1,1,44,0,0.1224764,"\int \frac{\cos ^4(x)}{a+a \csc (x)} \, dx","Int[Cos[x]^4/(a + a*Csc[x]),x]","-\frac{x}{8 a}-\frac{\cos ^3(x)}{3 a}+\frac{\sin (x) \cos ^3(x)}{4 a}-\frac{\sin (x) \cos (x)}{8 a}","-\frac{x}{8 a}-\frac{\cos ^3(x)}{3 a}+\frac{\sin (x) \cos ^3(x)}{4 a}-\frac{\sin (x) \cos (x)}{8 a}",1,"-x/(8*a) - Cos[x]^3/(3*a) - (Cos[x]*Sin[x])/(8*a) + (Cos[x]^3*Sin[x])/(4*a)","A",7,7,13,0.5385,1,"{3872, 2839, 2565, 30, 2568, 2635, 8}"
2,1,23,0,0.1007734,"\int \frac{\cos ^3(x)}{a+a \csc (x)} \, dx","Int[Cos[x]^3/(a + a*Csc[x]),x]","\frac{\sin ^2(x)}{2 a}-\frac{\sin ^3(x)}{3 a}","\frac{\sin ^2(x)}{2 a}-\frac{\sin ^3(x)}{3 a}",1,"Sin[x]^2/(2*a) - Sin[x]^3/(3*a)","A",6,4,13,0.3077,1,"{3872, 2835, 2564, 30}"
3,1,27,0,0.0911225,"\int \frac{\cos ^2(x)}{a+a \csc (x)} \, dx","Int[Cos[x]^2/(a + a*Csc[x]),x]","-\frac{x}{2 a}-\frac{\cos (x)}{a}+\frac{\sin (x) \cos (x)}{2 a}","-\frac{x}{2 a}-\frac{\cos (x)}{a}+\frac{\sin (x) \cos (x)}{2 a}",1,"-x/(2*a) - Cos[x]/a + (Cos[x]*Sin[x])/(2*a)","A",5,5,13,0.3846,1,"{3872, 2839, 2638, 2635, 8}"
4,1,17,0,0.0597008,"\int \frac{\cos (x)}{a+a \csc (x)} \, dx","Int[Cos[x]/(a + a*Csc[x]),x]","\frac{\sin (x)}{a}-\frac{\log (\sin (x)+1)}{a}","\frac{\sin (x)}{a}-\frac{\log (\sin (x)+1)}{a}",1,"-(Log[1 + Sin[x]]/a) + Sin[x]/a","A",5,4,11,0.3636,1,"{3872, 2833, 12, 43}"
5,1,33,0,0.072992,"\int \frac{\sec (x)}{a+a \csc (x)} \, dx","Int[Sec[x]/(a + a*Csc[x]),x]","\frac{\sec ^2(x)}{2 a}+\frac{\tanh ^{-1}(\sin (x))}{2 a}-\frac{\tan (x) \sec (x)}{2 a}","\frac{\sec ^2(x)}{2 a}+\frac{\tanh ^{-1}(\sin (x))}{2 a}-\frac{\tan (x) \sec (x)}{2 a}",1,"ArcTanh[Sin[x]]/(2*a) + Sec[x]^2/(2*a) - (Sec[x]*Tan[x])/(2*a)","A",6,6,11,0.5455,1,"{3872, 2706, 2606, 30, 2611, 3770}"
6,1,23,0,0.1071166,"\int \frac{\sec ^2(x)}{a+a \csc (x)} \, dx","Int[Sec[x]^2/(a + a*Csc[x]),x]","\frac{\sec ^3(x)}{3 a}-\frac{\tan ^3(x)}{3 a}","\frac{\sec ^3(x)}{3 a}-\frac{\tan ^3(x)}{3 a}",1,"Sec[x]^3/(3*a) - Tan[x]^3/(3*a)","A",6,5,13,0.3846,1,"{3872, 2839, 2606, 30, 2607}"
7,1,46,0,0.1268516,"\int \frac{\sec ^3(x)}{a+a \csc (x)} \, dx","Int[Sec[x]^3/(a + a*Csc[x]),x]","\frac{\sec ^4(x)}{4 a}+\frac{\tanh ^{-1}(\sin (x))}{8 a}-\frac{\tan (x) \sec ^3(x)}{4 a}+\frac{\tan (x) \sec (x)}{8 a}","\frac{\sec ^4(x)}{4 a}+\frac{\tanh ^{-1}(\sin (x))}{8 a}-\frac{\tan (x) \sec ^3(x)}{4 a}+\frac{\tan (x) \sec (x)}{8 a}",1,"ArcTanh[Sin[x]]/(8*a) + Sec[x]^4/(4*a) + (Sec[x]*Tan[x])/(8*a) - (Sec[x]^3*Tan[x])/(4*a)","A",7,7,13,0.5385,1,"{3872, 2835, 2606, 30, 2611, 3768, 3770}"
8,1,34,0,0.1207798,"\int \frac{\sec ^4(x)}{a+a \csc (x)} \, dx","Int[Sec[x]^4/(a + a*Csc[x]),x]","-\frac{\tan ^5(x)}{5 a}-\frac{\tan ^3(x)}{3 a}+\frac{\sec ^5(x)}{5 a}","-\frac{\tan ^5(x)}{5 a}-\frac{\tan ^3(x)}{3 a}+\frac{\sec ^5(x)}{5 a}",1,"Sec[x]^5/(5*a) - Tan[x]^3/(3*a) - Tan[x]^5/(5*a)","A",7,6,13,0.4615,1,"{3872, 2839, 2606, 30, 2607, 14}"
9,1,130,0,0.3254433,"\int \frac{\cos ^4(x)}{a+b \csc (x)} \, dx","Int[Cos[x]^4/(a + b*Csc[x]),x]","\frac{x \left(-12 a^2 b^2+3 a^4+8 b^4\right)}{8 a^5}-\frac{\cos (x) \left(8 b \left(a^2-b^2\right)-a \left(3 a^2-4 b^2\right) \sin (x)\right)}{8 a^4}+\frac{2 b \left(a^2-b^2\right)^{3/2} \tanh ^{-1}\left(\frac{a+b \tan \left(\frac{x}{2}\right)}{\sqrt{a^2-b^2}}\right)}{a^5}-\frac{\cos ^3(x) (4 b-3 a \sin (x))}{12 a^2}","\frac{x \left(-12 a^2 b^2+3 a^4+8 b^4\right)}{8 a^5}-\frac{\cos (x) \left(8 b \left(a^2-b^2\right)-a \left(3 a^2-4 b^2\right) \sin (x)\right)}{8 a^4}+\frac{2 b \left(a^2-b^2\right)^{3/2} \tanh ^{-1}\left(\frac{a+b \tan \left(\frac{x}{2}\right)}{\sqrt{a^2-b^2}}\right)}{a^5}-\frac{\cos ^3(x) (4 b-3 a \sin (x))}{12 a^2}",1,"((3*a^4 - 12*a^2*b^2 + 8*b^4)*x)/(8*a^5) + (2*b*(a^2 - b^2)^(3/2)*ArcTanh[(a + b*Tan[x/2])/Sqrt[a^2 - b^2]])/a^5 - (Cos[x]^3*(4*b - 3*a*Sin[x]))/(12*a^2) - (Cos[x]*(8*b*(a^2 - b^2) - a*(3*a^2 - 4*b^2)*Sin[x]))/(8*a^4)","A",7,6,13,0.4615,1,"{3872, 2865, 2735, 2660, 618, 206}"
10,1,61,0,0.1311118,"\int \frac{\cos ^3(x)}{a+b \csc (x)} \, dx","Int[Cos[x]^3/(a + b*Csc[x]),x]","\frac{\left(a^2-b^2\right) \sin (x)}{a^3}-\frac{b \left(a^2-b^2\right) \log (a \sin (x)+b)}{a^4}+\frac{b \sin ^2(x)}{2 a^2}-\frac{\sin ^3(x)}{3 a}","\frac{\left(a^2-b^2\right) \sin (x)}{a^3}-\frac{b \left(a^2-b^2\right) \log (a \sin (x)+b)}{a^4}+\frac{b \sin ^2(x)}{2 a^2}-\frac{\sin ^3(x)}{3 a}",1,"-((b*(a^2 - b^2)*Log[b + a*Sin[x]])/a^4) + ((a^2 - b^2)*Sin[x])/a^3 + (b*Sin[x]^2)/(2*a^2) - Sin[x]^3/(3*a)","A",5,4,13,0.3077,1,"{3872, 2837, 12, 772}"
11,1,80,0,0.189825,"\int \frac{\cos ^2(x)}{a+b \csc (x)} \, dx","Int[Cos[x]^2/(a + b*Csc[x]),x]","\frac{x \left(a^2-2 b^2\right)}{2 a^3}+\frac{2 b \sqrt{a^2-b^2} \tanh ^{-1}\left(\frac{a+b \tan \left(\frac{x}{2}\right)}{\sqrt{a^2-b^2}}\right)}{a^3}-\frac{\cos (x) (2 b-a \sin (x))}{2 a^2}","\frac{x \left(a^2-2 b^2\right)}{2 a^3}+\frac{2 b \sqrt{a^2-b^2} \tanh ^{-1}\left(\frac{a+b \tan \left(\frac{x}{2}\right)}{\sqrt{a^2-b^2}}\right)}{a^3}-\frac{\cos (x) (2 b-a \sin (x))}{2 a^2}",1,"((a^2 - 2*b^2)*x)/(2*a^3) + (2*b*Sqrt[a^2 - b^2]*ArcTanh[(a + b*Tan[x/2])/Sqrt[a^2 - b^2]])/a^3 - (Cos[x]*(2*b - a*Sin[x]))/(2*a^2)","A",6,6,13,0.4615,1,"{3872, 2865, 2735, 2660, 618, 206}"
12,1,20,0,0.0631133,"\int \frac{\cos (x)}{a+b \csc (x)} \, dx","Int[Cos[x]/(a + b*Csc[x]),x]","\frac{\sin (x)}{a}-\frac{b \log (a \sin (x)+b)}{a^2}","\frac{\sin (x)}{a}-\frac{b \log (a \sin (x)+b)}{a^2}",1,"-((b*Log[b + a*Sin[x]])/a^2) + Sin[x]/a","A",5,4,11,0.3636,1,"{3872, 2833, 12, 43}"
13,1,54,0,0.0842033,"\int \frac{\sec (x)}{a+b \csc (x)} \, dx","Int[Sec[x]/(a + b*Csc[x]),x]","-\frac{b \log (a \sin (x)+b)}{a^2-b^2}-\frac{\log (1-\sin (x))}{2 (a+b)}+\frac{\log (\sin (x)+1)}{2 (a-b)}","-\frac{b \log (a \sin (x)+b)}{a^2-b^2}-\frac{\log (1-\sin (x))}{2 (a+b)}+\frac{\log (\sin (x)+1)}{2 (a-b)}",1,"-Log[1 - Sin[x]]/(2*(a + b)) + Log[1 + Sin[x]]/(2*(a - b)) - (b*Log[b + a*Sin[x]])/(a^2 - b^2)","A",4,3,11,0.2727,1,"{3872, 2721, 801}"
14,1,65,0,0.1352166,"\int \frac{\sec ^2(x)}{a+b \csc (x)} \, dx","Int[Sec[x]^2/(a + b*Csc[x]),x]","\frac{2 a b \tanh ^{-1}\left(\frac{a+b \tan \left(\frac{x}{2}\right)}{\sqrt{a^2-b^2}}\right)}{\left(a^2-b^2\right)^{3/2}}-\frac{\sec (x) (b-a \sin (x))}{a^2-b^2}","\frac{2 a b \tanh ^{-1}\left(\frac{a+b \tan \left(\frac{x}{2}\right)}{\sqrt{a^2-b^2}}\right)}{\left(a^2-b^2\right)^{3/2}}-\frac{\sec (x) (b-a \sin (x))}{a^2-b^2}",1,"(2*a*b*ArcTanh[(a + b*Tan[x/2])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(3/2) - (Sec[x]*(b - a*Sin[x]))/(a^2 - b^2)","A",6,6,13,0.4615,1,"{3872, 2866, 12, 2660, 618, 206}"
15,1,85,0,0.1868686,"\int \frac{\sec ^3(x)}{a+b \csc (x)} \, dx","Int[Sec[x]^3/(a + b*Csc[x]),x]","-\frac{a^2 b \log (a \sin (x)+b)}{\left(a^2-b^2\right)^2}-\frac{\sec ^2(x) (b-a \sin (x))}{2 \left(a^2-b^2\right)}-\frac{a \log (1-\sin (x))}{4 (a+b)^2}+\frac{a \log (\sin (x)+1)}{4 (a-b)^2}","-\frac{a^2 b \log (a \sin (x)+b)}{\left(a^2-b^2\right)^2}-\frac{\sec ^2(x) (b-a \sin (x))}{2 \left(a^2-b^2\right)}-\frac{a \log (1-\sin (x))}{4 (a+b)^2}+\frac{a \log (\sin (x)+1)}{4 (a-b)^2}",1,"-(a*Log[1 - Sin[x]])/(4*(a + b)^2) + (a*Log[1 + Sin[x]])/(4*(a - b)^2) - (a^2*b*Log[b + a*Sin[x]])/(a^2 - b^2)^2 - (Sec[x]^2*(b - a*Sin[x]))/(2*(a^2 - b^2))","A",6,5,13,0.3846,1,"{3872, 2837, 12, 823, 801}"
16,1,109,0,0.2388329,"\int \frac{\sec ^4(x)}{a+b \csc (x)} \, dx","Int[Sec[x]^4/(a + b*Csc[x]),x]","\frac{2 a^3 b \tanh ^{-1}\left(\frac{a+b \tan \left(\frac{x}{2}\right)}{\sqrt{a^2-b^2}}\right)}{\left(a^2-b^2\right)^{5/2}}-\frac{\sec ^3(x) (b-a \sin (x))}{3 \left(a^2-b^2\right)}-\frac{\sec (x) \left(3 a^2 b-a \left(2 a^2+b^2\right) \sin (x)\right)}{3 \left(a^2-b^2\right)^2}","\frac{2 a^3 b \tanh ^{-1}\left(\frac{a+b \tan \left(\frac{x}{2}\right)}{\sqrt{a^2-b^2}}\right)}{\left(a^2-b^2\right)^{5/2}}-\frac{\sec ^3(x) (b-a \sin (x))}{3 \left(a^2-b^2\right)}-\frac{\sec (x) \left(3 a^2 b-a \left(2 a^2+b^2\right) \sin (x)\right)}{3 \left(a^2-b^2\right)^2}",1,"(2*a^3*b*ArcTanh[(a + b*Tan[x/2])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(5/2) - (Sec[x]^3*(b - a*Sin[x]))/(3*(a^2 - b^2)) - (Sec[x]*(3*a^2*b - a*(2*a^2 + b^2)*Sin[x]))/(3*(a^2 - b^2)^2)","A",7,6,13,0.4615,1,"{3872, 2866, 12, 2660, 618, 206}"