1,1,148,233,0.813067,"\int \frac{A+C \cot ^2(c+d x)}{\sqrt{b \tan (c+d x)}} \, dx","Integrate[(A + C*Cot[c + d*x]^2)/Sqrt[b*Tan[c + d*x]],x]","\frac{-3 \sqrt{2} (A-C) \sqrt{\tan (c+d x)} \left(2 \tan ^{-1}\left(1-\sqrt{2} \sqrt{\tan (c+d x)}\right)-2 \tan ^{-1}\left(\sqrt{2} \sqrt{\tan (c+d x)}+1\right)+\log \left(\tan (c+d x)-\sqrt{2} \sqrt{\tan (c+d x)}+1\right)-\log \left(\tan (c+d x)+\sqrt{2} \sqrt{\tan (c+d x)}+1\right)\right)-8 C \cot (c+d x)}{12 d \sqrt{b \tan (c+d x)}}","-\frac{(A-C) \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{b \tan (c+d x)}}{\sqrt{b}}\right)}{\sqrt{2} \sqrt{b} d}+\frac{(A-C) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{b \tan (c+d x)}}{\sqrt{b}}+1\right)}{\sqrt{2} \sqrt{b} d}-\frac{(A-C) \log \left(\sqrt{b} \tan (c+d x)-\sqrt{2} \sqrt{b \tan (c+d x)}+\sqrt{b}\right)}{2 \sqrt{2} \sqrt{b} d}+\frac{(A-C) \log \left(\sqrt{b} \tan (c+d x)+\sqrt{2} \sqrt{b \tan (c+d x)}+\sqrt{b}\right)}{2 \sqrt{2} \sqrt{b} d}-\frac{2 b C}{3 d (b \tan (c+d x))^{3/2}}",1,"(-8*C*Cot[c + d*x] - 3*Sqrt[2]*(A - C)*(2*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]] - 2*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]] + Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]] - Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])*Sqrt[Tan[c + d*x]])/(12*d*Sqrt[b*Tan[c + d*x]])","A",1
2,1,34,20,0.0226385,"\int \left(a+b \cot ^2(c+d x)\right) \, dx","Integrate[a + b*Cot[c + d*x]^2,x]","a x-\frac{b \cot (c+d x) \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};-\tan ^2(c+d x)\right)}{d}","a x-\frac{b \cot (c+d x)}{d}-b x",1,"a*x - (b*Cot[c + d*x]*Hypergeometric2F1[-1/2, 1, 1/2, -Tan[c + d*x]^2])/d","C",1
3,1,71,47,1.1710939,"\int \left(a+b \cot ^2(c+d x)\right)^2 \, dx","Integrate[(a + b*Cot[c + d*x]^2)^2,x]","-\frac{\cot (c+d x) \left(b \left(6 a+b \cot ^2(c+d x)-3 b\right)+3 (a-b)^2 \sqrt{-\tan ^2(c+d x)} \tanh ^{-1}\left(\sqrt{-\tan ^2(c+d x)}\right)\right)}{3 d}","-\frac{b (2 a-b) \cot (c+d x)}{d}+x (a-b)^2-\frac{b^2 \cot ^3(c+d x)}{3 d}",1,"-1/3*(Cot[c + d*x]*(b*(6*a - 3*b + b*Cot[c + d*x]^2) + 3*(a - b)^2*ArcTanh[Sqrt[-Tan[c + d*x]^2]]*Sqrt[-Tan[c + d*x]^2]))/d","A",1
4,1,111,78,2.7691377,"\int \left(a+b \cot ^2(c+d x)\right)^3 \, dx","Integrate[(a + b*Cot[c + d*x]^2)^3,x]","-\frac{\cot ^5(c+d x) \left(b \left(15 \left(3 a^2-3 a b+b^2\right) \tan ^4(c+d x)+5 b (3 a-b) \tan ^2(c+d x)+3 b^2\right)+\frac{15 (a-b)^3 \tan ^8(c+d x) \tanh ^{-1}\left(\sqrt{-\tan ^2(c+d x)}\right)}{\left(-\tan ^2(c+d x)\right)^{3/2}}\right)}{15 d}","-\frac{b \left(3 a^2-3 a b+b^2\right) \cot (c+d x)}{d}-\frac{b^2 (3 a-b) \cot ^3(c+d x)}{3 d}+x (a-b)^3-\frac{b^3 \cot ^5(c+d x)}{5 d}",1,"-1/15*(Cot[c + d*x]^5*((15*(a - b)^3*ArcTanh[Sqrt[-Tan[c + d*x]^2]]*Tan[c + d*x]^8)/(-Tan[c + d*x]^2)^(3/2) + b*(3*b^2 + 5*(3*a - b)*b*Tan[c + d*x]^2 + 15*(3*a^2 - 3*a*b + b^2)*Tan[c + d*x]^4)))/d","A",1
5,1,49,49,0.0569315,"\int \frac{1}{a+b \cot ^2(c+d x)} \, dx","Integrate[(a + b*Cot[c + d*x]^2)^(-1),x]","\frac{\tan ^{-1}(\tan (c+d x))-\frac{\sqrt{b} \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{b}}\right)}{\sqrt{a}}}{a d-b d}","\frac{\sqrt{b} \tan ^{-1}\left(\frac{\sqrt{b} \cot (c+d x)}{\sqrt{a}}\right)}{\sqrt{a} d (a-b)}+\frac{x}{a-b}",1,"(ArcTan[Tan[c + d*x]] - (Sqrt[b]*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[b]])/Sqrt[a])/(a*d - b*d)","A",1
6,1,90,97,0.9066238,"\int \frac{1}{\left(a+b \cot ^2(c+d x)\right)^2} \, dx","Integrate[(a + b*Cot[c + d*x]^2)^(-2),x]","\frac{\frac{\sqrt{b} (3 a-b) \tan ^{-1}\left(\frac{\sqrt{b} \cot (c+d x)}{\sqrt{a}}\right)}{a^{3/2}}+\frac{b (a-b) \cot (c+d x)}{a \left(a+b \cot ^2(c+d x)\right)}-2 \tan ^{-1}(\cot (c+d x))}{2 d (a-b)^2}","\frac{\sqrt{b} (3 a-b) \tan ^{-1}\left(\frac{\sqrt{b} \cot (c+d x)}{\sqrt{a}}\right)}{2 a^{3/2} d (a-b)^2}+\frac{b \cot (c+d x)}{2 a d (a-b) \left(a+b \cot ^2(c+d x)\right)}+\frac{x}{(a-b)^2}",1,"(-2*ArcTan[Cot[c + d*x]] + ((3*a - b)*Sqrt[b]*ArcTan[(Sqrt[b]*Cot[c + d*x])/Sqrt[a]])/a^(3/2) + ((a - b)*b*Cot[c + d*x])/(a*(a + b*Cot[c + d*x]^2)))/(2*(a - b)^2*d)","A",1
7,1,138,150,0.3033944,"\int \frac{1}{\left(a+b \cot ^2(c+d x)\right)^3} \, dx","Integrate[(a + b*Cot[c + d*x]^2)^(-3),x]","\frac{\frac{b (7 a-3 b) (a-b) \cot (c+d x)}{a^2 \left(a+b \cot ^2(c+d x)\right)}+\frac{\sqrt{b} \left(15 a^2-10 a b+3 b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \cot (c+d x)}{\sqrt{a}}\right)}{a^{5/2}}+\frac{2 b (a-b)^2 \cot (c+d x)}{a \left(a+b \cot ^2(c+d x)\right)^2}-8 \tan ^{-1}(\cot (c+d x))}{8 d (a-b)^3}","\frac{b (7 a-3 b) \cot (c+d x)}{8 a^2 d (a-b)^2 \left(a+b \cot ^2(c+d x)\right)}+\frac{\sqrt{b} \left(15 a^2-10 a b+3 b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \cot (c+d x)}{\sqrt{a}}\right)}{8 a^{5/2} d (a-b)^3}+\frac{b \cot (c+d x)}{4 a d (a-b) \left(a+b \cot ^2(c+d x)\right)^2}+\frac{x}{(a-b)^3}",1,"(-8*ArcTan[Cot[c + d*x]] + (Sqrt[b]*(15*a^2 - 10*a*b + 3*b^2)*ArcTan[(Sqrt[b]*Cot[c + d*x])/Sqrt[a]])/a^(5/2) + (2*(a - b)^2*b*Cot[c + d*x])/(a*(a + b*Cot[c + d*x]^2)^2) + ((7*a - 3*b)*(a - b)*b*Cot[c + d*x])/(a^2*(a + b*Cot[c + d*x]^2)))/(8*(a - b)^3*d)","A",1
8,1,51,22,0.0995552,"\int \left(1+\cot ^2(x)\right)^{3/2} \, dx","Integrate[(1 + Cot[x]^2)^(3/2),x]","\frac{1}{8} \sin (x) \sqrt{\csc ^2(x)} \left(-\csc ^2\left(\frac{x}{2}\right)+\sec ^2\left(\frac{x}{2}\right)+4 \log \left(\sin \left(\frac{x}{2}\right)\right)-4 \log \left(\cos \left(\frac{x}{2}\right)\right)\right)","-\frac{1}{2} \cot (x) \sqrt{\csc ^2(x)}-\frac{1}{2} \sinh ^{-1}(\cot (x))",1,"(Sqrt[Csc[x]^2]*(-Csc[x/2]^2 - 4*Log[Cos[x/2]] + 4*Log[Sin[x/2]] + Sec[x/2]^2)*Sin[x])/8","B",1
9,1,28,5,0.0131619,"\int \sqrt{1+\cot ^2(x)} \, dx","Integrate[Sqrt[1 + Cot[x]^2],x]","\sin (x) \sqrt{\csc ^2(x)} \left(\log \left(\sin \left(\frac{x}{2}\right)\right)-\log \left(\cos \left(\frac{x}{2}\right)\right)\right)","-\sinh ^{-1}(\cot (x))",1,"Sqrt[Csc[x]^2]*(-Log[Cos[x/2]] + Log[Sin[x/2]])*Sin[x]","B",1
10,1,12,12,0.0096073,"\int \frac{1}{\sqrt{1+\cot ^2(x)}} \, dx","Integrate[1/Sqrt[1 + Cot[x]^2],x]","-\frac{\cot (x)}{\sqrt{\csc ^2(x)}}","-\frac{\cot (x)}{\sqrt{\csc ^2(x)}}",1,"-(Cot[x]/Sqrt[Csc[x]^2])","A",1
11,1,48,35,0.0826107,"\int \left(-1-\cot ^2(x)\right)^{3/2} \, dx","Integrate[(-1 - Cot[x]^2)^(3/2),x]","-\frac{\csc \left(\frac{x}{2}\right) \sec \left(\frac{x}{2}\right) \left(-\log \left(\sin \left(\frac{x}{2}\right)\right)+\log \left(\cos \left(\frac{x}{2}\right)\right)+\cot (x) \csc (x)\right)}{4 \sqrt{-\csc ^2(x)}}","\frac{1}{2} \cot (x) \sqrt{-\csc ^2(x)}-\frac{1}{2} \tan ^{-1}\left(\frac{\cot (x)}{\sqrt{-\csc ^2(x)}}\right)",1,"-1/4*(Csc[x/2]*(Cot[x]*Csc[x] + Log[Cos[x/2]] - Log[Sin[x/2]])*Sec[x/2])/Sqrt[-Csc[x]^2]","A",1
12,1,30,14,0.0170974,"\int \sqrt{-1-\cot ^2(x)} \, dx","Integrate[Sqrt[-1 - Cot[x]^2],x]","\frac{\csc (x) \left(\log \left(\cos \left(\frac{x}{2}\right)\right)-\log \left(\sin \left(\frac{x}{2}\right)\right)\right)}{\sqrt{-\csc ^2(x)}}","\tan ^{-1}\left(\frac{\cot (x)}{\sqrt{-\csc ^2(x)}}\right)",1,"(Csc[x]*(Log[Cos[x/2]] - Log[Sin[x/2]]))/Sqrt[-Csc[x]^2]","B",1
13,1,14,14,0.0043447,"\int \frac{1}{\sqrt{-1-\cot ^2(x)}} \, dx","Integrate[1/Sqrt[-1 - Cot[x]^2],x]","-\frac{\cot (x)}{\sqrt{-\csc ^2(x)}}","-\frac{\cot (x)}{\sqrt{-\csc ^2(x)}}",1,"-(Cot[x]/Sqrt[-Csc[x]^2])","A",1
14,1,19,28,0.0240546,"\int \frac{\cot ^3(x)}{\sqrt{a+a \cot ^2(x)}} \, dx","Integrate[Cot[x]^3/Sqrt[a + a*Cot[x]^2],x]","\frac{-\csc ^2(x)-1}{\sqrt{a \csc ^2(x)}}","-\frac{\sqrt{a \csc ^2(x)}}{a}-\frac{1}{\sqrt{a \csc ^2(x)}}",1,"(-1 - Csc[x]^2)/Sqrt[a*Csc[x]^2]","A",1
15,1,32,31,0.0329468,"\int \frac{\cot ^2(x)}{\sqrt{a+a \cot ^2(x)}} \, dx","Integrate[Cot[x]^2/Sqrt[a + a*Cot[x]^2],x]","\frac{\csc (x) \left(\cos (x)+\log \left(\sin \left(\frac{x}{2}\right)\right)-\log \left(\cos \left(\frac{x}{2}\right)\right)\right)}{\sqrt{a \csc ^2(x)}}","\frac{\cot (x)}{\sqrt{a \csc ^2(x)}}-\frac{\csc (x) \tanh ^{-1}(\cos (x))}{\sqrt{a \csc ^2(x)}}",1,"(Csc[x]*(Cos[x] - Log[Cos[x/2]] + Log[Sin[x/2]]))/Sqrt[a*Csc[x]^2]","A",1
16,1,10,10,0.010728,"\int \frac{\cot (x)}{\sqrt{a+a \cot ^2(x)}} \, dx","Integrate[Cot[x]/Sqrt[a + a*Cot[x]^2],x]","\frac{1}{\sqrt{a \csc ^2(x)}}","\frac{1}{\sqrt{a \csc ^2(x)}}",1,"1/Sqrt[a*Csc[x]^2]","A",1
17,1,49,36,0.0377276,"\int \frac{\tan (x)}{\sqrt{a+a \cot ^2(x)}} \, dx","Integrate[Tan[x]/Sqrt[a + a*Cot[x]^2],x]","-\frac{\csc (x) \left(\sin (x)+\log \left(\cos \left(\frac{x}{2}\right)-\sin \left(\frac{x}{2}\right)\right)-\log \left(\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)\right)\right)}{\sqrt{a \csc ^2(x)}}","\frac{\tanh ^{-1}\left(\frac{\sqrt{a \csc ^2(x)}}{\sqrt{a}}\right)}{\sqrt{a}}-\frac{1}{\sqrt{a \csc ^2(x)}}",1,"-((Csc[x]*(Log[Cos[x/2] - Sin[x/2]] - Log[Cos[x/2] + Sin[x/2]] + Sin[x]))/Sqrt[a*Csc[x]^2])","A",1
18,1,19,29,0.0331166,"\int \frac{\tan ^2(x)}{\sqrt{a+a \cot ^2(x)}} \, dx","Integrate[Tan[x]^2/Sqrt[a + a*Cot[x]^2],x]","\frac{\cot (x)+\csc (x) \sec (x)}{\sqrt{a \csc ^2(x)}}","\frac{\cot (x)}{\sqrt{a \csc ^2(x)}}+\frac{\csc (x) \sec (x)}{\sqrt{a \csc ^2(x)}}",1,"(Cot[x] + Csc[x]*Sec[x])/Sqrt[a*Csc[x]^2]","A",1
19,1,65,66,0.1841548,"\int \cot ^3(x) \sqrt{a+b \cot ^2(x)} \, dx","Integrate[Cot[x]^3*Sqrt[a + b*Cot[x]^2],x]","-\frac{\sqrt{a+b \cot ^2(x)} \left(a+b \cot ^2(x)-3 b\right)}{3 b}-\sqrt{a-b} \tanh ^{-1}\left(\frac{\sqrt{a+b \cot ^2(x)}}{\sqrt{a-b}}\right)","-\frac{\left(a+b \cot ^2(x)\right)^{3/2}}{3 b}+\sqrt{a+b \cot ^2(x)}-\sqrt{a-b} \tanh ^{-1}\left(\frac{\sqrt{a+b \cot ^2(x)}}{\sqrt{a-b}}\right)",1,"-(Sqrt[a - b]*ArcTanh[Sqrt[a + b*Cot[x]^2]/Sqrt[a - b]]) - (Sqrt[a + b*Cot[x]^2]*(a - 3*b + b*Cot[x]^2))/(3*b)","A",1
20,1,48,48,0.0271,"\int \cot (x) \sqrt{a+b \cot ^2(x)} \, dx","Integrate[Cot[x]*Sqrt[a + b*Cot[x]^2],x]","\sqrt{a-b} \tanh ^{-1}\left(\frac{\sqrt{a+b \cot ^2(x)}}{\sqrt{a-b}}\right)-\sqrt{a+b \cot ^2(x)}","\sqrt{a-b} \tanh ^{-1}\left(\frac{\sqrt{a+b \cot ^2(x)}}{\sqrt{a-b}}\right)-\sqrt{a+b \cot ^2(x)}",1,"Sqrt[a - b]*ArcTanh[Sqrt[a + b*Cot[x]^2]/Sqrt[a - b]] - Sqrt[a + b*Cot[x]^2]","A",1
21,1,60,60,0.0266062,"\int \sqrt{a+b \cot ^2(x)} \tan (x) \, dx","Integrate[Sqrt[a + b*Cot[x]^2]*Tan[x],x]","\sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a+b \cot ^2(x)}}{\sqrt{a}}\right)-\sqrt{a-b} \tanh ^{-1}\left(\frac{\sqrt{a+b \cot ^2(x)}}{\sqrt{a-b}}\right)","\sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a+b \cot ^2(x)}}{\sqrt{a}}\right)-\sqrt{a-b} \tanh ^{-1}\left(\frac{\sqrt{a+b \cot ^2(x)}}{\sqrt{a-b}}\right)",1,"Sqrt[a]*ArcTanh[Sqrt[a + b*Cot[x]^2]/Sqrt[a]] - Sqrt[a - b]*ArcTanh[Sqrt[a + b*Cot[x]^2]/Sqrt[a - b]]","A",1
22,1,2105,89,22.3975662,"\int \cot ^2(x) \sqrt{a+b \cot ^2(x)} \, dx","Integrate[Cot[x]^2*Sqrt[a + b*Cot[x]^2],x]","\text{Result too large to show}","-\frac{1}{2} \cot (x) \sqrt{a+b \cot ^2(x)}+\sqrt{a-b} \tan ^{-1}\left(\frac{\sqrt{a-b} \cot (x)}{\sqrt{a+b \cot ^2(x)}}\right)-\frac{(a-2 b) \tanh ^{-1}\left(\frac{\sqrt{b} \cot (x)}{\sqrt{a+b \cot ^2(x)}}\right)}{2 \sqrt{b}}",1,"-1/2*(Sqrt[(-a - b + a*Cos[2*x] - b*Cos[2*x])/(-1 + Cos[2*x])]*Cot[x]) + ((-4*Sqrt[a - b]*Sqrt[b]*ArcTan[(Sqrt[a - b]*(-1 + Tan[x/2]^2))/Sqrt[b*Cos[x]^2*Sec[x/2]^4 + 4*a*Tan[x/2]^2]] - (a - 2*b)*ArcTanh[(Sqrt[2]*(a + (-a + b)*Cos[x])*Sec[x/2]^2)/(Sqrt[b]*Sqrt[(a + b + (-a + b)*Cos[2*x])*Sec[x/2]^4])] + (a - 2*b)*ArcTanh[(2*a + b*(-1 + Tan[x/2]^2))/(Sqrt[b]*Sqrt[b*Cos[x]^2*Sec[x/2]^4 + 4*a*Tan[x/2]^2])])*((b*Sqrt[-(a/(-1 + Cos[2*x])) - b/(-1 + Cos[2*x]) + (a*Cos[2*x])/(-1 + Cos[2*x]) - (b*Cos[2*x])/(-1 + Cos[2*x])])/(-a - b + a*Cos[2*x] - b*Cos[2*x]) - (a*Cos[2*x]*Sqrt[-(a/(-1 + Cos[2*x])) - b/(-1 + Cos[2*x]) + (a*Cos[2*x])/(-1 + Cos[2*x]) - (b*Cos[2*x])/(-1 + Cos[2*x])])/(-a - b + a*Cos[2*x] - b*Cos[2*x]) + (b*Cos[2*x]*Sqrt[-(a/(-1 + Cos[2*x])) - b/(-1 + Cos[2*x]) + (a*Cos[2*x])/(-1 + Cos[2*x]) - (b*Cos[2*x])/(-1 + Cos[2*x])])/(-a - b + a*Cos[2*x] - b*Cos[2*x]))*Sqrt[a + b*Cot[x]^2]*Tan[x/2])/(Sqrt[2]*Sqrt[b]*Sqrt[(a + b + (-a + b)*Cos[2*x])*Sec[x/2]^4]*(((-4*Sqrt[a - b]*Sqrt[b]*ArcTan[(Sqrt[a - b]*(-1 + Tan[x/2]^2))/Sqrt[b*Cos[x]^2*Sec[x/2]^4 + 4*a*Tan[x/2]^2]] - (a - 2*b)*ArcTanh[(Sqrt[2]*(a + (-a + b)*Cos[x])*Sec[x/2]^2)/(Sqrt[b]*Sqrt[(a + b + (-a + b)*Cos[2*x])*Sec[x/2]^4])] + (a - 2*b)*ArcTanh[(2*a + b*(-1 + Tan[x/2]^2))/(Sqrt[b]*Sqrt[b*Cos[x]^2*Sec[x/2]^4 + 4*a*Tan[x/2]^2])])*Sqrt[a + b*Cot[x]^2]*Sec[x/2]^2)/(2*Sqrt[2]*Sqrt[b]*Sqrt[(a + b + (-a + b)*Cos[2*x])*Sec[x/2]^4]) - (Sqrt[b]*(-4*Sqrt[a - b]*Sqrt[b]*ArcTan[(Sqrt[a - b]*(-1 + Tan[x/2]^2))/Sqrt[b*Cos[x]^2*Sec[x/2]^4 + 4*a*Tan[x/2]^2]] - (a - 2*b)*ArcTanh[(Sqrt[2]*(a + (-a + b)*Cos[x])*Sec[x/2]^2)/(Sqrt[b]*Sqrt[(a + b + (-a + b)*Cos[2*x])*Sec[x/2]^4])] + (a - 2*b)*ArcTanh[(2*a + b*(-1 + Tan[x/2]^2))/(Sqrt[b]*Sqrt[b*Cos[x]^2*Sec[x/2]^4 + 4*a*Tan[x/2]^2])])*Cot[x]*Csc[x]^2*Tan[x/2])/(Sqrt[2]*Sqrt[a + b*Cot[x]^2]*Sqrt[(a + b + (-a + b)*Cos[2*x])*Sec[x/2]^4]) - ((-4*Sqrt[a - b]*Sqrt[b]*ArcTan[(Sqrt[a - b]*(-1 + Tan[x/2]^2))/Sqrt[b*Cos[x]^2*Sec[x/2]^4 + 4*a*Tan[x/2]^2]] - (a - 2*b)*ArcTanh[(Sqrt[2]*(a + (-a + b)*Cos[x])*Sec[x/2]^2)/(Sqrt[b]*Sqrt[(a + b + (-a + b)*Cos[2*x])*Sec[x/2]^4])] + (a - 2*b)*ArcTanh[(2*a + b*(-1 + Tan[x/2]^2))/(Sqrt[b]*Sqrt[b*Cos[x]^2*Sec[x/2]^4 + 4*a*Tan[x/2]^2])])*Sqrt[a + b*Cot[x]^2]*Tan[x/2]*(-2*(-a + b)*Sec[x/2]^4*Sin[2*x] + 2*(a + b + (-a + b)*Cos[2*x])*Sec[x/2]^4*Tan[x/2]))/(2*Sqrt[2]*Sqrt[b]*((a + b + (-a + b)*Cos[2*x])*Sec[x/2]^4)^(3/2)) + (Sqrt[a + b*Cot[x]^2]*Tan[x/2]*(-(((a - 2*b)*(-((Sqrt[2]*(-a + b)*Sec[x/2]^2*Sin[x])/(Sqrt[b]*Sqrt[(a + b + (-a + b)*Cos[2*x])*Sec[x/2]^4])) + (Sqrt[2]*(a + (-a + b)*Cos[x])*Sec[x/2]^2*Tan[x/2])/(Sqrt[b]*Sqrt[(a + b + (-a + b)*Cos[2*x])*Sec[x/2]^4]) - ((a + (-a + b)*Cos[x])*Sec[x/2]^2*(-2*(-a + b)*Sec[x/2]^4*Sin[2*x] + 2*(a + b + (-a + b)*Cos[2*x])*Sec[x/2]^4*Tan[x/2]))/(Sqrt[2]*Sqrt[b]*((a + b + (-a + b)*Cos[2*x])*Sec[x/2]^4)^(3/2))))/(1 - (2*(a + (-a + b)*Cos[x])^2)/(b*(a + b + (-a + b)*Cos[2*x])))) - (4*Sqrt[a - b]*Sqrt[b]*(-1/2*(Sqrt[a - b]*(-2*b*Cos[x]*Sec[x/2]^4*Sin[x] + 4*a*Sec[x/2]^2*Tan[x/2] + 2*b*Cos[x]^2*Sec[x/2]^4*Tan[x/2])*(-1 + Tan[x/2]^2))/(b*Cos[x]^2*Sec[x/2]^4 + 4*a*Tan[x/2]^2)^(3/2) + (Sqrt[a - b]*Sec[x/2]^2*Tan[x/2])/Sqrt[b*Cos[x]^2*Sec[x/2]^4 + 4*a*Tan[x/2]^2]))/(1 + ((a - b)*(-1 + Tan[x/2]^2)^2)/(b*Cos[x]^2*Sec[x/2]^4 + 4*a*Tan[x/2]^2)) + ((a - 2*b)*((Sqrt[b]*Sec[x/2]^2*Tan[x/2])/Sqrt[b*Cos[x]^2*Sec[x/2]^4 + 4*a*Tan[x/2]^2] - ((-2*b*Cos[x]*Sec[x/2]^4*Sin[x] + 4*a*Sec[x/2]^2*Tan[x/2] + 2*b*Cos[x]^2*Sec[x/2]^4*Tan[x/2])*(2*a + b*(-1 + Tan[x/2]^2)))/(2*Sqrt[b]*(b*Cos[x]^2*Sec[x/2]^4 + 4*a*Tan[x/2]^2)^(3/2))))/(1 - (2*a + b*(-1 + Tan[x/2]^2))^2/(b*(b*Cos[x]^2*Sec[x/2]^4 + 4*a*Tan[x/2]^2)))))/(Sqrt[2]*Sqrt[b]*Sqrt[(a + b + (-a + b)*Cos[2*x])*Sec[x/2]^4])))","B",0
23,1,167,65,0.4081536,"\int \sqrt{a+b \cot ^2(x)} \, dx","Integrate[Sqrt[a + b*Cot[x]^2],x]","\frac{1}{2} i \left(\sqrt{a-b} \log \left(-\frac{4 i \left(\sqrt{a-b} \sqrt{a+b \cot ^2(x)}+a-i b \cot (x)\right)}{(a-b)^{3/2} (\cot (x)+i)}\right)-\sqrt{a-b} \log \left(\frac{4 i \left(\sqrt{a-b} \sqrt{a+b \cot ^2(x)}+a+i b \cot (x)\right)}{(a-b)^{3/2} (\cot (x)-i)}\right)+2 i \sqrt{b} \log \left(\sqrt{b} \sqrt{a+b \cot ^2(x)}+b \cot (x)\right)\right)","-\sqrt{a-b} \tan ^{-1}\left(\frac{\sqrt{a-b} \cot (x)}{\sqrt{a+b \cot ^2(x)}}\right)-\sqrt{b} \tanh ^{-1}\left(\frac{\sqrt{b} \cot (x)}{\sqrt{a+b \cot ^2(x)}}\right)",1,"(I/2)*(Sqrt[a - b]*Log[((-4*I)*(a - I*b*Cot[x] + Sqrt[a - b]*Sqrt[a + b*Cot[x]^2]))/((a - b)^(3/2)*(I + Cot[x]))] - Sqrt[a - b]*Log[((4*I)*(a + I*b*Cot[x] + Sqrt[a - b]*Sqrt[a + b*Cot[x]^2]))/((a - b)^(3/2)*(-I + Cot[x]))] + (2*I)*Sqrt[b]*Log[b*Cot[x] + Sqrt[b]*Sqrt[a + b*Cot[x]^2]])","C",1
24,1,44,51,0.0968882,"\int \sqrt{a+b \cot ^2(x)} \tan ^2(x) \, dx","Integrate[Sqrt[a + b*Cot[x]^2]*Tan[x]^2,x]","\tan (x) \sqrt{a+b \cot ^2(x)} \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};-\frac{(a-b) \cot ^2(x)}{b \cot ^2(x)+a}\right)","\sqrt{a-b} \tan ^{-1}\left(\frac{\sqrt{a-b} \cot (x)}{\sqrt{a+b \cot ^2(x)}}\right)+\tan (x) \sqrt{a+b \cot ^2(x)}",1,"Sqrt[a + b*Cot[x]^2]*Hypergeometric2F1[-1/2, 1, 1/2, -(((a - b)*Cot[x]^2)/(a + b*Cot[x]^2))]*Tan[x]","C",1
25,1,174,85,1.6469924,"\int \sqrt{a+b \cot ^2(x)} \tan ^4(x) \, dx","Integrate[Sqrt[a + b*Cot[x]^2]*Tan[x]^4,x]","\frac{1}{3} \sin ^2(x) \tan ^3(x) \sqrt{a+b \cot ^2(x)} \left(\frac{b \cot ^2(x)}{a}+1\right) \left(\frac{\csc ^2(x) \left(a-2 b \cot ^2(x)\right) \left(\sqrt{\frac{(a-b) \cos ^2(x)}{a}} \sin ^{-1}\left(\sqrt{\frac{(a-b) \cos ^2(x)}{a}}\right)+\sqrt{\frac{b \cos ^2(x)}{a}+\sin ^2(x)}\right)}{\left(a+b \cot ^2(x)\right) \sqrt{\frac{b \cos ^2(x)}{a}+\sin ^2(x)}}-\frac{4 (a-b) \cos ^2(x) \left(a+b \cot ^2(x)\right) \, _2F_1\left(2,2;\frac{3}{2};\frac{(a-b) \cos ^2(x)}{a}\right)}{a^2}\right)","-\sqrt{a-b} \tan ^{-1}\left(\frac{\sqrt{a-b} \cot (x)}{\sqrt{a+b \cot ^2(x)}}\right)+\frac{1}{3} \tan ^3(x) \sqrt{a+b \cot ^2(x)}-\frac{(3 a-b) \tan (x) \sqrt{a+b \cot ^2(x)}}{3 a}",1,"(Sqrt[a + b*Cot[x]^2]*(1 + (b*Cot[x]^2)/a)*Sin[x]^2*((-4*(a - b)*Cos[x]^2*(a + b*Cot[x]^2)*Hypergeometric2F1[2, 2, 3/2, ((a - b)*Cos[x]^2)/a])/a^2 + ((a - 2*b*Cot[x]^2)*Csc[x]^2*(ArcSin[Sqrt[((a - b)*Cos[x]^2)/a]]*Sqrt[((a - b)*Cos[x]^2)/a] + Sqrt[(b*Cos[x]^2)/a + Sin[x]^2]))/((a + b*Cot[x]^2)*Sqrt[(b*Cos[x]^2)/a + Sin[x]^2]))*Tan[x]^3)/3","C",0
26,1,91,88,0.5025568,"\int \cot ^3(x) \left(a+b \cot ^2(x)\right)^{3/2} \, dx","Integrate[Cot[x]^3*(a + b*Cot[x]^2)^(3/2),x]","(a-b)^{3/2} \left(-\tanh ^{-1}\left(\frac{\sqrt{a+b \cot ^2(x)}}{\sqrt{a-b}}\right)\right)-\frac{\sqrt{a+b \cot ^2(x)} \left(3 a^2+b (6 a-5 b) \cot ^2(x)-20 a b+3 b^2 \cot ^4(x)+15 b^2\right)}{15 b}","-\frac{\left(a+b \cot ^2(x)\right)^{5/2}}{5 b}+\frac{1}{3} \left(a+b \cot ^2(x)\right)^{3/2}+(a-b) \sqrt{a+b \cot ^2(x)}-(a-b)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a+b \cot ^2(x)}}{\sqrt{a-b}}\right)",1,"-((a - b)^(3/2)*ArcTanh[Sqrt[a + b*Cot[x]^2]/Sqrt[a - b]]) - (Sqrt[a + b*Cot[x]^2]*(3*a^2 - 20*a*b + 15*b^2 + (6*a - 5*b)*b*Cot[x]^2 + 3*b^2*Cot[x]^4))/(15*b)","A",1
27,1,253,127,1.1935775,"\int \cot ^2(x) \left(a+b \cot ^2(x)\right)^{3/2} \, dx","Integrate[Cot[x]^2*(a + b*Cot[x]^2)^(3/2),x]","\frac{\csc (x) \sqrt{(a-b) \cos (2 x)-a-b} \left(\sqrt{a-b} \left(\sqrt{-b} \cot (x) \csc (x) \sqrt{(a-b) \cos (2 x)-a-b} \left(5 a+2 b \csc ^2(x)-6 b\right)-\sqrt{2} \left(3 a^2-12 a b+8 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{-b} \cos (x)}{\sqrt{(a-b) \cos (2 x)-a-b}}\right)\right)+8 \sqrt{2} \sqrt{-b} (a-b)^2 \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{a-b} \cos (x)}{\sqrt{(a-b) \cos (2 x)-a-b}}\right)\right)}{8 \sqrt{2} \sqrt{-b} \sqrt{a-b} \sqrt{-\left(\csc ^2(x) ((a-b) \cos (2 x)-a-b)\right)}}","-\frac{\left(3 a^2-12 a b+8 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{b} \cot (x)}{\sqrt{a+b \cot ^2(x)}}\right)}{8 \sqrt{b}}-\frac{1}{8} (5 a-4 b) \cot (x) \sqrt{a+b \cot ^2(x)}-\frac{1}{4} b \cot ^3(x) \sqrt{a+b \cot ^2(x)}+(a-b)^{3/2} \tan ^{-1}\left(\frac{\sqrt{a-b} \cot (x)}{\sqrt{a+b \cot ^2(x)}}\right)",1,"(Sqrt[-a - b + (a - b)*Cos[2*x]]*Csc[x]*(8*Sqrt[2]*(a - b)^2*Sqrt[-b]*ArcTanh[(Sqrt[2]*Sqrt[a - b]*Cos[x])/Sqrt[-a - b + (a - b)*Cos[2*x]]] + Sqrt[a - b]*(-(Sqrt[2]*(3*a^2 - 12*a*b + 8*b^2)*ArcTanh[(Sqrt[2]*Sqrt[-b]*Cos[x])/Sqrt[-a - b + (a - b)*Cos[2*x]]]) + Sqrt[-b]*Sqrt[-a - b + (a - b)*Cos[2*x]]*Cot[x]*Csc[x]*(5*a - 6*b + 2*b*Csc[x]^2))))/(8*Sqrt[2]*Sqrt[a - b]*Sqrt[-b]*Sqrt[-((-a - b + (a - b)*Cos[2*x])*Csc[x]^2)])","A",1
28,1,63,69,0.1737221,"\int \cot (x) \left(a+b \cot ^2(x)\right)^{3/2} \, dx","Integrate[Cot[x]*(a + b*Cot[x]^2)^(3/2),x]","(a-b)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a+b \cot ^2(x)}}{\sqrt{a-b}}\right)-\frac{1}{3} \sqrt{a+b \cot ^2(x)} \left(4 a+b \cot ^2(x)-3 b\right)","-(a-b) \sqrt{a+b \cot ^2(x)}-\frac{1}{3} \left(a+b \cot ^2(x)\right)^{3/2}+(a-b)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a+b \cot ^2(x)}}{\sqrt{a-b}}\right)",1,"(a - b)^(3/2)*ArcTanh[Sqrt[a + b*Cot[x]^2]/Sqrt[a - b]] - (Sqrt[a + b*Cot[x]^2]*(4*a - 3*b + b*Cot[x]^2))/3","A",1
29,1,75,75,0.0731622,"\int \left(a+b \cot ^2(x)\right)^{3/2} \tan (x) \, dx","Integrate[(a + b*Cot[x]^2)^(3/2)*Tan[x],x]","a^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a+b \cot ^2(x)}}{\sqrt{a}}\right)-b \sqrt{a+b \cot ^2(x)}-(a-b)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a+b \cot ^2(x)}}{\sqrt{a-b}}\right)","a^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a+b \cot ^2(x)}}{\sqrt{a}}\right)-b \sqrt{a+b \cot ^2(x)}-(a-b)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a+b \cot ^2(x)}}{\sqrt{a-b}}\right)",1,"a^(3/2)*ArcTanh[Sqrt[a + b*Cot[x]^2]/Sqrt[a]] - (a - b)^(3/2)*ArcTanh[Sqrt[a + b*Cot[x]^2]/Sqrt[a - b]] - b*Sqrt[a + b*Cot[x]^2]","A",1
30,1,222,80,0.7408916,"\int \left(a+b \cot ^2(x)\right)^{3/2} \tan ^2(x) \, dx","Integrate[(a + b*Cot[x]^2)^(3/2)*Tan[x]^2,x]","\frac{\sin (x) \sqrt{-\left(\csc ^2(x) ((a-b) \cos (2 x)-a-b)\right)} \left(\sqrt{a-b} \left(\sqrt{2} b^2 \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{-b} \cos (x)}{\sqrt{(a-b) \cos (2 x)-a-b}}\right)+a \sqrt{-b} \sec (x) \sqrt{(a-b) \cos (2 x)-a-b}\right)-\sqrt{2} \sqrt{-b} (a-b)^2 \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{a-b} \cos (x)}{\sqrt{(a-b) \cos (2 x)-a-b}}\right)\right)}{\sqrt{2} \sqrt{-b} \sqrt{a-b} \sqrt{(a-b) \cos (2 x)-a-b}}","-b^{3/2} \tanh ^{-1}\left(\frac{\sqrt{b} \cot (x)}{\sqrt{a+b \cot ^2(x)}}\right)+(a-b)^{3/2} \tan ^{-1}\left(\frac{\sqrt{a-b} \cot (x)}{\sqrt{a+b \cot ^2(x)}}\right)+a \tan (x) \sqrt{a+b \cot ^2(x)}",1,"(Sqrt[-((-a - b + (a - b)*Cos[2*x])*Csc[x]^2)]*(-(Sqrt[2]*(a - b)^2*Sqrt[-b]*ArcTanh[(Sqrt[2]*Sqrt[a - b]*Cos[x])/Sqrt[-a - b + (a - b)*Cos[2*x]]]) + Sqrt[a - b]*(Sqrt[2]*b^2*ArcTanh[(Sqrt[2]*Sqrt[-b]*Cos[x])/Sqrt[-a - b + (a - b)*Cos[2*x]]] + a*Sqrt[-b]*Sqrt[-a - b + (a - b)*Cos[2*x]]*Sec[x]))*Sin[x])/(Sqrt[2]*Sqrt[a - b]*Sqrt[-b]*Sqrt[-a - b + (a - b)*Cos[2*x]])","B",1
31,1,259,171,1.6898643,"\int \left(a+b \cot ^2(c+d x)\right)^{5/2} \, dx","Integrate[(a + b*Cot[c + d*x]^2)^(5/2),x]","-\frac{\sqrt{b} \left(15 a^2-20 a b+8 b^2\right) \log \left(\sqrt{b} \sqrt{a+b \cot ^2(c+d x)}+b \cot (c+d x)\right)+b \cot (c+d x) \sqrt{a+b \cot ^2(c+d x)} \left(9 a+2 b \cot ^2(c+d x)-4 b\right)-4 i (a-b)^{5/2} \log \left(-\frac{4 i \left(\sqrt{a-b} \sqrt{a+b \cot ^2(c+d x)}+a-i b \cot (c+d x)\right)}{(a-b)^{7/2} (\cot (c+d x)+i)}\right)+4 i (a-b)^{5/2} \log \left(\frac{4 i \left(\sqrt{a-b} \sqrt{a+b \cot ^2(c+d x)}+a+i b \cot (c+d x)\right)}{(a-b)^{7/2} (\cot (c+d x)-i)}\right)}{8 d}","-\frac{\sqrt{b} \left(15 a^2-20 a b+8 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{b} \cot (c+d x)}{\sqrt{a+b \cot ^2(c+d x)}}\right)}{8 d}-\frac{b \cot (c+d x) \left(a+b \cot ^2(c+d x)\right)^{3/2}}{4 d}-\frac{b (7 a-4 b) \cot (c+d x) \sqrt{a+b \cot ^2(c+d x)}}{8 d}-\frac{(a-b)^{5/2} \tan ^{-1}\left(\frac{\sqrt{a-b} \cot (c+d x)}{\sqrt{a+b \cot ^2(c+d x)}}\right)}{d}",1,"-1/8*(b*Cot[c + d*x]*Sqrt[a + b*Cot[c + d*x]^2]*(9*a - 4*b + 2*b*Cot[c + d*x]^2) - (4*I)*(a - b)^(5/2)*Log[((-4*I)*(a - I*b*Cot[c + d*x] + Sqrt[a - b]*Sqrt[a + b*Cot[c + d*x]^2]))/((a - b)^(7/2)*(I + Cot[c + d*x]))] + (4*I)*(a - b)^(5/2)*Log[((4*I)*(a + I*b*Cot[c + d*x] + Sqrt[a - b]*Sqrt[a + b*Cot[c + d*x]^2]))/((a - b)^(7/2)*(-I + Cot[c + d*x]))] + Sqrt[b]*(15*a^2 - 20*a*b + 8*b^2)*Log[b*Cot[c + d*x] + Sqrt[b]*Sqrt[a + b*Cot[c + d*x]^2]])/d","C",1
32,1,234,126,1.375934,"\int \left(a+b \cot ^2(c+d x)\right)^{3/2} \, dx","Integrate[(a + b*Cot[c + d*x]^2)^(3/2),x]","\frac{-b \cot (c+d x) \sqrt{a+b \cot ^2(c+d x)}+i (a-b)^{3/2} \log \left(-\frac{4 i \left(\sqrt{a-b} \sqrt{a+b \cot ^2(c+d x)}+a-i b \cot (c+d x)\right)}{(a-b)^{5/2} (\cot (c+d x)+i)}\right)-i (a-b)^{3/2} \log \left(\frac{4 i \left(\sqrt{a-b} \sqrt{a+b \cot ^2(c+d x)}+a+i b \cot (c+d x)\right)}{(a-b)^{5/2} (\cot (c+d x)-i)}\right)+\sqrt{b} (2 b-3 a) \log \left(\sqrt{b} \sqrt{a+b \cot ^2(c+d x)}+b \cot (c+d x)\right)}{2 d}","-\frac{b \cot (c+d x) \sqrt{a+b \cot ^2(c+d x)}}{2 d}-\frac{(a-b)^{3/2} \tan ^{-1}\left(\frac{\sqrt{a-b} \cot (c+d x)}{\sqrt{a+b \cot ^2(c+d x)}}\right)}{d}-\frac{\sqrt{b} (3 a-2 b) \tanh ^{-1}\left(\frac{\sqrt{b} \cot (c+d x)}{\sqrt{a+b \cot ^2(c+d x)}}\right)}{2 d}",1,"(-(b*Cot[c + d*x]*Sqrt[a + b*Cot[c + d*x]^2]) + I*(a - b)^(3/2)*Log[((-4*I)*(a - I*b*Cot[c + d*x] + Sqrt[a - b]*Sqrt[a + b*Cot[c + d*x]^2]))/((a - b)^(5/2)*(I + Cot[c + d*x]))] - I*(a - b)^(3/2)*Log[((4*I)*(a + I*b*Cot[c + d*x] + Sqrt[a - b]*Sqrt[a + b*Cot[c + d*x]^2]))/((a - b)^(5/2)*(-I + Cot[c + d*x]))] + Sqrt[b]*(-3*a + 2*b)*Log[b*Cot[c + d*x] + Sqrt[b]*Sqrt[a + b*Cot[c + d*x]^2]])/(2*d)","C",1
33,1,202,87,0.6113443,"\int \sqrt{a+b \cot ^2(c+d x)} \, dx","Integrate[Sqrt[a + b*Cot[c + d*x]^2],x]","\frac{i \left(\sqrt{a-b} \log \left(-\frac{4 i \left(\sqrt{a-b} \sqrt{a+b \cot ^2(c+d x)}+a-i b \cot (c+d x)\right)}{(a-b)^{3/2} (\cot (c+d x)+i)}\right)-\sqrt{a-b} \log \left(\frac{4 i \left(\sqrt{a-b} \sqrt{a+b \cot ^2(c+d x)}+a+i b \cot (c+d x)\right)}{(a-b)^{3/2} (\cot (c+d x)-i)}\right)+2 i \sqrt{b} \log \left(\sqrt{b} \sqrt{a+b \cot ^2(c+d x)}+b \cot (c+d x)\right)\right)}{2 d}","-\frac{\sqrt{a-b} \tan ^{-1}\left(\frac{\sqrt{a-b} \cot (c+d x)}{\sqrt{a+b \cot ^2(c+d x)}}\right)}{d}-\frac{\sqrt{b} \tanh ^{-1}\left(\frac{\sqrt{b} \cot (c+d x)}{\sqrt{a+b \cot ^2(c+d x)}}\right)}{d}",1,"((I/2)*(Sqrt[a - b]*Log[((-4*I)*(a - I*b*Cot[c + d*x] + Sqrt[a - b]*Sqrt[a + b*Cot[c + d*x]^2]))/((a - b)^(3/2)*(I + Cot[c + d*x]))] - Sqrt[a - b]*Log[((4*I)*(a + I*b*Cot[c + d*x] + Sqrt[a - b]*Sqrt[a + b*Cot[c + d*x]^2]))/((a - b)^(3/2)*(-I + Cot[c + d*x]))] + (2*I)*Sqrt[b]*Log[b*Cot[c + d*x] + Sqrt[b]*Sqrt[a + b*Cot[c + d*x]^2]]))/d","C",1
34,1,111,47,0.4130954,"\int \frac{1}{\sqrt{a+b \cot ^2(c+d x)}} \, dx","Integrate[1/Sqrt[a + b*Cot[c + d*x]^2],x]","-\frac{\cot (c+d x) \sqrt{\frac{b \cot ^2(c+d x)}{a}+1} \tanh ^{-1}\left(\frac{\sqrt{-\frac{(a-b) \cot ^2(c+d x)}{a}}}{\sqrt{\frac{b \cot ^2(c+d x)}{a}+1}}\right)}{d \sqrt{-\frac{(a-b) \cot ^2(c+d x)}{a}} \sqrt{a+b \cot ^2(c+d x)}}","-\frac{\tan ^{-1}\left(\frac{\sqrt{a-b} \cot (c+d x)}{\sqrt{a+b \cot ^2(c+d x)}}\right)}{d \sqrt{a-b}}",1,"-((ArcTanh[Sqrt[-(((a - b)*Cot[c + d*x]^2)/a)]/Sqrt[1 + (b*Cot[c + d*x]^2)/a]]*Cot[c + d*x]*Sqrt[1 + (b*Cot[c + d*x]^2)/a])/(d*Sqrt[-(((a - b)*Cot[c + d*x]^2)/a)]*Sqrt[a + b*Cot[c + d*x]^2]))","B",1
35,1,231,85,3.6354692,"\int \frac{1}{\left(a+b \cot ^2(c+d x)\right)^{3/2}} \, dx","Integrate[(a + b*Cot[c + d*x]^2)^(-3/2),x]","-\frac{\cos ^2(c+d x) \cot (c+d x) \left(4 (a-b)^2 \cos ^2(c+d x) \left(a \tan ^2(c+d x)+b\right) \, _2F_1\left(2,2;\frac{7}{2};\frac{(a-b) \cos ^2(c+d x)}{a}\right)-\frac{15 a \left(3 a \tan ^2(c+d x)+2 b\right) \left(\left(a \tan ^2(c+d x)+b\right) \sin ^{-1}\left(\sqrt{\frac{(a-b) \cos ^2(c+d x)}{a}}\right)-a \sec ^2(c+d x) \sqrt{\frac{(a-b) \cos ^4(c+d x) \left(a \tan ^2(c+d x)+b\right)}{a^2}}\right)}{\sqrt{\frac{(a-b) \cos ^4(c+d x) \left(a \tan ^2(c+d x)+b\right)}{a^2}}}\right)}{15 a^3 d (a-b) \sqrt{a+b \cot ^2(c+d x)}}","\frac{b \cot (c+d x)}{a d (a-b) \sqrt{a+b \cot ^2(c+d x)}}-\frac{\tan ^{-1}\left(\frac{\sqrt{a-b} \cot (c+d x)}{\sqrt{a+b \cot ^2(c+d x)}}\right)}{d (a-b)^{3/2}}",1,"-1/15*(Cos[c + d*x]^2*Cot[c + d*x]*(4*(a - b)^2*Cos[c + d*x]^2*Hypergeometric2F1[2, 2, 7/2, ((a - b)*Cos[c + d*x]^2)/a]*(b + a*Tan[c + d*x]^2) - (15*a*(2*b + 3*a*Tan[c + d*x]^2)*(ArcSin[Sqrt[((a - b)*Cos[c + d*x]^2)/a]]*(b + a*Tan[c + d*x]^2) - a*Sec[c + d*x]^2*Sqrt[((a - b)*Cos[c + d*x]^4*(b + a*Tan[c + d*x]^2))/a^2]))/Sqrt[((a - b)*Cos[c + d*x]^4*(b + a*Tan[c + d*x]^2))/a^2]))/(a^3*(a - b)*d*Sqrt[a + b*Cot[c + d*x]^2])","C",0
36,1,367,135,7.9378664,"\int \frac{1}{\left(a+b \cot ^2(c+d x)\right)^{5/2}} \, dx","Integrate[(a + b*Cot[c + d*x]^2)^(-5/2),x]","-\frac{\cot ^5(c+d x) \left(24 (a-b)^3 \cos ^2(c+d x) \left(a \tan ^2(c+d x)+b\right)^2 \, _3F_2\left(2,2,2;1,\frac{9}{2};\frac{(a-b) \cos ^2(c+d x)}{a}\right)+24 (a-b)^3 \cos ^2(c+d x) \left(4 a^2 \tan ^4(c+d x)+7 a b \tan ^2(c+d x)+3 b^2\right) \, _2F_1\left(2,2;\frac{9}{2};\frac{(a-b) \cos ^2(c+d x)}{a}\right)-\frac{35 a \left(15 a^2 \tan ^4(c+d x)+20 a b \tan ^2(c+d x)+8 b^2\right) \left(a \sec ^2(c+d x) \left(a \left(3 \tan ^2(c+d x)-1\right)+4 b\right) \sqrt{\frac{(a-b) \cos ^4(c+d x) \left(a \tan ^2(c+d x)+b\right)}{a^2}}-3 \left(a \tan ^2(c+d x)+b\right)^2 \sin ^{-1}\left(\sqrt{\frac{(a-b) \cos ^2(c+d x)}{a}}\right)\right)}{\sqrt{\frac{(a-b) \cos ^4(c+d x) \left(a \tan ^2(c+d x)+b\right)}{a^2}}}\right)}{315 a^5 d (a-b)^2 \left(\cot ^2(c+d x)+1\right) \sqrt{a+b \cot ^2(c+d x)} \left(\frac{b \cot ^2(c+d x)}{a}+1\right)}","\frac{b (5 a-2 b) \cot (c+d x)}{3 a^2 d (a-b)^2 \sqrt{a+b \cot ^2(c+d x)}}+\frac{b \cot (c+d x)}{3 a d (a-b) \left(a+b \cot ^2(c+d x)\right)^{3/2}}-\frac{\tan ^{-1}\left(\frac{\sqrt{a-b} \cot (c+d x)}{\sqrt{a+b \cot ^2(c+d x)}}\right)}{d (a-b)^{5/2}}",1,"-1/315*(Cot[c + d*x]^5*(24*(a - b)^3*Cos[c + d*x]^2*HypergeometricPFQ[{2, 2, 2}, {1, 9/2}, ((a - b)*Cos[c + d*x]^2)/a]*(b + a*Tan[c + d*x]^2)^2 + 24*(a - b)^3*Cos[c + d*x]^2*Hypergeometric2F1[2, 2, 9/2, ((a - b)*Cos[c + d*x]^2)/a]*(3*b^2 + 7*a*b*Tan[c + d*x]^2 + 4*a^2*Tan[c + d*x]^4) - (35*a*(8*b^2 + 20*a*b*Tan[c + d*x]^2 + 15*a^2*Tan[c + d*x]^4)*(-3*ArcSin[Sqrt[((a - b)*Cos[c + d*x]^2)/a]]*(b + a*Tan[c + d*x]^2)^2 + a*Sec[c + d*x]^2*Sqrt[((a - b)*Cos[c + d*x]^4*(b + a*Tan[c + d*x]^2))/a^2]*(4*b + a*(-1 + 3*Tan[c + d*x]^2))))/Sqrt[((a - b)*Cos[c + d*x]^4*(b + a*Tan[c + d*x]^2))/a^2]))/(a^5*(a - b)^2*d*(1 + Cot[c + d*x]^2)*Sqrt[a + b*Cot[c + d*x]^2]*(1 + (b*Cot[c + d*x]^2)/a))","C",0
37,1,2553,190,14.6757189,"\int \frac{1}{\left(a+b \cot ^2(c+d x)\right)^{7/2}} \, dx","Integrate[(a + b*Cot[c + d*x]^2)^(-7/2),x]","\text{Result too large to show}","\frac{b (9 a-4 b) \cot (c+d x)}{15 a^2 d (a-b)^2 \left(a+b \cot ^2(c+d x)\right)^{3/2}}+\frac{b \left(33 a^2-26 a b+8 b^2\right) \cot (c+d x)}{15 a^3 d (a-b)^3 \sqrt{a+b \cot ^2(c+d x)}}+\frac{b \cot (c+d x)}{5 a d (a-b) \left(a+b \cot ^2(c+d x)\right)^{5/2}}-\frac{\tan ^{-1}\left(\frac{\sqrt{a-b} \cot (c+d x)}{\sqrt{a+b \cot ^2(c+d x)}}\right)}{d (a-b)^{7/2}}",1,"-1/4725*(Cot[c + d*x]*(-33075*ArcSin[Sqrt[((a - b)*Cos[c + d*x]^2)/a]] + (99225*(a - b)*ArcSin[Sqrt[((a - b)*Cos[c + d*x]^2)/a]]*Cos[c + d*x]^2)/a - (99225*(a - b)^2*ArcSin[Sqrt[((a - b)*Cos[c + d*x]^2)/a]]*Cos[c + d*x]^4)/a^2 + (33075*(a - b)^3*ArcSin[Sqrt[((a - b)*Cos[c + d*x]^2)/a]]*Cos[c + d*x]^6)/a^3 - (66150*b*ArcSin[Sqrt[((a - b)*Cos[c + d*x]^2)/a]]*Cot[c + d*x]^2)/a + (198450*(a - b)*b*ArcSin[Sqrt[((a - b)*Cos[c + d*x]^2)/a]]*Cos[c + d*x]^2*Cot[c + d*x]^2)/a^2 + (66150*(a - b)^3*b*ArcSin[Sqrt[((a - b)*Cos[c + d*x]^2)/a]]*Cos[c + d*x]^6*Cot[c + d*x]^2)/a^4 - (52920*b^2*ArcSin[Sqrt[((a - b)*Cos[c + d*x]^2)/a]]*Cot[c + d*x]^4)/a^2 + (158760*(a - b)*b^2*ArcSin[Sqrt[((a - b)*Cos[c + d*x]^2)/a]]*Cos[c + d*x]^2*Cot[c + d*x]^4)/a^3 - (158760*(a - b)^2*b^2*ArcSin[Sqrt[((a - b)*Cos[c + d*x]^2)/a]]*Cos[c + d*x]^4*Cot[c + d*x]^4)/a^4 + (52920*(a - b)^3*b^2*ArcSin[Sqrt[((a - b)*Cos[c + d*x]^2)/a]]*Cos[c + d*x]^6*Cot[c + d*x]^4)/a^5 - (15120*b^3*ArcSin[Sqrt[((a - b)*Cos[c + d*x]^2)/a]]*Cot[c + d*x]^6)/a^3 + (45360*(a - b)*b^3*ArcSin[Sqrt[((a - b)*Cos[c + d*x]^2)/a]]*Cos[c + d*x]^2*Cot[c + d*x]^6)/a^4 - (45360*(a - b)^2*b^3*ArcSin[Sqrt[((a - b)*Cos[c + d*x]^2)/a]]*Cos[c + d*x]^4*Cot[c + d*x]^6)/a^5 + (15120*(a - b)^3*b^3*ArcSin[Sqrt[((a - b)*Cos[c + d*x]^2)/a]]*Cos[c + d*x]^6*Cot[c + d*x]^6)/a^6 - 77175*(((a - b)*Cos[c + d*x]^2)/a)^(3/2)*Sqrt[(Cos[c + d*x]^2*(b + a*Tan[c + d*x]^2))/a] + 50715*(((a - b)*Cos[c + d*x]^2)/a)^(5/2)*Sqrt[(Cos[c + d*x]^2*(b + a*Tan[c + d*x]^2))/a] - (154350*b*(((a - b)*Cos[c + d*x]^2)/a)^(3/2)*Cot[c + d*x]^2*Sqrt[(Cos[c + d*x]^2*(b + a*Tan[c + d*x]^2))/a])/a + (101430*b*(((a - b)*Cos[c + d*x]^2)/a)^(5/2)*Cot[c + d*x]^2*Sqrt[(Cos[c + d*x]^2*(b + a*Tan[c + d*x]^2))/a])/a - (123480*b^2*(((a - b)*Cos[c + d*x]^2)/a)^(3/2)*Cot[c + d*x]^4*Sqrt[(Cos[c + d*x]^2*(b + a*Tan[c + d*x]^2))/a])/a^2 + (81144*b^2*(((a - b)*Cos[c + d*x]^2)/a)^(5/2)*Cot[c + d*x]^4*Sqrt[(Cos[c + d*x]^2*(b + a*Tan[c + d*x]^2))/a])/a^2 - (35280*b^3*(((a - b)*Cos[c + d*x]^2)/a)^(3/2)*Cot[c + d*x]^6*Sqrt[(Cos[c + d*x]^2*(b + a*Tan[c + d*x]^2))/a])/a^3 + (23184*b^3*(((a - b)*Cos[c + d*x]^2)/a)^(5/2)*Cot[c + d*x]^6*Sqrt[(Cos[c + d*x]^2*(b + a*Tan[c + d*x]^2))/a])/a^3 + 1420*(((a - b)*Cos[c + d*x]^2)/a)^(9/2)*Hypergeometric2F1[2, 2, 11/2, ((a - b)*Cos[c + d*x]^2)/a]*Sqrt[(Cos[c + d*x]^2*(b + a*Tan[c + d*x]^2))/a] + (3540*b*(((a - b)*Cos[c + d*x]^2)/a)^(9/2)*Cot[c + d*x]^2*Hypergeometric2F1[2, 2, 11/2, ((a - b)*Cos[c + d*x]^2)/a]*Sqrt[(Cos[c + d*x]^2*(b + a*Tan[c + d*x]^2))/a])/a + (3000*b^2*(((a - b)*Cos[c + d*x]^2)/a)^(9/2)*Cot[c + d*x]^4*Hypergeometric2F1[2, 2, 11/2, ((a - b)*Cos[c + d*x]^2)/a]*Sqrt[(Cos[c + d*x]^2*(b + a*Tan[c + d*x]^2))/a])/a^2 + (880*b^3*(((a - b)*Cos[c + d*x]^2)/a)^(9/2)*Cot[c + d*x]^6*Hypergeometric2F1[2, 2, 11/2, ((a - b)*Cos[c + d*x]^2)/a]*Sqrt[(Cos[c + d*x]^2*(b + a*Tan[c + d*x]^2))/a])/a^3 + 600*(((a - b)*Cos[c + d*x]^2)/a)^(9/2)*HypergeometricPFQ[{2, 2, 2}, {1, 11/2}, ((a - b)*Cos[c + d*x]^2)/a]*Sqrt[(Cos[c + d*x]^2*(b + a*Tan[c + d*x]^2))/a] + (1680*b*(((a - b)*Cos[c + d*x]^2)/a)^(9/2)*Cot[c + d*x]^2*HypergeometricPFQ[{2, 2, 2}, {1, 11/2}, ((a - b)*Cos[c + d*x]^2)/a]*Sqrt[(Cos[c + d*x]^2*(b + a*Tan[c + d*x]^2))/a])/a + (1560*b^2*(((a - b)*Cos[c + d*x]^2)/a)^(9/2)*Cot[c + d*x]^4*HypergeometricPFQ[{2, 2, 2}, {1, 11/2}, ((a - b)*Cos[c + d*x]^2)/a]*Sqrt[(Cos[c + d*x]^2*(b + a*Tan[c + d*x]^2))/a])/a^2 + (480*b^3*(((a - b)*Cos[c + d*x]^2)/a)^(9/2)*Cot[c + d*x]^6*HypergeometricPFQ[{2, 2, 2}, {1, 11/2}, ((a - b)*Cos[c + d*x]^2)/a]*Sqrt[(Cos[c + d*x]^2*(b + a*Tan[c + d*x]^2))/a])/a^3 + 80*(((a - b)*Cos[c + d*x]^2)/a)^(9/2)*HypergeometricPFQ[{2, 2, 2, 2}, {1, 1, 11/2}, ((a - b)*Cos[c + d*x]^2)/a]*Sqrt[(Cos[c + d*x]^2*(b + a*Tan[c + d*x]^2))/a] + (240*b*(((a - b)*Cos[c + d*x]^2)/a)^(9/2)*Cot[c + d*x]^2*HypergeometricPFQ[{2, 2, 2, 2}, {1, 1, 11/2}, ((a - b)*Cos[c + d*x]^2)/a]*Sqrt[(Cos[c + d*x]^2*(b + a*Tan[c + d*x]^2))/a])/a + (240*b^2*(((a - b)*Cos[c + d*x]^2)/a)^(9/2)*Cot[c + d*x]^4*HypergeometricPFQ[{2, 2, 2, 2}, {1, 1, 11/2}, ((a - b)*Cos[c + d*x]^2)/a]*Sqrt[(Cos[c + d*x]^2*(b + a*Tan[c + d*x]^2))/a])/a^2 + (80*b^3*(((a - b)*Cos[c + d*x]^2)/a)^(9/2)*Cot[c + d*x]^6*HypergeometricPFQ[{2, 2, 2, 2}, {1, 1, 11/2}, ((a - b)*Cos[c + d*x]^2)/a]*Sqrt[(Cos[c + d*x]^2*(b + a*Tan[c + d*x]^2))/a])/a^3 + 33075*Sqrt[((a - b)*Cos[c + d*x]^4*(b + a*Tan[c + d*x]^2))/a^2] + (66150*b*Cot[c + d*x]^2*Sqrt[((a - b)*Cos[c + d*x]^4*(b + a*Tan[c + d*x]^2))/a^2])/a + (52920*b^2*Cot[c + d*x]^4*Sqrt[((a - b)*Cos[c + d*x]^4*(b + a*Tan[c + d*x]^2))/a^2])/a^2 + (15120*b^3*Cot[c + d*x]^6*Sqrt[((a - b)*Cos[c + d*x]^4*(b + a*Tan[c + d*x]^2))/a^2])/a^3 - (198450*(a - b)^2*b*ArcSin[Sqrt[((a - b)*Cos[c + d*x]^2)/a]])/(a^3*(Tan[c + d*x] + Tan[c + d*x]^3)^2)))/(a^3*d*(((a - b)*Cos[c + d*x]^2)/a)^(7/2)*(1 + Cot[c + d*x]^2)*Sqrt[a + b*Cot[c + d*x]^2]*(1 + (b*Cot[c + d*x]^2)/a)^2*Sqrt[(Cos[c + d*x]^2*(b + a*Tan[c + d*x]^2))/a])","C",0
38,1,123,54,0.3923841,"\int \left(1-\cot ^2(x)\right)^{3/2} \, dx","Integrate[(1 - Cot[x]^2)^(3/2),x]","\frac{1}{2} \left(1-\cot ^2(x)\right)^{3/2} \sec ^2(2 x) \left(-\frac{1}{4} \sin (4 x)-4 \sqrt{2} \sin ^3(x) \sqrt{\cos (2 x)} \log \left(\sqrt{2} \cos (x)+\sqrt{\cos (2 x)}\right)+\sin ^3(x) \sqrt{-\cos (2 x)} \tan ^{-1}\left(\frac{\cos (x)}{\sqrt{-\cos (2 x)}}\right)+4 \sin ^3(x) \sqrt{\cos (2 x)} \tanh ^{-1}\left(\frac{\cos (x)}{\sqrt{\cos (2 x)}}\right)\right)","\frac{1}{2} \cot (x) \sqrt{1-\cot ^2(x)}-2 \sqrt{2} \tan ^{-1}\left(\frac{\sqrt{2} \cot (x)}{\sqrt{1-\cot ^2(x)}}\right)+\frac{5}{2} \sin ^{-1}(\cot (x))",1,"((1 - Cot[x]^2)^(3/2)*Sec[2*x]^2*(ArcTan[Cos[x]/Sqrt[-Cos[2*x]]]*Sqrt[-Cos[2*x]]*Sin[x]^3 + 4*ArcTanh[Cos[x]/Sqrt[Cos[2*x]]]*Sqrt[Cos[2*x]]*Sin[x]^3 - 4*Sqrt[2]*Sqrt[Cos[2*x]]*Log[Sqrt[2]*Cos[x] + Sqrt[Cos[2*x]]]*Sin[x]^3 - Sin[4*x]/4))/2","B",1
39,1,62,32,0.0695642,"\int \sqrt{1-\cot ^2(x)} \, dx","Integrate[Sqrt[1 - Cot[x]^2],x]","\frac{\sin (x) \sqrt{1-\cot ^2(x)} \left(\sqrt{2} \log \left(\sqrt{2} \cos (x)+\sqrt{\cos (2 x)}\right)-\tanh ^{-1}\left(\frac{\cos (x)}{\sqrt{\cos (2 x)}}\right)\right)}{\sqrt{\cos (2 x)}}","\sin ^{-1}(\cot (x))-\sqrt{2} \tan ^{-1}\left(\frac{\sqrt{2} \cot (x)}{\sqrt{1-\cot ^2(x)}}\right)",1,"(Sqrt[1 - Cot[x]^2]*(-ArcTanh[Cos[x]/Sqrt[Cos[2*x]]] + Sqrt[2]*Log[Sqrt[2]*Cos[x] + Sqrt[Cos[2*x]]])*Sin[x])/Sqrt[Cos[2*x]]","A",1
40,1,42,28,0.0629458,"\int \frac{1}{\sqrt{1-\cot ^2(x)}} \, dx","Integrate[1/Sqrt[1 - Cot[x]^2],x]","-\frac{\sqrt{\cos (2 x)} \csc (x) \log \left(\sqrt{2} \cos (x)+\sqrt{\cos (2 x)}\right)}{\sqrt{2-2 \cot ^2(x)}}","-\frac{\tan ^{-1}\left(\frac{\sqrt{2} \cot (x)}{\sqrt{1-\cot ^2(x)}}\right)}{\sqrt{2}}",1,"-((Sqrt[Cos[2*x]]*Csc[x]*Log[Sqrt[2]*Cos[x] + Sqrt[Cos[2*x]]])/Sqrt[2 - 2*Cot[x]^2])","A",1
41,1,121,61,0.1228678,"\int \left(-1+\cot ^2(x)\right)^{3/2} \, dx","Integrate[(-1 + Cot[x]^2)^(3/2),x]","\frac{1}{2} \left(\cot ^2(x)-1\right)^{3/2} \sec ^2(2 x) \left(-\frac{1}{4} \sin (4 x)-4 \sqrt{2} \sin ^3(x) \sqrt{\cos (2 x)} \log \left(\sqrt{2} \cos (x)+\sqrt{\cos (2 x)}\right)+\sin ^3(x) \sqrt{-\cos (2 x)} \tan ^{-1}\left(\frac{\cos (x)}{\sqrt{-\cos (2 x)}}\right)+4 \sin ^3(x) \sqrt{\cos (2 x)} \tanh ^{-1}\left(\frac{\cos (x)}{\sqrt{\cos (2 x)}}\right)\right)","-\frac{1}{2} \cot (x) \sqrt{\cot ^2(x)-1}+\frac{5}{2} \tanh ^{-1}\left(\frac{\cot (x)}{\sqrt{\cot ^2(x)-1}}\right)-2 \sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{2} \cot (x)}{\sqrt{\cot ^2(x)-1}}\right)",1,"((-1 + Cot[x]^2)^(3/2)*Sec[2*x]^2*(ArcTan[Cos[x]/Sqrt[-Cos[2*x]]]*Sqrt[-Cos[2*x]]*Sin[x]^3 + 4*ArcTanh[Cos[x]/Sqrt[Cos[2*x]]]*Sqrt[Cos[2*x]]*Sin[x]^3 - 4*Sqrt[2]*Sqrt[Cos[2*x]]*Log[Sqrt[2]*Cos[x] + Sqrt[Cos[2*x]]]*Sin[x]^3 - Sin[4*x]/4))/2","A",1
42,1,60,42,0.0431369,"\int \sqrt{-1+\cot ^2(x)} \, dx","Integrate[Sqrt[-1 + Cot[x]^2],x]","\frac{\sin (x) \sqrt{\cot ^2(x)-1} \left(\sqrt{2} \log \left(\sqrt{2} \cos (x)+\sqrt{\cos (2 x)}\right)-\tanh ^{-1}\left(\frac{\cos (x)}{\sqrt{\cos (2 x)}}\right)\right)}{\sqrt{\cos (2 x)}}","\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{2} \cot (x)}{\sqrt{\cot ^2(x)-1}}\right)-\tanh ^{-1}\left(\frac{\cot (x)}{\sqrt{\cot ^2(x)-1}}\right)",1,"(Sqrt[-1 + Cot[x]^2]*(-ArcTanh[Cos[x]/Sqrt[Cos[2*x]]] + Sqrt[2]*Log[Sqrt[2]*Cos[x] + Sqrt[Cos[2*x]]])*Sin[x])/Sqrt[Cos[2*x]]","A",1
43,1,45,26,0.0341709,"\int \frac{1}{\sqrt{-1+\cot ^2(x)}} \, dx","Integrate[1/Sqrt[-1 + Cot[x]^2],x]","-\frac{\sqrt{\cos (2 x)} \csc (x) \log \left(\sqrt{2} \cos (x)+\sqrt{\cos (2 x)}\right)}{\sqrt{2} \sqrt{\cot ^2(x)-1}}","-\frac{\tanh ^{-1}\left(\frac{\sqrt{2} \cot (x)}{\sqrt{\cot ^2(x)-1}}\right)}{\sqrt{2}}",1,"-((Sqrt[Cos[2*x]]*Csc[x]*Log[Sqrt[2]*Cos[x] + Sqrt[Cos[2*x]]])/(Sqrt[2]*Sqrt[-1 + Cot[x]^2]))","A",1
44,1,52,52,0.1426229,"\int \frac{\cot ^3(x)}{\sqrt{a+b \cot ^2(x)}} \, dx","Integrate[Cot[x]^3/Sqrt[a + b*Cot[x]^2],x]","-\frac{\sqrt{a+b \cot ^2(x)}+\frac{b \tanh ^{-1}\left(\frac{\sqrt{a+b \cot ^2(x)}}{\sqrt{a-b}}\right)}{\sqrt{a-b}}}{b}","-\frac{\sqrt{a+b \cot ^2(x)}}{b}-\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \cot ^2(x)}}{\sqrt{a-b}}\right)}{\sqrt{a-b}}",1,"-(((b*ArcTanh[Sqrt[a + b*Cot[x]^2]/Sqrt[a - b]])/Sqrt[a - b] + Sqrt[a + b*Cot[x]^2])/b)","A",1
45,1,158,64,0.2722882,"\int \frac{\cot ^2(x)}{\sqrt{a+b \cot ^2(x)}} \, dx","Integrate[Cot[x]^2/Sqrt[a + b*Cot[x]^2],x]","\frac{\sin (x) \sqrt{\csc ^2(x) ((b-a) \cos (2 x)+a+b)} \left(\sqrt{a-b} \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{-b} \cos (x)}{\sqrt{(a-b) \cos (2 x)-a-b}}\right)-\sqrt{-b} \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{a-b} \cos (x)}{\sqrt{(a-b) \cos (2 x)-a-b}}\right)\right)}{\sqrt{-b} \sqrt{a-b} \sqrt{(a-b) \cos (2 x)-a-b}}","\frac{\tan ^{-1}\left(\frac{\sqrt{a-b} \cot (x)}{\sqrt{a+b \cot ^2(x)}}\right)}{\sqrt{a-b}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{b} \cot (x)}{\sqrt{a+b \cot ^2(x)}}\right)}{\sqrt{b}}",1,"((-(Sqrt[-b]*ArcTanh[(Sqrt[2]*Sqrt[a - b]*Cos[x])/Sqrt[-a - b + (a - b)*Cos[2*x]]]) + Sqrt[a - b]*ArcTanh[(Sqrt[2]*Sqrt[-b]*Cos[x])/Sqrt[-a - b + (a - b)*Cos[2*x]]])*Sqrt[(a + b + (-a + b)*Cos[2*x])*Csc[x]^2]*Sin[x])/(Sqrt[a - b]*Sqrt[-b]*Sqrt[-a - b + (a - b)*Cos[2*x]])","B",1
46,1,33,33,0.0142187,"\int \frac{\cot (x)}{\sqrt{a+b \cot ^2(x)}} \, dx","Integrate[Cot[x]/Sqrt[a + b*Cot[x]^2],x]","\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \cot ^2(x)}}{\sqrt{a-b}}\right)}{\sqrt{a-b}}","\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \cot ^2(x)}}{\sqrt{a-b}}\right)}{\sqrt{a-b}}",1,"ArcTanh[Sqrt[a + b*Cot[x]^2]/Sqrt[a - b]]/Sqrt[a - b]","A",1
47,1,60,60,0.0453889,"\int \frac{\tan (x)}{\sqrt{a+b \cot ^2(x)}} \, dx","Integrate[Tan[x]/Sqrt[a + b*Cot[x]^2],x]","\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \cot ^2(x)}}{\sqrt{a}}\right)}{\sqrt{a}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \cot ^2(x)}}{\sqrt{a-b}}\right)}{\sqrt{a-b}}","\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \cot ^2(x)}}{\sqrt{a}}\right)}{\sqrt{a}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \cot ^2(x)}}{\sqrt{a-b}}\right)}{\sqrt{a-b}}",1,"ArcTanh[Sqrt[a + b*Cot[x]^2]/Sqrt[a]]/Sqrt[a] - ArcTanh[Sqrt[a + b*Cot[x]^2]/Sqrt[a - b]]/Sqrt[a - b]","A",1
48,1,134,54,1.5720318,"\int \frac{\tan ^2(x)}{\sqrt{a+b \cot ^2(x)}} \, dx","Integrate[Tan[x]^2/Sqrt[a + b*Cot[x]^2],x]","\frac{\sin ^2(x) \tan (x) \left(\frac{b \cot ^2(x)}{a}+1\right) \left(\frac{3 a \left(a+2 b \cot ^2(x)\right) \sin ^{-1}\left(\sqrt{\frac{(a-b) \cos ^2(x)}{a}}\right)}{\sqrt{\frac{(a-b) \sin ^2(x) \cos ^2(x) \left(a+b \cot ^2(x)\right)}{a^2}}}+4 (a-b) \cos ^2(x) \left(a+b \cot ^2(x)\right) \, _2F_1\left(2,2;\frac{5}{2};\frac{(a-b) \cos ^2(x)}{a}\right)\right)}{3 a^2 \sqrt{a+b \cot ^2(x)}}","\frac{\tan ^{-1}\left(\frac{\sqrt{a-b} \cot (x)}{\sqrt{a+b \cot ^2(x)}}\right)}{\sqrt{a-b}}+\frac{\tan (x) \sqrt{a+b \cot ^2(x)}}{a}",1,"((1 + (b*Cot[x]^2)/a)*Sin[x]^2*(4*(a - b)*Cos[x]^2*(a + b*Cot[x]^2)*Hypergeometric2F1[2, 2, 5/2, ((a - b)*Cos[x]^2)/a] + (3*a*ArcSin[Sqrt[((a - b)*Cos[x]^2)/a]]*(a + 2*b*Cot[x]^2))/Sqrt[((a - b)*Cos[x]^2*(a + b*Cot[x]^2)*Sin[x]^2)/a^2])*Tan[x])/(3*a^2*Sqrt[a + b*Cot[x]^2])","C",0
49,1,59,59,0.213367,"\int \frac{\cot ^3(x)}{\left(a+b \cot ^2(x)\right)^{3/2}} \, dx","Integrate[Cot[x]^3/(a + b*Cot[x]^2)^(3/2),x]","\frac{a}{b (a-b) \sqrt{a+b \cot ^2(x)}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \cot ^2(x)}}{\sqrt{a-b}}\right)}{(a-b)^{3/2}}","\frac{a}{b (a-b) \sqrt{a+b \cot ^2(x)}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \cot ^2(x)}}{\sqrt{a-b}}\right)}{(a-b)^{3/2}}",1,"-(ArcTanh[Sqrt[a + b*Cot[x]^2]/Sqrt[a - b]]/(a - b)^(3/2)) + a/((a - b)*b*Sqrt[a + b*Cot[x]^2])","A",1
50,1,137,59,0.7319007,"\int \frac{\cot ^2(x)}{\left(a+b \cot ^2(x)\right)^{3/2}} \, dx","Integrate[Cot[x]^2/(a + b*Cot[x]^2)^(3/2),x]","\frac{(b-a) \cot (x) \sqrt{\frac{b \cot ^2(x)}{a}+1}+\frac{1}{2} \csc (x) \sec (x) ((a-b) \cos (2 x)-a-b) \sqrt{-\frac{(a-b) \cot ^2(x)}{a}} \tanh ^{-1}\left(\frac{\sqrt{-\frac{(a-b) \cot ^2(x)}{a}}}{\sqrt{\frac{b \cot ^2(x)}{a}+1}}\right)}{(a-b)^2 \sqrt{a+b \cot ^2(x)} \sqrt{\frac{b \cot ^2(x)}{a}+1}}","\frac{\tan ^{-1}\left(\frac{\sqrt{a-b} \cot (x)}{\sqrt{a+b \cot ^2(x)}}\right)}{(a-b)^{3/2}}-\frac{\cot (x)}{(a-b) \sqrt{a+b \cot ^2(x)}}",1,"((-a + b)*Cot[x]*Sqrt[1 + (b*Cot[x]^2)/a] + (ArcTanh[Sqrt[-(((a - b)*Cot[x]^2)/a)]/Sqrt[1 + (b*Cot[x]^2)/a]]*(-a - b + (a - b)*Cos[2*x])*Sqrt[-(((a - b)*Cot[x]^2)/a)]*Csc[x]*Sec[x])/2)/((a - b)^2*Sqrt[a + b*Cot[x]^2]*Sqrt[1 + (b*Cot[x]^2)/a])","B",1
51,1,44,55,0.0411544,"\int \frac{\cot (x)}{\left(a+b \cot ^2(x)\right)^{3/2}} \, dx","Integrate[Cot[x]/(a + b*Cot[x]^2)^(3/2),x]","\frac{\, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\frac{b \cot ^2(x)+a}{a-b}\right)}{(b-a) \sqrt{a+b \cot ^2(x)}}","\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \cot ^2(x)}}{\sqrt{a-b}}\right)}{(a-b)^{3/2}}-\frac{1}{(a-b) \sqrt{a+b \cot ^2(x)}}",1,"Hypergeometric2F1[-1/2, 1, 1/2, (a + b*Cot[x]^2)/(a - b)]/((-a + b)*Sqrt[a + b*Cot[x]^2])","C",1
52,1,75,84,0.0575503,"\int \frac{\tan (x)}{\left(a+b \cot ^2(x)\right)^{3/2}} \, dx","Integrate[Tan[x]/(a + b*Cot[x]^2)^(3/2),x]","\frac{a \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\frac{b \cot ^2(x)+a}{a-b}\right)+(b-a) \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\frac{b \cot ^2(x)}{a}+1\right)}{a (a-b) \sqrt{a+b \cot ^2(x)}}","\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \cot ^2(x)}}{\sqrt{a}}\right)}{a^{3/2}}+\frac{b}{a (a-b) \sqrt{a+b \cot ^2(x)}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \cot ^2(x)}}{\sqrt{a-b}}\right)}{(a-b)^{3/2}}",1,"(a*Hypergeometric2F1[-1/2, 1, 1/2, (a + b*Cot[x]^2)/(a - b)] + (-a + b)*Hypergeometric2F1[-1/2, 1, 1/2, 1 + (b*Cot[x]^2)/a])/(a*(a - b)*Sqrt[a + b*Cot[x]^2])","C",1
53,1,674,92,6.8972181,"\int \frac{\tan ^2(x)}{\left(a+b \cot ^2(x)\right)^{3/2}} \, dx","Integrate[Tan[x]^2/(a + b*Cot[x]^2)^(3/2),x]","\frac{\sin ^2(x) \tan (x) \left(\frac{8 b^2 (a-b) \cos ^2(x) \cot ^4(x) \, _3F_2\left(2,2,2;1,\frac{7}{2};\frac{(a-b) \cos ^2(x)}{a}\right)}{15 a^3}+\frac{16 b (a-b) \cos ^2(x) \cot ^2(x) \, _3F_2\left(2,2,2;1,\frac{7}{2};\frac{(a-b) \cos ^2(x)}{a}\right)}{15 a^2}+\frac{8 (a-b) \cos ^2(x) \, _3F_2\left(2,2,2;1,\frac{7}{2};\frac{(a-b) \cos ^2(x)}{a}\right)}{15 a}+\frac{8 b^2 (a-b) \cos ^2(x) \cot ^4(x) \, _2F_1\left(2,2;\frac{7}{2};\frac{(a-b) \cos ^2(x)}{a}\right)}{5 a^3}-\frac{8 b^2 \cot ^4(x) \sin ^{-1}\left(\sqrt{\frac{(a-b) \cos ^2(x)}{a}}\right)}{a^2 \left(\frac{(a-b) \cos ^2(x)}{a}\right)^{3/2} \sqrt{\frac{\sin ^2(x) \left(a+b \cot ^2(x)\right)}{a}}}+\frac{8 b^2 \cot ^4(x) \sin ^{-1}\left(\sqrt{\frac{(a-b) \cos ^2(x)}{a}}\right)}{a^2 \sqrt{\frac{(a-b) \sin ^2(x) \cos ^2(x) \left(a+b \cot ^2(x)\right)}{a^2}}}+\frac{8 b (a-b) \cos ^2(x) \cot ^2(x) \, _2F_1\left(2,2;\frac{7}{2};\frac{(a-b) \cos ^2(x)}{a}\right)}{3 a^2}+\frac{12 b \cot ^2(x) \sin ^{-1}\left(\sqrt{\frac{(a-b) \cos ^2(x)}{a}}\right)}{a \sqrt{\frac{(a-b) \sin ^2(x) \cos ^2(x) \left(a+b \cot ^2(x)\right)}{a^2}}}+\frac{3 \sin ^{-1}\left(\sqrt{\frac{(a-b) \cos ^2(x)}{a}}\right)}{\sqrt{\frac{(a-b) \sin ^2(x) \cos ^2(x) \left(a+b \cot ^2(x)\right)}{a^2}}}+\frac{8 b^2 \cot ^2(x) \csc ^2(x)}{a (a-b)}+\frac{16 (a-b) \cos ^2(x) \, _2F_1\left(2,2;\frac{7}{2};\frac{(a-b) \cos ^2(x)}{a}\right)}{15 a}+\frac{12 b \csc ^2(x)}{a-b}+\frac{3 a \sec ^2(x)}{a-b}-\frac{12 b \cot ^2(x) \sin ^{-1}\left(\sqrt{\frac{(a-b) \cos ^2(x)}{a}}\right)}{a \left(\frac{(a-b) \cos ^2(x)}{a}\right)^{3/2} \sqrt{\frac{\sin ^2(x) \left(a+b \cot ^2(x)\right)}{a}}}-\frac{3 \sin ^{-1}\left(\sqrt{\frac{(a-b) \cos ^2(x)}{a}}\right)}{\left(\frac{(a-b) \cos ^2(x)}{a}\right)^{3/2} \sqrt{\frac{\sin ^2(x) \left(a+b \cot ^2(x)\right)}{a}}}\right)}{a \sqrt{a+b \cot ^2(x)}}","\frac{(a-2 b) \tan (x) \sqrt{a+b \cot ^2(x)}}{a^2 (a-b)}+\frac{\tan ^{-1}\left(\frac{\sqrt{a-b} \cot (x)}{\sqrt{a+b \cot ^2(x)}}\right)}{(a-b)^{3/2}}+\frac{b \tan (x)}{a (a-b) \sqrt{a+b \cot ^2(x)}}",1,"(Sin[x]^2*((12*b*Csc[x]^2)/(a - b) + (8*b^2*Cot[x]^2*Csc[x]^2)/(a*(a - b)) + (16*(a - b)*Cos[x]^2*Hypergeometric2F1[2, 2, 7/2, ((a - b)*Cos[x]^2)/a])/(15*a) + (8*(a - b)*b*Cos[x]^2*Cot[x]^2*Hypergeometric2F1[2, 2, 7/2, ((a - b)*Cos[x]^2)/a])/(3*a^2) + (8*(a - b)*b^2*Cos[x]^2*Cot[x]^4*Hypergeometric2F1[2, 2, 7/2, ((a - b)*Cos[x]^2)/a])/(5*a^3) + (8*(a - b)*Cos[x]^2*HypergeometricPFQ[{2, 2, 2}, {1, 7/2}, ((a - b)*Cos[x]^2)/a])/(15*a) + (16*(a - b)*b*Cos[x]^2*Cot[x]^2*HypergeometricPFQ[{2, 2, 2}, {1, 7/2}, ((a - b)*Cos[x]^2)/a])/(15*a^2) + (8*(a - b)*b^2*Cos[x]^2*Cot[x]^4*HypergeometricPFQ[{2, 2, 2}, {1, 7/2}, ((a - b)*Cos[x]^2)/a])/(15*a^3) + (3*a*Sec[x]^2)/(a - b) - (3*ArcSin[Sqrt[((a - b)*Cos[x]^2)/a]])/((((a - b)*Cos[x]^2)/a)^(3/2)*Sqrt[((a + b*Cot[x]^2)*Sin[x]^2)/a]) - (12*b*ArcSin[Sqrt[((a - b)*Cos[x]^2)/a]]*Cot[x]^2)/(a*(((a - b)*Cos[x]^2)/a)^(3/2)*Sqrt[((a + b*Cot[x]^2)*Sin[x]^2)/a]) - (8*b^2*ArcSin[Sqrt[((a - b)*Cos[x]^2)/a]]*Cot[x]^4)/(a^2*(((a - b)*Cos[x]^2)/a)^(3/2)*Sqrt[((a + b*Cot[x]^2)*Sin[x]^2)/a]) + (3*ArcSin[Sqrt[((a - b)*Cos[x]^2)/a]])/Sqrt[((a - b)*Cos[x]^2*(a + b*Cot[x]^2)*Sin[x]^2)/a^2] + (12*b*ArcSin[Sqrt[((a - b)*Cos[x]^2)/a]]*Cot[x]^2)/(a*Sqrt[((a - b)*Cos[x]^2*(a + b*Cot[x]^2)*Sin[x]^2)/a^2]) + (8*b^2*ArcSin[Sqrt[((a - b)*Cos[x]^2)/a]]*Cot[x]^4)/(a^2*Sqrt[((a - b)*Cos[x]^2*(a + b*Cot[x]^2)*Sin[x]^2)/a^2]))*Tan[x])/(a*Sqrt[a + b*Cot[x]^2])","C",0
54,1,69,82,0.098906,"\int \frac{\cot ^3(x)}{\left(a+b \cot ^2(x)\right)^{5/2}} \, dx","Integrate[Cot[x]^3/(a + b*Cot[x]^2)^(5/2),x]","\frac{3 b \left(a+b \cot ^2(x)\right) \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\frac{b \cot ^2(x)+a}{a-b}\right)+a (a-b)}{3 b (a-b)^2 \left(a+b \cot ^2(x)\right)^{3/2}}","\frac{a}{3 b (a-b) \left(a+b \cot ^2(x)\right)^{3/2}}+\frac{1}{(a-b)^2 \sqrt{a+b \cot ^2(x)}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \cot ^2(x)}}{\sqrt{a-b}}\right)}{(a-b)^{5/2}}",1,"(a*(a - b) + 3*b*(a + b*Cot[x]^2)*Hypergeometric2F1[-1/2, 1, 1/2, (a + b*Cot[x]^2)/(a - b)])/(3*(a - b)^2*b*(a + b*Cot[x]^2)^(3/2))","C",1
55,1,200,94,6.5130456,"\int \frac{\cot ^2(x)}{\left(a+b \cot ^2(x)\right)^{5/2}} \, dx","Integrate[Cot[x]^2/(a + b*Cot[x]^2)^(5/2),x]","\frac{\tan (x) \left(-\frac{35 a \sin ^2(x) \left(5 a+2 b \cot ^2(x)\right) \left(a \csc ^2(x) \left((a-4 b) \cot ^2(x)-3 a\right) \sqrt{\frac{(a-b) \sin ^2(x) \cos ^2(x) \left(a+b \cot ^2(x)\right)}{a^2}}+3 \left(a+b \cot ^2(x)\right)^2 \sin ^{-1}\left(\sqrt{\frac{(a-b) \cos ^2(x)}{a}}\right)\right)}{\sqrt{\frac{(a-b) \sin ^2(x) \cos ^2(x) \left(a+b \cot ^2(x)\right)}{a^2}}}-12 (a-b)^3 \cos ^4(x) \cot ^2(x) \left(a+b \cot ^2(x)\right) \, _2F_1\left(2,2;\frac{9}{2};\frac{(a-b) \cos ^2(x)}{a}\right)\right)}{315 a^3 (a-b)^2 \left(a+b \cot ^2(x)\right)^{3/2}}","-\frac{(2 a+b) \cot (x)}{3 a (a-b)^2 \sqrt{a+b \cot ^2(x)}}-\frac{\cot (x)}{3 (a-b) \left(a+b \cot ^2(x)\right)^{3/2}}+\frac{\tan ^{-1}\left(\frac{\sqrt{a-b} \cot (x)}{\sqrt{a+b \cot ^2(x)}}\right)}{(a-b)^{5/2}}",1,"((-12*(a - b)^3*Cos[x]^4*Cot[x]^2*(a + b*Cot[x]^2)*Hypergeometric2F1[2, 2, 9/2, ((a - b)*Cos[x]^2)/a] - (35*a*(5*a + 2*b*Cot[x]^2)*Sin[x]^2*(3*ArcSin[Sqrt[((a - b)*Cos[x]^2)/a]]*(a + b*Cot[x]^2)^2 + a*(-3*a + (a - 4*b)*Cot[x]^2)*Csc[x]^2*Sqrt[((a - b)*Cos[x]^2*(a + b*Cot[x]^2)*Sin[x]^2)/a^2]))/Sqrt[((a - b)*Cos[x]^2*(a + b*Cot[x]^2)*Sin[x]^2)/a^2])*Tan[x])/(315*a^3*(a - b)^2*(a + b*Cot[x]^2)^(3/2))","C",0
56,1,47,78,0.0410545,"\int \frac{\cot (x)}{\left(a+b \cot ^2(x)\right)^{5/2}} \, dx","Integrate[Cot[x]/(a + b*Cot[x]^2)^(5/2),x]","-\frac{\, _2F_1\left(-\frac{3}{2},1;-\frac{1}{2};\frac{b \cot ^2(x)+a}{a-b}\right)}{3 (a-b) \left(a+b \cot ^2(x)\right)^{3/2}}","-\frac{1}{(a-b)^2 \sqrt{a+b \cot ^2(x)}}-\frac{1}{3 (a-b) \left(a+b \cot ^2(x)\right)^{3/2}}+\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \cot ^2(x)}}{\sqrt{a-b}}\right)}{(a-b)^{5/2}}",1,"-1/3*Hypergeometric2F1[-3/2, 1, -1/2, (a + b*Cot[x]^2)/(a - b)]/((a - b)*(a + b*Cot[x]^2)^(3/2))","C",1
57,1,78,118,0.0478614,"\int \frac{\tan (x)}{\left(a+b \cot ^2(x)\right)^{5/2}} \, dx","Integrate[Tan[x]/(a + b*Cot[x]^2)^(5/2),x]","\frac{a \, _2F_1\left(-\frac{3}{2},1;-\frac{1}{2};\frac{b \cot ^2(x)+a}{a-b}\right)+(b-a) \, _2F_1\left(-\frac{3}{2},1;-\frac{1}{2};\frac{b \cot ^2(x)}{a}+1\right)}{3 a (a-b) \left(a+b \cot ^2(x)\right)^{3/2}}","\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \cot ^2(x)}}{\sqrt{a}}\right)}{a^{5/2}}+\frac{b (2 a-b)}{a^2 (a-b)^2 \sqrt{a+b \cot ^2(x)}}+\frac{b}{3 a (a-b) \left(a+b \cot ^2(x)\right)^{3/2}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \cot ^2(x)}}{\sqrt{a-b}}\right)}{(a-b)^{5/2}}",1,"(a*Hypergeometric2F1[-3/2, 1, -1/2, (a + b*Cot[x]^2)/(a - b)] + (-a + b)*Hypergeometric2F1[-3/2, 1, -1/2, 1 + (b*Cot[x]^2)/a])/(3*a*(a - b)*(a + b*Cot[x]^2)^(3/2))","C",1
58,1,1450,141,8.0171178,"\int \frac{\tan ^2(x)}{\left(a+b \cot ^2(x)\right)^{5/2}} \, dx","Integrate[Tan[x]^2/(a + b*Cot[x]^2)^(5/2),x]","\frac{\sin ^2(x) \left(\frac{176 (a-b) b^3 \cos ^2(x) \, _2F_1\left(2,2;\frac{9}{2};\frac{(a-b) \cos ^2(x)}{a}\right) \cot ^6(x)}{105 a^4}+\frac{32 (a-b) b^3 \cos ^2(x) \, _3F_2\left(2,2,2;1,\frac{9}{2};\frac{(a-b) \cos ^2(x)}{a}\right) \cot ^6(x)}{35 a^4}+\frac{16 (a-b) b^3 \cos ^2(x) \, _4F_3\left(2,2,2,2;1,1,\frac{9}{2};\frac{(a-b) \cos ^2(x)}{a}\right) \cot ^6(x)}{105 a^4}+\frac{16 b^3 \sin ^{-1}\left(\sqrt{\frac{(a-b) \cos ^2(x)}{a}}\right) \cot ^6(x)}{a^3 \left(\frac{(a-b) \cos ^2(x)}{a}\right)^{5/2} \sqrt{\frac{\left(b \cot ^2(x)+a\right) \sin ^2(x)}{a}}}+\frac{16 b^3 \sin ^{-1}\left(\sqrt{\frac{(a-b) \cos ^2(x)}{a}}\right) \cot ^6(x)}{a^3 \sqrt{\frac{(a-b) \cos ^2(x) \left(b \cot ^2(x)+a\right) \sin ^2(x)}{a^2}}}+\frac{64 b^3 \csc ^2(x) \cot ^4(x)}{3 a^2 (a-b)}+\frac{152 (a-b) b^2 \cos ^2(x) \, _2F_1\left(2,2;\frac{9}{2};\frac{(a-b) \cos ^2(x)}{a}\right) \cot ^4(x)}{35 a^3}+\frac{88 (a-b) b^2 \cos ^2(x) \, _3F_2\left(2,2,2;1,\frac{9}{2};\frac{(a-b) \cos ^2(x)}{a}\right) \cot ^4(x)}{35 a^3}+\frac{16 (a-b) b^2 \cos ^2(x) \, _4F_3\left(2,2,2,2;1,1,\frac{9}{2};\frac{(a-b) \cos ^2(x)}{a}\right) \cot ^4(x)}{35 a^3}+\frac{40 b^2 \sin ^{-1}\left(\sqrt{\frac{(a-b) \cos ^2(x)}{a}}\right) \cot ^4(x)}{a^2 \left(\frac{(a-b) \cos ^2(x)}{a}\right)^{5/2} \sqrt{\frac{\left(b \cot ^2(x)+a\right) \sin ^2(x)}{a}}}-\frac{32 b^3 \sin ^{-1}\left(\sqrt{\frac{(a-b) \cos ^2(x)}{a}}\right) \csc ^2(x) \cot ^4(x)}{a^2 (a-b) \sqrt{\frac{(a-b) \cos ^2(x) \left(b \cot ^2(x)+a\right) \sin ^2(x)}{a^2}}}+\frac{40 b^2 \sin ^{-1}\left(\sqrt{\frac{(a-b) \cos ^2(x)}{a}}\right) \cot ^4(x)}{a^2 \sqrt{\frac{(a-b) \cos ^2(x) \left(b \cot ^2(x)+a\right) \sin ^2(x)}{a^2}}}+\frac{160 b^2 \csc ^2(x) \cot ^2(x)}{3 a (a-b)}+\frac{124 (a-b) b \cos ^2(x) \, _2F_1\left(2,2;\frac{9}{2};\frac{(a-b) \cos ^2(x)}{a}\right) \cot ^2(x)}{35 a^2}+\frac{16 (a-b) b \cos ^2(x) \, _3F_2\left(2,2,2;1,\frac{9}{2};\frac{(a-b) \cos ^2(x)}{a}\right) \cot ^2(x)}{7 a^2}+\frac{16 (a-b) b \cos ^2(x) \, _4F_3\left(2,2,2,2;1,1,\frac{9}{2};\frac{(a-b) \cos ^2(x)}{a}\right) \cot ^2(x)}{35 a^2}+\frac{30 b \sin ^{-1}\left(\sqrt{\frac{(a-b) \cos ^2(x)}{a}}\right) \cot ^2(x)}{a \left(\frac{(a-b) \cos ^2(x)}{a}\right)^{5/2} \sqrt{\frac{\left(b \cot ^2(x)+a\right) \sin ^2(x)}{a}}}-\frac{80 b^2 \sin ^{-1}\left(\sqrt{\frac{(a-b) \cos ^2(x)}{a}}\right) \csc ^2(x) \cot ^2(x)}{a (a-b) \sqrt{\frac{(a-b) \cos ^2(x) \left(b \cot ^2(x)+a\right) \sin ^2(x)}{a^2}}}+\frac{30 b \sin ^{-1}\left(\sqrt{\frac{(a-b) \cos ^2(x)}{a}}\right) \cot ^2(x)}{a \sqrt{\frac{(a-b) \cos ^2(x) \left(b \cot ^2(x)+a\right) \sin ^2(x)}{a^2}}}-\frac{40 b^2 \csc ^4(x)}{(a-b)^2}-\frac{5 a^2 \sec ^4(x)}{(a-b)^2}-\frac{16 b^3 \left(\cot ^3(x)+\cot (x)\right)^2}{a (a-b)^2}+\frac{40 b \csc ^2(x)}{a-b}-\frac{30 a b \csc ^2(x) \sec ^2(x)}{(a-b)^2}+\frac{20 a \sec ^2(x)}{3 (a-b)}+\frac{92 (a-b) \cos ^2(x) \, _2F_1\left(2,2;\frac{9}{2};\frac{(a-b) \cos ^2(x)}{a}\right)}{105 a}+\frac{24 (a-b) \cos ^2(x) \, _3F_2\left(2,2,2;1,\frac{9}{2};\frac{(a-b) \cos ^2(x)}{a}\right)}{35 a}+\frac{16 (a-b) \cos ^2(x) \, _4F_3\left(2,2,2,2;1,1,\frac{9}{2};\frac{(a-b) \cos ^2(x)}{a}\right)}{105 a}+\frac{5 \sin ^{-1}\left(\sqrt{\frac{(a-b) \cos ^2(x)}{a}}\right)}{\left(\frac{(a-b) \cos ^2(x)}{a}\right)^{5/2} \sqrt{\frac{\left(b \cot ^2(x)+a\right) \sin ^2(x)}{a}}}-\frac{60 b \sin ^{-1}\left(\sqrt{\frac{(a-b) \cos ^2(x)}{a}}\right) \csc ^2(x)}{(a-b) \sqrt{\frac{(a-b) \cos ^2(x) \left(b \cot ^2(x)+a\right) \sin ^2(x)}{a^2}}}-\frac{10 a \sin ^{-1}\left(\sqrt{\frac{(a-b) \cos ^2(x)}{a}}\right) \sec ^2(x)}{(a-b) \sqrt{\frac{(a-b) \cos ^2(x) \left(b \cot ^2(x)+a\right) \sin ^2(x)}{a^2}}}+\frac{5 \sin ^{-1}\left(\sqrt{\frac{(a-b) \cos ^2(x)}{a}}\right)}{\sqrt{\frac{(a-b) \cos ^2(x) \left(b \cot ^2(x)+a\right) \sin ^2(x)}{a^2}}}\right) \tan (x)}{a^2 \sqrt{b \cot ^2(x)+a} \left(\frac{b \cot ^2(x)}{a}+1\right)}","\frac{(a-4 b) (3 a-2 b) \tan (x) \sqrt{a+b \cot ^2(x)}}{3 a^3 (a-b)^2}+\frac{b (7 a-4 b) \tan (x)}{3 a^2 (a-b)^2 \sqrt{a+b \cot ^2(x)}}+\frac{\tan ^{-1}\left(\frac{\sqrt{a-b} \cot (x)}{\sqrt{a+b \cot ^2(x)}}\right)}{(a-b)^{5/2}}+\frac{b \tan (x)}{3 a (a-b) \left(a+b \cot ^2(x)\right)^{3/2}}",1,"(Sin[x]^2*((-16*b^3*(Cot[x] + Cot[x]^3)^2)/(a*(a - b)^2) + (40*b*Csc[x]^2)/(a - b) + (160*b^2*Cot[x]^2*Csc[x]^2)/(3*a*(a - b)) + (64*b^3*Cot[x]^4*Csc[x]^2)/(3*a^2*(a - b)) - (40*b^2*Csc[x]^4)/(a - b)^2 + (92*(a - b)*Cos[x]^2*Hypergeometric2F1[2, 2, 9/2, ((a - b)*Cos[x]^2)/a])/(105*a) + (124*(a - b)*b*Cos[x]^2*Cot[x]^2*Hypergeometric2F1[2, 2, 9/2, ((a - b)*Cos[x]^2)/a])/(35*a^2) + (152*(a - b)*b^2*Cos[x]^2*Cot[x]^4*Hypergeometric2F1[2, 2, 9/2, ((a - b)*Cos[x]^2)/a])/(35*a^3) + (176*(a - b)*b^3*Cos[x]^2*Cot[x]^6*Hypergeometric2F1[2, 2, 9/2, ((a - b)*Cos[x]^2)/a])/(105*a^4) + (24*(a - b)*Cos[x]^2*HypergeometricPFQ[{2, 2, 2}, {1, 9/2}, ((a - b)*Cos[x]^2)/a])/(35*a) + (16*(a - b)*b*Cos[x]^2*Cot[x]^2*HypergeometricPFQ[{2, 2, 2}, {1, 9/2}, ((a - b)*Cos[x]^2)/a])/(7*a^2) + (88*(a - b)*b^2*Cos[x]^2*Cot[x]^4*HypergeometricPFQ[{2, 2, 2}, {1, 9/2}, ((a - b)*Cos[x]^2)/a])/(35*a^3) + (32*(a - b)*b^3*Cos[x]^2*Cot[x]^6*HypergeometricPFQ[{2, 2, 2}, {1, 9/2}, ((a - b)*Cos[x]^2)/a])/(35*a^4) + (16*(a - b)*Cos[x]^2*HypergeometricPFQ[{2, 2, 2, 2}, {1, 1, 9/2}, ((a - b)*Cos[x]^2)/a])/(105*a) + (16*(a - b)*b*Cos[x]^2*Cot[x]^2*HypergeometricPFQ[{2, 2, 2, 2}, {1, 1, 9/2}, ((a - b)*Cos[x]^2)/a])/(35*a^2) + (16*(a - b)*b^2*Cos[x]^2*Cot[x]^4*HypergeometricPFQ[{2, 2, 2, 2}, {1, 1, 9/2}, ((a - b)*Cos[x]^2)/a])/(35*a^3) + (16*(a - b)*b^3*Cos[x]^2*Cot[x]^6*HypergeometricPFQ[{2, 2, 2, 2}, {1, 1, 9/2}, ((a - b)*Cos[x]^2)/a])/(105*a^4) + (20*a*Sec[x]^2)/(3*(a - b)) - (30*a*b*Csc[x]^2*Sec[x]^2)/(a - b)^2 - (5*a^2*Sec[x]^4)/(a - b)^2 + (5*ArcSin[Sqrt[((a - b)*Cos[x]^2)/a]])/((((a - b)*Cos[x]^2)/a)^(5/2)*Sqrt[((a + b*Cot[x]^2)*Sin[x]^2)/a]) + (30*b*ArcSin[Sqrt[((a - b)*Cos[x]^2)/a]]*Cot[x]^2)/(a*(((a - b)*Cos[x]^2)/a)^(5/2)*Sqrt[((a + b*Cot[x]^2)*Sin[x]^2)/a]) + (40*b^2*ArcSin[Sqrt[((a - b)*Cos[x]^2)/a]]*Cot[x]^4)/(a^2*(((a - b)*Cos[x]^2)/a)^(5/2)*Sqrt[((a + b*Cot[x]^2)*Sin[x]^2)/a]) + (16*b^3*ArcSin[Sqrt[((a - b)*Cos[x]^2)/a]]*Cot[x]^6)/(a^3*(((a - b)*Cos[x]^2)/a)^(5/2)*Sqrt[((a + b*Cot[x]^2)*Sin[x]^2)/a]) + (5*ArcSin[Sqrt[((a - b)*Cos[x]^2)/a]])/Sqrt[((a - b)*Cos[x]^2*(a + b*Cot[x]^2)*Sin[x]^2)/a^2] + (30*b*ArcSin[Sqrt[((a - b)*Cos[x]^2)/a]]*Cot[x]^2)/(a*Sqrt[((a - b)*Cos[x]^2*(a + b*Cot[x]^2)*Sin[x]^2)/a^2]) + (40*b^2*ArcSin[Sqrt[((a - b)*Cos[x]^2)/a]]*Cot[x]^4)/(a^2*Sqrt[((a - b)*Cos[x]^2*(a + b*Cot[x]^2)*Sin[x]^2)/a^2]) + (16*b^3*ArcSin[Sqrt[((a - b)*Cos[x]^2)/a]]*Cot[x]^6)/(a^3*Sqrt[((a - b)*Cos[x]^2*(a + b*Cot[x]^2)*Sin[x]^2)/a^2]) - (60*b*ArcSin[Sqrt[((a - b)*Cos[x]^2)/a]]*Csc[x]^2)/((a - b)*Sqrt[((a - b)*Cos[x]^2*(a + b*Cot[x]^2)*Sin[x]^2)/a^2]) - (80*b^2*ArcSin[Sqrt[((a - b)*Cos[x]^2)/a]]*Cot[x]^2*Csc[x]^2)/(a*(a - b)*Sqrt[((a - b)*Cos[x]^2*(a + b*Cot[x]^2)*Sin[x]^2)/a^2]) - (32*b^3*ArcSin[Sqrt[((a - b)*Cos[x]^2)/a]]*Cot[x]^4*Csc[x]^2)/(a^2*(a - b)*Sqrt[((a - b)*Cos[x]^2*(a + b*Cot[x]^2)*Sin[x]^2)/a^2]) - (10*a*ArcSin[Sqrt[((a - b)*Cos[x]^2)/a]]*Sec[x]^2)/((a - b)*Sqrt[((a - b)*Cos[x]^2*(a + b*Cot[x]^2)*Sin[x]^2)/a^2]))*Tan[x])/(a^2*Sqrt[a + b*Cot[x]^2]*(1 + (b*Cot[x]^2)/a))","C",0
59,1,57,37,0.0358874,"\int \frac{1}{1+\cot ^3(x)} \, dx","Integrate[(1 + Cot[x]^3)^(-1),x]","\frac{1}{3} \log \left(\tan ^2(x)-\tan (x)+1\right)+\left(-\frac{1}{4}-\frac{i}{4}\right) \log (-\tan (x)+i)-\left(\frac{1}{4}-\frac{i}{4}\right) \log (\tan (x)+i)-\frac{1}{6} \log (\tan (x)+1)","\frac{x}{2}+\frac{1}{2} \log (\sin (x))+\frac{1}{3} \log \left(\cot ^2(x)-\cot (x)+1\right)-\frac{1}{6} \log (\cot (x)+1)",1,"(-1/4 - I/4)*Log[I - Tan[x]] - (1/4 - I/4)*Log[I + Tan[x]] - Log[1 + Tan[x]]/6 + Log[1 - Tan[x] + Tan[x]^2]/3","C",1
60,1,86,90,0.1541212,"\int \cot (x) \sqrt{a+b \cot ^4(x)} \, dx","Integrate[Cot[x]*Sqrt[a + b*Cot[x]^4],x]","\frac{1}{2} \left(-\sqrt{a+b \cot ^4(x)}+\sqrt{b} \tanh ^{-1}\left(\frac{\sqrt{b} \cot ^2(x)}{\sqrt{a+b \cot ^4(x)}}\right)+\sqrt{a+b} \tanh ^{-1}\left(\frac{a-b \cot ^2(x)}{\sqrt{a+b} \sqrt{a+b \cot ^4(x)}}\right)\right)","-\frac{1}{2} \sqrt{a+b \cot ^4(x)}+\frac{1}{2} \sqrt{b} \tanh ^{-1}\left(\frac{\sqrt{b} \cot ^2(x)}{\sqrt{a+b \cot ^4(x)}}\right)+\frac{1}{2} \sqrt{a+b} \tanh ^{-1}\left(\frac{a-b \cot ^2(x)}{\sqrt{a+b} \sqrt{a+b \cot ^4(x)}}\right)",1,"(Sqrt[b]*ArcTanh[(Sqrt[b]*Cot[x]^2)/Sqrt[a + b*Cot[x]^4]] + Sqrt[a + b]*ArcTanh[(a - b*Cot[x]^2)/(Sqrt[a + b]*Sqrt[a + b*Cot[x]^4])] - Sqrt[a + b*Cot[x]^4])/2","A",1
61,1,167,126,4.3964005,"\int \cot (x) \left(a+b \cot ^4(x)\right)^{3/2} \, dx","Integrate[Cot[x]*(a + b*Cot[x]^4)^(3/2),x]","\frac{1}{12} \left(-\sqrt{a+b \cot ^4(x)} \left(8 a+2 b \cot ^4(x)-3 b \cot ^2(x)+6 b\right)+\frac{3 \sqrt{a} \sqrt{b} \sqrt{a+b \cot ^4(x)} \sinh ^{-1}\left(\frac{\sqrt{b} \cot ^2(x)}{\sqrt{a}}\right)}{\sqrt{\frac{b \cot ^4(x)}{a}+1}}+6 (a+b)^{3/2} \tanh ^{-1}\left(\frac{a-b \cot ^2(x)}{\sqrt{a+b} \sqrt{a+b \cot ^4(x)}}\right)+6 \sqrt{b} (a+b) \tanh ^{-1}\left(\frac{\sqrt{b} \cot ^2(x)}{\sqrt{a+b \cot ^4(x)}}\right)\right)","-\frac{1}{6} \left(a+b \cot ^4(x)\right)^{3/2}-\frac{1}{4} \left(2 (a+b)-b \cot ^2(x)\right) \sqrt{a+b \cot ^4(x)}+\frac{1}{2} (a+b)^{3/2} \tanh ^{-1}\left(\frac{a-b \cot ^2(x)}{\sqrt{a+b} \sqrt{a+b \cot ^4(x)}}\right)+\frac{1}{4} \sqrt{b} (3 a+2 b) \tanh ^{-1}\left(\frac{\sqrt{b} \cot ^2(x)}{\sqrt{a+b \cot ^4(x)}}\right)",1,"(6*Sqrt[b]*(a + b)*ArcTanh[(Sqrt[b]*Cot[x]^2)/Sqrt[a + b*Cot[x]^4]] + 6*(a + b)^(3/2)*ArcTanh[(a - b*Cot[x]^2)/(Sqrt[a + b]*Sqrt[a + b*Cot[x]^4])] - Sqrt[a + b*Cot[x]^4]*(8*a + 6*b - 3*b*Cot[x]^2 + 2*b*Cot[x]^4) + (3*Sqrt[a]*Sqrt[b]*ArcSinh[(Sqrt[b]*Cot[x]^2)/Sqrt[a]]*Sqrt[a + b*Cot[x]^4])/Sqrt[1 + (b*Cot[x]^4)/a])/12","A",1
62,1,41,41,0.0191381,"\int \frac{\cot (x)}{\sqrt{a+b \cot ^4(x)}} \, dx","Integrate[Cot[x]/Sqrt[a + b*Cot[x]^4],x]","\frac{\tanh ^{-1}\left(\frac{a-b \cot ^2(x)}{\sqrt{a+b} \sqrt{a+b \cot ^4(x)}}\right)}{2 \sqrt{a+b}}","\frac{\tanh ^{-1}\left(\frac{a-b \cot ^2(x)}{\sqrt{a+b} \sqrt{a+b \cot ^4(x)}}\right)}{2 \sqrt{a+b}}",1,"ArcTanh[(a - b*Cot[x]^2)/(Sqrt[a + b]*Sqrt[a + b*Cot[x]^4])]/(2*Sqrt[a + b])","A",1
63,1,73,74,0.3024335,"\int \frac{\cot (x)}{\left(a+b \cot ^4(x)\right)^{3/2}} \, dx","Integrate[Cot[x]/(a + b*Cot[x]^4)^(3/2),x]","\frac{1}{2} \left(\frac{\tanh ^{-1}\left(\frac{a-b \cot ^2(x)}{\sqrt{a+b} \sqrt{a+b \cot ^4(x)}}\right)}{(a+b)^{3/2}}-\frac{a+b \cot ^2(x)}{a (a+b) \sqrt{a+b \cot ^4(x)}}\right)","\frac{\tanh ^{-1}\left(\frac{a-b \cot ^2(x)}{\sqrt{a+b} \sqrt{a+b \cot ^4(x)}}\right)}{2 (a+b)^{3/2}}-\frac{a+b \cot ^2(x)}{2 a (a+b) \sqrt{a+b \cot ^4(x)}}",1,"(ArcTanh[(a - b*Cot[x]^2)/(Sqrt[a + b]*Sqrt[a + b*Cot[x]^4])]/(a + b)^(3/2) - (a + b*Cot[x]^2)/(a*(a + b)*Sqrt[a + b*Cot[x]^4]))/2","A",1
64,1,114,117,0.7451967,"\int \frac{\cot (x)}{\left(a+b \cot ^4(x)\right)^{5/2}} \, dx","Integrate[Cot[x]/(a + b*Cot[x]^4)^(5/2),x]","\frac{\tanh ^{-1}\left(\frac{a-b \cot ^2(x)}{\sqrt{a+b} \sqrt{a+b \cot ^4(x)}}\right)}{2 (a+b)^{5/2}}-\frac{3 a^2 b \cot ^4(x)+a^2 (4 a+b)+b^2 (5 a+2 b) \cot ^6(x)+3 a b (2 a+b) \cot ^2(x)}{6 a^2 (a+b)^2 \left(a+b \cot ^4(x)\right)^{3/2}}","-\frac{3 a^2+b (5 a+2 b) \cot ^2(x)}{6 a^2 (a+b)^2 \sqrt{a+b \cot ^4(x)}}-\frac{a+b \cot ^2(x)}{6 a (a+b) \left(a+b \cot ^4(x)\right)^{3/2}}+\frac{\tanh ^{-1}\left(\frac{a-b \cot ^2(x)}{\sqrt{a+b} \sqrt{a+b \cot ^4(x)}}\right)}{2 (a+b)^{5/2}}",1,"ArcTanh[(a - b*Cot[x]^2)/(Sqrt[a + b]*Sqrt[a + b*Cot[x]^4])]/(2*(a + b)^(5/2)) - (a^2*(4*a + b) + 3*a*b*(2*a + b)*Cot[x]^2 + 3*a^2*b*Cot[x]^4 + b^2*(5*a + 2*b)*Cot[x]^6)/(6*a^2*(a + b)^2*(a + b*Cot[x]^4)^(3/2))","A",1