1,1,56,57,0.0990459,"\int \frac{A+B \sin (x)}{a+b \cos (x)} \, dx","Integrate[(A + B*Sin[x])/(a + b*Cos[x]),x]","-\frac{2 A \tanh ^{-1}\left(\frac{(a-b) \tan \left(\frac{x}{2}\right)}{\sqrt{b^2-a^2}}\right)}{\sqrt{b^2-a^2}}-\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*ArcTanh[((a - b)*Tan[x/2])/Sqrt[-a^2 + b^2]])/Sqrt[-a^2 + b^2] - (B*Log[a + b*Cos[x]])/b","A",1
2,1,19,19,0.0357273,"\int \frac{A+B \sin (x)}{1+\cos (x)} \, dx","Integrate[(A + B*Sin[x])/(1 + Cos[x]),x]","A \tan \left(\frac{x}{2}\right)-2 B \log \left(\cos \left(\frac{x}{2}\right)\right)","\frac{A \sin (x)}{\cos (x)+1}-B \log (\cos (x)+1)",1,"-2*B*Log[Cos[x/2]] + A*Tan[x/2]","A",1
3,1,20,23,0.0507512,"\int \frac{A+B \sin (x)}{1-\cos (x)} \, dx","Integrate[(A + B*Sin[x])/(1 - Cos[x]),x]","2 B \log \left(\sin \left(\frac{x}{2}\right)\right)-A \cot \left(\frac{x}{2}\right)","B \log (1-\cos (x))-\frac{A \sin (x)}{1-\cos (x)}",1,"-(A*Cot[x/2]) + 2*B*Log[Sin[x/2]]","A",1
4,1,57,58,0.0803394,"\int \frac{b+c+\sin (x)}{a+b \cos (x)} \, dx","Integrate[(b + c + Sin[x])/(a + b*Cos[x]),x]","-\frac{2 (b+c) \tanh ^{-1}\left(\frac{(a-b) \tan \left(\frac{x}{2}\right)}{\sqrt{b^2-a^2}}\right)}{\sqrt{b^2-a^2}}-\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)*ArcTanh[((a - b)*Tan[x/2])/Sqrt[-a^2 + b^2]])/Sqrt[-a^2 + b^2] - Log[a + b*Cos[x]]/b","A",1
5,1,55,58,0.0983509,"\int \frac{b+c+\sin (x)}{a-b \cos (x)} \, dx","Integrate[(b + c + Sin[x])/(a - b*Cos[x]),x]","\frac{\log (a-b \cos (x))}{b}-\frac{2 (b+c) \tanh ^{-1}\left(\frac{(a+b) \tan \left(\frac{x}{2}\right)}{\sqrt{b^2-a^2}}\right)}{\sqrt{b^2-a^2}}","\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)*ArcTanh[((a + b)*Tan[x/2])/Sqrt[-a^2 + b^2]])/Sqrt[-a^2 + b^2] + Log[a - b*Cos[x]]/b","A",1
6,1,61,65,0.1517146,"\int \frac{A+B \tan (x)}{a+b \cos (x)} \, dx","Integrate[(A + B*Tan[x])/(a + b*Cos[x]),x]","\frac{B (\log (a+b \cos (x))-\log (\cos (x)))}{a}-\frac{2 A \tanh ^{-1}\left(\frac{(a-b) \tan \left(\frac{x}{2}\right)}{\sqrt{b^2-a^2}}\right)}{\sqrt{b^2-a^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 (a+b \cos (x))}{a}-\frac{B \log (\cos (x))}{a}",1,"(-2*A*ArcTanh[((a - b)*Tan[x/2])/Sqrt[-a^2 + b^2]])/Sqrt[-a^2 + b^2] + (B*(-Log[Cos[x]] + Log[a + b*Cos[x]]))/a","A",1
7,1,134,100,0.2746664,"\int \frac{A+B \cot (x)}{a+b \cos (x)} \, dx","Integrate[(A + B*Cot[x])/(a + b*Cos[x]),x]","\frac{\sin (x) (A+B \cot (x)) \left(B \sqrt{b^2-a^2} \left((a-b) \log \left(\sin \left(\frac{x}{2}\right)\right)+(a+b) \log \left(\cos \left(\frac{x}{2}\right)\right)-a \log (a+b \cos (x))\right)-2 A \left(a^2-b^2\right) \tanh ^{-1}\left(\frac{(a-b) \tan \left(\frac{x}{2}\right)}{\sqrt{b^2-a^2}}\right)\right)}{(a-b) (a+b) \sqrt{b^2-a^2} (A \sin (x)+B \cos (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)}",1,"((A + B*Cot[x])*(-2*A*(a^2 - b^2)*ArcTanh[((a - b)*Tan[x/2])/Sqrt[-a^2 + b^2]] + Sqrt[-a^2 + b^2]*B*((a + b)*Log[Cos[x/2]] - a*Log[a + b*Cos[x]] + (a - b)*Log[Sin[x/2]]))*Sin[x])/((a - b)*(a + b)*Sqrt[-a^2 + b^2]*(B*Cos[x] + A*Sin[x]))","A",1
8,1,116,99,0.2088402,"\int \frac{A+B \csc (x)}{a+b \cos (x)} \, dx","Integrate[(A + B*Csc[x])/(a + b*Cos[x]),x]","\frac{-2 A \left(a^2-b^2\right) \tanh ^{-1}\left(\frac{(a-b) \tan \left(\frac{x}{2}\right)}{\sqrt{b^2-a^2}}\right)-B \sqrt{b^2-a^2} \left((b-a) \log \left(\sin \left(\frac{x}{2}\right)\right)+(a+b) \log \left(\cos \left(\frac{x}{2}\right)\right)-b \log (a+b \cos (x))\right)}{(a-b) (a+b) \sqrt{b^2-a^2}}","\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*(a^2 - b^2)*ArcTanh[((a - b)*Tan[x/2])/Sqrt[-a^2 + b^2]] - Sqrt[-a^2 + b^2]*B*((a + b)*Log[Cos[x/2]] - b*Log[a + b*Cos[x]] + (-a + b)*Log[Sin[x/2]]))/((a - b)*(a + b)*Sqrt[-a^2 + b^2])","A",1
9,1,526,247,4.0024573,"\int \frac{(c+d \sec (e+f x))^4}{a+b \cos (e+f x)} \, dx","Integrate[(c + d*Sec[e + f*x])^4/(a + b*Cos[e + f*x]),x]","\frac{\frac{2 a^3 d^4 \sin \left(\frac{1}{2} (e+f x)\right)}{\left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^3}+\frac{2 a^3 d^4 \sin \left(\frac{1}{2} (e+f x)\right)}{\left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^3}+\frac{4 a d^2 \left(2 a^2 \left(9 c^2+d^2\right)-12 a b c d+3 b^2 d^2\right) \sin \left(\frac{1}{2} (e+f x)\right)}{\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)}+\frac{4 a d^2 \left(2 a^2 \left(9 c^2+d^2\right)-12 a b c d+3 b^2 d^2\right) \sin \left(\frac{1}{2} (e+f x)\right)}{\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)}-\frac{24 (a c-b d)^4 \tanh ^{-1}\left(\frac{(a-b) \tan \left(\frac{1}{2} (e+f x)\right)}{\sqrt{b^2-a^2}}\right)}{\sqrt{b^2-a^2}}+\frac{a^2 d^3 (a (12 c+d)-3 b d)}{\left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^2}-\frac{a^2 d^3 (a (12 c+d)-3 b d)}{\left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^2}-6 d \left(4 a^3 c \left(2 c^2+d^2\right)-a^2 b d \left(12 c^2+d^2\right)+8 a b^2 c d^2-2 b^3 d^3\right) \log \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)-6 d \left(-4 a^3 c \left(2 c^2+d^2\right)+a^2 b d \left(12 c^2+d^2\right)-8 a b^2 c d^2+2 b^3 d^3\right) \log \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)}{12 a^4 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^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{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^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^4 \tan ^3(e+f x)}{3 a f}+\frac{d^4 \tan (e+f x)}{a f}",1,"((-24*(a*c - b*d)^4*ArcTanh[((a - b)*Tan[(e + f*x)/2])/Sqrt[-a^2 + b^2]])/Sqrt[-a^2 + b^2] - 6*d*(8*a*b^2*c*d^2 - 2*b^3*d^3 + 4*a^3*c*(2*c^2 + d^2) - a^2*b*d*(12*c^2 + d^2))*Log[Cos[(e + f*x)/2] - Sin[(e + f*x)/2]] - 6*d*(-8*a*b^2*c*d^2 + 2*b^3*d^3 - 4*a^3*c*(2*c^2 + d^2) + a^2*b*d*(12*c^2 + d^2))*Log[Cos[(e + f*x)/2] + Sin[(e + f*x)/2]] + (a^2*d^3*(-3*b*d + a*(12*c + d)))/(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^2 + (2*a^3*d^4*Sin[(e + f*x)/2])/(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^3 + (4*a*d^2*(-12*a*b*c*d + 3*b^2*d^2 + 2*a^2*(9*c^2 + d^2))*Sin[(e + f*x)/2])/(Cos[(e + f*x)/2] - Sin[(e + f*x)/2]) + (2*a^3*d^4*Sin[(e + f*x)/2])/(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3 - (a^2*d^3*(-3*b*d + a*(12*c + d)))/(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^2 + (4*a*d^2*(-12*a*b*c*d + 3*b^2*d^2 + 2*a^2*(9*c^2 + d^2))*Sin[(e + f*x)/2])/(Cos[(e + f*x)/2] + Sin[(e + f*x)/2]))/(12*a^4*f)","B",0
10,1,335,170,1.289432,"\int \frac{(c+d \sec (e+f x))^3}{a+b \cos (e+f x)} \, dx","Integrate[(c + d*Sec[e + f*x])^3/(a + b*Cos[e + f*x]),x]","\frac{-2 d \left(a^2 \left(6 c^2+d^2\right)-6 a b c d+2 b^2 d^2\right) \log \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)+2 d \left(a^2 \left(6 c^2+d^2\right)-6 a b c d+2 b^2 d^2\right) \log \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)-\frac{8 (a c-b d)^3 \tanh ^{-1}\left(\frac{(a-b) \tan \left(\frac{1}{2} (e+f x)\right)}{\sqrt{b^2-a^2}}\right)}{\sqrt{b^2-a^2}}+\frac{a^2 d^3}{\left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)^2}-\frac{a^2 d^3}{\left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)^2}+\frac{4 a d^2 (3 a c-b d) \sin \left(\frac{1}{2} (e+f x)\right)}{\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)}+\frac{4 a d^2 (3 a c-b d) \sin \left(\frac{1}{2} (e+f x)\right)}{\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)}}{4 a^3 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^2 (3 a c-b d) \tan (e+f x)}{a^2 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^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,"((-8*(a*c - b*d)^3*ArcTanh[((a - b)*Tan[(e + f*x)/2])/Sqrt[-a^2 + b^2]])/Sqrt[-a^2 + b^2] - 2*d*(-6*a*b*c*d + 2*b^2*d^2 + a^2*(6*c^2 + d^2))*Log[Cos[(e + f*x)/2] - Sin[(e + f*x)/2]] + 2*d*(-6*a*b*c*d + 2*b^2*d^2 + a^2*(6*c^2 + d^2))*Log[Cos[(e + f*x)/2] + Sin[(e + f*x)/2]] + (a^2*d^3)/(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^2 + (4*a*d^2*(3*a*c - b*d)*Sin[(e + f*x)/2])/(Cos[(e + f*x)/2] - Sin[(e + f*x)/2]) - (a^2*d^3)/(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^2 + (4*a*d^2*(3*a*c - b*d)*Sin[(e + f*x)/2])/(Cos[(e + f*x)/2] + Sin[(e + f*x)/2]))/(4*a^3*f)","A",1
11,1,135,103,0.8965041,"\int \frac{(c+d \sec (e+f x))^2}{a+b \cos (e+f x)} \, dx","Integrate[(c + d*Sec[e + f*x])^2/(a + b*Cos[e + f*x]),x]","\frac{d \left(a d \tan (e+f x)-(2 a c-b d) \left(\log \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)\right)\right)-\frac{2 (a c-b d)^2 \tanh ^{-1}\left(\frac{(a-b) \tan \left(\frac{1}{2} (e+f x)\right)}{\sqrt{b^2-a^2}}\right)}{\sqrt{b^2-a^2}}}{a^2 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*ArcTanh[((a - b)*Tan[(e + f*x)/2])/Sqrt[-a^2 + b^2]])/Sqrt[-a^2 + b^2] + d*(-((2*a*c - b*d)*(Log[Cos[(e + f*x)/2] - Sin[(e + f*x)/2]] - Log[Cos[(e + f*x)/2] + Sin[(e + f*x)/2]])) + a*d*Tan[e + f*x]))/(a^2*f)","A",1
12,1,112,76,0.1594965,"\int \frac{c+d \sec (e+f x)}{a+b \cos (e+f x)} \, dx","Integrate[(c + d*Sec[e + f*x])/(a + b*Cos[e + f*x]),x]","\frac{\frac{(2 b d-2 a c) \tanh ^{-1}\left(\frac{(a-b) \tan \left(\frac{1}{2} (e+f x)\right)}{\sqrt{b^2-a^2}}\right)}{\sqrt{b^2-a^2}}+d \left(\log \left(\sin \left(\frac{1}{2} (e+f x)\right)+\cos \left(\frac{1}{2} (e+f x)\right)\right)-\log \left(\cos \left(\frac{1}{2} (e+f x)\right)-\sin \left(\frac{1}{2} (e+f x)\right)\right)\right)}{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 + 2*b*d)*ArcTanh[((a - b)*Tan[(e + f*x)/2])/Sqrt[-a^2 + b^2]])/Sqrt[-a^2 + b^2] + d*(-Log[Cos[(e + f*x)/2] - Sin[(e + f*x)/2]] + Log[Cos[(e + f*x)/2] + Sin[(e + f*x)/2]]))/(a*f)","A",1
13,1,106,121,0.256969,"\int \frac{1}{(a+b \cos (e+f x)) (c+d \sec (e+f x))} \, dx","Integrate[1/((a + b*Cos[e + f*x])*(c + d*Sec[e + f*x])),x]","\frac{\frac{2 d \tanh ^{-1}\left(\frac{(d-c) \tan \left(\frac{1}{2} (e+f x)\right)}{\sqrt{c^2-d^2}}\right)}{\sqrt{c^2-d^2}}-\frac{2 a \tanh ^{-1}\left(\frac{(a-b) \tan \left(\frac{1}{2} (e+f x)\right)}{\sqrt{b^2-a^2}}\right)}{\sqrt{b^2-a^2}}}{a c f-b d f}","\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*ArcTanh[((a - b)*Tan[(e + f*x)/2])/Sqrt[-a^2 + b^2]])/Sqrt[-a^2 + b^2] + (2*d*ArcTanh[((-c + d)*Tan[(e + f*x)/2])/Sqrt[c^2 - d^2]])/Sqrt[c^2 - d^2])/(a*c*f - b*d*f)","A",1
14,1,205,187,0.8481852,"\int \frac{1}{(a+b \cos (e+f x)) (c+d \sec (e+f x))^2} \, dx","Integrate[1/((a + b*Cos[e + f*x])*(c + d*Sec[e + f*x])^2),x]","\frac{\sec ^2(e+f x) (c \cos (e+f x)+d) \left(-\frac{2 a^2 (c \cos (e+f x)+d) \tanh ^{-1}\left(\frac{(a-b) \tan \left(\frac{1}{2} (e+f x)\right)}{\sqrt{b^2-a^2}}\right)}{\sqrt{b^2-a^2}}-\frac{2 d \left(a \left(d^2-2 c^2\right)+b c d\right) (c \cos (e+f x)+d) \tanh ^{-1}\left(\frac{(d-c) \tan \left(\frac{1}{2} (e+f x)\right)}{\sqrt{c^2-d^2}}\right)}{\left(c^2-d^2\right)^{3/2}}+\frac{d^2 (a c-b d) \sin (e+f x)}{(c-d) (c+d)}\right)}{f (a c-b d)^2 (c+d \sec (e+f x))^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,"((d + c*Cos[e + f*x])*Sec[e + f*x]^2*((-2*a^2*ArcTanh[((a - b)*Tan[(e + f*x)/2])/Sqrt[-a^2 + b^2]]*(d + c*Cos[e + f*x]))/Sqrt[-a^2 + b^2] - (2*d*(b*c*d + a*(-2*c^2 + d^2))*ArcTanh[((-c + d)*Tan[(e + f*x)/2])/Sqrt[c^2 - d^2]]*(d + c*Cos[e + f*x]))/(c^2 - d^2)^(3/2) + (d^2*(a*c - b*d)*Sin[e + f*x])/((c - d)*(c + d))))/((a*c - b*d)^2*f*(c + d*Sec[e + f*x])^2)","A",1
15,1,319,458,2.4733099,"\int \frac{1}{(a+b \cos (e+f x)) (c+d \sec (e+f x))^3} \, dx","Integrate[1/((a + b*Cos[e + f*x])*(c + d*Sec[e + f*x])^3),x]","\frac{\sec ^3(e+f x) (c \cos (e+f x)+d) \left(\frac{2 d \left(a^2 \left(6 c^4-5 c^2 d^2+2 d^4\right)-6 a b c^3 d+b^2 d^2 \left(2 c^2+d^2\right)\right) (c \cos (e+f x)+d)^2 \tanh ^{-1}\left(\frac{(d-c) \tan \left(\frac{1}{2} (e+f x)\right)}{\sqrt{c^2-d^2}}\right)}{\left(c^2-d^2\right)^{5/2}}-\frac{4 a^3 (c \cos (e+f x)+d)^2 \tanh ^{-1}\left(\frac{(a-b) \tan \left(\frac{1}{2} (e+f x)\right)}{\sqrt{b^2-a^2}}\right)}{\sqrt{b^2-a^2}}+\frac{d^2 (a c-b d) \left(6 a c^3-3 a c d^2-4 b c^2 d+b d^3\right) \sin (e+f x) (c \cos (e+f x)+d)}{c (c-d)^2 (c+d)^2}-\frac{d^3 (a c-b d)^2 \sin (e+f x)}{c (c-d) (c+d)}\right)}{2 f (a c-b d)^3 (c+d \sec (e+f x))^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{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 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{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{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^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)}",1,"((d + c*Cos[e + f*x])*Sec[e + f*x]^3*((-4*a^3*ArcTanh[((a - b)*Tan[(e + f*x)/2])/Sqrt[-a^2 + b^2]]*(d + c*Cos[e + f*x])^2)/Sqrt[-a^2 + b^2] + (2*d*(-6*a*b*c^3*d + b^2*d^2*(2*c^2 + d^2) + a^2*(6*c^4 - 5*c^2*d^2 + 2*d^4))*ArcTanh[((-c + d)*Tan[(e + f*x)/2])/Sqrt[c^2 - d^2]]*(d + c*Cos[e + f*x])^2)/(c^2 - d^2)^(5/2) - (d^3*(a*c - b*d)^2*Sin[e + f*x])/(c*(c - d)*(c + d)) + (d^2*(a*c - b*d)*(6*a*c^3 - 4*b*c^2*d - 3*a*c*d^2 + b*d^3)*(d + c*Cos[e + f*x])*Sin[e + f*x])/(c*(c - d)^2*(c + d)^2)))/(2*(a*c - b*d)^3*f*(c + d*Sec[e + f*x])^3)","A",1
16,1,184,213,4.1581647,"\int \frac{\sqrt{c+d \sec (e+f x)}}{a+b \cos (e+f x)} \, dx","Integrate[Sqrt[c + d*Sec[e + f*x]]/(a + b*Cos[e + f*x]),x]","\frac{4 \cos ^2\left(\frac{1}{2} (e+f x)\right) \sqrt{\frac{\cos (e+f x)}{\cos (e+f x)+1}} \sqrt{\frac{c \cos (e+f x)+d}{(c+d) (\cos (e+f x)+1)}} \sqrt{c+d \sec (e+f x)} \left(2 (a c-b d) \Pi \left(\frac{b-a}{a+b};\sin ^{-1}\left(\tan \left(\frac{1}{2} (e+f x)\right)\right)|\frac{c-d}{c+d}\right)-(a+b) (c-d) F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (e+f x)\right)\right)|\frac{c-d}{c+d}\right)\right)}{f (a-b) (a+b) (c \cos (e+f x)+d)}","\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,"(4*Cos[(e + f*x)/2]^2*Sqrt[Cos[e + f*x]/(1 + Cos[e + f*x])]*Sqrt[(d + c*Cos[e + f*x])/((c + d)*(1 + Cos[e + f*x]))]*(-((a + b)*(c - d)*EllipticF[ArcSin[Tan[(e + f*x)/2]], (c - d)/(c + d)]) + 2*(a*c - b*d)*EllipticPi[(-a + b)/(a + b), ArcSin[Tan[(e + f*x)/2]], (c - d)/(c + d)])*Sqrt[c + d*Sec[e + f*x]])/((a - b)*(a + b)*f*(d + c*Cos[e + f*x]))","A",1
17,1,187,102,4.3981198,"\int \frac{1}{(a+b \cos (e+f x)) \sqrt{c+d \sec (e+f x)}} \, dx","Integrate[1/((a + b*Cos[e + f*x])*Sqrt[c + d*Sec[e + f*x]]),x]","-\frac{2 \sqrt{\sec (e+f x)} \sqrt{\sec (e+f x)+1} \sqrt{\cos (e+f x) \sec ^2\left(\frac{1}{2} (e+f x)\right)} \sqrt{\frac{c \cos (e+f x)+d}{(c+d) (\cos (e+f x)+1)}} \left((a+b) F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (e+f x)\right)\right)|\frac{c-d}{c+d}\right)-2 a \Pi \left(\frac{b-a}{a+b};\sin ^{-1}\left(\tan \left(\frac{1}{2} (e+f x)\right)\right)|\frac{c-d}{c+d}\right)\right)}{f (a-b) (a+b) \sqrt{\sec ^2\left(\frac{1}{2} (e+f x)\right)} \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*Sqrt[(d + c*Cos[e + f*x])/((c + d)*(1 + Cos[e + f*x]))]*((a + b)*EllipticF[ArcSin[Tan[(e + f*x)/2]], (c - d)/(c + d)] - 2*a*EllipticPi[(-a + b)/(a + b), ArcSin[Tan[(e + f*x)/2]], (c - d)/(c + d)])*Sqrt[Cos[e + f*x]*Sec[(e + f*x)/2]^2]*Sqrt[Sec[e + f*x]]*Sqrt[1 + Sec[e + f*x]])/((a - b)*(a + b)*f*Sqrt[Sec[(e + f*x)/2]^2]*Sqrt[c + d*Sec[e + f*x]])","A",1
18,1,82,87,0.2456847,"\int \frac{A+B \cos (d+e x)+C \sin (d+e x)}{a+b \cos (d+e x)} \, dx","Integrate[(A + B*Cos[d + e*x] + C*Sin[d + e*x])/(a + b*Cos[d + e*x]),x]","\frac{\frac{2 (a B-A b) \tanh ^{-1}\left(\frac{(a-b) \tan \left(\frac{1}{2} (d+e x)\right)}{\sqrt{b^2-a^2}}\right)}{\sqrt{b^2-a^2}}-C \log (a+b \cos (d+e x))+B (d+e x)}{b e}","\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*(d + e*x) + (2*(-(A*b) + a*B)*ArcTanh[((a - b)*Tan[(d + e*x)/2])/Sqrt[-a^2 + b^2]])/Sqrt[-a^2 + b^2] - C*Log[a + b*Cos[d + e*x]])/(b*e)","A",1
19,1,115,120,0.4299884,"\int \frac{A+B \cos (d+e x)+C \sin (d+e x)}{(a+b \cos (d+e x))^2} \, dx","Integrate[(A + B*Cos[d + e*x] + C*Sin[d + e*x])/(a + b*Cos[d + e*x])^2,x]","\frac{\frac{C \left(a^2-b^2\right)-b (A b-a B) \sin (d+e x)}{b (a-b) (a+b) (a+b \cos (d+e x))}+\frac{2 (a A-b B) \tanh ^{-1}\left(\frac{(a-b) \tan \left(\frac{1}{2} (d+e x)\right)}{\sqrt{b^2-a^2}}\right)}{\left(b^2-a^2\right)^{3/2}}}{e}","-\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)*ArcTanh[((a - b)*Tan[(d + e*x)/2])/Sqrt[-a^2 + b^2]])/(-a^2 + b^2)^(3/2) + ((a^2 - b^2)*C - b*(A*b - a*B)*Sin[d + e*x])/((a - b)*b*(a + b)*(a + b*Cos[d + e*x])))/e","A",1
20,1,175,187,0.8593312,"\int \frac{A+B \cos (d+e x)+C \sin (d+e x)}{(a+b \cos (d+e x))^3} \, dx","Integrate[(A + B*Cos[d + e*x] + C*Sin[d + e*x])/(a + b*Cos[d + e*x])^3,x]","\frac{\frac{C \left(a^2-b^2\right)-b (A b-a B) \sin (d+e x)}{b (a-b) (a+b) (a+b \cos (d+e x))^2}+\frac{\left(a^2 B-3 a A b+2 b^2 B\right) \sin (d+e x)}{(a-b)^2 (a+b)^2 (a+b \cos (d+e x))}-\frac{2 \left(2 a^2 A-3 a b B+A b^2\right) \tanh ^{-1}\left(\frac{(a-b) \tan \left(\frac{1}{2} (d+e x)\right)}{\sqrt{b^2-a^2}}\right)}{\left(b^2-a^2\right)^{5/2}}}{2 e}","\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*(2*a^2*A + A*b^2 - 3*a*b*B)*ArcTanh[((a - b)*Tan[(d + e*x)/2])/Sqrt[-a^2 + b^2]])/(-a^2 + b^2)^(5/2) + ((-3*a*A*b + a^2*B + 2*b^2*B)*Sin[d + e*x])/((a - b)^2*(a + b)^2*(a + b*Cos[d + e*x])) + ((a^2 - b^2)*C - b*(A*b - a*B)*Sin[d + e*x])/((a - b)*b*(a + b)*(a + b*Cos[d + e*x])^2))/(2*e)","A",1
21,1,302,260,1.1923078,"\int \frac{A+B \cos (d+e x)+C \sin (d+e x)}{(a+b \cos (d+e x))^4} \, dx","Integrate[(A + B*Cos[d + e*x] + C*Sin[d + e*x])/(a + b*Cos[d + e*x])^4,x]","\frac{\frac{24 \left(2 a^3 A-4 a^2 b B+3 a A b^2-b^3 B\right) \tanh ^{-1}\left(\frac{(a-b) \tan \left(\frac{1}{2} (d+e x)\right)}{\sqrt{b^2-a^2}}\right)}{\left(b^2-a^2\right)^{7/2}}+\frac{-8 a^6 C+24 a^4 b^2 C-2 a^3 b^3 B \sin (3 (d+e x))+11 a^2 A b^4 \sin (3 (d+e x))-24 a^2 b^4 C+6 b^2 \left(-2 a^4 B+9 a^3 A b-9 a^2 b^2 B+a A b^3+b^4 B\right) \sin (2 (d+e x))-3 b \left(8 a^5 B-24 a^4 A b+14 a^3 b^2 B+3 a^2 A b^3+3 a b^4 B-4 A b^5\right) \sin (d+e x)-13 a b^5 B \sin (3 (d+e x))+4 A b^6 \sin (3 (d+e x))+8 b^6 C}{b \left(b^2-a^2\right)^3 (a+b \cos (d+e x))^3}}{24 e}","-\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{\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(-2 a^3 B+11 a^2 A 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{C}{3 b e (a+b \cos (d+e x))^3}",1,"((24*(2*a^3*A + 3*a*A*b^2 - 4*a^2*b*B - b^3*B)*ArcTanh[((a - b)*Tan[(d + e*x)/2])/Sqrt[-a^2 + b^2]])/(-a^2 + b^2)^(7/2) + (-8*a^6*C + 24*a^4*b^2*C - 24*a^2*b^4*C + 8*b^6*C - 3*b*(-24*a^4*A*b + 3*a^2*A*b^3 - 4*A*b^5 + 8*a^5*B + 14*a^3*b^2*B + 3*a*b^4*B)*Sin[d + e*x] + 6*b^2*(9*a^3*A*b + a*A*b^3 - 2*a^4*B - 9*a^2*b^2*B + b^4*B)*Sin[2*(d + e*x)] + 11*a^2*A*b^4*Sin[3*(d + e*x)] + 4*A*b^6*Sin[3*(d + e*x)] - 2*a^3*b^3*B*Sin[3*(d + e*x)] - 13*a*b^5*B*Sin[3*(d + e*x)])/(b*(-a^2 + b^2)^3*(a + b*Cos[d + e*x])^3))/(24*e)","A",1