1,1,2494,182,2.507000," ","int((A+C*cot(d*x+c)^2)/(b*tan(d*x+c))^(1/2),x)","\text{Expression too large to display}"," ",0,"-1/6/d*(1+cos(d*x+c))^2*(-1+cos(d*x+c))^2*(3*I*C*cos(d*x+c)*EllipticPi((-(-sin(d*x+c)-1+cos(d*x+c))/sin(d*x+c))^(1/2),1/2-1/2*I,1/2*2^(1/2))*((-1+cos(d*x+c))/sin(d*x+c))^(1/2)*((-1+cos(d*x+c)+sin(d*x+c))/sin(d*x+c))^(1/2)*(-(-sin(d*x+c)-1+cos(d*x+c))/sin(d*x+c))^(1/2)*sin(d*x+c)-3*I*A*cos(d*x+c)*EllipticPi((-(-sin(d*x+c)-1+cos(d*x+c))/sin(d*x+c))^(1/2),1/2-1/2*I,1/2*2^(1/2))*((-1+cos(d*x+c))/sin(d*x+c))^(1/2)*((-1+cos(d*x+c)+sin(d*x+c))/sin(d*x+c))^(1/2)*(-(-sin(d*x+c)-1+cos(d*x+c))/sin(d*x+c))^(1/2)*sin(d*x+c)-3*I*A*EllipticPi((-(-sin(d*x+c)-1+cos(d*x+c))/sin(d*x+c))^(1/2),1/2-1/2*I,1/2*2^(1/2))*((-1+cos(d*x+c))/sin(d*x+c))^(1/2)*((-1+cos(d*x+c)+sin(d*x+c))/sin(d*x+c))^(1/2)*(-(-sin(d*x+c)-1+cos(d*x+c))/sin(d*x+c))^(1/2)*sin(d*x+c)-3*I*C*cos(d*x+c)*EllipticPi((-(-sin(d*x+c)-1+cos(d*x+c))/sin(d*x+c))^(1/2),1/2+1/2*I,1/2*2^(1/2))*((-1+cos(d*x+c))/sin(d*x+c))^(1/2)*((-1+cos(d*x+c)+sin(d*x+c))/sin(d*x+c))^(1/2)*(-(-sin(d*x+c)-1+cos(d*x+c))/sin(d*x+c))^(1/2)*sin(d*x+c)-3*A*cos(d*x+c)*EllipticPi((-(-sin(d*x+c)-1+cos(d*x+c))/sin(d*x+c))^(1/2),1/2+1/2*I,1/2*2^(1/2))*((-1+cos(d*x+c))/sin(d*x+c))^(1/2)*((-1+cos(d*x+c)+sin(d*x+c))/sin(d*x+c))^(1/2)*(-(-sin(d*x+c)-1+cos(d*x+c))/sin(d*x+c))^(1/2)*sin(d*x+c)-3*A*cos(d*x+c)*EllipticPi((-(-sin(d*x+c)-1+cos(d*x+c))/sin(d*x+c))^(1/2),1/2-1/2*I,1/2*2^(1/2))*((-1+cos(d*x+c))/sin(d*x+c))^(1/2)*((-1+cos(d*x+c)+sin(d*x+c))/sin(d*x+c))^(1/2)*(-(-sin(d*x+c)-1+cos(d*x+c))/sin(d*x+c))^(1/2)*sin(d*x+c)+6*A*cos(d*x+c)*EllipticF((-(-sin(d*x+c)-1+cos(d*x+c))/sin(d*x+c))^(1/2),1/2*2^(1/2))*((-1+cos(d*x+c))/sin(d*x+c))^(1/2)*((-1+cos(d*x+c)+sin(d*x+c))/sin(d*x+c))^(1/2)*(-(-sin(d*x+c)-1+cos(d*x+c))/sin(d*x+c))^(1/2)*sin(d*x+c)+3*I*A*cos(d*x+c)*EllipticPi((-(-sin(d*x+c)-1+cos(d*x+c))/sin(d*x+c))^(1/2),1/2+1/2*I,1/2*2^(1/2))*((-1+cos(d*x+c))/sin(d*x+c))^(1/2)*((-1+cos(d*x+c)+sin(d*x+c))/sin(d*x+c))^(1/2)*(-(-sin(d*x+c)-1+cos(d*x+c))/sin(d*x+c))^(1/2)*sin(d*x+c)-3*I*C*EllipticPi((-(-sin(d*x+c)-1+cos(d*x+c))/sin(d*x+c))^(1/2),1/2+1/2*I,1/2*2^(1/2))*((-1+cos(d*x+c))/sin(d*x+c))^(1/2)*((-1+cos(d*x+c)+sin(d*x+c))/sin(d*x+c))^(1/2)*(-(-sin(d*x+c)-1+cos(d*x+c))/sin(d*x+c))^(1/2)*sin(d*x+c)+3*C*cos(d*x+c)*EllipticPi((-(-sin(d*x+c)-1+cos(d*x+c))/sin(d*x+c))^(1/2),1/2+1/2*I,1/2*2^(1/2))*((-1+cos(d*x+c))/sin(d*x+c))^(1/2)*((-1+cos(d*x+c)+sin(d*x+c))/sin(d*x+c))^(1/2)*(-(-sin(d*x+c)-1+cos(d*x+c))/sin(d*x+c))^(1/2)*sin(d*x+c)+3*C*cos(d*x+c)*EllipticPi((-(-sin(d*x+c)-1+cos(d*x+c))/sin(d*x+c))^(1/2),1/2-1/2*I,1/2*2^(1/2))*((-1+cos(d*x+c))/sin(d*x+c))^(1/2)*((-1+cos(d*x+c)+sin(d*x+c))/sin(d*x+c))^(1/2)*(-(-sin(d*x+c)-1+cos(d*x+c))/sin(d*x+c))^(1/2)*sin(d*x+c)-6*C*cos(d*x+c)*EllipticF((-(-sin(d*x+c)-1+cos(d*x+c))/sin(d*x+c))^(1/2),1/2*2^(1/2))*((-1+cos(d*x+c))/sin(d*x+c))^(1/2)*((-1+cos(d*x+c)+sin(d*x+c))/sin(d*x+c))^(1/2)*(-(-sin(d*x+c)-1+cos(d*x+c))/sin(d*x+c))^(1/2)*sin(d*x+c)+3*I*A*EllipticPi((-(-sin(d*x+c)-1+cos(d*x+c))/sin(d*x+c))^(1/2),1/2+1/2*I,1/2*2^(1/2))*((-1+cos(d*x+c))/sin(d*x+c))^(1/2)*((-1+cos(d*x+c)+sin(d*x+c))/sin(d*x+c))^(1/2)*(-(-sin(d*x+c)-1+cos(d*x+c))/sin(d*x+c))^(1/2)*sin(d*x+c)+3*I*C*EllipticPi((-(-sin(d*x+c)-1+cos(d*x+c))/sin(d*x+c))^(1/2),1/2-1/2*I,1/2*2^(1/2))*((-1+cos(d*x+c))/sin(d*x+c))^(1/2)*((-1+cos(d*x+c)+sin(d*x+c))/sin(d*x+c))^(1/2)*(-(-sin(d*x+c)-1+cos(d*x+c))/sin(d*x+c))^(1/2)*sin(d*x+c)-3*A*EllipticPi((-(-sin(d*x+c)-1+cos(d*x+c))/sin(d*x+c))^(1/2),1/2+1/2*I,1/2*2^(1/2))*((-1+cos(d*x+c))/sin(d*x+c))^(1/2)*((-1+cos(d*x+c)+sin(d*x+c))/sin(d*x+c))^(1/2)*(-(-sin(d*x+c)-1+cos(d*x+c))/sin(d*x+c))^(1/2)*sin(d*x+c)-3*A*EllipticPi((-(-sin(d*x+c)-1+cos(d*x+c))/sin(d*x+c))^(1/2),1/2-1/2*I,1/2*2^(1/2))*((-1+cos(d*x+c))/sin(d*x+c))^(1/2)*((-1+cos(d*x+c)+sin(d*x+c))/sin(d*x+c))^(1/2)*(-(-sin(d*x+c)-1+cos(d*x+c))/sin(d*x+c))^(1/2)*sin(d*x+c)+6*A*EllipticF((-(-sin(d*x+c)-1+cos(d*x+c))/sin(d*x+c))^(1/2),1/2*2^(1/2))*((-1+cos(d*x+c))/sin(d*x+c))^(1/2)*((-1+cos(d*x+c)+sin(d*x+c))/sin(d*x+c))^(1/2)*(-(-sin(d*x+c)-1+cos(d*x+c))/sin(d*x+c))^(1/2)*sin(d*x+c)+3*C*EllipticPi((-(-sin(d*x+c)-1+cos(d*x+c))/sin(d*x+c))^(1/2),1/2+1/2*I,1/2*2^(1/2))*((-1+cos(d*x+c))/sin(d*x+c))^(1/2)*((-1+cos(d*x+c)+sin(d*x+c))/sin(d*x+c))^(1/2)*(-(-sin(d*x+c)-1+cos(d*x+c))/sin(d*x+c))^(1/2)*sin(d*x+c)+3*C*EllipticPi((-(-sin(d*x+c)-1+cos(d*x+c))/sin(d*x+c))^(1/2),1/2-1/2*I,1/2*2^(1/2))*((-1+cos(d*x+c))/sin(d*x+c))^(1/2)*((-1+cos(d*x+c)+sin(d*x+c))/sin(d*x+c))^(1/2)*(-(-sin(d*x+c)-1+cos(d*x+c))/sin(d*x+c))^(1/2)*sin(d*x+c)-6*C*EllipticF((-(-sin(d*x+c)-1+cos(d*x+c))/sin(d*x+c))^(1/2),1/2*2^(1/2))*((-1+cos(d*x+c))/sin(d*x+c))^(1/2)*((-1+cos(d*x+c)+sin(d*x+c))/sin(d*x+c))^(1/2)*(-(-sin(d*x+c)-1+cos(d*x+c))/sin(d*x+c))^(1/2)*sin(d*x+c)+2*C*2^(1/2)*cos(d*x+c)^2)/sin(d*x+c)^5/cos(d*x+c)/(b*sin(d*x+c)/cos(d*x+c))^(1/2)*2^(1/2)","C"
2,1,31,20,0.035000," ","int(a+b*cot(d*x+c)^2,x)","a x +\frac{b \left(-\cot \left(d x +c \right)+\frac{\pi}{2}-\mathrm{arccot}\left(\cot \left(d x +c \right)\right)\right)}{d}"," ",0,"a*x+b/d*(-cot(d*x+c)+1/2*Pi-arccot(cot(d*x+c)))","A"
3,1,68,45,0.033000," ","int((a+b*cot(d*x+c)^2)^2,x)","\frac{-\frac{\left(\cot^{3}\left(d x +c \right)\right) b^{2}}{3}-2 a b \cot \left(d x +c \right)+b^{2} \cot \left(d x +c \right)+\left(-a^{2}+2 a b -b^{2}\right) \left(\frac{\pi}{2}-\mathrm{arccot}\left(\cot \left(d x +c \right)\right)\right)}{d}"," ",0,"1/d*(-1/3*cot(d*x+c)^3*b^2-2*a*b*cot(d*x+c)+b^2*cot(d*x+c)+(-a^2+2*a*b-b^2)*(1/2*Pi-arccot(cot(d*x+c))))","A"
4,1,116,74,0.036000," ","int((a+b*cot(d*x+c)^2)^3,x)","\frac{-\frac{b^{3} \left(\cot^{5}\left(d x +c \right)\right)}{5}-\left(\cot^{3}\left(d x +c \right)\right) a \,b^{2}+\frac{b^{3} \left(\cot^{3}\left(d x +c \right)\right)}{3}-3 a^{2} b \cot \left(d x +c \right)+3 b^{2} a \cot \left(d x +c \right)-b^{3} \cot \left(d x +c \right)+\left(-a^{3}+3 a^{2} b -3 b^{2} a +b^{3}\right) \left(\frac{\pi}{2}-\mathrm{arccot}\left(\cot \left(d x +c \right)\right)\right)}{d}"," ",0,"1/d*(-1/5*b^3*cot(d*x+c)^5-cot(d*x+c)^3*a*b^2+1/3*b^3*cot(d*x+c)^3-3*a^2*b*cot(d*x+c)+3*b^2*a*cot(d*x+c)-b^3*cot(d*x+c)+(-a^3+3*a^2*b-3*a*b^2+b^3)*(1/2*Pi-arccot(cot(d*x+c))))","A"
5,1,64,41,0.326000," ","int(1/(a+b*cot(d*x+c)^2),x)","\frac{b \arctan \left(\frac{\cot \left(d x +c \right) b}{\sqrt{a b}}\right)}{d \left(a -b \right) \sqrt{a b}}-\frac{\pi}{2 d \left(a -b \right)}+\frac{\mathrm{arccot}\left(\cot \left(d x +c \right)\right)}{d \left(a -b \right)}"," ",0,"1/d*b/(a-b)/(a*b)^(1/2)*arctan(cot(d*x+c)*b/(a*b)^(1/2))-1/2/d/(a-b)*Pi+1/d/(a-b)*arccot(cot(d*x+c))","A"
6,1,173,85,0.345000," ","int(1/(a+b*cot(d*x+c)^2)^2,x)","\frac{b \cot \left(d x +c \right)}{2 d \left(a -b \right)^{2} \left(a +b \left(\cot^{2}\left(d x +c \right)\right)\right)}-\frac{b^{2} \cot \left(d x +c \right)}{2 d \left(a -b \right)^{2} a \left(a +b \left(\cot^{2}\left(d x +c \right)\right)\right)}+\frac{3 b \arctan \left(\frac{\cot \left(d x +c \right) b}{\sqrt{a b}}\right)}{2 d \left(a -b \right)^{2} \sqrt{a b}}-\frac{b^{2} \arctan \left(\frac{\cot \left(d x +c \right) b}{\sqrt{a b}}\right)}{2 d \left(a -b \right)^{2} a \sqrt{a b}}-\frac{\pi}{2 d \left(a -b \right)^{2}}+\frac{\mathrm{arccot}\left(\cot \left(d x +c \right)\right)}{d \left(a -b \right)^{2}}"," ",0,"1/2/d*b/(a-b)^2*cot(d*x+c)/(a+b*cot(d*x+c)^2)-1/2/d*b^2/(a-b)^2/a*cot(d*x+c)/(a+b*cot(d*x+c)^2)+3/2/d*b/(a-b)^2/(a*b)^(1/2)*arctan(cot(d*x+c)*b/(a*b)^(1/2))-1/2/d*b^2/(a-b)^2/a/(a*b)^(1/2)*arctan(cot(d*x+c)*b/(a*b)^(1/2))-1/2/d/(a-b)^2*Pi+1/d/(a-b)^2*arccot(cot(d*x+c))","B"
7,1,363,136,0.352000," ","int(1/(a+b*cot(d*x+c)^2)^3,x)","\frac{7 b^{2} \left(\cot^{3}\left(d x +c \right)\right)}{8 d \left(a -b \right)^{3} \left(a +b \left(\cot^{2}\left(d x +c \right)\right)\right)^{2}}-\frac{5 b^{3} \left(\cot^{3}\left(d x +c \right)\right)}{4 d \left(a -b \right)^{3} \left(a +b \left(\cot^{2}\left(d x +c \right)\right)\right)^{2} a}+\frac{3 b^{4} \left(\cot^{3}\left(d x +c \right)\right)}{8 d \left(a -b \right)^{3} \left(a +b \left(\cot^{2}\left(d x +c \right)\right)\right)^{2} a^{2}}+\frac{9 b a \cot \left(d x +c \right)}{8 d \left(a -b \right)^{3} \left(a +b \left(\cot^{2}\left(d x +c \right)\right)\right)^{2}}-\frac{7 b^{2} \cot \left(d x +c \right)}{4 d \left(a -b \right)^{3} \left(a +b \left(\cot^{2}\left(d x +c \right)\right)\right)^{2}}+\frac{5 b^{3} \cot \left(d x +c \right)}{8 d \left(a -b \right)^{3} \left(a +b \left(\cot^{2}\left(d x +c \right)\right)\right)^{2} a}+\frac{15 b \arctan \left(\frac{\cot \left(d x +c \right) b}{\sqrt{a b}}\right)}{8 d \left(a -b \right)^{3} \sqrt{a b}}-\frac{5 b^{2} \arctan \left(\frac{\cot \left(d x +c \right) b}{\sqrt{a b}}\right)}{4 d \left(a -b \right)^{3} a \sqrt{a b}}+\frac{3 b^{3} \arctan \left(\frac{\cot \left(d x +c \right) b}{\sqrt{a b}}\right)}{8 d \left(a -b \right)^{3} a^{2} \sqrt{a b}}-\frac{\pi}{2 d \left(a -b \right)^{3}}+\frac{\mathrm{arccot}\left(\cot \left(d x +c \right)\right)}{d \left(a -b \right)^{3}}"," ",0,"7/8/d*b^2/(a-b)^3/(a+b*cot(d*x+c)^2)^2*cot(d*x+c)^3-5/4/d*b^3/(a-b)^3/(a+b*cot(d*x+c)^2)^2/a*cot(d*x+c)^3+3/8/d*b^4/(a-b)^3/(a+b*cot(d*x+c)^2)^2/a^2*cot(d*x+c)^3+9/8/d*b/(a-b)^3/(a+b*cot(d*x+c)^2)^2*a*cot(d*x+c)-7/4/d*b^2/(a-b)^3/(a+b*cot(d*x+c)^2)^2*cot(d*x+c)+5/8/d*b^3/(a-b)^3/(a+b*cot(d*x+c)^2)^2/a*cot(d*x+c)+15/8/d*b/(a-b)^3/(a*b)^(1/2)*arctan(cot(d*x+c)*b/(a*b)^(1/2))-5/4/d*b^2/(a-b)^3/a/(a*b)^(1/2)*arctan(cot(d*x+c)*b/(a*b)^(1/2))+3/8/d*b^3/(a-b)^3/a^2/(a*b)^(1/2)*arctan(cot(d*x+c)*b/(a*b)^(1/2))-1/2/d/(a-b)^3*Pi+1/d/(a-b)^3*arccot(cot(d*x+c))","B"
8,1,19,16,0.274000," ","int((1+cot(x)^2)^(3/2),x)","-\frac{\cot \left(x \right) \sqrt{1+\cot^{2}\left(x \right)}}{2}-\frac{\arcsinh \left(\cot \left(x \right)\right)}{2}"," ",0,"-1/2*cot(x)*(1+cot(x)^2)^(1/2)-1/2*arcsinh(cot(x))","A"
9,1,6,5,0.229000," ","int((1+cot(x)^2)^(1/2),x)","-\arcsinh \left(\cot \left(x \right)\right)"," ",0,"-arcsinh(cot(x))","A"
10,1,13,10,0.179000," ","int(1/(1+cot(x)^2)^(1/2),x)","-\frac{\cot \left(x \right)}{\sqrt{1+\cot^{2}\left(x \right)}}"," ",0,"-cot(x)/(1+cot(x)^2)^(1/2)","A"
11,1,32,27,0.125000," ","int((-1-cot(x)^2)^(3/2),x)","\frac{\cot \left(x \right) \sqrt{-1-\left(\cot^{2}\left(x \right)\right)}}{2}-\frac{\arctan \left(\frac{\cot \left(x \right)}{\sqrt{-1-\left(\cot^{2}\left(x \right)\right)}}\right)}{2}"," ",0,"1/2*cot(x)*(-1-cot(x)^2)^(1/2)-1/2*arctan(cot(x)/(-1-cot(x)^2)^(1/2))","A"
12,1,15,12,0.128000," ","int((-1-cot(x)^2)^(1/2),x)","\arctan \left(\frac{\cot \left(x \right)}{\sqrt{-1-\left(\cot^{2}\left(x \right)\right)}}\right)"," ",0,"arctan(cot(x)/(-1-cot(x)^2)^(1/2))","A"
13,1,15,12,0.103000," ","int(1/(-1-cot(x)^2)^(1/2),x)","-\frac{\cot \left(x \right)}{\sqrt{-1-\left(\cot^{2}\left(x \right)\right)}}"," ",0,"-cot(x)/(-1-cot(x)^2)^(1/2)","A"
14,1,29,24,0.296000," ","int(cot(x)^3/(a+a*cot(x)^2)^(1/2),x)","-\frac{\sqrt{a +a \left(\cot^{2}\left(x \right)\right)}}{a}-\frac{1}{\sqrt{a +a \left(\cot^{2}\left(x \right)\right)}}"," ",0,"-1/a*(a+a*cot(x)^2)^(1/2)-1/(a+a*cot(x)^2)^(1/2)","A"
15,1,38,27,0.204000," ","int(cot(x)^2/(a+a*cot(x)^2)^(1/2),x)","-\frac{\ln \left(\sqrt{a}\, \cot \left(x \right)+\sqrt{a +a \left(\cot^{2}\left(x \right)\right)}\right)}{\sqrt{a}}+\frac{\cot \left(x \right)}{\sqrt{a +a \left(\cot^{2}\left(x \right)\right)}}"," ",0,"-ln(a^(1/2)*cot(x)+(a+a*cot(x)^2)^(1/2))/a^(1/2)+cot(x)/(a+a*cot(x)^2)^(1/2)","A"
16,1,11,8,0.186000," ","int(cot(x)/(a+a*cot(x)^2)^(1/2),x)","\frac{1}{\sqrt{a +a \left(\cot^{2}\left(x \right)\right)}}"," ",0,"1/(a+a*cot(x)^2)^(1/2)","A"
17,1,56,28,0.744000," ","int(tan(x)/(a+a*cot(x)^2)^(1/2),x)","-\frac{\left(\sin \left(x \right)+\ln \left(-\frac{-1+\cos \left(x \right)+\sin \left(x \right)}{\sin \left(x \right)}\right)-\ln \left(-\frac{-\sin \left(x \right)-1+\cos \left(x \right)}{\sin \left(x \right)}\right)\right) \sqrt{4}}{2 \sin \left(x \right) \sqrt{-\frac{a}{-1+\cos^{2}\left(x \right)}}}"," ",0,"-1/2*(sin(x)+ln(-(-1+cos(x)+sin(x))/sin(x))-ln(-(-sin(x)-1+cos(x))/sin(x)))/sin(x)/(-1/(-1+cos(x)^2)*a)^(1/2)*4^(1/2)","A"
18,1,33,25,0.573000," ","int(tan(x)^2/(a+a*cot(x)^2)^(1/2),x)","\frac{\left(\sin^{3}\left(x \right)\right) \sqrt{4}}{2 \sqrt{-\frac{a}{-1+\cos^{2}\left(x \right)}}\, \cos \left(x \right) \left(-1+\cos \left(x \right)\right)^{2}}"," ",0,"1/2*sin(x)^3/(-1/(-1+cos(x)^2)*a)^(1/2)/cos(x)/(-1+cos(x))^2*4^(1/2)","A"
19,1,84,54,0.192000," ","int(cot(x)^3*(a+b*cot(x)^2)^(1/2),x)","-\frac{\left(a +b \left(\cot^{2}\left(x \right)\right)\right)^{\frac{3}{2}}}{3 b}+\sqrt{a +b \left(\cot^{2}\left(x \right)\right)}-\frac{b \arctan \left(\frac{\sqrt{a +b \left(\cot^{2}\left(x \right)\right)}}{\sqrt{-a +b}}\right)}{\sqrt{-a +b}}+\frac{a \arctan \left(\frac{\sqrt{a +b \left(\cot^{2}\left(x \right)\right)}}{\sqrt{-a +b}}\right)}{\sqrt{-a +b}}"," ",0,"-1/3*(a+b*cot(x)^2)^(3/2)/b+(a+b*cot(x)^2)^(1/2)-b/(-a+b)^(1/2)*arctan((a+b*cot(x)^2)^(1/2)/(-a+b)^(1/2))+a/(-a+b)^(1/2)*arctan((a+b*cot(x)^2)^(1/2)/(-a+b)^(1/2))","A"
20,1,71,40,0.128000," ","int(cot(x)*(a+b*cot(x)^2)^(1/2),x)","-\sqrt{a +b \left(\cot^{2}\left(x \right)\right)}+\frac{b \arctan \left(\frac{\sqrt{a +b \left(\cot^{2}\left(x \right)\right)}}{\sqrt{-a +b}}\right)}{\sqrt{-a +b}}-\frac{a \arctan \left(\frac{\sqrt{a +b \left(\cot^{2}\left(x \right)\right)}}{\sqrt{-a +b}}\right)}{\sqrt{-a +b}}"," ",0,"-(a+b*cot(x)^2)^(1/2)+b/(-a+b)^(1/2)*arctan((a+b*cot(x)^2)^(1/2)/(-a+b)^(1/2))-a/(-a+b)^(1/2)*arctan((a+b*cot(x)^2)^(1/2)/(-a+b)^(1/2))","A"
21,1,591,48,1.074000," ","int((a+b*cot(x)^2)^(1/2)*tan(x),x)","\frac{\left(\EllipticF \left(\frac{\left(-1+\cos \left(x \right)\right) \sqrt{\frac{2 \sqrt{a}\, \sqrt{a -b}-2 a +b}{b}}}{\sin \left(x \right)}, \sqrt{\frac{8 a^{\frac{3}{2}} \sqrt{a -b}-4 \sqrt{a}\, \sqrt{a -b}\, b +8 a^{2}-8 a b +b^{2}}{b^{2}}}\right) b -2 \EllipticPi \left(\frac{\left(-1+\cos \left(x \right)\right) \sqrt{\frac{2 \sqrt{a}\, \sqrt{a -b}-2 a +b}{b}}}{\sin \left(x \right)}, \frac{b}{2 \sqrt{a}\, \sqrt{a -b}-2 a +b}, \frac{\sqrt{-\frac{2 \sqrt{a}\, \sqrt{a -b}+2 a -b}{b}}}{\sqrt{\frac{2 \sqrt{a}\, \sqrt{a -b}-2 a +b}{b}}}\right) a +2 \EllipticPi \left(\frac{\left(-1+\cos \left(x \right)\right) \sqrt{\frac{2 \sqrt{a}\, \sqrt{a -b}-2 a +b}{b}}}{\sin \left(x \right)}, -\frac{b}{2 \sqrt{a}\, \sqrt{a -b}-2 a +b}, \frac{\sqrt{-\frac{2 \sqrt{a}\, \sqrt{a -b}+2 a -b}{b}}}{\sqrt{\frac{2 \sqrt{a}\, \sqrt{a -b}-2 a +b}{b}}}\right) a -2 \EllipticPi \left(\frac{\left(-1+\cos \left(x \right)\right) \sqrt{\frac{2 \sqrt{a}\, \sqrt{a -b}-2 a +b}{b}}}{\sin \left(x \right)}, -\frac{b}{2 \sqrt{a}\, \sqrt{a -b}-2 a +b}, \frac{\sqrt{-\frac{2 \sqrt{a}\, \sqrt{a -b}+2 a -b}{b}}}{\sqrt{\frac{2 \sqrt{a}\, \sqrt{a -b}-2 a +b}{b}}}\right) b \right) \left(\sin^{3}\left(x \right)\right) \sqrt{-\frac{2 \left(\cos \left(x \right) \sqrt{a}\, \sqrt{a -b}-\sqrt{a}\, \sqrt{a -b}+a \cos \left(x \right)-b \cos \left(x \right)-a \right)}{\left(\cos \left(x \right)+1\right) b}}\, \sqrt{2}\, \sqrt{\frac{\cos \left(x \right) \sqrt{a}\, \sqrt{a -b}-\sqrt{a}\, \sqrt{a -b}-a \cos \left(x \right)+b \cos \left(x \right)+a}{\left(\cos \left(x \right)+1\right) b}}\, \sqrt{\frac{a \left(\cos^{2}\left(x \right)\right)-b \left(\cos^{2}\left(x \right)\right)-a}{-1+\cos^{2}\left(x \right)}}\, \sqrt{4}}{2 \left(-1+\cos \left(x \right)\right) \left(a \left(\cos^{2}\left(x \right)\right)-b \left(\cos^{2}\left(x \right)\right)-a \right) \sqrt{\frac{2 \sqrt{a}\, \sqrt{a -b}-2 a +b}{b}}}"," ",0,"1/2*(EllipticF((-1+cos(x))*((2*a^(1/2)*(a-b)^(1/2)-2*a+b)/b)^(1/2)/sin(x),((8*a^(3/2)*(a-b)^(1/2)-4*a^(1/2)*(a-b)^(1/2)*b+8*a^2-8*a*b+b^2)/b^2)^(1/2))*b-2*EllipticPi((-1+cos(x))*((2*a^(1/2)*(a-b)^(1/2)-2*a+b)/b)^(1/2)/sin(x),1/(2*a^(1/2)*(a-b)^(1/2)-2*a+b)*b,(-(2*a^(1/2)*(a-b)^(1/2)+2*a-b)/b)^(1/2)/((2*a^(1/2)*(a-b)^(1/2)-2*a+b)/b)^(1/2))*a+2*EllipticPi((-1+cos(x))*((2*a^(1/2)*(a-b)^(1/2)-2*a+b)/b)^(1/2)/sin(x),-1/(2*a^(1/2)*(a-b)^(1/2)-2*a+b)*b,(-(2*a^(1/2)*(a-b)^(1/2)+2*a-b)/b)^(1/2)/((2*a^(1/2)*(a-b)^(1/2)-2*a+b)/b)^(1/2))*a-2*EllipticPi((-1+cos(x))*((2*a^(1/2)*(a-b)^(1/2)-2*a+b)/b)^(1/2)/sin(x),-1/(2*a^(1/2)*(a-b)^(1/2)-2*a+b)*b,(-(2*a^(1/2)*(a-b)^(1/2)+2*a-b)/b)^(1/2)/((2*a^(1/2)*(a-b)^(1/2)-2*a+b)/b)^(1/2))*b)*sin(x)^3*(-2*(cos(x)*a^(1/2)*(a-b)^(1/2)-a^(1/2)*(a-b)^(1/2)+a*cos(x)-b*cos(x)-a)/(cos(x)+1)/b)^(1/2)*2^(1/2)*((cos(x)*a^(1/2)*(a-b)^(1/2)-a^(1/2)*(a-b)^(1/2)-a*cos(x)+b*cos(x)+a)/(cos(x)+1)/b)^(1/2)*((a*cos(x)^2-b*cos(x)^2-a)/(-1+cos(x)^2))^(1/2)/(-1+cos(x))/(a*cos(x)^2-b*cos(x)^2-a)*4^(1/2)/((2*a^(1/2)*(a-b)^(1/2)-2*a+b)/b)^(1/2)","C"
22,1,174,71,0.184000," ","int(cot(x)^2*(a+b*cot(x)^2)^(1/2),x)","-\frac{\cot \left(x \right) \sqrt{a +b \left(\cot^{2}\left(x \right)\right)}}{2}-\frac{a \ln \left(\cot \left(x \right) \sqrt{b}+\sqrt{a +b \left(\cot^{2}\left(x \right)\right)}\right)}{2 \sqrt{b}}+\sqrt{b}\, \ln \left(\cot \left(x \right) \sqrt{b}+\sqrt{a +b \left(\cot^{2}\left(x \right)\right)}\right)-\frac{\sqrt{b^{4} \left(a -b \right)}\, \arctan \left(\frac{\left(a -b \right) b^{2} \cot \left(x \right)}{\sqrt{b^{4} \left(a -b \right)}\, \sqrt{a +b \left(\cot^{2}\left(x \right)\right)}}\right)}{b \left(a -b \right)}+\frac{a \sqrt{b^{4} \left(a -b \right)}\, \arctan \left(\frac{\left(a -b \right) b^{2} \cot \left(x \right)}{\sqrt{b^{4} \left(a -b \right)}\, \sqrt{a +b \left(\cot^{2}\left(x \right)\right)}}\right)}{b^{2} \left(a -b \right)}"," ",0,"-1/2*cot(x)*(a+b*cot(x)^2)^(1/2)-1/2*a/b^(1/2)*ln(cot(x)*b^(1/2)+(a+b*cot(x)^2)^(1/2))+b^(1/2)*ln(cot(x)*b^(1/2)+(a+b*cot(x)^2)^(1/2))-(b^4*(a-b))^(1/2)/b/(a-b)*arctan((a-b)*b^2/(b^4*(a-b))^(1/2)/(a+b*cot(x)^2)^(1/2)*cot(x))+a*(b^4*(a-b))^(1/2)/b^2/(a-b)*arctan((a-b)*b^2/(b^4*(a-b))^(1/2)/(a+b*cot(x)^2)^(1/2)*cot(x))","B"
23,1,137,53,0.230000," ","int((a+b*cot(x)^2)^(1/2),x)","-\sqrt{b}\, \ln \left(\cot \left(x \right) \sqrt{b}+\sqrt{a +b \left(\cot^{2}\left(x \right)\right)}\right)+\frac{\sqrt{b^{4} \left(a -b \right)}\, \arctan \left(\frac{\left(a -b \right) b^{2} \cot \left(x \right)}{\sqrt{b^{4} \left(a -b \right)}\, \sqrt{a +b \left(\cot^{2}\left(x \right)\right)}}\right)}{b \left(a -b \right)}-\frac{a \sqrt{b^{4} \left(a -b \right)}\, \arctan \left(\frac{\left(a -b \right) b^{2} \cot \left(x \right)}{\sqrt{b^{4} \left(a -b \right)}\, \sqrt{a +b \left(\cot^{2}\left(x \right)\right)}}\right)}{b^{2} \left(a -b \right)}"," ",0,"-b^(1/2)*ln(cot(x)*b^(1/2)+(a+b*cot(x)^2)^(1/2))+(b^4*(a-b))^(1/2)/b/(a-b)*arctan((a-b)*b^2/(b^4*(a-b))^(1/2)/(a+b*cot(x)^2)^(1/2)*cot(x))-a*(b^4*(a-b))^(1/2)/b^2/(a-b)*arctan((a-b)*b^2/(b^4*(a-b))^(1/2)/(a+b*cot(x)^2)^(1/2)*cot(x))","B"
24,1,752,43,0.988000," ","int((a+b*cot(x)^2)^(1/2)*tan(x)^2,x)","\frac{\left(-1+\cos \left(x \right)\right) \left(\cos \left(x \right) b^{\frac{3}{2}} \ln \left(4 \cos \left(x \right) \sqrt{-a +b}\, \sqrt{-\frac{a \left(\cos^{2}\left(x \right)\right)-b \left(\cos^{2}\left(x \right)\right)-a}{\left(\cos \left(x \right)+1\right)^{2}}}-4 a \cos \left(x \right)+4 b \cos \left(x \right)+4 \sqrt{-a +b}\, \sqrt{-\frac{a \left(\cos^{2}\left(x \right)\right)-b \left(\cos^{2}\left(x \right)\right)-a}{\left(\cos \left(x \right)+1\right)^{2}}}\right)+\cos \left(x \right) \sqrt{-a +b}\, \ln \left(-\frac{4 \left(-1+\cos \left(x \right)\right) \left(\cos \left(x \right) \sqrt{b}\, \sqrt{-\frac{a \left(\cos^{2}\left(x \right)\right)-b \left(\cos^{2}\left(x \right)\right)-a}{\left(\cos \left(x \right)+1\right)^{2}}}+a \cos \left(x \right)-b \cos \left(x \right)+\sqrt{-\frac{a \left(\cos^{2}\left(x \right)\right)-b \left(\cos^{2}\left(x \right)\right)-a}{\left(\cos \left(x \right)+1\right)^{2}}}\, \sqrt{b}+a \right)}{\sin \left(x \right)^{2} \sqrt{b}}\right) a -\cos \left(x \right) \sqrt{-a +b}\, \ln \left(-\frac{4 \left(-1+\cos \left(x \right)\right) \left(\cos \left(x \right) \sqrt{b}\, \sqrt{-\frac{a \left(\cos^{2}\left(x \right)\right)-b \left(\cos^{2}\left(x \right)\right)-a}{\left(\cos \left(x \right)+1\right)^{2}}}+a \cos \left(x \right)-b \cos \left(x \right)+\sqrt{-\frac{a \left(\cos^{2}\left(x \right)\right)-b \left(\cos^{2}\left(x \right)\right)-a}{\left(\cos \left(x \right)+1\right)^{2}}}\, \sqrt{b}+a \right)}{\sin \left(x \right)^{2} \sqrt{b}}\right) b -\cos \left(x \right) \sqrt{-a +b}\, \ln \left(-\frac{2 \left(-1+\cos \left(x \right)\right) \left(\cos \left(x \right) \sqrt{b}\, \sqrt{-\frac{a \left(\cos^{2}\left(x \right)\right)-b \left(\cos^{2}\left(x \right)\right)-a}{\left(\cos \left(x \right)+1\right)^{2}}}+a \cos \left(x \right)-b \cos \left(x \right)+\sqrt{-\frac{a \left(\cos^{2}\left(x \right)\right)-b \left(\cos^{2}\left(x \right)\right)-a}{\left(\cos \left(x \right)+1\right)^{2}}}\, \sqrt{b}+a \right)}{\sin \left(x \right)^{2} \sqrt{b}}\right) a +\cos \left(x \right) \sqrt{-a +b}\, \ln \left(-\frac{2 \left(-1+\cos \left(x \right)\right) \left(\cos \left(x \right) \sqrt{b}\, \sqrt{-\frac{a \left(\cos^{2}\left(x \right)\right)-b \left(\cos^{2}\left(x \right)\right)-a}{\left(\cos \left(x \right)+1\right)^{2}}}+a \cos \left(x \right)-b \cos \left(x \right)+\sqrt{-\frac{a \left(\cos^{2}\left(x \right)\right)-b \left(\cos^{2}\left(x \right)\right)-a}{\left(\cos \left(x \right)+1\right)^{2}}}\, \sqrt{b}+a \right)}{\sin \left(x \right)^{2} \sqrt{b}}\right) b -\cos \left(x \right) \sqrt{b}\, \ln \left(4 \cos \left(x \right) \sqrt{-a +b}\, \sqrt{-\frac{a \left(\cos^{2}\left(x \right)\right)-b \left(\cos^{2}\left(x \right)\right)-a}{\left(\cos \left(x \right)+1\right)^{2}}}-4 a \cos \left(x \right)+4 b \cos \left(x \right)+4 \sqrt{-a +b}\, \sqrt{-\frac{a \left(\cos^{2}\left(x \right)\right)-b \left(\cos^{2}\left(x \right)\right)-a}{\left(\cos \left(x \right)+1\right)^{2}}}\right) a -\cos \left(x \right) \sqrt{b}\, \sqrt{-a +b}\, \sqrt{-\frac{a \left(\cos^{2}\left(x \right)\right)-b \left(\cos^{2}\left(x \right)\right)-a}{\left(\cos \left(x \right)+1\right)^{2}}}-\sqrt{-\frac{a \left(\cos^{2}\left(x \right)\right)-b \left(\cos^{2}\left(x \right)\right)-a}{\left(\cos \left(x \right)+1\right)^{2}}}\, \sqrt{-a +b}\, \sqrt{b}\right) \sqrt{\frac{a \left(\cos^{2}\left(x \right)\right)-b \left(\cos^{2}\left(x \right)\right)-a}{-1+\cos^{2}\left(x \right)}}\, \sqrt{4}}{2 \cos \left(x \right) \sin \left(x \right) \sqrt{-\frac{a \left(\cos^{2}\left(x \right)\right)-b \left(\cos^{2}\left(x \right)\right)-a}{\left(\cos \left(x \right)+1\right)^{2}}}\, \sqrt{-a +b}\, \sqrt{b}}"," ",0,"1/2*(-1+cos(x))*(cos(x)*b^(3/2)*ln(4*cos(x)*(-a+b)^(1/2)*(-(a*cos(x)^2-b*cos(x)^2-a)/(cos(x)+1)^2)^(1/2)-4*a*cos(x)+4*b*cos(x)+4*(-a+b)^(1/2)*(-(a*cos(x)^2-b*cos(x)^2-a)/(cos(x)+1)^2)^(1/2))+cos(x)*(-a+b)^(1/2)*ln(-4*(-1+cos(x))*(cos(x)*b^(1/2)*(-(a*cos(x)^2-b*cos(x)^2-a)/(cos(x)+1)^2)^(1/2)+a*cos(x)-b*cos(x)+(-(a*cos(x)^2-b*cos(x)^2-a)/(cos(x)+1)^2)^(1/2)*b^(1/2)+a)/sin(x)^2/b^(1/2))*a-cos(x)*(-a+b)^(1/2)*ln(-4*(-1+cos(x))*(cos(x)*b^(1/2)*(-(a*cos(x)^2-b*cos(x)^2-a)/(cos(x)+1)^2)^(1/2)+a*cos(x)-b*cos(x)+(-(a*cos(x)^2-b*cos(x)^2-a)/(cos(x)+1)^2)^(1/2)*b^(1/2)+a)/sin(x)^2/b^(1/2))*b-cos(x)*(-a+b)^(1/2)*ln(-2*(-1+cos(x))*(cos(x)*b^(1/2)*(-(a*cos(x)^2-b*cos(x)^2-a)/(cos(x)+1)^2)^(1/2)+a*cos(x)-b*cos(x)+(-(a*cos(x)^2-b*cos(x)^2-a)/(cos(x)+1)^2)^(1/2)*b^(1/2)+a)/sin(x)^2/b^(1/2))*a+cos(x)*(-a+b)^(1/2)*ln(-2*(-1+cos(x))*(cos(x)*b^(1/2)*(-(a*cos(x)^2-b*cos(x)^2-a)/(cos(x)+1)^2)^(1/2)+a*cos(x)-b*cos(x)+(-(a*cos(x)^2-b*cos(x)^2-a)/(cos(x)+1)^2)^(1/2)*b^(1/2)+a)/sin(x)^2/b^(1/2))*b-cos(x)*b^(1/2)*ln(4*cos(x)*(-a+b)^(1/2)*(-(a*cos(x)^2-b*cos(x)^2-a)/(cos(x)+1)^2)^(1/2)-4*a*cos(x)+4*b*cos(x)+4*(-a+b)^(1/2)*(-(a*cos(x)^2-b*cos(x)^2-a)/(cos(x)+1)^2)^(1/2))*a-cos(x)*b^(1/2)*(-a+b)^(1/2)*(-(a*cos(x)^2-b*cos(x)^2-a)/(cos(x)+1)^2)^(1/2)-(-(a*cos(x)^2-b*cos(x)^2-a)/(cos(x)+1)^2)^(1/2)*(-a+b)^(1/2)*b^(1/2))*((a*cos(x)^2-b*cos(x)^2-a)/(-1+cos(x)^2))^(1/2)/cos(x)/sin(x)/(-(a*cos(x)^2-b*cos(x)^2-a)/(cos(x)+1)^2)^(1/2)*4^(1/2)/(-a+b)^(1/2)/b^(1/2)","B"
25,1,951,71,0.770000," ","int((a+b*cot(x)^2)^(1/2)*tan(x)^4,x)","-\frac{\left(-1+\cos \left(x \right)\right) \left(\left(\cos^{3}\left(x \right)\right) \sqrt{-\frac{a \left(\cos^{2}\left(x \right)\right)-b \left(\cos^{2}\left(x \right)\right)-a}{\left(\cos \left(x \right)+1\right)^{2}}}\, \sqrt{-a +b}\, b^{\frac{3}{2}}+3 \left(\cos^{3}\left(x \right)\right) \ln \left(4 \cos \left(x \right) \sqrt{-a +b}\, \sqrt{-\frac{a \left(\cos^{2}\left(x \right)\right)-b \left(\cos^{2}\left(x \right)\right)-a}{\left(\cos \left(x \right)+1\right)^{2}}}-4 a \cos \left(x \right)+4 b \cos \left(x \right)+4 \sqrt{-a +b}\, \sqrt{-\frac{a \left(\cos^{2}\left(x \right)\right)-b \left(\cos^{2}\left(x \right)\right)-a}{\left(\cos \left(x \right)+1\right)^{2}}}\right) b^{\frac{3}{2}} a -4 \left(\cos^{3}\left(x \right)\right) \sqrt{-\frac{a \left(\cos^{2}\left(x \right)\right)-b \left(\cos^{2}\left(x \right)\right)-a}{\left(\cos \left(x \right)+1\right)^{2}}}\, \sqrt{-a +b}\, \sqrt{b}\, a +3 \left(\cos^{3}\left(x \right)\right) \ln \left(-\frac{4 \left(-1+\cos \left(x \right)\right) \left(\cos \left(x \right) \sqrt{b}\, \sqrt{-\frac{a \left(\cos^{2}\left(x \right)\right)-b \left(\cos^{2}\left(x \right)\right)-a}{\left(\cos \left(x \right)+1\right)^{2}}}+a \cos \left(x \right)-b \cos \left(x \right)+\sqrt{-\frac{a \left(\cos^{2}\left(x \right)\right)-b \left(\cos^{2}\left(x \right)\right)-a}{\left(\cos \left(x \right)+1\right)^{2}}}\, \sqrt{b}+a \right)}{\sin \left(x \right)^{2} \sqrt{b}}\right) \sqrt{-a +b}\, a^{2}-3 \left(\cos^{3}\left(x \right)\right) \ln \left(-\frac{4 \left(-1+\cos \left(x \right)\right) \left(\cos \left(x \right) \sqrt{b}\, \sqrt{-\frac{a \left(\cos^{2}\left(x \right)\right)-b \left(\cos^{2}\left(x \right)\right)-a}{\left(\cos \left(x \right)+1\right)^{2}}}+a \cos \left(x \right)-b \cos \left(x \right)+\sqrt{-\frac{a \left(\cos^{2}\left(x \right)\right)-b \left(\cos^{2}\left(x \right)\right)-a}{\left(\cos \left(x \right)+1\right)^{2}}}\, \sqrt{b}+a \right)}{\sin \left(x \right)^{2} \sqrt{b}}\right) \sqrt{-a +b}\, a b -3 \left(\cos^{3}\left(x \right)\right) \ln \left(-\frac{2 \left(-1+\cos \left(x \right)\right) \left(\cos \left(x \right) \sqrt{b}\, \sqrt{-\frac{a \left(\cos^{2}\left(x \right)\right)-b \left(\cos^{2}\left(x \right)\right)-a}{\left(\cos \left(x \right)+1\right)^{2}}}+a \cos \left(x \right)-b \cos \left(x \right)+\sqrt{-\frac{a \left(\cos^{2}\left(x \right)\right)-b \left(\cos^{2}\left(x \right)\right)-a}{\left(\cos \left(x \right)+1\right)^{2}}}\, \sqrt{b}+a \right)}{\sin \left(x \right)^{2} \sqrt{b}}\right) \sqrt{-a +b}\, a^{2}+3 \left(\cos^{3}\left(x \right)\right) \ln \left(-\frac{2 \left(-1+\cos \left(x \right)\right) \left(\cos \left(x \right) \sqrt{b}\, \sqrt{-\frac{a \left(\cos^{2}\left(x \right)\right)-b \left(\cos^{2}\left(x \right)\right)-a}{\left(\cos \left(x \right)+1\right)^{2}}}+a \cos \left(x \right)-b \cos \left(x \right)+\sqrt{-\frac{a \left(\cos^{2}\left(x \right)\right)-b \left(\cos^{2}\left(x \right)\right)-a}{\left(\cos \left(x \right)+1\right)^{2}}}\, \sqrt{b}+a \right)}{\sin \left(x \right)^{2} \sqrt{b}}\right) \sqrt{-a +b}\, a b -3 \left(\cos^{3}\left(x \right)\right) \ln \left(4 \cos \left(x \right) \sqrt{-a +b}\, \sqrt{-\frac{a \left(\cos^{2}\left(x \right)\right)-b \left(\cos^{2}\left(x \right)\right)-a}{\left(\cos \left(x \right)+1\right)^{2}}}-4 a \cos \left(x \right)+4 b \cos \left(x \right)+4 \sqrt{-a +b}\, \sqrt{-\frac{a \left(\cos^{2}\left(x \right)\right)-b \left(\cos^{2}\left(x \right)\right)-a}{\left(\cos \left(x \right)+1\right)^{2}}}\right) \sqrt{b}\, a^{2}+\left(\cos^{2}\left(x \right)\right) \sqrt{-\frac{a \left(\cos^{2}\left(x \right)\right)-b \left(\cos^{2}\left(x \right)\right)-a}{\left(\cos \left(x \right)+1\right)^{2}}}\, \sqrt{-a +b}\, b^{\frac{3}{2}}-4 \left(\cos^{2}\left(x \right)\right) \sqrt{-\frac{a \left(\cos^{2}\left(x \right)\right)-b \left(\cos^{2}\left(x \right)\right)-a}{\left(\cos \left(x \right)+1\right)^{2}}}\, \sqrt{-a +b}\, \sqrt{b}\, a +\cos \left(x \right) \sqrt{-\frac{a \left(\cos^{2}\left(x \right)\right)-b \left(\cos^{2}\left(x \right)\right)-a}{\left(\cos \left(x \right)+1\right)^{2}}}\, \sqrt{-a +b}\, \sqrt{b}\, a +\sqrt{-\frac{a \left(\cos^{2}\left(x \right)\right)-b \left(\cos^{2}\left(x \right)\right)-a}{\left(\cos \left(x \right)+1\right)^{2}}}\, \sqrt{-a +b}\, a \sqrt{b}\right) \sqrt{\frac{a \left(\cos^{2}\left(x \right)\right)-b \left(\cos^{2}\left(x \right)\right)-a}{-1+\cos^{2}\left(x \right)}}\, \sqrt{4}}{6 \cos \left(x \right)^{3} \sin \left(x \right) \sqrt{-\frac{a \left(\cos^{2}\left(x \right)\right)-b \left(\cos^{2}\left(x \right)\right)-a}{\left(\cos \left(x \right)+1\right)^{2}}}\, \sqrt{-a +b}\, a \sqrt{b}}"," ",0,"-1/6*(-1+cos(x))*(cos(x)^3*(-(a*cos(x)^2-b*cos(x)^2-a)/(cos(x)+1)^2)^(1/2)*(-a+b)^(1/2)*b^(3/2)+3*cos(x)^3*ln(4*cos(x)*(-a+b)^(1/2)*(-(a*cos(x)^2-b*cos(x)^2-a)/(cos(x)+1)^2)^(1/2)-4*a*cos(x)+4*b*cos(x)+4*(-a+b)^(1/2)*(-(a*cos(x)^2-b*cos(x)^2-a)/(cos(x)+1)^2)^(1/2))*b^(3/2)*a-4*cos(x)^3*(-(a*cos(x)^2-b*cos(x)^2-a)/(cos(x)+1)^2)^(1/2)*(-a+b)^(1/2)*b^(1/2)*a+3*cos(x)^3*ln(-4*(-1+cos(x))*(cos(x)*b^(1/2)*(-(a*cos(x)^2-b*cos(x)^2-a)/(cos(x)+1)^2)^(1/2)+a*cos(x)-b*cos(x)+(-(a*cos(x)^2-b*cos(x)^2-a)/(cos(x)+1)^2)^(1/2)*b^(1/2)+a)/sin(x)^2/b^(1/2))*(-a+b)^(1/2)*a^2-3*cos(x)^3*ln(-4*(-1+cos(x))*(cos(x)*b^(1/2)*(-(a*cos(x)^2-b*cos(x)^2-a)/(cos(x)+1)^2)^(1/2)+a*cos(x)-b*cos(x)+(-(a*cos(x)^2-b*cos(x)^2-a)/(cos(x)+1)^2)^(1/2)*b^(1/2)+a)/sin(x)^2/b^(1/2))*(-a+b)^(1/2)*a*b-3*cos(x)^3*ln(-2*(-1+cos(x))*(cos(x)*b^(1/2)*(-(a*cos(x)^2-b*cos(x)^2-a)/(cos(x)+1)^2)^(1/2)+a*cos(x)-b*cos(x)+(-(a*cos(x)^2-b*cos(x)^2-a)/(cos(x)+1)^2)^(1/2)*b^(1/2)+a)/sin(x)^2/b^(1/2))*(-a+b)^(1/2)*a^2+3*cos(x)^3*ln(-2*(-1+cos(x))*(cos(x)*b^(1/2)*(-(a*cos(x)^2-b*cos(x)^2-a)/(cos(x)+1)^2)^(1/2)+a*cos(x)-b*cos(x)+(-(a*cos(x)^2-b*cos(x)^2-a)/(cos(x)+1)^2)^(1/2)*b^(1/2)+a)/sin(x)^2/b^(1/2))*(-a+b)^(1/2)*a*b-3*cos(x)^3*ln(4*cos(x)*(-a+b)^(1/2)*(-(a*cos(x)^2-b*cos(x)^2-a)/(cos(x)+1)^2)^(1/2)-4*a*cos(x)+4*b*cos(x)+4*(-a+b)^(1/2)*(-(a*cos(x)^2-b*cos(x)^2-a)/(cos(x)+1)^2)^(1/2))*b^(1/2)*a^2+cos(x)^2*(-(a*cos(x)^2-b*cos(x)^2-a)/(cos(x)+1)^2)^(1/2)*(-a+b)^(1/2)*b^(3/2)-4*cos(x)^2*(-(a*cos(x)^2-b*cos(x)^2-a)/(cos(x)+1)^2)^(1/2)*(-a+b)^(1/2)*b^(1/2)*a+cos(x)*(-(a*cos(x)^2-b*cos(x)^2-a)/(cos(x)+1)^2)^(1/2)*(-a+b)^(1/2)*b^(1/2)*a+(-(a*cos(x)^2-b*cos(x)^2-a)/(cos(x)+1)^2)^(1/2)*(-a+b)^(1/2)*a*b^(1/2))*((a*cos(x)^2-b*cos(x)^2-a)/(-1+cos(x)^2))^(1/2)/cos(x)^3/sin(x)/(-(a*cos(x)^2-b*cos(x)^2-a)/(cos(x)+1)^2)^(1/2)*4^(1/2)/(-a+b)^(1/2)/a/b^(1/2)","B"
26,1,150,72,0.132000," ","int(cot(x)^3*(a+b*cot(x)^2)^(3/2),x)","-\frac{\left(a +b \left(\cot^{2}\left(x \right)\right)\right)^{\frac{5}{2}}}{5 b}+\frac{b \left(\cot^{2}\left(x \right)\right) \sqrt{a +b \left(\cot^{2}\left(x \right)\right)}}{3}+\frac{4 a \sqrt{a +b \left(\cot^{2}\left(x \right)\right)}}{3}-b \sqrt{a +b \left(\cot^{2}\left(x \right)\right)}+\frac{b^{2} \arctan \left(\frac{\sqrt{a +b \left(\cot^{2}\left(x \right)\right)}}{\sqrt{-a +b}}\right)}{\sqrt{-a +b}}-\frac{2 a b \arctan \left(\frac{\sqrt{a +b \left(\cot^{2}\left(x \right)\right)}}{\sqrt{-a +b}}\right)}{\sqrt{-a +b}}+\frac{a^{2} \arctan \left(\frac{\sqrt{a +b \left(\cot^{2}\left(x \right)\right)}}{\sqrt{-a +b}}\right)}{\sqrt{-a +b}}"," ",0,"-1/5*(a+b*cot(x)^2)^(5/2)/b+1/3*b*cot(x)^2*(a+b*cot(x)^2)^(1/2)+4/3*a*(a+b*cot(x)^2)^(1/2)-b*(a+b*cot(x)^2)^(1/2)+b^2/(-a+b)^(1/2)*arctan((a+b*cot(x)^2)^(1/2)/(-a+b)^(1/2))-2*a*b/(-a+b)^(1/2)*arctan((a+b*cot(x)^2)^(1/2)/(-a+b)^(1/2))+a^2/(-a+b)^(1/2)*arctan((a+b*cot(x)^2)^(1/2)/(-a+b)^(1/2))","B"
27,1,286,105,0.129000," ","int(cot(x)^2*(a+b*cot(x)^2)^(3/2),x)","-\frac{\cot \left(x \right) \left(a +b \left(\cot^{2}\left(x \right)\right)\right)^{\frac{3}{2}}}{4}-\frac{3 a \cot \left(x \right) \sqrt{a +b \left(\cot^{2}\left(x \right)\right)}}{8}-\frac{3 a^{2} \ln \left(\cot \left(x \right) \sqrt{b}+\sqrt{a +b \left(\cot^{2}\left(x \right)\right)}\right)}{8 \sqrt{b}}+\frac{b \cot \left(x \right) \sqrt{a +b \left(\cot^{2}\left(x \right)\right)}}{2}+\frac{3 \sqrt{b}\, a \ln \left(\cot \left(x \right) \sqrt{b}+\sqrt{a +b \left(\cot^{2}\left(x \right)\right)}\right)}{2}-b^{\frac{3}{2}} \ln \left(\cot \left(x \right) \sqrt{b}+\sqrt{a +b \left(\cot^{2}\left(x \right)\right)}\right)+\frac{\sqrt{b^{4} \left(a -b \right)}\, \arctan \left(\frac{\left(a -b \right) b^{2} \cot \left(x \right)}{\sqrt{b^{4} \left(a -b \right)}\, \sqrt{a +b \left(\cot^{2}\left(x \right)\right)}}\right)}{a -b}-\frac{2 a \sqrt{b^{4} \left(a -b \right)}\, \arctan \left(\frac{\left(a -b \right) b^{2} \cot \left(x \right)}{\sqrt{b^{4} \left(a -b \right)}\, \sqrt{a +b \left(\cot^{2}\left(x \right)\right)}}\right)}{b \left(a -b \right)}+\frac{a^{2} \sqrt{b^{4} \left(a -b \right)}\, \arctan \left(\frac{\left(a -b \right) b^{2} \cot \left(x \right)}{\sqrt{b^{4} \left(a -b \right)}\, \sqrt{a +b \left(\cot^{2}\left(x \right)\right)}}\right)}{b^{2} \left(a -b \right)}"," ",0,"-1/4*cot(x)*(a+b*cot(x)^2)^(3/2)-3/8*a*cot(x)*(a+b*cot(x)^2)^(1/2)-3/8*a^2/b^(1/2)*ln(cot(x)*b^(1/2)+(a+b*cot(x)^2)^(1/2))+1/2*b*cot(x)*(a+b*cot(x)^2)^(1/2)+3/2*b^(1/2)*a*ln(cot(x)*b^(1/2)+(a+b*cot(x)^2)^(1/2))-b^(3/2)*ln(cot(x)*b^(1/2)+(a+b*cot(x)^2)^(1/2))+(b^4*(a-b))^(1/2)/(a-b)*arctan((a-b)*b^2/(b^4*(a-b))^(1/2)/(a+b*cot(x)^2)^(1/2)*cot(x))-2*a/b*(b^4*(a-b))^(1/2)/(a-b)*arctan((a-b)*b^2/(b^4*(a-b))^(1/2)/(a+b*cot(x)^2)^(1/2)*cot(x))+a^2*(b^4*(a-b))^(1/2)/b^2/(a-b)*arctan((a-b)*b^2/(b^4*(a-b))^(1/2)/(a+b*cot(x)^2)^(1/2)*cot(x))","B"
28,1,136,57,0.093000," ","int(cot(x)*(a+b*cot(x)^2)^(3/2),x)","-\frac{b \left(\cot^{2}\left(x \right)\right) \sqrt{a +b \left(\cot^{2}\left(x \right)\right)}}{3}-\frac{4 a \sqrt{a +b \left(\cot^{2}\left(x \right)\right)}}{3}+b \sqrt{a +b \left(\cot^{2}\left(x \right)\right)}-\frac{b^{2} \arctan \left(\frac{\sqrt{a +b \left(\cot^{2}\left(x \right)\right)}}{\sqrt{-a +b}}\right)}{\sqrt{-a +b}}+\frac{2 a b \arctan \left(\frac{\sqrt{a +b \left(\cot^{2}\left(x \right)\right)}}{\sqrt{-a +b}}\right)}{\sqrt{-a +b}}-\frac{a^{2} \arctan \left(\frac{\sqrt{a +b \left(\cot^{2}\left(x \right)\right)}}{\sqrt{-a +b}}\right)}{\sqrt{-a +b}}"," ",0,"-1/3*b*cot(x)^2*(a+b*cot(x)^2)^(1/2)-4/3*a*(a+b*cot(x)^2)^(1/2)+b*(a+b*cot(x)^2)^(1/2)-b^2/(-a+b)^(1/2)*arctan((a+b*cot(x)^2)^(1/2)/(-a+b)^(1/2))+2*a*b/(-a+b)^(1/2)*arctan((a+b*cot(x)^2)^(1/2)/(-a+b)^(1/2))-a^2/(-a+b)^(1/2)*arctan((a+b*cot(x)^2)^(1/2)/(-a+b)^(1/2))","B"
29,1,2628,61,0.759000," ","int((a+b*cot(x)^2)^(3/2)*tan(x),x)","\text{Expression too large to display}"," ",0,"sin(x)^2*((a*cos(x)^2-b*cos(x)^2-a)/(-1+cos(x)^2))^(3/2)*(2*2^(1/2)*((cos(x)*a^(1/2)*(a-b)^(1/2)-a^(1/2)*(a-b)^(1/2)-a*cos(x)+b*cos(x)+a)/(cos(x)+1)/b)^(1/2)*(-2*(cos(x)*a^(1/2)*(a-b)^(1/2)-a^(1/2)*(a-b)^(1/2)+a*cos(x)-b*cos(x)-a)/(cos(x)+1)/b)^(1/2)*EllipticF((-1+cos(x))*((2*a^(1/2)*(a-b)^(1/2)-2*a+b)/b)^(1/2)/sin(x),((8*a^(3/2)*(a-b)^(1/2)-4*a^(1/2)*(a-b)^(1/2)*b+8*a^2-8*a*b+b^2)/b^2)^(1/2))*a*b*sin(x)*cos(x)-2^(1/2)*((cos(x)*a^(1/2)*(a-b)^(1/2)-a^(1/2)*(a-b)^(1/2)-a*cos(x)+b*cos(x)+a)/(cos(x)+1)/b)^(1/2)*(-2*(cos(x)*a^(1/2)*(a-b)^(1/2)-a^(1/2)*(a-b)^(1/2)+a*cos(x)-b*cos(x)-a)/(cos(x)+1)/b)^(1/2)*EllipticF((-1+cos(x))*((2*a^(1/2)*(a-b)^(1/2)-2*a+b)/b)^(1/2)/sin(x),((8*a^(3/2)*(a-b)^(1/2)-4*a^(1/2)*(a-b)^(1/2)*b+8*a^2-8*a*b+b^2)/b^2)^(1/2))*b^2*sin(x)*cos(x)+2*2^(1/2)*((cos(x)*a^(1/2)*(a-b)^(1/2)-a^(1/2)*(a-b)^(1/2)-a*cos(x)+b*cos(x)+a)/(cos(x)+1)/b)^(1/2)*(-2*(cos(x)*a^(1/2)*(a-b)^(1/2)-a^(1/2)*(a-b)^(1/2)+a*cos(x)-b*cos(x)-a)/(cos(x)+1)/b)^(1/2)*EllipticPi((-1+cos(x))*((2*a^(1/2)*(a-b)^(1/2)-2*a+b)/b)^(1/2)/sin(x),-1/(2*a^(1/2)*(a-b)^(1/2)-2*a+b)*b,(-(2*a^(1/2)*(a-b)^(1/2)+2*a-b)/b)^(1/2)/((2*a^(1/2)*(a-b)^(1/2)-2*a+b)/b)^(1/2))*a^2*sin(x)*cos(x)-4*2^(1/2)*((cos(x)*a^(1/2)*(a-b)^(1/2)-a^(1/2)*(a-b)^(1/2)-a*cos(x)+b*cos(x)+a)/(cos(x)+1)/b)^(1/2)*(-2*(cos(x)*a^(1/2)*(a-b)^(1/2)-a^(1/2)*(a-b)^(1/2)+a*cos(x)-b*cos(x)-a)/(cos(x)+1)/b)^(1/2)*EllipticPi((-1+cos(x))*((2*a^(1/2)*(a-b)^(1/2)-2*a+b)/b)^(1/2)/sin(x),-1/(2*a^(1/2)*(a-b)^(1/2)-2*a+b)*b,(-(2*a^(1/2)*(a-b)^(1/2)+2*a-b)/b)^(1/2)/((2*a^(1/2)*(a-b)^(1/2)-2*a+b)/b)^(1/2))*a*b*sin(x)*cos(x)+2*2^(1/2)*((cos(x)*a^(1/2)*(a-b)^(1/2)-a^(1/2)*(a-b)^(1/2)-a*cos(x)+b*cos(x)+a)/(cos(x)+1)/b)^(1/2)*(-2*(cos(x)*a^(1/2)*(a-b)^(1/2)-a^(1/2)*(a-b)^(1/2)+a*cos(x)-b*cos(x)-a)/(cos(x)+1)/b)^(1/2)*EllipticPi((-1+cos(x))*((2*a^(1/2)*(a-b)^(1/2)-2*a+b)/b)^(1/2)/sin(x),-1/(2*a^(1/2)*(a-b)^(1/2)-2*a+b)*b,(-(2*a^(1/2)*(a-b)^(1/2)+2*a-b)/b)^(1/2)/((2*a^(1/2)*(a-b)^(1/2)-2*a+b)/b)^(1/2))*b^2*sin(x)*cos(x)-2*2^(1/2)*((cos(x)*a^(1/2)*(a-b)^(1/2)-a^(1/2)*(a-b)^(1/2)-a*cos(x)+b*cos(x)+a)/(cos(x)+1)/b)^(1/2)*(-2*(cos(x)*a^(1/2)*(a-b)^(1/2)-a^(1/2)*(a-b)^(1/2)+a*cos(x)-b*cos(x)-a)/(cos(x)+1)/b)^(1/2)*EllipticPi((-1+cos(x))*((2*a^(1/2)*(a-b)^(1/2)-2*a+b)/b)^(1/2)/sin(x),1/(2*a^(1/2)*(a-b)^(1/2)-2*a+b)*b,(-(2*a^(1/2)*(a-b)^(1/2)+2*a-b)/b)^(1/2)/((2*a^(1/2)*(a-b)^(1/2)-2*a+b)/b)^(1/2))*a^2*sin(x)*cos(x)+2*2^(1/2)*((cos(x)*a^(1/2)*(a-b)^(1/2)-a^(1/2)*(a-b)^(1/2)-a*cos(x)+b*cos(x)+a)/(cos(x)+1)/b)^(1/2)*(-2*(cos(x)*a^(1/2)*(a-b)^(1/2)-a^(1/2)*(a-b)^(1/2)+a*cos(x)-b*cos(x)-a)/(cos(x)+1)/b)^(1/2)*EllipticF((-1+cos(x))*((2*a^(1/2)*(a-b)^(1/2)-2*a+b)/b)^(1/2)/sin(x),((8*a^(3/2)*(a-b)^(1/2)-4*a^(1/2)*(a-b)^(1/2)*b+8*a^2-8*a*b+b^2)/b^2)^(1/2))*a*b*sin(x)-2^(1/2)*((cos(x)*a^(1/2)*(a-b)^(1/2)-a^(1/2)*(a-b)^(1/2)-a*cos(x)+b*cos(x)+a)/(cos(x)+1)/b)^(1/2)*(-2*(cos(x)*a^(1/2)*(a-b)^(1/2)-a^(1/2)*(a-b)^(1/2)+a*cos(x)-b*cos(x)-a)/(cos(x)+1)/b)^(1/2)*EllipticF((-1+cos(x))*((2*a^(1/2)*(a-b)^(1/2)-2*a+b)/b)^(1/2)/sin(x),((8*a^(3/2)*(a-b)^(1/2)-4*a^(1/2)*(a-b)^(1/2)*b+8*a^2-8*a*b+b^2)/b^2)^(1/2))*b^2*sin(x)+2*2^(1/2)*((cos(x)*a^(1/2)*(a-b)^(1/2)-a^(1/2)*(a-b)^(1/2)-a*cos(x)+b*cos(x)+a)/(cos(x)+1)/b)^(1/2)*(-2*(cos(x)*a^(1/2)*(a-b)^(1/2)-a^(1/2)*(a-b)^(1/2)+a*cos(x)-b*cos(x)-a)/(cos(x)+1)/b)^(1/2)*EllipticPi((-1+cos(x))*((2*a^(1/2)*(a-b)^(1/2)-2*a+b)/b)^(1/2)/sin(x),-1/(2*a^(1/2)*(a-b)^(1/2)-2*a+b)*b,(-(2*a^(1/2)*(a-b)^(1/2)+2*a-b)/b)^(1/2)/((2*a^(1/2)*(a-b)^(1/2)-2*a+b)/b)^(1/2))*a^2*sin(x)-4*2^(1/2)*((cos(x)*a^(1/2)*(a-b)^(1/2)-a^(1/2)*(a-b)^(1/2)-a*cos(x)+b*cos(x)+a)/(cos(x)+1)/b)^(1/2)*(-2*(cos(x)*a^(1/2)*(a-b)^(1/2)-a^(1/2)*(a-b)^(1/2)+a*cos(x)-b*cos(x)-a)/(cos(x)+1)/b)^(1/2)*EllipticPi((-1+cos(x))*((2*a^(1/2)*(a-b)^(1/2)-2*a+b)/b)^(1/2)/sin(x),-1/(2*a^(1/2)*(a-b)^(1/2)-2*a+b)*b,(-(2*a^(1/2)*(a-b)^(1/2)+2*a-b)/b)^(1/2)/((2*a^(1/2)*(a-b)^(1/2)-2*a+b)/b)^(1/2))*a*b*sin(x)+2*2^(1/2)*((cos(x)*a^(1/2)*(a-b)^(1/2)-a^(1/2)*(a-b)^(1/2)-a*cos(x)+b*cos(x)+a)/(cos(x)+1)/b)^(1/2)*(-2*(cos(x)*a^(1/2)*(a-b)^(1/2)-a^(1/2)*(a-b)^(1/2)+a*cos(x)-b*cos(x)-a)/(cos(x)+1)/b)^(1/2)*EllipticPi((-1+cos(x))*((2*a^(1/2)*(a-b)^(1/2)-2*a+b)/b)^(1/2)/sin(x),-1/(2*a^(1/2)*(a-b)^(1/2)-2*a+b)*b,(-(2*a^(1/2)*(a-b)^(1/2)+2*a-b)/b)^(1/2)/((2*a^(1/2)*(a-b)^(1/2)-2*a+b)/b)^(1/2))*b^2*sin(x)-2*2^(1/2)*((cos(x)*a^(1/2)*(a-b)^(1/2)-a^(1/2)*(a-b)^(1/2)-a*cos(x)+b*cos(x)+a)/(cos(x)+1)/b)^(1/2)*(-2*(cos(x)*a^(1/2)*(a-b)^(1/2)-a^(1/2)*(a-b)^(1/2)+a*cos(x)-b*cos(x)-a)/(cos(x)+1)/b)^(1/2)*EllipticPi((-1+cos(x))*((2*a^(1/2)*(a-b)^(1/2)-2*a+b)/b)^(1/2)/sin(x),1/(2*a^(1/2)*(a-b)^(1/2)-2*a+b)*b,(-(2*a^(1/2)*(a-b)^(1/2)+2*a-b)/b)^(1/2)/((2*a^(1/2)*(a-b)^(1/2)-2*a+b)/b)^(1/2))*a^2*sin(x)+cos(x)^2*((2*a^(1/2)*(a-b)^(1/2)-2*a+b)/b)^(1/2)*a*b-cos(x)^2*((2*a^(1/2)*(a-b)^(1/2)-2*a+b)/b)^(1/2)*b^2-((2*a^(1/2)*(a-b)^(1/2)-2*a+b)/b)^(1/2)*a*b)/(a*cos(x)^2-b*cos(x)^2-a)^2/((2*a^(1/2)*(a-b)^(1/2)-2*a+b)/b)^(1/2)","C"
30,1,1276,66,0.691000," ","int((a+b*cot(x)^2)^(3/2)*tan(x)^2,x)","-\frac{\left(\frac{a \left(\cos^{2}\left(x \right)\right)-b \left(\cos^{2}\left(x \right)\right)-a}{-1+\cos^{2}\left(x \right)}\right)^{\frac{3}{2}} \left(-1+\cos \left(x \right)\right)^{3} \left(2 \cos \left(x \right) \ln \left(4 \cos \left(x \right) \sqrt{-a +b}\, \sqrt{-\frac{a \left(\cos^{2}\left(x \right)\right)-b \left(\cos^{2}\left(x \right)\right)-a}{\left(\cos \left(x \right)+1\right)^{2}}}-4 a \cos \left(x \right)+4 b \cos \left(x \right)+4 \sqrt{-a +b}\, \sqrt{-\frac{a \left(\cos^{2}\left(x \right)\right)-b \left(\cos^{2}\left(x \right)\right)-a}{\left(\cos \left(x \right)+1\right)^{2}}}\right) b^{\frac{9}{2}}-4 \cos \left(x \right) \ln \left(4 \cos \left(x \right) \sqrt{-a +b}\, \sqrt{-\frac{a \left(\cos^{2}\left(x \right)\right)-b \left(\cos^{2}\left(x \right)\right)-a}{\left(\cos \left(x \right)+1\right)^{2}}}-4 a \cos \left(x \right)+4 b \cos \left(x \right)+4 \sqrt{-a +b}\, \sqrt{-\frac{a \left(\cos^{2}\left(x \right)\right)-b \left(\cos^{2}\left(x \right)\right)-a}{\left(\cos \left(x \right)+1\right)^{2}}}\right) b^{\frac{7}{2}} a +2 \cos \left(x \right) \sqrt{-a +b}\, b^{\frac{5}{2}} \sqrt{-\frac{a \left(\cos^{2}\left(x \right)\right)-b \left(\cos^{2}\left(x \right)\right)-a}{\left(\cos \left(x \right)+1\right)^{2}}}\, a +2 \cos \left(x \right) \ln \left(4 \cos \left(x \right) \sqrt{-a +b}\, \sqrt{-\frac{a \left(\cos^{2}\left(x \right)\right)-b \left(\cos^{2}\left(x \right)\right)-a}{\left(\cos \left(x \right)+1\right)^{2}}}-4 a \cos \left(x \right)+4 b \cos \left(x \right)+4 \sqrt{-a +b}\, \sqrt{-\frac{a \left(\cos^{2}\left(x \right)\right)-b \left(\cos^{2}\left(x \right)\right)-a}{\left(\cos \left(x \right)+1\right)^{2}}}\right) b^{\frac{5}{2}} a^{2}+2 a \sqrt{-\frac{a \left(\cos^{2}\left(x \right)\right)-b \left(\cos^{2}\left(x \right)\right)-a}{\left(\cos \left(x \right)+1\right)^{2}}}\, b^{\frac{5}{2}} \sqrt{-a +b}-\cos \left(x \right) \sqrt{-a +b}\, \ln \left(-\frac{4 \left(\cos \left(x \right) \sqrt{b}\, \sqrt{-\frac{a \left(\cos^{2}\left(x \right)\right)-b \left(\cos^{2}\left(x \right)\right)-a}{\left(\cos \left(x \right)+1\right)^{2}}}-a \cos \left(x \right)+b \cos \left(x \right)+\sqrt{-\frac{a \left(\cos^{2}\left(x \right)\right)-b \left(\cos^{2}\left(x \right)\right)-a}{\left(\cos \left(x \right)+1\right)^{2}}}\, \sqrt{b}+a \right)}{-1+\cos \left(x \right)}\right) b^{4}-3 \cos \left(x \right) \sqrt{-a +b}\, \ln \left(-\frac{2 \left(-1+\cos \left(x \right)\right) \left(\cos \left(x \right) \sqrt{b}\, \sqrt{-\frac{a \left(\cos^{2}\left(x \right)\right)-b \left(\cos^{2}\left(x \right)\right)-a}{\left(\cos \left(x \right)+1\right)^{2}}}+a \cos \left(x \right)-b \cos \left(x \right)+\sqrt{-\frac{a \left(\cos^{2}\left(x \right)\right)-b \left(\cos^{2}\left(x \right)\right)-a}{\left(\cos \left(x \right)+1\right)^{2}}}\, \sqrt{b}+a \right)}{\sin \left(x \right)^{2} \sqrt{b}}\right) a^{3} b +6 \cos \left(x \right) \sqrt{-a +b}\, \ln \left(-\frac{2 \left(-1+\cos \left(x \right)\right) \left(\cos \left(x \right) \sqrt{b}\, \sqrt{-\frac{a \left(\cos^{2}\left(x \right)\right)-b \left(\cos^{2}\left(x \right)\right)-a}{\left(\cos \left(x \right)+1\right)^{2}}}+a \cos \left(x \right)-b \cos \left(x \right)+\sqrt{-\frac{a \left(\cos^{2}\left(x \right)\right)-b \left(\cos^{2}\left(x \right)\right)-a}{\left(\cos \left(x \right)+1\right)^{2}}}\, \sqrt{b}+a \right)}{\sin \left(x \right)^{2} \sqrt{b}}\right) a^{2} b^{2}-3 \cos \left(x \right) \sqrt{-a +b}\, \ln \left(-\frac{2 \left(-1+\cos \left(x \right)\right) \left(\cos \left(x \right) \sqrt{b}\, \sqrt{-\frac{a \left(\cos^{2}\left(x \right)\right)-b \left(\cos^{2}\left(x \right)\right)-a}{\left(\cos \left(x \right)+1\right)^{2}}}+a \cos \left(x \right)-b \cos \left(x \right)+\sqrt{-\frac{a \left(\cos^{2}\left(x \right)\right)-b \left(\cos^{2}\left(x \right)\right)-a}{\left(\cos \left(x \right)+1\right)^{2}}}\, \sqrt{b}+a \right)}{\sin \left(x \right)^{2} \sqrt{b}}\right) a \,b^{3}+\cos \left(x \right) \sqrt{-a +b}\, \ln \left(-\frac{2 \left(-1+\cos \left(x \right)\right) \left(\cos \left(x \right) \sqrt{b}\, \sqrt{-\frac{a \left(\cos^{2}\left(x \right)\right)-b \left(\cos^{2}\left(x \right)\right)-a}{\left(\cos \left(x \right)+1\right)^{2}}}+a \cos \left(x \right)-b \cos \left(x \right)+\sqrt{-\frac{a \left(\cos^{2}\left(x \right)\right)-b \left(\cos^{2}\left(x \right)\right)-a}{\left(\cos \left(x \right)+1\right)^{2}}}\, \sqrt{b}+a \right)}{\sin \left(x \right)^{2} \sqrt{b}}\right) b^{4}+3 \cos \left(x \right) \sqrt{-a +b}\, \ln \left(-\frac{4 \left(-1+\cos \left(x \right)\right) \left(\cos \left(x \right) \sqrt{b}\, \sqrt{-\frac{a \left(\cos^{2}\left(x \right)\right)-b \left(\cos^{2}\left(x \right)\right)-a}{\left(\cos \left(x \right)+1\right)^{2}}}+a \cos \left(x \right)-b \cos \left(x \right)+\sqrt{-\frac{a \left(\cos^{2}\left(x \right)\right)-b \left(\cos^{2}\left(x \right)\right)-a}{\left(\cos \left(x \right)+1\right)^{2}}}\, \sqrt{b}+a \right)}{\sin \left(x \right)^{2} \sqrt{b}}\right) a^{3} b -6 \cos \left(x \right) \sqrt{-a +b}\, \ln \left(-\frac{4 \left(-1+\cos \left(x \right)\right) \left(\cos \left(x \right) \sqrt{b}\, \sqrt{-\frac{a \left(\cos^{2}\left(x \right)\right)-b \left(\cos^{2}\left(x \right)\right)-a}{\left(\cos \left(x \right)+1\right)^{2}}}+a \cos \left(x \right)-b \cos \left(x \right)+\sqrt{-\frac{a \left(\cos^{2}\left(x \right)\right)-b \left(\cos^{2}\left(x \right)\right)-a}{\left(\cos \left(x \right)+1\right)^{2}}}\, \sqrt{b}+a \right)}{\sin \left(x \right)^{2} \sqrt{b}}\right) a^{2} b^{2}+3 \cos \left(x \right) \sqrt{-a +b}\, \ln \left(-\frac{4 \left(-1+\cos \left(x \right)\right) \left(\cos \left(x \right) \sqrt{b}\, \sqrt{-\frac{a \left(\cos^{2}\left(x \right)\right)-b \left(\cos^{2}\left(x \right)\right)-a}{\left(\cos \left(x \right)+1\right)^{2}}}+a \cos \left(x \right)-b \cos \left(x \right)+\sqrt{-\frac{a \left(\cos^{2}\left(x \right)\right)-b \left(\cos^{2}\left(x \right)\right)-a}{\left(\cos \left(x \right)+1\right)^{2}}}\, \sqrt{b}+a \right)}{\sin \left(x \right)^{2} \sqrt{b}}\right) a \,b^{3}\right)}{2 \cos \left(x \right) \sin \left(x \right)^{3} \left(-\frac{a \left(\cos^{2}\left(x \right)\right)-b \left(\cos^{2}\left(x \right)\right)-a}{\left(\cos \left(x \right)+1\right)^{2}}\right)^{\frac{3}{2}} b^{\frac{5}{2}} \sqrt{-a +b}}"," ",0,"-1/2*((a*cos(x)^2-b*cos(x)^2-a)/(-1+cos(x)^2))^(3/2)*(-1+cos(x))^3*(2*cos(x)*ln(4*cos(x)*(-a+b)^(1/2)*(-(a*cos(x)^2-b*cos(x)^2-a)/(cos(x)+1)^2)^(1/2)-4*a*cos(x)+4*b*cos(x)+4*(-a+b)^(1/2)*(-(a*cos(x)^2-b*cos(x)^2-a)/(cos(x)+1)^2)^(1/2))*b^(9/2)-4*cos(x)*ln(4*cos(x)*(-a+b)^(1/2)*(-(a*cos(x)^2-b*cos(x)^2-a)/(cos(x)+1)^2)^(1/2)-4*a*cos(x)+4*b*cos(x)+4*(-a+b)^(1/2)*(-(a*cos(x)^2-b*cos(x)^2-a)/(cos(x)+1)^2)^(1/2))*b^(7/2)*a+2*cos(x)*(-a+b)^(1/2)*b^(5/2)*(-(a*cos(x)^2-b*cos(x)^2-a)/(cos(x)+1)^2)^(1/2)*a+2*cos(x)*ln(4*cos(x)*(-a+b)^(1/2)*(-(a*cos(x)^2-b*cos(x)^2-a)/(cos(x)+1)^2)^(1/2)-4*a*cos(x)+4*b*cos(x)+4*(-a+b)^(1/2)*(-(a*cos(x)^2-b*cos(x)^2-a)/(cos(x)+1)^2)^(1/2))*b^(5/2)*a^2+2*a*(-(a*cos(x)^2-b*cos(x)^2-a)/(cos(x)+1)^2)^(1/2)*b^(5/2)*(-a+b)^(1/2)-cos(x)*(-a+b)^(1/2)*ln(-4*(cos(x)*b^(1/2)*(-(a*cos(x)^2-b*cos(x)^2-a)/(cos(x)+1)^2)^(1/2)-a*cos(x)+b*cos(x)+(-(a*cos(x)^2-b*cos(x)^2-a)/(cos(x)+1)^2)^(1/2)*b^(1/2)+a)/(-1+cos(x)))*b^4-3*cos(x)*(-a+b)^(1/2)*ln(-2*(-1+cos(x))*(cos(x)*b^(1/2)*(-(a*cos(x)^2-b*cos(x)^2-a)/(cos(x)+1)^2)^(1/2)+a*cos(x)-b*cos(x)+(-(a*cos(x)^2-b*cos(x)^2-a)/(cos(x)+1)^2)^(1/2)*b^(1/2)+a)/sin(x)^2/b^(1/2))*a^3*b+6*cos(x)*(-a+b)^(1/2)*ln(-2*(-1+cos(x))*(cos(x)*b^(1/2)*(-(a*cos(x)^2-b*cos(x)^2-a)/(cos(x)+1)^2)^(1/2)+a*cos(x)-b*cos(x)+(-(a*cos(x)^2-b*cos(x)^2-a)/(cos(x)+1)^2)^(1/2)*b^(1/2)+a)/sin(x)^2/b^(1/2))*a^2*b^2-3*cos(x)*(-a+b)^(1/2)*ln(-2*(-1+cos(x))*(cos(x)*b^(1/2)*(-(a*cos(x)^2-b*cos(x)^2-a)/(cos(x)+1)^2)^(1/2)+a*cos(x)-b*cos(x)+(-(a*cos(x)^2-b*cos(x)^2-a)/(cos(x)+1)^2)^(1/2)*b^(1/2)+a)/sin(x)^2/b^(1/2))*a*b^3+cos(x)*(-a+b)^(1/2)*ln(-2*(-1+cos(x))*(cos(x)*b^(1/2)*(-(a*cos(x)^2-b*cos(x)^2-a)/(cos(x)+1)^2)^(1/2)+a*cos(x)-b*cos(x)+(-(a*cos(x)^2-b*cos(x)^2-a)/(cos(x)+1)^2)^(1/2)*b^(1/2)+a)/sin(x)^2/b^(1/2))*b^4+3*cos(x)*(-a+b)^(1/2)*ln(-4*(-1+cos(x))*(cos(x)*b^(1/2)*(-(a*cos(x)^2-b*cos(x)^2-a)/(cos(x)+1)^2)^(1/2)+a*cos(x)-b*cos(x)+(-(a*cos(x)^2-b*cos(x)^2-a)/(cos(x)+1)^2)^(1/2)*b^(1/2)+a)/sin(x)^2/b^(1/2))*a^3*b-6*cos(x)*(-a+b)^(1/2)*ln(-4*(-1+cos(x))*(cos(x)*b^(1/2)*(-(a*cos(x)^2-b*cos(x)^2-a)/(cos(x)+1)^2)^(1/2)+a*cos(x)-b*cos(x)+(-(a*cos(x)^2-b*cos(x)^2-a)/(cos(x)+1)^2)^(1/2)*b^(1/2)+a)/sin(x)^2/b^(1/2))*a^2*b^2+3*cos(x)*(-a+b)^(1/2)*ln(-4*(-1+cos(x))*(cos(x)*b^(1/2)*(-(a*cos(x)^2-b*cos(x)^2-a)/(cos(x)+1)^2)^(1/2)+a*cos(x)-b*cos(x)+(-(a*cos(x)^2-b*cos(x)^2-a)/(cos(x)+1)^2)^(1/2)*b^(1/2)+a)/sin(x)^2/b^(1/2))*a*b^3)/cos(x)/sin(x)^3/(-(a*cos(x)^2-b*cos(x)^2-a)/(cos(x)+1)^2)^(3/2)/b^(5/2)/(-a+b)^(1/2)","B"
31,1,462,149,0.455000," ","int((a+b*cot(d*x+c)^2)^(5/2),x)","-\frac{b^{2} \left(\cot^{3}\left(d x +c \right)\right) \sqrt{a +b \left(\cot^{2}\left(d x +c \right)\right)}}{4 d}-\frac{9 b a \cot \left(d x +c \right) \sqrt{a +b \left(\cot^{2}\left(d x +c \right)\right)}}{8 d}-\frac{15 \sqrt{b}\, a^{2} \ln \left(\cot \left(d x +c \right) \sqrt{b}+\sqrt{a +b \left(\cot^{2}\left(d x +c \right)\right)}\right)}{8 d}+\frac{b^{2} \cot \left(d x +c \right) \sqrt{a +b \left(\cot^{2}\left(d x +c \right)\right)}}{2 d}+\frac{5 b^{\frac{3}{2}} a \ln \left(\cot \left(d x +c \right) \sqrt{b}+\sqrt{a +b \left(\cot^{2}\left(d x +c \right)\right)}\right)}{2 d}-\frac{b^{\frac{5}{2}} \ln \left(\cot \left(d x +c \right) \sqrt{b}+\sqrt{a +b \left(\cot^{2}\left(d x +c \right)\right)}\right)}{d}+\frac{b \sqrt{b^{4} \left(a -b \right)}\, \arctan \left(\frac{\left(a -b \right) b^{2} \cot \left(d x +c \right)}{\sqrt{b^{4} \left(a -b \right)}\, \sqrt{a +b \left(\cot^{2}\left(d x +c \right)\right)}}\right)}{d \left(a -b \right)}-\frac{3 a \sqrt{b^{4} \left(a -b \right)}\, \arctan \left(\frac{\left(a -b \right) b^{2} \cot \left(d x +c \right)}{\sqrt{b^{4} \left(a -b \right)}\, \sqrt{a +b \left(\cot^{2}\left(d x +c \right)\right)}}\right)}{d \left(a -b \right)}+\frac{3 a^{2} \sqrt{b^{4} \left(a -b \right)}\, \arctan \left(\frac{\left(a -b \right) b^{2} \cot \left(d x +c \right)}{\sqrt{b^{4} \left(a -b \right)}\, \sqrt{a +b \left(\cot^{2}\left(d x +c \right)\right)}}\right)}{d b \left(a -b \right)}-\frac{a^{3} \sqrt{b^{4} \left(a -b \right)}\, \arctan \left(\frac{\left(a -b \right) b^{2} \cot \left(d x +c \right)}{\sqrt{b^{4} \left(a -b \right)}\, \sqrt{a +b \left(\cot^{2}\left(d x +c \right)\right)}}\right)}{d \,b^{2} \left(a -b \right)}"," ",0,"-1/4/d*b^2*cot(d*x+c)^3*(a+b*cot(d*x+c)^2)^(1/2)-9/8/d*b*a*cot(d*x+c)*(a+b*cot(d*x+c)^2)^(1/2)-15/8/d*b^(1/2)*a^2*ln(cot(d*x+c)*b^(1/2)+(a+b*cot(d*x+c)^2)^(1/2))+1/2/d*b^2*cot(d*x+c)*(a+b*cot(d*x+c)^2)^(1/2)+5/2/d*b^(3/2)*a*ln(cot(d*x+c)*b^(1/2)+(a+b*cot(d*x+c)^2)^(1/2))-1/d*b^(5/2)*ln(cot(d*x+c)*b^(1/2)+(a+b*cot(d*x+c)^2)^(1/2))+1/d*b*(b^4*(a-b))^(1/2)/(a-b)*arctan((a-b)*b^2/(b^4*(a-b))^(1/2)/(a+b*cot(d*x+c)^2)^(1/2)*cot(d*x+c))-3/d*a*(b^4*(a-b))^(1/2)/(a-b)*arctan((a-b)*b^2/(b^4*(a-b))^(1/2)/(a+b*cot(d*x+c)^2)^(1/2)*cot(d*x+c))+3/d*a^2/b*(b^4*(a-b))^(1/2)/(a-b)*arctan((a-b)*b^2/(b^4*(a-b))^(1/2)/(a+b*cot(d*x+c)^2)^(1/2)*cot(d*x+c))-1/d*a^3*(b^4*(a-b))^(1/2)/b^2/(a-b)*arctan((a-b)*b^2/(b^4*(a-b))^(1/2)/(a+b*cot(d*x+c)^2)^(1/2)*cot(d*x+c))","B"
32,1,298,108,0.379000," ","int((a+b*cot(d*x+c)^2)^(3/2),x)","-\frac{b \cot \left(d x +c \right) \sqrt{a +b \left(\cot^{2}\left(d x +c \right)\right)}}{2 d}-\frac{3 \sqrt{b}\, a \ln \left(\cot \left(d x +c \right) \sqrt{b}+\sqrt{a +b \left(\cot^{2}\left(d x +c \right)\right)}\right)}{2 d}+\frac{b^{\frac{3}{2}} \ln \left(\cot \left(d x +c \right) \sqrt{b}+\sqrt{a +b \left(\cot^{2}\left(d x +c \right)\right)}\right)}{d}-\frac{\sqrt{b^{4} \left(a -b \right)}\, \arctan \left(\frac{\left(a -b \right) b^{2} \cot \left(d x +c \right)}{\sqrt{b^{4} \left(a -b \right)}\, \sqrt{a +b \left(\cot^{2}\left(d x +c \right)\right)}}\right)}{d \left(a -b \right)}+\frac{2 a \sqrt{b^{4} \left(a -b \right)}\, \arctan \left(\frac{\left(a -b \right) b^{2} \cot \left(d x +c \right)}{\sqrt{b^{4} \left(a -b \right)}\, \sqrt{a +b \left(\cot^{2}\left(d x +c \right)\right)}}\right)}{d b \left(a -b \right)}-\frac{a^{2} \sqrt{b^{4} \left(a -b \right)}\, \arctan \left(\frac{\left(a -b \right) b^{2} \cot \left(d x +c \right)}{\sqrt{b^{4} \left(a -b \right)}\, \sqrt{a +b \left(\cot^{2}\left(d x +c \right)\right)}}\right)}{d \,b^{2} \left(a -b \right)}"," ",0,"-1/2*b*cot(d*x+c)*(a+b*cot(d*x+c)^2)^(1/2)/d-3/2/d*b^(1/2)*a*ln(cot(d*x+c)*b^(1/2)+(a+b*cot(d*x+c)^2)^(1/2))+1/d*b^(3/2)*ln(cot(d*x+c)*b^(1/2)+(a+b*cot(d*x+c)^2)^(1/2))-1/d*(b^4*(a-b))^(1/2)/(a-b)*arctan((a-b)*b^2/(b^4*(a-b))^(1/2)/(a+b*cot(d*x+c)^2)^(1/2)*cot(d*x+c))+2/d*a/b*(b^4*(a-b))^(1/2)/(a-b)*arctan((a-b)*b^2/(b^4*(a-b))^(1/2)/(a+b*cot(d*x+c)^2)^(1/2)*cot(d*x+c))-1/d*a^2*(b^4*(a-b))^(1/2)/b^2/(a-b)*arctan((a-b)*b^2/(b^4*(a-b))^(1/2)/(a+b*cot(d*x+c)^2)^(1/2)*cot(d*x+c))","B"
33,1,170,75,0.420000," ","int((a+b*cot(d*x+c)^2)^(1/2),x)","-\frac{\sqrt{b}\, \ln \left(\cot \left(d x +c \right) \sqrt{b}+\sqrt{a +b \left(\cot^{2}\left(d x +c \right)\right)}\right)}{d}+\frac{\sqrt{b^{4} \left(a -b \right)}\, \arctan \left(\frac{\left(a -b \right) b^{2} \cot \left(d x +c \right)}{\sqrt{b^{4} \left(a -b \right)}\, \sqrt{a +b \left(\cot^{2}\left(d x +c \right)\right)}}\right)}{d b \left(a -b \right)}-\frac{a \sqrt{b^{4} \left(a -b \right)}\, \arctan \left(\frac{\left(a -b \right) b^{2} \cot \left(d x +c \right)}{\sqrt{b^{4} \left(a -b \right)}\, \sqrt{a +b \left(\cot^{2}\left(d x +c \right)\right)}}\right)}{d \,b^{2} \left(a -b \right)}"," ",0,"-1/d*b^(1/2)*ln(cot(d*x+c)*b^(1/2)+(a+b*cot(d*x+c)^2)^(1/2))+1/d*(b^4*(a-b))^(1/2)/b/(a-b)*arctan((a-b)*b^2/(b^4*(a-b))^(1/2)/(a+b*cot(d*x+c)^2)^(1/2)*cot(d*x+c))-1/d*a*(b^4*(a-b))^(1/2)/b^2/(a-b)*arctan((a-b)*b^2/(b^4*(a-b))^(1/2)/(a+b*cot(d*x+c)^2)^(1/2)*cot(d*x+c))","B"
34,1,68,41,0.394000," ","int(1/(a+b*cot(d*x+c)^2)^(1/2),x)","-\frac{\sqrt{b^{4} \left(a -b \right)}\, \arctan \left(\frac{\left(a -b \right) b^{2} \cot \left(d x +c \right)}{\sqrt{b^{4} \left(a -b \right)}\, \sqrt{a +b \left(\cot^{2}\left(d x +c \right)\right)}}\right)}{d \,b^{2} \left(a -b \right)}"," ",0,"-1/d*(b^4*(a-b))^(1/2)/b^2/(a-b)*arctan((a-b)*b^2/(b^4*(a-b))^(1/2)/(a+b*cot(d*x+c)^2)^(1/2)*cot(d*x+c))","A"
35,1,104,77,0.355000," ","int(1/(a+b*cot(d*x+c)^2)^(3/2),x)","\frac{b \cot \left(d x +c \right)}{a \left(a -b \right) d \sqrt{a +b \left(\cot^{2}\left(d x +c \right)\right)}}-\frac{\sqrt{b^{4} \left(a -b \right)}\, \arctan \left(\frac{\left(a -b \right) b^{2} \cot \left(d x +c \right)}{\sqrt{b^{4} \left(a -b \right)}\, \sqrt{a +b \left(\cot^{2}\left(d x +c \right)\right)}}\right)}{d \left(a -b \right)^{2} b^{2}}"," ",0,"b*cot(d*x+c)/a/(a-b)/d/(a+b*cot(d*x+c)^2)^(1/2)-1/d/(a-b)^2*(b^4*(a-b))^(1/2)/b^2*arctan((a-b)*b^2/(b^4*(a-b))^(1/2)/(a+b*cot(d*x+c)^2)^(1/2)*cot(d*x+c))","A"
36,1,176,121,0.382000," ","int(1/(a+b*cot(d*x+c)^2)^(5/2),x)","\frac{b \cot \left(d x +c \right)}{d \left(a -b \right)^{2} a \sqrt{a +b \left(\cot^{2}\left(d x +c \right)\right)}}+\frac{b \cot \left(d x +c \right)}{3 a \left(a -b \right) d \left(a +b \left(\cot^{2}\left(d x +c \right)\right)\right)^{\frac{3}{2}}}+\frac{2 b \cot \left(d x +c \right)}{3 d \left(a -b \right) a^{2} \sqrt{a +b \left(\cot^{2}\left(d x +c \right)\right)}}-\frac{\sqrt{b^{4} \left(a -b \right)}\, \arctan \left(\frac{\left(a -b \right) b^{2} \cot \left(d x +c \right)}{\sqrt{b^{4} \left(a -b \right)}\, \sqrt{a +b \left(\cot^{2}\left(d x +c \right)\right)}}\right)}{d \left(a -b \right)^{3} b^{2}}"," ",0,"1/d*b/(a-b)^2*cot(d*x+c)/a/(a+b*cot(d*x+c)^2)^(1/2)+1/3*b*cot(d*x+c)/a/(a-b)/d/(a+b*cot(d*x+c)^2)^(3/2)+2/3/d*b/(a-b)/a^2*cot(d*x+c)/(a+b*cot(d*x+c)^2)^(1/2)-1/d/(a-b)^3*(b^4*(a-b))^(1/2)/b^2*arctan((a-b)*b^2/(b^4*(a-b))^(1/2)/(a+b*cot(d*x+c)^2)^(1/2)*cot(d*x+c))","A"
37,1,284,172,0.381000," ","int(1/(a+b*cot(d*x+c)^2)^(7/2),x)","\frac{b \cot \left(d x +c \right)}{5 a \left(a -b \right) d \left(a +b \left(\cot^{2}\left(d x +c \right)\right)\right)^{\frac{5}{2}}}+\frac{4 b \cot \left(d x +c \right)}{15 d \left(a -b \right) a^{2} \left(a +b \left(\cot^{2}\left(d x +c \right)\right)\right)^{\frac{3}{2}}}+\frac{8 b \cot \left(d x +c \right)}{15 d \left(a -b \right) a^{3} \sqrt{a +b \left(\cot^{2}\left(d x +c \right)\right)}}+\frac{b \cot \left(d x +c \right)}{d \left(a -b \right)^{3} a \sqrt{a +b \left(\cot^{2}\left(d x +c \right)\right)}}+\frac{b \cot \left(d x +c \right)}{3 d \left(a -b \right)^{2} a \left(a +b \left(\cot^{2}\left(d x +c \right)\right)\right)^{\frac{3}{2}}}+\frac{2 b \cot \left(d x +c \right)}{3 d \left(a -b \right)^{2} a^{2} \sqrt{a +b \left(\cot^{2}\left(d x +c \right)\right)}}-\frac{\sqrt{b^{4} \left(a -b \right)}\, \arctan \left(\frac{\left(a -b \right) b^{2} \cot \left(d x +c \right)}{\sqrt{b^{4} \left(a -b \right)}\, \sqrt{a +b \left(\cot^{2}\left(d x +c \right)\right)}}\right)}{d \left(a -b \right)^{4} b^{2}}"," ",0,"1/5*b*cot(d*x+c)/a/(a-b)/d/(a+b*cot(d*x+c)^2)^(5/2)+4/15/d*b/(a-b)/a^2*cot(d*x+c)/(a+b*cot(d*x+c)^2)^(3/2)+8/15/d*b/(a-b)/a^3*cot(d*x+c)/(a+b*cot(d*x+c)^2)^(1/2)+1/d*b/(a-b)^3*cot(d*x+c)/a/(a+b*cot(d*x+c)^2)^(1/2)+1/3/d*b/(a-b)^2*cot(d*x+c)/a/(a+b*cot(d*x+c)^2)^(3/2)+2/3/d*b/(a-b)^2/a^2*cot(d*x+c)/(a+b*cot(d*x+c)^2)^(1/2)-1/d/(a-b)^4*(b^4*(a-b))^(1/2)/b^2*arctan((a-b)*b^2/(b^4*(a-b))^(1/2)/(a+b*cot(d*x+c)^2)^(1/2)*cot(d*x+c))","A"
38,1,51,42,0.230000," ","int((1-cot(x)^2)^(3/2),x)","\frac{\cot \left(x \right) \sqrt{1-\left(\cot^{2}\left(x \right)\right)}}{2}+\frac{5 \arcsin \left(\cot \left(x \right)\right)}{2}+2 \sqrt{2}\, \arctan \left(\frac{\sqrt{2}\, \sqrt{1-\left(\cot^{2}\left(x \right)\right)}\, \cot \left(x \right)}{-1+\cot^{2}\left(x \right)}\right)"," ",0,"1/2*cot(x)*(1-cot(x)^2)^(1/2)+5/2*arcsin(cot(x))+2*2^(1/2)*arctan(2^(1/2)*(1-cot(x)^2)^(1/2)/(-1+cot(x)^2)*cot(x))","A"
39,1,34,26,0.234000," ","int((1-cot(x)^2)^(1/2),x)","\arcsin \left(\cot \left(x \right)\right)+\sqrt{2}\, \arctan \left(\frac{\sqrt{2}\, \sqrt{1-\left(\cot^{2}\left(x \right)\right)}\, \cot \left(x \right)}{-1+\cot^{2}\left(x \right)}\right)"," ",0,"arcsin(cot(x))+2^(1/2)*arctan(2^(1/2)*(1-cot(x)^2)^(1/2)/(-1+cot(x)^2)*cot(x))","A"
40,1,31,22,0.258000," ","int(1/(1-cot(x)^2)^(1/2),x)","\frac{\sqrt{2}\, \arctan \left(\frac{\sqrt{2}\, \sqrt{1-\left(\cot^{2}\left(x \right)\right)}\, \cot \left(x \right)}{-1+\cot^{2}\left(x \right)}\right)}{2}"," ",0,"1/2*2^(1/2)*arctan(2^(1/2)*(1-cot(x)^2)^(1/2)/(-1+cot(x)^2)*cot(x))","A"
41,1,48,47,0.216000," ","int((-1+cot(x)^2)^(3/2),x)","-\frac{\cot \left(x \right) \sqrt{-1+\cot^{2}\left(x \right)}}{2}+\frac{5 \ln \left(\cot \left(x \right)+\sqrt{-1+\cot^{2}\left(x \right)}\right)}{2}-2 \arctanh \left(\frac{\cot \left(x \right) \sqrt{2}}{\sqrt{-1+\cot^{2}\left(x \right)}}\right) \sqrt{2}"," ",0,"-1/2*cot(x)*(-1+cot(x)^2)^(1/2)+5/2*ln(cot(x)+(-1+cot(x)^2)^(1/2))-2*arctanh(cot(x)*2^(1/2)/(-1+cot(x)^2)^(1/2))*2^(1/2)","A"
42,1,35,34,0.234000," ","int((-1+cot(x)^2)^(1/2),x)","-\ln \left(\cot \left(x \right)+\sqrt{-1+\cot^{2}\left(x \right)}\right)+\arctanh \left(\frac{\cot \left(x \right) \sqrt{2}}{\sqrt{-1+\cot^{2}\left(x \right)}}\right) \sqrt{2}"," ",0,"-ln(cot(x)+(-1+cot(x)^2)^(1/2))+arctanh(cot(x)*2^(1/2)/(-1+cot(x)^2)^(1/2))*2^(1/2)","A"
43,1,21,20,0.247000," ","int(1/(-1+cot(x)^2)^(1/2),x)","-\frac{\arctanh \left(\frac{\cot \left(x \right) \sqrt{2}}{\sqrt{-1+\cot^{2}\left(x \right)}}\right) \sqrt{2}}{2}"," ",0,"-1/2*arctanh(cot(x)*2^(1/2)/(-1+cot(x)^2)^(1/2))*2^(1/2)","A"
44,1,44,44,0.211000," ","int(cot(x)^3/(a+b*cot(x)^2)^(1/2),x)","-\frac{\sqrt{a +b \left(\cot^{2}\left(x \right)\right)}}{b}+\frac{\arctan \left(\frac{\sqrt{a +b \left(\cot^{2}\left(x \right)\right)}}{\sqrt{-a +b}}\right)}{\sqrt{-a +b}}"," ",0,"-(a+b*cot(x)^2)^(1/2)/b+1/(-a+b)^(1/2)*arctan((a+b*cot(x)^2)^(1/2)/(-a+b)^(1/2))","A"
45,1,80,52,0.192000," ","int(cot(x)^2/(a+b*cot(x)^2)^(1/2),x)","-\frac{\ln \left(\cot \left(x \right) \sqrt{b}+\sqrt{a +b \left(\cot^{2}\left(x \right)\right)}\right)}{\sqrt{b}}+\frac{\sqrt{b^{4} \left(a -b \right)}\, \arctan \left(\frac{\left(a -b \right) b^{2} \cot \left(x \right)}{\sqrt{b^{4} \left(a -b \right)}\, \sqrt{a +b \left(\cot^{2}\left(x \right)\right)}}\right)}{b^{2} \left(a -b \right)}"," ",0,"-ln(cot(x)*b^(1/2)+(a+b*cot(x)^2)^(1/2))/b^(1/2)+(b^4*(a-b))^(1/2)/b^2/(a-b)*arctan((a-b)*b^2/(b^4*(a-b))^(1/2)/(a+b*cot(x)^2)^(1/2)*cot(x))","A"
46,1,29,27,0.151000," ","int(cot(x)/(a+b*cot(x)^2)^(1/2),x)","-\frac{\arctan \left(\frac{\sqrt{a +b \left(\cot^{2}\left(x \right)\right)}}{\sqrt{-a +b}}\right)}{\sqrt{-a +b}}"," ",0,"-1/(-a+b)^(1/2)*arctan((a+b*cot(x)^2)^(1/2)/(-a+b)^(1/2))","A"
47,1,376,48,0.749000," ","int(tan(x)/(a+b*cot(x)^2)^(1/2),x)","\frac{2 \sqrt{2}\, \sqrt{\frac{\cos \left(x \right) \sqrt{a}\, \sqrt{a -b}-\sqrt{a}\, \sqrt{a -b}-a \cos \left(x \right)+b \cos \left(x \right)+a}{\left(\cos \left(x \right)+1\right) b}}\, \sqrt{-\frac{2 \left(\cos \left(x \right) \sqrt{a}\, \sqrt{a -b}-\sqrt{a}\, \sqrt{a -b}+a \cos \left(x \right)-b \cos \left(x \right)-a \right)}{\left(\cos \left(x \right)+1\right) b}}\, \left(\EllipticPi \left(\frac{\left(-1+\cos \left(x \right)\right) \sqrt{\frac{2 \sqrt{a}\, \sqrt{a -b}-2 a +b}{b}}}{\sin \left(x \right)}, \frac{b}{2 \sqrt{a}\, \sqrt{a -b}-2 a +b}, \frac{\sqrt{-\frac{2 \sqrt{a}\, \sqrt{a -b}+2 a -b}{b}}}{\sqrt{\frac{2 \sqrt{a}\, \sqrt{a -b}-2 a +b}{b}}}\right)-\EllipticPi \left(\frac{\left(-1+\cos \left(x \right)\right) \sqrt{\frac{2 \sqrt{a}\, \sqrt{a -b}-2 a +b}{b}}}{\sin \left(x \right)}, -\frac{b}{2 \sqrt{a}\, \sqrt{a -b}-2 a +b}, \frac{\sqrt{-\frac{2 \sqrt{a}\, \sqrt{a -b}+2 a -b}{b}}}{\sqrt{\frac{2 \sqrt{a}\, \sqrt{a -b}-2 a +b}{b}}}\right)\right) \sin \left(x \right)}{\sqrt{\frac{a \left(\cos^{2}\left(x \right)\right)-b \left(\cos^{2}\left(x \right)\right)-a}{-1+\cos^{2}\left(x \right)}}\, \left(-1+\cos \left(x \right)\right) \sqrt{\frac{2 \sqrt{a}\, \sqrt{a -b}-2 a +b}{b}}}"," ",0,"2*2^(1/2)*((cos(x)*a^(1/2)*(a-b)^(1/2)-a^(1/2)*(a-b)^(1/2)-a*cos(x)+b*cos(x)+a)/(cos(x)+1)/b)^(1/2)*(-2*(cos(x)*a^(1/2)*(a-b)^(1/2)-a^(1/2)*(a-b)^(1/2)+a*cos(x)-b*cos(x)-a)/(cos(x)+1)/b)^(1/2)*(EllipticPi((-1+cos(x))*((2*a^(1/2)*(a-b)^(1/2)-2*a+b)/b)^(1/2)/sin(x),1/(2*a^(1/2)*(a-b)^(1/2)-2*a+b)*b,(-(2*a^(1/2)*(a-b)^(1/2)+2*a-b)/b)^(1/2)/((2*a^(1/2)*(a-b)^(1/2)-2*a+b)/b)^(1/2))-EllipticPi((-1+cos(x))*((2*a^(1/2)*(a-b)^(1/2)-2*a+b)/b)^(1/2)/sin(x),-1/(2*a^(1/2)*(a-b)^(1/2)-2*a+b)*b,(-(2*a^(1/2)*(a-b)^(1/2)+2*a-b)/b)^(1/2)/((2*a^(1/2)*(a-b)^(1/2)-2*a+b)/b)^(1/2)))*sin(x)/((a*cos(x)^2-b*cos(x)^2-a)/(-1+cos(x)^2))^(1/2)/(-1+cos(x))/((2*a^(1/2)*(a-b)^(1/2)-2*a+b)/b)^(1/2)","C"
48,1,328,46,0.819000," ","int(tan(x)^2/(a+b*cot(x)^2)^(1/2),x)","-\frac{\sin \left(x \right) \left(\left(\cos^{2}\left(x \right)\right) \sqrt{-\frac{a \left(\cos^{2}\left(x \right)\right)-b \left(\cos^{2}\left(x \right)\right)-a}{\left(\cos \left(x \right)+1\right)^{2}}}\, \ln \left(4 \cos \left(x \right) \sqrt{-a +b}\, \sqrt{-\frac{a \left(\cos^{2}\left(x \right)\right)-b \left(\cos^{2}\left(x \right)\right)-a}{\left(\cos \left(x \right)+1\right)^{2}}}-4 a \cos \left(x \right)+4 b \cos \left(x \right)+4 \sqrt{-a +b}\, \sqrt{-\frac{a \left(\cos^{2}\left(x \right)\right)-b \left(\cos^{2}\left(x \right)\right)-a}{\left(\cos \left(x \right)+1\right)^{2}}}\right) a -\left(\cos^{2}\left(x \right)\right) \sqrt{-a +b}\, a +\left(\cos^{2}\left(x \right)\right) \sqrt{-a +b}\, b +\cos \left(x \right) \sqrt{-\frac{a \left(\cos^{2}\left(x \right)\right)-b \left(\cos^{2}\left(x \right)\right)-a}{\left(\cos \left(x \right)+1\right)^{2}}}\, \ln \left(4 \cos \left(x \right) \sqrt{-a +b}\, \sqrt{-\frac{a \left(\cos^{2}\left(x \right)\right)-b \left(\cos^{2}\left(x \right)\right)-a}{\left(\cos \left(x \right)+1\right)^{2}}}-4 a \cos \left(x \right)+4 b \cos \left(x \right)+4 \sqrt{-a +b}\, \sqrt{-\frac{a \left(\cos^{2}\left(x \right)\right)-b \left(\cos^{2}\left(x \right)\right)-a}{\left(\cos \left(x \right)+1\right)^{2}}}\right) a +\sqrt{-a +b}\, a \right)}{\cos \left(x \right) \sqrt{\frac{a \left(\cos^{2}\left(x \right)\right)-b \left(\cos^{2}\left(x \right)\right)-a}{-1+\cos^{2}\left(x \right)}}\, \left(-1+\cos^{2}\left(x \right)\right) a \sqrt{-a +b}}"," ",0,"-sin(x)*(cos(x)^2*(-(a*cos(x)^2-b*cos(x)^2-a)/(cos(x)+1)^2)^(1/2)*ln(4*cos(x)*(-a+b)^(1/2)*(-(a*cos(x)^2-b*cos(x)^2-a)/(cos(x)+1)^2)^(1/2)-4*a*cos(x)+4*b*cos(x)+4*(-a+b)^(1/2)*(-(a*cos(x)^2-b*cos(x)^2-a)/(cos(x)+1)^2)^(1/2))*a-cos(x)^2*(-a+b)^(1/2)*a+cos(x)^2*(-a+b)^(1/2)*b+cos(x)*(-(a*cos(x)^2-b*cos(x)^2-a)/(cos(x)+1)^2)^(1/2)*ln(4*cos(x)*(-a+b)^(1/2)*(-(a*cos(x)^2-b*cos(x)^2-a)/(cos(x)+1)^2)^(1/2)-4*a*cos(x)+4*b*cos(x)+4*(-a+b)^(1/2)*(-(a*cos(x)^2-b*cos(x)^2-a)/(cos(x)+1)^2)^(1/2))*a+(-a+b)^(1/2)*a)/cos(x)/((a*cos(x)^2-b*cos(x)^2-a)/(-1+cos(x)^2))^(1/2)/(-1+cos(x)^2)/a/(-a+b)^(1/2)","B"
49,1,68,51,0.166000," ","int(cot(x)^3/(a+b*cot(x)^2)^(3/2),x)","\frac{1}{b \sqrt{a +b \left(\cot^{2}\left(x \right)\right)}}+\frac{1}{\left(a -b \right) \sqrt{a +b \left(\cot^{2}\left(x \right)\right)}}+\frac{\arctan \left(\frac{\sqrt{a +b \left(\cot^{2}\left(x \right)\right)}}{\sqrt{-a +b}}\right)}{\left(a -b \right) \sqrt{-a +b}}"," ",0,"1/b/(a+b*cot(x)^2)^(1/2)+1/(a-b)/(a+b*cot(x)^2)^(1/2)+1/(a-b)/(-a+b)^(1/2)*arctan((a+b*cot(x)^2)^(1/2)/(-a+b)^(1/2))","A"
50,1,99,51,0.178000," ","int(cot(x)^2/(a+b*cot(x)^2)^(3/2),x)","-\frac{\cot \left(x \right)}{a \sqrt{a +b \left(\cot^{2}\left(x \right)\right)}}-\frac{b \cot \left(x \right)}{\left(a -b \right) a \sqrt{a +b \left(\cot^{2}\left(x \right)\right)}}+\frac{\sqrt{b^{4} \left(a -b \right)}\, \arctan \left(\frac{\left(a -b \right) b^{2} \cot \left(x \right)}{\sqrt{b^{4} \left(a -b \right)}\, \sqrt{a +b \left(\cot^{2}\left(x \right)\right)}}\right)}{\left(a -b \right)^{2} b^{2}}"," ",0,"-cot(x)/a/(a+b*cot(x)^2)^(1/2)-b/(a-b)*cot(x)/a/(a+b*cot(x)^2)^(1/2)+1/(a-b)^2*(b^4*(a-b))^(1/2)/b^2*arctan((a-b)*b^2/(b^4*(a-b))^(1/2)/(a+b*cot(x)^2)^(1/2)*cot(x))","A"
51,1,56,47,0.141000," ","int(cot(x)/(a+b*cot(x)^2)^(3/2),x)","-\frac{1}{\left(a -b \right) \sqrt{a +b \left(\cot^{2}\left(x \right)\right)}}-\frac{\arctan \left(\frac{\sqrt{a +b \left(\cot^{2}\left(x \right)\right)}}{\sqrt{-a +b}}\right)}{\left(a -b \right) \sqrt{-a +b}}"," ",0,"-1/(a-b)/(a+b*cot(x)^2)^(1/2)-1/(a-b)/(-a+b)^(1/2)*arctan((a+b*cot(x)^2)^(1/2)/(-a+b)^(1/2))","A"
52,1,962,70,0.898000," ","int(tan(x)/(a+b*cot(x)^2)^(3/2),x)","-\frac{\left(a \left(\cos^{2}\left(x \right)\right)-b \left(\cos^{2}\left(x \right)\right)-a \right) \left(\sqrt{2}\, \sqrt{\frac{\cos \left(x \right) \sqrt{a}\, \sqrt{a -b}-\sqrt{a}\, \sqrt{a -b}-a \cos \left(x \right)+b \cos \left(x \right)+a}{\left(\cos \left(x \right)+1\right) b}}\, \sqrt{-\frac{2 \left(\cos \left(x \right) \sqrt{a}\, \sqrt{a -b}-\sqrt{a}\, \sqrt{a -b}+a \cos \left(x \right)-b \cos \left(x \right)-a \right)}{\left(\cos \left(x \right)+1\right) b}}\, \EllipticF \left(\frac{\left(-1+\cos \left(x \right)\right) \sqrt{\frac{2 \sqrt{a}\, \sqrt{a -b}-2 a +b}{b}}}{\sin \left(x \right)}, \sqrt{\frac{8 a^{\frac{3}{2}} \sqrt{a -b}-4 \sqrt{a}\, \sqrt{a -b}\, b +8 a^{2}-8 a b +b^{2}}{b^{2}}}\right) b \sin \left(x \right)+2 \sqrt{2}\, \sqrt{\frac{\cos \left(x \right) \sqrt{a}\, \sqrt{a -b}-\sqrt{a}\, \sqrt{a -b}-a \cos \left(x \right)+b \cos \left(x \right)+a}{\left(\cos \left(x \right)+1\right) b}}\, \sqrt{-\frac{2 \left(\cos \left(x \right) \sqrt{a}\, \sqrt{a -b}-\sqrt{a}\, \sqrt{a -b}+a \cos \left(x \right)-b \cos \left(x \right)-a \right)}{\left(\cos \left(x \right)+1\right) b}}\, \EllipticPi \left(\frac{\left(-1+\cos \left(x \right)\right) \sqrt{\frac{2 \sqrt{a}\, \sqrt{a -b}-2 a +b}{b}}}{\sin \left(x \right)}, \frac{b}{2 \sqrt{a}\, \sqrt{a -b}-2 a +b}, \frac{\sqrt{-\frac{2 \sqrt{a}\, \sqrt{a -b}+2 a -b}{b}}}{\sqrt{\frac{2 \sqrt{a}\, \sqrt{a -b}-2 a +b}{b}}}\right) a \sin \left(x \right)-2 \sqrt{2}\, \sqrt{\frac{\cos \left(x \right) \sqrt{a}\, \sqrt{a -b}-\sqrt{a}\, \sqrt{a -b}-a \cos \left(x \right)+b \cos \left(x \right)+a}{\left(\cos \left(x \right)+1\right) b}}\, \sqrt{-\frac{2 \left(\cos \left(x \right) \sqrt{a}\, \sqrt{a -b}-\sqrt{a}\, \sqrt{a -b}+a \cos \left(x \right)-b \cos \left(x \right)-a \right)}{\left(\cos \left(x \right)+1\right) b}}\, \EllipticPi \left(\frac{\left(-1+\cos \left(x \right)\right) \sqrt{\frac{2 \sqrt{a}\, \sqrt{a -b}-2 a +b}{b}}}{\sin \left(x \right)}, \frac{b}{2 \sqrt{a}\, \sqrt{a -b}-2 a +b}, \frac{\sqrt{-\frac{2 \sqrt{a}\, \sqrt{a -b}+2 a -b}{b}}}{\sqrt{\frac{2 \sqrt{a}\, \sqrt{a -b}-2 a +b}{b}}}\right) b \sin \left(x \right)-2 \sqrt{2}\, \sqrt{\frac{\cos \left(x \right) \sqrt{a}\, \sqrt{a -b}-\sqrt{a}\, \sqrt{a -b}-a \cos \left(x \right)+b \cos \left(x \right)+a}{\left(\cos \left(x \right)+1\right) b}}\, \sqrt{-\frac{2 \left(\cos \left(x \right) \sqrt{a}\, \sqrt{a -b}-\sqrt{a}\, \sqrt{a -b}+a \cos \left(x \right)-b \cos \left(x \right)-a \right)}{\left(\cos \left(x \right)+1\right) b}}\, \EllipticPi \left(\frac{\left(-1+\cos \left(x \right)\right) \sqrt{\frac{2 \sqrt{a}\, \sqrt{a -b}-2 a +b}{b}}}{\sin \left(x \right)}, -\frac{b}{2 \sqrt{a}\, \sqrt{a -b}-2 a +b}, \frac{\sqrt{-\frac{2 \sqrt{a}\, \sqrt{a -b}+2 a -b}{b}}}{\sqrt{\frac{2 \sqrt{a}\, \sqrt{a -b}-2 a +b}{b}}}\right) a \sin \left(x \right)+\sqrt{\frac{2 \sqrt{a}\, \sqrt{a -b}-2 a +b}{b}}\, b \cos \left(x \right)-\sqrt{\frac{2 \sqrt{a}\, \sqrt{a -b}-2 a +b}{b}}\, b \right)}{\left(-1+\cos \left(x \right)\right) \sin \left(x \right)^{2} \left(\frac{a \left(\cos^{2}\left(x \right)\right)-b \left(\cos^{2}\left(x \right)\right)-a}{-1+\cos^{2}\left(x \right)}\right)^{\frac{3}{2}} \sqrt{\frac{2 \sqrt{a}\, \sqrt{a -b}-2 a +b}{b}}\, \left(a -b \right) a}"," ",0,"-(a*cos(x)^2-b*cos(x)^2-a)*(2^(1/2)*((cos(x)*a^(1/2)*(a-b)^(1/2)-a^(1/2)*(a-b)^(1/2)-a*cos(x)+b*cos(x)+a)/(cos(x)+1)/b)^(1/2)*(-2*(cos(x)*a^(1/2)*(a-b)^(1/2)-a^(1/2)*(a-b)^(1/2)+a*cos(x)-b*cos(x)-a)/(cos(x)+1)/b)^(1/2)*EllipticF((-1+cos(x))*((2*a^(1/2)*(a-b)^(1/2)-2*a+b)/b)^(1/2)/sin(x),((8*a^(3/2)*(a-b)^(1/2)-4*a^(1/2)*(a-b)^(1/2)*b+8*a^2-8*a*b+b^2)/b^2)^(1/2))*b*sin(x)+2*2^(1/2)*((cos(x)*a^(1/2)*(a-b)^(1/2)-a^(1/2)*(a-b)^(1/2)-a*cos(x)+b*cos(x)+a)/(cos(x)+1)/b)^(1/2)*(-2*(cos(x)*a^(1/2)*(a-b)^(1/2)-a^(1/2)*(a-b)^(1/2)+a*cos(x)-b*cos(x)-a)/(cos(x)+1)/b)^(1/2)*EllipticPi((-1+cos(x))*((2*a^(1/2)*(a-b)^(1/2)-2*a+b)/b)^(1/2)/sin(x),1/(2*a^(1/2)*(a-b)^(1/2)-2*a+b)*b,(-(2*a^(1/2)*(a-b)^(1/2)+2*a-b)/b)^(1/2)/((2*a^(1/2)*(a-b)^(1/2)-2*a+b)/b)^(1/2))*a*sin(x)-2*2^(1/2)*((cos(x)*a^(1/2)*(a-b)^(1/2)-a^(1/2)*(a-b)^(1/2)-a*cos(x)+b*cos(x)+a)/(cos(x)+1)/b)^(1/2)*(-2*(cos(x)*a^(1/2)*(a-b)^(1/2)-a^(1/2)*(a-b)^(1/2)+a*cos(x)-b*cos(x)-a)/(cos(x)+1)/b)^(1/2)*EllipticPi((-1+cos(x))*((2*a^(1/2)*(a-b)^(1/2)-2*a+b)/b)^(1/2)/sin(x),1/(2*a^(1/2)*(a-b)^(1/2)-2*a+b)*b,(-(2*a^(1/2)*(a-b)^(1/2)+2*a-b)/b)^(1/2)/((2*a^(1/2)*(a-b)^(1/2)-2*a+b)/b)^(1/2))*b*sin(x)-2*2^(1/2)*((cos(x)*a^(1/2)*(a-b)^(1/2)-a^(1/2)*(a-b)^(1/2)-a*cos(x)+b*cos(x)+a)/(cos(x)+1)/b)^(1/2)*(-2*(cos(x)*a^(1/2)*(a-b)^(1/2)-a^(1/2)*(a-b)^(1/2)+a*cos(x)-b*cos(x)-a)/(cos(x)+1)/b)^(1/2)*EllipticPi((-1+cos(x))*((2*a^(1/2)*(a-b)^(1/2)-2*a+b)/b)^(1/2)/sin(x),-1/(2*a^(1/2)*(a-b)^(1/2)-2*a+b)*b,(-(2*a^(1/2)*(a-b)^(1/2)+2*a-b)/b)^(1/2)/((2*a^(1/2)*(a-b)^(1/2)-2*a+b)/b)^(1/2))*a*sin(x)+((2*a^(1/2)*(a-b)^(1/2)-2*a+b)/b)^(1/2)*b*cos(x)-((2*a^(1/2)*(a-b)^(1/2)-2*a+b)/b)^(1/2)*b)/(-1+cos(x))/sin(x)^2/((a*cos(x)^2-b*cos(x)^2-a)/(-1+cos(x)^2))^(3/2)/((2*a^(1/2)*(a-b)^(1/2)-2*a+b)/b)^(1/2)/(a-b)/a","C"
53,1,421,82,0.955000," ","int(tan(x)^2/(a+b*cot(x)^2)^(3/2),x)","\frac{\left(-1+\cos \left(x \right)\right)^{2} \left(\cos \left(x \right)+1\right)^{2} \left(a \left(\cos^{2}\left(x \right)\right)-b \left(\cos^{2}\left(x \right)\right)-a \right) \left(-\left(\cos^{2}\left(x \right)\right) \sqrt{-\frac{a \left(\cos^{2}\left(x \right)\right)-b \left(\cos^{2}\left(x \right)\right)-a}{\left(\cos \left(x \right)+1\right)^{2}}}\, \ln \left(4 \cos \left(x \right) \sqrt{-a +b}\, \sqrt{-\frac{a \left(\cos^{2}\left(x \right)\right)-b \left(\cos^{2}\left(x \right)\right)-a}{\left(\cos \left(x \right)+1\right)^{2}}}-4 a \cos \left(x \right)+4 b \cos \left(x \right)+4 \sqrt{-a +b}\, \sqrt{-\frac{a \left(\cos^{2}\left(x \right)\right)-b \left(\cos^{2}\left(x \right)\right)-a}{\left(\cos \left(x \right)+1\right)^{2}}}\right) a^{2}+\left(\cos^{2}\left(x \right)\right) \sqrt{-a +b}\, a^{2}-2 \left(\cos^{2}\left(x \right)\right) \sqrt{-a +b}\, a b +2 \left(\cos^{2}\left(x \right)\right) \sqrt{-a +b}\, b^{2}-\cos \left(x \right) \sqrt{-\frac{a \left(\cos^{2}\left(x \right)\right)-b \left(\cos^{2}\left(x \right)\right)-a}{\left(\cos \left(x \right)+1\right)^{2}}}\, \ln \left(4 \cos \left(x \right) \sqrt{-a +b}\, \sqrt{-\frac{a \left(\cos^{2}\left(x \right)\right)-b \left(\cos^{2}\left(x \right)\right)-a}{\left(\cos \left(x \right)+1\right)^{2}}}-4 a \cos \left(x \right)+4 b \cos \left(x \right)+4 \sqrt{-a +b}\, \sqrt{-\frac{a \left(\cos^{2}\left(x \right)\right)-b \left(\cos^{2}\left(x \right)\right)-a}{\left(\cos \left(x \right)+1\right)^{2}}}\right) a^{2}-\sqrt{-a +b}\, a^{2}+\sqrt{-a +b}\, a b \right) b}{\cos \left(x \right) \left(\frac{a \left(\cos^{2}\left(x \right)\right)-b \left(\cos^{2}\left(x \right)\right)-a}{-1+\cos^{2}\left(x \right)}\right)^{\frac{3}{2}} \sin \left(x \right)^{7} \sqrt{-a +b}\, \left(\sqrt{a \left(a -b \right)}+a -b \right) \left(\sqrt{a \left(a -b \right)}-a +b \right) a^{2}}"," ",0,"(-1+cos(x))^2*(cos(x)+1)^2*(a*cos(x)^2-b*cos(x)^2-a)*(-cos(x)^2*(-(a*cos(x)^2-b*cos(x)^2-a)/(cos(x)+1)^2)^(1/2)*ln(4*cos(x)*(-a+b)^(1/2)*(-(a*cos(x)^2-b*cos(x)^2-a)/(cos(x)+1)^2)^(1/2)-4*a*cos(x)+4*b*cos(x)+4*(-a+b)^(1/2)*(-(a*cos(x)^2-b*cos(x)^2-a)/(cos(x)+1)^2)^(1/2))*a^2+cos(x)^2*(-a+b)^(1/2)*a^2-2*cos(x)^2*(-a+b)^(1/2)*a*b+2*cos(x)^2*(-a+b)^(1/2)*b^2-cos(x)*(-(a*cos(x)^2-b*cos(x)^2-a)/(cos(x)+1)^2)^(1/2)*ln(4*cos(x)*(-a+b)^(1/2)*(-(a*cos(x)^2-b*cos(x)^2-a)/(cos(x)+1)^2)^(1/2)-4*a*cos(x)+4*b*cos(x)+4*(-a+b)^(1/2)*(-(a*cos(x)^2-b*cos(x)^2-a)/(cos(x)+1)^2)^(1/2))*a^2-(-a+b)^(1/2)*a^2+(-a+b)^(1/2)*a*b)*b/cos(x)/((a*cos(x)^2-b*cos(x)^2-a)/(-1+cos(x)^2))^(3/2)/sin(x)^7/(-a+b)^(1/2)/((a*(a-b))^(1/2)+a-b)/((a*(a-b))^(1/2)-a+b)/a^2","B"
54,1,88,70,0.173000," ","int(cot(x)^3/(a+b*cot(x)^2)^(5/2),x)","\frac{1}{3 b \left(a +b \left(\cot^{2}\left(x \right)\right)\right)^{\frac{3}{2}}}+\frac{1}{\left(a -b \right)^{2} \sqrt{a +b \left(\cot^{2}\left(x \right)\right)}}+\frac{1}{3 \left(a -b \right) \left(a +b \left(\cot^{2}\left(x \right)\right)\right)^{\frac{3}{2}}}+\frac{\arctan \left(\frac{\sqrt{a +b \left(\cot^{2}\left(x \right)\right)}}{\sqrt{-a +b}}\right)}{\left(a -b \right)^{2} \sqrt{-a +b}}"," ",0,"1/3/b/(a+b*cot(x)^2)^(3/2)+1/(a-b)^2/(a+b*cot(x)^2)^(1/2)+1/3/(a-b)/(a+b*cot(x)^2)^(3/2)+1/(a-b)^2/(-a+b)^(1/2)*arctan((a+b*cot(x)^2)^(1/2)/(-a+b)^(1/2))","A"
55,1,166,80,0.199000," ","int(cot(x)^2/(a+b*cot(x)^2)^(5/2),x)","-\frac{\cot \left(x \right)}{3 a \left(a +b \left(\cot^{2}\left(x \right)\right)\right)^{\frac{3}{2}}}-\frac{2 \cot \left(x \right)}{3 a^{2} \sqrt{a +b \left(\cot^{2}\left(x \right)\right)}}-\frac{b \cot \left(x \right)}{\left(a -b \right)^{2} a \sqrt{a +b \left(\cot^{2}\left(x \right)\right)}}-\frac{b \cot \left(x \right)}{3 \left(a -b \right) a \left(a +b \left(\cot^{2}\left(x \right)\right)\right)^{\frac{3}{2}}}-\frac{2 b \cot \left(x \right)}{3 \left(a -b \right) a^{2} \sqrt{a +b \left(\cot^{2}\left(x \right)\right)}}+\frac{\sqrt{b^{4} \left(a -b \right)}\, \arctan \left(\frac{\left(a -b \right) b^{2} \cot \left(x \right)}{\sqrt{b^{4} \left(a -b \right)}\, \sqrt{a +b \left(\cot^{2}\left(x \right)\right)}}\right)}{\left(a -b \right)^{3} b^{2}}"," ",0,"-1/3*cot(x)/a/(a+b*cot(x)^2)^(3/2)-2/3/a^2*cot(x)/(a+b*cot(x)^2)^(1/2)-b/(a-b)^2*cot(x)/a/(a+b*cot(x)^2)^(1/2)-1/3*b/(a-b)*cot(x)/a/(a+b*cot(x)^2)^(3/2)-2/3*b/(a-b)/a^2*cot(x)/(a+b*cot(x)^2)^(1/2)+1/(a-b)^3*(b^4*(a-b))^(1/2)/b^2*arctan((a-b)*b^2/(b^4*(a-b))^(1/2)/(a+b*cot(x)^2)^(1/2)*cot(x))","B"
56,1,75,66,0.138000," ","int(cot(x)/(a+b*cot(x)^2)^(5/2),x)","-\frac{1}{\left(a -b \right)^{2} \sqrt{a +b \left(\cot^{2}\left(x \right)\right)}}-\frac{1}{3 \left(a -b \right) \left(a +b \left(\cot^{2}\left(x \right)\right)\right)^{\frac{3}{2}}}-\frac{\arctan \left(\frac{\sqrt{a +b \left(\cot^{2}\left(x \right)\right)}}{\sqrt{-a +b}}\right)}{\left(a -b \right)^{2} \sqrt{-a +b}}"," ",0,"-1/(a-b)^2/(a+b*cot(x)^2)^(1/2)-1/3/(a-b)/(a+b*cot(x)^2)^(3/2)-1/(a-b)^2/(-a+b)^(1/2)*arctan((a+b*cot(x)^2)^(1/2)/(-a+b)^(1/2))","A"
57,0,0,100,0.604000," ","int(tan(x)/(a+b*cot(x)^2)^(5/2),x)","\int \frac{\tan \left(x \right)}{\left(a +b \left(\cot^{2}\left(x \right)\right)\right)^{\frac{5}{2}}}\, dx"," ",0,"int(tan(x)/(a+b*cot(x)^2)^(5/2),x)","F"
58,1,1040,123,1.299000," ","int(tan(x)^2/(a+b*cot(x)^2)^(5/2),x)","\frac{\left(-1+\cos \left(x \right)\right)^{2} \left(\cos \left(x \right)+1\right)^{2} \left(a \left(\cos^{2}\left(x \right)\right)-b \left(\cos^{2}\left(x \right)\right)-a \right) \left(3 \left(\cos^{4}\left(x \right)\right) \sqrt{-\frac{a \left(\cos^{2}\left(x \right)\right)-b \left(\cos^{2}\left(x \right)\right)-a}{\left(\cos \left(x \right)+1\right)^{2}}}\, \ln \left(4 \cos \left(x \right) \sqrt{-a +b}\, \sqrt{-\frac{a \left(\cos^{2}\left(x \right)\right)-b \left(\cos^{2}\left(x \right)\right)-a}{\left(\cos \left(x \right)+1\right)^{2}}}-4 a \cos \left(x \right)+4 b \cos \left(x \right)+4 \sqrt{-a +b}\, \sqrt{-\frac{a \left(\cos^{2}\left(x \right)\right)-b \left(\cos^{2}\left(x \right)\right)-a}{\left(\cos \left(x \right)+1\right)^{2}}}\right) a^{4}-3 \left(\cos^{4}\left(x \right)\right) \sqrt{-\frac{a \left(\cos^{2}\left(x \right)\right)-b \left(\cos^{2}\left(x \right)\right)-a}{\left(\cos \left(x \right)+1\right)^{2}}}\, \ln \left(4 \cos \left(x \right) \sqrt{-a +b}\, \sqrt{-\frac{a \left(\cos^{2}\left(x \right)\right)-b \left(\cos^{2}\left(x \right)\right)-a}{\left(\cos \left(x \right)+1\right)^{2}}}-4 a \cos \left(x \right)+4 b \cos \left(x \right)+4 \sqrt{-a +b}\, \sqrt{-\frac{a \left(\cos^{2}\left(x \right)\right)-b \left(\cos^{2}\left(x \right)\right)-a}{\left(\cos \left(x \right)+1\right)^{2}}}\right) a^{3} b -3 \left(\cos^{4}\left(x \right)\right) \sqrt{-a +b}\, a^{4}+12 \left(\cos^{4}\left(x \right)\right) \sqrt{-a +b}\, a^{3} b -27 \left(\cos^{4}\left(x \right)\right) \sqrt{-a +b}\, a^{2} b^{2}+26 \left(\cos^{4}\left(x \right)\right) \sqrt{-a +b}\, a \,b^{3}-8 \left(\cos^{4}\left(x \right)\right) \sqrt{-a +b}\, b^{4}+3 \left(\cos^{3}\left(x \right)\right) \sqrt{-\frac{a \left(\cos^{2}\left(x \right)\right)-b \left(\cos^{2}\left(x \right)\right)-a}{\left(\cos \left(x \right)+1\right)^{2}}}\, \ln \left(4 \cos \left(x \right) \sqrt{-a +b}\, \sqrt{-\frac{a \left(\cos^{2}\left(x \right)\right)-b \left(\cos^{2}\left(x \right)\right)-a}{\left(\cos \left(x \right)+1\right)^{2}}}-4 a \cos \left(x \right)+4 b \cos \left(x \right)+4 \sqrt{-a +b}\, \sqrt{-\frac{a \left(\cos^{2}\left(x \right)\right)-b \left(\cos^{2}\left(x \right)\right)-a}{\left(\cos \left(x \right)+1\right)^{2}}}\right) a^{4}-3 \left(\cos^{3}\left(x \right)\right) \sqrt{-\frac{a \left(\cos^{2}\left(x \right)\right)-b \left(\cos^{2}\left(x \right)\right)-a}{\left(\cos \left(x \right)+1\right)^{2}}}\, \ln \left(4 \cos \left(x \right) \sqrt{-a +b}\, \sqrt{-\frac{a \left(\cos^{2}\left(x \right)\right)-b \left(\cos^{2}\left(x \right)\right)-a}{\left(\cos \left(x \right)+1\right)^{2}}}-4 a \cos \left(x \right)+4 b \cos \left(x \right)+4 \sqrt{-a +b}\, \sqrt{-\frac{a \left(\cos^{2}\left(x \right)\right)-b \left(\cos^{2}\left(x \right)\right)-a}{\left(\cos \left(x \right)+1\right)^{2}}}\right) a^{3} b -3 \left(\cos^{2}\left(x \right)\right) \sqrt{-\frac{a \left(\cos^{2}\left(x \right)\right)-b \left(\cos^{2}\left(x \right)\right)-a}{\left(\cos \left(x \right)+1\right)^{2}}}\, \ln \left(4 \cos \left(x \right) \sqrt{-a +b}\, \sqrt{-\frac{a \left(\cos^{2}\left(x \right)\right)-b \left(\cos^{2}\left(x \right)\right)-a}{\left(\cos \left(x \right)+1\right)^{2}}}-4 a \cos \left(x \right)+4 b \cos \left(x \right)+4 \sqrt{-a +b}\, \sqrt{-\frac{a \left(\cos^{2}\left(x \right)\right)-b \left(\cos^{2}\left(x \right)\right)-a}{\left(\cos \left(x \right)+1\right)^{2}}}\right) a^{4}+6 \left(\cos^{2}\left(x \right)\right) \sqrt{-a +b}\, a^{4}-18 \left(\cos^{2}\left(x \right)\right) \sqrt{-a +b}\, a^{3} b +27 \left(\cos^{2}\left(x \right)\right) \sqrt{-a +b}\, a^{2} b^{2}-12 \left(\cos^{2}\left(x \right)\right) \sqrt{-a +b}\, a \,b^{3}-3 \cos \left(x \right) \sqrt{-\frac{a \left(\cos^{2}\left(x \right)\right)-b \left(\cos^{2}\left(x \right)\right)-a}{\left(\cos \left(x \right)+1\right)^{2}}}\, \ln \left(4 \cos \left(x \right) \sqrt{-a +b}\, \sqrt{-\frac{a \left(\cos^{2}\left(x \right)\right)-b \left(\cos^{2}\left(x \right)\right)-a}{\left(\cos \left(x \right)+1\right)^{2}}}-4 a \cos \left(x \right)+4 b \cos \left(x \right)+4 \sqrt{-a +b}\, \sqrt{-\frac{a \left(\cos^{2}\left(x \right)\right)-b \left(\cos^{2}\left(x \right)\right)-a}{\left(\cos \left(x \right)+1\right)^{2}}}\right) a^{4}-3 \sqrt{-a +b}\, a^{4}+6 \sqrt{-a +b}\, a^{3} b -3 \sqrt{-a +b}\, a^{2} b^{2}\right) b^{2}}{3 \cos \left(x \right) \left(\frac{a \left(\cos^{2}\left(x \right)\right)-b \left(\cos^{2}\left(x \right)\right)-a}{-1+\cos^{2}\left(x \right)}\right)^{\frac{5}{2}} \sin \left(x \right)^{9} \left(\sqrt{a \left(a -b \right)}-a +b \right)^{2} a^{3} \left(\sqrt{a \left(a -b \right)}+a -b \right)^{2} \sqrt{-a +b}}"," ",0,"1/3*(-1+cos(x))^2*(cos(x)+1)^2*(a*cos(x)^2-b*cos(x)^2-a)*(3*cos(x)^4*(-(a*cos(x)^2-b*cos(x)^2-a)/(cos(x)+1)^2)^(1/2)*ln(4*cos(x)*(-a+b)^(1/2)*(-(a*cos(x)^2-b*cos(x)^2-a)/(cos(x)+1)^2)^(1/2)-4*a*cos(x)+4*b*cos(x)+4*(-a+b)^(1/2)*(-(a*cos(x)^2-b*cos(x)^2-a)/(cos(x)+1)^2)^(1/2))*a^4-3*cos(x)^4*(-(a*cos(x)^2-b*cos(x)^2-a)/(cos(x)+1)^2)^(1/2)*ln(4*cos(x)*(-a+b)^(1/2)*(-(a*cos(x)^2-b*cos(x)^2-a)/(cos(x)+1)^2)^(1/2)-4*a*cos(x)+4*b*cos(x)+4*(-a+b)^(1/2)*(-(a*cos(x)^2-b*cos(x)^2-a)/(cos(x)+1)^2)^(1/2))*a^3*b-3*cos(x)^4*(-a+b)^(1/2)*a^4+12*cos(x)^4*(-a+b)^(1/2)*a^3*b-27*cos(x)^4*(-a+b)^(1/2)*a^2*b^2+26*cos(x)^4*(-a+b)^(1/2)*a*b^3-8*cos(x)^4*(-a+b)^(1/2)*b^4+3*cos(x)^3*(-(a*cos(x)^2-b*cos(x)^2-a)/(cos(x)+1)^2)^(1/2)*ln(4*cos(x)*(-a+b)^(1/2)*(-(a*cos(x)^2-b*cos(x)^2-a)/(cos(x)+1)^2)^(1/2)-4*a*cos(x)+4*b*cos(x)+4*(-a+b)^(1/2)*(-(a*cos(x)^2-b*cos(x)^2-a)/(cos(x)+1)^2)^(1/2))*a^4-3*cos(x)^3*(-(a*cos(x)^2-b*cos(x)^2-a)/(cos(x)+1)^2)^(1/2)*ln(4*cos(x)*(-a+b)^(1/2)*(-(a*cos(x)^2-b*cos(x)^2-a)/(cos(x)+1)^2)^(1/2)-4*a*cos(x)+4*b*cos(x)+4*(-a+b)^(1/2)*(-(a*cos(x)^2-b*cos(x)^2-a)/(cos(x)+1)^2)^(1/2))*a^3*b-3*cos(x)^2*(-(a*cos(x)^2-b*cos(x)^2-a)/(cos(x)+1)^2)^(1/2)*ln(4*cos(x)*(-a+b)^(1/2)*(-(a*cos(x)^2-b*cos(x)^2-a)/(cos(x)+1)^2)^(1/2)-4*a*cos(x)+4*b*cos(x)+4*(-a+b)^(1/2)*(-(a*cos(x)^2-b*cos(x)^2-a)/(cos(x)+1)^2)^(1/2))*a^4+6*cos(x)^2*(-a+b)^(1/2)*a^4-18*cos(x)^2*(-a+b)^(1/2)*a^3*b+27*cos(x)^2*(-a+b)^(1/2)*a^2*b^2-12*cos(x)^2*(-a+b)^(1/2)*a*b^3-3*cos(x)*(-(a*cos(x)^2-b*cos(x)^2-a)/(cos(x)+1)^2)^(1/2)*ln(4*cos(x)*(-a+b)^(1/2)*(-(a*cos(x)^2-b*cos(x)^2-a)/(cos(x)+1)^2)^(1/2)-4*a*cos(x)+4*b*cos(x)+4*(-a+b)^(1/2)*(-(a*cos(x)^2-b*cos(x)^2-a)/(cos(x)+1)^2)^(1/2))*a^4-3*(-a+b)^(1/2)*a^4+6*(-a+b)^(1/2)*a^3*b-3*(-a+b)^(1/2)*a^2*b^2)*b^2/cos(x)/((a*cos(x)^2-b*cos(x)^2-a)/(-1+cos(x)^2))^(5/2)/sin(x)^9/((a*(a-b))^(1/2)-a+b)^2/a^3/((a*(a-b))^(1/2)+a-b)^2/(-a+b)^(1/2)","B"
59,1,37,29,0.151000," ","int(1/(1+cot(x)^3),x)","\frac{\ln \left(1-\cot \left(x \right)+\cot^{2}\left(x \right)\right)}{3}-\frac{\ln \left(1+\cot \left(x \right)\right)}{6}-\frac{\ln \left(1+\cot^{2}\left(x \right)\right)}{4}-\frac{\pi}{4}+\frac{x}{2}"," ",0,"1/3*ln(1-cot(x)+cot(x)^2)-1/6*ln(1+cot(x))-1/4*ln(1+cot(x)^2)-1/4*Pi+1/2*x","A"
60,1,139,70,0.348000," ","int(cot(x)*(a+b*cot(x)^4)^(1/2),x)","-\frac{\sqrt{\left(1+\cot^{2}\left(x \right)\right)^{2} b -2 \left(1+\cot^{2}\left(x \right)\right) b +a +b}}{2}+\frac{\sqrt{b}\, \ln \left(\frac{\left(1+\cot^{2}\left(x \right)\right) b -b}{\sqrt{b}}+\sqrt{\left(1+\cot^{2}\left(x \right)\right)^{2} b -2 \left(1+\cot^{2}\left(x \right)\right) b +a +b}\right)}{2}+\frac{\sqrt{a +b}\, \ln \left(\frac{2 a +2 b -2 \left(1+\cot^{2}\left(x \right)\right) b +2 \sqrt{a +b}\, \sqrt{\left(1+\cot^{2}\left(x \right)\right)^{2} b -2 \left(1+\cot^{2}\left(x \right)\right) b +a +b}}{1+\cot^{2}\left(x \right)}\right)}{2}"," ",0,"-1/2*((1+cot(x)^2)^2*b-2*(1+cot(x)^2)*b+a+b)^(1/2)+1/2*b^(1/2)*ln(((1+cot(x)^2)*b-b)/b^(1/2)+((1+cot(x)^2)^2*b-2*(1+cot(x)^2)*b+a+b)^(1/2))+1/2*(a+b)^(1/2)*ln((2*a+2*b-2*(1+cot(x)^2)*b+2*(a+b)^(1/2)*((1+cot(x)^2)^2*b-2*(1+cot(x)^2)*b+a+b)^(1/2))/(1+cot(x)^2))","A"
61,1,312,103,0.286000," ","int(cot(x)*(a+b*cot(x)^4)^(3/2),x)","-\frac{b \left(\cot^{4}\left(x \right)\right) \sqrt{a +b \left(\cot^{4}\left(x \right)\right)}}{6}-\frac{2 a \sqrt{a +b \left(\cot^{4}\left(x \right)\right)}}{3}+\frac{b \left(\cot^{2}\left(x \right)\right) \sqrt{a +b \left(\cot^{4}\left(x \right)\right)}}{4}+\frac{3 a \sqrt{b}\, \ln \left(\sqrt{b}\, \left(\cot^{2}\left(x \right)\right)+\sqrt{a +b \left(\cot^{4}\left(x \right)\right)}\right)}{4}-\frac{b \sqrt{a +b \left(\cot^{4}\left(x \right)\right)}}{2}+\frac{b^{\frac{3}{2}} \ln \left(\sqrt{b}\, \left(\cot^{2}\left(x \right)\right)+\sqrt{a +b \left(\cot^{4}\left(x \right)\right)}\right)}{2}+\frac{\ln \left(\frac{2 a +2 b -2 \left(1+\cot^{2}\left(x \right)\right) b +2 \sqrt{a +b}\, \sqrt{\left(1+\cot^{2}\left(x \right)\right)^{2} b -2 \left(1+\cot^{2}\left(x \right)\right) b +a +b}}{1+\cot^{2}\left(x \right)}\right) a^{2}}{2 \sqrt{a +b}}+\frac{\ln \left(\frac{2 a +2 b -2 \left(1+\cot^{2}\left(x \right)\right) b +2 \sqrt{a +b}\, \sqrt{\left(1+\cot^{2}\left(x \right)\right)^{2} b -2 \left(1+\cot^{2}\left(x \right)\right) b +a +b}}{1+\cot^{2}\left(x \right)}\right) a b}{\sqrt{a +b}}+\frac{\ln \left(\frac{2 a +2 b -2 \left(1+\cot^{2}\left(x \right)\right) b +2 \sqrt{a +b}\, \sqrt{\left(1+\cot^{2}\left(x \right)\right)^{2} b -2 \left(1+\cot^{2}\left(x \right)\right) b +a +b}}{1+\cot^{2}\left(x \right)}\right) b^{2}}{2 \sqrt{a +b}}"," ",0,"-1/6*b*cot(x)^4*(a+b*cot(x)^4)^(1/2)-2/3*a*(a+b*cot(x)^4)^(1/2)+1/4*b*cot(x)^2*(a+b*cot(x)^4)^(1/2)+3/4*a*b^(1/2)*ln(b^(1/2)*cot(x)^2+(a+b*cot(x)^4)^(1/2))-1/2*b*(a+b*cot(x)^4)^(1/2)+1/2*b^(3/2)*ln(b^(1/2)*cot(x)^2+(a+b*cot(x)^4)^(1/2))+1/2/(a+b)^(1/2)*ln((2*a+2*b-2*(1+cot(x)^2)*b+2*(a+b)^(1/2)*((1+cot(x)^2)^2*b-2*(1+cot(x)^2)*b+a+b)^(1/2))/(1+cot(x)^2))*a^2+1/(a+b)^(1/2)*ln((2*a+2*b-2*(1+cot(x)^2)*b+2*(a+b)^(1/2)*((1+cot(x)^2)^2*b-2*(1+cot(x)^2)*b+a+b)^(1/2))/(1+cot(x)^2))*a*b+1/2/(a+b)^(1/2)*ln((2*a+2*b-2*(1+cot(x)^2)*b+2*(a+b)^(1/2)*((1+cot(x)^2)^2*b-2*(1+cot(x)^2)*b+a+b)^(1/2))/(1+cot(x)^2))*b^2","B"
62,1,65,33,0.312000," ","int(cot(x)/(a+b*cot(x)^4)^(1/2),x)","\frac{\ln \left(\frac{2 a +2 b -2 \left(1+\cot^{2}\left(x \right)\right) b +2 \sqrt{a +b}\, \sqrt{\left(1+\cot^{2}\left(x \right)\right)^{2} b -2 \left(1+\cot^{2}\left(x \right)\right) b +a +b}}{1+\cot^{2}\left(x \right)}\right)}{2 \sqrt{a +b}}"," ",0,"1/2/(a+b)^(1/2)*ln((2*a+2*b-2*(1+cot(x)^2)*b+2*(a+b)^(1/2)*((1+cot(x)^2)^2*b-2*(1+cot(x)^2)*b+a+b)^(1/2))/(1+cot(x)^2))","A"
63,1,248,65,0.316000," ","int(cot(x)/(a+b*cot(x)^4)^(3/2),x)","-\frac{\sqrt{\left(\cot^{2}\left(x \right)-\frac{\sqrt{-a b}}{b}\right)^{2} b +2 \sqrt{-a b}\, \left(\cot^{2}\left(x \right)-\frac{\sqrt{-a b}}{b}\right)}}{4 \left(\sqrt{-a b}+b \right) a \left(\cot^{2}\left(x \right)-\frac{\sqrt{-a b}}{b}\right)}-\frac{b \ln \left(\frac{2 a +2 b -2 \left(1+\cot^{2}\left(x \right)\right) b +2 \sqrt{a +b}\, \sqrt{\left(1+\cot^{2}\left(x \right)\right)^{2} b -2 \left(1+\cot^{2}\left(x \right)\right) b +a +b}}{1+\cot^{2}\left(x \right)}\right)}{2 \left(\sqrt{-a b}+b \right) \left(\sqrt{-a b}-b \right) \sqrt{a +b}}+\frac{\sqrt{\left(\cot^{2}\left(x \right)+\frac{\sqrt{-a b}}{b}\right)^{2} b -2 \sqrt{-a b}\, \left(\cot^{2}\left(x \right)+\frac{\sqrt{-a b}}{b}\right)}}{4 \left(\sqrt{-a b}-b \right) a \left(\cot^{2}\left(x \right)+\frac{\sqrt{-a b}}{b}\right)}"," ",0,"-1/4/((-a*b)^(1/2)+b)/a/(cot(x)^2-(-a*b)^(1/2)/b)*((cot(x)^2-(-a*b)^(1/2)/b)^2*b+2*(-a*b)^(1/2)*(cot(x)^2-(-a*b)^(1/2)/b))^(1/2)-1/2*b/((-a*b)^(1/2)+b)/((-a*b)^(1/2)-b)/(a+b)^(1/2)*ln((2*a+2*b-2*(1+cot(x)^2)*b+2*(a+b)^(1/2)*((1+cot(x)^2)^2*b-2*(1+cot(x)^2)*b+a+b)^(1/2))/(1+cot(x)^2))+1/4/((-a*b)^(1/2)-b)/a/(cot(x)^2+(-a*b)^(1/2)/b)*((cot(x)^2+(-a*b)^(1/2)/b)^2*b-2*(-a*b)^(1/2)*(cot(x)^2+(-a*b)^(1/2)/b))^(1/2)","B"
64,1,602,105,0.333000," ","int(cot(x)/(a+b*cot(x)^4)^(5/2),x)","-\frac{\sqrt{\left(\cot^{2}\left(x \right)-\frac{\sqrt{-a b}}{b}\right)^{2} b +2 \sqrt{-a b}\, \left(\cot^{2}\left(x \right)-\frac{\sqrt{-a b}}{b}\right)}}{24 \left(\sqrt{-a b}+b \right) a \sqrt{-a b}\, \left(\cot^{2}\left(x \right)-\frac{\sqrt{-a b}}{b}\right)^{2}}-\frac{\sqrt{\left(\cot^{2}\left(x \right)-\frac{\sqrt{-a b}}{b}\right)^{2} b +2 \sqrt{-a b}\, \left(\cot^{2}\left(x \right)-\frac{\sqrt{-a b}}{b}\right)}}{24 \left(\sqrt{-a b}+b \right) a^{2} \left(\cot^{2}\left(x \right)-\frac{\sqrt{-a b}}{b}\right)}-\frac{\sqrt{\left(\cot^{2}\left(x \right)+\frac{\sqrt{-a b}}{b}\right)^{2} b -2 \sqrt{-a b}\, \left(\cot^{2}\left(x \right)+\frac{\sqrt{-a b}}{b}\right)}}{24 \left(\sqrt{-a b}-b \right) a \sqrt{-a b}\, \left(\cot^{2}\left(x \right)+\frac{\sqrt{-a b}}{b}\right)^{2}}+\frac{\sqrt{\left(\cot^{2}\left(x \right)+\frac{\sqrt{-a b}}{b}\right)^{2} b -2 \sqrt{-a b}\, \left(\cot^{2}\left(x \right)+\frac{\sqrt{-a b}}{b}\right)}}{24 \left(\sqrt{-a b}-b \right) a^{2} \left(\cot^{2}\left(x \right)+\frac{\sqrt{-a b}}{b}\right)}-\frac{\left(2 \sqrt{-a b}+b \right) \sqrt{\left(\cot^{2}\left(x \right)-\frac{\sqrt{-a b}}{b}\right)^{2} b +2 \sqrt{-a b}\, \left(\cot^{2}\left(x \right)-\frac{\sqrt{-a b}}{b}\right)}}{8 \left(\sqrt{-a b}+b \right)^{2} a^{2} \left(\cot^{2}\left(x \right)-\frac{\sqrt{-a b}}{b}\right)}+\frac{b^{2} \ln \left(\frac{2 a +2 b -2 \left(1+\cot^{2}\left(x \right)\right) b +2 \sqrt{a +b}\, \sqrt{\left(1+\cot^{2}\left(x \right)\right)^{2} b -2 \left(1+\cot^{2}\left(x \right)\right) b +a +b}}{1+\cot^{2}\left(x \right)}\right)}{2 \left(\sqrt{-a b}+b \right)^{2} \left(\sqrt{-a b}-b \right)^{2} \sqrt{a +b}}+\frac{\left(2 \sqrt{-a b}-b \right) \sqrt{\left(\cot^{2}\left(x \right)+\frac{\sqrt{-a b}}{b}\right)^{2} b -2 \sqrt{-a b}\, \left(\cot^{2}\left(x \right)+\frac{\sqrt{-a b}}{b}\right)}}{8 \left(\sqrt{-a b}-b \right)^{2} a^{2} \left(\cot^{2}\left(x \right)+\frac{\sqrt{-a b}}{b}\right)}"," ",0,"-1/24/((-a*b)^(1/2)+b)/a/(-a*b)^(1/2)/(cot(x)^2-(-a*b)^(1/2)/b)^2*((cot(x)^2-(-a*b)^(1/2)/b)^2*b+2*(-a*b)^(1/2)*(cot(x)^2-(-a*b)^(1/2)/b))^(1/2)-1/24/((-a*b)^(1/2)+b)/a^2/(cot(x)^2-(-a*b)^(1/2)/b)*((cot(x)^2-(-a*b)^(1/2)/b)^2*b+2*(-a*b)^(1/2)*(cot(x)^2-(-a*b)^(1/2)/b))^(1/2)-1/24/((-a*b)^(1/2)-b)/a/(-a*b)^(1/2)/(cot(x)^2+(-a*b)^(1/2)/b)^2*((cot(x)^2+(-a*b)^(1/2)/b)^2*b-2*(-a*b)^(1/2)*(cot(x)^2+(-a*b)^(1/2)/b))^(1/2)+1/24/((-a*b)^(1/2)-b)/a^2/(cot(x)^2+(-a*b)^(1/2)/b)*((cot(x)^2+(-a*b)^(1/2)/b)^2*b-2*(-a*b)^(1/2)*(cot(x)^2+(-a*b)^(1/2)/b))^(1/2)-1/8*(2*(-a*b)^(1/2)+b)/((-a*b)^(1/2)+b)^2/a^2/(cot(x)^2-(-a*b)^(1/2)/b)*((cot(x)^2-(-a*b)^(1/2)/b)^2*b+2*(-a*b)^(1/2)*(cot(x)^2-(-a*b)^(1/2)/b))^(1/2)+1/2*b^2/((-a*b)^(1/2)+b)^2/((-a*b)^(1/2)-b)^2/(a+b)^(1/2)*ln((2*a+2*b-2*(1+cot(x)^2)*b+2*(a+b)^(1/2)*((1+cot(x)^2)^2*b-2*(1+cot(x)^2)*b+a+b)^(1/2))/(1+cot(x)^2))+1/8*(2*(-a*b)^(1/2)-b)/((-a*b)^(1/2)-b)^2/a^2/(cot(x)^2+(-a*b)^(1/2)/b)*((cot(x)^2+(-a*b)^(1/2)/b)^2*b-2*(-a*b)^(1/2)*(cot(x)^2+(-a*b)^(1/2)/b))^(1/2)","B"