left(a^{2} - b^{2}\right)} d x + \sqrt{-a^{2} + b^{2}} b \arctan\left(-\frac{\sqrt{-a^{2} + b^{2}} {\left(b \sin\left(d x + c\right) + a\right)}}{{\left(a^{2} - b^{2}\right)} \cos\left(d x + c\right)}\right)}{{\left(a^{3} - a b^{2}\right)} d}\right]"," ",0,"[1/2*(2*(a^2 - b^2)*d*x + sqrt(a^2 - b^2)*b*log(((a^2 - 2*b^2)*cos(d*x + c)^2 + 2*a*b*sin(d*x + c) + a^2 + b^2 + 2*(b*cos(d*x + c)*sin(d*x + c) + a*cos(d*x + c))*sqrt(a^2 - b^2))/(a^2*cos(d*x + c)^2 - 2*a*b*sin(d*x + c) - a^2 - b^2)))/((a^3 - a*b^2)*d), ((a^2 - b^2)*d*x + sqrt(-a^2 + b^2)*b*arctan(-sqrt(-a^2 + b^2)*(b*sin(d*x + c) + a)/((a^2 - b^2)*cos(d*x + c))))/((a^3 - a*b^2)*d)]","A",0
45,1,235,0,0.491643," ","integrate(sin(x)/(a+b*csc(x)),x, algorithm=""fricas"")","\left[\frac{\sqrt{a^{2} - b^{2}} b^{2} \log\left(-\frac{{\left(a^{2} - 2 \, b^{2}\right)} \cos\left(x\right)^{2} + 2 \, a b \sin\left(x\right) + a^{2} + b^{2} - 2 \, {\left(b \cos\left(x\right) \sin\left(x\right) + a \cos\left(x\right)\right)} \sqrt{a^{2} - b^{2}}}{a^{2} \cos\left(x\right)^{2} - 2 \, a b \sin\left(x\right) - a^{2} - b^{2}}\right) - 2 \, {\left(a^{2} b - b^{3}\right)} x - 2 \, {\left(a^{3} - a b^{2}\right)} \cos\left(x\right)}{2 \, {\left(a^{4} - a^{2} b^{2}\right)}}, -\frac{\sqrt{-a^{2} + b^{2}} b^{2} \arctan\left(-\frac{\sqrt{-a^{2} + b^{2}} {\left(b \sin\left(x\right) + a\right)}}{{\left(a^{2} - b^{2}\right)} \cos\left(x\right)}\right) + {\left(a^{2} b - b^{3}\right)} x + {\left(a^{3} - a b^{2}\right)} \cos\left(x\right)}{a^{4} - a^{2} b^{2}}\right]"," ",0,"[1/2*(sqrt(a^2 - b^2)*b^2*log(-((a^2 - 2*b^2)*cos(x)^2 + 2*a*b*sin(x) + a^2 + b^2 - 2*(b*cos(x)*sin(x) + a*cos(x))*sqrt(a^2 - b^2))/(a^2*cos(x)^2 - 2*a*b*sin(x) - a^2 - b^2)) - 2*(a^2*b - b^3)*x - 2*(a^3 - a*b^2)*cos(x))/(a^4 - a^2*b^2), -(sqrt(-a^2 + b^2)*b^2*arctan(-sqrt(-a^2 + b^2)*(b*sin(x) + a)/((a^2 - b^2)*cos(x))) + (a^2*b - b^3)*x + (a^3 - a*b^2)*cos(x))/(a^4 - a^2*b^2)]","A",0
46,1,285,0,0.675606," ","integrate(sin(x)^2/(a+b*csc(x)),x, algorithm=""fricas"")","\left[\frac{\sqrt{a^{2} - b^{2}} b^{3} \log\left(\frac{{\left(a^{2} - 2 \, b^{2}\right)} \cos\left(x\right)^{2} + 2 \, a b \sin\left(x\right) + a^{2} + b^{2} + 2 \, {\left(b \cos\left(x\right) \sin\left(x\right) + a \cos\left(x\right)\right)} \sqrt{a^{2} - b^{2}}}{a^{2} \cos\left(x\right)^{2} - 2 \, a b \sin\left(x\right) - a^{2} - b^{2}}\right) - {\left(a^{4} - a^{2} b^{2}\right)} \cos\left(x\right) \sin\left(x\right) + {\left(a^{4} + a^{2} b^{2} - 2 \, b^{4}\right)} x + 2 \, {\left(a^{3} b - a b^{3}\right)} \cos\left(x\right)}{2 \, {\left(a^{5} - a^{3} b^{2}\right)}}, \frac{2 \, \sqrt{-a^{2} + b^{2}} b^{3} \arctan\left(-\frac{\sqrt{-a^{2} + b^{2}} {\left(b \sin\left(x\right) + a\right)}}{{\left(a^{2} - b^{2}\right)} \cos\left(x\right)}\right) - {\left(a^{4} - a^{2} b^{2}\right)} \cos\left(x\right) \sin\left(x\right) + {\left(a^{4} + a^{2} b^{2} - 2 \, b^{4}\right)} x + 2 \, {\left(a^{3} b - a b^{3}\right)} \cos\left(x\right)}{2 \, {\left(a^{5} - a^{3} b^{2}\right)}}\right]"," ",0,"[1/2*(sqrt(a^2 - b^2)*b^3*log(((a^2 - 2*b^2)*cos(x)^2 + 2*a*b*sin(x) + a^2 + b^2 + 2*(b*cos(x)*sin(x) + a*cos(x))*sqrt(a^2 - b^2))/(a^2*cos(x)^2 - 2*a*b*sin(x) - a^2 - b^2)) - (a^4 - a^2*b^2)*cos(x)*sin(x) + (a^4 + a^2*b^2 - 2*b^4)*x + 2*(a^3*b - a*b^3)*cos(x))/(a^5 - a^3*b^2), 1/2*(2*sqrt(-a^2 + b^2)*b^3*arctan(-sqrt(-a^2 + b^2)*(b*sin(x) + a)/((a^2 - b^2)*cos(x))) - (a^4 - a^2*b^2)*cos(x)*sin(x) + (a^4 + a^2*b^2 - 2*b^4)*x + 2*(a^3*b - a*b^3)*cos(x))/(a^5 - a^3*b^2)]","A",0
47,1,329,0,0.616610," ","integrate(sin(x)^3/(a+b*csc(x)),x, algorithm=""fricas"")","\left[\frac{3 \, \sqrt{a^{2} - b^{2}} b^{4} \log\left(-\frac{{\left(a^{2} - 2 \, b^{2}\right)} \cos\left(x\right)^{2} + 2 \, a b \sin\left(x\right) + a^{2} + b^{2} - 2 \, {\left(b \cos\left(x\right) \sin\left(x\right) + a \cos\left(x\right)\right)} \sqrt{a^{2} - b^{2}}}{a^{2} \cos\left(x\right)^{2} - 2 \, a b \sin\left(x\right) - a^{2} - b^{2}}\right) + 2 \, {\left(a^{5} - a^{3} b^{2}\right)} \cos\left(x\right)^{3} + 3 \, {\left(a^{4} b - a^{2} b^{3}\right)} \cos\left(x\right) \sin\left(x\right) - 3 \, {\left(a^{4} b + a^{2} b^{3} - 2 \, b^{5}\right)} x - 6 \, {\left(a^{5} - a b^{4}\right)} \cos\left(x\right)}{6 \, {\left(a^{6} - a^{4} b^{2}\right)}}, -\frac{6 \, \sqrt{-a^{2} + b^{2}} b^{4} \arctan\left(-\frac{\sqrt{-a^{2} + b^{2}} {\left(b \sin\left(x\right) + a\right)}}{{\left(a^{2} - b^{2}\right)} \cos\left(x\right)}\right) - 2 \, {\left(a^{5} - a^{3} b^{2}\right)} \cos\left(x\right)^{3} - 3 \, {\left(a^{4} b - a^{2} b^{3}\right)} \cos\left(x\right) \sin\left(x\right) + 3 \, {\left(a^{4} b + a^{2} b^{3} - 2 \, b^{5}\right)} x + 6 \, {\left(a^{5} - a b^{4}\right)} \cos\left(x\right)}{6 \, {\left(a^{6} - a^{4} b^{2}\right)}}\right]"," ",0,"[1/6*(3*sqrt(a^2 - b^2)*b^4*log(-((a^2 - 2*b^2)*cos(x)^2 + 2*a*b*sin(x) + a^2 + b^2 - 2*(b*cos(x)*sin(x) + a*cos(x))*sqrt(a^2 - b^2))/(a^2*cos(x)^2 - 2*a*b*sin(x) - a^2 - b^2)) + 2*(a^5 - a^3*b^2)*cos(x)^3 + 3*(a^4*b - a^2*b^3)*cos(x)*sin(x) - 3*(a^4*b + a^2*b^3 - 2*b^5)*x - 6*(a^5 - a*b^4)*cos(x))/(a^6 - a^4*b^2), -1/6*(6*sqrt(-a^2 + b^2)*b^4*arctan(-sqrt(-a^2 + b^2)*(b*sin(x) + a)/((a^2 - b^2)*cos(x))) - 2*(a^5 - a^3*b^2)*cos(x)^3 - 3*(a^4*b - a^2*b^3)*cos(x)*sin(x) + 3*(a^4*b + a^2*b^3 - 2*b^5)*x + 6*(a^5 - a*b^4)*cos(x))/(a^6 - a^4*b^2)]","A",0
48,1,410,0,1.130901," ","integrate(sin(x)^4/(a+b*csc(x)),x, algorithm=""fricas"")","\left[\frac{12 \, \sqrt{a^{2} - b^{2}} b^{5} \log\left(\frac{{\left(a^{2} - 2 \, b^{2}\right)} \cos\left(x\right)^{2} + 2 \, a b \sin\left(x\right) + a^{2} + b^{2} + 2 \, {\left(b \cos\left(x\right) \sin\left(x\right) + a \cos\left(x\right)\right)} \sqrt{a^{2} - b^{2}}}{a^{2} \cos\left(x\right)^{2} - 2 \, a b \sin\left(x\right) - a^{2} - b^{2}}\right) - 8 \, {\left(a^{5} b - a^{3} b^{3}\right)} \cos\left(x\right)^{3} + 3 \, {\left(3 \, a^{6} + a^{4} b^{2} + 4 \, a^{2} b^{4} - 8 \, b^{6}\right)} x + 24 \, {\left(a^{5} b - a b^{5}\right)} \cos\left(x\right) + 3 \, {\left(2 \, {\left(a^{6} - a^{4} b^{2}\right)} \cos\left(x\right)^{3} - {\left(5 \, a^{6} - a^{4} b^{2} - 4 \, a^{2} b^{4}\right)} \cos\left(x\right)\right)} \sin\left(x\right)}{24 \, {\left(a^{7} - a^{5} b^{2}\right)}}, \frac{24 \, \sqrt{-a^{2} + b^{2}} b^{5} \arctan\left(-\frac{\sqrt{-a^{2} + b^{2}} {\left(b \sin\left(x\right) + a\right)}}{{\left(a^{2} - b^{2}\right)} \cos\left(x\right)}\right) - 8 \, {\left(a^{5} b - a^{3} b^{3}\right)} \cos\left(x\right)^{3} + 3 \, {\left(3 \, a^{6} + a^{4} b^{2} + 4 \, a^{2} b^{4} - 8 \, b^{6}\right)} x + 24 \, {\left(a^{5} b - a b^{5}\right)} \cos\left(x\right) + 3 \, {\left(2 \, {\left(a^{6} - a^{4} b^{2}\right)} \cos\left(x\right)^{3} - {\left(5 \, a^{6} - a^{4} b^{2} - 4 \, a^{2} b^{4}\right)} \cos\left(x\right)\right)} \sin\left(x\right)}{24 \, {\left(a^{7} - a^{5} b^{2}\right)}}\right]"," ",0,"[1/24*(12*sqrt(a^2 - b^2)*b^5*log(((a^2 - 2*b^2)*cos(x)^2 + 2*a*b*sin(x) + a^2 + b^2 + 2*(b*cos(x)*sin(x) + a*cos(x))*sqrt(a^2 - b^2))/(a^2*cos(x)^2 - 2*a*b*sin(x) - a^2 - b^2)) - 8*(a^5*b - a^3*b^3)*cos(x)^3 + 3*(3*a^6 + a^4*b^2 + 4*a^2*b^4 - 8*b^6)*x + 24*(a^5*b - a*b^5)*cos(x) + 3*(2*(a^6 - a^4*b^2)*cos(x)^3 - (5*a^6 - a^4*b^2 - 4*a^2*b^4)*cos(x))*sin(x))/(a^7 - a^5*b^2), 1/24*(24*sqrt(-a^2 + b^2)*b^5*arctan(-sqrt(-a^2 + b^2)*(b*sin(x) + a)/((a^2 - b^2)*cos(x))) - 8*(a^5*b - a^3*b^3)*cos(x)^3 + 3*(3*a^6 + a^4*b^2 + 4*a^2*b^4 - 8*b^6)*x + 24*(a^5*b - a*b^5)*cos(x) + 3*(2*(a^6 - a^4*b^2)*cos(x)^3 - (5*a^6 - a^4*b^2 - 4*a^2*b^4)*cos(x))*sin(x))/(a^7 - a^5*b^2)]","A",0
49,1,493,0,0.611757," ","integrate(1/(a+b*csc(d*x+c))^2,x, algorithm=""fricas"")","\left[\frac{2 \, {\left(a^{5} - 2 \, a^{3} b^{2} + a b^{4}\right)} d x \sin\left(d x + c\right) + 2 \, {\left(a^{4} b - 2 \, a^{2} b^{3} + b^{5}\right)} d x + {\left(2 \, a^{2} b^{2} - b^{4} + {\left(2 \, a^{3} b - a b^{3}\right)} \sin\left(d x + c\right)\right)} \sqrt{a^{2} - b^{2}} \log\left(\frac{{\left(a^{2} - 2 \, b^{2}\right)} \cos\left(d x + c\right)^{2} + 2 \, a b \sin\left(d x + c\right) + a^{2} + b^{2} + 2 \, {\left(b \cos\left(d x + c\right) \sin\left(d x + c\right) + a \cos\left(d x + c\right)\right)} \sqrt{a^{2} - b^{2}}}{a^{2} \cos\left(d x + c\right)^{2} - 2 \, a b \sin\left(d x + c\right) - a^{2} - b^{2}}\right) - 2 \, {\left(a^{3} b^{2} - a b^{4}\right)} \cos\left(d x + c\right)}{2 \, {\left({\left(a^{7} - 2 \, a^{5} b^{2} + a^{3} b^{4}\right)} d \sin\left(d x + c\right) + {\left(a^{6} b - 2 \, a^{4} b^{3} + a^{2} b^{5}\right)} d\right)}}, \frac{{\left(a^{5} - 2 \, a^{3} b^{2} + a b^{4}\right)} d x \sin\left(d x + c\right) + {\left(a^{4} b - 2 \, a^{2} b^{3} + b^{5}\right)} d x + {\left(2 \, a^{2} b^{2} - b^{4} + {\left(2 \, a^{3} b - a b^{3}\right)} \sin\left(d x + c\right)\right)} \sqrt{-a^{2} + b^{2}} \arctan\left(-\frac{\sqrt{-a^{2} + b^{2}} {\left(b \sin\left(d x + c\right) + a\right)}}{{\left(a^{2} - b^{2}\right)} \cos\left(d x + c\right)}\right) - {\left(a^{3} b^{2} - a b^{4}\right)} \cos\left(d x + c\right)}{{\left(a^{7} - 2 \, a^{5} b^{2} + a^{3} b^{4}\right)} d \sin\left(d x + c\right) + {\left(a^{6} b - 2 \, a^{4} b^{3} + a^{2} b^{5}\right)} d}\right]"," ",0,"[1/2*(2*(a^5 - 2*a^3*b^2 + a*b^4)*d*x*sin(d*x + c) + 2*(a^4*b - 2*a^2*b^3 + b^5)*d*x + (2*a^2*b^2 - b^4 + (2*a^3*b - a*b^3)*sin(d*x + c))*sqrt(a^2 - b^2)*log(((a^2 - 2*b^2)*cos(d*x + c)^2 + 2*a*b*sin(d*x + c) + a^2 + b^2 + 2*(b*cos(d*x + c)*sin(d*x + c) + a*cos(d*x + c))*sqrt(a^2 - b^2))/(a^2*cos(d*x + c)^2 - 2*a*b*sin(d*x + c) - a^2 - b^2)) - 2*(a^3*b^2 - a*b^4)*cos(d*x + c))/((a^7 - 2*a^5*b^2 + a^3*b^4)*d*sin(d*x + c) + (a^6*b - 2*a^4*b^3 + a^2*b^5)*d), ((a^5 - 2*a^3*b^2 + a*b^4)*d*x*sin(d*x + c) + (a^4*b - 2*a^2*b^3 + b^5)*d*x + (2*a^2*b^2 - b^4 + (2*a^3*b - a*b^3)*sin(d*x + c))*sqrt(-a^2 + b^2)*arctan(-sqrt(-a^2 + b^2)*(b*sin(d*x + c) + a)/((a^2 - b^2)*cos(d*x + c))) - (a^3*b^2 - a*b^4)*cos(d*x + c))/((a^7 - 2*a^5*b^2 + a^3*b^4)*d*sin(d*x + c) + (a^6*b - 2*a^4*b^3 + a^2*b^5)*d)]","B",0
50,1,933,0,0.754484," ","integrate(1/(a+b*csc(d*x+c))^3,x, algorithm=""fricas"")","\left[\frac{4 \, {\left(a^{8} - 3 \, a^{6} b^{2} + 3 \, a^{4} b^{4} - a^{2} b^{6}\right)} d x \cos\left(d x + c\right)^{2} - 4 \, {\left(a^{8} - 2 \, a^{6} b^{2} + 2 \, a^{2} b^{6} - b^{8}\right)} d x - {\left(6 \, a^{6} b + a^{4} b^{3} - 3 \, a^{2} b^{5} + 2 \, b^{7} - {\left(6 \, a^{6} b - 5 \, a^{4} b^{3} + 2 \, a^{2} b^{5}\right)} \cos\left(d x + c\right)^{2} + 2 \, {\left(6 \, a^{5} b^{2} - 5 \, a^{3} b^{4} + 2 \, a b^{6}\right)} \sin\left(d x + c\right)\right)} \sqrt{a^{2} - b^{2}} \log\left(\frac{{\left(a^{2} - 2 \, b^{2}\right)} \cos\left(d x + c\right)^{2} + 2 \, a b \sin\left(d x + c\right) + a^{2} + b^{2} + 2 \, {\left(b \cos\left(d x + c\right) \sin\left(d x + c\right) + a \cos\left(d x + c\right)\right)} \sqrt{a^{2} - b^{2}}}{a^{2} \cos\left(d x + c\right)^{2} - 2 \, a b \sin\left(d x + c\right) - a^{2} - b^{2}}\right) + 2 \, {\left(5 \, a^{5} b^{3} - 7 \, a^{3} b^{5} + 2 \, a b^{7}\right)} \cos\left(d x + c\right) - 2 \, {\left(4 \, {\left(a^{7} b - 3 \, a^{5} b^{3} + 3 \, a^{3} b^{5} - a b^{7}\right)} d x - 3 \, {\left(2 \, a^{6} b^{2} - 3 \, a^{4} b^{4} + a^{2} b^{6}\right)} \cos\left(d x + c\right)\right)} \sin\left(d x + c\right)}{4 \, {\left({\left(a^{11} - 3 \, a^{9} b^{2} + 3 \, a^{7} b^{4} - a^{5} b^{6}\right)} d \cos\left(d x + c\right)^{2} - 2 \, {\left(a^{10} b - 3 \, a^{8} b^{3} + 3 \, a^{6} b^{5} - a^{4} b^{7}\right)} d \sin\left(d x + c\right) - {\left(a^{11} - 2 \, a^{9} b^{2} + 2 \, a^{5} b^{6} - a^{3} b^{8}\right)} d\right)}}, \frac{2 \, {\left(a^{8} - 3 \, a^{6} b^{2} + 3 \, a^{4} b^{4} - a^{2} b^{6}\right)} d x \cos\left(d x + c\right)^{2} - 2 \, {\left(a^{8} - 2 \, a^{6} b^{2} + 2 \, a^{2} b^{6} - b^{8}\right)} d x - {\left(6 \, a^{6} b + a^{4} b^{3} - 3 \, a^{2} b^{5} + 2 \, b^{7} - {\left(6 \, a^{6} b - 5 \, a^{4} b^{3} + 2 \, a^{2} b^{5}\right)} \cos\left(d x + c\right)^{2} + 2 \, {\left(6 \, a^{5} b^{2} - 5 \, a^{3} b^{4} + 2 \, a b^{6}\right)} \sin\left(d x + c\right)\right)} \sqrt{-a^{2} + b^{2}} \arctan\left(-\frac{\sqrt{-a^{2} + b^{2}} {\left(b \sin\left(d x + c\right) + a\right)}}{{\left(a^{2} - b^{2}\right)} \cos\left(d x + c\right)}\right) + {\left(5 \, a^{5} b^{3} - 7 \, a^{3} b^{5} + 2 \, a b^{7}\right)} \cos\left(d x + c\right) - {\left(4 \, {\left(a^{7} b - 3 \, a^{5} b^{3} + 3 \, a^{3} b^{5} - a b^{7}\right)} d x - 3 \, {\left(2 \, a^{6} b^{2} - 3 \, a^{4} b^{4} + a^{2} b^{6}\right)} \cos\left(d x + c\right)\right)} \sin\left(d x + c\right)}{2 \, {\left({\left(a^{11} - 3 \, a^{9} b^{2} + 3 \, a^{7} b^{4} - a^{5} b^{6}\right)} d \cos\left(d x + c\right)^{2} - 2 \, {\left(a^{10} b - 3 \, a^{8} b^{3} + 3 \, a^{6} b^{5} - a^{4} b^{7}\right)} d \sin\left(d x + c\right) - {\left(a^{11} - 2 \, a^{9} b^{2} + 2 \, a^{5} b^{6} - a^{3} b^{8}\right)} d\right)}}\right]"," ",0,"[1/4*(4*(a^8 - 3*a^6*b^2 + 3*a^4*b^4 - a^2*b^6)*d*x*cos(d*x + c)^2 - 4*(a^8 - 2*a^6*b^2 + 2*a^2*b^6 - b^8)*d*x - (6*a^6*b + a^4*b^3 - 3*a^2*b^5 + 2*b^7 - (6*a^6*b - 5*a^4*b^3 + 2*a^2*b^5)*cos(d*x + c)^2 + 2*(6*a^5*b^2 - 5*a^3*b^4 + 2*a*b^6)*sin(d*x + c))*sqrt(a^2 - b^2)*log(((a^2 - 2*b^2)*cos(d*x + c)^2 + 2*a*b*sin(d*x + c) + a^2 + b^2 + 2*(b*cos(d*x + c)*sin(d*x + c) + a*cos(d*x + c))*sqrt(a^2 - b^2))/(a^2*cos(d*x + c)^2 - 2*a*b*sin(d*x + c) - a^2 - b^2)) + 2*(5*a^5*b^3 - 7*a^3*b^5 + 2*a*b^7)*cos(d*x + c) - 2*(4*(a^7*b - 3*a^5*b^3 + 3*a^3*b^5 - a*b^7)*d*x - 3*(2*a^6*b^2 - 3*a^4*b^4 + a^2*b^6)*cos(d*x + c))*sin(d*x + c))/((a^11 - 3*a^9*b^2 + 3*a^7*b^4 - a^5*b^6)*d*cos(d*x + c)^2 - 2*(a^10*b - 3*a^8*b^3 + 3*a^6*b^5 - a^4*b^7)*d*sin(d*x + c) - (a^11 - 2*a^9*b^2 + 2*a^5*b^6 - a^3*b^8)*d), 1/2*(2*(a^8 - 3*a^6*b^2 + 3*a^4*b^4 - a^2*b^6)*d*x*cos(d*x + c)^2 - 2*(a^8 - 2*a^6*b^2 + 2*a^2*b^6 - b^8)*d*x - (6*a^6*b + a^4*b^3 - 3*a^2*b^5 + 2*b^7 - (6*a^6*b - 5*a^4*b^3 + 2*a^2*b^5)*cos(d*x + c)^2 + 2*(6*a^5*b^2 - 5*a^3*b^4 + 2*a*b^6)*sin(d*x + c))*sqrt(-a^2 + b^2)*arctan(-sqrt(-a^2 + b^2)*(b*sin(d*x + c) + a)/((a^2 - b^2)*cos(d*x + c))) + (5*a^5*b^3 - 7*a^3*b^5 + 2*a*b^7)*cos(d*x + c) - (4*(a^7*b - 3*a^5*b^3 + 3*a^3*b^5 - a*b^7)*d*x - 3*(2*a^6*b^2 - 3*a^4*b^4 + a^2*b^6)*cos(d*x + c))*sin(d*x + c))/((a^11 - 3*a^9*b^2 + 3*a^7*b^4 - a^5*b^6)*d*cos(d*x + c)^2 - 2*(a^10*b - 3*a^8*b^3 + 3*a^6*b^5 - a^4*b^7)*d*sin(d*x + c) - (a^11 - 2*a^9*b^2 + 2*a^5*b^6 - a^3*b^8)*d)]","B",0
51,1,1554,0,0.754383," ","integrate(1/(a+b*csc(d*x+c))^4,x, algorithm=""fricas"")","\left[\frac{36 \, {\left(a^{10} b - 4 \, a^{8} b^{3} + 6 \, a^{6} b^{5} - 4 \, a^{4} b^{7} + a^{2} b^{9}\right)} d x \cos\left(d x + c\right)^{2} - 2 \, {\left(36 \, a^{9} b^{2} - 68 \, a^{7} b^{4} + 43 \, a^{5} b^{6} - 11 \, a^{3} b^{8}\right)} \cos\left(d x + c\right)^{3} - 12 \, {\left(3 \, a^{10} b - 11 \, a^{8} b^{3} + 14 \, a^{6} b^{5} - 6 \, a^{4} b^{7} - a^{2} b^{9} + b^{11}\right)} d x - 3 \, {\left(24 \, a^{8} b^{2} - 16 \, a^{6} b^{4} + 13 \, a^{4} b^{6} + a^{2} b^{8} - 2 \, b^{10} - 3 \, {\left(8 \, a^{8} b^{2} - 8 \, a^{6} b^{4} + 7 \, a^{4} b^{6} - 2 \, a^{2} b^{8}\right)} \cos\left(d x + c\right)^{2} + {\left(8 \, a^{9} b + 16 \, a^{7} b^{3} - 17 \, a^{5} b^{5} + 19 \, a^{3} b^{7} - 6 \, a b^{9} - {\left(8 \, a^{9} b - 8 \, a^{7} b^{3} + 7 \, a^{5} b^{5} - 2 \, a^{3} b^{7}\right)} \cos\left(d x + c\right)^{2}\right)} \sin\left(d x + c\right)\right)} \sqrt{a^{2} - b^{2}} \log\left(\frac{{\left(a^{2} - 2 \, b^{2}\right)} \cos\left(d x + c\right)^{2} + 2 \, a b \sin\left(d x + c\right) + a^{2} + b^{2} + 2 \, {\left(b \cos\left(d x + c\right) \sin\left(d x + c\right) + a \cos\left(d x + c\right)\right)} \sqrt{a^{2} - b^{2}}}{a^{2} \cos\left(d x + c\right)^{2} - 2 \, a b \sin\left(d x + c\right) - a^{2} - b^{2}}\right) + 12 \, {\left(6 \, a^{9} b^{2} - 7 \, a^{7} b^{4} + 2 \, a^{3} b^{8} - a b^{10}\right)} \cos\left(d x + c\right) + 6 \, {\left(2 \, {\left(a^{11} - 4 \, a^{9} b^{2} + 6 \, a^{7} b^{4} - 4 \, a^{5} b^{6} + a^{3} b^{8}\right)} d x \cos\left(d x + c\right)^{2} - 2 \, {\left(a^{11} - a^{9} b^{2} - 6 \, a^{7} b^{4} + 14 \, a^{5} b^{6} - 11 \, a^{3} b^{8} + 3 \, a b^{10}\right)} d x + 5 \, {\left(4 \, a^{8} b^{3} - 7 \, a^{6} b^{5} + 4 \, a^{4} b^{7} - a^{2} b^{9}\right)} \cos\left(d x + c\right)\right)} \sin\left(d x + c\right)}{12 \, {\left(3 \, {\left(a^{14} b - 4 \, a^{12} b^{3} + 6 \, a^{10} b^{5} - 4 \, a^{8} b^{7} + a^{6} b^{9}\right)} d \cos\left(d x + c\right)^{2} - {\left(3 \, a^{14} b - 11 \, a^{12} b^{3} + 14 \, a^{10} b^{5} - 6 \, a^{8} b^{7} - a^{6} b^{9} + a^{4} b^{11}\right)} d + {\left({\left(a^{15} - 4 \, a^{13} b^{2} + 6 \, a^{11} b^{4} - 4 \, a^{9} b^{6} + a^{7} b^{8}\right)} d \cos\left(d x + c\right)^{2} - {\left(a^{15} - a^{13} b^{2} - 6 \, a^{11} b^{4} + 14 \, a^{9} b^{6} - 11 \, a^{7} b^{8} + 3 \, a^{5} b^{10}\right)} d\right)} \sin\left(d x + c\right)\right)}}, \frac{18 \, {\left(a^{10} b - 4 \, a^{8} b^{3} + 6 \, a^{6} b^{5} - 4 \, a^{4} b^{7} + a^{2} b^{9}\right)} d x \cos\left(d x + c\right)^{2} - {\left(36 \, a^{9} b^{2} - 68 \, a^{7} b^{4} + 43 \, a^{5} b^{6} - 11 \, a^{3} b^{8}\right)} \cos\left(d x + c\right)^{3} - 6 \, {\left(3 \, a^{10} b - 11 \, a^{8} b^{3} + 14 \, a^{6} b^{5} - 6 \, a^{4} b^{7} - a^{2} b^{9} + b^{11}\right)} d x - 3 \, {\left(24 \, a^{8} b^{2} - 16 \, a^{6} b^{4} + 13 \, a^{4} b^{6} + a^{2} b^{8} - 2 \, b^{10} - 3 \, {\left(8 \, a^{8} b^{2} - 8 \, a^{6} b^{4} + 7 \, a^{4} b^{6} - 2 \, a^{2} b^{8}\right)} \cos\left(d x + c\right)^{2} + {\left(8 \, a^{9} b + 16 \, a^{7} b^{3} - 17 \, a^{5} b^{5} + 19 \, a^{3} b^{7} - 6 \, a b^{9} - {\left(8 \, a^{9} b - 8 \, a^{7} b^{3} + 7 \, a^{5} b^{5} - 2 \, a^{3} b^{7}\right)} \cos\left(d x + c\right)^{2}\right)} \sin\left(d x + c\right)\right)} \sqrt{-a^{2} + b^{2}} \arctan\left(-\frac{\sqrt{-a^{2} + b^{2}} {\left(b \sin\left(d x + c\right) + a\right)}}{{\left(a^{2} - b^{2}\right)} \cos\left(d x + c\right)}\right) + 6 \, {\left(6 \, a^{9} b^{2} - 7 \, a^{7} b^{4} + 2 \, a^{3} b^{8} - a b^{10}\right)} \cos\left(d x + c\right) + 3 \, {\left(2 \, {\left(a^{11} - 4 \, a^{9} b^{2} + 6 \, a^{7} b^{4} - 4 \, a^{5} b^{6} + a^{3} b^{8}\right)} d x \cos\left(d x + c\right)^{2} - 2 \, {\left(a^{11} - a^{9} b^{2} - 6 \, a^{7} b^{4} + 14 \, a^{5} b^{6} - 11 \, a^{3} b^{8} + 3 \, a b^{10}\right)} d x + 5 \, {\left(4 \, a^{8} b^{3} - 7 \, a^{6} b^{5} + 4 \, a^{4} b^{7} - a^{2} b^{9}\right)} \cos\left(d x + c\right)\right)} \sin\left(d x + c\right)}{6 \, {\left(3 \, {\left(a^{14} b - 4 \, a^{12} b^{3} + 6 \, a^{10} b^{5} - 4 \, a^{8} b^{7} + a^{6} b^{9}\right)} d \cos\left(d x + c\right)^{2} - {\left(3 \, a^{14} b - 11 \, a^{12} b^{3} + 14 \, a^{10} b^{5} - 6 \, a^{8} b^{7} - a^{6} b^{9} + a^{4} b^{11}\right)} d + {\left({\left(a^{15} - 4 \, a^{13} b^{2} + 6 \, a^{11} b^{4} - 4 \, a^{9} b^{6} + a^{7} b^{8}\right)} d \cos\left(d x + c\right)^{2} - {\left(a^{15} - a^{13} b^{2} - 6 \, a^{11} b^{4} + 14 \, a^{9} b^{6} - 11 \, a^{7} b^{8} + 3 \, a^{5} b^{10}\right)} d\right)} \sin\left(d x + c\right)\right)}}\right]"," ",0,"[1/12*(36*(a^10*b - 4*a^8*b^3 + 6*a^6*b^5 - 4*a^4*b^7 + a^2*b^9)*d*x*cos(d*x + c)^2 - 2*(36*a^9*b^2 - 68*a^7*b^4 + 43*a^5*b^6 - 11*a^3*b^8)*cos(d*x + c)^3 - 12*(3*a^10*b - 11*a^8*b^3 + 14*a^6*b^5 - 6*a^4*b^7 - a^2*b^9 + b^11)*d*x - 3*(24*a^8*b^2 - 16*a^6*b^4 + 13*a^4*b^6 + a^2*b^8 - 2*b^10 - 3*(8*a^8*b^2 - 8*a^6*b^4 + 7*a^4*b^6 - 2*a^2*b^8)*cos(d*x + c)^2 + (8*a^9*b + 16*a^7*b^3 - 17*a^5*b^5 + 19*a^3*b^7 - 6*a*b^9 - (8*a^9*b - 8*a^7*b^3 + 7*a^5*b^5 - 2*a^3*b^7)*cos(d*x + c)^2)*sin(d*x + c))*sqrt(a^2 - b^2)*log(((a^2 - 2*b^2)*cos(d*x + c)^2 + 2*a*b*sin(d*x + c) + a^2 + b^2 + 2*(b*cos(d*x + c)*sin(d*x + c) + a*cos(d*x + c))*sqrt(a^2 - b^2))/(a^2*cos(d*x + c)^2 - 2*a*b*sin(d*x + c) - a^2 - b^2)) + 12*(6*a^9*b^2 - 7*a^7*b^4 + 2*a^3*b^8 - a*b^10)*cos(d*x + c) + 6*(2*(a^11 - 4*a^9*b^2 + 6*a^7*b^4 - 4*a^5*b^6 + a^3*b^8)*d*x*cos(d*x + c)^2 - 2*(a^11 - a^9*b^2 - 6*a^7*b^4 + 14*a^5*b^6 - 11*a^3*b^8 + 3*a*b^10)*d*x + 5*(4*a^8*b^3 - 7*a^6*b^5 + 4*a^4*b^7 - a^2*b^9)*cos(d*x + c))*sin(d*x + c))/(3*(a^14*b - 4*a^12*b^3 + 6*a^10*b^5 - 4*a^8*b^7 + a^6*b^9)*d*cos(d*x + c)^2 - (3*a^14*b - 11*a^12*b^3 + 14*a^10*b^5 - 6*a^8*b^7 - a^6*b^9 + a^4*b^11)*d + ((a^15 - 4*a^13*b^2 + 6*a^11*b^4 - 4*a^9*b^6 + a^7*b^8)*d*cos(d*x + c)^2 - (a^15 - a^13*b^2 - 6*a^11*b^4 + 14*a^9*b^6 - 11*a^7*b^8 + 3*a^5*b^10)*d)*sin(d*x + c)), 1/6*(18*(a^10*b - 4*a^8*b^3 + 6*a^6*b^5 - 4*a^4*b^7 + a^2*b^9)*d*x*cos(d*x + c)^2 - (36*a^9*b^2 - 68*a^7*b^4 + 43*a^5*b^6 - 11*a^3*b^8)*cos(d*x + c)^3 - 6*(3*a^10*b - 11*a^8*b^3 + 14*a^6*b^5 - 6*a^4*b^7 - a^2*b^9 + b^11)*d*x - 3*(24*a^8*b^2 - 16*a^6*b^4 + 13*a^4*b^6 + a^2*b^8 - 2*b^10 - 3*(8*a^8*b^2 - 8*a^6*b^4 + 7*a^4*b^6 - 2*a^2*b^8)*cos(d*x + c)^2 + (8*a^9*b + 16*a^7*b^3 - 17*a^5*b^5 + 19*a^3*b^7 - 6*a*b^9 - (8*a^9*b - 8*a^7*b^3 + 7*a^5*b^5 - 2*a^3*b^7)*cos(d*x + c)^2)*sin(d*x + c))*sqrt(-a^2 + b^2)*arctan(-sqrt(-a^2 + b^2)*(b*sin(d*x + c) + a)/((a^2 - b^2)*cos(d*x + c))) + 6*(6*a^9*b^2 - 7*a^7*b^4 + 2*a^3*b^8 - a*b^10)*cos(d*x + c) + 3*(2*(a^11 - 4*a^9*b^2 + 6*a^7*b^4 - 4*a^5*b^6 + a^3*b^8)*d*x*cos(d*x + c)^2 - 2*(a^11 - a^9*b^2 - 6*a^7*b^4 + 14*a^5*b^6 - 11*a^3*b^8 + 3*a*b^10)*d*x + 5*(4*a^8*b^3 - 7*a^6*b^5 + 4*a^4*b^7 - a^2*b^9)*cos(d*x + c))*sin(d*x + c))/(3*(a^14*b - 4*a^12*b^3 + 6*a^10*b^5 - 4*a^8*b^7 + a^6*b^9)*d*cos(d*x + c)^2 - (3*a^14*b - 11*a^12*b^3 + 14*a^10*b^5 - 6*a^8*b^7 - a^6*b^9 + a^4*b^11)*d + ((a^15 - 4*a^13*b^2 + 6*a^11*b^4 - 4*a^9*b^6 + a^7*b^8)*d*cos(d*x + c)^2 - (a^15 - a^13*b^2 - 6*a^11*b^4 + 14*a^9*b^6 - 11*a^7*b^8 + 3*a^5*b^10)*d)*sin(d*x + c))]","B",0
52,1,33,0,0.510043," ","integrate(1/(3+5*csc(d*x+c)),x, algorithm=""fricas"")","\frac{4 \, d x - 5 \, \arctan\left(\frac{5 \, \sin\left(d x + c\right) + 3}{4 \, \cos\left(d x + c\right)}\right)}{12 \, d}"," ",0,"1/12*(4*d*x - 5*arctan(1/4*(5*sin(d*x + c) + 3)/cos(d*x + c)))/d","A",0
53,1,52,0,0.538145," ","integrate(1/(5+3*csc(d*x+c)),x, algorithm=""fricas"")","\frac{8 \, d x + 3 \, \log\left(4 \, \cos\left(d x + c\right) + 3 \, \sin\left(d x + c\right) + 5\right) - 3 \, \log\left(-4 \, \cos\left(d x + c\right) + 3 \, \sin\left(d x + c\right) + 5\right)}{40 \, d}"," ",0,"1/40*(8*d*x + 3*log(4*cos(d*x + c) + 3*sin(d*x + c) + 5) - 3*log(-4*cos(d*x + c) + 3*sin(d*x + c) + 5))/d","A",0
54,0,0,0,0.622974," ","integrate(csc(f*x+e)^3*(a+b*csc(f*x+e))^m,x, algorithm=""fricas"")","{\rm integral}\left({\left(b \csc\left(f x + e\right) + a\right)}^{m} \csc\left(f x + e\right)^{3}, x\right)"," ",0,"integral((b*csc(f*x + e) + a)^m*csc(f*x + e)^3, x)","F",0
55,0,0,0,0.532365," ","integrate(csc(f*x+e)^2*(a+b*csc(f*x+e))^m,x, algorithm=""fricas"")","{\rm integral}\left({\left(b \csc\left(f x + e\right) + a\right)}^{m} \csc\left(f x + e\right)^{2}, x\right)"," ",0,"integral((b*csc(f*x + e) + a)^m*csc(f*x + e)^2, x)","F",0
56,0,0,0,0.507703," ","integrate(csc(f*x+e)*(a+b*csc(f*x+e))^m,x, algorithm=""fricas"")","{\rm integral}\left({\left(b \csc\left(f x + e\right) + a\right)}^{m} \csc\left(f x + e\right), x\right)"," ",0,"integral((b*csc(f*x + e) + a)^m*csc(f*x + e), x)","F",0
57,0,0,0,0.567985," ","integrate((a+b*csc(f*x+e))^m,x, algorithm=""fricas"")","{\rm integral}\left({\left(b \csc\left(f x + e\right) + a\right)}^{m}, x\right)"," ",0,"integral((b*csc(f*x + e) + a)^m, x)","F",0
58,0,0,0,0.504437," ","integrate((a+b*csc(f*x+e))^m*sin(f*x+e),x, algorithm=""fricas"")","{\rm integral}\left({\left(b \csc\left(f x + e\right) + a\right)}^{m} \sin\left(f x + e\right), x\right)"," ",0,"integral((b*csc(f*x + e) + a)^m*sin(f*x + e), x)","F",0
59,0,0,0,0.505194," ","integrate((a+b*csc(f*x+e))^m*sin(f*x+e)^2,x, algorithm=""fricas"")","{\rm integral}\left(-{\left(\cos\left(f x + e\right)^{2} - 1\right)} {\left(b \csc\left(f x + e\right) + a\right)}^{m}, x\right)"," ",0,"integral(-(cos(f*x + e)^2 - 1)*(b*csc(f*x + e) + a)^m, x)","F",0
