1,1,83,119,0.2583587,"\int (a+a \cos (c+d x))^{7/2} \, dx","Integrate[(a + a*Cos[c + d*x])^(7/2),x]","\frac{a^3 \left(1225 \sin \left(\frac{1}{2} (c+d x)\right)+245 \sin \left(\frac{3}{2} (c+d x)\right)+49 \sin \left(\frac{5}{2} (c+d x)\right)+5 \sin \left(\frac{7}{2} (c+d x)\right)\right) \sec \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\cos (c+d x)+1)}}{140 d}","\frac{256 a^4 \sin (c+d x)}{35 d \sqrt{a \cos (c+d x)+a}}+\frac{64 a^3 \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{35 d}+\frac{24 a^2 \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{35 d}+\frac{2 a \sin (c+d x) (a \cos (c+d x)+a)^{5/2}}{7 d}",1,"(a^3*Sqrt[a*(1 + Cos[c + d*x])]*Sec[(c + d*x)/2]*(1225*Sin[(c + d*x)/2] + 245*Sin[(3*(c + d*x))/2] + 49*Sin[(5*(c + d*x))/2] + 5*Sin[(7*(c + d*x))/2]))/(140*d)","A",1
2,1,71,89,0.1140284,"\int (a+a \cos (c+d x))^{5/2} \, dx","Integrate[(a + a*Cos[c + d*x])^(5/2),x]","\frac{a^2 \left(150 \sin \left(\frac{1}{2} (c+d x)\right)+25 \sin \left(\frac{3}{2} (c+d x)\right)+3 \sin \left(\frac{5}{2} (c+d x)\right)\right) \sec \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\cos (c+d x)+1)}}{30 d}","\frac{64 a^3 \sin (c+d x)}{15 d \sqrt{a \cos (c+d x)+a}}+\frac{16 a^2 \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{15 d}+\frac{2 a \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{5 d}",1,"(a^2*Sqrt[a*(1 + Cos[c + d*x])]*Sec[(c + d*x)/2]*(150*Sin[(c + d*x)/2] + 25*Sin[(3*(c + d*x))/2] + 3*Sin[(5*(c + d*x))/2]))/(30*d)","A",1
3,1,55,59,0.0657614,"\int (a+a \cos (c+d x))^{3/2} \, dx","Integrate[(a + a*Cos[c + d*x])^(3/2),x]","\frac{a \left(9 \sin \left(\frac{1}{2} (c+d x)\right)+\sin \left(\frac{3}{2} (c+d x)\right)\right) \sec \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\cos (c+d x)+1)}}{3 d}","\frac{8 a^2 \sin (c+d x)}{3 d \sqrt{a \cos (c+d x)+a}}+\frac{2 a \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{3 d}",1,"(a*Sqrt[a*(1 + Cos[c + d*x])]*Sec[(c + d*x)/2]*(9*Sin[(c + d*x)/2] + Sin[(3*(c + d*x))/2]))/(3*d)","A",1
4,1,29,26,0.0270462,"\int \sqrt{a+a \cos (c+d x)} \, dx","Integrate[Sqrt[a + a*Cos[c + d*x]],x]","\frac{2 \tan \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\cos (c+d x)+1)}}{d}","\frac{2 a \sin (c+d x)}{d \sqrt{a \cos (c+d x)+a}}",1,"(2*Sqrt[a*(1 + Cos[c + d*x])]*Tan[(c + d*x)/2])/d","A",1
5,1,40,46,0.0160762,"\int \frac{1}{\sqrt{a+a \cos (c+d x)}} \, dx","Integrate[1/Sqrt[a + a*Cos[c + d*x]],x]","\frac{2 \cos \left(\frac{1}{2} (c+d x)\right) \tanh ^{-1}\left(\sin \left(\frac{1}{2} (c+d x)\right)\right)}{d \sqrt{a (\cos (c+d x)+1)}}","\frac{\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{a \cos (c+d x)+a}}\right)}{\sqrt{a} d}",1,"(2*ArcTanh[Sin[(c + d*x)/2]]*Cos[(c + d*x)/2])/(d*Sqrt[a*(1 + Cos[c + d*x])])","A",1
6,1,63,77,0.0871092,"\int \frac{1}{(a+a \cos (c+d x))^{3/2}} \, dx","Integrate[(a + a*Cos[c + d*x])^(-3/2),x]","\frac{\cos ^2\left(\frac{1}{2} (c+d x)\right) \left(\tan \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right) \tanh ^{-1}\left(\sin \left(\frac{1}{2} (c+d x)\right)\right)\right)}{d (a (\cos (c+d x)+1))^{3/2}}","\frac{\tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{a \cos (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}+\frac{\sin (c+d x)}{2 d (a \cos (c+d x)+a)^{3/2}}",1,"(Cos[(c + d*x)/2]^2*(ArcTanh[Sin[(c + d*x)/2]]*Cos[(c + d*x)/2] + Tan[(c + d*x)/2]))/(d*(a*(1 + Cos[c + d*x]))^(3/2))","A",1
7,1,65,107,0.1905878,"\int \frac{1}{(a+a \cos (c+d x))^{5/2}} \, dx","Integrate[(a + a*Cos[c + d*x])^(-5/2),x]","\frac{14 \sin (c+d x)+3 \sin (2 (c+d x))+24 \cos ^5\left(\frac{1}{2} (c+d x)\right) \tanh ^{-1}\left(\sin \left(\frac{1}{2} (c+d x)\right)\right)}{32 d (a (\cos (c+d x)+1))^{5/2}}","\frac{3 \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{a \cos (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}+\frac{3 \sin (c+d x)}{16 a d (a \cos (c+d x)+a)^{3/2}}+\frac{\sin (c+d x)}{4 d (a \cos (c+d x)+a)^{5/2}}",1,"(24*ArcTanh[Sin[(c + d*x)/2]]*Cos[(c + d*x)/2]^5 + 14*Sin[c + d*x] + 3*Sin[2*(c + d*x)])/(32*d*(a*(1 + Cos[c + d*x]))^(5/2))","A",1
8,1,69,67,0.1006189,"\int (a+a \cos (c+d x))^{4/3} \, dx","Integrate[(a + a*Cos[c + d*x])^(4/3),x]","-\frac{6 \sqrt{\sin ^2\left(\frac{1}{2} (c+d x)\right)} \cot \left(\frac{1}{2} (c+d x)\right) (a (\cos (c+d x)+1))^{4/3} \, _2F_1\left(\frac{1}{2},\frac{11}{6};\frac{17}{6};\cos ^2\left(\frac{1}{2} (c+d x)\right)\right)}{11 d}","\frac{2\ 2^{5/6} a \sin (c+d x) \sqrt[3]{a \cos (c+d x)+a} \, _2F_1\left(-\frac{5}{6},\frac{1}{2};\frac{3}{2};\frac{1}{2} (1-\cos (c+d x))\right)}{d (\cos (c+d x)+1)^{5/6}}",1,"(-6*(a*(1 + Cos[c + d*x]))^(4/3)*Cot[(c + d*x)/2]*Hypergeometric2F1[1/2, 11/6, 17/6, Cos[(c + d*x)/2]^2]*Sqrt[Sin[(c + d*x)/2]^2])/(11*d)","A",1
9,1,69,66,0.0678691,"\int (a+a \cos (c+d x))^{2/3} \, dx","Integrate[(a + a*Cos[c + d*x])^(2/3),x]","-\frac{6 \sqrt{\sin ^2\left(\frac{1}{2} (c+d x)\right)} \cot \left(\frac{1}{2} (c+d x)\right) (a (\cos (c+d x)+1))^{2/3} \, _2F_1\left(\frac{1}{2},\frac{7}{6};\frac{13}{6};\cos ^2\left(\frac{1}{2} (c+d x)\right)\right)}{7 d}","\frac{2 \sqrt[6]{2} \sin (c+d x) (a \cos (c+d x)+a)^{2/3} \, _2F_1\left(-\frac{1}{6},\frac{1}{2};\frac{3}{2};\frac{1}{2} (1-\cos (c+d x))\right)}{d (\cos (c+d x)+1)^{7/6}}",1,"(-6*(a*(1 + Cos[c + d*x]))^(2/3)*Cot[(c + d*x)/2]*Hypergeometric2F1[1/2, 7/6, 13/6, Cos[(c + d*x)/2]^2]*Sqrt[Sin[(c + d*x)/2]^2])/(7*d)","A",1
10,1,69,65,0.0646004,"\int \sqrt[3]{a+a \cos (c+d x)} \, dx","Integrate[(a + a*Cos[c + d*x])^(1/3),x]","-\frac{6 \sqrt{\sin ^2\left(\frac{1}{2} (c+d x)\right)} \cot \left(\frac{1}{2} (c+d x)\right) \sqrt[3]{a (\cos (c+d x)+1)} \, _2F_1\left(\frac{1}{2},\frac{5}{6};\frac{11}{6};\cos ^2\left(\frac{1}{2} (c+d x)\right)\right)}{5 d}","\frac{2^{5/6} \sin (c+d x) \sqrt[3]{a \cos (c+d x)+a} \, _2F_1\left(\frac{1}{6},\frac{1}{2};\frac{3}{2};\frac{1}{2} (1-\cos (c+d x))\right)}{d (\cos (c+d x)+1)^{5/6}}",1,"(-6*(a*(1 + Cos[c + d*x]))^(1/3)*Cot[(c + d*x)/2]*Hypergeometric2F1[1/2, 5/6, 11/6, Cos[(c + d*x)/2]^2]*Sqrt[Sin[(c + d*x)/2]^2])/(5*d)","A",1
11,1,67,65,0.056002,"\int \frac{1}{\sqrt[3]{a+a \cos (c+d x)}} \, dx","Integrate[(a + a*Cos[c + d*x])^(-1/3),x]","-\frac{6 \sqrt{\sin ^2\left(\frac{1}{2} (c+d x)\right)} \cot \left(\frac{1}{2} (c+d x)\right) \, _2F_1\left(\frac{1}{6},\frac{1}{2};\frac{7}{6};\cos ^2\left(\frac{1}{2} (c+d x)\right)\right)}{d \sqrt[3]{a (\cos (c+d x)+1)}}","\frac{\sqrt[6]{2} \sin (c+d x) \, _2F_1\left(\frac{1}{2},\frac{5}{6};\frac{3}{2};\frac{1}{2} (1-\cos (c+d x))\right)}{d \sqrt[6]{\cos (c+d x)+1} \sqrt[3]{a \cos (c+d x)+a}}",1,"(-6*Cot[(c + d*x)/2]*Hypergeometric2F1[1/6, 1/2, 7/6, Cos[(c + d*x)/2]^2]*Sqrt[Sin[(c + d*x)/2]^2])/(d*(a*(1 + Cos[c + d*x]))^(1/3))","A",1
12,1,67,65,0.057186,"\int \frac{1}{(a+a \cos (c+d x))^{2/3}} \, dx","Integrate[(a + a*Cos[c + d*x])^(-2/3),x]","\frac{6 \sqrt{\sin ^2\left(\frac{1}{2} (c+d x)\right)} \cot \left(\frac{1}{2} (c+d x)\right) \, _2F_1\left(-\frac{1}{6},\frac{1}{2};\frac{5}{6};\cos ^2\left(\frac{1}{2} (c+d x)\right)\right)}{d (a (\cos (c+d x)+1))^{2/3}}","\frac{\sin (c+d x) \sqrt[6]{\cos (c+d x)+1} \, _2F_1\left(\frac{1}{2},\frac{7}{6};\frac{3}{2};\frac{1}{2} (1-\cos (c+d x))\right)}{\sqrt[6]{2} d (a \cos (c+d x)+a)^{2/3}}",1,"(6*Cot[(c + d*x)/2]*Hypergeometric2F1[-1/6, 1/2, 5/6, Cos[(c + d*x)/2]^2]*Sqrt[Sin[(c + d*x)/2]^2])/(d*(a*(1 + Cos[c + d*x]))^(2/3))","A",1
13,1,69,68,0.0649686,"\int \frac{1}{(a+a \cos (c+d x))^{4/3}} \, dx","Integrate[(a + a*Cos[c + d*x])^(-4/3),x]","\frac{6 \sqrt{\sin ^2\left(\frac{1}{2} (c+d x)\right)} \cot \left(\frac{1}{2} (c+d x)\right) \, _2F_1\left(-\frac{5}{6},\frac{1}{2};\frac{1}{6};\cos ^2\left(\frac{1}{2} (c+d x)\right)\right)}{5 d (a (\cos (c+d x)+1))^{4/3}}","\frac{\sin (c+d x) \, _2F_1\left(\frac{1}{2},\frac{11}{6};\frac{3}{2};\frac{1}{2} (1-\cos (c+d x))\right)}{2^{5/6} a d \sqrt[6]{\cos (c+d x)+1} \sqrt[3]{a \cos (c+d x)+a}}",1,"(6*Cot[(c + d*x)/2]*Hypergeometric2F1[-5/6, 1/2, 1/6, Cos[(c + d*x)/2]^2]*Sqrt[Sin[(c + d*x)/2]^2])/(5*d*(a*(1 + Cos[c + d*x]))^(4/3))","A",1
14,1,74,73,0.0715974,"\int (a+a \cos (c+d x))^n \, dx","Integrate[(a + a*Cos[c + d*x])^n,x]","-\frac{2 \sqrt{\sin ^2\left(\frac{1}{2} (c+d x)\right)} \cot \left(\frac{1}{2} (c+d x)\right) (a (\cos (c+d x)+1))^n \, _2F_1\left(\frac{1}{2},n+\frac{1}{2};n+\frac{3}{2};\cos ^2\left(\frac{1}{2} (c+d x)\right)\right)}{2 d n+d}","\frac{2^{n+\frac{1}{2}} \sin (c+d x) (\cos (c+d x)+1)^{-n-\frac{1}{2}} (a \cos (c+d x)+a)^n \, _2F_1\left(\frac{1}{2},\frac{1}{2}-n;\frac{3}{2};\frac{1}{2} (1-\cos (c+d x))\right)}{d}",1,"(-2*(a*(1 + Cos[c + d*x]))^n*Cot[(c + d*x)/2]*Hypergeometric2F1[1/2, 1/2 + n, 3/2 + n, Cos[(c + d*x)/2]^2]*Sqrt[Sin[(c + d*x)/2]^2])/(d + 2*d*n)","A",1
15,1,75,75,0.082277,"\int (a-a \cos (c+d x))^n \, dx","Integrate[(a - a*Cos[c + d*x])^n,x]","\frac{\sqrt{2} \sqrt{\cos (c+d x)+1} \tan \left(\frac{1}{2} (c+d x)\right) (a-a \cos (c+d x))^n \, _2F_1\left(\frac{1}{2},n+\frac{1}{2};n+\frac{3}{2};\sin ^2\left(\frac{1}{2} (c+d x)\right)\right)}{2 d n+d}","-\frac{2^{n+\frac{1}{2}} \sin (c+d x) (1-\cos (c+d x))^{-n-\frac{1}{2}} (a-a \cos (c+d x))^n \, _2F_1\left(\frac{1}{2},\frac{1}{2}-n;\frac{3}{2};\frac{1}{2} (\cos (c+d x)+1)\right)}{d}",1,"(Sqrt[2]*Sqrt[1 + Cos[c + d*x]]*(a - a*Cos[c + d*x])^n*Hypergeometric2F1[1/2, 1/2 + n, 3/2 + n, Sin[(c + d*x)/2]^2]*Tan[(c + d*x)/2])/(d + 2*d*n)","A",1
16,1,77,59,0.0933495,"\int (2+2 \cos (c+d x))^n \, dx","Integrate[(2 + 2*Cos[c + d*x])^n,x]","-\frac{2^{n+1} \sqrt{\sin ^2\left(\frac{1}{2} (c+d x)\right)} \cot \left(\frac{1}{2} (c+d x)\right) (\cos (c+d x)+1)^n \, _2F_1\left(\frac{1}{2},n+\frac{1}{2};n+\frac{3}{2};\cos ^2\left(\frac{1}{2} (c+d x)\right)\right)}{2 d n+d}","\frac{2^{2 n+\frac{1}{2}} \sin (c+d x) \, _2F_1\left(\frac{1}{2},\frac{1}{2}-n;\frac{3}{2};\frac{1}{2} (1-\cos (c+d x))\right)}{d \sqrt{\cos (c+d x)+1}}",1,"-((2^(1 + n)*(1 + Cos[c + d*x])^n*Cot[(c + d*x)/2]*Hypergeometric2F1[1/2, 1/2 + n, 3/2 + n, Cos[(c + d*x)/2]^2]*Sqrt[Sin[(c + d*x)/2]^2])/(d + 2*d*n))","A",1
17,1,74,60,0.0739701,"\int (2-2 \cos (c+d x))^n \, dx","Integrate[(2 - 2*Cos[c + d*x])^n,x]","\frac{\sqrt{2} \sqrt{\cos (c+d x)+1} \tan \left(\frac{1}{2} (c+d x)\right) (2-2 \cos (c+d x))^n \, _2F_1\left(\frac{1}{2},n+\frac{1}{2};n+\frac{3}{2};\sin ^2\left(\frac{1}{2} (c+d x)\right)\right)}{2 d n+d}","-\frac{2^{2 n+\frac{1}{2}} \sin (c+d x) \, _2F_1\left(\frac{1}{2},\frac{1}{2}-n;\frac{3}{2};\frac{1}{2} (\cos (c+d x)+1)\right)}{d \sqrt{1-\cos (c+d x)}}",1,"(Sqrt[2]*(2 - 2*Cos[c + d*x])^n*Sqrt[1 + Cos[c + d*x]]*Hypergeometric2F1[1/2, 1/2 + n, 3/2 + n, Sin[(c + d*x)/2]^2]*Tan[(c + d*x)/2])/(d + 2*d*n)","A",1
18,1,20,31,0.0278445,"\int \frac{1}{5+3 \cos (c+d x)} \, dx","Integrate[(5 + 3*Cos[c + d*x])^(-1),x]","-\frac{\tan ^{-1}\left(2 \cot \left(\frac{1}{2} (c+d x)\right)\right)}{2 d}","\frac{x}{4}-\frac{\tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)+3}\right)}{2 d}",1,"-1/2*ArcTan[2*Cot[(c + d*x)/2]]/d","A",1
19,1,43,56,0.0880697,"\int \frac{1}{(5+3 \cos (c+d x))^2} \, dx","Integrate[(5 + 3*Cos[c + d*x])^(-2),x]","-\frac{\frac{6 \sin (c+d x)}{3 \cos (c+d x)+5}+5 \tan ^{-1}\left(2 \cot \left(\frac{1}{2} (c+d x)\right)\right)}{32 d}","-\frac{3 \sin (c+d x)}{16 d (3 \cos (c+d x)+5)}-\frac{5 \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)+3}\right)}{32 d}+\frac{5 x}{64}",1,"-1/32*(5*ArcTan[2*Cot[(c + d*x)/2]] + (6*Sin[c + d*x])/(5 + 3*Cos[c + d*x]))/d","A",1
20,1,56,81,0.1918354,"\int \frac{1}{(5+3 \cos (c+d x))^3} \, dx","Integrate[(5 + 3*Cos[c + d*x])^(-3),x]","-\frac{\frac{3 (182 \sin (c+d x)+45 \sin (2 (c+d x)))}{(3 \cos (c+d x)+5)^2}+59 \tan ^{-1}\left(2 \cot \left(\frac{1}{2} (c+d x)\right)\right)}{1024 d}","-\frac{45 \sin (c+d x)}{512 d (3 \cos (c+d x)+5)}-\frac{3 \sin (c+d x)}{32 d (3 \cos (c+d x)+5)^2}-\frac{59 \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)+3}\right)}{1024 d}+\frac{59 x}{2048}",1,"-1/1024*(59*ArcTan[2*Cot[(c + d*x)/2]] + (3*(182*Sin[c + d*x] + 45*Sin[2*(c + d*x)]))/(5 + 3*Cos[c + d*x])^2)/d","A",1
21,1,66,106,0.2642279,"\int \frac{1}{(5+3 \cos (c+d x))^4} \, dx","Integrate[(5 + 3*Cos[c + d*x])^(-4),x]","-\frac{\frac{9 (4883 \sin (c+d x)+2340 \sin (2 (c+d x))+311 \sin (3 (c+d x)))}{(3 \cos (c+d x)+5)^3}+770 \tan ^{-1}\left(2 \cot \left(\frac{1}{2} (c+d x)\right)\right)}{32768 d}","-\frac{311 \sin (c+d x)}{8192 d (3 \cos (c+d x)+5)}-\frac{25 \sin (c+d x)}{512 d (3 \cos (c+d x)+5)^2}-\frac{\sin (c+d x)}{16 d (3 \cos (c+d x)+5)^3}-\frac{385 \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)+3}\right)}{16384 d}+\frac{385 x}{32768}",1,"-1/32768*(770*ArcTan[2*Cot[(c + d*x)/2]] + (9*(4883*Sin[c + d*x] + 2340*Sin[2*(c + d*x)] + 311*Sin[3*(c + d*x)]))/(5 + 3*Cos[c + d*x])^3)/d","A",1
22,1,20,33,0.0265662,"\int \frac{1}{5-3 \cos (c+d x)} \, dx","Integrate[(5 - 3*Cos[c + d*x])^(-1),x]","\frac{\tan ^{-1}\left(2 \tan \left(\frac{1}{2} (c+d x)\right)\right)}{2 d}","\frac{\tan ^{-1}\left(\frac{\sin (c+d x)}{3-\cos (c+d x)}\right)}{2 d}+\frac{x}{4}",1,"ArcTan[2*Tan[(c + d*x)/2]]/(2*d)","A",1
23,1,43,58,0.0772609,"\int \frac{1}{(5-3 \cos (c+d x))^2} \, dx","Integrate[(5 - 3*Cos[c + d*x])^(-2),x]","\frac{5 \tan ^{-1}\left(2 \tan \left(\frac{1}{2} (c+d x)\right)\right)-\frac{6 \sin (c+d x)}{3 \cos (c+d x)-5}}{32 d}","\frac{3 \sin (c+d x)}{16 d (5-3 \cos (c+d x))}+\frac{5 \tan ^{-1}\left(\frac{\sin (c+d x)}{3-\cos (c+d x)}\right)}{32 d}+\frac{5 x}{64}",1,"(5*ArcTan[2*Tan[(c + d*x)/2]] - (6*Sin[c + d*x])/(-5 + 3*Cos[c + d*x]))/(32*d)","A",1
24,1,65,83,0.1385412,"\int \frac{1}{(5-3 \cos (c+d x))^3} \, dx","Integrate[(5 - 3*Cos[c + d*x])^(-3),x]","\frac{546 \sin (c+d x)-135 \sin (2 (c+d x))+59 (5-3 \cos (c+d x))^2 \tan ^{-1}\left(2 \tan \left(\frac{1}{2} (c+d x)\right)\right)}{1024 d (5-3 \cos (c+d x))^2}","\frac{45 \sin (c+d x)}{512 d (5-3 \cos (c+d x))}+\frac{3 \sin (c+d x)}{32 d (5-3 \cos (c+d x))^2}+\frac{59 \tan ^{-1}\left(\frac{\sin (c+d x)}{3-\cos (c+d x)}\right)}{1024 d}+\frac{59 x}{2048}",1,"(59*ArcTan[2*Tan[(c + d*x)/2]]*(5 - 3*Cos[c + d*x])^2 + 546*Sin[c + d*x] - 135*Sin[2*(c + d*x)])/(1024*d*(5 - 3*Cos[c + d*x])^2)","A",1
25,1,66,108,0.2192826,"\int \frac{1}{(5-3 \cos (c+d x))^4} \, dx","Integrate[(5 - 3*Cos[c + d*x])^(-4),x]","\frac{770 \tan ^{-1}\left(2 \tan \left(\frac{1}{2} (c+d x)\right)\right)-\frac{9 (4883 \sin (c+d x)-2340 \sin (2 (c+d x))+311 \sin (3 (c+d x)))}{(3 \cos (c+d x)-5)^3}}{32768 d}","\frac{311 \sin (c+d x)}{8192 d (5-3 \cos (c+d x))}+\frac{25 \sin (c+d x)}{512 d (5-3 \cos (c+d x))^2}+\frac{\sin (c+d x)}{16 d (5-3 \cos (c+d x))^3}+\frac{385 \tan ^{-1}\left(\frac{\sin (c+d x)}{3-\cos (c+d x)}\right)}{16384 d}+\frac{385 x}{32768}",1,"(770*ArcTan[2*Tan[(c + d*x)/2]] - (9*(4883*Sin[c + d*x] - 2340*Sin[2*(c + d*x)] + 311*Sin[3*(c + d*x)]))/(-5 + 3*Cos[c + d*x])^3)/(32768*d)","A",1
26,1,20,33,0.0237646,"\int \frac{1}{-5+3 \cos (c+d x)} \, dx","Integrate[(-5 + 3*Cos[c + d*x])^(-1),x]","-\frac{\tan ^{-1}\left(2 \tan \left(\frac{1}{2} (c+d x)\right)\right)}{2 d}","-\frac{\tan ^{-1}\left(\frac{\sin (c+d x)}{3-\cos (c+d x)}\right)}{2 d}-\frac{x}{4}",1,"-1/2*ArcTan[2*Tan[(c + d*x)/2]]/d","A",1
27,1,43,58,0.0227331,"\int \frac{1}{(-5+3 \cos (c+d x))^2} \, dx","Integrate[(-5 + 3*Cos[c + d*x])^(-2),x]","\frac{5 \tan ^{-1}\left(2 \tan \left(\frac{1}{2} (c+d x)\right)\right)-\frac{6 \sin (c+d x)}{3 \cos (c+d x)-5}}{32 d}","\frac{3 \sin (c+d x)}{16 d (5-3 \cos (c+d x))}+\frac{5 \tan ^{-1}\left(\frac{\sin (c+d x)}{3-\cos (c+d x)}\right)}{32 d}+\frac{5 x}{64}",1,"(5*ArcTan[2*Tan[(c + d*x)/2]] - (6*Sin[c + d*x])/(-5 + 3*Cos[c + d*x]))/(32*d)","A",1
28,1,65,83,0.1031923,"\int \frac{1}{(-5+3 \cos (c+d x))^3} \, dx","Integrate[(-5 + 3*Cos[c + d*x])^(-3),x]","\frac{-546 \sin (c+d x)+135 \sin (2 (c+d x))-59 (5-3 \cos (c+d x))^2 \tan ^{-1}\left(2 \tan \left(\frac{1}{2} (c+d x)\right)\right)}{1024 d (5-3 \cos (c+d x))^2}","-\frac{45 \sin (c+d x)}{512 d (5-3 \cos (c+d x))}-\frac{3 \sin (c+d x)}{32 d (5-3 \cos (c+d x))^2}-\frac{59 \tan ^{-1}\left(\frac{\sin (c+d x)}{3-\cos (c+d x)}\right)}{1024 d}-\frac{59 x}{2048}",1,"(-59*ArcTan[2*Tan[(c + d*x)/2]]*(5 - 3*Cos[c + d*x])^2 - 546*Sin[c + d*x] + 135*Sin[2*(c + d*x)])/(1024*d*(5 - 3*Cos[c + d*x])^2)","A",1
29,1,66,108,0.0245222,"\int \frac{1}{(-5+3 \cos (c+d x))^4} \, dx","Integrate[(-5 + 3*Cos[c + d*x])^(-4),x]","\frac{770 \tan ^{-1}\left(2 \tan \left(\frac{1}{2} (c+d x)\right)\right)-\frac{9 (4883 \sin (c+d x)-2340 \sin (2 (c+d x))+311 \sin (3 (c+d x)))}{(3 \cos (c+d x)-5)^3}}{32768 d}","\frac{311 \sin (c+d x)}{8192 d (5-3 \cos (c+d x))}+\frac{25 \sin (c+d x)}{512 d (5-3 \cos (c+d x))^2}+\frac{\sin (c+d x)}{16 d (5-3 \cos (c+d x))^3}+\frac{385 \tan ^{-1}\left(\frac{\sin (c+d x)}{3-\cos (c+d x)}\right)}{16384 d}+\frac{385 x}{32768}",1,"(770*ArcTan[2*Tan[(c + d*x)/2]] - (9*(4883*Sin[c + d*x] - 2340*Sin[2*(c + d*x)] + 311*Sin[3*(c + d*x)]))/(-5 + 3*Cos[c + d*x])^3)/(32768*d)","A",1
30,1,20,31,0.0255457,"\int \frac{1}{-5-3 \cos (c+d x)} \, dx","Integrate[(-5 - 3*Cos[c + d*x])^(-1),x]","\frac{\tan ^{-1}\left(2 \cot \left(\frac{1}{2} (c+d x)\right)\right)}{2 d}","\frac{\tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)+3}\right)}{2 d}-\frac{x}{4}",1,"ArcTan[2*Cot[(c + d*x)/2]]/(2*d)","A",1
31,1,43,56,0.0229235,"\int \frac{1}{(-5-3 \cos (c+d x))^2} \, dx","Integrate[(-5 - 3*Cos[c + d*x])^(-2),x]","-\frac{\frac{6 \sin (c+d x)}{3 \cos (c+d x)+5}+5 \tan ^{-1}\left(2 \cot \left(\frac{1}{2} (c+d x)\right)\right)}{32 d}","-\frac{3 \sin (c+d x)}{16 d (3 \cos (c+d x)+5)}-\frac{5 \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)+3}\right)}{32 d}+\frac{5 x}{64}",1,"-1/32*(5*ArcTan[2*Cot[(c + d*x)/2]] + (6*Sin[c + d*x])/(5 + 3*Cos[c + d*x]))/d","A",1
32,1,65,81,0.0990293,"\int \frac{1}{(-5-3 \cos (c+d x))^3} \, dx","Integrate[(-5 - 3*Cos[c + d*x])^(-3),x]","\frac{546 \sin (c+d x)+135 \sin (2 (c+d x))+59 (3 \cos (c+d x)+5)^2 \tan ^{-1}\left(2 \cot \left(\frac{1}{2} (c+d x)\right)\right)}{1024 d (3 \cos (c+d x)+5)^2}","\frac{45 \sin (c+d x)}{512 d (3 \cos (c+d x)+5)}+\frac{3 \sin (c+d x)}{32 d (3 \cos (c+d x)+5)^2}+\frac{59 \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)+3}\right)}{1024 d}-\frac{59 x}{2048}",1,"(59*ArcTan[2*Cot[(c + d*x)/2]]*(5 + 3*Cos[c + d*x])^2 + 546*Sin[c + d*x] + 135*Sin[2*(c + d*x)])/(1024*d*(5 + 3*Cos[c + d*x])^2)","A",1
33,1,66,106,0.0250499,"\int \frac{1}{(-5-3 \cos (c+d x))^4} \, dx","Integrate[(-5 - 3*Cos[c + d*x])^(-4),x]","-\frac{\frac{9 (4883 \sin (c+d x)+2340 \sin (2 (c+d x))+311 \sin (3 (c+d x)))}{(3 \cos (c+d x)+5)^3}+770 \tan ^{-1}\left(2 \cot \left(\frac{1}{2} (c+d x)\right)\right)}{32768 d}","-\frac{311 \sin (c+d x)}{8192 d (3 \cos (c+d x)+5)}-\frac{25 \sin (c+d x)}{512 d (3 \cos (c+d x)+5)^2}-\frac{\sin (c+d x)}{16 d (3 \cos (c+d x)+5)^3}-\frac{385 \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)+3}\right)}{16384 d}+\frac{385 x}{32768}",1,"-1/32768*(770*ArcTan[2*Cot[(c + d*x)/2]] + (9*(4883*Sin[c + d*x] + 2340*Sin[2*(c + d*x)] + 311*Sin[3*(c + d*x)]))/(5 + 3*Cos[c + d*x])^3)/d","A",1
34,1,65,65,0.0246501,"\int \frac{1}{3+5 \cos (c+d x)} \, dx","Integrate[(3 + 5*Cos[c + d*x])^(-1),x]","\frac{\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+2 \cos \left(\frac{1}{2} (c+d x)\right)\right)}{4 d}-\frac{\log \left(2 \cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}{4 d}","\frac{\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+2 \cos \left(\frac{1}{2} (c+d x)\right)\right)}{4 d}-\frac{\log \left(2 \cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}{4 d}",1,"-1/4*Log[2*Cos[(c + d*x)/2] - Sin[(c + d*x)/2]]/d + Log[2*Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]/(4*d)","A",1
35,1,143,90,0.0823136,"\int \frac{1}{(3+5 \cos (c+d x))^2} \, dx","Integrate[(3 + 5*Cos[c + d*x])^(-2),x]","\frac{20 \sin (c+d x)+9 \log \left(2 \cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+15 \cos (c+d x) \left(\log \left(2 \cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+2 \cos \left(\frac{1}{2} (c+d x)\right)\right)\right)-9 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+2 \cos \left(\frac{1}{2} (c+d x)\right)\right)}{64 d (5 \cos (c+d x)+3)}","\frac{5 \sin (c+d x)}{16 d (5 \cos (c+d x)+3)}+\frac{3 \log \left(2 \cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}{64 d}-\frac{3 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+2 \cos \left(\frac{1}{2} (c+d x)\right)\right)}{64 d}",1,"(9*Log[2*Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 15*Cos[c + d*x]*(Log[2*Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[2*Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) - 9*Log[2*Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + 20*Sin[c + d*x])/(64*d*(3 + 5*Cos[c + d*x]))","A",1
36,1,217,115,0.1963544,"\int \frac{1}{(3+5 \cos (c+d x))^3} \, dx","Integrate[(3 + 5*Cos[c + d*x])^(-3),x]","-\frac{45 \sin \left(\frac{1}{2} (c+d x)\right)}{2048 d \left(2 \cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}-\frac{45 \sin \left(\frac{1}{2} (c+d x)\right)}{2048 d \left(\sin \left(\frac{1}{2} (c+d x)\right)+2 \cos \left(\frac{1}{2} (c+d x)\right)\right)}+\frac{5}{512 d \left(2 \cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^2}-\frac{5}{512 d \left(\sin \left(\frac{1}{2} (c+d x)\right)+2 \cos \left(\frac{1}{2} (c+d x)\right)\right)^2}-\frac{43 \log \left(2 \cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}{2048 d}+\frac{43 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+2 \cos \left(\frac{1}{2} (c+d x)\right)\right)}{2048 d}","-\frac{45 \sin (c+d x)}{512 d (5 \cos (c+d x)+3)}+\frac{5 \sin (c+d x)}{32 d (5 \cos (c+d x)+3)^2}-\frac{43 \log \left(2 \cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}{2048 d}+\frac{43 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+2 \cos \left(\frac{1}{2} (c+d x)\right)\right)}{2048 d}",1,"(-43*Log[2*Cos[(c + d*x)/2] - Sin[(c + d*x)/2]])/(2048*d) + (43*Log[2*Cos[(c + d*x)/2] + Sin[(c + d*x)/2]])/(2048*d) + 5/(512*d*(2*Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2) - (45*Sin[(c + d*x)/2])/(2048*d*(2*Cos[(c + d*x)/2] - Sin[(c + d*x)/2])) - 5/(512*d*(2*Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2) - (45*Sin[(c + d*x)/2])/(2048*d*(2*Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))","A",1
37,1,296,140,0.2460423,"\int \frac{1}{(3+5 \cos (c+d x))^4} \, dx","Integrate[(3 + 5*Cos[c + d*x])^(-4),x]","\frac{226140 \sin (c+d x)+190800 \sin (2 (c+d x))+99500 \sin (3 (c+d x))+104625 \cos (3 (c+d x)) \log \left(2 \cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+467046 \log \left(2 \cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+765855 \cos (c+d x) \left(\log \left(2 \cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+2 \cos \left(\frac{1}{2} (c+d x)\right)\right)\right)+376650 \cos (2 (c+d x)) \left(\log \left(2 \cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+2 \cos \left(\frac{1}{2} (c+d x)\right)\right)\right)-104625 \cos (3 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+2 \cos \left(\frac{1}{2} (c+d x)\right)\right)-467046 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+2 \cos \left(\frac{1}{2} (c+d x)\right)\right)}{393216 d (5 \cos (c+d x)+3)^3}","\frac{995 \sin (c+d x)}{24576 d (5 \cos (c+d x)+3)}-\frac{25 \sin (c+d x)}{512 d (5 \cos (c+d x)+3)^2}+\frac{5 \sin (c+d x)}{48 d (5 \cos (c+d x)+3)^3}+\frac{279 \log \left(2 \cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}{32768 d}-\frac{279 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+2 \cos \left(\frac{1}{2} (c+d x)\right)\right)}{32768 d}",1,"(467046*Log[2*Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 104625*Cos[3*(c + d*x)]*Log[2*Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 765855*Cos[c + d*x]*(Log[2*Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[2*Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) + 376650*Cos[2*(c + d*x)]*(Log[2*Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[2*Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) - 467046*Log[2*Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] - 104625*Cos[3*(c + d*x)]*Log[2*Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + 226140*Sin[c + d*x] + 190800*Sin[2*(c + d*x)] + 99500*Sin[3*(c + d*x)])/(393216*d*(3 + 5*Cos[c + d*x])^3)","B",1
38,1,63,63,0.0241819,"\int \frac{1}{3-5 \cos (c+d x)} \, dx","Integrate[(3 - 5*Cos[c + d*x])^(-1),x]","\frac{\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-2 \sin \left(\frac{1}{2} (c+d x)\right)\right)}{4 d}-\frac{\log \left(2 \sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}{4 d}","\frac{\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-2 \sin \left(\frac{1}{2} (c+d x)\right)\right)}{4 d}-\frac{\log \left(2 \sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}{4 d}",1,"Log[Cos[(c + d*x)/2] - 2*Sin[(c + d*x)/2]]/(4*d) - Log[Cos[(c + d*x)/2] + 2*Sin[(c + d*x)/2]]/(4*d)","A",1
39,1,139,88,0.0783711,"\int \frac{1}{(3-5 \cos (c+d x))^2} \, dx","Integrate[(3 - 5*Cos[c + d*x])^(-2),x]","\frac{20 \sin (c+d x)+9 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-2 \sin \left(\frac{1}{2} (c+d x)\right)\right)-15 \cos (c+d x) \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-2 \sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(2 \sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)-9 \log \left(2 \sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}{64 d (5 \cos (c+d x)-3)}","-\frac{5 \sin (c+d x)}{16 d (3-5 \cos (c+d x))}-\frac{3 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-2 \sin \left(\frac{1}{2} (c+d x)\right)\right)}{64 d}+\frac{3 \log \left(2 \sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}{64 d}",1,"(9*Log[Cos[(c + d*x)/2] - 2*Sin[(c + d*x)/2]] - 15*Cos[c + d*x]*(Log[Cos[(c + d*x)/2] - 2*Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + 2*Sin[(c + d*x)/2]]) - 9*Log[Cos[(c + d*x)/2] + 2*Sin[(c + d*x)/2]] + 20*Sin[c + d*x])/(64*d*(-3 + 5*Cos[c + d*x]))","A",1
40,1,211,113,0.1451958,"\int \frac{1}{(3-5 \cos (c+d x))^3} \, dx","Integrate[(3 - 5*Cos[c + d*x])^(-3),x]","-\frac{45 \sin \left(\frac{1}{2} (c+d x)\right)}{1024 d \left(\cos \left(\frac{1}{2} (c+d x)\right)-2 \sin \left(\frac{1}{2} (c+d x)\right)\right)}-\frac{45 \sin \left(\frac{1}{2} (c+d x)\right)}{1024 d \left(2 \sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}-\frac{5}{512 d \left(\cos \left(\frac{1}{2} (c+d x)\right)-2 \sin \left(\frac{1}{2} (c+d x)\right)\right)^2}+\frac{5}{512 d \left(2 \sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2}+\frac{43 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-2 \sin \left(\frac{1}{2} (c+d x)\right)\right)}{2048 d}-\frac{43 \log \left(2 \sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}{2048 d}","\frac{45 \sin (c+d x)}{512 d (3-5 \cos (c+d x))}-\frac{5 \sin (c+d x)}{32 d (3-5 \cos (c+d x))^2}+\frac{43 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-2 \sin \left(\frac{1}{2} (c+d x)\right)\right)}{2048 d}-\frac{43 \log \left(2 \sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}{2048 d}",1,"(43*Log[Cos[(c + d*x)/2] - 2*Sin[(c + d*x)/2]])/(2048*d) - (43*Log[Cos[(c + d*x)/2] + 2*Sin[(c + d*x)/2]])/(2048*d) - 5/(512*d*(Cos[(c + d*x)/2] - 2*Sin[(c + d*x)/2])^2) - (45*Sin[(c + d*x)/2])/(1024*d*(Cos[(c + d*x)/2] - 2*Sin[(c + d*x)/2])) + 5/(512*d*(Cos[(c + d*x)/2] + 2*Sin[(c + d*x)/2])^2) - (45*Sin[(c + d*x)/2])/(1024*d*(Cos[(c + d*x)/2] + 2*Sin[(c + d*x)/2]))","A",1
41,1,288,138,0.2350823,"\int \frac{1}{(3-5 \cos (c+d x))^4} \, dx","Integrate[(3 - 5*Cos[c + d*x])^(-4),x]","\frac{226140 \sin (c+d x)-190800 \sin (2 (c+d x))+99500 \sin (3 (c+d x))-104625 \cos (3 (c+d x)) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-2 \sin \left(\frac{1}{2} (c+d x)\right)\right)+467046 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-2 \sin \left(\frac{1}{2} (c+d x)\right)\right)-765855 \cos (c+d x) \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-2 \sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(2 \sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)+376650 \cos (2 (c+d x)) \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-2 \sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(2 \sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)+104625 \cos (3 (c+d x)) \log \left(2 \sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)-467046 \log \left(2 \sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}{393216 d (5 \cos (c+d x)-3)^3}","-\frac{995 \sin (c+d x)}{24576 d (3-5 \cos (c+d x))}+\frac{25 \sin (c+d x)}{512 d (3-5 \cos (c+d x))^2}-\frac{5 \sin (c+d x)}{48 d (3-5 \cos (c+d x))^3}-\frac{279 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-2 \sin \left(\frac{1}{2} (c+d x)\right)\right)}{32768 d}+\frac{279 \log \left(2 \sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}{32768 d}",1,"(467046*Log[Cos[(c + d*x)/2] - 2*Sin[(c + d*x)/2]] - 104625*Cos[3*(c + d*x)]*Log[Cos[(c + d*x)/2] - 2*Sin[(c + d*x)/2]] - 765855*Cos[c + d*x]*(Log[Cos[(c + d*x)/2] - 2*Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + 2*Sin[(c + d*x)/2]]) + 376650*Cos[2*(c + d*x)]*(Log[Cos[(c + d*x)/2] - 2*Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + 2*Sin[(c + d*x)/2]]) - 467046*Log[Cos[(c + d*x)/2] + 2*Sin[(c + d*x)/2]] + 104625*Cos[3*(c + d*x)]*Log[Cos[(c + d*x)/2] + 2*Sin[(c + d*x)/2]] + 226140*Sin[c + d*x] - 190800*Sin[2*(c + d*x)] + 99500*Sin[3*(c + d*x)])/(393216*d*(-3 + 5*Cos[c + d*x])^3)","B",1
42,1,63,63,0.0212177,"\int \frac{1}{-3+5 \cos (c+d x)} \, dx","Integrate[(-3 + 5*Cos[c + d*x])^(-1),x]","\frac{\log \left(2 \sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}{4 d}-\frac{\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-2 \sin \left(\frac{1}{2} (c+d x)\right)\right)}{4 d}","\frac{\log \left(2 \sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}{4 d}-\frac{\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-2 \sin \left(\frac{1}{2} (c+d x)\right)\right)}{4 d}",1,"-1/4*Log[Cos[(c + d*x)/2] - 2*Sin[(c + d*x)/2]]/d + Log[Cos[(c + d*x)/2] + 2*Sin[(c + d*x)/2]]/(4*d)","A",1
43,1,139,88,0.0209449,"\int \frac{1}{(-3+5 \cos (c+d x))^2} \, dx","Integrate[(-3 + 5*Cos[c + d*x])^(-2),x]","\frac{20 \sin (c+d x)+9 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-2 \sin \left(\frac{1}{2} (c+d x)\right)\right)-15 \cos (c+d x) \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-2 \sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(2 \sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)-9 \log \left(2 \sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}{64 d (5 \cos (c+d x)-3)}","-\frac{5 \sin (c+d x)}{16 d (3-5 \cos (c+d x))}-\frac{3 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-2 \sin \left(\frac{1}{2} (c+d x)\right)\right)}{64 d}+\frac{3 \log \left(2 \sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}{64 d}",1,"(9*Log[Cos[(c + d*x)/2] - 2*Sin[(c + d*x)/2]] - 15*Cos[c + d*x]*(Log[Cos[(c + d*x)/2] - 2*Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + 2*Sin[(c + d*x)/2]]) - 9*Log[Cos[(c + d*x)/2] + 2*Sin[(c + d*x)/2]] + 20*Sin[c + d*x])/(64*d*(-3 + 5*Cos[c + d*x]))","A",1
44,1,211,113,0.1245094,"\int \frac{1}{(-3+5 \cos (c+d x))^3} \, dx","Integrate[(-3 + 5*Cos[c + d*x])^(-3),x]","\frac{45 \sin \left(\frac{1}{2} (c+d x)\right)}{1024 d \left(\cos \left(\frac{1}{2} (c+d x)\right)-2 \sin \left(\frac{1}{2} (c+d x)\right)\right)}+\frac{45 \sin \left(\frac{1}{2} (c+d x)\right)}{1024 d \left(2 \sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}+\frac{5}{512 d \left(\cos \left(\frac{1}{2} (c+d x)\right)-2 \sin \left(\frac{1}{2} (c+d x)\right)\right)^2}-\frac{5}{512 d \left(2 \sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2}-\frac{43 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-2 \sin \left(\frac{1}{2} (c+d x)\right)\right)}{2048 d}+\frac{43 \log \left(2 \sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}{2048 d}","-\frac{45 \sin (c+d x)}{512 d (3-5 \cos (c+d x))}+\frac{5 \sin (c+d x)}{32 d (3-5 \cos (c+d x))^2}-\frac{43 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-2 \sin \left(\frac{1}{2} (c+d x)\right)\right)}{2048 d}+\frac{43 \log \left(2 \sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}{2048 d}",1,"(-43*Log[Cos[(c + d*x)/2] - 2*Sin[(c + d*x)/2]])/(2048*d) + (43*Log[Cos[(c + d*x)/2] + 2*Sin[(c + d*x)/2]])/(2048*d) + 5/(512*d*(Cos[(c + d*x)/2] - 2*Sin[(c + d*x)/2])^2) + (45*Sin[(c + d*x)/2])/(1024*d*(Cos[(c + d*x)/2] - 2*Sin[(c + d*x)/2])) - 5/(512*d*(Cos[(c + d*x)/2] + 2*Sin[(c + d*x)/2])^2) + (45*Sin[(c + d*x)/2])/(1024*d*(Cos[(c + d*x)/2] + 2*Sin[(c + d*x)/2]))","A",1
45,1,288,138,0.0223921,"\int \frac{1}{(-3+5 \cos (c+d x))^4} \, dx","Integrate[(-3 + 5*Cos[c + d*x])^(-4),x]","\frac{226140 \sin (c+d x)-190800 \sin (2 (c+d x))+99500 \sin (3 (c+d x))-104625 \cos (3 (c+d x)) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-2 \sin \left(\frac{1}{2} (c+d x)\right)\right)+467046 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-2 \sin \left(\frac{1}{2} (c+d x)\right)\right)-765855 \cos (c+d x) \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-2 \sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(2 \sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)+376650 \cos (2 (c+d x)) \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-2 \sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(2 \sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)+104625 \cos (3 (c+d x)) \log \left(2 \sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)-467046 \log \left(2 \sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}{393216 d (5 \cos (c+d x)-3)^3}","-\frac{995 \sin (c+d x)}{24576 d (3-5 \cos (c+d x))}+\frac{25 \sin (c+d x)}{512 d (3-5 \cos (c+d x))^2}-\frac{5 \sin (c+d x)}{48 d (3-5 \cos (c+d x))^3}-\frac{279 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-2 \sin \left(\frac{1}{2} (c+d x)\right)\right)}{32768 d}+\frac{279 \log \left(2 \sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}{32768 d}",1,"(467046*Log[Cos[(c + d*x)/2] - 2*Sin[(c + d*x)/2]] - 104625*Cos[3*(c + d*x)]*Log[Cos[(c + d*x)/2] - 2*Sin[(c + d*x)/2]] - 765855*Cos[c + d*x]*(Log[Cos[(c + d*x)/2] - 2*Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + 2*Sin[(c + d*x)/2]]) + 376650*Cos[2*(c + d*x)]*(Log[Cos[(c + d*x)/2] - 2*Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + 2*Sin[(c + d*x)/2]]) - 467046*Log[Cos[(c + d*x)/2] + 2*Sin[(c + d*x)/2]] + 104625*Cos[3*(c + d*x)]*Log[Cos[(c + d*x)/2] + 2*Sin[(c + d*x)/2]] + 226140*Sin[c + d*x] - 190800*Sin[2*(c + d*x)] + 99500*Sin[3*(c + d*x)])/(393216*d*(-3 + 5*Cos[c + d*x])^3)","B",1
46,1,65,65,0.0221695,"\int \frac{1}{-3-5 \cos (c+d x)} \, dx","Integrate[(-3 - 5*Cos[c + d*x])^(-1),x]","\frac{\log \left(2 \cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}{4 d}-\frac{\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+2 \cos \left(\frac{1}{2} (c+d x)\right)\right)}{4 d}","\frac{\log \left(2 \cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}{4 d}-\frac{\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+2 \cos \left(\frac{1}{2} (c+d x)\right)\right)}{4 d}",1,"Log[2*Cos[(c + d*x)/2] - Sin[(c + d*x)/2]]/(4*d) - Log[2*Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]/(4*d)","A",1
47,1,143,90,0.0193583,"\int \frac{1}{(-3-5 \cos (c+d x))^2} \, dx","Integrate[(-3 - 5*Cos[c + d*x])^(-2),x]","\frac{20 \sin (c+d x)+9 \log \left(2 \cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+15 \cos (c+d x) \left(\log \left(2 \cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+2 \cos \left(\frac{1}{2} (c+d x)\right)\right)\right)-9 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+2 \cos \left(\frac{1}{2} (c+d x)\right)\right)}{64 d (5 \cos (c+d x)+3)}","\frac{5 \sin (c+d x)}{16 d (5 \cos (c+d x)+3)}+\frac{3 \log \left(2 \cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}{64 d}-\frac{3 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+2 \cos \left(\frac{1}{2} (c+d x)\right)\right)}{64 d}",1,"(9*Log[2*Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 15*Cos[c + d*x]*(Log[2*Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[2*Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) - 9*Log[2*Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + 20*Sin[c + d*x])/(64*d*(3 + 5*Cos[c + d*x]))","A",1
48,1,217,115,0.1000122,"\int \frac{1}{(-3-5 \cos (c+d x))^3} \, dx","Integrate[(-3 - 5*Cos[c + d*x])^(-3),x]","\frac{45 \sin \left(\frac{1}{2} (c+d x)\right)}{2048 d \left(2 \cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}+\frac{45 \sin \left(\frac{1}{2} (c+d x)\right)}{2048 d \left(\sin \left(\frac{1}{2} (c+d x)\right)+2 \cos \left(\frac{1}{2} (c+d x)\right)\right)}-\frac{5}{512 d \left(2 \cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^2}+\frac{5}{512 d \left(\sin \left(\frac{1}{2} (c+d x)\right)+2 \cos \left(\frac{1}{2} (c+d x)\right)\right)^2}+\frac{43 \log \left(2 \cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}{2048 d}-\frac{43 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+2 \cos \left(\frac{1}{2} (c+d x)\right)\right)}{2048 d}","\frac{45 \sin (c+d x)}{512 d (5 \cos (c+d x)+3)}-\frac{5 \sin (c+d x)}{32 d (5 \cos (c+d x)+3)^2}+\frac{43 \log \left(2 \cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}{2048 d}-\frac{43 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+2 \cos \left(\frac{1}{2} (c+d x)\right)\right)}{2048 d}",1,"(43*Log[2*Cos[(c + d*x)/2] - Sin[(c + d*x)/2]])/(2048*d) - (43*Log[2*Cos[(c + d*x)/2] + Sin[(c + d*x)/2]])/(2048*d) - 5/(512*d*(2*Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2) + (45*Sin[(c + d*x)/2])/(2048*d*(2*Cos[(c + d*x)/2] - Sin[(c + d*x)/2])) + 5/(512*d*(2*Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2) + (45*Sin[(c + d*x)/2])/(2048*d*(2*Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))","A",1
49,1,296,140,0.022217,"\int \frac{1}{(-3-5 \cos (c+d x))^4} \, dx","Integrate[(-3 - 5*Cos[c + d*x])^(-4),x]","\frac{226140 \sin (c+d x)+190800 \sin (2 (c+d x))+99500 \sin (3 (c+d x))+104625 \cos (3 (c+d x)) \log \left(2 \cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+467046 \log \left(2 \cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+765855 \cos (c+d x) \left(\log \left(2 \cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+2 \cos \left(\frac{1}{2} (c+d x)\right)\right)\right)+376650 \cos (2 (c+d x)) \left(\log \left(2 \cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+2 \cos \left(\frac{1}{2} (c+d x)\right)\right)\right)-104625 \cos (3 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+2 \cos \left(\frac{1}{2} (c+d x)\right)\right)-467046 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+2 \cos \left(\frac{1}{2} (c+d x)\right)\right)}{393216 d (5 \cos (c+d x)+3)^3}","\frac{995 \sin (c+d x)}{24576 d (5 \cos (c+d x)+3)}-\frac{25 \sin (c+d x)}{512 d (5 \cos (c+d x)+3)^2}+\frac{5 \sin (c+d x)}{48 d (5 \cos (c+d x)+3)^3}+\frac{279 \log \left(2 \cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}{32768 d}-\frac{279 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+2 \cos \left(\frac{1}{2} (c+d x)\right)\right)}{32768 d}",1,"(467046*Log[2*Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 104625*Cos[3*(c + d*x)]*Log[2*Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 765855*Cos[c + d*x]*(Log[2*Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[2*Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) + 376650*Cos[2*(c + d*x)]*(Log[2*Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[2*Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) - 467046*Log[2*Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] - 104625*Cos[3*(c + d*x)]*Log[2*Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + 226140*Sin[c + d*x] + 190800*Sin[2*(c + d*x)] + 99500*Sin[3*(c + d*x)])/(393216*d*(3 + 5*Cos[c + d*x])^3)","B",1
50,1,177,197,0.7881809,"\int (a+b \cos (c+d x))^{5/2} \, dx","Integrate[(a + b*Cos[c + d*x])^(5/2),x]","\frac{b \sin (c+d x) \left(22 a^2+28 a b \cos (c+d x)+3 b^2 \cos (2 (c+d x))+3 b^2\right)-16 a \left(a^2-b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)+2 \left(23 a^3+23 a^2 b+9 a b^2+9 b^3\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{15 d \sqrt{a+b \cos (c+d x)}}","-\frac{16 a \left(a^2-b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{15 d \sqrt{a+b \cos (c+d x)}}+\frac{2 \left(23 a^2+9 b^2\right) \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{15 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{2 b \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}+\frac{16 a b \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{15 d}",1,"(2*(23*a^3 + 23*a^2*b + 9*a*b^2 + 9*b^3)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticE[(c + d*x)/2, (2*b)/(a + b)] - 16*a*(a^2 - b^2)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)] + b*(22*a^2 + 3*b^2 + 28*a*b*Cos[c + d*x] + 3*b^2*Cos[2*(c + d*x)])*Sin[c + d*x])/(15*d*Sqrt[a + b*Cos[c + d*x]])","A",1
51,1,134,157,0.5340899,"\int (a+b \cos (c+d x))^{3/2} \, dx","Integrate[(a + b*Cos[c + d*x])^(3/2),x]","\frac{-2 \left(a^2-b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)+2 b \sin (c+d x) (a+b \cos (c+d x))+8 a (a+b) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 d \sqrt{a+b \cos (c+d x)}}","-\frac{2 \left(a^2-b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 d \sqrt{a+b \cos (c+d x)}}+\frac{2 b \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{3 d}+\frac{8 a \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}",1,"(8*a*(a + b)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticE[(c + d*x)/2, (2*b)/(a + b)] - 2*(a^2 - b^2)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)] + 2*b*(a + b*Cos[c + d*x])*Sin[c + d*x])/(3*d*Sqrt[a + b*Cos[c + d*x]])","A",1
52,1,57,57,0.0687861,"\int \sqrt{a+b \cos (c+d x)} \, dx","Integrate[Sqrt[a + b*Cos[c + d*x]],x]","\frac{2 \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}","\frac{2 \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{d \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}",1,"(2*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(d*Sqrt[(a + b*Cos[c + d*x])/(a + b)])","A",1
53,1,57,57,0.048123,"\int \frac{1}{\sqrt{a+b \cos (c+d x)}} \, dx","Integrate[1/Sqrt[a + b*Cos[c + d*x]],x]","\frac{2 \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{d \sqrt{a+b \cos (c+d x)}}","\frac{2 \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{d \sqrt{a+b \cos (c+d x)}}",1,"(2*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(d*Sqrt[a + b*Cos[c + d*x]])","A",1
54,1,83,106,0.1991737,"\int \frac{1}{(a+b \cos (c+d x))^{3/2}} \, dx","Integrate[(a + b*Cos[c + d*x])^(-3/2),x]","\frac{2 (a+b) \sqrt{\frac{a+b \cos (c+d x)}{a+b}} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)-2 b \sin (c+d x)}{d (a-b) (a+b) \sqrt{a+b \cos (c+d x)}}","\frac{2 \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{d \left(a^2-b^2\right) \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}-\frac{2 b \sin (c+d x)}{d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}",1,"(2*(a + b)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticE[(c + d*x)/2, (2*b)/(a + b)] - 2*b*Sin[c + d*x])/((a - b)*(a + b)*d*Sqrt[a + b*Cos[c + d*x]])","A",1
55,1,158,221,0.9083196,"\int \frac{1}{(a+b \cos (c+d x))^{5/2}} \, dx","Integrate[(a + b*Cos[c + d*x])^(-5/2),x]","\frac{2 b \sin (c+d x) \left(-5 a^2-4 a b \cos (c+d x)+b^2\right)-2 (a-b) (a+b)^2 \left(\frac{a+b \cos (c+d x)}{a+b}\right)^{3/2} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)+8 a (a+b)^2 \left(\frac{a+b \cos (c+d x)}{a+b}\right)^{3/2} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 d (a-b)^2 (a+b)^2 (a+b \cos (c+d x))^{3/2}}","-\frac{8 a b \sin (c+d x)}{3 d \left(a^2-b^2\right)^2 \sqrt{a+b \cos (c+d x)}}-\frac{2 b \sin (c+d x)}{3 d \left(a^2-b^2\right) (a+b \cos (c+d x))^{3/2}}-\frac{2 \sqrt{\frac{a+b \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 d \left(a^2-b^2\right) \sqrt{a+b \cos (c+d x)}}+\frac{8 a \sqrt{a+b \cos (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 b}{a+b}\right)}{3 d \left(a^2-b^2\right)^2 \sqrt{\frac{a+b \cos (c+d x)}{a+b}}}",1,"(8*a*(a + b)^2*((a + b*Cos[c + d*x])/(a + b))^(3/2)*EllipticE[(c + d*x)/2, (2*b)/(a + b)] - 2*(a - b)*(a + b)^2*((a + b*Cos[c + d*x])/(a + b))^(3/2)*EllipticF[(c + d*x)/2, (2*b)/(a + b)] + 2*b*(-5*a^2 + b^2 - 4*a*b*Cos[c + d*x])*Sin[c + d*x])/(3*(a - b)^2*(a + b)^2*d*(a + b*Cos[c + d*x])^(3/2))","A",1
56,1,246,108,1.8664036,"\int (a+b \cos (c+d x))^{4/3} \, dx","Integrate[(a + b*Cos[c + d*x])^(4/3),x]","-\frac{3 \csc (c+d x) \sqrt[3]{a+b \cos (c+d x)} \left(4 \left(b^2-a^2\right) \sqrt{-\frac{b (\cos (c+d x)-1)}{a+b}} \sqrt{-\frac{b (\cos (c+d x)+1)}{a-b}} F_1\left(\frac{1}{3};\frac{1}{2},\frac{1}{2};\frac{4}{3};\frac{a+b \cos (c+d x)}{a-b},\frac{a+b \cos (c+d x)}{a+b}\right)+5 a \sqrt{-\frac{b (\cos (c+d x)-1)}{a+b}} \sqrt{-\frac{b (\cos (c+d x)+1)}{a-b}} (a+b \cos (c+d x)) F_1\left(\frac{4}{3};\frac{1}{2},\frac{1}{2};\frac{7}{3};\frac{a+b \cos (c+d x)}{a-b},\frac{a+b \cos (c+d x)}{a+b}\right)-4 b^2 \sin ^2(c+d x)\right)}{16 b d}","\frac{\sqrt{2} (a+b) \sin (c+d x) \sqrt[3]{a+b \cos (c+d x)} F_1\left(\frac{1}{2};\frac{1}{2},-\frac{4}{3};\frac{3}{2};\frac{1}{2} (1-\cos (c+d x)),\frac{b (1-\cos (c+d x))}{a+b}\right)}{d \sqrt{\cos (c+d x)+1} \sqrt[3]{\frac{a+b \cos (c+d x)}{a+b}}}",1,"(-3*(a + b*Cos[c + d*x])^(1/3)*Csc[c + d*x]*(4*(-a^2 + b^2)*AppellF1[1/3, 1/2, 1/2, 4/3, (a + b*Cos[c + d*x])/(a - b), (a + b*Cos[c + d*x])/(a + b)]*Sqrt[-((b*(-1 + Cos[c + d*x]))/(a + b))]*Sqrt[-((b*(1 + Cos[c + d*x]))/(a - b))] + 5*a*AppellF1[4/3, 1/2, 1/2, 7/3, (a + b*Cos[c + d*x])/(a - b), (a + b*Cos[c + d*x])/(a + b)]*Sqrt[-((b*(-1 + Cos[c + d*x]))/(a + b))]*Sqrt[-((b*(1 + Cos[c + d*x]))/(a - b))]*(a + b*Cos[c + d*x]) - 4*b^2*Sin[c + d*x]^2))/(16*b*d)","B",0
57,1,118,105,0.174319,"\int (a+b \cos (c+d x))^{2/3} \, dx","Integrate[(a + b*Cos[c + d*x])^(2/3),x]","-\frac{3 \csc (c+d x) \sqrt{-\frac{b (\cos (c+d x)-1)}{a+b}} \sqrt{\frac{b (\cos (c+d x)+1)}{b-a}} (a+b \cos (c+d x))^{5/3} F_1\left(\frac{5}{3};\frac{1}{2},\frac{1}{2};\frac{8}{3};\frac{a+b \cos (c+d x)}{a-b},\frac{a+b \cos (c+d x)}{a+b}\right)}{5 b d}","\frac{\sqrt{2} \sin (c+d x) (a+b \cos (c+d x))^{2/3} F_1\left(\frac{1}{2};\frac{1}{2},-\frac{2}{3};\frac{3}{2};\frac{1}{2} (1-\cos (c+d x)),\frac{b (1-\cos (c+d x))}{a+b}\right)}{d \sqrt{\cos (c+d x)+1} \left(\frac{a+b \cos (c+d x)}{a+b}\right)^{2/3}}",1,"(-3*AppellF1[5/3, 1/2, 1/2, 8/3, (a + b*Cos[c + d*x])/(a - b), (a + b*Cos[c + d*x])/(a + b)]*Sqrt[-((b*(-1 + Cos[c + d*x]))/(a + b))]*Sqrt[(b*(1 + Cos[c + d*x]))/(-a + b)]*(a + b*Cos[c + d*x])^(5/3)*Csc[c + d*x])/(5*b*d)","A",0
58,1,118,105,0.1626143,"\int \sqrt[3]{a+b \cos (c+d x)} \, dx","Integrate[(a + b*Cos[c + d*x])^(1/3),x]","-\frac{3 \csc (c+d x) \sqrt{-\frac{b (\cos (c+d x)-1)}{a+b}} \sqrt{\frac{b (\cos (c+d x)+1)}{b-a}} (a+b \cos (c+d x))^{4/3} F_1\left(\frac{4}{3};\frac{1}{2},\frac{1}{2};\frac{7}{3};\frac{a+b \cos (c+d x)}{a-b},\frac{a+b \cos (c+d x)}{a+b}\right)}{4 b d}","\frac{\sqrt{2} \sin (c+d x) \sqrt[3]{a+b \cos (c+d x)} F_1\left(\frac{1}{2};\frac{1}{2},-\frac{1}{3};\frac{3}{2};\frac{1}{2} (1-\cos (c+d x)),\frac{b (1-\cos (c+d x))}{a+b}\right)}{d \sqrt{\cos (c+d x)+1} \sqrt[3]{\frac{a+b \cos (c+d x)}{a+b}}}",1,"(-3*AppellF1[4/3, 1/2, 1/2, 7/3, (a + b*Cos[c + d*x])/(a - b), (a + b*Cos[c + d*x])/(a + b)]*Sqrt[-((b*(-1 + Cos[c + d*x]))/(a + b))]*Sqrt[(b*(1 + Cos[c + d*x]))/(-a + b)]*(a + b*Cos[c + d*x])^(4/3)*Csc[c + d*x])/(4*b*d)","A",0
59,1,118,105,0.1779151,"\int \frac{1}{\sqrt[3]{a+b \cos (c+d x)}} \, dx","Integrate[(a + b*Cos[c + d*x])^(-1/3),x]","-\frac{3 \csc (c+d x) \sqrt{-\frac{b (\cos (c+d x)-1)}{a+b}} \sqrt{\frac{b (\cos (c+d x)+1)}{b-a}} (a+b \cos (c+d x))^{2/3} F_1\left(\frac{2}{3};\frac{1}{2},\frac{1}{2};\frac{5}{3};\frac{a+b \cos (c+d x)}{a-b},\frac{a+b \cos (c+d x)}{a+b}\right)}{2 b d}","\frac{\sqrt{2} \sin (c+d x) \sqrt[3]{\frac{a+b \cos (c+d x)}{a+b}} F_1\left(\frac{1}{2};\frac{1}{2},\frac{1}{3};\frac{3}{2};\frac{1}{2} (1-\cos (c+d x)),\frac{b (1-\cos (c+d x))}{a+b}\right)}{d \sqrt{\cos (c+d x)+1} \sqrt[3]{a+b \cos (c+d x)}}",1,"(-3*AppellF1[2/3, 1/2, 1/2, 5/3, (a + b*Cos[c + d*x])/(a - b), (a + b*Cos[c + d*x])/(a + b)]*Sqrt[-((b*(-1 + Cos[c + d*x]))/(a + b))]*Sqrt[(b*(1 + Cos[c + d*x]))/(-a + b)]*(a + b*Cos[c + d*x])^(2/3)*Csc[c + d*x])/(2*b*d)","A",0
60,1,116,105,0.1661551,"\int \frac{1}{(a+b \cos (c+d x))^{2/3}} \, dx","Integrate[(a + b*Cos[c + d*x])^(-2/3),x]","-\frac{3 \csc (c+d x) \sqrt{-\frac{b (\cos (c+d x)-1)}{a+b}} \sqrt{\frac{b (\cos (c+d x)+1)}{b-a}} \sqrt[3]{a+b \cos (c+d x)} F_1\left(\frac{1}{3};\frac{1}{2},\frac{1}{2};\frac{4}{3};\frac{a+b \cos (c+d x)}{a-b},\frac{a+b \cos (c+d x)}{a+b}\right)}{b d}","\frac{\sqrt{2} \sin (c+d x) \left(\frac{a+b \cos (c+d x)}{a+b}\right)^{2/3} F_1\left(\frac{1}{2};\frac{1}{2},\frac{2}{3};\frac{3}{2};\frac{1}{2} (1-\cos (c+d x)),\frac{b (1-\cos (c+d x))}{a+b}\right)}{d \sqrt{\cos (c+d x)+1} (a+b \cos (c+d x))^{2/3}}",1,"(-3*AppellF1[1/3, 1/2, 1/2, 4/3, (a + b*Cos[c + d*x])/(a - b), (a + b*Cos[c + d*x])/(a + b)]*Sqrt[-((b*(-1 + Cos[c + d*x]))/(a + b))]*Sqrt[(b*(1 + Cos[c + d*x]))/(-a + b)]*(a + b*Cos[c + d*x])^(1/3)*Csc[c + d*x])/(b*d)","A",0
61,1,268,110,2.0274796,"\int \frac{1}{(a+b \cos (c+d x))^{4/3}} \, dx","Integrate[(a + b*Cos[c + d*x])^(-4/3),x]","\frac{15 a \csc (c+d x) \sqrt{-\frac{b (\cos (c+d x)-1)}{a+b}} \sqrt{-\frac{b (\cos (c+d x)+1)}{a-b}} (a+b \cos (c+d x)) F_1\left(\frac{2}{3};\frac{1}{2},\frac{1}{2};\frac{5}{3};\frac{a+b \cos (c+d x)}{a-b},\frac{a+b \cos (c+d x)}{a+b}\right)-6 \sin (c+d x) \left(2 \csc ^2(c+d x) \sqrt{-\frac{b (\cos (c+d x)-1)}{a+b}} \sqrt{\frac{b (\cos (c+d x)+1)}{b-a}} (a+b \cos (c+d x))^2 F_1\left(\frac{5}{3};\frac{1}{2},\frac{1}{2};\frac{8}{3};\frac{a+b \cos (c+d x)}{a-b},\frac{a+b \cos (c+d x)}{a+b}\right)+5 b^2\right)}{10 b d \left(a^2-b^2\right) \sqrt[3]{a+b \cos (c+d x)}}","\frac{\sqrt{2} \sin (c+d x) \sqrt[3]{\frac{a+b \cos (c+d x)}{a+b}} F_1\left(\frac{1}{2};\frac{1}{2},\frac{4}{3};\frac{3}{2};\frac{1}{2} (1-\cos (c+d x)),\frac{b (1-\cos (c+d x))}{a+b}\right)}{d (a+b) \sqrt{\cos (c+d x)+1} \sqrt[3]{a+b \cos (c+d x)}}",1,"(15*a*AppellF1[2/3, 1/2, 1/2, 5/3, (a + b*Cos[c + d*x])/(a - b), (a + b*Cos[c + d*x])/(a + b)]*Sqrt[-((b*(-1 + Cos[c + d*x]))/(a + b))]*Sqrt[-((b*(1 + Cos[c + d*x]))/(a - b))]*(a + b*Cos[c + d*x])*Csc[c + d*x] - 6*(5*b^2 + 2*AppellF1[5/3, 1/2, 1/2, 8/3, (a + b*Cos[c + d*x])/(a - b), (a + b*Cos[c + d*x])/(a + b)]*Sqrt[-((b*(-1 + Cos[c + d*x]))/(a + b))]*Sqrt[(b*(1 + Cos[c + d*x]))/(-a + b)]*(a + b*Cos[c + d*x])^2*Csc[c + d*x]^2)*Sin[c + d*x])/(10*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^(1/3))","B",0
62,1,121,103,0.2212178,"\int (a+b \cos (c+d x))^n \, dx","Integrate[(a + b*Cos[c + d*x])^n,x]","-\frac{\csc (c+d x) \sqrt{-\frac{b (\cos (c+d x)-1)}{a+b}} \sqrt{\frac{b (\cos (c+d x)+1)}{b-a}} (a+b \cos (c+d x))^{n+1} F_1\left(n+1;\frac{1}{2},\frac{1}{2};n+2;\frac{a+b \cos (c+d x)}{a-b},\frac{a+b \cos (c+d x)}{a+b}\right)}{b d (n+1)}","\frac{\sqrt{2} \sin (c+d x) (a+b \cos (c+d x))^n \left(\frac{a+b \cos (c+d x)}{a+b}\right)^{-n} F_1\left(\frac{1}{2};\frac{1}{2},-n;\frac{3}{2};\frac{1}{2} (1-\cos (c+d x)),\frac{b (1-\cos (c+d x))}{a+b}\right)}{d \sqrt{\cos (c+d x)+1}}",1,"-((AppellF1[1 + n, 1/2, 1/2, 2 + n, (a + b*Cos[c + d*x])/(a - b), (a + b*Cos[c + d*x])/(a + b)]*Sqrt[-((b*(-1 + Cos[c + d*x]))/(a + b))]*Sqrt[(b*(1 + Cos[c + d*x]))/(-a + b)]*(a + b*Cos[c + d*x])^(1 + n)*Csc[c + d*x])/(b*d*(1 + n)))","A",0