1,1,39,0,0.738288," ","integrate(cos(x)^4/(I+cot(x)),x, algorithm=""fricas"")","\frac{1}{384} \, {\left(-24 i \, x e^{\left(6 i \, x\right)} - 3 \, e^{\left(10 i \, x\right)} - 18 \, e^{\left(8 i \, x\right)} - 12 \, e^{\left(4 i \, x\right)} - 9 \, e^{\left(2 i \, x\right)} - 2\right)} e^{\left(-6 i \, x\right)}"," ",0,"1/384*(-24*I*x*e^(6*I*x) - 3*e^(10*I*x) - 18*e^(8*I*x) - 12*e^(4*I*x) - 9*e^(2*I*x) - 2)*e^(-6*I*x)","A",0
2,1,26,0,0.694104," ","integrate(cos(x)^3/(I+cot(x)),x, algorithm=""fricas"")","-\frac{1}{240} \, {\left(5 \, e^{\left(8 i \, x\right)} + 30 \, e^{\left(6 i \, x\right)} + 10 \, e^{\left(2 i \, x\right)} + 3\right)} e^{\left(-5 i \, x\right)}"," ",0,"-1/240*(5*e^(8*I*x) + 30*e^(6*I*x) + 10*e^(2*I*x) + 3)*e^(-5*I*x)","A",0
3,1,27,0,0.689100," ","integrate(cos(x)^2/(I+cot(x)),x, algorithm=""fricas"")","\frac{1}{32} \, {\left(-4 i \, x e^{\left(4 i \, x\right)} - 2 \, e^{\left(6 i \, x\right)} - 2 \, e^{\left(2 i \, x\right)} - 1\right)} e^{\left(-4 i \, x\right)}"," ",0,"1/32*(-4*I*x*e^(4*I*x) - 2*e^(6*I*x) - 2*e^(2*I*x) - 1)*e^(-4*I*x)","A",0
4,1,14,0,0.555840," ","integrate(cos(x)/(I+cot(x)),x, algorithm=""fricas"")","-\frac{1}{12} \, {\left(3 \, e^{\left(4 i \, x\right)} + 1\right)} e^{\left(-3 i \, x\right)}"," ",0,"-1/12*(3*e^(4*I*x) + 1)*e^(-3*I*x)","A",0
5,1,33,0,0.550911," ","integrate(sec(x)/(I+cot(x)),x, algorithm=""fricas"")","{\left(-i \, e^{\left(i \, x\right)} \log\left(e^{\left(i \, x\right)} + i\right) + i \, e^{\left(i \, x\right)} \log\left(e^{\left(i \, x\right)} - i\right) - 1\right)} e^{\left(-i \, x\right)}"," ",0,"(-I*e^(I*x)*log(e^(I*x) + I) + I*e^(I*x)*log(e^(I*x) - I) - 1)*e^(-I*x)","B",0
6,1,36,0,0.654922," ","integrate(sec(x)^2/(I+cot(x)),x, algorithm=""fricas"")","\frac{2 i \, x e^{\left(2 i \, x\right)} - {\left(e^{\left(2 i \, x\right)} + 1\right)} \log\left(e^{\left(2 i \, x\right)} + 1\right) + 2 i \, x + 2}{e^{\left(2 i \, x\right)} + 1}"," ",0,"(2*I*x*e^(2*I*x) - (e^(2*I*x) + 1)*log(e^(2*I*x) + 1) + 2*I*x + 2)/(e^(2*I*x) + 1)","B",0
7,1,73,0,0.617367," ","integrate(sec(x)^3/(I+cot(x)),x, algorithm=""fricas"")","\frac{{\left(i \, e^{\left(4 i \, x\right)} + 2 i \, e^{\left(2 i \, x\right)} + i\right)} \log\left(e^{\left(i \, x\right)} + i\right) + {\left(-i \, e^{\left(4 i \, x\right)} - 2 i \, e^{\left(2 i \, x\right)} - i\right)} \log\left(e^{\left(i \, x\right)} - i\right) + 2 \, e^{\left(3 i \, x\right)} + 6 \, e^{\left(i \, x\right)}}{2 \, {\left(e^{\left(4 i \, x\right)} + 2 \, e^{\left(2 i \, x\right)} + 1\right)}}"," ",0,"1/2*((I*e^(4*I*x) + 2*I*e^(2*I*x) + I)*log(e^(I*x) + I) + (-I*e^(4*I*x) - 2*I*e^(2*I*x) - I)*log(e^(I*x) - I) + 2*e^(3*I*x) + 6*e^(I*x))/(e^(4*I*x) + 2*e^(2*I*x) + 1)","B",0
8,1,30,0,0.950598," ","integrate(sec(x)^4/(I+cot(x)),x, algorithm=""fricas"")","\frac{2 \, {\left(3 \, e^{\left(2 i \, x\right)} - 1\right)}}{3 \, {\left(e^{\left(6 i \, x\right)} + 3 \, e^{\left(4 i \, x\right)} + 3 \, e^{\left(2 i \, x\right)} + 1\right)}}"," ",0,"2/3*(3*e^(2*I*x) - 1)/(e^(6*I*x) + 3*e^(4*I*x) + 3*e^(2*I*x) + 1)","B",0
9,1,121,0,0.771697," ","integrate(sec(x)^5/(I+cot(x)),x, algorithm=""fricas"")","\frac{{\left(3 i \, e^{\left(8 i \, x\right)} + 12 i \, e^{\left(6 i \, x\right)} + 18 i \, e^{\left(4 i \, x\right)} + 12 i \, e^{\left(2 i \, x\right)} + 3 i\right)} \log\left(e^{\left(i \, x\right)} + i\right) + {\left(-3 i \, e^{\left(8 i \, x\right)} - 12 i \, e^{\left(6 i \, x\right)} - 18 i \, e^{\left(4 i \, x\right)} - 12 i \, e^{\left(2 i \, x\right)} - 3 i\right)} \log\left(e^{\left(i \, x\right)} - i\right) + 6 \, e^{\left(7 i \, x\right)} + 22 \, e^{\left(5 i \, x\right)} + 106 \, e^{\left(3 i \, x\right)} - 6 \, e^{\left(i \, x\right)}}{24 \, {\left(e^{\left(8 i \, x\right)} + 4 \, e^{\left(6 i \, x\right)} + 6 \, e^{\left(4 i \, x\right)} + 4 \, e^{\left(2 i \, x\right)} + 1\right)}}"," ",0,"1/24*((3*I*e^(8*I*x) + 12*I*e^(6*I*x) + 18*I*e^(4*I*x) + 12*I*e^(2*I*x) + 3*I)*log(e^(I*x) + I) + (-3*I*e^(8*I*x) - 12*I*e^(6*I*x) - 18*I*e^(4*I*x) - 12*I*e^(2*I*x) - 3*I)*log(e^(I*x) - I) + 6*e^(7*I*x) + 22*e^(5*I*x) + 106*e^(3*I*x) - 6*e^(I*x))/(e^(8*I*x) + 4*e^(6*I*x) + 6*e^(4*I*x) + 4*e^(2*I*x) + 1)","B",0
10,1,48,0,0.802867," ","integrate(sec(x)^6/(I+cot(x)),x, algorithm=""fricas"")","\frac{4 \, {\left(20 \, e^{\left(4 i \, x\right)} - 5 \, e^{\left(2 i \, x\right)} - 1\right)}}{15 \, {\left(e^{\left(10 i \, x\right)} + 5 \, e^{\left(8 i \, x\right)} + 10 \, e^{\left(6 i \, x\right)} + 10 \, e^{\left(4 i \, x\right)} + 5 \, e^{\left(2 i \, x\right)} + 1\right)}}"," ",0,"4/15*(20*e^(4*I*x) - 5*e^(2*I*x) - 1)/(e^(10*I*x) + 5*e^(8*I*x) + 10*e^(6*I*x) + 10*e^(4*I*x) + 5*e^(2*I*x) + 1)","A",0
11,1,178,0,0.704379," ","integrate(cos(x)^4/(a+b*cot(x)),x, algorithm=""fricas"")","-\frac{4 \, a^{4} b \log\left(2 \, a b \cos\left(x\right) \sin\left(x\right) - {\left(a^{2} - b^{2}\right)} \cos\left(x\right)^{2} + a^{2}\right) + 2 \, {\left(a^{4} b + 2 \, a^{2} b^{3} + b^{5}\right)} \cos\left(x\right)^{4} + 4 \, {\left(a^{4} b + a^{2} b^{3}\right)} \cos\left(x\right)^{2} - {\left(3 \, a^{5} - 6 \, a^{3} b^{2} - a b^{4}\right)} x - {\left(2 \, {\left(a^{5} + 2 \, a^{3} b^{2} + a b^{4}\right)} \cos\left(x\right)^{3} + {\left(3 \, a^{5} + 2 \, a^{3} b^{2} - a b^{4}\right)} \cos\left(x\right)\right)} \sin\left(x\right)}{8 \, {\left(a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}\right)}}"," ",0,"-1/8*(4*a^4*b*log(2*a*b*cos(x)*sin(x) - (a^2 - b^2)*cos(x)^2 + a^2) + 2*(a^4*b + 2*a^2*b^3 + b^5)*cos(x)^4 + 4*(a^4*b + a^2*b^3)*cos(x)^2 - (3*a^5 - 6*a^3*b^2 - a*b^4)*x - (2*(a^5 + 2*a^3*b^2 + a*b^4)*cos(x)^3 + (3*a^5 + 2*a^3*b^2 - a*b^4)*cos(x))*sin(x))/(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)","A",0
12,1,214,0,0.618930," ","integrate(cos(x)^3/(a+b*cot(x)),x, algorithm=""fricas"")","\frac{3 \, \sqrt{a^{2} + b^{2}} a^{3} b \log\left(\frac{2 \, a b \cos\left(x\right) \sin\left(x\right) - {\left(a^{2} - b^{2}\right)} \cos\left(x\right)^{2} - a^{2} - 2 \, b^{2} - 2 \, \sqrt{a^{2} + b^{2}} {\left(a \cos\left(x\right) - b \sin\left(x\right)\right)}}{2 \, a b \cos\left(x\right) \sin\left(x\right) - {\left(a^{2} - b^{2}\right)} \cos\left(x\right)^{2} + a^{2}}\right) - 2 \, {\left(a^{4} b + 2 \, a^{2} b^{3} + b^{5}\right)} \cos\left(x\right)^{3} - 6 \, {\left(a^{4} b + a^{2} b^{3}\right)} \cos\left(x\right) + 2 \, {\left(2 \, a^{5} + a^{3} b^{2} - a b^{4} + {\left(a^{5} + 2 \, 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)}}"," ",0,"1/6*(3*sqrt(a^2 + b^2)*a^3*b*log((2*a*b*cos(x)*sin(x) - (a^2 - b^2)*cos(x)^2 - a^2 - 2*b^2 - 2*sqrt(a^2 + b^2)*(a*cos(x) - b*sin(x)))/(2*a*b*cos(x)*sin(x) - (a^2 - b^2)*cos(x)^2 + a^2)) - 2*(a^4*b + 2*a^2*b^3 + b^5)*cos(x)^3 - 6*(a^4*b + a^2*b^3)*cos(x) + 2*(2*a^5 + a^3*b^2 - a*b^4 + (a^5 + 2*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)","A",0
13,1,95,0,0.644964," ","integrate(cos(x)^2/(a+b*cot(x)),x, algorithm=""fricas"")","-\frac{a^{2} b \log\left(2 \, a b \cos\left(x\right) \sin\left(x\right) - {\left(a^{2} - b^{2}\right)} \cos\left(x\right)^{2} + a^{2}\right) + {\left(a^{2} b + b^{3}\right)} \cos\left(x\right)^{2} - {\left(a^{3} + a b^{2}\right)} \cos\left(x\right) \sin\left(x\right) - {\left(a^{3} - a b^{2}\right)} x}{2 \, {\left(a^{4} + 2 \, a^{2} b^{2} + b^{4}\right)}}"," ",0,"-1/2*(a^2*b*log(2*a*b*cos(x)*sin(x) - (a^2 - b^2)*cos(x)^2 + a^2) + (a^2*b + b^3)*cos(x)^2 - (a^3 + a*b^2)*cos(x)*sin(x) - (a^3 - a*b^2)*x)/(a^4 + 2*a^2*b^2 + b^4)","A",0
14,1,144,0,0.693660," ","integrate(cos(x)/(a+b*cot(x)),x, algorithm=""fricas"")","\frac{\sqrt{a^{2} + b^{2}} a b \log\left(\frac{2 \, a b \cos\left(x\right) \sin\left(x\right) - {\left(a^{2} - b^{2}\right)} \cos\left(x\right)^{2} - a^{2} - 2 \, b^{2} - 2 \, \sqrt{a^{2} + b^{2}} {\left(a \cos\left(x\right) - b \sin\left(x\right)\right)}}{2 \, a b \cos\left(x\right) \sin\left(x\right) - {\left(a^{2} - b^{2}\right)} \cos\left(x\right)^{2} + a^{2}}\right) - 2 \, {\left(a^{2} b + b^{3}\right)} \cos\left(x\right) + 2 \, {\left(a^{3} + a b^{2}\right)} \sin\left(x\right)}{2 \, {\left(a^{4} + 2 \, a^{2} b^{2} + b^{4}\right)}}"," ",0,"1/2*(sqrt(a^2 + b^2)*a*b*log((2*a*b*cos(x)*sin(x) - (a^2 - b^2)*cos(x)^2 - a^2 - 2*b^2 - 2*sqrt(a^2 + b^2)*(a*cos(x) - b*sin(x)))/(2*a*b*cos(x)*sin(x) - (a^2 - b^2)*cos(x)^2 + a^2)) - 2*(a^2*b + b^3)*cos(x) + 2*(a^3 + a*b^2)*sin(x))/(a^4 + 2*a^2*b^2 + b^4)","B",0
15,1,140,0,0.682007," ","integrate(sec(x)/(a+b*cot(x)),x, algorithm=""fricas"")","\frac{\sqrt{a^{2} + b^{2}} b \log\left(\frac{2 \, a b \cos\left(x\right) \sin\left(x\right) - {\left(a^{2} - b^{2}\right)} \cos\left(x\right)^{2} - a^{2} - 2 \, b^{2} - 2 \, \sqrt{a^{2} + b^{2}} {\left(a \cos\left(x\right) - b \sin\left(x\right)\right)}}{2 \, a b \cos\left(x\right) \sin\left(x\right) - {\left(a^{2} - b^{2}\right)} \cos\left(x\right)^{2} + a^{2}}\right) + {\left(a^{2} + b^{2}\right)} \log\left(\sin\left(x\right) + 1\right) - {\left(a^{2} + b^{2}\right)} \log\left(-\sin\left(x\right) + 1\right)}{2 \, {\left(a^{3} + a b^{2}\right)}}"," ",0,"1/2*(sqrt(a^2 + b^2)*b*log((2*a*b*cos(x)*sin(x) - (a^2 - b^2)*cos(x)^2 - a^2 - 2*b^2 - 2*sqrt(a^2 + b^2)*(a*cos(x) - b*sin(x)))/(2*a*b*cos(x)*sin(x) - (a^2 - b^2)*cos(x)^2 + a^2)) + (a^2 + b^2)*log(sin(x) + 1) - (a^2 + b^2)*log(-sin(x) + 1))/(a^3 + a*b^2)","B",0
16,1,57,0,0.713398," ","integrate(sec(x)^2/(a+b*cot(x)),x, algorithm=""fricas"")","-\frac{b \cos\left(x\right) \log\left(2 \, a b \cos\left(x\right) \sin\left(x\right) - {\left(a^{2} - b^{2}\right)} \cos\left(x\right)^{2} + a^{2}\right) - b \cos\left(x\right) \log\left(\cos\left(x\right)^{2}\right) - 2 \, a \sin\left(x\right)}{2 \, a^{2} \cos\left(x\right)}"," ",0,"-1/2*(b*cos(x)*log(2*a*b*cos(x)*sin(x) - (a^2 - b^2)*cos(x)^2 + a^2) - b*cos(x)*log(cos(x)^2) - 2*a*sin(x))/(a^2*cos(x))","A",0
17,1,166,0,0.719711," ","integrate(sec(x)^3/(a+b*cot(x)),x, algorithm=""fricas"")","\frac{2 \, \sqrt{a^{2} + b^{2}} b \cos\left(x\right)^{2} \log\left(\frac{2 \, a b \cos\left(x\right) \sin\left(x\right) - {\left(a^{2} - b^{2}\right)} \cos\left(x\right)^{2} - a^{2} - 2 \, b^{2} - 2 \, \sqrt{a^{2} + b^{2}} {\left(a \cos\left(x\right) - b \sin\left(x\right)\right)}}{2 \, a b \cos\left(x\right) \sin\left(x\right) - {\left(a^{2} - b^{2}\right)} \cos\left(x\right)^{2} + a^{2}}\right) + {\left(a^{2} + 2 \, b^{2}\right)} \cos\left(x\right)^{2} \log\left(\sin\left(x\right) + 1\right) - {\left(a^{2} + 2 \, b^{2}\right)} \cos\left(x\right)^{2} \log\left(-\sin\left(x\right) + 1\right) - 4 \, a b \cos\left(x\right) + 2 \, a^{2} \sin\left(x\right)}{4 \, a^{3} \cos\left(x\right)^{2}}"," ",0,"1/4*(2*sqrt(a^2 + b^2)*b*cos(x)^2*log((2*a*b*cos(x)*sin(x) - (a^2 - b^2)*cos(x)^2 - a^2 - 2*b^2 - 2*sqrt(a^2 + b^2)*(a*cos(x) - b*sin(x)))/(2*a*b*cos(x)*sin(x) - (a^2 - b^2)*cos(x)^2 + a^2)) + (a^2 + 2*b^2)*cos(x)^2*log(sin(x) + 1) - (a^2 + 2*b^2)*cos(x)^2*log(-sin(x) + 1) - 4*a*b*cos(x) + 2*a^2*sin(x))/(a^3*cos(x)^2)","B",0
18,1,106,0,1.006196," ","integrate(sec(x)^4/(a+b*cot(x)),x, algorithm=""fricas"")","-\frac{3 \, {\left(a^{2} b + b^{3}\right)} \cos\left(x\right)^{3} \log\left(2 \, a b \cos\left(x\right) \sin\left(x\right) - {\left(a^{2} - b^{2}\right)} \cos\left(x\right)^{2} + a^{2}\right) - 3 \, {\left(a^{2} b + b^{3}\right)} \cos\left(x\right)^{3} \log\left(\cos\left(x\right)^{2}\right) + 3 \, a^{2} b \cos\left(x\right) - 2 \, {\left(a^{3} + {\left(2 \, a^{3} + 3 \, a b^{2}\right)} \cos\left(x\right)^{2}\right)} \sin\left(x\right)}{6 \, a^{4} \cos\left(x\right)^{3}}"," ",0,"-1/6*(3*(a^2*b + b^3)*cos(x)^3*log(2*a*b*cos(x)*sin(x) - (a^2 - b^2)*cos(x)^2 + a^2) - 3*(a^2*b + b^3)*cos(x)^3*log(cos(x)^2) + 3*a^2*b*cos(x) - 2*(a^3 + (2*a^3 + 3*a*b^2)*cos(x)^2)*sin(x))/(a^4*cos(x)^3)","A",0
19,1,66,0,0.757064," ","integrate(sec(x)/(1+2*cot(x)),x, algorithm=""fricas"")","\frac{1}{5} \, \sqrt{5} \log\left(\frac{2 \, {\left(3 \, \cos\left(x\right)^{2} + 4 \, {\left(\sqrt{5} + \cos\left(x\right)\right)} \sin\left(x\right) - 2 \, \sqrt{5} \cos\left(x\right) - 9\right)}}{3 \, \cos\left(x\right)^{2} + 4 \, \cos\left(x\right) \sin\left(x\right) + 1}\right) + \frac{1}{2} \, \log\left(\sin\left(x\right) + 1\right) - \frac{1}{2} \, \log\left(-\sin\left(x\right) + 1\right)"," ",0,"1/5*sqrt(5)*log(2*(3*cos(x)^2 + 4*(sqrt(5) + cos(x))*sin(x) - 2*sqrt(5)*cos(x) - 9)/(3*cos(x)^2 + 4*cos(x)*sin(x) + 1)) + 1/2*log(sin(x) + 1) - 1/2*log(-sin(x) + 1)","B",0
