1,1,233,0,0.2990694,"\int \frac{A+C \cot ^2(c+d x)}{\sqrt{b \tan (c+d x)}} \, dx","Int[(A + C*Cot[c + d*x]^2)/Sqrt[b*Tan[c + d*x]],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}}","-\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,"-(((A - C)*ArcTan[1 - (Sqrt[2]*Sqrt[b*Tan[c + d*x]])/Sqrt[b]])/(Sqrt[2]*Sqrt[b]*d)) + ((A - C)*ArcTan[1 + (Sqrt[2]*Sqrt[b*Tan[c + d*x]])/Sqrt[b]])/(Sqrt[2]*Sqrt[b]*d) - ((A - C)*Log[Sqrt[b] + Sqrt[b]*Tan[c + d*x] - Sqrt[2]*Sqrt[b*Tan[c + d*x]]])/(2*Sqrt[2]*Sqrt[b]*d) + ((A - C)*Log[Sqrt[b] + Sqrt[b]*Tan[c + d*x] + Sqrt[2]*Sqrt[b*Tan[c + d*x]]])/(2*Sqrt[2]*Sqrt[b]*d) - (2*b*C)/(3*d*(b*Tan[c + d*x])^(3/2))","A",15,12,25,0.4800,1,"{3673, 3629, 12, 16, 3476, 329, 211, 1165, 628, 1162, 617, 204}"
2,1,20,0,0.0125953,"\int \left(a+b \cot ^2(c+d x)\right) \, dx","Int[a + b*Cot[c + d*x]^2,x]","a x-\frac{b \cot (c+d x)}{d}-b x","a x-\frac{b \cot (c+d x)}{d}-b x",1,"a*x - b*x - (b*Cot[c + d*x])/d","A",3,2,12,0.1667,1,"{3473, 8}"
3,1,47,0,0.0333153,"\int \left(a+b \cot ^2(c+d x)\right)^2 \, dx","Int[(a + b*Cot[c + d*x]^2)^2,x]","-\frac{b (2 a-b) \cot (c+d x)}{d}+x (a-b)^2-\frac{b^2 \cot ^3(c+d x)}{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,"(a - b)^2*x - ((2*a - b)*b*Cot[c + d*x])/d - (b^2*Cot[c + d*x]^3)/(3*d)","A",4,3,14,0.2143,1,"{3661, 390, 203}"
4,1,78,0,0.0472866,"\int \left(a+b \cot ^2(c+d x)\right)^3 \, dx","Int[(a + b*Cot[c + d*x]^2)^3,x]","-\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}","-\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,"(a - b)^3*x - (b*(3*a^2 - 3*a*b + b^2)*Cot[c + d*x])/d - ((3*a - b)*b^2*Cot[c + d*x]^3)/(3*d) - (b^3*Cot[c + d*x]^5)/(5*d)","A",4,3,14,0.2143,1,"{3661, 390, 203}"
5,1,49,0,0.0726474,"\int \frac{1}{a+b \cot ^2(c+d x)} \, dx","Int[(a + b*Cot[c + d*x]^2)^(-1),x]","\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}","\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,"x/(a - b) + (Sqrt[b]*ArcTan[(Sqrt[b]*Cot[c + d*x])/Sqrt[a]])/(Sqrt[a]*(a - b)*d)","A",3,3,14,0.2143,1,"{3660, 3675, 205}"
6,1,97,0,0.1003192,"\int \frac{1}{\left(a+b \cot ^2(c+d x)\right)^2} \, dx","Int[(a + b*Cot[c + d*x]^2)^(-2),x]","\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}","\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,"x/(a - b)^2 + ((3*a - b)*Sqrt[b]*ArcTan[(Sqrt[b]*Cot[c + d*x])/Sqrt[a]])/(2*a^(3/2)*(a - b)^2*d) + (b*Cot[c + d*x])/(2*a*(a - b)*d*(a + b*Cot[c + d*x]^2))","A",5,5,14,0.3571,1,"{3661, 414, 522, 203, 205}"
7,1,150,0,0.1569412,"\int \frac{1}{\left(a+b \cot ^2(c+d x)\right)^3} \, dx","Int[(a + b*Cot[c + d*x]^2)^(-3),x]","\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 (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{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}","\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 (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{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,"x/(a - b)^3 + (Sqrt[b]*(15*a^2 - 10*a*b + 3*b^2)*ArcTan[(Sqrt[b]*Cot[c + d*x])/Sqrt[a]])/(8*a^(5/2)*(a - b)^3*d) + (b*Cot[c + d*x])/(4*a*(a - b)*d*(a + b*Cot[c + d*x]^2)^2) + ((7*a - 3*b)*b*Cot[c + d*x])/(8*a^2*(a - b)^2*d*(a + b*Cot[c + d*x]^2))","A",6,6,14,0.4286,1,"{3661, 414, 527, 522, 203, 205}"
8,1,22,0,0.0184442,"\int \left(1+\cot ^2(x)\right)^{3/2} \, dx","Int[(1 + Cot[x]^2)^(3/2),x]","-\frac{1}{2} \cot (x) \sqrt{\csc ^2(x)}-\frac{1}{2} \sinh ^{-1}(\cot (x))","-\frac{1}{2} \cot (x) \sqrt{\csc ^2(x)}-\frac{1}{2} \sinh ^{-1}(\cot (x))",1,"-ArcSinh[Cot[x]]/2 - (Cot[x]*Sqrt[Csc[x]^2])/2","A",4,4,10,0.4000,1,"{3657, 4122, 195, 215}"
9,1,5,0,0.0140171,"\int \sqrt{1+\cot ^2(x)} \, dx","Int[Sqrt[1 + Cot[x]^2],x]","-\sinh ^{-1}(\cot (x))","-\sinh ^{-1}(\cot (x))",1,"-ArcSinh[Cot[x]]","A",3,3,10,0.3000,1,"{3657, 4122, 215}"
10,1,12,0,0.0205382,"\int \frac{1}{\sqrt{1+\cot ^2(x)}} \, dx","Int[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",3,3,10,0.3000,1,"{3657, 4122, 191}"
11,1,35,0,0.0255565,"\int \left(-1-\cot ^2(x)\right)^{3/2} \, dx","Int[(-1 - Cot[x]^2)^(3/2),x]","\frac{1}{2} \cot (x) \sqrt{-\csc ^2(x)}-\frac{1}{2} \tan ^{-1}\left(\frac{\cot (x)}{\sqrt{-\csc ^2(x)}}\right)","\frac{1}{2} \cot (x) \sqrt{-\csc ^2(x)}-\frac{1}{2} \tan ^{-1}\left(\frac{\cot (x)}{\sqrt{-\csc ^2(x)}}\right)",1,"-ArcTan[Cot[x]/Sqrt[-Csc[x]^2]]/2 + (Cot[x]*Sqrt[-Csc[x]^2])/2","A",5,5,12,0.4167,1,"{3657, 4122, 195, 217, 203}"
12,1,14,0,0.0202224,"\int \sqrt{-1-\cot ^2(x)} \, dx","Int[Sqrt[-1 - Cot[x]^2],x]","\tan ^{-1}\left(\frac{\cot (x)}{\sqrt{-\csc ^2(x)}}\right)","\tan ^{-1}\left(\frac{\cot (x)}{\sqrt{-\csc ^2(x)}}\right)",1,"ArcTan[Cot[x]/Sqrt[-Csc[x]^2]]","A",4,4,12,0.3333,1,"{3657, 4122, 217, 203}"
13,1,14,0,0.0226727,"\int \frac{1}{\sqrt{-1-\cot ^2(x)}} \, dx","Int[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",3,3,12,0.2500,1,"{3657, 4122, 191}"
14,1,28,0,0.0962848,"\int \frac{\cot ^3(x)}{\sqrt{a+a \cot ^2(x)}} \, dx","Int[Cot[x]^3/Sqrt[a + a*Cot[x]^2],x]","-\frac{\sqrt{a \csc ^2(x)}}{a}-\frac{1}{\sqrt{a \csc ^2(x)}}","-\frac{\sqrt{a \csc ^2(x)}}{a}-\frac{1}{\sqrt{a \csc ^2(x)}}",1,"-(1/Sqrt[a*Csc[x]^2]) - Sqrt[a*Csc[x]^2]/a","A",4,3,17,0.1765,1,"{3657, 4124, 43}"
15,1,31,0,0.0997412,"\int \frac{\cot ^2(x)}{\sqrt{a+a \cot ^2(x)}} \, dx","Int[Cot[x]^2/Sqrt[a + a*Cot[x]^2],x]","\frac{\cot (x)}{\sqrt{a \csc ^2(x)}}-\frac{\csc (x) \tanh ^{-1}(\cos (x))}{\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,"Cot[x]/Sqrt[a*Csc[x]^2] - (ArcTanh[Cos[x]]*Csc[x])/Sqrt[a*Csc[x]^2]","A",5,5,17,0.2941,1,"{3657, 4125, 2592, 321, 206}"
16,1,10,0,0.0482915,"\int \frac{\cot (x)}{\sqrt{a+a \cot ^2(x)}} \, dx","Int[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",3,3,15,0.2000,1,"{3657, 4124, 32}"
17,1,36,0,0.0874944,"\int \frac{\tan (x)}{\sqrt{a+a \cot ^2(x)}} \, dx","Int[Tan[x]/Sqrt[a + a*Cot[x]^2],x]","\frac{\tanh ^{-1}\left(\frac{\sqrt{a \csc ^2(x)}}{\sqrt{a}}\right)}{\sqrt{a}}-\frac{1}{\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,"ArcTanh[Sqrt[a*Csc[x]^2]/Sqrt[a]]/Sqrt[a] - 1/Sqrt[a*Csc[x]^2]","A",5,5,15,0.3333,1,"{3657, 4124, 51, 63, 207}"
18,1,29,0,0.1012825,"\int \frac{\tan ^2(x)}{\sqrt{a+a \cot ^2(x)}} \, dx","Int[Tan[x]^2/Sqrt[a + a*Cot[x]^2],x]","\frac{\cot (x)}{\sqrt{a \csc ^2(x)}}+\frac{\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]/Sqrt[a*Csc[x]^2] + (Csc[x]*Sec[x])/Sqrt[a*Csc[x]^2]","A",5,4,17,0.2353,1,"{3657, 4125, 2590, 14}"
19,1,66,0,0.1160807,"\int \cot ^3(x) \sqrt{a+b \cot ^2(x)} \, dx","Int[Cot[x]^3*Sqrt[a + b*Cot[x]^2],x]","-\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)","-\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 + b*Cot[x]^2)^(3/2)/(3*b)","A",6,6,17,0.3529,1,"{3670, 446, 80, 50, 63, 208}"
20,1,48,0,0.0667223,"\int \cot (x) \sqrt{a+b \cot ^2(x)} \, dx","Int[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",5,5,15,0.3333,1,"{3670, 444, 50, 63, 208}"
21,1,60,0,0.096891,"\int \sqrt{a+b \cot ^2(x)} \tan (x) \, dx","Int[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",7,5,15,0.3333,1,"{3670, 446, 83, 63, 208}"
22,1,89,0,0.1249895,"\int \cot ^2(x) \sqrt{a+b \cot ^2(x)} \, dx","Int[Cot[x]^2*Sqrt[a + b*Cot[x]^2],x]","-\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}}","-\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,"Sqrt[a - b]*ArcTan[(Sqrt[a - b]*Cot[x])/Sqrt[a + b*Cot[x]^2]] - ((a - 2*b)*ArcTanh[(Sqrt[b]*Cot[x])/Sqrt[a + b*Cot[x]^2]])/(2*Sqrt[b]) - (Cot[x]*Sqrt[a + b*Cot[x]^2])/2","A",7,7,17,0.4118,1,"{3670, 478, 523, 217, 206, 377, 203}"
23,1,65,0,0.0452154,"\int \sqrt{a+b \cot ^2(x)} \, dx","Int[Sqrt[a + b*Cot[x]^2],x]","-\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)","-\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,"-(Sqrt[a - b]*ArcTan[(Sqrt[a - b]*Cot[x])/Sqrt[a + b*Cot[x]^2]]) - Sqrt[b]*ArcTanh[(Sqrt[b]*Cot[x])/Sqrt[a + b*Cot[x]^2]]","A",6,6,12,0.5000,1,"{3661, 402, 217, 206, 377, 203}"
24,1,51,0,0.088678,"\int \sqrt{a+b \cot ^2(x)} \tan ^2(x) \, dx","Int[Sqrt[a + b*Cot[x]^2]*Tan[x]^2,x]","\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)}","\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]*ArcTan[(Sqrt[a - b]*Cot[x])/Sqrt[a + b*Cot[x]^2]] + Sqrt[a + b*Cot[x]^2]*Tan[x]","A",5,5,17,0.2941,1,"{3670, 475, 12, 377, 203}"
25,1,85,0,0.1408704,"\int \sqrt{a+b \cot ^2(x)} \tan ^4(x) \, dx","Int[Sqrt[a + b*Cot[x]^2]*Tan[x]^4,x]","\frac{1}{3} \tan ^3(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{(3 a-b) \tan (x) \sqrt{a+b \cot ^2(x)}}{3 a}","\frac{1}{3} \tan ^3(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{(3 a-b) \tan (x) \sqrt{a+b \cot ^2(x)}}{3 a}",1,"-(Sqrt[a - b]*ArcTan[(Sqrt[a - b]*Cot[x])/Sqrt[a + b*Cot[x]^2]]) - ((3*a - b)*Sqrt[a + b*Cot[x]^2]*Tan[x])/(3*a) + (Sqrt[a + b*Cot[x]^2]*Tan[x]^3)/3","A",6,6,17,0.3529,1,"{3670, 475, 583, 12, 377, 203}"
26,1,88,0,0.1368617,"\int \cot ^3(x) \left(a+b \cot ^2(x)\right)^{3/2} \, dx","Int[Cot[x]^3*(a + b*Cot[x]^2)^(3/2),x]","-\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)","-\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]]) + (a - b)*Sqrt[a + b*Cot[x]^2] + (a + b*Cot[x]^2)^(3/2)/3 - (a + b*Cot[x]^2)^(5/2)/(5*b)","A",7,6,17,0.3529,1,"{3670, 446, 80, 50, 63, 208}"
27,1,127,0,0.229799,"\int \cot ^2(x) \left(a+b \cot ^2(x)\right)^{3/2} \, dx","Int[Cot[x]^2*(a + b*Cot[x]^2)^(3/2),x]","-\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}{4} b \cot ^3(x) \sqrt{a+b \cot ^2(x)}-\frac{1}{8} (5 a-4 b) \cot (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)","-\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}{4} b \cot ^3(x) \sqrt{a+b \cot ^2(x)}-\frac{1}{8} (5 a-4 b) \cot (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,"(a - b)^(3/2)*ArcTan[(Sqrt[a - b]*Cot[x])/Sqrt[a + b*Cot[x]^2]] - ((3*a^2 - 12*a*b + 8*b^2)*ArcTanh[(Sqrt[b]*Cot[x])/Sqrt[a + b*Cot[x]^2]])/(8*Sqrt[b]) - ((5*a - 4*b)*Cot[x]*Sqrt[a + b*Cot[x]^2])/8 - (b*Cot[x]^3*Sqrt[a + b*Cot[x]^2])/4","A",8,8,17,0.4706,1,"{3670, 477, 582, 523, 217, 206, 377, 203}"
28,1,69,0,0.091423,"\int \cot (x) \left(a+b \cot ^2(x)\right)^{3/2} \, dx","Int[Cot[x]*(a + b*Cot[x]^2)^(3/2),x]","-(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)","-(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]] - (a - b)*Sqrt[a + b*Cot[x]^2] - (a + b*Cot[x]^2)^(3/2)/3","A",6,5,15,0.3333,1,"{3670, 444, 50, 63, 208}"
29,1,75,0,0.1278041,"\int \left(a+b \cot ^2(x)\right)^{3/2} \tan (x) \, dx","Int[(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",8,6,15,0.4000,1,"{3670, 446, 84, 156, 63, 208}"
30,1,80,0,0.1254827,"\int \left(a+b \cot ^2(x)\right)^{3/2} \tan ^2(x) \, dx","Int[(a + b*Cot[x]^2)^(3/2)*Tan[x]^2,x]","-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)}","-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,"(a - b)^(3/2)*ArcTan[(Sqrt[a - b]*Cot[x])/Sqrt[a + b*Cot[x]^2]] - b^(3/2)*ArcTanh[(Sqrt[b]*Cot[x])/Sqrt[a + b*Cot[x]^2]] + a*Sqrt[a + b*Cot[x]^2]*Tan[x]","A",7,7,17,0.4118,1,"{3670, 474, 523, 217, 206, 377, 203}"
31,1,171,0,0.1863505,"\int \left(a+b \cot ^2(c+d x)\right)^{5/2} \, dx","Int[(a + b*Cot[c + d*x]^2)^(5/2),x]","-\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}","-\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,"-(((a - b)^(5/2)*ArcTan[(Sqrt[a - b]*Cot[c + d*x])/Sqrt[a + b*Cot[c + d*x]^2]])/d) - (Sqrt[b]*(15*a^2 - 20*a*b + 8*b^2)*ArcTanh[(Sqrt[b]*Cot[c + d*x])/Sqrt[a + b*Cot[c + d*x]^2]])/(8*d) - ((7*a - 4*b)*b*Cot[c + d*x]*Sqrt[a + b*Cot[c + d*x]^2])/(8*d) - (b*Cot[c + d*x]*(a + b*Cot[c + d*x]^2)^(3/2))/(4*d)","A",8,8,16,0.5000,1,"{3661, 416, 528, 523, 217, 206, 377, 203}"
32,1,126,0,0.0978133,"\int \left(a+b \cot ^2(c+d x)\right)^{3/2} \, dx","Int[(a + b*Cot[c + d*x]^2)^(3/2),x]","-\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}","-\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,"-(((a - b)^(3/2)*ArcTan[(Sqrt[a - b]*Cot[c + d*x])/Sqrt[a + b*Cot[c + d*x]^2]])/d) - ((3*a - 2*b)*Sqrt[b]*ArcTanh[(Sqrt[b]*Cot[c + d*x])/Sqrt[a + b*Cot[c + d*x]^2]])/(2*d) - (b*Cot[c + d*x]*Sqrt[a + b*Cot[c + d*x]^2])/(2*d)","A",7,7,16,0.4375,1,"{3661, 416, 523, 217, 206, 377, 203}"
33,1,87,0,0.0549406,"\int \sqrt{a+b \cot ^2(c+d x)} \, dx","Int[Sqrt[a + b*Cot[c + d*x]^2],x]","-\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}","-\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,"-((Sqrt[a - b]*ArcTan[(Sqrt[a - b]*Cot[c + d*x])/Sqrt[a + b*Cot[c + d*x]^2]])/d) - (Sqrt[b]*ArcTanh[(Sqrt[b]*Cot[c + d*x])/Sqrt[a + b*Cot[c + d*x]^2]])/d","A",6,6,16,0.3750,1,"{3661, 402, 217, 206, 377, 203}"
34,1,47,0,0.0329305,"\int \frac{1}{\sqrt{a+b \cot ^2(c+d x)}} \, dx","Int[1/Sqrt[a + b*Cot[c + d*x]^2],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}}","-\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,"-(ArcTan[(Sqrt[a - b]*Cot[c + d*x])/Sqrt[a + b*Cot[c + d*x]^2]]/(Sqrt[a - b]*d))","A",3,3,16,0.1875,1,"{3661, 377, 203}"
35,1,85,0,0.0612232,"\int \frac{1}{\left(a+b \cot ^2(c+d x)\right)^{3/2}} \, dx","Int[(a + b*Cot[c + d*x]^2)^(-3/2),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}}","\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,"-(ArcTan[(Sqrt[a - b]*Cot[c + d*x])/Sqrt[a + b*Cot[c + d*x]^2]]/((a - b)^(3/2)*d)) + (b*Cot[c + d*x])/(a*(a - b)*d*Sqrt[a + b*Cot[c + d*x]^2])","A",4,4,16,0.2500,1,"{3661, 382, 377, 203}"
36,1,135,0,0.1095086,"\int \frac{1}{\left(a+b \cot ^2(c+d x)\right)^{5/2}} \, dx","Int[(a + b*Cot[c + d*x]^2)^(-5/2),x]","\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}}","\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,"-(ArcTan[(Sqrt[a - b]*Cot[c + d*x])/Sqrt[a + b*Cot[c + d*x]^2]]/((a - b)^(5/2)*d)) + (b*Cot[c + d*x])/(3*a*(a - b)*d*(a + b*Cot[c + d*x]^2)^(3/2)) + ((5*a - 2*b)*b*Cot[c + d*x])/(3*a^2*(a - b)^2*d*Sqrt[a + b*Cot[c + d*x]^2])","A",6,6,16,0.3750,1,"{3661, 414, 527, 12, 377, 203}"
37,1,190,0,0.1940826,"\int \frac{1}{\left(a+b \cot ^2(c+d x)\right)^{7/2}} \, dx","Int[(a + b*Cot[c + d*x]^2)^(-7/2),x]","\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 (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 \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}}","\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 (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 \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,"-(ArcTan[(Sqrt[a - b]*Cot[c + d*x])/Sqrt[a + b*Cot[c + d*x]^2]]/((a - b)^(7/2)*d)) + (b*Cot[c + d*x])/(5*a*(a - b)*d*(a + b*Cot[c + d*x]^2)^(5/2)) + ((9*a - 4*b)*b*Cot[c + d*x])/(15*a^2*(a - b)^2*d*(a + b*Cot[c + d*x]^2)^(3/2)) + (b*(33*a^2 - 26*a*b + 8*b^2)*Cot[c + d*x])/(15*a^3*(a - b)^3*d*Sqrt[a + b*Cot[c + d*x]^2])","A",7,6,16,0.3750,1,"{3661, 414, 527, 12, 377, 203}"
38,1,54,0,0.0459119,"\int \left(1-\cot ^2(x)\right)^{3/2} \, dx","Int[(1 - Cot[x]^2)^(3/2),x]","\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))","\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,"(5*ArcSin[Cot[x]])/2 - 2*Sqrt[2]*ArcTan[(Sqrt[2]*Cot[x])/Sqrt[1 - Cot[x]^2]] + (Cot[x]*Sqrt[1 - Cot[x]^2])/2","A",6,6,12,0.5000,1,"{3661, 416, 523, 216, 377, 203}"
39,1,32,0,0.0266071,"\int \sqrt{1-\cot ^2(x)} \, dx","Int[Sqrt[1 - Cot[x]^2],x]","\sin ^{-1}(\cot (x))-\sqrt{2} \tan ^{-1}\left(\frac{\sqrt{2} \cot (x)}{\sqrt{1-\cot ^2(x)}}\right)","\sin ^{-1}(\cot (x))-\sqrt{2} \tan ^{-1}\left(\frac{\sqrt{2} \cot (x)}{\sqrt{1-\cot ^2(x)}}\right)",1,"ArcSin[Cot[x]] - Sqrt[2]*ArcTan[(Sqrt[2]*Cot[x])/Sqrt[1 - Cot[x]^2]]","A",5,5,12,0.4167,1,"{3661, 402, 216, 377, 203}"
40,1,28,0,0.0191643,"\int \frac{1}{\sqrt{1-\cot ^2(x)}} \, dx","Int[1/Sqrt[1 - Cot[x]^2],x]","-\frac{\tan ^{-1}\left(\frac{\sqrt{2} \cot (x)}{\sqrt{1-\cot ^2(x)}}\right)}{\sqrt{2}}","-\frac{\tan ^{-1}\left(\frac{\sqrt{2} \cot (x)}{\sqrt{1-\cot ^2(x)}}\right)}{\sqrt{2}}",1,"-(ArcTan[(Sqrt[2]*Cot[x])/Sqrt[1 - Cot[x]^2]]/Sqrt[2])","A",3,3,12,0.2500,1,"{3661, 377, 203}"
41,1,61,0,0.0425351,"\int \left(-1+\cot ^2(x)\right)^{3/2} \, dx","Int[(-1 + Cot[x]^2)^(3/2),x]","-\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)","-\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,"(5*ArcTanh[Cot[x]/Sqrt[-1 + Cot[x]^2]])/2 - 2*Sqrt[2]*ArcTanh[(Sqrt[2]*Cot[x])/Sqrt[-1 + Cot[x]^2]] - (Cot[x]*Sqrt[-1 + Cot[x]^2])/2","A",7,6,10,0.6000,1,"{3661, 416, 523, 217, 206, 377}"
42,1,42,0,0.0254463,"\int \sqrt{-1+\cot ^2(x)} \, dx","Int[Sqrt[-1 + Cot[x]^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)","\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,"-ArcTanh[Cot[x]/Sqrt[-1 + Cot[x]^2]] + Sqrt[2]*ArcTanh[(Sqrt[2]*Cot[x])/Sqrt[-1 + Cot[x]^2]]","A",6,5,10,0.5000,1,"{3661, 402, 217, 206, 377}"
43,1,26,0,0.0172601,"\int \frac{1}{\sqrt{-1+\cot ^2(x)}} \, dx","Int[1/Sqrt[-1 + Cot[x]^2],x]","-\frac{\tanh ^{-1}\left(\frac{\sqrt{2} \cot (x)}{\sqrt{\cot ^2(x)-1}}\right)}{\sqrt{2}}","-\frac{\tanh ^{-1}\left(\frac{\sqrt{2} \cot (x)}{\sqrt{\cot ^2(x)-1}}\right)}{\sqrt{2}}",1,"-(ArcTanh[(Sqrt[2]*Cot[x])/Sqrt[-1 + Cot[x]^2]]/Sqrt[2])","A",3,3,10,0.3000,1,"{3661, 377, 206}"
44,1,52,0,0.097881,"\int \frac{\cot ^3(x)}{\sqrt{a+b \cot ^2(x)}} \, dx","Int[Cot[x]^3/Sqrt[a + b*Cot[x]^2],x]","-\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}}","-\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,"-(ArcTanh[Sqrt[a + b*Cot[x]^2]/Sqrt[a - b]]/Sqrt[a - b]) - Sqrt[a + b*Cot[x]^2]/b","A",5,5,17,0.2941,1,"{3670, 446, 80, 63, 208}"
45,1,64,0,0.0903149,"\int \frac{\cot ^2(x)}{\sqrt{a+b \cot ^2(x)}} \, dx","Int[Cot[x]^2/Sqrt[a + b*Cot[x]^2],x]","\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}}","\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,"ArcTan[(Sqrt[a - b]*Cot[x])/Sqrt[a + b*Cot[x]^2]]/Sqrt[a - b] - ArcTanh[(Sqrt[b]*Cot[x])/Sqrt[a + b*Cot[x]^2]]/Sqrt[b]","A",6,6,17,0.3529,1,"{3670, 483, 217, 206, 377, 203}"
46,1,33,0,0.0638457,"\int \frac{\cot (x)}{\sqrt{a+b \cot ^2(x)}} \, dx","Int[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",4,4,15,0.2667,1,"{3670, 444, 63, 208}"
47,1,60,0,0.0941914,"\int \frac{\tan (x)}{\sqrt{a+b \cot ^2(x)}} \, dx","Int[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",7,5,15,0.3333,1,"{3670, 446, 86, 63, 208}"
48,1,54,0,0.0912968,"\int \frac{\tan ^2(x)}{\sqrt{a+b \cot ^2(x)}} \, dx","Int[Tan[x]^2/Sqrt[a + b*Cot[x]^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}","\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,"ArcTan[(Sqrt[a - b]*Cot[x])/Sqrt[a + b*Cot[x]^2]]/Sqrt[a - b] + (Sqrt[a + b*Cot[x]^2]*Tan[x])/a","A",5,5,17,0.2941,1,"{3670, 480, 12, 377, 203}"
49,1,59,0,0.1105835,"\int \frac{\cot ^3(x)}{\left(a+b \cot ^2(x)\right)^{3/2}} \, dx","Int[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",5,5,17,0.2941,1,"{3670, 446, 78, 63, 208}"
50,1,59,0,0.0957071,"\int \frac{\cot ^2(x)}{\left(a+b \cot ^2(x)\right)^{3/2}} \, dx","Int[Cot[x]^2/(a + b*Cot[x]^2)^(3/2),x]","\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)}}","\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,"ArcTan[(Sqrt[a - b]*Cot[x])/Sqrt[a + b*Cot[x]^2]]/(a - b)^(3/2) - Cot[x]/((a - b)*Sqrt[a + b*Cot[x]^2])","A",4,4,17,0.2353,1,"{3670, 471, 377, 203}"
51,1,55,0,0.0750258,"\int \frac{\cot (x)}{\left(a+b \cot ^2(x)\right)^{3/2}} \, dx","Int[Cot[x]/(a + b*Cot[x]^2)^(3/2),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)}}","\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,"ArcTanh[Sqrt[a + b*Cot[x]^2]/Sqrt[a - b]]/(a - b)^(3/2) - 1/((a - b)*Sqrt[a + b*Cot[x]^2])","A",5,5,15,0.3333,1,"{3670, 444, 51, 63, 208}"
52,1,84,0,0.1307861,"\int \frac{\tan (x)}{\left(a+b \cot ^2(x)\right)^{3/2}} \, dx","Int[Tan[x]/(a + b*Cot[x]^2)^(3/2),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}}","\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,"ArcTanh[Sqrt[a + b*Cot[x]^2]/Sqrt[a]]/a^(3/2) - ArcTanh[Sqrt[a + b*Cot[x]^2]/Sqrt[a - b]]/(a - b)^(3/2) + b/(a*(a - b)*Sqrt[a + b*Cot[x]^2])","A",8,6,15,0.4000,1,"{3670, 446, 85, 156, 63, 208}"
53,1,92,0,0.1544104,"\int \frac{\tan ^2(x)}{\left(a+b \cot ^2(x)\right)^{3/2}} \, dx","Int[Tan[x]^2/(a + b*Cot[x]^2)^(3/2),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)}}","\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,"ArcTan[(Sqrt[a - b]*Cot[x])/Sqrt[a + b*Cot[x]^2]]/(a - b)^(3/2) + (b*Tan[x])/(a*(a - b)*Sqrt[a + b*Cot[x]^2]) + ((a - 2*b)*Sqrt[a + b*Cot[x]^2]*Tan[x])/(a^2*(a - b))","A",6,6,17,0.3529,1,"{3670, 472, 583, 12, 377, 203}"
54,1,82,0,0.1267725,"\int \frac{\cot ^3(x)}{\left(a+b \cot ^2(x)\right)^{5/2}} \, dx","Int[Cot[x]^3/(a + b*Cot[x]^2)^(5/2),x]","\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}}","\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,"-(ArcTanh[Sqrt[a + b*Cot[x]^2]/Sqrt[a - b]]/(a - b)^(5/2)) + a/(3*(a - b)*b*(a + b*Cot[x]^2)^(3/2)) + 1/((a - b)^2*Sqrt[a + b*Cot[x]^2])","A",6,6,17,0.3529,1,"{3670, 446, 78, 51, 63, 208}"
55,1,94,0,0.1241374,"\int \frac{\cot ^2(x)}{\left(a+b \cot ^2(x)\right)^{5/2}} \, dx","Int[Cot[x]^2/(a + b*Cot[x]^2)^(5/2),x]","-\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}}","-\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,"ArcTan[(Sqrt[a - b]*Cot[x])/Sqrt[a + b*Cot[x]^2]]/(a - b)^(5/2) - Cot[x]/(3*(a - b)*(a + b*Cot[x]^2)^(3/2)) - ((2*a + b)*Cot[x])/(3*a*(a - b)^2*Sqrt[a + b*Cot[x]^2])","A",6,6,17,0.3529,1,"{3670, 471, 527, 12, 377, 203}"
56,1,78,0,0.0915714,"\int \frac{\cot (x)}{\left(a+b \cot ^2(x)\right)^{5/2}} \, dx","Int[Cot[x]/(a + b*Cot[x]^2)^(5/2),x]","-\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}}","-\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,"ArcTanh[Sqrt[a + b*Cot[x]^2]/Sqrt[a - b]]/(a - b)^(5/2) - 1/(3*(a - b)*(a + b*Cot[x]^2)^(3/2)) - 1/((a - b)^2*Sqrt[a + b*Cot[x]^2])","A",6,5,15,0.3333,1,"{3670, 444, 51, 63, 208}"
57,1,118,0,0.187825,"\int \frac{\tan (x)}{\left(a+b \cot ^2(x)\right)^{5/2}} \, dx","Int[Tan[x]/(a + b*Cot[x]^2)^(5/2),x]","\frac{b (2 a-b)}{a^2 (a-b)^2 \sqrt{a+b \cot ^2(x)}}+\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \cot ^2(x)}}{\sqrt{a}}\right)}{a^{5/2}}+\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}}","\frac{b (2 a-b)}{a^2 (a-b)^2 \sqrt{a+b \cot ^2(x)}}+\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \cot ^2(x)}}{\sqrt{a}}\right)}{a^{5/2}}+\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,"ArcTanh[Sqrt[a + b*Cot[x]^2]/Sqrt[a]]/a^(5/2) - ArcTanh[Sqrt[a + b*Cot[x]^2]/Sqrt[a - b]]/(a - b)^(5/2) + b/(3*a*(a - b)*(a + b*Cot[x]^2)^(3/2)) + ((2*a - b)*b)/(a^2*(a - b)^2*Sqrt[a + b*Cot[x]^2])","A",9,7,15,0.4667,1,"{3670, 446, 85, 152, 156, 63, 208}"
58,1,141,0,0.2413361,"\int \frac{\tan ^2(x)}{\left(a+b \cot ^2(x)\right)^{5/2}} \, dx","Int[Tan[x]^2/(a + b*Cot[x]^2)^(5/2),x]","\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}}","\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,"ArcTan[(Sqrt[a - b]*Cot[x])/Sqrt[a + b*Cot[x]^2]]/(a - b)^(5/2) + (b*Tan[x])/(3*a*(a - b)*(a + b*Cot[x]^2)^(3/2)) + ((7*a - 4*b)*b*Tan[x])/(3*a^2*(a - b)^2*Sqrt[a + b*Cot[x]^2]) + ((a - 4*b)*(3*a - 2*b)*Sqrt[a + b*Cot[x]^2]*Tan[x])/(3*a^3*(a - b)^2)","A",7,7,17,0.4118,1,"{3670, 472, 579, 583, 12, 377, 203}"
59,1,37,0,0.0619308,"\int \frac{1}{1+\cot ^3(x)} \, dx","Int[(1 + Cot[x]^3)^(-1),x]","\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)","\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,"x/2 - Log[1 + Cot[x]]/6 + Log[1 - Cot[x] + Cot[x]^2]/3 + Log[Sin[x]]/2","A",7,6,8,0.7500,1,"{3661, 2074, 635, 203, 260, 628}"
60,1,90,0,0.1353197,"\int \cot (x) \sqrt{a+b \cot ^4(x)} \, dx","Int[Cot[x]*Sqrt[a + b*Cot[x]^4],x]","-\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)","-\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]])/2 + (Sqrt[a + b]*ArcTanh[(a - b*Cot[x]^2)/(Sqrt[a + b]*Sqrt[a + b*Cot[x]^4])])/2 - Sqrt[a + b*Cot[x]^4]/2","A",8,7,15,0.4667,1,"{3670, 1248, 735, 844, 217, 206, 725}"
61,1,126,0,0.2017104,"\int \cot (x) \left(a+b \cot ^4(x)\right)^{3/2} \, dx","Int[Cot[x]*(a + b*Cot[x]^4)^(3/2),x]","-\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)","-\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,"(Sqrt[b]*(3*a + 2*b)*ArcTanh[(Sqrt[b]*Cot[x]^2)/Sqrt[a + b*Cot[x]^4]])/4 + ((a + b)^(3/2)*ArcTanh[(a - b*Cot[x]^2)/(Sqrt[a + b]*Sqrt[a + b*Cot[x]^4])])/2 - ((2*(a + b) - b*Cot[x]^2)*Sqrt[a + b*Cot[x]^4])/4 - (a + b*Cot[x]^4)^(3/2)/6","A",9,8,15,0.5333,1,"{3670, 1248, 735, 815, 844, 217, 206, 725}"
62,1,41,0,0.0691749,"\int \frac{\cot (x)}{\sqrt{a+b \cot ^4(x)}} \, dx","Int[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",4,4,15,0.2667,1,"{3670, 1248, 725, 206}"
63,1,74,0,0.1107301,"\int \frac{\cot (x)}{\left(a+b \cot ^4(x)\right)^{3/2}} \, dx","Int[Cot[x]/(a + b*Cot[x]^4)^(3/2),x]","\frac{\tanh ^{-1}\left(\frac{a-b \cot ^2(x)}{\sqrt{a+b} \sqrt{a+b \cot ^4(x)}}\right)}{2 (a+b)^{3/2}}-\frac{a+b \cot ^2(x)}{2 a (a+b) \sqrt{a+b \cot ^4(x)}}","\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])]/(2*(a + b)^(3/2)) - (a + b*Cot[x]^2)/(2*a*(a + b)*Sqrt[a + b*Cot[x]^4])","A",6,6,15,0.4000,1,"{3670, 1248, 741, 12, 725, 206}"
64,1,117,0,0.1903168,"\int \frac{\cot (x)}{\left(a+b \cot ^4(x)\right)^{5/2}} \, dx","Int[Cot[x]/(a + b*Cot[x]^4)^(5/2),x]","-\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}}","-\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 + b*Cot[x]^2)/(6*a*(a + b)*(a + b*Cot[x]^4)^(3/2)) - (3*a^2 + b*(5*a + 2*b)*Cot[x]^2)/(6*a^2*(a + b)^2*Sqrt[a + b*Cot[x]^4])","A",7,7,15,0.4667,1,"{3670, 1248, 741, 823, 12, 725, 206}"