1,1,54,0,0.1316328,"\int \frac{A+B \cos (x)}{a+b \sin (x)} \, dx","Int[(A + B*Cos[x])/(a + b*Sin[x]),x]","\frac{2 A \tan ^{-1}\left(\frac{a \tan \left(\frac{x}{2}\right)+b}{\sqrt{a^2-b^2}}\right)}{\sqrt{a^2-b^2}}+\frac{B \log (a+b \sin (x))}{b}","\frac{2 A \tan ^{-1}\left(\frac{a \tan \left(\frac{x}{2}\right)+b}{\sqrt{a^2-b^2}}\right)}{\sqrt{a^2-b^2}}+\frac{B \log (a+b \sin (x))}{b}",1,"(2*A*ArcTan[(b + a*Tan[x/2])/Sqrt[a^2 - b^2]])/Sqrt[a^2 - b^2] + (B*Log[a + b*Sin[x]])/b","A",7,6,15,0.4000,1,"{4401, 2660, 618, 204, 2668, 31}"
2,1,19,0,0.0683109,"\int \frac{A+B \cos (x)}{1+\sin (x)} \, dx","Int[(A + B*Cos[x])/(1 + Sin[x]),x]","B \log (\sin (x)+1)-\frac{A \cos (x)}{\sin (x)+1}","B \log (\sin (x)+1)-\frac{A \cos (x)}{\sin (x)+1}",1,"B*Log[1 + Sin[x]] - (A*Cos[x])/(1 + Sin[x])","A",5,4,13,0.3077,1,"{4401, 2648, 2667, 31}"
3,1,23,0,0.077904,"\int \frac{A+B \cos (x)}{1-\sin (x)} \, dx","Int[(A + B*Cos[x])/(1 - Sin[x]),x]","\frac{A \cos (x)}{1-\sin (x)}-B \log (1-\sin (x))","\frac{A \cos (x)}{1-\sin (x)}-B \log (1-\sin (x))",1,"-(B*Log[1 - Sin[x]]) + (A*Cos[x])/(1 - Sin[x])","A",5,4,15,0.2667,1,"{4401, 2648, 2667, 31}"
4,1,55,0,0.1140243,"\int \frac{b+c+\cos (x)}{a+b \sin (x)} \, dx","Int[(b + c + Cos[x])/(a + b*Sin[x]),x]","\frac{2 (b+c) \tan ^{-1}\left(\frac{a \tan \left(\frac{x}{2}\right)+b}{\sqrt{a^2-b^2}}\right)}{\sqrt{a^2-b^2}}+\frac{\log (a+b \sin (x))}{b}","\frac{2 (b+c) \tan ^{-1}\left(\frac{a \tan \left(\frac{x}{2}\right)+b}{\sqrt{a^2-b^2}}\right)}{\sqrt{a^2-b^2}}+\frac{\log (a+b \sin (x))}{b}",1,"(2*(b + c)*ArcTan[(b + a*Tan[x/2])/Sqrt[a^2 - b^2]])/Sqrt[a^2 - b^2] + Log[a + b*Sin[x]]/b","A",7,6,14,0.4286,1,"{4401, 2660, 618, 204, 2668, 31}"
5,1,58,0,0.1387683,"\int \frac{b+c+\cos (x)}{a-b \sin (x)} \, dx","Int[(b + c + Cos[x])/(a - b*Sin[x]),x]","-\frac{2 (b+c) \tan ^{-1}\left(\frac{b-a \tan \left(\frac{x}{2}\right)}{\sqrt{a^2-b^2}}\right)}{\sqrt{a^2-b^2}}-\frac{\log (a-b \sin (x))}{b}","-\frac{2 (b+c) \tan ^{-1}\left(\frac{b-a \tan \left(\frac{x}{2}\right)}{\sqrt{a^2-b^2}}\right)}{\sqrt{a^2-b^2}}-\frac{\log (a-b \sin (x))}{b}",1,"(-2*(b + c)*ArcTan[(b - a*Tan[x/2])/Sqrt[a^2 - b^2]])/Sqrt[a^2 - b^2] - Log[a - b*Sin[x]]/b","A",7,6,15,0.4000,1,"{4401, 2660, 618, 204, 2668, 31}"
6,1,97,0,0.1682953,"\int \frac{A+B \tan (x)}{a+b \sin (x)} \, dx","Int[(A + B*Tan[x])/(a + b*Sin[x]),x]","\frac{2 A \tan ^{-1}\left(\frac{a \tan \left(\frac{x}{2}\right)+b}{\sqrt{a^2-b^2}}\right)}{\sqrt{a^2-b^2}}+\frac{a B \log (a+b \sin (x))}{a^2-b^2}-\frac{B \log (1-\sin (x))}{2 (a+b)}-\frac{B \log (\sin (x)+1)}{2 (a-b)}","\frac{2 A \tan ^{-1}\left(\frac{a \tan \left(\frac{x}{2}\right)+b}{\sqrt{a^2-b^2}}\right)}{\sqrt{a^2-b^2}}+\frac{a B \log (a+b \sin (x))}{a^2-b^2}-\frac{B \log (1-\sin (x))}{2 (a+b)}-\frac{B \log (\sin (x)+1)}{2 (a-b)}",1,"(2*A*ArcTan[(b + a*Tan[x/2])/Sqrt[a^2 - b^2]])/Sqrt[a^2 - b^2] - (B*Log[1 - Sin[x]])/(2*(a + b)) - (B*Log[1 + Sin[x]])/(2*(a - b)) + (a*B*Log[a + b*Sin[x]])/(a^2 - b^2)","A",8,6,15,0.4000,1,"{4401, 2660, 618, 204, 2721, 801}"
7,1,63,0,0.1475158,"\int \frac{A+B \cot (x)}{a+b \sin (x)} \, dx","Int[(A + B*Cot[x])/(a + b*Sin[x]),x]","\frac{2 A \tan ^{-1}\left(\frac{a \tan \left(\frac{x}{2}\right)+b}{\sqrt{a^2-b^2}}\right)}{\sqrt{a^2-b^2}}-\frac{B \log (a+b \sin (x))}{a}+\frac{B \log (\sin (x))}{a}","\frac{2 A \tan ^{-1}\left(\frac{a \tan \left(\frac{x}{2}\right)+b}{\sqrt{a^2-b^2}}\right)}{\sqrt{a^2-b^2}}-\frac{B \log (a+b \sin (x))}{a}+\frac{B \log (\sin (x))}{a}",1,"(2*A*ArcTan[(b + a*Tan[x/2])/Sqrt[a^2 - b^2]])/Sqrt[a^2 - b^2] + (B*Log[Sin[x]])/a - (B*Log[a + b*Sin[x]])/a","A",9,8,15,0.5333,1,"{4401, 2660, 618, 204, 2721, 36, 29, 31}"
8,1,98,0,0.2583052,"\int \frac{A+B \sec (x)}{a+b \sin (x)} \, dx","Int[(A + B*Sec[x])/(a + b*Sin[x]),x]","\frac{2 A \tan ^{-1}\left(\frac{a \tan \left(\frac{x}{2}\right)+b}{\sqrt{a^2-b^2}}\right)}{\sqrt{a^2-b^2}}-\frac{b B \log (a+b \sin (x))}{a^2-b^2}-\frac{B \log (1-\sin (x))}{2 (a+b)}+\frac{B \log (\sin (x)+1)}{2 (a-b)}","\frac{2 A \tan ^{-1}\left(\frac{a \tan \left(\frac{x}{2}\right)+b}{\sqrt{a^2-b^2}}\right)}{\sqrt{a^2-b^2}}-\frac{b B \log (a+b \sin (x))}{a^2-b^2}-\frac{B \log (1-\sin (x))}{2 (a+b)}+\frac{B \log (\sin (x)+1)}{2 (a-b)}",1,"(2*A*ArcTan[(b + a*Tan[x/2])/Sqrt[a^2 - b^2]])/Sqrt[a^2 - b^2] - (B*Log[1 - Sin[x]])/(2*(a + b)) + (B*Log[1 + Sin[x]])/(2*(a - b)) - (b*B*Log[a + b*Sin[x]])/(a^2 - b^2)","A",12,9,15,0.6000,1,"{4226, 4401, 2660, 618, 204, 2668, 706, 31, 633}"
9,1,61,0,0.130843,"\int \frac{A+B \csc (x)}{a+b \sin (x)} \, dx","Int[(A + B*Csc[x])/(a + b*Sin[x]),x]","\frac{2 (a A-b B) \tan ^{-1}\left(\frac{a \tan \left(\frac{x}{2}\right)+b}{\sqrt{a^2-b^2}}\right)}{a \sqrt{a^2-b^2}}-\frac{B \tanh ^{-1}(\cos (x))}{a}","\frac{2 (a A-b B) \tan ^{-1}\left(\frac{a \tan \left(\frac{x}{2}\right)+b}{\sqrt{a^2-b^2}}\right)}{a \sqrt{a^2-b^2}}-\frac{B \tanh ^{-1}(\cos (x))}{a}",1,"(2*(a*A - b*B)*ArcTan[(b + a*Tan[x/2])/Sqrt[a^2 - b^2]])/(a*Sqrt[a^2 - b^2]) - (B*ArcTanh[Cos[x]])/a","A",6,6,15,0.4000,1,"{2828, 3001, 3770, 2660, 618, 204}"