1,1,54,54,0.0931238,"\int \frac{A+B \cos (x)}{a+b \sin (x)} \, dx","Integrate[(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",1
2,1,42,19,0.0488813,"\int \frac{A+B \cos (x)}{1+\sin (x)} \, dx","Integrate[(A + B*Cos[x])/(1 + Sin[x]),x]","\frac{2 A \sin \left(\frac{x}{2}\right)}{\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)}+2 B \log \left(\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)\right)","B \log (\sin (x)+1)-\frac{A \cos (x)}{\sin (x)+1}",1,"2*B*Log[Cos[x/2] + Sin[x/2]] + (2*A*Sin[x/2])/(Cos[x/2] + Sin[x/2])","B",1
3,1,46,23,0.0501484,"\int \frac{A+B \cos (x)}{1-\sin (x)} \, dx","Integrate[(A + B*Cos[x])/(1 - Sin[x]),x]","\frac{2 A \sin \left(\frac{x}{2}\right)}{\cos \left(\frac{x}{2}\right)-\sin \left(\frac{x}{2}\right)}-2 B \log \left(\cos \left(\frac{x}{2}\right)-\sin \left(\frac{x}{2}\right)\right)","\frac{A \cos (x)}{1-\sin (x)}-B \log (1-\sin (x))",1,"-2*B*Log[Cos[x/2] - Sin[x/2]] + (2*A*Sin[x/2])/(Cos[x/2] - Sin[x/2])","A",1
4,1,55,55,0.0929007,"\int \frac{b+c+\cos (x)}{a+b \sin (x)} \, dx","Integrate[(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",1
5,1,59,58,0.1137731,"\int \frac{b+c+\cos (x)}{a-b \sin (x)} \, dx","Integrate[(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 (b \sin (x)-a)}{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",1
6,1,150,97,0.364049,"\int \frac{A+B \tan (x)}{a+b \sin (x)} \, dx","Integrate[(A + B*Tan[x])/(a + b*Sin[x]),x]","\frac{\cos (x) (A+B \tan (x)) \left(2 A \left(a^2-b^2\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{x}{2}\right)+b}{\sqrt{a^2-b^2}}\right)-B \sqrt{a^2-b^2} \left(-a \log (a+b \sin (x))+(a-b) \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)\right)\right)}{(a-b) (a+b) \sqrt{a^2-b^2} (A \cos (x)+B \sin (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)}",1,"(Cos[x]*(2*A*(a^2 - b^2)*ArcTan[(b + a*Tan[x/2])/Sqrt[a^2 - b^2]] - Sqrt[a^2 - b^2]*B*((a - b)*Log[Cos[x/2] - Sin[x/2]] + (a + b)*Log[Cos[x/2] + Sin[x/2]] - a*Log[a + b*Sin[x]]))*(A + B*Tan[x]))/((a - b)*(a + b)*Sqrt[a^2 - b^2]*(A*Cos[x] + B*Sin[x]))","A",1
7,1,60,63,0.1596918,"\int \frac{A+B \cot (x)}{a+b \sin (x)} \, dx","Integrate[(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 (\sin (x))-\log (a+b \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]] - Log[a + b*Sin[x]]))/a","A",1
8,1,108,98,0.2416035,"\int \frac{A+B \sec (x)}{a+b \sin (x)} \, dx","Integrate[(A + B*Sec[x])/(a + b*Sin[x]),x]","\frac{2 A \sqrt{a^2-b^2} \tan ^{-1}\left(\frac{a \tan \left(\frac{x}{2}\right)+b}{\sqrt{a^2-b^2}}\right)+B \left(-b \log (a+b \sin (x))+(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)\right)}{(a-b) (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*Sqrt[a^2 - b^2]*ArcTan[(b + a*Tan[x/2])/Sqrt[a^2 - b^2]] + B*((-a + b)*Log[Cos[x/2] - Sin[x/2]] + (a + b)*Log[Cos[x/2] + Sin[x/2]] - b*Log[a + b*Sin[x]]))/((a - b)*(a + b))","A",1
9,1,72,61,0.1218487,"\int \frac{A+B \csc (x)}{a+b \sin (x)} \, dx","Integrate[(A + B*Csc[x])/(a + b*Sin[x]),x]","\frac{\frac{2 (a A-b B) \tan ^{-1}\left(\frac{a \tan \left(\frac{x}{2}\right)+b}{\sqrt{a^2-b^2}}\right)}{\sqrt{a^2-b^2}}+B \left(\log \left(\sin \left(\frac{x}{2}\right)\right)-\log \left(\cos \left(\frac{x}{2}\right)\right)\right)}{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]])/Sqrt[a^2 - b^2] + B*(-Log[Cos[x/2]] + Log[Sin[x/2]]))/a","A",1