1,1,111,55,0.1640382,"\int \frac{\tan ^4(x)}{a+a \csc (x)} \, dx","Integrate[Tan[x]^4/(a + a*Csc[x]),x]","\frac{-64 \sin (x)+240 x \sin (2 x)-178 \sin (2 x)-128 \sin (3 x)+120 x \sin (4 x)-89 \sin (4 x)+6 (120 x-89) \cos (x)+128 \cos (2 x)+240 x \cos (3 x)-178 \cos (3 x)+184 \cos (4 x)+200}{960 a \left(\cos \left(\frac{x}{2}\right)-\sin \left(\frac{x}{2}\right)\right)^3 \left(\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)\right)^5}","\frac{x}{a}-\frac{\tan ^5(x) (1-\csc (x))}{5 a}+\frac{\tan ^3(x) (5-4 \csc (x))}{15 a}-\frac{\tan (x) (15-8 \csc (x))}{15 a}",1,"(200 + 6*(-89 + 120*x)*Cos[x] + 128*Cos[2*x] - 178*Cos[3*x] + 240*x*Cos[3*x] + 184*Cos[4*x] - 64*Sin[x] - 178*Sin[2*x] + 240*x*Sin[2*x] - 128*Sin[3*x] - 89*Sin[4*x] + 120*x*Sin[4*x])/(960*a*(Cos[x/2] - Sin[x/2])^3*(Cos[x/2] + Sin[x/2])^5)","B",1
2,1,50,68,0.1352899,"\int \frac{\tan ^3(x)}{a+a \csc (x)} \, dx","Integrate[Tan[x]^3/(a + a*Csc[x]),x]","\frac{\frac{2 \left(5 \sin ^2(x)-3 \sin (x)-6\right)}{(\sin (x)-1) (\sin (x)+1)^2}+5 \log (1-\sin (x))+11 \log (\sin (x)+1)}{16 a}","\frac{1}{8 a (1-\sin (x))}+\frac{3}{4 a (\sin (x)+1)}-\frac{1}{8 a (\sin (x)+1)^2}+\frac{5 \log (1-\sin (x))}{16 a}+\frac{11 \log (\sin (x)+1)}{16 a}",1,"(5*Log[1 - Sin[x]] + 11*Log[1 + Sin[x]] + (2*(-6 - 3*Sin[x] + 5*Sin[x]^2))/((-1 + Sin[x])*(1 + Sin[x])^2))/(16*a)","A",1
3,1,62,39,0.1076674,"\int \frac{\tan ^2(x)}{a+a \csc (x)} \, dx","Integrate[Tan[x]^2/(a + a*Csc[x]),x]","-\frac{-2 \sin (x)+4 \cos (2 x)+(6 x-5) (\sin (x)+1) \cos (x)}{6 a \left(\cos \left(\frac{x}{2}\right)-\sin \left(\frac{x}{2}\right)\right) \left(\sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)\right)^3}","-\frac{x}{a}-\frac{\tan ^3(x) (1-\csc (x))}{3 a}+\frac{\tan (x) (3-2 \csc (x))}{3 a}",1,"-1/6*(4*Cos[2*x] - 2*Sin[x] + (-5 + 6*x)*Cos[x]*(1 + Sin[x]))/(a*(Cos[x/2] - Sin[x/2])*(Cos[x/2] + Sin[x/2])^3)","A",1
4,1,30,40,0.0544378,"\int \frac{\tan (x)}{a+a \csc (x)} \, dx","Integrate[Tan[x]/(a + a*Csc[x]),x]","-\frac{\frac{2}{\sin (x)+1}+\log (1-\sin (x))+3 \log (\sin (x)+1)}{4 a}","-\frac{1}{2 a (\sin (x)+1)}-\frac{\log (1-\sin (x))}{4 a}-\frac{3 \log (\sin (x)+1)}{4 a}",1,"-1/4*(Log[1 - Sin[x]] + 3*Log[1 + Sin[x]] + 2/(1 + Sin[x]))/a","A",1
5,1,9,9,0.0090902,"\int \frac{\cot (x)}{a+a \csc (x)} \, dx","Integrate[Cot[x]/(a + a*Csc[x]),x]","\frac{\log (\sin (x)+1)}{a}","\frac{\log (\sin (x)+1)}{a}",1,"Log[1 + Sin[x]]/a","A",1
6,1,30,15,0.0417048,"\int \frac{\cot ^2(x)}{a+a \csc (x)} \, dx","Integrate[Cot[x]^2/(a + a*Csc[x]),x]","-\frac{x}{a}+\frac{\log \left(\sin \left(\frac{x}{2}\right)\right)}{a}-\frac{\log \left(\cos \left(\frac{x}{2}\right)\right)}{a}","-\frac{x}{a}-\frac{\tanh ^{-1}(\cos (x))}{a}",1,"-(x/a) - Log[Cos[x/2]]/a + Log[Sin[x/2]]/a","A",1
7,1,11,16,0.0199906,"\int \frac{\cot ^3(x)}{a+a \csc (x)} \, dx","Integrate[Cot[x]^3/(a + a*Csc[x]),x]","-\frac{\csc (x)+\log (\sin (x))}{a}","-\frac{\csc (x)}{a}-\frac{\log (\sin (x))}{a}",1,"-((Csc[x] + Log[Sin[x]])/a)","A",1
8,1,90,31,0.0456662,"\int \frac{\cot ^4(x)}{a+a \csc (x)} \, dx","Integrate[Cot[x]^4/(a + a*Csc[x]),x]","\frac{x}{a}-\frac{\tan \left(\frac{x}{2}\right)}{2 a}+\frac{\cot \left(\frac{x}{2}\right)}{2 a}-\frac{\csc ^2\left(\frac{x}{2}\right)}{8 a}+\frac{\sec ^2\left(\frac{x}{2}\right)}{8 a}-\frac{\log \left(\sin \left(\frac{x}{2}\right)\right)}{2 a}+\frac{\log \left(\cos \left(\frac{x}{2}\right)\right)}{2 a}","\frac{x}{a}+\frac{\tanh ^{-1}(\cos (x))}{2 a}+\frac{\cot (x) (2-\csc (x))}{2 a}",1,"x/a + Cot[x/2]/(2*a) - Csc[x/2]^2/(8*a) + Log[Cos[x/2]]/(2*a) - Log[Sin[x/2]]/(2*a) + Sec[x/2]^2/(8*a) - Tan[x/2]/(2*a)","B",1
9,1,26,36,0.027917,"\int \frac{\cot ^5(x)}{a+a \csc (x)} \, dx","Integrate[Cot[x]^5/(a + a*Csc[x]),x]","\frac{-\frac{1}{3} \csc ^3(x)+\frac{\csc ^2(x)}{2}+\csc (x)+\log (\sin (x))}{a}","-\frac{\csc ^3(x)}{3 a}+\frac{\csc ^2(x)}{2 a}+\frac{\csc (x)}{a}+\frac{\log (\sin (x))}{a}",1,"(Csc[x] + Csc[x]^2/2 - Csc[x]^3/3 + Log[Sin[x]])/a","A",1
10,1,163,49,0.0523499,"\int \frac{\cot ^6(x)}{a+a \csc (x)} \, dx","Integrate[Cot[x]^6/(a + a*Csc[x]),x]","-\frac{x}{a}+\frac{2 \tan \left(\frac{x}{2}\right)}{3 a}-\frac{2 \cot \left(\frac{x}{2}\right)}{3 a}-\frac{\csc ^4\left(\frac{x}{2}\right)}{64 a}+\frac{5 \csc ^2\left(\frac{x}{2}\right)}{32 a}+\frac{\sec ^4\left(\frac{x}{2}\right)}{64 a}-\frac{5 \sec ^2\left(\frac{x}{2}\right)}{32 a}+\frac{3 \log \left(\sin \left(\frac{x}{2}\right)\right)}{8 a}-\frac{3 \log \left(\cos \left(\frac{x}{2}\right)\right)}{8 a}+\frac{\cot \left(\frac{x}{2}\right) \csc ^2\left(\frac{x}{2}\right)}{24 a}-\frac{\tan \left(\frac{x}{2}\right) \sec ^2\left(\frac{x}{2}\right)}{24 a}","-\frac{x}{a}-\frac{3 \tanh ^{-1}(\cos (x))}{8 a}+\frac{\cot ^3(x) (4-3 \csc (x))}{12 a}-\frac{\cot (x) (8-3 \csc (x))}{8 a}",1,"-(x/a) - (2*Cot[x/2])/(3*a) + (5*Csc[x/2]^2)/(32*a) + (Cot[x/2]*Csc[x/2]^2)/(24*a) - Csc[x/2]^4/(64*a) - (3*Log[Cos[x/2]])/(8*a) + (3*Log[Sin[x/2]])/(8*a) - (5*Sec[x/2]^2)/(32*a) + Sec[x/2]^4/(64*a) + (2*Tan[x/2])/(3*a) - (Sec[x/2]^2*Tan[x/2])/(24*a)","B",1
11,1,39,58,0.0512442,"\int \frac{\cot ^7(x)}{a+a \csc (x)} \, dx","Integrate[Cot[x]^7/(a + a*Csc[x]),x]","-\frac{\frac{\csc ^5(x)}{5}-\frac{\csc ^4(x)}{4}-\frac{2 \csc ^3(x)}{3}+\csc ^2(x)+\csc (x)+\log (\sin (x))}{a}","-\frac{\csc ^5(x)}{5 a}+\frac{\csc ^4(x)}{4 a}+\frac{2 \csc ^3(x)}{3 a}-\frac{\csc ^2(x)}{a}-\frac{\csc (x)}{a}-\frac{\log (\sin (x))}{a}",1,"-((Csc[x] + Csc[x]^2 - (2*Csc[x]^3)/3 - Csc[x]^4/4 + Csc[x]^5/5 + Log[Sin[x]])/a)","A",1
12,1,172,178,0.7866696,"\int \frac{\tan ^5(x)}{a+b \csc (x)} \, dx","Integrate[Tan[x]^5/(a + b*Csc[x]),x]","\frac{\csc (x) (a \sin (x)+b) \left(-\frac{\left(8 a^2+21 a b+15 b^2\right) \log (1-\sin (x))}{(a+b)^3}-\frac{\left(8 a^2-21 a b+15 b^2\right) \log (\sin (x)+1)}{(a-b)^3}+\frac{16 b^6 \log (a \sin (x)+b)}{a (a-b)^3 (a+b)^3}+\frac{7 a+9 b}{(a+b)^2 (\sin (x)-1)}+\frac{9 b-7 a}{(a-b)^2 (\sin (x)+1)}+\frac{1}{(a+b) (\sin (x)-1)^2}+\frac{1}{(a-b) (\sin (x)+1)^2}\right)}{16 (a+b \csc (x))}","-\frac{\left(8 a^2+21 a b+15 b^2\right) \log (1-\csc (x))}{16 (a+b)^3}-\frac{\left(8 a^2-21 a b+15 b^2\right) \log (\csc (x)+1)}{16 (a-b)^3}+\frac{b^6 \log (a+b \csc (x))}{a \left(a^2-b^2\right)^3}+\frac{5 a+7 b}{16 (a+b)^2 (1-\csc (x))}+\frac{5 a-7 b}{16 (a-b)^2 (\csc (x)+1)}+\frac{1}{16 (a+b) (1-\csc (x))^2}+\frac{1}{16 (a-b) (\csc (x)+1)^2}-\frac{\log (\sin (x))}{a}",1,"(Csc[x]*(b + a*Sin[x])*(-(((8*a^2 + 21*a*b + 15*b^2)*Log[1 - Sin[x]])/(a + b)^3) - ((8*a^2 - 21*a*b + 15*b^2)*Log[1 + Sin[x]])/(a - b)^3 + (16*b^6*Log[b + a*Sin[x]])/(a*(a - b)^3*(a + b)^3) + 1/((a + b)*(-1 + Sin[x])^2) + (7*a + 9*b)/((a + b)^2*(-1 + Sin[x])) + 1/((a - b)*(1 + Sin[x])^2) + (-7*a + 9*b)/((a - b)^2*(1 + Sin[x]))))/(16*(a + b*Csc[x]))","A",1
13,1,115,113,0.48846,"\int \frac{\tan ^3(x)}{a+b \csc (x)} \, dx","Integrate[Tan[x]^3/(a + b*Csc[x]),x]","\frac{\csc (x) (a \sin (x)+b) \left(\frac{4 b^4 \log (a \sin (x)+b)}{a (a-b)^2 (a+b)^2}-\frac{1}{(a+b) (\sin (x)-1)}+\frac{1}{(a-b) (\sin (x)+1)}+\frac{(2 a+3 b) \log (1-\sin (x))}{(a+b)^2}+\frac{(2 a-3 b) \log (\sin (x)+1)}{(a-b)^2}\right)}{4 (a+b \csc (x))}","\frac{b^4 \log (a+b \csc (x))}{a \left(a^2-b^2\right)^2}-\frac{1}{4 (a+b) (1-\csc (x))}-\frac{1}{4 (a-b) (\csc (x)+1)}+\frac{(2 a+3 b) \log (1-\csc (x))}{4 (a+b)^2}+\frac{(2 a-3 b) \log (\csc (x)+1)}{4 (a-b)^2}+\frac{\log (\sin (x))}{a}",1,"(Csc[x]*(b + a*Sin[x])*(((2*a + 3*b)*Log[1 - Sin[x]])/(a + b)^2 + ((2*a - 3*b)*Log[1 + Sin[x]])/(a - b)^2 + (4*b^4*Log[b + a*Sin[x]])/(a*(a - b)^2*(a + b)^2) - 1/((a + b)*(-1 + Sin[x])) + 1/((a - b)*(1 + Sin[x]))))/(4*(a + b*Csc[x]))","A",1
14,1,56,66,0.0609464,"\int \frac{\tan (x)}{a+b \csc (x)} \, dx","Integrate[Tan[x]/(a + b*Csc[x]),x]","-\frac{-2 b^2 \log (a \sin (x)+b)+a (a-b) \log (1-\sin (x))+a (a+b) \log (\sin (x)+1)}{2 a (a-b) (a+b)}","\frac{b^2 \log (a+b \csc (x))}{a \left(a^2-b^2\right)}-\frac{\log (1-\csc (x))}{2 (a+b)}-\frac{\log (\csc (x)+1)}{2 (a-b)}-\frac{\log (\sin (x))}{a}",1,"-1/2*(a*(a - b)*Log[1 - Sin[x]] + a*(a + b)*Log[1 + Sin[x]] - 2*b^2*Log[b + a*Sin[x]])/(a*(a - b)*(a + b))","A",1
15,1,11,19,0.0077566,"\int \frac{\cot (x)}{a+b \csc (x)} \, dx","Integrate[Cot[x]/(a + b*Csc[x]),x]","\frac{\log (a \sin (x)+b)}{a}","\frac{\log (a+b \csc (x))}{a}+\frac{\log (\sin (x))}{a}",1,"Log[b + a*Sin[x]]/a","A",1
16,1,39,38,0.0492435,"\int \frac{\cot ^3(x)}{a+b \csc (x)} \, dx","Integrate[Cot[x]^3/(a + b*Csc[x]),x]","\frac{\left(a^2-b^2\right) \log (a \sin (x)+b)+a^2 (-\log (\sin (x)))-a b \csc (x)}{a b^2}","-\frac{\left(1-\frac{a^2}{b^2}\right) \log (a+b \csc (x))}{a}-\frac{\log (\sin (x))}{a}-\frac{\csc (x)}{b}",1,"(-(a*b*Csc[x]) - a^2*Log[Sin[x]] + (a^2 - b^2)*Log[b + a*Sin[x]])/(a*b^2)","A",1
17,1,85,72,0.1354064,"\int \frac{\cot ^5(x)}{a+b \csc (x)} \, dx","Integrate[Cot[x]^5/(a + b*Csc[x]),x]","\frac{3 a^2 b^2 \csc ^2(x)-6 a b \left(a^2-2 b^2\right) \csc (x)-6 a^2 \left(a^2-2 b^2\right) \log (\sin (x))+6 \left(a^2-b^2\right)^2 \log (a \sin (x)+b)-2 a b^3 \csc ^3(x)}{6 a b^4}","\frac{\left(a^2-b^2\right)^2 \log (a+b \csc (x))}{a b^4}-\frac{\left(a^2-2 b^2\right) \csc (x)}{b^3}+\frac{a \csc ^2(x)}{2 b^2}+\frac{\log (\sin (x))}{a}-\frac{\csc ^3(x)}{3 b}",1,"(-6*a*b*(a^2 - 2*b^2)*Csc[x] + 3*a^2*b^2*Csc[x]^2 - 2*a*b^3*Csc[x]^3 - 6*a^2*(a^2 - 2*b^2)*Log[Sin[x]] + 6*(a^2 - b^2)^2*Log[b + a*Sin[x]])/(6*a*b^4)","A",1
18,1,132,122,0.2818327,"\int \frac{\cot ^7(x)}{a+b \csc (x)} \, dx","Integrate[Cot[x]^7/(a + b*Csc[x]),x]","\frac{30 a b^2 \left(a^2-3 b^2\right) \csc ^2(x)+\frac{60 \left(a^2-b^2\right)^3 \log (a \sin (x)+b)}{a}-20 b^3 \left(a^2-3 b^2\right) \csc ^3(x)-60 b \left(a^4-3 a^2 b^2+3 b^4\right) \csc (x)-60 a \left(a^4-3 a^2 b^2+3 b^4\right) \log (\sin (x))+15 a b^4 \csc ^4(x)-12 b^5 \csc ^5(x)}{60 b^6}","\frac{\left(a^2-b^2\right)^3 \log (a+b \csc (x))}{a b^6}+\frac{a \left(a^2-3 b^2\right) \csc ^2(x)}{2 b^4}-\frac{\left(a^2-3 b^2\right) \csc ^3(x)}{3 b^3}-\frac{\left(a^4-3 a^2 b^2+3 b^4\right) \csc (x)}{b^5}+\frac{a \csc ^4(x)}{4 b^2}-\frac{\log (\sin (x))}{a}-\frac{\csc ^5(x)}{5 b}",1,"(-60*b*(a^4 - 3*a^2*b^2 + 3*b^4)*Csc[x] + 30*a*b^2*(a^2 - 3*b^2)*Csc[x]^2 - 20*b^3*(a^2 - 3*b^2)*Csc[x]^3 + 15*a*b^4*Csc[x]^4 - 12*b^5*Csc[x]^5 - 60*a*(a^4 - 3*a^2*b^2 + 3*b^4)*Log[Sin[x]] + (60*(a^2 - b^2)^3*Log[b + a*Sin[x]])/a)/(60*b^6)","A",1
19,1,194,159,1.0718029,"\int \frac{\tan ^4(x)}{a+b \csc (x)} \, dx","Integrate[Tan[x]^4/(a + b*Csc[x]),x]","\frac{\csc (x) (a \sin (x)+b) \left(\frac{\sec ^3(x) \left(-4 a^4 \sin (3 x)+3 a^4 x \cos (3 x)+2 a^3 b+3 a^2 b^2 \sin (x)+7 a^2 b^2 \sin (3 x)-6 a^2 b^2 x \cos (3 x)+6 a b \left(a^2-2 b^2\right) \cos (2 x)+9 x \left(a^2-b^2\right)^2 \cos (x)-8 a b^3+3 b^4 x \cos (3 x)\right)}{(a-b)^2 (a+b)^2}-\frac{24 b^5 \tan ^{-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}}\right)}{12 a (a+b \csc (x))}","\frac{a \tan ^3(x)}{3 \left(a^2-b^2\right)}+\frac{a b^2 \tan (x)}{\left(a^2-b^2\right)^2}-\frac{a \tan (x)}{a^2-b^2}-\frac{b \sec ^3(x)}{3 \left(a^2-b^2\right)}+\frac{b \sec (x)}{a^2-b^2}+\frac{2 b^5 \tanh ^{-1}\left(\frac{a+b \tan \left(\frac{x}{2}\right)}{\sqrt{a^2-b^2}}\right)}{a \left(a^2-b^2\right)^{5/2}}-\frac{b^3 \sec (x)}{\left(a^2-b^2\right)^2}+\frac{x}{a}",1,"(Csc[x]*(b + a*Sin[x])*((-24*b^5*ArcTan[(a + b*Tan[x/2])/Sqrt[-a^2 + b^2]])/(-a^2 + b^2)^(5/2) + (Sec[x]^3*(2*a^3*b - 8*a*b^3 + 9*(a^2 - b^2)^2*x*Cos[x] + 6*a*b*(a^2 - 2*b^2)*Cos[2*x] + 3*a^4*x*Cos[3*x] - 6*a^2*b^2*x*Cos[3*x] + 3*b^4*x*Cos[3*x] + 3*a^2*b^2*Sin[x] - 4*a^4*Sin[3*x] + 7*a^2*b^2*Sin[3*x]))/((a - b)^2*(a + b)^2)))/(12*a*(a + b*Csc[x]))","A",1
20,1,87,84,0.2670308,"\int \frac{\tan ^2(x)}{a+b \csc (x)} \, dx","Integrate[Tan[x]^2/(a + b*Csc[x]),x]","-\frac{\sqrt{b^2-a^2} \left(a^2 (-x)+a^2 \tan (x)-a b \sec (x)+b^2 x\right)-2 b^3 \tan ^{-1}\left(\frac{a+b \tan \left(\frac{x}{2}\right)}{\sqrt{b^2-a^2}}\right)}{a \left(b^2-a^2\right)^{3/2}}","\frac{a \tan (x)}{a^2-b^2}-\frac{b \sec (x)}{a^2-b^2}+\frac{2 b^3 \tanh ^{-1}\left(\frac{a+b \tan \left(\frac{x}{2}\right)}{\sqrt{a^2-b^2}}\right)}{a \left(a^2-b^2\right)^{3/2}}-\frac{x}{a}",1,"-((-2*b^3*ArcTan[(a + b*Tan[x/2])/Sqrt[-a^2 + b^2]] + Sqrt[-a^2 + b^2]*(-(a^2*x) + b^2*x - a*b*Sec[x] + a^2*Tan[x]))/(a*(-a^2 + b^2)^(3/2)))","A",1
21,1,71,61,0.0643103,"\int \frac{\cot ^2(x)}{a+b \csc (x)} \, dx","Integrate[Cot[x]^2/(a + b*Csc[x]),x]","\frac{2 \sqrt{b^2-a^2} \tan ^{-1}\left(\frac{a+b \tan \left(\frac{x}{2}\right)}{\sqrt{b^2-a^2}}\right)+a \log \left(\sin \left(\frac{x}{2}\right)\right)-a \log \left(\cos \left(\frac{x}{2}\right)\right)-b x}{a b}","\frac{2 \sqrt{a^2-b^2} \tanh ^{-1}\left(\frac{a+b \tan \left(\frac{x}{2}\right)}{\sqrt{a^2-b^2}}\right)}{a b}-\frac{x}{a}-\frac{\tanh ^{-1}(\cos (x))}{b}",1,"(-(b*x) + 2*Sqrt[-a^2 + b^2]*ArcTan[(a + b*Tan[x/2])/Sqrt[-a^2 + b^2]] - a*Log[Cos[x/2]] + a*Log[Sin[x/2]])/(a*b)","A",1
22,1,158,91,0.4132606,"\int \frac{\cot ^4(x)}{a+b \csc (x)} \, dx","Integrate[Cot[x]^4/(a + b*Csc[x]),x]","\frac{8 a^3 \log \left(\sin \left(\frac{x}{2}\right)\right)-8 a^3 \log \left(\cos \left(\frac{x}{2}\right)\right)-16 \left(b^2-a^2\right)^{3/2} \tan ^{-1}\left(\frac{a+b \tan \left(\frac{x}{2}\right)}{\sqrt{b^2-a^2}}\right)-4 a^2 b \tan \left(\frac{x}{2}\right)+4 a^2 b \cot \left(\frac{x}{2}\right)-a b^2 \csc ^2\left(\frac{x}{2}\right)+a b^2 \sec ^2\left(\frac{x}{2}\right)-12 a b^2 \log \left(\sin \left(\frac{x}{2}\right)\right)+12 a b^2 \log \left(\cos \left(\frac{x}{2}\right)\right)+8 b^3 x}{8 a b^3}","-\frac{\left(2 a^2-3 b^2\right) \tanh ^{-1}(\cos (x))}{2 b^3}+\frac{2 \left(a^2-b^2\right)^{3/2} \tanh ^{-1}\left(\frac{a+b \tan \left(\frac{x}{2}\right)}{\sqrt{a^2-b^2}}\right)}{a b^3}+\frac{a \cot (x)}{b^2}+\frac{x}{a}-\frac{\cot (x) \csc (x)}{2 b}",1,"(8*b^3*x - 16*(-a^2 + b^2)^(3/2)*ArcTan[(a + b*Tan[x/2])/Sqrt[-a^2 + b^2]] + 4*a^2*b*Cot[x/2] - a*b^2*Csc[x/2]^2 - 8*a^3*Log[Cos[x/2]] + 12*a*b^2*Log[Cos[x/2]] + 8*a^3*Log[Sin[x/2]] - 12*a*b^2*Log[Sin[x/2]] + a*b^2*Sec[x/2]^2 - 4*a^2*b*Tan[x/2])/(8*a*b^3)","A",1
23,1,312,186,0.9222669,"\int \frac{\cot ^6(x)}{a+b \csc (x)} \, dx","Integrate[Cot[x]^6/(a + b*Csc[x]),x]","\frac{192 a^5 \log \left(\sin \left(\frac{x}{2}\right)\right)-192 a^5 \log \left(\cos \left(\frac{x}{2}\right)\right)-96 a^4 b \tan \left(\frac{x}{2}\right)-24 a^3 b^2 \csc ^2\left(\frac{x}{2}\right)+24 a^3 b^2 \sec ^2\left(\frac{x}{2}\right)-480 a^3 b^2 \log \left(\sin \left(\frac{x}{2}\right)\right)+480 a^3 b^2 \log \left(\cos \left(\frac{x}{2}\right)\right)+224 a^2 b^3 \tan \left(\frac{x}{2}\right)+4 a^2 b^3 \sin (x) \csc ^4\left(\frac{x}{2}\right)-64 a^2 b^3 \sin ^4\left(\frac{x}{2}\right) \csc ^3(x)+384 \left(b^2-a^2\right)^{5/2} \tan ^{-1}\left(\frac{a+b \tan \left(\frac{x}{2}\right)}{\sqrt{b^2-a^2}}\right)+32 a^2 b \left(3 a^2-7 b^2\right) \cot \left(\frac{x}{2}\right)-3 a b^4 \csc ^4\left(\frac{x}{2}\right)+54 a b^4 \csc ^2\left(\frac{x}{2}\right)+3 a b^4 \sec ^4\left(\frac{x}{2}\right)-54 a b^4 \sec ^2\left(\frac{x}{2}\right)+360 a b^4 \log \left(\sin \left(\frac{x}{2}\right)\right)-360 a b^4 \log \left(\cos \left(\frac{x}{2}\right)\right)-192 b^5 x}{192 a b^5}","\frac{2 \left(a^2-b^2\right)^{5/2} \tanh ^{-1}\left(\frac{a+b \tan \left(\frac{x}{2}\right)}{\sqrt{a^2-b^2}}\right)}{a b^5}+\frac{a \left(a^2-3 b^2\right) \cot (x)}{b^4}-\frac{\left(a^2-3 b^2\right) \tanh ^{-1}(\cos (x))}{2 b^3}-\frac{\left(a^2-3 b^2\right) \cot (x) \csc (x)}{2 b^3}-\frac{\left(a^4-3 a^2 b^2+3 b^4\right) \tanh ^{-1}(\cos (x))}{b^5}+\frac{a \cot ^3(x)}{3 b^2}+\frac{a \cot (x)}{b^2}-\frac{x}{a}-\frac{3 \tanh ^{-1}(\cos (x))}{8 b}-\frac{\cot (x) \csc ^3(x)}{4 b}-\frac{3 \cot (x) \csc (x)}{8 b}",1,"(-192*b^5*x + 384*(-a^2 + b^2)^(5/2)*ArcTan[(a + b*Tan[x/2])/Sqrt[-a^2 + b^2]] + 32*a^2*b*(3*a^2 - 7*b^2)*Cot[x/2] - 24*a^3*b^2*Csc[x/2]^2 + 54*a*b^4*Csc[x/2]^2 - 3*a*b^4*Csc[x/2]^4 - 192*a^5*Log[Cos[x/2]] + 480*a^3*b^2*Log[Cos[x/2]] - 360*a*b^4*Log[Cos[x/2]] + 192*a^5*Log[Sin[x/2]] - 480*a^3*b^2*Log[Sin[x/2]] + 360*a*b^4*Log[Sin[x/2]] + 24*a^3*b^2*Sec[x/2]^2 - 54*a*b^4*Sec[x/2]^2 + 3*a*b^4*Sec[x/2]^4 - 64*a^2*b^3*Csc[x]^3*Sin[x/2]^4 + 4*a^2*b^3*Csc[x/2]^4*Sin[x] - 96*a^4*b*Tan[x/2] + 224*a^2*b^3*Tan[x/2])/(192*a*b^5)","A",1