1,1,1236,0,1.148728," ","integrate((A+C*cot(d*x+c)^2)/(b*tan(d*x+c))^(1/2),x, algorithm=""fricas"")","\frac{8 \, C \sqrt{\frac{b \sin\left(d x + c\right)}{\cos\left(d x + c\right)}} \cos\left(d x + c\right)^{2} + 12 \, {\left(\sqrt{2} b d \cos\left(d x + c\right)^{2} - \sqrt{2} b d\right)} \left(\frac{A^{4} - 4 \, A^{3} C + 6 \, A^{2} C^{2} - 4 \, A C^{3} + C^{4}}{b^{2} d^{4}}\right)^{\frac{1}{4}} \arctan\left(\frac{\sqrt{2} {\left(A - C\right)} b d^{3} \sqrt{\frac{b \sin\left(d x + c\right)}{\cos\left(d x + c\right)}} \left(\frac{A^{4} - 4 \, A^{3} C + 6 \, A^{2} C^{2} - 4 \, A C^{3} + C^{4}}{b^{2} d^{4}}\right)^{\frac{3}{4}} + \sqrt{2} b d^{3} \sqrt{\frac{b^{2} d^{2} \sqrt{\frac{A^{4} - 4 \, A^{3} C + 6 \, A^{2} C^{2} - 4 \, A C^{3} + C^{4}}{b^{2} d^{4}}} \cos\left(d x + c\right) + \sqrt{2} {\left(A - C\right)} b d \sqrt{\frac{b \sin\left(d x + c\right)}{\cos\left(d x + c\right)}} \left(\frac{A^{4} - 4 \, A^{3} C + 6 \, A^{2} C^{2} - 4 \, A C^{3} + C^{4}}{b^{2} d^{4}}\right)^{\frac{1}{4}} \cos\left(d x + c\right) + {\left(A^{2} - 2 \, A C + C^{2}\right)} b \sin\left(d x + c\right)}{\cos\left(d x + c\right)}} \left(\frac{A^{4} - 4 \, A^{3} C + 6 \, A^{2} C^{2} - 4 \, A C^{3} + C^{4}}{b^{2} d^{4}}\right)^{\frac{3}{4}} + A^{4} - 4 \, A^{3} C + 6 \, A^{2} C^{2} - 4 \, A C^{3} + C^{4}}{A^{4} - 4 \, A^{3} C + 6 \, A^{2} C^{2} - 4 \, A C^{3} + C^{4}}\right) + 12 \, {\left(\sqrt{2} b d \cos\left(d x + c\right)^{2} - \sqrt{2} b d\right)} \left(\frac{A^{4} - 4 \, A^{3} C + 6 \, A^{2} C^{2} - 4 \, A C^{3} + C^{4}}{b^{2} d^{4}}\right)^{\frac{1}{4}} \arctan\left(\frac{\sqrt{2} {\left(A - C\right)} b d^{3} \sqrt{\frac{b \sin\left(d x + c\right)}{\cos\left(d x + c\right)}} \left(\frac{A^{4} - 4 \, A^{3} C + 6 \, A^{2} C^{2} - 4 \, A C^{3} + C^{4}}{b^{2} d^{4}}\right)^{\frac{3}{4}} + \sqrt{2} b d^{3} \sqrt{\frac{b^{2} d^{2} \sqrt{\frac{A^{4} - 4 \, A^{3} C + 6 \, A^{2} C^{2} - 4 \, A C^{3} + C^{4}}{b^{2} d^{4}}} \cos\left(d x + c\right) - \sqrt{2} {\left(A - C\right)} b d \sqrt{\frac{b \sin\left(d x + c\right)}{\cos\left(d x + c\right)}} \left(\frac{A^{4} - 4 \, A^{3} C + 6 \, A^{2} C^{2} - 4 \, A C^{3} + C^{4}}{b^{2} d^{4}}\right)^{\frac{1}{4}} \cos\left(d x + c\right) + {\left(A^{2} - 2 \, A C + C^{2}\right)} b \sin\left(d x + c\right)}{\cos\left(d x + c\right)}} \left(\frac{A^{4} - 4 \, A^{3} C + 6 \, A^{2} C^{2} - 4 \, A C^{3} + C^{4}}{b^{2} d^{4}}\right)^{\frac{3}{4}} - A^{4} + 4 \, A^{3} C - 6 \, A^{2} C^{2} + 4 \, A C^{3} - C^{4}}{A^{4} - 4 \, A^{3} C + 6 \, A^{2} C^{2} - 4 \, A C^{3} + C^{4}}\right) + 3 \, {\left(\sqrt{2} b d \cos\left(d x + c\right)^{2} - \sqrt{2} b d\right)} \left(\frac{A^{4} - 4 \, A^{3} C + 6 \, A^{2} C^{2} - 4 \, A C^{3} + C^{4}}{b^{2} d^{4}}\right)^{\frac{1}{4}} \log\left(\frac{b^{2} d^{2} \sqrt{\frac{A^{4} - 4 \, A^{3} C + 6 \, A^{2} C^{2} - 4 \, A C^{3} + C^{4}}{b^{2} d^{4}}} \cos\left(d x + c\right) + \sqrt{2} {\left(A - C\right)} b d \sqrt{\frac{b \sin\left(d x + c\right)}{\cos\left(d x + c\right)}} \left(\frac{A^{4} - 4 \, A^{3} C + 6 \, A^{2} C^{2} - 4 \, A C^{3} + C^{4}}{b^{2} d^{4}}\right)^{\frac{1}{4}} \cos\left(d x + c\right) + {\left(A^{2} - 2 \, A C + C^{2}\right)} b \sin\left(d x + c\right)}{\cos\left(d x + c\right)}\right) - 3 \, {\left(\sqrt{2} b d \cos\left(d x + c\right)^{2} - \sqrt{2} b d\right)} \left(\frac{A^{4} - 4 \, A^{3} C + 6 \, A^{2} C^{2} - 4 \, A C^{3} + C^{4}}{b^{2} d^{4}}\right)^{\frac{1}{4}} \log\left(\frac{b^{2} d^{2} \sqrt{\frac{A^{4} - 4 \, A^{3} C + 6 \, A^{2} C^{2} - 4 \, A C^{3} + C^{4}}{b^{2} d^{4}}} \cos\left(d x + c\right) - \sqrt{2} {\left(A - C\right)} b d \sqrt{\frac{b \sin\left(d x + c\right)}{\cos\left(d x + c\right)}} \left(\frac{A^{4} - 4 \, A^{3} C + 6 \, A^{2} C^{2} - 4 \, A C^{3} + C^{4}}{b^{2} d^{4}}\right)^{\frac{1}{4}} \cos\left(d x + c\right) + {\left(A^{2} - 2 \, A C + C^{2}\right)} b \sin\left(d x + c\right)}{\cos\left(d x + c\right)}\right)}{12 \, {\left(b d \cos\left(d x + c\right)^{2} - b d\right)}}"," ",0,"1/12*(8*C*sqrt(b*sin(d*x + c)/cos(d*x + c))*cos(d*x + c)^2 + 12*(sqrt(2)*b*d*cos(d*x + c)^2 - sqrt(2)*b*d)*((A^4 - 4*A^3*C + 6*A^2*C^2 - 4*A*C^3 + C^4)/(b^2*d^4))^(1/4)*arctan((sqrt(2)*(A - C)*b*d^3*sqrt(b*sin(d*x + c)/cos(d*x + c))*((A^4 - 4*A^3*C + 6*A^2*C^2 - 4*A*C^3 + C^4)/(b^2*d^4))^(3/4) + sqrt(2)*b*d^3*sqrt((b^2*d^2*sqrt((A^4 - 4*A^3*C + 6*A^2*C^2 - 4*A*C^3 + C^4)/(b^2*d^4))*cos(d*x + c) + sqrt(2)*(A - C)*b*d*sqrt(b*sin(d*x + c)/cos(d*x + c))*((A^4 - 4*A^3*C + 6*A^2*C^2 - 4*A*C^3 + C^4)/(b^2*d^4))^(1/4)*cos(d*x + c) + (A^2 - 2*A*C + C^2)*b*sin(d*x + c))/cos(d*x + c))*((A^4 - 4*A^3*C + 6*A^2*C^2 - 4*A*C^3 + C^4)/(b^2*d^4))^(3/4) + A^4 - 4*A^3*C + 6*A^2*C^2 - 4*A*C^3 + C^4)/(A^4 - 4*A^3*C + 6*A^2*C^2 - 4*A*C^3 + C^4)) + 12*(sqrt(2)*b*d*cos(d*x + c)^2 - sqrt(2)*b*d)*((A^4 - 4*A^3*C + 6*A^2*C^2 - 4*A*C^3 + C^4)/(b^2*d^4))^(1/4)*arctan((sqrt(2)*(A - C)*b*d^3*sqrt(b*sin(d*x + c)/cos(d*x + c))*((A^4 - 4*A^3*C + 6*A^2*C^2 - 4*A*C^3 + C^4)/(b^2*d^4))^(3/4) + sqrt(2)*b*d^3*sqrt((b^2*d^2*sqrt((A^4 - 4*A^3*C + 6*A^2*C^2 - 4*A*C^3 + C^4)/(b^2*d^4))*cos(d*x + c) - sqrt(2)*(A - C)*b*d*sqrt(b*sin(d*x + c)/cos(d*x + c))*((A^4 - 4*A^3*C + 6*A^2*C^2 - 4*A*C^3 + C^4)/(b^2*d^4))^(1/4)*cos(d*x + c) + (A^2 - 2*A*C + C^2)*b*sin(d*x + c))/cos(d*x + c))*((A^4 - 4*A^3*C + 6*A^2*C^2 - 4*A*C^3 + C^4)/(b^2*d^4))^(3/4) - A^4 + 4*A^3*C - 6*A^2*C^2 + 4*A*C^3 - C^4)/(A^4 - 4*A^3*C + 6*A^2*C^2 - 4*A*C^3 + C^4)) + 3*(sqrt(2)*b*d*cos(d*x + c)^2 - sqrt(2)*b*d)*((A^4 - 4*A^3*C + 6*A^2*C^2 - 4*A*C^3 + C^4)/(b^2*d^4))^(1/4)*log((b^2*d^2*sqrt((A^4 - 4*A^3*C + 6*A^2*C^2 - 4*A*C^3 + C^4)/(b^2*d^4))*cos(d*x + c) + sqrt(2)*(A - C)*b*d*sqrt(b*sin(d*x + c)/cos(d*x + c))*((A^4 - 4*A^3*C + 6*A^2*C^2 - 4*A*C^3 + C^4)/(b^2*d^4))^(1/4)*cos(d*x + c) + (A^2 - 2*A*C + C^2)*b*sin(d*x + c))/cos(d*x + c)) - 3*(sqrt(2)*b*d*cos(d*x + c)^2 - sqrt(2)*b*d)*((A^4 - 4*A^3*C + 6*A^2*C^2 - 4*A*C^3 + C^4)/(b^2*d^4))^(1/4)*log((b^2*d^2*sqrt((A^4 - 4*A^3*C + 6*A^2*C^2 - 4*A*C^3 + C^4)/(b^2*d^4))*cos(d*x + c) - sqrt(2)*(A - C)*b*d*sqrt(b*sin(d*x + c)/cos(d*x + c))*((A^4 - 4*A^3*C + 6*A^2*C^2 - 4*A*C^3 + C^4)/(b^2*d^4))^(1/4)*cos(d*x + c) + (A^2 - 2*A*C + C^2)*b*sin(d*x + c))/cos(d*x + c)))/(b*d*cos(d*x + c)^2 - b*d)","B",0
2,1,48,0,0.837194," ","integrate(a+b*cot(d*x+c)^2,x, algorithm=""fricas"")","\frac{{\left(a - b\right)} d x \sin\left(2 \, d x + 2 \, c\right) - b \cos\left(2 \, d x + 2 \, c\right) - b}{d \sin\left(2 \, d x + 2 \, c\right)}"," ",0,"((a - b)*d*x*sin(2*d*x + 2*c) - b*cos(2*d*x + 2*c) - b)/(d*sin(2*d*x + 2*c))","B",0
3,1,127,0,0.817336," ","integrate((a+b*cot(d*x+c)^2)^2,x, algorithm=""fricas"")","\frac{2 \, b^{2} \cos\left(2 \, d x + 2 \, c\right) - 2 \, {\left(3 \, a b - 2 \, b^{2}\right)} \cos\left(2 \, d x + 2 \, c\right)^{2} + 6 \, a b - 2 \, b^{2} + 3 \, {\left({\left(a^{2} - 2 \, a b + b^{2}\right)} d x \cos\left(2 \, d x + 2 \, c\right) - {\left(a^{2} - 2 \, a b + b^{2}\right)} d x\right)} \sin\left(2 \, d x + 2 \, c\right)}{3 \, {\left(d \cos\left(2 \, d x + 2 \, c\right) - d\right)} \sin\left(2 \, d x + 2 \, c\right)}"," ",0,"1/3*(2*b^2*cos(2*d*x + 2*c) - 2*(3*a*b - 2*b^2)*cos(2*d*x + 2*c)^2 + 6*a*b - 2*b^2 + 3*((a^2 - 2*a*b + b^2)*d*x*cos(2*d*x + 2*c) - (a^2 - 2*a*b + b^2)*d*x)*sin(2*d*x + 2*c))/((d*cos(2*d*x + 2*c) - d)*sin(2*d*x + 2*c))","B",0
4,1,253,0,0.488590," ","integrate((a+b*cot(d*x+c)^2)^3,x, algorithm=""fricas"")","-\frac{{\left(45 \, a^{2} b - 60 \, a b^{2} + 23 \, b^{3}\right)} \cos\left(2 \, d x + 2 \, c\right)^{3} + 45 \, a^{2} b - 30 \, a b^{2} + 13 \, b^{3} - {\left(45 \, a^{2} b - 30 \, a b^{2} + b^{3}\right)} \cos\left(2 \, d x + 2 \, c\right)^{2} - {\left(45 \, a^{2} b - 60 \, a b^{2} + 11 \, b^{3}\right)} \cos\left(2 \, d x + 2 \, c\right) - 15 \, {\left({\left(a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}\right)} d x \cos\left(2 \, d x + 2 \, c\right)^{2} - 2 \, {\left(a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}\right)} d x \cos\left(2 \, d x + 2 \, c\right) + {\left(a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}\right)} d x\right)} \sin\left(2 \, d x + 2 \, c\right)}{15 \, {\left(d \cos\left(2 \, d x + 2 \, c\right)^{2} - 2 \, d \cos\left(2 \, d x + 2 \, c\right) + d\right)} \sin\left(2 \, d x + 2 \, c\right)}"," ",0,"-1/15*((45*a^2*b - 60*a*b^2 + 23*b^3)*cos(2*d*x + 2*c)^3 + 45*a^2*b - 30*a*b^2 + 13*b^3 - (45*a^2*b - 30*a*b^2 + b^3)*cos(2*d*x + 2*c)^2 - (45*a^2*b - 60*a*b^2 + 11*b^3)*cos(2*d*x + 2*c) - 15*((a^3 - 3*a^2*b + 3*a*b^2 - b^3)*d*x*cos(2*d*x + 2*c)^2 - 2*(a^3 - 3*a^2*b + 3*a*b^2 - b^3)*d*x*cos(2*d*x + 2*c) + (a^3 - 3*a^2*b + 3*a*b^2 - b^3)*d*x)*sin(2*d*x + 2*c))/((d*cos(2*d*x + 2*c)^2 - 2*d*cos(2*d*x + 2*c) + d)*sin(2*d*x + 2*c))","B",0
5,1,252,0,0.461224," ","integrate(1/(a+b*cot(d*x+c)^2),x, algorithm=""fricas"")","\left[\frac{4 \, d x - \sqrt{-\frac{b}{a}} \log\left(\frac{{\left(a^{2} + 6 \, a b + b^{2}\right)} \cos\left(2 \, d x + 2 \, c\right)^{2} + 4 \, {\left(a^{2} - a b - {\left(a^{2} + a b\right)} \cos\left(2 \, d x + 2 \, c\right)\right)} \sqrt{-\frac{b}{a}} \sin\left(2 \, d x + 2 \, c\right) + a^{2} - 6 \, a b + b^{2} - 2 \, {\left(a^{2} - b^{2}\right)} \cos\left(2 \, d x + 2 \, c\right)}{{\left(a^{2} - 2 \, a b + b^{2}\right)} \cos\left(2 \, d x + 2 \, c\right)^{2} + a^{2} + 2 \, a b + b^{2} - 2 \, {\left(a^{2} - b^{2}\right)} \cos\left(2 \, d x + 2 \, c\right)}\right)}{4 \, {\left(a - b\right)} d}, \frac{2 \, d x + \sqrt{\frac{b}{a}} \arctan\left(\frac{{\left({\left(a + b\right)} \cos\left(2 \, d x + 2 \, c\right) - a + b\right)} \sqrt{\frac{b}{a}}}{2 \, b \sin\left(2 \, d x + 2 \, c\right)}\right)}{2 \, {\left(a - b\right)} d}\right]"," ",0,"[1/4*(4*d*x - sqrt(-b/a)*log(((a^2 + 6*a*b + b^2)*cos(2*d*x + 2*c)^2 + 4*(a^2 - a*b - (a^2 + a*b)*cos(2*d*x + 2*c))*sqrt(-b/a)*sin(2*d*x + 2*c) + a^2 - 6*a*b + b^2 - 2*(a^2 - b^2)*cos(2*d*x + 2*c))/((a^2 - 2*a*b + b^2)*cos(2*d*x + 2*c)^2 + a^2 + 2*a*b + b^2 - 2*(a^2 - b^2)*cos(2*d*x + 2*c))))/((a - b)*d), 1/2*(2*d*x + sqrt(b/a)*arctan(1/2*((a + b)*cos(2*d*x + 2*c) - a + b)*sqrt(b/a)/(b*sin(2*d*x + 2*c))))/((a - b)*d)]","A",0
6,1,534,0,0.510605," ","integrate(1/(a+b*cot(d*x+c)^2)^2,x, algorithm=""fricas"")","\left[\frac{8 \, {\left(a^{2} - a b\right)} d x \cos\left(2 \, d x + 2 \, c\right) - 8 \, {\left(a^{2} + a b\right)} d x + {\left(3 \, a^{2} + 2 \, a b - b^{2} - {\left(3 \, a^{2} - 4 \, a b + b^{2}\right)} \cos\left(2 \, d x + 2 \, c\right)\right)} \sqrt{-\frac{b}{a}} \log\left(\frac{{\left(a^{2} + 6 \, a b + b^{2}\right)} \cos\left(2 \, d x + 2 \, c\right)^{2} + 4 \, {\left(a^{2} - a b - {\left(a^{2} + a b\right)} \cos\left(2 \, d x + 2 \, c\right)\right)} \sqrt{-\frac{b}{a}} \sin\left(2 \, d x + 2 \, c\right) + a^{2} - 6 \, a b + b^{2} - 2 \, {\left(a^{2} - b^{2}\right)} \cos\left(2 \, d x + 2 \, c\right)}{{\left(a^{2} - 2 \, a b + b^{2}\right)} \cos\left(2 \, d x + 2 \, c\right)^{2} + a^{2} + 2 \, a b + b^{2} - 2 \, {\left(a^{2} - b^{2}\right)} \cos\left(2 \, d x + 2 \, c\right)}\right) - 4 \, {\left(a b - b^{2}\right)} \sin\left(2 \, d x + 2 \, c\right)}{8 \, {\left({\left(a^{4} - 3 \, a^{3} b + 3 \, a^{2} b^{2} - a b^{3}\right)} d \cos\left(2 \, d x + 2 \, c\right) - {\left(a^{4} - a^{3} b - a^{2} b^{2} + a b^{3}\right)} d\right)}}, \frac{4 \, {\left(a^{2} - a b\right)} d x \cos\left(2 \, d x + 2 \, c\right) - 4 \, {\left(a^{2} + a b\right)} d x - {\left(3 \, a^{2} + 2 \, a b - b^{2} - {\left(3 \, a^{2} - 4 \, a b + b^{2}\right)} \cos\left(2 \, d x + 2 \, c\right)\right)} \sqrt{\frac{b}{a}} \arctan\left(\frac{{\left({\left(a + b\right)} \cos\left(2 \, d x + 2 \, c\right) - a + b\right)} \sqrt{\frac{b}{a}}}{2 \, b \sin\left(2 \, d x + 2 \, c\right)}\right) - 2 \, {\left(a b - b^{2}\right)} \sin\left(2 \, d x + 2 \, c\right)}{4 \, {\left({\left(a^{4} - 3 \, a^{3} b + 3 \, a^{2} b^{2} - a b^{3}\right)} d \cos\left(2 \, d x + 2 \, c\right) - {\left(a^{4} - a^{3} b - a^{2} b^{2} + a b^{3}\right)} d\right)}}\right]"," ",0,"[1/8*(8*(a^2 - a*b)*d*x*cos(2*d*x + 2*c) - 8*(a^2 + a*b)*d*x + (3*a^2 + 2*a*b - b^2 - (3*a^2 - 4*a*b + b^2)*cos(2*d*x + 2*c))*sqrt(-b/a)*log(((a^2 + 6*a*b + b^2)*cos(2*d*x + 2*c)^2 + 4*(a^2 - a*b - (a^2 + a*b)*cos(2*d*x + 2*c))*sqrt(-b/a)*sin(2*d*x + 2*c) + a^2 - 6*a*b + b^2 - 2*(a^2 - b^2)*cos(2*d*x + 2*c))/((a^2 - 2*a*b + b^2)*cos(2*d*x + 2*c)^2 + a^2 + 2*a*b + b^2 - 2*(a^2 - b^2)*cos(2*d*x + 2*c))) - 4*(a*b - b^2)*sin(2*d*x + 2*c))/((a^4 - 3*a^3*b + 3*a^2*b^2 - a*b^3)*d*cos(2*d*x + 2*c) - (a^4 - a^3*b - a^2*b^2 + a*b^3)*d), 1/4*(4*(a^2 - a*b)*d*x*cos(2*d*x + 2*c) - 4*(a^2 + a*b)*d*x - (3*a^2 + 2*a*b - b^2 - (3*a^2 - 4*a*b + b^2)*cos(2*d*x + 2*c))*sqrt(b/a)*arctan(1/2*((a + b)*cos(2*d*x + 2*c) - a + b)*sqrt(b/a)/(b*sin(2*d*x + 2*c))) - 2*(a*b - b^2)*sin(2*d*x + 2*c))/((a^4 - 3*a^3*b + 3*a^2*b^2 - a*b^3)*d*cos(2*d*x + 2*c) - (a^4 - a^3*b - a^2*b^2 + a*b^3)*d)]","B",0
7,1,1068,0,0.555066," ","integrate(1/(a+b*cot(d*x+c)^2)^3,x, algorithm=""fricas"")","\left[\frac{32 \, {\left(a^{4} - 2 \, a^{3} b + a^{2} b^{2}\right)} d x \cos\left(2 \, d x + 2 \, c\right)^{2} - 64 \, {\left(a^{4} - a^{2} b^{2}\right)} d x \cos\left(2 \, d x + 2 \, c\right) + 32 \, {\left(a^{4} + 2 \, a^{3} b + a^{2} b^{2}\right)} d x - {\left(15 \, a^{4} + 20 \, a^{3} b - 2 \, a^{2} b^{2} - 4 \, a b^{3} + 3 \, b^{4} + {\left(15 \, a^{4} - 40 \, a^{3} b + 38 \, a^{2} b^{2} - 16 \, a b^{3} + 3 \, b^{4}\right)} \cos\left(2 \, d x + 2 \, c\right)^{2} - 2 \, {\left(15 \, a^{4} - 10 \, a^{3} b - 12 \, a^{2} b^{2} + 10 \, a b^{3} - 3 \, b^{4}\right)} \cos\left(2 \, d x + 2 \, c\right)\right)} \sqrt{-\frac{b}{a}} \log\left(\frac{{\left(a^{2} + 6 \, a b + b^{2}\right)} \cos\left(2 \, d x + 2 \, c\right)^{2} + 4 \, {\left(a^{2} - a b - {\left(a^{2} + a b\right)} \cos\left(2 \, d x + 2 \, c\right)\right)} \sqrt{-\frac{b}{a}} \sin\left(2 \, d x + 2 \, c\right) + a^{2} - 6 \, a b + b^{2} - 2 \, {\left(a^{2} - b^{2}\right)} \cos\left(2 \, d x + 2 \, c\right)}{{\left(a^{2} - 2 \, a b + b^{2}\right)} \cos\left(2 \, d x + 2 \, c\right)^{2} + a^{2} + 2 \, a b + b^{2} - 2 \, {\left(a^{2} - b^{2}\right)} \cos\left(2 \, d x + 2 \, c\right)}\right) + 4 \, {\left(9 \, a^{3} b - 7 \, a^{2} b^{2} - 5 \, a b^{3} + 3 \, b^{4} - 3 \, {\left(3 \, a^{3} b - 7 \, a^{2} b^{2} + 5 \, a b^{3} - b^{4}\right)} \cos\left(2 \, d x + 2 \, c\right)\right)} \sin\left(2 \, d x + 2 \, c\right)}{32 \, {\left({\left(a^{7} - 5 \, a^{6} b + 10 \, a^{5} b^{2} - 10 \, a^{4} b^{3} + 5 \, a^{3} b^{4} - a^{2} b^{5}\right)} d \cos\left(2 \, d x + 2 \, c\right)^{2} - 2 \, {\left(a^{7} - 3 \, a^{6} b + 2 \, a^{5} b^{2} + 2 \, a^{4} b^{3} - 3 \, a^{3} b^{4} + a^{2} b^{5}\right)} d \cos\left(2 \, d x + 2 \, c\right) + {\left(a^{7} - a^{6} b - 2 \, a^{5} b^{2} + 2 \, a^{4} b^{3} + a^{3} b^{4} - a^{2} b^{5}\right)} d\right)}}, \frac{16 \, {\left(a^{4} - 2 \, a^{3} b + a^{2} b^{2}\right)} d x \cos\left(2 \, d x + 2 \, c\right)^{2} - 32 \, {\left(a^{4} - a^{2} b^{2}\right)} d x \cos\left(2 \, d x + 2 \, c\right) + 16 \, {\left(a^{4} + 2 \, a^{3} b + a^{2} b^{2}\right)} d x + {\left(15 \, a^{4} + 20 \, a^{3} b - 2 \, a^{2} b^{2} - 4 \, a b^{3} + 3 \, b^{4} + {\left(15 \, a^{4} - 40 \, a^{3} b + 38 \, a^{2} b^{2} - 16 \, a b^{3} + 3 \, b^{4}\right)} \cos\left(2 \, d x + 2 \, c\right)^{2} - 2 \, {\left(15 \, a^{4} - 10 \, a^{3} b - 12 \, a^{2} b^{2} + 10 \, a b^{3} - 3 \, b^{4}\right)} \cos\left(2 \, d x + 2 \, c\right)\right)} \sqrt{\frac{b}{a}} \arctan\left(\frac{{\left({\left(a + b\right)} \cos\left(2 \, d x + 2 \, c\right) - a + b\right)} \sqrt{\frac{b}{a}}}{2 \, b \sin\left(2 \, d x + 2 \, c\right)}\right) + 2 \, {\left(9 \, a^{3} b - 7 \, a^{2} b^{2} - 5 \, a b^{3} + 3 \, b^{4} - 3 \, {\left(3 \, a^{3} b - 7 \, a^{2} b^{2} + 5 \, a b^{3} - b^{4}\right)} \cos\left(2 \, d x + 2 \, c\right)\right)} \sin\left(2 \, d x + 2 \, c\right)}{16 \, {\left({\left(a^{7} - 5 \, a^{6} b + 10 \, a^{5} b^{2} - 10 \, a^{4} b^{3} + 5 \, a^{3} b^{4} - a^{2} b^{5}\right)} d \cos\left(2 \, d x + 2 \, c\right)^{2} - 2 \, {\left(a^{7} - 3 \, a^{6} b + 2 \, a^{5} b^{2} + 2 \, a^{4} b^{3} - 3 \, a^{3} b^{4} + a^{2} b^{5}\right)} d \cos\left(2 \, d x + 2 \, c\right) + {\left(a^{7} - a^{6} b - 2 \, a^{5} b^{2} + 2 \, a^{4} b^{3} + a^{3} b^{4} - a^{2} b^{5}\right)} d\right)}}\right]"," ",0,"[1/32*(32*(a^4 - 2*a^3*b + a^2*b^2)*d*x*cos(2*d*x + 2*c)^2 - 64*(a^4 - a^2*b^2)*d*x*cos(2*d*x + 2*c) + 32*(a^4 + 2*a^3*b + a^2*b^2)*d*x - (15*a^4 + 20*a^3*b - 2*a^2*b^2 - 4*a*b^3 + 3*b^4 + (15*a^4 - 40*a^3*b + 38*a^2*b^2 - 16*a*b^3 + 3*b^4)*cos(2*d*x + 2*c)^2 - 2*(15*a^4 - 10*a^3*b - 12*a^2*b^2 + 10*a*b^3 - 3*b^4)*cos(2*d*x + 2*c))*sqrt(-b/a)*log(((a^2 + 6*a*b + b^2)*cos(2*d*x + 2*c)^2 + 4*(a^2 - a*b - (a^2 + a*b)*cos(2*d*x + 2*c))*sqrt(-b/a)*sin(2*d*x + 2*c) + a^2 - 6*a*b + b^2 - 2*(a^2 - b^2)*cos(2*d*x + 2*c))/((a^2 - 2*a*b + b^2)*cos(2*d*x + 2*c)^2 + a^2 + 2*a*b + b^2 - 2*(a^2 - b^2)*cos(2*d*x + 2*c))) + 4*(9*a^3*b - 7*a^2*b^2 - 5*a*b^3 + 3*b^4 - 3*(3*a^3*b - 7*a^2*b^2 + 5*a*b^3 - b^4)*cos(2*d*x + 2*c))*sin(2*d*x + 2*c))/((a^7 - 5*a^6*b + 10*a^5*b^2 - 10*a^4*b^3 + 5*a^3*b^4 - a^2*b^5)*d*cos(2*d*x + 2*c)^2 - 2*(a^7 - 3*a^6*b + 2*a^5*b^2 + 2*a^4*b^3 - 3*a^3*b^4 + a^2*b^5)*d*cos(2*d*x + 2*c) + (a^7 - a^6*b - 2*a^5*b^2 + 2*a^4*b^3 + a^3*b^4 - a^2*b^5)*d), 1/16*(16*(a^4 - 2*a^3*b + a^2*b^2)*d*x*cos(2*d*x + 2*c)^2 - 32*(a^4 - a^2*b^2)*d*x*cos(2*d*x + 2*c) + 16*(a^4 + 2*a^3*b + a^2*b^2)*d*x + (15*a^4 + 20*a^3*b - 2*a^2*b^2 - 4*a*b^3 + 3*b^4 + (15*a^4 - 40*a^3*b + 38*a^2*b^2 - 16*a*b^3 + 3*b^4)*cos(2*d*x + 2*c)^2 - 2*(15*a^4 - 10*a^3*b - 12*a^2*b^2 + 10*a*b^3 - 3*b^4)*cos(2*d*x + 2*c))*sqrt(b/a)*arctan(1/2*((a + b)*cos(2*d*x + 2*c) - a + b)*sqrt(b/a)/(b*sin(2*d*x + 2*c))) + 2*(9*a^3*b - 7*a^2*b^2 - 5*a*b^3 + 3*b^4 - 3*(3*a^3*b - 7*a^2*b^2 + 5*a*b^3 - b^4)*cos(2*d*x + 2*c))*sin(2*d*x + 2*c))/((a^7 - 5*a^6*b + 10*a^5*b^2 - 10*a^4*b^3 + 5*a^3*b^4 - a^2*b^5)*d*cos(2*d*x + 2*c)^2 - 2*(a^7 - 3*a^6*b + 2*a^5*b^2 + 2*a^4*b^3 - 3*a^3*b^4 + a^2*b^5)*d*cos(2*d*x + 2*c) + (a^7 - a^6*b - 2*a^5*b^2 + 2*a^4*b^3 + a^3*b^4 - a^2*b^5)*d)]","B",0
8,1,91,0,0.488674," ","integrate((1+cot(x)^2)^(3/2),x, algorithm=""fricas"")","-\frac{2 \, \sqrt{2} \sqrt{-\frac{1}{\cos\left(2 \, x\right) - 1}} {\left(\cos\left(2 \, x\right) + 1\right)} + \log\left(\frac{1}{2} \, \sqrt{2} \sqrt{-\frac{1}{\cos\left(2 \, x\right) - 1}} \sin\left(2 \, x\right) + 1\right) \sin\left(2 \, x\right) - \log\left(-\frac{1}{2} \, \sqrt{2} \sqrt{-\frac{1}{\cos\left(2 \, x\right) - 1}} \sin\left(2 \, x\right) + 1\right) \sin\left(2 \, x\right)}{4 \, \sin\left(2 \, x\right)}"," ",0,"-1/4*(2*sqrt(2)*sqrt(-1/(cos(2*x) - 1))*(cos(2*x) + 1) + log(1/2*sqrt(2)*sqrt(-1/(cos(2*x) - 1))*sin(2*x) + 1)*sin(2*x) - log(-1/2*sqrt(2)*sqrt(-1/(cos(2*x) - 1))*sin(2*x) + 1)*sin(2*x))/sin(2*x)","B",0
9,1,53,0,0.518080," ","integrate((1+cot(x)^2)^(1/2),x, algorithm=""fricas"")","-\frac{1}{2} \, \log\left(\frac{1}{2} \, \sqrt{2} \sqrt{-\frac{1}{\cos\left(2 \, x\right) - 1}} \sin\left(2 \, x\right) + 1\right) + \frac{1}{2} \, \log\left(-\frac{1}{2} \, \sqrt{2} \sqrt{-\frac{1}{\cos\left(2 \, x\right) - 1}} \sin\left(2 \, x\right) + 1\right)"," ",0,"-1/2*log(1/2*sqrt(2)*sqrt(-1/(cos(2*x) - 1))*sin(2*x) + 1) + 1/2*log(-1/2*sqrt(2)*sqrt(-1/(cos(2*x) - 1))*sin(2*x) + 1)","B",0
10,1,21,0,0.426275," ","integrate(1/(1+cot(x)^2)^(1/2),x, algorithm=""fricas"")","-\frac{1}{2} \, \sqrt{2} \sqrt{-\frac{1}{\cos\left(2 \, x\right) - 1}} \sin\left(2 \, x\right)"," ",0,"-1/2*sqrt(2)*sqrt(-1/(cos(2*x) - 1))*sin(2*x)","B",0
11,1,73,0,0.495277," ","integrate((-1-cot(x)^2)^(3/2),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)} + 1\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)} - 1\right) + 2 i \, e^{\left(3 i \, x\right)} + 2 i \, 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) + 1) + (I*e^(4*I*x) - 2*I*e^(2*I*x) + I)*log(e^(I*x) - 1) + 2*I*e^(3*I*x) + 2*I*e^(I*x))/(e^(4*I*x) - 2*e^(2*I*x) + 1)","C",0
12,1,19,0,0.574067," ","integrate((-1-cot(x)^2)^(1/2),x, algorithm=""fricas"")","i \, \log\left(e^{\left(i \, x\right)} + 1\right) - i \, \log\left(e^{\left(i \, x\right)} - 1\right)"," ",0,"I*log(e^(I*x) + 1) - I*log(e^(I*x) - 1)","C",0
13,1,14,0,0.841203," ","integrate(1/(-1-cot(x)^2)^(1/2),x, algorithm=""fricas"")","\frac{1}{2} \, {\left(-i \, e^{\left(2 i \, x\right)} - i\right)} e^{\left(-i \, x\right)}"," ",0,"1/2*(-I*e^(2*I*x) - I)*e^(-I*x)","C",0
14,1,27,0,0.430364," ","integrate(cot(x)^3/(a+a*cot(x)^2)^(1/2),x, algorithm=""fricas"")","\frac{\sqrt{2} \sqrt{-\frac{a}{\cos\left(2 \, x\right) - 1}} {\left(\cos\left(2 \, x\right) - 3\right)}}{2 \, a}"," ",0,"1/2*sqrt(2)*sqrt(-a/(cos(2*x) - 1))*(cos(2*x) - 3)/a","A",0
15,1,77,0,0.749313," ","integrate(cot(x)^2/(a+a*cot(x)^2)^(1/2),x, algorithm=""fricas"")","\frac{\sqrt{2} \sqrt{-\frac{a}{\cos\left(2 \, x\right) - 1}} \sin\left(2 \, x\right) + \sqrt{a} \log\left(\frac{2 \, \sqrt{2} \sqrt{a} \sqrt{-\frac{a}{\cos\left(2 \, x\right) - 1}} \sin\left(2 \, x\right) - a \cos\left(2 \, x\right) - 3 \, a}{\cos\left(2 \, x\right) - 1}\right)}{2 \, a}"," ",0,"1/2*(sqrt(2)*sqrt(-a/(cos(2*x) - 1))*sin(2*x) + sqrt(a)*log((2*sqrt(2)*sqrt(a)*sqrt(-a/(cos(2*x) - 1))*sin(2*x) - a*cos(2*x) - 3*a)/(cos(2*x) - 1)))/a","B",0
16,1,27,0,0.521870," ","integrate(cot(x)/(a+a*cot(x)^2)^(1/2),x, algorithm=""fricas"")","-\frac{\sqrt{2} \sqrt{-\frac{a}{\cos\left(2 \, x\right) - 1}} {\left(\cos\left(2 \, x\right) - 1\right)}}{2 \, a}"," ",0,"-1/2*sqrt(2)*sqrt(-a/(cos(2*x) - 1))*(cos(2*x) - 1)/a","B",0
17,1,78,0,0.442550," ","integrate(tan(x)/(a+a*cot(x)^2)^(1/2),x, algorithm=""fricas"")","\frac{{\left(\tan\left(x\right)^{2} + 1\right)} \sqrt{a} \log\left(2 \, a \tan\left(x\right)^{2} + 2 \, \sqrt{a} \sqrt{\frac{a \tan\left(x\right)^{2} + a}{\tan\left(x\right)^{2}}} \tan\left(x\right)^{2} + a\right) - 2 \, \sqrt{\frac{a \tan\left(x\right)^{2} + a}{\tan\left(x\right)^{2}}} \tan\left(x\right)^{2}}{2 \, {\left(a \tan\left(x\right)^{2} + a\right)}}"," ",0,"1/2*((tan(x)^2 + 1)*sqrt(a)*log(2*a*tan(x)^2 + 2*sqrt(a)*sqrt((a*tan(x)^2 + a)/tan(x)^2)*tan(x)^2 + a) - 2*sqrt((a*tan(x)^2 + a)/tan(x)^2)*tan(x)^2)/(a*tan(x)^2 + a)","B",0
18,1,35,0,0.412956," ","integrate(tan(x)^2/(a+a*cot(x)^2)^(1/2),x, algorithm=""fricas"")","\frac{{\left(\tan\left(x\right)^{3} + 2 \, \tan\left(x\right)\right)} \sqrt{\frac{a \tan\left(x\right)^{2} + a}{\tan\left(x\right)^{2}}}}{a \tan\left(x\right)^{2} + a}"," ",0,"(tan(x)^3 + 2*tan(x))*sqrt((a*tan(x)^2 + a)/tan(x)^2)/(a*tan(x)^2 + a)","A",0
19,1,330,0,0.724303," ","integrate(cot(x)^3*(a+b*cot(x)^2)^(1/2),x, algorithm=""fricas"")","\left[\frac{3 \, {\left(b \cos\left(2 \, x\right) - b\right)} \sqrt{a - b} \log\left(-2 \, {\left(a^{2} - 2 \, a b + b^{2}\right)} \cos\left(2 \, x\right)^{2} - 2 \, a^{2} + b^{2} + 2 \, {\left({\left(a - b\right)} \cos\left(2 \, x\right)^{2} - {\left(2 \, a - b\right)} \cos\left(2 \, x\right) + a\right)} \sqrt{a - b} \sqrt{\frac{{\left(a - b\right)} \cos\left(2 \, x\right) - a - b}{\cos\left(2 \, x\right) - 1}} + 4 \, {\left(a^{2} - a b\right)} \cos\left(2 \, x\right)\right) - 4 \, {\left({\left(a - 4 \, b\right)} \cos\left(2 \, x\right) - a + 2 \, b\right)} \sqrt{\frac{{\left(a - b\right)} \cos\left(2 \, x\right) - a - b}{\cos\left(2 \, x\right) - 1}}}{12 \, {\left(b \cos\left(2 \, x\right) - b\right)}}, -\frac{3 \, {\left(b \cos\left(2 \, x\right) - b\right)} \sqrt{-a + b} \arctan\left(-\frac{\sqrt{-a + b} \sqrt{\frac{{\left(a - b\right)} \cos\left(2 \, x\right) - a - b}{\cos\left(2 \, x\right) - 1}} {\left(\cos\left(2 \, x\right) - 1\right)}}{{\left(a - b\right)} \cos\left(2 \, x\right) - a}\right) + 2 \, {\left({\left(a - 4 \, b\right)} \cos\left(2 \, x\right) - a + 2 \, b\right)} \sqrt{\frac{{\left(a - b\right)} \cos\left(2 \, x\right) - a - b}{\cos\left(2 \, x\right) - 1}}}{6 \, {\left(b \cos\left(2 \, x\right) - b\right)}}\right]"," ",0,"[1/12*(3*(b*cos(2*x) - b)*sqrt(a - b)*log(-2*(a^2 - 2*a*b + b^2)*cos(2*x)^2 - 2*a^2 + b^2 + 2*((a - b)*cos(2*x)^2 - (2*a - b)*cos(2*x) + a)*sqrt(a - b)*sqrt(((a - b)*cos(2*x) - a - b)/(cos(2*x) - 1)) + 4*(a^2 - a*b)*cos(2*x)) - 4*((a - 4*b)*cos(2*x) - a + 2*b)*sqrt(((a - b)*cos(2*x) - a - b)/(cos(2*x) - 1)))/(b*cos(2*x) - b), -1/6*(3*(b*cos(2*x) - b)*sqrt(-a + b)*arctan(-sqrt(-a + b)*sqrt(((a - b)*cos(2*x) - a - b)/(cos(2*x) - 1))*(cos(2*x) - 1)/((a - b)*cos(2*x) - a)) + 2*((a - 4*b)*cos(2*x) - a + 2*b)*sqrt(((a - b)*cos(2*x) - a - b)/(cos(2*x) - 1)))/(b*cos(2*x) - b)]","B",0
20,1,248,0,1.511291," ","integrate(cot(x)*(a+b*cot(x)^2)^(1/2),x, algorithm=""fricas"")","\left[\frac{1}{4} \, \sqrt{a - b} \log\left(-2 \, {\left(a^{2} - 2 \, a b + b^{2}\right)} \cos\left(2 \, x\right)^{2} - 2 \, a^{2} + b^{2} - 2 \, {\left({\left(a - b\right)} \cos\left(2 \, x\right)^{2} - {\left(2 \, a - b\right)} \cos\left(2 \, x\right) + a\right)} \sqrt{a - b} \sqrt{\frac{{\left(a - b\right)} \cos\left(2 \, x\right) - a - b}{\cos\left(2 \, x\right) - 1}} + 4 \, {\left(a^{2} - a b\right)} \cos\left(2 \, x\right)\right) - \sqrt{\frac{{\left(a - b\right)} \cos\left(2 \, x\right) - a - b}{\cos\left(2 \, x\right) - 1}}, \frac{1}{2} \, \sqrt{-a + b} \arctan\left(-\frac{\sqrt{-a + b} \sqrt{\frac{{\left(a - b\right)} \cos\left(2 \, x\right) - a - b}{\cos\left(2 \, x\right) - 1}} {\left(\cos\left(2 \, x\right) - 1\right)}}{{\left(a - b\right)} \cos\left(2 \, x\right) - a}\right) - \sqrt{\frac{{\left(a - b\right)} \cos\left(2 \, x\right) - a - b}{\cos\left(2 \, x\right) - 1}}\right]"," ",0,"[1/4*sqrt(a - b)*log(-2*(a^2 - 2*a*b + b^2)*cos(2*x)^2 - 2*a^2 + b^2 - 2*((a - b)*cos(2*x)^2 - (2*a - b)*cos(2*x) + a)*sqrt(a - b)*sqrt(((a - b)*cos(2*x) - a - b)/(cos(2*x) - 1)) + 4*(a^2 - a*b)*cos(2*x)) - sqrt(((a - b)*cos(2*x) - a - b)/(cos(2*x) - 1)), 1/2*sqrt(-a + b)*arctan(-sqrt(-a + b)*sqrt(((a - b)*cos(2*x) - a - b)/(cos(2*x) - 1))*(cos(2*x) - 1)/((a - b)*cos(2*x) - a)) - sqrt(((a - b)*cos(2*x) - a - b)/(cos(2*x) - 1))]","B",0
21,1,351,0,0.502330," ","integrate((a+b*cot(x)^2)^(1/2)*tan(x),x, algorithm=""fricas"")","\left[\frac{1}{2} \, \sqrt{a} \log\left(2 \, a \tan\left(x\right)^{2} + 2 \, \sqrt{a} \sqrt{\frac{a \tan\left(x\right)^{2} + b}{\tan\left(x\right)^{2}}} \tan\left(x\right)^{2} + b\right) + \frac{1}{2} \, \sqrt{a - b} \log\left(\frac{{\left(2 \, a - b\right)} \tan\left(x\right)^{2} - 2 \, \sqrt{a - b} \sqrt{\frac{a \tan\left(x\right)^{2} + b}{\tan\left(x\right)^{2}}} \tan\left(x\right)^{2} + b}{\tan\left(x\right)^{2} + 1}\right), -\sqrt{-a + b} \arctan\left(-\frac{\sqrt{-a + b} \sqrt{\frac{a \tan\left(x\right)^{2} + b}{\tan\left(x\right)^{2}}}}{a - b}\right) + \frac{1}{2} \, \sqrt{a} \log\left(2 \, a \tan\left(x\right)^{2} + 2 \, \sqrt{a} \sqrt{\frac{a \tan\left(x\right)^{2} + b}{\tan\left(x\right)^{2}}} \tan\left(x\right)^{2} + b\right), -\sqrt{-a} \arctan\left(\frac{\sqrt{-a} \sqrt{\frac{a \tan\left(x\right)^{2} + b}{\tan\left(x\right)^{2}}}}{a}\right) + \frac{1}{2} \, \sqrt{a - b} \log\left(\frac{{\left(2 \, a - b\right)} \tan\left(x\right)^{2} - 2 \, \sqrt{a - b} \sqrt{\frac{a \tan\left(x\right)^{2} + b}{\tan\left(x\right)^{2}}} \tan\left(x\right)^{2} + b}{\tan\left(x\right)^{2} + 1}\right), -\sqrt{-a} \arctan\left(\frac{\sqrt{-a} \sqrt{\frac{a \tan\left(x\right)^{2} + b}{\tan\left(x\right)^{2}}}}{a}\right) - \sqrt{-a + b} \arctan\left(-\frac{\sqrt{-a + b} \sqrt{\frac{a \tan\left(x\right)^{2} + b}{\tan\left(x\right)^{2}}}}{a - b}\right)\right]"," ",0,"[1/2*sqrt(a)*log(2*a*tan(x)^2 + 2*sqrt(a)*sqrt((a*tan(x)^2 + b)/tan(x)^2)*tan(x)^2 + b) + 1/2*sqrt(a - b)*log(((2*a - b)*tan(x)^2 - 2*sqrt(a - b)*sqrt((a*tan(x)^2 + b)/tan(x)^2)*tan(x)^2 + b)/(tan(x)^2 + 1)), -sqrt(-a + b)*arctan(-sqrt(-a + b)*sqrt((a*tan(x)^2 + b)/tan(x)^2)/(a - b)) + 1/2*sqrt(a)*log(2*a*tan(x)^2 + 2*sqrt(a)*sqrt((a*tan(x)^2 + b)/tan(x)^2)*tan(x)^2 + b), -sqrt(-a)*arctan(sqrt(-a)*sqrt((a*tan(x)^2 + b)/tan(x)^2)/a) + 1/2*sqrt(a - b)*log(((2*a - b)*tan(x)^2 - 2*sqrt(a - b)*sqrt((a*tan(x)^2 + b)/tan(x)^2)*tan(x)^2 + b)/(tan(x)^2 + 1)), -sqrt(-a)*arctan(sqrt(-a)*sqrt((a*tan(x)^2 + b)/tan(x)^2)/a) - sqrt(-a + b)*arctan(-sqrt(-a + b)*sqrt((a*tan(x)^2 + b)/tan(x)^2)/(a - b))]","A",0
22,1,768,0,0.493934," ","integrate(cot(x)^2*(a+b*cot(x)^2)^(1/2),x, algorithm=""fricas"")","\left[\frac{2 \, \sqrt{-a + b} b \log\left(-{\left(a - b\right)} \cos\left(2 \, x\right) - \sqrt{-a + b} \sqrt{\frac{{\left(a - b\right)} \cos\left(2 \, x\right) - a - b}{\cos\left(2 \, x\right) - 1}} \sin\left(2 \, x\right) + b\right) \sin\left(2 \, x\right) - {\left(a - 2 \, b\right)} \sqrt{b} \log\left(\frac{{\left(a - 2 \, b\right)} \cos\left(2 \, x\right) - 2 \, \sqrt{b} \sqrt{\frac{{\left(a - b\right)} \cos\left(2 \, x\right) - a - b}{\cos\left(2 \, x\right) - 1}} \sin\left(2 \, x\right) - a - 2 \, b}{\cos\left(2 \, x\right) - 1}\right) \sin\left(2 \, x\right) - 2 \, {\left(b \cos\left(2 \, x\right) + b\right)} \sqrt{\frac{{\left(a - b\right)} \cos\left(2 \, x\right) - a - b}{\cos\left(2 \, x\right) - 1}}}{4 \, b \sin\left(2 \, x\right)}, \frac{4 \, \sqrt{a - b} b \arctan\left(-\frac{\sqrt{a - b} \sqrt{\frac{{\left(a - b\right)} \cos\left(2 \, x\right) - a - b}{\cos\left(2 \, x\right) - 1}} \sin\left(2 \, x\right)}{{\left(a - b\right)} \cos\left(2 \, x\right) + a - b}\right) \sin\left(2 \, x\right) - {\left(a - 2 \, b\right)} \sqrt{b} \log\left(\frac{{\left(a - 2 \, b\right)} \cos\left(2 \, x\right) - 2 \, \sqrt{b} \sqrt{\frac{{\left(a - b\right)} \cos\left(2 \, x\right) - a - b}{\cos\left(2 \, x\right) - 1}} \sin\left(2 \, x\right) - a - 2 \, b}{\cos\left(2 \, x\right) - 1}\right) \sin\left(2 \, x\right) - 2 \, {\left(b \cos\left(2 \, x\right) + b\right)} \sqrt{\frac{{\left(a - b\right)} \cos\left(2 \, x\right) - a - b}{\cos\left(2 \, x\right) - 1}}}{4 \, b \sin\left(2 \, x\right)}, \frac{{\left(a - 2 \, b\right)} \sqrt{-b} \arctan\left(\frac{\sqrt{-b} \sqrt{\frac{{\left(a - b\right)} \cos\left(2 \, x\right) - a - b}{\cos\left(2 \, x\right) - 1}} \sin\left(2 \, x\right)}{b \cos\left(2 \, x\right) + b}\right) \sin\left(2 \, x\right) + \sqrt{-a + b} b \log\left(-{\left(a - b\right)} \cos\left(2 \, x\right) - \sqrt{-a + b} \sqrt{\frac{{\left(a - b\right)} \cos\left(2 \, x\right) - a - b}{\cos\left(2 \, x\right) - 1}} \sin\left(2 \, x\right) + b\right) \sin\left(2 \, x\right) - {\left(b \cos\left(2 \, x\right) + b\right)} \sqrt{\frac{{\left(a - b\right)} \cos\left(2 \, x\right) - a - b}{\cos\left(2 \, x\right) - 1}}}{2 \, b \sin\left(2 \, x\right)}, \frac{2 \, \sqrt{a - b} b \arctan\left(-\frac{\sqrt{a - b} \sqrt{\frac{{\left(a - b\right)} \cos\left(2 \, x\right) - a - b}{\cos\left(2 \, x\right) - 1}} \sin\left(2 \, x\right)}{{\left(a - b\right)} \cos\left(2 \, x\right) + a - b}\right) \sin\left(2 \, x\right) + {\left(a - 2 \, b\right)} \sqrt{-b} \arctan\left(\frac{\sqrt{-b} \sqrt{\frac{{\left(a - b\right)} \cos\left(2 \, x\right) - a - b}{\cos\left(2 \, x\right) - 1}} \sin\left(2 \, x\right)}{b \cos\left(2 \, x\right) + b}\right) \sin\left(2 \, x\right) - {\left(b \cos\left(2 \, x\right) + b\right)} \sqrt{\frac{{\left(a - b\right)} \cos\left(2 \, x\right) - a - b}{\cos\left(2 \, x\right) - 1}}}{2 \, b \sin\left(2 \, x\right)}\right]"," ",0,"[1/4*(2*sqrt(-a + b)*b*log(-(a - b)*cos(2*x) - sqrt(-a + b)*sqrt(((a - b)*cos(2*x) - a - b)/(cos(2*x) - 1))*sin(2*x) + b)*sin(2*x) - (a - 2*b)*sqrt(b)*log(((a - 2*b)*cos(2*x) - 2*sqrt(b)*sqrt(((a - b)*cos(2*x) - a - b)/(cos(2*x) - 1))*sin(2*x) - a - 2*b)/(cos(2*x) - 1))*sin(2*x) - 2*(b*cos(2*x) + b)*sqrt(((a - b)*cos(2*x) - a - b)/(cos(2*x) - 1)))/(b*sin(2*x)), 1/4*(4*sqrt(a - b)*b*arctan(-sqrt(a - b)*sqrt(((a - b)*cos(2*x) - a - b)/(cos(2*x) - 1))*sin(2*x)/((a - b)*cos(2*x) + a - b))*sin(2*x) - (a - 2*b)*sqrt(b)*log(((a - 2*b)*cos(2*x) - 2*sqrt(b)*sqrt(((a - b)*cos(2*x) - a - b)/(cos(2*x) - 1))*sin(2*x) - a - 2*b)/(cos(2*x) - 1))*sin(2*x) - 2*(b*cos(2*x) + b)*sqrt(((a - b)*cos(2*x) - a - b)/(cos(2*x) - 1)))/(b*sin(2*x)), 1/2*((a - 2*b)*sqrt(-b)*arctan(sqrt(-b)*sqrt(((a - b)*cos(2*x) - a - b)/(cos(2*x) - 1))*sin(2*x)/(b*cos(2*x) + b))*sin(2*x) + sqrt(-a + b)*b*log(-(a - b)*cos(2*x) - sqrt(-a + b)*sqrt(((a - b)*cos(2*x) - a - b)/(cos(2*x) - 1))*sin(2*x) + b)*sin(2*x) - (b*cos(2*x) + b)*sqrt(((a - b)*cos(2*x) - a - b)/(cos(2*x) - 1)))/(b*sin(2*x)), 1/2*(2*sqrt(a - b)*b*arctan(-sqrt(a - b)*sqrt(((a - b)*cos(2*x) - a - b)/(cos(2*x) - 1))*sin(2*x)/((a - b)*cos(2*x) + a - b))*sin(2*x) + (a - 2*b)*sqrt(-b)*arctan(sqrt(-b)*sqrt(((a - b)*cos(2*x) - a - b)/(cos(2*x) - 1))*sin(2*x)/(b*cos(2*x) + b))*sin(2*x) - (b*cos(2*x) + b)*sqrt(((a - b)*cos(2*x) - a - b)/(cos(2*x) - 1)))/(b*sin(2*x))]","B",0
23,1,515,0,0.505399," ","integrate((a+b*cot(x)^2)^(1/2),x, algorithm=""fricas"")","\left[\frac{1}{2} \, \sqrt{-a + b} \log\left(-{\left(a - b\right)} \cos\left(2 \, x\right) + \sqrt{-a + b} \sqrt{\frac{{\left(a - b\right)} \cos\left(2 \, x\right) - a - b}{\cos\left(2 \, x\right) - 1}} \sin\left(2 \, x\right) + b\right) + \frac{1}{2} \, \sqrt{b} \log\left(\frac{{\left(a - 2 \, b\right)} \cos\left(2 \, x\right) + 2 \, \sqrt{b} \sqrt{\frac{{\left(a - b\right)} \cos\left(2 \, x\right) - a - b}{\cos\left(2 \, x\right) - 1}} \sin\left(2 \, x\right) - a - 2 \, b}{\cos\left(2 \, x\right) - 1}\right), -\sqrt{a - b} \arctan\left(-\frac{\sqrt{a - b} \sqrt{\frac{{\left(a - b\right)} \cos\left(2 \, x\right) - a - b}{\cos\left(2 \, x\right) - 1}} \sin\left(2 \, x\right)}{{\left(a - b\right)} \cos\left(2 \, x\right) + a - b}\right) + \frac{1}{2} \, \sqrt{b} \log\left(\frac{{\left(a - 2 \, b\right)} \cos\left(2 \, x\right) + 2 \, \sqrt{b} \sqrt{\frac{{\left(a - b\right)} \cos\left(2 \, x\right) - a - b}{\cos\left(2 \, x\right) - 1}} \sin\left(2 \, x\right) - a - 2 \, b}{\cos\left(2 \, x\right) - 1}\right), \sqrt{-b} \arctan\left(\frac{\sqrt{-b} \sqrt{\frac{{\left(a - b\right)} \cos\left(2 \, x\right) - a - b}{\cos\left(2 \, x\right) - 1}} \sin\left(2 \, x\right)}{b \cos\left(2 \, x\right) + b}\right) + \frac{1}{2} \, \sqrt{-a + b} \log\left(-{\left(a - b\right)} \cos\left(2 \, x\right) + \sqrt{-a + b} \sqrt{\frac{{\left(a - b\right)} \cos\left(2 \, x\right) - a - b}{\cos\left(2 \, x\right) - 1}} \sin\left(2 \, x\right) + b\right), -\sqrt{a - b} \arctan\left(-\frac{\sqrt{a - b} \sqrt{\frac{{\left(a - b\right)} \cos\left(2 \, x\right) - a - b}{\cos\left(2 \, x\right) - 1}} \sin\left(2 \, x\right)}{{\left(a - b\right)} \cos\left(2 \, x\right) + a - b}\right) + \sqrt{-b} \arctan\left(\frac{\sqrt{-b} \sqrt{\frac{{\left(a - b\right)} \cos\left(2 \, x\right) - a - b}{\cos\left(2 \, x\right) - 1}} \sin\left(2 \, x\right)}{b \cos\left(2 \, x\right) + b}\right)\right]"," ",0,"[1/2*sqrt(-a + b)*log(-(a - b)*cos(2*x) + sqrt(-a + b)*sqrt(((a - b)*cos(2*x) - a - b)/(cos(2*x) - 1))*sin(2*x) + b) + 1/2*sqrt(b)*log(((a - 2*b)*cos(2*x) + 2*sqrt(b)*sqrt(((a - b)*cos(2*x) - a - b)/(cos(2*x) - 1))*sin(2*x) - a - 2*b)/(cos(2*x) - 1)), -sqrt(a - b)*arctan(-sqrt(a - b)*sqrt(((a - b)*cos(2*x) - a - b)/(cos(2*x) - 1))*sin(2*x)/((a - b)*cos(2*x) + a - b)) + 1/2*sqrt(b)*log(((a - 2*b)*cos(2*x) + 2*sqrt(b)*sqrt(((a - b)*cos(2*x) - a - b)/(cos(2*x) - 1))*sin(2*x) - a - 2*b)/(cos(2*x) - 1)), sqrt(-b)*arctan(sqrt(-b)*sqrt(((a - b)*cos(2*x) - a - b)/(cos(2*x) - 1))*sin(2*x)/(b*cos(2*x) + b)) + 1/2*sqrt(-a + b)*log(-(a - b)*cos(2*x) + sqrt(-a + b)*sqrt(((a - b)*cos(2*x) - a - b)/(cos(2*x) - 1))*sin(2*x) + b), -sqrt(a - b)*arctan(-sqrt(a - b)*sqrt(((a - b)*cos(2*x) - a - b)/(cos(2*x) - 1))*sin(2*x)/((a - b)*cos(2*x) + a - b)) + sqrt(-b)*arctan(sqrt(-b)*sqrt(((a - b)*cos(2*x) - a - b)/(cos(2*x) - 1))*sin(2*x)/(b*cos(2*x) + b))]","B",0
24,1,193,0,0.576595," ","integrate((a+b*cot(x)^2)^(1/2)*tan(x)^2,x, algorithm=""fricas"")","\left[\frac{1}{4} \, \sqrt{-a + b} \log\left(-\frac{a^{2} \tan\left(x\right)^{4} - 2 \, {\left(3 \, a^{2} - 4 \, a b\right)} \tan\left(x\right)^{2} + a^{2} - 8 \, a b + 8 \, b^{2} - 4 \, {\left(a \tan\left(x\right)^{3} - {\left(a - 2 \, b\right)} \tan\left(x\right)\right)} \sqrt{-a + b} \sqrt{\frac{a \tan\left(x\right)^{2} + b}{\tan\left(x\right)^{2}}}}{\tan\left(x\right)^{4} + 2 \, \tan\left(x\right)^{2} + 1}\right) + \sqrt{\frac{a \tan\left(x\right)^{2} + b}{\tan\left(x\right)^{2}}} \tan\left(x\right), \frac{1}{2} \, \sqrt{a - b} \arctan\left(\frac{2 \, \sqrt{a - b} \sqrt{\frac{a \tan\left(x\right)^{2} + b}{\tan\left(x\right)^{2}}} \tan\left(x\right)}{a \tan\left(x\right)^{2} - a + 2 \, b}\right) + \sqrt{\frac{a \tan\left(x\right)^{2} + b}{\tan\left(x\right)^{2}}} \tan\left(x\right)\right]"," ",0,"[1/4*sqrt(-a + b)*log(-(a^2*tan(x)^4 - 2*(3*a^2 - 4*a*b)*tan(x)^2 + a^2 - 8*a*b + 8*b^2 - 4*(a*tan(x)^3 - (a - 2*b)*tan(x))*sqrt(-a + b)*sqrt((a*tan(x)^2 + b)/tan(x)^2))/(tan(x)^4 + 2*tan(x)^2 + 1)) + sqrt((a*tan(x)^2 + b)/tan(x)^2)*tan(x), 1/2*sqrt(a - b)*arctan(2*sqrt(a - b)*sqrt((a*tan(x)^2 + b)/tan(x)^2)*tan(x)/(a*tan(x)^2 - a + 2*b)) + sqrt((a*tan(x)^2 + b)/tan(x)^2)*tan(x)]","A",0
25,1,239,0,0.585561," ","integrate((a+b*cot(x)^2)^(1/2)*tan(x)^4,x, algorithm=""fricas"")","\left[\frac{3 \, a \sqrt{-a + b} \log\left(-\frac{a^{2} \tan\left(x\right)^{4} - 2 \, {\left(3 \, a^{2} - 4 \, a b\right)} \tan\left(x\right)^{2} + a^{2} - 8 \, a b + 8 \, b^{2} + 4 \, {\left(a \tan\left(x\right)^{3} - {\left(a - 2 \, b\right)} \tan\left(x\right)\right)} \sqrt{-a + b} \sqrt{\frac{a \tan\left(x\right)^{2} + b}{\tan\left(x\right)^{2}}}}{\tan\left(x\right)^{4} + 2 \, \tan\left(x\right)^{2} + 1}\right) + 4 \, {\left(a \tan\left(x\right)^{3} - {\left(3 \, a - b\right)} \tan\left(x\right)\right)} \sqrt{\frac{a \tan\left(x\right)^{2} + b}{\tan\left(x\right)^{2}}}}{12 \, a}, -\frac{3 \, \sqrt{a - b} a \arctan\left(\frac{2 \, \sqrt{a - b} \sqrt{\frac{a \tan\left(x\right)^{2} + b}{\tan\left(x\right)^{2}}} \tan\left(x\right)}{a \tan\left(x\right)^{2} - a + 2 \, b}\right) - 2 \, {\left(a \tan\left(x\right)^{3} - {\left(3 \, a - b\right)} \tan\left(x\right)\right)} \sqrt{\frac{a \tan\left(x\right)^{2} + b}{\tan\left(x\right)^{2}}}}{6 \, a}\right]"," ",0,"[1/12*(3*a*sqrt(-a + b)*log(-(a^2*tan(x)^4 - 2*(3*a^2 - 4*a*b)*tan(x)^2 + a^2 - 8*a*b + 8*b^2 + 4*(a*tan(x)^3 - (a - 2*b)*tan(x))*sqrt(-a + b)*sqrt((a*tan(x)^2 + b)/tan(x)^2))/(tan(x)^4 + 2*tan(x)^2 + 1)) + 4*(a*tan(x)^3 - (3*a - b)*tan(x))*sqrt((a*tan(x)^2 + b)/tan(x)^2))/a, -1/6*(3*sqrt(a - b)*a*arctan(2*sqrt(a - b)*sqrt((a*tan(x)^2 + b)/tan(x)^2)*tan(x)/(a*tan(x)^2 - a + 2*b)) - 2*(a*tan(x)^3 - (3*a - b)*tan(x))*sqrt((a*tan(x)^2 + b)/tan(x)^2))/a]","A",0
26,1,486,0,0.606777," ","integrate(cot(x)^3*(a+b*cot(x)^2)^(3/2),x, algorithm=""fricas"")","\left[-\frac{15 \, {\left({\left(a b - b^{2}\right)} \cos\left(2 \, x\right)^{2} + a b - b^{2} - 2 \, {\left(a b - b^{2}\right)} \cos\left(2 \, x\right)\right)} \sqrt{a - b} \log\left(-2 \, {\left(a^{2} - 2 \, a b + b^{2}\right)} \cos\left(2 \, x\right)^{2} - 2 \, a^{2} + b^{2} - 2 \, {\left({\left(a - b\right)} \cos\left(2 \, x\right)^{2} - {\left(2 \, a - b\right)} \cos\left(2 \, x\right) + a\right)} \sqrt{a - b} \sqrt{\frac{{\left(a - b\right)} \cos\left(2 \, x\right) - a - b}{\cos\left(2 \, x\right) - 1}} + 4 \, {\left(a^{2} - a b\right)} \cos\left(2 \, x\right)\right) + 4 \, {\left({\left(3 \, a^{2} - 26 \, a b + 23 \, b^{2}\right)} \cos\left(2 \, x\right)^{2} + 3 \, a^{2} - 14 \, a b + 13 \, b^{2} - 2 \, {\left(3 \, a^{2} - 20 \, a b + 12 \, b^{2}\right)} \cos\left(2 \, x\right)\right)} \sqrt{\frac{{\left(a - b\right)} \cos\left(2 \, x\right) - a - b}{\cos\left(2 \, x\right) - 1}}}{60 \, {\left(b \cos\left(2 \, x\right)^{2} - 2 \, b \cos\left(2 \, x\right) + b\right)}}, -\frac{15 \, {\left({\left(a b - b^{2}\right)} \cos\left(2 \, x\right)^{2} + a b - b^{2} - 2 \, {\left(a b - b^{2}\right)} \cos\left(2 \, x\right)\right)} \sqrt{-a + b} \arctan\left(-\frac{\sqrt{-a + b} \sqrt{\frac{{\left(a - b\right)} \cos\left(2 \, x\right) - a - b}{\cos\left(2 \, x\right) - 1}} {\left(\cos\left(2 \, x\right) - 1\right)}}{{\left(a - b\right)} \cos\left(2 \, x\right) - a}\right) + 2 \, {\left({\left(3 \, a^{2} - 26 \, a b + 23 \, b^{2}\right)} \cos\left(2 \, x\right)^{2} + 3 \, a^{2} - 14 \, a b + 13 \, b^{2} - 2 \, {\left(3 \, a^{2} - 20 \, a b + 12 \, b^{2}\right)} \cos\left(2 \, x\right)\right)} \sqrt{\frac{{\left(a - b\right)} \cos\left(2 \, x\right) - a - b}{\cos\left(2 \, x\right) - 1}}}{30 \, {\left(b \cos\left(2 \, x\right)^{2} - 2 \, b \cos\left(2 \, x\right) + b\right)}}\right]"," ",0,"[-1/60*(15*((a*b - b^2)*cos(2*x)^2 + a*b - b^2 - 2*(a*b - b^2)*cos(2*x))*sqrt(a - b)*log(-2*(a^2 - 2*a*b + b^2)*cos(2*x)^2 - 2*a^2 + b^2 - 2*((a - b)*cos(2*x)^2 - (2*a - b)*cos(2*x) + a)*sqrt(a - b)*sqrt(((a - b)*cos(2*x) - a - b)/(cos(2*x) - 1)) + 4*(a^2 - a*b)*cos(2*x)) + 4*((3*a^2 - 26*a*b + 23*b^2)*cos(2*x)^2 + 3*a^2 - 14*a*b + 13*b^2 - 2*(3*a^2 - 20*a*b + 12*b^2)*cos(2*x))*sqrt(((a - b)*cos(2*x) - a - b)/(cos(2*x) - 1)))/(b*cos(2*x)^2 - 2*b*cos(2*x) + b), -1/30*(15*((a*b - b^2)*cos(2*x)^2 + a*b - b^2 - 2*(a*b - b^2)*cos(2*x))*sqrt(-a + b)*arctan(-sqrt(-a + b)*sqrt(((a - b)*cos(2*x) - a - b)/(cos(2*x) - 1))*(cos(2*x) - 1)/((a - b)*cos(2*x) - a)) + 2*((3*a^2 - 26*a*b + 23*b^2)*cos(2*x)^2 + 3*a^2 - 14*a*b + 13*b^2 - 2*(3*a^2 - 20*a*b + 12*b^2)*cos(2*x))*sqrt(((a - b)*cos(2*x) - a - b)/(cos(2*x) - 1)))/(b*cos(2*x)^2 - 2*b*cos(2*x) + b)]","B",0
27,1,1134,0,0.680978," ","integrate(cot(x)^2*(a+b*cot(x)^2)^(3/2),x, algorithm=""fricas"")","\left[\frac{8 \, {\left(a b - b^{2} - {\left(a b - b^{2}\right)} \cos\left(2 \, x\right)\right)} \sqrt{-a + b} \log\left(-{\left(a - b\right)} \cos\left(2 \, x\right) + \sqrt{-a + b} \sqrt{\frac{{\left(a - b\right)} \cos\left(2 \, x\right) - a - b}{\cos\left(2 \, x\right) - 1}} \sin\left(2 \, x\right) + b\right) \sin\left(2 \, x\right) - {\left(3 \, a^{2} - 12 \, a b + 8 \, b^{2} - {\left(3 \, a^{2} - 12 \, a b + 8 \, b^{2}\right)} \cos\left(2 \, x\right)\right)} \sqrt{b} \log\left(\frac{{\left(a - 2 \, b\right)} \cos\left(2 \, x\right) + 2 \, \sqrt{b} \sqrt{\frac{{\left(a - b\right)} \cos\left(2 \, x\right) - a - b}{\cos\left(2 \, x\right) - 1}} \sin\left(2 \, x\right) - a - 2 \, b}{\cos\left(2 \, x\right) - 1}\right) \sin\left(2 \, x\right) + 2 \, {\left(4 \, b^{2} \cos\left(2 \, x\right) - {\left(5 \, a b - 6 \, b^{2}\right)} \cos\left(2 \, x\right)^{2} + 5 \, a b - 2 \, b^{2}\right)} \sqrt{\frac{{\left(a - b\right)} \cos\left(2 \, x\right) - a - b}{\cos\left(2 \, x\right) - 1}}}{16 \, {\left(b \cos\left(2 \, x\right) - b\right)} \sin\left(2 \, x\right)}, -\frac{{\left(3 \, a^{2} - 12 \, a b + 8 \, b^{2} - {\left(3 \, a^{2} - 12 \, a b + 8 \, b^{2}\right)} \cos\left(2 \, x\right)\right)} \sqrt{-b} \arctan\left(\frac{\sqrt{-b} \sqrt{\frac{{\left(a - b\right)} \cos\left(2 \, x\right) - a - b}{\cos\left(2 \, x\right) - 1}} \sin\left(2 \, x\right)}{b \cos\left(2 \, x\right) + b}\right) \sin\left(2 \, x\right) - 4 \, {\left(a b - b^{2} - {\left(a b - b^{2}\right)} \cos\left(2 \, x\right)\right)} \sqrt{-a + b} \log\left(-{\left(a - b\right)} \cos\left(2 \, x\right) + \sqrt{-a + b} \sqrt{\frac{{\left(a - b\right)} \cos\left(2 \, x\right) - a - b}{\cos\left(2 \, x\right) - 1}} \sin\left(2 \, x\right) + b\right) \sin\left(2 \, x\right) - {\left(4 \, b^{2} \cos\left(2 \, x\right) - {\left(5 \, a b - 6 \, b^{2}\right)} \cos\left(2 \, x\right)^{2} + 5 \, a b - 2 \, b^{2}\right)} \sqrt{\frac{{\left(a - b\right)} \cos\left(2 \, x\right) - a - b}{\cos\left(2 \, x\right) - 1}}}{8 \, {\left(b \cos\left(2 \, x\right) - b\right)} \sin\left(2 \, x\right)}, -\frac{16 \, {\left(a b - b^{2} - {\left(a b - b^{2}\right)} \cos\left(2 \, x\right)\right)} \sqrt{a - b} \arctan\left(-\frac{\sqrt{a - b} \sqrt{\frac{{\left(a - b\right)} \cos\left(2 \, x\right) - a - b}{\cos\left(2 \, x\right) - 1}} \sin\left(2 \, x\right)}{{\left(a - b\right)} \cos\left(2 \, x\right) + a - b}\right) \sin\left(2 \, x\right) + {\left(3 \, a^{2} - 12 \, a b + 8 \, b^{2} - {\left(3 \, a^{2} - 12 \, a b + 8 \, b^{2}\right)} \cos\left(2 \, x\right)\right)} \sqrt{b} \log\left(\frac{{\left(a - 2 \, b\right)} \cos\left(2 \, x\right) + 2 \, \sqrt{b} \sqrt{\frac{{\left(a - b\right)} \cos\left(2 \, x\right) - a - b}{\cos\left(2 \, x\right) - 1}} \sin\left(2 \, x\right) - a - 2 \, b}{\cos\left(2 \, x\right) - 1}\right) \sin\left(2 \, x\right) - 2 \, {\left(4 \, b^{2} \cos\left(2 \, x\right) - {\left(5 \, a b - 6 \, b^{2}\right)} \cos\left(2 \, x\right)^{2} + 5 \, a b - 2 \, b^{2}\right)} \sqrt{\frac{{\left(a - b\right)} \cos\left(2 \, x\right) - a - b}{\cos\left(2 \, x\right) - 1}}}{16 \, {\left(b \cos\left(2 \, x\right) - b\right)} \sin\left(2 \, x\right)}, -\frac{8 \, {\left(a b - b^{2} - {\left(a b - b^{2}\right)} \cos\left(2 \, x\right)\right)} \sqrt{a - b} \arctan\left(-\frac{\sqrt{a - b} \sqrt{\frac{{\left(a - b\right)} \cos\left(2 \, x\right) - a - b}{\cos\left(2 \, x\right) - 1}} \sin\left(2 \, x\right)}{{\left(a - b\right)} \cos\left(2 \, x\right) + a - b}\right) \sin\left(2 \, x\right) + {\left(3 \, a^{2} - 12 \, a b + 8 \, b^{2} - {\left(3 \, a^{2} - 12 \, a b + 8 \, b^{2}\right)} \cos\left(2 \, x\right)\right)} \sqrt{-b} \arctan\left(\frac{\sqrt{-b} \sqrt{\frac{{\left(a - b\right)} \cos\left(2 \, x\right) - a - b}{\cos\left(2 \, x\right) - 1}} \sin\left(2 \, x\right)}{b \cos\left(2 \, x\right) + b}\right) \sin\left(2 \, x\right) - {\left(4 \, b^{2} \cos\left(2 \, x\right) - {\left(5 \, a b - 6 \, b^{2}\right)} \cos\left(2 \, x\right)^{2} + 5 \, a b - 2 \, b^{2}\right)} \sqrt{\frac{{\left(a - b\right)} \cos\left(2 \, x\right) - a - b}{\cos\left(2 \, x\right) - 1}}}{8 \, {\left(b \cos\left(2 \, x\right) - b\right)} \sin\left(2 \, x\right)}\right]"," ",0,"[1/16*(8*(a*b - b^2 - (a*b - b^2)*cos(2*x))*sqrt(-a + b)*log(-(a - b)*cos(2*x) + sqrt(-a + b)*sqrt(((a - b)*cos(2*x) - a - b)/(cos(2*x) - 1))*sin(2*x) + b)*sin(2*x) - (3*a^2 - 12*a*b + 8*b^2 - (3*a^2 - 12*a*b + 8*b^2)*cos(2*x))*sqrt(b)*log(((a - 2*b)*cos(2*x) + 2*sqrt(b)*sqrt(((a - b)*cos(2*x) - a - b)/(cos(2*x) - 1))*sin(2*x) - a - 2*b)/(cos(2*x) - 1))*sin(2*x) + 2*(4*b^2*cos(2*x) - (5*a*b - 6*b^2)*cos(2*x)^2 + 5*a*b - 2*b^2)*sqrt(((a - b)*cos(2*x) - a - b)/(cos(2*x) - 1)))/((b*cos(2*x) - b)*sin(2*x)), -1/8*((3*a^2 - 12*a*b + 8*b^2 - (3*a^2 - 12*a*b + 8*b^2)*cos(2*x))*sqrt(-b)*arctan(sqrt(-b)*sqrt(((a - b)*cos(2*x) - a - b)/(cos(2*x) - 1))*sin(2*x)/(b*cos(2*x) + b))*sin(2*x) - 4*(a*b - b^2 - (a*b - b^2)*cos(2*x))*sqrt(-a + b)*log(-(a - b)*cos(2*x) + sqrt(-a + b)*sqrt(((a - b)*cos(2*x) - a - b)/(cos(2*x) - 1))*sin(2*x) + b)*sin(2*x) - (4*b^2*cos(2*x) - (5*a*b - 6*b^2)*cos(2*x)^2 + 5*a*b - 2*b^2)*sqrt(((a - b)*cos(2*x) - a - b)/(cos(2*x) - 1)))/((b*cos(2*x) - b)*sin(2*x)), -1/16*(16*(a*b - b^2 - (a*b - b^2)*cos(2*x))*sqrt(a - b)*arctan(-sqrt(a - b)*sqrt(((a - b)*cos(2*x) - a - b)/(cos(2*x) - 1))*sin(2*x)/((a - b)*cos(2*x) + a - b))*sin(2*x) + (3*a^2 - 12*a*b + 8*b^2 - (3*a^2 - 12*a*b + 8*b^2)*cos(2*x))*sqrt(b)*log(((a - 2*b)*cos(2*x) + 2*sqrt(b)*sqrt(((a - b)*cos(2*x) - a - b)/(cos(2*x) - 1))*sin(2*x) - a - 2*b)/(cos(2*x) - 1))*sin(2*x) - 2*(4*b^2*cos(2*x) - (5*a*b - 6*b^2)*cos(2*x)^2 + 5*a*b - 2*b^2)*sqrt(((a - b)*cos(2*x) - a - b)/(cos(2*x) - 1)))/((b*cos(2*x) - b)*sin(2*x)), -1/8*(8*(a*b - b^2 - (a*b - b^2)*cos(2*x))*sqrt(a - b)*arctan(-sqrt(a - b)*sqrt(((a - b)*cos(2*x) - a - b)/(cos(2*x) - 1))*sin(2*x)/((a - b)*cos(2*x) + a - b))*sin(2*x) + (3*a^2 - 12*a*b + 8*b^2 - (3*a^2 - 12*a*b + 8*b^2)*cos(2*x))*sqrt(-b)*arctan(sqrt(-b)*sqrt(((a - b)*cos(2*x) - a - b)/(cos(2*x) - 1))*sin(2*x)/(b*cos(2*x) + b))*sin(2*x) - (4*b^2*cos(2*x) - (5*a*b - 6*b^2)*cos(2*x)^2 + 5*a*b - 2*b^2)*sqrt(((a - b)*cos(2*x) - a - b)/(cos(2*x) - 1)))/((b*cos(2*x) - b)*sin(2*x))]","B",0
28,1,330,0,0.674885," ","integrate(cot(x)*(a+b*cot(x)^2)^(3/2),x, algorithm=""fricas"")","\left[-\frac{3 \, {\left({\left(a - b\right)} \cos\left(2 \, x\right) - a + b\right)} \sqrt{a - b} \log\left(-2 \, {\left(a^{2} - 2 \, a b + b^{2}\right)} \cos\left(2 \, x\right)^{2} - 2 \, a^{2} + b^{2} + 2 \, {\left({\left(a - b\right)} \cos\left(2 \, x\right)^{2} - {\left(2 \, a - b\right)} \cos\left(2 \, x\right) + a\right)} \sqrt{a - b} \sqrt{\frac{{\left(a - b\right)} \cos\left(2 \, x\right) - a - b}{\cos\left(2 \, x\right) - 1}} + 4 \, {\left(a^{2} - a b\right)} \cos\left(2 \, x\right)\right) + 8 \, {\left(2 \, {\left(a - b\right)} \cos\left(2 \, x\right) - 2 \, a + b\right)} \sqrt{\frac{{\left(a - b\right)} \cos\left(2 \, x\right) - a - b}{\cos\left(2 \, x\right) - 1}}}{12 \, {\left(\cos\left(2 \, x\right) - 1\right)}}, \frac{3 \, {\left({\left(a - b\right)} \cos\left(2 \, x\right) - a + b\right)} \sqrt{-a + b} \arctan\left(-\frac{\sqrt{-a + b} \sqrt{\frac{{\left(a - b\right)} \cos\left(2 \, x\right) - a - b}{\cos\left(2 \, x\right) - 1}} {\left(\cos\left(2 \, x\right) - 1\right)}}{{\left(a - b\right)} \cos\left(2 \, x\right) - a}\right) - 4 \, {\left(2 \, {\left(a - b\right)} \cos\left(2 \, x\right) - 2 \, a + b\right)} \sqrt{\frac{{\left(a - b\right)} \cos\left(2 \, x\right) - a - b}{\cos\left(2 \, x\right) - 1}}}{6 \, {\left(\cos\left(2 \, x\right) - 1\right)}}\right]"," ",0,"[-1/12*(3*((a - b)*cos(2*x) - a + b)*sqrt(a - b)*log(-2*(a^2 - 2*a*b + b^2)*cos(2*x)^2 - 2*a^2 + b^2 + 2*((a - b)*cos(2*x)^2 - (2*a - b)*cos(2*x) + a)*sqrt(a - b)*sqrt(((a - b)*cos(2*x) - a - b)/(cos(2*x) - 1)) + 4*(a^2 - a*b)*cos(2*x)) + 8*(2*(a - b)*cos(2*x) - 2*a + b)*sqrt(((a - b)*cos(2*x) - a - b)/(cos(2*x) - 1)))/(cos(2*x) - 1), 1/6*(3*((a - b)*cos(2*x) - a + b)*sqrt(-a + b)*arctan(-sqrt(-a + b)*sqrt(((a - b)*cos(2*x) - a - b)/(cos(2*x) - 1))*(cos(2*x) - 1)/((a - b)*cos(2*x) - a)) - 4*(2*(a - b)*cos(2*x) - 2*a + b)*sqrt(((a - b)*cos(2*x) - a - b)/(cos(2*x) - 1)))/(cos(2*x) - 1)]","B",0
29,1,565,0,1.778567," ","integrate((a+b*cot(x)^2)^(3/2)*tan(x),x, algorithm=""fricas"")","\left[\frac{1}{2} \, a^{\frac{3}{2}} \log\left(2 \, a \tan\left(x\right)^{2} + 2 \, \sqrt{a} \sqrt{\frac{a \tan\left(x\right)^{2} + b}{\tan\left(x\right)^{2}}} \tan\left(x\right)^{2} + b\right) - \frac{1}{4} \, {\left(a - b\right)}^{\frac{3}{2}} \log\left(-\frac{{\left(8 \, a^{2} - 8 \, a b + b^{2}\right)} \tan\left(x\right)^{4} + 2 \, {\left(4 \, a b - 3 \, b^{2}\right)} \tan\left(x\right)^{2} + b^{2} + 4 \, {\left({\left(2 \, a - b\right)} \tan\left(x\right)^{4} + b \tan\left(x\right)^{2}\right)} \sqrt{a - b} \sqrt{\frac{a \tan\left(x\right)^{2} + b}{\tan\left(x\right)^{2}}}}{\tan\left(x\right)^{4} + 2 \, \tan\left(x\right)^{2} + 1}\right) - b \sqrt{\frac{a \tan\left(x\right)^{2} + b}{\tan\left(x\right)^{2}}}, -\sqrt{-a} a \arctan\left(\frac{\sqrt{-a} \sqrt{\frac{a \tan\left(x\right)^{2} + b}{\tan\left(x\right)^{2}}} \tan\left(x\right)^{2}}{a \tan\left(x\right)^{2} + b}\right) - \frac{1}{4} \, {\left(a - b\right)}^{\frac{3}{2}} \log\left(-\frac{{\left(8 \, a^{2} - 8 \, a b + b^{2}\right)} \tan\left(x\right)^{4} + 2 \, {\left(4 \, a b - 3 \, b^{2}\right)} \tan\left(x\right)^{2} + b^{2} + 4 \, {\left({\left(2 \, a - b\right)} \tan\left(x\right)^{4} + b \tan\left(x\right)^{2}\right)} \sqrt{a - b} \sqrt{\frac{a \tan\left(x\right)^{2} + b}{\tan\left(x\right)^{2}}}}{\tan\left(x\right)^{4} + 2 \, \tan\left(x\right)^{2} + 1}\right) - b \sqrt{\frac{a \tan\left(x\right)^{2} + b}{\tan\left(x\right)^{2}}}, \frac{1}{2} \, {\left(-a + b\right)}^{\frac{3}{2}} \arctan\left(-\frac{2 \, \sqrt{-a + b} \sqrt{\frac{a \tan\left(x\right)^{2} + b}{\tan\left(x\right)^{2}}} \tan\left(x\right)^{2}}{{\left(2 \, a - b\right)} \tan\left(x\right)^{2} + b}\right) + \frac{1}{2} \, a^{\frac{3}{2}} \log\left(2 \, a \tan\left(x\right)^{2} + 2 \, \sqrt{a} \sqrt{\frac{a \tan\left(x\right)^{2} + b}{\tan\left(x\right)^{2}}} \tan\left(x\right)^{2} + b\right) - b \sqrt{\frac{a \tan\left(x\right)^{2} + b}{\tan\left(x\right)^{2}}}, -\sqrt{-a} a \arctan\left(\frac{\sqrt{-a} \sqrt{\frac{a \tan\left(x\right)^{2} + b}{\tan\left(x\right)^{2}}} \tan\left(x\right)^{2}}{a \tan\left(x\right)^{2} + b}\right) + \frac{1}{2} \, {\left(-a + b\right)}^{\frac{3}{2}} \arctan\left(-\frac{2 \, \sqrt{-a + b} \sqrt{\frac{a \tan\left(x\right)^{2} + b}{\tan\left(x\right)^{2}}} \tan\left(x\right)^{2}}{{\left(2 \, a - b\right)} \tan\left(x\right)^{2} + b}\right) - b \sqrt{\frac{a \tan\left(x\right)^{2} + b}{\tan\left(x\right)^{2}}}\right]"," ",0,"[1/2*a^(3/2)*log(2*a*tan(x)^2 + 2*sqrt(a)*sqrt((a*tan(x)^2 + b)/tan(x)^2)*tan(x)^2 + b) - 1/4*(a - b)^(3/2)*log(-((8*a^2 - 8*a*b + b^2)*tan(x)^4 + 2*(4*a*b - 3*b^2)*tan(x)^2 + b^2 + 4*((2*a - b)*tan(x)^4 + b*tan(x)^2)*sqrt(a - b)*sqrt((a*tan(x)^2 + b)/tan(x)^2))/(tan(x)^4 + 2*tan(x)^2 + 1)) - b*sqrt((a*tan(x)^2 + b)/tan(x)^2), -sqrt(-a)*a*arctan(sqrt(-a)*sqrt((a*tan(x)^2 + b)/tan(x)^2)*tan(x)^2/(a*tan(x)^2 + b)) - 1/4*(a - b)^(3/2)*log(-((8*a^2 - 8*a*b + b^2)*tan(x)^4 + 2*(4*a*b - 3*b^2)*tan(x)^2 + b^2 + 4*((2*a - b)*tan(x)^4 + b*tan(x)^2)*sqrt(a - b)*sqrt((a*tan(x)^2 + b)/tan(x)^2))/(tan(x)^4 + 2*tan(x)^2 + 1)) - b*sqrt((a*tan(x)^2 + b)/tan(x)^2), 1/2*(-a + b)^(3/2)*arctan(-2*sqrt(-a + b)*sqrt((a*tan(x)^2 + b)/tan(x)^2)*tan(x)^2/((2*a - b)*tan(x)^2 + b)) + 1/2*a^(3/2)*log(2*a*tan(x)^2 + 2*sqrt(a)*sqrt((a*tan(x)^2 + b)/tan(x)^2)*tan(x)^2 + b) - b*sqrt((a*tan(x)^2 + b)/tan(x)^2), -sqrt(-a)*a*arctan(sqrt(-a)*sqrt((a*tan(x)^2 + b)/tan(x)^2)*tan(x)^2/(a*tan(x)^2 + b)) + 1/2*(-a + b)^(3/2)*arctan(-2*sqrt(-a + b)*sqrt((a*tan(x)^2 + b)/tan(x)^2)*tan(x)^2/((2*a - b)*tan(x)^2 + b)) - b*sqrt((a*tan(x)^2 + b)/tan(x)^2)]","A",0
30,1,543,0,1.433164," ","integrate((a+b*cot(x)^2)^(3/2)*tan(x)^2,x, algorithm=""fricas"")","\left[\frac{1}{4} \, {\left(-a + b\right)}^{\frac{3}{2}} \log\left(-\frac{a^{2} \tan\left(x\right)^{4} - 2 \, {\left(3 \, a^{2} - 4 \, a b\right)} \tan\left(x\right)^{2} + a^{2} - 8 \, a b + 8 \, b^{2} + 4 \, {\left(a \tan\left(x\right)^{3} - {\left(a - 2 \, b\right)} \tan\left(x\right)\right)} \sqrt{-a + b} \sqrt{\frac{a \tan\left(x\right)^{2} + b}{\tan\left(x\right)^{2}}}}{\tan\left(x\right)^{4} + 2 \, \tan\left(x\right)^{2} + 1}\right) + \frac{1}{2} \, b^{\frac{3}{2}} \log\left(\frac{a \tan\left(x\right)^{2} - 2 \, \sqrt{b} \sqrt{\frac{a \tan\left(x\right)^{2} + b}{\tan\left(x\right)^{2}}} \tan\left(x\right) + 2 \, b}{\tan\left(x\right)^{2}}\right) + a \sqrt{\frac{a \tan\left(x\right)^{2} + b}{\tan\left(x\right)^{2}}} \tan\left(x\right), \sqrt{-b} b \arctan\left(\frac{\sqrt{-b} \sqrt{\frac{a \tan\left(x\right)^{2} + b}{\tan\left(x\right)^{2}}} \tan\left(x\right)}{b}\right) + \frac{1}{4} \, {\left(-a + b\right)}^{\frac{3}{2}} \log\left(-\frac{a^{2} \tan\left(x\right)^{4} - 2 \, {\left(3 \, a^{2} - 4 \, a b\right)} \tan\left(x\right)^{2} + a^{2} - 8 \, a b + 8 \, b^{2} + 4 \, {\left(a \tan\left(x\right)^{3} - {\left(a - 2 \, b\right)} \tan\left(x\right)\right)} \sqrt{-a + b} \sqrt{\frac{a \tan\left(x\right)^{2} + b}{\tan\left(x\right)^{2}}}}{\tan\left(x\right)^{4} + 2 \, \tan\left(x\right)^{2} + 1}\right) + a \sqrt{\frac{a \tan\left(x\right)^{2} + b}{\tan\left(x\right)^{2}}} \tan\left(x\right), \frac{1}{2} \, {\left(a - b\right)}^{\frac{3}{2}} \arctan\left(\frac{2 \, \sqrt{a - b} \sqrt{\frac{a \tan\left(x\right)^{2} + b}{\tan\left(x\right)^{2}}} \tan\left(x\right)}{a \tan\left(x\right)^{2} - a + 2 \, b}\right) + \frac{1}{2} \, b^{\frac{3}{2}} \log\left(\frac{a \tan\left(x\right)^{2} - 2 \, \sqrt{b} \sqrt{\frac{a \tan\left(x\right)^{2} + b}{\tan\left(x\right)^{2}}} \tan\left(x\right) + 2 \, b}{\tan\left(x\right)^{2}}\right) + a \sqrt{\frac{a \tan\left(x\right)^{2} + b}{\tan\left(x\right)^{2}}} \tan\left(x\right), \frac{1}{2} \, {\left(a - b\right)}^{\frac{3}{2}} \arctan\left(\frac{2 \, \sqrt{a - b} \sqrt{\frac{a \tan\left(x\right)^{2} + b}{\tan\left(x\right)^{2}}} \tan\left(x\right)}{a \tan\left(x\right)^{2} - a + 2 \, b}\right) + \sqrt{-b} b \arctan\left(\frac{\sqrt{-b} \sqrt{\frac{a \tan\left(x\right)^{2} + b}{\tan\left(x\right)^{2}}} \tan\left(x\right)}{b}\right) + a \sqrt{\frac{a \tan\left(x\right)^{2} + b}{\tan\left(x\right)^{2}}} \tan\left(x\right)\right]"," ",0,"[1/4*(-a + b)^(3/2)*log(-(a^2*tan(x)^4 - 2*(3*a^2 - 4*a*b)*tan(x)^2 + a^2 - 8*a*b + 8*b^2 + 4*(a*tan(x)^3 - (a - 2*b)*tan(x))*sqrt(-a + b)*sqrt((a*tan(x)^2 + b)/tan(x)^2))/(tan(x)^4 + 2*tan(x)^2 + 1)) + 1/2*b^(3/2)*log((a*tan(x)^2 - 2*sqrt(b)*sqrt((a*tan(x)^2 + b)/tan(x)^2)*tan(x) + 2*b)/tan(x)^2) + a*sqrt((a*tan(x)^2 + b)/tan(x)^2)*tan(x), sqrt(-b)*b*arctan(sqrt(-b)*sqrt((a*tan(x)^2 + b)/tan(x)^2)*tan(x)/b) + 1/4*(-a + b)^(3/2)*log(-(a^2*tan(x)^4 - 2*(3*a^2 - 4*a*b)*tan(x)^2 + a^2 - 8*a*b + 8*b^2 + 4*(a*tan(x)^3 - (a - 2*b)*tan(x))*sqrt(-a + b)*sqrt((a*tan(x)^2 + b)/tan(x)^2))/(tan(x)^4 + 2*tan(x)^2 + 1)) + a*sqrt((a*tan(x)^2 + b)/tan(x)^2)*tan(x), 1/2*(a - b)^(3/2)*arctan(2*sqrt(a - b)*sqrt((a*tan(x)^2 + b)/tan(x)^2)*tan(x)/(a*tan(x)^2 - a + 2*b)) + 1/2*b^(3/2)*log((a*tan(x)^2 - 2*sqrt(b)*sqrt((a*tan(x)^2 + b)/tan(x)^2)*tan(x) + 2*b)/tan(x)^2) + a*sqrt((a*tan(x)^2 + b)/tan(x)^2)*tan(x), 1/2*(a - b)^(3/2)*arctan(2*sqrt(a - b)*sqrt((a*tan(x)^2 + b)/tan(x)^2)*tan(x)/(a*tan(x)^2 - a + 2*b)) + sqrt(-b)*b*arctan(sqrt(-b)*sqrt((a*tan(x)^2 + b)/tan(x)^2)*tan(x)/b) + a*sqrt((a*tan(x)^2 + b)/tan(x)^2)*tan(x)]","A",0
31,1,1520,0,0.527256," ","integrate((a+b*cot(d*x+c)^2)^(5/2),x, algorithm=""fricas"")","\left[-\frac{8 \, {\left(a^{2} - 2 \, a b + b^{2} - {\left(a^{2} - 2 \, a b + b^{2}\right)} \cos\left(2 \, d x + 2 \, c\right)\right)} \sqrt{-a + b} \log\left(-{\left(a - b\right)} \cos\left(2 \, d x + 2 \, c\right) + \sqrt{-a + b} \sqrt{\frac{{\left(a - b\right)} \cos\left(2 \, d x + 2 \, c\right) - a - b}{\cos\left(2 \, d x + 2 \, c\right) - 1}} \sin\left(2 \, d x + 2 \, c\right) + b\right) \sin\left(2 \, d x + 2 \, c\right) + {\left(15 \, a^{2} - 20 \, a b + 8 \, b^{2} - {\left(15 \, a^{2} - 20 \, a b + 8 \, b^{2}\right)} \cos\left(2 \, d x + 2 \, c\right)\right)} \sqrt{b} \log\left(\frac{{\left(a - 2 \, b\right)} \cos\left(2 \, d x + 2 \, c\right) + 2 \, \sqrt{b} \sqrt{\frac{{\left(a - b\right)} \cos\left(2 \, d x + 2 \, c\right) - a - b}{\cos\left(2 \, d x + 2 \, c\right) - 1}} \sin\left(2 \, d x + 2 \, c\right) - a - 2 \, b}{\cos\left(2 \, d x + 2 \, c\right) - 1}\right) \sin\left(2 \, d x + 2 \, c\right) - 2 \, {\left(4 \, b^{2} \cos\left(2 \, d x + 2 \, c\right) - 3 \, {\left(3 \, a b - 2 \, b^{2}\right)} \cos\left(2 \, d x + 2 \, c\right)^{2} + 9 \, a b - 2 \, b^{2}\right)} \sqrt{\frac{{\left(a - b\right)} \cos\left(2 \, d x + 2 \, c\right) - a - b}{\cos\left(2 \, d x + 2 \, c\right) - 1}}}{16 \, {\left(d \cos\left(2 \, d x + 2 \, c\right) - d\right)} \sin\left(2 \, d x + 2 \, c\right)}, \frac{16 \, {\left(a^{2} - 2 \, a b + b^{2} - {\left(a^{2} - 2 \, a b + b^{2}\right)} \cos\left(2 \, d x + 2 \, c\right)\right)} \sqrt{a - b} \arctan\left(-\frac{\sqrt{a - b} \sqrt{\frac{{\left(a - b\right)} \cos\left(2 \, d x + 2 \, c\right) - a - b}{\cos\left(2 \, d x + 2 \, c\right) - 1}} \sin\left(2 \, d x + 2 \, c\right)}{{\left(a - b\right)} \cos\left(2 \, d x + 2 \, c\right) + a - b}\right) \sin\left(2 \, d x + 2 \, c\right) - {\left(15 \, a^{2} - 20 \, a b + 8 \, b^{2} - {\left(15 \, a^{2} - 20 \, a b + 8 \, b^{2}\right)} \cos\left(2 \, d x + 2 \, c\right)\right)} \sqrt{b} \log\left(\frac{{\left(a - 2 \, b\right)} \cos\left(2 \, d x + 2 \, c\right) + 2 \, \sqrt{b} \sqrt{\frac{{\left(a - b\right)} \cos\left(2 \, d x + 2 \, c\right) - a - b}{\cos\left(2 \, d x + 2 \, c\right) - 1}} \sin\left(2 \, d x + 2 \, c\right) - a - 2 \, b}{\cos\left(2 \, d x + 2 \, c\right) - 1}\right) \sin\left(2 \, d x + 2 \, c\right) + 2 \, {\left(4 \, b^{2} \cos\left(2 \, d x + 2 \, c\right) - 3 \, {\left(3 \, a b - 2 \, b^{2}\right)} \cos\left(2 \, d x + 2 \, c\right)^{2} + 9 \, a b - 2 \, b^{2}\right)} \sqrt{\frac{{\left(a - b\right)} \cos\left(2 \, d x + 2 \, c\right) - a - b}{\cos\left(2 \, d x + 2 \, c\right) - 1}}}{16 \, {\left(d \cos\left(2 \, d x + 2 \, c\right) - d\right)} \sin\left(2 \, d x + 2 \, c\right)}, -\frac{{\left(15 \, a^{2} - 20 \, a b + 8 \, b^{2} - {\left(15 \, a^{2} - 20 \, a b + 8 \, b^{2}\right)} \cos\left(2 \, d x + 2 \, c\right)\right)} \sqrt{-b} \arctan\left(\frac{\sqrt{-b} \sqrt{\frac{{\left(a - b\right)} \cos\left(2 \, d x + 2 \, c\right) - a - b}{\cos\left(2 \, d x + 2 \, c\right) - 1}} \sin\left(2 \, d x + 2 \, c\right)}{b \cos\left(2 \, d x + 2 \, c\right) + b}\right) \sin\left(2 \, d x + 2 \, c\right) + 4 \, {\left(a^{2} - 2 \, a b + b^{2} - {\left(a^{2} - 2 \, a b + b^{2}\right)} \cos\left(2 \, d x + 2 \, c\right)\right)} \sqrt{-a + b} \log\left(-{\left(a - b\right)} \cos\left(2 \, d x + 2 \, c\right) + \sqrt{-a + b} \sqrt{\frac{{\left(a - b\right)} \cos\left(2 \, d x + 2 \, c\right) - a - b}{\cos\left(2 \, d x + 2 \, c\right) - 1}} \sin\left(2 \, d x + 2 \, c\right) + b\right) \sin\left(2 \, d x + 2 \, c\right) - {\left(4 \, b^{2} \cos\left(2 \, d x + 2 \, c\right) - 3 \, {\left(3 \, a b - 2 \, b^{2}\right)} \cos\left(2 \, d x + 2 \, c\right)^{2} + 9 \, a b - 2 \, b^{2}\right)} \sqrt{\frac{{\left(a - b\right)} \cos\left(2 \, d x + 2 \, c\right) - a - b}{\cos\left(2 \, d x + 2 \, c\right) - 1}}}{8 \, {\left(d \cos\left(2 \, d x + 2 \, c\right) - d\right)} \sin\left(2 \, d x + 2 \, c\right)}, \frac{8 \, {\left(a^{2} - 2 \, a b + b^{2} - {\left(a^{2} - 2 \, a b + b^{2}\right)} \cos\left(2 \, d x + 2 \, c\right)\right)} \sqrt{a - b} \arctan\left(-\frac{\sqrt{a - b} \sqrt{\frac{{\left(a - b\right)} \cos\left(2 \, d x + 2 \, c\right) - a - b}{\cos\left(2 \, d x + 2 \, c\right) - 1}} \sin\left(2 \, d x + 2 \, c\right)}{{\left(a - b\right)} \cos\left(2 \, d x + 2 \, c\right) + a - b}\right) \sin\left(2 \, d x + 2 \, c\right) - {\left(15 \, a^{2} - 20 \, a b + 8 \, b^{2} - {\left(15 \, a^{2} - 20 \, a b + 8 \, b^{2}\right)} \cos\left(2 \, d x + 2 \, c\right)\right)} \sqrt{-b} \arctan\left(\frac{\sqrt{-b} \sqrt{\frac{{\left(a - b\right)} \cos\left(2 \, d x + 2 \, c\right) - a - b}{\cos\left(2 \, d x + 2 \, c\right) - 1}} \sin\left(2 \, d x + 2 \, c\right)}{b \cos\left(2 \, d x + 2 \, c\right) + b}\right) \sin\left(2 \, d x + 2 \, c\right) + {\left(4 \, b^{2} \cos\left(2 \, d x + 2 \, c\right) - 3 \, {\left(3 \, a b - 2 \, b^{2}\right)} \cos\left(2 \, d x + 2 \, c\right)^{2} + 9 \, a b - 2 \, b^{2}\right)} \sqrt{\frac{{\left(a - b\right)} \cos\left(2 \, d x + 2 \, c\right) - a - b}{\cos\left(2 \, d x + 2 \, c\right) - 1}}}{8 \, {\left(d \cos\left(2 \, d x + 2 \, c\right) - d\right)} \sin\left(2 \, d x + 2 \, c\right)}\right]"," ",0,"[-1/16*(8*(a^2 - 2*a*b + b^2 - (a^2 - 2*a*b + b^2)*cos(2*d*x + 2*c))*sqrt(-a + b)*log(-(a - b)*cos(2*d*x + 2*c) + sqrt(-a + b)*sqrt(((a - b)*cos(2*d*x + 2*c) - a - b)/(cos(2*d*x + 2*c) - 1))*sin(2*d*x + 2*c) + b)*sin(2*d*x + 2*c) + (15*a^2 - 20*a*b + 8*b^2 - (15*a^2 - 20*a*b + 8*b^2)*cos(2*d*x + 2*c))*sqrt(b)*log(((a - 2*b)*cos(2*d*x + 2*c) + 2*sqrt(b)*sqrt(((a - b)*cos(2*d*x + 2*c) - a - b)/(cos(2*d*x + 2*c) - 1))*sin(2*d*x + 2*c) - a - 2*b)/(cos(2*d*x + 2*c) - 1))*sin(2*d*x + 2*c) - 2*(4*b^2*cos(2*d*x + 2*c) - 3*(3*a*b - 2*b^2)*cos(2*d*x + 2*c)^2 + 9*a*b - 2*b^2)*sqrt(((a - b)*cos(2*d*x + 2*c) - a - b)/(cos(2*d*x + 2*c) - 1)))/((d*cos(2*d*x + 2*c) - d)*sin(2*d*x + 2*c)), 1/16*(16*(a^2 - 2*a*b + b^2 - (a^2 - 2*a*b + b^2)*cos(2*d*x + 2*c))*sqrt(a - b)*arctan(-sqrt(a - b)*sqrt(((a - b)*cos(2*d*x + 2*c) - a - b)/(cos(2*d*x + 2*c) - 1))*sin(2*d*x + 2*c)/((a - b)*cos(2*d*x + 2*c) + a - b))*sin(2*d*x + 2*c) - (15*a^2 - 20*a*b + 8*b^2 - (15*a^2 - 20*a*b + 8*b^2)*cos(2*d*x + 2*c))*sqrt(b)*log(((a - 2*b)*cos(2*d*x + 2*c) + 2*sqrt(b)*sqrt(((a - b)*cos(2*d*x + 2*c) - a - b)/(cos(2*d*x + 2*c) - 1))*sin(2*d*x + 2*c) - a - 2*b)/(cos(2*d*x + 2*c) - 1))*sin(2*d*x + 2*c) + 2*(4*b^2*cos(2*d*x + 2*c) - 3*(3*a*b - 2*b^2)*cos(2*d*x + 2*c)^2 + 9*a*b - 2*b^2)*sqrt(((a - b)*cos(2*d*x + 2*c) - a - b)/(cos(2*d*x + 2*c) - 1)))/((d*cos(2*d*x + 2*c) - d)*sin(2*d*x + 2*c)), -1/8*((15*a^2 - 20*a*b + 8*b^2 - (15*a^2 - 20*a*b + 8*b^2)*cos(2*d*x + 2*c))*sqrt(-b)*arctan(sqrt(-b)*sqrt(((a - b)*cos(2*d*x + 2*c) - a - b)/(cos(2*d*x + 2*c) - 1))*sin(2*d*x + 2*c)/(b*cos(2*d*x + 2*c) + b))*sin(2*d*x + 2*c) + 4*(a^2 - 2*a*b + b^2 - (a^2 - 2*a*b + b^2)*cos(2*d*x + 2*c))*sqrt(-a + b)*log(-(a - b)*cos(2*d*x + 2*c) + sqrt(-a + b)*sqrt(((a - b)*cos(2*d*x + 2*c) - a - b)/(cos(2*d*x + 2*c) - 1))*sin(2*d*x + 2*c) + b)*sin(2*d*x + 2*c) - (4*b^2*cos(2*d*x + 2*c) - 3*(3*a*b - 2*b^2)*cos(2*d*x + 2*c)^2 + 9*a*b - 2*b^2)*sqrt(((a - b)*cos(2*d*x + 2*c) - a - b)/(cos(2*d*x + 2*c) - 1)))/((d*cos(2*d*x + 2*c) - d)*sin(2*d*x + 2*c)), 1/8*(8*(a^2 - 2*a*b + b^2 - (a^2 - 2*a*b + b^2)*cos(2*d*x + 2*c))*sqrt(a - b)*arctan(-sqrt(a - b)*sqrt(((a - b)*cos(2*d*x + 2*c) - a - b)/(cos(2*d*x + 2*c) - 1))*sin(2*d*x + 2*c)/((a - b)*cos(2*d*x + 2*c) + a - b))*sin(2*d*x + 2*c) - (15*a^2 - 20*a*b + 8*b^2 - (15*a^2 - 20*a*b + 8*b^2)*cos(2*d*x + 2*c))*sqrt(-b)*arctan(sqrt(-b)*sqrt(((a - b)*cos(2*d*x + 2*c) - a - b)/(cos(2*d*x + 2*c) - 1))*sin(2*d*x + 2*c)/(b*cos(2*d*x + 2*c) + b))*sin(2*d*x + 2*c) + (4*b^2*cos(2*d*x + 2*c) - 3*(3*a*b - 2*b^2)*cos(2*d*x + 2*c)^2 + 9*a*b - 2*b^2)*sqrt(((a - b)*cos(2*d*x + 2*c) - a - b)/(cos(2*d*x + 2*c) - 1)))/((d*cos(2*d*x + 2*c) - d)*sin(2*d*x + 2*c))]","B",0
32,1,1071,0,0.488385," ","integrate((a+b*cot(d*x+c)^2)^(3/2),x, algorithm=""fricas"")","\left[-\frac{2 \, {\left(a - b\right)} \sqrt{-a + b} \log\left(-{\left(a - b\right)} \cos\left(2 \, d x + 2 \, c\right) - \sqrt{-a + b} \sqrt{\frac{{\left(a - b\right)} \cos\left(2 \, d x + 2 \, c\right) - a - b}{\cos\left(2 \, d x + 2 \, c\right) - 1}} \sin\left(2 \, d x + 2 \, c\right) + b\right) \sin\left(2 \, d x + 2 \, c\right) + {\left(3 \, a - 2 \, b\right)} \sqrt{b} \log\left(\frac{{\left(a - 2 \, b\right)} \cos\left(2 \, d x + 2 \, c\right) - 2 \, \sqrt{b} \sqrt{\frac{{\left(a - b\right)} \cos\left(2 \, d x + 2 \, c\right) - a - b}{\cos\left(2 \, d x + 2 \, c\right) - 1}} \sin\left(2 \, d x + 2 \, c\right) - a - 2 \, b}{\cos\left(2 \, d x + 2 \, c\right) - 1}\right) \sin\left(2 \, d x + 2 \, c\right) + 2 \, {\left(b \cos\left(2 \, d x + 2 \, c\right) + b\right)} \sqrt{\frac{{\left(a - b\right)} \cos\left(2 \, d x + 2 \, c\right) - a - b}{\cos\left(2 \, d x + 2 \, c\right) - 1}}}{4 \, d \sin\left(2 \, d x + 2 \, c\right)}, \frac{{\left(3 \, a - 2 \, b\right)} \sqrt{-b} \arctan\left(\frac{\sqrt{-b} \sqrt{\frac{{\left(a - b\right)} \cos\left(2 \, d x + 2 \, c\right) - a - b}{\cos\left(2 \, d x + 2 \, c\right) - 1}} \sin\left(2 \, d x + 2 \, c\right)}{b \cos\left(2 \, d x + 2 \, c\right) + b}\right) \sin\left(2 \, d x + 2 \, c\right) - {\left(a - b\right)} \sqrt{-a + b} \log\left(-{\left(a - b\right)} \cos\left(2 \, d x + 2 \, c\right) - \sqrt{-a + b} \sqrt{\frac{{\left(a - b\right)} \cos\left(2 \, d x + 2 \, c\right) - a - b}{\cos\left(2 \, d x + 2 \, c\right) - 1}} \sin\left(2 \, d x + 2 \, c\right) + b\right) \sin\left(2 \, d x + 2 \, c\right) - {\left(b \cos\left(2 \, d x + 2 \, c\right) + b\right)} \sqrt{\frac{{\left(a - b\right)} \cos\left(2 \, d x + 2 \, c\right) - a - b}{\cos\left(2 \, d x + 2 \, c\right) - 1}}}{2 \, d \sin\left(2 \, d x + 2 \, c\right)}, -\frac{4 \, {\left(a - b\right)}^{\frac{3}{2}} \arctan\left(-\frac{\sqrt{a - b} \sqrt{\frac{{\left(a - b\right)} \cos\left(2 \, d x + 2 \, c\right) - a - b}{\cos\left(2 \, d x + 2 \, c\right) - 1}} \sin\left(2 \, d x + 2 \, c\right)}{{\left(a - b\right)} \cos\left(2 \, d x + 2 \, c\right) + a - b}\right) \sin\left(2 \, d x + 2 \, c\right) + {\left(3 \, a - 2 \, b\right)} \sqrt{b} \log\left(\frac{{\left(a - 2 \, b\right)} \cos\left(2 \, d x + 2 \, c\right) - 2 \, \sqrt{b} \sqrt{\frac{{\left(a - b\right)} \cos\left(2 \, d x + 2 \, c\right) - a - b}{\cos\left(2 \, d x + 2 \, c\right) - 1}} \sin\left(2 \, d x + 2 \, c\right) - a - 2 \, b}{\cos\left(2 \, d x + 2 \, c\right) - 1}\right) \sin\left(2 \, d x + 2 \, c\right) + 2 \, {\left(b \cos\left(2 \, d x + 2 \, c\right) + b\right)} \sqrt{\frac{{\left(a - b\right)} \cos\left(2 \, d x + 2 \, c\right) - a - b}{\cos\left(2 \, d x + 2 \, c\right) - 1}}}{4 \, d \sin\left(2 \, d x + 2 \, c\right)}, -\frac{2 \, {\left(a - b\right)}^{\frac{3}{2}} \arctan\left(-\frac{\sqrt{a - b} \sqrt{\frac{{\left(a - b\right)} \cos\left(2 \, d x + 2 \, c\right) - a - b}{\cos\left(2 \, d x + 2 \, c\right) - 1}} \sin\left(2 \, d x + 2 \, c\right)}{{\left(a - b\right)} \cos\left(2 \, d x + 2 \, c\right) + a - b}\right) \sin\left(2 \, d x + 2 \, c\right) - {\left(3 \, a - 2 \, b\right)} \sqrt{-b} \arctan\left(\frac{\sqrt{-b} \sqrt{\frac{{\left(a - b\right)} \cos\left(2 \, d x + 2 \, c\right) - a - b}{\cos\left(2 \, d x + 2 \, c\right) - 1}} \sin\left(2 \, d x + 2 \, c\right)}{b \cos\left(2 \, d x + 2 \, c\right) + b}\right) \sin\left(2 \, d x + 2 \, c\right) + {\left(b \cos\left(2 \, d x + 2 \, c\right) + b\right)} \sqrt{\frac{{\left(a - b\right)} \cos\left(2 \, d x + 2 \, c\right) - a - b}{\cos\left(2 \, d x + 2 \, c\right) - 1}}}{2 \, d \sin\left(2 \, d x + 2 \, c\right)}\right]"," ",0,"[-1/4*(2*(a - b)*sqrt(-a + b)*log(-(a - b)*cos(2*d*x + 2*c) - sqrt(-a + b)*sqrt(((a - b)*cos(2*d*x + 2*c) - a - b)/(cos(2*d*x + 2*c) - 1))*sin(2*d*x + 2*c) + b)*sin(2*d*x + 2*c) + (3*a - 2*b)*sqrt(b)*log(((a - 2*b)*cos(2*d*x + 2*c) - 2*sqrt(b)*sqrt(((a - b)*cos(2*d*x + 2*c) - a - b)/(cos(2*d*x + 2*c) - 1))*sin(2*d*x + 2*c) - a - 2*b)/(cos(2*d*x + 2*c) - 1))*sin(2*d*x + 2*c) + 2*(b*cos(2*d*x + 2*c) + b)*sqrt(((a - b)*cos(2*d*x + 2*c) - a - b)/(cos(2*d*x + 2*c) - 1)))/(d*sin(2*d*x + 2*c)), 1/2*((3*a - 2*b)*sqrt(-b)*arctan(sqrt(-b)*sqrt(((a - b)*cos(2*d*x + 2*c) - a - b)/(cos(2*d*x + 2*c) - 1))*sin(2*d*x + 2*c)/(b*cos(2*d*x + 2*c) + b))*sin(2*d*x + 2*c) - (a - b)*sqrt(-a + b)*log(-(a - b)*cos(2*d*x + 2*c) - sqrt(-a + b)*sqrt(((a - b)*cos(2*d*x + 2*c) - a - b)/(cos(2*d*x + 2*c) - 1))*sin(2*d*x + 2*c) + b)*sin(2*d*x + 2*c) - (b*cos(2*d*x + 2*c) + b)*sqrt(((a - b)*cos(2*d*x + 2*c) - a - b)/(cos(2*d*x + 2*c) - 1)))/(d*sin(2*d*x + 2*c)), -1/4*(4*(a - b)^(3/2)*arctan(-sqrt(a - b)*sqrt(((a - b)*cos(2*d*x + 2*c) - a - b)/(cos(2*d*x + 2*c) - 1))*sin(2*d*x + 2*c)/((a - b)*cos(2*d*x + 2*c) + a - b))*sin(2*d*x + 2*c) + (3*a - 2*b)*sqrt(b)*log(((a - 2*b)*cos(2*d*x + 2*c) - 2*sqrt(b)*sqrt(((a - b)*cos(2*d*x + 2*c) - a - b)/(cos(2*d*x + 2*c) - 1))*sin(2*d*x + 2*c) - a - 2*b)/(cos(2*d*x + 2*c) - 1))*sin(2*d*x + 2*c) + 2*(b*cos(2*d*x + 2*c) + b)*sqrt(((a - b)*cos(2*d*x + 2*c) - a - b)/(cos(2*d*x + 2*c) - 1)))/(d*sin(2*d*x + 2*c)), -1/2*(2*(a - b)^(3/2)*arctan(-sqrt(a - b)*sqrt(((a - b)*cos(2*d*x + 2*c) - a - b)/(cos(2*d*x + 2*c) - 1))*sin(2*d*x + 2*c)/((a - b)*cos(2*d*x + 2*c) + a - b))*sin(2*d*x + 2*c) - (3*a - 2*b)*sqrt(-b)*arctan(sqrt(-b)*sqrt(((a - b)*cos(2*d*x + 2*c) - a - b)/(cos(2*d*x + 2*c) - 1))*sin(2*d*x + 2*c)/(b*cos(2*d*x + 2*c) + b))*sin(2*d*x + 2*c) + (b*cos(2*d*x + 2*c) + b)*sqrt(((a - b)*cos(2*d*x + 2*c) - a - b)/(cos(2*d*x + 2*c) - 1)))/(d*sin(2*d*x + 2*c))]","B",0
33,1,703,0,0.937981," ","integrate((a+b*cot(d*x+c)^2)^(1/2),x, algorithm=""fricas"")","\left[\frac{\sqrt{-a + b} \log\left(-{\left(a - b\right)} \cos\left(2 \, d x + 2 \, c\right) + \sqrt{-a + b} \sqrt{\frac{{\left(a - b\right)} \cos\left(2 \, d x + 2 \, c\right) - a - b}{\cos\left(2 \, d x + 2 \, c\right) - 1}} \sin\left(2 \, d x + 2 \, c\right) + b\right) + \sqrt{b} \log\left(\frac{{\left(a - 2 \, b\right)} \cos\left(2 \, d x + 2 \, c\right) + 2 \, \sqrt{b} \sqrt{\frac{{\left(a - b\right)} \cos\left(2 \, d x + 2 \, c\right) - a - b}{\cos\left(2 \, d x + 2 \, c\right) - 1}} \sin\left(2 \, d x + 2 \, c\right) - a - 2 \, b}{\cos\left(2 \, d x + 2 \, c\right) - 1}\right)}{2 \, d}, -\frac{2 \, \sqrt{a - b} \arctan\left(-\frac{\sqrt{a - b} \sqrt{\frac{{\left(a - b\right)} \cos\left(2 \, d x + 2 \, c\right) - a - b}{\cos\left(2 \, d x + 2 \, c\right) - 1}} \sin\left(2 \, d x + 2 \, c\right)}{{\left(a - b\right)} \cos\left(2 \, d x + 2 \, c\right) + a - b}\right) - \sqrt{b} \log\left(\frac{{\left(a - 2 \, b\right)} \cos\left(2 \, d x + 2 \, c\right) + 2 \, \sqrt{b} \sqrt{\frac{{\left(a - b\right)} \cos\left(2 \, d x + 2 \, c\right) - a - b}{\cos\left(2 \, d x + 2 \, c\right) - 1}} \sin\left(2 \, d x + 2 \, c\right) - a - 2 \, b}{\cos\left(2 \, d x + 2 \, c\right) - 1}\right)}{2 \, d}, \frac{2 \, \sqrt{-b} \arctan\left(\frac{\sqrt{-b} \sqrt{\frac{{\left(a - b\right)} \cos\left(2 \, d x + 2 \, c\right) - a - b}{\cos\left(2 \, d x + 2 \, c\right) - 1}} \sin\left(2 \, d x + 2 \, c\right)}{b \cos\left(2 \, d x + 2 \, c\right) + b}\right) + \sqrt{-a + b} \log\left(-{\left(a - b\right)} \cos\left(2 \, d x + 2 \, c\right) + \sqrt{-a + b} \sqrt{\frac{{\left(a - b\right)} \cos\left(2 \, d x + 2 \, c\right) - a - b}{\cos\left(2 \, d x + 2 \, c\right) - 1}} \sin\left(2 \, d x + 2 \, c\right) + b\right)}{2 \, d}, -\frac{\sqrt{a - b} \arctan\left(-\frac{\sqrt{a - b} \sqrt{\frac{{\left(a - b\right)} \cos\left(2 \, d x + 2 \, c\right) - a - b}{\cos\left(2 \, d x + 2 \, c\right) - 1}} \sin\left(2 \, d x + 2 \, c\right)}{{\left(a - b\right)} \cos\left(2 \, d x + 2 \, c\right) + a - b}\right) - \sqrt{-b} \arctan\left(\frac{\sqrt{-b} \sqrt{\frac{{\left(a - b\right)} \cos\left(2 \, d x + 2 \, c\right) - a - b}{\cos\left(2 \, d x + 2 \, c\right) - 1}} \sin\left(2 \, d x + 2 \, c\right)}{b \cos\left(2 \, d x + 2 \, c\right) + b}\right)}{d}\right]"," ",0,"[1/2*(sqrt(-a + b)*log(-(a - b)*cos(2*d*x + 2*c) + sqrt(-a + b)*sqrt(((a - b)*cos(2*d*x + 2*c) - a - b)/(cos(2*d*x + 2*c) - 1))*sin(2*d*x + 2*c) + b) + sqrt(b)*log(((a - 2*b)*cos(2*d*x + 2*c) + 2*sqrt(b)*sqrt(((a - b)*cos(2*d*x + 2*c) - a - b)/(cos(2*d*x + 2*c) - 1))*sin(2*d*x + 2*c) - a - 2*b)/(cos(2*d*x + 2*c) - 1)))/d, -1/2*(2*sqrt(a - b)*arctan(-sqrt(a - b)*sqrt(((a - b)*cos(2*d*x + 2*c) - a - b)/(cos(2*d*x + 2*c) - 1))*sin(2*d*x + 2*c)/((a - b)*cos(2*d*x + 2*c) + a - b)) - sqrt(b)*log(((a - 2*b)*cos(2*d*x + 2*c) + 2*sqrt(b)*sqrt(((a - b)*cos(2*d*x + 2*c) - a - b)/(cos(2*d*x + 2*c) - 1))*sin(2*d*x + 2*c) - a - 2*b)/(cos(2*d*x + 2*c) - 1)))/d, 1/2*(2*sqrt(-b)*arctan(sqrt(-b)*sqrt(((a - b)*cos(2*d*x + 2*c) - a - b)/(cos(2*d*x + 2*c) - 1))*sin(2*d*x + 2*c)/(b*cos(2*d*x + 2*c) + b)) + sqrt(-a + b)*log(-(a - b)*cos(2*d*x + 2*c) + sqrt(-a + b)*sqrt(((a - b)*cos(2*d*x + 2*c) - a - b)/(cos(2*d*x + 2*c) - 1))*sin(2*d*x + 2*c) + b))/d, -(sqrt(a - b)*arctan(-sqrt(a - b)*sqrt(((a - b)*cos(2*d*x + 2*c) - a - b)/(cos(2*d*x + 2*c) - 1))*sin(2*d*x + 2*c)/((a - b)*cos(2*d*x + 2*c) + a - b)) - sqrt(-b)*arctan(sqrt(-b)*sqrt(((a - b)*cos(2*d*x + 2*c) - a - b)/(cos(2*d*x + 2*c) - 1))*sin(2*d*x + 2*c)/(b*cos(2*d*x + 2*c) + b)))/d]","B",0
34,1,239,0,0.478644," ","integrate(1/(a+b*cot(d*x+c)^2)^(1/2),x, algorithm=""fricas"")","\left[-\frac{\sqrt{-a + b} \log\left(-2 \, {\left(a^{2} - 2 \, a b + b^{2}\right)} \cos\left(2 \, d x + 2 \, c\right)^{2} - 2 \, {\left({\left(a - b\right)} \cos\left(2 \, d x + 2 \, c\right) - b\right)} \sqrt{-a + b} \sqrt{\frac{{\left(a - b\right)} \cos\left(2 \, d x + 2 \, c\right) - a - b}{\cos\left(2 \, d x + 2 \, c\right) - 1}} \sin\left(2 \, d x + 2 \, c\right) + a^{2} - 2 \, b^{2} + 4 \, {\left(a b - b^{2}\right)} \cos\left(2 \, d x + 2 \, c\right)\right)}{4 \, {\left(a - b\right)} d}, -\frac{\arctan\left(-\frac{\sqrt{a - b} \sqrt{\frac{{\left(a - b\right)} \cos\left(2 \, d x + 2 \, c\right) - a - b}{\cos\left(2 \, d x + 2 \, c\right) - 1}} \sin\left(2 \, d x + 2 \, c\right)}{{\left(a - b\right)} \cos\left(2 \, d x + 2 \, c\right) - b}\right)}{2 \, \sqrt{a - b} d}\right]"," ",0,"[-1/4*sqrt(-a + b)*log(-2*(a^2 - 2*a*b + b^2)*cos(2*d*x + 2*c)^2 - 2*((a - b)*cos(2*d*x + 2*c) - b)*sqrt(-a + b)*sqrt(((a - b)*cos(2*d*x + 2*c) - a - b)/(cos(2*d*x + 2*c) - 1))*sin(2*d*x + 2*c) + a^2 - 2*b^2 + 4*(a*b - b^2)*cos(2*d*x + 2*c))/((a - b)*d), -1/2*arctan(-sqrt(a - b)*sqrt(((a - b)*cos(2*d*x + 2*c) - a - b)/(cos(2*d*x + 2*c) - 1))*sin(2*d*x + 2*c)/((a - b)*cos(2*d*x + 2*c) - b))/(sqrt(a - b)*d)]","B",0
35,1,526,0,0.624146," ","integrate(1/(a+b*cot(d*x+c)^2)^(3/2),x, algorithm=""fricas"")","\left[-\frac{{\left(a^{2} + a b - {\left(a^{2} - a b\right)} \cos\left(2 \, d x + 2 \, c\right)\right)} \sqrt{-a + b} \log\left(-2 \, {\left(a^{2} - 2 \, a b + b^{2}\right)} \cos\left(2 \, d x + 2 \, c\right)^{2} + 2 \, {\left({\left(a - b\right)} \cos\left(2 \, d x + 2 \, c\right) - b\right)} \sqrt{-a + b} \sqrt{\frac{{\left(a - b\right)} \cos\left(2 \, d x + 2 \, c\right) - a - b}{\cos\left(2 \, d x + 2 \, c\right) - 1}} \sin\left(2 \, d x + 2 \, c\right) + a^{2} - 2 \, b^{2} + 4 \, {\left(a b - b^{2}\right)} \cos\left(2 \, d x + 2 \, c\right)\right) + 4 \, {\left(a b - b^{2}\right)} \sqrt{\frac{{\left(a - b\right)} \cos\left(2 \, d x + 2 \, c\right) - a - b}{\cos\left(2 \, d x + 2 \, c\right) - 1}} \sin\left(2 \, d x + 2 \, c\right)}{4 \, {\left({\left(a^{4} - 3 \, a^{3} b + 3 \, a^{2} b^{2} - a b^{3}\right)} d \cos\left(2 \, d x + 2 \, c\right) - {\left(a^{4} - a^{3} b - a^{2} b^{2} + a b^{3}\right)} d\right)}}, \frac{{\left(a^{2} + a b - {\left(a^{2} - a b\right)} \cos\left(2 \, d x + 2 \, c\right)\right)} \sqrt{a - b} \arctan\left(-\frac{\sqrt{a - b} \sqrt{\frac{{\left(a - b\right)} \cos\left(2 \, d x + 2 \, c\right) - a - b}{\cos\left(2 \, d x + 2 \, c\right) - 1}} \sin\left(2 \, d x + 2 \, c\right)}{{\left(a - b\right)} \cos\left(2 \, d x + 2 \, c\right) - b}\right) - 2 \, {\left(a b - b^{2}\right)} \sqrt{\frac{{\left(a - b\right)} \cos\left(2 \, d x + 2 \, c\right) - a - b}{\cos\left(2 \, d x + 2 \, c\right) - 1}} \sin\left(2 \, d x + 2 \, c\right)}{2 \, {\left({\left(a^{4} - 3 \, a^{3} b + 3 \, a^{2} b^{2} - a b^{3}\right)} d \cos\left(2 \, d x + 2 \, c\right) - {\left(a^{4} - a^{3} b - a^{2} b^{2} + a b^{3}\right)} d\right)}}\right]"," ",0,"[-1/4*((a^2 + a*b - (a^2 - a*b)*cos(2*d*x + 2*c))*sqrt(-a + b)*log(-2*(a^2 - 2*a*b + b^2)*cos(2*d*x + 2*c)^2 + 2*((a - b)*cos(2*d*x + 2*c) - b)*sqrt(-a + b)*sqrt(((a - b)*cos(2*d*x + 2*c) - a - b)/(cos(2*d*x + 2*c) - 1))*sin(2*d*x + 2*c) + a^2 - 2*b^2 + 4*(a*b - b^2)*cos(2*d*x + 2*c)) + 4*(a*b - b^2)*sqrt(((a - b)*cos(2*d*x + 2*c) - a - b)/(cos(2*d*x + 2*c) - 1))*sin(2*d*x + 2*c))/((a^4 - 3*a^3*b + 3*a^2*b^2 - a*b^3)*d*cos(2*d*x + 2*c) - (a^4 - a^3*b - a^2*b^2 + a*b^3)*d), 1/2*((a^2 + a*b - (a^2 - a*b)*cos(2*d*x + 2*c))*sqrt(a - b)*arctan(-sqrt(a - b)*sqrt(((a - b)*cos(2*d*x + 2*c) - a - b)/(cos(2*d*x + 2*c) - 1))*sin(2*d*x + 2*c)/((a - b)*cos(2*d*x + 2*c) - b)) - 2*(a*b - b^2)*sqrt(((a - b)*cos(2*d*x + 2*c) - a - b)/(cos(2*d*x + 2*c) - 1))*sin(2*d*x + 2*c))/((a^4 - 3*a^3*b + 3*a^2*b^2 - a*b^3)*d*cos(2*d*x + 2*c) - (a^4 - a^3*b - a^2*b^2 + a*b^3)*d)]","B",0
36,1,898,0,0.678167," ","integrate(1/(a+b*cot(d*x+c)^2)^(5/2),x, algorithm=""fricas"")","\left[-\frac{3 \, {\left(a^{4} + 2 \, a^{3} b + a^{2} b^{2} + {\left(a^{4} - 2 \, a^{3} b + a^{2} b^{2}\right)} \cos\left(2 \, d x + 2 \, c\right)^{2} - 2 \, {\left(a^{4} - a^{2} b^{2}\right)} \cos\left(2 \, d x + 2 \, c\right)\right)} \sqrt{-a + b} \log\left(-2 \, {\left(a^{2} - 2 \, a b + b^{2}\right)} \cos\left(2 \, d x + 2 \, c\right)^{2} - 2 \, {\left({\left(a - b\right)} \cos\left(2 \, d x + 2 \, c\right) - b\right)} \sqrt{-a + b} \sqrt{\frac{{\left(a - b\right)} \cos\left(2 \, d x + 2 \, c\right) - a - b}{\cos\left(2 \, d x + 2 \, c\right) - 1}} \sin\left(2 \, d x + 2 \, c\right) + a^{2} - 2 \, b^{2} + 4 \, {\left(a b - b^{2}\right)} \cos\left(2 \, d x + 2 \, c\right)\right) - 8 \, {\left(3 \, a^{3} b - 2 \, a^{2} b^{2} - 2 \, a b^{3} + b^{4} - {\left(3 \, a^{3} b - 7 \, a^{2} b^{2} + 5 \, a b^{3} - b^{4}\right)} \cos\left(2 \, d x + 2 \, c\right)\right)} \sqrt{\frac{{\left(a - b\right)} \cos\left(2 \, d x + 2 \, c\right) - a - b}{\cos\left(2 \, d x + 2 \, c\right) - 1}} \sin\left(2 \, d x + 2 \, c\right)}{12 \, {\left({\left(a^{7} - 5 \, a^{6} b + 10 \, a^{5} b^{2} - 10 \, a^{4} b^{3} + 5 \, a^{3} b^{4} - a^{2} b^{5}\right)} d \cos\left(2 \, d x + 2 \, c\right)^{2} - 2 \, {\left(a^{7} - 3 \, a^{6} b + 2 \, a^{5} b^{2} + 2 \, a^{4} b^{3} - 3 \, a^{3} b^{4} + a^{2} b^{5}\right)} d \cos\left(2 \, d x + 2 \, c\right) + {\left(a^{7} - a^{6} b - 2 \, a^{5} b^{2} + 2 \, a^{4} b^{3} + a^{3} b^{4} - a^{2} b^{5}\right)} d\right)}}, -\frac{3 \, {\left(a^{4} + 2 \, a^{3} b + a^{2} b^{2} + {\left(a^{4} - 2 \, a^{3} b + a^{2} b^{2}\right)} \cos\left(2 \, d x + 2 \, c\right)^{2} - 2 \, {\left(a^{4} - a^{2} b^{2}\right)} \cos\left(2 \, d x + 2 \, c\right)\right)} \sqrt{a - b} \arctan\left(-\frac{\sqrt{a - b} \sqrt{\frac{{\left(a - b\right)} \cos\left(2 \, d x + 2 \, c\right) - a - b}{\cos\left(2 \, d x + 2 \, c\right) - 1}} \sin\left(2 \, d x + 2 \, c\right)}{{\left(a - b\right)} \cos\left(2 \, d x + 2 \, c\right) - b}\right) - 4 \, {\left(3 \, a^{3} b - 2 \, a^{2} b^{2} - 2 \, a b^{3} + b^{4} - {\left(3 \, a^{3} b - 7 \, a^{2} b^{2} + 5 \, a b^{3} - b^{4}\right)} \cos\left(2 \, d x + 2 \, c\right)\right)} \sqrt{\frac{{\left(a - b\right)} \cos\left(2 \, d x + 2 \, c\right) - a - b}{\cos\left(2 \, d x + 2 \, c\right) - 1}} \sin\left(2 \, d x + 2 \, c\right)}{6 \, {\left({\left(a^{7} - 5 \, a^{6} b + 10 \, a^{5} b^{2} - 10 \, a^{4} b^{3} + 5 \, a^{3} b^{4} - a^{2} b^{5}\right)} d \cos\left(2 \, d x + 2 \, c\right)^{2} - 2 \, {\left(a^{7} - 3 \, a^{6} b + 2 \, a^{5} b^{2} + 2 \, a^{4} b^{3} - 3 \, a^{3} b^{4} + a^{2} b^{5}\right)} d \cos\left(2 \, d x + 2 \, c\right) + {\left(a^{7} - a^{6} b - 2 \, a^{5} b^{2} + 2 \, a^{4} b^{3} + a^{3} b^{4} - a^{2} b^{5}\right)} d\right)}}\right]"," ",0,"[-1/12*(3*(a^4 + 2*a^3*b + a^2*b^2 + (a^4 - 2*a^3*b + a^2*b^2)*cos(2*d*x + 2*c)^2 - 2*(a^4 - a^2*b^2)*cos(2*d*x + 2*c))*sqrt(-a + b)*log(-2*(a^2 - 2*a*b + b^2)*cos(2*d*x + 2*c)^2 - 2*((a - b)*cos(2*d*x + 2*c) - b)*sqrt(-a + b)*sqrt(((a - b)*cos(2*d*x + 2*c) - a - b)/(cos(2*d*x + 2*c) - 1))*sin(2*d*x + 2*c) + a^2 - 2*b^2 + 4*(a*b - b^2)*cos(2*d*x + 2*c)) - 8*(3*a^3*b - 2*a^2*b^2 - 2*a*b^3 + b^4 - (3*a^3*b - 7*a^2*b^2 + 5*a*b^3 - b^4)*cos(2*d*x + 2*c))*sqrt(((a - b)*cos(2*d*x + 2*c) - a - b)/(cos(2*d*x + 2*c) - 1))*sin(2*d*x + 2*c))/((a^7 - 5*a^6*b + 10*a^5*b^2 - 10*a^4*b^3 + 5*a^3*b^4 - a^2*b^5)*d*cos(2*d*x + 2*c)^2 - 2*(a^7 - 3*a^6*b + 2*a^5*b^2 + 2*a^4*b^3 - 3*a^3*b^4 + a^2*b^5)*d*cos(2*d*x + 2*c) + (a^7 - a^6*b - 2*a^5*b^2 + 2*a^4*b^3 + a^3*b^4 - a^2*b^5)*d), -1/6*(3*(a^4 + 2*a^3*b + a^2*b^2 + (a^4 - 2*a^3*b + a^2*b^2)*cos(2*d*x + 2*c)^2 - 2*(a^4 - a^2*b^2)*cos(2*d*x + 2*c))*sqrt(a - b)*arctan(-sqrt(a - b)*sqrt(((a - b)*cos(2*d*x + 2*c) - a - b)/(cos(2*d*x + 2*c) - 1))*sin(2*d*x + 2*c)/((a - b)*cos(2*d*x + 2*c) - b)) - 4*(3*a^3*b - 2*a^2*b^2 - 2*a*b^3 + b^4 - (3*a^3*b - 7*a^2*b^2 + 5*a*b^3 - b^4)*cos(2*d*x + 2*c))*sqrt(((a - b)*cos(2*d*x + 2*c) - a - b)/(cos(2*d*x + 2*c) - 1))*sin(2*d*x + 2*c))/((a^7 - 5*a^6*b + 10*a^5*b^2 - 10*a^4*b^3 + 5*a^3*b^4 - a^2*b^5)*d*cos(2*d*x + 2*c)^2 - 2*(a^7 - 3*a^6*b + 2*a^5*b^2 + 2*a^4*b^3 - 3*a^3*b^4 + a^2*b^5)*d*cos(2*d*x + 2*c) + (a^7 - a^6*b - 2*a^5*b^2 + 2*a^4*b^3 + a^3*b^4 - a^2*b^5)*d)]","B",0
37,1,1452,0,0.629209," ","integrate(1/(a+b*cot(d*x+c)^2)^(7/2),x, algorithm=""fricas"")","\left[-\frac{15 \, {\left(a^{6} + 3 \, a^{5} b + 3 \, a^{4} b^{2} + a^{3} b^{3} - {\left(a^{6} - 3 \, a^{5} b + 3 \, a^{4} b^{2} - a^{3} b^{3}\right)} \cos\left(2 \, d x + 2 \, c\right)^{3} + 3 \, {\left(a^{6} - a^{5} b - a^{4} b^{2} + a^{3} b^{3}\right)} \cos\left(2 \, d x + 2 \, c\right)^{2} - 3 \, {\left(a^{6} + a^{5} b - a^{4} b^{2} - a^{3} b^{3}\right)} \cos\left(2 \, d x + 2 \, c\right)\right)} \sqrt{-a + b} \log\left(-2 \, {\left(a^{2} - 2 \, a b + b^{2}\right)} \cos\left(2 \, d x + 2 \, c\right)^{2} + 2 \, {\left({\left(a - b\right)} \cos\left(2 \, d x + 2 \, c\right) - b\right)} \sqrt{-a + b} \sqrt{\frac{{\left(a - b\right)} \cos\left(2 \, d x + 2 \, c\right) - a - b}{\cos\left(2 \, d x + 2 \, c\right) - 1}} \sin\left(2 \, d x + 2 \, c\right) + a^{2} - 2 \, b^{2} + 4 \, {\left(a b - b^{2}\right)} \cos\left(2 \, d x + 2 \, c\right)\right) + 4 \, {\left(45 \, a^{5} b - 15 \, a^{4} b^{2} - 47 \, a^{3} b^{3} + 11 \, a^{2} b^{4} + 14 \, a b^{5} - 8 \, b^{6} + {\left(45 \, a^{5} b - 165 \, a^{4} b^{2} + 233 \, a^{3} b^{3} - 159 \, a^{2} b^{4} + 54 \, a b^{5} - 8 \, b^{6}\right)} \cos\left(2 \, d x + 2 \, c\right)^{2} - 2 \, {\left(45 \, a^{5} b - 90 \, a^{4} b^{2} + 27 \, a^{3} b^{3} + 44 \, a^{2} b^{4} - 34 \, a b^{5} + 8 \, b^{6}\right)} \cos\left(2 \, d x + 2 \, c\right)\right)} \sqrt{\frac{{\left(a - b\right)} \cos\left(2 \, d x + 2 \, c\right) - a - b}{\cos\left(2 \, d x + 2 \, c\right) - 1}} \sin\left(2 \, d x + 2 \, c\right)}{60 \, {\left({\left(a^{10} - 7 \, a^{9} b + 21 \, a^{8} b^{2} - 35 \, a^{7} b^{3} + 35 \, a^{6} b^{4} - 21 \, a^{5} b^{5} + 7 \, a^{4} b^{6} - a^{3} b^{7}\right)} d \cos\left(2 \, d x + 2 \, c\right)^{3} - 3 \, {\left(a^{10} - 5 \, a^{9} b + 9 \, a^{8} b^{2} - 5 \, a^{7} b^{3} - 5 \, a^{6} b^{4} + 9 \, a^{5} b^{5} - 5 \, a^{4} b^{6} + a^{3} b^{7}\right)} d \cos\left(2 \, d x + 2 \, c\right)^{2} + 3 \, {\left(a^{10} - 3 \, a^{9} b + a^{8} b^{2} + 5 \, a^{7} b^{3} - 5 \, a^{6} b^{4} - a^{5} b^{5} + 3 \, a^{4} b^{6} - a^{3} b^{7}\right)} d \cos\left(2 \, d x + 2 \, c\right) - {\left(a^{10} - a^{9} b - 3 \, a^{8} b^{2} + 3 \, a^{7} b^{3} + 3 \, a^{6} b^{4} - 3 \, a^{5} b^{5} - a^{4} b^{6} + a^{3} b^{7}\right)} d\right)}}, \frac{15 \, {\left(a^{6} + 3 \, a^{5} b + 3 \, a^{4} b^{2} + a^{3} b^{3} - {\left(a^{6} - 3 \, a^{5} b + 3 \, a^{4} b^{2} - a^{3} b^{3}\right)} \cos\left(2 \, d x + 2 \, c\right)^{3} + 3 \, {\left(a^{6} - a^{5} b - a^{4} b^{2} + a^{3} b^{3}\right)} \cos\left(2 \, d x + 2 \, c\right)^{2} - 3 \, {\left(a^{6} + a^{5} b - a^{4} b^{2} - a^{3} b^{3}\right)} \cos\left(2 \, d x + 2 \, c\right)\right)} \sqrt{a - b} \arctan\left(-\frac{\sqrt{a - b} \sqrt{\frac{{\left(a - b\right)} \cos\left(2 \, d x + 2 \, c\right) - a - b}{\cos\left(2 \, d x + 2 \, c\right) - 1}} \sin\left(2 \, d x + 2 \, c\right)}{{\left(a - b\right)} \cos\left(2 \, d x + 2 \, c\right) - b}\right) - 2 \, {\left(45 \, a^{5} b - 15 \, a^{4} b^{2} - 47 \, a^{3} b^{3} + 11 \, a^{2} b^{4} + 14 \, a b^{5} - 8 \, b^{6} + {\left(45 \, a^{5} b - 165 \, a^{4} b^{2} + 233 \, a^{3} b^{3} - 159 \, a^{2} b^{4} + 54 \, a b^{5} - 8 \, b^{6}\right)} \cos\left(2 \, d x + 2 \, c\right)^{2} - 2 \, {\left(45 \, a^{5} b - 90 \, a^{4} b^{2} + 27 \, a^{3} b^{3} + 44 \, a^{2} b^{4} - 34 \, a b^{5} + 8 \, b^{6}\right)} \cos\left(2 \, d x + 2 \, c\right)\right)} \sqrt{\frac{{\left(a - b\right)} \cos\left(2 \, d x + 2 \, c\right) - a - b}{\cos\left(2 \, d x + 2 \, c\right) - 1}} \sin\left(2 \, d x + 2 \, c\right)}{30 \, {\left({\left(a^{10} - 7 \, a^{9} b + 21 \, a^{8} b^{2} - 35 \, a^{7} b^{3} + 35 \, a^{6} b^{4} - 21 \, a^{5} b^{5} + 7 \, a^{4} b^{6} - a^{3} b^{7}\right)} d \cos\left(2 \, d x + 2 \, c\right)^{3} - 3 \, {\left(a^{10} - 5 \, a^{9} b + 9 \, a^{8} b^{2} - 5 \, a^{7} b^{3} - 5 \, a^{6} b^{4} + 9 \, a^{5} b^{5} - 5 \, a^{4} b^{6} + a^{3} b^{7}\right)} d \cos\left(2 \, d x + 2 \, c\right)^{2} + 3 \, {\left(a^{10} - 3 \, a^{9} b + a^{8} b^{2} + 5 \, a^{7} b^{3} - 5 \, a^{6} b^{4} - a^{5} b^{5} + 3 \, a^{4} b^{6} - a^{3} b^{7}\right)} d \cos\left(2 \, d x + 2 \, c\right) - {\left(a^{10} - a^{9} b - 3 \, a^{8} b^{2} + 3 \, a^{7} b^{3} + 3 \, a^{6} b^{4} - 3 \, a^{5} b^{5} - a^{4} b^{6} + a^{3} b^{7}\right)} d\right)}}\right]"," ",0,"[-1/60*(15*(a^6 + 3*a^5*b + 3*a^4*b^2 + a^3*b^3 - (a^6 - 3*a^5*b + 3*a^4*b^2 - a^3*b^3)*cos(2*d*x + 2*c)^3 + 3*(a^6 - a^5*b - a^4*b^2 + a^3*b^3)*cos(2*d*x + 2*c)^2 - 3*(a^6 + a^5*b - a^4*b^2 - a^3*b^3)*cos(2*d*x + 2*c))*sqrt(-a + b)*log(-2*(a^2 - 2*a*b + b^2)*cos(2*d*x + 2*c)^2 + 2*((a - b)*cos(2*d*x + 2*c) - b)*sqrt(-a + b)*sqrt(((a - b)*cos(2*d*x + 2*c) - a - b)/(cos(2*d*x + 2*c) - 1))*sin(2*d*x + 2*c) + a^2 - 2*b^2 + 4*(a*b - b^2)*cos(2*d*x + 2*c)) + 4*(45*a^5*b - 15*a^4*b^2 - 47*a^3*b^3 + 11*a^2*b^4 + 14*a*b^5 - 8*b^6 + (45*a^5*b - 165*a^4*b^2 + 233*a^3*b^3 - 159*a^2*b^4 + 54*a*b^5 - 8*b^6)*cos(2*d*x + 2*c)^2 - 2*(45*a^5*b - 90*a^4*b^2 + 27*a^3*b^3 + 44*a^2*b^4 - 34*a*b^5 + 8*b^6)*cos(2*d*x + 2*c))*sqrt(((a - b)*cos(2*d*x + 2*c) - a - b)/(cos(2*d*x + 2*c) - 1))*sin(2*d*x + 2*c))/((a^10 - 7*a^9*b + 21*a^8*b^2 - 35*a^7*b^3 + 35*a^6*b^4 - 21*a^5*b^5 + 7*a^4*b^6 - a^3*b^7)*d*cos(2*d*x + 2*c)^3 - 3*(a^10 - 5*a^9*b + 9*a^8*b^2 - 5*a^7*b^3 - 5*a^6*b^4 + 9*a^5*b^5 - 5*a^4*b^6 + a^3*b^7)*d*cos(2*d*x + 2*c)^2 + 3*(a^10 - 3*a^9*b + a^8*b^2 + 5*a^7*b^3 - 5*a^6*b^4 - a^5*b^5 + 3*a^4*b^6 - a^3*b^7)*d*cos(2*d*x + 2*c) - (a^10 - a^9*b - 3*a^8*b^2 + 3*a^7*b^3 + 3*a^6*b^4 - 3*a^5*b^5 - a^4*b^6 + a^3*b^7)*d), 1/30*(15*(a^6 + 3*a^5*b + 3*a^4*b^2 + a^3*b^3 - (a^6 - 3*a^5*b + 3*a^4*b^2 - a^3*b^3)*cos(2*d*x + 2*c)^3 + 3*(a^6 - a^5*b - a^4*b^2 + a^3*b^3)*cos(2*d*x + 2*c)^2 - 3*(a^6 + a^5*b - a^4*b^2 - a^3*b^3)*cos(2*d*x + 2*c))*sqrt(a - b)*arctan(-sqrt(a - b)*sqrt(((a - b)*cos(2*d*x + 2*c) - a - b)/(cos(2*d*x + 2*c) - 1))*sin(2*d*x + 2*c)/((a - b)*cos(2*d*x + 2*c) - b)) - 2*(45*a^5*b - 15*a^4*b^2 - 47*a^3*b^3 + 11*a^2*b^4 + 14*a*b^5 - 8*b^6 + (45*a^5*b - 165*a^4*b^2 + 233*a^3*b^3 - 159*a^2*b^4 + 54*a*b^5 - 8*b^6)*cos(2*d*x + 2*c)^2 - 2*(45*a^5*b - 90*a^4*b^2 + 27*a^3*b^3 + 44*a^2*b^4 - 34*a*b^5 + 8*b^6)*cos(2*d*x + 2*c))*sqrt(((a - b)*cos(2*d*x + 2*c) - a - b)/(cos(2*d*x + 2*c) - 1))*sin(2*d*x + 2*c))/((a^10 - 7*a^9*b + 21*a^8*b^2 - 35*a^7*b^3 + 35*a^6*b^4 - 21*a^5*b^5 + 7*a^4*b^6 - a^3*b^7)*d*cos(2*d*x + 2*c)^3 - 3*(a^10 - 5*a^9*b + 9*a^8*b^2 - 5*a^7*b^3 - 5*a^6*b^4 + 9*a^5*b^5 - 5*a^4*b^6 + a^3*b^7)*d*cos(2*d*x + 2*c)^2 + 3*(a^10 - 3*a^9*b + a^8*b^2 + 5*a^7*b^3 - 5*a^6*b^4 - a^5*b^5 + 3*a^4*b^6 - a^3*b^7)*d*cos(2*d*x + 2*c) - (a^10 - a^9*b - 3*a^8*b^2 + 3*a^7*b^3 + 3*a^6*b^4 - 3*a^5*b^5 - a^4*b^6 + a^3*b^7)*d)]","B",0
38,1,110,0,1.082599," ","integrate((1-cot(x)^2)^(3/2),x, algorithm=""fricas"")","\frac{4 \, \sqrt{2} \arctan\left(\frac{\sqrt{\frac{\cos\left(2 \, x\right)}{\cos\left(2 \, x\right) - 1}} \sin\left(2 \, x\right)}{\cos\left(2 \, x\right) + 1}\right) \sin\left(2 \, x\right) + \sqrt{2} \sqrt{\frac{\cos\left(2 \, x\right)}{\cos\left(2 \, x\right) - 1}} {\left(\cos\left(2 \, x\right) + 1\right)} - 5 \, \arctan\left(\frac{\sqrt{2} \sqrt{\frac{\cos\left(2 \, x\right)}{\cos\left(2 \, x\right) - 1}} \sin\left(2 \, x\right)}{\cos\left(2 \, x\right) + 1}\right) \sin\left(2 \, x\right)}{2 \, \sin\left(2 \, x\right)}"," ",0,"1/2*(4*sqrt(2)*arctan(sqrt(cos(2*x)/(cos(2*x) - 1))*sin(2*x)/(cos(2*x) + 1))*sin(2*x) + sqrt(2)*sqrt(cos(2*x)/(cos(2*x) - 1))*(cos(2*x) + 1) - 5*arctan(sqrt(2)*sqrt(cos(2*x)/(cos(2*x) - 1))*sin(2*x)/(cos(2*x) + 1))*sin(2*x))/sin(2*x)","B",0
39,1,68,0,0.408140," ","integrate((1-cot(x)^2)^(1/2),x, algorithm=""fricas"")","\sqrt{2} \arctan\left(\frac{\sqrt{\frac{\cos\left(2 \, x\right)}{\cos\left(2 \, x\right) - 1}} \sin\left(2 \, x\right)}{\cos\left(2 \, x\right) + 1}\right) - \arctan\left(\frac{\sqrt{2} \sqrt{\frac{\cos\left(2 \, x\right)}{\cos\left(2 \, x\right) - 1}} \sin\left(2 \, x\right)}{\cos\left(2 \, x\right) + 1}\right)"," ",0,"sqrt(2)*arctan(sqrt(cos(2*x)/(cos(2*x) - 1))*sin(2*x)/(cos(2*x) + 1)) - arctan(sqrt(2)*sqrt(cos(2*x)/(cos(2*x) - 1))*sin(2*x)/(cos(2*x) + 1))","B",0
40,1,56,0,0.441261," ","integrate(1/(1-cot(x)^2)^(1/2),x, algorithm=""fricas"")","\frac{1}{4} \, \sqrt{2} \arctan\left(\frac{\sqrt{2} {\left(2 \, \sqrt{2} \cos\left(2 \, x\right) + \sqrt{2}\right)} \sqrt{\frac{\cos\left(2 \, x\right)}{\cos\left(2 \, x\right) - 1}} \sin\left(2 \, x\right)}{4 \, {\left(\cos\left(2 \, x\right)^{2} + \cos\left(2 \, x\right)\right)}}\right)"," ",0,"1/4*sqrt(2)*arctan(1/4*sqrt(2)*(2*sqrt(2)*cos(2*x) + sqrt(2))*sqrt(cos(2*x)/(cos(2*x) - 1))*sin(2*x)/(cos(2*x)^2 + cos(2*x)))","B",0
41,1,170,0,0.523855," ","integrate((-1+cot(x)^2)^(3/2),x, algorithm=""fricas"")","\frac{4 \, \sqrt{2} \log\left(2 \, \sqrt{-\frac{\cos\left(2 \, x\right)}{\cos\left(2 \, x\right) - 1}} \sin\left(2 \, x\right) - 2 \, \cos\left(2 \, x\right) - 1\right) \sin\left(2 \, x\right) - 2 \, \sqrt{2} \sqrt{-\frac{\cos\left(2 \, x\right)}{\cos\left(2 \, x\right) - 1}} {\left(\cos\left(2 \, x\right) + 1\right)} + 5 \, \log\left(\frac{\sqrt{2} \sqrt{-\frac{\cos\left(2 \, x\right)}{\cos\left(2 \, x\right) - 1}} \sin\left(2 \, x\right) + \cos\left(2 \, x\right) + 1}{\cos\left(2 \, x\right) + 1}\right) \sin\left(2 \, x\right) - 5 \, \log\left(\frac{\sqrt{2} \sqrt{-\frac{\cos\left(2 \, x\right)}{\cos\left(2 \, x\right) - 1}} \sin\left(2 \, x\right) - \cos\left(2 \, x\right) - 1}{\cos\left(2 \, x\right) + 1}\right) \sin\left(2 \, x\right)}{4 \, \sin\left(2 \, x\right)}"," ",0,"1/4*(4*sqrt(2)*log(2*sqrt(-cos(2*x)/(cos(2*x) - 1))*sin(2*x) - 2*cos(2*x) - 1)*sin(2*x) - 2*sqrt(2)*sqrt(-cos(2*x)/(cos(2*x) - 1))*(cos(2*x) + 1) + 5*log((sqrt(2)*sqrt(-cos(2*x)/(cos(2*x) - 1))*sin(2*x) + cos(2*x) + 1)/(cos(2*x) + 1))*sin(2*x) - 5*log((sqrt(2)*sqrt(-cos(2*x)/(cos(2*x) - 1))*sin(2*x) - cos(2*x) - 1)/(cos(2*x) + 1))*sin(2*x))/sin(2*x)","B",0
42,1,123,0,1.557574," ","integrate((-1+cot(x)^2)^(1/2),x, algorithm=""fricas"")","\frac{1}{2} \, \sqrt{2} \log\left(-2 \, \sqrt{-\frac{\cos\left(2 \, x\right)}{\cos\left(2 \, x\right) - 1}} \sin\left(2 \, x\right) - 2 \, \cos\left(2 \, x\right) - 1\right) - \frac{1}{2} \, \log\left(\frac{\sqrt{2} \sqrt{-\frac{\cos\left(2 \, x\right)}{\cos\left(2 \, x\right) - 1}} \sin\left(2 \, x\right) + \cos\left(2 \, x\right) + 1}{\cos\left(2 \, x\right) + 1}\right) + \frac{1}{2} \, \log\left(\frac{\sqrt{2} \sqrt{-\frac{\cos\left(2 \, x\right)}{\cos\left(2 \, x\right) - 1}} \sin\left(2 \, x\right) - \cos\left(2 \, x\right) - 1}{\cos\left(2 \, x\right) + 1}\right)"," ",0,"1/2*sqrt(2)*log(-2*sqrt(-cos(2*x)/(cos(2*x) - 1))*sin(2*x) - 2*cos(2*x) - 1) - 1/2*log((sqrt(2)*sqrt(-cos(2*x)/(cos(2*x) - 1))*sin(2*x) + cos(2*x) + 1)/(cos(2*x) + 1)) + 1/2*log((sqrt(2)*sqrt(-cos(2*x)/(cos(2*x) - 1))*sin(2*x) - cos(2*x) - 1)/(cos(2*x) + 1))","B",0
43,1,60,0,0.524102," ","integrate(1/(-1+cot(x)^2)^(1/2),x, algorithm=""fricas"")","\frac{1}{8} \, \sqrt{2} \log\left(2 \, \sqrt{2} {\left(2 \, \sqrt{2} \cos\left(2 \, x\right) + \sqrt{2}\right)} \sqrt{-\frac{\cos\left(2 \, x\right)}{\cos\left(2 \, x\right) - 1}} \sin\left(2 \, x\right) - 8 \, \cos\left(2 \, x\right)^{2} - 8 \, \cos\left(2 \, x\right) - 1\right)"," ",0,"1/8*sqrt(2)*log(2*sqrt(2)*(2*sqrt(2)*cos(2*x) + sqrt(2))*sqrt(-cos(2*x)/(cos(2*x) - 1))*sin(2*x) - 8*cos(2*x)^2 - 8*cos(2*x) - 1)","B",0
44,1,284,0,0.660166," ","integrate(cot(x)^3/(a+b*cot(x)^2)^(1/2),x, algorithm=""fricas"")","\left[\frac{\sqrt{a - b} b \log\left(-2 \, {\left(a^{2} - 2 \, a b + b^{2}\right)} \cos\left(2 \, x\right)^{2} - 2 \, a^{2} + b^{2} + 2 \, {\left({\left(a - b\right)} \cos\left(2 \, x\right)^{2} - {\left(2 \, a - b\right)} \cos\left(2 \, x\right) + a\right)} \sqrt{a - b} \sqrt{\frac{{\left(a - b\right)} \cos\left(2 \, x\right) - a - b}{\cos\left(2 \, x\right) - 1}} + 4 \, {\left(a^{2} - a b\right)} \cos\left(2 \, x\right)\right) - 4 \, {\left(a - b\right)} \sqrt{\frac{{\left(a - b\right)} \cos\left(2 \, x\right) - a - b}{\cos\left(2 \, x\right) - 1}}}{4 \, {\left(a b - b^{2}\right)}}, -\frac{\sqrt{-a + b} b \arctan\left(-\frac{\sqrt{-a + b} \sqrt{\frac{{\left(a - b\right)} \cos\left(2 \, x\right) - a - b}{\cos\left(2 \, x\right) - 1}} {\left(\cos\left(2 \, x\right) - 1\right)}}{{\left(a - b\right)} \cos\left(2 \, x\right) - a}\right) + 2 \, {\left(a - b\right)} \sqrt{\frac{{\left(a - b\right)} \cos\left(2 \, x\right) - a - b}{\cos\left(2 \, x\right) - 1}}}{2 \, {\left(a b - b^{2}\right)}}\right]"," ",0,"[1/4*(sqrt(a - b)*b*log(-2*(a^2 - 2*a*b + b^2)*cos(2*x)^2 - 2*a^2 + b^2 + 2*((a - b)*cos(2*x)^2 - (2*a - b)*cos(2*x) + a)*sqrt(a - b)*sqrt(((a - b)*cos(2*x) - a - b)/(cos(2*x) - 1)) + 4*(a^2 - a*b)*cos(2*x)) - 4*(a - b)*sqrt(((a - b)*cos(2*x) - a - b)/(cos(2*x) - 1)))/(a*b - b^2), -1/2*(sqrt(-a + b)*b*arctan(-sqrt(-a + b)*sqrt(((a - b)*cos(2*x) - a - b)/(cos(2*x) - 1))*(cos(2*x) - 1)/((a - b)*cos(2*x) - a)) + 2*(a - b)*sqrt(((a - b)*cos(2*x) - a - b)/(cos(2*x) - 1)))/(a*b - b^2)]","B",0
45,1,588,0,0.637091," ","integrate(cot(x)^2/(a+b*cot(x)^2)^(1/2),x, algorithm=""fricas"")","\left[-\frac{\sqrt{-a + b} b \log\left(-{\left(a - b\right)} \cos\left(2 \, x\right) + \sqrt{-a + b} \sqrt{\frac{{\left(a - b\right)} \cos\left(2 \, x\right) - a - b}{\cos\left(2 \, x\right) - 1}} \sin\left(2 \, x\right) + b\right) - {\left(a - b\right)} \sqrt{b} \log\left(\frac{{\left(a - 2 \, b\right)} \cos\left(2 \, x\right) + 2 \, \sqrt{b} \sqrt{\frac{{\left(a - b\right)} \cos\left(2 \, x\right) - a - b}{\cos\left(2 \, x\right) - 1}} \sin\left(2 \, x\right) - a - 2 \, b}{\cos\left(2 \, x\right) - 1}\right)}{2 \, {\left(a b - b^{2}\right)}}, \frac{2 \, {\left(a - b\right)} \sqrt{-b} \arctan\left(\frac{\sqrt{-b} \sqrt{\frac{{\left(a - b\right)} \cos\left(2 \, x\right) - a - b}{\cos\left(2 \, x\right) - 1}} \sin\left(2 \, x\right)}{b \cos\left(2 \, x\right) + b}\right) - \sqrt{-a + b} b \log\left(-{\left(a - b\right)} \cos\left(2 \, x\right) + \sqrt{-a + b} \sqrt{\frac{{\left(a - b\right)} \cos\left(2 \, x\right) - a - b}{\cos\left(2 \, x\right) - 1}} \sin\left(2 \, x\right) + b\right)}{2 \, {\left(a b - b^{2}\right)}}, \frac{2 \, \sqrt{a - b} b \arctan\left(-\frac{\sqrt{a - b} \sqrt{\frac{{\left(a - b\right)} \cos\left(2 \, x\right) - a - b}{\cos\left(2 \, x\right) - 1}} \sin\left(2 \, x\right)}{{\left(a - b\right)} \cos\left(2 \, x\right) + a - b}\right) + {\left(a - b\right)} \sqrt{b} \log\left(\frac{{\left(a - 2 \, b\right)} \cos\left(2 \, x\right) + 2 \, \sqrt{b} \sqrt{\frac{{\left(a - b\right)} \cos\left(2 \, x\right) - a - b}{\cos\left(2 \, x\right) - 1}} \sin\left(2 \, x\right) - a - 2 \, b}{\cos\left(2 \, x\right) - 1}\right)}{2 \, {\left(a b - b^{2}\right)}}, \frac{\sqrt{a - b} b \arctan\left(-\frac{\sqrt{a - b} \sqrt{\frac{{\left(a - b\right)} \cos\left(2 \, x\right) - a - b}{\cos\left(2 \, x\right) - 1}} \sin\left(2 \, x\right)}{{\left(a - b\right)} \cos\left(2 \, x\right) + a - b}\right) + {\left(a - b\right)} \sqrt{-b} \arctan\left(\frac{\sqrt{-b} \sqrt{\frac{{\left(a - b\right)} \cos\left(2 \, x\right) - a - b}{\cos\left(2 \, x\right) - 1}} \sin\left(2 \, x\right)}{b \cos\left(2 \, x\right) + b}\right)}{a b - b^{2}}\right]"," ",0,"[-1/2*(sqrt(-a + b)*b*log(-(a - b)*cos(2*x) + sqrt(-a + b)*sqrt(((a - b)*cos(2*x) - a - b)/(cos(2*x) - 1))*sin(2*x) + b) - (a - b)*sqrt(b)*log(((a - 2*b)*cos(2*x) + 2*sqrt(b)*sqrt(((a - b)*cos(2*x) - a - b)/(cos(2*x) - 1))*sin(2*x) - a - 2*b)/(cos(2*x) - 1)))/(a*b - b^2), 1/2*(2*(a - b)*sqrt(-b)*arctan(sqrt(-b)*sqrt(((a - b)*cos(2*x) - a - b)/(cos(2*x) - 1))*sin(2*x)/(b*cos(2*x) + b)) - sqrt(-a + b)*b*log(-(a - b)*cos(2*x) + sqrt(-a + b)*sqrt(((a - b)*cos(2*x) - a - b)/(cos(2*x) - 1))*sin(2*x) + b))/(a*b - b^2), 1/2*(2*sqrt(a - b)*b*arctan(-sqrt(a - b)*sqrt(((a - b)*cos(2*x) - a - b)/(cos(2*x) - 1))*sin(2*x)/((a - b)*cos(2*x) + a - b)) + (a - b)*sqrt(b)*log(((a - 2*b)*cos(2*x) + 2*sqrt(b)*sqrt(((a - b)*cos(2*x) - a - b)/(cos(2*x) - 1))*sin(2*x) - a - 2*b)/(cos(2*x) - 1)))/(a*b - b^2), (sqrt(a - b)*b*arctan(-sqrt(a - b)*sqrt(((a - b)*cos(2*x) - a - b)/(cos(2*x) - 1))*sin(2*x)/((a - b)*cos(2*x) + a - b)) + (a - b)*sqrt(-b)*arctan(sqrt(-b)*sqrt(((a - b)*cos(2*x) - a - b)/(cos(2*x) - 1))*sin(2*x)/(b*cos(2*x) + b)))/(a*b - b^2)]","B",0
46,1,127,0,0.515201," ","integrate(cot(x)/(a+b*cot(x)^2)^(1/2),x, algorithm=""fricas"")","\left[\frac{\log\left(-\sqrt{a - b} \sqrt{\frac{{\left(a - b\right)} \cos\left(2 \, x\right) - a - b}{\cos\left(2 \, x\right) - 1}} {\left(\cos\left(2 \, x\right) - 1\right)} - {\left(a - b\right)} \cos\left(2 \, x\right) + a\right)}{2 \, \sqrt{a - b}}, \frac{\sqrt{-a + b} \arctan\left(-\frac{\sqrt{-a + b} \sqrt{\frac{{\left(a - b\right)} \cos\left(2 \, x\right) - a - b}{\cos\left(2 \, x\right) - 1}}}{a - b}\right)}{a - b}\right]"," ",0,"[1/2*log(-sqrt(a - b)*sqrt(((a - b)*cos(2*x) - a - b)/(cos(2*x) - 1))*(cos(2*x) - 1) - (a - b)*cos(2*x) + a)/sqrt(a - b), sqrt(-a + b)*arctan(-sqrt(-a + b)*sqrt(((a - b)*cos(2*x) - a - b)/(cos(2*x) - 1))/(a - b))/(a - b)]","B",0
47,1,419,0,0.695657," ","integrate(tan(x)/(a+b*cot(x)^2)^(1/2),x, algorithm=""fricas"")","\left[\frac{{\left(a - b\right)} \sqrt{a} \log\left(2 \, a \tan\left(x\right)^{2} + 2 \, \sqrt{a} \sqrt{\frac{a \tan\left(x\right)^{2} + b}{\tan\left(x\right)^{2}}} \tan\left(x\right)^{2} + b\right) + \sqrt{a - b} a \log\left(\frac{{\left(2 \, a - b\right)} \tan\left(x\right)^{2} - 2 \, \sqrt{a - b} \sqrt{\frac{a \tan\left(x\right)^{2} + b}{\tan\left(x\right)^{2}}} \tan\left(x\right)^{2} + b}{\tan\left(x\right)^{2} + 1}\right)}{2 \, {\left(a^{2} - a b\right)}}, -\frac{2 \, a \sqrt{-a + b} \arctan\left(-\frac{\sqrt{-a + b} \sqrt{\frac{a \tan\left(x\right)^{2} + b}{\tan\left(x\right)^{2}}}}{a - b}\right) - {\left(a - b\right)} \sqrt{a} \log\left(2 \, a \tan\left(x\right)^{2} + 2 \, \sqrt{a} \sqrt{\frac{a \tan\left(x\right)^{2} + b}{\tan\left(x\right)^{2}}} \tan\left(x\right)^{2} + b\right)}{2 \, {\left(a^{2} - a b\right)}}, -\frac{2 \, \sqrt{-a} {\left(a - b\right)} \arctan\left(\frac{\sqrt{-a} \sqrt{\frac{a \tan\left(x\right)^{2} + b}{\tan\left(x\right)^{2}}}}{a}\right) - \sqrt{a - b} a \log\left(\frac{{\left(2 \, a - b\right)} \tan\left(x\right)^{2} - 2 \, \sqrt{a - b} \sqrt{\frac{a \tan\left(x\right)^{2} + b}{\tan\left(x\right)^{2}}} \tan\left(x\right)^{2} + b}{\tan\left(x\right)^{2} + 1}\right)}{2 \, {\left(a^{2} - a b\right)}}, -\frac{\sqrt{-a} {\left(a - b\right)} \arctan\left(\frac{\sqrt{-a} \sqrt{\frac{a \tan\left(x\right)^{2} + b}{\tan\left(x\right)^{2}}}}{a}\right) + a \sqrt{-a + b} \arctan\left(-\frac{\sqrt{-a + b} \sqrt{\frac{a \tan\left(x\right)^{2} + b}{\tan\left(x\right)^{2}}}}{a - b}\right)}{a^{2} - a b}\right]"," ",0,"[1/2*((a - b)*sqrt(a)*log(2*a*tan(x)^2 + 2*sqrt(a)*sqrt((a*tan(x)^2 + b)/tan(x)^2)*tan(x)^2 + b) + sqrt(a - b)*a*log(((2*a - b)*tan(x)^2 - 2*sqrt(a - b)*sqrt((a*tan(x)^2 + b)/tan(x)^2)*tan(x)^2 + b)/(tan(x)^2 + 1)))/(a^2 - a*b), -1/2*(2*a*sqrt(-a + b)*arctan(-sqrt(-a + b)*sqrt((a*tan(x)^2 + b)/tan(x)^2)/(a - b)) - (a - b)*sqrt(a)*log(2*a*tan(x)^2 + 2*sqrt(a)*sqrt((a*tan(x)^2 + b)/tan(x)^2)*tan(x)^2 + b))/(a^2 - a*b), -1/2*(2*sqrt(-a)*(a - b)*arctan(sqrt(-a)*sqrt((a*tan(x)^2 + b)/tan(x)^2)/a) - sqrt(a - b)*a*log(((2*a - b)*tan(x)^2 - 2*sqrt(a - b)*sqrt((a*tan(x)^2 + b)/tan(x)^2)*tan(x)^2 + b)/(tan(x)^2 + 1)))/(a^2 - a*b), -(sqrt(-a)*(a - b)*arctan(sqrt(-a)*sqrt((a*tan(x)^2 + b)/tan(x)^2)/a) + a*sqrt(-a + b)*arctan(-sqrt(-a + b)*sqrt((a*tan(x)^2 + b)/tan(x)^2)/(a - b)))/(a^2 - a*b)]","A",0
48,1,229,0,1.105455," ","integrate(tan(x)^2/(a+b*cot(x)^2)^(1/2),x, algorithm=""fricas"")","\left[-\frac{a \sqrt{-a + b} \log\left(-\frac{a^{2} \tan\left(x\right)^{4} - 2 \, {\left(3 \, a^{2} - 4 \, a b\right)} \tan\left(x\right)^{2} + a^{2} - 8 \, a b + 8 \, b^{2} + 4 \, {\left(a \tan\left(x\right)^{3} - {\left(a - 2 \, b\right)} \tan\left(x\right)\right)} \sqrt{-a + b} \sqrt{\frac{a \tan\left(x\right)^{2} + b}{\tan\left(x\right)^{2}}}}{\tan\left(x\right)^{4} + 2 \, \tan\left(x\right)^{2} + 1}\right) - 4 \, {\left(a - b\right)} \sqrt{\frac{a \tan\left(x\right)^{2} + b}{\tan\left(x\right)^{2}}} \tan\left(x\right)}{4 \, {\left(a^{2} - a b\right)}}, \frac{\sqrt{a - b} a \arctan\left(\frac{2 \, \sqrt{a - b} \sqrt{\frac{a \tan\left(x\right)^{2} + b}{\tan\left(x\right)^{2}}} \tan\left(x\right)}{a \tan\left(x\right)^{2} - a + 2 \, b}\right) + 2 \, {\left(a - b\right)} \sqrt{\frac{a \tan\left(x\right)^{2} + b}{\tan\left(x\right)^{2}}} \tan\left(x\right)}{2 \, {\left(a^{2} - a b\right)}}\right]"," ",0,"[-1/4*(a*sqrt(-a + b)*log(-(a^2*tan(x)^4 - 2*(3*a^2 - 4*a*b)*tan(x)^2 + a^2 - 8*a*b + 8*b^2 + 4*(a*tan(x)^3 - (a - 2*b)*tan(x))*sqrt(-a + b)*sqrt((a*tan(x)^2 + b)/tan(x)^2))/(tan(x)^4 + 2*tan(x)^2 + 1)) - 4*(a - b)*sqrt((a*tan(x)^2 + b)/tan(x)^2)*tan(x))/(a^2 - a*b), 1/2*(sqrt(a - b)*a*arctan(2*sqrt(a - b)*sqrt((a*tan(x)^2 + b)/tan(x)^2)*tan(x)/(a*tan(x)^2 - a + 2*b)) + 2*(a - b)*sqrt((a*tan(x)^2 + b)/tan(x)^2)*tan(x))/(a^2 - a*b)]","A",0
49,1,385,0,0.453005," ","integrate(cot(x)^3/(a+b*cot(x)^2)^(3/2),x, algorithm=""fricas"")","\left[-\frac{{\left(a b + b^{2} - {\left(a b - b^{2}\right)} \cos\left(2 \, x\right)\right)} \sqrt{a - b} \log\left(-\sqrt{a - b} \sqrt{\frac{{\left(a - b\right)} \cos\left(2 \, x\right) - a - b}{\cos\left(2 \, x\right) - 1}} {\left(\cos\left(2 \, x\right) - 1\right)} - {\left(a - b\right)} \cos\left(2 \, x\right) + a\right) - 2 \, {\left(a^{2} - a b - {\left(a^{2} - a b\right)} \cos\left(2 \, x\right)\right)} \sqrt{\frac{{\left(a - b\right)} \cos\left(2 \, x\right) - a - b}{\cos\left(2 \, x\right) - 1}}}{2 \, {\left(a^{3} b - a^{2} b^{2} - a b^{3} + b^{4} - {\left(a^{3} b - 3 \, a^{2} b^{2} + 3 \, a b^{3} - b^{4}\right)} \cos\left(2 \, x\right)\right)}}, -\frac{{\left(a b + b^{2} - {\left(a b - b^{2}\right)} \cos\left(2 \, x\right)\right)} \sqrt{-a + b} \arctan\left(-\frac{\sqrt{-a + b} \sqrt{\frac{{\left(a - b\right)} \cos\left(2 \, x\right) - a - b}{\cos\left(2 \, x\right) - 1}}}{a - b}\right) - {\left(a^{2} - a b - {\left(a^{2} - a b\right)} \cos\left(2 \, x\right)\right)} \sqrt{\frac{{\left(a - b\right)} \cos\left(2 \, x\right) - a - b}{\cos\left(2 \, x\right) - 1}}}{a^{3} b - a^{2} b^{2} - a b^{3} + b^{4} - {\left(a^{3} b - 3 \, a^{2} b^{2} + 3 \, a b^{3} - b^{4}\right)} \cos\left(2 \, x\right)}\right]"," ",0,"[-1/2*((a*b + b^2 - (a*b - b^2)*cos(2*x))*sqrt(a - b)*log(-sqrt(a - b)*sqrt(((a - b)*cos(2*x) - a - b)/(cos(2*x) - 1))*(cos(2*x) - 1) - (a - b)*cos(2*x) + a) - 2*(a^2 - a*b - (a^2 - a*b)*cos(2*x))*sqrt(((a - b)*cos(2*x) - a - b)/(cos(2*x) - 1)))/(a^3*b - a^2*b^2 - a*b^3 + b^4 - (a^3*b - 3*a^2*b^2 + 3*a*b^3 - b^4)*cos(2*x)), -((a*b + b^2 - (a*b - b^2)*cos(2*x))*sqrt(-a + b)*arctan(-sqrt(-a + b)*sqrt(((a - b)*cos(2*x) - a - b)/(cos(2*x) - 1))/(a - b)) - (a^2 - a*b - (a^2 - a*b)*cos(2*x))*sqrt(((a - b)*cos(2*x) - a - b)/(cos(2*x) - 1)))/(a^3*b - a^2*b^2 - a*b^3 + b^4 - (a^3*b - 3*a^2*b^2 + 3*a*b^3 - b^4)*cos(2*x))]","B",0
50,1,388,0,0.744068," ","integrate(cot(x)^2/(a+b*cot(x)^2)^(3/2),x, algorithm=""fricas"")","\left[-\frac{{\left({\left(a - b\right)} \cos\left(2 \, x\right) - a - b\right)} \sqrt{-a + b} \log\left(-2 \, {\left(a^{2} - 2 \, a b + b^{2}\right)} \cos\left(2 \, x\right)^{2} - 2 \, {\left({\left(a - b\right)} \cos\left(2 \, x\right) - b\right)} \sqrt{-a + b} \sqrt{\frac{{\left(a - b\right)} \cos\left(2 \, x\right) - a - b}{\cos\left(2 \, x\right) - 1}} \sin\left(2 \, x\right) + a^{2} - 2 \, b^{2} + 4 \, {\left(a b - b^{2}\right)} \cos\left(2 \, x\right)\right) + 4 \, {\left(a - b\right)} \sqrt{\frac{{\left(a - b\right)} \cos\left(2 \, x\right) - a - b}{\cos\left(2 \, x\right) - 1}} \sin\left(2 \, x\right)}{4 \, {\left(a^{3} - a^{2} b - a b^{2} + b^{3} - {\left(a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}\right)} \cos\left(2 \, x\right)\right)}}, -\frac{{\left({\left(a - b\right)} \cos\left(2 \, x\right) - a - b\right)} \sqrt{a - b} \arctan\left(-\frac{\sqrt{a - b} \sqrt{\frac{{\left(a - b\right)} \cos\left(2 \, x\right) - a - b}{\cos\left(2 \, x\right) - 1}} \sin\left(2 \, x\right)}{{\left(a - b\right)} \cos\left(2 \, x\right) - b}\right) + 2 \, {\left(a - b\right)} \sqrt{\frac{{\left(a - b\right)} \cos\left(2 \, x\right) - a - b}{\cos\left(2 \, x\right) - 1}} \sin\left(2 \, x\right)}{2 \, {\left(a^{3} - a^{2} b - a b^{2} + b^{3} - {\left(a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}\right)} \cos\left(2 \, x\right)\right)}}\right]"," ",0,"[-1/4*(((a - b)*cos(2*x) - a - b)*sqrt(-a + b)*log(-2*(a^2 - 2*a*b + b^2)*cos(2*x)^2 - 2*((a - b)*cos(2*x) - b)*sqrt(-a + b)*sqrt(((a - b)*cos(2*x) - a - b)/(cos(2*x) - 1))*sin(2*x) + a^2 - 2*b^2 + 4*(a*b - b^2)*cos(2*x)) + 4*(a - b)*sqrt(((a - b)*cos(2*x) - a - b)/(cos(2*x) - 1))*sin(2*x))/(a^3 - a^2*b - a*b^2 + b^3 - (a^3 - 3*a^2*b + 3*a*b^2 - b^3)*cos(2*x)), -1/2*(((a - b)*cos(2*x) - a - b)*sqrt(a - b)*arctan(-sqrt(a - b)*sqrt(((a - b)*cos(2*x) - a - b)/(cos(2*x) - 1))*sin(2*x)/((a - b)*cos(2*x) - b)) + 2*(a - b)*sqrt(((a - b)*cos(2*x) - a - b)/(cos(2*x) - 1))*sin(2*x))/(a^3 - a^2*b - a*b^2 + b^3 - (a^3 - 3*a^2*b + 3*a*b^2 - b^3)*cos(2*x))]","B",0
51,1,344,0,0.473722," ","integrate(cot(x)/(a+b*cot(x)^2)^(3/2),x, algorithm=""fricas"")","\left[\frac{{\left({\left(a - b\right)} \cos\left(2 \, x\right) - a - b\right)} \sqrt{a - b} \log\left(\sqrt{a - b} \sqrt{\frac{{\left(a - b\right)} \cos\left(2 \, x\right) - a - b}{\cos\left(2 \, x\right) - 1}} {\left(\cos\left(2 \, x\right) - 1\right)} - {\left(a - b\right)} \cos\left(2 \, x\right) + a\right) + 2 \, {\left({\left(a - b\right)} \cos\left(2 \, x\right) - a + b\right)} \sqrt{\frac{{\left(a - b\right)} \cos\left(2 \, x\right) - a - b}{\cos\left(2 \, x\right) - 1}}}{2 \, {\left(a^{3} - a^{2} b - a b^{2} + b^{3} - {\left(a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}\right)} \cos\left(2 \, x\right)\right)}}, -\frac{{\left({\left(a - b\right)} \cos\left(2 \, x\right) - a - b\right)} \sqrt{-a + b} \arctan\left(-\frac{\sqrt{-a + b} \sqrt{\frac{{\left(a - b\right)} \cos\left(2 \, x\right) - a - b}{\cos\left(2 \, x\right) - 1}}}{a - b}\right) - {\left({\left(a - b\right)} \cos\left(2 \, x\right) - a + b\right)} \sqrt{\frac{{\left(a - b\right)} \cos\left(2 \, x\right) - a - b}{\cos\left(2 \, x\right) - 1}}}{a^{3} - a^{2} b - a b^{2} + b^{3} - {\left(a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}\right)} \cos\left(2 \, x\right)}\right]"," ",0,"[1/2*(((a - b)*cos(2*x) - a - b)*sqrt(a - b)*log(sqrt(a - b)*sqrt(((a - b)*cos(2*x) - a - b)/(cos(2*x) - 1))*(cos(2*x) - 1) - (a - b)*cos(2*x) + a) + 2*((a - b)*cos(2*x) - a + b)*sqrt(((a - b)*cos(2*x) - a - b)/(cos(2*x) - 1)))/(a^3 - a^2*b - a*b^2 + b^3 - (a^3 - 3*a^2*b + 3*a*b^2 - b^3)*cos(2*x)), -(((a - b)*cos(2*x) - a - b)*sqrt(-a + b)*arctan(-sqrt(-a + b)*sqrt(((a - b)*cos(2*x) - a - b)/(cos(2*x) - 1))/(a - b)) - ((a - b)*cos(2*x) - a + b)*sqrt(((a - b)*cos(2*x) - a - b)/(cos(2*x) - 1)))/(a^3 - a^2*b - a*b^2 + b^3 - (a^3 - 3*a^2*b + 3*a*b^2 - b^3)*cos(2*x))]","B",0
52,1,863,0,1.006877," ","integrate(tan(x)/(a+b*cot(x)^2)^(3/2),x, algorithm=""fricas"")","\left[\frac{2 \, {\left(a^{2} b - a b^{2}\right)} \sqrt{\frac{a \tan\left(x\right)^{2} + b}{\tan\left(x\right)^{2}}} \tan\left(x\right)^{2} + {\left(a^{2} b - 2 \, a b^{2} + b^{3} + {\left(a^{3} - 2 \, a^{2} b + a b^{2}\right)} \tan\left(x\right)^{2}\right)} \sqrt{a} \log\left(2 \, a \tan\left(x\right)^{2} + 2 \, \sqrt{a} \sqrt{\frac{a \tan\left(x\right)^{2} + b}{\tan\left(x\right)^{2}}} \tan\left(x\right)^{2} + b\right) - {\left(a^{3} \tan\left(x\right)^{2} + a^{2} b\right)} \sqrt{a - b} \log\left(\frac{{\left(2 \, a - b\right)} \tan\left(x\right)^{2} + 2 \, \sqrt{a - b} \sqrt{\frac{a \tan\left(x\right)^{2} + b}{\tan\left(x\right)^{2}}} \tan\left(x\right)^{2} + b}{\tan\left(x\right)^{2} + 1}\right)}{2 \, {\left(a^{4} b - 2 \, a^{3} b^{2} + a^{2} b^{3} + {\left(a^{5} - 2 \, a^{4} b + a^{3} b^{2}\right)} \tan\left(x\right)^{2}\right)}}, \frac{2 \, {\left(a^{2} b - a b^{2}\right)} \sqrt{\frac{a \tan\left(x\right)^{2} + b}{\tan\left(x\right)^{2}}} \tan\left(x\right)^{2} - 2 \, {\left(a^{3} \tan\left(x\right)^{2} + a^{2} b\right)} \sqrt{-a + b} \arctan\left(-\frac{\sqrt{-a + b} \sqrt{\frac{a \tan\left(x\right)^{2} + b}{\tan\left(x\right)^{2}}}}{a - b}\right) + {\left(a^{2} b - 2 \, a b^{2} + b^{3} + {\left(a^{3} - 2 \, a^{2} b + a b^{2}\right)} \tan\left(x\right)^{2}\right)} \sqrt{a} \log\left(2 \, a \tan\left(x\right)^{2} + 2 \, \sqrt{a} \sqrt{\frac{a \tan\left(x\right)^{2} + b}{\tan\left(x\right)^{2}}} \tan\left(x\right)^{2} + b\right)}{2 \, {\left(a^{4} b - 2 \, a^{3} b^{2} + a^{2} b^{3} + {\left(a^{5} - 2 \, a^{4} b + a^{3} b^{2}\right)} \tan\left(x\right)^{2}\right)}}, \frac{2 \, {\left(a^{2} b - a b^{2}\right)} \sqrt{\frac{a \tan\left(x\right)^{2} + b}{\tan\left(x\right)^{2}}} \tan\left(x\right)^{2} - 2 \, {\left(a^{2} b - 2 \, a b^{2} + b^{3} + {\left(a^{3} - 2 \, a^{2} b + a b^{2}\right)} \tan\left(x\right)^{2}\right)} \sqrt{-a} \arctan\left(\frac{\sqrt{-a} \sqrt{\frac{a \tan\left(x\right)^{2} + b}{\tan\left(x\right)^{2}}}}{a}\right) - {\left(a^{3} \tan\left(x\right)^{2} + a^{2} b\right)} \sqrt{a - b} \log\left(\frac{{\left(2 \, a - b\right)} \tan\left(x\right)^{2} + 2 \, \sqrt{a - b} \sqrt{\frac{a \tan\left(x\right)^{2} + b}{\tan\left(x\right)^{2}}} \tan\left(x\right)^{2} + b}{\tan\left(x\right)^{2} + 1}\right)}{2 \, {\left(a^{4} b - 2 \, a^{3} b^{2} + a^{2} b^{3} + {\left(a^{5} - 2 \, a^{4} b + a^{3} b^{2}\right)} \tan\left(x\right)^{2}\right)}}, \frac{{\left(a^{2} b - a b^{2}\right)} \sqrt{\frac{a \tan\left(x\right)^{2} + b}{\tan\left(x\right)^{2}}} \tan\left(x\right)^{2} - {\left(a^{2} b - 2 \, a b^{2} + b^{3} + {\left(a^{3} - 2 \, a^{2} b + a b^{2}\right)} \tan\left(x\right)^{2}\right)} \sqrt{-a} \arctan\left(\frac{\sqrt{-a} \sqrt{\frac{a \tan\left(x\right)^{2} + b}{\tan\left(x\right)^{2}}}}{a}\right) - {\left(a^{3} \tan\left(x\right)^{2} + a^{2} b\right)} \sqrt{-a + b} \arctan\left(-\frac{\sqrt{-a + b} \sqrt{\frac{a \tan\left(x\right)^{2} + b}{\tan\left(x\right)^{2}}}}{a - b}\right)}{a^{4} b - 2 \, a^{3} b^{2} + a^{2} b^{3} + {\left(a^{5} - 2 \, a^{4} b + a^{3} b^{2}\right)} \tan\left(x\right)^{2}}\right]"," ",0,"[1/2*(2*(a^2*b - a*b^2)*sqrt((a*tan(x)^2 + b)/tan(x)^2)*tan(x)^2 + (a^2*b - 2*a*b^2 + b^3 + (a^3 - 2*a^2*b + a*b^2)*tan(x)^2)*sqrt(a)*log(2*a*tan(x)^2 + 2*sqrt(a)*sqrt((a*tan(x)^2 + b)/tan(x)^2)*tan(x)^2 + b) - (a^3*tan(x)^2 + a^2*b)*sqrt(a - b)*log(((2*a - b)*tan(x)^2 + 2*sqrt(a - b)*sqrt((a*tan(x)^2 + b)/tan(x)^2)*tan(x)^2 + b)/(tan(x)^2 + 1)))/(a^4*b - 2*a^3*b^2 + a^2*b^3 + (a^5 - 2*a^4*b + a^3*b^2)*tan(x)^2), 1/2*(2*(a^2*b - a*b^2)*sqrt((a*tan(x)^2 + b)/tan(x)^2)*tan(x)^2 - 2*(a^3*tan(x)^2 + a^2*b)*sqrt(-a + b)*arctan(-sqrt(-a + b)*sqrt((a*tan(x)^2 + b)/tan(x)^2)/(a - b)) + (a^2*b - 2*a*b^2 + b^3 + (a^3 - 2*a^2*b + a*b^2)*tan(x)^2)*sqrt(a)*log(2*a*tan(x)^2 + 2*sqrt(a)*sqrt((a*tan(x)^2 + b)/tan(x)^2)*tan(x)^2 + b))/(a^4*b - 2*a^3*b^2 + a^2*b^3 + (a^5 - 2*a^4*b + a^3*b^2)*tan(x)^2), 1/2*(2*(a^2*b - a*b^2)*sqrt((a*tan(x)^2 + b)/tan(x)^2)*tan(x)^2 - 2*(a^2*b - 2*a*b^2 + b^3 + (a^3 - 2*a^2*b + a*b^2)*tan(x)^2)*sqrt(-a)*arctan(sqrt(-a)*sqrt((a*tan(x)^2 + b)/tan(x)^2)/a) - (a^3*tan(x)^2 + a^2*b)*sqrt(a - b)*log(((2*a - b)*tan(x)^2 + 2*sqrt(a - b)*sqrt((a*tan(x)^2 + b)/tan(x)^2)*tan(x)^2 + b)/(tan(x)^2 + 1)))/(a^4*b - 2*a^3*b^2 + a^2*b^3 + (a^5 - 2*a^4*b + a^3*b^2)*tan(x)^2), ((a^2*b - a*b^2)*sqrt((a*tan(x)^2 + b)/tan(x)^2)*tan(x)^2 - (a^2*b - 2*a*b^2 + b^3 + (a^3 - 2*a^2*b + a*b^2)*tan(x)^2)*sqrt(-a)*arctan(sqrt(-a)*sqrt((a*tan(x)^2 + b)/tan(x)^2)/a) - (a^3*tan(x)^2 + a^2*b)*sqrt(-a + b)*arctan(-sqrt(-a + b)*sqrt((a*tan(x)^2 + b)/tan(x)^2)/(a - b)))/(a^4*b - 2*a^3*b^2 + a^2*b^3 + (a^5 - 2*a^4*b + a^3*b^2)*tan(x)^2)]","B",0
53,1,393,0,0.904870," ","integrate(tan(x)^2/(a+b*cot(x)^2)^(3/2),x, algorithm=""fricas"")","\left[\frac{{\left(a^{3} \tan\left(x\right)^{2} + a^{2} b\right)} \sqrt{-a + b} \log\left(-\frac{a^{2} \tan\left(x\right)^{4} - 2 \, {\left(3 \, a^{2} - 4 \, a b\right)} \tan\left(x\right)^{2} + a^{2} - 8 \, a b + 8 \, b^{2} - 4 \, {\left(a \tan\left(x\right)^{3} - {\left(a - 2 \, b\right)} \tan\left(x\right)\right)} \sqrt{-a + b} \sqrt{\frac{a \tan\left(x\right)^{2} + b}{\tan\left(x\right)^{2}}}}{\tan\left(x\right)^{4} + 2 \, \tan\left(x\right)^{2} + 1}\right) + 4 \, {\left({\left(a^{3} - 2 \, a^{2} b + a b^{2}\right)} \tan\left(x\right)^{3} + {\left(a^{2} b - 3 \, a b^{2} + 2 \, b^{3}\right)} \tan\left(x\right)\right)} \sqrt{\frac{a \tan\left(x\right)^{2} + b}{\tan\left(x\right)^{2}}}}{4 \, {\left(a^{4} b - 2 \, a^{3} b^{2} + a^{2} b^{3} + {\left(a^{5} - 2 \, a^{4} b + a^{3} b^{2}\right)} \tan\left(x\right)^{2}\right)}}, \frac{{\left(a^{3} \tan\left(x\right)^{2} + a^{2} b\right)} \sqrt{a - b} \arctan\left(\frac{2 \, \sqrt{a - b} \sqrt{\frac{a \tan\left(x\right)^{2} + b}{\tan\left(x\right)^{2}}} \tan\left(x\right)}{a \tan\left(x\right)^{2} - a + 2 \, b}\right) + 2 \, {\left({\left(a^{3} - 2 \, a^{2} b + a b^{2}\right)} \tan\left(x\right)^{3} + {\left(a^{2} b - 3 \, a b^{2} + 2 \, b^{3}\right)} \tan\left(x\right)\right)} \sqrt{\frac{a \tan\left(x\right)^{2} + b}{\tan\left(x\right)^{2}}}}{2 \, {\left(a^{4} b - 2 \, a^{3} b^{2} + a^{2} b^{3} + {\left(a^{5} - 2 \, a^{4} b + a^{3} b^{2}\right)} \tan\left(x\right)^{2}\right)}}\right]"," ",0,"[1/4*((a^3*tan(x)^2 + a^2*b)*sqrt(-a + b)*log(-(a^2*tan(x)^4 - 2*(3*a^2 - 4*a*b)*tan(x)^2 + a^2 - 8*a*b + 8*b^2 - 4*(a*tan(x)^3 - (a - 2*b)*tan(x))*sqrt(-a + b)*sqrt((a*tan(x)^2 + b)/tan(x)^2))/(tan(x)^4 + 2*tan(x)^2 + 1)) + 4*((a^3 - 2*a^2*b + a*b^2)*tan(x)^3 + (a^2*b - 3*a*b^2 + 2*b^3)*tan(x))*sqrt((a*tan(x)^2 + b)/tan(x)^2))/(a^4*b - 2*a^3*b^2 + a^2*b^3 + (a^5 - 2*a^4*b + a^3*b^2)*tan(x)^2), 1/2*((a^3*tan(x)^2 + a^2*b)*sqrt(a - b)*arctan(2*sqrt(a - b)*sqrt((a*tan(x)^2 + b)/tan(x)^2)*tan(x)/(a*tan(x)^2 - a + 2*b)) + 2*((a^3 - 2*a^2*b + a*b^2)*tan(x)^3 + (a^2*b - 3*a*b^2 + 2*b^3)*tan(x))*sqrt((a*tan(x)^2 + b)/tan(x)^2))/(a^4*b - 2*a^3*b^2 + a^2*b^3 + (a^5 - 2*a^4*b + a^3*b^2)*tan(x)^2)]","B",0
54,1,698,0,0.680329," ","integrate(cot(x)^3/(a+b*cot(x)^2)^(5/2),x, algorithm=""fricas"")","\left[\frac{3 \, {\left(a^{2} b + 2 \, a b^{2} + b^{3} + {\left(a^{2} b - 2 \, a b^{2} + b^{3}\right)} \cos\left(2 \, x\right)^{2} - 2 \, {\left(a^{2} b - b^{3}\right)} \cos\left(2 \, x\right)\right)} \sqrt{a - b} \log\left(\sqrt{a - b} \sqrt{\frac{{\left(a - b\right)} \cos\left(2 \, x\right) - a - b}{\cos\left(2 \, x\right) - 1}} {\left(\cos\left(2 \, x\right) - 1\right)} - {\left(a - b\right)} \cos\left(2 \, x\right) + a\right) + 2 \, {\left(a^{3} + a^{2} b + a b^{2} - 3 \, b^{3} + {\left(a^{3} + a^{2} b - 5 \, a b^{2} + 3 \, b^{3}\right)} \cos\left(2 \, x\right)^{2} - 2 \, {\left(a^{3} + a^{2} b - 2 \, a b^{2}\right)} \cos\left(2 \, x\right)\right)} \sqrt{\frac{{\left(a - b\right)} \cos\left(2 \, x\right) - a - b}{\cos\left(2 \, x\right) - 1}}}{6 \, {\left(a^{5} b - a^{4} b^{2} - 2 \, a^{3} b^{3} + 2 \, a^{2} b^{4} + a b^{5} - b^{6} + {\left(a^{5} b - 5 \, a^{4} b^{2} + 10 \, a^{3} b^{3} - 10 \, a^{2} b^{4} + 5 \, a b^{5} - b^{6}\right)} \cos\left(2 \, x\right)^{2} - 2 \, {\left(a^{5} b - 3 \, a^{4} b^{2} + 2 \, a^{3} b^{3} + 2 \, a^{2} b^{4} - 3 \, a b^{5} + b^{6}\right)} \cos\left(2 \, x\right)\right)}}, -\frac{3 \, {\left(a^{2} b + 2 \, a b^{2} + b^{3} + {\left(a^{2} b - 2 \, a b^{2} + b^{3}\right)} \cos\left(2 \, x\right)^{2} - 2 \, {\left(a^{2} b - b^{3}\right)} \cos\left(2 \, x\right)\right)} \sqrt{-a + b} \arctan\left(-\frac{\sqrt{-a + b} \sqrt{\frac{{\left(a - b\right)} \cos\left(2 \, x\right) - a - b}{\cos\left(2 \, x\right) - 1}}}{a - b}\right) - {\left(a^{3} + a^{2} b + a b^{2} - 3 \, b^{3} + {\left(a^{3} + a^{2} b - 5 \, a b^{2} + 3 \, b^{3}\right)} \cos\left(2 \, x\right)^{2} - 2 \, {\left(a^{3} + a^{2} b - 2 \, a b^{2}\right)} \cos\left(2 \, x\right)\right)} \sqrt{\frac{{\left(a - b\right)} \cos\left(2 \, x\right) - a - b}{\cos\left(2 \, x\right) - 1}}}{3 \, {\left(a^{5} b - a^{4} b^{2} - 2 \, a^{3} b^{3} + 2 \, a^{2} b^{4} + a b^{5} - b^{6} + {\left(a^{5} b - 5 \, a^{4} b^{2} + 10 \, a^{3} b^{3} - 10 \, a^{2} b^{4} + 5 \, a b^{5} - b^{6}\right)} \cos\left(2 \, x\right)^{2} - 2 \, {\left(a^{5} b - 3 \, a^{4} b^{2} + 2 \, a^{3} b^{3} + 2 \, a^{2} b^{4} - 3 \, a b^{5} + b^{6}\right)} \cos\left(2 \, x\right)\right)}}\right]"," ",0,"[1/6*(3*(a^2*b + 2*a*b^2 + b^3 + (a^2*b - 2*a*b^2 + b^3)*cos(2*x)^2 - 2*(a^2*b - b^3)*cos(2*x))*sqrt(a - b)*log(sqrt(a - b)*sqrt(((a - b)*cos(2*x) - a - b)/(cos(2*x) - 1))*(cos(2*x) - 1) - (a - b)*cos(2*x) + a) + 2*(a^3 + a^2*b + a*b^2 - 3*b^3 + (a^3 + a^2*b - 5*a*b^2 + 3*b^3)*cos(2*x)^2 - 2*(a^3 + a^2*b - 2*a*b^2)*cos(2*x))*sqrt(((a - b)*cos(2*x) - a - b)/(cos(2*x) - 1)))/(a^5*b - a^4*b^2 - 2*a^3*b^3 + 2*a^2*b^4 + a*b^5 - b^6 + (a^5*b - 5*a^4*b^2 + 10*a^3*b^3 - 10*a^2*b^4 + 5*a*b^5 - b^6)*cos(2*x)^2 - 2*(a^5*b - 3*a^4*b^2 + 2*a^3*b^3 + 2*a^2*b^4 - 3*a*b^5 + b^6)*cos(2*x)), -1/3*(3*(a^2*b + 2*a*b^2 + b^3 + (a^2*b - 2*a*b^2 + b^3)*cos(2*x)^2 - 2*(a^2*b - b^3)*cos(2*x))*sqrt(-a + b)*arctan(-sqrt(-a + b)*sqrt(((a - b)*cos(2*x) - a - b)/(cos(2*x) - 1))/(a - b)) - (a^3 + a^2*b + a*b^2 - 3*b^3 + (a^3 + a^2*b - 5*a*b^2 + 3*b^3)*cos(2*x)^2 - 2*(a^3 + a^2*b - 2*a*b^2)*cos(2*x))*sqrt(((a - b)*cos(2*x) - a - b)/(cos(2*x) - 1)))/(a^5*b - a^4*b^2 - 2*a^3*b^3 + 2*a^2*b^4 + a*b^5 - b^6 + (a^5*b - 5*a^4*b^2 + 10*a^3*b^3 - 10*a^2*b^4 + 5*a*b^5 - b^6)*cos(2*x)^2 - 2*(a^5*b - 3*a^4*b^2 + 2*a^3*b^3 + 2*a^2*b^4 - 3*a*b^5 + b^6)*cos(2*x))]","B",0
55,1,720,0,1.346863," ","integrate(cot(x)^2/(a+b*cot(x)^2)^(5/2),x, algorithm=""fricas"")","\left[-\frac{3 \, {\left(a^{3} + 2 \, a^{2} b + a b^{2} + {\left(a^{3} - 2 \, a^{2} b + a b^{2}\right)} \cos\left(2 \, x\right)^{2} - 2 \, {\left(a^{3} - a b^{2}\right)} \cos\left(2 \, x\right)\right)} \sqrt{-a + b} \log\left(-2 \, {\left(a^{2} - 2 \, a b + b^{2}\right)} \cos\left(2 \, x\right)^{2} + 2 \, {\left({\left(a - b\right)} \cos\left(2 \, x\right) - b\right)} \sqrt{-a + b} \sqrt{\frac{{\left(a - b\right)} \cos\left(2 \, x\right) - a - b}{\cos\left(2 \, x\right) - 1}} \sin\left(2 \, x\right) + a^{2} - 2 \, b^{2} + 4 \, {\left(a b - b^{2}\right)} \cos\left(2 \, x\right)\right) + 4 \, {\left(3 \, a^{3} - a^{2} b - a b^{2} - b^{3} - {\left(3 \, a^{3} - 5 \, a^{2} b + a b^{2} + b^{3}\right)} \cos\left(2 \, x\right)\right)} \sqrt{\frac{{\left(a - b\right)} \cos\left(2 \, x\right) - a - b}{\cos\left(2 \, x\right) - 1}} \sin\left(2 \, x\right)}{12 \, {\left(a^{6} - a^{5} b - 2 \, a^{4} b^{2} + 2 \, a^{3} b^{3} + a^{2} b^{4} - a b^{5} + {\left(a^{6} - 5 \, a^{5} b + 10 \, a^{4} b^{2} - 10 \, a^{3} b^{3} + 5 \, a^{2} b^{4} - a b^{5}\right)} \cos\left(2 \, x\right)^{2} - 2 \, {\left(a^{6} - 3 \, a^{5} b + 2 \, a^{4} b^{2} + 2 \, a^{3} b^{3} - 3 \, a^{2} b^{4} + a b^{5}\right)} \cos\left(2 \, x\right)\right)}}, \frac{3 \, {\left(a^{3} + 2 \, a^{2} b + a b^{2} + {\left(a^{3} - 2 \, a^{2} b + a b^{2}\right)} \cos\left(2 \, x\right)^{2} - 2 \, {\left(a^{3} - a b^{2}\right)} \cos\left(2 \, x\right)\right)} \sqrt{a - b} \arctan\left(-\frac{\sqrt{a - b} \sqrt{\frac{{\left(a - b\right)} \cos\left(2 \, x\right) - a - b}{\cos\left(2 \, x\right) - 1}} \sin\left(2 \, x\right)}{{\left(a - b\right)} \cos\left(2 \, x\right) - b}\right) - 2 \, {\left(3 \, a^{3} - a^{2} b - a b^{2} - b^{3} - {\left(3 \, a^{3} - 5 \, a^{2} b + a b^{2} + b^{3}\right)} \cos\left(2 \, x\right)\right)} \sqrt{\frac{{\left(a - b\right)} \cos\left(2 \, x\right) - a - b}{\cos\left(2 \, x\right) - 1}} \sin\left(2 \, x\right)}{6 \, {\left(a^{6} - a^{5} b - 2 \, a^{4} b^{2} + 2 \, a^{3} b^{3} + a^{2} b^{4} - a b^{5} + {\left(a^{6} - 5 \, a^{5} b + 10 \, a^{4} b^{2} - 10 \, a^{3} b^{3} + 5 \, a^{2} b^{4} - a b^{5}\right)} \cos\left(2 \, x\right)^{2} - 2 \, {\left(a^{6} - 3 \, a^{5} b + 2 \, a^{4} b^{2} + 2 \, a^{3} b^{3} - 3 \, a^{2} b^{4} + a b^{5}\right)} \cos\left(2 \, x\right)\right)}}\right]"," ",0,"[-1/12*(3*(a^3 + 2*a^2*b + a*b^2 + (a^3 - 2*a^2*b + a*b^2)*cos(2*x)^2 - 2*(a^3 - a*b^2)*cos(2*x))*sqrt(-a + b)*log(-2*(a^2 - 2*a*b + b^2)*cos(2*x)^2 + 2*((a - b)*cos(2*x) - b)*sqrt(-a + b)*sqrt(((a - b)*cos(2*x) - a - b)/(cos(2*x) - 1))*sin(2*x) + a^2 - 2*b^2 + 4*(a*b - b^2)*cos(2*x)) + 4*(3*a^3 - a^2*b - a*b^2 - b^3 - (3*a^3 - 5*a^2*b + a*b^2 + b^3)*cos(2*x))*sqrt(((a - b)*cos(2*x) - a - b)/(cos(2*x) - 1))*sin(2*x))/(a^6 - a^5*b - 2*a^4*b^2 + 2*a^3*b^3 + a^2*b^4 - a*b^5 + (a^6 - 5*a^5*b + 10*a^4*b^2 - 10*a^3*b^3 + 5*a^2*b^4 - a*b^5)*cos(2*x)^2 - 2*(a^6 - 3*a^5*b + 2*a^4*b^2 + 2*a^3*b^3 - 3*a^2*b^4 + a*b^5)*cos(2*x)), 1/6*(3*(a^3 + 2*a^2*b + a*b^2 + (a^3 - 2*a^2*b + a*b^2)*cos(2*x)^2 - 2*(a^3 - a*b^2)*cos(2*x))*sqrt(a - b)*arctan(-sqrt(a - b)*sqrt(((a - b)*cos(2*x) - a - b)/(cos(2*x) - 1))*sin(2*x)/((a - b)*cos(2*x) - b)) - 2*(3*a^3 - a^2*b - a*b^2 - b^3 - (3*a^3 - 5*a^2*b + a*b^2 + b^3)*cos(2*x))*sqrt(((a - b)*cos(2*x) - a - b)/(cos(2*x) - 1))*sin(2*x))/(a^6 - a^5*b - 2*a^4*b^2 + 2*a^3*b^3 + a^2*b^4 - a*b^5 + (a^6 - 5*a^5*b + 10*a^4*b^2 - 10*a^3*b^3 + 5*a^2*b^4 - a*b^5)*cos(2*x)^2 - 2*(a^6 - 3*a^5*b + 2*a^4*b^2 + 2*a^3*b^3 - 3*a^2*b^4 + a*b^5)*cos(2*x))]","B",0
56,1,627,0,0.827393," ","integrate(cot(x)/(a+b*cot(x)^2)^(5/2),x, algorithm=""fricas"")","\left[\frac{3 \, {\left({\left(a^{2} - 2 \, a b + b^{2}\right)} \cos\left(2 \, x\right)^{2} + a^{2} + 2 \, a b + b^{2} - 2 \, {\left(a^{2} - b^{2}\right)} \cos\left(2 \, x\right)\right)} \sqrt{a - b} \log\left(-\sqrt{a - b} \sqrt{\frac{{\left(a - b\right)} \cos\left(2 \, x\right) - a - b}{\cos\left(2 \, x\right) - 1}} {\left(\cos\left(2 \, x\right) - 1\right)} - {\left(a - b\right)} \cos\left(2 \, x\right) + a\right) - 4 \, {\left(2 \, {\left(a^{2} - 2 \, a b + b^{2}\right)} \cos\left(2 \, x\right)^{2} + 2 \, a^{2} - a b - b^{2} - {\left(4 \, a^{2} - 5 \, a b + b^{2}\right)} \cos\left(2 \, x\right)\right)} \sqrt{\frac{{\left(a - b\right)} \cos\left(2 \, x\right) - a - b}{\cos\left(2 \, x\right) - 1}}}{6 \, {\left(a^{5} - a^{4} b - 2 \, a^{3} b^{2} + 2 \, a^{2} b^{3} + a b^{4} - b^{5} + {\left(a^{5} - 5 \, a^{4} b + 10 \, a^{3} b^{2} - 10 \, a^{2} b^{3} + 5 \, a b^{4} - b^{5}\right)} \cos\left(2 \, x\right)^{2} - 2 \, {\left(a^{5} - 3 \, a^{4} b + 2 \, a^{3} b^{2} + 2 \, a^{2} b^{3} - 3 \, a b^{4} + b^{5}\right)} \cos\left(2 \, x\right)\right)}}, \frac{3 \, {\left({\left(a^{2} - 2 \, a b + b^{2}\right)} \cos\left(2 \, x\right)^{2} + a^{2} + 2 \, a b + b^{2} - 2 \, {\left(a^{2} - b^{2}\right)} \cos\left(2 \, x\right)\right)} \sqrt{-a + b} \arctan\left(-\frac{\sqrt{-a + b} \sqrt{\frac{{\left(a - b\right)} \cos\left(2 \, x\right) - a - b}{\cos\left(2 \, x\right) - 1}}}{a - b}\right) - 2 \, {\left(2 \, {\left(a^{2} - 2 \, a b + b^{2}\right)} \cos\left(2 \, x\right)^{2} + 2 \, a^{2} - a b - b^{2} - {\left(4 \, a^{2} - 5 \, a b + b^{2}\right)} \cos\left(2 \, x\right)\right)} \sqrt{\frac{{\left(a - b\right)} \cos\left(2 \, x\right) - a - b}{\cos\left(2 \, x\right) - 1}}}{3 \, {\left(a^{5} - a^{4} b - 2 \, a^{3} b^{2} + 2 \, a^{2} b^{3} + a b^{4} - b^{5} + {\left(a^{5} - 5 \, a^{4} b + 10 \, a^{3} b^{2} - 10 \, a^{2} b^{3} + 5 \, a b^{4} - b^{5}\right)} \cos\left(2 \, x\right)^{2} - 2 \, {\left(a^{5} - 3 \, a^{4} b + 2 \, a^{3} b^{2} + 2 \, a^{2} b^{3} - 3 \, a b^{4} + b^{5}\right)} \cos\left(2 \, x\right)\right)}}\right]"," ",0,"[1/6*(3*((a^2 - 2*a*b + b^2)*cos(2*x)^2 + a^2 + 2*a*b + b^2 - 2*(a^2 - b^2)*cos(2*x))*sqrt(a - b)*log(-sqrt(a - b)*sqrt(((a - b)*cos(2*x) - a - b)/(cos(2*x) - 1))*(cos(2*x) - 1) - (a - b)*cos(2*x) + a) - 4*(2*(a^2 - 2*a*b + b^2)*cos(2*x)^2 + 2*a^2 - a*b - b^2 - (4*a^2 - 5*a*b + b^2)*cos(2*x))*sqrt(((a - b)*cos(2*x) - a - b)/(cos(2*x) - 1)))/(a^5 - a^4*b - 2*a^3*b^2 + 2*a^2*b^3 + a*b^4 - b^5 + (a^5 - 5*a^4*b + 10*a^3*b^2 - 10*a^2*b^3 + 5*a*b^4 - b^5)*cos(2*x)^2 - 2*(a^5 - 3*a^4*b + 2*a^3*b^2 + 2*a^2*b^3 - 3*a*b^4 + b^5)*cos(2*x)), 1/3*(3*((a^2 - 2*a*b + b^2)*cos(2*x)^2 + a^2 + 2*a*b + b^2 - 2*(a^2 - b^2)*cos(2*x))*sqrt(-a + b)*arctan(-sqrt(-a + b)*sqrt(((a - b)*cos(2*x) - a - b)/(cos(2*x) - 1))/(a - b)) - 2*(2*(a^2 - 2*a*b + b^2)*cos(2*x)^2 + 2*a^2 - a*b - b^2 - (4*a^2 - 5*a*b + b^2)*cos(2*x))*sqrt(((a - b)*cos(2*x) - a - b)/(cos(2*x) - 1)))/(a^5 - a^4*b - 2*a^3*b^2 + 2*a^2*b^3 + a*b^4 - b^5 + (a^5 - 5*a^4*b + 10*a^3*b^2 - 10*a^2*b^3 + 5*a*b^4 - b^5)*cos(2*x)^2 - 2*(a^5 - 3*a^4*b + 2*a^3*b^2 + 2*a^2*b^3 - 3*a*b^4 + b^5)*cos(2*x))]","B",0
57,1,1531,0,0.817165," ","integrate(tan(x)/(a+b*cot(x)^2)^(5/2),x, algorithm=""fricas"")","\left[\frac{3 \, {\left(a^{3} b^{2} - 3 \, a^{2} b^{3} + 3 \, a b^{4} - b^{5} + {\left(a^{5} - 3 \, a^{4} b + 3 \, a^{3} b^{2} - a^{2} b^{3}\right)} \tan\left(x\right)^{4} + 2 \, {\left(a^{4} b - 3 \, a^{3} b^{2} + 3 \, a^{2} b^{3} - a b^{4}\right)} \tan\left(x\right)^{2}\right)} \sqrt{a} \log\left(2 \, a \tan\left(x\right)^{2} + 2 \, \sqrt{a} \sqrt{\frac{a \tan\left(x\right)^{2} + b}{\tan\left(x\right)^{2}}} \tan\left(x\right)^{2} + b\right) + 3 \, {\left(a^{5} \tan\left(x\right)^{4} + 2 \, a^{4} b \tan\left(x\right)^{2} + a^{3} b^{2}\right)} \sqrt{a - b} \log\left(\frac{{\left(2 \, a - b\right)} \tan\left(x\right)^{2} - 2 \, \sqrt{a - b} \sqrt{\frac{a \tan\left(x\right)^{2} + b}{\tan\left(x\right)^{2}}} \tan\left(x\right)^{2} + b}{\tan\left(x\right)^{2} + 1}\right) + 2 \, {\left({\left(7 \, a^{4} b - 11 \, a^{3} b^{2} + 4 \, a^{2} b^{3}\right)} \tan\left(x\right)^{4} + 3 \, {\left(2 \, a^{3} b^{2} - 3 \, a^{2} b^{3} + a b^{4}\right)} \tan\left(x\right)^{2}\right)} \sqrt{\frac{a \tan\left(x\right)^{2} + b}{\tan\left(x\right)^{2}}}}{6 \, {\left(a^{6} b^{2} - 3 \, a^{5} b^{3} + 3 \, a^{4} b^{4} - a^{3} b^{5} + {\left(a^{8} - 3 \, a^{7} b + 3 \, a^{6} b^{2} - a^{5} b^{3}\right)} \tan\left(x\right)^{4} + 2 \, {\left(a^{7} b - 3 \, a^{6} b^{2} + 3 \, a^{5} b^{3} - a^{4} b^{4}\right)} \tan\left(x\right)^{2}\right)}}, -\frac{6 \, {\left(a^{5} \tan\left(x\right)^{4} + 2 \, a^{4} b \tan\left(x\right)^{2} + a^{3} b^{2}\right)} \sqrt{-a + b} \arctan\left(-\frac{\sqrt{-a + b} \sqrt{\frac{a \tan\left(x\right)^{2} + b}{\tan\left(x\right)^{2}}}}{a - b}\right) - 3 \, {\left(a^{3} b^{2} - 3 \, a^{2} b^{3} + 3 \, a b^{4} - b^{5} + {\left(a^{5} - 3 \, a^{4} b + 3 \, a^{3} b^{2} - a^{2} b^{3}\right)} \tan\left(x\right)^{4} + 2 \, {\left(a^{4} b - 3 \, a^{3} b^{2} + 3 \, a^{2} b^{3} - a b^{4}\right)} \tan\left(x\right)^{2}\right)} \sqrt{a} \log\left(2 \, a \tan\left(x\right)^{2} + 2 \, \sqrt{a} \sqrt{\frac{a \tan\left(x\right)^{2} + b}{\tan\left(x\right)^{2}}} \tan\left(x\right)^{2} + b\right) - 2 \, {\left({\left(7 \, a^{4} b - 11 \, a^{3} b^{2} + 4 \, a^{2} b^{3}\right)} \tan\left(x\right)^{4} + 3 \, {\left(2 \, a^{3} b^{2} - 3 \, a^{2} b^{3} + a b^{4}\right)} \tan\left(x\right)^{2}\right)} \sqrt{\frac{a \tan\left(x\right)^{2} + b}{\tan\left(x\right)^{2}}}}{6 \, {\left(a^{6} b^{2} - 3 \, a^{5} b^{3} + 3 \, a^{4} b^{4} - a^{3} b^{5} + {\left(a^{8} - 3 \, a^{7} b + 3 \, a^{6} b^{2} - a^{5} b^{3}\right)} \tan\left(x\right)^{4} + 2 \, {\left(a^{7} b - 3 \, a^{6} b^{2} + 3 \, a^{5} b^{3} - a^{4} b^{4}\right)} \tan\left(x\right)^{2}\right)}}, -\frac{6 \, {\left(a^{3} b^{2} - 3 \, a^{2} b^{3} + 3 \, a b^{4} - b^{5} + {\left(a^{5} - 3 \, a^{4} b + 3 \, a^{3} b^{2} - a^{2} b^{3}\right)} \tan\left(x\right)^{4} + 2 \, {\left(a^{4} b - 3 \, a^{3} b^{2} + 3 \, a^{2} b^{3} - a b^{4}\right)} \tan\left(x\right)^{2}\right)} \sqrt{-a} \arctan\left(\frac{\sqrt{-a} \sqrt{\frac{a \tan\left(x\right)^{2} + b}{\tan\left(x\right)^{2}}}}{a}\right) - 3 \, {\left(a^{5} \tan\left(x\right)^{4} + 2 \, a^{4} b \tan\left(x\right)^{2} + a^{3} b^{2}\right)} \sqrt{a - b} \log\left(\frac{{\left(2 \, a - b\right)} \tan\left(x\right)^{2} - 2 \, \sqrt{a - b} \sqrt{\frac{a \tan\left(x\right)^{2} + b}{\tan\left(x\right)^{2}}} \tan\left(x\right)^{2} + b}{\tan\left(x\right)^{2} + 1}\right) - 2 \, {\left({\left(7 \, a^{4} b - 11 \, a^{3} b^{2} + 4 \, a^{2} b^{3}\right)} \tan\left(x\right)^{4} + 3 \, {\left(2 \, a^{3} b^{2} - 3 \, a^{2} b^{3} + a b^{4}\right)} \tan\left(x\right)^{2}\right)} \sqrt{\frac{a \tan\left(x\right)^{2} + b}{\tan\left(x\right)^{2}}}}{6 \, {\left(a^{6} b^{2} - 3 \, a^{5} b^{3} + 3 \, a^{4} b^{4} - a^{3} b^{5} + {\left(a^{8} - 3 \, a^{7} b + 3 \, a^{6} b^{2} - a^{5} b^{3}\right)} \tan\left(x\right)^{4} + 2 \, {\left(a^{7} b - 3 \, a^{6} b^{2} + 3 \, a^{5} b^{3} - a^{4} b^{4}\right)} \tan\left(x\right)^{2}\right)}}, -\frac{3 \, {\left(a^{3} b^{2} - 3 \, a^{2} b^{3} + 3 \, a b^{4} - b^{5} + {\left(a^{5} - 3 \, a^{4} b + 3 \, a^{3} b^{2} - a^{2} b^{3}\right)} \tan\left(x\right)^{4} + 2 \, {\left(a^{4} b - 3 \, a^{3} b^{2} + 3 \, a^{2} b^{3} - a b^{4}\right)} \tan\left(x\right)^{2}\right)} \sqrt{-a} \arctan\left(\frac{\sqrt{-a} \sqrt{\frac{a \tan\left(x\right)^{2} + b}{\tan\left(x\right)^{2}}}}{a}\right) + 3 \, {\left(a^{5} \tan\left(x\right)^{4} + 2 \, a^{4} b \tan\left(x\right)^{2} + a^{3} b^{2}\right)} \sqrt{-a + b} \arctan\left(-\frac{\sqrt{-a + b} \sqrt{\frac{a \tan\left(x\right)^{2} + b}{\tan\left(x\right)^{2}}}}{a - b}\right) - {\left({\left(7 \, a^{4} b - 11 \, a^{3} b^{2} + 4 \, a^{2} b^{3}\right)} \tan\left(x\right)^{4} + 3 \, {\left(2 \, a^{3} b^{2} - 3 \, a^{2} b^{3} + a b^{4}\right)} \tan\left(x\right)^{2}\right)} \sqrt{\frac{a \tan\left(x\right)^{2} + b}{\tan\left(x\right)^{2}}}}{3 \, {\left(a^{6} b^{2} - 3 \, a^{5} b^{3} + 3 \, a^{4} b^{4} - a^{3} b^{5} + {\left(a^{8} - 3 \, a^{7} b + 3 \, a^{6} b^{2} - a^{5} b^{3}\right)} \tan\left(x\right)^{4} + 2 \, {\left(a^{7} b - 3 \, a^{6} b^{2} + 3 \, a^{5} b^{3} - a^{4} b^{4}\right)} \tan\left(x\right)^{2}\right)}}\right]"," ",0,"[1/6*(3*(a^3*b^2 - 3*a^2*b^3 + 3*a*b^4 - b^5 + (a^5 - 3*a^4*b + 3*a^3*b^2 - a^2*b^3)*tan(x)^4 + 2*(a^4*b - 3*a^3*b^2 + 3*a^2*b^3 - a*b^4)*tan(x)^2)*sqrt(a)*log(2*a*tan(x)^2 + 2*sqrt(a)*sqrt((a*tan(x)^2 + b)/tan(x)^2)*tan(x)^2 + b) + 3*(a^5*tan(x)^4 + 2*a^4*b*tan(x)^2 + a^3*b^2)*sqrt(a - b)*log(((2*a - b)*tan(x)^2 - 2*sqrt(a - b)*sqrt((a*tan(x)^2 + b)/tan(x)^2)*tan(x)^2 + b)/(tan(x)^2 + 1)) + 2*((7*a^4*b - 11*a^3*b^2 + 4*a^2*b^3)*tan(x)^4 + 3*(2*a^3*b^2 - 3*a^2*b^3 + a*b^4)*tan(x)^2)*sqrt((a*tan(x)^2 + b)/tan(x)^2))/(a^6*b^2 - 3*a^5*b^3 + 3*a^4*b^4 - a^3*b^5 + (a^8 - 3*a^7*b + 3*a^6*b^2 - a^5*b^3)*tan(x)^4 + 2*(a^7*b - 3*a^6*b^2 + 3*a^5*b^3 - a^4*b^4)*tan(x)^2), -1/6*(6*(a^5*tan(x)^4 + 2*a^4*b*tan(x)^2 + a^3*b^2)*sqrt(-a + b)*arctan(-sqrt(-a + b)*sqrt((a*tan(x)^2 + b)/tan(x)^2)/(a - b)) - 3*(a^3*b^2 - 3*a^2*b^3 + 3*a*b^4 - b^5 + (a^5 - 3*a^4*b + 3*a^3*b^2 - a^2*b^3)*tan(x)^4 + 2*(a^4*b - 3*a^3*b^2 + 3*a^2*b^3 - a*b^4)*tan(x)^2)*sqrt(a)*log(2*a*tan(x)^2 + 2*sqrt(a)*sqrt((a*tan(x)^2 + b)/tan(x)^2)*tan(x)^2 + b) - 2*((7*a^4*b - 11*a^3*b^2 + 4*a^2*b^3)*tan(x)^4 + 3*(2*a^3*b^2 - 3*a^2*b^3 + a*b^4)*tan(x)^2)*sqrt((a*tan(x)^2 + b)/tan(x)^2))/(a^6*b^2 - 3*a^5*b^3 + 3*a^4*b^4 - a^3*b^5 + (a^8 - 3*a^7*b + 3*a^6*b^2 - a^5*b^3)*tan(x)^4 + 2*(a^7*b - 3*a^6*b^2 + 3*a^5*b^3 - a^4*b^4)*tan(x)^2), -1/6*(6*(a^3*b^2 - 3*a^2*b^3 + 3*a*b^4 - b^5 + (a^5 - 3*a^4*b + 3*a^3*b^2 - a^2*b^3)*tan(x)^4 + 2*(a^4*b - 3*a^3*b^2 + 3*a^2*b^3 - a*b^4)*tan(x)^2)*sqrt(-a)*arctan(sqrt(-a)*sqrt((a*tan(x)^2 + b)/tan(x)^2)/a) - 3*(a^5*tan(x)^4 + 2*a^4*b*tan(x)^2 + a^3*b^2)*sqrt(a - b)*log(((2*a - b)*tan(x)^2 - 2*sqrt(a - b)*sqrt((a*tan(x)^2 + b)/tan(x)^2)*tan(x)^2 + b)/(tan(x)^2 + 1)) - 2*((7*a^4*b - 11*a^3*b^2 + 4*a^2*b^3)*tan(x)^4 + 3*(2*a^3*b^2 - 3*a^2*b^3 + a*b^4)*tan(x)^2)*sqrt((a*tan(x)^2 + b)/tan(x)^2))/(a^6*b^2 - 3*a^5*b^3 + 3*a^4*b^4 - a^3*b^5 + (a^8 - 3*a^7*b + 3*a^6*b^2 - a^5*b^3)*tan(x)^4 + 2*(a^7*b - 3*a^6*b^2 + 3*a^5*b^3 - a^4*b^4)*tan(x)^2), -1/3*(3*(a^3*b^2 - 3*a^2*b^3 + 3*a*b^4 - b^5 + (a^5 - 3*a^4*b + 3*a^3*b^2 - a^2*b^3)*tan(x)^4 + 2*(a^4*b - 3*a^3*b^2 + 3*a^2*b^3 - a*b^4)*tan(x)^2)*sqrt(-a)*arctan(sqrt(-a)*sqrt((a*tan(x)^2 + b)/tan(x)^2)/a) + 3*(a^5*tan(x)^4 + 2*a^4*b*tan(x)^2 + a^3*b^2)*sqrt(-a + b)*arctan(-sqrt(-a + b)*sqrt((a*tan(x)^2 + b)/tan(x)^2)/(a - b)) - ((7*a^4*b - 11*a^3*b^2 + 4*a^2*b^3)*tan(x)^4 + 3*(2*a^3*b^2 - 3*a^2*b^3 + a*b^4)*tan(x)^2)*sqrt((a*tan(x)^2 + b)/tan(x)^2))/(a^6*b^2 - 3*a^5*b^3 + 3*a^4*b^4 - a^3*b^5 + (a^8 - 3*a^7*b + 3*a^6*b^2 - a^5*b^3)*tan(x)^4 + 2*(a^7*b - 3*a^6*b^2 + 3*a^5*b^3 - a^4*b^4)*tan(x)^2)]","B",0
58,1,647,0,0.700781," ","integrate(tan(x)^2/(a+b*cot(x)^2)^(5/2),x, algorithm=""fricas"")","\left[-\frac{3 \, {\left(a^{5} \tan\left(x\right)^{4} + 2 \, a^{4} b \tan\left(x\right)^{2} + a^{3} b^{2}\right)} \sqrt{-a + b} \log\left(-\frac{a^{2} \tan\left(x\right)^{4} - 2 \, {\left(3 \, a^{2} - 4 \, a b\right)} \tan\left(x\right)^{2} + a^{2} - 8 \, a b + 8 \, b^{2} + 4 \, {\left(a \tan\left(x\right)^{3} - {\left(a - 2 \, b\right)} \tan\left(x\right)\right)} \sqrt{-a + b} \sqrt{\frac{a \tan\left(x\right)^{2} + b}{\tan\left(x\right)^{2}}}}{\tan\left(x\right)^{4} + 2 \, \tan\left(x\right)^{2} + 1}\right) - 4 \, {\left(3 \, {\left(a^{5} - 3 \, a^{4} b + 3 \, a^{3} b^{2} - a^{2} b^{3}\right)} \tan\left(x\right)^{5} + 3 \, {\left(2 \, a^{4} b - 9 \, a^{3} b^{2} + 11 \, a^{2} b^{3} - 4 \, a b^{4}\right)} \tan\left(x\right)^{3} + {\left(3 \, a^{3} b^{2} - 17 \, a^{2} b^{3} + 22 \, a b^{4} - 8 \, b^{5}\right)} \tan\left(x\right)\right)} \sqrt{\frac{a \tan\left(x\right)^{2} + b}{\tan\left(x\right)^{2}}}}{12 \, {\left(a^{6} b^{2} - 3 \, a^{5} b^{3} + 3 \, a^{4} b^{4} - a^{3} b^{5} + {\left(a^{8} - 3 \, a^{7} b + 3 \, a^{6} b^{2} - a^{5} b^{3}\right)} \tan\left(x\right)^{4} + 2 \, {\left(a^{7} b - 3 \, a^{6} b^{2} + 3 \, a^{5} b^{3} - a^{4} b^{4}\right)} \tan\left(x\right)^{2}\right)}}, \frac{3 \, {\left(a^{5} \tan\left(x\right)^{4} + 2 \, a^{4} b \tan\left(x\right)^{2} + a^{3} b^{2}\right)} \sqrt{a - b} \arctan\left(\frac{2 \, \sqrt{a - b} \sqrt{\frac{a \tan\left(x\right)^{2} + b}{\tan\left(x\right)^{2}}} \tan\left(x\right)}{a \tan\left(x\right)^{2} - a + 2 \, b}\right) + 2 \, {\left(3 \, {\left(a^{5} - 3 \, a^{4} b + 3 \, a^{3} b^{2} - a^{2} b^{3}\right)} \tan\left(x\right)^{5} + 3 \, {\left(2 \, a^{4} b - 9 \, a^{3} b^{2} + 11 \, a^{2} b^{3} - 4 \, a b^{4}\right)} \tan\left(x\right)^{3} + {\left(3 \, a^{3} b^{2} - 17 \, a^{2} b^{3} + 22 \, a b^{4} - 8 \, b^{5}\right)} \tan\left(x\right)\right)} \sqrt{\frac{a \tan\left(x\right)^{2} + b}{\tan\left(x\right)^{2}}}}{6 \, {\left(a^{6} b^{2} - 3 \, a^{5} b^{3} + 3 \, a^{4} b^{4} - a^{3} b^{5} + {\left(a^{8} - 3 \, a^{7} b + 3 \, a^{6} b^{2} - a^{5} b^{3}\right)} \tan\left(x\right)^{4} + 2 \, {\left(a^{7} b - 3 \, a^{6} b^{2} + 3 \, a^{5} b^{3} - a^{4} b^{4}\right)} \tan\left(x\right)^{2}\right)}}\right]"," ",0,"[-1/12*(3*(a^5*tan(x)^4 + 2*a^4*b*tan(x)^2 + a^3*b^2)*sqrt(-a + b)*log(-(a^2*tan(x)^4 - 2*(3*a^2 - 4*a*b)*tan(x)^2 + a^2 - 8*a*b + 8*b^2 + 4*(a*tan(x)^3 - (a - 2*b)*tan(x))*sqrt(-a + b)*sqrt((a*tan(x)^2 + b)/tan(x)^2))/(tan(x)^4 + 2*tan(x)^2 + 1)) - 4*(3*(a^5 - 3*a^4*b + 3*a^3*b^2 - a^2*b^3)*tan(x)^5 + 3*(2*a^4*b - 9*a^3*b^2 + 11*a^2*b^3 - 4*a*b^4)*tan(x)^3 + (3*a^3*b^2 - 17*a^2*b^3 + 22*a*b^4 - 8*b^5)*tan(x))*sqrt((a*tan(x)^2 + b)/tan(x)^2))/(a^6*b^2 - 3*a^5*b^3 + 3*a^4*b^4 - a^3*b^5 + (a^8 - 3*a^7*b + 3*a^6*b^2 - a^5*b^3)*tan(x)^4 + 2*(a^7*b - 3*a^6*b^2 + 3*a^5*b^3 - a^4*b^4)*tan(x)^2), 1/6*(3*(a^5*tan(x)^4 + 2*a^4*b*tan(x)^2 + a^3*b^2)*sqrt(a - b)*arctan(2*sqrt(a - b)*sqrt((a*tan(x)^2 + b)/tan(x)^2)*tan(x)/(a*tan(x)^2 - a + 2*b)) + 2*(3*(a^5 - 3*a^4*b + 3*a^3*b^2 - a^2*b^3)*tan(x)^5 + 3*(2*a^4*b - 9*a^3*b^2 + 11*a^2*b^3 - 4*a*b^4)*tan(x)^3 + (3*a^3*b^2 - 17*a^2*b^3 + 22*a*b^4 - 8*b^5)*tan(x))*sqrt((a*tan(x)^2 + b)/tan(x)^2))/(a^6*b^2 - 3*a^5*b^3 + 3*a^4*b^4 - a^3*b^5 + (a^8 - 3*a^7*b + 3*a^6*b^2 - a^5*b^3)*tan(x)^4 + 2*(a^7*b - 3*a^6*b^2 + 3*a^5*b^3 - a^4*b^4)*tan(x)^2)]","B",0
59,1,24,0,0.462367," ","integrate(1/(1+cot(x)^3),x, algorithm=""fricas"")","\frac{1}{2} \, x - \frac{1}{12} \, \log\left(\sin\left(2 \, x\right) + 1\right) + \frac{1}{3} \, \log\left(-\frac{1}{2} \, \sin\left(2 \, x\right) + 1\right)"," ",0,"1/2*x - 1/12*log(sin(2*x) + 1) + 1/3*log(-1/2*sin(2*x) + 1)","A",0
60,1,1063,0,0.830235," ","integrate(cot(x)*(a+b*cot(x)^4)^(1/2),x, algorithm=""fricas"")","\left[\frac{1}{4} \, \sqrt{a + b} \log\left(\frac{1}{2} \, {\left(a^{2} + 2 \, a b + b^{2}\right)} \cos\left(2 \, x\right)^{2} + \frac{1}{2} \, a^{2} + \frac{1}{2} \, b^{2} + \frac{1}{2} \, {\left({\left(a + b\right)} \cos\left(2 \, x\right)^{2} - 2 \, a \cos\left(2 \, x\right) + a - b\right)} \sqrt{a + b} \sqrt{\frac{{\left(a + b\right)} \cos\left(2 \, x\right)^{2} - 2 \, {\left(a - b\right)} \cos\left(2 \, x\right) + a + b}{\cos\left(2 \, x\right)^{2} - 2 \, \cos\left(2 \, x\right) + 1}} - {\left(a^{2} - b^{2}\right)} \cos\left(2 \, x\right)\right) + \frac{1}{4} \, \sqrt{b} \log\left(-\frac{{\left(a + 2 \, b\right)} \cos\left(2 \, x\right)^{2} - 2 \, {\left(\cos\left(2 \, x\right)^{2} - 1\right)} \sqrt{b} \sqrt{\frac{{\left(a + b\right)} \cos\left(2 \, x\right)^{2} - 2 \, {\left(a - b\right)} \cos\left(2 \, x\right) + a + b}{\cos\left(2 \, x\right)^{2} - 2 \, \cos\left(2 \, x\right) + 1}} - 2 \, {\left(a - 2 \, b\right)} \cos\left(2 \, x\right) + a + 2 \, b}{\cos\left(2 \, x\right)^{2} - 2 \, \cos\left(2 \, x\right) + 1}\right) - \frac{1}{2} \, \sqrt{\frac{{\left(a + b\right)} \cos\left(2 \, x\right)^{2} - 2 \, {\left(a - b\right)} \cos\left(2 \, x\right) + a + b}{\cos\left(2 \, x\right)^{2} - 2 \, \cos\left(2 \, x\right) + 1}}, \frac{1}{2} \, \sqrt{-b} \arctan\left(\frac{\sqrt{-b} \sqrt{\frac{{\left(a + b\right)} \cos\left(2 \, x\right)^{2} - 2 \, {\left(a - b\right)} \cos\left(2 \, x\right) + a + b}{\cos\left(2 \, x\right)^{2} - 2 \, \cos\left(2 \, x\right) + 1}} {\left(\cos\left(2 \, x\right) - 1\right)}}{b \cos\left(2 \, x\right) + b}\right) + \frac{1}{4} \, \sqrt{a + b} \log\left(\frac{1}{2} \, {\left(a^{2} + 2 \, a b + b^{2}\right)} \cos\left(2 \, x\right)^{2} + \frac{1}{2} \, a^{2} + \frac{1}{2} \, b^{2} + \frac{1}{2} \, {\left({\left(a + b\right)} \cos\left(2 \, x\right)^{2} - 2 \, a \cos\left(2 \, x\right) + a - b\right)} \sqrt{a + b} \sqrt{\frac{{\left(a + b\right)} \cos\left(2 \, x\right)^{2} - 2 \, {\left(a - b\right)} \cos\left(2 \, x\right) + a + b}{\cos\left(2 \, x\right)^{2} - 2 \, \cos\left(2 \, x\right) + 1}} - {\left(a^{2} - b^{2}\right)} \cos\left(2 \, x\right)\right) - \frac{1}{2} \, \sqrt{\frac{{\left(a + b\right)} \cos\left(2 \, x\right)^{2} - 2 \, {\left(a - b\right)} \cos\left(2 \, x\right) + a + b}{\cos\left(2 \, x\right)^{2} - 2 \, \cos\left(2 \, x\right) + 1}}, -\frac{1}{2} \, \sqrt{-a - b} \arctan\left(\frac{{\left({\left(a + b\right)} \cos\left(2 \, x\right)^{2} - 2 \, a \cos\left(2 \, x\right) + a - b\right)} \sqrt{-a - b} \sqrt{\frac{{\left(a + b\right)} \cos\left(2 \, x\right)^{2} - 2 \, {\left(a - b\right)} \cos\left(2 \, x\right) + a + b}{\cos\left(2 \, x\right)^{2} - 2 \, \cos\left(2 \, x\right) + 1}}}{{\left(a^{2} + 2 \, a b + b^{2}\right)} \cos\left(2 \, x\right)^{2} + a^{2} + 2 \, a b + b^{2} - 2 \, {\left(a^{2} - b^{2}\right)} \cos\left(2 \, x\right)}\right) + \frac{1}{4} \, \sqrt{b} \log\left(-\frac{{\left(a + 2 \, b\right)} \cos\left(2 \, x\right)^{2} - 2 \, {\left(\cos\left(2 \, x\right)^{2} - 1\right)} \sqrt{b} \sqrt{\frac{{\left(a + b\right)} \cos\left(2 \, x\right)^{2} - 2 \, {\left(a - b\right)} \cos\left(2 \, x\right) + a + b}{\cos\left(2 \, x\right)^{2} - 2 \, \cos\left(2 \, x\right) + 1}} - 2 \, {\left(a - 2 \, b\right)} \cos\left(2 \, x\right) + a + 2 \, b}{\cos\left(2 \, x\right)^{2} - 2 \, \cos\left(2 \, x\right) + 1}\right) - \frac{1}{2} \, \sqrt{\frac{{\left(a + b\right)} \cos\left(2 \, x\right)^{2} - 2 \, {\left(a - b\right)} \cos\left(2 \, x\right) + a + b}{\cos\left(2 \, x\right)^{2} - 2 \, \cos\left(2 \, x\right) + 1}}, -\frac{1}{2} \, \sqrt{-a - b} \arctan\left(\frac{{\left({\left(a + b\right)} \cos\left(2 \, x\right)^{2} - 2 \, a \cos\left(2 \, x\right) + a - b\right)} \sqrt{-a - b} \sqrt{\frac{{\left(a + b\right)} \cos\left(2 \, x\right)^{2} - 2 \, {\left(a - b\right)} \cos\left(2 \, x\right) + a + b}{\cos\left(2 \, x\right)^{2} - 2 \, \cos\left(2 \, x\right) + 1}}}{{\left(a^{2} + 2 \, a b + b^{2}\right)} \cos\left(2 \, x\right)^{2} + a^{2} + 2 \, a b + b^{2} - 2 \, {\left(a^{2} - b^{2}\right)} \cos\left(2 \, x\right)}\right) + \frac{1}{2} \, \sqrt{-b} \arctan\left(\frac{\sqrt{-b} \sqrt{\frac{{\left(a + b\right)} \cos\left(2 \, x\right)^{2} - 2 \, {\left(a - b\right)} \cos\left(2 \, x\right) + a + b}{\cos\left(2 \, x\right)^{2} - 2 \, \cos\left(2 \, x\right) + 1}} {\left(\cos\left(2 \, x\right) - 1\right)}}{b \cos\left(2 \, x\right) + b}\right) - \frac{1}{2} \, \sqrt{\frac{{\left(a + b\right)} \cos\left(2 \, x\right)^{2} - 2 \, {\left(a - b\right)} \cos\left(2 \, x\right) + a + b}{\cos\left(2 \, x\right)^{2} - 2 \, \cos\left(2 \, x\right) + 1}}\right]"," ",0,"[1/4*sqrt(a + b)*log(1/2*(a^2 + 2*a*b + b^2)*cos(2*x)^2 + 1/2*a^2 + 1/2*b^2 + 1/2*((a + b)*cos(2*x)^2 - 2*a*cos(2*x) + a - b)*sqrt(a + b)*sqrt(((a + b)*cos(2*x)^2 - 2*(a - b)*cos(2*x) + a + b)/(cos(2*x)^2 - 2*cos(2*x) + 1)) - (a^2 - b^2)*cos(2*x)) + 1/4*sqrt(b)*log(-((a + 2*b)*cos(2*x)^2 - 2*(cos(2*x)^2 - 1)*sqrt(b)*sqrt(((a + b)*cos(2*x)^2 - 2*(a - b)*cos(2*x) + a + b)/(cos(2*x)^2 - 2*cos(2*x) + 1)) - 2*(a - 2*b)*cos(2*x) + a + 2*b)/(cos(2*x)^2 - 2*cos(2*x) + 1)) - 1/2*sqrt(((a + b)*cos(2*x)^2 - 2*(a - b)*cos(2*x) + a + b)/(cos(2*x)^2 - 2*cos(2*x) + 1)), 1/2*sqrt(-b)*arctan(sqrt(-b)*sqrt(((a + b)*cos(2*x)^2 - 2*(a - b)*cos(2*x) + a + b)/(cos(2*x)^2 - 2*cos(2*x) + 1))*(cos(2*x) - 1)/(b*cos(2*x) + b)) + 1/4*sqrt(a + b)*log(1/2*(a^2 + 2*a*b + b^2)*cos(2*x)^2 + 1/2*a^2 + 1/2*b^2 + 1/2*((a + b)*cos(2*x)^2 - 2*a*cos(2*x) + a - b)*sqrt(a + b)*sqrt(((a + b)*cos(2*x)^2 - 2*(a - b)*cos(2*x) + a + b)/(cos(2*x)^2 - 2*cos(2*x) + 1)) - (a^2 - b^2)*cos(2*x)) - 1/2*sqrt(((a + b)*cos(2*x)^2 - 2*(a - b)*cos(2*x) + a + b)/(cos(2*x)^2 - 2*cos(2*x) + 1)), -1/2*sqrt(-a - b)*arctan(((a + b)*cos(2*x)^2 - 2*a*cos(2*x) + a - b)*sqrt(-a - b)*sqrt(((a + b)*cos(2*x)^2 - 2*(a - b)*cos(2*x) + a + b)/(cos(2*x)^2 - 2*cos(2*x) + 1))/((a^2 + 2*a*b + b^2)*cos(2*x)^2 + a^2 + 2*a*b + b^2 - 2*(a^2 - b^2)*cos(2*x))) + 1/4*sqrt(b)*log(-((a + 2*b)*cos(2*x)^2 - 2*(cos(2*x)^2 - 1)*sqrt(b)*sqrt(((a + b)*cos(2*x)^2 - 2*(a - b)*cos(2*x) + a + b)/(cos(2*x)^2 - 2*cos(2*x) + 1)) - 2*(a - 2*b)*cos(2*x) + a + 2*b)/(cos(2*x)^2 - 2*cos(2*x) + 1)) - 1/2*sqrt(((a + b)*cos(2*x)^2 - 2*(a - b)*cos(2*x) + a + b)/(cos(2*x)^2 - 2*cos(2*x) + 1)), -1/2*sqrt(-a - b)*arctan(((a + b)*cos(2*x)^2 - 2*a*cos(2*x) + a - b)*sqrt(-a - b)*sqrt(((a + b)*cos(2*x)^2 - 2*(a - b)*cos(2*x) + a + b)/(cos(2*x)^2 - 2*cos(2*x) + 1))/((a^2 + 2*a*b + b^2)*cos(2*x)^2 + a^2 + 2*a*b + b^2 - 2*(a^2 - b^2)*cos(2*x))) + 1/2*sqrt(-b)*arctan(sqrt(-b)*sqrt(((a + b)*cos(2*x)^2 - 2*(a - b)*cos(2*x) + a + b)/(cos(2*x)^2 - 2*cos(2*x) + 1))*(cos(2*x) - 1)/(b*cos(2*x) + b)) - 1/2*sqrt(((a + b)*cos(2*x)^2 - 2*(a - b)*cos(2*x) + a + b)/(cos(2*x)^2 - 2*cos(2*x) + 1))]","B",0
61,1,1486,0,0.851292," ","integrate(cot(x)*(a+b*cot(x)^4)^(3/2),x, algorithm=""fricas"")","\left[\frac{6 \, {\left({\left(a + b\right)} \cos\left(2 \, x\right)^{2} - 2 \, {\left(a + b\right)} \cos\left(2 \, x\right) + a + b\right)} \sqrt{a + b} \log\left(\frac{1}{2} \, {\left(a^{2} + 2 \, a b + b^{2}\right)} \cos\left(2 \, x\right)^{2} + \frac{1}{2} \, a^{2} + \frac{1}{2} \, b^{2} + \frac{1}{2} \, {\left({\left(a + b\right)} \cos\left(2 \, x\right)^{2} - 2 \, a \cos\left(2 \, x\right) + a - b\right)} \sqrt{a + b} \sqrt{\frac{{\left(a + b\right)} \cos\left(2 \, x\right)^{2} - 2 \, {\left(a - b\right)} \cos\left(2 \, x\right) + a + b}{\cos\left(2 \, x\right)^{2} - 2 \, \cos\left(2 \, x\right) + 1}} - {\left(a^{2} - b^{2}\right)} \cos\left(2 \, x\right)\right) + 3 \, {\left({\left(3 \, a + 2 \, b\right)} \cos\left(2 \, x\right)^{2} - 2 \, {\left(3 \, a + 2 \, b\right)} \cos\left(2 \, x\right) + 3 \, a + 2 \, b\right)} \sqrt{b} \log\left(-\frac{{\left(a + 2 \, b\right)} \cos\left(2 \, x\right)^{2} - 2 \, {\left(\cos\left(2 \, x\right)^{2} - 1\right)} \sqrt{b} \sqrt{\frac{{\left(a + b\right)} \cos\left(2 \, x\right)^{2} - 2 \, {\left(a - b\right)} \cos\left(2 \, x\right) + a + b}{\cos\left(2 \, x\right)^{2} - 2 \, \cos\left(2 \, x\right) + 1}} - 2 \, {\left(a - 2 \, b\right)} \cos\left(2 \, x\right) + a + 2 \, b}{\cos\left(2 \, x\right)^{2} - 2 \, \cos\left(2 \, x\right) + 1}\right) - 2 \, {\left({\left(8 \, a + 11 \, b\right)} \cos\left(2 \, x\right)^{2} - 8 \, {\left(2 \, a + b\right)} \cos\left(2 \, x\right) + 8 \, a + 5 \, b\right)} \sqrt{\frac{{\left(a + b\right)} \cos\left(2 \, x\right)^{2} - 2 \, {\left(a - b\right)} \cos\left(2 \, x\right) + a + b}{\cos\left(2 \, x\right)^{2} - 2 \, \cos\left(2 \, x\right) + 1}}}{24 \, {\left(\cos\left(2 \, x\right)^{2} - 2 \, \cos\left(2 \, x\right) + 1\right)}}, \frac{3 \, {\left({\left(3 \, a + 2 \, b\right)} \cos\left(2 \, x\right)^{2} - 2 \, {\left(3 \, a + 2 \, b\right)} \cos\left(2 \, x\right) + 3 \, a + 2 \, b\right)} \sqrt{-b} \arctan\left(\frac{\sqrt{-b} \sqrt{\frac{{\left(a + b\right)} \cos\left(2 \, x\right)^{2} - 2 \, {\left(a - b\right)} \cos\left(2 \, x\right) + a + b}{\cos\left(2 \, x\right)^{2} - 2 \, \cos\left(2 \, x\right) + 1}} {\left(\cos\left(2 \, x\right) - 1\right)}}{b \cos\left(2 \, x\right) + b}\right) + 3 \, {\left({\left(a + b\right)} \cos\left(2 \, x\right)^{2} - 2 \, {\left(a + b\right)} \cos\left(2 \, x\right) + a + b\right)} \sqrt{a + b} \log\left(\frac{1}{2} \, {\left(a^{2} + 2 \, a b + b^{2}\right)} \cos\left(2 \, x\right)^{2} + \frac{1}{2} \, a^{2} + \frac{1}{2} \, b^{2} + \frac{1}{2} \, {\left({\left(a + b\right)} \cos\left(2 \, x\right)^{2} - 2 \, a \cos\left(2 \, x\right) + a - b\right)} \sqrt{a + b} \sqrt{\frac{{\left(a + b\right)} \cos\left(2 \, x\right)^{2} - 2 \, {\left(a - b\right)} \cos\left(2 \, x\right) + a + b}{\cos\left(2 \, x\right)^{2} - 2 \, \cos\left(2 \, x\right) + 1}} - {\left(a^{2} - b^{2}\right)} \cos\left(2 \, x\right)\right) - {\left({\left(8 \, a + 11 \, b\right)} \cos\left(2 \, x\right)^{2} - 8 \, {\left(2 \, a + b\right)} \cos\left(2 \, x\right) + 8 \, a + 5 \, b\right)} \sqrt{\frac{{\left(a + b\right)} \cos\left(2 \, x\right)^{2} - 2 \, {\left(a - b\right)} \cos\left(2 \, x\right) + a + b}{\cos\left(2 \, x\right)^{2} - 2 \, \cos\left(2 \, x\right) + 1}}}{12 \, {\left(\cos\left(2 \, x\right)^{2} - 2 \, \cos\left(2 \, x\right) + 1\right)}}, -\frac{12 \, {\left({\left(a + b\right)} \cos\left(2 \, x\right)^{2} - 2 \, {\left(a + b\right)} \cos\left(2 \, x\right) + a + b\right)} \sqrt{-a - b} \arctan\left(\frac{{\left({\left(a + b\right)} \cos\left(2 \, x\right)^{2} - 2 \, a \cos\left(2 \, x\right) + a - b\right)} \sqrt{-a - b} \sqrt{\frac{{\left(a + b\right)} \cos\left(2 \, x\right)^{2} - 2 \, {\left(a - b\right)} \cos\left(2 \, x\right) + a + b}{\cos\left(2 \, x\right)^{2} - 2 \, \cos\left(2 \, x\right) + 1}}}{{\left(a^{2} + 2 \, a b + b^{2}\right)} \cos\left(2 \, x\right)^{2} + a^{2} + 2 \, a b + b^{2} - 2 \, {\left(a^{2} - b^{2}\right)} \cos\left(2 \, x\right)}\right) - 3 \, {\left({\left(3 \, a + 2 \, b\right)} \cos\left(2 \, x\right)^{2} - 2 \, {\left(3 \, a + 2 \, b\right)} \cos\left(2 \, x\right) + 3 \, a + 2 \, b\right)} \sqrt{b} \log\left(-\frac{{\left(a + 2 \, b\right)} \cos\left(2 \, x\right)^{2} - 2 \, {\left(\cos\left(2 \, x\right)^{2} - 1\right)} \sqrt{b} \sqrt{\frac{{\left(a + b\right)} \cos\left(2 \, x\right)^{2} - 2 \, {\left(a - b\right)} \cos\left(2 \, x\right) + a + b}{\cos\left(2 \, x\right)^{2} - 2 \, \cos\left(2 \, x\right) + 1}} - 2 \, {\left(a - 2 \, b\right)} \cos\left(2 \, x\right) + a + 2 \, b}{\cos\left(2 \, x\right)^{2} - 2 \, \cos\left(2 \, x\right) + 1}\right) + 2 \, {\left({\left(8 \, a + 11 \, b\right)} \cos\left(2 \, x\right)^{2} - 8 \, {\left(2 \, a + b\right)} \cos\left(2 \, x\right) + 8 \, a + 5 \, b\right)} \sqrt{\frac{{\left(a + b\right)} \cos\left(2 \, x\right)^{2} - 2 \, {\left(a - b\right)} \cos\left(2 \, x\right) + a + b}{\cos\left(2 \, x\right)^{2} - 2 \, \cos\left(2 \, x\right) + 1}}}{24 \, {\left(\cos\left(2 \, x\right)^{2} - 2 \, \cos\left(2 \, x\right) + 1\right)}}, -\frac{6 \, {\left({\left(a + b\right)} \cos\left(2 \, x\right)^{2} - 2 \, {\left(a + b\right)} \cos\left(2 \, x\right) + a + b\right)} \sqrt{-a - b} \arctan\left(\frac{{\left({\left(a + b\right)} \cos\left(2 \, x\right)^{2} - 2 \, a \cos\left(2 \, x\right) + a - b\right)} \sqrt{-a - b} \sqrt{\frac{{\left(a + b\right)} \cos\left(2 \, x\right)^{2} - 2 \, {\left(a - b\right)} \cos\left(2 \, x\right) + a + b}{\cos\left(2 \, x\right)^{2} - 2 \, \cos\left(2 \, x\right) + 1}}}{{\left(a^{2} + 2 \, a b + b^{2}\right)} \cos\left(2 \, x\right)^{2} + a^{2} + 2 \, a b + b^{2} - 2 \, {\left(a^{2} - b^{2}\right)} \cos\left(2 \, x\right)}\right) - 3 \, {\left({\left(3 \, a + 2 \, b\right)} \cos\left(2 \, x\right)^{2} - 2 \, {\left(3 \, a + 2 \, b\right)} \cos\left(2 \, x\right) + 3 \, a + 2 \, b\right)} \sqrt{-b} \arctan\left(\frac{\sqrt{-b} \sqrt{\frac{{\left(a + b\right)} \cos\left(2 \, x\right)^{2} - 2 \, {\left(a - b\right)} \cos\left(2 \, x\right) + a + b}{\cos\left(2 \, x\right)^{2} - 2 \, \cos\left(2 \, x\right) + 1}} {\left(\cos\left(2 \, x\right) - 1\right)}}{b \cos\left(2 \, x\right) + b}\right) + {\left({\left(8 \, a + 11 \, b\right)} \cos\left(2 \, x\right)^{2} - 8 \, {\left(2 \, a + b\right)} \cos\left(2 \, x\right) + 8 \, a + 5 \, b\right)} \sqrt{\frac{{\left(a + b\right)} \cos\left(2 \, x\right)^{2} - 2 \, {\left(a - b\right)} \cos\left(2 \, x\right) + a + b}{\cos\left(2 \, x\right)^{2} - 2 \, \cos\left(2 \, x\right) + 1}}}{12 \, {\left(\cos\left(2 \, x\right)^{2} - 2 \, \cos\left(2 \, x\right) + 1\right)}}\right]"," ",0,"[1/24*(6*((a + b)*cos(2*x)^2 - 2*(a + b)*cos(2*x) + a + b)*sqrt(a + b)*log(1/2*(a^2 + 2*a*b + b^2)*cos(2*x)^2 + 1/2*a^2 + 1/2*b^2 + 1/2*((a + b)*cos(2*x)^2 - 2*a*cos(2*x) + a - b)*sqrt(a + b)*sqrt(((a + b)*cos(2*x)^2 - 2*(a - b)*cos(2*x) + a + b)/(cos(2*x)^2 - 2*cos(2*x) + 1)) - (a^2 - b^2)*cos(2*x)) + 3*((3*a + 2*b)*cos(2*x)^2 - 2*(3*a + 2*b)*cos(2*x) + 3*a + 2*b)*sqrt(b)*log(-((a + 2*b)*cos(2*x)^2 - 2*(cos(2*x)^2 - 1)*sqrt(b)*sqrt(((a + b)*cos(2*x)^2 - 2*(a - b)*cos(2*x) + a + b)/(cos(2*x)^2 - 2*cos(2*x) + 1)) - 2*(a - 2*b)*cos(2*x) + a + 2*b)/(cos(2*x)^2 - 2*cos(2*x) + 1)) - 2*((8*a + 11*b)*cos(2*x)^2 - 8*(2*a + b)*cos(2*x) + 8*a + 5*b)*sqrt(((a + b)*cos(2*x)^2 - 2*(a - b)*cos(2*x) + a + b)/(cos(2*x)^2 - 2*cos(2*x) + 1)))/(cos(2*x)^2 - 2*cos(2*x) + 1), 1/12*(3*((3*a + 2*b)*cos(2*x)^2 - 2*(3*a + 2*b)*cos(2*x) + 3*a + 2*b)*sqrt(-b)*arctan(sqrt(-b)*sqrt(((a + b)*cos(2*x)^2 - 2*(a - b)*cos(2*x) + a + b)/(cos(2*x)^2 - 2*cos(2*x) + 1))*(cos(2*x) - 1)/(b*cos(2*x) + b)) + 3*((a + b)*cos(2*x)^2 - 2*(a + b)*cos(2*x) + a + b)*sqrt(a + b)*log(1/2*(a^2 + 2*a*b + b^2)*cos(2*x)^2 + 1/2*a^2 + 1/2*b^2 + 1/2*((a + b)*cos(2*x)^2 - 2*a*cos(2*x) + a - b)*sqrt(a + b)*sqrt(((a + b)*cos(2*x)^2 - 2*(a - b)*cos(2*x) + a + b)/(cos(2*x)^2 - 2*cos(2*x) + 1)) - (a^2 - b^2)*cos(2*x)) - ((8*a + 11*b)*cos(2*x)^2 - 8*(2*a + b)*cos(2*x) + 8*a + 5*b)*sqrt(((a + b)*cos(2*x)^2 - 2*(a - b)*cos(2*x) + a + b)/(cos(2*x)^2 - 2*cos(2*x) + 1)))/(cos(2*x)^2 - 2*cos(2*x) + 1), -1/24*(12*((a + b)*cos(2*x)^2 - 2*(a + b)*cos(2*x) + a + b)*sqrt(-a - b)*arctan(((a + b)*cos(2*x)^2 - 2*a*cos(2*x) + a - b)*sqrt(-a - b)*sqrt(((a + b)*cos(2*x)^2 - 2*(a - b)*cos(2*x) + a + b)/(cos(2*x)^2 - 2*cos(2*x) + 1))/((a^2 + 2*a*b + b^2)*cos(2*x)^2 + a^2 + 2*a*b + b^2 - 2*(a^2 - b^2)*cos(2*x))) - 3*((3*a + 2*b)*cos(2*x)^2 - 2*(3*a + 2*b)*cos(2*x) + 3*a + 2*b)*sqrt(b)*log(-((a + 2*b)*cos(2*x)^2 - 2*(cos(2*x)^2 - 1)*sqrt(b)*sqrt(((a + b)*cos(2*x)^2 - 2*(a - b)*cos(2*x) + a + b)/(cos(2*x)^2 - 2*cos(2*x) + 1)) - 2*(a - 2*b)*cos(2*x) + a + 2*b)/(cos(2*x)^2 - 2*cos(2*x) + 1)) + 2*((8*a + 11*b)*cos(2*x)^2 - 8*(2*a + b)*cos(2*x) + 8*a + 5*b)*sqrt(((a + b)*cos(2*x)^2 - 2*(a - b)*cos(2*x) + a + b)/(cos(2*x)^2 - 2*cos(2*x) + 1)))/(cos(2*x)^2 - 2*cos(2*x) + 1), -1/12*(6*((a + b)*cos(2*x)^2 - 2*(a + b)*cos(2*x) + a + b)*sqrt(-a - b)*arctan(((a + b)*cos(2*x)^2 - 2*a*cos(2*x) + a - b)*sqrt(-a - b)*sqrt(((a + b)*cos(2*x)^2 - 2*(a - b)*cos(2*x) + a + b)/(cos(2*x)^2 - 2*cos(2*x) + 1))/((a^2 + 2*a*b + b^2)*cos(2*x)^2 + a^2 + 2*a*b + b^2 - 2*(a^2 - b^2)*cos(2*x))) - 3*((3*a + 2*b)*cos(2*x)^2 - 2*(3*a + 2*b)*cos(2*x) + 3*a + 2*b)*sqrt(-b)*arctan(sqrt(-b)*sqrt(((a + b)*cos(2*x)^2 - 2*(a - b)*cos(2*x) + a + b)/(cos(2*x)^2 - 2*cos(2*x) + 1))*(cos(2*x) - 1)/(b*cos(2*x) + b)) + ((8*a + 11*b)*cos(2*x)^2 - 8*(2*a + b)*cos(2*x) + 8*a + 5*b)*sqrt(((a + b)*cos(2*x)^2 - 2*(a - b)*cos(2*x) + a + b)/(cos(2*x)^2 - 2*cos(2*x) + 1)))/(cos(2*x)^2 - 2*cos(2*x) + 1)]","B",0
62,1,264,0,0.648382," ","integrate(cot(x)/(a+b*cot(x)^4)^(1/2),x, algorithm=""fricas"")","\left[\frac{\log\left(\frac{1}{2} \, {\left(a^{2} + 2 \, a b + b^{2}\right)} \cos\left(2 \, x\right)^{2} + \frac{1}{2} \, a^{2} + \frac{1}{2} \, b^{2} + \frac{1}{2} \, {\left({\left(a + b\right)} \cos\left(2 \, x\right)^{2} - 2 \, a \cos\left(2 \, x\right) + a - b\right)} \sqrt{a + b} \sqrt{\frac{{\left(a + b\right)} \cos\left(2 \, x\right)^{2} - 2 \, {\left(a - b\right)} \cos\left(2 \, x\right) + a + b}{\cos\left(2 \, x\right)^{2} - 2 \, \cos\left(2 \, x\right) + 1}} - {\left(a^{2} - b^{2}\right)} \cos\left(2 \, x\right)\right)}{4 \, \sqrt{a + b}}, -\frac{\sqrt{-a - b} \arctan\left(\frac{{\left({\left(a + b\right)} \cos\left(2 \, x\right)^{2} - 2 \, a \cos\left(2 \, x\right) + a - b\right)} \sqrt{-a - b} \sqrt{\frac{{\left(a + b\right)} \cos\left(2 \, x\right)^{2} - 2 \, {\left(a - b\right)} \cos\left(2 \, x\right) + a + b}{\cos\left(2 \, x\right)^{2} - 2 \, \cos\left(2 \, x\right) + 1}}}{{\left(a^{2} + 2 \, a b + b^{2}\right)} \cos\left(2 \, x\right)^{2} + a^{2} + 2 \, a b + b^{2} - 2 \, {\left(a^{2} - b^{2}\right)} \cos\left(2 \, x\right)}\right)}{2 \, {\left(a + b\right)}}\right]"," ",0,"[1/4*log(1/2*(a^2 + 2*a*b + b^2)*cos(2*x)^2 + 1/2*a^2 + 1/2*b^2 + 1/2*((a + b)*cos(2*x)^2 - 2*a*cos(2*x) + a - b)*sqrt(a + b)*sqrt(((a + b)*cos(2*x)^2 - 2*(a - b)*cos(2*x) + a + b)/(cos(2*x)^2 - 2*cos(2*x) + 1)) - (a^2 - b^2)*cos(2*x))/sqrt(a + b), -1/2*sqrt(-a - b)*arctan(((a + b)*cos(2*x)^2 - 2*a*cos(2*x) + a - b)*sqrt(-a - b)*sqrt(((a + b)*cos(2*x)^2 - 2*(a - b)*cos(2*x) + a + b)/(cos(2*x)^2 - 2*cos(2*x) + 1))/((a^2 + 2*a*b + b^2)*cos(2*x)^2 + a^2 + 2*a*b + b^2 - 2*(a^2 - b^2)*cos(2*x)))/(a + b)]","B",0
63,1,670,0,1.355640," ","integrate(cot(x)/(a+b*cot(x)^4)^(3/2),x, algorithm=""fricas"")","\left[\frac{{\left({\left(a^{2} + a b\right)} \cos\left(2 \, x\right)^{2} + a^{2} + a b - 2 \, {\left(a^{2} - a b\right)} \cos\left(2 \, x\right)\right)} \sqrt{a + b} \log\left(\frac{1}{2} \, {\left(a^{2} + 2 \, a b + b^{2}\right)} \cos\left(2 \, x\right)^{2} + \frac{1}{2} \, a^{2} + \frac{1}{2} \, b^{2} + \frac{1}{2} \, {\left({\left(a + b\right)} \cos\left(2 \, x\right)^{2} - 2 \, a \cos\left(2 \, x\right) + a - b\right)} \sqrt{a + b} \sqrt{\frac{{\left(a + b\right)} \cos\left(2 \, x\right)^{2} - 2 \, {\left(a - b\right)} \cos\left(2 \, x\right) + a + b}{\cos\left(2 \, x\right)^{2} - 2 \, \cos\left(2 \, x\right) + 1}} - {\left(a^{2} - b^{2}\right)} \cos\left(2 \, x\right)\right) - 2 \, {\left({\left(a^{2} - b^{2}\right)} \cos\left(2 \, x\right)^{2} + a^{2} + 2 \, a b + b^{2} - 2 \, {\left(a^{2} + a b\right)} \cos\left(2 \, x\right)\right)} \sqrt{\frac{{\left(a + b\right)} \cos\left(2 \, x\right)^{2} - 2 \, {\left(a - b\right)} \cos\left(2 \, x\right) + a + b}{\cos\left(2 \, x\right)^{2} - 2 \, \cos\left(2 \, x\right) + 1}}}{4 \, {\left(a^{4} + 3 \, a^{3} b + 3 \, a^{2} b^{2} + a b^{3} + {\left(a^{4} + 3 \, a^{3} b + 3 \, a^{2} b^{2} + a b^{3}\right)} \cos\left(2 \, x\right)^{2} - 2 \, {\left(a^{4} + a^{3} b - a^{2} b^{2} - a b^{3}\right)} \cos\left(2 \, x\right)\right)}}, -\frac{{\left({\left(a^{2} + a b\right)} \cos\left(2 \, x\right)^{2} + a^{2} + a b - 2 \, {\left(a^{2} - a b\right)} \cos\left(2 \, x\right)\right)} \sqrt{-a - b} \arctan\left(\frac{{\left({\left(a + b\right)} \cos\left(2 \, x\right)^{2} - 2 \, a \cos\left(2 \, x\right) + a - b\right)} \sqrt{-a - b} \sqrt{\frac{{\left(a + b\right)} \cos\left(2 \, x\right)^{2} - 2 \, {\left(a - b\right)} \cos\left(2 \, x\right) + a + b}{\cos\left(2 \, x\right)^{2} - 2 \, \cos\left(2 \, x\right) + 1}}}{{\left(a^{2} + 2 \, a b + b^{2}\right)} \cos\left(2 \, x\right)^{2} + a^{2} + 2 \, a b + b^{2} - 2 \, {\left(a^{2} - b^{2}\right)} \cos\left(2 \, x\right)}\right) + {\left({\left(a^{2} - b^{2}\right)} \cos\left(2 \, x\right)^{2} + a^{2} + 2 \, a b + b^{2} - 2 \, {\left(a^{2} + a b\right)} \cos\left(2 \, x\right)\right)} \sqrt{\frac{{\left(a + b\right)} \cos\left(2 \, x\right)^{2} - 2 \, {\left(a - b\right)} \cos\left(2 \, x\right) + a + b}{\cos\left(2 \, x\right)^{2} - 2 \, \cos\left(2 \, x\right) + 1}}}{2 \, {\left(a^{4} + 3 \, a^{3} b + 3 \, a^{2} b^{2} + a b^{3} + {\left(a^{4} + 3 \, a^{3} b + 3 \, a^{2} b^{2} + a b^{3}\right)} \cos\left(2 \, x\right)^{2} - 2 \, {\left(a^{4} + a^{3} b - a^{2} b^{2} - a b^{3}\right)} \cos\left(2 \, x\right)\right)}}\right]"," ",0,"[1/4*(((a^2 + a*b)*cos(2*x)^2 + a^2 + a*b - 2*(a^2 - a*b)*cos(2*x))*sqrt(a + b)*log(1/2*(a^2 + 2*a*b + b^2)*cos(2*x)^2 + 1/2*a^2 + 1/2*b^2 + 1/2*((a + b)*cos(2*x)^2 - 2*a*cos(2*x) + a - b)*sqrt(a + b)*sqrt(((a + b)*cos(2*x)^2 - 2*(a - b)*cos(2*x) + a + b)/(cos(2*x)^2 - 2*cos(2*x) + 1)) - (a^2 - b^2)*cos(2*x)) - 2*((a^2 - b^2)*cos(2*x)^2 + a^2 + 2*a*b + b^2 - 2*(a^2 + a*b)*cos(2*x))*sqrt(((a + b)*cos(2*x)^2 - 2*(a - b)*cos(2*x) + a + b)/(cos(2*x)^2 - 2*cos(2*x) + 1)))/(a^4 + 3*a^3*b + 3*a^2*b^2 + a*b^3 + (a^4 + 3*a^3*b + 3*a^2*b^2 + a*b^3)*cos(2*x)^2 - 2*(a^4 + a^3*b - a^2*b^2 - a*b^3)*cos(2*x)), -1/2*(((a^2 + a*b)*cos(2*x)^2 + a^2 + a*b - 2*(a^2 - a*b)*cos(2*x))*sqrt(-a - b)*arctan(((a + b)*cos(2*x)^2 - 2*a*cos(2*x) + a - b)*sqrt(-a - b)*sqrt(((a + b)*cos(2*x)^2 - 2*(a - b)*cos(2*x) + a + b)/(cos(2*x)^2 - 2*cos(2*x) + 1))/((a^2 + 2*a*b + b^2)*cos(2*x)^2 + a^2 + 2*a*b + b^2 - 2*(a^2 - b^2)*cos(2*x))) + ((a^2 - b^2)*cos(2*x)^2 + a^2 + 2*a*b + b^2 - 2*(a^2 + a*b)*cos(2*x))*sqrt(((a + b)*cos(2*x)^2 - 2*(a - b)*cos(2*x) + a + b)/(cos(2*x)^2 - 2*cos(2*x) + 1)))/(a^4 + 3*a^3*b + 3*a^2*b^2 + a*b^3 + (a^4 + 3*a^3*b + 3*a^2*b^2 + a*b^3)*cos(2*x)^2 - 2*(a^4 + a^3*b - a^2*b^2 - a*b^3)*cos(2*x))]","B",0
64,1,1365,0,0.737159," ","integrate(cot(x)/(a+b*cot(x)^4)^(5/2),x, algorithm=""fricas"")","\left[\frac{3 \, {\left({\left(a^{4} + 2 \, a^{3} b + a^{2} b^{2}\right)} \cos\left(2 \, x\right)^{4} + a^{4} + 2 \, a^{3} b + a^{2} b^{2} - 4 \, {\left(a^{4} - a^{2} b^{2}\right)} \cos\left(2 \, x\right)^{3} + 2 \, {\left(3 \, a^{4} - 2 \, a^{3} b + 3 \, a^{2} b^{2}\right)} \cos\left(2 \, x\right)^{2} - 4 \, {\left(a^{4} - a^{2} b^{2}\right)} \cos\left(2 \, x\right)\right)} \sqrt{a + b} \log\left(\frac{1}{2} \, {\left(a^{2} + 2 \, a b + b^{2}\right)} \cos\left(2 \, x\right)^{2} + \frac{1}{2} \, a^{2} + \frac{1}{2} \, b^{2} + \frac{1}{2} \, {\left({\left(a + b\right)} \cos\left(2 \, x\right)^{2} - 2 \, a \cos\left(2 \, x\right) + a - b\right)} \sqrt{a + b} \sqrt{\frac{{\left(a + b\right)} \cos\left(2 \, x\right)^{2} - 2 \, {\left(a - b\right)} \cos\left(2 \, x\right) + a + b}{\cos\left(2 \, x\right)^{2} - 2 \, \cos\left(2 \, x\right) + 1}} - {\left(a^{2} - b^{2}\right)} \cos\left(2 \, x\right)\right) - 4 \, {\left({\left(2 \, a^{4} + a^{3} b - 5 \, a^{2} b^{2} - 5 \, a b^{3} - b^{4}\right)} \cos\left(2 \, x\right)^{4} + 2 \, a^{4} + 7 \, a^{3} b + 9 \, a^{2} b^{2} + 5 \, a b^{3} + b^{4} - 2 \, {\left(4 \, a^{4} + 2 \, a^{3} b - a^{2} b^{2} + 2 \, a b^{3} + b^{4}\right)} \cos\left(2 \, x\right)^{3} + 12 \, {\left(a^{4} + a^{3} b\right)} \cos\left(2 \, x\right)^{2} - 2 \, {\left(4 \, a^{4} + 8 \, a^{3} b + 3 \, a^{2} b^{2} - 2 \, a b^{3} - b^{4}\right)} \cos\left(2 \, x\right)\right)} \sqrt{\frac{{\left(a + b\right)} \cos\left(2 \, x\right)^{2} - 2 \, {\left(a - b\right)} \cos\left(2 \, x\right) + a + b}{\cos\left(2 \, x\right)^{2} - 2 \, \cos\left(2 \, x\right) + 1}}}{12 \, {\left(a^{7} + 5 \, a^{6} b + 10 \, a^{5} b^{2} + 10 \, a^{4} b^{3} + 5 \, a^{3} b^{4} + a^{2} b^{5} + {\left(a^{7} + 5 \, a^{6} b + 10 \, a^{5} b^{2} + 10 \, a^{4} b^{3} + 5 \, a^{3} b^{4} + a^{2} b^{5}\right)} \cos\left(2 \, x\right)^{4} - 4 \, {\left(a^{7} + 3 \, a^{6} b + 2 \, a^{5} b^{2} - 2 \, a^{4} b^{3} - 3 \, a^{3} b^{4} - a^{2} b^{5}\right)} \cos\left(2 \, x\right)^{3} + 2 \, {\left(3 \, a^{7} + 7 \, a^{6} b + 6 \, a^{5} b^{2} + 6 \, a^{4} b^{3} + 7 \, a^{3} b^{4} + 3 \, a^{2} b^{5}\right)} \cos\left(2 \, x\right)^{2} - 4 \, {\left(a^{7} + 3 \, a^{6} b + 2 \, a^{5} b^{2} - 2 \, a^{4} b^{3} - 3 \, a^{3} b^{4} - a^{2} b^{5}\right)} \cos\left(2 \, x\right)\right)}}, -\frac{3 \, {\left({\left(a^{4} + 2 \, a^{3} b + a^{2} b^{2}\right)} \cos\left(2 \, x\right)^{4} + a^{4} + 2 \, a^{3} b + a^{2} b^{2} - 4 \, {\left(a^{4} - a^{2} b^{2}\right)} \cos\left(2 \, x\right)^{3} + 2 \, {\left(3 \, a^{4} - 2 \, a^{3} b + 3 \, a^{2} b^{2}\right)} \cos\left(2 \, x\right)^{2} - 4 \, {\left(a^{4} - a^{2} b^{2}\right)} \cos\left(2 \, x\right)\right)} \sqrt{-a - b} \arctan\left(\frac{{\left({\left(a + b\right)} \cos\left(2 \, x\right)^{2} - 2 \, a \cos\left(2 \, x\right) + a - b\right)} \sqrt{-a - b} \sqrt{\frac{{\left(a + b\right)} \cos\left(2 \, x\right)^{2} - 2 \, {\left(a - b\right)} \cos\left(2 \, x\right) + a + b}{\cos\left(2 \, x\right)^{2} - 2 \, \cos\left(2 \, x\right) + 1}}}{{\left(a^{2} + 2 \, a b + b^{2}\right)} \cos\left(2 \, x\right)^{2} + a^{2} + 2 \, a b + b^{2} - 2 \, {\left(a^{2} - b^{2}\right)} \cos\left(2 \, x\right)}\right) + 2 \, {\left({\left(2 \, a^{4} + a^{3} b - 5 \, a^{2} b^{2} - 5 \, a b^{3} - b^{4}\right)} \cos\left(2 \, x\right)^{4} + 2 \, a^{4} + 7 \, a^{3} b + 9 \, a^{2} b^{2} + 5 \, a b^{3} + b^{4} - 2 \, {\left(4 \, a^{4} + 2 \, a^{3} b - a^{2} b^{2} + 2 \, a b^{3} + b^{4}\right)} \cos\left(2 \, x\right)^{3} + 12 \, {\left(a^{4} + a^{3} b\right)} \cos\left(2 \, x\right)^{2} - 2 \, {\left(4 \, a^{4} + 8 \, a^{3} b + 3 \, a^{2} b^{2} - 2 \, a b^{3} - b^{4}\right)} \cos\left(2 \, x\right)\right)} \sqrt{\frac{{\left(a + b\right)} \cos\left(2 \, x\right)^{2} - 2 \, {\left(a - b\right)} \cos\left(2 \, x\right) + a + b}{\cos\left(2 \, x\right)^{2} - 2 \, \cos\left(2 \, x\right) + 1}}}{6 \, {\left(a^{7} + 5 \, a^{6} b + 10 \, a^{5} b^{2} + 10 \, a^{4} b^{3} + 5 \, a^{3} b^{4} + a^{2} b^{5} + {\left(a^{7} + 5 \, a^{6} b + 10 \, a^{5} b^{2} + 10 \, a^{4} b^{3} + 5 \, a^{3} b^{4} + a^{2} b^{5}\right)} \cos\left(2 \, x\right)^{4} - 4 \, {\left(a^{7} + 3 \, a^{6} b + 2 \, a^{5} b^{2} - 2 \, a^{4} b^{3} - 3 \, a^{3} b^{4} - a^{2} b^{5}\right)} \cos\left(2 \, x\right)^{3} + 2 \, {\left(3 \, a^{7} + 7 \, a^{6} b + 6 \, a^{5} b^{2} + 6 \, a^{4} b^{3} + 7 \, a^{3} b^{4} + 3 \, a^{2} b^{5}\right)} \cos\left(2 \, x\right)^{2} - 4 \, {\left(a^{7} + 3 \, a^{6} b + 2 \, a^{5} b^{2} - 2 \, a^{4} b^{3} - 3 \, a^{3} b^{4} - a^{2} b^{5}\right)} \cos\left(2 \, x\right)\right)}}\right]"," ",0,"[1/12*(3*((a^4 + 2*a^3*b + a^2*b^2)*cos(2*x)^4 + a^4 + 2*a^3*b + a^2*b^2 - 4*(a^4 - a^2*b^2)*cos(2*x)^3 + 2*(3*a^4 - 2*a^3*b + 3*a^2*b^2)*cos(2*x)^2 - 4*(a^4 - a^2*b^2)*cos(2*x))*sqrt(a + b)*log(1/2*(a^2 + 2*a*b + b^2)*cos(2*x)^2 + 1/2*a^2 + 1/2*b^2 + 1/2*((a + b)*cos(2*x)^2 - 2*a*cos(2*x) + a - b)*sqrt(a + b)*sqrt(((a + b)*cos(2*x)^2 - 2*(a - b)*cos(2*x) + a + b)/(cos(2*x)^2 - 2*cos(2*x) + 1)) - (a^2 - b^2)*cos(2*x)) - 4*((2*a^4 + a^3*b - 5*a^2*b^2 - 5*a*b^3 - b^4)*cos(2*x)^4 + 2*a^4 + 7*a^3*b + 9*a^2*b^2 + 5*a*b^3 + b^4 - 2*(4*a^4 + 2*a^3*b - a^2*b^2 + 2*a*b^3 + b^4)*cos(2*x)^3 + 12*(a^4 + a^3*b)*cos(2*x)^2 - 2*(4*a^4 + 8*a^3*b + 3*a^2*b^2 - 2*a*b^3 - b^4)*cos(2*x))*sqrt(((a + b)*cos(2*x)^2 - 2*(a - b)*cos(2*x) + a + b)/(cos(2*x)^2 - 2*cos(2*x) + 1)))/(a^7 + 5*a^6*b + 10*a^5*b^2 + 10*a^4*b^3 + 5*a^3*b^4 + a^2*b^5 + (a^7 + 5*a^6*b + 10*a^5*b^2 + 10*a^4*b^3 + 5*a^3*b^4 + a^2*b^5)*cos(2*x)^4 - 4*(a^7 + 3*a^6*b + 2*a^5*b^2 - 2*a^4*b^3 - 3*a^3*b^4 - a^2*b^5)*cos(2*x)^3 + 2*(3*a^7 + 7*a^6*b + 6*a^5*b^2 + 6*a^4*b^3 + 7*a^3*b^4 + 3*a^2*b^5)*cos(2*x)^2 - 4*(a^7 + 3*a^6*b + 2*a^5*b^2 - 2*a^4*b^3 - 3*a^3*b^4 - a^2*b^5)*cos(2*x)), -1/6*(3*((a^4 + 2*a^3*b + a^2*b^2)*cos(2*x)^4 + a^4 + 2*a^3*b + a^2*b^2 - 4*(a^4 - a^2*b^2)*cos(2*x)^3 + 2*(3*a^4 - 2*a^3*b + 3*a^2*b^2)*cos(2*x)^2 - 4*(a^4 - a^2*b^2)*cos(2*x))*sqrt(-a - b)*arctan(((a + b)*cos(2*x)^2 - 2*a*cos(2*x) + a - b)*sqrt(-a - b)*sqrt(((a + b)*cos(2*x)^2 - 2*(a - b)*cos(2*x) + a + b)/(cos(2*x)^2 - 2*cos(2*x) + 1))/((a^2 + 2*a*b + b^2)*cos(2*x)^2 + a^2 + 2*a*b + b^2 - 2*(a^2 - b^2)*cos(2*x))) + 2*((2*a^4 + a^3*b - 5*a^2*b^2 - 5*a*b^3 - b^4)*cos(2*x)^4 + 2*a^4 + 7*a^3*b + 9*a^2*b^2 + 5*a*b^3 + b^4 - 2*(4*a^4 + 2*a^3*b - a^2*b^2 + 2*a*b^3 + b^4)*cos(2*x)^3 + 12*(a^4 + a^3*b)*cos(2*x)^2 - 2*(4*a^4 + 8*a^3*b + 3*a^2*b^2 - 2*a*b^3 - b^4)*cos(2*x))*sqrt(((a + b)*cos(2*x)^2 - 2*(a - b)*cos(2*x) + a + b)/(cos(2*x)^2 - 2*cos(2*x) + 1)))/(a^7 + 5*a^6*b + 10*a^5*b^2 + 10*a^4*b^3 + 5*a^3*b^4 + a^2*b^5 + (a^7 + 5*a^6*b + 10*a^5*b^2 + 10*a^4*b^3 + 5*a^3*b^4 + a^2*b^5)*cos(2*x)^4 - 4*(a^7 + 3*a^6*b + 2*a^5*b^2 - 2*a^4*b^3 - 3*a^3*b^4 - a^2*b^5)*cos(2*x)^3 + 2*(3*a^7 + 7*a^6*b + 6*a^5*b^2 + 6*a^4*b^3 + 7*a^3*b^4 + 3*a^2*b^5)*cos(2*x)^2 - 4*(a^7 + 3*a^6*b + 2*a^5*b^2 - 2*a^4*b^3 - 3*a^3*b^4 - a^2*b^5)*cos(2*x))]","B",0
