1,1,57,0,0.1270842,"\int \frac{A+B \sin (x)}{a+b \cos (x)} \, dx","Int[(A + B*Sin[x])/(a + b*Cos[x]),x]","\frac{2 A \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{x}{2}\right)}{\sqrt{a+b}}\right)}{\sqrt{a-b} \sqrt{a+b}}-\frac{B \log (a+b \cos (x))}{b}","\frac{2 A \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{x}{2}\right)}{\sqrt{a+b}}\right)}{\sqrt{a-b} \sqrt{a+b}}-\frac{B \log (a+b \cos (x))}{b}",1,"(2*A*ArcTan[(Sqrt[a - b]*Tan[x/2])/Sqrt[a + b]])/(Sqrt[a - b]*Sqrt[a + b]) - (B*Log[a + b*Cos[x]])/b","A",6,5,15,0.3333,1,"{4401, 2659, 205, 2668, 31}"
2,1,19,0,0.0677288,"\int \frac{A+B \sin (x)}{1+\cos (x)} \, dx","Int[(A + B*Sin[x])/(1 + Cos[x]),x]","\frac{A \sin (x)}{\cos (x)+1}-B \log (\cos (x)+1)","\frac{A \sin (x)}{\cos (x)+1}-B \log (\cos (x)+1)",1,"-(B*Log[1 + Cos[x]]) + (A*Sin[x])/(1 + Cos[x])","A",5,4,13,0.3077,1,"{4401, 2648, 2667, 31}"
3,1,23,0,0.0783222,"\int \frac{A+B \sin (x)}{1-\cos (x)} \, dx","Int[(A + B*Sin[x])/(1 - Cos[x]),x]","B \log (1-\cos (x))-\frac{A \sin (x)}{1-\cos (x)}","B \log (1-\cos (x))-\frac{A \sin (x)}{1-\cos (x)}",1,"B*Log[1 - Cos[x]] - (A*Sin[x])/(1 - Cos[x])","A",5,4,15,0.2667,1,"{4401, 2648, 2667, 31}"
4,1,58,0,0.1146856,"\int \frac{b+c+\sin (x)}{a+b \cos (x)} \, dx","Int[(b + c + Sin[x])/(a + b*Cos[x]),x]","\frac{2 (b+c) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{x}{2}\right)}{\sqrt{a+b}}\right)}{\sqrt{a-b} \sqrt{a+b}}-\frac{\log (a+b \cos (x))}{b}","\frac{2 (b+c) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{x}{2}\right)}{\sqrt{a+b}}\right)}{\sqrt{a-b} \sqrt{a+b}}-\frac{\log (a+b \cos (x))}{b}",1,"(2*(b + c)*ArcTan[(Sqrt[a - b]*Tan[x/2])/Sqrt[a + b]])/(Sqrt[a - b]*Sqrt[a + b]) - Log[a + b*Cos[x]]/b","A",6,5,14,0.3571,1,"{4401, 2659, 205, 2668, 31}"
5,1,58,0,0.1304027,"\int \frac{b+c+\sin (x)}{a-b \cos (x)} \, dx","Int[(b + c + Sin[x])/(a - b*Cos[x]),x]","\frac{2 (b+c) \tan ^{-1}\left(\frac{\sqrt{a+b} \tan \left(\frac{x}{2}\right)}{\sqrt{a-b}}\right)}{\sqrt{a-b} \sqrt{a+b}}+\frac{\log (a-b \cos (x))}{b}","\frac{2 (b+c) \tan ^{-1}\left(\frac{\sqrt{a+b} \tan \left(\frac{x}{2}\right)}{\sqrt{a-b}}\right)}{\sqrt{a-b} \sqrt{a+b}}+\frac{\log (a-b \cos (x))}{b}",1,"(2*(b + c)*ArcTan[(Sqrt[a + b]*Tan[x/2])/Sqrt[a - b]])/(Sqrt[a - b]*Sqrt[a + b]) + Log[a - b*Cos[x]]/b","A",6,5,15,0.3333,1,"{4401, 2659, 205, 2668, 31}"
6,1,65,0,0.1369451,"\int \frac{A+B \tan (x)}{a+b \cos (x)} \, dx","Int[(A + B*Tan[x])/(a + b*Cos[x]),x]","\frac{2 A \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{x}{2}\right)}{\sqrt{a+b}}\right)}{\sqrt{a-b} \sqrt{a+b}}+\frac{B \log (a+b \cos (x))}{a}-\frac{B \log (\cos (x))}{a}","\frac{2 A \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{x}{2}\right)}{\sqrt{a+b}}\right)}{\sqrt{a-b} \sqrt{a+b}}+\frac{B \log (a+b \cos (x))}{a}-\frac{B \log (\cos (x))}{a}",1,"(2*A*ArcTan[(Sqrt[a - b]*Tan[x/2])/Sqrt[a + b]])/(Sqrt[a - b]*Sqrt[a + b]) - (B*Log[Cos[x]])/a + (B*Log[a + b*Cos[x]])/a","A",8,7,15,0.4667,1,"{4401, 2659, 205, 2721, 36, 29, 31}"
7,1,100,0,0.1563655,"\int \frac{A+B \cot (x)}{a+b \cos (x)} \, dx","Int[(A + B*Cot[x])/(a + b*Cos[x]),x]","-\frac{a B \log (a+b \cos (x))}{a^2-b^2}+\frac{2 A \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{x}{2}\right)}{\sqrt{a+b}}\right)}{\sqrt{a-b} \sqrt{a+b}}+\frac{B \log (1-\cos (x))}{2 (a+b)}+\frac{B \log (\cos (x)+1)}{2 (a-b)}","-\frac{a B \log (a+b \cos (x))}{a^2-b^2}+\frac{2 A \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{x}{2}\right)}{\sqrt{a+b}}\right)}{\sqrt{a-b} \sqrt{a+b}}+\frac{B \log (1-\cos (x))}{2 (a+b)}+\frac{B \log (\cos (x)+1)}{2 (a-b)}",1,"(2*A*ArcTan[(Sqrt[a - b]*Tan[x/2])/Sqrt[a + b]])/(Sqrt[a - b]*Sqrt[a + b]) + (B*Log[1 - Cos[x]])/(2*(a + b)) + (B*Log[1 + Cos[x]])/(2*(a - b)) - (a*B*Log[a + b*Cos[x]])/(a^2 - b^2)","A",7,5,15,0.3333,1,"{4401, 2659, 205, 2721, 801}"
8,1,99,0,0.2501721,"\int \frac{A+B \csc (x)}{a+b \cos (x)} \, dx","Int[(A + B*Csc[x])/(a + b*Cos[x]),x]","\frac{b B \log (a+b \cos (x))}{a^2-b^2}+\frac{2 A \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{x}{2}\right)}{\sqrt{a+b}}\right)}{\sqrt{a-b} \sqrt{a+b}}+\frac{B \log (1-\cos (x))}{2 (a+b)}-\frac{B \log (\cos (x)+1)}{2 (a-b)}","\frac{b B \log (a+b \cos (x))}{a^2-b^2}+\frac{2 A \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{x}{2}\right)}{\sqrt{a+b}}\right)}{\sqrt{a-b} \sqrt{a+b}}+\frac{B \log (1-\cos (x))}{2 (a+b)}-\frac{B \log (\cos (x)+1)}{2 (a-b)}",1,"(2*A*ArcTan[(Sqrt[a - b]*Tan[x/2])/Sqrt[a + b]])/(Sqrt[a - b]*Sqrt[a + b]) + (B*Log[1 - Cos[x]])/(2*(a + b)) - (B*Log[1 + Cos[x]])/(2*(a - b)) + (b*B*Log[a + b*Cos[x]])/(a^2 - b^2)","A",11,8,15,0.5333,1,"{4225, 4401, 2659, 205, 2668, 706, 31, 633}"
9,1,247,0,0.4085119,"\int \frac{(c+d \sec (e+f x))^4}{a+b \cos (e+f x)} \, dx","Int[(c + d*Sec[e + f*x])^4/(a + b*Cos[e + f*x]),x]","\frac{d^2 \left(6 a^2 c^2-4 a b c d+b^2 d^2\right) \tan (e+f x)}{a^3 f}+\frac{d (2 a c-b d) \left(2 a^2 c^2-2 a b c d+b^2 d^2\right) \tanh ^{-1}(\sin (e+f x))}{a^4 f}+\frac{d^3 (4 a c-b d) \tanh ^{-1}(\sin (e+f x))}{2 a^2 f}+\frac{d^3 (4 a c-b d) \tan (e+f x) \sec (e+f x)}{2 a^2 f}+\frac{2 (a c-b d)^4 \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (e+f x)\right)}{\sqrt{a+b}}\right)}{a^4 f \sqrt{a-b} \sqrt{a+b}}+\frac{d^4 \tan ^3(e+f x)}{3 a f}+\frac{d^4 \tan (e+f x)}{a f}","\frac{d^2 \left(6 a^2 c^2-4 a b c d+b^2 d^2\right) \tan (e+f x)}{a^3 f}+\frac{d (2 a c-b d) \left(2 a^2 c^2-2 a b c d+b^2 d^2\right) \tanh ^{-1}(\sin (e+f x))}{a^4 f}+\frac{d^3 (4 a c-b d) \tanh ^{-1}(\sin (e+f x))}{2 a^2 f}+\frac{d^3 (4 a c-b d) \tan (e+f x) \sec (e+f x)}{2 a^2 f}+\frac{2 (a c-b d)^4 \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (e+f x)\right)}{\sqrt{a+b}}\right)}{a^4 f \sqrt{a-b} \sqrt{a+b}}+\frac{d^4 \tan ^3(e+f x)}{3 a f}+\frac{d^4 \tan (e+f x)}{a f}",1,"(2*(a*c - b*d)^4*ArcTan[(Sqrt[a - b]*Tan[(e + f*x)/2])/Sqrt[a + b]])/(a^4*Sqrt[a - b]*Sqrt[a + b]*f) + (d^3*(4*a*c - b*d)*ArcTanh[Sin[e + f*x]])/(2*a^2*f) + (d*(2*a*c - b*d)*(2*a^2*c^2 - 2*a*b*c*d + b^2*d^2)*ArcTanh[Sin[e + f*x]])/(a^4*f) + (d^4*Tan[e + f*x])/(a*f) + (d^2*(6*a^2*c^2 - 4*a*b*c*d + b^2*d^2)*Tan[e + f*x])/(a^3*f) + (d^3*(4*a*c - b*d)*Sec[e + f*x]*Tan[e + f*x])/(2*a^2*f) + (d^4*Tan[e + f*x]^3)/(3*a*f)","A",12,8,25,0.3200,1,"{2828, 2952, 2659, 205, 3770, 3767, 8, 3768}"
10,1,170,0,0.3268455,"\int \frac{(c+d \sec (e+f x))^3}{a+b \cos (e+f x)} \, dx","Int[(c + d*Sec[e + f*x])^3/(a + b*Cos[e + f*x]),x]","\frac{d \left(3 a^2 c^2-3 a b c d+b^2 d^2\right) \tanh ^{-1}(\sin (e+f x))}{a^3 f}+\frac{d^2 (3 a c-b d) \tan (e+f x)}{a^2 f}+\frac{2 (a c-b d)^3 \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (e+f x)\right)}{\sqrt{a+b}}\right)}{a^3 f \sqrt{a-b} \sqrt{a+b}}+\frac{d^3 \tanh ^{-1}(\sin (e+f x))}{2 a f}+\frac{d^3 \tan (e+f x) \sec (e+f x)}{2 a f}","\frac{d \left(3 a^2 c^2-3 a b c d+b^2 d^2\right) \tanh ^{-1}(\sin (e+f x))}{a^3 f}+\frac{d^2 (3 a c-b d) \tan (e+f x)}{a^2 f}+\frac{2 (a c-b d)^3 \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (e+f x)\right)}{\sqrt{a+b}}\right)}{a^3 f \sqrt{a-b} \sqrt{a+b}}+\frac{d^3 \tanh ^{-1}(\sin (e+f x))}{2 a f}+\frac{d^3 \tan (e+f x) \sec (e+f x)}{2 a f}",1,"(2*(a*c - b*d)^3*ArcTan[(Sqrt[a - b]*Tan[(e + f*x)/2])/Sqrt[a + b]])/(a^3*Sqrt[a - b]*Sqrt[a + b]*f) + (d^3*ArcTanh[Sin[e + f*x]])/(2*a*f) + (d*(3*a^2*c^2 - 3*a*b*c*d + b^2*d^2)*ArcTanh[Sin[e + f*x]])/(a^3*f) + (d^2*(3*a*c - b*d)*Tan[e + f*x])/(a^2*f) + (d^3*Sec[e + f*x]*Tan[e + f*x])/(2*a*f)","A",10,8,25,0.3200,1,"{2828, 2952, 2659, 205, 3770, 3767, 8, 3768}"
11,1,103,0,0.2706275,"\int \frac{(c+d \sec (e+f x))^2}{a+b \cos (e+f x)} \, dx","Int[(c + d*Sec[e + f*x])^2/(a + b*Cos[e + f*x]),x]","\frac{2 (a c-b d)^2 \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (e+f x)\right)}{\sqrt{a+b}}\right)}{a^2 f \sqrt{a-b} \sqrt{a+b}}+\frac{d (2 a c-b d) \tanh ^{-1}(\sin (e+f x))}{a^2 f}+\frac{d^2 \tan (e+f x)}{a f}","\frac{2 (a c-b d)^2 \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (e+f x)\right)}{\sqrt{a+b}}\right)}{a^2 f \sqrt{a-b} \sqrt{a+b}}+\frac{d (2 a c-b d) \tanh ^{-1}(\sin (e+f x))}{a^2 f}+\frac{d^2 \tan (e+f x)}{a f}",1,"(2*(a*c - b*d)^2*ArcTan[(Sqrt[a - b]*Tan[(e + f*x)/2])/Sqrt[a + b]])/(a^2*Sqrt[a - b]*Sqrt[a + b]*f) + (d*(2*a*c - b*d)*ArcTanh[Sin[e + f*x]])/(a^2*f) + (d^2*Tan[e + f*x])/(a*f)","A",8,7,25,0.2800,1,"{2828, 2952, 2659, 205, 3770, 3767, 8}"
12,1,76,0,0.1372975,"\int \frac{c+d \sec (e+f x)}{a+b \cos (e+f x)} \, dx","Int[(c + d*Sec[e + f*x])/(a + b*Cos[e + f*x]),x]","\frac{2 (a c-b d) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (e+f x)\right)}{\sqrt{a+b}}\right)}{a f \sqrt{a-b} \sqrt{a+b}}+\frac{d \tanh ^{-1}(\sin (e+f x))}{a f}","\frac{2 (a c-b d) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (e+f x)\right)}{\sqrt{a+b}}\right)}{a f \sqrt{a-b} \sqrt{a+b}}+\frac{d \tanh ^{-1}(\sin (e+f x))}{a f}",1,"(2*(a*c - b*d)*ArcTan[(Sqrt[a - b]*Tan[(e + f*x)/2])/Sqrt[a + b]])/(a*Sqrt[a - b]*Sqrt[a + b]*f) + (d*ArcTanh[Sin[e + f*x]])/(a*f)","A",5,5,23,0.2174,1,"{2828, 3001, 3770, 2659, 205}"
13,1,121,0,0.2658113,"\int \frac{1}{(a+b \cos (e+f x)) (c+d \sec (e+f x))} \, dx","Int[1/((a + b*Cos[e + f*x])*(c + d*Sec[e + f*x])),x]","\frac{2 a \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (e+f x)\right)}{\sqrt{a+b}}\right)}{f \sqrt{a-b} \sqrt{a+b} (a c-b d)}-\frac{2 d \tanh ^{-1}\left(\frac{\sqrt{c-d} \tan \left(\frac{1}{2} (e+f x)\right)}{\sqrt{c+d}}\right)}{f \sqrt{c-d} \sqrt{c+d} (a c-b d)}","\frac{2 a \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (e+f x)\right)}{\sqrt{a+b}}\right)}{f \sqrt{a-b} \sqrt{a+b} (a c-b d)}-\frac{2 d \tanh ^{-1}\left(\frac{\sqrt{c-d} \tan \left(\frac{1}{2} (e+f x)\right)}{\sqrt{c+d}}\right)}{f \sqrt{c-d} \sqrt{c+d} (a c-b d)}",1,"(2*a*ArcTan[(Sqrt[a - b]*Tan[(e + f*x)/2])/Sqrt[a + b]])/(Sqrt[a - b]*Sqrt[a + b]*(a*c - b*d)*f) - (2*d*ArcTanh[(Sqrt[c - d]*Tan[(e + f*x)/2])/Sqrt[c + d]])/(Sqrt[c - d]*Sqrt[c + d]*(a*c - b*d)*f)","A",6,5,25,0.2000,1,"{2828, 3001, 2659, 205, 208}"
14,1,187,0,0.6311872,"\int \frac{1}{(a+b \cos (e+f x)) (c+d \sec (e+f x))^2} \, dx","Int[1/((a + b*Cos[e + f*x])*(c + d*Sec[e + f*x])^2),x]","\frac{2 a^2 \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (e+f x)\right)}{\sqrt{a+b}}\right)}{f \sqrt{a-b} \sqrt{a+b} (a c-b d)^2}+\frac{d^2 \sin (e+f x)}{f \left(c^2-d^2\right) (a c-b d) (c \cos (e+f x)+d)}-\frac{2 d \left(2 a c^2-a d^2-b c d\right) \tanh ^{-1}\left(\frac{\sqrt{c-d} \tan \left(\frac{1}{2} (e+f x)\right)}{\sqrt{c+d}}\right)}{f (c-d)^{3/2} (c+d)^{3/2} (a c-b d)^2}","\frac{2 a^2 \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (e+f x)\right)}{\sqrt{a+b}}\right)}{f \sqrt{a-b} \sqrt{a+b} (a c-b d)^2}+\frac{d^2 \sin (e+f x)}{f \left(c^2-d^2\right) (a c-b d) (c \cos (e+f x)+d)}-\frac{2 d \left(2 a c^2-a d^2-b c d\right) \tanh ^{-1}\left(\frac{\sqrt{c-d} \tan \left(\frac{1}{2} (e+f x)\right)}{\sqrt{c+d}}\right)}{f (c-d)^{3/2} (c+d)^{3/2} (a c-b d)^2}",1,"(2*a^2*ArcTan[(Sqrt[a - b]*Tan[(e + f*x)/2])/Sqrt[a + b]])/(Sqrt[a - b]*Sqrt[a + b]*(a*c - b*d)^2*f) - (2*d*(2*a*c^2 - b*c*d - a*d^2)*ArcTanh[(Sqrt[c - d]*Tan[(e + f*x)/2])/Sqrt[c + d]])/((c - d)^(3/2)*(c + d)^(3/2)*(a*c - b*d)^2*f) + (d^2*Sin[e + f*x])/((a*c - b*d)*(c^2 - d^2)*f*(d + c*Cos[e + f*x]))","A",7,6,25,0.2400,1,"{2828, 3056, 3001, 2659, 205, 208}"
15,1,458,0,0.9708451,"\int \frac{1}{(a+b \cos (e+f x)) (c+d \sec (e+f x))^3} \, dx","Int[1/((a + b*Cos[e + f*x])*(c + d*Sec[e + f*x])^3),x]","-\frac{2 d \left(3 a^2 c^2-3 a b c d+b^2 d^2\right) \tanh ^{-1}\left(\frac{\sqrt{c-d} \tan \left(\frac{1}{2} (e+f x)\right)}{\sqrt{c+d}}\right)}{c^2 f \sqrt{c-d} \sqrt{c+d} (a c-b d)^3}+\frac{2 a^3 \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (e+f x)\right)}{\sqrt{a+b}}\right)}{f \sqrt{a-b} \sqrt{a+b} (a c-b d)^3}+\frac{3 d^4 \sin (e+f x)}{2 c f \left(c^2-d^2\right)^2 (a c-b d) (c \cos (e+f x)+d)}-\frac{d^3 \sin (e+f x)}{2 c f \left(c^2-d^2\right) (a c-b d) (c \cos (e+f x)+d)^2}+\frac{d^2 (3 a c-2 b d) \sin (e+f x)}{c f \left(c^2-d^2\right) (a c-b d)^2 (c \cos (e+f x)+d)}-\frac{d^3 \left(c^2+2 d^2\right) \tanh ^{-1}\left(\frac{\sqrt{c-d} \tan \left(\frac{1}{2} (e+f x)\right)}{\sqrt{c+d}}\right)}{c^2 f (c-d)^{5/2} (c+d)^{5/2} (a c-b d)}-\frac{2 d^3 (3 a c-2 b d) \tanh ^{-1}\left(\frac{\sqrt{c-d} \tan \left(\frac{1}{2} (e+f x)\right)}{\sqrt{c+d}}\right)}{c^2 f (c-d)^{3/2} (c+d)^{3/2} (a c-b d)^2}","-\frac{2 d \left(3 a^2 c^2-3 a b c d+b^2 d^2\right) \tanh ^{-1}\left(\frac{\sqrt{c-d} \tan \left(\frac{1}{2} (e+f x)\right)}{\sqrt{c+d}}\right)}{c^2 f \sqrt{c-d} \sqrt{c+d} (a c-b d)^3}+\frac{2 a^3 \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (e+f x)\right)}{\sqrt{a+b}}\right)}{f \sqrt{a-b} \sqrt{a+b} (a c-b d)^3}+\frac{3 d^4 \sin (e+f x)}{2 c f \left(c^2-d^2\right)^2 (a c-b d) (c \cos (e+f x)+d)}-\frac{d^3 \sin (e+f x)}{2 c f \left(c^2-d^2\right) (a c-b d) (c \cos (e+f x)+d)^2}+\frac{d^2 (3 a c-2 b d) \sin (e+f x)}{c f \left(c^2-d^2\right) (a c-b d)^2 (c \cos (e+f x)+d)}-\frac{d^3 \left(c^2+2 d^2\right) \tanh ^{-1}\left(\frac{\sqrt{c-d} \tan \left(\frac{1}{2} (e+f x)\right)}{\sqrt{c+d}}\right)}{c^2 f (c-d)^{5/2} (c+d)^{5/2} (a c-b d)}-\frac{2 d^3 (3 a c-2 b d) \tanh ^{-1}\left(\frac{\sqrt{c-d} \tan \left(\frac{1}{2} (e+f x)\right)}{\sqrt{c+d}}\right)}{c^2 f (c-d)^{3/2} (c+d)^{3/2} (a c-b d)^2}",1,"(2*a^3*ArcTan[(Sqrt[a - b]*Tan[(e + f*x)/2])/Sqrt[a + b]])/(Sqrt[a - b]*Sqrt[a + b]*(a*c - b*d)^3*f) - (2*d^3*(3*a*c - 2*b*d)*ArcTanh[(Sqrt[c - d]*Tan[(e + f*x)/2])/Sqrt[c + d]])/(c^2*(c - d)^(3/2)*(c + d)^(3/2)*(a*c - b*d)^2*f) - (d^3*(c^2 + 2*d^2)*ArcTanh[(Sqrt[c - d]*Tan[(e + f*x)/2])/Sqrt[c + d]])/(c^2*(c - d)^(5/2)*(c + d)^(5/2)*(a*c - b*d)*f) - (2*d*(3*a^2*c^2 - 3*a*b*c*d + b^2*d^2)*ArcTanh[(Sqrt[c - d]*Tan[(e + f*x)/2])/Sqrt[c + d]])/(c^2*Sqrt[c - d]*Sqrt[c + d]*(a*c - b*d)^3*f) - (d^3*Sin[e + f*x])/(2*c*(a*c - b*d)*(c^2 - d^2)*f*(d + c*Cos[e + f*x])^2) + (3*d^4*Sin[e + f*x])/(2*c*(a*c - b*d)*(c^2 - d^2)^2*f*(d + c*Cos[e + f*x])) + (d^2*(3*a*c - 2*b*d)*Sin[e + f*x])/(c*(a*c - b*d)^2*(c^2 - d^2)*f*(d + c*Cos[e + f*x]))","A",16,8,25,0.3200,1,"{2828, 2952, 2659, 205, 2664, 2754, 12, 208}"
16,1,213,0,0.3846449,"\int \frac{\sqrt{c+d \sec (e+f x)}}{a+b \cos (e+f x)} \, dx","Int[Sqrt[c + d*Sec[e + f*x]]/(a + b*Cos[e + f*x]),x]","\frac{2 (a c-b d) \tan (e+f x) \sqrt{\frac{c+d \sec (e+f x)}{c+d}} \Pi \left(\frac{2 a}{a+b};\sin ^{-1}\left(\frac{\sqrt{1-\sec (e+f x)}}{\sqrt{2}}\right)|\frac{2 d}{c+d}\right)}{a f (a+b) \sqrt{-\tan ^2(e+f x)} \sqrt{c+d \sec (e+f x)}}+\frac{2 \sqrt{c+d} \cot (e+f x) \sqrt{\frac{d (1-\sec (e+f x))}{c+d}} \sqrt{-\frac{d (\sec (e+f x)+1)}{c-d}} F\left(\sin ^{-1}\left(\frac{\sqrt{c+d \sec (e+f x)}}{\sqrt{c+d}}\right)|\frac{c+d}{c-d}\right)}{a f}","\frac{2 (a c-b d) \tan (e+f x) \sqrt{\frac{c+d \sec (e+f x)}{c+d}} \Pi \left(\frac{2 a}{a+b};\sin ^{-1}\left(\frac{\sqrt{1-\sec (e+f x)}}{\sqrt{2}}\right)|\frac{2 d}{c+d}\right)}{a f (a+b) \sqrt{-\tan ^2(e+f x)} \sqrt{c+d \sec (e+f x)}}+\frac{2 \sqrt{c+d} \cot (e+f x) \sqrt{\frac{d (1-\sec (e+f x))}{c+d}} \sqrt{-\frac{d (\sec (e+f x)+1)}{c-d}} F\left(\sin ^{-1}\left(\frac{\sqrt{c+d \sec (e+f x)}}{\sqrt{c+d}}\right)|\frac{c+d}{c-d}\right)}{a f}",1,"(2*Sqrt[c + d]*Cot[e + f*x]*EllipticF[ArcSin[Sqrt[c + d*Sec[e + f*x]]/Sqrt[c + d]], (c + d)/(c - d)]*Sqrt[(d*(1 - Sec[e + f*x]))/(c + d)]*Sqrt[-((d*(1 + Sec[e + f*x]))/(c - d))])/(a*f) + (2*(a*c - b*d)*EllipticPi[(2*a)/(a + b), ArcSin[Sqrt[1 - Sec[e + f*x]]/Sqrt[2]], (2*d)/(c + d)]*Sqrt[(c + d*Sec[e + f*x])/(c + d)]*Tan[e + f*x])/(a*(a + b)*f*Sqrt[c + d*Sec[e + f*x]]*Sqrt[-Tan[e + f*x]^2])","A",4,4,27,0.1481,1,"{2829, 3969, 3832, 3973}"
17,1,102,0,0.2021703,"\int \frac{1}{(a+b \cos (e+f x)) \sqrt{c+d \sec (e+f x)}} \, dx","Int[1/((a + b*Cos[e + f*x])*Sqrt[c + d*Sec[e + f*x]]),x]","\frac{2 \tan (e+f x) \sqrt{\frac{c+d \sec (e+f x)}{c+d}} \Pi \left(\frac{2 a}{a+b};\sin ^{-1}\left(\frac{\sqrt{1-\sec (e+f x)}}{\sqrt{2}}\right)|\frac{2 d}{c+d}\right)}{f (a+b) \sqrt{-\tan ^2(e+f x)} \sqrt{c+d \sec (e+f x)}}","\frac{2 \tan (e+f x) \sqrt{\frac{c+d \sec (e+f x)}{c+d}} \Pi \left(\frac{2 a}{a+b};\sin ^{-1}\left(\frac{\sqrt{1-\sec (e+f x)}}{\sqrt{2}}\right)|\frac{2 d}{c+d}\right)}{f (a+b) \sqrt{-\tan ^2(e+f x)} \sqrt{c+d \sec (e+f x)}}",1,"(2*EllipticPi[(2*a)/(a + b), ArcSin[Sqrt[1 - Sec[e + f*x]]/Sqrt[2]], (2*d)/(c + d)]*Sqrt[(c + d*Sec[e + f*x])/(c + d)]*Tan[e + f*x])/((a + b)*f*Sqrt[c + d*Sec[e + f*x]]*Sqrt[-Tan[e + f*x]^2])","A",2,2,27,0.07407,1,"{2829, 3973}"
18,1,87,0,0.1437015,"\int \frac{A+B \cos (d+e x)+C \sin (d+e x)}{a+b \cos (d+e x)} \, dx","Int[(A + B*Cos[d + e*x] + C*Sin[d + e*x])/(a + b*Cos[d + e*x]),x]","\frac{2 (A b-a B) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (d+e x)\right)}{\sqrt{a+b}}\right)}{b e \sqrt{a-b} \sqrt{a+b}}-\frac{C \log (a+b \cos (d+e x))}{b e}+\frac{B x}{b}","\frac{2 (A b-a B) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (d+e x)\right)}{\sqrt{a+b}}\right)}{b e \sqrt{a-b} \sqrt{a+b}}-\frac{C \log (a+b \cos (d+e x))}{b e}+\frac{B x}{b}",1,"(B*x)/b + (2*(A*b - a*B)*ArcTan[(Sqrt[a - b]*Tan[(d + e*x)/2])/Sqrt[a + b]])/(Sqrt[a - b]*b*Sqrt[a + b]*e) - (C*Log[a + b*Cos[d + e*x]])/(b*e)","A",6,6,31,0.1935,1,"{4377, 2735, 2659, 205, 2668, 31}"
19,1,120,0,0.1716443,"\int \frac{A+B \cos (d+e x)+C \sin (d+e x)}{(a+b \cos (d+e x))^2} \, dx","Int[(A + B*Cos[d + e*x] + C*Sin[d + e*x])/(a + b*Cos[d + e*x])^2,x]","-\frac{(A b-a B) \sin (d+e x)}{e \left(a^2-b^2\right) (a+b \cos (d+e x))}+\frac{2 (a A-b B) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (d+e x)\right)}{\sqrt{a+b}}\right)}{e (a-b)^{3/2} (a+b)^{3/2}}+\frac{C}{b e (a+b \cos (d+e x))}","-\frac{(A b-a B) \sin (d+e x)}{e \left(a^2-b^2\right) (a+b \cos (d+e x))}+\frac{2 (a A-b B) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (d+e x)\right)}{\sqrt{a+b}}\right)}{e (a-b)^{3/2} (a+b)^{3/2}}+\frac{C}{b e (a+b \cos (d+e x))}",1,"(2*(a*A - b*B)*ArcTan[(Sqrt[a - b]*Tan[(d + e*x)/2])/Sqrt[a + b]])/((a - b)^(3/2)*(a + b)^(3/2)*e) + C/(b*e*(a + b*Cos[d + e*x])) - ((A*b - a*B)*Sin[d + e*x])/((a^2 - b^2)*e*(a + b*Cos[d + e*x]))","A",7,7,31,0.2258,1,"{4377, 2754, 12, 2659, 205, 2668, 32}"
20,1,187,0,0.2841887,"\int \frac{A+B \cos (d+e x)+C \sin (d+e x)}{(a+b \cos (d+e x))^3} \, dx","Int[(A + B*Cos[d + e*x] + C*Sin[d + e*x])/(a + b*Cos[d + e*x])^3,x]","\frac{\left(2 a^2 A-3 a b B+A b^2\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (d+e x)\right)}{\sqrt{a+b}}\right)}{e (a-b)^{5/2} (a+b)^{5/2}}-\frac{\left(a^2 (-B)+3 a A b-2 b^2 B\right) \sin (d+e x)}{2 e \left(a^2-b^2\right)^2 (a+b \cos (d+e x))}-\frac{(A b-a B) \sin (d+e x)}{2 e \left(a^2-b^2\right) (a+b \cos (d+e x))^2}+\frac{C}{2 b e (a+b \cos (d+e x))^2}","\frac{\left(2 a^2 A-3 a b B+A b^2\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (d+e x)\right)}{\sqrt{a+b}}\right)}{e (a-b)^{5/2} (a+b)^{5/2}}-\frac{\left(a^2 (-B)+3 a A b-2 b^2 B\right) \sin (d+e x)}{2 e \left(a^2-b^2\right)^2 (a+b \cos (d+e x))}-\frac{(A b-a B) \sin (d+e x)}{2 e \left(a^2-b^2\right) (a+b \cos (d+e x))^2}+\frac{C}{2 b e (a+b \cos (d+e x))^2}",1,"((2*a^2*A + A*b^2 - 3*a*b*B)*ArcTan[(Sqrt[a - b]*Tan[(d + e*x)/2])/Sqrt[a + b]])/((a - b)^(5/2)*(a + b)^(5/2)*e) + C/(2*b*e*(a + b*Cos[d + e*x])^2) - ((A*b - a*B)*Sin[d + e*x])/(2*(a^2 - b^2)*e*(a + b*Cos[d + e*x])^2) - ((3*a*A*b - a^2*B - 2*b^2*B)*Sin[d + e*x])/(2*(a^2 - b^2)^2*e*(a + b*Cos[d + e*x]))","A",8,7,31,0.2258,1,"{4377, 2754, 12, 2659, 205, 2668, 32}"
21,1,260,0,0.4992835,"\int \frac{A+B \cos (d+e x)+C \sin (d+e x)}{(a+b \cos (d+e x))^4} \, dx","Int[(A + B*Cos[d + e*x] + C*Sin[d + e*x])/(a + b*Cos[d + e*x])^4,x]","\frac{\left(2 a^3 A-4 a^2 b B+3 a A b^2-b^3 B\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (d+e x)\right)}{\sqrt{a+b}}\right)}{e (a-b)^{7/2} (a+b)^{7/2}}-\frac{\left(11 a^2 A b-2 a^3 B-13 a b^2 B+4 A b^3\right) \sin (d+e x)}{6 e \left(a^2-b^2\right)^3 (a+b \cos (d+e x))}-\frac{\left(-2 a^2 B+5 a A b-3 b^2 B\right) \sin (d+e x)}{6 e \left(a^2-b^2\right)^2 (a+b \cos (d+e x))^2}-\frac{(A b-a B) \sin (d+e x)}{3 e \left(a^2-b^2\right) (a+b \cos (d+e x))^3}+\frac{C}{3 b e (a+b \cos (d+e x))^3}","\frac{\left(2 a^3 A-4 a^2 b B+3 a A b^2-b^3 B\right) \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (d+e x)\right)}{\sqrt{a+b}}\right)}{e (a-b)^{7/2} (a+b)^{7/2}}-\frac{\left(11 a^2 A b-2 a^3 B-13 a b^2 B+4 A b^3\right) \sin (d+e x)}{6 e \left(a^2-b^2\right)^3 (a+b \cos (d+e x))}-\frac{\left(-2 a^2 B+5 a A b-3 b^2 B\right) \sin (d+e x)}{6 e \left(a^2-b^2\right)^2 (a+b \cos (d+e x))^2}-\frac{(A b-a B) \sin (d+e x)}{3 e \left(a^2-b^2\right) (a+b \cos (d+e x))^3}+\frac{C}{3 b e (a+b \cos (d+e x))^3}",1,"((2*a^3*A + 3*a*A*b^2 - 4*a^2*b*B - b^3*B)*ArcTan[(Sqrt[a - b]*Tan[(d + e*x)/2])/Sqrt[a + b]])/((a - b)^(7/2)*(a + b)^(7/2)*e) + C/(3*b*e*(a + b*Cos[d + e*x])^3) - ((A*b - a*B)*Sin[d + e*x])/(3*(a^2 - b^2)*e*(a + b*Cos[d + e*x])^3) - ((5*a*A*b - 2*a^2*B - 3*b^2*B)*Sin[d + e*x])/(6*(a^2 - b^2)^2*e*(a + b*Cos[d + e*x])^2) - ((11*a^2*A*b + 4*A*b^3 - 2*a^3*B - 13*a*b^2*B)*Sin[d + e*x])/(6*(a^2 - b^2)^3*e*(a + b*Cos[d + e*x]))","A",9,7,31,0.2258,1,"{4377, 2754, 12, 2659, 205, 2668, 32}"