1,1,33,0,0.0805143,"\int \frac{\tan ^4(x)}{a+a \cos (x)} \, dx","Int[Tan[x]^4/(a + a*Cos[x]),x]","\frac{\tan ^3(x)}{3 a}+\frac{\tanh ^{-1}(\sin (x))}{2 a}-\frac{\tan (x) \sec (x)}{2 a}","\frac{\tan ^3(x)}{3 a}+\frac{\tanh ^{-1}(\sin (x))}{2 a}-\frac{\tan (x) \sec (x)}{2 a}",1,"ArcTanh[Sin[x]]/(2*a) - (Sec[x]*Tan[x])/(2*a) + Tan[x]^3/(3*a)","A",5,5,13,0.3846,1,"{2706, 2607, 30, 2611, 3770}"
2,1,19,0,0.0596611,"\int \frac{\tan ^3(x)}{a+a \cos (x)} \, dx","Int[Tan[x]^3/(a + a*Cos[x]),x]","\frac{\sec ^2(x)}{2 a}-\frac{\sec (x)}{a}","\frac{\sec ^2(x)}{2 a}-\frac{\sec (x)}{a}",1,"-(Sec[x]/a) + Sec[x]^2/(2*a)","A",5,4,13,0.3077,1,"{2706, 2606, 30, 8}"
3,1,15,0,0.0501199,"\int \frac{\tan ^2(x)}{a+a \cos (x)} \, dx","Int[Tan[x]^2/(a + a*Cos[x]),x]","\frac{\tan (x)}{a}-\frac{\tanh ^{-1}(\sin (x))}{a}","\frac{\tan (x)}{a}-\frac{\tanh ^{-1}(\sin (x))}{a}",1,"-(ArcTanh[Sin[x]]/a) + Tan[x]/a","A",4,4,13,0.3077,1,"{2706, 3767, 8, 3770}"
4,1,18,0,0.0355734,"\int \frac{\tan (x)}{a+a \cos (x)} \, dx","Int[Tan[x]/(a + a*Cos[x]),x]","\frac{\log (\cos (x)+1)}{a}-\frac{\log (\cos (x))}{a}","\frac{\log (\cos (x)+1)}{a}-\frac{\log (\cos (x))}{a}",1,"-(Log[Cos[x]]/a) + Log[1 + Cos[x]]/a","A",4,4,11,0.3636,1,"{2707, 36, 29, 31}"
5,1,33,0,0.0548422,"\int \frac{\cot (x)}{a+a \cos (x)} \, dx","Int[Cot[x]/(a + a*Cos[x]),x]","-\frac{\csc ^2(x)}{2 a}-\frac{\tanh ^{-1}(\cos (x))}{2 a}+\frac{\cot (x) \csc (x)}{2 a}","-\frac{\csc ^2(x)}{2 a}-\frac{\tanh ^{-1}(\cos (x))}{2 a}+\frac{\cot (x) \csc (x)}{2 a}",1,"-ArcTanh[Cos[x]]/(2*a) + (Cot[x]*Csc[x])/(2*a) - Csc[x]^2/(2*a)","A",5,5,11,0.4545,1,"{2706, 2606, 30, 2611, 3770}"
6,1,30,0,0.0799864,"\int \frac{\cot ^2(x)}{a+a \cos (x)} \, dx","Int[Cot[x]^2/(a + a*Cos[x]),x]","-\frac{\cot ^3(x)}{3 a}+\frac{\csc ^3(x)}{3 a}-\frac{\csc (x)}{a}","-\frac{\cot ^3(x)}{3 a}+\frac{\csc ^3(x)}{3 a}-\frac{\csc (x)}{a}",1,"-Cot[x]^3/(3*a) - Csc[x]/a + Csc[x]^3/(3*a)","A",5,4,13,0.3077,1,"{2706, 2607, 30, 2606}"
7,1,46,0,0.0962216,"\int \frac{\cot ^3(x)}{a+a \cos (x)} \, dx","Int[Cot[x]^3/(a + a*Cos[x]),x]","-\frac{\cot ^4(x)}{4 a}+\frac{3 \tanh ^{-1}(\cos (x))}{8 a}+\frac{\cot ^3(x) \csc (x)}{4 a}-\frac{3 \cot (x) \csc (x)}{8 a}","-\frac{\cot ^4(x)}{4 a}+\frac{3 \tanh ^{-1}(\cos (x))}{8 a}+\frac{\cot ^3(x) \csc (x)}{4 a}-\frac{3 \cot (x) \csc (x)}{8 a}",1,"(3*ArcTanh[Cos[x]])/(8*a) - Cot[x]^4/(4*a) - (3*Cot[x]*Csc[x])/(8*a) + (Cot[x]^3*Csc[x])/(4*a)","A",6,5,13,0.3846,1,"{2706, 2607, 30, 2611, 3770}"
8,1,40,0,0.0805112,"\int \frac{\cot ^4(x)}{a+a \cos (x)} \, dx","Int[Cot[x]^4/(a + a*Cos[x]),x]","-\frac{\cot ^5(x)}{5 a}+\frac{\csc ^5(x)}{5 a}-\frac{2 \csc ^3(x)}{3 a}+\frac{\csc (x)}{a}","-\frac{\cot ^5(x)}{5 a}+\frac{\csc ^5(x)}{5 a}-\frac{2 \csc ^3(x)}{3 a}+\frac{\csc (x)}{a}",1,"-Cot[x]^5/(5*a) + Csc[x]/a - (2*Csc[x]^3)/(3*a) + Csc[x]^5/(5*a)","A",6,5,13,0.3846,1,"{2706, 2607, 30, 2606, 194}"
9,1,33,0,0.0359734,"\int \frac{\tan (3 x)}{(1+\cos (3 x))^2} \, dx","Int[Tan[3*x]/(1 + Cos[3*x])^2,x]","-\frac{1}{3 (\cos (3 x)+1)}-\frac{1}{3} \log (\cos (3 x))+\frac{1}{3} \log (\cos (3 x)+1)","-\frac{1}{3 (\cos (3 x)+1)}-\frac{1}{3} \log (\cos (3 x))+\frac{1}{3} \log (\cos (3 x)+1)",1,"-1/(3*(1 + Cos[3*x])) - Log[Cos[3*x]]/3 + Log[1 + Cos[3*x]]/3","A",3,2,13,0.1538,1,"{2707, 44}"
10,1,113,0,0.4182548,"\int \frac{\tan ^4(x)}{a+b \cos (x)} \, dx","Int[Tan[x]^4/(a + b*Cos[x]),x]","-\frac{\left(4 a^2-3 b^2\right) \tan (x)}{3 a^3}+\frac{b \left(3 a^2-2 b^2\right) \tanh ^{-1}(\sin (x))}{2 a^4}+\frac{2 (a-b)^{3/2} (a+b)^{3/2} \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{x}{2}\right)}{\sqrt{a+b}}\right)}{a^4}-\frac{b \tan (x) \sec (x)}{2 a^2}+\frac{\tan (x) \sec ^2(x)}{3 a}","-\frac{\left(4 a^2-3 b^2\right) \tan (x)}{3 a^3}+\frac{b \left(3 a^2-2 b^2\right) \tanh ^{-1}(\sin (x))}{2 a^4}+\frac{2 (a-b)^{3/2} (a+b)^{3/2} \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{x}{2}\right)}{\sqrt{a+b}}\right)}{a^4}-\frac{b \tan (x) \sec (x)}{2 a^2}+\frac{\tan (x) \sec ^2(x)}{3 a}",1,"(2*(a - b)^(3/2)*(a + b)^(3/2)*ArcTan[(Sqrt[a - b]*Tan[x/2])/Sqrt[a + b]])/a^4 + (b*(3*a^2 - 2*b^2)*ArcTanh[Sin[x]])/(2*a^4) - ((4*a^2 - 3*b^2)*Tan[x])/(3*a^3) - (b*Sec[x]*Tan[x])/(2*a^2) + (Sec[x]^2*Tan[x])/(3*a)","A",6,6,13,0.4615,1,"{2725, 3055, 3001, 3770, 2659, 205}"
11,1,57,0,0.0835597,"\int \frac{\tan ^3(x)}{a+b \cos (x)} \, dx","Int[Tan[x]^3/(a + b*Cos[x]),x]","\frac{\left(a^2-b^2\right) \log (\cos (x))}{a^3}-\frac{\left(a^2-b^2\right) \log (a+b \cos (x))}{a^3}-\frac{b \sec (x)}{a^2}+\frac{\sec ^2(x)}{2 a}","\frac{\left(a^2-b^2\right) \log (\cos (x))}{a^3}-\frac{\left(a^2-b^2\right) \log (a+b \cos (x))}{a^3}-\frac{b \sec (x)}{a^2}+\frac{\sec ^2(x)}{2 a}",1,"((a^2 - b^2)*Log[Cos[x]])/a^3 - ((a^2 - b^2)*Log[a + b*Cos[x]])/a^3 - (b*Sec[x])/a^2 + Sec[x]^2/(2*a)","A",3,2,13,0.1538,1,"{2721, 894}"
12,1,61,0,0.2241744,"\int \frac{\tan ^2(x)}{a+b \cos (x)} \, dx","Int[Tan[x]^2/(a + b*Cos[x]),x]","-\frac{2 \sqrt{a-b} \sqrt{a+b} \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{x}{2}\right)}{\sqrt{a+b}}\right)}{a^2}-\frac{b \tanh ^{-1}(\sin (x))}{a^2}+\frac{\tan (x)}{a}","-\frac{2 \sqrt{a-b} \sqrt{a+b} \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{x}{2}\right)}{\sqrt{a+b}}\right)}{a^2}-\frac{b \tanh ^{-1}(\sin (x))}{a^2}+\frac{\tan (x)}{a}",1,"(-2*Sqrt[a - b]*Sqrt[a + b]*ArcTan[(Sqrt[a - b]*Tan[x/2])/Sqrt[a + b]])/a^2 - (b*ArcTanh[Sin[x]])/a^2 + Tan[x]/a","A",6,6,13,0.4615,1,"{2723, 3056, 3001, 3770, 2659, 205}"
13,1,20,0,0.0372535,"\int \frac{\tan (x)}{a+b \cos (x)} \, dx","Int[Tan[x]/(a + b*Cos[x]),x]","\frac{\log (a+b \cos (x))}{a}-\frac{\log (\cos (x))}{a}","\frac{\log (a+b \cos (x))}{a}-\frac{\log (\cos (x))}{a}",1,"-(Log[Cos[x]]/a) + Log[a + b*Cos[x]]/a","A",4,4,11,0.3636,1,"{2721, 36, 29, 31}"
14,1,54,0,0.0579725,"\int \frac{\cot (x)}{a+b \cos (x)} \, dx","Int[Cot[x]/(a + b*Cos[x]),x]","-\frac{a \log (a+b \cos (x))}{a^2-b^2}+\frac{\log (1-\cos (x))}{2 (a+b)}+\frac{\log (\cos (x)+1)}{2 (a-b)}","-\frac{a \log (a+b \cos (x))}{a^2-b^2}+\frac{\log (1-\cos (x))}{2 (a+b)}+\frac{\log (\cos (x)+1)}{2 (a-b)}",1,"Log[1 - Cos[x]]/(2*(a + b)) + Log[1 + Cos[x]]/(2*(a - b)) - (a*Log[a + b*Cos[x]])/(a^2 - b^2)","A",3,2,11,0.1818,1,"{2721, 801}"
15,1,77,0,0.1015054,"\int \frac{\cot ^2(x)}{a+b \cos (x)} \, dx","Int[Cot[x]^2/(a + b*Cos[x]),x]","-\frac{a \cot (x)}{a^2-b^2}+\frac{b \csc (x)}{a^2-b^2}-\frac{2 a^2 \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{x}{2}\right)}{\sqrt{a+b}}\right)}{(a-b)^{3/2} (a+b)^{3/2}}","-\frac{a \cot (x)}{a^2-b^2}+\frac{b \csc (x)}{a^2-b^2}-\frac{2 a^2 \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{x}{2}\right)}{\sqrt{a+b}}\right)}{(a-b)^{3/2} (a+b)^{3/2}}",1,"(-2*a^2*ArcTan[(Sqrt[a - b]*Tan[x/2])/Sqrt[a + b]])/((a - b)^(3/2)*(a + b)^(3/2)) - (a*Cot[x])/(a^2 - b^2) + (b*Csc[x])/(a^2 - b^2)","A",7,6,13,0.4615,1,"{2727, 3767, 8, 2606, 2659, 205}"
16,1,93,0,0.1828525,"\int \frac{\cot ^3(x)}{a+b \cos (x)} \, dx","Int[Cot[x]^3/(a + b*Cos[x]),x]","\frac{a^3 \log (a+b \cos (x))}{\left(a^2-b^2\right)^2}-\frac{\csc ^2(x) (a-b \cos (x))}{2 \left(a^2-b^2\right)}-\frac{(2 a+b) \log (1-\cos (x))}{4 (a+b)^2}-\frac{(2 a-b) \log (\cos (x)+1)}{4 (a-b)^2}","\frac{a^3 \log (a+b \cos (x))}{\left(a^2-b^2\right)^2}-\frac{\csc ^2(x) (a-b \cos (x))}{2 \left(a^2-b^2\right)}-\frac{(2 a+b) \log (1-\cos (x))}{4 (a+b)^2}-\frac{(2 a-b) \log (\cos (x)+1)}{4 (a-b)^2}",1,"-((a - b*Cos[x])*Csc[x]^2)/(2*(a^2 - b^2)) - ((2*a + b)*Log[1 - Cos[x]])/(4*(a + b)^2) - ((2*a - b)*Log[1 + Cos[x]])/(4*(a - b)^2) + (a^3*Log[a + b*Cos[x]])/(a^2 - b^2)^2","A",4,3,13,0.2308,1,"{2721, 1647, 801}"
17,1,138,0,0.2060362,"\int \frac{\cot ^4(x)}{a+b \cos (x)} \, dx","Int[Cot[x]^4/(a + b*Cos[x]),x]","-\frac{a \cot ^3(x)}{3 \left(a^2-b^2\right)}+\frac{a^3 \cot (x)}{\left(a^2-b^2\right)^2}+\frac{b \csc ^3(x)}{3 \left(a^2-b^2\right)}-\frac{a^2 b \csc (x)}{\left(a^2-b^2\right)^2}-\frac{b \csc (x)}{a^2-b^2}+\frac{2 a^4 \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{x}{2}\right)}{\sqrt{a+b}}\right)}{(a-b)^{5/2} (a+b)^{5/2}}","-\frac{a \cot ^3(x)}{3 \left(a^2-b^2\right)}+\frac{a^3 \cot (x)}{\left(a^2-b^2\right)^2}+\frac{b \csc ^3(x)}{3 \left(a^2-b^2\right)}-\frac{a^2 b \csc (x)}{\left(a^2-b^2\right)^2}-\frac{b \csc (x)}{a^2-b^2}+\frac{2 a^4 \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{x}{2}\right)}{\sqrt{a+b}}\right)}{(a-b)^{5/2} (a+b)^{5/2}}",1,"(2*a^4*ArcTan[(Sqrt[a - b]*Tan[x/2])/Sqrt[a + b]])/((a - b)^(5/2)*(a + b)^(5/2)) + (a^3*Cot[x])/(a^2 - b^2)^2 - (a*Cot[x]^3)/(3*(a^2 - b^2)) - (a^2*b*Csc[x])/(a^2 - b^2)^2 - (b*Csc[x])/(a^2 - b^2) + (b*Csc[x]^3)/(3*(a^2 - b^2))","A",12,8,13,0.6154,1,"{2727, 2607, 30, 2606, 3767, 8, 2659, 205}"
18,1,44,0,0.0494116,"\int \frac{\cot (x)}{\sqrt{3-\cos (x)}} \, dx","Int[Cot[x]/Sqrt[3 - Cos[x]],x]","-\frac{1}{2} \tanh ^{-1}\left(\frac{1}{2} \sqrt{3-\cos (x)}\right)-\frac{\tanh ^{-1}\left(\frac{\sqrt{3-\cos (x)}}{\sqrt{2}}\right)}{\sqrt{2}}","-\frac{1}{2} \tanh ^{-1}\left(\frac{1}{2} \sqrt{3-\cos (x)}\right)-\frac{\tanh ^{-1}\left(\frac{\sqrt{3-\cos (x)}}{\sqrt{2}}\right)}{\sqrt{2}}",1,"-ArcTanh[Sqrt[3 - Cos[x]]/2]/2 - ArcTanh[Sqrt[3 - Cos[x]]/Sqrt[2]]/Sqrt[2]","A",5,4,13,0.3077,1,"{2721, 827, 1166, 206}"
19,1,37,0,0.0574767,"\int \sqrt{a+b \cos (x)} \tan (x) \, dx","Int[Sqrt[a + b*Cos[x]]*Tan[x],x]","2 \sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a+b \cos (x)}}{\sqrt{a}}\right)-2 \sqrt{a+b \cos (x)}","2 \sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a+b \cos (x)}}{\sqrt{a}}\right)-2 \sqrt{a+b \cos (x)}",1,"2*Sqrt[a]*ArcTanh[Sqrt[a + b*Cos[x]]/Sqrt[a]] - 2*Sqrt[a + b*Cos[x]]","A",4,4,13,0.3077,1,"{2721, 50, 63, 207}"
20,1,24,0,0.0579263,"\int \frac{\tan (x)}{\sqrt{a+b \cos (x)}} \, dx","Int[Tan[x]/Sqrt[a + b*Cos[x]],x]","\frac{2 \tanh ^{-1}\left(\frac{\sqrt{a+b \cos (x)}}{\sqrt{a}}\right)}{\sqrt{a}}","\frac{2 \tanh ^{-1}\left(\frac{\sqrt{a+b \cos (x)}}{\sqrt{a}}\right)}{\sqrt{a}}",1,"(2*ArcTanh[Sqrt[a + b*Cos[x]]/Sqrt[a]])/Sqrt[a]","A",3,3,13,0.2308,1,"{2721, 63, 207}"
21,1,204,0,0.5830389,"\int \frac{\sqrt{e \tan (c+d x)}}{a+b \cos (c+d x)} \, dx","Int[Sqrt[e*Tan[c + d*x]]/(a + b*Cos[c + d*x]),x]","\frac{2 \sqrt{2} \sqrt{\cos (c+d x)} \sqrt{e \tan (c+d x)} \Pi \left(\frac{\sqrt{b-a}}{\sqrt{a+b}};\left.\sin ^{-1}\left(\frac{\sqrt{\sin (c+d x)}}{\sqrt{\cos (c+d x)+1}}\right)\right|-1\right)}{d \sqrt{b-a} \sqrt{a+b} \sqrt{\sin (c+d x)}}-\frac{2 \sqrt{2} \sqrt{\cos (c+d x)} \sqrt{e \tan (c+d x)} \Pi \left(-\frac{\sqrt{b-a}}{\sqrt{a+b}};\left.\sin ^{-1}\left(\frac{\sqrt{\sin (c+d x)}}{\sqrt{\cos (c+d x)+1}}\right)\right|-1\right)}{d \sqrt{b-a} \sqrt{a+b} \sqrt{\sin (c+d x)}}","\frac{2 \sqrt{2} \sqrt{\cos (c+d x)} \sqrt{e \tan (c+d x)} \Pi \left(\frac{\sqrt{b-a}}{\sqrt{a+b}};\left.\sin ^{-1}\left(\frac{\sqrt{\sin (c+d x)}}{\sqrt{\cos (c+d x)+1}}\right)\right|-1\right)}{d \sqrt{b-a} \sqrt{a+b} \sqrt{\sin (c+d x)}}-\frac{2 \sqrt{2} \sqrt{\cos (c+d x)} \sqrt{e \tan (c+d x)} \Pi \left(-\frac{\sqrt{b-a}}{\sqrt{a+b}};\left.\sin ^{-1}\left(\frac{\sqrt{\sin (c+d x)}}{\sqrt{\cos (c+d x)+1}}\right)\right|-1\right)}{d \sqrt{b-a} \sqrt{a+b} \sqrt{\sin (c+d x)}}",1,"(-2*Sqrt[2]*Sqrt[Cos[c + d*x]]*EllipticPi[-(Sqrt[-a + b]/Sqrt[a + b]), ArcSin[Sqrt[Sin[c + d*x]]/Sqrt[1 + Cos[c + d*x]]], -1]*Sqrt[e*Tan[c + d*x]])/(Sqrt[-a + b]*Sqrt[a + b]*d*Sqrt[Sin[c + d*x]]) + (2*Sqrt[2]*Sqrt[Cos[c + d*x]]*EllipticPi[Sqrt[-a + b]/Sqrt[a + b], ArcSin[Sqrt[Sin[c + d*x]]/Sqrt[1 + Cos[c + d*x]]], -1]*Sqrt[e*Tan[c + d*x]])/(Sqrt[-a + b]*Sqrt[a + b]*d*Sqrt[Sin[c + d*x]])","A",9,7,25,0.2800,1,"{2733, 2730, 2906, 2905, 490, 1213, 537}"
22,0,0,0,0.105059,"\int (a+b \cos (e+f x))^m (g \tan (e+f x))^p \, dx","Int[(a + b*Cos[e + f*x])^m*(g*Tan[e + f*x])^p,x]","\int (a+b \cos (e+f x))^m (g \tan (e+f x))^p \, dx","(g \tan (e+f x))^p (g \cot (e+f x))^p \text{Int}\left((g \cot (e+f x))^{-p} (a+b \cos (e+f x))^m,x\right)",0,"(g*Cot[e + f*x])^p*(g*Tan[e + f*x])^p*Defer[Int][(a + b*Cos[e + f*x])^m/(g*Cot[e + f*x])^p, x]","A",0,0,0,0,-1,"{}"