1,1,87,0,0.0555053,"\int \cos ^7(c+d x) (a+a \sin (c+d x)) \, dx","Int[Cos[c + d*x]^7*(a + a*Sin[c + d*x]),x]","-\frac{(a \sin (c+d x)+a)^8}{8 a^7 d}+\frac{6 (a \sin (c+d x)+a)^7}{7 a^6 d}-\frac{2 (a \sin (c+d x)+a)^6}{a^5 d}+\frac{8 (a \sin (c+d x)+a)^5}{5 a^4 d}","-\frac{(a \sin (c+d x)+a)^8}{8 a^7 d}+\frac{6 (a \sin (c+d x)+a)^7}{7 a^6 d}-\frac{2 (a \sin (c+d x)+a)^6}{a^5 d}+\frac{8 (a \sin (c+d x)+a)^5}{5 a^4 d}",1,"(8*(a + a*Sin[c + d*x])^5)/(5*a^4*d) - (2*(a + a*Sin[c + d*x])^6)/(a^5*d) + (6*(a + a*Sin[c + d*x])^7)/(7*a^6*d) - (a + a*Sin[c + d*x])^8/(8*a^7*d)","A",3,2,19,0.1053,1,"{2667, 43}"
2,1,87,0,0.0610808,"\int \cos ^6(c+d x) (a+a \sin (c+d x)) \, dx","Int[Cos[c + d*x]^6*(a + a*Sin[c + d*x]),x]","-\frac{a \cos ^7(c+d x)}{7 d}+\frac{a \sin (c+d x) \cos ^5(c+d x)}{6 d}+\frac{5 a \sin (c+d x) \cos ^3(c+d x)}{24 d}+\frac{5 a \sin (c+d x) \cos (c+d x)}{16 d}+\frac{5 a x}{16}","-\frac{a \cos ^7(c+d x)}{7 d}+\frac{a \sin (c+d x) \cos ^5(c+d x)}{6 d}+\frac{5 a \sin (c+d x) \cos ^3(c+d x)}{24 d}+\frac{5 a \sin (c+d x) \cos (c+d x)}{16 d}+\frac{5 a x}{16}",1,"(5*a*x)/16 - (a*Cos[c + d*x]^7)/(7*d) + (5*a*Cos[c + d*x]*Sin[c + d*x])/(16*d) + (5*a*Cos[c + d*x]^3*Sin[c + d*x])/(24*d) + (a*Cos[c + d*x]^5*Sin[c + d*x])/(6*d)","A",5,3,19,0.1579,1,"{2669, 2635, 8}"
3,1,64,0,0.0444546,"\int \cos ^5(c+d x) (a+a \sin (c+d x)) \, dx","Int[Cos[c + d*x]^5*(a + a*Sin[c + d*x]),x]","\frac{(a \sin (c+d x)+a)^6}{6 a^5 d}-\frac{4 (a \sin (c+d x)+a)^5}{5 a^4 d}+\frac{(a \sin (c+d x)+a)^4}{a^3 d}","\frac{(a \sin (c+d x)+a)^6}{6 a^5 d}-\frac{4 (a \sin (c+d x)+a)^5}{5 a^4 d}+\frac{(a \sin (c+d x)+a)^4}{a^3 d}",1,"(a + a*Sin[c + d*x])^4/(a^3*d) - (4*(a + a*Sin[c + d*x])^5)/(5*a^4*d) + (a + a*Sin[c + d*x])^6/(6*a^5*d)","A",3,2,19,0.1053,1,"{2667, 43}"
4,1,65,0,0.0453441,"\int \cos ^4(c+d x) (a+a \sin (c+d x)) \, dx","Int[Cos[c + d*x]^4*(a + a*Sin[c + d*x]),x]","-\frac{a \cos ^5(c+d x)}{5 d}+\frac{a \sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac{3 a \sin (c+d x) \cos (c+d x)}{8 d}+\frac{3 a x}{8}","-\frac{a \cos ^5(c+d x)}{5 d}+\frac{a \sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac{3 a \sin (c+d x) \cos (c+d x)}{8 d}+\frac{3 a x}{8}",1,"(3*a*x)/8 - (a*Cos[c + d*x]^5)/(5*d) + (3*a*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (a*Cos[c + d*x]^3*Sin[c + d*x])/(4*d)","A",4,3,19,0.1579,1,"{2669, 2635, 8}"
5,1,45,0,0.0365742,"\int \cos ^3(c+d x) (a+a \sin (c+d x)) \, dx","Int[Cos[c + d*x]^3*(a + a*Sin[c + d*x]),x]","\frac{2 (a \sin (c+d x)+a)^3}{3 a^2 d}-\frac{(a \sin (c+d x)+a)^4}{4 a^3 d}","\frac{2 (a \sin (c+d x)+a)^3}{3 a^2 d}-\frac{(a \sin (c+d x)+a)^4}{4 a^3 d}",1,"(2*(a + a*Sin[c + d*x])^3)/(3*a^2*d) - (a + a*Sin[c + d*x])^4/(4*a^3*d)","A",3,2,19,0.1053,1,"{2667, 43}"
6,1,43,0,0.0330301,"\int \cos ^2(c+d x) (a+a \sin (c+d x)) \, dx","Int[Cos[c + d*x]^2*(a + a*Sin[c + d*x]),x]","-\frac{a \cos ^3(c+d x)}{3 d}+\frac{a \sin (c+d x) \cos (c+d x)}{2 d}+\frac{a x}{2}","-\frac{a \cos ^3(c+d x)}{3 d}+\frac{a \sin (c+d x) \cos (c+d x)}{2 d}+\frac{a x}{2}",1,"(a*x)/2 - (a*Cos[c + d*x]^3)/(3*d) + (a*Cos[c + d*x]*Sin[c + d*x])/(2*d)","A",3,3,19,0.1579,1,"{2669, 2635, 8}"
7,1,28,0,0.0163306,"\int \cos (c+d x) (a+a \sin (c+d x)) \, dx","Int[Cos[c + d*x]*(a + a*Sin[c + d*x]),x]","\frac{a \sin ^2(c+d x)}{2 d}+\frac{a \sin (c+d x)}{d}","\frac{(a \sin (c+d x)+a)^2}{2 a d}",1,"(a*Sin[c + d*x])/d + (a*Sin[c + d*x]^2)/(2*d)","A",2,1,17,0.05882,1,"{2667}"
8,1,17,0,0.0199603,"\int \sec (c+d x) (a+a \sin (c+d x)) \, dx","Int[Sec[c + d*x]*(a + a*Sin[c + d*x]),x]","-\frac{a \log (1-\sin (c+d x))}{d}","-\frac{a \log (1-\sin (c+d x))}{d}",1,"-((a*Log[1 - Sin[c + d*x]])/d)","A",2,2,17,0.1176,1,"{2667, 31}"
9,1,23,0,0.0321409,"\int \sec ^2(c+d x) (a+a \sin (c+d x)) \, dx","Int[Sec[c + d*x]^2*(a + a*Sin[c + d*x]),x]","\frac{a \tan (c+d x)}{d}+\frac{a \sec (c+d x)}{d}","\frac{a \tan (c+d x)}{d}+\frac{a \sec (c+d x)}{d}",1,"(a*Sec[c + d*x])/d + (a*Tan[c + d*x])/d","A",3,3,19,0.1579,1,"{2669, 3767, 8}"
10,1,39,0,0.0417776,"\int \sec ^3(c+d x) (a+a \sin (c+d x)) \, dx","Int[Sec[c + d*x]^3*(a + a*Sin[c + d*x]),x]","\frac{a^2}{2 d (a-a \sin (c+d x))}+\frac{a \tanh ^{-1}(\sin (c+d x))}{2 d}","\frac{a^2}{2 d (a-a \sin (c+d x))}+\frac{a \tanh ^{-1}(\sin (c+d x))}{2 d}",1,"(a*ArcTanh[Sin[c + d*x]])/(2*d) + a^2/(2*d*(a - a*Sin[c + d*x]))","A",4,3,19,0.1579,1,"{2667, 44, 206}"
11,1,44,0,0.036244,"\int \sec ^4(c+d x) (a+a \sin (c+d x)) \, dx","Int[Sec[c + d*x]^4*(a + a*Sin[c + d*x]),x]","\frac{a \tan ^3(c+d x)}{3 d}+\frac{a \tan (c+d x)}{d}+\frac{a \sec ^3(c+d x)}{3 d}","\frac{a \tan ^3(c+d x)}{3 d}+\frac{a \tan (c+d x)}{d}+\frac{a \sec ^3(c+d x)}{3 d}",1,"(a*Sec[c + d*x]^3)/(3*d) + (a*Tan[c + d*x])/d + (a*Tan[c + d*x]^3)/(3*d)","A",3,2,19,0.1053,1,"{2669, 3767}"
12,1,84,0,0.063694,"\int \sec ^5(c+d x) (a+a \sin (c+d x)) \, dx","Int[Sec[c + d*x]^5*(a + a*Sin[c + d*x]),x]","\frac{a^3}{8 d (a-a \sin (c+d x))^2}+\frac{a^2}{4 d (a-a \sin (c+d x))}-\frac{a^2}{8 d (a \sin (c+d x)+a)}+\frac{3 a \tanh ^{-1}(\sin (c+d x))}{8 d}","\frac{a^3}{8 d (a-a \sin (c+d x))^2}+\frac{a^2}{4 d (a-a \sin (c+d x))}-\frac{a^2}{8 d (a \sin (c+d x)+a)}+\frac{3 a \tanh ^{-1}(\sin (c+d x))}{8 d}",1,"(3*a*ArcTanh[Sin[c + d*x]])/(8*d) + a^3/(8*d*(a - a*Sin[c + d*x])^2) + a^2/(4*d*(a - a*Sin[c + d*x])) - a^2/(8*d*(a + a*Sin[c + d*x]))","A",4,3,19,0.1579,1,"{2667, 44, 206}"
13,1,126,0,0.1067511,"\int \cos ^6(c+d x) (a+a \sin (c+d x))^2 \, dx","Int[Cos[c + d*x]^6*(a + a*Sin[c + d*x])^2,x]","-\frac{9 a^2 \cos ^7(c+d x)}{56 d}-\frac{\cos ^7(c+d x) \left(a^2 \sin (c+d x)+a^2\right)}{8 d}+\frac{3 a^2 \sin (c+d x) \cos ^5(c+d x)}{16 d}+\frac{15 a^2 \sin (c+d x) \cos ^3(c+d x)}{64 d}+\frac{45 a^2 \sin (c+d x) \cos (c+d x)}{128 d}+\frac{45 a^2 x}{128}","-\frac{9 a^2 \cos ^7(c+d x)}{56 d}-\frac{\cos ^7(c+d x) \left(a^2 \sin (c+d x)+a^2\right)}{8 d}+\frac{3 a^2 \sin (c+d x) \cos ^5(c+d x)}{16 d}+\frac{15 a^2 \sin (c+d x) \cos ^3(c+d x)}{64 d}+\frac{45 a^2 \sin (c+d x) \cos (c+d x)}{128 d}+\frac{45 a^2 x}{128}",1,"(45*a^2*x)/128 - (9*a^2*Cos[c + d*x]^7)/(56*d) + (45*a^2*Cos[c + d*x]*Sin[c + d*x])/(128*d) + (15*a^2*Cos[c + d*x]^3*Sin[c + d*x])/(64*d) + (3*a^2*Cos[c + d*x]^5*Sin[c + d*x])/(16*d) - (Cos[c + d*x]^7*(a^2 + a^2*Sin[c + d*x]))/(8*d)","A",6,4,21,0.1905,1,"{2678, 2669, 2635, 8}"
14,1,67,0,0.0610944,"\int \cos ^5(c+d x) (a+a \sin (c+d x))^2 \, dx","Int[Cos[c + d*x]^5*(a + a*Sin[c + d*x])^2,x]","\frac{(a \sin (c+d x)+a)^7}{7 a^5 d}-\frac{2 (a \sin (c+d x)+a)^6}{3 a^4 d}+\frac{4 (a \sin (c+d x)+a)^5}{5 a^3 d}","\frac{(a \sin (c+d x)+a)^7}{7 a^5 d}-\frac{2 (a \sin (c+d x)+a)^6}{3 a^4 d}+\frac{4 (a \sin (c+d x)+a)^5}{5 a^3 d}",1,"(4*(a + a*Sin[c + d*x])^5)/(5*a^3*d) - (2*(a + a*Sin[c + d*x])^6)/(3*a^4*d) + (a + a*Sin[c + d*x])^7/(7*a^5*d)","A",3,2,21,0.09524,1,"{2667, 43}"
15,1,102,0,0.0925856,"\int \cos ^4(c+d x) (a+a \sin (c+d x))^2 \, dx","Int[Cos[c + d*x]^4*(a + a*Sin[c + d*x])^2,x]","-\frac{7 a^2 \cos ^5(c+d x)}{30 d}-\frac{\cos ^5(c+d x) \left(a^2 \sin (c+d x)+a^2\right)}{6 d}+\frac{7 a^2 \sin (c+d x) \cos ^3(c+d x)}{24 d}+\frac{7 a^2 \sin (c+d x) \cos (c+d x)}{16 d}+\frac{7 a^2 x}{16}","-\frac{7 a^2 \cos ^5(c+d x)}{30 d}-\frac{\cos ^5(c+d x) \left(a^2 \sin (c+d x)+a^2\right)}{6 d}+\frac{7 a^2 \sin (c+d x) \cos ^3(c+d x)}{24 d}+\frac{7 a^2 \sin (c+d x) \cos (c+d x)}{16 d}+\frac{7 a^2 x}{16}",1,"(7*a^2*x)/16 - (7*a^2*Cos[c + d*x]^5)/(30*d) + (7*a^2*Cos[c + d*x]*Sin[c + d*x])/(16*d) + (7*a^2*Cos[c + d*x]^3*Sin[c + d*x])/(24*d) - (Cos[c + d*x]^5*(a^2 + a^2*Sin[c + d*x]))/(6*d)","A",5,4,21,0.1905,1,"{2678, 2669, 2635, 8}"
16,1,45,0,0.046204,"\int \cos ^3(c+d x) (a+a \sin (c+d x))^2 \, dx","Int[Cos[c + d*x]^3*(a + a*Sin[c + d*x])^2,x]","\frac{(a \sin (c+d x)+a)^4}{2 a^2 d}-\frac{(a \sin (c+d x)+a)^5}{5 a^3 d}","\frac{(a \sin (c+d x)+a)^4}{2 a^2 d}-\frac{(a \sin (c+d x)+a)^5}{5 a^3 d}",1,"(a + a*Sin[c + d*x])^4/(2*a^2*d) - (a + a*Sin[c + d*x])^5/(5*a^3*d)","A",3,2,21,0.09524,1,"{2667, 43}"
17,1,78,0,0.0885705,"\int \cos ^2(c+d x) (a+a \sin (c+d x))^2 \, dx","Int[Cos[c + d*x]^2*(a + a*Sin[c + d*x])^2,x]","-\frac{5 a^2 \cos ^3(c+d x)}{12 d}-\frac{\cos ^3(c+d x) \left(a^2 \sin (c+d x)+a^2\right)}{4 d}+\frac{5 a^2 \sin (c+d x) \cos (c+d x)}{8 d}+\frac{5 a^2 x}{8}","-\frac{5 a^2 \cos ^3(c+d x)}{12 d}-\frac{\cos ^3(c+d x) \left(a^2 \sin (c+d x)+a^2\right)}{4 d}+\frac{5 a^2 \sin (c+d x) \cos (c+d x)}{8 d}+\frac{5 a^2 x}{8}",1,"(5*a^2*x)/8 - (5*a^2*Cos[c + d*x]^3)/(12*d) + (5*a^2*Cos[c + d*x]*Sin[c + d*x])/(8*d) - (Cos[c + d*x]^3*(a^2 + a^2*Sin[c + d*x]))/(4*d)","A",4,4,21,0.1905,1,"{2678, 2669, 2635, 8}"
18,1,22,0,0.0237991,"\int \cos (c+d x) (a+a \sin (c+d x))^2 \, dx","Int[Cos[c + d*x]*(a + a*Sin[c + d*x])^2,x]","\frac{(a \sin (c+d x)+a)^3}{3 a d}","\frac{(a \sin (c+d x)+a)^3}{3 a d}",1,"(a + a*Sin[c + d*x])^3/(3*a*d)","A",2,2,19,0.1053,1,"{2667, 32}"
19,1,34,0,0.0416049,"\int \sec (c+d x) (a+a \sin (c+d x))^2 \, dx","Int[Sec[c + d*x]*(a + a*Sin[c + d*x])^2,x]","-\frac{a^2 \sin (c+d x)}{d}-\frac{2 a^2 \log (1-\sin (c+d x))}{d}","-\frac{a^2 \sin (c+d x)}{d}-\frac{2 a^2 \log (1-\sin (c+d x))}{d}",1,"(-2*a^2*Log[1 - Sin[c + d*x]])/d - (a^2*Sin[c + d*x])/d","A",3,2,19,0.1053,1,"{2667, 43}"
20,1,38,0,0.0817753,"\int \sec ^2(c+d x) (a+a \sin (c+d x))^2 \, dx","Int[Sec[c + d*x]^2*(a + a*Sin[c + d*x])^2,x]","\frac{2 a^4 \cos (c+d x)}{d \left(a^2-a^2 \sin (c+d x)\right)}-a^2 x","\frac{2 a^4 \cos (c+d x)}{d \left(a^2-a^2 \sin (c+d x)\right)}-a^2 x",1,"-(a^2*x) + (2*a^4*Cos[c + d*x])/(d*(a^2 - a^2*Sin[c + d*x]))","A",3,3,21,0.1429,1,"{2670, 2680, 8}"
21,1,20,0,0.0376204,"\int \sec ^3(c+d x) (a+a \sin (c+d x))^2 \, dx","Int[Sec[c + d*x]^3*(a + a*Sin[c + d*x])^2,x]","\frac{a^3}{d (a-a \sin (c+d x))}","\frac{a^3}{d (a-a \sin (c+d x))}",1,"a^3/(d*(a - a*Sin[c + d*x]))","A",2,2,21,0.09524,1,"{2667, 32}"
22,1,63,0,0.0691645,"\int \sec ^4(c+d x) (a+a \sin (c+d x))^2 \, dx","Int[Sec[c + d*x]^4*(a + a*Sin[c + d*x])^2,x]","\frac{a^4 \cos (c+d x)}{3 d \left(a^2-a^2 \sin (c+d x)\right)}+\frac{a^4 \cos (c+d x)}{3 d (a-a \sin (c+d x))^2}","\frac{a^4 \cos (c+d x)}{3 d \left(a^2-a^2 \sin (c+d x)\right)}+\frac{a^4 \cos (c+d x)}{3 d (a-a \sin (c+d x))^2}",1,"(a^4*Cos[c + d*x])/(3*d*(a - a*Sin[c + d*x])^2) + (a^4*Cos[c + d*x])/(3*d*(a^2 - a^2*Sin[c + d*x]))","A",3,3,21,0.1429,1,"{2670, 2650, 2648}"
23,1,64,0,0.0663574,"\int \sec ^5(c+d x) (a+a \sin (c+d x))^2 \, dx","Int[Sec[c + d*x]^5*(a + a*Sin[c + d*x])^2,x]","\frac{a^4}{4 d (a-a \sin (c+d x))^2}+\frac{a^3}{4 d (a-a \sin (c+d x))}+\frac{a^2 \tanh ^{-1}(\sin (c+d x))}{4 d}","\frac{a^4}{4 d (a-a \sin (c+d x))^2}+\frac{a^3}{4 d (a-a \sin (c+d x))}+\frac{a^2 \tanh ^{-1}(\sin (c+d x))}{4 d}",1,"(a^2*ArcTanh[Sin[c + d*x]])/(4*d) + a^4/(4*d*(a - a*Sin[c + d*x])^2) + a^3/(4*d*(a - a*Sin[c + d*x]))","A",4,3,21,0.1429,1,"{2667, 44, 206}"
24,1,64,0,0.0562102,"\int \sec ^6(c+d x) (a+a \sin (c+d x))^2 \, dx","Int[Sec[c + d*x]^6*(a + a*Sin[c + d*x])^2,x]","\frac{a^2 \tan ^3(c+d x)}{5 d}+\frac{3 a^2 \tan (c+d x)}{5 d}+\frac{2 \sec ^5(c+d x) \left(a^2 \sin (c+d x)+a^2\right)}{5 d}","\frac{a^2 \tan ^3(c+d x)}{5 d}+\frac{3 a^2 \tan (c+d x)}{5 d}+\frac{2 \sec ^5(c+d x) \left(a^2 \sin (c+d x)+a^2\right)}{5 d}",1,"(2*Sec[c + d*x]^5*(a^2 + a^2*Sin[c + d*x]))/(5*d) + (3*a^2*Tan[c + d*x])/(5*d) + (a^2*Tan[c + d*x]^3)/(5*d)","A",3,2,21,0.09524,1,"{2676, 3767}"
25,1,109,0,0.0886078,"\int \sec ^7(c+d x) (a+a \sin (c+d x))^2 \, dx","Int[Sec[c + d*x]^7*(a + a*Sin[c + d*x])^2,x]","\frac{a^5}{12 d (a-a \sin (c+d x))^3}+\frac{a^4}{8 d (a-a \sin (c+d x))^2}+\frac{3 a^3}{16 d (a-a \sin (c+d x))}-\frac{a^3}{16 d (a \sin (c+d x)+a)}+\frac{a^2 \tanh ^{-1}(\sin (c+d x))}{4 d}","\frac{a^5}{12 d (a-a \sin (c+d x))^3}+\frac{a^4}{8 d (a-a \sin (c+d x))^2}+\frac{3 a^3}{16 d (a-a \sin (c+d x))}-\frac{a^3}{16 d (a \sin (c+d x)+a)}+\frac{a^2 \tanh ^{-1}(\sin (c+d x))}{4 d}",1,"(a^2*ArcTanh[Sin[c + d*x]])/(4*d) + a^5/(12*d*(a - a*Sin[c + d*x])^3) + a^4/(8*d*(a - a*Sin[c + d*x])^2) + (3*a^3)/(16*d*(a - a*Sin[c + d*x])) - a^3/(16*d*(a + a*Sin[c + d*x]))","A",4,3,21,0.1429,1,"{2667, 44, 206}"
26,1,82,0,0.058504,"\int \sec ^8(c+d x) (a+a \sin (c+d x))^2 \, dx","Int[Sec[c + d*x]^8*(a + a*Sin[c + d*x])^2,x]","\frac{a^2 \tan ^5(c+d x)}{7 d}+\frac{10 a^2 \tan ^3(c+d x)}{21 d}+\frac{5 a^2 \tan (c+d x)}{7 d}+\frac{2 \sec ^7(c+d x) \left(a^2 \sin (c+d x)+a^2\right)}{7 d}","\frac{a^2 \tan ^5(c+d x)}{7 d}+\frac{10 a^2 \tan ^3(c+d x)}{21 d}+\frac{5 a^2 \tan (c+d x)}{7 d}+\frac{2 \sec ^7(c+d x) \left(a^2 \sin (c+d x)+a^2\right)}{7 d}",1,"(2*Sec[c + d*x]^7*(a^2 + a^2*Sin[c + d*x]))/(7*d) + (5*a^2*Tan[c + d*x])/(7*d) + (10*a^2*Tan[c + d*x]^3)/(21*d) + (a^2*Tan[c + d*x]^5)/(7*d)","A",3,2,21,0.09524,1,"{2676, 3767}"
27,1,154,0,0.1537378,"\int \cos ^6(c+d x) (a+a \sin (c+d x))^3 \, dx","Int[Cos[c + d*x]^6*(a + a*Sin[c + d*x])^3,x]","-\frac{11 a^3 \cos ^7(c+d x)}{56 d}-\frac{11 \cos ^7(c+d x) \left(a^3 \sin (c+d x)+a^3\right)}{72 d}+\frac{11 a^3 \sin (c+d x) \cos ^5(c+d x)}{48 d}+\frac{55 a^3 \sin (c+d x) \cos ^3(c+d x)}{192 d}+\frac{55 a^3 \sin (c+d x) \cos (c+d x)}{128 d}+\frac{55 a^3 x}{128}-\frac{a \cos ^7(c+d x) (a \sin (c+d x)+a)^2}{9 d}","-\frac{11 a^3 \cos ^7(c+d x)}{56 d}-\frac{11 \cos ^7(c+d x) \left(a^3 \sin (c+d x)+a^3\right)}{72 d}+\frac{11 a^3 \sin (c+d x) \cos ^5(c+d x)}{48 d}+\frac{55 a^3 \sin (c+d x) \cos ^3(c+d x)}{192 d}+\frac{55 a^3 \sin (c+d x) \cos (c+d x)}{128 d}+\frac{55 a^3 x}{128}-\frac{a \cos ^7(c+d x) (a \sin (c+d x)+a)^2}{9 d}",1,"(55*a^3*x)/128 - (11*a^3*Cos[c + d*x]^7)/(56*d) + (55*a^3*Cos[c + d*x]*Sin[c + d*x])/(128*d) + (55*a^3*Cos[c + d*x]^3*Sin[c + d*x])/(192*d) + (11*a^3*Cos[c + d*x]^5*Sin[c + d*x])/(48*d) - (a*Cos[c + d*x]^7*(a + a*Sin[c + d*x])^2)/(9*d) - (11*Cos[c + d*x]^7*(a^3 + a^3*Sin[c + d*x]))/(72*d)","A",7,4,21,0.1905,1,"{2678, 2669, 2635, 8}"
28,1,67,0,0.0656316,"\int \cos ^5(c+d x) (a+a \sin (c+d x))^3 \, dx","Int[Cos[c + d*x]^5*(a + a*Sin[c + d*x])^3,x]","\frac{(a \sin (c+d x)+a)^8}{8 a^5 d}-\frac{4 (a \sin (c+d x)+a)^7}{7 a^4 d}+\frac{2 (a \sin (c+d x)+a)^6}{3 a^3 d}","\frac{(a \sin (c+d x)+a)^8}{8 a^5 d}-\frac{4 (a \sin (c+d x)+a)^7}{7 a^4 d}+\frac{2 (a \sin (c+d x)+a)^6}{3 a^3 d}",1,"(2*(a + a*Sin[c + d*x])^6)/(3*a^3*d) - (4*(a + a*Sin[c + d*x])^7)/(7*a^4*d) + (a + a*Sin[c + d*x])^8/(8*a^5*d)","A",3,2,21,0.09524,1,"{2667, 43}"
29,1,130,0,0.1401126,"\int \cos ^4(c+d x) (a+a \sin (c+d x))^3 \, dx","Int[Cos[c + d*x]^4*(a + a*Sin[c + d*x])^3,x]","-\frac{3 a^3 \cos ^5(c+d x)}{10 d}-\frac{3 \cos ^5(c+d x) \left(a^3 \sin (c+d x)+a^3\right)}{14 d}+\frac{3 a^3 \sin (c+d x) \cos ^3(c+d x)}{8 d}+\frac{9 a^3 \sin (c+d x) \cos (c+d x)}{16 d}+\frac{9 a^3 x}{16}-\frac{a \cos ^5(c+d x) (a \sin (c+d x)+a)^2}{7 d}","-\frac{3 a^3 \cos ^5(c+d x)}{10 d}-\frac{3 \cos ^5(c+d x) \left(a^3 \sin (c+d x)+a^3\right)}{14 d}+\frac{3 a^3 \sin (c+d x) \cos ^3(c+d x)}{8 d}+\frac{9 a^3 \sin (c+d x) \cos (c+d x)}{16 d}+\frac{9 a^3 x}{16}-\frac{a \cos ^5(c+d x) (a \sin (c+d x)+a)^2}{7 d}",1,"(9*a^3*x)/16 - (3*a^3*Cos[c + d*x]^5)/(10*d) + (9*a^3*Cos[c + d*x]*Sin[c + d*x])/(16*d) + (3*a^3*Cos[c + d*x]^3*Sin[c + d*x])/(8*d) - (a*Cos[c + d*x]^5*(a + a*Sin[c + d*x])^2)/(7*d) - (3*Cos[c + d*x]^5*(a^3 + a^3*Sin[c + d*x]))/(14*d)","A",6,4,21,0.1905,1,"{2678, 2669, 2635, 8}"
30,1,45,0,0.0471518,"\int \cos ^3(c+d x) (a+a \sin (c+d x))^3 \, dx","Int[Cos[c + d*x]^3*(a + a*Sin[c + d*x])^3,x]","\frac{2 (a \sin (c+d x)+a)^5}{5 a^2 d}-\frac{(a \sin (c+d x)+a)^6}{6 a^3 d}","\frac{2 (a \sin (c+d x)+a)^5}{5 a^2 d}-\frac{(a \sin (c+d x)+a)^6}{6 a^3 d}",1,"(2*(a + a*Sin[c + d*x])^5)/(5*a^2*d) - (a + a*Sin[c + d*x])^6/(6*a^3*d)","A",3,2,21,0.09524,1,"{2667, 43}"
31,1,106,0,0.1238796,"\int \cos ^2(c+d x) (a+a \sin (c+d x))^3 \, dx","Int[Cos[c + d*x]^2*(a + a*Sin[c + d*x])^3,x]","-\frac{7 a^3 \cos ^3(c+d x)}{12 d}-\frac{7 \cos ^3(c+d x) \left(a^3 \sin (c+d x)+a^3\right)}{20 d}+\frac{7 a^3 \sin (c+d x) \cos (c+d x)}{8 d}+\frac{7 a^3 x}{8}-\frac{a \cos ^3(c+d x) (a \sin (c+d x)+a)^2}{5 d}","-\frac{7 a^3 \cos ^3(c+d x)}{12 d}-\frac{7 \cos ^3(c+d x) \left(a^3 \sin (c+d x)+a^3\right)}{20 d}+\frac{7 a^3 \sin (c+d x) \cos (c+d x)}{8 d}+\frac{7 a^3 x}{8}-\frac{a \cos ^3(c+d x) (a \sin (c+d x)+a)^2}{5 d}",1,"(7*a^3*x)/8 - (7*a^3*Cos[c + d*x]^3)/(12*d) + (7*a^3*Cos[c + d*x]*Sin[c + d*x])/(8*d) - (a*Cos[c + d*x]^3*(a + a*Sin[c + d*x])^2)/(5*d) - (7*Cos[c + d*x]^3*(a^3 + a^3*Sin[c + d*x]))/(20*d)","A",5,4,21,0.1905,1,"{2678, 2669, 2635, 8}"
32,1,22,0,0.0253023,"\int \cos (c+d x) (a+a \sin (c+d x))^3 \, dx","Int[Cos[c + d*x]*(a + a*Sin[c + d*x])^3,x]","\frac{(a \sin (c+d x)+a)^4}{4 a d}","\frac{(a \sin (c+d x)+a)^4}{4 a d}",1,"(a + a*Sin[c + d*x])^4/(4*a*d)","A",2,2,19,0.1053,1,"{2667, 32}"
33,1,52,0,0.0469152,"\int \sec (c+d x) (a+a \sin (c+d x))^3 \, dx","Int[Sec[c + d*x]*(a + a*Sin[c + d*x])^3,x]","-\frac{a^3 \sin ^2(c+d x)}{2 d}-\frac{3 a^3 \sin (c+d x)}{d}-\frac{4 a^3 \log (1-\sin (c+d x))}{d}","-\frac{a^3 \sin ^2(c+d x)}{2 d}-\frac{3 a^3 \sin (c+d x)}{d}-\frac{4 a^3 \log (1-\sin (c+d x))}{d}",1,"(-4*a^3*Log[1 - Sin[c + d*x]])/d - (3*a^3*Sin[c + d*x])/d - (a^3*Sin[c + d*x]^2)/(2*d)","A",3,2,19,0.1053,1,"{2667, 43}"
34,1,50,0,0.1362536,"\int \sec ^2(c+d x) (a+a \sin (c+d x))^3 \, dx","Int[Sec[c + d*x]^2*(a + a*Sin[c + d*x])^3,x]","\frac{3 a^3 \cos (c+d x)}{d}+\frac{2 a^5 \cos ^3(c+d x)}{d (a-a \sin (c+d x))^2}-3 a^3 x","\frac{3 a^3 \cos (c+d x)}{d}+\frac{2 a^5 \cos ^3(c+d x)}{d (a-a \sin (c+d x))^2}-3 a^3 x",1,"-3*a^3*x + (3*a^3*Cos[c + d*x])/d + (2*a^5*Cos[c + d*x]^3)/(d*(a - a*Sin[c + d*x])^2)","A",4,4,21,0.1905,1,"{2670, 2680, 2682, 8}"
35,1,40,0,0.0522718,"\int \sec ^3(c+d x) (a+a \sin (c+d x))^3 \, dx","Int[Sec[c + d*x]^3*(a + a*Sin[c + d*x])^3,x]","\frac{2 a^4}{d (a-a \sin (c+d x))}+\frac{a^3 \log (1-\sin (c+d x))}{d}","\frac{2 a^4}{d (a-a \sin (c+d x))}+\frac{a^3 \log (1-\sin (c+d x))}{d}",1,"(a^3*Log[1 - Sin[c + d*x]])/d + (2*a^4)/(d*(a - a*Sin[c + d*x]))","A",3,2,21,0.09524,1,"{2667, 43}"
36,1,31,0,0.0847633,"\int \sec ^4(c+d x) (a+a \sin (c+d x))^3 \, dx","Int[Sec[c + d*x]^4*(a + a*Sin[c + d*x])^3,x]","\frac{a^6 \cos ^3(c+d x)}{3 d (a-a \sin (c+d x))^3}","\frac{a^6 \cos ^3(c+d x)}{3 d (a-a \sin (c+d x))^3}",1,"(a^6*Cos[c + d*x]^3)/(3*d*(a - a*Sin[c + d*x])^3)","A",2,2,21,0.09524,1,"{2670, 2671}"
37,1,23,0,0.0399265,"\int \sec ^5(c+d x) (a+a \sin (c+d x))^3 \, dx","Int[Sec[c + d*x]^5*(a + a*Sin[c + d*x])^3,x]","\frac{a^5}{2 d (a-a \sin (c+d x))^2}","\frac{a^5}{2 d (a-a \sin (c+d x))^2}",1,"a^5/(2*d*(a - a*Sin[c + d*x])^2)","A",2,2,21,0.09524,1,"{2667, 32}"
38,1,92,0,0.0934226,"\int \sec ^6(c+d x) (a+a \sin (c+d x))^3 \, dx","Int[Sec[c + d*x]^6*(a + a*Sin[c + d*x])^3,x]","\frac{2 a^6 \cos (c+d x)}{15 d \left(a^3-a^3 \sin (c+d x)\right)}+\frac{a^6 \cos (c+d x)}{5 d (a-a \sin (c+d x))^3}+\frac{2 a^5 \cos (c+d x)}{15 d (a-a \sin (c+d x))^2}","\frac{2 a^6 \cos (c+d x)}{15 d \left(a^3-a^3 \sin (c+d x)\right)}+\frac{a^6 \cos (c+d x)}{5 d (a-a \sin (c+d x))^3}+\frac{2 a^5 \cos (c+d x)}{15 d (a-a \sin (c+d x))^2}",1,"(a^6*Cos[c + d*x])/(5*d*(a - a*Sin[c + d*x])^3) + (2*a^5*Cos[c + d*x])/(15*d*(a - a*Sin[c + d*x])^2) + (2*a^6*Cos[c + d*x])/(15*d*(a^3 - a^3*Sin[c + d*x]))","A",4,3,21,0.1429,1,"{2670, 2650, 2648}"
39,1,87,0,0.0724086,"\int \sec ^7(c+d x) (a+a \sin (c+d x))^3 \, dx","Int[Sec[c + d*x]^7*(a + a*Sin[c + d*x])^3,x]","\frac{a^6}{6 d (a-a \sin (c+d x))^3}+\frac{a^5}{8 d (a-a \sin (c+d x))^2}+\frac{a^4}{8 d (a-a \sin (c+d x))}+\frac{a^3 \tanh ^{-1}(\sin (c+d x))}{8 d}","\frac{a^6}{6 d (a-a \sin (c+d x))^3}+\frac{a^5}{8 d (a-a \sin (c+d x))^2}+\frac{a^4}{8 d (a-a \sin (c+d x))}+\frac{a^3 \tanh ^{-1}(\sin (c+d x))}{8 d}",1,"(a^3*ArcTanh[Sin[c + d*x]])/(8*d) + a^6/(6*d*(a - a*Sin[c + d*x])^3) + a^5/(8*d*(a - a*Sin[c + d*x])^2) + a^4/(8*d*(a - a*Sin[c + d*x]))","A",4,3,21,0.1429,1,"{2667, 44, 206}"
40,1,99,0,0.083488,"\int \sec ^8(c+d x) (a+a \sin (c+d x))^3 \, dx","Int[Sec[c + d*x]^8*(a + a*Sin[c + d*x])^3,x]","\frac{3 a^3 \tan ^5(c+d x)}{35 d}+\frac{2 a^3 \tan ^3(c+d x)}{7 d}+\frac{3 a^3 \tan (c+d x)}{7 d}+\frac{3 a^3 \sec ^5(c+d x)}{35 d}+\frac{2 a \sec ^7(c+d x) (a \sin (c+d x)+a)^2}{7 d}","\frac{3 a^3 \tan ^5(c+d x)}{35 d}+\frac{2 a^3 \tan ^3(c+d x)}{7 d}+\frac{3 a^3 \tan (c+d x)}{7 d}+\frac{3 a^3 \sec ^5(c+d x)}{35 d}+\frac{2 a \sec ^7(c+d x) (a \sin (c+d x)+a)^2}{7 d}",1,"(3*a^3*Sec[c + d*x]^5)/(35*d) + (2*a*Sec[c + d*x]^7*(a + a*Sin[c + d*x])^2)/(7*d) + (3*a^3*Tan[c + d*x])/(7*d) + (2*a^3*Tan[c + d*x]^3)/(7*d) + (3*a^3*Tan[c + d*x]^5)/(35*d)","A",4,3,21,0.1429,1,"{2676, 2669, 3767}"
41,1,67,0,0.084119,"\int \cos ^5(c+d x) (a+a \sin (c+d x))^8 \, dx","Int[Cos[c + d*x]^5*(a + a*Sin[c + d*x])^8,x]","\frac{(a \sin (c+d x)+a)^{13}}{13 a^5 d}-\frac{(a \sin (c+d x)+a)^{12}}{3 a^4 d}+\frac{4 (a \sin (c+d x)+a)^{11}}{11 a^3 d}","\frac{(a \sin (c+d x)+a)^{13}}{13 a^5 d}-\frac{(a \sin (c+d x)+a)^{12}}{3 a^4 d}+\frac{4 (a \sin (c+d x)+a)^{11}}{11 a^3 d}",1,"(4*(a + a*Sin[c + d*x])^11)/(11*a^3*d) - (a + a*Sin[c + d*x])^12/(3*a^4*d) + (a + a*Sin[c + d*x])^13/(13*a^5*d)","A",3,2,21,0.09524,1,"{2667, 43}"
42,1,286,0,0.4034416,"\int \cos ^4(c+d x) (a+a \sin (c+d x))^8 \, dx","Int[Cos[c + d*x]^4*(a + a*Sin[c + d*x])^8,x]","-\frac{4199 a^8 \cos ^5(c+d x)}{1920 d}+\frac{4199 a^8 \sin (c+d x) \cos ^3(c+d x)}{1536 d}-\frac{323 a^3 \cos ^5(c+d x) (a \sin (c+d x)+a)^5}{1320 d}-\frac{19 a^2 \cos ^5(c+d x) (a \sin (c+d x)+a)^6}{132 d}-\frac{4199 a^2 \cos ^5(c+d x) \left(a^2 \sin (c+d x)+a^2\right)^3}{6336 d}-\frac{323 \cos ^5(c+d x) \left(a^2 \sin (c+d x)+a^2\right)^4}{792 d}-\frac{4199 \cos ^5(c+d x) \left(a^4 \sin (c+d x)+a^4\right)^2}{4032 d}-\frac{4199 \cos ^5(c+d x) \left(a^8 \sin (c+d x)+a^8\right)}{2688 d}+\frac{4199 a^8 \sin (c+d x) \cos (c+d x)}{1024 d}+\frac{4199 a^8 x}{1024}-\frac{a \cos ^5(c+d x) (a \sin (c+d x)+a)^7}{12 d}","-\frac{4199 a^8 \cos ^5(c+d x)}{1920 d}+\frac{4199 a^8 \sin (c+d x) \cos ^3(c+d x)}{1536 d}-\frac{323 a^3 \cos ^5(c+d x) (a \sin (c+d x)+a)^5}{1320 d}-\frac{19 a^2 \cos ^5(c+d x) (a \sin (c+d x)+a)^6}{132 d}-\frac{4199 a^2 \cos ^5(c+d x) \left(a^2 \sin (c+d x)+a^2\right)^3}{6336 d}-\frac{323 \cos ^5(c+d x) \left(a^2 \sin (c+d x)+a^2\right)^4}{792 d}-\frac{4199 \cos ^5(c+d x) \left(a^4 \sin (c+d x)+a^4\right)^2}{4032 d}-\frac{4199 \cos ^5(c+d x) \left(a^8 \sin (c+d x)+a^8\right)}{2688 d}+\frac{4199 a^8 \sin (c+d x) \cos (c+d x)}{1024 d}+\frac{4199 a^8 x}{1024}-\frac{a \cos ^5(c+d x) (a \sin (c+d x)+a)^7}{12 d}",1,"(4199*a^8*x)/1024 - (4199*a^8*Cos[c + d*x]^5)/(1920*d) + (4199*a^8*Cos[c + d*x]*Sin[c + d*x])/(1024*d) + (4199*a^8*Cos[c + d*x]^3*Sin[c + d*x])/(1536*d) - (323*a^3*Cos[c + d*x]^5*(a + a*Sin[c + d*x])^5)/(1320*d) - (19*a^2*Cos[c + d*x]^5*(a + a*Sin[c + d*x])^6)/(132*d) - (a*Cos[c + d*x]^5*(a + a*Sin[c + d*x])^7)/(12*d) - (4199*a^2*Cos[c + d*x]^5*(a^2 + a^2*Sin[c + d*x])^3)/(6336*d) - (323*Cos[c + d*x]^5*(a^2 + a^2*Sin[c + d*x])^4)/(792*d) - (4199*Cos[c + d*x]^5*(a^4 + a^4*Sin[c + d*x])^2)/(4032*d) - (4199*Cos[c + d*x]^5*(a^8 + a^8*Sin[c + d*x]))/(2688*d)","A",11,4,21,0.1905,1,"{2678, 2669, 2635, 8}"
43,1,45,0,0.0471371,"\int \cos ^3(c+d x) (a+a \sin (c+d x))^8 \, dx","Int[Cos[c + d*x]^3*(a + a*Sin[c + d*x])^8,x]","\frac{(a \sin (c+d x)+a)^{10}}{5 a^2 d}-\frac{(a \sin (c+d x)+a)^{11}}{11 a^3 d}","\frac{(a \sin (c+d x)+a)^{10}}{5 a^2 d}-\frac{(a \sin (c+d x)+a)^{11}}{11 a^3 d}",1,"(a + a*Sin[c + d*x])^10/(5*a^2*d) - (a + a*Sin[c + d*x])^11/(11*a^3*d)","A",3,2,21,0.09524,1,"{2667, 43}"
44,1,262,0,0.3743432,"\int \cos ^2(c+d x) (a+a \sin (c+d x))^8 \, dx","Int[Cos[c + d*x]^2*(a + a*Sin[c + d*x])^8,x]","-\frac{2431 a^8 \cos ^3(c+d x)}{384 d}-\frac{17 a^3 \cos ^3(c+d x) (a \sin (c+d x)+a)^5}{48 d}-\frac{17 a^2 \cos ^3(c+d x) (a \sin (c+d x)+a)^6}{90 d}-\frac{2431 a^2 \cos ^3(c+d x) \left(a^2 \sin (c+d x)+a^2\right)^3}{2016 d}-\frac{221 \cos ^3(c+d x) \left(a^2 \sin (c+d x)+a^2\right)^4}{336 d}-\frac{2431 \cos ^3(c+d x) \left(a^4 \sin (c+d x)+a^4\right)^2}{1120 d}-\frac{2431 \cos ^3(c+d x) \left(a^8 \sin (c+d x)+a^8\right)}{640 d}+\frac{2431 a^8 \sin (c+d x) \cos (c+d x)}{256 d}+\frac{2431 a^8 x}{256}-\frac{a \cos ^3(c+d x) (a \sin (c+d x)+a)^7}{10 d}","-\frac{2431 a^8 \cos ^3(c+d x)}{384 d}-\frac{17 a^3 \cos ^3(c+d x) (a \sin (c+d x)+a)^5}{48 d}-\frac{17 a^2 \cos ^3(c+d x) (a \sin (c+d x)+a)^6}{90 d}-\frac{2431 a^2 \cos ^3(c+d x) \left(a^2 \sin (c+d x)+a^2\right)^3}{2016 d}-\frac{221 \cos ^3(c+d x) \left(a^2 \sin (c+d x)+a^2\right)^4}{336 d}-\frac{2431 \cos ^3(c+d x) \left(a^4 \sin (c+d x)+a^4\right)^2}{1120 d}-\frac{2431 \cos ^3(c+d x) \left(a^8 \sin (c+d x)+a^8\right)}{640 d}+\frac{2431 a^8 \sin (c+d x) \cos (c+d x)}{256 d}+\frac{2431 a^8 x}{256}-\frac{a \cos ^3(c+d x) (a \sin (c+d x)+a)^7}{10 d}",1,"(2431*a^8*x)/256 - (2431*a^8*Cos[c + d*x]^3)/(384*d) + (2431*a^8*Cos[c + d*x]*Sin[c + d*x])/(256*d) - (17*a^3*Cos[c + d*x]^3*(a + a*Sin[c + d*x])^5)/(48*d) - (17*a^2*Cos[c + d*x]^3*(a + a*Sin[c + d*x])^6)/(90*d) - (a*Cos[c + d*x]^3*(a + a*Sin[c + d*x])^7)/(10*d) - (2431*a^2*Cos[c + d*x]^3*(a^2 + a^2*Sin[c + d*x])^3)/(2016*d) - (221*Cos[c + d*x]^3*(a^2 + a^2*Sin[c + d*x])^4)/(336*d) - (2431*Cos[c + d*x]^3*(a^4 + a^4*Sin[c + d*x])^2)/(1120*d) - (2431*Cos[c + d*x]^3*(a^8 + a^8*Sin[c + d*x]))/(640*d)","A",10,4,21,0.1905,1,"{2678, 2669, 2635, 8}"
45,1,22,0,0.0243345,"\int \cos (c+d x) (a+a \sin (c+d x))^8 \, dx","Int[Cos[c + d*x]*(a + a*Sin[c + d*x])^8,x]","\frac{(a \sin (c+d x)+a)^9}{9 a d}","\frac{(a \sin (c+d x)+a)^9}{9 a d}",1,"(a + a*Sin[c + d*x])^9/(9*a*d)","A",2,2,19,0.1053,1,"{2667, 32}"
46,1,162,0,0.0768911,"\int \sec (c+d x) (a+a \sin (c+d x))^8 \, dx","Int[Sec[c + d*x]*(a + a*Sin[c + d*x])^8,x]","-\frac{64 a^8 \sin (c+d x)}{d}-\frac{16 a^5 (a \sin (c+d x)+a)^3}{3 d}-\frac{4 a^3 (a \sin (c+d x)+a)^5}{5 d}-\frac{a^2 (a \sin (c+d x)+a)^6}{3 d}-\frac{2 \left(a^2 \sin (c+d x)+a^2\right)^4}{d}-\frac{16 \left(a^4 \sin (c+d x)+a^4\right)^2}{d}-\frac{128 a^8 \log (1-\sin (c+d x))}{d}-\frac{a (a \sin (c+d x)+a)^7}{7 d}","-\frac{64 a^8 \sin (c+d x)}{d}-\frac{16 a^5 (a \sin (c+d x)+a)^3}{3 d}-\frac{4 a^3 (a \sin (c+d x)+a)^5}{5 d}-\frac{a^2 (a \sin (c+d x)+a)^6}{3 d}-\frac{2 \left(a^2 \sin (c+d x)+a^2\right)^4}{d}-\frac{16 \left(a^4 \sin (c+d x)+a^4\right)^2}{d}-\frac{128 a^8 \log (1-\sin (c+d x))}{d}-\frac{a (a \sin (c+d x)+a)^7}{7 d}",1,"(-128*a^8*Log[1 - Sin[c + d*x]])/d - (64*a^8*Sin[c + d*x])/d - (16*a^5*(a + a*Sin[c + d*x])^3)/(3*d) - (4*a^3*(a + a*Sin[c + d*x])^5)/(5*d) - (a^2*(a + a*Sin[c + d*x])^6)/(3*d) - (a*(a + a*Sin[c + d*x])^7)/(7*d) - (2*(a^2 + a^2*Sin[c + d*x])^4)/d - (16*(a^4 + a^4*Sin[c + d*x])^2)/d","A",3,2,19,0.1053,1,"{2667, 43}"
47,1,201,0,0.3419122,"\int \sec ^2(c+d x) (a+a \sin (c+d x))^8 \, dx","Int[Sec[c + d*x]^2*(a + a*Sin[c + d*x])^8,x]","\frac{1001 a^8 \cos ^5(c+d x)}{10 d}+\frac{143 a^{16} \cos ^7(c+d x)}{2 d \left(a^8-a^8 \sin (c+d x)\right)}+\frac{2 a^{15} \cos ^{13}(c+d x)}{d (a-a \sin (c+d x))^7}+\frac{286 a^{14} \cos ^9(c+d x)}{3 d \left(a^2-a^2 \sin (c+d x)\right)^3}+\frac{26 a^{13} \cos ^{11}(c+d x)}{d (a-a \sin (c+d x))^5}-\frac{1001 a^8 \sin (c+d x) \cos ^3(c+d x)}{8 d}-\frac{3003 a^8 \sin (c+d x) \cos (c+d x)}{16 d}-\frac{3003 a^8 x}{16}","\frac{1001 a^8 \cos ^5(c+d x)}{10 d}+\frac{143 a^{16} \cos ^7(c+d x)}{2 d \left(a^8-a^8 \sin (c+d x)\right)}+\frac{2 a^{15} \cos ^{13}(c+d x)}{d (a-a \sin (c+d x))^7}+\frac{286 a^{14} \cos ^9(c+d x)}{3 d \left(a^2-a^2 \sin (c+d x)\right)^3}+\frac{26 a^{13} \cos ^{11}(c+d x)}{d (a-a \sin (c+d x))^5}-\frac{1001 a^8 \sin (c+d x) \cos ^3(c+d x)}{8 d}-\frac{3003 a^8 \sin (c+d x) \cos (c+d x)}{16 d}-\frac{3003 a^8 x}{16}",1,"(-3003*a^8*x)/16 + (1001*a^8*Cos[c + d*x]^5)/(10*d) - (3003*a^8*Cos[c + d*x]*Sin[c + d*x])/(16*d) - (1001*a^8*Cos[c + d*x]^3*Sin[c + d*x])/(8*d) + (2*a^15*Cos[c + d*x]^13)/(d*(a - a*Sin[c + d*x])^7) + (26*a^13*Cos[c + d*x]^11)/(d*(a - a*Sin[c + d*x])^5) + (286*a^14*Cos[c + d*x]^9)/(3*d*(a^2 - a^2*Sin[c + d*x])^3) + (143*a^16*Cos[c + d*x]^7)/(2*d*(a^8 - a^8*Sin[c + d*x]))","A",9,6,21,0.2857,1,"{2670, 2680, 2679, 2682, 2635, 8}"
48,1,121,0,0.0940941,"\int \sec ^3(c+d x) (a+a \sin (c+d x))^8 \, dx","Int[Sec[c + d*x]^3*(a + a*Sin[c + d*x])^8,x]","\frac{a^8 \sin ^5(c+d x)}{5 d}+\frac{2 a^8 \sin ^4(c+d x)}{d}+\frac{10 a^8 \sin ^3(c+d x)}{d}+\frac{36 a^8 \sin ^2(c+d x)}{d}+\frac{64 a^9}{d (a-a \sin (c+d x))}+\frac{129 a^8 \sin (c+d x)}{d}+\frac{192 a^8 \log (1-\sin (c+d x))}{d}","\frac{a^8 \sin ^5(c+d x)}{5 d}+\frac{2 a^8 \sin ^4(c+d x)}{d}+\frac{10 a^8 \sin ^3(c+d x)}{d}+\frac{36 a^8 \sin ^2(c+d x)}{d}+\frac{64 a^9}{d (a-a \sin (c+d x))}+\frac{129 a^8 \sin (c+d x)}{d}+\frac{192 a^8 \log (1-\sin (c+d x))}{d}",1,"(192*a^8*Log[1 - Sin[c + d*x]])/d + (129*a^8*Sin[c + d*x])/d + (36*a^8*Sin[c + d*x]^2)/d + (10*a^8*Sin[c + d*x]^3)/d + (2*a^8*Sin[c + d*x]^4)/d + (a^8*Sin[c + d*x]^5)/(5*d) + (64*a^9)/(d*(a - a*Sin[c + d*x]))","A",3,2,21,0.09524,1,"{2667, 43}"
49,1,179,0,0.3186565,"\int \sec ^4(c+d x) (a+a \sin (c+d x))^8 \, dx","Int[Sec[c + d*x]^4*(a + a*Sin[c + d*x])^8,x]","-\frac{385 a^8 \cos ^3(c+d x)}{4 d}-\frac{231 a^{16} \cos ^5(c+d x)}{4 d \left(a^8-a^8 \sin (c+d x)\right)}+\frac{2 a^{15} \cos ^{11}(c+d x)}{3 d (a-a \sin (c+d x))^7}-\frac{66 a^{14} \cos ^7(c+d x)}{d \left(a^2-a^2 \sin (c+d x)\right)^3}-\frac{22 a^{13} \cos ^9(c+d x)}{3 d (a-a \sin (c+d x))^5}+\frac{1155 a^8 \sin (c+d x) \cos (c+d x)}{8 d}+\frac{1155 a^8 x}{8}","-\frac{385 a^8 \cos ^3(c+d x)}{4 d}-\frac{231 a^{16} \cos ^5(c+d x)}{4 d \left(a^8-a^8 \sin (c+d x)\right)}+\frac{2 a^{15} \cos ^{11}(c+d x)}{3 d (a-a \sin (c+d x))^7}-\frac{66 a^{14} \cos ^7(c+d x)}{d \left(a^2-a^2 \sin (c+d x)\right)^3}-\frac{22 a^{13} \cos ^9(c+d x)}{3 d (a-a \sin (c+d x))^5}+\frac{1155 a^8 \sin (c+d x) \cos (c+d x)}{8 d}+\frac{1155 a^8 x}{8}",1,"(1155*a^8*x)/8 - (385*a^8*Cos[c + d*x]^3)/(4*d) + (1155*a^8*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (2*a^15*Cos[c + d*x]^11)/(3*d*(a - a*Sin[c + d*x])^7) - (22*a^13*Cos[c + d*x]^9)/(3*d*(a - a*Sin[c + d*x])^5) - (66*a^14*Cos[c + d*x]^7)/(d*(a^2 - a^2*Sin[c + d*x])^3) - (231*a^16*Cos[c + d*x]^5)/(4*d*(a^8 - a^8*Sin[c + d*x]))","A",8,6,21,0.2857,1,"{2670, 2680, 2679, 2682, 2635, 8}"
50,1,110,0,0.091071,"\int \sec ^5(c+d x) (a+a \sin (c+d x))^8 \, dx","Int[Sec[c + d*x]^5*(a + a*Sin[c + d*x])^8,x]","-\frac{a^8 \sin ^3(c+d x)}{3 d}-\frac{4 a^8 \sin ^2(c+d x)}{d}+\frac{16 a^{10}}{d (a-a \sin (c+d x))^2}-\frac{80 a^9}{d (a-a \sin (c+d x))}-\frac{31 a^8 \sin (c+d x)}{d}-\frac{80 a^8 \log (1-\sin (c+d x))}{d}","-\frac{a^8 \sin ^3(c+d x)}{3 d}-\frac{4 a^8 \sin ^2(c+d x)}{d}+\frac{16 a^{10}}{d (a-a \sin (c+d x))^2}-\frac{80 a^9}{d (a-a \sin (c+d x))}-\frac{31 a^8 \sin (c+d x)}{d}-\frac{80 a^8 \log (1-\sin (c+d x))}{d}",1,"(-80*a^8*Log[1 - Sin[c + d*x]])/d - (31*a^8*Sin[c + d*x])/d - (4*a^8*Sin[c + d*x]^2)/d - (a^8*Sin[c + d*x]^3)/(3*d) + (16*a^10)/(d*(a - a*Sin[c + d*x])^2) - (80*a^9)/(d*(a - a*Sin[c + d*x]))","A",3,2,21,0.09524,1,"{2667, 43}"
51,1,73,0,0.0682715,"\int \frac{\cos ^6(c+d x)}{a+a \sin (c+d x)} \, dx","Int[Cos[c + d*x]^6/(a + a*Sin[c + d*x]),x]","\frac{\cos ^5(c+d x)}{5 a d}+\frac{\sin (c+d x) \cos ^3(c+d x)}{4 a d}+\frac{3 \sin (c+d x) \cos (c+d x)}{8 a d}+\frac{3 x}{8 a}","\frac{\cos ^5(c+d x)}{5 a d}+\frac{\sin (c+d x) \cos ^3(c+d x)}{4 a d}+\frac{3 \sin (c+d x) \cos (c+d x)}{8 a d}+\frac{3 x}{8 a}",1,"(3*x)/(8*a) + Cos[c + d*x]^5/(5*a*d) + (3*Cos[c + d*x]*Sin[c + d*x])/(8*a*d) + (Cos[c + d*x]^3*Sin[c + d*x])/(4*a*d)","A",4,3,21,0.1429,1,"{2682, 2635, 8}"
52,1,47,0,0.0562803,"\int \frac{\cos ^5(c+d x)}{a+a \sin (c+d x)} \, dx","Int[Cos[c + d*x]^5/(a + a*Sin[c + d*x]),x]","\frac{(a-a \sin (c+d x))^4}{4 a^5 d}-\frac{2 (a-a \sin (c+d x))^3}{3 a^4 d}","\frac{(a-a \sin (c+d x))^4}{4 a^5 d}-\frac{2 (a-a \sin (c+d x))^3}{3 a^4 d}",1,"(-2*(a - a*Sin[c + d*x])^3)/(3*a^4*d) + (a - a*Sin[c + d*x])^4/(4*a^5*d)","A",3,2,21,0.09524,1,"{2667, 43}"
53,1,49,0,0.0550841,"\int \frac{\cos ^4(c+d x)}{a+a \sin (c+d x)} \, dx","Int[Cos[c + d*x]^4/(a + a*Sin[c + d*x]),x]","\frac{\cos ^3(c+d x)}{3 a d}+\frac{\sin (c+d x) \cos (c+d x)}{2 a d}+\frac{x}{2 a}","\frac{\cos ^3(c+d x)}{3 a d}+\frac{\sin (c+d x) \cos (c+d x)}{2 a d}+\frac{x}{2 a}",1,"x/(2*a) + Cos[c + d*x]^3/(3*a*d) + (Cos[c + d*x]*Sin[c + d*x])/(2*a*d)","A",3,3,21,0.1429,1,"{2682, 2635, 8}"
54,1,32,0,0.0457101,"\int \frac{\cos ^3(c+d x)}{a+a \sin (c+d x)} \, dx","Int[Cos[c + d*x]^3/(a + a*Sin[c + d*x]),x]","\frac{\sin (c+d x)}{a d}-\frac{\sin ^2(c+d x)}{2 a d}","\frac{\sin (c+d x)}{a d}-\frac{\sin ^2(c+d x)}{2 a d}",1,"Sin[c + d*x]/(a*d) - Sin[c + d*x]^2/(2*a*d)","A",2,1,21,0.04762,1,"{2667}"
55,1,19,0,0.0430879,"\int \frac{\cos ^2(c+d x)}{a+a \sin (c+d x)} \, dx","Int[Cos[c + d*x]^2/(a + a*Sin[c + d*x]),x]","\frac{\cos (c+d x)}{a d}+\frac{x}{a}","\frac{\cos (c+d x)}{a d}+\frac{x}{a}",1,"x/a + Cos[c + d*x]/(a*d)","A",2,2,21,0.09524,1,"{2682, 8}"
56,1,16,0,0.0255872,"\int \frac{\cos (c+d x)}{a+a \sin (c+d x)} \, dx","Int[Cos[c + d*x]/(a + a*Sin[c + d*x]),x]","\frac{\log (\sin (c+d x)+1)}{a d}","\frac{\log (\sin (c+d x)+1)}{a d}",1,"Log[1 + Sin[c + d*x]]/(a*d)","A",2,2,19,0.1053,1,"{2667, 31}"
57,1,37,0,0.0505354,"\int \frac{\sec (c+d x)}{a+a \sin (c+d x)} \, dx","Int[Sec[c + d*x]/(a + a*Sin[c + d*x]),x]","\frac{\tanh ^{-1}(\sin (c+d x))}{2 a d}-\frac{1}{2 d (a \sin (c+d x)+a)}","\frac{\tanh ^{-1}(\sin (c+d x))}{2 a d}-\frac{1}{2 d (a \sin (c+d x)+a)}",1,"ArcTanh[Sin[c + d*x]]/(2*a*d) - 1/(2*d*(a + a*Sin[c + d*x]))","A",4,3,19,0.1579,1,"{2667, 44, 206}"
58,1,42,0,0.0522441,"\int \frac{\sec ^2(c+d x)}{a+a \sin (c+d x)} \, dx","Int[Sec[c + d*x]^2/(a + a*Sin[c + d*x]),x]","\frac{2 \tan (c+d x)}{3 a d}-\frac{\sec (c+d x)}{3 d (a \sin (c+d x)+a)}","\frac{2 \tan (c+d x)}{3 a d}-\frac{\sec (c+d x)}{3 d (a \sin (c+d x)+a)}",1,"-Sec[c + d*x]/(3*d*(a + a*Sin[c + d*x])) + (2*Tan[c + d*x])/(3*a*d)","A",3,3,21,0.1429,1,"{2672, 3767, 8}"
59,1,77,0,0.0757208,"\int \frac{\sec ^3(c+d x)}{a+a \sin (c+d x)} \, dx","Int[Sec[c + d*x]^3/(a + a*Sin[c + d*x]),x]","-\frac{a}{8 d (a \sin (c+d x)+a)^2}+\frac{1}{8 d (a-a \sin (c+d x))}-\frac{1}{4 d (a \sin (c+d x)+a)}+\frac{3 \tanh ^{-1}(\sin (c+d x))}{8 a d}","-\frac{a}{8 d (a \sin (c+d x)+a)^2}+\frac{1}{8 d (a-a \sin (c+d x))}-\frac{1}{4 d (a \sin (c+d x)+a)}+\frac{3 \tanh ^{-1}(\sin (c+d x))}{8 a d}",1,"(3*ArcTanh[Sin[c + d*x]])/(8*a*d) + 1/(8*d*(a - a*Sin[c + d*x])) - a/(8*d*(a + a*Sin[c + d*x])^2) - 1/(4*d*(a + a*Sin[c + d*x]))","A",4,3,21,0.1429,1,"{2667, 44, 206}"
60,1,62,0,0.0593119,"\int \frac{\sec ^4(c+d x)}{a+a \sin (c+d x)} \, dx","Int[Sec[c + d*x]^4/(a + a*Sin[c + d*x]),x]","\frac{4 \tan ^3(c+d x)}{15 a d}+\frac{4 \tan (c+d x)}{5 a d}-\frac{\sec ^3(c+d x)}{5 d (a \sin (c+d x)+a)}","\frac{4 \tan ^3(c+d x)}{15 a d}+\frac{4 \tan (c+d x)}{5 a d}-\frac{\sec ^3(c+d x)}{5 d (a \sin (c+d x)+a)}",1,"-Sec[c + d*x]^3/(5*d*(a + a*Sin[c + d*x])) + (4*Tan[c + d*x])/(5*a*d) + (4*Tan[c + d*x]^3)/(15*a*d)","A",3,2,21,0.09524,1,"{2672, 3767}"
61,1,120,0,0.1068393,"\int \frac{\sec ^5(c+d x)}{a+a \sin (c+d x)} \, dx","Int[Sec[c + d*x]^5/(a + a*Sin[c + d*x]),x]","-\frac{a^2}{24 d (a \sin (c+d x)+a)^3}+\frac{a}{32 d (a-a \sin (c+d x))^2}-\frac{3 a}{32 d (a \sin (c+d x)+a)^2}+\frac{1}{8 d (a-a \sin (c+d x))}-\frac{3}{16 d (a \sin (c+d x)+a)}+\frac{5 \tanh ^{-1}(\sin (c+d x))}{16 a d}","-\frac{a^2}{24 d (a \sin (c+d x)+a)^3}+\frac{a}{32 d (a-a \sin (c+d x))^2}-\frac{3 a}{32 d (a \sin (c+d x)+a)^2}+\frac{1}{8 d (a-a \sin (c+d x))}-\frac{3}{16 d (a \sin (c+d x)+a)}+\frac{5 \tanh ^{-1}(\sin (c+d x))}{16 a d}",1,"(5*ArcTanh[Sin[c + d*x]])/(16*a*d) + a/(32*d*(a - a*Sin[c + d*x])^2) + 1/(8*d*(a - a*Sin[c + d*x])) - a^2/(24*d*(a + a*Sin[c + d*x])^3) - (3*a)/(32*d*(a + a*Sin[c + d*x])^2) - 3/(16*d*(a + a*Sin[c + d*x]))","A",4,3,21,0.1429,1,"{2667, 44, 206}"
62,1,104,0,0.1124898,"\int \frac{\cos ^8(c+d x)}{(a+a \sin (c+d x))^2} \, dx","Int[Cos[c + d*x]^8/(a + a*Sin[c + d*x])^2,x]","\frac{7 \cos ^5(c+d x)}{30 a^2 d}+\frac{\cos ^7(c+d x)}{6 d \left(a^2 \sin (c+d x)+a^2\right)}+\frac{7 \sin (c+d x) \cos ^3(c+d x)}{24 a^2 d}+\frac{7 \sin (c+d x) \cos (c+d x)}{16 a^2 d}+\frac{7 x}{16 a^2}","\frac{7 \cos ^5(c+d x)}{30 a^2 d}+\frac{\cos ^7(c+d x)}{6 d \left(a^2 \sin (c+d x)+a^2\right)}+\frac{7 \sin (c+d x) \cos ^3(c+d x)}{24 a^2 d}+\frac{7 \sin (c+d x) \cos (c+d x)}{16 a^2 d}+\frac{7 x}{16 a^2}",1,"(7*x)/(16*a^2) + (7*Cos[c + d*x]^5)/(30*a^2*d) + (7*Cos[c + d*x]*Sin[c + d*x])/(16*a^2*d) + (7*Cos[c + d*x]^3*Sin[c + d*x])/(24*a^2*d) + Cos[c + d*x]^7/(6*d*(a^2 + a^2*Sin[c + d*x]))","A",5,4,21,0.1905,1,"{2679, 2682, 2635, 8}"
63,1,47,0,0.0520617,"\int \frac{\cos ^7(c+d x)}{(a+a \sin (c+d x))^2} \, dx","Int[Cos[c + d*x]^7/(a + a*Sin[c + d*x])^2,x]","\frac{(a-a \sin (c+d x))^5}{5 a^7 d}-\frac{(a-a \sin (c+d x))^4}{2 a^6 d}","\frac{(a-a \sin (c+d x))^5}{5 a^7 d}-\frac{(a-a \sin (c+d x))^4}{2 a^6 d}",1,"-(a - a*Sin[c + d*x])^4/(2*a^6*d) + (a - a*Sin[c + d*x])^5/(5*a^7*d)","A",3,2,21,0.09524,1,"{2667, 43}"
64,1,80,0,0.1004512,"\int \frac{\cos ^6(c+d x)}{(a+a \sin (c+d x))^2} \, dx","Int[Cos[c + d*x]^6/(a + a*Sin[c + d*x])^2,x]","\frac{5 \cos ^3(c+d x)}{12 a^2 d}+\frac{\cos ^5(c+d x)}{4 d \left(a^2 \sin (c+d x)+a^2\right)}+\frac{5 \sin (c+d x) \cos (c+d x)}{8 a^2 d}+\frac{5 x}{8 a^2}","\frac{5 \cos ^3(c+d x)}{12 a^2 d}+\frac{\cos ^5(c+d x)}{4 d \left(a^2 \sin (c+d x)+a^2\right)}+\frac{5 \sin (c+d x) \cos (c+d x)}{8 a^2 d}+\frac{5 x}{8 a^2}",1,"(5*x)/(8*a^2) + (5*Cos[c + d*x]^3)/(12*a^2*d) + (5*Cos[c + d*x]*Sin[c + d*x])/(8*a^2*d) + Cos[c + d*x]^5/(4*d*(a^2 + a^2*Sin[c + d*x]))","A",4,4,21,0.1905,1,"{2679, 2682, 2635, 8}"
65,1,23,0,0.0431779,"\int \frac{\cos ^5(c+d x)}{(a+a \sin (c+d x))^2} \, dx","Int[Cos[c + d*x]^5/(a + a*Sin[c + d*x])^2,x]","-\frac{(a-a \sin (c+d x))^3}{3 a^5 d}","-\frac{(a-a \sin (c+d x))^3}{3 a^5 d}",1,"-(a - a*Sin[c + d*x])^3/(3*a^5*d)","A",2,2,21,0.09524,1,"{2667, 32}"
66,1,56,0,0.0847453,"\int \frac{\cos ^4(c+d x)}{(a+a \sin (c+d x))^2} \, dx","Int[Cos[c + d*x]^4/(a + a*Sin[c + d*x])^2,x]","\frac{3 \cos (c+d x)}{2 a^2 d}+\frac{\cos ^3(c+d x)}{2 d \left(a^2 \sin (c+d x)+a^2\right)}+\frac{3 x}{2 a^2}","\frac{3 \cos (c+d x)}{2 a^2 d}+\frac{\cos ^3(c+d x)}{2 d \left(a^2 \sin (c+d x)+a^2\right)}+\frac{3 x}{2 a^2}",1,"(3*x)/(2*a^2) + (3*Cos[c + d*x])/(2*a^2*d) + Cos[c + d*x]^3/(2*d*(a^2 + a^2*Sin[c + d*x]))","A",3,3,21,0.1429,1,"{2679, 2682, 8}"
67,1,32,0,0.0494738,"\int \frac{\cos ^3(c+d x)}{(a+a \sin (c+d x))^2} \, dx","Int[Cos[c + d*x]^3/(a + a*Sin[c + d*x])^2,x]","\frac{2 \log (\sin (c+d x)+1)}{a^2 d}-\frac{\sin (c+d x)}{a^2 d}","\frac{2 \log (\sin (c+d x)+1)}{a^2 d}-\frac{\sin (c+d x)}{a^2 d}",1,"(2*Log[1 + Sin[c + d*x]])/(a^2*d) - Sin[c + d*x]/(a^2*d)","A",3,2,21,0.09524,1,"{2667, 43}"
68,1,34,0,0.0427926,"\int \frac{\cos ^2(c+d x)}{(a+a \sin (c+d x))^2} \, dx","Int[Cos[c + d*x]^2/(a + a*Sin[c + d*x])^2,x]","-\frac{2 \cos (c+d x)}{d \left(a^2 \sin (c+d x)+a^2\right)}-\frac{x}{a^2}","-\frac{2 \cos (c+d x)}{d \left(a^2 \sin (c+d x)+a^2\right)}-\frac{x}{a^2}",1,"-(x/a^2) - (2*Cos[c + d*x])/(d*(a^2 + a^2*Sin[c + d*x]))","A",2,2,21,0.09524,1,"{2680, 8}"
69,1,21,0,0.0261668,"\int \frac{\cos (c+d x)}{(a+a \sin (c+d x))^2} \, dx","Int[Cos[c + d*x]/(a + a*Sin[c + d*x])^2,x]","-\frac{1}{d \left(a^2 \sin (c+d x)+a^2\right)}","-\frac{1}{d \left(a^2 \sin (c+d x)+a^2\right)}",1,"-(1/(d*(a^2 + a^2*Sin[c + d*x])))","A",2,2,19,0.1053,1,"{2667, 32}"
70,1,60,0,0.0582109,"\int \frac{\sec (c+d x)}{(a+a \sin (c+d x))^2} \, dx","Int[Sec[c + d*x]/(a + a*Sin[c + d*x])^2,x]","-\frac{1}{4 d \left(a^2 \sin (c+d x)+a^2\right)}+\frac{\tanh ^{-1}(\sin (c+d x))}{4 a^2 d}-\frac{1}{4 d (a \sin (c+d x)+a)^2}","-\frac{1}{4 d \left(a^2 \sin (c+d x)+a^2\right)}+\frac{\tanh ^{-1}(\sin (c+d x))}{4 a^2 d}-\frac{1}{4 d (a \sin (c+d x)+a)^2}",1,"ArcTanh[Sin[c + d*x]]/(4*a^2*d) - 1/(4*d*(a + a*Sin[c + d*x])^2) - 1/(4*d*(a^2 + a^2*Sin[c + d*x]))","A",4,3,19,0.1579,1,"{2667, 44, 206}"
71,1,71,0,0.09312,"\int \frac{\sec ^2(c+d x)}{(a+a \sin (c+d x))^2} \, dx","Int[Sec[c + d*x]^2/(a + a*Sin[c + d*x])^2,x]","\frac{2 \tan (c+d x)}{5 a^2 d}-\frac{\sec (c+d x)}{5 d \left(a^2 \sin (c+d x)+a^2\right)}-\frac{\sec (c+d x)}{5 d (a \sin (c+d x)+a)^2}","\frac{2 \tan (c+d x)}{5 a^2 d}-\frac{\sec (c+d x)}{5 d \left(a^2 \sin (c+d x)+a^2\right)}-\frac{\sec (c+d x)}{5 d (a \sin (c+d x)+a)^2}",1,"-Sec[c + d*x]/(5*d*(a + a*Sin[c + d*x])^2) - Sec[c + d*x]/(5*d*(a^2 + a^2*Sin[c + d*x])) + (2*Tan[c + d*x])/(5*a^2*d)","A",4,3,21,0.1429,1,"{2672, 3767, 8}"
72,1,104,0,0.0820373,"\int \frac{\sec ^3(c+d x)}{(a+a \sin (c+d x))^2} \, dx","Int[Sec[c + d*x]^3/(a + a*Sin[c + d*x])^2,x]","\frac{1}{16 d \left(a^2-a^2 \sin (c+d x)\right)}-\frac{3}{16 d \left(a^2 \sin (c+d x)+a^2\right)}+\frac{\tanh ^{-1}(\sin (c+d x))}{4 a^2 d}-\frac{a}{12 d (a \sin (c+d x)+a)^3}-\frac{1}{8 d (a \sin (c+d x)+a)^2}","\frac{1}{16 d \left(a^2-a^2 \sin (c+d x)\right)}-\frac{3}{16 d \left(a^2 \sin (c+d x)+a^2\right)}+\frac{\tanh ^{-1}(\sin (c+d x))}{4 a^2 d}-\frac{a}{12 d (a \sin (c+d x)+a)^3}-\frac{1}{8 d (a \sin (c+d x)+a)^2}",1,"ArcTanh[Sin[c + d*x]]/(4*a^2*d) - a/(12*d*(a + a*Sin[c + d*x])^3) - 1/(8*d*(a + a*Sin[c + d*x])^2) + 1/(16*d*(a^2 - a^2*Sin[c + d*x])) - 3/(16*d*(a^2 + a^2*Sin[c + d*x]))","A",4,3,21,0.1429,1,"{2667, 44, 206}"
73,1,93,0,0.0975248,"\int \frac{\sec ^4(c+d x)}{(a+a \sin (c+d x))^2} \, dx","Int[Sec[c + d*x]^4/(a + a*Sin[c + d*x])^2,x]","\frac{4 \tan ^3(c+d x)}{21 a^2 d}+\frac{4 \tan (c+d x)}{7 a^2 d}-\frac{\sec ^3(c+d x)}{7 d \left(a^2 \sin (c+d x)+a^2\right)}-\frac{\sec ^3(c+d x)}{7 d (a \sin (c+d x)+a)^2}","\frac{4 \tan ^3(c+d x)}{21 a^2 d}+\frac{4 \tan (c+d x)}{7 a^2 d}-\frac{\sec ^3(c+d x)}{7 d \left(a^2 \sin (c+d x)+a^2\right)}-\frac{\sec ^3(c+d x)}{7 d (a \sin (c+d x)+a)^2}",1,"-Sec[c + d*x]^3/(7*d*(a + a*Sin[c + d*x])^2) - Sec[c + d*x]^3/(7*d*(a^2 + a^2*Sin[c + d*x])) + (4*Tan[c + d*x])/(7*a^2*d) + (4*Tan[c + d*x]^3)/(21*a^2*d)","A",4,2,21,0.09524,1,"{2672, 3767}"
74,1,146,0,0.1098897,"\int \frac{\sec ^5(c+d x)}{(a+a \sin (c+d x))^2} \, dx","Int[Sec[c + d*x]^5/(a + a*Sin[c + d*x])^2,x]","-\frac{a^2}{32 d (a \sin (c+d x)+a)^4}+\frac{5}{64 d \left(a^2-a^2 \sin (c+d x)\right)}-\frac{5}{32 d \left(a^2 \sin (c+d x)+a^2\right)}+\frac{15 \tanh ^{-1}(\sin (c+d x))}{64 a^2 d}-\frac{a}{16 d (a \sin (c+d x)+a)^3}+\frac{1}{64 d (a-a \sin (c+d x))^2}-\frac{3}{32 d (a \sin (c+d x)+a)^2}","-\frac{a^2}{32 d (a \sin (c+d x)+a)^4}+\frac{5}{64 d \left(a^2-a^2 \sin (c+d x)\right)}-\frac{5}{32 d \left(a^2 \sin (c+d x)+a^2\right)}+\frac{15 \tanh ^{-1}(\sin (c+d x))}{64 a^2 d}-\frac{a}{16 d (a \sin (c+d x)+a)^3}+\frac{1}{64 d (a-a \sin (c+d x))^2}-\frac{3}{32 d (a \sin (c+d x)+a)^2}",1,"(15*ArcTanh[Sin[c + d*x]])/(64*a^2*d) + 1/(64*d*(a - a*Sin[c + d*x])^2) - a^2/(32*d*(a + a*Sin[c + d*x])^4) - a/(16*d*(a + a*Sin[c + d*x])^3) - 3/(32*d*(a + a*Sin[c + d*x])^2) + 5/(64*d*(a^2 - a^2*Sin[c + d*x])) - 5/(32*d*(a^2 + a^2*Sin[c + d*x]))","A",4,3,21,0.1429,1,"{2667, 44, 206}"
75,1,103,0,0.109426,"\int \frac{\cos ^8(c+d x)}{(a+a \sin (c+d x))^3} \, dx","Int[Cos[c + d*x]^8/(a + a*Sin[c + d*x])^3,x]","\frac{7 \cos ^5(c+d x)}{15 a^3 d}+\frac{7 \sin (c+d x) \cos ^3(c+d x)}{12 a^3 d}+\frac{7 \sin (c+d x) \cos (c+d x)}{8 a^3 d}+\frac{7 x}{8 a^3}+\frac{2 \cos ^7(c+d x)}{3 a d (a \sin (c+d x)+a)^2}","\frac{7 \cos ^5(c+d x)}{15 a^3 d}+\frac{7 \sin (c+d x) \cos ^3(c+d x)}{12 a^3 d}+\frac{7 \sin (c+d x) \cos (c+d x)}{8 a^3 d}+\frac{7 x}{8 a^3}+\frac{2 \cos ^7(c+d x)}{3 a d (a \sin (c+d x)+a)^2}",1,"(7*x)/(8*a^3) + (7*Cos[c + d*x]^5)/(15*a^3*d) + (7*Cos[c + d*x]*Sin[c + d*x])/(8*a^3*d) + (7*Cos[c + d*x]^3*Sin[c + d*x])/(12*a^3*d) + (2*Cos[c + d*x]^7)/(3*a*d*(a + a*Sin[c + d*x])^2)","A",5,4,21,0.1905,1,"{2680, 2682, 2635, 8}"
76,1,23,0,0.0428404,"\int \frac{\cos ^7(c+d x)}{(a+a \sin (c+d x))^3} \, dx","Int[Cos[c + d*x]^7/(a + a*Sin[c + d*x])^3,x]","-\frac{(a-a \sin (c+d x))^4}{4 a^7 d}","-\frac{(a-a \sin (c+d x))^4}{4 a^7 d}",1,"-(a - a*Sin[c + d*x])^4/(4*a^7*d)","A",2,2,21,0.09524,1,"{2667, 32}"
77,1,77,0,0.0999435,"\int \frac{\cos ^6(c+d x)}{(a+a \sin (c+d x))^3} \, dx","Int[Cos[c + d*x]^6/(a + a*Sin[c + d*x])^3,x]","\frac{5 \cos ^3(c+d x)}{3 a^3 d}+\frac{5 \sin (c+d x) \cos (c+d x)}{2 a^3 d}+\frac{5 x}{2 a^3}+\frac{2 \cos ^5(c+d x)}{a d (a \sin (c+d x)+a)^2}","\frac{5 \cos ^3(c+d x)}{3 a^3 d}+\frac{5 \sin (c+d x) \cos (c+d x)}{2 a^3 d}+\frac{5 x}{2 a^3}+\frac{2 \cos ^5(c+d x)}{a d (a \sin (c+d x)+a)^2}",1,"(5*x)/(2*a^3) + (5*Cos[c + d*x]^3)/(3*a^3*d) + (5*Cos[c + d*x]*Sin[c + d*x])/(2*a^3*d) + (2*Cos[c + d*x]^5)/(a*d*(a + a*Sin[c + d*x])^2)","A",4,4,21,0.1905,1,"{2680, 2682, 2635, 8}"
78,1,50,0,0.0499715,"\int \frac{\cos ^5(c+d x)}{(a+a \sin (c+d x))^3} \, dx","Int[Cos[c + d*x]^5/(a + a*Sin[c + d*x])^3,x]","\frac{\sin ^2(c+d x)}{2 a^3 d}-\frac{3 \sin (c+d x)}{a^3 d}+\frac{4 \log (\sin (c+d x)+1)}{a^3 d}","\frac{\sin ^2(c+d x)}{2 a^3 d}-\frac{3 \sin (c+d x)}{a^3 d}+\frac{4 \log (\sin (c+d x)+1)}{a^3 d}",1,"(4*Log[1 + Sin[c + d*x]])/(a^3*d) - (3*Sin[c + d*x])/(a^3*d) + Sin[c + d*x]^2/(2*a^3*d)","A",3,2,21,0.09524,1,"{2667, 43}"
79,1,49,0,0.0850498,"\int \frac{\cos ^4(c+d x)}{(a+a \sin (c+d x))^3} \, dx","Int[Cos[c + d*x]^4/(a + a*Sin[c + d*x])^3,x]","-\frac{3 \cos (c+d x)}{a^3 d}-\frac{3 x}{a^3}-\frac{2 \cos ^3(c+d x)}{a d (a \sin (c+d x)+a)^2}","-\frac{3 \cos (c+d x)}{a^3 d}-\frac{3 x}{a^3}-\frac{2 \cos ^3(c+d x)}{a d (a \sin (c+d x)+a)^2}",1,"(-3*x)/a^3 - (3*Cos[c + d*x])/(a^3*d) - (2*Cos[c + d*x]^3)/(a*d*(a + a*Sin[c + d*x])^2)","A",3,3,21,0.1429,1,"{2680, 2682, 8}"
80,1,39,0,0.0505272,"\int \frac{\cos ^3(c+d x)}{(a+a \sin (c+d x))^3} \, dx","Int[Cos[c + d*x]^3/(a + a*Sin[c + d*x])^3,x]","-\frac{2}{d \left(a^3 \sin (c+d x)+a^3\right)}-\frac{\log (\sin (c+d x)+1)}{a^3 d}","-\frac{2}{d \left(a^3 \sin (c+d x)+a^3\right)}-\frac{\log (\sin (c+d x)+1)}{a^3 d}",1,"-(Log[1 + Sin[c + d*x]]/(a^3*d)) - 2/(d*(a^3 + a^3*Sin[c + d*x]))","A",3,2,21,0.09524,1,"{2667, 43}"
81,1,27,0,0.038762,"\int \frac{\cos ^2(c+d x)}{(a+a \sin (c+d x))^3} \, dx","Int[Cos[c + d*x]^2/(a + a*Sin[c + d*x])^3,x]","-\frac{\cos ^3(c+d x)}{3 d (a \sin (c+d x)+a)^3}","-\frac{\cos ^3(c+d x)}{3 d (a \sin (c+d x)+a)^3}",1,"-Cos[c + d*x]^3/(3*d*(a + a*Sin[c + d*x])^3)","A",1,1,21,0.04762,1,"{2671}"
82,1,22,0,0.0246114,"\int \frac{\cos (c+d x)}{(a+a \sin (c+d x))^3} \, dx","Int[Cos[c + d*x]/(a + a*Sin[c + d*x])^3,x]","-\frac{1}{2 a d (a \sin (c+d x)+a)^2}","-\frac{1}{2 a d (a \sin (c+d x)+a)^2}",1,"-1/(2*a*d*(a + a*Sin[c + d*x])^2)","A",2,2,19,0.1053,1,"{2667, 32}"
83,1,82,0,0.0631045,"\int \frac{\sec (c+d x)}{(a+a \sin (c+d x))^3} \, dx","Int[Sec[c + d*x]/(a + a*Sin[c + d*x])^3,x]","-\frac{1}{8 d \left(a^3 \sin (c+d x)+a^3\right)}+\frac{\tanh ^{-1}(\sin (c+d x))}{8 a^3 d}-\frac{1}{8 a d (a \sin (c+d x)+a)^2}-\frac{1}{6 d (a \sin (c+d x)+a)^3}","-\frac{1}{8 d \left(a^3 \sin (c+d x)+a^3\right)}+\frac{\tanh ^{-1}(\sin (c+d x))}{8 a^3 d}-\frac{1}{8 a d (a \sin (c+d x)+a)^2}-\frac{1}{6 d (a \sin (c+d x)+a)^3}",1,"ArcTanh[Sin[c + d*x]]/(8*a^3*d) - 1/(6*d*(a + a*Sin[c + d*x])^3) - 1/(8*a*d*(a + a*Sin[c + d*x])^2) - 1/(8*d*(a^3 + a^3*Sin[c + d*x]))","A",4,3,19,0.1579,1,"{2667, 44, 206}"
84,1,99,0,0.1348476,"\int \frac{\sec ^2(c+d x)}{(a+a \sin (c+d x))^3} \, dx","Int[Sec[c + d*x]^2/(a + a*Sin[c + d*x])^3,x]","\frac{8 \tan (c+d x)}{35 a^3 d}-\frac{4 \sec (c+d x)}{35 d \left(a^3 \sin (c+d x)+a^3\right)}-\frac{4 \sec (c+d x)}{35 a d (a \sin (c+d x)+a)^2}-\frac{\sec (c+d x)}{7 d (a \sin (c+d x)+a)^3}","\frac{8 \tan (c+d x)}{35 a^3 d}-\frac{4 \sec (c+d x)}{35 d \left(a^3 \sin (c+d x)+a^3\right)}-\frac{4 \sec (c+d x)}{35 a d (a \sin (c+d x)+a)^2}-\frac{\sec (c+d x)}{7 d (a \sin (c+d x)+a)^3}",1,"-Sec[c + d*x]/(7*d*(a + a*Sin[c + d*x])^3) - (4*Sec[c + d*x])/(35*a*d*(a + a*Sin[c + d*x])^2) - (4*Sec[c + d*x])/(35*d*(a^3 + a^3*Sin[c + d*x])) + (8*Tan[c + d*x])/(35*a^3*d)","A",5,3,21,0.1429,1,"{2672, 3767, 8}"
85,1,126,0,0.0946766,"\int \frac{\sec ^3(c+d x)}{(a+a \sin (c+d x))^3} \, dx","Int[Sec[c + d*x]^3/(a + a*Sin[c + d*x])^3,x]","\frac{1}{32 d \left(a^3-a^3 \sin (c+d x)\right)}-\frac{1}{8 d \left(a^3 \sin (c+d x)+a^3\right)}+\frac{5 \tanh ^{-1}(\sin (c+d x))}{32 a^3 d}-\frac{a}{16 d (a \sin (c+d x)+a)^4}-\frac{1}{12 d (a \sin (c+d x)+a)^3}-\frac{3}{32 a d (a \sin (c+d x)+a)^2}","\frac{1}{32 d \left(a^3-a^3 \sin (c+d x)\right)}-\frac{1}{8 d \left(a^3 \sin (c+d x)+a^3\right)}+\frac{5 \tanh ^{-1}(\sin (c+d x))}{32 a^3 d}-\frac{a}{16 d (a \sin (c+d x)+a)^4}-\frac{1}{12 d (a \sin (c+d x)+a)^3}-\frac{3}{32 a d (a \sin (c+d x)+a)^2}",1,"(5*ArcTanh[Sin[c + d*x]])/(32*a^3*d) - a/(16*d*(a + a*Sin[c + d*x])^4) - 1/(12*d*(a + a*Sin[c + d*x])^3) - 3/(32*a*d*(a + a*Sin[c + d*x])^2) + 1/(32*d*(a^3 - a^3*Sin[c + d*x])) - 1/(8*d*(a^3 + a^3*Sin[c + d*x]))","A",4,3,21,0.1429,1,"{2667, 44, 206}"
86,1,123,0,0.145029,"\int \frac{\sec ^4(c+d x)}{(a+a \sin (c+d x))^3} \, dx","Int[Sec[c + d*x]^4/(a + a*Sin[c + d*x])^3,x]","\frac{8 \tan ^3(c+d x)}{63 a^3 d}+\frac{8 \tan (c+d x)}{21 a^3 d}-\frac{2 \sec ^3(c+d x)}{21 d \left(a^3 \sin (c+d x)+a^3\right)}-\frac{2 \sec ^3(c+d x)}{21 a d (a \sin (c+d x)+a)^2}-\frac{\sec ^3(c+d x)}{9 d (a \sin (c+d x)+a)^3}","\frac{8 \tan ^3(c+d x)}{63 a^3 d}+\frac{8 \tan (c+d x)}{21 a^3 d}-\frac{2 \sec ^3(c+d x)}{21 d \left(a^3 \sin (c+d x)+a^3\right)}-\frac{2 \sec ^3(c+d x)}{21 a d (a \sin (c+d x)+a)^2}-\frac{\sec ^3(c+d x)}{9 d (a \sin (c+d x)+a)^3}",1,"-Sec[c + d*x]^3/(9*d*(a + a*Sin[c + d*x])^3) - (2*Sec[c + d*x]^3)/(21*a*d*(a + a*Sin[c + d*x])^2) - (2*Sec[c + d*x]^3)/(21*d*(a^3 + a^3*Sin[c + d*x])) + (8*Tan[c + d*x])/(21*a^3*d) + (8*Tan[c + d*x]^3)/(63*a^3*d)","A",5,2,21,0.09524,1,"{2672, 3767}"
87,1,171,0,0.1284178,"\int \frac{\sec ^5(c+d x)}{(a+a \sin (c+d x))^3} \, dx","Int[Sec[c + d*x]^5/(a + a*Sin[c + d*x])^3,x]","-\frac{a^2}{40 d (a \sin (c+d x)+a)^5}+\frac{3}{64 d \left(a^3-a^3 \sin (c+d x)\right)}-\frac{15}{128 d \left(a^3 \sin (c+d x)+a^3\right)}+\frac{21 \tanh ^{-1}(\sin (c+d x))}{128 a^3 d}-\frac{3 a}{64 d (a \sin (c+d x)+a)^4}-\frac{1}{16 d (a \sin (c+d x)+a)^3}+\frac{1}{128 a d (a-a \sin (c+d x))^2}-\frac{5}{64 a d (a \sin (c+d x)+a)^2}","-\frac{a^2}{40 d (a \sin (c+d x)+a)^5}+\frac{3}{64 d \left(a^3-a^3 \sin (c+d x)\right)}-\frac{15}{128 d \left(a^3 \sin (c+d x)+a^3\right)}+\frac{21 \tanh ^{-1}(\sin (c+d x))}{128 a^3 d}-\frac{3 a}{64 d (a \sin (c+d x)+a)^4}-\frac{1}{16 d (a \sin (c+d x)+a)^3}+\frac{1}{128 a d (a-a \sin (c+d x))^2}-\frac{5}{64 a d (a \sin (c+d x)+a)^2}",1,"(21*ArcTanh[Sin[c + d*x]])/(128*a^3*d) + 1/(128*a*d*(a - a*Sin[c + d*x])^2) - a^2/(40*d*(a + a*Sin[c + d*x])^5) - (3*a)/(64*d*(a + a*Sin[c + d*x])^4) - 1/(16*d*(a + a*Sin[c + d*x])^3) - 5/(64*a*d*(a + a*Sin[c + d*x])^2) + 3/(64*d*(a^3 - a^3*Sin[c + d*x])) - 15/(128*d*(a^3 + a^3*Sin[c + d*x]))","A",4,3,21,0.1429,1,"{2667, 44, 206}"
88,1,127,0,0.1823565,"\int \frac{\cos ^8(c+d x)}{(a+a \sin (c+d x))^8} \, dx","Int[Cos[c + d*x]^8/(a + a*Sin[c + d*x])^8,x]","\frac{2 \cos ^5(c+d x)}{5 a^3 d (a \sin (c+d x)+a)^5}-\frac{2 \cos ^3(c+d x)}{3 a^2 d \left(a^2 \sin (c+d x)+a^2\right)^3}+\frac{2 \cos (c+d x)}{d \left(a^8 \sin (c+d x)+a^8\right)}+\frac{x}{a^8}-\frac{2 \cos ^7(c+d x)}{7 a d (a \sin (c+d x)+a)^7}","\frac{2 \cos ^5(c+d x)}{5 a^3 d (a \sin (c+d x)+a)^5}-\frac{2 \cos ^3(c+d x)}{3 a^2 d \left(a^2 \sin (c+d x)+a^2\right)^3}+\frac{2 \cos (c+d x)}{d \left(a^8 \sin (c+d x)+a^8\right)}+\frac{x}{a^8}-\frac{2 \cos ^7(c+d x)}{7 a d (a \sin (c+d x)+a)^7}",1,"x/a^8 - (2*Cos[c + d*x]^7)/(7*a*d*(a + a*Sin[c + d*x])^7) + (2*Cos[c + d*x]^5)/(5*a^3*d*(a + a*Sin[c + d*x])^5) - (2*Cos[c + d*x]^3)/(3*a^2*d*(a^2 + a^2*Sin[c + d*x])^3) + (2*Cos[c + d*x])/(d*(a^8 + a^8*Sin[c + d*x]))","A",5,2,21,0.09524,1,"{2680, 8}"
89,1,36,0,0.0461098,"\int \frac{\cos ^7(c+d x)}{(a+a \sin (c+d x))^8} \, dx","Int[Cos[c + d*x]^7/(a + a*Sin[c + d*x])^8,x]","-\frac{(a-a \sin (c+d x))^4}{8 d \left(a^3 \sin (c+d x)+a^3\right)^4}","-\frac{(a-a \sin (c+d x))^4}{8 d \left(a^3 \sin (c+d x)+a^3\right)^4}",1,"-(a - a*Sin[c + d*x])^4/(8*d*(a^3 + a^3*Sin[c + d*x])^4)","A",2,2,21,0.09524,1,"{2667, 37}"
90,1,58,0,0.0803006,"\int \frac{\cos ^6(c+d x)}{(a+a \sin (c+d x))^8} \, dx","Int[Cos[c + d*x]^6/(a + a*Sin[c + d*x])^8,x]","-\frac{\cos ^7(c+d x)}{63 a d (a \sin (c+d x)+a)^7}-\frac{\cos ^7(c+d x)}{9 d (a \sin (c+d x)+a)^8}","-\frac{\cos ^7(c+d x)}{63 a d (a \sin (c+d x)+a)^7}-\frac{\cos ^7(c+d x)}{9 d (a \sin (c+d x)+a)^8}",1,"-Cos[c + d*x]^7/(9*d*(a + a*Sin[c + d*x])^8) - Cos[c + d*x]^7/(63*a*d*(a + a*Sin[c + d*x])^7)","A",2,2,21,0.09524,1,"{2672, 2671}"
91,1,65,0,0.0586843,"\int \frac{\cos ^5(c+d x)}{(a+a \sin (c+d x))^8} \, dx","Int[Cos[c + d*x]^5/(a + a*Sin[c + d*x])^8,x]","-\frac{1}{3 a^5 d (a \sin (c+d x)+a)^3}+\frac{1}{d \left(a^2 \sin (c+d x)+a^2\right)^4}-\frac{4}{5 a^3 d (a \sin (c+d x)+a)^5}","-\frac{1}{3 a^5 d (a \sin (c+d x)+a)^3}+\frac{1}{d \left(a^2 \sin (c+d x)+a^2\right)^4}-\frac{4}{5 a^3 d (a \sin (c+d x)+a)^5}",1,"-4/(5*a^3*d*(a + a*Sin[c + d*x])^5) - 1/(3*a^5*d*(a + a*Sin[c + d*x])^3) + 1/(d*(a^2 + a^2*Sin[c + d*x])^4)","A",3,2,21,0.09524,1,"{2667, 43}"
92,1,118,0,0.1675703,"\int \frac{\cos ^4(c+d x)}{(a+a \sin (c+d x))^8} \, dx","Int[Cos[c + d*x]^4/(a + a*Sin[c + d*x])^8,x]","-\frac{2 \cos ^5(c+d x)}{1155 a^3 d (a \sin (c+d x)+a)^5}-\frac{2 \cos ^5(c+d x)}{231 a^2 d (a \sin (c+d x)+a)^6}-\frac{\cos ^5(c+d x)}{33 a d (a \sin (c+d x)+a)^7}-\frac{\cos ^5(c+d x)}{11 d (a \sin (c+d x)+a)^8}","-\frac{2 \cos ^5(c+d x)}{1155 a^3 d (a \sin (c+d x)+a)^5}-\frac{2 \cos ^5(c+d x)}{231 a^2 d (a \sin (c+d x)+a)^6}-\frac{\cos ^5(c+d x)}{33 a d (a \sin (c+d x)+a)^7}-\frac{\cos ^5(c+d x)}{11 d (a \sin (c+d x)+a)^8}",1,"-Cos[c + d*x]^5/(11*d*(a + a*Sin[c + d*x])^8) - Cos[c + d*x]^5/(33*a*d*(a + a*Sin[c + d*x])^7) - (2*Cos[c + d*x]^5)/(231*a^2*d*(a + a*Sin[c + d*x])^6) - (2*Cos[c + d*x]^5)/(1155*a^3*d*(a + a*Sin[c + d*x])^5)","A",4,2,21,0.09524,1,"{2672, 2671}"
93,1,45,0,0.0525286,"\int \frac{\cos ^3(c+d x)}{(a+a \sin (c+d x))^8} \, dx","Int[Cos[c + d*x]^3/(a + a*Sin[c + d*x])^8,x]","\frac{1}{5 a^3 d (a \sin (c+d x)+a)^5}-\frac{1}{3 a^2 d (a \sin (c+d x)+a)^6}","\frac{1}{5 a^3 d (a \sin (c+d x)+a)^5}-\frac{1}{3 a^2 d (a \sin (c+d x)+a)^6}",1,"-1/(3*a^2*d*(a + a*Sin[c + d*x])^6) + 1/(5*a^3*d*(a + a*Sin[c + d*x])^5)","A",3,2,21,0.09524,1,"{2667, 43}"
94,1,183,0,0.2717029,"\int \frac{\cos ^2(c+d x)}{(a+a \sin (c+d x))^8} \, dx","Int[Cos[c + d*x]^2/(a + a*Sin[c + d*x])^8,x]","-\frac{8 \cos ^3(c+d x)}{9009 a^2 d \left(a^2 \sin (c+d x)+a^2\right)^3}-\frac{8 \cos ^3(c+d x)}{3003 d \left(a^2 \sin (c+d x)+a^2\right)^4}-\frac{20 \cos ^3(c+d x)}{3003 a^3 d (a \sin (c+d x)+a)^5}-\frac{20 \cos ^3(c+d x)}{1287 a^2 d (a \sin (c+d x)+a)^6}-\frac{5 \cos ^3(c+d x)}{143 a d (a \sin (c+d x)+a)^7}-\frac{\cos ^3(c+d x)}{13 d (a \sin (c+d x)+a)^8}","-\frac{8 \cos ^3(c+d x)}{9009 a^2 d \left(a^2 \sin (c+d x)+a^2\right)^3}-\frac{8 \cos ^3(c+d x)}{3003 d \left(a^2 \sin (c+d x)+a^2\right)^4}-\frac{20 \cos ^3(c+d x)}{3003 a^3 d (a \sin (c+d x)+a)^5}-\frac{20 \cos ^3(c+d x)}{1287 a^2 d (a \sin (c+d x)+a)^6}-\frac{5 \cos ^3(c+d x)}{143 a d (a \sin (c+d x)+a)^7}-\frac{\cos ^3(c+d x)}{13 d (a \sin (c+d x)+a)^8}",1,"-Cos[c + d*x]^3/(13*d*(a + a*Sin[c + d*x])^8) - (5*Cos[c + d*x]^3)/(143*a*d*(a + a*Sin[c + d*x])^7) - (20*Cos[c + d*x]^3)/(1287*a^2*d*(a + a*Sin[c + d*x])^6) - (20*Cos[c + d*x]^3)/(3003*a^3*d*(a + a*Sin[c + d*x])^5) - (8*Cos[c + d*x]^3)/(3003*d*(a^2 + a^2*Sin[c + d*x])^4) - (8*Cos[c + d*x]^3)/(9009*a^2*d*(a^2 + a^2*Sin[c + d*x])^3)","A",6,2,21,0.09524,1,"{2672, 2671}"
95,1,22,0,0.0250511,"\int \frac{\cos (c+d x)}{(a+a \sin (c+d x))^8} \, dx","Int[Cos[c + d*x]/(a + a*Sin[c + d*x])^8,x]","-\frac{1}{7 a d (a \sin (c+d x)+a)^7}","-\frac{1}{7 a d (a \sin (c+d x)+a)^7}",1,"-1/(7*a*d*(a + a*Sin[c + d*x])^7)","A",2,2,19,0.1053,1,"{2667, 32}"
96,1,194,0,0.1122546,"\int \frac{\sec (c+d x)}{(a+a \sin (c+d x))^8} \, dx","Int[Sec[c + d*x]/(a + a*Sin[c + d*x])^8,x]","-\frac{1}{256 d \left(a^8 \sin (c+d x)+a^8\right)}-\frac{1}{256 d \left(a^4 \sin (c+d x)+a^4\right)^2}-\frac{1}{192 a^5 d (a \sin (c+d x)+a)^3}-\frac{1}{128 d \left(a^2 \sin (c+d x)+a^2\right)^4}-\frac{1}{80 a^3 d (a \sin (c+d x)+a)^5}-\frac{1}{48 a^2 d (a \sin (c+d x)+a)^6}+\frac{\tanh ^{-1}(\sin (c+d x))}{256 a^8 d}-\frac{1}{28 a d (a \sin (c+d x)+a)^7}-\frac{1}{16 d (a \sin (c+d x)+a)^8}","-\frac{1}{256 d \left(a^8 \sin (c+d x)+a^8\right)}-\frac{1}{256 d \left(a^4 \sin (c+d x)+a^4\right)^2}-\frac{1}{192 a^5 d (a \sin (c+d x)+a)^3}-\frac{1}{128 d \left(a^2 \sin (c+d x)+a^2\right)^4}-\frac{1}{80 a^3 d (a \sin (c+d x)+a)^5}-\frac{1}{48 a^2 d (a \sin (c+d x)+a)^6}+\frac{\tanh ^{-1}(\sin (c+d x))}{256 a^8 d}-\frac{1}{28 a d (a \sin (c+d x)+a)^7}-\frac{1}{16 d (a \sin (c+d x)+a)^8}",1,"ArcTanh[Sin[c + d*x]]/(256*a^8*d) - 1/(16*d*(a + a*Sin[c + d*x])^8) - 1/(28*a*d*(a + a*Sin[c + d*x])^7) - 1/(48*a^2*d*(a + a*Sin[c + d*x])^6) - 1/(80*a^3*d*(a + a*Sin[c + d*x])^5) - 1/(192*a^5*d*(a + a*Sin[c + d*x])^3) - 1/(128*d*(a^2 + a^2*Sin[c + d*x])^4) - 1/(256*d*(a^4 + a^4*Sin[c + d*x])^2) - 1/(256*d*(a^8 + a^8*Sin[c + d*x]))","A",4,3,19,0.1579,1,"{2667, 44, 206}"
97,1,245,0,0.40397,"\int \frac{\sec ^2(c+d x)}{(a+a \sin (c+d x))^8} \, dx","Int[Sec[c + d*x]^2/(a + a*Sin[c + d*x])^8,x]","\frac{128 \tan (c+d x)}{12155 a^8 d}-\frac{64 \sec (c+d x)}{12155 d \left(a^8 \sin (c+d x)+a^8\right)}-\frac{64 \sec (c+d x)}{12155 d \left(a^4 \sin (c+d x)+a^4\right)^2}-\frac{16 \sec (c+d x)}{2431 a^2 d \left(a^2 \sin (c+d x)+a^2\right)^3}-\frac{112 \sec (c+d x)}{12155 d \left(a^2 \sin (c+d x)+a^2\right)^4}-\frac{168 \sec (c+d x)}{12155 a^3 d (a \sin (c+d x)+a)^5}-\frac{24 \sec (c+d x)}{1105 a^2 d (a \sin (c+d x)+a)^6}-\frac{3 \sec (c+d x)}{85 a d (a \sin (c+d x)+a)^7}-\frac{\sec (c+d x)}{17 d (a \sin (c+d x)+a)^8}","\frac{128 \tan (c+d x)}{12155 a^8 d}-\frac{64 \sec (c+d x)}{12155 d \left(a^8 \sin (c+d x)+a^8\right)}-\frac{64 \sec (c+d x)}{12155 d \left(a^4 \sin (c+d x)+a^4\right)^2}-\frac{16 \sec (c+d x)}{2431 a^2 d \left(a^2 \sin (c+d x)+a^2\right)^3}-\frac{112 \sec (c+d x)}{12155 d \left(a^2 \sin (c+d x)+a^2\right)^4}-\frac{168 \sec (c+d x)}{12155 a^3 d (a \sin (c+d x)+a)^5}-\frac{24 \sec (c+d x)}{1105 a^2 d (a \sin (c+d x)+a)^6}-\frac{3 \sec (c+d x)}{85 a d (a \sin (c+d x)+a)^7}-\frac{\sec (c+d x)}{17 d (a \sin (c+d x)+a)^8}",1,"-Sec[c + d*x]/(17*d*(a + a*Sin[c + d*x])^8) - (3*Sec[c + d*x])/(85*a*d*(a + a*Sin[c + d*x])^7) - (24*Sec[c + d*x])/(1105*a^2*d*(a + a*Sin[c + d*x])^6) - (168*Sec[c + d*x])/(12155*a^3*d*(a + a*Sin[c + d*x])^5) - (112*Sec[c + d*x])/(12155*d*(a^2 + a^2*Sin[c + d*x])^4) - (16*Sec[c + d*x])/(2431*a^2*d*(a^2 + a^2*Sin[c + d*x])^3) - (64*Sec[c + d*x])/(12155*d*(a^4 + a^4*Sin[c + d*x])^2) - (64*Sec[c + d*x])/(12155*d*(a^8 + a^8*Sin[c + d*x])) + (128*Tan[c + d*x])/(12155*a^8*d)","A",10,3,21,0.1429,1,"{2672, 3767, 8}"
98,1,238,0,0.1712667,"\int \frac{\sec ^3(c+d x)}{(a+a \sin (c+d x))^8} \, dx","Int[Sec[c + d*x]^3/(a + a*Sin[c + d*x])^8,x]","\frac{1}{1024 d \left(a^8-a^8 \sin (c+d x)\right)}-\frac{9}{1024 d \left(a^8 \sin (c+d x)+a^8\right)}-\frac{1}{128 d \left(a^4 \sin (c+d x)+a^4\right)^2}-\frac{3}{256 d \left(a^2 \sin (c+d x)+a^2\right)^4}-\frac{1}{48 a^2 d (a \sin (c+d x)+a)^6}-\frac{1}{64 a^3 d (a \sin (c+d x)+a)^5}-\frac{7}{768 a^5 d (a \sin (c+d x)+a)^3}+\frac{5 \tanh ^{-1}(\sin (c+d x))}{512 a^8 d}-\frac{a}{36 d (a \sin (c+d x)+a)^9}-\frac{1}{32 d (a \sin (c+d x)+a)^8}-\frac{3}{112 a d (a \sin (c+d x)+a)^7}","\frac{1}{1024 d \left(a^8-a^8 \sin (c+d x)\right)}-\frac{9}{1024 d \left(a^8 \sin (c+d x)+a^8\right)}-\frac{1}{128 d \left(a^4 \sin (c+d x)+a^4\right)^2}-\frac{3}{256 d \left(a^2 \sin (c+d x)+a^2\right)^4}-\frac{1}{48 a^2 d (a \sin (c+d x)+a)^6}-\frac{1}{64 a^3 d (a \sin (c+d x)+a)^5}-\frac{7}{768 a^5 d (a \sin (c+d x)+a)^3}+\frac{5 \tanh ^{-1}(\sin (c+d x))}{512 a^8 d}-\frac{a}{36 d (a \sin (c+d x)+a)^9}-\frac{1}{32 d (a \sin (c+d x)+a)^8}-\frac{3}{112 a d (a \sin (c+d x)+a)^7}",1,"(5*ArcTanh[Sin[c + d*x]])/(512*a^8*d) - a/(36*d*(a + a*Sin[c + d*x])^9) - 1/(32*d*(a + a*Sin[c + d*x])^8) - 3/(112*a*d*(a + a*Sin[c + d*x])^7) - 1/(48*a^2*d*(a + a*Sin[c + d*x])^6) - 1/(64*a^3*d*(a + a*Sin[c + d*x])^5) - 7/(768*a^5*d*(a + a*Sin[c + d*x])^3) - 3/(256*d*(a^2 + a^2*Sin[c + d*x])^4) - 1/(128*d*(a^4 + a^4*Sin[c + d*x])^2) + 1/(1024*d*(a^8 - a^8*Sin[c + d*x])) - 9/(1024*d*(a^8 + a^8*Sin[c + d*x]))","A",4,3,21,0.1429,1,"{2667, 44, 206}"
99,1,279,0,0.4185729,"\int \frac{\sec ^4(c+d x)}{(a+a \sin (c+d x))^8} \, dx","Int[Sec[c + d*x]^4/(a + a*Sin[c + d*x])^8,x]","\frac{128 \tan ^3(c+d x)}{12597 a^8 d}+\frac{128 \tan (c+d x)}{4199 a^8 d}-\frac{32 \sec ^3(c+d x)}{4199 d \left(a^8 \sin (c+d x)+a^8\right)}-\frac{32 \sec ^3(c+d x)}{4199 d \left(a^4 \sin (c+d x)+a^4\right)^2}-\frac{112 \sec ^3(c+d x)}{12597 a^2 d \left(a^2 \sin (c+d x)+a^2\right)^3}-\frac{48 \sec ^3(c+d x)}{4199 d \left(a^2 \sin (c+d x)+a^2\right)^4}-\frac{66 \sec ^3(c+d x)}{4199 a^3 d (a \sin (c+d x)+a)^5}-\frac{22 \sec ^3(c+d x)}{969 a^2 d (a \sin (c+d x)+a)^6}-\frac{11 \sec ^3(c+d x)}{323 a d (a \sin (c+d x)+a)^7}-\frac{\sec ^3(c+d x)}{19 d (a \sin (c+d x)+a)^8}","\frac{128 \tan ^3(c+d x)}{12597 a^8 d}+\frac{128 \tan (c+d x)}{4199 a^8 d}-\frac{32 \sec ^3(c+d x)}{4199 d \left(a^8 \sin (c+d x)+a^8\right)}-\frac{32 \sec ^3(c+d x)}{4199 d \left(a^4 \sin (c+d x)+a^4\right)^2}-\frac{112 \sec ^3(c+d x)}{12597 a^2 d \left(a^2 \sin (c+d x)+a^2\right)^3}-\frac{48 \sec ^3(c+d x)}{4199 d \left(a^2 \sin (c+d x)+a^2\right)^4}-\frac{66 \sec ^3(c+d x)}{4199 a^3 d (a \sin (c+d x)+a)^5}-\frac{22 \sec ^3(c+d x)}{969 a^2 d (a \sin (c+d x)+a)^6}-\frac{11 \sec ^3(c+d x)}{323 a d (a \sin (c+d x)+a)^7}-\frac{\sec ^3(c+d x)}{19 d (a \sin (c+d x)+a)^8}",1,"-Sec[c + d*x]^3/(19*d*(a + a*Sin[c + d*x])^8) - (11*Sec[c + d*x]^3)/(323*a*d*(a + a*Sin[c + d*x])^7) - (22*Sec[c + d*x]^3)/(969*a^2*d*(a + a*Sin[c + d*x])^6) - (66*Sec[c + d*x]^3)/(4199*a^3*d*(a + a*Sin[c + d*x])^5) - (48*Sec[c + d*x]^3)/(4199*d*(a^2 + a^2*Sin[c + d*x])^4) - (112*Sec[c + d*x]^3)/(12597*a^2*d*(a^2 + a^2*Sin[c + d*x])^3) - (32*Sec[c + d*x]^3)/(4199*d*(a^4 + a^4*Sin[c + d*x])^2) - (32*Sec[c + d*x]^3)/(4199*d*(a^8 + a^8*Sin[c + d*x])) + (128*Tan[c + d*x])/(4199*a^8*d) + (128*Tan[c + d*x]^3)/(12597*a^8*d)","A",10,2,21,0.09524,1,"{2672, 3767}"
100,1,284,0,0.2156328,"\int \frac{\sec ^5(c+d x)}{(a+a \sin (c+d x))^8} \, dx","Int[Sec[c + d*x]^5/(a + a*Sin[c + d*x])^8,x]","-\frac{a^2}{80 d (a \sin (c+d x)+a)^{10}}+\frac{11}{4096 d \left(a^8-a^8 \sin (c+d x)\right)}-\frac{55}{4096 d \left(a^8 \sin (c+d x)+a^8\right)}+\frac{1}{4096 d \left(a^4-a^4 \sin (c+d x)\right)^2}-\frac{45}{4096 d \left(a^4 \sin (c+d x)+a^4\right)^2}-\frac{7}{512 d \left(a^2 \sin (c+d x)+a^2\right)^4}-\frac{5}{256 a^2 d (a \sin (c+d x)+a)^6}-\frac{21}{1280 a^3 d (a \sin (c+d x)+a)^5}-\frac{3}{256 a^5 d (a \sin (c+d x)+a)^3}+\frac{33 \tanh ^{-1}(\sin (c+d x))}{2048 a^8 d}-\frac{a}{48 d (a \sin (c+d x)+a)^9}-\frac{3}{128 d (a \sin (c+d x)+a)^8}-\frac{5}{224 a d (a \sin (c+d x)+a)^7}","-\frac{a^2}{80 d (a \sin (c+d x)+a)^{10}}+\frac{11}{4096 d \left(a^8-a^8 \sin (c+d x)\right)}-\frac{55}{4096 d \left(a^8 \sin (c+d x)+a^8\right)}+\frac{1}{4096 d \left(a^4-a^4 \sin (c+d x)\right)^2}-\frac{45}{4096 d \left(a^4 \sin (c+d x)+a^4\right)^2}-\frac{7}{512 d \left(a^2 \sin (c+d x)+a^2\right)^4}-\frac{5}{256 a^2 d (a \sin (c+d x)+a)^6}-\frac{21}{1280 a^3 d (a \sin (c+d x)+a)^5}-\frac{3}{256 a^5 d (a \sin (c+d x)+a)^3}+\frac{33 \tanh ^{-1}(\sin (c+d x))}{2048 a^8 d}-\frac{a}{48 d (a \sin (c+d x)+a)^9}-\frac{3}{128 d (a \sin (c+d x)+a)^8}-\frac{5}{224 a d (a \sin (c+d x)+a)^7}",1,"(33*ArcTanh[Sin[c + d*x]])/(2048*a^8*d) - a^2/(80*d*(a + a*Sin[c + d*x])^10) - a/(48*d*(a + a*Sin[c + d*x])^9) - 3/(128*d*(a + a*Sin[c + d*x])^8) - 5/(224*a*d*(a + a*Sin[c + d*x])^7) - 5/(256*a^2*d*(a + a*Sin[c + d*x])^6) - 21/(1280*a^3*d*(a + a*Sin[c + d*x])^5) - 3/(256*a^5*d*(a + a*Sin[c + d*x])^3) - 7/(512*d*(a^2 + a^2*Sin[c + d*x])^4) + 1/(4096*d*(a^4 - a^4*Sin[c + d*x])^2) - 45/(4096*d*(a^4 + a^4*Sin[c + d*x])^2) + 11/(4096*d*(a^8 - a^8*Sin[c + d*x])) - 55/(4096*d*(a^8 + a^8*Sin[c + d*x]))","A",4,3,21,0.1429,1,"{2667, 44, 206}"
101,1,97,0,0.0829884,"\int \cos ^7(c+d x) \sqrt{a+a \sin (c+d x)} \, dx","Int[Cos[c + d*x]^7*Sqrt[a + a*Sin[c + d*x]],x]","-\frac{2 (a \sin (c+d x)+a)^{15/2}}{15 a^7 d}+\frac{12 (a \sin (c+d x)+a)^{13/2}}{13 a^6 d}-\frac{24 (a \sin (c+d x)+a)^{11/2}}{11 a^5 d}+\frac{16 (a \sin (c+d x)+a)^{9/2}}{9 a^4 d}","-\frac{2 (a \sin (c+d x)+a)^{15/2}}{15 a^7 d}+\frac{12 (a \sin (c+d x)+a)^{13/2}}{13 a^6 d}-\frac{24 (a \sin (c+d x)+a)^{11/2}}{11 a^5 d}+\frac{16 (a \sin (c+d x)+a)^{9/2}}{9 a^4 d}",1,"(16*(a + a*Sin[c + d*x])^(9/2))/(9*a^4*d) - (24*(a + a*Sin[c + d*x])^(11/2))/(11*a^5*d) + (12*(a + a*Sin[c + d*x])^(13/2))/(13*a^6*d) - (2*(a + a*Sin[c + d*x])^(15/2))/(15*a^7*d)","A",3,2,23,0.08696,1,"{2667, 43}"
102,1,127,0,0.2583616,"\int \cos ^6(c+d x) \sqrt{a+a \sin (c+d x)} \, dx","Int[Cos[c + d*x]^6*Sqrt[a + a*Sin[c + d*x]],x]","-\frac{24 a^2 \cos ^7(c+d x)}{143 d (a \sin (c+d x)+a)^{3/2}}-\frac{64 a^3 \cos ^7(c+d x)}{429 d (a \sin (c+d x)+a)^{5/2}}-\frac{256 a^4 \cos ^7(c+d x)}{3003 d (a \sin (c+d x)+a)^{7/2}}-\frac{2 a \cos ^7(c+d x)}{13 d \sqrt{a \sin (c+d x)+a}}","-\frac{24 a^2 \cos ^7(c+d x)}{143 d (a \sin (c+d x)+a)^{3/2}}-\frac{64 a^3 \cos ^7(c+d x)}{429 d (a \sin (c+d x)+a)^{5/2}}-\frac{256 a^4 \cos ^7(c+d x)}{3003 d (a \sin (c+d x)+a)^{7/2}}-\frac{2 a \cos ^7(c+d x)}{13 d \sqrt{a \sin (c+d x)+a}}",1,"(-256*a^4*Cos[c + d*x]^7)/(3003*d*(a + a*Sin[c + d*x])^(7/2)) - (64*a^3*Cos[c + d*x]^7)/(429*d*(a + a*Sin[c + d*x])^(5/2)) - (24*a^2*Cos[c + d*x]^7)/(143*d*(a + a*Sin[c + d*x])^(3/2)) - (2*a*Cos[c + d*x]^7)/(13*d*Sqrt[a + a*Sin[c + d*x]])","A",4,2,23,0.08696,1,"{2674, 2673}"
103,1,73,0,0.0726169,"\int \cos ^5(c+d x) \sqrt{a+a \sin (c+d x)} \, dx","Int[Cos[c + d*x]^5*Sqrt[a + a*Sin[c + d*x]],x]","\frac{2 (a \sin (c+d x)+a)^{11/2}}{11 a^5 d}-\frac{8 (a \sin (c+d x)+a)^{9/2}}{9 a^4 d}+\frac{8 (a \sin (c+d x)+a)^{7/2}}{7 a^3 d}","\frac{2 (a \sin (c+d x)+a)^{11/2}}{11 a^5 d}-\frac{8 (a \sin (c+d x)+a)^{9/2}}{9 a^4 d}+\frac{8 (a \sin (c+d x)+a)^{7/2}}{7 a^3 d}",1,"(8*(a + a*Sin[c + d*x])^(7/2))/(7*a^3*d) - (8*(a + a*Sin[c + d*x])^(9/2))/(9*a^4*d) + (2*(a + a*Sin[c + d*x])^(11/2))/(11*a^5*d)","A",3,2,23,0.08696,1,"{2667, 43}"
104,1,95,0,0.1785182,"\int \cos ^4(c+d x) \sqrt{a+a \sin (c+d x)} \, dx","Int[Cos[c + d*x]^4*Sqrt[a + a*Sin[c + d*x]],x]","-\frac{16 a^2 \cos ^5(c+d x)}{63 d (a \sin (c+d x)+a)^{3/2}}-\frac{64 a^3 \cos ^5(c+d x)}{315 d (a \sin (c+d x)+a)^{5/2}}-\frac{2 a \cos ^5(c+d x)}{9 d \sqrt{a \sin (c+d x)+a}}","-\frac{16 a^2 \cos ^5(c+d x)}{63 d (a \sin (c+d x)+a)^{3/2}}-\frac{64 a^3 \cos ^5(c+d x)}{315 d (a \sin (c+d x)+a)^{5/2}}-\frac{2 a \cos ^5(c+d x)}{9 d \sqrt{a \sin (c+d x)+a}}",1,"(-64*a^3*Cos[c + d*x]^5)/(315*d*(a + a*Sin[c + d*x])^(5/2)) - (16*a^2*Cos[c + d*x]^5)/(63*d*(a + a*Sin[c + d*x])^(3/2)) - (2*a*Cos[c + d*x]^5)/(9*d*Sqrt[a + a*Sin[c + d*x]])","A",3,2,23,0.08696,1,"{2674, 2673}"
105,1,49,0,0.0649647,"\int \cos ^3(c+d x) \sqrt{a+a \sin (c+d x)} \, dx","Int[Cos[c + d*x]^3*Sqrt[a + a*Sin[c + d*x]],x]","\frac{4 (a \sin (c+d x)+a)^{5/2}}{5 a^2 d}-\frac{2 (a \sin (c+d x)+a)^{7/2}}{7 a^3 d}","\frac{4 (a \sin (c+d x)+a)^{5/2}}{5 a^2 d}-\frac{2 (a \sin (c+d x)+a)^{7/2}}{7 a^3 d}",1,"(4*(a + a*Sin[c + d*x])^(5/2))/(5*a^2*d) - (2*(a + a*Sin[c + d*x])^(7/2))/(7*a^3*d)","A",3,2,23,0.08696,1,"{2667, 43}"
106,1,63,0,0.1107746,"\int \cos ^2(c+d x) \sqrt{a+a \sin (c+d x)} \, dx","Int[Cos[c + d*x]^2*Sqrt[a + a*Sin[c + d*x]],x]","-\frac{8 a^2 \cos ^3(c+d x)}{15 d (a \sin (c+d x)+a)^{3/2}}-\frac{2 a \cos ^3(c+d x)}{5 d \sqrt{a \sin (c+d x)+a}}","-\frac{8 a^2 \cos ^3(c+d x)}{15 d (a \sin (c+d x)+a)^{3/2}}-\frac{2 a \cos ^3(c+d x)}{5 d \sqrt{a \sin (c+d x)+a}}",1,"(-8*a^2*Cos[c + d*x]^3)/(15*d*(a + a*Sin[c + d*x])^(3/2)) - (2*a*Cos[c + d*x]^3)/(5*d*Sqrt[a + a*Sin[c + d*x]])","A",2,2,23,0.08696,1,"{2674, 2673}"
107,1,24,0,0.0327388,"\int \cos (c+d x) \sqrt{a+a \sin (c+d x)} \, dx","Int[Cos[c + d*x]*Sqrt[a + a*Sin[c + d*x]],x]","\frac{2 (a \sin (c+d x)+a)^{3/2}}{3 a d}","\frac{2 (a \sin (c+d x)+a)^{3/2}}{3 a d}",1,"(2*(a + a*Sin[c + d*x])^(3/2))/(3*a*d)","A",2,2,21,0.09524,1,"{2667, 32}"
108,1,40,0,0.0604097,"\int \sec (c+d x) \sqrt{a+a \sin (c+d x)} \, dx","Int[Sec[c + d*x]*Sqrt[a + a*Sin[c + d*x]],x]","\frac{\sqrt{2} \sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a \sin (c+d x)+a}}{\sqrt{2} \sqrt{a}}\right)}{d}","\frac{\sqrt{2} \sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a \sin (c+d x)+a}}{\sqrt{2} \sqrt{a}}\right)}{d}",1,"(Sqrt[2]*Sqrt[a]*ArcTanh[Sqrt[a + a*Sin[c + d*x]]/(Sqrt[2]*Sqrt[a])])/d","A",3,3,21,0.1429,1,"{2667, 63, 206}"
109,1,72,0,0.0793447,"\int \sec ^2(c+d x) \sqrt{a+a \sin (c+d x)} \, dx","Int[Sec[c + d*x]^2*Sqrt[a + a*Sin[c + d*x]],x]","\frac{\sec (c+d x) \sqrt{a \sin (c+d x)+a}}{d}-\frac{\sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a \sin (c+d x)+a}}\right)}{\sqrt{2} d}","\frac{\sec (c+d x) \sqrt{a \sin (c+d x)+a}}{d}-\frac{\sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a \sin (c+d x)+a}}\right)}{\sqrt{2} d}",1,"-((Sqrt[a]*ArcTanh[(Sqrt[a]*Cos[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sin[c + d*x]])])/(Sqrt[2]*d)) + (Sec[c + d*x]*Sqrt[a + a*Sin[c + d*x]])/d","A",3,3,23,0.1304,1,"{2675, 2649, 206}"
110,1,95,0,0.1234471,"\int \sec ^3(c+d x) \sqrt{a+a \sin (c+d x)} \, dx","Int[Sec[c + d*x]^3*Sqrt[a + a*Sin[c + d*x]],x]","-\frac{3 a}{4 d \sqrt{a \sin (c+d x)+a}}+\frac{3 \sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a \sin (c+d x)+a}}{\sqrt{2} \sqrt{a}}\right)}{4 \sqrt{2} d}+\frac{\sec ^2(c+d x) \sqrt{a \sin (c+d x)+a}}{2 d}","-\frac{3 a}{4 d \sqrt{a \sin (c+d x)+a}}+\frac{3 \sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a \sin (c+d x)+a}}{\sqrt{2} \sqrt{a}}\right)}{4 \sqrt{2} d}+\frac{\sec ^2(c+d x) \sqrt{a \sin (c+d x)+a}}{2 d}",1,"(3*Sqrt[a]*ArcTanh[Sqrt[a + a*Sin[c + d*x]]/(Sqrt[2]*Sqrt[a])])/(4*Sqrt[2]*d) - (3*a)/(4*d*Sqrt[a + a*Sin[c + d*x]]) + (Sec[c + d*x]^2*Sqrt[a + a*Sin[c + d*x]])/(2*d)","A",5,5,23,0.2174,1,"{2675, 2667, 51, 63, 206}"
111,1,137,0,0.1587929,"\int \sec ^4(c+d x) \sqrt{a+a \sin (c+d x)} \, dx","Int[Sec[c + d*x]^4*Sqrt[a + a*Sin[c + d*x]],x]","-\frac{5 a^2 \cos (c+d x)}{8 d (a \sin (c+d x)+a)^{3/2}}+\frac{\sec ^3(c+d x) \sqrt{a \sin (c+d x)+a}}{3 d}+\frac{5 a \sec (c+d x)}{6 d \sqrt{a \sin (c+d x)+a}}-\frac{5 \sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a \sin (c+d x)+a}}\right)}{8 \sqrt{2} d}","-\frac{5 a^2 \cos (c+d x)}{8 d (a \sin (c+d x)+a)^{3/2}}+\frac{\sec ^3(c+d x) \sqrt{a \sin (c+d x)+a}}{3 d}+\frac{5 a \sec (c+d x)}{6 d \sqrt{a \sin (c+d x)+a}}-\frac{5 \sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a \sin (c+d x)+a}}\right)}{8 \sqrt{2} d}",1,"(-5*Sqrt[a]*ArcTanh[(Sqrt[a]*Cos[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sin[c + d*x]])])/(8*Sqrt[2]*d) - (5*a^2*Cos[c + d*x])/(8*d*(a + a*Sin[c + d*x])^(3/2)) + (5*a*Sec[c + d*x])/(6*d*Sqrt[a + a*Sin[c + d*x]]) + (Sec[c + d*x]^3*Sqrt[a + a*Sin[c + d*x]])/(3*d)","A",5,5,23,0.2174,1,"{2675, 2687, 2650, 2649, 206}"
112,1,149,0,0.2033733,"\int \sec ^5(c+d x) \sqrt{a+a \sin (c+d x)} \, dx","Int[Sec[c + d*x]^5*Sqrt[a + a*Sin[c + d*x]],x]","-\frac{35 a^2}{96 d (a \sin (c+d x)+a)^{3/2}}-\frac{35 a}{64 d \sqrt{a \sin (c+d x)+a}}+\frac{35 \sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a \sin (c+d x)+a}}{\sqrt{2} \sqrt{a}}\right)}{64 \sqrt{2} d}+\frac{\sec ^4(c+d x) \sqrt{a \sin (c+d x)+a}}{4 d}+\frac{7 a \sec ^2(c+d x)}{16 d \sqrt{a \sin (c+d x)+a}}","-\frac{35 a^2}{96 d (a \sin (c+d x)+a)^{3/2}}-\frac{35 a}{64 d \sqrt{a \sin (c+d x)+a}}+\frac{35 \sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a \sin (c+d x)+a}}{\sqrt{2} \sqrt{a}}\right)}{64 \sqrt{2} d}+\frac{\sec ^4(c+d x) \sqrt{a \sin (c+d x)+a}}{4 d}+\frac{7 a \sec ^2(c+d x)}{16 d \sqrt{a \sin (c+d x)+a}}",1,"(35*Sqrt[a]*ArcTanh[Sqrt[a + a*Sin[c + d*x]]/(Sqrt[2]*Sqrt[a])])/(64*Sqrt[2]*d) - (35*a^2)/(96*d*(a + a*Sin[c + d*x])^(3/2)) - (35*a)/(64*d*Sqrt[a + a*Sin[c + d*x]]) + (7*a*Sec[c + d*x]^2)/(16*d*Sqrt[a + a*Sin[c + d*x]]) + (Sec[c + d*x]^4*Sqrt[a + a*Sin[c + d*x]])/(4*d)","A",7,6,23,0.2609,1,"{2675, 2687, 2667, 51, 63, 206}"
113,1,197,0,0.293853,"\int \sec ^6(c+d x) \sqrt{a+a \sin (c+d x)} \, dx","Int[Sec[c + d*x]^6*Sqrt[a + a*Sin[c + d*x]],x]","-\frac{63 a^2 \cos (c+d x)}{128 d (a \sin (c+d x)+a)^{3/2}}-\frac{21 a^2 \sec (c+d x)}{80 d (a \sin (c+d x)+a)^{3/2}}+\frac{\sec ^5(c+d x) \sqrt{a \sin (c+d x)+a}}{5 d}+\frac{3 a \sec ^3(c+d x)}{10 d \sqrt{a \sin (c+d x)+a}}+\frac{21 a \sec (c+d x)}{32 d \sqrt{a \sin (c+d x)+a}}-\frac{63 \sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a \sin (c+d x)+a}}\right)}{128 \sqrt{2} d}","-\frac{63 a^2 \cos (c+d x)}{128 d (a \sin (c+d x)+a)^{3/2}}-\frac{21 a^2 \sec (c+d x)}{80 d (a \sin (c+d x)+a)^{3/2}}+\frac{\sec ^5(c+d x) \sqrt{a \sin (c+d x)+a}}{5 d}+\frac{3 a \sec ^3(c+d x)}{10 d \sqrt{a \sin (c+d x)+a}}+\frac{21 a \sec (c+d x)}{32 d \sqrt{a \sin (c+d x)+a}}-\frac{63 \sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a \sin (c+d x)+a}}\right)}{128 \sqrt{2} d}",1,"(-63*Sqrt[a]*ArcTanh[(Sqrt[a]*Cos[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sin[c + d*x]])])/(128*Sqrt[2]*d) - (63*a^2*Cos[c + d*x])/(128*d*(a + a*Sin[c + d*x])^(3/2)) - (21*a^2*Sec[c + d*x])/(80*d*(a + a*Sin[c + d*x])^(3/2)) + (21*a*Sec[c + d*x])/(32*d*Sqrt[a + a*Sin[c + d*x]]) + (3*a*Sec[c + d*x]^3)/(10*d*Sqrt[a + a*Sin[c + d*x]]) + (Sec[c + d*x]^5*Sqrt[a + a*Sin[c + d*x]])/(5*d)","A",7,6,23,0.2609,1,"{2675, 2687, 2681, 2650, 2649, 206}"
114,1,97,0,0.0861733,"\int \cos ^7(c+d x) (a+a \sin (c+d x))^{3/2} \, dx","Int[Cos[c + d*x]^7*(a + a*Sin[c + d*x])^(3/2),x]","-\frac{2 (a \sin (c+d x)+a)^{17/2}}{17 a^7 d}+\frac{4 (a \sin (c+d x)+a)^{15/2}}{5 a^6 d}-\frac{24 (a \sin (c+d x)+a)^{13/2}}{13 a^5 d}+\frac{16 (a \sin (c+d x)+a)^{11/2}}{11 a^4 d}","-\frac{2 (a \sin (c+d x)+a)^{17/2}}{17 a^7 d}+\frac{4 (a \sin (c+d x)+a)^{15/2}}{5 a^6 d}-\frac{24 (a \sin (c+d x)+a)^{13/2}}{13 a^5 d}+\frac{16 (a \sin (c+d x)+a)^{11/2}}{11 a^4 d}",1,"(16*(a + a*Sin[c + d*x])^(11/2))/(11*a^4*d) - (24*(a + a*Sin[c + d*x])^(13/2))/(13*a^5*d) + (4*(a + a*Sin[c + d*x])^(15/2))/(5*a^6*d) - (2*(a + a*Sin[c + d*x])^(17/2))/(17*a^7*d)","A",3,2,23,0.08696,1,"{2667, 43}"
115,1,159,0,0.3021724,"\int \cos ^6(c+d x) (a+a \sin (c+d x))^{3/2} \, dx","Int[Cos[c + d*x]^6*(a + a*Sin[c + d*x])^(3/2),x]","-\frac{32 a^2 \cos ^7(c+d x)}{195 d \sqrt{a \sin (c+d x)+a}}-\frac{128 a^3 \cos ^7(c+d x)}{715 d (a \sin (c+d x)+a)^{3/2}}-\frac{1024 a^4 \cos ^7(c+d x)}{6435 d (a \sin (c+d x)+a)^{5/2}}-\frac{4096 a^5 \cos ^7(c+d x)}{45045 d (a \sin (c+d x)+a)^{7/2}}-\frac{2 a \cos ^7(c+d x) \sqrt{a \sin (c+d x)+a}}{15 d}","-\frac{32 a^2 \cos ^7(c+d x)}{195 d \sqrt{a \sin (c+d x)+a}}-\frac{128 a^3 \cos ^7(c+d x)}{715 d (a \sin (c+d x)+a)^{3/2}}-\frac{1024 a^4 \cos ^7(c+d x)}{6435 d (a \sin (c+d x)+a)^{5/2}}-\frac{4096 a^5 \cos ^7(c+d x)}{45045 d (a \sin (c+d x)+a)^{7/2}}-\frac{2 a \cos ^7(c+d x) \sqrt{a \sin (c+d x)+a}}{15 d}",1,"(-4096*a^5*Cos[c + d*x]^7)/(45045*d*(a + a*Sin[c + d*x])^(7/2)) - (1024*a^4*Cos[c + d*x]^7)/(6435*d*(a + a*Sin[c + d*x])^(5/2)) - (128*a^3*Cos[c + d*x]^7)/(715*d*(a + a*Sin[c + d*x])^(3/2)) - (32*a^2*Cos[c + d*x]^7)/(195*d*Sqrt[a + a*Sin[c + d*x]]) - (2*a*Cos[c + d*x]^7*Sqrt[a + a*Sin[c + d*x]])/(15*d)","A",5,2,23,0.08696,1,"{2674, 2673}"
116,1,73,0,0.0764606,"\int \cos ^5(c+d x) (a+a \sin (c+d x))^{3/2} \, dx","Int[Cos[c + d*x]^5*(a + a*Sin[c + d*x])^(3/2),x]","\frac{2 (a \sin (c+d x)+a)^{13/2}}{13 a^5 d}-\frac{8 (a \sin (c+d x)+a)^{11/2}}{11 a^4 d}+\frac{8 (a \sin (c+d x)+a)^{9/2}}{9 a^3 d}","\frac{2 (a \sin (c+d x)+a)^{13/2}}{13 a^5 d}-\frac{8 (a \sin (c+d x)+a)^{11/2}}{11 a^4 d}+\frac{8 (a \sin (c+d x)+a)^{9/2}}{9 a^3 d}",1,"(8*(a + a*Sin[c + d*x])^(9/2))/(9*a^3*d) - (8*(a + a*Sin[c + d*x])^(11/2))/(11*a^4*d) + (2*(a + a*Sin[c + d*x])^(13/2))/(13*a^5*d)","A",3,2,23,0.08696,1,"{2667, 43}"
117,1,127,0,0.2347394,"\int \cos ^4(c+d x) (a+a \sin (c+d x))^{3/2} \, dx","Int[Cos[c + d*x]^4*(a + a*Sin[c + d*x])^(3/2),x]","-\frac{8 a^2 \cos ^5(c+d x)}{33 d \sqrt{a \sin (c+d x)+a}}-\frac{64 a^3 \cos ^5(c+d x)}{231 d (a \sin (c+d x)+a)^{3/2}}-\frac{256 a^4 \cos ^5(c+d x)}{1155 d (a \sin (c+d x)+a)^{5/2}}-\frac{2 a \cos ^5(c+d x) \sqrt{a \sin (c+d x)+a}}{11 d}","-\frac{8 a^2 \cos ^5(c+d x)}{33 d \sqrt{a \sin (c+d x)+a}}-\frac{64 a^3 \cos ^5(c+d x)}{231 d (a \sin (c+d x)+a)^{3/2}}-\frac{256 a^4 \cos ^5(c+d x)}{1155 d (a \sin (c+d x)+a)^{5/2}}-\frac{2 a \cos ^5(c+d x) \sqrt{a \sin (c+d x)+a}}{11 d}",1,"(-256*a^4*Cos[c + d*x]^5)/(1155*d*(a + a*Sin[c + d*x])^(5/2)) - (64*a^3*Cos[c + d*x]^5)/(231*d*(a + a*Sin[c + d*x])^(3/2)) - (8*a^2*Cos[c + d*x]^5)/(33*d*Sqrt[a + a*Sin[c + d*x]]) - (2*a*Cos[c + d*x]^5*Sqrt[a + a*Sin[c + d*x]])/(11*d)","A",4,2,23,0.08696,1,"{2674, 2673}"
118,1,49,0,0.0677168,"\int \cos ^3(c+d x) (a+a \sin (c+d x))^{3/2} \, dx","Int[Cos[c + d*x]^3*(a + a*Sin[c + d*x])^(3/2),x]","\frac{4 (a \sin (c+d x)+a)^{7/2}}{7 a^2 d}-\frac{2 (a \sin (c+d x)+a)^{9/2}}{9 a^3 d}","\frac{4 (a \sin (c+d x)+a)^{7/2}}{7 a^2 d}-\frac{2 (a \sin (c+d x)+a)^{9/2}}{9 a^3 d}",1,"(4*(a + a*Sin[c + d*x])^(7/2))/(7*a^2*d) - (2*(a + a*Sin[c + d*x])^(9/2))/(9*a^3*d)","A",3,2,23,0.08696,1,"{2667, 43}"
119,1,95,0,0.1672305,"\int \cos ^2(c+d x) (a+a \sin (c+d x))^{3/2} \, dx","Int[Cos[c + d*x]^2*(a + a*Sin[c + d*x])^(3/2),x]","-\frac{16 a^2 \cos ^3(c+d x)}{35 d \sqrt{a \sin (c+d x)+a}}-\frac{64 a^3 \cos ^3(c+d x)}{105 d (a \sin (c+d x)+a)^{3/2}}-\frac{2 a \cos ^3(c+d x) \sqrt{a \sin (c+d x)+a}}{7 d}","-\frac{16 a^2 \cos ^3(c+d x)}{35 d \sqrt{a \sin (c+d x)+a}}-\frac{64 a^3 \cos ^3(c+d x)}{105 d (a \sin (c+d x)+a)^{3/2}}-\frac{2 a \cos ^3(c+d x) \sqrt{a \sin (c+d x)+a}}{7 d}",1,"(-64*a^3*Cos[c + d*x]^3)/(105*d*(a + a*Sin[c + d*x])^(3/2)) - (16*a^2*Cos[c + d*x]^3)/(35*d*Sqrt[a + a*Sin[c + d*x]]) - (2*a*Cos[c + d*x]^3*Sqrt[a + a*Sin[c + d*x]])/(7*d)","A",3,2,23,0.08696,1,"{2674, 2673}"
120,1,24,0,0.0342689,"\int \cos (c+d x) (a+a \sin (c+d x))^{3/2} \, dx","Int[Cos[c + d*x]*(a + a*Sin[c + d*x])^(3/2),x]","\frac{2 (a \sin (c+d x)+a)^{5/2}}{5 a d}","\frac{2 (a \sin (c+d x)+a)^{5/2}}{5 a d}",1,"(2*(a + a*Sin[c + d*x])^(5/2))/(5*a*d)","A",2,2,21,0.09524,1,"{2667, 32}"
121,1,62,0,0.0682787,"\int \sec (c+d x) (a+a \sin (c+d x))^{3/2} \, dx","Int[Sec[c + d*x]*(a + a*Sin[c + d*x])^(3/2),x]","\frac{2 \sqrt{2} a^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a \sin (c+d x)+a}}{\sqrt{2} \sqrt{a}}\right)}{d}-\frac{2 a \sqrt{a \sin (c+d x)+a}}{d}","\frac{2 \sqrt{2} a^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a \sin (c+d x)+a}}{\sqrt{2} \sqrt{a}}\right)}{d}-\frac{2 a \sqrt{a \sin (c+d x)+a}}{d}",1,"(2*Sqrt[2]*a^(3/2)*ArcTanh[Sqrt[a + a*Sin[c + d*x]]/(Sqrt[2]*Sqrt[a])])/d - (2*a*Sqrt[a + a*Sin[c + d*x]])/d","A",4,4,21,0.1905,1,"{2667, 50, 63, 206}"
122,1,26,0,0.0575865,"\int \sec ^2(c+d x) (a+a \sin (c+d x))^{3/2} \, dx","Int[Sec[c + d*x]^2*(a + a*Sin[c + d*x])^(3/2),x]","\frac{2 a \sec (c+d x) \sqrt{a \sin (c+d x)+a}}{d}","\frac{2 a \sec (c+d x) \sqrt{a \sin (c+d x)+a}}{d}",1,"(2*a*Sec[c + d*x]*Sqrt[a + a*Sin[c + d*x]])/d","A",1,1,23,0.04348,1,"{2673}"
123,1,73,0,0.1121902,"\int \sec ^3(c+d x) (a+a \sin (c+d x))^{3/2} \, dx","Int[Sec[c + d*x]^3*(a + a*Sin[c + d*x])^(3/2),x]","\frac{a^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a \sin (c+d x)+a}}{\sqrt{2} \sqrt{a}}\right)}{2 \sqrt{2} d}+\frac{\sec ^2(c+d x) (a \sin (c+d x)+a)^{3/2}}{2 d}","\frac{a^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a \sin (c+d x)+a}}{\sqrt{2} \sqrt{a}}\right)}{2 \sqrt{2} d}+\frac{\sec ^2(c+d x) (a \sin (c+d x)+a)^{3/2}}{2 d}",1,"(a^(3/2)*ArcTanh[Sqrt[a + a*Sin[c + d*x]]/(Sqrt[2]*Sqrt[a])])/(2*Sqrt[2]*d) + (Sec[c + d*x]^2*(a + a*Sin[c + d*x])^(3/2))/(2*d)","A",4,4,23,0.1739,1,"{2675, 2667, 63, 206}"
124,1,107,0,0.1349847,"\int \sec ^4(c+d x) (a+a \sin (c+d x))^{3/2} \, dx","Int[Sec[c + d*x]^4*(a + a*Sin[c + d*x])^(3/2),x]","-\frac{a^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a \sin (c+d x)+a}}\right)}{2 \sqrt{2} d}+\frac{\sec ^3(c+d x) (a \sin (c+d x)+a)^{3/2}}{3 d}+\frac{a \sec (c+d x) \sqrt{a \sin (c+d x)+a}}{2 d}","-\frac{a^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a \sin (c+d x)+a}}\right)}{2 \sqrt{2} d}+\frac{\sec ^3(c+d x) (a \sin (c+d x)+a)^{3/2}}{3 d}+\frac{a \sec (c+d x) \sqrt{a \sin (c+d x)+a}}{2 d}",1,"-(a^(3/2)*ArcTanh[(Sqrt[a]*Cos[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sin[c + d*x]])])/(2*Sqrt[2]*d) + (a*Sec[c + d*x]*Sqrt[a + a*Sin[c + d*x]])/(2*d) + (Sec[c + d*x]^3*(a + a*Sin[c + d*x])^(3/2))/(3*d)","A",4,3,23,0.1304,1,"{2675, 2649, 206}"
125,1,127,0,0.1812118,"\int \sec ^5(c+d x) (a+a \sin (c+d x))^{3/2} \, dx","Int[Sec[c + d*x]^5*(a + a*Sin[c + d*x])^(3/2),x]","-\frac{15 a^2}{32 d \sqrt{a \sin (c+d x)+a}}+\frac{15 a^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a \sin (c+d x)+a}}{\sqrt{2} \sqrt{a}}\right)}{32 \sqrt{2} d}+\frac{\sec ^4(c+d x) (a \sin (c+d x)+a)^{3/2}}{4 d}+\frac{5 a \sec ^2(c+d x) \sqrt{a \sin (c+d x)+a}}{16 d}","-\frac{15 a^2}{32 d \sqrt{a \sin (c+d x)+a}}+\frac{15 a^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a \sin (c+d x)+a}}{\sqrt{2} \sqrt{a}}\right)}{32 \sqrt{2} d}+\frac{\sec ^4(c+d x) (a \sin (c+d x)+a)^{3/2}}{4 d}+\frac{5 a \sec ^2(c+d x) \sqrt{a \sin (c+d x)+a}}{16 d}",1,"(15*a^(3/2)*ArcTanh[Sqrt[a + a*Sin[c + d*x]]/(Sqrt[2]*Sqrt[a])])/(32*Sqrt[2]*d) - (15*a^2)/(32*d*Sqrt[a + a*Sin[c + d*x]]) + (5*a*Sec[c + d*x]^2*Sqrt[a + a*Sin[c + d*x]])/(16*d) + (Sec[c + d*x]^4*(a + a*Sin[c + d*x])^(3/2))/(4*d)","A",6,5,23,0.2174,1,"{2675, 2667, 51, 63, 206}"
126,1,169,0,0.2213888,"\int \sec ^6(c+d x) (a+a \sin (c+d x))^{3/2} \, dx","Int[Sec[c + d*x]^6*(a + a*Sin[c + d*x])^(3/2),x]","-\frac{7 a^3 \cos (c+d x)}{16 d (a \sin (c+d x)+a)^{3/2}}+\frac{7 a^2 \sec (c+d x)}{12 d \sqrt{a \sin (c+d x)+a}}-\frac{7 a^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a \sin (c+d x)+a}}\right)}{16 \sqrt{2} d}+\frac{\sec ^5(c+d x) (a \sin (c+d x)+a)^{3/2}}{5 d}+\frac{7 a \sec ^3(c+d x) \sqrt{a \sin (c+d x)+a}}{30 d}","-\frac{7 a^3 \cos (c+d x)}{16 d (a \sin (c+d x)+a)^{3/2}}+\frac{7 a^2 \sec (c+d x)}{12 d \sqrt{a \sin (c+d x)+a}}-\frac{7 a^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a \sin (c+d x)+a}}\right)}{16 \sqrt{2} d}+\frac{\sec ^5(c+d x) (a \sin (c+d x)+a)^{3/2}}{5 d}+\frac{7 a \sec ^3(c+d x) \sqrt{a \sin (c+d x)+a}}{30 d}",1,"(-7*a^(3/2)*ArcTanh[(Sqrt[a]*Cos[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sin[c + d*x]])])/(16*Sqrt[2]*d) - (7*a^3*Cos[c + d*x])/(16*d*(a + a*Sin[c + d*x])^(3/2)) + (7*a^2*Sec[c + d*x])/(12*d*Sqrt[a + a*Sin[c + d*x]]) + (7*a*Sec[c + d*x]^3*Sqrt[a + a*Sin[c + d*x]])/(30*d) + (Sec[c + d*x]^5*(a + a*Sin[c + d*x])^(3/2))/(5*d)","A",6,5,23,0.2174,1,"{2675, 2687, 2650, 2649, 206}"
127,1,73,0,0.0735895,"\int \cos ^5(c+d x) (a+a \sin (c+d x))^{5/2} \, dx","Int[Cos[c + d*x]^5*(a + a*Sin[c + d*x])^(5/2),x]","\frac{2 (a \sin (c+d x)+a)^{15/2}}{15 a^5 d}-\frac{8 (a \sin (c+d x)+a)^{13/2}}{13 a^4 d}+\frac{8 (a \sin (c+d x)+a)^{11/2}}{11 a^3 d}","\frac{2 (a \sin (c+d x)+a)^{15/2}}{15 a^5 d}-\frac{8 (a \sin (c+d x)+a)^{13/2}}{13 a^4 d}+\frac{8 (a \sin (c+d x)+a)^{11/2}}{11 a^3 d}",1,"(8*(a + a*Sin[c + d*x])^(11/2))/(11*a^3*d) - (8*(a + a*Sin[c + d*x])^(13/2))/(13*a^4*d) + (2*(a + a*Sin[c + d*x])^(15/2))/(15*a^5*d)","A",3,2,23,0.08696,1,"{2667, 43}"
128,1,159,0,0.2933383,"\int \cos ^4(c+d x) (a+a \sin (c+d x))^{5/2} \, dx","Int[Cos[c + d*x]^4*(a + a*Sin[c + d*x])^(5/2),x]","-\frac{32 a^2 \cos ^5(c+d x) \sqrt{a \sin (c+d x)+a}}{143 d}-\frac{128 a^3 \cos ^5(c+d x)}{429 d \sqrt{a \sin (c+d x)+a}}-\frac{1024 a^4 \cos ^5(c+d x)}{3003 d (a \sin (c+d x)+a)^{3/2}}-\frac{4096 a^5 \cos ^5(c+d x)}{15015 d (a \sin (c+d x)+a)^{5/2}}-\frac{2 a \cos ^5(c+d x) (a \sin (c+d x)+a)^{3/2}}{13 d}","-\frac{32 a^2 \cos ^5(c+d x) \sqrt{a \sin (c+d x)+a}}{143 d}-\frac{128 a^3 \cos ^5(c+d x)}{429 d \sqrt{a \sin (c+d x)+a}}-\frac{1024 a^4 \cos ^5(c+d x)}{3003 d (a \sin (c+d x)+a)^{3/2}}-\frac{4096 a^5 \cos ^5(c+d x)}{15015 d (a \sin (c+d x)+a)^{5/2}}-\frac{2 a \cos ^5(c+d x) (a \sin (c+d x)+a)^{3/2}}{13 d}",1,"(-4096*a^5*Cos[c + d*x]^5)/(15015*d*(a + a*Sin[c + d*x])^(5/2)) - (1024*a^4*Cos[c + d*x]^5)/(3003*d*(a + a*Sin[c + d*x])^(3/2)) - (128*a^3*Cos[c + d*x]^5)/(429*d*Sqrt[a + a*Sin[c + d*x]]) - (32*a^2*Cos[c + d*x]^5*Sqrt[a + a*Sin[c + d*x]])/(143*d) - (2*a*Cos[c + d*x]^5*(a + a*Sin[c + d*x])^(3/2))/(13*d)","A",5,2,23,0.08696,1,"{2674, 2673}"
129,1,49,0,0.0665718,"\int \cos ^3(c+d x) (a+a \sin (c+d x))^{5/2} \, dx","Int[Cos[c + d*x]^3*(a + a*Sin[c + d*x])^(5/2),x]","\frac{4 (a \sin (c+d x)+a)^{9/2}}{9 a^2 d}-\frac{2 (a \sin (c+d x)+a)^{11/2}}{11 a^3 d}","\frac{4 (a \sin (c+d x)+a)^{9/2}}{9 a^2 d}-\frac{2 (a \sin (c+d x)+a)^{11/2}}{11 a^3 d}",1,"(4*(a + a*Sin[c + d*x])^(9/2))/(9*a^2*d) - (2*(a + a*Sin[c + d*x])^(11/2))/(11*a^3*d)","A",3,2,23,0.08696,1,"{2667, 43}"
130,1,127,0,0.2241748,"\int \cos ^2(c+d x) (a+a \sin (c+d x))^{5/2} \, dx","Int[Cos[c + d*x]^2*(a + a*Sin[c + d*x])^(5/2),x]","-\frac{256 a^4 \cos ^3(c+d x)}{315 d (a \sin (c+d x)+a)^{3/2}}-\frac{64 a^3 \cos ^3(c+d x)}{105 d \sqrt{a \sin (c+d x)+a}}-\frac{8 a^2 \cos ^3(c+d x) \sqrt{a \sin (c+d x)+a}}{21 d}-\frac{2 a \cos ^3(c+d x) (a \sin (c+d x)+a)^{3/2}}{9 d}","-\frac{256 a^4 \cos ^3(c+d x)}{315 d (a \sin (c+d x)+a)^{3/2}}-\frac{64 a^3 \cos ^3(c+d x)}{105 d \sqrt{a \sin (c+d x)+a}}-\frac{8 a^2 \cos ^3(c+d x) \sqrt{a \sin (c+d x)+a}}{21 d}-\frac{2 a \cos ^3(c+d x) (a \sin (c+d x)+a)^{3/2}}{9 d}",1,"(-256*a^4*Cos[c + d*x]^3)/(315*d*(a + a*Sin[c + d*x])^(3/2)) - (64*a^3*Cos[c + d*x]^3)/(105*d*Sqrt[a + a*Sin[c + d*x]]) - (8*a^2*Cos[c + d*x]^3*Sqrt[a + a*Sin[c + d*x]])/(21*d) - (2*a*Cos[c + d*x]^3*(a + a*Sin[c + d*x])^(3/2))/(9*d)","A",4,2,23,0.08696,1,"{2674, 2673}"
131,1,24,0,0.0337825,"\int \cos (c+d x) (a+a \sin (c+d x))^{5/2} \, dx","Int[Cos[c + d*x]*(a + a*Sin[c + d*x])^(5/2),x]","\frac{2 (a \sin (c+d x)+a)^{7/2}}{7 a d}","\frac{2 (a \sin (c+d x)+a)^{7/2}}{7 a d}",1,"(2*(a + a*Sin[c + d*x])^(7/2))/(7*a*d)","A",2,2,21,0.09524,1,"{2667, 32}"
132,1,86,0,0.0747303,"\int \sec (c+d x) (a+a \sin (c+d x))^{5/2} \, dx","Int[Sec[c + d*x]*(a + a*Sin[c + d*x])^(5/2),x]","-\frac{4 a^2 \sqrt{a \sin (c+d x)+a}}{d}+\frac{4 \sqrt{2} a^{5/2} \tanh ^{-1}\left(\frac{\sqrt{a \sin (c+d x)+a}}{\sqrt{2} \sqrt{a}}\right)}{d}-\frac{2 a (a \sin (c+d x)+a)^{3/2}}{3 d}","-\frac{4 a^2 \sqrt{a \sin (c+d x)+a}}{d}+\frac{4 \sqrt{2} a^{5/2} \tanh ^{-1}\left(\frac{\sqrt{a \sin (c+d x)+a}}{\sqrt{2} \sqrt{a}}\right)}{d}-\frac{2 a (a \sin (c+d x)+a)^{3/2}}{3 d}",1,"(4*Sqrt[2]*a^(5/2)*ArcTanh[Sqrt[a + a*Sin[c + d*x]]/(Sqrt[2]*Sqrt[a])])/d - (4*a^2*Sqrt[a + a*Sin[c + d*x]])/d - (2*a*(a + a*Sin[c + d*x])^(3/2))/(3*d)","A",5,4,21,0.1905,1,"{2667, 50, 63, 206}"
133,1,55,0,0.1167144,"\int \sec ^2(c+d x) (a+a \sin (c+d x))^{5/2} \, dx","Int[Sec[c + d*x]^2*(a + a*Sin[c + d*x])^(5/2),x]","\frac{8 a^2 \sec (c+d x) \sqrt{a \sin (c+d x)+a}}{d}-\frac{2 a \sec (c+d x) (a \sin (c+d x)+a)^{3/2}}{d}","\frac{8 a^2 \sec (c+d x) \sqrt{a \sin (c+d x)+a}}{d}-\frac{2 a \sec (c+d x) (a \sin (c+d x)+a)^{3/2}}{d}",1,"(8*a^2*Sec[c + d*x]*Sqrt[a + a*Sin[c + d*x]])/d - (2*a*Sec[c + d*x]*(a + a*Sin[c + d*x])^(3/2))/d","A",2,2,23,0.08696,1,"{2674, 2673}"
134,1,69,0,0.1099755,"\int \sec ^3(c+d x) (a+a \sin (c+d x))^{5/2} \, dx","Int[Sec[c + d*x]^3*(a + a*Sin[c + d*x])^(5/2),x]","\frac{a \sec ^2(c+d x) (a \sin (c+d x)+a)^{3/2}}{d}-\frac{a^{5/2} \tanh ^{-1}\left(\frac{\sqrt{a \sin (c+d x)+a}}{\sqrt{2} \sqrt{a}}\right)}{\sqrt{2} d}","\frac{a \sec ^2(c+d x) (a \sin (c+d x)+a)^{3/2}}{d}-\frac{a^{5/2} \tanh ^{-1}\left(\frac{\sqrt{a \sin (c+d x)+a}}{\sqrt{2} \sqrt{a}}\right)}{\sqrt{2} d}",1,"-((a^(5/2)*ArcTanh[Sqrt[a + a*Sin[c + d*x]]/(Sqrt[2]*Sqrt[a])])/(Sqrt[2]*d)) + (a*Sec[c + d*x]^2*(a + a*Sin[c + d*x])^(3/2))/d","A",4,4,23,0.1739,1,"{2676, 2667, 63, 206}"
135,1,30,0,0.058527,"\int \sec ^4(c+d x) (a+a \sin (c+d x))^{5/2} \, dx","Int[Sec[c + d*x]^4*(a + a*Sin[c + d*x])^(5/2),x]","\frac{2 a \sec ^3(c+d x) (a \sin (c+d x)+a)^{3/2}}{3 d}","\frac{2 a \sec ^3(c+d x) (a \sin (c+d x)+a)^{3/2}}{3 d}",1,"(2*a*Sec[c + d*x]^3*(a + a*Sin[c + d*x])^(3/2))/(3*d)","A",1,1,23,0.04348,1,"{2673}"
136,1,103,0,0.1721294,"\int \sec ^5(c+d x) (a+a \sin (c+d x))^{5/2} \, dx","Int[Sec[c + d*x]^5*(a + a*Sin[c + d*x])^(5/2),x]","\frac{3 a^{5/2} \tanh ^{-1}\left(\frac{\sqrt{a \sin (c+d x)+a}}{\sqrt{2} \sqrt{a}}\right)}{16 \sqrt{2} d}+\frac{\sec ^4(c+d x) (a \sin (c+d x)+a)^{5/2}}{4 d}+\frac{3 a \sec ^2(c+d x) (a \sin (c+d x)+a)^{3/2}}{16 d}","\frac{3 a^{5/2} \tanh ^{-1}\left(\frac{\sqrt{a \sin (c+d x)+a}}{\sqrt{2} \sqrt{a}}\right)}{16 \sqrt{2} d}+\frac{\sec ^4(c+d x) (a \sin (c+d x)+a)^{5/2}}{4 d}+\frac{3 a \sec ^2(c+d x) (a \sin (c+d x)+a)^{3/2}}{16 d}",1,"(3*a^(5/2)*ArcTanh[Sqrt[a + a*Sin[c + d*x]]/(Sqrt[2]*Sqrt[a])])/(16*Sqrt[2]*d) + (3*a*Sec[c + d*x]^2*(a + a*Sin[c + d*x])^(3/2))/(16*d) + (Sec[c + d*x]^4*(a + a*Sin[c + d*x])^(5/2))/(4*d)","A",5,4,23,0.1739,1,"{2675, 2667, 63, 206}"
137,1,139,0,0.1972698,"\int \sec ^6(c+d x) (a+a \sin (c+d x))^{5/2} \, dx","Int[Sec[c + d*x]^6*(a + a*Sin[c + d*x])^(5/2),x]","\frac{a^2 \sec (c+d x) \sqrt{a \sin (c+d x)+a}}{4 d}-\frac{a^{5/2} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a \sin (c+d x)+a}}\right)}{4 \sqrt{2} d}+\frac{\sec ^5(c+d x) (a \sin (c+d x)+a)^{5/2}}{5 d}+\frac{a \sec ^3(c+d x) (a \sin (c+d x)+a)^{3/2}}{6 d}","\frac{a^2 \sec (c+d x) \sqrt{a \sin (c+d x)+a}}{4 d}-\frac{a^{5/2} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a \sin (c+d x)+a}}\right)}{4 \sqrt{2} d}+\frac{\sec ^5(c+d x) (a \sin (c+d x)+a)^{5/2}}{5 d}+\frac{a \sec ^3(c+d x) (a \sin (c+d x)+a)^{3/2}}{6 d}",1,"-(a^(5/2)*ArcTanh[(Sqrt[a]*Cos[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sin[c + d*x]])])/(4*Sqrt[2]*d) + (a^2*Sec[c + d*x]*Sqrt[a + a*Sin[c + d*x]])/(4*d) + (a*Sec[c + d*x]^3*(a + a*Sin[c + d*x])^(3/2))/(6*d) + (Sec[c + d*x]^5*(a + a*Sin[c + d*x])^(5/2))/(5*d)","A",5,3,23,0.1304,1,"{2675, 2649, 206}"
138,1,159,0,0.2418281,"\int \sec ^7(c+d x) (a+a \sin (c+d x))^{5/2} \, dx","Int[Sec[c + d*x]^7*(a + a*Sin[c + d*x])^(5/2),x]","-\frac{35 a^3}{128 d \sqrt{a \sin (c+d x)+a}}+\frac{35 a^{5/2} \tanh ^{-1}\left(\frac{\sqrt{a \sin (c+d x)+a}}{\sqrt{2} \sqrt{a}}\right)}{128 \sqrt{2} d}+\frac{35 a^2 \sec ^2(c+d x) \sqrt{a \sin (c+d x)+a}}{192 d}+\frac{\sec ^6(c+d x) (a \sin (c+d x)+a)^{5/2}}{6 d}+\frac{7 a \sec ^4(c+d x) (a \sin (c+d x)+a)^{3/2}}{48 d}","-\frac{35 a^3}{128 d \sqrt{a \sin (c+d x)+a}}+\frac{35 a^{5/2} \tanh ^{-1}\left(\frac{\sqrt{a \sin (c+d x)+a}}{\sqrt{2} \sqrt{a}}\right)}{128 \sqrt{2} d}+\frac{35 a^2 \sec ^2(c+d x) \sqrt{a \sin (c+d x)+a}}{192 d}+\frac{\sec ^6(c+d x) (a \sin (c+d x)+a)^{5/2}}{6 d}+\frac{7 a \sec ^4(c+d x) (a \sin (c+d x)+a)^{3/2}}{48 d}",1,"(35*a^(5/2)*ArcTanh[Sqrt[a + a*Sin[c + d*x]]/(Sqrt[2]*Sqrt[a])])/(128*Sqrt[2]*d) - (35*a^3)/(128*d*Sqrt[a + a*Sin[c + d*x]]) + (35*a^2*Sec[c + d*x]^2*Sqrt[a + a*Sin[c + d*x]])/(192*d) + (7*a*Sec[c + d*x]^4*(a + a*Sin[c + d*x])^(3/2))/(48*d) + (Sec[c + d*x]^6*(a + a*Sin[c + d*x])^(5/2))/(6*d)","A",7,5,23,0.2174,1,"{2675, 2667, 51, 63, 206}"
139,1,97,0,0.079213,"\int \cos ^7(c+d x) (a+a \sin (c+d x))^{7/2} \, dx","Int[Cos[c + d*x]^7*(a + a*Sin[c + d*x])^(7/2),x]","-\frac{2 (a \sin (c+d x)+a)^{21/2}}{21 a^7 d}+\frac{12 (a \sin (c+d x)+a)^{19/2}}{19 a^6 d}-\frac{24 (a \sin (c+d x)+a)^{17/2}}{17 a^5 d}+\frac{16 (a \sin (c+d x)+a)^{15/2}}{15 a^4 d}","-\frac{2 (a \sin (c+d x)+a)^{21/2}}{21 a^7 d}+\frac{12 (a \sin (c+d x)+a)^{19/2}}{19 a^6 d}-\frac{24 (a \sin (c+d x)+a)^{17/2}}{17 a^5 d}+\frac{16 (a \sin (c+d x)+a)^{15/2}}{15 a^4 d}",1,"(16*(a + a*Sin[c + d*x])^(15/2))/(15*a^4*d) - (24*(a + a*Sin[c + d*x])^(17/2))/(17*a^5*d) + (12*(a + a*Sin[c + d*x])^(19/2))/(19*a^6*d) - (2*(a + a*Sin[c + d*x])^(21/2))/(21*a^7*d)","A",3,2,23,0.08696,1,"{2667, 43}"
140,1,223,0,0.4295086,"\int \cos ^6(c+d x) (a+a \sin (c+d x))^{7/2} \, dx","Int[Cos[c + d*x]^6*(a + a*Sin[c + d*x])^(7/2),x]","-\frac{48 a^2 \cos ^7(c+d x) (a \sin (c+d x)+a)^{3/2}}{323 d}-\frac{64 a^3 \cos ^7(c+d x) \sqrt{a \sin (c+d x)+a}}{323 d}-\frac{1024 a^4 \cos ^7(c+d x)}{4199 d \sqrt{a \sin (c+d x)+a}}-\frac{12288 a^5 \cos ^7(c+d x)}{46189 d (a \sin (c+d x)+a)^{3/2}}-\frac{32768 a^6 \cos ^7(c+d x)}{138567 d (a \sin (c+d x)+a)^{5/2}}-\frac{131072 a^7 \cos ^7(c+d x)}{969969 d (a \sin (c+d x)+a)^{7/2}}-\frac{2 a \cos ^7(c+d x) (a \sin (c+d x)+a)^{5/2}}{19 d}","-\frac{48 a^2 \cos ^7(c+d x) (a \sin (c+d x)+a)^{3/2}}{323 d}-\frac{64 a^3 \cos ^7(c+d x) \sqrt{a \sin (c+d x)+a}}{323 d}-\frac{1024 a^4 \cos ^7(c+d x)}{4199 d \sqrt{a \sin (c+d x)+a}}-\frac{12288 a^5 \cos ^7(c+d x)}{46189 d (a \sin (c+d x)+a)^{3/2}}-\frac{32768 a^6 \cos ^7(c+d x)}{138567 d (a \sin (c+d x)+a)^{5/2}}-\frac{131072 a^7 \cos ^7(c+d x)}{969969 d (a \sin (c+d x)+a)^{7/2}}-\frac{2 a \cos ^7(c+d x) (a \sin (c+d x)+a)^{5/2}}{19 d}",1,"(-131072*a^7*Cos[c + d*x]^7)/(969969*d*(a + a*Sin[c + d*x])^(7/2)) - (32768*a^6*Cos[c + d*x]^7)/(138567*d*(a + a*Sin[c + d*x])^(5/2)) - (12288*a^5*Cos[c + d*x]^7)/(46189*d*(a + a*Sin[c + d*x])^(3/2)) - (1024*a^4*Cos[c + d*x]^7)/(4199*d*Sqrt[a + a*Sin[c + d*x]]) - (64*a^3*Cos[c + d*x]^7*Sqrt[a + a*Sin[c + d*x]])/(323*d) - (48*a^2*Cos[c + d*x]^7*(a + a*Sin[c + d*x])^(3/2))/(323*d) - (2*a*Cos[c + d*x]^7*(a + a*Sin[c + d*x])^(5/2))/(19*d)","A",7,2,23,0.08696,1,"{2674, 2673}"
141,1,73,0,0.0746197,"\int \cos ^5(c+d x) (a+a \sin (c+d x))^{7/2} \, dx","Int[Cos[c + d*x]^5*(a + a*Sin[c + d*x])^(7/2),x]","\frac{2 (a \sin (c+d x)+a)^{17/2}}{17 a^5 d}-\frac{8 (a \sin (c+d x)+a)^{15/2}}{15 a^4 d}+\frac{8 (a \sin (c+d x)+a)^{13/2}}{13 a^3 d}","\frac{2 (a \sin (c+d x)+a)^{17/2}}{17 a^5 d}-\frac{8 (a \sin (c+d x)+a)^{15/2}}{15 a^4 d}+\frac{8 (a \sin (c+d x)+a)^{13/2}}{13 a^3 d}",1,"(8*(a + a*Sin[c + d*x])^(13/2))/(13*a^3*d) - (8*(a + a*Sin[c + d*x])^(15/2))/(15*a^4*d) + (2*(a + a*Sin[c + d*x])^(17/2))/(17*a^5*d)","A",3,2,23,0.08696,1,"{2667, 43}"
142,1,191,0,0.3652843,"\int \cos ^4(c+d x) (a+a \sin (c+d x))^{7/2} \, dx","Int[Cos[c + d*x]^4*(a + a*Sin[c + d*x])^(7/2),x]","-\frac{16384 a^6 \cos ^5(c+d x)}{45045 d (a \sin (c+d x)+a)^{5/2}}-\frac{4096 a^5 \cos ^5(c+d x)}{9009 d (a \sin (c+d x)+a)^{3/2}}-\frac{512 a^4 \cos ^5(c+d x)}{1287 d \sqrt{a \sin (c+d x)+a}}-\frac{128 a^3 \cos ^5(c+d x) \sqrt{a \sin (c+d x)+a}}{429 d}-\frac{8 a^2 \cos ^5(c+d x) (a \sin (c+d x)+a)^{3/2}}{39 d}-\frac{2 a \cos ^5(c+d x) (a \sin (c+d x)+a)^{5/2}}{15 d}","-\frac{16384 a^6 \cos ^5(c+d x)}{45045 d (a \sin (c+d x)+a)^{5/2}}-\frac{4096 a^5 \cos ^5(c+d x)}{9009 d (a \sin (c+d x)+a)^{3/2}}-\frac{512 a^4 \cos ^5(c+d x)}{1287 d \sqrt{a \sin (c+d x)+a}}-\frac{128 a^3 \cos ^5(c+d x) \sqrt{a \sin (c+d x)+a}}{429 d}-\frac{8 a^2 \cos ^5(c+d x) (a \sin (c+d x)+a)^{3/2}}{39 d}-\frac{2 a \cos ^5(c+d x) (a \sin (c+d x)+a)^{5/2}}{15 d}",1,"(-16384*a^6*Cos[c + d*x]^5)/(45045*d*(a + a*Sin[c + d*x])^(5/2)) - (4096*a^5*Cos[c + d*x]^5)/(9009*d*(a + a*Sin[c + d*x])^(3/2)) - (512*a^4*Cos[c + d*x]^5)/(1287*d*Sqrt[a + a*Sin[c + d*x]]) - (128*a^3*Cos[c + d*x]^5*Sqrt[a + a*Sin[c + d*x]])/(429*d) - (8*a^2*Cos[c + d*x]^5*(a + a*Sin[c + d*x])^(3/2))/(39*d) - (2*a*Cos[c + d*x]^5*(a + a*Sin[c + d*x])^(5/2))/(15*d)","A",6,2,23,0.08696,1,"{2674, 2673}"
143,1,49,0,0.0661054,"\int \cos ^3(c+d x) (a+a \sin (c+d x))^{7/2} \, dx","Int[Cos[c + d*x]^3*(a + a*Sin[c + d*x])^(7/2),x]","\frac{4 (a \sin (c+d x)+a)^{11/2}}{11 a^2 d}-\frac{2 (a \sin (c+d x)+a)^{13/2}}{13 a^3 d}","\frac{4 (a \sin (c+d x)+a)^{11/2}}{11 a^2 d}-\frac{2 (a \sin (c+d x)+a)^{13/2}}{13 a^3 d}",1,"(4*(a + a*Sin[c + d*x])^(11/2))/(11*a^2*d) - (2*(a + a*Sin[c + d*x])^(13/2))/(13*a^3*d)","A",3,2,23,0.08696,1,"{2667, 43}"
144,1,159,0,0.2920996,"\int \cos ^2(c+d x) (a+a \sin (c+d x))^{7/2} \, dx","Int[Cos[c + d*x]^2*(a + a*Sin[c + d*x])^(7/2),x]","-\frac{4096 a^5 \cos ^3(c+d x)}{3465 d (a \sin (c+d x)+a)^{3/2}}-\frac{1024 a^4 \cos ^3(c+d x)}{1155 d \sqrt{a \sin (c+d x)+a}}-\frac{128 a^3 \cos ^3(c+d x) \sqrt{a \sin (c+d x)+a}}{231 d}-\frac{32 a^2 \cos ^3(c+d x) (a \sin (c+d x)+a)^{3/2}}{99 d}-\frac{2 a \cos ^3(c+d x) (a \sin (c+d x)+a)^{5/2}}{11 d}","-\frac{4096 a^5 \cos ^3(c+d x)}{3465 d (a \sin (c+d x)+a)^{3/2}}-\frac{1024 a^4 \cos ^3(c+d x)}{1155 d \sqrt{a \sin (c+d x)+a}}-\frac{128 a^3 \cos ^3(c+d x) \sqrt{a \sin (c+d x)+a}}{231 d}-\frac{32 a^2 \cos ^3(c+d x) (a \sin (c+d x)+a)^{3/2}}{99 d}-\frac{2 a \cos ^3(c+d x) (a \sin (c+d x)+a)^{5/2}}{11 d}",1,"(-4096*a^5*Cos[c + d*x]^3)/(3465*d*(a + a*Sin[c + d*x])^(3/2)) - (1024*a^4*Cos[c + d*x]^3)/(1155*d*Sqrt[a + a*Sin[c + d*x]]) - (128*a^3*Cos[c + d*x]^3*Sqrt[a + a*Sin[c + d*x]])/(231*d) - (32*a^2*Cos[c + d*x]^3*(a + a*Sin[c + d*x])^(3/2))/(99*d) - (2*a*Cos[c + d*x]^3*(a + a*Sin[c + d*x])^(5/2))/(11*d)","A",5,2,23,0.08696,1,"{2674, 2673}"
145,1,24,0,0.0332088,"\int \cos (c+d x) (a+a \sin (c+d x))^{7/2} \, dx","Int[Cos[c + d*x]*(a + a*Sin[c + d*x])^(7/2),x]","\frac{2 (a \sin (c+d x)+a)^{9/2}}{9 a d}","\frac{2 (a \sin (c+d x)+a)^{9/2}}{9 a d}",1,"(2*(a + a*Sin[c + d*x])^(9/2))/(9*a*d)","A",2,2,21,0.09524,1,"{2667, 32}"
146,1,110,0,0.0852477,"\int \sec (c+d x) (a+a \sin (c+d x))^{7/2} \, dx","Int[Sec[c + d*x]*(a + a*Sin[c + d*x])^(7/2),x]","-\frac{8 a^3 \sqrt{a \sin (c+d x)+a}}{d}-\frac{4 a^2 (a \sin (c+d x)+a)^{3/2}}{3 d}+\frac{8 \sqrt{2} a^{7/2} \tanh ^{-1}\left(\frac{\sqrt{a \sin (c+d x)+a}}{\sqrt{2} \sqrt{a}}\right)}{d}-\frac{2 a (a \sin (c+d x)+a)^{5/2}}{5 d}","-\frac{8 a^3 \sqrt{a \sin (c+d x)+a}}{d}-\frac{4 a^2 (a \sin (c+d x)+a)^{3/2}}{3 d}+\frac{8 \sqrt{2} a^{7/2} \tanh ^{-1}\left(\frac{\sqrt{a \sin (c+d x)+a}}{\sqrt{2} \sqrt{a}}\right)}{d}-\frac{2 a (a \sin (c+d x)+a)^{5/2}}{5 d}",1,"(8*Sqrt[2]*a^(7/2)*ArcTanh[Sqrt[a + a*Sin[c + d*x]]/(Sqrt[2]*Sqrt[a])])/d - (8*a^3*Sqrt[a + a*Sin[c + d*x]])/d - (4*a^2*(a + a*Sin[c + d*x])^(3/2))/(3*d) - (2*a*(a + a*Sin[c + d*x])^(5/2))/(5*d)","A",6,4,21,0.1905,1,"{2667, 50, 63, 206}"
147,1,89,0,0.1736594,"\int \sec ^2(c+d x) (a+a \sin (c+d x))^{7/2} \, dx","Int[Sec[c + d*x]^2*(a + a*Sin[c + d*x])^(7/2),x]","\frac{64 a^3 \sec (c+d x) \sqrt{a \sin (c+d x)+a}}{3 d}-\frac{16 a^2 \sec (c+d x) (a \sin (c+d x)+a)^{3/2}}{3 d}-\frac{2 a \sec (c+d x) (a \sin (c+d x)+a)^{5/2}}{3 d}","\frac{64 a^3 \sec (c+d x) \sqrt{a \sin (c+d x)+a}}{3 d}-\frac{16 a^2 \sec (c+d x) (a \sin (c+d x)+a)^{3/2}}{3 d}-\frac{2 a \sec (c+d x) (a \sin (c+d x)+a)^{5/2}}{3 d}",1,"(64*a^3*Sec[c + d*x]*Sqrt[a + a*Sin[c + d*x]])/(3*d) - (16*a^2*Sec[c + d*x]*(a + a*Sin[c + d*x])^(3/2))/(3*d) - (2*a*Sec[c + d*x]*(a + a*Sin[c + d*x])^(5/2))/(3*d)","A",3,2,23,0.08696,1,"{2674, 2673}"
148,1,91,0,0.1272454,"\int \sec ^3(c+d x) (a+a \sin (c+d x))^{7/2} \, dx","Int[Sec[c + d*x]^3*(a + a*Sin[c + d*x])^(7/2),x]","\frac{3 a^3 \sqrt{a \sin (c+d x)+a}}{d}-\frac{3 \sqrt{2} a^{7/2} \tanh ^{-1}\left(\frac{\sqrt{a \sin (c+d x)+a}}{\sqrt{2} \sqrt{a}}\right)}{d}+\frac{a \sec ^2(c+d x) (a \sin (c+d x)+a)^{5/2}}{d}","\frac{3 a^3 \sqrt{a \sin (c+d x)+a}}{d}-\frac{3 \sqrt{2} a^{7/2} \tanh ^{-1}\left(\frac{\sqrt{a \sin (c+d x)+a}}{\sqrt{2} \sqrt{a}}\right)}{d}+\frac{a \sec ^2(c+d x) (a \sin (c+d x)+a)^{5/2}}{d}",1,"(-3*Sqrt[2]*a^(7/2)*ArcTanh[Sqrt[a + a*Sin[c + d*x]]/(Sqrt[2]*Sqrt[a])])/d + (3*a^3*Sqrt[a + a*Sin[c + d*x]])/d + (a*Sec[c + d*x]^2*(a + a*Sin[c + d*x])^(5/2))/d","A",5,5,23,0.2174,1,"{2676, 2667, 50, 63, 206}"
149,1,61,0,0.113317,"\int \sec ^4(c+d x) (a+a \sin (c+d x))^{7/2} \, dx","Int[Sec[c + d*x]^4*(a + a*Sin[c + d*x])^(7/2),x]","\frac{2 a \sec ^3(c+d x) (a \sin (c+d x)+a)^{5/2}}{d}-\frac{8 a^2 \sec ^3(c+d x) (a \sin (c+d x)+a)^{3/2}}{3 d}","\frac{2 a \sec ^3(c+d x) (a \sin (c+d x)+a)^{5/2}}{d}-\frac{8 a^2 \sec ^3(c+d x) (a \sin (c+d x)+a)^{3/2}}{3 d}",1,"(-8*a^2*Sec[c + d*x]^3*(a + a*Sin[c + d*x])^(3/2))/(3*d) + (2*a*Sec[c + d*x]^3*(a + a*Sin[c + d*x])^(5/2))/d","A",2,2,23,0.08696,1,"{2674, 2673}"
150,1,106,0,0.1712818,"\int \sec ^5(c+d x) (a+a \sin (c+d x))^{7/2} \, dx","Int[Sec[c + d*x]^5*(a + a*Sin[c + d*x])^(7/2),x]","-\frac{a^{7/2} \tanh ^{-1}\left(\frac{\sqrt{a \sin (c+d x)+a}}{\sqrt{2} \sqrt{a}}\right)}{8 \sqrt{2} d}-\frac{a^2 \sec ^2(c+d x) (a \sin (c+d x)+a)^{3/2}}{8 d}+\frac{a \sec ^4(c+d x) (a \sin (c+d x)+a)^{5/2}}{2 d}","-\frac{a^{7/2} \tanh ^{-1}\left(\frac{\sqrt{a \sin (c+d x)+a}}{\sqrt{2} \sqrt{a}}\right)}{8 \sqrt{2} d}-\frac{a^2 \sec ^2(c+d x) (a \sin (c+d x)+a)^{3/2}}{8 d}+\frac{a \sec ^4(c+d x) (a \sin (c+d x)+a)^{5/2}}{2 d}",1,"-(a^(7/2)*ArcTanh[Sqrt[a + a*Sin[c + d*x]]/(Sqrt[2]*Sqrt[a])])/(8*Sqrt[2]*d) - (a^2*Sec[c + d*x]^2*(a + a*Sin[c + d*x])^(3/2))/(8*d) + (a*Sec[c + d*x]^4*(a + a*Sin[c + d*x])^(5/2))/(2*d)","A",5,5,23,0.2174,1,"{2676, 2675, 2667, 63, 206}"
151,1,30,0,0.056813,"\int \sec ^6(c+d x) (a+a \sin (c+d x))^{7/2} \, dx","Int[Sec[c + d*x]^6*(a + a*Sin[c + d*x])^(7/2),x]","\frac{2 a \sec ^5(c+d x) (a \sin (c+d x)+a)^{5/2}}{5 d}","\frac{2 a \sec ^5(c+d x) (a \sin (c+d x)+a)^{5/2}}{5 d}",1,"(2*a*Sec[c + d*x]^5*(a + a*Sin[c + d*x])^(5/2))/(5*d)","A",1,1,23,0.04348,1,"{2673}"
152,1,135,0,0.2320924,"\int \sec ^7(c+d x) (a+a \sin (c+d x))^{7/2} \, dx","Int[Sec[c + d*x]^7*(a + a*Sin[c + d*x])^(7/2),x]","\frac{5 a^{7/2} \tanh ^{-1}\left(\frac{\sqrt{a \sin (c+d x)+a}}{\sqrt{2} \sqrt{a}}\right)}{64 \sqrt{2} d}+\frac{5 a^2 \sec ^2(c+d x) (a \sin (c+d x)+a)^{3/2}}{64 d}+\frac{\sec ^6(c+d x) (a \sin (c+d x)+a)^{7/2}}{6 d}+\frac{5 a \sec ^4(c+d x) (a \sin (c+d x)+a)^{5/2}}{48 d}","\frac{5 a^{7/2} \tanh ^{-1}\left(\frac{\sqrt{a \sin (c+d x)+a}}{\sqrt{2} \sqrt{a}}\right)}{64 \sqrt{2} d}+\frac{5 a^2 \sec ^2(c+d x) (a \sin (c+d x)+a)^{3/2}}{64 d}+\frac{\sec ^6(c+d x) (a \sin (c+d x)+a)^{7/2}}{6 d}+\frac{5 a \sec ^4(c+d x) (a \sin (c+d x)+a)^{5/2}}{48 d}",1,"(5*a^(7/2)*ArcTanh[Sqrt[a + a*Sin[c + d*x]]/(Sqrt[2]*Sqrt[a])])/(64*Sqrt[2]*d) + (5*a^2*Sec[c + d*x]^2*(a + a*Sin[c + d*x])^(3/2))/(64*d) + (5*a*Sec[c + d*x]^4*(a + a*Sin[c + d*x])^(5/2))/(48*d) + (Sec[c + d*x]^6*(a + a*Sin[c + d*x])^(7/2))/(6*d)","A",6,4,23,0.1739,1,"{2675, 2667, 63, 206}"
153,1,171,0,0.267813,"\int \sec ^8(c+d x) (a+a \sin (c+d x))^{7/2} \, dx","Int[Sec[c + d*x]^8*(a + a*Sin[c + d*x])^(7/2),x]","\frac{a^2 \sec ^3(c+d x) (a \sin (c+d x)+a)^{3/2}}{12 d}+\frac{a^3 \sec (c+d x) \sqrt{a \sin (c+d x)+a}}{8 d}-\frac{a^{7/2} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a \sin (c+d x)+a}}\right)}{8 \sqrt{2} d}+\frac{\sec ^7(c+d x) (a \sin (c+d x)+a)^{7/2}}{7 d}+\frac{a \sec ^5(c+d x) (a \sin (c+d x)+a)^{5/2}}{10 d}","\frac{a^2 \sec ^3(c+d x) (a \sin (c+d x)+a)^{3/2}}{12 d}+\frac{a^3 \sec (c+d x) \sqrt{a \sin (c+d x)+a}}{8 d}-\frac{a^{7/2} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a \sin (c+d x)+a}}\right)}{8 \sqrt{2} d}+\frac{\sec ^7(c+d x) (a \sin (c+d x)+a)^{7/2}}{7 d}+\frac{a \sec ^5(c+d x) (a \sin (c+d x)+a)^{5/2}}{10 d}",1,"-(a^(7/2)*ArcTanh[(Sqrt[a]*Cos[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sin[c + d*x]])])/(8*Sqrt[2]*d) + (a^3*Sec[c + d*x]*Sqrt[a + a*Sin[c + d*x]])/(8*d) + (a^2*Sec[c + d*x]^3*(a + a*Sin[c + d*x])^(3/2))/(12*d) + (a*Sec[c + d*x]^5*(a + a*Sin[c + d*x])^(5/2))/(10*d) + (Sec[c + d*x]^7*(a + a*Sin[c + d*x])^(7/2))/(7*d)","A",6,3,23,0.1304,1,"{2675, 2649, 206}"
154,1,191,0,0.3122833,"\int \sec ^9(c+d x) (a+a \sin (c+d x))^{7/2} \, dx","Int[Sec[c + d*x]^9*(a + a*Sin[c + d*x])^(7/2),x]","-\frac{315 a^4}{2048 d \sqrt{a \sin (c+d x)+a}}+\frac{315 a^{7/2} \tanh ^{-1}\left(\frac{\sqrt{a \sin (c+d x)+a}}{\sqrt{2} \sqrt{a}}\right)}{2048 \sqrt{2} d}+\frac{21 a^2 \sec ^4(c+d x) (a \sin (c+d x)+a)^{3/2}}{256 d}+\frac{105 a^3 \sec ^2(c+d x) \sqrt{a \sin (c+d x)+a}}{1024 d}+\frac{\sec ^8(c+d x) (a \sin (c+d x)+a)^{7/2}}{8 d}+\frac{3 a \sec ^6(c+d x) (a \sin (c+d x)+a)^{5/2}}{32 d}","-\frac{315 a^4}{2048 d \sqrt{a \sin (c+d x)+a}}+\frac{315 a^{7/2} \tanh ^{-1}\left(\frac{\sqrt{a \sin (c+d x)+a}}{\sqrt{2} \sqrt{a}}\right)}{2048 \sqrt{2} d}+\frac{21 a^2 \sec ^4(c+d x) (a \sin (c+d x)+a)^{3/2}}{256 d}+\frac{105 a^3 \sec ^2(c+d x) \sqrt{a \sin (c+d x)+a}}{1024 d}+\frac{\sec ^8(c+d x) (a \sin (c+d x)+a)^{7/2}}{8 d}+\frac{3 a \sec ^6(c+d x) (a \sin (c+d x)+a)^{5/2}}{32 d}",1,"(315*a^(7/2)*ArcTanh[Sqrt[a + a*Sin[c + d*x]]/(Sqrt[2]*Sqrt[a])])/(2048*Sqrt[2]*d) - (315*a^4)/(2048*d*Sqrt[a + a*Sin[c + d*x]]) + (105*a^3*Sec[c + d*x]^2*Sqrt[a + a*Sin[c + d*x]])/(1024*d) + (21*a^2*Sec[c + d*x]^4*(a + a*Sin[c + d*x])^(3/2))/(256*d) + (3*a*Sec[c + d*x]^6*(a + a*Sin[c + d*x])^(5/2))/(32*d) + (Sec[c + d*x]^8*(a + a*Sin[c + d*x])^(7/2))/(8*d)","A",8,5,23,0.2174,1,"{2675, 2667, 51, 63, 206}"
155,1,233,0,0.3543775,"\int \sec ^{10}(c+d x) (a+a \sin (c+d x))^{7/2} \, dx","Int[Sec[c + d*x]^10*(a + a*Sin[c + d*x])^(7/2),x]","-\frac{11 a^5 \cos (c+d x)}{64 d (a \sin (c+d x)+a)^{3/2}}+\frac{11 a^2 \sec ^5(c+d x) (a \sin (c+d x)+a)^{3/2}}{140 d}+\frac{11 a^3 \sec ^3(c+d x) \sqrt{a \sin (c+d x)+a}}{120 d}+\frac{11 a^4 \sec (c+d x)}{48 d \sqrt{a \sin (c+d x)+a}}-\frac{11 a^{7/2} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a \sin (c+d x)+a}}\right)}{64 \sqrt{2} d}+\frac{\sec ^9(c+d x) (a \sin (c+d x)+a)^{7/2}}{9 d}+\frac{11 a \sec ^7(c+d x) (a \sin (c+d x)+a)^{5/2}}{126 d}","-\frac{11 a^5 \cos (c+d x)}{64 d (a \sin (c+d x)+a)^{3/2}}+\frac{11 a^2 \sec ^5(c+d x) (a \sin (c+d x)+a)^{3/2}}{140 d}+\frac{11 a^3 \sec ^3(c+d x) \sqrt{a \sin (c+d x)+a}}{120 d}+\frac{11 a^4 \sec (c+d x)}{48 d \sqrt{a \sin (c+d x)+a}}-\frac{11 a^{7/2} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a \sin (c+d x)+a}}\right)}{64 \sqrt{2} d}+\frac{\sec ^9(c+d x) (a \sin (c+d x)+a)^{7/2}}{9 d}+\frac{11 a \sec ^7(c+d x) (a \sin (c+d x)+a)^{5/2}}{126 d}",1,"(-11*a^(7/2)*ArcTanh[(Sqrt[a]*Cos[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sin[c + d*x]])])/(64*Sqrt[2]*d) - (11*a^5*Cos[c + d*x])/(64*d*(a + a*Sin[c + d*x])^(3/2)) + (11*a^4*Sec[c + d*x])/(48*d*Sqrt[a + a*Sin[c + d*x]]) + (11*a^3*Sec[c + d*x]^3*Sqrt[a + a*Sin[c + d*x]])/(120*d) + (11*a^2*Sec[c + d*x]^5*(a + a*Sin[c + d*x])^(3/2))/(140*d) + (11*a*Sec[c + d*x]^7*(a + a*Sin[c + d*x])^(5/2))/(126*d) + (Sec[c + d*x]^9*(a + a*Sin[c + d*x])^(7/2))/(9*d)","A",8,5,23,0.2174,1,"{2675, 2687, 2650, 2649, 206}"
156,1,97,0,0.075736,"\int \frac{\cos ^7(c+d x)}{\sqrt{a+a \sin (c+d x)}} \, dx","Int[Cos[c + d*x]^7/Sqrt[a + a*Sin[c + d*x]],x]","-\frac{2 (a \sin (c+d x)+a)^{13/2}}{13 a^7 d}+\frac{12 (a \sin (c+d x)+a)^{11/2}}{11 a^6 d}-\frac{8 (a \sin (c+d x)+a)^{9/2}}{3 a^5 d}+\frac{16 (a \sin (c+d x)+a)^{7/2}}{7 a^4 d}","-\frac{2 (a \sin (c+d x)+a)^{13/2}}{13 a^7 d}+\frac{12 (a \sin (c+d x)+a)^{11/2}}{11 a^6 d}-\frac{8 (a \sin (c+d x)+a)^{9/2}}{3 a^5 d}+\frac{16 (a \sin (c+d x)+a)^{7/2}}{7 a^4 d}",1,"(16*(a + a*Sin[c + d*x])^(7/2))/(7*a^4*d) - (8*(a + a*Sin[c + d*x])^(9/2))/(3*a^5*d) + (12*(a + a*Sin[c + d*x])^(11/2))/(11*a^6*d) - (2*(a + a*Sin[c + d*x])^(13/2))/(13*a^7*d)","A",3,2,23,0.08696,1,"{2667, 43}"
157,1,95,0,0.1704335,"\int \frac{\cos ^6(c+d x)}{\sqrt{a+a \sin (c+d x)}} \, dx","Int[Cos[c + d*x]^6/Sqrt[a + a*Sin[c + d*x]],x]","-\frac{16 a^2 \cos ^7(c+d x)}{99 d (a \sin (c+d x)+a)^{5/2}}-\frac{64 a^3 \cos ^7(c+d x)}{693 d (a \sin (c+d x)+a)^{7/2}}-\frac{2 a \cos ^7(c+d x)}{11 d (a \sin (c+d x)+a)^{3/2}}","-\frac{16 a^2 \cos ^7(c+d x)}{99 d (a \sin (c+d x)+a)^{5/2}}-\frac{64 a^3 \cos ^7(c+d x)}{693 d (a \sin (c+d x)+a)^{7/2}}-\frac{2 a \cos ^7(c+d x)}{11 d (a \sin (c+d x)+a)^{3/2}}",1,"(-64*a^3*Cos[c + d*x]^7)/(693*d*(a + a*Sin[c + d*x])^(7/2)) - (16*a^2*Cos[c + d*x]^7)/(99*d*(a + a*Sin[c + d*x])^(5/2)) - (2*a*Cos[c + d*x]^7)/(11*d*(a + a*Sin[c + d*x])^(3/2))","A",3,2,23,0.08696,1,"{2674, 2673}"
158,1,73,0,0.0672223,"\int \frac{\cos ^5(c+d x)}{\sqrt{a+a \sin (c+d x)}} \, dx","Int[Cos[c + d*x]^5/Sqrt[a + a*Sin[c + d*x]],x]","\frac{2 (a \sin (c+d x)+a)^{9/2}}{9 a^5 d}-\frac{8 (a \sin (c+d x)+a)^{7/2}}{7 a^4 d}+\frac{8 (a \sin (c+d x)+a)^{5/2}}{5 a^3 d}","\frac{2 (a \sin (c+d x)+a)^{9/2}}{9 a^5 d}-\frac{8 (a \sin (c+d x)+a)^{7/2}}{7 a^4 d}+\frac{8 (a \sin (c+d x)+a)^{5/2}}{5 a^3 d}",1,"(8*(a + a*Sin[c + d*x])^(5/2))/(5*a^3*d) - (8*(a + a*Sin[c + d*x])^(7/2))/(7*a^4*d) + (2*(a + a*Sin[c + d*x])^(9/2))/(9*a^5*d)","A",3,2,23,0.08696,1,"{2667, 43}"
159,1,63,0,0.1111403,"\int \frac{\cos ^4(c+d x)}{\sqrt{a+a \sin (c+d x)}} \, dx","Int[Cos[c + d*x]^4/Sqrt[a + a*Sin[c + d*x]],x]","-\frac{8 a^2 \cos ^5(c+d x)}{35 d (a \sin (c+d x)+a)^{5/2}}-\frac{2 a \cos ^5(c+d x)}{7 d (a \sin (c+d x)+a)^{3/2}}","-\frac{8 a^2 \cos ^5(c+d x)}{35 d (a \sin (c+d x)+a)^{5/2}}-\frac{2 a \cos ^5(c+d x)}{7 d (a \sin (c+d x)+a)^{3/2}}",1,"(-8*a^2*Cos[c + d*x]^5)/(35*d*(a + a*Sin[c + d*x])^(5/2)) - (2*a*Cos[c + d*x]^5)/(7*d*(a + a*Sin[c + d*x])^(3/2))","A",2,2,23,0.08696,1,"{2674, 2673}"
160,1,49,0,0.0625084,"\int \frac{\cos ^3(c+d x)}{\sqrt{a+a \sin (c+d x)}} \, dx","Int[Cos[c + d*x]^3/Sqrt[a + a*Sin[c + d*x]],x]","\frac{4 (a \sin (c+d x)+a)^{3/2}}{3 a^2 d}-\frac{2 (a \sin (c+d x)+a)^{5/2}}{5 a^3 d}","\frac{4 (a \sin (c+d x)+a)^{3/2}}{3 a^2 d}-\frac{2 (a \sin (c+d x)+a)^{5/2}}{5 a^3 d}",1,"(4*(a + a*Sin[c + d*x])^(3/2))/(3*a^2*d) - (2*(a + a*Sin[c + d*x])^(5/2))/(5*a^3*d)","A",3,2,23,0.08696,1,"{2667, 43}"
161,1,30,0,0.0512926,"\int \frac{\cos ^2(c+d x)}{\sqrt{a+a \sin (c+d x)}} \, dx","Int[Cos[c + d*x]^2/Sqrt[a + a*Sin[c + d*x]],x]","-\frac{2 a \cos ^3(c+d x)}{3 d (a \sin (c+d x)+a)^{3/2}}","-\frac{2 a \cos ^3(c+d x)}{3 d (a \sin (c+d x)+a)^{3/2}}",1,"(-2*a*Cos[c + d*x]^3)/(3*d*(a + a*Sin[c + d*x])^(3/2))","A",1,1,23,0.04348,1,"{2673}"
162,1,22,0,0.0300435,"\int \frac{\cos (c+d x)}{\sqrt{a+a \sin (c+d x)}} \, dx","Int[Cos[c + d*x]/Sqrt[a + a*Sin[c + d*x]],x]","\frac{2 \sqrt{a \sin (c+d x)+a}}{a d}","\frac{2 \sqrt{a \sin (c+d x)+a}}{a d}",1,"(2*Sqrt[a + a*Sin[c + d*x]])/(a*d)","A",2,2,21,0.09524,1,"{2667, 32}"
163,1,60,0,0.0621285,"\int \frac{\sec (c+d x)}{\sqrt{a+a \sin (c+d x)}} \, dx","Int[Sec[c + d*x]/Sqrt[a + a*Sin[c + d*x]],x]","\frac{\tanh ^{-1}\left(\frac{\sqrt{a \sin (c+d x)+a}}{\sqrt{2} \sqrt{a}}\right)}{\sqrt{2} \sqrt{a} d}-\frac{1}{d \sqrt{a \sin (c+d x)+a}}","\frac{\tanh ^{-1}\left(\frac{\sqrt{a \sin (c+d x)+a}}{\sqrt{2} \sqrt{a}}\right)}{\sqrt{2} \sqrt{a} d}-\frac{1}{d \sqrt{a \sin (c+d x)+a}}",1,"ArcTanh[Sqrt[a + a*Sin[c + d*x]]/(Sqrt[2]*Sqrt[a])]/(Sqrt[2]*Sqrt[a]*d) - 1/(d*Sqrt[a + a*Sin[c + d*x]])","A",4,4,21,0.1905,1,"{2667, 51, 63, 206}"
164,1,102,0,0.0934787,"\int \frac{\sec ^2(c+d x)}{\sqrt{a+a \sin (c+d x)}} \, dx","Int[Sec[c + d*x]^2/Sqrt[a + a*Sin[c + d*x]],x]","-\frac{3 a \cos (c+d x)}{4 d (a \sin (c+d x)+a)^{3/2}}+\frac{\sec (c+d x)}{d \sqrt{a \sin (c+d x)+a}}-\frac{3 \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a \sin (c+d x)+a}}\right)}{4 \sqrt{2} \sqrt{a} d}","-\frac{3 a \cos (c+d x)}{4 d (a \sin (c+d x)+a)^{3/2}}+\frac{\sec (c+d x)}{d \sqrt{a \sin (c+d x)+a}}-\frac{3 \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a \sin (c+d x)+a}}\right)}{4 \sqrt{2} \sqrt{a} d}",1,"(-3*ArcTanh[(Sqrt[a]*Cos[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sin[c + d*x]])])/(4*Sqrt[2]*Sqrt[a]*d) - (3*a*Cos[c + d*x])/(4*d*(a + a*Sin[c + d*x])^(3/2)) + Sec[c + d*x]/(d*Sqrt[a + a*Sin[c + d*x]])","A",4,4,23,0.1739,1,"{2687, 2650, 2649, 206}"
165,1,116,0,0.1333646,"\int \frac{\sec ^3(c+d x)}{\sqrt{a+a \sin (c+d x)}} \, dx","Int[Sec[c + d*x]^3/Sqrt[a + a*Sin[c + d*x]],x]","-\frac{5}{8 d \sqrt{a \sin (c+d x)+a}}-\frac{5 a}{12 d (a \sin (c+d x)+a)^{3/2}}+\frac{5 \tanh ^{-1}\left(\frac{\sqrt{a \sin (c+d x)+a}}{\sqrt{2} \sqrt{a}}\right)}{8 \sqrt{2} \sqrt{a} d}+\frac{\sec ^2(c+d x)}{2 d \sqrt{a \sin (c+d x)+a}}","-\frac{5}{8 d \sqrt{a \sin (c+d x)+a}}-\frac{5 a}{12 d (a \sin (c+d x)+a)^{3/2}}+\frac{5 \tanh ^{-1}\left(\frac{\sqrt{a \sin (c+d x)+a}}{\sqrt{2} \sqrt{a}}\right)}{8 \sqrt{2} \sqrt{a} d}+\frac{\sec ^2(c+d x)}{2 d \sqrt{a \sin (c+d x)+a}}",1,"(5*ArcTanh[Sqrt[a + a*Sin[c + d*x]]/(Sqrt[2]*Sqrt[a])])/(8*Sqrt[2]*Sqrt[a]*d) - (5*a)/(12*d*(a + a*Sin[c + d*x])^(3/2)) - 5/(8*d*Sqrt[a + a*Sin[c + d*x]]) + Sec[c + d*x]^2/(2*d*Sqrt[a + a*Sin[c + d*x]])","A",6,5,23,0.2174,1,"{2687, 2667, 51, 63, 206}"
166,1,162,0,0.2176016,"\int \frac{\sec ^4(c+d x)}{\sqrt{a+a \sin (c+d x)}} \, dx","Int[Sec[c + d*x]^4/Sqrt[a + a*Sin[c + d*x]],x]","-\frac{35 a \cos (c+d x)}{64 d (a \sin (c+d x)+a)^{3/2}}+\frac{\sec ^3(c+d x)}{3 d \sqrt{a \sin (c+d x)+a}}+\frac{35 \sec (c+d x)}{48 d \sqrt{a \sin (c+d x)+a}}-\frac{7 a \sec (c+d x)}{24 d (a \sin (c+d x)+a)^{3/2}}-\frac{35 \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a \sin (c+d x)+a}}\right)}{64 \sqrt{2} \sqrt{a} d}","-\frac{35 a \cos (c+d x)}{64 d (a \sin (c+d x)+a)^{3/2}}+\frac{\sec ^3(c+d x)}{3 d \sqrt{a \sin (c+d x)+a}}+\frac{35 \sec (c+d x)}{48 d \sqrt{a \sin (c+d x)+a}}-\frac{7 a \sec (c+d x)}{24 d (a \sin (c+d x)+a)^{3/2}}-\frac{35 \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a \sin (c+d x)+a}}\right)}{64 \sqrt{2} \sqrt{a} d}",1,"(-35*ArcTanh[(Sqrt[a]*Cos[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sin[c + d*x]])])/(64*Sqrt[2]*Sqrt[a]*d) - (35*a*Cos[c + d*x])/(64*d*(a + a*Sin[c + d*x])^(3/2)) - (7*a*Sec[c + d*x])/(24*d*(a + a*Sin[c + d*x])^(3/2)) + (35*Sec[c + d*x])/(48*d*Sqrt[a + a*Sin[c + d*x]]) + Sec[c + d*x]^3/(3*d*Sqrt[a + a*Sin[c + d*x]])","A",6,5,23,0.2174,1,"{2687, 2681, 2650, 2649, 206}"
167,1,175,0,0.2629057,"\int \frac{\sec ^5(c+d x)}{\sqrt{a+a \sin (c+d x)}} \, dx","Int[Sec[c + d*x]^5/Sqrt[a + a*Sin[c + d*x]],x]","-\frac{63}{128 d \sqrt{a \sin (c+d x)+a}}-\frac{21 a}{64 d (a \sin (c+d x)+a)^{3/2}}+\frac{63 \tanh ^{-1}\left(\frac{\sqrt{a \sin (c+d x)+a}}{\sqrt{2} \sqrt{a}}\right)}{128 \sqrt{2} \sqrt{a} d}+\frac{\sec ^4(c+d x)}{4 d \sqrt{a \sin (c+d x)+a}}+\frac{63 \sec ^2(c+d x)}{160 d \sqrt{a \sin (c+d x)+a}}-\frac{9 a \sec ^2(c+d x)}{40 d (a \sin (c+d x)+a)^{3/2}}","-\frac{63}{128 d \sqrt{a \sin (c+d x)+a}}-\frac{21 a}{64 d (a \sin (c+d x)+a)^{3/2}}+\frac{63 \tanh ^{-1}\left(\frac{\sqrt{a \sin (c+d x)+a}}{\sqrt{2} \sqrt{a}}\right)}{128 \sqrt{2} \sqrt{a} d}+\frac{\sec ^4(c+d x)}{4 d \sqrt{a \sin (c+d x)+a}}+\frac{63 \sec ^2(c+d x)}{160 d \sqrt{a \sin (c+d x)+a}}-\frac{9 a \sec ^2(c+d x)}{40 d (a \sin (c+d x)+a)^{3/2}}",1,"(63*ArcTanh[Sqrt[a + a*Sin[c + d*x]]/(Sqrt[2]*Sqrt[a])])/(128*Sqrt[2]*Sqrt[a]*d) - (21*a)/(64*d*(a + a*Sin[c + d*x])^(3/2)) - (9*a*Sec[c + d*x]^2)/(40*d*(a + a*Sin[c + d*x])^(3/2)) - 63/(128*d*Sqrt[a + a*Sin[c + d*x]]) + (63*Sec[c + d*x]^2)/(160*d*Sqrt[a + a*Sin[c + d*x]]) + Sec[c + d*x]^4/(4*d*Sqrt[a + a*Sin[c + d*x]])","A",8,6,23,0.2609,1,"{2687, 2681, 2667, 51, 63, 206}"
168,1,221,0,0.3568796,"\int \frac{\sec ^6(c+d x)}{\sqrt{a+a \sin (c+d x)}} \, dx","Int[Sec[c + d*x]^6/Sqrt[a + a*Sin[c + d*x]],x]","-\frac{231 a \cos (c+d x)}{512 d (a \sin (c+d x)+a)^{3/2}}+\frac{\sec ^5(c+d x)}{5 d \sqrt{a \sin (c+d x)+a}}+\frac{11 \sec ^3(c+d x)}{40 d \sqrt{a \sin (c+d x)+a}}-\frac{11 a \sec ^3(c+d x)}{60 d (a \sin (c+d x)+a)^{3/2}}+\frac{77 \sec (c+d x)}{128 d \sqrt{a \sin (c+d x)+a}}-\frac{77 a \sec (c+d x)}{320 d (a \sin (c+d x)+a)^{3/2}}-\frac{231 \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a \sin (c+d x)+a}}\right)}{512 \sqrt{2} \sqrt{a} d}","-\frac{231 a \cos (c+d x)}{512 d (a \sin (c+d x)+a)^{3/2}}+\frac{\sec ^5(c+d x)}{5 d \sqrt{a \sin (c+d x)+a}}+\frac{11 \sec ^3(c+d x)}{40 d \sqrt{a \sin (c+d x)+a}}-\frac{11 a \sec ^3(c+d x)}{60 d (a \sin (c+d x)+a)^{3/2}}+\frac{77 \sec (c+d x)}{128 d \sqrt{a \sin (c+d x)+a}}-\frac{77 a \sec (c+d x)}{320 d (a \sin (c+d x)+a)^{3/2}}-\frac{231 \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a \sin (c+d x)+a}}\right)}{512 \sqrt{2} \sqrt{a} d}",1,"(-231*ArcTanh[(Sqrt[a]*Cos[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sin[c + d*x]])])/(512*Sqrt[2]*Sqrt[a]*d) - (231*a*Cos[c + d*x])/(512*d*(a + a*Sin[c + d*x])^(3/2)) - (77*a*Sec[c + d*x])/(320*d*(a + a*Sin[c + d*x])^(3/2)) - (11*a*Sec[c + d*x]^3)/(60*d*(a + a*Sin[c + d*x])^(3/2)) + (77*Sec[c + d*x])/(128*d*Sqrt[a + a*Sin[c + d*x]]) + (11*Sec[c + d*x]^3)/(40*d*Sqrt[a + a*Sin[c + d*x]]) + Sec[c + d*x]^5/(5*d*Sqrt[a + a*Sin[c + d*x]])","A",8,5,23,0.2174,1,"{2687, 2681, 2650, 2649, 206}"
169,1,97,0,0.0822959,"\int \frac{\cos ^7(c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx","Int[Cos[c + d*x]^7/(a + a*Sin[c + d*x])^(3/2),x]","-\frac{2 (a \sin (c+d x)+a)^{11/2}}{11 a^7 d}+\frac{4 (a \sin (c+d x)+a)^{9/2}}{3 a^6 d}-\frac{24 (a \sin (c+d x)+a)^{7/2}}{7 a^5 d}+\frac{16 (a \sin (c+d x)+a)^{5/2}}{5 a^4 d}","-\frac{2 (a \sin (c+d x)+a)^{11/2}}{11 a^7 d}+\frac{4 (a \sin (c+d x)+a)^{9/2}}{3 a^6 d}-\frac{24 (a \sin (c+d x)+a)^{7/2}}{7 a^5 d}+\frac{16 (a \sin (c+d x)+a)^{5/2}}{5 a^4 d}",1,"(16*(a + a*Sin[c + d*x])^(5/2))/(5*a^4*d) - (24*(a + a*Sin[c + d*x])^(7/2))/(7*a^5*d) + (4*(a + a*Sin[c + d*x])^(9/2))/(3*a^6*d) - (2*(a + a*Sin[c + d*x])^(11/2))/(11*a^7*d)","A",3,2,23,0.08696,1,"{2667, 43}"
170,1,63,0,0.11561,"\int \frac{\cos ^6(c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx","Int[Cos[c + d*x]^6/(a + a*Sin[c + d*x])^(3/2),x]","-\frac{8 a^2 \cos ^7(c+d x)}{63 d (a \sin (c+d x)+a)^{7/2}}-\frac{2 a \cos ^7(c+d x)}{9 d (a \sin (c+d x)+a)^{5/2}}","-\frac{8 a^2 \cos ^7(c+d x)}{63 d (a \sin (c+d x)+a)^{7/2}}-\frac{2 a \cos ^7(c+d x)}{9 d (a \sin (c+d x)+a)^{5/2}}",1,"(-8*a^2*Cos[c + d*x]^7)/(63*d*(a + a*Sin[c + d*x])^(7/2)) - (2*a*Cos[c + d*x]^7)/(9*d*(a + a*Sin[c + d*x])^(5/2))","A",2,2,23,0.08696,1,"{2674, 2673}"
171,1,73,0,0.0745764,"\int \frac{\cos ^5(c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx","Int[Cos[c + d*x]^5/(a + a*Sin[c + d*x])^(3/2),x]","\frac{2 (a \sin (c+d x)+a)^{7/2}}{7 a^5 d}-\frac{8 (a \sin (c+d x)+a)^{5/2}}{5 a^4 d}+\frac{8 (a \sin (c+d x)+a)^{3/2}}{3 a^3 d}","\frac{2 (a \sin (c+d x)+a)^{7/2}}{7 a^5 d}-\frac{8 (a \sin (c+d x)+a)^{5/2}}{5 a^4 d}+\frac{8 (a \sin (c+d x)+a)^{3/2}}{3 a^3 d}",1,"(8*(a + a*Sin[c + d*x])^(3/2))/(3*a^3*d) - (8*(a + a*Sin[c + d*x])^(5/2))/(5*a^4*d) + (2*(a + a*Sin[c + d*x])^(7/2))/(7*a^5*d)","A",3,2,23,0.08696,1,"{2667, 43}"
172,1,30,0,0.0581363,"\int \frac{\cos ^4(c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx","Int[Cos[c + d*x]^4/(a + a*Sin[c + d*x])^(3/2),x]","-\frac{2 a \cos ^5(c+d x)}{5 d (a \sin (c+d x)+a)^{5/2}}","-\frac{2 a \cos ^5(c+d x)}{5 d (a \sin (c+d x)+a)^{5/2}}",1,"(-2*a*Cos[c + d*x]^5)/(5*d*(a + a*Sin[c + d*x])^(5/2))","A",1,1,23,0.04348,1,"{2673}"
173,1,47,0,0.0665267,"\int \frac{\cos ^3(c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx","Int[Cos[c + d*x]^3/(a + a*Sin[c + d*x])^(3/2),x]","\frac{4 \sqrt{a \sin (c+d x)+a}}{a^2 d}-\frac{2 (a \sin (c+d x)+a)^{3/2}}{3 a^3 d}","\frac{4 \sqrt{a \sin (c+d x)+a}}{a^2 d}-\frac{2 (a \sin (c+d x)+a)^{3/2}}{3 a^3 d}",1,"(4*Sqrt[a + a*Sin[c + d*x]])/(a^2*d) - (2*(a + a*Sin[c + d*x])^(3/2))/(3*a^3*d)","A",3,2,23,0.08696,1,"{2667, 43}"
174,1,76,0,0.0806601,"\int \frac{\cos ^2(c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx","Int[Cos[c + d*x]^2/(a + a*Sin[c + d*x])^(3/2),x]","\frac{2 \cos (c+d x)}{a d \sqrt{a \sin (c+d x)+a}}-\frac{2 \sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a \sin (c+d x)+a}}\right)}{a^{3/2} d}","\frac{2 \cos (c+d x)}{a d \sqrt{a \sin (c+d x)+a}}-\frac{2 \sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a \sin (c+d x)+a}}\right)}{a^{3/2} d}",1,"(-2*Sqrt[2]*ArcTanh[(Sqrt[a]*Cos[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sin[c + d*x]])])/(a^(3/2)*d) + (2*Cos[c + d*x])/(a*d*Sqrt[a + a*Sin[c + d*x]])","A",3,3,23,0.1304,1,"{2679, 2649, 206}"
175,1,22,0,0.0335983,"\int \frac{\cos (c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx","Int[Cos[c + d*x]/(a + a*Sin[c + d*x])^(3/2),x]","-\frac{2}{a d \sqrt{a \sin (c+d x)+a}}","-\frac{2}{a d \sqrt{a \sin (c+d x)+a}}",1,"-2/(a*d*Sqrt[a + a*Sin[c + d*x]])","A",2,2,21,0.09524,1,"{2667, 32}"
176,1,89,0,0.0768569,"\int \frac{\sec (c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx","Int[Sec[c + d*x]/(a + a*Sin[c + d*x])^(3/2),x]","\frac{\tanh ^{-1}\left(\frac{\sqrt{a \sin (c+d x)+a}}{\sqrt{2} \sqrt{a}}\right)}{2 \sqrt{2} a^{3/2} d}-\frac{1}{2 a d \sqrt{a \sin (c+d x)+a}}-\frac{1}{3 d (a \sin (c+d x)+a)^{3/2}}","\frac{\tanh ^{-1}\left(\frac{\sqrt{a \sin (c+d x)+a}}{\sqrt{2} \sqrt{a}}\right)}{2 \sqrt{2} a^{3/2} d}-\frac{1}{2 a d \sqrt{a \sin (c+d x)+a}}-\frac{1}{3 d (a \sin (c+d x)+a)^{3/2}}",1,"ArcTanh[Sqrt[a + a*Sin[c + d*x]]/(Sqrt[2]*Sqrt[a])]/(2*Sqrt[2]*a^(3/2)*d) - 1/(3*d*(a + a*Sin[c + d*x])^(3/2)) - 1/(2*a*d*Sqrt[a + a*Sin[c + d*x]])","A",5,4,21,0.1905,1,"{2667, 51, 63, 206}"
177,1,134,0,0.1615349,"\int \frac{\sec ^2(c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx","Int[Sec[c + d*x]^2/(a + a*Sin[c + d*x])^(3/2),x]","-\frac{15 \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a \sin (c+d x)+a}}\right)}{32 \sqrt{2} a^{3/2} d}-\frac{15 \cos (c+d x)}{32 d (a \sin (c+d x)+a)^{3/2}}+\frac{5 \sec (c+d x)}{8 a d \sqrt{a \sin (c+d x)+a}}-\frac{\sec (c+d x)}{4 d (a \sin (c+d x)+a)^{3/2}}","-\frac{15 \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a \sin (c+d x)+a}}\right)}{32 \sqrt{2} a^{3/2} d}-\frac{15 \cos (c+d x)}{32 d (a \sin (c+d x)+a)^{3/2}}+\frac{5 \sec (c+d x)}{8 a d \sqrt{a \sin (c+d x)+a}}-\frac{\sec (c+d x)}{4 d (a \sin (c+d x)+a)^{3/2}}",1,"(-15*ArcTanh[(Sqrt[a]*Cos[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sin[c + d*x]])])/(32*Sqrt[2]*a^(3/2)*d) - (15*Cos[c + d*x])/(32*d*(a + a*Sin[c + d*x])^(3/2)) - Sec[c + d*x]/(4*d*(a + a*Sin[c + d*x])^(3/2)) + (5*Sec[c + d*x])/(8*a*d*Sqrt[a + a*Sin[c + d*x]])","A",5,5,23,0.2174,1,"{2681, 2687, 2650, 2649, 206}"
178,1,150,0,0.2024781,"\int \frac{\sec ^3(c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx","Int[Sec[c + d*x]^3/(a + a*Sin[c + d*x])^(3/2),x]","\frac{7 \tanh ^{-1}\left(\frac{\sqrt{a \sin (c+d x)+a}}{\sqrt{2} \sqrt{a}}\right)}{16 \sqrt{2} a^{3/2} d}-\frac{7}{16 a d \sqrt{a \sin (c+d x)+a}}-\frac{7}{24 d (a \sin (c+d x)+a)^{3/2}}+\frac{7 \sec ^2(c+d x)}{20 a d \sqrt{a \sin (c+d x)+a}}-\frac{\sec ^2(c+d x)}{5 d (a \sin (c+d x)+a)^{3/2}}","\frac{7 \tanh ^{-1}\left(\frac{\sqrt{a \sin (c+d x)+a}}{\sqrt{2} \sqrt{a}}\right)}{16 \sqrt{2} a^{3/2} d}-\frac{7}{16 a d \sqrt{a \sin (c+d x)+a}}-\frac{7}{24 d (a \sin (c+d x)+a)^{3/2}}+\frac{7 \sec ^2(c+d x)}{20 a d \sqrt{a \sin (c+d x)+a}}-\frac{\sec ^2(c+d x)}{5 d (a \sin (c+d x)+a)^{3/2}}",1,"(7*ArcTanh[Sqrt[a + a*Sin[c + d*x]]/(Sqrt[2]*Sqrt[a])])/(16*Sqrt[2]*a^(3/2)*d) - 7/(24*d*(a + a*Sin[c + d*x])^(3/2)) - Sec[c + d*x]^2/(5*d*(a + a*Sin[c + d*x])^(3/2)) - 7/(16*a*d*Sqrt[a + a*Sin[c + d*x]]) + (7*Sec[c + d*x]^2)/(20*a*d*Sqrt[a + a*Sin[c + d*x]])","A",7,6,23,0.2609,1,"{2681, 2687, 2667, 51, 63, 206}"
179,1,195,0,0.2905301,"\int \frac{\sec ^4(c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx","Int[Sec[c + d*x]^4/(a + a*Sin[c + d*x])^(3/2),x]","-\frac{105 \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a \sin (c+d x)+a}}\right)}{256 \sqrt{2} a^{3/2} d}-\frac{105 \cos (c+d x)}{256 d (a \sin (c+d x)+a)^{3/2}}+\frac{\sec ^3(c+d x)}{4 a d \sqrt{a \sin (c+d x)+a}}-\frac{\sec ^3(c+d x)}{6 d (a \sin (c+d x)+a)^{3/2}}+\frac{35 \sec (c+d x)}{64 a d \sqrt{a \sin (c+d x)+a}}-\frac{7 \sec (c+d x)}{32 d (a \sin (c+d x)+a)^{3/2}}","-\frac{105 \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a \sin (c+d x)+a}}\right)}{256 \sqrt{2} a^{3/2} d}-\frac{105 \cos (c+d x)}{256 d (a \sin (c+d x)+a)^{3/2}}+\frac{\sec ^3(c+d x)}{4 a d \sqrt{a \sin (c+d x)+a}}-\frac{\sec ^3(c+d x)}{6 d (a \sin (c+d x)+a)^{3/2}}+\frac{35 \sec (c+d x)}{64 a d \sqrt{a \sin (c+d x)+a}}-\frac{7 \sec (c+d x)}{32 d (a \sin (c+d x)+a)^{3/2}}",1,"(-105*ArcTanh[(Sqrt[a]*Cos[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sin[c + d*x]])])/(256*Sqrt[2]*a^(3/2)*d) - (105*Cos[c + d*x])/(256*d*(a + a*Sin[c + d*x])^(3/2)) - (7*Sec[c + d*x])/(32*d*(a + a*Sin[c + d*x])^(3/2)) - Sec[c + d*x]^3/(6*d*(a + a*Sin[c + d*x])^(3/2)) + (35*Sec[c + d*x])/(64*a*d*Sqrt[a + a*Sin[c + d*x]]) + Sec[c + d*x]^3/(4*a*d*Sqrt[a + a*Sin[c + d*x]])","A",7,5,23,0.2174,1,"{2681, 2687, 2650, 2649, 206}"
180,1,211,0,0.3427638,"\int \frac{\sec ^5(c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx","Int[Sec[c + d*x]^5/(a + a*Sin[c + d*x])^(3/2),x]","\frac{99 \tanh ^{-1}\left(\frac{\sqrt{a \sin (c+d x)+a}}{\sqrt{2} \sqrt{a}}\right)}{256 \sqrt{2} a^{3/2} d}-\frac{99}{256 a d \sqrt{a \sin (c+d x)+a}}-\frac{33}{128 d (a \sin (c+d x)+a)^{3/2}}+\frac{11 \sec ^4(c+d x)}{56 a d \sqrt{a \sin (c+d x)+a}}-\frac{\sec ^4(c+d x)}{7 d (a \sin (c+d x)+a)^{3/2}}+\frac{99 \sec ^2(c+d x)}{320 a d \sqrt{a \sin (c+d x)+a}}-\frac{99 \sec ^2(c+d x)}{560 d (a \sin (c+d x)+a)^{3/2}}","\frac{99 \tanh ^{-1}\left(\frac{\sqrt{a \sin (c+d x)+a}}{\sqrt{2} \sqrt{a}}\right)}{256 \sqrt{2} a^{3/2} d}-\frac{99}{256 a d \sqrt{a \sin (c+d x)+a}}-\frac{33}{128 d (a \sin (c+d x)+a)^{3/2}}+\frac{11 \sec ^4(c+d x)}{56 a d \sqrt{a \sin (c+d x)+a}}-\frac{\sec ^4(c+d x)}{7 d (a \sin (c+d x)+a)^{3/2}}+\frac{99 \sec ^2(c+d x)}{320 a d \sqrt{a \sin (c+d x)+a}}-\frac{99 \sec ^2(c+d x)}{560 d (a \sin (c+d x)+a)^{3/2}}",1,"(99*ArcTanh[Sqrt[a + a*Sin[c + d*x]]/(Sqrt[2]*Sqrt[a])])/(256*Sqrt[2]*a^(3/2)*d) - 33/(128*d*(a + a*Sin[c + d*x])^(3/2)) - (99*Sec[c + d*x]^2)/(560*d*(a + a*Sin[c + d*x])^(3/2)) - Sec[c + d*x]^4/(7*d*(a + a*Sin[c + d*x])^(3/2)) - 99/(256*a*d*Sqrt[a + a*Sin[c + d*x]]) + (99*Sec[c + d*x]^2)/(320*a*d*Sqrt[a + a*Sin[c + d*x]]) + (11*Sec[c + d*x]^4)/(56*a*d*Sqrt[a + a*Sin[c + d*x]])","A",9,6,23,0.2609,1,"{2681, 2687, 2667, 51, 63, 206}"
181,1,256,0,0.4290091,"\int \frac{\sec ^6(c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx","Int[Sec[c + d*x]^6/(a + a*Sin[c + d*x])^(3/2),x]","-\frac{3003 \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a \sin (c+d x)+a}}\right)}{8192 \sqrt{2} a^{3/2} d}-\frac{3003 \cos (c+d x)}{8192 d (a \sin (c+d x)+a)^{3/2}}+\frac{13 \sec ^5(c+d x)}{80 a d \sqrt{a \sin (c+d x)+a}}-\frac{\sec ^5(c+d x)}{8 d (a \sin (c+d x)+a)^{3/2}}+\frac{143 \sec ^3(c+d x)}{640 a d \sqrt{a \sin (c+d x)+a}}-\frac{143 \sec ^3(c+d x)}{960 d (a \sin (c+d x)+a)^{3/2}}+\frac{1001 \sec (c+d x)}{2048 a d \sqrt{a \sin (c+d x)+a}}-\frac{1001 \sec (c+d x)}{5120 d (a \sin (c+d x)+a)^{3/2}}","-\frac{3003 \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a \sin (c+d x)+a}}\right)}{8192 \sqrt{2} a^{3/2} d}-\frac{3003 \cos (c+d x)}{8192 d (a \sin (c+d x)+a)^{3/2}}+\frac{13 \sec ^5(c+d x)}{80 a d \sqrt{a \sin (c+d x)+a}}-\frac{\sec ^5(c+d x)}{8 d (a \sin (c+d x)+a)^{3/2}}+\frac{143 \sec ^3(c+d x)}{640 a d \sqrt{a \sin (c+d x)+a}}-\frac{143 \sec ^3(c+d x)}{960 d (a \sin (c+d x)+a)^{3/2}}+\frac{1001 \sec (c+d x)}{2048 a d \sqrt{a \sin (c+d x)+a}}-\frac{1001 \sec (c+d x)}{5120 d (a \sin (c+d x)+a)^{3/2}}",1,"(-3003*ArcTanh[(Sqrt[a]*Cos[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sin[c + d*x]])])/(8192*Sqrt[2]*a^(3/2)*d) - (3003*Cos[c + d*x])/(8192*d*(a + a*Sin[c + d*x])^(3/2)) - (1001*Sec[c + d*x])/(5120*d*(a + a*Sin[c + d*x])^(3/2)) - (143*Sec[c + d*x]^3)/(960*d*(a + a*Sin[c + d*x])^(3/2)) - Sec[c + d*x]^5/(8*d*(a + a*Sin[c + d*x])^(3/2)) + (1001*Sec[c + d*x])/(2048*a*d*Sqrt[a + a*Sin[c + d*x]]) + (143*Sec[c + d*x]^3)/(640*a*d*Sqrt[a + a*Sin[c + d*x]]) + (13*Sec[c + d*x]^5)/(80*a*d*Sqrt[a + a*Sin[c + d*x]])","A",9,5,23,0.2174,1,"{2681, 2687, 2650, 2649, 206}"
182,1,95,0,0.194634,"\int \frac{\cos ^{10}(c+d x)}{(a+a \sin (c+d x))^{5/2}} \, dx","Int[Cos[c + d*x]^10/(a + a*Sin[c + d*x])^(5/2),x]","-\frac{16 a^2 \cos ^{11}(c+d x)}{195 d (a \sin (c+d x)+a)^{9/2}}-\frac{64 a^3 \cos ^{11}(c+d x)}{2145 d (a \sin (c+d x)+a)^{11/2}}-\frac{2 a \cos ^{11}(c+d x)}{15 d (a \sin (c+d x)+a)^{7/2}}","-\frac{16 a^2 \cos ^{11}(c+d x)}{195 d (a \sin (c+d x)+a)^{9/2}}-\frac{64 a^3 \cos ^{11}(c+d x)}{2145 d (a \sin (c+d x)+a)^{11/2}}-\frac{2 a \cos ^{11}(c+d x)}{15 d (a \sin (c+d x)+a)^{7/2}}",1,"(-64*a^3*Cos[c + d*x]^11)/(2145*d*(a + a*Sin[c + d*x])^(11/2)) - (16*a^2*Cos[c + d*x]^11)/(195*d*(a + a*Sin[c + d*x])^(9/2)) - (2*a*Cos[c + d*x]^11)/(15*d*(a + a*Sin[c + d*x])^(7/2))","A",3,2,23,0.08696,1,"{2674, 2673}"
183,1,121,0,0.0900153,"\int \frac{\cos ^9(c+d x)}{(a+a \sin (c+d x))^{5/2}} \, dx","Int[Cos[c + d*x]^9/(a + a*Sin[c + d*x])^(5/2),x]","\frac{2 (a \sin (c+d x)+a)^{13/2}}{13 a^9 d}-\frac{16 (a \sin (c+d x)+a)^{11/2}}{11 a^8 d}+\frac{16 (a \sin (c+d x)+a)^{9/2}}{3 a^7 d}-\frac{64 (a \sin (c+d x)+a)^{7/2}}{7 a^6 d}+\frac{32 (a \sin (c+d x)+a)^{5/2}}{5 a^5 d}","\frac{2 (a \sin (c+d x)+a)^{13/2}}{13 a^9 d}-\frac{16 (a \sin (c+d x)+a)^{11/2}}{11 a^8 d}+\frac{16 (a \sin (c+d x)+a)^{9/2}}{3 a^7 d}-\frac{64 (a \sin (c+d x)+a)^{7/2}}{7 a^6 d}+\frac{32 (a \sin (c+d x)+a)^{5/2}}{5 a^5 d}",1,"(32*(a + a*Sin[c + d*x])^(5/2))/(5*a^5*d) - (64*(a + a*Sin[c + d*x])^(7/2))/(7*a^6*d) + (16*(a + a*Sin[c + d*x])^(9/2))/(3*a^7*d) - (16*(a + a*Sin[c + d*x])^(11/2))/(11*a^8*d) + (2*(a + a*Sin[c + d*x])^(13/2))/(13*a^9*d)","A",3,2,23,0.08696,1,"{2667, 43}"
184,1,63,0,0.1188397,"\int \frac{\cos ^8(c+d x)}{(a+a \sin (c+d x))^{5/2}} \, dx","Int[Cos[c + d*x]^8/(a + a*Sin[c + d*x])^(5/2),x]","-\frac{8 a^2 \cos ^9(c+d x)}{99 d (a \sin (c+d x)+a)^{9/2}}-\frac{2 a \cos ^9(c+d x)}{11 d (a \sin (c+d x)+a)^{7/2}}","-\frac{8 a^2 \cos ^9(c+d x)}{99 d (a \sin (c+d x)+a)^{9/2}}-\frac{2 a \cos ^9(c+d x)}{11 d (a \sin (c+d x)+a)^{7/2}}",1,"(-8*a^2*Cos[c + d*x]^9)/(99*d*(a + a*Sin[c + d*x])^(9/2)) - (2*a*Cos[c + d*x]^9)/(11*d*(a + a*Sin[c + d*x])^(7/2))","A",2,2,23,0.08696,1,"{2674, 2673}"
185,1,97,0,0.0831496,"\int \frac{\cos ^7(c+d x)}{(a+a \sin (c+d x))^{5/2}} \, dx","Int[Cos[c + d*x]^7/(a + a*Sin[c + d*x])^(5/2),x]","-\frac{2 (a \sin (c+d x)+a)^{9/2}}{9 a^7 d}+\frac{12 (a \sin (c+d x)+a)^{7/2}}{7 a^6 d}-\frac{24 (a \sin (c+d x)+a)^{5/2}}{5 a^5 d}+\frac{16 (a \sin (c+d x)+a)^{3/2}}{3 a^4 d}","-\frac{2 (a \sin (c+d x)+a)^{9/2}}{9 a^7 d}+\frac{12 (a \sin (c+d x)+a)^{7/2}}{7 a^6 d}-\frac{24 (a \sin (c+d x)+a)^{5/2}}{5 a^5 d}+\frac{16 (a \sin (c+d x)+a)^{3/2}}{3 a^4 d}",1,"(16*(a + a*Sin[c + d*x])^(3/2))/(3*a^4*d) - (24*(a + a*Sin[c + d*x])^(5/2))/(5*a^5*d) + (12*(a + a*Sin[c + d*x])^(7/2))/(7*a^6*d) - (2*(a + a*Sin[c + d*x])^(9/2))/(9*a^7*d)","A",3,2,23,0.08696,1,"{2667, 43}"
186,1,30,0,0.0571351,"\int \frac{\cos ^6(c+d x)}{(a+a \sin (c+d x))^{5/2}} \, dx","Int[Cos[c + d*x]^6/(a + a*Sin[c + d*x])^(5/2),x]","-\frac{2 a \cos ^7(c+d x)}{7 d (a \sin (c+d x)+a)^{7/2}}","-\frac{2 a \cos ^7(c+d x)}{7 d (a \sin (c+d x)+a)^{7/2}}",1,"(-2*a*Cos[c + d*x]^7)/(7*d*(a + a*Sin[c + d*x])^(7/2))","A",1,1,23,0.04348,1,"{2673}"
187,1,71,0,0.074642,"\int \frac{\cos ^5(c+d x)}{(a+a \sin (c+d x))^{5/2}} \, dx","Int[Cos[c + d*x]^5/(a + a*Sin[c + d*x])^(5/2),x]","\frac{2 (a \sin (c+d x)+a)^{5/2}}{5 a^5 d}-\frac{8 (a \sin (c+d x)+a)^{3/2}}{3 a^4 d}+\frac{8 \sqrt{a \sin (c+d x)+a}}{a^3 d}","\frac{2 (a \sin (c+d x)+a)^{5/2}}{5 a^5 d}-\frac{8 (a \sin (c+d x)+a)^{3/2}}{3 a^4 d}+\frac{8 \sqrt{a \sin (c+d x)+a}}{a^3 d}",1,"(8*Sqrt[a + a*Sin[c + d*x]])/(a^3*d) - (8*(a + a*Sin[c + d*x])^(3/2))/(3*a^4*d) + (2*(a + a*Sin[c + d*x])^(5/2))/(5*a^5*d)","A",3,2,23,0.08696,1,"{2667, 43}"
188,1,108,0,0.1440762,"\int \frac{\cos ^4(c+d x)}{(a+a \sin (c+d x))^{5/2}} \, dx","Int[Cos[c + d*x]^4/(a + a*Sin[c + d*x])^(5/2),x]","\frac{4 \cos (c+d x)}{a^2 d \sqrt{a \sin (c+d x)+a}}-\frac{4 \sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a \sin (c+d x)+a}}\right)}{a^{5/2} d}+\frac{2 \cos ^3(c+d x)}{3 a d (a \sin (c+d x)+a)^{3/2}}","\frac{4 \cos (c+d x)}{a^2 d \sqrt{a \sin (c+d x)+a}}-\frac{4 \sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a \sin (c+d x)+a}}\right)}{a^{5/2} d}+\frac{2 \cos ^3(c+d x)}{3 a d (a \sin (c+d x)+a)^{3/2}}",1,"(-4*Sqrt[2]*ArcTanh[(Sqrt[a]*Cos[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sin[c + d*x]])])/(a^(5/2)*d) + (2*Cos[c + d*x]^3)/(3*a*d*(a + a*Sin[c + d*x])^(3/2)) + (4*Cos[c + d*x])/(a^2*d*Sqrt[a + a*Sin[c + d*x]])","A",4,3,23,0.1304,1,"{2679, 2649, 206}"
189,1,45,0,0.0676261,"\int \frac{\cos ^3(c+d x)}{(a+a \sin (c+d x))^{5/2}} \, dx","Int[Cos[c + d*x]^3/(a + a*Sin[c + d*x])^(5/2),x]","-\frac{2 \sqrt{a \sin (c+d x)+a}}{a^3 d}-\frac{4}{a^2 d \sqrt{a \sin (c+d x)+a}}","-\frac{2 \sqrt{a \sin (c+d x)+a}}{a^3 d}-\frac{4}{a^2 d \sqrt{a \sin (c+d x)+a}}",1,"-4/(a^2*d*Sqrt[a + a*Sin[c + d*x]]) - (2*Sqrt[a + a*Sin[c + d*x]])/(a^3*d)","A",3,2,23,0.08696,1,"{2667, 43}"
190,1,75,0,0.0815447,"\int \frac{\cos ^2(c+d x)}{(a+a \sin (c+d x))^{5/2}} \, dx","Int[Cos[c + d*x]^2/(a + a*Sin[c + d*x])^(5/2),x]","\frac{\tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a \sin (c+d x)+a}}\right)}{\sqrt{2} a^{5/2} d}-\frac{\cos (c+d x)}{a d (a \sin (c+d x)+a)^{3/2}}","\frac{\tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a \sin (c+d x)+a}}\right)}{\sqrt{2} a^{5/2} d}-\frac{\cos (c+d x)}{a d (a \sin (c+d x)+a)^{3/2}}",1,"ArcTanh[(Sqrt[a]*Cos[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sin[c + d*x]])]/(Sqrt[2]*a^(5/2)*d) - Cos[c + d*x]/(a*d*(a + a*Sin[c + d*x])^(3/2))","A",3,3,23,0.1304,1,"{2680, 2649, 206}"
191,1,24,0,0.0348205,"\int \frac{\cos (c+d x)}{(a+a \sin (c+d x))^{5/2}} \, dx","Int[Cos[c + d*x]/(a + a*Sin[c + d*x])^(5/2),x]","-\frac{2}{3 a d (a \sin (c+d x)+a)^{3/2}}","-\frac{2}{3 a d (a \sin (c+d x)+a)^{3/2}}",1,"-2/(3*a*d*(a + a*Sin[c + d*x])^(3/2))","A",2,2,21,0.09524,1,"{2667, 32}"
192,1,113,0,0.0886034,"\int \frac{\sec (c+d x)}{(a+a \sin (c+d x))^{5/2}} \, dx","Int[Sec[c + d*x]/(a + a*Sin[c + d*x])^(5/2),x]","-\frac{1}{4 a^2 d \sqrt{a \sin (c+d x)+a}}+\frac{\tanh ^{-1}\left(\frac{\sqrt{a \sin (c+d x)+a}}{\sqrt{2} \sqrt{a}}\right)}{4 \sqrt{2} a^{5/2} d}-\frac{1}{6 a d (a \sin (c+d x)+a)^{3/2}}-\frac{1}{5 d (a \sin (c+d x)+a)^{5/2}}","-\frac{1}{4 a^2 d \sqrt{a \sin (c+d x)+a}}+\frac{\tanh ^{-1}\left(\frac{\sqrt{a \sin (c+d x)+a}}{\sqrt{2} \sqrt{a}}\right)}{4 \sqrt{2} a^{5/2} d}-\frac{1}{6 a d (a \sin (c+d x)+a)^{3/2}}-\frac{1}{5 d (a \sin (c+d x)+a)^{5/2}}",1,"ArcTanh[Sqrt[a + a*Sin[c + d*x]]/(Sqrt[2]*Sqrt[a])]/(4*Sqrt[2]*a^(5/2)*d) - 1/(5*d*(a + a*Sin[c + d*x])^(5/2)) - 1/(6*a*d*(a + a*Sin[c + d*x])^(3/2)) - 1/(4*a^2*d*Sqrt[a + a*Sin[c + d*x]])","A",6,4,21,0.1905,1,"{2667, 51, 63, 206}"
193,1,167,0,0.2304179,"\int \frac{\sec ^2(c+d x)}{(a+a \sin (c+d x))^{5/2}} \, dx","Int[Sec[c + d*x]^2/(a + a*Sin[c + d*x])^(5/2),x]","\frac{35 \sec (c+d x)}{96 a^2 d \sqrt{a \sin (c+d x)+a}}-\frac{35 \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a \sin (c+d x)+a}}\right)}{128 \sqrt{2} a^{5/2} d}-\frac{35 \cos (c+d x)}{128 a d (a \sin (c+d x)+a)^{3/2}}-\frac{7 \sec (c+d x)}{48 a d (a \sin (c+d x)+a)^{3/2}}-\frac{\sec (c+d x)}{6 d (a \sin (c+d x)+a)^{5/2}}","\frac{35 \sec (c+d x)}{96 a^2 d \sqrt{a \sin (c+d x)+a}}-\frac{35 \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a \sin (c+d x)+a}}\right)}{128 \sqrt{2} a^{5/2} d}-\frac{35 \cos (c+d x)}{128 a d (a \sin (c+d x)+a)^{3/2}}-\frac{7 \sec (c+d x)}{48 a d (a \sin (c+d x)+a)^{3/2}}-\frac{\sec (c+d x)}{6 d (a \sin (c+d x)+a)^{5/2}}",1,"(-35*ArcTanh[(Sqrt[a]*Cos[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sin[c + d*x]])])/(128*Sqrt[2]*a^(5/2)*d) - Sec[c + d*x]/(6*d*(a + a*Sin[c + d*x])^(5/2)) - (35*Cos[c + d*x])/(128*a*d*(a + a*Sin[c + d*x])^(3/2)) - (7*Sec[c + d*x])/(48*a*d*(a + a*Sin[c + d*x])^(3/2)) + (35*Sec[c + d*x])/(96*a^2*d*Sqrt[a + a*Sin[c + d*x]])","A",6,5,23,0.2174,1,"{2681, 2687, 2650, 2649, 206}"
194,1,185,0,0.27491,"\int \frac{\sec ^3(c+d x)}{(a+a \sin (c+d x))^{5/2}} \, dx","Int[Sec[c + d*x]^3/(a + a*Sin[c + d*x])^(5/2),x]","-\frac{9}{32 a^2 d \sqrt{a \sin (c+d x)+a}}+\frac{9 \tanh ^{-1}\left(\frac{\sqrt{a \sin (c+d x)+a}}{\sqrt{2} \sqrt{a}}\right)}{32 \sqrt{2} a^{5/2} d}+\frac{9 \sec ^2(c+d x)}{40 a^2 d \sqrt{a \sin (c+d x)+a}}-\frac{3}{16 a d (a \sin (c+d x)+a)^{3/2}}-\frac{9 \sec ^2(c+d x)}{70 a d (a \sin (c+d x)+a)^{3/2}}-\frac{\sec ^2(c+d x)}{7 d (a \sin (c+d x)+a)^{5/2}}","-\frac{9}{32 a^2 d \sqrt{a \sin (c+d x)+a}}+\frac{9 \tanh ^{-1}\left(\frac{\sqrt{a \sin (c+d x)+a}}{\sqrt{2} \sqrt{a}}\right)}{32 \sqrt{2} a^{5/2} d}+\frac{9 \sec ^2(c+d x)}{40 a^2 d \sqrt{a \sin (c+d x)+a}}-\frac{3}{16 a d (a \sin (c+d x)+a)^{3/2}}-\frac{9 \sec ^2(c+d x)}{70 a d (a \sin (c+d x)+a)^{3/2}}-\frac{\sec ^2(c+d x)}{7 d (a \sin (c+d x)+a)^{5/2}}",1,"(9*ArcTanh[Sqrt[a + a*Sin[c + d*x]]/(Sqrt[2]*Sqrt[a])])/(32*Sqrt[2]*a^(5/2)*d) - Sec[c + d*x]^2/(7*d*(a + a*Sin[c + d*x])^(5/2)) - 3/(16*a*d*(a + a*Sin[c + d*x])^(3/2)) - (9*Sec[c + d*x]^2)/(70*a*d*(a + a*Sin[c + d*x])^(3/2)) - 9/(32*a^2*d*Sqrt[a + a*Sin[c + d*x]]) + (9*Sec[c + d*x]^2)/(40*a^2*d*Sqrt[a + a*Sin[c + d*x]])","A",8,6,23,0.2609,1,"{2681, 2687, 2667, 51, 63, 206}"
195,1,233,0,0.3638382,"\int \frac{\sec ^4(c+d x)}{(a+a \sin (c+d x))^{5/2}} \, dx","Int[Sec[c + d*x]^4/(a + a*Sin[c + d*x])^(5/2),x]","\frac{11 \sec ^3(c+d x)}{64 a^2 d \sqrt{a \sin (c+d x)+a}}+\frac{385 \sec (c+d x)}{1024 a^2 d \sqrt{a \sin (c+d x)+a}}-\frac{1155 \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a \sin (c+d x)+a}}\right)}{4096 \sqrt{2} a^{5/2} d}-\frac{1155 \cos (c+d x)}{4096 a d (a \sin (c+d x)+a)^{3/2}}-\frac{11 \sec ^3(c+d x)}{96 a d (a \sin (c+d x)+a)^{3/2}}-\frac{\sec ^3(c+d x)}{8 d (a \sin (c+d x)+a)^{5/2}}-\frac{77 \sec (c+d x)}{512 a d (a \sin (c+d x)+a)^{3/2}}","\frac{11 \sec ^3(c+d x)}{64 a^2 d \sqrt{a \sin (c+d x)+a}}+\frac{385 \sec (c+d x)}{1024 a^2 d \sqrt{a \sin (c+d x)+a}}-\frac{1155 \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a \sin (c+d x)+a}}\right)}{4096 \sqrt{2} a^{5/2} d}-\frac{1155 \cos (c+d x)}{4096 a d (a \sin (c+d x)+a)^{3/2}}-\frac{11 \sec ^3(c+d x)}{96 a d (a \sin (c+d x)+a)^{3/2}}-\frac{\sec ^3(c+d x)}{8 d (a \sin (c+d x)+a)^{5/2}}-\frac{77 \sec (c+d x)}{512 a d (a \sin (c+d x)+a)^{3/2}}",1,"(-1155*ArcTanh[(Sqrt[a]*Cos[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sin[c + d*x]])])/(4096*Sqrt[2]*a^(5/2)*d) - Sec[c + d*x]^3/(8*d*(a + a*Sin[c + d*x])^(5/2)) - (1155*Cos[c + d*x])/(4096*a*d*(a + a*Sin[c + d*x])^(3/2)) - (77*Sec[c + d*x])/(512*a*d*(a + a*Sin[c + d*x])^(3/2)) - (11*Sec[c + d*x]^3)/(96*a*d*(a + a*Sin[c + d*x])^(3/2)) + (385*Sec[c + d*x])/(1024*a^2*d*Sqrt[a + a*Sin[c + d*x]]) + (11*Sec[c + d*x]^3)/(64*a^2*d*Sqrt[a + a*Sin[c + d*x]])","A",8,5,23,0.2174,1,"{2681, 2687, 2650, 2649, 206}"
196,1,124,0,0.0842286,"\int (e \cos (c+d x))^{7/2} (a+a \sin (c+d x)) \, dx","Int[(e*Cos[c + d*x])^(7/2)*(a + a*Sin[c + d*x]),x]","\frac{10 a e^3 \sin (c+d x) \sqrt{e \cos (c+d x)}}{21 d}+\frac{10 a e^4 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d \sqrt{e \cos (c+d x)}}-\frac{2 a (e \cos (c+d x))^{9/2}}{9 d e}+\frac{2 a e \sin (c+d x) (e \cos (c+d x))^{5/2}}{7 d}","\frac{10 a e^3 \sin (c+d x) \sqrt{e \cos (c+d x)}}{21 d}+\frac{10 a e^4 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d \sqrt{e \cos (c+d x)}}-\frac{2 a (e \cos (c+d x))^{9/2}}{9 d e}+\frac{2 a e \sin (c+d x) (e \cos (c+d x))^{5/2}}{7 d}",1,"(-2*a*(e*Cos[c + d*x])^(9/2))/(9*d*e) + (10*a*e^4*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(21*d*Sqrt[e*Cos[c + d*x]]) + (10*a*e^3*Sqrt[e*Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (2*a*e*(e*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(7*d)","A",5,4,23,0.1739,1,"{2669, 2635, 2642, 2641}"
197,1,95,0,0.0671417,"\int (e \cos (c+d x))^{5/2} (a+a \sin (c+d x)) \, dx","Int[(e*Cos[c + d*x])^(5/2)*(a + a*Sin[c + d*x]),x]","\frac{6 a e^2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{5 d \sqrt{\cos (c+d x)}}-\frac{2 a (e \cos (c+d x))^{7/2}}{7 d e}+\frac{2 a e \sin (c+d x) (e \cos (c+d x))^{3/2}}{5 d}","\frac{6 a e^2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{5 d \sqrt{\cos (c+d x)}}-\frac{2 a (e \cos (c+d x))^{7/2}}{7 d e}+\frac{2 a e \sin (c+d x) (e \cos (c+d x))^{3/2}}{5 d}",1,"(-2*a*(e*Cos[c + d*x])^(7/2))/(7*d*e) + (6*a*e^2*Sqrt[e*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(5*d*Sqrt[Cos[c + d*x]]) + (2*a*e*(e*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*d)","A",4,4,23,0.1739,1,"{2669, 2635, 2640, 2639}"
198,1,95,0,0.0649123,"\int (e \cos (c+d x))^{3/2} (a+a \sin (c+d x)) \, dx","Int[(e*Cos[c + d*x])^(3/2)*(a + a*Sin[c + d*x]),x]","\frac{2 a e^2 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d \sqrt{e \cos (c+d x)}}-\frac{2 a (e \cos (c+d x))^{5/2}}{5 d e}+\frac{2 a e \sin (c+d x) \sqrt{e \cos (c+d x)}}{3 d}","\frac{2 a e^2 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d \sqrt{e \cos (c+d x)}}-\frac{2 a (e \cos (c+d x))^{5/2}}{5 d e}+\frac{2 a e \sin (c+d x) \sqrt{e \cos (c+d x)}}{3 d}",1,"(-2*a*(e*Cos[c + d*x])^(5/2))/(5*d*e) + (2*a*e^2*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(3*d*Sqrt[e*Cos[c + d*x]]) + (2*a*e*Sqrt[e*Cos[c + d*x]]*Sin[c + d*x])/(3*d)","A",4,4,23,0.1739,1,"{2669, 2635, 2642, 2641}"
199,1,63,0,0.0460988,"\int \sqrt{e \cos (c+d x)} (a+a \sin (c+d x)) \, dx","Int[Sqrt[e*Cos[c + d*x]]*(a + a*Sin[c + d*x]),x]","\frac{2 a E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{d \sqrt{\cos (c+d x)}}-\frac{2 a (e \cos (c+d x))^{3/2}}{3 d e}","\frac{2 a E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{d \sqrt{\cos (c+d x)}}-\frac{2 a (e \cos (c+d x))^{3/2}}{3 d e}",1,"(-2*a*(e*Cos[c + d*x])^(3/2))/(3*d*e) + (2*a*Sqrt[e*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(d*Sqrt[Cos[c + d*x]])","A",3,3,23,0.1304,1,"{2669, 2640, 2639}"
200,1,61,0,0.0463531,"\int \frac{a+a \sin (c+d x)}{\sqrt{e \cos (c+d x)}} \, dx","Int[(a + a*Sin[c + d*x])/Sqrt[e*Cos[c + d*x]],x]","\frac{2 a \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d \sqrt{e \cos (c+d x)}}-\frac{2 a \sqrt{e \cos (c+d x)}}{d e}","\frac{2 a \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d \sqrt{e \cos (c+d x)}}-\frac{2 a \sqrt{e \cos (c+d x)}}{d e}",1,"(-2*a*Sqrt[e*Cos[c + d*x]])/(d*e) + (2*a*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(d*Sqrt[e*Cos[c + d*x]])","A",3,3,23,0.1304,1,"{2669, 2642, 2641}"
201,1,91,0,0.0676036,"\int \frac{a+a \sin (c+d x)}{(e \cos (c+d x))^{3/2}} \, dx","Int[(a + a*Sin[c + d*x])/(e*Cos[c + d*x])^(3/2),x]","-\frac{2 a E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{d e^2 \sqrt{\cos (c+d x)}}+\frac{2 a}{d e \sqrt{e \cos (c+d x)}}+\frac{2 a \sin (c+d x)}{d e \sqrt{e \cos (c+d x)}}","-\frac{2 a E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{d e^2 \sqrt{\cos (c+d x)}}+\frac{2 a}{d e \sqrt{e \cos (c+d x)}}+\frac{2 a \sin (c+d x)}{d e \sqrt{e \cos (c+d x)}}",1,"(2*a)/(d*e*Sqrt[e*Cos[c + d*x]]) - (2*a*Sqrt[e*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(d*e^2*Sqrt[Cos[c + d*x]]) + (2*a*Sin[c + d*x])/(d*e*Sqrt[e*Cos[c + d*x]])","A",4,4,23,0.1739,1,"{2669, 2636, 2640, 2639}"
202,1,97,0,0.0652218,"\int \frac{a+a \sin (c+d x)}{(e \cos (c+d x))^{5/2}} \, dx","Int[(a + a*Sin[c + d*x])/(e*Cos[c + d*x])^(5/2),x]","\frac{2 a \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d e^2 \sqrt{e \cos (c+d x)}}+\frac{2 a}{3 d e (e \cos (c+d x))^{3/2}}+\frac{2 a \sin (c+d x)}{3 d e (e \cos (c+d x))^{3/2}}","\frac{2 a \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d e^2 \sqrt{e \cos (c+d x)}}+\frac{2 a}{3 d e (e \cos (c+d x))^{3/2}}+\frac{2 a \sin (c+d x)}{3 d e (e \cos (c+d x))^{3/2}}",1,"(2*a)/(3*d*e*(e*Cos[c + d*x])^(3/2)) + (2*a*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(3*d*e^2*Sqrt[e*Cos[c + d*x]]) + (2*a*Sin[c + d*x])/(3*d*e*(e*Cos[c + d*x])^(3/2))","A",4,4,23,0.1739,1,"{2669, 2636, 2642, 2641}"
203,1,126,0,0.0863084,"\int \frac{a+a \sin (c+d x)}{(e \cos (c+d x))^{7/2}} \, dx","Int[(a + a*Sin[c + d*x])/(e*Cos[c + d*x])^(7/2),x]","\frac{6 a \sin (c+d x)}{5 d e^3 \sqrt{e \cos (c+d x)}}-\frac{6 a E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{5 d e^4 \sqrt{\cos (c+d x)}}+\frac{2 a}{5 d e (e \cos (c+d x))^{5/2}}+\frac{2 a \sin (c+d x)}{5 d e (e \cos (c+d x))^{5/2}}","\frac{6 a \sin (c+d x)}{5 d e^3 \sqrt{e \cos (c+d x)}}-\frac{6 a E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{5 d e^4 \sqrt{\cos (c+d x)}}+\frac{2 a}{5 d e (e \cos (c+d x))^{5/2}}+\frac{2 a \sin (c+d x)}{5 d e (e \cos (c+d x))^{5/2}}",1,"(2*a)/(5*d*e*(e*Cos[c + d*x])^(5/2)) - (6*a*Sqrt[e*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(5*d*e^4*Sqrt[Cos[c + d*x]]) + (2*a*Sin[c + d*x])/(5*d*e*(e*Cos[c + d*x])^(5/2)) + (6*a*Sin[c + d*x])/(5*d*e^3*Sqrt[e*Cos[c + d*x]])","A",5,4,23,0.1739,1,"{2669, 2636, 2640, 2639}"
204,1,168,0,0.144141,"\int (e \cos (c+d x))^{7/2} (a+a \sin (c+d x))^2 \, dx","Int[(e*Cos[c + d*x])^(7/2)*(a + a*Sin[c + d*x])^2,x]","\frac{130 a^2 e^3 \sin (c+d x) \sqrt{e \cos (c+d x)}}{231 d}+\frac{130 a^2 e^4 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{231 d \sqrt{e \cos (c+d x)}}-\frac{26 a^2 (e \cos (c+d x))^{9/2}}{99 d e}-\frac{2 \left(a^2 \sin (c+d x)+a^2\right) (e \cos (c+d x))^{9/2}}{11 d e}+\frac{26 a^2 e \sin (c+d x) (e \cos (c+d x))^{5/2}}{77 d}","\frac{130 a^2 e^3 \sin (c+d x) \sqrt{e \cos (c+d x)}}{231 d}+\frac{130 a^2 e^4 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{231 d \sqrt{e \cos (c+d x)}}-\frac{26 a^2 (e \cos (c+d x))^{9/2}}{99 d e}-\frac{2 \left(a^2 \sin (c+d x)+a^2\right) (e \cos (c+d x))^{9/2}}{11 d e}+\frac{26 a^2 e \sin (c+d x) (e \cos (c+d x))^{5/2}}{77 d}",1,"(-26*a^2*(e*Cos[c + d*x])^(9/2))/(99*d*e) + (130*a^2*e^4*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(231*d*Sqrt[e*Cos[c + d*x]]) + (130*a^2*e^3*Sqrt[e*Cos[c + d*x]]*Sin[c + d*x])/(231*d) + (26*a^2*e*(e*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(77*d) - (2*(e*Cos[c + d*x])^(9/2)*(a^2 + a^2*Sin[c + d*x]))/(11*d*e)","A",6,5,25,0.2000,1,"{2678, 2669, 2635, 2642, 2641}"
205,1,137,0,0.1220077,"\int (e \cos (c+d x))^{5/2} (a+a \sin (c+d x))^2 \, dx","Int[(e*Cos[c + d*x])^(5/2)*(a + a*Sin[c + d*x])^2,x]","\frac{22 a^2 e^2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{15 d \sqrt{\cos (c+d x)}}-\frac{22 a^2 (e \cos (c+d x))^{7/2}}{63 d e}-\frac{2 \left(a^2 \sin (c+d x)+a^2\right) (e \cos (c+d x))^{7/2}}{9 d e}+\frac{22 a^2 e \sin (c+d x) (e \cos (c+d x))^{3/2}}{45 d}","\frac{22 a^2 e^2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{15 d \sqrt{\cos (c+d x)}}-\frac{22 a^2 (e \cos (c+d x))^{7/2}}{63 d e}-\frac{2 \left(a^2 \sin (c+d x)+a^2\right) (e \cos (c+d x))^{7/2}}{9 d e}+\frac{22 a^2 e \sin (c+d x) (e \cos (c+d x))^{3/2}}{45 d}",1,"(-22*a^2*(e*Cos[c + d*x])^(7/2))/(63*d*e) + (22*a^2*e^2*Sqrt[e*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(15*d*Sqrt[Cos[c + d*x]]) + (22*a^2*e*(e*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(45*d) - (2*(e*Cos[c + d*x])^(7/2)*(a^2 + a^2*Sin[c + d*x]))/(9*d*e)","A",5,5,25,0.2000,1,"{2678, 2669, 2635, 2640, 2639}"
206,1,137,0,0.1247659,"\int (e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^2 \, dx","Int[(e*Cos[c + d*x])^(3/2)*(a + a*Sin[c + d*x])^2,x]","\frac{6 a^2 e^2 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{7 d \sqrt{e \cos (c+d x)}}-\frac{18 a^2 (e \cos (c+d x))^{5/2}}{35 d e}-\frac{2 \left(a^2 \sin (c+d x)+a^2\right) (e \cos (c+d x))^{5/2}}{7 d e}+\frac{6 a^2 e \sin (c+d x) \sqrt{e \cos (c+d x)}}{7 d}","\frac{6 a^2 e^2 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{7 d \sqrt{e \cos (c+d x)}}-\frac{18 a^2 (e \cos (c+d x))^{5/2}}{35 d e}-\frac{2 \left(a^2 \sin (c+d x)+a^2\right) (e \cos (c+d x))^{5/2}}{7 d e}+\frac{6 a^2 e \sin (c+d x) \sqrt{e \cos (c+d x)}}{7 d}",1,"(-18*a^2*(e*Cos[c + d*x])^(5/2))/(35*d*e) + (6*a^2*e^2*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(7*d*Sqrt[e*Cos[c + d*x]]) + (6*a^2*e*Sqrt[e*Cos[c + d*x]]*Sin[c + d*x])/(7*d) - (2*(e*Cos[c + d*x])^(5/2)*(a^2 + a^2*Sin[c + d*x]))/(7*d*e)","A",5,5,25,0.2000,1,"{2678, 2669, 2635, 2642, 2641}"
207,1,105,0,0.091795,"\int \sqrt{e \cos (c+d x)} (a+a \sin (c+d x))^2 \, dx","Int[Sqrt[e*Cos[c + d*x]]*(a + a*Sin[c + d*x])^2,x]","-\frac{14 a^2 (e \cos (c+d x))^{3/2}}{15 d e}-\frac{2 \left(a^2 \sin (c+d x)+a^2\right) (e \cos (c+d x))^{3/2}}{5 d e}+\frac{14 a^2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{5 d \sqrt{\cos (c+d x)}}","-\frac{14 a^2 (e \cos (c+d x))^{3/2}}{15 d e}-\frac{2 \left(a^2 \sin (c+d x)+a^2\right) (e \cos (c+d x))^{3/2}}{5 d e}+\frac{14 a^2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{5 d \sqrt{\cos (c+d x)}}",1,"(-14*a^2*(e*Cos[c + d*x])^(3/2))/(15*d*e) + (14*a^2*Sqrt[e*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(5*d*Sqrt[Cos[c + d*x]]) - (2*(e*Cos[c + d*x])^(3/2)*(a^2 + a^2*Sin[c + d*x]))/(5*d*e)","A",4,4,25,0.1600,1,"{2678, 2669, 2640, 2639}"
208,1,105,0,0.0951575,"\int \frac{(a+a \sin (c+d x))^2}{\sqrt{e \cos (c+d x)}} \, dx","Int[(a + a*Sin[c + d*x])^2/Sqrt[e*Cos[c + d*x]],x]","-\frac{10 a^2 \sqrt{e \cos (c+d x)}}{3 d e}-\frac{2 \left(a^2 \sin (c+d x)+a^2\right) \sqrt{e \cos (c+d x)}}{3 d e}+\frac{10 a^2 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d \sqrt{e \cos (c+d x)}}","-\frac{10 a^2 \sqrt{e \cos (c+d x)}}{3 d e}-\frac{2 \left(a^2 \sin (c+d x)+a^2\right) \sqrt{e \cos (c+d x)}}{3 d e}+\frac{10 a^2 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d \sqrt{e \cos (c+d x)}}",1,"(-10*a^2*Sqrt[e*Cos[c + d*x]])/(3*d*e) + (10*a^2*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(3*d*Sqrt[e*Cos[c + d*x]]) - (2*Sqrt[e*Cos[c + d*x]]*(a^2 + a^2*Sin[c + d*x]))/(3*d*e)","A",4,4,25,0.1600,1,"{2678, 2669, 2642, 2641}"
209,1,85,0,0.1323916,"\int \frac{(a+a \sin (c+d x))^2}{(e \cos (c+d x))^{3/2}} \, dx","Int[(a + a*Sin[c + d*x])^2/(e*Cos[c + d*x])^(3/2),x]","\frac{4 a^4 (e \cos (c+d x))^{3/2}}{d e^3 \left(a^2-a^2 \sin (c+d x)\right)}-\frac{6 a^2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{d e^2 \sqrt{\cos (c+d x)}}","\frac{4 a^4 (e \cos (c+d x))^{3/2}}{d e^3 \left(a^2-a^2 \sin (c+d x)\right)}-\frac{6 a^2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{d e^2 \sqrt{\cos (c+d x)}}",1,"(-6*a^2*Sqrt[e*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(d*e^2*Sqrt[Cos[c + d*x]]) + (4*a^4*(e*Cos[c + d*x])^(3/2))/(d*e^3*(a^2 - a^2*Sin[c + d*x]))","A",4,4,25,0.1600,1,"{2670, 2680, 2640, 2639}"
210,1,89,0,0.1293707,"\int \frac{(a+a \sin (c+d x))^2}{(e \cos (c+d x))^{5/2}} \, dx","Int[(a + a*Sin[c + d*x])^2/(e*Cos[c + d*x])^(5/2),x]","\frac{4 a^4 \sqrt{e \cos (c+d x)}}{3 d e^3 \left(a^2-a^2 \sin (c+d x)\right)}-\frac{2 a^2 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d e^2 \sqrt{e \cos (c+d x)}}","\frac{4 a^4 \sqrt{e \cos (c+d x)}}{3 d e^3 \left(a^2-a^2 \sin (c+d x)\right)}-\frac{2 a^2 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d e^2 \sqrt{e \cos (c+d x)}}",1,"(-2*a^2*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(3*d*e^2*Sqrt[e*Cos[c + d*x]]) + (4*a^4*Sqrt[e*Cos[c + d*x]])/(3*d*e^3*(a^2 - a^2*Sin[c + d*x]))","A",4,4,25,0.1600,1,"{2670, 2680, 2642, 2641}"
211,1,127,0,0.180943,"\int \frac{(a+a \sin (c+d x))^2}{(e \cos (c+d x))^{7/2}} \, dx","Int[(a + a*Sin[c + d*x])^2/(e*Cos[c + d*x])^(7/2),x]","\frac{2 a^4 (e \cos (c+d x))^{3/2}}{5 d e^5 \left(a^2-a^2 \sin (c+d x)\right)}+\frac{2 a^4 (e \cos (c+d x))^{3/2}}{5 d e^5 (a-a \sin (c+d x))^2}-\frac{2 a^2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{5 d e^4 \sqrt{\cos (c+d x)}}","\frac{2 a^4 (e \cos (c+d x))^{3/2}}{5 d e^5 \left(a^2-a^2 \sin (c+d x)\right)}+\frac{2 a^4 (e \cos (c+d x))^{3/2}}{5 d e^5 (a-a \sin (c+d x))^2}-\frac{2 a^2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{5 d e^4 \sqrt{\cos (c+d x)}}",1,"(-2*a^2*Sqrt[e*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(5*d*e^4*Sqrt[Cos[c + d*x]]) + (2*a^4*(e*Cos[c + d*x])^(3/2))/(5*d*e^5*(a - a*Sin[c + d*x])^2) + (2*a^4*(e*Cos[c + d*x])^(3/2))/(5*d*e^5*(a^2 - a^2*Sin[c + d*x]))","A",5,5,25,0.2000,1,"{2670, 2681, 2683, 2640, 2639}"
212,1,114,0,0.0914094,"\int \frac{(a+a \sin (c+d x))^2}{(e \cos (c+d x))^{9/2}} \, dx","Int[(a + a*Sin[c + d*x])^2/(e*Cos[c + d*x])^(9/2),x]","\frac{2 a^2 \sin (c+d x)}{7 d e^3 (e \cos (c+d x))^{3/2}}+\frac{2 a^2 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{7 d e^4 \sqrt{e \cos (c+d x)}}+\frac{4 \left(a^2 \sin (c+d x)+a^2\right)}{7 d e (e \cos (c+d x))^{7/2}}","\frac{2 a^2 \sin (c+d x)}{7 d e^3 (e \cos (c+d x))^{3/2}}+\frac{2 a^2 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{7 d e^4 \sqrt{e \cos (c+d x)}}+\frac{4 \left(a^2 \sin (c+d x)+a^2\right)}{7 d e (e \cos (c+d x))^{7/2}}",1,"(2*a^2*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(7*d*e^4*Sqrt[e*Cos[c + d*x]]) + (2*a^2*Sin[c + d*x])/(7*d*e^3*(e*Cos[c + d*x])^(3/2)) + (4*(a^2 + a^2*Sin[c + d*x]))/(7*d*e*(e*Cos[c + d*x])^(7/2))","A",4,4,25,0.1600,1,"{2676, 2636, 2642, 2641}"
213,1,145,0,0.1120721,"\int \frac{(a+a \sin (c+d x))^2}{(e \cos (c+d x))^{11/2}} \, dx","Int[(a + a*Sin[c + d*x])^2/(e*Cos[c + d*x])^(11/2),x]","\frac{2 a^2 \sin (c+d x)}{3 d e^5 \sqrt{e \cos (c+d x)}}+\frac{2 a^2 \sin (c+d x)}{9 d e^3 (e \cos (c+d x))^{5/2}}-\frac{2 a^2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{3 d e^6 \sqrt{\cos (c+d x)}}+\frac{4 \left(a^2 \sin (c+d x)+a^2\right)}{9 d e (e \cos (c+d x))^{9/2}}","\frac{2 a^2 \sin (c+d x)}{3 d e^5 \sqrt{e \cos (c+d x)}}+\frac{2 a^2 \sin (c+d x)}{9 d e^3 (e \cos (c+d x))^{5/2}}-\frac{2 a^2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{3 d e^6 \sqrt{\cos (c+d x)}}+\frac{4 \left(a^2 \sin (c+d x)+a^2\right)}{9 d e (e \cos (c+d x))^{9/2}}",1,"(-2*a^2*Sqrt[e*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(3*d*e^6*Sqrt[Cos[c + d*x]]) + (2*a^2*Sin[c + d*x])/(9*d*e^3*(e*Cos[c + d*x])^(5/2)) + (2*a^2*Sin[c + d*x])/(3*d*e^5*Sqrt[e*Cos[c + d*x]]) + (4*(a^2 + a^2*Sin[c + d*x]))/(9*d*e*(e*Cos[c + d*x])^(9/2))","A",5,4,25,0.1600,1,"{2676, 2636, 2640, 2639}"
214,1,203,0,0.2112563,"\int (e \cos (c+d x))^{7/2} (a+a \sin (c+d x))^3 \, dx","Int[(e*Cos[c + d*x])^(7/2)*(a + a*Sin[c + d*x])^3,x]","\frac{170 a^3 e^3 \sin (c+d x) \sqrt{e \cos (c+d x)}}{231 d}+\frac{170 a^3 e^4 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{231 d \sqrt{e \cos (c+d x)}}-\frac{34 a^3 (e \cos (c+d x))^{9/2}}{99 d e}-\frac{34 \left(a^3 \sin (c+d x)+a^3\right) (e \cos (c+d x))^{9/2}}{143 d e}+\frac{34 a^3 e \sin (c+d x) (e \cos (c+d x))^{5/2}}{77 d}-\frac{2 a (a \sin (c+d x)+a)^2 (e \cos (c+d x))^{9/2}}{13 d e}","\frac{170 a^3 e^3 \sin (c+d x) \sqrt{e \cos (c+d x)}}{231 d}+\frac{170 a^3 e^4 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{231 d \sqrt{e \cos (c+d x)}}-\frac{34 a^3 (e \cos (c+d x))^{9/2}}{99 d e}-\frac{34 \left(a^3 \sin (c+d x)+a^3\right) (e \cos (c+d x))^{9/2}}{143 d e}+\frac{34 a^3 e \sin (c+d x) (e \cos (c+d x))^{5/2}}{77 d}-\frac{2 a (a \sin (c+d x)+a)^2 (e \cos (c+d x))^{9/2}}{13 d e}",1,"(-34*a^3*(e*Cos[c + d*x])^(9/2))/(99*d*e) + (170*a^3*e^4*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(231*d*Sqrt[e*Cos[c + d*x]]) + (170*a^3*e^3*Sqrt[e*Cos[c + d*x]]*Sin[c + d*x])/(231*d) + (34*a^3*e*(e*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(77*d) - (2*a*(e*Cos[c + d*x])^(9/2)*(a + a*Sin[c + d*x])^2)/(13*d*e) - (34*(e*Cos[c + d*x])^(9/2)*(a^3 + a^3*Sin[c + d*x]))/(143*d*e)","A",7,5,25,0.2000,1,"{2678, 2669, 2635, 2642, 2641}"
215,1,170,0,0.1894654,"\int (e \cos (c+d x))^{5/2} (a+a \sin (c+d x))^3 \, dx","Int[(e*Cos[c + d*x])^(5/2)*(a + a*Sin[c + d*x])^3,x]","\frac{2 a^3 e^2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{d \sqrt{\cos (c+d x)}}-\frac{10 a^3 (e \cos (c+d x))^{7/2}}{21 d e}-\frac{10 \left(a^3 \sin (c+d x)+a^3\right) (e \cos (c+d x))^{7/2}}{33 d e}+\frac{2 a^3 e \sin (c+d x) (e \cos (c+d x))^{3/2}}{3 d}-\frac{2 a (a \sin (c+d x)+a)^2 (e \cos (c+d x))^{7/2}}{11 d e}","\frac{2 a^3 e^2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{d \sqrt{\cos (c+d x)}}-\frac{10 a^3 (e \cos (c+d x))^{7/2}}{21 d e}-\frac{10 \left(a^3 \sin (c+d x)+a^3\right) (e \cos (c+d x))^{7/2}}{33 d e}+\frac{2 a^3 e \sin (c+d x) (e \cos (c+d x))^{3/2}}{3 d}-\frac{2 a (a \sin (c+d x)+a)^2 (e \cos (c+d x))^{7/2}}{11 d e}",1,"(-10*a^3*(e*Cos[c + d*x])^(7/2))/(21*d*e) + (2*a^3*e^2*Sqrt[e*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(d*Sqrt[Cos[c + d*x]]) + (2*a^3*e*(e*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(3*d) - (2*a*(e*Cos[c + d*x])^(7/2)*(a + a*Sin[c + d*x])^2)/(11*d*e) - (10*(e*Cos[c + d*x])^(7/2)*(a^3 + a^3*Sin[c + d*x]))/(33*d*e)","A",6,5,25,0.2000,1,"{2678, 2669, 2635, 2640, 2639}"
216,1,172,0,0.1884794,"\int (e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^3 \, dx","Int[(e*Cos[c + d*x])^(3/2)*(a + a*Sin[c + d*x])^3,x]","\frac{26 a^3 e^2 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d \sqrt{e \cos (c+d x)}}-\frac{26 a^3 (e \cos (c+d x))^{5/2}}{35 d e}+\frac{26 a^3 e \sin (c+d x) \sqrt{e \cos (c+d x)}}{21 d}-\frac{26 \left(a^3 \sin (c+d x)+a^3\right) (e \cos (c+d x))^{5/2}}{63 d e}-\frac{2 a (a \sin (c+d x)+a)^2 (e \cos (c+d x))^{5/2}}{9 d e}","\frac{26 a^3 e^2 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d \sqrt{e \cos (c+d x)}}-\frac{26 a^3 (e \cos (c+d x))^{5/2}}{35 d e}+\frac{26 a^3 e \sin (c+d x) \sqrt{e \cos (c+d x)}}{21 d}-\frac{26 \left(a^3 \sin (c+d x)+a^3\right) (e \cos (c+d x))^{5/2}}{63 d e}-\frac{2 a (a \sin (c+d x)+a)^2 (e \cos (c+d x))^{5/2}}{9 d e}",1,"(-26*a^3*(e*Cos[c + d*x])^(5/2))/(35*d*e) + (26*a^3*e^2*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(21*d*Sqrt[e*Cos[c + d*x]]) + (26*a^3*e*Sqrt[e*Cos[c + d*x]]*Sin[c + d*x])/(21*d) - (2*a*(e*Cos[c + d*x])^(5/2)*(a + a*Sin[c + d*x])^2)/(9*d*e) - (26*(e*Cos[c + d*x])^(5/2)*(a^3 + a^3*Sin[c + d*x]))/(63*d*e)","A",6,5,25,0.2000,1,"{2678, 2669, 2635, 2642, 2641}"
217,1,140,0,0.1469042,"\int \sqrt{e \cos (c+d x)} (a+a \sin (c+d x))^3 \, dx","Int[Sqrt[e*Cos[c + d*x]]*(a + a*Sin[c + d*x])^3,x]","-\frac{22 a^3 (e \cos (c+d x))^{3/2}}{15 d e}-\frac{22 \left(a^3 \sin (c+d x)+a^3\right) (e \cos (c+d x))^{3/2}}{35 d e}+\frac{22 a^3 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{5 d \sqrt{\cos (c+d x)}}-\frac{2 a (a \sin (c+d x)+a)^2 (e \cos (c+d x))^{3/2}}{7 d e}","-\frac{22 a^3 (e \cos (c+d x))^{3/2}}{15 d e}-\frac{22 \left(a^3 \sin (c+d x)+a^3\right) (e \cos (c+d x))^{3/2}}{35 d e}+\frac{22 a^3 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{5 d \sqrt{\cos (c+d x)}}-\frac{2 a (a \sin (c+d x)+a)^2 (e \cos (c+d x))^{3/2}}{7 d e}",1,"(-22*a^3*(e*Cos[c + d*x])^(3/2))/(15*d*e) + (22*a^3*Sqrt[e*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(5*d*Sqrt[Cos[c + d*x]]) - (2*a*(e*Cos[c + d*x])^(3/2)*(a + a*Sin[c + d*x])^2)/(7*d*e) - (22*(e*Cos[c + d*x])^(3/2)*(a^3 + a^3*Sin[c + d*x]))/(35*d*e)","A",5,4,25,0.1600,1,"{2678, 2669, 2640, 2639}"
218,1,136,0,0.1473565,"\int \frac{(a+a \sin (c+d x))^3}{\sqrt{e \cos (c+d x)}} \, dx","Int[(a + a*Sin[c + d*x])^3/Sqrt[e*Cos[c + d*x]],x]","-\frac{6 a^3 \sqrt{e \cos (c+d x)}}{d e}-\frac{6 \left(a^3 \sin (c+d x)+a^3\right) \sqrt{e \cos (c+d x)}}{5 d e}+\frac{6 a^3 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d \sqrt{e \cos (c+d x)}}-\frac{2 a (a \sin (c+d x)+a)^2 \sqrt{e \cos (c+d x)}}{5 d e}","-\frac{6 a^3 \sqrt{e \cos (c+d x)}}{d e}-\frac{6 \left(a^3 \sin (c+d x)+a^3\right) \sqrt{e \cos (c+d x)}}{5 d e}+\frac{6 a^3 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d \sqrt{e \cos (c+d x)}}-\frac{2 a (a \sin (c+d x)+a)^2 \sqrt{e \cos (c+d x)}}{5 d e}",1,"(-6*a^3*Sqrt[e*Cos[c + d*x]])/(d*e) + (6*a^3*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(d*Sqrt[e*Cos[c + d*x]]) - (2*a*Sqrt[e*Cos[c + d*x]]*(a + a*Sin[c + d*x])^2)/(5*d*e) - (6*Sqrt[e*Cos[c + d*x]]*(a^3 + a^3*Sin[c + d*x]))/(5*d*e)","A",5,4,25,0.1600,1,"{2678, 2669, 2642, 2641}"
219,1,106,0,0.1975962,"\int \frac{(a+a \sin (c+d x))^3}{(e \cos (c+d x))^{3/2}} \, dx","Int[(a + a*Sin[c + d*x])^3/(e*Cos[c + d*x])^(3/2),x]","\frac{14 a^3 (e \cos (c+d x))^{3/2}}{3 d e^3}+\frac{4 a^5 (e \cos (c+d x))^{7/2}}{d e^5 (a-a \sin (c+d x))^2}-\frac{14 a^3 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{d e^2 \sqrt{\cos (c+d x)}}","\frac{14 a^3 (e \cos (c+d x))^{3/2}}{3 d e^3}+\frac{4 a^5 (e \cos (c+d x))^{7/2}}{d e^5 (a-a \sin (c+d x))^2}-\frac{14 a^3 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{d e^2 \sqrt{\cos (c+d x)}}",1,"(14*a^3*(e*Cos[c + d*x])^(3/2))/(3*d*e^3) - (14*a^3*Sqrt[e*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(d*e^2*Sqrt[Cos[c + d*x]]) + (4*a^5*(e*Cos[c + d*x])^(7/2))/(d*e^5*(a - a*Sin[c + d*x])^2)","A",5,5,25,0.2000,1,"{2670, 2680, 2682, 2640, 2639}"
220,1,110,0,0.2023048,"\int \frac{(a+a \sin (c+d x))^3}{(e \cos (c+d x))^{5/2}} \, dx","Int[(a + a*Sin[c + d*x])^3/(e*Cos[c + d*x])^(5/2),x]","\frac{10 a^3 \sqrt{e \cos (c+d x)}}{3 d e^3}+\frac{4 a^5 (e \cos (c+d x))^{5/2}}{3 d e^5 (a-a \sin (c+d x))^2}-\frac{10 a^3 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d e^2 \sqrt{e \cos (c+d x)}}","\frac{10 a^3 \sqrt{e \cos (c+d x)}}{3 d e^3}+\frac{4 a^5 (e \cos (c+d x))^{5/2}}{3 d e^5 (a-a \sin (c+d x))^2}-\frac{10 a^3 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d e^2 \sqrt{e \cos (c+d x)}}",1,"(10*a^3*Sqrt[e*Cos[c + d*x]])/(3*d*e^3) - (10*a^3*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(3*d*e^2*Sqrt[e*Cos[c + d*x]]) + (4*a^5*(e*Cos[c + d*x])^(5/2))/(3*d*e^5*(a - a*Sin[c + d*x])^2)","A",5,5,25,0.2000,1,"{2670, 2680, 2682, 2642, 2641}"
221,1,127,0,0.2023822,"\int \frac{(a+a \sin (c+d x))^3}{(e \cos (c+d x))^{7/2}} \, dx","Int[(a + a*Sin[c + d*x])^3/(e*Cos[c + d*x])^(7/2),x]","-\frac{6 a^6 (e \cos (c+d x))^{3/2}}{5 d e^5 \left(a^3-a^3 \sin (c+d x)\right)}+\frac{4 a^5 (e \cos (c+d x))^{3/2}}{5 d e^5 (a-a \sin (c+d x))^2}+\frac{6 a^3 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{5 d e^4 \sqrt{\cos (c+d x)}}","-\frac{6 a^6 (e \cos (c+d x))^{3/2}}{5 d e^5 \left(a^3-a^3 \sin (c+d x)\right)}+\frac{4 a^5 (e \cos (c+d x))^{3/2}}{5 d e^5 (a-a \sin (c+d x))^2}+\frac{6 a^3 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{5 d e^4 \sqrt{\cos (c+d x)}}",1,"(6*a^3*Sqrt[e*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(5*d*e^4*Sqrt[Cos[c + d*x]]) + (4*a^5*(e*Cos[c + d*x])^(3/2))/(5*d*e^5*(a - a*Sin[c + d*x])^2) - (6*a^6*(e*Cos[c + d*x])^(3/2))/(5*d*e^5*(a^3 - a^3*Sin[c + d*x]))","A",5,5,25,0.2000,1,"{2670, 2680, 2683, 2640, 2639}"
222,1,127,0,0.1976077,"\int \frac{(a+a \sin (c+d x))^3}{(e \cos (c+d x))^{9/2}} \, dx","Int[(a + a*Sin[c + d*x])^3/(e*Cos[c + d*x])^(9/2),x]","-\frac{2 a^6 \sqrt{e \cos (c+d x)}}{21 d e^5 \left(a^3-a^3 \sin (c+d x)\right)}+\frac{4 a^5 \sqrt{e \cos (c+d x)}}{7 d e^5 (a-a \sin (c+d x))^2}-\frac{2 a^3 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d e^4 \sqrt{e \cos (c+d x)}}","-\frac{2 a^6 \sqrt{e \cos (c+d x)}}{21 d e^5 \left(a^3-a^3 \sin (c+d x)\right)}+\frac{4 a^5 \sqrt{e \cos (c+d x)}}{7 d e^5 (a-a \sin (c+d x))^2}-\frac{2 a^3 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d e^4 \sqrt{e \cos (c+d x)}}",1,"(-2*a^3*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(21*d*e^4*Sqrt[e*Cos[c + d*x]]) + (4*a^5*Sqrt[e*Cos[c + d*x]])/(7*d*e^5*(a - a*Sin[c + d*x])^2) - (2*a^6*Sqrt[e*Cos[c + d*x]])/(21*d*e^5*(a^3 - a^3*Sin[c + d*x]))","A",5,5,25,0.2000,1,"{2670, 2680, 2683, 2642, 2641}"
223,1,165,0,0.2431651,"\int \frac{(a+a \sin (c+d x))^3}{(e \cos (c+d x))^{11/2}} \, dx","Int[(a + a*Sin[c + d*x])^3/(e*Cos[c + d*x])^(11/2),x]","\frac{2 a^6 (e \cos (c+d x))^{3/2}}{15 d e^7 \left(a^3-a^3 \sin (c+d x)\right)}+\frac{2 a^6 (e \cos (c+d x))^{3/2}}{9 d e^7 (a-a \sin (c+d x))^3}+\frac{2 a^5 (e \cos (c+d x))^{3/2}}{15 d e^7 (a-a \sin (c+d x))^2}-\frac{2 a^3 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{15 d e^6 \sqrt{\cos (c+d x)}}","\frac{2 a^6 (e \cos (c+d x))^{3/2}}{15 d e^7 \left(a^3-a^3 \sin (c+d x)\right)}+\frac{2 a^6 (e \cos (c+d x))^{3/2}}{9 d e^7 (a-a \sin (c+d x))^3}+\frac{2 a^5 (e \cos (c+d x))^{3/2}}{15 d e^7 (a-a \sin (c+d x))^2}-\frac{2 a^3 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{15 d e^6 \sqrt{\cos (c+d x)}}",1,"(-2*a^3*Sqrt[e*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(15*d*e^6*Sqrt[Cos[c + d*x]]) + (2*a^6*(e*Cos[c + d*x])^(3/2))/(9*d*e^7*(a - a*Sin[c + d*x])^3) + (2*a^5*(e*Cos[c + d*x])^(3/2))/(15*d*e^7*(a - a*Sin[c + d*x])^2) + (2*a^6*(e*Cos[c + d*x])^(3/2))/(15*d*e^7*(a^3 - a^3*Sin[c + d*x]))","A",6,5,25,0.2000,1,"{2670, 2681, 2683, 2640, 2639}"
224,1,210,0,0.247785,"\int (e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^4 \, dx","Int[(e*Cos[c + d*x])^(3/2)*(a + a*Sin[c + d*x])^4,x]","\frac{442 a^4 e^2 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{231 d \sqrt{e \cos (c+d x)}}-\frac{442 a^4 (e \cos (c+d x))^{5/2}}{385 d e}+\frac{442 a^4 e \sin (c+d x) \sqrt{e \cos (c+d x)}}{231 d}-\frac{34 \left(a^2 \sin (c+d x)+a^2\right)^2 (e \cos (c+d x))^{5/2}}{99 d e}-\frac{442 \left(a^4 \sin (c+d x)+a^4\right) (e \cos (c+d x))^{5/2}}{693 d e}-\frac{2 a (a \sin (c+d x)+a)^3 (e \cos (c+d x))^{5/2}}{11 d e}","\frac{442 a^4 e^2 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{231 d \sqrt{e \cos (c+d x)}}-\frac{442 a^4 (e \cos (c+d x))^{5/2}}{385 d e}+\frac{442 a^4 e \sin (c+d x) \sqrt{e \cos (c+d x)}}{231 d}-\frac{34 \left(a^2 \sin (c+d x)+a^2\right)^2 (e \cos (c+d x))^{5/2}}{99 d e}-\frac{442 \left(a^4 \sin (c+d x)+a^4\right) (e \cos (c+d x))^{5/2}}{693 d e}-\frac{2 a (a \sin (c+d x)+a)^3 (e \cos (c+d x))^{5/2}}{11 d e}",1,"(-442*a^4*(e*Cos[c + d*x])^(5/2))/(385*d*e) + (442*a^4*e^2*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(231*d*Sqrt[e*Cos[c + d*x]]) + (442*a^4*e*Sqrt[e*Cos[c + d*x]]*Sin[c + d*x])/(231*d) - (2*a*(e*Cos[c + d*x])^(5/2)*(a + a*Sin[c + d*x])^3)/(11*d*e) - (34*(e*Cos[c + d*x])^(5/2)*(a^2 + a^2*Sin[c + d*x])^2)/(99*d*e) - (442*(e*Cos[c + d*x])^(5/2)*(a^4 + a^4*Sin[c + d*x]))/(693*d*e)","A",7,5,25,0.2000,1,"{2678, 2669, 2635, 2642, 2641}"
225,1,178,0,0.2007011,"\int \sqrt{e \cos (c+d x)} (a+a \sin (c+d x))^4 \, dx","Int[Sqrt[e*Cos[c + d*x]]*(a + a*Sin[c + d*x])^4,x]","-\frac{22 a^4 (e \cos (c+d x))^{3/2}}{9 d e}-\frac{10 \left(a^2 \sin (c+d x)+a^2\right)^2 (e \cos (c+d x))^{3/2}}{21 d e}-\frac{22 \left(a^4 \sin (c+d x)+a^4\right) (e \cos (c+d x))^{3/2}}{21 d e}+\frac{22 a^4 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{3 d \sqrt{\cos (c+d x)}}-\frac{2 a (a \sin (c+d x)+a)^3 (e \cos (c+d x))^{3/2}}{9 d e}","-\frac{22 a^4 (e \cos (c+d x))^{3/2}}{9 d e}-\frac{10 \left(a^2 \sin (c+d x)+a^2\right)^2 (e \cos (c+d x))^{3/2}}{21 d e}-\frac{22 \left(a^4 \sin (c+d x)+a^4\right) (e \cos (c+d x))^{3/2}}{21 d e}+\frac{22 a^4 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{3 d \sqrt{\cos (c+d x)}}-\frac{2 a (a \sin (c+d x)+a)^3 (e \cos (c+d x))^{3/2}}{9 d e}",1,"(-22*a^4*(e*Cos[c + d*x])^(3/2))/(9*d*e) + (22*a^4*Sqrt[e*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(3*d*Sqrt[Cos[c + d*x]]) - (2*a*(e*Cos[c + d*x])^(3/2)*(a + a*Sin[c + d*x])^3)/(9*d*e) - (10*(e*Cos[c + d*x])^(3/2)*(a^2 + a^2*Sin[c + d*x])^2)/(21*d*e) - (22*(e*Cos[c + d*x])^(3/2)*(a^4 + a^4*Sin[c + d*x]))/(21*d*e)","A",6,4,25,0.1600,1,"{2678, 2669, 2640, 2639}"
226,1,178,0,0.21279,"\int \frac{(a+a \sin (c+d x))^4}{\sqrt{e \cos (c+d x)}} \, dx","Int[(a + a*Sin[c + d*x])^4/Sqrt[e*Cos[c + d*x]],x]","-\frac{78 a^4 \sqrt{e \cos (c+d x)}}{7 d e}-\frac{26 \left(a^2 \sin (c+d x)+a^2\right)^2 \sqrt{e \cos (c+d x)}}{35 d e}-\frac{78 \left(a^4 \sin (c+d x)+a^4\right) \sqrt{e \cos (c+d x)}}{35 d e}+\frac{78 a^4 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{7 d \sqrt{e \cos (c+d x)}}-\frac{2 a (a \sin (c+d x)+a)^3 \sqrt{e \cos (c+d x)}}{7 d e}","-\frac{78 a^4 \sqrt{e \cos (c+d x)}}{7 d e}-\frac{26 \left(a^2 \sin (c+d x)+a^2\right)^2 \sqrt{e \cos (c+d x)}}{35 d e}-\frac{78 \left(a^4 \sin (c+d x)+a^4\right) \sqrt{e \cos (c+d x)}}{35 d e}+\frac{78 a^4 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{7 d \sqrt{e \cos (c+d x)}}-\frac{2 a (a \sin (c+d x)+a)^3 \sqrt{e \cos (c+d x)}}{7 d e}",1,"(-78*a^4*Sqrt[e*Cos[c + d*x]])/(7*d*e) + (78*a^4*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(7*d*Sqrt[e*Cos[c + d*x]]) - (2*a*Sqrt[e*Cos[c + d*x]]*(a + a*Sin[c + d*x])^3)/(7*d*e) - (26*Sqrt[e*Cos[c + d*x]]*(a^2 + a^2*Sin[c + d*x])^2)/(35*d*e) - (78*Sqrt[e*Cos[c + d*x]]*(a^4 + a^4*Sin[c + d*x]))/(35*d*e)","A",6,4,25,0.1600,1,"{2678, 2669, 2642, 2641}"
227,1,156,0,0.2340534,"\int \frac{(a+a \sin (c+d x))^4}{(e \cos (c+d x))^{3/2}} \, dx","Int[(a + a*Sin[c + d*x])^4/(e*Cos[c + d*x])^(3/2),x]","\frac{44 a^8 (e \cos (c+d x))^{7/2}}{3 d e^5 \left(a^4-a^4 \sin (c+d x)\right)}+\frac{4 a^7 (e \cos (c+d x))^{11/2}}{d e^7 (a-a \sin (c+d x))^3}-\frac{154 a^4 \sin (c+d x) (e \cos (c+d x))^{3/2}}{15 d e^3}-\frac{154 a^4 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{5 d e^2 \sqrt{\cos (c+d x)}}","\frac{44 a^8 (e \cos (c+d x))^{7/2}}{3 d e^5 \left(a^4-a^4 \sin (c+d x)\right)}+\frac{4 a^7 (e \cos (c+d x))^{11/2}}{d e^7 (a-a \sin (c+d x))^3}-\frac{154 a^4 \sin (c+d x) (e \cos (c+d x))^{3/2}}{15 d e^3}-\frac{154 a^4 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{5 d e^2 \sqrt{\cos (c+d x)}}",1,"(-154*a^4*Sqrt[e*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(5*d*e^2*Sqrt[Cos[c + d*x]]) - (154*a^4*(e*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(15*d*e^3) + (4*a^7*(e*Cos[c + d*x])^(11/2))/(d*e^7*(a - a*Sin[c + d*x])^3) + (44*a^8*(e*Cos[c + d*x])^(7/2))/(3*d*e^5*(a^4 - a^4*Sin[c + d*x]))","A",6,5,25,0.2000,1,"{2670, 2680, 2635, 2640, 2639}"
228,1,152,0,0.2235856,"\int \frac{(a+a \sin (c+d x))^4}{(e \cos (c+d x))^{5/2}} \, dx","Int[(a + a*Sin[c + d*x])^4/(e*Cos[c + d*x])^(5/2),x]","\frac{12 a^8 (e \cos (c+d x))^{5/2}}{d e^5 \left(a^4-a^4 \sin (c+d x)\right)}+\frac{4 a^7 (e \cos (c+d x))^{9/2}}{3 d e^7 (a-a \sin (c+d x))^3}-\frac{10 a^4 \sin (c+d x) \sqrt{e \cos (c+d x)}}{d e^3}-\frac{10 a^4 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d e^2 \sqrt{e \cos (c+d x)}}","\frac{12 a^8 (e \cos (c+d x))^{5/2}}{d e^5 \left(a^4-a^4 \sin (c+d x)\right)}+\frac{4 a^7 (e \cos (c+d x))^{9/2}}{3 d e^7 (a-a \sin (c+d x))^3}-\frac{10 a^4 \sin (c+d x) \sqrt{e \cos (c+d x)}}{d e^3}-\frac{10 a^4 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d e^2 \sqrt{e \cos (c+d x)}}",1,"(-10*a^4*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(d*e^2*Sqrt[e*Cos[c + d*x]]) - (10*a^4*Sqrt[e*Cos[c + d*x]]*Sin[c + d*x])/(d*e^3) + (4*a^7*(e*Cos[c + d*x])^(9/2))/(3*d*e^7*(a - a*Sin[c + d*x])^3) + (12*a^8*(e*Cos[c + d*x])^(5/2))/(d*e^5*(a^4 - a^4*Sin[c + d*x]))","A",6,5,25,0.2000,1,"{2670, 2680, 2635, 2642, 2641}"
229,1,127,0,0.1994323,"\int \frac{(a+a \sin (c+d x))^4}{(e \cos (c+d x))^{7/2}} \, dx","Int[(a + a*Sin[c + d*x])^4/(e*Cos[c + d*x])^(7/2),x]","-\frac{28 a^8 (e \cos (c+d x))^{3/2}}{5 d e^5 \left(a^4-a^4 \sin (c+d x)\right)}+\frac{4 a^7 (e \cos (c+d x))^{7/2}}{5 d e^7 (a-a \sin (c+d x))^3}+\frac{42 a^4 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{5 d e^4 \sqrt{\cos (c+d x)}}","-\frac{28 a^8 (e \cos (c+d x))^{3/2}}{5 d e^5 \left(a^4-a^4 \sin (c+d x)\right)}+\frac{4 a^7 (e \cos (c+d x))^{7/2}}{5 d e^7 (a-a \sin (c+d x))^3}+\frac{42 a^4 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{5 d e^4 \sqrt{\cos (c+d x)}}",1,"(42*a^4*Sqrt[e*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(5*d*e^4*Sqrt[Cos[c + d*x]]) + (4*a^7*(e*Cos[c + d*x])^(7/2))/(5*d*e^7*(a - a*Sin[c + d*x])^3) - (28*a^8*(e*Cos[c + d*x])^(3/2))/(5*d*e^5*(a^4 - a^4*Sin[c + d*x]))","A",5,4,25,0.1600,1,"{2670, 2680, 2640, 2639}"
230,1,127,0,0.1984811,"\int \frac{(a+a \sin (c+d x))^4}{(e \cos (c+d x))^{9/2}} \, dx","Int[(a + a*Sin[c + d*x])^4/(e*Cos[c + d*x])^(9/2),x]","-\frac{20 a^8 \sqrt{e \cos (c+d x)}}{21 d e^5 \left(a^4-a^4 \sin (c+d x)\right)}+\frac{4 a^7 (e \cos (c+d x))^{5/2}}{7 d e^7 (a-a \sin (c+d x))^3}+\frac{10 a^4 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d e^4 \sqrt{e \cos (c+d x)}}","-\frac{20 a^8 \sqrt{e \cos (c+d x)}}{21 d e^5 \left(a^4-a^4 \sin (c+d x)\right)}+\frac{4 a^7 (e \cos (c+d x))^{5/2}}{7 d e^7 (a-a \sin (c+d x))^3}+\frac{10 a^4 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d e^4 \sqrt{e \cos (c+d x)}}",1,"(10*a^4*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(21*d*e^4*Sqrt[e*Cos[c + d*x]]) + (4*a^7*(e*Cos[c + d*x])^(5/2))/(7*d*e^7*(a - a*Sin[c + d*x])^3) - (20*a^8*Sqrt[e*Cos[c + d*x]])/(21*d*e^5*(a^4 - a^4*Sin[c + d*x]))","A",5,4,25,0.1600,1,"{2670, 2680, 2642, 2641}"
231,1,169,0,0.2499454,"\int \frac{(a+a \sin (c+d x))^4}{(e \cos (c+d x))^{11/2}} \, dx","Int[(a + a*Sin[c + d*x])^4/(e*Cos[c + d*x])^(11/2),x]","-\frac{2 a^8 (e \cos (c+d x))^{3/2}}{15 d e^7 \left(a^4-a^4 \sin (c+d x)\right)}-\frac{2 a^8 (e \cos (c+d x))^{3/2}}{15 d e^7 \left(a^2-a^2 \sin (c+d x)\right)^2}+\frac{4 a^7 (e \cos (c+d x))^{3/2}}{9 d e^7 (a-a \sin (c+d x))^3}+\frac{2 a^4 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{15 d e^6 \sqrt{\cos (c+d x)}}","-\frac{2 a^8 (e \cos (c+d x))^{3/2}}{15 d e^7 \left(a^4-a^4 \sin (c+d x)\right)}-\frac{2 a^8 (e \cos (c+d x))^{3/2}}{15 d e^7 \left(a^2-a^2 \sin (c+d x)\right)^2}+\frac{4 a^7 (e \cos (c+d x))^{3/2}}{9 d e^7 (a-a \sin (c+d x))^3}+\frac{2 a^4 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{15 d e^6 \sqrt{\cos (c+d x)}}",1,"(2*a^4*Sqrt[e*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(15*d*e^6*Sqrt[Cos[c + d*x]]) + (4*a^7*(e*Cos[c + d*x])^(3/2))/(9*d*e^7*(a - a*Sin[c + d*x])^3) - (2*a^8*(e*Cos[c + d*x])^(3/2))/(15*d*e^7*(a^2 - a^2*Sin[c + d*x])^2) - (2*a^8*(e*Cos[c + d*x])^(3/2))/(15*d*e^7*(a^4 - a^4*Sin[c + d*x]))","A",6,6,25,0.2400,1,"{2670, 2680, 2681, 2683, 2640, 2639}"
232,1,169,0,0.2594349,"\int \frac{(a+a \sin (c+d x))^4}{(e \cos (c+d x))^{13/2}} \, dx","Int[(a + a*Sin[c + d*x])^4/(e*Cos[c + d*x])^(13/2),x]","-\frac{2 a^8 \sqrt{e \cos (c+d x)}}{77 d e^7 \left(a^4-a^4 \sin (c+d x)\right)}-\frac{2 a^8 \sqrt{e \cos (c+d x)}}{77 d e^7 \left(a^2-a^2 \sin (c+d x)\right)^2}+\frac{4 a^7 \sqrt{e \cos (c+d x)}}{11 d e^7 (a-a \sin (c+d x))^3}-\frac{2 a^4 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{77 d e^6 \sqrt{e \cos (c+d x)}}","-\frac{2 a^8 \sqrt{e \cos (c+d x)}}{77 d e^7 \left(a^4-a^4 \sin (c+d x)\right)}-\frac{2 a^8 \sqrt{e \cos (c+d x)}}{77 d e^7 \left(a^2-a^2 \sin (c+d x)\right)^2}+\frac{4 a^7 \sqrt{e \cos (c+d x)}}{11 d e^7 (a-a \sin (c+d x))^3}-\frac{2 a^4 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{77 d e^6 \sqrt{e \cos (c+d x)}}",1,"(-2*a^4*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(77*d*e^6*Sqrt[e*Cos[c + d*x]]) + (4*a^7*Sqrt[e*Cos[c + d*x]])/(11*d*e^7*(a - a*Sin[c + d*x])^3) - (2*a^8*Sqrt[e*Cos[c + d*x]])/(77*d*e^7*(a^2 - a^2*Sin[c + d*x])^2) - (2*a^8*Sqrt[e*Cos[c + d*x]])/(77*d*e^7*(a^4 - a^4*Sin[c + d*x]))","A",6,6,25,0.2400,1,"{2670, 2680, 2681, 2683, 2642, 2641}"
233,1,132,0,0.1174886,"\int \frac{(e \cos (c+d x))^{11/2}}{a+a \sin (c+d x)} \, dx","Int[(e*Cos[c + d*x])^(11/2)/(a + a*Sin[c + d*x]),x]","\frac{10 e^5 \sin (c+d x) \sqrt{e \cos (c+d x)}}{21 a d}+\frac{2 e^3 \sin (c+d x) (e \cos (c+d x))^{5/2}}{7 a d}+\frac{10 e^6 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 a d \sqrt{e \cos (c+d x)}}+\frac{2 e (e \cos (c+d x))^{9/2}}{9 a d}","\frac{10 e^5 \sin (c+d x) \sqrt{e \cos (c+d x)}}{21 a d}+\frac{2 e^3 \sin (c+d x) (e \cos (c+d x))^{5/2}}{7 a d}+\frac{10 e^6 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 a d \sqrt{e \cos (c+d x)}}+\frac{2 e (e \cos (c+d x))^{9/2}}{9 a d}",1,"(2*e*(e*Cos[c + d*x])^(9/2))/(9*a*d) + (10*e^6*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(21*a*d*Sqrt[e*Cos[c + d*x]]) + (10*e^5*Sqrt[e*Cos[c + d*x]]*Sin[c + d*x])/(21*a*d) + (2*e^3*(e*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(7*a*d)","A",5,4,25,0.1600,1,"{2682, 2635, 2642, 2641}"
234,1,101,0,0.0954738,"\int \frac{(e \cos (c+d x))^{9/2}}{a+a \sin (c+d x)} \, dx","Int[(e*Cos[c + d*x])^(9/2)/(a + a*Sin[c + d*x]),x]","\frac{2 e^3 \sin (c+d x) (e \cos (c+d x))^{3/2}}{5 a d}+\frac{6 e^4 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{5 a d \sqrt{\cos (c+d x)}}+\frac{2 e (e \cos (c+d x))^{7/2}}{7 a d}","\frac{2 e^3 \sin (c+d x) (e \cos (c+d x))^{3/2}}{5 a d}+\frac{6 e^4 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{5 a d \sqrt{\cos (c+d x)}}+\frac{2 e (e \cos (c+d x))^{7/2}}{7 a d}",1,"(2*e*(e*Cos[c + d*x])^(7/2))/(7*a*d) + (6*e^4*Sqrt[e*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(5*a*d*Sqrt[Cos[c + d*x]]) + (2*e^3*(e*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*a*d)","A",4,4,25,0.1600,1,"{2682, 2635, 2640, 2639}"
235,1,101,0,0.0949842,"\int \frac{(e \cos (c+d x))^{7/2}}{a+a \sin (c+d x)} \, dx","Int[(e*Cos[c + d*x])^(7/2)/(a + a*Sin[c + d*x]),x]","\frac{2 e^3 \sin (c+d x) \sqrt{e \cos (c+d x)}}{3 a d}+\frac{2 e^4 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a d \sqrt{e \cos (c+d x)}}+\frac{2 e (e \cos (c+d x))^{5/2}}{5 a d}","\frac{2 e^3 \sin (c+d x) \sqrt{e \cos (c+d x)}}{3 a d}+\frac{2 e^4 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a d \sqrt{e \cos (c+d x)}}+\frac{2 e (e \cos (c+d x))^{5/2}}{5 a d}",1,"(2*e*(e*Cos[c + d*x])^(5/2))/(5*a*d) + (2*e^4*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(3*a*d*Sqrt[e*Cos[c + d*x]]) + (2*e^3*Sqrt[e*Cos[c + d*x]]*Sin[c + d*x])/(3*a*d)","A",4,4,25,0.1600,1,"{2682, 2635, 2642, 2641}"
236,1,68,0,0.0754318,"\int \frac{(e \cos (c+d x))^{5/2}}{a+a \sin (c+d x)} \, dx","Int[(e*Cos[c + d*x])^(5/2)/(a + a*Sin[c + d*x]),x]","\frac{2 e^2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{a d \sqrt{\cos (c+d x)}}+\frac{2 e (e \cos (c+d x))^{3/2}}{3 a d}","\frac{2 e^2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{a d \sqrt{\cos (c+d x)}}+\frac{2 e (e \cos (c+d x))^{3/2}}{3 a d}",1,"(2*e*(e*Cos[c + d*x])^(3/2))/(3*a*d) + (2*e^2*Sqrt[e*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(a*d*Sqrt[Cos[c + d*x]])","A",3,3,25,0.1200,1,"{2682, 2640, 2639}"
237,1,66,0,0.0756069,"\int \frac{(e \cos (c+d x))^{3/2}}{a+a \sin (c+d x)} \, dx","Int[(e*Cos[c + d*x])^(3/2)/(a + a*Sin[c + d*x]),x]","\frac{2 e^2 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d \sqrt{e \cos (c+d x)}}+\frac{2 e \sqrt{e \cos (c+d x)}}{a d}","\frac{2 e^2 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d \sqrt{e \cos (c+d x)}}+\frac{2 e \sqrt{e \cos (c+d x)}}{a d}",1,"(2*e*Sqrt[e*Cos[c + d*x]])/(a*d) + (2*e^2*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(a*d*Sqrt[e*Cos[c + d*x]])","A",3,3,25,0.1200,1,"{2682, 2642, 2641}"
238,1,74,0,0.0679212,"\int \frac{\sqrt{e \cos (c+d x)}}{a+a \sin (c+d x)} \, dx","Int[Sqrt[e*Cos[c + d*x]]/(a + a*Sin[c + d*x]),x]","-\frac{2 (e \cos (c+d x))^{3/2}}{d e (a \sin (c+d x)+a)}-\frac{2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{a d \sqrt{\cos (c+d x)}}","-\frac{2 (e \cos (c+d x))^{3/2}}{d e (a \sin (c+d x)+a)}-\frac{2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{a d \sqrt{\cos (c+d x)}}",1,"(-2*Sqrt[e*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(a*d*Sqrt[Cos[c + d*x]]) - (2*(e*Cos[c + d*x])^(3/2))/(d*e*(a + a*Sin[c + d*x]))","A",3,3,25,0.1200,1,"{2683, 2640, 2639}"
239,1,78,0,0.0724614,"\int \frac{1}{\sqrt{e \cos (c+d x)} (a+a \sin (c+d x))} \, dx","Int[1/(Sqrt[e*Cos[c + d*x]]*(a + a*Sin[c + d*x])),x]","\frac{2 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a d \sqrt{e \cos (c+d x)}}-\frac{2 \sqrt{e \cos (c+d x)}}{3 d e (a \sin (c+d x)+a)}","\frac{2 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a d \sqrt{e \cos (c+d x)}}-\frac{2 \sqrt{e \cos (c+d x)}}{3 d e (a \sin (c+d x)+a)}",1,"(2*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(3*a*d*Sqrt[e*Cos[c + d*x]]) - (2*Sqrt[e*Cos[c + d*x]])/(3*d*e*(a + a*Sin[c + d*x]))","A",3,3,25,0.1200,1,"{2683, 2642, 2641}"
240,1,112,0,0.0968475,"\int \frac{1}{(e \cos (c+d x))^{3/2} (a+a \sin (c+d x))} \, dx","Int[1/((e*Cos[c + d*x])^(3/2)*(a + a*Sin[c + d*x])),x]","-\frac{6 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{5 a d e^2 \sqrt{\cos (c+d x)}}+\frac{6 \sin (c+d x)}{5 a d e \sqrt{e \cos (c+d x)}}-\frac{2}{5 d e (a \sin (c+d x)+a) \sqrt{e \cos (c+d x)}}","-\frac{6 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{5 a d e^2 \sqrt{\cos (c+d x)}}+\frac{6 \sin (c+d x)}{5 a d e \sqrt{e \cos (c+d x)}}-\frac{2}{5 d e (a \sin (c+d x)+a) \sqrt{e \cos (c+d x)}}",1,"(-6*Sqrt[e*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(5*a*d*e^2*Sqrt[Cos[c + d*x]]) + (6*Sin[c + d*x])/(5*a*d*e*Sqrt[e*Cos[c + d*x]]) - 2/(5*d*e*Sqrt[e*Cos[c + d*x]]*(a + a*Sin[c + d*x]))","A",4,4,25,0.1600,1,"{2683, 2636, 2640, 2639}"
241,1,112,0,0.0943606,"\int \frac{1}{(e \cos (c+d x))^{5/2} (a+a \sin (c+d x))} \, dx","Int[1/((e*Cos[c + d*x])^(5/2)*(a + a*Sin[c + d*x])),x]","\frac{10 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 a d e^2 \sqrt{e \cos (c+d x)}}+\frac{10 \sin (c+d x)}{21 a d e (e \cos (c+d x))^{3/2}}-\frac{2}{7 d e (a \sin (c+d x)+a) (e \cos (c+d x))^{3/2}}","\frac{10 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 a d e^2 \sqrt{e \cos (c+d x)}}+\frac{10 \sin (c+d x)}{21 a d e (e \cos (c+d x))^{3/2}}-\frac{2}{7 d e (a \sin (c+d x)+a) (e \cos (c+d x))^{3/2}}",1,"(10*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(21*a*d*e^2*Sqrt[e*Cos[c + d*x]]) + (10*Sin[c + d*x])/(21*a*d*e*(e*Cos[c + d*x])^(3/2)) - 2/(7*d*e*(e*Cos[c + d*x])^(3/2)*(a + a*Sin[c + d*x]))","A",4,4,25,0.1600,1,"{2683, 2636, 2642, 2641}"
242,1,143,0,0.114257,"\int \frac{1}{(e \cos (c+d x))^{7/2} (a+a \sin (c+d x))} \, dx","Int[1/((e*Cos[c + d*x])^(7/2)*(a + a*Sin[c + d*x])),x]","\frac{14 \sin (c+d x)}{15 a d e^3 \sqrt{e \cos (c+d x)}}-\frac{14 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{15 a d e^4 \sqrt{\cos (c+d x)}}+\frac{14 \sin (c+d x)}{45 a d e (e \cos (c+d x))^{5/2}}-\frac{2}{9 d e (a \sin (c+d x)+a) (e \cos (c+d x))^{5/2}}","\frac{14 \sin (c+d x)}{15 a d e^3 \sqrt{e \cos (c+d x)}}-\frac{14 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{15 a d e^4 \sqrt{\cos (c+d x)}}+\frac{14 \sin (c+d x)}{45 a d e (e \cos (c+d x))^{5/2}}-\frac{2}{9 d e (a \sin (c+d x)+a) (e \cos (c+d x))^{5/2}}",1,"(-14*Sqrt[e*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(15*a*d*e^4*Sqrt[Cos[c + d*x]]) + (14*Sin[c + d*x])/(45*a*d*e*(e*Cos[c + d*x])^(5/2)) + (14*Sin[c + d*x])/(15*a*d*e^3*Sqrt[e*Cos[c + d*x]]) - 2/(9*d*e*(e*Cos[c + d*x])^(5/2)*(a + a*Sin[c + d*x]))","A",5,4,25,0.1600,1,"{2683, 2636, 2640, 2639}"
243,1,145,0,0.1095621,"\int \frac{(e \cos (c+d x))^{11/2}}{(a+a \sin (c+d x))^2} \, dx","Int[(e*Cos[c + d*x])^(11/2)/(a + a*Sin[c + d*x])^2,x]","\frac{6 e^5 \sin (c+d x) \sqrt{e \cos (c+d x)}}{7 a^2 d}+\frac{18 e^3 \sin (c+d x) (e \cos (c+d x))^{5/2}}{35 a^2 d}+\frac{6 e^6 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{7 a^2 d \sqrt{e \cos (c+d x)}}+\frac{4 e (e \cos (c+d x))^{9/2}}{5 d \left(a^2 \sin (c+d x)+a^2\right)}","\frac{6 e^5 \sin (c+d x) \sqrt{e \cos (c+d x)}}{7 a^2 d}+\frac{18 e^3 \sin (c+d x) (e \cos (c+d x))^{5/2}}{35 a^2 d}+\frac{6 e^6 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{7 a^2 d \sqrt{e \cos (c+d x)}}+\frac{4 e (e \cos (c+d x))^{9/2}}{5 d \left(a^2 \sin (c+d x)+a^2\right)}",1,"(6*e^6*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(7*a^2*d*Sqrt[e*Cos[c + d*x]]) + (6*e^5*Sqrt[e*Cos[c + d*x]]*Sin[c + d*x])/(7*a^2*d) + (18*e^3*(e*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(35*a^2*d) + (4*e*(e*Cos[c + d*x])^(9/2))/(5*d*(a^2 + a^2*Sin[c + d*x]))","A",5,4,25,0.1600,1,"{2680, 2635, 2642, 2641}"
244,1,114,0,0.0922453,"\int \frac{(e \cos (c+d x))^{9/2}}{(a+a \sin (c+d x))^2} \, dx","Int[(e*Cos[c + d*x])^(9/2)/(a + a*Sin[c + d*x])^2,x]","\frac{14 e^3 \sin (c+d x) (e \cos (c+d x))^{3/2}}{15 a^2 d}+\frac{14 e^4 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{5 a^2 d \sqrt{\cos (c+d x)}}+\frac{4 e (e \cos (c+d x))^{7/2}}{3 d \left(a^2 \sin (c+d x)+a^2\right)}","\frac{14 e^3 \sin (c+d x) (e \cos (c+d x))^{3/2}}{15 a^2 d}+\frac{14 e^4 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{5 a^2 d \sqrt{\cos (c+d x)}}+\frac{4 e (e \cos (c+d x))^{7/2}}{3 d \left(a^2 \sin (c+d x)+a^2\right)}",1,"(14*e^4*Sqrt[e*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(5*a^2*d*Sqrt[Cos[c + d*x]]) + (14*e^3*(e*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(15*a^2*d) + (4*e*(e*Cos[c + d*x])^(7/2))/(3*d*(a^2 + a^2*Sin[c + d*x]))","A",4,4,25,0.1600,1,"{2680, 2635, 2640, 2639}"
245,1,112,0,0.0970945,"\int \frac{(e \cos (c+d x))^{7/2}}{(a+a \sin (c+d x))^2} \, dx","Int[(e*Cos[c + d*x])^(7/2)/(a + a*Sin[c + d*x])^2,x]","\frac{10 e^3 \sin (c+d x) \sqrt{e \cos (c+d x)}}{3 a^2 d}+\frac{10 e^4 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^2 d \sqrt{e \cos (c+d x)}}+\frac{4 e (e \cos (c+d x))^{5/2}}{d \left(a^2 \sin (c+d x)+a^2\right)}","\frac{10 e^3 \sin (c+d x) \sqrt{e \cos (c+d x)}}{3 a^2 d}+\frac{10 e^4 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^2 d \sqrt{e \cos (c+d x)}}+\frac{4 e (e \cos (c+d x))^{5/2}}{d \left(a^2 \sin (c+d x)+a^2\right)}",1,"(10*e^4*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(3*a^2*d*Sqrt[e*Cos[c + d*x]]) + (10*e^3*Sqrt[e*Cos[c + d*x]]*Sin[c + d*x])/(3*a^2*d) + (4*e*(e*Cos[c + d*x])^(5/2))/(d*(a^2 + a^2*Sin[c + d*x]))","A",4,4,25,0.1600,1,"{2680, 2635, 2642, 2641}"
246,1,79,0,0.0779328,"\int \frac{(e \cos (c+d x))^{5/2}}{(a+a \sin (c+d x))^2} \, dx","Int[(e*Cos[c + d*x])^(5/2)/(a + a*Sin[c + d*x])^2,x]","-\frac{6 e^2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{a^2 d \sqrt{\cos (c+d x)}}-\frac{4 e (e \cos (c+d x))^{3/2}}{d \left(a^2 \sin (c+d x)+a^2\right)}","-\frac{6 e^2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{a^2 d \sqrt{\cos (c+d x)}}-\frac{4 e (e \cos (c+d x))^{3/2}}{d \left(a^2 \sin (c+d x)+a^2\right)}",1,"(-6*e^2*Sqrt[e*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(a^2*d*Sqrt[Cos[c + d*x]]) - (4*e*(e*Cos[c + d*x])^(3/2))/(d*(a^2 + a^2*Sin[c + d*x]))","A",3,3,25,0.1200,1,"{2680, 2640, 2639}"
247,1,83,0,0.074943,"\int \frac{(e \cos (c+d x))^{3/2}}{(a+a \sin (c+d x))^2} \, dx","Int[(e*Cos[c + d*x])^(3/2)/(a + a*Sin[c + d*x])^2,x]","-\frac{2 e^2 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^2 d \sqrt{e \cos (c+d x)}}-\frac{4 e \sqrt{e \cos (c+d x)}}{3 d \left(a^2 \sin (c+d x)+a^2\right)}","-\frac{2 e^2 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^2 d \sqrt{e \cos (c+d x)}}-\frac{4 e \sqrt{e \cos (c+d x)}}{3 d \left(a^2 \sin (c+d x)+a^2\right)}",1,"(-2*e^2*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(3*a^2*d*Sqrt[e*Cos[c + d*x]]) - (4*e*Sqrt[e*Cos[c + d*x]])/(3*d*(a^2 + a^2*Sin[c + d*x]))","A",3,3,25,0.1200,1,"{2680, 2642, 2641}"
248,1,116,0,0.1193962,"\int \frac{\sqrt{e \cos (c+d x)}}{(a+a \sin (c+d x))^2} \, dx","Int[Sqrt[e*Cos[c + d*x]]/(a + a*Sin[c + d*x])^2,x]","-\frac{2 (e \cos (c+d x))^{3/2}}{5 d e \left(a^2 \sin (c+d x)+a^2\right)}-\frac{2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{5 a^2 d \sqrt{\cos (c+d x)}}-\frac{2 (e \cos (c+d x))^{3/2}}{5 d e (a \sin (c+d x)+a)^2}","-\frac{2 (e \cos (c+d x))^{3/2}}{5 d e \left(a^2 \sin (c+d x)+a^2\right)}-\frac{2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{5 a^2 d \sqrt{\cos (c+d x)}}-\frac{2 (e \cos (c+d x))^{3/2}}{5 d e (a \sin (c+d x)+a)^2}",1,"(-2*Sqrt[e*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(5*a^2*d*Sqrt[Cos[c + d*x]]) - (2*(e*Cos[c + d*x])^(3/2))/(5*d*e*(a + a*Sin[c + d*x])^2) - (2*(e*Cos[c + d*x])^(3/2))/(5*d*e*(a^2 + a^2*Sin[c + d*x]))","A",4,4,25,0.1600,1,"{2681, 2683, 2640, 2639}"
249,1,116,0,0.1270725,"\int \frac{1}{\sqrt{e \cos (c+d x)} (a+a \sin (c+d x))^2} \, dx","Int[1/(Sqrt[e*Cos[c + d*x]]*(a + a*Sin[c + d*x])^2),x]","-\frac{2 \sqrt{e \cos (c+d x)}}{7 d e \left(a^2 \sin (c+d x)+a^2\right)}+\frac{2 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{7 a^2 d \sqrt{e \cos (c+d x)}}-\frac{2 \sqrt{e \cos (c+d x)}}{7 d e (a \sin (c+d x)+a)^2}","-\frac{2 \sqrt{e \cos (c+d x)}}{7 d e \left(a^2 \sin (c+d x)+a^2\right)}+\frac{2 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{7 a^2 d \sqrt{e \cos (c+d x)}}-\frac{2 \sqrt{e \cos (c+d x)}}{7 d e (a \sin (c+d x)+a)^2}",1,"(2*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(7*a^2*d*Sqrt[e*Cos[c + d*x]]) - (2*Sqrt[e*Cos[c + d*x]])/(7*d*e*(a + a*Sin[c + d*x])^2) - (2*Sqrt[e*Cos[c + d*x]])/(7*d*e*(a^2 + a^2*Sin[c + d*x]))","A",4,4,25,0.1600,1,"{2681, 2683, 2642, 2641}"
250,1,150,0,0.1574003,"\int \frac{1}{(e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^2} \, dx","Int[1/((e*Cos[c + d*x])^(3/2)*(a + a*Sin[c + d*x])^2),x]","-\frac{2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{3 a^2 d e^2 \sqrt{\cos (c+d x)}}+\frac{2 \sin (c+d x)}{3 a^2 d e \sqrt{e \cos (c+d x)}}-\frac{2}{9 d e \left(a^2 \sin (c+d x)+a^2\right) \sqrt{e \cos (c+d x)}}-\frac{2}{9 d e (a \sin (c+d x)+a)^2 \sqrt{e \cos (c+d x)}}","-\frac{2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{3 a^2 d e^2 \sqrt{\cos (c+d x)}}+\frac{2 \sin (c+d x)}{3 a^2 d e \sqrt{e \cos (c+d x)}}-\frac{2}{9 d e \left(a^2 \sin (c+d x)+a^2\right) \sqrt{e \cos (c+d x)}}-\frac{2}{9 d e (a \sin (c+d x)+a)^2 \sqrt{e \cos (c+d x)}}",1,"(-2*Sqrt[e*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(3*a^2*d*e^2*Sqrt[Cos[c + d*x]]) + (2*Sin[c + d*x])/(3*a^2*d*e*Sqrt[e*Cos[c + d*x]]) - 2/(9*d*e*Sqrt[e*Cos[c + d*x]]*(a + a*Sin[c + d*x])^2) - 2/(9*d*e*Sqrt[e*Cos[c + d*x]]*(a^2 + a^2*Sin[c + d*x]))","A",5,5,25,0.2000,1,"{2681, 2683, 2636, 2640, 2639}"
251,1,150,0,0.1619272,"\int \frac{1}{(e \cos (c+d x))^{5/2} (a+a \sin (c+d x))^2} \, dx","Int[1/((e*Cos[c + d*x])^(5/2)*(a + a*Sin[c + d*x])^2),x]","\frac{10 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{33 a^2 d e^2 \sqrt{e \cos (c+d x)}}+\frac{10 \sin (c+d x)}{33 a^2 d e (e \cos (c+d x))^{3/2}}-\frac{2}{11 d e \left(a^2 \sin (c+d x)+a^2\right) (e \cos (c+d x))^{3/2}}-\frac{2}{11 d e (a \sin (c+d x)+a)^2 (e \cos (c+d x))^{3/2}}","\frac{10 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{33 a^2 d e^2 \sqrt{e \cos (c+d x)}}+\frac{10 \sin (c+d x)}{33 a^2 d e (e \cos (c+d x))^{3/2}}-\frac{2}{11 d e \left(a^2 \sin (c+d x)+a^2\right) (e \cos (c+d x))^{3/2}}-\frac{2}{11 d e (a \sin (c+d x)+a)^2 (e \cos (c+d x))^{3/2}}",1,"(10*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(33*a^2*d*e^2*Sqrt[e*Cos[c + d*x]]) + (10*Sin[c + d*x])/(33*a^2*d*e*(e*Cos[c + d*x])^(3/2)) - 2/(11*d*e*(e*Cos[c + d*x])^(3/2)*(a + a*Sin[c + d*x])^2) - 2/(11*d*e*(e*Cos[c + d*x])^(3/2)*(a^2 + a^2*Sin[c + d*x]))","A",5,5,25,0.2000,1,"{2681, 2683, 2636, 2642, 2641}"
252,1,181,0,0.1830305,"\int \frac{1}{(e \cos (c+d x))^{7/2} (a+a \sin (c+d x))^2} \, dx","Int[1/((e*Cos[c + d*x])^(7/2)*(a + a*Sin[c + d*x])^2),x]","\frac{42 \sin (c+d x)}{65 a^2 d e^3 \sqrt{e \cos (c+d x)}}-\frac{42 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{65 a^2 d e^4 \sqrt{\cos (c+d x)}}+\frac{14 \sin (c+d x)}{65 a^2 d e (e \cos (c+d x))^{5/2}}-\frac{2}{13 d e \left(a^2 \sin (c+d x)+a^2\right) (e \cos (c+d x))^{5/2}}-\frac{2}{13 d e (a \sin (c+d x)+a)^2 (e \cos (c+d x))^{5/2}}","\frac{42 \sin (c+d x)}{65 a^2 d e^3 \sqrt{e \cos (c+d x)}}-\frac{42 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{65 a^2 d e^4 \sqrt{\cos (c+d x)}}+\frac{14 \sin (c+d x)}{65 a^2 d e (e \cos (c+d x))^{5/2}}-\frac{2}{13 d e \left(a^2 \sin (c+d x)+a^2\right) (e \cos (c+d x))^{5/2}}-\frac{2}{13 d e (a \sin (c+d x)+a)^2 (e \cos (c+d x))^{5/2}}",1,"(-42*Sqrt[e*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(65*a^2*d*e^4*Sqrt[Cos[c + d*x]]) + (14*Sin[c + d*x])/(65*a^2*d*e*(e*Cos[c + d*x])^(5/2)) + (42*Sin[c + d*x])/(65*a^2*d*e^3*Sqrt[e*Cos[c + d*x]]) - 2/(13*d*e*(e*Cos[c + d*x])^(5/2)*(a + a*Sin[c + d*x])^2) - 2/(13*d*e*(e*Cos[c + d*x])^(5/2)*(a^2 + a^2*Sin[c + d*x]))","A",6,5,25,0.2000,1,"{2681, 2683, 2636, 2640, 2639}"
253,1,169,0,0.1790507,"\int \frac{(e \cos (c+d x))^{15/2}}{(a+a \sin (c+d x))^3} \, dx","Int[(e*Cos[c + d*x])^(15/2)/(a + a*Sin[c + d*x])^3,x]","\frac{26 e^3 (e \cos (c+d x))^{9/2}}{45 a^3 d}+\frac{26 e^7 \sin (c+d x) \sqrt{e \cos (c+d x)}}{21 a^3 d}+\frac{26 e^5 \sin (c+d x) (e \cos (c+d x))^{5/2}}{35 a^3 d}+\frac{26 e^8 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 a^3 d \sqrt{e \cos (c+d x)}}+\frac{4 e (e \cos (c+d x))^{13/2}}{5 a d (a \sin (c+d x)+a)^2}","\frac{26 e^3 (e \cos (c+d x))^{9/2}}{45 a^3 d}+\frac{26 e^7 \sin (c+d x) \sqrt{e \cos (c+d x)}}{21 a^3 d}+\frac{26 e^5 \sin (c+d x) (e \cos (c+d x))^{5/2}}{35 a^3 d}+\frac{26 e^8 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 a^3 d \sqrt{e \cos (c+d x)}}+\frac{4 e (e \cos (c+d x))^{13/2}}{5 a d (a \sin (c+d x)+a)^2}",1,"(26*e^3*(e*Cos[c + d*x])^(9/2))/(45*a^3*d) + (26*e^8*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(21*a^3*d*Sqrt[e*Cos[c + d*x]]) + (26*e^7*Sqrt[e*Cos[c + d*x]]*Sin[c + d*x])/(21*a^3*d) + (26*e^5*(e*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(35*a^3*d) + (4*e*(e*Cos[c + d*x])^(13/2))/(5*a*d*(a + a*Sin[c + d*x])^2)","A",6,5,25,0.2000,1,"{2680, 2682, 2635, 2642, 2641}"
254,1,138,0,0.1520683,"\int \frac{(e \cos (c+d x))^{13/2}}{(a+a \sin (c+d x))^3} \, dx","Int[(e*Cos[c + d*x])^(13/2)/(a + a*Sin[c + d*x])^3,x]","\frac{22 e^3 (e \cos (c+d x))^{7/2}}{21 a^3 d}+\frac{22 e^5 \sin (c+d x) (e \cos (c+d x))^{3/2}}{15 a^3 d}+\frac{22 e^6 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{5 a^3 d \sqrt{\cos (c+d x)}}+\frac{4 e (e \cos (c+d x))^{11/2}}{3 a d (a \sin (c+d x)+a)^2}","\frac{22 e^3 (e \cos (c+d x))^{7/2}}{21 a^3 d}+\frac{22 e^5 \sin (c+d x) (e \cos (c+d x))^{3/2}}{15 a^3 d}+\frac{22 e^6 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{5 a^3 d \sqrt{\cos (c+d x)}}+\frac{4 e (e \cos (c+d x))^{11/2}}{3 a d (a \sin (c+d x)+a)^2}",1,"(22*e^3*(e*Cos[c + d*x])^(7/2))/(21*a^3*d) + (22*e^6*Sqrt[e*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(5*a^3*d*Sqrt[Cos[c + d*x]]) + (22*e^5*(e*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(15*a^3*d) + (4*e*(e*Cos[c + d*x])^(11/2))/(3*a*d*(a + a*Sin[c + d*x])^2)","A",5,5,25,0.2000,1,"{2680, 2682, 2635, 2640, 2639}"
255,1,132,0,0.1509779,"\int \frac{(e \cos (c+d x))^{11/2}}{(a+a \sin (c+d x))^3} \, dx","Int[(e*Cos[c + d*x])^(11/2)/(a + a*Sin[c + d*x])^3,x]","\frac{18 e^3 (e \cos (c+d x))^{5/2}}{5 a^3 d}+\frac{6 e^5 \sin (c+d x) \sqrt{e \cos (c+d x)}}{a^3 d}+\frac{6 e^6 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^3 d \sqrt{e \cos (c+d x)}}+\frac{4 e (e \cos (c+d x))^{9/2}}{a d (a \sin (c+d x)+a)^2}","\frac{18 e^3 (e \cos (c+d x))^{5/2}}{5 a^3 d}+\frac{6 e^5 \sin (c+d x) \sqrt{e \cos (c+d x)}}{a^3 d}+\frac{6 e^6 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^3 d \sqrt{e \cos (c+d x)}}+\frac{4 e (e \cos (c+d x))^{9/2}}{a d (a \sin (c+d x)+a)^2}",1,"(18*e^3*(e*Cos[c + d*x])^(5/2))/(5*a^3*d) + (6*e^6*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(a^3*d*Sqrt[e*Cos[c + d*x]]) + (6*e^5*Sqrt[e*Cos[c + d*x]]*Sin[c + d*x])/(a^3*d) + (4*e*(e*Cos[c + d*x])^(9/2))/(a*d*(a + a*Sin[c + d*x])^2)","A",5,5,25,0.2000,1,"{2680, 2682, 2635, 2642, 2641}"
256,1,103,0,0.1548473,"\int \frac{(e \cos (c+d x))^{9/2}}{(a+a \sin (c+d x))^3} \, dx","Int[(e*Cos[c + d*x])^(9/2)/(a + a*Sin[c + d*x])^3,x]","-\frac{14 e^3 (e \cos (c+d x))^{3/2}}{3 a^3 d}-\frac{14 e^4 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{a^3 d \sqrt{\cos (c+d x)}}-\frac{4 e (e \cos (c+d x))^{7/2}}{a d (a \sin (c+d x)+a)^2}","-\frac{14 e^3 (e \cos (c+d x))^{3/2}}{3 a^3 d}-\frac{14 e^4 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{a^3 d \sqrt{\cos (c+d x)}}-\frac{4 e (e \cos (c+d x))^{7/2}}{a d (a \sin (c+d x)+a)^2}",1,"(-14*e^3*(e*Cos[c + d*x])^(3/2))/(3*a^3*d) - (14*e^4*Sqrt[e*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(a^3*d*Sqrt[Cos[c + d*x]]) - (4*e*(e*Cos[c + d*x])^(7/2))/(a*d*(a + a*Sin[c + d*x])^2)","A",4,4,25,0.1600,1,"{2680, 2682, 2640, 2639}"
257,1,107,0,0.1360386,"\int \frac{(e \cos (c+d x))^{7/2}}{(a+a \sin (c+d x))^3} \, dx","Int[(e*Cos[c + d*x])^(7/2)/(a + a*Sin[c + d*x])^3,x]","-\frac{10 e^3 \sqrt{e \cos (c+d x)}}{3 a^3 d}-\frac{10 e^4 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^3 d \sqrt{e \cos (c+d x)}}-\frac{4 e (e \cos (c+d x))^{5/2}}{3 a d (a \sin (c+d x)+a)^2}","-\frac{10 e^3 \sqrt{e \cos (c+d x)}}{3 a^3 d}-\frac{10 e^4 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^3 d \sqrt{e \cos (c+d x)}}-\frac{4 e (e \cos (c+d x))^{5/2}}{3 a d (a \sin (c+d x)+a)^2}",1,"(-10*e^3*Sqrt[e*Cos[c + d*x]])/(3*a^3*d) - (10*e^4*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(3*a^3*d*Sqrt[e*Cos[c + d*x]]) - (4*e*(e*Cos[c + d*x])^(5/2))/(3*a*d*(a + a*Sin[c + d*x])^2)","A",4,4,25,0.1600,1,"{2680, 2682, 2642, 2641}"
258,1,118,0,0.1306749,"\int \frac{(e \cos (c+d x))^{5/2}}{(a+a \sin (c+d x))^3} \, dx","Int[(e*Cos[c + d*x])^(5/2)/(a + a*Sin[c + d*x])^3,x]","\frac{6 e^2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{5 a^3 d \sqrt{\cos (c+d x)}}+\frac{6 e (e \cos (c+d x))^{3/2}}{5 d \left(a^3 \sin (c+d x)+a^3\right)}-\frac{4 e (e \cos (c+d x))^{3/2}}{5 a d (a \sin (c+d x)+a)^2}","\frac{6 e^2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{5 a^3 d \sqrt{\cos (c+d x)}}+\frac{6 e (e \cos (c+d x))^{3/2}}{5 d \left(a^3 \sin (c+d x)+a^3\right)}-\frac{4 e (e \cos (c+d x))^{3/2}}{5 a d (a \sin (c+d x)+a)^2}",1,"(6*e^2*Sqrt[e*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(5*a^3*d*Sqrt[Cos[c + d*x]]) - (4*e*(e*Cos[c + d*x])^(3/2))/(5*a*d*(a + a*Sin[c + d*x])^2) + (6*e*(e*Cos[c + d*x])^(3/2))/(5*d*(a^3 + a^3*Sin[c + d*x]))","A",4,4,25,0.1600,1,"{2680, 2683, 2640, 2639}"
259,1,118,0,0.1338178,"\int \frac{(e \cos (c+d x))^{3/2}}{(a+a \sin (c+d x))^3} \, dx","Int[(e*Cos[c + d*x])^(3/2)/(a + a*Sin[c + d*x])^3,x]","-\frac{2 e^2 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 a^3 d \sqrt{e \cos (c+d x)}}+\frac{2 e \sqrt{e \cos (c+d x)}}{21 d \left(a^3 \sin (c+d x)+a^3\right)}-\frac{4 e \sqrt{e \cos (c+d x)}}{7 a d (a \sin (c+d x)+a)^2}","-\frac{2 e^2 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 a^3 d \sqrt{e \cos (c+d x)}}+\frac{2 e \sqrt{e \cos (c+d x)}}{21 d \left(a^3 \sin (c+d x)+a^3\right)}-\frac{4 e \sqrt{e \cos (c+d x)}}{7 a d (a \sin (c+d x)+a)^2}",1,"(-2*e^2*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(21*a^3*d*Sqrt[e*Cos[c + d*x]]) - (4*e*Sqrt[e*Cos[c + d*x]])/(7*a*d*(a + a*Sin[c + d*x])^2) + (2*e*Sqrt[e*Cos[c + d*x]])/(21*d*(a^3 + a^3*Sin[c + d*x]))","A",4,4,25,0.1600,1,"{2680, 2683, 2642, 2641}"
260,1,153,0,0.1733752,"\int \frac{\sqrt{e \cos (c+d x)}}{(a+a \sin (c+d x))^3} \, dx","Int[Sqrt[e*Cos[c + d*x]]/(a + a*Sin[c + d*x])^3,x]","-\frac{2 (e \cos (c+d x))^{3/2}}{15 d e \left(a^3 \sin (c+d x)+a^3\right)}-\frac{2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{15 a^3 d \sqrt{\cos (c+d x)}}-\frac{2 (e \cos (c+d x))^{3/2}}{15 a d e (a \sin (c+d x)+a)^2}-\frac{2 (e \cos (c+d x))^{3/2}}{9 d e (a \sin (c+d x)+a)^3}","-\frac{2 (e \cos (c+d x))^{3/2}}{15 d e \left(a^3 \sin (c+d x)+a^3\right)}-\frac{2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{15 a^3 d \sqrt{\cos (c+d x)}}-\frac{2 (e \cos (c+d x))^{3/2}}{15 a d e (a \sin (c+d x)+a)^2}-\frac{2 (e \cos (c+d x))^{3/2}}{9 d e (a \sin (c+d x)+a)^3}",1,"(-2*Sqrt[e*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(15*a^3*d*Sqrt[Cos[c + d*x]]) - (2*(e*Cos[c + d*x])^(3/2))/(9*d*e*(a + a*Sin[c + d*x])^3) - (2*(e*Cos[c + d*x])^(3/2))/(15*a*d*e*(a + a*Sin[c + d*x])^2) - (2*(e*Cos[c + d*x])^(3/2))/(15*d*e*(a^3 + a^3*Sin[c + d*x]))","A",5,4,25,0.1600,1,"{2681, 2683, 2640, 2639}"
261,1,153,0,0.1841464,"\int \frac{1}{\sqrt{e \cos (c+d x)} (a+a \sin (c+d x))^3} \, dx","Int[1/(Sqrt[e*Cos[c + d*x]]*(a + a*Sin[c + d*x])^3),x]","-\frac{10 \sqrt{e \cos (c+d x)}}{77 d e \left(a^3 \sin (c+d x)+a^3\right)}+\frac{10 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{77 a^3 d \sqrt{e \cos (c+d x)}}-\frac{10 \sqrt{e \cos (c+d x)}}{77 a d e (a \sin (c+d x)+a)^2}-\frac{2 \sqrt{e \cos (c+d x)}}{11 d e (a \sin (c+d x)+a)^3}","-\frac{10 \sqrt{e \cos (c+d x)}}{77 d e \left(a^3 \sin (c+d x)+a^3\right)}+\frac{10 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{77 a^3 d \sqrt{e \cos (c+d x)}}-\frac{10 \sqrt{e \cos (c+d x)}}{77 a d e (a \sin (c+d x)+a)^2}-\frac{2 \sqrt{e \cos (c+d x)}}{11 d e (a \sin (c+d x)+a)^3}",1,"(10*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(77*a^3*d*Sqrt[e*Cos[c + d*x]]) - (2*Sqrt[e*Cos[c + d*x]])/(11*d*e*(a + a*Sin[c + d*x])^3) - (10*Sqrt[e*Cos[c + d*x]])/(77*a*d*e*(a + a*Sin[c + d*x])^2) - (10*Sqrt[e*Cos[c + d*x]])/(77*d*e*(a^3 + a^3*Sin[c + d*x]))","A",5,4,25,0.1600,1,"{2681, 2683, 2642, 2641}"
262,1,187,0,0.2247712,"\int \frac{1}{(e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^3} \, dx","Int[1/((e*Cos[c + d*x])^(3/2)*(a + a*Sin[c + d*x])^3),x]","-\frac{14 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{39 a^3 d e^2 \sqrt{\cos (c+d x)}}+\frac{14 \sin (c+d x)}{39 a^3 d e \sqrt{e \cos (c+d x)}}-\frac{14}{117 d e \left(a^3 \sin (c+d x)+a^3\right) \sqrt{e \cos (c+d x)}}-\frac{14}{117 a d e (a \sin (c+d x)+a)^2 \sqrt{e \cos (c+d x)}}-\frac{2}{13 d e (a \sin (c+d x)+a)^3 \sqrt{e \cos (c+d x)}}","-\frac{14 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{39 a^3 d e^2 \sqrt{\cos (c+d x)}}+\frac{14 \sin (c+d x)}{39 a^3 d e \sqrt{e \cos (c+d x)}}-\frac{14}{117 d e \left(a^3 \sin (c+d x)+a^3\right) \sqrt{e \cos (c+d x)}}-\frac{14}{117 a d e (a \sin (c+d x)+a)^2 \sqrt{e \cos (c+d x)}}-\frac{2}{13 d e (a \sin (c+d x)+a)^3 \sqrt{e \cos (c+d x)}}",1,"(-14*Sqrt[e*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(39*a^3*d*e^2*Sqrt[Cos[c + d*x]]) + (14*Sin[c + d*x])/(39*a^3*d*e*Sqrt[e*Cos[c + d*x]]) - 2/(13*d*e*Sqrt[e*Cos[c + d*x]]*(a + a*Sin[c + d*x])^3) - 14/(117*a*d*e*Sqrt[e*Cos[c + d*x]]*(a + a*Sin[c + d*x])^2) - 14/(117*d*e*Sqrt[e*Cos[c + d*x]]*(a^3 + a^3*Sin[c + d*x]))","A",6,5,25,0.2000,1,"{2681, 2683, 2636, 2640, 2639}"
263,1,180,0,0.1753367,"\int \frac{(e \cos (c+d x))^{15/2}}{(a+a \sin (c+d x))^4} \, dx","Int[(e*Cos[c + d*x])^(15/2)/(a + a*Sin[c + d*x])^4,x]","\frac{78 e^7 \sin (c+d x) \sqrt{e \cos (c+d x)}}{7 a^4 d}+\frac{234 e^5 \sin (c+d x) (e \cos (c+d x))^{5/2}}{35 a^4 d}+\frac{52 e^3 (e \cos (c+d x))^{9/2}}{5 d \left(a^4 \sin (c+d x)+a^4\right)}+\frac{78 e^8 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{7 a^4 d \sqrt{e \cos (c+d x)}}+\frac{4 e (e \cos (c+d x))^{13/2}}{a d (a \sin (c+d x)+a)^3}","\frac{78 e^7 \sin (c+d x) \sqrt{e \cos (c+d x)}}{7 a^4 d}+\frac{234 e^5 \sin (c+d x) (e \cos (c+d x))^{5/2}}{35 a^4 d}+\frac{52 e^3 (e \cos (c+d x))^{9/2}}{5 d \left(a^4 \sin (c+d x)+a^4\right)}+\frac{78 e^8 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{7 a^4 d \sqrt{e \cos (c+d x)}}+\frac{4 e (e \cos (c+d x))^{13/2}}{a d (a \sin (c+d x)+a)^3}",1,"(78*e^8*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(7*a^4*d*Sqrt[e*Cos[c + d*x]]) + (78*e^7*Sqrt[e*Cos[c + d*x]]*Sin[c + d*x])/(7*a^4*d) + (234*e^5*(e*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(35*a^4*d) + (4*e*(e*Cos[c + d*x])^(13/2))/(a*d*(a + a*Sin[c + d*x])^3) + (52*e^3*(e*Cos[c + d*x])^(9/2))/(5*d*(a^4 + a^4*Sin[c + d*x]))","A",6,4,25,0.1600,1,"{2680, 2635, 2642, 2641}"
264,1,149,0,0.1531246,"\int \frac{(e \cos (c+d x))^{13/2}}{(a+a \sin (c+d x))^4} \, dx","Int[(e*Cos[c + d*x])^(13/2)/(a + a*Sin[c + d*x])^4,x]","-\frac{154 e^5 \sin (c+d x) (e \cos (c+d x))^{3/2}}{15 a^4 d}-\frac{44 e^3 (e \cos (c+d x))^{7/2}}{3 d \left(a^4 \sin (c+d x)+a^4\right)}-\frac{154 e^6 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{5 a^4 d \sqrt{\cos (c+d x)}}-\frac{4 e (e \cos (c+d x))^{11/2}}{a d (a \sin (c+d x)+a)^3}","-\frac{154 e^5 \sin (c+d x) (e \cos (c+d x))^{3/2}}{15 a^4 d}-\frac{44 e^3 (e \cos (c+d x))^{7/2}}{3 d \left(a^4 \sin (c+d x)+a^4\right)}-\frac{154 e^6 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{5 a^4 d \sqrt{\cos (c+d x)}}-\frac{4 e (e \cos (c+d x))^{11/2}}{a d (a \sin (c+d x)+a)^3}",1,"(-154*e^6*Sqrt[e*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(5*a^4*d*Sqrt[Cos[c + d*x]]) - (154*e^5*(e*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(15*a^4*d) - (4*e*(e*Cos[c + d*x])^(11/2))/(a*d*(a + a*Sin[c + d*x])^3) - (44*e^3*(e*Cos[c + d*x])^(7/2))/(3*d*(a^4 + a^4*Sin[c + d*x]))","A",5,4,25,0.1600,1,"{2680, 2635, 2640, 2639}"
265,1,145,0,0.1521415,"\int \frac{(e \cos (c+d x))^{11/2}}{(a+a \sin (c+d x))^4} \, dx","Int[(e*Cos[c + d*x])^(11/2)/(a + a*Sin[c + d*x])^4,x]","-\frac{10 e^5 \sin (c+d x) \sqrt{e \cos (c+d x)}}{a^4 d}-\frac{12 e^3 (e \cos (c+d x))^{5/2}}{d \left(a^4 \sin (c+d x)+a^4\right)}-\frac{10 e^6 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^4 d \sqrt{e \cos (c+d x)}}-\frac{4 e (e \cos (c+d x))^{9/2}}{3 a d (a \sin (c+d x)+a)^3}","-\frac{10 e^5 \sin (c+d x) \sqrt{e \cos (c+d x)}}{a^4 d}-\frac{12 e^3 (e \cos (c+d x))^{5/2}}{d \left(a^4 \sin (c+d x)+a^4\right)}-\frac{10 e^6 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^4 d \sqrt{e \cos (c+d x)}}-\frac{4 e (e \cos (c+d x))^{9/2}}{3 a d (a \sin (c+d x)+a)^3}",1,"(-10*e^6*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(a^4*d*Sqrt[e*Cos[c + d*x]]) - (10*e^5*Sqrt[e*Cos[c + d*x]]*Sin[c + d*x])/(a^4*d) - (4*e*(e*Cos[c + d*x])^(9/2))/(3*a*d*(a + a*Sin[c + d*x])^3) - (12*e^3*(e*Cos[c + d*x])^(5/2))/(d*(a^4 + a^4*Sin[c + d*x]))","A",5,4,25,0.1600,1,"{2680, 2635, 2642, 2641}"
266,1,120,0,0.1338172,"\int \frac{(e \cos (c+d x))^{9/2}}{(a+a \sin (c+d x))^4} \, dx","Int[(e*Cos[c + d*x])^(9/2)/(a + a*Sin[c + d*x])^4,x]","\frac{28 e^3 (e \cos (c+d x))^{3/2}}{5 d \left(a^4 \sin (c+d x)+a^4\right)}+\frac{42 e^4 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{5 a^4 d \sqrt{\cos (c+d x)}}-\frac{4 e (e \cos (c+d x))^{7/2}}{5 a d (a \sin (c+d x)+a)^3}","\frac{28 e^3 (e \cos (c+d x))^{3/2}}{5 d \left(a^4 \sin (c+d x)+a^4\right)}+\frac{42 e^4 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{5 a^4 d \sqrt{\cos (c+d x)}}-\frac{4 e (e \cos (c+d x))^{7/2}}{5 a d (a \sin (c+d x)+a)^3}",1,"(42*e^4*Sqrt[e*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(5*a^4*d*Sqrt[Cos[c + d*x]]) - (4*e*(e*Cos[c + d*x])^(7/2))/(5*a*d*(a + a*Sin[c + d*x])^3) + (28*e^3*(e*Cos[c + d*x])^(3/2))/(5*d*(a^4 + a^4*Sin[c + d*x]))","A",4,3,25,0.1200,1,"{2680, 2640, 2639}"
267,1,120,0,0.1335445,"\int \frac{(e \cos (c+d x))^{7/2}}{(a+a \sin (c+d x))^4} \, dx","Int[(e*Cos[c + d*x])^(7/2)/(a + a*Sin[c + d*x])^4,x]","\frac{20 e^3 \sqrt{e \cos (c+d x)}}{21 d \left(a^4 \sin (c+d x)+a^4\right)}+\frac{10 e^4 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 a^4 d \sqrt{e \cos (c+d x)}}-\frac{4 e (e \cos (c+d x))^{5/2}}{7 a d (a \sin (c+d x)+a)^3}","\frac{20 e^3 \sqrt{e \cos (c+d x)}}{21 d \left(a^4 \sin (c+d x)+a^4\right)}+\frac{10 e^4 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 a^4 d \sqrt{e \cos (c+d x)}}-\frac{4 e (e \cos (c+d x))^{5/2}}{7 a d (a \sin (c+d x)+a)^3}",1,"(10*e^4*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(21*a^4*d*Sqrt[e*Cos[c + d*x]]) - (4*e*(e*Cos[c + d*x])^(5/2))/(7*a*d*(a + a*Sin[c + d*x])^3) + (20*e^3*Sqrt[e*Cos[c + d*x]])/(21*d*(a^4 + a^4*Sin[c + d*x]))","A",4,3,25,0.1200,1,"{2680, 2642, 2641}"
268,1,154,0,0.182129,"\int \frac{(e \cos (c+d x))^{5/2}}{(a+a \sin (c+d x))^4} \, dx","Int[(e*Cos[c + d*x])^(5/2)/(a + a*Sin[c + d*x])^4,x]","\frac{2 e^2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{15 a^4 d \sqrt{\cos (c+d x)}}+\frac{2 e (e \cos (c+d x))^{3/2}}{15 d \left(a^4 \sin (c+d x)+a^4\right)}+\frac{2 e (e \cos (c+d x))^{3/2}}{15 d \left(a^2 \sin (c+d x)+a^2\right)^2}-\frac{4 e (e \cos (c+d x))^{3/2}}{9 a d (a \sin (c+d x)+a)^3}","\frac{2 e^2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{15 a^4 d \sqrt{\cos (c+d x)}}+\frac{2 e (e \cos (c+d x))^{3/2}}{15 d \left(a^4 \sin (c+d x)+a^4\right)}+\frac{2 e (e \cos (c+d x))^{3/2}}{15 d \left(a^2 \sin (c+d x)+a^2\right)^2}-\frac{4 e (e \cos (c+d x))^{3/2}}{9 a d (a \sin (c+d x)+a)^3}",1,"(2*e^2*Sqrt[e*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(15*a^4*d*Sqrt[Cos[c + d*x]]) - (4*e*(e*Cos[c + d*x])^(3/2))/(9*a*d*(a + a*Sin[c + d*x])^3) + (2*e*(e*Cos[c + d*x])^(3/2))/(15*d*(a^2 + a^2*Sin[c + d*x])^2) + (2*e*(e*Cos[c + d*x])^(3/2))/(15*d*(a^4 + a^4*Sin[c + d*x]))","A",5,5,25,0.2000,1,"{2680, 2681, 2683, 2640, 2639}"
269,1,154,0,0.1848097,"\int \frac{(e \cos (c+d x))^{3/2}}{(a+a \sin (c+d x))^4} \, dx","Int[(e*Cos[c + d*x])^(3/2)/(a + a*Sin[c + d*x])^4,x]","-\frac{2 e^2 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{77 a^4 d \sqrt{e \cos (c+d x)}}+\frac{2 e \sqrt{e \cos (c+d x)}}{77 d \left(a^4 \sin (c+d x)+a^4\right)}+\frac{2 e \sqrt{e \cos (c+d x)}}{77 d \left(a^2 \sin (c+d x)+a^2\right)^2}-\frac{4 e \sqrt{e \cos (c+d x)}}{11 a d (a \sin (c+d x)+a)^3}","-\frac{2 e^2 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{77 a^4 d \sqrt{e \cos (c+d x)}}+\frac{2 e \sqrt{e \cos (c+d x)}}{77 d \left(a^4 \sin (c+d x)+a^4\right)}+\frac{2 e \sqrt{e \cos (c+d x)}}{77 d \left(a^2 \sin (c+d x)+a^2\right)^2}-\frac{4 e \sqrt{e \cos (c+d x)}}{11 a d (a \sin (c+d x)+a)^3}",1,"(-2*e^2*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(77*a^4*d*Sqrt[e*Cos[c + d*x]]) - (4*e*Sqrt[e*Cos[c + d*x]])/(11*a*d*(a + a*Sin[c + d*x])^3) + (2*e*Sqrt[e*Cos[c + d*x]])/(77*d*(a^2 + a^2*Sin[c + d*x])^2) + (2*e*Sqrt[e*Cos[c + d*x]])/(77*d*(a^4 + a^4*Sin[c + d*x]))","A",5,5,25,0.2000,1,"{2680, 2681, 2683, 2642, 2641}"
270,1,191,0,0.2284979,"\int \frac{\sqrt{e \cos (c+d x)}}{(a+a \sin (c+d x))^4} \, dx","Int[Sqrt[e*Cos[c + d*x]]/(a + a*Sin[c + d*x])^4,x]","-\frac{2 (e \cos (c+d x))^{3/2}}{39 d e \left(a^4 \sin (c+d x)+a^4\right)}-\frac{2 (e \cos (c+d x))^{3/2}}{39 d e \left(a^2 \sin (c+d x)+a^2\right)^2}-\frac{2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{39 a^4 d \sqrt{\cos (c+d x)}}-\frac{10 (e \cos (c+d x))^{3/2}}{117 a d e (a \sin (c+d x)+a)^3}-\frac{2 (e \cos (c+d x))^{3/2}}{13 d e (a \sin (c+d x)+a)^4}","-\frac{2 (e \cos (c+d x))^{3/2}}{39 d e \left(a^4 \sin (c+d x)+a^4\right)}-\frac{2 (e \cos (c+d x))^{3/2}}{39 d e \left(a^2 \sin (c+d x)+a^2\right)^2}-\frac{2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{39 a^4 d \sqrt{\cos (c+d x)}}-\frac{10 (e \cos (c+d x))^{3/2}}{117 a d e (a \sin (c+d x)+a)^3}-\frac{2 (e \cos (c+d x))^{3/2}}{13 d e (a \sin (c+d x)+a)^4}",1,"(-2*Sqrt[e*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(39*a^4*d*Sqrt[Cos[c + d*x]]) - (2*(e*Cos[c + d*x])^(3/2))/(13*d*e*(a + a*Sin[c + d*x])^4) - (10*(e*Cos[c + d*x])^(3/2))/(117*a*d*e*(a + a*Sin[c + d*x])^3) - (2*(e*Cos[c + d*x])^(3/2))/(39*d*e*(a^2 + a^2*Sin[c + d*x])^2) - (2*(e*Cos[c + d*x])^(3/2))/(39*d*e*(a^4 + a^4*Sin[c + d*x]))","A",6,4,25,0.1600,1,"{2681, 2683, 2640, 2639}"
271,1,191,0,0.2415437,"\int \frac{1}{\sqrt{e \cos (c+d x)} (a+a \sin (c+d x))^4} \, dx","Int[1/(Sqrt[e*Cos[c + d*x]]*(a + a*Sin[c + d*x])^4),x]","-\frac{2 \sqrt{e \cos (c+d x)}}{33 d e \left(a^4 \sin (c+d x)+a^4\right)}-\frac{2 \sqrt{e \cos (c+d x)}}{33 d e \left(a^2 \sin (c+d x)+a^2\right)^2}+\frac{2 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{33 a^4 d \sqrt{e \cos (c+d x)}}-\frac{14 \sqrt{e \cos (c+d x)}}{165 a d e (a \sin (c+d x)+a)^3}-\frac{2 \sqrt{e \cos (c+d x)}}{15 d e (a \sin (c+d x)+a)^4}","-\frac{2 \sqrt{e \cos (c+d x)}}{33 d e \left(a^4 \sin (c+d x)+a^4\right)}-\frac{2 \sqrt{e \cos (c+d x)}}{33 d e \left(a^2 \sin (c+d x)+a^2\right)^2}+\frac{2 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{33 a^4 d \sqrt{e \cos (c+d x)}}-\frac{14 \sqrt{e \cos (c+d x)}}{165 a d e (a \sin (c+d x)+a)^3}-\frac{2 \sqrt{e \cos (c+d x)}}{15 d e (a \sin (c+d x)+a)^4}",1,"(2*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(33*a^4*d*Sqrt[e*Cos[c + d*x]]) - (2*Sqrt[e*Cos[c + d*x]])/(15*d*e*(a + a*Sin[c + d*x])^4) - (14*Sqrt[e*Cos[c + d*x]])/(165*a*d*e*(a + a*Sin[c + d*x])^3) - (2*Sqrt[e*Cos[c + d*x]])/(33*d*e*(a^2 + a^2*Sin[c + d*x])^2) - (2*Sqrt[e*Cos[c + d*x]])/(33*d*e*(a^4 + a^4*Sin[c + d*x]))","A",6,4,25,0.1600,1,"{2681, 2683, 2642, 2641}"
272,1,225,0,0.2990814,"\int \frac{1}{(e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^4} \, dx","Int[1/((e*Cos[c + d*x])^(3/2)*(a + a*Sin[c + d*x])^4),x]","-\frac{42 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{221 a^4 d e^2 \sqrt{\cos (c+d x)}}+\frac{42 \sin (c+d x)}{221 a^4 d e \sqrt{e \cos (c+d x)}}-\frac{14}{221 d e \left(a^4 \sin (c+d x)+a^4\right) \sqrt{e \cos (c+d x)}}-\frac{14}{221 d e \left(a^2 \sin (c+d x)+a^2\right)^2 \sqrt{e \cos (c+d x)}}-\frac{18}{221 a d e (a \sin (c+d x)+a)^3 \sqrt{e \cos (c+d x)}}-\frac{2}{17 d e (a \sin (c+d x)+a)^4 \sqrt{e \cos (c+d x)}}","-\frac{42 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{221 a^4 d e^2 \sqrt{\cos (c+d x)}}+\frac{42 \sin (c+d x)}{221 a^4 d e \sqrt{e \cos (c+d x)}}-\frac{14}{221 d e \left(a^4 \sin (c+d x)+a^4\right) \sqrt{e \cos (c+d x)}}-\frac{14}{221 d e \left(a^2 \sin (c+d x)+a^2\right)^2 \sqrt{e \cos (c+d x)}}-\frac{18}{221 a d e (a \sin (c+d x)+a)^3 \sqrt{e \cos (c+d x)}}-\frac{2}{17 d e (a \sin (c+d x)+a)^4 \sqrt{e \cos (c+d x)}}",1,"(-42*Sqrt[e*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(221*a^4*d*e^2*Sqrt[Cos[c + d*x]]) + (42*Sin[c + d*x])/(221*a^4*d*e*Sqrt[e*Cos[c + d*x]]) - 2/(17*d*e*Sqrt[e*Cos[c + d*x]]*(a + a*Sin[c + d*x])^4) - 18/(221*a*d*e*Sqrt[e*Cos[c + d*x]]*(a + a*Sin[c + d*x])^3) - 14/(221*d*e*Sqrt[e*Cos[c + d*x]]*(a^2 + a^2*Sin[c + d*x])^2) - 14/(221*d*e*Sqrt[e*Cos[c + d*x]]*(a^4 + a^4*Sin[c + d*x]))","A",7,5,25,0.2000,1,"{2681, 2683, 2636, 2640, 2639}"
273,1,236,0,0.3570755,"\int (e \cos (c+d x))^{3/2} \sqrt{a+a \sin (c+d x)} \, dx","Int[(e*Cos[c + d*x])^(3/2)*Sqrt[a + a*Sin[c + d*x]],x]","\frac{3 e^{3/2} \sqrt{\cos (c+d x)+1} \sqrt{a \sin (c+d x)+a} \tan ^{-1}\left(\frac{\sqrt{e} \sin (c+d x)}{\sqrt{\cos (c+d x)+1} \sqrt{e \cos (c+d x)}}\right)}{4 d (\sin (c+d x)+\cos (c+d x)+1)}-\frac{3 e^{3/2} \sqrt{\cos (c+d x)+1} \sqrt{a \sin (c+d x)+a} \sinh ^{-1}\left(\frac{\sqrt{e \cos (c+d x)}}{\sqrt{e}}\right)}{4 d (\sin (c+d x)+\cos (c+d x)+1)}-\frac{a (e \cos (c+d x))^{5/2}}{2 d e \sqrt{a \sin (c+d x)+a}}+\frac{3 e \sqrt{a \sin (c+d x)+a} \sqrt{e \cos (c+d x)}}{4 d}","\frac{3 e^{3/2} \sqrt{\cos (c+d x)+1} \sqrt{a \sin (c+d x)+a} \tan ^{-1}\left(\frac{\sqrt{e} \sin (c+d x)}{\sqrt{\cos (c+d x)+1} \sqrt{e \cos (c+d x)}}\right)}{4 d (\sin (c+d x)+\cos (c+d x)+1)}-\frac{3 e^{3/2} \sqrt{\cos (c+d x)+1} \sqrt{a \sin (c+d x)+a} \sinh ^{-1}\left(\frac{\sqrt{e \cos (c+d x)}}{\sqrt{e}}\right)}{4 d (\sin (c+d x)+\cos (c+d x)+1)}-\frac{a (e \cos (c+d x))^{5/2}}{2 d e \sqrt{a \sin (c+d x)+a}}+\frac{3 e \sqrt{a \sin (c+d x)+a} \sqrt{e \cos (c+d x)}}{4 d}",1,"-(a*(e*Cos[c + d*x])^(5/2))/(2*d*e*Sqrt[a + a*Sin[c + d*x]]) + (3*e*Sqrt[e*Cos[c + d*x]]*Sqrt[a + a*Sin[c + d*x]])/(4*d) - (3*e^(3/2)*ArcSinh[Sqrt[e*Cos[c + d*x]]/Sqrt[e]]*Sqrt[1 + Cos[c + d*x]]*Sqrt[a + a*Sin[c + d*x]])/(4*d*(1 + Cos[c + d*x] + Sin[c + d*x])) + (3*e^(3/2)*ArcTan[(Sqrt[e]*Sin[c + d*x])/(Sqrt[e*Cos[c + d*x]]*Sqrt[1 + Cos[c + d*x]])]*Sqrt[1 + Cos[c + d*x]]*Sqrt[a + a*Sin[c + d*x]])/(4*d*(1 + Cos[c + d*x] + Sin[c + d*x]))","A",8,8,27,0.2963,1,"{2678, 2685, 2677, 2775, 203, 2833, 63, 215}"
274,1,194,0,0.2695837,"\int \sqrt{e \cos (c+d x)} \sqrt{a+a \sin (c+d x)} \, dx","Int[Sqrt[e*Cos[c + d*x]]*Sqrt[a + a*Sin[c + d*x]],x]","-\frac{a (e \cos (c+d x))^{3/2}}{d e \sqrt{a \sin (c+d x)+a}}+\frac{\sqrt{e} \sqrt{\cos (c+d x)+1} \sqrt{a \sin (c+d x)+a} \tan ^{-1}\left(\frac{\sqrt{e} \sin (c+d x)}{\sqrt{\cos (c+d x)+1} \sqrt{e \cos (c+d x)}}\right)}{d (\sin (c+d x)+\cos (c+d x)+1)}+\frac{\sqrt{e} \sqrt{\cos (c+d x)+1} \sqrt{a \sin (c+d x)+a} \sinh ^{-1}\left(\frac{\sqrt{e \cos (c+d x)}}{\sqrt{e}}\right)}{d (\sin (c+d x)+\cos (c+d x)+1)}","-\frac{a (e \cos (c+d x))^{3/2}}{d e \sqrt{a \sin (c+d x)+a}}+\frac{\sqrt{e} \sqrt{\cos (c+d x)+1} \sqrt{a \sin (c+d x)+a} \tan ^{-1}\left(\frac{\sqrt{e} \sin (c+d x)}{\sqrt{\cos (c+d x)+1} \sqrt{e \cos (c+d x)}}\right)}{d (\sin (c+d x)+\cos (c+d x)+1)}+\frac{\sqrt{e} \sqrt{\cos (c+d x)+1} \sqrt{a \sin (c+d x)+a} \sinh ^{-1}\left(\frac{\sqrt{e \cos (c+d x)}}{\sqrt{e}}\right)}{d (\sin (c+d x)+\cos (c+d x)+1)}",1,"-((a*(e*Cos[c + d*x])^(3/2))/(d*e*Sqrt[a + a*Sin[c + d*x]])) + (Sqrt[e]*ArcSinh[Sqrt[e*Cos[c + d*x]]/Sqrt[e]]*Sqrt[1 + Cos[c + d*x]]*Sqrt[a + a*Sin[c + d*x]])/(d*(1 + Cos[c + d*x] + Sin[c + d*x])) + (Sqrt[e]*ArcTan[(Sqrt[e]*Sin[c + d*x])/(Sqrt[e*Cos[c + d*x]]*Sqrt[1 + Cos[c + d*x]])]*Sqrt[1 + Cos[c + d*x]]*Sqrt[a + a*Sin[c + d*x]])/(d*(1 + Cos[c + d*x] + Sin[c + d*x]))","A",7,7,27,0.2593,1,"{2678, 2684, 2775, 203, 2833, 63, 215}"
275,1,161,0,0.2091452,"\int \frac{\sqrt{a+a \sin (c+d x)}}{\sqrt{e \cos (c+d x)}} \, dx","Int[Sqrt[a + a*Sin[c + d*x]]/Sqrt[e*Cos[c + d*x]],x]","\frac{2 \sqrt{\cos (c+d x)+1} \sqrt{a \sin (c+d x)+a} \tan ^{-1}\left(\frac{\sqrt{e} \sin (c+d x)}{\sqrt{\cos (c+d x)+1} \sqrt{e \cos (c+d x)}}\right)}{d \sqrt{e} (\sin (c+d x)+\cos (c+d x)+1)}-\frac{2 \sqrt{\cos (c+d x)+1} \sqrt{a \sin (c+d x)+a} \sinh ^{-1}\left(\frac{\sqrt{e \cos (c+d x)}}{\sqrt{e}}\right)}{d \sqrt{e} (\sin (c+d x)+\cos (c+d x)+1)}","\frac{2 \sqrt{\cos (c+d x)+1} \sqrt{a \sin (c+d x)+a} \tan ^{-1}\left(\frac{\sqrt{e} \sin (c+d x)}{\sqrt{\cos (c+d x)+1} \sqrt{e \cos (c+d x)}}\right)}{d \sqrt{e} (\sin (c+d x)+\cos (c+d x)+1)}-\frac{2 \sqrt{\cos (c+d x)+1} \sqrt{a \sin (c+d x)+a} \sinh ^{-1}\left(\frac{\sqrt{e \cos (c+d x)}}{\sqrt{e}}\right)}{d \sqrt{e} (\sin (c+d x)+\cos (c+d x)+1)}",1,"(-2*ArcSinh[Sqrt[e*Cos[c + d*x]]/Sqrt[e]]*Sqrt[1 + Cos[c + d*x]]*Sqrt[a + a*Sin[c + d*x]])/(d*Sqrt[e]*(1 + Cos[c + d*x] + Sin[c + d*x])) + (2*ArcTan[(Sqrt[e]*Sin[c + d*x])/(Sqrt[e*Cos[c + d*x]]*Sqrt[1 + Cos[c + d*x]])]*Sqrt[1 + Cos[c + d*x]]*Sqrt[a + a*Sin[c + d*x]])/(d*Sqrt[e]*(1 + Cos[c + d*x] + Sin[c + d*x]))","A",6,6,27,0.2222,1,"{2677, 2775, 203, 2833, 63, 215}"
276,1,34,0,0.0681025,"\int \frac{\sqrt{a+a \sin (c+d x)}}{(e \cos (c+d x))^{3/2}} \, dx","Int[Sqrt[a + a*Sin[c + d*x]]/(e*Cos[c + d*x])^(3/2),x]","\frac{2 \sqrt{a \sin (c+d x)+a}}{d e \sqrt{e \cos (c+d x)}}","\frac{2 \sqrt{a \sin (c+d x)+a}}{d e \sqrt{e \cos (c+d x)}}",1,"(2*Sqrt[a + a*Sin[c + d*x]])/(d*e*Sqrt[e*Cos[c + d*x]])","A",1,1,27,0.03704,1,"{2671}"
277,1,74,0,0.1459516,"\int \frac{\sqrt{a+a \sin (c+d x)}}{(e \cos (c+d x))^{5/2}} \, dx","Int[Sqrt[a + a*Sin[c + d*x]]/(e*Cos[c + d*x])^(5/2),x]","\frac{4 (a \sin (c+d x)+a)^{3/2}}{3 a d e (e \cos (c+d x))^{3/2}}-\frac{2 \sqrt{a \sin (c+d x)+a}}{d e (e \cos (c+d x))^{3/2}}","\frac{4 (a \sin (c+d x)+a)^{3/2}}{3 a d e (e \cos (c+d x))^{3/2}}-\frac{2 \sqrt{a \sin (c+d x)+a}}{d e (e \cos (c+d x))^{3/2}}",1,"(-2*Sqrt[a + a*Sin[c + d*x]])/(d*e*(e*Cos[c + d*x])^(3/2)) + (4*(a + a*Sin[c + d*x])^(3/2))/(3*a*d*e*(e*Cos[c + d*x])^(3/2))","A",2,2,27,0.07407,1,"{2672, 2671}"
278,1,115,0,0.223558,"\int \frac{\sqrt{a+a \sin (c+d x)}}{(e \cos (c+d x))^{7/2}} \, dx","Int[Sqrt[a + a*Sin[c + d*x]]/(e*Cos[c + d*x])^(7/2),x]","-\frac{16 (a \sin (c+d x)+a)^{5/2}}{15 a^2 d e (e \cos (c+d x))^{5/2}}+\frac{8 (a \sin (c+d x)+a)^{3/2}}{3 a d e (e \cos (c+d x))^{5/2}}-\frac{2 \sqrt{a \sin (c+d x)+a}}{3 d e (e \cos (c+d x))^{5/2}}","-\frac{16 (a \sin (c+d x)+a)^{5/2}}{15 a^2 d e (e \cos (c+d x))^{5/2}}+\frac{8 (a \sin (c+d x)+a)^{3/2}}{3 a d e (e \cos (c+d x))^{5/2}}-\frac{2 \sqrt{a \sin (c+d x)+a}}{3 d e (e \cos (c+d x))^{5/2}}",1,"(-2*Sqrt[a + a*Sin[c + d*x]])/(3*d*e*(e*Cos[c + d*x])^(5/2)) + (8*(a + a*Sin[c + d*x])^(3/2))/(3*a*d*e*(e*Cos[c + d*x])^(5/2)) - (16*(a + a*Sin[c + d*x])^(5/2))/(15*a^2*d*e*(e*Cos[c + d*x])^(5/2))","A",3,2,27,0.07407,1,"{2672, 2671}"
279,1,154,0,0.3074863,"\int \frac{\sqrt{a+a \sin (c+d x)}}{(e \cos (c+d x))^{9/2}} \, dx","Int[Sqrt[a + a*Sin[c + d*x]]/(e*Cos[c + d*x])^(9/2),x]","-\frac{32 (a \sin (c+d x)+a)^{7/2}}{35 a^3 d e (e \cos (c+d x))^{7/2}}+\frac{16 (a \sin (c+d x)+a)^{5/2}}{5 a^2 d e (e \cos (c+d x))^{7/2}}-\frac{12 (a \sin (c+d x)+a)^{3/2}}{5 a d e (e \cos (c+d x))^{7/2}}-\frac{2 \sqrt{a \sin (c+d x)+a}}{5 d e (e \cos (c+d x))^{7/2}}","-\frac{32 (a \sin (c+d x)+a)^{7/2}}{35 a^3 d e (e \cos (c+d x))^{7/2}}+\frac{16 (a \sin (c+d x)+a)^{5/2}}{5 a^2 d e (e \cos (c+d x))^{7/2}}-\frac{12 (a \sin (c+d x)+a)^{3/2}}{5 a d e (e \cos (c+d x))^{7/2}}-\frac{2 \sqrt{a \sin (c+d x)+a}}{5 d e (e \cos (c+d x))^{7/2}}",1,"(-2*Sqrt[a + a*Sin[c + d*x]])/(5*d*e*(e*Cos[c + d*x])^(7/2)) - (12*(a + a*Sin[c + d*x])^(3/2))/(5*a*d*e*(e*Cos[c + d*x])^(7/2)) + (16*(a + a*Sin[c + d*x])^(5/2))/(5*a^2*d*e*(e*Cos[c + d*x])^(7/2)) - (32*(a + a*Sin[c + d*x])^(7/2))/(35*a^3*d*e*(e*Cos[c + d*x])^(7/2))","A",4,2,27,0.07407,1,"{2672, 2671}"
280,1,319,0,0.5642437,"\int (e \cos (c+d x))^{5/2} (a+a \sin (c+d x))^{3/2} \, dx","Int[(e*Cos[c + d*x])^(5/2)*(a + a*Sin[c + d*x])^(3/2),x]","-\frac{3 a^2 (e \cos (c+d x))^{7/2}}{8 d e \sqrt{a \sin (c+d x)+a}}-\frac{15 a^3 (e \cos (c+d x))^{7/2}}{32 d e (a \sin (c+d x)+a)^{3/2}}+\frac{15 a^2 e (e \cos (c+d x))^{3/2}}{64 d \sqrt{a \sin (c+d x)+a}}+\frac{45 a e^{5/2} \sqrt{\cos (c+d x)+1} \sqrt{a \sin (c+d x)+a} \tan ^{-1}\left(\frac{\sqrt{e} \sin (c+d x)}{\sqrt{\cos (c+d x)+1} \sqrt{e \cos (c+d x)}}\right)}{64 d (\sin (c+d x)+\cos (c+d x)+1)}+\frac{45 a e^{5/2} \sqrt{\cos (c+d x)+1} \sqrt{a \sin (c+d x)+a} \sinh ^{-1}\left(\frac{\sqrt{e \cos (c+d x)}}{\sqrt{e}}\right)}{64 d (\sin (c+d x)+\cos (c+d x)+1)}-\frac{a \sqrt{a \sin (c+d x)+a} (e \cos (c+d x))^{7/2}}{4 d e}","-\frac{3 a^2 (e \cos (c+d x))^{7/2}}{8 d e \sqrt{a \sin (c+d x)+a}}-\frac{15 a^3 (e \cos (c+d x))^{7/2}}{32 d e (a \sin (c+d x)+a)^{3/2}}+\frac{15 a^2 e (e \cos (c+d x))^{3/2}}{64 d \sqrt{a \sin (c+d x)+a}}+\frac{45 a e^{5/2} \sqrt{\cos (c+d x)+1} \sqrt{a \sin (c+d x)+a} \tan ^{-1}\left(\frac{\sqrt{e} \sin (c+d x)}{\sqrt{\cos (c+d x)+1} \sqrt{e \cos (c+d x)}}\right)}{64 d (\sin (c+d x)+\cos (c+d x)+1)}+\frac{45 a e^{5/2} \sqrt{\cos (c+d x)+1} \sqrt{a \sin (c+d x)+a} \sinh ^{-1}\left(\frac{\sqrt{e \cos (c+d x)}}{\sqrt{e}}\right)}{64 d (\sin (c+d x)+\cos (c+d x)+1)}-\frac{a \sqrt{a \sin (c+d x)+a} (e \cos (c+d x))^{7/2}}{4 d e}",1,"(-15*a^3*(e*Cos[c + d*x])^(7/2))/(32*d*e*(a + a*Sin[c + d*x])^(3/2)) + (15*a^2*e*(e*Cos[c + d*x])^(3/2))/(64*d*Sqrt[a + a*Sin[c + d*x]]) - (3*a^2*(e*Cos[c + d*x])^(7/2))/(8*d*e*Sqrt[a + a*Sin[c + d*x]]) - (a*(e*Cos[c + d*x])^(7/2)*Sqrt[a + a*Sin[c + d*x]])/(4*d*e) + (45*a*e^(5/2)*ArcSinh[Sqrt[e*Cos[c + d*x]]/Sqrt[e]]*Sqrt[1 + Cos[c + d*x]]*Sqrt[a + a*Sin[c + d*x]])/(64*d*(1 + Cos[c + d*x] + Sin[c + d*x])) + (45*a*e^(5/2)*ArcTan[(Sqrt[e]*Sin[c + d*x])/(Sqrt[e*Cos[c + d*x]]*Sqrt[1 + Cos[c + d*x]])]*Sqrt[1 + Cos[c + d*x]]*Sqrt[a + a*Sin[c + d*x]])/(64*d*(1 + Cos[c + d*x] + Sin[c + d*x]))","A",10,9,27,0.3333,1,"{2678, 2686, 2679, 2684, 2775, 203, 2833, 63, 215}"
281,1,278,0,0.4747272,"\int (e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^{3/2} \, dx","Int[(e*Cos[c + d*x])^(3/2)*(a + a*Sin[c + d*x])^(3/2),x]","-\frac{7 a^2 (e \cos (c+d x))^{5/2}}{12 d e \sqrt{a \sin (c+d x)+a}}+\frac{7 a e^{3/2} \sqrt{\cos (c+d x)+1} \sqrt{a \sin (c+d x)+a} \tan ^{-1}\left(\frac{\sqrt{e} \sin (c+d x)}{\sqrt{\cos (c+d x)+1} \sqrt{e \cos (c+d x)}}\right)}{8 d (\sin (c+d x)+\cos (c+d x)+1)}-\frac{7 a e^{3/2} \sqrt{\cos (c+d x)+1} \sqrt{a \sin (c+d x)+a} \sinh ^{-1}\left(\frac{\sqrt{e \cos (c+d x)}}{\sqrt{e}}\right)}{8 d (\sin (c+d x)+\cos (c+d x)+1)}-\frac{a \sqrt{a \sin (c+d x)+a} (e \cos (c+d x))^{5/2}}{3 d e}+\frac{7 a e \sqrt{a \sin (c+d x)+a} \sqrt{e \cos (c+d x)}}{8 d}","-\frac{7 a^2 (e \cos (c+d x))^{5/2}}{12 d e \sqrt{a \sin (c+d x)+a}}+\frac{7 a e^{3/2} \sqrt{\cos (c+d x)+1} \sqrt{a \sin (c+d x)+a} \tan ^{-1}\left(\frac{\sqrt{e} \sin (c+d x)}{\sqrt{\cos (c+d x)+1} \sqrt{e \cos (c+d x)}}\right)}{8 d (\sin (c+d x)+\cos (c+d x)+1)}-\frac{7 a e^{3/2} \sqrt{\cos (c+d x)+1} \sqrt{a \sin (c+d x)+a} \sinh ^{-1}\left(\frac{\sqrt{e \cos (c+d x)}}{\sqrt{e}}\right)}{8 d (\sin (c+d x)+\cos (c+d x)+1)}-\frac{a \sqrt{a \sin (c+d x)+a} (e \cos (c+d x))^{5/2}}{3 d e}+\frac{7 a e \sqrt{a \sin (c+d x)+a} \sqrt{e \cos (c+d x)}}{8 d}",1,"(-7*a^2*(e*Cos[c + d*x])^(5/2))/(12*d*e*Sqrt[a + a*Sin[c + d*x]]) + (7*a*e*Sqrt[e*Cos[c + d*x]]*Sqrt[a + a*Sin[c + d*x]])/(8*d) - (a*(e*Cos[c + d*x])^(5/2)*Sqrt[a + a*Sin[c + d*x]])/(3*d*e) - (7*a*e^(3/2)*ArcSinh[Sqrt[e*Cos[c + d*x]]/Sqrt[e]]*Sqrt[1 + Cos[c + d*x]]*Sqrt[a + a*Sin[c + d*x]])/(8*d*(1 + Cos[c + d*x] + Sin[c + d*x])) + (7*a*e^(3/2)*ArcTan[(Sqrt[e]*Sin[c + d*x])/(Sqrt[e*Cos[c + d*x]]*Sqrt[1 + Cos[c + d*x]])]*Sqrt[1 + Cos[c + d*x]]*Sqrt[a + a*Sin[c + d*x]])/(8*d*(1 + Cos[c + d*x] + Sin[c + d*x]))","A",9,8,27,0.2963,1,"{2678, 2685, 2677, 2775, 203, 2833, 63, 215}"
282,1,243,0,0.3583372,"\int \sqrt{e \cos (c+d x)} (a+a \sin (c+d x))^{3/2} \, dx","Int[Sqrt[e*Cos[c + d*x]]*(a + a*Sin[c + d*x])^(3/2),x]","-\frac{5 a^2 (e \cos (c+d x))^{3/2}}{4 d e \sqrt{a \sin (c+d x)+a}}-\frac{a \sqrt{a \sin (c+d x)+a} (e \cos (c+d x))^{3/2}}{2 d e}+\frac{5 a \sqrt{e} \sqrt{\cos (c+d x)+1} \sqrt{a \sin (c+d x)+a} \tan ^{-1}\left(\frac{\sqrt{e} \sin (c+d x)}{\sqrt{\cos (c+d x)+1} \sqrt{e \cos (c+d x)}}\right)}{4 d (\sin (c+d x)+\cos (c+d x)+1)}+\frac{5 a \sqrt{e} \sqrt{\cos (c+d x)+1} \sqrt{a \sin (c+d x)+a} \sinh ^{-1}\left(\frac{\sqrt{e \cos (c+d x)}}{\sqrt{e}}\right)}{4 d (\sin (c+d x)+\cos (c+d x)+1)}","-\frac{5 a^2 (e \cos (c+d x))^{3/2}}{4 d e \sqrt{a \sin (c+d x)+a}}-\frac{a \sqrt{a \sin (c+d x)+a} (e \cos (c+d x))^{3/2}}{2 d e}+\frac{5 a \sqrt{e} \sqrt{\cos (c+d x)+1} \sqrt{a \sin (c+d x)+a} \tan ^{-1}\left(\frac{\sqrt{e} \sin (c+d x)}{\sqrt{\cos (c+d x)+1} \sqrt{e \cos (c+d x)}}\right)}{4 d (\sin (c+d x)+\cos (c+d x)+1)}+\frac{5 a \sqrt{e} \sqrt{\cos (c+d x)+1} \sqrt{a \sin (c+d x)+a} \sinh ^{-1}\left(\frac{\sqrt{e \cos (c+d x)}}{\sqrt{e}}\right)}{4 d (\sin (c+d x)+\cos (c+d x)+1)}",1,"(-5*a^2*(e*Cos[c + d*x])^(3/2))/(4*d*e*Sqrt[a + a*Sin[c + d*x]]) - (a*(e*Cos[c + d*x])^(3/2)*Sqrt[a + a*Sin[c + d*x]])/(2*d*e) + (5*a*Sqrt[e]*ArcSinh[Sqrt[e*Cos[c + d*x]]/Sqrt[e]]*Sqrt[1 + Cos[c + d*x]]*Sqrt[a + a*Sin[c + d*x]])/(4*d*(1 + Cos[c + d*x] + Sin[c + d*x])) + (5*a*Sqrt[e]*ArcTan[(Sqrt[e]*Sin[c + d*x])/(Sqrt[e*Cos[c + d*x]]*Sqrt[1 + Cos[c + d*x]])]*Sqrt[1 + Cos[c + d*x]]*Sqrt[a + a*Sin[c + d*x]])/(4*d*(1 + Cos[c + d*x] + Sin[c + d*x]))","A",8,7,27,0.2593,1,"{2678, 2684, 2775, 203, 2833, 63, 215}"
283,1,198,0,0.2842021,"\int \frac{(a+a \sin (c+d x))^{3/2}}{\sqrt{e \cos (c+d x)}} \, dx","Int[(a + a*Sin[c + d*x])^(3/2)/Sqrt[e*Cos[c + d*x]],x]","-\frac{a \sqrt{a \sin (c+d x)+a} \sqrt{e \cos (c+d x)}}{d e}+\frac{3 a \sqrt{\cos (c+d x)+1} \sqrt{a \sin (c+d x)+a} \tan ^{-1}\left(\frac{\sqrt{e} \sin (c+d x)}{\sqrt{\cos (c+d x)+1} \sqrt{e \cos (c+d x)}}\right)}{d \sqrt{e} (\sin (c+d x)+\cos (c+d x)+1)}-\frac{3 a \sqrt{\cos (c+d x)+1} \sqrt{a \sin (c+d x)+a} \sinh ^{-1}\left(\frac{\sqrt{e \cos (c+d x)}}{\sqrt{e}}\right)}{d \sqrt{e} (\sin (c+d x)+\cos (c+d x)+1)}","-\frac{a \sqrt{a \sin (c+d x)+a} \sqrt{e \cos (c+d x)}}{d e}+\frac{3 a \sqrt{\cos (c+d x)+1} \sqrt{a \sin (c+d x)+a} \tan ^{-1}\left(\frac{\sqrt{e} \sin (c+d x)}{\sqrt{\cos (c+d x)+1} \sqrt{e \cos (c+d x)}}\right)}{d \sqrt{e} (\sin (c+d x)+\cos (c+d x)+1)}-\frac{3 a \sqrt{\cos (c+d x)+1} \sqrt{a \sin (c+d x)+a} \sinh ^{-1}\left(\frac{\sqrt{e \cos (c+d x)}}{\sqrt{e}}\right)}{d \sqrt{e} (\sin (c+d x)+\cos (c+d x)+1)}",1,"-((a*Sqrt[e*Cos[c + d*x]]*Sqrt[a + a*Sin[c + d*x]])/(d*e)) - (3*a*ArcSinh[Sqrt[e*Cos[c + d*x]]/Sqrt[e]]*Sqrt[1 + Cos[c + d*x]]*Sqrt[a + a*Sin[c + d*x]])/(d*Sqrt[e]*(1 + Cos[c + d*x] + Sin[c + d*x])) + (3*a*ArcTan[(Sqrt[e]*Sin[c + d*x])/(Sqrt[e*Cos[c + d*x]]*Sqrt[1 + Cos[c + d*x]])]*Sqrt[1 + Cos[c + d*x]]*Sqrt[a + a*Sin[c + d*x]])/(d*Sqrt[e]*(1 + Cos[c + d*x] + Sin[c + d*x]))","A",7,7,27,0.2593,1,"{2678, 2677, 2775, 203, 2833, 63, 215}"
284,1,210,0,0.2934318,"\int \frac{(a+a \sin (c+d x))^{3/2}}{(e \cos (c+d x))^{3/2}} \, dx","Int[(a + a*Sin[c + d*x])^(3/2)/(e*Cos[c + d*x])^(3/2),x]","-\frac{2 a^2 \sqrt{\cos (c+d x)+1} \sqrt{a \sin (c+d x)+a} \tan ^{-1}\left(\frac{\sqrt{e} \sin (c+d x)}{\sqrt{\cos (c+d x)+1} \sqrt{e \cos (c+d x)}}\right)}{d e^{3/2} (a \sin (c+d x)+a \cos (c+d x)+a)}-\frac{2 a^2 \sqrt{\cos (c+d x)+1} \sqrt{a \sin (c+d x)+a} \sinh ^{-1}\left(\frac{\sqrt{e \cos (c+d x)}}{\sqrt{e}}\right)}{d e^{3/2} (a \sin (c+d x)+a \cos (c+d x)+a)}+\frac{4 a \sqrt{a \sin (c+d x)+a}}{d e \sqrt{e \cos (c+d x)}}","-\frac{2 a^2 \sqrt{\cos (c+d x)+1} \sqrt{a \sin (c+d x)+a} \tan ^{-1}\left(\frac{\sqrt{e} \sin (c+d x)}{\sqrt{\cos (c+d x)+1} \sqrt{e \cos (c+d x)}}\right)}{d e^{3/2} (a \sin (c+d x)+a \cos (c+d x)+a)}-\frac{2 a^2 \sqrt{\cos (c+d x)+1} \sqrt{a \sin (c+d x)+a} \sinh ^{-1}\left(\frac{\sqrt{e \cos (c+d x)}}{\sqrt{e}}\right)}{d e^{3/2} (a \sin (c+d x)+a \cos (c+d x)+a)}+\frac{4 a \sqrt{a \sin (c+d x)+a}}{d e \sqrt{e \cos (c+d x)}}",1,"(4*a*Sqrt[a + a*Sin[c + d*x]])/(d*e*Sqrt[e*Cos[c + d*x]]) - (2*a^2*ArcSinh[Sqrt[e*Cos[c + d*x]]/Sqrt[e]]*Sqrt[1 + Cos[c + d*x]]*Sqrt[a + a*Sin[c + d*x]])/(d*e^(3/2)*(a + a*Cos[c + d*x] + a*Sin[c + d*x])) - (2*a^2*ArcTan[(Sqrt[e]*Sin[c + d*x])/(Sqrt[e*Cos[c + d*x]]*Sqrt[1 + Cos[c + d*x]])]*Sqrt[1 + Cos[c + d*x]]*Sqrt[a + a*Sin[c + d*x]])/(d*e^(3/2)*(a + a*Cos[c + d*x] + a*Sin[c + d*x]))","A",7,7,27,0.2593,1,"{2676, 2684, 2775, 203, 2833, 63, 215}"
285,1,36,0,0.0725385,"\int \frac{(a+a \sin (c+d x))^{3/2}}{(e \cos (c+d x))^{5/2}} \, dx","Int[(a + a*Sin[c + d*x])^(3/2)/(e*Cos[c + d*x])^(5/2),x]","\frac{2 (a \sin (c+d x)+a)^{3/2}}{3 d e (e \cos (c+d x))^{3/2}}","\frac{2 (a \sin (c+d x)+a)^{3/2}}{3 d e (e \cos (c+d x))^{3/2}}",1,"(2*(a + a*Sin[c + d*x])^(3/2))/(3*d*e*(e*Cos[c + d*x])^(3/2))","A",1,1,27,0.03704,1,"{2671}"
286,1,74,0,0.1480828,"\int \frac{(a+a \sin (c+d x))^{3/2}}{(e \cos (c+d x))^{7/2}} \, dx","Int[(a + a*Sin[c + d*x])^(3/2)/(e*Cos[c + d*x])^(7/2),x]","\frac{2 (a \sin (c+d x)+a)^{3/2}}{d e (e \cos (c+d x))^{5/2}}-\frac{4 (a \sin (c+d x)+a)^{5/2}}{5 a d e (e \cos (c+d x))^{5/2}}","\frac{2 (a \sin (c+d x)+a)^{3/2}}{d e (e \cos (c+d x))^{5/2}}-\frac{4 (a \sin (c+d x)+a)^{5/2}}{5 a d e (e \cos (c+d x))^{5/2}}",1,"(2*(a + a*Sin[c + d*x])^(3/2))/(d*e*(e*Cos[c + d*x])^(5/2)) - (4*(a + a*Sin[c + d*x])^(5/2))/(5*a*d*e*(e*Cos[c + d*x])^(5/2))","A",2,2,27,0.07407,1,"{2672, 2671}"
287,1,113,0,0.2299428,"\int \frac{(a+a \sin (c+d x))^{3/2}}{(e \cos (c+d x))^{9/2}} \, dx","Int[(a + a*Sin[c + d*x])^(3/2)/(e*Cos[c + d*x])^(9/2),x]","-\frac{16 (a \sin (c+d x)+a)^{7/2}}{21 a^2 d e (e \cos (c+d x))^{7/2}}+\frac{8 (a \sin (c+d x)+a)^{5/2}}{3 a d e (e \cos (c+d x))^{7/2}}-\frac{2 (a \sin (c+d x)+a)^{3/2}}{d e (e \cos (c+d x))^{7/2}}","-\frac{16 (a \sin (c+d x)+a)^{7/2}}{21 a^2 d e (e \cos (c+d x))^{7/2}}+\frac{8 (a \sin (c+d x)+a)^{5/2}}{3 a d e (e \cos (c+d x))^{7/2}}-\frac{2 (a \sin (c+d x)+a)^{3/2}}{d e (e \cos (c+d x))^{7/2}}",1,"(-2*(a + a*Sin[c + d*x])^(3/2))/(d*e*(e*Cos[c + d*x])^(7/2)) + (8*(a + a*Sin[c + d*x])^(5/2))/(3*a*d*e*(e*Cos[c + d*x])^(7/2)) - (16*(a + a*Sin[c + d*x])^(7/2))/(21*a^2*d*e*(e*Cos[c + d*x])^(7/2))","A",3,2,27,0.07407,1,"{2672, 2671}"
288,1,152,0,0.3088014,"\int \frac{(a+a \sin (c+d x))^{3/2}}{(e \cos (c+d x))^{11/2}} \, dx","Int[(a + a*Sin[c + d*x])^(3/2)/(e*Cos[c + d*x])^(11/2),x]","\frac{32 (a \sin (c+d x)+a)^{9/2}}{45 a^3 d e (e \cos (c+d x))^{9/2}}-\frac{16 (a \sin (c+d x)+a)^{7/2}}{5 a^2 d e (e \cos (c+d x))^{9/2}}+\frac{4 (a \sin (c+d x)+a)^{5/2}}{a d e (e \cos (c+d x))^{9/2}}-\frac{2 (a \sin (c+d x)+a)^{3/2}}{3 d e (e \cos (c+d x))^{9/2}}","\frac{32 (a \sin (c+d x)+a)^{9/2}}{45 a^3 d e (e \cos (c+d x))^{9/2}}-\frac{16 (a \sin (c+d x)+a)^{7/2}}{5 a^2 d e (e \cos (c+d x))^{9/2}}+\frac{4 (a \sin (c+d x)+a)^{5/2}}{a d e (e \cos (c+d x))^{9/2}}-\frac{2 (a \sin (c+d x)+a)^{3/2}}{3 d e (e \cos (c+d x))^{9/2}}",1,"(-2*(a + a*Sin[c + d*x])^(3/2))/(3*d*e*(e*Cos[c + d*x])^(9/2)) + (4*(a + a*Sin[c + d*x])^(5/2))/(a*d*e*(e*Cos[c + d*x])^(9/2)) - (16*(a + a*Sin[c + d*x])^(7/2))/(5*a^2*d*e*(e*Cos[c + d*x])^(9/2)) + (32*(a + a*Sin[c + d*x])^(9/2))/(45*a^3*d*e*(e*Cos[c + d*x])^(9/2))","A",4,2,27,0.07407,1,"{2672, 2671}"
289,1,323,0,0.5430579,"\int (e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^{5/2} \, dx","Int[(e*Cos[c + d*x])^(3/2)*(a + a*Sin[c + d*x])^(5/2),x]","\frac{77 a^2 e^{3/2} \sqrt{\cos (c+d x)+1} \sqrt{a \sin (c+d x)+a} \tan ^{-1}\left(\frac{\sqrt{e} \sin (c+d x)}{\sqrt{\cos (c+d x)+1} \sqrt{e \cos (c+d x)}}\right)}{64 d (\sin (c+d x)+\cos (c+d x)+1)}-\frac{77 a^2 e^{3/2} \sqrt{\cos (c+d x)+1} \sqrt{a \sin (c+d x)+a} \sinh ^{-1}\left(\frac{\sqrt{e \cos (c+d x)}}{\sqrt{e}}\right)}{64 d (\sin (c+d x)+\cos (c+d x)+1)}-\frac{77 a^3 (e \cos (c+d x))^{5/2}}{96 d e \sqrt{a \sin (c+d x)+a}}-\frac{11 a^2 \sqrt{a \sin (c+d x)+a} (e \cos (c+d x))^{5/2}}{24 d e}+\frac{77 a^2 e \sqrt{a \sin (c+d x)+a} \sqrt{e \cos (c+d x)}}{64 d}-\frac{a (a \sin (c+d x)+a)^{3/2} (e \cos (c+d x))^{5/2}}{4 d e}","\frac{77 a^2 e^{3/2} \sqrt{\cos (c+d x)+1} \sqrt{a \sin (c+d x)+a} \tan ^{-1}\left(\frac{\sqrt{e} \sin (c+d x)}{\sqrt{\cos (c+d x)+1} \sqrt{e \cos (c+d x)}}\right)}{64 d (\sin (c+d x)+\cos (c+d x)+1)}-\frac{77 a^2 e^{3/2} \sqrt{\cos (c+d x)+1} \sqrt{a \sin (c+d x)+a} \sinh ^{-1}\left(\frac{\sqrt{e \cos (c+d x)}}{\sqrt{e}}\right)}{64 d (\sin (c+d x)+\cos (c+d x)+1)}-\frac{77 a^3 (e \cos (c+d x))^{5/2}}{96 d e \sqrt{a \sin (c+d x)+a}}-\frac{11 a^2 \sqrt{a \sin (c+d x)+a} (e \cos (c+d x))^{5/2}}{24 d e}+\frac{77 a^2 e \sqrt{a \sin (c+d x)+a} \sqrt{e \cos (c+d x)}}{64 d}-\frac{a (a \sin (c+d x)+a)^{3/2} (e \cos (c+d x))^{5/2}}{4 d e}",1,"(-77*a^3*(e*Cos[c + d*x])^(5/2))/(96*d*e*Sqrt[a + a*Sin[c + d*x]]) + (77*a^2*e*Sqrt[e*Cos[c + d*x]]*Sqrt[a + a*Sin[c + d*x]])/(64*d) - (11*a^2*(e*Cos[c + d*x])^(5/2)*Sqrt[a + a*Sin[c + d*x]])/(24*d*e) - (77*a^2*e^(3/2)*ArcSinh[Sqrt[e*Cos[c + d*x]]/Sqrt[e]]*Sqrt[1 + Cos[c + d*x]]*Sqrt[a + a*Sin[c + d*x]])/(64*d*(1 + Cos[c + d*x] + Sin[c + d*x])) + (77*a^2*e^(3/2)*ArcTan[(Sqrt[e]*Sin[c + d*x])/(Sqrt[e*Cos[c + d*x]]*Sqrt[1 + Cos[c + d*x]])]*Sqrt[1 + Cos[c + d*x]]*Sqrt[a + a*Sin[c + d*x]])/(64*d*(1 + Cos[c + d*x] + Sin[c + d*x])) - (a*(e*Cos[c + d*x])^(5/2)*(a + a*Sin[c + d*x])^(3/2))/(4*d*e)","A",10,8,27,0.2963,1,"{2678, 2685, 2677, 2775, 203, 2833, 63, 215}"
290,1,286,0,0.4365062,"\int \sqrt{e \cos (c+d x)} (a+a \sin (c+d x))^{5/2} \, dx","Int[Sqrt[e*Cos[c + d*x]]*(a + a*Sin[c + d*x])^(5/2),x]","-\frac{15 a^3 (e \cos (c+d x))^{3/2}}{8 d e \sqrt{a \sin (c+d x)+a}}-\frac{3 a^2 \sqrt{a \sin (c+d x)+a} (e \cos (c+d x))^{3/2}}{4 d e}+\frac{15 a^2 \sqrt{e} \sqrt{\cos (c+d x)+1} \sqrt{a \sin (c+d x)+a} \tan ^{-1}\left(\frac{\sqrt{e} \sin (c+d x)}{\sqrt{\cos (c+d x)+1} \sqrt{e \cos (c+d x)}}\right)}{8 d (\sin (c+d x)+\cos (c+d x)+1)}+\frac{15 a^2 \sqrt{e} \sqrt{\cos (c+d x)+1} \sqrt{a \sin (c+d x)+a} \sinh ^{-1}\left(\frac{\sqrt{e \cos (c+d x)}}{\sqrt{e}}\right)}{8 d (\sin (c+d x)+\cos (c+d x)+1)}-\frac{a (a \sin (c+d x)+a)^{3/2} (e \cos (c+d x))^{3/2}}{3 d e}","-\frac{15 a^3 (e \cos (c+d x))^{3/2}}{8 d e \sqrt{a \sin (c+d x)+a}}-\frac{3 a^2 \sqrt{a \sin (c+d x)+a} (e \cos (c+d x))^{3/2}}{4 d e}+\frac{15 a^2 \sqrt{e} \sqrt{\cos (c+d x)+1} \sqrt{a \sin (c+d x)+a} \tan ^{-1}\left(\frac{\sqrt{e} \sin (c+d x)}{\sqrt{\cos (c+d x)+1} \sqrt{e \cos (c+d x)}}\right)}{8 d (\sin (c+d x)+\cos (c+d x)+1)}+\frac{15 a^2 \sqrt{e} \sqrt{\cos (c+d x)+1} \sqrt{a \sin (c+d x)+a} \sinh ^{-1}\left(\frac{\sqrt{e \cos (c+d x)}}{\sqrt{e}}\right)}{8 d (\sin (c+d x)+\cos (c+d x)+1)}-\frac{a (a \sin (c+d x)+a)^{3/2} (e \cos (c+d x))^{3/2}}{3 d e}",1,"(-15*a^3*(e*Cos[c + d*x])^(3/2))/(8*d*e*Sqrt[a + a*Sin[c + d*x]]) - (3*a^2*(e*Cos[c + d*x])^(3/2)*Sqrt[a + a*Sin[c + d*x]])/(4*d*e) + (15*a^2*Sqrt[e]*ArcSinh[Sqrt[e*Cos[c + d*x]]/Sqrt[e]]*Sqrt[1 + Cos[c + d*x]]*Sqrt[a + a*Sin[c + d*x]])/(8*d*(1 + Cos[c + d*x] + Sin[c + d*x])) + (15*a^2*Sqrt[e]*ArcTan[(Sqrt[e]*Sin[c + d*x])/(Sqrt[e*Cos[c + d*x]]*Sqrt[1 + Cos[c + d*x]])]*Sqrt[1 + Cos[c + d*x]]*Sqrt[a + a*Sin[c + d*x]])/(8*d*(1 + Cos[c + d*x] + Sin[c + d*x])) - (a*(e*Cos[c + d*x])^(3/2)*(a + a*Sin[c + d*x])^(3/2))/(3*d*e)","A",9,7,27,0.2593,1,"{2678, 2684, 2775, 203, 2833, 63, 215}"
291,1,247,0,0.3586914,"\int \frac{(a+a \sin (c+d x))^{5/2}}{\sqrt{e \cos (c+d x)}} \, dx","Int[(a + a*Sin[c + d*x])^(5/2)/Sqrt[e*Cos[c + d*x]],x]","-\frac{7 a^2 \sqrt{a \sin (c+d x)+a} \sqrt{e \cos (c+d x)}}{4 d e}+\frac{21 a^2 \sqrt{\cos (c+d x)+1} \sqrt{a \sin (c+d x)+a} \tan ^{-1}\left(\frac{\sqrt{e} \sin (c+d x)}{\sqrt{\cos (c+d x)+1} \sqrt{e \cos (c+d x)}}\right)}{4 d \sqrt{e} (\sin (c+d x)+\cos (c+d x)+1)}-\frac{21 a^2 \sqrt{\cos (c+d x)+1} \sqrt{a \sin (c+d x)+a} \sinh ^{-1}\left(\frac{\sqrt{e \cos (c+d x)}}{\sqrt{e}}\right)}{4 d \sqrt{e} (\sin (c+d x)+\cos (c+d x)+1)}-\frac{a (a \sin (c+d x)+a)^{3/2} \sqrt{e \cos (c+d x)}}{2 d e}","-\frac{7 a^2 \sqrt{a \sin (c+d x)+a} \sqrt{e \cos (c+d x)}}{4 d e}+\frac{21 a^2 \sqrt{\cos (c+d x)+1} \sqrt{a \sin (c+d x)+a} \tan ^{-1}\left(\frac{\sqrt{e} \sin (c+d x)}{\sqrt{\cos (c+d x)+1} \sqrt{e \cos (c+d x)}}\right)}{4 d \sqrt{e} (\sin (c+d x)+\cos (c+d x)+1)}-\frac{21 a^2 \sqrt{\cos (c+d x)+1} \sqrt{a \sin (c+d x)+a} \sinh ^{-1}\left(\frac{\sqrt{e \cos (c+d x)}}{\sqrt{e}}\right)}{4 d \sqrt{e} (\sin (c+d x)+\cos (c+d x)+1)}-\frac{a (a \sin (c+d x)+a)^{3/2} \sqrt{e \cos (c+d x)}}{2 d e}",1,"(-7*a^2*Sqrt[e*Cos[c + d*x]]*Sqrt[a + a*Sin[c + d*x]])/(4*d*e) - (21*a^2*ArcSinh[Sqrt[e*Cos[c + d*x]]/Sqrt[e]]*Sqrt[1 + Cos[c + d*x]]*Sqrt[a + a*Sin[c + d*x]])/(4*d*Sqrt[e]*(1 + Cos[c + d*x] + Sin[c + d*x])) + (21*a^2*ArcTan[(Sqrt[e]*Sin[c + d*x])/(Sqrt[e*Cos[c + d*x]]*Sqrt[1 + Cos[c + d*x]])]*Sqrt[1 + Cos[c + d*x]]*Sqrt[a + a*Sin[c + d*x]])/(4*d*Sqrt[e]*(1 + Cos[c + d*x] + Sin[c + d*x])) - (a*Sqrt[e*Cos[c + d*x]]*(a + a*Sin[c + d*x])^(3/2))/(2*d*e)","A",8,7,27,0.2593,1,"{2678, 2677, 2775, 203, 2833, 63, 215}"
292,1,239,0,0.3643324,"\int \frac{(a+a \sin (c+d x))^{5/2}}{(e \cos (c+d x))^{3/2}} \, dx","Int[(a + a*Sin[c + d*x])^(5/2)/(e*Cos[c + d*x])^(3/2),x]","\frac{5 a^3 (e \cos (c+d x))^{3/2}}{d e^3 \sqrt{a \sin (c+d x)+a}}-\frac{5 a^2 \sqrt{\cos (c+d x)+1} \sqrt{a \sin (c+d x)+a} \tan ^{-1}\left(\frac{\sqrt{e} \sin (c+d x)}{\sqrt{\cos (c+d x)+1} \sqrt{e \cos (c+d x)}}\right)}{d e^{3/2} (\sin (c+d x)+\cos (c+d x)+1)}-\frac{5 a^2 \sqrt{\cos (c+d x)+1} \sqrt{a \sin (c+d x)+a} \sinh ^{-1}\left(\frac{\sqrt{e \cos (c+d x)}}{\sqrt{e}}\right)}{d e^{3/2} (\sin (c+d x)+\cos (c+d x)+1)}+\frac{4 a (a \sin (c+d x)+a)^{3/2}}{d e \sqrt{e \cos (c+d x)}}","\frac{5 a^3 (e \cos (c+d x))^{3/2}}{d e^3 \sqrt{a \sin (c+d x)+a}}-\frac{5 a^2 \sqrt{\cos (c+d x)+1} \sqrt{a \sin (c+d x)+a} \tan ^{-1}\left(\frac{\sqrt{e} \sin (c+d x)}{\sqrt{\cos (c+d x)+1} \sqrt{e \cos (c+d x)}}\right)}{d e^{3/2} (\sin (c+d x)+\cos (c+d x)+1)}-\frac{5 a^2 \sqrt{\cos (c+d x)+1} \sqrt{a \sin (c+d x)+a} \sinh ^{-1}\left(\frac{\sqrt{e \cos (c+d x)}}{\sqrt{e}}\right)}{d e^{3/2} (\sin (c+d x)+\cos (c+d x)+1)}+\frac{4 a (a \sin (c+d x)+a)^{3/2}}{d e \sqrt{e \cos (c+d x)}}",1,"(5*a^3*(e*Cos[c + d*x])^(3/2))/(d*e^3*Sqrt[a + a*Sin[c + d*x]]) - (5*a^2*ArcSinh[Sqrt[e*Cos[c + d*x]]/Sqrt[e]]*Sqrt[1 + Cos[c + d*x]]*Sqrt[a + a*Sin[c + d*x]])/(d*e^(3/2)*(1 + Cos[c + d*x] + Sin[c + d*x])) - (5*a^2*ArcTan[(Sqrt[e]*Sin[c + d*x])/(Sqrt[e*Cos[c + d*x]]*Sqrt[1 + Cos[c + d*x]])]*Sqrt[1 + Cos[c + d*x]]*Sqrt[a + a*Sin[c + d*x]])/(d*e^(3/2)*(1 + Cos[c + d*x] + Sin[c + d*x])) + (4*a*(a + a*Sin[c + d*x])^(3/2))/(d*e*Sqrt[e*Cos[c + d*x]])","A",8,8,27,0.2963,1,"{2676, 2678, 2684, 2775, 203, 2833, 63, 215}"
293,1,204,0,0.2960198,"\int \frac{(a+a \sin (c+d x))^{5/2}}{(e \cos (c+d x))^{5/2}} \, dx","Int[(a + a*Sin[c + d*x])^(5/2)/(e*Cos[c + d*x])^(5/2),x]","-\frac{2 a^2 \sqrt{\cos (c+d x)+1} \sqrt{a \sin (c+d x)+a} \tan ^{-1}\left(\frac{\sqrt{e} \sin (c+d x)}{\sqrt{\cos (c+d x)+1} \sqrt{e \cos (c+d x)}}\right)}{d e^{5/2} (\sin (c+d x)+\cos (c+d x)+1)}+\frac{2 a^2 \sqrt{\cos (c+d x)+1} \sqrt{a \sin (c+d x)+a} \sinh ^{-1}\left(\frac{\sqrt{e \cos (c+d x)}}{\sqrt{e}}\right)}{d e^{5/2} (\sin (c+d x)+\cos (c+d x)+1)}+\frac{4 a (a \sin (c+d x)+a)^{3/2}}{3 d e (e \cos (c+d x))^{3/2}}","-\frac{2 a^2 \sqrt{\cos (c+d x)+1} \sqrt{a \sin (c+d x)+a} \tan ^{-1}\left(\frac{\sqrt{e} \sin (c+d x)}{\sqrt{\cos (c+d x)+1} \sqrt{e \cos (c+d x)}}\right)}{d e^{5/2} (\sin (c+d x)+\cos (c+d x)+1)}+\frac{2 a^2 \sqrt{\cos (c+d x)+1} \sqrt{a \sin (c+d x)+a} \sinh ^{-1}\left(\frac{\sqrt{e \cos (c+d x)}}{\sqrt{e}}\right)}{d e^{5/2} (\sin (c+d x)+\cos (c+d x)+1)}+\frac{4 a (a \sin (c+d x)+a)^{3/2}}{3 d e (e \cos (c+d x))^{3/2}}",1,"(2*a^2*ArcSinh[Sqrt[e*Cos[c + d*x]]/Sqrt[e]]*Sqrt[1 + Cos[c + d*x]]*Sqrt[a + a*Sin[c + d*x]])/(d*e^(5/2)*(1 + Cos[c + d*x] + Sin[c + d*x])) - (2*a^2*ArcTan[(Sqrt[e]*Sin[c + d*x])/(Sqrt[e*Cos[c + d*x]]*Sqrt[1 + Cos[c + d*x]])]*Sqrt[1 + Cos[c + d*x]]*Sqrt[a + a*Sin[c + d*x]])/(d*e^(5/2)*(1 + Cos[c + d*x] + Sin[c + d*x])) + (4*a*(a + a*Sin[c + d*x])^(3/2))/(3*d*e*(e*Cos[c + d*x])^(3/2))","A",7,7,27,0.2593,1,"{2676, 2677, 2775, 203, 2833, 63, 215}"
294,1,36,0,0.0746067,"\int \frac{(a+a \sin (c+d x))^{5/2}}{(e \cos (c+d x))^{7/2}} \, dx","Int[(a + a*Sin[c + d*x])^(5/2)/(e*Cos[c + d*x])^(7/2),x]","\frac{2 (a \sin (c+d x)+a)^{5/2}}{5 d e (e \cos (c+d x))^{5/2}}","\frac{2 (a \sin (c+d x)+a)^{5/2}}{5 d e (e \cos (c+d x))^{5/2}}",1,"(2*(a + a*Sin[c + d*x])^(5/2))/(5*d*e*(e*Cos[c + d*x])^(5/2))","A",1,1,27,0.03704,1,"{2671}"
295,1,76,0,0.1484374,"\int \frac{(a+a \sin (c+d x))^{5/2}}{(e \cos (c+d x))^{9/2}} \, dx","Int[(a + a*Sin[c + d*x])^(5/2)/(e*Cos[c + d*x])^(9/2),x]","\frac{2 (a \sin (c+d x)+a)^{5/2}}{3 d e (e \cos (c+d x))^{7/2}}-\frac{4 (a \sin (c+d x)+a)^{7/2}}{21 a d e (e \cos (c+d x))^{7/2}}","\frac{2 (a \sin (c+d x)+a)^{5/2}}{3 d e (e \cos (c+d x))^{7/2}}-\frac{4 (a \sin (c+d x)+a)^{7/2}}{21 a d e (e \cos (c+d x))^{7/2}}",1,"(2*(a + a*Sin[c + d*x])^(5/2))/(3*d*e*(e*Cos[c + d*x])^(7/2)) - (4*(a + a*Sin[c + d*x])^(7/2))/(21*a*d*e*(e*Cos[c + d*x])^(7/2))","A",2,2,27,0.07407,1,"{2672, 2671}"
296,1,113,0,0.2233808,"\int \frac{(a+a \sin (c+d x))^{5/2}}{(e \cos (c+d x))^{11/2}} \, dx","Int[(a + a*Sin[c + d*x])^(5/2)/(e*Cos[c + d*x])^(11/2),x]","\frac{16 (a \sin (c+d x)+a)^{9/2}}{45 a^2 d e (e \cos (c+d x))^{9/2}}-\frac{8 (a \sin (c+d x)+a)^{7/2}}{5 a d e (e \cos (c+d x))^{9/2}}+\frac{2 (a \sin (c+d x)+a)^{5/2}}{d e (e \cos (c+d x))^{9/2}}","\frac{16 (a \sin (c+d x)+a)^{9/2}}{45 a^2 d e (e \cos (c+d x))^{9/2}}-\frac{8 (a \sin (c+d x)+a)^{7/2}}{5 a d e (e \cos (c+d x))^{9/2}}+\frac{2 (a \sin (c+d x)+a)^{5/2}}{d e (e \cos (c+d x))^{9/2}}",1,"(2*(a + a*Sin[c + d*x])^(5/2))/(d*e*(e*Cos[c + d*x])^(9/2)) - (8*(a + a*Sin[c + d*x])^(7/2))/(5*a*d*e*(e*Cos[c + d*x])^(9/2)) + (16*(a + a*Sin[c + d*x])^(9/2))/(45*a^2*d*e*(e*Cos[c + d*x])^(9/2))","A",3,2,27,0.07407,1,"{2672, 2671}"
297,1,150,0,0.3051979,"\int \frac{(a+a \sin (c+d x))^{5/2}}{(e \cos (c+d x))^{13/2}} \, dx","Int[(a + a*Sin[c + d*x])^(5/2)/(e*Cos[c + d*x])^(13/2),x]","\frac{32 (a \sin (c+d x)+a)^{11/2}}{77 a^3 d e (e \cos (c+d x))^{11/2}}-\frac{16 (a \sin (c+d x)+a)^{9/2}}{7 a^2 d e (e \cos (c+d x))^{11/2}}+\frac{4 (a \sin (c+d x)+a)^{7/2}}{a d e (e \cos (c+d x))^{11/2}}-\frac{2 (a \sin (c+d x)+a)^{5/2}}{d e (e \cos (c+d x))^{11/2}}","\frac{32 (a \sin (c+d x)+a)^{11/2}}{77 a^3 d e (e \cos (c+d x))^{11/2}}-\frac{16 (a \sin (c+d x)+a)^{9/2}}{7 a^2 d e (e \cos (c+d x))^{11/2}}+\frac{4 (a \sin (c+d x)+a)^{7/2}}{a d e (e \cos (c+d x))^{11/2}}-\frac{2 (a \sin (c+d x)+a)^{5/2}}{d e (e \cos (c+d x))^{11/2}}",1,"(-2*(a + a*Sin[c + d*x])^(5/2))/(d*e*(e*Cos[c + d*x])^(11/2)) + (4*(a + a*Sin[c + d*x])^(7/2))/(a*d*e*(e*Cos[c + d*x])^(11/2)) - (16*(a + a*Sin[c + d*x])^(9/2))/(7*a^2*d*e*(e*Cos[c + d*x])^(11/2)) + (32*(a + a*Sin[c + d*x])^(11/2))/(77*a^3*d*e*(e*Cos[c + d*x])^(11/2))","A",4,2,27,0.07407,1,"{2672, 2671}"
298,1,244,0,0.3666955,"\int \frac{(e \cos (c+d x))^{5/2}}{\sqrt{a+a \sin (c+d x)}} \, dx","Int[(e*Cos[c + d*x])^(5/2)/Sqrt[a + a*Sin[c + d*x]],x]","\frac{3 e^{5/2} \sqrt{\cos (c+d x)+1} \sqrt{a \sin (c+d x)+a} \tan ^{-1}\left(\frac{\sqrt{e} \sin (c+d x)}{\sqrt{\cos (c+d x)+1} \sqrt{e \cos (c+d x)}}\right)}{4 d (a \sin (c+d x)+a \cos (c+d x)+a)}+\frac{3 e^{5/2} \sqrt{\cos (c+d x)+1} \sqrt{a \sin (c+d x)+a} \sinh ^{-1}\left(\frac{\sqrt{e \cos (c+d x)}}{\sqrt{e}}\right)}{4 d (a \sin (c+d x)+a \cos (c+d x)+a)}-\frac{a (e \cos (c+d x))^{7/2}}{2 d e (a \sin (c+d x)+a)^{3/2}}+\frac{e (e \cos (c+d x))^{3/2}}{4 d \sqrt{a \sin (c+d x)+a}}","\frac{3 e^{5/2} \sqrt{\cos (c+d x)+1} \sqrt{a \sin (c+d x)+a} \tan ^{-1}\left(\frac{\sqrt{e} \sin (c+d x)}{\sqrt{\cos (c+d x)+1} \sqrt{e \cos (c+d x)}}\right)}{4 d (a \sin (c+d x)+a \cos (c+d x)+a)}+\frac{3 e^{5/2} \sqrt{\cos (c+d x)+1} \sqrt{a \sin (c+d x)+a} \sinh ^{-1}\left(\frac{\sqrt{e \cos (c+d x)}}{\sqrt{e}}\right)}{4 d (a \sin (c+d x)+a \cos (c+d x)+a)}-\frac{a (e \cos (c+d x))^{7/2}}{2 d e (a \sin (c+d x)+a)^{3/2}}+\frac{e (e \cos (c+d x))^{3/2}}{4 d \sqrt{a \sin (c+d x)+a}}",1,"-(a*(e*Cos[c + d*x])^(7/2))/(2*d*e*(a + a*Sin[c + d*x])^(3/2)) + (e*(e*Cos[c + d*x])^(3/2))/(4*d*Sqrt[a + a*Sin[c + d*x]]) + (3*e^(5/2)*ArcSinh[Sqrt[e*Cos[c + d*x]]/Sqrt[e]]*Sqrt[1 + Cos[c + d*x]]*Sqrt[a + a*Sin[c + d*x]])/(4*d*(a + a*Cos[c + d*x] + a*Sin[c + d*x])) + (3*e^(5/2)*ArcTan[(Sqrt[e]*Sin[c + d*x])/(Sqrt[e*Cos[c + d*x]]*Sqrt[1 + Cos[c + d*x]])]*Sqrt[1 + Cos[c + d*x]]*Sqrt[a + a*Sin[c + d*x]])/(4*d*(a + a*Cos[c + d*x] + a*Sin[c + d*x]))","A",8,8,27,0.2963,1,"{2686, 2679, 2684, 2775, 203, 2833, 63, 215}"
299,1,200,0,0.2787854,"\int \frac{(e \cos (c+d x))^{3/2}}{\sqrt{a+a \sin (c+d x)}} \, dx","Int[(e*Cos[c + d*x])^(3/2)/Sqrt[a + a*Sin[c + d*x]],x]","\frac{e^{3/2} \sqrt{\cos (c+d x)+1} \sqrt{a \sin (c+d x)+a} \tan ^{-1}\left(\frac{\sqrt{e} \sin (c+d x)}{\sqrt{\cos (c+d x)+1} \sqrt{e \cos (c+d x)}}\right)}{a d (\sin (c+d x)+\cos (c+d x)+1)}-\frac{e^{3/2} \sqrt{\cos (c+d x)+1} \sqrt{a \sin (c+d x)+a} \sinh ^{-1}\left(\frac{\sqrt{e \cos (c+d x)}}{\sqrt{e}}\right)}{a d (\sin (c+d x)+\cos (c+d x)+1)}+\frac{e \sqrt{a \sin (c+d x)+a} \sqrt{e \cos (c+d x)}}{a d}","\frac{e^{3/2} \sqrt{\cos (c+d x)+1} \sqrt{a \sin (c+d x)+a} \tan ^{-1}\left(\frac{\sqrt{e} \sin (c+d x)}{\sqrt{\cos (c+d x)+1} \sqrt{e \cos (c+d x)}}\right)}{a d (\sin (c+d x)+\cos (c+d x)+1)}-\frac{e^{3/2} \sqrt{\cos (c+d x)+1} \sqrt{a \sin (c+d x)+a} \sinh ^{-1}\left(\frac{\sqrt{e \cos (c+d x)}}{\sqrt{e}}\right)}{a d (\sin (c+d x)+\cos (c+d x)+1)}+\frac{e \sqrt{a \sin (c+d x)+a} \sqrt{e \cos (c+d x)}}{a d}",1,"(e*Sqrt[e*Cos[c + d*x]]*Sqrt[a + a*Sin[c + d*x]])/(a*d) - (e^(3/2)*ArcSinh[Sqrt[e*Cos[c + d*x]]/Sqrt[e]]*Sqrt[1 + Cos[c + d*x]]*Sqrt[a + a*Sin[c + d*x]])/(a*d*(1 + Cos[c + d*x] + Sin[c + d*x])) + (e^(3/2)*ArcTan[(Sqrt[e]*Sin[c + d*x])/(Sqrt[e*Cos[c + d*x]]*Sqrt[1 + Cos[c + d*x]])]*Sqrt[1 + Cos[c + d*x]]*Sqrt[a + a*Sin[c + d*x]])/(a*d*(1 + Cos[c + d*x] + Sin[c + d*x]))","A",7,7,27,0.2593,1,"{2685, 2677, 2775, 203, 2833, 63, 215}"
300,1,169,0,0.1962347,"\int \frac{\sqrt{e \cos (c+d x)}}{\sqrt{a+a \sin (c+d x)}} \, dx","Int[Sqrt[e*Cos[c + d*x]]/Sqrt[a + a*Sin[c + d*x]],x]","\frac{2 \sqrt{e} \sqrt{\cos (c+d x)+1} \sqrt{a \sin (c+d x)+a} \tan ^{-1}\left(\frac{\sqrt{e} \sin (c+d x)}{\sqrt{\cos (c+d x)+1} \sqrt{e \cos (c+d x)}}\right)}{d (a \sin (c+d x)+a \cos (c+d x)+a)}+\frac{2 \sqrt{e} \sqrt{\cos (c+d x)+1} \sqrt{a \sin (c+d x)+a} \sinh ^{-1}\left(\frac{\sqrt{e \cos (c+d x)}}{\sqrt{e}}\right)}{d (a \sin (c+d x)+a \cos (c+d x)+a)}","\frac{2 \sqrt{e} \sqrt{\cos (c+d x)+1} \sqrt{a \sin (c+d x)+a} \tan ^{-1}\left(\frac{\sqrt{e} \sin (c+d x)}{\sqrt{\cos (c+d x)+1} \sqrt{e \cos (c+d x)}}\right)}{d (a \sin (c+d x)+a \cos (c+d x)+a)}+\frac{2 \sqrt{e} \sqrt{\cos (c+d x)+1} \sqrt{a \sin (c+d x)+a} \sinh ^{-1}\left(\frac{\sqrt{e \cos (c+d x)}}{\sqrt{e}}\right)}{d (a \sin (c+d x)+a \cos (c+d x)+a)}",1,"(2*Sqrt[e]*ArcSinh[Sqrt[e*Cos[c + d*x]]/Sqrt[e]]*Sqrt[1 + Cos[c + d*x]]*Sqrt[a + a*Sin[c + d*x]])/(d*(a + a*Cos[c + d*x] + a*Sin[c + d*x])) + (2*Sqrt[e]*ArcTan[(Sqrt[e]*Sin[c + d*x])/(Sqrt[e*Cos[c + d*x]]*Sqrt[1 + Cos[c + d*x]])]*Sqrt[1 + Cos[c + d*x]]*Sqrt[a + a*Sin[c + d*x]])/(d*(a + a*Cos[c + d*x] + a*Sin[c + d*x]))","A",6,6,27,0.2222,1,"{2684, 2775, 203, 2833, 63, 215}"
301,1,34,0,0.0604187,"\int \frac{1}{\sqrt{e \cos (c+d x)} \sqrt{a+a \sin (c+d x)}} \, dx","Int[1/(Sqrt[e*Cos[c + d*x]]*Sqrt[a + a*Sin[c + d*x]]),x]","-\frac{2 \sqrt{e \cos (c+d x)}}{d e \sqrt{a \sin (c+d x)+a}}","-\frac{2 \sqrt{e \cos (c+d x)}}{d e \sqrt{a \sin (c+d x)+a}}",1,"(-2*Sqrt[e*Cos[c + d*x]])/(d*e*Sqrt[a + a*Sin[c + d*x]])","A",1,1,27,0.03704,1,"{2671}"
302,1,76,0,0.132634,"\int \frac{1}{(e \cos (c+d x))^{3/2} \sqrt{a+a \sin (c+d x)}} \, dx","Int[1/((e*Cos[c + d*x])^(3/2)*Sqrt[a + a*Sin[c + d*x]]),x]","\frac{4 \sqrt{a \sin (c+d x)+a}}{3 a d e \sqrt{e \cos (c+d x)}}-\frac{2}{3 d e \sqrt{a \sin (c+d x)+a} \sqrt{e \cos (c+d x)}}","\frac{4 \sqrt{a \sin (c+d x)+a}}{3 a d e \sqrt{e \cos (c+d x)}}-\frac{2}{3 d e \sqrt{a \sin (c+d x)+a} \sqrt{e \cos (c+d x)}}",1,"-2/(3*d*e*Sqrt[e*Cos[c + d*x]]*Sqrt[a + a*Sin[c + d*x]]) + (4*Sqrt[a + a*Sin[c + d*x]])/(3*a*d*e*Sqrt[e*Cos[c + d*x]])","A",2,2,27,0.07407,1,"{2672, 2671}"
303,1,115,0,0.2085821,"\int \frac{1}{(e \cos (c+d x))^{5/2} \sqrt{a+a \sin (c+d x)}} \, dx","Int[1/((e*Cos[c + d*x])^(5/2)*Sqrt[a + a*Sin[c + d*x]]),x]","\frac{16 (a \sin (c+d x)+a)^{3/2}}{15 a^2 d e (e \cos (c+d x))^{3/2}}-\frac{8 \sqrt{a \sin (c+d x)+a}}{5 a d e (e \cos (c+d x))^{3/2}}-\frac{2}{5 d e \sqrt{a \sin (c+d x)+a} (e \cos (c+d x))^{3/2}}","\frac{16 (a \sin (c+d x)+a)^{3/2}}{15 a^2 d e (e \cos (c+d x))^{3/2}}-\frac{8 \sqrt{a \sin (c+d x)+a}}{5 a d e (e \cos (c+d x))^{3/2}}-\frac{2}{5 d e \sqrt{a \sin (c+d x)+a} (e \cos (c+d x))^{3/2}}",1,"-2/(5*d*e*(e*Cos[c + d*x])^(3/2)*Sqrt[a + a*Sin[c + d*x]]) - (8*Sqrt[a + a*Sin[c + d*x]])/(5*a*d*e*(e*Cos[c + d*x])^(3/2)) + (16*(a + a*Sin[c + d*x])^(3/2))/(15*a^2*d*e*(e*Cos[c + d*x])^(3/2))","A",3,2,27,0.07407,1,"{2672, 2671}"
304,1,154,0,0.2876674,"\int \frac{1}{(e \cos (c+d x))^{7/2} \sqrt{a+a \sin (c+d x)}} \, dx","Int[1/((e*Cos[c + d*x])^(7/2)*Sqrt[a + a*Sin[c + d*x]]),x]","-\frac{32 (a \sin (c+d x)+a)^{5/2}}{35 a^3 d e (e \cos (c+d x))^{5/2}}+\frac{16 (a \sin (c+d x)+a)^{3/2}}{7 a^2 d e (e \cos (c+d x))^{5/2}}-\frac{4 \sqrt{a \sin (c+d x)+a}}{7 a d e (e \cos (c+d x))^{5/2}}-\frac{2}{7 d e \sqrt{a \sin (c+d x)+a} (e \cos (c+d x))^{5/2}}","-\frac{32 (a \sin (c+d x)+a)^{5/2}}{35 a^3 d e (e \cos (c+d x))^{5/2}}+\frac{16 (a \sin (c+d x)+a)^{3/2}}{7 a^2 d e (e \cos (c+d x))^{5/2}}-\frac{4 \sqrt{a \sin (c+d x)+a}}{7 a d e (e \cos (c+d x))^{5/2}}-\frac{2}{7 d e \sqrt{a \sin (c+d x)+a} (e \cos (c+d x))^{5/2}}",1,"-2/(7*d*e*(e*Cos[c + d*x])^(5/2)*Sqrt[a + a*Sin[c + d*x]]) - (4*Sqrt[a + a*Sin[c + d*x]])/(7*a*d*e*(e*Cos[c + d*x])^(5/2)) + (16*(a + a*Sin[c + d*x])^(3/2))/(7*a^2*d*e*(e*Cos[c + d*x])^(5/2)) - (32*(a + a*Sin[c + d*x])^(5/2))/(35*a^3*d*e*(e*Cos[c + d*x])^(5/2))","A",4,2,27,0.07407,1,"{2672, 2671}"
305,1,247,0,0.3717362,"\int \frac{(e \cos (c+d x))^{7/2}}{(a+a \sin (c+d x))^{3/2}} \, dx","Int[(e*Cos[c + d*x])^(7/2)/(a + a*Sin[c + d*x])^(3/2),x]","\frac{5 e^3 \sqrt{a \sin (c+d x)+a} \sqrt{e \cos (c+d x)}}{4 a^2 d}+\frac{5 e^{7/2} \sqrt{\cos (c+d x)+1} \sqrt{a \sin (c+d x)+a} \tan ^{-1}\left(\frac{\sqrt{e} \sin (c+d x)}{\sqrt{\cos (c+d x)+1} \sqrt{e \cos (c+d x)}}\right)}{4 a^2 d (\sin (c+d x)+\cos (c+d x)+1)}-\frac{5 e^{7/2} \sqrt{\cos (c+d x)+1} \sqrt{a \sin (c+d x)+a} \sinh ^{-1}\left(\frac{\sqrt{e \cos (c+d x)}}{\sqrt{e}}\right)}{4 a^2 d (\sin (c+d x)+\cos (c+d x)+1)}+\frac{e (e \cos (c+d x))^{5/2}}{2 a d \sqrt{a \sin (c+d x)+a}}","\frac{5 e^3 \sqrt{a \sin (c+d x)+a} \sqrt{e \cos (c+d x)}}{4 a^2 d}+\frac{5 e^{7/2} \sqrt{\cos (c+d x)+1} \sqrt{a \sin (c+d x)+a} \tan ^{-1}\left(\frac{\sqrt{e} \sin (c+d x)}{\sqrt{\cos (c+d x)+1} \sqrt{e \cos (c+d x)}}\right)}{4 a^2 d (\sin (c+d x)+\cos (c+d x)+1)}-\frac{5 e^{7/2} \sqrt{\cos (c+d x)+1} \sqrt{a \sin (c+d x)+a} \sinh ^{-1}\left(\frac{\sqrt{e \cos (c+d x)}}{\sqrt{e}}\right)}{4 a^2 d (\sin (c+d x)+\cos (c+d x)+1)}+\frac{e (e \cos (c+d x))^{5/2}}{2 a d \sqrt{a \sin (c+d x)+a}}",1,"(e*(e*Cos[c + d*x])^(5/2))/(2*a*d*Sqrt[a + a*Sin[c + d*x]]) + (5*e^3*Sqrt[e*Cos[c + d*x]]*Sqrt[a + a*Sin[c + d*x]])/(4*a^2*d) - (5*e^(7/2)*ArcSinh[Sqrt[e*Cos[c + d*x]]/Sqrt[e]]*Sqrt[1 + Cos[c + d*x]]*Sqrt[a + a*Sin[c + d*x]])/(4*a^2*d*(1 + Cos[c + d*x] + Sin[c + d*x])) + (5*e^(7/2)*ArcTan[(Sqrt[e]*Sin[c + d*x])/(Sqrt[e*Cos[c + d*x]]*Sqrt[1 + Cos[c + d*x]])]*Sqrt[1 + Cos[c + d*x]]*Sqrt[a + a*Sin[c + d*x]])/(4*a^2*d*(1 + Cos[c + d*x] + Sin[c + d*x]))","A",8,8,27,0.2963,1,"{2679, 2685, 2677, 2775, 203, 2833, 63, 215}"
306,1,215,0,0.2869135,"\int \frac{(e \cos (c+d x))^{5/2}}{(a+a \sin (c+d x))^{3/2}} \, dx","Int[(e*Cos[c + d*x])^(5/2)/(a + a*Sin[c + d*x])^(3/2),x]","\frac{3 e^{5/2} \sqrt{\cos (c+d x)+1} \sqrt{a \sin (c+d x)+a} \tan ^{-1}\left(\frac{\sqrt{e} \sin (c+d x)}{\sqrt{\cos (c+d x)+1} \sqrt{e \cos (c+d x)}}\right)}{d \left(a^2 \sin (c+d x)+a^2 \cos (c+d x)+a^2\right)}+\frac{3 e^{5/2} \sqrt{\cos (c+d x)+1} \sqrt{a \sin (c+d x)+a} \sinh ^{-1}\left(\frac{\sqrt{e \cos (c+d x)}}{\sqrt{e}}\right)}{d \left(a^2 \sin (c+d x)+a^2 \cos (c+d x)+a^2\right)}+\frac{e (e \cos (c+d x))^{3/2}}{a d \sqrt{a \sin (c+d x)+a}}","\frac{3 e^{5/2} \sqrt{\cos (c+d x)+1} \sqrt{a \sin (c+d x)+a} \tan ^{-1}\left(\frac{\sqrt{e} \sin (c+d x)}{\sqrt{\cos (c+d x)+1} \sqrt{e \cos (c+d x)}}\right)}{d \left(a^2 \sin (c+d x)+a^2 \cos (c+d x)+a^2\right)}+\frac{3 e^{5/2} \sqrt{\cos (c+d x)+1} \sqrt{a \sin (c+d x)+a} \sinh ^{-1}\left(\frac{\sqrt{e \cos (c+d x)}}{\sqrt{e}}\right)}{d \left(a^2 \sin (c+d x)+a^2 \cos (c+d x)+a^2\right)}+\frac{e (e \cos (c+d x))^{3/2}}{a d \sqrt{a \sin (c+d x)+a}}",1,"(e*(e*Cos[c + d*x])^(3/2))/(a*d*Sqrt[a + a*Sin[c + d*x]]) + (3*e^(5/2)*ArcSinh[Sqrt[e*Cos[c + d*x]]/Sqrt[e]]*Sqrt[1 + Cos[c + d*x]]*Sqrt[a + a*Sin[c + d*x]])/(d*(a^2 + a^2*Cos[c + d*x] + a^2*Sin[c + d*x])) + (3*e^(5/2)*ArcTan[(Sqrt[e]*Sin[c + d*x])/(Sqrt[e*Cos[c + d*x]]*Sqrt[1 + Cos[c + d*x]])]*Sqrt[1 + Cos[c + d*x]]*Sqrt[a + a*Sin[c + d*x]])/(d*(a^2 + a^2*Cos[c + d*x] + a^2*Sin[c + d*x]))","A",7,7,27,0.2593,1,"{2679, 2684, 2775, 203, 2833, 63, 215}"
307,1,236,0,0.3621829,"\int \frac{(e \cos (c+d x))^{3/2}}{(a+a \sin (c+d x))^{3/2}} \, dx","Int[(e*Cos[c + d*x])^(3/2)/(a + a*Sin[c + d*x])^(3/2),x]","-\frac{2 e^{3/2} \sqrt{\cos (c+d x)+1} \sqrt{a \sin (c+d x)+a} \tan ^{-1}\left(\frac{\sqrt{e} \sin (c+d x)}{\sqrt{\cos (c+d x)+1} \sqrt{e \cos (c+d x)}}\right)}{a^2 d (\sin (c+d x)+\cos (c+d x)+1)}+\frac{2 e^{3/2} \sqrt{\cos (c+d x)+1} \sqrt{a \sin (c+d x)+a} \sinh ^{-1}\left(\frac{\sqrt{e \cos (c+d x)}}{\sqrt{e}}\right)}{a^2 d (\sin (c+d x)+\cos (c+d x)+1)}-\frac{2 e \sqrt{a \sin (c+d x)+a} \sqrt{e \cos (c+d x)}}{a^2 d}-\frac{2 (e \cos (c+d x))^{5/2}}{d e (a \sin (c+d x)+a)^{3/2}}","-\frac{2 e^{3/2} \sqrt{\cos (c+d x)+1} \sqrt{a \sin (c+d x)+a} \tan ^{-1}\left(\frac{\sqrt{e} \sin (c+d x)}{\sqrt{\cos (c+d x)+1} \sqrt{e \cos (c+d x)}}\right)}{a^2 d (\sin (c+d x)+\cos (c+d x)+1)}+\frac{2 e^{3/2} \sqrt{\cos (c+d x)+1} \sqrt{a \sin (c+d x)+a} \sinh ^{-1}\left(\frac{\sqrt{e \cos (c+d x)}}{\sqrt{e}}\right)}{a^2 d (\sin (c+d x)+\cos (c+d x)+1)}-\frac{2 e \sqrt{a \sin (c+d x)+a} \sqrt{e \cos (c+d x)}}{a^2 d}-\frac{2 (e \cos (c+d x))^{5/2}}{d e (a \sin (c+d x)+a)^{3/2}}",1,"(-2*(e*Cos[c + d*x])^(5/2))/(d*e*(a + a*Sin[c + d*x])^(3/2)) - (2*e*Sqrt[e*Cos[c + d*x]]*Sqrt[a + a*Sin[c + d*x]])/(a^2*d) + (2*e^(3/2)*ArcSinh[Sqrt[e*Cos[c + d*x]]/Sqrt[e]]*Sqrt[1 + Cos[c + d*x]]*Sqrt[a + a*Sin[c + d*x]])/(a^2*d*(1 + Cos[c + d*x] + Sin[c + d*x])) - (2*e^(3/2)*ArcTan[(Sqrt[e]*Sin[c + d*x])/(Sqrt[e*Cos[c + d*x]]*Sqrt[1 + Cos[c + d*x]])]*Sqrt[1 + Cos[c + d*x]]*Sqrt[a + a*Sin[c + d*x]])/(a^2*d*(1 + Cos[c + d*x] + Sin[c + d*x]))","A",8,8,27,0.2963,1,"{2681, 2685, 2677, 2775, 203, 2833, 63, 215}"
308,1,36,0,0.06364,"\int \frac{\sqrt{e \cos (c+d x)}}{(a+a \sin (c+d x))^{3/2}} \, dx","Int[Sqrt[e*Cos[c + d*x]]/(a + a*Sin[c + d*x])^(3/2),x]","-\frac{2 (e \cos (c+d x))^{3/2}}{3 d e (a \sin (c+d x)+a)^{3/2}}","-\frac{2 (e \cos (c+d x))^{3/2}}{3 d e (a \sin (c+d x)+a)^{3/2}}",1,"(-2*(e*Cos[c + d*x])^(3/2))/(3*d*e*(a + a*Sin[c + d*x])^(3/2))","A",1,1,27,0.03704,1,"{2671}"
309,1,76,0,0.129558,"\int \frac{1}{\sqrt{e \cos (c+d x)} (a+a \sin (c+d x))^{3/2}} \, dx","Int[1/(Sqrt[e*Cos[c + d*x]]*(a + a*Sin[c + d*x])^(3/2)),x]","-\frac{4 \sqrt{e \cos (c+d x)}}{5 a d e \sqrt{a \sin (c+d x)+a}}-\frac{2 \sqrt{e \cos (c+d x)}}{5 d e (a \sin (c+d x)+a)^{3/2}}","-\frac{4 \sqrt{e \cos (c+d x)}}{5 a d e \sqrt{a \sin (c+d x)+a}}-\frac{2 \sqrt{e \cos (c+d x)}}{5 d e (a \sin (c+d x)+a)^{3/2}}",1,"(-2*Sqrt[e*Cos[c + d*x]])/(5*d*e*(a + a*Sin[c + d*x])^(3/2)) - (4*Sqrt[e*Cos[c + d*x]])/(5*a*d*e*Sqrt[a + a*Sin[c + d*x]])","A",2,2,27,0.07407,1,"{2672, 2671}"
310,1,115,0,0.2097607,"\int \frac{1}{(e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^{3/2}} \, dx","Int[1/((e*Cos[c + d*x])^(3/2)*(a + a*Sin[c + d*x])^(3/2)),x]","\frac{16 \sqrt{a \sin (c+d x)+a}}{21 a^2 d e \sqrt{e \cos (c+d x)}}-\frac{8}{21 a d e \sqrt{a \sin (c+d x)+a} \sqrt{e \cos (c+d x)}}-\frac{2}{7 d e (a \sin (c+d x)+a)^{3/2} \sqrt{e \cos (c+d x)}}","\frac{16 \sqrt{a \sin (c+d x)+a}}{21 a^2 d e \sqrt{e \cos (c+d x)}}-\frac{8}{21 a d e \sqrt{a \sin (c+d x)+a} \sqrt{e \cos (c+d x)}}-\frac{2}{7 d e (a \sin (c+d x)+a)^{3/2} \sqrt{e \cos (c+d x)}}",1,"-2/(7*d*e*Sqrt[e*Cos[c + d*x]]*(a + a*Sin[c + d*x])^(3/2)) - 8/(21*a*d*e*Sqrt[e*Cos[c + d*x]]*Sqrt[a + a*Sin[c + d*x]]) + (16*Sqrt[a + a*Sin[c + d*x]])/(21*a^2*d*e*Sqrt[e*Cos[c + d*x]])","A",3,2,27,0.07407,1,"{2672, 2671}"
311,1,154,0,0.2887342,"\int \frac{1}{(e \cos (c+d x))^{5/2} (a+a \sin (c+d x))^{3/2}} \, dx","Int[1/((e*Cos[c + d*x])^(5/2)*(a + a*Sin[c + d*x])^(3/2)),x]","\frac{32 (a \sin (c+d x)+a)^{3/2}}{45 a^3 d e (e \cos (c+d x))^{3/2}}-\frac{16 \sqrt{a \sin (c+d x)+a}}{15 a^2 d e (e \cos (c+d x))^{3/2}}-\frac{4}{15 a d e \sqrt{a \sin (c+d x)+a} (e \cos (c+d x))^{3/2}}-\frac{2}{9 d e (a \sin (c+d x)+a)^{3/2} (e \cos (c+d x))^{3/2}}","\frac{32 (a \sin (c+d x)+a)^{3/2}}{45 a^3 d e (e \cos (c+d x))^{3/2}}-\frac{16 \sqrt{a \sin (c+d x)+a}}{15 a^2 d e (e \cos (c+d x))^{3/2}}-\frac{4}{15 a d e \sqrt{a \sin (c+d x)+a} (e \cos (c+d x))^{3/2}}-\frac{2}{9 d e (a \sin (c+d x)+a)^{3/2} (e \cos (c+d x))^{3/2}}",1,"-2/(9*d*e*(e*Cos[c + d*x])^(3/2)*(a + a*Sin[c + d*x])^(3/2)) - 4/(15*a*d*e*(e*Cos[c + d*x])^(3/2)*Sqrt[a + a*Sin[c + d*x]]) - (16*Sqrt[a + a*Sin[c + d*x]])/(15*a^2*d*e*(e*Cos[c + d*x])^(3/2)) + (32*(a + a*Sin[c + d*x])^(3/2))/(45*a^3*d*e*(e*Cos[c + d*x])^(3/2))","A",4,2,27,0.07407,1,"{2672, 2671}"
312,1,193,0,0.3716009,"\int \frac{1}{(e \cos (c+d x))^{7/2} (a+a \sin (c+d x))^{3/2}} \, dx","Int[1/((e*Cos[c + d*x])^(7/2)*(a + a*Sin[c + d*x])^(3/2)),x]","-\frac{256 (a \sin (c+d x)+a)^{5/2}}{385 a^4 d e (e \cos (c+d x))^{5/2}}+\frac{128 (a \sin (c+d x)+a)^{3/2}}{77 a^3 d e (e \cos (c+d x))^{5/2}}-\frac{32 \sqrt{a \sin (c+d x)+a}}{77 a^2 d e (e \cos (c+d x))^{5/2}}-\frac{16}{77 a d e \sqrt{a \sin (c+d x)+a} (e \cos (c+d x))^{5/2}}-\frac{2}{11 d e (a \sin (c+d x)+a)^{3/2} (e \cos (c+d x))^{5/2}}","-\frac{256 (a \sin (c+d x)+a)^{5/2}}{385 a^4 d e (e \cos (c+d x))^{5/2}}+\frac{128 (a \sin (c+d x)+a)^{3/2}}{77 a^3 d e (e \cos (c+d x))^{5/2}}-\frac{32 \sqrt{a \sin (c+d x)+a}}{77 a^2 d e (e \cos (c+d x))^{5/2}}-\frac{16}{77 a d e \sqrt{a \sin (c+d x)+a} (e \cos (c+d x))^{5/2}}-\frac{2}{11 d e (a \sin (c+d x)+a)^{3/2} (e \cos (c+d x))^{5/2}}",1,"-2/(11*d*e*(e*Cos[c + d*x])^(5/2)*(a + a*Sin[c + d*x])^(3/2)) - 16/(77*a*d*e*(e*Cos[c + d*x])^(5/2)*Sqrt[a + a*Sin[c + d*x]]) - (32*Sqrt[a + a*Sin[c + d*x]])/(77*a^2*d*e*(e*Cos[c + d*x])^(5/2)) + (128*(a + a*Sin[c + d*x])^(3/2))/(77*a^3*d*e*(e*Cos[c + d*x])^(5/2)) - (256*(a + a*Sin[c + d*x])^(5/2))/(385*a^4*d*e*(e*Cos[c + d*x])^(5/2))","A",5,2,27,0.07407,1,"{2672, 2671}"
313,1,261,0,0.4687401,"\int \frac{(e \cos (c+d x))^{9/2}}{(a+a \sin (c+d x))^{5/2}} \, dx","Int[(e*Cos[c + d*x])^(9/2)/(a + a*Sin[c + d*x])^(5/2),x]","\frac{7 e^3 (e \cos (c+d x))^{3/2}}{4 a^2 d \sqrt{a \sin (c+d x)+a}}+\frac{21 e^{9/2} \sqrt{\cos (c+d x)+1} \sqrt{a \sin (c+d x)+a} \tan ^{-1}\left(\frac{\sqrt{e} \sin (c+d x)}{\sqrt{\cos (c+d x)+1} \sqrt{e \cos (c+d x)}}\right)}{4 d \left(a^3 \sin (c+d x)+a^3 \cos (c+d x)+a^3\right)}+\frac{21 e^{9/2} \sqrt{\cos (c+d x)+1} \sqrt{a \sin (c+d x)+a} \sinh ^{-1}\left(\frac{\sqrt{e \cos (c+d x)}}{\sqrt{e}}\right)}{4 d \left(a^3 \sin (c+d x)+a^3 \cos (c+d x)+a^3\right)}+\frac{e (e \cos (c+d x))^{7/2}}{2 a d (a \sin (c+d x)+a)^{3/2}}","\frac{7 e^3 (e \cos (c+d x))^{3/2}}{4 a^2 d \sqrt{a \sin (c+d x)+a}}+\frac{21 e^{9/2} \sqrt{\cos (c+d x)+1} \sqrt{a \sin (c+d x)+a} \tan ^{-1}\left(\frac{\sqrt{e} \sin (c+d x)}{\sqrt{\cos (c+d x)+1} \sqrt{e \cos (c+d x)}}\right)}{4 d \left(a^3 \sin (c+d x)+a^3 \cos (c+d x)+a^3\right)}+\frac{21 e^{9/2} \sqrt{\cos (c+d x)+1} \sqrt{a \sin (c+d x)+a} \sinh ^{-1}\left(\frac{\sqrt{e \cos (c+d x)}}{\sqrt{e}}\right)}{4 d \left(a^3 \sin (c+d x)+a^3 \cos (c+d x)+a^3\right)}+\frac{e (e \cos (c+d x))^{7/2}}{2 a d (a \sin (c+d x)+a)^{3/2}}",1,"(e*(e*Cos[c + d*x])^(7/2))/(2*a*d*(a + a*Sin[c + d*x])^(3/2)) + (7*e^3*(e*Cos[c + d*x])^(3/2))/(4*a^2*d*Sqrt[a + a*Sin[c + d*x]]) + (21*e^(9/2)*ArcSinh[Sqrt[e*Cos[c + d*x]]/Sqrt[e]]*Sqrt[1 + Cos[c + d*x]]*Sqrt[a + a*Sin[c + d*x]])/(4*d*(a^3 + a^3*Cos[c + d*x] + a^3*Sin[c + d*x])) + (21*e^(9/2)*ArcTan[(Sqrt[e]*Sin[c + d*x])/(Sqrt[e*Cos[c + d*x]]*Sqrt[1 + Cos[c + d*x]])]*Sqrt[1 + Cos[c + d*x]]*Sqrt[a + a*Sin[c + d*x]])/(4*d*(a^3 + a^3*Cos[c + d*x] + a^3*Sin[c + d*x]))","A",9,9,27,0.3333,1,"{2680, 2686, 2679, 2684, 2775, 203, 2833, 63, 215}"
314,1,239,0,0.3664823,"\int \frac{(e \cos (c+d x))^{7/2}}{(a+a \sin (c+d x))^{5/2}} \, dx","Int[(e*Cos[c + d*x])^(7/2)/(a + a*Sin[c + d*x])^(5/2),x]","-\frac{5 e^3 \sqrt{a \sin (c+d x)+a} \sqrt{e \cos (c+d x)}}{a^3 d}-\frac{5 e^{7/2} \sqrt{\cos (c+d x)+1} \sqrt{a \sin (c+d x)+a} \tan ^{-1}\left(\frac{\sqrt{e} \sin (c+d x)}{\sqrt{\cos (c+d x)+1} \sqrt{e \cos (c+d x)}}\right)}{a^3 d (\sin (c+d x)+\cos (c+d x)+1)}+\frac{5 e^{7/2} \sqrt{\cos (c+d x)+1} \sqrt{a \sin (c+d x)+a} \sinh ^{-1}\left(\frac{\sqrt{e \cos (c+d x)}}{\sqrt{e}}\right)}{a^3 d (\sin (c+d x)+\cos (c+d x)+1)}-\frac{4 e (e \cos (c+d x))^{5/2}}{a d (a \sin (c+d x)+a)^{3/2}}","-\frac{5 e^3 \sqrt{a \sin (c+d x)+a} \sqrt{e \cos (c+d x)}}{a^3 d}-\frac{5 e^{7/2} \sqrt{\cos (c+d x)+1} \sqrt{a \sin (c+d x)+a} \tan ^{-1}\left(\frac{\sqrt{e} \sin (c+d x)}{\sqrt{\cos (c+d x)+1} \sqrt{e \cos (c+d x)}}\right)}{a^3 d (\sin (c+d x)+\cos (c+d x)+1)}+\frac{5 e^{7/2} \sqrt{\cos (c+d x)+1} \sqrt{a \sin (c+d x)+a} \sinh ^{-1}\left(\frac{\sqrt{e \cos (c+d x)}}{\sqrt{e}}\right)}{a^3 d (\sin (c+d x)+\cos (c+d x)+1)}-\frac{4 e (e \cos (c+d x))^{5/2}}{a d (a \sin (c+d x)+a)^{3/2}}",1,"(-4*e*(e*Cos[c + d*x])^(5/2))/(a*d*(a + a*Sin[c + d*x])^(3/2)) - (5*e^3*Sqrt[e*Cos[c + d*x]]*Sqrt[a + a*Sin[c + d*x]])/(a^3*d) + (5*e^(7/2)*ArcSinh[Sqrt[e*Cos[c + d*x]]/Sqrt[e]]*Sqrt[1 + Cos[c + d*x]]*Sqrt[a + a*Sin[c + d*x]])/(a^3*d*(1 + Cos[c + d*x] + Sin[c + d*x])) - (5*e^(7/2)*ArcTan[(Sqrt[e]*Sin[c + d*x])/(Sqrt[e*Cos[c + d*x]]*Sqrt[1 + Cos[c + d*x]])]*Sqrt[1 + Cos[c + d*x]]*Sqrt[a + a*Sin[c + d*x]])/(a^3*d*(1 + Cos[c + d*x] + Sin[c + d*x]))","A",8,8,27,0.2963,1,"{2680, 2685, 2677, 2775, 203, 2833, 63, 215}"
315,1,218,0,0.295897,"\int \frac{(e \cos (c+d x))^{5/2}}{(a+a \sin (c+d x))^{5/2}} \, dx","Int[(e*Cos[c + d*x])^(5/2)/(a + a*Sin[c + d*x])^(5/2),x]","-\frac{2 e^{5/2} \sqrt{\cos (c+d x)+1} \sqrt{a \sin (c+d x)+a} \tan ^{-1}\left(\frac{\sqrt{e} \sin (c+d x)}{\sqrt{\cos (c+d x)+1} \sqrt{e \cos (c+d x)}}\right)}{d \left(a^3 \sin (c+d x)+a^3 \cos (c+d x)+a^3\right)}-\frac{2 e^{5/2} \sqrt{\cos (c+d x)+1} \sqrt{a \sin (c+d x)+a} \sinh ^{-1}\left(\frac{\sqrt{e \cos (c+d x)}}{\sqrt{e}}\right)}{d \left(a^3 \sin (c+d x)+a^3 \cos (c+d x)+a^3\right)}-\frac{4 e (e \cos (c+d x))^{3/2}}{3 a d (a \sin (c+d x)+a)^{3/2}}","-\frac{2 e^{5/2} \sqrt{\cos (c+d x)+1} \sqrt{a \sin (c+d x)+a} \tan ^{-1}\left(\frac{\sqrt{e} \sin (c+d x)}{\sqrt{\cos (c+d x)+1} \sqrt{e \cos (c+d x)}}\right)}{d \left(a^3 \sin (c+d x)+a^3 \cos (c+d x)+a^3\right)}-\frac{2 e^{5/2} \sqrt{\cos (c+d x)+1} \sqrt{a \sin (c+d x)+a} \sinh ^{-1}\left(\frac{\sqrt{e \cos (c+d x)}}{\sqrt{e}}\right)}{d \left(a^3 \sin (c+d x)+a^3 \cos (c+d x)+a^3\right)}-\frac{4 e (e \cos (c+d x))^{3/2}}{3 a d (a \sin (c+d x)+a)^{3/2}}",1,"(-4*e*(e*Cos[c + d*x])^(3/2))/(3*a*d*(a + a*Sin[c + d*x])^(3/2)) - (2*e^(5/2)*ArcSinh[Sqrt[e*Cos[c + d*x]]/Sqrt[e]]*Sqrt[1 + Cos[c + d*x]]*Sqrt[a + a*Sin[c + d*x]])/(d*(a^3 + a^3*Cos[c + d*x] + a^3*Sin[c + d*x])) - (2*e^(5/2)*ArcTan[(Sqrt[e]*Sin[c + d*x])/(Sqrt[e*Cos[c + d*x]]*Sqrt[1 + Cos[c + d*x]])]*Sqrt[1 + Cos[c + d*x]]*Sqrt[a + a*Sin[c + d*x]])/(d*(a^3 + a^3*Cos[c + d*x] + a^3*Sin[c + d*x]))","A",7,7,27,0.2593,1,"{2680, 2684, 2775, 203, 2833, 63, 215}"
316,1,36,0,0.0705864,"\int \frac{(e \cos (c+d x))^{3/2}}{(a+a \sin (c+d x))^{5/2}} \, dx","Int[(e*Cos[c + d*x])^(3/2)/(a + a*Sin[c + d*x])^(5/2),x]","-\frac{2 (e \cos (c+d x))^{5/2}}{5 d e (a \sin (c+d x)+a)^{5/2}}","-\frac{2 (e \cos (c+d x))^{5/2}}{5 d e (a \sin (c+d x)+a)^{5/2}}",1,"(-2*(e*Cos[c + d*x])^(5/2))/(5*d*e*(a + a*Sin[c + d*x])^(5/2))","A",1,1,27,0.03704,1,"{2671}"
317,1,76,0,0.1313803,"\int \frac{\sqrt{e \cos (c+d x)}}{(a+a \sin (c+d x))^{5/2}} \, dx","Int[Sqrt[e*Cos[c + d*x]]/(a + a*Sin[c + d*x])^(5/2),x]","-\frac{4 (e \cos (c+d x))^{3/2}}{21 a d e (a \sin (c+d x)+a)^{3/2}}-\frac{2 (e \cos (c+d x))^{3/2}}{7 d e (a \sin (c+d x)+a)^{5/2}}","-\frac{4 (e \cos (c+d x))^{3/2}}{21 a d e (a \sin (c+d x)+a)^{3/2}}-\frac{2 (e \cos (c+d x))^{3/2}}{7 d e (a \sin (c+d x)+a)^{5/2}}",1,"(-2*(e*Cos[c + d*x])^(3/2))/(7*d*e*(a + a*Sin[c + d*x])^(5/2)) - (4*(e*Cos[c + d*x])^(3/2))/(21*a*d*e*(a + a*Sin[c + d*x])^(3/2))","A",2,2,27,0.07407,1,"{2672, 2671}"
318,1,115,0,0.2021199,"\int \frac{1}{\sqrt{e \cos (c+d x)} (a+a \sin (c+d x))^{5/2}} \, dx","Int[1/(Sqrt[e*Cos[c + d*x]]*(a + a*Sin[c + d*x])^(5/2)),x]","-\frac{16 \sqrt{e \cos (c+d x)}}{45 a^2 d e \sqrt{a \sin (c+d x)+a}}-\frac{8 \sqrt{e \cos (c+d x)}}{45 a d e (a \sin (c+d x)+a)^{3/2}}-\frac{2 \sqrt{e \cos (c+d x)}}{9 d e (a \sin (c+d x)+a)^{5/2}}","-\frac{16 \sqrt{e \cos (c+d x)}}{45 a^2 d e \sqrt{a \sin (c+d x)+a}}-\frac{8 \sqrt{e \cos (c+d x)}}{45 a d e (a \sin (c+d x)+a)^{3/2}}-\frac{2 \sqrt{e \cos (c+d x)}}{9 d e (a \sin (c+d x)+a)^{5/2}}",1,"(-2*Sqrt[e*Cos[c + d*x]])/(9*d*e*(a + a*Sin[c + d*x])^(5/2)) - (8*Sqrt[e*Cos[c + d*x]])/(45*a*d*e*(a + a*Sin[c + d*x])^(3/2)) - (16*Sqrt[e*Cos[c + d*x]])/(45*a^2*d*e*Sqrt[a + a*Sin[c + d*x]])","A",3,2,27,0.07407,1,"{2672, 2671}"
319,1,154,0,0.2966737,"\int \frac{1}{(e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^{5/2}} \, dx","Int[1/((e*Cos[c + d*x])^(3/2)*(a + a*Sin[c + d*x])^(5/2)),x]","\frac{32 \sqrt{a \sin (c+d x)+a}}{77 a^3 d e \sqrt{e \cos (c+d x)}}-\frac{16}{77 a^2 d e \sqrt{a \sin (c+d x)+a} \sqrt{e \cos (c+d x)}}-\frac{12}{77 a d e (a \sin (c+d x)+a)^{3/2} \sqrt{e \cos (c+d x)}}-\frac{2}{11 d e (a \sin (c+d x)+a)^{5/2} \sqrt{e \cos (c+d x)}}","\frac{32 \sqrt{a \sin (c+d x)+a}}{77 a^3 d e \sqrt{e \cos (c+d x)}}-\frac{16}{77 a^2 d e \sqrt{a \sin (c+d x)+a} \sqrt{e \cos (c+d x)}}-\frac{12}{77 a d e (a \sin (c+d x)+a)^{3/2} \sqrt{e \cos (c+d x)}}-\frac{2}{11 d e (a \sin (c+d x)+a)^{5/2} \sqrt{e \cos (c+d x)}}",1,"-2/(11*d*e*Sqrt[e*Cos[c + d*x]]*(a + a*Sin[c + d*x])^(5/2)) - 12/(77*a*d*e*Sqrt[e*Cos[c + d*x]]*(a + a*Sin[c + d*x])^(3/2)) - 16/(77*a^2*d*e*Sqrt[e*Cos[c + d*x]]*Sqrt[a + a*Sin[c + d*x]]) + (32*Sqrt[a + a*Sin[c + d*x]])/(77*a^3*d*e*Sqrt[e*Cos[c + d*x]])","A",4,2,27,0.07407,1,"{2672, 2671}"
320,1,193,0,0.3753883,"\int \frac{1}{(e \cos (c+d x))^{5/2} (a+a \sin (c+d x))^{5/2}} \, dx","Int[1/((e*Cos[c + d*x])^(5/2)*(a + a*Sin[c + d*x])^(5/2)),x]","\frac{256 (a \sin (c+d x)+a)^{3/2}}{585 a^4 d e (e \cos (c+d x))^{3/2}}-\frac{128 \sqrt{a \sin (c+d x)+a}}{195 a^3 d e (e \cos (c+d x))^{3/2}}-\frac{32}{195 a^2 d e \sqrt{a \sin (c+d x)+a} (e \cos (c+d x))^{3/2}}-\frac{16}{117 a d e (a \sin (c+d x)+a)^{3/2} (e \cos (c+d x))^{3/2}}-\frac{2}{13 d e (a \sin (c+d x)+a)^{5/2} (e \cos (c+d x))^{3/2}}","\frac{256 (a \sin (c+d x)+a)^{3/2}}{585 a^4 d e (e \cos (c+d x))^{3/2}}-\frac{128 \sqrt{a \sin (c+d x)+a}}{195 a^3 d e (e \cos (c+d x))^{3/2}}-\frac{32}{195 a^2 d e \sqrt{a \sin (c+d x)+a} (e \cos (c+d x))^{3/2}}-\frac{16}{117 a d e (a \sin (c+d x)+a)^{3/2} (e \cos (c+d x))^{3/2}}-\frac{2}{13 d e (a \sin (c+d x)+a)^{5/2} (e \cos (c+d x))^{3/2}}",1,"-2/(13*d*e*(e*Cos[c + d*x])^(3/2)*(a + a*Sin[c + d*x])^(5/2)) - 16/(117*a*d*e*(e*Cos[c + d*x])^(3/2)*(a + a*Sin[c + d*x])^(3/2)) - 32/(195*a^2*d*e*(e*Cos[c + d*x])^(3/2)*Sqrt[a + a*Sin[c + d*x]]) - (128*Sqrt[a + a*Sin[c + d*x]])/(195*a^3*d*e*(e*Cos[c + d*x])^(3/2)) + (256*(a + a*Sin[c + d*x])^(3/2))/(585*a^4*d*e*(e*Cos[c + d*x])^(3/2))","A",5,2,27,0.07407,1,"{2672, 2671}"
321,1,78,0,0.0954889,"\int \frac{(e \cos (c+d x))^{7/3}}{\sqrt{a+a \sin (c+d x)}} \, dx","Int[(e*Cos[c + d*x])^(7/3)/Sqrt[a + a*Sin[c + d*x]],x]","-\frac{3 \sqrt[6]{2} a (e \cos (c+d x))^{10/3} \, _2F_1\left(-\frac{1}{6},\frac{5}{3};\frac{8}{3};\frac{1}{2} (1-\sin (c+d x))\right)}{5 d e \sqrt[6]{\sin (c+d x)+1} (a \sin (c+d x)+a)^{3/2}}","-\frac{3 \sqrt[6]{2} a (e \cos (c+d x))^{10/3} \, _2F_1\left(-\frac{1}{6},\frac{5}{3};\frac{8}{3};\frac{1}{2} (1-\sin (c+d x))\right)}{5 d e \sqrt[6]{\sin (c+d x)+1} (a \sin (c+d x)+a)^{3/2}}",1,"(-3*2^(1/6)*a*(e*Cos[c + d*x])^(10/3)*Hypergeometric2F1[-1/6, 5/3, 8/3, (1 - Sin[c + d*x])/2])/(5*d*e*(1 + Sin[c + d*x])^(1/6)*(a + a*Sin[c + d*x])^(3/2))","A",3,3,27,0.1111,1,"{2689, 70, 69}"
322,1,78,0,0.0973489,"\int \frac{(e \cos (c+d x))^{5/3}}{\sqrt{a+a \sin (c+d x)}} \, dx","Int[(e*Cos[c + d*x])^(5/3)/Sqrt[a + a*Sin[c + d*x]],x]","-\frac{3 a \sqrt[6]{\sin (c+d x)+1} (e \cos (c+d x))^{8/3} \, _2F_1\left(\frac{1}{6},\frac{4}{3};\frac{7}{3};\frac{1}{2} (1-\sin (c+d x))\right)}{4 \sqrt[6]{2} d e (a \sin (c+d x)+a)^{3/2}}","-\frac{3 a \sqrt[6]{\sin (c+d x)+1} (e \cos (c+d x))^{8/3} \, _2F_1\left(\frac{1}{6},\frac{4}{3};\frac{7}{3};\frac{1}{2} (1-\sin (c+d x))\right)}{4 \sqrt[6]{2} d e (a \sin (c+d x)+a)^{3/2}}",1,"(-3*a*(e*Cos[c + d*x])^(8/3)*Hypergeometric2F1[1/6, 4/3, 7/3, (1 - Sin[c + d*x])/2]*(1 + Sin[c + d*x])^(1/6))/(4*2^(1/6)*d*e*(a + a*Sin[c + d*x])^(3/2))","A",3,3,27,0.1111,1,"{2689, 70, 69}"
323,1,78,0,0.0970896,"\int \frac{(e \cos (c+d x))^{2/3}}{\sqrt{a+a \sin (c+d x)}} \, dx","Int[(e*Cos[c + d*x])^(2/3)/Sqrt[a + a*Sin[c + d*x]],x]","-\frac{3 \sqrt[3]{2} a (\sin (c+d x)+1)^{2/3} (e \cos (c+d x))^{5/3} \, _2F_1\left(\frac{2}{3},\frac{5}{6};\frac{11}{6};\frac{1}{2} (1-\sin (c+d x))\right)}{5 d e (a \sin (c+d x)+a)^{3/2}}","-\frac{3 \sqrt[3]{2} a (\sin (c+d x)+1)^{2/3} (e \cos (c+d x))^{5/3} \, _2F_1\left(\frac{2}{3},\frac{5}{6};\frac{11}{6};\frac{1}{2} (1-\sin (c+d x))\right)}{5 d e (a \sin (c+d x)+a)^{3/2}}",1,"(-3*2^(1/3)*a*(e*Cos[c + d*x])^(5/3)*Hypergeometric2F1[2/3, 5/6, 11/6, (1 - Sin[c + d*x])/2]*(1 + Sin[c + d*x])^(2/3))/(5*d*e*(a + a*Sin[c + d*x])^(3/2))","A",3,3,27,0.1111,1,"{2689, 70, 69}"
324,1,78,0,0.0855781,"\int \frac{\sqrt[3]{e \cos (c+d x)}}{\sqrt{a+a \sin (c+d x)}} \, dx","Int[(e*Cos[c + d*x])^(1/3)/Sqrt[a + a*Sin[c + d*x]],x]","-\frac{3 a (\sin (c+d x)+1)^{5/6} (e \cos (c+d x))^{4/3} \, _2F_1\left(\frac{2}{3},\frac{5}{6};\frac{5}{3};\frac{1}{2} (1-\sin (c+d x))\right)}{2\ 2^{5/6} d e (a \sin (c+d x)+a)^{3/2}}","-\frac{3 a (\sin (c+d x)+1)^{5/6} (e \cos (c+d x))^{4/3} \, _2F_1\left(\frac{2}{3},\frac{5}{6};\frac{5}{3};\frac{1}{2} (1-\sin (c+d x))\right)}{2\ 2^{5/6} d e (a \sin (c+d x)+a)^{3/2}}",1,"(-3*a*(e*Cos[c + d*x])^(4/3)*Hypergeometric2F1[2/3, 5/6, 5/3, (1 - Sin[c + d*x])/2]*(1 + Sin[c + d*x])^(5/6))/(2*2^(5/6)*d*e*(a + a*Sin[c + d*x])^(3/2))","A",3,3,27,0.1111,1,"{2689, 70, 69}"
325,1,77,0,0.0872455,"\int \frac{1}{\sqrt[3]{e \cos (c+d x)} \sqrt{a+a \sin (c+d x)}} \, dx","Int[1/((e*Cos[c + d*x])^(1/3)*Sqrt[a + a*Sin[c + d*x]]),x]","-\frac{3 \sqrt[6]{\sin (c+d x)+1} (e \cos (c+d x))^{2/3} \, _2F_1\left(\frac{1}{3},\frac{7}{6};\frac{4}{3};\frac{1}{2} (1-\sin (c+d x))\right)}{2 \sqrt[6]{2} d e \sqrt{a \sin (c+d x)+a}}","-\frac{3 \sqrt[6]{\sin (c+d x)+1} (e \cos (c+d x))^{2/3} \, _2F_1\left(\frac{1}{3},\frac{7}{6};\frac{4}{3};\frac{1}{2} (1-\sin (c+d x))\right)}{2 \sqrt[6]{2} d e \sqrt{a \sin (c+d x)+a}}",1,"(-3*(e*Cos[c + d*x])^(2/3)*Hypergeometric2F1[1/3, 7/6, 4/3, (1 - Sin[c + d*x])/2]*(1 + Sin[c + d*x])^(1/6))/(2*2^(1/6)*d*e*Sqrt[a + a*Sin[c + d*x]])","A",3,3,27,0.1111,1,"{2689, 70, 69}"
326,1,75,0,0.0945724,"\int \frac{1}{(e \cos (c+d x))^{4/3} \sqrt{a+a \sin (c+d x)}} \, dx","Int[1/((e*Cos[c + d*x])^(4/3)*Sqrt[a + a*Sin[c + d*x]]),x]","\frac{3 (\sin (c+d x)+1)^{2/3} \, _2F_1\left(-\frac{1}{6},\frac{5}{3};\frac{5}{6};\frac{1}{2} (1-\sin (c+d x))\right)}{2^{2/3} d e \sqrt{a \sin (c+d x)+a} \sqrt[3]{e \cos (c+d x)}}","\frac{3 (\sin (c+d x)+1)^{2/3} \, _2F_1\left(-\frac{1}{6},\frac{5}{3};\frac{5}{6};\frac{1}{2} (1-\sin (c+d x))\right)}{2^{2/3} d e \sqrt{a \sin (c+d x)+a} \sqrt[3]{e \cos (c+d x)}}",1,"(3*Hypergeometric2F1[-1/6, 5/3, 5/6, (1 - Sin[c + d*x])/2]*(1 + Sin[c + d*x])^(2/3))/(2^(2/3)*d*e*(e*Cos[c + d*x])^(1/3)*Sqrt[a + a*Sin[c + d*x]])","A",3,3,27,0.1111,1,"{2689, 70, 69}"
327,1,95,0,0.0809274,"\int (e \cos (c+d x))^p (a+a \sin (c+d x))^8 \, dx","Int[(e*Cos[c + d*x])^p*(a + a*Sin[c + d*x])^8,x]","-\frac{a^8 2^{\frac{p}{2}+\frac{17}{2}} (\sin (c+d x)+1)^{\frac{1}{2} (-p-1)} (e \cos (c+d x))^{p+1} \, _2F_1\left(\frac{1}{2} (-p-15),\frac{p+1}{2};\frac{p+3}{2};\frac{1}{2} (1-\sin (c+d x))\right)}{d e (p+1)}","-\frac{a^8 2^{\frac{p}{2}+\frac{17}{2}} (\sin (c+d x)+1)^{\frac{1}{2} (-p-1)} (e \cos (c+d x))^{p+1} \, _2F_1\left(\frac{1}{2} (-p-15),\frac{p+1}{2};\frac{p+3}{2};\frac{1}{2} (1-\sin (c+d x))\right)}{d e (p+1)}",1,"-((2^(17/2 + p/2)*a^8*(e*Cos[c + d*x])^(1 + p)*Hypergeometric2F1[(-15 - p)/2, (1 + p)/2, (3 + p)/2, (1 - Sin[c + d*x])/2]*(1 + Sin[c + d*x])^((-1 - p)/2))/(d*e*(1 + p)))","A",2,2,23,0.08696,1,"{2688, 69}"
328,1,95,0,0.0801343,"\int (e \cos (c+d x))^p (a+a \sin (c+d x))^3 \, dx","Int[(e*Cos[c + d*x])^p*(a + a*Sin[c + d*x])^3,x]","-\frac{a^3 2^{\frac{p}{2}+\frac{7}{2}} (\sin (c+d x)+1)^{\frac{1}{2} (-p-1)} (e \cos (c+d x))^{p+1} \, _2F_1\left(\frac{1}{2} (-p-5),\frac{p+1}{2};\frac{p+3}{2};\frac{1}{2} (1-\sin (c+d x))\right)}{d e (p+1)}","-\frac{a^3 2^{\frac{p}{2}+\frac{7}{2}} (\sin (c+d x)+1)^{\frac{1}{2} (-p-1)} (e \cos (c+d x))^{p+1} \, _2F_1\left(\frac{1}{2} (-p-5),\frac{p+1}{2};\frac{p+3}{2};\frac{1}{2} (1-\sin (c+d x))\right)}{d e (p+1)}",1,"-((2^(7/2 + p/2)*a^3*(e*Cos[c + d*x])^(1 + p)*Hypergeometric2F1[(-5 - p)/2, (1 + p)/2, (3 + p)/2, (1 - Sin[c + d*x])/2]*(1 + Sin[c + d*x])^((-1 - p)/2))/(d*e*(1 + p)))","A",2,2,23,0.08696,1,"{2688, 69}"
329,1,95,0,0.0790328,"\int (e \cos (c+d x))^p (a+a \sin (c+d x))^2 \, dx","Int[(e*Cos[c + d*x])^p*(a + a*Sin[c + d*x])^2,x]","-\frac{a^2 2^{\frac{p}{2}+\frac{5}{2}} (\sin (c+d x)+1)^{\frac{1}{2} (-p-1)} (e \cos (c+d x))^{p+1} \, _2F_1\left(\frac{1}{2} (-p-3),\frac{p+1}{2};\frac{p+3}{2};\frac{1}{2} (1-\sin (c+d x))\right)}{d e (p+1)}","-\frac{a^2 2^{\frac{p}{2}+\frac{5}{2}} (\sin (c+d x)+1)^{\frac{1}{2} (-p-1)} (e \cos (c+d x))^{p+1} \, _2F_1\left(\frac{1}{2} (-p-3),\frac{p+1}{2};\frac{p+3}{2};\frac{1}{2} (1-\sin (c+d x))\right)}{d e (p+1)}",1,"-((2^(5/2 + p/2)*a^2*(e*Cos[c + d*x])^(1 + p)*Hypergeometric2F1[(-3 - p)/2, (1 + p)/2, (3 + p)/2, (1 - Sin[c + d*x])/2]*(1 + Sin[c + d*x])^((-1 - p)/2))/(d*e*(1 + p)))","A",2,2,23,0.08696,1,"{2688, 69}"
330,1,93,0,0.0568303,"\int (e \cos (c+d x))^p (a+a \sin (c+d x)) \, dx","Int[(e*Cos[c + d*x])^p*(a + a*Sin[c + d*x]),x]","-\frac{a 2^{\frac{p}{2}+\frac{3}{2}} (\sin (c+d x)+1)^{\frac{1}{2} (-p-1)} (e \cos (c+d x))^{p+1} \, _2F_1\left(\frac{1}{2} (-p-1),\frac{p+1}{2};\frac{p+3}{2};\frac{1}{2} (1-\sin (c+d x))\right)}{d e (p+1)}","-\frac{a 2^{\frac{p}{2}+\frac{3}{2}} (\sin (c+d x)+1)^{\frac{1}{2} (-p-1)} (e \cos (c+d x))^{p+1} \, _2F_1\left(\frac{1}{2} (-p-1),\frac{p+1}{2};\frac{p+3}{2};\frac{1}{2} (1-\sin (c+d x))\right)}{d e (p+1)}",1,"-((2^(3/2 + p/2)*a*(e*Cos[c + d*x])^(1 + p)*Hypergeometric2F1[(-1 - p)/2, (1 + p)/2, (3 + p)/2, (1 - Sin[c + d*x])/2]*(1 + Sin[c + d*x])^((-1 - p)/2))/(d*e*(1 + p)))","A",2,2,21,0.09524,1,"{2688, 69}"
331,1,95,0,0.0990199,"\int \frac{(e \cos (c+d x))^p}{a+a \sin (c+d x)} \, dx","Int[(e*Cos[c + d*x])^p/(a + a*Sin[c + d*x]),x]","-\frac{2^{\frac{p}{2}-\frac{1}{2}} (\sin (c+d x)+1)^{\frac{1}{2} (-p-1)} (e \cos (c+d x))^{p+1} \, _2F_1\left(\frac{3-p}{2},\frac{p+1}{2};\frac{p+3}{2};\frac{1}{2} (1-\sin (c+d x))\right)}{a d e (p+1)}","-\frac{2^{\frac{p}{2}-\frac{1}{2}} (\sin (c+d x)+1)^{\frac{1}{2} (-p-1)} (e \cos (c+d x))^{p+1} \, _2F_1\left(\frac{3-p}{2},\frac{p+1}{2};\frac{p+3}{2};\frac{1}{2} (1-\sin (c+d x))\right)}{a d e (p+1)}",1,"-((2^(-1/2 + p/2)*(e*Cos[c + d*x])^(1 + p)*Hypergeometric2F1[(3 - p)/2, (1 + p)/2, (3 + p)/2, (1 - Sin[c + d*x])/2]*(1 + Sin[c + d*x])^((-1 - p)/2))/(a*d*e*(1 + p)))","A",2,2,23,0.08696,1,"{2688, 69}"
332,1,93,0,0.0901636,"\int \frac{(e \cos (c+d x))^p}{(a+a \sin (c+d x))^2} \, dx","Int[(e*Cos[c + d*x])^p/(a + a*Sin[c + d*x])^2,x]","-\frac{2^{\frac{p-3}{2}} (\sin (c+d x)+1)^{\frac{1}{2} (-p-1)} (e \cos (c+d x))^{p+1} \, _2F_1\left(\frac{5-p}{2},\frac{p+1}{2};\frac{p+3}{2};\frac{1}{2} (1-\sin (c+d x))\right)}{a^2 d e (p+1)}","-\frac{2^{\frac{p-3}{2}} (\sin (c+d x)+1)^{\frac{1}{2} (-p-1)} (e \cos (c+d x))^{p+1} \, _2F_1\left(\frac{5-p}{2},\frac{p+1}{2};\frac{p+3}{2};\frac{1}{2} (1-\sin (c+d x))\right)}{a^2 d e (p+1)}",1,"-((2^((-3 + p)/2)*(e*Cos[c + d*x])^(1 + p)*Hypergeometric2F1[(5 - p)/2, (1 + p)/2, (3 + p)/2, (1 - Sin[c + d*x])/2]*(1 + Sin[c + d*x])^((-1 - p)/2))/(a^2*d*e*(1 + p)))","A",2,2,23,0.08696,1,"{2688, 69}"
333,1,93,0,0.0921136,"\int \frac{(e \cos (c+d x))^p}{(a+a \sin (c+d x))^3} \, dx","Int[(e*Cos[c + d*x])^p/(a + a*Sin[c + d*x])^3,x]","-\frac{2^{\frac{p-5}{2}} (\sin (c+d x)+1)^{\frac{1}{2} (-p-1)} (e \cos (c+d x))^{p+1} \, _2F_1\left(\frac{7-p}{2},\frac{p+1}{2};\frac{p+3}{2};\frac{1}{2} (1-\sin (c+d x))\right)}{a^3 d e (p+1)}","-\frac{2^{\frac{p-5}{2}} (\sin (c+d x)+1)^{\frac{1}{2} (-p-1)} (e \cos (c+d x))^{p+1} \, _2F_1\left(\frac{7-p}{2},\frac{p+1}{2};\frac{p+3}{2};\frac{1}{2} (1-\sin (c+d x))\right)}{a^3 d e (p+1)}",1,"-((2^((-5 + p)/2)*(e*Cos[c + d*x])^(1 + p)*Hypergeometric2F1[(7 - p)/2, (1 + p)/2, (3 + p)/2, (1 - Sin[c + d*x])/2]*(1 + Sin[c + d*x])^((-1 - p)/2))/(a^3*d*e*(1 + p)))","A",2,2,23,0.08696,1,"{2688, 69}"
334,1,93,0,0.0913771,"\int \frac{(e \cos (c+d x))^p}{(a+a \sin (c+d x))^8} \, dx","Int[(e*Cos[c + d*x])^p/(a + a*Sin[c + d*x])^8,x]","-\frac{2^{\frac{p-15}{2}} (\sin (c+d x)+1)^{\frac{1}{2} (-p-1)} (e \cos (c+d x))^{p+1} \, _2F_1\left(\frac{17-p}{2},\frac{p+1}{2};\frac{p+3}{2};\frac{1}{2} (1-\sin (c+d x))\right)}{a^8 d e (p+1)}","-\frac{2^{\frac{p-15}{2}} (\sin (c+d x)+1)^{\frac{1}{2} (-p-1)} (e \cos (c+d x))^{p+1} \, _2F_1\left(\frac{17-p}{2},\frac{p+1}{2};\frac{p+3}{2};\frac{1}{2} (1-\sin (c+d x))\right)}{a^8 d e (p+1)}",1,"-((2^((-15 + p)/2)*(e*Cos[c + d*x])^(1 + p)*Hypergeometric2F1[(17 - p)/2, (1 + p)/2, (3 + p)/2, (1 - Sin[c + d*x])/2]*(1 + Sin[c + d*x])^((-1 - p)/2))/(a^8*d*e*(1 + p)))","A",2,2,23,0.08696,1,"{2688, 69}"
335,1,103,0,0.1231967,"\int (e \cos (c+d x))^p (a+a \sin (c+d x))^{7/2} \, dx","Int[(e*Cos[c + d*x])^p*(a + a*Sin[c + d*x])^(7/2),x]","-\frac{a^4 2^{\frac{p}{2}+4} (\sin (c+d x)+1)^{-p/2} (e \cos (c+d x))^{p+1} \, _2F_1\left(\frac{1}{2} (-p-6),\frac{p+1}{2};\frac{p+3}{2};\frac{1}{2} (1-\sin (c+d x))\right)}{d e (p+1) \sqrt{a \sin (c+d x)+a}}","-\frac{a^4 2^{\frac{p}{2}+4} (\sin (c+d x)+1)^{-p/2} (e \cos (c+d x))^{p+1} \, _2F_1\left(\frac{1}{2} (-p-6),\frac{p+1}{2};\frac{p+3}{2};\frac{1}{2} (1-\sin (c+d x))\right)}{d e (p+1) \sqrt{a \sin (c+d x)+a}}",1,"-((2^(4 + p/2)*a^4*(e*Cos[c + d*x])^(1 + p)*Hypergeometric2F1[(-6 - p)/2, (1 + p)/2, (3 + p)/2, (1 - Sin[c + d*x])/2])/(d*e*(1 + p)*(1 + Sin[c + d*x])^(p/2)*Sqrt[a + a*Sin[c + d*x]]))","A",3,3,25,0.1200,1,"{2689, 70, 69}"
336,1,103,0,0.1168025,"\int (e \cos (c+d x))^p (a+a \sin (c+d x))^{5/2} \, dx","Int[(e*Cos[c + d*x])^p*(a + a*Sin[c + d*x])^(5/2),x]","-\frac{a^3 2^{\frac{p}{2}+3} (\sin (c+d x)+1)^{-p/2} (e \cos (c+d x))^{p+1} \, _2F_1\left(\frac{1}{2} (-p-4),\frac{p+1}{2};\frac{p+3}{2};\frac{1}{2} (1-\sin (c+d x))\right)}{d e (p+1) \sqrt{a \sin (c+d x)+a}}","-\frac{a^3 2^{\frac{p}{2}+3} (\sin (c+d x)+1)^{-p/2} (e \cos (c+d x))^{p+1} \, _2F_1\left(\frac{1}{2} (-p-4),\frac{p+1}{2};\frac{p+3}{2};\frac{1}{2} (1-\sin (c+d x))\right)}{d e (p+1) \sqrt{a \sin (c+d x)+a}}",1,"-((2^(3 + p/2)*a^3*(e*Cos[c + d*x])^(1 + p)*Hypergeometric2F1[(-4 - p)/2, (1 + p)/2, (3 + p)/2, (1 - Sin[c + d*x])/2])/(d*e*(1 + p)*(1 + Sin[c + d*x])^(p/2)*Sqrt[a + a*Sin[c + d*x]]))","A",3,3,25,0.1200,1,"{2689, 70, 69}"
337,1,103,0,0.1138549,"\int (e \cos (c+d x))^p (a+a \sin (c+d x))^{3/2} \, dx","Int[(e*Cos[c + d*x])^p*(a + a*Sin[c + d*x])^(3/2),x]","-\frac{a^2 2^{\frac{p}{2}+2} (\sin (c+d x)+1)^{-p/2} (e \cos (c+d x))^{p+1} \, _2F_1\left(\frac{1}{2} (-p-2),\frac{p+1}{2};\frac{p+3}{2};\frac{1}{2} (1-\sin (c+d x))\right)}{d e (p+1) \sqrt{a \sin (c+d x)+a}}","-\frac{a^2 2^{\frac{p}{2}+2} (\sin (c+d x)+1)^{-p/2} (e \cos (c+d x))^{p+1} \, _2F_1\left(\frac{1}{2} (-p-2),\frac{p+1}{2};\frac{p+3}{2};\frac{1}{2} (1-\sin (c+d x))\right)}{d e (p+1) \sqrt{a \sin (c+d x)+a}}",1,"-((2^(2 + p/2)*a^2*(e*Cos[c + d*x])^(1 + p)*Hypergeometric2F1[(-2 - p)/2, (1 + p)/2, (3 + p)/2, (1 - Sin[c + d*x])/2])/(d*e*(1 + p)*(1 + Sin[c + d*x])^(p/2)*Sqrt[a + a*Sin[c + d*x]]))","A",3,3,25,0.1200,1,"{2689, 70, 69}"
338,1,97,0,0.104876,"\int (e \cos (c+d x))^p \sqrt{a+a \sin (c+d x)} \, dx","Int[(e*Cos[c + d*x])^p*Sqrt[a + a*Sin[c + d*x]],x]","-\frac{a 2^{\frac{p}{2}+1} (\sin (c+d x)+1)^{-p/2} (e \cos (c+d x))^{p+1} \, _2F_1\left(-\frac{p}{2},\frac{p+1}{2};\frac{p+3}{2};\frac{1}{2} (1-\sin (c+d x))\right)}{d e (p+1) \sqrt{a \sin (c+d x)+a}}","-\frac{a 2^{\frac{p}{2}+1} (\sin (c+d x)+1)^{-p/2} (e \cos (c+d x))^{p+1} \, _2F_1\left(-\frac{p}{2},\frac{p+1}{2};\frac{p+3}{2};\frac{1}{2} (1-\sin (c+d x))\right)}{d e (p+1) \sqrt{a \sin (c+d x)+a}}",1,"-((2^(1 + p/2)*a*(e*Cos[c + d*x])^(1 + p)*Hypergeometric2F1[-p/2, (1 + p)/2, (3 + p)/2, (1 - Sin[c + d*x])/2])/(d*e*(1 + p)*(1 + Sin[c + d*x])^(p/2)*Sqrt[a + a*Sin[c + d*x]]))","A",3,3,25,0.1200,1,"{2689, 70, 69}"
339,1,101,0,0.1088963,"\int \frac{(e \cos (c+d x))^p}{\sqrt{a+a \sin (c+d x)}} \, dx","Int[(e*Cos[c + d*x])^p/Sqrt[a + a*Sin[c + d*x]],x]","-\frac{a 2^{p/2} (\sin (c+d x)+1)^{1-\frac{p}{2}} (e \cos (c+d x))^{p+1} \, _2F_1\left(\frac{2-p}{2},\frac{p+1}{2};\frac{p+3}{2};\frac{1}{2} (1-\sin (c+d x))\right)}{d e (p+1) (a \sin (c+d x)+a)^{3/2}}","-\frac{a 2^{p/2} (\sin (c+d x)+1)^{1-\frac{p}{2}} (e \cos (c+d x))^{p+1} \, _2F_1\left(\frac{2-p}{2},\frac{p+1}{2};\frac{p+3}{2};\frac{1}{2} (1-\sin (c+d x))\right)}{d e (p+1) (a \sin (c+d x)+a)^{3/2}}",1,"-((2^(p/2)*a*(e*Cos[c + d*x])^(1 + p)*Hypergeometric2F1[(2 - p)/2, (1 + p)/2, (3 + p)/2, (1 - Sin[c + d*x])/2]*(1 + Sin[c + d*x])^(1 - p/2))/(d*e*(1 + p)*(a + a*Sin[c + d*x])^(3/2)))","A",3,3,25,0.1200,1,"{2689, 70, 69}"
340,1,102,0,0.1213179,"\int \frac{(e \cos (c+d x))^p}{(a+a \sin (c+d x))^{3/2}} \, dx","Int[(e*Cos[c + d*x])^p/(a + a*Sin[c + d*x])^(3/2),x]","-\frac{2^{\frac{p}{2}-1} (\sin (c+d x)+1)^{1-\frac{p}{2}} (e \cos (c+d x))^{p+1} \, _2F_1\left(\frac{4-p}{2},\frac{p+1}{2};\frac{p+3}{2};\frac{1}{2} (1-\sin (c+d x))\right)}{d e (p+1) (a \sin (c+d x)+a)^{3/2}}","-\frac{2^{\frac{p}{2}-1} (\sin (c+d x)+1)^{1-\frac{p}{2}} (e \cos (c+d x))^{p+1} \, _2F_1\left(\frac{4-p}{2},\frac{p+1}{2};\frac{p+3}{2};\frac{1}{2} (1-\sin (c+d x))\right)}{d e (p+1) (a \sin (c+d x)+a)^{3/2}}",1,"-((2^(-1 + p/2)*(e*Cos[c + d*x])^(1 + p)*Hypergeometric2F1[(4 - p)/2, (1 + p)/2, (3 + p)/2, (1 - Sin[c + d*x])/2]*(1 + Sin[c + d*x])^(1 - p/2))/(d*e*(1 + p)*(a + a*Sin[c + d*x])^(3/2)))","A",3,3,25,0.1200,1,"{2689, 70, 69}"
341,1,105,0,0.1230949,"\int \frac{(e \cos (c+d x))^p}{(a+a \sin (c+d x))^{5/2}} \, dx","Int[(e*Cos[c + d*x])^p/(a + a*Sin[c + d*x])^(5/2),x]","-\frac{2^{\frac{p}{2}-2} (\sin (c+d x)+1)^{1-\frac{p}{2}} (e \cos (c+d x))^{p+1} \, _2F_1\left(\frac{6-p}{2},\frac{p+1}{2};\frac{p+3}{2};\frac{1}{2} (1-\sin (c+d x))\right)}{a d e (p+1) (a \sin (c+d x)+a)^{3/2}}","-\frac{2^{\frac{p}{2}-2} (\sin (c+d x)+1)^{1-\frac{p}{2}} (e \cos (c+d x))^{p+1} \, _2F_1\left(\frac{6-p}{2},\frac{p+1}{2};\frac{p+3}{2};\frac{1}{2} (1-\sin (c+d x))\right)}{a d e (p+1) (a \sin (c+d x)+a)^{3/2}}",1,"-((2^(-2 + p/2)*(e*Cos[c + d*x])^(1 + p)*Hypergeometric2F1[(6 - p)/2, (1 + p)/2, (3 + p)/2, (1 - Sin[c + d*x])/2]*(1 + Sin[c + d*x])^(1 - p/2))/(a*d*e*(1 + p)*(a + a*Sin[c + d*x])^(3/2)))","A",3,3,25,0.1200,1,"{2689, 70, 69}"
342,1,114,0,0.1149999,"\int (e \cos (c+d x))^p (a+a \sin (c+d x))^m \, dx","Int[(e*Cos[c + d*x])^p*(a + a*Sin[c + d*x])^m,x]","-\frac{a 2^{m+\frac{p}{2}+\frac{1}{2}} (a \sin (c+d x)+a)^{m-1} (e \cos (c+d x))^{p+1} (\sin (c+d x)+1)^{\frac{1}{2} (-2 m-p+1)} \, _2F_1\left(\frac{1}{2} (-2 m-p+1),\frac{p+1}{2};\frac{p+3}{2};\frac{1}{2} (1-\sin (c+d x))\right)}{d e (p+1)}","-\frac{a 2^{m+\frac{p}{2}+\frac{1}{2}} (a \sin (c+d x)+a)^{m-1} (e \cos (c+d x))^{p+1} (\sin (c+d x)+1)^{\frac{1}{2} (-2 m-p+1)} \, _2F_1\left(\frac{1}{2} (-2 m-p+1),\frac{p+1}{2};\frac{p+3}{2};\frac{1}{2} (1-\sin (c+d x))\right)}{d e (p+1)}",1,"-((2^(1/2 + m + p/2)*a*(e*Cos[c + d*x])^(1 + p)*Hypergeometric2F1[(1 - 2*m - p)/2, (1 + p)/2, (3 + p)/2, (1 - Sin[c + d*x])/2]*(1 + Sin[c + d*x])^((1 - 2*m - p)/2)*(a + a*Sin[c + d*x])^(-1 + m))/(d*e*(1 + p)))","A",3,3,23,0.1304,1,"{2689, 70, 69}"
343,1,109,0,0.0857299,"\int \cos ^7(c+d x) (a+a \sin (c+d x))^m \, dx","Int[Cos[c + d*x]^7*(a + a*Sin[c + d*x])^m,x]","\frac{8 (a \sin (c+d x)+a)^{m+4}}{a^4 d (m+4)}-\frac{12 (a \sin (c+d x)+a)^{m+5}}{a^5 d (m+5)}+\frac{6 (a \sin (c+d x)+a)^{m+6}}{a^6 d (m+6)}-\frac{(a \sin (c+d x)+a)^{m+7}}{a^7 d (m+7)}","\frac{8 (a \sin (c+d x)+a)^{m+4}}{a^4 d (m+4)}-\frac{12 (a \sin (c+d x)+a)^{m+5}}{a^5 d (m+5)}+\frac{6 (a \sin (c+d x)+a)^{m+6}}{a^6 d (m+6)}-\frac{(a \sin (c+d x)+a)^{m+7}}{a^7 d (m+7)}",1,"(8*(a + a*Sin[c + d*x])^(4 + m))/(a^4*d*(4 + m)) - (12*(a + a*Sin[c + d*x])^(5 + m))/(a^5*d*(5 + m)) + (6*(a + a*Sin[c + d*x])^(6 + m))/(a^6*d*(6 + m)) - (a + a*Sin[c + d*x])^(7 + m)/(a^7*d*(7 + m))","A",3,2,21,0.09524,1,"{2667, 43}"
344,1,81,0,0.0690724,"\int \cos ^5(c+d x) (a+a \sin (c+d x))^m \, dx","Int[Cos[c + d*x]^5*(a + a*Sin[c + d*x])^m,x]","\frac{4 (a \sin (c+d x)+a)^{m+3}}{a^3 d (m+3)}-\frac{4 (a \sin (c+d x)+a)^{m+4}}{a^4 d (m+4)}+\frac{(a \sin (c+d x)+a)^{m+5}}{a^5 d (m+5)}","\frac{4 (a \sin (c+d x)+a)^{m+3}}{a^3 d (m+3)}-\frac{4 (a \sin (c+d x)+a)^{m+4}}{a^4 d (m+4)}+\frac{(a \sin (c+d x)+a)^{m+5}}{a^5 d (m+5)}",1,"(4*(a + a*Sin[c + d*x])^(3 + m))/(a^3*d*(3 + m)) - (4*(a + a*Sin[c + d*x])^(4 + m))/(a^4*d*(4 + m)) + (a + a*Sin[c + d*x])^(5 + m)/(a^5*d*(5 + m))","A",3,2,21,0.09524,1,"{2667, 43}"
345,1,55,0,0.0576335,"\int \cos ^3(c+d x) (a+a \sin (c+d x))^m \, dx","Int[Cos[c + d*x]^3*(a + a*Sin[c + d*x])^m,x]","\frac{2 (a \sin (c+d x)+a)^{m+2}}{a^2 d (m+2)}-\frac{(a \sin (c+d x)+a)^{m+3}}{a^3 d (m+3)}","\frac{2 (a \sin (c+d x)+a)^{m+2}}{a^2 d (m+2)}-\frac{(a \sin (c+d x)+a)^{m+3}}{a^3 d (m+3)}",1,"(2*(a + a*Sin[c + d*x])^(2 + m))/(a^2*d*(2 + m)) - (a + a*Sin[c + d*x])^(3 + m)/(a^3*d*(3 + m))","A",3,2,21,0.09524,1,"{2667, 43}"
346,1,26,0,0.0275411,"\int \cos (c+d x) (a+a \sin (c+d x))^m \, dx","Int[Cos[c + d*x]*(a + a*Sin[c + d*x])^m,x]","\frac{(a \sin (c+d x)+a)^{m+1}}{a d (m+1)}","\frac{(a \sin (c+d x)+a)^{m+1}}{a d (m+1)}",1,"(a + a*Sin[c + d*x])^(1 + m)/(a*d*(1 + m))","A",2,2,19,0.1053,1,"{2667, 32}"
347,1,40,0,0.0487234,"\int \sec (c+d x) (a+a \sin (c+d x))^m \, dx","Int[Sec[c + d*x]*(a + a*Sin[c + d*x])^m,x]","\frac{(a \sin (c+d x)+a)^m \, _2F_1\left(1,m;m+1;\frac{1}{2} (\sin (c+d x)+1)\right)}{2 d m}","\frac{(a \sin (c+d x)+a)^m \, _2F_1\left(1,m;m+1;\frac{1}{2} (\sin (c+d x)+1)\right)}{2 d m}",1,"(Hypergeometric2F1[1, m, 1 + m, (1 + Sin[c + d*x])/2]*(a + a*Sin[c + d*x])^m)/(2*d*m)","A",2,2,19,0.1053,1,"{2667, 68}"
348,1,47,0,0.0545228,"\int \sec ^3(c+d x) (a+a \sin (c+d x))^m \, dx","Int[Sec[c + d*x]^3*(a + a*Sin[c + d*x])^m,x]","-\frac{a (a \sin (c+d x)+a)^{m-1} \, _2F_1\left(2,m-1;m;\frac{1}{2} (\sin (c+d x)+1)\right)}{4 d (1-m)}","-\frac{a (a \sin (c+d x)+a)^{m-1} \, _2F_1\left(2,m-1;m;\frac{1}{2} (\sin (c+d x)+1)\right)}{4 d (1-m)}",1,"-(a*Hypergeometric2F1[2, -1 + m, m, (1 + Sin[c + d*x])/2]*(a + a*Sin[c + d*x])^(-1 + m))/(4*d*(1 - m))","A",2,2,21,0.09524,1,"{2667, 68}"
349,1,51,0,0.0561663,"\int \sec ^5(c+d x) (a+a \sin (c+d x))^m \, dx","Int[Sec[c + d*x]^5*(a + a*Sin[c + d*x])^m,x]","-\frac{a^2 (a \sin (c+d x)+a)^{m-2} \, _2F_1\left(3,m-2;m-1;\frac{1}{2} (\sin (c+d x)+1)\right)}{8 d (2-m)}","-\frac{a^2 (a \sin (c+d x)+a)^{m-2} \, _2F_1\left(3,m-2;m-1;\frac{1}{2} (\sin (c+d x)+1)\right)}{8 d (2-m)}",1,"-(a^2*Hypergeometric2F1[3, -2 + m, -1 + m, (1 + Sin[c + d*x])/2]*(a + a*Sin[c + d*x])^(-2 + m))/(8*d*(2 - m))","A",2,2,21,0.09524,1,"{2667, 68}"
350,1,83,0,0.0751118,"\int \cos ^4(c+d x) (a+a \sin (c+d x))^m \, dx","Int[Cos[c + d*x]^4*(a + a*Sin[c + d*x])^m,x]","-\frac{a^2 2^{m+\frac{5}{2}} \cos ^5(c+d x) (\sin (c+d x)+1)^{-m-\frac{1}{2}} (a \sin (c+d x)+a)^{m-2} \, _2F_1\left(\frac{5}{2},-m-\frac{3}{2};\frac{7}{2};\frac{1}{2} (1-\sin (c+d x))\right)}{5 d}","-\frac{a^2 2^{m+\frac{5}{2}} \cos ^5(c+d x) (\sin (c+d x)+1)^{-m-\frac{1}{2}} (a \sin (c+d x)+a)^{m-2} \, _2F_1\left(\frac{5}{2},-m-\frac{3}{2};\frac{7}{2};\frac{1}{2} (1-\sin (c+d x))\right)}{5 d}",1,"-(2^(5/2 + m)*a^2*Cos[c + d*x]^5*Hypergeometric2F1[5/2, -3/2 - m, 7/2, (1 - Sin[c + d*x])/2]*(1 + Sin[c + d*x])^(-1/2 - m)*(a + a*Sin[c + d*x])^(-2 + m))/(5*d)","A",3,3,21,0.1429,1,"{2689, 70, 69}"
351,1,81,0,0.0744746,"\int \cos ^2(c+d x) (a+a \sin (c+d x))^m \, dx","Int[Cos[c + d*x]^2*(a + a*Sin[c + d*x])^m,x]","-\frac{a 2^{m+\frac{3}{2}} \cos ^3(c+d x) (\sin (c+d x)+1)^{-m-\frac{1}{2}} (a \sin (c+d x)+a)^{m-1} \, _2F_1\left(\frac{3}{2},-m-\frac{1}{2};\frac{5}{2};\frac{1}{2} (1-\sin (c+d x))\right)}{3 d}","-\frac{a 2^{m+\frac{3}{2}} \cos ^3(c+d x) (\sin (c+d x)+1)^{-m-\frac{1}{2}} (a \sin (c+d x)+a)^{m-1} \, _2F_1\left(\frac{3}{2},-m-\frac{1}{2};\frac{5}{2};\frac{1}{2} (1-\sin (c+d x))\right)}{3 d}",1,"-(2^(3/2 + m)*a*Cos[c + d*x]^3*Hypergeometric2F1[3/2, -1/2 - m, 5/2, (1 - Sin[c + d*x])/2]*(1 + Sin[c + d*x])^(-1/2 - m)*(a + a*Sin[c + d*x])^(-1 + m))/(3*d)","A",3,3,21,0.1429,1,"{2689, 70, 69}"
352,1,73,0,0.083032,"\int \sec ^2(c+d x) (a+a \sin (c+d x))^m \, dx","Int[Sec[c + d*x]^2*(a + a*Sin[c + d*x])^m,x]","\frac{2^{m-\frac{1}{2}} \sec (c+d x) (\sin (c+d x)+1)^{\frac{1}{2}-m} (a \sin (c+d x)+a)^m \, _2F_1\left(-\frac{1}{2},\frac{3}{2}-m;\frac{1}{2};\frac{1}{2} (1-\sin (c+d x))\right)}{d}","\frac{2^{m-\frac{1}{2}} \sec (c+d x) (\sin (c+d x)+1)^{\frac{1}{2}-m} (a \sin (c+d x)+a)^m \, _2F_1\left(-\frac{1}{2},\frac{3}{2}-m;\frac{1}{2};\frac{1}{2} (1-\sin (c+d x))\right)}{d}",1,"(2^(-1/2 + m)*Hypergeometric2F1[-1/2, 3/2 - m, 1/2, (1 - Sin[c + d*x])/2]*Sec[c + d*x]*(1 + Sin[c + d*x])^(1/2 - m)*(a + a*Sin[c + d*x])^m)/d","A",3,3,21,0.1429,1,"{2689, 70, 69}"
353,1,83,0,0.0841911,"\int \sec ^4(c+d x) (a+a \sin (c+d x))^m \, dx","Int[Sec[c + d*x]^4*(a + a*Sin[c + d*x])^m,x]","\frac{2^{m-\frac{3}{2}} \sec ^3(c+d x) (\sin (c+d x)+1)^{\frac{1}{2}-m} (a \sin (c+d x)+a)^{m+1} \, _2F_1\left(-\frac{3}{2},\frac{5}{2}-m;-\frac{1}{2};\frac{1}{2} (1-\sin (c+d x))\right)}{3 a d}","\frac{2^{m-\frac{3}{2}} \sec ^3(c+d x) (\sin (c+d x)+1)^{\frac{1}{2}-m} (a \sin (c+d x)+a)^{m+1} \, _2F_1\left(-\frac{3}{2},\frac{5}{2}-m;-\frac{1}{2};\frac{1}{2} (1-\sin (c+d x))\right)}{3 a d}",1,"(2^(-3/2 + m)*Hypergeometric2F1[-3/2, 5/2 - m, -1/2, (1 - Sin[c + d*x])/2]*Sec[c + d*x]^3*(1 + Sin[c + d*x])^(1/2 - m)*(a + a*Sin[c + d*x])^(1 + m))/(3*a*d)","A",3,3,21,0.1429,1,"{2689, 70, 69}"
354,1,88,0,0.096131,"\int (e \cos (c+d x))^{5/2} (a+a \sin (c+d x))^m \, dx","Int[(e*Cos[c + d*x])^(5/2)*(a + a*Sin[c + d*x])^m,x]","-\frac{a 2^{m+\frac{11}{4}} (e \cos (c+d x))^{7/2} (\sin (c+d x)+1)^{-m-\frac{3}{4}} (a \sin (c+d x)+a)^{m-1} \, _2F_1\left(\frac{7}{4},-m-\frac{3}{4};\frac{11}{4};\frac{1}{2} (1-\sin (c+d x))\right)}{7 d e}","-\frac{a 2^{m+\frac{11}{4}} (e \cos (c+d x))^{7/2} (\sin (c+d x)+1)^{-m-\frac{3}{4}} (a \sin (c+d x)+a)^{m-1} \, _2F_1\left(\frac{7}{4},-m-\frac{3}{4};\frac{11}{4};\frac{1}{2} (1-\sin (c+d x))\right)}{7 d e}",1,"-(2^(11/4 + m)*a*(e*Cos[c + d*x])^(7/2)*Hypergeometric2F1[7/4, -3/4 - m, 11/4, (1 - Sin[c + d*x])/2]*(1 + Sin[c + d*x])^(-3/4 - m)*(a + a*Sin[c + d*x])^(-1 + m))/(7*d*e)","A",3,3,25,0.1200,1,"{2689, 70, 69}"
355,1,88,0,0.0933461,"\int (e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^m \, dx","Int[(e*Cos[c + d*x])^(3/2)*(a + a*Sin[c + d*x])^m,x]","-\frac{a 2^{m+\frac{9}{4}} (e \cos (c+d x))^{5/2} (\sin (c+d x)+1)^{-m-\frac{1}{4}} (a \sin (c+d x)+a)^{m-1} \, _2F_1\left(\frac{5}{4},-m-\frac{1}{4};\frac{9}{4};\frac{1}{2} (1-\sin (c+d x))\right)}{5 d e}","-\frac{a 2^{m+\frac{9}{4}} (e \cos (c+d x))^{5/2} (\sin (c+d x)+1)^{-m-\frac{1}{4}} (a \sin (c+d x)+a)^{m-1} \, _2F_1\left(\frac{5}{4},-m-\frac{1}{4};\frac{9}{4};\frac{1}{2} (1-\sin (c+d x))\right)}{5 d e}",1,"-(2^(9/4 + m)*a*(e*Cos[c + d*x])^(5/2)*Hypergeometric2F1[5/4, -1/4 - m, 9/4, (1 - Sin[c + d*x])/2]*(1 + Sin[c + d*x])^(-1/4 - m)*(a + a*Sin[c + d*x])^(-1 + m))/(5*d*e)","A",3,3,25,0.1200,1,"{2689, 70, 69}"
356,1,88,0,0.0992657,"\int \sqrt{e \cos (c+d x)} (a+a \sin (c+d x))^m \, dx","Int[Sqrt[e*Cos[c + d*x]]*(a + a*Sin[c + d*x])^m,x]","-\frac{a 2^{m+\frac{7}{4}} (e \cos (c+d x))^{3/2} (\sin (c+d x)+1)^{\frac{1}{4}-m} (a \sin (c+d x)+a)^{m-1} \, _2F_1\left(\frac{3}{4},\frac{1}{4}-m;\frac{7}{4};\frac{1}{2} (1-\sin (c+d x))\right)}{3 d e}","-\frac{a 2^{m+\frac{7}{4}} (e \cos (c+d x))^{3/2} (\sin (c+d x)+1)^{\frac{1}{4}-m} (a \sin (c+d x)+a)^{m-1} \, _2F_1\left(\frac{3}{4},\frac{1}{4}-m;\frac{7}{4};\frac{1}{2} (1-\sin (c+d x))\right)}{3 d e}",1,"-(2^(7/4 + m)*a*(e*Cos[c + d*x])^(3/2)*Hypergeometric2F1[3/4, 1/4 - m, 7/4, (1 - Sin[c + d*x])/2]*(1 + Sin[c + d*x])^(1/4 - m)*(a + a*Sin[c + d*x])^(-1 + m))/(3*d*e)","A",3,3,25,0.1200,1,"{2689, 70, 69}"
357,1,86,0,0.0884103,"\int \frac{(a+a \sin (c+d x))^m}{\sqrt{e \cos (c+d x)}} \, dx","Int[(a + a*Sin[c + d*x])^m/Sqrt[e*Cos[c + d*x]],x]","-\frac{a 2^{m+\frac{5}{4}} \sqrt{e \cos (c+d x)} (\sin (c+d x)+1)^{\frac{3}{4}-m} (a \sin (c+d x)+a)^{m-1} \, _2F_1\left(\frac{1}{4},\frac{3}{4}-m;\frac{5}{4};\frac{1}{2} (1-\sin (c+d x))\right)}{d e}","-\frac{a 2^{m+\frac{5}{4}} \sqrt{e \cos (c+d x)} (\sin (c+d x)+1)^{\frac{3}{4}-m} (a \sin (c+d x)+a)^{m-1} \, _2F_1\left(\frac{1}{4},\frac{3}{4}-m;\frac{5}{4};\frac{1}{2} (1-\sin (c+d x))\right)}{d e}",1,"-((2^(5/4 + m)*a*Sqrt[e*Cos[c + d*x]]*Hypergeometric2F1[1/4, 3/4 - m, 5/4, (1 - Sin[c + d*x])/2]*(1 + Sin[c + d*x])^(3/4 - m)*(a + a*Sin[c + d*x])^(-1 + m))/(d*e))","A",3,3,25,0.1200,1,"{2689, 70, 69}"
358,1,82,0,0.0950441,"\int \frac{(a+a \sin (c+d x))^m}{(e \cos (c+d x))^{3/2}} \, dx","Int[(a + a*Sin[c + d*x])^m/(e*Cos[c + d*x])^(3/2),x]","\frac{2^{m+\frac{3}{4}} (\sin (c+d x)+1)^{\frac{1}{4}-m} (a \sin (c+d x)+a)^m \, _2F_1\left(-\frac{1}{4},\frac{5}{4}-m;\frac{3}{4};\frac{1}{2} (1-\sin (c+d x))\right)}{d e \sqrt{e \cos (c+d x)}}","\frac{2^{m+\frac{3}{4}} (\sin (c+d x)+1)^{\frac{1}{4}-m} (a \sin (c+d x)+a)^m \, _2F_1\left(-\frac{1}{4},\frac{5}{4}-m;\frac{3}{4};\frac{1}{2} (1-\sin (c+d x))\right)}{d e \sqrt{e \cos (c+d x)}}",1,"(2^(3/4 + m)*Hypergeometric2F1[-1/4, 5/4 - m, 3/4, (1 - Sin[c + d*x])/2]*(1 + Sin[c + d*x])^(1/4 - m)*(a + a*Sin[c + d*x])^m)/(d*e*Sqrt[e*Cos[c + d*x]])","A",3,3,25,0.1200,1,"{2689, 70, 69}"
359,1,85,0,0.0941946,"\int \frac{(a+a \sin (c+d x))^m}{(e \cos (c+d x))^{5/2}} \, dx","Int[(a + a*Sin[c + d*x])^m/(e*Cos[c + d*x])^(5/2),x]","\frac{2^{m+\frac{1}{4}} (\sin (c+d x)+1)^{\frac{3}{4}-m} (a \sin (c+d x)+a)^m \, _2F_1\left(-\frac{3}{4},\frac{7}{4}-m;\frac{1}{4};\frac{1}{2} (1-\sin (c+d x))\right)}{3 d e (e \cos (c+d x))^{3/2}}","\frac{2^{m+\frac{1}{4}} (\sin (c+d x)+1)^{\frac{3}{4}-m} (a \sin (c+d x)+a)^m \, _2F_1\left(-\frac{3}{4},\frac{7}{4}-m;\frac{1}{4};\frac{1}{2} (1-\sin (c+d x))\right)}{3 d e (e \cos (c+d x))^{3/2}}",1,"(2^(1/4 + m)*Hypergeometric2F1[-3/4, 7/4 - m, 1/4, (1 - Sin[c + d*x])/2]*(1 + Sin[c + d*x])^(3/4 - m)*(a + a*Sin[c + d*x])^m)/(3*d*e*(e*Cos[c + d*x])^(3/2))","A",3,3,25,0.1200,1,"{2689, 70, 69}"
360,1,201,0,0.3212276,"\int (e \cos (c+d x))^{-4-m} (a+a \sin (c+d x))^m \, dx","Int[(e*Cos[c + d*x])^(-4 - m)*(a + a*Sin[c + d*x])^m,x]","\frac{6 (a \sin (c+d x)+a)^{m+2} (e \cos (c+d x))^{-m-3}}{a^2 d e (3-m) \left(1-m^2\right)}-\frac{6 (a \sin (c+d x)+a)^{m+3} (e \cos (c+d x))^{-m-3}}{a^3 d e \left(m^4-10 m^2+9\right)}-\frac{(a \sin (c+d x)+a)^m (e \cos (c+d x))^{-m-3}}{d e (3-m)}-\frac{3 (a \sin (c+d x)+a)^{m+1} (e \cos (c+d x))^{-m-3}}{a d e (1-m) (3-m)}","\frac{6 (a \sin (c+d x)+a)^{m+2} (e \cos (c+d x))^{-m-3}}{a^2 d e (3-m) \left(1-m^2\right)}-\frac{6 (a \sin (c+d x)+a)^{m+3} (e \cos (c+d x))^{-m-3}}{a^3 d e \left(m^4-10 m^2+9\right)}-\frac{(a \sin (c+d x)+a)^m (e \cos (c+d x))^{-m-3}}{d e (3-m)}-\frac{3 (a \sin (c+d x)+a)^{m+1} (e \cos (c+d x))^{-m-3}}{a d e (1-m) (3-m)}",1,"-(((e*Cos[c + d*x])^(-3 - m)*(a + a*Sin[c + d*x])^m)/(d*e*(3 - m))) - (3*(e*Cos[c + d*x])^(-3 - m)*(a + a*Sin[c + d*x])^(1 + m))/(a*d*e*(1 - m)*(3 - m)) + (6*(e*Cos[c + d*x])^(-3 - m)*(a + a*Sin[c + d*x])^(2 + m))/(a^2*d*e*(3 - m)*(1 - m^2)) - (6*(e*Cos[c + d*x])^(-3 - m)*(a + a*Sin[c + d*x])^(3 + m))/(a^3*d*e*(9 - 10*m^2 + m^4))","A",4,2,27,0.07407,1,"{2672, 2671}"
361,1,142,0,0.2224826,"\int (e \cos (c+d x))^{-3-m} (a+a \sin (c+d x))^m \, dx","Int[(e*Cos[c + d*x])^(-3 - m)*(a + a*Sin[c + d*x])^m,x]","-\frac{2 (a \sin (c+d x)+a)^{m+2} (e \cos (c+d x))^{-m-2}}{a^2 d e m \left(4-m^2\right)}-\frac{(a \sin (c+d x)+a)^m (e \cos (c+d x))^{-m-2}}{d e (2-m)}+\frac{2 (a \sin (c+d x)+a)^{m+1} (e \cos (c+d x))^{-m-2}}{a d e (2-m) m}","-\frac{2 (a \sin (c+d x)+a)^{m+2} (e \cos (c+d x))^{-m-2}}{a^2 d e m \left(4-m^2\right)}-\frac{(a \sin (c+d x)+a)^m (e \cos (c+d x))^{-m-2}}{d e (2-m)}+\frac{2 (a \sin (c+d x)+a)^{m+1} (e \cos (c+d x))^{-m-2}}{a d e (2-m) m}",1,"-(((e*Cos[c + d*x])^(-2 - m)*(a + a*Sin[c + d*x])^m)/(d*e*(2 - m))) + (2*(e*Cos[c + d*x])^(-2 - m)*(a + a*Sin[c + d*x])^(1 + m))/(a*d*e*(2 - m)*m) - (2*(e*Cos[c + d*x])^(-2 - m)*(a + a*Sin[c + d*x])^(2 + m))/(a^2*d*e*m*(4 - m^2))","A",3,2,27,0.07407,1,"{2672, 2671}"
362,1,89,0,0.1240961,"\int (e \cos (c+d x))^{-2-m} (a+a \sin (c+d x))^m \, dx","Int[(e*Cos[c + d*x])^(-2 - m)*(a + a*Sin[c + d*x])^m,x]","\frac{(a \sin (c+d x)+a)^{m+1} (e \cos (c+d x))^{-m-1}}{a d e \left(1-m^2\right)}-\frac{(a \sin (c+d x)+a)^m (e \cos (c+d x))^{-m-1}}{d e (1-m)}","\frac{(a \sin (c+d x)+a)^{m+1} (e \cos (c+d x))^{-m-1}}{a d e \left(1-m^2\right)}-\frac{(a \sin (c+d x)+a)^m (e \cos (c+d x))^{-m-1}}{d e (1-m)}",1,"-(((e*Cos[c + d*x])^(-1 - m)*(a + a*Sin[c + d*x])^m)/(d*e*(1 - m))) + ((e*Cos[c + d*x])^(-1 - m)*(a + a*Sin[c + d*x])^(1 + m))/(a*d*e*(1 - m^2))","A",2,2,27,0.07407,1,"{2672, 2671}"
363,1,34,0,0.0506929,"\int (e \cos (c+d x))^{-1-m} (a+a \sin (c+d x))^m \, dx","Int[(e*Cos[c + d*x])^(-1 - m)*(a + a*Sin[c + d*x])^m,x]","\frac{(a \sin (c+d x)+a)^m (e \cos (c+d x))^{-m}}{d e m}","\frac{(a \sin (c+d x)+a)^m (e \cos (c+d x))^{-m}}{d e m}",1,"(a + a*Sin[c + d*x])^m/(d*e*m*(e*Cos[c + d*x])^m)","A",1,1,27,0.03704,1,"{2671}"
364,1,115,0,0.1088658,"\int (e \cos (c+d x))^{-m} (a+a \sin (c+d x))^m \, dx","Int[(a + a*Sin[c + d*x])^m/(e*Cos[c + d*x])^m,x]","-\frac{a 2^{\frac{m}{2}+\frac{1}{2}} (\sin (c+d x)+1)^{\frac{1-m}{2}} (a \sin (c+d x)+a)^{m-1} (e \cos (c+d x))^{1-m} \, _2F_1\left(\frac{1-m}{2},\frac{1-m}{2};\frac{3-m}{2};\frac{1}{2} (1-\sin (c+d x))\right)}{d e (1-m)}","-\frac{a 2^{\frac{m}{2}+\frac{1}{2}} (\sin (c+d x)+1)^{\frac{1-m}{2}} (a \sin (c+d x)+a)^{m-1} (e \cos (c+d x))^{1-m} \, _2F_1\left(\frac{1-m}{2},\frac{1-m}{2};\frac{3-m}{2};\frac{1}{2} (1-\sin (c+d x))\right)}{d e (1-m)}",1,"-((2^(1/2 + m/2)*a*(e*Cos[c + d*x])^(1 - m)*Hypergeometric2F1[(1 - m)/2, (1 - m)/2, (3 - m)/2, (1 - Sin[c + d*x])/2]*(1 + Sin[c + d*x])^((1 - m)/2)*(a + a*Sin[c + d*x])^(-1 + m))/(d*e*(1 - m)))","A",3,3,25,0.1200,1,"{2689, 70, 69}"
365,1,97,0,0.1061282,"\int (e \cos (c+d x))^{1-m} (a+a \sin (c+d x))^m \, dx","Int[(e*Cos[c + d*x])^(1 - m)*(a + a*Sin[c + d*x])^m,x]","\frac{2^{1-\frac{m}{2}} (1-\sin (c+d x))^{\frac{m}{2}-1} (a \sin (c+d x)+a)^m (e \cos (c+d x))^{2-m} \, _2F_1\left(\frac{m}{2},\frac{m+2}{2};\frac{m+4}{2};\frac{1}{2} (\sin (c+d x)+1)\right)}{d e (m+2)}","\frac{2^{1-\frac{m}{2}} (1-\sin (c+d x))^{\frac{m}{2}-1} (a \sin (c+d x)+a)^m (e \cos (c+d x))^{2-m} \, _2F_1\left(\frac{m}{2},\frac{m+2}{2};\frac{m+4}{2};\frac{1}{2} (\sin (c+d x)+1)\right)}{d e (m+2)}",1,"(2^(1 - m/2)*(e*Cos[c + d*x])^(2 - m)*Hypergeometric2F1[m/2, (2 + m)/2, (4 + m)/2, (1 + Sin[c + d*x])/2]*(1 - Sin[c + d*x])^(-1 + m/2)*(a + a*Sin[c + d*x])^m)/(d*e*(2 + m))","A",3,3,27,0.1111,1,"{2689, 70, 69}"
366,1,115,0,0.1169319,"\int (e \cos (c+d x))^{2-m} (a+a \sin (c+d x))^m \, dx","Int[(e*Cos[c + d*x])^(2 - m)*(a + a*Sin[c + d*x])^m,x]","-\frac{a 2^{\frac{m}{2}+\frac{3}{2}} (\sin (c+d x)+1)^{\frac{1}{2} (-m-1)} (a \sin (c+d x)+a)^{m-1} (e \cos (c+d x))^{3-m} \, _2F_1\left(\frac{1}{2} (-m-1),\frac{3-m}{2};\frac{5-m}{2};\frac{1}{2} (1-\sin (c+d x))\right)}{d e (3-m)}","-\frac{a 2^{\frac{m}{2}+\frac{3}{2}} (\sin (c+d x)+1)^{\frac{1}{2} (-m-1)} (a \sin (c+d x)+a)^{m-1} (e \cos (c+d x))^{3-m} \, _2F_1\left(\frac{1}{2} (-m-1),\frac{3-m}{2};\frac{5-m}{2};\frac{1}{2} (1-\sin (c+d x))\right)}{d e (3-m)}",1,"-((2^(3/2 + m/2)*a*(e*Cos[c + d*x])^(3 - m)*Hypergeometric2F1[(-1 - m)/2, (3 - m)/2, (5 - m)/2, (1 - Sin[c + d*x])/2]*(1 + Sin[c + d*x])^((-1 - m)/2)*(a + a*Sin[c + d*x])^(-1 + m))/(d*e*(3 - m)))","A",3,3,27,0.1111,1,"{2689, 70, 69}"
367,1,150,0,0.2404355,"\int (e \cos (c+d x))^{5-2 m} (a+a \sin (c+d x))^m \, dx","Int[(e*Cos[c + d*x])^(5 - 2*m)*(a + a*Sin[c + d*x])^m,x]","-\frac{8 a^3 (a \sin (c+d x)+a)^{m-3} (e \cos (c+d x))^{6-2 m}}{d e (5-m) \left(m^2-7 m+12\right)}-\frac{4 a^2 (a \sin (c+d x)+a)^{m-2} (e \cos (c+d x))^{6-2 m}}{d e \left(m^2-9 m+20\right)}-\frac{a (a \sin (c+d x)+a)^{m-1} (e \cos (c+d x))^{6-2 m}}{d e (5-m)}","-\frac{8 a^3 (a \sin (c+d x)+a)^{m-3} (e \cos (c+d x))^{6-2 m}}{d e (5-m) \left(m^2-7 m+12\right)}-\frac{4 a^2 (a \sin (c+d x)+a)^{m-2} (e \cos (c+d x))^{6-2 m}}{d e \left(m^2-9 m+20\right)}-\frac{a (a \sin (c+d x)+a)^{m-1} (e \cos (c+d x))^{6-2 m}}{d e (5-m)}",1,"(-8*a^3*(e*Cos[c + d*x])^(6 - 2*m)*(a + a*Sin[c + d*x])^(-3 + m))/(d*e*(5 - m)*(12 - 7*m + m^2)) - (4*a^2*(e*Cos[c + d*x])^(6 - 2*m)*(a + a*Sin[c + d*x])^(-2 + m))/(d*e*(20 - 9*m + m^2)) - (a*(e*Cos[c + d*x])^(6 - 2*m)*(a + a*Sin[c + d*x])^(-1 + m))/(d*e*(5 - m))","A",3,2,27,0.07407,1,"{2674, 2673}"
368,1,94,0,0.1406411,"\int (e \cos (c+d x))^{3-2 m} (a+a \sin (c+d x))^m \, dx","Int[(e*Cos[c + d*x])^(3 - 2*m)*(a + a*Sin[c + d*x])^m,x]","-\frac{2 a^2 (a \sin (c+d x)+a)^{m-2} (e \cos (c+d x))^{4-2 m}}{d e \left(m^2-5 m+6\right)}-\frac{a (a \sin (c+d x)+a)^{m-1} (e \cos (c+d x))^{4-2 m}}{d e (3-m)}","-\frac{2 a^2 (a \sin (c+d x)+a)^{m-2} (e \cos (c+d x))^{4-2 m}}{d e \left(m^2-5 m+6\right)}-\frac{a (a \sin (c+d x)+a)^{m-1} (e \cos (c+d x))^{4-2 m}}{d e (3-m)}",1,"(-2*a^2*(e*Cos[c + d*x])^(4 - 2*m)*(a + a*Sin[c + d*x])^(-2 + m))/(d*e*(6 - 5*m + m^2)) - (a*(e*Cos[c + d*x])^(4 - 2*m)*(a + a*Sin[c + d*x])^(-1 + m))/(d*e*(3 - m))","A",2,2,27,0.07407,1,"{2674, 2673}"
369,1,44,0,0.0554556,"\int (e \cos (c+d x))^{1-2 m} (a+a \sin (c+d x))^m \, dx","Int[(e*Cos[c + d*x])^(1 - 2*m)*(a + a*Sin[c + d*x])^m,x]","-\frac{a (a \sin (c+d x)+a)^{m-1} (e \cos (c+d x))^{2-2 m}}{d e (1-m)}","-\frac{a (a \sin (c+d x)+a)^{m-1} (e \cos (c+d x))^{2-2 m}}{d e (1-m)}",1,"-((a*(e*Cos[c + d*x])^(2 - 2*m)*(a + a*Sin[c + d*x])^(-1 + m))/(d*e*(1 - m)))","A",1,1,27,0.03704,1,"{2673}"
370,1,61,0,0.067174,"\int (e \cos (c+d x))^{-1-2 m} (a+a \sin (c+d x))^m \, dx","Int[(e*Cos[c + d*x])^(-1 - 2*m)*(a + a*Sin[c + d*x])^m,x]","\frac{(a \sin (c+d x)+a)^m (e \cos (c+d x))^{-2 m} \, _2F_1\left(1,-m;1-m;\frac{1}{2} (1-\sin (c+d x))\right)}{2 d e m}","\frac{(a \sin (c+d x)+a)^m (e \cos (c+d x))^{-2 m} \, _2F_1\left(1,-m;1-m;\frac{1}{2} (1-\sin (c+d x))\right)}{2 d e m}",1,"(Hypergeometric2F1[1, -m, 1 - m, (1 - Sin[c + d*x])/2]*(a + a*Sin[c + d*x])^m)/(2*d*e*m*(e*Cos[c + d*x])^(2*m))","A",3,3,27,0.1111,1,"{2689, 7, 68}"
371,1,70,0,0.0751064,"\int (e \cos (c+d x))^{-3-2 m} (a+a \sin (c+d x))^m \, dx","Int[(e*Cos[c + d*x])^(-3 - 2*m)*(a + a*Sin[c + d*x])^m,x]","\frac{(a \sin (c+d x)+a)^{m+1} (e \cos (c+d x))^{-2 (m+1)} \, _2F_1\left(2,-m-1;-m;\frac{1}{2} (1-\sin (c+d x))\right)}{4 a d e (m+1)}","\frac{(a \sin (c+d x)+a)^{m+1} (e \cos (c+d x))^{-2 (m+1)} \, _2F_1\left(2,-m-1;-m;\frac{1}{2} (1-\sin (c+d x))\right)}{4 a d e (m+1)}",1,"(Hypergeometric2F1[2, -1 - m, -m, (1 - Sin[c + d*x])/2]*(a + a*Sin[c + d*x])^(1 + m))/(4*a*d*e*(1 + m)*(e*Cos[c + d*x])^(2*(1 + m)))","A",3,3,27,0.1111,1,"{2689, 7, 68}"
372,1,89,0,0.0964986,"\int (e \cos (c+d x))^{4-2 m} (a+a \sin (c+d x))^m \, dx","Int[(e*Cos[c + d*x])^(4 - 2*m)*(a + a*Sin[c + d*x])^m,x]","\frac{2^{\frac{5}{2}-m} (1-\sin (c+d x))^{m-\frac{5}{2}} (a \sin (c+d x)+a)^m (e \cos (c+d x))^{5-2 m} \, _2F_1\left(\frac{5}{2},\frac{1}{2} (2 m-3);\frac{7}{2};\frac{1}{2} (\sin (c+d x)+1)\right)}{5 d e}","\frac{2^{\frac{5}{2}-m} (1-\sin (c+d x))^{m-\frac{5}{2}} (a \sin (c+d x)+a)^m (e \cos (c+d x))^{5-2 m} \, _2F_1\left(\frac{5}{2},\frac{1}{2} (2 m-3);\frac{7}{2};\frac{1}{2} (\sin (c+d x)+1)\right)}{5 d e}",1,"(2^(5/2 - m)*(e*Cos[c + d*x])^(5 - 2*m)*Hypergeometric2F1[5/2, (-3 + 2*m)/2, 7/2, (1 + Sin[c + d*x])/2]*(1 - Sin[c + d*x])^(-5/2 + m)*(a + a*Sin[c + d*x])^m)/(5*d*e)","A",4,4,27,0.1481,1,"{2689, 7, 70, 69}"
373,1,89,0,0.0992501,"\int (e \cos (c+d x))^{2-2 m} (a+a \sin (c+d x))^m \, dx","Int[(e*Cos[c + d*x])^(2 - 2*m)*(a + a*Sin[c + d*x])^m,x]","\frac{2^{\frac{3}{2}-m} (1-\sin (c+d x))^{m-\frac{3}{2}} (a \sin (c+d x)+a)^m (e \cos (c+d x))^{3-2 m} \, _2F_1\left(\frac{3}{2},\frac{1}{2} (2 m-1);\frac{5}{2};\frac{1}{2} (\sin (c+d x)+1)\right)}{3 d e}","\frac{2^{\frac{3}{2}-m} (1-\sin (c+d x))^{m-\frac{3}{2}} (a \sin (c+d x)+a)^m (e \cos (c+d x))^{3-2 m} \, _2F_1\left(\frac{3}{2},\frac{1}{2} (2 m-1);\frac{5}{2};\frac{1}{2} (\sin (c+d x)+1)\right)}{3 d e}",1,"(2^(3/2 - m)*(e*Cos[c + d*x])^(3 - 2*m)*Hypergeometric2F1[3/2, (-1 + 2*m)/2, 5/2, (1 + Sin[c + d*x])/2]*(1 - Sin[c + d*x])^(-3/2 + m)*(a + a*Sin[c + d*x])^m)/(3*d*e)","A",4,4,27,0.1481,1,"{2689, 7, 70, 69}"
374,1,86,0,0.0904185,"\int (e \cos (c+d x))^{-2 m} (a+a \sin (c+d x))^m \, dx","Int[(a + a*Sin[c + d*x])^m/(e*Cos[c + d*x])^(2*m),x]","\frac{2^{\frac{1}{2}-m} (1-\sin (c+d x))^{m-\frac{1}{2}} (a \sin (c+d x)+a)^m (e \cos (c+d x))^{1-2 m} \, _2F_1\left(\frac{1}{2},\frac{1}{2} (2 m+1);\frac{3}{2};\frac{1}{2} (\sin (c+d x)+1)\right)}{d e}","\frac{2^{\frac{1}{2}-m} (1-\sin (c+d x))^{m-\frac{1}{2}} (a \sin (c+d x)+a)^m (e \cos (c+d x))^{1-2 m} \, _2F_1\left(\frac{1}{2},\frac{1}{2} (2 m+1);\frac{3}{2};\frac{1}{2} (\sin (c+d x)+1)\right)}{d e}",1,"(2^(1/2 - m)*(e*Cos[c + d*x])^(1 - 2*m)*Hypergeometric2F1[1/2, (1 + 2*m)/2, 3/2, (1 + Sin[c + d*x])/2]*(1 - Sin[c + d*x])^(-1/2 + m)*(a + a*Sin[c + d*x])^m)/(d*e)","A",4,4,25,0.1600,1,"{2689, 7, 70, 69}"
375,1,87,0,0.0953067,"\int (e \cos (c+d x))^{-2-2 m} (a+a \sin (c+d x))^m \, dx","Int[(e*Cos[c + d*x])^(-2 - 2*m)*(a + a*Sin[c + d*x])^m,x]","-\frac{2^{-m-\frac{1}{2}} (1-\sin (c+d x))^{m+\frac{1}{2}} (a \sin (c+d x)+a)^m (e \cos (c+d x))^{-2 m-1} \, _2F_1\left(-\frac{1}{2},\frac{1}{2} (2 m+3);\frac{1}{2};\frac{1}{2} (\sin (c+d x)+1)\right)}{d e}","-\frac{2^{-m-\frac{1}{2}} (1-\sin (c+d x))^{m+\frac{1}{2}} (a \sin (c+d x)+a)^m (e \cos (c+d x))^{-2 m-1} \, _2F_1\left(-\frac{1}{2},\frac{1}{2} (2 m+3);\frac{1}{2};\frac{1}{2} (\sin (c+d x)+1)\right)}{d e}",1,"-((2^(-1/2 - m)*(e*Cos[c + d*x])^(-1 - 2*m)*Hypergeometric2F1[-1/2, (3 + 2*m)/2, 1/2, (1 + Sin[c + d*x])/2]*(1 - Sin[c + d*x])^(1/2 + m)*(a + a*Sin[c + d*x])^m)/(d*e))","A",4,4,27,0.1481,1,"{2689, 7, 70, 69}"
376,1,60,0,0.0422384,"\int \cos ^5(c+d x) (a+b \sin (c+d x)) \, dx","Int[Cos[c + d*x]^5*(a + b*Sin[c + d*x]),x]","\frac{a \sin ^5(c+d x)}{5 d}-\frac{2 a \sin ^3(c+d x)}{3 d}+\frac{a \sin (c+d x)}{d}-\frac{b \cos ^6(c+d x)}{6 d}","\frac{a \sin ^5(c+d x)}{5 d}-\frac{2 a \sin ^3(c+d x)}{3 d}+\frac{a \sin (c+d x)}{d}-\frac{b \cos ^6(c+d x)}{6 d}",1,"-(b*Cos[c + d*x]^6)/(6*d) + (a*Sin[c + d*x])/d - (2*a*Sin[c + d*x]^3)/(3*d) + (a*Sin[c + d*x]^5)/(5*d)","A",4,3,19,0.1579,1,"{2668, 641, 194}"
377,1,44,0,0.0318842,"\int \cos ^3(c+d x) (a+b \sin (c+d x)) \, dx","Int[Cos[c + d*x]^3*(a + b*Sin[c + d*x]),x]","-\frac{a \sin ^3(c+d x)}{3 d}+\frac{a \sin (c+d x)}{d}-\frac{b \cos ^4(c+d x)}{4 d}","-\frac{a \sin ^3(c+d x)}{3 d}+\frac{a \sin (c+d x)}{d}-\frac{b \cos ^4(c+d x)}{4 d}",1,"-(b*Cos[c + d*x]^4)/(4*d) + (a*Sin[c + d*x])/d - (a*Sin[c + d*x]^3)/(3*d)","A",3,2,19,0.1053,1,"{2668, 641}"
378,1,28,0,0.0164447,"\int \cos (c+d x) (a+b \sin (c+d x)) \, dx","Int[Cos[c + d*x]*(a + b*Sin[c + d*x]),x]","\frac{a \sin (c+d x)}{d}+\frac{b \sin ^2(c+d x)}{2 d}","\frac{(a+b \sin (c+d x))^2}{2 b d}",1,"(a*Sin[c + d*x])/d + (b*Sin[c + d*x]^2)/(2*d)","A",2,1,17,0.05882,1,"{2668}"
379,1,43,0,0.0400481,"\int \sec (c+d x) (a+b \sin (c+d x)) \, dx","Int[Sec[c + d*x]*(a + b*Sin[c + d*x]),x]","\frac{(a-b) \log (\sin (c+d x)+1)}{2 d}-\frac{(a+b) \log (1-\sin (c+d x))}{2 d}","\frac{(a-b) \log (\sin (c+d x)+1)}{2 d}-\frac{(a+b) \log (1-\sin (c+d x))}{2 d}",1,"-((a + b)*Log[1 - Sin[c + d*x]])/(2*d) + ((a - b)*Log[1 + Sin[c + d*x]])/(2*d)","A",4,3,17,0.1765,1,"{2668, 633, 31}"
380,1,41,0,0.037927,"\int \sec ^3(c+d x) (a+b \sin (c+d x)) \, dx","Int[Sec[c + d*x]^3*(a + b*Sin[c + d*x]),x]","\frac{\sec ^2(c+d x) (a \sin (c+d x)+b)}{2 d}+\frac{a \tanh ^{-1}(\sin (c+d x))}{2 d}","\frac{\sec ^2(c+d x) (a \sin (c+d x)+b)}{2 d}+\frac{a \tanh ^{-1}(\sin (c+d x))}{2 d}",1,"(a*ArcTanh[Sin[c + d*x]])/(2*d) + (Sec[c + d*x]^2*(b + a*Sin[c + d*x]))/(2*d)","A",3,3,19,0.1579,1,"{2668, 639, 206}"
381,1,61,0,0.0438092,"\int \sec ^5(c+d x) (a+b \sin (c+d x)) \, dx","Int[Sec[c + d*x]^5*(a + b*Sin[c + d*x]),x]","\frac{\sec ^4(c+d x) (a \sin (c+d x)+b)}{4 d}+\frac{3 a \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{3 a \tan (c+d x) \sec (c+d x)}{8 d}","\frac{\sec ^4(c+d x) (a \sin (c+d x)+b)}{4 d}+\frac{3 a \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{3 a \tan (c+d x) \sec (c+d x)}{8 d}",1,"(3*a*ArcTanh[Sin[c + d*x]])/(8*d) + (Sec[c + d*x]^4*(b + a*Sin[c + d*x]))/(4*d) + (3*a*Sec[c + d*x]*Tan[c + d*x])/(8*d)","A",4,4,19,0.2105,1,"{2668, 639, 199, 206}"
382,1,65,0,0.0449726,"\int \cos ^4(c+d x) (a+b \sin (c+d x)) \, dx","Int[Cos[c + d*x]^4*(a + b*Sin[c + d*x]),x]","\frac{a \sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac{3 a \sin (c+d x) \cos (c+d x)}{8 d}+\frac{3 a x}{8}-\frac{b \cos ^5(c+d x)}{5 d}","\frac{a \sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac{3 a \sin (c+d x) \cos (c+d x)}{8 d}+\frac{3 a x}{8}-\frac{b \cos ^5(c+d x)}{5 d}",1,"(3*a*x)/8 - (b*Cos[c + d*x]^5)/(5*d) + (3*a*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (a*Cos[c + d*x]^3*Sin[c + d*x])/(4*d)","A",4,3,19,0.1579,1,"{2669, 2635, 8}"
383,1,43,0,0.0327304,"\int \cos ^2(c+d x) (a+b \sin (c+d x)) \, dx","Int[Cos[c + d*x]^2*(a + b*Sin[c + d*x]),x]","\frac{a \sin (c+d x) \cos (c+d x)}{2 d}+\frac{a x}{2}-\frac{b \cos ^3(c+d x)}{3 d}","\frac{a \sin (c+d x) \cos (c+d x)}{2 d}+\frac{a x}{2}-\frac{b \cos ^3(c+d x)}{3 d}",1,"(a*x)/2 - (b*Cos[c + d*x]^3)/(3*d) + (a*Cos[c + d*x]*Sin[c + d*x])/(2*d)","A",3,3,19,0.1579,1,"{2669, 2635, 8}"
384,1,23,0,0.0311392,"\int \sec ^2(c+d x) (a+b \sin (c+d x)) \, dx","Int[Sec[c + d*x]^2*(a + b*Sin[c + d*x]),x]","\frac{a \tan (c+d x)}{d}+\frac{b \sec (c+d x)}{d}","\frac{a \tan (c+d x)}{d}+\frac{b \sec (c+d x)}{d}",1,"(b*Sec[c + d*x])/d + (a*Tan[c + d*x])/d","A",3,3,19,0.1579,1,"{2669, 3767, 8}"
385,1,44,0,0.035826,"\int \sec ^4(c+d x) (a+b \sin (c+d x)) \, dx","Int[Sec[c + d*x]^4*(a + b*Sin[c + d*x]),x]","\frac{a \tan ^3(c+d x)}{3 d}+\frac{a \tan (c+d x)}{d}+\frac{b \sec ^3(c+d x)}{3 d}","\frac{a \tan ^3(c+d x)}{3 d}+\frac{a \tan (c+d x)}{d}+\frac{b \sec ^3(c+d x)}{3 d}",1,"(b*Sec[c + d*x]^3)/(3*d) + (a*Tan[c + d*x])/d + (a*Tan[c + d*x]^3)/(3*d)","A",3,2,19,0.1053,1,"{2669, 3767}"
386,1,60,0,0.0400507,"\int \sec ^6(c+d x) (a+b \sin (c+d x)) \, dx","Int[Sec[c + d*x]^6*(a + b*Sin[c + d*x]),x]","\frac{a \tan ^5(c+d x)}{5 d}+\frac{2 a \tan ^3(c+d x)}{3 d}+\frac{a \tan (c+d x)}{d}+\frac{b \sec ^5(c+d x)}{5 d}","\frac{a \tan ^5(c+d x)}{5 d}+\frac{2 a \tan ^3(c+d x)}{3 d}+\frac{a \tan (c+d x)}{d}+\frac{b \sec ^5(c+d x)}{5 d}",1,"(b*Sec[c + d*x]^5)/(5*d) + (a*Tan[c + d*x])/d + (2*a*Tan[c + d*x]^3)/(3*d) + (a*Tan[c + d*x]^5)/(5*d)","A",3,2,19,0.1053,1,"{2669, 3767}"
387,1,99,0,0.089251,"\int \cos ^5(c+d x) (a+b \sin (c+d x))^2 \, dx","Int[Cos[c + d*x]^5*(a + b*Sin[c + d*x])^2,x]","\frac{\left(a^2-2 b^2\right) \sin ^5(c+d x)}{5 d}-\frac{\left(2 a^2-b^2\right) \sin ^3(c+d x)}{3 d}+\frac{a^2 \sin (c+d x)}{d}-\frac{a b \cos ^6(c+d x)}{3 d}+\frac{b^2 \sin ^7(c+d x)}{7 d}","\frac{\left(a^2-2 b^2\right) \sin ^5(c+d x)}{5 d}-\frac{\left(2 a^2-b^2\right) \sin ^3(c+d x)}{3 d}+\frac{a^2 \sin (c+d x)}{d}-\frac{a b \cos ^6(c+d x)}{3 d}+\frac{b^2 \sin ^7(c+d x)}{7 d}",1,"-(a*b*Cos[c + d*x]^6)/(3*d) + (a^2*Sin[c + d*x])/d - ((2*a^2 - b^2)*Sin[c + d*x]^3)/(3*d) + ((a^2 - 2*b^2)*Sin[c + d*x]^5)/(5*d) + (b^2*Sin[c + d*x]^7)/(7*d)","A",4,3,21,0.1429,1,"{2668, 696, 1810}"
388,1,77,0,0.0707126,"\int \cos ^3(c+d x) (a+b \sin (c+d x))^2 \, dx","Int[Cos[c + d*x]^3*(a + b*Sin[c + d*x])^2,x]","-\frac{\left(a^2-b^2\right) (a+b \sin (c+d x))^3}{3 b^3 d}-\frac{(a+b \sin (c+d x))^5}{5 b^3 d}+\frac{a (a+b \sin (c+d x))^4}{2 b^3 d}","-\frac{\left(a^2-b^2\right) (a+b \sin (c+d x))^3}{3 b^3 d}-\frac{(a+b \sin (c+d x))^5}{5 b^3 d}+\frac{a (a+b \sin (c+d x))^4}{2 b^3 d}",1,"-((a^2 - b^2)*(a + b*Sin[c + d*x])^3)/(3*b^3*d) + (a*(a + b*Sin[c + d*x])^4)/(2*b^3*d) - (a + b*Sin[c + d*x])^5/(5*b^3*d)","A",3,2,21,0.09524,1,"{2668, 697}"
389,1,22,0,0.0269699,"\int \cos (c+d x) (a+b \sin (c+d x))^2 \, dx","Int[Cos[c + d*x]*(a + b*Sin[c + d*x])^2,x]","\frac{(a+b \sin (c+d x))^3}{3 b d}","\frac{(a+b \sin (c+d x))^3}{3 b d}",1,"(a + b*Sin[c + d*x])^3/(3*b*d)","A",2,2,19,0.1053,1,"{2668, 32}"
390,1,61,0,0.0818299,"\int \sec (c+d x) (a+b \sin (c+d x))^2 \, dx","Int[Sec[c + d*x]*(a + b*Sin[c + d*x])^2,x]","\frac{(a-b)^2 \log (\sin (c+d x)+1)}{2 d}-\frac{(a+b)^2 \log (1-\sin (c+d x))}{2 d}-\frac{b^2 \sin (c+d x)}{d}","\frac{(a-b)^2 \log (\sin (c+d x)+1)}{2 d}-\frac{(a+b)^2 \log (1-\sin (c+d x))}{2 d}-\frac{b^2 \sin (c+d x)}{d}",1,"-((a + b)^2*Log[1 - Sin[c + d*x]])/(2*d) + ((a - b)^2*Log[1 + Sin[c + d*x]])/(2*d) - (b^2*Sin[c + d*x])/d","A",6,4,19,0.2105,1,"{2668, 702, 633, 31}"
391,1,59,0,0.0612642,"\int \sec ^3(c+d x) (a+b \sin (c+d x))^2 \, dx","Int[Sec[c + d*x]^3*(a + b*Sin[c + d*x])^2,x]","\frac{\left(a^2-b^2\right) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{\sec ^2(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))}{2 d}","\frac{\left(a^2-b^2\right) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{\sec ^2(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))}{2 d}",1,"((a^2 - b^2)*ArcTanh[Sin[c + d*x]])/(2*d) + (Sec[c + d*x]^2*(b + a*Sin[c + d*x])*(a + b*Sin[c + d*x]))/(2*d)","A",3,3,21,0.1429,1,"{2668, 723, 206}"
392,1,99,0,0.0878322,"\int \sec ^5(c+d x) (a+b \sin (c+d x))^2 \, dx","Int[Sec[c + d*x]^5*(a + b*Sin[c + d*x])^2,x]","\frac{\left(3 a^2-b^2\right) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{\sec ^2(c+d x) \left(\left(3 a^2-b^2\right) \sin (c+d x)+2 a b\right)}{8 d}+\frac{\sec ^4(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))}{4 d}","\frac{\left(3 a^2-b^2\right) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{\sec ^2(c+d x) \left(\left(3 a^2-b^2\right) \sin (c+d x)+2 a b\right)}{8 d}+\frac{\sec ^4(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))}{4 d}",1,"((3*a^2 - b^2)*ArcTanh[Sin[c + d*x]])/(8*d) + (Sec[c + d*x]^4*(b + a*Sin[c + d*x])*(a + b*Sin[c + d*x]))/(4*d) + (Sec[c + d*x]^2*(2*a*b + (3*a^2 - b^2)*Sin[c + d*x]))/(8*d)","A",4,4,21,0.1905,1,"{2668, 739, 639, 206}"
393,1,146,0,0.1335608,"\int \cos ^6(c+d x) (a+b \sin (c+d x))^2 \, dx","Int[Cos[c + d*x]^6*(a + b*Sin[c + d*x])^2,x]","\frac{\left(8 a^2+b^2\right) \sin (c+d x) \cos ^5(c+d x)}{48 d}+\frac{5 \left(8 a^2+b^2\right) \sin (c+d x) \cos ^3(c+d x)}{192 d}+\frac{5 \left(8 a^2+b^2\right) \sin (c+d x) \cos (c+d x)}{128 d}+\frac{5}{128} x \left(8 a^2+b^2\right)-\frac{9 a b \cos ^7(c+d x)}{56 d}-\frac{b \cos ^7(c+d x) (a+b \sin (c+d x))}{8 d}","\frac{\left(8 a^2+b^2\right) \sin (c+d x) \cos ^5(c+d x)}{48 d}+\frac{5 \left(8 a^2+b^2\right) \sin (c+d x) \cos ^3(c+d x)}{192 d}+\frac{5 \left(8 a^2+b^2\right) \sin (c+d x) \cos (c+d x)}{128 d}+\frac{5}{128} x \left(8 a^2+b^2\right)-\frac{9 a b \cos ^7(c+d x)}{56 d}-\frac{b \cos ^7(c+d x) (a+b \sin (c+d x))}{8 d}",1,"(5*(8*a^2 + b^2)*x)/128 - (9*a*b*Cos[c + d*x]^7)/(56*d) + (5*(8*a^2 + b^2)*Cos[c + d*x]*Sin[c + d*x])/(128*d) + (5*(8*a^2 + b^2)*Cos[c + d*x]^3*Sin[c + d*x])/(192*d) + ((8*a^2 + b^2)*Cos[c + d*x]^5*Sin[c + d*x])/(48*d) - (b*Cos[c + d*x]^7*(a + b*Sin[c + d*x]))/(8*d)","A",6,4,21,0.1905,1,"{2692, 2669, 2635, 8}"
394,1,116,0,0.1158153,"\int \cos ^4(c+d x) (a+b \sin (c+d x))^2 \, dx","Int[Cos[c + d*x]^4*(a + b*Sin[c + d*x])^2,x]","\frac{\left(6 a^2+b^2\right) \sin (c+d x) \cos ^3(c+d x)}{24 d}+\frac{\left(6 a^2+b^2\right) \sin (c+d x) \cos (c+d x)}{16 d}+\frac{1}{16} x \left(6 a^2+b^2\right)-\frac{7 a b \cos ^5(c+d x)}{30 d}-\frac{b \cos ^5(c+d x) (a+b \sin (c+d x))}{6 d}","\frac{\left(6 a^2+b^2\right) \sin (c+d x) \cos ^3(c+d x)}{24 d}+\frac{\left(6 a^2+b^2\right) \sin (c+d x) \cos (c+d x)}{16 d}+\frac{1}{16} x \left(6 a^2+b^2\right)-\frac{7 a b \cos ^5(c+d x)}{30 d}-\frac{b \cos ^5(c+d x) (a+b \sin (c+d x))}{6 d}",1,"((6*a^2 + b^2)*x)/16 - (7*a*b*Cos[c + d*x]^5)/(30*d) + ((6*a^2 + b^2)*Cos[c + d*x]*Sin[c + d*x])/(16*d) + ((6*a^2 + b^2)*Cos[c + d*x]^3*Sin[c + d*x])/(24*d) - (b*Cos[c + d*x]^5*(a + b*Sin[c + d*x]))/(6*d)","A",5,4,21,0.1905,1,"{2692, 2669, 2635, 8}"
395,1,86,0,0.0954186,"\int \cos ^2(c+d x) (a+b \sin (c+d x))^2 \, dx","Int[Cos[c + d*x]^2*(a + b*Sin[c + d*x])^2,x]","\frac{\left(4 a^2+b^2\right) \sin (c+d x) \cos (c+d x)}{8 d}+\frac{1}{8} x \left(4 a^2+b^2\right)-\frac{5 a b \cos ^3(c+d x)}{12 d}-\frac{b \cos ^3(c+d x) (a+b \sin (c+d x))}{4 d}","\frac{\left(4 a^2+b^2\right) \sin (c+d x) \cos (c+d x)}{8 d}+\frac{1}{8} x \left(4 a^2+b^2\right)-\frac{5 a b \cos ^3(c+d x)}{12 d}-\frac{b \cos ^3(c+d x) (a+b \sin (c+d x))}{4 d}",1,"((4*a^2 + b^2)*x)/8 - (5*a*b*Cos[c + d*x]^3)/(12*d) + ((4*a^2 + b^2)*Cos[c + d*x]*Sin[c + d*x])/(8*d) - (b*Cos[c + d*x]^3*(a + b*Sin[c + d*x]))/(4*d)","A",4,4,21,0.1905,1,"{2692, 2669, 2635, 8}"
396,1,49,0,0.0484278,"\int \sec ^2(c+d x) (a+b \sin (c+d x))^2 \, dx","Int[Sec[c + d*x]^2*(a + b*Sin[c + d*x])^2,x]","\frac{a b \cos (c+d x)}{d}+\frac{\sec (c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))}{d}+b^2 (-x)","\frac{a b \cos (c+d x)}{d}+\frac{\sec (c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))}{d}+b^2 (-x)",1,"-(b^2*x) + (a*b*Cos[c + d*x])/d + (Sec[c + d*x]*(b + a*Sin[c + d*x])*(a + b*Sin[c + d*x]))/d","A",3,2,21,0.09524,1,"{2691, 2638}"
397,1,75,0,0.0962427,"\int \sec ^4(c+d x) (a+b \sin (c+d x))^2 \, dx","Int[Sec[c + d*x]^4*(a + b*Sin[c + d*x])^2,x]","\frac{\left(2 a^2-b^2\right) \tan (c+d x)}{3 d}+\frac{a b \sec (c+d x)}{3 d}+\frac{\sec ^3(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))}{3 d}","\frac{\left(2 a^2-b^2\right) \tan (c+d x)}{3 d}+\frac{a b \sec (c+d x)}{3 d}+\frac{\sec ^3(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))}{3 d}",1,"(a*b*Sec[c + d*x])/(3*d) + (Sec[c + d*x]^3*(b + a*Sin[c + d*x])*(a + b*Sin[c + d*x]))/(3*d) + ((2*a^2 - b^2)*Tan[c + d*x])/(3*d)","A",4,4,21,0.1905,1,"{2691, 2669, 3767, 8}"
398,1,103,0,0.101227,"\int \sec ^6(c+d x) (a+b \sin (c+d x))^2 \, dx","Int[Sec[c + d*x]^6*(a + b*Sin[c + d*x])^2,x]","\frac{\left(4 a^2-b^2\right) \tan ^3(c+d x)}{15 d}+\frac{\left(4 a^2-b^2\right) \tan (c+d x)}{5 d}+\frac{a b \sec ^3(c+d x)}{5 d}+\frac{\sec ^5(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))}{5 d}","\frac{\left(4 a^2-b^2\right) \tan ^3(c+d x)}{15 d}+\frac{\left(4 a^2-b^2\right) \tan (c+d x)}{5 d}+\frac{a b \sec ^3(c+d x)}{5 d}+\frac{\sec ^5(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))}{5 d}",1,"(a*b*Sec[c + d*x]^3)/(5*d) + (Sec[c + d*x]^5*(b + a*Sin[c + d*x])*(a + b*Sin[c + d*x]))/(5*d) + ((4*a^2 - b^2)*Tan[c + d*x])/(5*d) + ((4*a^2 - b^2)*Tan[c + d*x]^3)/(15*d)","A",4,3,21,0.1429,1,"{2691, 2669, 3767}"
399,1,129,0,0.1228622,"\int \sec ^8(c+d x) (a+b \sin (c+d x))^2 \, dx","Int[Sec[c + d*x]^8*(a + b*Sin[c + d*x])^2,x]","\frac{\left(6 a^2-b^2\right) \tan ^5(c+d x)}{35 d}+\frac{2 \left(6 a^2-b^2\right) \tan ^3(c+d x)}{21 d}+\frac{\left(6 a^2-b^2\right) \tan (c+d x)}{7 d}+\frac{a b \sec ^5(c+d x)}{7 d}+\frac{\sec ^7(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))}{7 d}","\frac{\left(6 a^2-b^2\right) \tan ^5(c+d x)}{35 d}+\frac{2 \left(6 a^2-b^2\right) \tan ^3(c+d x)}{21 d}+\frac{\left(6 a^2-b^2\right) \tan (c+d x)}{7 d}+\frac{a b \sec ^5(c+d x)}{7 d}+\frac{\sec ^7(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))}{7 d}",1,"(a*b*Sec[c + d*x]^5)/(7*d) + (Sec[c + d*x]^7*(b + a*Sin[c + d*x])*(a + b*Sin[c + d*x]))/(7*d) + ((6*a^2 - b^2)*Tan[c + d*x])/(7*d) + (2*(6*a^2 - b^2)*Tan[c + d*x]^3)/(21*d) + ((6*a^2 - b^2)*Tan[c + d*x]^5)/(35*d)","A",4,3,21,0.1429,1,"{2691, 2669, 3767}"
400,1,144,0,0.1327021,"\int \cos ^5(c+d x) (a+b \sin (c+d x))^3 \, dx","Int[Cos[c + d*x]^5*(a + b*Sin[c + d*x])^3,x]","\frac{\left(3 a^2-b^2\right) (a+b \sin (c+d x))^6}{3 b^5 d}-\frac{4 a \left(a^2-b^2\right) (a+b \sin (c+d x))^5}{5 b^5 d}+\frac{\left(a^2-b^2\right)^2 (a+b \sin (c+d x))^4}{4 b^5 d}+\frac{(a+b \sin (c+d x))^8}{8 b^5 d}-\frac{4 a (a+b \sin (c+d x))^7}{7 b^5 d}","\frac{\left(3 a^2-b^2\right) (a+b \sin (c+d x))^6}{3 b^5 d}-\frac{4 a \left(a^2-b^2\right) (a+b \sin (c+d x))^5}{5 b^5 d}+\frac{\left(a^2-b^2\right)^2 (a+b \sin (c+d x))^4}{4 b^5 d}+\frac{(a+b \sin (c+d x))^8}{8 b^5 d}-\frac{4 a (a+b \sin (c+d x))^7}{7 b^5 d}",1,"((a^2 - b^2)^2*(a + b*Sin[c + d*x])^4)/(4*b^5*d) - (4*a*(a^2 - b^2)*(a + b*Sin[c + d*x])^5)/(5*b^5*d) + ((3*a^2 - b^2)*(a + b*Sin[c + d*x])^6)/(3*b^5*d) - (4*a*(a + b*Sin[c + d*x])^7)/(7*b^5*d) + (a + b*Sin[c + d*x])^8/(8*b^5*d)","A",3,2,21,0.09524,1,"{2668, 697}"
401,1,77,0,0.0802826,"\int \cos ^3(c+d x) (a+b \sin (c+d x))^3 \, dx","Int[Cos[c + d*x]^3*(a + b*Sin[c + d*x])^3,x]","-\frac{\left(a^2-b^2\right) (a+b \sin (c+d x))^4}{4 b^3 d}-\frac{(a+b \sin (c+d x))^6}{6 b^3 d}+\frac{2 a (a+b \sin (c+d x))^5}{5 b^3 d}","-\frac{\left(a^2-b^2\right) (a+b \sin (c+d x))^4}{4 b^3 d}-\frac{(a+b \sin (c+d x))^6}{6 b^3 d}+\frac{2 a (a+b \sin (c+d x))^5}{5 b^3 d}",1,"-((a^2 - b^2)*(a + b*Sin[c + d*x])^4)/(4*b^3*d) + (2*a*(a + b*Sin[c + d*x])^5)/(5*b^3*d) - (a + b*Sin[c + d*x])^6/(6*b^3*d)","A",3,2,21,0.09524,1,"{2668, 697}"
402,1,22,0,0.0263097,"\int \cos (c+d x) (a+b \sin (c+d x))^3 \, dx","Int[Cos[c + d*x]*(a + b*Sin[c + d*x])^3,x]","\frac{(a+b \sin (c+d x))^4}{4 b d}","\frac{(a+b \sin (c+d x))^4}{4 b d}",1,"(a + b*Sin[c + d*x])^4/(4*b*d)","A",2,2,19,0.1053,1,"{2668, 32}"
403,1,80,0,0.1054915,"\int \sec (c+d x) (a+b \sin (c+d x))^3 \, dx","Int[Sec[c + d*x]*(a + b*Sin[c + d*x])^3,x]","-\frac{3 a b^2 \sin (c+d x)}{d}+\frac{(a-b)^3 \log (\sin (c+d x)+1)}{2 d}-\frac{(a+b)^3 \log (1-\sin (c+d x))}{2 d}-\frac{b^3 \sin ^2(c+d x)}{2 d}","-\frac{3 a b^2 \sin (c+d x)}{d}+\frac{(a-b)^3 \log (\sin (c+d x)+1)}{2 d}-\frac{(a+b)^3 \log (1-\sin (c+d x))}{2 d}-\frac{b^3 \sin ^2(c+d x)}{2 d}",1,"-((a + b)^3*Log[1 - Sin[c + d*x]])/(2*d) + ((a - b)^3*Log[1 + Sin[c + d*x]])/(2*d) - (3*a*b^2*Sin[c + d*x])/d - (b^3*Sin[c + d*x]^2)/(2*d)","A",6,4,19,0.2105,1,"{2668, 702, 633, 31}"
404,1,111,0,0.1342947,"\int \sec ^3(c+d x) (a+b \sin (c+d x))^3 \, dx","Int[Sec[c + d*x]^3*(a + b*Sin[c + d*x])^3,x]","\frac{a b^2 \sin (c+d x)}{2 d}+\frac{(a+2 b) (a-b)^2 \log (\sin (c+d x)+1)}{4 d}-\frac{(a-2 b) (a+b)^2 \log (1-\sin (c+d x))}{4 d}+\frac{\sec ^2(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^2}{2 d}","\frac{a b^2 \sin (c+d x)}{2 d}+\frac{(a+2 b) (a-b)^2 \log (\sin (c+d x)+1)}{4 d}-\frac{(a-2 b) (a+b)^2 \log (1-\sin (c+d x))}{4 d}+\frac{\sec ^2(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^2}{2 d}",1,"-((a - 2*b)*(a + b)^2*Log[1 - Sin[c + d*x]])/(4*d) + ((a - b)^2*(a + 2*b)*Log[1 + Sin[c + d*x]])/(4*d) + (a*b^2*Sin[c + d*x])/(2*d) + (Sec[c + d*x]^2*(b + a*Sin[c + d*x])*(a + b*Sin[c + d*x])^2)/(2*d)","A",6,5,21,0.2381,1,"{2668, 739, 774, 633, 31}"
405,1,94,0,0.0800275,"\int \sec ^5(c+d x) (a+b \sin (c+d x))^3 \, dx","Int[Sec[c + d*x]^5*(a + b*Sin[c + d*x])^3,x]","\frac{3 a \left(a^2-b^2\right) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{3 a \sec ^2(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))}{8 d}+\frac{\tan (c+d x) \sec ^3(c+d x) (a+b \sin (c+d x))^3}{4 d}","\frac{3 a \left(a^2-b^2\right) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{3 a \sec ^2(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))}{8 d}+\frac{\tan (c+d x) \sec ^3(c+d x) (a+b \sin (c+d x))^3}{4 d}",1,"(3*a*(a^2 - b^2)*ArcTanh[Sin[c + d*x]])/(8*d) + (3*a*Sec[c + d*x]^2*(b + a*Sin[c + d*x])*(a + b*Sin[c + d*x]))/(8*d) + (Sec[c + d*x]^3*(a + b*Sin[c + d*x])^3*Tan[c + d*x])/(4*d)","A",4,4,21,0.1905,1,"{2668, 729, 723, 206}"
406,1,158,0,0.2167481,"\int \cos ^4(c+d x) (a+b \sin (c+d x))^3 \, dx","Int[Cos[c + d*x]^4*(a + b*Sin[c + d*x])^3,x]","-\frac{b \left(17 a^2+4 b^2\right) \cos ^5(c+d x)}{70 d}+\frac{a \left(2 a^2+b^2\right) \sin (c+d x) \cos ^3(c+d x)}{8 d}+\frac{3 a \left(2 a^2+b^2\right) \sin (c+d x) \cos (c+d x)}{16 d}+\frac{3}{16} a x \left(2 a^2+b^2\right)-\frac{b \cos ^5(c+d x) (a+b \sin (c+d x))^2}{7 d}-\frac{3 a b \cos ^5(c+d x) (a+b \sin (c+d x))}{14 d}","-\frac{b \left(17 a^2+4 b^2\right) \cos ^5(c+d x)}{70 d}+\frac{a \left(2 a^2+b^2\right) \sin (c+d x) \cos ^3(c+d x)}{8 d}+\frac{3 a \left(2 a^2+b^2\right) \sin (c+d x) \cos (c+d x)}{16 d}+\frac{3}{16} a x \left(2 a^2+b^2\right)-\frac{b \cos ^5(c+d x) (a+b \sin (c+d x))^2}{7 d}-\frac{3 a b \cos ^5(c+d x) (a+b \sin (c+d x))}{14 d}",1,"(3*a*(2*a^2 + b^2)*x)/16 - (b*(17*a^2 + 4*b^2)*Cos[c + d*x]^5)/(70*d) + (3*a*(2*a^2 + b^2)*Cos[c + d*x]*Sin[c + d*x])/(16*d) + (a*(2*a^2 + b^2)*Cos[c + d*x]^3*Sin[c + d*x])/(8*d) - (3*a*b*Cos[c + d*x]^5*(a + b*Sin[c + d*x]))/(14*d) - (b*Cos[c + d*x]^5*(a + b*Sin[c + d*x])^2)/(7*d)","A",6,5,21,0.2381,1,"{2692, 2862, 2669, 2635, 8}"
407,1,131,0,0.1935779,"\int \cos ^2(c+d x) (a+b \sin (c+d x))^3 \, dx","Int[Cos[c + d*x]^2*(a + b*Sin[c + d*x])^3,x]","-\frac{b \left(27 a^2+8 b^2\right) \cos ^3(c+d x)}{60 d}+\frac{a \left(4 a^2+3 b^2\right) \sin (c+d x) \cos (c+d x)}{8 d}+\frac{1}{8} a x \left(4 a^2+3 b^2\right)-\frac{b \cos ^3(c+d x) (a+b \sin (c+d x))^2}{5 d}-\frac{7 a b \cos ^3(c+d x) (a+b \sin (c+d x))}{20 d}","-\frac{b \left(27 a^2+8 b^2\right) \cos ^3(c+d x)}{60 d}+\frac{a \left(4 a^2+3 b^2\right) \sin (c+d x) \cos (c+d x)}{8 d}+\frac{1}{8} a x \left(4 a^2+3 b^2\right)-\frac{b \cos ^3(c+d x) (a+b \sin (c+d x))^2}{5 d}-\frac{7 a b \cos ^3(c+d x) (a+b \sin (c+d x))}{20 d}",1,"(a*(4*a^2 + 3*b^2)*x)/8 - (b*(27*a^2 + 8*b^2)*Cos[c + d*x]^3)/(60*d) + (a*(4*a^2 + 3*b^2)*Cos[c + d*x]*Sin[c + d*x])/(8*d) - (7*a*b*Cos[c + d*x]^3*(a + b*Sin[c + d*x]))/(20*d) - (b*Cos[c + d*x]^3*(a + b*Sin[c + d*x])^2)/(5*d)","A",5,5,21,0.2381,1,"{2692, 2862, 2669, 2635, 8}"
408,1,79,0,0.0704697,"\int \sec ^2(c+d x) (a+b \sin (c+d x))^3 \, dx","Int[Sec[c + d*x]^2*(a + b*Sin[c + d*x])^3,x]","\frac{2 b \left(a^2+b^2\right) \cos (c+d x)}{d}+\frac{a b^2 \sin (c+d x) \cos (c+d x)}{d}-3 a b^2 x+\frac{\sec (c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^2}{d}","\frac{2 b \left(a^2+b^2\right) \cos (c+d x)}{d}+\frac{a b^2 \sin (c+d x) \cos (c+d x)}{d}-3 a b^2 x+\frac{\sec (c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^2}{d}",1,"-3*a*b^2*x + (2*b*(a^2 + b^2)*Cos[c + d*x])/d + (a*b^2*Cos[c + d*x]*Sin[c + d*x])/d + (Sec[c + d*x]*(b + a*Sin[c + d*x])*(a + b*Sin[c + d*x])^2)/d","A",2,2,21,0.09524,1,"{2691, 2734}"
409,1,84,0,0.088474,"\int \sec ^4(c+d x) (a+b \sin (c+d x))^3 \, dx","Int[Sec[c + d*x]^4*(a + b*Sin[c + d*x])^3,x]","\frac{2 a \left(a^2-b^2\right) \tan (c+d x)}{3 d}+\frac{2 b \left(a^2-b^2\right) \sec (c+d x)}{3 d}+\frac{\sec ^3(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^2}{3 d}","\frac{2 a \left(a^2-b^2\right) \tan (c+d x)}{3 d}+\frac{2 b \left(a^2-b^2\right) \sec (c+d x)}{3 d}+\frac{\sec ^3(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^2}{3 d}",1,"(2*b*(a^2 - b^2)*Sec[c + d*x])/(3*d) + (Sec[c + d*x]^3*(b + a*Sin[c + d*x])*(a + b*Sin[c + d*x])^2)/(3*d) + (2*a*(a^2 - b^2)*Tan[c + d*x])/(3*d)","A",5,5,21,0.2381,1,"{2691, 12, 2669, 3767, 8}"
410,1,135,0,0.1921595,"\int \sec ^6(c+d x) (a+b \sin (c+d x))^3 \, dx","Int[Sec[c + d*x]^6*(a + b*Sin[c + d*x])^3,x]","\frac{2 a \left(4 a^2-3 b^2\right) \tan (c+d x)}{15 d}+\frac{2 b \left(2 a^2-b^2\right) \sec (c+d x)}{15 d}+\frac{2 \sec ^3(c+d x) (a+b \sin (c+d x)) \left(\left(2 a^2-b^2\right) \sin (c+d x)+a b\right)}{15 d}+\frac{\sec ^5(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^2}{5 d}","\frac{2 a \left(4 a^2-3 b^2\right) \tan (c+d x)}{15 d}+\frac{2 b \left(2 a^2-b^2\right) \sec (c+d x)}{15 d}+\frac{2 \sec ^3(c+d x) (a+b \sin (c+d x)) \left(\left(2 a^2-b^2\right) \sin (c+d x)+a b\right)}{15 d}+\frac{\sec ^5(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^2}{5 d}",1,"(2*b*(2*a^2 - b^2)*Sec[c + d*x])/(15*d) + (Sec[c + d*x]^5*(b + a*Sin[c + d*x])*(a + b*Sin[c + d*x])^2)/(5*d) + (2*Sec[c + d*x]^3*(a + b*Sin[c + d*x])*(a*b + (2*a^2 - b^2)*Sin[c + d*x]))/(15*d) + (2*a*(4*a^2 - 3*b^2)*Tan[c + d*x])/(15*d)","A",5,5,21,0.2381,1,"{2691, 2861, 2669, 3767, 8}"
411,1,165,0,0.2070017,"\int \sec ^8(c+d x) (a+b \sin (c+d x))^3 \, dx","Int[Sec[c + d*x]^8*(a + b*Sin[c + d*x])^3,x]","\frac{4 a \left(2 a^2-b^2\right) \tan ^3(c+d x)}{35 d}+\frac{12 a \left(2 a^2-b^2\right) \tan (c+d x)}{35 d}+\frac{2 b \left(3 a^2-b^2\right) \sec ^3(c+d x)}{35 d}+\frac{2 \sec ^5(c+d x) (a+b \sin (c+d x)) \left(\left(3 a^2-b^2\right) \sin (c+d x)+2 a b\right)}{35 d}+\frac{\sec ^7(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^2}{7 d}","\frac{4 a \left(2 a^2-b^2\right) \tan ^3(c+d x)}{35 d}+\frac{12 a \left(2 a^2-b^2\right) \tan (c+d x)}{35 d}+\frac{2 b \left(3 a^2-b^2\right) \sec ^3(c+d x)}{35 d}+\frac{2 \sec ^5(c+d x) (a+b \sin (c+d x)) \left(\left(3 a^2-b^2\right) \sin (c+d x)+2 a b\right)}{35 d}+\frac{\sec ^7(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^2}{7 d}",1,"(2*b*(3*a^2 - b^2)*Sec[c + d*x]^3)/(35*d) + (Sec[c + d*x]^7*(b + a*Sin[c + d*x])*(a + b*Sin[c + d*x])^2)/(7*d) + (2*Sec[c + d*x]^5*(a + b*Sin[c + d*x])*(2*a*b + (3*a^2 - b^2)*Sin[c + d*x]))/(35*d) + (12*a*(2*a^2 - b^2)*Tan[c + d*x])/(35*d) + (4*a*(2*a^2 - b^2)*Tan[c + d*x]^3)/(35*d)","A",5,4,21,0.1905,1,"{2691, 2861, 2669, 3767}"
412,1,192,0,0.2213968,"\int \sec ^{10}(c+d x) (a+b \sin (c+d x))^3 \, dx","Int[Sec[c + d*x]^10*(a + b*Sin[c + d*x])^3,x]","\frac{2 a \left(8 a^2-3 b^2\right) \tan ^5(c+d x)}{105 d}+\frac{4 a \left(8 a^2-3 b^2\right) \tan ^3(c+d x)}{63 d}+\frac{2 a \left(8 a^2-3 b^2\right) \tan (c+d x)}{21 d}+\frac{2 b \left(4 a^2-b^2\right) \sec ^5(c+d x)}{63 d}+\frac{2 \sec ^7(c+d x) (a+b \sin (c+d x)) \left(\left(4 a^2-b^2\right) \sin (c+d x)+3 a b\right)}{63 d}+\frac{\sec ^9(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^2}{9 d}","\frac{2 a \left(8 a^2-3 b^2\right) \tan ^5(c+d x)}{105 d}+\frac{4 a \left(8 a^2-3 b^2\right) \tan ^3(c+d x)}{63 d}+\frac{2 a \left(8 a^2-3 b^2\right) \tan (c+d x)}{21 d}+\frac{2 b \left(4 a^2-b^2\right) \sec ^5(c+d x)}{63 d}+\frac{2 \sec ^7(c+d x) (a+b \sin (c+d x)) \left(\left(4 a^2-b^2\right) \sin (c+d x)+3 a b\right)}{63 d}+\frac{\sec ^9(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^2}{9 d}",1,"(2*b*(4*a^2 - b^2)*Sec[c + d*x]^5)/(63*d) + (Sec[c + d*x]^9*(b + a*Sin[c + d*x])*(a + b*Sin[c + d*x])^2)/(9*d) + (2*Sec[c + d*x]^7*(a + b*Sin[c + d*x])*(3*a*b + (4*a^2 - b^2)*Sin[c + d*x]))/(63*d) + (2*a*(8*a^2 - 3*b^2)*Tan[c + d*x])/(21*d) + (4*a*(8*a^2 - 3*b^2)*Tan[c + d*x]^3)/(63*d) + (2*a*(8*a^2 - 3*b^2)*Tan[c + d*x]^5)/(105*d)","A",5,4,21,0.1905,1,"{2691, 2861, 2669, 3767}"
413,1,144,0,0.221048,"\int \cos ^5(c+d x) (a+b \sin (c+d x))^8 \, dx","Int[Cos[c + d*x]^5*(a + b*Sin[c + d*x])^8,x]","\frac{2 \left(3 a^2-b^2\right) (a+b \sin (c+d x))^{11}}{11 b^5 d}-\frac{2 a \left(a^2-b^2\right) (a+b \sin (c+d x))^{10}}{5 b^5 d}+\frac{\left(a^2-b^2\right)^2 (a+b \sin (c+d x))^9}{9 b^5 d}+\frac{(a+b \sin (c+d x))^{13}}{13 b^5 d}-\frac{a (a+b \sin (c+d x))^{12}}{3 b^5 d}","\frac{2 \left(3 a^2-b^2\right) (a+b \sin (c+d x))^{11}}{11 b^5 d}-\frac{2 a \left(a^2-b^2\right) (a+b \sin (c+d x))^{10}}{5 b^5 d}+\frac{\left(a^2-b^2\right)^2 (a+b \sin (c+d x))^9}{9 b^5 d}+\frac{(a+b \sin (c+d x))^{13}}{13 b^5 d}-\frac{a (a+b \sin (c+d x))^{12}}{3 b^5 d}",1,"((a^2 - b^2)^2*(a + b*Sin[c + d*x])^9)/(9*b^5*d) - (2*a*(a^2 - b^2)*(a + b*Sin[c + d*x])^10)/(5*b^5*d) + (2*(3*a^2 - b^2)*(a + b*Sin[c + d*x])^11)/(11*b^5*d) - (a*(a + b*Sin[c + d*x])^12)/(3*b^5*d) + (a + b*Sin[c + d*x])^13/(13*b^5*d)","A",3,2,21,0.09524,1,"{2668, 697}"
414,1,77,0,0.1509836,"\int \cos ^3(c+d x) (a+b \sin (c+d x))^8 \, dx","Int[Cos[c + d*x]^3*(a + b*Sin[c + d*x])^8,x]","-\frac{\left(a^2-b^2\right) (a+b \sin (c+d x))^9}{9 b^3 d}-\frac{(a+b \sin (c+d x))^{11}}{11 b^3 d}+\frac{a (a+b \sin (c+d x))^{10}}{5 b^3 d}","-\frac{\left(a^2-b^2\right) (a+b \sin (c+d x))^9}{9 b^3 d}-\frac{(a+b \sin (c+d x))^{11}}{11 b^3 d}+\frac{a (a+b \sin (c+d x))^{10}}{5 b^3 d}",1,"-((a^2 - b^2)*(a + b*Sin[c + d*x])^9)/(9*b^3*d) + (a*(a + b*Sin[c + d*x])^10)/(5*b^3*d) - (a + b*Sin[c + d*x])^11/(11*b^3*d)","A",3,2,21,0.09524,1,"{2668, 697}"
415,1,22,0,0.0263247,"\int \cos (c+d x) (a+b \sin (c+d x))^8 \, dx","Int[Cos[c + d*x]*(a + b*Sin[c + d*x])^8,x]","\frac{(a+b \sin (c+d x))^9}{9 b d}","\frac{(a+b \sin (c+d x))^9}{9 b d}",1,"(a + b*Sin[c + d*x])^9/(9*b*d)","A",2,2,19,0.1053,1,"{2668, 32}"
416,1,245,0,0.1822917,"\int \sec (c+d x) (a+b \sin (c+d x))^8 \, dx","Int[Sec[c + d*x]*(a + b*Sin[c + d*x])^8,x]","-\frac{b^6 \left(28 a^2+b^2\right) \sin ^5(c+d x)}{5 d}-\frac{2 a b^5 \left(7 a^2+b^2\right) \sin ^4(c+d x)}{d}-\frac{b^4 \left(28 a^2 b^2+70 a^4+b^4\right) \sin ^3(c+d x)}{3 d}-\frac{4 a b^3 \left(7 a^2 b^2+7 a^4+b^4\right) \sin ^2(c+d x)}{d}-\frac{b^2 \left(70 a^4 b^2+28 a^2 b^4+28 a^6+b^6\right) \sin (c+d x)}{d}-\frac{4 a b^7 \sin ^6(c+d x)}{3 d}+\frac{(a-b)^8 \log (\sin (c+d x)+1)}{2 d}-\frac{(a+b)^8 \log (1-\sin (c+d x))}{2 d}-\frac{b^8 \sin ^7(c+d x)}{7 d}","-\frac{b^6 \left(28 a^2+b^2\right) \sin ^5(c+d x)}{5 d}-\frac{2 a b^5 \left(7 a^2+b^2\right) \sin ^4(c+d x)}{d}-\frac{b^4 \left(28 a^2 b^2+70 a^4+b^4\right) \sin ^3(c+d x)}{3 d}-\frac{4 a b^3 \left(7 a^2 b^2+7 a^4+b^4\right) \sin ^2(c+d x)}{d}-\frac{b^2 \left(70 a^4 b^2+28 a^2 b^4+28 a^6+b^6\right) \sin (c+d x)}{d}-\frac{4 a b^7 \sin ^6(c+d x)}{3 d}+\frac{(a-b)^8 \log (\sin (c+d x)+1)}{2 d}-\frac{(a+b)^8 \log (1-\sin (c+d x))}{2 d}-\frac{b^8 \sin ^7(c+d x)}{7 d}",1,"-((a + b)^8*Log[1 - Sin[c + d*x]])/(2*d) + ((a - b)^8*Log[1 + Sin[c + d*x]])/(2*d) - (b^2*(28*a^6 + 70*a^4*b^2 + 28*a^2*b^4 + b^6)*Sin[c + d*x])/d - (4*a*b^3*(7*a^4 + 7*a^2*b^2 + b^4)*Sin[c + d*x]^2)/d - (b^4*(70*a^4 + 28*a^2*b^2 + b^4)*Sin[c + d*x]^3)/(3*d) - (2*a*b^5*(7*a^2 + b^2)*Sin[c + d*x]^4)/d - (b^6*(28*a^2 + b^2)*Sin[c + d*x]^5)/(5*d) - (4*a*b^7*Sin[c + d*x]^6)/(3*d) - (b^8*Sin[c + d*x]^7)/(7*d)","A",6,4,19,0.2105,1,"{2668, 702, 633, 31}"
417,1,284,0,0.2421268,"\int \sec ^3(c+d x) (a+b \sin (c+d x))^8 \, dx","Int[Sec[c + d*x]^3*(a + b*Sin[c + d*x])^8,x]","\frac{7 b^6 \left(5 a^2+b^2\right) \sin ^5(c+d x)}{10 d}+\frac{3 a b^5 \left(7 a^2+4 b^2\right) \sin ^4(c+d x)}{2 d}+\frac{7 b^4 \left(20 a^2 b^2+15 a^4+b^4\right) \sin ^3(c+d x)}{6 d}+\frac{a b^3 \left(112 a^2 b^2+35 a^4+24 b^4\right) \sin ^2(c+d x)}{2 d}+\frac{7 b^2 \left(30 a^4 b^2+20 a^2 b^4+3 a^6+b^6\right) \sin (c+d x)}{2 d}+\frac{a b^7 \sin ^6(c+d x)}{2 d}+\frac{(a+7 b) (a-b)^7 \log (\sin (c+d x)+1)}{4 d}-\frac{(a-7 b) (a+b)^7 \log (1-\sin (c+d x))}{4 d}+\frac{\sec ^2(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^7}{2 d}","\frac{7 b^6 \left(5 a^2+b^2\right) \sin ^5(c+d x)}{10 d}+\frac{3 a b^5 \left(7 a^2+4 b^2\right) \sin ^4(c+d x)}{2 d}+\frac{7 b^4 \left(20 a^2 b^2+15 a^4+b^4\right) \sin ^3(c+d x)}{6 d}+\frac{a b^3 \left(112 a^2 b^2+35 a^4+24 b^4\right) \sin ^2(c+d x)}{2 d}+\frac{7 b^2 \left(30 a^4 b^2+20 a^2 b^4+3 a^6+b^6\right) \sin (c+d x)}{2 d}+\frac{a b^7 \sin ^6(c+d x)}{2 d}+\frac{(a+7 b) (a-b)^7 \log (\sin (c+d x)+1)}{4 d}-\frac{(a-7 b) (a+b)^7 \log (1-\sin (c+d x))}{4 d}+\frac{\sec ^2(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^7}{2 d}",1,"-((a - 7*b)*(a + b)^7*Log[1 - Sin[c + d*x]])/(4*d) + ((a - b)^7*(a + 7*b)*Log[1 + Sin[c + d*x]])/(4*d) + (7*b^2*(3*a^6 + 30*a^4*b^2 + 20*a^2*b^4 + b^6)*Sin[c + d*x])/(2*d) + (a*b^3*(35*a^4 + 112*a^2*b^2 + 24*b^4)*Sin[c + d*x]^2)/(2*d) + (7*b^4*(15*a^4 + 20*a^2*b^2 + b^4)*Sin[c + d*x]^3)/(6*d) + (3*a*b^5*(7*a^2 + 4*b^2)*Sin[c + d*x]^4)/(2*d) + (7*b^6*(5*a^2 + b^2)*Sin[c + d*x]^5)/(10*d) + (a*b^7*Sin[c + d*x]^6)/(2*d) + (Sec[c + d*x]^2*(b + a*Sin[c + d*x])*(a + b*Sin[c + d*x])^7)/(2*d)","A",7,5,21,0.2381,1,"{2668, 739, 801, 633, 31}"
418,1,320,0,0.3038288,"\int \sec ^5(c+d x) (a+b \sin (c+d x))^8 \, dx","Int[Sec[c + d*x]^5*(a + b*Sin[c + d*x])^8,x]","-\frac{a b^7 \left(13-\frac{3 a^2}{b^2}\right) \sin ^4(c+d x)}{8 d}+\frac{5 b^4 \left(-42 a^2 b^2+9 a^4-7 b^4\right) \sin ^3(c+d x)}{24 d}+\frac{a b^3 \left(-77 a^2 b^2+15 a^4-48 b^4\right) \sin ^2(c+d x)}{4 d}+\frac{5 b^2 \left(-35 a^4 b^2-84 a^2 b^4+6 a^6-7 b^6\right) \sin (c+d x)}{8 d}-\frac{(a+b)^6 \left(3 a^2-18 a b+35 b^2\right) \log (1-\sin (c+d x))}{16 d}+\frac{(a-b)^6 \left(3 a^2+18 a b+35 b^2\right) \log (\sin (c+d x)+1)}{16 d}-\frac{\sec ^2(c+d x) (a+b \sin (c+d x))^5 \left(b \left(a^2+7 b^2\right)-a \left(3 a^2-11 b^2\right) \sin (c+d x)\right)}{8 d}+\frac{\sec ^4(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^7}{4 d}","-\frac{a b^7 \left(13-\frac{3 a^2}{b^2}\right) \sin ^4(c+d x)}{8 d}+\frac{5 b^4 \left(-42 a^2 b^2+9 a^4-7 b^4\right) \sin ^3(c+d x)}{24 d}+\frac{a b^3 \left(-77 a^2 b^2+15 a^4-48 b^4\right) \sin ^2(c+d x)}{4 d}+\frac{5 b^2 \left(-35 a^4 b^2-84 a^2 b^4+6 a^6-7 b^6\right) \sin (c+d x)}{8 d}-\frac{(a+b)^6 \left(3 a^2-18 a b+35 b^2\right) \log (1-\sin (c+d x))}{16 d}+\frac{(a-b)^6 \left(3 a^2+18 a b+35 b^2\right) \log (\sin (c+d x)+1)}{16 d}-\frac{\sec ^2(c+d x) (a+b \sin (c+d x))^5 \left(b \left(a^2+7 b^2\right)-a \left(3 a^2-11 b^2\right) \sin (c+d x)\right)}{8 d}+\frac{\sec ^4(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^7}{4 d}",1,"-((a + b)^6*(3*a^2 - 18*a*b + 35*b^2)*Log[1 - Sin[c + d*x]])/(16*d) + ((a - b)^6*(3*a^2 + 18*a*b + 35*b^2)*Log[1 + Sin[c + d*x]])/(16*d) + (5*b^2*(6*a^6 - 35*a^4*b^2 - 84*a^2*b^4 - 7*b^6)*Sin[c + d*x])/(8*d) + (a*b^3*(15*a^4 - 77*a^2*b^2 - 48*b^4)*Sin[c + d*x]^2)/(4*d) + (5*b^4*(9*a^4 - 42*a^2*b^2 - 7*b^4)*Sin[c + d*x]^3)/(24*d) - (a*(13 - (3*a^2)/b^2)*b^7*Sin[c + d*x]^4)/(8*d) + (Sec[c + d*x]^4*(b + a*Sin[c + d*x])*(a + b*Sin[c + d*x])^7)/(4*d) - (Sec[c + d*x]^2*(a + b*Sin[c + d*x])^5*(b*(a^2 + 7*b^2) - a*(3*a^2 - 11*b^2)*Sin[c + d*x]))/(8*d)","A",8,6,21,0.2857,1,"{2668, 739, 819, 801, 633, 31}"
419,1,423,0,1.2170591,"\int \cos ^2(c+d x) (a+b \sin (c+d x))^8 \, dx","Int[Cos[c + d*x]^2*(a + b*Sin[c + d*x])^8,x]","-\frac{11 a b \left(10536 a^4 b^2+9588 a^2 b^4+1792 a^6+1289 b^6\right) \cos ^3(c+d x)}{40320 d}-\frac{b \left(64 a^2+21 b^2\right) \cos ^3(c+d x) (a+b \sin (c+d x))^5}{240 d}-\frac{a b \left(112 a^2+109 b^2\right) \cos ^3(c+d x) (a+b \sin (c+d x))^4}{336 d}-\frac{b \left(1500 a^2 b^2+784 a^4+147 b^4\right) \cos ^3(c+d x) (a+b \sin (c+d x))^3}{2016 d}-\frac{13 a b \left(348 a^2 b^2+112 a^4+101 b^4\right) \cos ^3(c+d x) (a+b \sin (c+d x))^2}{3360 d}-\frac{b \left(28088 a^4 b^2+15956 a^2 b^4+6272 a^6+735 b^6\right) \cos ^3(c+d x) (a+b \sin (c+d x))}{13440 d}+\frac{\left(896 a^6 b^2+1120 a^4 b^4+280 a^2 b^6+128 a^8+7 b^8\right) \sin (c+d x) \cos (c+d x)}{256 d}+\frac{1}{256} x \left(896 a^6 b^2+1120 a^4 b^4+280 a^2 b^6+128 a^8+7 b^8\right)-\frac{b \cos ^3(c+d x) (a+b \sin (c+d x))^7}{10 d}-\frac{17 a b \cos ^3(c+d x) (a+b \sin (c+d x))^6}{90 d}","-\frac{11 a b \left(10536 a^4 b^2+9588 a^2 b^4+1792 a^6+1289 b^6\right) \cos ^3(c+d x)}{40320 d}-\frac{b \left(64 a^2+21 b^2\right) \cos ^3(c+d x) (a+b \sin (c+d x))^5}{240 d}-\frac{a b \left(112 a^2+109 b^2\right) \cos ^3(c+d x) (a+b \sin (c+d x))^4}{336 d}-\frac{b \left(1500 a^2 b^2+784 a^4+147 b^4\right) \cos ^3(c+d x) (a+b \sin (c+d x))^3}{2016 d}-\frac{13 a b \left(348 a^2 b^2+112 a^4+101 b^4\right) \cos ^3(c+d x) (a+b \sin (c+d x))^2}{3360 d}-\frac{b \left(28088 a^4 b^2+15956 a^2 b^4+6272 a^6+735 b^6\right) \cos ^3(c+d x) (a+b \sin (c+d x))}{13440 d}+\frac{\left(896 a^6 b^2+1120 a^4 b^4+280 a^2 b^6+128 a^8+7 b^8\right) \sin (c+d x) \cos (c+d x)}{256 d}+\frac{1}{256} x \left(896 a^6 b^2+1120 a^4 b^4+280 a^2 b^6+128 a^8+7 b^8\right)-\frac{b \cos ^3(c+d x) (a+b \sin (c+d x))^7}{10 d}-\frac{17 a b \cos ^3(c+d x) (a+b \sin (c+d x))^6}{90 d}",1,"((128*a^8 + 896*a^6*b^2 + 1120*a^4*b^4 + 280*a^2*b^6 + 7*b^8)*x)/256 - (11*a*b*(1792*a^6 + 10536*a^4*b^2 + 9588*a^2*b^4 + 1289*b^6)*Cos[c + d*x]^3)/(40320*d) + ((128*a^8 + 896*a^6*b^2 + 1120*a^4*b^4 + 280*a^2*b^6 + 7*b^8)*Cos[c + d*x]*Sin[c + d*x])/(256*d) - (b*(6272*a^6 + 28088*a^4*b^2 + 15956*a^2*b^4 + 735*b^6)*Cos[c + d*x]^3*(a + b*Sin[c + d*x]))/(13440*d) - (13*a*b*(112*a^4 + 348*a^2*b^2 + 101*b^4)*Cos[c + d*x]^3*(a + b*Sin[c + d*x])^2)/(3360*d) - (b*(784*a^4 + 1500*a^2*b^2 + 147*b^4)*Cos[c + d*x]^3*(a + b*Sin[c + d*x])^3)/(2016*d) - (a*b*(112*a^2 + 109*b^2)*Cos[c + d*x]^3*(a + b*Sin[c + d*x])^4)/(336*d) - (b*(64*a^2 + 21*b^2)*Cos[c + d*x]^3*(a + b*Sin[c + d*x])^5)/(240*d) - (17*a*b*Cos[c + d*x]^3*(a + b*Sin[c + d*x])^6)/(90*d) - (b*Cos[c + d*x]^3*(a + b*Sin[c + d*x])^7)/(10*d)","A",10,5,21,0.2381,1,"{2692, 2862, 2669, 2635, 8}"
420,1,349,0,0.563405,"\int \sec ^2(c+d x) (a+b \sin (c+d x))^8 \, dx","Int[Sec[c + d*x]^2*(a + b*Sin[c + d*x])^8,x]","\frac{a b \left(1664 a^4 b^2+2789 a^2 b^4+40 a^6+512 b^6\right) \cos (c+d x)}{20 d}+\frac{b \left(6 a^2+7 b^2\right) \cos (c+d x) (a+b \sin (c+d x))^5}{6 d}+\frac{a b \left(30 a^2+113 b^2\right) \cos (c+d x) (a+b \sin (c+d x))^4}{30 d}+\frac{b \left(992 a^2 b^2+120 a^4+175 b^4\right) \cos (c+d x) (a+b \sin (c+d x))^3}{120 d}+\frac{a b \left(624 a^2 b^2+40 a^4+337 b^4\right) \cos (c+d x) (a+b \sin (c+d x))^2}{40 d}+\frac{b^2 \left(2248 a^4 b^2+2502 a^2 b^4+80 a^6+175 b^6\right) \sin (c+d x) \cos (c+d x)}{80 d}-\frac{7}{16} b^2 x \left(240 a^4 b^2+120 a^2 b^4+64 a^6+5 b^6\right)+\frac{a b \cos (c+d x) (a+b \sin (c+d x))^6}{d}+\frac{\sec (c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^7}{d}","\frac{a b \left(1664 a^4 b^2+2789 a^2 b^4+40 a^6+512 b^6\right) \cos (c+d x)}{20 d}+\frac{b \left(6 a^2+7 b^2\right) \cos (c+d x) (a+b \sin (c+d x))^5}{6 d}+\frac{a b \left(30 a^2+113 b^2\right) \cos (c+d x) (a+b \sin (c+d x))^4}{30 d}+\frac{b \left(992 a^2 b^2+120 a^4+175 b^4\right) \cos (c+d x) (a+b \sin (c+d x))^3}{120 d}+\frac{a b \left(624 a^2 b^2+40 a^4+337 b^4\right) \cos (c+d x) (a+b \sin (c+d x))^2}{40 d}+\frac{b^2 \left(2248 a^4 b^2+2502 a^2 b^4+80 a^6+175 b^6\right) \sin (c+d x) \cos (c+d x)}{80 d}-\frac{7}{16} b^2 x \left(240 a^4 b^2+120 a^2 b^4+64 a^6+5 b^6\right)+\frac{a b \cos (c+d x) (a+b \sin (c+d x))^6}{d}+\frac{\sec (c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^7}{d}",1,"(-7*b^2*(64*a^6 + 240*a^4*b^2 + 120*a^2*b^4 + 5*b^6)*x)/16 + (a*b*(40*a^6 + 1664*a^4*b^2 + 2789*a^2*b^4 + 512*b^6)*Cos[c + d*x])/(20*d) + (b^2*(80*a^6 + 2248*a^4*b^2 + 2502*a^2*b^4 + 175*b^6)*Cos[c + d*x]*Sin[c + d*x])/(80*d) + (a*b*(40*a^4 + 624*a^2*b^2 + 337*b^4)*Cos[c + d*x]*(a + b*Sin[c + d*x])^2)/(40*d) + (b*(120*a^4 + 992*a^2*b^2 + 175*b^4)*Cos[c + d*x]*(a + b*Sin[c + d*x])^3)/(120*d) + (a*b*(30*a^2 + 113*b^2)*Cos[c + d*x]*(a + b*Sin[c + d*x])^4)/(30*d) + (b*(6*a^2 + 7*b^2)*Cos[c + d*x]*(a + b*Sin[c + d*x])^5)/(6*d) + (a*b*Cos[c + d*x]*(a + b*Sin[c + d*x])^6)/d + (Sec[c + d*x]*(b + a*Sin[c + d*x])*(a + b*Sin[c + d*x])^7)/d","A",7,3,21,0.1429,1,"{2691, 2753, 2734}"
421,1,369,0,0.6457078,"\int \sec ^4(c+d x) (a+b \sin (c+d x))^8 \, dx","Int[Sec[c + d*x]^4*(a + b*Sin[c + d*x])^8,x]","\frac{a b \left(-104 a^4 b^2-803 a^2 b^4+8 a^6-256 b^6\right) \cos (c+d x)}{6 d}+\frac{b \left(2 a^2-7 b^2\right) \cos (c+d x) (a+b \sin (c+d x))^5}{3 d}+\frac{a b \left(2 a^2-13 b^2\right) \cos (c+d x) (a+b \sin (c+d x))^4}{3 d}+\frac{b \left(-72 a^2 b^2+8 a^4-35 b^4\right) \cos (c+d x) (a+b \sin (c+d x))^3}{12 d}+\frac{a b \left(-88 a^2 b^2+8 a^4-151 b^4\right) \cos (c+d x) (a+b \sin (c+d x))^2}{12 d}+\frac{b^2 \left(-200 a^4 b^2-866 a^2 b^4+16 a^6-105 b^6\right) \sin (c+d x) \cos (c+d x)}{24 d}-\frac{\sec (c+d x) \left(5 a b-\left(2 a^2-7 b^2\right) \sin (c+d x)\right) (a+b \sin (c+d x))^6}{3 d}+\frac{35}{8} b^4 x \left(16 a^2 b^2+16 a^4+b^4\right)+\frac{\sec ^3(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^7}{3 d}","\frac{a b \left(-104 a^4 b^2-803 a^2 b^4+8 a^6-256 b^6\right) \cos (c+d x)}{6 d}+\frac{b \left(2 a^2-7 b^2\right) \cos (c+d x) (a+b \sin (c+d x))^5}{3 d}+\frac{a b \left(2 a^2-13 b^2\right) \cos (c+d x) (a+b \sin (c+d x))^4}{3 d}+\frac{b \left(-72 a^2 b^2+8 a^4-35 b^4\right) \cos (c+d x) (a+b \sin (c+d x))^3}{12 d}+\frac{a b \left(-88 a^2 b^2+8 a^4-151 b^4\right) \cos (c+d x) (a+b \sin (c+d x))^2}{12 d}+\frac{b^2 \left(-200 a^4 b^2-866 a^2 b^4+16 a^6-105 b^6\right) \sin (c+d x) \cos (c+d x)}{24 d}-\frac{\sec (c+d x) \left(5 a b-\left(2 a^2-7 b^2\right) \sin (c+d x)\right) (a+b \sin (c+d x))^6}{3 d}+\frac{35}{8} b^4 x \left(16 a^2 b^2+16 a^4+b^4\right)+\frac{\sec ^3(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^7}{3 d}",1,"(35*b^4*(16*a^4 + 16*a^2*b^2 + b^4)*x)/8 + (a*b*(8*a^6 - 104*a^4*b^2 - 803*a^2*b^4 - 256*b^6)*Cos[c + d*x])/(6*d) + (b^2*(16*a^6 - 200*a^4*b^2 - 866*a^2*b^4 - 105*b^6)*Cos[c + d*x]*Sin[c + d*x])/(24*d) + (a*b*(8*a^4 - 88*a^2*b^2 - 151*b^4)*Cos[c + d*x]*(a + b*Sin[c + d*x])^2)/(12*d) + (b*(8*a^4 - 72*a^2*b^2 - 35*b^4)*Cos[c + d*x]*(a + b*Sin[c + d*x])^3)/(12*d) + (a*b*(2*a^2 - 13*b^2)*Cos[c + d*x]*(a + b*Sin[c + d*x])^4)/(3*d) + (b*(2*a^2 - 7*b^2)*Cos[c + d*x]*(a + b*Sin[c + d*x])^5)/(3*d) + (Sec[c + d*x]^3*(b + a*Sin[c + d*x])*(a + b*Sin[c + d*x])^7)/(3*d) - (Sec[c + d*x]*(a + b*Sin[c + d*x])^6*(5*a*b - (2*a^2 - 7*b^2)*Sin[c + d*x]))/(3*d)","A",7,4,21,0.1905,1,"{2691, 2861, 2753, 2734}"
422,1,381,0,0.7235761,"\int \sec ^6(c+d x) (a+b \sin (c+d x))^8 \, dx","Int[Sec[c + d*x]^6*(a + b*Sin[c + d*x])^8,x]","\frac{2 a b \left(-48 a^4 b^2+163 a^2 b^4+8 a^6+192 b^6\right) \cos (c+d x)}{15 d}+\frac{4 a b \left(2 a^2+b^2\right) \cos (c+d x) (a+b \sin (c+d x))^4}{15 d}+\frac{b \left(-16 a^2 b^2+8 a^4+35 b^4\right) \cos (c+d x) (a+b \sin (c+d x))^3}{15 d}+\frac{a b \left(-32 a^2 b^2+8 a^4+87 b^4\right) \cos (c+d x) (a+b \sin (c+d x))^2}{15 d}+\frac{b^2 \left(-88 a^4 b^2+282 a^2 b^4+16 a^6+105 b^6\right) \sin (c+d x) \cos (c+d x)}{30 d}-\frac{\sec ^3(c+d x) \left(3 a b-\left(4 a^2-7 b^2\right) \sin (c+d x)\right) (a+b \sin (c+d x))^6}{15 d}-\frac{4 \sec (c+d x) \left(b \left(4 a^2-7 b^2\right)-a \left(2 a^2+b^2\right) \sin (c+d x)\right) (a+b \sin (c+d x))^5}{15 d}-\frac{7}{2} b^6 x \left(8 a^2+b^2\right)+\frac{\sec ^5(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^7}{5 d}","\frac{2 a b \left(-48 a^4 b^2+163 a^2 b^4+8 a^6+192 b^6\right) \cos (c+d x)}{15 d}+\frac{4 a b \left(2 a^2+b^2\right) \cos (c+d x) (a+b \sin (c+d x))^4}{15 d}+\frac{b \left(-16 a^2 b^2+8 a^4+35 b^4\right) \cos (c+d x) (a+b \sin (c+d x))^3}{15 d}+\frac{a b \left(-32 a^2 b^2+8 a^4+87 b^4\right) \cos (c+d x) (a+b \sin (c+d x))^2}{15 d}+\frac{b^2 \left(-88 a^4 b^2+282 a^2 b^4+16 a^6+105 b^6\right) \sin (c+d x) \cos (c+d x)}{30 d}-\frac{\sec ^3(c+d x) \left(3 a b-\left(4 a^2-7 b^2\right) \sin (c+d x)\right) (a+b \sin (c+d x))^6}{15 d}-\frac{4 \sec (c+d x) \left(b \left(4 a^2-7 b^2\right)-a \left(2 a^2+b^2\right) \sin (c+d x)\right) (a+b \sin (c+d x))^5}{15 d}-\frac{7}{2} b^6 x \left(8 a^2+b^2\right)+\frac{\sec ^5(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^7}{5 d}",1,"(-7*b^6*(8*a^2 + b^2)*x)/2 + (2*a*b*(8*a^6 - 48*a^4*b^2 + 163*a^2*b^4 + 192*b^6)*Cos[c + d*x])/(15*d) + (b^2*(16*a^6 - 88*a^4*b^2 + 282*a^2*b^4 + 105*b^6)*Cos[c + d*x]*Sin[c + d*x])/(30*d) + (a*b*(8*a^4 - 32*a^2*b^2 + 87*b^4)*Cos[c + d*x]*(a + b*Sin[c + d*x])^2)/(15*d) + (b*(8*a^4 - 16*a^2*b^2 + 35*b^4)*Cos[c + d*x]*(a + b*Sin[c + d*x])^3)/(15*d) + (4*a*b*(2*a^2 + b^2)*Cos[c + d*x]*(a + b*Sin[c + d*x])^4)/(15*d) + (Sec[c + d*x]^5*(b + a*Sin[c + d*x])*(a + b*Sin[c + d*x])^7)/(5*d) - (Sec[c + d*x]^3*(a + b*Sin[c + d*x])^6*(3*a*b - (4*a^2 - 7*b^2)*Sin[c + d*x]))/(15*d) - (4*Sec[c + d*x]*(a + b*Sin[c + d*x])^5*(b*(4*a^2 - 7*b^2) - a*(2*a^2 + b^2)*Sin[c + d*x]))/(15*d)","A",7,4,21,0.1905,1,"{2691, 2861, 2753, 2734}"
423,1,404,0,0.8205472,"\int \sec ^8(c+d x) (a+b \sin (c+d x))^8 \, dx","Int[Sec[c + d*x]^8*(a + b*Sin[c + d*x])^8,x]","\frac{4 a b \left(-88 a^4 b^2+125 a^2 b^4+24 a^6-96 b^6\right) \cos (c+d x)}{105 d}+\frac{b^2 \left(-152 a^4 b^2+174 a^2 b^4+48 a^6-105 b^6\right) \sin (c+d x) \cos (c+d x)}{105 d}+\frac{2 b \left(8 a^2 b^2+24 a^4-35 b^4\right) \cos (c+d x) (a+b \sin (c+d x))^3}{105 d}+\frac{2 a b \left(-40 a^2 b^2+24 a^4+9 b^4\right) \cos (c+d x) (a+b \sin (c+d x))^2}{105 d}-\frac{2 \sec ^3(c+d x) (a+b \sin (c+d x))^5 \left(b \left(6 a^2-7 b^2\right)-a \left(12 a^2-11 b^2\right) \sin (c+d x)\right)}{105 d}-\frac{\sec ^5(c+d x) (a+b \sin (c+d x))^6 \left(a b-\left(6 a^2-7 b^2\right) \sin (c+d x)\right)}{35 d}-\frac{2 \sec (c+d x) (a+b \sin (c+d x))^4 \left(3 a b \left(12 a^2-11 b^2\right)-\left(8 a^2 b^2+24 a^4-35 b^4\right) \sin (c+d x)\right)}{105 d}+\frac{\sec ^7(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^7}{7 d}+b^8 x","\frac{4 a b \left(-88 a^4 b^2+125 a^2 b^4+24 a^6-96 b^6\right) \cos (c+d x)}{105 d}+\frac{b^2 \left(-152 a^4 b^2+174 a^2 b^4+48 a^6-105 b^6\right) \sin (c+d x) \cos (c+d x)}{105 d}+\frac{2 b \left(8 a^2 b^2+24 a^4-35 b^4\right) \cos (c+d x) (a+b \sin (c+d x))^3}{105 d}+\frac{2 a b \left(-40 a^2 b^2+24 a^4+9 b^4\right) \cos (c+d x) (a+b \sin (c+d x))^2}{105 d}-\frac{2 \sec ^3(c+d x) (a+b \sin (c+d x))^5 \left(b \left(6 a^2-7 b^2\right)-a \left(12 a^2-11 b^2\right) \sin (c+d x)\right)}{105 d}-\frac{\sec ^5(c+d x) (a+b \sin (c+d x))^6 \left(a b-\left(6 a^2-7 b^2\right) \sin (c+d x)\right)}{35 d}-\frac{2 \sec (c+d x) (a+b \sin (c+d x))^4 \left(3 a b \left(12 a^2-11 b^2\right)-\left(8 a^2 b^2+24 a^4-35 b^4\right) \sin (c+d x)\right)}{105 d}+\frac{\sec ^7(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^7}{7 d}+b^8 x",1,"b^8*x + (4*a*b*(24*a^6 - 88*a^4*b^2 + 125*a^2*b^4 - 96*b^6)*Cos[c + d*x])/(105*d) + (b^2*(48*a^6 - 152*a^4*b^2 + 174*a^2*b^4 - 105*b^6)*Cos[c + d*x]*Sin[c + d*x])/(105*d) + (2*a*b*(24*a^4 - 40*a^2*b^2 + 9*b^4)*Cos[c + d*x]*(a + b*Sin[c + d*x])^2)/(105*d) + (2*b*(24*a^4 + 8*a^2*b^2 - 35*b^4)*Cos[c + d*x]*(a + b*Sin[c + d*x])^3)/(105*d) + (Sec[c + d*x]^7*(b + a*Sin[c + d*x])*(a + b*Sin[c + d*x])^7)/(7*d) - (2*Sec[c + d*x]^3*(a + b*Sin[c + d*x])^5*(b*(6*a^2 - 7*b^2) - a*(12*a^2 - 11*b^2)*Sin[c + d*x]))/(105*d) - (Sec[c + d*x]^5*(a + b*Sin[c + d*x])^6*(a*b - (6*a^2 - 7*b^2)*Sin[c + d*x]))/(35*d) - (2*Sec[c + d*x]*(a + b*Sin[c + d*x])^4*(3*a*b*(12*a^2 - 11*b^2) - (24*a^4 + 8*a^2*b^2 - 35*b^4)*Sin[c + d*x]))/(105*d)","A",7,4,21,0.1905,1,"{2691, 2861, 2753, 2734}"
424,1,236,0,0.3849361,"\int \sec ^{10}(c+d x) (a+b \sin (c+d x))^8 \, dx","Int[Sec[c + d*x]^10*(a + b*Sin[c + d*x])^8,x]","\frac{128 a^2 \left(a^2-b^2\right)^3 \tan (c+d x)}{315 d}+\frac{128 a b \left(a^2-b^2\right)^3 \sec (c+d x)}{315 d}+\frac{\sec ^7(c+d x) (a+b \sin (c+d x))^6 \left(\left(8 a^2-7 b^2\right) \sin (c+d x)+a b\right)}{63 d}+\frac{16 a \left(a^2-b^2\right) \sec ^5(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^4}{105 d}+\frac{64 a \left(a^2-b^2\right)^2 \sec ^3(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^2}{315 d}+\frac{\sec ^9(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^7}{9 d}","\frac{128 a^2 \left(a^2-b^2\right)^3 \tan (c+d x)}{315 d}+\frac{128 a b \left(a^2-b^2\right)^3 \sec (c+d x)}{315 d}+\frac{\sec ^7(c+d x) (a+b \sin (c+d x))^6 \left(\left(8 a^2-7 b^2\right) \sin (c+d x)+a b\right)}{63 d}+\frac{16 a \left(a^2-b^2\right) \sec ^5(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^4}{105 d}+\frac{64 a \left(a^2-b^2\right)^2 \sec ^3(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^2}{315 d}+\frac{\sec ^9(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^7}{9 d}",1,"(128*a*b*(a^2 - b^2)^3*Sec[c + d*x])/(315*d) + (64*a*(a^2 - b^2)^2*Sec[c + d*x]^3*(b + a*Sin[c + d*x])*(a + b*Sin[c + d*x])^2)/(315*d) + (16*a*(a^2 - b^2)*Sec[c + d*x]^5*(b + a*Sin[c + d*x])*(a + b*Sin[c + d*x])^4)/(105*d) + (Sec[c + d*x]^9*(b + a*Sin[c + d*x])*(a + b*Sin[c + d*x])^7)/(9*d) + (Sec[c + d*x]^7*(a + b*Sin[c + d*x])^6*(a*b + (8*a^2 - 7*b^2)*Sin[c + d*x]))/(63*d) + (128*a^2*(a^2 - b^2)^3*Tan[c + d*x])/(315*d)","A",10,6,21,0.2857,1,"{2691, 2861, 12, 2669, 3767, 8}"
425,1,118,0,0.1053132,"\int \frac{\cos ^5(c+d x)}{a+b \sin (c+d x)} \, dx","Int[Cos[c + d*x]^5/(a + b*Sin[c + d*x]),x]","\frac{\left(a^2-2 b^2\right) \sin ^2(c+d x)}{2 b^3 d}-\frac{a \left(a^2-2 b^2\right) \sin (c+d x)}{b^4 d}+\frac{\left(a^2-b^2\right)^2 \log (a+b \sin (c+d x))}{b^5 d}-\frac{a \sin ^3(c+d x)}{3 b^2 d}+\frac{\sin ^4(c+d x)}{4 b d}","\frac{\left(a^2-2 b^2\right) \sin ^2(c+d x)}{2 b^3 d}-\frac{a \left(a^2-2 b^2\right) \sin (c+d x)}{b^4 d}+\frac{\left(a^2-b^2\right)^2 \log (a+b \sin (c+d x))}{b^5 d}-\frac{a \sin ^3(c+d x)}{3 b^2 d}+\frac{\sin ^4(c+d x)}{4 b d}",1,"((a^2 - b^2)^2*Log[a + b*Sin[c + d*x]])/(b^5*d) - (a*(a^2 - 2*b^2)*Sin[c + d*x])/(b^4*d) + ((a^2 - 2*b^2)*Sin[c + d*x]^2)/(2*b^3*d) - (a*Sin[c + d*x]^3)/(3*b^2*d) + Sin[c + d*x]^4/(4*b*d)","A",3,2,21,0.09524,1,"{2668, 697}"
426,1,61,0,0.066362,"\int \frac{\cos ^3(c+d x)}{a+b \sin (c+d x)} \, dx","Int[Cos[c + d*x]^3/(a + b*Sin[c + d*x]),x]","-\frac{\left(a^2-b^2\right) \log (a+b \sin (c+d x))}{b^3 d}+\frac{a \sin (c+d x)}{b^2 d}-\frac{\sin ^2(c+d x)}{2 b d}","-\frac{\left(a^2-b^2\right) \log (a+b \sin (c+d x))}{b^3 d}+\frac{a \sin (c+d x)}{b^2 d}-\frac{\sin ^2(c+d x)}{2 b d}",1,"-(((a^2 - b^2)*Log[a + b*Sin[c + d*x]])/(b^3*d)) + (a*Sin[c + d*x])/(b^2*d) - Sin[c + d*x]^2/(2*b*d)","A",3,2,21,0.09524,1,"{2668, 697}"
427,1,18,0,0.0267588,"\int \frac{\cos (c+d x)}{a+b \sin (c+d x)} \, dx","Int[Cos[c + d*x]/(a + b*Sin[c + d*x]),x]","\frac{\log (a+b \sin (c+d x))}{b d}","\frac{\log (a+b \sin (c+d x))}{b d}",1,"Log[a + b*Sin[c + d*x]]/(b*d)","A",2,2,19,0.1053,1,"{2668, 31}"
428,1,75,0,0.0829564,"\int \frac{\sec (c+d x)}{a+b \sin (c+d x)} \, dx","Int[Sec[c + d*x]/(a + b*Sin[c + d*x]),x]","-\frac{b \log (a+b \sin (c+d x))}{d \left(a^2-b^2\right)}-\frac{\log (1-\sin (c+d x))}{2 d (a+b)}+\frac{\log (\sin (c+d x)+1)}{2 d (a-b)}","-\frac{b \log (a+b \sin (c+d x))}{d \left(a^2-b^2\right)}-\frac{\log (1-\sin (c+d x))}{2 d (a+b)}+\frac{\log (\sin (c+d x)+1)}{2 d (a-b)}",1,"-Log[1 - Sin[c + d*x]]/(2*(a + b)*d) + Log[1 + Sin[c + d*x]]/(2*(a - b)*d) - (b*Log[a + b*Sin[c + d*x]])/((a^2 - b^2)*d)","A",6,4,19,0.2105,1,"{2668, 706, 31, 633}"
429,1,123,0,0.163535,"\int \frac{\sec ^3(c+d x)}{a+b \sin (c+d x)} \, dx","Int[Sec[c + d*x]^3/(a + b*Sin[c + d*x]),x]","\frac{b^3 \log (a+b \sin (c+d x))}{d \left(a^2-b^2\right)^2}-\frac{\sec ^2(c+d x) (b-a \sin (c+d x))}{2 d \left(a^2-b^2\right)}-\frac{(a+2 b) \log (1-\sin (c+d x))}{4 d (a+b)^2}+\frac{(a-2 b) \log (\sin (c+d x)+1)}{4 d (a-b)^2}","\frac{b^3 \log (a+b \sin (c+d x))}{d \left(a^2-b^2\right)^2}-\frac{\sec ^2(c+d x) (b-a \sin (c+d x))}{2 d \left(a^2-b^2\right)}-\frac{(a+2 b) \log (1-\sin (c+d x))}{4 d (a+b)^2}+\frac{(a-2 b) \log (\sin (c+d x)+1)}{4 d (a-b)^2}",1,"-((a + 2*b)*Log[1 - Sin[c + d*x]])/(4*(a + b)^2*d) + ((a - 2*b)*Log[1 + Sin[c + d*x]])/(4*(a - b)^2*d) + (b^3*Log[a + b*Sin[c + d*x]])/((a^2 - b^2)^2*d) - (Sec[c + d*x]^2*(b - a*Sin[c + d*x]))/(2*(a^2 - b^2)*d)","A",4,3,21,0.1429,1,"{2668, 741, 801}"
430,1,195,0,0.2548623,"\int \frac{\sec ^5(c+d x)}{a+b \sin (c+d x)} \, dx","Int[Sec[c + d*x]^5/(a + b*Sin[c + d*x]),x]","-\frac{b^5 \log (a+b \sin (c+d x))}{d \left(a^2-b^2\right)^3}-\frac{\left(3 a^2+9 a b+8 b^2\right) \log (1-\sin (c+d x))}{16 d (a+b)^3}+\frac{\left(3 a^2-9 a b+8 b^2\right) \log (\sin (c+d x)+1)}{16 d (a-b)^3}-\frac{\sec ^4(c+d x) (b-a \sin (c+d x))}{4 d \left(a^2-b^2\right)}+\frac{\sec ^2(c+d x) \left(a \left(3 a^2-7 b^2\right) \sin (c+d x)+4 b^3\right)}{8 d \left(a^2-b^2\right)^2}","-\frac{b^5 \log (a+b \sin (c+d x))}{d \left(a^2-b^2\right)^3}-\frac{\left(3 a^2+9 a b+8 b^2\right) \log (1-\sin (c+d x))}{16 d (a+b)^3}+\frac{\left(3 a^2-9 a b+8 b^2\right) \log (\sin (c+d x)+1)}{16 d (a-b)^3}-\frac{\sec ^4(c+d x) (b-a \sin (c+d x))}{4 d \left(a^2-b^2\right)}+\frac{\sec ^2(c+d x) \left(a \left(3 a^2-7 b^2\right) \sin (c+d x)+4 b^3\right)}{8 d \left(a^2-b^2\right)^2}",1,"-((3*a^2 + 9*a*b + 8*b^2)*Log[1 - Sin[c + d*x]])/(16*(a + b)^3*d) + ((3*a^2 - 9*a*b + 8*b^2)*Log[1 + Sin[c + d*x]])/(16*(a - b)^3*d) - (b^5*Log[a + b*Sin[c + d*x]])/((a^2 - b^2)^3*d) - (Sec[c + d*x]^4*(b - a*Sin[c + d*x]))/(4*(a^2 - b^2)*d) + (Sec[c + d*x]^2*(4*b^3 + a*(3*a^2 - 7*b^2)*Sin[c + d*x]))/(8*(a^2 - b^2)^2*d)","A",5,4,21,0.1905,1,"{2668, 741, 823, 801}"
431,1,188,0,0.461873,"\int \frac{\cos ^6(c+d x)}{a+b \sin (c+d x)} \, dx","Int[Cos[c + d*x]^6/(a + b*Sin[c + d*x]),x]","-\frac{2 \left(a^2-b^2\right)^{5/2} \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{b^6 d}-\frac{\cos ^3(c+d x) \left(4 \left(a^2-b^2\right)-3 a b \sin (c+d x)\right)}{12 b^3 d}+\frac{\cos (c+d x) \left(8 \left(a^2-b^2\right)^2-a b \left(4 a^2-7 b^2\right) \sin (c+d x)\right)}{8 b^5 d}+\frac{a x \left(-20 a^2 b^2+8 a^4+15 b^4\right)}{8 b^6}+\frac{\cos ^5(c+d x)}{5 b d}","-\frac{2 \left(a^2-b^2\right)^{5/2} \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{b^6 d}-\frac{\cos ^3(c+d x) \left(4 \left(a^2-b^2\right)-3 a b \sin (c+d x)\right)}{12 b^3 d}+\frac{\cos (c+d x) \left(8 \left(a^2-b^2\right)^2-a b \left(4 a^2-7 b^2\right) \sin (c+d x)\right)}{8 b^5 d}+\frac{a x \left(-20 a^2 b^2+8 a^4+15 b^4\right)}{8 b^6}+\frac{\cos ^5(c+d x)}{5 b d}",1,"(a*(8*a^4 - 20*a^2*b^2 + 15*b^4)*x)/(8*b^6) - (2*(a^2 - b^2)^(5/2)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(b^6*d) + Cos[c + d*x]^5/(5*b*d) - (Cos[c + d*x]^3*(4*(a^2 - b^2) - 3*a*b*Sin[c + d*x]))/(12*b^3*d) + (Cos[c + d*x]*(8*(a^2 - b^2)^2 - a*b*(4*a^2 - 7*b^2)*Sin[c + d*x]))/(8*b^5*d)","A",7,6,21,0.2857,1,"{2695, 2865, 2735, 2660, 618, 204}"
432,1,127,0,0.2522205,"\int \frac{\cos ^4(c+d x)}{a+b \sin (c+d x)} \, dx","Int[Cos[c + d*x]^4/(a + b*Sin[c + d*x]),x]","\frac{2 \left(a^2-b^2\right)^{3/2} \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{b^4 d}-\frac{\cos (c+d x) \left(2 \left(a^2-b^2\right)-a b \sin (c+d x)\right)}{2 b^3 d}-\frac{a x \left(2 a^2-3 b^2\right)}{2 b^4}+\frac{\cos ^3(c+d x)}{3 b d}","\frac{2 \left(a^2-b^2\right)^{3/2} \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{b^4 d}-\frac{\cos (c+d x) \left(2 \left(a^2-b^2\right)-a b \sin (c+d x)\right)}{2 b^3 d}-\frac{a x \left(2 a^2-3 b^2\right)}{2 b^4}+\frac{\cos ^3(c+d x)}{3 b d}",1,"-(a*(2*a^2 - 3*b^2)*x)/(2*b^4) + (2*(a^2 - b^2)^(3/2)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(b^4*d) + Cos[c + d*x]^3/(3*b*d) - (Cos[c + d*x]*(2*(a^2 - b^2) - a*b*Sin[c + d*x]))/(2*b^3*d)","A",6,6,21,0.2857,1,"{2695, 2865, 2735, 2660, 618, 204}"
433,1,70,0,0.114138,"\int \frac{\cos ^2(c+d x)}{a+b \sin (c+d x)} \, dx","Int[Cos[c + d*x]^2/(a + b*Sin[c + d*x]),x]","-\frac{2 \sqrt{a^2-b^2} \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{b^2 d}+\frac{a x}{b^2}+\frac{\cos (c+d x)}{b d}","-\frac{2 \sqrt{a^2-b^2} \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{b^2 d}+\frac{a x}{b^2}+\frac{\cos (c+d x)}{b d}",1,"(a*x)/b^2 - (2*Sqrt[a^2 - b^2]*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(b^2*d) + Cos[c + d*x]/(b*d)","A",5,5,21,0.2381,1,"{2695, 2735, 2660, 618, 204}"
434,1,84,0,0.0939213,"\int \frac{\sec ^2(c+d x)}{a+b \sin (c+d x)} \, dx","Int[Sec[c + d*x]^2/(a + b*Sin[c + d*x]),x]","-\frac{2 b^2 \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{d \left(a^2-b^2\right)^{3/2}}-\frac{\sec (c+d x) (b-a \sin (c+d x))}{d \left(a^2-b^2\right)}","-\frac{2 b^2 \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{d \left(a^2-b^2\right)^{3/2}}-\frac{\sec (c+d x) (b-a \sin (c+d x))}{d \left(a^2-b^2\right)}",1,"(-2*b^2*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/((a^2 - b^2)^(3/2)*d) - (Sec[c + d*x]*(b - a*Sin[c + d*x]))/((a^2 - b^2)*d)","A",5,5,21,0.2381,1,"{2696, 12, 2660, 618, 204}"
435,1,137,0,0.2509848,"\int \frac{\sec ^4(c+d x)}{a+b \sin (c+d x)} \, dx","Int[Sec[c + d*x]^4/(a + b*Sin[c + d*x]),x]","\frac{2 b^4 \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{d \left(a^2-b^2\right)^{5/2}}-\frac{\sec ^3(c+d x) (b-a \sin (c+d x))}{3 d \left(a^2-b^2\right)}+\frac{\sec (c+d x) \left(a \left(2 a^2-5 b^2\right) \sin (c+d x)+3 b^3\right)}{3 d \left(a^2-b^2\right)^2}","\frac{2 b^4 \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{d \left(a^2-b^2\right)^{5/2}}-\frac{\sec ^3(c+d x) (b-a \sin (c+d x))}{3 d \left(a^2-b^2\right)}+\frac{\sec (c+d x) \left(a \left(2 a^2-5 b^2\right) \sin (c+d x)+3 b^3\right)}{3 d \left(a^2-b^2\right)^2}",1,"(2*b^4*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/((a^2 - b^2)^(5/2)*d) - (Sec[c + d*x]^3*(b - a*Sin[c + d*x]))/(3*(a^2 - b^2)*d) + (Sec[c + d*x]*(3*b^3 + a*(2*a^2 - 5*b^2)*Sin[c + d*x]))/(3*(a^2 - b^2)^2*d)","A",6,6,21,0.2857,1,"{2696, 2866, 12, 2660, 618, 204}"
436,1,197,0,0.4953774,"\int \frac{\sec ^6(c+d x)}{a+b \sin (c+d x)} \, dx","Int[Sec[c + d*x]^6/(a + b*Sin[c + d*x]),x]","-\frac{2 b^6 \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{d \left(a^2-b^2\right)^{7/2}}-\frac{\sec ^5(c+d x) (b-a \sin (c+d x))}{5 d \left(a^2-b^2\right)}+\frac{\sec ^3(c+d x) \left(a \left(4 a^2-9 b^2\right) \sin (c+d x)+5 b^3\right)}{15 d \left(a^2-b^2\right)^2}-\frac{\sec (c+d x) \left(15 b^5-a \left(-26 a^2 b^2+8 a^4+33 b^4\right) \sin (c+d x)\right)}{15 d \left(a^2-b^2\right)^3}","-\frac{2 b^6 \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{d \left(a^2-b^2\right)^{7/2}}-\frac{\sec ^5(c+d x) (b-a \sin (c+d x))}{5 d \left(a^2-b^2\right)}+\frac{\sec ^3(c+d x) \left(a \left(4 a^2-9 b^2\right) \sin (c+d x)+5 b^3\right)}{15 d \left(a^2-b^2\right)^2}-\frac{\sec (c+d x) \left(15 b^5-a \left(-26 a^2 b^2+8 a^4+33 b^4\right) \sin (c+d x)\right)}{15 d \left(a^2-b^2\right)^3}",1,"(-2*b^6*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/((a^2 - b^2)^(7/2)*d) - (Sec[c + d*x]^5*(b - a*Sin[c + d*x]))/(5*(a^2 - b^2)*d) + (Sec[c + d*x]^3*(5*b^3 + a*(4*a^2 - 9*b^2)*Sin[c + d*x]))/(15*(a^2 - b^2)^2*d) - (Sec[c + d*x]*(15*b^5 - a*(8*a^4 - 26*a^2*b^2 + 33*b^4)*Sin[c + d*x]))/(15*(a^2 - b^2)^3*d)","A",7,6,21,0.2857,1,"{2696, 2866, 12, 2660, 618, 204}"
437,1,184,0,0.1728944,"\int \frac{\cos ^7(c+d x)}{(a+b \sin (c+d x))^2} \, dx","Int[Cos[c + d*x]^7/(a + b*Sin[c + d*x])^2,x]","-\frac{\left(a^2-b^2\right) \sin ^3(c+d x)}{b^4 d}+\frac{a \left(2 a^2-3 b^2\right) \sin ^2(c+d x)}{b^5 d}-\frac{\left(-9 a^2 b^2+5 a^4+3 b^4\right) \sin (c+d x)}{b^6 d}+\frac{\left(a^2-b^2\right)^3}{b^7 d (a+b \sin (c+d x))}+\frac{6 a \left(a^2-b^2\right)^2 \log (a+b \sin (c+d x))}{b^7 d}+\frac{a \sin ^4(c+d x)}{2 b^3 d}-\frac{\sin ^5(c+d x)}{5 b^2 d}","-\frac{\left(a^2-b^2\right) \sin ^3(c+d x)}{b^4 d}+\frac{a \left(2 a^2-3 b^2\right) \sin ^2(c+d x)}{b^5 d}-\frac{\left(-9 a^2 b^2+5 a^4+3 b^4\right) \sin (c+d x)}{b^6 d}+\frac{\left(a^2-b^2\right)^3}{b^7 d (a+b \sin (c+d x))}+\frac{6 a \left(a^2-b^2\right)^2 \log (a+b \sin (c+d x))}{b^7 d}+\frac{a \sin ^4(c+d x)}{2 b^3 d}-\frac{\sin ^5(c+d x)}{5 b^2 d}",1,"(6*a*(a^2 - b^2)^2*Log[a + b*Sin[c + d*x]])/(b^7*d) - ((5*a^4 - 9*a^2*b^2 + 3*b^4)*Sin[c + d*x])/(b^6*d) + (a*(2*a^2 - 3*b^2)*Sin[c + d*x]^2)/(b^5*d) - ((a^2 - b^2)*Sin[c + d*x]^3)/(b^4*d) + (a*Sin[c + d*x]^4)/(2*b^3*d) - Sin[c + d*x]^5/(5*b^2*d) + (a^2 - b^2)^3/(b^7*d*(a + b*Sin[c + d*x]))","A",3,2,21,0.09524,1,"{2668, 697}"
438,1,120,0,0.1012751,"\int \frac{\cos ^5(c+d x)}{(a+b \sin (c+d x))^2} \, dx","Int[Cos[c + d*x]^5/(a + b*Sin[c + d*x])^2,x]","\frac{\left(3 a^2-2 b^2\right) \sin (c+d x)}{b^4 d}-\frac{\left(a^2-b^2\right)^2}{b^5 d (a+b \sin (c+d x))}-\frac{4 a \left(a^2-b^2\right) \log (a+b \sin (c+d x))}{b^5 d}-\frac{a \sin ^2(c+d x)}{b^3 d}+\frac{\sin ^3(c+d x)}{3 b^2 d}","\frac{\left(3 a^2-2 b^2\right) \sin (c+d x)}{b^4 d}-\frac{\left(a^2-b^2\right)^2}{b^5 d (a+b \sin (c+d x))}-\frac{4 a \left(a^2-b^2\right) \log (a+b \sin (c+d x))}{b^5 d}-\frac{a \sin ^2(c+d x)}{b^3 d}+\frac{\sin ^3(c+d x)}{3 b^2 d}",1,"(-4*a*(a^2 - b^2)*Log[a + b*Sin[c + d*x]])/(b^5*d) + ((3*a^2 - 2*b^2)*Sin[c + d*x])/(b^4*d) - (a*Sin[c + d*x]^2)/(b^3*d) + Sin[c + d*x]^3/(3*b^2*d) - (a^2 - b^2)^2/(b^5*d*(a + b*Sin[c + d*x]))","A",3,2,21,0.09524,1,"{2668, 697}"
439,1,63,0,0.0651441,"\int \frac{\cos ^3(c+d x)}{(a+b \sin (c+d x))^2} \, dx","Int[Cos[c + d*x]^3/(a + b*Sin[c + d*x])^2,x]","\frac{a^2-b^2}{b^3 d (a+b \sin (c+d x))}+\frac{2 a \log (a+b \sin (c+d x))}{b^3 d}-\frac{\sin (c+d x)}{b^2 d}","\frac{a^2-b^2}{b^3 d (a+b \sin (c+d x))}+\frac{2 a \log (a+b \sin (c+d x))}{b^3 d}-\frac{\sin (c+d x)}{b^2 d}",1,"(2*a*Log[a + b*Sin[c + d*x]])/(b^3*d) - Sin[c + d*x]/(b^2*d) + (a^2 - b^2)/(b^3*d*(a + b*Sin[c + d*x]))","A",3,2,21,0.09524,1,"{2668, 697}"
440,1,20,0,0.0266825,"\int \frac{\cos (c+d x)}{(a+b \sin (c+d x))^2} \, dx","Int[Cos[c + d*x]/(a + b*Sin[c + d*x])^2,x]","-\frac{1}{b d (a+b \sin (c+d x))}","-\frac{1}{b d (a+b \sin (c+d x))}",1,"-(1/(b*d*(a + b*Sin[c + d*x])))","A",2,2,19,0.1053,1,"{2668, 32}"
441,1,104,0,0.1146536,"\int \frac{\sec (c+d x)}{(a+b \sin (c+d x))^2} \, dx","Int[Sec[c + d*x]/(a + b*Sin[c + d*x])^2,x]","\frac{b}{d \left(a^2-b^2\right) (a+b \sin (c+d x))}-\frac{2 a b \log (a+b \sin (c+d x))}{d \left(a^2-b^2\right)^2}-\frac{\log (1-\sin (c+d x))}{2 d (a+b)^2}+\frac{\log (\sin (c+d x)+1)}{2 d (a-b)^2}","\frac{b}{d \left(a^2-b^2\right) (a+b \sin (c+d x))}-\frac{2 a b \log (a+b \sin (c+d x))}{d \left(a^2-b^2\right)^2}-\frac{\log (1-\sin (c+d x))}{2 d (a+b)^2}+\frac{\log (\sin (c+d x)+1)}{2 d (a-b)^2}",1,"-Log[1 - Sin[c + d*x]]/(2*(a + b)^2*d) + Log[1 + Sin[c + d*x]]/(2*(a - b)^2*d) - (2*a*b*Log[a + b*Sin[c + d*x]])/((a^2 - b^2)^2*d) + b/((a^2 - b^2)*d*(a + b*Sin[c + d*x]))","A",4,3,19,0.1579,1,"{2668, 710, 801}"
442,1,177,0,0.2088395,"\int \frac{\sec ^3(c+d x)}{(a+b \sin (c+d x))^2} \, dx","Int[Sec[c + d*x]^3/(a + b*Sin[c + d*x])^2,x]","-\frac{b \left(a^2+3 b^2\right)}{2 d \left(a^2-b^2\right)^2 (a+b \sin (c+d x))}+\frac{4 a b^3 \log (a+b \sin (c+d x))}{d \left(a^2-b^2\right)^3}-\frac{\sec ^2(c+d x) (b-a \sin (c+d x))}{2 d \left(a^2-b^2\right) (a+b \sin (c+d x))}-\frac{(a+3 b) \log (1-\sin (c+d x))}{4 d (a+b)^3}+\frac{(a-3 b) \log (\sin (c+d x)+1)}{4 d (a-b)^3}","-\frac{b \left(a^2+3 b^2\right)}{2 d \left(a^2-b^2\right)^2 (a+b \sin (c+d x))}+\frac{4 a b^3 \log (a+b \sin (c+d x))}{d \left(a^2-b^2\right)^3}-\frac{\sec ^2(c+d x) (b-a \sin (c+d x))}{2 d \left(a^2-b^2\right) (a+b \sin (c+d x))}-\frac{(a+3 b) \log (1-\sin (c+d x))}{4 d (a+b)^3}+\frac{(a-3 b) \log (\sin (c+d x)+1)}{4 d (a-b)^3}",1,"-((a + 3*b)*Log[1 - Sin[c + d*x]])/(4*(a + b)^3*d) + ((a - 3*b)*Log[1 + Sin[c + d*x]])/(4*(a - b)^3*d) + (4*a*b^3*Log[a + b*Sin[c + d*x]])/((a^2 - b^2)^3*d) - (b*(a^2 + 3*b^2))/(2*(a^2 - b^2)^2*d*(a + b*Sin[c + d*x])) - (Sec[c + d*x]^2*(b - a*Sin[c + d*x]))/(2*(a^2 - b^2)*d*(a + b*Sin[c + d*x]))","A",4,3,21,0.1429,1,"{2668, 741, 801}"
443,1,269,0,0.3214006,"\int \frac{\sec ^5(c+d x)}{(a+b \sin (c+d x))^2} \, dx","Int[Sec[c + d*x]^5/(a + b*Sin[c + d*x])^2,x]","-\frac{3 b \left(-4 a^2 b^2+a^4-5 b^4\right)}{8 d \left(a^2-b^2\right)^3 (a+b \sin (c+d x))}-\frac{6 a b^5 \log (a+b \sin (c+d x))}{d \left(a^2-b^2\right)^4}-\frac{3 \left(a^2+4 a b+5 b^2\right) \log (1-\sin (c+d x))}{16 d (a+b)^4}+\frac{3 \left(a^2-4 a b+5 b^2\right) \log (\sin (c+d x)+1)}{16 d (a-b)^4}+\frac{\sec ^2(c+d x) \left(3 a \left(a^2-3 b^2\right) \sin (c+d x)+b \left(a^2+5 b^2\right)\right)}{8 d \left(a^2-b^2\right)^2 (a+b \sin (c+d x))}-\frac{\sec ^4(c+d x) (b-a \sin (c+d x))}{4 d \left(a^2-b^2\right) (a+b \sin (c+d x))}","-\frac{3 b \left(-4 a^2 b^2+a^4-5 b^4\right)}{8 d \left(a^2-b^2\right)^3 (a+b \sin (c+d x))}-\frac{6 a b^5 \log (a+b \sin (c+d x))}{d \left(a^2-b^2\right)^4}-\frac{3 \left(a^2+4 a b+5 b^2\right) \log (1-\sin (c+d x))}{16 d (a+b)^4}+\frac{3 \left(a^2-4 a b+5 b^2\right) \log (\sin (c+d x)+1)}{16 d (a-b)^4}+\frac{\sec ^2(c+d x) \left(3 a \left(a^2-3 b^2\right) \sin (c+d x)+b \left(a^2+5 b^2\right)\right)}{8 d \left(a^2-b^2\right)^2 (a+b \sin (c+d x))}-\frac{\sec ^4(c+d x) (b-a \sin (c+d x))}{4 d \left(a^2-b^2\right) (a+b \sin (c+d x))}",1,"(-3*(a^2 + 4*a*b + 5*b^2)*Log[1 - Sin[c + d*x]])/(16*(a + b)^4*d) + (3*(a^2 - 4*a*b + 5*b^2)*Log[1 + Sin[c + d*x]])/(16*(a - b)^4*d) - (6*a*b^5*Log[a + b*Sin[c + d*x]])/((a^2 - b^2)^4*d) - (3*b*(a^4 - 4*a^2*b^2 - 5*b^4))/(8*(a^2 - b^2)^3*d*(a + b*Sin[c + d*x])) - (Sec[c + d*x]^4*(b - a*Sin[c + d*x]))/(4*(a^2 - b^2)*d*(a + b*Sin[c + d*x])) + (Sec[c + d*x]^2*(b*(a^2 + 5*b^2) + 3*a*(a^2 - 3*b^2)*Sin[c + d*x]))/(8*(a^2 - b^2)^2*d*(a + b*Sin[c + d*x]))","A",5,4,21,0.1905,1,"{2668, 741, 823, 801}"
444,1,187,0,0.3713671,"\int \frac{\cos ^6(c+d x)}{(a+b \sin (c+d x))^2} \, dx","Int[Cos[c + d*x]^6/(a + b*Sin[c + d*x])^2,x]","\frac{10 a \left(a^2-b^2\right)^{3/2} \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{b^6 d}-\frac{5 \cos (c+d x) \left(8 a \left(a^2-b^2\right)-b \left(4 a^2-3 b^2\right) \sin (c+d x)\right)}{8 b^5 d}-\frac{5 x \left(-12 a^2 b^2+8 a^4+3 b^4\right)}{8 b^6}+\frac{5 \cos ^3(c+d x) (4 a-3 b \sin (c+d x))}{12 b^3 d}-\frac{\cos ^5(c+d x)}{b d (a+b \sin (c+d x))}","\frac{10 a \left(a^2-b^2\right)^{3/2} \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{b^6 d}-\frac{5 \cos (c+d x) \left(8 a \left(a^2-b^2\right)-b \left(4 a^2-3 b^2\right) \sin (c+d x)\right)}{8 b^5 d}-\frac{5 x \left(-12 a^2 b^2+8 a^4+3 b^4\right)}{8 b^6}+\frac{5 \cos ^3(c+d x) (4 a-3 b \sin (c+d x))}{12 b^3 d}-\frac{\cos ^5(c+d x)}{b d (a+b \sin (c+d x))}",1,"(-5*(8*a^4 - 12*a^2*b^2 + 3*b^4)*x)/(8*b^6) + (10*a*(a^2 - b^2)^(3/2)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(b^6*d) + (5*Cos[c + d*x]^3*(4*a - 3*b*Sin[c + d*x]))/(12*b^3*d) - Cos[c + d*x]^5/(b*d*(a + b*Sin[c + d*x])) - (5*Cos[c + d*x]*(8*a*(a^2 - b^2) - b*(4*a^2 - 3*b^2)*Sin[c + d*x]))/(8*b^5*d)","A",7,6,21,0.2857,1,"{2693, 2865, 2735, 2660, 618, 204}"
445,1,128,0,0.2132903,"\int \frac{\cos ^4(c+d x)}{(a+b \sin (c+d x))^2} \, dx","Int[Cos[c + d*x]^4/(a + b*Sin[c + d*x])^2,x]","-\frac{6 a \sqrt{a^2-b^2} \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{b^4 d}+\frac{3 x \left(2 a^2-b^2\right)}{2 b^4}+\frac{3 \cos (c+d x) (2 a-b \sin (c+d x))}{2 b^3 d}-\frac{\cos ^3(c+d x)}{b d (a+b \sin (c+d x))}","-\frac{6 a \sqrt{a^2-b^2} \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{b^4 d}+\frac{3 x \left(2 a^2-b^2\right)}{2 b^4}+\frac{3 \cos (c+d x) (2 a-b \sin (c+d x))}{2 b^3 d}-\frac{\cos ^3(c+d x)}{b d (a+b \sin (c+d x))}",1,"(3*(2*a^2 - b^2)*x)/(2*b^4) - (6*a*Sqrt[a^2 - b^2]*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(b^4*d) + (3*Cos[c + d*x]*(2*a - b*Sin[c + d*x]))/(2*b^3*d) - Cos[c + d*x]^3/(b*d*(a + b*Sin[c + d*x]))","A",6,6,21,0.2857,1,"{2693, 2865, 2735, 2660, 618, 204}"
446,1,84,0,0.1101575,"\int \frac{\cos ^2(c+d x)}{(a+b \sin (c+d x))^2} \, dx","Int[Cos[c + d*x]^2/(a + b*Sin[c + d*x])^2,x]","\frac{2 a \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{b^2 d \sqrt{a^2-b^2}}-\frac{\cos (c+d x)}{b d (a+b \sin (c+d x))}-\frac{x}{b^2}","\frac{2 a \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{b^2 d \sqrt{a^2-b^2}}-\frac{\cos (c+d x)}{b d (a+b \sin (c+d x))}-\frac{x}{b^2}",1,"-(x/b^2) + (2*a*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(b^2*Sqrt[a^2 - b^2]*d) - Cos[c + d*x]/(b*d*(a + b*Sin[c + d*x]))","A",5,5,21,0.2381,1,"{2693, 2735, 2660, 618, 204}"
447,1,130,0,0.2081698,"\int \frac{\sec ^2(c+d x)}{(a+b \sin (c+d x))^2} \, dx","Int[Sec[c + d*x]^2/(a + b*Sin[c + d*x])^2,x]","-\frac{6 a b^2 \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{d \left(a^2-b^2\right)^{5/2}}+\frac{b \sec (c+d x)}{d \left(a^2-b^2\right) (a+b \sin (c+d x))}-\frac{\sec (c+d x) \left(3 a b-\left(a^2+2 b^2\right) \sin (c+d x)\right)}{d \left(a^2-b^2\right)^2}","-\frac{6 a b^2 \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{d \left(a^2-b^2\right)^{5/2}}+\frac{b \sec (c+d x)}{d \left(a^2-b^2\right) (a+b \sin (c+d x))}-\frac{\sec (c+d x) \left(3 a b-\left(a^2+2 b^2\right) \sin (c+d x)\right)}{d \left(a^2-b^2\right)^2}",1,"(-6*a*b^2*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/((a^2 - b^2)^(5/2)*d) + (b*Sec[c + d*x])/((a^2 - b^2)*d*(a + b*Sin[c + d*x])) - (Sec[c + d*x]*(3*a*b - (a^2 + 2*b^2)*Sin[c + d*x]))/((a^2 - b^2)^2*d)","A",6,6,21,0.2857,1,"{2694, 2866, 12, 2660, 618, 204}"
448,1,193,0,0.3667796,"\int \frac{\sec ^4(c+d x)}{(a+b \sin (c+d x))^2} \, dx","Int[Sec[c + d*x]^4/(a + b*Sin[c + d*x])^2,x]","\frac{10 a b^4 \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{d \left(a^2-b^2\right)^{7/2}}+\frac{b \sec ^3(c+d x)}{d \left(a^2-b^2\right) (a+b \sin (c+d x))}-\frac{\sec ^3(c+d x) \left(5 a b-\left(a^2+4 b^2\right) \sin (c+d x)\right)}{3 d \left(a^2-b^2\right)^2}+\frac{\sec (c+d x) \left(\left(-9 a^2 b^2+2 a^4-8 b^4\right) \sin (c+d x)+15 a b^3\right)}{3 d \left(a^2-b^2\right)^3}","\frac{10 a b^4 \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{d \left(a^2-b^2\right)^{7/2}}+\frac{b \sec ^3(c+d x)}{d \left(a^2-b^2\right) (a+b \sin (c+d x))}-\frac{\sec ^3(c+d x) \left(5 a b-\left(a^2+4 b^2\right) \sin (c+d x)\right)}{3 d \left(a^2-b^2\right)^2}+\frac{\sec (c+d x) \left(\left(-9 a^2 b^2+2 a^4-8 b^4\right) \sin (c+d x)+15 a b^3\right)}{3 d \left(a^2-b^2\right)^3}",1,"(10*a*b^4*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/((a^2 - b^2)^(7/2)*d) + (b*Sec[c + d*x]^3)/((a^2 - b^2)*d*(a + b*Sin[c + d*x])) - (Sec[c + d*x]^3*(5*a*b - (a^2 + 4*b^2)*Sin[c + d*x]))/(3*(a^2 - b^2)^2*d) + (Sec[c + d*x]*(15*a*b^3 + (2*a^4 - 9*a^2*b^2 - 8*b^4)*Sin[c + d*x]))/(3*(a^2 - b^2)^3*d)","A",7,6,21,0.2857,1,"{2694, 2866, 12, 2660, 618, 204}"
449,1,190,0,0.1585815,"\int \frac{\cos ^7(c+d x)}{(a+b \sin (c+d x))^3} \, dx","Int[Cos[c + d*x]^7/(a + b*Sin[c + d*x])^3,x]","-\frac{3 \left(2 a^2-b^2\right) \sin ^2(c+d x)}{2 b^5 d}+\frac{a \left(10 a^2-9 b^2\right) \sin (c+d x)}{b^6 d}-\frac{6 a \left(a^2-b^2\right)^2}{b^7 d (a+b \sin (c+d x))}+\frac{\left(a^2-b^2\right)^3}{2 b^7 d (a+b \sin (c+d x))^2}-\frac{3 \left(-6 a^2 b^2+5 a^4+b^4\right) \log (a+b \sin (c+d x))}{b^7 d}+\frac{a \sin ^3(c+d x)}{b^4 d}-\frac{\sin ^4(c+d x)}{4 b^3 d}","-\frac{3 \left(2 a^2-b^2\right) \sin ^2(c+d x)}{2 b^5 d}+\frac{a \left(10 a^2-9 b^2\right) \sin (c+d x)}{b^6 d}-\frac{6 a \left(a^2-b^2\right)^2}{b^7 d (a+b \sin (c+d x))}+\frac{\left(a^2-b^2\right)^3}{2 b^7 d (a+b \sin (c+d x))^2}-\frac{3 \left(-6 a^2 b^2+5 a^4+b^4\right) \log (a+b \sin (c+d x))}{b^7 d}+\frac{a \sin ^3(c+d x)}{b^4 d}-\frac{\sin ^4(c+d x)}{4 b^3 d}",1,"(-3*(5*a^4 - 6*a^2*b^2 + b^4)*Log[a + b*Sin[c + d*x]])/(b^7*d) + (a*(10*a^2 - 9*b^2)*Sin[c + d*x])/(b^6*d) - (3*(2*a^2 - b^2)*Sin[c + d*x]^2)/(2*b^5*d) + (a*Sin[c + d*x]^3)/(b^4*d) - Sin[c + d*x]^4/(4*b^3*d) + (a^2 - b^2)^3/(2*b^7*d*(a + b*Sin[c + d*x])^2) - (6*a*(a^2 - b^2)^2)/(b^7*d*(a + b*Sin[c + d*x]))","A",3,2,21,0.09524,1,"{2668, 697}"
450,1,127,0,0.1049382,"\int \frac{\cos ^5(c+d x)}{(a+b \sin (c+d x))^3} \, dx","Int[Cos[c + d*x]^5/(a + b*Sin[c + d*x])^3,x]","-\frac{\left(a^2-b^2\right)^2}{2 b^5 d (a+b \sin (c+d x))^2}+\frac{4 a \left(a^2-b^2\right)}{b^5 d (a+b \sin (c+d x))}+\frac{2 \left(3 a^2-b^2\right) \log (a+b \sin (c+d x))}{b^5 d}-\frac{3 a \sin (c+d x)}{b^4 d}+\frac{\sin ^2(c+d x)}{2 b^3 d}","-\frac{\left(a^2-b^2\right)^2}{2 b^5 d (a+b \sin (c+d x))^2}+\frac{4 a \left(a^2-b^2\right)}{b^5 d (a+b \sin (c+d x))}+\frac{2 \left(3 a^2-b^2\right) \log (a+b \sin (c+d x))}{b^5 d}-\frac{3 a \sin (c+d x)}{b^4 d}+\frac{\sin ^2(c+d x)}{2 b^3 d}",1,"(2*(3*a^2 - b^2)*Log[a + b*Sin[c + d*x]])/(b^5*d) - (3*a*Sin[c + d*x])/(b^4*d) + Sin[c + d*x]^2/(2*b^3*d) - (a^2 - b^2)^2/(2*b^5*d*(a + b*Sin[c + d*x])^2) + (4*a*(a^2 - b^2))/(b^5*d*(a + b*Sin[c + d*x]))","A",3,2,21,0.09524,1,"{2668, 697}"
451,1,72,0,0.071842,"\int \frac{\cos ^3(c+d x)}{(a+b \sin (c+d x))^3} \, dx","Int[Cos[c + d*x]^3/(a + b*Sin[c + d*x])^3,x]","\frac{a^2-b^2}{2 b^3 d (a+b \sin (c+d x))^2}-\frac{2 a}{b^3 d (a+b \sin (c+d x))}-\frac{\log (a+b \sin (c+d x))}{b^3 d}","\frac{a^2-b^2}{2 b^3 d (a+b \sin (c+d x))^2}-\frac{2 a}{b^3 d (a+b \sin (c+d x))}-\frac{\log (a+b \sin (c+d x))}{b^3 d}",1,"-(Log[a + b*Sin[c + d*x]]/(b^3*d)) + (a^2 - b^2)/(2*b^3*d*(a + b*Sin[c + d*x])^2) - (2*a)/(b^3*d*(a + b*Sin[c + d*x]))","A",3,2,21,0.09524,1,"{2668, 697}"
452,1,22,0,0.0263888,"\int \frac{\cos (c+d x)}{(a+b \sin (c+d x))^3} \, dx","Int[Cos[c + d*x]/(a + b*Sin[c + d*x])^3,x]","-\frac{1}{2 b d (a+b \sin (c+d x))^2}","-\frac{1}{2 b d (a+b \sin (c+d x))^2}",1,"-1/(2*b*d*(a + b*Sin[c + d*x])^2)","A",2,2,19,0.1053,1,"{2668, 32}"
453,1,145,0,0.1539613,"\int \frac{\sec (c+d x)}{(a+b \sin (c+d x))^3} \, dx","Int[Sec[c + d*x]/(a + b*Sin[c + d*x])^3,x]","\frac{2 a b}{d \left(a^2-b^2\right)^2 (a+b \sin (c+d x))}+\frac{b}{2 d \left(a^2-b^2\right) (a+b \sin (c+d x))^2}-\frac{b \left(3 a^2+b^2\right) \log (a+b \sin (c+d x))}{d \left(a^2-b^2\right)^3}-\frac{\log (1-\sin (c+d x))}{2 d (a+b)^3}+\frac{\log (\sin (c+d x)+1)}{2 d (a-b)^3}","\frac{2 a b}{d \left(a^2-b^2\right)^2 (a+b \sin (c+d x))}+\frac{b}{2 d \left(a^2-b^2\right) (a+b \sin (c+d x))^2}-\frac{b \left(3 a^2+b^2\right) \log (a+b \sin (c+d x))}{d \left(a^2-b^2\right)^3}-\frac{\log (1-\sin (c+d x))}{2 d (a+b)^3}+\frac{\log (\sin (c+d x)+1)}{2 d (a-b)^3}",1,"-Log[1 - Sin[c + d*x]]/(2*(a + b)^3*d) + Log[1 + Sin[c + d*x]]/(2*(a - b)^3*d) - (b*(3*a^2 + b^2)*Log[a + b*Sin[c + d*x]])/((a^2 - b^2)^3*d) + b/(2*(a^2 - b^2)*d*(a + b*Sin[c + d*x])^2) + (2*a*b)/((a^2 - b^2)^2*d*(a + b*Sin[c + d*x]))","A",4,3,19,0.1579,1,"{2668, 710, 801}"
454,1,226,0,0.2765353,"\int \frac{\sec ^3(c+d x)}{(a+b \sin (c+d x))^3} \, dx","Int[Sec[c + d*x]^3/(a + b*Sin[c + d*x])^3,x]","-\frac{a b \left(a^2+11 b^2\right)}{2 d \left(a^2-b^2\right)^3 (a+b \sin (c+d x))}-\frac{b \left(a^2+2 b^2\right)}{2 d \left(a^2-b^2\right)^2 (a+b \sin (c+d x))^2}+\frac{2 b^3 \left(5 a^2+b^2\right) \log (a+b \sin (c+d x))}{d \left(a^2-b^2\right)^4}-\frac{\sec ^2(c+d x) (b-a \sin (c+d x))}{2 d \left(a^2-b^2\right) (a+b \sin (c+d x))^2}-\frac{(a+4 b) \log (1-\sin (c+d x))}{4 d (a+b)^4}+\frac{(a-4 b) \log (\sin (c+d x)+1)}{4 d (a-b)^4}","-\frac{a b \left(a^2+11 b^2\right)}{2 d \left(a^2-b^2\right)^3 (a+b \sin (c+d x))}-\frac{b \left(a^2+2 b^2\right)}{2 d \left(a^2-b^2\right)^2 (a+b \sin (c+d x))^2}+\frac{2 b^3 \left(5 a^2+b^2\right) \log (a+b \sin (c+d x))}{d \left(a^2-b^2\right)^4}-\frac{\sec ^2(c+d x) (b-a \sin (c+d x))}{2 d \left(a^2-b^2\right) (a+b \sin (c+d x))^2}-\frac{(a+4 b) \log (1-\sin (c+d x))}{4 d (a+b)^4}+\frac{(a-4 b) \log (\sin (c+d x)+1)}{4 d (a-b)^4}",1,"-((a + 4*b)*Log[1 - Sin[c + d*x]])/(4*(a + b)^4*d) + ((a - 4*b)*Log[1 + Sin[c + d*x]])/(4*(a - b)^4*d) + (2*b^3*(5*a^2 + b^2)*Log[a + b*Sin[c + d*x]])/((a^2 - b^2)^4*d) - (b*(a^2 + 2*b^2))/(2*(a^2 - b^2)^2*d*(a + b*Sin[c + d*x])^2) - (Sec[c + d*x]^2*(b - a*Sin[c + d*x]))/(2*(a^2 - b^2)*d*(a + b*Sin[c + d*x])^2) - (a*b*(a^2 + 11*b^2))/(2*(a^2 - b^2)^3*d*(a + b*Sin[c + d*x]))","A",4,3,21,0.1429,1,"{2668, 741, 801}"
455,1,328,0,0.4201856,"\int \frac{\sec ^5(c+d x)}{(a+b \sin (c+d x))^3} \, dx","Int[Sec[c + d*x]^5/(a + b*Sin[c + d*x])^3,x]","-\frac{3 a b \left(-6 a^2 b^2+a^4-27 b^4\right)}{8 d \left(a^2-b^2\right)^4 (a+b \sin (c+d x))}-\frac{3 b \left(-5 a^2 b^2+a^4-4 b^4\right)}{8 d \left(a^2-b^2\right)^3 (a+b \sin (c+d x))^2}-\frac{3 b^5 \left(7 a^2+b^2\right) \log (a+b \sin (c+d x))}{d \left(a^2-b^2\right)^5}-\frac{3 \left(a^2+5 a b+8 b^2\right) \log (1-\sin (c+d x))}{16 d (a+b)^5}+\frac{3 \left(a^2-5 a b+8 b^2\right) \log (\sin (c+d x)+1)}{16 d (a-b)^5}+\frac{\sec ^2(c+d x) \left(a \left(3 a^2-11 b^2\right) \sin (c+d x)+2 b \left(a^2+3 b^2\right)\right)}{8 d \left(a^2-b^2\right)^2 (a+b \sin (c+d x))^2}-\frac{\sec ^4(c+d x) (b-a \sin (c+d x))}{4 d \left(a^2-b^2\right) (a+b \sin (c+d x))^2}","-\frac{3 a b \left(-6 a^2 b^2+a^4-27 b^4\right)}{8 d \left(a^2-b^2\right)^4 (a+b \sin (c+d x))}-\frac{3 b \left(-5 a^2 b^2+a^4-4 b^4\right)}{8 d \left(a^2-b^2\right)^3 (a+b \sin (c+d x))^2}-\frac{3 b^5 \left(7 a^2+b^2\right) \log (a+b \sin (c+d x))}{d \left(a^2-b^2\right)^5}-\frac{3 \left(a^2+5 a b+8 b^2\right) \log (1-\sin (c+d x))}{16 d (a+b)^5}+\frac{3 \left(a^2-5 a b+8 b^2\right) \log (\sin (c+d x)+1)}{16 d (a-b)^5}+\frac{\sec ^2(c+d x) \left(a \left(3 a^2-11 b^2\right) \sin (c+d x)+2 b \left(a^2+3 b^2\right)\right)}{8 d \left(a^2-b^2\right)^2 (a+b \sin (c+d x))^2}-\frac{\sec ^4(c+d x) (b-a \sin (c+d x))}{4 d \left(a^2-b^2\right) (a+b \sin (c+d x))^2}",1,"(-3*(a^2 + 5*a*b + 8*b^2)*Log[1 - Sin[c + d*x]])/(16*(a + b)^5*d) + (3*(a^2 - 5*a*b + 8*b^2)*Log[1 + Sin[c + d*x]])/(16*(a - b)^5*d) - (3*b^5*(7*a^2 + b^2)*Log[a + b*Sin[c + d*x]])/((a^2 - b^2)^5*d) - (3*b*(a^4 - 5*a^2*b^2 - 4*b^4))/(8*(a^2 - b^2)^3*d*(a + b*Sin[c + d*x])^2) - (Sec[c + d*x]^4*(b - a*Sin[c + d*x]))/(4*(a^2 - b^2)*d*(a + b*Sin[c + d*x])^2) - (3*a*b*(a^4 - 6*a^2*b^2 - 27*b^4))/(8*(a^2 - b^2)^4*d*(a + b*Sin[c + d*x])) + (Sec[c + d*x]^2*(2*b*(a^2 + 3*b^2) + a*(3*a^2 - 11*b^2)*Sin[c + d*x]))/(8*(a^2 - b^2)^2*d*(a + b*Sin[c + d*x])^2)","A",5,4,21,0.1905,1,"{2668, 741, 823, 801}"
456,1,197,0,0.3668806,"\int \frac{\cos ^6(c+d x)}{(a+b \sin (c+d x))^3} \, dx","Int[Cos[c + d*x]^6/(a + b*Sin[c + d*x])^3,x]","-\frac{5 \left(-5 a^2 b^2+4 a^4+b^4\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{b^6 d \sqrt{a^2-b^2}}+\frac{5 \cos (c+d x) \left(4 a^2-2 a b \sin (c+d x)-b^2\right)}{2 b^5 d}+\frac{5 a x \left(4 a^2-3 b^2\right)}{2 b^6}-\frac{5 \cos ^3(c+d x) (4 a+b \sin (c+d x))}{6 b^3 d (a+b \sin (c+d x))}-\frac{\cos ^5(c+d x)}{2 b d (a+b \sin (c+d x))^2}","-\frac{5 \left(-5 a^2 b^2+4 a^4+b^4\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{b^6 d \sqrt{a^2-b^2}}+\frac{5 \cos (c+d x) \left(4 a^2-2 a b \sin (c+d x)-b^2\right)}{2 b^5 d}+\frac{5 a x \left(4 a^2-3 b^2\right)}{2 b^6}-\frac{5 \cos ^3(c+d x) (4 a+b \sin (c+d x))}{6 b^3 d (a+b \sin (c+d x))}-\frac{\cos ^5(c+d x)}{2 b d (a+b \sin (c+d x))^2}",1,"(5*a*(4*a^2 - 3*b^2)*x)/(2*b^6) - (5*(4*a^4 - 5*a^2*b^2 + b^4)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(b^6*Sqrt[a^2 - b^2]*d) - Cos[c + d*x]^5/(2*b*d*(a + b*Sin[c + d*x])^2) - (5*Cos[c + d*x]^3*(4*a + b*Sin[c + d*x]))/(6*b^3*d*(a + b*Sin[c + d*x])) + (5*Cos[c + d*x]*(4*a^2 - b^2 - 2*a*b*Sin[c + d*x]))/(2*b^5*d)","A",7,7,21,0.3333,1,"{2693, 2863, 2865, 2735, 2660, 618, 204}"
457,1,139,0,0.2083447,"\int \frac{\cos ^4(c+d x)}{(a+b \sin (c+d x))^3} \, dx","Int[Cos[c + d*x]^4/(a + b*Sin[c + d*x])^3,x]","\frac{3 \left(2 a^2-b^2\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{b^4 d \sqrt{a^2-b^2}}-\frac{3 \cos (c+d x) (2 a+b \sin (c+d x))}{2 b^3 d (a+b \sin (c+d x))}-\frac{3 a x}{b^4}-\frac{\cos ^3(c+d x)}{2 b d (a+b \sin (c+d x))^2}","\frac{3 \left(2 a^2-b^2\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{b^4 d \sqrt{a^2-b^2}}-\frac{3 \cos (c+d x) (2 a+b \sin (c+d x))}{2 b^3 d (a+b \sin (c+d x))}-\frac{3 a x}{b^4}-\frac{\cos ^3(c+d x)}{2 b d (a+b \sin (c+d x))^2}",1,"(-3*a*x)/b^4 + (3*(2*a^2 - b^2)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(b^4*Sqrt[a^2 - b^2]*d) - Cos[c + d*x]^3/(2*b*d*(a + b*Sin[c + d*x])^2) - (3*Cos[c + d*x]*(2*a + b*Sin[c + d*x]))/(2*b^3*d*(a + b*Sin[c + d*x]))","A",6,6,21,0.2857,1,"{2693, 2863, 2735, 2660, 618, 204}"
458,1,115,0,0.1278623,"\int \frac{\cos ^2(c+d x)}{(a+b \sin (c+d x))^3} \, dx","Int[Cos[c + d*x]^2/(a + b*Sin[c + d*x])^3,x]","\frac{\tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{d \left(a^2-b^2\right)^{3/2}}+\frac{a \cos (c+d x)}{2 b d \left(a^2-b^2\right) (a+b \sin (c+d x))}-\frac{\cos (c+d x)}{2 b d (a+b \sin (c+d x))^2}","\frac{\tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{d \left(a^2-b^2\right)^{3/2}}+\frac{a \cos (c+d x)}{2 b d \left(a^2-b^2\right) (a+b \sin (c+d x))}-\frac{\cos (c+d x)}{2 b d (a+b \sin (c+d x))^2}",1,"ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]]/((a^2 - b^2)^(3/2)*d) - Cos[c + d*x]/(2*b*d*(a + b*Sin[c + d*x])^2) + (a*Cos[c + d*x])/(2*b*(a^2 - b^2)*d*(a + b*Sin[c + d*x]))","A",6,6,21,0.2857,1,"{2693, 2754, 12, 2660, 618, 204}"
459,1,192,0,0.3912005,"\int \frac{\sec ^2(c+d x)}{(a+b \sin (c+d x))^3} \, dx","Int[Sec[c + d*x]^2/(a + b*Sin[c + d*x])^3,x]","-\frac{3 b^2 \left(4 a^2+b^2\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{d \left(a^2-b^2\right)^{7/2}}+\frac{5 a b \sec (c+d x)}{2 d \left(a^2-b^2\right)^2 (a+b \sin (c+d x))}+\frac{b \sec (c+d x)}{2 d \left(a^2-b^2\right) (a+b \sin (c+d x))^2}-\frac{\sec (c+d x) \left(3 b \left(4 a^2+b^2\right)-a \left(2 a^2+13 b^2\right) \sin (c+d x)\right)}{2 d \left(a^2-b^2\right)^3}","-\frac{3 b^2 \left(4 a^2+b^2\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{d \left(a^2-b^2\right)^{7/2}}+\frac{5 a b \sec (c+d x)}{2 d \left(a^2-b^2\right)^2 (a+b \sin (c+d x))}+\frac{b \sec (c+d x)}{2 d \left(a^2-b^2\right) (a+b \sin (c+d x))^2}-\frac{\sec (c+d x) \left(3 b \left(4 a^2+b^2\right)-a \left(2 a^2+13 b^2\right) \sin (c+d x)\right)}{2 d \left(a^2-b^2\right)^3}",1,"(-3*b^2*(4*a^2 + b^2)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/((a^2 - b^2)^(7/2)*d) + (b*Sec[c + d*x])/(2*(a^2 - b^2)*d*(a + b*Sin[c + d*x])^2) + (5*a*b*Sec[c + d*x])/(2*(a^2 - b^2)^2*d*(a + b*Sin[c + d*x])) - (Sec[c + d*x]*(3*b*(4*a^2 + b^2) - a*(2*a^2 + 13*b^2)*Sin[c + d*x]))/(2*(a^2 - b^2)^3*d)","A",7,7,21,0.3333,1,"{2694, 2864, 2866, 12, 2660, 618, 204}"
460,1,264,0,0.6401526,"\int \frac{\sec ^4(c+d x)}{(a+b \sin (c+d x))^3} \, dx","Int[Sec[c + d*x]^4/(a + b*Sin[c + d*x])^3,x]","\frac{5 b^4 \left(6 a^2+b^2\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{d \left(a^2-b^2\right)^{9/2}}+\frac{7 a b \sec ^3(c+d x)}{2 d \left(a^2-b^2\right)^2 (a+b \sin (c+d x))}+\frac{b \sec ^3(c+d x)}{2 d \left(a^2-b^2\right) (a+b \sin (c+d x))^2}-\frac{\sec ^3(c+d x) \left(5 b \left(6 a^2+b^2\right)-a \left(2 a^2+33 b^2\right) \sin (c+d x)\right)}{6 d \left(a^2-b^2\right)^3}+\frac{\sec (c+d x) \left(a \left(-28 a^2 b^2+4 a^4-81 b^4\right) \sin (c+d x)+15 b^3 \left(6 a^2+b^2\right)\right)}{6 d \left(a^2-b^2\right)^4}","\frac{5 b^4 \left(6 a^2+b^2\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{d \left(a^2-b^2\right)^{9/2}}+\frac{7 a b \sec ^3(c+d x)}{2 d \left(a^2-b^2\right)^2 (a+b \sin (c+d x))}+\frac{b \sec ^3(c+d x)}{2 d \left(a^2-b^2\right) (a+b \sin (c+d x))^2}-\frac{\sec ^3(c+d x) \left(5 b \left(6 a^2+b^2\right)-a \left(2 a^2+33 b^2\right) \sin (c+d x)\right)}{6 d \left(a^2-b^2\right)^3}+\frac{\sec (c+d x) \left(a \left(-28 a^2 b^2+4 a^4-81 b^4\right) \sin (c+d x)+15 b^3 \left(6 a^2+b^2\right)\right)}{6 d \left(a^2-b^2\right)^4}",1,"(5*b^4*(6*a^2 + b^2)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/((a^2 - b^2)^(9/2)*d) + (b*Sec[c + d*x]^3)/(2*(a^2 - b^2)*d*(a + b*Sin[c + d*x])^2) + (7*a*b*Sec[c + d*x]^3)/(2*(a^2 - b^2)^2*d*(a + b*Sin[c + d*x])) - (Sec[c + d*x]^3*(5*b*(6*a^2 + b^2) - a*(2*a^2 + 33*b^2)*Sin[c + d*x]))/(6*(a^2 - b^2)^3*d) + (Sec[c + d*x]*(15*b^3*(6*a^2 + b^2) + a*(4*a^4 - 28*a^2*b^2 - 81*b^4)*Sin[c + d*x]))/(6*(a^2 - b^2)^4*d)","A",8,7,21,0.3333,1,"{2694, 2864, 2866, 12, 2660, 618, 204}"
461,1,207,0,0.1711684,"\int \frac{\cos ^7(c+d x)}{(a+b \sin (c+d x))^8} \, dx","Int[Cos[c + d*x]^7/(a + b*Sin[c + d*x])^8,x]","\frac{\left(a^2-b^2\right)^3}{7 b^7 d (a+b \sin (c+d x))^7}-\frac{a \left(a^2-b^2\right)^2}{b^7 d (a+b \sin (c+d x))^6}+\frac{5 a^2-b^2}{b^7 d (a+b \sin (c+d x))^3}-\frac{a \left(5 a^2-3 b^2\right)}{b^7 d (a+b \sin (c+d x))^4}+\frac{3 \left(-6 a^2 b^2+5 a^4+b^4\right)}{5 b^7 d (a+b \sin (c+d x))^5}+\frac{1}{b^7 d (a+b \sin (c+d x))}-\frac{3 a}{b^7 d (a+b \sin (c+d x))^2}","\frac{\left(a^2-b^2\right)^3}{7 b^7 d (a+b \sin (c+d x))^7}-\frac{a \left(a^2-b^2\right)^2}{b^7 d (a+b \sin (c+d x))^6}+\frac{5 a^2-b^2}{b^7 d (a+b \sin (c+d x))^3}-\frac{a \left(5 a^2-3 b^2\right)}{b^7 d (a+b \sin (c+d x))^4}+\frac{3 \left(-6 a^2 b^2+5 a^4+b^4\right)}{5 b^7 d (a+b \sin (c+d x))^5}+\frac{1}{b^7 d (a+b \sin (c+d x))}-\frac{3 a}{b^7 d (a+b \sin (c+d x))^2}",1,"(a^2 - b^2)^3/(7*b^7*d*(a + b*Sin[c + d*x])^7) - (a*(a^2 - b^2)^2)/(b^7*d*(a + b*Sin[c + d*x])^6) + (3*(5*a^4 - 6*a^2*b^2 + b^4))/(5*b^7*d*(a + b*Sin[c + d*x])^5) - (a*(5*a^2 - 3*b^2))/(b^7*d*(a + b*Sin[c + d*x])^4) + (5*a^2 - b^2)/(b^7*d*(a + b*Sin[c + d*x])^3) - (3*a)/(b^7*d*(a + b*Sin[c + d*x])^2) + 1/(b^7*d*(a + b*Sin[c + d*x]))","A",3,2,21,0.09524,1,"{2668, 697}"
462,1,141,0,0.1087657,"\int \frac{\cos ^5(c+d x)}{(a+b \sin (c+d x))^8} \, dx","Int[Cos[c + d*x]^5/(a + b*Sin[c + d*x])^8,x]","-\frac{\left(a^2-b^2\right)^2}{7 b^5 d (a+b \sin (c+d x))^7}+\frac{2 a \left(a^2-b^2\right)}{3 b^5 d (a+b \sin (c+d x))^6}-\frac{2 \left(3 a^2-b^2\right)}{5 b^5 d (a+b \sin (c+d x))^5}-\frac{1}{3 b^5 d (a+b \sin (c+d x))^3}+\frac{a}{b^5 d (a+b \sin (c+d x))^4}","-\frac{\left(a^2-b^2\right)^2}{7 b^5 d (a+b \sin (c+d x))^7}+\frac{2 a \left(a^2-b^2\right)}{3 b^5 d (a+b \sin (c+d x))^6}-\frac{2 \left(3 a^2-b^2\right)}{5 b^5 d (a+b \sin (c+d x))^5}-\frac{1}{3 b^5 d (a+b \sin (c+d x))^3}+\frac{a}{b^5 d (a+b \sin (c+d x))^4}",1,"-(a^2 - b^2)^2/(7*b^5*d*(a + b*Sin[c + d*x])^7) + (2*a*(a^2 - b^2))/(3*b^5*d*(a + b*Sin[c + d*x])^6) - (2*(3*a^2 - b^2))/(5*b^5*d*(a + b*Sin[c + d*x])^5) + a/(b^5*d*(a + b*Sin[c + d*x])^4) - 1/(3*b^5*d*(a + b*Sin[c + d*x])^3)","A",3,2,21,0.09524,1,"{2668, 697}"
463,1,77,0,0.0715896,"\int \frac{\cos ^3(c+d x)}{(a+b \sin (c+d x))^8} \, dx","Int[Cos[c + d*x]^3/(a + b*Sin[c + d*x])^8,x]","\frac{a^2-b^2}{7 b^3 d (a+b \sin (c+d x))^7}-\frac{a}{3 b^3 d (a+b \sin (c+d x))^6}+\frac{1}{5 b^3 d (a+b \sin (c+d x))^5}","\frac{a^2-b^2}{7 b^3 d (a+b \sin (c+d x))^7}-\frac{a}{3 b^3 d (a+b \sin (c+d x))^6}+\frac{1}{5 b^3 d (a+b \sin (c+d x))^5}",1,"(a^2 - b^2)/(7*b^3*d*(a + b*Sin[c + d*x])^7) - a/(3*b^3*d*(a + b*Sin[c + d*x])^6) + 1/(5*b^3*d*(a + b*Sin[c + d*x])^5)","A",3,2,21,0.09524,1,"{2668, 697}"
464,1,22,0,0.0269824,"\int \frac{\cos (c+d x)}{(a+b \sin (c+d x))^8} \, dx","Int[Cos[c + d*x]/(a + b*Sin[c + d*x])^8,x]","-\frac{1}{7 b d (a+b \sin (c+d x))^7}","-\frac{1}{7 b d (a+b \sin (c+d x))^7}",1,"-1/(7*b*d*(a + b*Sin[c + d*x])^7)","A",2,2,19,0.1053,1,"{2668, 32}"
465,1,385,0,0.5280992,"\int \frac{\sec (c+d x)}{(a+b \sin (c+d x))^8} \, dx","Int[Sec[c + d*x]/(a + b*Sin[c + d*x])^8,x]","\frac{b \left(35 a^4 b^2+21 a^2 b^4+7 a^6+b^6\right)}{d \left(a^2-b^2\right)^7 (a+b \sin (c+d x))}+\frac{a b \left(3 a^2+b^2\right) \left(a^2+3 b^2\right)}{d \left(a^2-b^2\right)^6 (a+b \sin (c+d x))^2}+\frac{b \left(10 a^2 b^2+5 a^4+b^4\right)}{3 d \left(a^2-b^2\right)^5 (a+b \sin (c+d x))^3}+\frac{a b \left(a^2+b^2\right)}{d \left(a^2-b^2\right)^4 (a+b \sin (c+d x))^4}+\frac{b \left(3 a^2+b^2\right)}{5 d \left(a^2-b^2\right)^3 (a+b \sin (c+d x))^5}+\frac{a b}{3 d \left(a^2-b^2\right)^2 (a+b \sin (c+d x))^6}+\frac{b}{7 d \left(a^2-b^2\right) (a+b \sin (c+d x))^7}-\frac{8 a b \left(a^2+b^2\right) \left(6 a^2 b^2+a^4+b^4\right) \log (a+b \sin (c+d x))}{d \left(a^2-b^2\right)^8}-\frac{\log (1-\sin (c+d x))}{2 d (a+b)^8}+\frac{\log (\sin (c+d x)+1)}{2 d (a-b)^8}","\frac{b \left(35 a^4 b^2+21 a^2 b^4+7 a^6+b^6\right)}{d \left(a^2-b^2\right)^7 (a+b \sin (c+d x))}+\frac{a b \left(3 a^2+b^2\right) \left(a^2+3 b^2\right)}{d \left(a^2-b^2\right)^6 (a+b \sin (c+d x))^2}+\frac{b \left(10 a^2 b^2+5 a^4+b^4\right)}{3 d \left(a^2-b^2\right)^5 (a+b \sin (c+d x))^3}+\frac{a b \left(a^2+b^2\right)}{d \left(a^2-b^2\right)^4 (a+b \sin (c+d x))^4}+\frac{b \left(3 a^2+b^2\right)}{5 d \left(a^2-b^2\right)^3 (a+b \sin (c+d x))^5}+\frac{a b}{3 d \left(a^2-b^2\right)^2 (a+b \sin (c+d x))^6}+\frac{b}{7 d \left(a^2-b^2\right) (a+b \sin (c+d x))^7}-\frac{8 a b \left(a^2+b^2\right) \left(6 a^2 b^2+a^4+b^4\right) \log (a+b \sin (c+d x))}{d \left(a^2-b^2\right)^8}-\frac{\log (1-\sin (c+d x))}{2 d (a+b)^8}+\frac{\log (\sin (c+d x)+1)}{2 d (a-b)^8}",1,"-Log[1 - Sin[c + d*x]]/(2*(a + b)^8*d) + Log[1 + Sin[c + d*x]]/(2*(a - b)^8*d) - (8*a*b*(a^2 + b^2)*(a^4 + 6*a^2*b^2 + b^4)*Log[a + b*Sin[c + d*x]])/((a^2 - b^2)^8*d) + b/(7*(a^2 - b^2)*d*(a + b*Sin[c + d*x])^7) + (a*b)/(3*(a^2 - b^2)^2*d*(a + b*Sin[c + d*x])^6) + (b*(3*a^2 + b^2))/(5*(a^2 - b^2)^3*d*(a + b*Sin[c + d*x])^5) + (a*b*(a^2 + b^2))/((a^2 - b^2)^4*d*(a + b*Sin[c + d*x])^4) + (b*(5*a^4 + 10*a^2*b^2 + b^4))/(3*(a^2 - b^2)^5*d*(a + b*Sin[c + d*x])^3) + (a*b*(3*a^2 + b^2)*(a^2 + 3*b^2))/((a^2 - b^2)^6*d*(a + b*Sin[c + d*x])^2) + (b*(7*a^6 + 35*a^4*b^2 + 21*a^2*b^4 + b^6))/((a^2 - b^2)^7*d*(a + b*Sin[c + d*x]))","A",4,3,19,0.1579,1,"{2668, 710, 801}"
466,1,527,0,0.7356195,"\int \frac{\sec ^3(c+d x)}{(a+b \sin (c+d x))^8} \, dx","Int[Sec[c + d*x]^3/(a + b*Sin[c + d*x])^8,x]","-\frac{b \left(196 a^6 b^2+574 a^4 b^4+244 a^2 b^6+a^8+9 b^8\right)}{2 d \left(a^2-b^2\right)^8 (a+b \sin (c+d x))}-\frac{a b \left(77 a^4 b^2+147 a^2 b^4+a^6+31 b^6\right)}{2 d \left(a^2-b^2\right)^7 (a+b \sin (c+d x))^2}-\frac{b \left(115 a^4 b^2+129 a^2 b^4+3 a^6+9 b^6\right)}{6 d \left(a^2-b^2\right)^6 (a+b \sin (c+d x))^3}-\frac{a b \left(20 a^2 b^2+a^4+11 b^4\right)}{2 d \left(a^2-b^2\right)^5 (a+b \sin (c+d x))^4}-\frac{b \left(50 a^2 b^2+5 a^4+9 b^4\right)}{10 d \left(a^2-b^2\right)^4 (a+b \sin (c+d x))^5}-\frac{a b \left(3 a^2+13 b^2\right)}{6 d \left(a^2-b^2\right)^3 (a+b \sin (c+d x))^6}-\frac{b \left(7 a^2+9 b^2\right)}{14 d \left(a^2-b^2\right)^2 (a+b \sin (c+d x))^7}+\frac{8 a b^3 \left(63 a^4 b^2+45 a^2 b^4+15 a^6+5 b^6\right) \log (a+b \sin (c+d x))}{d \left(a^2-b^2\right)^9}-\frac{\sec ^2(c+d x) (b-a \sin (c+d x))}{2 d \left(a^2-b^2\right) (a+b \sin (c+d x))^7}-\frac{(a+9 b) \log (1-\sin (c+d x))}{4 d (a+b)^9}+\frac{(a-9 b) \log (\sin (c+d x)+1)}{4 d (a-b)^9}","-\frac{b \left(196 a^6 b^2+574 a^4 b^4+244 a^2 b^6+a^8+9 b^8\right)}{2 d \left(a^2-b^2\right)^8 (a+b \sin (c+d x))}-\frac{a b \left(77 a^4 b^2+147 a^2 b^4+a^6+31 b^6\right)}{2 d \left(a^2-b^2\right)^7 (a+b \sin (c+d x))^2}-\frac{b \left(115 a^4 b^2+129 a^2 b^4+3 a^6+9 b^6\right)}{6 d \left(a^2-b^2\right)^6 (a+b \sin (c+d x))^3}-\frac{a b \left(20 a^2 b^2+a^4+11 b^4\right)}{2 d \left(a^2-b^2\right)^5 (a+b \sin (c+d x))^4}-\frac{b \left(50 a^2 b^2+5 a^4+9 b^4\right)}{10 d \left(a^2-b^2\right)^4 (a+b \sin (c+d x))^5}-\frac{a b \left(3 a^2+13 b^2\right)}{6 d \left(a^2-b^2\right)^3 (a+b \sin (c+d x))^6}-\frac{b \left(7 a^2+9 b^2\right)}{14 d \left(a^2-b^2\right)^2 (a+b \sin (c+d x))^7}+\frac{8 a b^3 \left(63 a^4 b^2+45 a^2 b^4+15 a^6+5 b^6\right) \log (a+b \sin (c+d x))}{d \left(a^2-b^2\right)^9}-\frac{\sec ^2(c+d x) (b-a \sin (c+d x))}{2 d \left(a^2-b^2\right) (a+b \sin (c+d x))^7}-\frac{(a+9 b) \log (1-\sin (c+d x))}{4 d (a+b)^9}+\frac{(a-9 b) \log (\sin (c+d x)+1)}{4 d (a-b)^9}",1,"-((a + 9*b)*Log[1 - Sin[c + d*x]])/(4*(a + b)^9*d) + ((a - 9*b)*Log[1 + Sin[c + d*x]])/(4*(a - b)^9*d) + (8*a*b^3*(15*a^6 + 63*a^4*b^2 + 45*a^2*b^4 + 5*b^6)*Log[a + b*Sin[c + d*x]])/((a^2 - b^2)^9*d) - (b*(7*a^2 + 9*b^2))/(14*(a^2 - b^2)^2*d*(a + b*Sin[c + d*x])^7) - (Sec[c + d*x]^2*(b - a*Sin[c + d*x]))/(2*(a^2 - b^2)*d*(a + b*Sin[c + d*x])^7) - (a*b*(3*a^2 + 13*b^2))/(6*(a^2 - b^2)^3*d*(a + b*Sin[c + d*x])^6) - (b*(5*a^4 + 50*a^2*b^2 + 9*b^4))/(10*(a^2 - b^2)^4*d*(a + b*Sin[c + d*x])^5) - (a*b*(a^4 + 20*a^2*b^2 + 11*b^4))/(2*(a^2 - b^2)^5*d*(a + b*Sin[c + d*x])^4) - (b*(3*a^6 + 115*a^4*b^2 + 129*a^2*b^4 + 9*b^6))/(6*(a^2 - b^2)^6*d*(a + b*Sin[c + d*x])^3) - (a*b*(a^6 + 77*a^4*b^2 + 147*a^2*b^4 + 31*b^6))/(2*(a^2 - b^2)^7*d*(a + b*Sin[c + d*x])^2) - (b*(a^8 + 196*a^6*b^2 + 574*a^4*b^4 + 244*a^2*b^6 + 9*b^8))/(2*(a^2 - b^2)^8*d*(a + b*Sin[c + d*x]))","A",4,3,21,0.1429,1,"{2668, 741, 801}"
467,1,491,0,1.2738246,"\int \frac{\cos ^8(c+d x)}{(a+b \sin (c+d x))^8} \, dx","Int[Cos[c + d*x]^8/(a + b*Sin[c + d*x])^8,x]","-\frac{a \left(-56 a^4 b^2+70 a^2 b^4+16 a^6-35 b^6\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{8 b^8 d \left(a^2-b^2\right)^{7/2}}+\frac{a \cos ^7(c+d x)}{6 b d \left(a^2-b^2\right) (a+b \sin (c+d x))^6}+\frac{\cos ^5(c+d x) \left(6 \left(a^2-b^2\right)+5 a b \sin (c+d x)\right)}{30 b^3 d \left(a^2-b^2\right) (a+b \sin (c+d x))^5}-\frac{a \left(6 a^2-11 b^2\right) \cos ^5(c+d x)}{24 b^3 d \left(a^2-b^2\right)^2 (a+b \sin (c+d x))^4}-\frac{\cos ^3(c+d x) \left(a b \left(6 a^2-11 b^2\right) \sin (c+d x)+8 \left(a^2-b^2\right)^2\right)}{24 b^5 d \left(a^2-b^2\right)^2 (a+b \sin (c+d x))^3}+\frac{a \left(-22 a^2 b^2+8 a^4+19 b^4\right) \cos ^3(c+d x)}{16 b^5 d \left(a^2-b^2\right)^3 (a+b \sin (c+d x))^2}+\frac{\cos (c+d x) \left(a b \left(-22 a^2 b^2+8 a^4+19 b^4\right) \sin (c+d x)+16 \left(a^2-b^2\right)^3\right)}{16 b^7 d \left(a^2-b^2\right)^3 (a+b \sin (c+d x))}-\frac{\cos ^7(c+d x)}{7 b d (a+b \sin (c+d x))^7}+\frac{x}{b^8}","-\frac{a \left(-56 a^4 b^2+70 a^2 b^4+16 a^6-35 b^6\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{8 b^8 d \left(a^2-b^2\right)^{7/2}}+\frac{a \cos ^7(c+d x)}{6 b d \left(a^2-b^2\right) (a+b \sin (c+d x))^6}+\frac{\cos ^5(c+d x) \left(6 \left(a^2-b^2\right)+5 a b \sin (c+d x)\right)}{30 b^3 d \left(a^2-b^2\right) (a+b \sin (c+d x))^5}-\frac{a \left(6 a^2-11 b^2\right) \cos ^5(c+d x)}{24 b^3 d \left(a^2-b^2\right)^2 (a+b \sin (c+d x))^4}-\frac{\cos ^3(c+d x) \left(a b \left(6 a^2-11 b^2\right) \sin (c+d x)+8 \left(a^2-b^2\right)^2\right)}{24 b^5 d \left(a^2-b^2\right)^2 (a+b \sin (c+d x))^3}+\frac{a \left(-22 a^2 b^2+8 a^4+19 b^4\right) \cos ^3(c+d x)}{16 b^5 d \left(a^2-b^2\right)^3 (a+b \sin (c+d x))^2}+\frac{\cos (c+d x) \left(a b \left(-22 a^2 b^2+8 a^4+19 b^4\right) \sin (c+d x)+16 \left(a^2-b^2\right)^3\right)}{16 b^7 d \left(a^2-b^2\right)^3 (a+b \sin (c+d x))}-\frac{\cos ^7(c+d x)}{7 b d (a+b \sin (c+d x))^7}+\frac{x}{b^8}",1,"x/b^8 - (a*(16*a^6 - 56*a^4*b^2 + 70*a^2*b^4 - 35*b^6)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(8*b^8*(a^2 - b^2)^(7/2)*d) - Cos[c + d*x]^7/(7*b*d*(a + b*Sin[c + d*x])^7) + (a*Cos[c + d*x]^7)/(6*b*(a^2 - b^2)*d*(a + b*Sin[c + d*x])^6) - (a*(6*a^2 - 11*b^2)*Cos[c + d*x]^5)/(24*b^3*(a^2 - b^2)^2*d*(a + b*Sin[c + d*x])^4) + (a*(8*a^4 - 22*a^2*b^2 + 19*b^4)*Cos[c + d*x]^3)/(16*b^5*(a^2 - b^2)^3*d*(a + b*Sin[c + d*x])^2) + (Cos[c + d*x]^5*(6*(a^2 - b^2) + 5*a*b*Sin[c + d*x]))/(30*b^3*(a^2 - b^2)*d*(a + b*Sin[c + d*x])^5) - (Cos[c + d*x]^3*(8*(a^2 - b^2)^2 + a*b*(6*a^2 - 11*b^2)*Sin[c + d*x]))/(24*b^5*(a^2 - b^2)^2*d*(a + b*Sin[c + d*x])^3) + (Cos[c + d*x]*(16*(a^2 - b^2)^3 + a*b*(8*a^4 - 22*a^2*b^2 + 19*b^4)*Sin[c + d*x]))/(16*b^7*(a^2 - b^2)^3*d*(a + b*Sin[c + d*x]))","A",11,7,21,0.3333,1,"{2693, 2864, 2863, 2735, 2660, 618, 204}"
468,1,407,0,0.7915195,"\int \frac{\cos ^6(c+d x)}{(a+b \sin (c+d x))^8} \, dx","Int[Cos[c + d*x]^6/(a + b*Sin[c + d*x])^8,x]","\frac{5 a \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{8 d \left(a^2-b^2\right)^{9/2}}-\frac{\cos (c+d x) \left(4 a^2+10 a b \sin (c+d x)+9 b^2\right)}{42 b^5 d (a+b \sin (c+d x))^5}+\frac{\left(-38 a^4 b^2+87 a^2 b^4+8 a^6+48 b^6\right) \cos (c+d x)}{336 b^5 d \left(a^2-b^2\right)^4 (a+b \sin (c+d x))}+\frac{a \left(-30 a^2 b^2+8 a^4+57 b^4\right) \cos (c+d x)}{336 b^5 d \left(a^2-b^2\right)^3 (a+b \sin (c+d x))^2}+\frac{\left(-9 a^2 b^2+4 a^4+12 b^4\right) \cos (c+d x)}{168 b^5 d \left(a^2-b^2\right)^2 (a+b \sin (c+d x))^3}+\frac{a \left(4 a^2-b^2\right) \cos (c+d x)}{168 b^5 d \left(a^2-b^2\right) (a+b \sin (c+d x))^4}+\frac{5 \cos ^3(c+d x) (2 a+3 b \sin (c+d x))}{42 b^3 d (a+b \sin (c+d x))^6}-\frac{\cos ^5(c+d x)}{7 b d (a+b \sin (c+d x))^7}","\frac{5 a \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{8 d \left(a^2-b^2\right)^{9/2}}-\frac{\cos (c+d x) \left(4 a^2+10 a b \sin (c+d x)+9 b^2\right)}{42 b^5 d (a+b \sin (c+d x))^5}+\frac{\left(-38 a^4 b^2+87 a^2 b^4+8 a^6+48 b^6\right) \cos (c+d x)}{336 b^5 d \left(a^2-b^2\right)^4 (a+b \sin (c+d x))}+\frac{a \left(-30 a^2 b^2+8 a^4+57 b^4\right) \cos (c+d x)}{336 b^5 d \left(a^2-b^2\right)^3 (a+b \sin (c+d x))^2}+\frac{\left(-9 a^2 b^2+4 a^4+12 b^4\right) \cos (c+d x)}{168 b^5 d \left(a^2-b^2\right)^2 (a+b \sin (c+d x))^3}+\frac{a \left(4 a^2-b^2\right) \cos (c+d x)}{168 b^5 d \left(a^2-b^2\right) (a+b \sin (c+d x))^4}+\frac{5 \cos ^3(c+d x) (2 a+3 b \sin (c+d x))}{42 b^3 d (a+b \sin (c+d x))^6}-\frac{\cos ^5(c+d x)}{7 b d (a+b \sin (c+d x))^7}",1,"(5*a*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(8*(a^2 - b^2)^(9/2)*d) - Cos[c + d*x]^5/(7*b*d*(a + b*Sin[c + d*x])^7) + (a*(4*a^2 - b^2)*Cos[c + d*x])/(168*b^5*(a^2 - b^2)*d*(a + b*Sin[c + d*x])^4) + ((4*a^4 - 9*a^2*b^2 + 12*b^4)*Cos[c + d*x])/(168*b^5*(a^2 - b^2)^2*d*(a + b*Sin[c + d*x])^3) + (a*(8*a^4 - 30*a^2*b^2 + 57*b^4)*Cos[c + d*x])/(336*b^5*(a^2 - b^2)^3*d*(a + b*Sin[c + d*x])^2) + ((8*a^6 - 38*a^4*b^2 + 87*a^2*b^4 + 48*b^6)*Cos[c + d*x])/(336*b^5*(a^2 - b^2)^4*d*(a + b*Sin[c + d*x])) + (5*Cos[c + d*x]^3*(2*a + 3*b*Sin[c + d*x]))/(42*b^3*d*(a + b*Sin[c + d*x])^6) - (Cos[c + d*x]*(4*a^2 + 9*b^2 + 10*a*b*Sin[c + d*x]))/(42*b^5*d*(a + b*Sin[c + d*x])^5)","A",11,7,21,0.3333,1,"{2693, 2863, 2754, 12, 2660, 618, 204}"
469,1,411,0,0.7906442,"\int \frac{\cos ^4(c+d x)}{(a+b \sin (c+d x))^8} \, dx","Int[Cos[c + d*x]^4/(a + b*Sin[c + d*x])^8,x]","\frac{3 a \left(2 a^2+b^2\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{8 d \left(a^2-b^2\right)^{11/2}}-\frac{\left(-40 a^4 b^2-247 a^2 b^4+4 a^6-32 b^6\right) \cos (c+d x)}{560 b^3 d \left(a^2-b^2\right)^5 (a+b \sin (c+d x))}-\frac{a \left(-36 a^2 b^2+4 a^4-73 b^4\right) \cos (c+d x)}{560 b^3 d \left(a^2-b^2\right)^4 (a+b \sin (c+d x))^2}-\frac{\left(-15 a^2 b^2+2 a^4-8 b^4\right) \cos (c+d x)}{280 b^3 d \left(a^2-b^2\right)^3 (a+b \sin (c+d x))^3}-\frac{a \left(2 a^2-11 b^2\right) \cos (c+d x)}{280 b^3 d \left(a^2-b^2\right)^2 (a+b \sin (c+d x))^4}-\frac{\left(a^2-3 b^2\right) \cos (c+d x)}{140 b^3 d \left(a^2-b^2\right) (a+b \sin (c+d x))^5}+\frac{\cos (c+d x) (a+3 b \sin (c+d x))}{28 b^3 d (a+b \sin (c+d x))^6}-\frac{\cos ^3(c+d x)}{7 b d (a+b \sin (c+d x))^7}","\frac{3 a \left(2 a^2+b^2\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{8 d \left(a^2-b^2\right)^{11/2}}-\frac{\left(-40 a^4 b^2-247 a^2 b^4+4 a^6-32 b^6\right) \cos (c+d x)}{560 b^3 d \left(a^2-b^2\right)^5 (a+b \sin (c+d x))}-\frac{a \left(-36 a^2 b^2+4 a^4-73 b^4\right) \cos (c+d x)}{560 b^3 d \left(a^2-b^2\right)^4 (a+b \sin (c+d x))^2}-\frac{\left(-15 a^2 b^2+2 a^4-8 b^4\right) \cos (c+d x)}{280 b^3 d \left(a^2-b^2\right)^3 (a+b \sin (c+d x))^3}-\frac{a \left(2 a^2-11 b^2\right) \cos (c+d x)}{280 b^3 d \left(a^2-b^2\right)^2 (a+b \sin (c+d x))^4}-\frac{\left(a^2-3 b^2\right) \cos (c+d x)}{140 b^3 d \left(a^2-b^2\right) (a+b \sin (c+d x))^5}+\frac{\cos (c+d x) (a+3 b \sin (c+d x))}{28 b^3 d (a+b \sin (c+d x))^6}-\frac{\cos ^3(c+d x)}{7 b d (a+b \sin (c+d x))^7}",1,"(3*a*(2*a^2 + b^2)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(8*(a^2 - b^2)^(11/2)*d) - Cos[c + d*x]^3/(7*b*d*(a + b*Sin[c + d*x])^7) - ((a^2 - 3*b^2)*Cos[c + d*x])/(140*b^3*(a^2 - b^2)*d*(a + b*Sin[c + d*x])^5) - (a*(2*a^2 - 11*b^2)*Cos[c + d*x])/(280*b^3*(a^2 - b^2)^2*d*(a + b*Sin[c + d*x])^4) - ((2*a^4 - 15*a^2*b^2 - 8*b^4)*Cos[c + d*x])/(280*b^3*(a^2 - b^2)^3*d*(a + b*Sin[c + d*x])^3) - (a*(4*a^4 - 36*a^2*b^2 - 73*b^4)*Cos[c + d*x])/(560*b^3*(a^2 - b^2)^4*d*(a + b*Sin[c + d*x])^2) - ((4*a^6 - 40*a^4*b^2 - 247*a^2*b^4 - 32*b^6)*Cos[c + d*x])/(560*b^3*(a^2 - b^2)^5*d*(a + b*Sin[c + d*x])) + (Cos[c + d*x]*(a + 3*b*Sin[c + d*x]))/(28*b^3*d*(a + b*Sin[c + d*x])^6)","A",11,7,21,0.3333,1,"{2693, 2863, 2754, 12, 2660, 618, 204}"
470,1,422,0,0.7459747,"\int \frac{\cos ^2(c+d x)}{(a+b \sin (c+d x))^8} \, dx","Int[Cos[c + d*x]^2/(a + b*Sin[c + d*x])^8,x]","\frac{a \left(20 a^2 b^2+8 a^4+5 b^4\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{8 d \left(a^2-b^2\right)^{13/2}}+\frac{\left(1518 a^4 b^2+1779 a^2 b^4+40 a^6+128 b^6\right) \cos (c+d x)}{1680 b d \left(a^2-b^2\right)^6 (a+b \sin (c+d x))}+\frac{a \left(718 a^2 b^2+40 a^4+397 b^4\right) \cos (c+d x)}{1680 b d \left(a^2-b^2\right)^5 (a+b \sin (c+d x))^2}+\frac{\left(179 a^2 b^2+20 a^4+32 b^4\right) \cos (c+d x)}{840 b d \left(a^2-b^2\right)^4 (a+b \sin (c+d x))^3}+\frac{a \left(20 a^2+79 b^2\right) \cos (c+d x)}{840 b d \left(a^2-b^2\right)^3 (a+b \sin (c+d x))^4}+\frac{\left(5 a^2+6 b^2\right) \cos (c+d x)}{210 b d \left(a^2-b^2\right)^2 (a+b \sin (c+d x))^5}+\frac{a \cos (c+d x)}{42 b d \left(a^2-b^2\right) (a+b \sin (c+d x))^6}-\frac{\cos (c+d x)}{7 b d (a+b \sin (c+d x))^7}","\frac{a \left(20 a^2 b^2+8 a^4+5 b^4\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{8 d \left(a^2-b^2\right)^{13/2}}+\frac{\left(1518 a^4 b^2+1779 a^2 b^4+40 a^6+128 b^6\right) \cos (c+d x)}{1680 b d \left(a^2-b^2\right)^6 (a+b \sin (c+d x))}+\frac{a \left(718 a^2 b^2+40 a^4+397 b^4\right) \cos (c+d x)}{1680 b d \left(a^2-b^2\right)^5 (a+b \sin (c+d x))^2}+\frac{\left(179 a^2 b^2+20 a^4+32 b^4\right) \cos (c+d x)}{840 b d \left(a^2-b^2\right)^4 (a+b \sin (c+d x))^3}+\frac{a \left(20 a^2+79 b^2\right) \cos (c+d x)}{840 b d \left(a^2-b^2\right)^3 (a+b \sin (c+d x))^4}+\frac{\left(5 a^2+6 b^2\right) \cos (c+d x)}{210 b d \left(a^2-b^2\right)^2 (a+b \sin (c+d x))^5}+\frac{a \cos (c+d x)}{42 b d \left(a^2-b^2\right) (a+b \sin (c+d x))^6}-\frac{\cos (c+d x)}{7 b d (a+b \sin (c+d x))^7}",1,"(a*(8*a^4 + 20*a^2*b^2 + 5*b^4)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(8*(a^2 - b^2)^(13/2)*d) - Cos[c + d*x]/(7*b*d*(a + b*Sin[c + d*x])^7) + (a*Cos[c + d*x])/(42*b*(a^2 - b^2)*d*(a + b*Sin[c + d*x])^6) + ((5*a^2 + 6*b^2)*Cos[c + d*x])/(210*b*(a^2 - b^2)^2*d*(a + b*Sin[c + d*x])^5) + (a*(20*a^2 + 79*b^2)*Cos[c + d*x])/(840*b*(a^2 - b^2)^3*d*(a + b*Sin[c + d*x])^4) + ((20*a^4 + 179*a^2*b^2 + 32*b^4)*Cos[c + d*x])/(840*b*(a^2 - b^2)^4*d*(a + b*Sin[c + d*x])^3) + (a*(40*a^4 + 718*a^2*b^2 + 397*b^4)*Cos[c + d*x])/(1680*b*(a^2 - b^2)^5*d*(a + b*Sin[c + d*x])^2) + ((40*a^6 + 1518*a^4*b^2 + 1779*a^2*b^4 + 128*b^6)*Cos[c + d*x])/(1680*b*(a^2 - b^2)^6*d*(a + b*Sin[c + d*x]))","A",11,6,21,0.2857,1,"{2693, 2754, 12, 2660, 618, 204}"
471,1,529,0,1.7652601,"\int \frac{\sec ^2(c+d x)}{(a+b \sin (c+d x))^8} \, dx","Int[Sec[c + d*x]^2/(a + b*Sin[c + d*x])^8,x]","-\frac{9 a b^2 \left(336 a^4 b^2+280 a^2 b^4+64 a^6+35 b^6\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{8 d \left(a^2-b^2\right)^{17/2}}+\frac{b \left(41484 a^4 b^2+22767 a^2 b^4+9800 a^6+1024 b^6\right) \sec (c+d x)}{560 d \left(a^2-b^2\right)^7 (a+b \sin (c+d x))}+\frac{11 a b \left(844 a^2 b^2+280 a^4+241 b^4\right) \sec (c+d x)}{560 d \left(a^2-b^2\right)^6 (a+b \sin (c+d x))^2}+\frac{b \left(1317 a^2 b^2+700 a^4+128 b^4\right) \sec (c+d x)}{280 d \left(a^2-b^2\right)^5 (a+b \sin (c+d x))^3}+\frac{13 a b \left(28 a^2+27 b^2\right) \sec (c+d x)}{280 d \left(a^2-b^2\right)^4 (a+b \sin (c+d x))^4}+\frac{b \left(49 a^2+16 b^2\right) \sec (c+d x)}{70 d \left(a^2-b^2\right)^3 (a+b \sin (c+d x))^5}+\frac{5 a b \sec (c+d x)}{14 d \left(a^2-b^2\right)^2 (a+b \sin (c+d x))^6}+\frac{b \sec (c+d x)}{7 d \left(a^2-b^2\right) (a+b \sin (c+d x))^7}-\frac{\sec (c+d x) \left(315 a b \left(336 a^4 b^2+280 a^2 b^4+64 a^6+35 b^6\right)-\left(42472 a^6 b^2+125634 a^4 b^4+54511 a^2 b^6+560 a^8+2048 b^8\right) \sin (c+d x)\right)}{560 d \left(a^2-b^2\right)^8}","-\frac{9 a b^2 \left(336 a^4 b^2+280 a^2 b^4+64 a^6+35 b^6\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{8 d \left(a^2-b^2\right)^{17/2}}+\frac{b \left(41484 a^4 b^2+22767 a^2 b^4+9800 a^6+1024 b^6\right) \sec (c+d x)}{560 d \left(a^2-b^2\right)^7 (a+b \sin (c+d x))}+\frac{11 a b \left(844 a^2 b^2+280 a^4+241 b^4\right) \sec (c+d x)}{560 d \left(a^2-b^2\right)^6 (a+b \sin (c+d x))^2}+\frac{b \left(1317 a^2 b^2+700 a^4+128 b^4\right) \sec (c+d x)}{280 d \left(a^2-b^2\right)^5 (a+b \sin (c+d x))^3}+\frac{13 a b \left(28 a^2+27 b^2\right) \sec (c+d x)}{280 d \left(a^2-b^2\right)^4 (a+b \sin (c+d x))^4}+\frac{b \left(49 a^2+16 b^2\right) \sec (c+d x)}{70 d \left(a^2-b^2\right)^3 (a+b \sin (c+d x))^5}+\frac{5 a b \sec (c+d x)}{14 d \left(a^2-b^2\right)^2 (a+b \sin (c+d x))^6}+\frac{b \sec (c+d x)}{7 d \left(a^2-b^2\right) (a+b \sin (c+d x))^7}-\frac{\sec (c+d x) \left(315 a b \left(336 a^4 b^2+280 a^2 b^4+64 a^6+35 b^6\right)-\left(42472 a^6 b^2+125634 a^4 b^4+54511 a^2 b^6+560 a^8+2048 b^8\right) \sin (c+d x)\right)}{560 d \left(a^2-b^2\right)^8}",1,"(-9*a*b^2*(64*a^6 + 336*a^4*b^2 + 280*a^2*b^4 + 35*b^6)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(8*(a^2 - b^2)^(17/2)*d) + (b*Sec[c + d*x])/(7*(a^2 - b^2)*d*(a + b*Sin[c + d*x])^7) + (5*a*b*Sec[c + d*x])/(14*(a^2 - b^2)^2*d*(a + b*Sin[c + d*x])^6) + (b*(49*a^2 + 16*b^2)*Sec[c + d*x])/(70*(a^2 - b^2)^3*d*(a + b*Sin[c + d*x])^5) + (13*a*b*(28*a^2 + 27*b^2)*Sec[c + d*x])/(280*(a^2 - b^2)^4*d*(a + b*Sin[c + d*x])^4) + (b*(700*a^4 + 1317*a^2*b^2 + 128*b^4)*Sec[c + d*x])/(280*(a^2 - b^2)^5*d*(a + b*Sin[c + d*x])^3) + (11*a*b*(280*a^4 + 844*a^2*b^2 + 241*b^4)*Sec[c + d*x])/(560*(a^2 - b^2)^6*d*(a + b*Sin[c + d*x])^2) + (b*(9800*a^6 + 41484*a^4*b^2 + 22767*a^2*b^4 + 1024*b^6)*Sec[c + d*x])/(560*(a^2 - b^2)^7*d*(a + b*Sin[c + d*x])) - (Sec[c + d*x]*(315*a*b*(64*a^6 + 336*a^4*b^2 + 280*a^2*b^4 + 35*b^6) - (560*a^8 + 42472*a^6*b^2 + 125634*a^4*b^4 + 54511*a^2*b^6 + 2048*b^8)*Sin[c + d*x]))/(560*(a^2 - b^2)^8*d)","A",12,7,21,0.3333,1,"{2694, 2864, 2866, 12, 2660, 618, 204}"
472,1,653,0,2.1369979,"\int \frac{\sec ^4(c+d x)}{(a+b \sin (c+d x))^8} \, dx","Int[Sec[c + d*x]^4/(a + b*Sin[c + d*x])^8,x]","\frac{165 a b^4 \left(112 a^4 b^2+70 a^2 b^4+32 a^6+7 b^6\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{8 d \left(a^2-b^2\right)^{19/2}}+\frac{b \left(28420 a^4 b^2+12907 a^2 b^4+9212 a^6+512 b^6\right) \sec ^3(c+d x)}{112 d \left(a^2-b^2\right)^7 (a+b \sin (c+d x))}+\frac{13 a b \left(336 a^2 b^2+140 a^4+85 b^4\right) \sec ^3(c+d x)}{112 d \left(a^2-b^2\right)^6 (a+b \sin (c+d x))^2}+\frac{b \left(1421 a^2 b^2+882 a^4+128 b^4\right) \sec ^3(c+d x)}{168 d \left(a^2-b^2\right)^5 (a+b \sin (c+d x))^3}+\frac{a b \left(118 a^2+103 b^2\right) \sec ^3(c+d x)}{56 d \left(a^2-b^2\right)^4 (a+b \sin (c+d x))^4}+\frac{b \left(13 a^2+4 b^2\right) \sec ^3(c+d x)}{14 d \left(a^2-b^2\right)^3 (a+b \sin (c+d x))^5}+\frac{17 a b \sec ^3(c+d x)}{42 d \left(a^2-b^2\right)^2 (a+b \sin (c+d x))^6}+\frac{b \sec ^3(c+d x)}{7 d \left(a^2-b^2\right) (a+b \sin (c+d x))^7}-\frac{\sec ^3(c+d x) \left(1155 a b \left(112 a^4 b^2+70 a^2 b^4+32 a^6+7 b^6\right)-\left(52528 a^6 b^2+142902 a^4 b^4+57665 a^2 b^6+112 a^8+2048 b^8\right) \sin (c+d x)\right)}{336 d \left(a^2-b^2\right)^8}+\frac{\sec (c+d x) \left(\left(-6048 a^8 b^2-207332 a^6 b^4-413024 a^4 b^6-135489 a^2 b^8+224 a^{10}-4096 b^{10}\right) \sin (c+d x)+3465 a b^3 \left(112 a^4 b^2+70 a^2 b^4+32 a^6+7 b^6\right)\right)}{336 d \left(a^2-b^2\right)^9}","\frac{165 a b^4 \left(112 a^4 b^2+70 a^2 b^4+32 a^6+7 b^6\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{8 d \left(a^2-b^2\right)^{19/2}}+\frac{b \left(28420 a^4 b^2+12907 a^2 b^4+9212 a^6+512 b^6\right) \sec ^3(c+d x)}{112 d \left(a^2-b^2\right)^7 (a+b \sin (c+d x))}+\frac{13 a b \left(336 a^2 b^2+140 a^4+85 b^4\right) \sec ^3(c+d x)}{112 d \left(a^2-b^2\right)^6 (a+b \sin (c+d x))^2}+\frac{b \left(1421 a^2 b^2+882 a^4+128 b^4\right) \sec ^3(c+d x)}{168 d \left(a^2-b^2\right)^5 (a+b \sin (c+d x))^3}+\frac{a b \left(118 a^2+103 b^2\right) \sec ^3(c+d x)}{56 d \left(a^2-b^2\right)^4 (a+b \sin (c+d x))^4}+\frac{b \left(13 a^2+4 b^2\right) \sec ^3(c+d x)}{14 d \left(a^2-b^2\right)^3 (a+b \sin (c+d x))^5}+\frac{17 a b \sec ^3(c+d x)}{42 d \left(a^2-b^2\right)^2 (a+b \sin (c+d x))^6}+\frac{b \sec ^3(c+d x)}{7 d \left(a^2-b^2\right) (a+b \sin (c+d x))^7}-\frac{\sec ^3(c+d x) \left(1155 a b \left(112 a^4 b^2+70 a^2 b^4+32 a^6+7 b^6\right)-\left(52528 a^6 b^2+142902 a^4 b^4+57665 a^2 b^6+112 a^8+2048 b^8\right) \sin (c+d x)\right)}{336 d \left(a^2-b^2\right)^8}+\frac{\sec (c+d x) \left(\left(-6048 a^8 b^2-207332 a^6 b^4-413024 a^4 b^6-135489 a^2 b^8+224 a^{10}-4096 b^{10}\right) \sin (c+d x)+3465 a b^3 \left(112 a^4 b^2+70 a^2 b^4+32 a^6+7 b^6\right)\right)}{336 d \left(a^2-b^2\right)^9}",1,"(165*a*b^4*(32*a^6 + 112*a^4*b^2 + 70*a^2*b^4 + 7*b^6)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(8*(a^2 - b^2)^(19/2)*d) + (b*Sec[c + d*x]^3)/(7*(a^2 - b^2)*d*(a + b*Sin[c + d*x])^7) + (17*a*b*Sec[c + d*x]^3)/(42*(a^2 - b^2)^2*d*(a + b*Sin[c + d*x])^6) + (b*(13*a^2 + 4*b^2)*Sec[c + d*x]^3)/(14*(a^2 - b^2)^3*d*(a + b*Sin[c + d*x])^5) + (a*b*(118*a^2 + 103*b^2)*Sec[c + d*x]^3)/(56*(a^2 - b^2)^4*d*(a + b*Sin[c + d*x])^4) + (b*(882*a^4 + 1421*a^2*b^2 + 128*b^4)*Sec[c + d*x]^3)/(168*(a^2 - b^2)^5*d*(a + b*Sin[c + d*x])^3) + (13*a*b*(140*a^4 + 336*a^2*b^2 + 85*b^4)*Sec[c + d*x]^3)/(112*(a^2 - b^2)^6*d*(a + b*Sin[c + d*x])^2) + (b*(9212*a^6 + 28420*a^4*b^2 + 12907*a^2*b^4 + 512*b^6)*Sec[c + d*x]^3)/(112*(a^2 - b^2)^7*d*(a + b*Sin[c + d*x])) - (Sec[c + d*x]^3*(1155*a*b*(32*a^6 + 112*a^4*b^2 + 70*a^2*b^4 + 7*b^6) - (112*a^8 + 52528*a^6*b^2 + 142902*a^4*b^4 + 57665*a^2*b^6 + 2048*b^8)*Sin[c + d*x]))/(336*(a^2 - b^2)^8*d) + (Sec[c + d*x]*(3465*a*b^3*(32*a^6 + 112*a^4*b^2 + 70*a^2*b^4 + 7*b^6) + (224*a^10 - 6048*a^8*b^2 - 207332*a^6*b^4 - 413024*a^4*b^6 - 135489*a^2*b^8 - 4096*b^10)*Sin[c + d*x]))/(336*(a^2 - b^2)^9*d)","A",13,7,21,0.3333,1,"{2694, 2864, 2866, 12, 2660, 618, 204}"
473,1,154,0,0.1165895,"\int \cos ^5(c+d x) \sqrt{a+b \sin (c+d x)} \, dx","Int[Cos[c + d*x]^5*Sqrt[a + b*Sin[c + d*x]],x]","\frac{4 \left(3 a^2-b^2\right) (a+b \sin (c+d x))^{7/2}}{7 b^5 d}-\frac{8 a \left(a^2-b^2\right) (a+b \sin (c+d x))^{5/2}}{5 b^5 d}+\frac{2 \left(a^2-b^2\right)^2 (a+b \sin (c+d x))^{3/2}}{3 b^5 d}+\frac{2 (a+b \sin (c+d x))^{11/2}}{11 b^5 d}-\frac{8 a (a+b \sin (c+d x))^{9/2}}{9 b^5 d}","\frac{4 \left(3 a^2-b^2\right) (a+b \sin (c+d x))^{7/2}}{7 b^5 d}-\frac{8 a \left(a^2-b^2\right) (a+b \sin (c+d x))^{5/2}}{5 b^5 d}+\frac{2 \left(a^2-b^2\right)^2 (a+b \sin (c+d x))^{3/2}}{3 b^5 d}+\frac{2 (a+b \sin (c+d x))^{11/2}}{11 b^5 d}-\frac{8 a (a+b \sin (c+d x))^{9/2}}{9 b^5 d}",1,"(2*(a^2 - b^2)^2*(a + b*Sin[c + d*x])^(3/2))/(3*b^5*d) - (8*a*(a^2 - b^2)*(a + b*Sin[c + d*x])^(5/2))/(5*b^5*d) + (4*(3*a^2 - b^2)*(a + b*Sin[c + d*x])^(7/2))/(7*b^5*d) - (8*a*(a + b*Sin[c + d*x])^(9/2))/(9*b^5*d) + (2*(a + b*Sin[c + d*x])^(11/2))/(11*b^5*d)","A",3,2,23,0.08696,1,"{2668, 697}"
474,1,83,0,0.0819508,"\int \cos ^3(c+d x) \sqrt{a+b \sin (c+d x)} \, dx","Int[Cos[c + d*x]^3*Sqrt[a + b*Sin[c + d*x]],x]","-\frac{2 \left(a^2-b^2\right) (a+b \sin (c+d x))^{3/2}}{3 b^3 d}-\frac{2 (a+b \sin (c+d x))^{7/2}}{7 b^3 d}+\frac{4 a (a+b \sin (c+d x))^{5/2}}{5 b^3 d}","-\frac{2 \left(a^2-b^2\right) (a+b \sin (c+d x))^{3/2}}{3 b^3 d}-\frac{2 (a+b \sin (c+d x))^{7/2}}{7 b^3 d}+\frac{4 a (a+b \sin (c+d x))^{5/2}}{5 b^3 d}",1,"(-2*(a^2 - b^2)*(a + b*Sin[c + d*x])^(3/2))/(3*b^3*d) + (4*a*(a + b*Sin[c + d*x])^(5/2))/(5*b^3*d) - (2*(a + b*Sin[c + d*x])^(7/2))/(7*b^3*d)","A",3,2,23,0.08696,1,"{2668, 697}"
475,1,24,0,0.0357776,"\int \cos (c+d x) \sqrt{a+b \sin (c+d x)} \, dx","Int[Cos[c + d*x]*Sqrt[a + b*Sin[c + d*x]],x]","\frac{2 (a+b \sin (c+d x))^{3/2}}{3 b d}","\frac{2 (a+b \sin (c+d x))^{3/2}}{3 b d}",1,"(2*(a + b*Sin[c + d*x])^(3/2))/(3*b*d)","A",2,2,21,0.09524,1,"{2668, 32}"
476,1,74,0,0.1164402,"\int \sec (c+d x) \sqrt{a+b \sin (c+d x)} \, dx","Int[Sec[c + d*x]*Sqrt[a + b*Sin[c + d*x]],x]","\frac{\sqrt{a+b} \tanh ^{-1}\left(\frac{\sqrt{a+b \sin (c+d x)}}{\sqrt{a+b}}\right)}{d}-\frac{\sqrt{a-b} \tanh ^{-1}\left(\frac{\sqrt{a+b \sin (c+d x)}}{\sqrt{a-b}}\right)}{d}","\frac{\sqrt{a+b} \tanh ^{-1}\left(\frac{\sqrt{a+b \sin (c+d x)}}{\sqrt{a+b}}\right)}{d}-\frac{\sqrt{a-b} \tanh ^{-1}\left(\frac{\sqrt{a+b \sin (c+d x)}}{\sqrt{a-b}}\right)}{d}",1,"-((Sqrt[a - b]*ArcTanh[Sqrt[a + b*Sin[c + d*x]]/Sqrt[a - b]])/d) + (Sqrt[a + b]*ArcTanh[Sqrt[a + b*Sin[c + d*x]]/Sqrt[a + b]])/d","A",5,4,21,0.1905,1,"{2668, 700, 1130, 206}"
477,1,124,0,0.1669411,"\int \sec ^3(c+d x) \sqrt{a+b \sin (c+d x)} \, dx","Int[Sec[c + d*x]^3*Sqrt[a + b*Sin[c + d*x]],x]","-\frac{(2 a-b) \tanh ^{-1}\left(\frac{\sqrt{a+b \sin (c+d x)}}{\sqrt{a-b}}\right)}{4 d \sqrt{a-b}}+\frac{(2 a+b) \tanh ^{-1}\left(\frac{\sqrt{a+b \sin (c+d x)}}{\sqrt{a+b}}\right)}{4 d \sqrt{a+b}}+\frac{\tan (c+d x) \sec (c+d x) \sqrt{a+b \sin (c+d x)}}{2 d}","-\frac{(2 a-b) \tanh ^{-1}\left(\frac{\sqrt{a+b \sin (c+d x)}}{\sqrt{a-b}}\right)}{4 d \sqrt{a-b}}+\frac{(2 a+b) \tanh ^{-1}\left(\frac{\sqrt{a+b \sin (c+d x)}}{\sqrt{a+b}}\right)}{4 d \sqrt{a+b}}+\frac{\tan (c+d x) \sec (c+d x) \sqrt{a+b \sin (c+d x)}}{2 d}",1,"-((2*a - b)*ArcTanh[Sqrt[a + b*Sin[c + d*x]]/Sqrt[a - b]])/(4*Sqrt[a - b]*d) + ((2*a + b)*ArcTanh[Sqrt[a + b*Sin[c + d*x]]/Sqrt[a + b]])/(4*Sqrt[a + b]*d) + (Sec[c + d*x]*Sqrt[a + b*Sin[c + d*x]]*Tan[c + d*x])/(2*d)","A",6,5,23,0.2174,1,"{2668, 737, 827, 1166, 206}"
478,1,207,0,0.3233325,"\int \sec ^5(c+d x) \sqrt{a+b \sin (c+d x)} \, dx","Int[Sec[c + d*x]^5*Sqrt[a + b*Sin[c + d*x]],x]","-\frac{\left(12 a^2-18 a b+5 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a+b \sin (c+d x)}}{\sqrt{a-b}}\right)}{32 d (a-b)^{3/2}}+\frac{\left(12 a^2+18 a b+5 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a+b \sin (c+d x)}}{\sqrt{a+b}}\right)}{32 d (a+b)^{3/2}}-\frac{\sec ^2(c+d x) \sqrt{a+b \sin (c+d x)} \left(a b-\left(6 a^2-5 b^2\right) \sin (c+d x)\right)}{16 d \left(a^2-b^2\right)}+\frac{\tan (c+d x) \sec ^3(c+d x) \sqrt{a+b \sin (c+d x)}}{4 d}","-\frac{\left(12 a^2-18 a b+5 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a+b \sin (c+d x)}}{\sqrt{a-b}}\right)}{32 d (a-b)^{3/2}}+\frac{\left(12 a^2+18 a b+5 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a+b \sin (c+d x)}}{\sqrt{a+b}}\right)}{32 d (a+b)^{3/2}}-\frac{\sec ^2(c+d x) \sqrt{a+b \sin (c+d x)} \left(a b-\left(6 a^2-5 b^2\right) \sin (c+d x)\right)}{16 d \left(a^2-b^2\right)}+\frac{\tan (c+d x) \sec ^3(c+d x) \sqrt{a+b \sin (c+d x)}}{4 d}",1,"-((12*a^2 - 18*a*b + 5*b^2)*ArcTanh[Sqrt[a + b*Sin[c + d*x]]/Sqrt[a - b]])/(32*(a - b)^(3/2)*d) + ((12*a^2 + 18*a*b + 5*b^2)*ArcTanh[Sqrt[a + b*Sin[c + d*x]]/Sqrt[a + b]])/(32*(a + b)^(3/2)*d) - (Sec[c + d*x]^2*Sqrt[a + b*Sin[c + d*x]]*(a*b - (6*a^2 - 5*b^2)*Sin[c + d*x]))/(16*(a^2 - b^2)*d) + (Sec[c + d*x]^3*Sqrt[a + b*Sin[c + d*x]]*Tan[c + d*x])/(4*d)","A",7,6,23,0.2609,1,"{2668, 737, 823, 827, 1166, 206}"
479,1,298,0,0.5653005,"\int \cos ^4(c+d x) \sqrt{a+b \sin (c+d x)} \, dx","Int[Cos[c + d*x]^4*Sqrt[a + b*Sin[c + d*x]],x]","-\frac{4 \cos (c+d x) \sqrt{a+b \sin (c+d x)} \left(4 a \left(a^2-3 b^2\right)-3 b \left(a^2+7 b^2\right) \sin (c+d x)\right)}{315 b^3 d}+\frac{32 a \left(-4 a^2 b^2+a^4+3 b^4\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{315 b^4 d \sqrt{a+b \sin (c+d x)}}-\frac{8 \left(-15 a^2 b^2+4 a^4-21 b^4\right) \sqrt{a+b \sin (c+d x)} E\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{315 b^4 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}+\frac{2 \cos ^3(c+d x) (a+b \sin (c+d x))^{3/2}}{9 b d}-\frac{4 a \cos ^3(c+d x) \sqrt{a+b \sin (c+d x)}}{21 b d}","-\frac{4 \cos (c+d x) \sqrt{a+b \sin (c+d x)} \left(4 a \left(a^2-3 b^2\right)-3 b \left(a^2+7 b^2\right) \sin (c+d x)\right)}{315 b^3 d}+\frac{32 a \left(-4 a^2 b^2+a^4+3 b^4\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{315 b^4 d \sqrt{a+b \sin (c+d x)}}-\frac{8 \left(-15 a^2 b^2+4 a^4-21 b^4\right) \sqrt{a+b \sin (c+d x)} E\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{315 b^4 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}+\frac{2 \cos ^3(c+d x) (a+b \sin (c+d x))^{3/2}}{9 b d}-\frac{4 a \cos ^3(c+d x) \sqrt{a+b \sin (c+d x)}}{21 b d}",1,"(-4*a*Cos[c + d*x]^3*Sqrt[a + b*Sin[c + d*x]])/(21*b*d) + (2*Cos[c + d*x]^3*(a + b*Sin[c + d*x])^(3/2))/(9*b*d) - (8*(4*a^4 - 15*a^2*b^2 - 21*b^4)*EllipticE[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[a + b*Sin[c + d*x]])/(315*b^4*d*Sqrt[(a + b*Sin[c + d*x])/(a + b)]) + (32*a*(a^4 - 4*a^2*b^2 + 3*b^4)*EllipticF[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(315*b^4*d*Sqrt[a + b*Sin[c + d*x]]) - (4*Cos[c + d*x]*Sqrt[a + b*Sin[c + d*x]]*(4*a*(a^2 - 3*b^2) - 3*b*(a^2 + 7*b^2)*Sin[c + d*x]))/(315*b^3*d)","A",8,8,23,0.3478,1,"{2695, 2862, 2865, 2752, 2663, 2661, 2655, 2653}"
480,1,215,0,0.2605526,"\int \cos ^2(c+d x) \sqrt{a+b \sin (c+d x)} \, dx","Int[Cos[c + d*x]^2*Sqrt[a + b*Sin[c + d*x]],x]","-\frac{4 a \left(a^2-b^2\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{15 b^2 d \sqrt{a+b \sin (c+d x)}}+\frac{4 \left(a^2+3 b^2\right) \sqrt{a+b \sin (c+d x)} E\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{15 b^2 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}+\frac{2 \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{5 b d}-\frac{4 a \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{15 b d}","-\frac{4 a \left(a^2-b^2\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{15 b^2 d \sqrt{a+b \sin (c+d x)}}+\frac{4 \left(a^2+3 b^2\right) \sqrt{a+b \sin (c+d x)} E\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{15 b^2 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}+\frac{2 \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{5 b d}-\frac{4 a \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{15 b d}",1,"(-4*a*Cos[c + d*x]*Sqrt[a + b*Sin[c + d*x]])/(15*b*d) + (2*Cos[c + d*x]*(a + b*Sin[c + d*x])^(3/2))/(5*b*d) + (4*(a^2 + 3*b^2)*EllipticE[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[a + b*Sin[c + d*x]])/(15*b^2*d*Sqrt[(a + b*Sin[c + d*x])/(a + b)]) - (4*a*(a^2 - b^2)*EllipticF[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(15*b^2*d*Sqrt[a + b*Sin[c + d*x]])","A",7,7,23,0.3043,1,"{2695, 2753, 2752, 2663, 2661, 2655, 2653}"
481,1,149,0,0.1706378,"\int \sec ^2(c+d x) \sqrt{a+b \sin (c+d x)} \, dx","Int[Sec[c + d*x]^2*Sqrt[a + b*Sin[c + d*x]],x]","\frac{\tan (c+d x) \sqrt{a+b \sin (c+d x)}}{d}+\frac{a \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{d \sqrt{a+b \sin (c+d x)}}-\frac{\sqrt{a+b \sin (c+d x)} E\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}","\frac{\tan (c+d x) \sqrt{a+b \sin (c+d x)}}{d}+\frac{a \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{d \sqrt{a+b \sin (c+d x)}}-\frac{\sqrt{a+b \sin (c+d x)} E\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}",1,"-((EllipticE[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[a + b*Sin[c + d*x]])/(d*Sqrt[(a + b*Sin[c + d*x])/(a + b)])) + (a*EllipticF[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(d*Sqrt[a + b*Sin[c + d*x]]) + (Sqrt[a + b*Sin[c + d*x]]*Tan[c + d*x])/d","A",7,7,23,0.3043,1,"{2690, 12, 2752, 2663, 2661, 2655, 2653}"
482,1,248,0,0.3714298,"\int \sec ^4(c+d x) \sqrt{a+b \sin (c+d x)} \, dx","Int[Sec[c + d*x]^4*Sqrt[a + b*Sin[c + d*x]],x]","-\frac{\sec (c+d x) \sqrt{a+b \sin (c+d x)} \left(a b-\left(4 a^2-3 b^2\right) \sin (c+d x)\right)}{6 d \left(a^2-b^2\right)}-\frac{\left(4 a^2-3 b^2\right) \sqrt{a+b \sin (c+d x)} E\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{6 d \left(a^2-b^2\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}+\frac{2 a \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{3 d \sqrt{a+b \sin (c+d x)}}+\frac{\tan (c+d x) \sec ^2(c+d x) \sqrt{a+b \sin (c+d x)}}{3 d}","-\frac{\sec (c+d x) \sqrt{a+b \sin (c+d x)} \left(a b-\left(4 a^2-3 b^2\right) \sin (c+d x)\right)}{6 d \left(a^2-b^2\right)}-\frac{\left(4 a^2-3 b^2\right) \sqrt{a+b \sin (c+d x)} E\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{6 d \left(a^2-b^2\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}+\frac{2 a \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{3 d \sqrt{a+b \sin (c+d x)}}+\frac{\tan (c+d x) \sec ^2(c+d x) \sqrt{a+b \sin (c+d x)}}{3 d}",1,"-((4*a^2 - 3*b^2)*EllipticE[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[a + b*Sin[c + d*x]])/(6*(a^2 - b^2)*d*Sqrt[(a + b*Sin[c + d*x])/(a + b)]) + (2*a*EllipticF[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(3*d*Sqrt[a + b*Sin[c + d*x]]) - (Sec[c + d*x]*Sqrt[a + b*Sin[c + d*x]]*(a*b - (4*a^2 - 3*b^2)*Sin[c + d*x]))/(6*(a^2 - b^2)*d) + (Sec[c + d*x]^2*Sqrt[a + b*Sin[c + d*x]]*Tan[c + d*x])/(3*d)","A",7,7,23,0.3043,1,"{2690, 2866, 2752, 2663, 2661, 2655, 2653}"
483,1,154,0,0.1225706,"\int \cos ^5(c+d x) (a+b \sin (c+d x))^{3/2} \, dx","Int[Cos[c + d*x]^5*(a + b*Sin[c + d*x])^(3/2),x]","\frac{4 \left(3 a^2-b^2\right) (a+b \sin (c+d x))^{9/2}}{9 b^5 d}-\frac{8 a \left(a^2-b^2\right) (a+b \sin (c+d x))^{7/2}}{7 b^5 d}+\frac{2 \left(a^2-b^2\right)^2 (a+b \sin (c+d x))^{5/2}}{5 b^5 d}+\frac{2 (a+b \sin (c+d x))^{13/2}}{13 b^5 d}-\frac{8 a (a+b \sin (c+d x))^{11/2}}{11 b^5 d}","\frac{4 \left(3 a^2-b^2\right) (a+b \sin (c+d x))^{9/2}}{9 b^5 d}-\frac{8 a \left(a^2-b^2\right) (a+b \sin (c+d x))^{7/2}}{7 b^5 d}+\frac{2 \left(a^2-b^2\right)^2 (a+b \sin (c+d x))^{5/2}}{5 b^5 d}+\frac{2 (a+b \sin (c+d x))^{13/2}}{13 b^5 d}-\frac{8 a (a+b \sin (c+d x))^{11/2}}{11 b^5 d}",1,"(2*(a^2 - b^2)^2*(a + b*Sin[c + d*x])^(5/2))/(5*b^5*d) - (8*a*(a^2 - b^2)*(a + b*Sin[c + d*x])^(7/2))/(7*b^5*d) + (4*(3*a^2 - b^2)*(a + b*Sin[c + d*x])^(9/2))/(9*b^5*d) - (8*a*(a + b*Sin[c + d*x])^(11/2))/(11*b^5*d) + (2*(a + b*Sin[c + d*x])^(13/2))/(13*b^5*d)","A",3,2,23,0.08696,1,"{2668, 697}"
484,1,83,0,0.0913597,"\int \cos ^3(c+d x) (a+b \sin (c+d x))^{3/2} \, dx","Int[Cos[c + d*x]^3*(a + b*Sin[c + d*x])^(3/2),x]","-\frac{2 \left(a^2-b^2\right) (a+b \sin (c+d x))^{5/2}}{5 b^3 d}-\frac{2 (a+b \sin (c+d x))^{9/2}}{9 b^3 d}+\frac{4 a (a+b \sin (c+d x))^{7/2}}{7 b^3 d}","-\frac{2 \left(a^2-b^2\right) (a+b \sin (c+d x))^{5/2}}{5 b^3 d}-\frac{2 (a+b \sin (c+d x))^{9/2}}{9 b^3 d}+\frac{4 a (a+b \sin (c+d x))^{7/2}}{7 b^3 d}",1,"(-2*(a^2 - b^2)*(a + b*Sin[c + d*x])^(5/2))/(5*b^3*d) + (4*a*(a + b*Sin[c + d*x])^(7/2))/(7*b^3*d) - (2*(a + b*Sin[c + d*x])^(9/2))/(9*b^3*d)","A",3,2,23,0.08696,1,"{2668, 697}"
485,1,24,0,0.0387544,"\int \cos (c+d x) (a+b \sin (c+d x))^{3/2} \, dx","Int[Cos[c + d*x]*(a + b*Sin[c + d*x])^(3/2),x]","\frac{2 (a+b \sin (c+d x))^{5/2}}{5 b d}","\frac{2 (a+b \sin (c+d x))^{5/2}}{5 b d}",1,"(2*(a + b*Sin[c + d*x])^(5/2))/(5*b*d)","A",2,2,21,0.09524,1,"{2668, 32}"
486,1,94,0,0.1674403,"\int \sec (c+d x) (a+b \sin (c+d x))^{3/2} \, dx","Int[Sec[c + d*x]*(a + b*Sin[c + d*x])^(3/2),x]","-\frac{2 b \sqrt{a+b \sin (c+d x)}}{d}-\frac{(a-b)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a+b \sin (c+d x)}}{\sqrt{a-b}}\right)}{d}+\frac{(a+b)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a+b \sin (c+d x)}}{\sqrt{a+b}}\right)}{d}","-\frac{2 b \sqrt{a+b \sin (c+d x)}}{d}-\frac{(a-b)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a+b \sin (c+d x)}}{\sqrt{a-b}}\right)}{d}+\frac{(a+b)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a+b \sin (c+d x)}}{\sqrt{a+b}}\right)}{d}",1,"-(((a - b)^(3/2)*ArcTanh[Sqrt[a + b*Sin[c + d*x]]/Sqrt[a - b]])/d) + ((a + b)^(3/2)*ArcTanh[Sqrt[a + b*Sin[c + d*x]]/Sqrt[a + b]])/d - (2*b*Sqrt[a + b*Sin[c + d*x]])/d","A",6,5,21,0.2381,1,"{2668, 704, 827, 1166, 206}"
487,1,130,0,0.2742217,"\int \sec ^3(c+d x) (a+b \sin (c+d x))^{3/2} \, dx","Int[Sec[c + d*x]^3*(a + b*Sin[c + d*x])^(3/2),x]","-\frac{\sqrt{a-b} (2 a+b) \tanh ^{-1}\left(\frac{\sqrt{a+b \sin (c+d x)}}{\sqrt{a-b}}\right)}{4 d}+\frac{(2 a-b) \sqrt{a+b} \tanh ^{-1}\left(\frac{\sqrt{a+b \sin (c+d x)}}{\sqrt{a+b}}\right)}{4 d}+\frac{\sec ^2(c+d x) (a \sin (c+d x)+b) \sqrt{a+b \sin (c+d x)}}{2 d}","-\frac{\sqrt{a-b} (2 a+b) \tanh ^{-1}\left(\frac{\sqrt{a+b \sin (c+d x)}}{\sqrt{a-b}}\right)}{4 d}+\frac{(2 a-b) \sqrt{a+b} \tanh ^{-1}\left(\frac{\sqrt{a+b \sin (c+d x)}}{\sqrt{a+b}}\right)}{4 d}+\frac{\sec ^2(c+d x) (a \sin (c+d x)+b) \sqrt{a+b \sin (c+d x)}}{2 d}",1,"-(Sqrt[a - b]*(2*a + b)*ArcTanh[Sqrt[a + b*Sin[c + d*x]]/Sqrt[a - b]])/(4*d) + ((2*a - b)*Sqrt[a + b]*ArcTanh[Sqrt[a + b*Sin[c + d*x]]/Sqrt[a + b]])/(4*d) + (Sec[c + d*x]^2*(b + a*Sin[c + d*x])*Sqrt[a + b*Sin[c + d*x]])/(2*d)","A",6,5,23,0.2174,1,"{2668, 739, 827, 1166, 206}"
488,1,188,0,0.3184608,"\int \sec ^5(c+d x) (a+b \sin (c+d x))^{3/2} \, dx","Int[Sec[c + d*x]^5*(a + b*Sin[c + d*x])^(3/2),x]","-\frac{3 \left(4 a^2-2 a b-b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a+b \sin (c+d x)}}{\sqrt{a-b}}\right)}{32 d \sqrt{a-b}}+\frac{3 \left(4 a^2+2 a b-b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a+b \sin (c+d x)}}{\sqrt{a+b}}\right)}{32 d \sqrt{a+b}}+\frac{\sec ^4(c+d x) (a \sin (c+d x)+b) \sqrt{a+b \sin (c+d x)}}{4 d}-\frac{\sec ^2(c+d x) (b-6 a \sin (c+d x)) \sqrt{a+b \sin (c+d x)}}{16 d}","-\frac{3 \left(4 a^2-2 a b-b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a+b \sin (c+d x)}}{\sqrt{a-b}}\right)}{32 d \sqrt{a-b}}+\frac{3 \left(4 a^2+2 a b-b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a+b \sin (c+d x)}}{\sqrt{a+b}}\right)}{32 d \sqrt{a+b}}+\frac{\sec ^4(c+d x) (a \sin (c+d x)+b) \sqrt{a+b \sin (c+d x)}}{4 d}-\frac{\sec ^2(c+d x) (b-6 a \sin (c+d x)) \sqrt{a+b \sin (c+d x)}}{16 d}",1,"(-3*(4*a^2 - 2*a*b - b^2)*ArcTanh[Sqrt[a + b*Sin[c + d*x]]/Sqrt[a - b]])/(32*Sqrt[a - b]*d) + (3*(4*a^2 + 2*a*b - b^2)*ArcTanh[Sqrt[a + b*Sin[c + d*x]]/Sqrt[a + b]])/(32*Sqrt[a + b]*d) - (Sec[c + d*x]^2*(b - 6*a*Sin[c + d*x])*Sqrt[a + b*Sin[c + d*x]])/(16*d) + (Sec[c + d*x]^4*(b + a*Sin[c + d*x])*Sqrt[a + b*Sin[c + d*x]])/(4*d)","A",7,6,23,0.2609,1,"{2668, 739, 823, 827, 1166, 206}"
489,1,329,0,0.6916357,"\int \cos ^4(c+d x) (a+b \sin (c+d x))^{3/2} \, dx","Int[Cos[c + d*x]^4*(a + b*Sin[c + d*x])^(3/2),x]","\frac{2 \cos ^3(c+d x) \sqrt{a+b \sin (c+d x)} \left(a^2+28 a b \sin (c+d x)+3 b^2\right)}{231 b d}-\frac{4 \cos (c+d x) \sqrt{a+b \sin (c+d x)} \left(-3 a b \left(a^2+31 b^2\right) \sin (c+d x)-21 a^2 b^2+4 a^4-15 b^4\right)}{1155 b^3 d}+\frac{8 \left(-25 a^4 b^2+6 a^2 b^4+4 a^6+15 b^6\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{1155 b^4 d \sqrt{a+b \sin (c+d x)}}-\frac{32 a \left(-6 a^2 b^2+a^4-27 b^4\right) \sqrt{a+b \sin (c+d x)} E\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{1155 b^4 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}-\frac{2 b \cos ^5(c+d x) \sqrt{a+b \sin (c+d x)}}{11 d}","\frac{2 \cos ^3(c+d x) \sqrt{a+b \sin (c+d x)} \left(a^2+28 a b \sin (c+d x)+3 b^2\right)}{231 b d}-\frac{4 \cos (c+d x) \sqrt{a+b \sin (c+d x)} \left(-3 a b \left(a^2+31 b^2\right) \sin (c+d x)-21 a^2 b^2+4 a^4-15 b^4\right)}{1155 b^3 d}+\frac{8 \left(-25 a^4 b^2+6 a^2 b^4+4 a^6+15 b^6\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{1155 b^4 d \sqrt{a+b \sin (c+d x)}}-\frac{32 a \left(-6 a^2 b^2+a^4-27 b^4\right) \sqrt{a+b \sin (c+d x)} E\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{1155 b^4 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}-\frac{2 b \cos ^5(c+d x) \sqrt{a+b \sin (c+d x)}}{11 d}",1,"(-2*b*Cos[c + d*x]^5*Sqrt[a + b*Sin[c + d*x]])/(11*d) - (32*a*(a^4 - 6*a^2*b^2 - 27*b^4)*EllipticE[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[a + b*Sin[c + d*x]])/(1155*b^4*d*Sqrt[(a + b*Sin[c + d*x])/(a + b)]) + (8*(4*a^6 - 25*a^4*b^2 + 6*a^2*b^4 + 15*b^6)*EllipticF[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(1155*b^4*d*Sqrt[a + b*Sin[c + d*x]]) + (2*Cos[c + d*x]^3*Sqrt[a + b*Sin[c + d*x]]*(a^2 + 3*b^2 + 28*a*b*Sin[c + d*x]))/(231*b*d) - (4*Cos[c + d*x]*Sqrt[a + b*Sin[c + d*x]]*(4*a^4 - 21*a^2*b^2 - 15*b^4 - 3*a*b*(a^2 + 31*b^2)*Sin[c + d*x]))/(1155*b^3*d)","A",8,7,23,0.3043,1,"{2692, 2865, 2752, 2663, 2661, 2655, 2653}"
490,1,247,0,0.4621355,"\int \cos ^2(c+d x) (a+b \sin (c+d x))^{3/2} \, dx","Int[Cos[c + d*x]^2*(a + b*Sin[c + d*x])^(3/2),x]","\frac{2 \cos (c+d x) \sqrt{a+b \sin (c+d x)} \left(3 a^2+24 a b \sin (c+d x)+5 b^2\right)}{105 b d}-\frac{4 \left(2 a^2 b^2+3 a^4-5 b^4\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{105 b^2 d \sqrt{a+b \sin (c+d x)}}+\frac{4 a \left(3 a^2+29 b^2\right) \sqrt{a+b \sin (c+d x)} E\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{105 b^2 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}-\frac{2 b \cos ^3(c+d x) \sqrt{a+b \sin (c+d x)}}{7 d}","\frac{2 \cos (c+d x) \sqrt{a+b \sin (c+d x)} \left(3 a^2+24 a b \sin (c+d x)+5 b^2\right)}{105 b d}-\frac{4 \left(2 a^2 b^2+3 a^4-5 b^4\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{105 b^2 d \sqrt{a+b \sin (c+d x)}}+\frac{4 a \left(3 a^2+29 b^2\right) \sqrt{a+b \sin (c+d x)} E\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{105 b^2 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}-\frac{2 b \cos ^3(c+d x) \sqrt{a+b \sin (c+d x)}}{7 d}",1,"(-2*b*Cos[c + d*x]^3*Sqrt[a + b*Sin[c + d*x]])/(7*d) + (4*a*(3*a^2 + 29*b^2)*EllipticE[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[a + b*Sin[c + d*x]])/(105*b^2*d*Sqrt[(a + b*Sin[c + d*x])/(a + b)]) - (4*(3*a^4 + 2*a^2*b^2 - 5*b^4)*EllipticF[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(105*b^2*d*Sqrt[a + b*Sin[c + d*x]]) + (2*Cos[c + d*x]*Sqrt[a + b*Sin[c + d*x]]*(3*a^2 + 5*b^2 + 24*a*b*Sin[c + d*x]))/(105*b*d)","A",7,7,23,0.3043,1,"{2692, 2865, 2752, 2663, 2661, 2655, 2653}"
491,1,168,0,0.2031093,"\int \sec ^2(c+d x) (a+b \sin (c+d x))^{3/2} \, dx","Int[Sec[c + d*x]^2*(a + b*Sin[c + d*x])^(3/2),x]","\frac{\left(a^2-b^2\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{d \sqrt{a+b \sin (c+d x)}}+\frac{\sec (c+d x) (a \sin (c+d x)+b) \sqrt{a+b \sin (c+d x)}}{d}-\frac{a \sqrt{a+b \sin (c+d x)} E\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}","\frac{\left(a^2-b^2\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{d \sqrt{a+b \sin (c+d x)}}+\frac{\sec (c+d x) (a \sin (c+d x)+b) \sqrt{a+b \sin (c+d x)}}{d}-\frac{a \sqrt{a+b \sin (c+d x)} E\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}",1,"(Sec[c + d*x]*(b + a*Sin[c + d*x])*Sqrt[a + b*Sin[c + d*x]])/d - (a*EllipticE[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[a + b*Sin[c + d*x]])/(d*Sqrt[(a + b*Sin[c + d*x])/(a + b)]) + ((a^2 - b^2)*EllipticF[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(d*Sqrt[a + b*Sin[c + d*x]])","A",6,6,23,0.2609,1,"{2691, 2752, 2663, 2661, 2655, 2653}"
492,1,218,0,0.4380239,"\int \sec ^4(c+d x) (a+b \sin (c+d x))^{3/2} \, dx","Int[Sec[c + d*x]^4*(a + b*Sin[c + d*x])^(3/2),x]","\frac{\left(4 a^2-b^2\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{6 d \sqrt{a+b \sin (c+d x)}}+\frac{\sec ^3(c+d x) (a \sin (c+d x)+b) \sqrt{a+b \sin (c+d x)}}{3 d}-\frac{\sec (c+d x) (b-4 a \sin (c+d x)) \sqrt{a+b \sin (c+d x)}}{6 d}-\frac{2 a \sqrt{a+b \sin (c+d x)} E\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{3 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}","\frac{\left(4 a^2-b^2\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{6 d \sqrt{a+b \sin (c+d x)}}+\frac{\sec ^3(c+d x) (a \sin (c+d x)+b) \sqrt{a+b \sin (c+d x)}}{3 d}-\frac{\sec (c+d x) (b-4 a \sin (c+d x)) \sqrt{a+b \sin (c+d x)}}{6 d}-\frac{2 a \sqrt{a+b \sin (c+d x)} E\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{3 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}",1,"-(Sec[c + d*x]*(b - 4*a*Sin[c + d*x])*Sqrt[a + b*Sin[c + d*x]])/(6*d) + (Sec[c + d*x]^3*(b + a*Sin[c + d*x])*Sqrt[a + b*Sin[c + d*x]])/(3*d) - (2*a*EllipticE[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[a + b*Sin[c + d*x]])/(3*d*Sqrt[(a + b*Sin[c + d*x])/(a + b)]) + ((4*a^2 - b^2)*EllipticF[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(6*d*Sqrt[a + b*Sin[c + d*x]])","A",7,7,23,0.3043,1,"{2691, 2866, 2752, 2663, 2661, 2655, 2653}"
493,1,330,0,0.6975387,"\int \sec ^6(c+d x) (a+b \sin (c+d x))^{3/2} \, dx","Int[Sec[c + d*x]^6*(a + b*Sin[c + d*x])^(3/2),x]","-\frac{\sec (c+d x) \sqrt{a+b \sin (c+d x)} \left(b \left(-13 a^2 b^2+8 a^4+5 b^4\right)-a \left(-61 a^2 b^2+32 a^4+29 b^4\right) \sin (c+d x)\right)}{60 d \left(a^2-b^2\right)^2}+\frac{\left(32 a^2-5 b^2\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{60 d \sqrt{a+b \sin (c+d x)}}-\frac{a \left(32 a^2-29 b^2\right) \sqrt{a+b \sin (c+d x)} E\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{60 d \left(a^2-b^2\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}+\frac{\sec ^5(c+d x) (a \sin (c+d x)+b) \sqrt{a+b \sin (c+d x)}}{5 d}-\frac{\sec ^3(c+d x) (b-8 a \sin (c+d x)) \sqrt{a+b \sin (c+d x)}}{30 d}","-\frac{\sec (c+d x) \sqrt{a+b \sin (c+d x)} \left(b \left(-13 a^2 b^2+8 a^4+5 b^4\right)-a \left(-61 a^2 b^2+32 a^4+29 b^4\right) \sin (c+d x)\right)}{60 d \left(a^2-b^2\right)^2}+\frac{\left(32 a^2-5 b^2\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{60 d \sqrt{a+b \sin (c+d x)}}-\frac{a \left(32 a^2-29 b^2\right) \sqrt{a+b \sin (c+d x)} E\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{60 d \left(a^2-b^2\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}+\frac{\sec ^5(c+d x) (a \sin (c+d x)+b) \sqrt{a+b \sin (c+d x)}}{5 d}-\frac{\sec ^3(c+d x) (b-8 a \sin (c+d x)) \sqrt{a+b \sin (c+d x)}}{30 d}",1,"-(Sec[c + d*x]^3*(b - 8*a*Sin[c + d*x])*Sqrt[a + b*Sin[c + d*x]])/(30*d) + (Sec[c + d*x]^5*(b + a*Sin[c + d*x])*Sqrt[a + b*Sin[c + d*x]])/(5*d) - (a*(32*a^2 - 29*b^2)*EllipticE[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[a + b*Sin[c + d*x]])/(60*(a^2 - b^2)*d*Sqrt[(a + b*Sin[c + d*x])/(a + b)]) + ((32*a^2 - 5*b^2)*EllipticF[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(60*d*Sqrt[a + b*Sin[c + d*x]]) - (Sec[c + d*x]*Sqrt[a + b*Sin[c + d*x]]*(b*(8*a^4 - 13*a^2*b^2 + 5*b^4) - a*(32*a^4 - 61*a^2*b^2 + 29*b^4)*Sin[c + d*x]))/(60*(a^2 - b^2)^2*d)","A",8,7,23,0.3043,1,"{2691, 2866, 2752, 2663, 2661, 2655, 2653}"
494,1,154,0,0.1199445,"\int \cos ^5(c+d x) (a+b \sin (c+d x))^{5/2} \, dx","Int[Cos[c + d*x]^5*(a + b*Sin[c + d*x])^(5/2),x]","\frac{4 \left(3 a^2-b^2\right) (a+b \sin (c+d x))^{11/2}}{11 b^5 d}-\frac{8 a \left(a^2-b^2\right) (a+b \sin (c+d x))^{9/2}}{9 b^5 d}+\frac{2 \left(a^2-b^2\right)^2 (a+b \sin (c+d x))^{7/2}}{7 b^5 d}+\frac{2 (a+b \sin (c+d x))^{15/2}}{15 b^5 d}-\frac{8 a (a+b \sin (c+d x))^{13/2}}{13 b^5 d}","\frac{4 \left(3 a^2-b^2\right) (a+b \sin (c+d x))^{11/2}}{11 b^5 d}-\frac{8 a \left(a^2-b^2\right) (a+b \sin (c+d x))^{9/2}}{9 b^5 d}+\frac{2 \left(a^2-b^2\right)^2 (a+b \sin (c+d x))^{7/2}}{7 b^5 d}+\frac{2 (a+b \sin (c+d x))^{15/2}}{15 b^5 d}-\frac{8 a (a+b \sin (c+d x))^{13/2}}{13 b^5 d}",1,"(2*(a^2 - b^2)^2*(a + b*Sin[c + d*x])^(7/2))/(7*b^5*d) - (8*a*(a^2 - b^2)*(a + b*Sin[c + d*x])^(9/2))/(9*b^5*d) + (4*(3*a^2 - b^2)*(a + b*Sin[c + d*x])^(11/2))/(11*b^5*d) - (8*a*(a + b*Sin[c + d*x])^(13/2))/(13*b^5*d) + (2*(a + b*Sin[c + d*x])^(15/2))/(15*b^5*d)","A",3,2,23,0.08696,1,"{2668, 697}"
495,1,83,0,0.0910357,"\int \cos ^3(c+d x) (a+b \sin (c+d x))^{5/2} \, dx","Int[Cos[c + d*x]^3*(a + b*Sin[c + d*x])^(5/2),x]","-\frac{2 \left(a^2-b^2\right) (a+b \sin (c+d x))^{7/2}}{7 b^3 d}-\frac{2 (a+b \sin (c+d x))^{11/2}}{11 b^3 d}+\frac{4 a (a+b \sin (c+d x))^{9/2}}{9 b^3 d}","-\frac{2 \left(a^2-b^2\right) (a+b \sin (c+d x))^{7/2}}{7 b^3 d}-\frac{2 (a+b \sin (c+d x))^{11/2}}{11 b^3 d}+\frac{4 a (a+b \sin (c+d x))^{9/2}}{9 b^3 d}",1,"(-2*(a^2 - b^2)*(a + b*Sin[c + d*x])^(7/2))/(7*b^3*d) + (4*a*(a + b*Sin[c + d*x])^(9/2))/(9*b^3*d) - (2*(a + b*Sin[c + d*x])^(11/2))/(11*b^3*d)","A",3,2,23,0.08696,1,"{2668, 697}"
496,1,24,0,0.0374452,"\int \cos (c+d x) (a+b \sin (c+d x))^{5/2} \, dx","Int[Cos[c + d*x]*(a + b*Sin[c + d*x])^(5/2),x]","\frac{2 (a+b \sin (c+d x))^{7/2}}{7 b d}","\frac{2 (a+b \sin (c+d x))^{7/2}}{7 b d}",1,"(2*(a + b*Sin[c + d*x])^(7/2))/(7*b*d)","A",2,2,21,0.09524,1,"{2668, 32}"
497,1,117,0,0.2322516,"\int \sec (c+d x) (a+b \sin (c+d x))^{5/2} \, dx","Int[Sec[c + d*x]*(a + b*Sin[c + d*x])^(5/2),x]","-\frac{2 b (a+b \sin (c+d x))^{3/2}}{3 d}-\frac{4 a b \sqrt{a+b \sin (c+d x)}}{d}-\frac{(a-b)^{5/2} \tanh ^{-1}\left(\frac{\sqrt{a+b \sin (c+d x)}}{\sqrt{a-b}}\right)}{d}+\frac{(a+b)^{5/2} \tanh ^{-1}\left(\frac{\sqrt{a+b \sin (c+d x)}}{\sqrt{a+b}}\right)}{d}","-\frac{2 b (a+b \sin (c+d x))^{3/2}}{3 d}-\frac{4 a b \sqrt{a+b \sin (c+d x)}}{d}-\frac{(a-b)^{5/2} \tanh ^{-1}\left(\frac{\sqrt{a+b \sin (c+d x)}}{\sqrt{a-b}}\right)}{d}+\frac{(a+b)^{5/2} \tanh ^{-1}\left(\frac{\sqrt{a+b \sin (c+d x)}}{\sqrt{a+b}}\right)}{d}",1,"-(((a - b)^(5/2)*ArcTanh[Sqrt[a + b*Sin[c + d*x]]/Sqrt[a - b]])/d) + ((a + b)^(5/2)*ArcTanh[Sqrt[a + b*Sin[c + d*x]]/Sqrt[a + b]])/d - (4*a*b*Sqrt[a + b*Sin[c + d*x]])/d - (2*b*(a + b*Sin[c + d*x])^(3/2))/(3*d)","A",7,6,21,0.2857,1,"{2668, 704, 825, 827, 1166, 206}"
498,1,155,0,0.2697159,"\int \sec ^3(c+d x) (a+b \sin (c+d x))^{5/2} \, dx","Int[Sec[c + d*x]^3*(a + b*Sin[c + d*x])^(5/2),x]","\frac{a b \sqrt{a+b \sin (c+d x)}}{2 d}-\frac{(a-b)^{3/2} (2 a+3 b) \tanh ^{-1}\left(\frac{\sqrt{a+b \sin (c+d x)}}{\sqrt{a-b}}\right)}{4 d}+\frac{(2 a-3 b) (a+b)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a+b \sin (c+d x)}}{\sqrt{a+b}}\right)}{4 d}+\frac{\sec ^2(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^{3/2}}{2 d}","\frac{a b \sqrt{a+b \sin (c+d x)}}{2 d}-\frac{(a-b)^{3/2} (2 a+3 b) \tanh ^{-1}\left(\frac{\sqrt{a+b \sin (c+d x)}}{\sqrt{a-b}}\right)}{4 d}+\frac{(2 a-3 b) (a+b)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a+b \sin (c+d x)}}{\sqrt{a+b}}\right)}{4 d}+\frac{\sec ^2(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^{3/2}}{2 d}",1,"-((a - b)^(3/2)*(2*a + 3*b)*ArcTanh[Sqrt[a + b*Sin[c + d*x]]/Sqrt[a - b]])/(4*d) + ((2*a - 3*b)*(a + b)^(3/2)*ArcTanh[Sqrt[a + b*Sin[c + d*x]]/Sqrt[a + b]])/(4*d) + (a*b*Sqrt[a + b*Sin[c + d*x]])/(2*d) + (Sec[c + d*x]^2*(b + a*Sin[c + d*x])*(a + b*Sin[c + d*x])^(3/2))/(2*d)","A",7,6,23,0.2609,1,"{2668, 739, 825, 827, 1166, 206}"
499,1,199,0,0.282019,"\int \sec ^5(c+d x) (a+b \sin (c+d x))^{5/2} \, dx","Int[Sec[c + d*x]^5*(a + b*Sin[c + d*x])^(5/2),x]","-\frac{3 \sqrt{a-b} \left(4 a^2+2 a b-b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a+b \sin (c+d x)}}{\sqrt{a-b}}\right)}{32 d}+\frac{3 \sqrt{a+b} \left(4 a^2-2 a b-b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a+b \sin (c+d x)}}{\sqrt{a+b}}\right)}{32 d}+\frac{3 \sec ^2(c+d x) \sqrt{a+b \sin (c+d x)} \left(\left(2 a^2-b^2\right) \sin (c+d x)+a b\right)}{16 d}+\frac{\sec ^4(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^{3/2}}{4 d}","-\frac{3 \sqrt{a-b} \left(4 a^2+2 a b-b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a+b \sin (c+d x)}}{\sqrt{a-b}}\right)}{32 d}+\frac{3 \sqrt{a+b} \left(4 a^2-2 a b-b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a+b \sin (c+d x)}}{\sqrt{a+b}}\right)}{32 d}+\frac{3 \sec ^2(c+d x) \sqrt{a+b \sin (c+d x)} \left(\left(2 a^2-b^2\right) \sin (c+d x)+a b\right)}{16 d}+\frac{\sec ^4(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^{3/2}}{4 d}",1,"(-3*Sqrt[a - b]*(4*a^2 + 2*a*b - b^2)*ArcTanh[Sqrt[a + b*Sin[c + d*x]]/Sqrt[a - b]])/(32*d) + (3*Sqrt[a + b]*(4*a^2 - 2*a*b - b^2)*ArcTanh[Sqrt[a + b*Sin[c + d*x]]/Sqrt[a + b]])/(32*d) + (Sec[c + d*x]^4*(b + a*Sin[c + d*x])*(a + b*Sin[c + d*x])^(3/2))/(4*d) + (3*Sec[c + d*x]^2*Sqrt[a + b*Sin[c + d*x]]*(a*b + (2*a^2 - b^2)*Sin[c + d*x]))/(16*d)","A",7,6,23,0.2609,1,"{2668, 739, 821, 827, 1166, 206}"
500,1,398,0,0.9365004,"\int \cos ^4(c+d x) (a+b \sin (c+d x))^{5/2} \, dx","Int[Cos[c + d*x]^4*(a + b*Sin[c + d*x])^(5/2),x]","\frac{2 \cos ^3(c+d x) \sqrt{a+b \sin (c+d x)} \left(7 b \left(53 a^2+11 b^2\right) \sin (c+d x)+a \left(5 a^2+59 b^2\right)\right)}{3003 b d}-\frac{4 \cos (c+d x) \sqrt{a+b \sin (c+d x)} \left(4 a \left(-40 a^2 b^2+5 a^4-93 b^4\right)-3 b \left(430 a^2 b^2+5 a^4+77 b^4\right) \sin (c+d x)\right)}{15015 b^3 d}+\frac{32 a \left(-45 a^4 b^2-53 a^2 b^4+5 a^6+93 b^6\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{15015 b^4 d \sqrt{a+b \sin (c+d x)}}-\frac{8 \left(-175 a^4 b^2-1662 a^2 b^4+20 a^6-231 b^6\right) \sqrt{a+b \sin (c+d x)} E\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{15015 b^4 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}-\frac{2 b \cos ^5(c+d x) (a+b \sin (c+d x))^{3/2}}{13 d}-\frac{32 a b \cos ^5(c+d x) \sqrt{a+b \sin (c+d x)}}{143 d}","\frac{2 \cos ^3(c+d x) \sqrt{a+b \sin (c+d x)} \left(7 b \left(53 a^2+11 b^2\right) \sin (c+d x)+a \left(5 a^2+59 b^2\right)\right)}{3003 b d}-\frac{4 \cos (c+d x) \sqrt{a+b \sin (c+d x)} \left(4 a \left(-40 a^2 b^2+5 a^4-93 b^4\right)-3 b \left(430 a^2 b^2+5 a^4+77 b^4\right) \sin (c+d x)\right)}{15015 b^3 d}+\frac{32 a \left(-45 a^4 b^2-53 a^2 b^4+5 a^6+93 b^6\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{15015 b^4 d \sqrt{a+b \sin (c+d x)}}-\frac{8 \left(-175 a^4 b^2-1662 a^2 b^4+20 a^6-231 b^6\right) \sqrt{a+b \sin (c+d x)} E\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{15015 b^4 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}-\frac{2 b \cos ^5(c+d x) (a+b \sin (c+d x))^{3/2}}{13 d}-\frac{32 a b \cos ^5(c+d x) \sqrt{a+b \sin (c+d x)}}{143 d}",1,"(-32*a*b*Cos[c + d*x]^5*Sqrt[a + b*Sin[c + d*x]])/(143*d) - (2*b*Cos[c + d*x]^5*(a + b*Sin[c + d*x])^(3/2))/(13*d) - (8*(20*a^6 - 175*a^4*b^2 - 1662*a^2*b^4 - 231*b^6)*EllipticE[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[a + b*Sin[c + d*x]])/(15015*b^4*d*Sqrt[(a + b*Sin[c + d*x])/(a + b)]) + (32*a*(5*a^6 - 45*a^4*b^2 - 53*a^2*b^4 + 93*b^6)*EllipticF[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(15015*b^4*d*Sqrt[a + b*Sin[c + d*x]]) + (2*Cos[c + d*x]^3*Sqrt[a + b*Sin[c + d*x]]*(a*(5*a^2 + 59*b^2) + 7*b*(53*a^2 + 11*b^2)*Sin[c + d*x]))/(3003*b*d) - (4*Cos[c + d*x]*Sqrt[a + b*Sin[c + d*x]]*(4*a*(5*a^4 - 40*a^2*b^2 - 93*b^4) - 3*b*(5*a^4 + 430*a^2*b^2 + 77*b^4)*Sin[c + d*x]))/(15015*b^3*d)","A",9,8,23,0.3478,1,"{2692, 2862, 2865, 2752, 2663, 2661, 2655, 2653}"
501,1,299,0,0.6692935,"\int \cos ^2(c+d x) (a+b \sin (c+d x))^{5/2} \, dx","Int[Cos[c + d*x]^2*(a + b*Sin[c + d*x])^(5/2),x]","\frac{2 \cos (c+d x) \sqrt{a+b \sin (c+d x)} \left(3 b \left(25 a^2+7 b^2\right) \sin (c+d x)+a \left(5 a^2+27 b^2\right)\right)}{315 b d}-\frac{4 a \left(22 a^2 b^2+5 a^4-27 b^4\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{315 b^2 d \sqrt{a+b \sin (c+d x)}}+\frac{4 \left(102 a^2 b^2+5 a^4+21 b^4\right) \sqrt{a+b \sin (c+d x)} E\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{315 b^2 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}-\frac{2 b \cos ^3(c+d x) (a+b \sin (c+d x))^{3/2}}{9 d}-\frac{8 a b \cos ^3(c+d x) \sqrt{a+b \sin (c+d x)}}{21 d}","\frac{2 \cos (c+d x) \sqrt{a+b \sin (c+d x)} \left(3 b \left(25 a^2+7 b^2\right) \sin (c+d x)+a \left(5 a^2+27 b^2\right)\right)}{315 b d}-\frac{4 a \left(22 a^2 b^2+5 a^4-27 b^4\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{315 b^2 d \sqrt{a+b \sin (c+d x)}}+\frac{4 \left(102 a^2 b^2+5 a^4+21 b^4\right) \sqrt{a+b \sin (c+d x)} E\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{315 b^2 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}-\frac{2 b \cos ^3(c+d x) (a+b \sin (c+d x))^{3/2}}{9 d}-\frac{8 a b \cos ^3(c+d x) \sqrt{a+b \sin (c+d x)}}{21 d}",1,"(-8*a*b*Cos[c + d*x]^3*Sqrt[a + b*Sin[c + d*x]])/(21*d) - (2*b*Cos[c + d*x]^3*(a + b*Sin[c + d*x])^(3/2))/(9*d) + (4*(5*a^4 + 102*a^2*b^2 + 21*b^4)*EllipticE[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[a + b*Sin[c + d*x]])/(315*b^2*d*Sqrt[(a + b*Sin[c + d*x])/(a + b)]) - (4*a*(5*a^4 + 22*a^2*b^2 - 27*b^4)*EllipticF[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(315*b^2*d*Sqrt[a + b*Sin[c + d*x]]) + (2*Cos[c + d*x]*Sqrt[a + b*Sin[c + d*x]]*(a*(5*a^2 + 27*b^2) + 3*b*(25*a^2 + 7*b^2)*Sin[c + d*x]))/(315*b*d)","A",8,8,23,0.3478,1,"{2692, 2862, 2865, 2752, 2663, 2661, 2655, 2653}"
502,1,203,0,0.280267,"\int \sec ^2(c+d x) (a+b \sin (c+d x))^{5/2} \, dx","Int[Sec[c + d*x]^2*(a + b*Sin[c + d*x])^(5/2),x]","\frac{a \left(a^2-b^2\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{d \sqrt{a+b \sin (c+d x)}}-\frac{\left(a^2+3 b^2\right) \sqrt{a+b \sin (c+d x)} E\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}+\frac{a b \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{d}+\frac{\sec (c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^{3/2}}{d}","\frac{a \left(a^2-b^2\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{d \sqrt{a+b \sin (c+d x)}}-\frac{\left(a^2+3 b^2\right) \sqrt{a+b \sin (c+d x)} E\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}+\frac{a b \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{d}+\frac{\sec (c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^{3/2}}{d}",1,"(a*b*Cos[c + d*x]*Sqrt[a + b*Sin[c + d*x]])/d + (Sec[c + d*x]*(b + a*Sin[c + d*x])*(a + b*Sin[c + d*x])^(3/2))/d - ((a^2 + 3*b^2)*EllipticE[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[a + b*Sin[c + d*x]])/(d*Sqrt[(a + b*Sin[c + d*x])/(a + b)]) + (a*(a^2 - b^2)*EllipticF[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(d*Sqrt[a + b*Sin[c + d*x]])","A",7,7,23,0.3043,1,"{2691, 2753, 2752, 2663, 2661, 2655, 2653}"
503,1,238,0,0.3936964,"\int \sec ^4(c+d x) (a+b \sin (c+d x))^{5/2} \, dx","Int[Sec[c + d*x]^4*(a + b*Sin[c + d*x])^(5/2),x]","\frac{\sec (c+d x) \sqrt{a+b \sin (c+d x)} \left(\left(4 a^2-3 b^2\right) \sin (c+d x)+a b\right)}{6 d}+\frac{2 a \left(a^2-b^2\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{3 d \sqrt{a+b \sin (c+d x)}}-\frac{\left(4 a^2-3 b^2\right) \sqrt{a+b \sin (c+d x)} E\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{6 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}+\frac{\sec ^3(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^{3/2}}{3 d}","\frac{\sec (c+d x) \sqrt{a+b \sin (c+d x)} \left(\left(4 a^2-3 b^2\right) \sin (c+d x)+a b\right)}{6 d}+\frac{2 a \left(a^2-b^2\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{3 d \sqrt{a+b \sin (c+d x)}}-\frac{\left(4 a^2-3 b^2\right) \sqrt{a+b \sin (c+d x)} E\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{6 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}+\frac{\sec ^3(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^{3/2}}{3 d}",1,"(Sec[c + d*x]^3*(b + a*Sin[c + d*x])*(a + b*Sin[c + d*x])^(3/2))/(3*d) - ((4*a^2 - 3*b^2)*EllipticE[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[a + b*Sin[c + d*x]])/(6*d*Sqrt[(a + b*Sin[c + d*x])/(a + b)]) + (2*a*(a^2 - b^2)*EllipticF[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(3*d*Sqrt[a + b*Sin[c + d*x]]) + (Sec[c + d*x]*Sqrt[a + b*Sin[c + d*x]]*(a*b + (4*a^2 - 3*b^2)*Sin[c + d*x]))/(6*d)","A",7,7,23,0.3043,1,"{2691, 2861, 2752, 2663, 2661, 2655, 2653}"
504,1,322,0,0.6724087,"\int \sec ^6(c+d x) (a+b \sin (c+d x))^{5/2} \, dx","Int[Sec[c + d*x]^6*(a + b*Sin[c + d*x])^(5/2),x]","\frac{\sec ^3(c+d x) \sqrt{a+b \sin (c+d x)} \left(\left(8 a^2-3 b^2\right) \sin (c+d x)+5 a b\right)}{30 d}-\frac{\sec (c+d x) \sqrt{a+b \sin (c+d x)} \left(8 a b \left(a^2-b^2\right)-\left(-41 a^2 b^2+32 a^4+9 b^4\right) \sin (c+d x)\right)}{60 d \left(a^2-b^2\right)}+\frac{a \left(32 a^2-17 b^2\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{60 d \sqrt{a+b \sin (c+d x)}}-\frac{\left(32 a^2-9 b^2\right) \sqrt{a+b \sin (c+d x)} E\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{60 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}+\frac{\sec ^5(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^{3/2}}{5 d}","\frac{\sec ^3(c+d x) \sqrt{a+b \sin (c+d x)} \left(\left(8 a^2-3 b^2\right) \sin (c+d x)+5 a b\right)}{30 d}-\frac{\sec (c+d x) \sqrt{a+b \sin (c+d x)} \left(8 a b \left(a^2-b^2\right)-\left(-41 a^2 b^2+32 a^4+9 b^4\right) \sin (c+d x)\right)}{60 d \left(a^2-b^2\right)}+\frac{a \left(32 a^2-17 b^2\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{60 d \sqrt{a+b \sin (c+d x)}}-\frac{\left(32 a^2-9 b^2\right) \sqrt{a+b \sin (c+d x)} E\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{60 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}+\frac{\sec ^5(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^{3/2}}{5 d}",1,"(Sec[c + d*x]^5*(b + a*Sin[c + d*x])*(a + b*Sin[c + d*x])^(3/2))/(5*d) - ((32*a^2 - 9*b^2)*EllipticE[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[a + b*Sin[c + d*x]])/(60*d*Sqrt[(a + b*Sin[c + d*x])/(a + b)]) + (a*(32*a^2 - 17*b^2)*EllipticF[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(60*d*Sqrt[a + b*Sin[c + d*x]]) + (Sec[c + d*x]^3*Sqrt[a + b*Sin[c + d*x]]*(5*a*b + (8*a^2 - 3*b^2)*Sin[c + d*x]))/(30*d) - (Sec[c + d*x]*Sqrt[a + b*Sin[c + d*x]]*(8*a*b*(a^2 - b^2) - (32*a^4 - 41*a^2*b^2 + 9*b^4)*Sin[c + d*x]))/(60*(a^2 - b^2)*d)","A",8,8,23,0.3478,1,"{2691, 2861, 2866, 2752, 2663, 2661, 2655, 2653}"
505,1,439,0,0.9437912,"\int \sec ^8(c+d x) (a+b \sin (c+d x))^{5/2} \, dx","Int[Sec[c + d*x]^8*(a + b*Sin[c + d*x])^(5/2),x]","\frac{3 \sec ^5(c+d x) \sqrt{a+b \sin (c+d x)} \left(\left(4 a^2-b^2\right) \sin (c+d x)+3 a b\right)}{70 d}-\frac{\sec ^3(c+d x) \sqrt{a+b \sin (c+d x)} \left(4 a b \left(a^2-b^2\right)-\left(-39 a^2 b^2+32 a^4+7 b^4\right) \sin (c+d x)\right)}{140 d \left(a^2-b^2\right)}-\frac{\sec (c+d x) \sqrt{a+b \sin (c+d x)} \left(a b \left(-59 a^2 b^2+32 a^4+27 b^4\right)-\left(-272 a^4 b^2+165 a^2 b^4+128 a^6-21 b^6\right) \sin (c+d x)\right)}{280 d \left(a^2-b^2\right)^2}+\frac{2 a \left(8 a^2-3 b^2\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{35 d \sqrt{a+b \sin (c+d x)}}-\frac{\left(-144 a^2 b^2+128 a^4+21 b^4\right) \sqrt{a+b \sin (c+d x)} E\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{280 d \left(a^2-b^2\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}+\frac{\sec ^7(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^{3/2}}{7 d}","\frac{3 \sec ^5(c+d x) \sqrt{a+b \sin (c+d x)} \left(\left(4 a^2-b^2\right) \sin (c+d x)+3 a b\right)}{70 d}-\frac{\sec ^3(c+d x) \sqrt{a+b \sin (c+d x)} \left(4 a b \left(a^2-b^2\right)-\left(-39 a^2 b^2+32 a^4+7 b^4\right) \sin (c+d x)\right)}{140 d \left(a^2-b^2\right)}-\frac{\sec (c+d x) \sqrt{a+b \sin (c+d x)} \left(a b \left(-59 a^2 b^2+32 a^4+27 b^4\right)-\left(-272 a^4 b^2+165 a^2 b^4+128 a^6-21 b^6\right) \sin (c+d x)\right)}{280 d \left(a^2-b^2\right)^2}+\frac{2 a \left(8 a^2-3 b^2\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{35 d \sqrt{a+b \sin (c+d x)}}-\frac{\left(-144 a^2 b^2+128 a^4+21 b^4\right) \sqrt{a+b \sin (c+d x)} E\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{280 d \left(a^2-b^2\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}+\frac{\sec ^7(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^{3/2}}{7 d}",1,"(Sec[c + d*x]^7*(b + a*Sin[c + d*x])*(a + b*Sin[c + d*x])^(3/2))/(7*d) - ((128*a^4 - 144*a^2*b^2 + 21*b^4)*EllipticE[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[a + b*Sin[c + d*x]])/(280*(a^2 - b^2)*d*Sqrt[(a + b*Sin[c + d*x])/(a + b)]) + (2*a*(8*a^2 - 3*b^2)*EllipticF[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(35*d*Sqrt[a + b*Sin[c + d*x]]) + (3*Sec[c + d*x]^5*Sqrt[a + b*Sin[c + d*x]]*(3*a*b + (4*a^2 - b^2)*Sin[c + d*x]))/(70*d) - (Sec[c + d*x]^3*Sqrt[a + b*Sin[c + d*x]]*(4*a*b*(a^2 - b^2) - (32*a^4 - 39*a^2*b^2 + 7*b^4)*Sin[c + d*x]))/(140*(a^2 - b^2)*d) - (Sec[c + d*x]*Sqrt[a + b*Sin[c + d*x]]*(a*b*(32*a^4 - 59*a^2*b^2 + 27*b^4) - (128*a^6 - 272*a^4*b^2 + 165*a^2*b^4 - 21*b^6)*Sin[c + d*x]))/(280*(a^2 - b^2)^2*d)","A",9,8,23,0.3478,1,"{2691, 2861, 2866, 2752, 2663, 2661, 2655, 2653}"
506,1,152,0,0.1140306,"\int \frac{\cos ^5(c+d x)}{\sqrt{a+b \sin (c+d x)}} \, dx","Int[Cos[c + d*x]^5/Sqrt[a + b*Sin[c + d*x]],x]","\frac{4 \left(3 a^2-b^2\right) (a+b \sin (c+d x))^{5/2}}{5 b^5 d}-\frac{8 a \left(a^2-b^2\right) (a+b \sin (c+d x))^{3/2}}{3 b^5 d}+\frac{2 \left(a^2-b^2\right)^2 \sqrt{a+b \sin (c+d x)}}{b^5 d}+\frac{2 (a+b \sin (c+d x))^{9/2}}{9 b^5 d}-\frac{8 a (a+b \sin (c+d x))^{7/2}}{7 b^5 d}","\frac{4 \left(3 a^2-b^2\right) (a+b \sin (c+d x))^{5/2}}{5 b^5 d}-\frac{8 a \left(a^2-b^2\right) (a+b \sin (c+d x))^{3/2}}{3 b^5 d}+\frac{2 \left(a^2-b^2\right)^2 \sqrt{a+b \sin (c+d x)}}{b^5 d}+\frac{2 (a+b \sin (c+d x))^{9/2}}{9 b^5 d}-\frac{8 a (a+b \sin (c+d x))^{7/2}}{7 b^5 d}",1,"(2*(a^2 - b^2)^2*Sqrt[a + b*Sin[c + d*x]])/(b^5*d) - (8*a*(a^2 - b^2)*(a + b*Sin[c + d*x])^(3/2))/(3*b^5*d) + (4*(3*a^2 - b^2)*(a + b*Sin[c + d*x])^(5/2))/(5*b^5*d) - (8*a*(a + b*Sin[c + d*x])^(7/2))/(7*b^5*d) + (2*(a + b*Sin[c + d*x])^(9/2))/(9*b^5*d)","A",3,2,23,0.08696,1,"{2668, 697}"
507,1,81,0,0.0848283,"\int \frac{\cos ^3(c+d x)}{\sqrt{a+b \sin (c+d x)}} \, dx","Int[Cos[c + d*x]^3/Sqrt[a + b*Sin[c + d*x]],x]","-\frac{2 \left(a^2-b^2\right) \sqrt{a+b \sin (c+d x)}}{b^3 d}-\frac{2 (a+b \sin (c+d x))^{5/2}}{5 b^3 d}+\frac{4 a (a+b \sin (c+d x))^{3/2}}{3 b^3 d}","-\frac{2 \left(a^2-b^2\right) \sqrt{a+b \sin (c+d x)}}{b^3 d}-\frac{2 (a+b \sin (c+d x))^{5/2}}{5 b^3 d}+\frac{4 a (a+b \sin (c+d x))^{3/2}}{3 b^3 d}",1,"(-2*(a^2 - b^2)*Sqrt[a + b*Sin[c + d*x]])/(b^3*d) + (4*a*(a + b*Sin[c + d*x])^(3/2))/(3*b^3*d) - (2*(a + b*Sin[c + d*x])^(5/2))/(5*b^3*d)","A",3,2,23,0.08696,1,"{2668, 697}"
508,1,22,0,0.0353723,"\int \frac{\cos (c+d x)}{\sqrt{a+b \sin (c+d x)}} \, dx","Int[Cos[c + d*x]/Sqrt[a + b*Sin[c + d*x]],x]","\frac{2 \sqrt{a+b \sin (c+d x)}}{b d}","\frac{2 \sqrt{a+b \sin (c+d x)}}{b d}",1,"(2*Sqrt[a + b*Sin[c + d*x]])/(b*d)","A",2,2,21,0.09524,1,"{2668, 32}"
509,1,74,0,0.0942083,"\int \frac{\sec (c+d x)}{\sqrt{a+b \sin (c+d x)}} \, dx","Int[Sec[c + d*x]/Sqrt[a + b*Sin[c + d*x]],x]","\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \sin (c+d x)}}{\sqrt{a+b}}\right)}{d \sqrt{a+b}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \sin (c+d x)}}{\sqrt{a-b}}\right)}{d \sqrt{a-b}}","\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \sin (c+d x)}}{\sqrt{a+b}}\right)}{d \sqrt{a+b}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \sin (c+d x)}}{\sqrt{a-b}}\right)}{d \sqrt{a-b}}",1,"-(ArcTanh[Sqrt[a + b*Sin[c + d*x]]/Sqrt[a - b]]/(Sqrt[a - b]*d)) + ArcTanh[Sqrt[a + b*Sin[c + d*x]]/Sqrt[a + b]]/(Sqrt[a + b]*d)","A",5,4,21,0.1905,1,"{2668, 708, 1093, 206}"
510,1,144,0,0.3054683,"\int \frac{\sec ^3(c+d x)}{\sqrt{a+b \sin (c+d x)}} \, dx","Int[Sec[c + d*x]^3/Sqrt[a + b*Sin[c + d*x]],x]","-\frac{\sec ^2(c+d x) (b-a \sin (c+d x)) \sqrt{a+b \sin (c+d x)}}{2 d \left(a^2-b^2\right)}-\frac{(2 a-3 b) \tanh ^{-1}\left(\frac{\sqrt{a+b \sin (c+d x)}}{\sqrt{a-b}}\right)}{4 d (a-b)^{3/2}}+\frac{(2 a+3 b) \tanh ^{-1}\left(\frac{\sqrt{a+b \sin (c+d x)}}{\sqrt{a+b}}\right)}{4 d (a+b)^{3/2}}","-\frac{\sec ^2(c+d x) (b-a \sin (c+d x)) \sqrt{a+b \sin (c+d x)}}{2 d \left(a^2-b^2\right)}-\frac{(2 a-3 b) \tanh ^{-1}\left(\frac{\sqrt{a+b \sin (c+d x)}}{\sqrt{a-b}}\right)}{4 d (a-b)^{3/2}}+\frac{(2 a+3 b) \tanh ^{-1}\left(\frac{\sqrt{a+b \sin (c+d x)}}{\sqrt{a+b}}\right)}{4 d (a+b)^{3/2}}",1,"-((2*a - 3*b)*ArcTanh[Sqrt[a + b*Sin[c + d*x]]/Sqrt[a - b]])/(4*(a - b)^(3/2)*d) + ((2*a + 3*b)*ArcTanh[Sqrt[a + b*Sin[c + d*x]]/Sqrt[a + b]])/(4*(a + b)^(3/2)*d) - (Sec[c + d*x]^2*(b - a*Sin[c + d*x])*Sqrt[a + b*Sin[c + d*x]])/(2*(a^2 - b^2)*d)","A",6,5,23,0.2174,1,"{2668, 741, 827, 1166, 206}"
511,1,230,0,0.3894787,"\int \frac{\sec ^5(c+d x)}{\sqrt{a+b \sin (c+d x)}} \, dx","Int[Sec[c + d*x]^5/Sqrt[a + b*Sin[c + d*x]],x]","-\frac{3 \left(4 a^2-10 a b+7 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a+b \sin (c+d x)}}{\sqrt{a-b}}\right)}{32 d (a-b)^{5/2}}+\frac{3 \left(4 a^2+10 a b+7 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a+b \sin (c+d x)}}{\sqrt{a+b}}\right)}{32 d (a+b)^{5/2}}-\frac{\sec ^4(c+d x) (b-a \sin (c+d x)) \sqrt{a+b \sin (c+d x)}}{4 d \left(a^2-b^2\right)}-\frac{\sec ^2(c+d x) \sqrt{a+b \sin (c+d x)} \left(b \left(a^2-7 b^2\right)-6 a \left(a^2-2 b^2\right) \sin (c+d x)\right)}{16 d \left(a^2-b^2\right)^2}","-\frac{3 \left(4 a^2-10 a b+7 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a+b \sin (c+d x)}}{\sqrt{a-b}}\right)}{32 d (a-b)^{5/2}}+\frac{3 \left(4 a^2+10 a b+7 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a+b \sin (c+d x)}}{\sqrt{a+b}}\right)}{32 d (a+b)^{5/2}}-\frac{\sec ^4(c+d x) (b-a \sin (c+d x)) \sqrt{a+b \sin (c+d x)}}{4 d \left(a^2-b^2\right)}-\frac{\sec ^2(c+d x) \sqrt{a+b \sin (c+d x)} \left(b \left(a^2-7 b^2\right)-6 a \left(a^2-2 b^2\right) \sin (c+d x)\right)}{16 d \left(a^2-b^2\right)^2}",1,"(-3*(4*a^2 - 10*a*b + 7*b^2)*ArcTanh[Sqrt[a + b*Sin[c + d*x]]/Sqrt[a - b]])/(32*(a - b)^(5/2)*d) + (3*(4*a^2 + 10*a*b + 7*b^2)*ArcTanh[Sqrt[a + b*Sin[c + d*x]]/Sqrt[a + b]])/(32*(a + b)^(5/2)*d) - (Sec[c + d*x]^4*(b - a*Sin[c + d*x])*Sqrt[a + b*Sin[c + d*x]])/(4*(a^2 - b^2)*d) - (Sec[c + d*x]^2*Sqrt[a + b*Sin[c + d*x]]*(b*(a^2 - 7*b^2) - 6*a*(a^2 - 2*b^2)*Sin[c + d*x]))/(16*(a^2 - b^2)^2*d)","A",7,6,23,0.2609,1,"{2668, 741, 823, 827, 1166, 206}"
512,1,247,0,0.3654504,"\int \frac{\cos ^4(c+d x)}{\sqrt{a+b \sin (c+d x)}} \, dx","Int[Cos[c + d*x]^4/Sqrt[a + b*Sin[c + d*x]],x]","-\frac{4 \cos (c+d x) \sqrt{a+b \sin (c+d x)} \left(4 a^2-3 a b \sin (c+d x)-5 b^2\right)}{35 b^3 d}+\frac{8 \left(-9 a^2 b^2+4 a^4+5 b^4\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{35 b^4 d \sqrt{a+b \sin (c+d x)}}-\frac{32 a \left(a^2-2 b^2\right) \sqrt{a+b \sin (c+d x)} E\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{35 b^4 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}+\frac{2 \cos ^3(c+d x) \sqrt{a+b \sin (c+d x)}}{7 b d}","-\frac{4 \cos (c+d x) \sqrt{a+b \sin (c+d x)} \left(4 a^2-3 a b \sin (c+d x)-5 b^2\right)}{35 b^3 d}+\frac{8 \left(-9 a^2 b^2+4 a^4+5 b^4\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{35 b^4 d \sqrt{a+b \sin (c+d x)}}-\frac{32 a \left(a^2-2 b^2\right) \sqrt{a+b \sin (c+d x)} E\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{35 b^4 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}+\frac{2 \cos ^3(c+d x) \sqrt{a+b \sin (c+d x)}}{7 b d}",1,"(2*Cos[c + d*x]^3*Sqrt[a + b*Sin[c + d*x]])/(7*b*d) - (32*a*(a^2 - 2*b^2)*EllipticE[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[a + b*Sin[c + d*x]])/(35*b^4*d*Sqrt[(a + b*Sin[c + d*x])/(a + b)]) + (8*(4*a^4 - 9*a^2*b^2 + 5*b^4)*EllipticF[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(35*b^4*d*Sqrt[a + b*Sin[c + d*x]]) - (4*Cos[c + d*x]*Sqrt[a + b*Sin[c + d*x]]*(4*a^2 - 5*b^2 - 3*a*b*Sin[c + d*x]))/(35*b^3*d)","A",7,7,23,0.3043,1,"{2695, 2865, 2752, 2663, 2661, 2655, 2653}"
513,1,175,0,0.1879624,"\int \frac{\cos ^2(c+d x)}{\sqrt{a+b \sin (c+d x)}} \, dx","Int[Cos[c + d*x]^2/Sqrt[a + b*Sin[c + d*x]],x]","-\frac{4 \left(a^2-b^2\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{3 b^2 d \sqrt{a+b \sin (c+d x)}}+\frac{4 a \sqrt{a+b \sin (c+d x)} E\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{3 b^2 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}+\frac{2 \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{3 b d}","-\frac{4 \left(a^2-b^2\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{3 b^2 d \sqrt{a+b \sin (c+d x)}}+\frac{4 a \sqrt{a+b \sin (c+d x)} E\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{3 b^2 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}+\frac{2 \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{3 b d}",1,"(2*Cos[c + d*x]*Sqrt[a + b*Sin[c + d*x]])/(3*b*d) + (4*a*EllipticE[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[a + b*Sin[c + d*x]])/(3*b^2*d*Sqrt[(a + b*Sin[c + d*x])/(a + b)]) - (4*(a^2 - b^2)*EllipticF[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(3*b^2*d*Sqrt[a + b*Sin[c + d*x]])","A",6,6,23,0.2609,1,"{2695, 2752, 2663, 2661, 2655, 2653}"
514,1,183,0,0.2033311,"\int \frac{\sec ^2(c+d x)}{\sqrt{a+b \sin (c+d x)}} \, dx","Int[Sec[c + d*x]^2/Sqrt[a + b*Sin[c + d*x]],x]","-\frac{\sec (c+d x) (b-a \sin (c+d x)) \sqrt{a+b \sin (c+d x)}}{d \left(a^2-b^2\right)}-\frac{a \sqrt{a+b \sin (c+d x)} E\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{d \left(a^2-b^2\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}+\frac{\sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{d \sqrt{a+b \sin (c+d x)}}","-\frac{\sec (c+d x) (b-a \sin (c+d x)) \sqrt{a+b \sin (c+d x)}}{d \left(a^2-b^2\right)}-\frac{a \sqrt{a+b \sin (c+d x)} E\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{d \left(a^2-b^2\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}+\frac{\sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{d \sqrt{a+b \sin (c+d x)}}",1,"-((Sec[c + d*x]*(b - a*Sin[c + d*x])*Sqrt[a + b*Sin[c + d*x]])/((a^2 - b^2)*d)) - (a*EllipticE[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[a + b*Sin[c + d*x]])/((a^2 - b^2)*d*Sqrt[(a + b*Sin[c + d*x])/(a + b)]) + (EllipticF[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(d*Sqrt[a + b*Sin[c + d*x]])","A",6,6,23,0.2609,1,"{2696, 2752, 2663, 2661, 2655, 2653}"
515,1,291,0,0.4390489,"\int \frac{\sec ^4(c+d x)}{\sqrt{a+b \sin (c+d x)}} \, dx","Int[Sec[c + d*x]^4/Sqrt[a + b*Sin[c + d*x]],x]","-\frac{\sec ^3(c+d x) (b-a \sin (c+d x)) \sqrt{a+b \sin (c+d x)}}{3 d \left(a^2-b^2\right)}-\frac{\sec (c+d x) \sqrt{a+b \sin (c+d x)} \left(b \left(a^2-5 b^2\right)-4 a \left(a^2-2 b^2\right) \sin (c+d x)\right)}{6 d \left(a^2-b^2\right)^2}+\frac{\left(4 a^2-5 b^2\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{6 d \left(a^2-b^2\right) \sqrt{a+b \sin (c+d x)}}-\frac{2 a \left(a^2-2 b^2\right) \sqrt{a+b \sin (c+d x)} E\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{3 d \left(a^2-b^2\right)^2 \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}","-\frac{\sec ^3(c+d x) (b-a \sin (c+d x)) \sqrt{a+b \sin (c+d x)}}{3 d \left(a^2-b^2\right)}-\frac{\sec (c+d x) \sqrt{a+b \sin (c+d x)} \left(b \left(a^2-5 b^2\right)-4 a \left(a^2-2 b^2\right) \sin (c+d x)\right)}{6 d \left(a^2-b^2\right)^2}+\frac{\left(4 a^2-5 b^2\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{6 d \left(a^2-b^2\right) \sqrt{a+b \sin (c+d x)}}-\frac{2 a \left(a^2-2 b^2\right) \sqrt{a+b \sin (c+d x)} E\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{3 d \left(a^2-b^2\right)^2 \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}",1,"-(Sec[c + d*x]^3*(b - a*Sin[c + d*x])*Sqrt[a + b*Sin[c + d*x]])/(3*(a^2 - b^2)*d) - (2*a*(a^2 - 2*b^2)*EllipticE[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[a + b*Sin[c + d*x]])/(3*(a^2 - b^2)^2*d*Sqrt[(a + b*Sin[c + d*x])/(a + b)]) + ((4*a^2 - 5*b^2)*EllipticF[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(6*(a^2 - b^2)*d*Sqrt[a + b*Sin[c + d*x]]) - (Sec[c + d*x]*Sqrt[a + b*Sin[c + d*x]]*(b*(a^2 - 5*b^2) - 4*a*(a^2 - 2*b^2)*Sin[c + d*x]))/(6*(a^2 - b^2)^2*d)","A",7,7,23,0.3043,1,"{2696, 2866, 2752, 2663, 2661, 2655, 2653}"
516,1,150,0,0.1206825,"\int \frac{\cos ^5(c+d x)}{(a+b \sin (c+d x))^{3/2}} \, dx","Int[Cos[c + d*x]^5/(a + b*Sin[c + d*x])^(3/2),x]","\frac{4 \left(3 a^2-b^2\right) (a+b \sin (c+d x))^{3/2}}{3 b^5 d}-\frac{8 a \left(a^2-b^2\right) \sqrt{a+b \sin (c+d x)}}{b^5 d}-\frac{2 \left(a^2-b^2\right)^2}{b^5 d \sqrt{a+b \sin (c+d x)}}+\frac{2 (a+b \sin (c+d x))^{7/2}}{7 b^5 d}-\frac{8 a (a+b \sin (c+d x))^{5/2}}{5 b^5 d}","\frac{4 \left(3 a^2-b^2\right) (a+b \sin (c+d x))^{3/2}}{3 b^5 d}-\frac{8 a \left(a^2-b^2\right) \sqrt{a+b \sin (c+d x)}}{b^5 d}-\frac{2 \left(a^2-b^2\right)^2}{b^5 d \sqrt{a+b \sin (c+d x)}}+\frac{2 (a+b \sin (c+d x))^{7/2}}{7 b^5 d}-\frac{8 a (a+b \sin (c+d x))^{5/2}}{5 b^5 d}",1,"(-2*(a^2 - b^2)^2)/(b^5*d*Sqrt[a + b*Sin[c + d*x]]) - (8*a*(a^2 - b^2)*Sqrt[a + b*Sin[c + d*x]])/(b^5*d) + (4*(3*a^2 - b^2)*(a + b*Sin[c + d*x])^(3/2))/(3*b^5*d) - (8*a*(a + b*Sin[c + d*x])^(5/2))/(5*b^5*d) + (2*(a + b*Sin[c + d*x])^(7/2))/(7*b^5*d)","A",3,2,23,0.08696,1,"{2668, 697}"
517,1,79,0,0.0935693,"\int \frac{\cos ^3(c+d x)}{(a+b \sin (c+d x))^{3/2}} \, dx","Int[Cos[c + d*x]^3/(a + b*Sin[c + d*x])^(3/2),x]","\frac{2 \left(a^2-b^2\right)}{b^3 d \sqrt{a+b \sin (c+d x)}}-\frac{2 (a+b \sin (c+d x))^{3/2}}{3 b^3 d}+\frac{4 a \sqrt{a+b \sin (c+d x)}}{b^3 d}","\frac{2 \left(a^2-b^2\right)}{b^3 d \sqrt{a+b \sin (c+d x)}}-\frac{2 (a+b \sin (c+d x))^{3/2}}{3 b^3 d}+\frac{4 a \sqrt{a+b \sin (c+d x)}}{b^3 d}",1,"(2*(a^2 - b^2))/(b^3*d*Sqrt[a + b*Sin[c + d*x]]) + (4*a*Sqrt[a + b*Sin[c + d*x]])/(b^3*d) - (2*(a + b*Sin[c + d*x])^(3/2))/(3*b^3*d)","A",3,2,23,0.08696,1,"{2668, 697}"
518,1,22,0,0.0377284,"\int \frac{\cos (c+d x)}{(a+b \sin (c+d x))^{3/2}} \, dx","Int[Cos[c + d*x]/(a + b*Sin[c + d*x])^(3/2),x]","-\frac{2}{b d \sqrt{a+b \sin (c+d x)}}","-\frac{2}{b d \sqrt{a+b \sin (c+d x)}}",1,"-2/(b*d*Sqrt[a + b*Sin[c + d*x]])","A",2,2,21,0.09524,1,"{2668, 32}"
519,1,105,0,0.1576255,"\int \frac{\sec (c+d x)}{(a+b \sin (c+d x))^{3/2}} \, dx","Int[Sec[c + d*x]/(a + b*Sin[c + d*x])^(3/2),x]","\frac{2 b}{d \left(a^2-b^2\right) \sqrt{a+b \sin (c+d x)}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \sin (c+d x)}}{\sqrt{a-b}}\right)}{d (a-b)^{3/2}}+\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \sin (c+d x)}}{\sqrt{a+b}}\right)}{d (a+b)^{3/2}}","\frac{2 b}{d \left(a^2-b^2\right) \sqrt{a+b \sin (c+d x)}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \sin (c+d x)}}{\sqrt{a-b}}\right)}{d (a-b)^{3/2}}+\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \sin (c+d x)}}{\sqrt{a+b}}\right)}{d (a+b)^{3/2}}",1,"-(ArcTanh[Sqrt[a + b*Sin[c + d*x]]/Sqrt[a - b]]/((a - b)^(3/2)*d)) + ArcTanh[Sqrt[a + b*Sin[c + d*x]]/Sqrt[a + b]]/((a + b)^(3/2)*d) + (2*b)/((a^2 - b^2)*d*Sqrt[a + b*Sin[c + d*x]])","A",6,5,21,0.2381,1,"{2668, 710, 827, 1166, 206}"
520,1,186,0,0.3343003,"\int \frac{\sec ^3(c+d x)}{(a+b \sin (c+d x))^{3/2}} \, dx","Int[Sec[c + d*x]^3/(a + b*Sin[c + d*x])^(3/2),x]","-\frac{b \left(a^2+5 b^2\right)}{2 d \left(a^2-b^2\right)^2 \sqrt{a+b \sin (c+d x)}}-\frac{\sec ^2(c+d x) (b-a \sin (c+d x))}{2 d \left(a^2-b^2\right) \sqrt{a+b \sin (c+d x)}}-\frac{(2 a-5 b) \tanh ^{-1}\left(\frac{\sqrt{a+b \sin (c+d x)}}{\sqrt{a-b}}\right)}{4 d (a-b)^{5/2}}+\frac{(2 a+5 b) \tanh ^{-1}\left(\frac{\sqrt{a+b \sin (c+d x)}}{\sqrt{a+b}}\right)}{4 d (a+b)^{5/2}}","-\frac{b \left(a^2+5 b^2\right)}{2 d \left(a^2-b^2\right)^2 \sqrt{a+b \sin (c+d x)}}-\frac{\sec ^2(c+d x) (b-a \sin (c+d x))}{2 d \left(a^2-b^2\right) \sqrt{a+b \sin (c+d x)}}-\frac{(2 a-5 b) \tanh ^{-1}\left(\frac{\sqrt{a+b \sin (c+d x)}}{\sqrt{a-b}}\right)}{4 d (a-b)^{5/2}}+\frac{(2 a+5 b) \tanh ^{-1}\left(\frac{\sqrt{a+b \sin (c+d x)}}{\sqrt{a+b}}\right)}{4 d (a+b)^{5/2}}",1,"-((2*a - 5*b)*ArcTanh[Sqrt[a + b*Sin[c + d*x]]/Sqrt[a - b]])/(4*(a - b)^(5/2)*d) + ((2*a + 5*b)*ArcTanh[Sqrt[a + b*Sin[c + d*x]]/Sqrt[a + b]])/(4*(a + b)^(5/2)*d) - (b*(a^2 + 5*b^2))/(2*(a^2 - b^2)^2*d*Sqrt[a + b*Sin[c + d*x]]) - (Sec[c + d*x]^2*(b - a*Sin[c + d*x]))/(2*(a^2 - b^2)*d*Sqrt[a + b*Sin[c + d*x]])","A",7,6,23,0.2609,1,"{2668, 741, 829, 827, 1166, 206}"
521,1,284,0,0.5205484,"\int \frac{\sec ^5(c+d x)}{(a+b \sin (c+d x))^{3/2}} \, dx","Int[Sec[c + d*x]^5/(a + b*Sin[c + d*x])^(3/2),x]","-\frac{3 b \left(-7 a^2 b^2+2 a^4-15 b^4\right)}{16 d \left(a^2-b^2\right)^3 \sqrt{a+b \sin (c+d x)}}-\frac{3 \left(4 a^2-14 a b+15 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a+b \sin (c+d x)}}{\sqrt{a-b}}\right)}{32 d (a-b)^{7/2}}+\frac{3 \left(4 a^2+14 a b+15 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a+b \sin (c+d x)}}{\sqrt{a+b}}\right)}{32 d (a+b)^{7/2}}-\frac{\sec ^4(c+d x) (b-a \sin (c+d x))}{4 d \left(a^2-b^2\right) \sqrt{a+b \sin (c+d x)}}+\frac{\sec ^2(c+d x) \left(2 a \left(3 a^2-8 b^2\right) \sin (c+d x)+b \left(a^2+9 b^2\right)\right)}{16 d \left(a^2-b^2\right)^2 \sqrt{a+b \sin (c+d x)}}","-\frac{3 b \left(-7 a^2 b^2+2 a^4-15 b^4\right)}{16 d \left(a^2-b^2\right)^3 \sqrt{a+b \sin (c+d x)}}-\frac{3 \left(4 a^2-14 a b+15 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a+b \sin (c+d x)}}{\sqrt{a-b}}\right)}{32 d (a-b)^{7/2}}+\frac{3 \left(4 a^2+14 a b+15 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a+b \sin (c+d x)}}{\sqrt{a+b}}\right)}{32 d (a+b)^{7/2}}-\frac{\sec ^4(c+d x) (b-a \sin (c+d x))}{4 d \left(a^2-b^2\right) \sqrt{a+b \sin (c+d x)}}+\frac{\sec ^2(c+d x) \left(2 a \left(3 a^2-8 b^2\right) \sin (c+d x)+b \left(a^2+9 b^2\right)\right)}{16 d \left(a^2-b^2\right)^2 \sqrt{a+b \sin (c+d x)}}",1,"(-3*(4*a^2 - 14*a*b + 15*b^2)*ArcTanh[Sqrt[a + b*Sin[c + d*x]]/Sqrt[a - b]])/(32*(a - b)^(7/2)*d) + (3*(4*a^2 + 14*a*b + 15*b^2)*ArcTanh[Sqrt[a + b*Sin[c + d*x]]/Sqrt[a + b]])/(32*(a + b)^(7/2)*d) - (3*b*(2*a^4 - 7*a^2*b^2 - 15*b^4))/(16*(a^2 - b^2)^3*d*Sqrt[a + b*Sin[c + d*x]]) - (Sec[c + d*x]^4*(b - a*Sin[c + d*x]))/(4*(a^2 - b^2)*d*Sqrt[a + b*Sin[c + d*x]]) + (Sec[c + d*x]^2*(b*(a^2 + 9*b^2) + 2*a*(3*a^2 - 8*b^2)*Sin[c + d*x]))/(16*(a^2 - b^2)^2*d*Sqrt[a + b*Sin[c + d*x]])","A",8,7,23,0.3043,1,"{2668, 741, 823, 829, 827, 1166, 206}"
522,1,313,0,0.5392542,"\int \frac{\cos ^6(c+d x)}{(a+b \sin (c+d x))^{3/2}} \, dx","Int[Cos[c + d*x]^6/(a + b*Sin[c + d*x])^(3/2),x]","-\frac{8 \cos (c+d x) \sqrt{a+b \sin (c+d x)} \left(a \left(32 a^2-33 b^2\right)-3 b \left(8 a^2-7 b^2\right) \sin (c+d x)\right)}{63 b^5 d}+\frac{16 a \left(-65 a^2 b^2+32 a^4+33 b^4\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{63 b^6 d \sqrt{a+b \sin (c+d x)}}-\frac{16 \left(-57 a^2 b^2+32 a^4+21 b^4\right) \sqrt{a+b \sin (c+d x)} E\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{63 b^6 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}+\frac{20 \cos ^3(c+d x) (8 a-7 b \sin (c+d x)) \sqrt{a+b \sin (c+d x)}}{63 b^3 d}-\frac{2 \cos ^5(c+d x)}{b d \sqrt{a+b \sin (c+d x)}}","-\frac{8 \cos (c+d x) \sqrt{a+b \sin (c+d x)} \left(a \left(32 a^2-33 b^2\right)-3 b \left(8 a^2-7 b^2\right) \sin (c+d x)\right)}{63 b^5 d}+\frac{16 a \left(-65 a^2 b^2+32 a^4+33 b^4\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{63 b^6 d \sqrt{a+b \sin (c+d x)}}-\frac{16 \left(-57 a^2 b^2+32 a^4+21 b^4\right) \sqrt{a+b \sin (c+d x)} E\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{63 b^6 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}+\frac{20 \cos ^3(c+d x) (8 a-7 b \sin (c+d x)) \sqrt{a+b \sin (c+d x)}}{63 b^3 d}-\frac{2 \cos ^5(c+d x)}{b d \sqrt{a+b \sin (c+d x)}}",1,"(-2*Cos[c + d*x]^5)/(b*d*Sqrt[a + b*Sin[c + d*x]]) + (20*Cos[c + d*x]^3*(8*a - 7*b*Sin[c + d*x])*Sqrt[a + b*Sin[c + d*x]])/(63*b^3*d) - (16*(32*a^4 - 57*a^2*b^2 + 21*b^4)*EllipticE[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[a + b*Sin[c + d*x]])/(63*b^6*d*Sqrt[(a + b*Sin[c + d*x])/(a + b)]) + (16*a*(32*a^4 - 65*a^2*b^2 + 33*b^4)*EllipticF[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(63*b^6*d*Sqrt[a + b*Sin[c + d*x]]) - (8*Cos[c + d*x]*Sqrt[a + b*Sin[c + d*x]]*(a*(32*a^2 - 33*b^2) - 3*b*(8*a^2 - 7*b^2)*Sin[c + d*x]))/(63*b^5*d)","A",8,7,23,0.3043,1,"{2693, 2865, 2752, 2663, 2661, 2655, 2653}"
523,1,229,0,0.3326582,"\int \frac{\cos ^4(c+d x)}{(a+b \sin (c+d x))^{3/2}} \, dx","Int[Cos[c + d*x]^4/(a + b*Sin[c + d*x])^(3/2),x]","-\frac{32 a \left(a^2-b^2\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{5 b^4 d \sqrt{a+b \sin (c+d x)}}+\frac{8 \left(4 a^2-3 b^2\right) \sqrt{a+b \sin (c+d x)} E\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{5 b^4 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}+\frac{4 \cos (c+d x) (4 a-3 b \sin (c+d x)) \sqrt{a+b \sin (c+d x)}}{5 b^3 d}-\frac{2 \cos ^3(c+d x)}{b d \sqrt{a+b \sin (c+d x)}}","-\frac{32 a \left(a^2-b^2\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{5 b^4 d \sqrt{a+b \sin (c+d x)}}+\frac{8 \left(4 a^2-3 b^2\right) \sqrt{a+b \sin (c+d x)} E\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{5 b^4 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}+\frac{4 \cos (c+d x) (4 a-3 b \sin (c+d x)) \sqrt{a+b \sin (c+d x)}}{5 b^3 d}-\frac{2 \cos ^3(c+d x)}{b d \sqrt{a+b \sin (c+d x)}}",1,"(-2*Cos[c + d*x]^3)/(b*d*Sqrt[a + b*Sin[c + d*x]]) + (4*Cos[c + d*x]*(4*a - 3*b*Sin[c + d*x])*Sqrt[a + b*Sin[c + d*x]])/(5*b^3*d) + (8*(4*a^2 - 3*b^2)*EllipticE[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[a + b*Sin[c + d*x]])/(5*b^4*d*Sqrt[(a + b*Sin[c + d*x])/(a + b)]) - (32*a*(a^2 - b^2)*EllipticF[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(5*b^4*d*Sqrt[a + b*Sin[c + d*x]])","A",7,7,23,0.3043,1,"{2693, 2865, 2752, 2663, 2661, 2655, 2653}"
524,1,160,0,0.1873067,"\int \frac{\cos ^2(c+d x)}{(a+b \sin (c+d x))^{3/2}} \, dx","Int[Cos[c + d*x]^2/(a + b*Sin[c + d*x])^(3/2),x]","\frac{4 a \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{b^2 d \sqrt{a+b \sin (c+d x)}}-\frac{4 \sqrt{a+b \sin (c+d x)} E\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{b^2 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}-\frac{2 \cos (c+d x)}{b d \sqrt{a+b \sin (c+d x)}}","\frac{4 a \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{b^2 d \sqrt{a+b \sin (c+d x)}}-\frac{4 \sqrt{a+b \sin (c+d x)} E\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{b^2 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}-\frac{2 \cos (c+d x)}{b d \sqrt{a+b \sin (c+d x)}}",1,"(-2*Cos[c + d*x])/(b*d*Sqrt[a + b*Sin[c + d*x]]) - (4*EllipticE[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[a + b*Sin[c + d*x]])/(b^2*d*Sqrt[(a + b*Sin[c + d*x])/(a + b)]) + (4*a*EllipticF[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(b^2*d*Sqrt[a + b*Sin[c + d*x]])","A",6,6,23,0.2609,1,"{2693, 2752, 2663, 2661, 2655, 2653}"
525,1,251,0,0.365161,"\int \frac{\sec ^2(c+d x)}{(a+b \sin (c+d x))^{3/2}} \, dx","Int[Sec[c + d*x]^2/(a + b*Sin[c + d*x])^(3/2),x]","-\frac{\sec (c+d x) \sqrt{a+b \sin (c+d x)} \left(4 a b-\left(a^2+3 b^2\right) \sin (c+d x)\right)}{d \left(a^2-b^2\right)^2}+\frac{2 b \sec (c+d x)}{d \left(a^2-b^2\right) \sqrt{a+b \sin (c+d x)}}+\frac{a \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{d \left(a^2-b^2\right) \sqrt{a+b \sin (c+d x)}}-\frac{\left(a^2+3 b^2\right) \sqrt{a+b \sin (c+d x)} E\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{d \left(a^2-b^2\right)^2 \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}","-\frac{\sec (c+d x) \sqrt{a+b \sin (c+d x)} \left(4 a b-\left(a^2+3 b^2\right) \sin (c+d x)\right)}{d \left(a^2-b^2\right)^2}+\frac{2 b \sec (c+d x)}{d \left(a^2-b^2\right) \sqrt{a+b \sin (c+d x)}}+\frac{a \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{d \left(a^2-b^2\right) \sqrt{a+b \sin (c+d x)}}-\frac{\left(a^2+3 b^2\right) \sqrt{a+b \sin (c+d x)} E\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{d \left(a^2-b^2\right)^2 \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}",1,"(2*b*Sec[c + d*x])/((a^2 - b^2)*d*Sqrt[a + b*Sin[c + d*x]]) - ((a^2 + 3*b^2)*EllipticE[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[a + b*Sin[c + d*x]])/((a^2 - b^2)^2*d*Sqrt[(a + b*Sin[c + d*x])/(a + b)]) + (a*EllipticF[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/((a^2 - b^2)*d*Sqrt[a + b*Sin[c + d*x]]) - (Sec[c + d*x]*Sqrt[a + b*Sin[c + d*x]]*(4*a*b - (a^2 + 3*b^2)*Sin[c + d*x]))/((a^2 - b^2)^2*d)","A",7,7,23,0.3043,1,"{2694, 2866, 2752, 2663, 2661, 2655, 2653}"
526,1,359,0,0.6134704,"\int \frac{\sec ^4(c+d x)}{(a+b \sin (c+d x))^{3/2}} \, dx","Int[Sec[c + d*x]^4/(a + b*Sin[c + d*x])^(3/2),x]","-\frac{\sec ^3(c+d x) \sqrt{a+b \sin (c+d x)} \left(8 a b-\left(a^2+7 b^2\right) \sin (c+d x)\right)}{3 d \left(a^2-b^2\right)^2}+\frac{2 b \sec ^3(c+d x)}{d \left(a^2-b^2\right) \sqrt{a+b \sin (c+d x)}}-\frac{\sec (c+d x) \sqrt{a+b \sin (c+d x)} \left(a b \left(a^2-33 b^2\right)-\left(-15 a^2 b^2+4 a^4-21 b^4\right) \sin (c+d x)\right)}{6 d \left(a^2-b^2\right)^3}+\frac{2 a \left(a^2-3 b^2\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{3 d \left(a^2-b^2\right)^2 \sqrt{a+b \sin (c+d x)}}-\frac{\left(-15 a^2 b^2+4 a^4-21 b^4\right) \sqrt{a+b \sin (c+d x)} E\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{6 d \left(a^2-b^2\right)^3 \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}","-\frac{\sec ^3(c+d x) \sqrt{a+b \sin (c+d x)} \left(8 a b-\left(a^2+7 b^2\right) \sin (c+d x)\right)}{3 d \left(a^2-b^2\right)^2}+\frac{2 b \sec ^3(c+d x)}{d \left(a^2-b^2\right) \sqrt{a+b \sin (c+d x)}}-\frac{\sec (c+d x) \sqrt{a+b \sin (c+d x)} \left(a b \left(a^2-33 b^2\right)-\left(-15 a^2 b^2+4 a^4-21 b^4\right) \sin (c+d x)\right)}{6 d \left(a^2-b^2\right)^3}+\frac{2 a \left(a^2-3 b^2\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{3 d \left(a^2-b^2\right)^2 \sqrt{a+b \sin (c+d x)}}-\frac{\left(-15 a^2 b^2+4 a^4-21 b^4\right) \sqrt{a+b \sin (c+d x)} E\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{6 d \left(a^2-b^2\right)^3 \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}",1,"(2*b*Sec[c + d*x]^3)/((a^2 - b^2)*d*Sqrt[a + b*Sin[c + d*x]]) - ((4*a^4 - 15*a^2*b^2 - 21*b^4)*EllipticE[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[a + b*Sin[c + d*x]])/(6*(a^2 - b^2)^3*d*Sqrt[(a + b*Sin[c + d*x])/(a + b)]) + (2*a*(a^2 - 3*b^2)*EllipticF[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(3*(a^2 - b^2)^2*d*Sqrt[a + b*Sin[c + d*x]]) - (Sec[c + d*x]^3*Sqrt[a + b*Sin[c + d*x]]*(8*a*b - (a^2 + 7*b^2)*Sin[c + d*x]))/(3*(a^2 - b^2)^2*d) - (Sec[c + d*x]*Sqrt[a + b*Sin[c + d*x]]*(a*b*(a^2 - 33*b^2) - (4*a^4 - 15*a^2*b^2 - 21*b^4)*Sin[c + d*x]))/(6*(a^2 - b^2)^3*d)","A",8,7,23,0.3043,1,"{2694, 2866, 2752, 2663, 2661, 2655, 2653}"
527,1,150,0,0.1224096,"\int \frac{\cos ^5(c+d x)}{(a+b \sin (c+d x))^{5/2}} \, dx","Int[Cos[c + d*x]^5/(a + b*Sin[c + d*x])^(5/2),x]","\frac{4 \left(3 a^2-b^2\right) \sqrt{a+b \sin (c+d x)}}{b^5 d}+\frac{8 a \left(a^2-b^2\right)}{b^5 d \sqrt{a+b \sin (c+d x)}}-\frac{2 \left(a^2-b^2\right)^2}{3 b^5 d (a+b \sin (c+d x))^{3/2}}+\frac{2 (a+b \sin (c+d x))^{5/2}}{5 b^5 d}-\frac{8 a (a+b \sin (c+d x))^{3/2}}{3 b^5 d}","\frac{4 \left(3 a^2-b^2\right) \sqrt{a+b \sin (c+d x)}}{b^5 d}+\frac{8 a \left(a^2-b^2\right)}{b^5 d \sqrt{a+b \sin (c+d x)}}-\frac{2 \left(a^2-b^2\right)^2}{3 b^5 d (a+b \sin (c+d x))^{3/2}}+\frac{2 (a+b \sin (c+d x))^{5/2}}{5 b^5 d}-\frac{8 a (a+b \sin (c+d x))^{3/2}}{3 b^5 d}",1,"(-2*(a^2 - b^2)^2)/(3*b^5*d*(a + b*Sin[c + d*x])^(3/2)) + (8*a*(a^2 - b^2))/(b^5*d*Sqrt[a + b*Sin[c + d*x]]) + (4*(3*a^2 - b^2)*Sqrt[a + b*Sin[c + d*x]])/(b^5*d) - (8*a*(a + b*Sin[c + d*x])^(3/2))/(3*b^5*d) + (2*(a + b*Sin[c + d*x])^(5/2))/(5*b^5*d)","A",3,2,23,0.08696,1,"{2668, 697}"
528,1,79,0,0.0939744,"\int \frac{\cos ^3(c+d x)}{(a+b \sin (c+d x))^{5/2}} \, dx","Int[Cos[c + d*x]^3/(a + b*Sin[c + d*x])^(5/2),x]","\frac{2 \left(a^2-b^2\right)}{3 b^3 d (a+b \sin (c+d x))^{3/2}}-\frac{4 a}{b^3 d \sqrt{a+b \sin (c+d x)}}-\frac{2 \sqrt{a+b \sin (c+d x)}}{b^3 d}","\frac{2 \left(a^2-b^2\right)}{3 b^3 d (a+b \sin (c+d x))^{3/2}}-\frac{4 a}{b^3 d \sqrt{a+b \sin (c+d x)}}-\frac{2 \sqrt{a+b \sin (c+d x)}}{b^3 d}",1,"(2*(a^2 - b^2))/(3*b^3*d*(a + b*Sin[c + d*x])^(3/2)) - (4*a)/(b^3*d*Sqrt[a + b*Sin[c + d*x]]) - (2*Sqrt[a + b*Sin[c + d*x]])/(b^3*d)","A",3,2,23,0.08696,1,"{2668, 697}"
529,1,24,0,0.0422203,"\int \frac{\cos (c+d x)}{(a+b \sin (c+d x))^{5/2}} \, dx","Int[Cos[c + d*x]/(a + b*Sin[c + d*x])^(5/2),x]","-\frac{2}{3 b d (a+b \sin (c+d x))^{3/2}}","-\frac{2}{3 b d (a+b \sin (c+d x))^{3/2}}",1,"-2/(3*b*d*(a + b*Sin[c + d*x])^(3/2))","A",2,2,21,0.09524,1,"{2668, 32}"
530,1,139,0,0.2426256,"\int \frac{\sec (c+d x)}{(a+b \sin (c+d x))^{5/2}} \, dx","Int[Sec[c + d*x]/(a + b*Sin[c + d*x])^(5/2),x]","\frac{4 a b}{d \left(a^2-b^2\right)^2 \sqrt{a+b \sin (c+d x)}}+\frac{2 b}{3 d \left(a^2-b^2\right) (a+b \sin (c+d x))^{3/2}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \sin (c+d x)}}{\sqrt{a-b}}\right)}{d (a-b)^{5/2}}+\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \sin (c+d x)}}{\sqrt{a+b}}\right)}{d (a+b)^{5/2}}","\frac{4 a b}{d \left(a^2-b^2\right)^2 \sqrt{a+b \sin (c+d x)}}+\frac{2 b}{3 d \left(a^2-b^2\right) (a+b \sin (c+d x))^{3/2}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \sin (c+d x)}}{\sqrt{a-b}}\right)}{d (a-b)^{5/2}}+\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \sin (c+d x)}}{\sqrt{a+b}}\right)}{d (a+b)^{5/2}}",1,"-(ArcTanh[Sqrt[a + b*Sin[c + d*x]]/Sqrt[a - b]]/((a - b)^(5/2)*d)) + ArcTanh[Sqrt[a + b*Sin[c + d*x]]/Sqrt[a + b]]/((a + b)^(5/2)*d) + (2*b)/(3*(a^2 - b^2)*d*(a + b*Sin[c + d*x])^(3/2)) + (4*a*b)/((a^2 - b^2)^2*d*Sqrt[a + b*Sin[c + d*x]])","A",7,6,21,0.2857,1,"{2668, 710, 829, 827, 1166, 206}"
531,1,231,0,0.4159602,"\int \frac{\sec ^3(c+d x)}{(a+b \sin (c+d x))^{5/2}} \, dx","Int[Sec[c + d*x]^3/(a + b*Sin[c + d*x])^(5/2),x]","-\frac{a b \left(a^2+19 b^2\right)}{2 d \left(a^2-b^2\right)^3 \sqrt{a+b \sin (c+d x)}}-\frac{b \left(3 a^2+7 b^2\right)}{6 d \left(a^2-b^2\right)^2 (a+b \sin (c+d x))^{3/2}}-\frac{\sec ^2(c+d x) (b-a \sin (c+d x))}{2 d \left(a^2-b^2\right) (a+b \sin (c+d x))^{3/2}}-\frac{(2 a-7 b) \tanh ^{-1}\left(\frac{\sqrt{a+b \sin (c+d x)}}{\sqrt{a-b}}\right)}{4 d (a-b)^{7/2}}+\frac{(2 a+7 b) \tanh ^{-1}\left(\frac{\sqrt{a+b \sin (c+d x)}}{\sqrt{a+b}}\right)}{4 d (a+b)^{7/2}}","-\frac{a b \left(a^2+19 b^2\right)}{2 d \left(a^2-b^2\right)^3 \sqrt{a+b \sin (c+d x)}}-\frac{b \left(3 a^2+7 b^2\right)}{6 d \left(a^2-b^2\right)^2 (a+b \sin (c+d x))^{3/2}}-\frac{\sec ^2(c+d x) (b-a \sin (c+d x))}{2 d \left(a^2-b^2\right) (a+b \sin (c+d x))^{3/2}}-\frac{(2 a-7 b) \tanh ^{-1}\left(\frac{\sqrt{a+b \sin (c+d x)}}{\sqrt{a-b}}\right)}{4 d (a-b)^{7/2}}+\frac{(2 a+7 b) \tanh ^{-1}\left(\frac{\sqrt{a+b \sin (c+d x)}}{\sqrt{a+b}}\right)}{4 d (a+b)^{7/2}}",1,"-((2*a - 7*b)*ArcTanh[Sqrt[a + b*Sin[c + d*x]]/Sqrt[a - b]])/(4*(a - b)^(7/2)*d) + ((2*a + 7*b)*ArcTanh[Sqrt[a + b*Sin[c + d*x]]/Sqrt[a + b]])/(4*(a + b)^(7/2)*d) - (b*(3*a^2 + 7*b^2))/(6*(a^2 - b^2)^2*d*(a + b*Sin[c + d*x])^(3/2)) - (Sec[c + d*x]^2*(b - a*Sin[c + d*x]))/(2*(a^2 - b^2)*d*(a + b*Sin[c + d*x])^(3/2)) - (a*b*(a^2 + 19*b^2))/(2*(a^2 - b^2)^3*d*Sqrt[a + b*Sin[c + d*x]])","A",8,6,23,0.2609,1,"{2668, 741, 829, 827, 1166, 206}"
532,1,339,0,0.6449664,"\int \frac{\sec ^5(c+d x)}{(a+b \sin (c+d x))^{5/2}} \, dx","Int[Sec[c + d*x]^5/(a + b*Sin[c + d*x])^(5/2),x]","-\frac{a b \left(-16 a^2 b^2+3 a^4-127 b^4\right)}{8 d \left(a^2-b^2\right)^4 \sqrt{a+b \sin (c+d x)}}-\frac{b \left(-81 a^2 b^2+18 a^4-77 b^4\right)}{48 d \left(a^2-b^2\right)^3 (a+b \sin (c+d x))^{3/2}}-\frac{\left(12 a^2-54 a b+77 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a+b \sin (c+d x)}}{\sqrt{a-b}}\right)}{32 d (a-b)^{9/2}}+\frac{\left(12 a^2+54 a b+77 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a+b \sin (c+d x)}}{\sqrt{a+b}}\right)}{32 d (a+b)^{9/2}}-\frac{\sec ^4(c+d x) (b-a \sin (c+d x))}{4 d \left(a^2-b^2\right) (a+b \sin (c+d x))^{3/2}}+\frac{\sec ^2(c+d x) \left(2 a \left(3 a^2-10 b^2\right) \sin (c+d x)+b \left(3 a^2+11 b^2\right)\right)}{16 d \left(a^2-b^2\right)^2 (a+b \sin (c+d x))^{3/2}}","-\frac{a b \left(-16 a^2 b^2+3 a^4-127 b^4\right)}{8 d \left(a^2-b^2\right)^4 \sqrt{a+b \sin (c+d x)}}-\frac{b \left(-81 a^2 b^2+18 a^4-77 b^4\right)}{48 d \left(a^2-b^2\right)^3 (a+b \sin (c+d x))^{3/2}}-\frac{\left(12 a^2-54 a b+77 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a+b \sin (c+d x)}}{\sqrt{a-b}}\right)}{32 d (a-b)^{9/2}}+\frac{\left(12 a^2+54 a b+77 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a+b \sin (c+d x)}}{\sqrt{a+b}}\right)}{32 d (a+b)^{9/2}}-\frac{\sec ^4(c+d x) (b-a \sin (c+d x))}{4 d \left(a^2-b^2\right) (a+b \sin (c+d x))^{3/2}}+\frac{\sec ^2(c+d x) \left(2 a \left(3 a^2-10 b^2\right) \sin (c+d x)+b \left(3 a^2+11 b^2\right)\right)}{16 d \left(a^2-b^2\right)^2 (a+b \sin (c+d x))^{3/2}}",1,"-((12*a^2 - 54*a*b + 77*b^2)*ArcTanh[Sqrt[a + b*Sin[c + d*x]]/Sqrt[a - b]])/(32*(a - b)^(9/2)*d) + ((12*a^2 + 54*a*b + 77*b^2)*ArcTanh[Sqrt[a + b*Sin[c + d*x]]/Sqrt[a + b]])/(32*(a + b)^(9/2)*d) - (b*(18*a^4 - 81*a^2*b^2 - 77*b^4))/(48*(a^2 - b^2)^3*d*(a + b*Sin[c + d*x])^(3/2)) - (Sec[c + d*x]^4*(b - a*Sin[c + d*x]))/(4*(a^2 - b^2)*d*(a + b*Sin[c + d*x])^(3/2)) - (a*b*(3*a^4 - 16*a^2*b^2 - 127*b^4))/(8*(a^2 - b^2)^4*d*Sqrt[a + b*Sin[c + d*x]]) + (Sec[c + d*x]^2*(b*(3*a^2 + 11*b^2) + 2*a*(3*a^2 - 10*b^2)*Sin[c + d*x]))/(16*(a^2 - b^2)^2*d*(a + b*Sin[c + d*x])^(3/2))","A",9,7,23,0.3043,1,"{2668, 741, 823, 829, 827, 1166, 206}"
533,1,384,0,0.761083,"\int \frac{\cos ^8(c+d x)}{(a+b \sin (c+d x))^{5/2}} \, dx","Int[Cos[c + d*x]^8/(a + b*Sin[c + d*x])^(5/2),x]","\frac{40 \cos ^3(c+d x) \sqrt{a+b \sin (c+d x)} \left(32 a^2-28 a b \sin (c+d x)-3 b^2\right)}{99 b^5 d}-\frac{16 \cos (c+d x) \sqrt{a+b \sin (c+d x)} \left(-3 a b \left(32 a^2-31 b^2\right) \sin (c+d x)-144 a^2 b^2+128 a^4+15 b^4\right)}{99 b^7 d}+\frac{32 \left(-272 a^4 b^2+159 a^2 b^4+128 a^6-15 b^6\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{99 b^8 d \sqrt{a+b \sin (c+d x)}}-\frac{128 a \left(8 a^2-9 b^2\right) \left(4 a^2-3 b^2\right) \sqrt{a+b \sin (c+d x)} E\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{99 b^8 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}-\frac{28 \cos ^5(c+d x) (12 a+b \sin (c+d x))}{33 b^3 d \sqrt{a+b \sin (c+d x)}}-\frac{2 \cos ^7(c+d x)}{3 b d (a+b \sin (c+d x))^{3/2}}","\frac{40 \cos ^3(c+d x) \sqrt{a+b \sin (c+d x)} \left(32 a^2-28 a b \sin (c+d x)-3 b^2\right)}{99 b^5 d}-\frac{16 \cos (c+d x) \sqrt{a+b \sin (c+d x)} \left(-3 a b \left(32 a^2-31 b^2\right) \sin (c+d x)-144 a^2 b^2+128 a^4+15 b^4\right)}{99 b^7 d}+\frac{32 \left(-272 a^4 b^2+159 a^2 b^4+128 a^6-15 b^6\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{99 b^8 d \sqrt{a+b \sin (c+d x)}}-\frac{128 a \left(8 a^2-9 b^2\right) \left(4 a^2-3 b^2\right) \sqrt{a+b \sin (c+d x)} E\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{99 b^8 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}-\frac{28 \cos ^5(c+d x) (12 a+b \sin (c+d x))}{33 b^3 d \sqrt{a+b \sin (c+d x)}}-\frac{2 \cos ^7(c+d x)}{3 b d (a+b \sin (c+d x))^{3/2}}",1,"(-2*Cos[c + d*x]^7)/(3*b*d*(a + b*Sin[c + d*x])^(3/2)) - (128*a*(8*a^2 - 9*b^2)*(4*a^2 - 3*b^2)*EllipticE[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[a + b*Sin[c + d*x]])/(99*b^8*d*Sqrt[(a + b*Sin[c + d*x])/(a + b)]) + (32*(128*a^6 - 272*a^4*b^2 + 159*a^2*b^4 - 15*b^6)*EllipticF[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(99*b^8*d*Sqrt[a + b*Sin[c + d*x]]) - (28*Cos[c + d*x]^5*(12*a + b*Sin[c + d*x]))/(33*b^3*d*Sqrt[a + b*Sin[c + d*x]]) + (40*Cos[c + d*x]^3*Sqrt[a + b*Sin[c + d*x]]*(32*a^2 - 3*b^2 - 28*a*b*Sin[c + d*x]))/(99*b^5*d) - (16*Cos[c + d*x]*Sqrt[a + b*Sin[c + d*x]]*(128*a^4 - 144*a^2*b^2 + 15*b^4 - 3*a*b*(32*a^2 - 31*b^2)*Sin[c + d*x]))/(99*b^7*d)","A",9,8,23,0.3478,1,"{2693, 2863, 2865, 2752, 2663, 2661, 2655, 2653}"
534,1,293,0,0.512548,"\int \frac{\cos ^6(c+d x)}{(a+b \sin (c+d x))^{5/2}} \, dx","Int[Cos[c + d*x]^6/(a + b*Sin[c + d*x])^(5/2),x]","\frac{8 \cos (c+d x) \sqrt{a+b \sin (c+d x)} \left(32 a^2-24 a b \sin (c+d x)-5 b^2\right)}{21 b^5 d}-\frac{16 \left(-37 a^2 b^2+32 a^4+5 b^4\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{21 b^6 d \sqrt{a+b \sin (c+d x)}}+\frac{16 a \left(32 a^2-29 b^2\right) \sqrt{a+b \sin (c+d x)} E\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{21 b^6 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}-\frac{20 \cos ^3(c+d x) (8 a+b \sin (c+d x))}{21 b^3 d \sqrt{a+b \sin (c+d x)}}-\frac{2 \cos ^5(c+d x)}{3 b d (a+b \sin (c+d x))^{3/2}}","\frac{8 \cos (c+d x) \sqrt{a+b \sin (c+d x)} \left(32 a^2-24 a b \sin (c+d x)-5 b^2\right)}{21 b^5 d}-\frac{16 \left(-37 a^2 b^2+32 a^4+5 b^4\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{21 b^6 d \sqrt{a+b \sin (c+d x)}}+\frac{16 a \left(32 a^2-29 b^2\right) \sqrt{a+b \sin (c+d x)} E\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{21 b^6 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}-\frac{20 \cos ^3(c+d x) (8 a+b \sin (c+d x))}{21 b^3 d \sqrt{a+b \sin (c+d x)}}-\frac{2 \cos ^5(c+d x)}{3 b d (a+b \sin (c+d x))^{3/2}}",1,"(-2*Cos[c + d*x]^5)/(3*b*d*(a + b*Sin[c + d*x])^(3/2)) + (16*a*(32*a^2 - 29*b^2)*EllipticE[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[a + b*Sin[c + d*x]])/(21*b^6*d*Sqrt[(a + b*Sin[c + d*x])/(a + b)]) - (16*(32*a^4 - 37*a^2*b^2 + 5*b^4)*EllipticF[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(21*b^6*d*Sqrt[a + b*Sin[c + d*x]]) - (20*Cos[c + d*x]^3*(8*a + b*Sin[c + d*x]))/(21*b^3*d*Sqrt[a + b*Sin[c + d*x]]) + (8*Cos[c + d*x]*Sqrt[a + b*Sin[c + d*x]]*(32*a^2 - 5*b^2 - 24*a*b*Sin[c + d*x]))/(21*b^5*d)","A",8,8,23,0.3478,1,"{2693, 2863, 2865, 2752, 2663, 2661, 2655, 2653}"
535,1,221,0,0.3195775,"\int \frac{\cos ^4(c+d x)}{(a+b \sin (c+d x))^{5/2}} \, dx","Int[Cos[c + d*x]^4/(a + b*Sin[c + d*x])^(5/2),x]","\frac{8 \left(4 a^2-b^2\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{3 b^4 d \sqrt{a+b \sin (c+d x)}}-\frac{4 \cos (c+d x) (4 a+b \sin (c+d x))}{3 b^3 d \sqrt{a+b \sin (c+d x)}}-\frac{32 a \sqrt{a+b \sin (c+d x)} E\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{3 b^4 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}-\frac{2 \cos ^3(c+d x)}{3 b d (a+b \sin (c+d x))^{3/2}}","\frac{8 \left(4 a^2-b^2\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{3 b^4 d \sqrt{a+b \sin (c+d x)}}-\frac{4 \cos (c+d x) (4 a+b \sin (c+d x))}{3 b^3 d \sqrt{a+b \sin (c+d x)}}-\frac{32 a \sqrt{a+b \sin (c+d x)} E\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{3 b^4 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}-\frac{2 \cos ^3(c+d x)}{3 b d (a+b \sin (c+d x))^{3/2}}",1,"(-2*Cos[c + d*x]^3)/(3*b*d*(a + b*Sin[c + d*x])^(3/2)) - (32*a*EllipticE[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[a + b*Sin[c + d*x]])/(3*b^4*d*Sqrt[(a + b*Sin[c + d*x])/(a + b)]) + (8*(4*a^2 - b^2)*EllipticF[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(3*b^4*d*Sqrt[a + b*Sin[c + d*x]]) - (4*Cos[c + d*x]*(4*a + b*Sin[c + d*x]))/(3*b^3*d*Sqrt[a + b*Sin[c + d*x]])","A",7,7,23,0.3043,1,"{2693, 2863, 2752, 2663, 2661, 2655, 2653}"
536,1,219,0,0.262706,"\int \frac{\cos ^2(c+d x)}{(a+b \sin (c+d x))^{5/2}} \, dx","Int[Cos[c + d*x]^2/(a + b*Sin[c + d*x])^(5/2),x]","\frac{4 a \cos (c+d x)}{3 b d \left(a^2-b^2\right) \sqrt{a+b \sin (c+d x)}}+\frac{4 a \sqrt{a+b \sin (c+d x)} E\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{3 b^2 d \left(a^2-b^2\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}-\frac{4 \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{3 b^2 d \sqrt{a+b \sin (c+d x)}}-\frac{2 \cos (c+d x)}{3 b d (a+b \sin (c+d x))^{3/2}}","\frac{4 a \cos (c+d x)}{3 b d \left(a^2-b^2\right) \sqrt{a+b \sin (c+d x)}}+\frac{4 a \sqrt{a+b \sin (c+d x)} E\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{3 b^2 d \left(a^2-b^2\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}-\frac{4 \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{3 b^2 d \sqrt{a+b \sin (c+d x)}}-\frac{2 \cos (c+d x)}{3 b d (a+b \sin (c+d x))^{3/2}}",1,"(-2*Cos[c + d*x])/(3*b*d*(a + b*Sin[c + d*x])^(3/2)) + (4*a*Cos[c + d*x])/(3*b*(a^2 - b^2)*d*Sqrt[a + b*Sin[c + d*x]]) + (4*a*EllipticE[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[a + b*Sin[c + d*x]])/(3*b^2*(a^2 - b^2)*d*Sqrt[(a + b*Sin[c + d*x])/(a + b)]) - (4*EllipticF[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(3*b^2*d*Sqrt[a + b*Sin[c + d*x]])","A",7,7,23,0.3043,1,"{2693, 2754, 2752, 2663, 2661, 2655, 2653}"
537,1,325,0,0.6094878,"\int \frac{\sec ^2(c+d x)}{(a+b \sin (c+d x))^{5/2}} \, dx","Int[Sec[c + d*x]^2/(a + b*Sin[c + d*x])^(5/2),x]","-\frac{\sec (c+d x) \sqrt{a+b \sin (c+d x)} \left(b \left(27 a^2+5 b^2\right)-a \left(3 a^2+29 b^2\right) \sin (c+d x)\right)}{3 d \left(a^2-b^2\right)^3}+\frac{16 a b \sec (c+d x)}{3 d \left(a^2-b^2\right)^2 \sqrt{a+b \sin (c+d x)}}+\frac{2 b \sec (c+d x)}{3 d \left(a^2-b^2\right) (a+b \sin (c+d x))^{3/2}}+\frac{\left(3 a^2+5 b^2\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{3 d \left(a^2-b^2\right)^2 \sqrt{a+b \sin (c+d x)}}-\frac{a \left(3 a^2+29 b^2\right) \sqrt{a+b \sin (c+d x)} E\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{3 d \left(a^2-b^2\right)^3 \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}","-\frac{\sec (c+d x) \sqrt{a+b \sin (c+d x)} \left(b \left(27 a^2+5 b^2\right)-a \left(3 a^2+29 b^2\right) \sin (c+d x)\right)}{3 d \left(a^2-b^2\right)^3}+\frac{16 a b \sec (c+d x)}{3 d \left(a^2-b^2\right)^2 \sqrt{a+b \sin (c+d x)}}+\frac{2 b \sec (c+d x)}{3 d \left(a^2-b^2\right) (a+b \sin (c+d x))^{3/2}}+\frac{\left(3 a^2+5 b^2\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{3 d \left(a^2-b^2\right)^2 \sqrt{a+b \sin (c+d x)}}-\frac{a \left(3 a^2+29 b^2\right) \sqrt{a+b \sin (c+d x)} E\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{3 d \left(a^2-b^2\right)^3 \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}",1,"(2*b*Sec[c + d*x])/(3*(a^2 - b^2)*d*(a + b*Sin[c + d*x])^(3/2)) + (16*a*b*Sec[c + d*x])/(3*(a^2 - b^2)^2*d*Sqrt[a + b*Sin[c + d*x]]) - (a*(3*a^2 + 29*b^2)*EllipticE[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[a + b*Sin[c + d*x]])/(3*(a^2 - b^2)^3*d*Sqrt[(a + b*Sin[c + d*x])/(a + b)]) + ((3*a^2 + 5*b^2)*EllipticF[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(3*(a^2 - b^2)^2*d*Sqrt[a + b*Sin[c + d*x]]) - (Sec[c + d*x]*Sqrt[a + b*Sin[c + d*x]]*(b*(27*a^2 + 5*b^2) - a*(3*a^2 + 29*b^2)*Sin[c + d*x]))/(3*(a^2 - b^2)^3*d)","A",8,8,23,0.3478,1,"{2694, 2864, 2866, 2752, 2663, 2661, 2655, 2653}"
538,1,425,0,0.8761697,"\int \frac{\sec ^4(c+d x)}{(a+b \sin (c+d x))^{5/2}} \, dx","Int[Sec[c + d*x]^4/(a + b*Sin[c + d*x])^(5/2),x]","-\frac{\sec ^3(c+d x) \sqrt{a+b \sin (c+d x)} \left(b \left(29 a^2+3 b^2\right)-a \left(a^2+31 b^2\right) \sin (c+d x)\right)}{3 d \left(a^2-b^2\right)^3}+\frac{8 a b \sec ^3(c+d x)}{d \left(a^2-b^2\right)^2 \sqrt{a+b \sin (c+d x)}}+\frac{2 b \sec ^3(c+d x)}{3 d \left(a^2-b^2\right) (a+b \sin (c+d x))^{3/2}}-\frac{\sec (c+d x) \sqrt{a+b \sin (c+d x)} \left(b \left(-114 a^2 b^2+a^4-15 b^4\right)-4 a \left(-6 a^2 b^2+a^4-27 b^4\right) \sin (c+d x)\right)}{6 d \left(a^2-b^2\right)^4}+\frac{\left(-21 a^2 b^2+4 a^4-15 b^4\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{6 d \left(a^2-b^2\right)^3 \sqrt{a+b \sin (c+d x)}}-\frac{2 a \left(-6 a^2 b^2+a^4-27 b^4\right) \sqrt{a+b \sin (c+d x)} E\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{3 d \left(a^2-b^2\right)^4 \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}","-\frac{\sec ^3(c+d x) \sqrt{a+b \sin (c+d x)} \left(b \left(29 a^2+3 b^2\right)-a \left(a^2+31 b^2\right) \sin (c+d x)\right)}{3 d \left(a^2-b^2\right)^3}+\frac{8 a b \sec ^3(c+d x)}{d \left(a^2-b^2\right)^2 \sqrt{a+b \sin (c+d x)}}+\frac{2 b \sec ^3(c+d x)}{3 d \left(a^2-b^2\right) (a+b \sin (c+d x))^{3/2}}-\frac{\sec (c+d x) \sqrt{a+b \sin (c+d x)} \left(b \left(-114 a^2 b^2+a^4-15 b^4\right)-4 a \left(-6 a^2 b^2+a^4-27 b^4\right) \sin (c+d x)\right)}{6 d \left(a^2-b^2\right)^4}+\frac{\left(-21 a^2 b^2+4 a^4-15 b^4\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{6 d \left(a^2-b^2\right)^3 \sqrt{a+b \sin (c+d x)}}-\frac{2 a \left(-6 a^2 b^2+a^4-27 b^4\right) \sqrt{a+b \sin (c+d x)} E\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{3 d \left(a^2-b^2\right)^4 \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}",1,"(2*b*Sec[c + d*x]^3)/(3*(a^2 - b^2)*d*(a + b*Sin[c + d*x])^(3/2)) + (8*a*b*Sec[c + d*x]^3)/((a^2 - b^2)^2*d*Sqrt[a + b*Sin[c + d*x]]) - (2*a*(a^4 - 6*a^2*b^2 - 27*b^4)*EllipticE[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[a + b*Sin[c + d*x]])/(3*(a^2 - b^2)^4*d*Sqrt[(a + b*Sin[c + d*x])/(a + b)]) + ((4*a^4 - 21*a^2*b^2 - 15*b^4)*EllipticF[(c - Pi/2 + d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(6*(a^2 - b^2)^3*d*Sqrt[a + b*Sin[c + d*x]]) - (Sec[c + d*x]^3*Sqrt[a + b*Sin[c + d*x]]*(b*(29*a^2 + 3*b^2) - a*(a^2 + 31*b^2)*Sin[c + d*x]))/(3*(a^2 - b^2)^3*d) - (Sec[c + d*x]*Sqrt[a + b*Sin[c + d*x]]*(b*(a^4 - 114*a^2*b^2 - 15*b^4) - 4*a*(a^4 - 6*a^2*b^2 - 27*b^4)*Sin[c + d*x]))/(6*(a^2 - b^2)^4*d)","A",9,8,23,0.3478,1,"{2694, 2864, 2866, 2752, 2663, 2661, 2655, 2653}"
539,1,124,0,0.0921037,"\int (e \cos (c+d x))^{7/2} (a+b \sin (c+d x)) \, dx","Int[(e*Cos[c + d*x])^(7/2)*(a + b*Sin[c + d*x]),x]","\frac{10 a e^3 \sin (c+d x) \sqrt{e \cos (c+d x)}}{21 d}+\frac{10 a e^4 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d \sqrt{e \cos (c+d x)}}+\frac{2 a e \sin (c+d x) (e \cos (c+d x))^{5/2}}{7 d}-\frac{2 b (e \cos (c+d x))^{9/2}}{9 d e}","\frac{10 a e^3 \sin (c+d x) \sqrt{e \cos (c+d x)}}{21 d}+\frac{10 a e^4 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d \sqrt{e \cos (c+d x)}}+\frac{2 a e \sin (c+d x) (e \cos (c+d x))^{5/2}}{7 d}-\frac{2 b (e \cos (c+d x))^{9/2}}{9 d e}",1,"(-2*b*(e*Cos[c + d*x])^(9/2))/(9*d*e) + (10*a*e^4*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(21*d*Sqrt[e*Cos[c + d*x]]) + (10*a*e^3*Sqrt[e*Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (2*a*e*(e*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(7*d)","A",5,4,23,0.1739,1,"{2669, 2635, 2642, 2641}"
540,1,95,0,0.0730842,"\int (e \cos (c+d x))^{5/2} (a+b \sin (c+d x)) \, dx","Int[(e*Cos[c + d*x])^(5/2)*(a + b*Sin[c + d*x]),x]","\frac{6 a e^2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{5 d \sqrt{\cos (c+d x)}}+\frac{2 a e \sin (c+d x) (e \cos (c+d x))^{3/2}}{5 d}-\frac{2 b (e \cos (c+d x))^{7/2}}{7 d e}","\frac{6 a e^2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{5 d \sqrt{\cos (c+d x)}}+\frac{2 a e \sin (c+d x) (e \cos (c+d x))^{3/2}}{5 d}-\frac{2 b (e \cos (c+d x))^{7/2}}{7 d e}",1,"(-2*b*(e*Cos[c + d*x])^(7/2))/(7*d*e) + (6*a*e^2*Sqrt[e*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(5*d*Sqrt[Cos[c + d*x]]) + (2*a*e*(e*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*d)","A",4,4,23,0.1739,1,"{2669, 2635, 2640, 2639}"
541,1,95,0,0.071392,"\int (e \cos (c+d x))^{3/2} (a+b \sin (c+d x)) \, dx","Int[(e*Cos[c + d*x])^(3/2)*(a + b*Sin[c + d*x]),x]","\frac{2 a e^2 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d \sqrt{e \cos (c+d x)}}+\frac{2 a e \sin (c+d x) \sqrt{e \cos (c+d x)}}{3 d}-\frac{2 b (e \cos (c+d x))^{5/2}}{5 d e}","\frac{2 a e^2 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d \sqrt{e \cos (c+d x)}}+\frac{2 a e \sin (c+d x) \sqrt{e \cos (c+d x)}}{3 d}-\frac{2 b (e \cos (c+d x))^{5/2}}{5 d e}",1,"(-2*b*(e*Cos[c + d*x])^(5/2))/(5*d*e) + (2*a*e^2*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(3*d*Sqrt[e*Cos[c + d*x]]) + (2*a*e*Sqrt[e*Cos[c + d*x]]*Sin[c + d*x])/(3*d)","A",4,4,23,0.1739,1,"{2669, 2635, 2642, 2641}"
542,1,63,0,0.0508737,"\int \sqrt{e \cos (c+d x)} (a+b \sin (c+d x)) \, dx","Int[Sqrt[e*Cos[c + d*x]]*(a + b*Sin[c + d*x]),x]","\frac{2 a E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{d \sqrt{\cos (c+d x)}}-\frac{2 b (e \cos (c+d x))^{3/2}}{3 d e}","\frac{2 a E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{d \sqrt{\cos (c+d x)}}-\frac{2 b (e \cos (c+d x))^{3/2}}{3 d e}",1,"(-2*b*(e*Cos[c + d*x])^(3/2))/(3*d*e) + (2*a*Sqrt[e*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(d*Sqrt[Cos[c + d*x]])","A",3,3,23,0.1304,1,"{2669, 2640, 2639}"
543,1,61,0,0.0517239,"\int \frac{a+b \sin (c+d x)}{\sqrt{e \cos (c+d x)}} \, dx","Int[(a + b*Sin[c + d*x])/Sqrt[e*Cos[c + d*x]],x]","\frac{2 a \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d \sqrt{e \cos (c+d x)}}-\frac{2 b \sqrt{e \cos (c+d x)}}{d e}","\frac{2 a \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d \sqrt{e \cos (c+d x)}}-\frac{2 b \sqrt{e \cos (c+d x)}}{d e}",1,"(-2*b*Sqrt[e*Cos[c + d*x]])/(d*e) + (2*a*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(d*Sqrt[e*Cos[c + d*x]])","A",3,3,23,0.1304,1,"{2669, 2642, 2641}"
544,1,91,0,0.0736677,"\int \frac{a+b \sin (c+d x)}{(e \cos (c+d x))^{3/2}} \, dx","Int[(a + b*Sin[c + d*x])/(e*Cos[c + d*x])^(3/2),x]","-\frac{2 a E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{d e^2 \sqrt{\cos (c+d x)}}+\frac{2 a \sin (c+d x)}{d e \sqrt{e \cos (c+d x)}}+\frac{2 b}{d e \sqrt{e \cos (c+d x)}}","-\frac{2 a E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{d e^2 \sqrt{\cos (c+d x)}}+\frac{2 a \sin (c+d x)}{d e \sqrt{e \cos (c+d x)}}+\frac{2 b}{d e \sqrt{e \cos (c+d x)}}",1,"(2*b)/(d*e*Sqrt[e*Cos[c + d*x]]) - (2*a*Sqrt[e*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(d*e^2*Sqrt[Cos[c + d*x]]) + (2*a*Sin[c + d*x])/(d*e*Sqrt[e*Cos[c + d*x]])","A",4,4,23,0.1739,1,"{2669, 2636, 2640, 2639}"
545,1,97,0,0.0716748,"\int \frac{a+b \sin (c+d x)}{(e \cos (c+d x))^{5/2}} \, dx","Int[(a + b*Sin[c + d*x])/(e*Cos[c + d*x])^(5/2),x]","\frac{2 a \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d e^2 \sqrt{e \cos (c+d x)}}+\frac{2 a \sin (c+d x)}{3 d e (e \cos (c+d x))^{3/2}}+\frac{2 b}{3 d e (e \cos (c+d x))^{3/2}}","\frac{2 a \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d e^2 \sqrt{e \cos (c+d x)}}+\frac{2 a \sin (c+d x)}{3 d e (e \cos (c+d x))^{3/2}}+\frac{2 b}{3 d e (e \cos (c+d x))^{3/2}}",1,"(2*b)/(3*d*e*(e*Cos[c + d*x])^(3/2)) + (2*a*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(3*d*e^2*Sqrt[e*Cos[c + d*x]]) + (2*a*Sin[c + d*x])/(3*d*e*(e*Cos[c + d*x])^(3/2))","A",4,4,23,0.1739,1,"{2669, 2636, 2642, 2641}"
546,1,126,0,0.0914813,"\int \frac{a+b \sin (c+d x)}{(e \cos (c+d x))^{7/2}} \, dx","Int[(a + b*Sin[c + d*x])/(e*Cos[c + d*x])^(7/2),x]","\frac{6 a \sin (c+d x)}{5 d e^3 \sqrt{e \cos (c+d x)}}-\frac{6 a E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{5 d e^4 \sqrt{\cos (c+d x)}}+\frac{2 a \sin (c+d x)}{5 d e (e \cos (c+d x))^{5/2}}+\frac{2 b}{5 d e (e \cos (c+d x))^{5/2}}","\frac{6 a \sin (c+d x)}{5 d e^3 \sqrt{e \cos (c+d x)}}-\frac{6 a E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{5 d e^4 \sqrt{\cos (c+d x)}}+\frac{2 a \sin (c+d x)}{5 d e (e \cos (c+d x))^{5/2}}+\frac{2 b}{5 d e (e \cos (c+d x))^{5/2}}",1,"(2*b)/(5*d*e*(e*Cos[c + d*x])^(5/2)) - (6*a*Sqrt[e*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(5*d*e^4*Sqrt[Cos[c + d*x]]) + (2*a*Sin[c + d*x])/(5*d*e*(e*Cos[c + d*x])^(5/2)) + (6*a*Sin[c + d*x])/(5*d*e^3*Sqrt[e*Cos[c + d*x]])","A",5,4,23,0.1739,1,"{2669, 2636, 2640, 2639}"
547,1,188,0,0.1912991,"\int (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))^2 \, dx","Int[(e*Cos[c + d*x])^(7/2)*(a + b*Sin[c + d*x])^2,x]","\frac{10 e^3 \left(11 a^2+2 b^2\right) \sin (c+d x) \sqrt{e \cos (c+d x)}}{231 d}+\frac{10 e^4 \left(11 a^2+2 b^2\right) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{231 d \sqrt{e \cos (c+d x)}}+\frac{2 e \left(11 a^2+2 b^2\right) \sin (c+d x) (e \cos (c+d x))^{5/2}}{77 d}-\frac{26 a b (e \cos (c+d x))^{9/2}}{99 d e}-\frac{2 b (e \cos (c+d x))^{9/2} (a+b \sin (c+d x))}{11 d e}","\frac{10 e^3 \left(11 a^2+2 b^2\right) \sin (c+d x) \sqrt{e \cos (c+d x)}}{231 d}+\frac{10 e^4 \left(11 a^2+2 b^2\right) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{231 d \sqrt{e \cos (c+d x)}}+\frac{2 e \left(11 a^2+2 b^2\right) \sin (c+d x) (e \cos (c+d x))^{5/2}}{77 d}-\frac{26 a b (e \cos (c+d x))^{9/2}}{99 d e}-\frac{2 b (e \cos (c+d x))^{9/2} (a+b \sin (c+d x))}{11 d e}",1,"(-26*a*b*(e*Cos[c + d*x])^(9/2))/(99*d*e) + (10*(11*a^2 + 2*b^2)*e^4*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(231*d*Sqrt[e*Cos[c + d*x]]) + (10*(11*a^2 + 2*b^2)*e^3*Sqrt[e*Cos[c + d*x]]*Sin[c + d*x])/(231*d) + (2*(11*a^2 + 2*b^2)*e*(e*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(77*d) - (2*b*(e*Cos[c + d*x])^(9/2)*(a + b*Sin[c + d*x]))/(11*d*e)","A",6,5,25,0.2000,1,"{2692, 2669, 2635, 2642, 2641}"
548,1,149,0,0.1668293,"\int (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))^2 \, dx","Int[(e*Cos[c + d*x])^(5/2)*(a + b*Sin[c + d*x])^2,x]","\frac{2 e^2 \left(9 a^2+2 b^2\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{15 d \sqrt{\cos (c+d x)}}+\frac{2 e \left(9 a^2+2 b^2\right) \sin (c+d x) (e \cos (c+d x))^{3/2}}{45 d}-\frac{22 a b (e \cos (c+d x))^{7/2}}{63 d e}-\frac{2 b (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))}{9 d e}","\frac{2 e^2 \left(9 a^2+2 b^2\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{15 d \sqrt{\cos (c+d x)}}+\frac{2 e \left(9 a^2+2 b^2\right) \sin (c+d x) (e \cos (c+d x))^{3/2}}{45 d}-\frac{22 a b (e \cos (c+d x))^{7/2}}{63 d e}-\frac{2 b (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))}{9 d e}",1,"(-22*a*b*(e*Cos[c + d*x])^(7/2))/(63*d*e) + (2*(9*a^2 + 2*b^2)*e^2*Sqrt[e*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(15*d*Sqrt[Cos[c + d*x]]) + (2*(9*a^2 + 2*b^2)*e*(e*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(45*d) - (2*b*(e*Cos[c + d*x])^(7/2)*(a + b*Sin[c + d*x]))/(9*d*e)","A",5,5,25,0.2000,1,"{2692, 2669, 2635, 2640, 2639}"
549,1,149,0,0.1640137,"\int (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^2 \, dx","Int[(e*Cos[c + d*x])^(3/2)*(a + b*Sin[c + d*x])^2,x]","\frac{2 e^2 \left(7 a^2+2 b^2\right) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d \sqrt{e \cos (c+d x)}}+\frac{2 e \left(7 a^2+2 b^2\right) \sin (c+d x) \sqrt{e \cos (c+d x)}}{21 d}-\frac{18 a b (e \cos (c+d x))^{5/2}}{35 d e}-\frac{2 b (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))}{7 d e}","\frac{2 e^2 \left(7 a^2+2 b^2\right) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d \sqrt{e \cos (c+d x)}}+\frac{2 e \left(7 a^2+2 b^2\right) \sin (c+d x) \sqrt{e \cos (c+d x)}}{21 d}-\frac{18 a b (e \cos (c+d x))^{5/2}}{35 d e}-\frac{2 b (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))}{7 d e}",1,"(-18*a*b*(e*Cos[c + d*x])^(5/2))/(35*d*e) + (2*(7*a^2 + 2*b^2)*e^2*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(21*d*Sqrt[e*Cos[c + d*x]]) + (2*(7*a^2 + 2*b^2)*e*Sqrt[e*Cos[c + d*x]]*Sin[c + d*x])/(21*d) - (2*b*(e*Cos[c + d*x])^(5/2)*(a + b*Sin[c + d*x]))/(7*d*e)","A",5,5,25,0.2000,1,"{2692, 2669, 2635, 2642, 2641}"
550,1,109,0,0.1296012,"\int \sqrt{e \cos (c+d x)} (a+b \sin (c+d x))^2 \, dx","Int[Sqrt[e*Cos[c + d*x]]*(a + b*Sin[c + d*x])^2,x]","\frac{2 \left(5 a^2+2 b^2\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{5 d \sqrt{\cos (c+d x)}}-\frac{14 a b (e \cos (c+d x))^{3/2}}{15 d e}-\frac{2 b (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))}{5 d e}","\frac{2 \left(5 a^2+2 b^2\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{5 d \sqrt{\cos (c+d x)}}-\frac{14 a b (e \cos (c+d x))^{3/2}}{15 d e}-\frac{2 b (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))}{5 d e}",1,"(-14*a*b*(e*Cos[c + d*x])^(3/2))/(15*d*e) + (2*(5*a^2 + 2*b^2)*Sqrt[e*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(5*d*Sqrt[Cos[c + d*x]]) - (2*b*(e*Cos[c + d*x])^(3/2)*(a + b*Sin[c + d*x]))/(5*d*e)","A",4,4,25,0.1600,1,"{2692, 2669, 2640, 2639}"
551,1,109,0,0.1288764,"\int \frac{(a+b \sin (c+d x))^2}{\sqrt{e \cos (c+d x)}} \, dx","Int[(a + b*Sin[c + d*x])^2/Sqrt[e*Cos[c + d*x]],x]","\frac{2 \left(3 a^2+2 b^2\right) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d \sqrt{e \cos (c+d x)}}-\frac{10 a b \sqrt{e \cos (c+d x)}}{3 d e}-\frac{2 b \sqrt{e \cos (c+d x)} (a+b \sin (c+d x))}{3 d e}","\frac{2 \left(3 a^2+2 b^2\right) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d \sqrt{e \cos (c+d x)}}-\frac{10 a b \sqrt{e \cos (c+d x)}}{3 d e}-\frac{2 b \sqrt{e \cos (c+d x)} (a+b \sin (c+d x))}{3 d e}",1,"(-10*a*b*Sqrt[e*Cos[c + d*x]])/(3*d*e) + (2*(3*a^2 + 2*b^2)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(3*d*Sqrt[e*Cos[c + d*x]]) - (2*b*Sqrt[e*Cos[c + d*x]]*(a + b*Sin[c + d*x]))/(3*d*e)","A",4,4,25,0.1600,1,"{2692, 2669, 2642, 2641}"
552,1,113,0,0.1352374,"\int \frac{(a+b \sin (c+d x))^2}{(e \cos (c+d x))^{3/2}} \, dx","Int[(a + b*Sin[c + d*x])^2/(e*Cos[c + d*x])^(3/2),x]","-\frac{2 \left(a^2+2 b^2\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{d e^2 \sqrt{\cos (c+d x)}}+\frac{2 a b (e \cos (c+d x))^{3/2}}{d e^3}+\frac{2 (a \sin (c+d x)+b) (a+b \sin (c+d x))}{d e \sqrt{e \cos (c+d x)}}","-\frac{2 \left(a^2+2 b^2\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{d e^2 \sqrt{\cos (c+d x)}}+\frac{2 a b (e \cos (c+d x))^{3/2}}{d e^3}+\frac{2 (a \sin (c+d x)+b) (a+b \sin (c+d x))}{d e \sqrt{e \cos (c+d x)}}",1,"(2*a*b*(e*Cos[c + d*x])^(3/2))/(d*e^3) - (2*(a^2 + 2*b^2)*Sqrt[e*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(d*e^2*Sqrt[Cos[c + d*x]]) + (2*(b + a*Sin[c + d*x])*(a + b*Sin[c + d*x]))/(d*e*Sqrt[e*Cos[c + d*x]])","A",4,4,25,0.1600,1,"{2691, 2669, 2640, 2639}"
553,1,119,0,0.1393952,"\int \frac{(a+b \sin (c+d x))^2}{(e \cos (c+d x))^{5/2}} \, dx","Int[(a + b*Sin[c + d*x])^2/(e*Cos[c + d*x])^(5/2),x]","\frac{2 \left(a^2-2 b^2\right) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d e^2 \sqrt{e \cos (c+d x)}}+\frac{2 a b \sqrt{e \cos (c+d x)}}{3 d e^3}+\frac{2 (a \sin (c+d x)+b) (a+b \sin (c+d x))}{3 d e (e \cos (c+d x))^{3/2}}","\frac{2 \left(a^2-2 b^2\right) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d e^2 \sqrt{e \cos (c+d x)}}+\frac{2 a b \sqrt{e \cos (c+d x)}}{3 d e^3}+\frac{2 (a \sin (c+d x)+b) (a+b \sin (c+d x))}{3 d e (e \cos (c+d x))^{3/2}}",1,"(2*a*b*Sqrt[e*Cos[c + d*x]])/(3*d*e^3) + (2*(a^2 - 2*b^2)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(3*d*e^2*Sqrt[e*Cos[c + d*x]]) + (2*(b + a*Sin[c + d*x])*(a + b*Sin[c + d*x]))/(3*d*e*(e*Cos[c + d*x])^(3/2))","A",4,4,25,0.1600,1,"{2691, 2669, 2642, 2641}"
554,1,160,0,0.1723506,"\int \frac{(a+b \sin (c+d x))^2}{(e \cos (c+d x))^{7/2}} \, dx","Int[(a + b*Sin[c + d*x])^2/(e*Cos[c + d*x])^(7/2),x]","\frac{2 \left(3 a^2-2 b^2\right) \sin (c+d x)}{5 d e^3 \sqrt{e \cos (c+d x)}}-\frac{2 \left(3 a^2-2 b^2\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{5 d e^4 \sqrt{\cos (c+d x)}}+\frac{2 a b}{5 d e^3 \sqrt{e \cos (c+d x)}}+\frac{2 (a \sin (c+d x)+b) (a+b \sin (c+d x))}{5 d e (e \cos (c+d x))^{5/2}}","\frac{2 \left(3 a^2-2 b^2\right) \sin (c+d x)}{5 d e^3 \sqrt{e \cos (c+d x)}}-\frac{2 \left(3 a^2-2 b^2\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{5 d e^4 \sqrt{\cos (c+d x)}}+\frac{2 a b}{5 d e^3 \sqrt{e \cos (c+d x)}}+\frac{2 (a \sin (c+d x)+b) (a+b \sin (c+d x))}{5 d e (e \cos (c+d x))^{5/2}}",1,"(2*a*b)/(5*d*e^3*Sqrt[e*Cos[c + d*x]]) - (2*(3*a^2 - 2*b^2)*Sqrt[e*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(5*d*e^4*Sqrt[Cos[c + d*x]]) + (2*(3*a^2 - 2*b^2)*Sin[c + d*x])/(5*d*e^3*Sqrt[e*Cos[c + d*x]]) + (2*(b + a*Sin[c + d*x])*(a + b*Sin[c + d*x]))/(5*d*e*(e*Cos[c + d*x])^(5/2))","A",5,5,25,0.2000,1,"{2691, 2669, 2636, 2640, 2639}"
555,1,237,0,0.3079949,"\int (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))^3 \, dx","Int[(e*Cos[c + d*x])^(7/2)*(a + b*Sin[c + d*x])^3,x]","\frac{10 a e^3 \left(11 a^2+6 b^2\right) \sin (c+d x) \sqrt{e \cos (c+d x)}}{231 d}+\frac{10 a e^4 \left(11 a^2+6 b^2\right) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{231 d \sqrt{e \cos (c+d x)}}-\frac{2 b \left(177 a^2+44 b^2\right) (e \cos (c+d x))^{9/2}}{1287 d e}+\frac{2 a e \left(11 a^2+6 b^2\right) \sin (c+d x) (e \cos (c+d x))^{5/2}}{77 d}-\frac{2 b (e \cos (c+d x))^{9/2} (a+b \sin (c+d x))^2}{13 d e}-\frac{34 a b (e \cos (c+d x))^{9/2} (a+b \sin (c+d x))}{143 d e}","\frac{10 a e^3 \left(11 a^2+6 b^2\right) \sin (c+d x) \sqrt{e \cos (c+d x)}}{231 d}+\frac{10 a e^4 \left(11 a^2+6 b^2\right) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{231 d \sqrt{e \cos (c+d x)}}-\frac{2 b \left(177 a^2+44 b^2\right) (e \cos (c+d x))^{9/2}}{1287 d e}+\frac{2 a e \left(11 a^2+6 b^2\right) \sin (c+d x) (e \cos (c+d x))^{5/2}}{77 d}-\frac{2 b (e \cos (c+d x))^{9/2} (a+b \sin (c+d x))^2}{13 d e}-\frac{34 a b (e \cos (c+d x))^{9/2} (a+b \sin (c+d x))}{143 d e}",1,"(-2*b*(177*a^2 + 44*b^2)*(e*Cos[c + d*x])^(9/2))/(1287*d*e) + (10*a*(11*a^2 + 6*b^2)*e^4*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(231*d*Sqrt[e*Cos[c + d*x]]) + (10*a*(11*a^2 + 6*b^2)*e^3*Sqrt[e*Cos[c + d*x]]*Sin[c + d*x])/(231*d) + (2*a*(11*a^2 + 6*b^2)*e*(e*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(77*d) - (34*a*b*(e*Cos[c + d*x])^(9/2)*(a + b*Sin[c + d*x]))/(143*d*e) - (2*b*(e*Cos[c + d*x])^(9/2)*(a + b*Sin[c + d*x])^2)/(13*d*e)","A",7,6,25,0.2400,1,"{2692, 2862, 2669, 2635, 2642, 2641}"
556,1,197,0,0.286019,"\int (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))^3 \, dx","Int[(e*Cos[c + d*x])^(5/2)*(a + b*Sin[c + d*x])^3,x]","\frac{2 a e^2 \left(3 a^2+2 b^2\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{5 d \sqrt{\cos (c+d x)}}-\frac{2 b \left(43 a^2+12 b^2\right) (e \cos (c+d x))^{7/2}}{231 d e}+\frac{2 a e \left(3 a^2+2 b^2\right) \sin (c+d x) (e \cos (c+d x))^{3/2}}{15 d}-\frac{2 b (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))^2}{11 d e}-\frac{10 a b (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))}{33 d e}","\frac{2 a e^2 \left(3 a^2+2 b^2\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{5 d \sqrt{\cos (c+d x)}}-\frac{2 b \left(43 a^2+12 b^2\right) (e \cos (c+d x))^{7/2}}{231 d e}+\frac{2 a e \left(3 a^2+2 b^2\right) \sin (c+d x) (e \cos (c+d x))^{3/2}}{15 d}-\frac{2 b (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))^2}{11 d e}-\frac{10 a b (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))}{33 d e}",1,"(-2*b*(43*a^2 + 12*b^2)*(e*Cos[c + d*x])^(7/2))/(231*d*e) + (2*a*(3*a^2 + 2*b^2)*e^2*Sqrt[e*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(5*d*Sqrt[Cos[c + d*x]]) + (2*a*(3*a^2 + 2*b^2)*e*(e*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(15*d) - (10*a*b*(e*Cos[c + d*x])^(7/2)*(a + b*Sin[c + d*x]))/(33*d*e) - (2*b*(e*Cos[c + d*x])^(7/2)*(a + b*Sin[c + d*x])^2)/(11*d*e)","A",6,6,25,0.2400,1,"{2692, 2862, 2669, 2635, 2640, 2639}"
557,1,197,0,0.288608,"\int (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^3 \, dx","Int[(e*Cos[c + d*x])^(3/2)*(a + b*Sin[c + d*x])^3,x]","\frac{2 a e^2 \left(7 a^2+6 b^2\right) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d \sqrt{e \cos (c+d x)}}-\frac{2 b \left(89 a^2+28 b^2\right) (e \cos (c+d x))^{5/2}}{315 d e}+\frac{2 a e \left(7 a^2+6 b^2\right) \sin (c+d x) \sqrt{e \cos (c+d x)}}{21 d}-\frac{2 b (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))^2}{9 d e}-\frac{26 a b (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))}{63 d e}","\frac{2 a e^2 \left(7 a^2+6 b^2\right) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d \sqrt{e \cos (c+d x)}}-\frac{2 b \left(89 a^2+28 b^2\right) (e \cos (c+d x))^{5/2}}{315 d e}+\frac{2 a e \left(7 a^2+6 b^2\right) \sin (c+d x) \sqrt{e \cos (c+d x)}}{21 d}-\frac{2 b (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))^2}{9 d e}-\frac{26 a b (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))}{63 d e}",1,"(-2*b*(89*a^2 + 28*b^2)*(e*Cos[c + d*x])^(5/2))/(315*d*e) + (2*a*(7*a^2 + 6*b^2)*e^2*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(21*d*Sqrt[e*Cos[c + d*x]]) + (2*a*(7*a^2 + 6*b^2)*e*Sqrt[e*Cos[c + d*x]]*Sin[c + d*x])/(21*d) - (26*a*b*(e*Cos[c + d*x])^(5/2)*(a + b*Sin[c + d*x]))/(63*d*e) - (2*b*(e*Cos[c + d*x])^(5/2)*(a + b*Sin[c + d*x])^2)/(9*d*e)","A",6,6,25,0.2400,1,"{2692, 2862, 2669, 2635, 2642, 2641}"
558,1,156,0,0.2407111,"\int \sqrt{e \cos (c+d x)} (a+b \sin (c+d x))^3 \, dx","Int[Sqrt[e*Cos[c + d*x]]*(a + b*Sin[c + d*x])^3,x]","-\frac{2 b \left(57 a^2+20 b^2\right) (e \cos (c+d x))^{3/2}}{105 d e}+\frac{2 a \left(5 a^2+6 b^2\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{5 d \sqrt{\cos (c+d x)}}-\frac{2 b (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^2}{7 d e}-\frac{22 a b (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))}{35 d e}","-\frac{2 b \left(57 a^2+20 b^2\right) (e \cos (c+d x))^{3/2}}{105 d e}+\frac{2 a \left(5 a^2+6 b^2\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{5 d \sqrt{\cos (c+d x)}}-\frac{2 b (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^2}{7 d e}-\frac{22 a b (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))}{35 d e}",1,"(-2*b*(57*a^2 + 20*b^2)*(e*Cos[c + d*x])^(3/2))/(105*d*e) + (2*a*(5*a^2 + 6*b^2)*Sqrt[e*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(5*d*Sqrt[Cos[c + d*x]]) - (22*a*b*(e*Cos[c + d*x])^(3/2)*(a + b*Sin[c + d*x]))/(35*d*e) - (2*b*(e*Cos[c + d*x])^(3/2)*(a + b*Sin[c + d*x])^2)/(7*d*e)","A",5,5,25,0.2000,1,"{2692, 2862, 2669, 2640, 2639}"
559,1,152,0,0.2404201,"\int \frac{(a+b \sin (c+d x))^3}{\sqrt{e \cos (c+d x)}} \, dx","Int[(a + b*Sin[c + d*x])^3/Sqrt[e*Cos[c + d*x]],x]","-\frac{2 b \left(11 a^2+4 b^2\right) \sqrt{e \cos (c+d x)}}{5 d e}+\frac{2 a \left(a^2+2 b^2\right) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d \sqrt{e \cos (c+d x)}}-\frac{2 b \sqrt{e \cos (c+d x)} (a+b \sin (c+d x))^2}{5 d e}-\frac{6 a b \sqrt{e \cos (c+d x)} (a+b \sin (c+d x))}{5 d e}","-\frac{2 b \left(11 a^2+4 b^2\right) \sqrt{e \cos (c+d x)}}{5 d e}+\frac{2 a \left(a^2+2 b^2\right) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d \sqrt{e \cos (c+d x)}}-\frac{2 b \sqrt{e \cos (c+d x)} (a+b \sin (c+d x))^2}{5 d e}-\frac{6 a b \sqrt{e \cos (c+d x)} (a+b \sin (c+d x))}{5 d e}",1,"(-2*b*(11*a^2 + 4*b^2)*Sqrt[e*Cos[c + d*x]])/(5*d*e) + (2*a*(a^2 + 2*b^2)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(d*Sqrt[e*Cos[c + d*x]]) - (6*a*b*Sqrt[e*Cos[c + d*x]]*(a + b*Sin[c + d*x]))/(5*d*e) - (2*b*Sqrt[e*Cos[c + d*x]]*(a + b*Sin[c + d*x])^2)/(5*d*e)","A",5,5,25,0.2000,1,"{2692, 2862, 2669, 2642, 2641}"
560,1,160,0,0.2410242,"\int \frac{(a+b \sin (c+d x))^3}{(e \cos (c+d x))^{3/2}} \, dx","Int[(a + b*Sin[c + d*x])^3/(e*Cos[c + d*x])^(3/2),x]","\frac{2 b \left(3 a^2+4 b^2\right) (e \cos (c+d x))^{3/2}}{3 d e^3}-\frac{2 a \left(a^2+6 b^2\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{d e^2 \sqrt{\cos (c+d x)}}+\frac{2 a b (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))}{d e^3}+\frac{2 (a \sin (c+d x)+b) (a+b \sin (c+d x))^2}{d e \sqrt{e \cos (c+d x)}}","\frac{2 b \left(3 a^2+4 b^2\right) (e \cos (c+d x))^{3/2}}{3 d e^3}-\frac{2 a \left(a^2+6 b^2\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{d e^2 \sqrt{\cos (c+d x)}}+\frac{2 a b (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))}{d e^3}+\frac{2 (a \sin (c+d x)+b) (a+b \sin (c+d x))^2}{d e \sqrt{e \cos (c+d x)}}",1,"(2*b*(3*a^2 + 4*b^2)*(e*Cos[c + d*x])^(3/2))/(3*d*e^3) - (2*a*(a^2 + 6*b^2)*Sqrt[e*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(d*e^2*Sqrt[Cos[c + d*x]]) + (2*a*b*(e*Cos[c + d*x])^(3/2)*(a + b*Sin[c + d*x]))/(d*e^3) + (2*(b + a*Sin[c + d*x])*(a + b*Sin[c + d*x])^2)/(d*e*Sqrt[e*Cos[c + d*x]])","A",5,5,25,0.2000,1,"{2691, 2862, 2669, 2640, 2639}"
561,1,164,0,0.2493314,"\int \frac{(a+b \sin (c+d x))^3}{(e \cos (c+d x))^{5/2}} \, dx","Int[(a + b*Sin[c + d*x])^3/(e*Cos[c + d*x])^(5/2),x]","\frac{2 b \left(a^2+4 b^2\right) \sqrt{e \cos (c+d x)}}{3 d e^3}+\frac{2 a \left(a^2-6 b^2\right) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d e^2 \sqrt{e \cos (c+d x)}}+\frac{2 a b \sqrt{e \cos (c+d x)} (a+b \sin (c+d x))}{3 d e^3}+\frac{2 (a \sin (c+d x)+b) (a+b \sin (c+d x))^2}{3 d e (e \cos (c+d x))^{3/2}}","\frac{2 b \left(a^2+4 b^2\right) \sqrt{e \cos (c+d x)}}{3 d e^3}+\frac{2 a \left(a^2-6 b^2\right) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d e^2 \sqrt{e \cos (c+d x)}}+\frac{2 a b \sqrt{e \cos (c+d x)} (a+b \sin (c+d x))}{3 d e^3}+\frac{2 (a \sin (c+d x)+b) (a+b \sin (c+d x))^2}{3 d e (e \cos (c+d x))^{3/2}}",1,"(2*b*(a^2 + 4*b^2)*Sqrt[e*Cos[c + d*x]])/(3*d*e^3) + (2*a*(a^2 - 6*b^2)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(3*d*e^2*Sqrt[e*Cos[c + d*x]]) + (2*a*b*Sqrt[e*Cos[c + d*x]]*(a + b*Sin[c + d*x]))/(3*d*e^3) + (2*(b + a*Sin[c + d*x])*(a + b*Sin[c + d*x])^2)/(3*d*e*(e*Cos[c + d*x])^(3/2))","A",5,5,25,0.2000,1,"{2691, 2862, 2669, 2642, 2641}"
562,1,187,0,0.2640791,"\int \frac{(a+b \sin (c+d x))^3}{(e \cos (c+d x))^{7/2}} \, dx","Int[(a + b*Sin[c + d*x])^3/(e*Cos[c + d*x])^(7/2),x]","\frac{2 b \left(3 a^2-4 b^2\right) (e \cos (c+d x))^{3/2}}{5 d e^5}-\frac{2 \left(a b-\left(3 a^2-4 b^2\right) \sin (c+d x)\right) (a+b \sin (c+d x))}{5 d e^3 \sqrt{e \cos (c+d x)}}-\frac{6 a \left(a^2-2 b^2\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{5 d e^4 \sqrt{\cos (c+d x)}}+\frac{2 (a \sin (c+d x)+b) (a+b \sin (c+d x))^2}{5 d e (e \cos (c+d x))^{5/2}}","\frac{2 b \left(3 a^2-4 b^2\right) (e \cos (c+d x))^{3/2}}{5 d e^5}-\frac{2 \left(a b-\left(3 a^2-4 b^2\right) \sin (c+d x)\right) (a+b \sin (c+d x))}{5 d e^3 \sqrt{e \cos (c+d x)}}-\frac{6 a \left(a^2-2 b^2\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{5 d e^4 \sqrt{\cos (c+d x)}}+\frac{2 (a \sin (c+d x)+b) (a+b \sin (c+d x))^2}{5 d e (e \cos (c+d x))^{5/2}}",1,"(2*b*(3*a^2 - 4*b^2)*(e*Cos[c + d*x])^(3/2))/(5*d*e^5) - (6*a*(a^2 - 2*b^2)*Sqrt[e*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(5*d*e^4*Sqrt[Cos[c + d*x]]) + (2*(b + a*Sin[c + d*x])*(a + b*Sin[c + d*x])^2)/(5*d*e*(e*Cos[c + d*x])^(5/2)) - (2*(a + b*Sin[c + d*x])*(a*b - (3*a^2 - 4*b^2)*Sin[c + d*x]))/(5*d*e^3*Sqrt[e*Cos[c + d*x]])","A",5,5,25,0.2000,1,"{2691, 2861, 2669, 2640, 2639}"
563,1,188,0,0.2673736,"\int \frac{(a+b \sin (c+d x))^3}{(e \cos (c+d x))^{9/2}} \, dx","Int[(a + b*Sin[c + d*x])^3/(e*Cos[c + d*x])^(9/2),x]","\frac{2 b \left(5 a^2-4 b^2\right) \sqrt{e \cos (c+d x)}}{21 d e^5}+\frac{2 \left(\left(5 a^2-4 b^2\right) \sin (c+d x)+a b\right) (a+b \sin (c+d x))}{21 d e^3 (e \cos (c+d x))^{3/2}}+\frac{2 a \left(5 a^2-6 b^2\right) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d e^4 \sqrt{e \cos (c+d x)}}+\frac{2 (a \sin (c+d x)+b) (a+b \sin (c+d x))^2}{7 d e (e \cos (c+d x))^{7/2}}","\frac{2 b \left(5 a^2-4 b^2\right) \sqrt{e \cos (c+d x)}}{21 d e^5}+\frac{2 \left(\left(5 a^2-4 b^2\right) \sin (c+d x)+a b\right) (a+b \sin (c+d x))}{21 d e^3 (e \cos (c+d x))^{3/2}}+\frac{2 a \left(5 a^2-6 b^2\right) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d e^4 \sqrt{e \cos (c+d x)}}+\frac{2 (a \sin (c+d x)+b) (a+b \sin (c+d x))^2}{7 d e (e \cos (c+d x))^{7/2}}",1,"(2*b*(5*a^2 - 4*b^2)*Sqrt[e*Cos[c + d*x]])/(21*d*e^5) + (2*a*(5*a^2 - 6*b^2)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(21*d*e^4*Sqrt[e*Cos[c + d*x]]) + (2*(b + a*Sin[c + d*x])*(a + b*Sin[c + d*x])^2)/(7*d*e*(e*Cos[c + d*x])^(7/2)) + (2*(a + b*Sin[c + d*x])*(a*b + (5*a^2 - 4*b^2)*Sin[c + d*x]))/(21*d*e^3*(e*Cos[c + d*x])^(3/2))","A",5,5,25,0.2000,1,"{2691, 2861, 2669, 2642, 2641}"
564,1,305,0,0.551403,"\int (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))^4 \, dx","Int[(e*Cos[c + d*x])^(7/2)*(a + b*Sin[c + d*x])^4,x]","\frac{2 e^3 \left(60 a^2 b^2+55 a^4+4 b^4\right) \sin (c+d x) \sqrt{e \cos (c+d x)}}{231 d}+\frac{2 e^4 \left(60 a^2 b^2+55 a^4+4 b^4\right) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{231 d \sqrt{e \cos (c+d x)}}-\frac{34 a b \left(53 a^2+38 b^2\right) (e \cos (c+d x))^{9/2}}{6435 d e}-\frac{2 b \left(93 a^2+26 b^2\right) (e \cos (c+d x))^{9/2} (a+b \sin (c+d x))}{715 d e}+\frac{2 e \left(60 a^2 b^2+55 a^4+4 b^4\right) \sin (c+d x) (e \cos (c+d x))^{5/2}}{385 d}-\frac{2 b (e \cos (c+d x))^{9/2} (a+b \sin (c+d x))^3}{15 d e}-\frac{14 a b (e \cos (c+d x))^{9/2} (a+b \sin (c+d x))^2}{65 d e}","\frac{2 e^3 \left(60 a^2 b^2+55 a^4+4 b^4\right) \sin (c+d x) \sqrt{e \cos (c+d x)}}{231 d}+\frac{2 e^4 \left(60 a^2 b^2+55 a^4+4 b^4\right) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{231 d \sqrt{e \cos (c+d x)}}-\frac{34 a b \left(53 a^2+38 b^2\right) (e \cos (c+d x))^{9/2}}{6435 d e}-\frac{2 b \left(93 a^2+26 b^2\right) (e \cos (c+d x))^{9/2} (a+b \sin (c+d x))}{715 d e}+\frac{2 e \left(60 a^2 b^2+55 a^4+4 b^4\right) \sin (c+d x) (e \cos (c+d x))^{5/2}}{385 d}-\frac{2 b (e \cos (c+d x))^{9/2} (a+b \sin (c+d x))^3}{15 d e}-\frac{14 a b (e \cos (c+d x))^{9/2} (a+b \sin (c+d x))^2}{65 d e}",1,"(-34*a*b*(53*a^2 + 38*b^2)*(e*Cos[c + d*x])^(9/2))/(6435*d*e) + (2*(55*a^4 + 60*a^2*b^2 + 4*b^4)*e^4*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(231*d*Sqrt[e*Cos[c + d*x]]) + (2*(55*a^4 + 60*a^2*b^2 + 4*b^4)*e^3*Sqrt[e*Cos[c + d*x]]*Sin[c + d*x])/(231*d) + (2*(55*a^4 + 60*a^2*b^2 + 4*b^4)*e*(e*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(385*d) - (2*b*(93*a^2 + 26*b^2)*(e*Cos[c + d*x])^(9/2)*(a + b*Sin[c + d*x]))/(715*d*e) - (14*a*b*(e*Cos[c + d*x])^(9/2)*(a + b*Sin[c + d*x])^2)/(65*d*e) - (2*b*(e*Cos[c + d*x])^(9/2)*(a + b*Sin[c + d*x])^3)/(15*d*e)","A",8,6,25,0.2400,1,"{2692, 2862, 2669, 2635, 2642, 2641}"
565,1,258,0,0.5082047,"\int (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))^4 \, dx","Int[(e*Cos[c + d*x])^(5/2)*(a + b*Sin[c + d*x])^4,x]","\frac{2 e^2 \left(52 a^2 b^2+39 a^4+4 b^4\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{65 d \sqrt{\cos (c+d x)}}-\frac{10 a b \left(115 a^2+94 b^2\right) (e \cos (c+d x))^{7/2}}{3003 d e}-\frac{2 b \left(73 a^2+22 b^2\right) (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))}{429 d e}+\frac{2 e \left(52 a^2 b^2+39 a^4+4 b^4\right) \sin (c+d x) (e \cos (c+d x))^{3/2}}{195 d}-\frac{2 b (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))^3}{13 d e}-\frac{38 a b (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))^2}{143 d e}","\frac{2 e^2 \left(52 a^2 b^2+39 a^4+4 b^4\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{65 d \sqrt{\cos (c+d x)}}-\frac{10 a b \left(115 a^2+94 b^2\right) (e \cos (c+d x))^{7/2}}{3003 d e}-\frac{2 b \left(73 a^2+22 b^2\right) (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))}{429 d e}+\frac{2 e \left(52 a^2 b^2+39 a^4+4 b^4\right) \sin (c+d x) (e \cos (c+d x))^{3/2}}{195 d}-\frac{2 b (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))^3}{13 d e}-\frac{38 a b (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))^2}{143 d e}",1,"(-10*a*b*(115*a^2 + 94*b^2)*(e*Cos[c + d*x])^(7/2))/(3003*d*e) + (2*(39*a^4 + 52*a^2*b^2 + 4*b^4)*e^2*Sqrt[e*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(65*d*Sqrt[Cos[c + d*x]]) + (2*(39*a^4 + 52*a^2*b^2 + 4*b^4)*e*(e*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(195*d) - (2*b*(73*a^2 + 22*b^2)*(e*Cos[c + d*x])^(7/2)*(a + b*Sin[c + d*x]))/(429*d*e) - (38*a*b*(e*Cos[c + d*x])^(7/2)*(a + b*Sin[c + d*x])^2)/(143*d*e) - (2*b*(e*Cos[c + d*x])^(7/2)*(a + b*Sin[c + d*x])^3)/(13*d*e)","A",7,6,25,0.2400,1,"{2692, 2862, 2669, 2635, 2640, 2639}"
566,1,258,0,0.5094083,"\int (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^4 \, dx","Int[(e*Cos[c + d*x])^(3/2)*(a + b*Sin[c + d*x])^4,x]","\frac{2 e^2 \left(132 a^2 b^2+77 a^4+12 b^4\right) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{231 d \sqrt{e \cos (c+d x)}}-\frac{26 a b \left(79 a^2+74 b^2\right) (e \cos (c+d x))^{5/2}}{3465 d e}-\frac{2 b \left(167 a^2+54 b^2\right) (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))}{693 d e}+\frac{2 e \left(132 a^2 b^2+77 a^4+12 b^4\right) \sin (c+d x) \sqrt{e \cos (c+d x)}}{231 d}-\frac{2 b (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))^3}{11 d e}-\frac{34 a b (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))^2}{99 d e}","\frac{2 e^2 \left(132 a^2 b^2+77 a^4+12 b^4\right) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{231 d \sqrt{e \cos (c+d x)}}-\frac{26 a b \left(79 a^2+74 b^2\right) (e \cos (c+d x))^{5/2}}{3465 d e}-\frac{2 b \left(167 a^2+54 b^2\right) (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))}{693 d e}+\frac{2 e \left(132 a^2 b^2+77 a^4+12 b^4\right) \sin (c+d x) \sqrt{e \cos (c+d x)}}{231 d}-\frac{2 b (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))^3}{11 d e}-\frac{34 a b (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))^2}{99 d e}",1,"(-26*a*b*(79*a^2 + 74*b^2)*(e*Cos[c + d*x])^(5/2))/(3465*d*e) + (2*(77*a^4 + 132*a^2*b^2 + 12*b^4)*e^2*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(231*d*Sqrt[e*Cos[c + d*x]]) + (2*(77*a^4 + 132*a^2*b^2 + 12*b^4)*e*Sqrt[e*Cos[c + d*x]]*Sin[c + d*x])/(231*d) - (2*b*(167*a^2 + 54*b^2)*(e*Cos[c + d*x])^(5/2)*(a + b*Sin[c + d*x]))/(693*d*e) - (34*a*b*(e*Cos[c + d*x])^(5/2)*(a + b*Sin[c + d*x])^2)/(99*d*e) - (2*b*(e*Cos[c + d*x])^(5/2)*(a + b*Sin[c + d*x])^3)/(11*d*e)","A",7,6,25,0.2400,1,"{2692, 2862, 2669, 2635, 2642, 2641}"
567,1,210,0,0.4417759,"\int \sqrt{e \cos (c+d x)} (a+b \sin (c+d x))^4 \, dx","Int[Sqrt[e*Cos[c + d*x]]*(a + b*Sin[c + d*x])^4,x]","-\frac{22 a b \left(17 a^2+18 b^2\right) (e \cos (c+d x))^{3/2}}{315 d e}-\frac{2 b \left(41 a^2+14 b^2\right) (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))}{105 d e}+\frac{2 \left(36 a^2 b^2+15 a^4+4 b^4\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{15 d \sqrt{\cos (c+d x)}}-\frac{2 b (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^3}{9 d e}-\frac{10 a b (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^2}{21 d e}","-\frac{22 a b \left(17 a^2+18 b^2\right) (e \cos (c+d x))^{3/2}}{315 d e}-\frac{2 b \left(41 a^2+14 b^2\right) (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))}{105 d e}+\frac{2 \left(36 a^2 b^2+15 a^4+4 b^4\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{15 d \sqrt{\cos (c+d x)}}-\frac{2 b (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^3}{9 d e}-\frac{10 a b (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^2}{21 d e}",1,"(-22*a*b*(17*a^2 + 18*b^2)*(e*Cos[c + d*x])^(3/2))/(315*d*e) + (2*(15*a^4 + 36*a^2*b^2 + 4*b^4)*Sqrt[e*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(15*d*Sqrt[Cos[c + d*x]]) - (2*b*(41*a^2 + 14*b^2)*(e*Cos[c + d*x])^(3/2)*(a + b*Sin[c + d*x]))/(105*d*e) - (10*a*b*(e*Cos[c + d*x])^(3/2)*(a + b*Sin[c + d*x])^2)/(21*d*e) - (2*b*(e*Cos[c + d*x])^(3/2)*(a + b*Sin[c + d*x])^3)/(9*d*e)","A",6,5,25,0.2000,1,"{2692, 2862, 2669, 2640, 2639}"
568,1,210,0,0.4450166,"\int \frac{(a+b \sin (c+d x))^4}{\sqrt{e \cos (c+d x)}} \, dx","Int[(a + b*Sin[c + d*x])^4/Sqrt[e*Cos[c + d*x]],x]","-\frac{6 a b \left(31 a^2+34 b^2\right) \sqrt{e \cos (c+d x)}}{35 d e}-\frac{2 b \left(29 a^2+10 b^2\right) \sqrt{e \cos (c+d x)} (a+b \sin (c+d x))}{35 d e}+\frac{2 \left(28 a^2 b^2+7 a^4+4 b^4\right) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{7 d \sqrt{e \cos (c+d x)}}-\frac{2 b \sqrt{e \cos (c+d x)} (a+b \sin (c+d x))^3}{7 d e}-\frac{26 a b \sqrt{e \cos (c+d x)} (a+b \sin (c+d x))^2}{35 d e}","-\frac{6 a b \left(31 a^2+34 b^2\right) \sqrt{e \cos (c+d x)}}{35 d e}-\frac{2 b \left(29 a^2+10 b^2\right) \sqrt{e \cos (c+d x)} (a+b \sin (c+d x))}{35 d e}+\frac{2 \left(28 a^2 b^2+7 a^4+4 b^4\right) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{7 d \sqrt{e \cos (c+d x)}}-\frac{2 b \sqrt{e \cos (c+d x)} (a+b \sin (c+d x))^3}{7 d e}-\frac{26 a b \sqrt{e \cos (c+d x)} (a+b \sin (c+d x))^2}{35 d e}",1,"(-6*a*b*(31*a^2 + 34*b^2)*Sqrt[e*Cos[c + d*x]])/(35*d*e) + (2*(7*a^4 + 28*a^2*b^2 + 4*b^4)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(7*d*Sqrt[e*Cos[c + d*x]]) - (2*b*(29*a^2 + 10*b^2)*Sqrt[e*Cos[c + d*x]]*(a + b*Sin[c + d*x]))/(35*d*e) - (26*a*b*Sqrt[e*Cos[c + d*x]]*(a + b*Sin[c + d*x])^2)/(35*d*e) - (2*b*Sqrt[e*Cos[c + d*x]]*(a + b*Sin[c + d*x])^3)/(7*d*e)","A",6,5,25,0.2000,1,"{2692, 2862, 2669, 2642, 2641}"
569,1,218,0,0.4381021,"\int \frac{(a+b \sin (c+d x))^4}{(e \cos (c+d x))^{3/2}} \, dx","Int[(a + b*Sin[c + d*x])^4/(e*Cos[c + d*x])^(3/2),x]","\frac{2 a b \left(15 a^2+62 b^2\right) (e \cos (c+d x))^{3/2}}{15 d e^3}+\frac{2 b \left(5 a^2+6 b^2\right) (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))}{5 d e^3}-\frac{2 \left(60 a^2 b^2+5 a^4+12 b^4\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{5 d e^2 \sqrt{\cos (c+d x)}}+\frac{2 a b (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^2}{d e^3}+\frac{2 (a \sin (c+d x)+b) (a+b \sin (c+d x))^3}{d e \sqrt{e \cos (c+d x)}}","\frac{2 a b \left(15 a^2+62 b^2\right) (e \cos (c+d x))^{3/2}}{15 d e^3}+\frac{2 b \left(5 a^2+6 b^2\right) (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))}{5 d e^3}-\frac{2 \left(60 a^2 b^2+5 a^4+12 b^4\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{5 d e^2 \sqrt{\cos (c+d x)}}+\frac{2 a b (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^2}{d e^3}+\frac{2 (a \sin (c+d x)+b) (a+b \sin (c+d x))^3}{d e \sqrt{e \cos (c+d x)}}",1,"(2*a*b*(15*a^2 + 62*b^2)*(e*Cos[c + d*x])^(3/2))/(15*d*e^3) - (2*(5*a^4 + 60*a^2*b^2 + 12*b^4)*Sqrt[e*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(5*d*e^2*Sqrt[Cos[c + d*x]]) + (2*b*(5*a^2 + 6*b^2)*(e*Cos[c + d*x])^(3/2)*(a + b*Sin[c + d*x]))/(5*d*e^3) + (2*a*b*(e*Cos[c + d*x])^(3/2)*(a + b*Sin[c + d*x])^2)/(d*e^3) + (2*(b + a*Sin[c + d*x])*(a + b*Sin[c + d*x])^3)/(d*e*Sqrt[e*Cos[c + d*x]])","A",6,5,25,0.2000,1,"{2691, 2862, 2669, 2640, 2639}"
570,1,216,0,0.4452778,"\int \frac{(a+b \sin (c+d x))^4}{(e \cos (c+d x))^{5/2}} \, dx","Int[(a + b*Sin[c + d*x])^4/(e*Cos[c + d*x])^(5/2),x]","\frac{2 a b \left(a^2+14 b^2\right) \sqrt{e \cos (c+d x)}}{3 d e^3}+\frac{2 b \left(a^2+2 b^2\right) \sqrt{e \cos (c+d x)} (a+b \sin (c+d x))}{3 d e^3}+\frac{2 \left(-12 a^2 b^2+a^4-4 b^4\right) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d e^2 \sqrt{e \cos (c+d x)}}+\frac{2 a b \sqrt{e \cos (c+d x)} (a+b \sin (c+d x))^2}{3 d e^3}+\frac{2 (a \sin (c+d x)+b) (a+b \sin (c+d x))^3}{3 d e (e \cos (c+d x))^{3/2}}","\frac{2 a b \left(a^2+14 b^2\right) \sqrt{e \cos (c+d x)}}{3 d e^3}+\frac{2 b \left(a^2+2 b^2\right) \sqrt{e \cos (c+d x)} (a+b \sin (c+d x))}{3 d e^3}+\frac{2 \left(-12 a^2 b^2+a^4-4 b^4\right) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d e^2 \sqrt{e \cos (c+d x)}}+\frac{2 a b \sqrt{e \cos (c+d x)} (a+b \sin (c+d x))^2}{3 d e^3}+\frac{2 (a \sin (c+d x)+b) (a+b \sin (c+d x))^3}{3 d e (e \cos (c+d x))^{3/2}}",1,"(2*a*b*(a^2 + 14*b^2)*Sqrt[e*Cos[c + d*x]])/(3*d*e^3) + (2*(a^4 - 12*a^2*b^2 - 4*b^4)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(3*d*e^2*Sqrt[e*Cos[c + d*x]]) + (2*b*(a^2 + 2*b^2)*Sqrt[e*Cos[c + d*x]]*(a + b*Sin[c + d*x]))/(3*d*e^3) + (2*a*b*Sqrt[e*Cos[c + d*x]]*(a + b*Sin[c + d*x])^2)/(3*d*e^3) + (2*(b + a*Sin[c + d*x])*(a + b*Sin[c + d*x])^3)/(3*d*e*(e*Cos[c + d*x])^(3/2))","A",6,5,25,0.2000,1,"{2691, 2862, 2669, 2642, 2641}"
571,1,237,0,0.4653065,"\int \frac{(a+b \sin (c+d x))^4}{(e \cos (c+d x))^{7/2}} \, dx","Int[(a + b*Sin[c + d*x])^4/(e*Cos[c + d*x])^(7/2),x]","\frac{2 a b \left(3 a^2-10 b^2\right) (e \cos (c+d x))^{3/2}}{5 d e^5}-\frac{6 \left(a b-\left(a^2-2 b^2\right) \sin (c+d x)\right) (a+b \sin (c+d x))^2}{5 d e^3 \sqrt{e \cos (c+d x)}}+\frac{6 b \left(a^2-2 b^2\right) (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))}{5 d e^5}-\frac{6 \left(-4 a^2 b^2+a^4-4 b^4\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{5 d e^4 \sqrt{\cos (c+d x)}}+\frac{2 (a \sin (c+d x)+b) (a+b \sin (c+d x))^3}{5 d e (e \cos (c+d x))^{5/2}}","\frac{2 a b \left(3 a^2-10 b^2\right) (e \cos (c+d x))^{3/2}}{5 d e^5}-\frac{6 \left(a b-\left(a^2-2 b^2\right) \sin (c+d x)\right) (a+b \sin (c+d x))^2}{5 d e^3 \sqrt{e \cos (c+d x)}}+\frac{6 b \left(a^2-2 b^2\right) (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))}{5 d e^5}-\frac{6 \left(-4 a^2 b^2+a^4-4 b^4\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{5 d e^4 \sqrt{\cos (c+d x)}}+\frac{2 (a \sin (c+d x)+b) (a+b \sin (c+d x))^3}{5 d e (e \cos (c+d x))^{5/2}}",1,"(2*a*b*(3*a^2 - 10*b^2)*(e*Cos[c + d*x])^(3/2))/(5*d*e^5) - (6*(a^4 - 4*a^2*b^2 - 4*b^4)*Sqrt[e*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(5*d*e^4*Sqrt[Cos[c + d*x]]) + (6*b*(a^2 - 2*b^2)*(e*Cos[c + d*x])^(3/2)*(a + b*Sin[c + d*x]))/(5*d*e^5) + (2*(b + a*Sin[c + d*x])*(a + b*Sin[c + d*x])^3)/(5*d*e*(e*Cos[c + d*x])^(5/2)) - (6*(a + b*Sin[c + d*x])^2*(a*b - (a^2 - 2*b^2)*Sin[c + d*x]))/(5*d*e^3*Sqrt[e*Cos[c + d*x]])","A",6,6,25,0.2400,1,"{2691, 2861, 2862, 2669, 2640, 2639}"
572,1,241,0,0.4628518,"\int \frac{(a+b \sin (c+d x))^4}{(e \cos (c+d x))^{9/2}} \, dx","Int[(a + b*Sin[c + d*x])^4/(e*Cos[c + d*x])^(9/2),x]","\frac{10 a b \left(a^2-2 b^2\right) \sqrt{e \cos (c+d x)}}{21 d e^5}-\frac{2 \left(a b-\left(5 a^2-6 b^2\right) \sin (c+d x)\right) (a+b \sin (c+d x))^2}{21 d e^3 (e \cos (c+d x))^{3/2}}+\frac{2 b \left(5 a^2-6 b^2\right) \sqrt{e \cos (c+d x)} (a+b \sin (c+d x))}{21 d e^5}+\frac{2 \left(-12 a^2 b^2+5 a^4+12 b^4\right) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d e^4 \sqrt{e \cos (c+d x)}}+\frac{2 (a \sin (c+d x)+b) (a+b \sin (c+d x))^3}{7 d e (e \cos (c+d x))^{7/2}}","\frac{10 a b \left(a^2-2 b^2\right) \sqrt{e \cos (c+d x)}}{21 d e^5}-\frac{2 \left(a b-\left(5 a^2-6 b^2\right) \sin (c+d x)\right) (a+b \sin (c+d x))^2}{21 d e^3 (e \cos (c+d x))^{3/2}}+\frac{2 b \left(5 a^2-6 b^2\right) \sqrt{e \cos (c+d x)} (a+b \sin (c+d x))}{21 d e^5}+\frac{2 \left(-12 a^2 b^2+5 a^4+12 b^4\right) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d e^4 \sqrt{e \cos (c+d x)}}+\frac{2 (a \sin (c+d x)+b) (a+b \sin (c+d x))^3}{7 d e (e \cos (c+d x))^{7/2}}",1,"(10*a*b*(a^2 - 2*b^2)*Sqrt[e*Cos[c + d*x]])/(21*d*e^5) + (2*(5*a^4 - 12*a^2*b^2 + 12*b^4)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(21*d*e^4*Sqrt[e*Cos[c + d*x]]) + (2*b*(5*a^2 - 6*b^2)*Sqrt[e*Cos[c + d*x]]*(a + b*Sin[c + d*x]))/(21*d*e^5) + (2*(b + a*Sin[c + d*x])*(a + b*Sin[c + d*x])^3)/(7*d*e*(e*Cos[c + d*x])^(7/2)) - (2*(a + b*Sin[c + d*x])^2*(a*b - (5*a^2 - 6*b^2)*Sin[c + d*x]))/(21*d*e^3*(e*Cos[c + d*x])^(3/2))","A",6,6,25,0.2400,1,"{2691, 2861, 2862, 2669, 2642, 2641}"
573,1,264,0,0.4790279,"\int \frac{(a+b \sin (c+d x))^4}{(e \cos (c+d x))^{11/2}} \, dx","Int[(a + b*Sin[c + d*x])^4/(e*Cos[c + d*x])^(11/2),x]","\frac{2 a b \left(21 a^2-22 b^2\right) (e \cos (c+d x))^{3/2}}{45 d e^7}+\frac{2 \left(\left(7 a^2-6 b^2\right) \sin (c+d x)+a b\right) (a+b \sin (c+d x))^2}{45 d e^3 (e \cos (c+d x))^{5/2}}-\frac{2 \left(b \left(7 a^2-6 b^2\right)-a \left(21 a^2-22 b^2\right) \sin (c+d x)\right) (a+b \sin (c+d x))}{45 d e^5 \sqrt{e \cos (c+d x)}}-\frac{2 \left(-12 a^2 b^2+7 a^4+4 b^4\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{15 d e^6 \sqrt{\cos (c+d x)}}+\frac{2 (a \sin (c+d x)+b) (a+b \sin (c+d x))^3}{9 d e (e \cos (c+d x))^{9/2}}","\frac{2 a b \left(21 a^2-22 b^2\right) (e \cos (c+d x))^{3/2}}{45 d e^7}+\frac{2 \left(\left(7 a^2-6 b^2\right) \sin (c+d x)+a b\right) (a+b \sin (c+d x))^2}{45 d e^3 (e \cos (c+d x))^{5/2}}-\frac{2 \left(b \left(7 a^2-6 b^2\right)-a \left(21 a^2-22 b^2\right) \sin (c+d x)\right) (a+b \sin (c+d x))}{45 d e^5 \sqrt{e \cos (c+d x)}}-\frac{2 \left(-12 a^2 b^2+7 a^4+4 b^4\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{15 d e^6 \sqrt{\cos (c+d x)}}+\frac{2 (a \sin (c+d x)+b) (a+b \sin (c+d x))^3}{9 d e (e \cos (c+d x))^{9/2}}",1,"(2*a*b*(21*a^2 - 22*b^2)*(e*Cos[c + d*x])^(3/2))/(45*d*e^7) - (2*(7*a^4 - 12*a^2*b^2 + 4*b^4)*Sqrt[e*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(15*d*e^6*Sqrt[Cos[c + d*x]]) + (2*(b + a*Sin[c + d*x])*(a + b*Sin[c + d*x])^3)/(9*d*e*(e*Cos[c + d*x])^(9/2)) - (2*(a + b*Sin[c + d*x])*(b*(7*a^2 - 6*b^2) - a*(21*a^2 - 22*b^2)*Sin[c + d*x]))/(45*d*e^5*Sqrt[e*Cos[c + d*x]]) + (2*(a + b*Sin[c + d*x])^2*(a*b + (7*a^2 - 6*b^2)*Sin[c + d*x]))/(45*d*e^3*(e*Cos[c + d*x])^(5/2))","A",6,5,25,0.2000,1,"{2691, 2861, 2669, 2640, 2639}"
574,1,531,0,1.9079637,"\int \frac{(e \cos (c+d x))^{11/2}}{a+b \sin (c+d x)} \, dx","Int[(e*Cos[c + d*x])^(11/2)/(a + b*Sin[c + d*x]),x]","\frac{2 e^5 \sqrt{e \cos (c+d x)} \left(21 \left(a^2-b^2\right)^2-a b \left(7 a^2-12 b^2\right) \sin (c+d x)\right)}{21 b^5 d}-\frac{2 e^3 (e \cos (c+d x))^{5/2} \left(7 \left(a^2-b^2\right)-5 a b \sin (c+d x)\right)}{35 b^3 d}-\frac{e^{11/2} \left(b^2-a^2\right)^{9/4} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{b^{11/2} d}-\frac{e^{11/2} \left(b^2-a^2\right)^{9/4} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{b^{11/2} d}+\frac{2 a e^6 \left(-49 a^2 b^2+21 a^4+33 b^4\right) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 b^6 d \sqrt{e \cos (c+d x)}}-\frac{a e^6 \left(a^2-b^2\right)^3 \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{b^6 d \left(a^2-b \left(b-\sqrt{b^2-a^2}\right)\right) \sqrt{e \cos (c+d x)}}-\frac{a e^6 \left(a^2-b^2\right)^3 \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{b^6 d \left(a^2-b \left(\sqrt{b^2-a^2}+b\right)\right) \sqrt{e \cos (c+d x)}}+\frac{2 e (e \cos (c+d x))^{9/2}}{9 b d}","\frac{2 e^5 \sqrt{e \cos (c+d x)} \left(21 \left(a^2-b^2\right)^2-a b \left(7 a^2-12 b^2\right) \sin (c+d x)\right)}{21 b^5 d}-\frac{2 e^3 (e \cos (c+d x))^{5/2} \left(7 \left(a^2-b^2\right)-5 a b \sin (c+d x)\right)}{35 b^3 d}-\frac{e^{11/2} \left(b^2-a^2\right)^{9/4} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{b^{11/2} d}-\frac{e^{11/2} \left(b^2-a^2\right)^{9/4} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{b^{11/2} d}+\frac{2 a e^6 \left(-49 a^2 b^2+21 a^4+33 b^4\right) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 b^6 d \sqrt{e \cos (c+d x)}}-\frac{a e^6 \left(a^2-b^2\right)^3 \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{b^6 d \left(a^2-b \left(b-\sqrt{b^2-a^2}\right)\right) \sqrt{e \cos (c+d x)}}-\frac{a e^6 \left(a^2-b^2\right)^3 \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{b^6 d \left(a^2-b \left(\sqrt{b^2-a^2}+b\right)\right) \sqrt{e \cos (c+d x)}}+\frac{2 e (e \cos (c+d x))^{9/2}}{9 b d}",1,"-(((-a^2 + b^2)^(9/4)*e^(11/2)*ArcTan[(Sqrt[b]*Sqrt[e*Cos[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(b^(11/2)*d)) - ((-a^2 + b^2)^(9/4)*e^(11/2)*ArcTanh[(Sqrt[b]*Sqrt[e*Cos[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(b^(11/2)*d) + (2*e*(e*Cos[c + d*x])^(9/2))/(9*b*d) + (2*a*(21*a^4 - 49*a^2*b^2 + 33*b^4)*e^6*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(21*b^6*d*Sqrt[e*Cos[c + d*x]]) - (a*(a^2 - b^2)^3*e^6*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (c + d*x)/2, 2])/(b^6*(a^2 - b*(b - Sqrt[-a^2 + b^2]))*d*Sqrt[e*Cos[c + d*x]]) - (a*(a^2 - b^2)^3*e^6*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (c + d*x)/2, 2])/(b^6*(a^2 - b*(b + Sqrt[-a^2 + b^2]))*d*Sqrt[e*Cos[c + d*x]]) - (2*e^3*(e*Cos[c + d*x])^(5/2)*(7*(a^2 - b^2) - 5*a*b*Sin[c + d*x]))/(35*b^3*d) + (2*e^5*Sqrt[e*Cos[c + d*x]]*(21*(a^2 - b^2)^2 - a*b*(7*a^2 - 12*b^2)*Sin[c + d*x]))/(21*b^5*d)","A",15,12,25,0.4800,1,"{2695, 2865, 2867, 2642, 2641, 2702, 2807, 2805, 329, 212, 208, 205}"
575,1,446,0,1.285779,"\int \frac{(e \cos (c+d x))^{9/2}}{a+b \sin (c+d x)} \, dx","Int[(e*Cos[c + d*x])^(9/2)/(a + b*Sin[c + d*x]),x]","-\frac{2 e^3 (e \cos (c+d x))^{3/2} \left(5 \left(a^2-b^2\right)-3 a b \sin (c+d x)\right)}{15 b^3 d}+\frac{e^{9/2} \left(b^2-a^2\right)^{7/4} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{b^{9/2} d}-\frac{e^{9/2} \left(b^2-a^2\right)^{7/4} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{b^{9/2} d}-\frac{2 a e^4 \left(5 a^2-8 b^2\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{5 b^4 d \sqrt{\cos (c+d x)}}+\frac{a e^5 \left(a^2-b^2\right)^2 \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{b^5 d \left(b-\sqrt{b^2-a^2}\right) \sqrt{e \cos (c+d x)}}+\frac{a e^5 \left(a^2-b^2\right)^2 \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{b^5 d \left(\sqrt{b^2-a^2}+b\right) \sqrt{e \cos (c+d x)}}+\frac{2 e (e \cos (c+d x))^{7/2}}{7 b d}","-\frac{2 e^3 (e \cos (c+d x))^{3/2} \left(5 \left(a^2-b^2\right)-3 a b \sin (c+d x)\right)}{15 b^3 d}+\frac{e^{9/2} \left(b^2-a^2\right)^{7/4} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{b^{9/2} d}-\frac{e^{9/2} \left(b^2-a^2\right)^{7/4} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{b^{9/2} d}-\frac{2 a e^4 \left(5 a^2-8 b^2\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{5 b^4 d \sqrt{\cos (c+d x)}}+\frac{a e^5 \left(a^2-b^2\right)^2 \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{b^5 d \left(b-\sqrt{b^2-a^2}\right) \sqrt{e \cos (c+d x)}}+\frac{a e^5 \left(a^2-b^2\right)^2 \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{b^5 d \left(\sqrt{b^2-a^2}+b\right) \sqrt{e \cos (c+d x)}}+\frac{2 e (e \cos (c+d x))^{7/2}}{7 b d}",1,"((-a^2 + b^2)^(7/4)*e^(9/2)*ArcTan[(Sqrt[b]*Sqrt[e*Cos[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(b^(9/2)*d) - ((-a^2 + b^2)^(7/4)*e^(9/2)*ArcTanh[(Sqrt[b]*Sqrt[e*Cos[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(b^(9/2)*d) + (2*e*(e*Cos[c + d*x])^(7/2))/(7*b*d) - (2*a*(5*a^2 - 8*b^2)*e^4*Sqrt[e*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(5*b^4*d*Sqrt[Cos[c + d*x]]) + (a*(a^2 - b^2)^2*e^5*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (c + d*x)/2, 2])/(b^5*(b - Sqrt[-a^2 + b^2])*d*Sqrt[e*Cos[c + d*x]]) + (a*(a^2 - b^2)^2*e^5*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (c + d*x)/2, 2])/(b^5*(b + Sqrt[-a^2 + b^2])*d*Sqrt[e*Cos[c + d*x]]) - (2*e^3*(e*Cos[c + d*x])^(3/2)*(5*(a^2 - b^2) - 3*a*b*Sin[c + d*x]))/(15*b^3*d)","A",14,12,25,0.4800,1,"{2695, 2865, 2867, 2640, 2639, 2701, 2807, 2805, 329, 298, 205, 208}"
576,1,461,0,1.3229351,"\int \frac{(e \cos (c+d x))^{7/2}}{a+b \sin (c+d x)} \, dx","Int[(e*Cos[c + d*x])^(7/2)/(a + b*Sin[c + d*x]),x]","-\frac{2 e^3 \sqrt{e \cos (c+d x)} \left(3 \left(a^2-b^2\right)-a b \sin (c+d x)\right)}{3 b^3 d}-\frac{e^{7/2} \left(b^2-a^2\right)^{5/4} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{b^{7/2} d}-\frac{e^{7/2} \left(b^2-a^2\right)^{5/4} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{b^{7/2} d}-\frac{2 a e^4 \left(3 a^2-4 b^2\right) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 b^4 d \sqrt{e \cos (c+d x)}}+\frac{a e^4 \left(a^2-b^2\right)^2 \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{b^4 d \left(a^2-b \left(b-\sqrt{b^2-a^2}\right)\right) \sqrt{e \cos (c+d x)}}+\frac{a e^4 \left(a^2-b^2\right)^2 \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{b^4 d \left(a^2-b \left(\sqrt{b^2-a^2}+b\right)\right) \sqrt{e \cos (c+d x)}}+\frac{2 e (e \cos (c+d x))^{5/2}}{5 b d}","-\frac{2 e^3 \sqrt{e \cos (c+d x)} \left(3 \left(a^2-b^2\right)-a b \sin (c+d x)\right)}{3 b^3 d}-\frac{e^{7/2} \left(b^2-a^2\right)^{5/4} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{b^{7/2} d}-\frac{e^{7/2} \left(b^2-a^2\right)^{5/4} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{b^{7/2} d}-\frac{2 a e^4 \left(3 a^2-4 b^2\right) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 b^4 d \sqrt{e \cos (c+d x)}}+\frac{a e^4 \left(a^2-b^2\right)^2 \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{b^4 d \left(a^2-b \left(b-\sqrt{b^2-a^2}\right)\right) \sqrt{e \cos (c+d x)}}+\frac{a e^4 \left(a^2-b^2\right)^2 \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{b^4 d \left(a^2-b \left(\sqrt{b^2-a^2}+b\right)\right) \sqrt{e \cos (c+d x)}}+\frac{2 e (e \cos (c+d x))^{5/2}}{5 b d}",1,"-(((-a^2 + b^2)^(5/4)*e^(7/2)*ArcTan[(Sqrt[b]*Sqrt[e*Cos[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(b^(7/2)*d)) - ((-a^2 + b^2)^(5/4)*e^(7/2)*ArcTanh[(Sqrt[b]*Sqrt[e*Cos[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(b^(7/2)*d) + (2*e*(e*Cos[c + d*x])^(5/2))/(5*b*d) - (2*a*(3*a^2 - 4*b^2)*e^4*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(3*b^4*d*Sqrt[e*Cos[c + d*x]]) + (a*(a^2 - b^2)^2*e^4*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (c + d*x)/2, 2])/(b^4*(a^2 - b*(b - Sqrt[-a^2 + b^2]))*d*Sqrt[e*Cos[c + d*x]]) + (a*(a^2 - b^2)^2*e^4*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (c + d*x)/2, 2])/(b^4*(a^2 - b*(b + Sqrt[-a^2 + b^2]))*d*Sqrt[e*Cos[c + d*x]]) - (2*e^3*Sqrt[e*Cos[c + d*x]]*(3*(a^2 - b^2) - a*b*Sin[c + d*x]))/(3*b^3*d)","A",14,12,25,0.4800,1,"{2695, 2865, 2867, 2642, 2641, 2702, 2807, 2805, 329, 212, 208, 205}"
577,1,384,0,0.8699766,"\int \frac{(e \cos (c+d x))^{5/2}}{a+b \sin (c+d x)} \, dx","Int[(e*Cos[c + d*x])^(5/2)/(a + b*Sin[c + d*x]),x]","\frac{e^{5/2} \left(b^2-a^2\right)^{3/4} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{b^{5/2} d}-\frac{e^{5/2} \left(b^2-a^2\right)^{3/4} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{b^{5/2} d}-\frac{a e^3 \left(a^2-b^2\right) \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{b^3 d \left(b-\sqrt{b^2-a^2}\right) \sqrt{e \cos (c+d x)}}-\frac{a e^3 \left(a^2-b^2\right) \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{b^3 d \left(\sqrt{b^2-a^2}+b\right) \sqrt{e \cos (c+d x)}}+\frac{2 a e^2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{b^2 d \sqrt{\cos (c+d x)}}+\frac{2 e (e \cos (c+d x))^{3/2}}{3 b d}","\frac{e^{5/2} \left(b^2-a^2\right)^{3/4} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{b^{5/2} d}-\frac{e^{5/2} \left(b^2-a^2\right)^{3/4} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{b^{5/2} d}-\frac{a e^3 \left(a^2-b^2\right) \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{b^3 d \left(b-\sqrt{b^2-a^2}\right) \sqrt{e \cos (c+d x)}}-\frac{a e^3 \left(a^2-b^2\right) \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{b^3 d \left(\sqrt{b^2-a^2}+b\right) \sqrt{e \cos (c+d x)}}+\frac{2 a e^2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{b^2 d \sqrt{\cos (c+d x)}}+\frac{2 e (e \cos (c+d x))^{3/2}}{3 b d}",1,"((-a^2 + b^2)^(3/4)*e^(5/2)*ArcTan[(Sqrt[b]*Sqrt[e*Cos[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(b^(5/2)*d) - ((-a^2 + b^2)^(3/4)*e^(5/2)*ArcTanh[(Sqrt[b]*Sqrt[e*Cos[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(b^(5/2)*d) + (2*e*(e*Cos[c + d*x])^(3/2))/(3*b*d) + (2*a*e^2*Sqrt[e*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(b^2*d*Sqrt[Cos[c + d*x]]) - (a*(a^2 - b^2)*e^3*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (c + d*x)/2, 2])/(b^3*(b - Sqrt[-a^2 + b^2])*d*Sqrt[e*Cos[c + d*x]]) - (a*(a^2 - b^2)*e^3*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (c + d*x)/2, 2])/(b^3*(b + Sqrt[-a^2 + b^2])*d*Sqrt[e*Cos[c + d*x]])","A",13,11,25,0.4400,1,"{2695, 2867, 2640, 2639, 2701, 2807, 2805, 329, 298, 205, 208}"
578,1,397,0,0.8819623,"\int \frac{(e \cos (c+d x))^{3/2}}{a+b \sin (c+d x)} \, dx","Int[(e*Cos[c + d*x])^(3/2)/(a + b*Sin[c + d*x]),x]","-\frac{e^{3/2} \sqrt[4]{b^2-a^2} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{b^{3/2} d}-\frac{e^{3/2} \sqrt[4]{b^2-a^2} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{b^{3/2} d}-\frac{a e^2 \left(a^2-b^2\right) \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{b^2 d \left(a^2-b \left(b-\sqrt{b^2-a^2}\right)\right) \sqrt{e \cos (c+d x)}}-\frac{a e^2 \left(a^2-b^2\right) \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{b^2 d \left(a^2-b \left(\sqrt{b^2-a^2}+b\right)\right) \sqrt{e \cos (c+d x)}}+\frac{2 a e^2 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b^2 d \sqrt{e \cos (c+d x)}}+\frac{2 e \sqrt{e \cos (c+d x)}}{b d}","-\frac{e^{3/2} \sqrt[4]{b^2-a^2} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{b^{3/2} d}-\frac{e^{3/2} \sqrt[4]{b^2-a^2} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{b^{3/2} d}-\frac{a e^2 \left(a^2-b^2\right) \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{b^2 d \left(a^2-b \left(b-\sqrt{b^2-a^2}\right)\right) \sqrt{e \cos (c+d x)}}-\frac{a e^2 \left(a^2-b^2\right) \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{b^2 d \left(a^2-b \left(\sqrt{b^2-a^2}+b\right)\right) \sqrt{e \cos (c+d x)}}+\frac{2 a e^2 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b^2 d \sqrt{e \cos (c+d x)}}+\frac{2 e \sqrt{e \cos (c+d x)}}{b d}",1,"-(((-a^2 + b^2)^(1/4)*e^(3/2)*ArcTan[(Sqrt[b]*Sqrt[e*Cos[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(b^(3/2)*d)) - ((-a^2 + b^2)^(1/4)*e^(3/2)*ArcTanh[(Sqrt[b]*Sqrt[e*Cos[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(b^(3/2)*d) + (2*e*Sqrt[e*Cos[c + d*x]])/(b*d) + (2*a*e^2*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(b^2*d*Sqrt[e*Cos[c + d*x]]) - (a*(a^2 - b^2)*e^2*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (c + d*x)/2, 2])/(b^2*(a^2 - b*(b - Sqrt[-a^2 + b^2]))*d*Sqrt[e*Cos[c + d*x]]) - (a*(a^2 - b^2)*e^2*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (c + d*x)/2, 2])/(b^2*(a^2 - b*(b + Sqrt[-a^2 + b^2]))*d*Sqrt[e*Cos[c + d*x]])","A",13,11,25,0.4400,1,"{2695, 2867, 2642, 2641, 2702, 2807, 2805, 329, 212, 208, 205}"
579,1,292,0,0.582375,"\int \frac{\sqrt{e \cos (c+d x)}}{a+b \sin (c+d x)} \, dx","Int[Sqrt[e*Cos[c + d*x]]/(a + b*Sin[c + d*x]),x]","\frac{\sqrt{e} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{\sqrt{b} d \sqrt[4]{b^2-a^2}}-\frac{\sqrt{e} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{\sqrt{b} d \sqrt[4]{b^2-a^2}}+\frac{a e \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{b d \left(b-\sqrt{b^2-a^2}\right) \sqrt{e \cos (c+d x)}}+\frac{a e \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{b d \left(\sqrt{b^2-a^2}+b\right) \sqrt{e \cos (c+d x)}}","\frac{\sqrt{e} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{\sqrt{b} d \sqrt[4]{b^2-a^2}}-\frac{\sqrt{e} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{\sqrt{b} d \sqrt[4]{b^2-a^2}}+\frac{a e \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{b d \left(b-\sqrt{b^2-a^2}\right) \sqrt{e \cos (c+d x)}}+\frac{a e \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{b d \left(\sqrt{b^2-a^2}+b\right) \sqrt{e \cos (c+d x)}}",1,"(Sqrt[e]*ArcTan[(Sqrt[b]*Sqrt[e*Cos[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(Sqrt[b]*(-a^2 + b^2)^(1/4)*d) - (Sqrt[e]*ArcTanh[(Sqrt[b]*Sqrt[e*Cos[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(Sqrt[b]*(-a^2 + b^2)^(1/4)*d) + (a*e*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (c + d*x)/2, 2])/(b*(b - Sqrt[-a^2 + b^2])*d*Sqrt[e*Cos[c + d*x]]) + (a*e*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (c + d*x)/2, 2])/(b*(b + Sqrt[-a^2 + b^2])*d*Sqrt[e*Cos[c + d*x]])","A",9,7,25,0.2800,1,"{2701, 2807, 2805, 329, 298, 205, 208}"
580,1,299,0,0.5742797,"\int \frac{1}{\sqrt{e \cos (c+d x)} (a+b \sin (c+d x))} \, dx","Int[1/(Sqrt[e*Cos[c + d*x]]*(a + b*Sin[c + d*x])),x]","-\frac{\sqrt{b} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{d \sqrt{e} \left(b^2-a^2\right)^{3/4}}-\frac{\sqrt{b} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{d \sqrt{e} \left(b^2-a^2\right)^{3/4}}+\frac{a \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{d \left(a^2-b \left(b-\sqrt{b^2-a^2}\right)\right) \sqrt{e \cos (c+d x)}}+\frac{a \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{d \left(a^2-b \left(\sqrt{b^2-a^2}+b\right)\right) \sqrt{e \cos (c+d x)}}","-\frac{\sqrt{b} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{d \sqrt{e} \left(b^2-a^2\right)^{3/4}}-\frac{\sqrt{b} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{d \sqrt{e} \left(b^2-a^2\right)^{3/4}}+\frac{a \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{d \left(a^2-b \left(b-\sqrt{b^2-a^2}\right)\right) \sqrt{e \cos (c+d x)}}+\frac{a \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{d \left(a^2-b \left(\sqrt{b^2-a^2}+b\right)\right) \sqrt{e \cos (c+d x)}}",1,"-((Sqrt[b]*ArcTan[(Sqrt[b]*Sqrt[e*Cos[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/((-a^2 + b^2)^(3/4)*d*Sqrt[e])) - (Sqrt[b]*ArcTanh[(Sqrt[b]*Sqrt[e*Cos[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/((-a^2 + b^2)^(3/4)*d*Sqrt[e]) + (a*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (c + d*x)/2, 2])/((a^2 - b*(b - Sqrt[-a^2 + b^2]))*d*Sqrt[e*Cos[c + d*x]]) + (a*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (c + d*x)/2, 2])/((a^2 - b*(b + Sqrt[-a^2 + b^2]))*d*Sqrt[e*Cos[c + d*x]])","A",9,7,25,0.2800,1,"{2702, 2807, 2805, 329, 212, 208, 205}"
581,1,411,0,0.9284259,"\int \frac{1}{(e \cos (c+d x))^{3/2} (a+b \sin (c+d x))} \, dx","Int[1/((e*Cos[c + d*x])^(3/2)*(a + b*Sin[c + d*x])),x]","\frac{b^{3/2} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{d e^{3/2} \left(b^2-a^2\right)^{5/4}}-\frac{b^{3/2} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{d e^{3/2} \left(b^2-a^2\right)^{5/4}}-\frac{2 a E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{d e^2 \left(a^2-b^2\right) \sqrt{\cos (c+d x)}}-\frac{2 (b-a \sin (c+d x))}{d e \left(a^2-b^2\right) \sqrt{e \cos (c+d x)}}-\frac{a b \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{d e \left(a^2-b^2\right) \left(b-\sqrt{b^2-a^2}\right) \sqrt{e \cos (c+d x)}}-\frac{a b \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{d e \left(a^2-b^2\right) \left(\sqrt{b^2-a^2}+b\right) \sqrt{e \cos (c+d x)}}","\frac{b^{3/2} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{d e^{3/2} \left(b^2-a^2\right)^{5/4}}-\frac{b^{3/2} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{d e^{3/2} \left(b^2-a^2\right)^{5/4}}-\frac{2 a E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{d e^2 \left(a^2-b^2\right) \sqrt{\cos (c+d x)}}-\frac{2 (b-a \sin (c+d x))}{d e \left(a^2-b^2\right) \sqrt{e \cos (c+d x)}}-\frac{a b \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{d e \left(a^2-b^2\right) \left(b-\sqrt{b^2-a^2}\right) \sqrt{e \cos (c+d x)}}-\frac{a b \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{d e \left(a^2-b^2\right) \left(\sqrt{b^2-a^2}+b\right) \sqrt{e \cos (c+d x)}}",1,"(b^(3/2)*ArcTan[(Sqrt[b]*Sqrt[e*Cos[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/((-a^2 + b^2)^(5/4)*d*e^(3/2)) - (b^(3/2)*ArcTanh[(Sqrt[b]*Sqrt[e*Cos[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/((-a^2 + b^2)^(5/4)*d*e^(3/2)) - (2*a*Sqrt[e*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/((a^2 - b^2)*d*e^2*Sqrt[Cos[c + d*x]]) - (a*b*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (c + d*x)/2, 2])/((a^2 - b^2)*(b - Sqrt[-a^2 + b^2])*d*e*Sqrt[e*Cos[c + d*x]]) - (a*b*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (c + d*x)/2, 2])/((a^2 - b^2)*(b + Sqrt[-a^2 + b^2])*d*e*Sqrt[e*Cos[c + d*x]]) - (2*(b - a*Sin[c + d*x]))/((a^2 - b^2)*d*e*Sqrt[e*Cos[c + d*x]])","A",13,11,25,0.4400,1,"{2696, 2867, 2640, 2639, 2701, 2807, 2805, 329, 298, 205, 208}"
582,1,434,0,0.9989023,"\int \frac{1}{(e \cos (c+d x))^{5/2} (a+b \sin (c+d x))} \, dx","Int[1/((e*Cos[c + d*x])^(5/2)*(a + b*Sin[c + d*x])),x]","-\frac{b^{5/2} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{d e^{5/2} \left(b^2-a^2\right)^{7/4}}-\frac{b^{5/2} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{d e^{5/2} \left(b^2-a^2\right)^{7/4}}+\frac{2 a \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d e^2 \left(a^2-b^2\right) \sqrt{e \cos (c+d x)}}-\frac{a b^2 \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{d e^2 \left(a^2-b^2\right) \left(a^2-b \left(b-\sqrt{b^2-a^2}\right)\right) \sqrt{e \cos (c+d x)}}-\frac{a b^2 \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{d e^2 \left(a^2-b^2\right) \left(a^2-b \left(\sqrt{b^2-a^2}+b\right)\right) \sqrt{e \cos (c+d x)}}-\frac{2 (b-a \sin (c+d x))}{3 d e \left(a^2-b^2\right) (e \cos (c+d x))^{3/2}}","-\frac{b^{5/2} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{d e^{5/2} \left(b^2-a^2\right)^{7/4}}-\frac{b^{5/2} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{d e^{5/2} \left(b^2-a^2\right)^{7/4}}+\frac{2 a \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d e^2 \left(a^2-b^2\right) \sqrt{e \cos (c+d x)}}-\frac{a b^2 \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{d e^2 \left(a^2-b^2\right) \left(a^2-b \left(b-\sqrt{b^2-a^2}\right)\right) \sqrt{e \cos (c+d x)}}-\frac{a b^2 \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{d e^2 \left(a^2-b^2\right) \left(a^2-b \left(\sqrt{b^2-a^2}+b\right)\right) \sqrt{e \cos (c+d x)}}-\frac{2 (b-a \sin (c+d x))}{3 d e \left(a^2-b^2\right) (e \cos (c+d x))^{3/2}}",1,"-((b^(5/2)*ArcTan[(Sqrt[b]*Sqrt[e*Cos[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/((-a^2 + b^2)^(7/4)*d*e^(5/2))) - (b^(5/2)*ArcTanh[(Sqrt[b]*Sqrt[e*Cos[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/((-a^2 + b^2)^(7/4)*d*e^(5/2)) + (2*a*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(3*(a^2 - b^2)*d*e^2*Sqrt[e*Cos[c + d*x]]) - (a*b^2*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (c + d*x)/2, 2])/((a^2 - b^2)*(a^2 - b*(b - Sqrt[-a^2 + b^2]))*d*e^2*Sqrt[e*Cos[c + d*x]]) - (a*b^2*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (c + d*x)/2, 2])/((a^2 - b^2)*(a^2 - b*(b + Sqrt[-a^2 + b^2]))*d*e^2*Sqrt[e*Cos[c + d*x]]) - (2*(b - a*Sin[c + d*x]))/(3*(a^2 - b^2)*d*e*(e*Cos[c + d*x])^(3/2))","A",13,11,25,0.4400,1,"{2696, 2867, 2642, 2641, 2702, 2807, 2805, 329, 212, 208, 205}"
583,1,486,0,1.3284789,"\int \frac{1}{(e \cos (c+d x))^{7/2} (a+b \sin (c+d x))} \, dx","Int[1/((e*Cos[c + d*x])^(7/2)*(a + b*Sin[c + d*x])),x]","\frac{2 \left(a \left(3 a^2-8 b^2\right) \sin (c+d x)+5 b^3\right)}{5 d e^3 \left(a^2-b^2\right)^2 \sqrt{e \cos (c+d x)}}+\frac{b^{7/2} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{d e^{7/2} \left(b^2-a^2\right)^{9/4}}-\frac{b^{7/2} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{d e^{7/2} \left(b^2-a^2\right)^{9/4}}-\frac{2 a \left(3 a^2-8 b^2\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{5 d e^4 \left(a^2-b^2\right)^2 \sqrt{\cos (c+d x)}}+\frac{a b^3 \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{d e^3 \left(a^2-b^2\right)^2 \left(b-\sqrt{b^2-a^2}\right) \sqrt{e \cos (c+d x)}}+\frac{a b^3 \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{d e^3 \left(a^2-b^2\right)^2 \left(\sqrt{b^2-a^2}+b\right) \sqrt{e \cos (c+d x)}}-\frac{2 (b-a \sin (c+d x))}{5 d e \left(a^2-b^2\right) (e \cos (c+d x))^{5/2}}","\frac{2 \left(a \left(3 a^2-8 b^2\right) \sin (c+d x)+5 b^3\right)}{5 d e^3 \left(a^2-b^2\right)^2 \sqrt{e \cos (c+d x)}}+\frac{b^{7/2} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{d e^{7/2} \left(b^2-a^2\right)^{9/4}}-\frac{b^{7/2} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{d e^{7/2} \left(b^2-a^2\right)^{9/4}}-\frac{2 a \left(3 a^2-8 b^2\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{5 d e^4 \left(a^2-b^2\right)^2 \sqrt{\cos (c+d x)}}+\frac{a b^3 \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{d e^3 \left(a^2-b^2\right)^2 \left(b-\sqrt{b^2-a^2}\right) \sqrt{e \cos (c+d x)}}+\frac{a b^3 \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{d e^3 \left(a^2-b^2\right)^2 \left(\sqrt{b^2-a^2}+b\right) \sqrt{e \cos (c+d x)}}-\frac{2 (b-a \sin (c+d x))}{5 d e \left(a^2-b^2\right) (e \cos (c+d x))^{5/2}}",1,"(b^(7/2)*ArcTan[(Sqrt[b]*Sqrt[e*Cos[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/((-a^2 + b^2)^(9/4)*d*e^(7/2)) - (b^(7/2)*ArcTanh[(Sqrt[b]*Sqrt[e*Cos[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/((-a^2 + b^2)^(9/4)*d*e^(7/2)) - (2*a*(3*a^2 - 8*b^2)*Sqrt[e*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(5*(a^2 - b^2)^2*d*e^4*Sqrt[Cos[c + d*x]]) + (a*b^3*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (c + d*x)/2, 2])/((a^2 - b^2)^2*(b - Sqrt[-a^2 + b^2])*d*e^3*Sqrt[e*Cos[c + d*x]]) + (a*b^3*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (c + d*x)/2, 2])/((a^2 - b^2)^2*(b + Sqrt[-a^2 + b^2])*d*e^3*Sqrt[e*Cos[c + d*x]]) - (2*(b - a*Sin[c + d*x]))/(5*(a^2 - b^2)*d*e*(e*Cos[c + d*x])^(5/2)) + (2*(5*b^3 + a*(3*a^2 - 8*b^2)*Sin[c + d*x]))/(5*(a^2 - b^2)^2*d*e^3*Sqrt[e*Cos[c + d*x]])","A",14,12,25,0.4800,1,"{2696, 2866, 2867, 2640, 2639, 2701, 2807, 2805, 329, 298, 205, 208}"
584,1,543,0,1.5189026,"\int \frac{(e \cos (c+d x))^{11/2}}{(a+b \sin (c+d x))^2} \, dx","Int[(e*Cos[c + d*x])^(11/2)/(a + b*Sin[c + d*x])^2,x]","-\frac{3 e^5 \sqrt{e \cos (c+d x)} \left(21 a \left(a^2-b^2\right)-b \left(7 a^2-5 b^2\right) \sin (c+d x)\right)}{7 b^5 d}-\frac{9 a e^{11/2} \left(b^2-a^2\right)^{5/4} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{2 b^{11/2} d}-\frac{9 a e^{11/2} \left(b^2-a^2\right)^{5/4} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{2 b^{11/2} d}-\frac{3 e^6 \left(-28 a^2 b^2+21 a^4+5 b^4\right) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{7 b^6 d \sqrt{e \cos (c+d x)}}+\frac{9 a^2 e^6 \left(a^2-b^2\right)^2 \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{2 b^6 d \left(a^2-b \left(b-\sqrt{b^2-a^2}\right)\right) \sqrt{e \cos (c+d x)}}+\frac{9 a^2 e^6 \left(a^2-b^2\right)^2 \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{2 b^6 d \left(a^2-b \left(\sqrt{b^2-a^2}+b\right)\right) \sqrt{e \cos (c+d x)}}+\frac{9 e^3 (e \cos (c+d x))^{5/2} (7 a-5 b \sin (c+d x))}{35 b^3 d}-\frac{e (e \cos (c+d x))^{9/2}}{b d (a+b \sin (c+d x))}","-\frac{3 e^5 \sqrt{e \cos (c+d x)} \left(21 a \left(a^2-b^2\right)-b \left(7 a^2-5 b^2\right) \sin (c+d x)\right)}{7 b^5 d}-\frac{9 a e^{11/2} \left(b^2-a^2\right)^{5/4} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{2 b^{11/2} d}-\frac{9 a e^{11/2} \left(b^2-a^2\right)^{5/4} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{2 b^{11/2} d}-\frac{3 e^6 \left(-28 a^2 b^2+21 a^4+5 b^4\right) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{7 b^6 d \sqrt{e \cos (c+d x)}}+\frac{9 a^2 e^6 \left(a^2-b^2\right)^2 \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{2 b^6 d \left(a^2-b \left(b-\sqrt{b^2-a^2}\right)\right) \sqrt{e \cos (c+d x)}}+\frac{9 a^2 e^6 \left(a^2-b^2\right)^2 \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{2 b^6 d \left(a^2-b \left(\sqrt{b^2-a^2}+b\right)\right) \sqrt{e \cos (c+d x)}}+\frac{9 e^3 (e \cos (c+d x))^{5/2} (7 a-5 b \sin (c+d x))}{35 b^3 d}-\frac{e (e \cos (c+d x))^{9/2}}{b d (a+b \sin (c+d x))}",1,"(-9*a*(-a^2 + b^2)^(5/4)*e^(11/2)*ArcTan[(Sqrt[b]*Sqrt[e*Cos[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(2*b^(11/2)*d) - (9*a*(-a^2 + b^2)^(5/4)*e^(11/2)*ArcTanh[(Sqrt[b]*Sqrt[e*Cos[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(2*b^(11/2)*d) - (3*(21*a^4 - 28*a^2*b^2 + 5*b^4)*e^6*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(7*b^6*d*Sqrt[e*Cos[c + d*x]]) + (9*a^2*(a^2 - b^2)^2*e^6*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (c + d*x)/2, 2])/(2*b^6*(a^2 - b*(b - Sqrt[-a^2 + b^2]))*d*Sqrt[e*Cos[c + d*x]]) + (9*a^2*(a^2 - b^2)^2*e^6*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (c + d*x)/2, 2])/(2*b^6*(a^2 - b*(b + Sqrt[-a^2 + b^2]))*d*Sqrt[e*Cos[c + d*x]]) + (9*e^3*(e*Cos[c + d*x])^(5/2)*(7*a - 5*b*Sin[c + d*x]))/(35*b^3*d) - (e*(e*Cos[c + d*x])^(9/2))/(b*d*(a + b*Sin[c + d*x])) - (3*e^5*Sqrt[e*Cos[c + d*x]]*(21*a*(a^2 - b^2) - b*(7*a^2 - 5*b^2)*Sin[c + d*x]))/(7*b^5*d)","A",15,12,25,0.4800,1,"{2693, 2865, 2867, 2642, 2641, 2702, 2807, 2805, 329, 212, 208, 205}"
585,1,459,0,1.120323,"\int \frac{(e \cos (c+d x))^{9/2}}{(a+b \sin (c+d x))^2} \, dx","Int[(e*Cos[c + d*x])^(9/2)/(a + b*Sin[c + d*x])^2,x]","\frac{7 a e^{9/2} \left(b^2-a^2\right)^{3/4} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{2 b^{9/2} d}-\frac{7 a e^{9/2} \left(b^2-a^2\right)^{3/4} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{2 b^{9/2} d}+\frac{7 e^4 \left(5 a^2-3 b^2\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{5 b^4 d \sqrt{\cos (c+d x)}}-\frac{7 a^2 e^5 \left(a^2-b^2\right) \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{2 b^5 d \left(b-\sqrt{b^2-a^2}\right) \sqrt{e \cos (c+d x)}}-\frac{7 a^2 e^5 \left(a^2-b^2\right) \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{2 b^5 d \left(\sqrt{b^2-a^2}+b\right) \sqrt{e \cos (c+d x)}}+\frac{7 e^3 (e \cos (c+d x))^{3/2} (5 a-3 b \sin (c+d x))}{15 b^3 d}-\frac{e (e \cos (c+d x))^{7/2}}{b d (a+b \sin (c+d x))}","\frac{7 a e^{9/2} \left(b^2-a^2\right)^{3/4} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{2 b^{9/2} d}-\frac{7 a e^{9/2} \left(b^2-a^2\right)^{3/4} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{2 b^{9/2} d}+\frac{7 e^4 \left(5 a^2-3 b^2\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{5 b^4 d \sqrt{\cos (c+d x)}}-\frac{7 a^2 e^5 \left(a^2-b^2\right) \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{2 b^5 d \left(b-\sqrt{b^2-a^2}\right) \sqrt{e \cos (c+d x)}}-\frac{7 a^2 e^5 \left(a^2-b^2\right) \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{2 b^5 d \left(\sqrt{b^2-a^2}+b\right) \sqrt{e \cos (c+d x)}}+\frac{7 e^3 (e \cos (c+d x))^{3/2} (5 a-3 b \sin (c+d x))}{15 b^3 d}-\frac{e (e \cos (c+d x))^{7/2}}{b d (a+b \sin (c+d x))}",1,"(7*a*(-a^2 + b^2)^(3/4)*e^(9/2)*ArcTan[(Sqrt[b]*Sqrt[e*Cos[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(2*b^(9/2)*d) - (7*a*(-a^2 + b^2)^(3/4)*e^(9/2)*ArcTanh[(Sqrt[b]*Sqrt[e*Cos[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(2*b^(9/2)*d) + (7*(5*a^2 - 3*b^2)*e^4*Sqrt[e*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(5*b^4*d*Sqrt[Cos[c + d*x]]) - (7*a^2*(a^2 - b^2)*e^5*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (c + d*x)/2, 2])/(2*b^5*(b - Sqrt[-a^2 + b^2])*d*Sqrt[e*Cos[c + d*x]]) - (7*a^2*(a^2 - b^2)*e^5*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (c + d*x)/2, 2])/(2*b^5*(b + Sqrt[-a^2 + b^2])*d*Sqrt[e*Cos[c + d*x]]) + (7*e^3*(e*Cos[c + d*x])^(3/2)*(5*a - 3*b*Sin[c + d*x]))/(15*b^3*d) - (e*(e*Cos[c + d*x])^(7/2))/(b*d*(a + b*Sin[c + d*x]))","A",14,12,25,0.4800,1,"{2693, 2865, 2867, 2640, 2639, 2701, 2807, 2805, 329, 298, 205, 208}"
586,1,473,0,1.1184407,"\int \frac{(e \cos (c+d x))^{7/2}}{(a+b \sin (c+d x))^2} \, dx","Int[(e*Cos[c + d*x])^(7/2)/(a + b*Sin[c + d*x])^2,x]","-\frac{5 a e^{7/2} \sqrt[4]{b^2-a^2} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{2 b^{7/2} d}-\frac{5 a e^{7/2} \sqrt[4]{b^2-a^2} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{2 b^{7/2} d}+\frac{5 e^4 \left(3 a^2-b^2\right) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 b^4 d \sqrt{e \cos (c+d x)}}-\frac{5 a^2 e^4 \left(a^2-b^2\right) \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{2 b^4 d \left(a^2-b \left(b-\sqrt{b^2-a^2}\right)\right) \sqrt{e \cos (c+d x)}}-\frac{5 a^2 e^4 \left(a^2-b^2\right) \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{2 b^4 d \left(a^2-b \left(\sqrt{b^2-a^2}+b\right)\right) \sqrt{e \cos (c+d x)}}+\frac{5 e^3 \sqrt{e \cos (c+d x)} (3 a-b \sin (c+d x))}{3 b^3 d}-\frac{e (e \cos (c+d x))^{5/2}}{b d (a+b \sin (c+d x))}","-\frac{5 a e^{7/2} \sqrt[4]{b^2-a^2} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{2 b^{7/2} d}-\frac{5 a e^{7/2} \sqrt[4]{b^2-a^2} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{2 b^{7/2} d}+\frac{5 e^4 \left(3 a^2-b^2\right) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 b^4 d \sqrt{e \cos (c+d x)}}-\frac{5 a^2 e^4 \left(a^2-b^2\right) \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{2 b^4 d \left(a^2-b \left(b-\sqrt{b^2-a^2}\right)\right) \sqrt{e \cos (c+d x)}}-\frac{5 a^2 e^4 \left(a^2-b^2\right) \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{2 b^4 d \left(a^2-b \left(\sqrt{b^2-a^2}+b\right)\right) \sqrt{e \cos (c+d x)}}+\frac{5 e^3 \sqrt{e \cos (c+d x)} (3 a-b \sin (c+d x))}{3 b^3 d}-\frac{e (e \cos (c+d x))^{5/2}}{b d (a+b \sin (c+d x))}",1,"(-5*a*(-a^2 + b^2)^(1/4)*e^(7/2)*ArcTan[(Sqrt[b]*Sqrt[e*Cos[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(2*b^(7/2)*d) - (5*a*(-a^2 + b^2)^(1/4)*e^(7/2)*ArcTanh[(Sqrt[b]*Sqrt[e*Cos[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(2*b^(7/2)*d) + (5*(3*a^2 - b^2)*e^4*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(3*b^4*d*Sqrt[e*Cos[c + d*x]]) - (5*a^2*(a^2 - b^2)*e^4*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (c + d*x)/2, 2])/(2*b^4*(a^2 - b*(b - Sqrt[-a^2 + b^2]))*d*Sqrt[e*Cos[c + d*x]]) - (5*a^2*(a^2 - b^2)*e^4*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (c + d*x)/2, 2])/(2*b^4*(a^2 - b*(b + Sqrt[-a^2 + b^2]))*d*Sqrt[e*Cos[c + d*x]]) + (5*e^3*Sqrt[e*Cos[c + d*x]]*(3*a - b*Sin[c + d*x]))/(3*b^3*d) - (e*(e*Cos[c + d*x])^(5/2))/(b*d*(a + b*Sin[c + d*x]))","A",14,12,25,0.4800,1,"{2693, 2865, 2867, 2642, 2641, 2702, 2807, 2805, 329, 212, 208, 205}"
587,1,390,0,0.8191432,"\int \frac{(e \cos (c+d x))^{5/2}}{(a+b \sin (c+d x))^2} \, dx","Int[(e*Cos[c + d*x])^(5/2)/(a + b*Sin[c + d*x])^2,x]","\frac{3 a e^{5/2} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{2 b^{5/2} d \sqrt[4]{b^2-a^2}}-\frac{3 a e^{5/2} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{2 b^{5/2} d \sqrt[4]{b^2-a^2}}+\frac{3 a^2 e^3 \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{2 b^3 d \left(b-\sqrt{b^2-a^2}\right) \sqrt{e \cos (c+d x)}}+\frac{3 a^2 e^3 \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{2 b^3 d \left(\sqrt{b^2-a^2}+b\right) \sqrt{e \cos (c+d x)}}-\frac{e (e \cos (c+d x))^{3/2}}{b d (a+b \sin (c+d x))}-\frac{3 e^2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{b^2 d \sqrt{\cos (c+d x)}}","\frac{3 a e^{5/2} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{2 b^{5/2} d \sqrt[4]{b^2-a^2}}-\frac{3 a e^{5/2} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{2 b^{5/2} d \sqrt[4]{b^2-a^2}}+\frac{3 a^2 e^3 \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{2 b^3 d \left(b-\sqrt{b^2-a^2}\right) \sqrt{e \cos (c+d x)}}+\frac{3 a^2 e^3 \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{2 b^3 d \left(\sqrt{b^2-a^2}+b\right) \sqrt{e \cos (c+d x)}}-\frac{e (e \cos (c+d x))^{3/2}}{b d (a+b \sin (c+d x))}-\frac{3 e^2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{b^2 d \sqrt{\cos (c+d x)}}",1,"(3*a*e^(5/2)*ArcTan[(Sqrt[b]*Sqrt[e*Cos[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(2*b^(5/2)*(-a^2 + b^2)^(1/4)*d) - (3*a*e^(5/2)*ArcTanh[(Sqrt[b]*Sqrt[e*Cos[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(2*b^(5/2)*(-a^2 + b^2)^(1/4)*d) - (3*e^2*Sqrt[e*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(b^2*d*Sqrt[Cos[c + d*x]]) + (3*a^2*e^3*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (c + d*x)/2, 2])/(2*b^3*(b - Sqrt[-a^2 + b^2])*d*Sqrt[e*Cos[c + d*x]]) + (3*a^2*e^3*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (c + d*x)/2, 2])/(2*b^3*(b + Sqrt[-a^2 + b^2])*d*Sqrt[e*Cos[c + d*x]]) - (e*(e*Cos[c + d*x])^(3/2))/(b*d*(a + b*Sin[c + d*x]))","A",13,11,25,0.4400,1,"{2693, 2867, 2640, 2639, 2701, 2807, 2805, 329, 298, 205, 208}"
588,1,404,0,0.8922476,"\int \frac{(e \cos (c+d x))^{3/2}}{(a+b \sin (c+d x))^2} \, dx","Int[(e*Cos[c + d*x])^(3/2)/(a + b*Sin[c + d*x])^2,x]","-\frac{a e^{3/2} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{2 b^{3/2} d \left(b^2-a^2\right)^{3/4}}-\frac{a e^{3/2} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{2 b^{3/2} d \left(b^2-a^2\right)^{3/4}}+\frac{a^2 e^2 \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{2 b^2 d \left(a^2-b \left(b-\sqrt{b^2-a^2}\right)\right) \sqrt{e \cos (c+d x)}}+\frac{a^2 e^2 \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{2 b^2 d \left(a^2-b \left(\sqrt{b^2-a^2}+b\right)\right) \sqrt{e \cos (c+d x)}}-\frac{e \sqrt{e \cos (c+d x)}}{b d (a+b \sin (c+d x))}-\frac{e^2 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b^2 d \sqrt{e \cos (c+d x)}}","-\frac{a e^{3/2} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{2 b^{3/2} d \left(b^2-a^2\right)^{3/4}}-\frac{a e^{3/2} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{2 b^{3/2} d \left(b^2-a^2\right)^{3/4}}+\frac{a^2 e^2 \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{2 b^2 d \left(a^2-b \left(b-\sqrt{b^2-a^2}\right)\right) \sqrt{e \cos (c+d x)}}+\frac{a^2 e^2 \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{2 b^2 d \left(a^2-b \left(\sqrt{b^2-a^2}+b\right)\right) \sqrt{e \cos (c+d x)}}-\frac{e \sqrt{e \cos (c+d x)}}{b d (a+b \sin (c+d x))}-\frac{e^2 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b^2 d \sqrt{e \cos (c+d x)}}",1,"-(a*e^(3/2)*ArcTan[(Sqrt[b]*Sqrt[e*Cos[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(2*b^(3/2)*(-a^2 + b^2)^(3/4)*d) - (a*e^(3/2)*ArcTanh[(Sqrt[b]*Sqrt[e*Cos[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(2*b^(3/2)*(-a^2 + b^2)^(3/4)*d) - (e^2*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(b^2*d*Sqrt[e*Cos[c + d*x]]) + (a^2*e^2*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (c + d*x)/2, 2])/(2*b^2*(a^2 - b*(b - Sqrt[-a^2 + b^2]))*d*Sqrt[e*Cos[c + d*x]]) + (a^2*e^2*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (c + d*x)/2, 2])/(2*b^2*(a^2 - b*(b + Sqrt[-a^2 + b^2]))*d*Sqrt[e*Cos[c + d*x]]) - (e*Sqrt[e*Cos[c + d*x]])/(b*d*(a + b*Sin[c + d*x]))","A",13,11,25,0.4400,1,"{2693, 2867, 2642, 2641, 2702, 2807, 2805, 329, 212, 208, 205}"
589,1,422,0,0.868491,"\int \frac{\sqrt{e \cos (c+d x)}}{(a+b \sin (c+d x))^2} \, dx","Int[Sqrt[e*Cos[c + d*x]]/(a + b*Sin[c + d*x])^2,x]","\frac{b (e \cos (c+d x))^{3/2}}{d e \left(a^2-b^2\right) (a+b \sin (c+d x))}-\frac{a \sqrt{e} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{2 \sqrt{b} d \left(b^2-a^2\right)^{5/4}}+\frac{a \sqrt{e} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{2 \sqrt{b} d \left(b^2-a^2\right)^{5/4}}+\frac{E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{d \left(a^2-b^2\right) \sqrt{\cos (c+d x)}}+\frac{a^2 e \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{2 b d \left(a^2-b^2\right) \left(b-\sqrt{b^2-a^2}\right) \sqrt{e \cos (c+d x)}}+\frac{a^2 e \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{2 b d \left(a^2-b^2\right) \left(\sqrt{b^2-a^2}+b\right) \sqrt{e \cos (c+d x)}}","\frac{b (e \cos (c+d x))^{3/2}}{d e \left(a^2-b^2\right) (a+b \sin (c+d x))}-\frac{a \sqrt{e} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{2 \sqrt{b} d \left(b^2-a^2\right)^{5/4}}+\frac{a \sqrt{e} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{2 \sqrt{b} d \left(b^2-a^2\right)^{5/4}}+\frac{E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{d \left(a^2-b^2\right) \sqrt{\cos (c+d x)}}+\frac{a^2 e \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{2 b d \left(a^2-b^2\right) \left(b-\sqrt{b^2-a^2}\right) \sqrt{e \cos (c+d x)}}+\frac{a^2 e \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{2 b d \left(a^2-b^2\right) \left(\sqrt{b^2-a^2}+b\right) \sqrt{e \cos (c+d x)}}",1,"-(a*Sqrt[e]*ArcTan[(Sqrt[b]*Sqrt[e*Cos[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(2*Sqrt[b]*(-a^2 + b^2)^(5/4)*d) + (a*Sqrt[e]*ArcTanh[(Sqrt[b]*Sqrt[e*Cos[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(2*Sqrt[b]*(-a^2 + b^2)^(5/4)*d) + (Sqrt[e*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/((a^2 - b^2)*d*Sqrt[Cos[c + d*x]]) + (a^2*e*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (c + d*x)/2, 2])/(2*b*(a^2 - b^2)*(b - Sqrt[-a^2 + b^2])*d*Sqrt[e*Cos[c + d*x]]) + (a^2*e*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (c + d*x)/2, 2])/(2*b*(a^2 - b^2)*(b + Sqrt[-a^2 + b^2])*d*Sqrt[e*Cos[c + d*x]]) + (b*(e*Cos[c + d*x])^(3/2))/((a^2 - b^2)*d*e*(a + b*Sin[c + d*x]))","A",13,11,25,0.4400,1,"{2694, 2867, 2640, 2639, 2701, 2807, 2805, 329, 298, 205, 208}"
590,1,429,0,0.8980042,"\int \frac{1}{\sqrt{e \cos (c+d x)} (a+b \sin (c+d x))^2} \, dx","Int[1/(Sqrt[e*Cos[c + d*x]]*(a + b*Sin[c + d*x])^2),x]","\frac{b \sqrt{e \cos (c+d x)}}{d e \left(a^2-b^2\right) (a+b \sin (c+d x))}+\frac{3 a \sqrt{b} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{2 d \sqrt{e} \left(b^2-a^2\right)^{7/4}}+\frac{3 a \sqrt{b} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{2 d \sqrt{e} \left(b^2-a^2\right)^{7/4}}-\frac{\sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d \left(a^2-b^2\right) \sqrt{e \cos (c+d x)}}+\frac{3 a^2 \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{2 d \left(a^2-b^2\right) \left(a^2-b \left(b-\sqrt{b^2-a^2}\right)\right) \sqrt{e \cos (c+d x)}}+\frac{3 a^2 \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{2 d \left(a^2-b^2\right) \left(a^2-b \left(\sqrt{b^2-a^2}+b\right)\right) \sqrt{e \cos (c+d x)}}","\frac{b \sqrt{e \cos (c+d x)}}{d e \left(a^2-b^2\right) (a+b \sin (c+d x))}+\frac{3 a \sqrt{b} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{2 d \sqrt{e} \left(b^2-a^2\right)^{7/4}}+\frac{3 a \sqrt{b} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{2 d \sqrt{e} \left(b^2-a^2\right)^{7/4}}-\frac{\sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d \left(a^2-b^2\right) \sqrt{e \cos (c+d x)}}+\frac{3 a^2 \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{2 d \left(a^2-b^2\right) \left(a^2-b \left(b-\sqrt{b^2-a^2}\right)\right) \sqrt{e \cos (c+d x)}}+\frac{3 a^2 \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{2 d \left(a^2-b^2\right) \left(a^2-b \left(\sqrt{b^2-a^2}+b\right)\right) \sqrt{e \cos (c+d x)}}",1,"(3*a*Sqrt[b]*ArcTan[(Sqrt[b]*Sqrt[e*Cos[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(2*(-a^2 + b^2)^(7/4)*d*Sqrt[e]) + (3*a*Sqrt[b]*ArcTanh[(Sqrt[b]*Sqrt[e*Cos[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(2*(-a^2 + b^2)^(7/4)*d*Sqrt[e]) - (Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/((a^2 - b^2)*d*Sqrt[e*Cos[c + d*x]]) + (3*a^2*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (c + d*x)/2, 2])/(2*(a^2 - b^2)*(a^2 - b*(b - Sqrt[-a^2 + b^2]))*d*Sqrt[e*Cos[c + d*x]]) + (3*a^2*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (c + d*x)/2, 2])/(2*(a^2 - b^2)*(a^2 - b*(b + Sqrt[-a^2 + b^2]))*d*Sqrt[e*Cos[c + d*x]]) + (b*Sqrt[e*Cos[c + d*x]])/((a^2 - b^2)*d*e*(a + b*Sin[c + d*x]))","A",13,11,25,0.4400,1,"{2694, 2867, 2642, 2641, 2702, 2807, 2805, 329, 212, 208, 205}"
591,1,492,0,1.2185296,"\int \frac{1}{(e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^2} \, dx","Int[1/((e*Cos[c + d*x])^(3/2)*(a + b*Sin[c + d*x])^2),x]","-\frac{5 a b^{3/2} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{2 d e^{3/2} \left(b^2-a^2\right)^{9/4}}+\frac{5 a b^{3/2} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{2 d e^{3/2} \left(b^2-a^2\right)^{9/4}}-\frac{\left(2 a^2+3 b^2\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{d e^2 \left(a^2-b^2\right)^2 \sqrt{\cos (c+d x)}}-\frac{5 a b-\left(2 a^2+3 b^2\right) \sin (c+d x)}{d e \left(a^2-b^2\right)^2 \sqrt{e \cos (c+d x)}}+\frac{b}{d e \left(a^2-b^2\right) \sqrt{e \cos (c+d x)} (a+b \sin (c+d x))}-\frac{5 a^2 b \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{2 d e \left(a^2-b^2\right)^2 \left(b-\sqrt{b^2-a^2}\right) \sqrt{e \cos (c+d x)}}-\frac{5 a^2 b \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{2 d e \left(a^2-b^2\right)^2 \left(\sqrt{b^2-a^2}+b\right) \sqrt{e \cos (c+d x)}}","-\frac{5 a b^{3/2} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{2 d e^{3/2} \left(b^2-a^2\right)^{9/4}}+\frac{5 a b^{3/2} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{2 d e^{3/2} \left(b^2-a^2\right)^{9/4}}-\frac{\left(2 a^2+3 b^2\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{d e^2 \left(a^2-b^2\right)^2 \sqrt{\cos (c+d x)}}-\frac{5 a b-\left(2 a^2+3 b^2\right) \sin (c+d x)}{d e \left(a^2-b^2\right)^2 \sqrt{e \cos (c+d x)}}+\frac{b}{d e \left(a^2-b^2\right) \sqrt{e \cos (c+d x)} (a+b \sin (c+d x))}-\frac{5 a^2 b \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{2 d e \left(a^2-b^2\right)^2 \left(b-\sqrt{b^2-a^2}\right) \sqrt{e \cos (c+d x)}}-\frac{5 a^2 b \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{2 d e \left(a^2-b^2\right)^2 \left(\sqrt{b^2-a^2}+b\right) \sqrt{e \cos (c+d x)}}",1,"(-5*a*b^(3/2)*ArcTan[(Sqrt[b]*Sqrt[e*Cos[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(2*(-a^2 + b^2)^(9/4)*d*e^(3/2)) + (5*a*b^(3/2)*ArcTanh[(Sqrt[b]*Sqrt[e*Cos[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(2*(-a^2 + b^2)^(9/4)*d*e^(3/2)) - ((2*a^2 + 3*b^2)*Sqrt[e*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/((a^2 - b^2)^2*d*e^2*Sqrt[Cos[c + d*x]]) - (5*a^2*b*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (c + d*x)/2, 2])/(2*(a^2 - b^2)^2*(b - Sqrt[-a^2 + b^2])*d*e*Sqrt[e*Cos[c + d*x]]) - (5*a^2*b*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (c + d*x)/2, 2])/(2*(a^2 - b^2)^2*(b + Sqrt[-a^2 + b^2])*d*e*Sqrt[e*Cos[c + d*x]]) + b/((a^2 - b^2)*d*e*Sqrt[e*Cos[c + d*x]]*(a + b*Sin[c + d*x])) - (5*a*b - (2*a^2 + 3*b^2)*Sin[c + d*x])/((a^2 - b^2)^2*d*e*Sqrt[e*Cos[c + d*x]])","A",14,12,25,0.4800,1,"{2694, 2866, 2867, 2640, 2639, 2701, 2807, 2805, 329, 298, 205, 208}"
592,1,514,0,1.3091979,"\int \frac{1}{(e \cos (c+d x))^{5/2} (a+b \sin (c+d x))^2} \, dx","Int[1/((e*Cos[c + d*x])^(5/2)*(a + b*Sin[c + d*x])^2),x]","\frac{7 a b^{5/2} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{2 d e^{5/2} \left(b^2-a^2\right)^{11/4}}+\frac{7 a b^{5/2} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{2 d e^{5/2} \left(b^2-a^2\right)^{11/4}}+\frac{\left(2 a^2+5 b^2\right) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d e^2 \left(a^2-b^2\right)^2 \sqrt{e \cos (c+d x)}}-\frac{7 a^2 b^2 \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{2 d e^2 \left(a^2-b^2\right)^2 \left(a^2-b \left(b-\sqrt{b^2-a^2}\right)\right) \sqrt{e \cos (c+d x)}}-\frac{7 a^2 b^2 \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{2 d e^2 \left(a^2-b^2\right)^2 \left(a^2-b \left(\sqrt{b^2-a^2}+b\right)\right) \sqrt{e \cos (c+d x)}}+\frac{b}{d e \left(a^2-b^2\right) (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))}-\frac{7 a b-\left(2 a^2+5 b^2\right) \sin (c+d x)}{3 d e \left(a^2-b^2\right)^2 (e \cos (c+d x))^{3/2}}","\frac{7 a b^{5/2} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{2 d e^{5/2} \left(b^2-a^2\right)^{11/4}}+\frac{7 a b^{5/2} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{2 d e^{5/2} \left(b^2-a^2\right)^{11/4}}+\frac{\left(2 a^2+5 b^2\right) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d e^2 \left(a^2-b^2\right)^2 \sqrt{e \cos (c+d x)}}-\frac{7 a^2 b^2 \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{2 d e^2 \left(a^2-b^2\right)^2 \left(a^2-b \left(b-\sqrt{b^2-a^2}\right)\right) \sqrt{e \cos (c+d x)}}-\frac{7 a^2 b^2 \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{2 d e^2 \left(a^2-b^2\right)^2 \left(a^2-b \left(\sqrt{b^2-a^2}+b\right)\right) \sqrt{e \cos (c+d x)}}+\frac{b}{d e \left(a^2-b^2\right) (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))}-\frac{7 a b-\left(2 a^2+5 b^2\right) \sin (c+d x)}{3 d e \left(a^2-b^2\right)^2 (e \cos (c+d x))^{3/2}}",1,"(7*a*b^(5/2)*ArcTan[(Sqrt[b]*Sqrt[e*Cos[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(2*(-a^2 + b^2)^(11/4)*d*e^(5/2)) + (7*a*b^(5/2)*ArcTanh[(Sqrt[b]*Sqrt[e*Cos[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(2*(-a^2 + b^2)^(11/4)*d*e^(5/2)) + ((2*a^2 + 5*b^2)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(3*(a^2 - b^2)^2*d*e^2*Sqrt[e*Cos[c + d*x]]) - (7*a^2*b^2*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (c + d*x)/2, 2])/(2*(a^2 - b^2)^2*(a^2 - b*(b - Sqrt[-a^2 + b^2]))*d*e^2*Sqrt[e*Cos[c + d*x]]) - (7*a^2*b^2*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (c + d*x)/2, 2])/(2*(a^2 - b^2)^2*(a^2 - b*(b + Sqrt[-a^2 + b^2]))*d*e^2*Sqrt[e*Cos[c + d*x]]) + b/((a^2 - b^2)*d*e*(e*Cos[c + d*x])^(3/2)*(a + b*Sin[c + d*x])) - (7*a*b - (2*a^2 + 5*b^2)*Sin[c + d*x])/(3*(a^2 - b^2)^2*d*e*(e*Cos[c + d*x])^(3/2))","A",14,12,25,0.4800,1,"{2694, 2866, 2867, 2642, 2641, 2702, 2807, 2805, 329, 212, 208, 205}"
593,1,574,0,1.6232989,"\int \frac{1}{(e \cos (c+d x))^{7/2} (a+b \sin (c+d x))^2} \, dx","Int[1/((e*Cos[c + d*x])^(7/2)*(a + b*Sin[c + d*x])^2),x]","\frac{3 \left(\left(-10 a^2 b^2+2 a^4-7 b^4\right) \sin (c+d x)+15 a b^3\right)}{5 d e^3 \left(a^2-b^2\right)^3 \sqrt{e \cos (c+d x)}}-\frac{9 a b^{7/2} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{2 d e^{7/2} \left(b^2-a^2\right)^{13/4}}+\frac{9 a b^{7/2} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{2 d e^{7/2} \left(b^2-a^2\right)^{13/4}}-\frac{3 \left(-10 a^2 b^2+2 a^4-7 b^4\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{5 d e^4 \left(a^2-b^2\right)^3 \sqrt{\cos (c+d x)}}+\frac{9 a^2 b^3 \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{2 d e^3 \left(a^2-b^2\right)^3 \left(b-\sqrt{b^2-a^2}\right) \sqrt{e \cos (c+d x)}}+\frac{9 a^2 b^3 \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{2 d e^3 \left(a^2-b^2\right)^3 \left(\sqrt{b^2-a^2}+b\right) \sqrt{e \cos (c+d x)}}+\frac{b}{d e \left(a^2-b^2\right) (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))}-\frac{9 a b-\left(2 a^2+7 b^2\right) \sin (c+d x)}{5 d e \left(a^2-b^2\right)^2 (e \cos (c+d x))^{5/2}}","\frac{3 \left(\left(-10 a^2 b^2+2 a^4-7 b^4\right) \sin (c+d x)+15 a b^3\right)}{5 d e^3 \left(a^2-b^2\right)^3 \sqrt{e \cos (c+d x)}}-\frac{9 a b^{7/2} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{2 d e^{7/2} \left(b^2-a^2\right)^{13/4}}+\frac{9 a b^{7/2} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{2 d e^{7/2} \left(b^2-a^2\right)^{13/4}}-\frac{3 \left(-10 a^2 b^2+2 a^4-7 b^4\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{5 d e^4 \left(a^2-b^2\right)^3 \sqrt{\cos (c+d x)}}+\frac{9 a^2 b^3 \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{2 d e^3 \left(a^2-b^2\right)^3 \left(b-\sqrt{b^2-a^2}\right) \sqrt{e \cos (c+d x)}}+\frac{9 a^2 b^3 \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{2 d e^3 \left(a^2-b^2\right)^3 \left(\sqrt{b^2-a^2}+b\right) \sqrt{e \cos (c+d x)}}+\frac{b}{d e \left(a^2-b^2\right) (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))}-\frac{9 a b-\left(2 a^2+7 b^2\right) \sin (c+d x)}{5 d e \left(a^2-b^2\right)^2 (e \cos (c+d x))^{5/2}}",1,"(-9*a*b^(7/2)*ArcTan[(Sqrt[b]*Sqrt[e*Cos[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(2*(-a^2 + b^2)^(13/4)*d*e^(7/2)) + (9*a*b^(7/2)*ArcTanh[(Sqrt[b]*Sqrt[e*Cos[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(2*(-a^2 + b^2)^(13/4)*d*e^(7/2)) - (3*(2*a^4 - 10*a^2*b^2 - 7*b^4)*Sqrt[e*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(5*(a^2 - b^2)^3*d*e^4*Sqrt[Cos[c + d*x]]) + (9*a^2*b^3*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (c + d*x)/2, 2])/(2*(a^2 - b^2)^3*(b - Sqrt[-a^2 + b^2])*d*e^3*Sqrt[e*Cos[c + d*x]]) + (9*a^2*b^3*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (c + d*x)/2, 2])/(2*(a^2 - b^2)^3*(b + Sqrt[-a^2 + b^2])*d*e^3*Sqrt[e*Cos[c + d*x]]) + b/((a^2 - b^2)*d*e*(e*Cos[c + d*x])^(5/2)*(a + b*Sin[c + d*x])) - (9*a*b - (2*a^2 + 7*b^2)*Sin[c + d*x])/(5*(a^2 - b^2)^2*d*e*(e*Cos[c + d*x])^(5/2)) + (3*(15*a*b^3 + (2*a^4 - 10*a^2*b^2 - 7*b^4)*Sin[c + d*x]))/(5*(a^2 - b^2)^3*d*e^3*Sqrt[e*Cos[c + d*x]])","A",15,12,25,0.4800,1,"{2694, 2866, 2867, 2640, 2639, 2701, 2807, 2805, 329, 298, 205, 208}"
594,1,575,0,1.4147497,"\int \frac{(e \cos (c+d x))^{13/2}}{(a+b \sin (c+d x))^3} \, dx","Int[(e*Cos[c + d*x])^(13/2)/(a + b*Sin[c + d*x])^3,x]","\frac{11 e^5 (e \cos (c+d x))^{3/2} \left(5 \left(9 a^2-2 b^2\right)-27 a b \sin (c+d x)\right)}{60 b^5 d}-\frac{11 e^{13/2} \left(-11 a^2 b^2+9 a^4+2 b^4\right) \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{8 b^{13/2} d \sqrt[4]{b^2-a^2}}+\frac{11 e^{13/2} \left(-11 a^2 b^2+9 a^4+2 b^4\right) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{8 b^{13/2} d \sqrt[4]{b^2-a^2}}+\frac{11 a e^6 \left(45 a^2-37 b^2\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{20 b^6 d \sqrt{\cos (c+d x)}}-\frac{11 a e^7 \left(-11 a^2 b^2+9 a^4+2 b^4\right) \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{8 b^7 d \left(b-\sqrt{b^2-a^2}\right) \sqrt{e \cos (c+d x)}}-\frac{11 a e^7 \left(-11 a^2 b^2+9 a^4+2 b^4\right) \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{8 b^7 d \left(\sqrt{b^2-a^2}+b\right) \sqrt{e \cos (c+d x)}}-\frac{11 e^3 (e \cos (c+d x))^{7/2} (9 a+2 b \sin (c+d x))}{28 b^3 d (a+b \sin (c+d x))}-\frac{e (e \cos (c+d x))^{11/2}}{2 b d (a+b \sin (c+d x))^2}","\frac{11 e^5 (e \cos (c+d x))^{3/2} \left(5 \left(9 a^2-2 b^2\right)-27 a b \sin (c+d x)\right)}{60 b^5 d}-\frac{11 e^{13/2} \left(-11 a^2 b^2+9 a^4+2 b^4\right) \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{8 b^{13/2} d \sqrt[4]{b^2-a^2}}+\frac{11 e^{13/2} \left(-11 a^2 b^2+9 a^4+2 b^4\right) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{8 b^{13/2} d \sqrt[4]{b^2-a^2}}+\frac{11 a e^6 \left(45 a^2-37 b^2\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{20 b^6 d \sqrt{\cos (c+d x)}}-\frac{11 a e^7 \left(-11 a^2 b^2+9 a^4+2 b^4\right) \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{8 b^7 d \left(b-\sqrt{b^2-a^2}\right) \sqrt{e \cos (c+d x)}}-\frac{11 a e^7 \left(-11 a^2 b^2+9 a^4+2 b^4\right) \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{8 b^7 d \left(\sqrt{b^2-a^2}+b\right) \sqrt{e \cos (c+d x)}}-\frac{11 e^3 (e \cos (c+d x))^{7/2} (9 a+2 b \sin (c+d x))}{28 b^3 d (a+b \sin (c+d x))}-\frac{e (e \cos (c+d x))^{11/2}}{2 b d (a+b \sin (c+d x))^2}",1,"(-11*(9*a^4 - 11*a^2*b^2 + 2*b^4)*e^(13/2)*ArcTan[(Sqrt[b]*Sqrt[e*Cos[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(8*b^(13/2)*(-a^2 + b^2)^(1/4)*d) + (11*(9*a^4 - 11*a^2*b^2 + 2*b^4)*e^(13/2)*ArcTanh[(Sqrt[b]*Sqrt[e*Cos[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(8*b^(13/2)*(-a^2 + b^2)^(1/4)*d) + (11*a*(45*a^2 - 37*b^2)*e^6*Sqrt[e*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(20*b^6*d*Sqrt[Cos[c + d*x]]) - (11*a*(9*a^4 - 11*a^2*b^2 + 2*b^4)*e^7*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (c + d*x)/2, 2])/(8*b^7*(b - Sqrt[-a^2 + b^2])*d*Sqrt[e*Cos[c + d*x]]) - (11*a*(9*a^4 - 11*a^2*b^2 + 2*b^4)*e^7*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (c + d*x)/2, 2])/(8*b^7*(b + Sqrt[-a^2 + b^2])*d*Sqrt[e*Cos[c + d*x]]) - (e*(e*Cos[c + d*x])^(11/2))/(2*b*d*(a + b*Sin[c + d*x])^2) - (11*e^3*(e*Cos[c + d*x])^(7/2)*(9*a + 2*b*Sin[c + d*x]))/(28*b^3*d*(a + b*Sin[c + d*x])) + (11*e^5*(e*Cos[c + d*x])^(3/2)*(5*(9*a^2 - 2*b^2) - 27*a*b*Sin[c + d*x]))/(60*b^5*d)","A",15,13,25,0.5200,1,"{2693, 2863, 2865, 2867, 2640, 2639, 2701, 2807, 2805, 329, 298, 205, 208}"
595,1,589,0,1.4868279,"\int \frac{(e \cos (c+d x))^{11/2}}{(a+b \sin (c+d x))^3} \, dx","Int[(e*Cos[c + d*x])^(11/2)/(a + b*Sin[c + d*x])^3,x]","\frac{3 e^5 \sqrt{e \cos (c+d x)} \left(3 \left(7 a^2-2 b^2\right)-7 a b \sin (c+d x)\right)}{4 b^5 d}+\frac{9 e^{11/2} \left(-9 a^2 b^2+7 a^4+2 b^4\right) \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{8 b^{11/2} d \left(b^2-a^2\right)^{3/4}}+\frac{9 e^{11/2} \left(-9 a^2 b^2+7 a^4+2 b^4\right) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{8 b^{11/2} d \left(b^2-a^2\right)^{3/4}}+\frac{3 a e^6 \left(21 a^2-13 b^2\right) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{4 b^6 d \sqrt{e \cos (c+d x)}}-\frac{9 a e^6 \left(-9 a^2 b^2+7 a^4+2 b^4\right) \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{8 b^6 d \left(a^2-b \left(b-\sqrt{b^2-a^2}\right)\right) \sqrt{e \cos (c+d x)}}-\frac{9 a e^6 \left(-9 a^2 b^2+7 a^4+2 b^4\right) \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{8 b^6 d \left(a^2-b \left(\sqrt{b^2-a^2}+b\right)\right) \sqrt{e \cos (c+d x)}}-\frac{9 e^3 (e \cos (c+d x))^{5/2} (7 a+2 b \sin (c+d x))}{20 b^3 d (a+b \sin (c+d x))}-\frac{e (e \cos (c+d x))^{9/2}}{2 b d (a+b \sin (c+d x))^2}","\frac{3 e^5 \sqrt{e \cos (c+d x)} \left(3 \left(7 a^2-2 b^2\right)-7 a b \sin (c+d x)\right)}{4 b^5 d}+\frac{9 e^{11/2} \left(-9 a^2 b^2+7 a^4+2 b^4\right) \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{8 b^{11/2} d \left(b^2-a^2\right)^{3/4}}+\frac{9 e^{11/2} \left(-9 a^2 b^2+7 a^4+2 b^4\right) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{8 b^{11/2} d \left(b^2-a^2\right)^{3/4}}+\frac{3 a e^6 \left(21 a^2-13 b^2\right) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{4 b^6 d \sqrt{e \cos (c+d x)}}-\frac{9 a e^6 \left(-9 a^2 b^2+7 a^4+2 b^4\right) \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{8 b^6 d \left(a^2-b \left(b-\sqrt{b^2-a^2}\right)\right) \sqrt{e \cos (c+d x)}}-\frac{9 a e^6 \left(-9 a^2 b^2+7 a^4+2 b^4\right) \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{8 b^6 d \left(a^2-b \left(\sqrt{b^2-a^2}+b\right)\right) \sqrt{e \cos (c+d x)}}-\frac{9 e^3 (e \cos (c+d x))^{5/2} (7 a+2 b \sin (c+d x))}{20 b^3 d (a+b \sin (c+d x))}-\frac{e (e \cos (c+d x))^{9/2}}{2 b d (a+b \sin (c+d x))^2}",1,"(9*(7*a^4 - 9*a^2*b^2 + 2*b^4)*e^(11/2)*ArcTan[(Sqrt[b]*Sqrt[e*Cos[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(8*b^(11/2)*(-a^2 + b^2)^(3/4)*d) + (9*(7*a^4 - 9*a^2*b^2 + 2*b^4)*e^(11/2)*ArcTanh[(Sqrt[b]*Sqrt[e*Cos[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(8*b^(11/2)*(-a^2 + b^2)^(3/4)*d) + (3*a*(21*a^2 - 13*b^2)*e^6*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(4*b^6*d*Sqrt[e*Cos[c + d*x]]) - (9*a*(7*a^4 - 9*a^2*b^2 + 2*b^4)*e^6*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (c + d*x)/2, 2])/(8*b^6*(a^2 - b*(b - Sqrt[-a^2 + b^2]))*d*Sqrt[e*Cos[c + d*x]]) - (9*a*(7*a^4 - 9*a^2*b^2 + 2*b^4)*e^6*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (c + d*x)/2, 2])/(8*b^6*(a^2 - b*(b + Sqrt[-a^2 + b^2]))*d*Sqrt[e*Cos[c + d*x]]) - (e*(e*Cos[c + d*x])^(9/2))/(2*b*d*(a + b*Sin[c + d*x])^2) - (9*e^3*(e*Cos[c + d*x])^(5/2)*(7*a + 2*b*Sin[c + d*x]))/(20*b^3*d*(a + b*Sin[c + d*x])) + (3*e^5*Sqrt[e*Cos[c + d*x]]*(3*(7*a^2 - 2*b^2) - 7*a*b*Sin[c + d*x]))/(4*b^5*d)","A",15,13,25,0.5200,1,"{2693, 2863, 2865, 2867, 2642, 2641, 2702, 2807, 2805, 329, 212, 208, 205}"
596,1,483,0,1.0753863,"\int \frac{(e \cos (c+d x))^{9/2}}{(a+b \sin (c+d x))^3} \, dx","Int[(e*Cos[c + d*x])^(9/2)/(a + b*Sin[c + d*x])^3,x]","\frac{7 e^{9/2} \left(5 a^2-2 b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{8 b^{9/2} d \sqrt[4]{b^2-a^2}}-\frac{7 e^{9/2} \left(5 a^2-2 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{8 b^{9/2} d \sqrt[4]{b^2-a^2}}+\frac{7 a e^5 \left(5 a^2-2 b^2\right) \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{8 b^5 d \left(b-\sqrt{b^2-a^2}\right) \sqrt{e \cos (c+d x)}}+\frac{7 a e^5 \left(5 a^2-2 b^2\right) \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{8 b^5 d \left(\sqrt{b^2-a^2}+b\right) \sqrt{e \cos (c+d x)}}-\frac{7 e^3 (e \cos (c+d x))^{3/2} (5 a+2 b \sin (c+d x))}{12 b^3 d (a+b \sin (c+d x))}-\frac{35 a e^4 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{4 b^4 d \sqrt{\cos (c+d x)}}-\frac{e (e \cos (c+d x))^{7/2}}{2 b d (a+b \sin (c+d x))^2}","\frac{7 e^{9/2} \left(5 a^2-2 b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{8 b^{9/2} d \sqrt[4]{b^2-a^2}}-\frac{7 e^{9/2} \left(5 a^2-2 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{8 b^{9/2} d \sqrt[4]{b^2-a^2}}+\frac{7 a e^5 \left(5 a^2-2 b^2\right) \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{8 b^5 d \left(b-\sqrt{b^2-a^2}\right) \sqrt{e \cos (c+d x)}}+\frac{7 a e^5 \left(5 a^2-2 b^2\right) \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{8 b^5 d \left(\sqrt{b^2-a^2}+b\right) \sqrt{e \cos (c+d x)}}-\frac{7 e^3 (e \cos (c+d x))^{3/2} (5 a+2 b \sin (c+d x))}{12 b^3 d (a+b \sin (c+d x))}-\frac{35 a e^4 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{4 b^4 d \sqrt{\cos (c+d x)}}-\frac{e (e \cos (c+d x))^{7/2}}{2 b d (a+b \sin (c+d x))^2}",1,"(7*(5*a^2 - 2*b^2)*e^(9/2)*ArcTan[(Sqrt[b]*Sqrt[e*Cos[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(8*b^(9/2)*(-a^2 + b^2)^(1/4)*d) - (7*(5*a^2 - 2*b^2)*e^(9/2)*ArcTanh[(Sqrt[b]*Sqrt[e*Cos[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(8*b^(9/2)*(-a^2 + b^2)^(1/4)*d) - (35*a*e^4*Sqrt[e*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(4*b^4*d*Sqrt[Cos[c + d*x]]) + (7*a*(5*a^2 - 2*b^2)*e^5*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (c + d*x)/2, 2])/(8*b^5*(b - Sqrt[-a^2 + b^2])*d*Sqrt[e*Cos[c + d*x]]) + (7*a*(5*a^2 - 2*b^2)*e^5*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (c + d*x)/2, 2])/(8*b^5*(b + Sqrt[-a^2 + b^2])*d*Sqrt[e*Cos[c + d*x]]) - (e*(e*Cos[c + d*x])^(7/2))/(2*b*d*(a + b*Sin[c + d*x])^2) - (7*e^3*(e*Cos[c + d*x])^(3/2)*(5*a + 2*b*Sin[c + d*x]))/(12*b^3*d*(a + b*Sin[c + d*x]))","A",14,12,25,0.4800,1,"{2693, 2863, 2867, 2640, 2639, 2701, 2807, 2805, 329, 298, 205, 208}"
597,1,497,0,1.0785605,"\int \frac{(e \cos (c+d x))^{7/2}}{(a+b \sin (c+d x))^3} \, dx","Int[(e*Cos[c + d*x])^(7/2)/(a + b*Sin[c + d*x])^3,x]","-\frac{5 e^{7/2} \left(3 a^2-2 b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{8 b^{7/2} d \left(b^2-a^2\right)^{3/4}}-\frac{5 e^{7/2} \left(3 a^2-2 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{8 b^{7/2} d \left(b^2-a^2\right)^{3/4}}+\frac{5 a e^4 \left(3 a^2-2 b^2\right) \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{8 b^4 d \left(a^2-b \left(b-\sqrt{b^2-a^2}\right)\right) \sqrt{e \cos (c+d x)}}+\frac{5 a e^4 \left(3 a^2-2 b^2\right) \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{8 b^4 d \left(a^2-b \left(\sqrt{b^2-a^2}+b\right)\right) \sqrt{e \cos (c+d x)}}-\frac{5 e^3 \sqrt{e \cos (c+d x)} (3 a+2 b \sin (c+d x))}{4 b^3 d (a+b \sin (c+d x))}-\frac{15 a e^4 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{4 b^4 d \sqrt{e \cos (c+d x)}}-\frac{e (e \cos (c+d x))^{5/2}}{2 b d (a+b \sin (c+d x))^2}","-\frac{5 e^{7/2} \left(3 a^2-2 b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{8 b^{7/2} d \left(b^2-a^2\right)^{3/4}}-\frac{5 e^{7/2} \left(3 a^2-2 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{8 b^{7/2} d \left(b^2-a^2\right)^{3/4}}+\frac{5 a e^4 \left(3 a^2-2 b^2\right) \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{8 b^4 d \left(a^2-b \left(b-\sqrt{b^2-a^2}\right)\right) \sqrt{e \cos (c+d x)}}+\frac{5 a e^4 \left(3 a^2-2 b^2\right) \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{8 b^4 d \left(a^2-b \left(\sqrt{b^2-a^2}+b\right)\right) \sqrt{e \cos (c+d x)}}-\frac{5 e^3 \sqrt{e \cos (c+d x)} (3 a+2 b \sin (c+d x))}{4 b^3 d (a+b \sin (c+d x))}-\frac{15 a e^4 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{4 b^4 d \sqrt{e \cos (c+d x)}}-\frac{e (e \cos (c+d x))^{5/2}}{2 b d (a+b \sin (c+d x))^2}",1,"(-5*(3*a^2 - 2*b^2)*e^(7/2)*ArcTan[(Sqrt[b]*Sqrt[e*Cos[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(8*b^(7/2)*(-a^2 + b^2)^(3/4)*d) - (5*(3*a^2 - 2*b^2)*e^(7/2)*ArcTanh[(Sqrt[b]*Sqrt[e*Cos[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(8*b^(7/2)*(-a^2 + b^2)^(3/4)*d) - (15*a*e^4*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(4*b^4*d*Sqrt[e*Cos[c + d*x]]) + (5*a*(3*a^2 - 2*b^2)*e^4*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (c + d*x)/2, 2])/(8*b^4*(a^2 - b*(b - Sqrt[-a^2 + b^2]))*d*Sqrt[e*Cos[c + d*x]]) + (5*a*(3*a^2 - 2*b^2)*e^4*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (c + d*x)/2, 2])/(8*b^4*(a^2 - b*(b + Sqrt[-a^2 + b^2]))*d*Sqrt[e*Cos[c + d*x]]) - (e*(e*Cos[c + d*x])^(5/2))/(2*b*d*(a + b*Sin[c + d*x])^2) - (5*e^3*Sqrt[e*Cos[c + d*x]]*(3*a + 2*b*Sin[c + d*x]))/(4*b^3*d*(a + b*Sin[c + d*x]))","A",14,12,25,0.4800,1,"{2693, 2863, 2867, 2642, 2641, 2702, 2807, 2805, 329, 212, 208, 205}"
598,1,505,0,1.1047297,"\int \frac{(e \cos (c+d x))^{5/2}}{(a+b \sin (c+d x))^3} \, dx","Int[(e*Cos[c + d*x])^(5/2)/(a + b*Sin[c + d*x])^3,x]","\frac{3 e^{5/2} \left(a^2-2 b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{8 b^{5/2} d \left(b^2-a^2\right)^{5/4}}-\frac{3 e^{5/2} \left(a^2-2 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{8 b^{5/2} d \left(b^2-a^2\right)^{5/4}}+\frac{3 a e^2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{4 b^2 d \left(a^2-b^2\right) \sqrt{\cos (c+d x)}}-\frac{3 a e^3 \left(a^2-2 b^2\right) \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{8 b^3 d \left(a^2-b^2\right) \left(b-\sqrt{b^2-a^2}\right) \sqrt{e \cos (c+d x)}}-\frac{3 a e^3 \left(a^2-2 b^2\right) \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{8 b^3 d \left(a^2-b^2\right) \left(\sqrt{b^2-a^2}+b\right) \sqrt{e \cos (c+d x)}}+\frac{3 a e (e \cos (c+d x))^{3/2}}{4 b d \left(a^2-b^2\right) (a+b \sin (c+d x))}-\frac{e (e \cos (c+d x))^{3/2}}{2 b d (a+b \sin (c+d x))^2}","\frac{3 e^{5/2} \left(a^2-2 b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{8 b^{5/2} d \left(b^2-a^2\right)^{5/4}}-\frac{3 e^{5/2} \left(a^2-2 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{8 b^{5/2} d \left(b^2-a^2\right)^{5/4}}+\frac{3 a e^2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{4 b^2 d \left(a^2-b^2\right) \sqrt{\cos (c+d x)}}-\frac{3 a e^3 \left(a^2-2 b^2\right) \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{8 b^3 d \left(a^2-b^2\right) \left(b-\sqrt{b^2-a^2}\right) \sqrt{e \cos (c+d x)}}-\frac{3 a e^3 \left(a^2-2 b^2\right) \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{8 b^3 d \left(a^2-b^2\right) \left(\sqrt{b^2-a^2}+b\right) \sqrt{e \cos (c+d x)}}+\frac{3 a e (e \cos (c+d x))^{3/2}}{4 b d \left(a^2-b^2\right) (a+b \sin (c+d x))}-\frac{e (e \cos (c+d x))^{3/2}}{2 b d (a+b \sin (c+d x))^2}",1,"(3*(a^2 - 2*b^2)*e^(5/2)*ArcTan[(Sqrt[b]*Sqrt[e*Cos[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(8*b^(5/2)*(-a^2 + b^2)^(5/4)*d) - (3*(a^2 - 2*b^2)*e^(5/2)*ArcTanh[(Sqrt[b]*Sqrt[e*Cos[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(8*b^(5/2)*(-a^2 + b^2)^(5/4)*d) + (3*a*e^2*Sqrt[e*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(4*b^2*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]]) - (3*a*(a^2 - 2*b^2)*e^3*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (c + d*x)/2, 2])/(8*b^3*(a^2 - b^2)*(b - Sqrt[-a^2 + b^2])*d*Sqrt[e*Cos[c + d*x]]) - (3*a*(a^2 - 2*b^2)*e^3*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (c + d*x)/2, 2])/(8*b^3*(a^2 - b^2)*(b + Sqrt[-a^2 + b^2])*d*Sqrt[e*Cos[c + d*x]]) - (e*(e*Cos[c + d*x])^(3/2))/(2*b*d*(a + b*Sin[c + d*x])^2) + (3*a*e*(e*Cos[c + d*x])^(3/2))/(4*b*(a^2 - b^2)*d*(a + b*Sin[c + d*x]))","A",14,12,25,0.4800,1,"{2693, 2864, 2867, 2640, 2639, 2701, 2807, 2805, 329, 298, 205, 208}"
599,1,519,0,1.1384013,"\int \frac{(e \cos (c+d x))^{3/2}}{(a+b \sin (c+d x))^3} \, dx","Int[(e*Cos[c + d*x])^(3/2)/(a + b*Sin[c + d*x])^3,x]","\frac{e^{3/2} \left(a^2+2 b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{8 b^{3/2} d \left(b^2-a^2\right)^{7/4}}+\frac{e^{3/2} \left(a^2+2 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{8 b^{3/2} d \left(b^2-a^2\right)^{7/4}}-\frac{a e^2 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{4 b^2 d \left(a^2-b^2\right) \sqrt{e \cos (c+d x)}}+\frac{a e^2 \left(a^2+2 b^2\right) \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{8 b^2 d \left(a^2-b^2\right) \left(a^2-b \left(b-\sqrt{b^2-a^2}\right)\right) \sqrt{e \cos (c+d x)}}+\frac{a e^2 \left(a^2+2 b^2\right) \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{8 b^2 d \left(a^2-b^2\right) \left(a^2-b \left(\sqrt{b^2-a^2}+b\right)\right) \sqrt{e \cos (c+d x)}}+\frac{a e \sqrt{e \cos (c+d x)}}{4 b d \left(a^2-b^2\right) (a+b \sin (c+d x))}-\frac{e \sqrt{e \cos (c+d x)}}{2 b d (a+b \sin (c+d x))^2}","\frac{e^{3/2} \left(a^2+2 b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{8 b^{3/2} d \left(b^2-a^2\right)^{7/4}}+\frac{e^{3/2} \left(a^2+2 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{8 b^{3/2} d \left(b^2-a^2\right)^{7/4}}-\frac{a e^2 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{4 b^2 d \left(a^2-b^2\right) \sqrt{e \cos (c+d x)}}+\frac{a e^2 \left(a^2+2 b^2\right) \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{8 b^2 d \left(a^2-b^2\right) \left(a^2-b \left(b-\sqrt{b^2-a^2}\right)\right) \sqrt{e \cos (c+d x)}}+\frac{a e^2 \left(a^2+2 b^2\right) \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{8 b^2 d \left(a^2-b^2\right) \left(a^2-b \left(\sqrt{b^2-a^2}+b\right)\right) \sqrt{e \cos (c+d x)}}+\frac{a e \sqrt{e \cos (c+d x)}}{4 b d \left(a^2-b^2\right) (a+b \sin (c+d x))}-\frac{e \sqrt{e \cos (c+d x)}}{2 b d (a+b \sin (c+d x))^2}",1,"((a^2 + 2*b^2)*e^(3/2)*ArcTan[(Sqrt[b]*Sqrt[e*Cos[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(8*b^(3/2)*(-a^2 + b^2)^(7/4)*d) + ((a^2 + 2*b^2)*e^(3/2)*ArcTanh[(Sqrt[b]*Sqrt[e*Cos[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(8*b^(3/2)*(-a^2 + b^2)^(7/4)*d) - (a*e^2*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(4*b^2*(a^2 - b^2)*d*Sqrt[e*Cos[c + d*x]]) + (a*(a^2 + 2*b^2)*e^2*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (c + d*x)/2, 2])/(8*b^2*(a^2 - b^2)*(a^2 - b*(b - Sqrt[-a^2 + b^2]))*d*Sqrt[e*Cos[c + d*x]]) + (a*(a^2 + 2*b^2)*e^2*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (c + d*x)/2, 2])/(8*b^2*(a^2 - b^2)*(a^2 - b*(b + Sqrt[-a^2 + b^2]))*d*Sqrt[e*Cos[c + d*x]]) - (e*Sqrt[e*Cos[c + d*x]])/(2*b*d*(a + b*Sin[c + d*x])^2) + (a*e*Sqrt[e*Cos[c + d*x]])/(4*b*(a^2 - b^2)*d*(a + b*Sin[c + d*x]))","A",14,12,25,0.4800,1,"{2693, 2864, 2867, 2642, 2641, 2702, 2807, 2805, 329, 212, 208, 205}"
600,1,514,0,1.1869588,"\int \frac{\sqrt{e \cos (c+d x)}}{(a+b \sin (c+d x))^3} \, dx","Int[Sqrt[e*Cos[c + d*x]]/(a + b*Sin[c + d*x])^3,x]","\frac{5 a b (e \cos (c+d x))^{3/2}}{4 d e \left(a^2-b^2\right)^2 (a+b \sin (c+d x))}+\frac{b (e \cos (c+d x))^{3/2}}{2 d e \left(a^2-b^2\right) (a+b \sin (c+d x))^2}+\frac{\sqrt{e} \left(3 a^2+2 b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{8 \sqrt{b} d \left(b^2-a^2\right)^{9/4}}-\frac{\sqrt{e} \left(3 a^2+2 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{8 \sqrt{b} d \left(b^2-a^2\right)^{9/4}}+\frac{5 a E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{4 d \left(a^2-b^2\right)^2 \sqrt{\cos (c+d x)}}+\frac{a e \left(3 a^2+2 b^2\right) \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{8 b d \left(a^2-b^2\right)^2 \left(b-\sqrt{b^2-a^2}\right) \sqrt{e \cos (c+d x)}}+\frac{a e \left(3 a^2+2 b^2\right) \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{8 b d \left(a^2-b^2\right)^2 \left(\sqrt{b^2-a^2}+b\right) \sqrt{e \cos (c+d x)}}","\frac{5 a b (e \cos (c+d x))^{3/2}}{4 d e \left(a^2-b^2\right)^2 (a+b \sin (c+d x))}+\frac{b (e \cos (c+d x))^{3/2}}{2 d e \left(a^2-b^2\right) (a+b \sin (c+d x))^2}+\frac{\sqrt{e} \left(3 a^2+2 b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{8 \sqrt{b} d \left(b^2-a^2\right)^{9/4}}-\frac{\sqrt{e} \left(3 a^2+2 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{8 \sqrt{b} d \left(b^2-a^2\right)^{9/4}}+\frac{5 a E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{4 d \left(a^2-b^2\right)^2 \sqrt{\cos (c+d x)}}+\frac{a e \left(3 a^2+2 b^2\right) \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{8 b d \left(a^2-b^2\right)^2 \left(b-\sqrt{b^2-a^2}\right) \sqrt{e \cos (c+d x)}}+\frac{a e \left(3 a^2+2 b^2\right) \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{8 b d \left(a^2-b^2\right)^2 \left(\sqrt{b^2-a^2}+b\right) \sqrt{e \cos (c+d x)}}",1,"((3*a^2 + 2*b^2)*Sqrt[e]*ArcTan[(Sqrt[b]*Sqrt[e*Cos[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(8*Sqrt[b]*(-a^2 + b^2)^(9/4)*d) - ((3*a^2 + 2*b^2)*Sqrt[e]*ArcTanh[(Sqrt[b]*Sqrt[e*Cos[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(8*Sqrt[b]*(-a^2 + b^2)^(9/4)*d) + (5*a*Sqrt[e*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(4*(a^2 - b^2)^2*d*Sqrt[Cos[c + d*x]]) + (a*(3*a^2 + 2*b^2)*e*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (c + d*x)/2, 2])/(8*b*(a^2 - b^2)^2*(b - Sqrt[-a^2 + b^2])*d*Sqrt[e*Cos[c + d*x]]) + (a*(3*a^2 + 2*b^2)*e*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (c + d*x)/2, 2])/(8*b*(a^2 - b^2)^2*(b + Sqrt[-a^2 + b^2])*d*Sqrt[e*Cos[c + d*x]]) + (b*(e*Cos[c + d*x])^(3/2))/(2*(a^2 - b^2)*d*e*(a + b*Sin[c + d*x])^2) + (5*a*b*(e*Cos[c + d*x])^(3/2))/(4*(a^2 - b^2)^2*d*e*(a + b*Sin[c + d*x]))","A",14,12,25,0.4800,1,"{2694, 2864, 2867, 2640, 2639, 2701, 2807, 2805, 329, 298, 205, 208}"
601,1,520,0,1.2260779,"\int \frac{1}{\sqrt{e \cos (c+d x)} (a+b \sin (c+d x))^3} \, dx","Int[1/(Sqrt[e*Cos[c + d*x]]*(a + b*Sin[c + d*x])^3),x]","\frac{7 a b \sqrt{e \cos (c+d x)}}{4 d e \left(a^2-b^2\right)^2 (a+b \sin (c+d x))}+\frac{b \sqrt{e \cos (c+d x)}}{2 d e \left(a^2-b^2\right) (a+b \sin (c+d x))^2}-\frac{3 \sqrt{b} \left(5 a^2+2 b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{8 d \sqrt{e} \left(b^2-a^2\right)^{11/4}}-\frac{3 \sqrt{b} \left(5 a^2+2 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{8 d \sqrt{e} \left(b^2-a^2\right)^{11/4}}-\frac{7 a \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{4 d \left(a^2-b^2\right)^2 \sqrt{e \cos (c+d x)}}+\frac{3 a \left(5 a^2+2 b^2\right) \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{8 d \left(a^2-b^2\right)^2 \left(a^2-b \left(b-\sqrt{b^2-a^2}\right)\right) \sqrt{e \cos (c+d x)}}+\frac{3 a \left(5 a^2+2 b^2\right) \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{8 d \left(a^2-b^2\right)^2 \left(a^2-b \left(\sqrt{b^2-a^2}+b\right)\right) \sqrt{e \cos (c+d x)}}","\frac{7 a b \sqrt{e \cos (c+d x)}}{4 d e \left(a^2-b^2\right)^2 (a+b \sin (c+d x))}+\frac{b \sqrt{e \cos (c+d x)}}{2 d e \left(a^2-b^2\right) (a+b \sin (c+d x))^2}-\frac{3 \sqrt{b} \left(5 a^2+2 b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{8 d \sqrt{e} \left(b^2-a^2\right)^{11/4}}-\frac{3 \sqrt{b} \left(5 a^2+2 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{8 d \sqrt{e} \left(b^2-a^2\right)^{11/4}}-\frac{7 a \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{4 d \left(a^2-b^2\right)^2 \sqrt{e \cos (c+d x)}}+\frac{3 a \left(5 a^2+2 b^2\right) \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{8 d \left(a^2-b^2\right)^2 \left(a^2-b \left(b-\sqrt{b^2-a^2}\right)\right) \sqrt{e \cos (c+d x)}}+\frac{3 a \left(5 a^2+2 b^2\right) \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{8 d \left(a^2-b^2\right)^2 \left(a^2-b \left(\sqrt{b^2-a^2}+b\right)\right) \sqrt{e \cos (c+d x)}}",1,"(-3*Sqrt[b]*(5*a^2 + 2*b^2)*ArcTan[(Sqrt[b]*Sqrt[e*Cos[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(8*(-a^2 + b^2)^(11/4)*d*Sqrt[e]) - (3*Sqrt[b]*(5*a^2 + 2*b^2)*ArcTanh[(Sqrt[b]*Sqrt[e*Cos[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(8*(-a^2 + b^2)^(11/4)*d*Sqrt[e]) - (7*a*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(4*(a^2 - b^2)^2*d*Sqrt[e*Cos[c + d*x]]) + (3*a*(5*a^2 + 2*b^2)*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (c + d*x)/2, 2])/(8*(a^2 - b^2)^2*(a^2 - b*(b - Sqrt[-a^2 + b^2]))*d*Sqrt[e*Cos[c + d*x]]) + (3*a*(5*a^2 + 2*b^2)*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (c + d*x)/2, 2])/(8*(a^2 - b^2)^2*(a^2 - b*(b + Sqrt[-a^2 + b^2]))*d*Sqrt[e*Cos[c + d*x]]) + (b*Sqrt[e*Cos[c + d*x]])/(2*(a^2 - b^2)*d*e*(a + b*Sin[c + d*x])^2) + (7*a*b*Sqrt[e*Cos[c + d*x]])/(4*(a^2 - b^2)^2*d*e*(a + b*Sin[c + d*x]))","A",14,12,25,0.4800,1,"{2694, 2864, 2867, 2642, 2641, 2702, 2807, 2805, 329, 212, 208, 205}"
602,1,596,0,1.5932213,"\int \frac{1}{(e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^3} \, dx","Int[1/((e*Cos[c + d*x])^(3/2)*(a + b*Sin[c + d*x])^3),x]","\frac{5 b^{3/2} \left(7 a^2+2 b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{8 d e^{3/2} \left(b^2-a^2\right)^{13/4}}-\frac{5 b^{3/2} \left(7 a^2+2 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{8 d e^{3/2} \left(b^2-a^2\right)^{13/4}}-\frac{a \left(8 a^2+37 b^2\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{4 d e^2 \left(a^2-b^2\right)^3 \sqrt{\cos (c+d x)}}+\frac{9 a b}{4 d e \left(a^2-b^2\right)^2 \sqrt{e \cos (c+d x)} (a+b \sin (c+d x))}+\frac{b}{2 d e \left(a^2-b^2\right) \sqrt{e \cos (c+d x)} (a+b \sin (c+d x))^2}-\frac{5 b \left(7 a^2+2 b^2\right)-a \left(8 a^2+37 b^2\right) \sin (c+d x)}{4 d e \left(a^2-b^2\right)^3 \sqrt{e \cos (c+d x)}}-\frac{5 a b \left(7 a^2+2 b^2\right) \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{8 d e \left(a^2-b^2\right)^3 \left(b-\sqrt{b^2-a^2}\right) \sqrt{e \cos (c+d x)}}-\frac{5 a b \left(7 a^2+2 b^2\right) \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{8 d e \left(a^2-b^2\right)^3 \left(\sqrt{b^2-a^2}+b\right) \sqrt{e \cos (c+d x)}}","\frac{5 b^{3/2} \left(7 a^2+2 b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{8 d e^{3/2} \left(b^2-a^2\right)^{13/4}}-\frac{5 b^{3/2} \left(7 a^2+2 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{8 d e^{3/2} \left(b^2-a^2\right)^{13/4}}-\frac{a \left(8 a^2+37 b^2\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{4 d e^2 \left(a^2-b^2\right)^3 \sqrt{\cos (c+d x)}}+\frac{9 a b}{4 d e \left(a^2-b^2\right)^2 \sqrt{e \cos (c+d x)} (a+b \sin (c+d x))}+\frac{b}{2 d e \left(a^2-b^2\right) \sqrt{e \cos (c+d x)} (a+b \sin (c+d x))^2}-\frac{5 b \left(7 a^2+2 b^2\right)-a \left(8 a^2+37 b^2\right) \sin (c+d x)}{4 d e \left(a^2-b^2\right)^3 \sqrt{e \cos (c+d x)}}-\frac{5 a b \left(7 a^2+2 b^2\right) \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{8 d e \left(a^2-b^2\right)^3 \left(b-\sqrt{b^2-a^2}\right) \sqrt{e \cos (c+d x)}}-\frac{5 a b \left(7 a^2+2 b^2\right) \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{8 d e \left(a^2-b^2\right)^3 \left(\sqrt{b^2-a^2}+b\right) \sqrt{e \cos (c+d x)}}",1,"(5*b^(3/2)*(7*a^2 + 2*b^2)*ArcTan[(Sqrt[b]*Sqrt[e*Cos[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(8*(-a^2 + b^2)^(13/4)*d*e^(3/2)) - (5*b^(3/2)*(7*a^2 + 2*b^2)*ArcTanh[(Sqrt[b]*Sqrt[e*Cos[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(8*(-a^2 + b^2)^(13/4)*d*e^(3/2)) - (a*(8*a^2 + 37*b^2)*Sqrt[e*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(4*(a^2 - b^2)^3*d*e^2*Sqrt[Cos[c + d*x]]) - (5*a*b*(7*a^2 + 2*b^2)*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (c + d*x)/2, 2])/(8*(a^2 - b^2)^3*(b - Sqrt[-a^2 + b^2])*d*e*Sqrt[e*Cos[c + d*x]]) - (5*a*b*(7*a^2 + 2*b^2)*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (c + d*x)/2, 2])/(8*(a^2 - b^2)^3*(b + Sqrt[-a^2 + b^2])*d*e*Sqrt[e*Cos[c + d*x]]) + b/(2*(a^2 - b^2)*d*e*Sqrt[e*Cos[c + d*x]]*(a + b*Sin[c + d*x])^2) + (9*a*b)/(4*(a^2 - b^2)^2*d*e*Sqrt[e*Cos[c + d*x]]*(a + b*Sin[c + d*x])) - (5*b*(7*a^2 + 2*b^2) - a*(8*a^2 + 37*b^2)*Sin[c + d*x])/(4*(a^2 - b^2)^3*d*e*Sqrt[e*Cos[c + d*x]])","A",15,13,25,0.5200,1,"{2694, 2864, 2866, 2867, 2640, 2639, 2701, 2807, 2805, 329, 298, 205, 208}"
603,1,614,0,1.7214497,"\int \frac{1}{(e \cos (c+d x))^{5/2} (a+b \sin (c+d x))^3} \, dx","Int[1/((e*Cos[c + d*x])^(5/2)*(a + b*Sin[c + d*x])^3),x]","-\frac{7 b^{5/2} \left(9 a^2+2 b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{8 d e^{5/2} \left(b^2-a^2\right)^{15/4}}-\frac{7 b^{5/2} \left(9 a^2+2 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{8 d e^{5/2} \left(b^2-a^2\right)^{15/4}}+\frac{a \left(8 a^2+69 b^2\right) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{12 d e^2 \left(a^2-b^2\right)^3 \sqrt{e \cos (c+d x)}}-\frac{7 a b^2 \left(9 a^2+2 b^2\right) \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{8 d e^2 \left(a^2-b^2\right)^3 \left(a^2-b \left(b-\sqrt{b^2-a^2}\right)\right) \sqrt{e \cos (c+d x)}}-\frac{7 a b^2 \left(9 a^2+2 b^2\right) \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{8 d e^2 \left(a^2-b^2\right)^3 \left(a^2-b \left(\sqrt{b^2-a^2}+b\right)\right) \sqrt{e \cos (c+d x)}}+\frac{11 a b}{4 d e \left(a^2-b^2\right)^2 (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))}+\frac{b}{2 d e \left(a^2-b^2\right) (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^2}-\frac{7 b \left(9 a^2+2 b^2\right)-a \left(8 a^2+69 b^2\right) \sin (c+d x)}{12 d e \left(a^2-b^2\right)^3 (e \cos (c+d x))^{3/2}}","-\frac{7 b^{5/2} \left(9 a^2+2 b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{8 d e^{5/2} \left(b^2-a^2\right)^{15/4}}-\frac{7 b^{5/2} \left(9 a^2+2 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{8 d e^{5/2} \left(b^2-a^2\right)^{15/4}}+\frac{a \left(8 a^2+69 b^2\right) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{12 d e^2 \left(a^2-b^2\right)^3 \sqrt{e \cos (c+d x)}}-\frac{7 a b^2 \left(9 a^2+2 b^2\right) \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{8 d e^2 \left(a^2-b^2\right)^3 \left(a^2-b \left(b-\sqrt{b^2-a^2}\right)\right) \sqrt{e \cos (c+d x)}}-\frac{7 a b^2 \left(9 a^2+2 b^2\right) \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{8 d e^2 \left(a^2-b^2\right)^3 \left(a^2-b \left(\sqrt{b^2-a^2}+b\right)\right) \sqrt{e \cos (c+d x)}}+\frac{11 a b}{4 d e \left(a^2-b^2\right)^2 (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))}+\frac{b}{2 d e \left(a^2-b^2\right) (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^2}-\frac{7 b \left(9 a^2+2 b^2\right)-a \left(8 a^2+69 b^2\right) \sin (c+d x)}{12 d e \left(a^2-b^2\right)^3 (e \cos (c+d x))^{3/2}}",1,"(-7*b^(5/2)*(9*a^2 + 2*b^2)*ArcTan[(Sqrt[b]*Sqrt[e*Cos[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(8*(-a^2 + b^2)^(15/4)*d*e^(5/2)) - (7*b^(5/2)*(9*a^2 + 2*b^2)*ArcTanh[(Sqrt[b]*Sqrt[e*Cos[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(8*(-a^2 + b^2)^(15/4)*d*e^(5/2)) + (a*(8*a^2 + 69*b^2)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(12*(a^2 - b^2)^3*d*e^2*Sqrt[e*Cos[c + d*x]]) - (7*a*b^2*(9*a^2 + 2*b^2)*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (c + d*x)/2, 2])/(8*(a^2 - b^2)^3*(a^2 - b*(b - Sqrt[-a^2 + b^2]))*d*e^2*Sqrt[e*Cos[c + d*x]]) - (7*a*b^2*(9*a^2 + 2*b^2)*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (c + d*x)/2, 2])/(8*(a^2 - b^2)^3*(a^2 - b*(b + Sqrt[-a^2 + b^2]))*d*e^2*Sqrt[e*Cos[c + d*x]]) + b/(2*(a^2 - b^2)*d*e*(e*Cos[c + d*x])^(3/2)*(a + b*Sin[c + d*x])^2) + (11*a*b)/(4*(a^2 - b^2)^2*d*e*(e*Cos[c + d*x])^(3/2)*(a + b*Sin[c + d*x])) - (7*b*(9*a^2 + 2*b^2) - a*(8*a^2 + 69*b^2)*Sin[c + d*x])/(12*(a^2 - b^2)^3*d*e*(e*Cos[c + d*x])^(3/2))","A",15,13,25,0.5200,1,"{2694, 2864, 2866, 2867, 2642, 2641, 2702, 2807, 2805, 329, 212, 208, 205}"
604,1,685,0,2.0281313,"\int \frac{1}{(e \cos (c+d x))^{7/2} (a+b \sin (c+d x))^3} \, dx","Int[1/((e*Cos[c + d*x])^(7/2)*(a + b*Sin[c + d*x])^3),x]","\frac{3 \left(a \left(-64 a^2 b^2+8 a^4-139 b^4\right) \sin (c+d x)+15 b^3 \left(11 a^2+2 b^2\right)\right)}{20 d e^3 \left(a^2-b^2\right)^4 \sqrt{e \cos (c+d x)}}+\frac{9 b^{7/2} \left(11 a^2+2 b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{8 d e^{7/2} \left(b^2-a^2\right)^{17/4}}-\frac{9 b^{7/2} \left(11 a^2+2 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{8 d e^{7/2} \left(b^2-a^2\right)^{17/4}}-\frac{3 a \left(-64 a^2 b^2+8 a^4-139 b^4\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{20 d e^4 \left(a^2-b^2\right)^4 \sqrt{\cos (c+d x)}}+\frac{9 a b^3 \left(11 a^2+2 b^2\right) \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{8 d e^3 \left(a^2-b^2\right)^4 \left(b-\sqrt{b^2-a^2}\right) \sqrt{e \cos (c+d x)}}+\frac{9 a b^3 \left(11 a^2+2 b^2\right) \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{8 d e^3 \left(a^2-b^2\right)^4 \left(\sqrt{b^2-a^2}+b\right) \sqrt{e \cos (c+d x)}}+\frac{13 a b}{4 d e \left(a^2-b^2\right)^2 (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))}+\frac{b}{2 d e \left(a^2-b^2\right) (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))^2}-\frac{9 b \left(11 a^2+2 b^2\right)-a \left(8 a^2+109 b^2\right) \sin (c+d x)}{20 d e \left(a^2-b^2\right)^3 (e \cos (c+d x))^{5/2}}","\frac{3 \left(a \left(-64 a^2 b^2+8 a^4-139 b^4\right) \sin (c+d x)+15 b^3 \left(11 a^2+2 b^2\right)\right)}{20 d e^3 \left(a^2-b^2\right)^4 \sqrt{e \cos (c+d x)}}+\frac{9 b^{7/2} \left(11 a^2+2 b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{8 d e^{7/2} \left(b^2-a^2\right)^{17/4}}-\frac{9 b^{7/2} \left(11 a^2+2 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{8 d e^{7/2} \left(b^2-a^2\right)^{17/4}}-\frac{3 a \left(-64 a^2 b^2+8 a^4-139 b^4\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{20 d e^4 \left(a^2-b^2\right)^4 \sqrt{\cos (c+d x)}}+\frac{9 a b^3 \left(11 a^2+2 b^2\right) \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{8 d e^3 \left(a^2-b^2\right)^4 \left(b-\sqrt{b^2-a^2}\right) \sqrt{e \cos (c+d x)}}+\frac{9 a b^3 \left(11 a^2+2 b^2\right) \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{8 d e^3 \left(a^2-b^2\right)^4 \left(\sqrt{b^2-a^2}+b\right) \sqrt{e \cos (c+d x)}}+\frac{13 a b}{4 d e \left(a^2-b^2\right)^2 (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))}+\frac{b}{2 d e \left(a^2-b^2\right) (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))^2}-\frac{9 b \left(11 a^2+2 b^2\right)-a \left(8 a^2+109 b^2\right) \sin (c+d x)}{20 d e \left(a^2-b^2\right)^3 (e \cos (c+d x))^{5/2}}",1,"(9*b^(7/2)*(11*a^2 + 2*b^2)*ArcTan[(Sqrt[b]*Sqrt[e*Cos[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(8*(-a^2 + b^2)^(17/4)*d*e^(7/2)) - (9*b^(7/2)*(11*a^2 + 2*b^2)*ArcTanh[(Sqrt[b]*Sqrt[e*Cos[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(8*(-a^2 + b^2)^(17/4)*d*e^(7/2)) - (3*a*(8*a^4 - 64*a^2*b^2 - 139*b^4)*Sqrt[e*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(20*(a^2 - b^2)^4*d*e^4*Sqrt[Cos[c + d*x]]) + (9*a*b^3*(11*a^2 + 2*b^2)*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (c + d*x)/2, 2])/(8*(a^2 - b^2)^4*(b - Sqrt[-a^2 + b^2])*d*e^3*Sqrt[e*Cos[c + d*x]]) + (9*a*b^3*(11*a^2 + 2*b^2)*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (c + d*x)/2, 2])/(8*(a^2 - b^2)^4*(b + Sqrt[-a^2 + b^2])*d*e^3*Sqrt[e*Cos[c + d*x]]) + b/(2*(a^2 - b^2)*d*e*(e*Cos[c + d*x])^(5/2)*(a + b*Sin[c + d*x])^2) + (13*a*b)/(4*(a^2 - b^2)^2*d*e*(e*Cos[c + d*x])^(5/2)*(a + b*Sin[c + d*x])) - (9*b*(11*a^2 + 2*b^2) - a*(8*a^2 + 109*b^2)*Sin[c + d*x])/(20*(a^2 - b^2)^3*d*e*(e*Cos[c + d*x])^(5/2)) + (3*(15*b^3*(11*a^2 + 2*b^2) + a*(8*a^4 - 64*a^2*b^2 - 139*b^4)*Sin[c + d*x]))/(20*(a^2 - b^2)^4*d*e^3*Sqrt[e*Cos[c + d*x]])","A",16,13,25,0.5200,1,"{2694, 2864, 2866, 2867, 2640, 2639, 2701, 2807, 2805, 329, 298, 205, 208}"
605,1,671,0,1.8316146,"\int \frac{(e \cos (c+d x))^{15/2}}{(a+b \sin (c+d x))^4} \, dx","Int[(e*Cos[c + d*x])^(15/2)/(a + b*Sin[c + d*x])^4,x]","\frac{13 e^7 \sqrt{e \cos (c+d x)} \left(21 a \left(11 a^2-6 b^2\right)-b \left(77 a^2-20 b^2\right) \sin (c+d x)\right)}{56 b^7 d}-\frac{39 e^5 (e \cos (c+d x))^{5/2} \left(77 a^2+22 a b \sin (c+d x)-20 b^2\right)}{280 b^5 d (a+b \sin (c+d x))}+\frac{39 a e^{15/2} \left(-17 a^2 b^2+11 a^4+6 b^4\right) \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{16 b^{15/2} d \left(b^2-a^2\right)^{3/4}}+\frac{39 a e^{15/2} \left(-17 a^2 b^2+11 a^4+6 b^4\right) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{16 b^{15/2} d \left(b^2-a^2\right)^{3/4}}+\frac{13 e^8 \left(-203 a^2 b^2+231 a^4+20 b^4\right) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{56 b^8 d \sqrt{e \cos (c+d x)}}-\frac{39 a^2 e^8 \left(-17 a^2 b^2+11 a^4+6 b^4\right) \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{16 b^8 d \left(a^2-b \left(b-\sqrt{b^2-a^2}\right)\right) \sqrt{e \cos (c+d x)}}-\frac{39 a^2 e^8 \left(-17 a^2 b^2+11 a^4+6 b^4\right) \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{16 b^8 d \left(a^2-b \left(\sqrt{b^2-a^2}+b\right)\right) \sqrt{e \cos (c+d x)}}-\frac{13 e^3 (e \cos (c+d x))^{9/2} (11 a+4 b \sin (c+d x))}{84 b^3 d (a+b \sin (c+d x))^2}-\frac{e (e \cos (c+d x))^{13/2}}{3 b d (a+b \sin (c+d x))^3}","\frac{13 e^7 \sqrt{e \cos (c+d x)} \left(21 a \left(11 a^2-6 b^2\right)-b \left(77 a^2-20 b^2\right) \sin (c+d x)\right)}{56 b^7 d}-\frac{39 e^5 (e \cos (c+d x))^{5/2} \left(77 a^2+22 a b \sin (c+d x)-20 b^2\right)}{280 b^5 d (a+b \sin (c+d x))}+\frac{39 a e^{15/2} \left(-17 a^2 b^2+11 a^4+6 b^4\right) \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{16 b^{15/2} d \left(b^2-a^2\right)^{3/4}}+\frac{39 a e^{15/2} \left(-17 a^2 b^2+11 a^4+6 b^4\right) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{16 b^{15/2} d \left(b^2-a^2\right)^{3/4}}+\frac{13 e^8 \left(-203 a^2 b^2+231 a^4+20 b^4\right) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{56 b^8 d \sqrt{e \cos (c+d x)}}-\frac{39 a^2 e^8 \left(-17 a^2 b^2+11 a^4+6 b^4\right) \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{16 b^8 d \left(a^2-b \left(b-\sqrt{b^2-a^2}\right)\right) \sqrt{e \cos (c+d x)}}-\frac{39 a^2 e^8 \left(-17 a^2 b^2+11 a^4+6 b^4\right) \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{16 b^8 d \left(a^2-b \left(\sqrt{b^2-a^2}+b\right)\right) \sqrt{e \cos (c+d x)}}-\frac{13 e^3 (e \cos (c+d x))^{9/2} (11 a+4 b \sin (c+d x))}{84 b^3 d (a+b \sin (c+d x))^2}-\frac{e (e \cos (c+d x))^{13/2}}{3 b d (a+b \sin (c+d x))^3}",1,"(39*a*(11*a^4 - 17*a^2*b^2 + 6*b^4)*e^(15/2)*ArcTan[(Sqrt[b]*Sqrt[e*Cos[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(16*b^(15/2)*(-a^2 + b^2)^(3/4)*d) + (39*a*(11*a^4 - 17*a^2*b^2 + 6*b^4)*e^(15/2)*ArcTanh[(Sqrt[b]*Sqrt[e*Cos[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(16*b^(15/2)*(-a^2 + b^2)^(3/4)*d) + (13*(231*a^4 - 203*a^2*b^2 + 20*b^4)*e^8*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(56*b^8*d*Sqrt[e*Cos[c + d*x]]) - (39*a^2*(11*a^4 - 17*a^2*b^2 + 6*b^4)*e^8*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (c + d*x)/2, 2])/(16*b^8*(a^2 - b*(b - Sqrt[-a^2 + b^2]))*d*Sqrt[e*Cos[c + d*x]]) - (39*a^2*(11*a^4 - 17*a^2*b^2 + 6*b^4)*e^8*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (c + d*x)/2, 2])/(16*b^8*(a^2 - b*(b + Sqrt[-a^2 + b^2]))*d*Sqrt[e*Cos[c + d*x]]) - (e*(e*Cos[c + d*x])^(13/2))/(3*b*d*(a + b*Sin[c + d*x])^3) - (13*e^3*(e*Cos[c + d*x])^(9/2)*(11*a + 4*b*Sin[c + d*x]))/(84*b^3*d*(a + b*Sin[c + d*x])^2) - (39*e^5*(e*Cos[c + d*x])^(5/2)*(77*a^2 - 20*b^2 + 22*a*b*Sin[c + d*x]))/(280*b^5*d*(a + b*Sin[c + d*x])) + (13*e^7*Sqrt[e*Cos[c + d*x]]*(21*a*(11*a^2 - 6*b^2) - b*(77*a^2 - 20*b^2)*Sin[c + d*x]))/(56*b^7*d)","A",16,13,25,0.5200,1,"{2693, 2863, 2865, 2867, 2642, 2641, 2702, 2807, 2805, 329, 212, 208, 205}"
606,1,557,0,1.3640682,"\int \frac{(e \cos (c+d x))^{13/2}}{(a+b \sin (c+d x))^4} \, dx","Int[(e*Cos[c + d*x])^(13/2)/(a + b*Sin[c + d*x])^4,x]","-\frac{77 e^5 (e \cos (c+d x))^{3/2} \left(15 a^2+6 a b \sin (c+d x)-4 b^2\right)}{120 b^5 d (a+b \sin (c+d x))}+\frac{77 a e^{13/2} \left(3 a^2-2 b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{16 b^{13/2} d \sqrt[4]{b^2-a^2}}-\frac{77 a e^{13/2} \left(3 a^2-2 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{16 b^{13/2} d \sqrt[4]{b^2-a^2}}-\frac{77 e^6 \left(15 a^2-4 b^2\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{40 b^6 d \sqrt{\cos (c+d x)}}+\frac{77 a^2 e^7 \left(3 a^2-2 b^2\right) \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{16 b^7 d \left(b-\sqrt{b^2-a^2}\right) \sqrt{e \cos (c+d x)}}+\frac{77 a^2 e^7 \left(3 a^2-2 b^2\right) \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{16 b^7 d \left(\sqrt{b^2-a^2}+b\right) \sqrt{e \cos (c+d x)}}-\frac{11 e^3 (e \cos (c+d x))^{7/2} (9 a+4 b \sin (c+d x))}{60 b^3 d (a+b \sin (c+d x))^2}-\frac{e (e \cos (c+d x))^{11/2}}{3 b d (a+b \sin (c+d x))^3}","-\frac{77 e^5 (e \cos (c+d x))^{3/2} \left(15 a^2+6 a b \sin (c+d x)-4 b^2\right)}{120 b^5 d (a+b \sin (c+d x))}+\frac{77 a e^{13/2} \left(3 a^2-2 b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{16 b^{13/2} d \sqrt[4]{b^2-a^2}}-\frac{77 a e^{13/2} \left(3 a^2-2 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{16 b^{13/2} d \sqrt[4]{b^2-a^2}}-\frac{77 e^6 \left(15 a^2-4 b^2\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{40 b^6 d \sqrt{\cos (c+d x)}}+\frac{77 a^2 e^7 \left(3 a^2-2 b^2\right) \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{16 b^7 d \left(b-\sqrt{b^2-a^2}\right) \sqrt{e \cos (c+d x)}}+\frac{77 a^2 e^7 \left(3 a^2-2 b^2\right) \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{16 b^7 d \left(\sqrt{b^2-a^2}+b\right) \sqrt{e \cos (c+d x)}}-\frac{11 e^3 (e \cos (c+d x))^{7/2} (9 a+4 b \sin (c+d x))}{60 b^3 d (a+b \sin (c+d x))^2}-\frac{e (e \cos (c+d x))^{11/2}}{3 b d (a+b \sin (c+d x))^3}",1,"(77*a*(3*a^2 - 2*b^2)*e^(13/2)*ArcTan[(Sqrt[b]*Sqrt[e*Cos[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(16*b^(13/2)*(-a^2 + b^2)^(1/4)*d) - (77*a*(3*a^2 - 2*b^2)*e^(13/2)*ArcTanh[(Sqrt[b]*Sqrt[e*Cos[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(16*b^(13/2)*(-a^2 + b^2)^(1/4)*d) - (77*(15*a^2 - 4*b^2)*e^6*Sqrt[e*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(40*b^6*d*Sqrt[Cos[c + d*x]]) + (77*a^2*(3*a^2 - 2*b^2)*e^7*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (c + d*x)/2, 2])/(16*b^7*(b - Sqrt[-a^2 + b^2])*d*Sqrt[e*Cos[c + d*x]]) + (77*a^2*(3*a^2 - 2*b^2)*e^7*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (c + d*x)/2, 2])/(16*b^7*(b + Sqrt[-a^2 + b^2])*d*Sqrt[e*Cos[c + d*x]]) - (e*(e*Cos[c + d*x])^(11/2))/(3*b*d*(a + b*Sin[c + d*x])^3) - (11*e^3*(e*Cos[c + d*x])^(7/2)*(9*a + 4*b*Sin[c + d*x]))/(60*b^3*d*(a + b*Sin[c + d*x])^2) - (77*e^5*(e*Cos[c + d*x])^(3/2)*(15*a^2 - 4*b^2 + 6*a*b*Sin[c + d*x]))/(120*b^5*d*(a + b*Sin[c + d*x]))","A",15,12,25,0.4800,1,"{2693, 2863, 2867, 2640, 2639, 2701, 2807, 2805, 329, 298, 205, 208}"
607,1,571,0,1.3778643,"\int \frac{(e \cos (c+d x))^{11/2}}{(a+b \sin (c+d x))^4} \, dx","Int[(e*Cos[c + d*x])^(11/2)/(a + b*Sin[c + d*x])^4,x]","-\frac{5 e^5 \sqrt{e \cos (c+d x)} \left(21 a^2+14 a b \sin (c+d x)-4 b^2\right)}{8 b^5 d (a+b \sin (c+d x))}-\frac{15 a e^{11/2} \left(7 a^2-6 b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{16 b^{11/2} d \left(b^2-a^2\right)^{3/4}}-\frac{15 a e^{11/2} \left(7 a^2-6 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{16 b^{11/2} d \left(b^2-a^2\right)^{3/4}}-\frac{5 e^6 \left(21 a^2-4 b^2\right) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{8 b^6 d \sqrt{e \cos (c+d x)}}+\frac{15 a^2 e^6 \left(7 a^2-6 b^2\right) \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{16 b^6 d \left(a^2-b \left(b-\sqrt{b^2-a^2}\right)\right) \sqrt{e \cos (c+d x)}}+\frac{15 a^2 e^6 \left(7 a^2-6 b^2\right) \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{16 b^6 d \left(a^2-b \left(\sqrt{b^2-a^2}+b\right)\right) \sqrt{e \cos (c+d x)}}-\frac{e^3 (e \cos (c+d x))^{5/2} (7 a+4 b \sin (c+d x))}{4 b^3 d (a+b \sin (c+d x))^2}-\frac{e (e \cos (c+d x))^{9/2}}{3 b d (a+b \sin (c+d x))^3}","-\frac{5 e^5 \sqrt{e \cos (c+d x)} \left(21 a^2+14 a b \sin (c+d x)-4 b^2\right)}{8 b^5 d (a+b \sin (c+d x))}-\frac{15 a e^{11/2} \left(7 a^2-6 b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{16 b^{11/2} d \left(b^2-a^2\right)^{3/4}}-\frac{15 a e^{11/2} \left(7 a^2-6 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{16 b^{11/2} d \left(b^2-a^2\right)^{3/4}}-\frac{5 e^6 \left(21 a^2-4 b^2\right) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{8 b^6 d \sqrt{e \cos (c+d x)}}+\frac{15 a^2 e^6 \left(7 a^2-6 b^2\right) \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{16 b^6 d \left(a^2-b \left(b-\sqrt{b^2-a^2}\right)\right) \sqrt{e \cos (c+d x)}}+\frac{15 a^2 e^6 \left(7 a^2-6 b^2\right) \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{16 b^6 d \left(a^2-b \left(\sqrt{b^2-a^2}+b\right)\right) \sqrt{e \cos (c+d x)}}-\frac{e^3 (e \cos (c+d x))^{5/2} (7 a+4 b \sin (c+d x))}{4 b^3 d (a+b \sin (c+d x))^2}-\frac{e (e \cos (c+d x))^{9/2}}{3 b d (a+b \sin (c+d x))^3}",1,"(-15*a*(7*a^2 - 6*b^2)*e^(11/2)*ArcTan[(Sqrt[b]*Sqrt[e*Cos[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(16*b^(11/2)*(-a^2 + b^2)^(3/4)*d) - (15*a*(7*a^2 - 6*b^2)*e^(11/2)*ArcTanh[(Sqrt[b]*Sqrt[e*Cos[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(16*b^(11/2)*(-a^2 + b^2)^(3/4)*d) - (5*(21*a^2 - 4*b^2)*e^6*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(8*b^6*d*Sqrt[e*Cos[c + d*x]]) + (15*a^2*(7*a^2 - 6*b^2)*e^6*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (c + d*x)/2, 2])/(16*b^6*(a^2 - b*(b - Sqrt[-a^2 + b^2]))*d*Sqrt[e*Cos[c + d*x]]) + (15*a^2*(7*a^2 - 6*b^2)*e^6*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (c + d*x)/2, 2])/(16*b^6*(a^2 - b*(b + Sqrt[-a^2 + b^2]))*d*Sqrt[e*Cos[c + d*x]]) - (e*(e*Cos[c + d*x])^(9/2))/(3*b*d*(a + b*Sin[c + d*x])^3) - (e^3*(e*Cos[c + d*x])^(5/2)*(7*a + 4*b*Sin[c + d*x]))/(4*b^3*d*(a + b*Sin[c + d*x])^2) - (5*e^5*Sqrt[e*Cos[c + d*x]]*(21*a^2 - 4*b^2 + 14*a*b*Sin[c + d*x]))/(8*b^5*d*(a + b*Sin[c + d*x]))","A",15,12,25,0.4800,1,"{2693, 2863, 2867, 2642, 2641, 2702, 2807, 2805, 329, 212, 208, 205}"
608,1,591,0,1.4300872,"\int \frac{(e \cos (c+d x))^{9/2}}{(a+b \sin (c+d x))^4} \, dx","Int[(e*Cos[c + d*x])^(9/2)/(a + b*Sin[c + d*x])^4,x]","\frac{7 e^3 \left(5 a^2-4 b^2\right) (e \cos (c+d x))^{3/2}}{8 b^3 d \left(a^2-b^2\right) (a+b \sin (c+d x))}+\frac{7 a e^{9/2} \left(5 a^2-6 b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{16 b^{9/2} d \left(b^2-a^2\right)^{5/4}}-\frac{7 a e^{9/2} \left(5 a^2-6 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{16 b^{9/2} d \left(b^2-a^2\right)^{5/4}}+\frac{7 e^4 \left(5 a^2-4 b^2\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{8 b^4 d \left(a^2-b^2\right) \sqrt{\cos (c+d x)}}-\frac{7 a^2 e^5 \left(5 a^2-6 b^2\right) \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{16 b^5 d \left(a^2-b^2\right) \left(b-\sqrt{b^2-a^2}\right) \sqrt{e \cos (c+d x)}}-\frac{7 a^2 e^5 \left(5 a^2-6 b^2\right) \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{16 b^5 d \left(a^2-b^2\right) \left(\sqrt{b^2-a^2}+b\right) \sqrt{e \cos (c+d x)}}-\frac{7 e^3 (e \cos (c+d x))^{3/2} (5 a+4 b \sin (c+d x))}{12 b^3 d (a+b \sin (c+d x))^2}-\frac{e (e \cos (c+d x))^{7/2}}{3 b d (a+b \sin (c+d x))^3}","\frac{7 e^3 \left(5 a^2-4 b^2\right) (e \cos (c+d x))^{3/2}}{8 b^3 d \left(a^2-b^2\right) (a+b \sin (c+d x))}+\frac{7 a e^{9/2} \left(5 a^2-6 b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{16 b^{9/2} d \left(b^2-a^2\right)^{5/4}}-\frac{7 a e^{9/2} \left(5 a^2-6 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{16 b^{9/2} d \left(b^2-a^2\right)^{5/4}}+\frac{7 e^4 \left(5 a^2-4 b^2\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{8 b^4 d \left(a^2-b^2\right) \sqrt{\cos (c+d x)}}-\frac{7 a^2 e^5 \left(5 a^2-6 b^2\right) \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{16 b^5 d \left(a^2-b^2\right) \left(b-\sqrt{b^2-a^2}\right) \sqrt{e \cos (c+d x)}}-\frac{7 a^2 e^5 \left(5 a^2-6 b^2\right) \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{16 b^5 d \left(a^2-b^2\right) \left(\sqrt{b^2-a^2}+b\right) \sqrt{e \cos (c+d x)}}-\frac{7 e^3 (e \cos (c+d x))^{3/2} (5 a+4 b \sin (c+d x))}{12 b^3 d (a+b \sin (c+d x))^2}-\frac{e (e \cos (c+d x))^{7/2}}{3 b d (a+b \sin (c+d x))^3}",1,"(7*a*(5*a^2 - 6*b^2)*e^(9/2)*ArcTan[(Sqrt[b]*Sqrt[e*Cos[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(16*b^(9/2)*(-a^2 + b^2)^(5/4)*d) - (7*a*(5*a^2 - 6*b^2)*e^(9/2)*ArcTanh[(Sqrt[b]*Sqrt[e*Cos[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(16*b^(9/2)*(-a^2 + b^2)^(5/4)*d) + (7*(5*a^2 - 4*b^2)*e^4*Sqrt[e*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(8*b^4*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]]) - (7*a^2*(5*a^2 - 6*b^2)*e^5*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (c + d*x)/2, 2])/(16*b^5*(a^2 - b^2)*(b - Sqrt[-a^2 + b^2])*d*Sqrt[e*Cos[c + d*x]]) - (7*a^2*(5*a^2 - 6*b^2)*e^5*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (c + d*x)/2, 2])/(16*b^5*(a^2 - b^2)*(b + Sqrt[-a^2 + b^2])*d*Sqrt[e*Cos[c + d*x]]) - (e*(e*Cos[c + d*x])^(7/2))/(3*b*d*(a + b*Sin[c + d*x])^3) + (7*(5*a^2 - 4*b^2)*e^3*(e*Cos[c + d*x])^(3/2))/(8*b^3*(a^2 - b^2)*d*(a + b*Sin[c + d*x])) - (7*e^3*(e*Cos[c + d*x])^(3/2)*(5*a + 4*b*Sin[c + d*x]))/(12*b^3*d*(a + b*Sin[c + d*x])^2)","A",15,13,25,0.5200,1,"{2693, 2863, 2864, 2867, 2640, 2639, 2701, 2807, 2805, 329, 298, 205, 208}"
609,1,597,0,1.5203202,"\int \frac{(e \cos (c+d x))^{7/2}}{(a+b \sin (c+d x))^4} \, dx","Int[(e*Cos[c + d*x])^(7/2)/(a + b*Sin[c + d*x])^4,x]","-\frac{5 e^3 \left(3 a^2-4 b^2\right) \sqrt{e \cos (c+d x)}}{24 b^3 d \left(a^2-b^2\right) (a+b \sin (c+d x))}-\frac{5 a e^{7/2} \left(a^2-2 b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{16 b^{7/2} d \left(b^2-a^2\right)^{7/4}}-\frac{5 a e^{7/2} \left(a^2-2 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{16 b^{7/2} d \left(b^2-a^2\right)^{7/4}}+\frac{5 e^4 \left(3 a^2-4 b^2\right) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{24 b^4 d \left(a^2-b^2\right) \sqrt{e \cos (c+d x)}}-\frac{5 a^2 e^4 \left(a^2-2 b^2\right) \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{16 b^4 d \left(a^2-b^2\right) \left(a^2-b \left(b-\sqrt{b^2-a^2}\right)\right) \sqrt{e \cos (c+d x)}}-\frac{5 a^2 e^4 \left(a^2-2 b^2\right) \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{16 b^4 d \left(a^2-b^2\right) \left(a^2-b \left(\sqrt{b^2-a^2}+b\right)\right) \sqrt{e \cos (c+d x)}}+\frac{5 e^3 \sqrt{e \cos (c+d x)} (3 a+4 b \sin (c+d x))}{12 b^3 d (a+b \sin (c+d x))^2}-\frac{e (e \cos (c+d x))^{5/2}}{3 b d (a+b \sin (c+d x))^3}","-\frac{5 e^3 \left(3 a^2-4 b^2\right) \sqrt{e \cos (c+d x)}}{24 b^3 d \left(a^2-b^2\right) (a+b \sin (c+d x))}-\frac{5 a e^{7/2} \left(a^2-2 b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{16 b^{7/2} d \left(b^2-a^2\right)^{7/4}}-\frac{5 a e^{7/2} \left(a^2-2 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{16 b^{7/2} d \left(b^2-a^2\right)^{7/4}}+\frac{5 e^4 \left(3 a^2-4 b^2\right) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{24 b^4 d \left(a^2-b^2\right) \sqrt{e \cos (c+d x)}}-\frac{5 a^2 e^4 \left(a^2-2 b^2\right) \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{16 b^4 d \left(a^2-b^2\right) \left(a^2-b \left(b-\sqrt{b^2-a^2}\right)\right) \sqrt{e \cos (c+d x)}}-\frac{5 a^2 e^4 \left(a^2-2 b^2\right) \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{16 b^4 d \left(a^2-b^2\right) \left(a^2-b \left(\sqrt{b^2-a^2}+b\right)\right) \sqrt{e \cos (c+d x)}}+\frac{5 e^3 \sqrt{e \cos (c+d x)} (3 a+4 b \sin (c+d x))}{12 b^3 d (a+b \sin (c+d x))^2}-\frac{e (e \cos (c+d x))^{5/2}}{3 b d (a+b \sin (c+d x))^3}",1,"(-5*a*(a^2 - 2*b^2)*e^(7/2)*ArcTan[(Sqrt[b]*Sqrt[e*Cos[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(16*b^(7/2)*(-a^2 + b^2)^(7/4)*d) - (5*a*(a^2 - 2*b^2)*e^(7/2)*ArcTanh[(Sqrt[b]*Sqrt[e*Cos[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(16*b^(7/2)*(-a^2 + b^2)^(7/4)*d) + (5*(3*a^2 - 4*b^2)*e^4*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(24*b^4*(a^2 - b^2)*d*Sqrt[e*Cos[c + d*x]]) - (5*a^2*(a^2 - 2*b^2)*e^4*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (c + d*x)/2, 2])/(16*b^4*(a^2 - b^2)*(a^2 - b*(b - Sqrt[-a^2 + b^2]))*d*Sqrt[e*Cos[c + d*x]]) - (5*a^2*(a^2 - 2*b^2)*e^4*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (c + d*x)/2, 2])/(16*b^4*(a^2 - b^2)*(a^2 - b*(b + Sqrt[-a^2 + b^2]))*d*Sqrt[e*Cos[c + d*x]]) - (e*(e*Cos[c + d*x])^(5/2))/(3*b*d*(a + b*Sin[c + d*x])^3) - (5*(3*a^2 - 4*b^2)*e^3*Sqrt[e*Cos[c + d*x]])/(24*b^3*(a^2 - b^2)*d*(a + b*Sin[c + d*x])) + (5*e^3*Sqrt[e*Cos[c + d*x]]*(3*a + 4*b*Sin[c + d*x]))/(12*b^3*d*(a + b*Sin[c + d*x])^2)","A",15,13,25,0.5200,1,"{2693, 2863, 2864, 2867, 2642, 2641, 2702, 2807, 2805, 329, 212, 208, 205}"
610,1,574,0,1.4470846,"\int \frac{(e \cos (c+d x))^{5/2}}{(a+b \sin (c+d x))^4} \, dx","Int[(e*Cos[c + d*x])^(5/2)/(a + b*Sin[c + d*x])^4,x]","-\frac{a e^{5/2} \left(a^2-6 b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{16 b^{5/2} d \left(b^2-a^2\right)^{9/4}}+\frac{a e^{5/2} \left(a^2-6 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{16 b^{5/2} d \left(b^2-a^2\right)^{9/4}}+\frac{e^2 \left(a^2+4 b^2\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{8 b^2 d \left(a^2-b^2\right)^2 \sqrt{\cos (c+d x)}}-\frac{a^2 e^3 \left(a^2-6 b^2\right) \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{16 b^3 d \left(a^2-b^2\right)^2 \left(b-\sqrt{b^2-a^2}\right) \sqrt{e \cos (c+d x)}}-\frac{a^2 e^3 \left(a^2-6 b^2\right) \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{16 b^3 d \left(a^2-b^2\right)^2 \left(\sqrt{b^2-a^2}+b\right) \sqrt{e \cos (c+d x)}}+\frac{e \left(a^2+4 b^2\right) (e \cos (c+d x))^{3/2}}{8 b d \left(a^2-b^2\right)^2 (a+b \sin (c+d x))}+\frac{a e (e \cos (c+d x))^{3/2}}{4 b d \left(a^2-b^2\right) (a+b \sin (c+d x))^2}-\frac{e (e \cos (c+d x))^{3/2}}{3 b d (a+b \sin (c+d x))^3}","-\frac{a e^{5/2} \left(a^2-6 b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{16 b^{5/2} d \left(b^2-a^2\right)^{9/4}}+\frac{a e^{5/2} \left(a^2-6 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{16 b^{5/2} d \left(b^2-a^2\right)^{9/4}}+\frac{e^2 \left(a^2+4 b^2\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{8 b^2 d \left(a^2-b^2\right)^2 \sqrt{\cos (c+d x)}}-\frac{a^2 e^3 \left(a^2-6 b^2\right) \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{16 b^3 d \left(a^2-b^2\right)^2 \left(b-\sqrt{b^2-a^2}\right) \sqrt{e \cos (c+d x)}}-\frac{a^2 e^3 \left(a^2-6 b^2\right) \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{16 b^3 d \left(a^2-b^2\right)^2 \left(\sqrt{b^2-a^2}+b\right) \sqrt{e \cos (c+d x)}}+\frac{e \left(a^2+4 b^2\right) (e \cos (c+d x))^{3/2}}{8 b d \left(a^2-b^2\right)^2 (a+b \sin (c+d x))}+\frac{a e (e \cos (c+d x))^{3/2}}{4 b d \left(a^2-b^2\right) (a+b \sin (c+d x))^2}-\frac{e (e \cos (c+d x))^{3/2}}{3 b d (a+b \sin (c+d x))^3}",1,"-(a*(a^2 - 6*b^2)*e^(5/2)*ArcTan[(Sqrt[b]*Sqrt[e*Cos[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(16*b^(5/2)*(-a^2 + b^2)^(9/4)*d) + (a*(a^2 - 6*b^2)*e^(5/2)*ArcTanh[(Sqrt[b]*Sqrt[e*Cos[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(16*b^(5/2)*(-a^2 + b^2)^(9/4)*d) + ((a^2 + 4*b^2)*e^2*Sqrt[e*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(8*b^2*(a^2 - b^2)^2*d*Sqrt[Cos[c + d*x]]) - (a^2*(a^2 - 6*b^2)*e^3*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (c + d*x)/2, 2])/(16*b^3*(a^2 - b^2)^2*(b - Sqrt[-a^2 + b^2])*d*Sqrt[e*Cos[c + d*x]]) - (a^2*(a^2 - 6*b^2)*e^3*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (c + d*x)/2, 2])/(16*b^3*(a^2 - b^2)^2*(b + Sqrt[-a^2 + b^2])*d*Sqrt[e*Cos[c + d*x]]) - (e*(e*Cos[c + d*x])^(3/2))/(3*b*d*(a + b*Sin[c + d*x])^3) + (a*e*(e*Cos[c + d*x])^(3/2))/(4*b*(a^2 - b^2)*d*(a + b*Sin[c + d*x])^2) + ((a^2 + 4*b^2)*e*(e*Cos[c + d*x])^(3/2))/(8*b*(a^2 - b^2)^2*d*(a + b*Sin[c + d*x]))","A",15,12,25,0.4800,1,"{2693, 2864, 2867, 2640, 2639, 2701, 2807, 2805, 329, 298, 205, 208}"
611,1,592,0,1.4710261,"\int \frac{(e \cos (c+d x))^{3/2}}{(a+b \sin (c+d x))^4} \, dx","Int[(e*Cos[c + d*x])^(3/2)/(a + b*Sin[c + d*x])^4,x]","-\frac{a e^{3/2} \left(a^2+6 b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{16 b^{3/2} d \left(b^2-a^2\right)^{11/4}}-\frac{a e^{3/2} \left(a^2+6 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{16 b^{3/2} d \left(b^2-a^2\right)^{11/4}}-\frac{e^2 \left(3 a^2+4 b^2\right) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{24 b^2 d \left(a^2-b^2\right)^2 \sqrt{e \cos (c+d x)}}+\frac{a^2 e^2 \left(a^2+6 b^2\right) \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{16 b^2 d \left(a^2-b^2\right)^2 \left(a^2-b \left(b-\sqrt{b^2-a^2}\right)\right) \sqrt{e \cos (c+d x)}}+\frac{a^2 e^2 \left(a^2+6 b^2\right) \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{16 b^2 d \left(a^2-b^2\right)^2 \left(a^2-b \left(\sqrt{b^2-a^2}+b\right)\right) \sqrt{e \cos (c+d x)}}+\frac{e \left(3 a^2+4 b^2\right) \sqrt{e \cos (c+d x)}}{24 b d \left(a^2-b^2\right)^2 (a+b \sin (c+d x))}+\frac{a e \sqrt{e \cos (c+d x)}}{12 b d \left(a^2-b^2\right) (a+b \sin (c+d x))^2}-\frac{e \sqrt{e \cos (c+d x)}}{3 b d (a+b \sin (c+d x))^3}","-\frac{a e^{3/2} \left(a^2+6 b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{16 b^{3/2} d \left(b^2-a^2\right)^{11/4}}-\frac{a e^{3/2} \left(a^2+6 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{16 b^{3/2} d \left(b^2-a^2\right)^{11/4}}-\frac{e^2 \left(3 a^2+4 b^2\right) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{24 b^2 d \left(a^2-b^2\right)^2 \sqrt{e \cos (c+d x)}}+\frac{a^2 e^2 \left(a^2+6 b^2\right) \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{16 b^2 d \left(a^2-b^2\right)^2 \left(a^2-b \left(b-\sqrt{b^2-a^2}\right)\right) \sqrt{e \cos (c+d x)}}+\frac{a^2 e^2 \left(a^2+6 b^2\right) \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{16 b^2 d \left(a^2-b^2\right)^2 \left(a^2-b \left(\sqrt{b^2-a^2}+b\right)\right) \sqrt{e \cos (c+d x)}}+\frac{e \left(3 a^2+4 b^2\right) \sqrt{e \cos (c+d x)}}{24 b d \left(a^2-b^2\right)^2 (a+b \sin (c+d x))}+\frac{a e \sqrt{e \cos (c+d x)}}{12 b d \left(a^2-b^2\right) (a+b \sin (c+d x))^2}-\frac{e \sqrt{e \cos (c+d x)}}{3 b d (a+b \sin (c+d x))^3}",1,"-(a*(a^2 + 6*b^2)*e^(3/2)*ArcTan[(Sqrt[b]*Sqrt[e*Cos[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(16*b^(3/2)*(-a^2 + b^2)^(11/4)*d) - (a*(a^2 + 6*b^2)*e^(3/2)*ArcTanh[(Sqrt[b]*Sqrt[e*Cos[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(16*b^(3/2)*(-a^2 + b^2)^(11/4)*d) - ((3*a^2 + 4*b^2)*e^2*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(24*b^2*(a^2 - b^2)^2*d*Sqrt[e*Cos[c + d*x]]) + (a^2*(a^2 + 6*b^2)*e^2*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (c + d*x)/2, 2])/(16*b^2*(a^2 - b^2)^2*(a^2 - b*(b - Sqrt[-a^2 + b^2]))*d*Sqrt[e*Cos[c + d*x]]) + (a^2*(a^2 + 6*b^2)*e^2*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (c + d*x)/2, 2])/(16*b^2*(a^2 - b^2)^2*(a^2 - b*(b + Sqrt[-a^2 + b^2]))*d*Sqrt[e*Cos[c + d*x]]) - (e*Sqrt[e*Cos[c + d*x]])/(3*b*d*(a + b*Sin[c + d*x])^3) + (a*e*Sqrt[e*Cos[c + d*x]])/(12*b*(a^2 - b^2)*d*(a + b*Sin[c + d*x])^2) + ((3*a^2 + 4*b^2)*e*Sqrt[e*Cos[c + d*x]])/(24*b*(a^2 - b^2)^2*d*(a + b*Sin[c + d*x]))","A",15,12,25,0.4800,1,"{2693, 2864, 2867, 2642, 2641, 2702, 2807, 2805, 329, 212, 208, 205}"
612,1,579,0,1.5317034,"\int \frac{\sqrt{e \cos (c+d x)}}{(a+b \sin (c+d x))^4} \, dx","Int[Sqrt[e*Cos[c + d*x]]/(a + b*Sin[c + d*x])^4,x]","\frac{3 a b (e \cos (c+d x))^{3/2}}{4 d e \left(a^2-b^2\right)^2 (a+b \sin (c+d x))^2}+\frac{b \left(11 a^2+4 b^2\right) (e \cos (c+d x))^{3/2}}{8 d e \left(a^2-b^2\right)^3 (a+b \sin (c+d x))}+\frac{b (e \cos (c+d x))^{3/2}}{3 d e \left(a^2-b^2\right) (a+b \sin (c+d x))^3}-\frac{5 a \sqrt{e} \left(a^2+2 b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{16 \sqrt{b} d \left(b^2-a^2\right)^{13/4}}+\frac{5 a \sqrt{e} \left(a^2+2 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{16 \sqrt{b} d \left(b^2-a^2\right)^{13/4}}+\frac{\left(11 a^2+4 b^2\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{8 d \left(a^2-b^2\right)^3 \sqrt{\cos (c+d x)}}+\frac{5 a^2 e \left(a^2+2 b^2\right) \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{16 b d \left(a^2-b^2\right)^3 \left(b-\sqrt{b^2-a^2}\right) \sqrt{e \cos (c+d x)}}+\frac{5 a^2 e \left(a^2+2 b^2\right) \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{16 b d \left(a^2-b^2\right)^3 \left(\sqrt{b^2-a^2}+b\right) \sqrt{e \cos (c+d x)}}","\frac{3 a b (e \cos (c+d x))^{3/2}}{4 d e \left(a^2-b^2\right)^2 (a+b \sin (c+d x))^2}+\frac{b \left(11 a^2+4 b^2\right) (e \cos (c+d x))^{3/2}}{8 d e \left(a^2-b^2\right)^3 (a+b \sin (c+d x))}+\frac{b (e \cos (c+d x))^{3/2}}{3 d e \left(a^2-b^2\right) (a+b \sin (c+d x))^3}-\frac{5 a \sqrt{e} \left(a^2+2 b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{16 \sqrt{b} d \left(b^2-a^2\right)^{13/4}}+\frac{5 a \sqrt{e} \left(a^2+2 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{16 \sqrt{b} d \left(b^2-a^2\right)^{13/4}}+\frac{\left(11 a^2+4 b^2\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{8 d \left(a^2-b^2\right)^3 \sqrt{\cos (c+d x)}}+\frac{5 a^2 e \left(a^2+2 b^2\right) \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{16 b d \left(a^2-b^2\right)^3 \left(b-\sqrt{b^2-a^2}\right) \sqrt{e \cos (c+d x)}}+\frac{5 a^2 e \left(a^2+2 b^2\right) \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{16 b d \left(a^2-b^2\right)^3 \left(\sqrt{b^2-a^2}+b\right) \sqrt{e \cos (c+d x)}}",1,"(-5*a*(a^2 + 2*b^2)*Sqrt[e]*ArcTan[(Sqrt[b]*Sqrt[e*Cos[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(16*Sqrt[b]*(-a^2 + b^2)^(13/4)*d) + (5*a*(a^2 + 2*b^2)*Sqrt[e]*ArcTanh[(Sqrt[b]*Sqrt[e*Cos[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(16*Sqrt[b]*(-a^2 + b^2)^(13/4)*d) + ((11*a^2 + 4*b^2)*Sqrt[e*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(8*(a^2 - b^2)^3*d*Sqrt[Cos[c + d*x]]) + (5*a^2*(a^2 + 2*b^2)*e*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (c + d*x)/2, 2])/(16*b*(a^2 - b^2)^3*(b - Sqrt[-a^2 + b^2])*d*Sqrt[e*Cos[c + d*x]]) + (5*a^2*(a^2 + 2*b^2)*e*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (c + d*x)/2, 2])/(16*b*(a^2 - b^2)^3*(b + Sqrt[-a^2 + b^2])*d*Sqrt[e*Cos[c + d*x]]) + (b*(e*Cos[c + d*x])^(3/2))/(3*(a^2 - b^2)*d*e*(a + b*Sin[c + d*x])^3) + (3*a*b*(e*Cos[c + d*x])^(3/2))/(4*(a^2 - b^2)^2*d*e*(a + b*Sin[c + d*x])^2) + (b*(11*a^2 + 4*b^2)*(e*Cos[c + d*x])^(3/2))/(8*(a^2 - b^2)^3*d*e*(a + b*Sin[c + d*x]))","A",15,12,25,0.4800,1,"{2694, 2864, 2867, 2640, 2639, 2701, 2807, 2805, 329, 298, 205, 208}"
613,1,593,0,1.5700165,"\int \frac{1}{\sqrt{e \cos (c+d x)} (a+b \sin (c+d x))^4} \, dx","Int[1/(Sqrt[e*Cos[c + d*x]]*(a + b*Sin[c + d*x])^4),x]","\frac{11 a b \sqrt{e \cos (c+d x)}}{12 d e \left(a^2-b^2\right)^2 (a+b \sin (c+d x))^2}+\frac{b \left(57 a^2+20 b^2\right) \sqrt{e \cos (c+d x)}}{24 d e \left(a^2-b^2\right)^3 (a+b \sin (c+d x))}+\frac{b \sqrt{e \cos (c+d x)}}{3 d e \left(a^2-b^2\right) (a+b \sin (c+d x))^3}+\frac{7 a \sqrt{b} \left(5 a^2+6 b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{16 d \sqrt{e} \left(b^2-a^2\right)^{15/4}}+\frac{7 a \sqrt{b} \left(5 a^2+6 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{16 d \sqrt{e} \left(b^2-a^2\right)^{15/4}}-\frac{\left(57 a^2+20 b^2\right) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{24 d \left(a^2-b^2\right)^3 \sqrt{e \cos (c+d x)}}+\frac{7 a^2 \left(5 a^2+6 b^2\right) \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{16 d \left(a^2-b^2\right)^3 \left(a^2-b \left(b-\sqrt{b^2-a^2}\right)\right) \sqrt{e \cos (c+d x)}}+\frac{7 a^2 \left(5 a^2+6 b^2\right) \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{16 d \left(a^2-b^2\right)^3 \left(a^2-b \left(\sqrt{b^2-a^2}+b\right)\right) \sqrt{e \cos (c+d x)}}","\frac{11 a b \sqrt{e \cos (c+d x)}}{12 d e \left(a^2-b^2\right)^2 (a+b \sin (c+d x))^2}+\frac{b \left(57 a^2+20 b^2\right) \sqrt{e \cos (c+d x)}}{24 d e \left(a^2-b^2\right)^3 (a+b \sin (c+d x))}+\frac{b \sqrt{e \cos (c+d x)}}{3 d e \left(a^2-b^2\right) (a+b \sin (c+d x))^3}+\frac{7 a \sqrt{b} \left(5 a^2+6 b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{16 d \sqrt{e} \left(b^2-a^2\right)^{15/4}}+\frac{7 a \sqrt{b} \left(5 a^2+6 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{16 d \sqrt{e} \left(b^2-a^2\right)^{15/4}}-\frac{\left(57 a^2+20 b^2\right) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{24 d \left(a^2-b^2\right)^3 \sqrt{e \cos (c+d x)}}+\frac{7 a^2 \left(5 a^2+6 b^2\right) \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{16 d \left(a^2-b^2\right)^3 \left(a^2-b \left(b-\sqrt{b^2-a^2}\right)\right) \sqrt{e \cos (c+d x)}}+\frac{7 a^2 \left(5 a^2+6 b^2\right) \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{16 d \left(a^2-b^2\right)^3 \left(a^2-b \left(\sqrt{b^2-a^2}+b\right)\right) \sqrt{e \cos (c+d x)}}",1,"(7*a*Sqrt[b]*(5*a^2 + 6*b^2)*ArcTan[(Sqrt[b]*Sqrt[e*Cos[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(16*(-a^2 + b^2)^(15/4)*d*Sqrt[e]) + (7*a*Sqrt[b]*(5*a^2 + 6*b^2)*ArcTanh[(Sqrt[b]*Sqrt[e*Cos[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(16*(-a^2 + b^2)^(15/4)*d*Sqrt[e]) - ((57*a^2 + 20*b^2)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(24*(a^2 - b^2)^3*d*Sqrt[e*Cos[c + d*x]]) + (7*a^2*(5*a^2 + 6*b^2)*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (c + d*x)/2, 2])/(16*(a^2 - b^2)^3*(a^2 - b*(b - Sqrt[-a^2 + b^2]))*d*Sqrt[e*Cos[c + d*x]]) + (7*a^2*(5*a^2 + 6*b^2)*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (c + d*x)/2, 2])/(16*(a^2 - b^2)^3*(a^2 - b*(b + Sqrt[-a^2 + b^2]))*d*Sqrt[e*Cos[c + d*x]]) + (b*Sqrt[e*Cos[c + d*x]])/(3*(a^2 - b^2)*d*e*(a + b*Sin[c + d*x])^3) + (11*a*b*Sqrt[e*Cos[c + d*x]])/(12*(a^2 - b^2)^2*d*e*(a + b*Sin[c + d*x])^2) + (b*(57*a^2 + 20*b^2)*Sqrt[e*Cos[c + d*x]])/(24*(a^2 - b^2)^3*d*e*(a + b*Sin[c + d*x]))","A",15,12,25,0.4800,1,"{2694, 2864, 2867, 2642, 2641, 2702, 2807, 2805, 329, 212, 208, 205}"
614,1,674,0,1.9523504,"\int \frac{1}{(e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^4} \, dx","Int[1/((e*Cos[c + d*x])^(3/2)*(a + b*Sin[c + d*x])^4),x]","-\frac{15 a b^{3/2} \left(7 a^2+6 b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{16 d e^{3/2} \left(b^2-a^2\right)^{17/4}}+\frac{15 a b^{3/2} \left(7 a^2+6 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{16 d e^{3/2} \left(b^2-a^2\right)^{17/4}}-\frac{\left(151 a^2 b^2+16 a^4+28 b^4\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{8 d e^2 \left(a^2-b^2\right)^4 \sqrt{\cos (c+d x)}}+\frac{13 a b}{12 d e \left(a^2-b^2\right)^2 \sqrt{e \cos (c+d x)} (a+b \sin (c+d x))^2}-\frac{15 a b \left(7 a^2+6 b^2\right)-\left(151 a^2 b^2+16 a^4+28 b^4\right) \sin (c+d x)}{8 d e \left(a^2-b^2\right)^4 \sqrt{e \cos (c+d x)}}+\frac{b \left(89 a^2+28 b^2\right)}{24 d e \left(a^2-b^2\right)^3 \sqrt{e \cos (c+d x)} (a+b \sin (c+d x))}+\frac{b}{3 d e \left(a^2-b^2\right) \sqrt{e \cos (c+d x)} (a+b \sin (c+d x))^3}-\frac{15 a^2 b \left(7 a^2+6 b^2\right) \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{16 d e \left(a^2-b^2\right)^4 \left(b-\sqrt{b^2-a^2}\right) \sqrt{e \cos (c+d x)}}-\frac{15 a^2 b \left(7 a^2+6 b^2\right) \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{16 d e \left(a^2-b^2\right)^4 \left(\sqrt{b^2-a^2}+b\right) \sqrt{e \cos (c+d x)}}","-\frac{15 a b^{3/2} \left(7 a^2+6 b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{16 d e^{3/2} \left(b^2-a^2\right)^{17/4}}+\frac{15 a b^{3/2} \left(7 a^2+6 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{16 d e^{3/2} \left(b^2-a^2\right)^{17/4}}-\frac{\left(151 a^2 b^2+16 a^4+28 b^4\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{8 d e^2 \left(a^2-b^2\right)^4 \sqrt{\cos (c+d x)}}+\frac{13 a b}{12 d e \left(a^2-b^2\right)^2 \sqrt{e \cos (c+d x)} (a+b \sin (c+d x))^2}-\frac{15 a b \left(7 a^2+6 b^2\right)-\left(151 a^2 b^2+16 a^4+28 b^4\right) \sin (c+d x)}{8 d e \left(a^2-b^2\right)^4 \sqrt{e \cos (c+d x)}}+\frac{b \left(89 a^2+28 b^2\right)}{24 d e \left(a^2-b^2\right)^3 \sqrt{e \cos (c+d x)} (a+b \sin (c+d x))}+\frac{b}{3 d e \left(a^2-b^2\right) \sqrt{e \cos (c+d x)} (a+b \sin (c+d x))^3}-\frac{15 a^2 b \left(7 a^2+6 b^2\right) \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{16 d e \left(a^2-b^2\right)^4 \left(b-\sqrt{b^2-a^2}\right) \sqrt{e \cos (c+d x)}}-\frac{15 a^2 b \left(7 a^2+6 b^2\right) \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{16 d e \left(a^2-b^2\right)^4 \left(\sqrt{b^2-a^2}+b\right) \sqrt{e \cos (c+d x)}}",1,"(-15*a*b^(3/2)*(7*a^2 + 6*b^2)*ArcTan[(Sqrt[b]*Sqrt[e*Cos[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(16*(-a^2 + b^2)^(17/4)*d*e^(3/2)) + (15*a*b^(3/2)*(7*a^2 + 6*b^2)*ArcTanh[(Sqrt[b]*Sqrt[e*Cos[c + d*x]])/((-a^2 + b^2)^(1/4)*Sqrt[e])])/(16*(-a^2 + b^2)^(17/4)*d*e^(3/2)) - ((16*a^4 + 151*a^2*b^2 + 28*b^4)*Sqrt[e*Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2])/(8*(a^2 - b^2)^4*d*e^2*Sqrt[Cos[c + d*x]]) - (15*a^2*b*(7*a^2 + 6*b^2)*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(b - Sqrt[-a^2 + b^2]), (c + d*x)/2, 2])/(16*(a^2 - b^2)^4*(b - Sqrt[-a^2 + b^2])*d*e*Sqrt[e*Cos[c + d*x]]) - (15*a^2*b*(7*a^2 + 6*b^2)*Sqrt[Cos[c + d*x]]*EllipticPi[(2*b)/(b + Sqrt[-a^2 + b^2]), (c + d*x)/2, 2])/(16*(a^2 - b^2)^4*(b + Sqrt[-a^2 + b^2])*d*e*Sqrt[e*Cos[c + d*x]]) + b/(3*(a^2 - b^2)*d*e*Sqrt[e*Cos[c + d*x]]*(a + b*Sin[c + d*x])^3) + (13*a*b)/(12*(a^2 - b^2)^2*d*e*Sqrt[e*Cos[c + d*x]]*(a + b*Sin[c + d*x])^2) + (b*(89*a^2 + 28*b^2))/(24*(a^2 - b^2)^3*d*e*Sqrt[e*Cos[c + d*x]]*(a + b*Sin[c + d*x])) - (15*a*b*(7*a^2 + 6*b^2) - (16*a^4 + 151*a^2*b^2 + 28*b^4)*Sin[c + d*x])/(8*(a^2 - b^2)^4*d*e*Sqrt[e*Cos[c + d*x]])","A",16,13,25,0.5200,1,"{2694, 2864, 2866, 2867, 2640, 2639, 2701, 2807, 2805, 329, 298, 205, 208}"
615,1,374,0,0.4275066,"\int \frac{1}{\sqrt{c \cos (e+f x)} \sqrt{a+b \sin (e+f x)}} \, dx","Int[1/(Sqrt[c*Cos[e + f*x]]*Sqrt[a + b*Sin[e + f*x]]),x]","\frac{\sqrt{2} \sqrt[4]{a-b} \sqrt{c \cos (e+f x)} \sqrt{\frac{a+b \sin (e+f x)}{(a-b) (1-\sin (e+f x))}} \sqrt{\frac{a+b \sin (e+f x)}{(a-b) (\sin (e) (-\cos (f x))-\cos (e) \sin (f x)+1) \left(\frac{\sqrt{a+b} (\sin (e+f x)+\cos (e+f x)+1)}{\sqrt{a-b} (-\sin (e+f x)+\cos (e+f x)+1)}+1\right)^2}} \left(\frac{\sqrt{a+b} (\sin (e+f x)+\cos (e+f x)+1)}{\sqrt{a-b} (-\sin (e+f x)+\cos (e+f x)+1)}+1\right) F\left(2 \tan ^{-1}\left(\frac{\sqrt[4]{a+b} \sqrt{\frac{\cos (e+f x)+\sin (e+f x)+1}{\cos (e+f x)-\sin (e+f x)+1}}}{\sqrt[4]{a-b}}\right)|\frac{1}{2}\right)}{c f \sqrt[4]{a+b} \sqrt{\frac{\sin (e+f x)+\cos (e+f x)+1}{-\sin (e+f x)+\cos (e+f x)+1}} \sqrt{a+b \sin (e+f x)} \sqrt{\frac{a+b \sin (e+f x)}{(a-b) (\sin (e) (-\cos (f x))-\cos (e) \sin (f x)+1)}}}","\frac{2 \sqrt{2} \sqrt[4]{b-a} \sqrt{c \cos (e+f x)} \sqrt{\frac{a+b \sin (e+f x)}{(a-b) (1-\sin (e+f x))}} F\left(\left.\sin ^{-1}\left(\frac{\sqrt[4]{a+b} \sqrt{\frac{\cos (e+f x)+\sin (e+f x)+1}{\cos (e+f x)-\sin (e+f x)+1}}}{\sqrt[4]{b-a}}\right)\right|-1\right)}{c f \sqrt[4]{a+b} \sqrt{\frac{\sin (e+f x)+\cos (e+f x)+1}{-\sin (e+f x)+\cos (e+f x)+1}} \sqrt{a+b \sin (e+f x)}}",1,"(Sqrt[2]*(a - b)^(1/4)*Sqrt[c*Cos[e + f*x]]*EllipticF[2*ArcTan[((a + b)^(1/4)*Sqrt[(1 + Cos[e + f*x] + Sin[e + f*x])/(1 + Cos[e + f*x] - Sin[e + f*x])])/(a - b)^(1/4)], 1/2]*Sqrt[(a + b*Sin[e + f*x])/((a - b)*(1 - Sin[e + f*x]))]*Sqrt[(a + b*Sin[e + f*x])/((a - b)*(1 - Cos[f*x]*Sin[e] - Cos[e]*Sin[f*x])*(1 + (Sqrt[a + b]*(1 + Cos[e + f*x] + Sin[e + f*x]))/(Sqrt[a - b]*(1 + Cos[e + f*x] - Sin[e + f*x])))^2)]*(1 + (Sqrt[a + b]*(1 + Cos[e + f*x] + Sin[e + f*x]))/(Sqrt[a - b]*(1 + Cos[e + f*x] - Sin[e + f*x]))))/((a + b)^(1/4)*c*f*Sqrt[(1 + Cos[e + f*x] + Sin[e + f*x])/(1 + Cos[e + f*x] - Sin[e + f*x])]*Sqrt[a + b*Sin[e + f*x]]*Sqrt[(a + b*Sin[e + f*x])/((a - b)*(1 - Cos[f*x]*Sin[e] - Cos[e]*Sin[f*x]))])","B",2,2,27,0.07407,0,"{2697, 220}"
616,1,229,0,0.3697077,"\int (e \cos (c+d x))^p (a+b \sin (c+d x))^3 \, dx","Int[(e*Cos[c + d*x])^p*(a + b*Sin[c + d*x])^3,x]","-\frac{a \left(a^2 (p+2)+3 b^2\right) \sin (c+d x) (e \cos (c+d x))^{p+1} \, _2F_1\left(\frac{1}{2},\frac{p+1}{2};\frac{p+3}{2};\cos ^2(c+d x)\right)}{d e (p+1) (p+2) \sqrt{\sin ^2(c+d x)}}-\frac{b \left(a^2 \left(p^2+6 p+11\right)+2 b^2 (p+2)\right) (e \cos (c+d x))^{p+1}}{d e (p+1) (p+2) (p+3)}-\frac{b (a+b \sin (c+d x))^2 (e \cos (c+d x))^{p+1}}{d e (p+3)}-\frac{a b (p+5) (a+b \sin (c+d x)) (e \cos (c+d x))^{p+1}}{d e (p+2) (p+3)}","-\frac{a \left(a^2 (p+2)+3 b^2\right) \sin (c+d x) (e \cos (c+d x))^{p+1} \, _2F_1\left(\frac{1}{2},\frac{p+1}{2};\frac{p+3}{2};\cos ^2(c+d x)\right)}{d e (p+1) (p+2) \sqrt{\sin ^2(c+d x)}}-\frac{b \left(a^2 \left(p^2+6 p+11\right)+2 b^2 (p+2)\right) (e \cos (c+d x))^{p+1}}{d e (p+1) (p+2) (p+3)}-\frac{b (a+b \sin (c+d x))^2 (e \cos (c+d x))^{p+1}}{d e (p+3)}-\frac{a b (p+5) (a+b \sin (c+d x)) (e \cos (c+d x))^{p+1}}{d e (p+2) (p+3)}",1,"-((b*(2*b^2*(2 + p) + a^2*(11 + 6*p + p^2))*(e*Cos[c + d*x])^(1 + p))/(d*e*(1 + p)*(2 + p)*(3 + p))) - (a*(3*b^2 + a^2*(2 + p))*(e*Cos[c + d*x])^(1 + p)*Hypergeometric2F1[1/2, (1 + p)/2, (3 + p)/2, Cos[c + d*x]^2]*Sin[c + d*x])/(d*e*(1 + p)*(2 + p)*Sqrt[Sin[c + d*x]^2]) - (a*b*(5 + p)*(e*Cos[c + d*x])^(1 + p)*(a + b*Sin[c + d*x]))/(d*e*(2 + p)*(3 + p)) - (b*(e*Cos[c + d*x])^(1 + p)*(a + b*Sin[c + d*x])^2)/(d*e*(3 + p))","A",4,4,23,0.1739,1,"{2692, 2862, 2669, 2643}"
617,1,157,0,0.1492406,"\int (e \cos (c+d x))^p (a+b \sin (c+d x))^2 \, dx","Int[(e*Cos[c + d*x])^p*(a + b*Sin[c + d*x])^2,x]","-\frac{\left(a^2 (p+2)+b^2\right) \sin (c+d x) (e \cos (c+d x))^{p+1} \, _2F_1\left(\frac{1}{2},\frac{p+1}{2};\frac{p+3}{2};\cos ^2(c+d x)\right)}{d e (p+1) (p+2) \sqrt{\sin ^2(c+d x)}}-\frac{a b (p+3) (e \cos (c+d x))^{p+1}}{d e (p+1) (p+2)}-\frac{b (a+b \sin (c+d x)) (e \cos (c+d x))^{p+1}}{d e (p+2)}","-\frac{\left(a^2 (p+2)+b^2\right) \sin (c+d x) (e \cos (c+d x))^{p+1} \, _2F_1\left(\frac{1}{2},\frac{p+1}{2};\frac{p+3}{2};\cos ^2(c+d x)\right)}{d e (p+1) (p+2) \sqrt{\sin ^2(c+d x)}}-\frac{a b (p+3) (e \cos (c+d x))^{p+1}}{d e (p+1) (p+2)}-\frac{b (a+b \sin (c+d x)) (e \cos (c+d x))^{p+1}}{d e (p+2)}",1,"-((a*b*(3 + p)*(e*Cos[c + d*x])^(1 + p))/(d*e*(1 + p)*(2 + p))) - ((b^2 + a^2*(2 + p))*(e*Cos[c + d*x])^(1 + p)*Hypergeometric2F1[1/2, (1 + p)/2, (3 + p)/2, Cos[c + d*x]^2]*Sin[c + d*x])/(d*e*(1 + p)*(2 + p)*Sqrt[Sin[c + d*x]^2]) - (b*(e*Cos[c + d*x])^(1 + p)*(a + b*Sin[c + d*x]))/(d*e*(2 + p))","A",3,3,23,0.1304,1,"{2692, 2669, 2643}"
618,1,97,0,0.0520769,"\int (e \cos (c+d x))^p (a+b \sin (c+d x)) \, dx","Int[(e*Cos[c + d*x])^p*(a + b*Sin[c + d*x]),x]","-\frac{a \sin (c+d x) (e \cos (c+d x))^{p+1} \, _2F_1\left(\frac{1}{2},\frac{p+1}{2};\frac{p+3}{2};\cos ^2(c+d x)\right)}{d e (p+1) \sqrt{\sin ^2(c+d x)}}-\frac{b (e \cos (c+d x))^{p+1}}{d e (p+1)}","-\frac{a \sin (c+d x) (e \cos (c+d x))^{p+1} \, _2F_1\left(\frac{1}{2},\frac{p+1}{2};\frac{p+3}{2};\cos ^2(c+d x)\right)}{d e (p+1) \sqrt{\sin ^2(c+d x)}}-\frac{b (e \cos (c+d x))^{p+1}}{d e (p+1)}",1,"-((b*(e*Cos[c + d*x])^(1 + p))/(d*e*(1 + p))) - (a*(e*Cos[c + d*x])^(1 + p)*Hypergeometric2F1[1/2, (1 + p)/2, (3 + p)/2, Cos[c + d*x]^2]*Sin[c + d*x])/(d*e*(1 + p)*Sqrt[Sin[c + d*x]^2])","A",2,2,21,0.09524,1,"{2669, 2643}"
619,1,158,0,0.0793573,"\int \frac{(e \cos (c+d x))^p}{a+b \sin (c+d x)} \, dx","Int[(e*Cos[c + d*x])^p/(a + b*Sin[c + d*x]),x]","-\frac{e (e \cos (c+d x))^{p-1} \left(-\frac{b (1-\sin (c+d x))}{a+b \sin (c+d x)}\right)^{\frac{1-p}{2}} \left(\frac{b (\sin (c+d x)+1)}{a+b \sin (c+d x)}\right)^{\frac{1-p}{2}} F_1\left(1-p;\frac{1-p}{2},\frac{1-p}{2};2-p;\frac{a+b}{a+b \sin (c+d x)},\frac{a-b}{a+b \sin (c+d x)}\right)}{b d (1-p)}","-\frac{e (e \cos (c+d x))^{p-1} \left(-\frac{b (1-\sin (c+d x))}{a+b \sin (c+d x)}\right)^{\frac{1-p}{2}} \left(\frac{b (\sin (c+d x)+1)}{a+b \sin (c+d x)}\right)^{\frac{1-p}{2}} F_1\left(1-p;\frac{1-p}{2},\frac{1-p}{2};2-p;\frac{a+b}{a+b \sin (c+d x)},\frac{a-b}{a+b \sin (c+d x)}\right)}{b d (1-p)}",1,"-((e*AppellF1[1 - p, (1 - p)/2, (1 - p)/2, 2 - p, (a + b)/(a + b*Sin[c + d*x]), (a - b)/(a + b*Sin[c + d*x])]*(e*Cos[c + d*x])^(-1 + p)*(-((b*(1 - Sin[c + d*x]))/(a + b*Sin[c + d*x])))^((1 - p)/2)*((b*(1 + Sin[c + d*x]))/(a + b*Sin[c + d*x]))^((1 - p)/2))/(b*d*(1 - p)))","A",1,1,23,0.04348,1,"{2703}"
620,1,170,0,0.0694385,"\int \frac{(e \cos (c+d x))^p}{(a+b \sin (c+d x))^2} \, dx","Int[(e*Cos[c + d*x])^p/(a + b*Sin[c + d*x])^2,x]","-\frac{e (e \cos (c+d x))^{p-1} \left(-\frac{b (1-\sin (c+d x))}{a+b \sin (c+d x)}\right)^{\frac{1-p}{2}} \left(\frac{b (\sin (c+d x)+1)}{a+b \sin (c+d x)}\right)^{\frac{1-p}{2}} F_1\left(2-p;\frac{1-p}{2},\frac{1-p}{2};3-p;\frac{a+b}{a+b \sin (c+d x)},\frac{a-b}{a+b \sin (c+d x)}\right)}{b d (2-p) (a+b \sin (c+d x))}","-\frac{e (e \cos (c+d x))^{p-1} \left(-\frac{b (1-\sin (c+d x))}{a+b \sin (c+d x)}\right)^{\frac{1-p}{2}} \left(\frac{b (\sin (c+d x)+1)}{a+b \sin (c+d x)}\right)^{\frac{1-p}{2}} F_1\left(2-p;\frac{1-p}{2},\frac{1-p}{2};3-p;\frac{a+b}{a+b \sin (c+d x)},\frac{a-b}{a+b \sin (c+d x)}\right)}{b d (2-p) (a+b \sin (c+d x))}",1,"-((e*AppellF1[2 - p, (1 - p)/2, (1 - p)/2, 3 - p, (a + b)/(a + b*Sin[c + d*x]), (a - b)/(a + b*Sin[c + d*x])]*(e*Cos[c + d*x])^(-1 + p)*(-((b*(1 - Sin[c + d*x]))/(a + b*Sin[c + d*x])))^((1 - p)/2)*((b*(1 + Sin[c + d*x]))/(a + b*Sin[c + d*x]))^((1 - p)/2))/(b*d*(2 - p)*(a + b*Sin[c + d*x])))","A",1,1,23,0.04348,1,"{2703}"
621,1,170,0,0.0689775,"\int \frac{(e \cos (c+d x))^p}{(a+b \sin (c+d x))^3} \, dx","Int[(e*Cos[c + d*x])^p/(a + b*Sin[c + d*x])^3,x]","-\frac{e (e \cos (c+d x))^{p-1} \left(-\frac{b (1-\sin (c+d x))}{a+b \sin (c+d x)}\right)^{\frac{1-p}{2}} \left(\frac{b (\sin (c+d x)+1)}{a+b \sin (c+d x)}\right)^{\frac{1-p}{2}} F_1\left(3-p;\frac{1-p}{2},\frac{1-p}{2};4-p;\frac{a+b}{a+b \sin (c+d x)},\frac{a-b}{a+b \sin (c+d x)}\right)}{b d (3-p) (a+b \sin (c+d x))^2}","-\frac{e (e \cos (c+d x))^{p-1} \left(-\frac{b (1-\sin (c+d x))}{a+b \sin (c+d x)}\right)^{\frac{1-p}{2}} \left(\frac{b (\sin (c+d x)+1)}{a+b \sin (c+d x)}\right)^{\frac{1-p}{2}} F_1\left(3-p;\frac{1-p}{2},\frac{1-p}{2};4-p;\frac{a+b}{a+b \sin (c+d x)},\frac{a-b}{a+b \sin (c+d x)}\right)}{b d (3-p) (a+b \sin (c+d x))^2}",1,"-((e*AppellF1[3 - p, (1 - p)/2, (1 - p)/2, 4 - p, (a + b)/(a + b*Sin[c + d*x]), (a - b)/(a + b*Sin[c + d*x])]*(e*Cos[c + d*x])^(-1 + p)*(-((b*(1 - Sin[c + d*x]))/(a + b*Sin[c + d*x])))^((1 - p)/2)*((b*(1 + Sin[c + d*x]))/(a + b*Sin[c + d*x]))^((1 - p)/2))/(b*d*(3 - p)*(a + b*Sin[c + d*x])^2))","A",1,1,23,0.04348,1,"{2703}"
622,1,170,0,0.0704151,"\int \frac{(e \cos (c+d x))^p}{(a+b \sin (c+d x))^8} \, dx","Int[(e*Cos[c + d*x])^p/(a + b*Sin[c + d*x])^8,x]","-\frac{e (e \cos (c+d x))^{p-1} \left(-\frac{b (1-\sin (c+d x))}{a+b \sin (c+d x)}\right)^{\frac{1-p}{2}} \left(\frac{b (\sin (c+d x)+1)}{a+b \sin (c+d x)}\right)^{\frac{1-p}{2}} F_1\left(8-p;\frac{1-p}{2},\frac{1-p}{2};9-p;\frac{a+b}{a+b \sin (c+d x)},\frac{a-b}{a+b \sin (c+d x)}\right)}{b d (8-p) (a+b \sin (c+d x))^7}","-\frac{e (e \cos (c+d x))^{p-1} \left(-\frac{b (1-\sin (c+d x))}{a+b \sin (c+d x)}\right)^{\frac{1-p}{2}} \left(\frac{b (\sin (c+d x)+1)}{a+b \sin (c+d x)}\right)^{\frac{1-p}{2}} F_1\left(8-p;\frac{1-p}{2},\frac{1-p}{2};9-p;\frac{a+b}{a+b \sin (c+d x)},\frac{a-b}{a+b \sin (c+d x)}\right)}{b d (8-p) (a+b \sin (c+d x))^7}",1,"-((e*AppellF1[8 - p, (1 - p)/2, (1 - p)/2, 9 - p, (a + b)/(a + b*Sin[c + d*x]), (a - b)/(a + b*Sin[c + d*x])]*(e*Cos[c + d*x])^(-1 + p)*(-((b*(1 - Sin[c + d*x]))/(a + b*Sin[c + d*x])))^((1 - p)/2)*((b*(1 + Sin[c + d*x]))/(a + b*Sin[c + d*x]))^((1 - p)/2))/(b*d*(8 - p)*(a + b*Sin[c + d*x])^7))","A",1,1,23,0.04348,1,"{2703}"
623,1,156,0,0.1337578,"\int (e \cos (c+d x))^p (a+b \sin (c+d x))^{5/2} \, dx","Int[(e*Cos[c + d*x])^p*(a + b*Sin[c + d*x])^(5/2),x]","\frac{2 e (a+b \sin (c+d x))^{7/2} (e \cos (c+d x))^{p-1} \left(1-\frac{a+b \sin (c+d x)}{a-b}\right)^{\frac{1-p}{2}} \left(1-\frac{a+b \sin (c+d x)}{a+b}\right)^{\frac{1-p}{2}} F_1\left(\frac{7}{2};\frac{1-p}{2},\frac{1-p}{2};\frac{9}{2};\frac{a+b \sin (c+d x)}{a-b},\frac{a+b \sin (c+d x)}{a+b}\right)}{7 b d}","\frac{2 e (a+b \sin (c+d x))^{7/2} (e \cos (c+d x))^{p-1} \left(1-\frac{a+b \sin (c+d x)}{a-b}\right)^{\frac{1-p}{2}} \left(1-\frac{a+b \sin (c+d x)}{a+b}\right)^{\frac{1-p}{2}} F_1\left(\frac{7}{2};\frac{1-p}{2},\frac{1-p}{2};\frac{9}{2};\frac{a+b \sin (c+d x)}{a-b},\frac{a+b \sin (c+d x)}{a+b}\right)}{7 b d}",1,"(2*e*AppellF1[7/2, (1 - p)/2, (1 - p)/2, 9/2, (a + b*Sin[c + d*x])/(a - b), (a + b*Sin[c + d*x])/(a + b)]*(e*Cos[c + d*x])^(-1 + p)*(a + b*Sin[c + d*x])^(7/2)*(1 - (a + b*Sin[c + d*x])/(a - b))^((1 - p)/2)*(1 - (a + b*Sin[c + d*x])/(a + b))^((1 - p)/2))/(7*b*d)","A",2,2,25,0.08000,1,"{2704, 138}"
624,1,156,0,0.1156627,"\int (e \cos (c+d x))^p (a+b \sin (c+d x))^{3/2} \, dx","Int[(e*Cos[c + d*x])^p*(a + b*Sin[c + d*x])^(3/2),x]","\frac{2 e (a+b \sin (c+d x))^{5/2} (e \cos (c+d x))^{p-1} \left(1-\frac{a+b \sin (c+d x)}{a-b}\right)^{\frac{1-p}{2}} \left(1-\frac{a+b \sin (c+d x)}{a+b}\right)^{\frac{1-p}{2}} F_1\left(\frac{5}{2};\frac{1-p}{2},\frac{1-p}{2};\frac{7}{2};\frac{a+b \sin (c+d x)}{a-b},\frac{a+b \sin (c+d x)}{a+b}\right)}{5 b d}","\frac{2 e (a+b \sin (c+d x))^{5/2} (e \cos (c+d x))^{p-1} \left(1-\frac{a+b \sin (c+d x)}{a-b}\right)^{\frac{1-p}{2}} \left(1-\frac{a+b \sin (c+d x)}{a+b}\right)^{\frac{1-p}{2}} F_1\left(\frac{5}{2};\frac{1-p}{2},\frac{1-p}{2};\frac{7}{2};\frac{a+b \sin (c+d x)}{a-b},\frac{a+b \sin (c+d x)}{a+b}\right)}{5 b d}",1,"(2*e*AppellF1[5/2, (1 - p)/2, (1 - p)/2, 7/2, (a + b*Sin[c + d*x])/(a - b), (a + b*Sin[c + d*x])/(a + b)]*(e*Cos[c + d*x])^(-1 + p)*(a + b*Sin[c + d*x])^(5/2)*(1 - (a + b*Sin[c + d*x])/(a - b))^((1 - p)/2)*(1 - (a + b*Sin[c + d*x])/(a + b))^((1 - p)/2))/(5*b*d)","A",2,2,25,0.08000,1,"{2704, 138}"
625,1,156,0,0.1060509,"\int (e \cos (c+d x))^p \sqrt{a+b \sin (c+d x)} \, dx","Int[(e*Cos[c + d*x])^p*Sqrt[a + b*Sin[c + d*x]],x]","\frac{2 e (a+b \sin (c+d x))^{3/2} (e \cos (c+d x))^{p-1} \left(1-\frac{a+b \sin (c+d x)}{a-b}\right)^{\frac{1-p}{2}} \left(1-\frac{a+b \sin (c+d x)}{a+b}\right)^{\frac{1-p}{2}} F_1\left(\frac{3}{2};\frac{1-p}{2},\frac{1-p}{2};\frac{5}{2};\frac{a+b \sin (c+d x)}{a-b},\frac{a+b \sin (c+d x)}{a+b}\right)}{3 b d}","\frac{2 e (a+b \sin (c+d x))^{3/2} (e \cos (c+d x))^{p-1} \left(1-\frac{a+b \sin (c+d x)}{a-b}\right)^{\frac{1-p}{2}} \left(1-\frac{a+b \sin (c+d x)}{a+b}\right)^{\frac{1-p}{2}} F_1\left(\frac{3}{2};\frac{1-p}{2},\frac{1-p}{2};\frac{5}{2};\frac{a+b \sin (c+d x)}{a-b},\frac{a+b \sin (c+d x)}{a+b}\right)}{3 b d}",1,"(2*e*AppellF1[3/2, (1 - p)/2, (1 - p)/2, 5/2, (a + b*Sin[c + d*x])/(a - b), (a + b*Sin[c + d*x])/(a + b)]*(e*Cos[c + d*x])^(-1 + p)*(a + b*Sin[c + d*x])^(3/2)*(1 - (a + b*Sin[c + d*x])/(a - b))^((1 - p)/2)*(1 - (a + b*Sin[c + d*x])/(a + b))^((1 - p)/2))/(3*b*d)","A",2,2,25,0.08000,1,"{2704, 138}"
626,1,154,0,0.116675,"\int \frac{(e \cos (c+d x))^p}{\sqrt{a+b \sin (c+d x)}} \, dx","Int[(e*Cos[c + d*x])^p/Sqrt[a + b*Sin[c + d*x]],x]","\frac{2 e \sqrt{a+b \sin (c+d x)} (e \cos (c+d x))^{p-1} \left(1-\frac{a+b \sin (c+d x)}{a-b}\right)^{\frac{1-p}{2}} \left(1-\frac{a+b \sin (c+d x)}{a+b}\right)^{\frac{1-p}{2}} F_1\left(\frac{1}{2};\frac{1-p}{2},\frac{1-p}{2};\frac{3}{2};\frac{a+b \sin (c+d x)}{a-b},\frac{a+b \sin (c+d x)}{a+b}\right)}{b d}","\frac{2 e \sqrt{a+b \sin (c+d x)} (e \cos (c+d x))^{p-1} \left(1-\frac{a+b \sin (c+d x)}{a-b}\right)^{\frac{1-p}{2}} \left(1-\frac{a+b \sin (c+d x)}{a+b}\right)^{\frac{1-p}{2}} F_1\left(\frac{1}{2};\frac{1-p}{2},\frac{1-p}{2};\frac{3}{2};\frac{a+b \sin (c+d x)}{a-b},\frac{a+b \sin (c+d x)}{a+b}\right)}{b d}",1,"(2*e*AppellF1[1/2, (1 - p)/2, (1 - p)/2, 3/2, (a + b*Sin[c + d*x])/(a - b), (a + b*Sin[c + d*x])/(a + b)]*(e*Cos[c + d*x])^(-1 + p)*Sqrt[a + b*Sin[c + d*x]]*(1 - (a + b*Sin[c + d*x])/(a - b))^((1 - p)/2)*(1 - (a + b*Sin[c + d*x])/(a + b))^((1 - p)/2))/(b*d)","A",2,2,25,0.08000,1,"{2704, 138}"
627,1,154,0,0.1165095,"\int \frac{(e \cos (c+d x))^p}{(a+b \sin (c+d x))^{3/2}} \, dx","Int[(e*Cos[c + d*x])^p/(a + b*Sin[c + d*x])^(3/2),x]","-\frac{2 e (e \cos (c+d x))^{p-1} \left(1-\frac{a+b \sin (c+d x)}{a-b}\right)^{\frac{1-p}{2}} \left(1-\frac{a+b \sin (c+d x)}{a+b}\right)^{\frac{1-p}{2}} F_1\left(-\frac{1}{2};\frac{1-p}{2},\frac{1-p}{2};\frac{1}{2};\frac{a+b \sin (c+d x)}{a-b},\frac{a+b \sin (c+d x)}{a+b}\right)}{b d \sqrt{a+b \sin (c+d x)}}","-\frac{2 e (e \cos (c+d x))^{p-1} \left(1-\frac{a+b \sin (c+d x)}{a-b}\right)^{\frac{1-p}{2}} \left(1-\frac{a+b \sin (c+d x)}{a+b}\right)^{\frac{1-p}{2}} F_1\left(-\frac{1}{2};\frac{1-p}{2},\frac{1-p}{2};\frac{1}{2};\frac{a+b \sin (c+d x)}{a-b},\frac{a+b \sin (c+d x)}{a+b}\right)}{b d \sqrt{a+b \sin (c+d x)}}",1,"(-2*e*AppellF1[-1/2, (1 - p)/2, (1 - p)/2, 1/2, (a + b*Sin[c + d*x])/(a - b), (a + b*Sin[c + d*x])/(a + b)]*(e*Cos[c + d*x])^(-1 + p)*(1 - (a + b*Sin[c + d*x])/(a - b))^((1 - p)/2)*(1 - (a + b*Sin[c + d*x])/(a + b))^((1 - p)/2))/(b*d*Sqrt[a + b*Sin[c + d*x]])","A",2,2,25,0.08000,1,"{2704, 138}"
628,1,156,0,0.1177124,"\int \frac{(e \cos (c+d x))^p}{(a+b \sin (c+d x))^{5/2}} \, dx","Int[(e*Cos[c + d*x])^p/(a + b*Sin[c + d*x])^(5/2),x]","-\frac{2 e (e \cos (c+d x))^{p-1} \left(1-\frac{a+b \sin (c+d x)}{a-b}\right)^{\frac{1-p}{2}} \left(1-\frac{a+b \sin (c+d x)}{a+b}\right)^{\frac{1-p}{2}} F_1\left(-\frac{3}{2};\frac{1-p}{2},\frac{1-p}{2};-\frac{1}{2};\frac{a+b \sin (c+d x)}{a-b},\frac{a+b \sin (c+d x)}{a+b}\right)}{3 b d (a+b \sin (c+d x))^{3/2}}","-\frac{2 e (e \cos (c+d x))^{p-1} \left(1-\frac{a+b \sin (c+d x)}{a-b}\right)^{\frac{1-p}{2}} \left(1-\frac{a+b \sin (c+d x)}{a+b}\right)^{\frac{1-p}{2}} F_1\left(-\frac{3}{2};\frac{1-p}{2},\frac{1-p}{2};-\frac{1}{2};\frac{a+b \sin (c+d x)}{a-b},\frac{a+b \sin (c+d x)}{a+b}\right)}{3 b d (a+b \sin (c+d x))^{3/2}}",1,"(-2*e*AppellF1[-3/2, (1 - p)/2, (1 - p)/2, -1/2, (a + b*Sin[c + d*x])/(a - b), (a + b*Sin[c + d*x])/(a + b)]*(e*Cos[c + d*x])^(-1 + p)*(1 - (a + b*Sin[c + d*x])/(a - b))^((1 - p)/2)*(1 - (a + b*Sin[c + d*x])/(a + b))^((1 - p)/2))/(3*b*d*(a + b*Sin[c + d*x])^(3/2))","A",2,2,25,0.08000,1,"{2704, 138}"
629,1,158,0,0.0985978,"\int (e \cos (c+d x))^p (a+b \sin (c+d x))^m \, dx","Int[(e*Cos[c + d*x])^p*(a + b*Sin[c + d*x])^m,x]","\frac{e (e \cos (c+d x))^{p-1} (a+b \sin (c+d x))^{m+1} \left(1-\frac{a+b \sin (c+d x)}{a-b}\right)^{\frac{1-p}{2}} \left(1-\frac{a+b \sin (c+d x)}{a+b}\right)^{\frac{1-p}{2}} F_1\left(m+1;\frac{1-p}{2},\frac{1-p}{2};m+2;\frac{a+b \sin (c+d x)}{a-b},\frac{a+b \sin (c+d x)}{a+b}\right)}{b d (m+1)}","\frac{e (e \cos (c+d x))^{p-1} (a+b \sin (c+d x))^{m+1} \left(1-\frac{a+b \sin (c+d x)}{a-b}\right)^{\frac{1-p}{2}} \left(1-\frac{a+b \sin (c+d x)}{a+b}\right)^{\frac{1-p}{2}} F_1\left(m+1;\frac{1-p}{2},\frac{1-p}{2};m+2;\frac{a+b \sin (c+d x)}{a-b},\frac{a+b \sin (c+d x)}{a+b}\right)}{b d (m+1)}",1,"(e*AppellF1[1 + m, (1 - p)/2, (1 - p)/2, 2 + m, (a + b*Sin[c + d*x])/(a - b), (a + b*Sin[c + d*x])/(a + b)]*(e*Cos[c + d*x])^(-1 + p)*(a + b*Sin[c + d*x])^(1 + m)*(1 - (a + b*Sin[c + d*x])/(a - b))^((1 - p)/2)*(1 - (a + b*Sin[c + d*x])/(a + b))^((1 - p)/2))/(b*d*(1 + m))","A",2,2,23,0.08696,1,"{2704, 138}"
630,1,254,0,0.1639904,"\int \cos ^7(c+d x) (a+b \sin (c+d x))^m \, dx","Int[Cos[c + d*x]^7*(a + b*Sin[c + d*x])^m,x]","-\frac{\left(a^2-b^2\right)^3 (a+b \sin (c+d x))^{m+1}}{b^7 d (m+1)}+\frac{6 a \left(a^2-b^2\right)^2 (a+b \sin (c+d x))^{m+2}}{b^7 d (m+2)}-\frac{3 \left(-6 a^2 b^2+5 a^4+b^4\right) (a+b \sin (c+d x))^{m+3}}{b^7 d (m+3)}+\frac{4 a \left(5 a^2-3 b^2\right) (a+b \sin (c+d x))^{m+4}}{b^7 d (m+4)}-\frac{3 \left(5 a^2-b^2\right) (a+b \sin (c+d x))^{m+5}}{b^7 d (m+5)}+\frac{6 a (a+b \sin (c+d x))^{m+6}}{b^7 d (m+6)}-\frac{(a+b \sin (c+d x))^{m+7}}{b^7 d (m+7)}","-\frac{\left(a^2-b^2\right)^3 (a+b \sin (c+d x))^{m+1}}{b^7 d (m+1)}+\frac{6 a \left(a^2-b^2\right)^2 (a+b \sin (c+d x))^{m+2}}{b^7 d (m+2)}-\frac{3 \left(-6 a^2 b^2+5 a^4+b^4\right) (a+b \sin (c+d x))^{m+3}}{b^7 d (m+3)}+\frac{4 a \left(5 a^2-3 b^2\right) (a+b \sin (c+d x))^{m+4}}{b^7 d (m+4)}-\frac{3 \left(5 a^2-b^2\right) (a+b \sin (c+d x))^{m+5}}{b^7 d (m+5)}+\frac{6 a (a+b \sin (c+d x))^{m+6}}{b^7 d (m+6)}-\frac{(a+b \sin (c+d x))^{m+7}}{b^7 d (m+7)}",1,"-(((a^2 - b^2)^3*(a + b*Sin[c + d*x])^(1 + m))/(b^7*d*(1 + m))) + (6*a*(a^2 - b^2)^2*(a + b*Sin[c + d*x])^(2 + m))/(b^7*d*(2 + m)) - (3*(5*a^4 - 6*a^2*b^2 + b^4)*(a + b*Sin[c + d*x])^(3 + m))/(b^7*d*(3 + m)) + (4*a*(5*a^2 - 3*b^2)*(a + b*Sin[c + d*x])^(4 + m))/(b^7*d*(4 + m)) - (3*(5*a^2 - b^2)*(a + b*Sin[c + d*x])^(5 + m))/(b^7*d*(5 + m)) + (6*a*(a + b*Sin[c + d*x])^(6 + m))/(b^7*d*(6 + m)) - (a + b*Sin[c + d*x])^(7 + m)/(b^7*d*(7 + m))","A",3,2,21,0.09524,1,"{2668, 697}"
631,1,167,0,0.1120143,"\int \cos ^5(c+d x) (a+b \sin (c+d x))^m \, dx","Int[Cos[c + d*x]^5*(a + b*Sin[c + d*x])^m,x]","\frac{\left(a^2-b^2\right)^2 (a+b \sin (c+d x))^{m+1}}{b^5 d (m+1)}-\frac{4 a \left(a^2-b^2\right) (a+b \sin (c+d x))^{m+2}}{b^5 d (m+2)}+\frac{2 \left(3 a^2-b^2\right) (a+b \sin (c+d x))^{m+3}}{b^5 d (m+3)}-\frac{4 a (a+b \sin (c+d x))^{m+4}}{b^5 d (m+4)}+\frac{(a+b \sin (c+d x))^{m+5}}{b^5 d (m+5)}","\frac{\left(a^2-b^2\right)^2 (a+b \sin (c+d x))^{m+1}}{b^5 d (m+1)}-\frac{4 a \left(a^2-b^2\right) (a+b \sin (c+d x))^{m+2}}{b^5 d (m+2)}+\frac{2 \left(3 a^2-b^2\right) (a+b \sin (c+d x))^{m+3}}{b^5 d (m+3)}-\frac{4 a (a+b \sin (c+d x))^{m+4}}{b^5 d (m+4)}+\frac{(a+b \sin (c+d x))^{m+5}}{b^5 d (m+5)}",1,"((a^2 - b^2)^2*(a + b*Sin[c + d*x])^(1 + m))/(b^5*d*(1 + m)) - (4*a*(a^2 - b^2)*(a + b*Sin[c + d*x])^(2 + m))/(b^5*d*(2 + m)) + (2*(3*a^2 - b^2)*(a + b*Sin[c + d*x])^(3 + m))/(b^5*d*(3 + m)) - (4*a*(a + b*Sin[c + d*x])^(4 + m))/(b^5*d*(4 + m)) + (a + b*Sin[c + d*x])^(5 + m)/(b^5*d*(5 + m))","A",3,2,21,0.09524,1,"{2668, 697}"
632,1,92,0,0.0719789,"\int \cos ^3(c+d x) (a+b \sin (c+d x))^m \, dx","Int[Cos[c + d*x]^3*(a + b*Sin[c + d*x])^m,x]","-\frac{\left(a^2-b^2\right) (a+b \sin (c+d x))^{m+1}}{b^3 d (m+1)}+\frac{2 a (a+b \sin (c+d x))^{m+2}}{b^3 d (m+2)}-\frac{(a+b \sin (c+d x))^{m+3}}{b^3 d (m+3)}","-\frac{\left(a^2-b^2\right) (a+b \sin (c+d x))^{m+1}}{b^3 d (m+1)}+\frac{2 a (a+b \sin (c+d x))^{m+2}}{b^3 d (m+2)}-\frac{(a+b \sin (c+d x))^{m+3}}{b^3 d (m+3)}",1,"-(((a^2 - b^2)*(a + b*Sin[c + d*x])^(1 + m))/(b^3*d*(1 + m))) + (2*a*(a + b*Sin[c + d*x])^(2 + m))/(b^3*d*(2 + m)) - (a + b*Sin[c + d*x])^(3 + m)/(b^3*d*(3 + m))","A",3,2,21,0.09524,1,"{2668, 697}"
633,1,26,0,0.026876,"\int \cos (c+d x) (a+b \sin (c+d x))^m \, dx","Int[Cos[c + d*x]*(a + b*Sin[c + d*x])^m,x]","\frac{(a+b \sin (c+d x))^{m+1}}{b d (m+1)}","\frac{(a+b \sin (c+d x))^{m+1}}{b d (m+1)}",1,"(a + b*Sin[c + d*x])^(1 + m)/(b*d*(1 + m))","A",2,2,19,0.1053,1,"{2668, 32}"
634,1,115,0,0.1146426,"\int \sec (c+d x) (a+b \sin (c+d x))^m \, dx","Int[Sec[c + d*x]*(a + b*Sin[c + d*x])^m,x]","\frac{(a+b \sin (c+d x))^{m+1} \, _2F_1\left(1,m+1;m+2;\frac{a+b \sin (c+d x)}{a+b}\right)}{2 d (m+1) (a+b)}-\frac{(a+b \sin (c+d x))^{m+1} \, _2F_1\left(1,m+1;m+2;\frac{a+b \sin (c+d x)}{a-b}\right)}{2 d (m+1) (a-b)}","\frac{(a+b \sin (c+d x))^{m+1} \, _2F_1\left(1,m+1;m+2;\frac{a+b \sin (c+d x)}{a+b}\right)}{2 d (m+1) (a+b)}-\frac{(a+b \sin (c+d x))^{m+1} \, _2F_1\left(1,m+1;m+2;\frac{a+b \sin (c+d x)}{a-b}\right)}{2 d (m+1) (a-b)}",1,"-(Hypergeometric2F1[1, 1 + m, 2 + m, (a + b*Sin[c + d*x])/(a - b)]*(a + b*Sin[c + d*x])^(1 + m))/(2*(a - b)*d*(1 + m)) + (Hypergeometric2F1[1, 1 + m, 2 + m, (a + b*Sin[c + d*x])/(a + b)]*(a + b*Sin[c + d*x])^(1 + m))/(2*(a + b)*d*(1 + m))","A",5,3,19,0.1579,1,"{2668, 712, 68}"
635,1,183,0,0.226047,"\int \sec ^3(c+d x) (a+b \sin (c+d x))^m \, dx","Int[Sec[c + d*x]^3*(a + b*Sin[c + d*x])^m,x]","-\frac{\sec ^2(c+d x) (b-a \sin (c+d x)) (a+b \sin (c+d x))^{m+1}}{2 d \left(a^2-b^2\right)}-\frac{(a-b (1-m)) (a+b \sin (c+d x))^{m+1} \, _2F_1\left(1,m+1;m+2;\frac{a+b \sin (c+d x)}{a-b}\right)}{4 d (m+1) (a-b)^2}+\frac{(a-b m+b) (a+b \sin (c+d x))^{m+1} \, _2F_1\left(1,m+1;m+2;\frac{a+b \sin (c+d x)}{a+b}\right)}{4 d (m+1) (a+b)^2}","-\frac{\sec ^2(c+d x) (b-a \sin (c+d x)) (a+b \sin (c+d x))^{m+1}}{2 d \left(a^2-b^2\right)}-\frac{(a-b (1-m)) (a+b \sin (c+d x))^{m+1} \, _2F_1\left(1,m+1;m+2;\frac{a+b \sin (c+d x)}{a-b}\right)}{4 d (m+1) (a-b)^2}+\frac{(a-b m+b) (a+b \sin (c+d x))^{m+1} \, _2F_1\left(1,m+1;m+2;\frac{a+b \sin (c+d x)}{a+b}\right)}{4 d (m+1) (a+b)^2}",1,"-((a - b*(1 - m))*Hypergeometric2F1[1, 1 + m, 2 + m, (a + b*Sin[c + d*x])/(a - b)]*(a + b*Sin[c + d*x])^(1 + m))/(4*(a - b)^2*d*(1 + m)) + ((a + b - b*m)*Hypergeometric2F1[1, 1 + m, 2 + m, (a + b*Sin[c + d*x])/(a + b)]*(a + b*Sin[c + d*x])^(1 + m))/(4*(a + b)^2*d*(1 + m)) - (Sec[c + d*x]^2*(b - a*Sin[c + d*x])*(a + b*Sin[c + d*x])^(1 + m))/(2*(a^2 - b^2)*d)","A",6,4,21,0.1905,1,"{2668, 741, 831, 68}"
636,1,305,0,0.420683,"\int \sec ^5(c+d x) (a+b \sin (c+d x))^m \, dx","Int[Sec[c + d*x]^5*(a + b*Sin[c + d*x])^m,x]","-\frac{\left(3 a^2-3 a b (2-m)+b^2 \left(m^2-4 m+3\right)\right) (a+b \sin (c+d x))^{m+1} \, _2F_1\left(1,m+1;m+2;\frac{a+b \sin (c+d x)}{a-b}\right)}{16 d (m+1) (a-b)^3}+\frac{\left(3 a^2+3 a b (2-m)+b^2 \left(m^2-4 m+3\right)\right) (a+b \sin (c+d x))^{m+1} \, _2F_1\left(1,m+1;m+2;\frac{a+b \sin (c+d x)}{a+b}\right)}{16 d (m+1) (a+b)^3}-\frac{\sec ^4(c+d x) (b-a \sin (c+d x)) (a+b \sin (c+d x))^{m+1}}{4 d \left(a^2-b^2\right)}+\frac{\sec ^2(c+d x) \left(a \left(3 a^2-b^2 (5-2 m)\right) \sin (c+d x)+b \left(b^2 (3-m)-a^2 (m+1)\right)\right) (a+b \sin (c+d x))^{m+1}}{8 d \left(a^2-b^2\right)^2}","-\frac{\left(3 a^2-3 a b (2-m)+b^2 \left(m^2-4 m+3\right)\right) (a+b \sin (c+d x))^{m+1} \, _2F_1\left(1,m+1;m+2;\frac{a+b \sin (c+d x)}{a-b}\right)}{16 d (m+1) (a-b)^3}+\frac{\left(3 a^2+3 a b (2-m)+b^2 \left(m^2-4 m+3\right)\right) (a+b \sin (c+d x))^{m+1} \, _2F_1\left(1,m+1;m+2;\frac{a+b \sin (c+d x)}{a+b}\right)}{16 d (m+1) (a+b)^3}-\frac{\sec ^4(c+d x) (b-a \sin (c+d x)) (a+b \sin (c+d x))^{m+1}}{4 d \left(a^2-b^2\right)}+\frac{\sec ^2(c+d x) \left(a \left(3 a^2-b^2 (5-2 m)\right) \sin (c+d x)+b \left(b^2 (3-m)-a^2 (m+1)\right)\right) (a+b \sin (c+d x))^{m+1}}{8 d \left(a^2-b^2\right)^2}",1,"-((3*a^2 - 3*a*b*(2 - m) + b^2*(3 - 4*m + m^2))*Hypergeometric2F1[1, 1 + m, 2 + m, (a + b*Sin[c + d*x])/(a - b)]*(a + b*Sin[c + d*x])^(1 + m))/(16*(a - b)^3*d*(1 + m)) + ((3*a^2 + 3*a*b*(2 - m) + b^2*(3 - 4*m + m^2))*Hypergeometric2F1[1, 1 + m, 2 + m, (a + b*Sin[c + d*x])/(a + b)]*(a + b*Sin[c + d*x])^(1 + m))/(16*(a + b)^3*d*(1 + m)) - (Sec[c + d*x]^4*(b - a*Sin[c + d*x])*(a + b*Sin[c + d*x])^(1 + m))/(4*(a^2 - b^2)*d) + (Sec[c + d*x]^2*(a + b*Sin[c + d*x])^(1 + m)*(b*(b^2*(3 - m) - a^2*(1 + m)) + a*(3*a^2 - b^2*(5 - 2*m))*Sin[c + d*x]))/(8*(a^2 - b^2)^2*d)","A",7,5,21,0.2381,1,"{2668, 741, 823, 831, 68}"
637,1,129,0,0.0940427,"\int \cos ^4(c+d x) (a+b \sin (c+d x))^m \, dx","Int[Cos[c + d*x]^4*(a + b*Sin[c + d*x])^m,x]","\frac{\cos ^3(c+d x) (a+b \sin (c+d x))^{m+1} F_1\left(m+1;-\frac{3}{2},-\frac{3}{2};m+2;\frac{a+b \sin (c+d x)}{a-b},\frac{a+b \sin (c+d x)}{a+b}\right)}{b d (m+1) \left(1-\frac{a+b \sin (c+d x)}{a-b}\right)^{3/2} \left(1-\frac{a+b \sin (c+d x)}{a+b}\right)^{3/2}}","\frac{\cos ^3(c+d x) (a+b \sin (c+d x))^{m+1} F_1\left(m+1;-\frac{3}{2},-\frac{3}{2};m+2;\frac{a+b \sin (c+d x)}{a-b},\frac{a+b \sin (c+d x)}{a+b}\right)}{b d (m+1) \left(1-\frac{a+b \sin (c+d x)}{a-b}\right)^{3/2} \left(1-\frac{a+b \sin (c+d x)}{a+b}\right)^{3/2}}",1,"(AppellF1[1 + m, -3/2, -3/2, 2 + m, (a + b*Sin[c + d*x])/(a - b), (a + b*Sin[c + d*x])/(a + b)]*Cos[c + d*x]^3*(a + b*Sin[c + d*x])^(1 + m))/(b*d*(1 + m)*(1 - (a + b*Sin[c + d*x])/(a - b))^(3/2)*(1 - (a + b*Sin[c + d*x])/(a + b))^(3/2))","A",2,2,21,0.09524,1,"{2704, 138}"
638,1,127,0,0.0889027,"\int \cos ^2(c+d x) (a+b \sin (c+d x))^m \, dx","Int[Cos[c + d*x]^2*(a + b*Sin[c + d*x])^m,x]","\frac{\cos (c+d x) (a+b \sin (c+d x))^{m+1} F_1\left(m+1;-\frac{1}{2},-\frac{1}{2};m+2;\frac{a+b \sin (c+d x)}{a-b},\frac{a+b \sin (c+d x)}{a+b}\right)}{b d (m+1) \sqrt{1-\frac{a+b \sin (c+d x)}{a-b}} \sqrt{1-\frac{a+b \sin (c+d x)}{a+b}}}","\frac{\cos (c+d x) (a+b \sin (c+d x))^{m+1} F_1\left(m+1;-\frac{1}{2},-\frac{1}{2};m+2;\frac{a+b \sin (c+d x)}{a-b},\frac{a+b \sin (c+d x)}{a+b}\right)}{b d (m+1) \sqrt{1-\frac{a+b \sin (c+d x)}{a-b}} \sqrt{1-\frac{a+b \sin (c+d x)}{a+b}}}",1,"(AppellF1[1 + m, -1/2, -1/2, 2 + m, (a + b*Sin[c + d*x])/(a - b), (a + b*Sin[c + d*x])/(a + b)]*Cos[c + d*x]*(a + b*Sin[c + d*x])^(1 + m))/(b*d*(1 + m)*Sqrt[1 - (a + b*Sin[c + d*x])/(a - b)]*Sqrt[1 - (a + b*Sin[c + d*x])/(a + b)])","A",2,2,21,0.09524,1,"{2704, 138}"
639,1,129,0,0.086305,"\int \sec ^2(c+d x) (a+b \sin (c+d x))^m \, dx","Int[Sec[c + d*x]^2*(a + b*Sin[c + d*x])^m,x]","\frac{\sec ^3(c+d x) \left(1-\frac{a+b \sin (c+d x)}{a-b}\right)^{3/2} \left(1-\frac{a+b \sin (c+d x)}{a+b}\right)^{3/2} (a+b \sin (c+d x))^{m+1} F_1\left(m+1;\frac{3}{2},\frac{3}{2};m+2;\frac{a+b \sin (c+d x)}{a-b},\frac{a+b \sin (c+d x)}{a+b}\right)}{b d (m+1)}","\frac{\sec ^3(c+d x) \left(1-\frac{a+b \sin (c+d x)}{a-b}\right)^{3/2} \left(1-\frac{a+b \sin (c+d x)}{a+b}\right)^{3/2} (a+b \sin (c+d x))^{m+1} F_1\left(m+1;\frac{3}{2},\frac{3}{2};m+2;\frac{a+b \sin (c+d x)}{a-b},\frac{a+b \sin (c+d x)}{a+b}\right)}{b d (m+1)}",1,"(AppellF1[1 + m, 3/2, 3/2, 2 + m, (a + b*Sin[c + d*x])/(a - b), (a + b*Sin[c + d*x])/(a + b)]*Sec[c + d*x]^3*(a + b*Sin[c + d*x])^(1 + m)*(1 - (a + b*Sin[c + d*x])/(a - b))^(3/2)*(1 - (a + b*Sin[c + d*x])/(a + b))^(3/2))/(b*d*(1 + m))","A",2,2,21,0.09524,1,"{2704, 138}"
640,1,129,0,0.0859064,"\int \sec ^4(c+d x) (a+b \sin (c+d x))^m \, dx","Int[Sec[c + d*x]^4*(a + b*Sin[c + d*x])^m,x]","\frac{\sec ^5(c+d x) \left(1-\frac{a+b \sin (c+d x)}{a-b}\right)^{5/2} \left(1-\frac{a+b \sin (c+d x)}{a+b}\right)^{5/2} (a+b \sin (c+d x))^{m+1} F_1\left(m+1;\frac{5}{2},\frac{5}{2};m+2;\frac{a+b \sin (c+d x)}{a-b},\frac{a+b \sin (c+d x)}{a+b}\right)}{b d (m+1)}","\frac{\sec ^5(c+d x) \left(1-\frac{a+b \sin (c+d x)}{a-b}\right)^{5/2} \left(1-\frac{a+b \sin (c+d x)}{a+b}\right)^{5/2} (a+b \sin (c+d x))^{m+1} F_1\left(m+1;\frac{5}{2},\frac{5}{2};m+2;\frac{a+b \sin (c+d x)}{a-b},\frac{a+b \sin (c+d x)}{a+b}\right)}{b d (m+1)}",1,"(AppellF1[1 + m, 5/2, 5/2, 2 + m, (a + b*Sin[c + d*x])/(a - b), (a + b*Sin[c + d*x])/(a + b)]*Sec[c + d*x]^5*(a + b*Sin[c + d*x])^(1 + m)*(1 - (a + b*Sin[c + d*x])/(a - b))^(5/2)*(1 - (a + b*Sin[c + d*x])/(a + b))^(5/2))/(b*d*(1 + m))","A",2,2,21,0.09524,1,"{2704, 138}"
641,1,134,0,0.1009953,"\int (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))^m \, dx","Int[(e*Cos[c + d*x])^(5/2)*(a + b*Sin[c + d*x])^m,x]","\frac{e (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^{m+1} F_1\left(m+1;-\frac{3}{4},-\frac{3}{4};m+2;\frac{a+b \sin (c+d x)}{a-b},\frac{a+b \sin (c+d x)}{a+b}\right)}{b d (m+1) \left(1-\frac{a+b \sin (c+d x)}{a-b}\right)^{3/4} \left(1-\frac{a+b \sin (c+d x)}{a+b}\right)^{3/4}}","\frac{e (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^{m+1} F_1\left(m+1;-\frac{3}{4},-\frac{3}{4};m+2;\frac{a+b \sin (c+d x)}{a-b},\frac{a+b \sin (c+d x)}{a+b}\right)}{b d (m+1) \left(1-\frac{a+b \sin (c+d x)}{a-b}\right)^{3/4} \left(1-\frac{a+b \sin (c+d x)}{a+b}\right)^{3/4}}",1,"(e*AppellF1[1 + m, -3/4, -3/4, 2 + m, (a + b*Sin[c + d*x])/(a - b), (a + b*Sin[c + d*x])/(a + b)]*(e*Cos[c + d*x])^(3/2)*(a + b*Sin[c + d*x])^(1 + m))/(b*d*(1 + m)*(1 - (a + b*Sin[c + d*x])/(a - b))^(3/4)*(1 - (a + b*Sin[c + d*x])/(a + b))^(3/4))","A",2,2,25,0.08000,1,"{2704, 138}"
642,1,134,0,0.0990883,"\int (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^m \, dx","Int[(e*Cos[c + d*x])^(3/2)*(a + b*Sin[c + d*x])^m,x]","\frac{e \sqrt{e \cos (c+d x)} (a+b \sin (c+d x))^{m+1} F_1\left(m+1;-\frac{1}{4},-\frac{1}{4};m+2;\frac{a+b \sin (c+d x)}{a-b},\frac{a+b \sin (c+d x)}{a+b}\right)}{b d (m+1) \sqrt[4]{1-\frac{a+b \sin (c+d x)}{a-b}} \sqrt[4]{1-\frac{a+b \sin (c+d x)}{a+b}}}","\frac{e \sqrt{e \cos (c+d x)} (a+b \sin (c+d x))^{m+1} F_1\left(m+1;-\frac{1}{4},-\frac{1}{4};m+2;\frac{a+b \sin (c+d x)}{a-b},\frac{a+b \sin (c+d x)}{a+b}\right)}{b d (m+1) \sqrt[4]{1-\frac{a+b \sin (c+d x)}{a-b}} \sqrt[4]{1-\frac{a+b \sin (c+d x)}{a+b}}}",1,"(e*AppellF1[1 + m, -1/4, -1/4, 2 + m, (a + b*Sin[c + d*x])/(a - b), (a + b*Sin[c + d*x])/(a + b)]*Sqrt[e*Cos[c + d*x]]*(a + b*Sin[c + d*x])^(1 + m))/(b*d*(1 + m)*(1 - (a + b*Sin[c + d*x])/(a - b))^(1/4)*(1 - (a + b*Sin[c + d*x])/(a + b))^(1/4))","A",2,2,25,0.08000,1,"{2704, 138}"
643,1,134,0,0.0905221,"\int \sqrt{e \cos (c+d x)} (a+b \sin (c+d x))^m \, dx","Int[Sqrt[e*Cos[c + d*x]]*(a + b*Sin[c + d*x])^m,x]","\frac{e \sqrt[4]{1-\frac{a+b \sin (c+d x)}{a-b}} \sqrt[4]{1-\frac{a+b \sin (c+d x)}{a+b}} (a+b \sin (c+d x))^{m+1} F_1\left(m+1;\frac{1}{4},\frac{1}{4};m+2;\frac{a+b \sin (c+d x)}{a-b},\frac{a+b \sin (c+d x)}{a+b}\right)}{b d (m+1) \sqrt{e \cos (c+d x)}}","\frac{e \sqrt[4]{1-\frac{a+b \sin (c+d x)}{a-b}} \sqrt[4]{1-\frac{a+b \sin (c+d x)}{a+b}} (a+b \sin (c+d x))^{m+1} F_1\left(m+1;\frac{1}{4},\frac{1}{4};m+2;\frac{a+b \sin (c+d x)}{a-b},\frac{a+b \sin (c+d x)}{a+b}\right)}{b d (m+1) \sqrt{e \cos (c+d x)}}",1,"(e*AppellF1[1 + m, 1/4, 1/4, 2 + m, (a + b*Sin[c + d*x])/(a - b), (a + b*Sin[c + d*x])/(a + b)]*(a + b*Sin[c + d*x])^(1 + m)*(1 - (a + b*Sin[c + d*x])/(a - b))^(1/4)*(1 - (a + b*Sin[c + d*x])/(a + b))^(1/4))/(b*d*(1 + m)*Sqrt[e*Cos[c + d*x]])","A",2,2,25,0.08000,1,"{2704, 138}"
644,1,134,0,0.0943236,"\int \frac{(a+b \sin (c+d x))^m}{\sqrt{e \cos (c+d x)}} \, dx","Int[(a + b*Sin[c + d*x])^m/Sqrt[e*Cos[c + d*x]],x]","\frac{e \left(1-\frac{a+b \sin (c+d x)}{a-b}\right)^{3/4} \left(1-\frac{a+b \sin (c+d x)}{a+b}\right)^{3/4} (a+b \sin (c+d x))^{m+1} F_1\left(m+1;\frac{3}{4},\frac{3}{4};m+2;\frac{a+b \sin (c+d x)}{a-b},\frac{a+b \sin (c+d x)}{a+b}\right)}{b d (m+1) (e \cos (c+d x))^{3/2}}","\frac{e \left(1-\frac{a+b \sin (c+d x)}{a-b}\right)^{3/4} \left(1-\frac{a+b \sin (c+d x)}{a+b}\right)^{3/4} (a+b \sin (c+d x))^{m+1} F_1\left(m+1;\frac{3}{4},\frac{3}{4};m+2;\frac{a+b \sin (c+d x)}{a-b},\frac{a+b \sin (c+d x)}{a+b}\right)}{b d (m+1) (e \cos (c+d x))^{3/2}}",1,"(e*AppellF1[1 + m, 3/4, 3/4, 2 + m, (a + b*Sin[c + d*x])/(a - b), (a + b*Sin[c + d*x])/(a + b)]*(a + b*Sin[c + d*x])^(1 + m)*(1 - (a + b*Sin[c + d*x])/(a - b))^(3/4)*(1 - (a + b*Sin[c + d*x])/(a + b))^(3/4))/(b*d*(1 + m)*(e*Cos[c + d*x])^(3/2))","A",2,2,25,0.08000,1,"{2704, 138}"
645,1,134,0,0.099244,"\int \frac{(a+b \sin (c+d x))^m}{(e \cos (c+d x))^{3/2}} \, dx","Int[(a + b*Sin[c + d*x])^m/(e*Cos[c + d*x])^(3/2),x]","\frac{e \left(1-\frac{a+b \sin (c+d x)}{a-b}\right)^{5/4} \left(1-\frac{a+b \sin (c+d x)}{a+b}\right)^{5/4} (a+b \sin (c+d x))^{m+1} F_1\left(m+1;\frac{5}{4},\frac{5}{4};m+2;\frac{a+b \sin (c+d x)}{a-b},\frac{a+b \sin (c+d x)}{a+b}\right)}{b d (m+1) (e \cos (c+d x))^{5/2}}","\frac{e \left(1-\frac{a+b \sin (c+d x)}{a-b}\right)^{5/4} \left(1-\frac{a+b \sin (c+d x)}{a+b}\right)^{5/4} (a+b \sin (c+d x))^{m+1} F_1\left(m+1;\frac{5}{4},\frac{5}{4};m+2;\frac{a+b \sin (c+d x)}{a-b},\frac{a+b \sin (c+d x)}{a+b}\right)}{b d (m+1) (e \cos (c+d x))^{5/2}}",1,"(e*AppellF1[1 + m, 5/4, 5/4, 2 + m, (a + b*Sin[c + d*x])/(a - b), (a + b*Sin[c + d*x])/(a + b)]*(a + b*Sin[c + d*x])^(1 + m)*(1 - (a + b*Sin[c + d*x])/(a - b))^(5/4)*(1 - (a + b*Sin[c + d*x])/(a + b))^(5/4))/(b*d*(1 + m)*(e*Cos[c + d*x])^(5/2))","A",2,2,25,0.08000,1,"{2704, 138}"
646,1,134,0,0.1003486,"\int \frac{(a+b \sin (c+d x))^m}{(e \cos (c+d x))^{5/2}} \, dx","Int[(a + b*Sin[c + d*x])^m/(e*Cos[c + d*x])^(5/2),x]","\frac{e \left(1-\frac{a+b \sin (c+d x)}{a-b}\right)^{7/4} \left(1-\frac{a+b \sin (c+d x)}{a+b}\right)^{7/4} (a+b \sin (c+d x))^{m+1} F_1\left(m+1;\frac{7}{4},\frac{7}{4};m+2;\frac{a+b \sin (c+d x)}{a-b},\frac{a+b \sin (c+d x)}{a+b}\right)}{b d (m+1) (e \cos (c+d x))^{7/2}}","\frac{e \left(1-\frac{a+b \sin (c+d x)}{a-b}\right)^{7/4} \left(1-\frac{a+b \sin (c+d x)}{a+b}\right)^{7/4} (a+b \sin (c+d x))^{m+1} F_1\left(m+1;\frac{7}{4},\frac{7}{4};m+2;\frac{a+b \sin (c+d x)}{a-b},\frac{a+b \sin (c+d x)}{a+b}\right)}{b d (m+1) (e \cos (c+d x))^{7/2}}",1,"(e*AppellF1[1 + m, 7/4, 7/4, 2 + m, (a + b*Sin[c + d*x])/(a - b), (a + b*Sin[c + d*x])/(a + b)]*(a + b*Sin[c + d*x])^(1 + m)*(1 - (a + b*Sin[c + d*x])/(a - b))^(7/4)*(1 - (a + b*Sin[c + d*x])/(a + b))^(7/4))/(b*d*(1 + m)*(e*Cos[c + d*x])^(7/2))","A",2,2,25,0.08000,1,"{2704, 138}"
647,1,598,0,1.0181148,"\int (e \cos (c+d x))^{-4-m} (a+b \sin (c+d x))^m \, dx","Int[(e*Cos[c + d*x])^(-4 - m)*(a + b*Sin[c + d*x])^m,x]","-\frac{a 2^{-\frac{m}{2}-\frac{1}{2}} \left(a^2 (m+2)+2 a b-b^2\right) (1-\sin (c+d x))^2 (e \cos (c+d x))^{-m-3} \left(\frac{(a+b) (\sin (c+d x)+1)}{a+b \sin (c+d x)}\right)^{\frac{m+3}{2}} (a+b \sin (c+d x))^{m+1} \, _2F_1\left(\frac{1-m}{2},\frac{m+3}{2};\frac{3-m}{2};\frac{(a-b) (1-\sin (c+d x))}{2 (a+b \sin (c+d x))}\right)}{d e (1-m) (m+3) (a-b) (a+b)^3}+\frac{a (\sin (c+d x)+1) (e \cos (c+d x))^{-m-3} (a+b \sin (c+d x))^{m+1}}{d e (m+3) \left(a^2-b^2\right)}-\frac{a b 2^{\frac{3}{2}-\frac{m}{2}} (e \cos (c+d x))^{-m-1} \left(\frac{(a+b) (\sin (c+d x)+1)}{a+b \sin (c+d x)}\right)^{\frac{m+1}{2}} (a+b \sin (c+d x))^{m+1} \, _2F_1\left(\frac{1}{2} (-m-1),\frac{m+1}{2};\frac{1-m}{2};\frac{(a-b) (1-\sin (c+d x))}{2 (a+b \sin (c+d x))}\right)}{d e^3 (m+1) (m+3) (a-b)^2 (a+b)}+\frac{2 b (e \cos (c+d x))^{-m-1} (a+b \sin (c+d x))^{m+1}}{d e^3 (m+1) (m+3) (a-b)^2}+\frac{a (a (m+2)+3 b) (1-\sin (c+d x)) (\sin (c+d x)+1) (e \cos (c+d x))^{-m-3} (a+b \sin (c+d x))^{m+1}}{d e (m+1) (m+3) (a-b) (a+b)^2}-\frac{(e \cos (c+d x))^{-m-3} (a+b \sin (c+d x))^{m+1}}{d e (m+3) (a-b)}","-\frac{a 2^{-\frac{m}{2}-\frac{1}{2}} \left(a^2 (m+2)+2 a b-b^2\right) (1-\sin (c+d x))^2 (e \cos (c+d x))^{-m-3} \left(\frac{(a+b) (\sin (c+d x)+1)}{a+b \sin (c+d x)}\right)^{\frac{m+3}{2}} (a+b \sin (c+d x))^{m+1} \, _2F_1\left(\frac{1-m}{2},\frac{m+3}{2};\frac{3-m}{2};\frac{(a-b) (1-\sin (c+d x))}{2 (a+b \sin (c+d x))}\right)}{d e (1-m) (m+3) (a-b) (a+b)^3}+\frac{a (\sin (c+d x)+1) (e \cos (c+d x))^{-m-3} (a+b \sin (c+d x))^{m+1}}{d e (m+3) \left(a^2-b^2\right)}-\frac{a b 2^{\frac{3}{2}-\frac{m}{2}} (e \cos (c+d x))^{-m-1} \left(\frac{(a+b) (\sin (c+d x)+1)}{a+b \sin (c+d x)}\right)^{\frac{m+1}{2}} (a+b \sin (c+d x))^{m+1} \, _2F_1\left(\frac{1}{2} (-m-1),\frac{m+1}{2};\frac{1-m}{2};\frac{(a-b) (1-\sin (c+d x))}{2 (a+b \sin (c+d x))}\right)}{d e^3 (m+1) (m+3) (a-b)^2 (a+b)}+\frac{2 b (e \cos (c+d x))^{-m-1} (a+b \sin (c+d x))^{m+1}}{d e^3 (m+1) (m+3) (a-b)^2}+\frac{a (a (m+2)+3 b) (1-\sin (c+d x)) (\sin (c+d x)+1) (e \cos (c+d x))^{-m-3} (a+b \sin (c+d x))^{m+1}}{d e (m+1) (m+3) (a-b) (a+b)^2}-\frac{(e \cos (c+d x))^{-m-3} (a+b \sin (c+d x))^{m+1}}{d e (m+3) (a-b)}",1,"-(((e*Cos[c + d*x])^(-3 - m)*(a + b*Sin[c + d*x])^(1 + m))/((a - b)*d*e*(3 + m))) + (2*b*(e*Cos[c + d*x])^(-1 - m)*(a + b*Sin[c + d*x])^(1 + m))/((a - b)^2*d*e^3*(1 + m)*(3 + m)) + (a*(e*Cos[c + d*x])^(-3 - m)*(1 + Sin[c + d*x])*(a + b*Sin[c + d*x])^(1 + m))/((a^2 - b^2)*d*e*(3 + m)) + (a*(3*b + a*(2 + m))*(e*Cos[c + d*x])^(-3 - m)*(1 - Sin[c + d*x])*(1 + Sin[c + d*x])*(a + b*Sin[c + d*x])^(1 + m))/((a - b)*(a + b)^2*d*e*(1 + m)*(3 + m)) - (2^(3/2 - m/2)*a*b*(e*Cos[c + d*x])^(-1 - m)*Hypergeometric2F1[(-1 - m)/2, (1 + m)/2, (1 - m)/2, ((a - b)*(1 - Sin[c + d*x]))/(2*(a + b*Sin[c + d*x]))]*(((a + b)*(1 + Sin[c + d*x]))/(a + b*Sin[c + d*x]))^((1 + m)/2)*(a + b*Sin[c + d*x])^(1 + m))/((a - b)^2*(a + b)*d*e^3*(1 + m)*(3 + m)) - (2^(-1/2 - m/2)*a*(2*a*b - b^2 + a^2*(2 + m))*(e*Cos[c + d*x])^(-3 - m)*Hypergeometric2F1[(1 - m)/2, (3 + m)/2, (3 - m)/2, ((a - b)*(1 - Sin[c + d*x]))/(2*(a + b*Sin[c + d*x]))]*(1 - Sin[c + d*x])^2*(((a + b)*(1 + Sin[c + d*x]))/(a + b*Sin[c + d*x]))^((3 + m)/2)*(a + b*Sin[c + d*x])^(1 + m))/((a - b)*(a + b)^3*d*e*(1 - m)*(3 + m))","A",9,7,27,0.2593,1,"{2700, 2699, 2920, 132, 129, 155, 12}"
648,1,420,0,0.5118805,"\int (e \cos (c+d x))^{-3-m} (a+b \sin (c+d x))^m \, dx","Int[(e*Cos[c + d*x])^(-3 - m)*(a + b*Sin[c + d*x])^m,x]","-\frac{b (1-\sin (c+d x)) (e \cos (c+d x))^{-m-2} \left(-\frac{(a-b) (1-\sin (c+d x))}{(a+b) (\sin (c+d x)+1)}\right)^{m/2} (a+b \sin (c+d x))^{m+1} \, _2F_1\left(m+1,\frac{m+2}{2};m+2;\frac{2 (a+b \sin (c+d x))}{(a+b) (\sin (c+d x)+1)}\right)}{d e (m+1) (m+2) \left(a^2-b^2\right)}+\frac{a (\sin (c+d x)+1) (e \cos (c+d x))^{-m-2} (a+b \sin (c+d x))^{m+1}}{d e (m+2) \left(a^2-b^2\right)}+\frac{a 2^{-m/2} (a m+a+b) (1-\sin (c+d x)) (e \cos (c+d x))^{-m-2} \left(\frac{(a+b) (\sin (c+d x)+1)}{a+b \sin (c+d x)}\right)^{\frac{m+2}{2}} (a+b \sin (c+d x))^{m+1} \, _2F_1\left(-\frac{m}{2},\frac{m+2}{2};\frac{2-m}{2};\frac{(a-b) (1-\sin (c+d x))}{2 (a+b \sin (c+d x))}\right)}{d e m (m+2) (a-b) (a+b)^2}-\frac{(e \cos (c+d x))^{-m-2} (a+b \sin (c+d x))^{m+1}}{d e (m+2) (a-b)}","-\frac{\left(a^2 (m+1)-b^2\right) (\sin (c+d x)+1)^3 \sec ^4(c+d x) (e \cos (c+d x))^{-m} (a+b \sin (c+d x))^{m+1} \left(\frac{(a+b) (\sin (c+d x)+1)}{(a-b) (\sin (c+d x)-1)}\right)^{\frac{m-2}{2}} \, _2F_1\left(\frac{m}{2},m+1;m+2;-\frac{2 (a+b \sin (c+d x))}{(a-b) (\sin (c+d x)-1)}\right)}{d e^3 m (m+1) (a-b)^3}+\frac{(a (m+2)-2 b) (\sin (c+d x)-1) (\sin (c+d x)+1)^2 \sec ^4(c+d x) (e \cos (c+d x))^{-m} (a+b \sin (c+d x))^{m+1}}{d e^3 m (m+2) (a-b)^2}+\frac{(\sin (c+d x)-1) (\sin (c+d x)+1) \sec ^4(c+d x) (e \cos (c+d x))^{-m} (a+b \sin (c+d x))^{m+1}}{d e^3 (m+2) (a-b)}",1,"-(((e*Cos[c + d*x])^(-2 - m)*(a + b*Sin[c + d*x])^(1 + m))/((a - b)*d*e*(2 + m))) - (b*(e*Cos[c + d*x])^(-2 - m)*Hypergeometric2F1[1 + m, (2 + m)/2, 2 + m, (2*(a + b*Sin[c + d*x]))/((a + b)*(1 + Sin[c + d*x]))]*(1 - Sin[c + d*x])*(-(((a - b)*(1 - Sin[c + d*x]))/((a + b)*(1 + Sin[c + d*x]))))^(m/2)*(a + b*Sin[c + d*x])^(1 + m))/((a^2 - b^2)*d*e*(1 + m)*(2 + m)) + (a*(e*Cos[c + d*x])^(-2 - m)*(1 + Sin[c + d*x])*(a + b*Sin[c + d*x])^(1 + m))/((a^2 - b^2)*d*e*(2 + m)) + (a*(a + b + a*m)*(e*Cos[c + d*x])^(-2 - m)*Hypergeometric2F1[-m/2, (2 + m)/2, (2 - m)/2, ((a - b)*(1 - Sin[c + d*x]))/(2*(a + b*Sin[c + d*x]))]*(1 - Sin[c + d*x])*(((a + b)*(1 + Sin[c + d*x]))/(a + b*Sin[c + d*x]))^((2 + m)/2)*(a + b*Sin[c + d*x])^(1 + m))/(2^(m/2)*(a - b)*(a + b)^2*d*e*m*(2 + m))","A",5,5,27,0.1852,0,"{2700, 2698, 2920, 96, 132}"
649,1,201,0,0.292773,"\int (e \cos (c+d x))^{-2-m} (a+b \sin (c+d x))^m \, dx","Int[(e*Cos[c + d*x])^(-2 - m)*(a + b*Sin[c + d*x])^m,x]","\frac{a 2^{\frac{1}{2}-\frac{m}{2}} (e \cos (c+d x))^{-m-1} \left(\frac{(a+b) (\sin (c+d x)+1)}{a+b \sin (c+d x)}\right)^{\frac{m+1}{2}} (a+b \sin (c+d x))^{m+1} \, _2F_1\left(\frac{1}{2} (-m-1),\frac{m+1}{2};\frac{1-m}{2};\frac{(a-b) (1-\sin (c+d x))}{2 (a+b \sin (c+d x))}\right)}{d e (m+1) \left(a^2-b^2\right)}-\frac{(e \cos (c+d x))^{-m-1} (a+b \sin (c+d x))^{m+1}}{d e (m+1) (a-b)}","\frac{a 2^{\frac{1}{2}-\frac{m}{2}} (e \cos (c+d x))^{-m-1} \left(\frac{(a+b) (\sin (c+d x)+1)}{a+b \sin (c+d x)}\right)^{\frac{m+1}{2}} (a+b \sin (c+d x))^{m+1} \, _2F_1\left(\frac{1}{2} (-m-1),\frac{m+1}{2};\frac{1-m}{2};\frac{(a-b) (1-\sin (c+d x))}{2 (a+b \sin (c+d x))}\right)}{d e (m+1) \left(a^2-b^2\right)}-\frac{(e \cos (c+d x))^{-m-1} (a+b \sin (c+d x))^{m+1}}{d e (m+1) (a-b)}",1,"-(((e*Cos[c + d*x])^(-1 - m)*(a + b*Sin[c + d*x])^(1 + m))/((a - b)*d*e*(1 + m))) + (2^(1/2 - m/2)*a*(e*Cos[c + d*x])^(-1 - m)*Hypergeometric2F1[(-1 - m)/2, (1 + m)/2, (1 - m)/2, ((a - b)*(1 - Sin[c + d*x]))/(2*(a + b*Sin[c + d*x]))]*(((a + b)*(1 + Sin[c + d*x]))/(a + b*Sin[c + d*x]))^((1 + m)/2)*(a + b*Sin[c + d*x])^(1 + m))/((a^2 - b^2)*d*e*(1 + m))","A",3,3,27,0.1111,1,"{2699, 2920, 132}"
650,1,132,0,0.0654538,"\int (e \cos (c+d x))^{-1-m} (a+b \sin (c+d x))^m \, dx","Int[(e*Cos[c + d*x])^(-1 - m)*(a + b*Sin[c + d*x])^m,x]","\frac{e (1-\sin (c+d x)) (e \cos (c+d x))^{-m-2} \left(-\frac{(a-b) (1-\sin (c+d x))}{(a+b) (\sin (c+d x)+1)}\right)^{m/2} (a+b \sin (c+d x))^{m+1} \, _2F_1\left(m+1,\frac{m+2}{2};m+2;\frac{2 (a+b \sin (c+d x))}{(a+b) (\sin (c+d x)+1)}\right)}{d (m+1) (a+b)}","\frac{e (1-\sin (c+d x)) (e \cos (c+d x))^{-m-2} \left(-\frac{(a-b) (1-\sin (c+d x))}{(a+b) (\sin (c+d x)+1)}\right)^{m/2} (a+b \sin (c+d x))^{m+1} \, _2F_1\left(m+1,\frac{m+2}{2};m+2;\frac{2 (a+b \sin (c+d x))}{(a+b) (\sin (c+d x)+1)}\right)}{d (m+1) (a+b)}",1,"(e*(e*Cos[c + d*x])^(-2 - m)*Hypergeometric2F1[1 + m, (2 + m)/2, 2 + m, (2*(a + b*Sin[c + d*x]))/((a + b)*(1 + Sin[c + d*x]))]*(1 - Sin[c + d*x])*(-(((a - b)*(1 - Sin[c + d*x]))/((a + b)*(1 + Sin[c + d*x]))))^(m/2)*(a + b*Sin[c + d*x])^(1 + m))/((a + b)*d*(1 + m))","A",1,1,27,0.03704,1,"{2698}"
651,1,152,0,0.1032086,"\int (e \cos (c+d x))^{-m} (a+b \sin (c+d x))^m \, dx","Int[(a + b*Sin[c + d*x])^m/(e*Cos[c + d*x])^m,x]","\frac{e (e \cos (c+d x))^{-m-1} (a+b \sin (c+d x))^{m+1} \left(1-\frac{a+b \sin (c+d x)}{a-b}\right)^{\frac{m+1}{2}} \left(1-\frac{a+b \sin (c+d x)}{a+b}\right)^{\frac{m+1}{2}} F_1\left(m+1;\frac{m+1}{2},\frac{m+1}{2};m+2;\frac{a+b \sin (c+d x)}{a-b},\frac{a+b \sin (c+d x)}{a+b}\right)}{b d (m+1)}","\frac{e (e \cos (c+d x))^{-m-1} (a+b \sin (c+d x))^{m+1} \left(1-\frac{a+b \sin (c+d x)}{a-b}\right)^{\frac{m+1}{2}} \left(1-\frac{a+b \sin (c+d x)}{a+b}\right)^{\frac{m+1}{2}} F_1\left(m+1;\frac{m+1}{2},\frac{m+1}{2};m+2;\frac{a+b \sin (c+d x)}{a-b},\frac{a+b \sin (c+d x)}{a+b}\right)}{b d (m+1)}",1,"(e*AppellF1[1 + m, (1 + m)/2, (1 + m)/2, 2 + m, (a + b*Sin[c + d*x])/(a - b), (a + b*Sin[c + d*x])/(a + b)]*(e*Cos[c + d*x])^(-1 - m)*(a + b*Sin[c + d*x])^(1 + m)*(1 - (a + b*Sin[c + d*x])/(a - b))^((1 + m)/2)*(1 - (a + b*Sin[c + d*x])/(a + b))^((1 + m)/2))/(b*d*(1 + m))","A",2,2,25,0.08000,1,"{2704, 138}"
652,1,142,0,0.0966598,"\int (e \cos (c+d x))^{1-m} (a+b \sin (c+d x))^m \, dx","Int[(e*Cos[c + d*x])^(1 - m)*(a + b*Sin[c + d*x])^m,x]","\frac{e (e \cos (c+d x))^{-m} (a+b \sin (c+d x))^{m+1} \left(1-\frac{a+b \sin (c+d x)}{a-b}\right)^{m/2} \left(1-\frac{a+b \sin (c+d x)}{a+b}\right)^{m/2} F_1\left(m+1;\frac{m}{2},\frac{m}{2};m+2;\frac{a+b \sin (c+d x)}{a-b},\frac{a+b \sin (c+d x)}{a+b}\right)}{b d (m+1)}","\frac{e (e \cos (c+d x))^{-m} (a+b \sin (c+d x))^{m+1} \left(1-\frac{a+b \sin (c+d x)}{a-b}\right)^{m/2} \left(1-\frac{a+b \sin (c+d x)}{a+b}\right)^{m/2} F_1\left(m+1;\frac{m}{2},\frac{m}{2};m+2;\frac{a+b \sin (c+d x)}{a-b},\frac{a+b \sin (c+d x)}{a+b}\right)}{b d (m+1)}",1,"(e*AppellF1[1 + m, m/2, m/2, 2 + m, (a + b*Sin[c + d*x])/(a - b), (a + b*Sin[c + d*x])/(a + b)]*(a + b*Sin[c + d*x])^(1 + m)*(1 - (a + b*Sin[c + d*x])/(a - b))^(m/2)*(1 - (a + b*Sin[c + d*x])/(a + b))^(m/2))/(b*d*(1 + m)*(e*Cos[c + d*x])^m)","A",2,2,27,0.07407,1,"{2704, 138}"
653,1,152,0,0.1037632,"\int (e \cos (c+d x))^{2-m} (a+b \sin (c+d x))^m \, dx","Int[(e*Cos[c + d*x])^(2 - m)*(a + b*Sin[c + d*x])^m,x]","\frac{e (e \cos (c+d x))^{1-m} (a+b \sin (c+d x))^{m+1} \left(1-\frac{a+b \sin (c+d x)}{a-b}\right)^{\frac{m-1}{2}} \left(1-\frac{a+b \sin (c+d x)}{a+b}\right)^{\frac{m-1}{2}} F_1\left(m+1;\frac{m-1}{2},\frac{m-1}{2};m+2;\frac{a+b \sin (c+d x)}{a-b},\frac{a+b \sin (c+d x)}{a+b}\right)}{b d (m+1)}","\frac{e (e \cos (c+d x))^{1-m} (a+b \sin (c+d x))^{m+1} \left(1-\frac{a+b \sin (c+d x)}{a-b}\right)^{\frac{m-1}{2}} \left(1-\frac{a+b \sin (c+d x)}{a+b}\right)^{\frac{m-1}{2}} F_1\left(m+1;\frac{m-1}{2},\frac{m-1}{2};m+2;\frac{a+b \sin (c+d x)}{a-b},\frac{a+b \sin (c+d x)}{a+b}\right)}{b d (m+1)}",1,"(e*AppellF1[1 + m, (-1 + m)/2, (-1 + m)/2, 2 + m, (a + b*Sin[c + d*x])/(a - b), (a + b*Sin[c + d*x])/(a + b)]*(e*Cos[c + d*x])^(1 - m)*(a + b*Sin[c + d*x])^(1 + m)*(1 - (a + b*Sin[c + d*x])/(a - b))^((-1 + m)/2)*(1 - (a + b*Sin[c + d*x])/(a + b))^((-1 + m)/2))/(b*d*(1 + m))","A",2,2,27,0.07407,1,"{2704, 138}"