1,1,105,33,0.1329975,"\int \frac{\tan ^4(x)}{a+a \cos (x)} \, dx","Integrate[Tan[x]^4/(a + a*Cos[x]),x]","-\frac{\sec ^3(x) \left(2 (-3 \sin (x)+3 \sin (2 x)+\sin (3 x))+9 \cos (x) \left(\log \left(\cos \left(\frac{x}{2}\right)-\sin \left(\frac{x}{2}\right)\right)-\log \left(\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)\right)\right)+3 \cos (3 x) \left(\log \left(\cos \left(\frac{x}{2}\right)-\sin \left(\frac{x}{2}\right)\right)-\log \left(\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)\right)\right)\right)}{24 a}","\frac{\tan ^3(x)}{3 a}+\frac{\tanh ^{-1}(\sin (x))}{2 a}-\frac{\tan (x) \sec (x)}{2 a}",1,"-1/24*(Sec[x]^3*(9*Cos[x]*(Log[Cos[x/2] - Sin[x/2]] - Log[Cos[x/2] + Sin[x/2]]) + 3*Cos[3*x]*(Log[Cos[x/2] - Sin[x/2]] - Log[Cos[x/2] + Sin[x/2]]) + 2*(-3*Sin[x] + 3*Sin[2*x] + Sin[3*x])))/a","B",1
2,1,17,19,0.0222282,"\int \frac{\tan ^3(x)}{a+a \cos (x)} \, dx","Integrate[Tan[x]^3/(a + a*Cos[x]),x]","\frac{2 \sin ^4\left(\frac{x}{2}\right) \sec ^2(x)}{a}","\frac{\sec ^2(x)}{2 a}-\frac{\sec (x)}{a}",1,"(2*Sec[x]^2*Sin[x/2]^4)/a","A",1
3,1,39,15,0.0702925,"\int \frac{\tan ^2(x)}{a+a \cos (x)} \, dx","Integrate[Tan[x]^2/(a + a*Cos[x]),x]","\frac{\tan (x)+\log \left(\cos \left(\frac{x}{2}\right)-\sin \left(\frac{x}{2}\right)\right)-\log \left(\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)\right)}{a}","\frac{\tan (x)}{a}-\frac{\tanh ^{-1}(\sin (x))}{a}",1,"(Log[Cos[x/2] - Sin[x/2]] - Log[Cos[x/2] + Sin[x/2]] + Tan[x])/a","B",1
4,1,12,18,0.0168829,"\int \frac{\tan (x)}{a+a \cos (x)} \, dx","Integrate[Tan[x]/(a + a*Cos[x]),x]","\frac{2 \tanh ^{-1}(2 \cos (x)+1)}{a}","\frac{\log (\cos (x)+1)}{a}-\frac{\log (\cos (x))}{a}",1,"(2*ArcTanh[1 + 2*Cos[x]])/a","A",1
5,1,42,33,0.0402163,"\int \frac{\cot (x)}{a+a \cos (x)} \, dx","Integrate[Cot[x]/(a + a*Cos[x]),x]","-\frac{2 \cos ^2\left(\frac{x}{2}\right) \left(\log \left(\cos \left(\frac{x}{2}\right)\right)-\log \left(\sin \left(\frac{x}{2}\right)\right)\right)+1}{2 a (\cos (x)+1)}","-\frac{\csc ^2(x)}{2 a}-\frac{\tanh ^{-1}(\cos (x))}{2 a}+\frac{\cot (x) \csc (x)}{2 a}",1,"-1/2*(1 + 2*Cos[x/2]^2*(Log[Cos[x/2]] - Log[Sin[x/2]]))/(a*(1 + Cos[x]))","A",1
6,1,25,30,0.0505441,"\int \frac{\cot ^2(x)}{a+a \cos (x)} \, dx","Integrate[Cot[x]^2/(a + a*Cos[x]),x]","\frac{(-4 \cos (x)+\cos (2 x)-3) \csc (x)}{6 a (\cos (x)+1)}","-\frac{\cot ^3(x)}{3 a}+\frac{\csc ^3(x)}{3 a}-\frac{\csc (x)}{a}",1,"((-3 - 4*Cos[x] + Cos[2*x])*Csc[x])/(6*a*(1 + Cos[x]))","A",1
7,1,60,46,0.1561888,"\int \frac{\cot ^3(x)}{a+a \cos (x)} \, dx","Integrate[Cot[x]^3/(a + a*Cos[x]),x]","-\frac{2 \cot ^2\left(\frac{x}{2}\right)+\sec ^2\left(\frac{x}{2}\right)-12 \cos ^2\left(\frac{x}{2}\right) \left(\log \left(\cos \left(\frac{x}{2}\right)\right)-\log \left(\sin \left(\frac{x}{2}\right)\right)\right)-8}{16 a (\cos (x)+1)}","-\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,"-1/16*(-8 + 2*Cot[x/2]^2 - 12*Cos[x/2]^2*(Log[Cos[x/2]] - Log[Sin[x/2]]) + Sec[x/2]^2)/(a*(1 + Cos[x]))","A",1
8,1,41,40,0.0817991,"\int \frac{\cot ^4(x)}{a+a \cos (x)} \, dx","Integrate[Cot[x]^4/(a + a*Cos[x]),x]","-\frac{(8 \cos (x)+36 \cos (2 x)+24 \cos (3 x)-3 \cos (4 x)-25) \csc ^3(x)}{120 a (\cos (x)+1)}","-\frac{\cot ^5(x)}{5 a}+\frac{\csc ^5(x)}{5 a}-\frac{2 \csc ^3(x)}{3 a}+\frac{\csc (x)}{a}",1,"-1/120*((-25 + 8*Cos[x] + 36*Cos[2*x] + 24*Cos[3*x] - 3*Cos[4*x])*Csc[x]^3)/(a*(1 + Cos[x]))","A",1
9,1,49,33,0.0723345,"\int \frac{\tan (3 x)}{(1+\cos (3 x))^2} \, dx","Integrate[Tan[3*x]/(1 + Cos[3*x])^2,x]","\frac{\cos ^4\left(\frac{3 x}{2}\right) \left(8 \log \left(\cos \left(\frac{3 x}{2}\right)\right)-4 \log (\cos (3 x))\right)-2 \cos ^2\left(\frac{3 x}{2}\right)}{3 (\cos (3 x)+1)^2}","-\frac{1}{3 (\cos (3 x)+1)}-\frac{1}{3} \log (\cos (3 x))+\frac{1}{3} \log (\cos (3 x)+1)",1,"(-2*Cos[(3*x)/2]^2 + Cos[(3*x)/2]^4*(8*Log[Cos[(3*x)/2]] - 4*Log[Cos[3*x]]))/(3*(1 + Cos[3*x])^2)","A",1
10,1,190,113,1.134088,"\int \frac{\tan ^4(x)}{a+b \cos (x)} \, dx","Integrate[Tan[x]^4/(a + b*Cos[x]),x]","-\frac{48 \left(b^2-a^2\right)^{3/2} \tanh ^{-1}\left(\frac{(a-b) \tan \left(\frac{x}{2}\right)}{\sqrt{b^2-a^2}}\right)+\sec ^3(x) \left(2 a \left(\left(4 a^2-3 b^2\right) \sin (3 x)+3 a b \sin (2 x)-3 b^2 \sin (x)\right)+9 b \left(3 a^2-2 b^2\right) \cos (x) \left(\log \left(\cos \left(\frac{x}{2}\right)-\sin \left(\frac{x}{2}\right)\right)-\log \left(\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)\right)\right)+3 b \left(3 a^2-2 b^2\right) \cos (3 x) \left(\log \left(\cos \left(\frac{x}{2}\right)-\sin \left(\frac{x}{2}\right)\right)-\log \left(\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)\right)\right)\right)}{24 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{b \left(3 a^2-2 b^2\right) \tanh ^{-1}(\sin (x))}{2 a^4}-\frac{\left(4 a^2-3 b^2\right) \tan (x)}{3 a^3}+\frac{\tan (x) \sec ^2(x)}{3 a}",1,"-1/24*(48*(-a^2 + b^2)^(3/2)*ArcTanh[((a - b)*Tan[x/2])/Sqrt[-a^2 + b^2]] + Sec[x]^3*(9*b*(3*a^2 - 2*b^2)*Cos[x]*(Log[Cos[x/2] - Sin[x/2]] - Log[Cos[x/2] + Sin[x/2]]) + 3*b*(3*a^2 - 2*b^2)*Cos[3*x]*(Log[Cos[x/2] - Sin[x/2]] - Log[Cos[x/2] + Sin[x/2]]) + 2*a*(-3*b^2*Sin[x] + 3*a*b*Sin[2*x] + (4*a^2 - 3*b^2)*Sin[3*x])))/a^4","A",1
11,1,46,57,0.0958579,"\int \frac{\tan ^3(x)}{a+b \cos (x)} \, dx","Integrate[Tan[x]^3/(a + b*Cos[x]),x]","\frac{2 \left(a^2-b^2\right) (\log (\cos (x))-\log (a+b \cos (x)))+a^2 \sec ^2(x)-2 a b \sec (x)}{2 a^3}","-\frac{b \sec (x)}{a^2}+\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{\sec ^2(x)}{2 a}",1,"(2*(a^2 - b^2)*(Log[Cos[x]] - Log[a + b*Cos[x]]) - 2*a*b*Sec[x] + a^2*Sec[x]^2)/(2*a^3)","A",1
12,1,85,61,0.1768298,"\int \frac{\tan ^2(x)}{a+b \cos (x)} \, dx","Integrate[Tan[x]^2/(a + b*Cos[x]),x]","\frac{-2 \sqrt{b^2-a^2} \tanh ^{-1}\left(\frac{(a-b) \tan \left(\frac{x}{2}\right)}{\sqrt{b^2-a^2}}\right)+a \tan (x)+b \left(\log \left(\cos \left(\frac{x}{2}\right)-\sin \left(\frac{x}{2}\right)\right)-\log \left(\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)\right)\right)}{a^2}","-\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^2 + b^2]*ArcTanh[((a - b)*Tan[x/2])/Sqrt[-a^2 + b^2]] + b*(Log[Cos[x/2] - Sin[x/2]] - Log[Cos[x/2] + Sin[x/2]]) + a*Tan[x])/a^2","A",1
13,1,20,20,0.0085186,"\int \frac{\tan (x)}{a+b \cos (x)} \, dx","Integrate[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",1
14,1,50,54,0.0682099,"\int \frac{\cot (x)}{a+b \cos (x)} \, dx","Integrate[Cot[x]/(a + b*Cos[x]),x]","-\frac{a \log (a+b \cos (x))}{a^2-b^2}+\frac{\log \left(\sin \left(\frac{x}{2}\right)\right)}{a+b}+\frac{\log \left(\cos \left(\frac{x}{2}\right)\right)}{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[Cos[x/2]]/(a - b) - (a*Log[a + b*Cos[x]])/(a^2 - b^2) + Log[Sin[x/2]]/(a + b)","A",1
15,1,67,77,0.3261978,"\int \frac{\cot ^2(x)}{a+b \cos (x)} \, dx","Integrate[Cot[x]^2/(a + b*Cos[x]),x]","\frac{b \csc (x)-a \cot (x)}{a^2-b^2}-\frac{2 a^2 \tanh ^{-1}\left(\frac{(a-b) \tan \left(\frac{x}{2}\right)}{\sqrt{b^2-a^2}}\right)}{\left(b^2-a^2\right)^{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*ArcTanh[((a - b)*Tan[x/2])/Sqrt[-a^2 + b^2]])/(-a^2 + b^2)^(3/2) + (-(a*Cot[x]) + b*Csc[x])/(a^2 - b^2)","A",1
16,1,100,93,0.5653017,"\int \frac{\cot ^3(x)}{a+b \cos (x)} \, dx","Integrate[Cot[x]^3/(a + b*Cos[x]),x]","\frac{1}{8} \left(\frac{8 a^3 \log (a+b \cos (x))}{\left(a^2-b^2\right)^2}-\frac{\csc ^2\left(\frac{x}{2}\right)}{a+b}-\frac{\sec ^2\left(\frac{x}{2}\right)}{a-b}-\frac{4 (2 a+b) \log \left(\sin \left(\frac{x}{2}\right)\right)}{(a+b)^2}+\frac{4 (b-2 a) \log \left(\cos \left(\frac{x}{2}\right)\right)}{(a-b)^2}\right)","-\frac{\csc ^2(x) (a-b \cos (x))}{2 \left(a^2-b^2\right)}+\frac{a^3 \log (a+b \cos (x))}{\left(a^2-b^2\right)^2}-\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,"(-(Csc[x/2]^2/(a + b)) + (4*(-2*a + b)*Log[Cos[x/2]])/(a - b)^2 + (8*a^3*Log[a + b*Cos[x]])/(a^2 - b^2)^2 - (4*(2*a + b)*Log[Sin[x/2]])/(a + b)^2 - Sec[x/2]^2/(a - b))/8","A",1
17,1,112,138,0.6492673,"\int \frac{\cot ^4(x)}{a+b \cos (x)} \, dx","Integrate[Cot[x]^4/(a + b*Cos[x]),x]","-\frac{\csc ^3(x) \left(6 b \left(b^2-2 a^2\right) \cos (2 x)+\left(4 a^2-b^2\right) (a \cos (3 x)+2 b)-3 a b^2 \cos (x)\right)}{12 (a-b)^2 (a+b)^2}-\frac{2 a^4 \tanh ^{-1}\left(\frac{(a-b) \tan \left(\frac{x}{2}\right)}{\sqrt{b^2-a^2}}\right)}{\left(b^2-a^2\right)^{5/2}}","\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{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{a^3 \cot (x)}{\left(a^2-b^2\right)^2}",1,"(-2*a^4*ArcTanh[((a - b)*Tan[x/2])/Sqrt[-a^2 + b^2]])/(-a^2 + b^2)^(5/2) - ((-3*a*b^2*Cos[x] + 6*b*(-2*a^2 + b^2)*Cos[2*x] + (4*a^2 - b^2)*(2*b + a*Cos[3*x]))*Csc[x]^3)/(12*(a - b)^2*(a + b)^2)","A",1
18,1,44,44,0.0722904,"\int \frac{\cot (x)}{\sqrt{3-\cos (x)}} \, dx","Integrate[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,"-1/2*ArcTanh[Sqrt[3 - Cos[x]]/2] - ArcTanh[Sqrt[3 - Cos[x]]/Sqrt[2]]/Sqrt[2]","A",1
19,1,37,37,0.0200272,"\int \sqrt{a+b \cos (x)} \tan (x) \, dx","Integrate[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",1
20,1,24,24,0.0126604,"\int \frac{\tan (x)}{\sqrt{a+b \cos (x)}} \, dx","Integrate[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",1
21,1,363,204,2.7807121,"\int \frac{\sqrt{e \tan (c+d x)}}{a+b \cos (c+d x)} \, dx","Integrate[Sqrt[e*Tan[c + d*x]]/(a + b*Cos[c + d*x]),x]","\frac{2 \sqrt{e \tan (c+d x)} \left(a \sqrt{\sec ^2(c+d x)}+b\right) \left(\frac{b \tan ^{\frac{3}{2}}(c+d x) F_1\left(\frac{3}{4};\frac{1}{2},1;\frac{7}{4};-\tan ^2(c+d x),-\frac{a^2 \tan ^2(c+d x)}{a^2-b^2}\right)}{3 \left(b^2-a^2\right)}+\frac{-2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt[4]{a^2-b^2}}\right)+2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt[4]{a^2-b^2}}+1\right)+\log \left(-\sqrt{2} \sqrt{a} \sqrt[4]{a^2-b^2} \sqrt{\tan (c+d x)}+\sqrt{a^2-b^2}+a \tan (c+d x)\right)-\log \left(\sqrt{2} \sqrt{a} \sqrt[4]{a^2-b^2} \sqrt{\tan (c+d x)}+\sqrt{a^2-b^2}+a \tan (c+d x)\right)}{4 \sqrt{2} \sqrt{a} \sqrt[4]{a^2-b^2}}\right)}{d \sqrt{\tan (c+d x)} \sqrt{\sec ^2(c+d x)} (a+b \cos (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*(b + a*Sqrt[Sec[c + d*x]^2])*Sqrt[e*Tan[c + d*x]]*((-2*ArcTan[1 - (Sqrt[2]*Sqrt[a]*Sqrt[Tan[c + d*x]])/(a^2 - b^2)^(1/4)] + 2*ArcTan[1 + (Sqrt[2]*Sqrt[a]*Sqrt[Tan[c + d*x]])/(a^2 - b^2)^(1/4)] + Log[Sqrt[a^2 - b^2] - Sqrt[2]*Sqrt[a]*(a^2 - b^2)^(1/4)*Sqrt[Tan[c + d*x]] + a*Tan[c + d*x]] - Log[Sqrt[a^2 - b^2] + Sqrt[2]*Sqrt[a]*(a^2 - b^2)^(1/4)*Sqrt[Tan[c + d*x]] + a*Tan[c + d*x]])/(4*Sqrt[2]*Sqrt[a]*(a^2 - b^2)^(1/4)) + (b*AppellF1[3/4, 1/2, 1, 7/4, -Tan[c + d*x]^2, -((a^2*Tan[c + d*x]^2)/(a^2 - b^2))]*Tan[c + d*x]^(3/2))/(3*(-a^2 + b^2))))/(d*(a + b*Cos[c + d*x])*Sqrt[Sec[c + d*x]^2]*Sqrt[Tan[c + d*x]])","C",0
22,0,0,49,2.4686228,"\int (a+b \cos (e+f x))^m (g \tan (e+f x))^p \, dx","Integrate[(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,"Integrate[(a + b*Cos[e + f*x])^m*(g*Tan[e + f*x])^p, x]","A",-1