1,1,11,16,0.0154382,"\int \frac{3}{5-4 \cos (x)} \, dx","Integrate[3/(5 - 4*Cos[x]),x]","2 \tan ^{-1}\left(3 \tan \left(\frac{x}{2}\right)\right)","x+2 \tan ^{-1}\left(\frac{\sin (x)}{2-\cos (x)}\right)",1,"2*ArcTan[3*Tan[x/2]]","A",1
2,1,46,19,0.1355134,"\int \frac{1+\cos (x)+2 \sin (x)}{3+\cos ^2(x)+2 \sin (x)-2 \cos (x) \sin (x)} \, dx","Integrate[(1 + Cos[x] + 2*Sin[x])/(3 + Cos[x]^2 + 2*Sin[x] - 2*Cos[x]*Sin[x]),x]","\frac{1}{2} \tan ^{-1}\left(\frac{\cos (x)+1}{-\sin (x)+\cos (x)-1}\right)-\frac{1}{2} \tan ^{-1}\left(\frac{1}{2} \sec ^2\left(\frac{x}{2}\right) (-\sin (x)+\cos (x)-1)\right)","-\tan ^{-1}\left(\frac{2 \cos (x)-\sin (x)}{\sin (x)+2}\right)",1,"ArcTan[(1 + Cos[x])/(-1 + Cos[x] - Sin[x])]/2 - ArcTan[(Sec[x/2]^2*(-1 + Cos[x] - Sin[x]))/2]/2","B",1
3,1,19,19,0.0979267,"\int \frac{2+\cos (x)+5 \sin (x)}{4 \cos (x)-2 \sin (x)+\cos (x) \sin (x)-2 \sin ^2(x)} \, dx","Integrate[(2 + Cos[x] + 5*Sin[x])/(4*Cos[x] - 2*Sin[x] + Cos[x]*Sin[x] - 2*Sin[x]^2),x]","\log (\sin (x)+\cos (x)+3)-\log (\sin (x)-3 \cos (x)+1)","\log (\sin (x)+\cos (x)+3)-\log (\sin (x)-3 \cos (x)+1)",1,"-Log[1 - 3*Cos[x] + Sin[x]] + Log[3 + Cos[x] + Sin[x]]","A",1
4,1,19,19,0.1811954,"\int \frac{3+7 \cos (x)+2 \sin (x)}{1+4 \cos (x)+3 \cos ^2(x)-5 \sin (x)-\cos (x) \sin (x)} \, dx","Integrate[(3 + 7*Cos[x] + 2*Sin[x])/(1 + 4*Cos[x] + 3*Cos[x]^2 - 5*Sin[x] - Cos[x]*Sin[x]),x]","\log (\sin (x)+\cos (x)+3)-\log (-2 \sin (x)+\cos (x)+1)","\log (\sin (x)+\cos (x)+3)-\log (-2 \sin (x)+\cos (x)+1)",1,"-Log[1 + Cos[x] - 2*Sin[x]] + Log[3 + Cos[x] + Sin[x]]","A",1
5,1,61,43,0.129515,"\int \frac{-1+4 \cos (x)+5 \cos ^2(x)}{-1-4 \cos (x)-3 \cos ^2(x)+4 \cos ^3(x)} \, dx","Integrate[(-1 + 4*Cos[x] + 5*Cos[x]^2)/(-1 - 4*Cos[x] - 3*Cos[x]^2 + 4*Cos[x]^3),x]","\tan ^{-1}\left(\frac{1}{4} \left(\sin \left(\frac{x}{2}\right)-3 \sin \left(\frac{3 x}{2}\right)\right) \sec ^3\left(\frac{x}{2}\right)\right)-\tan ^{-1}\left(\frac{1}{4} \left(3 \sin \left(\frac{3 x}{2}\right)-\sin \left(\frac{x}{2}\right)\right) \sec ^3\left(\frac{x}{2}\right)\right)","x-2 \tan ^{-1}\left(\frac{3 \sin (x)+7 \sin (x) \cos (x)}{5 \cos ^2(x)+2 \cos (x)+1}\right)-2 \tan ^{-1}\left(\frac{\sin (x)}{\cos (x)+3}\right)",1,"ArcTan[(Sec[x/2]^3*(Sin[x/2] - 3*Sin[(3*x)/2]))/4] - ArcTan[(Sec[x/2]^3*(-Sin[x/2] + 3*Sin[(3*x)/2]))/4]","A",1
6,1,63,25,0.1364761,"\int \frac{-5+2 \cos (x)+7 \cos ^2(x)}{-1+2 \cos (x)-9 \cos ^2(x)+4 \cos ^3(x)} \, dx","Integrate[(-5 + 2*Cos[x] + 7*Cos[x]^2)/(-1 + 2*Cos[x] - 9*Cos[x]^2 + 4*Cos[x]^3),x]","\tan ^{-1}\left(\frac{1}{4} \left(5 \sin \left(\frac{x}{2}\right)-3 \sin \left(\frac{3 x}{2}\right)\right) \sec ^3\left(\frac{x}{2}\right)\right)-\tan ^{-1}\left(\frac{1}{4} \left(3 \sin \left(\frac{3 x}{2}\right)-5 \sin \left(\frac{x}{2}\right)\right) \sec ^3\left(\frac{x}{2}\right)\right)","x-2 \tan ^{-1}\left(\frac{2 \sin (x) \cos (x)}{2 \cos ^2(x)-\cos (x)+1}\right)",1,"ArcTan[(Sec[x/2]^3*(5*Sin[x/2] - 3*Sin[(3*x)/2]))/4] - ArcTan[(Sec[x/2]^3*(-5*Sin[x/2] + 3*Sin[(3*x)/2]))/4]","B",1
7,1,79,14,0.0162537,"\int \frac{3}{5+4 \sin (x)} \, dx","Integrate[3/(5 + 4*Sin[x]),x]","3 \left(\frac{1}{3} \tan ^{-1}\left(\frac{2 \sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)}{\sin \left(\frac{x}{2}\right)+2 \cos \left(\frac{x}{2}\right)}\right)-\frac{1}{3} \tan ^{-1}\left(\frac{\sin \left(\frac{x}{2}\right)+2 \cos \left(\frac{x}{2}\right)}{2 \sin \left(\frac{x}{2}\right)+\cos \left(\frac{x}{2}\right)}\right)\right)","x+2 \tan ^{-1}\left(\frac{\cos (x)}{\sin (x)+2}\right)",1,"3*(-1/3*ArcTan[(2*Cos[x/2] + Sin[x/2])/(Cos[x/2] + 2*Sin[x/2])] + ArcTan[(Cos[x/2] + 2*Sin[x/2])/(2*Cos[x/2] + Sin[x/2])]/3)","B",1
8,1,15,34,0.0203963,"\int \frac{2}{1+\cos ^2(x)} \, dx","Integrate[2/(1 + Cos[x]^2),x]","\sqrt{2} \tan ^{-1}\left(\frac{\tan (x)}{\sqrt{2}}\right)","\sqrt{2} x-\sqrt{2} \tan ^{-1}\left(\frac{\sin (x) \cos (x)}{\cos ^2(x)+\sqrt{2}+1}\right)",1,"Sqrt[2]*ArcTan[Tan[x]/Sqrt[2]]","A",1
9,1,50,54,0.0859206,"\int \frac{1}{p+q \cos (x)+r \sin (x)} \, dx","Integrate[(p + q*Cos[x] + r*Sin[x])^(-1),x]","-\frac{2 \tanh ^{-1}\left(\frac{(p-q) \tan \left(\frac{x}{2}\right)+r}{\sqrt{-p^2+q^2+r^2}}\right)}{\sqrt{-p^2+q^2+r^2}}","\frac{2 \tan ^{-1}\left(\frac{(p-q) \tan \left(\frac{x}{2}\right)+r}{\sqrt{p^2-q^2-r^2}}\right)}{\sqrt{p^2-q^2-r^2}}",1,"(-2*ArcTanh[(r + (p - q)*Tan[x/2])/Sqrt[-p^2 + q^2 + r^2]])/Sqrt[-p^2 + q^2 + r^2]","A",1