1,1,26,33,0.0049093,"\int \frac{\sin ^6(x)}{a-a \cos ^2(x)} \, dx","Integrate[Sin[x]^6/(a - a*Cos[x]^2),x]","\frac{\frac{3 x}{8}-\frac{1}{4} \sin (2 x)+\frac{1}{32} \sin (4 x)}{a}","\frac{3 x}{8 a}-\frac{\sin ^3(x) \cos (x)}{4 a}-\frac{3 \sin (x) \cos (x)}{8 a}",1,"((3*x)/8 - Sin[2*x]/4 + Sin[4*x]/32)/a","A",1
2,1,19,19,0.0032564,"\int \frac{\sin ^5(x)}{a-a \cos ^2(x)} \, dx","Integrate[Sin[x]^5/(a - a*Cos[x]^2),x]","\frac{\frac{1}{12} \cos (3 x)-\frac{3 \cos (x)}{4}}{a}","\frac{\cos ^3(x)}{3 a}-\frac{\cos (x)}{a}",1,"((-3*Cos[x])/4 + Cos[3*x]/12)/a","A",1
3,1,18,20,0.002893,"\int \frac{\sin ^4(x)}{a-a \cos ^2(x)} \, dx","Integrate[Sin[x]^4/(a - a*Cos[x]^2),x]","\frac{\frac{x}{2}-\frac{1}{4} \sin (2 x)}{a}","\frac{x}{2 a}-\frac{\sin (x) \cos (x)}{2 a}",1,"(x/2 - Sin[2*x]/4)/a","A",1
4,1,7,7,0.0023989,"\int \frac{\sin ^3(x)}{a-a \cos ^2(x)} \, dx","Integrate[Sin[x]^3/(a - a*Cos[x]^2),x]","-\frac{\cos (x)}{a}","-\frac{\cos (x)}{a}",1,"-(Cos[x]/a)","A",1
5,1,5,5,0.0008397,"\int \frac{\sin ^2(x)}{a-a \cos ^2(x)} \, dx","Integrate[Sin[x]^2/(a - a*Cos[x]^2),x]","\frac{x}{a}","\frac{x}{a}",1,"x/a","A",1
6,1,21,8,0.005659,"\int \frac{\sin (x)}{a-a \cos ^2(x)} \, dx","Integrate[Sin[x]/(a - a*Cos[x]^2),x]","\frac{\log \left(\sin \left(\frac{x}{2}\right)\right)-\log \left(\cos \left(\frac{x}{2}\right)\right)}{a}","-\frac{\tanh ^{-1}(\cos (x))}{a}",1,"(-Log[Cos[x/2]] + Log[Sin[x/2]])/a","B",1
7,1,51,22,0.0081969,"\int \frac{\csc (x)}{a-a \cos ^2(x)} \, dx","Integrate[Csc[x]/(a - a*Cos[x]^2),x]","\frac{-\frac{1}{8} \csc ^2\left(\frac{x}{2}\right)+\frac{1}{8} \sec ^2\left(\frac{x}{2}\right)+\frac{1}{2} \log \left(\sin \left(\frac{x}{2}\right)\right)-\frac{1}{2} \log \left(\cos \left(\frac{x}{2}\right)\right)}{a}","-\frac{\tanh ^{-1}(\cos (x))}{2 a}-\frac{\cot (x) \csc (x)}{2 a}",1,"(-1/8*Csc[x/2]^2 - Log[Cos[x/2]]/2 + Log[Sin[x/2]]/2 + Sec[x/2]^2/8)/a","B",1
8,1,21,19,0.003777,"\int \frac{\csc ^2(x)}{a-a \cos ^2(x)} \, dx","Integrate[Csc[x]^2/(a - a*Cos[x]^2),x]","\frac{-\frac{2 \cot (x)}{3}-\frac{1}{3} \cot (x) \csc ^2(x)}{a}","-\frac{\cot ^3(x)}{3 a}-\frac{\cot (x)}{a}",1,"((-2*Cot[x])/3 - (Cot[x]*Csc[x]^2)/3)/a","A",1
9,1,75,35,0.0070028,"\int \frac{\csc ^3(x)}{a-a \cos ^2(x)} \, dx","Integrate[Csc[x]^3/(a - a*Cos[x]^2),x]","\frac{-\frac{1}{64} \csc ^4\left(\frac{x}{2}\right)-\frac{3}{32} \csc ^2\left(\frac{x}{2}\right)+\frac{1}{64} \sec ^4\left(\frac{x}{2}\right)+\frac{3}{32} \sec ^2\left(\frac{x}{2}\right)+\frac{3}{8} \log \left(\sin \left(\frac{x}{2}\right)\right)-\frac{3}{8} \log \left(\cos \left(\frac{x}{2}\right)\right)}{a}","-\frac{3 \tanh ^{-1}(\cos (x))}{8 a}-\frac{\cot (x) \csc ^3(x)}{4 a}-\frac{3 \cot (x) \csc (x)}{8 a}",1,"((-3*Csc[x/2]^2)/32 - Csc[x/2]^4/64 - (3*Log[Cos[x/2]])/8 + (3*Log[Sin[x/2]])/8 + (3*Sec[x/2]^2)/32 + Sec[x/2]^4/64)/a","B",1
10,1,143,78,0.2561794,"\int \frac{\sin ^7(x)}{a+b \cos ^2(x)} \, dx","Integrate[Sin[x]^7/(a + b*Cos[x]^2),x]","\frac{\left(8 a^2+22 a b+19 b^2\right) \cos (x)}{8 b^3}-\frac{(a+b)^3 \tan ^{-1}\left(\frac{\sqrt{b}-\sqrt{a+b} \tan \left(\frac{x}{2}\right)}{\sqrt{a}}\right)}{\sqrt{a} b^{7/2}}-\frac{(a+b)^3 \tan ^{-1}\left(\frac{\sqrt{a+b} \tan \left(\frac{x}{2}\right)+\sqrt{b}}{\sqrt{a}}\right)}{\sqrt{a} b^{7/2}}-\frac{(4 a+9 b) \cos (3 x)}{48 b^2}+\frac{\cos (5 x)}{80 b}","\frac{\left(a^2+3 a b+3 b^2\right) \cos (x)}{b^3}-\frac{(a+b)^3 \tan ^{-1}\left(\frac{\sqrt{b} \cos (x)}{\sqrt{a}}\right)}{\sqrt{a} b^{7/2}}-\frac{(a+3 b) \cos ^3(x)}{3 b^2}+\frac{\cos ^5(x)}{5 b}",1,"-(((a + b)^3*ArcTan[(Sqrt[b] - Sqrt[a + b]*Tan[x/2])/Sqrt[a]])/(Sqrt[a]*b^(7/2))) - ((a + b)^3*ArcTan[(Sqrt[b] + Sqrt[a + b]*Tan[x/2])/Sqrt[a]])/(Sqrt[a]*b^(7/2)) + ((8*a^2 + 22*a*b + 19*b^2)*Cos[x])/(8*b^3) - ((4*a + 9*b)*Cos[3*x])/(48*b^2) + Cos[5*x]/(80*b)","A",1
11,1,116,54,0.1822002,"\int \frac{\sin ^5(x)}{a+b \cos ^2(x)} \, dx","Integrate[Sin[x]^5/(a + b*Cos[x]^2),x]","\frac{3 \sqrt{b} (4 a+7 b) \cos (x)-\frac{12 (a+b)^2 \tan ^{-1}\left(\frac{\sqrt{b}-\sqrt{a+b} \tan \left(\frac{x}{2}\right)}{\sqrt{a}}\right)}{\sqrt{a}}-\frac{12 (a+b)^2 \tan ^{-1}\left(\frac{\sqrt{a+b} \tan \left(\frac{x}{2}\right)+\sqrt{b}}{\sqrt{a}}\right)}{\sqrt{a}}-b^{3/2} \cos (3 x)}{12 b^{5/2}}","-\frac{(a+b)^2 \tan ^{-1}\left(\frac{\sqrt{b} \cos (x)}{\sqrt{a}}\right)}{\sqrt{a} b^{5/2}}+\frac{(a+2 b) \cos (x)}{b^2}-\frac{\cos ^3(x)}{3 b}",1,"((-12*(a + b)^2*ArcTan[(Sqrt[b] - Sqrt[a + b]*Tan[x/2])/Sqrt[a]])/Sqrt[a] - (12*(a + b)^2*ArcTan[(Sqrt[b] + Sqrt[a + b]*Tan[x/2])/Sqrt[a]])/Sqrt[a] + 3*Sqrt[b]*(4*a + 7*b)*Cos[x] - b^(3/2)*Cos[3*x])/(12*b^(5/2))","B",1
12,1,90,36,0.1805265,"\int \frac{\sin ^3(x)}{a+b \cos ^2(x)} \, dx","Integrate[Sin[x]^3/(a + b*Cos[x]^2),x]","\frac{\sqrt{a} \sqrt{b} \cos (x)-\left((a+b) \tan ^{-1}\left(\frac{\sqrt{b}-\sqrt{a+b} \tan \left(\frac{x}{2}\right)}{\sqrt{a}}\right)\right)-(a+b) \tan ^{-1}\left(\frac{\sqrt{a+b} \tan \left(\frac{x}{2}\right)+\sqrt{b}}{\sqrt{a}}\right)}{\sqrt{a} b^{3/2}}","\frac{\cos (x)}{b}-\frac{(a+b) \tan ^{-1}\left(\frac{\sqrt{b} \cos (x)}{\sqrt{a}}\right)}{\sqrt{a} b^{3/2}}",1,"(-((a + b)*ArcTan[(Sqrt[b] - Sqrt[a + b]*Tan[x/2])/Sqrt[a]]) - (a + b)*ArcTan[(Sqrt[b] + Sqrt[a + b]*Tan[x/2])/Sqrt[a]] + Sqrt[a]*Sqrt[b]*Cos[x])/(Sqrt[a]*b^(3/2))","B",1
13,1,26,26,0.020689,"\int \frac{\sin (x)}{a+b \cos ^2(x)} \, dx","Integrate[Sin[x]/(a + b*Cos[x]^2),x]","-\frac{\tan ^{-1}\left(\frac{\sqrt{b} \cos (x)}{\sqrt{a}}\right)}{\sqrt{a} \sqrt{b}}","-\frac{\tan ^{-1}\left(\frac{\sqrt{b} \cos (x)}{\sqrt{a}}\right)}{\sqrt{a} \sqrt{b}}",1,"-(ArcTan[(Sqrt[b]*Cos[x])/Sqrt[a]]/(Sqrt[a]*Sqrt[b]))","A",1
14,1,50,42,0.0480945,"\int \frac{\csc (x)}{a+b \cos ^2(x)} \, dx","Integrate[Csc[x]/(a + b*Cos[x]^2),x]","\frac{-\frac{2 \sqrt{b} \tan ^{-1}\left(\frac{\sqrt{b} \cos (x)}{\sqrt{a}}\right)}{\sqrt{a}}+\log (1-\cos (x))-\log (\cos (x)+1)}{2 (a+b)}","-\frac{\sqrt{b} \tan ^{-1}\left(\frac{\sqrt{b} \cos (x)}{\sqrt{a}}\right)}{\sqrt{a} (a+b)}-\frac{\tanh ^{-1}(\cos (x))}{a+b}",1,"((-2*Sqrt[b]*ArcTan[(Sqrt[b]*Cos[x])/Sqrt[a]])/Sqrt[a] + Log[1 - Cos[x]] - Log[1 + Cos[x]])/(2*(a + b))","A",1
15,1,140,62,0.5501035,"\int \frac{\csc ^3(x)}{a+b \cos ^2(x)} \, dx","Integrate[Csc[x]^3/(a + b*Cos[x]^2),x]","\frac{-8 b^{3/2} \tan ^{-1}\left(\frac{\sqrt{b}-\sqrt{a+b} \tan \left(\frac{x}{2}\right)}{\sqrt{a}}\right)-8 b^{3/2} \tan ^{-1}\left(\frac{\sqrt{a+b} \tan \left(\frac{x}{2}\right)+\sqrt{b}}{\sqrt{a}}\right)+\sqrt{a} \left(-\left((a+b) \csc ^2\left(\frac{x}{2}\right)\right)+(a+b) \sec ^2\left(\frac{x}{2}\right)-4 (a+3 b) \left(\log \left(\cos \left(\frac{x}{2}\right)\right)-\log \left(\sin \left(\frac{x}{2}\right)\right)\right)\right)}{8 \sqrt{a} (a+b)^2}","-\frac{b^{3/2} \tan ^{-1}\left(\frac{\sqrt{b} \cos (x)}{\sqrt{a}}\right)}{\sqrt{a} (a+b)^2}-\frac{(a+3 b) \tanh ^{-1}(\cos (x))}{2 (a+b)^2}-\frac{\cot (x) \csc (x)}{2 (a+b)}",1,"(-8*b^(3/2)*ArcTan[(Sqrt[b] - Sqrt[a + b]*Tan[x/2])/Sqrt[a]] - 8*b^(3/2)*ArcTan[(Sqrt[b] + Sqrt[a + b]*Tan[x/2])/Sqrt[a]] + Sqrt[a]*(-((a + b)*Csc[x/2]^2) - 4*(a + 3*b)*(Log[Cos[x/2]] - Log[Sin[x/2]]) + (a + b)*Sec[x/2]^2))/(8*Sqrt[a]*(a + b)^2)","B",1
16,1,204,94,1.4655591,"\int \frac{\csc ^5(x)}{a+b \cos ^2(x)} \, dx","Integrate[Csc[x]^5/(a + b*Cos[x]^2),x]","\frac{\sqrt{a} \left(-2 \left(3 a^2+10 a b+7 b^2\right) \csc ^2\left(\frac{x}{2}\right)+2 \left(3 a^2+10 a b+7 b^2\right) \sec ^2\left(\frac{x}{2}\right)-8 \left(3 a^2+10 a b+15 b^2\right) \left(\log \left(\cos \left(\frac{x}{2}\right)\right)-\log \left(\sin \left(\frac{x}{2}\right)\right)\right)-(a+b)^2 \csc ^4\left(\frac{x}{2}\right)+(a+b)^2 \sec ^4\left(\frac{x}{2}\right)\right)-64 b^{5/2} \tan ^{-1}\left(\frac{\sqrt{b}-\sqrt{a+b} \tan \left(\frac{x}{2}\right)}{\sqrt{a}}\right)-64 b^{5/2} \tan ^{-1}\left(\frac{\sqrt{a+b} \tan \left(\frac{x}{2}\right)+\sqrt{b}}{\sqrt{a}}\right)}{64 \sqrt{a} (a+b)^3}","-\frac{\left(3 a^2+10 a b+15 b^2\right) \tanh ^{-1}(\cos (x))}{8 (a+b)^3}-\frac{b^{5/2} \tan ^{-1}\left(\frac{\sqrt{b} \cos (x)}{\sqrt{a}}\right)}{\sqrt{a} (a+b)^3}-\frac{\cot (x) \csc ^3(x)}{4 (a+b)}-\frac{(3 a+7 b) \cot (x) \csc (x)}{8 (a+b)^2}",1,"(-64*b^(5/2)*ArcTan[(Sqrt[b] - Sqrt[a + b]*Tan[x/2])/Sqrt[a]] - 64*b^(5/2)*ArcTan[(Sqrt[b] + Sqrt[a + b]*Tan[x/2])/Sqrt[a]] + Sqrt[a]*(-2*(3*a^2 + 10*a*b + 7*b^2)*Csc[x/2]^2 - (a + b)^2*Csc[x/2]^4 - 8*(3*a^2 + 10*a*b + 15*b^2)*(Log[Cos[x/2]] - Log[Sin[x/2]]) + 2*(3*a^2 + 10*a*b + 7*b^2)*Sec[x/2]^2 + (a + b)^2*Sec[x/2]^4))/(64*Sqrt[a]*(a + b)^3)","B",1
17,1,77,88,0.195595,"\int \frac{\sin ^6(x)}{a+b \cos ^2(x)} \, dx","Integrate[Sin[x]^6/(a + b*Cos[x]^2),x]","\frac{-4 x \left(8 a^2+20 a b+15 b^2\right)+8 b (a+2 b) \sin (2 x)+\frac{32 (a+b)^{5/2} \tan ^{-1}\left(\frac{\sqrt{a} \tan (x)}{\sqrt{a+b}}\right)}{\sqrt{a}}-b^2 \sin (4 x)}{32 b^3}","-\frac{x \left(8 a^2+20 a b+15 b^2\right)}{8 b^3}-\frac{(a+b)^{5/2} \tan ^{-1}\left(\frac{\sqrt{a+b} \cot (x)}{\sqrt{a}}\right)}{\sqrt{a} b^3}+\frac{(4 a+7 b) \sin (x) \cos (x)}{8 b^2}+\frac{\sin ^3(x) \cos (x)}{4 b}",1,"(-4*(8*a^2 + 20*a*b + 15*b^2)*x + (32*(a + b)^(5/2)*ArcTan[(Sqrt[a]*Tan[x])/Sqrt[a + b]])/Sqrt[a] + 8*b*(a + 2*b)*Sin[2*x] - b^2*Sin[4*x])/(32*b^3)","A",1
18,1,52,60,0.1034252,"\int \frac{\sin ^4(x)}{a+b \cos ^2(x)} \, dx","Integrate[Sin[x]^4/(a + b*Cos[x]^2),x]","\frac{\frac{4 (a+b)^{3/2} \tan ^{-1}\left(\frac{\sqrt{a} \tan (x)}{\sqrt{a+b}}\right)}{\sqrt{a}}-4 a x-6 b x+b \sin (2 x)}{4 b^2}","-\frac{x (2 a+3 b)}{2 b^2}-\frac{(a+b)^{3/2} \tan ^{-1}\left(\frac{\sqrt{a+b} \cot (x)}{\sqrt{a}}\right)}{\sqrt{a} b^2}+\frac{\sin (x) \cos (x)}{2 b}",1,"(-4*a*x - 6*b*x + (4*(a + b)^(3/2)*ArcTan[(Sqrt[a]*Tan[x])/Sqrt[a + b]])/Sqrt[a] + b*Sin[2*x])/(4*b^2)","A",1
19,1,37,40,0.0716223,"\int \frac{\sin ^2(x)}{a+b \cos ^2(x)} \, dx","Integrate[Sin[x]^2/(a + b*Cos[x]^2),x]","\frac{\frac{\sqrt{a+b} \tan ^{-1}\left(\frac{\sqrt{a} \tan (x)}{\sqrt{a+b}}\right)}{\sqrt{a}}-x}{b}","-\frac{\sqrt{a+b} \tan ^{-1}\left(\frac{\sqrt{a+b} \cot (x)}{\sqrt{a}}\right)}{\sqrt{a} b}-\frac{x}{b}",1,"(-x + (Sqrt[a + b]*ArcTan[(Sqrt[a]*Tan[x])/Sqrt[a + b]])/Sqrt[a])/b","A",1
20,1,29,30,0.0594409,"\int \frac{1}{a+b \cos ^2(x)} \, dx","Integrate[(a + b*Cos[x]^2)^(-1),x]","\frac{\tan ^{-1}\left(\frac{\sqrt{a} \tan (x)}{\sqrt{a+b}}\right)}{\sqrt{a} \sqrt{a+b}}","-\frac{\tan ^{-1}\left(\frac{\sqrt{a+b} \cot (x)}{\sqrt{a}}\right)}{\sqrt{a} \sqrt{a+b}}",1,"ArcTan[(Sqrt[a]*Tan[x])/Sqrt[a + b]]/(Sqrt[a]*Sqrt[a + b])","A",1
21,1,40,41,0.0929269,"\int \frac{\csc ^2(x)}{a+b \cos ^2(x)} \, dx","Integrate[Csc[x]^2/(a + b*Cos[x]^2),x]","\frac{b \tan ^{-1}\left(\frac{\sqrt{a} \tan (x)}{\sqrt{a+b}}\right)}{\sqrt{a} (a+b)^{3/2}}-\frac{\cot (x)}{a+b}","-\frac{\cot (x)}{a+b}-\frac{b \tan ^{-1}\left(\frac{\sqrt{a+b} \cot (x)}{\sqrt{a}}\right)}{\sqrt{a} (a+b)^{3/2}}",1,"(b*ArcTan[(Sqrt[a]*Tan[x])/Sqrt[a + b]])/(Sqrt[a]*(a + b)^(3/2)) - Cot[x]/(a + b)","A",1
22,1,59,61,0.2148949,"\int \frac{\csc ^4(x)}{a+b \cos ^2(x)} \, dx","Integrate[Csc[x]^4/(a + b*Cos[x]^2),x]","\frac{b^2 \tan ^{-1}\left(\frac{\sqrt{a} \tan (x)}{\sqrt{a+b}}\right)}{\sqrt{a} (a+b)^{5/2}}-\frac{\cot (x) \left((a+b) \csc ^2(x)+2 a+5 b\right)}{3 (a+b)^2}","-\frac{b^2 \tan ^{-1}\left(\frac{\sqrt{a+b} \cot (x)}{\sqrt{a}}\right)}{\sqrt{a} (a+b)^{5/2}}-\frac{\cot ^3(x)}{3 (a+b)}-\frac{(a+2 b) \cot (x)}{(a+b)^2}",1,"(b^2*ArcTan[(Sqrt[a]*Tan[x])/Sqrt[a + b]])/(Sqrt[a]*(a + b)^(5/2)) - (Cot[x]*(2*a + 5*b + (a + b)*Csc[x]^2))/(3*(a + b)^2)","A",1
23,1,90,89,0.3816939,"\int \frac{\csc ^6(x)}{a+b \cos ^2(x)} \, dx","Integrate[Csc[x]^6/(a + b*Cos[x]^2),x]","\frac{b^3 \tan ^{-1}\left(\frac{\sqrt{a} \tan (x)}{\sqrt{a+b}}\right)}{\sqrt{a} (a+b)^{7/2}}-\frac{\cot (x) \left(\left(4 a^2+13 a b+9 b^2\right) \csc ^2(x)+8 a^2+3 (a+b)^2 \csc ^4(x)+26 a b+33 b^2\right)}{15 (a+b)^3}","-\frac{\left(a^2+3 a b+3 b^2\right) \cot (x)}{(a+b)^3}-\frac{b^3 \tan ^{-1}\left(\frac{\sqrt{a+b} \cot (x)}{\sqrt{a}}\right)}{\sqrt{a} (a+b)^{7/2}}-\frac{\cot ^5(x)}{5 (a+b)}-\frac{(2 a+3 b) \cot ^3(x)}{3 (a+b)^2}",1,"(b^3*ArcTan[(Sqrt[a]*Tan[x])/Sqrt[a + b]])/(Sqrt[a]*(a + b)^(7/2)) - (Cot[x]*(8*a^2 + 26*a*b + 33*b^2 + (4*a^2 + 13*a*b + 9*b^2)*Csc[x]^2 + 3*(a + b)^2*Csc[x]^4))/(15*(a + b)^3)","A",1
24,1,79,98,0.0907298,"\int \frac{\sin (x)}{4-3 \cos ^3(x)} \, dx","Integrate[Sin[x]/(4 - 3*Cos[x]^3),x]","\frac{1}{72} \left(6^{2/3} \left(2 \log \left(2-\sqrt[3]{6} \cos (x)\right)-\log \left(6^{2/3} \cos ^2(x)+2 \sqrt[3]{6} \cos (x)+4\right)\right)-6\ 2^{2/3} \sqrt[6]{3} \tan ^{-1}\left(\frac{\sqrt[3]{6} \cos (x)+1}{\sqrt{3}}\right)\right)","-\frac{\log \left(3^{2/3} \cos ^2(x)+2^{2/3} \sqrt[3]{3} \cos (x)+2 \sqrt[3]{2}\right)}{12 \sqrt[3]{6}}+\frac{\log \left(2^{2/3}-\sqrt[3]{3} \cos (x)\right)}{6 \sqrt[3]{6}}-\frac{\tan ^{-1}\left(\frac{\sqrt[3]{6} \cos (x)+1}{\sqrt{3}}\right)}{2 \sqrt[3]{2} 3^{5/6}}",1,"(-6*2^(2/3)*3^(1/6)*ArcTan[(1 + 6^(1/3)*Cos[x])/Sqrt[3]] + 6^(2/3)*(2*Log[2 - 6^(1/3)*Cos[x]] - Log[4 + 2*6^(1/3)*Cos[x] + 6^(2/3)*Cos[x]^2]))/72","A",1
25,1,4,4,0.0025103,"\int \frac{1}{1-\cos ^2(x)} \, dx","Integrate[(1 - Cos[x]^2)^(-1),x]","-\cot (x)","-\cot (x)",1,"-Cot[x]","A",1
26,1,17,13,0.0035574,"\int \frac{1}{\left(1-\cos ^2(x)\right)^2} \, dx","Integrate[(1 - Cos[x]^2)^(-2),x]","-\frac{2 \cot (x)}{3}-\frac{1}{3} \cot (x) \csc ^2(x)","-\frac{1}{3} \cot ^3(x)-\cot (x)",1,"(-2*Cot[x])/3 - (Cot[x]*Csc[x]^2)/3","A",1
27,1,27,21,0.0038518,"\int \frac{1}{\left(1-\cos ^2(x)\right)^3} \, dx","Integrate[(1 - Cos[x]^2)^(-3),x]","-\frac{8 \cot (x)}{15}-\frac{1}{5} \cot (x) \csc ^4(x)-\frac{4}{15} \cot (x) \csc ^2(x)","-\frac{1}{5} \cot ^5(x)-\frac{2 \cot ^3(x)}{3}-\cot (x)",1,"(-8*Cot[x])/15 - (4*Cot[x]*Csc[x]^2)/15 - (Cot[x]*Csc[x]^4)/5","A",1
28,1,111,78,0.4221918,"\int \frac{\cos ^7(x)}{a+b \cos ^2(x)} \, dx","Integrate[Cos[x]^7/(a + b*Cos[x]^2),x]","\frac{a^3 \left(\log \left(\sqrt{a+b}-\sqrt{b} \sin (x)\right)-\log \left(\sqrt{a+b}+\sqrt{b} \sin (x)\right)\right)}{2 b^{7/2} \sqrt{a+b}}+\frac{\left(8 a^2-6 a b+5 b^2\right) \sin (x)}{8 b^3}+\frac{(5 b-4 a) \sin (3 x)}{48 b^2}+\frac{\sin (5 x)}{80 b}","-\frac{a^3 \tanh ^{-1}\left(\frac{\sqrt{b} \sin (x)}{\sqrt{a+b}}\right)}{b^{7/2} \sqrt{a+b}}+\frac{\left(a^2-a b+b^2\right) \sin (x)}{b^3}+\frac{(a-2 b) \sin ^3(x)}{3 b^2}+\frac{\sin ^5(x)}{5 b}",1,"(a^3*(Log[Sqrt[a + b] - Sqrt[b]*Sin[x]] - Log[Sqrt[a + b] + Sqrt[b]*Sin[x]]))/(2*b^(7/2)*Sqrt[a + b]) + ((8*a^2 - 6*a*b + 5*b^2)*Sin[x])/(8*b^3) + ((-4*a + 5*b)*Sin[3*x])/(48*b^2) + Sin[5*x]/(80*b)","A",1
29,1,86,56,0.1963349,"\int \frac{\cos ^5(x)}{a+b \cos ^2(x)} \, dx","Integrate[Cos[x]^5/(a + b*Cos[x]^2),x]","\frac{\frac{6 a^2 \left(\log \left(\sqrt{a+b}+\sqrt{b} \sin (x)\right)-\log \left(\sqrt{a+b}-\sqrt{b} \sin (x)\right)\right)}{\sqrt{a+b}}+3 \sqrt{b} (3 b-4 a) \sin (x)+b^{3/2} \sin (3 x)}{12 b^{5/2}}","\frac{a^2 \tanh ^{-1}\left(\frac{\sqrt{b} \sin (x)}{\sqrt{a+b}}\right)}{b^{5/2} \sqrt{a+b}}-\frac{(a-b) \sin (x)}{b^2}-\frac{\sin ^3(x)}{3 b}",1,"((6*a^2*(-Log[Sqrt[a + b] - Sqrt[b]*Sin[x]] + Log[Sqrt[a + b] + Sqrt[b]*Sin[x]]))/Sqrt[a + b] + 3*Sqrt[b]*(-4*a + 3*b)*Sin[x] + b^(3/2)*Sin[3*x])/(12*b^(5/2))","A",1
30,1,38,38,0.0325952,"\int \frac{\cos ^3(x)}{a+b \cos ^2(x)} \, dx","Integrate[Cos[x]^3/(a + b*Cos[x]^2),x]","\frac{\sin (x)}{b}-\frac{a \tanh ^{-1}\left(\frac{\sqrt{b} \sin (x)}{\sqrt{a+b}}\right)}{b^{3/2} \sqrt{a+b}}","\frac{\sin (x)}{b}-\frac{a \tanh ^{-1}\left(\frac{\sqrt{b} \sin (x)}{\sqrt{a+b}}\right)}{b^{3/2} \sqrt{a+b}}",1,"-((a*ArcTanh[(Sqrt[b]*Sin[x])/Sqrt[a + b]])/(b^(3/2)*Sqrt[a + b])) + Sin[x]/b","A",1
31,1,29,29,0.0115453,"\int \frac{\cos (x)}{a+b \cos ^2(x)} \, dx","Integrate[Cos[x]/(a + b*Cos[x]^2),x]","\frac{\tanh ^{-1}\left(\frac{\sqrt{b} \sin (x)}{\sqrt{a+b}}\right)}{\sqrt{b} \sqrt{a+b}}","\frac{\tanh ^{-1}\left(\frac{\sqrt{b} \sin (x)}{\sqrt{a+b}}\right)}{\sqrt{b} \sqrt{a+b}}",1,"ArcTanh[(Sqrt[b]*Sin[x])/Sqrt[a + b]]/(Sqrt[b]*Sqrt[a + b])","A",1
32,1,38,41,0.0572948,"\int \frac{\sec (x)}{a+b \cos ^2(x)} \, dx","Integrate[Sec[x]/(a + b*Cos[x]^2),x]","\frac{\tanh ^{-1}(\sin (x))-\frac{\sqrt{b} \tanh ^{-1}\left(\frac{\sqrt{b} \sin (x)}{\sqrt{a+b}}\right)}{\sqrt{a+b}}}{a}","\frac{\tanh ^{-1}(\sin (x))}{a}-\frac{\sqrt{b} \tanh ^{-1}\left(\frac{\sqrt{b} \sin (x)}{\sqrt{a+b}}\right)}{a \sqrt{a+b}}",1,"(ArcTanh[Sin[x]] - (Sqrt[b]*ArcTanh[(Sqrt[b]*Sin[x])/Sqrt[a + b]])/Sqrt[a + b])/a","A",1
33,1,152,59,0.3881166,"\int \frac{\sec ^3(x)}{a+b \cos ^2(x)} \, dx","Integrate[Sec[x]^3/(a + b*Cos[x]^2),x]","\frac{-\frac{2 b^{3/2} \log \left(\sqrt{a+b}-\sqrt{b} \sin (x)\right)}{\sqrt{a+b}}+\frac{2 b^{3/2} \log \left(\sqrt{a+b}+\sqrt{b} \sin (x)\right)}{\sqrt{a+b}}-2 (a-2 b) \log \left(\cos \left(\frac{x}{2}\right)-\sin \left(\frac{x}{2}\right)\right)+2 (a-2 b) \log \left(\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)\right)+\frac{a}{\left(\cos \left(\frac{x}{2}\right)-\sin \left(\frac{x}{2}\right)\right)^2}-\frac{a}{\left(\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)\right)^2}}{4 a^2}","\frac{b^{3/2} \tanh ^{-1}\left(\frac{\sqrt{b} \sin (x)}{\sqrt{a+b}}\right)}{a^2 \sqrt{a+b}}+\frac{(a-2 b) \tanh ^{-1}(\sin (x))}{2 a^2}+\frac{\tan (x) \sec (x)}{2 a}",1,"(-2*(a - 2*b)*Log[Cos[x/2] - Sin[x/2]] + 2*(a - 2*b)*Log[Cos[x/2] + Sin[x/2]] - (2*b^(3/2)*Log[Sqrt[a + b] - Sqrt[b]*Sin[x]])/Sqrt[a + b] + (2*b^(3/2)*Log[Sqrt[a + b] + Sqrt[b]*Sin[x]])/Sqrt[a + b] + a/(Cos[x/2] - Sin[x/2])^2 - a/(Cos[x/2] + Sin[x/2])^2)/(4*a^2)","B",1
34,1,215,90,1.1844818,"\int \frac{\sec ^5(x)}{a+b \cos ^2(x)} \, dx","Integrate[Sec[x]^5/(a + b*Cos[x]^2),x]","\frac{-2 \left(3 a^2-4 a b+8 b^2\right) \log \left(\cos \left(\frac{x}{2}\right)-\sin \left(\frac{x}{2}\right)\right)+2 \left(3 a^2-4 a b+8 b^2\right) \log \left(\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)\right)+\frac{a^2}{\left(\cos \left(\frac{x}{2}\right)-\sin \left(\frac{x}{2}\right)\right)^4}-\frac{a^2}{\left(\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)\right)^4}+\frac{8 b^{5/2} \log \left(\sqrt{a+b}-\sqrt{b} \sin (x)\right)}{\sqrt{a+b}}-\frac{8 b^{5/2} \log \left(\sqrt{a+b}+\sqrt{b} \sin (x)\right)}{\sqrt{a+b}}+\frac{a (4 b-3 a)}{\sin (x)-1}+\frac{a (4 b-3 a)}{\left(\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)\right)^2}}{16 a^3}","-\frac{b^{5/2} \tanh ^{-1}\left(\frac{\sqrt{b} \sin (x)}{\sqrt{a+b}}\right)}{a^3 \sqrt{a+b}}+\frac{(3 a-4 b) \tan (x) \sec (x)}{8 a^2}+\frac{\left(3 a^2-4 a b+8 b^2\right) \tanh ^{-1}(\sin (x))}{8 a^3}+\frac{\tan (x) \sec ^3(x)}{4 a}",1,"(-2*(3*a^2 - 4*a*b + 8*b^2)*Log[Cos[x/2] - Sin[x/2]] + 2*(3*a^2 - 4*a*b + 8*b^2)*Log[Cos[x/2] + Sin[x/2]] + (8*b^(5/2)*Log[Sqrt[a + b] - Sqrt[b]*Sin[x]])/Sqrt[a + b] - (8*b^(5/2)*Log[Sqrt[a + b] + Sqrt[b]*Sin[x]])/Sqrt[a + b] + a^2/(Cos[x/2] - Sin[x/2])^4 - a^2/(Cos[x/2] + Sin[x/2])^4 + (a*(-3*a + 4*b))/(Cos[x/2] + Sin[x/2])^2 + (a*(-3*a + 4*b))/(-1 + Sin[x]))/(16*a^3)","B",1
35,1,76,87,0.2266406,"\int \frac{\cos ^6(x)}{a+b \cos ^2(x)} \, dx","Integrate[Cos[x]^6/(a + b*Cos[x]^2),x]","\frac{-\frac{32 a^{5/2} \tan ^{-1}\left(\frac{\sqrt{a} \tan (x)}{\sqrt{a+b}}\right)}{\sqrt{a+b}}+4 x \left(8 a^2-4 a b+3 b^2\right)-8 b (a-b) \sin (2 x)+b^2 \sin (4 x)}{32 b^3}","\frac{a^{5/2} \tan ^{-1}\left(\frac{\sqrt{a+b} \cot (x)}{\sqrt{a}}\right)}{b^3 \sqrt{a+b}}+\frac{x \left(8 a^2-4 a b+3 b^2\right)}{8 b^3}-\frac{(4 a-3 b) \sin (x) \cos (x)}{8 b^2}+\frac{\sin (x) \cos ^3(x)}{4 b}",1,"(4*(8*a^2 - 4*a*b + 3*b^2)*x - (32*a^(5/2)*ArcTan[(Sqrt[a]*Tan[x])/Sqrt[a + b]])/Sqrt[a + b] - 8*(a - b)*b*Sin[2*x] + b^2*Sin[4*x])/(32*b^3)","A",1
36,1,52,60,0.1271133,"\int \frac{\cos ^4(x)}{a+b \cos ^2(x)} \, dx","Integrate[Cos[x]^4/(a + b*Cos[x]^2),x]","\frac{\frac{4 a^{3/2} \tan ^{-1}\left(\frac{\sqrt{a} \tan (x)}{\sqrt{a+b}}\right)}{\sqrt{a+b}}+2 x (b-2 a)+b \sin (2 x)}{4 b^2}","-\frac{a^{3/2} \tan ^{-1}\left(\frac{\sqrt{a+b} \cot (x)}{\sqrt{a}}\right)}{b^2 \sqrt{a+b}}-\frac{x (2 a-b)}{2 b^2}+\frac{\sin (x) \cos (x)}{2 b}",1,"(2*(-2*a + b)*x + (4*a^(3/2)*ArcTan[(Sqrt[a]*Tan[x])/Sqrt[a + b]])/Sqrt[a + b] + b*Sin[2*x])/(4*b^2)","A",1
37,1,36,38,0.0852422,"\int \frac{\cos ^2(x)}{a+b \cos ^2(x)} \, dx","Integrate[Cos[x]^2/(a + b*Cos[x]^2),x]","\frac{x-\frac{\sqrt{a} \tan ^{-1}\left(\frac{\sqrt{a} \tan (x)}{\sqrt{a+b}}\right)}{\sqrt{a+b}}}{b}","\frac{\sqrt{a} \tan ^{-1}\left(\frac{\sqrt{a+b} \cot (x)}{\sqrt{a}}\right)}{b \sqrt{a+b}}+\frac{x}{b}",1,"(x - (Sqrt[a]*ArcTan[(Sqrt[a]*Tan[x])/Sqrt[a + b]])/Sqrt[a + b])/b","A",1
38,1,29,30,0.0518799,"\int \frac{1}{a+b \cos ^2(x)} \, dx","Integrate[(a + b*Cos[x]^2)^(-1),x]","\frac{\tan ^{-1}\left(\frac{\sqrt{a} \tan (x)}{\sqrt{a+b}}\right)}{\sqrt{a} \sqrt{a+b}}","-\frac{\tan ^{-1}\left(\frac{\sqrt{a+b} \cot (x)}{\sqrt{a}}\right)}{\sqrt{a} \sqrt{a+b}}",1,"ArcTan[(Sqrt[a]*Tan[x])/Sqrt[a + b]]/(Sqrt[a]*Sqrt[a + b])","A",1
39,1,38,37,0.0825744,"\int \frac{\sec ^2(x)}{a+b \cos ^2(x)} \, dx","Integrate[Sec[x]^2/(a + b*Cos[x]^2),x]","\frac{\tan (x)}{a}-\frac{b \tan ^{-1}\left(\frac{\sqrt{a} \tan (x)}{\sqrt{a+b}}\right)}{a^{3/2} \sqrt{a+b}}","\frac{b \tan ^{-1}\left(\frac{\sqrt{a+b} \cot (x)}{\sqrt{a}}\right)}{a^{3/2} \sqrt{a+b}}+\frac{\tan (x)}{a}",1,"-((b*ArcTan[(Sqrt[a]*Tan[x])/Sqrt[a + b]])/(a^(3/2)*Sqrt[a + b])) + Tan[x]/a","A",1
40,1,55,56,0.1503575,"\int \frac{\sec ^4(x)}{a+b \cos ^2(x)} \, dx","Integrate[Sec[x]^4/(a + b*Cos[x]^2),x]","\frac{b^2 \tan ^{-1}\left(\frac{\sqrt{a} \tan (x)}{\sqrt{a+b}}\right)}{a^{5/2} \sqrt{a+b}}+\frac{\tan (x) \left(a \sec ^2(x)+2 a-3 b\right)}{3 a^2}","-\frac{b^2 \tan ^{-1}\left(\frac{\sqrt{a+b} \cot (x)}{\sqrt{a}}\right)}{a^{5/2} \sqrt{a+b}}+\frac{(a-b) \tan (x)}{a^2}+\frac{\tan ^3(x)}{3 a}",1,"(b^2*ArcTan[(Sqrt[a]*Tan[x])/Sqrt[a + b]])/(a^(5/2)*Sqrt[a + b]) + ((2*a - 3*b + a*Sec[x]^2)*Tan[x])/(3*a^2)","A",1
41,1,80,79,0.3472582,"\int \frac{\sec ^6(x)}{a+b \cos ^2(x)} \, dx","Integrate[Sec[x]^6/(a + b*Cos[x]^2),x]","\frac{\tan (x) \left(3 a^2 \sec ^4(x)+8 a^2+a (4 a-5 b) \sec ^2(x)-10 a b+15 b^2\right)}{15 a^3}-\frac{b^3 \tan ^{-1}\left(\frac{\sqrt{a} \tan (x)}{\sqrt{a+b}}\right)}{a^{7/2} \sqrt{a+b}}","\frac{b^3 \tan ^{-1}\left(\frac{\sqrt{a+b} \cot (x)}{\sqrt{a}}\right)}{a^{7/2} \sqrt{a+b}}+\frac{(2 a-b) \tan ^3(x)}{3 a^2}+\frac{\left(a^2-a b+b^2\right) \tan (x)}{a^3}+\frac{\tan ^5(x)}{5 a}",1,"-((b^3*ArcTan[(Sqrt[a]*Tan[x])/Sqrt[a + b]])/(a^(7/2)*Sqrt[a + b])) + ((8*a^2 - 10*a*b + 15*b^2 + a*(4*a - 5*b)*Sec[x]^2 + 3*a^2*Sec[x]^4)*Tan[x])/(15*a^3)","A",1
42,1,70,65,0.2425502,"\int \frac{1}{\left(a+b \cos ^2(x)\right)^2} \, dx","Integrate[(a + b*Cos[x]^2)^(-2),x]","-\frac{(-2 a-b) \tan ^{-1}\left(\frac{\sqrt{a} \tan (x)}{\sqrt{a+b}}\right)}{2 a^{3/2} (a+b)^{3/2}}-\frac{b \sin (2 x)}{2 a (a+b) (2 a+b \cos (2 x)+b)}","-\frac{(2 a+b) \tan ^{-1}\left(\frac{\sqrt{a+b} \cot (x)}{\sqrt{a}}\right)}{2 a^{3/2} (a+b)^{3/2}}-\frac{b \sin (x) \cos (x)}{2 a (a+b) \left(a+b \cos ^2(x)\right)}",1,"-1/2*((-2*a - b)*ArcTan[(Sqrt[a]*Tan[x])/Sqrt[a + b]])/(a^(3/2)*(a + b)^(3/2)) - (b*Sin[2*x])/(2*a*(a + b)*(2*a + b + b*Cos[2*x]))","A",1
43,1,106,107,0.6977462,"\int \frac{1}{\left(a+b \cos ^2(x)\right)^3} \, dx","Integrate[(a + b*Cos[x]^2)^(-3),x]","\frac{\frac{\left(8 a^2+8 a b+3 b^2\right) \tan ^{-1}\left(\frac{\sqrt{a} \tan (x)}{\sqrt{a+b}}\right)}{(a+b)^{5/2}}-\frac{\sqrt{a} b \sin (2 x) \left(16 a^2+3 b (2 a+b) \cos (2 x)+16 a b+3 b^2\right)}{(a+b)^2 (2 a+b \cos (2 x)+b)^2}}{8 a^{5/2}}","-\frac{3 b (2 a+b) \sin (x) \cos (x)}{8 a^2 (a+b)^2 \left(a+b \cos ^2(x)\right)}-\frac{\left(8 a^2+8 a b+3 b^2\right) \tan ^{-1}\left(\frac{\sqrt{a+b} \cot (x)}{\sqrt{a}}\right)}{8 a^{5/2} (a+b)^{5/2}}-\frac{b \sin (x) \cos (x)}{4 a (a+b) \left(a+b \cos ^2(x)\right)^2}",1,"(((8*a^2 + 8*a*b + 3*b^2)*ArcTan[(Sqrt[a]*Tan[x])/Sqrt[a + b]])/(a + b)^(5/2) - (Sqrt[a]*b*(16*a^2 + 16*a*b + 3*b^2 + 3*b*(2*a + b)*Cos[2*x])*Sin[2*x])/((a + b)^2*(2*a + b + b*Cos[2*x])^2))/(8*a^(5/2))","A",1
44,1,15,34,0.0252571,"\int \frac{1}{1+\cos ^2(x)} \, dx","Integrate[(1 + Cos[x]^2)^(-1),x]","\frac{\tan ^{-1}\left(\frac{\tan (x)}{\sqrt{2}}\right)}{\sqrt{2}}","\frac{x}{\sqrt{2}}-\frac{\tan ^{-1}\left(\frac{\sin (x) \cos (x)}{\cos ^2(x)+\sqrt{2}+1}\right)}{\sqrt{2}}",1,"ArcTan[Tan[x]/Sqrt[2]]/Sqrt[2]","A",1
45,1,35,55,0.0794215,"\int \frac{1}{\left(1+\cos ^2(x)\right)^2} \, dx","Integrate[(1 + Cos[x]^2)^(-2),x]","\frac{3 \tan ^{-1}\left(\frac{\tan (x)}{\sqrt{2}}\right)}{4 \sqrt{2}}-\frac{\sin (2 x)}{4 (\cos (2 x)+3)}","\frac{3 x}{4 \sqrt{2}}-\frac{\sin (x) \cos (x)}{4 \left(\cos ^2(x)+1\right)}-\frac{3 \tan ^{-1}\left(\frac{\sin (x) \cos (x)}{\cos ^2(x)+\sqrt{2}+1}\right)}{4 \sqrt{2}}",1,"(3*ArcTan[Tan[x]/Sqrt[2]])/(4*Sqrt[2]) - Sin[2*x]/(4*(3 + Cos[2*x]))","A",1
46,1,51,71,0.1286247,"\int \frac{1}{\left(1+\cos ^2(x)\right)^3} \, dx","Integrate[(1 + Cos[x]^2)^(-3),x]","\frac{19 \tan ^{-1}\left(\frac{\tan (x)}{\sqrt{2}}\right)}{32 \sqrt{2}}-\frac{9 \sin (2 x)}{32 (\cos (2 x)+3)}-\frac{\sin (2 x)}{4 (\cos (2 x)+3)^2}","\frac{19 x}{32 \sqrt{2}}-\frac{9 \sin (x) \cos (x)}{32 \left(\cos ^2(x)+1\right)}-\frac{\sin (x) \cos (x)}{8 \left(\cos ^2(x)+1\right)^2}-\frac{19 \tan ^{-1}\left(\frac{\sin (x) \cos (x)}{\cos ^2(x)+\sqrt{2}+1}\right)}{32 \sqrt{2}}",1,"(19*ArcTan[Tan[x]/Sqrt[2]])/(32*Sqrt[2]) - Sin[2*x]/(4*(3 + Cos[2*x])^2) - (9*Sin[2*x])/(32*(3 + Cos[2*x]))","A",1
47,1,12,12,0.0065764,"\int \sqrt{1-\cos ^2(x)} \, dx","Integrate[Sqrt[1 - Cos[x]^2],x]","\sqrt{\sin ^2(x)} (-\cot (x))","\sqrt{\sin ^2(x)} (-\cot (x))",1,"-(Cot[x]*Sqrt[Sin[x]^2])","A",1
48,1,14,14,0.0057267,"\int \sqrt{-1+\cos ^2(x)} \, dx","Integrate[Sqrt[-1 + Cos[x]^2],x]","\sqrt{-\sin ^2(x)} (-\cot (x))","\sqrt{-\sin ^2(x)} (-\cot (x))",1,"-(Cot[x]*Sqrt[-Sin[x]^2])","A",1
49,1,23,29,0.0269059,"\int \left(1-\cos ^2(x)\right)^{3/2} \, dx","Integrate[(1 - Cos[x]^2)^(3/2),x]","\frac{1}{12} \sqrt{\sin ^2(x)} (\cos (3 x)-9 \cos (x)) \csc (x)","-\frac{1}{3} \sin ^2(x)^{3/2} \cot (x)-\frac{2}{3} \sqrt{\sin ^2(x)} \cot (x)",1,"((-9*Cos[x] + Cos[3*x])*Csc[x]*Sqrt[Sin[x]^2])/12","A",1
50,1,25,33,0.0280931,"\int \left(-1+\cos ^2(x)\right)^{3/2} \, dx","Integrate[(-1 + Cos[x]^2)^(3/2),x]","-\frac{1}{12} \sqrt{-\sin ^2(x)} (\cos (3 x)-9 \cos (x)) \csc (x)","\frac{2}{3} \sqrt{-\sin ^2(x)} \cot (x)-\frac{1}{3} \left(-\sin ^2(x)\right)^{3/2} \cot (x)",1,"-1/12*((-9*Cos[x] + Cos[3*x])*Csc[x]*Sqrt[-Sin[x]^2])","A",1
51,1,28,15,0.0172071,"\int \frac{1}{\sqrt{1-\cos ^2(x)}} \, dx","Integrate[1/Sqrt[1 - Cos[x]^2],x]","\frac{\sin (x) \left(\log \left(\sin \left(\frac{x}{2}\right)\right)-\log \left(\cos \left(\frac{x}{2}\right)\right)\right)}{\sqrt{\sin ^2(x)}}","-\frac{\sin (x) \tanh ^{-1}(\cos (x))}{\sqrt{\sin ^2(x)}}",1,"((-Log[Cos[x/2]] + Log[Sin[x/2]])*Sin[x])/Sqrt[Sin[x]^2]","A",1
52,1,30,17,0.0103041,"\int \frac{1}{\sqrt{-1+\cos ^2(x)}} \, dx","Integrate[1/Sqrt[-1 + Cos[x]^2],x]","\frac{\sin (x) \left(\log \left(\sin \left(\frac{x}{2}\right)\right)-\log \left(\cos \left(\frac{x}{2}\right)\right)\right)}{\sqrt{-\sin ^2(x)}}","-\frac{\sin (x) \tanh ^{-1}(\cos (x))}{\sqrt{-\sin ^2(x)}}",1,"((-Log[Cos[x/2]] + Log[Sin[x/2]])*Sin[x])/Sqrt[-Sin[x]^2]","A",1
53,1,51,32,0.0495898,"\int \frac{1}{\left(1-\cos ^2(x)\right)^{3/2}} \, dx","Integrate[(1 - Cos[x]^2)^(-3/2),x]","-\frac{\sin (x) \left(\csc ^2\left(\frac{x}{2}\right)-\sec ^2\left(\frac{x}{2}\right)-4 \log \left(\sin \left(\frac{x}{2}\right)\right)+4 \log \left(\cos \left(\frac{x}{2}\right)\right)\right)}{8 \sqrt{\sin ^2(x)}}","-\frac{\cot (x)}{2 \sqrt{\sin ^2(x)}}-\frac{\sin (x) \tanh ^{-1}(\cos (x))}{2 \sqrt{\sin ^2(x)}}",1,"-1/8*((Csc[x/2]^2 + 4*Log[Cos[x/2]] - 4*Log[Sin[x/2]] - Sec[x/2]^2)*Sin[x])/Sqrt[Sin[x]^2]","A",1
54,1,53,36,0.0327137,"\int \frac{1}{\left(-1+\cos ^2(x)\right)^{3/2}} \, dx","Integrate[(-1 + Cos[x]^2)^(-3/2),x]","\frac{\sin (x) \left(\csc ^2\left(\frac{x}{2}\right)-\sec ^2\left(\frac{x}{2}\right)-4 \log \left(\sin \left(\frac{x}{2}\right)\right)+4 \log \left(\cos \left(\frac{x}{2}\right)\right)\right)}{8 \sqrt{-\sin ^2(x)}}","\frac{\cot (x)}{2 \sqrt{-\sin ^2(x)}}+\frac{\sin (x) \tanh ^{-1}(\cos (x))}{2 \sqrt{-\sin ^2(x)}}",1,"((Csc[x/2]^2 + 4*Log[Cos[x/2]] - 4*Log[Sin[x/2]] - Sec[x/2]^2)*Sin[x])/(8*Sqrt[-Sin[x]^2])","A",1
55,1,11,9,0.0234642,"\int \sqrt{1+\cos ^2(x)} \, dx","Integrate[Sqrt[1 + Cos[x]^2],x]","\sqrt{2} E\left(x\left|\frac{1}{2}\right.\right)","E\left(\left.x+\frac{\pi }{2}\right|-1\right)",1,"Sqrt[2]*EllipticE[x, 1/2]","A",1
56,1,34,32,0.038837,"\int \sqrt{-1-\cos ^2(x)} \, dx","Integrate[Sqrt[-1 - Cos[x]^2],x]","-\frac{\sqrt{2} \sqrt{\cos (2 x)+3} E\left(x\left|\frac{1}{2}\right.\right)}{\sqrt{-\cos (2 x)-3}}","\frac{\sqrt{-\cos ^2(x)-1} E\left(\left.x+\frac{\pi }{2}\right|-1\right)}{\sqrt{\cos ^2(x)+1}}",1,"-((Sqrt[2]*Sqrt[3 + Cos[2*x]]*EllipticE[x, 1/2])/Sqrt[-3 - Cos[2*x]])","A",1
57,1,46,42,0.0783131,"\int \sqrt{a+b \cos ^2(x)} \, dx","Integrate[Sqrt[a + b*Cos[x]^2],x]","\frac{\sqrt{2 a+b \cos (2 x)+b} E\left(x\left|\frac{b}{a+b}\right.\right)}{\sqrt{\frac{2 a+b \cos (2 x)+b}{a+b}}}","\frac{\sqrt{a+b \cos ^2(x)} E\left(x+\frac{\pi }{2}|-\frac{b}{a}\right)}{\sqrt{\frac{b \cos ^2(x)}{a}+1}}",1,"(Sqrt[2*a + b + b*Cos[2*x]]*EllipticE[x, b/(a + b)])/Sqrt[(2*a + b + b*Cos[2*x])/(a + b)]","A",1
58,1,39,43,0.0524192,"\int \left(1+\cos ^2(x)\right)^{3/2} \, dx","Integrate[(1 + Cos[x]^2)^(3/2),x]","\frac{-4 F\left(x\left|\frac{1}{2}\right.\right)+24 E\left(x\left|\frac{1}{2}\right.\right)+\sin (2 x) \sqrt{\cos (2 x)+3}}{6 \sqrt{2}}","-\frac{2}{3} F\left(\left.x+\frac{\pi }{2}\right|-1\right)+2 E\left(\left.x+\frac{\pi }{2}\right|-1\right)+\frac{1}{3} \sin (x) \cos (x) \sqrt{\cos ^2(x)+1}",1,"(24*EllipticE[x, 1/2] - 4*EllipticF[x, 1/2] + Sqrt[3 + Cos[2*x]]*Sin[2*x])/(6*Sqrt[2])","A",1
59,1,66,89,0.07295,"\int \left(-1-\cos ^2(x)\right)^{3/2} \, dx","Integrate[(-1 - Cos[x]^2)^(3/2),x]","\frac{6 \sin (2 x)+\sin (4 x)-8 \sqrt{\cos (2 x)+3} F\left(x\left|\frac{1}{2}\right.\right)+48 \sqrt{\cos (2 x)+3} E\left(x\left|\frac{1}{2}\right.\right)}{12 \sqrt{2} \sqrt{-\cos (2 x)-3}}","-\frac{1}{3} \sin (x) \cos (x) \sqrt{-\cos ^2(x)-1}-\frac{2 \sqrt{\cos ^2(x)+1} F\left(\left.x+\frac{\pi }{2}\right|-1\right)}{3 \sqrt{-\cos ^2(x)-1}}-\frac{2 \sqrt{-\cos ^2(x)-1} E\left(\left.x+\frac{\pi }{2}\right|-1\right)}{\sqrt{\cos ^2(x)+1}}",1,"(48*Sqrt[3 + Cos[2*x]]*EllipticE[x, 1/2] - 8*Sqrt[3 + Cos[2*x]]*EllipticF[x, 1/2] + 6*Sin[2*x] + Sin[4*x])/(12*Sqrt[2]*Sqrt[-3 - Cos[2*x]])","A",1
60,1,123,121,0.4813389,"\int \left(a+b \cos ^2(x)\right)^{3/2} \, dx","Integrate[(a + b*Cos[x]^2)^(3/2),x]","\frac{8 \left(2 a^2+3 a b+b^2\right) \sqrt{\frac{2 a+b \cos (2 x)+b}{a+b}} E\left(x\left|\frac{b}{a+b}\right.\right)+\sqrt{2} b \sin (2 x) (2 a+b \cos (2 x)+b)-4 a (a+b) \sqrt{\frac{2 a+b \cos (2 x)+b}{a+b}} F\left(x\left|\frac{b}{a+b}\right.\right)}{12 \sqrt{2 a+b \cos (2 x)+b}}","\frac{1}{3} b \sin (x) \cos (x) \sqrt{a+b \cos ^2(x)}-\frac{a (a+b) \sqrt{\frac{b \cos ^2(x)}{a}+1} F\left(x+\frac{\pi }{2}|-\frac{b}{a}\right)}{3 \sqrt{a+b \cos ^2(x)}}+\frac{2 (2 a+b) \sqrt{a+b \cos ^2(x)} E\left(x+\frac{\pi }{2}|-\frac{b}{a}\right)}{3 \sqrt{\frac{b \cos ^2(x)}{a}+1}}",1,"(8*(2*a^2 + 3*a*b + b^2)*Sqrt[(2*a + b + b*Cos[2*x])/(a + b)]*EllipticE[x, b/(a + b)] - 4*a*(a + b)*Sqrt[(2*a + b + b*Cos[2*x])/(a + b)]*EllipticF[x, b/(a + b)] + Sqrt[2]*b*(2*a + b + b*Cos[2*x])*Sin[2*x])/(12*Sqrt[2*a + b + b*Cos[2*x]])","A",1
61,1,11,9,0.0349085,"\int \frac{1}{\sqrt{1+\cos ^2(x)}} \, dx","Integrate[1/Sqrt[1 + Cos[x]^2],x]","\frac{F\left(x\left|\frac{1}{2}\right.\right)}{\sqrt{2}}","F\left(\left.x+\frac{\pi }{2}\right|-1\right)",1,"EllipticF[x, 1/2]/Sqrt[2]","A",1
62,1,33,32,0.0402612,"\int \frac{1}{\sqrt{-1-\cos ^2(x)}} \, dx","Integrate[1/Sqrt[-1 - Cos[x]^2],x]","\frac{\sqrt{\cos (2 x)+3} F\left(x\left|\frac{1}{2}\right.\right)}{\sqrt{2} \sqrt{-\cos (2 x)-3}}","\frac{\sqrt{\cos ^2(x)+1} F\left(\left.x+\frac{\pi }{2}\right|-1\right)}{\sqrt{-\cos ^2(x)-1}}",1,"(Sqrt[3 + Cos[2*x]]*EllipticF[x, 1/2])/(Sqrt[2]*Sqrt[-3 - Cos[2*x]])","A",1
63,1,46,42,0.0711672,"\int \frac{1}{\sqrt{a+b \cos ^2(x)}} \, dx","Integrate[1/Sqrt[a + b*Cos[x]^2],x]","\frac{\sqrt{\frac{2 a+b \cos (2 x)+b}{a+b}} F\left(x\left|\frac{b}{a+b}\right.\right)}{\sqrt{2 a+b \cos (2 x)+b}}","\frac{\sqrt{\frac{b \cos ^2(x)}{a}+1} F\left(x+\frac{\pi }{2}|-\frac{b}{a}\right)}{\sqrt{a+b \cos ^2(x)}}",1,"(Sqrt[(2*a + b + b*Cos[2*x])/(a + b)]*EllipticF[x, b/(a + b)])/Sqrt[2*a + b + b*Cos[2*x]]","A",1
64,1,35,32,0.0631193,"\int \frac{1}{\left(1+\cos ^2(x)\right)^{3/2}} \, dx","Integrate[(1 + Cos[x]^2)^(-3/2),x]","\frac{E\left(x\left|\frac{1}{2}\right.\right)}{\sqrt{2}}-\frac{\sin (2 x)}{2 \sqrt{2} \sqrt{\cos (2 x)+3}}","\frac{1}{2} E\left(\left.x+\frac{\pi }{2}\right|-1\right)-\frac{\sin (x) \cos (x)}{2 \sqrt{\cos ^2(x)+1}}",1,"EllipticE[x, 1/2]/Sqrt[2] - Sin[2*x]/(2*Sqrt[2]*Sqrt[3 + Cos[2*x]])","A",1
65,1,43,56,0.0439787,"\int \frac{1}{\left(-1-\cos ^2(x)\right)^{3/2}} \, dx","Integrate[(-1 - Cos[x]^2)^(-3/2),x]","\frac{\sin (2 x)-2 \sqrt{\cos (2 x)+3} E\left(x\left|\frac{1}{2}\right.\right)}{2 \sqrt{2} \sqrt{-\cos (2 x)-3}}","\frac{\sin (x) \cos (x)}{2 \sqrt{-\cos ^2(x)-1}}+\frac{\sqrt{-\cos ^2(x)-1} E\left(\left.x+\frac{\pi }{2}\right|-1\right)}{2 \sqrt{\cos ^2(x)+1}}",1,"(-2*Sqrt[3 + Cos[2*x]]*EllipticE[x, 1/2] + Sin[2*x])/(2*Sqrt[2]*Sqrt[-3 - Cos[2*x]])","A",1
66,1,75,78,0.1851196,"\int \frac{1}{\left(a+b \cos ^2(x)\right)^{3/2}} \, dx","Integrate[(a + b*Cos[x]^2)^(-3/2),x]","\frac{2 (a+b) \sqrt{\frac{2 a+b \cos (2 x)+b}{a+b}} E\left(x\left|\frac{b}{a+b}\right.\right)-\sqrt{2} b \sin (2 x)}{2 a (a+b) \sqrt{2 a+b \cos (2 x)+b}}","\frac{\sqrt{a+b \cos ^2(x)} E\left(x+\frac{\pi }{2}|-\frac{b}{a}\right)}{a (a+b) \sqrt{\frac{b \cos ^2(x)}{a}+1}}-\frac{b \sin (x) \cos (x)}{a (a+b) \sqrt{a+b \cos ^2(x)}}",1,"(2*(a + b)*Sqrt[(2*a + b + b*Cos[2*x])/(a + b)]*EllipticE[x, b/(a + b)] - Sqrt[2]*b*Sin[2*x])/(2*a*(a + b)*Sqrt[2*a + b + b*Cos[2*x]])","A",1
67,1,9,9,0.0077094,"\int \frac{\cos (x)}{\sqrt{1+\cos ^2(x)}} \, dx","Integrate[Cos[x]/Sqrt[1 + Cos[x]^2],x]","\sin ^{-1}\left(\frac{\sin (x)}{\sqrt{2}}\right)","\sin ^{-1}\left(\frac{\sin (x)}{\sqrt{2}}\right)",1,"ArcSin[Sin[x]/Sqrt[2]]","A",1
68,1,15,15,0.0259648,"\int \frac{\cos (5+3 x)}{\sqrt{3+\cos ^2(5+3 x)}} \, dx","Integrate[Cos[5 + 3*x]/Sqrt[3 + Cos[5 + 3*x]^2],x]","\frac{1}{3} \sin ^{-1}\left(\frac{1}{2} \sin (3 x+5)\right)","\frac{1}{3} \sin ^{-1}\left(\frac{1}{2} \sin (3 x+5)\right)",1,"ArcSin[Sin[5 + 3*x]/2]/3","A",1
69,1,9,9,0.009602,"\int \frac{\cos (x)}{\sqrt{4-\cos ^2(x)}} \, dx","Integrate[Cos[x]/Sqrt[4 - Cos[x]^2],x]","\sinh ^{-1}\left(\frac{\sin (x)}{\sqrt{3}}\right)","\sinh ^{-1}\left(\frac{\sin (x)}{\sqrt{3}}\right)",1,"ArcSinh[Sin[x]/Sqrt[3]]","A",1
70,1,121,487,0.2419376,"\int \frac{1}{a+b \cos ^4(x)} \, dx","Integrate[(a + b*Cos[x]^4)^(-1),x]","\frac{\tan ^{-1}\left(\frac{\sqrt{a} \tan (x)}{\sqrt{a+i \sqrt{a} \sqrt{b}}}\right)}{2 \sqrt{a} \sqrt{a+i \sqrt{a} \sqrt{b}}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{a} \tan (x)}{\sqrt{-a+i \sqrt{a} \sqrt{b}}}\right)}{2 \sqrt{a} \sqrt{-a+i \sqrt{a} \sqrt{b}}}","-\frac{\left(\sqrt{a}-\sqrt{a+b}\right) \log \left((a+b)^{3/4} \cot ^2(x)-\sqrt{2} \sqrt[4]{a} \sqrt{-\sqrt{a} \sqrt{a+b}+a+b} \cot (x)+\sqrt{a} \sqrt[4]{a+b}\right)}{4 \sqrt{2} a^{3/4} \sqrt[4]{a+b} \sqrt{-\sqrt{a} \sqrt{a+b}+a+b}}+\frac{\left(\sqrt{a}-\sqrt{a+b}\right) \log \left((a+b)^{3/4} \cot ^2(x)+\sqrt{2} \sqrt[4]{a} \sqrt{-\sqrt{a} \sqrt{a+b}+a+b} \cot (x)+\sqrt{a} \sqrt[4]{a+b}\right)}{4 \sqrt{2} a^{3/4} \sqrt[4]{a+b} \sqrt{-\sqrt{a} \sqrt{a+b}+a+b}}+\frac{\left(\sqrt{a+b}+\sqrt{a}\right) \tan ^{-1}\left(\frac{\sqrt[4]{a} \sqrt{-\sqrt{a} \sqrt{a+b}+a+b}-\sqrt{2} (a+b)^{3/4} \cot (x)}{\sqrt[4]{a} \sqrt{\sqrt{a} \sqrt{a+b}+a+b}}\right)}{2 \sqrt{2} a^{3/4} \sqrt[4]{a+b} \sqrt{\sqrt{a} \sqrt{a+b}+a+b}}-\frac{\left(\sqrt{a+b}+\sqrt{a}\right) \tan ^{-1}\left(\frac{\sqrt{2} (a+b)^{3/4} \cot (x)+\sqrt[4]{a} \sqrt{-\sqrt{a} \sqrt{a+b}+a+b}}{\sqrt[4]{a} \sqrt{\sqrt{a} \sqrt{a+b}+a+b}}\right)}{2 \sqrt{2} a^{3/4} \sqrt[4]{a+b} \sqrt{\sqrt{a} \sqrt{a+b}+a+b}}",1,"ArcTan[(Sqrt[a]*Tan[x])/Sqrt[a + I*Sqrt[a]*Sqrt[b]]]/(2*Sqrt[a]*Sqrt[a + I*Sqrt[a]*Sqrt[b]]) - ArcTanh[(Sqrt[a]*Tan[x])/Sqrt[-a + I*Sqrt[a]*Sqrt[b]]]/(2*Sqrt[a]*Sqrt[-a + I*Sqrt[a]*Sqrt[b]])","C",1
71,1,109,101,0.1975833,"\int \frac{1}{a-b \cos ^4(x)} \, dx","Integrate[(a - b*Cos[x]^4)^(-1),x]","\frac{\tan ^{-1}\left(\frac{\sqrt{a} \tan (x)}{\sqrt{\sqrt{a} \sqrt{b}+a}}\right)}{2 \sqrt{a} \sqrt{\sqrt{a} \sqrt{b}+a}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{a} \tan (x)}{\sqrt{\sqrt{a} \sqrt{b}-a}}\right)}{2 \sqrt{a} \sqrt{\sqrt{a} \sqrt{b}-a}}","-\frac{\tan ^{-1}\left(\frac{\sqrt{\sqrt{a}-\sqrt{b}} \cot (x)}{\sqrt[4]{a}}\right)}{2 a^{3/4} \sqrt{\sqrt{a}-\sqrt{b}}}-\frac{\tan ^{-1}\left(\frac{\sqrt{\sqrt{a}+\sqrt{b}} \cot (x)}{\sqrt[4]{a}}\right)}{2 a^{3/4} \sqrt{\sqrt{a}+\sqrt{b}}}",1,"ArcTan[(Sqrt[a]*Tan[x])/Sqrt[a + Sqrt[a]*Sqrt[b]]]/(2*Sqrt[a]*Sqrt[a + Sqrt[a]*Sqrt[b]]) - ArcTanh[(Sqrt[a]*Tan[x])/Sqrt[-a + Sqrt[a]*Sqrt[b]]]/(2*Sqrt[a]*Sqrt[-a + Sqrt[a]*Sqrt[b]])","A",1
72,1,45,292,0.0770131,"\int \frac{1}{1+\cos ^4(x)} \, dx","Integrate[(1 + Cos[x]^4)^(-1),x]","\frac{\tan ^{-1}\left(\frac{\tan (x)}{\sqrt{1-i}}\right)}{2 \sqrt{1-i}}+\frac{\tan ^{-1}\left(\frac{\tan (x)}{\sqrt{1+i}}\right)}{2 \sqrt{1+i}}","\frac{x}{2 \sqrt{\sqrt{2}-1}}+\frac{1}{8} \sqrt{\sqrt{2}-1} \log \left(2 \cot ^2(x)-2 \sqrt{\sqrt{2}-1} \cot (x)+\sqrt{2}\right)-\frac{1}{8} \sqrt{\sqrt{2}-1} \log \left(\sqrt{2} \cot ^2(x)+\sqrt{2 \left(\sqrt{2}-1\right)} \cot (x)+1\right)+\frac{\tan ^{-1}\left(\frac{\sqrt{\sqrt{2}-1} \left(1-2 \sin ^2(x)\right)+\left(\sqrt{2}-2\right) \sin (x) \cos (x)}{\left(\sqrt{2}-2\right) \sin ^2(x)+2 \sqrt{\sqrt{2}-1} \sin (x) \cos (x)+\sqrt{1+\sqrt{2}}+2}\right)}{4 \sqrt{\sqrt{2}-1}}+\frac{\tan ^{-1}\left(\frac{\sqrt{\sqrt{2}-1} \left(2 \sin ^2(x)-1\right)+\left(\sqrt{2}-2\right) \sin (x) \cos (x)}{\left(\sqrt{2}-2\right) \sin ^2(x)-2 \sqrt{\sqrt{2}-1} \sin (x) \cos (x)+\sqrt{1+\sqrt{2}}+2}\right)}{4 \sqrt{\sqrt{2}-1}}",1,"ArcTan[Tan[x]/Sqrt[1 - I]]/(2*Sqrt[1 - I]) + ArcTan[Tan[x]/Sqrt[1 + I]]/(2*Sqrt[1 + I])","C",1
73,1,24,45,0.05816,"\int \frac{1}{1-\cos ^4(x)} \, dx","Integrate[(1 - Cos[x]^4)^(-1),x]","\frac{1}{4} \left(\sqrt{2} \tan ^{-1}\left(\frac{\tan (x)}{\sqrt{2}}\right)-2 \cot (x)\right)","\frac{x}{2 \sqrt{2}}-\frac{\cot (x)}{2}-\frac{\tan ^{-1}\left(\frac{\sin (x) \cos (x)}{\cos ^2(x)+\sqrt{2}+1}\right)}{2 \sqrt{2}}",1,"(Sqrt[2]*ArcTan[Tan[x]/Sqrt[2]] - 2*Cot[x])/4","A",1
74,1,130,494,0.2494475,"\int \frac{1}{a+b \cos ^5(x)} \, dx","Integrate[(a + b*Cos[x]^5)^(-1),x]","\frac{8}{5} \text{RootSum}\left[\text{$\#$1}^{10} b+5 \text{$\#$1}^8 b+10 \text{$\#$1}^6 b+32 \text{$\#$1}^5 a+10 \text{$\#$1}^4 b+5 \text{$\#$1}^2 b+b\&,\frac{2 \text{$\#$1}^3 \tan ^{-1}\left(\frac{\sin (x)}{\cos (x)-\text{$\#$1}}\right)-i \text{$\#$1}^3 \log \left(\text{$\#$1}^2-2 \text{$\#$1} \cos (x)+1\right)}{\text{$\#$1}^8 b+4 \text{$\#$1}^6 b+6 \text{$\#$1}^4 b+16 \text{$\#$1}^3 a+4 \text{$\#$1}^2 b+b}\&\right]","\frac{2 \tan ^{-1}\left(\frac{\sqrt{\sqrt[5]{a}-\sqrt[5]{b}} \tan \left(\frac{x}{2}\right)}{\sqrt{\sqrt[5]{a}+\sqrt[5]{b}}}\right)}{5 a^{4/5} \sqrt{\sqrt[5]{a}-\sqrt[5]{b}} \sqrt{\sqrt[5]{a}+\sqrt[5]{b}}}+\frac{2 \tan ^{-1}\left(\frac{\sqrt{\sqrt[5]{a}+\sqrt[5]{-1} \sqrt[5]{b}} \tan \left(\frac{x}{2}\right)}{\sqrt{\sqrt[5]{a}-\sqrt[5]{-1} \sqrt[5]{b}}}\right)}{5 a^{4/5} \sqrt{\sqrt[5]{a}-\sqrt[5]{-1} \sqrt[5]{b}} \sqrt{\sqrt[5]{a}+\sqrt[5]{-1} \sqrt[5]{b}}}+\frac{2 \tan ^{-1}\left(\frac{\sqrt{\sqrt[5]{a}-(-1)^{2/5} \sqrt[5]{b}} \tan \left(\frac{x}{2}\right)}{\sqrt{\sqrt[5]{a}+(-1)^{2/5} \sqrt[5]{b}}}\right)}{5 a^{4/5} \sqrt{\sqrt[5]{a}-(-1)^{2/5} \sqrt[5]{b}} \sqrt{\sqrt[5]{a}+(-1)^{2/5} \sqrt[5]{b}}}+\frac{2 \tan ^{-1}\left(\frac{\sqrt{\sqrt[5]{a}+(-1)^{3/5} \sqrt[5]{b}} \tan \left(\frac{x}{2}\right)}{\sqrt{\sqrt[5]{a}-(-1)^{3/5} \sqrt[5]{b}}}\right)}{5 a^{4/5} \sqrt{\sqrt[5]{a}-(-1)^{3/5} \sqrt[5]{b}} \sqrt{\sqrt[5]{a}+(-1)^{3/5} \sqrt[5]{b}}}+\frac{2 \tan ^{-1}\left(\frac{\sqrt{\sqrt[5]{a}-(-1)^{4/5} \sqrt[5]{b}} \tan \left(\frac{x}{2}\right)}{\sqrt{\sqrt[5]{a}+(-1)^{4/5} \sqrt[5]{b}}}\right)}{5 a^{4/5} \sqrt{\sqrt[5]{a}-(-1)^{4/5} \sqrt[5]{b}} \sqrt{\sqrt[5]{a}+(-1)^{4/5} \sqrt[5]{b}}}",1,"(8*RootSum[b + 5*b*#1^2 + 10*b*#1^4 + 32*a*#1^5 + 10*b*#1^6 + 5*b*#1^8 + b*#1^10 & , (2*ArcTan[Sin[x]/(Cos[x] - #1)]*#1^3 - I*Log[1 - 2*Cos[x]*#1 + #1^2]*#1^3)/(b + 4*b*#1^2 + 16*a*#1^3 + 6*b*#1^4 + 4*b*#1^6 + b*#1^8) & ])/5","C",0
75,1,146,171,0.2275425,"\int \frac{1}{a+b \cos ^6(x)} \, dx","Integrate[(a + b*Cos[x]^6)^(-1),x]","\frac{8}{3} \text{RootSum}\left[\text{$\#$1}^6 b+6 \text{$\#$1}^5 b+15 \text{$\#$1}^4 b+64 \text{$\#$1}^3 a+20 \text{$\#$1}^3 b+15 \text{$\#$1}^2 b+6 \text{$\#$1} b+b\&,\frac{2 \text{$\#$1}^2 \tan ^{-1}\left(\frac{\sin (2 x)}{\cos (2 x)-\text{$\#$1}}\right)-i \text{$\#$1}^2 \log \left(\text{$\#$1}^2-2 \text{$\#$1} \cos (2 x)+1\right)}{\text{$\#$1}^5 b+5 \text{$\#$1}^4 b+10 \text{$\#$1}^3 b+32 \text{$\#$1}^2 a+10 \text{$\#$1}^2 b+5 \text{$\#$1} b+b}\&\right]","-\frac{\tan ^{-1}\left(\frac{\sqrt{\sqrt[3]{a}+\sqrt[3]{b}} \cot (x)}{\sqrt[6]{a}}\right)}{3 a^{5/6} \sqrt{\sqrt[3]{a}+\sqrt[3]{b}}}-\frac{\tan ^{-1}\left(\frac{\sqrt{\sqrt[3]{a}-\sqrt[3]{-1} \sqrt[3]{b}} \cot (x)}{\sqrt[6]{a}}\right)}{3 a^{5/6} \sqrt{\sqrt[3]{a}-\sqrt[3]{-1} \sqrt[3]{b}}}-\frac{\tan ^{-1}\left(\frac{\sqrt{\sqrt[3]{a}+(-1)^{2/3} \sqrt[3]{b}} \cot (x)}{\sqrt[6]{a}}\right)}{3 a^{5/6} \sqrt{\sqrt[3]{a}+(-1)^{2/3} \sqrt[3]{b}}}",1,"(8*RootSum[b + 6*b*#1 + 15*b*#1^2 + 64*a*#1^3 + 20*b*#1^3 + 15*b*#1^4 + 6*b*#1^5 + b*#1^6 & , (2*ArcTan[Sin[2*x]/(Cos[2*x] - #1)]*#1^2 - I*Log[1 - 2*Cos[2*x]*#1 + #1^2]*#1^2)/(b + 5*b*#1 + 32*a*#1^2 + 10*b*#1^2 + 10*b*#1^3 + 5*b*#1^4 + b*#1^5) & ])/3","C",1
76,1,172,245,0.2742261,"\int \frac{1}{a+b \cos ^8(x)} \, dx","Integrate[(a + b*Cos[x]^8)^(-1),x]","8 \text{RootSum}\left[\text{$\#$1}^8 b+8 \text{$\#$1}^7 b+28 \text{$\#$1}^6 b+56 \text{$\#$1}^5 b+256 \text{$\#$1}^4 a+70 \text{$\#$1}^4 b+56 \text{$\#$1}^3 b+28 \text{$\#$1}^2 b+8 \text{$\#$1} b+b\&,\frac{2 \text{$\#$1}^3 \tan ^{-1}\left(\frac{\sin (2 x)}{\cos (2 x)-\text{$\#$1}}\right)-i \text{$\#$1}^3 \log \left(\text{$\#$1}^2-2 \text{$\#$1} \cos (2 x)+1\right)}{\text{$\#$1}^7 b+7 \text{$\#$1}^6 b+21 \text{$\#$1}^5 b+35 \text{$\#$1}^4 b+128 \text{$\#$1}^3 a+35 \text{$\#$1}^3 b+21 \text{$\#$1}^2 b+7 \text{$\#$1} b+b}\&\right]","\frac{\tan ^{-1}\left(\frac{\sqrt{\sqrt[4]{-a}-\sqrt[4]{b}} \cot (x)}{\sqrt[8]{-a}}\right)}{4 (-a)^{7/8} \sqrt{\sqrt[4]{-a}-\sqrt[4]{b}}}+\frac{\tan ^{-1}\left(\frac{\sqrt{\sqrt[4]{-a}-i \sqrt[4]{b}} \cot (x)}{\sqrt[8]{-a}}\right)}{4 (-a)^{7/8} \sqrt{\sqrt[4]{-a}-i \sqrt[4]{b}}}+\frac{\tan ^{-1}\left(\frac{\sqrt{\sqrt[4]{-a}+i \sqrt[4]{b}} \cot (x)}{\sqrt[8]{-a}}\right)}{4 (-a)^{7/8} \sqrt{\sqrt[4]{-a}+i \sqrt[4]{b}}}+\frac{\tan ^{-1}\left(\frac{\sqrt{\sqrt[4]{-a}+\sqrt[4]{b}} \cot (x)}{\sqrt[8]{-a}}\right)}{4 (-a)^{7/8} \sqrt{\sqrt[4]{-a}+\sqrt[4]{b}}}",1,"8*RootSum[b + 8*b*#1 + 28*b*#1^2 + 56*b*#1^3 + 256*a*#1^4 + 70*b*#1^4 + 56*b*#1^5 + 28*b*#1^6 + 8*b*#1^7 + b*#1^8 & , (2*ArcTan[Sin[2*x]/(Cos[2*x] - #1)]*#1^3 - I*Log[1 - 2*Cos[2*x]*#1 + #1^2]*#1^3)/(b + 7*b*#1 + 21*b*#1^2 + 128*a*#1^3 + 35*b*#1^3 + 35*b*#1^4 + 21*b*#1^5 + 7*b*#1^6 + b*#1^7) & ]","C",0
77,1,130,494,0.18484,"\int \frac{1}{a-b \cos ^5(x)} \, dx","Integrate[(a - b*Cos[x]^5)^(-1),x]","-\frac{8}{5} \text{RootSum}\left[\text{$\#$1}^{10} b+5 \text{$\#$1}^8 b+10 \text{$\#$1}^6 b-32 \text{$\#$1}^5 a+10 \text{$\#$1}^4 b+5 \text{$\#$1}^2 b+b\&,\frac{2 \text{$\#$1}^3 \tan ^{-1}\left(\frac{\sin (x)}{\cos (x)-\text{$\#$1}}\right)-i \text{$\#$1}^3 \log \left(\text{$\#$1}^2-2 \text{$\#$1} \cos (x)+1\right)}{\text{$\#$1}^8 b+4 \text{$\#$1}^6 b+6 \text{$\#$1}^4 b-16 \text{$\#$1}^3 a+4 \text{$\#$1}^2 b+b}\&\right]","\frac{2 \tan ^{-1}\left(\frac{\sqrt{\sqrt[5]{a}+\sqrt[5]{b}} \tan \left(\frac{x}{2}\right)}{\sqrt{\sqrt[5]{a}-\sqrt[5]{b}}}\right)}{5 a^{4/5} \sqrt{\sqrt[5]{a}-\sqrt[5]{b}} \sqrt{\sqrt[5]{a}+\sqrt[5]{b}}}+\frac{2 \tan ^{-1}\left(\frac{\sqrt{\sqrt[5]{a}-\sqrt[5]{-1} \sqrt[5]{b}} \tan \left(\frac{x}{2}\right)}{\sqrt{\sqrt[5]{a}+\sqrt[5]{-1} \sqrt[5]{b}}}\right)}{5 a^{4/5} \sqrt{\sqrt[5]{a}-\sqrt[5]{-1} \sqrt[5]{b}} \sqrt{\sqrt[5]{a}+\sqrt[5]{-1} \sqrt[5]{b}}}+\frac{2 \tan ^{-1}\left(\frac{\sqrt{\sqrt[5]{a}+(-1)^{2/5} \sqrt[5]{b}} \tan \left(\frac{x}{2}\right)}{\sqrt{\sqrt[5]{a}-(-1)^{2/5} \sqrt[5]{b}}}\right)}{5 a^{4/5} \sqrt{\sqrt[5]{a}-(-1)^{2/5} \sqrt[5]{b}} \sqrt{\sqrt[5]{a}+(-1)^{2/5} \sqrt[5]{b}}}+\frac{2 \tan ^{-1}\left(\frac{\sqrt{\sqrt[5]{a}-(-1)^{3/5} \sqrt[5]{b}} \tan \left(\frac{x}{2}\right)}{\sqrt{\sqrt[5]{a}+(-1)^{3/5} \sqrt[5]{b}}}\right)}{5 a^{4/5} \sqrt{\sqrt[5]{a}-(-1)^{3/5} \sqrt[5]{b}} \sqrt{\sqrt[5]{a}+(-1)^{3/5} \sqrt[5]{b}}}+\frac{2 \tan ^{-1}\left(\frac{\sqrt{\sqrt[5]{a}+(-1)^{4/5} \sqrt[5]{b}} \tan \left(\frac{x}{2}\right)}{\sqrt{\sqrt[5]{a}-(-1)^{4/5} \sqrt[5]{b}}}\right)}{5 a^{4/5} \sqrt{\sqrt[5]{a}-(-1)^{4/5} \sqrt[5]{b}} \sqrt{\sqrt[5]{a}+(-1)^{4/5} \sqrt[5]{b}}}",1,"(-8*RootSum[b + 5*b*#1^2 + 10*b*#1^4 - 32*a*#1^5 + 10*b*#1^6 + 5*b*#1^8 + b*#1^10 & , (2*ArcTan[Sin[x]/(Cos[x] - #1)]*#1^3 - I*Log[1 - 2*Cos[x]*#1 + #1^2]*#1^3)/(b + 4*b*#1^2 - 16*a*#1^3 + 6*b*#1^4 + 4*b*#1^6 + b*#1^8) & ])/5","C",0
78,1,146,175,0.1803532,"\int \frac{1}{a-b \cos ^6(x)} \, dx","Integrate[(a - b*Cos[x]^6)^(-1),x]","-\frac{8}{3} \text{RootSum}\left[\text{$\#$1}^6 b+6 \text{$\#$1}^5 b+15 \text{$\#$1}^4 b-64 \text{$\#$1}^3 a+20 \text{$\#$1}^3 b+15 \text{$\#$1}^2 b+6 \text{$\#$1} b+b\&,\frac{2 \text{$\#$1}^2 \tan ^{-1}\left(\frac{\sin (2 x)}{\cos (2 x)-\text{$\#$1}}\right)-i \text{$\#$1}^2 \log \left(\text{$\#$1}^2-2 \text{$\#$1} \cos (2 x)+1\right)}{\text{$\#$1}^5 b+5 \text{$\#$1}^4 b+10 \text{$\#$1}^3 b-32 \text{$\#$1}^2 a+10 \text{$\#$1}^2 b+5 \text{$\#$1} b+b}\&\right]","-\frac{\tan ^{-1}\left(\frac{\sqrt{\sqrt[3]{a}-\sqrt[3]{b}} \cot (x)}{\sqrt[6]{a}}\right)}{3 a^{5/6} \sqrt{\sqrt[3]{a}-\sqrt[3]{b}}}-\frac{\tan ^{-1}\left(\frac{\sqrt{\sqrt[3]{a}+\sqrt[3]{-1} \sqrt[3]{b}} \cot (x)}{\sqrt[6]{a}}\right)}{3 a^{5/6} \sqrt{\sqrt[3]{a}+\sqrt[3]{-1} \sqrt[3]{b}}}-\frac{\tan ^{-1}\left(\frac{\sqrt{\sqrt[3]{a}-(-1)^{2/3} \sqrt[3]{b}} \cot (x)}{\sqrt[6]{a}}\right)}{3 a^{5/6} \sqrt{\sqrt[3]{a}-(-1)^{2/3} \sqrt[3]{b}}}",1,"(-8*RootSum[b + 6*b*#1 + 15*b*#1^2 - 64*a*#1^3 + 20*b*#1^3 + 15*b*#1^4 + 6*b*#1^5 + b*#1^6 & , (2*ArcTan[Sin[2*x]/(Cos[2*x] - #1)]*#1^2 - I*Log[1 - 2*Cos[2*x]*#1 + #1^2]*#1^2)/(b + 5*b*#1 - 32*a*#1^2 + 10*b*#1^2 + 10*b*#1^3 + 5*b*#1^4 + b*#1^5) & ])/3","C",1
79,1,172,213,0.2241662,"\int \frac{1}{a-b \cos ^8(x)} \, dx","Integrate[(a - b*Cos[x]^8)^(-1),x]","-8 \text{RootSum}\left[\text{$\#$1}^8 b+8 \text{$\#$1}^7 b+28 \text{$\#$1}^6 b+56 \text{$\#$1}^5 b-256 \text{$\#$1}^4 a+70 \text{$\#$1}^4 b+56 \text{$\#$1}^3 b+28 \text{$\#$1}^2 b+8 \text{$\#$1} b+b\&,\frac{2 \text{$\#$1}^3 \tan ^{-1}\left(\frac{\sin (2 x)}{\cos (2 x)-\text{$\#$1}}\right)-i \text{$\#$1}^3 \log \left(\text{$\#$1}^2-2 \text{$\#$1} \cos (2 x)+1\right)}{\text{$\#$1}^7 b+7 \text{$\#$1}^6 b+21 \text{$\#$1}^5 b+35 \text{$\#$1}^4 b-128 \text{$\#$1}^3 a+35 \text{$\#$1}^3 b+21 \text{$\#$1}^2 b+7 \text{$\#$1} b+b}\&\right]","-\frac{\tan ^{-1}\left(\frac{\sqrt{\sqrt[4]{a}-\sqrt[4]{b}} \cot (x)}{\sqrt[8]{a}}\right)}{4 a^{7/8} \sqrt{\sqrt[4]{a}-\sqrt[4]{b}}}-\frac{\tan ^{-1}\left(\frac{\sqrt{\sqrt[4]{a}-i \sqrt[4]{b}} \cot (x)}{\sqrt[8]{a}}\right)}{4 a^{7/8} \sqrt{\sqrt[4]{a}-i \sqrt[4]{b}}}-\frac{\tan ^{-1}\left(\frac{\sqrt{\sqrt[4]{a}+i \sqrt[4]{b}} \cot (x)}{\sqrt[8]{a}}\right)}{4 a^{7/8} \sqrt{\sqrt[4]{a}+i \sqrt[4]{b}}}-\frac{\tan ^{-1}\left(\frac{\sqrt{\sqrt[4]{a}+\sqrt[4]{b}} \cot (x)}{\sqrt[8]{a}}\right)}{4 a^{7/8} \sqrt{\sqrt[4]{a}+\sqrt[4]{b}}}",1,"-8*RootSum[b + 8*b*#1 + 28*b*#1^2 + 56*b*#1^3 - 256*a*#1^4 + 70*b*#1^4 + 56*b*#1^5 + 28*b*#1^6 + 8*b*#1^7 + b*#1^8 & , (2*ArcTan[Sin[2*x]/(Cos[2*x] - #1)]*#1^3 - I*Log[1 - 2*Cos[2*x]*#1 + #1^2]*#1^3)/(b + 7*b*#1 + 21*b*#1^2 - 128*a*#1^3 + 35*b*#1^3 + 35*b*#1^4 + 21*b*#1^5 + 7*b*#1^6 + b*#1^7) & ]","C",0
80,1,378,223,0.1265149,"\int \frac{1}{1+\cos ^5(x)} \, dx","Integrate[(1 + Cos[x]^5)^(-1),x]","\frac{1}{5} \tan \left(\frac{x}{2}\right)-\frac{1}{10} \text{RootSum}\left[\text{$\#$1}^8-2 \text{$\#$1}^7+8 \text{$\#$1}^6-14 \text{$\#$1}^5+30 \text{$\#$1}^4-14 \text{$\#$1}^3+8 \text{$\#$1}^2-2 \text{$\#$1}+1\&,\frac{2 \text{$\#$1}^6 \tan ^{-1}\left(\frac{\sin (x)}{\cos (x)-\text{$\#$1}}\right)-8 \text{$\#$1}^5 \tan ^{-1}\left(\frac{\sin (x)}{\cos (x)-\text{$\#$1}}\right)+30 \text{$\#$1}^4 \tan ^{-1}\left(\frac{\sin (x)}{\cos (x)-\text{$\#$1}}\right)-80 \text{$\#$1}^3 \tan ^{-1}\left(\frac{\sin (x)}{\cos (x)-\text{$\#$1}}\right)-15 i \text{$\#$1}^2 \log \left(\text{$\#$1}^2-2 \text{$\#$1} \cos (x)+1\right)+4 i \text{$\#$1} \log \left(\text{$\#$1}^2-2 \text{$\#$1} \cos (x)+1\right)-i \log \left(\text{$\#$1}^2-2 \text{$\#$1} \cos (x)+1\right)+30 \text{$\#$1}^2 \tan ^{-1}\left(\frac{\sin (x)}{\cos (x)-\text{$\#$1}}\right)-i \text{$\#$1}^6 \log \left(\text{$\#$1}^2-2 \text{$\#$1} \cos (x)+1\right)+4 i \text{$\#$1}^5 \log \left(\text{$\#$1}^2-2 \text{$\#$1} \cos (x)+1\right)-15 i \text{$\#$1}^4 \log \left(\text{$\#$1}^2-2 \text{$\#$1} \cos (x)+1\right)+40 i \text{$\#$1}^3 \log \left(\text{$\#$1}^2-2 \text{$\#$1} \cos (x)+1\right)-8 \text{$\#$1} \tan ^{-1}\left(\frac{\sin (x)}{\cos (x)-\text{$\#$1}}\right)+2 \tan ^{-1}\left(\frac{\sin (x)}{\cos (x)-\text{$\#$1}}\right)}{4 \text{$\#$1}^7-7 \text{$\#$1}^6+24 \text{$\#$1}^5-35 \text{$\#$1}^4+60 \text{$\#$1}^3-21 \text{$\#$1}^2+8 \text{$\#$1}-1}\&\right]","\frac{2 \tan ^{-1}\left(\sqrt{\frac{1-(-1)^{2/5}}{1+(-1)^{2/5}}} \tan \left(\frac{x}{2}\right)\right)}{5 \sqrt{1-(-1)^{4/5}}}+\frac{2 \tan ^{-1}\left(\sqrt{\frac{1-(-1)^{4/5}}{1+(-1)^{4/5}}} \tan \left(\frac{x}{2}\right)\right)}{5 \sqrt{1+(-1)^{3/5}}}+\frac{\sin (x)}{5 (\cos (x)+1)}-\frac{2 \tanh ^{-1}\left(\frac{\tan \left(\frac{x}{2}\right)}{\sqrt{-\frac{1-\sqrt[5]{-1}}{1+\sqrt[5]{-1}}}}\right)}{5 \sqrt{(-1)^{2/5}-1}}-\frac{2 \sqrt{-\frac{1+(-1)^{3/5}}{1-(-1)^{3/5}}} \tanh ^{-1}\left(\sqrt{-\frac{1+(-1)^{3/5}}{1-(-1)^{3/5}}} \tan \left(\frac{x}{2}\right)\right)}{5 \left(1+(-1)^{3/5}\right)}",1,"-1/10*RootSum[1 - 2*#1 + 8*#1^2 - 14*#1^3 + 30*#1^4 - 14*#1^5 + 8*#1^6 - 2*#1^7 + #1^8 & , (2*ArcTan[Sin[x]/(Cos[x] - #1)] - I*Log[1 - 2*Cos[x]*#1 + #1^2] - 8*ArcTan[Sin[x]/(Cos[x] - #1)]*#1 + (4*I)*Log[1 - 2*Cos[x]*#1 + #1^2]*#1 + 30*ArcTan[Sin[x]/(Cos[x] - #1)]*#1^2 - (15*I)*Log[1 - 2*Cos[x]*#1 + #1^2]*#1^2 - 80*ArcTan[Sin[x]/(Cos[x] - #1)]*#1^3 + (40*I)*Log[1 - 2*Cos[x]*#1 + #1^2]*#1^3 + 30*ArcTan[Sin[x]/(Cos[x] - #1)]*#1^4 - (15*I)*Log[1 - 2*Cos[x]*#1 + #1^2]*#1^4 - 8*ArcTan[Sin[x]/(Cos[x] - #1)]*#1^5 + (4*I)*Log[1 - 2*Cos[x]*#1 + #1^2]*#1^5 + 2*ArcTan[Sin[x]/(Cos[x] - #1)]*#1^6 - I*Log[1 - 2*Cos[x]*#1 + #1^2]*#1^6)/(-1 + 8*#1 - 21*#1^2 + 60*#1^3 - 35*#1^4 + 24*#1^5 - 7*#1^6 + 4*#1^7) & ] + Tan[x/2]/5","C",1
81,1,79,83,0.1745296,"\int \frac{1}{1+\cos ^6(x)} \, dx","Integrate[(1 + Cos[x]^6)^(-1),x]","\frac{1}{12} \left(-2 \sqrt{3} \tan ^{-1}\left(\frac{1-2 \tan (x)}{\sqrt{3}}\right)+2 \sqrt{2} \tan ^{-1}\left(\frac{\tan (x)}{\sqrt{2}}\right)+2 \sqrt{3} \tan ^{-1}\left(\frac{2 \tan (x)+1}{\sqrt{3}}\right)+\log (2-\sin (2 x))-\log (\sin (2 x)+2)\right)","\frac{\tan ^{-1}\left(\frac{\tan (x)}{\sqrt{2}}\right)}{3 \sqrt{2}}+\frac{\tan ^{-1}\left(\frac{\tan (x)}{\sqrt{1-\sqrt[3]{-1}}}\right)}{3 \sqrt{1-\sqrt[3]{-1}}}+\frac{\tan ^{-1}\left(\frac{\tan (x)}{\sqrt{1+(-1)^{2/3}}}\right)}{3 \sqrt{1+(-1)^{2/3}}}",1,"(-2*Sqrt[3]*ArcTan[(1 - 2*Tan[x])/Sqrt[3]] + 2*Sqrt[2]*ArcTan[Tan[x]/Sqrt[2]] + 2*Sqrt[3]*ArcTan[(1 + 2*Tan[x])/Sqrt[3]] + Log[2 - Sin[2*x]] - Log[2 + Sin[2*x]])/12","A",1
82,1,141,129,0.1522407,"\int \frac{1}{1+\cos ^8(x)} \, dx","Integrate[(1 + Cos[x]^8)^(-1),x]","8 \text{RootSum}\left[\text{$\#$1}^8+8 \text{$\#$1}^7+28 \text{$\#$1}^6+56 \text{$\#$1}^5+326 \text{$\#$1}^4+56 \text{$\#$1}^3+28 \text{$\#$1}^2+8 \text{$\#$1}+1\&,\frac{2 \text{$\#$1}^3 \tan ^{-1}\left(\frac{\sin (2 x)}{\cos (2 x)-\text{$\#$1}}\right)-i \text{$\#$1}^3 \log \left(\text{$\#$1}^2-2 \text{$\#$1} \cos (2 x)+1\right)}{\text{$\#$1}^7+7 \text{$\#$1}^6+21 \text{$\#$1}^5+35 \text{$\#$1}^4+163 \text{$\#$1}^3+21 \text{$\#$1}^2+7 \text{$\#$1}+1}\&\right]","-\frac{\tan ^{-1}\left(\sqrt{1-\sqrt[4]{-1}} \cot (x)\right)}{4 \sqrt{1-\sqrt[4]{-1}}}-\frac{\tan ^{-1}\left(\sqrt{1+\sqrt[4]{-1}} \cot (x)\right)}{4 \sqrt{1+\sqrt[4]{-1}}}-\frac{\tan ^{-1}\left(\sqrt{1-(-1)^{3/4}} \cot (x)\right)}{4 \sqrt{1-(-1)^{3/4}}}-\frac{\tan ^{-1}\left(\sqrt{1+(-1)^{3/4}} \cot (x)\right)}{4 \sqrt{1+(-1)^{3/4}}}",1,"8*RootSum[1 + 8*#1 + 28*#1^2 + 56*#1^3 + 326*#1^4 + 56*#1^5 + 28*#1^6 + 8*#1^7 + #1^8 & , (2*ArcTan[Sin[2*x]/(Cos[2*x] - #1)]*#1^3 - I*Log[1 - 2*Cos[2*x]*#1 + #1^2]*#1^3)/(1 + 7*#1 + 21*#1^2 + 163*#1^3 + 35*#1^4 + 21*#1^5 + 7*#1^6 + #1^7) & ]","C",1
83,1,378,205,0.1215808,"\int \frac{1}{1-\cos ^5(x)} \, dx","Integrate[(1 - Cos[x]^5)^(-1),x]","-\frac{1}{5} \cot \left(\frac{x}{2}\right)+\frac{1}{10} \text{RootSum}\left[\text{$\#$1}^8+2 \text{$\#$1}^7+8 \text{$\#$1}^6+14 \text{$\#$1}^5+30 \text{$\#$1}^4+14 \text{$\#$1}^3+8 \text{$\#$1}^2+2 \text{$\#$1}+1\&,\frac{2 \text{$\#$1}^6 \tan ^{-1}\left(\frac{\sin (x)}{\cos (x)-\text{$\#$1}}\right)+8 \text{$\#$1}^5 \tan ^{-1}\left(\frac{\sin (x)}{\cos (x)-\text{$\#$1}}\right)+30 \text{$\#$1}^4 \tan ^{-1}\left(\frac{\sin (x)}{\cos (x)-\text{$\#$1}}\right)+80 \text{$\#$1}^3 \tan ^{-1}\left(\frac{\sin (x)}{\cos (x)-\text{$\#$1}}\right)-15 i \text{$\#$1}^2 \log \left(\text{$\#$1}^2-2 \text{$\#$1} \cos (x)+1\right)-4 i \text{$\#$1} \log \left(\text{$\#$1}^2-2 \text{$\#$1} \cos (x)+1\right)-i \log \left(\text{$\#$1}^2-2 \text{$\#$1} \cos (x)+1\right)+30 \text{$\#$1}^2 \tan ^{-1}\left(\frac{\sin (x)}{\cos (x)-\text{$\#$1}}\right)-i \text{$\#$1}^6 \log \left(\text{$\#$1}^2-2 \text{$\#$1} \cos (x)+1\right)-4 i \text{$\#$1}^5 \log \left(\text{$\#$1}^2-2 \text{$\#$1} \cos (x)+1\right)-15 i \text{$\#$1}^4 \log \left(\text{$\#$1}^2-2 \text{$\#$1} \cos (x)+1\right)-40 i \text{$\#$1}^3 \log \left(\text{$\#$1}^2-2 \text{$\#$1} \cos (x)+1\right)+8 \text{$\#$1} \tan ^{-1}\left(\frac{\sin (x)}{\cos (x)-\text{$\#$1}}\right)+2 \tan ^{-1}\left(\frac{\sin (x)}{\cos (x)-\text{$\#$1}}\right)}{4 \text{$\#$1}^7+7 \text{$\#$1}^6+24 \text{$\#$1}^5+35 \text{$\#$1}^4+60 \text{$\#$1}^3+21 \text{$\#$1}^2+8 \text{$\#$1}+1}\&\right]","\frac{2 \tan ^{-1}\left(\sqrt{\frac{1-\sqrt[5]{-1}}{1+\sqrt[5]{-1}}} \tan \left(\frac{x}{2}\right)\right)}{5 \sqrt{1-(-1)^{2/5}}}+\frac{2 \tan ^{-1}\left(\sqrt{\frac{1-(-1)^{3/5}}{1+(-1)^{3/5}}} \tan \left(\frac{x}{2}\right)\right)}{5 \sqrt{1+\sqrt[5]{-1}}}-\frac{\sin (x)}{5 (1-\cos (x))}-\frac{2 \tanh ^{-1}\left(\frac{\tan \left(\frac{x}{2}\right)}{\sqrt{-\frac{1-(-1)^{2/5}}{1+(-1)^{2/5}}}}\right)}{5 \sqrt{(-1)^{4/5}-1}}+\frac{2 \tanh ^{-1}\left(\sqrt{-\frac{1+(-1)^{4/5}}{1-(-1)^{4/5}}} \tan \left(\frac{x}{2}\right)\right)}{5 \sqrt{-1-(-1)^{3/5}}}",1,"-1/5*Cot[x/2] + RootSum[1 + 2*#1 + 8*#1^2 + 14*#1^3 + 30*#1^4 + 14*#1^5 + 8*#1^6 + 2*#1^7 + #1^8 & , (2*ArcTan[Sin[x]/(Cos[x] - #1)] - I*Log[1 - 2*Cos[x]*#1 + #1^2] + 8*ArcTan[Sin[x]/(Cos[x] - #1)]*#1 - (4*I)*Log[1 - 2*Cos[x]*#1 + #1^2]*#1 + 30*ArcTan[Sin[x]/(Cos[x] - #1)]*#1^2 - (15*I)*Log[1 - 2*Cos[x]*#1 + #1^2]*#1^2 + 80*ArcTan[Sin[x]/(Cos[x] - #1)]*#1^3 - (40*I)*Log[1 - 2*Cos[x]*#1 + #1^2]*#1^3 + 30*ArcTan[Sin[x]/(Cos[x] - #1)]*#1^4 - (15*I)*Log[1 - 2*Cos[x]*#1 + #1^2]*#1^4 + 8*ArcTan[Sin[x]/(Cos[x] - #1)]*#1^5 - (4*I)*Log[1 - 2*Cos[x]*#1 + #1^2]*#1^5 + 2*ArcTan[Sin[x]/(Cos[x] - #1)]*#1^6 - I*Log[1 - 2*Cos[x]*#1 + #1^2]*#1^6)/(1 + 8*#1 + 21*#1^2 + 60*#1^3 + 35*#1^4 + 24*#1^5 + 7*#1^6 + 4*#1^7) & ]/10","C",1
84,1,117,71,0.2935594,"\int \frac{1}{1-\cos ^6(x)} \, dx","Integrate[(1 - Cos[x]^6)^(-1),x]","\frac{\sin (x) (8 \cos (2 x)+\cos (4 x)+15) \left(6 \cos (x)+i \sqrt[4]{-3} \left(\sqrt{3}+3 i\right) \sin (x) \tan ^{-1}\left(\frac{1}{2} \sqrt[4]{-\frac{1}{3}} \left(\sqrt{3}-i\right) \tan (x)\right)+\sqrt[4]{-3} \left(\sqrt{3}-3 i\right) \sin (x) \tan ^{-1}\left(\frac{(-1)^{3/4} \left(\sqrt{3}+i\right) \tan (x)}{2 \sqrt[4]{3}}\right)\right)}{144 \left(\cos ^6(x)-1\right)}","-\frac{\cot (x)}{3}-\frac{\tan ^{-1}\left(\sqrt{1+\sqrt[3]{-1}} \cot (x)\right)}{3 \sqrt{1+\sqrt[3]{-1}}}-\frac{\tan ^{-1}\left(\sqrt{1-(-1)^{2/3}} \cot (x)\right)}{3 \sqrt{1-(-1)^{2/3}}}",1,"((15 + 8*Cos[2*x] + Cos[4*x])*Sin[x]*(6*Cos[x] + I*(-3)^(1/4)*(3*I + Sqrt[3])*ArcTan[((-1/3)^(1/4)*(-I + Sqrt[3])*Tan[x])/2]*Sin[x] + (-3)^(1/4)*(-3*I + Sqrt[3])*ArcTan[((-1)^(3/4)*(I + Sqrt[3])*Tan[x])/(2*3^(1/4))]*Sin[x]))/(144*(-1 + Cos[x]^6))","C",1
85,1,64,89,0.1616568,"\int \frac{1}{1-\cos ^8(x)} \, dx","Integrate[(1 - Cos[x]^8)^(-1),x]","\frac{1}{8} \left(\frac{2 \tan ^{-1}\left(\frac{\tan (x)}{\sqrt{1-i}}\right)}{\sqrt{1-i}}+\frac{2 \tan ^{-1}\left(\frac{\tan (x)}{\sqrt{1+i}}\right)}{\sqrt{1+i}}+\sqrt{2} \tan ^{-1}\left(\frac{\tan (x)}{\sqrt{2}}\right)-2 \cot (x)\right)","\frac{x}{4 \sqrt{2}}-\frac{\cot (x)}{4}-\frac{\tan ^{-1}\left(\sqrt{1-i} \cot (x)\right)}{4 \sqrt{1-i}}-\frac{\tan ^{-1}\left(\sqrt{1+i} \cot (x)\right)}{4 \sqrt{1+i}}-\frac{\tan ^{-1}\left(\frac{\sin (x) \cos (x)}{\cos ^2(x)+\sqrt{2}+1}\right)}{4 \sqrt{2}}",1,"((2*ArcTan[Tan[x]/Sqrt[1 - I]])/Sqrt[1 - I] + (2*ArcTan[Tan[x]/Sqrt[1 + I]])/Sqrt[1 + I] + Sqrt[2]*ArcTan[Tan[x]/Sqrt[2]] - 2*Cot[x])/8","A",1
86,1,17,17,0.0089079,"\int \frac{\tan (x)}{1+\cos ^2(x)} \, dx","Integrate[Tan[x]/(1 + Cos[x]^2),x]","\frac{1}{2} \log \left(\cos ^2(x)+1\right)-\log (\cos (x))","\frac{1}{2} \log \left(\cos ^2(x)+1\right)-\log (\cos (x))",1,"-Log[Cos[x]] + Log[1 + Cos[x]^2]/2","A",1
87,1,40,40,0.0286633,"\int \sqrt{a+b \cos ^2(x)} \tan (x) \, dx","Integrate[Sqrt[a + b*Cos[x]^2]*Tan[x],x]","\sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a+b \cos ^2(x)}}{\sqrt{a}}\right)-\sqrt{a+b \cos ^2(x)}","\sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a+b \cos ^2(x)}}{\sqrt{a}}\right)-\sqrt{a+b \cos ^2(x)}",1,"Sqrt[a]*ArcTanh[Sqrt[a + b*Cos[x]^2]/Sqrt[a]] - Sqrt[a + b*Cos[x]^2]","A",1
88,1,47,20,0.0371362,"\int \sqrt{1-\cos ^2(x)} \tan (x) \, dx","Integrate[Sqrt[1 - Cos[x]^2]*Tan[x],x]","\sqrt{\sin ^2(x)} (-\csc (x)) \left(\sin (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)\right)","\tanh ^{-1}\left(\sqrt{\sin ^2(x)}\right)-\sqrt{\sin ^2(x)}",1,"-(Csc[x]*Sqrt[Sin[x]^2]*(Log[Cos[x/2] - Sin[x/2]] - Log[Cos[x/2] + Sin[x/2]] + Sin[x]))","B",1
89,1,25,25,0.0158731,"\int \frac{\tan (x)}{\sqrt{a+b \cos ^2(x)}} \, dx","Integrate[Tan[x]/Sqrt[a + b*Cos[x]^2],x]","\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \cos ^2(x)}}{\sqrt{a}}\right)}{\sqrt{a}}","\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \cos ^2(x)}}{\sqrt{a}}\right)}{\sqrt{a}}",1,"ArcTanh[Sqrt[a + b*Cos[x]^2]/Sqrt[a]]/Sqrt[a]","A",1
90,1,11,11,0.0092303,"\int \frac{\tan (x)}{\sqrt{1+\cos ^2(x)}} \, dx","Integrate[Tan[x]/Sqrt[1 + Cos[x]^2],x]","\tanh ^{-1}\left(\sqrt{\cos ^2(x)+1}\right)","\tanh ^{-1}\left(\sqrt{\cos ^2(x)+1}\right)",1,"ArcTanh[Sqrt[1 + Cos[x]^2]]","A",1
91,1,44,9,0.0238634,"\int \frac{\tan (x)}{\sqrt{1-\cos ^2(x)}} \, dx","Integrate[Tan[x]/Sqrt[1 - Cos[x]^2],x]","\frac{\sin (x) \left(\log \left(\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)\right)-\log \left(\cos \left(\frac{x}{2}\right)-\sin \left(\frac{x}{2}\right)\right)\right)}{\sqrt{\sin ^2(x)}}","\tanh ^{-1}\left(\sqrt{\sin ^2(x)}\right)",1,"((-Log[Cos[x/2] - Sin[x/2]] + Log[Cos[x/2] + Sin[x/2]])*Sin[x])/Sqrt[Sin[x]^2]","B",1
92,1,217,153,0.3006712,"\int \frac{\tan ^3(x)}{a+b \cos ^3(x)} \, dx","Integrate[Tan[x]^3/(a + b*Cos[x]^3),x]","\frac{-2 \text{RootSum}\left[\text{$\#$1}^3 a-\text{$\#$1}^3 b+3 \text{$\#$1}^2 a+3 \text{$\#$1}^2 b+3 \text{$\#$1} a-3 \text{$\#$1} b+a+b\&,\frac{\text{$\#$1}^2 a \log \left(\tan ^2\left(\frac{x}{2}\right)-\text{$\#$1}\right)-\text{$\#$1}^2 b \log \left(\tan ^2\left(\frac{x}{2}\right)-\text{$\#$1}\right)+2 \text{$\#$1} a \log \left(\tan ^2\left(\frac{x}{2}\right)-\text{$\#$1}\right)+a \log \left(\tan ^2\left(\frac{x}{2}\right)-\text{$\#$1}\right)+4 \text{$\#$1} b \log \left(\tan ^2\left(\frac{x}{2}\right)-\text{$\#$1}\right)+b \log \left(\tan ^2\left(\frac{x}{2}\right)-\text{$\#$1}\right)}{\text{$\#$1}^2 a-\text{$\#$1}^2 b+2 \text{$\#$1} a+2 \text{$\#$1} b+a-b}\&\right]+3 \sec ^2(x)+6 \left(\log \left(\sec ^2\left(\frac{x}{2}\right)\right)+\log (\cos (x))\right)}{6 a}","-\frac{b^{2/3} \log \left(a^{2/3}-\sqrt[3]{a} \sqrt[3]{b} \cos (x)+b^{2/3} \cos ^2(x)\right)}{6 a^{5/3}}+\frac{b^{2/3} \log \left(\sqrt[3]{a}+\sqrt[3]{b} \cos (x)\right)}{3 a^{5/3}}-\frac{b^{2/3} \tan ^{-1}\left(\frac{\sqrt[3]{a}-2 \sqrt[3]{b} \cos (x)}{\sqrt{3} \sqrt[3]{a}}\right)}{\sqrt{3} a^{5/3}}-\frac{\log \left(a+b \cos ^3(x)\right)}{3 a}+\frac{\sec ^2(x)}{2 a}+\frac{\log (\cos (x))}{a}",1,"(6*(Log[Cos[x]] + Log[Sec[x/2]^2]) - 2*RootSum[a + b + 3*a*#1 - 3*b*#1 + 3*a*#1^2 + 3*b*#1^2 + a*#1^3 - b*#1^3 & , (a*Log[-#1 + Tan[x/2]^2] + b*Log[-#1 + Tan[x/2]^2] + 2*a*Log[-#1 + Tan[x/2]^2]*#1 + 4*b*Log[-#1 + Tan[x/2]^2]*#1 + a*Log[-#1 + Tan[x/2]^2]*#1^2 - b*Log[-#1 + Tan[x/2]^2]*#1^2)/(a - b + 2*a*#1 + 2*b*#1 + a*#1^2 - b*#1^2) & ] + 3*Sec[x]^2)/(6*a)","C",1
93,1,45,45,0.0267939,"\int \sqrt{a+b \cos ^3(x)} \tan (x) \, dx","Integrate[Sqrt[a + b*Cos[x]^3]*Tan[x],x]","\frac{2}{3} \sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a+b \cos ^3(x)}}{\sqrt{a}}\right)-\frac{2}{3} \sqrt{a+b \cos ^3(x)}","\frac{2}{3} \sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a+b \cos ^3(x)}}{\sqrt{a}}\right)-\frac{2}{3} \sqrt{a+b \cos ^3(x)}",1,"(2*Sqrt[a]*ArcTanh[Sqrt[a + b*Cos[x]^3]/Sqrt[a]])/3 - (2*Sqrt[a + b*Cos[x]^3])/3","A",1
94,1,28,28,0.0174719,"\int \frac{\tan (x)}{\sqrt{a+b \cos ^3(x)}} \, dx","Integrate[Tan[x]/Sqrt[a + b*Cos[x]^3],x]","\frac{2 \tanh ^{-1}\left(\frac{\sqrt{a+b \cos ^3(x)}}{\sqrt{a}}\right)}{3 \sqrt{a}}","\frac{2 \tanh ^{-1}\left(\frac{\sqrt{a+b \cos ^3(x)}}{\sqrt{a}}\right)}{3 \sqrt{a}}",1,"(2*ArcTanh[Sqrt[a + b*Cos[x]^3]/Sqrt[a]])/(3*Sqrt[a])","A",1
95,1,45,45,0.034364,"\int \sqrt{a+b \cos ^4(x)} \tan (x) \, dx","Integrate[Sqrt[a + b*Cos[x]^4]*Tan[x],x]","\frac{1}{2} \sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a+b \cos ^4(x)}}{\sqrt{a}}\right)-\frac{1}{2} \sqrt{a+b \cos ^4(x)}","\frac{1}{2} \sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a+b \cos ^4(x)}}{\sqrt{a}}\right)-\frac{1}{2} \sqrt{a+b \cos ^4(x)}",1,"(Sqrt[a]*ArcTanh[Sqrt[a + b*Cos[x]^4]/Sqrt[a]])/2 - Sqrt[a + b*Cos[x]^4]/2","A",1
96,1,28,28,0.0177084,"\int \frac{\tan (x)}{\sqrt{a+b \cos ^4(x)}} \, dx","Integrate[Tan[x]/Sqrt[a + b*Cos[x]^4],x]","\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \cos ^4(x)}}{\sqrt{a}}\right)}{2 \sqrt{a}}","\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \cos ^4(x)}}{\sqrt{a}}\right)}{2 \sqrt{a}}",1,"ArcTanh[Sqrt[a + b*Cos[x]^4]/Sqrt[a]]/(2*Sqrt[a])","A",1
97,1,46,47,0.0323729,"\int \sqrt{a+b \cos ^n(x)} \tan (x) \, dx","Integrate[Sqrt[a + b*Cos[x]^n]*Tan[x],x]","-\frac{2 \sqrt{a+b \cos ^n(x)}-2 \sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a+b \cos ^n(x)}}{\sqrt{a}}\right)}{n}","\frac{2 \sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a+b \cos ^n(x)}}{\sqrt{a}}\right)}{n}-\frac{2 \sqrt{a+b \cos ^n(x)}}{n}",1,"-((-2*Sqrt[a]*ArcTanh[Sqrt[a + b*Cos[x]^n]/Sqrt[a]] + 2*Sqrt[a + b*Cos[x]^n])/n)","A",1
98,1,29,29,0.0171457,"\int \frac{\tan (x)}{\sqrt{a+b \cos ^n(x)}} \, dx","Integrate[Tan[x]/Sqrt[a + b*Cos[x]^n],x]","\frac{2 \tanh ^{-1}\left(\frac{\sqrt{a+b \cos ^n(x)}}{\sqrt{a}}\right)}{\sqrt{a} n}","\frac{2 \tanh ^{-1}\left(\frac{\sqrt{a+b \cos ^n(x)}}{\sqrt{a}}\right)}{\sqrt{a} n}",1,"(2*ArcTanh[Sqrt[a + b*Cos[x]^n]/Sqrt[a]])/(Sqrt[a]*n)","A",1