1,1,47,61,0.4769096,"\int \sin (a+b x) \sin ^7(2 a+2 b x) \, dx","Integrate[Sin[a + b*x]*Sin[2*a + 2*b*x]^7,x]","\frac{4 \sin ^9(a+b x) (10755 \cos (2 (a+b x))+3366 \cos (4 (a+b x))+429 \cos (6 (a+b x))+8330)}{6435 b}","-\frac{128 \sin ^{15}(a+b x)}{15 b}+\frac{384 \sin ^{13}(a+b x)}{13 b}-\frac{384 \sin ^{11}(a+b x)}{11 b}+\frac{128 \sin ^9(a+b x)}{9 b}",1,"(4*(8330 + 10755*Cos[2*(a + b*x)] + 3366*Cos[4*(a + b*x)] + 429*Cos[6*(a + b*x)])*Sin[a + b*x]^9)/(6435*b)","A",1
2,1,47,61,0.3327469,"\int \sin (a+b x) \sin ^6(2 a+2 b x) \, dx","Integrate[Sin[a + b*x]*Sin[2*a + 2*b*x]^6,x]","\frac{2 \cos ^7(a+b x) (6377 \cos (2 (a+b x))-1890 \cos (4 (a+b x))+231 \cos (6 (a+b x))-5230)}{3003 b}","\frac{64 \cos ^{13}(a+b x)}{13 b}-\frac{192 \cos ^{11}(a+b x)}{11 b}+\frac{64 \cos ^9(a+b x)}{3 b}-\frac{64 \cos ^7(a+b x)}{7 b}",1,"(2*Cos[a + b*x]^7*(-5230 + 6377*Cos[2*(a + b*x)] - 1890*Cos[4*(a + b*x)] + 231*Cos[6*(a + b*x)]))/(3003*b)","A",1
3,1,37,46,0.2574118,"\int \sin (a+b x) \sin ^5(2 a+2 b x) \, dx","Integrate[Sin[a + b*x]*Sin[2*a + 2*b*x]^5,x]","\frac{4 \sin ^7(a+b x) (364 \cos (2 (a+b x))+63 \cos (4 (a+b x))+365)}{693 b}","\frac{32 \sin ^{11}(a+b x)}{11 b}-\frac{64 \sin ^9(a+b x)}{9 b}+\frac{32 \sin ^7(a+b x)}{7 b}",1,"(4*(365 + 364*Cos[2*(a + b*x)] + 63*Cos[4*(a + b*x)])*Sin[a + b*x]^7)/(693*b)","A",1
4,1,37,46,0.1498939,"\int \sin (a+b x) \sin ^4(2 a+2 b x) \, dx","Integrate[Sin[a + b*x]*Sin[2*a + 2*b*x]^4,x]","\frac{2 \cos ^5(a+b x) (220 \cos (2 (a+b x))-35 \cos (4 (a+b x))-249)}{315 b}","-\frac{16 \cos ^9(a+b x)}{9 b}+\frac{32 \cos ^7(a+b x)}{7 b}-\frac{16 \cos ^5(a+b x)}{5 b}",1,"(2*Cos[a + b*x]^5*(-249 + 220*Cos[2*(a + b*x)] - 35*Cos[4*(a + b*x)]))/(315*b)","A",1
5,1,27,31,0.0944578,"\int \sin (a+b x) \sin ^3(2 a+2 b x) \, dx","Integrate[Sin[a + b*x]*Sin[2*a + 2*b*x]^3,x]","\frac{4 \sin ^5(a+b x) (5 \cos (2 (a+b x))+9)}{35 b}","\frac{8 \sin ^5(a+b x)}{5 b}-\frac{8 \sin ^7(a+b x)}{7 b}",1,"(4*(9 + 5*Cos[2*(a + b*x)])*Sin[a + b*x]^5)/(35*b)","A",1
6,1,27,31,0.0666398,"\int \sin (a+b x) \sin ^2(2 a+2 b x) \, dx","Integrate[Sin[a + b*x]*Sin[2*a + 2*b*x]^2,x]","\frac{2 \cos ^3(a+b x) (3 \cos (2 (a+b x))-7)}{15 b}","\frac{4 \cos ^5(a+b x)}{5 b}-\frac{4 \cos ^3(a+b x)}{3 b}",1,"(2*Cos[a + b*x]^3*(-7 + 3*Cos[2*(a + b*x)]))/(15*b)","A",1
7,1,15,30,0.0335612,"\int \sin (a+b x) \sin (2 a+2 b x) \, dx","Integrate[Sin[a + b*x]*Sin[2*a + 2*b*x],x]","\frac{2 \sin ^3(a+b x)}{3 b}","\frac{\sin (a+b x)}{2 b}-\frac{\sin (3 a+3 b x)}{6 b}",1,"(2*Sin[a + b*x]^3)/(3*b)","A",1
8,1,14,14,0.004623,"\int \csc (2 a+2 b x) \sin (a+b x) \, dx","Integrate[Csc[2*a + 2*b*x]*Sin[a + b*x],x]","\frac{\tanh ^{-1}(\sin (a+b x))}{2 b}","\frac{\tanh ^{-1}(\sin (a+b x))}{2 b}",1,"ArcTanh[Sin[a + b*x]]/(2*b)","A",1
9,1,50,28,0.0408643,"\int \csc ^2(2 a+2 b x) \sin (a+b x) \, dx","Integrate[Csc[2*a + 2*b*x]^2*Sin[a + b*x],x]","\frac{\sec (a+b x)}{4 b}+\frac{\log \left(\sin \left(\frac{1}{2} (a+b x)\right)\right)}{4 b}-\frac{\log \left(\cos \left(\frac{1}{2} (a+b x)\right)\right)}{4 b}","\frac{\sec (a+b x)}{4 b}-\frac{\tanh ^{-1}(\cos (a+b x))}{4 b}",1,"-1/4*Log[Cos[(a + b*x)/2]]/b + Log[Sin[(a + b*x)/2]]/(4*b) + Sec[a + b*x]/(4*b)","A",1
10,1,29,49,0.0189634,"\int \csc ^3(2 a+2 b x) \sin (a+b x) \, dx","Integrate[Csc[2*a + 2*b*x]^3*Sin[a + b*x],x]","-\frac{\csc (a+b x) \, _2F_1\left(-\frac{1}{2},2;\frac{1}{2};\sin ^2(a+b x)\right)}{8 b}","-\frac{3 \csc (a+b x)}{16 b}+\frac{3 \tanh ^{-1}(\sin (a+b x))}{16 b}+\frac{\csc (a+b x) \sec ^2(a+b x)}{16 b}",1,"-1/8*(Csc[a + b*x]*Hypergeometric2F1[-1/2, 2, 1/2, Sin[a + b*x]^2])/b","C",1
11,1,205,66,0.4801387,"\int \csc ^4(2 a+2 b x) \sin (a+b x) \, dx","Integrate[Csc[2*a + 2*b*x]^4*Sin[a + b*x],x]","\frac{\csc ^8(a+b x) \left(-40 \cos (2 (a+b x))+13 \cos (3 (a+b x))-30 \cos (4 (a+b x))+13 \cos (5 (a+b x))+15 \cos (3 (a+b x)) \log \left(\cos \left(\frac{1}{2} (a+b x)\right)\right)+15 \cos (5 (a+b x)) \log \left(\cos \left(\frac{1}{2} (a+b x)\right)\right)-15 \cos (3 (a+b x)) \log \left(\sin \left(\frac{1}{2} (a+b x)\right)\right)-15 \cos (5 (a+b x)) \log \left(\sin \left(\frac{1}{2} (a+b x)\right)\right)+\cos (a+b x) \left(30 \log \left(\sin \left(\frac{1}{2} (a+b x)\right)\right)-30 \log \left(\cos \left(\frac{1}{2} (a+b x)\right)\right)-26\right)+22\right)}{24 b \left(\csc ^2\left(\frac{1}{2} (a+b x)\right)-\sec ^2\left(\frac{1}{2} (a+b x)\right)\right)^3}","\frac{5 \sec ^3(a+b x)}{96 b}+\frac{5 \sec (a+b x)}{32 b}-\frac{5 \tanh ^{-1}(\cos (a+b x))}{32 b}-\frac{\csc ^2(a+b x) \sec ^3(a+b x)}{32 b}",1,"(Csc[a + b*x]^8*(22 - 40*Cos[2*(a + b*x)] + 13*Cos[3*(a + b*x)] - 30*Cos[4*(a + b*x)] + 13*Cos[5*(a + b*x)] + 15*Cos[3*(a + b*x)]*Log[Cos[(a + b*x)/2]] + 15*Cos[5*(a + b*x)]*Log[Cos[(a + b*x)/2]] - 15*Cos[3*(a + b*x)]*Log[Sin[(a + b*x)/2]] - 15*Cos[5*(a + b*x)]*Log[Sin[(a + b*x)/2]] + Cos[a + b*x]*(-26 - 30*Log[Cos[(a + b*x)/2]] + 30*Log[Sin[(a + b*x)/2]])))/(24*b*(Csc[(a + b*x)/2]^2 - Sec[(a + b*x)/2]^2)^3)","B",1
12,1,31,89,0.0307118,"\int \csc ^5(2 a+2 b x) \sin (a+b x) \, dx","Integrate[Csc[2*a + 2*b*x]^5*Sin[a + b*x],x]","-\frac{\csc ^3(a+b x) \, _2F_1\left(-\frac{3}{2},3;-\frac{1}{2};\sin ^2(a+b x)\right)}{96 b}","-\frac{35 \csc ^3(a+b x)}{768 b}-\frac{35 \csc (a+b x)}{256 b}+\frac{35 \tanh ^{-1}(\sin (a+b x))}{256 b}+\frac{\csc ^3(a+b x) \sec ^4(a+b x)}{128 b}+\frac{7 \csc ^3(a+b x) \sec ^2(a+b x)}{256 b}",1,"-1/96*(Csc[a + b*x]^3*Hypergeometric2F1[-3/2, 3, -1/2, Sin[a + b*x]^2])/b","C",1
13,1,68,44,0.3567672,"\int \sin ^2(a+b x) \sin ^5(2 a+2 b x) \, dx","Integrate[Sin[a + b*x]^2*Sin[2*a + 2*b*x]^5,x]","\frac{-600 \cos (2 (a+b x))+75 \cos (4 (a+b x))+100 \cos (6 (a+b x))-30 \cos (8 (a+b x))-12 \cos (10 (a+b x))+5 \cos (12 (a+b x))}{3840 b}","\frac{8 \sin ^{12}(a+b x)}{3 b}-\frac{32 \sin ^{10}(a+b x)}{5 b}+\frac{4 \sin ^8(a+b x)}{b}",1,"(-600*Cos[2*(a + b*x)] + 75*Cos[4*(a + b*x)] + 100*Cos[6*(a + b*x)] - 30*Cos[8*(a + b*x)] - 12*Cos[10*(a + b*x)] + 5*Cos[12*(a + b*x)])/(3840*b)","A",1
14,1,62,76,0.1955679,"\int \sin ^2(a+b x) \sin ^4(2 a+2 b x) \, dx","Integrate[Sin[a + b*x]^2*Sin[2*a + 2*b*x]^4,x]","\frac{-20 \sin (2 (a+b x))-40 \sin (4 (a+b x))+10 \sin (6 (a+b x))+5 \sin (8 (a+b x))-2 \sin (10 (a+b x))+120 b x}{640 b}","-\frac{\sin ^5(2 a+2 b x)}{20 b}-\frac{\sin ^3(2 a+2 b x) \cos (2 a+2 b x)}{16 b}-\frac{3 \sin (2 a+2 b x) \cos (2 a+2 b x)}{32 b}+\frac{3 x}{16}",1,"(120*b*x - 20*Sin[2*(a + b*x)] - 40*Sin[4*(a + b*x)] + 10*Sin[6*(a + b*x)] + 5*Sin[8*(a + b*x)] - 2*Sin[10*(a + b*x)])/(640*b)","A",1
15,1,48,29,0.1169635,"\int \sin ^2(a+b x) \sin ^3(2 a+2 b x) \, dx","Integrate[Sin[a + b*x]^2*Sin[2*a + 2*b*x]^3,x]","\frac{-72 \cos (2 (a+b x))+12 \cos (4 (a+b x))+8 \cos (6 (a+b x))-3 \cos (8 (a+b x))}{384 b}","\frac{4 \sin ^6(a+b x)}{3 b}-\frac{\sin ^8(a+b x)}{b}",1,"(-72*Cos[2*(a + b*x)] + 12*Cos[4*(a + b*x)] + 8*Cos[6*(a + b*x)] - 3*Cos[8*(a + b*x)])/(384*b)","A",1
16,1,40,49,0.0747464,"\int \sin ^2(a+b x) \sin ^2(2 a+2 b x) \, dx","Integrate[Sin[a + b*x]^2*Sin[2*a + 2*b*x]^2,x]","\frac{-3 \sin (2 (a+b x))-3 \sin (4 (a+b x))+\sin (6 (a+b x))+12 b x}{48 b}","-\frac{\sin ^3(2 a+2 b x)}{12 b}-\frac{\sin (2 a+2 b x) \cos (2 a+2 b x)}{8 b}+\frac{x}{4}",1,"(12*b*x - 3*Sin[2*(a + b*x)] - 3*Sin[4*(a + b*x)] + Sin[6*(a + b*x)])/(48*b)","A",1
17,1,15,15,0.0053468,"\int \sin ^2(a+b x) \sin (2 a+2 b x) \, dx","Integrate[Sin[a + b*x]^2*Sin[2*a + 2*b*x],x]","\frac{\sin ^4(a+b x)}{2 b}","\frac{\sin ^4(a+b x)}{2 b}",1,"Sin[a + b*x]^4/(2*b)","A",1
18,1,14,14,0.0127942,"\int \csc (2 a+2 b x) \sin ^2(a+b x) \, dx","Integrate[Csc[2*a + 2*b*x]*Sin[a + b*x]^2,x]","-\frac{\log (\cos (a+b x))}{2 b}","-\frac{\log (\cos (a+b x))}{2 b}",1,"-1/2*Log[Cos[a + b*x]]/b","A",1
19,1,13,13,0.0080332,"\int \csc ^2(2 a+2 b x) \sin ^2(a+b x) \, dx","Integrate[Csc[2*a + 2*b*x]^2*Sin[a + b*x]^2,x]","\frac{\tan (a+b x)}{4 b}","\frac{\tan (a+b x)}{4 b}",1,"Tan[a + b*x]/(4*b)","A",1
20,1,36,30,0.0375217,"\int \csc ^3(2 a+2 b x) \sin ^2(a+b x) \, dx","Integrate[Csc[2*a + 2*b*x]^3*Sin[a + b*x]^2,x]","-\frac{-\sec ^2(a+b x)-2 \log (\sin (a+b x))+2 \log (\cos (a+b x))}{16 b}","\frac{\tan ^2(a+b x)}{16 b}+\frac{\log (\tan (a+b x))}{8 b}",1,"-1/16*(2*Log[Cos[a + b*x]] - 2*Log[Sin[a + b*x]] - Sec[a + b*x]^2)/b","A",1
21,1,48,42,0.0555857,"\int \csc ^4(2 a+2 b x) \sin ^2(a+b x) \, dx","Integrate[Csc[2*a + 2*b*x]^4*Sin[a + b*x]^2,x]","\frac{5 \tan (a+b x)}{48 b}-\frac{\cot (a+b x)}{16 b}+\frac{\tan (a+b x) \sec ^2(a+b x)}{48 b}","\frac{\tan ^3(a+b x)}{48 b}+\frac{\tan (a+b x)}{8 b}-\frac{\cot (a+b x)}{16 b}",1,"-1/16*Cot[a + b*x]/b + (5*Tan[a + b*x])/(48*b) + (Sec[a + b*x]^2*Tan[a + b*x])/(48*b)","A",1
22,1,56,60,0.2479782,"\int \csc ^5(2 a+2 b x) \sin ^2(a+b x) \, dx","Integrate[Csc[2*a + 2*b*x]^5*Sin[a + b*x]^2,x]","-\frac{2 \csc ^2(a+b x)-\sec ^4(a+b x)-4 \sec ^2(a+b x)-12 \log (\sin (a+b x))+12 \log (\cos (a+b x))}{128 b}","\frac{\tan ^4(a+b x)}{128 b}+\frac{3 \tan ^2(a+b x)}{64 b}-\frac{\cot ^2(a+b x)}{64 b}+\frac{3 \log (\tan (a+b x))}{32 b}",1,"-1/128*(2*Csc[a + b*x]^2 + 12*Log[Cos[a + b*x]] - 12*Log[Sin[a + b*x]] - 4*Sec[a + b*x]^2 - Sec[a + b*x]^4)/b","A",1
23,1,37,46,0.3691181,"\int \sin ^3(a+b x) \sin ^5(2 a+2 b x) \, dx","Integrate[Sin[a + b*x]^3*Sin[2*a + 2*b*x]^5,x]","\frac{4 \sin ^9(a+b x) (540 \cos (2 (a+b x))+99 \cos (4 (a+b x))+505)}{1287 b}","\frac{32 \sin ^{13}(a+b x)}{13 b}-\frac{64 \sin ^{11}(a+b x)}{11 b}+\frac{32 \sin ^9(a+b x)}{9 b}",1,"(4*(505 + 540*Cos[2*(a + b*x)] + 99*Cos[4*(a + b*x)])*Sin[a + b*x]^9)/(1287*b)","A",1
24,1,47,61,0.2276341,"\int \sin ^3(a+b x) \sin ^4(2 a+2 b x) \, dx","Integrate[Sin[a + b*x]^3*Sin[2*a + 2*b*x]^4,x]","\frac{\cos ^5(a+b x) (3335 \cos (2 (a+b x))-910 \cos (4 (a+b x))+105 \cos (6 (a+b x))-3042)}{2310 b}","\frac{16 \cos ^{11}(a+b x)}{11 b}-\frac{16 \cos ^9(a+b x)}{3 b}+\frac{48 \cos ^7(a+b x)}{7 b}-\frac{16 \cos ^5(a+b x)}{5 b}",1,"(Cos[a + b*x]^5*(-3042 + 3335*Cos[2*(a + b*x)] - 910*Cos[4*(a + b*x)] + 105*Cos[6*(a + b*x)]))/(2310*b)","A",1
25,1,27,31,0.1883214,"\int \sin ^3(a+b x) \sin ^3(2 a+2 b x) \, dx","Integrate[Sin[a + b*x]^3*Sin[2*a + 2*b*x]^3,x]","\frac{4 \sin ^7(a+b x) (7 \cos (2 (a+b x))+11)}{63 b}","\frac{8 \sin ^7(a+b x)}{7 b}-\frac{8 \sin ^9(a+b x)}{9 b}",1,"(4*(11 + 7*Cos[2*(a + b*x)])*Sin[a + b*x]^7)/(63*b)","A",1
26,1,37,46,0.1133574,"\int \sin ^3(a+b x) \sin ^2(2 a+2 b x) \, dx","Integrate[Sin[a + b*x]^3*Sin[2*a + 2*b*x]^2,x]","\frac{\cos ^3(a+b x) (108 \cos (2 (a+b x))-15 \cos (4 (a+b x))-157)}{210 b}","-\frac{4 \cos ^7(a+b x)}{7 b}+\frac{8 \cos ^5(a+b x)}{5 b}-\frac{4 \cos ^3(a+b x)}{3 b}",1,"(Cos[a + b*x]^3*(-157 + 108*Cos[2*(a + b*x)] - 15*Cos[4*(a + b*x)]))/(210*b)","A",1
27,1,15,15,0.0075551,"\int \sin ^3(a+b x) \sin (2 a+2 b x) \, dx","Integrate[Sin[a + b*x]^3*Sin[2*a + 2*b*x],x]","\frac{2 \sin ^5(a+b x)}{5 b}","\frac{2 \sin ^5(a+b x)}{5 b}",1,"(2*Sin[a + b*x]^5)/(5*b)","A",1
28,1,27,28,0.0136963,"\int \csc (2 a+2 b x) \sin ^3(a+b x) \, dx","Integrate[Csc[2*a + 2*b*x]*Sin[a + b*x]^3,x]","\frac{1}{2} \left(\frac{\tanh ^{-1}(\sin (a+b x))}{b}-\frac{\sin (a+b x)}{b}\right)","\frac{\tanh ^{-1}(\sin (a+b x))}{2 b}-\frac{\sin (a+b x)}{2 b}",1,"(ArcTanh[Sin[a + b*x]]/b - Sin[a + b*x]/b)/2","A",1
29,1,13,13,0.0104795,"\int \csc ^2(2 a+2 b x) \sin ^3(a+b x) \, dx","Integrate[Csc[2*a + 2*b*x]^2*Sin[a + b*x]^3,x]","\frac{\sec (a+b x)}{4 b}","\frac{\sec (a+b x)}{4 b}",1,"Sec[a + b*x]/(4*b)","A",1
30,1,38,34,0.0118342,"\int \csc ^3(2 a+2 b x) \sin ^3(a+b x) \, dx","Integrate[Csc[2*a + 2*b*x]^3*Sin[a + b*x]^3,x]","\frac{1}{8} \left(\frac{\tanh ^{-1}(\sin (a+b x))}{2 b}+\frac{\tan (a+b x) \sec (a+b x)}{2 b}\right)","\frac{\tanh ^{-1}(\sin (a+b x))}{16 b}+\frac{\tan (a+b x) \sec (a+b x)}{16 b}",1,"(ArcTanh[Sin[a + b*x]]/(2*b) + (Sec[a + b*x]*Tan[a + b*x])/(2*b))/8","A",1
31,1,61,43,0.0290107,"\int \csc ^4(2 a+2 b x) \sin ^3(a+b x) \, dx","Integrate[Csc[2*a + 2*b*x]^4*Sin[a + b*x]^3,x]","\frac{1}{16} \left(\frac{\sec ^3(a+b x)}{3 b}+\frac{\sec (a+b x)}{b}+\frac{\log \left(\sin \left(\frac{1}{2} (a+b x)\right)\right)}{b}-\frac{\log \left(\cos \left(\frac{1}{2} (a+b x)\right)\right)}{b}\right)","\frac{\sec ^3(a+b x)}{48 b}+\frac{\sec (a+b x)}{16 b}-\frac{\tanh ^{-1}(\cos (a+b x))}{16 b}",1,"(-(Log[Cos[(a + b*x)/2]]/b) + Log[Sin[(a + b*x)/2]]/b + Sec[a + b*x]/b + Sec[a + b*x]^3/(3*b))/16","A",1
32,1,29,70,0.0358367,"\int \csc ^5(2 a+2 b x) \sin ^3(a+b x) \, dx","Integrate[Csc[2*a + 2*b*x]^5*Sin[a + b*x]^3,x]","-\frac{\csc (a+b x) \, _2F_1\left(-\frac{1}{2},3;\frac{1}{2};\sin ^2(a+b x)\right)}{32 b}","-\frac{15 \csc (a+b x)}{256 b}+\frac{15 \tanh ^{-1}(\sin (a+b x))}{256 b}+\frac{\csc (a+b x) \sec ^4(a+b x)}{128 b}+\frac{5 \csc (a+b x) \sec ^2(a+b x)}{256 b}",1,"-1/32*(Csc[a + b*x]*Hypergeometric2F1[-1/2, 3, 1/2, Sin[a + b*x]^2])/b","C",1
33,1,119,61,0.0965725,"\int \csc (a+b x) \sin ^8(2 a+2 b x) \, dx","Integrate[Csc[a + b*x]*Sin[2*a + 2*b*x]^8,x]","-\frac{35 \cos (a+b x)}{64 b}-\frac{35 \cos (3 (a+b x))}{192 b}+\frac{21 \cos (5 (a+b x))}{320 b}+\frac{3 \cos (7 (a+b x))}{64 b}-\frac{7 \cos (9 (a+b x))}{576 b}-\frac{7 \cos (11 (a+b x))}{704 b}+\frac{\cos (13 (a+b x))}{832 b}+\frac{\cos (15 (a+b x))}{960 b}","\frac{256 \cos ^{15}(a+b x)}{15 b}-\frac{768 \cos ^{13}(a+b x)}{13 b}+\frac{768 \cos ^{11}(a+b x)}{11 b}-\frac{256 \cos ^9(a+b x)}{9 b}",1,"(-35*Cos[a + b*x])/(64*b) - (35*Cos[3*(a + b*x)])/(192*b) + (21*Cos[5*(a + b*x)])/(320*b) + (3*Cos[7*(a + b*x)])/(64*b) - (7*Cos[9*(a + b*x)])/(576*b) - (7*Cos[11*(a + b*x)])/(704*b) + Cos[13*(a + b*x)]/(832*b) + Cos[15*(a + b*x)]/(960*b)","A",1
34,1,48,61,0.2174558,"\int \csc (a+b x) \sin ^7(2 a+2 b x) \, dx","Integrate[Csc[a + b*x]*Sin[2*a + 2*b*x]^7,x]","\frac{128 \left(-231 \sin ^{13}(a+b x)+819 \sin ^{11}(a+b x)-1001 \sin ^9(a+b x)+429 \sin ^7(a+b x)\right)}{3003 b}","-\frac{128 \sin ^{13}(a+b x)}{13 b}+\frac{384 \sin ^{11}(a+b x)}{11 b}-\frac{128 \sin ^9(a+b x)}{3 b}+\frac{128 \sin ^7(a+b x)}{7 b}",1,"(128*(429*Sin[a + b*x]^7 - 1001*Sin[a + b*x]^9 + 819*Sin[a + b*x]^11 - 231*Sin[a + b*x]^13))/(3003*b)","A",1
35,1,89,46,0.0582863,"\int \csc (a+b x) \sin ^6(2 a+2 b x) \, dx","Integrate[Csc[a + b*x]*Sin[2*a + 2*b*x]^6,x]","-\frac{5 \cos (a+b x)}{8 b}-\frac{5 \cos (3 (a+b x))}{24 b}+\frac{\cos (5 (a+b x))}{16 b}+\frac{5 \cos (7 (a+b x))}{112 b}-\frac{\cos (9 (a+b x))}{144 b}-\frac{\cos (11 (a+b x))}{176 b}","-\frac{64 \cos ^{11}(a+b x)}{11 b}+\frac{128 \cos ^9(a+b x)}{9 b}-\frac{64 \cos ^7(a+b x)}{7 b}",1,"(-5*Cos[a + b*x])/(8*b) - (5*Cos[3*(a + b*x)])/(24*b) + Cos[5*(a + b*x)]/(16*b) + (5*Cos[7*(a + b*x)])/(112*b) - Cos[9*(a + b*x)]/(144*b) - Cos[11*(a + b*x)]/(176*b)","A",1
36,1,38,46,0.1006847,"\int \csc (a+b x) \sin ^5(2 a+2 b x) \, dx","Integrate[Csc[a + b*x]*Sin[2*a + 2*b*x]^5,x]","\frac{32 \left(35 \sin ^9(a+b x)-90 \sin ^7(a+b x)+63 \sin ^5(a+b x)\right)}{315 b}","\frac{32 \sin ^9(a+b x)}{9 b}-\frac{64 \sin ^7(a+b x)}{7 b}+\frac{32 \sin ^5(a+b x)}{5 b}",1,"(32*(63*Sin[a + b*x]^5 - 90*Sin[a + b*x]^7 + 35*Sin[a + b*x]^9))/(315*b)","A",1
37,1,59,31,0.0395856,"\int \csc (a+b x) \sin ^4(2 a+2 b x) \, dx","Integrate[Csc[a + b*x]*Sin[2*a + 2*b*x]^4,x]","-\frac{3 \cos (a+b x)}{4 b}-\frac{\cos (3 (a+b x))}{4 b}+\frac{\cos (5 (a+b x))}{20 b}+\frac{\cos (7 (a+b x))}{28 b}","\frac{16 \cos ^7(a+b x)}{7 b}-\frac{16 \cos ^5(a+b x)}{5 b}",1,"(-3*Cos[a + b*x])/(4*b) - Cos[3*(a + b*x)]/(4*b) + Cos[5*(a + b*x)]/(20*b) + Cos[7*(a + b*x)]/(28*b)","A",1
38,1,28,31,0.0494991,"\int \csc (a+b x) \sin ^3(2 a+2 b x) \, dx","Integrate[Csc[a + b*x]*Sin[2*a + 2*b*x]^3,x]","\frac{8 \left(5 \sin ^3(a+b x)-3 \sin ^5(a+b x)\right)}{15 b}","\frac{8 \sin ^3(a+b x)}{3 b}-\frac{8 \sin ^5(a+b x)}{5 b}",1,"(8*(5*Sin[a + b*x]^3 - 3*Sin[a + b*x]^5))/(15*b)","A",1
39,1,15,15,0.006907,"\int \csc (a+b x) \sin ^2(2 a+2 b x) \, dx","Integrate[Csc[a + b*x]*Sin[2*a + 2*b*x]^2,x]","-\frac{4 \cos ^3(a+b x)}{3 b}","-\frac{4 \cos ^3(a+b x)}{3 b}",1,"(-4*Cos[a + b*x]^3)/(3*b)","A",1
40,1,23,11,0.0088248,"\int \csc (a+b x) \sin (2 a+2 b x) \, dx","Integrate[Csc[a + b*x]*Sin[2*a + 2*b*x],x]","2 \left(\frac{\sin (a) \cos (b x)}{b}+\frac{\cos (a) \sin (b x)}{b}\right)","\frac{2 \sin (a+b x)}{b}",1,"2*((Cos[b*x]*Sin[a])/b + (Cos[a]*Sin[b*x])/b)","B",1
41,1,29,28,0.0183312,"\int \csc (a+b x) \csc (2 a+2 b x) \, dx","Integrate[Csc[a + b*x]*Csc[2*a + 2*b*x],x]","-\frac{\csc (a+b x) \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\sin ^2(a+b x)\right)}{2 b}","\frac{\tanh ^{-1}(\sin (a+b x))}{2 b}-\frac{\csc (a+b x)}{2 b}",1,"-1/2*(Csc[a + b*x]*Hypergeometric2F1[-1/2, 1, 1/2, Sin[a + b*x]^2])/b","C",1
42,1,143,49,0.2616427,"\int \csc (a+b x) \csc ^2(2 a+2 b x) \, dx","Integrate[Csc[a + b*x]*Csc[2*a + 2*b*x]^2,x]","\frac{\csc ^4(a+b x) \left(-6 \cos (2 (a+b x))+2 \cos (3 (a+b x))+3 \cos (3 (a+b x)) \log \left(\cos \left(\frac{1}{2} (a+b x)\right)\right)-3 \cos (3 (a+b x)) \log \left(\sin \left(\frac{1}{2} (a+b x)\right)\right)+\cos (a+b x) \left(3 \log \left(\sin \left(\frac{1}{2} (a+b x)\right)\right)-3 \log \left(\cos \left(\frac{1}{2} (a+b x)\right)\right)-2\right)+2\right)}{8 b \left(\csc ^2\left(\frac{1}{2} (a+b x)\right)-\sec ^2\left(\frac{1}{2} (a+b x)\right)\right)}","\frac{3 \sec (a+b x)}{8 b}-\frac{3 \tanh ^{-1}(\cos (a+b x))}{8 b}-\frac{\csc ^2(a+b x) \sec (a+b x)}{8 b}",1,"(Csc[a + b*x]^4*(2 - 6*Cos[2*(a + b*x)] + 2*Cos[3*(a + b*x)] + 3*Cos[3*(a + b*x)]*Log[Cos[(a + b*x)/2]] - 3*Cos[3*(a + b*x)]*Log[Sin[(a + b*x)/2]] + Cos[a + b*x]*(-2 - 3*Log[Cos[(a + b*x)/2]] + 3*Log[Sin[(a + b*x)/2]])))/(8*b*(Csc[(a + b*x)/2]^2 - Sec[(a + b*x)/2]^2))","B",1
43,1,31,66,0.0210776,"\int \csc (a+b x) \csc ^3(2 a+2 b x) \, dx","Integrate[Csc[a + b*x]*Csc[2*a + 2*b*x]^3,x]","-\frac{\csc ^3(a+b x) \, _2F_1\left(-\frac{3}{2},2;-\frac{1}{2};\sin ^2(a+b x)\right)}{24 b}","-\frac{5 \csc ^3(a+b x)}{48 b}-\frac{5 \csc (a+b x)}{16 b}+\frac{5 \tanh ^{-1}(\sin (a+b x))}{16 b}+\frac{\csc ^3(a+b x) \sec ^2(a+b x)}{16 b}",1,"-1/24*(Csc[a + b*x]^3*Hypergeometric2F1[-3/2, 2, -1/2, Sin[a + b*x]^2])/b","C",1
44,1,268,89,0.5142682,"\int \csc (a+b x) \csc ^4(2 a+2 b x) \, dx","Integrate[Csc[a + b*x]*Csc[2*a + 2*b*x]^4,x]","-\frac{\csc ^{10}(a+b x) \left(658 \cos (2 (a+b x))-228 \cos (3 (a+b x))+140 \cos (4 (a+b x))-76 \cos (5 (a+b x))-210 \cos (6 (a+b x))+76 \cos (7 (a+b x))-315 \cos (3 (a+b x)) \log \left(\cos \left(\frac{1}{2} (a+b x)\right)\right)-105 \cos (5 (a+b x)) \log \left(\cos \left(\frac{1}{2} (a+b x)\right)\right)+105 \cos (7 (a+b x)) \log \left(\cos \left(\frac{1}{2} (a+b x)\right)\right)+3 \cos (a+b x) \left(-105 \log \left(\sin \left(\frac{1}{2} (a+b x)\right)\right)+105 \log \left(\cos \left(\frac{1}{2} (a+b x)\right)\right)+76\right)+315 \cos (3 (a+b x)) \log \left(\sin \left(\frac{1}{2} (a+b x)\right)\right)+105 \cos (5 (a+b x)) \log \left(\sin \left(\frac{1}{2} (a+b x)\right)\right)-105 \cos (7 (a+b x)) \log \left(\sin \left(\frac{1}{2} (a+b x)\right)\right)-204\right)}{384 b \left(\csc ^2\left(\frac{1}{2} (a+b x)\right)-\sec ^2\left(\frac{1}{2} (a+b x)\right)\right)^3}","\frac{35 \sec ^3(a+b x)}{384 b}+\frac{35 \sec (a+b x)}{128 b}-\frac{35 \tanh ^{-1}(\cos (a+b x))}{128 b}-\frac{\csc ^4(a+b x) \sec ^3(a+b x)}{64 b}-\frac{7 \csc ^2(a+b x) \sec ^3(a+b x)}{128 b}",1,"-1/384*(Csc[a + b*x]^10*(-204 + 658*Cos[2*(a + b*x)] - 228*Cos[3*(a + b*x)] + 140*Cos[4*(a + b*x)] - 76*Cos[5*(a + b*x)] - 210*Cos[6*(a + b*x)] + 76*Cos[7*(a + b*x)] - 315*Cos[3*(a + b*x)]*Log[Cos[(a + b*x)/2]] - 105*Cos[5*(a + b*x)]*Log[Cos[(a + b*x)/2]] + 105*Cos[7*(a + b*x)]*Log[Cos[(a + b*x)/2]] + 3*Cos[a + b*x]*(76 + 105*Log[Cos[(a + b*x)/2]] - 105*Log[Sin[(a + b*x)/2]]) + 315*Cos[3*(a + b*x)]*Log[Sin[(a + b*x)/2]] + 105*Cos[5*(a + b*x)]*Log[Sin[(a + b*x)/2]] - 105*Cos[7*(a + b*x)]*Log[Sin[(a + b*x)/2]]))/(b*(Csc[(a + b*x)/2]^2 - Sec[(a + b*x)/2]^2)^3)","B",1
45,1,85,155,0.255238,"\int \csc ^2(a+b x) \sin ^8(2 a+2 b x) \, dx","Integrate[Csc[a + b*x]^2*Sin[2*a + 2*b*x]^8,x]","\frac{105 \sin (2 (a+b x))-315 \sin (4 (a+b x))-63 \sin (6 (a+b x))+63 \sin (8 (a+b x))+21 \sin (10 (a+b x))-7 \sin (12 (a+b x))-3 \sin (14 (a+b x))+840 a+840 b x}{1344 b}","-\frac{128 \sin ^5(a+b x) \cos ^9(a+b x)}{7 b}-\frac{160 \sin ^3(a+b x) \cos ^9(a+b x)}{21 b}-\frac{16 \sin (a+b x) \cos ^9(a+b x)}{7 b}+\frac{2 \sin (a+b x) \cos ^7(a+b x)}{7 b}+\frac{\sin (a+b x) \cos ^5(a+b x)}{3 b}+\frac{5 \sin (a+b x) \cos ^3(a+b x)}{12 b}+\frac{5 \sin (a+b x) \cos (a+b x)}{8 b}+\frac{5 x}{8}",1,"(840*a + 840*b*x + 105*Sin[2*(a + b*x)] - 315*Sin[4*(a + b*x)] - 63*Sin[6*(a + b*x)] + 63*Sin[8*(a + b*x)] + 21*Sin[10*(a + b*x)] - 7*Sin[12*(a + b*x)] - 3*Sin[14*(a + b*x)])/(1344*b)","A",1
46,1,48,44,0.1752094,"\int \csc ^2(a+b x) \sin ^7(2 a+2 b x) \, dx","Integrate[Csc[a + b*x]^2*Sin[2*a + 2*b*x]^7,x]","\frac{16 \left(-10 \sin ^{12}(a+b x)+36 \sin ^{10}(a+b x)-45 \sin ^8(a+b x)+20 \sin ^6(a+b x)\right)}{15 b}","-\frac{32 \cos ^{12}(a+b x)}{3 b}+\frac{128 \cos ^{10}(a+b x)}{5 b}-\frac{16 \cos ^8(a+b x)}{b}",1,"(16*(20*Sin[a + b*x]^6 - 45*Sin[a + b*x]^8 + 36*Sin[a + b*x]^10 - 10*Sin[a + b*x]^12))/(15*b)","A",1
47,1,62,111,0.1986059,"\int \csc ^2(a+b x) \sin ^6(2 a+2 b x) \, dx","Integrate[Csc[a + b*x]^2*Sin[2*a + 2*b*x]^6,x]","\frac{20 \sin (2 (a+b x))-40 \sin (4 (a+b x))-10 \sin (6 (a+b x))+5 \sin (8 (a+b x))+2 \sin (10 (a+b x))+120 b x}{160 b}","-\frac{32 \sin ^3(a+b x) \cos ^7(a+b x)}{5 b}-\frac{12 \sin (a+b x) \cos ^7(a+b x)}{5 b}+\frac{2 \sin (a+b x) \cos ^5(a+b x)}{5 b}+\frac{\sin (a+b x) \cos ^3(a+b x)}{2 b}+\frac{3 \sin (a+b x) \cos (a+b x)}{4 b}+\frac{3 x}{4}",1,"(120*b*x + 20*Sin[2*(a + b*x)] - 40*Sin[4*(a + b*x)] - 10*Sin[6*(a + b*x)] + 5*Sin[8*(a + b*x)] + 2*Sin[10*(a + b*x)])/(160*b)","A",1
48,1,48,29,0.1250936,"\int \csc ^2(a+b x) \sin ^5(2 a+2 b x) \, dx","Integrate[Csc[a + b*x]^2*Sin[2*a + 2*b*x]^5,x]","\frac{-72 \cos (2 (a+b x))-12 \cos (4 (a+b x))+8 \cos (6 (a+b x))+3 \cos (8 (a+b x))}{96 b}","\frac{4 \cos ^8(a+b x)}{b}-\frac{16 \cos ^6(a+b x)}{3 b}",1,"(-72*Cos[2*(a + b*x)] - 12*Cos[4*(a + b*x)] + 8*Cos[6*(a + b*x)] + 3*Cos[8*(a + b*x)])/(96*b)","A",1
49,1,40,60,0.1026398,"\int \csc ^2(a+b x) \sin ^4(2 a+2 b x) \, dx","Integrate[Csc[a + b*x]^2*Sin[2*a + 2*b*x]^4,x]","-\frac{-3 \sin (2 (a+b x))+3 \sin (4 (a+b x))+\sin (6 (a+b x))-12 b x}{12 b}","-\frac{8 \sin (a+b x) \cos ^5(a+b x)}{3 b}+\frac{2 \sin (a+b x) \cos ^3(a+b x)}{3 b}+\frac{\sin (a+b x) \cos (a+b x)}{b}+x",1,"-1/12*(-12*b*x - 3*Sin[2*(a + b*x)] + 3*Sin[4*(a + b*x)] + Sin[6*(a + b*x)])/b","A",1
50,1,13,13,0.0070622,"\int \csc ^2(a+b x) \sin ^3(2 a+2 b x) \, dx","Integrate[Csc[a + b*x]^2*Sin[2*a + 2*b*x]^3,x]","-\frac{2 \cos ^4(a+b x)}{b}","-\frac{2 \cos ^4(a+b x)}{b}",1,"(-2*Cos[a + b*x]^4)/b","A",1
51,1,20,21,0.0221377,"\int \csc ^2(a+b x) \sin ^2(2 a+2 b x) \, dx","Integrate[Csc[a + b*x]^2*Sin[2*a + 2*b*x]^2,x]","\frac{2 (a+b x)+\sin (2 (a+b x))}{b}","\frac{2 \sin (a+b x) \cos (a+b x)}{b}+2 x",1,"(2*(a + b*x) + Sin[2*(a + b*x)])/b","A",1
52,1,20,12,0.0177977,"\int \csc ^2(a+b x) \sin (2 a+2 b x) \, dx","Integrate[Csc[a + b*x]^2*Sin[2*a + 2*b*x],x]","\frac{2 (\log (\tan (a+b x))+\log (\cos (a+b x)))}{b}","\frac{2 \log (\sin (a+b x))}{b}",1,"(2*(Log[Cos[a + b*x]] + Log[Tan[a + b*x]]))/b","A",1
53,1,34,30,0.0547937,"\int \csc ^2(a+b x) \csc (2 a+2 b x) \, dx","Integrate[Csc[a + b*x]^2*Csc[2*a + 2*b*x],x]","-\frac{\csc ^2(a+b x)-2 \log (\sin (a+b x))+2 \log (\cos (a+b x))}{4 b}","\frac{\log (\tan (a+b x))}{2 b}-\frac{\cot ^2(a+b x)}{4 b}",1,"-1/4*(Csc[a + b*x]^2 + 2*Log[Cos[a + b*x]] - 2*Log[Sin[a + b*x]])/b","A",1
54,1,48,42,0.0803898,"\int \csc ^2(a+b x) \csc ^2(2 a+2 b x) \, dx","Integrate[Csc[a + b*x]^2*Csc[2*a + 2*b*x]^2,x]","\frac{\tan (a+b x)}{4 b}-\frac{5 \cot (a+b x)}{12 b}-\frac{\cot (a+b x) \csc ^2(a+b x)}{12 b}","\frac{\tan (a+b x)}{4 b}-\frac{\cot ^3(a+b x)}{12 b}-\frac{\cot (a+b x)}{2 b}",1,"(-5*Cot[a + b*x])/(12*b) - (Cot[a + b*x]*Csc[a + b*x]^2)/(12*b) + Tan[a + b*x]/(4*b)","A",1
55,1,54,60,0.3609973,"\int \csc ^2(a+b x) \csc ^3(2 a+2 b x) \, dx","Integrate[Csc[a + b*x]^2*Csc[2*a + 2*b*x]^3,x]","-\frac{\csc ^4(a+b x)+4 \csc ^2(a+b x)-2 \sec ^2(a+b x)-12 \log (\sin (a+b x))+12 \log (\cos (a+b x))}{32 b}","\frac{\tan ^2(a+b x)}{16 b}-\frac{\cot ^4(a+b x)}{32 b}-\frac{3 \cot ^2(a+b x)}{16 b}+\frac{3 \log (\tan (a+b x))}{8 b}",1,"-1/32*(4*Csc[a + b*x]^2 + Csc[a + b*x]^4 + 12*Log[Cos[a + b*x]] - 12*Log[Sin[a + b*x]] - 2*Sec[a + b*x]^2)/b","A",1
56,1,90,72,0.054687,"\int \csc ^2(a+b x) \csc ^4(2 a+2 b x) \, dx","Integrate[Csc[a + b*x]^2*Csc[2*a + 2*b*x]^4,x]","\frac{11 \tan (a+b x)}{48 b}-\frac{73 \cot (a+b x)}{240 b}-\frac{\cot (a+b x) \csc ^4(a+b x)}{80 b}-\frac{7 \cot (a+b x) \csc ^2(a+b x)}{120 b}+\frac{\tan (a+b x) \sec ^2(a+b x)}{48 b}","\frac{\tan ^3(a+b x)}{48 b}+\frac{\tan (a+b x)}{4 b}-\frac{\cot ^5(a+b x)}{80 b}-\frac{\cot ^3(a+b x)}{12 b}-\frac{3 \cot (a+b x)}{8 b}",1,"(-73*Cot[a + b*x])/(240*b) - (7*Cot[a + b*x]*Csc[a + b*x]^2)/(120*b) - (Cot[a + b*x]*Csc[a + b*x]^4)/(80*b) + (11*Tan[a + b*x])/(48*b) + (Sec[a + b*x]^2*Tan[a + b*x])/(48*b)","A",1
57,1,76,90,0.4248518,"\int \csc ^2(a+b x) \csc ^5(2 a+2 b x) \, dx","Integrate[Csc[a + b*x]^2*Csc[2*a + 2*b*x]^5,x]","-\frac{2 \csc ^6(a+b x)+9 \csc ^4(a+b x)+36 \csc ^2(a+b x)-3 \sec ^4(a+b x)-24 \sec ^2(a+b x)-120 \log (\sin (a+b x))+120 \log (\cos (a+b x))}{384 b}","\frac{\tan ^4(a+b x)}{128 b}+\frac{5 \tan ^2(a+b x)}{64 b}-\frac{\cot ^6(a+b x)}{192 b}-\frac{5 \cot ^4(a+b x)}{128 b}-\frac{5 \cot ^2(a+b x)}{32 b}+\frac{5 \log (\tan (a+b x))}{16 b}",1,"-1/384*(36*Csc[a + b*x]^2 + 9*Csc[a + b*x]^4 + 2*Csc[a + b*x]^6 + 120*Log[Cos[a + b*x]] - 120*Log[Sin[a + b*x]] - 24*Sec[a + b*x]^2 - 3*Sec[a + b*x]^4)/b","A",1
58,1,132,102,0.0727272,"\int \csc ^2(a+b x) \csc ^6(2 a+2 b x) \, dx","Integrate[Csc[a + b*x]^2*Csc[2*a + 2*b*x]^6,x]","\frac{33 \tan (a+b x)}{160 b}-\frac{281 \cot (a+b x)}{1120 b}-\frac{\cot (a+b x) \csc ^6(a+b x)}{448 b}-\frac{27 \cot (a+b x) \csc ^4(a+b x)}{2240 b}-\frac{53 \cot (a+b x) \csc ^2(a+b x)}{1120 b}+\frac{\tan (a+b x) \sec ^4(a+b x)}{320 b}+\frac{\tan (a+b x) \sec ^2(a+b x)}{40 b}","\frac{\tan ^5(a+b x)}{320 b}+\frac{\tan ^3(a+b x)}{32 b}+\frac{15 \tan (a+b x)}{64 b}-\frac{\cot ^7(a+b x)}{448 b}-\frac{3 \cot ^5(a+b x)}{160 b}-\frac{5 \cot ^3(a+b x)}{64 b}-\frac{5 \cot (a+b x)}{16 b}",1,"(-281*Cot[a + b*x])/(1120*b) - (53*Cot[a + b*x]*Csc[a + b*x]^2)/(1120*b) - (27*Cot[a + b*x]*Csc[a + b*x]^4)/(2240*b) - (Cot[a + b*x]*Csc[a + b*x]^6)/(448*b) + (33*Tan[a + b*x])/(160*b) + (Sec[a + b*x]^2*Tan[a + b*x])/(40*b) + (Sec[a + b*x]^4*Tan[a + b*x])/(320*b)","A",1
59,1,119,61,0.1520223,"\int \csc ^3(a+b x) \sin ^{10}(2 a+2 b x) \, dx","Integrate[Csc[a + b*x]^3*Sin[2*a + 2*b*x]^10,x]","-\frac{35 \cos (a+b x)}{32 b}-\frac{7 \cos (3 (a+b x))}{16 b}+\frac{7 \cos (5 (a+b x))}{80 b}+\frac{\cos (7 (a+b x))}{8 b}-\frac{5 \cos (11 (a+b x))}{176 b}-\frac{\cos (13 (a+b x))}{208 b}+\frac{\cos (15 (a+b x))}{320 b}+\frac{\cos (17 (a+b x))}{1088 b}","\frac{1024 \cos ^{17}(a+b x)}{17 b}-\frac{1024 \cos ^{15}(a+b x)}{5 b}+\frac{3072 \cos ^{13}(a+b x)}{13 b}-\frac{1024 \cos ^{11}(a+b x)}{11 b}",1,"(-35*Cos[a + b*x])/(32*b) - (7*Cos[3*(a + b*x)])/(16*b) + (7*Cos[5*(a + b*x)])/(80*b) + Cos[7*(a + b*x)]/(8*b) - (5*Cos[11*(a + b*x)])/(176*b) - Cos[13*(a + b*x)]/(208*b) + Cos[15*(a + b*x)]/(320*b) + Cos[17*(a + b*x)]/(1088*b)","A",1
60,1,58,76,0.327488,"\int \csc ^3(a+b x) \sin ^9(2 a+2 b x) \, dx","Integrate[Csc[a + b*x]^3*Sin[2*a + 2*b*x]^9,x]","\frac{512 \left(3003 \sin ^{15}(a+b x)-13860 \sin ^{13}(a+b x)+24570 \sin ^{11}(a+b x)-20020 \sin ^9(a+b x)+6435 \sin ^7(a+b x)\right)}{45045 b}","\frac{512 \sin ^{15}(a+b x)}{15 b}-\frac{2048 \sin ^{13}(a+b x)}{13 b}+\frac{3072 \sin ^{11}(a+b x)}{11 b}-\frac{2048 \sin ^9(a+b x)}{9 b}+\frac{512 \sin ^7(a+b x)}{7 b}",1,"(512*(6435*Sin[a + b*x]^7 - 20020*Sin[a + b*x]^9 + 24570*Sin[a + b*x]^11 - 13860*Sin[a + b*x]^13 + 3003*Sin[a + b*x]^15))/(45045*b)","A",1
61,1,104,46,0.1014758,"\int \csc ^3(a+b x) \sin ^8(2 a+2 b x) \, dx","Integrate[Csc[a + b*x]^3*Sin[2*a + 2*b*x]^8,x]","-\frac{5 \cos (a+b x)}{4 b}-\frac{25 \cos (3 (a+b x))}{48 b}+\frac{\cos (5 (a+b x))}{16 b}+\frac{\cos (7 (a+b x))}{8 b}+\frac{\cos (9 (a+b x))}{72 b}-\frac{3 \cos (11 (a+b x))}{176 b}-\frac{\cos (13 (a+b x))}{208 b}","-\frac{256 \cos ^{13}(a+b x)}{13 b}+\frac{512 \cos ^{11}(a+b x)}{11 b}-\frac{256 \cos ^9(a+b x)}{9 b}",1,"(-5*Cos[a + b*x])/(4*b) - (25*Cos[3*(a + b*x)])/(48*b) + Cos[5*(a + b*x)]/(16*b) + Cos[7*(a + b*x)]/(8*b) + Cos[9*(a + b*x)]/(72*b) - (3*Cos[11*(a + b*x)])/(176*b) - Cos[13*(a + b*x)]/(208*b)","B",1
62,1,48,61,0.1688646,"\int \csc ^3(a+b x) \sin ^7(2 a+2 b x) \, dx","Integrate[Csc[a + b*x]^3*Sin[2*a + 2*b*x]^7,x]","\frac{128 \left(-105 \sin ^{11}(a+b x)+385 \sin ^9(a+b x)-495 \sin ^7(a+b x)+231 \sin ^5(a+b x)\right)}{1155 b}","-\frac{128 \sin ^{11}(a+b x)}{11 b}+\frac{128 \sin ^9(a+b x)}{3 b}-\frac{384 \sin ^7(a+b x)}{7 b}+\frac{128 \sin ^5(a+b x)}{5 b}",1,"(128*(231*Sin[a + b*x]^5 - 495*Sin[a + b*x]^7 + 385*Sin[a + b*x]^9 - 105*Sin[a + b*x]^11))/(1155*b)","A",1
63,1,27,31,0.1512543,"\int \csc ^3(a+b x) \sin ^6(2 a+2 b x) \, dx","Integrate[Csc[a + b*x]^3*Sin[2*a + 2*b*x]^6,x]","\frac{32 \cos ^7(a+b x) (7 \cos (2 (a+b x))-11)}{63 b}","\frac{64 \cos ^9(a+b x)}{9 b}-\frac{64 \cos ^7(a+b x)}{7 b}",1,"(32*Cos[a + b*x]^7*(-11 + 7*Cos[2*(a + b*x)]))/(63*b)","A",1
64,1,37,46,0.110692,"\int \csc ^3(a+b x) \sin ^5(2 a+2 b x) \, dx","Integrate[Csc[a + b*x]^3*Sin[2*a + 2*b*x]^5,x]","\frac{4 \sin ^3(a+b x) (108 \cos (2 (a+b x))+15 \cos (4 (a+b x))+157)}{105 b}","\frac{32 \sin ^7(a+b x)}{7 b}-\frac{64 \sin ^5(a+b x)}{5 b}+\frac{32 \sin ^3(a+b x)}{3 b}",1,"(4*(157 + 108*Cos[2*(a + b*x)] + 15*Cos[4*(a + b*x)])*Sin[a + b*x]^3)/(105*b)","A",1
65,1,15,15,0.0091515,"\int \csc ^3(a+b x) \sin ^4(2 a+2 b x) \, dx","Integrate[Csc[a + b*x]^3*Sin[2*a + 2*b*x]^4,x]","-\frac{16 \cos ^5(a+b x)}{5 b}","-\frac{16 \cos ^5(a+b x)}{5 b}",1,"(-16*Cos[a + b*x]^5)/(5*b)","A",1
66,1,28,27,0.0114744,"\int \csc ^3(a+b x) \sin ^3(2 a+2 b x) \, dx","Integrate[Csc[a + b*x]^3*Sin[2*a + 2*b*x]^3,x]","8 \left(\frac{\sin (a+b x)}{b}-\frac{\sin ^3(a+b x)}{3 b}\right)","\frac{8 \sin (a+b x)}{b}-\frac{8 \sin ^3(a+b x)}{3 b}",1,"8*(Sin[a + b*x]/b - Sin[a + b*x]^3/(3*b))","A",1
67,1,44,24,0.0229732,"\int \csc ^3(a+b x) \sin ^2(2 a+2 b x) \, dx","Integrate[Csc[a + b*x]^3*Sin[2*a + 2*b*x]^2,x]","4 \left(\frac{\cos (a+b x)}{b}+\frac{\log \left(\sin \left(\frac{1}{2} (a+b x)\right)\right)}{b}-\frac{\log \left(\cos \left(\frac{1}{2} (a+b x)\right)\right)}{b}\right)","\frac{4 \cos (a+b x)}{b}-\frac{4 \tanh ^{-1}(\cos (a+b x))}{b}",1,"4*(Cos[a + b*x]/b - Log[Cos[(a + b*x)/2]]/b + Log[Sin[(a + b*x)/2]]/b)","A",1
68,1,11,11,0.0105855,"\int \csc ^3(a+b x) \sin (2 a+2 b x) \, dx","Integrate[Csc[a + b*x]^3*Sin[2*a + 2*b*x],x]","-\frac{2 \csc (a+b x)}{b}","-\frac{2 \csc (a+b x)}{b}",1,"(-2*Csc[a + b*x])/b","A",1
69,1,31,43,0.0177719,"\int \csc ^3(a+b x) \csc (2 a+2 b x) \, dx","Integrate[Csc[a + b*x]^3*Csc[2*a + 2*b*x],x]","-\frac{\csc ^3(a+b x) \, _2F_1\left(-\frac{3}{2},1;-\frac{1}{2};\sin ^2(a+b x)\right)}{6 b}","-\frac{\csc ^3(a+b x)}{6 b}-\frac{\csc (a+b x)}{2 b}+\frac{\tanh ^{-1}(\sin (a+b x))}{2 b}",1,"-1/6*(Csc[a + b*x]^3*Hypergeometric2F1[-3/2, 1, -1/2, Sin[a + b*x]^2])/b","C",1
70,1,129,70,4.4702881,"\int \csc ^3(a+b x) \csc ^2(2 a+2 b x) \, dx","Integrate[Csc[a + b*x]^3*Csc[2*a + 2*b*x]^2,x]","-\frac{\csc ^4\left(\frac{1}{2} (a+b x)\right)+14 \csc ^2\left(\frac{1}{2} (a+b x)\right)+\frac{\sec ^2\left(\frac{1}{2} (a+b x)\right) \left(-14 \tan ^2\left(\frac{1}{2} (a+b x)\right)+\cos (a+b x) \left(\sec ^4\left(\frac{1}{2} (a+b x)\right)-8 \left(-15 \log \left(\sin \left(\frac{1}{2} (a+b x)\right)\right)+15 \log \left(\cos \left(\frac{1}{2} (a+b x)\right)\right)+8\right)\right)+78\right)}{\tan ^2\left(\frac{1}{2} (a+b x)\right)-1}}{256 b}","\frac{15 \sec (a+b x)}{32 b}-\frac{15 \tanh ^{-1}(\cos (a+b x))}{32 b}-\frac{\csc ^4(a+b x) \sec (a+b x)}{16 b}-\frac{5 \csc ^2(a+b x) \sec (a+b x)}{32 b}",1,"-1/256*(14*Csc[(a + b*x)/2]^2 + Csc[(a + b*x)/2]^4 + (Sec[(a + b*x)/2]^2*(78 + Cos[a + b*x]*(-8*(8 + 15*Log[Cos[(a + b*x)/2]] - 15*Log[Sin[(a + b*x)/2]]) + Sec[(a + b*x)/2]^4) - 14*Tan[(a + b*x)/2]^2))/(-1 + Tan[(a + b*x)/2]^2))/b","A",1
71,1,31,81,0.035941,"\int \csc ^3(a+b x) \csc ^3(2 a+2 b x) \, dx","Integrate[Csc[a + b*x]^3*Csc[2*a + 2*b*x]^3,x]","-\frac{\csc ^5(a+b x) \, _2F_1\left(-\frac{5}{2},2;-\frac{3}{2};\sin ^2(a+b x)\right)}{40 b}","-\frac{7 \csc ^5(a+b x)}{80 b}-\frac{7 \csc ^3(a+b x)}{48 b}-\frac{7 \csc (a+b x)}{16 b}+\frac{7 \tanh ^{-1}(\sin (a+b x))}{16 b}+\frac{\csc ^5(a+b x) \sec ^2(a+b x)}{16 b}",1,"-1/40*(Csc[a + b*x]^5*Hypergeometric2F1[-5/2, 2, -3/2, Sin[a + b*x]^2])/b","C",1
72,1,278,112,0.8457746,"\int \csc ^3(a+b x) \csc ^4(2 a+2 b x) \, dx","Integrate[Csc[a + b*x]^3*Csc[2*a + 2*b*x]^4,x]","\frac{\csc ^{12}(a+b x) \left(-4752 \cos (2 (a+b x))+1600 \cos (3 (a+b x))+504 \cos (4 (a+b x))+1680 \cos (6 (a+b x))-600 \cos (7 (a+b x))-630 \cos (8 (a+b x))+200 \cos (9 (a+b x))+2520 \cos (3 (a+b x)) \log \left(\cos \left(\frac{1}{2} (a+b x)\right)\right)-945 \cos (7 (a+b x)) \log \left(\cos \left(\frac{1}{2} (a+b x)\right)\right)+315 \cos (9 (a+b x)) \log \left(\cos \left(\frac{1}{2} (a+b x)\right)\right)-30 \cos (a+b x) \left(-63 \log \left(\sin \left(\frac{1}{2} (a+b x)\right)\right)+63 \log \left(\cos \left(\frac{1}{2} (a+b x)\right)\right)+40\right)-2520 \cos (3 (a+b x)) \log \left(\sin \left(\frac{1}{2} (a+b x)\right)\right)+945 \cos (7 (a+b x)) \log \left(\sin \left(\frac{1}{2} (a+b x)\right)\right)-315 \cos (9 (a+b x)) \log \left(\sin \left(\frac{1}{2} (a+b x)\right)\right)+1150\right)}{3072 b \left(\csc ^2\left(\frac{1}{2} (a+b x)\right)-\sec ^2\left(\frac{1}{2} (a+b x)\right)\right)^3}","\frac{35 \sec ^3(a+b x)}{256 b}+\frac{105 \sec (a+b x)}{256 b}-\frac{105 \tanh ^{-1}(\cos (a+b x))}{256 b}-\frac{\csc ^6(a+b x) \sec ^3(a+b x)}{96 b}-\frac{3 \csc ^4(a+b x) \sec ^3(a+b x)}{128 b}-\frac{21 \csc ^2(a+b x) \sec ^3(a+b x)}{256 b}",1,"(Csc[a + b*x]^12*(1150 - 4752*Cos[2*(a + b*x)] + 1600*Cos[3*(a + b*x)] + 504*Cos[4*(a + b*x)] + 1680*Cos[6*(a + b*x)] - 600*Cos[7*(a + b*x)] - 630*Cos[8*(a + b*x)] + 200*Cos[9*(a + b*x)] + 2520*Cos[3*(a + b*x)]*Log[Cos[(a + b*x)/2]] - 945*Cos[7*(a + b*x)]*Log[Cos[(a + b*x)/2]] + 315*Cos[9*(a + b*x)]*Log[Cos[(a + b*x)/2]] - 30*Cos[a + b*x]*(40 + 63*Log[Cos[(a + b*x)/2]] - 63*Log[Sin[(a + b*x)/2]]) - 2520*Cos[3*(a + b*x)]*Log[Sin[(a + b*x)/2]] + 945*Cos[7*(a + b*x)]*Log[Sin[(a + b*x)/2]] - 315*Cos[9*(a + b*x)]*Log[Sin[(a + b*x)/2]]))/(3072*b*(Csc[(a + b*x)/2]^2 - Sec[(a + b*x)/2]^2)^3)","B",1
73,1,98,136,0.2966369,"\int \sin (a+b x) \sin ^{\frac{5}{2}}(2 a+2 b x) \, dx","Integrate[Sin[a + b*x]*Sin[2*a + 2*b*x]^(5/2),x]","\frac{15 \left(\log \left(\sin (a+b x)+\sqrt{\sin (2 (a+b x))}+\cos (a+b x)\right)-\sin ^{-1}(\cos (a+b x)-\sin (a+b x))\right)-2 \sqrt{\sin (2 (a+b x))} (14 \cos (a+b x)+3 \cos (3 (a+b x))-2 \cos (5 (a+b x)))}{96 b}","\frac{5 \sin (a+b x) \sin ^{\frac{3}{2}}(2 a+2 b x)}{24 b}-\frac{5 \sin ^{-1}(\cos (a+b x)-\sin (a+b x))}{32 b}-\frac{\sin ^{\frac{5}{2}}(2 a+2 b x) \cos (a+b x)}{6 b}-\frac{5 \sqrt{\sin (2 a+2 b x)} \cos (a+b x)}{16 b}+\frac{5 \log \left(\sin (a+b x)+\sqrt{\sin (2 a+2 b x)}+\cos (a+b x)\right)}{32 b}",1,"(15*(-ArcSin[Cos[a + b*x] - Sin[a + b*x]] + Log[Cos[a + b*x] + Sin[a + b*x] + Sqrt[Sin[2*(a + b*x)]]]) - 2*(14*Cos[a + b*x] + 3*Cos[3*(a + b*x)] - 2*Cos[5*(a + b*x)])*Sqrt[Sin[2*(a + b*x)]])/(96*b)","A",1
74,1,86,110,0.1992183,"\int \sin (a+b x) \sin ^{\frac{3}{2}}(2 a+2 b x) \, dx","Integrate[Sin[a + b*x]*Sin[2*a + 2*b*x]^(3/2),x]","\frac{2 \sqrt{\sin (2 (a+b x))} (2 \sin (a+b x)-\sin (3 (a+b x)))-3 \left(\sin ^{-1}(\cos (a+b x)-\sin (a+b x))+\log \left(\sin (a+b x)+\sqrt{\sin (2 (a+b x))}+\cos (a+b x)\right)\right)}{16 b}","\frac{3 \sin (a+b x) \sqrt{\sin (2 a+2 b x)}}{8 b}-\frac{3 \sin ^{-1}(\cos (a+b x)-\sin (a+b x))}{16 b}-\frac{\sin ^{\frac{3}{2}}(2 a+2 b x) \cos (a+b x)}{4 b}-\frac{3 \log \left(\sin (a+b x)+\sqrt{\sin (2 a+2 b x)}+\cos (a+b x)\right)}{16 b}",1,"(-3*(ArcSin[Cos[a + b*x] - Sin[a + b*x]] + Log[Cos[a + b*x] + Sin[a + b*x] + Sqrt[Sin[2*(a + b*x)]]]) + 2*Sqrt[Sin[2*(a + b*x)]]*(2*Sin[a + b*x] - Sin[3*(a + b*x)]))/(16*b)","A",1
75,1,72,84,0.0763048,"\int \sin (a+b x) \sqrt{\sin (2 a+2 b x)} \, dx","Integrate[Sin[a + b*x]*Sqrt[Sin[2*a + 2*b*x]],x]","\frac{-\sin ^{-1}(\cos (a+b x)-\sin (a+b x))-2 \sqrt{\sin (2 (a+b x))} \cos (a+b x)+\log \left(\sin (a+b x)+\sqrt{\sin (2 (a+b x))}+\cos (a+b x)\right)}{4 b}","-\frac{\sin ^{-1}(\cos (a+b x)-\sin (a+b x))}{4 b}-\frac{\sqrt{\sin (2 a+2 b x)} \cos (a+b x)}{2 b}+\frac{\log \left(\sin (a+b x)+\sqrt{\sin (2 a+2 b x)}+\cos (a+b x)\right)}{4 b}",1,"(-ArcSin[Cos[a + b*x] - Sin[a + b*x]] + Log[Cos[a + b*x] + Sin[a + b*x] + Sqrt[Sin[2*(a + b*x)]]] - 2*Cos[a + b*x]*Sqrt[Sin[2*(a + b*x)]])/(4*b)","A",1
76,1,50,58,0.0526351,"\int \frac{\sin (a+b x)}{\sqrt{\sin (2 a+2 b x)}} \, dx","Integrate[Sin[a + b*x]/Sqrt[Sin[2*a + 2*b*x]],x]","-\frac{\sin ^{-1}(\cos (a+b x)-\sin (a+b x))+\log \left(\sin (a+b x)+\sqrt{\sin (2 (a+b x))}+\cos (a+b x)\right)}{2 b}","-\frac{\sin ^{-1}(\cos (a+b x)-\sin (a+b x))}{2 b}-\frac{\log \left(\sin (a+b x)+\sqrt{\sin (2 a+2 b x)}+\cos (a+b x)\right)}{2 b}",1,"-1/2*(ArcSin[Cos[a + b*x] - Sin[a + b*x]] + Log[Cos[a + b*x] + Sin[a + b*x] + Sqrt[Sin[2*(a + b*x)]]])/b","A",1
77,1,22,23,0.0196333,"\int \frac{\sin (a+b x)}{\sin ^{\frac{3}{2}}(2 a+2 b x)} \, dx","Integrate[Sin[a + b*x]/Sin[2*a + 2*b*x]^(3/2),x]","\frac{\sin (a+b x)}{b \sqrt{\sin (2 (a+b x))}}","\frac{\sin (a+b x)}{b \sqrt{\sin (2 a+2 b x)}}",1,"Sin[a + b*x]/(b*Sqrt[Sin[2*(a + b*x)]])","A",1
78,1,43,53,0.1088953,"\int \frac{\sin (a+b x)}{\sin ^{\frac{5}{2}}(2 a+2 b x)} \, dx","Integrate[Sin[a + b*x]/Sin[2*a + 2*b*x]^(5/2),x]","\frac{\sqrt{\sin (2 (a+b x))} \left(\frac{1}{12} \tan (a+b x) \sec (a+b x)-\frac{1}{4} \csc (a+b x)\right)}{b}","\frac{\sin (a+b x)}{3 b \sin ^{\frac{3}{2}}(2 a+2 b x)}-\frac{2 \cos (a+b x)}{3 b \sqrt{\sin (2 a+2 b x)}}",1,"(Sqrt[Sin[2*(a + b*x)]]*(-1/4*Csc[a + b*x] + (Sec[a + b*x]*Tan[a + b*x])/12))/b","A",1
79,1,52,79,0.1954642,"\int \frac{\sin (a+b x)}{\sin ^{\frac{7}{2}}(2 a+2 b x)} \, dx","Integrate[Sin[a + b*x]/Sin[2*a + 2*b*x]^(7/2),x]","\frac{\sqrt{\sin (2 (a+b x))} \left(3 \sec (a+b x) \left(\sec ^2(a+b x)+9\right)-5 \cot (a+b x) \csc (a+b x)\right)}{120 b}","\frac{\sin (a+b x)}{5 b \sin ^{\frac{5}{2}}(2 a+2 b x)}+\frac{8 \sin (a+b x)}{15 b \sqrt{\sin (2 a+2 b x)}}-\frac{4 \cos (a+b x)}{15 b \sin ^{\frac{3}{2}}(2 a+2 b x)}",1,"((-5*Cot[a + b*x]*Csc[a + b*x] + 3*Sec[a + b*x]*(9 + Sec[a + b*x]^2))*Sqrt[Sin[2*(a + b*x)]])/(120*b)","A",1
80,1,67,105,0.146454,"\int \frac{\sin (a+b x)}{\sin ^{\frac{9}{2}}(2 a+2 b x)} \, dx","Integrate[Sin[a + b*x]/Sin[2*a + 2*b*x]^(9/2),x]","\frac{\sqrt{\sin (2 (a+b x))} (-10 \cos (2 (a+b x))+4 \cos (4 (a+b x))+4 \cos (6 (a+b x))-5) \csc ^3(a+b x) \sec ^4(a+b x)}{560 b}","\frac{8 \sin (a+b x)}{35 b \sin ^{\frac{3}{2}}(2 a+2 b x)}+\frac{\sin (a+b x)}{7 b \sin ^{\frac{7}{2}}(2 a+2 b x)}-\frac{6 \cos (a+b x)}{35 b \sin ^{\frac{5}{2}}(2 a+2 b x)}-\frac{16 \cos (a+b x)}{35 b \sqrt{\sin (2 a+2 b x)}}",1,"((-5 - 10*Cos[2*(a + b*x)] + 4*Cos[4*(a + b*x)] + 4*Cos[6*(a + b*x)])*Csc[a + b*x]^3*Sec[a + b*x]^4*Sqrt[Sin[2*(a + b*x)]])/(560*b)","A",1
81,1,96,98,0.4073431,"\int \sin ^2(a+b x) \sin ^{\frac{7}{2}}(2 a+2 b x) \, dx","Integrate[Sin[a + b*x]^2*Sin[2*a + 2*b*x]^(7/2),x]","\frac{-70 \sin (2 (a+b x))-156 \sin (4 (a+b x))+35 \sin (6 (a+b x))+18 \sin (8 (a+b x))-7 \sin (10 (a+b x))+240 \sqrt{\sin (2 (a+b x))} F\left(\left.a+b x-\frac{\pi }{4}\right|2\right)}{2016 b \sqrt{\sin (2 (a+b x))}}","-\frac{\sin ^{\frac{9}{2}}(2 a+2 b x)}{18 b}+\frac{5 F\left(\left.a+b x-\frac{\pi }{4}\right|2\right)}{42 b}-\frac{\sin ^{\frac{5}{2}}(2 a+2 b x) \cos (2 a+2 b x)}{14 b}-\frac{5 \sqrt{\sin (2 a+2 b x)} \cos (2 a+2 b x)}{42 b}",1,"(240*EllipticF[a - Pi/4 + b*x, 2]*Sqrt[Sin[2*(a + b*x)]] - 70*Sin[2*(a + b*x)] - 156*Sin[4*(a + b*x)] + 35*Sin[6*(a + b*x)] + 18*Sin[8*(a + b*x)] - 7*Sin[10*(a + b*x)])/(2016*b*Sqrt[Sin[2*(a + b*x)]])","A",1
82,1,66,69,0.2203426,"\int \sin ^2(a+b x) \sin ^{\frac{5}{2}}(2 a+2 b x) \, dx","Integrate[Sin[a + b*x]^2*Sin[2*a + 2*b*x]^(5/2),x]","\frac{\sqrt{\sin (2 (a+b x))} (-15 \sin (2 (a+b x))-14 \sin (4 (a+b x))+5 \sin (6 (a+b x)))+84 E\left(\left.a+b x-\frac{\pi }{4}\right|2\right)}{280 b}","-\frac{\sin ^{\frac{7}{2}}(2 a+2 b x)}{14 b}+\frac{3 E\left(\left.a+b x-\frac{\pi }{4}\right|2\right)}{10 b}-\frac{\sin ^{\frac{3}{2}}(2 a+2 b x) \cos (2 a+2 b x)}{10 b}",1,"(84*EllipticE[a - Pi/4 + b*x, 2] + Sqrt[Sin[2*(a + b*x)]]*(-15*Sin[2*(a + b*x)] - 14*Sin[4*(a + b*x)] + 5*Sin[6*(a + b*x)]))/(280*b)","A",1
83,1,76,69,0.3586821,"\int \sin ^2(a+b x) \sin ^{\frac{3}{2}}(2 a+2 b x) \, dx","Integrate[Sin[a + b*x]^2*Sin[2*a + 2*b*x]^(3/2),x]","\frac{-9 \sin (2 (a+b x))-10 \sin (4 (a+b x))+3 \sin (6 (a+b x))+20 \sqrt{\sin (2 (a+b x))} F\left(\left.a+b x-\frac{\pi }{4}\right|2\right)}{120 b \sqrt{\sin (2 (a+b x))}}","-\frac{\sin ^{\frac{5}{2}}(2 a+2 b x)}{10 b}+\frac{F\left(\left.a+b x-\frac{\pi }{4}\right|2\right)}{6 b}-\frac{\sqrt{\sin (2 a+2 b x)} \cos (2 a+2 b x)}{6 b}",1,"(20*EllipticF[a - Pi/4 + b*x, 2]*Sqrt[Sin[2*(a + b*x)]] - 9*Sin[2*(a + b*x)] - 10*Sin[4*(a + b*x)] + 3*Sin[6*(a + b*x)])/(120*b*Sqrt[Sin[2*(a + b*x)]])","A",1
84,1,34,40,0.0867137,"\int \sin ^2(a+b x) \sqrt{\sin (2 a+2 b x)} \, dx","Integrate[Sin[a + b*x]^2*Sqrt[Sin[2*a + 2*b*x]],x]","-\frac{\sin ^{\frac{3}{2}}(2 (a+b x))-3 E\left(\left.a+b x-\frac{\pi }{4}\right|2\right)}{6 b}","\frac{E\left(\left.a+b x-\frac{\pi }{4}\right|2\right)}{2 b}-\frac{\sin ^{\frac{3}{2}}(2 a+2 b x)}{6 b}",1,"-1/6*(-3*EllipticE[a - Pi/4 + b*x, 2] + Sin[2*(a + b*x)]^(3/2))/b","A",1
85,1,75,40,0.2451887,"\int \frac{\sin ^2(a+b x)}{\sqrt{\sin (2 a+2 b x)}} \, dx","Integrate[Sin[a + b*x]^2/Sqrt[Sin[2*a + 2*b*x]],x]","-\frac{2 \sqrt{\sin (2 (a+b x))}+\frac{\sqrt{2} (\sin (a+b x)+\cos (a+b x)) F\left(\sin ^{-1}(\cos (a+b x)-\sin (a+b x))|\frac{1}{2}\right)}{\sqrt{\sin (2 (a+b x))+1}}}{4 b}","\frac{F\left(\left.a+b x-\frac{\pi }{4}\right|2\right)}{2 b}-\frac{\sqrt{\sin (2 a+2 b x)}}{2 b}",1,"-1/4*(2*Sqrt[Sin[2*(a + b*x)]] + (Sqrt[2]*EllipticF[ArcSin[Cos[a + b*x] - Sin[a + b*x]], 1/2]*(Cos[a + b*x] + Sin[a + b*x]))/Sqrt[1 + Sin[2*(a + b*x)]])/b","A",1
86,1,41,45,0.0981531,"\int \frac{\sin ^2(a+b x)}{\sin ^{\frac{3}{2}}(2 a+2 b x)} \, dx","Integrate[Sin[a + b*x]^2/Sin[2*a + 2*b*x]^(3/2),x]","\frac{\sqrt{\sin (2 (a+b x))} \tan (a+b x)-E\left(\left.a+b x-\frac{\pi }{4}\right|2\right)}{2 b}","\frac{\sin ^2(a+b x)}{b \sqrt{\sin (2 a+2 b x)}}-\frac{E\left(\left.a+b x-\frac{\pi }{4}\right|2\right)}{2 b}",1,"(-EllipticE[a - Pi/4 + b*x, 2] + Sqrt[Sin[2*(a + b*x)]]*Tan[a + b*x])/(2*b)","A",1
87,1,83,48,0.1943707,"\int \frac{\sin ^2(a+b x)}{\sin ^{\frac{5}{2}}(2 a+2 b x)} \, dx","Integrate[Sin[a + b*x]^2/Sin[2*a + 2*b*x]^(5/2),x]","\frac{\sqrt{\sin (2 (a+b x))} \sec ^2(a+b x)-\frac{\sqrt{2} (\sin (a+b x)+\cos (a+b x)) F\left(\sin ^{-1}(\cos (a+b x)-\sin (a+b x))|\frac{1}{2}\right)}{\sqrt{\sin (2 (a+b x))+1}}}{12 b}","\frac{\sin ^2(a+b x)}{3 b \sin ^{\frac{3}{2}}(2 a+2 b x)}+\frac{F\left(\left.a+b x-\frac{\pi }{4}\right|2\right)}{6 b}",1,"(Sec[a + b*x]^2*Sqrt[Sin[2*(a + b*x)]] - (Sqrt[2]*EllipticF[ArcSin[Cos[a + b*x] - Sin[a + b*x]], 1/2]*(Cos[a + b*x] + Sin[a + b*x]))/Sqrt[1 + Sin[2*(a + b*x)]])/(12*b)","A",1
88,1,66,77,0.8099573,"\int \frac{\sin ^2(a+b x)}{\sin ^{\frac{7}{2}}(2 a+2 b x)} \, dx","Integrate[Sin[a + b*x]^2/Sin[2*a + 2*b*x]^(7/2),x]","-\frac{12 E\left(\left.a+b x-\frac{\pi }{4}\right|2\right)+\frac{4 \sin ^2(a+b x) (6 \cos (2 (a+b x))+3 \cos (4 (a+b x))+1)}{\sin ^{\frac{5}{2}}(2 (a+b x))}}{40 b}","\frac{\sin ^2(a+b x)}{5 b \sin ^{\frac{5}{2}}(2 a+2 b x)}-\frac{3 E\left(\left.a+b x-\frac{\pi }{4}\right|2\right)}{10 b}-\frac{3 \cos (2 a+2 b x)}{10 b \sqrt{\sin (2 a+2 b x)}}",1,"-1/40*(12*EllipticE[a - Pi/4 + b*x, 2] + (4*(1 + 6*Cos[2*(a + b*x)] + 3*Cos[4*(a + b*x)])*Sin[a + b*x]^2)/Sin[2*(a + b*x)]^(5/2))/b","A",1
89,1,98,136,0.3469303,"\int \sin ^3(a+b x) \sin ^{\frac{3}{2}}(2 a+2 b x) \, dx","Integrate[Sin[a + b*x]^3*Sin[2*a + 2*b*x]^(3/2),x]","\frac{\frac{2}{3} \sqrt{\sin (2 (a+b x))} (10 \sin (a+b x)-9 \sin (3 (a+b x))+2 \sin (5 (a+b x)))-7 \left(\sin ^{-1}(\cos (a+b x)-\sin (a+b x))+\log \left(\sin (a+b x)+\sqrt{\sin (2 (a+b x))}+\cos (a+b x)\right)\right)}{64 b}","-\frac{\sin (a+b x) \sin ^{\frac{5}{2}}(2 a+2 b x)}{12 b}+\frac{7 \sin (a+b x) \sqrt{\sin (2 a+2 b x)}}{32 b}-\frac{7 \sin ^{-1}(\cos (a+b x)-\sin (a+b x))}{64 b}-\frac{7 \sin ^{\frac{3}{2}}(2 a+2 b x) \cos (a+b x)}{48 b}-\frac{7 \log \left(\sin (a+b x)+\sqrt{\sin (2 a+2 b x)}+\cos (a+b x)\right)}{64 b}",1,"(-7*(ArcSin[Cos[a + b*x] - Sin[a + b*x]] + Log[Cos[a + b*x] + Sin[a + b*x] + Sqrt[Sin[2*(a + b*x)]]]) + (2*Sqrt[Sin[2*(a + b*x)]]*(10*Sin[a + b*x] - 9*Sin[3*(a + b*x)] + 2*Sin[5*(a + b*x)]))/3)/(64*b)","A",1
90,1,86,110,0.2200962,"\int \sin ^3(a+b x) \sqrt{\sin (2 a+2 b x)} \, dx","Integrate[Sin[a + b*x]^3*Sqrt[Sin[2*a + 2*b*x]],x]","\frac{2 \sqrt{\sin (2 (a+b x))} (\cos (3 (a+b x))-6 \cos (a+b x))+5 \left(\log \left(\sin (a+b x)+\sqrt{\sin (2 (a+b x))}+\cos (a+b x)\right)-\sin ^{-1}(\cos (a+b x)-\sin (a+b x))\right)}{32 b}","-\frac{\sin (a+b x) \sin ^{\frac{3}{2}}(2 a+2 b x)}{8 b}-\frac{5 \sin ^{-1}(\cos (a+b x)-\sin (a+b x))}{32 b}-\frac{5 \sqrt{\sin (2 a+2 b x)} \cos (a+b x)}{16 b}+\frac{5 \log \left(\sin (a+b x)+\sqrt{\sin (2 a+2 b x)}+\cos (a+b x)\right)}{32 b}",1,"(5*(-ArcSin[Cos[a + b*x] - Sin[a + b*x]] + Log[Cos[a + b*x] + Sin[a + b*x] + Sqrt[Sin[2*(a + b*x)]]]) + 2*(-6*Cos[a + b*x] + Cos[3*(a + b*x)])*Sqrt[Sin[2*(a + b*x)]])/(32*b)","A",1
91,1,74,84,0.1443167,"\int \frac{\sin ^3(a+b x)}{\sqrt{\sin (2 a+2 b x)}} \, dx","Integrate[Sin[a + b*x]^3/Sqrt[Sin[2*a + 2*b*x]],x]","-\frac{2 \sin (a+b x) \sqrt{\sin (2 (a+b x))}+3 \sin ^{-1}(\cos (a+b x)-\sin (a+b x))+3 \log \left(\sin (a+b x)+\sqrt{\sin (2 (a+b x))}+\cos (a+b x)\right)}{8 b}","-\frac{\sin (a+b x) \sqrt{\sin (2 a+2 b x)}}{4 b}-\frac{3 \sin ^{-1}(\cos (a+b x)-\sin (a+b x))}{8 b}-\frac{3 \log \left(\sin (a+b x)+\sqrt{\sin (2 a+2 b x)}+\cos (a+b x)\right)}{8 b}",1,"-1/8*(3*ArcSin[Cos[a + b*x] - Sin[a + b*x]] + 3*Log[Cos[a + b*x] + Sin[a + b*x] + Sqrt[Sin[2*(a + b*x)]]] + 2*Sin[a + b*x]*Sqrt[Sin[2*(a + b*x)]])/b","A",1
92,1,72,81,0.095057,"\int \frac{\sin ^3(a+b x)}{\sin ^{\frac{3}{2}}(2 a+2 b x)} \, dx","Integrate[Sin[a + b*x]^3/Sin[2*a + 2*b*x]^(3/2),x]","\frac{\sin ^{-1}(\cos (a+b x)-\sin (a+b x))+2 \sqrt{\sin (2 (a+b x))} \sec (a+b x)-\log \left(\sin (a+b x)+\sqrt{\sin (2 (a+b x))}+\cos (a+b x)\right)}{4 b}","\frac{\sin (a+b x)}{b \sqrt{\sin (2 a+2 b x)}}+\frac{\sin ^{-1}(\cos (a+b x)-\sin (a+b x))}{4 b}-\frac{\log \left(\sin (a+b x)+\sqrt{\sin (2 a+2 b x)}+\cos (a+b x)\right)}{4 b}",1,"(ArcSin[Cos[a + b*x] - Sin[a + b*x]] - Log[Cos[a + b*x] + Sin[a + b*x] + Sqrt[Sin[2*(a + b*x)]]] + 2*Sec[a + b*x]*Sqrt[Sin[2*(a + b*x)]])/(4*b)","A",1
93,1,27,28,0.0552345,"\int \frac{\sin ^3(a+b x)}{\sin ^{\frac{5}{2}}(2 a+2 b x)} \, dx","Integrate[Sin[a + b*x]^3/Sin[2*a + 2*b*x]^(5/2),x]","\frac{\sin ^3(a+b x)}{3 b \sin ^{\frac{3}{2}}(2 (a+b x))}","\frac{\sin ^3(a+b x)}{3 b \sin ^{\frac{3}{2}}(2 a+2 b x)}",1,"Sin[a + b*x]^3/(3*b*Sin[2*(a + b*x)]^(3/2))","A",1
94,1,35,55,0.0950005,"\int \frac{\sin ^3(a+b x)}{\sin ^{\frac{7}{2}}(2 a+2 b x)} \, dx","Integrate[Sin[a + b*x]^3/Sin[2*a + 2*b*x]^(7/2),x]","\frac{\sqrt{\sin (2 (a+b x))} \sec (a+b x) \left(\sec ^2(a+b x)+4\right)}{40 b}","\frac{\sin ^3(a+b x)}{5 b \sin ^{\frac{5}{2}}(2 a+2 b x)}+\frac{\sin (a+b x)}{5 b \sqrt{\sin (2 a+2 b x)}}",1,"(Sec[a + b*x]*(4 + Sec[a + b*x]^2)*Sqrt[Sin[2*(a + b*x)]])/(40*b)","A",1
95,1,55,81,0.1090172,"\int \frac{\sin ^3(a+b x)}{\sin ^{\frac{9}{2}}(2 a+2 b x)} \, dx","Integrate[Sin[a + b*x]^3/Sin[2*a + 2*b*x]^(9/2),x]","-\frac{\sqrt{\sin (2 (a+b x))} (12 \cos (2 (a+b x))+4 \cos (4 (a+b x))+5) \csc (a+b x) \sec ^4(a+b x)}{336 b}","\frac{2 \sin (a+b x)}{21 b \sin ^{\frac{3}{2}}(2 a+2 b x)}+\frac{\sin ^3(a+b x)}{7 b \sin ^{\frac{7}{2}}(2 a+2 b x)}-\frac{4 \cos (a+b x)}{21 b \sqrt{\sin (2 a+2 b x)}}",1,"-1/336*((5 + 12*Cos[2*(a + b*x)] + 4*Cos[4*(a + b*x)])*Csc[a + b*x]*Sec[a + b*x]^4*Sqrt[Sin[2*(a + b*x)]])/b","A",1
96,1,62,107,0.1672533,"\int \frac{\sin ^3(a+b x)}{\sin ^{\frac{11}{2}}(2 a+2 b x)} \, dx","Integrate[Sin[a + b*x]^3/Sin[2*a + 2*b*x]^(11/2),x]","\frac{\sqrt{\sin (2 (a+b x))} \left(5 \sec ^5(a+b x)+17 \sec ^3(a+b x)+113 \sec (a+b x)-15 \cot (a+b x) \csc (a+b x)\right)}{1440 b}","\frac{\sin (a+b x)}{15 b \sin ^{\frac{5}{2}}(2 a+2 b x)}+\frac{\sin ^3(a+b x)}{9 b \sin ^{\frac{9}{2}}(2 a+2 b x)}+\frac{8 \sin (a+b x)}{45 b \sqrt{\sin (2 a+2 b x)}}-\frac{4 \cos (a+b x)}{45 b \sin ^{\frac{3}{2}}(2 a+2 b x)}",1,"((-15*Cot[a + b*x]*Csc[a + b*x] + 113*Sec[a + b*x] + 17*Sec[a + b*x]^3 + 5*Sec[a + b*x]^5)*Sqrt[Sin[2*(a + b*x)]])/(1440*b)","A",1
97,1,98,136,0.3256479,"\int \csc (a+b x) \sin ^{\frac{7}{2}}(2 a+2 b x) \, dx","Integrate[Csc[a + b*x]*Sin[2*a + 2*b*x]^(7/2),x]","\frac{\frac{2}{3} \sqrt{\sin (2 (a+b x))} (14 \sin (a+b x)-3 \sin (3 (a+b x))-2 \sin (5 (a+b x)))-5 \left(\sin ^{-1}(\cos (a+b x)-\sin (a+b x))+\log \left(\sin (a+b x)+\sqrt{\sin (2 (a+b x))}+\cos (a+b x)\right)\right)}{16 b}","\frac{\sin (a+b x) \sin ^{\frac{5}{2}}(2 a+2 b x)}{3 b}+\frac{5 \sin (a+b x) \sqrt{\sin (2 a+2 b x)}}{8 b}-\frac{5 \sin ^{-1}(\cos (a+b x)-\sin (a+b x))}{16 b}-\frac{5 \sin ^{\frac{3}{2}}(2 a+2 b x) \cos (a+b x)}{12 b}-\frac{5 \log \left(\sin (a+b x)+\sqrt{\sin (2 a+2 b x)}+\cos (a+b x)\right)}{16 b}",1,"(-5*(ArcSin[Cos[a + b*x] - Sin[a + b*x]] + Log[Cos[a + b*x] + Sin[a + b*x] + Sqrt[Sin[2*(a + b*x)]]]) + (2*Sqrt[Sin[2*(a + b*x)]]*(14*Sin[a + b*x] - 3*Sin[3*(a + b*x)] - 2*Sin[5*(a + b*x)]))/3)/(16*b)","A",1
98,1,86,110,0.202335,"\int \csc (a+b x) \sin ^{\frac{5}{2}}(2 a+2 b x) \, dx","Integrate[Csc[a + b*x]*Sin[2*a + 2*b*x]^(5/2),x]","\frac{3 \left(\log \left(\sin (a+b x)+\sqrt{\sin (2 (a+b x))}+\cos (a+b x)\right)-\sin ^{-1}(\cos (a+b x)-\sin (a+b x))\right)-2 \sqrt{\sin (2 (a+b x))} (2 \cos (a+b x)+\cos (3 (a+b x)))}{8 b}","\frac{\sin (a+b x) \sin ^{\frac{3}{2}}(2 a+2 b x)}{2 b}-\frac{3 \sin ^{-1}(\cos (a+b x)-\sin (a+b x))}{8 b}-\frac{3 \sqrt{\sin (2 a+2 b x)} \cos (a+b x)}{4 b}+\frac{3 \log \left(\sin (a+b x)+\sqrt{\sin (2 a+2 b x)}+\cos (a+b x)\right)}{8 b}",1,"(3*(-ArcSin[Cos[a + b*x] - Sin[a + b*x]] + Log[Cos[a + b*x] + Sin[a + b*x] + Sqrt[Sin[2*(a + b*x)]]]) - 2*(2*Cos[a + b*x] + Cos[3*(a + b*x)])*Sqrt[Sin[2*(a + b*x)]])/(8*b)","A",1
99,1,70,81,0.0863389,"\int \csc (a+b x) \sin ^{\frac{3}{2}}(2 a+2 b x) \, dx","Integrate[Csc[a + b*x]*Sin[2*a + 2*b*x]^(3/2),x]","-\frac{-2 \sin (a+b x) \sqrt{\sin (2 (a+b x))}+\sin ^{-1}(\cos (a+b x)-\sin (a+b x))+\log \left(\sin (a+b x)+\sqrt{\sin (2 (a+b x))}+\cos (a+b x)\right)}{2 b}","\frac{\sin (a+b x) \sqrt{\sin (2 a+2 b x)}}{b}-\frac{\sin ^{-1}(\cos (a+b x)-\sin (a+b x))}{2 b}-\frac{\log \left(\sin (a+b x)+\sqrt{\sin (2 a+2 b x)}+\cos (a+b x)\right)}{2 b}",1,"-1/2*(ArcSin[Cos[a + b*x] - Sin[a + b*x]] + Log[Cos[a + b*x] + Sin[a + b*x] + Sqrt[Sin[2*(a + b*x)]]] - 2*Sin[a + b*x]*Sqrt[Sin[2*(a + b*x)]])/b","A",1
100,1,52,53,0.0439332,"\int \csc (a+b x) \sqrt{\sin (2 a+2 b x)} \, dx","Integrate[Csc[a + b*x]*Sqrt[Sin[2*a + 2*b*x]],x]","\frac{\log \left(\sin (a+b x)+\sqrt{\sin (2 (a+b x))}+\cos (a+b x)\right)}{b}-\frac{\sin ^{-1}(\cos (a+b x)-\sin (a+b x))}{b}","\frac{\log \left(\sin (a+b x)+\sqrt{\sin (2 a+2 b x)}+\cos (a+b x)\right)}{b}-\frac{\sin ^{-1}(\cos (a+b x)-\sin (a+b x))}{b}",1,"-(ArcSin[Cos[a + b*x] - Sin[a + b*x]]/b) + Log[Cos[a + b*x] + Sin[a + b*x] + Sqrt[Sin[2*(a + b*x)]]]/b","A",1
101,1,23,24,0.0457719,"\int \frac{\csc (a+b x)}{\sqrt{\sin (2 a+2 b x)}} \, dx","Integrate[Csc[a + b*x]/Sqrt[Sin[2*a + 2*b*x]],x]","-\frac{\sqrt{\sin (2 (a+b x))} \csc (a+b x)}{b}","-\frac{\sqrt{\sin (2 a+2 b x)} \csc (a+b x)}{b}",1,"-((Csc[a + b*x]*Sqrt[Sin[2*(a + b*x)]])/b)","A",1
102,1,43,53,0.1063758,"\int \frac{\csc (a+b x)}{\sin ^{\frac{3}{2}}(2 a+2 b x)} \, dx","Integrate[Csc[a + b*x]/Sin[2*a + 2*b*x]^(3/2),x]","\frac{\sqrt{\sin (2 (a+b x))} \left(\frac{1}{2} \sec (a+b x)-\frac{1}{6} \cot (a+b x) \csc (a+b x)\right)}{b}","\frac{4 \sin (a+b x)}{3 b \sqrt{\sin (2 a+2 b x)}}-\frac{2 \cos (a+b x)}{3 b \sin ^{\frac{3}{2}}(2 a+2 b x)}",1,"((-1/6*(Cot[a + b*x]*Csc[a + b*x]) + Sec[a + b*x]/2)*Sqrt[Sin[2*(a + b*x)]])/b","A",1
103,1,52,79,0.1247234,"\int \frac{\csc (a+b x)}{\sin ^{\frac{5}{2}}(2 a+2 b x)} \, dx","Integrate[Csc[a + b*x]/Sin[2*a + 2*b*x]^(5/2),x]","-\frac{\sqrt{\sin (2 (a+b x))} \left(3 \csc ^3(a+b x)+27 \csc (a+b x)-5 \tan (a+b x) \sec (a+b x)\right)}{60 b}","\frac{8 \sin (a+b x)}{15 b \sin ^{\frac{3}{2}}(2 a+2 b x)}-\frac{2 \cos (a+b x)}{5 b \sin ^{\frac{5}{2}}(2 a+2 b x)}-\frac{16 \cos (a+b x)}{15 b \sqrt{\sin (2 a+2 b x)}}",1,"-1/60*(Sqrt[Sin[2*(a + b*x)]]*(27*Csc[a + b*x] + 3*Csc[a + b*x]^3 - 5*Sec[a + b*x]*Tan[a + b*x]))/b","A",1
104,1,67,105,0.1420746,"\int \frac{\csc (a+b x)}{\sin ^{\frac{7}{2}}(2 a+2 b x)} \, dx","Integrate[Csc[a + b*x]/Sin[2*a + 2*b*x]^(7/2),x]","\frac{\sqrt{\sin (2 (a+b x))} (-10 \cos (2 (a+b x))-4 \cos (4 (a+b x))+4 \cos (6 (a+b x))+5) \csc ^4(a+b x) \sec ^3(a+b x)}{280 b}","\frac{12 \sin (a+b x)}{35 b \sin ^{\frac{5}{2}}(2 a+2 b x)}+\frac{32 \sin (a+b x)}{35 b \sqrt{\sin (2 a+2 b x)}}-\frac{16 \cos (a+b x)}{35 b \sin ^{\frac{3}{2}}(2 a+2 b x)}-\frac{2 \cos (a+b x)}{7 b \sin ^{\frac{7}{2}}(2 a+2 b x)}",1,"((5 - 10*Cos[2*(a + b*x)] - 4*Cos[4*(a + b*x)] + 4*Cos[6*(a + b*x)])*Csc[a + b*x]^4*Sec[a + b*x]^3*Sqrt[Sin[2*(a + b*x)]])/(280*b)","A",1
105,1,66,106,0.2453116,"\int \csc ^2(a+b x) \sin ^{\frac{9}{2}}(2 a+2 b x) \, dx","Integrate[Csc[a + b*x]^2*Sin[2*a + 2*b*x]^(9/2),x]","\frac{\sqrt{\sin (2 (a+b x))} (15 \sin (2 (a+b x))-14 \sin (4 (a+b x))-5 \sin (6 (a+b x)))+84 E\left(\left.a+b x-\frac{\pi }{4}\right|2\right)}{70 b}","\frac{6 E\left(\left.a+b x-\frac{\pi }{4}\right|2\right)}{5 b}-\frac{2 \sin ^{\frac{7}{2}}(2 a+2 b x) \cos (2 a+2 b x)}{7 b}-\frac{2 \sin ^{\frac{3}{2}}(2 a+2 b x) \cos (2 a+2 b x)}{5 b}+\frac{\sin ^{\frac{11}{2}}(2 a+2 b x) \csc ^2(a+b x)}{7 b}",1,"(84*EllipticE[a - Pi/4 + b*x, 2] + Sqrt[Sin[2*(a + b*x)]]*(15*Sin[2*(a + b*x)] - 14*Sin[4*(a + b*x)] - 5*Sin[6*(a + b*x)]))/(70*b)","A",1
106,1,76,106,0.2681249,"\int \csc ^2(a+b x) \sin ^{\frac{7}{2}}(2 a+2 b x) \, dx","Integrate[Csc[a + b*x]^2*Sin[2*a + 2*b*x]^(7/2),x]","\frac{9 \sin (2 (a+b x))-10 \sin (4 (a+b x))-3 \sin (6 (a+b x))+20 \sqrt{\sin (2 (a+b x))} F\left(\left.a+b x-\frac{\pi }{4}\right|2\right)}{30 b \sqrt{\sin (2 (a+b x))}}","\frac{2 F\left(\left.a+b x-\frac{\pi }{4}\right|2\right)}{3 b}-\frac{2 \sin ^{\frac{5}{2}}(2 a+2 b x) \cos (2 a+2 b x)}{5 b}-\frac{2 \sqrt{\sin (2 a+2 b x)} \cos (2 a+2 b x)}{3 b}+\frac{\sin ^{\frac{9}{2}}(2 a+2 b x) \csc ^2(a+b x)}{5 b}",1,"(20*EllipticF[a - Pi/4 + b*x, 2]*Sqrt[Sin[2*(a + b*x)]] + 9*Sin[2*(a + b*x)] - 10*Sin[4*(a + b*x)] - 3*Sin[6*(a + b*x)])/(30*b*Sqrt[Sin[2*(a + b*x)]])","A",1
107,1,34,75,0.0827126,"\int \csc ^2(a+b x) \sin ^{\frac{5}{2}}(2 a+2 b x) \, dx","Integrate[Csc[a + b*x]^2*Sin[2*a + 2*b*x]^(5/2),x]","\frac{2 \left(\sin ^{\frac{3}{2}}(2 (a+b x))+3 E\left(\left.a+b x-\frac{\pi }{4}\right|2\right)\right)}{3 b}","\frac{2 E\left(\left.a+b x-\frac{\pi }{4}\right|2\right)}{b}-\frac{2 \sin ^{\frac{3}{2}}(2 a+2 b x) \cos (2 a+2 b x)}{3 b}+\frac{\sin ^{\frac{7}{2}}(2 a+2 b x) \csc ^2(a+b x)}{3 b}",1,"(2*(3*EllipticE[a - Pi/4 + b*x, 2] + Sin[2*(a + b*x)]^(3/2)))/(3*b)","A",1
108,1,73,70,0.9231744,"\int \csc ^2(a+b x) \sin ^{\frac{3}{2}}(2 a+2 b x) \, dx","Integrate[Csc[a + b*x]^2*Sin[2*a + 2*b*x]^(3/2),x]","\frac{2 \sqrt{\sin (2 (a+b x))}-\frac{\sqrt{2} (\sin (a+b x)+\cos (a+b x)) F\left(\sin ^{-1}(\cos (a+b x)-\sin (a+b x))|\frac{1}{2}\right)}{\sqrt{\sin (2 (a+b x))+1}}}{b}","\frac{2 F\left(\left.a+b x-\frac{\pi }{4}\right|2\right)}{b}-\frac{2 \sqrt{\sin (2 a+2 b x)} \cos (2 a+2 b x)}{b}+\frac{\sin ^{\frac{5}{2}}(2 a+2 b x) \csc ^2(a+b x)}{b}",1,"(2*Sqrt[Sin[2*(a + b*x)]] - (Sqrt[2]*EllipticF[ArcSin[Cos[a + b*x] - Sin[a + b*x]], 1/2]*(Cos[a + b*x] + Sin[a + b*x]))/Sqrt[1 + Sin[2*(a + b*x)]])/b","A",1
109,1,37,44,0.1312753,"\int \csc ^2(a+b x) \sqrt{\sin (2 a+2 b x)} \, dx","Integrate[Csc[a + b*x]^2*Sqrt[Sin[2*a + 2*b*x]],x]","-\frac{2 \left(E\left(\left.a+b x-\frac{\pi }{4}\right|2\right)+\sqrt{\sin (2 (a+b x))} \cot (a+b x)\right)}{b}","-\frac{2 E\left(\left.a+b x-\frac{\pi }{4}\right|2\right)}{b}-\frac{\sin ^{\frac{3}{2}}(2 a+2 b x) \csc ^2(a+b x)}{b}",1,"(-2*(EllipticE[a - Pi/4 + b*x, 2] + Cot[a + b*x]*Sqrt[Sin[2*(a + b*x)]]))/b","A",1
110,1,82,48,0.9925344,"\int \frac{\csc ^2(a+b x)}{\sqrt{\sin (2 a+2 b x)}} \, dx","Integrate[Csc[a + b*x]^2/Sqrt[Sin[2*a + 2*b*x]],x]","-\frac{\sqrt{\sin (2 (a+b x))} \csc ^2(a+b x)+\frac{\sqrt{2} (\sin (a+b x)+\cos (a+b x)) F\left(\sin ^{-1}(\cos (a+b x)-\sin (a+b x))|\frac{1}{2}\right)}{\sqrt{\sin (2 (a+b x))+1}}}{3 b}","\frac{2 F\left(\left.a+b x-\frac{\pi }{4}\right|2\right)}{3 b}-\frac{\sqrt{\sin (2 a+2 b x)} \csc ^2(a+b x)}{3 b}",1,"-1/3*(Csc[a + b*x]^2*Sqrt[Sin[2*(a + b*x)]] + (Sqrt[2]*EllipticF[ArcSin[Cos[a + b*x] - Sin[a + b*x]], 1/2]*(Cos[a + b*x] + Sin[a + b*x]))/Sqrt[1 + Sin[2*(a + b*x)]])/b","A",1
111,1,64,77,0.5912174,"\int \frac{\csc ^2(a+b x)}{\sin ^{\frac{3}{2}}(2 a+2 b x)} \, dx","Integrate[Csc[a + b*x]^2/Sin[2*a + 2*b*x]^(3/2),x]","\frac{\frac{2 (-6 \cos (2 (a+b x))+3 \cos (4 (a+b x))+1) \cot (a+b x)}{\sin ^{\frac{3}{2}}(2 (a+b x))}-12 E\left(\left.a+b x-\frac{\pi }{4}\right|2\right)}{10 b}","-\frac{6 E\left(\left.a+b x-\frac{\pi }{4}\right|2\right)}{5 b}-\frac{6 \cos (2 a+2 b x)}{5 b \sqrt{\sin (2 a+2 b x)}}-\frac{\csc ^2(a+b x)}{5 b \sqrt{\sin (2 a+2 b x)}}",1,"(-12*EllipticE[a - Pi/4 + b*x, 2] + (2*(1 - 6*Cos[2*(a + b*x)] + 3*Cos[4*(a + b*x)])*Cot[a + b*x])/Sin[2*(a + b*x)]^(3/2))/(10*b)","A",1
112,1,66,77,0.4688624,"\int \frac{\csc ^2(a+b x)}{\sin ^{\frac{5}{2}}(2 a+2 b x)} \, dx","Integrate[Csc[a + b*x]^2/Sin[2*a + 2*b*x]^(5/2),x]","\frac{40 F\left(\left.a+b x-\frac{\pi }{4}\right|2\right)+\sqrt{\sin (2 (a+b x))} \left(-3 \csc ^4(a+b x)-13 \csc ^2(a+b x)+7 \sec ^2(a+b x)\right)}{84 b}","\frac{10 F\left(\left.a+b x-\frac{\pi }{4}\right|2\right)}{21 b}-\frac{10 \cos (2 a+2 b x)}{21 b \sin ^{\frac{3}{2}}(2 a+2 b x)}-\frac{\csc ^2(a+b x)}{7 b \sin ^{\frac{3}{2}}(2 a+2 b x)}",1,"(40*EllipticF[a - Pi/4 + b*x, 2] + (-13*Csc[a + b*x]^2 - 3*Csc[a + b*x]^4 + 7*Sec[a + b*x]^2)*Sqrt[Sin[2*(a + b*x)]])/(84*b)","A",1
113,1,85,106,0.8271818,"\int \frac{\csc ^2(a+b x)}{\sin ^{\frac{7}{2}}(2 a+2 b x)} \, dx","Integrate[Csc[a + b*x]^2/Sin[2*a + 2*b*x]^(7/2),x]","-\frac{336 E\left(\left.a+b x-\frac{\pi }{4}\right|2\right)+\frac{(98 \cos (2 (a+b x))-28 \cos (4 (a+b x))-42 \cos (6 (a+b x))+21 \cos (8 (a+b x))-9) \csc ^2(a+b x)}{\sin ^{\frac{5}{2}}(2 (a+b x))}}{360 b}","-\frac{14 E\left(\left.a+b x-\frac{\pi }{4}\right|2\right)}{15 b}-\frac{14 \cos (2 a+2 b x)}{45 b \sin ^{\frac{5}{2}}(2 a+2 b x)}-\frac{14 \cos (2 a+2 b x)}{15 b \sqrt{\sin (2 a+2 b x)}}-\frac{\csc ^2(a+b x)}{9 b \sin ^{\frac{5}{2}}(2 a+2 b x)}",1,"-1/360*(336*EllipticE[a - Pi/4 + b*x, 2] + ((-9 + 98*Cos[2*(a + b*x)] - 28*Cos[4*(a + b*x)] - 42*Cos[6*(a + b*x)] + 21*Cos[8*(a + b*x)])*Csc[a + b*x]^2)/Sin[2*(a + b*x)]^(5/2))/b","A",1
114,1,86,106,0.3638509,"\int \frac{\csc ^2(a+b x)}{\sin ^{\frac{9}{2}}(2 a+2 b x)} \, dx","Integrate[Csc[a + b*x]^2/Sin[2*a + 2*b*x]^(9/2),x]","\frac{480 F\left(\left.a+b x-\frac{\pi }{4}\right|2\right)+\sqrt{\sin (2 (a+b x))} \left(-7 \csc ^6(a+b x)-32 \csc ^4(a+b x)-141 \csc ^2(a+b x)+11 \sec ^2(a+b x) \left(\sec ^2(a+b x)+9\right)\right)}{1232 b}","\frac{30 F\left(\left.a+b x-\frac{\pi }{4}\right|2\right)}{77 b}-\frac{30 \cos (2 a+2 b x)}{77 b \sin ^{\frac{3}{2}}(2 a+2 b x)}-\frac{18 \cos (2 a+2 b x)}{77 b \sin ^{\frac{7}{2}}(2 a+2 b x)}-\frac{\csc ^2(a+b x)}{11 b \sin ^{\frac{7}{2}}(2 a+2 b x)}",1,"(480*EllipticF[a - Pi/4 + b*x, 2] + (-141*Csc[a + b*x]^2 - 32*Csc[a + b*x]^4 - 7*Csc[a + b*x]^6 + 11*Sec[a + b*x]^2*(9 + Sec[a + b*x]^2))*Sqrt[Sin[2*(a + b*x)]])/(1232*b)","A",1
115,1,100,190,0.4380405,"\int \csc ^3(a+b x) \sin ^{\frac{9}{2}}(2 a+2 b x) \, dx","Integrate[Csc[a + b*x]^3*Sin[2*a + 2*b*x]^(9/2),x]","\frac{7 \left(\log \left(\sin (a+b x)+\sqrt{\sin (2 (a+b x))}+\cos (a+b x)\right)-\sin ^{-1}(\cos (a+b x)-\sin (a+b x))\right)-\frac{2}{3} \sqrt{\sin (2 (a+b x))} (10 \cos (a+b x)+9 \cos (3 (a+b x))+2 \cos (5 (a+b x)))}{8 b}","\frac{4 \sin (a+b x) \sin ^{\frac{7}{2}}(2 a+2 b x)}{5 b}+\frac{7 \sin (a+b x) \sin ^{\frac{3}{2}}(2 a+2 b x)}{6 b}-\frac{7 \sin ^{-1}(\cos (a+b x)-\sin (a+b x))}{8 b}-\frac{14 \sin ^{\frac{5}{2}}(2 a+2 b x) \cos (a+b x)}{15 b}-\frac{7 \sqrt{\sin (2 a+2 b x)} \cos (a+b x)}{4 b}+\frac{\sin ^{\frac{11}{2}}(2 a+2 b x) \csc ^3(a+b x)}{5 b}+\frac{7 \log \left(\sin (a+b x)+\sqrt{\sin (2 a+2 b x)}+\cos (a+b x)\right)}{8 b}",1,"(7*(-ArcSin[Cos[a + b*x] - Sin[a + b*x]] + Log[Cos[a + b*x] + Sin[a + b*x] + Sqrt[Sin[2*(a + b*x)]]]) - (2*(10*Cos[a + b*x] + 9*Cos[3*(a + b*x)] + 2*Cos[5*(a + b*x)])*Sqrt[Sin[2*(a + b*x)]])/3)/(8*b)","A",1
116,1,84,164,0.2267096,"\int \csc ^3(a+b x) \sin ^{\frac{7}{2}}(2 a+2 b x) \, dx","Integrate[Csc[a + b*x]^3*Sin[2*a + 2*b*x]^(7/2),x]","\frac{2 \sqrt{\sin (2 (a+b x))} (6 \sin (a+b x)+\sin (3 (a+b x)))-5 \left(\sin ^{-1}(\cos (a+b x)-\sin (a+b x))+\log \left(\sin (a+b x)+\sqrt{\sin (2 (a+b x))}+\cos (a+b x)\right)\right)}{4 b}","\frac{4 \sin (a+b x) \sin ^{\frac{5}{2}}(2 a+2 b x)}{3 b}+\frac{5 \sin (a+b x) \sqrt{\sin (2 a+2 b x)}}{2 b}-\frac{5 \sin ^{-1}(\cos (a+b x)-\sin (a+b x))}{4 b}-\frac{5 \sin ^{\frac{3}{2}}(2 a+2 b x) \cos (a+b x)}{3 b}+\frac{\sin ^{\frac{9}{2}}(2 a+2 b x) \csc ^3(a+b x)}{3 b}-\frac{5 \log \left(\sin (a+b x)+\sqrt{\sin (2 a+2 b x)}+\cos (a+b x)\right)}{4 b}",1,"(-5*(ArcSin[Cos[a + b*x] - Sin[a + b*x]] + Log[Cos[a + b*x] + Sin[a + b*x] + Sqrt[Sin[2*(a + b*x)]]]) + 2*Sqrt[Sin[2*(a + b*x)]]*(6*Sin[a + b*x] + Sin[3*(a + b*x)]))/(4*b)","A",1
117,1,70,127,0.1387194,"\int \csc ^3(a+b x) \sin ^{\frac{5}{2}}(2 a+2 b x) \, dx","Integrate[Csc[a + b*x]^3*Sin[2*a + 2*b*x]^(5/2),x]","\frac{-3 \sin ^{-1}(\cos (a+b x)-\sin (a+b x))+\sin ^{\frac{3}{2}}(2 (a+b x)) \csc (a+b x)+3 \log \left(\sin (a+b x)+\sqrt{\sin (2 (a+b x))}+\cos (a+b x)\right)}{b}","\frac{4 \sin (a+b x) \sin ^{\frac{3}{2}}(2 a+2 b x)}{b}-\frac{3 \sin ^{-1}(\cos (a+b x)-\sin (a+b x))}{b}-\frac{6 \sqrt{\sin (2 a+2 b x)} \cos (a+b x)}{b}+\frac{\sin ^{\frac{7}{2}}(2 a+2 b x) \csc ^3(a+b x)}{b}+\frac{3 \log \left(\sin (a+b x)+\sqrt{\sin (2 a+2 b x)}+\cos (a+b x)\right)}{b}",1,"(-3*ArcSin[Cos[a + b*x] - Sin[a + b*x]] + 3*Log[Cos[a + b*x] + Sin[a + b*x] + Sqrt[Sin[2*(a + b*x)]]] + Csc[a + b*x]*Sin[2*(a + b*x)]^(3/2))/b","A",1
118,1,68,104,0.0989893,"\int \csc ^3(a+b x) \sin ^{\frac{3}{2}}(2 a+2 b x) \, dx","Integrate[Csc[a + b*x]^3*Sin[2*a + 2*b*x]^(3/2),x]","\frac{2 \left(\sin ^{-1}(\cos (a+b x)-\sin (a+b x))-2 \sqrt{\sin (2 (a+b x))} \csc (a+b x)+\log \left(\sin (a+b x)+\sqrt{\sin (2 (a+b x))}+\cos (a+b x)\right)\right)}{b}","-\frac{4 \sin (a+b x) \sqrt{\sin (2 a+2 b x)}}{b}+\frac{2 \sin ^{-1}(\cos (a+b x)-\sin (a+b x))}{b}-\frac{\sin ^{\frac{5}{2}}(2 a+2 b x) \csc ^3(a+b x)}{b}+\frac{2 \log \left(\sin (a+b x)+\sqrt{\sin (2 a+2 b x)}+\cos (a+b x)\right)}{b}",1,"(2*(ArcSin[Cos[a + b*x] - Sin[a + b*x]] + Log[Cos[a + b*x] + Sin[a + b*x] + Sqrt[Sin[2*(a + b*x)]]] - 2*Csc[a + b*x]*Sqrt[Sin[2*(a + b*x)]]))/b","A",1
119,1,27,28,0.0509009,"\int \csc ^3(a+b x) \sqrt{\sin (2 a+2 b x)} \, dx","Integrate[Csc[a + b*x]^3*Sqrt[Sin[2*a + 2*b*x]],x]","-\frac{\sin ^{\frac{3}{2}}(2 (a+b x)) \csc ^3(a+b x)}{3 b}","-\frac{\sin ^{\frac{3}{2}}(2 a+2 b x) \csc ^3(a+b x)}{3 b}",1,"-1/3*(Csc[a + b*x]^3*Sin[2*(a + b*x)]^(3/2))/b","A",1
120,1,35,55,0.0933308,"\int \frac{\csc ^3(a+b x)}{\sqrt{\sin (2 a+2 b x)}} \, dx","Integrate[Csc[a + b*x]^3/Sqrt[Sin[2*a + 2*b*x]],x]","-\frac{\sqrt{\sin (2 (a+b x))} \csc (a+b x) \left(\csc ^2(a+b x)+4\right)}{5 b}","-\frac{\sqrt{\sin (2 a+2 b x)} \csc ^3(a+b x)}{5 b}-\frac{4 \sqrt{\sin (2 a+2 b x)} \csc (a+b x)}{5 b}",1,"-1/5*(Csc[a + b*x]*(4 + Csc[a + b*x]^2)*Sqrt[Sin[2*(a + b*x)]])/b","A",1
121,1,55,81,0.1158571,"\int \frac{\csc ^3(a+b x)}{\sin ^{\frac{3}{2}}(2 a+2 b x)} \, dx","Integrate[Csc[a + b*x]^3/Sin[2*a + 2*b*x]^(3/2),x]","\frac{\sqrt{\sin (2 (a+b x))} (-12 \cos (2 (a+b x))+4 \cos (4 (a+b x))+5) \csc ^4(a+b x) \sec (a+b x)}{42 b}","\frac{32 \sin (a+b x)}{21 b \sqrt{\sin (2 a+2 b x)}}-\frac{16 \cos (a+b x)}{21 b \sin ^{\frac{3}{2}}(2 a+2 b x)}-\frac{\csc ^3(a+b x)}{7 b \sqrt{\sin (2 a+2 b x)}}",1,"((5 - 12*Cos[2*(a + b*x)] + 4*Cos[4*(a + b*x)])*Csc[a + b*x]^4*Sec[a + b*x]*Sqrt[Sin[2*(a + b*x)]])/(42*b)","A",1
122,1,62,107,0.0971032,"\int \frac{\csc ^3(a+b x)}{\sin ^{\frac{5}{2}}(2 a+2 b x)} \, dx","Integrate[Csc[a + b*x]^3/Sin[2*a + 2*b*x]^(5/2),x]","-\frac{\sqrt{\sin (2 (a+b x))} \left(5 \csc ^5(a+b x)+17 \csc ^3(a+b x)+113 \csc (a+b x)-15 \tan (a+b x) \sec (a+b x)\right)}{180 b}","\frac{32 \sin (a+b x)}{45 b \sin ^{\frac{3}{2}}(2 a+2 b x)}-\frac{8 \cos (a+b x)}{15 b \sin ^{\frac{5}{2}}(2 a+2 b x)}-\frac{64 \cos (a+b x)}{45 b \sqrt{\sin (2 a+2 b x)}}-\frac{\csc ^3(a+b x)}{9 b \sin ^{\frac{3}{2}}(2 a+2 b x)}",1,"-1/180*(Sqrt[Sin[2*(a + b*x)]]*(113*Csc[a + b*x] + 17*Csc[a + b*x]^3 + 5*Csc[a + b*x]^5 - 15*Sec[a + b*x]*Tan[a + b*x]))/b","A",1
123,1,602,84,5.7255949,"\int \sin ^3(a+b x) \sin ^m(2 a+2 b x) \, dx","Integrate[Sin[a + b*x]^3*Sin[2*a + 2*b*x]^m,x]","\frac{32 (m+4) \sin ^4\left(\frac{1}{2} (a+b x)\right) \cos ^6\left(\frac{1}{2} (a+b x)\right) \sin ^m(2 (a+b x)) \left(F_1\left(\frac{m}{2}+1;-m,2 m+3;\frac{m}{2}+2;\tan ^2\left(\frac{1}{2} (a+b x)\right),-\tan ^2\left(\frac{1}{2} (a+b x)\right)\right)-F_1\left(\frac{m}{2}+1;-m,2 m+4;\frac{m}{2}+2;\tan ^2\left(\frac{1}{2} (a+b x)\right),-\tan ^2\left(\frac{1}{2} (a+b x)\right)\right)\right)}{b (m+2) \left(-2 (m+4) \cos ^2\left(\frac{1}{2} (a+b x)\right) F_1\left(\frac{m}{2}+1;-m,2 m+4;\frac{m}{2}+2;\tan ^2\left(\frac{1}{2} (a+b x)\right),-\tan ^2\left(\frac{1}{2} (a+b x)\right)\right)+2 (\cos (a+b x)-1) \left(m F_1\left(\frac{m}{2}+2;1-m,2 m+3;\frac{m}{2}+3;\tan ^2\left(\frac{1}{2} (a+b x)\right),-\tan ^2\left(\frac{1}{2} (a+b x)\right)\right)-m F_1\left(\frac{m}{2}+2;1-m,2 m+4;\frac{m}{2}+3;\tan ^2\left(\frac{1}{2} (a+b x)\right),-\tan ^2\left(\frac{1}{2} (a+b x)\right)\right)+2 m F_1\left(\frac{m}{2}+2;-m,2 m+4;\frac{m}{2}+3;\tan ^2\left(\frac{1}{2} (a+b x)\right),-\tan ^2\left(\frac{1}{2} (a+b x)\right)\right)+3 F_1\left(\frac{m}{2}+2;-m,2 m+4;\frac{m}{2}+3;\tan ^2\left(\frac{1}{2} (a+b x)\right),-\tan ^2\left(\frac{1}{2} (a+b x)\right)\right)-2 m F_1\left(\frac{m}{2}+2;-m,2 m+5;\frac{m}{2}+3;\tan ^2\left(\frac{1}{2} (a+b x)\right),-\tan ^2\left(\frac{1}{2} (a+b x)\right)\right)-4 F_1\left(\frac{m}{2}+2;-m,2 m+5;\frac{m}{2}+3;\tan ^2\left(\frac{1}{2} (a+b x)\right),-\tan ^2\left(\frac{1}{2} (a+b x)\right)\right)\right)+(m+4) (\cos (a+b x)+1) F_1\left(\frac{m}{2}+1;-m,2 m+3;\frac{m}{2}+2;\tan ^2\left(\frac{1}{2} (a+b x)\right),-\tan ^2\left(\frac{1}{2} (a+b x)\right)\right)\right)}","\frac{\sin ^3(a+b x) \tan (a+b x) \sin ^m(2 a+2 b x) \cos ^2(a+b x)^{\frac{1-m}{2}} \, _2F_1\left(\frac{1-m}{2},\frac{m+4}{2};\frac{m+6}{2};\sin ^2(a+b x)\right)}{b (m+4)}",1,"(32*(4 + m)*(AppellF1[1 + m/2, -m, 3 + 2*m, 2 + m/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2] - AppellF1[1 + m/2, -m, 4 + 2*m, 2 + m/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2])*Cos[(a + b*x)/2]^6*Sin[(a + b*x)/2]^4*Sin[2*(a + b*x)]^m)/(b*(2 + m)*(-2*(4 + m)*AppellF1[1 + m/2, -m, 4 + 2*m, 2 + m/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2]*Cos[(a + b*x)/2]^2 + 2*(m*AppellF1[2 + m/2, 1 - m, 3 + 2*m, 3 + m/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2] - m*AppellF1[2 + m/2, 1 - m, 4 + 2*m, 3 + m/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2] + 3*AppellF1[2 + m/2, -m, 4 + 2*m, 3 + m/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2] + 2*m*AppellF1[2 + m/2, -m, 4 + 2*m, 3 + m/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2] - 4*AppellF1[2 + m/2, -m, 5 + 2*m, 3 + m/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2] - 2*m*AppellF1[2 + m/2, -m, 5 + 2*m, 3 + m/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2])*(-1 + Cos[a + b*x]) + (4 + m)*AppellF1[1 + m/2, -m, 3 + 2*m, 2 + m/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2]*(1 + Cos[a + b*x])))","C",0
124,1,602,84,3.5087396,"\int \sin ^2(a+b x) \sin ^m(2 a+2 b x) \, dx","Integrate[Sin[a + b*x]^2*Sin[2*a + 2*b*x]^m,x]","\frac{16 (m+3) \sin ^3\left(\frac{1}{2} (a+b x)\right) \cos ^5\left(\frac{1}{2} (a+b x)\right) \sin ^m(2 (a+b x)) \left(F_1\left(\frac{m+1}{2};-m,2 (m+1);\frac{m+3}{2};\tan ^2\left(\frac{1}{2} (a+b x)\right),-\tan ^2\left(\frac{1}{2} (a+b x)\right)\right)-F_1\left(\frac{m+1}{2};-m,2 m+3;\frac{m+3}{2};\tan ^2\left(\frac{1}{2} (a+b x)\right),-\tan ^2\left(\frac{1}{2} (a+b x)\right)\right)\right)}{b (m+1) \left(-2 (m+3) \cos ^2\left(\frac{1}{2} (a+b x)\right) F_1\left(\frac{m+1}{2};-m,2 m+3;\frac{m+3}{2};\tan ^2\left(\frac{1}{2} (a+b x)\right),-\tan ^2\left(\frac{1}{2} (a+b x)\right)\right)+2 (\cos (a+b x)-1) \left(m F_1\left(\frac{m+3}{2};1-m,2 (m+1);\frac{m+5}{2};\tan ^2\left(\frac{1}{2} (a+b x)\right),-\tan ^2\left(\frac{1}{2} (a+b x)\right)\right)-m F_1\left(\frac{m+3}{2};1-m,2 m+3;\frac{m+5}{2};\tan ^2\left(\frac{1}{2} (a+b x)\right),-\tan ^2\left(\frac{1}{2} (a+b x)\right)\right)-2 m F_1\left(\frac{m+3}{2};-m,2 (m+2);\frac{m+5}{2};\tan ^2\left(\frac{1}{2} (a+b x)\right),-\tan ^2\left(\frac{1}{2} (a+b x)\right)\right)-3 F_1\left(\frac{m+3}{2};-m,2 (m+2);\frac{m+5}{2};\tan ^2\left(\frac{1}{2} (a+b x)\right),-\tan ^2\left(\frac{1}{2} (a+b x)\right)\right)+2 m F_1\left(\frac{m+3}{2};-m,2 m+3;\frac{m+5}{2};\tan ^2\left(\frac{1}{2} (a+b x)\right),-\tan ^2\left(\frac{1}{2} (a+b x)\right)\right)+2 F_1\left(\frac{m+3}{2};-m,2 m+3;\frac{m+5}{2};\tan ^2\left(\frac{1}{2} (a+b x)\right),-\tan ^2\left(\frac{1}{2} (a+b x)\right)\right)\right)+(m+3) (\cos (a+b x)+1) F_1\left(\frac{m+1}{2};-m,2 (m+1);\frac{m+3}{2};\tan ^2\left(\frac{1}{2} (a+b x)\right),-\tan ^2\left(\frac{1}{2} (a+b x)\right)\right)\right)}","\frac{\sin ^2(a+b x) \tan (a+b x) \sin ^m(2 a+2 b x) \cos ^2(a+b x)^{\frac{1-m}{2}} \, _2F_1\left(\frac{1-m}{2},\frac{m+3}{2};\frac{m+5}{2};\sin ^2(a+b x)\right)}{b (m+3)}",1,"(16*(3 + m)*(AppellF1[(1 + m)/2, -m, 2*(1 + m), (3 + m)/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2] - AppellF1[(1 + m)/2, -m, 3 + 2*m, (3 + m)/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2])*Cos[(a + b*x)/2]^5*Sin[(a + b*x)/2]^3*Sin[2*(a + b*x)]^m)/(b*(1 + m)*(-2*(3 + m)*AppellF1[(1 + m)/2, -m, 3 + 2*m, (3 + m)/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2]*Cos[(a + b*x)/2]^2 + 2*(m*AppellF1[(3 + m)/2, 1 - m, 2*(1 + m), (5 + m)/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2] - m*AppellF1[(3 + m)/2, 1 - m, 3 + 2*m, (5 + m)/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2] - 3*AppellF1[(3 + m)/2, -m, 2*(2 + m), (5 + m)/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2] - 2*m*AppellF1[(3 + m)/2, -m, 2*(2 + m), (5 + m)/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2] + 2*AppellF1[(3 + m)/2, -m, 3 + 2*m, (5 + m)/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2] + 2*m*AppellF1[(3 + m)/2, -m, 3 + 2*m, (5 + m)/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2])*(-1 + Cos[a + b*x]) + (3 + m)*AppellF1[(1 + m)/2, -m, 2*(1 + m), (3 + m)/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2]*(1 + Cos[a + b*x])))","C",0
125,1,152,82,0.2636795,"\int \sin (a+b x) \sin ^m(2 a+2 b x) \, dx","Integrate[Sin[a + b*x]*Sin[2*a + 2*b*x]^m,x]","-\frac{i 2^{-m-1} e^{i (a+b x)} \left(-i e^{-2 i (a+b x)} \left(-1+e^{4 i (a+b x)}\right)\right)^{m+1} \left((1-2 m) \, _2F_1\left(1,\frac{1}{4} (2 m+3);\frac{1}{4} (3-2 m);e^{4 i (a+b x)}\right)+(2 m+1) e^{2 i (a+b x)} \, _2F_1\left(1,\frac{1}{4} (2 m+5);\frac{1}{4} (5-2 m);e^{4 i (a+b x)}\right)\right)}{b \left(4 m^2-1\right)}","\frac{\sin (a+b x) \tan (a+b x) \sin ^m(2 a+2 b x) \cos ^2(a+b x)^{\frac{1-m}{2}} \, _2F_1\left(\frac{1-m}{2},\frac{m+2}{2};\frac{m+4}{2};\sin ^2(a+b x)\right)}{b (m+2)}",1,"((-I)*2^(-1 - m)*E^(I*(a + b*x))*(((-I)*(-1 + E^((4*I)*(a + b*x))))/E^((2*I)*(a + b*x)))^(1 + m)*((1 - 2*m)*Hypergeometric2F1[1, (3 + 2*m)/4, (3 - 2*m)/4, E^((4*I)*(a + b*x))] + E^((2*I)*(a + b*x))*(1 + 2*m)*Hypergeometric2F1[1, (5 + 2*m)/4, (5 - 2*m)/4, E^((4*I)*(a + b*x))]))/(b*(-1 + 4*m^2))","C",0
126,1,254,72,0.8662377,"\int \csc (a+b x) \sin ^m(2 a+2 b x) \, dx","Integrate[Csc[a + b*x]*Sin[2*a + 2*b*x]^m,x]","\frac{2 (m+2) \cos ^2\left(\frac{1}{2} (a+b x)\right) \sin ^m(2 (a+b x)) F_1\left(\frac{m}{2};-m,2 m;\frac{m+2}{2};\tan ^2\left(\frac{1}{2} (a+b x)\right),-\tan ^2\left(\frac{1}{2} (a+b x)\right)\right)}{b m \left((m+2) (\cos (a+b x)+1) F_1\left(\frac{m}{2};-m,2 m;\frac{m+2}{2};\tan ^2\left(\frac{1}{2} (a+b x)\right),-\tan ^2\left(\frac{1}{2} (a+b x)\right)\right)-4 m \sin ^2\left(\frac{1}{2} (a+b x)\right) \left(F_1\left(\frac{m+2}{2};1-m,2 m;\frac{m+4}{2};\tan ^2\left(\frac{1}{2} (a+b x)\right),-\tan ^2\left(\frac{1}{2} (a+b x)\right)\right)+2 F_1\left(\frac{m+2}{2};-m,2 m+1;\frac{m+4}{2};\tan ^2\left(\frac{1}{2} (a+b x)\right),-\tan ^2\left(\frac{1}{2} (a+b x)\right)\right)\right)\right)}","\frac{\sec (a+b x) \sin ^m(2 a+2 b x) \cos ^2(a+b x)^{\frac{1-m}{2}} \, _2F_1\left(\frac{1-m}{2},\frac{m}{2};\frac{m+2}{2};\sin ^2(a+b x)\right)}{b m}",1,"(2*(2 + m)*AppellF1[m/2, -m, 2*m, (2 + m)/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2]*Cos[(a + b*x)/2]^2*Sin[2*(a + b*x)]^m)/(b*m*((2 + m)*AppellF1[m/2, -m, 2*m, (2 + m)/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2]*(1 + Cos[a + b*x]) - 4*m*(AppellF1[(2 + m)/2, 1 - m, 2*m, (4 + m)/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2] + 2*AppellF1[(2 + m)/2, -m, 1 + 2*m, (4 + m)/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2])*Sin[(a + b*x)/2]^2))","C",0
127,1,938,85,5.4160524,"\int \csc ^2(a+b x) \sin ^m(2 a+2 b x) \, dx","Integrate[Csc[a + b*x]^2*Sin[2*a + 2*b*x]^m,x]","\frac{2 \left((m+1) F_1\left(\frac{m-1}{2};-m,2 m;\frac{m+1}{2};\tan ^2\left(\frac{1}{2} (a+b x)\right),-\tan ^2\left(\frac{1}{2} (a+b x)\right)\right) \cot ^2\left(\frac{1}{2} (a+b x)\right)+(m-1) F_1\left(\frac{m+1}{2};-m,2 m;\frac{m+3}{2};\tan ^2\left(\frac{1}{2} (a+b x)\right),-\tan ^2\left(\frac{1}{2} (a+b x)\right)\right)\right) \csc ^2(a+b x) \sin ^m(2 (a+b x)) \tan \left(\frac{1}{2} (a+b x)\right)}{b \left(m (m+1) F_1\left(\frac{m-1}{2};-m,2 m;\frac{m+1}{2};\tan ^2\left(\frac{1}{2} (a+b x)\right),-\tan ^2\left(\frac{1}{2} (a+b x)\right)\right) (3 \cos (a+b x)-2) \sec (a+b x) \cot ^2\left(\frac{1}{2} (a+b x)\right)+2 m (m+1) F_1\left(\frac{m-1}{2};-m,2 m;\frac{m+1}{2};\tan ^2\left(\frac{1}{2} (a+b x)\right),-\tan ^2\left(\frac{1}{2} (a+b x)\right)\right) \tan (a+b x) \cot \left(\frac{1}{2} (a+b x)\right)-(m+1) F_1\left(\frac{m-1}{2};-m,2 m;\frac{m+1}{2};\tan ^2\left(\frac{1}{2} (a+b x)\right),-\tan ^2\left(\frac{1}{2} (a+b x)\right)\right) \csc ^2\left(\frac{1}{2} (a+b x)\right)+(m-1) F_1\left(\frac{m+1}{2};-m,2 m;\frac{m+3}{2};\tan ^2\left(\frac{1}{2} (a+b x)\right),-\tan ^2\left(\frac{1}{2} (a+b x)\right)\right) \sec ^2\left(\frac{1}{2} (a+b x)\right)-2 (m-1) m \left(F_1\left(\frac{m+1}{2};1-m,2 m;\frac{m+3}{2};\tan ^2\left(\frac{1}{2} (a+b x)\right),-\tan ^2\left(\frac{1}{2} (a+b x)\right)\right)+2 F_1\left(\frac{m+1}{2};-m,2 m+1;\frac{m+3}{2};\tan ^2\left(\frac{1}{2} (a+b x)\right),-\tan ^2\left(\frac{1}{2} (a+b x)\right)\right)\right) \sec ^2\left(\frac{1}{2} (a+b x)\right)-\frac{2 (m-1) m (m+1) \left(F_1\left(\frac{m+3}{2};1-m,2 m;\frac{m+5}{2};\tan ^2\left(\frac{1}{2} (a+b x)\right),-\tan ^2\left(\frac{1}{2} (a+b x)\right)\right)+2 F_1\left(\frac{m+3}{2};-m,2 m+1;\frac{m+5}{2};\tan ^2\left(\frac{1}{2} (a+b x)\right),-\tan ^2\left(\frac{1}{2} (a+b x)\right)\right)\right) \sec ^2\left(\frac{1}{2} (a+b x)\right) \tan ^2\left(\frac{1}{2} (a+b x)\right)}{m+3}+(m-1) m F_1\left(\frac{m+1}{2};-m,2 m;\frac{m+3}{2};\tan ^2\left(\frac{1}{2} (a+b x)\right),-\tan ^2\left(\frac{1}{2} (a+b x)\right)\right) \tan ^2\left(\frac{1}{2} (a+b x)\right)+m (m+1) F_1\left(\frac{m-1}{2};-m,2 m;\frac{m+1}{2};\tan ^2\left(\frac{1}{2} (a+b x)\right),-\tan ^2\left(\frac{1}{2} (a+b x)\right)\right)+(m-1) m F_1\left(\frac{m+1}{2};-m,2 m;\frac{m+3}{2};\tan ^2\left(\frac{1}{2} (a+b x)\right),-\tan ^2\left(\frac{1}{2} (a+b x)\right)\right) (3 \cos (a+b x)-2) \sec (a+b x)+2 (m-1) m F_1\left(\frac{m+1}{2};-m,2 m;\frac{m+3}{2};\tan ^2\left(\frac{1}{2} (a+b x)\right),-\tan ^2\left(\frac{1}{2} (a+b x)\right)\right) \tan \left(\frac{1}{2} (a+b x)\right) \tan (a+b x)\right)}","-\frac{\csc (a+b x) \sec (a+b x) \sin ^m(2 a+2 b x) \cos ^2(a+b x)^{\frac{1-m}{2}} \, _2F_1\left(\frac{1-m}{2},\frac{m-1}{2};\frac{m+1}{2};\sin ^2(a+b x)\right)}{b (1-m)}",1,"(2*((-1 + m)*AppellF1[(1 + m)/2, -m, 2*m, (3 + m)/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2] + (1 + m)*AppellF1[(-1 + m)/2, -m, 2*m, (1 + m)/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2]*Cot[(a + b*x)/2]^2)*Csc[a + b*x]^2*Sin[2*(a + b*x)]^m*Tan[(a + b*x)/2])/(b*(m*(1 + m)*AppellF1[(-1 + m)/2, -m, 2*m, (1 + m)/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2] - (1 + m)*AppellF1[(-1 + m)/2, -m, 2*m, (1 + m)/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2]*Csc[(a + b*x)/2]^2 + (-1 + m)*AppellF1[(1 + m)/2, -m, 2*m, (3 + m)/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2]*Sec[(a + b*x)/2]^2 - 2*(-1 + m)*m*(AppellF1[(1 + m)/2, 1 - m, 2*m, (3 + m)/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2] + 2*AppellF1[(1 + m)/2, -m, 1 + 2*m, (3 + m)/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2])*Sec[(a + b*x)/2]^2 + (-1 + m)*m*AppellF1[(1 + m)/2, -m, 2*m, (3 + m)/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2]*(-2 + 3*Cos[a + b*x])*Sec[a + b*x] + m*(1 + m)*AppellF1[(-1 + m)/2, -m, 2*m, (1 + m)/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2]*(-2 + 3*Cos[a + b*x])*Cot[(a + b*x)/2]^2*Sec[a + b*x] + (-1 + m)*m*AppellF1[(1 + m)/2, -m, 2*m, (3 + m)/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2]*Tan[(a + b*x)/2]^2 - (2*(-1 + m)*m*(1 + m)*(AppellF1[(3 + m)/2, 1 - m, 2*m, (5 + m)/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2] + 2*AppellF1[(3 + m)/2, -m, 1 + 2*m, (5 + m)/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2])*Sec[(a + b*x)/2]^2*Tan[(a + b*x)/2]^2)/(3 + m) + 2*m*(1 + m)*AppellF1[(-1 + m)/2, -m, 2*m, (1 + m)/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2]*Cot[(a + b*x)/2]*Tan[a + b*x] + 2*(-1 + m)*m*AppellF1[(1 + m)/2, -m, 2*m, (3 + m)/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2]*Tan[(a + b*x)/2]*Tan[a + b*x]))","C",0
128,1,2308,85,19.219737,"\int \csc ^3(a+b x) \sin ^m(2 a+2 b x) \, dx","Integrate[Csc[a + b*x]^3*Sin[2*a + 2*b*x]^m,x]","\text{Result too large to show}","-\frac{\csc ^2(a+b x) \sec (a+b x) \sin ^m(2 a+2 b x) \cos ^2(a+b x)^{\frac{1-m}{2}} \, _2F_1\left(\frac{1-m}{2},\frac{m-2}{2};\frac{m}{2};\sin ^2(a+b x)\right)}{b (2-m)}",1,"(AppellF1[-1 + m/2, -m, 2*m, m/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2]*Cos[(a + b*x)/2]^2*Cot[(a + b*x)/2]^2*Sin[2*(a + b*x)]^m)/(2*b*(-2 + m)*(2*(AppellF1[m/2, 1 - m, 2*m, 1 + m/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2] + 2*AppellF1[m/2, -m, 1 + 2*m, 1 + m/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2])*(-1 + Cos[a + b*x]) + AppellF1[-1 + m/2, -m, 2*m, m/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2]*(1 + Cos[a + b*x]))) + ((4 + m)*AppellF1[1 + m/2, -m, 2*m, 2 + m/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2]*Sec[a + b*x]*Sin[(a + b*x)/2]^2*Sin[2*(a + b*x)]^m)/(2*b*(2 + m)*((4 + m)*AppellF1[1 + m/2, -m, 2*m, 2 + m/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2]*(1 + Sec[a + b*x]) - 4*m*(AppellF1[2 + m/2, 1 - m, 2*m, 3 + m/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2] + 2*AppellF1[2 + m/2, -m, 1 + 2*m, 3 + m/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2])*Sec[a + b*x]*Sin[(a + b*x)/2]^2)) + ((4 + m)*AppellF1[(2 + m)/2, -m, 1 + 2*m, (4 + m)/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2]*Sin[a + b*x]^2*Sin[2*(a + b*x)]^m)/(4*b*(2 + m)*(2*(m*AppellF1[2 + m/2, 1 - m, 1 + 2*m, 3 + m/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2] + (1 + 2*m)*AppellF1[2 + m/2, -m, 2 + 2*m, 3 + m/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2])*(-1 + Cos[a + b*x]) + (4 + m)*AppellF1[(2 + m)/2, -m, 1 + 2*m, (4 + m)/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2]*(1 + Cos[a + b*x]))) + (2^(-3 + m)*Cot[(a + b*x)/2]*(Sec[(a + b*x)/2]^2)^(2*m)*(Cos[(a + b*x)/2]*(-Sin[(a + b*x)/2] + Sin[(3*(a + b*x))/2]))^m*Sin[2*(a + b*x)]^m*((2 + m)*AppellF1[m/2, -m, 2*m, 1 + m/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2] - m*AppellF1[1 + m/2, -m, 1 + 2*m, 2 + m/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2]*Tan[(a + b*x)/2]^2))/(b*m*(2 + m)*(Cos[a + b*x]*Sec[(a + b*x)/2]^2)^m*((2^(-1 + m)*(Sec[(a + b*x)/2]^2)^(2*m)*(Cos[(a + b*x)/2]*(-Sin[(a + b*x)/2] + Sin[(3*(a + b*x))/2]))^(-1 + m)*(Cos[(a + b*x)/2]*(-1/2*Cos[(a + b*x)/2] + (3*Cos[(3*(a + b*x))/2])/2) - (Sin[(a + b*x)/2]*(-Sin[(a + b*x)/2] + Sin[(3*(a + b*x))/2]))/2)*((2 + m)*AppellF1[m/2, -m, 2*m, 1 + m/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2] - m*AppellF1[1 + m/2, -m, 1 + 2*m, 2 + m/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2]*Tan[(a + b*x)/2]^2))/((2 + m)*(Cos[a + b*x]*Sec[(a + b*x)/2]^2)^m) + (2^m*(Sec[(a + b*x)/2]^2)^(2*m)*(Cos[(a + b*x)/2]*(-Sin[(a + b*x)/2] + Sin[(3*(a + b*x))/2]))^m*Tan[(a + b*x)/2]*((2 + m)*AppellF1[m/2, -m, 2*m, 1 + m/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2] - m*AppellF1[1 + m/2, -m, 1 + 2*m, 2 + m/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2]*Tan[(a + b*x)/2]^2))/((2 + m)*(Cos[a + b*x]*Sec[(a + b*x)/2]^2)^m) - (2^(-1 + m)*(Sec[(a + b*x)/2]^2)^(2*m)*(Cos[a + b*x]*Sec[(a + b*x)/2]^2)^(-1 - m)*(Cos[(a + b*x)/2]*(-Sin[(a + b*x)/2] + Sin[(3*(a + b*x))/2]))^m*(-(Sec[(a + b*x)/2]^2*Sin[a + b*x]) + Cos[a + b*x]*Sec[(a + b*x)/2]^2*Tan[(a + b*x)/2])*((2 + m)*AppellF1[m/2, -m, 2*m, 1 + m/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2] - m*AppellF1[1 + m/2, -m, 1 + 2*m, 2 + m/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2]*Tan[(a + b*x)/2]^2))/(2 + m) + (2^(-1 + m)*(Sec[(a + b*x)/2]^2)^(2*m)*(Cos[(a + b*x)/2]*(-Sin[(a + b*x)/2] + Sin[(3*(a + b*x))/2]))^m*(-(m*AppellF1[1 + m/2, -m, 1 + 2*m, 2 + m/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2]*Sec[(a + b*x)/2]^2*Tan[(a + b*x)/2]) + (2 + m)*(-1/2*(m^2*AppellF1[1 + m/2, 1 - m, 2*m, 2 + m/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2]*Sec[(a + b*x)/2]^2*Tan[(a + b*x)/2])/(1 + m/2) - (m^2*AppellF1[1 + m/2, -m, 1 + 2*m, 2 + m/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2]*Sec[(a + b*x)/2]^2*Tan[(a + b*x)/2])/(1 + m/2)) - m*Tan[(a + b*x)/2]^2*(-(((1 + m/2)*m*AppellF1[2 + m/2, 1 - m, 1 + 2*m, 3 + m/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2]*Sec[(a + b*x)/2]^2*Tan[(a + b*x)/2])/(2 + m/2)) - ((1 + m/2)*(1 + 2*m)*AppellF1[2 + m/2, -m, 2 + 2*m, 3 + m/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2]*Sec[(a + b*x)/2]^2*Tan[(a + b*x)/2])/(2 + m/2))))/(m*(2 + m)*(Cos[a + b*x]*Sec[(a + b*x)/2]^2)^m)))","C",0
129,1,47,61,0.4388566,"\int \cos (a+b x) \sin ^7(2 a+2 b x) \, dx","Integrate[Cos[a + b*x]*Sin[2*a + 2*b*x]^7,x]","\frac{4 \cos ^9(a+b x) (10755 \cos (2 (a+b x))-3366 \cos (4 (a+b x))+429 \cos (6 (a+b x))-8330)}{6435 b}","\frac{128 \cos ^{15}(a+b x)}{15 b}-\frac{384 \cos ^{13}(a+b x)}{13 b}+\frac{384 \cos ^{11}(a+b x)}{11 b}-\frac{128 \cos ^9(a+b x)}{9 b}",1,"(4*Cos[a + b*x]^9*(-8330 + 10755*Cos[2*(a + b*x)] - 3366*Cos[4*(a + b*x)] + 429*Cos[6*(a + b*x)]))/(6435*b)","A",1
130,1,47,61,0.2906997,"\int \cos (a+b x) \sin ^6(2 a+2 b x) \, dx","Integrate[Cos[a + b*x]*Sin[2*a + 2*b*x]^6,x]","\frac{2 \sin ^7(a+b x) (6377 \cos (2 (a+b x))+1890 \cos (4 (a+b x))+231 \cos (6 (a+b x))+5230)}{3003 b}","-\frac{64 \sin ^{13}(a+b x)}{13 b}+\frac{192 \sin ^{11}(a+b x)}{11 b}-\frac{64 \sin ^9(a+b x)}{3 b}+\frac{64 \sin ^7(a+b x)}{7 b}",1,"(2*(5230 + 6377*Cos[2*(a + b*x)] + 1890*Cos[4*(a + b*x)] + 231*Cos[6*(a + b*x)])*Sin[a + b*x]^7)/(3003*b)","A",1
131,1,37,46,0.2543572,"\int \cos (a+b x) \sin ^5(2 a+2 b x) \, dx","Integrate[Cos[a + b*x]*Sin[2*a + 2*b*x]^5,x]","\frac{4 \cos ^7(a+b x) (364 \cos (2 (a+b x))-63 \cos (4 (a+b x))-365)}{693 b}","-\frac{32 \cos ^{11}(a+b x)}{11 b}+\frac{64 \cos ^9(a+b x)}{9 b}-\frac{32 \cos ^7(a+b x)}{7 b}",1,"(4*Cos[a + b*x]^7*(-365 + 364*Cos[2*(a + b*x)] - 63*Cos[4*(a + b*x)]))/(693*b)","A",1
132,1,37,46,0.1350587,"\int \cos (a+b x) \sin ^4(2 a+2 b x) \, dx","Integrate[Cos[a + b*x]*Sin[2*a + 2*b*x]^4,x]","\frac{2 \sin ^5(a+b x) (220 \cos (2 (a+b x))+35 \cos (4 (a+b x))+249)}{315 b}","\frac{16 \sin ^9(a+b x)}{9 b}-\frac{32 \sin ^7(a+b x)}{7 b}+\frac{16 \sin ^5(a+b x)}{5 b}",1,"(2*(249 + 220*Cos[2*(a + b*x)] + 35*Cos[4*(a + b*x)])*Sin[a + b*x]^5)/(315*b)","A",1
133,1,27,31,0.0927197,"\int \cos (a+b x) \sin ^3(2 a+2 b x) \, dx","Integrate[Cos[a + b*x]*Sin[2*a + 2*b*x]^3,x]","\frac{4 \cos ^5(a+b x) (5 \cos (2 (a+b x))-9)}{35 b}","\frac{8 \cos ^7(a+b x)}{7 b}-\frac{8 \cos ^5(a+b x)}{5 b}",1,"(4*Cos[a + b*x]^5*(-9 + 5*Cos[2*(a + b*x)]))/(35*b)","A",1
134,1,27,31,0.0644072,"\int \cos (a+b x) \sin ^2(2 a+2 b x) \, dx","Integrate[Cos[a + b*x]*Sin[2*a + 2*b*x]^2,x]","\frac{2 \sin ^3(a+b x) (3 \cos (2 (a+b x))+7)}{15 b}","\frac{4 \sin ^3(a+b x)}{3 b}-\frac{4 \sin ^5(a+b x)}{5 b}",1,"(2*(7 + 3*Cos[2*(a + b*x)])*Sin[a + b*x]^3)/(15*b)","A",1
135,1,15,30,0.0063843,"\int \cos (a+b x) \sin (2 a+2 b x) \, dx","Integrate[Cos[a + b*x]*Sin[2*a + 2*b*x],x]","-\frac{2 \cos ^3(a+b x)}{3 b}","-\frac{\cos (a+b x)}{2 b}-\frac{\cos (3 a+3 b x)}{6 b}",1,"(-2*Cos[a + b*x]^3)/(3*b)","A",1
136,1,42,14,0.0122948,"\int \cos (a+b x) \csc (2 a+2 b x) \, dx","Integrate[Cos[a + b*x]*Csc[2*a + 2*b*x],x]","\frac{1}{2} \left(\frac{\log \left(\sin \left(\frac{a}{2}+\frac{b x}{2}\right)\right)}{b}-\frac{\log \left(\cos \left(\frac{a}{2}+\frac{b x}{2}\right)\right)}{b}\right)","-\frac{\tanh ^{-1}(\cos (a+b x))}{2 b}",1,"(-(Log[Cos[a/2 + (b*x)/2]]/b) + Log[Sin[a/2 + (b*x)/2]]/b)/2","B",1
137,1,29,28,0.0190262,"\int \cos (a+b x) \csc ^2(2 a+2 b x) \, dx","Integrate[Cos[a + b*x]*Csc[2*a + 2*b*x]^2,x]","-\frac{\csc (a+b x) \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\sin ^2(a+b x)\right)}{4 b}","\frac{\tanh ^{-1}(\sin (a+b x))}{4 b}-\frac{\csc (a+b x)}{4 b}",1,"-1/4*(Csc[a + b*x]*Hypergeometric2F1[-1/2, 1, 1/2, Sin[a + b*x]^2])/b","C",1
138,1,143,49,0.2548864,"\int \cos (a+b x) \csc ^3(2 a+2 b x) \, dx","Integrate[Cos[a + b*x]*Csc[2*a + 2*b*x]^3,x]","\frac{\csc ^4(a+b x) \left(-6 \cos (2 (a+b x))+2 \cos (3 (a+b x))+3 \cos (3 (a+b x)) \log \left(\cos \left(\frac{1}{2} (a+b x)\right)\right)-3 \cos (3 (a+b x)) \log \left(\sin \left(\frac{1}{2} (a+b x)\right)\right)+\cos (a+b x) \left(3 \log \left(\sin \left(\frac{1}{2} (a+b x)\right)\right)-3 \log \left(\cos \left(\frac{1}{2} (a+b x)\right)\right)-2\right)+2\right)}{16 b \left(\csc ^2\left(\frac{1}{2} (a+b x)\right)-\sec ^2\left(\frac{1}{2} (a+b x)\right)\right)}","\frac{3 \sec (a+b x)}{16 b}-\frac{3 \tanh ^{-1}(\cos (a+b x))}{16 b}-\frac{\csc ^2(a+b x) \sec (a+b x)}{16 b}",1,"(Csc[a + b*x]^4*(2 - 6*Cos[2*(a + b*x)] + 2*Cos[3*(a + b*x)] + 3*Cos[3*(a + b*x)]*Log[Cos[(a + b*x)/2]] - 3*Cos[3*(a + b*x)]*Log[Sin[(a + b*x)/2]] + Cos[a + b*x]*(-2 - 3*Log[Cos[(a + b*x)/2]] + 3*Log[Sin[(a + b*x)/2]])))/(16*b*(Csc[(a + b*x)/2]^2 - Sec[(a + b*x)/2]^2))","B",1
139,1,31,66,0.0228673,"\int \cos (a+b x) \csc ^4(2 a+2 b x) \, dx","Integrate[Cos[a + b*x]*Csc[2*a + 2*b*x]^4,x]","-\frac{\csc ^3(a+b x) \, _2F_1\left(-\frac{3}{2},2;-\frac{1}{2};\sin ^2(a+b x)\right)}{48 b}","-\frac{5 \csc ^3(a+b x)}{96 b}-\frac{5 \csc (a+b x)}{32 b}+\frac{5 \tanh ^{-1}(\sin (a+b x))}{32 b}+\frac{\csc ^3(a+b x) \sec ^2(a+b x)}{32 b}",1,"-1/48*(Csc[a + b*x]^3*Hypergeometric2F1[-3/2, 2, -1/2, Sin[a + b*x]^2])/b","C",1
140,1,268,89,0.4758976,"\int \cos (a+b x) \csc ^5(2 a+2 b x) \, dx","Integrate[Cos[a + b*x]*Csc[2*a + 2*b*x]^5,x]","-\frac{\csc ^{10}(a+b x) \left(658 \cos (2 (a+b x))-228 \cos (3 (a+b x))+140 \cos (4 (a+b x))-76 \cos (5 (a+b x))-210 \cos (6 (a+b x))+76 \cos (7 (a+b x))-315 \cos (3 (a+b x)) \log \left(\cos \left(\frac{1}{2} (a+b x)\right)\right)-105 \cos (5 (a+b x)) \log \left(\cos \left(\frac{1}{2} (a+b x)\right)\right)+105 \cos (7 (a+b x)) \log \left(\cos \left(\frac{1}{2} (a+b x)\right)\right)+3 \cos (a+b x) \left(-105 \log \left(\sin \left(\frac{1}{2} (a+b x)\right)\right)+105 \log \left(\cos \left(\frac{1}{2} (a+b x)\right)\right)+76\right)+315 \cos (3 (a+b x)) \log \left(\sin \left(\frac{1}{2} (a+b x)\right)\right)+105 \cos (5 (a+b x)) \log \left(\sin \left(\frac{1}{2} (a+b x)\right)\right)-105 \cos (7 (a+b x)) \log \left(\sin \left(\frac{1}{2} (a+b x)\right)\right)-204\right)}{768 b \left(\csc ^2\left(\frac{1}{2} (a+b x)\right)-\sec ^2\left(\frac{1}{2} (a+b x)\right)\right)^3}","\frac{35 \sec ^3(a+b x)}{768 b}+\frac{35 \sec (a+b x)}{256 b}-\frac{35 \tanh ^{-1}(\cos (a+b x))}{256 b}-\frac{\csc ^4(a+b x) \sec ^3(a+b x)}{128 b}-\frac{7 \csc ^2(a+b x) \sec ^3(a+b x)}{256 b}",1,"-1/768*(Csc[a + b*x]^10*(-204 + 658*Cos[2*(a + b*x)] - 228*Cos[3*(a + b*x)] + 140*Cos[4*(a + b*x)] - 76*Cos[5*(a + b*x)] - 210*Cos[6*(a + b*x)] + 76*Cos[7*(a + b*x)] - 315*Cos[3*(a + b*x)]*Log[Cos[(a + b*x)/2]] - 105*Cos[5*(a + b*x)]*Log[Cos[(a + b*x)/2]] + 105*Cos[7*(a + b*x)]*Log[Cos[(a + b*x)/2]] + 3*Cos[a + b*x]*(76 + 105*Log[Cos[(a + b*x)/2]] - 105*Log[Sin[(a + b*x)/2]]) + 315*Cos[3*(a + b*x)]*Log[Sin[(a + b*x)/2]] + 105*Cos[5*(a + b*x)]*Log[Sin[(a + b*x)/2]] - 105*Cos[7*(a + b*x)]*Log[Sin[(a + b*x)/2]]))/(b*(Csc[(a + b*x)/2]^2 - Sec[(a + b*x)/2]^2)^3)","B",1
141,1,68,44,0.4116692,"\int \cos ^2(a+b x) \sin ^5(2 a+2 b x) \, dx","Integrate[Cos[a + b*x]^2*Sin[2*a + 2*b*x]^5,x]","-\frac{600 \cos (2 (a+b x))+75 \cos (4 (a+b x))-100 \cos (6 (a+b x))-30 \cos (8 (a+b x))+12 \cos (10 (a+b x))+5 \cos (12 (a+b x))}{3840 b}","-\frac{8 \cos ^{12}(a+b x)}{3 b}+\frac{32 \cos ^{10}(a+b x)}{5 b}-\frac{4 \cos ^8(a+b x)}{b}",1,"-1/3840*(600*Cos[2*(a + b*x)] + 75*Cos[4*(a + b*x)] - 100*Cos[6*(a + b*x)] - 30*Cos[8*(a + b*x)] + 12*Cos[10*(a + b*x)] + 5*Cos[12*(a + b*x)])/b","A",1
142,1,62,76,0.185342,"\int \cos ^2(a+b x) \sin ^4(2 a+2 b x) \, dx","Integrate[Cos[a + b*x]^2*Sin[2*a + 2*b*x]^4,x]","\frac{20 \sin (2 (a+b x))-40 \sin (4 (a+b x))-10 \sin (6 (a+b x))+5 \sin (8 (a+b x))+2 \sin (10 (a+b x))+120 b x}{640 b}","\frac{\sin ^5(2 a+2 b x)}{20 b}-\frac{\sin ^3(2 a+2 b x) \cos (2 a+2 b x)}{16 b}-\frac{3 \sin (2 a+2 b x) \cos (2 a+2 b x)}{32 b}+\frac{3 x}{16}",1,"(120*b*x + 20*Sin[2*(a + b*x)] - 40*Sin[4*(a + b*x)] - 10*Sin[6*(a + b*x)] + 5*Sin[8*(a + b*x)] + 2*Sin[10*(a + b*x)])/(640*b)","A",1
143,1,48,28,0.1162377,"\int \cos ^2(a+b x) \sin ^3(2 a+2 b x) \, dx","Integrate[Cos[a + b*x]^2*Sin[2*a + 2*b*x]^3,x]","\frac{-72 \cos (2 (a+b x))-12 \cos (4 (a+b x))+8 \cos (6 (a+b x))+3 \cos (8 (a+b x))}{384 b}","\frac{\cos ^8(a+b x)}{b}-\frac{4 \cos ^6(a+b x)}{3 b}",1,"(-72*Cos[2*(a + b*x)] - 12*Cos[4*(a + b*x)] + 8*Cos[6*(a + b*x)] + 3*Cos[8*(a + b*x)])/(384*b)","A",1
144,1,40,49,0.0955972,"\int \cos ^2(a+b x) \sin ^2(2 a+2 b x) \, dx","Integrate[Cos[a + b*x]^2*Sin[2*a + 2*b*x]^2,x]","-\frac{-3 \sin (2 (a+b x))+3 \sin (4 (a+b x))+\sin (6 (a+b x))-12 b x}{48 b}","\frac{\sin ^3(2 a+2 b x)}{12 b}-\frac{\sin (2 a+2 b x) \cos (2 a+2 b x)}{8 b}+\frac{x}{4}",1,"-1/48*(-12*b*x - 3*Sin[2*(a + b*x)] + 3*Sin[4*(a + b*x)] + Sin[6*(a + b*x)])/b","A",1
145,1,15,15,0.0046515,"\int \cos ^2(a+b x) \sin (2 a+2 b x) \, dx","Integrate[Cos[a + b*x]^2*Sin[2*a + 2*b*x],x]","-\frac{\cos ^4(a+b x)}{2 b}","-\frac{\cos ^4(a+b x)}{2 b}",1,"-1/2*Cos[a + b*x]^4/b","A",1
146,1,22,14,0.0162972,"\int \cos ^2(a+b x) \csc (2 a+2 b x) \, dx","Integrate[Cos[a + b*x]^2*Csc[2*a + 2*b*x],x]","\frac{\log (\tan (a+b x))+\log (\cos (a+b x))}{2 b}","\frac{\log (\sin (a+b x))}{2 b}",1,"(Log[Cos[a + b*x]] + Log[Tan[a + b*x]])/(2*b)","A",1
147,1,13,13,0.0128319,"\int \cos ^2(a+b x) \csc ^2(2 a+2 b x) \, dx","Integrate[Cos[a + b*x]^2*Csc[2*a + 2*b*x]^2,x]","-\frac{\cot (a+b x)}{4 b}","-\frac{\cot (a+b x)}{4 b}",1,"-1/4*Cot[a + b*x]/b","A",1
148,1,34,30,0.0485244,"\int \cos ^2(a+b x) \csc ^3(2 a+2 b x) \, dx","Integrate[Cos[a + b*x]^2*Csc[2*a + 2*b*x]^3,x]","-\frac{\csc ^2(a+b x)-2 \log (\sin (a+b x))+2 \log (\cos (a+b x))}{16 b}","\frac{\log (\tan (a+b x))}{8 b}-\frac{\cot ^2(a+b x)}{16 b}",1,"-1/16*(Csc[a + b*x]^2 + 2*Log[Cos[a + b*x]] - 2*Log[Sin[a + b*x]])/b","A",1
149,1,48,42,0.0518197,"\int \cos ^2(a+b x) \csc ^4(2 a+2 b x) \, dx","Integrate[Cos[a + b*x]^2*Csc[2*a + 2*b*x]^4,x]","\frac{\tan (a+b x)}{16 b}-\frac{5 \cot (a+b x)}{48 b}-\frac{\cot (a+b x) \csc ^2(a+b x)}{48 b}","\frac{\tan (a+b x)}{16 b}-\frac{\cot ^3(a+b x)}{48 b}-\frac{\cot (a+b x)}{8 b}",1,"(-5*Cot[a + b*x])/(48*b) - (Cot[a + b*x]*Csc[a + b*x]^2)/(48*b) + Tan[a + b*x]/(16*b)","A",1
150,1,54,60,0.3514015,"\int \cos ^2(a+b x) \csc ^5(2 a+2 b x) \, dx","Integrate[Cos[a + b*x]^2*Csc[2*a + 2*b*x]^5,x]","-\frac{\csc ^4(a+b x)+4 \csc ^2(a+b x)-2 \sec ^2(a+b x)-12 \log (\sin (a+b x))+12 \log (\cos (a+b x))}{128 b}","\frac{\tan ^2(a+b x)}{64 b}-\frac{\cot ^4(a+b x)}{128 b}-\frac{3 \cot ^2(a+b x)}{64 b}+\frac{3 \log (\tan (a+b x))}{32 b}",1,"-1/128*(4*Csc[a + b*x]^2 + Csc[a + b*x]^4 + 12*Log[Cos[a + b*x]] - 12*Log[Sin[a + b*x]] - 2*Sec[a + b*x]^2)/b","A",1
151,1,37,46,0.389824,"\int \cos ^3(a+b x) \sin ^5(2 a+2 b x) \, dx","Integrate[Cos[a + b*x]^3*Sin[2*a + 2*b*x]^5,x]","\frac{4 \cos ^9(a+b x) (540 \cos (2 (a+b x))-99 \cos (4 (a+b x))-505)}{1287 b}","-\frac{32 \cos ^{13}(a+b x)}{13 b}+\frac{64 \cos ^{11}(a+b x)}{11 b}-\frac{32 \cos ^9(a+b x)}{9 b}",1,"(4*Cos[a + b*x]^9*(-505 + 540*Cos[2*(a + b*x)] - 99*Cos[4*(a + b*x)]))/(1287*b)","A",1
152,1,47,61,0.2130531,"\int \cos ^3(a+b x) \sin ^4(2 a+2 b x) \, dx","Integrate[Cos[a + b*x]^3*Sin[2*a + 2*b*x]^4,x]","\frac{\sin ^5(a+b x) (3335 \cos (2 (a+b x))+910 \cos (4 (a+b x))+105 \cos (6 (a+b x))+3042)}{2310 b}","-\frac{16 \sin ^{11}(a+b x)}{11 b}+\frac{16 \sin ^9(a+b x)}{3 b}-\frac{48 \sin ^7(a+b x)}{7 b}+\frac{16 \sin ^5(a+b x)}{5 b}",1,"((3042 + 3335*Cos[2*(a + b*x)] + 910*Cos[4*(a + b*x)] + 105*Cos[6*(a + b*x)])*Sin[a + b*x]^5)/(2310*b)","A",1
153,1,27,31,0.1372496,"\int \cos ^3(a+b x) \sin ^3(2 a+2 b x) \, dx","Integrate[Cos[a + b*x]^3*Sin[2*a + 2*b*x]^3,x]","\frac{4 \cos ^7(a+b x) (7 \cos (2 (a+b x))-11)}{63 b}","\frac{8 \cos ^9(a+b x)}{9 b}-\frac{8 \cos ^7(a+b x)}{7 b}",1,"(4*Cos[a + b*x]^7*(-11 + 7*Cos[2*(a + b*x)]))/(63*b)","A",1
154,1,37,46,0.0925386,"\int \cos ^3(a+b x) \sin ^2(2 a+2 b x) \, dx","Integrate[Cos[a + b*x]^3*Sin[2*a + 2*b*x]^2,x]","\frac{\sin ^3(a+b x) (108 \cos (2 (a+b x))+15 \cos (4 (a+b x))+157)}{210 b}","\frac{4 \sin ^7(a+b x)}{7 b}-\frac{8 \sin ^5(a+b x)}{5 b}+\frac{4 \sin ^3(a+b x)}{3 b}",1,"((157 + 108*Cos[2*(a + b*x)] + 15*Cos[4*(a + b*x)])*Sin[a + b*x]^3)/(210*b)","A",1
155,1,15,15,0.0080603,"\int \cos ^3(a+b x) \sin (2 a+2 b x) \, dx","Integrate[Cos[a + b*x]^3*Sin[2*a + 2*b*x],x]","-\frac{2 \cos ^5(a+b x)}{5 b}","-\frac{2 \cos ^5(a+b x)}{5 b}",1,"(-2*Cos[a + b*x]^5)/(5*b)","A",1
156,1,46,28,0.0203396,"\int \cos ^3(a+b x) \csc (2 a+2 b x) \, dx","Integrate[Cos[a + b*x]^3*Csc[2*a + 2*b*x],x]","\frac{1}{2} \left(\frac{\cos (a+b x)}{b}+\frac{\log \left(\sin \left(\frac{1}{2} (a+b x)\right)\right)}{b}-\frac{\log \left(\cos \left(\frac{1}{2} (a+b x)\right)\right)}{b}\right)","\frac{\cos (a+b x)}{2 b}-\frac{\tanh ^{-1}(\cos (a+b x))}{2 b}",1,"(Cos[a + b*x]/b - Log[Cos[(a + b*x)/2]]/b + Log[Sin[(a + b*x)/2]]/b)/2","A",1
157,1,13,13,0.0119965,"\int \cos ^3(a+b x) \csc ^2(2 a+2 b x) \, dx","Integrate[Cos[a + b*x]^3*Csc[2*a + 2*b*x]^2,x]","-\frac{\csc (a+b x)}{4 b}","-\frac{\csc (a+b x)}{4 b}",1,"-1/4*Csc[a + b*x]/b","A",1
158,1,79,34,0.0153289,"\int \cos ^3(a+b x) \csc ^3(2 a+2 b x) \, dx","Integrate[Cos[a + b*x]^3*Csc[2*a + 2*b*x]^3,x]","\frac{1}{8} \left(-\frac{\csc ^2\left(\frac{1}{2} (a+b x)\right)}{8 b}+\frac{\sec ^2\left(\frac{1}{2} (a+b x)\right)}{8 b}+\frac{\log \left(\sin \left(\frac{1}{2} (a+b x)\right)\right)}{2 b}-\frac{\log \left(\cos \left(\frac{1}{2} (a+b x)\right)\right)}{2 b}\right)","-\frac{\tanh ^{-1}(\cos (a+b x))}{16 b}-\frac{\cot (a+b x) \csc (a+b x)}{16 b}",1,"(-1/8*Csc[(a + b*x)/2]^2/b - Log[Cos[(a + b*x)/2]]/(2*b) + Log[Sin[(a + b*x)/2]]/(2*b) + Sec[(a + b*x)/2]^2/(8*b))/8","B",1
159,1,31,43,0.017949,"\int \cos ^3(a+b x) \csc ^4(2 a+2 b x) \, dx","Integrate[Cos[a + b*x]^3*Csc[2*a + 2*b*x]^4,x]","-\frac{\csc ^3(a+b x) \, _2F_1\left(-\frac{3}{2},1;-\frac{1}{2};\sin ^2(a+b x)\right)}{48 b}","-\frac{\csc ^3(a+b x)}{48 b}-\frac{\csc (a+b x)}{16 b}+\frac{\tanh ^{-1}(\sin (a+b x))}{16 b}",1,"-1/48*(Csc[a + b*x]^3*Hypergeometric2F1[-3/2, 1, -1/2, Sin[a + b*x]^2])/b","C",1
160,1,195,70,0.3330665,"\int \cos ^3(a+b x) \csc ^5(2 a+2 b x) \, dx","Integrate[Cos[a + b*x]^3*Csc[2*a + 2*b*x]^5,x]","-\frac{\csc ^4\left(\frac{1}{2} (a+b x)\right)}{2048 b}-\frac{7 \csc ^2\left(\frac{1}{2} (a+b x)\right)}{1024 b}+\frac{\sec ^4\left(\frac{1}{2} (a+b x)\right)}{2048 b}+\frac{7 \sec ^2\left(\frac{1}{2} (a+b x)\right)}{1024 b}+\frac{15 \log \left(\sin \left(\frac{1}{2} (a+b x)\right)\right)}{256 b}-\frac{15 \log \left(\cos \left(\frac{1}{2} (a+b x)\right)\right)}{256 b}+\frac{\sin \left(\frac{1}{2} (a+b x)\right)}{32 b \left(\cos \left(\frac{1}{2} (a+b x)\right)-\sin \left(\frac{1}{2} (a+b x)\right)\right)}-\frac{\sin \left(\frac{1}{2} (a+b x)\right)}{32 b \left(\sin \left(\frac{1}{2} (a+b x)\right)+\cos \left(\frac{1}{2} (a+b x)\right)\right)}","\frac{15 \sec (a+b x)}{256 b}-\frac{15 \tanh ^{-1}(\cos (a+b x))}{256 b}-\frac{\csc ^4(a+b x) \sec (a+b x)}{128 b}-\frac{5 \csc ^2(a+b x) \sec (a+b x)}{256 b}",1,"(-7*Csc[(a + b*x)/2]^2)/(1024*b) - Csc[(a + b*x)/2]^4/(2048*b) - (15*Log[Cos[(a + b*x)/2]])/(256*b) + (15*Log[Sin[(a + b*x)/2]])/(256*b) + (7*Sec[(a + b*x)/2]^2)/(1024*b) + Sec[(a + b*x)/2]^4/(2048*b) + Sin[(a + b*x)/2]/(32*b*(Cos[(a + b*x)/2] - Sin[(a + b*x)/2])) - Sin[(a + b*x)/2]/(32*b*(Cos[(a + b*x)/2] + Sin[(a + b*x)/2]))","B",1
161,1,98,136,0.34457,"\int \cos (a+b x) \sin ^{\frac{5}{2}}(2 a+2 b x) \, dx","Integrate[Cos[a + b*x]*Sin[2*a + 2*b*x]^(5/2),x]","\frac{\frac{2}{3} \sqrt{\sin (2 (a+b x))} (14 \sin (a+b x)-3 \sin (3 (a+b x))-2 \sin (5 (a+b x)))-5 \left(\sin ^{-1}(\cos (a+b x)-\sin (a+b x))+\log \left(\sin (a+b x)+\sqrt{\sin (2 (a+b x))}+\cos (a+b x)\right)\right)}{32 b}","\frac{\sin (a+b x) \sin ^{\frac{5}{2}}(2 a+2 b x)}{6 b}+\frac{5 \sin (a+b x) \sqrt{\sin (2 a+2 b x)}}{16 b}-\frac{5 \sin ^{-1}(\cos (a+b x)-\sin (a+b x))}{32 b}-\frac{5 \sin ^{\frac{3}{2}}(2 a+2 b x) \cos (a+b x)}{24 b}-\frac{5 \log \left(\sin (a+b x)+\sqrt{\sin (2 a+2 b x)}+\cos (a+b x)\right)}{32 b}",1,"(-5*(ArcSin[Cos[a + b*x] - Sin[a + b*x]] + Log[Cos[a + b*x] + Sin[a + b*x] + Sqrt[Sin[2*(a + b*x)]]]) + (2*Sqrt[Sin[2*(a + b*x)]]*(14*Sin[a + b*x] - 3*Sin[3*(a + b*x)] - 2*Sin[5*(a + b*x)]))/3)/(32*b)","A",1
162,1,86,110,0.1900921,"\int \cos (a+b x) \sin ^{\frac{3}{2}}(2 a+2 b x) \, dx","Integrate[Cos[a + b*x]*Sin[2*a + 2*b*x]^(3/2),x]","\frac{3 \left(\log \left(\sin (a+b x)+\sqrt{\sin (2 (a+b x))}+\cos (a+b x)\right)-\sin ^{-1}(\cos (a+b x)-\sin (a+b x))\right)-2 \sqrt{\sin (2 (a+b x))} (2 \cos (a+b x)+\cos (3 (a+b x)))}{16 b}","\frac{\sin (a+b x) \sin ^{\frac{3}{2}}(2 a+2 b x)}{4 b}-\frac{3 \sin ^{-1}(\cos (a+b x)-\sin (a+b x))}{16 b}-\frac{3 \sqrt{\sin (2 a+2 b x)} \cos (a+b x)}{8 b}+\frac{3 \log \left(\sin (a+b x)+\sqrt{\sin (2 a+2 b x)}+\cos (a+b x)\right)}{16 b}",1,"(3*(-ArcSin[Cos[a + b*x] - Sin[a + b*x]] + Log[Cos[a + b*x] + Sin[a + b*x] + Sqrt[Sin[2*(a + b*x)]]]) - 2*(2*Cos[a + b*x] + Cos[3*(a + b*x)])*Sqrt[Sin[2*(a + b*x)]])/(16*b)","A",1
163,1,70,84,0.0983627,"\int \cos (a+b x) \sqrt{\sin (2 a+2 b x)} \, dx","Integrate[Cos[a + b*x]*Sqrt[Sin[2*a + 2*b*x]],x]","-\frac{-2 \sin (a+b x) \sqrt{\sin (2 (a+b x))}+\sin ^{-1}(\cos (a+b x)-\sin (a+b x))+\log \left(\sin (a+b x)+\sqrt{\sin (2 (a+b x))}+\cos (a+b x)\right)}{4 b}","\frac{\sin (a+b x) \sqrt{\sin (2 a+2 b x)}}{2 b}-\frac{\sin ^{-1}(\cos (a+b x)-\sin (a+b x))}{4 b}-\frac{\log \left(\sin (a+b x)+\sqrt{\sin (2 a+2 b x)}+\cos (a+b x)\right)}{4 b}",1,"-1/4*(ArcSin[Cos[a + b*x] - Sin[a + b*x]] + Log[Cos[a + b*x] + Sin[a + b*x] + Sqrt[Sin[2*(a + b*x)]]] - 2*Sin[a + b*x]*Sqrt[Sin[2*(a + b*x)]])/b","A",1
164,1,52,58,0.0491486,"\int \frac{\cos (a+b x)}{\sqrt{\sin (2 a+2 b x)}} \, dx","Integrate[Cos[a + b*x]/Sqrt[Sin[2*a + 2*b*x]],x]","\frac{\log \left(\sin (a+b x)+\sqrt{\sin (2 (a+b x))}+\cos (a+b x)\right)-\sin ^{-1}(\cos (a+b x)-\sin (a+b x))}{2 b}","\frac{\log \left(\sin (a+b x)+\sqrt{\sin (2 a+2 b x)}+\cos (a+b x)\right)}{2 b}-\frac{\sin ^{-1}(\cos (a+b x)-\sin (a+b x))}{2 b}",1,"(-ArcSin[Cos[a + b*x] - Sin[a + b*x]] + Log[Cos[a + b*x] + Sin[a + b*x] + Sqrt[Sin[2*(a + b*x)]]])/(2*b)","A",1
165,1,23,24,0.0233225,"\int \frac{\cos (a+b x)}{\sin ^{\frac{3}{2}}(2 a+2 b x)} \, dx","Integrate[Cos[a + b*x]/Sin[2*a + 2*b*x]^(3/2),x]","-\frac{\cos (a+b x)}{b \sqrt{\sin (2 (a+b x))}}","-\frac{\cos (a+b x)}{b \sqrt{\sin (2 a+2 b x)}}",1,"-(Cos[a + b*x]/(b*Sqrt[Sin[2*(a + b*x)]]))","A",1
166,1,43,53,0.107675,"\int \frac{\cos (a+b x)}{\sin ^{\frac{5}{2}}(2 a+2 b x)} \, dx","Integrate[Cos[a + b*x]/Sin[2*a + 2*b*x]^(5/2),x]","\frac{\sqrt{\sin (2 (a+b x))} \left(\frac{1}{4} \sec (a+b x)-\frac{1}{12} \cot (a+b x) \csc (a+b x)\right)}{b}","\frac{2 \sin (a+b x)}{3 b \sqrt{\sin (2 a+2 b x)}}-\frac{\cos (a+b x)}{3 b \sin ^{\frac{3}{2}}(2 a+2 b x)}",1,"((-1/12*(Cot[a + b*x]*Csc[a + b*x]) + Sec[a + b*x]/4)*Sqrt[Sin[2*(a + b*x)]])/b","A",1
167,1,52,79,0.1253366,"\int \frac{\cos (a+b x)}{\sin ^{\frac{7}{2}}(2 a+2 b x)} \, dx","Integrate[Cos[a + b*x]/Sin[2*a + 2*b*x]^(7/2),x]","-\frac{\sqrt{\sin (2 (a+b x))} \left(3 \csc ^3(a+b x)+27 \csc (a+b x)-5 \tan (a+b x) \sec (a+b x)\right)}{120 b}","\frac{4 \sin (a+b x)}{15 b \sin ^{\frac{3}{2}}(2 a+2 b x)}-\frac{\cos (a+b x)}{5 b \sin ^{\frac{5}{2}}(2 a+2 b x)}-\frac{8 \cos (a+b x)}{15 b \sqrt{\sin (2 a+2 b x)}}",1,"-1/120*(Sqrt[Sin[2*(a + b*x)]]*(27*Csc[a + b*x] + 3*Csc[a + b*x]^3 - 5*Sec[a + b*x]*Tan[a + b*x]))/b","A",1
168,1,67,105,0.1426929,"\int \frac{\cos (a+b x)}{\sin ^{\frac{9}{2}}(2 a+2 b x)} \, dx","Integrate[Cos[a + b*x]/Sin[2*a + 2*b*x]^(9/2),x]","\frac{\sqrt{\sin (2 (a+b x))} (-10 \cos (2 (a+b x))-4 \cos (4 (a+b x))+4 \cos (6 (a+b x))+5) \csc ^4(a+b x) \sec ^3(a+b x)}{560 b}","\frac{6 \sin (a+b x)}{35 b \sin ^{\frac{5}{2}}(2 a+2 b x)}+\frac{16 \sin (a+b x)}{35 b \sqrt{\sin (2 a+2 b x)}}-\frac{8 \cos (a+b x)}{35 b \sin ^{\frac{3}{2}}(2 a+2 b x)}-\frac{\cos (a+b x)}{7 b \sin ^{\frac{7}{2}}(2 a+2 b x)}",1,"((5 - 10*Cos[2*(a + b*x)] - 4*Cos[4*(a + b*x)] + 4*Cos[6*(a + b*x)])*Csc[a + b*x]^4*Sec[a + b*x]^3*Sqrt[Sin[2*(a + b*x)]])/(560*b)","A",1
169,1,96,98,0.3897509,"\int \cos ^2(a+b x) \sin ^{\frac{7}{2}}(2 a+2 b x) \, dx","Integrate[Cos[a + b*x]^2*Sin[2*a + 2*b*x]^(7/2),x]","\frac{70 \sin (2 (a+b x))-156 \sin (4 (a+b x))-35 \sin (6 (a+b x))+18 \sin (8 (a+b x))+7 \sin (10 (a+b x))+240 \sqrt{\sin (2 (a+b x))} F\left(\left.a+b x-\frac{\pi }{4}\right|2\right)}{2016 b \sqrt{\sin (2 (a+b x))}}","\frac{\sin ^{\frac{9}{2}}(2 a+2 b x)}{18 b}+\frac{5 F\left(\left.a+b x-\frac{\pi }{4}\right|2\right)}{42 b}-\frac{\sin ^{\frac{5}{2}}(2 a+2 b x) \cos (2 a+2 b x)}{14 b}-\frac{5 \sqrt{\sin (2 a+2 b x)} \cos (2 a+2 b x)}{42 b}",1,"(240*EllipticF[a - Pi/4 + b*x, 2]*Sqrt[Sin[2*(a + b*x)]] + 70*Sin[2*(a + b*x)] - 156*Sin[4*(a + b*x)] - 35*Sin[6*(a + b*x)] + 18*Sin[8*(a + b*x)] + 7*Sin[10*(a + b*x)])/(2016*b*Sqrt[Sin[2*(a + b*x)]])","A",1
170,1,66,69,0.2045976,"\int \cos ^2(a+b x) \sin ^{\frac{5}{2}}(2 a+2 b x) \, dx","Integrate[Cos[a + b*x]^2*Sin[2*a + 2*b*x]^(5/2),x]","\frac{\sqrt{\sin (2 (a+b x))} (15 \sin (2 (a+b x))-14 \sin (4 (a+b x))-5 \sin (6 (a+b x)))+84 E\left(\left.a+b x-\frac{\pi }{4}\right|2\right)}{280 b}","\frac{\sin ^{\frac{7}{2}}(2 a+2 b x)}{14 b}+\frac{3 E\left(\left.a+b x-\frac{\pi }{4}\right|2\right)}{10 b}-\frac{\sin ^{\frac{3}{2}}(2 a+2 b x) \cos (2 a+2 b x)}{10 b}",1,"(84*EllipticE[a - Pi/4 + b*x, 2] + Sqrt[Sin[2*(a + b*x)]]*(15*Sin[2*(a + b*x)] - 14*Sin[4*(a + b*x)] - 5*Sin[6*(a + b*x)]))/(280*b)","A",1
171,1,76,69,0.3478367,"\int \cos ^2(a+b x) \sin ^{\frac{3}{2}}(2 a+2 b x) \, dx","Integrate[Cos[a + b*x]^2*Sin[2*a + 2*b*x]^(3/2),x]","\frac{9 \sin (2 (a+b x))-10 \sin (4 (a+b x))-3 \sin (6 (a+b x))+20 \sqrt{\sin (2 (a+b x))} F\left(\left.a+b x-\frac{\pi }{4}\right|2\right)}{120 b \sqrt{\sin (2 (a+b x))}}","\frac{\sin ^{\frac{5}{2}}(2 a+2 b x)}{10 b}+\frac{F\left(\left.a+b x-\frac{\pi }{4}\right|2\right)}{6 b}-\frac{\sqrt{\sin (2 a+2 b x)} \cos (2 a+2 b x)}{6 b}",1,"(20*EllipticF[a - Pi/4 + b*x, 2]*Sqrt[Sin[2*(a + b*x)]] + 9*Sin[2*(a + b*x)] - 10*Sin[4*(a + b*x)] - 3*Sin[6*(a + b*x)])/(120*b*Sqrt[Sin[2*(a + b*x)]])","A",1
172,1,34,40,0.0570945,"\int \cos ^2(a+b x) \sqrt{\sin (2 a+2 b x)} \, dx","Integrate[Cos[a + b*x]^2*Sqrt[Sin[2*a + 2*b*x]],x]","\frac{\sin ^{\frac{3}{2}}(2 (a+b x))+3 E\left(\left.a+b x-\frac{\pi }{4}\right|2\right)}{6 b}","\frac{\sin ^{\frac{3}{2}}(2 a+2 b x)}{6 b}+\frac{E\left(\left.a+b x-\frac{\pi }{4}\right|2\right)}{2 b}",1,"(3*EllipticE[a - Pi/4 + b*x, 2] + Sin[2*(a + b*x)]^(3/2))/(6*b)","A",1
173,1,76,40,0.8901125,"\int \frac{\cos ^2(a+b x)}{\sqrt{\sin (2 a+2 b x)}} \, dx","Integrate[Cos[a + b*x]^2/Sqrt[Sin[2*a + 2*b*x]],x]","\frac{2 \sqrt{\sin (2 (a+b x))}-\frac{\sqrt{2} (\sin (a+b x)+\cos (a+b x)) F\left(\sin ^{-1}(\cos (a+b x)-\sin (a+b x))|\frac{1}{2}\right)}{\sqrt{\sin (2 (a+b x))+1}}}{4 b}","\frac{\sqrt{\sin (2 a+2 b x)}}{2 b}+\frac{F\left(\left.a+b x-\frac{\pi }{4}\right|2\right)}{2 b}",1,"(2*Sqrt[Sin[2*(a + b*x)]] - (Sqrt[2]*EllipticF[ArcSin[Cos[a + b*x] - Sin[a + b*x]], 1/2]*(Cos[a + b*x] + Sin[a + b*x]))/Sqrt[1 + Sin[2*(a + b*x)]])/(4*b)","A",1
174,1,39,46,0.1116584,"\int \frac{\cos ^2(a+b x)}{\sin ^{\frac{3}{2}}(2 a+2 b x)} \, dx","Integrate[Cos[a + b*x]^2/Sin[2*a + 2*b*x]^(3/2),x]","-\frac{E\left(\left.a+b x-\frac{\pi }{4}\right|2\right)+\sqrt{\sin (2 (a+b x))} \cot (a+b x)}{2 b}","-\frac{E\left(\left.a+b x-\frac{\pi }{4}\right|2\right)}{2 b}-\frac{\cos ^2(a+b x)}{b \sqrt{\sin (2 a+2 b x)}}",1,"-1/2*(EllipticE[a - Pi/4 + b*x, 2] + Cot[a + b*x]*Sqrt[Sin[2*(a + b*x)]])/b","A",1
175,1,82,48,1.002284,"\int \frac{\cos ^2(a+b x)}{\sin ^{\frac{5}{2}}(2 a+2 b x)} \, dx","Integrate[Cos[a + b*x]^2/Sin[2*a + 2*b*x]^(5/2),x]","-\frac{\sqrt{\sin (2 (a+b x))} \csc ^2(a+b x)+\frac{\sqrt{2} (\sin (a+b x)+\cos (a+b x)) F\left(\sin ^{-1}(\cos (a+b x)-\sin (a+b x))|\frac{1}{2}\right)}{\sqrt{\sin (2 (a+b x))+1}}}{12 b}","\frac{F\left(\left.a+b x-\frac{\pi }{4}\right|2\right)}{6 b}-\frac{\cos ^2(a+b x)}{3 b \sin ^{\frac{3}{2}}(2 a+2 b x)}",1,"-1/12*(Csc[a + b*x]^2*Sqrt[Sin[2*(a + b*x)]] + (Sqrt[2]*EllipticF[ArcSin[Cos[a + b*x] - Sin[a + b*x]], 1/2]*(Cos[a + b*x] + Sin[a + b*x]))/Sqrt[1 + Sin[2*(a + b*x)]])/b","A",1
176,1,64,77,0.5610374,"\int \frac{\cos ^2(a+b x)}{\sin ^{\frac{7}{2}}(2 a+2 b x)} \, dx","Integrate[Cos[a + b*x]^2/Sin[2*a + 2*b*x]^(7/2),x]","\frac{\frac{2 (-6 \cos (2 (a+b x))+3 \cos (4 (a+b x))+1) \cot (a+b x)}{\sin ^{\frac{3}{2}}(2 (a+b x))}-12 E\left(\left.a+b x-\frac{\pi }{4}\right|2\right)}{40 b}","-\frac{3 E\left(\left.a+b x-\frac{\pi }{4}\right|2\right)}{10 b}-\frac{\cos ^2(a+b x)}{5 b \sin ^{\frac{5}{2}}(2 a+2 b x)}-\frac{3 \cos (2 a+2 b x)}{10 b \sqrt{\sin (2 a+2 b x)}}",1,"(-12*EllipticE[a - Pi/4 + b*x, 2] + (2*(1 - 6*Cos[2*(a + b*x)] + 3*Cos[4*(a + b*x)])*Cot[a + b*x])/Sin[2*(a + b*x)]^(3/2))/(40*b)","A",1
177,1,99,136,0.3364595,"\int \cos ^3(a+b x) \sin ^{\frac{3}{2}}(2 a+2 b x) \, dx","Integrate[Cos[a + b*x]^3*Sin[2*a + 2*b*x]^(3/2),x]","\frac{-7 \sin ^{-1}(\cos (a+b x)-\sin (a+b x))-\frac{2}{3} \sqrt{\sin (2 (a+b x))} (10 \cos (a+b x)+9 \cos (3 (a+b x))+2 \cos (5 (a+b x)))+7 \log \left(\sin (a+b x)+\sqrt{\sin (2 (a+b x))}+\cos (a+b x)\right)}{64 b}","\frac{7 \sin (a+b x) \sin ^{\frac{3}{2}}(2 a+2 b x)}{48 b}-\frac{7 \sin ^{-1}(\cos (a+b x)-\sin (a+b x))}{64 b}+\frac{\sin ^{\frac{5}{2}}(2 a+2 b x) \cos (a+b x)}{12 b}-\frac{7 \sqrt{\sin (2 a+2 b x)} \cos (a+b x)}{32 b}+\frac{7 \log \left(\sin (a+b x)+\sqrt{\sin (2 a+2 b x)}+\cos (a+b x)\right)}{64 b}",1,"(-7*ArcSin[Cos[a + b*x] - Sin[a + b*x]] + 7*Log[Cos[a + b*x] + Sin[a + b*x] + Sqrt[Sin[2*(a + b*x)]]] - (2*(10*Cos[a + b*x] + 9*Cos[3*(a + b*x)] + 2*Cos[5*(a + b*x)])*Sqrt[Sin[2*(a + b*x)]])/3)/(64*b)","A",1
178,1,84,110,0.186712,"\int \cos ^3(a+b x) \sqrt{\sin (2 a+2 b x)} \, dx","Integrate[Cos[a + b*x]^3*Sqrt[Sin[2*a + 2*b*x]],x]","\frac{2 \sqrt{\sin (2 (a+b x))} (6 \sin (a+b x)+\sin (3 (a+b x)))-5 \left(\sin ^{-1}(\cos (a+b x)-\sin (a+b x))+\log \left(\sin (a+b x)+\sqrt{\sin (2 (a+b x))}+\cos (a+b x)\right)\right)}{32 b}","\frac{5 \sin (a+b x) \sqrt{\sin (2 a+2 b x)}}{16 b}-\frac{5 \sin ^{-1}(\cos (a+b x)-\sin (a+b x))}{32 b}+\frac{\sin ^{\frac{3}{2}}(2 a+2 b x) \cos (a+b x)}{8 b}-\frac{5 \log \left(\sin (a+b x)+\sqrt{\sin (2 a+2 b x)}+\cos (a+b x)\right)}{32 b}",1,"(-5*(ArcSin[Cos[a + b*x] - Sin[a + b*x]] + Log[Cos[a + b*x] + Sin[a + b*x] + Sqrt[Sin[2*(a + b*x)]]]) + 2*Sqrt[Sin[2*(a + b*x)]]*(6*Sin[a + b*x] + Sin[3*(a + b*x)]))/(32*b)","A",1
179,1,73,84,0.1065811,"\int \frac{\cos ^3(a+b x)}{\sqrt{\sin (2 a+2 b x)}} \, dx","Integrate[Cos[a + b*x]^3/Sqrt[Sin[2*a + 2*b*x]],x]","\frac{-3 \sin ^{-1}(\cos (a+b x)-\sin (a+b x))+\sin ^{\frac{3}{2}}(2 (a+b x)) \csc (a+b x)+3 \log \left(\sin (a+b x)+\sqrt{\sin (2 (a+b x))}+\cos (a+b x)\right)}{8 b}","-\frac{3 \sin ^{-1}(\cos (a+b x)-\sin (a+b x))}{8 b}+\frac{\sqrt{\sin (2 a+2 b x)} \cos (a+b x)}{4 b}+\frac{3 \log \left(\sin (a+b x)+\sqrt{\sin (2 a+2 b x)}+\cos (a+b x)\right)}{8 b}",1,"(-3*ArcSin[Cos[a + b*x] - Sin[a + b*x]] + 3*Log[Cos[a + b*x] + Sin[a + b*x] + Sqrt[Sin[2*(a + b*x)]]] + Csc[a + b*x]*Sin[2*(a + b*x)]^(3/2))/(8*b)","A",1
180,1,70,82,0.092462,"\int \frac{\cos ^3(a+b x)}{\sin ^{\frac{3}{2}}(2 a+2 b x)} \, dx","Integrate[Cos[a + b*x]^3/Sin[2*a + 2*b*x]^(3/2),x]","\frac{\sin ^{-1}(\cos (a+b x)-\sin (a+b x))-2 \sqrt{\sin (2 (a+b x))} \csc (a+b x)+\log \left(\sin (a+b x)+\sqrt{\sin (2 (a+b x))}+\cos (a+b x)\right)}{4 b}","\frac{\sin ^{-1}(\cos (a+b x)-\sin (a+b x))}{4 b}-\frac{\cos (a+b x)}{b \sqrt{\sin (2 a+2 b x)}}+\frac{\log \left(\sin (a+b x)+\sqrt{\sin (2 a+2 b x)}+\cos (a+b x)\right)}{4 b}",1,"(ArcSin[Cos[a + b*x] - Sin[a + b*x]] + Log[Cos[a + b*x] + Sin[a + b*x] + Sqrt[Sin[2*(a + b*x)]]] - 2*Csc[a + b*x]*Sqrt[Sin[2*(a + b*x)]])/(4*b)","A",1
181,1,27,28,0.0540184,"\int \frac{\cos ^3(a+b x)}{\sin ^{\frac{5}{2}}(2 a+2 b x)} \, dx","Integrate[Cos[a + b*x]^3/Sin[2*a + 2*b*x]^(5/2),x]","-\frac{\sin ^{\frac{3}{2}}(2 (a+b x)) \csc ^3(a+b x)}{24 b}","-\frac{\cos ^3(a+b x)}{3 b \sin ^{\frac{3}{2}}(2 a+2 b x)}",1,"-1/24*(Csc[a + b*x]^3*Sin[2*(a + b*x)]^(3/2))/b","A",1
182,1,35,55,0.0921785,"\int \frac{\cos ^3(a+b x)}{\sin ^{\frac{7}{2}}(2 a+2 b x)} \, dx","Integrate[Cos[a + b*x]^3/Sin[2*a + 2*b*x]^(7/2),x]","-\frac{\sqrt{\sin (2 (a+b x))} \csc (a+b x) \left(\csc ^2(a+b x)+4\right)}{40 b}","-\frac{\cos ^3(a+b x)}{5 b \sin ^{\frac{5}{2}}(2 a+2 b x)}-\frac{\cos (a+b x)}{5 b \sqrt{\sin (2 a+2 b x)}}",1,"-1/40*(Csc[a + b*x]*(4 + Csc[a + b*x]^2)*Sqrt[Sin[2*(a + b*x)]])/b","A",1
183,1,55,81,0.1155576,"\int \frac{\cos ^3(a+b x)}{\sin ^{\frac{9}{2}}(2 a+2 b x)} \, dx","Integrate[Cos[a + b*x]^3/Sin[2*a + 2*b*x]^(9/2),x]","\frac{\sqrt{\sin (2 (a+b x))} (-12 \cos (2 (a+b x))+4 \cos (4 (a+b x))+5) \csc ^4(a+b x) \sec (a+b x)}{336 b}","\frac{4 \sin (a+b x)}{21 b \sqrt{\sin (2 a+2 b x)}}-\frac{\cos ^3(a+b x)}{7 b \sin ^{\frac{7}{2}}(2 a+2 b x)}-\frac{2 \cos (a+b x)}{21 b \sin ^{\frac{3}{2}}(2 a+2 b x)}",1,"((5 - 12*Cos[2*(a + b*x)] + 4*Cos[4*(a + b*x)])*Csc[a + b*x]^4*Sec[a + b*x]*Sqrt[Sin[2*(a + b*x)]])/(336*b)","A",1
184,1,62,107,0.0932904,"\int \frac{\cos ^3(a+b x)}{\sin ^{\frac{11}{2}}(2 a+2 b x)} \, dx","Integrate[Cos[a + b*x]^3/Sin[2*a + 2*b*x]^(11/2),x]","-\frac{\sqrt{\sin (2 (a+b x))} \left(5 \csc ^5(a+b x)+17 \csc ^3(a+b x)+113 \csc (a+b x)-15 \tan (a+b x) \sec (a+b x)\right)}{1440 b}","\frac{4 \sin (a+b x)}{45 b \sin ^{\frac{3}{2}}(2 a+2 b x)}-\frac{\cos ^3(a+b x)}{9 b \sin ^{\frac{9}{2}}(2 a+2 b x)}-\frac{\cos (a+b x)}{15 b \sin ^{\frac{5}{2}}(2 a+2 b x)}-\frac{8 \cos (a+b x)}{45 b \sqrt{\sin (2 a+2 b x)}}",1,"-1/1440*(Sqrt[Sin[2*(a + b*x)]]*(113*Csc[a + b*x] + 17*Csc[a + b*x]^3 + 5*Csc[a + b*x]^5 - 15*Sec[a + b*x]*Tan[a + b*x]))/b","A",1
185,1,29,31,0.0246624,"\int \frac{\cos (x)}{\sqrt{\sin (2 x)}} \, dx","Integrate[Cos[x]/Sqrt[Sin[2*x]],x]","\frac{1}{2} \left(\log \left(\sin (x)+\sqrt{\sin (2 x)}+\cos (x)\right)-\sin ^{-1}(\cos (x)-\sin (x))\right)","\frac{1}{2} \log \left(\sin (x)+\sqrt{\sin (2 x)}+\cos (x)\right)-\frac{1}{2} \sin ^{-1}(\cos (x)-\sin (x))",1,"(-ArcSin[Cos[x] - Sin[x]] + Log[Cos[x] + Sin[x] + Sqrt[Sin[2*x]]])/2","A",1
186,1,25,25,0.0158482,"\int \csc (x) \sqrt{\sin (2 x)} \, dx","Integrate[Csc[x]*Sqrt[Sin[2*x]],x]","\log \left(\sin (x)+\sqrt{\sin (2 x)}+\cos (x)\right)-\sin ^{-1}(\cos (x)-\sin (x))","\log \left(\sin (x)+\sqrt{\sin (2 x)}+\cos (x)\right)-\sin ^{-1}(\cos (x)-\sin (x))",1,"-ArcSin[Cos[x] - Sin[x]] + Log[Cos[x] + Sin[x] + Sqrt[Sin[2*x]]]","A",1
187,1,2472,85,13.247689,"\int \cos ^3(a+b x) \sin ^m(2 a+2 b x) \, dx","Integrate[Cos[a + b*x]^3*Sin[2*a + 2*b*x]^m,x]","\text{Result too large to show}","-\frac{\cos ^3(a+b x) \cot (a+b x) \sin ^2(a+b x)^{\frac{1-m}{2}} \sin ^m(2 a+2 b x) \, _2F_1\left(\frac{1-m}{2},\frac{m+4}{2};\frac{m+6}{2};\cos ^2(a+b x)\right)}{b (m+4)}",1,"(2^(1 + m)*(6*AppellF1[(1 + m)/2, -m, 2*(1 + m), (3 + m)/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2] + 8*AppellF1[(1 + m)/2, -m, 2*(2 + m), (3 + m)/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2] - AppellF1[(1 + m)/2, -m, 1 + 2*m, (3 + m)/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2] - 12*AppellF1[(1 + m)/2, -m, 3 + 2*m, (3 + m)/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2])*Cos[a + b*x]^3*(Sec[(a + b*x)/2]^2)^(2*m)*(Cos[(a + b*x)/2]*(-Sin[(a + b*x)/2] + Sin[(3*(a + b*x))/2]))^m*Sin[2*(a + b*x)]^m*Tan[(a + b*x)/2])/(b*(1 + m)*(Cos[a + b*x]*Sec[(a + b*x)/2]^2)^m*((2^m*(6*AppellF1[(1 + m)/2, -m, 2*(1 + m), (3 + m)/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2] + 8*AppellF1[(1 + m)/2, -m, 2*(2 + m), (3 + m)/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2] - AppellF1[(1 + m)/2, -m, 1 + 2*m, (3 + m)/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2] - 12*AppellF1[(1 + m)/2, -m, 3 + 2*m, (3 + m)/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2])*(Sec[(a + b*x)/2]^2)^(1 + 2*m)*(Cos[(a + b*x)/2]*(-Sin[(a + b*x)/2] + Sin[(3*(a + b*x))/2]))^m)/((1 + m)*(Cos[a + b*x]*Sec[(a + b*x)/2]^2)^m) + (2^(1 + m)*m*(6*AppellF1[(1 + m)/2, -m, 2*(1 + m), (3 + m)/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2] + 8*AppellF1[(1 + m)/2, -m, 2*(2 + m), (3 + m)/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2] - AppellF1[(1 + m)/2, -m, 1 + 2*m, (3 + m)/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2] - 12*AppellF1[(1 + m)/2, -m, 3 + 2*m, (3 + m)/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2])*(Sec[(a + b*x)/2]^2)^(2*m)*(Cos[(a + b*x)/2]*(-Sin[(a + b*x)/2] + Sin[(3*(a + b*x))/2]))^(-1 + m)*(Cos[(a + b*x)/2]*(-1/2*Cos[(a + b*x)/2] + (3*Cos[(3*(a + b*x))/2])/2) - (Sin[(a + b*x)/2]*(-Sin[(a + b*x)/2] + Sin[(3*(a + b*x))/2]))/2)*Tan[(a + b*x)/2])/((1 + m)*(Cos[a + b*x]*Sec[(a + b*x)/2]^2)^m) + (2^(2 + m)*m*(6*AppellF1[(1 + m)/2, -m, 2*(1 + m), (3 + m)/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2] + 8*AppellF1[(1 + m)/2, -m, 2*(2 + m), (3 + m)/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2] - AppellF1[(1 + m)/2, -m, 1 + 2*m, (3 + m)/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2] - 12*AppellF1[(1 + m)/2, -m, 3 + 2*m, (3 + m)/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2])*(Sec[(a + b*x)/2]^2)^(2*m)*(Cos[(a + b*x)/2]*(-Sin[(a + b*x)/2] + Sin[(3*(a + b*x))/2]))^m*Tan[(a + b*x)/2]^2)/((1 + m)*(Cos[a + b*x]*Sec[(a + b*x)/2]^2)^m) - (2^(1 + m)*m*(6*AppellF1[(1 + m)/2, -m, 2*(1 + m), (3 + m)/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2] + 8*AppellF1[(1 + m)/2, -m, 2*(2 + m), (3 + m)/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2] - AppellF1[(1 + m)/2, -m, 1 + 2*m, (3 + m)/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2] - 12*AppellF1[(1 + m)/2, -m, 3 + 2*m, (3 + m)/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2])*(Sec[(a + b*x)/2]^2)^(2*m)*(Cos[a + b*x]*Sec[(a + b*x)/2]^2)^(-1 - m)*(Cos[(a + b*x)/2]*(-Sin[(a + b*x)/2] + Sin[(3*(a + b*x))/2]))^m*Tan[(a + b*x)/2]*(-(Sec[(a + b*x)/2]^2*Sin[a + b*x]) + Cos[a + b*x]*Sec[(a + b*x)/2]^2*Tan[(a + b*x)/2]))/(1 + m) + (2^(1 + m)*(Sec[(a + b*x)/2]^2)^(2*m)*(Cos[(a + b*x)/2]*(-Sin[(a + b*x)/2] + Sin[(3*(a + b*x))/2]))^m*Tan[(a + b*x)/2]*((m*(1 + m)*AppellF1[1 + (1 + m)/2, 1 - m, 1 + 2*m, 1 + (3 + m)/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2]*Sec[(a + b*x)/2]^2*Tan[(a + b*x)/2])/(3 + m) + ((1 + m)*(1 + 2*m)*AppellF1[1 + (1 + m)/2, -m, 2 + 2*m, 1 + (3 + m)/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2]*Sec[(a + b*x)/2]^2*Tan[(a + b*x)/2])/(3 + m) - 12*(-((m*(1 + m)*AppellF1[1 + (1 + m)/2, 1 - m, 3 + 2*m, 1 + (3 + m)/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2]*Sec[(a + b*x)/2]^2*Tan[(a + b*x)/2])/(3 + m)) - ((1 + m)*(3 + 2*m)*AppellF1[1 + (1 + m)/2, -m, 4 + 2*m, 1 + (3 + m)/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2]*Sec[(a + b*x)/2]^2*Tan[(a + b*x)/2])/(3 + m)) + 6*(-((m*(1 + m)*AppellF1[1 + (1 + m)/2, 1 - m, 2*(1 + m), 1 + (3 + m)/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2]*Sec[(a + b*x)/2]^2*Tan[(a + b*x)/2])/(3 + m)) - (2*(1 + m)^2*AppellF1[1 + (1 + m)/2, -m, 1 + 2*(1 + m), 1 + (3 + m)/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2]*Sec[(a + b*x)/2]^2*Tan[(a + b*x)/2])/(3 + m)) + 8*(-((m*(1 + m)*AppellF1[1 + (1 + m)/2, 1 - m, 2*(2 + m), 1 + (3 + m)/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2]*Sec[(a + b*x)/2]^2*Tan[(a + b*x)/2])/(3 + m)) - (2*(1 + m)*(2 + m)*AppellF1[1 + (1 + m)/2, -m, 1 + 2*(2 + m), 1 + (3 + m)/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2]*Sec[(a + b*x)/2]^2*Tan[(a + b*x)/2])/(3 + m))))/((1 + m)*(Cos[a + b*x]*Sec[(a + b*x)/2]^2)^m)))","C",0
188,1,890,85,7.8086924,"\int \cos ^2(a+b x) \sin ^m(2 a+2 b x) \, dx","Integrate[Cos[a + b*x]^2*Sin[2*a + 2*b*x]^m,x]","\frac{4 (m+3) \left(4 F_1\left(\frac{m+1}{2};-m,2 (m+1);\frac{m+3}{2};\tan ^2\left(\frac{1}{2} (a+b x)\right),-\tan ^2\left(\frac{1}{2} (a+b x)\right)\right)-F_1\left(\frac{m+1}{2};-m,2 m+1;\frac{m+3}{2};\tan ^2\left(\frac{1}{2} (a+b x)\right),-\tan ^2\left(\frac{1}{2} (a+b x)\right)\right)-4 F_1\left(\frac{m+1}{2};-m,2 m+3;\frac{m+3}{2};\tan ^2\left(\frac{1}{2} (a+b x)\right),-\tan ^2\left(\frac{1}{2} (a+b x)\right)\right)\right) \cos ^3\left(\frac{1}{2} (a+b x)\right) \cos ^2(a+b x) \sin \left(\frac{1}{2} (a+b x)\right) \sin ^m(2 (a+b x))}{b (m+1) \left(8 (m+3) F_1\left(\frac{m+1}{2};-m,2 (m+1);\frac{m+3}{2};\tan ^2\left(\frac{1}{2} (a+b x)\right),-\tan ^2\left(\frac{1}{2} (a+b x)\right)\right) \cos ^2\left(\frac{1}{2} (a+b x)\right)-2 (m+3) F_1\left(\frac{m+1}{2};-m,2 m+1;\frac{m+3}{2};\tan ^2\left(\frac{1}{2} (a+b x)\right),-\tan ^2\left(\frac{1}{2} (a+b x)\right)\right) \cos ^2\left(\frac{1}{2} (a+b x)\right)-8 (m+3) F_1\left(\frac{m+1}{2};-m,2 m+3;\frac{m+3}{2};\tan ^2\left(\frac{1}{2} (a+b x)\right),-\tan ^2\left(\frac{1}{2} (a+b x)\right)\right) \cos ^2\left(\frac{1}{2} (a+b x)\right)+2 \left(4 m F_1\left(\frac{m+3}{2};1-m,2 (m+1);\frac{m+5}{2};\tan ^2\left(\frac{1}{2} (a+b x)\right),-\tan ^2\left(\frac{1}{2} (a+b x)\right)\right)-m F_1\left(\frac{m+3}{2};1-m,2 m+1;\frac{m+5}{2};\tan ^2\left(\frac{1}{2} (a+b x)\right),-\tan ^2\left(\frac{1}{2} (a+b x)\right)\right)-4 m F_1\left(\frac{m+3}{2};1-m,2 m+3;\frac{m+5}{2};\tan ^2\left(\frac{1}{2} (a+b x)\right),-\tan ^2\left(\frac{1}{2} (a+b x)\right)\right)-2 m F_1\left(\frac{m+3}{2};-m,2 (m+1);\frac{m+5}{2};\tan ^2\left(\frac{1}{2} (a+b x)\right),-\tan ^2\left(\frac{1}{2} (a+b x)\right)\right)-F_1\left(\frac{m+3}{2};-m,2 (m+1);\frac{m+5}{2};\tan ^2\left(\frac{1}{2} (a+b x)\right),-\tan ^2\left(\frac{1}{2} (a+b x)\right)\right)-8 m F_1\left(\frac{m+3}{2};-m,2 (m+2);\frac{m+5}{2};\tan ^2\left(\frac{1}{2} (a+b x)\right),-\tan ^2\left(\frac{1}{2} (a+b x)\right)\right)-12 F_1\left(\frac{m+3}{2};-m,2 (m+2);\frac{m+5}{2};\tan ^2\left(\frac{1}{2} (a+b x)\right),-\tan ^2\left(\frac{1}{2} (a+b x)\right)\right)+8 m F_1\left(\frac{m+3}{2};-m,2 m+3;\frac{m+5}{2};\tan ^2\left(\frac{1}{2} (a+b x)\right),-\tan ^2\left(\frac{1}{2} (a+b x)\right)\right)+8 F_1\left(\frac{m+3}{2};-m,2 m+3;\frac{m+5}{2};\tan ^2\left(\frac{1}{2} (a+b x)\right),-\tan ^2\left(\frac{1}{2} (a+b x)\right)\right)\right) (\cos (a+b x)-1)\right)}","-\frac{\cos ^2(a+b x) \cot (a+b x) \sin ^2(a+b x)^{\frac{1-m}{2}} \sin ^m(2 a+2 b x) \, _2F_1\left(\frac{1-m}{2},\frac{m+3}{2};\frac{m+5}{2};\cos ^2(a+b x)\right)}{b (m+3)}",1,"(4*(3 + m)*(4*AppellF1[(1 + m)/2, -m, 2*(1 + m), (3 + m)/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2] - AppellF1[(1 + m)/2, -m, 1 + 2*m, (3 + m)/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2] - 4*AppellF1[(1 + m)/2, -m, 3 + 2*m, (3 + m)/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2])*Cos[(a + b*x)/2]^3*Cos[a + b*x]^2*Sin[(a + b*x)/2]*Sin[2*(a + b*x)]^m)/(b*(1 + m)*(8*(3 + m)*AppellF1[(1 + m)/2, -m, 2*(1 + m), (3 + m)/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2]*Cos[(a + b*x)/2]^2 - 2*(3 + m)*AppellF1[(1 + m)/2, -m, 1 + 2*m, (3 + m)/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2]*Cos[(a + b*x)/2]^2 - 8*(3 + m)*AppellF1[(1 + m)/2, -m, 3 + 2*m, (3 + m)/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2]*Cos[(a + b*x)/2]^2 + 2*(4*m*AppellF1[(3 + m)/2, 1 - m, 2*(1 + m), (5 + m)/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2] - m*AppellF1[(3 + m)/2, 1 - m, 1 + 2*m, (5 + m)/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2] - 4*m*AppellF1[(3 + m)/2, 1 - m, 3 + 2*m, (5 + m)/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2] - AppellF1[(3 + m)/2, -m, 2*(1 + m), (5 + m)/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2] - 2*m*AppellF1[(3 + m)/2, -m, 2*(1 + m), (5 + m)/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2] - 12*AppellF1[(3 + m)/2, -m, 2*(2 + m), (5 + m)/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2] - 8*m*AppellF1[(3 + m)/2, -m, 2*(2 + m), (5 + m)/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2] + 8*AppellF1[(3 + m)/2, -m, 3 + 2*m, (5 + m)/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2] + 8*m*AppellF1[(3 + m)/2, -m, 3 + 2*m, (5 + m)/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2])*(-1 + Cos[a + b*x])))","C",0
189,1,149,83,0.2580637,"\int \cos (a+b x) \sin ^m(2 a+2 b x) \, dx","Integrate[Cos[a + b*x]*Sin[2*a + 2*b*x]^m,x]","\frac{2^{-m-1} e^{i (a+b x)} \left(-i e^{-2 i (a+b x)} \left(-1+e^{4 i (a+b x)}\right)\right)^{m+1} \left((2 m-1) \, _2F_1\left(1,\frac{1}{4} (2 m+3);\frac{1}{4} (3-2 m);e^{4 i (a+b x)}\right)+(2 m+1) e^{2 i (a+b x)} \, _2F_1\left(1,\frac{1}{4} (2 m+5);\frac{1}{4} (5-2 m);e^{4 i (a+b x)}\right)\right)}{b \left(4 m^2-1\right)}","-\frac{\cos (a+b x) \cot (a+b x) \sin ^2(a+b x)^{\frac{1-m}{2}} \sin ^m(2 a+2 b x) \, _2F_1\left(\frac{1-m}{2},\frac{m+2}{2};\frac{m+4}{2};\cos ^2(a+b x)\right)}{b (m+2)}",1,"(2^(-1 - m)*E^(I*(a + b*x))*(((-I)*(-1 + E^((4*I)*(a + b*x))))/E^((2*I)*(a + b*x)))^(1 + m)*((-1 + 2*m)*Hypergeometric2F1[1, (3 + 2*m)/4, (3 - 2*m)/4, E^((4*I)*(a + b*x))] + E^((2*I)*(a + b*x))*(1 + 2*m)*Hypergeometric2F1[1, (5 + 2*m)/4, (5 - 2*m)/4, E^((4*I)*(a + b*x))]))/(b*(-1 + 4*m^2))","C",0
190,1,37,46,0.1559431,"\int \cos ^2(a+b x) \sin ^3(a+b x) \sin ^2(2 a+2 b x) \, dx","Integrate[Cos[a + b*x]^2*Sin[a + b*x]^3*Sin[2*a + 2*b*x]^2,x]","\frac{\cos ^5(a+b x) (220 \cos (2 (a+b x))-35 \cos (4 (a+b x))-249)}{630 b}","-\frac{4 \cos ^9(a+b x)}{9 b}+\frac{8 \cos ^7(a+b x)}{7 b}-\frac{4 \cos ^5(a+b x)}{5 b}",1,"(Cos[a + b*x]^5*(-249 + 220*Cos[2*(a + b*x)] - 35*Cos[4*(a + b*x)]))/(630*b)","A",1
191,1,209,293,0.8951655,"\int \sin (a+b x) \sin ^n(c+d x) \, dx","Integrate[Sin[a + b*x]*Sin[c + d*x]^n,x]","\frac{2^{-n-1} e^{-i x (b+d)} \left(-1+e^{2 i (c+d x)}\right) \left(-i e^{-i (c+d x)} \left(-1+e^{2 i (c+d x)}\right)\right)^n \left(e^{i d x} (\cos (a)-i \sin (a)) (b-d n) \, _2F_1\left(1,\frac{1}{2} \left(-\frac{b}{d}+n+2\right);-\frac{b+d (n-2)}{2 d};e^{2 i (c+d x)}\right)+(\cos (a)+i \sin (a)) (b+d n) e^{i x (2 b+d)} \, _2F_1\left(1,\frac{b+d (n+2)}{2 d};\frac{1}{2} \left(\frac{b}{d}-n+2\right);e^{2 i (c+d x)}\right)\right)}{(b-d n) (b+d n)}","-\frac{2^{-n-1} \left(i e^{-i (c+d x)}-i e^{i (c+d x)}\right)^n \left(1-e^{2 i c+2 i d x}\right)^{-n} \, _2F_1\left(-n,\frac{b-d n}{2 d};\frac{1}{2} \left(\frac{b}{d}-n+2\right);e^{2 i (c+d x)}\right) \exp (i (a-c n)+i x (b-d n)+i n (c+d x))}{b-d n}-\frac{2^{-n-1} \left(i e^{-i (c+d x)}-i e^{i (c+d x)}\right)^n \left(1-e^{2 i c+2 i d x}\right)^{-n} \, _2F_1\left(-n,-\frac{b+d n}{2 d};1-\frac{b+d n}{2 d};e^{2 i (c+d x)}\right) \exp (-i (a+c n)-i x (b+d n)+i n (c+d x))}{b+d n}",1,"(2^(-1 - n)*(-1 + E^((2*I)*(c + d*x)))*(((-I)*(-1 + E^((2*I)*(c + d*x))))/E^(I*(c + d*x)))^n*(E^(I*d*x)*(b - d*n)*Hypergeometric2F1[1, (2 - b/d + n)/2, -1/2*(b + d*(-2 + n))/d, E^((2*I)*(c + d*x))]*(Cos[a] - I*Sin[a]) + E^(I*(2*b + d)*x)*(b + d*n)*Hypergeometric2F1[1, (b + d*(2 + n))/(2*d), (2 + b/d - n)/2, E^((2*I)*(c + d*x))]*(Cos[a] + I*Sin[a])))/(E^(I*(b + d)*x)*(b - d*n)*(b + d*n))","A",0
192,1,86,91,0.544775,"\int \sin (a+b x) \sin ^3(c+d x) \, dx","Integrate[Sin[a + b*x]*Sin[c + d*x]^3,x]","\frac{1}{8} \left(-\frac{\sin (a+b x-3 c-3 d x)}{b-3 d}+\frac{3 \sin (a+b x-c-d x)}{b-d}+\frac{\sin (a+b x+3 c+3 d x)}{b+3 d}-\frac{3 \sin (a+x (b+d)+c)}{b+d}\right)","-\frac{\sin (a+x (b-3 d)-3 c)}{8 (b-3 d)}+\frac{3 \sin (a+x (b-d)-c)}{8 (b-d)}-\frac{3 \sin (a+x (b+d)+c)}{8 (b+d)}+\frac{\sin (a+x (b+3 d)+3 c)}{8 (b+3 d)}",1,"(-(Sin[a - 3*c + b*x - 3*d*x]/(b - 3*d)) + (3*Sin[a - c + b*x - d*x])/(b - d) + Sin[a + 3*c + b*x + 3*d*x]/(b + 3*d) - (3*Sin[a + c + (b + d)*x])/(b + d))/8","A",1
193,1,69,62,0.7143901,"\int \sin (a+b x) \sin ^2(c+d x) \, dx","Integrate[Sin[a + b*x]*Sin[c + d*x]^2,x]","\frac{1}{4} \left(\frac{\cos (a+b x-2 c-2 d x)}{b-2 d}+\frac{\cos (a+b x+2 c+2 d x)}{b+2 d}+\frac{2 \sin (a) \sin (b x)}{b}-\frac{2 \cos (a) \cos (b x)}{b}\right)","\frac{\cos (a+x (b-2 d)-2 c)}{4 (b-2 d)}+\frac{\cos (a+x (b+2 d)+2 c)}{4 (b+2 d)}-\frac{\cos (a+b x)}{2 b}",1,"((-2*Cos[a]*Cos[b*x])/b + Cos[a - 2*c + b*x - 2*d*x]/(b - 2*d) + Cos[a + 2*c + b*x + 2*d*x]/(b + 2*d) + (2*Sin[a]*Sin[b*x])/b)/4","A",1
194,1,43,43,0.1939527,"\int \sin (a+b x) \sin (c+d x) \, dx","Integrate[Sin[a + b*x]*Sin[c + d*x],x]","\frac{\sin (a+x (b-d)-c)}{2 (b-d)}-\frac{\sin (a+x (b+d)+c)}{2 (b+d)}","\frac{\sin (a+x (b-d)-c)}{2 (b-d)}-\frac{\sin (a+x (b+d)+c)}{2 (b+d)}",1,"Sin[a - c + (b - d)*x]/(2*(b - d)) - Sin[a + c + (b + d)*x]/(2*(b + d))","A",1
195,1,26,26,0.1511946,"\int \csc (c+b x) \sin (a+b x) \, dx","Integrate[Csc[c + b*x]*Sin[a + b*x],x]","\frac{\sin (a-c) \log (\sin (b x+c))}{b}+x \cos (a-c)","\frac{\sin (a-c) \log (\sin (b x+c))}{b}+x \cos (a-c)",1,"x*Cos[a - c] + (Log[Sin[c + b*x]]*Sin[a - c])/b","A",1
196,1,90,36,0.0993325,"\int \csc ^2(c+b x) \sin (a+b x) \, dx","Integrate[Csc[c + b*x]^2*Sin[a + b*x],x]","-\frac{\sin (a-c) \csc (b x+c)}{b}-\frac{2 i \cos (a-c) \tan ^{-1}\left(\frac{(\cos (c)-i \sin (c)) \left(\cos (c) \cos \left(\frac{b x}{2}\right)-\sin (c) \sin \left(\frac{b x}{2}\right)\right)}{\sin (c) \cos \left(\frac{b x}{2}\right)+i \cos (c) \cos \left(\frac{b x}{2}\right)}\right)}{b}","-\frac{\cos (a-c) \tanh ^{-1}(\cos (b x+c))}{b}-\frac{\sin (a-c) \csc (b x+c)}{b}",1,"((-2*I)*ArcTan[((Cos[c] - I*Sin[c])*(Cos[c]*Cos[(b*x)/2] - Sin[c]*Sin[(b*x)/2]))/(I*Cos[c]*Cos[(b*x)/2] + Cos[(b*x)/2]*Sin[c])]*Cos[a - c])/b - (Csc[c + b*x]*Sin[a - c])/b","C",1
197,1,35,39,0.2021498,"\int \csc ^3(c+b x) \sin (a+b x) \, dx","Integrate[Csc[c + b*x]^3*Sin[a + b*x],x]","\frac{\csc (c) \csc ^2(b x+c) (\cos (a)-\cos (a-c) \cos (2 b x+c))}{2 b}","-\frac{\cos (a-c) \cot (b x+c)}{b}-\frac{\sin (a-c) \csc ^2(b x+c)}{2 b}",1,"((Cos[a] - Cos[a - c]*Cos[c + 2*b*x])*Csc[c]*Csc[c + b*x]^2)/(2*b)","A",1
198,1,67,67,0.5763218,"\int \csc ^4(c+b x) \sin (a+b x) \, dx","Integrate[Csc[c + b*x]^4*Sin[a + b*x],x]","-\frac{2 \sin (a-c) \csc ^3(b x+c)+3 \cos (a-c) \cot (b x+c) \csc (b x+c)+6 \cos (a-c) \tanh ^{-1}\left(\cos (c)-\sin (c) \tan \left(\frac{b x}{2}\right)\right)}{6 b}","-\frac{\cos (a-c) \tanh ^{-1}(\cos (b x+c))}{2 b}-\frac{\sin (a-c) \csc ^3(b x+c)}{3 b}-\frac{\cos (a-c) \cot (b x+c) \csc (b x+c)}{2 b}",1,"-1/6*(6*ArcTanh[Cos[c] - Sin[c]*Tan[(b*x)/2]]*Cos[a - c] + 3*Cos[a - c]*Cot[c + b*x]*Csc[c + b*x] + 2*Csc[c + b*x]^3*Sin[a - c])/b","A",1
199,1,58,60,0.3810371,"\int \csc ^5(c+b x) \sin (a+b x) \, dx","Integrate[Csc[c + b*x]^5*Sin[a + b*x],x]","\frac{\csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \csc ^4(b x+c) (\cos (a-c) (\cos (4 b x+3 c)-4 \cos (2 b x+c))+3 \cos (a))}{24 b}","-\frac{\cos (a-c) \cot ^3(b x+c)}{3 b}-\frac{\cos (a-c) \cot (b x+c)}{b}-\frac{\sin (a-c) \csc ^4(b x+c)}{4 b}",1,"((3*Cos[a] + Cos[a - c]*(-4*Cos[c + 2*b*x] + Cos[3*c + 4*b*x]))*Csc[c/2]*Csc[c + b*x]^4*Sec[c/2])/(24*b)","A",1
200,1,79,94,1.1603443,"\int \csc ^6(c+b x) \sin (a+b x) \, dx","Integrate[Csc[c + b*x]^6*Sin[a + b*x],x]","-\frac{2 \csc ^5(b x+c) (5 \cos (a-c) (14 \sin (2 (b x+c))-3 \sin (4 (b x+c)))+64 \sin (a-c))+480 \cos (a-c) \tanh ^{-1}\left(\cos (c)-\sin (c) \tan \left(\frac{b x}{2}\right)\right)}{640 b}","-\frac{3 \cos (a-c) \tanh ^{-1}(\cos (b x+c))}{8 b}-\frac{\sin (a-c) \csc ^5(b x+c)}{5 b}-\frac{\cos (a-c) \cot (b x+c) \csc ^3(b x+c)}{4 b}-\frac{3 \cos (a-c) \cot (b x+c) \csc (b x+c)}{8 b}",1,"-1/640*(480*ArcTanh[Cos[c] - Sin[c]*Tan[(b*x)/2]]*Cos[a - c] + 2*Csc[c + b*x]^5*(64*Sin[a - c] + 5*Cos[a - c]*(14*Sin[2*(c + b*x)] - 3*Sin[4*(c + b*x)])))/b","A",1
201,0,0,410,0.4030345,"\int \sin ^2(a+b x) \sin ^n(c+d x) \, dx","Integrate[Sin[a + b*x]^2*Sin[c + d*x]^n,x]","\int \sin ^2(a+b x) \sin ^n(c+d x) \, dx","-\frac{i 2^{-n-2} \left(i e^{-i (c+d x)}-i e^{i (c+d x)}\right)^n \left(1-e^{2 i c+2 i d x}\right)^{-n} \, _2F_1\left(\frac{1}{2} \left(-\frac{2 b}{d}-n\right),-n;\frac{1}{2} \left(-\frac{2 b}{d}-n+2\right);e^{2 i (c+d x)}\right) \exp (-i (2 a+c n)-i x (2 b+d n)+i n (c+d x))}{2 b+d n}+\frac{i 2^{-n-2} \left(i e^{-i (c+d x)}-i e^{i (c+d x)}\right)^n \left(1-e^{2 i c+2 i d x}\right)^{-n} \, _2F_1\left(\frac{1}{2} \left(\frac{2 b}{d}-n\right),-n;\frac{1}{2} \left(\frac{2 b}{d}-n+2\right);e^{2 i (c+d x)}\right) \exp (i (2 a-c n)+i x (2 b-d n)+i n (c+d x))}{2 b-d n}+\frac{i 2^{-n-1} \left(i e^{-i (c+d x)}-i e^{i (c+d x)}\right)^n \left(1-e^{2 i (c+d x)}\right)^{-n} \, _2F_1\left(-n,-\frac{n}{2};1-\frac{n}{2};e^{2 i (c+d x)}\right)}{d n}",1,"Integrate[Sin[a + b*x]^2*Sin[c + d*x]^n, x]","F",-1
202,1,80,68,0.3423428,"\int \sin ^2(a+b x) \sin (c+d x) \, dx","Integrate[Sin[a + b*x]^2*Sin[c + d*x],x]","-\frac{\cos (2 a+2 b x-c-d x)}{4 (2 b-d)}+\frac{\cos (2 a+x (2 b+d)+c)}{4 (2 b+d)}+\frac{1}{2} \left(\frac{\sin (c) \sin (d x)}{d}-\frac{\cos (c) \cos (d x)}{d}\right)","-\frac{\cos (2 a+x (2 b-d)-c)}{4 (2 b-d)}+\frac{\cos (2 a+x (2 b+d)+c)}{4 (2 b+d)}-\frac{\cos (c+d x)}{2 d}",1,"-1/4*Cos[2*a - c + 2*b*x - d*x]/(2*b - d) + Cos[2*a + c + (2*b + d)*x]/(4*(2*b + d)) + (-((Cos[c]*Cos[d*x])/d) + (Sin[c]*Sin[d*x])/d)/2","A",1
203,1,106,88,0.7610056,"\int \sin ^2(a+b x) \sin ^2(c+d x) \, dx","Integrate[Sin[a + b*x]^2*Sin[c + d*x]^2,x]","\frac{\left(2 d^3-2 b^2 d\right) \sin (2 (a+b x))+b d (b+d) \sin (2 (a+x (b-d)-c))+b (b-d) (d (\sin (2 (a+x (b+d)+c))+4 x (b+d))-2 (b+d) \sin (2 (c+d x)))}{16 b d (b-d) (b+d)}","\frac{\sin (2 (a-c)+2 x (b-d))}{16 (b-d)}+\frac{\sin (2 (a+c)+2 x (b+d))}{16 (b+d)}-\frac{\sin (2 a+2 b x)}{8 b}-\frac{\sin (2 c+2 d x)}{8 d}+\frac{x}{4}",1,"((-2*b^2*d + 2*d^3)*Sin[2*(a + b*x)] + b*d*(b + d)*Sin[2*(a - c + (b - d)*x)] + b*(b - d)*(-2*(b + d)*Sin[2*(c + d*x)] + d*(4*(b + d)*x + Sin[2*(a + c + (b + d)*x)])))/(16*b*(b - d)*d*(b + d))","A",1
204,1,158,144,1.5681039,"\int \sin ^2(a+b x) \sin ^3(c+d x) \, dx","Integrate[Sin[a + b*x]^2*Sin[c + d*x]^3,x]","\frac{1}{48} \left(\frac{3 \cos (2 a+2 b x-3 c-3 d x)}{2 b-3 d}-\frac{9 \cos (2 a+2 b x-c-d x)}{2 b-d}+\frac{9 \cos (2 a+2 b x+c+d x)}{2 b+d}-\frac{3 \cos (2 a+2 b x+3 c+3 d x)}{2 b+3 d}+\frac{18 \sin (c) \sin (d x)}{d}-\frac{2 \sin (3 c) \sin (3 d x)}{d}-\frac{18 \cos (c) \cos (d x)}{d}+\frac{2 \cos (3 c) \cos (3 d x)}{d}\right)","\frac{\cos (2 a+x (2 b-3 d)-3 c)}{16 (2 b-3 d)}-\frac{3 \cos (2 a+x (2 b-d)-c)}{16 (2 b-d)}+\frac{3 \cos (2 a+x (2 b+d)+c)}{16 (2 b+d)}-\frac{\cos (2 a+x (2 b+3 d)+3 c)}{16 (2 b+3 d)}-\frac{3 \cos (c+d x)}{8 d}+\frac{\cos (3 c+3 d x)}{24 d}",1,"((-18*Cos[c]*Cos[d*x])/d + (2*Cos[3*c]*Cos[3*d*x])/d + (3*Cos[2*a - 3*c + 2*b*x - 3*d*x])/(2*b - 3*d) - (9*Cos[2*a - c + 2*b*x - d*x])/(2*b - d) + (9*Cos[2*a + c + 2*b*x + d*x])/(2*b + d) - (3*Cos[2*a + 3*c + 2*b*x + 3*d*x])/(2*b + 3*d) + (18*Sin[c]*Sin[d*x])/d - (2*Sin[3*c]*Sin[3*d*x])/d)/48","A",1
205,0,0,600,0.5794324,"\int \sin ^3(a+b x) \sin ^n(c+d x) \, dx","Integrate[Sin[a + b*x]^3*Sin[c + d*x]^n,x]","\int \sin ^3(a+b x) \sin ^n(c+d x) \, dx","\frac{2^{-n-3} \left(i e^{-i (c+d x)}-i e^{i (c+d x)}\right)^n \left(1-e^{2 i c+2 i d x}\right)^{-n} \, _2F_1\left(\frac{1}{2} \left(\frac{3 b}{d}-n\right),-n;\frac{1}{2} \left(\frac{3 b}{d}-n+2\right);e^{2 i (c+d x)}\right) \exp (i (3 a-c n)+i x (3 b-d n)+i n (c+d x))}{3 b-d n}-\frac{3\ 2^{-n-3} \left(i e^{-i (c+d x)}-i e^{i (c+d x)}\right)^n \left(1-e^{2 i c+2 i d x}\right)^{-n} \, _2F_1\left(-n,\frac{b-d n}{2 d};\frac{1}{2} \left(\frac{b}{d}-n+2\right);e^{2 i (c+d x)}\right) \exp (i (a-c n)+i x (b-d n)+i n (c+d x))}{b-d n}-\frac{3\ 2^{-n-3} \left(i e^{-i (c+d x)}-i e^{i (c+d x)}\right)^n \left(1-e^{2 i c+2 i d x}\right)^{-n} \, _2F_1\left(-n,-\frac{b+d n}{2 d};1-\frac{b+d n}{2 d};e^{2 i (c+d x)}\right) \exp (-i (a+c n)-i x (b+d n)+i n (c+d x))}{b+d n}+\frac{2^{-n-3} \left(i e^{-i (c+d x)}-i e^{i (c+d x)}\right)^n \left(1-e^{2 i c+2 i d x}\right)^{-n} \, _2F_1\left(-n,-\frac{3 b+d n}{2 d};\frac{1}{2} \left(-\frac{3 b}{d}-n+2\right);e^{2 i (c+d x)}\right) \exp (-i (3 a+c n)-i x (3 b+d n)+i n (c+d x))}{3 b+d n}",1,"Integrate[Sin[a + b*x]^3*Sin[c + d*x]^n, x]","F",-1
206,1,91,97,0.5371341,"\int \sin ^3(a+b x) \sin (c+d x) \, dx","Integrate[Sin[a + b*x]^3*Sin[c + d*x],x]","\frac{1}{8} \left(\frac{3 \sin (a+b x-c-d x)}{b-d}-\frac{\sin (3 a+3 b x-c-d x)}{3 b-d}+\frac{\sin (3 a+3 b x+c+d x)}{3 b+d}-\frac{3 \sin (a+x (b+d)+c)}{b+d}\right)","\frac{3 \sin (a+x (b-d)-c)}{8 (b-d)}-\frac{\sin (3 a+x (3 b-d)-c)}{8 (3 b-d)}-\frac{3 \sin (a+x (b+d)+c)}{8 (b+d)}+\frac{\sin (3 a+x (3 b+d)+c)}{8 (3 b+d)}",1,"((3*Sin[a - c + b*x - d*x])/(b - d) - Sin[3*a - c + 3*b*x - d*x]/(3*b - d) + Sin[3*a + c + 3*b*x + d*x]/(3*b + d) - (3*Sin[a + c + (b + d)*x])/(b + d))/8","A",1
207,1,153,138,1.5982147,"\int \sin ^3(a+b x) \sin ^2(c+d x) \, dx","Integrate[Sin[a + b*x]^3*Sin[c + d*x]^2,x]","\frac{1}{48} \left(\frac{9 \cos (a+b x-2 c-2 d x)}{b-2 d}-\frac{3 \cos (3 a+3 b x-2 c-2 d x)}{3 b-2 d}+\frac{9 \cos (a+b x+2 c+2 d x)}{b+2 d}-\frac{3 \cos (3 a+3 b x+2 c+2 d x)}{3 b+2 d}+\frac{18 \sin (a) \sin (b x)}{b}-\frac{2 \sin (3 a) \sin (3 b x)}{b}-\frac{18 \cos (a) \cos (b x)}{b}+\frac{2 \cos (3 a) \cos (3 b x)}{b}\right)","\frac{3 \cos (a+x (b-2 d)-2 c)}{16 (b-2 d)}-\frac{\cos (3 a+x (3 b-2 d)-2 c)}{16 (3 b-2 d)}+\frac{3 \cos (a+x (b+2 d)+2 c)}{16 (b+2 d)}-\frac{\cos (3 a+x (3 b+2 d)+2 c)}{16 (3 b+2 d)}-\frac{3 \cos (a+b x)}{8 b}+\frac{\cos (3 a+3 b x)}{24 b}",1,"((-18*Cos[a]*Cos[b*x])/b + (2*Cos[3*a]*Cos[3*b*x])/b + (9*Cos[a - 2*c + b*x - 2*d*x])/(b - 2*d) - (3*Cos[3*a - 2*c + 3*b*x - 2*d*x])/(3*b - 2*d) + (9*Cos[a + 2*c + b*x + 2*d*x])/(b + 2*d) - (3*Cos[3*a + 2*c + 3*b*x + 2*d*x])/(3*b + 2*d) + (18*Sin[a]*Sin[b*x])/b - (2*Sin[3*a]*Sin[3*b*x])/b)/48","A",1
208,1,177,195,1.6605214,"\int \sin ^3(a+b x) \sin ^3(c+d x) \, dx","Integrate[Sin[a + b*x]^3*Sin[c + d*x]^3,x]","\frac{1}{96} \left(-\frac{9 \sin (a+b x-3 c-3 d x)}{b-3 d}+\frac{27 \sin (a+b x-c-d x)}{b-d}+\frac{\sin (3 (a+b x-c-d x))}{b-d}-\frac{9 \sin (3 a+3 b x-c-d x)}{3 b-d}+\frac{9 \sin (3 a+3 b x+c+d x)}{3 b+d}+\frac{9 \sin (a+b x+3 c+3 d x)}{b+3 d}-\frac{27 \sin (a+x (b+d)+c)}{b+d}-\frac{\sin (3 (a+x (b+d)+c))}{b+d}\right)","-\frac{3 \sin (a+x (b-3 d)-3 c)}{32 (b-3 d)}+\frac{9 \sin (a+x (b-d)-c)}{32 (b-d)}+\frac{\sin (3 (a-c)+3 x (b-d))}{96 (b-d)}-\frac{3 \sin (3 a+x (3 b-d)-c)}{32 (3 b-d)}-\frac{9 \sin (a+x (b+d)+c)}{32 (b+d)}-\frac{\sin (3 (a+c)+3 x (b+d))}{96 (b+d)}+\frac{3 \sin (3 a+x (3 b+d)+c)}{32 (3 b+d)}+\frac{3 \sin (a+x (b+3 d)+3 c)}{32 (b+3 d)}",1,"((-9*Sin[a - 3*c + b*x - 3*d*x])/(b - 3*d) + (27*Sin[a - c + b*x - d*x])/(b - d) + Sin[3*(a - c + b*x - d*x)]/(b - d) - (9*Sin[3*a - c + 3*b*x - d*x])/(3*b - d) + (9*Sin[3*a + c + 3*b*x + d*x])/(3*b + d) + (9*Sin[a + 3*c + b*x + 3*d*x])/(b + 3*d) - (27*Sin[a + c + (b + d)*x])/(b + d) - Sin[3*(a + c + (b + d)*x)]/(b + d))/96","A",1
209,1,202,277,1.0247208,"\int \cos ^n(c+d x) \sin (a+b x) \, dx","Integrate[Cos[c + d*x]^n*Sin[a + b*x],x]","-\frac{2^{-n-1} e^{i (c-b x)} \left(e^{-i (c+d x)} \left(1+e^{2 i (c+d x)}\right)\right)^{n+1} \left(e^{i d x} (\cos (a)-i \sin (a)) (b-d n) \, _2F_1\left(1,\frac{1}{2} \left(-\frac{b}{d}+n+2\right);-\frac{b+d (n-2)}{2 d};-e^{2 i (c+d x)}\right)+(\cos (a)+i \sin (a)) (b+d n) e^{i x (2 b+d)} \, _2F_1\left(1,\frac{b+d (n+2)}{2 d};\frac{1}{2} \left(\frac{b}{d}-n+2\right);-e^{2 i (c+d x)}\right)\right)}{(b-d n) (b+d n)}","-\frac{2^{-n-1} \left(e^{-i (c+d x)}+e^{i (c+d x)}\right)^n \left(1+e^{2 i c+2 i d x}\right)^{-n} \, _2F_1\left(-n,\frac{b-d n}{2 d};\frac{1}{2} \left(\frac{b}{d}-n+2\right);-e^{2 i (c+d x)}\right) \exp (i (a-c n)+i x (b-d n)+i n (c+d x))}{b-d n}-\frac{2^{-n-1} \left(e^{-i (c+d x)}+e^{i (c+d x)}\right)^n \left(1+e^{2 i c+2 i d x}\right)^{-n} \, _2F_1\left(-n,-\frac{b+d n}{2 d};1-\frac{b+d n}{2 d};-e^{2 i (c+d x)}\right) \exp (-i (a+c n)-i x (b+d n)+i n (c+d x))}{b+d n}",1,"-((2^(-1 - n)*E^(I*(c - b*x))*((1 + E^((2*I)*(c + d*x)))/E^(I*(c + d*x)))^(1 + n)*(E^(I*d*x)*(b - d*n)*Hypergeometric2F1[1, (2 - b/d + n)/2, -1/2*(b + d*(-2 + n))/d, -E^((2*I)*(c + d*x))]*(Cos[a] - I*Sin[a]) + E^(I*(2*b + d)*x)*(b + d*n)*Hypergeometric2F1[1, (b + d*(2 + n))/(2*d), (2 + b/d - n)/2, -E^((2*I)*(c + d*x))]*(Cos[a] + I*Sin[a])))/((b - d*n)*(b + d*n)))","A",0
210,1,87,91,0.504644,"\int \cos ^3(c+d x) \sin (a+b x) \, dx","Integrate[Cos[c + d*x]^3*Sin[a + b*x],x]","\frac{1}{8} \left(-\frac{\cos (a+b x-3 c-3 d x)}{b-3 d}-\frac{3 \cos (a+b x-c-d x)}{b-d}-\frac{\cos (a+b x+3 c+3 d x)}{b+3 d}-\frac{3 \cos (a+x (b+d)+c)}{b+d}\right)","-\frac{\cos (a+x (b-3 d)-3 c)}{8 (b-3 d)}-\frac{3 \cos (a+x (b-d)-c)}{8 (b-d)}-\frac{3 \cos (a+x (b+d)+c)}{8 (b+d)}-\frac{\cos (a+x (b+3 d)+3 c)}{8 (b+3 d)}",1,"(-(Cos[a - 3*c + b*x - 3*d*x]/(b - 3*d)) - (3*Cos[a - c + b*x - d*x])/(b - d) - Cos[a + 3*c + b*x + 3*d*x]/(b + 3*d) - (3*Cos[a + c + (b + d)*x])/(b + d))/8","A",1
211,1,71,62,0.7776759,"\int \cos ^2(c+d x) \sin (a+b x) \, dx","Integrate[Cos[c + d*x]^2*Sin[a + b*x],x]","\frac{1}{4} \left(-\frac{\cos (a+b x-2 c-2 d x)}{b-2 d}-\frac{\cos (a+b x+2 c+2 d x)}{b+2 d}+\frac{2 \sin (a) \sin (b x)}{b}-\frac{2 \cos (a) \cos (b x)}{b}\right)","-\frac{\cos (a+x (b-2 d)-2 c)}{4 (b-2 d)}-\frac{\cos (a+x (b+2 d)+2 c)}{4 (b+2 d)}-\frac{\cos (a+b x)}{2 b}",1,"((-2*Cos[a]*Cos[b*x])/b - Cos[a - 2*c + b*x - 2*d*x]/(b - 2*d) - Cos[a + 2*c + b*x + 2*d*x]/(b + 2*d) + (2*Sin[a]*Sin[b*x])/b)/4","A",1
212,1,43,43,0.1872946,"\int \cos (c+d x) \sin (a+b x) \, dx","Integrate[Cos[c + d*x]*Sin[a + b*x],x]","-\frac{\cos (a+x (b-d)-c)}{2 (b-d)}-\frac{\cos (a+x (b+d)+c)}{2 (b+d)}","-\frac{\cos (a+x (b-d)-c)}{2 (b-d)}-\frac{\cos (a+x (b+d)+c)}{2 (b+d)}",1,"-1/2*Cos[a - c + (b - d)*x]/(b - d) - Cos[a + c + (b + d)*x]/(2*(b + d))","A",1
213,1,27,27,0.1498066,"\int \sec (c+b x) \sin (a+b x) \, dx","Integrate[Sec[c + b*x]*Sin[a + b*x],x]","x \sin (a-c)-\frac{\cos (a-c) \log (\cos (b x+c))}{b}","x \sin (a-c)-\frac{\cos (a-c) \log (\cos (b x+c))}{b}",1,"-((Cos[a - c]*Log[Cos[c + b*x]])/b) + x*Sin[a - c]","A",1
214,1,88,34,0.0998429,"\int \sec ^2(c+b x) \sin (a+b x) \, dx","Integrate[Sec[c + b*x]^2*Sin[a + b*x],x]","\frac{\cos (a-c) \sec (b x+c)}{b}-\frac{2 i \sin (a-c) \tan ^{-1}\left(\frac{(\sin (c)+i \cos (c)) \left(\sin (c) \cos \left(\frac{b x}{2}\right)+\cos (c) \sin \left(\frac{b x}{2}\right)\right)}{\cos (c) \cos \left(\frac{b x}{2}\right)-i \sin (c) \cos \left(\frac{b x}{2}\right)}\right)}{b}","\frac{\sin (a-c) \tanh ^{-1}(\sin (b x+c))}{b}+\frac{\cos (a-c) \sec (b x+c)}{b}",1,"(Cos[a - c]*Sec[c + b*x])/b - ((2*I)*ArcTan[((I*Cos[c] + Sin[c])*(Cos[(b*x)/2]*Sin[c] + Cos[c]*Sin[(b*x)/2]))/(Cos[c]*Cos[(b*x)/2] - I*Cos[(b*x)/2]*Sin[c])]*Sin[a - c])/b","C",1
215,1,34,38,0.1764702,"\int \sec ^3(c+b x) \sin (a+b x) \, dx","Integrate[Sec[c + b*x]^3*Sin[a + b*x],x]","\frac{\sec (c) \sec ^2(b x+c) (\sin (a-c) \sin (2 b x+c)+\cos (a))}{2 b}","\frac{\sin (a-c) \tan (b x+c)}{b}+\frac{\cos (a-c) \sec ^2(b x+c)}{2 b}",1,"(Sec[c]*Sec[c + b*x]^2*(Cos[a] + Sin[a - c]*Sin[c + 2*b*x]))/(2*b)","A",1
216,1,64,67,0.4427068,"\int \sec ^4(c+b x) \sin (a+b x) \, dx","Integrate[Sec[c + b*x]^4*Sin[a + b*x],x]","\frac{\sec ^3(b x+c) (3 \sin (a-c) \sin (2 (b x+c))+4 \cos (a-c))+12 \sin (a-c) \tanh ^{-1}\left(\cos (c) \tan \left(\frac{b x}{2}\right)+\sin (c)\right)}{12 b}","\frac{\sin (a-c) \tanh ^{-1}(\sin (b x+c))}{2 b}+\frac{\cos (a-c) \sec ^3(b x+c)}{3 b}+\frac{\sin (a-c) \tan (b x+c) \sec (b x+c)}{2 b}",1,"(12*ArcTanh[Sin[c] + Cos[c]*Tan[(b*x)/2]]*Sin[a - c] + Sec[c + b*x]^3*(4*Cos[a - c] + 3*Sin[a - c]*Sin[2*(c + b*x)]))/(12*b)","A",1
217,1,48,59,0.3428134,"\int \sec ^5(c+b x) \sin (a+b x) \, dx","Integrate[Sec[c + b*x]^5*Sin[a + b*x],x]","\frac{\sec (c) \sec ^4(b x+c) (\sin (a-c) (4 \sin (2 b x+c)+\sin (4 b x+3 c))+3 \cos (a))}{12 b}","\frac{\sin (a-c) \tan ^3(b x+c)}{3 b}+\frac{\sin (a-c) \tan (b x+c)}{b}+\frac{\cos (a-c) \sec ^4(b x+c)}{4 b}",1,"(Sec[c]*Sec[c + b*x]^4*(3*Cos[a] + Sin[a - c]*(4*Sin[c + 2*b*x] + Sin[3*c + 4*b*x])))/(12*b)","A",1
218,1,78,94,0.9861915,"\int \sec ^6(c+b x) \sin (a+b x) \, dx","Integrate[Sec[c + b*x]^6*Sin[a + b*x],x]","\frac{2 \sec ^5(b x+c) (5 \sin (a-c) (14 \sin (2 (b x+c))+3 \sin (4 (b x+c)))+64 \cos (a-c))+480 \sin (a-c) \tanh ^{-1}\left(\cos (c) \tan \left(\frac{b x}{2}\right)+\sin (c)\right)}{640 b}","\frac{3 \sin (a-c) \tanh ^{-1}(\sin (b x+c))}{8 b}+\frac{\cos (a-c) \sec ^5(b x+c)}{5 b}+\frac{\sin (a-c) \tan (b x+c) \sec ^3(b x+c)}{4 b}+\frac{3 \sin (a-c) \tan (b x+c) \sec (b x+c)}{8 b}",1,"(480*ArcTanh[Sin[c] + Cos[c]*Tan[(b*x)/2]]*Sin[a - c] + 2*Sec[c + b*x]^5*(64*Cos[a - c] + 5*Sin[a - c]*(14*Sin[2*(c + b*x)] + 3*Sin[4*(c + b*x)])))/(640*b)","A",1
219,1,242,386,1.832227,"\int \cos ^n(c+d x) \sin ^2(a+b x) \, dx","Integrate[Cos[c + d*x]^n*Sin[a + b*x]^2,x]","-\frac{i 2^{-n-2} \left(e^{-i (c+d x)} \left(1+e^{2 i (c+d x)}\right)\right)^{n+1} e^{i (c+d x)-2 i (a+b x)} \left(e^{2 i (a+b x)} (2 b+d n) \left(d n e^{2 i (a+b x)} \, _2F_1\left(1,\frac{b}{d}+\frac{n}{2}+1;\frac{b}{d}-\frac{n}{2}+1;-e^{2 i (c+d x)}\right)+2 (2 b-d n) \, _2F_1\left(1,\frac{n+2}{2};1-\frac{n}{2};-e^{2 i (c+d x)}\right)\right)+d n (d n-2 b) \, _2F_1\left(1,-\frac{b}{d}+\frac{n}{2}+1;-\frac{b}{d}-\frac{n}{2}+1;-e^{2 i (c+d x)}\right)\right)}{d^3 n^3-4 b^2 d n}","-\frac{i 2^{-n-2} \left(e^{-i (c+d x)}+e^{i (c+d x)}\right)^n \left(1+e^{2 i c+2 i d x}\right)^{-n} \, _2F_1\left(\frac{1}{2} \left(-\frac{2 b}{d}-n\right),-n;\frac{1}{2} \left(-\frac{2 b}{d}-n+2\right);-e^{2 i (c+d x)}\right) \exp (-i (2 a+c n)-i x (2 b+d n)+i n (c+d x))}{2 b+d n}+\frac{i 2^{-n-2} \left(e^{-i (c+d x)}+e^{i (c+d x)}\right)^n \left(1+e^{2 i c+2 i d x}\right)^{-n} \, _2F_1\left(\frac{1}{2} \left(\frac{2 b}{d}-n\right),-n;\frac{1}{2} \left(\frac{2 b}{d}-n+2\right);-e^{2 i (c+d x)}\right) \exp (i (2 a-c n)+i x (2 b-d n)+i n (c+d x))}{2 b-d n}+\frac{i 2^{-n-1} \left(e^{-i (c+d x)}+e^{i (c+d x)}\right)^n \left(1+e^{2 i (c+d x)}\right)^{-n} \, _2F_1\left(-n,-\frac{n}{2};1-\frac{n}{2};-e^{2 i (c+d x)}\right)}{d n}",1,"((-I)*2^(-2 - n)*E^((-2*I)*(a + b*x) + I*(c + d*x))*((1 + E^((2*I)*(c + d*x)))/E^(I*(c + d*x)))^(1 + n)*(d*n*(-2*b + d*n)*Hypergeometric2F1[1, 1 - b/d + n/2, 1 - b/d - n/2, -E^((2*I)*(c + d*x))] + E^((2*I)*(a + b*x))*(2*b + d*n)*(d*E^((2*I)*(a + b*x))*n*Hypergeometric2F1[1, 1 + b/d + n/2, 1 + b/d - n/2, -E^((2*I)*(c + d*x))] + 2*(2*b - d*n)*Hypergeometric2F1[1, (2 + n)/2, 1 - n/2, -E^((2*I)*(c + d*x))])))/(-4*b^2*d*n + d^3*n^3)","A",0
220,1,76,68,0.792986,"\int \cos (c+d x) \sin ^2(a+b x) \, dx","Integrate[Cos[c + d*x]*Sin[a + b*x]^2,x]","\frac{1}{4} \left(-\frac{\sin (2 a+2 b x-c-d x)}{2 b-d}-\frac{\sin (2 a+2 b x+c+d x)}{2 b+d}+\frac{2 \sin (c) \cos (d x)}{d}+\frac{2 \cos (c) \sin (d x)}{d}\right)","-\frac{\sin (2 a+x (2 b-d)-c)}{4 (2 b-d)}-\frac{\sin (2 a+x (2 b+d)+c)}{4 (2 b+d)}+\frac{\sin (c+d x)}{2 d}",1,"((2*Cos[d*x]*Sin[c])/d + (2*Cos[c]*Sin[d*x])/d - Sin[2*a - c + 2*b*x - d*x]/(2*b - d) - Sin[2*a + c + 2*b*x + d*x]/(2*b + d))/4","A",1
221,1,108,88,0.7301647,"\int \cos ^2(c+d x) \sin ^2(a+b x) \, dx","Integrate[Cos[c + d*x]^2*Sin[a + b*x]^2,x]","\frac{\left(2 d^3-2 b^2 d\right) \sin (2 (a+b x))-b d (b+d) \sin (2 (a+x (b-d)-c))+b (b-d) (-d \sin (2 (a+x (b+d)+c))+2 (b+d) \sin (2 (c+d x))+4 d x (b+d))}{16 b d (b-d) (b+d)}","-\frac{\sin (2 (a-c)+2 x (b-d))}{16 (b-d)}-\frac{\sin (2 (a+c)+2 x (b+d))}{16 (b+d)}-\frac{\sin (2 a+2 b x)}{8 b}+\frac{\sin (2 c+2 d x)}{8 d}+\frac{x}{4}",1,"((-2*b^2*d + 2*d^3)*Sin[2*(a + b*x)] - b*d*(b + d)*Sin[2*(a - c + (b - d)*x)] + b*(b - d)*(4*d*(b + d)*x + 2*(b + d)*Sin[2*(c + d*x)] - d*Sin[2*(a + c + (b + d)*x)]))/(16*b*(b - d)*d*(b + d))","A",1
222,1,158,144,1.7593523,"\int \cos ^3(c+d x) \sin ^2(a+b x) \, dx","Integrate[Cos[c + d*x]^3*Sin[a + b*x]^2,x]","\frac{1}{48} \left(-\frac{3 \sin (2 a+2 b x-3 c-3 d x)}{2 b-3 d}-\frac{9 \sin (2 a+2 b x-c-d x)}{2 b-d}-\frac{9 \sin (2 a+2 b x+c+d x)}{2 b+d}-\frac{3 \sin (2 a+2 b x+3 c+3 d x)}{2 b+3 d}+\frac{18 \sin (c) \cos (d x)}{d}+\frac{2 \sin (3 c) \cos (3 d x)}{d}+\frac{18 \cos (c) \sin (d x)}{d}+\frac{2 \cos (3 c) \sin (3 d x)}{d}\right)","-\frac{\sin (2 a+x (2 b-3 d)-3 c)}{16 (2 b-3 d)}-\frac{3 \sin (2 a+x (2 b-d)-c)}{16 (2 b-d)}-\frac{3 \sin (2 a+x (2 b+d)+c)}{16 (2 b+d)}-\frac{\sin (2 a+x (2 b+3 d)+3 c)}{16 (2 b+3 d)}+\frac{3 \sin (c+d x)}{8 d}+\frac{\sin (3 c+3 d x)}{24 d}",1,"((18*Cos[d*x]*Sin[c])/d + (2*Cos[3*d*x]*Sin[3*c])/d + (18*Cos[c]*Sin[d*x])/d + (2*Cos[3*c]*Sin[3*d*x])/d - (3*Sin[2*a - 3*c + 2*b*x - 3*d*x])/(2*b - 3*d) - (9*Sin[2*a - c + 2*b*x - d*x])/(2*b - d) - (9*Sin[2*a + c + 2*b*x + d*x])/(2*b + d) - (3*Sin[2*a + 3*c + 2*b*x + 3*d*x])/(2*b + 3*d))/48","A",1
223,1,329,568,23.938035,"\int \cos ^n(c+d x) \sin ^3(a+b x) \, dx","Integrate[Cos[c + d*x]^n*Sin[a + b*x]^3,x]","2^{-n-3} \left(e^{-i (c+d x)} \left(1+e^{2 i (c+d x)}\right)\right)^{n+1} e^{i (-3 a+c+d (n+1) x)} \left(-\frac{3 e^{2 i a-i x (b+d n)} \, _2F_1\left(1,\frac{1}{2} \left(-\frac{b}{d}+n+2\right);-\frac{b+d (n-2)}{2 d};-e^{2 i (c+d x)}\right)}{b+d n}+e^{i (4 a+b x-d n x)} \left(\frac{e^{2 i (a+b x)} \, _2F_1\left(1,\frac{1}{2} \left(\frac{3 b}{d}+n+2\right);\frac{3 b}{2 d}-\frac{n}{2}+1;-e^{2 i (c+d x)}\right)}{3 b-d n}-\frac{3 \, _2F_1\left(1,\frac{b+d (n+2)}{2 d};\frac{1}{2} \left(\frac{b}{d}-n+2\right);-e^{2 i (c+d x)}\right)}{b-d n}\right)+\frac{e^{-i x (3 b+d n)} \, _2F_1\left(1,\frac{1}{2} \left(-\frac{3 b}{d}+n+2\right);-\frac{3 b}{2 d}-\frac{n}{2}+1;-e^{2 i (c+d x)}\right)}{3 b+d n}\right)","\frac{2^{-n-3} \left(e^{-i (c+d x)}+e^{i (c+d x)}\right)^n \left(1+e^{2 i c+2 i d x}\right)^{-n} \, _2F_1\left(\frac{1}{2} \left(\frac{3 b}{d}-n\right),-n;\frac{1}{2} \left(\frac{3 b}{d}-n+2\right);-e^{2 i (c+d x)}\right) \exp (i (3 a-c n)+i x (3 b-d n)+i n (c+d x))}{3 b-d n}-\frac{3\ 2^{-n-3} \left(e^{-i (c+d x)}+e^{i (c+d x)}\right)^n \left(1+e^{2 i c+2 i d x}\right)^{-n} \, _2F_1\left(-n,\frac{b-d n}{2 d};\frac{1}{2} \left(\frac{b}{d}-n+2\right);-e^{2 i (c+d x)}\right) \exp (i (a-c n)+i x (b-d n)+i n (c+d x))}{b-d n}-\frac{3\ 2^{-n-3} \left(e^{-i (c+d x)}+e^{i (c+d x)}\right)^n \left(1+e^{2 i c+2 i d x}\right)^{-n} \, _2F_1\left(-n,-\frac{b+d n}{2 d};1-\frac{b+d n}{2 d};-e^{2 i (c+d x)}\right) \exp (-i (a+c n)-i x (b+d n)+i n (c+d x))}{b+d n}+\frac{2^{-n-3} \left(e^{-i (c+d x)}+e^{i (c+d x)}\right)^n \left(1+e^{2 i c+2 i d x}\right)^{-n} \, _2F_1\left(-n,-\frac{3 b+d n}{2 d};\frac{1}{2} \left(-\frac{3 b}{d}-n+2\right);-e^{2 i (c+d x)}\right) \exp (-i (3 a+c n)-i x (3 b+d n)+i n (c+d x))}{3 b+d n}",1,"2^(-3 - n)*E^(I*(-3*a + c + d*(1 + n)*x))*((1 + E^((2*I)*(c + d*x)))/E^(I*(c + d*x)))^(1 + n)*(Hypergeometric2F1[1, (2 - (3*b)/d + n)/2, 1 - (3*b)/(2*d) - n/2, -E^((2*I)*(c + d*x))]/(E^(I*(3*b + d*n)*x)*(3*b + d*n)) - (3*E^((2*I)*a - I*(b + d*n)*x)*Hypergeometric2F1[1, (2 - b/d + n)/2, -1/2*(b + d*(-2 + n))/d, -E^((2*I)*(c + d*x))])/(b + d*n) + E^(I*(4*a + b*x - d*n*x))*((E^((2*I)*(a + b*x))*Hypergeometric2F1[1, (2 + (3*b)/d + n)/2, 1 + (3*b)/(2*d) - n/2, -E^((2*I)*(c + d*x))])/(3*b - d*n) - (3*Hypergeometric2F1[1, (b + d*(2 + n))/(2*d), (2 + b/d - n)/2, -E^((2*I)*(c + d*x))])/(b - d*n)))","A",0
224,1,90,97,0.5273589,"\int \cos (c+d x) \sin ^3(a+b x) \, dx","Integrate[Cos[c + d*x]*Sin[a + b*x]^3,x]","\frac{1}{8} \left(-\frac{3 \cos (a+b x-c-d x)}{b-d}+\frac{\cos (3 a+3 b x-c-d x)}{3 b-d}+\frac{\cos (3 a+3 b x+c+d x)}{3 b+d}-\frac{3 \cos (a+x (b+d)+c)}{b+d}\right)","-\frac{3 \cos (a+x (b-d)-c)}{8 (b-d)}+\frac{\cos (3 a+x (3 b-d)-c)}{8 (3 b-d)}-\frac{3 \cos (a+x (b+d)+c)}{8 (b+d)}+\frac{\cos (3 a+x (3 b+d)+c)}{8 (3 b+d)}",1,"((-3*Cos[a - c + b*x - d*x])/(b - d) + Cos[3*a - c + 3*b*x - d*x]/(3*b - d) + Cos[3*a + c + 3*b*x + d*x]/(3*b + d) - (3*Cos[a + c + (b + d)*x])/(b + d))/8","A",1
225,1,153,138,1.5741555,"\int \cos ^2(c+d x) \sin ^3(a+b x) \, dx","Integrate[Cos[c + d*x]^2*Sin[a + b*x]^3,x]","\frac{1}{48} \left(-\frac{9 \cos (a+b x-2 c-2 d x)}{b-2 d}+\frac{3 \cos (3 a+3 b x-2 c-2 d x)}{3 b-2 d}-\frac{9 \cos (a+b x+2 c+2 d x)}{b+2 d}+\frac{3 \cos (3 a+3 b x+2 c+2 d x)}{3 b+2 d}+\frac{18 \sin (a) \sin (b x)}{b}-\frac{2 \sin (3 a) \sin (3 b x)}{b}-\frac{18 \cos (a) \cos (b x)}{b}+\frac{2 \cos (3 a) \cos (3 b x)}{b}\right)","-\frac{3 \cos (a+x (b-2 d)-2 c)}{16 (b-2 d)}+\frac{\cos (3 a+x (3 b-2 d)-2 c)}{16 (3 b-2 d)}-\frac{3 \cos (a+x (b+2 d)+2 c)}{16 (b+2 d)}+\frac{\cos (3 a+x (3 b+2 d)+2 c)}{16 (3 b+2 d)}-\frac{3 \cos (a+b x)}{8 b}+\frac{\cos (3 a+3 b x)}{24 b}",1,"((-18*Cos[a]*Cos[b*x])/b + (2*Cos[3*a]*Cos[3*b*x])/b - (9*Cos[a - 2*c + b*x - 2*d*x])/(b - 2*d) + (3*Cos[3*a - 2*c + 3*b*x - 2*d*x])/(3*b - 2*d) - (9*Cos[a + 2*c + b*x + 2*d*x])/(b + 2*d) + (3*Cos[3*a + 2*c + 3*b*x + 2*d*x])/(3*b + 2*d) + (18*Sin[a]*Sin[b*x])/b - (2*Sin[3*a]*Sin[3*b*x])/b)/48","A",1
226,1,176,195,1.5457334,"\int \cos ^3(c+d x) \sin ^3(a+b x) \, dx","Integrate[Cos[c + d*x]^3*Sin[a + b*x]^3,x]","\frac{1}{96} \left(-\frac{9 \cos (a+b x-3 c-3 d x)}{b-3 d}-\frac{27 \cos (a+b x-c-d x)}{b-d}+\frac{\cos (3 (a+b x-c-d x))}{b-d}+\frac{9 \cos (3 a+3 b x-c-d x)}{3 b-d}+\frac{9 \cos (3 a+3 b x+c+d x)}{3 b+d}-\frac{9 \cos (a+b x+3 c+3 d x)}{b+3 d}-\frac{27 \cos (a+x (b+d)+c)}{b+d}+\frac{\cos (3 (a+x (b+d)+c))}{b+d}\right)","-\frac{3 \cos (a+x (b-3 d)-3 c)}{32 (b-3 d)}-\frac{9 \cos (a+x (b-d)-c)}{32 (b-d)}+\frac{\cos (3 (a-c)+3 x (b-d))}{96 (b-d)}+\frac{3 \cos (3 a+x (3 b-d)-c)}{32 (3 b-d)}-\frac{9 \cos (a+x (b+d)+c)}{32 (b+d)}+\frac{\cos (3 (a+c)+3 x (b+d))}{96 (b+d)}+\frac{3 \cos (3 a+x (3 b+d)+c)}{32 (3 b+d)}-\frac{3 \cos (a+x (b+3 d)+3 c)}{32 (b+3 d)}",1,"((-9*Cos[a - 3*c + b*x - 3*d*x])/(b - 3*d) - (27*Cos[a - c + b*x - d*x])/(b - d) + Cos[3*(a - c + b*x - d*x)]/(b - d) + (9*Cos[3*a - c + 3*b*x - d*x])/(3*b - d) + (9*Cos[3*a + c + 3*b*x + d*x])/(3*b + d) - (9*Cos[a + 3*c + b*x + 3*d*x])/(b + 3*d) - (27*Cos[a + c + (b + d)*x])/(b + d) + Cos[3*(a + c + (b + d)*x)]/(b + d))/96","A",1
227,1,58,27,0.1788342,"\int \cos (a+b x) \csc (c+b x) \, dx","Integrate[Cos[a + b*x]*Csc[c + b*x],x]","\frac{-2 b x \sin (a-c)-2 i \cos (a-c) \tan ^{-1}(\tan (b x+c))+\cos (a-c) \left(\log \left(\sin ^2(b x+c)\right)+2 i b x\right)}{2 b}","\frac{\cos (a-c) \log (\sin (b x+c))}{b}-x \sin (a-c)",1,"((-2*I)*ArcTan[Tan[c + b*x]]*Cos[a - c] + Cos[a - c]*((2*I)*b*x + Log[Sin[c + b*x]^2]) - 2*b*x*Sin[a - c])/(2*b)","C",1
228,1,90,35,0.1018561,"\int \cos (a+b x) \csc ^2(c+b x) \, dx","Integrate[Cos[a + b*x]*Csc[c + b*x]^2,x]","-\frac{\cos (a-c) \csc (b x+c)}{b}+\frac{2 i \sin (a-c) \tan ^{-1}\left(\frac{(\cos (c)-i \sin (c)) \left(\cos (c) \cos \left(\frac{b x}{2}\right)-\sin (c) \sin \left(\frac{b x}{2}\right)\right)}{\sin (c) \cos \left(\frac{b x}{2}\right)+i \cos (c) \cos \left(\frac{b x}{2}\right)}\right)}{b}","\frac{\sin (a-c) \tanh ^{-1}(\cos (b x+c))}{b}-\frac{\cos (a-c) \csc (b x+c)}{b}",1,"-((Cos[a - c]*Csc[c + b*x])/b) + ((2*I)*ArcTan[((Cos[c] - I*Sin[c])*(Cos[c]*Cos[(b*x)/2] - Sin[c]*Sin[(b*x)/2]))/(I*Cos[c]*Cos[(b*x)/2] + Cos[(b*x)/2]*Sin[c])]*Sin[a - c])/b","C",1
229,1,35,38,0.2101496,"\int \cos (a+b x) \csc ^3(c+b x) \, dx","Integrate[Cos[a + b*x]*Csc[c + b*x]^3,x]","-\frac{\csc (c) \csc ^2(b x+c) (\sin (a)-\sin (a-c) \cos (2 b x+c))}{2 b}","\frac{\sin (a-c) \cot (b x+c)}{b}-\frac{\cos (a-c) \csc ^2(b x+c)}{2 b}",1,"-1/2*(Csc[c]*Csc[c + b*x]^2*(Sin[a] - Cos[c + 2*b*x]*Sin[a - c]))/b","A",1
230,1,70,72,0.3717924,"\int \sin (a+b x) \tan ^3(c+b x) \, dx","Integrate[Sin[a + b*x]*Tan[c + b*x]^3,x]","\frac{\sec ^2(b x+c) (2 \sin (a-b x-2 c)+\sin (a+3 b x+2 c)+5 \sin (a+b x))-12 \cos (a-c) \tanh ^{-1}\left(\cos (c) \tan \left(\frac{b x}{2}\right)+\sin (c)\right)}{4 b}","\frac{\sin (a-c) \sec (b x+c)}{b}-\frac{3 \cos (a-c) \tanh ^{-1}(\sin (b x+c))}{2 b}+\frac{\cos (a-c) \tan (b x+c) \sec (b x+c)}{2 b}+\frac{\sin (a+b x)}{b}",1,"(-12*ArcTanh[Sin[c] + Cos[c]*Tan[(b*x)/2]]*Cos[a - c] + Sec[c + b*x]^2*(2*Sin[a - 2*c - b*x] + 5*Sin[a + b*x] + Sin[a + 2*c + 3*b*x]))/(4*b)","A",1
231,1,109,44,0.0959056,"\int \sin (a+b x) \tan ^2(c+b x) \, dx","Integrate[Sin[a + b*x]*Tan[c + b*x]^2,x]","\frac{\cos (a-c) \sec (b x+c)}{b}-\frac{2 i \sin (a-c) \tan ^{-1}\left(\frac{(\sin (c)+i \cos (c)) \left(\sin (c) \cos \left(\frac{b x}{2}\right)+\cos (c) \sin \left(\frac{b x}{2}\right)\right)}{\cos (c) \cos \left(\frac{b x}{2}\right)-i \sin (c) \cos \left(\frac{b x}{2}\right)}\right)}{b}-\frac{\sin (a) \sin (b x)}{b}+\frac{\cos (a) \cos (b x)}{b}","\frac{\sin (a-c) \tanh ^{-1}(\sin (b x+c))}{b}+\frac{\cos (a-c) \sec (b x+c)}{b}+\frac{\cos (a+b x)}{b}",1,"(Cos[a]*Cos[b*x])/b + (Cos[a - c]*Sec[c + b*x])/b - ((2*I)*ArcTan[((I*Cos[c] + Sin[c])*(Cos[(b*x)/2]*Sin[c] + Cos[c]*Sin[(b*x)/2]))/(Cos[c]*Cos[(b*x)/2] - I*Cos[(b*x)/2]*Sin[c])]*Sin[a - c])/b - (Sin[a]*Sin[b*x])/b","C",1
232,1,94,29,0.0482075,"\int \sin (a+b x) \tan (c+b x) \, dx","Integrate[Sin[a + b*x]*Tan[c + b*x],x]","-\frac{2 i \cos (a-c) \tan ^{-1}\left(\frac{(\sin (c)+i \cos (c)) \left(\sin (c) \cos \left(\frac{b x}{2}\right)+\cos (c) \sin \left(\frac{b x}{2}\right)\right)}{\cos (c) \cos \left(\frac{b x}{2}\right)-i \sin (c) \cos \left(\frac{b x}{2}\right)}\right)}{b}-\frac{\sin (a) \cos (b x)}{b}-\frac{\cos (a) \sin (b x)}{b}","\frac{\cos (a-c) \tanh ^{-1}(\sin (b x+c))}{b}-\frac{\sin (a+b x)}{b}",1,"((-2*I)*ArcTan[((I*Cos[c] + Sin[c])*(Cos[(b*x)/2]*Sin[c] + Cos[c]*Sin[(b*x)/2]))/(Cos[c]*Cos[(b*x)/2] - I*Cos[(b*x)/2]*Sin[c])]*Cos[a - c])/b - (Cos[b*x]*Sin[a])/b - (Cos[a]*Sin[b*x])/b","C",1
233,1,93,29,0.0574794,"\int \cot (c+b x) \sin (a+b x) \, dx","Integrate[Cot[c + b*x]*Sin[a + b*x],x]","-\frac{2 i \sin (a-c) \tan ^{-1}\left(\frac{(\cos (c)-i \sin (c)) \left(\cos (c) \cos \left(\frac{b x}{2}\right)-\sin (c) \sin \left(\frac{b x}{2}\right)\right)}{\sin (c) \cos \left(\frac{b x}{2}\right)+i \cos (c) \cos \left(\frac{b x}{2}\right)}\right)}{b}+\frac{\sin (a) \cos (b x)}{b}+\frac{\cos (a) \sin (b x)}{b}","\frac{\sin (a+b x)}{b}-\frac{\sin (a-c) \tanh ^{-1}(\cos (b x+c))}{b}",1,"(Cos[b*x]*Sin[a])/b - ((2*I)*ArcTan[((Cos[c] - I*Sin[c])*(Cos[c]*Cos[(b*x)/2] - Sin[c]*Sin[(b*x)/2]))/(I*Cos[c]*Cos[(b*x)/2] + Cos[(b*x)/2]*Sin[c])]*Sin[a - c])/b + (Cos[a]*Sin[b*x])/b","C",1
234,1,111,46,0.096172,"\int \cot ^2(c+b x) \sin (a+b x) \, dx","Integrate[Cot[c + b*x]^2*Sin[a + b*x],x]","-\frac{\sin (a-c) \csc (b x+c)}{b}-\frac{2 i \cos (a-c) \tan ^{-1}\left(\frac{(\cos (c)-i \sin (c)) \left(\cos (c) \cos \left(\frac{b x}{2}\right)-\sin (c) \sin \left(\frac{b x}{2}\right)\right)}{\sin (c) \cos \left(\frac{b x}{2}\right)+i \cos (c) \cos \left(\frac{b x}{2}\right)}\right)}{b}-\frac{\sin (a) \sin (b x)}{b}+\frac{\cos (a) \cos (b x)}{b}","-\frac{\cos (a-c) \tanh ^{-1}(\cos (b x+c))}{b}-\frac{\sin (a-c) \csc (b x+c)}{b}+\frac{\cos (a+b x)}{b}",1,"((-2*I)*ArcTan[((Cos[c] - I*Sin[c])*(Cos[c]*Cos[(b*x)/2] - Sin[c]*Sin[(b*x)/2]))/(I*Cos[c]*Cos[(b*x)/2] + Cos[(b*x)/2]*Sin[c])]*Cos[a - c])/b + (Cos[a]*Cos[b*x])/b - (Csc[c + b*x]*Sin[a - c])/b - (Sin[a]*Sin[b*x])/b","C",1
235,1,71,74,0.3737402,"\int \cot ^3(c+b x) \sin (a+b x) \, dx","Integrate[Cot[c + b*x]^3*Sin[a + b*x],x]","\frac{\csc ^2(b x+c) (2 \sin (a-b x-2 c)+\sin (a+3 b x+2 c)-5 \sin (a+b x))+12 \sin (a-c) \tanh ^{-1}\left(\cos (c)-\sin (c) \tan \left(\frac{b x}{2}\right)\right)}{4 b}","-\frac{\cos (a-c) \csc (b x+c)}{b}+\frac{3 \sin (a-c) \tanh ^{-1}(\cos (b x+c))}{2 b}-\frac{\sin (a-c) \cot (b x+c) \csc (b x+c)}{2 b}-\frac{\sin (a+b x)}{b}",1,"(12*ArcTanh[Cos[c] - Sin[c]*Tan[(b*x)/2]]*Sin[a - c] + Csc[c + b*x]^2*(2*Sin[a - 2*c - b*x] - 5*Sin[a + b*x] + Sin[a + 2*c + 3*b*x]))/(4*b)","A",1
236,1,116,143,1.8058612,"\int \sin (a+b x) \tan (c+d x) \, dx","Integrate[Sin[a + b*x]*Tan[c + d*x],x]","-\frac{i e^{-i (a+b x)} \left(2 e^{2 i (a+b x)} \, _2F_1\left(1,\frac{b}{2 d};\frac{b}{2 d}+1;-e^{2 i (c+d x)}\right)-e^{2 i (a+b x)}+2 \, _2F_1\left(1,-\frac{b}{2 d};1-\frac{b}{2 d};-e^{2 i (c+d x)}\right)-1\right)}{2 b}","-\frac{i e^{-i (a+b x)} \, _2F_1\left(1,-\frac{b}{2 d};1-\frac{b}{2 d};-e^{2 i (c+d x)}\right)}{b}-\frac{i e^{i (a+b x)} \, _2F_1\left(1,\frac{b}{2 d};\frac{b}{2 d}+1;-e^{2 i (c+d x)}\right)}{b}+\frac{i e^{-i (a+b x)}}{2 b}+\frac{i e^{i (a+b x)}}{2 b}",1,"((-1/2*I)*(-1 - E^((2*I)*(a + b*x)) + 2*Hypergeometric2F1[1, -1/2*b/d, 1 - b/(2*d), -E^((2*I)*(c + d*x))] + 2*E^((2*I)*(a + b*x))*Hypergeometric2F1[1, b/(2*d), 1 + b/(2*d), -E^((2*I)*(c + d*x))]))/(b*E^(I*(a + b*x)))","A",1
237,1,260,139,3.6154015,"\int \cot (c+d x) \sin (a+b x) \, dx","Integrate[Cot[c + d*x]*Sin[a + b*x],x]","\frac{-\frac{i e^{-i (a+b x-2 c)} \left(b e^{2 i d x} \, _2F_1\left(1,1-\frac{b}{2 d};2-\frac{b}{2 d};e^{2 i (c+d x)}\right)-(b-2 d) \, _2F_1\left(1,-\frac{b}{2 d};1-\frac{b}{2 d};e^{2 i (c+d x)}\right)\right)}{\left(-1+e^{2 i c}\right) (b-2 d)}-\frac{i e^{i (a+b x+2 c)} \left(b e^{2 i d x} \, _2F_1\left(1,\frac{b}{2 d}+1;\frac{b}{2 d}+2;e^{2 i (c+d x)}\right)-(b+2 d) \, _2F_1\left(1,\frac{b}{2 d};\frac{b}{2 d}+1;e^{2 i (c+d x)}\right)\right)}{\left(-1+e^{2 i c}\right) (b+2 d)}-\cos (a) \cot (c) \cos (b x)+\sin (a) \cot (c) \sin (b x)}{b}","\frac{i e^{-i (a+b x)} \, _2F_1\left(1,-\frac{b}{2 d};1-\frac{b}{2 d};e^{2 i (c+d x)}\right)}{b}+\frac{i e^{i (a+b x)} \, _2F_1\left(1,\frac{b}{2 d};\frac{b}{2 d}+1;e^{2 i (c+d x)}\right)}{b}-\frac{i e^{-i (a+b x)}}{2 b}-\frac{i e^{i (a+b x)}}{2 b}",1,"(-(Cos[a]*Cos[b*x]*Cot[c]) - (I*(b*E^((2*I)*d*x)*Hypergeometric2F1[1, 1 - b/(2*d), 2 - b/(2*d), E^((2*I)*(c + d*x))] - (b - 2*d)*Hypergeometric2F1[1, -1/2*b/d, 1 - b/(2*d), E^((2*I)*(c + d*x))]))/((b - 2*d)*E^(I*(a - 2*c + b*x))*(-1 + E^((2*I)*c))) - (I*E^(I*(a + 2*c + b*x))*(b*E^((2*I)*d*x)*Hypergeometric2F1[1, 1 + b/(2*d), 2 + b/(2*d), E^((2*I)*(c + d*x))] - (b + 2*d)*Hypergeometric2F1[1, b/(2*d), 1 + b/(2*d), E^((2*I)*(c + d*x))]))/((b + 2*d)*(-1 + E^((2*I)*c))) + Cot[c]*Sin[a]*Sin[b*x])/b","A",1
238,1,85,91,0.517773,"\int \cos (a+b x) \cos ^3(c+d x) \, dx","Integrate[Cos[a + b*x]*Cos[c + d*x]^3,x]","\frac{1}{8} \left(\frac{\sin (a+b x-3 c-3 d x)}{b-3 d}+\frac{3 \sin (a+b x-c-d x)}{b-d}+\frac{\sin (a+b x+3 c+3 d x)}{b+3 d}+\frac{3 \sin (a+x (b+d)+c)}{b+d}\right)","\frac{\sin (a+x (b-3 d)-3 c)}{8 (b-3 d)}+\frac{3 \sin (a+x (b-d)-c)}{8 (b-d)}+\frac{3 \sin (a+x (b+d)+c)}{8 (b+d)}+\frac{\sin (a+x (b+3 d)+3 c)}{8 (b+3 d)}",1,"(Sin[a - 3*c + b*x - 3*d*x]/(b - 3*d) + (3*Sin[a - c + b*x - d*x])/(b - d) + Sin[a + 3*c + b*x + 3*d*x]/(b + 3*d) + (3*Sin[a + c + (b + d)*x])/(b + d))/8","A",1
239,1,69,62,0.755652,"\int \cos (a+b x) \cos ^2(c+d x) \, dx","Integrate[Cos[a + b*x]*Cos[c + d*x]^2,x]","\frac{1}{4} \left(\frac{\sin (a+b x-2 c-2 d x)}{b-2 d}+\frac{\sin (a+b x+2 c+2 d x)}{b+2 d}+\frac{2 \sin (a) \cos (b x)}{b}+\frac{2 \cos (a) \sin (b x)}{b}\right)","\frac{\sin (a+x (b-2 d)-2 c)}{4 (b-2 d)}+\frac{\sin (a+x (b+2 d)+2 c)}{4 (b+2 d)}+\frac{\sin (a+b x)}{2 b}",1,"((2*Cos[b*x]*Sin[a])/b + (2*Cos[a]*Sin[b*x])/b + Sin[a - 2*c + b*x - 2*d*x]/(b - 2*d) + Sin[a + 2*c + b*x + 2*d*x]/(b + 2*d))/4","A",1
240,1,43,43,0.1839711,"\int \cos (a+b x) \cos (c+d x) \, dx","Integrate[Cos[a + b*x]*Cos[c + d*x],x]","\frac{\sin (a+x (b-d)-c)}{2 (b-d)}+\frac{\sin (a+x (b+d)+c)}{2 (b+d)}","\frac{\sin (a+x (b-d)-c)}{2 (b-d)}+\frac{\sin (a+x (b+d)+c)}{2 (b+d)}",1,"Sin[a - c + (b - d)*x]/(2*(b - d)) + Sin[a + c + (b + d)*x]/(2*(b + d))","A",1
241,1,26,26,0.1214073,"\int \cos (a+b x) \sec (c+b x) \, dx","Integrate[Cos[a + b*x]*Sec[c + b*x],x]","\frac{\sin (a-c) \log (\cos (b x+c))}{b}+x \cos (a-c)","\frac{\sin (a-c) \log (\cos (b x+c))}{b}+x \cos (a-c)",1,"x*Cos[a - c] + (Log[Cos[c + b*x]]*Sin[a - c])/b","A",1
242,1,89,35,0.089976,"\int \cos (a+b x) \sec ^2(c+b x) \, dx","Integrate[Cos[a + b*x]*Sec[c + b*x]^2,x]","-\frac{\sin (a-c) \sec (b x+c)}{b}-\frac{2 i \cos (a-c) \tan ^{-1}\left(\frac{(\sin (c)+i \cos (c)) \left(\sin (c) \cos \left(\frac{b x}{2}\right)+\cos (c) \sin \left(\frac{b x}{2}\right)\right)}{\cos (c) \cos \left(\frac{b x}{2}\right)-i \sin (c) \cos \left(\frac{b x}{2}\right)}\right)}{b}","\frac{\cos (a-c) \tanh ^{-1}(\sin (b x+c))}{b}-\frac{\sin (a-c) \sec (b x+c)}{b}",1,"((-2*I)*ArcTan[((I*Cos[c] + Sin[c])*(Cos[(b*x)/2]*Sin[c] + Cos[c]*Sin[(b*x)/2]))/(Cos[c]*Cos[(b*x)/2] - I*Cos[(b*x)/2]*Sin[c])]*Cos[a - c])/b - (Sec[c + b*x]*Sin[a - c])/b","C",1
243,1,35,38,0.1969316,"\int \cos (a+b x) \sec ^3(c+b x) \, dx","Integrate[Cos[a + b*x]*Sec[c + b*x]^3,x]","-\frac{\sec (c) \sec ^2(b x+c) (\sin (a)-\cos (a-c) \sin (2 b x+c))}{2 b}","\frac{\cos (a-c) \tan (b x+c)}{b}-\frac{\sin (a-c) \sec ^2(b x+c)}{2 b}",1,"-1/2*(Sec[c]*Sec[c + b*x]^2*(Sin[a] - Cos[a - c]*Sin[c + 2*b*x]))/b","A",1
244,1,158,144,1.5737195,"\int \cos ^2(a+b x) \cos ^3(c+d x) \, dx","Integrate[Cos[a + b*x]^2*Cos[c + d*x]^3,x]","\frac{1}{48} \left(\frac{3 \sin (2 a+2 b x-3 c-3 d x)}{2 b-3 d}+\frac{9 \sin (2 a+2 b x-c-d x)}{2 b-d}+\frac{9 \sin (2 a+2 b x+c+d x)}{2 b+d}+\frac{3 \sin (2 a+2 b x+3 c+3 d x)}{2 b+3 d}+\frac{18 \sin (c) \cos (d x)}{d}+\frac{2 \sin (3 c) \cos (3 d x)}{d}+\frac{18 \cos (c) \sin (d x)}{d}+\frac{2 \cos (3 c) \sin (3 d x)}{d}\right)","\frac{\sin (2 a+x (2 b-3 d)-3 c)}{16 (2 b-3 d)}+\frac{3 \sin (2 a+x (2 b-d)-c)}{16 (2 b-d)}+\frac{3 \sin (2 a+x (2 b+d)+c)}{16 (2 b+d)}+\frac{\sin (2 a+x (2 b+3 d)+3 c)}{16 (2 b+3 d)}+\frac{3 \sin (c+d x)}{8 d}+\frac{\sin (3 c+3 d x)}{24 d}",1,"((18*Cos[d*x]*Sin[c])/d + (2*Cos[3*d*x]*Sin[3*c])/d + (18*Cos[c]*Sin[d*x])/d + (2*Cos[3*c]*Sin[3*d*x])/d + (3*Sin[2*a - 3*c + 2*b*x - 3*d*x])/(2*b - 3*d) + (9*Sin[2*a - c + 2*b*x - d*x])/(2*b - d) + (9*Sin[2*a + c + 2*b*x + d*x])/(2*b + d) + (3*Sin[2*a + 3*c + 2*b*x + 3*d*x])/(2*b + 3*d))/48","A",1
245,1,105,88,0.723456,"\int \cos ^2(a+b x) \cos ^2(c+d x) \, dx","Integrate[Cos[a + b*x]^2*Cos[c + d*x]^2,x]","\frac{2 d \left(b^2-d^2\right) \sin (2 (a+b x))+b d (b+d) \sin (2 (a+x (b-d)-c))+b (b-d) (d (\sin (2 (a+x (b+d)+c))+4 x (b+d))+2 (b+d) \sin (2 (c+d x)))}{16 b d (b-d) (b+d)}","\frac{\sin (2 (a-c)+2 x (b-d))}{16 (b-d)}+\frac{\sin (2 (a+c)+2 x (b+d))}{16 (b+d)}+\frac{\sin (2 a+2 b x)}{8 b}+\frac{\sin (2 c+2 d x)}{8 d}+\frac{x}{4}",1,"(2*d*(b^2 - d^2)*Sin[2*(a + b*x)] + b*d*(b + d)*Sin[2*(a - c + (b - d)*x)] + b*(b - d)*(2*(b + d)*Sin[2*(c + d*x)] + d*(4*(b + d)*x + Sin[2*(a + c + (b + d)*x)])))/(16*b*(b - d)*d*(b + d))","A",1
246,1,176,195,1.6369585,"\int \cos ^3(a+b x) \cos ^3(c+d x) \, dx","Integrate[Cos[a + b*x]^3*Cos[c + d*x]^3,x]","\frac{1}{96} \left(\frac{9 \sin (a+b x-3 c-3 d x)}{b-3 d}+\frac{27 \sin (a+b x-c-d x)}{b-d}+\frac{\sin (3 (a+b x-c-d x))}{b-d}+\frac{9 \sin (3 a+3 b x-c-d x)}{3 b-d}+\frac{9 \sin (3 a+3 b x+c+d x)}{3 b+d}+\frac{9 \sin (a+b x+3 c+3 d x)}{b+3 d}+\frac{27 \sin (a+x (b+d)+c)}{b+d}+\frac{\sin (3 (a+x (b+d)+c))}{b+d}\right)","\frac{3 \sin (a+x (b-3 d)-3 c)}{32 (b-3 d)}+\frac{9 \sin (a+x (b-d)-c)}{32 (b-d)}+\frac{\sin (3 (a-c)+3 x (b-d))}{96 (b-d)}+\frac{3 \sin (3 a+x (3 b-d)-c)}{32 (3 b-d)}+\frac{9 \sin (a+x (b+d)+c)}{32 (b+d)}+\frac{\sin (3 (a+c)+3 x (b+d))}{96 (b+d)}+\frac{3 \sin (3 a+x (3 b+d)+c)}{32 (3 b+d)}+\frac{3 \sin (a+x (b+3 d)+3 c)}{32 (b+3 d)}",1,"((9*Sin[a - 3*c + b*x - 3*d*x])/(b - 3*d) + (27*Sin[a - c + b*x - d*x])/(b - d) + Sin[3*(a - c + b*x - d*x)]/(b - d) + (9*Sin[3*a - c + 3*b*x - d*x])/(3*b - d) + (9*Sin[3*a + c + 3*b*x + d*x])/(3*b + d) + (9*Sin[a + 3*c + b*x + 3*d*x])/(b + 3*d) + (27*Sin[a + c + (b + d)*x])/(b + d) + Sin[3*(a + c + (b + d)*x)]/(b + d))/96","A",1
247,1,70,72,0.3718522,"\int \cos (a+b x) \tan ^3(c+b x) \, dx","Integrate[Cos[a + b*x]*Tan[c + b*x]^3,x]","\frac{\sec ^2(b x+c) (2 \cos (a-b x-2 c)+\cos (a+3 b x+2 c)+5 \cos (a+b x))+12 \sin (a-c) \tanh ^{-1}\left(\cos (c) \tan \left(\frac{b x}{2}\right)+\sin (c)\right)}{4 b}","\frac{3 \sin (a-c) \tanh ^{-1}(\sin (b x+c))}{2 b}+\frac{\cos (a-c) \sec (b x+c)}{b}-\frac{\sin (a-c) \tan (b x+c) \sec (b x+c)}{2 b}+\frac{\cos (a+b x)}{b}",1,"((2*Cos[a - 2*c - b*x] + 5*Cos[a + b*x] + Cos[a + 2*c + 3*b*x])*Sec[c + b*x]^2 + 12*ArcTanh[Sin[c] + Cos[c]*Tan[(b*x)/2]]*Sin[a - c])/(4*b)","A",1
248,1,111,46,0.0942387,"\int \cos (a+b x) \tan ^2(c+b x) \, dx","Integrate[Cos[a + b*x]*Tan[c + b*x]^2,x]","-\frac{\sin (a-c) \sec (b x+c)}{b}-\frac{2 i \cos (a-c) \tan ^{-1}\left(\frac{(\sin (c)+i \cos (c)) \left(\sin (c) \cos \left(\frac{b x}{2}\right)+\cos (c) \sin \left(\frac{b x}{2}\right)\right)}{\cos (c) \cos \left(\frac{b x}{2}\right)-i \sin (c) \cos \left(\frac{b x}{2}\right)}\right)}{b}-\frac{\sin (a) \cos (b x)}{b}-\frac{\cos (a) \sin (b x)}{b}","-\frac{\sin (a-c) \sec (b x+c)}{b}+\frac{\cos (a-c) \tanh ^{-1}(\sin (b x+c))}{b}-\frac{\sin (a+b x)}{b}",1,"((-2*I)*ArcTan[((I*Cos[c] + Sin[c])*(Cos[(b*x)/2]*Sin[c] + Cos[c]*Sin[(b*x)/2]))/(Cos[c]*Cos[(b*x)/2] - I*Cos[(b*x)/2]*Sin[c])]*Cos[a - c])/b - (Cos[b*x]*Sin[a])/b - (Sec[c + b*x]*Sin[a - c])/b - (Cos[a]*Sin[b*x])/b","C",1
249,1,93,30,0.0531554,"\int \cos (a+b x) \tan (c+b x) \, dx","Integrate[Cos[a + b*x]*Tan[c + b*x],x]","\frac{2 i \sin (a-c) \tan ^{-1}\left(\frac{(\sin (c)+i \cos (c)) \left(\sin (c) \cos \left(\frac{b x}{2}\right)+\cos (c) \sin \left(\frac{b x}{2}\right)\right)}{\cos (c) \cos \left(\frac{b x}{2}\right)-i \sin (c) \cos \left(\frac{b x}{2}\right)}\right)}{b}+\frac{\sin (a) \sin (b x)}{b}-\frac{\cos (a) \cos (b x)}{b}","-\frac{\sin (a-c) \tanh ^{-1}(\sin (b x+c))}{b}-\frac{\cos (a+b x)}{b}",1,"-((Cos[a]*Cos[b*x])/b) + ((2*I)*ArcTan[((I*Cos[c] + Sin[c])*(Cos[(b*x)/2]*Sin[c] + Cos[c]*Sin[(b*x)/2]))/(Cos[c]*Cos[(b*x)/2] - I*Cos[(b*x)/2]*Sin[c])]*Sin[a - c])/b + (Sin[a]*Sin[b*x])/b","C",1
250,1,94,29,0.0520242,"\int \cos (a+b x) \cot (c+b x) \, dx","Integrate[Cos[a + b*x]*Cot[c + b*x],x]","-\frac{2 i \cos (a-c) \tan ^{-1}\left(\frac{(\cos (c)-i \sin (c)) \left(\cos (c) \cos \left(\frac{b x}{2}\right)-\sin (c) \sin \left(\frac{b x}{2}\right)\right)}{\sin (c) \cos \left(\frac{b x}{2}\right)+i \cos (c) \cos \left(\frac{b x}{2}\right)}\right)}{b}-\frac{\sin (a) \sin (b x)}{b}+\frac{\cos (a) \cos (b x)}{b}","\frac{\cos (a+b x)}{b}-\frac{\cos (a-c) \tanh ^{-1}(\cos (b x+c))}{b}",1,"((-2*I)*ArcTan[((Cos[c] - I*Sin[c])*(Cos[c]*Cos[(b*x)/2] - Sin[c]*Sin[(b*x)/2]))/(I*Cos[c]*Cos[(b*x)/2] + Cos[(b*x)/2]*Sin[c])]*Cos[a - c])/b + (Cos[a]*Cos[b*x])/b - (Sin[a]*Sin[b*x])/b","C",1
251,1,112,46,0.1005651,"\int \cos (a+b x) \cot ^2(c+b x) \, dx","Integrate[Cos[a + b*x]*Cot[c + b*x]^2,x]","-\frac{\cos (a-c) \csc (b x+c)}{b}+\frac{2 i \sin (a-c) \tan ^{-1}\left(\frac{(\cos (c)-i \sin (c)) \left(\cos (c) \cos \left(\frac{b x}{2}\right)-\sin (c) \sin \left(\frac{b x}{2}\right)\right)}{\sin (c) \cos \left(\frac{b x}{2}\right)+i \cos (c) \cos \left(\frac{b x}{2}\right)}\right)}{b}-\frac{\sin (a) \cos (b x)}{b}-\frac{\cos (a) \sin (b x)}{b}","-\frac{\cos (a-c) \csc (b x+c)}{b}+\frac{\sin (a-c) \tanh ^{-1}(\cos (b x+c))}{b}-\frac{\sin (a+b x)}{b}",1,"-((Cos[a - c]*Csc[c + b*x])/b) - (Cos[b*x]*Sin[a])/b + ((2*I)*ArcTan[((Cos[c] - I*Sin[c])*(Cos[c]*Cos[(b*x)/2] - Sin[c]*Sin[(b*x)/2]))/(I*Cos[c]*Cos[(b*x)/2] + Cos[(b*x)/2]*Sin[c])]*Sin[a - c])/b - (Cos[a]*Sin[b*x])/b","C",1
252,1,71,73,0.3420606,"\int \cos (a+b x) \cot ^3(c+b x) \, dx","Integrate[Cos[a + b*x]*Cot[c + b*x]^3,x]","\frac{\csc ^2(b x+c) (2 \cos (a-b x-2 c)+\cos (a+3 b x+2 c)-5 \cos (a+b x))+12 \cos (a-c) \tanh ^{-1}\left(\cos (c)-\sin (c) \tan \left(\frac{b x}{2}\right)\right)}{4 b}","\frac{3 \cos (a-c) \tanh ^{-1}(\cos (b x+c))}{2 b}+\frac{\sin (a-c) \csc (b x+c)}{b}-\frac{\cos (a-c) \cot (b x+c) \csc (b x+c)}{2 b}-\frac{\cos (a+b x)}{b}",1,"(12*ArcTanh[Cos[c] - Sin[c]*Tan[(b*x)/2]]*Cos[a - c] + (2*Cos[a - 2*c - b*x] - 5*Cos[a + b*x] + Cos[a + 2*c + 3*b*x])*Csc[c + b*x]^2)/(4*b)","A",1
253,1,114,134,1.6597675,"\int \cos (a+b x) \tan (c+d x) \, dx","Integrate[Cos[a + b*x]*Tan[c + d*x],x]","\frac{e^{-i (a+b x)} \left(2 e^{2 i (a+b x)} \, _2F_1\left(1,\frac{b}{2 d};\frac{b}{2 d}+1;-e^{2 i (c+d x)}\right)-e^{2 i (a+b x)}-2 \, _2F_1\left(1,-\frac{b}{2 d};1-\frac{b}{2 d};-e^{2 i (c+d x)}\right)+1\right)}{2 b}","-\frac{e^{-i (a+b x)} \, _2F_1\left(1,-\frac{b}{2 d};1-\frac{b}{2 d};-e^{2 i (c+d x)}\right)}{b}+\frac{e^{i (a+b x)} \, _2F_1\left(1,\frac{b}{2 d};\frac{b}{2 d}+1;-e^{2 i (c+d x)}\right)}{b}+\frac{e^{-i (a+b x)}}{2 b}-\frac{e^{i (a+b x)}}{2 b}",1,"(1 - E^((2*I)*(a + b*x)) - 2*Hypergeometric2F1[1, -1/2*b/d, 1 - b/(2*d), -E^((2*I)*(c + d*x))] + 2*E^((2*I)*(a + b*x))*Hypergeometric2F1[1, b/(2*d), 1 + b/(2*d), -E^((2*I)*(c + d*x))])/(2*b*E^(I*(a + b*x)))","A",1
254,1,108,130,1.79493,"\int \cos (a+b x) \cot (c+d x) \, dx","Integrate[Cos[a + b*x]*Cot[c + d*x],x]","\frac{e^{-i (a+b x)} \left(-2 e^{2 i (a+b x)} \, _2F_1\left(1,\frac{b}{2 d};\frac{b}{2 d}+1;e^{2 i (c+d x)}\right)+e^{2 i (a+b x)}+2 \, _2F_1\left(1,-\frac{b}{2 d};1-\frac{b}{2 d};e^{2 i (c+d x)}\right)-1\right)}{2 b}","\frac{e^{-i (a+b x)} \, _2F_1\left(1,-\frac{b}{2 d};1-\frac{b}{2 d};e^{2 i (c+d x)}\right)}{b}-\frac{e^{i (a+b x)} \, _2F_1\left(1,\frac{b}{2 d};\frac{b}{2 d}+1;e^{2 i (c+d x)}\right)}{b}-\frac{e^{-i (a+b x)}}{2 b}+\frac{e^{i (a+b x)}}{2 b}",1,"(-1 + E^((2*I)*(a + b*x)) + 2*Hypergeometric2F1[1, -1/2*b/d, 1 - b/(2*d), E^((2*I)*(c + d*x))] - 2*E^((2*I)*(a + b*x))*Hypergeometric2F1[1, b/(2*d), 1 + b/(2*d), E^((2*I)*(c + d*x))])/(2*b*E^(I*(a + b*x)))","A",1