1,1,96,0,0.333952," ","integrate(csc(x)^5/(a+a*csc(x)),x, algorithm=""giac"")","-\frac{3 \, \log\left({\left| \tan\left(\frac{1}{2} \, x\right) \right|}\right)}{2 \, a} + \frac{a^{2} \tan\left(\frac{1}{2} \, x\right)^{3} - 3 \, a^{2} \tan\left(\frac{1}{2} \, x\right)^{2} + 21 \, a^{2} \tan\left(\frac{1}{2} \, x\right)}{24 \, a^{3}} - \frac{2}{a {\left(\tan\left(\frac{1}{2} \, x\right) + 1\right)}} + \frac{66 \, \tan\left(\frac{1}{2} \, x\right)^{3} - 21 \, \tan\left(\frac{1}{2} \, x\right)^{2} + 3 \, \tan\left(\frac{1}{2} \, x\right) - 1}{24 \, a \tan\left(\frac{1}{2} \, x\right)^{3}}"," ",0,"-3/2*log(abs(tan(1/2*x)))/a + 1/24*(a^2*tan(1/2*x)^3 - 3*a^2*tan(1/2*x)^2 + 21*a^2*tan(1/2*x))/a^3 - 2/(a*(tan(1/2*x) + 1)) + 1/24*(66*tan(1/2*x)^3 - 21*tan(1/2*x)^2 + 3*tan(1/2*x) - 1)/(a*tan(1/2*x)^3)","A",0
2,1,73,0,0.365100," ","integrate(csc(x)^4/(a+a*csc(x)),x, algorithm=""giac"")","\frac{3 \, \log\left({\left| \tan\left(\frac{1}{2} \, x\right) \right|}\right)}{2 \, a} + \frac{a \tan\left(\frac{1}{2} \, x\right)^{2} - 4 \, a \tan\left(\frac{1}{2} \, x\right)}{8 \, a^{2}} + \frac{2}{a {\left(\tan\left(\frac{1}{2} \, x\right) + 1\right)}} - \frac{18 \, \tan\left(\frac{1}{2} \, x\right)^{2} - 4 \, \tan\left(\frac{1}{2} \, x\right) + 1}{8 \, a \tan\left(\frac{1}{2} \, x\right)^{2}}"," ",0,"3/2*log(abs(tan(1/2*x)))/a + 1/8*(a*tan(1/2*x)^2 - 4*a*tan(1/2*x))/a^2 + 2/(a*(tan(1/2*x) + 1)) - 1/8*(18*tan(1/2*x)^2 - 4*tan(1/2*x) + 1)/(a*tan(1/2*x)^2)","A",0
3,1,53,0,0.560071," ","integrate(csc(x)^3/(a+a*csc(x)),x, algorithm=""giac"")","-\frac{\log\left({\left| \tan\left(\frac{1}{2} \, x\right) \right|}\right)}{a} + \frac{\tan\left(\frac{1}{2} \, x\right)}{2 \, a} + \frac{\tan\left(\frac{1}{2} \, x\right)^{2} - 4 \, \tan\left(\frac{1}{2} \, x\right) - 1}{2 \, {\left(\tan\left(\frac{1}{2} \, x\right)^{2} + \tan\left(\frac{1}{2} \, x\right)\right)} a}"," ",0,"-log(abs(tan(1/2*x)))/a + 1/2*tan(1/2*x)/a + 1/2*(tan(1/2*x)^2 - 4*tan(1/2*x) - 1)/((tan(1/2*x)^2 + tan(1/2*x))*a)","A",0
4,1,24,0,0.596816," ","integrate(csc(x)^2/(a+a*csc(x)),x, algorithm=""giac"")","\frac{\log\left({\left| \tan\left(\frac{1}{2} \, x\right) \right|}\right)}{a} + \frac{2}{a {\left(\tan\left(\frac{1}{2} \, x\right) + 1\right)}}"," ",0,"log(abs(tan(1/2*x)))/a + 2/(a*(tan(1/2*x) + 1))","A",0
5,1,13,0,0.445417," ","integrate(csc(x)/(a+a*csc(x)),x, algorithm=""giac"")","-\frac{2}{a {\left(\tan\left(\frac{1}{2} \, x\right) + 1\right)}}"," ",0,"-2/(a*(tan(1/2*x) + 1))","A",0
6,1,32,0,0.422430," ","integrate(1/(a+a*csc(d*x+c)),x, algorithm=""giac"")","\frac{\frac{d x + c}{a} + \frac{2}{a {\left(\tan\left(\frac{1}{2} \, d x + \frac{1}{2} \, c\right) + 1\right)}}}{d}"," ",0,"((d*x + c)/a + 2/(a*(tan(1/2*d*x + 1/2*c) + 1)))/d","A",0
7,1,44,0,0.534056," ","integrate(sin(x)/(a+a*csc(x)),x, algorithm=""giac"")","-\frac{x}{a} - \frac{2 \, {\left(\tan\left(\frac{1}{2} \, x\right)^{2} + \tan\left(\frac{1}{2} \, x\right) + 2\right)}}{{\left(\tan\left(\frac{1}{2} \, x\right)^{3} + \tan\left(\frac{1}{2} \, x\right)^{2} + \tan\left(\frac{1}{2} \, x\right) + 1\right)} a}"," ",0,"-x/a - 2*(tan(1/2*x)^2 + tan(1/2*x) + 2)/((tan(1/2*x)^3 + tan(1/2*x)^2 + tan(1/2*x) + 1)*a)","A",0
8,1,56,0,0.685243," ","integrate(sin(x)^2/(a+a*csc(x)),x, algorithm=""giac"")","\frac{3 \, 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} + \frac{2}{a {\left(\tan\left(\frac{1}{2} \, x\right) + 1\right)}}"," ",0,"3/2*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) + 2/(a*(tan(1/2*x) + 1))","A",0
9,1,67,0,0.288909," ","integrate(sin(x)^3/(a+a*csc(x)),x, algorithm=""giac"")","-\frac{3 \, x}{2 \, a} - \frac{2}{a {\left(\tan\left(\frac{1}{2} \, x\right) + 1\right)}} - \frac{3 \, \tan\left(\frac{1}{2} \, x\right)^{5} + 6 \, \tan\left(\frac{1}{2} \, x\right)^{4} + 24 \, \tan\left(\frac{1}{2} \, x\right)^{2} - 3 \, \tan\left(\frac{1}{2} \, x\right) + 10}{3 \, {\left(\tan\left(\frac{1}{2} \, x\right)^{2} + 1\right)}^{3} a}"," ",0,"-3/2*x/a - 2/(a*(tan(1/2*x) + 1)) - 1/3*(3*tan(1/2*x)^5 + 6*tan(1/2*x)^4 + 24*tan(1/2*x)^2 - 3*tan(1/2*x) + 10)/((tan(1/2*x)^2 + 1)^3*a)","A",0
10,1,91,0,0.343829," ","integrate(sin(x)^4/(a+a*csc(x)),x, algorithm=""giac"")","\frac{15 \, x}{8 \, a} + \frac{2}{a {\left(\tan\left(\frac{1}{2} \, x\right) + 1\right)}} + \frac{21 \, \tan\left(\frac{1}{2} \, x\right)^{7} + 24 \, \tan\left(\frac{1}{2} \, x\right)^{6} + 45 \, \tan\left(\frac{1}{2} \, x\right)^{5} + 120 \, \tan\left(\frac{1}{2} \, x\right)^{4} - 45 \, \tan\left(\frac{1}{2} \, x\right)^{3} + 136 \, \tan\left(\frac{1}{2} \, x\right)^{2} - 21 \, \tan\left(\frac{1}{2} \, x\right) + 40}{12 \, {\left(\tan\left(\frac{1}{2} \, x\right)^{2} + 1\right)}^{4} a}"," ",0,"15/8*x/a + 2/(a*(tan(1/2*x) + 1)) + 1/12*(21*tan(1/2*x)^7 + 24*tan(1/2*x)^6 + 45*tan(1/2*x)^5 + 120*tan(1/2*x)^4 - 45*tan(1/2*x)^3 + 136*tan(1/2*x)^2 - 21*tan(1/2*x) + 40)/((tan(1/2*x)^2 + 1)^4*a)","A",0
11,1,60,0,0.735532," ","integrate(1/(a+a*csc(d*x+c))^2,x, algorithm=""giac"")","\frac{\frac{3 \, {\left(d x + c\right)}}{a^{2}} + \frac{2 \, {\left(3 \, \tan\left(\frac{1}{2} \, d x + \frac{1}{2} \, c\right)^{2} + 9 \, \tan\left(\frac{1}{2} \, d x + \frac{1}{2} \, c\right) + 4\right)}}{a^{2} {\left(\tan\left(\frac{1}{2} \, d x + \frac{1}{2} \, c\right) + 1\right)}^{3}}}{3 \, d}"," ",0,"1/3*(3*(d*x + c)/a^2 + 2*(3*tan(1/2*d*x + 1/2*c)^2 + 9*tan(1/2*d*x + 1/2*c) + 4)/(a^2*(tan(1/2*d*x + 1/2*c) + 1)^3))/d","A",0
12,1,86,0,0.448794," ","integrate(1/(a+a*csc(d*x+c))^3,x, algorithm=""giac"")","\frac{\frac{15 \, {\left(d x + c\right)}}{a^{3}} + \frac{2 \, {\left(15 \, \tan\left(\frac{1}{2} \, d x + \frac{1}{2} \, c\right)^{4} + 75 \, \tan\left(\frac{1}{2} \, d x + \frac{1}{2} \, c\right)^{3} + 145 \, \tan\left(\frac{1}{2} \, d x + \frac{1}{2} \, c\right)^{2} + 95 \, \tan\left(\frac{1}{2} \, d x + \frac{1}{2} \, c\right) + 22\right)}}{a^{3} {\left(\tan\left(\frac{1}{2} \, d x + \frac{1}{2} \, c\right) + 1\right)}^{5}}}{15 \, d}"," ",0,"1/15*(15*(d*x + c)/a^3 + 2*(15*tan(1/2*d*x + 1/2*c)^4 + 75*tan(1/2*d*x + 1/2*c)^3 + 145*tan(1/2*d*x + 1/2*c)^2 + 95*tan(1/2*d*x + 1/2*c) + 22)/(a^3*(tan(1/2*d*x + 1/2*c) + 1)^5))/d","A",0
13,1,250,0,1.010117," ","integrate((a+a*csc(x))^(5/2),x, algorithm=""giac"")","{\left(a^{2} \sqrt{{\left| a \right|}} + a {\left| a \right|}^{\frac{3}{2}}\right)} \arctan\left(\frac{\sqrt{2} {\left(\sqrt{2} \sqrt{{\left| a \right|}} + 2 \, \sqrt{a \tan\left(\frac{1}{2} \, x\right)}\right)}}{2 \, \sqrt{{\left| a \right|}}}\right) + {\left(a^{2} \sqrt{{\left| a \right|}} + a {\left| a \right|}^{\frac{3}{2}}\right)} \arctan\left(-\frac{\sqrt{2} {\left(\sqrt{2} \sqrt{{\left| a \right|}} - 2 \, \sqrt{a \tan\left(\frac{1}{2} \, x\right)}\right)}}{2 \, \sqrt{{\left| a \right|}}}\right) + \frac{1}{2} \, {\left(a^{2} \sqrt{{\left| a \right|}} - a {\left| a \right|}^{\frac{3}{2}}\right)} \log\left(a \tan\left(\frac{1}{2} \, x\right) + \sqrt{2} \sqrt{a \tan\left(\frac{1}{2} \, x\right)} \sqrt{{\left| a \right|}} + {\left| a \right|}\right) - \frac{1}{2} \, {\left(a^{2} \sqrt{{\left| a \right|}} - a {\left| a \right|}^{\frac{3}{2}}\right)} \log\left(a \tan\left(\frac{1}{2} \, x\right) - \sqrt{2} \sqrt{a \tan\left(\frac{1}{2} \, x\right)} \sqrt{{\left| a \right|}} + {\left| a \right|}\right) + \frac{1}{6} \, \sqrt{2} {\left(\sqrt{a \tan\left(\frac{1}{2} \, x\right)} a^{2} \tan\left(\frac{1}{2} \, x\right) + 15 \, \sqrt{a \tan\left(\frac{1}{2} \, x\right)} a^{2}\right)} - \frac{\sqrt{2} {\left(15 \, a^{4} \tan\left(\frac{1}{2} \, x\right) + a^{4}\right)}}{6 \, \sqrt{a \tan\left(\frac{1}{2} \, x\right)} a \tan\left(\frac{1}{2} \, x\right)}"," ",0,"(a^2*sqrt(abs(a)) + a*abs(a)^(3/2))*arctan(1/2*sqrt(2)*(sqrt(2)*sqrt(abs(a)) + 2*sqrt(a*tan(1/2*x)))/sqrt(abs(a))) + (a^2*sqrt(abs(a)) + a*abs(a)^(3/2))*arctan(-1/2*sqrt(2)*(sqrt(2)*sqrt(abs(a)) - 2*sqrt(a*tan(1/2*x)))/sqrt(abs(a))) + 1/2*(a^2*sqrt(abs(a)) - a*abs(a)^(3/2))*log(a*tan(1/2*x) + sqrt(2)*sqrt(a*tan(1/2*x))*sqrt(abs(a)) + abs(a)) - 1/2*(a^2*sqrt(abs(a)) - a*abs(a)^(3/2))*log(a*tan(1/2*x) - sqrt(2)*sqrt(a*tan(1/2*x))*sqrt(abs(a)) + abs(a)) + 1/6*sqrt(2)*(sqrt(a*tan(1/2*x))*a^2*tan(1/2*x) + 15*sqrt(a*tan(1/2*x))*a^2) - 1/6*sqrt(2)*(15*a^4*tan(1/2*x) + a^4)/(sqrt(a*tan(1/2*x))*a*tan(1/2*x))","B",0
14,1,195,0,1.162295," ","integrate((a+a*csc(x))^(3/2),x, algorithm=""giac"")","\sqrt{2} \sqrt{a \tan\left(\frac{1}{2} \, x\right)} a - \frac{\sqrt{2} a^{2}}{\sqrt{a \tan\left(\frac{1}{2} \, x\right)}} + {\left(a \sqrt{{\left| a \right|}} + {\left| a \right|}^{\frac{3}{2}}\right)} \arctan\left(\frac{\sqrt{2} {\left(\sqrt{2} \sqrt{{\left| a \right|}} + 2 \, \sqrt{a \tan\left(\frac{1}{2} \, x\right)}\right)}}{2 \, \sqrt{{\left| a \right|}}}\right) + {\left(a \sqrt{{\left| a \right|}} + {\left| a \right|}^{\frac{3}{2}}\right)} \arctan\left(-\frac{\sqrt{2} {\left(\sqrt{2} \sqrt{{\left| a \right|}} - 2 \, \sqrt{a \tan\left(\frac{1}{2} \, x\right)}\right)}}{2 \, \sqrt{{\left| a \right|}}}\right) + \frac{1}{2} \, {\left(a \sqrt{{\left| a \right|}} - {\left| a \right|}^{\frac{3}{2}}\right)} \log\left(a \tan\left(\frac{1}{2} \, x\right) + \sqrt{2} \sqrt{a \tan\left(\frac{1}{2} \, x\right)} \sqrt{{\left| a \right|}} + {\left| a \right|}\right) - \frac{1}{2} \, {\left(a \sqrt{{\left| a \right|}} - {\left| a \right|}^{\frac{3}{2}}\right)} \log\left(a \tan\left(\frac{1}{2} \, x\right) - \sqrt{2} \sqrt{a \tan\left(\frac{1}{2} \, x\right)} \sqrt{{\left| a \right|}} + {\left| a \right|}\right)"," ",0,"sqrt(2)*sqrt(a*tan(1/2*x))*a - sqrt(2)*a^2/sqrt(a*tan(1/2*x)) + (a*sqrt(abs(a)) + abs(a)^(3/2))*arctan(1/2*sqrt(2)*(sqrt(2)*sqrt(abs(a)) + 2*sqrt(a*tan(1/2*x)))/sqrt(abs(a))) + (a*sqrt(abs(a)) + abs(a)^(3/2))*arctan(-1/2*sqrt(2)*(sqrt(2)*sqrt(abs(a)) - 2*sqrt(a*tan(1/2*x)))/sqrt(abs(a))) + 1/2*(a*sqrt(abs(a)) - abs(a)^(3/2))*log(a*tan(1/2*x) + sqrt(2)*sqrt(a*tan(1/2*x))*sqrt(abs(a)) + abs(a)) - 1/2*(a*sqrt(abs(a)) - abs(a)^(3/2))*log(a*tan(1/2*x) - sqrt(2)*sqrt(a*tan(1/2*x))*sqrt(abs(a)) + abs(a))","B",0
15,1,353,0,1.055428," ","integrate((a+a*csc(x))^(1/2),x, algorithm=""giac"")","\frac{1}{4} \, \sqrt{2} {\left(\frac{2 \, \sqrt{2} {\left(a \sqrt{{\left| a \right|}} \mathrm{sgn}\left(\tan\left(\frac{1}{2} \, x\right)^{3} + \tan\left(\frac{1}{2} \, x\right)^{2} + \tan\left(\frac{1}{2} \, x\right) + 1\right) + {\left| a \right|}^{\frac{3}{2}} \mathrm{sgn}\left(\tan\left(\frac{1}{2} \, x\right)^{3} + \tan\left(\frac{1}{2} \, x\right)^{2} + \tan\left(\frac{1}{2} \, x\right) + 1\right)\right)} \arctan\left(\frac{\sqrt{2} {\left(\sqrt{2} \sqrt{{\left| a \right|}} + 2 \, \sqrt{a \tan\left(\frac{1}{2} \, x\right)}\right)}}{2 \, \sqrt{{\left| a \right|}}}\right)}{a} + \frac{2 \, \sqrt{2} {\left(a \sqrt{{\left| a \right|}} \mathrm{sgn}\left(\tan\left(\frac{1}{2} \, x\right)^{3} + \tan\left(\frac{1}{2} \, x\right)^{2} + \tan\left(\frac{1}{2} \, x\right) + 1\right) + {\left| a \right|}^{\frac{3}{2}} \mathrm{sgn}\left(\tan\left(\frac{1}{2} \, x\right)^{3} + \tan\left(\frac{1}{2} \, x\right)^{2} + \tan\left(\frac{1}{2} \, x\right) + 1\right)\right)} \arctan\left(-\frac{\sqrt{2} {\left(\sqrt{2} \sqrt{{\left| a \right|}} - 2 \, \sqrt{a \tan\left(\frac{1}{2} \, x\right)}\right)}}{2 \, \sqrt{{\left| a \right|}}}\right)}{a} + \frac{\sqrt{2} {\left(a \sqrt{{\left| a \right|}} \mathrm{sgn}\left(\tan\left(\frac{1}{2} \, x\right)^{3} + \tan\left(\frac{1}{2} \, x\right)^{2} + \tan\left(\frac{1}{2} \, x\right) + 1\right) - {\left| a \right|}^{\frac{3}{2}} \mathrm{sgn}\left(\tan\left(\frac{1}{2} \, x\right)^{3} + \tan\left(\frac{1}{2} \, x\right)^{2} + \tan\left(\frac{1}{2} \, x\right) + 1\right)\right)} \log\left(a \tan\left(\frac{1}{2} \, x\right) + \sqrt{2} \sqrt{a \tan\left(\frac{1}{2} \, x\right)} \sqrt{{\left| a \right|}} + {\left| a \right|}\right)}{a} - \frac{\sqrt{2} {\left(a \sqrt{{\left| a \right|}} \mathrm{sgn}\left(\tan\left(\frac{1}{2} \, x\right)^{3} + \tan\left(\frac{1}{2} \, x\right)^{2} + \tan\left(\frac{1}{2} \, x\right) + 1\right) - {\left| a \right|}^{\frac{3}{2}} \mathrm{sgn}\left(\tan\left(\frac{1}{2} \, x\right)^{3} + \tan\left(\frac{1}{2} \, x\right)^{2} + \tan\left(\frac{1}{2} \, x\right) + 1\right)\right)} \log\left(a \tan\left(\frac{1}{2} \, x\right) - \sqrt{2} \sqrt{a \tan\left(\frac{1}{2} \, x\right)} \sqrt{{\left| a \right|}} + {\left| a \right|}\right)}{a}\right)} \mathrm{sgn}\left(\sin\left(x\right)\right)"," ",0,"1/4*sqrt(2)*(2*sqrt(2)*(a*sqrt(abs(a))*sgn(tan(1/2*x)^3 + tan(1/2*x)^2 + tan(1/2*x) + 1) + abs(a)^(3/2)*sgn(tan(1/2*x)^3 + tan(1/2*x)^2 + tan(1/2*x) + 1))*arctan(1/2*sqrt(2)*(sqrt(2)*sqrt(abs(a)) + 2*sqrt(a*tan(1/2*x)))/sqrt(abs(a)))/a + 2*sqrt(2)*(a*sqrt(abs(a))*sgn(tan(1/2*x)^3 + tan(1/2*x)^2 + tan(1/2*x) + 1) + abs(a)^(3/2)*sgn(tan(1/2*x)^3 + tan(1/2*x)^2 + tan(1/2*x) + 1))*arctan(-1/2*sqrt(2)*(sqrt(2)*sqrt(abs(a)) - 2*sqrt(a*tan(1/2*x)))/sqrt(abs(a)))/a + sqrt(2)*(a*sqrt(abs(a))*sgn(tan(1/2*x)^3 + tan(1/2*x)^2 + tan(1/2*x) + 1) - abs(a)^(3/2)*sgn(tan(1/2*x)^3 + tan(1/2*x)^2 + tan(1/2*x) + 1))*log(a*tan(1/2*x) + sqrt(2)*sqrt(a*tan(1/2*x))*sqrt(abs(a)) + abs(a))/a - sqrt(2)*(a*sqrt(abs(a))*sgn(tan(1/2*x)^3 + tan(1/2*x)^2 + tan(1/2*x) + 1) - abs(a)^(3/2)*sgn(tan(1/2*x)^3 + tan(1/2*x)^2 + tan(1/2*x) + 1))*log(a*tan(1/2*x) - sqrt(2)*sqrt(a*tan(1/2*x))*sqrt(abs(a)) + abs(a))/a)*sgn(sin(x))","B",0
16,0,0,0,0.000000," ","integrate(1/(a+a*csc(x))^(1/2),x, algorithm=""giac"")","\int \frac{1}{\sqrt{a \csc\left(x\right) + a}}\,{d x}"," ",0,"integrate(1/sqrt(a*csc(x) + a), x)","F",0
17,1,304,0,1.930204," ","integrate(1/(a+a*csc(x))^(3/2),x, algorithm=""giac"")","-\frac{1}{2} \, {\left(\frac{5 \, \sqrt{2} \arctan\left(\frac{\sqrt{a \tan\left(\frac{1}{2} \, x\right)}}{\sqrt{a}}\right)}{a^{\frac{3}{2}} \mathrm{sgn}\left(\tan\left(\frac{1}{2} \, x\right) + 1\right)} - \frac{2 \, {\left(a \sqrt{{\left| a \right|}} + {\left| a \right|}^{\frac{3}{2}}\right)} \arctan\left(\frac{\sqrt{2} {\left(\sqrt{2} \sqrt{{\left| a \right|}} + 2 \, \sqrt{a \tan\left(\frac{1}{2} \, x\right)}\right)}}{2 \, \sqrt{{\left| a \right|}}}\right)}{a^{3} \mathrm{sgn}\left(\tan\left(\frac{1}{2} \, x\right) + 1\right)} - \frac{2 \, {\left(a \sqrt{{\left| a \right|}} + {\left| a \right|}^{\frac{3}{2}}\right)} \arctan\left(-\frac{\sqrt{2} {\left(\sqrt{2} \sqrt{{\left| a \right|}} - 2 \, \sqrt{a \tan\left(\frac{1}{2} \, x\right)}\right)}}{2 \, \sqrt{{\left| a \right|}}}\right)}{a^{3} \mathrm{sgn}\left(\tan\left(\frac{1}{2} \, x\right) + 1\right)} - \frac{{\left(a \sqrt{{\left| a \right|}} - {\left| a \right|}^{\frac{3}{2}}\right)} \log\left(a \tan\left(\frac{1}{2} \, x\right) + \sqrt{2} \sqrt{a \tan\left(\frac{1}{2} \, x\right)} \sqrt{{\left| a \right|}} + {\left| a \right|}\right)}{a^{3} \mathrm{sgn}\left(\tan\left(\frac{1}{2} \, x\right) + 1\right)} + \frac{{\left(a \sqrt{{\left| a \right|}} - {\left| a \right|}^{\frac{3}{2}}\right)} \log\left(a \tan\left(\frac{1}{2} \, x\right) - \sqrt{2} \sqrt{a \tan\left(\frac{1}{2} \, x\right)} \sqrt{{\left| a \right|}} + {\left| a \right|}\right)}{a^{3} \mathrm{sgn}\left(\tan\left(\frac{1}{2} \, x\right) + 1\right)} + \frac{\sqrt{2} {\left(\sqrt{a \tan\left(\frac{1}{2} \, x\right)} a \tan\left(\frac{1}{2} \, x\right) - \sqrt{a \tan\left(\frac{1}{2} \, x\right)} a\right)}}{{\left(a \tan\left(\frac{1}{2} \, x\right) + a\right)}^{2} a \mathrm{sgn}\left(\tan\left(\frac{1}{2} \, x\right) + 1\right)}\right)} \mathrm{sgn}\left(\tan\left(\frac{1}{2} \, x\right)\right)"," ",0,"-1/2*(5*sqrt(2)*arctan(sqrt(a*tan(1/2*x))/sqrt(a))/(a^(3/2)*sgn(tan(1/2*x) + 1)) - 2*(a*sqrt(abs(a)) + abs(a)^(3/2))*arctan(1/2*sqrt(2)*(sqrt(2)*sqrt(abs(a)) + 2*sqrt(a*tan(1/2*x)))/sqrt(abs(a)))/(a^3*sgn(tan(1/2*x) + 1)) - 2*(a*sqrt(abs(a)) + abs(a)^(3/2))*arctan(-1/2*sqrt(2)*(sqrt(2)*sqrt(abs(a)) - 2*sqrt(a*tan(1/2*x)))/sqrt(abs(a)))/(a^3*sgn(tan(1/2*x) + 1)) - (a*sqrt(abs(a)) - abs(a)^(3/2))*log(a*tan(1/2*x) + sqrt(2)*sqrt(a*tan(1/2*x))*sqrt(abs(a)) + abs(a))/(a^3*sgn(tan(1/2*x) + 1)) + (a*sqrt(abs(a)) - abs(a)^(3/2))*log(a*tan(1/2*x) - sqrt(2)*sqrt(a*tan(1/2*x))*sqrt(abs(a)) + abs(a))/(a^3*sgn(tan(1/2*x) + 1)) + sqrt(2)*(sqrt(a*tan(1/2*x))*a*tan(1/2*x) - sqrt(a*tan(1/2*x))*a)/((a*tan(1/2*x) + a)^2*a*sgn(tan(1/2*x) + 1)))*sgn(tan(1/2*x))","B",0
18,1,348,0,1.615300," ","integrate(1/(a+a*csc(x))^(5/2),x, algorithm=""giac"")","-\frac{1}{16} \, {\left(\frac{43 \, \sqrt{2} \arctan\left(\frac{\sqrt{a \tan\left(\frac{1}{2} \, x\right)}}{\sqrt{a}}\right)}{a^{\frac{5}{2}} \mathrm{sgn}\left(\tan\left(\frac{1}{2} \, x\right) + 1\right)} - \frac{16 \, {\left(a \sqrt{{\left| a \right|}} + {\left| a \right|}^{\frac{3}{2}}\right)} \arctan\left(\frac{\sqrt{2} {\left(\sqrt{2} \sqrt{{\left| a \right|}} + 2 \, \sqrt{a \tan\left(\frac{1}{2} \, x\right)}\right)}}{2 \, \sqrt{{\left| a \right|}}}\right)}{a^{4} \mathrm{sgn}\left(\tan\left(\frac{1}{2} \, x\right) + 1\right)} - \frac{16 \, {\left(a \sqrt{{\left| a \right|}} + {\left| a \right|}^{\frac{3}{2}}\right)} \arctan\left(-\frac{\sqrt{2} {\left(\sqrt{2} \sqrt{{\left| a \right|}} - 2 \, \sqrt{a \tan\left(\frac{1}{2} \, x\right)}\right)}}{2 \, \sqrt{{\left| a \right|}}}\right)}{a^{4} \mathrm{sgn}\left(\tan\left(\frac{1}{2} \, x\right) + 1\right)} - \frac{8 \, {\left(a \sqrt{{\left| a \right|}} - {\left| a \right|}^{\frac{3}{2}}\right)} \log\left(a \tan\left(\frac{1}{2} \, x\right) + \sqrt{2} \sqrt{a \tan\left(\frac{1}{2} \, x\right)} \sqrt{{\left| a \right|}} + {\left| a \right|}\right)}{a^{4} \mathrm{sgn}\left(\tan\left(\frac{1}{2} \, x\right) + 1\right)} + \frac{8 \, {\left(a \sqrt{{\left| a \right|}} - {\left| a \right|}^{\frac{3}{2}}\right)} \log\left(a \tan\left(\frac{1}{2} \, x\right) - \sqrt{2} \sqrt{a \tan\left(\frac{1}{2} \, x\right)} \sqrt{{\left| a \right|}} + {\left| a \right|}\right)}{a^{4} \mathrm{sgn}\left(\tan\left(\frac{1}{2} \, x\right) + 1\right)} + \frac{\sqrt{2} {\left(11 \, \sqrt{a \tan\left(\frac{1}{2} \, x\right)} a^{3} \tan\left(\frac{1}{2} \, x\right)^{3} + 19 \, \sqrt{a \tan\left(\frac{1}{2} \, x\right)} a^{3} \tan\left(\frac{1}{2} \, x\right)^{2} - 19 \, \sqrt{a \tan\left(\frac{1}{2} \, x\right)} a^{3} \tan\left(\frac{1}{2} \, x\right) - 11 \, \sqrt{a \tan\left(\frac{1}{2} \, x\right)} a^{3}\right)}}{{\left(a \tan\left(\frac{1}{2} \, x\right) + a\right)}^{4} a^{2} \mathrm{sgn}\left(\tan\left(\frac{1}{2} \, x\right) + 1\right)}\right)} \mathrm{sgn}\left(\tan\left(\frac{1}{2} \, x\right)\right)"," ",0,"-1/16*(43*sqrt(2)*arctan(sqrt(a*tan(1/2*x))/sqrt(a))/(a^(5/2)*sgn(tan(1/2*x) + 1)) - 16*(a*sqrt(abs(a)) + abs(a)^(3/2))*arctan(1/2*sqrt(2)*(sqrt(2)*sqrt(abs(a)) + 2*sqrt(a*tan(1/2*x)))/sqrt(abs(a)))/(a^4*sgn(tan(1/2*x) + 1)) - 16*(a*sqrt(abs(a)) + abs(a)^(3/2))*arctan(-1/2*sqrt(2)*(sqrt(2)*sqrt(abs(a)) - 2*sqrt(a*tan(1/2*x)))/sqrt(abs(a)))/(a^4*sgn(tan(1/2*x) + 1)) - 8*(a*sqrt(abs(a)) - abs(a)^(3/2))*log(a*tan(1/2*x) + sqrt(2)*sqrt(a*tan(1/2*x))*sqrt(abs(a)) + abs(a))/(a^4*sgn(tan(1/2*x) + 1)) + 8*(a*sqrt(abs(a)) - abs(a)^(3/2))*log(a*tan(1/2*x) - sqrt(2)*sqrt(a*tan(1/2*x))*sqrt(abs(a)) + abs(a))/(a^4*sgn(tan(1/2*x) + 1)) + sqrt(2)*(11*sqrt(a*tan(1/2*x))*a^3*tan(1/2*x)^3 + 19*sqrt(a*tan(1/2*x))*a^3*tan(1/2*x)^2 - 19*sqrt(a*tan(1/2*x))*a^3*tan(1/2*x) - 11*sqrt(a*tan(1/2*x))*a^3)/((a*tan(1/2*x) + a)^4*a^2*sgn(tan(1/2*x) + 1)))*sgn(tan(1/2*x))","B",0
19,-2,0,0,0.000000," ","integrate(csc(f*x+e)^(1/2)*(a+a*csc(f*x+e))^(1/2),x, algorithm=""giac"")","\text{Exception raised: NotImplementedError}"," ",0,"Exception raised: NotImplementedError >> Unable to parse Giac output: Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)2*a^2*(a*atan((sqrt(a^3+a*(a*tan(1/2*(f*x+exp(1))))^2)-sqrt(a)*a*tan(1/2*(f*x+exp(1))))/sqrt(-a)/a)/sqrt(-a)/a-1/2*sqrt(a)*ln(abs(sqrt(a^3+a*(a*tan(1/2*(f*x+exp(1))))^2)-sqrt(a)*a*tan(1/2*(f*x+exp(1)))))/a)*sign(sin(f*x+exp(1)))/a/f","F(-2)",0
20,-2,0,0,0.000000," ","integrate((-csc(f*x+e))^(1/2)*(a-a*csc(f*x+e))^(1/2),x, algorithm=""giac"")","\text{Exception raised: NotImplementedError}"," ",0,"Exception raised: NotImplementedError >> Unable to parse Giac output: Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)8*(1/8*sqrt(a)*ln(abs(sqrt(a^3+a*(-a*tan(1/2*(f*x+exp(1))))^2)+sqrt(a)*a*tan(1/2*(f*x+exp(1)))))-1/4*a^2*atan((sqrt(a^3+a*(-a*tan(1/2*(f*x+exp(1))))^2)+sqrt(a)*a*tan(1/2*(f*x+exp(1))))/sqrt(-a)/a)/sqrt(-a)/a)/f","F(-2)",0
21,0,0,0,0.000000," ","integrate(csc(d*x+c)^(4/3)*(a+a*csc(d*x+c))^(1/2),x, algorithm=""giac"")","\int \sqrt{a \csc\left(d x + c\right) + a} \csc\left(d x + c\right)^{\frac{4}{3}}\,{d x}"," ",0,"integrate(sqrt(a*csc(d*x + c) + a)*csc(d*x + c)^(4/3), x)","F",0
22,0,0,0,0.000000," ","integrate(csc(d*x+c)^(1/3)*(a+a*csc(d*x+c))^(1/2),x, algorithm=""giac"")","\int \sqrt{a \csc\left(d x + c\right) + a} \csc\left(d x + c\right)^{\frac{1}{3}}\,{d x}"," ",0,"integrate(sqrt(a*csc(d*x + c) + a)*csc(d*x + c)^(1/3), x)","F",0
23,0,0,0,0.000000," ","integrate((a+a*csc(d*x+c))^(1/2)/csc(d*x+c)^(2/3),x, algorithm=""giac"")","\int \frac{\sqrt{a \csc\left(d x + c\right) + a}}{\csc\left(d x + c\right)^{\frac{2}{3}}}\,{d x}"," ",0,"integrate(sqrt(a*csc(d*x + c) + a)/csc(d*x + c)^(2/3), x)","F",0
24,0,0,0,0.000000," ","integrate(csc(d*x+c)^(5/3)*(a+a*csc(d*x+c))^(1/2),x, algorithm=""giac"")","\int \sqrt{a \csc\left(d x + c\right) + a} \csc\left(d x + c\right)^{\frac{5}{3}}\,{d x}"," ",0,"integrate(sqrt(a*csc(d*x + c) + a)*csc(d*x + c)^(5/3), x)","F",0
25,0,0,0,0.000000," ","integrate(csc(d*x+c)^(2/3)*(a+a*csc(d*x+c))^(1/2),x, algorithm=""giac"")","\int \sqrt{a \csc\left(d x + c\right) + a} \csc\left(d x + c\right)^{\frac{2}{3}}\,{d x}"," ",0,"integrate(sqrt(a*csc(d*x + c) + a)*csc(d*x + c)^(2/3), x)","F",0
26,0,0,0,0.000000," ","integrate((a+a*csc(d*x+c))^(1/2)/csc(d*x+c)^(1/3),x, algorithm=""giac"")","\int \frac{\sqrt{a \csc\left(d x + c\right) + a}}{\csc\left(d x + c\right)^{\frac{1}{3}}}\,{d x}"," ",0,"integrate(sqrt(a*csc(d*x + c) + a)/csc(d*x + c)^(1/3), x)","F",0
27,0,0,0,0.000000," ","integrate((a+a*csc(d*x+c))^(1/2)/csc(d*x+c)^(4/3),x, algorithm=""giac"")","\int \frac{\sqrt{a \csc\left(d x + c\right) + a}}{\csc\left(d x + c\right)^{\frac{4}{3}}}\,{d x}"," ",0,"integrate(sqrt(a*csc(d*x + c) + a)/csc(d*x + c)^(4/3), x)","F",0
28,0,0,0,0.000000," ","integrate(csc(d*x+c)^n*(a+a*csc(d*x+c))^(1/2),x, algorithm=""giac"")","\int \sqrt{a \csc\left(d x + c\right) + a} \csc\left(d x + c\right)^{n}\,{d x}"," ",0,"integrate(sqrt(a*csc(d*x + c) + a)*csc(d*x + c)^n, x)","F",0
29,0,0,0,0.000000," ","integrate(csc(d*x+c)^n*(a-a*csc(d*x+c))^(1/2),x, algorithm=""giac"")","\int \sqrt{-a \csc\left(d x + c\right) + a} \csc\left(d x + c\right)^{n}\,{d x}"," ",0,"integrate(sqrt(-a*csc(d*x + c) + a)*csc(d*x + c)^n, x)","F",0
30,0,0,0,0.000000," ","integrate(csc(f*x+e)^3*(a+a*csc(f*x+e))^m,x, algorithm=""giac"")","\int {\left(a \csc\left(f x + e\right) + a\right)}^{m} \csc\left(f x + e\right)^{3}\,{d x}"," ",0,"integrate((a*csc(f*x + e) + a)^m*csc(f*x + e)^3, x)","F",0
31,0,0,0,0.000000," ","integrate(csc(f*x+e)^2*(a+a*csc(f*x+e))^m,x, algorithm=""giac"")","\int {\left(a \csc\left(f x + e\right) + a\right)}^{m} \csc\left(f x + e\right)^{2}\,{d x}"," ",0,"integrate((a*csc(f*x + e) + a)^m*csc(f*x + e)^2, x)","F",0
32,0,0,0,0.000000," ","integrate(csc(f*x+e)*(a+a*csc(f*x+e))^m,x, algorithm=""giac"")","\int {\left(a \csc\left(f x + e\right) + a\right)}^{m} \csc\left(f x + e\right)\,{d x}"," ",0,"integrate((a*csc(f*x + e) + a)^m*csc(f*x + e), x)","F",0
33,0,0,0,0.000000," ","integrate((a+a*csc(f*x+e))^m,x, algorithm=""giac"")","\int {\left(a \csc\left(f x + e\right) + a\right)}^{m}\,{d x}"," ",0,"integrate((a*csc(f*x + e) + a)^m, x)","F",0
34,0,0,0,0.000000," ","integrate((a+a*csc(f*x+e))^m*sin(f*x+e),x, algorithm=""giac"")","\int {\left(a \csc\left(f x + e\right) + a\right)}^{m} \sin\left(f x + e\right)\,{d x}"," ",0,"integrate((a*csc(f*x + e) + a)^m*sin(f*x + e), x)","F",0
35,0,0,0,0.000000," ","integrate((a+a*csc(f*x+e))^m*sin(f*x+e)^2,x, algorithm=""giac"")","\int {\left(a \csc\left(f x + e\right) + a\right)}^{m} \sin\left(f x + e\right)^{2}\,{d x}"," ",0,"integrate((a*csc(f*x + e) + a)^m*sin(f*x + e)^2, x)","F",0
36,1,205,0,0.781857," ","integrate((a+b*csc(d*x+c))^4,x, algorithm=""giac"")","\frac{b^{4} \tan\left(\frac{1}{2} \, d x + \frac{1}{2} \, c\right)^{3} + 12 \, a b^{3} \tan\left(\frac{1}{2} \, d x + \frac{1}{2} \, c\right)^{2} + 24 \, {\left(d x + c\right)} a^{4} + 72 \, a^{2} b^{2} \tan\left(\frac{1}{2} \, d x + \frac{1}{2} \, c\right) + 9 \, b^{4} \tan\left(\frac{1}{2} \, d x + \frac{1}{2} \, c\right) + 48 \, {\left(2 \, a^{3} b + a b^{3}\right)} \log\left({\left| \tan\left(\frac{1}{2} \, d x + \frac{1}{2} \, c\right) \right|}\right) - \frac{176 \, a^{3} b \tan\left(\frac{1}{2} \, d x + \frac{1}{2} \, c\right)^{3} + 88 \, a b^{3} \tan\left(\frac{1}{2} \, d x + \frac{1}{2} \, c\right)^{3} + 72 \, a^{2} b^{2} \tan\left(\frac{1}{2} \, d x + \frac{1}{2} \, c\right)^{2} + 9 \, b^{4} \tan\left(\frac{1}{2} \, d x + \frac{1}{2} \, c\right)^{2} + 12 \, a b^{3} \tan\left(\frac{1}{2} \, d x + \frac{1}{2} \, c\right) + b^{4}}{\tan\left(\frac{1}{2} \, d x + \frac{1}{2} \, c\right)^{3}}}{24 \, d}"," ",0,"1/24*(b^4*tan(1/2*d*x + 1/2*c)^3 + 12*a*b^3*tan(1/2*d*x + 1/2*c)^2 + 24*(d*x + c)*a^4 + 72*a^2*b^2*tan(1/2*d*x + 1/2*c) + 9*b^4*tan(1/2*d*x + 1/2*c) + 48*(2*a^3*b + a*b^3)*log(abs(tan(1/2*d*x + 1/2*c))) - (176*a^3*b*tan(1/2*d*x + 1/2*c)^3 + 88*a*b^3*tan(1/2*d*x + 1/2*c)^3 + 72*a^2*b^2*tan(1/2*d*x + 1/2*c)^2 + 9*b^4*tan(1/2*d*x + 1/2*c)^2 + 12*a*b^3*tan(1/2*d*x + 1/2*c) + b^4)/tan(1/2*d*x + 1/2*c)^3)/d","B",0
37,1,134,0,0.473671," ","integrate((a+b*csc(d*x+c))^3,x, algorithm=""giac"")","\frac{b^{3} \tan\left(\frac{1}{2} \, d x + \frac{1}{2} \, c\right)^{2} + 8 \, {\left(d x + c\right)} a^{3} + 12 \, a b^{2} \tan\left(\frac{1}{2} \, d x + \frac{1}{2} \, c\right) + 4 \, {\left(6 \, a^{2} b + b^{3}\right)} \log\left({\left| \tan\left(\frac{1}{2} \, d x + \frac{1}{2} \, c\right) \right|}\right) - \frac{36 \, a^{2} b \tan\left(\frac{1}{2} \, d x + \frac{1}{2} \, c\right)^{2} + 6 \, b^{3} \tan\left(\frac{1}{2} \, d x + \frac{1}{2} \, c\right)^{2} + 12 \, a b^{2} \tan\left(\frac{1}{2} \, d x + \frac{1}{2} \, c\right) + b^{3}}{\tan\left(\frac{1}{2} \, d x + \frac{1}{2} \, c\right)^{2}}}{8 \, d}"," ",0,"1/8*(b^3*tan(1/2*d*x + 1/2*c)^2 + 8*(d*x + c)*a^3 + 12*a*b^2*tan(1/2*d*x + 1/2*c) + 4*(6*a^2*b + b^3)*log(abs(tan(1/2*d*x + 1/2*c))) - (36*a^2*b*tan(1/2*d*x + 1/2*c)^2 + 6*b^3*tan(1/2*d*x + 1/2*c)^2 + 12*a*b^2*tan(1/2*d*x + 1/2*c) + b^3)/tan(1/2*d*x + 1/2*c)^2)/d","A",0
38,1,74,0,0.859910," ","integrate((a+b*csc(d*x+c))^2,x, algorithm=""giac"")","\frac{2 \, {\left(d x + c\right)} a^{2} + 4 \, a b \log\left({\left| \tan\left(\frac{1}{2} \, d x + \frac{1}{2} \, c\right) \right|}\right) + b^{2} \tan\left(\frac{1}{2} \, d x + \frac{1}{2} \, c\right) - \frac{4 \, a b \tan\left(\frac{1}{2} \, d x + \frac{1}{2} \, c\right) + b^{2}}{\tan\left(\frac{1}{2} \, d x + \frac{1}{2} \, c\right)}}{2 \, d}"," ",0,"1/2*(2*(d*x + c)*a^2 + 4*a*b*log(abs(tan(1/2*d*x + 1/2*c))) + b^2*tan(1/2*d*x + 1/2*c) - (4*a*b*tan(1/2*d*x + 1/2*c) + b^2)/tan(1/2*d*x + 1/2*c))/d","B",0
39,1,194,0,0.656731," ","integrate(csc(x)^5/(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^{4}}{\sqrt{-a^{2} + b^{2}} b^{4}} + \frac{b^{2} \tan\left(\frac{1}{2} \, x\right)^{3} - 3 \, a b \tan\left(\frac{1}{2} \, x\right)^{2} + 12 \, a^{2} \tan\left(\frac{1}{2} \, x\right) + 9 \, b^{2} \tan\left(\frac{1}{2} \, x\right)}{24 \, b^{3}} - \frac{{\left(2 \, a^{3} + a b^{2}\right)} \log\left({\left| \tan\left(\frac{1}{2} \, x\right) \right|}\right)}{2 \, b^{4}} + \frac{44 \, a^{3} \tan\left(\frac{1}{2} \, x\right)^{3} + 22 \, a b^{2} \tan\left(\frac{1}{2} \, x\right)^{3} - 12 \, a^{2} b \tan\left(\frac{1}{2} \, x\right)^{2} - 9 \, b^{3} \tan\left(\frac{1}{2} \, x\right)^{2} + 3 \, a b^{2} \tan\left(\frac{1}{2} \, x\right) - b^{3}}{24 \, b^{4} \tan\left(\frac{1}{2} \, x\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^4/(sqrt(-a^2 + b^2)*b^4) + 1/24*(b^2*tan(1/2*x)^3 - 3*a*b*tan(1/2*x)^2 + 12*a^2*tan(1/2*x) + 9*b^2*tan(1/2*x))/b^3 - 1/2*(2*a^3 + a*b^2)*log(abs(tan(1/2*x)))/b^4 + 1/24*(44*a^3*tan(1/2*x)^3 + 22*a*b^2*tan(1/2*x)^3 - 12*a^2*b*tan(1/2*x)^2 - 9*b^3*tan(1/2*x)^2 + 3*a*b^2*tan(1/2*x) - b^3)/(b^4*tan(1/2*x)^3)","A",0
40,1,141,0,0.525177," ","integrate(csc(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}}{\sqrt{-a^{2} + b^{2}} b^{3}} + \frac{b \tan\left(\frac{1}{2} \, x\right)^{2} - 4 \, a \tan\left(\frac{1}{2} \, x\right)}{8 \, b^{2}} + \frac{{\left(2 \, a^{2} + b^{2}\right)} \log\left({\left| \tan\left(\frac{1}{2} \, x\right) \right|}\right)}{2 \, b^{3}} - \frac{12 \, a^{2} \tan\left(\frac{1}{2} \, x\right)^{2} + 6 \, b^{2} \tan\left(\frac{1}{2} \, x\right)^{2} - 4 \, a b \tan\left(\frac{1}{2} \, x\right) + b^{2}}{8 \, b^{3} \tan\left(\frac{1}{2} \, x\right)^{2}}"," ",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/(sqrt(-a^2 + b^2)*b^3) + 1/8*(b*tan(1/2*x)^2 - 4*a*tan(1/2*x))/b^2 + 1/2*(2*a^2 + b^2)*log(abs(tan(1/2*x)))/b^3 - 1/8*(12*a^2*tan(1/2*x)^2 + 6*b^2*tan(1/2*x)^2 - 4*a*b*tan(1/2*x) + b^2)/(b^3*tan(1/2*x)^2)","A",0
41,1,98,0,0.346294," ","integrate(csc(x)^3/(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^{2}}{\sqrt{-a^{2} + b^{2}} b^{2}} - \frac{a \log\left({\left| \tan\left(\frac{1}{2} \, x\right) \right|}\right)}{b^{2}} + \frac{\tan\left(\frac{1}{2} \, x\right)}{2 \, b} + \frac{2 \, a \tan\left(\frac{1}{2} \, x\right) - b}{2 \, b^{2} \tan\left(\frac{1}{2} \, x\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^2/(sqrt(-a^2 + b^2)*b^2) - a*log(abs(tan(1/2*x)))/b^2 + 1/2*tan(1/2*x)/b + 1/2*(2*a*tan(1/2*x) - b)/(b^2*tan(1/2*x))","A",0
42,1,63,0,0.534018," ","integrate(csc(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}{\sqrt{-a^{2} + b^{2}} b} + \frac{\log\left({\left| \tan\left(\frac{1}{2} \, x\right) \right|}\right)}{b}"," ",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/(sqrt(-a^2 + b^2)*b) + log(abs(tan(1/2*x)))/b","A",0
43,1,48,0,0.450992," ","integrate(csc(x)/(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)}}{\sqrt{-a^{2} + b^{2}}}"," ",0,"2*(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",0
44,1,77,0,0.315766," ","integrate(1/(a+b*csc(d*x+c)),x, algorithm=""giac"")","-\frac{\frac{2 \, {\left(\pi \left \lfloor \frac{d x + c}{2 \, \pi} + \frac{1}{2} \right \rfloor \mathrm{sgn}\left(b\right) + \arctan\left(\frac{b \tan\left(\frac{1}{2} \, d x + \frac{1}{2} \, c\right) + a}{\sqrt{-a^{2} + b^{2}}}\right)\right)} b}{\sqrt{-a^{2} + b^{2}} a} - \frac{d x + c}{a}}{d}"," ",0,"-(2*(pi*floor(1/2*(d*x + c)/pi + 1/2)*sgn(b) + arctan((b*tan(1/2*d*x + 1/2*c) + a)/sqrt(-a^2 + b^2)))*b/(sqrt(-a^2 + b^2)*a) - (d*x + c)/a)/d","A",0
45,1,77,0,0.520343," ","integrate(sin(x)/(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)} b^{2}}{\sqrt{-a^{2} + b^{2}} a^{2}} - \frac{b x}{a^{2}} - \frac{2}{{\left(\tan\left(\frac{1}{2} \, x\right)^{2} + 1\right)} a}"," ",0,"2*(pi*floor(1/2*x/pi + 1/2)*sgn(b) + arctan((b*tan(1/2*x) + a)/sqrt(-a^2 + b^2)))*b^2/(sqrt(-a^2 + b^2)*a^2) - b*x/a^2 - 2/((tan(1/2*x)^2 + 1)*a)","A",0
46,1,112,0,0.637452," ","integrate(sin(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)} b^{3}}{\sqrt{-a^{2} + b^{2}} a^{3}} + \frac{{\left(a^{2} + 2 \, b^{2}\right)} x}{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,"-2*(pi*floor(1/2*x/pi + 1/2)*sgn(b) + arctan((b*tan(1/2*x) + a)/sqrt(-a^2 + b^2)))*b^3/(sqrt(-a^2 + b^2)*a^3) + 1/2*(a^2 + 2*b^2)*x/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
47,1,149,0,0.608693," ","integrate(sin(x)^3/(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)} b^{4}}{\sqrt{-a^{2} + b^{2}} a^{4}} - \frac{{\left(a^{2} b + 2 \, b^{3}\right)} x}{2 \, a^{4}} - \frac{3 \, a b \tan\left(\frac{1}{2} \, x\right)^{5} + 6 \, b^{2} \tan\left(\frac{1}{2} \, x\right)^{4} + 12 \, a^{2} \tan\left(\frac{1}{2} \, x\right)^{2} + 12 \, b^{2} \tan\left(\frac{1}{2} \, x\right)^{2} - 3 \, a b \tan\left(\frac{1}{2} \, x\right) + 4 \, a^{2} + 6 \, b^{2}}{3 \, {\left(\tan\left(\frac{1}{2} \, x\right)^{2} + 1\right)}^{3} a^{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)))*b^4/(sqrt(-a^2 + b^2)*a^4) - 1/2*(a^2*b + 2*b^3)*x/a^4 - 1/3*(3*a*b*tan(1/2*x)^5 + 6*b^2*tan(1/2*x)^4 + 12*a^2*tan(1/2*x)^2 + 12*b^2*tan(1/2*x)^2 - 3*a*b*tan(1/2*x) + 4*a^2 + 6*b^2)/((tan(1/2*x)^2 + 1)^3*a^3)","A",0
48,1,252,0,0.662290," ","integrate(sin(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)} b^{5}}{\sqrt{-a^{2} + b^{2}} a^{5}} + \frac{{\left(3 \, a^{4} + 4 \, a^{2} b^{2} + 8 \, b^{4}\right)} x}{8 \, a^{5}} + \frac{9 \, a^{3} \tan\left(\frac{1}{2} \, x\right)^{7} + 12 \, a b^{2} \tan\left(\frac{1}{2} \, x\right)^{7} + 24 \, b^{3} \tan\left(\frac{1}{2} \, x\right)^{6} + 33 \, a^{3} \tan\left(\frac{1}{2} \, x\right)^{5} + 12 \, a b^{2} \tan\left(\frac{1}{2} \, x\right)^{5} + 48 \, a^{2} b \tan\left(\frac{1}{2} \, x\right)^{4} + 72 \, b^{3} \tan\left(\frac{1}{2} \, x\right)^{4} - 33 \, a^{3} \tan\left(\frac{1}{2} \, x\right)^{3} - 12 \, a b^{2} \tan\left(\frac{1}{2} \, x\right)^{3} + 64 \, a^{2} b \tan\left(\frac{1}{2} \, x\right)^{2} + 72 \, b^{3} \tan\left(\frac{1}{2} \, x\right)^{2} - 9 \, a^{3} \tan\left(\frac{1}{2} \, x\right) - 12 \, a b^{2} \tan\left(\frac{1}{2} \, x\right) + 16 \, a^{2} b + 24 \, b^{3}}{12 \, {\left(\tan\left(\frac{1}{2} \, x\right)^{2} + 1\right)}^{4} a^{4}}"," ",0,"-2*(pi*floor(1/2*x/pi + 1/2)*sgn(b) + arctan((b*tan(1/2*x) + a)/sqrt(-a^2 + b^2)))*b^5/(sqrt(-a^2 + b^2)*a^5) + 1/8*(3*a^4 + 4*a^2*b^2 + 8*b^4)*x/a^5 + 1/12*(9*a^3*tan(1/2*x)^7 + 12*a*b^2*tan(1/2*x)^7 + 24*b^3*tan(1/2*x)^6 + 33*a^3*tan(1/2*x)^5 + 12*a*b^2*tan(1/2*x)^5 + 48*a^2*b*tan(1/2*x)^4 + 72*b^3*tan(1/2*x)^4 - 33*a^3*tan(1/2*x)^3 - 12*a*b^2*tan(1/2*x)^3 + 64*a^2*b*tan(1/2*x)^2 + 72*b^3*tan(1/2*x)^2 - 9*a^3*tan(1/2*x) - 12*a*b^2*tan(1/2*x) + 16*a^2*b + 24*b^3)/((tan(1/2*x)^2 + 1)^4*a^4)","A",0
49,1,158,0,0.529783," ","integrate(1/(a+b*csc(d*x+c))^2,x, algorithm=""giac"")","-\frac{\frac{2 \, {\left(2 \, a^{2} b - b^{3}\right)} {\left(\pi \left \lfloor \frac{d x + c}{2 \, \pi} + \frac{1}{2} \right \rfloor \mathrm{sgn}\left(b\right) + \arctan\left(\frac{b \tan\left(\frac{1}{2} \, d x + \frac{1}{2} \, c\right) + a}{\sqrt{-a^{2} + b^{2}}}\right)\right)}}{{\left(a^{4} - a^{2} b^{2}\right)} \sqrt{-a^{2} + b^{2}}} + \frac{2 \, {\left(a b \tan\left(\frac{1}{2} \, d x + \frac{1}{2} \, c\right) + b^{2}\right)}}{{\left(a^{3} - a b^{2}\right)} {\left(b \tan\left(\frac{1}{2} \, d x + \frac{1}{2} \, c\right)^{2} + 2 \, a \tan\left(\frac{1}{2} \, d x + \frac{1}{2} \, c\right) + b\right)}} - \frac{d x + c}{a^{2}}}{d}"," ",0,"-(2*(2*a^2*b - b^3)*(pi*floor(1/2*(d*x + c)/pi + 1/2)*sgn(b) + arctan((b*tan(1/2*d*x + 1/2*c) + a)/sqrt(-a^2 + b^2)))/((a^4 - a^2*b^2)*sqrt(-a^2 + b^2)) + 2*(a*b*tan(1/2*d*x + 1/2*c) + b^2)/((a^3 - a*b^2)*(b*tan(1/2*d*x + 1/2*c)^2 + 2*a*tan(1/2*d*x + 1/2*c) + b)) - (d*x + c)/a^2)/d","A",0
50,1,297,0,0.380542," ","integrate(1/(a+b*csc(d*x+c))^3,x, algorithm=""giac"")","-\frac{\frac{{\left(6 \, a^{4} b - 5 \, a^{2} b^{3} + 2 \, b^{5}\right)} {\left(\pi \left \lfloor \frac{d x + c}{2 \, \pi} + \frac{1}{2} \right \rfloor \mathrm{sgn}\left(b\right) + \arctan\left(\frac{b \tan\left(\frac{1}{2} \, d x + \frac{1}{2} \, c\right) + a}{\sqrt{-a^{2} + b^{2}}}\right)\right)}}{{\left(a^{7} - 2 \, a^{5} b^{2} + a^{3} b^{4}\right)} \sqrt{-a^{2} + b^{2}}} + \frac{4 \, a^{3} b^{2} \tan\left(\frac{1}{2} \, d x + \frac{1}{2} \, c\right)^{3} - a b^{4} \tan\left(\frac{1}{2} \, d x + \frac{1}{2} \, c\right)^{3} + 10 \, a^{4} b \tan\left(\frac{1}{2} \, d x + \frac{1}{2} \, c\right)^{2} + a^{2} b^{3} \tan\left(\frac{1}{2} \, d x + \frac{1}{2} \, c\right)^{2} - 2 \, b^{5} \tan\left(\frac{1}{2} \, d x + \frac{1}{2} \, c\right)^{2} + 16 \, a^{3} b^{2} \tan\left(\frac{1}{2} \, d x + \frac{1}{2} \, c\right) - 7 \, a b^{4} \tan\left(\frac{1}{2} \, d x + \frac{1}{2} \, c\right) + 5 \, a^{2} b^{3} - 2 \, b^{5}}{{\left(a^{6} - 2 \, a^{4} b^{2} + a^{2} b^{4}\right)} {\left(b \tan\left(\frac{1}{2} \, d x + \frac{1}{2} \, c\right)^{2} + 2 \, a \tan\left(\frac{1}{2} \, d x + \frac{1}{2} \, c\right) + b\right)}^{2}} - \frac{d x + c}{a^{3}}}{d}"," ",0,"-((6*a^4*b - 5*a^2*b^3 + 2*b^5)*(pi*floor(1/2*(d*x + c)/pi + 1/2)*sgn(b) + arctan((b*tan(1/2*d*x + 1/2*c) + a)/sqrt(-a^2 + b^2)))/((a^7 - 2*a^5*b^2 + a^3*b^4)*sqrt(-a^2 + b^2)) + (4*a^3*b^2*tan(1/2*d*x + 1/2*c)^3 - a*b^4*tan(1/2*d*x + 1/2*c)^3 + 10*a^4*b*tan(1/2*d*x + 1/2*c)^2 + a^2*b^3*tan(1/2*d*x + 1/2*c)^2 - 2*b^5*tan(1/2*d*x + 1/2*c)^2 + 16*a^3*b^2*tan(1/2*d*x + 1/2*c) - 7*a*b^4*tan(1/2*d*x + 1/2*c) + 5*a^2*b^3 - 2*b^5)/((a^6 - 2*a^4*b^2 + a^2*b^4)*(b*tan(1/2*d*x + 1/2*c)^2 + 2*a*tan(1/2*d*x + 1/2*c) + b)^2) - (d*x + c)/a^3)/d","A",0
51,1,535,0,0.378294," ","integrate(1/(a+b*csc(d*x+c))^4,x, algorithm=""giac"")","-\frac{\frac{3 \, {\left(8 \, a^{6} b - 8 \, a^{4} b^{3} + 7 \, a^{2} b^{5} - 2 \, b^{7}\right)} {\left(\pi \left \lfloor \frac{d x + c}{2 \, \pi} + \frac{1}{2} \right \rfloor \mathrm{sgn}\left(b\right) + \arctan\left(\frac{b \tan\left(\frac{1}{2} \, d x + \frac{1}{2} \, c\right) + a}{\sqrt{-a^{2} + b^{2}}}\right)\right)}}{{\left(a^{10} - 3 \, a^{8} b^{2} + 3 \, a^{6} b^{4} - a^{4} b^{6}\right)} \sqrt{-a^{2} + b^{2}}} + \frac{18 \, a^{5} b^{3} \tan\left(\frac{1}{2} \, d x + \frac{1}{2} \, c\right)^{5} - 6 \, a^{3} b^{5} \tan\left(\frac{1}{2} \, d x + \frac{1}{2} \, c\right)^{5} + 3 \, a b^{7} \tan\left(\frac{1}{2} \, d x + \frac{1}{2} \, c\right)^{5} + 84 \, a^{6} b^{2} \tan\left(\frac{1}{2} \, d x + \frac{1}{2} \, c\right)^{4} - 12 \, a^{4} b^{4} \tan\left(\frac{1}{2} \, d x + \frac{1}{2} \, c\right)^{4} - 3 \, a^{2} b^{6} \tan\left(\frac{1}{2} \, d x + \frac{1}{2} \, c\right)^{4} + 6 \, b^{8} \tan\left(\frac{1}{2} \, d x + \frac{1}{2} \, c\right)^{4} + 104 \, a^{7} b \tan\left(\frac{1}{2} \, d x + \frac{1}{2} \, c\right)^{3} + 88 \, a^{5} b^{3} \tan\left(\frac{1}{2} \, d x + \frac{1}{2} \, c\right)^{3} - 78 \, a^{3} b^{5} \tan\left(\frac{1}{2} \, d x + \frac{1}{2} \, c\right)^{3} + 36 \, a b^{7} \tan\left(\frac{1}{2} \, d x + \frac{1}{2} \, c\right)^{3} + 228 \, a^{6} b^{2} \tan\left(\frac{1}{2} \, d x + \frac{1}{2} \, c\right)^{2} - 114 \, a^{4} b^{4} \tan\left(\frac{1}{2} \, d x + \frac{1}{2} \, c\right)^{2} + 24 \, a^{2} b^{6} \tan\left(\frac{1}{2} \, d x + \frac{1}{2} \, c\right)^{2} + 12 \, b^{8} \tan\left(\frac{1}{2} \, d x + \frac{1}{2} \, c\right)^{2} + 138 \, a^{5} b^{3} \tan\left(\frac{1}{2} \, d x + \frac{1}{2} \, c\right) - 96 \, a^{3} b^{5} \tan\left(\frac{1}{2} \, d x + \frac{1}{2} \, c\right) + 33 \, a b^{7} \tan\left(\frac{1}{2} \, d x + \frac{1}{2} \, c\right) + 26 \, a^{4} b^{4} - 17 \, a^{2} b^{6} + 6 \, b^{8}}{{\left(a^{9} - 3 \, a^{7} b^{2} + 3 \, a^{5} b^{4} - a^{3} b^{6}\right)} {\left(b \tan\left(\frac{1}{2} \, d x + \frac{1}{2} \, c\right)^{2} + 2 \, a \tan\left(\frac{1}{2} \, d x + \frac{1}{2} \, c\right) + b\right)}^{3}} - \frac{3 \, {\left(d x + c\right)}}{a^{4}}}{3 \, d}"," ",0,"-1/3*(3*(8*a^6*b - 8*a^4*b^3 + 7*a^2*b^5 - 2*b^7)*(pi*floor(1/2*(d*x + c)/pi + 1/2)*sgn(b) + arctan((b*tan(1/2*d*x + 1/2*c) + a)/sqrt(-a^2 + b^2)))/((a^10 - 3*a^8*b^2 + 3*a^6*b^4 - a^4*b^6)*sqrt(-a^2 + b^2)) + (18*a^5*b^3*tan(1/2*d*x + 1/2*c)^5 - 6*a^3*b^5*tan(1/2*d*x + 1/2*c)^5 + 3*a*b^7*tan(1/2*d*x + 1/2*c)^5 + 84*a^6*b^2*tan(1/2*d*x + 1/2*c)^4 - 12*a^4*b^4*tan(1/2*d*x + 1/2*c)^4 - 3*a^2*b^6*tan(1/2*d*x + 1/2*c)^4 + 6*b^8*tan(1/2*d*x + 1/2*c)^4 + 104*a^7*b*tan(1/2*d*x + 1/2*c)^3 + 88*a^5*b^3*tan(1/2*d*x + 1/2*c)^3 - 78*a^3*b^5*tan(1/2*d*x + 1/2*c)^3 + 36*a*b^7*tan(1/2*d*x + 1/2*c)^3 + 228*a^6*b^2*tan(1/2*d*x + 1/2*c)^2 - 114*a^4*b^4*tan(1/2*d*x + 1/2*c)^2 + 24*a^2*b^6*tan(1/2*d*x + 1/2*c)^2 + 12*b^8*tan(1/2*d*x + 1/2*c)^2 + 138*a^5*b^3*tan(1/2*d*x + 1/2*c) - 96*a^3*b^5*tan(1/2*d*x + 1/2*c) + 33*a*b^7*tan(1/2*d*x + 1/2*c) + 26*a^4*b^4 - 17*a^2*b^6 + 6*b^8)/((a^9 - 3*a^7*b^2 + 3*a^5*b^4 - a^3*b^6)*(b*tan(1/2*d*x + 1/2*c)^2 + 2*a*tan(1/2*d*x + 1/2*c) + b)^3) - 3*(d*x + c)/a^4)/d","B",0
52,1,49,0,0.586074," ","integrate(1/(3+5*csc(d*x+c)),x, algorithm=""giac"")","-\frac{d x + c + 10 \, \arctan\left(-\frac{3 \, \cos\left(d x + c\right) + \sin\left(d x + c\right) + 3}{\cos\left(d x + c\right) - 3 \, \sin\left(d x + c\right) - 9}\right)}{12 \, d}"," ",0,"-1/12*(d*x + c + 10*arctan(-(3*cos(d*x + c) + sin(d*x + c) + 3)/(cos(d*x + c) - 3*sin(d*x + c) - 9)))/d","A",0
53,1,45,0,0.508867," ","integrate(1/(5+3*csc(d*x+c)),x, algorithm=""giac"")","\frac{4 \, d x + 4 \, c - 3 \, \log\left({\left| 3 \, \tan\left(\frac{1}{2} \, d x + \frac{1}{2} \, c\right) + 1 \right|}\right) + 3 \, \log\left({\left| \tan\left(\frac{1}{2} \, d x + \frac{1}{2} \, c\right) + 3 \right|}\right)}{20 \, d}"," ",0,"1/20*(4*d*x + 4*c - 3*log(abs(3*tan(1/2*d*x + 1/2*c) + 1)) + 3*log(abs(tan(1/2*d*x + 1/2*c) + 3)))/d","A",0
54,0,0,0,0.000000," ","integrate(csc(f*x+e)^3*(a+b*csc(f*x+e))^m,x, algorithm=""giac"")","\int {\left(b \csc\left(f x + e\right) + a\right)}^{m} \csc\left(f x + e\right)^{3}\,{d x}"," ",0,"integrate((b*csc(f*x + e) + a)^m*csc(f*x + e)^3, x)","F",0
55,0,0,0,0.000000," ","integrate(csc(f*x+e)^2*(a+b*csc(f*x+e))^m,x, algorithm=""giac"")","\int {\left(b \csc\left(f x + e\right) + a\right)}^{m} \csc\left(f x + e\right)^{2}\,{d x}"," ",0,"integrate((b*csc(f*x + e) + a)^m*csc(f*x + e)^2, x)","F",0
56,0,0,0,0.000000," ","integrate(csc(f*x+e)*(a+b*csc(f*x+e))^m,x, algorithm=""giac"")","\int {\left(b \csc\left(f x + e\right) + a\right)}^{m} \csc\left(f x + e\right)\,{d x}"," ",0,"integrate((b*csc(f*x + e) + a)^m*csc(f*x + e), x)","F",0
57,0,0,0,0.000000," ","integrate((a+b*csc(f*x+e))^m,x, algorithm=""giac"")","\int {\left(b \csc\left(f x + e\right) + a\right)}^{m}\,{d x}"," ",0,"integrate((b*csc(f*x + e) + a)^m, x)","F",0
58,0,0,0,0.000000," ","integrate((a+b*csc(f*x+e))^m*sin(f*x+e),x, algorithm=""giac"")","\int {\left(b \csc\left(f x + e\right) + a\right)}^{m} \sin\left(f x + e\right)\,{d x}"," ",0,"integrate((b*csc(f*x + e) + a)^m*sin(f*x + e), x)","F",0
59,0,0,0,0.000000," ","integrate((a+b*csc(f*x+e))^m*sin(f*x+e)^2,x, algorithm=""giac"")","\int {\left(b \csc\left(f x + e\right) + a\right)}^{m} \sin\left(f x + e\right)^{2}\,{d x}"," ",0,"integrate((b*csc(f*x + e) + a)^m*sin(f*x + e)^2, x)","F",0
