1,1,40,44,0.0421497,"\int \frac{\cos ^4(x)}{a+a \csc (x)} \, dx","Integrate[Cos[x]^4/(a + a*Csc[x]),x]","-\frac{x}{8 a}+\frac{\sin (4 x)}{32 a}-\frac{\cos (x)}{4 a}-\frac{\cos (3 x)}{12 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,"-1/8*x/a - Cos[x]/(4*a) - Cos[3*x]/(12*a) + Sin[4*x]/(32*a)","A",1
2,1,17,23,0.0165571,"\int \frac{\cos ^3(x)}{a+a \csc (x)} \, dx","Integrate[Cos[x]^3/(a + a*Csc[x]),x]","\frac{(3-2 \sin (x)) \sin ^2(x)}{6 a}","\frac{\sin ^2(x)}{2 a}-\frac{\sin ^3(x)}{3 a}",1,"((3 - 2*Sin[x])*Sin[x]^2)/(6*a)","A",1
3,1,27,27,0.0348046,"\int \frac{\cos ^2(x)}{a+a \csc (x)} \, dx","Integrate[Cos[x]^2/(a + a*Csc[x]),x]","-\frac{x}{2 a}+\frac{\sin (2 x)}{4 a}-\frac{\cos (x)}{a}","-\frac{x}{2 a}-\frac{\cos (x)}{a}+\frac{\sin (x) \cos (x)}{2 a}",1,"-1/2*x/a - Cos[x]/a + Sin[2*x]/(4*a)","A",1
4,1,17,17,0.0134259,"\int \frac{\cos (x)}{a+a \csc (x)} \, dx","Integrate[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",1
5,1,17,33,0.026104,"\int \frac{\sec (x)}{a+a \csc (x)} \, dx","Integrate[Sec[x]/(a + a*Csc[x]),x]","\frac{\frac{1}{\sin (x)+1}+\tanh ^{-1}(\sin (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]] + (1 + Sin[x])^(-1))/(2*a)","A",1
6,1,56,23,0.0822758,"\int \frac{\sec ^2(x)}{a+a \csc (x)} \, dx","Integrate[Sec[x]^2/(a + a*Csc[x]),x]","-\frac{-2 \sin (x)+\cos (2 x)+(\sin (x)+1) \cos (x)-3}{6 a \left(\cos \left(\frac{x}{2}\right)-\sin \left(\frac{x}{2}\right)\right) \left(\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)\right)^3}","\frac{\sec ^3(x)}{3 a}-\frac{\tan ^3(x)}{3 a}",1,"-1/6*(-3 + Cos[2*x] - 2*Sin[x] + Cos[x]*(1 + Sin[x]))/(a*(Cos[x/2] - Sin[x/2])*(Cos[x/2] + Sin[x/2])^3)","B",1
7,1,25,46,0.0552398,"\int \frac{\sec ^3(x)}{a+a \csc (x)} \, dx","Integrate[Sec[x]^3/(a + a*Csc[x]),x]","\frac{\frac{1}{1-\sin (x)}+\frac{1}{(\sin (x)+1)^2}+\tanh ^{-1}(\sin (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]] + (1 - Sin[x])^(-1) + (1 + Sin[x])^(-2))/(8*a)","A",1
8,1,85,34,0.1492894,"\int \frac{\sec ^4(x)}{a+a \csc (x)} \, dx","Integrate[Sec[x]^4/(a + a*Csc[x]),x]","-\frac{-96 \sin (x)+18 \sin (2 x)-32 \sin (3 x)+9 \sin (4 x)+54 \cos (x)+32 \cos (2 x)+18 \cos (3 x)+16 \cos (4 x)-240}{960 a \left(\cos \left(\frac{x}{2}\right)-\sin \left(\frac{x}{2}\right)\right)^3 \left(\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)\right)^5}","-\frac{\tan ^5(x)}{5 a}-\frac{\tan ^3(x)}{3 a}+\frac{\sec ^5(x)}{5 a}",1,"-1/960*(-240 + 54*Cos[x] + 32*Cos[2*x] + 18*Cos[3*x] + 16*Cos[4*x] - 96*Sin[x] + 18*Sin[2*x] - 32*Sin[3*x] + 9*Sin[4*x])/(a*(Cos[x/2] - Sin[x/2])^3*(Cos[x/2] + Sin[x/2])^5)","B",1
9,1,127,130,0.7381484,"\int \frac{\cos ^4(x)}{a+b \csc (x)} \, dx","Integrate[Cos[x]^4/(a + b*Csc[x]),x]","-\frac{8 a^3 b \cos (3 x)+24 a b \left(5 a^2-4 b^2\right) \cos (x)+192 b \left(b^2-a^2\right)^{3/2} \tan ^{-1}\left(\frac{a+b \tan \left(\frac{x}{2}\right)}{\sqrt{b^2-a^2}}\right)-3 \left(a^4 \sin (4 x)+8 a^2 \left(a^2-b^2\right) \sin (2 x)+4 x \left(3 a^4-12 a^2 b^2+8 b^4\right)\right)}{96 a^5}","-\frac{\cos ^3(x) (4 b-3 a \sin (x))}{12 a^2}+\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 (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{x \left(3 a^4-12 a^2 b^2+8 b^4\right)}{8 a^5}",1,"-1/96*(192*b*(-a^2 + b^2)^(3/2)*ArcTan[(a + b*Tan[x/2])/Sqrt[-a^2 + b^2]] + 24*a*b*(5*a^2 - 4*b^2)*Cos[x] + 8*a^3*b*Cos[3*x] - 3*(4*(3*a^4 - 12*a^2*b^2 + 8*b^4)*x + 8*a^2*(a^2 - b^2)*Sin[2*x] + a^4*Sin[4*x]))/a^5","A",1
10,1,60,61,0.1188802,"\int \frac{\cos ^3(x)}{a+b \csc (x)} \, dx","Integrate[Cos[x]^3/(a + b*Csc[x]),x]","\frac{-2 a^3 \sin ^3(x)+6 a \left(a^2-b^2\right) \sin (x)+6 b \left(b^2-a^2\right) \log (a \sin (x)+b)+3 a^2 b \sin ^2(x)}{6 a^4}","\frac{b \sin ^2(x)}{2 a^2}-\frac{b \left(a^2-b^2\right) \log (a \sin (x)+b)}{a^4}+\frac{\left(a^2-b^2\right) \sin (x)}{a^3}-\frac{\sin ^3(x)}{3 a}",1,"(6*b*(-a^2 + b^2)*Log[b + a*Sin[x]] + 6*a*(a^2 - b^2)*Sin[x] + 3*a^2*b*Sin[x]^2 - 2*a^3*Sin[x]^3)/(6*a^4)","A",1
11,1,75,80,0.121104,"\int \frac{\cos ^2(x)}{a+b \csc (x)} \, dx","Integrate[Cos[x]^2/(a + b*Csc[x]),x]","\frac{8 b \sqrt{b^2-a^2} \tan ^{-1}\left(\frac{a+b \tan \left(\frac{x}{2}\right)}{\sqrt{b^2-a^2}}\right)+2 a^2 x+a^2 \sin (2 x)-4 a b \cos (x)-4 b^2 x}{4 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}",1,"(2*a^2*x - 4*b^2*x + 8*b*Sqrt[-a^2 + b^2]*ArcTan[(a + b*Tan[x/2])/Sqrt[-a^2 + b^2]] - 4*a*b*Cos[x] + a^2*Sin[2*x])/(4*a^3)","A",1
12,1,19,20,0.0121476,"\int \frac{\cos (x)}{a+b \csc (x)} \, dx","Integrate[Cos[x]/(a + b*Csc[x]),x]","\frac{a \sin (x)-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*Sin[x])/a^2","A",1
13,1,64,54,0.0696032,"\int \frac{\sec (x)}{a+b \csc (x)} \, dx","Integrate[Sec[x]/(a + b*Csc[x]),x]","\frac{-b \log (a \sin (x)+b)+(b-a) \log \left(\cos \left(\frac{x}{2}\right)-\sin \left(\frac{x}{2}\right)\right)+(a+b) \log \left(\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)\right)}{(a-b) (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,"((-a + b)*Log[Cos[x/2] - Sin[x/2]] + (a + b)*Log[Cos[x/2] + Sin[x/2]] - b*Log[b + a*Sin[x]])/((a - b)*(a + b))","A",1
14,1,97,65,0.3546585,"\int \frac{\sec ^2(x)}{a+b \csc (x)} \, dx","Integrate[Sec[x]^2/(a + b*Csc[x]),x]","\frac{2 a b \tan ^{-1}\left(\frac{a+b \tan \left(\frac{x}{2}\right)}{\sqrt{b^2-a^2}}\right)}{\left(b^2-a^2\right)^{3/2}}+\sin \left(\frac{x}{2}\right) \left(\frac{1}{(a-b) \left(\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)\right)}+\frac{1}{(a+b) \left(\cos \left(\frac{x}{2}\right)-\sin \left(\frac{x}{2}\right)\right)}\right)","\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*ArcTan[(a + b*Tan[x/2])/Sqrt[-a^2 + b^2]])/(-a^2 + b^2)^(3/2) + Sin[x/2]*(1/((a + b)*(Cos[x/2] - Sin[x/2])) + 1/((a - b)*(Cos[x/2] + Sin[x/2])))","A",1
15,1,139,85,0.4065743,"\int \frac{\sec ^3(x)}{a+b \csc (x)} \, dx","Integrate[Sec[x]^3/(a + b*Csc[x]),x]","\frac{\csc (x) (a \sin (x)+b) \left(-\frac{4 a^2 b \log (a \sin (x)+b)}{\left(a^2-b^2\right)^2}+\frac{1}{(a+b) \left(\cos \left(\frac{x}{2}\right)-\sin \left(\frac{x}{2}\right)\right)^2}+\frac{1}{(b-a) \left(\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)\right)^2}-\frac{2 a \log \left(\cos \left(\frac{x}{2}\right)-\sin \left(\frac{x}{2}\right)\right)}{(a+b)^2}+\frac{2 a \log \left(\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)\right)}{(a-b)^2}\right)}{4 (a+b \csc (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}",1,"(Csc[x]*((-2*a*Log[Cos[x/2] - Sin[x/2]])/(a + b)^2 + (2*a*Log[Cos[x/2] + Sin[x/2]])/(a - b)^2 - (4*a^2*b*Log[b + a*Sin[x]])/(a^2 - b^2)^2 + 1/((a + b)*(Cos[x/2] - Sin[x/2])^2) + 1/((-a + b)*(Cos[x/2] + Sin[x/2])^2))*(b + a*Sin[x]))/(4*(a + b*Csc[x]))","A",1
16,1,121,109,0.809883,"\int \frac{\sec ^4(x)}{a+b \csc (x)} \, dx","Integrate[Sec[x]^4/(a + b*Csc[x]),x]","\frac{\sec ^3(x) \left(6 a^3 \sin (x)+2 a^3 \sin (3 x)-6 a^2 b \cos (2 x)-10 a^2 b-3 a b^2 \sin (x)+a b^2 \sin (3 x)+4 b^3\right)-\frac{24 a^3 b \tan ^{-1}\left(\frac{a+b \tan \left(\frac{x}{2}\right)}{\sqrt{b^2-a^2}}\right)}{\sqrt{b^2-a^2}}}{12 (a-b)^2 (a+b)^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}}",1,"((-24*a^3*b*ArcTan[(a + b*Tan[x/2])/Sqrt[-a^2 + b^2]])/Sqrt[-a^2 + b^2] + Sec[x]^3*(-10*a^2*b + 4*b^3 - 6*a^2*b*Cos[2*x] + 6*a^3*Sin[x] - 3*a*b^2*Sin[x] + 2*a^3*Sin[3*x] + a*b^2*Sin[3*x]))/(12*(a - b)^2*(a + b)^2)","A",1