1,1,63,0,0.386837," ","integrate(cos(x)^4/(I+cot(x)),x, algorithm=""giac"")","\frac{3 \, \tan\left(x\right)^{2} + 10 i \, \tan\left(x\right) - 9}{64 \, {\left(-i \, \tan\left(x\right) + 1\right)}^{2}} + \frac{11 \, \tan\left(x\right)^{3} - 33 i \, \tan\left(x\right)^{2} - 27 \, \tan\left(x\right) - 3 i}{192 \, {\left(\tan\left(x\right) - i\right)}^{3}} + \frac{1}{32} \, \log\left(\tan\left(x\right) + i\right) - \frac{1}{32} \, \log\left(\tan\left(x\right) - i\right)"," ",0,"1/64*(3*tan(x)^2 + 10*I*tan(x) - 9)/(-I*tan(x) + 1)^2 + 1/192*(11*tan(x)^3 - 33*I*tan(x)^2 - 27*tan(x) - 3*I)/(tan(x) - I)^3 + 1/32*log(tan(x) + I) - 1/32*log(tan(x) - I)","A",0
2,1,69,0,0.768174," ","integrate(cos(x)^3/(I+cot(x)),x, algorithm=""giac"")","-\frac{9 i \, \tan\left(\frac{1}{2} \, x\right)^{2} - 12 \, \tan\left(\frac{1}{2} \, x\right) - 7 i}{24 \, {\left(\tan\left(\frac{1}{2} \, x\right) + i\right)}^{3}} - \frac{-45 i \, \tan\left(\frac{1}{2} \, x\right)^{4} - 60 \, \tan\left(\frac{1}{2} \, x\right)^{3} + 70 i \, \tan\left(\frac{1}{2} \, x\right)^{2} + 20 \, \tan\left(\frac{1}{2} \, x\right) - 13 i}{120 \, {\left(\tan\left(\frac{1}{2} \, x\right) - i\right)}^{5}}"," ",0,"-1/24*(9*I*tan(1/2*x)^2 - 12*tan(1/2*x) - 7*I)/(tan(1/2*x) + I)^3 - 1/120*(-45*I*tan(1/2*x)^4 - 60*tan(1/2*x)^3 + 70*I*tan(1/2*x)^2 + 20*tan(1/2*x) - 13*I)/(tan(1/2*x) - I)^5","B",0
3,1,51,0,0.500465," ","integrate(cos(x)^2/(I+cot(x)),x, algorithm=""giac"")","-\frac{-i \, \tan\left(x\right) + 3}{16 \, {\left(-i \, \tan\left(x\right) + 1\right)}} + \frac{3 \, \tan\left(x\right)^{2} - 6 i \, \tan\left(x\right) + 1}{32 \, {\left(\tan\left(x\right) - i\right)}^{2}} + \frac{1}{16} \, \log\left(\tan\left(x\right) + i\right) - \frac{1}{16} \, \log\left(\tan\left(x\right) - i\right)"," ",0,"-1/16*(-I*tan(x) + 3)/(-I*tan(x) + 1) + 1/32*(3*tan(x)^2 - 6*I*tan(x) + 1)/(tan(x) - I)^2 + 1/16*log(tan(x) + I) - 1/16*log(tan(x) - I)","A",0
4,1,31,0,0.371843," ","integrate(cos(x)/(I+cot(x)),x, algorithm=""giac"")","-\frac{i}{2 \, {\left(\tan\left(\frac{1}{2} \, x\right) + i\right)}} - \frac{-3 i \, \tan\left(\frac{1}{2} \, x\right)^{2} + i}{6 \, {\left(\tan\left(\frac{1}{2} \, x\right) - i\right)}^{3}}"," ",0,"-1/2*I/(tan(1/2*x) + I) - 1/6*(-3*I*tan(1/2*x)^2 + I)/(tan(1/2*x) - I)^3","B",0
5,1,29,0,0.388951," ","integrate(sec(x)/(I+cot(x)),x, algorithm=""giac"")","\frac{2 i}{\tan\left(\frac{1}{2} \, x\right) - i} - i \, \log\left(\tan\left(\frac{1}{2} \, x\right) + 1\right) + i \, \log\left(\tan\left(\frac{1}{2} \, x\right) - 1\right)"," ",0,"2*I/(tan(1/2*x) - I) - I*log(tan(1/2*x) + 1) + I*log(tan(1/2*x) - 1)","B",0
6,1,53,0,0.415662," ","integrate(sec(x)^2/(I+cot(x)),x, algorithm=""giac"")","\frac{\tan\left(\frac{1}{2} \, x\right)^{2} + 2 i \, \tan\left(\frac{1}{2} \, x\right) - 1}{\tan\left(\frac{1}{2} \, x\right)^{2} - 1} - \log\left(\tan\left(\frac{1}{2} \, x\right) + 1\right) + 2 \, \log\left(\tan\left(\frac{1}{2} \, x\right) - i\right) - \log\left(\tan\left(\frac{1}{2} \, x\right) - 1\right)"," ",0,"(tan(1/2*x)^2 + 2*I*tan(1/2*x) - 1)/(tan(1/2*x)^2 - 1) - log(tan(1/2*x) + 1) + 2*log(tan(1/2*x) - I) - log(tan(1/2*x) - 1)","B",0
7,1,55,0,0.412982," ","integrate(sec(x)^3/(I+cot(x)),x, algorithm=""giac"")","-\frac{i \, \tan\left(\frac{1}{2} \, x\right)^{3} + 2 \, \tan\left(\frac{1}{2} \, x\right)^{2} + i \, \tan\left(\frac{1}{2} \, x\right) - 2}{{\left(\tan\left(\frac{1}{2} \, x\right)^{2} - 1\right)}^{2}} + \frac{1}{2} i \, \log\left(\tan\left(\frac{1}{2} \, x\right) + 1\right) - \frac{1}{2} i \, \log\left(\tan\left(\frac{1}{2} \, x\right) - 1\right)"," ",0,"-(I*tan(1/2*x)^3 + 2*tan(1/2*x)^2 + I*tan(1/2*x) - 2)/(tan(1/2*x)^2 - 1)^2 + 1/2*I*log(tan(1/2*x) + 1) - 1/2*I*log(tan(1/2*x) - 1)","B",0
8,1,13,0,0.377488," ","integrate(sec(x)^4/(I+cot(x)),x, algorithm=""giac"")","-\frac{1}{3} i \, \tan\left(x\right)^{3} + \frac{1}{2} \, \tan\left(x\right)^{2}"," ",0,"-1/3*I*tan(x)^3 + 1/2*tan(x)^2","A",0
9,1,87,0,0.526754," ","integrate(sec(x)^5/(I+cot(x)),x, algorithm=""giac"")","-\frac{3 i \, \tan\left(\frac{1}{2} \, x\right)^{7} + 24 \, \tan\left(\frac{1}{2} \, x\right)^{6} + 21 i \, \tan\left(\frac{1}{2} \, x\right)^{5} - 24 \, \tan\left(\frac{1}{2} \, x\right)^{4} + 21 i \, \tan\left(\frac{1}{2} \, x\right)^{3} + 8 \, \tan\left(\frac{1}{2} \, x\right)^{2} + 3 i \, \tan\left(\frac{1}{2} \, x\right) - 8}{12 \, {\left(\tan\left(\frac{1}{2} \, x\right)^{2} - 1\right)}^{4}} + \frac{1}{8} i \, \log\left(\tan\left(\frac{1}{2} \, x\right) + 1\right) - \frac{1}{8} i \, \log\left(\tan\left(\frac{1}{2} \, x\right) - 1\right)"," ",0,"-1/12*(3*I*tan(1/2*x)^7 + 24*tan(1/2*x)^6 + 21*I*tan(1/2*x)^5 - 24*tan(1/2*x)^4 + 21*I*tan(1/2*x)^3 + 8*tan(1/2*x)^2 + 3*I*tan(1/2*x) - 8)/(tan(1/2*x)^2 - 1)^4 + 1/8*I*log(tan(1/2*x) + 1) - 1/8*I*log(tan(1/2*x) - 1)","B",0
10,1,25,0,0.620027," ","integrate(sec(x)^6/(I+cot(x)),x, algorithm=""giac"")","-\frac{1}{5} i \, \tan\left(x\right)^{5} + \frac{1}{4} \, \tan\left(x\right)^{4} - \frac{1}{3} i \, \tan\left(x\right)^{3} + \frac{1}{2} \, \tan\left(x\right)^{2}"," ",0,"-1/5*I*tan(x)^5 + 1/4*tan(x)^4 - 1/3*I*tan(x)^3 + 1/2*tan(x)^2","A",0
11,1,270,0,0.433685," ","integrate(cos(x)^4/(a+b*cot(x)),x, algorithm=""giac"")","-\frac{a^{5} b \log\left({\left| a \tan\left(x\right) + b \right|}\right)}{a^{7} + 3 \, a^{5} b^{2} + 3 \, a^{3} b^{4} + a b^{6}} + \frac{a^{4} b \log\left(\tan\left(x\right)^{2} + 1\right)}{2 \, {\left(a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}\right)}} + \frac{{\left(3 \, a^{5} - 6 \, a^{3} b^{2} - a b^{4}\right)} x}{8 \, {\left(a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}\right)}} - \frac{6 \, a^{4} b \tan\left(x\right)^{4} - 3 \, a^{5} \tan\left(x\right)^{3} - 2 \, a^{3} b^{2} \tan\left(x\right)^{3} + a b^{4} \tan\left(x\right)^{3} + 16 \, a^{4} b \tan\left(x\right)^{2} + 4 \, a^{2} b^{3} \tan\left(x\right)^{2} - 5 \, a^{5} \tan\left(x\right) - 6 \, a^{3} b^{2} \tan\left(x\right) - a b^{4} \tan\left(x\right) + 12 \, a^{4} b + 8 \, a^{2} b^{3} + 2 \, b^{5}}{8 \, {\left(a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}\right)} {\left(\tan\left(x\right)^{2} + 1\right)}^{2}}"," ",0,"-a^5*b*log(abs(a*tan(x) + b))/(a^7 + 3*a^5*b^2 + 3*a^3*b^4 + a*b^6) + 1/2*a^4*b*log(tan(x)^2 + 1)/(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6) + 1/8*(3*a^5 - 6*a^3*b^2 - a*b^4)*x/(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6) - 1/8*(6*a^4*b*tan(x)^4 - 3*a^5*tan(x)^3 - 2*a^3*b^2*tan(x)^3 + a*b^4*tan(x)^3 + 16*a^4*b*tan(x)^2 + 4*a^2*b^3*tan(x)^2 - 5*a^5*tan(x) - 6*a^3*b^2*tan(x) - a*b^4*tan(x) + 12*a^4*b + 8*a^2*b^3 + 2*b^5)/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*(tan(x)^2 + 1)^2)","B",0
12,1,201,0,0.745138," ","integrate(cos(x)^3/(a+b*cot(x)),x, algorithm=""giac"")","\frac{a^{3} b \log\left(\frac{{\left| 2 \, b \tan\left(\frac{1}{2} \, x\right) - 2 \, a - 2 \, \sqrt{a^{2} + b^{2}} \right|}}{{\left| 2 \, b \tan\left(\frac{1}{2} \, x\right) - 2 \, a + 2 \, \sqrt{a^{2} + b^{2}} \right|}}\right)}{{\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,"a^3*b*log(abs(2*b*tan(1/2*x) - 2*a - 2*sqrt(a^2 + b^2))/abs(2*b*tan(1/2*x) - 2*a + 2*sqrt(a^2 + b^2)))/((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
13,1,155,0,0.383009," ","integrate(cos(x)^2/(a+b*cot(x)),x, algorithm=""giac"")","-\frac{a^{3} b \log\left({\left| a \tan\left(x\right) + b \right|}\right)}{a^{5} + 2 \, a^{3} b^{2} + a b^{4}} + \frac{a^{2} b \log\left(\tan\left(x\right)^{2} + 1\right)}{2 \, {\left(a^{4} + 2 \, a^{2} b^{2} + b^{4}\right)}} + \frac{{\left(a^{3} - a b^{2}\right)} x}{2 \, {\left(a^{4} + 2 \, a^{2} b^{2} + b^{4}\right)}} - \frac{a^{2} b \tan\left(x\right)^{2} - a^{3} \tan\left(x\right) - a b^{2} \tan\left(x\right) + 2 \, a^{2} b + b^{3}}{2 \, {\left(a^{4} + 2 \, a^{2} b^{2} + b^{4}\right)} {\left(\tan\left(x\right)^{2} + 1\right)}}"," ",0,"-a^3*b*log(abs(a*tan(x) + b))/(a^5 + 2*a^3*b^2 + a*b^4) + 1/2*a^2*b*log(tan(x)^2 + 1)/(a^4 + 2*a^2*b^2 + b^4) + 1/2*(a^3 - a*b^2)*x/(a^4 + 2*a^2*b^2 + b^4) - 1/2*(a^2*b*tan(x)^2 - a^3*tan(x) - a*b^2*tan(x) + 2*a^2*b + b^3)/((a^4 + 2*a^2*b^2 + b^4)*(tan(x)^2 + 1))","B",0
14,1,94,0,0.458815," ","integrate(cos(x)/(a+b*cot(x)),x, algorithm=""giac"")","\frac{a b \log\left(\frac{{\left| 2 \, b \tan\left(\frac{1}{2} \, x\right) - 2 \, a - 2 \, \sqrt{a^{2} + b^{2}} \right|}}{{\left| 2 \, b \tan\left(\frac{1}{2} \, x\right) - 2 \, a + 2 \, \sqrt{a^{2} + b^{2}} \right|}}\right)}{{\left(a^{2} + b^{2}\right)}^{\frac{3}{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,"a*b*log(abs(2*b*tan(1/2*x) - 2*a - 2*sqrt(a^2 + b^2))/abs(2*b*tan(1/2*x) - 2*a + 2*sqrt(a^2 + b^2)))/(a^2 + b^2)^(3/2) + 2*(a*tan(1/2*x) - b)/((a^2 + b^2)*(tan(1/2*x)^2 + 1))","A",0
15,1,90,0,1.278990," ","integrate(sec(x)/(a+b*cot(x)),x, algorithm=""giac"")","\frac{b \log\left(\frac{{\left| 2 \, b \tan\left(\frac{1}{2} \, x\right) - 2 \, a - 2 \, \sqrt{a^{2} + b^{2}} \right|}}{{\left| 2 \, b \tan\left(\frac{1}{2} \, x\right) - 2 \, a + 2 \, \sqrt{a^{2} + b^{2}} \right|}}\right)}{\sqrt{a^{2} + b^{2}} a} + \frac{\log\left({\left| \tan\left(\frac{1}{2} \, x\right) + 1 \right|}\right)}{a} - \frac{\log\left({\left| \tan\left(\frac{1}{2} \, x\right) - 1 \right|}\right)}{a}"," ",0,"b*log(abs(2*b*tan(1/2*x) - 2*a - 2*sqrt(a^2 + b^2))/abs(2*b*tan(1/2*x) - 2*a + 2*sqrt(a^2 + b^2)))/(sqrt(a^2 + b^2)*a) + log(abs(tan(1/2*x) + 1))/a - log(abs(tan(1/2*x) - 1))/a","B",0
16,1,21,0,0.423996," ","integrate(sec(x)^2/(a+b*cot(x)),x, algorithm=""giac"")","-\frac{b \log\left({\left| a \tan\left(x\right) + b \right|}\right)}{a^{2}} + \frac{\tan\left(x\right)}{a}"," ",0,"-b*log(abs(a*tan(x) + b))/a^2 + tan(x)/a","A",0
17,1,158,0,1.014021," ","integrate(sec(x)^3/(a+b*cot(x)),x, algorithm=""giac"")","\frac{{\left(a^{2} + 2 \, b^{2}\right)} \log\left({\left| \tan\left(\frac{1}{2} \, x\right) + 1 \right|}\right)}{2 \, a^{3}} - \frac{{\left(a^{2} + 2 \, b^{2}\right)} \log\left({\left| \tan\left(\frac{1}{2} \, x\right) - 1 \right|}\right)}{2 \, a^{3}} + \frac{{\left(a^{2} b + b^{3}\right)} \log\left(\frac{{\left| 2 \, b \tan\left(\frac{1}{2} \, x\right) - 2 \, a - 2 \, \sqrt{a^{2} + b^{2}} \right|}}{{\left| 2 \, b \tan\left(\frac{1}{2} \, x\right) - 2 \, a + 2 \, \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)*log(abs(tan(1/2*x) + 1))/a^3 - 1/2*(a^2 + 2*b^2)*log(abs(tan(1/2*x) - 1))/a^3 + (a^2*b + b^3)*log(abs(2*b*tan(1/2*x) - 2*a - 2*sqrt(a^2 + b^2))/abs(2*b*tan(1/2*x) - 2*a + 2*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)","B",0
18,1,60,0,0.352053," ","integrate(sec(x)^4/(a+b*cot(x)),x, algorithm=""giac"")","\frac{2 \, a^{2} \tan\left(x\right)^{3} - 3 \, a b \tan\left(x\right)^{2} + 6 \, a^{2} \tan\left(x\right) + 6 \, b^{2} \tan\left(x\right)}{6 \, a^{3}} - \frac{{\left(a^{2} b + b^{3}\right)} \log\left({\left| a \tan\left(x\right) + b \right|}\right)}{a^{4}}"," ",0,"1/6*(2*a^2*tan(x)^3 - 3*a*b*tan(x)^2 + 6*a^2*tan(x) + 6*b^2*tan(x))/a^3 - (a^2*b + b^3)*log(abs(a*tan(x) + b))/a^4","A",0
19,1,54,0,0.518250," ","integrate(sec(x)/(1+2*cot(x)),x, algorithm=""giac"")","\frac{2}{5} \, \sqrt{5} \log\left(\frac{{\left| -\sqrt{5} + 2 \, \tan\left(\frac{1}{2} \, x\right) - 1 \right|}}{{\left| \sqrt{5} + 2 \, \tan\left(\frac{1}{2} \, x\right) - 1 \right|}}\right) + \log\left({\left| \tan\left(\frac{1}{2} \, x\right) + 1 \right|}\right) - \log\left({\left| \tan\left(\frac{1}{2} \, x\right) - 1 \right|}\right)"," ",0,"2/5*sqrt(5)*log(abs(-sqrt(5) + 2*tan(1/2*x) - 1)/abs(sqrt(5) + 2*tan(1/2*x) - 1)) + log(abs(tan(1/2*x) + 1)) - log(abs(tan(1/2*x) - 1))","B",0
