1,1,78,0,0.291801," ","integrate(cos(x)^4/(a+a*csc(x)),x, algorithm=""giac"")","-\frac{x}{8 \, a} - \frac{3 \, \tan\left(\frac{1}{2} \, x\right)^{7} + 24 \, \tan\left(\frac{1}{2} \, x\right)^{6} - 21 \, \tan\left(\frac{1}{2} \, x\right)^{5} + 24 \, \tan\left(\frac{1}{2} \, x\right)^{4} + 21 \, \tan\left(\frac{1}{2} \, x\right)^{3} + 8 \, \tan\left(\frac{1}{2} \, x\right)^{2} - 3 \, \tan\left(\frac{1}{2} \, x\right) + 8}{12 \, {\left(\tan\left(\frac{1}{2} \, x\right)^{2} + 1\right)}^{4} a}"," ",0,"-1/8*x/a - 1/12*(3*tan(1/2*x)^7 + 24*tan(1/2*x)^6 - 21*tan(1/2*x)^5 + 24*tan(1/2*x)^4 + 21*tan(1/2*x)^3 + 8*tan(1/2*x)^2 - 3*tan(1/2*x) + 8)/((tan(1/2*x)^2 + 1)^4*a)","B",0
2,1,18,0,0.427095," ","integrate(cos(x)^3/(a+a*csc(x)),x, algorithm=""giac"")","-\frac{2 \, \sin\left(x\right)^{3} - 3 \, \sin\left(x\right)^{2}}{6 \, a}"," ",0,"-1/6*(2*sin(x)^3 - 3*sin(x)^2)/a","A",0
3,1,44,0,0.577450," ","integrate(cos(x)^2/(a+a*csc(x)),x, algorithm=""giac"")","-\frac{x}{2 \, a} - \frac{\tan\left(\frac{1}{2} \, x\right)^{3} + 2 \, \tan\left(\frac{1}{2} \, x\right)^{2} - \tan\left(\frac{1}{2} \, x\right) + 2}{{\left(\tan\left(\frac{1}{2} \, x\right)^{2} + 1\right)}^{2} a}"," ",0,"-1/2*x/a - (tan(1/2*x)^3 + 2*tan(1/2*x)^2 - tan(1/2*x) + 2)/((tan(1/2*x)^2 + 1)^2*a)","A",0
4,1,17,0,0.910511," ","integrate(cos(x)/(a+a*csc(x)),x, algorithm=""giac"")","-\frac{\log\left(\sin\left(x\right) + 1\right)}{a} + \frac{\sin\left(x\right)}{a}"," ",0,"-log(sin(x) + 1)/a + sin(x)/a","A",0
5,1,34,0,0.753250," ","integrate(sec(x)/(a+a*csc(x)),x, algorithm=""giac"")","\frac{\log\left(\sin\left(x\right) + 1\right)}{4 \, a} - \frac{\log\left(-\sin\left(x\right) + 1\right)}{4 \, a} + \frac{1}{2 \, a {\left(\sin\left(x\right) + 1\right)}}"," ",0,"1/4*log(sin(x) + 1)/a - 1/4*log(-sin(x) + 1)/a + 1/2/(a*(sin(x) + 1))","A",0
6,1,37,0,0.552634," ","integrate(sec(x)^2/(a+a*csc(x)),x, algorithm=""giac"")","-\frac{1}{2 \, a {\left(\tan\left(\frac{1}{2} \, x\right) - 1\right)}} + \frac{3 \, \tan\left(\frac{1}{2} \, x\right)^{2} + 1}{6 \, a {\left(\tan\left(\frac{1}{2} \, x\right) + 1\right)}^{3}}"," ",0,"-1/2/(a*(tan(1/2*x) - 1)) + 1/6*(3*tan(1/2*x)^2 + 1)/(a*(tan(1/2*x) + 1)^3)","A",0
7,1,48,0,0.571771," ","integrate(sec(x)^3/(a+a*csc(x)),x, algorithm=""giac"")","\frac{\log\left(\sin\left(x\right) + 1\right)}{16 \, a} - \frac{\log\left(-\sin\left(x\right) + 1\right)}{16 \, a} - \frac{\sin\left(x\right)^{2} + \sin\left(x\right) + 2}{8 \, a {\left(\sin\left(x\right) + 1\right)}^{2} {\left(\sin\left(x\right) - 1\right)}}"," ",0,"1/16*log(sin(x) + 1)/a - 1/16*log(-sin(x) + 1)/a - 1/8*(sin(x)^2 + sin(x) + 2)/(a*(sin(x) + 1)^2*(sin(x) - 1))","A",0
8,1,75,0,0.362912," ","integrate(sec(x)^4/(a+a*csc(x)),x, algorithm=""giac"")","-\frac{9 \, \tan\left(\frac{1}{2} \, x\right)^{2} - 12 \, \tan\left(\frac{1}{2} \, x\right) + 7}{24 \, a {\left(\tan\left(\frac{1}{2} \, x\right) - 1\right)}^{3}} + \frac{45 \, \tan\left(\frac{1}{2} \, x\right)^{4} + 60 \, \tan\left(\frac{1}{2} \, x\right)^{3} + 70 \, \tan\left(\frac{1}{2} \, x\right)^{2} + 20 \, \tan\left(\frac{1}{2} \, x\right) + 13}{120 \, a {\left(\tan\left(\frac{1}{2} \, x\right) + 1\right)}^{5}}"," ",0,"-1/24*(9*tan(1/2*x)^2 - 12*tan(1/2*x) + 7)/(a*(tan(1/2*x) - 1)^3) + 1/120*(45*tan(1/2*x)^4 + 60*tan(1/2*x)^3 + 70*tan(1/2*x)^2 + 20*tan(1/2*x) + 13)/(a*(tan(1/2*x) + 1)^5)","B",0
9,1,278,0,0.635861," ","integrate(cos(x)^4/(a+b*csc(x)),x, algorithm=""giac"")","\frac{{\left(3 \, a^{4} - 12 \, a^{2} b^{2} + 8 \, b^{4}\right)} x}{8 \, a^{5}} - \frac{2 \, {\left(a^{4} b - 2 \, a^{2} b^{3} + b^{5}\right)} {\left(\pi \left \lfloor \frac{x}{2 \, \pi} + \frac{1}{2} \right \rfloor \mathrm{sgn}\left(b\right) + \arctan\left(\frac{b \tan\left(\frac{1}{2} \, x\right) + a}{\sqrt{-a^{2} + b^{2}}}\right)\right)}}{\sqrt{-a^{2} + b^{2}} a^{5}} - \frac{15 \, a^{3} \tan\left(\frac{1}{2} \, x\right)^{7} - 12 \, a b^{2} \tan\left(\frac{1}{2} \, x\right)^{7} + 48 \, a^{2} b \tan\left(\frac{1}{2} \, x\right)^{6} - 24 \, b^{3} \tan\left(\frac{1}{2} \, x\right)^{6} - 9 \, a^{3} \tan\left(\frac{1}{2} \, x\right)^{5} - 12 \, a b^{2} \tan\left(\frac{1}{2} \, x\right)^{5} + 96 \, a^{2} b \tan\left(\frac{1}{2} \, x\right)^{4} - 72 \, b^{3} \tan\left(\frac{1}{2} \, x\right)^{4} + 9 \, a^{3} \tan\left(\frac{1}{2} \, x\right)^{3} + 12 \, a b^{2} \tan\left(\frac{1}{2} \, x\right)^{3} + 80 \, a^{2} b \tan\left(\frac{1}{2} \, x\right)^{2} - 72 \, b^{3} \tan\left(\frac{1}{2} \, x\right)^{2} - 15 \, a^{3} \tan\left(\frac{1}{2} \, x\right) + 12 \, a b^{2} \tan\left(\frac{1}{2} \, x\right) + 32 \, a^{2} b - 24 \, b^{3}}{12 \, {\left(\tan\left(\frac{1}{2} \, x\right)^{2} + 1\right)}^{4} a^{4}}"," ",0,"1/8*(3*a^4 - 12*a^2*b^2 + 8*b^4)*x/a^5 - 2*(a^4*b - 2*a^2*b^3 + b^5)*(pi*floor(1/2*x/pi + 1/2)*sgn(b) + arctan((b*tan(1/2*x) + a)/sqrt(-a^2 + b^2)))/(sqrt(-a^2 + b^2)*a^5) - 1/12*(15*a^3*tan(1/2*x)^7 - 12*a*b^2*tan(1/2*x)^7 + 48*a^2*b*tan(1/2*x)^6 - 24*b^3*tan(1/2*x)^6 - 9*a^3*tan(1/2*x)^5 - 12*a*b^2*tan(1/2*x)^5 + 96*a^2*b*tan(1/2*x)^4 - 72*b^3*tan(1/2*x)^4 + 9*a^3*tan(1/2*x)^3 + 12*a*b^2*tan(1/2*x)^3 + 80*a^2*b*tan(1/2*x)^2 - 72*b^3*tan(1/2*x)^2 - 15*a^3*tan(1/2*x) + 12*a*b^2*tan(1/2*x) + 32*a^2*b - 24*b^3)/((tan(1/2*x)^2 + 1)^4*a^4)","B",0
10,1,62,0,0.533124," ","integrate(cos(x)^3/(a+b*csc(x)),x, algorithm=""giac"")","-\frac{2 \, a^{2} \sin\left(x\right)^{3} - 3 \, a b \sin\left(x\right)^{2} - 6 \, a^{2} \sin\left(x\right) + 6 \, b^{2} \sin\left(x\right)}{6 \, a^{3}} - \frac{{\left(a^{2} b - b^{3}\right)} \log\left({\left| a \sin\left(x\right) + b \right|}\right)}{a^{4}}"," ",0,"-1/6*(2*a^2*sin(x)^3 - 3*a*b*sin(x)^2 - 6*a^2*sin(x) + 6*b^2*sin(x))/a^3 - (a^2*b - b^3)*log(abs(a*sin(x) + b))/a^4","A",0
11,1,121,0,0.358338," ","integrate(cos(x)^2/(a+b*csc(x)),x, algorithm=""giac"")","\frac{{\left(a^{2} - 2 \, b^{2}\right)} x}{2 \, a^{3}} - \frac{2 \, {\left(a^{2} b - b^{3}\right)} {\left(\pi \left \lfloor \frac{x}{2 \, \pi} + \frac{1}{2} \right \rfloor \mathrm{sgn}\left(b\right) + \arctan\left(\frac{b \tan\left(\frac{1}{2} \, x\right) + a}{\sqrt{-a^{2} + b^{2}}}\right)\right)}}{\sqrt{-a^{2} + b^{2}} a^{3}} - \frac{a \tan\left(\frac{1}{2} \, x\right)^{3} + 2 \, b \tan\left(\frac{1}{2} \, x\right)^{2} - a \tan\left(\frac{1}{2} \, x\right) + 2 \, b}{{\left(\tan\left(\frac{1}{2} \, x\right)^{2} + 1\right)}^{2} a^{2}}"," ",0,"1/2*(a^2 - 2*b^2)*x/a^3 - 2*(a^2*b - b^3)*(pi*floor(1/2*x/pi + 1/2)*sgn(b) + arctan((b*tan(1/2*x) + a)/sqrt(-a^2 + b^2)))/(sqrt(-a^2 + b^2)*a^3) - (a*tan(1/2*x)^3 + 2*b*tan(1/2*x)^2 - a*tan(1/2*x) + 2*b)/((tan(1/2*x)^2 + 1)^2*a^2)","A",0
12,1,21,0,0.270481," ","integrate(cos(x)/(a+b*csc(x)),x, algorithm=""giac"")","-\frac{b \log\left({\left| a \sin\left(x\right) + b \right|}\right)}{a^{2}} + \frac{\sin\left(x\right)}{a}"," ",0,"-b*log(abs(a*sin(x) + b))/a^2 + sin(x)/a","A",0
13,1,53,0,0.430592," ","integrate(sec(x)/(a+b*csc(x)),x, algorithm=""giac"")","-\frac{a b \log\left({\left| a \sin\left(x\right) + b \right|}\right)}{a^{3} - a b^{2}} + \frac{\log\left(\sin\left(x\right) + 1\right)}{2 \, {\left(a - b\right)}} - \frac{\log\left(-\sin\left(x\right) + 1\right)}{2 \, {\left(a + b\right)}}"," ",0,"-a*b*log(abs(a*sin(x) + b))/(a^3 - a*b^2) + 1/2*log(sin(x) + 1)/(a - b) - 1/2*log(-sin(x) + 1)/(a + b)","A",0
14,1,95,0,0.770290," ","integrate(sec(x)^2/(a+b*csc(x)),x, algorithm=""giac"")","-\frac{2 \, {\left(\pi \left \lfloor \frac{x}{2 \, \pi} + \frac{1}{2} \right \rfloor \mathrm{sgn}\left(b\right) + \arctan\left(\frac{b \tan\left(\frac{1}{2} \, x\right) + a}{\sqrt{-a^{2} + b^{2}}}\right)\right)} a b}{{\left(a^{2} - b^{2}\right)} \sqrt{-a^{2} + b^{2}}} - \frac{2 \, {\left(a \tan\left(\frac{1}{2} \, x\right) - b\right)}}{{\left(a^{2} - b^{2}\right)} {\left(\tan\left(\frac{1}{2} \, x\right)^{2} - 1\right)}}"," ",0,"-2*(pi*floor(1/2*x/pi + 1/2)*sgn(b) + arctan((b*tan(1/2*x) + a)/sqrt(-a^2 + b^2)))*a*b/((a^2 - b^2)*sqrt(-a^2 + b^2)) - 2*(a*tan(1/2*x) - b)/((a^2 - b^2)*(tan(1/2*x)^2 - 1))","A",0
15,1,129,0,1.493431," ","integrate(sec(x)^3/(a+b*csc(x)),x, algorithm=""giac"")","-\frac{a^{3} b \log\left({\left| a \sin\left(x\right) + b \right|}\right)}{a^{5} - 2 \, a^{3} b^{2} + a b^{4}} + \frac{a \log\left(\sin\left(x\right) + 1\right)}{4 \, {\left(a^{2} - 2 \, a b + b^{2}\right)}} - \frac{a \log\left(-\sin\left(x\right) + 1\right)}{4 \, {\left(a^{2} + 2 \, a b + b^{2}\right)}} + \frac{a^{2} b - b^{3} - {\left(a^{3} - a b^{2}\right)} \sin\left(x\right)}{2 \, {\left(a + b\right)}^{2} {\left(a - b\right)}^{2} {\left(\sin\left(x\right) + 1\right)} {\left(\sin\left(x\right) - 1\right)}}"," ",0,"-a^3*b*log(abs(a*sin(x) + b))/(a^5 - 2*a^3*b^2 + a*b^4) + 1/4*a*log(sin(x) + 1)/(a^2 - 2*a*b + b^2) - 1/4*a*log(-sin(x) + 1)/(a^2 + 2*a*b + b^2) + 1/2*(a^2*b - b^3 - (a^3 - a*b^2)*sin(x))/((a + b)^2*(a - b)^2*(sin(x) + 1)*(sin(x) - 1))","A",0
16,1,187,0,0.732937," ","integrate(sec(x)^4/(a+b*csc(x)),x, algorithm=""giac"")","-\frac{2 \, {\left(\pi \left \lfloor \frac{x}{2 \, \pi} + \frac{1}{2} \right \rfloor \mathrm{sgn}\left(b\right) + \arctan\left(\frac{b \tan\left(\frac{1}{2} \, x\right) + a}{\sqrt{-a^{2} + b^{2}}}\right)\right)} a^{3} b}{{\left(a^{4} - 2 \, a^{2} b^{2} + b^{4}\right)} \sqrt{-a^{2} + b^{2}}} - \frac{2 \, {\left(3 \, a^{3} \tan\left(\frac{1}{2} \, x\right)^{5} - 6 \, a^{2} b \tan\left(\frac{1}{2} \, x\right)^{4} + 3 \, b^{3} \tan\left(\frac{1}{2} \, x\right)^{4} - 2 \, a^{3} \tan\left(\frac{1}{2} \, x\right)^{3} - 4 \, a b^{2} \tan\left(\frac{1}{2} \, x\right)^{3} + 6 \, a^{2} b \tan\left(\frac{1}{2} \, x\right)^{2} + 3 \, a^{3} \tan\left(\frac{1}{2} \, x\right) - 4 \, a^{2} b + b^{3}\right)}}{3 \, {\left(a^{4} - 2 \, a^{2} b^{2} + b^{4}\right)} {\left(\tan\left(\frac{1}{2} \, x\right)^{2} - 1\right)}^{3}}"," ",0,"-2*(pi*floor(1/2*x/pi + 1/2)*sgn(b) + arctan((b*tan(1/2*x) + a)/sqrt(-a^2 + b^2)))*a^3*b/((a^4 - 2*a^2*b^2 + b^4)*sqrt(-a^2 + b^2)) - 2/3*(3*a^3*tan(1/2*x)^5 - 6*a^2*b*tan(1/2*x)^4 + 3*b^3*tan(1/2*x)^4 - 2*a^3*tan(1/2*x)^3 - 4*a*b^2*tan(1/2*x)^3 + 6*a^2*b*tan(1/2*x)^2 + 3*a^3*tan(1/2*x) - 4*a^2*b + b^3)/((a^4 - 2*a^2*b^2 + b^4)*(tan(1/2*x)^2 - 1)^3)","A",0
