1,1,55,0,0.0714401,"\int \frac{\csc ^5(x)}{a+a \csc (x)} \, dx","Int[Csc[x]^5/(a + a*Csc[x]),x]","-\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}","-\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,"(3*ArcTanh[Cos[x]])/(2*a) - (4*Cot[x])/a - (4*Cot[x]^3)/(3*a) + (3*Cot[x]*Csc[x])/(2*a) + (Cot[x]*Csc[x]^3)/(a + a*Csc[x])","A",6,5,13,0.3846,1,"{3818, 3787, 3768, 3770, 3767}"
2,1,44,0,0.0659157,"\int \frac{\csc ^4(x)}{a+a \csc (x)} \, dx","Int[Csc[x]^4/(a + a*Csc[x]),x]","\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}","\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,"(-3*ArcTanh[Cos[x]])/(2*a) + (2*Cot[x])/a - (3*Cot[x]*Csc[x])/(2*a) + (Cot[x]*Csc[x]^2)/(a + a*Csc[x])","A",6,6,13,0.4615,1,"{3818, 3787, 3767, 8, 3768, 3770}"
3,1,27,0,0.0867268,"\int \frac{\csc ^3(x)}{a+a \csc (x)} \, dx","Int[Csc[x]^3/(a + a*Csc[x]),x]","-\frac{\cot (x)}{a}+\frac{\tanh ^{-1}(\cos (x))}{a}-\frac{\cot (x)}{a \csc (x)+a}","-\frac{\cot (x)}{a}+\frac{\tanh ^{-1}(\cos (x))}{a}-\frac{\cot (x)}{a \csc (x)+a}",1,"ArcTanh[Cos[x]]/a - Cot[x]/a - Cot[x]/(a + a*Csc[x])","A",4,4,13,0.3077,1,"{3790, 3789, 3770, 3794}"
4,1,20,0,0.0560358,"\int \frac{\csc ^2(x)}{a+a \csc (x)} \, dx","Int[Csc[x]^2/(a + a*Csc[x]),x]","\frac{\cot (x)}{a \csc (x)+a}-\frac{\tanh ^{-1}(\cos (x))}{a}","\frac{\cot (x)}{a \csc (x)+a}-\frac{\tanh ^{-1}(\cos (x))}{a}",1,"-(ArcTanh[Cos[x]]/a) + Cot[x]/(a + a*Csc[x])","A",3,3,13,0.2308,1,"{3789, 3770, 3794}"
5,1,12,0,0.0198414,"\int \frac{\csc (x)}{a+a \csc (x)} \, dx","Int[Csc[x]/(a + a*Csc[x]),x]","-\frac{\cot (x)}{a \csc (x)+a}","-\frac{\cot (x)}{a \csc (x)+a}",1,"-(Cot[x]/(a + a*Csc[x]))","A",1,1,11,0.09091,1,"{3794}"
6,1,28,0,0.0126391,"\int \frac{1}{a+a \csc (c+d x)} \, dx","Int[(a + a*Csc[c + d*x])^(-1),x]","\frac{\cot (c+d x)}{d (a \csc (c+d x)+a)}+\frac{x}{a}","\frac{\cot (c+d x)}{d (a \csc (c+d x)+a)}+\frac{x}{a}",1,"x/a + Cot[c + d*x]/(d*(a + a*Csc[c + d*x]))","A",2,2,12,0.1667,1,"{3777, 8}"
7,1,25,0,0.0468814,"\int \frac{\sin (x)}{a+a \csc (x)} \, dx","Int[Sin[x]/(a + a*Csc[x]),x]","-\frac{x}{a}-\frac{2 \cos (x)}{a}+\frac{\cos (x)}{a \csc (x)+a}","-\frac{x}{a}-\frac{2 \cos (x)}{a}+\frac{\cos (x)}{a \csc (x)+a}",1,"-(x/a) - (2*Cos[x])/a + Cos[x]/(a + a*Csc[x])","A",4,4,11,0.3636,1,"{3819, 3787, 2638, 8}"
8,1,40,0,0.0612801,"\int \frac{\sin ^2(x)}{a+a \csc (x)} \, dx","Int[Sin[x]^2/(a + a*Csc[x]),x]","\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}","\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,"(3*x)/(2*a) + (2*Cos[x])/a - (3*Cos[x]*Sin[x])/(2*a) + (Cos[x]*Sin[x])/(a + a*Csc[x])","A",5,5,13,0.3846,1,"{3819, 3787, 2635, 8, 2638}"
9,1,53,0,0.0670422,"\int \frac{\sin ^3(x)}{a+a \csc (x)} \, dx","Int[Sin[x]^3/(a + a*Csc[x]),x]","-\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}","-\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,"(-3*x)/(2*a) - (4*Cos[x])/a + (4*Cos[x]^3)/(3*a) + (3*Cos[x]*Sin[x])/(2*a) + (Cos[x]*Sin[x]^2)/(a + a*Csc[x])","A",6,5,13,0.3846,1,"{3819, 3787, 2633, 2635, 8}"
10,1,66,0,0.0733532,"\int \frac{\sin ^4(x)}{a+a \csc (x)} \, dx","Int[Sin[x]^4/(a + a*Csc[x]),x]","\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}","\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,"(15*x)/(8*a) + (4*Cos[x])/a - (4*Cos[x]^3)/(3*a) - (15*Cos[x]*Sin[x])/(8*a) - (5*Cos[x]*Sin[x]^3)/(4*a) + (Cos[x]*Sin[x]^3)/(a + a*Csc[x])","A",7,5,13,0.3846,1,"{3819, 3787, 2635, 8, 2633}"
11,1,57,0,0.066061,"\int \frac{1}{(a+a \csc (c+d x))^2} \, dx","Int[(a + a*Csc[c + d*x])^(-2),x]","\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}","\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,"x/a^2 + (4*Cot[c + d*x])/(3*a^2*d*(1 + Csc[c + d*x])) + Cot[c + d*x]/(3*d*(a + a*Csc[c + d*x])^2)","A",3,3,12,0.2500,1,"{3777, 3919, 3794}"
12,1,88,0,0.1096785,"\int \frac{1}{(a+a \csc (c+d x))^3} \, dx","Int[(a + a*Csc[c + d*x])^(-3),x]","\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}","\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,"x/a^3 + Cot[c + d*x]/(5*d*(a + a*Csc[c + d*x])^3) + (7*Cot[c + d*x])/(15*a*d*(a + a*Csc[c + d*x])^2) + (22*Cot[c + d*x])/(15*d*(a^3 + a^3*Csc[c + d*x]))","A",4,4,12,0.3333,1,"{3777, 3922, 3919, 3794}"
13,1,65,0,0.0913834,"\int (a+a \csc (x))^{5/2} \, dx","Int[(a + a*Csc[x])^(5/2),x]","-\frac{14 a^3 \cot (x)}{3 \sqrt{a \csc (x)+a}}-\frac{2}{3} a^2 \cot (x) \sqrt{a \csc (x)+a}-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}-2 a^{5/2} \tan ^{-1}\left(\frac{\sqrt{a} \cot (x)}{\sqrt{a \csc (x)+a}}\right)",1,"-2*a^(5/2)*ArcTan[(Sqrt[a]*Cot[x])/Sqrt[a + a*Csc[x]]] - (14*a^3*Cot[x])/(3*Sqrt[a + a*Csc[x]]) - (2*a^2*Cot[x]*Sqrt[a + a*Csc[x]])/3","A",5,5,10,0.5000,1,"{3775, 3915, 3774, 203, 3792}"
14,1,44,0,0.0304638,"\int (a+a \csc (x))^{3/2} \, dx","Int[(a + a*Csc[x])^(3/2),x]","-\frac{2 a^2 \cot (x)}{\sqrt{a \csc (x)+a}}-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}}-2 a^{3/2} \tan ^{-1}\left(\frac{\sqrt{a} \cot (x)}{\sqrt{a \csc (x)+a}}\right)",1,"-2*a^(3/2)*ArcTan[(Sqrt[a]*Cot[x])/Sqrt[a + a*Csc[x]]] - (2*a^2*Cot[x])/Sqrt[a + a*Csc[x]]","A",4,4,10,0.4000,1,"{3775, 21, 3774, 203}"
15,1,26,0,0.0155253,"\int \sqrt{a+a \csc (x)} \, dx","Int[Sqrt[a + a*Csc[x]],x]","-2 \sqrt{a} \tan ^{-1}\left(\frac{\sqrt{a} \cot (x)}{\sqrt{a \csc (x)+a}}\right)","-2 \sqrt{a} \tan ^{-1}\left(\frac{\sqrt{a} \cot (x)}{\sqrt{a \csc (x)+a}}\right)",1,"-2*Sqrt[a]*ArcTan[(Sqrt[a]*Cot[x])/Sqrt[a + a*Csc[x]]]","A",2,2,10,0.2000,1,"{3774, 203}"
16,1,62,0,0.0575193,"\int \frac{1}{\sqrt{a+a \csc (x)}} \, dx","Int[1/Sqrt[a + a*Csc[x]],x]","\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}}","\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[a]*Cot[x])/Sqrt[a + a*Csc[x]]])/Sqrt[a] + (Sqrt[2]*ArcTan[(Sqrt[a]*Cot[x])/(Sqrt[2]*Sqrt[a + a*Csc[x]])])/Sqrt[a]","A",5,4,10,0.4000,1,"{3776, 3774, 203, 3795}"
17,1,81,0,0.1018981,"\int \frac{1}{(a+a \csc (x))^{3/2}} \, dx","Int[(a + a*Csc[x])^(-3/2),x]","-\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}}","-\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,"(-2*ArcTan[(Sqrt[a]*Cot[x])/Sqrt[a + a*Csc[x]]])/a^(3/2) + (5*ArcTan[(Sqrt[a]*Cot[x])/(Sqrt[2]*Sqrt[a + a*Csc[x]])])/(2*Sqrt[2]*a^(3/2)) + Cot[x]/(2*(a + a*Csc[x])^(3/2))","A",6,5,10,0.5000,1,"{3777, 3920, 3774, 203, 3795}"
18,1,100,0,0.1481735,"\int \frac{1}{(a+a \csc (x))^{5/2}} \, dx","Int[(a + a*Csc[x])^(-5/2),x]","-\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}}","-\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,"(-2*ArcTan[(Sqrt[a]*Cot[x])/Sqrt[a + a*Csc[x]]])/a^(5/2) + (43*ArcTan[(Sqrt[a]*Cot[x])/(Sqrt[2]*Sqrt[a + a*Csc[x]])])/(16*Sqrt[2]*a^(5/2)) + Cot[x]/(4*(a + a*Csc[x])^(5/2)) + (11*Cot[x])/(16*a*(a + a*Csc[x])^(3/2))","A",7,6,10,0.6000,1,"{3777, 3922, 3920, 3774, 203, 3795}"
19,1,37,0,0.0585123,"\int \sqrt{\csc (e+f x)} \sqrt{a+a \csc (e+f x)} \, dx","Int[Sqrt[Csc[e + f*x]]*Sqrt[a + a*Csc[e + f*x]],x]","-\frac{2 \sqrt{a} \sinh ^{-1}\left(\frac{\sqrt{a} \cot (e+f x)}{\sqrt{a \csc (e+f x)+a}}\right)}{f}","-\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*Sqrt[a]*ArcSinh[(Sqrt[a]*Cot[e + f*x])/Sqrt[a + a*Csc[e + f*x]]])/f","A",2,2,25,0.08000,1,"{3801, 215}"
20,1,38,0,0.0708915,"\int \sqrt{-\csc (e+f x)} \sqrt{a-a \csc (e+f x)} \, dx","Int[Sqrt[-Csc[e + f*x]]*Sqrt[a - a*Csc[e + f*x]],x]","-\frac{2 \sqrt{a} \sinh ^{-1}\left(\frac{\sqrt{a} \cot (e+f x)}{\sqrt{a-a \csc (e+f x)}}\right)}{f}","-\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*Sqrt[a]*ArcSinh[(Sqrt[a]*Cot[e + f*x])/Sqrt[a - a*Csc[e + f*x]]])/f","A",2,2,28,0.07143,1,"{3801, 215}"
21,1,254,0,0.2805584,"\int \csc ^{\frac{4}{3}}(c+d x) \sqrt{a+a \csc (c+d x)} \, dx","Int[Csc[c + d*x]^(4/3)*Sqrt[a + a*Csc[c + d*x]],x]","-\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}}","-\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,"(-6*a*Cos[c + d*x]*Csc[c + d*x]^(4/3))/(5*d*Sqrt[a + a*Csc[c + d*x]]) - (4*3^(3/4)*Sqrt[2 + Sqrt[3]]*a^2*Cot[c + d*x]*(1 - Csc[c + d*x]^(1/3))*Sqrt[(1 + Csc[c + d*x]^(1/3) + Csc[c + d*x]^(2/3))/(1 + Sqrt[3] - Csc[c + d*x]^(1/3))^2]*EllipticF[ArcSin[(1 - Sqrt[3] - Csc[c + d*x]^(1/3))/(1 + Sqrt[3] - Csc[c + d*x]^(1/3))], -7 - 4*Sqrt[3]])/(5*d*Sqrt[(1 - Csc[c + d*x]^(1/3))/(1 + Sqrt[3] - Csc[c + d*x]^(1/3))^2]*(a - a*Csc[c + d*x])*Sqrt[a + a*Csc[c + d*x]])","A",4,4,25,0.1600,1,"{3806, 50, 63, 218}"
22,1,213,0,0.1277549,"\int \sqrt[3]{\csc (c+d x)} \sqrt{a+a \csc (c+d x)} \, dx","Int[Csc[c + d*x]^(1/3)*Sqrt[a + a*Csc[c + d*x]],x]","-\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}}","-\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*3^(3/4)*Sqrt[2 + Sqrt[3]]*a^2*Cot[c + d*x]*(1 - Csc[c + d*x]^(1/3))*Sqrt[(1 + Csc[c + d*x]^(1/3) + Csc[c + d*x]^(2/3))/(1 + Sqrt[3] - Csc[c + d*x]^(1/3))^2]*EllipticF[ArcSin[(1 - Sqrt[3] - Csc[c + d*x]^(1/3))/(1 + Sqrt[3] - Csc[c + d*x]^(1/3))], -7 - 4*Sqrt[3]])/(d*Sqrt[(1 - Csc[c + d*x]^(1/3))/(1 + Sqrt[3] - Csc[c + d*x]^(1/3))^2]*(a - a*Csc[c + d*x])*Sqrt[a + a*Csc[c + d*x]])","A",3,3,25,0.1200,1,"{3806, 63, 218}"
23,1,254,0,0.1449307,"\int \frac{\sqrt{a+a \csc (c+d x)}}{\csc ^{\frac{2}{3}}(c+d x)} \, dx","Int[Sqrt[a + a*Csc[c + d*x]]/Csc[c + d*x]^(2/3),x]","-\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}}","-\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,"(-3*a*Cos[c + d*x]*Csc[c + d*x]^(1/3))/(2*d*Sqrt[a + a*Csc[c + d*x]]) - (3^(3/4)*Sqrt[2 + Sqrt[3]]*a^2*Cot[c + d*x]*(1 - Csc[c + d*x]^(1/3))*Sqrt[(1 + Csc[c + d*x]^(1/3) + Csc[c + d*x]^(2/3))/(1 + Sqrt[3] - Csc[c + d*x]^(1/3))^2]*EllipticF[ArcSin[(1 - Sqrt[3] - Csc[c + d*x]^(1/3))/(1 + Sqrt[3] - Csc[c + d*x]^(1/3))], -7 - 4*Sqrt[3]])/(2*d*Sqrt[(1 - Csc[c + d*x]^(1/3))/(1 + Sqrt[3] - Csc[c + d*x]^(1/3))^2]*(a - a*Csc[c + d*x])*Sqrt[a + a*Csc[c + d*x]])","A",4,4,25,0.1600,1,"{3806, 51, 63, 218}"
24,1,514,0,0.2985014,"\int \csc ^{\frac{5}{3}}(c+d x) \sqrt{a+a \csc (c+d x)} \, dx","Int[Csc[c + d*x]^(5/3)*Sqrt[a + a*Csc[c + d*x]],x]","\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}}","\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,"(24*a*Cot[c + d*x])/(7*d*(1 + Sqrt[3] - Csc[c + d*x]^(1/3))*Sqrt[a + a*Csc[c + d*x]]) - (6*a*Cos[c + d*x]*Csc[c + d*x]^(5/3))/(7*d*Sqrt[a + a*Csc[c + d*x]]) - (12*3^(1/4)*Sqrt[2 - Sqrt[3]]*a^2*Cot[c + d*x]*(1 - Csc[c + d*x]^(1/3))*Sqrt[(1 + Csc[c + d*x]^(1/3) + Csc[c + d*x]^(2/3))/(1 + Sqrt[3] - Csc[c + d*x]^(1/3))^2]*EllipticE[ArcSin[(1 - Sqrt[3] - Csc[c + d*x]^(1/3))/(1 + Sqrt[3] - Csc[c + d*x]^(1/3))], -7 - 4*Sqrt[3]])/(7*d*Sqrt[(1 - Csc[c + d*x]^(1/3))/(1 + Sqrt[3] - Csc[c + d*x]^(1/3))^2]*(a - a*Csc[c + d*x])*Sqrt[a + a*Csc[c + d*x]]) + (8*Sqrt[2]*3^(3/4)*a^2*Cot[c + d*x]*(1 - Csc[c + d*x]^(1/3))*Sqrt[(1 + Csc[c + d*x]^(1/3) + Csc[c + d*x]^(2/3))/(1 + Sqrt[3] - Csc[c + d*x]^(1/3))^2]*EllipticF[ArcSin[(1 - Sqrt[3] - Csc[c + d*x]^(1/3))/(1 + Sqrt[3] - Csc[c + d*x]^(1/3))], -7 - 4*Sqrt[3]])/(7*d*Sqrt[(1 - Csc[c + d*x]^(1/3))/(1 + Sqrt[3] - Csc[c + d*x]^(1/3))^2]*(a - a*Csc[c + d*x])*Sqrt[a + a*Csc[c + d*x]])","A",6,6,25,0.2400,1,"{3806, 50, 63, 303, 218, 1877}"
25,1,470,0,0.2532671,"\int \csc ^{\frac{2}{3}}(c+d x) \sqrt{a+a \csc (c+d x)} \, dx","Int[Csc[c + d*x]^(2/3)*Sqrt[a + a*Csc[c + d*x]],x]","\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}}","\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,"(6*a*Cot[c + d*x])/(d*(1 + Sqrt[3] - Csc[c + d*x]^(1/3))*Sqrt[a + a*Csc[c + d*x]]) - (3*3^(1/4)*Sqrt[2 - Sqrt[3]]*a^2*Cot[c + d*x]*(1 - Csc[c + d*x]^(1/3))*Sqrt[(1 + Csc[c + d*x]^(1/3) + Csc[c + d*x]^(2/3))/(1 + Sqrt[3] - Csc[c + d*x]^(1/3))^2]*EllipticE[ArcSin[(1 - Sqrt[3] - Csc[c + d*x]^(1/3))/(1 + Sqrt[3] - Csc[c + d*x]^(1/3))], -7 - 4*Sqrt[3]])/(d*Sqrt[(1 - Csc[c + d*x]^(1/3))/(1 + Sqrt[3] - Csc[c + d*x]^(1/3))^2]*(a - a*Csc[c + d*x])*Sqrt[a + a*Csc[c + d*x]]) + (2*Sqrt[2]*3^(3/4)*a^2*Cot[c + d*x]*(1 - Csc[c + d*x]^(1/3))*Sqrt[(1 + Csc[c + d*x]^(1/3) + Csc[c + d*x]^(2/3))/(1 + Sqrt[3] - Csc[c + d*x]^(1/3))^2]*EllipticF[ArcSin[(1 - Sqrt[3] - Csc[c + d*x]^(1/3))/(1 + Sqrt[3] - Csc[c + d*x]^(1/3))], -7 - 4*Sqrt[3]])/(d*Sqrt[(1 - Csc[c + d*x]^(1/3))/(1 + Sqrt[3] - Csc[c + d*x]^(1/3))^2]*(a - a*Csc[c + d*x])*Sqrt[a + a*Csc[c + d*x]])","A",5,5,25,0.2000,1,"{3806, 63, 303, 218, 1877}"
26,1,508,0,0.274672,"\int \frac{\sqrt{a+a \csc (c+d x)}}{\sqrt[3]{\csc (c+d x)}} \, dx","Int[Sqrt[a + a*Csc[c + d*x]]/Csc[c + d*x]^(1/3),x]","-\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}}","-\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,"(-3*a*Cot[c + d*x])/(d*(1 + Sqrt[3] - Csc[c + d*x]^(1/3))*Sqrt[a + a*Csc[c + d*x]]) - (3*a*Cos[c + d*x]*Csc[c + d*x]^(2/3))/(d*Sqrt[a + a*Csc[c + d*x]]) + (3*3^(1/4)*Sqrt[2 - Sqrt[3]]*a^2*Cot[c + d*x]*(1 - Csc[c + d*x]^(1/3))*Sqrt[(1 + Csc[c + d*x]^(1/3) + Csc[c + d*x]^(2/3))/(1 + Sqrt[3] - Csc[c + d*x]^(1/3))^2]*EllipticE[ArcSin[(1 - Sqrt[3] - Csc[c + d*x]^(1/3))/(1 + Sqrt[3] - Csc[c + d*x]^(1/3))], -7 - 4*Sqrt[3]])/(2*d*Sqrt[(1 - Csc[c + d*x]^(1/3))/(1 + Sqrt[3] - Csc[c + d*x]^(1/3))^2]*(a - a*Csc[c + d*x])*Sqrt[a + a*Csc[c + d*x]]) - (Sqrt[2]*3^(3/4)*a^2*Cot[c + d*x]*(1 - Csc[c + d*x]^(1/3))*Sqrt[(1 + Csc[c + d*x]^(1/3) + Csc[c + d*x]^(2/3))/(1 + Sqrt[3] - Csc[c + d*x]^(1/3))^2]*EllipticF[ArcSin[(1 - Sqrt[3] - Csc[c + d*x]^(1/3))/(1 + Sqrt[3] - Csc[c + d*x]^(1/3))], -7 - 4*Sqrt[3]])/(d*Sqrt[(1 - Csc[c + d*x]^(1/3))/(1 + Sqrt[3] - Csc[c + d*x]^(1/3))^2]*(a - a*Csc[c + d*x])*Sqrt[a + a*Csc[c + d*x]])","A",6,6,25,0.2400,1,"{3806, 51, 63, 303, 218, 1877}"
27,1,552,0,0.3066938,"\int \frac{\sqrt{a+a \csc (c+d x)}}{\csc ^{\frac{4}{3}}(c+d x)} \, dx","Int[Sqrt[a + a*Csc[c + d*x]]/Csc[c + d*x]^(4/3),x]","-\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}}","-\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,"(-15*a*Cot[c + d*x])/(8*d*(1 + Sqrt[3] - Csc[c + d*x]^(1/3))*Sqrt[a + a*Csc[c + d*x]]) - (3*a*Cos[c + d*x])/(4*d*Csc[c + d*x]^(1/3)*Sqrt[a + a*Csc[c + d*x]]) - (15*a*Cos[c + d*x]*Csc[c + d*x]^(2/3))/(8*d*Sqrt[a + a*Csc[c + d*x]]) + (15*3^(1/4)*Sqrt[2 - Sqrt[3]]*a^2*Cot[c + d*x]*(1 - Csc[c + d*x]^(1/3))*Sqrt[(1 + Csc[c + d*x]^(1/3) + Csc[c + d*x]^(2/3))/(1 + Sqrt[3] - Csc[c + d*x]^(1/3))^2]*EllipticE[ArcSin[(1 - Sqrt[3] - Csc[c + d*x]^(1/3))/(1 + Sqrt[3] - Csc[c + d*x]^(1/3))], -7 - 4*Sqrt[3]])/(16*d*Sqrt[(1 - Csc[c + d*x]^(1/3))/(1 + Sqrt[3] - Csc[c + d*x]^(1/3))^2]*(a - a*Csc[c + d*x])*Sqrt[a + a*Csc[c + d*x]]) - (5*3^(3/4)*a^2*Cot[c + d*x]*(1 - Csc[c + d*x]^(1/3))*Sqrt[(1 + Csc[c + d*x]^(1/3) + Csc[c + d*x]^(2/3))/(1 + Sqrt[3] - Csc[c + d*x]^(1/3))^2]*EllipticF[ArcSin[(1 - Sqrt[3] - Csc[c + d*x]^(1/3))/(1 + Sqrt[3] - Csc[c + d*x]^(1/3))], -7 - 4*Sqrt[3]])/(4*Sqrt[2]*d*Sqrt[(1 - Csc[c + d*x]^(1/3))/(1 + Sqrt[3] - Csc[c + d*x]^(1/3))^2]*(a - a*Csc[c + d*x])*Sqrt[a + a*Csc[c + d*x]])","A",7,6,25,0.2400,1,"{3806, 51, 63, 303, 218, 1877}"
28,1,48,0,0.0584631,"\int \csc ^n(c+d x) \sqrt{a+a \csc (c+d x)} \, dx","Int[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)+a}}","-\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 + a*Csc[c + d*x]])","A",2,2,23,0.08696,1,"{3806, 65}"
29,1,69,0,0.0701908,"\int \csc ^n(c+d x) \sqrt{a-a \csc (c+d x)} \, dx","Int[Csc[c + d*x]^n*Sqrt[a - a*Csc[c + d*x]],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)}}","-\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 + n)*Hypergeometric2F1[1/2, 1 - n, 3/2, 1 + Csc[c + d*x]])/(d*(-Csc[c + d*x])^n*Sqrt[a - a*Csc[c + d*x]])","A",3,3,24,0.1250,1,"{3806, 67, 65}"
30,1,156,0,0.1896154,"\int \csc ^3(e+f x) (a+a \csc (e+f x))^m \, dx","Int[Csc[e + f*x]^3*(a + a*Csc[e + f*x])^m,x]","-\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)}","-\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,"(Cot[e + f*x]*(a + a*Csc[e + f*x])^m)/(f*(2 + 3*m + m^2)) - (Cot[e + f*x]*(a + a*Csc[e + f*x])^(1 + m))/(a*f*(2 + m)) - (2^(1/2 + m)*(1 + m + m^2)*Cot[e + f*x]*(1 + Csc[e + f*x])^(-1/2 - m)*(a + a*Csc[e + f*x])^m*Hypergeometric2F1[1/2, 1/2 - m, 3/2, (1 - Csc[e + f*x])/2])/(f*(1 + m)*(2 + m))","A",5,5,21,0.2381,1,"{3800, 4001, 3828, 3827, 69}"
31,1,109,0,0.1031764,"\int \csc ^2(e+f x) (a+a \csc (e+f x))^m \, dx","Int[Csc[e + f*x]^2*(a + a*Csc[e + f*x])^m,x]","-\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)}","-\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,"-((Cot[e + f*x]*(a + a*Csc[e + f*x])^m)/(f*(1 + m))) - (2^(1/2 + m)*m*Cot[e + f*x]*(1 + Csc[e + f*x])^(-1/2 - m)*(a + a*Csc[e + f*x])^m*Hypergeometric2F1[1/2, 1/2 - m, 3/2, (1 - Csc[e + f*x])/2])/(f*(1 + m))","A",4,4,21,0.1905,1,"{3798, 3828, 3827, 69}"
32,1,74,0,0.0564489,"\int \csc (e+f x) (a+a \csc (e+f x))^m \, dx","Int[Csc[e + f*x]*(a + a*Csc[e + f*x])^m,x]","-\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}","-\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,"-((2^(1/2 + m)*Cot[e + f*x]*(1 + Csc[e + f*x])^(-1/2 - m)*(a + a*Csc[e + f*x])^m*Hypergeometric2F1[1/2, 1/2 - m, 3/2, (1 - Csc[e + f*x])/2])/f)","A",3,3,19,0.1579,1,"{3828, 3827, 69}"
33,1,84,0,0.0574375,"\int (a+a \csc (e+f x))^m \, dx","Int[(a + a*Csc[e + f*x])^m,x]","-\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)}}","-\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,"-((Sqrt[2]*AppellF1[1/2 + m, 1/2, 1, 3/2 + m, (1 + Csc[e + f*x])/2, 1 + Csc[e + f*x]]*Cot[e + f*x]*(a + a*Csc[e + f*x])^m)/(f*(1 + 2*m)*Sqrt[1 - Csc[e + f*x]]))","A",3,3,12,0.2500,1,"{3779, 3778, 136}"
34,1,83,0,0.0895867,"\int (a+a \csc (e+f x))^m \sin (e+f x) \, dx","Int[(a + a*Csc[e + f*x])^m*Sin[e + f*x],x]","\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)}}","\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,"(Sqrt[2]*AppellF1[1/2 + m, 1/2, 2, 3/2 + m, (1 + Csc[e + f*x])/2, 1 + Csc[e + f*x]]*Cot[e + f*x]*(a + a*Csc[e + f*x])^m)/(f*(1 + 2*m)*Sqrt[1 - Csc[e + f*x]])","A",3,3,19,0.1579,1,"{3828, 3827, 136}"
35,1,84,0,0.1050515,"\int (a+a \csc (e+f x))^m \sin ^2(e+f x) \, dx","Int[(a + a*Csc[e + f*x])^m*Sin[e + f*x]^2,x]","-\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)}}","-\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,"-((Sqrt[2]*AppellF1[1/2 + m, 1/2, 3, 3/2 + m, (1 + Csc[e + f*x])/2, 1 + Csc[e + f*x]]*Cot[e + f*x]*(a + a*Csc[e + f*x])^m)/(f*(1 + 2*m)*Sqrt[1 - Csc[e + f*x]]))","A",3,3,21,0.1429,1,"{3828, 3827, 136}"
36,1,107,0,0.1097706,"\int (a+b \csc (c+d x))^4 \, dx","Int[(a + b*Csc[c + d*x])^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}+a^4 x-\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}","-\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}+a^4 x-\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*x - (2*a*b*(2*a^2 + b^2)*ArcTanh[Cos[c + d*x]])/d - (b^2*(17*a^2 + 2*b^2)*Cot[c + d*x])/(3*d) - (4*a*b^3*Cot[c + d*x]*Csc[c + d*x])/(3*d) - (b^2*Cot[c + d*x]*(a + b*Csc[c + d*x])^2)/(3*d)","A",6,5,12,0.4167,1,"{3782, 4048, 3770, 3767, 8}"
37,1,73,0,0.0469065,"\int (a+b \csc (c+d x))^3 \, dx","Int[(a + b*Csc[c + d*x])^3,x]","-\frac{b \left(6 a^2+b^2\right) \tanh ^{-1}(\cos (c+d x))}{2 d}+a^3 x-\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}","-\frac{b \left(6 a^2+b^2\right) \tanh ^{-1}(\cos (c+d x))}{2 d}+a^3 x-\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,"a^3*x - (b*(6*a^2 + b^2)*ArcTanh[Cos[c + d*x]])/(2*d) - (5*a*b^2*Cot[c + d*x])/(2*d) - (b^2*Cot[c + d*x]*(a + b*Csc[c + d*x]))/(2*d)","A",5,4,12,0.3333,1,"{3782, 3770, 3767, 8}"
38,1,34,0,0.025757,"\int (a+b \csc (c+d x))^2 \, dx","Int[(a + b*Csc[c + d*x])^2,x]","a^2 x-\frac{2 a b \tanh ^{-1}(\cos (c+d x))}{d}-\frac{b^2 \cot (c+d x)}{d}","a^2 x-\frac{2 a b \tanh ^{-1}(\cos (c+d x))}{d}-\frac{b^2 \cot (c+d x)}{d}",1,"a^2*x - (2*a*b*ArcTanh[Cos[c + d*x]])/d - (b^2*Cot[c + d*x])/d","A",4,4,12,0.3333,1,"{3773, 3770, 3767, 8}"
39,1,112,0,0.4066855,"\int \frac{\csc ^5(x)}{a+b \csc (x)} \, dx","Int[Csc[x]^5/(a + b*Csc[x]),x]","-\frac{\left(3 a^2+2 b^2\right) \cot (x)}{3 b^3}+\frac{a \left(2 a^2+b^2\right) \tanh ^{-1}(\cos (x))}{2 b^4}-\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}","-\frac{\left(3 a^2+2 b^2\right) \cot (x)}{3 b^3}+\frac{a \left(2 a^2+b^2\right) \tanh ^{-1}(\cos (x))}{2 b^4}-\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,"(a*(2*a^2 + b^2)*ArcTanh[Cos[x]])/(2*b^4) - (2*a^4*ArcTanh[(a + b*Tan[x/2])/Sqrt[a^2 - b^2]])/(b^4*Sqrt[a^2 - b^2]) - ((3*a^2 + 2*b^2)*Cot[x])/(3*b^3) + (a*Cot[x]*Csc[x])/(2*b^2) - (Cot[x]*Csc[x]^2)/(3*b)","A",9,9,13,0.6923,1,"{3851, 4092, 4082, 3998, 3770, 3831, 2660, 618, 206}"
40,1,84,0,0.252198,"\int \frac{\csc ^4(x)}{a+b \csc (x)} \, dx","Int[Csc[x]^4/(a + b*Csc[x]),x]","-\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}","-\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,"-((2*a^2 + b^2)*ArcTanh[Cos[x]])/(2*b^3) + (2*a^3*ArcTanh[(a + b*Tan[x/2])/Sqrt[a^2 - b^2]])/(b^3*Sqrt[a^2 - b^2]) + (a*Cot[x])/b^2 - (Cot[x]*Csc[x])/(2*b)","A",8,8,13,0.6154,1,"{3851, 4082, 3998, 3770, 3831, 2660, 618, 206}"
41,1,62,0,0.154549,"\int \frac{\csc ^3(x)}{a+b \csc (x)} \, dx","Int[Csc[x]^3/(a + b*Csc[x]),x]","-\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}","-\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,"(a*ArcTanh[Cos[x]])/b^2 - (2*a^2*ArcTanh[(a + b*Tan[x/2])/Sqrt[a^2 - b^2]])/(b^2*Sqrt[a^2 - b^2]) - Cot[x]/b","A",7,7,13,0.5385,1,"{3790, 3789, 3770, 3831, 2660, 618, 206}"
42,1,53,0,0.1128663,"\int \frac{\csc ^2(x)}{a+b \csc (x)} \, dx","Int[Csc[x]^2/(a + b*Csc[x]),x]","\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}","\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,"-(ArcTanh[Cos[x]]/b) + (2*a*ArcTanh[(a + b*Tan[x/2])/Sqrt[a^2 - b^2]])/(b*Sqrt[a^2 - b^2])","A",6,6,13,0.4615,1,"{3789, 3770, 3831, 2660, 618, 206}"
43,1,40,0,0.0702075,"\int \frac{\csc (x)}{a+b \csc (x)} \, dx","Int[Csc[x]/(a + b*Csc[x]),x]","-\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}}","-\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*ArcTanh[(a + b*Tan[x/2])/Sqrt[a^2 - b^2]])/Sqrt[a^2 - b^2]","A",4,4,11,0.3636,1,"{3831, 2660, 618, 206}"
44,1,57,0,0.064512,"\int \frac{1}{a+b \csc (c+d x)} \, dx","Int[(a + b*Csc[c + d*x])^(-1),x]","\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}","\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,"x/a + (2*b*ArcTanh[(a + b*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a*Sqrt[a^2 - b^2]*d)","A",4,4,12,0.3333,1,"{3783, 2660, 618, 206}"
45,1,61,0,0.1066083,"\int \frac{\sin (x)}{a+b \csc (x)} \, dx","Int[Sin[x]/(a + b*Csc[x]),x]","-\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}","-\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)/a^2) - (2*b^2*ArcTanh[(a + b*Tan[x/2])/Sqrt[a^2 - b^2]])/(a^2*Sqrt[a^2 - b^2]) - Cos[x]/a","A",6,6,11,0.5455,1,"{3853, 12, 3783, 2660, 618, 206}"
46,1,82,0,0.2611078,"\int \frac{\sin ^2(x)}{a+b \csc (x)} \, dx","Int[Sin[x]^2/(a + b*Csc[x]),x]","\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{b \cos (x)}{a^2}-\frac{\sin (x) \cos (x)}{2 a}","\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{b \cos (x)}{a^2}-\frac{\sin (x) \cos (x)}{2 a}",1,"((a^2 + 2*b^2)*x)/(2*a^3) + (2*b^3*ArcTanh[(a + b*Tan[x/2])/Sqrt[a^2 - b^2]])/(a^3*Sqrt[a^2 - b^2]) + (b*Cos[x])/a^2 - (Cos[x]*Sin[x])/(2*a)","A",7,7,13,0.5385,1,"{3853, 4104, 3919, 3831, 2660, 618, 206}"
47,1,110,0,0.3982447,"\int \frac{\sin ^3(x)}{a+b \csc (x)} \, dx","Int[Sin[x]^3/(a + b*Csc[x]),x]","-\frac{b x \left(a^2+2 b^2\right)}{2 a^4}-\frac{\left(2 a^2+3 b^2\right) \cos (x)}{3 a^3}-\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{b \sin (x) \cos (x)}{2 a^2}-\frac{\sin ^2(x) \cos (x)}{3 a}","-\frac{b x \left(a^2+2 b^2\right)}{2 a^4}-\frac{\left(2 a^2+3 b^2\right) \cos (x)}{3 a^3}-\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{b \sin (x) \cos (x)}{2 a^2}-\frac{\sin ^2(x) \cos (x)}{3 a}",1,"-(b*(a^2 + 2*b^2)*x)/(2*a^4) - (2*b^4*ArcTanh[(a + b*Tan[x/2])/Sqrt[a^2 - b^2]])/(a^4*Sqrt[a^2 - b^2]) - ((2*a^2 + 3*b^2)*Cos[x])/(3*a^3) + (b*Cos[x]*Sin[x])/(2*a^2) - (Cos[x]*Sin[x]^2)/(3*a)","A",8,7,13,0.5385,1,"{3853, 4104, 3919, 3831, 2660, 618, 206}"
48,1,144,0,0.5884632,"\int \frac{\sin ^4(x)}{a+b \csc (x)} \, dx","Int[Sin[x]^4/(a + b*Csc[x]),x]","\frac{x \left(4 a^2 b^2+3 a^4+8 b^4\right)}{8 a^5}+\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{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 \sin ^2(x) \cos (x)}{3 a^2}-\frac{\sin ^3(x) \cos (x)}{4 a}","\frac{x \left(4 a^2 b^2+3 a^4+8 b^4\right)}{8 a^5}+\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{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 \sin ^2(x) \cos (x)}{3 a^2}-\frac{\sin ^3(x) \cos (x)}{4 a}",1,"((3*a^4 + 4*a^2*b^2 + 8*b^4)*x)/(8*a^5) + (2*b^5*ArcTanh[(a + b*Tan[x/2])/Sqrt[a^2 - b^2]])/(a^5*Sqrt[a^2 - b^2]) + (b*(2*a^2 + 3*b^2)*Cos[x])/(3*a^4) - ((3*a^2 + 4*b^2)*Cos[x]*Sin[x])/(8*a^3) + (b*Cos[x]*Sin[x]^2)/(3*a^2) - (Cos[x]*Sin[x]^3)/(4*a)","A",9,7,13,0.5385,1,"{3853, 4104, 3919, 3831, 2660, 618, 206}"
49,1,108,0,0.1713991,"\int \frac{1}{(a+b \csc (c+d x))^2} \, dx","Int[(a + b*Csc[c + d*x])^(-2),x]","\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}","\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,"x/a^2 + (2*b*(2*a^2 - b^2)*ArcTanh[(a + b*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^2*(a^2 - b^2)^(3/2)*d) - (b^2*Cot[c + d*x])/(a*(a^2 - b^2)*d*(a + b*Csc[c + d*x]))","A",6,6,12,0.5000,1,"{3785, 3919, 3831, 2660, 618, 206}"
50,1,170,0,0.3189675,"\int \frac{1}{(a+b \csc (c+d x))^3} \, dx","Int[(a + b*Csc[c + d*x])^(-3),x]","\frac{b \left(-5 a^2 b^2+6 a^4+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}}-\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{x}{a^3}","\frac{b \left(-5 a^2 b^2+6 a^4+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}}-\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{x}{a^3}",1,"x/a^3 + (b*(6*a^4 - 5*a^2*b^2 + 2*b^4)*ArcTanh[(a + b*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^3*(a^2 - b^2)^(5/2)*d) - (b^2*Cot[c + d*x])/(2*a*(a^2 - b^2)*d*(a + b*Csc[c + d*x])^2) - (b^2*(5*a^2 - 2*b^2)*Cot[c + d*x])/(2*a^2*(a^2 - b^2)^2*d*(a + b*Csc[c + d*x]))","A",7,7,12,0.5833,1,"{3785, 4060, 3919, 3831, 2660, 618, 206}"
51,1,239,0,0.5042495,"\int \frac{1}{(a+b \csc (c+d x))^4} \, dx","Int[(a + b*Csc[c + d*x])^(-4),x]","\frac{b \left(-8 a^4 b^2+7 a^2 b^4+8 a^6-2 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(-17 a^2 b^2+26 a^4+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))}-\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{x}{a^4}","\frac{b \left(-8 a^4 b^2+7 a^2 b^4+8 a^6-2 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(-17 a^2 b^2+26 a^4+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))}-\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{x}{a^4}",1,"x/a^4 + (b*(8*a^6 - 8*a^4*b^2 + 7*a^2*b^4 - 2*b^6)*ArcTanh[(a + b*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^4*(a^2 - b^2)^(7/2)*d) - (b^2*Cot[c + d*x])/(3*a*(a^2 - b^2)*d*(a + b*Csc[c + d*x])^3) - (b^2*(8*a^2 - 3*b^2)*Cot[c + d*x])/(6*a^2*(a^2 - b^2)^2*d*(a + b*Csc[c + d*x])^2) - (b^2*(26*a^4 - 17*a^2*b^2 + 6*b^4)*Cot[c + d*x])/(6*a^3*(a^2 - b^2)^3*d*(a + b*Csc[c + d*x]))","A",8,7,12,0.5833,1,"{3785, 4060, 3919, 3831, 2660, 618, 206}"
52,1,31,0,0.029382,"\int \frac{1}{3+5 \csc (c+d x)} \, dx","Int[(3 + 5*Csc[c + d*x])^(-1),x]","-\frac{5 \tan ^{-1}\left(\frac{\cos (c+d x)}{\sin (c+d x)+3}\right)}{6 d}-\frac{x}{12}","-\frac{5 \tan ^{-1}\left(\frac{\cos (c+d x)}{\sin (c+d x)+3}\right)}{6 d}-\frac{x}{12}",1,"-x/12 - (5*ArcTan[Cos[c + d*x]/(3 + Sin[c + d*x])])/(6*d)","A",2,2,12,0.1667,1,"{3783, 2657}"
53,1,68,0,0.0390231,"\int \frac{1}{5+3 \csc (c+d x)} \, dx","Int[(5 + 3*Csc[c + d*x])^(-1),x]","\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}","\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,"x/5 + (3*Log[3*Cos[(c + d*x)/2] + Sin[(c + d*x)/2]])/(20*d) - (3*Log[Cos[(c + d*x)/2] + 3*Sin[(c + d*x)/2]])/(20*d)","A",5,4,12,0.3333,1,"{3783, 2660, 616, 31}"
54,1,274,0,0.3426596,"\int \csc ^3(e+f x) (a+b \csc (e+f x))^m \, dx","Int[Csc[e + f*x]^3*(a + b*Csc[e + f*x])^m,x]","-\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)}","-\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,"-((Cot[e + f*x]*(a + b*Csc[e + f*x])^(1 + m))/(b*f*(2 + m))) + (Sqrt[2]*a*(a + b)*AppellF1[1/2, 1/2, -1 - m, 3/2, (1 - Csc[e + f*x])/2, (b*(1 - Csc[e + f*x]))/(a + b)]*Cot[e + f*x]*(a + b*Csc[e + f*x])^m)/(b^2*f*(2 + m)*Sqrt[1 + Csc[e + f*x]]*((a + b*Csc[e + f*x])/(a + b))^m) - (Sqrt[2]*(a^2 + b^2*(1 + m))*AppellF1[1/2, 1/2, -m, 3/2, (1 - Csc[e + f*x])/2, (b*(1 - Csc[e + f*x]))/(a + b)]*Cot[e + f*x]*(a + b*Csc[e + f*x])^m)/(b^2*f*(2 + m)*Sqrt[1 + Csc[e + f*x]]*((a + b*Csc[e + f*x])/(a + b))^m)","A",8,5,21,0.2381,1,"{3840, 4007, 3834, 139, 138}"
55,1,220,0,0.2256041,"\int \csc ^2(e+f x) (a+b \csc (e+f x))^m \, dx","Int[Csc[e + f*x]^2*(a + b*Csc[e + f*x])^m,x]","\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}}","\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,"-((Sqrt[2]*(a + b)*AppellF1[1/2, 1/2, -1 - m, 3/2, (1 - Csc[e + f*x])/2, (b*(1 - Csc[e + f*x]))/(a + b)]*Cot[e + f*x]*(a + b*Csc[e + f*x])^m)/(b*f*Sqrt[1 + Csc[e + f*x]]*((a + b*Csc[e + f*x])/(a + b))^m)) + (Sqrt[2]*a*AppellF1[1/2, 1/2, -m, 3/2, (1 - Csc[e + f*x])/2, (b*(1 - Csc[e + f*x]))/(a + b)]*Cot[e + f*x]*(a + b*Csc[e + f*x])^m)/(b*f*Sqrt[1 + Csc[e + f*x]]*((a + b*Csc[e + f*x])/(a + b))^m)","A",7,4,21,0.1905,1,"{3838, 3834, 139, 138}"
56,1,104,0,0.0738236,"\int \csc (e+f x) (a+b \csc (e+f x))^m \, dx","Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^m,x]","-\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}}","-\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,"-((Sqrt[2]*AppellF1[1/2, 1/2, -m, 3/2, (1 - Csc[e + f*x])/2, (b*(1 - Csc[e + f*x]))/(a + b)]*Cot[e + f*x]*(a + b*Csc[e + f*x])^m)/(f*Sqrt[1 + Csc[e + f*x]]*((a + b*Csc[e + f*x])/(a + b))^m))","A",3,3,19,0.1579,1,"{3834, 139, 138}"
57,0,0,0,0.0095191,"\int (a+b \csc (e+f x))^m \, dx","Int[(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,"Defer[Int][(a + b*Csc[e + f*x])^m, x]","A",0,0,0,0,-1,"{}"
58,0,0,0,0.0318065,"\int (a+b \csc (e+f x))^m \sin (e+f x) \, dx","Int[(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,"Defer[Int][(a + b*Csc[e + f*x])^m*Sin[e + f*x], x]","A",0,0,0,0,-1,"{}"
59,0,0,0,0.040237,"\int (a+b \csc (e+f x))^m \sin ^2(e+f x) \, dx","Int[(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,"Defer[Int][(a + b*Csc[e + f*x])^m*Sin[e + f*x]^2, x]","A",0,0,0,0,-1,"{}"