1,1,30,0,0.784180," ","integrate(cos(x)^4/(a+a*csc(x)),x, algorithm=""fricas"")","-\frac{8 \, \cos\left(x\right)^{3} - 3 \, {\left(2 \, \cos\left(x\right)^{3} - \cos\left(x\right)\right)} \sin\left(x\right) + 3 \, x}{24 \, a}"," ",0,"-1/24*(8*cos(x)^3 - 3*(2*cos(x)^3 - cos(x))*sin(x) + 3*x)/a","A",0
2,1,22,0,0.641010," ","integrate(cos(x)^3/(a+a*csc(x)),x, algorithm=""fricas"")","-\frac{3 \, \cos\left(x\right)^{2} - 2 \, {\left(\cos\left(x\right)^{2} - 1\right)} \sin\left(x\right)}{6 \, a}"," ",0,"-1/6*(3*cos(x)^2 - 2*(cos(x)^2 - 1)*sin(x))/a","A",0
3,1,18,0,0.870144," ","integrate(cos(x)^2/(a+a*csc(x)),x, algorithm=""fricas"")","\frac{\cos\left(x\right) \sin\left(x\right) - x - 2 \, \cos\left(x\right)}{2 \, a}"," ",0,"1/2*(cos(x)*sin(x) - x - 2*cos(x))/a","A",0
4,1,15,0,0.633230," ","integrate(cos(x)/(a+a*csc(x)),x, algorithm=""fricas"")","-\frac{\log\left(\sin\left(x\right) + 1\right) - \sin\left(x\right)}{a}"," ",0,"-(log(sin(x) + 1) - sin(x))/a","A",0
5,1,35,0,0.538152," ","integrate(sec(x)/(a+a*csc(x)),x, algorithm=""fricas"")","\frac{{\left(\sin\left(x\right) + 1\right)} \log\left(\sin\left(x\right) + 1\right) - {\left(\sin\left(x\right) + 1\right)} \log\left(-\sin\left(x\right) + 1\right) + 2}{4 \, {\left(a \sin\left(x\right) + a\right)}}"," ",0,"1/4*((sin(x) + 1)*log(sin(x) + 1) - (sin(x) + 1)*log(-sin(x) + 1) + 2)/(a*sin(x) + a)","A",0
6,1,25,0,0.407195," ","integrate(sec(x)^2/(a+a*csc(x)),x, algorithm=""fricas"")","-\frac{\cos\left(x\right)^{2} - \sin\left(x\right) - 2}{3 \, {\left(a \cos\left(x\right) \sin\left(x\right) + a \cos\left(x\right)\right)}}"," ",0,"-1/3*(cos(x)^2 - sin(x) - 2)/(a*cos(x)*sin(x) + a*cos(x))","A",0
7,1,70,0,0.600994," ","integrate(sec(x)^3/(a+a*csc(x)),x, algorithm=""fricas"")","-\frac{2 \, \cos\left(x\right)^{2} - {\left(\cos\left(x\right)^{2} \sin\left(x\right) + \cos\left(x\right)^{2}\right)} \log\left(\sin\left(x\right) + 1\right) + {\left(\cos\left(x\right)^{2} \sin\left(x\right) + \cos\left(x\right)^{2}\right)} \log\left(-\sin\left(x\right) + 1\right) - 2 \, \sin\left(x\right) - 6}{16 \, {\left(a \cos\left(x\right)^{2} \sin\left(x\right) + a \cos\left(x\right)^{2}\right)}}"," ",0,"-1/16*(2*cos(x)^2 - (cos(x)^2*sin(x) + cos(x)^2)*log(sin(x) + 1) + (cos(x)^2*sin(x) + cos(x)^2)*log(-sin(x) + 1) - 2*sin(x) - 6)/(a*cos(x)^2*sin(x) + a*cos(x)^2)","A",0
8,1,45,0,0.481589," ","integrate(sec(x)^4/(a+a*csc(x)),x, algorithm=""fricas"")","-\frac{2 \, \cos\left(x\right)^{4} - \cos\left(x\right)^{2} - {\left(2 \, \cos\left(x\right)^{2} + 1\right)} \sin\left(x\right) - 4}{15 \, {\left(a \cos\left(x\right)^{3} \sin\left(x\right) + a \cos\left(x\right)^{3}\right)}}"," ",0,"-1/15*(2*cos(x)^4 - cos(x)^2 - (2*cos(x)^2 + 1)*sin(x) - 4)/(a*cos(x)^3*sin(x) + a*cos(x)^3)","A",0
9,1,335,0,0.513745," ","integrate(cos(x)^4/(a+b*csc(x)),x, algorithm=""fricas"")","\left[-\frac{8 \, a^{3} b \cos\left(x\right)^{3} + 12 \, {\left(a^{2} b - b^{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) - 3 \, {\left(3 \, a^{4} - 12 \, a^{2} b^{2} + 8 \, b^{4}\right)} x + 24 \, {\left(a^{3} b - a b^{3}\right)} \cos\left(x\right) - 3 \, {\left(2 \, a^{4} \cos\left(x\right)^{3} + {\left(3 \, a^{4} - 4 \, a^{2} b^{2}\right)} \cos\left(x\right)\right)} \sin\left(x\right)}{24 \, a^{5}}, -\frac{8 \, a^{3} b \cos\left(x\right)^{3} - 24 \, {\left(a^{2} b - b^{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) - 3 \, {\left(3 \, a^{4} - 12 \, a^{2} b^{2} + 8 \, b^{4}\right)} x + 24 \, {\left(a^{3} b - a b^{3}\right)} \cos\left(x\right) - 3 \, {\left(2 \, a^{4} \cos\left(x\right)^{3} + {\left(3 \, a^{4} - 4 \, a^{2} b^{2}\right)} \cos\left(x\right)\right)} \sin\left(x\right)}{24 \, a^{5}}\right]"," ",0,"[-1/24*(8*a^3*b*cos(x)^3 + 12*(a^2*b - b^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)) - 3*(3*a^4 - 12*a^2*b^2 + 8*b^4)*x + 24*(a^3*b - a*b^3)*cos(x) - 3*(2*a^4*cos(x)^3 + (3*a^4 - 4*a^2*b^2)*cos(x))*sin(x))/a^5, -1/24*(8*a^3*b*cos(x)^3 - 24*(a^2*b - b^3)*sqrt(-a^2 + b^2)*arctan(-sqrt(-a^2 + b^2)*(b*sin(x) + a)/((a^2 - b^2)*cos(x))) - 3*(3*a^4 - 12*a^2*b^2 + 8*b^4)*x + 24*(a^3*b - a*b^3)*cos(x) - 3*(2*a^4*cos(x)^3 + (3*a^4 - 4*a^2*b^2)*cos(x))*sin(x))/a^5]","A",0
10,1,60,0,0.530622," ","integrate(cos(x)^3/(a+b*csc(x)),x, algorithm=""fricas"")","-\frac{3 \, a^{2} b \cos\left(x\right)^{2} + 6 \, {\left(a^{2} b - b^{3}\right)} \log\left(a \sin\left(x\right) + b\right) - 2 \, {\left(a^{3} \cos\left(x\right)^{2} + 2 \, a^{3} - 3 \, a b^{2}\right)} \sin\left(x\right)}{6 \, a^{4}}"," ",0,"-1/6*(3*a^2*b*cos(x)^2 + 6*(a^2*b - b^3)*log(a*sin(x) + b) - 2*(a^3*cos(x)^2 + 2*a^3 - 3*a*b^2)*sin(x))/a^4","A",0
11,1,205,0,0.534311," ","integrate(cos(x)^2/(a+b*csc(x)),x, algorithm=""fricas"")","\left[\frac{a^{2} \cos\left(x\right) \sin\left(x\right) - 2 \, a b \cos\left(x\right) + \sqrt{a^{2} - b^{2}} b \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} - 2 \, b^{2}\right)} x}{2 \, a^{3}}, \frac{a^{2} \cos\left(x\right) \sin\left(x\right) - 2 \, a b \cos\left(x\right) + 2 \, \sqrt{-a^{2} + b^{2}} b \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} - 2 \, b^{2}\right)} x}{2 \, a^{3}}\right]"," ",0,"[1/2*(a^2*cos(x)*sin(x) - 2*a*b*cos(x) + sqrt(a^2 - b^2)*b*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 - 2*b^2)*x)/a^3, 1/2*(a^2*cos(x)*sin(x) - 2*a*b*cos(x) + 2*sqrt(-a^2 + b^2)*b*arctan(-sqrt(-a^2 + b^2)*(b*sin(x) + a)/((a^2 - b^2)*cos(x))) + (a^2 - 2*b^2)*x)/a^3]","A",0
12,1,20,0,0.536465," ","integrate(cos(x)/(a+b*csc(x)),x, algorithm=""fricas"")","-\frac{b \log\left(a \sin\left(x\right) + b\right) - a \sin\left(x\right)}{a^{2}}"," ",0,"-(b*log(a*sin(x) + b) - a*sin(x))/a^2","A",0
13,1,47,0,1.693690," ","integrate(sec(x)/(a+b*csc(x)),x, algorithm=""fricas"")","-\frac{2 \, b \log\left(a \sin\left(x\right) + b\right) - {\left(a + b\right)} \log\left(\sin\left(x\right) + 1\right) + {\left(a - b\right)} \log\left(-\sin\left(x\right) + 1\right)}{2 \, {\left(a^{2} - b^{2}\right)}}"," ",0,"-1/2*(2*b*log(a*sin(x) + b) - (a + b)*log(sin(x) + 1) + (a - b)*log(-sin(x) + 1))/(a^2 - b^2)","A",0
14,1,243,0,0.674913," ","integrate(sec(x)^2/(a+b*csc(x)),x, algorithm=""fricas"")","\left[-\frac{\sqrt{a^{2} - b^{2}} a b \cos\left(x\right) \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 \, a^{2} b - 2 \, b^{3} - 2 \, {\left(a^{3} - a b^{2}\right)} \sin\left(x\right)}{2 \, {\left(a^{4} - 2 \, a^{2} b^{2} + b^{4}\right)} \cos\left(x\right)}, \frac{\sqrt{-a^{2} + b^{2}} a b \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) \cos\left(x\right) - a^{2} b + b^{3} + {\left(a^{3} - a b^{2}\right)} \sin\left(x\right)}{{\left(a^{4} - 2 \, a^{2} b^{2} + b^{4}\right)} \cos\left(x\right)}\right]"," ",0,"[-1/2*(sqrt(a^2 - b^2)*a*b*cos(x)*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^3 - 2*(a^3 - a*b^2)*sin(x))/((a^4 - 2*a^2*b^2 + b^4)*cos(x)), (sqrt(-a^2 + b^2)*a*b*arctan(-sqrt(-a^2 + b^2)*(b*sin(x) + a)/((a^2 - b^2)*cos(x)))*cos(x) - a^2*b + b^3 + (a^3 - a*b^2)*sin(x))/((a^4 - 2*a^2*b^2 + b^4)*cos(x))]","A",0
15,1,119,0,0.731295," ","integrate(sec(x)^3/(a+b*csc(x)),x, algorithm=""fricas"")","-\frac{4 \, a^{2} b \cos\left(x\right)^{2} \log\left(a \sin\left(x\right) + b\right) - {\left(a^{3} + 2 \, a^{2} b + a b^{2}\right)} \cos\left(x\right)^{2} \log\left(\sin\left(x\right) + 1\right) + {\left(a^{3} - 2 \, a^{2} b + a b^{2}\right)} \cos\left(x\right)^{2} \log\left(-\sin\left(x\right) + 1\right) + 2 \, a^{2} b - 2 \, b^{3} - 2 \, {\left(a^{3} - a b^{2}\right)} \sin\left(x\right)}{4 \, {\left(a^{4} - 2 \, a^{2} b^{2} + b^{4}\right)} \cos\left(x\right)^{2}}"," ",0,"-1/4*(4*a^2*b*cos(x)^2*log(a*sin(x) + b) - (a^3 + 2*a^2*b + a*b^2)*cos(x)^2*log(sin(x) + 1) + (a^3 - 2*a^2*b + a*b^2)*cos(x)^2*log(-sin(x) + 1) + 2*a^2*b - 2*b^3 - 2*(a^3 - a*b^2)*sin(x))/((a^4 - 2*a^2*b^2 + b^4)*cos(x)^2)","A",0
16,1,395,0,0.558908," ","integrate(sec(x)^4/(a+b*csc(x)),x, algorithm=""fricas"")","\left[\frac{3 \, \sqrt{a^{2} - b^{2}} a^{3} b \cos\left(x\right)^{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) - 2 \, a^{4} b + 4 \, a^{2} b^{3} - 2 \, b^{5} - 6 \, {\left(a^{4} b - a^{2} b^{3}\right)} \cos\left(x\right)^{2} + 2 \, {\left(a^{5} - 2 \, 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)} \sin\left(x\right)}{6 \, {\left(a^{6} - 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} - b^{6}\right)} \cos\left(x\right)^{3}}, \frac{3 \, \sqrt{-a^{2} + b^{2}} a^{3} b \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) \cos\left(x\right)^{3} - a^{4} b + 2 \, a^{2} b^{3} - b^{5} - 3 \, {\left(a^{4} b - a^{2} b^{3}\right)} \cos\left(x\right)^{2} + {\left(a^{5} - 2 \, 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)} \sin\left(x\right)}{3 \, {\left(a^{6} - 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} - b^{6}\right)} \cos\left(x\right)^{3}}\right]"," ",0,"[1/6*(3*sqrt(a^2 - b^2)*a^3*b*cos(x)^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)) - 2*a^4*b + 4*a^2*b^3 - 2*b^5 - 6*(a^4*b - a^2*b^3)*cos(x)^2 + 2*(a^5 - 2*a^3*b^2 + a*b^4 + (2*a^5 - a^3*b^2 - a*b^4)*cos(x)^2)*sin(x))/((a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*cos(x)^3), 1/3*(3*sqrt(-a^2 + b^2)*a^3*b*arctan(-sqrt(-a^2 + b^2)*(b*sin(x) + a)/((a^2 - b^2)*cos(x)))*cos(x)^3 - a^4*b + 2*a^2*b^3 - b^5 - 3*(a^4*b - a^2*b^3)*cos(x)^2 + (a^5 - 2*a^3*b^2 + a*b^4 + (2*a^5 - a^3*b^2 - a*b^4)*cos(x)^2)*sin(x))/((a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*cos(x)^3)]","A",0
