1,1,113,55,0.8169432,"\int \frac{\csc ^5(x)}{a+a \csc (x)} \, dx","Integrate[Csc[x]^5/(a + a*Csc[x]),x]","\frac{20 \tan \left(\frac{x}{2}\right)-20 \cot \left(\frac{x}{2}\right)+3 \csc ^2\left(\frac{x}{2}\right)-3 \sec ^2\left(\frac{x}{2}\right)-36 \log \left(\sin \left(\frac{x}{2}\right)\right)+36 \log \left(\cos \left(\frac{x}{2}\right)\right)+\frac{48 \sin \left(\frac{x}{2}\right)}{\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)}-\frac{1}{2} \sin (x) \csc ^4\left(\frac{x}{2}\right)+8 \sin ^4\left(\frac{x}{2}\right) \csc ^3(x)}{24 a}","-\frac{4 \cot ^3(x)}{3 a}-\frac{4 \cot (x)}{a}+\frac{3 \tanh ^{-1}(\cos (x))}{2 a}+\frac{\cot (x) \csc ^3(x)}{a \csc (x)+a}+\frac{3 \cot (x) \csc (x)}{2 a}",1,"(-20*Cot[x/2] + 3*Csc[x/2]^2 + 36*Log[Cos[x/2]] - 36*Log[Sin[x/2]] - 3*Sec[x/2]^2 + 8*Csc[x]^3*Sin[x/2]^4 + (48*Sin[x/2])/(Cos[x/2] + Sin[x/2]) - (Csc[x/2]^4*Sin[x])/2 + 20*Tan[x/2])/(24*a)","B",1
2,1,83,44,0.3453812,"\int \frac{\csc ^4(x)}{a+a \csc (x)} \, dx","Integrate[Csc[x]^4/(a + a*Csc[x]),x]","\frac{-4 \tan \left(\frac{x}{2}\right)+4 \cot \left(\frac{x}{2}\right)-\csc ^2\left(\frac{x}{2}\right)+\sec ^2\left(\frac{x}{2}\right)+12 \log \left(\sin \left(\frac{x}{2}\right)\right)-12 \log \left(\cos \left(\frac{x}{2}\right)\right)-\frac{16 \sin \left(\frac{x}{2}\right)}{\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)}}{8 a}","\frac{2 \cot (x)}{a}-\frac{3 \tanh ^{-1}(\cos (x))}{2 a}+\frac{\cot (x) \csc ^2(x)}{a \csc (x)+a}-\frac{3 \cot (x) \csc (x)}{2 a}",1,"(4*Cot[x/2] - Csc[x/2]^2 - 12*Log[Cos[x/2]] + 12*Log[Sin[x/2]] + Sec[x/2]^2 - (16*Sin[x/2])/(Cos[x/2] + Sin[x/2]) - 4*Tan[x/2])/(8*a)","A",1
3,1,63,27,0.1593265,"\int \frac{\csc ^3(x)}{a+a \csc (x)} \, dx","Integrate[Csc[x]^3/(a + a*Csc[x]),x]","\frac{\tan \left(\frac{x}{2}\right)-\cot \left(\frac{x}{2}\right)-2 \log \left(\sin \left(\frac{x}{2}\right)\right)+2 \log \left(\cos \left(\frac{x}{2}\right)\right)+\frac{4 \sin \left(\frac{x}{2}\right)}{\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)}}{2 a}","-\frac{\cot (x)}{a}+\frac{\tanh ^{-1}(\cos (x))}{a}-\frac{\cot (x)}{a \csc (x)+a}",1,"(-Cot[x/2] + 2*Log[Cos[x/2]] - 2*Log[Sin[x/2]] + (4*Sin[x/2])/(Cos[x/2] + Sin[x/2]) + Tan[x/2])/(2*a)","B",1
4,1,44,20,0.0542645,"\int \frac{\csc ^2(x)}{a+a \csc (x)} \, dx","Integrate[Csc[x]^2/(a + a*Csc[x]),x]","\frac{\log \left(\sin \left(\frac{x}{2}\right)\right)-\log \left(\cos \left(\frac{x}{2}\right)\right)-\frac{2 \sin \left(\frac{x}{2}\right)}{\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)}}{a}","\frac{\cot (x)}{a \csc (x)+a}-\frac{\tanh ^{-1}(\cos (x))}{a}",1,"(-Log[Cos[x/2]] + Log[Sin[x/2]] - (2*Sin[x/2])/(Cos[x/2] + Sin[x/2]))/a","B",1
5,1,26,12,0.0243561,"\int \frac{\csc (x)}{a+a \csc (x)} \, dx","Integrate[Csc[x]/(a + a*Csc[x]),x]","\frac{2 \sin \left(\frac{x}{2}\right)}{a \left(\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)\right)}","-\frac{\cot (x)}{a \csc (x)+a}",1,"(2*Sin[x/2])/(a*(Cos[x/2] + Sin[x/2]))","B",1
6,1,47,28,0.0925256,"\int \frac{1}{a+a \csc (c+d x)} \, dx","Integrate[(a + a*Csc[c + d*x])^(-1),x]","\frac{-\frac{2 \sin \left(\frac{1}{2} (c+d x)\right)}{\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)}+c+d x}{a d}","\frac{\cot (c+d x)}{d (a \csc (c+d x)+a)}+\frac{x}{a}",1,"(c + d*x - (2*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))/(a*d)","A",1
7,1,32,25,0.0766438,"\int \frac{\sin (x)}{a+a \csc (x)} \, dx","Integrate[Sin[x]/(a + a*Csc[x]),x]","-\frac{x+\cos (x)-\frac{2 \sin \left(\frac{x}{2}\right)}{\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)}}{a}","-\frac{x}{a}-\frac{2 \cos (x)}{a}+\frac{\cos (x)}{a \csc (x)+a}",1,"-((x + Cos[x] - (2*Sin[x/2])/(Cos[x/2] + Sin[x/2]))/a)","A",1
8,1,42,40,0.1309368,"\int \frac{\sin ^2(x)}{a+a \csc (x)} \, dx","Integrate[Sin[x]^2/(a + a*Csc[x]),x]","-\frac{-6 x+\sin (2 x)-4 \cos (x)+\frac{8 \sin \left(\frac{x}{2}\right)}{\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)}}{4 a}","\frac{3 x}{2 a}+\frac{2 \cos (x)}{a}-\frac{3 \sin (x) \cos (x)}{2 a}+\frac{\sin (x) \cos (x)}{a \csc (x)+a}",1,"-1/4*(-6*x - 4*Cos[x] + (8*Sin[x/2])/(Cos[x/2] + Sin[x/2]) + Sin[2*x])/a","A",1
9,1,49,53,0.1717562,"\int \frac{\sin ^3(x)}{a+a \csc (x)} \, dx","Integrate[Sin[x]^3/(a + a*Csc[x]),x]","\frac{-21 \cos (x)+\cos (3 x)+3 \left(-6 x+\sin (2 x)+\frac{8 \sin \left(\frac{x}{2}\right)}{\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)}\right)}{12 a}","-\frac{3 x}{2 a}+\frac{4 \cos ^3(x)}{3 a}-\frac{4 \cos (x)}{a}+\frac{3 \sin (x) \cos (x)}{2 a}+\frac{\sin ^2(x) \cos (x)}{a \csc (x)+a}",1,"(-21*Cos[x] + Cos[3*x] + 3*(-6*x + (8*Sin[x/2])/(Cos[x/2] + Sin[x/2]) + Sin[2*x]))/(12*a)","A",1
10,1,57,66,0.2514661,"\int \frac{\sin ^4(x)}{a+a \csc (x)} \, dx","Integrate[Sin[x]^4/(a + a*Csc[x]),x]","\frac{168 \cos (x)-8 \cos (3 x)+3 \left(60 x-16 \sin (2 x)+\sin (4 x)-\frac{64 \sin \left(\frac{x}{2}\right)}{\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)}\right)}{96 a}","\frac{15 x}{8 a}-\frac{4 \cos ^3(x)}{3 a}+\frac{4 \cos (x)}{a}-\frac{5 \sin ^3(x) \cos (x)}{4 a}-\frac{15 \sin (x) \cos (x)}{8 a}+\frac{\sin ^3(x) \cos (x)}{a \csc (x)+a}",1,"(168*Cos[x] - 8*Cos[3*x] + 3*(60*x - (64*Sin[x/2])/(Cos[x/2] + Sin[x/2]) - 16*Sin[2*x] + Sin[4*x]))/(96*a)","A",1
11,1,108,57,0.3250552,"\int \frac{1}{(a+a \csc (c+d x))^2} \, dx","Integrate[(a + a*Csc[c + d*x])^(-2),x]","\frac{3 (3 c+3 d x-4) \cos \left(\frac{1}{2} (c+d x)\right)+(-3 c-3 d x+10) \cos \left(\frac{3}{2} (c+d x)\right)+6 \sin \left(\frac{1}{2} (c+d x)\right) ((c+d x) \cos (c+d x)+2 c+2 d x-3)}{6 a^2 d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^3}","\frac{4 \cot (c+d x)}{3 a^2 d (\csc (c+d x)+1)}+\frac{x}{a^2}+\frac{\cot (c+d x)}{3 d (a \csc (c+d x)+a)^2}",1,"(3*(-4 + 3*c + 3*d*x)*Cos[(c + d*x)/2] + (10 - 3*c - 3*d*x)*Cos[(3*(c + d*x))/2] + 6*(-3 + 2*c + 2*d*x + (c + d*x)*Cos[c + d*x])*Sin[(c + d*x)/2])/(6*a^2*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3)","A",1
12,1,123,88,1.0381669,"\int \frac{1}{(a+a \csc (c+d x))^3} \, dx","Integrate[(a + a*Csc[c + d*x])^(-3),x]","\frac{\frac{2 \sin \left(\frac{1}{2} (c+d x)\right) (-51 \sin (c+d x)+16 \cos (2 (c+d x))-38)}{\left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^5}-\frac{13}{\left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2}+\frac{3}{\left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^4}+15 c+15 d x}{15 a^3 d}","\frac{22 \cot (c+d x)}{15 d \left(a^3 \csc (c+d x)+a^3\right)}+\frac{x}{a^3}+\frac{7 \cot (c+d x)}{15 a d (a \csc (c+d x)+a)^2}+\frac{\cot (c+d x)}{5 d (a \csc (c+d x)+a)^3}",1,"(15*c + 15*d*x + 3/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^4 - 13/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2 + (2*Sin[(c + d*x)/2]*(-38 + 16*Cos[2*(c + d*x)] - 51*Sin[c + d*x]))/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^5)/(15*a^3*d)","A",1
13,1,80,65,1.9017757,"\int (a+a \csc (x))^{5/2} \, dx","Integrate[(a + a*Csc[x])^(5/2),x]","-\frac{2 a^2 \sqrt{a (\csc (x)+1)} \left(\cos \left(\frac{x}{2}\right)-\sin \left(\frac{x}{2}\right)\right) \left(\sqrt{\csc (x)-1} (\csc (x)+8)+3 \tan ^{-1}\left(\sqrt{\csc (x)-1}\right)\right)}{3 \sqrt{\csc (x)-1} \left(\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)\right)}","-2 a^{5/2} \tan ^{-1}\left(\frac{\sqrt{a} \cot (x)}{\sqrt{a \csc (x)+a}}\right)-\frac{14 a^3 \cot (x)}{3 \sqrt{a \csc (x)+a}}-\frac{2}{3} a^2 \cot (x) \sqrt{a \csc (x)+a}",1,"(-2*a^2*Sqrt[a*(1 + Csc[x])]*(3*ArcTan[Sqrt[-1 + Csc[x]]] + Sqrt[-1 + Csc[x]]*(8 + Csc[x]))*(Cos[x/2] - Sin[x/2]))/(3*Sqrt[-1 + Csc[x]]*(Cos[x/2] + Sin[x/2]))","A",1
14,1,69,44,0.089247,"\int (a+a \csc (x))^{3/2} \, dx","Integrate[(a + a*Csc[x])^(3/2),x]","-\frac{2 a \sqrt{a (\csc (x)+1)} \left(\cos \left(\frac{x}{2}\right)-\sin \left(\frac{x}{2}\right)\right) \left(\sqrt{\csc (x)-1}+\tan ^{-1}\left(\sqrt{\csc (x)-1}\right)\right)}{\sqrt{\csc (x)-1} \left(\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)\right)}","-2 a^{3/2} \tan ^{-1}\left(\frac{\sqrt{a} \cot (x)}{\sqrt{a \csc (x)+a}}\right)-\frac{2 a^2 \cot (x)}{\sqrt{a \csc (x)+a}}",1,"(-2*a*(ArcTan[Sqrt[-1 + Csc[x]]] + Sqrt[-1 + Csc[x]])*Sqrt[a*(1 + Csc[x])]*(Cos[x/2] - Sin[x/2]))/(Sqrt[-1 + Csc[x]]*(Cos[x/2] + Sin[x/2]))","A",1
15,1,32,26,0.0538394,"\int \sqrt{a+a \csc (x)} \, dx","Integrate[Sqrt[a + a*Csc[x]],x]","-\frac{2 a \cot (x) \tan ^{-1}\left(\sqrt{\csc (x)-1}\right)}{\sqrt{\csc (x)-1} \sqrt{a (\csc (x)+1)}}","-2 \sqrt{a} \tan ^{-1}\left(\frac{\sqrt{a} \cot (x)}{\sqrt{a \csc (x)+a}}\right)",1,"(-2*a*ArcTan[Sqrt[-1 + Csc[x]]]*Cot[x])/(Sqrt[-1 + Csc[x]]*Sqrt[a*(1 + Csc[x])])","A",1
16,1,54,62,0.1361955,"\int \frac{1}{\sqrt{a+a \csc (x)}} \, dx","Integrate[1/Sqrt[a + a*Csc[x]],x]","\frac{\cot (x) \left(\sqrt{2} \tan ^{-1}\left(\frac{\sqrt{\csc (x)-1}}{\sqrt{2}}\right)-2 \tan ^{-1}\left(\sqrt{\csc (x)-1}\right)\right)}{\sqrt{\csc (x)-1} \sqrt{a (\csc (x)+1)}}","\frac{\sqrt{2} \tan ^{-1}\left(\frac{\sqrt{a} \cot (x)}{\sqrt{2} \sqrt{a \csc (x)+a}}\right)}{\sqrt{a}}-\frac{2 \tan ^{-1}\left(\frac{\sqrt{a} \cot (x)}{\sqrt{a \csc (x)+a}}\right)}{\sqrt{a}}",1,"((-2*ArcTan[Sqrt[-1 + Csc[x]]] + Sqrt[2]*ArcTan[Sqrt[-1 + Csc[x]]/Sqrt[2]])*Cot[x])/(Sqrt[-1 + Csc[x]]*Sqrt[a*(1 + Csc[x])])","A",1
17,1,129,81,0.4278012,"\int \frac{1}{(a+a \csc (x))^{3/2}} \, dx","Integrate[(a + a*Csc[x])^(-3/2),x]","-\frac{\left(\cos \left(\frac{x}{2}\right)-\sin \left(\frac{x}{2}\right)\right) \left(-2 \csc (x)+8 \sqrt{\csc (x)-1} (\csc (x)+1) \tan ^{-1}\left(\sqrt{\csc (x)-1}\right)-5 \sqrt{2} \sqrt{\csc (x)-1} \csc (x) \left(\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)\right)^2 \tan ^{-1}\left(\frac{\sqrt{\csc (x)-1}}{\sqrt{2}}\right)+2\right)}{4 a (\csc (x)-1) \sqrt{a (\csc (x)+1)} \left(\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)\right)}","-\frac{2 \tan ^{-1}\left(\frac{\sqrt{a} \cot (x)}{\sqrt{a \csc (x)+a}}\right)}{a^{3/2}}+\frac{5 \tan ^{-1}\left(\frac{\sqrt{a} \cot (x)}{\sqrt{2} \sqrt{a \csc (x)+a}}\right)}{2 \sqrt{2} a^{3/2}}+\frac{\cot (x)}{2 (a \csc (x)+a)^{3/2}}",1,"-1/4*((Cos[x/2] - Sin[x/2])*(2 - 2*Csc[x] + 8*ArcTan[Sqrt[-1 + Csc[x]]]*Sqrt[-1 + Csc[x]]*(1 + Csc[x]) - 5*Sqrt[2]*ArcTan[Sqrt[-1 + Csc[x]]/Sqrt[2]]*Sqrt[-1 + Csc[x]]*Csc[x]*(Cos[x/2] + Sin[x/2])^2))/(a*(-1 + Csc[x])*Sqrt[a*(1 + Csc[x])]*(Cos[x/2] + Sin[x/2]))","A",1
18,1,139,100,0.5069848,"\int \frac{1}{(a+a \csc (x))^{5/2}} \, dx","Integrate[(a + a*Csc[x])^(-5/2),x]","\frac{\csc ^2(x) \left(\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)\right) \left(8 \sin (x)+15 \cos (2 x)-64 \sqrt{\csc (x)-1} \left(\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)\right)^4 \tan ^{-1}\left(\sqrt{\csc (x)-1}\right)+43 \sqrt{2} \sqrt{\csc (x)-1} \left(\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)\right)^4 \tan ^{-1}\left(\frac{\sqrt{\csc (x)-1}}{\sqrt{2}}\right)+7\right)}{32 (a (\csc (x)+1))^{5/2} \left(\cos \left(\frac{x}{2}\right)-\sin \left(\frac{x}{2}\right)\right)}","-\frac{2 \tan ^{-1}\left(\frac{\sqrt{a} \cot (x)}{\sqrt{a \csc (x)+a}}\right)}{a^{5/2}}+\frac{43 \tan ^{-1}\left(\frac{\sqrt{a} \cot (x)}{\sqrt{2} \sqrt{a \csc (x)+a}}\right)}{16 \sqrt{2} a^{5/2}}+\frac{11 \cot (x)}{16 a (a \csc (x)+a)^{3/2}}+\frac{\cot (x)}{4 (a \csc (x)+a)^{5/2}}",1,"(Csc[x]^2*(Cos[x/2] + Sin[x/2])*(7 + 15*Cos[2*x] - 64*ArcTan[Sqrt[-1 + Csc[x]]]*Sqrt[-1 + Csc[x]]*(Cos[x/2] + Sin[x/2])^4 + 43*Sqrt[2]*ArcTan[Sqrt[-1 + Csc[x]]/Sqrt[2]]*Sqrt[-1 + Csc[x]]*(Cos[x/2] + Sin[x/2])^4 + 8*Sin[x]))/(32*(a*(1 + Csc[x]))^(5/2)*(Cos[x/2] - Sin[x/2]))","A",1
19,1,108,37,0.4002394,"\int \sqrt{\csc (e+f x)} \sqrt{a+a \csc (e+f x)} \, dx","Integrate[Sqrt[Csc[e + f*x]]*Sqrt[a + a*Csc[e + f*x]],x]","\frac{2 \cot (e+f x) \sqrt{a (\csc (e+f x)+1)} \left(\log (\csc (e+f x)+1)-\log \left(\csc ^{\frac{3}{2}}(e+f x)+\sqrt{\csc (e+f x)}+\sqrt{\cot ^2(e+f x)} \sqrt{\csc (e+f x)+1}\right)\right)}{f \sqrt{\cot ^2(e+f x)} \sqrt{\csc (e+f x)+1}}","-\frac{2 \sqrt{a} \sinh ^{-1}\left(\frac{\sqrt{a} \cot (e+f x)}{\sqrt{a \csc (e+f x)+a}}\right)}{f}",1,"(2*Cot[e + f*x]*Sqrt[a*(1 + Csc[e + f*x])]*(Log[1 + Csc[e + f*x]] - Log[Sqrt[Csc[e + f*x]] + Csc[e + f*x]^(3/2) + Sqrt[Cot[e + f*x]^2]*Sqrt[1 + Csc[e + f*x]]]))/(f*Sqrt[Cot[e + f*x]^2]*Sqrt[1 + Csc[e + f*x]])","B",1
20,1,101,38,0.7936126,"\int \sqrt{-\csc (e+f x)} \sqrt{a-a \csc (e+f x)} \, dx","Integrate[Sqrt[-Csc[e + f*x]]*Sqrt[a - a*Csc[e + f*x]],x]","\frac{2 \tan \left(\frac{1}{2} (e+f x)\right) \sqrt{-\csc (e+f x)} \sqrt{a-a \csc (e+f x)} \left(\tanh ^{-1}\left(\sqrt{\sec ^2\left(\frac{1}{2} (e+f x)\right)}\right)+\sinh ^{-1}\left(\tan \left(\frac{1}{2} (e+f x)\right)\right)\right)}{f \left(\tan \left(\frac{1}{2} (e+f x)\right)-1\right) \sqrt{\sec ^2\left(\frac{1}{2} (e+f x)\right)}}","-\frac{2 \sqrt{a} \sinh ^{-1}\left(\frac{\sqrt{a} \cot (e+f x)}{\sqrt{a-a \csc (e+f x)}}\right)}{f}",1,"(2*(ArcSinh[Tan[(e + f*x)/2]] + ArcTanh[Sqrt[Sec[(e + f*x)/2]^2]])*Sqrt[-Csc[e + f*x]]*Sqrt[a - a*Csc[e + f*x]]*Tan[(e + f*x)/2])/(f*Sqrt[Sec[(e + f*x)/2]^2]*(-1 + Tan[(e + f*x)/2]))","B",1
21,1,102,254,0.3828241,"\int \csc ^{\frac{4}{3}}(c+d x) \sqrt{a+a \csc (c+d x)} \, dx","Integrate[Csc[c + d*x]^(4/3)*Sqrt[a + a*Csc[c + d*x]],x]","-\frac{2 \sqrt{a (\csc (c+d x)+1)} \left(2 \, _2F_1\left(\frac{1}{2},\frac{2}{3};\frac{3}{2};1-\csc (c+d x)\right)+3 \sqrt[3]{\csc (c+d x)}\right) \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}{5 d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}","-\frac{4\ 3^{3/4} \sqrt{2+\sqrt{3}} a^2 \cot (c+d x) \left(1-\sqrt[3]{\csc (c+d x)}\right) \sqrt{\frac{\csc ^{\frac{2}{3}}(c+d x)+\sqrt[3]{\csc (c+d x)}+1}{\left(-\sqrt[3]{\csc (c+d x)}+\sqrt{3}+1\right)^2}} F\left(\sin ^{-1}\left(\frac{-\sqrt[3]{\csc (c+d x)}-\sqrt{3}+1}{-\sqrt[3]{\csc (c+d x)}+\sqrt{3}+1}\right)|-7-4 \sqrt{3}\right)}{5 d \sqrt{\frac{1-\sqrt[3]{\csc (c+d x)}}{\left(-\sqrt[3]{\csc (c+d x)}+\sqrt{3}+1\right)^2}} (a-a \csc (c+d x)) \sqrt{a \csc (c+d x)+a}}-\frac{6 a \cos (c+d x) \csc ^{\frac{4}{3}}(c+d x)}{5 d \sqrt{a \csc (c+d x)+a}}",1,"(-2*Sqrt[a*(1 + Csc[c + d*x])]*(3*Csc[c + d*x]^(1/3) + 2*Hypergeometric2F1[1/2, 2/3, 3/2, 1 - Csc[c + d*x]])*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2]))/(5*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))","C",1
22,1,46,213,0.2298573,"\int \sqrt[3]{\csc (c+d x)} \sqrt{a+a \csc (c+d x)} \, dx","Integrate[Csc[c + d*x]^(1/3)*Sqrt[a + a*Csc[c + d*x]],x]","-\frac{2 a \cot (c+d x) \, _2F_1\left(\frac{1}{2},\frac{2}{3};\frac{3}{2};1-\csc (c+d x)\right)}{d \sqrt{a (\csc (c+d x)+1)}}","-\frac{2\ 3^{3/4} \sqrt{2+\sqrt{3}} a^2 \cot (c+d x) \left(1-\sqrt[3]{\csc (c+d x)}\right) \sqrt{\frac{\csc ^{\frac{2}{3}}(c+d x)+\sqrt[3]{\csc (c+d x)}+1}{\left(-\sqrt[3]{\csc (c+d x)}+\sqrt{3}+1\right)^2}} F\left(\sin ^{-1}\left(\frac{-\sqrt[3]{\csc (c+d x)}-\sqrt{3}+1}{-\sqrt[3]{\csc (c+d x)}+\sqrt{3}+1}\right)|-7-4 \sqrt{3}\right)}{d \sqrt{\frac{1-\sqrt[3]{\csc (c+d x)}}{\left(-\sqrt[3]{\csc (c+d x)}+\sqrt{3}+1\right)^2}} (a-a \csc (c+d x)) \sqrt{a \csc (c+d x)+a}}",1,"(-2*a*Cot[c + d*x]*Hypergeometric2F1[1/2, 2/3, 3/2, 1 - Csc[c + d*x]])/(d*Sqrt[a*(1 + Csc[c + d*x])])","C",1
23,1,110,254,0.4596977,"\int \frac{\sqrt{a+a \csc (c+d x)}}{\csc ^{\frac{2}{3}}(c+d x)} \, dx","Integrate[Sqrt[a + a*Csc[c + d*x]]/Csc[c + d*x]^(2/3),x]","-\frac{\sqrt{a (\csc (c+d x)+1)} \left(\csc ^{\frac{2}{3}}(c+d x) \, _2F_1\left(\frac{1}{2},\frac{2}{3};\frac{3}{2};1-\csc (c+d x)\right)+3\right) \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}{2 d \csc ^{\frac{2}{3}}(c+d x) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}","-\frac{3^{3/4} \sqrt{2+\sqrt{3}} a^2 \cot (c+d x) \left(1-\sqrt[3]{\csc (c+d x)}\right) \sqrt{\frac{\csc ^{\frac{2}{3}}(c+d x)+\sqrt[3]{\csc (c+d x)}+1}{\left(-\sqrt[3]{\csc (c+d x)}+\sqrt{3}+1\right)^2}} F\left(\sin ^{-1}\left(\frac{-\sqrt[3]{\csc (c+d x)}-\sqrt{3}+1}{-\sqrt[3]{\csc (c+d x)}+\sqrt{3}+1}\right)|-7-4 \sqrt{3}\right)}{2 d \sqrt{\frac{1-\sqrt[3]{\csc (c+d x)}}{\left(-\sqrt[3]{\csc (c+d x)}+\sqrt{3}+1\right)^2}} (a-a \csc (c+d x)) \sqrt{a \csc (c+d x)+a}}-\frac{3 a \cos (c+d x) \sqrt[3]{\csc (c+d x)}}{2 d \sqrt{a \csc (c+d x)+a}}",1,"-1/2*(Sqrt[a*(1 + Csc[c + d*x])]*(3 + Csc[c + d*x]^(2/3)*Hypergeometric2F1[1/2, 2/3, 3/2, 1 - Csc[c + d*x]])*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2]))/(d*Csc[c + d*x]^(2/3)*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))","C",1
24,1,120,514,1.178183,"\int \csc ^{\frac{5}{3}}(c+d x) \sqrt{a+a \csc (c+d x)} \, dx","Integrate[Csc[c + d*x]^(5/3)*Sqrt[a + a*Csc[c + d*x]],x]","-\frac{2 \sqrt{a (\csc (c+d x)+1)} \left(3 (\csc (c+d x)+4)-8 \sqrt[3]{\csc (c+d x)} \, _2F_1\left(\frac{1}{2},\frac{4}{3};\frac{3}{2};1-\csc (c+d x)\right)\right) \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}{7 d \sqrt[3]{\csc (c+d x)} \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}","\frac{8 \sqrt{2} 3^{3/4} a^2 \cot (c+d x) \left(1-\sqrt[3]{\csc (c+d x)}\right) \sqrt{\frac{\csc ^{\frac{2}{3}}(c+d x)+\sqrt[3]{\csc (c+d x)}+1}{\left(-\sqrt[3]{\csc (c+d x)}+\sqrt{3}+1\right)^2}} F\left(\sin ^{-1}\left(\frac{-\sqrt[3]{\csc (c+d x)}-\sqrt{3}+1}{-\sqrt[3]{\csc (c+d x)}+\sqrt{3}+1}\right)|-7-4 \sqrt{3}\right)}{7 d \sqrt{\frac{1-\sqrt[3]{\csc (c+d x)}}{\left(-\sqrt[3]{\csc (c+d x)}+\sqrt{3}+1\right)^2}} (a-a \csc (c+d x)) \sqrt{a \csc (c+d x)+a}}-\frac{12 \sqrt[4]{3} \sqrt{2-\sqrt{3}} a^2 \cot (c+d x) \left(1-\sqrt[3]{\csc (c+d x)}\right) \sqrt{\frac{\csc ^{\frac{2}{3}}(c+d x)+\sqrt[3]{\csc (c+d x)}+1}{\left(-\sqrt[3]{\csc (c+d x)}+\sqrt{3}+1\right)^2}} E\left(\sin ^{-1}\left(\frac{-\sqrt[3]{\csc (c+d x)}-\sqrt{3}+1}{-\sqrt[3]{\csc (c+d x)}+\sqrt{3}+1}\right)|-7-4 \sqrt{3}\right)}{7 d \sqrt{\frac{1-\sqrt[3]{\csc (c+d x)}}{\left(-\sqrt[3]{\csc (c+d x)}+\sqrt{3}+1\right)^2}} (a-a \csc (c+d x)) \sqrt{a \csc (c+d x)+a}}-\frac{6 a \cos (c+d x) \csc ^{\frac{5}{3}}(c+d x)}{7 d \sqrt{a \csc (c+d x)+a}}+\frac{24 a \cot (c+d x)}{7 d \left(-\sqrt[3]{\csc (c+d x)}+\sqrt{3}+1\right) \sqrt{a \csc (c+d x)+a}}",1,"(-2*Sqrt[a*(1 + Csc[c + d*x])]*(3*(4 + Csc[c + d*x]) - 8*Csc[c + d*x]^(1/3)*Hypergeometric2F1[1/2, 4/3, 3/2, 1 - Csc[c + d*x]])*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2]))/(7*d*Csc[c + d*x]^(1/3)*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))","C",1
25,1,109,470,0.9678341,"\int \csc ^{\frac{2}{3}}(c+d x) \sqrt{a+a \csc (c+d x)} \, dx","Integrate[Csc[c + d*x]^(2/3)*Sqrt[a + a*Csc[c + d*x]],x]","\frac{2 \sqrt{a (\csc (c+d x)+1)} \left(2 \sqrt[3]{\csc (c+d x)} \, _2F_1\left(\frac{1}{2},\frac{4}{3};\frac{3}{2};1-\csc (c+d x)\right)-3\right) \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}{d \sqrt[3]{\csc (c+d x)} \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}","\frac{2 \sqrt{2} 3^{3/4} a^2 \cot (c+d x) \left(1-\sqrt[3]{\csc (c+d x)}\right) \sqrt{\frac{\csc ^{\frac{2}{3}}(c+d x)+\sqrt[3]{\csc (c+d x)}+1}{\left(-\sqrt[3]{\csc (c+d x)}+\sqrt{3}+1\right)^2}} F\left(\sin ^{-1}\left(\frac{-\sqrt[3]{\csc (c+d x)}-\sqrt{3}+1}{-\sqrt[3]{\csc (c+d x)}+\sqrt{3}+1}\right)|-7-4 \sqrt{3}\right)}{d \sqrt{\frac{1-\sqrt[3]{\csc (c+d x)}}{\left(-\sqrt[3]{\csc (c+d x)}+\sqrt{3}+1\right)^2}} (a-a \csc (c+d x)) \sqrt{a \csc (c+d x)+a}}-\frac{3 \sqrt[4]{3} \sqrt{2-\sqrt{3}} a^2 \cot (c+d x) \left(1-\sqrt[3]{\csc (c+d x)}\right) \sqrt{\frac{\csc ^{\frac{2}{3}}(c+d x)+\sqrt[3]{\csc (c+d x)}+1}{\left(-\sqrt[3]{\csc (c+d x)}+\sqrt{3}+1\right)^2}} E\left(\sin ^{-1}\left(\frac{-\sqrt[3]{\csc (c+d x)}-\sqrt{3}+1}{-\sqrt[3]{\csc (c+d x)}+\sqrt{3}+1}\right)|-7-4 \sqrt{3}\right)}{d \sqrt{\frac{1-\sqrt[3]{\csc (c+d x)}}{\left(-\sqrt[3]{\csc (c+d x)}+\sqrt{3}+1\right)^2}} (a-a \csc (c+d x)) \sqrt{a \csc (c+d x)+a}}+\frac{6 a \cot (c+d x)}{d \left(-\sqrt[3]{\csc (c+d x)}+\sqrt{3}+1\right) \sqrt{a \csc (c+d x)+a}}",1,"(2*Sqrt[a*(1 + Csc[c + d*x])]*(-3 + 2*Csc[c + d*x]^(1/3)*Hypergeometric2F1[1/2, 4/3, 3/2, 1 - Csc[c + d*x]])*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2]))/(d*Csc[c + d*x]^(1/3)*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))","C",1
26,1,46,508,0.7564095,"\int \frac{\sqrt{a+a \csc (c+d x)}}{\sqrt[3]{\csc (c+d x)}} \, dx","Integrate[Sqrt[a + a*Csc[c + d*x]]/Csc[c + d*x]^(1/3),x]","-\frac{2 a \cot (c+d x) \, _2F_1\left(\frac{1}{2},\frac{4}{3};\frac{3}{2};1-\csc (c+d x)\right)}{d \sqrt{a (\csc (c+d x)+1)}}","-\frac{\sqrt{2} 3^{3/4} a^2 \cot (c+d x) \left(1-\sqrt[3]{\csc (c+d x)}\right) \sqrt{\frac{\csc ^{\frac{2}{3}}(c+d x)+\sqrt[3]{\csc (c+d x)}+1}{\left(-\sqrt[3]{\csc (c+d x)}+\sqrt{3}+1\right)^2}} F\left(\sin ^{-1}\left(\frac{-\sqrt[3]{\csc (c+d x)}-\sqrt{3}+1}{-\sqrt[3]{\csc (c+d x)}+\sqrt{3}+1}\right)|-7-4 \sqrt{3}\right)}{d \sqrt{\frac{1-\sqrt[3]{\csc (c+d x)}}{\left(-\sqrt[3]{\csc (c+d x)}+\sqrt{3}+1\right)^2}} (a-a \csc (c+d x)) \sqrt{a \csc (c+d x)+a}}+\frac{3 \sqrt[4]{3} \sqrt{2-\sqrt{3}} a^2 \cot (c+d x) \left(1-\sqrt[3]{\csc (c+d x)}\right) \sqrt{\frac{\csc ^{\frac{2}{3}}(c+d x)+\sqrt[3]{\csc (c+d x)}+1}{\left(-\sqrt[3]{\csc (c+d x)}+\sqrt{3}+1\right)^2}} E\left(\sin ^{-1}\left(\frac{-\sqrt[3]{\csc (c+d x)}-\sqrt{3}+1}{-\sqrt[3]{\csc (c+d x)}+\sqrt{3}+1}\right)|-7-4 \sqrt{3}\right)}{2 d \sqrt{\frac{1-\sqrt[3]{\csc (c+d x)}}{\left(-\sqrt[3]{\csc (c+d x)}+\sqrt{3}+1\right)^2}} (a-a \csc (c+d x)) \sqrt{a \csc (c+d x)+a}}-\frac{3 a \cos (c+d x) \csc ^{\frac{2}{3}}(c+d x)}{d \sqrt{a \csc (c+d x)+a}}-\frac{3 a \cot (c+d x)}{d \left(-\sqrt[3]{\csc (c+d x)}+\sqrt{3}+1\right) \sqrt{a \csc (c+d x)+a}}",1,"(-2*a*Cot[c + d*x]*Hypergeometric2F1[1/2, 4/3, 3/2, 1 - Csc[c + d*x]])/(d*Sqrt[a*(1 + Csc[c + d*x])])","C",1
27,1,72,552,1.2178282,"\int \frac{\sqrt{a+a \csc (c+d x)}}{\csc ^{\frac{4}{3}}(c+d x)} \, dx","Integrate[Sqrt[a + a*Csc[c + d*x]]/Csc[c + d*x]^(4/3),x]","-\frac{a \cos (c+d x) \left(5 \csc ^{\frac{4}{3}}(c+d x) \, _2F_1\left(\frac{1}{2},\frac{4}{3};\frac{3}{2};1-\csc (c+d x)\right)+3\right)}{4 d \sqrt[3]{\csc (c+d x)} \sqrt{a (\csc (c+d x)+1)}}","-\frac{5\ 3^{3/4} a^2 \cot (c+d x) \left(1-\sqrt[3]{\csc (c+d x)}\right) \sqrt{\frac{\csc ^{\frac{2}{3}}(c+d x)+\sqrt[3]{\csc (c+d x)}+1}{\left(-\sqrt[3]{\csc (c+d x)}+\sqrt{3}+1\right)^2}} F\left(\sin ^{-1}\left(\frac{-\sqrt[3]{\csc (c+d x)}-\sqrt{3}+1}{-\sqrt[3]{\csc (c+d x)}+\sqrt{3}+1}\right)|-7-4 \sqrt{3}\right)}{4 \sqrt{2} d \sqrt{\frac{1-\sqrt[3]{\csc (c+d x)}}{\left(-\sqrt[3]{\csc (c+d x)}+\sqrt{3}+1\right)^2}} (a-a \csc (c+d x)) \sqrt{a \csc (c+d x)+a}}+\frac{15 \sqrt[4]{3} \sqrt{2-\sqrt{3}} a^2 \cot (c+d x) \left(1-\sqrt[3]{\csc (c+d x)}\right) \sqrt{\frac{\csc ^{\frac{2}{3}}(c+d x)+\sqrt[3]{\csc (c+d x)}+1}{\left(-\sqrt[3]{\csc (c+d x)}+\sqrt{3}+1\right)^2}} E\left(\sin ^{-1}\left(\frac{-\sqrt[3]{\csc (c+d x)}-\sqrt{3}+1}{-\sqrt[3]{\csc (c+d x)}+\sqrt{3}+1}\right)|-7-4 \sqrt{3}\right)}{16 d \sqrt{\frac{1-\sqrt[3]{\csc (c+d x)}}{\left(-\sqrt[3]{\csc (c+d x)}+\sqrt{3}+1\right)^2}} (a-a \csc (c+d x)) \sqrt{a \csc (c+d x)+a}}-\frac{15 a \cos (c+d x) \csc ^{\frac{2}{3}}(c+d x)}{8 d \sqrt{a \csc (c+d x)+a}}-\frac{3 a \cos (c+d x)}{4 d \sqrt[3]{\csc (c+d x)} \sqrt{a \csc (c+d x)+a}}-\frac{15 a \cot (c+d x)}{8 d \left(-\sqrt[3]{\csc (c+d x)}+\sqrt{3}+1\right) \sqrt{a \csc (c+d x)+a}}",1,"-1/4*(a*Cos[c + d*x]*(3 + 5*Csc[c + d*x]^(4/3)*Hypergeometric2F1[1/2, 4/3, 3/2, 1 - Csc[c + d*x]]))/(d*Csc[c + d*x]^(1/3)*Sqrt[a*(1 + Csc[c + d*x])])","C",1
28,1,48,48,0.169892,"\int \csc ^n(c+d x) \sqrt{a+a \csc (c+d x)} \, dx","Integrate[Csc[c + d*x]^n*Sqrt[a + a*Csc[c + d*x]],x]","-\frac{2 a \cot (c+d x) \, _2F_1\left(\frac{1}{2},1-n;\frac{3}{2};1-\csc (c+d x)\right)}{d \sqrt{a (\csc (c+d x)+1)}}","-\frac{2 a \cot (c+d x) \, _2F_1\left(\frac{1}{2},1-n;\frac{3}{2};1-\csc (c+d x)\right)}{d \sqrt{a \csc (c+d x)+a}}",1,"(-2*a*Cot[c + d*x]*Hypergeometric2F1[1/2, 1 - n, 3/2, 1 - Csc[c + d*x]])/(d*Sqrt[a*(1 + Csc[c + d*x])])","A",1
29,1,73,69,0.9904818,"\int \csc ^n(c+d x) \sqrt{a-a \csc (c+d x)} \, dx","Integrate[Csc[c + d*x]^n*Sqrt[a - a*Csc[c + d*x]],x]","-\frac{2 a \cos (c+d x) \csc ^{2 n+1}(c+d x) \left(-\csc ^2(c+d x)\right)^{-n} \, _2F_1\left(\frac{1}{2},1-n;\frac{3}{2};\csc (c+d x)+1\right)}{d \sqrt{a-a \csc (c+d x)}}","-\frac{2 a \cos (c+d x) (-\csc (c+d x))^{-n} \csc ^{n+1}(c+d x) \, _2F_1\left(\frac{1}{2},1-n;\frac{3}{2};\csc (c+d x)+1\right)}{d \sqrt{a-a \csc (c+d x)}}",1,"(-2*a*Cos[c + d*x]*Csc[c + d*x]^(1 + 2*n)*Hypergeometric2F1[1/2, 1 - n, 3/2, 1 + Csc[c + d*x]])/(d*(-Csc[c + d*x]^2)^n*Sqrt[a - a*Csc[c + d*x]])","A",1
30,1,178,156,4.6133914,"\int \csc ^3(e+f x) (a+a \csc (e+f x))^m \, dx","Integrate[Csc[e + f*x]^3*(a + a*Csc[e + f*x])^m,x]","-\frac{\tan ^2\left(\frac{1}{2} (e+f x)\right) \left(\tan \left(\frac{1}{2} (e+f x)\right)+1\right)^{-2 m} (a (\csc (e+f x)+1))^m \left((m-2) m \cot ^4\left(\frac{1}{2} (e+f x)\right) \, _2F_1\left(-m-2,-2 m;-m-1;-\tan \left(\frac{1}{2} (e+f x)\right)\right)+(m+2) \left(m \, _2F_1\left(2-m,-2 m;3-m;-\tan \left(\frac{1}{2} (e+f x)\right)\right)+2 (m-2) \cot ^2\left(\frac{1}{2} (e+f x)\right) \, _2F_1\left(-2 m,-m;1-m;-\tan \left(\frac{1}{2} (e+f x)\right)\right)\right)\right)}{4 f (m-2) m (m+2)}","-\frac{2^{m+\frac{1}{2}} \left(m^2+m+1\right) \cot (e+f x) (\csc (e+f x)+1)^{-m-\frac{1}{2}} (a \csc (e+f x)+a)^m \, _2F_1\left(\frac{1}{2},\frac{1}{2}-m;\frac{3}{2};\frac{1}{2} (1-\csc (e+f x))\right)}{f (m+1) (m+2)}+\frac{\cot (e+f x) (a \csc (e+f x)+a)^m}{f \left(m^2+3 m+2\right)}-\frac{\cot (e+f x) (a \csc (e+f x)+a)^{m+1}}{a f (m+2)}",1,"-1/4*((a*(1 + Csc[e + f*x]))^m*((-2 + m)*m*Cot[(e + f*x)/2]^4*Hypergeometric2F1[-2 - m, -2*m, -1 - m, -Tan[(e + f*x)/2]] + (2 + m)*(m*Hypergeometric2F1[2 - m, -2*m, 3 - m, -Tan[(e + f*x)/2]] + 2*(-2 + m)*Cot[(e + f*x)/2]^2*Hypergeometric2F1[-2*m, -m, 1 - m, -Tan[(e + f*x)/2]]))*Tan[(e + f*x)/2]^2)/(f*(-2 + m)*m*(2 + m)*(1 + Tan[(e + f*x)/2])^(2*m))","A",1
31,1,126,109,1.3484441,"\int \csc ^2(e+f x) (a+a \csc (e+f x))^m \, dx","Integrate[Csc[e + f*x]^2*(a + a*Csc[e + f*x])^m,x]","-\frac{\tan \left(\frac{1}{2} (e+f x)\right) \left(\tan \left(\frac{1}{2} (e+f x)\right)+1\right)^{-2 m} (a (\csc (e+f x)+1))^m \left((m+1) \, _2F_1\left(1-m,-2 m;2-m;-\tan \left(\frac{1}{2} (e+f x)\right)\right)+(m-1) \cot ^2\left(\frac{1}{2} (e+f x)\right) \, _2F_1\left(-m-1,-2 m;-m;-\tan \left(\frac{1}{2} (e+f x)\right)\right)\right)}{2 f (m-1) (m+1)}","-\frac{2^{m+\frac{1}{2}} m \cot (e+f x) (\csc (e+f x)+1)^{-m-\frac{1}{2}} (a \csc (e+f x)+a)^m \, _2F_1\left(\frac{1}{2},\frac{1}{2}-m;\frac{3}{2};\frac{1}{2} (1-\csc (e+f x))\right)}{f (m+1)}-\frac{\cot (e+f x) (a \csc (e+f x)+a)^m}{f (m+1)}",1,"-1/2*((a*(1 + Csc[e + f*x]))^m*((-1 + m)*Cot[(e + f*x)/2]^2*Hypergeometric2F1[-1 - m, -2*m, -m, -Tan[(e + f*x)/2]] + (1 + m)*Hypergeometric2F1[1 - m, -2*m, 2 - m, -Tan[(e + f*x)/2]])*Tan[(e + f*x)/2])/(f*(-1 + m)*(1 + m)*(1 + Tan[(e + f*x)/2])^(2*m))","A",1
32,1,60,74,0.204692,"\int \csc (e+f x) (a+a \csc (e+f x))^m \, dx","Integrate[Csc[e + f*x]*(a + a*Csc[e + f*x])^m,x]","-\frac{\left(\tan \left(\frac{1}{2} (e+f x)\right)+1\right)^{-2 m} (a (\csc (e+f x)+1))^m \, _2F_1\left(-2 m,-m;1-m;-\tan \left(\frac{1}{2} (e+f x)\right)\right)}{f m}","-\frac{2^{m+\frac{1}{2}} \cot (e+f x) (\csc (e+f x)+1)^{-m-\frac{1}{2}} (a \csc (e+f x)+a)^m \, _2F_1\left(\frac{1}{2},\frac{1}{2}-m;\frac{3}{2};\frac{1}{2} (1-\csc (e+f x))\right)}{f}",1,"-(((a*(1 + Csc[e + f*x]))^m*Hypergeometric2F1[-2*m, -m, 1 - m, -Tan[(e + f*x)/2]])/(f*m*(1 + Tan[(e + f*x)/2])^(2*m)))","A",1
33,0,0,84,0.6148152,"\int (a+a \csc (e+f x))^m \, dx","Integrate[(a + a*Csc[e + f*x])^m,x]","\int (a+a \csc (e+f x))^m \, dx","-\frac{\sqrt{2} \cot (e+f x) (a \csc (e+f x)+a)^m F_1\left(m+\frac{1}{2};\frac{1}{2},1;m+\frac{3}{2};\frac{1}{2} (\csc (e+f x)+1),\csc (e+f x)+1\right)}{f (2 m+1) \sqrt{1-\csc (e+f x)}}",1,"Integrate[(a + a*Csc[e + f*x])^m, x]","F",-1
34,0,0,83,3.7712825,"\int (a+a \csc (e+f x))^m \sin (e+f x) \, dx","Integrate[(a + a*Csc[e + f*x])^m*Sin[e + f*x],x]","\int (a+a \csc (e+f x))^m \sin (e+f x) \, dx","\frac{\sqrt{2} \cot (e+f x) (a \csc (e+f x)+a)^m F_1\left(m+\frac{1}{2};\frac{1}{2},2;m+\frac{3}{2};\frac{1}{2} (\csc (e+f x)+1),\csc (e+f x)+1\right)}{f (2 m+1) \sqrt{1-\csc (e+f x)}}",1,"Integrate[(a + a*Csc[e + f*x])^m*Sin[e + f*x], x]","F",-1
35,0,0,84,7.7107379,"\int (a+a \csc (e+f x))^m \sin ^2(e+f x) \, dx","Integrate[(a + a*Csc[e + f*x])^m*Sin[e + f*x]^2,x]","\int (a+a \csc (e+f x))^m \sin ^2(e+f x) \, dx","-\frac{\sqrt{2} \cot (e+f x) (a \csc (e+f x)+a)^m F_1\left(m+\frac{1}{2};\frac{1}{2},3;m+\frac{3}{2};\frac{1}{2} (\csc (e+f x)+1),\csc (e+f x)+1\right)}{f (2 m+1) \sqrt{1-\csc (e+f x)}}",1,"Integrate[(a + a*Csc[e + f*x])^m*Sin[e + f*x]^2, x]","F",-1
36,1,568,107,6.2548111,"\int (a+b \csc (c+d x))^4 \, dx","Integrate[(a + b*Csc[c + d*x])^4,x]","\frac{a^4 (c+d x) \sin ^4(c+d x) (a+b \csc (c+d x))^4}{d (a \sin (c+d x)+b)^4}+\frac{2 \left(2 a^3 b+a b^3\right) \sin ^4(c+d x) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right) (a+b \csc (c+d x))^4}{d (a \sin (c+d x)+b)^4}-\frac{2 \left(2 a^3 b+a b^3\right) \sin ^4(c+d x) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right) (a+b \csc (c+d x))^4}{d (a \sin (c+d x)+b)^4}+\frac{\sin ^4(c+d x) \csc \left(\frac{1}{2} (c+d x)\right) \left(b^4 \left(-\cos \left(\frac{1}{2} (c+d x)\right)\right)-9 a^2 b^2 \cos \left(\frac{1}{2} (c+d x)\right)\right) (a+b \csc (c+d x))^4}{3 d (a \sin (c+d x)+b)^4}+\frac{\sin ^4(c+d x) \sec \left(\frac{1}{2} (c+d x)\right) \left(9 a^2 b^2 \sin \left(\frac{1}{2} (c+d x)\right)+b^4 \sin \left(\frac{1}{2} (c+d x)\right)\right) (a+b \csc (c+d x))^4}{3 d (a \sin (c+d x)+b)^4}-\frac{b^4 \sin ^4(c+d x) \cot \left(\frac{1}{2} (c+d x)\right) \csc ^2\left(\frac{1}{2} (c+d x)\right) (a+b \csc (c+d x))^4}{24 d (a \sin (c+d x)+b)^4}+\frac{b^4 \sin ^4(c+d x) \tan \left(\frac{1}{2} (c+d x)\right) \sec ^2\left(\frac{1}{2} (c+d x)\right) (a+b \csc (c+d x))^4}{24 d (a \sin (c+d x)+b)^4}-\frac{a b^3 \sin ^4(c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right) (a+b \csc (c+d x))^4}{2 d (a \sin (c+d x)+b)^4}+\frac{a b^3 \sin ^4(c+d x) \sec ^2\left(\frac{1}{2} (c+d x)\right) (a+b \csc (c+d x))^4}{2 d (a \sin (c+d x)+b)^4}","a^4 x-\frac{b^2 \left(17 a^2+2 b^2\right) \cot (c+d x)}{3 d}-\frac{2 a b \left(2 a^2+b^2\right) \tanh ^{-1}(\cos (c+d x))}{d}-\frac{4 a b^3 \cot (c+d x) \csc (c+d x)}{3 d}-\frac{b^2 \cot (c+d x) (a+b \csc (c+d x))^2}{3 d}",1,"(a^4*(c + d*x)*(a + b*Csc[c + d*x])^4*Sin[c + d*x]^4)/(d*(b + a*Sin[c + d*x])^4) + ((-9*a^2*b^2*Cos[(c + d*x)/2] - b^4*Cos[(c + d*x)/2])*Csc[(c + d*x)/2]*(a + b*Csc[c + d*x])^4*Sin[c + d*x]^4)/(3*d*(b + a*Sin[c + d*x])^4) - (a*b^3*Csc[(c + d*x)/2]^2*(a + b*Csc[c + d*x])^4*Sin[c + d*x]^4)/(2*d*(b + a*Sin[c + d*x])^4) - (b^4*Cot[(c + d*x)/2]*Csc[(c + d*x)/2]^2*(a + b*Csc[c + d*x])^4*Sin[c + d*x]^4)/(24*d*(b + a*Sin[c + d*x])^4) - (2*(2*a^3*b + a*b^3)*(a + b*Csc[c + d*x])^4*Log[Cos[(c + d*x)/2]]*Sin[c + d*x]^4)/(d*(b + a*Sin[c + d*x])^4) + (2*(2*a^3*b + a*b^3)*(a + b*Csc[c + d*x])^4*Log[Sin[(c + d*x)/2]]*Sin[c + d*x]^4)/(d*(b + a*Sin[c + d*x])^4) + (a*b^3*(a + b*Csc[c + d*x])^4*Sec[(c + d*x)/2]^2*Sin[c + d*x]^4)/(2*d*(b + a*Sin[c + d*x])^4) + ((a + b*Csc[c + d*x])^4*Sec[(c + d*x)/2]*(9*a^2*b^2*Sin[(c + d*x)/2] + b^4*Sin[(c + d*x)/2])*Sin[c + d*x]^4)/(3*d*(b + a*Sin[c + d*x])^4) + (b^4*(a + b*Csc[c + d*x])^4*Sec[(c + d*x)/2]^2*Sin[c + d*x]^4*Tan[(c + d*x)/2])/(24*d*(b + a*Sin[c + d*x])^4)","B",1
37,1,152,73,0.6275432,"\int (a+b \csc (c+d x))^3 \, dx","Integrate[(a + b*Csc[c + d*x])^3,x]","\frac{8 a^3 c+8 a^3 d x+24 a^2 b \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-24 a^2 b \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)+12 a b^2 \tan \left(\frac{1}{2} (c+d x)\right)-12 a b^2 \cot \left(\frac{1}{2} (c+d x)\right)-b^3 \csc ^2\left(\frac{1}{2} (c+d x)\right)+b^3 \sec ^2\left(\frac{1}{2} (c+d x)\right)+4 b^3 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-4 b^3 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)}{8 d}","a^3 x-\frac{b \left(6 a^2+b^2\right) \tanh ^{-1}(\cos (c+d x))}{2 d}-\frac{5 a b^2 \cot (c+d x)}{2 d}-\frac{b^2 \cot (c+d x) (a+b \csc (c+d x))}{2 d}",1,"(8*a^3*c + 8*a^3*d*x - 12*a*b^2*Cot[(c + d*x)/2] - b^3*Csc[(c + d*x)/2]^2 - 24*a^2*b*Log[Cos[(c + d*x)/2]] - 4*b^3*Log[Cos[(c + d*x)/2]] + 24*a^2*b*Log[Sin[(c + d*x)/2]] + 4*b^3*Log[Sin[(c + d*x)/2]] + b^3*Sec[(c + d*x)/2]^2 + 12*a*b^2*Tan[(c + d*x)/2])/(8*d)","B",1
38,1,76,34,0.1826814,"\int (a+b \csc (c+d x))^2 \, dx","Integrate[(a + b*Csc[c + d*x])^2,x]","\frac{2 a \left(a c+a d x+2 b \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-2 b \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)+b^2 \tan \left(\frac{1}{2} (c+d x)\right)+b^2 \left(-\cot \left(\frac{1}{2} (c+d x)\right)\right)}{2 d}","a^2 x-\frac{2 a b \tanh ^{-1}(\cos (c+d x))}{d}-\frac{b^2 \cot (c+d x)}{d}",1,"(-(b^2*Cot[(c + d*x)/2]) + 2*a*(a*c + a*d*x - 2*b*Log[Cos[(c + d*x)/2]] + 2*b*Log[Sin[(c + d*x)/2]]) + b^2*Tan[(c + d*x)/2])/(2*d)","B",1
39,1,125,112,1.7269324,"\int \frac{\csc ^5(x)}{a+b \csc (x)} \, dx","Integrate[Csc[x]^5/(a + b*Csc[x]),x]","\frac{b \left(3 a^2+2 b^2\right) \cos (3 x) \csc ^3(x)-3 b \cot (x) \csc (x) \left(\left(a^2+2 b^2\right) \csc (x)-2 a b\right)+6 a \left(2 a^2+b^2\right) \left(\log \left(\cos \left(\frac{x}{2}\right)\right)-\log \left(\sin \left(\frac{x}{2}\right)\right)\right)+\frac{24 a^4 \tan ^{-1}\left(\frac{a+b \tan \left(\frac{x}{2}\right)}{\sqrt{b^2-a^2}}\right)}{\sqrt{b^2-a^2}}}{12 b^4}","\frac{a \left(2 a^2+b^2\right) \tanh ^{-1}(\cos (x))}{2 b^4}-\frac{\left(3 a^2+2 b^2\right) \cot (x)}{3 b^3}-\frac{2 a^4 \tanh ^{-1}\left(\frac{a+b \tan \left(\frac{x}{2}\right)}{\sqrt{a^2-b^2}}\right)}{b^4 \sqrt{a^2-b^2}}+\frac{a \cot (x) \csc (x)}{2 b^2}-\frac{\cot (x) \csc ^2(x)}{3 b}",1,"((24*a^4*ArcTan[(a + b*Tan[x/2])/Sqrt[-a^2 + b^2]])/Sqrt[-a^2 + b^2] + b*(3*a^2 + 2*b^2)*Cos[3*x]*Csc[x]^3 - 3*b*Cot[x]*Csc[x]*(-2*a*b + (a^2 + 2*b^2)*Csc[x]) + 6*a*(2*a^2 + b^2)*(Log[Cos[x/2]] - Log[Sin[x/2]]))/(12*b^4)","A",1
40,1,144,84,0.4950871,"\int \frac{\csc ^4(x)}{a+b \csc (x)} \, dx","Integrate[Csc[x]^4/(a + b*Csc[x]),x]","\frac{8 a^2 \log \left(\sin \left(\frac{x}{2}\right)\right)-8 a^2 \log \left(\cos \left(\frac{x}{2}\right)\right)-\frac{16 a^3 \tan ^{-1}\left(\frac{a+b \tan \left(\frac{x}{2}\right)}{\sqrt{b^2-a^2}}\right)}{\sqrt{b^2-a^2}}-4 a b \tan \left(\frac{x}{2}\right)+4 a b \cot \left(\frac{x}{2}\right)-b^2 \csc ^2\left(\frac{x}{2}\right)+b^2 \sec ^2\left(\frac{x}{2}\right)+4 b^2 \log \left(\sin \left(\frac{x}{2}\right)\right)-4 b^2 \log \left(\cos \left(\frac{x}{2}\right)\right)}{8 b^3}","-\frac{\left(2 a^2+b^2\right) \tanh ^{-1}(\cos (x))}{2 b^3}+\frac{2 a^3 \tanh ^{-1}\left(\frac{a+b \tan \left(\frac{x}{2}\right)}{\sqrt{a^2-b^2}}\right)}{b^3 \sqrt{a^2-b^2}}+\frac{a \cot (x)}{b^2}-\frac{\cot (x) \csc (x)}{2 b}",1,"((-16*a^3*ArcTan[(a + b*Tan[x/2])/Sqrt[-a^2 + b^2]])/Sqrt[-a^2 + b^2] + 4*a*b*Cot[x/2] - b^2*Csc[x/2]^2 - 8*a^2*Log[Cos[x/2]] - 4*b^2*Log[Cos[x/2]] + 8*a^2*Log[Sin[x/2]] + 4*b^2*Log[Sin[x/2]] + b^2*Sec[x/2]^2 - 4*a*b*Tan[x/2])/(8*b^3)","A",1
41,1,106,62,0.2049547,"\int \frac{\csc ^3(x)}{a+b \csc (x)} \, dx","Integrate[Csc[x]^3/(a + b*Csc[x]),x]","\frac{\csc \left(\frac{x}{2}\right) \sec \left(\frac{x}{2}\right) \left(2 a^2 \sin (x) \tan ^{-1}\left(\frac{a+b \tan \left(\frac{x}{2}\right)}{\sqrt{b^2-a^2}}\right)+\sqrt{b^2-a^2} \left(a \sin (x) \left(\log \left(\cos \left(\frac{x}{2}\right)\right)-\log \left(\sin \left(\frac{x}{2}\right)\right)\right)-b \cos (x)\right)\right)}{2 b^2 \sqrt{b^2-a^2}}","-\frac{2 a^2 \tanh ^{-1}\left(\frac{a+b \tan \left(\frac{x}{2}\right)}{\sqrt{a^2-b^2}}\right)}{b^2 \sqrt{a^2-b^2}}+\frac{a \tanh ^{-1}(\cos (x))}{b^2}-\frac{\cot (x)}{b}",1,"(Csc[x/2]*Sec[x/2]*(2*a^2*ArcTan[(a + b*Tan[x/2])/Sqrt[-a^2 + b^2]]*Sin[x] + Sqrt[-a^2 + b^2]*(-(b*Cos[x]) + a*(Log[Cos[x/2]] - Log[Sin[x/2]])*Sin[x])))/(2*b^2*Sqrt[-a^2 + b^2])","A",1
42,1,62,53,0.0611378,"\int \frac{\csc ^2(x)}{a+b \csc (x)} \, dx","Integrate[Csc[x]^2/(a + b*Csc[x]),x]","\frac{-\frac{2 a \tan ^{-1}\left(\frac{a+b \tan \left(\frac{x}{2}\right)}{\sqrt{b^2-a^2}}\right)}{\sqrt{b^2-a^2}}+\log \left(\sin \left(\frac{x}{2}\right)\right)-\log \left(\cos \left(\frac{x}{2}\right)\right)}{b}","\frac{2 a \tanh ^{-1}\left(\frac{a+b \tan \left(\frac{x}{2}\right)}{\sqrt{a^2-b^2}}\right)}{b \sqrt{a^2-b^2}}-\frac{\tanh ^{-1}(\cos (x))}{b}",1,"((-2*a*ArcTan[(a + b*Tan[x/2])/Sqrt[-a^2 + b^2]])/Sqrt[-a^2 + b^2] - Log[Cos[x/2]] + Log[Sin[x/2]])/b","A",1
43,1,40,40,0.0236433,"\int \frac{\csc (x)}{a+b \csc (x)} \, dx","Integrate[Csc[x]/(a + b*Csc[x]),x]","\frac{2 \tan ^{-1}\left(\frac{a+b \tan \left(\frac{x}{2}\right)}{\sqrt{b^2-a^2}}\right)}{\sqrt{b^2-a^2}}","-\frac{2 \tanh ^{-1}\left(\frac{a+b \tan \left(\frac{x}{2}\right)}{\sqrt{a^2-b^2}}\right)}{\sqrt{a^2-b^2}}",1,"(2*ArcTan[(a + b*Tan[x/2])/Sqrt[-a^2 + b^2]])/Sqrt[-a^2 + b^2]","A",1
44,1,59,57,0.1058721,"\int \frac{1}{a+b \csc (c+d x)} \, dx","Integrate[(a + b*Csc[c + d*x])^(-1),x]","\frac{-\frac{2 b \tan ^{-1}\left(\frac{a+b \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{b^2-a^2}}\right)}{d \sqrt{b^2-a^2}}+\frac{c}{d}+x}{a}","\frac{2 b \tanh ^{-1}\left(\frac{a+b \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a^2-b^2}}\right)}{a d \sqrt{a^2-b^2}}+\frac{x}{a}",1,"(c/d + x - (2*b*ArcTan[(a + b*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]])/(Sqrt[-a^2 + b^2]*d))/a","A",1
45,1,56,61,0.0951011,"\int \frac{\sin (x)}{a+b \csc (x)} \, dx","Integrate[Sin[x]/(a + b*Csc[x]),x]","-\frac{-\frac{2 b^2 \tan ^{-1}\left(\frac{a+b \tan \left(\frac{x}{2}\right)}{\sqrt{b^2-a^2}}\right)}{\sqrt{b^2-a^2}}+a \cos (x)+b x}{a^2}","-\frac{2 b^2 \tanh ^{-1}\left(\frac{a+b \tan \left(\frac{x}{2}\right)}{\sqrt{a^2-b^2}}\right)}{a^2 \sqrt{a^2-b^2}}-\frac{b x}{a^2}-\frac{\cos (x)}{a}",1,"-((b*x - (2*b^2*ArcTan[(a + b*Tan[x/2])/Sqrt[-a^2 + b^2]])/Sqrt[-a^2 + b^2] + a*Cos[x])/a^2)","A",1
46,1,78,82,0.1134363,"\int \frac{\sin ^2(x)}{a+b \csc (x)} \, dx","Integrate[Sin[x]^2/(a + b*Csc[x]),x]","\frac{-\frac{8 b^3 \tan ^{-1}\left(\frac{a+b \tan \left(\frac{x}{2}\right)}{\sqrt{b^2-a^2}}\right)}{\sqrt{b^2-a^2}}+2 a^2 x-a^2 \sin (2 x)+4 a b \cos (x)+4 b^2 x}{4 a^3}","\frac{b \cos (x)}{a^2}+\frac{x \left(a^2+2 b^2\right)}{2 a^3}+\frac{2 b^3 \tanh ^{-1}\left(\frac{a+b \tan \left(\frac{x}{2}\right)}{\sqrt{a^2-b^2}}\right)}{a^3 \sqrt{a^2-b^2}}-\frac{\sin (x) \cos (x)}{2 a}",1,"(2*a^2*x + 4*b^2*x - (8*b^3*ArcTan[(a + b*Tan[x/2])/Sqrt[-a^2 + b^2]])/Sqrt[-a^2 + b^2] + 4*a*b*Cos[x] - a^2*Sin[2*x])/(4*a^3)","A",1
47,1,98,110,0.2319658,"\int \frac{\sin ^3(x)}{a+b \csc (x)} \, dx","Integrate[Sin[x]^3/(a + b*Csc[x]),x]","\frac{a^3 \cos (3 x)-6 b x \left(a^2+2 b^2\right)-3 a \left(3 a^2+4 b^2\right) \cos (x)+\frac{24 b^4 \tan ^{-1}\left(\frac{a+b \tan \left(\frac{x}{2}\right)}{\sqrt{b^2-a^2}}\right)}{\sqrt{b^2-a^2}}+3 a^2 b \sin (2 x)}{12 a^4}","\frac{b \sin (x) \cos (x)}{2 a^2}-\frac{b x \left(a^2+2 b^2\right)}{2 a^4}-\frac{2 b^4 \tanh ^{-1}\left(\frac{a+b \tan \left(\frac{x}{2}\right)}{\sqrt{a^2-b^2}}\right)}{a^4 \sqrt{a^2-b^2}}-\frac{\left(2 a^2+3 b^2\right) \cos (x)}{3 a^3}-\frac{\sin ^2(x) \cos (x)}{3 a}",1,"(-6*b*(a^2 + 2*b^2)*x + (24*b^4*ArcTan[(a + b*Tan[x/2])/Sqrt[-a^2 + b^2]])/Sqrt[-a^2 + b^2] - 3*a*(3*a^2 + 4*b^2)*Cos[x] + a^3*Cos[3*x] + 3*a^2*b*Sin[2*x])/(12*a^4)","A",1
48,1,129,144,0.3079239,"\int \frac{\sin ^4(x)}{a+b \csc (x)} \, dx","Integrate[Sin[x]^4/(a + b*Csc[x]),x]","\frac{36 a^4 x-24 a^4 \sin (2 x)+3 a^4 \sin (4 x)-8 a^3 b \cos (3 x)+48 a^2 b^2 x-24 a^2 b^2 \sin (2 x)+24 a b \left(3 a^2+4 b^2\right) \cos (x)-\frac{192 b^5 \tan ^{-1}\left(\frac{a+b \tan \left(\frac{x}{2}\right)}{\sqrt{b^2-a^2}}\right)}{\sqrt{b^2-a^2}}+96 b^4 x}{96 a^5}","\frac{b \sin ^2(x) \cos (x)}{3 a^2}+\frac{2 b^5 \tanh ^{-1}\left(\frac{a+b \tan \left(\frac{x}{2}\right)}{\sqrt{a^2-b^2}}\right)}{a^5 \sqrt{a^2-b^2}}+\frac{b \left(2 a^2+3 b^2\right) \cos (x)}{3 a^4}-\frac{\left(3 a^2+4 b^2\right) \sin (x) \cos (x)}{8 a^3}+\frac{x \left(3 a^4+4 a^2 b^2+8 b^4\right)}{8 a^5}-\frac{\sin ^3(x) \cos (x)}{4 a}",1,"(36*a^4*x + 48*a^2*b^2*x + 96*b^4*x - (192*b^5*ArcTan[(a + b*Tan[x/2])/Sqrt[-a^2 + b^2]])/Sqrt[-a^2 + b^2] + 24*a*b*(3*a^2 + 4*b^2)*Cos[x] - 8*a^3*b*Cos[3*x] - 24*a^4*Sin[2*x] - 24*a^2*b^2*Sin[2*x] + 3*a^4*Sin[4*x])/(96*a^5)","A",1
49,1,139,108,0.4449436,"\int \frac{1}{(a+b \csc (c+d x))^2} \, dx","Integrate[(a + b*Csc[c + d*x])^(-2),x]","\frac{\csc (c+d x) (a \sin (c+d x)+b) \left(-\frac{2 b \left(b^2-2 a^2\right) \tan ^{-1}\left(\frac{a+b \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{b^2-a^2}}\right) (a+b \csc (c+d x))}{\left(b^2-a^2\right)^{3/2}}+\frac{a b^2 \cot (c+d x)}{(b-a) (a+b)}+(c+d x) (a+b \csc (c+d x))\right)}{a^2 d (a+b \csc (c+d x))^2}","\frac{2 b \left(2 a^2-b^2\right) \tanh ^{-1}\left(\frac{a+b \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a^2-b^2}}\right)}{a^2 d \left(a^2-b^2\right)^{3/2}}-\frac{b^2 \cot (c+d x)}{a d \left(a^2-b^2\right) (a+b \csc (c+d x))}+\frac{x}{a^2}",1,"(Csc[c + d*x]*((a*b^2*Cot[c + d*x])/((-a + b)*(a + b)) + (c + d*x)*(a + b*Csc[c + d*x]) - (2*b*(-2*a^2 + b^2)*ArcTan[(a + b*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]]*(a + b*Csc[c + d*x]))/(-a^2 + b^2)^(3/2))*(b + a*Sin[c + d*x]))/(a^2*d*(a + b*Csc[c + d*x])^2)","A",1
50,1,216,170,1.0789224,"\int \frac{1}{(a+b \csc (c+d x))^3} \, dx","Integrate[(a + b*Csc[c + d*x])^(-3),x]","\frac{\csc ^2(c+d x) (a \sin (c+d x)+b) \left(-\frac{3 a b^2 \left(2 a^2-b^2\right) \cot (c+d x) (a \sin (c+d x)+b)}{(a-b)^2 (a+b)^2}-\frac{2 b \left(6 a^4-5 a^2 b^2+2 b^4\right) \csc (c+d x) (a \sin (c+d x)+b)^2 \tan ^{-1}\left(\frac{a+b \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{b^2-a^2}}\right)}{\left(b^2-a^2\right)^{5/2}}+\frac{a b^3 \cot (c+d x)}{(a-b) (a+b)}+2 (c+d x) \csc (c+d x) (a \sin (c+d x)+b)^2\right)}{2 a^3 d (a+b \csc (c+d x))^3}","\frac{x}{a^3}-\frac{b^2 \left(5 a^2-2 b^2\right) \cot (c+d x)}{2 a^2 d \left(a^2-b^2\right)^2 (a+b \csc (c+d x))}-\frac{b^2 \cot (c+d x)}{2 a d \left(a^2-b^2\right) (a+b \csc (c+d x))^2}+\frac{b \left(6 a^4-5 a^2 b^2+2 b^4\right) \tanh ^{-1}\left(\frac{a+b \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a^2-b^2}}\right)}{a^3 d \left(a^2-b^2\right)^{5/2}}",1,"(Csc[c + d*x]^2*(b + a*Sin[c + d*x])*((a*b^3*Cot[c + d*x])/((a - b)*(a + b)) - (3*a*b^2*(2*a^2 - b^2)*Cot[c + d*x]*(b + a*Sin[c + d*x]))/((a - b)^2*(a + b)^2) + 2*(c + d*x)*Csc[c + d*x]*(b + a*Sin[c + d*x])^2 - (2*b*(6*a^4 - 5*a^2*b^2 + 2*b^4)*ArcTan[(a + b*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]]*Csc[c + d*x]*(b + a*Sin[c + d*x])^2)/(-a^2 + b^2)^(5/2)))/(2*a^3*d*(a + b*Csc[c + d*x])^3)","A",1
51,1,279,239,2.0249304,"\int \frac{1}{(a+b \csc (c+d x))^4} \, dx","Integrate[(a + b*Csc[c + d*x])^(-4),x]","\frac{\csc ^3(c+d x) (a \sin (c+d x)+b) \left(\frac{a b^3 \left(12 a^2-7 b^2\right) \cot (c+d x) (a \sin (c+d x)+b)}{(a-b)^2 (a+b)^2}-\frac{a b^2 \left(36 a^4-32 a^2 b^2+11 b^4\right) \cot (c+d x) (a \sin (c+d x)+b)^2}{(a-b)^3 (a+b)^3}-\frac{6 b \left(-8 a^6+8 a^4 b^2-7 a^2 b^4+2 b^6\right) \csc (c+d x) (a \sin (c+d x)+b)^3 \tan ^{-1}\left(\frac{a+b \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{b^2-a^2}}\right)}{\left(b^2-a^2\right)^{7/2}}+\frac{2 a b^4 \cot (c+d x)}{(b-a) (a+b)}+6 (c+d x) \csc (c+d x) (a \sin (c+d x)+b)^3\right)}{6 a^4 d (a+b \csc (c+d x))^4}","\frac{x}{a^4}-\frac{b^2 \left(8 a^2-3 b^2\right) \cot (c+d x)}{6 a^2 d \left(a^2-b^2\right)^2 (a+b \csc (c+d x))^2}-\frac{b^2 \cot (c+d x)}{3 a d \left(a^2-b^2\right) (a+b \csc (c+d x))^3}+\frac{b \left(8 a^6-8 a^4 b^2+7 a^2 b^4-2 b^6\right) \tanh ^{-1}\left(\frac{a+b \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a^2-b^2}}\right)}{a^4 d \left(a^2-b^2\right)^{7/2}}-\frac{b^2 \left(26 a^4-17 a^2 b^2+6 b^4\right) \cot (c+d x)}{6 a^3 d \left(a^2-b^2\right)^3 (a+b \csc (c+d x))}",1,"(Csc[c + d*x]^3*(b + a*Sin[c + d*x])*((2*a*b^4*Cot[c + d*x])/((-a + b)*(a + b)) + (a*b^3*(12*a^2 - 7*b^2)*Cot[c + d*x]*(b + a*Sin[c + d*x]))/((a - b)^2*(a + b)^2) - (a*b^2*(36*a^4 - 32*a^2*b^2 + 11*b^4)*Cot[c + d*x]*(b + a*Sin[c + d*x])^2)/((a - b)^3*(a + b)^3) + 6*(c + d*x)*Csc[c + d*x]*(b + a*Sin[c + d*x])^3 - (6*b*(-8*a^6 + 8*a^4*b^2 - 7*a^2*b^4 + 2*b^6)*ArcTan[(a + b*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]]*Csc[c + d*x]*(b + a*Sin[c + d*x])^3)/(-a^2 + b^2)^(7/2)))/(6*a^4*d*(a + b*Csc[c + d*x])^4)","A",1
52,1,66,31,0.0508956,"\int \frac{1}{3+5 \csc (c+d x)} \, dx","Integrate[(3 + 5*Csc[c + d*x])^(-1),x]","\frac{2 (c+d x)-5 \tan ^{-1}\left(\frac{2 \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}{\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)}\right)}{6 d}","-\frac{5 \tan ^{-1}\left(\frac{\cos (c+d x)}{\sin (c+d x)+3}\right)}{6 d}-\frac{x}{12}",1,"(2*(c + d*x) - 5*ArcTan[(2*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])])/(6*d)","B",1
53,1,67,68,0.0450048,"\int \frac{1}{5+3 \csc (c+d x)} \, dx","Integrate[(5 + 3*Csc[c + d*x])^(-1),x]","\frac{4 (c+d x)+3 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+3 \cos \left(\frac{1}{2} (c+d x)\right)\right)-3 \log \left(3 \sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}{20 d}","\frac{3 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+3 \cos \left(\frac{1}{2} (c+d x)\right)\right)}{20 d}-\frac{3 \log \left(3 \sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}{20 d}+\frac{x}{5}",1,"(4*(c + d*x) + 3*Log[3*Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] - 3*Log[Cos[(c + d*x)/2] + 3*Sin[(c + d*x)/2]])/(20*d)","A",1
54,0,0,274,4.5142682,"\int \csc ^3(e+f x) (a+b \csc (e+f x))^m \, dx","Integrate[Csc[e + f*x]^3*(a + b*Csc[e + f*x])^m,x]","\int \csc ^3(e+f x) (a+b \csc (e+f x))^m \, dx","-\frac{\sqrt{2} \left(a^2+b^2 (m+1)\right) \cot (e+f x) (a+b \csc (e+f x))^m \left(\frac{a+b \csc (e+f x)}{a+b}\right)^{-m} F_1\left(\frac{1}{2};\frac{1}{2},-m;\frac{3}{2};\frac{1}{2} (1-\csc (e+f x)),\frac{b (1-\csc (e+f x))}{a+b}\right)}{b^2 f (m+2) \sqrt{\csc (e+f x)+1}}+\frac{\sqrt{2} a (a+b) \cot (e+f x) (a+b \csc (e+f x))^m \left(\frac{a+b \csc (e+f x)}{a+b}\right)^{-m} F_1\left(\frac{1}{2};\frac{1}{2},-m-1;\frac{3}{2};\frac{1}{2} (1-\csc (e+f x)),\frac{b (1-\csc (e+f x))}{a+b}\right)}{b^2 f (m+2) \sqrt{\csc (e+f x)+1}}-\frac{\cot (e+f x) (a+b \csc (e+f x))^{m+1}}{b f (m+2)}",1,"Integrate[Csc[e + f*x]^3*(a + b*Csc[e + f*x])^m, x]","F",-1
55,0,0,220,2.7698198,"\int \csc ^2(e+f x) (a+b \csc (e+f x))^m \, dx","Integrate[Csc[e + f*x]^2*(a + b*Csc[e + f*x])^m,x]","\int \csc ^2(e+f x) (a+b \csc (e+f x))^m \, dx","\frac{\sqrt{2} a \cot (e+f x) (a+b \csc (e+f x))^m \left(\frac{a+b \csc (e+f x)}{a+b}\right)^{-m} F_1\left(\frac{1}{2};\frac{1}{2},-m;\frac{3}{2};\frac{1}{2} (1-\csc (e+f x)),\frac{b (1-\csc (e+f x))}{a+b}\right)}{b f \sqrt{\csc (e+f x)+1}}-\frac{\sqrt{2} (a+b) \cot (e+f x) (a+b \csc (e+f x))^m \left(\frac{a+b \csc (e+f x)}{a+b}\right)^{-m} F_1\left(\frac{1}{2};\frac{1}{2},-m-1;\frac{3}{2};\frac{1}{2} (1-\csc (e+f x)),\frac{b (1-\csc (e+f x))}{a+b}\right)}{b f \sqrt{\csc (e+f x)+1}}",1,"Integrate[Csc[e + f*x]^2*(a + b*Csc[e + f*x])^m, x]","F",-1
56,0,0,104,1.9717634,"\int \csc (e+f x) (a+b \csc (e+f x))^m \, dx","Integrate[Csc[e + f*x]*(a + b*Csc[e + f*x])^m,x]","\int \csc (e+f x) (a+b \csc (e+f x))^m \, dx","-\frac{\sqrt{2} \cot (e+f x) (a+b \csc (e+f x))^m \left(\frac{a+b \csc (e+f x)}{a+b}\right)^{-m} F_1\left(\frac{1}{2};\frac{1}{2},-m;\frac{3}{2};\frac{1}{2} (1-\csc (e+f x)),\frac{b (1-\csc (e+f x))}{a+b}\right)}{f \sqrt{\csc (e+f x)+1}}",1,"Integrate[Csc[e + f*x]*(a + b*Csc[e + f*x])^m, x]","F",-1
57,0,0,15,1.5037791,"\int (a+b \csc (e+f x))^m \, dx","Integrate[(a + b*Csc[e + f*x])^m,x]","\int (a+b \csc (e+f x))^m \, dx","\text{Int}\left((a+b \csc (e+f x))^m,x\right)",0,"Integrate[(a + b*Csc[e + f*x])^m, x]","A",-1
58,0,0,22,6.3622838,"\int (a+b \csc (e+f x))^m \sin (e+f x) \, dx","Integrate[(a + b*Csc[e + f*x])^m*Sin[e + f*x],x]","\int (a+b \csc (e+f x))^m \sin (e+f x) \, dx","\text{Int}\left(\sin (e+f x) (a+b \csc (e+f x))^m,x\right)",0,"Integrate[(a + b*Csc[e + f*x])^m*Sin[e + f*x], x]","A",-1
59,0,0,24,6.2381037,"\int (a+b \csc (e+f x))^m \sin ^2(e+f x) \, dx","Integrate[(a + b*Csc[e + f*x])^m*Sin[e + f*x]^2,x]","\int (a+b \csc (e+f x))^m \sin ^2(e+f x) \, dx","\text{Int}\left(\sin ^2(e+f x) (a+b \csc (e+f x))^m,x\right)",0,"Integrate[(a + b*Csc[e + f*x])^m*Sin[e + f*x]^2, x]","A",-1