1,1,133,92,0.0529225,"\int \cos ^7(c+d x) \left(A+C \cos ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^7*(A + C*Cos[c + d*x]^2),x]","-\frac{A \sin ^7(c+d x)}{7 d}+\frac{3 A \sin ^5(c+d x)}{5 d}-\frac{A \sin ^3(c+d x)}{d}+\frac{A \sin (c+d x)}{d}+\frac{C \sin ^9(c+d x)}{9 d}-\frac{4 C \sin ^7(c+d x)}{7 d}+\frac{6 C \sin ^5(c+d x)}{5 d}-\frac{4 C \sin ^3(c+d x)}{3 d}+\frac{C \sin (c+d x)}{d}","-\frac{(A+4 C) \sin ^7(c+d x)}{7 d}+\frac{3 (A+2 C) \sin ^5(c+d x)}{5 d}-\frac{(3 A+4 C) \sin ^3(c+d x)}{3 d}+\frac{(A+C) \sin (c+d x)}{d}+\frac{C \sin ^9(c+d x)}{9 d}",1,"(A*Sin[c + d*x])/d + (C*Sin[c + d*x])/d - (A*Sin[c + d*x]^3)/d - (4*C*Sin[c + d*x]^3)/(3*d) + (3*A*Sin[c + d*x]^5)/(5*d) + (6*C*Sin[c + d*x]^5)/(5*d) - (A*Sin[c + d*x]^7)/(7*d) - (4*C*Sin[c + d*x]^7)/(7*d) + (C*Sin[c + d*x]^9)/(9*d)","A",1
2,1,101,72,0.0271312,"\int \cos ^5(c+d x) \left(A+C \cos ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^5*(A + C*Cos[c + d*x]^2),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{C \sin ^7(c+d x)}{7 d}+\frac{3 C \sin ^5(c+d x)}{5 d}-\frac{C \sin ^3(c+d x)}{d}+\frac{C \sin (c+d x)}{d}","\frac{(A+3 C) \sin ^5(c+d x)}{5 d}-\frac{(2 A+3 C) \sin ^3(c+d x)}{3 d}+\frac{(A+C) \sin (c+d x)}{d}-\frac{C \sin ^7(c+d x)}{7 d}",1,"(A*Sin[c + d*x])/d + (C*Sin[c + d*x])/d - (2*A*Sin[c + d*x]^3)/(3*d) - (C*Sin[c + d*x]^3)/d + (A*Sin[c + d*x]^5)/(5*d) + (3*C*Sin[c + d*x]^5)/(5*d) - (C*Sin[c + d*x]^7)/(7*d)","A",1
3,1,71,50,0.0191634,"\int \cos ^3(c+d x) \left(A+C \cos ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^3*(A + C*Cos[c + d*x]^2),x]","-\frac{A \sin ^3(c+d x)}{3 d}+\frac{A \sin (c+d x)}{d}+\frac{C \sin ^5(c+d x)}{5 d}-\frac{2 C \sin ^3(c+d x)}{3 d}+\frac{C \sin (c+d x)}{d}","-\frac{(A+2 C) \sin ^3(c+d x)}{3 d}+\frac{(A+C) \sin (c+d x)}{d}+\frac{C \sin ^5(c+d x)}{5 d}",1,"(A*Sin[c + d*x])/d + (C*Sin[c + d*x])/d - (A*Sin[c + d*x]^3)/(3*d) - (2*C*Sin[c + d*x]^3)/(3*d) + (C*Sin[c + d*x]^5)/(5*d)","A",1
4,1,50,30,0.0166429,"\int \cos (c+d x) \left(A+C \cos ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]*(A + C*Cos[c + d*x]^2),x]","\frac{A \sin (c) \cos (d x)}{d}+\frac{A \cos (c) \sin (d x)}{d}-\frac{C \sin ^3(c+d x)}{3 d}+\frac{C \sin (c+d x)}{d}","\frac{(A+C) \sin (c+d x)}{d}-\frac{C \sin ^3(c+d x)}{3 d}",1,"(A*Cos[d*x]*Sin[c])/d + (A*Cos[c]*Sin[d*x])/d + (C*Sin[c + d*x])/d - (C*Sin[c + d*x]^3)/(3*d)","A",1
5,1,35,24,0.0171287,"\int \left(A+C \cos ^2(c+d x)\right) \sec (c+d x) \, dx","Integrate[(A + C*Cos[c + d*x]^2)*Sec[c + d*x],x]","\frac{A \tanh ^{-1}(\sin (c+d x))}{d}+\frac{C \sin (c) \cos (d x)}{d}+\frac{C \cos (c) \sin (d x)}{d}","\frac{A \tanh ^{-1}(\sin (c+d x))}{d}+\frac{C \sin (c+d x)}{d}",1,"(A*ArcTanh[Sin[c + d*x]])/d + (C*Cos[d*x]*Sin[c])/d + (C*Cos[c]*Sin[d*x])/d","A",1
6,1,48,40,0.0296826,"\int \left(A+C \cos ^2(c+d x)\right) \sec ^3(c+d x) \, dx","Integrate[(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^3,x]","\frac{A \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{A \tan (c+d x) \sec (c+d x)}{2 d}+\frac{C \tanh ^{-1}(\sin (c+d x))}{d}","\frac{(A+2 C) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{A \tan (c+d x) \sec (c+d x)}{2 d}",1,"(A*ArcTanh[Sin[c + d*x]])/(2*d) + (C*ArcTanh[Sin[c + d*x]])/d + (A*Sec[c + d*x]*Tan[c + d*x])/(2*d)","A",1
7,1,54,70,0.136233,"\int \left(A+C \cos ^2(c+d x)\right) \sec ^5(c+d x) \, dx","Integrate[(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^5,x]","\frac{(3 A+4 C) \tanh ^{-1}(\sin (c+d x))+\tan (c+d x) \sec (c+d x) \left(2 A \sec ^2(c+d x)+3 A+4 C\right)}{8 d}","\frac{(3 A+4 C) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{(3 A+4 C) \tan (c+d x) \sec (c+d x)}{8 d}+\frac{A \tan (c+d x) \sec ^3(c+d x)}{4 d}",1,"((3*A + 4*C)*ArcTanh[Sin[c + d*x]] + Sec[c + d*x]*(3*A + 4*C + 2*A*Sec[c + d*x]^2)*Tan[c + d*x])/(8*d)","A",1
8,1,75,98,0.3632561,"\int \left(A+C \cos ^2(c+d x)\right) \sec ^7(c+d x) \, dx","Integrate[(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^7,x]","\frac{3 (5 A+6 C) \tanh ^{-1}(\sin (c+d x))+\tan (c+d x) \sec (c+d x) \left(2 (5 A+6 C) \sec ^2(c+d x)+8 A \sec ^4(c+d x)+3 (5 A+6 C)\right)}{48 d}","\frac{(5 A+6 C) \tanh ^{-1}(\sin (c+d x))}{16 d}+\frac{(5 A+6 C) \tan (c+d x) \sec ^3(c+d x)}{24 d}+\frac{(5 A+6 C) \tan (c+d x) \sec (c+d x)}{16 d}+\frac{A \tan (c+d x) \sec ^5(c+d x)}{6 d}",1,"(3*(5*A + 6*C)*ArcTanh[Sin[c + d*x]] + Sec[c + d*x]*(3*(5*A + 6*C) + 2*(5*A + 6*C)*Sec[c + d*x]^2 + 8*A*Sec[c + d*x]^4)*Tan[c + d*x])/(48*d)","A",1
9,1,93,117,0.1727698,"\int \cos ^6(c+d x) \left(A+C \cos ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^6*(A + C*Cos[c + d*x]^2),x]","\frac{48 (15 A+14 C) \sin (2 (c+d x))+24 (6 A+7 C) \sin (4 (c+d x))+16 A \sin (6 (c+d x))+960 A c+960 A d x+32 C \sin (6 (c+d x))+3 C \sin (8 (c+d x))+840 c C+840 C d x}{3072 d}","\frac{(8 A+7 C) \sin (c+d x) \cos ^5(c+d x)}{48 d}+\frac{5 (8 A+7 C) \sin (c+d x) \cos ^3(c+d x)}{192 d}+\frac{5 (8 A+7 C) \sin (c+d x) \cos (c+d x)}{128 d}+\frac{5}{128} x (8 A+7 C)+\frac{C \sin (c+d x) \cos ^7(c+d x)}{8 d}",1,"(960*A*c + 840*c*C + 960*A*d*x + 840*C*d*x + 48*(15*A + 14*C)*Sin[2*(c + d*x)] + 24*(6*A + 7*C)*Sin[4*(c + d*x)] + 16*A*Sin[6*(c + d*x)] + 32*C*Sin[6*(c + d*x)] + 3*C*Sin[8*(c + d*x)])/(3072*d)","A",1
10,1,68,89,0.105783,"\int \cos ^4(c+d x) \left(A+C \cos ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^4*(A + C*Cos[c + d*x]^2),x]","\frac{(48 A+45 C) \sin (2 (c+d x))+(6 A+9 C) \sin (4 (c+d x))+72 A c+72 A d x+C \sin (6 (c+d x))+60 c C+60 C d x}{192 d}","\frac{(6 A+5 C) \sin (c+d x) \cos ^3(c+d x)}{24 d}+\frac{(6 A+5 C) \sin (c+d x) \cos (c+d x)}{16 d}+\frac{1}{16} x (6 A+5 C)+\frac{C \sin (c+d x) \cos ^5(c+d x)}{6 d}",1,"(72*A*c + 60*c*C + 72*A*d*x + 60*C*d*x + (48*A + 45*C)*Sin[2*(c + d*x)] + (6*A + 9*C)*Sin[4*(c + d*x)] + C*Sin[6*(c + d*x)])/(192*d)","A",1
11,1,45,61,0.0940636,"\int \cos ^2(c+d x) \left(A+C \cos ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^2*(A + C*Cos[c + d*x]^2),x]","\frac{4 (4 A+3 C) (c+d x)+8 (A+C) \sin (2 (c+d x))+C \sin (4 (c+d x))}{32 d}","\frac{(4 A+3 C) \sin (c+d x) \cos (c+d x)}{8 d}+\frac{1}{8} x (4 A+3 C)+\frac{C \sin (c+d x) \cos ^3(c+d x)}{4 d}",1,"(4*(4*A + 3*C)*(c + d*x) + 8*(A + C)*Sin[2*(c + d*x)] + C*Sin[4*(c + d*x)])/(32*d)","A",1
12,1,15,15,0.0094283,"\int \left(A+C \cos ^2(c+d x)\right) \sec ^2(c+d x) \, dx","Integrate[(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^2,x]","\frac{A \tan (c+d x)}{d}+C x","\frac{A \tan (c+d x)}{d}+C x",1,"C*x + (A*Tan[c + d*x])/d","A",1
13,1,36,43,0.0966821,"\int \left(A+C \cos ^2(c+d x)\right) \sec ^4(c+d x) \, dx","Integrate[(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^4,x]","\frac{A \left(\frac{1}{3} \tan ^3(c+d x)+\tan (c+d x)\right)}{d}+\frac{C \tan (c+d x)}{d}","\frac{(2 A+3 C) \tan (c+d x)}{3 d}+\frac{A \tan (c+d x) \sec ^2(c+d x)}{3 d}",1,"(C*Tan[c + d*x])/d + (A*(Tan[c + d*x] + Tan[c + d*x]^3/3))/d","A",1
14,1,61,65,0.2222796,"\int \left(A+C \cos ^2(c+d x)\right) \sec ^6(c+d x) \, dx","Integrate[(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^6,x]","\frac{A \left(\frac{1}{5} \tan ^5(c+d x)+\frac{2}{3} \tan ^3(c+d x)+\tan (c+d x)\right)}{d}+\frac{C \left(\frac{1}{3} \tan ^3(c+d x)+\tan (c+d x)\right)}{d}","\frac{(4 A+5 C) \tan ^3(c+d x)}{15 d}+\frac{(4 A+5 C) \tan (c+d x)}{5 d}+\frac{A \tan (c+d x) \sec ^4(c+d x)}{5 d}",1,"(C*(Tan[c + d*x] + Tan[c + d*x]^3/3))/d + (A*(Tan[c + d*x] + (2*Tan[c + d*x]^3)/3 + Tan[c + d*x]^5/5))/d","A",1
15,1,81,87,0.3291421,"\int \left(A+C \cos ^2(c+d x)\right) \sec ^8(c+d x) \, dx","Integrate[(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^8,x]","\frac{A \left(\frac{1}{7} \tan ^7(c+d x)+\frac{3}{5} \tan ^5(c+d x)+\tan ^3(c+d x)+\tan (c+d x)\right)}{d}+\frac{C \left(\frac{1}{5} \tan ^5(c+d x)+\frac{2}{3} \tan ^3(c+d x)+\tan (c+d x)\right)}{d}","\frac{(6 A+7 C) \tan ^5(c+d x)}{35 d}+\frac{2 (6 A+7 C) \tan ^3(c+d x)}{21 d}+\frac{(6 A+7 C) \tan (c+d x)}{7 d}+\frac{A \tan (c+d x) \sec ^6(c+d x)}{7 d}",1,"(C*(Tan[c + d*x] + (2*Tan[c + d*x]^3)/3 + Tan[c + d*x]^5/5))/d + (A*(Tan[c + d*x] + Tan[c + d*x]^3 + (3*Tan[c + d*x]^5)/5 + Tan[c + d*x]^7/7))/d","A",1
16,1,88,113,0.4250629,"\int (b \cos (c+d x))^{5/2} \left(A+C \cos ^2(c+d x)\right) \, dx","Integrate[(b*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2),x]","\frac{(b \cos (c+d x))^{5/2} \left(24 (9 A+7 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+2 \sin (2 (c+d x)) \sqrt{\cos (c+d x)} (18 A+5 C \cos (2 (c+d x))+19 C)\right)}{180 d \cos ^{\frac{5}{2}}(c+d x)}","\frac{2 b^2 (9 A+7 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \cos (c+d x)}}{15 d \sqrt{\cos (c+d x)}}+\frac{2 b (9 A+7 C) \sin (c+d x) (b \cos (c+d x))^{3/2}}{45 d}+\frac{2 C \sin (c+d x) (b \cos (c+d x))^{7/2}}{9 b d}",1,"((b*Cos[c + d*x])^(5/2)*(24*(9*A + 7*C)*EllipticE[(c + d*x)/2, 2] + 2*Sqrt[Cos[c + d*x]]*(18*A + 19*C + 5*C*Cos[2*(c + d*x)])*Sin[2*(c + d*x)]))/(180*d*Cos[c + d*x]^(5/2))","A",1
17,1,86,113,0.3900489,"\int (b \cos (c+d x))^{3/2} \left(A+C \cos ^2(c+d x)\right) \, dx","Integrate[(b*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2),x]","\frac{(b \cos (c+d x))^{3/2} \left(4 (7 A+5 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+2 \sin (c+d x) \sqrt{\cos (c+d x)} (14 A+3 C \cos (2 (c+d x))+13 C)\right)}{42 d \cos ^{\frac{3}{2}}(c+d x)}","\frac{2 b^2 (7 A+5 C) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d \sqrt{b \cos (c+d x)}}+\frac{2 b (7 A+5 C) \sin (c+d x) \sqrt{b \cos (c+d x)}}{21 d}+\frac{2 C \sin (c+d x) (b \cos (c+d x))^{5/2}}{7 b d}",1,"((b*Cos[c + d*x])^(3/2)*(4*(7*A + 5*C)*EllipticF[(c + d*x)/2, 2] + 2*Sqrt[Cos[c + d*x]]*(14*A + 13*C + 3*C*Cos[2*(c + d*x)])*Sin[c + d*x]))/(42*d*Cos[c + d*x]^(3/2))","A",1
18,1,70,77,0.1423422,"\int \sqrt{b \cos (c+d x)} \left(A+C \cos ^2(c+d x)\right) \, dx","Integrate[Sqrt[b*Cos[c + d*x]]*(A + C*Cos[c + d*x]^2),x]","\frac{\sqrt{b \cos (c+d x)} \left(2 (5 A+3 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+C \sin (2 (c+d x)) \sqrt{\cos (c+d x)}\right)}{5 d \sqrt{\cos (c+d x)}}","\frac{2 (5 A+3 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \cos (c+d x)}}{5 d \sqrt{\cos (c+d x)}}+\frac{2 C \sin (c+d x) (b \cos (c+d x))^{3/2}}{5 b d}",1,"(Sqrt[b*Cos[c + d*x]]*(2*(5*A + 3*C)*EllipticE[(c + d*x)/2, 2] + C*Sqrt[Cos[c + d*x]]*Sin[2*(c + d*x)]))/(5*d*Sqrt[Cos[c + d*x]])","A",1
19,1,58,75,0.1559427,"\int \frac{A+C \cos ^2(c+d x)}{\sqrt{b \cos (c+d x)}} \, dx","Integrate[(A + C*Cos[c + d*x]^2)/Sqrt[b*Cos[c + d*x]],x]","\frac{2 (3 A+C) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+C \sin (2 (c+d x))}{3 d \sqrt{b \cos (c+d x)}}","\frac{2 (3 A+C) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d \sqrt{b \cos (c+d x)}}+\frac{2 C \sin (c+d x) \sqrt{b \cos (c+d x)}}{3 b d}",1,"(2*(3*A + C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + C*Sin[2*(c + d*x)])/(3*d*Sqrt[b*Cos[c + d*x]])","A",1
20,1,57,74,0.1618408,"\int \frac{A+C \cos ^2(c+d x)}{(b \cos (c+d x))^{3/2}} \, dx","Integrate[(A + C*Cos[c + d*x]^2)/(b*Cos[c + d*x])^(3/2),x]","\frac{2 A \sin (c+d x)-2 (A-C) \sqrt{\cos (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b d \sqrt{b \cos (c+d x)}}","\frac{2 A \sin (c+d x)}{b d \sqrt{b \cos (c+d x)}}-\frac{2 (A-C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \cos (c+d x)}}{b^2 d \sqrt{\cos (c+d x)}}",1,"(-2*(A - C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2] + 2*A*Sin[c + d*x])/(b*d*Sqrt[b*Cos[c + d*x]])","A",1
21,1,58,78,0.228732,"\int \frac{A+C \cos ^2(c+d x)}{(b \cos (c+d x))^{5/2}} \, dx","Integrate[(A + C*Cos[c + d*x]^2)/(b*Cos[c + d*x])^(5/2),x]","\frac{2 \left((A+3 C) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+A \tan (c+d x)\right)}{3 b^2 d \sqrt{b \cos (c+d x)}}","\frac{2 (A+3 C) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 b^2 d \sqrt{b \cos (c+d x)}}+\frac{2 A \sin (c+d x)}{3 b d (b \cos (c+d x))^{3/2}}",1,"(2*((A + 3*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + A*Tan[c + d*x]))/(3*b^2*d*Sqrt[b*Cos[c + d*x]])","A",1
22,1,81,115,0.2959294,"\int \frac{A+C \cos ^2(c+d x)}{(b \cos (c+d x))^{7/2}} \, dx","Integrate[(A + C*Cos[c + d*x]^2)/(b*Cos[c + d*x])^(7/2),x]","\frac{2 \left((3 A+5 C) \sin (c+d x)-\left((3 A+5 C) \sqrt{\cos (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)+A \tan (c+d x) \sec (c+d x)\right)}{5 b^3 d \sqrt{b \cos (c+d x)}}","-\frac{2 (3 A+5 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \cos (c+d x)}}{5 b^4 d \sqrt{\cos (c+d x)}}+\frac{2 (3 A+5 C) \sin (c+d x)}{5 b^3 d \sqrt{b \cos (c+d x)}}+\frac{2 A \sin (c+d x)}{5 b d (b \cos (c+d x))^{5/2}}",1,"(2*(-((3*A + 5*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]) + (3*A + 5*C)*Sin[c + d*x] + A*Sec[c + d*x]*Tan[c + d*x]))/(5*b^3*d*Sqrt[b*Cos[c + d*x]])","A",1
23,1,77,115,0.4158999,"\int \frac{A+C \cos ^2(c+d x)}{(b \cos (c+d x))^{9/2}} \, dx","Integrate[(A + C*Cos[c + d*x]^2)/(b*Cos[c + d*x])^(9/2),x]","\frac{2 \left((5 A+7 C) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+\tan (c+d x) \left(3 A \sec ^2(c+d x)+5 A+7 C\right)\right)}{21 b^4 d \sqrt{b \cos (c+d x)}}","\frac{2 (5 A+7 C) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 b^4 d \sqrt{b \cos (c+d x)}}+\frac{2 (5 A+7 C) \sin (c+d x)}{21 b^3 d (b \cos (c+d x))^{3/2}}+\frac{2 A \sin (c+d x)}{7 b d (b \cos (c+d x))^{7/2}}",1,"(2*((5*A + 7*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + (5*A + 7*C + 3*A*Sec[c + d*x]^2)*Tan[c + d*x]))/(21*b^4*d*Sqrt[b*Cos[c + d*x]])","A",1
24,1,23,21,0.0629893,"\int \sqrt{\cos (c+d x)} \left(3-5 \cos ^2(c+d x)\right) \, dx","Integrate[Sqrt[Cos[c + d*x]]*(3 - 5*Cos[c + d*x]^2),x]","-\frac{\sin (2 (c+d x)) \sqrt{\cos (c+d x)}}{d}","-\frac{2 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{d}",1,"-((Sqrt[Cos[c + d*x]]*Sin[2*(c + d*x)])/d)","A",1
25,1,21,21,0.0639147,"\int \frac{1-3 \cos ^2(c+d x)}{\sqrt{\cos (c+d x)}} \, dx","Integrate[(1 - 3*Cos[c + d*x]^2)/Sqrt[Cos[c + d*x]],x]","-\frac{2 \sin (c+d x) \sqrt{\cos (c+d x)}}{d}","-\frac{2 \sin (c+d x) \sqrt{\cos (c+d x)}}{d}",1,"(-2*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/d","A",1
26,1,78,115,0.8699118,"\int \left(A+C \cos ^2(c+d x)\right) (b \sec (c+d x))^{9/2} \, dx","Integrate[(A + C*Cos[c + d*x]^2)*(b*Sec[c + d*x])^(9/2),x]","\frac{b^2 (b \sec (c+d x))^{5/2} \left((5 A+7 C) \sin (2 (c+d x))+2 (5 A+7 C) \cos ^{\frac{5}{2}}(c+d x) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+6 A \tan (c+d x)\right)}{21 d}","\frac{2 b^4 (5 A+7 C) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \sec (c+d x)}}{21 d}+\frac{2 b^3 (5 A+7 C) \sin (c+d x) (b \sec (c+d x))^{3/2}}{21 d}+\frac{2 A b^2 \tan (c+d x) (b \sec (c+d x))^{5/2}}{7 d}",1,"(b^2*(b*Sec[c + d*x])^(5/2)*(2*(5*A + 7*C)*Cos[c + d*x]^(5/2)*EllipticF[(c + d*x)/2, 2] + (5*A + 7*C)*Sin[2*(c + d*x)] + 6*A*Tan[c + d*x]))/(21*d)","A",1
27,1,79,115,0.5151811,"\int \left(A+C \cos ^2(c+d x)\right) (b \sec (c+d x))^{7/2} \, dx","Integrate[(A + C*Cos[c + d*x]^2)*(b*Sec[c + d*x])^(7/2),x]","-\frac{b^2 (b \sec (c+d x))^{3/2} \left(-(3 A+5 C) \sin (2 (c+d x))+2 (3 A+5 C) \cos ^{\frac{3}{2}}(c+d x) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)-2 A \tan (c+d x)\right)}{5 d}","-\frac{2 b^4 (3 A+5 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d \sqrt{\cos (c+d x)} \sqrt{b \sec (c+d x)}}+\frac{2 b^3 (3 A+5 C) \sin (c+d x) \sqrt{b \sec (c+d x)}}{5 d}+\frac{2 A b^2 \tan (c+d x) (b \sec (c+d x))^{3/2}}{5 d}",1,"-1/5*(b^2*(b*Sec[c + d*x])^(3/2)*(2*(3*A + 5*C)*Cos[c + d*x]^(3/2)*EllipticE[(c + d*x)/2, 2] - (3*A + 5*C)*Sin[2*(c + d*x)] - 2*A*Tan[c + d*x]))/d","A",1
28,1,58,78,0.2599463,"\int \left(A+C \cos ^2(c+d x)\right) (b \sec (c+d x))^{5/2} \, dx","Integrate[(A + C*Cos[c + d*x]^2)*(b*Sec[c + d*x])^(5/2),x]","\frac{2 b^2 \sqrt{b \sec (c+d x)} \left((A+3 C) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+A \tan (c+d x)\right)}{3 d}","\frac{2 b^2 (A+3 C) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \sec (c+d x)}}{3 d}+\frac{2 A b^2 \tan (c+d x) \sqrt{b \sec (c+d x)}}{3 d}",1,"(2*b^2*Sqrt[b*Sec[c + d*x]]*((A + 3*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + A*Tan[c + d*x]))/(3*d)","A",1
29,1,55,74,0.1690776,"\int \left(A+C \cos ^2(c+d x)\right) (b \sec (c+d x))^{3/2} \, dx","Integrate[(A + C*Cos[c + d*x]^2)*(b*Sec[c + d*x])^(3/2),x]","\frac{2 b \sqrt{b \sec (c+d x)} \left(A \sin (c+d x)-(A-C) \sqrt{\cos (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{d}","\frac{2 A b^2 \tan (c+d x)}{d \sqrt{b \sec (c+d x)}}-\frac{2 b^2 (A-C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d \sqrt{\cos (c+d x)} \sqrt{b \sec (c+d x)}}",1,"(2*b*Sqrt[b*Sec[c + d*x]]*(-((A - C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]) + A*Sin[c + d*x]))/d","A",1
30,1,58,75,0.137407,"\int \left(A+C \cos ^2(c+d x)\right) \sqrt{b \sec (c+d x)} \, dx","Integrate[(A + C*Cos[c + d*x]^2)*Sqrt[b*Sec[c + d*x]],x]","\frac{\sqrt{b \sec (c+d x)} \left(2 (3 A+C) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+C \sin (2 (c+d x))\right)}{3 d}","\frac{2 (3 A+C) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \sec (c+d x)}}{3 d}+\frac{2 b^2 C \tan (c+d x)}{3 d (b \sec (c+d x))^{3/2}}",1,"(Sqrt[b*Sec[c + d*x]]*(2*(3*A + C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + C*Sin[2*(c + d*x)]))/(3*d)","A",1
31,1,61,77,0.2517635,"\int \frac{A+C \cos ^2(c+d x)}{\sqrt{b \sec (c+d x)}} \, dx","Integrate[(A + C*Cos[c + d*x]^2)/Sqrt[b*Sec[c + d*x]],x]","\frac{\frac{4 (5 A+3 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{\sqrt{\cos (c+d x)}}+2 C \sin (2 (c+d x))}{10 d \sqrt{b \sec (c+d x)}}","\frac{2 (5 A+3 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d \sqrt{\cos (c+d x)} \sqrt{b \sec (c+d x)}}+\frac{2 b^2 C \tan (c+d x)}{5 d (b \sec (c+d x))^{5/2}}",1,"((4*(5*A + 3*C)*EllipticE[(c + d*x)/2, 2])/Sqrt[Cos[c + d*x]] + 2*C*Sin[2*(c + d*x)])/(10*d*Sqrt[b*Sec[c + d*x]])","A",1
32,1,79,115,0.5637876,"\int \frac{A+C \cos ^2(c+d x)}{(b \sec (c+d x))^{3/2}} \, dx","Integrate[(A + C*Cos[c + d*x]^2)/(b*Sec[c + d*x])^(3/2),x]","\frac{2 \sin (c+d x) (14 A+3 C \cos (2 (c+d x))+13 C)+\frac{4 (7 A+5 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{\sqrt{\cos (c+d x)}}}{42 b d \sqrt{b \sec (c+d x)}}","\frac{2 (7 A+5 C) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \sec (c+d x)}}{21 b^2 d}+\frac{2 (7 A+5 C) \sin (c+d x)}{21 b d \sqrt{b \sec (c+d x)}}+\frac{2 b^2 C \tan (c+d x)}{7 d (b \sec (c+d x))^{7/2}}",1,"((4*(7*A + 5*C)*EllipticF[(c + d*x)/2, 2])/Sqrt[Cos[c + d*x]] + 2*(14*A + 13*C + 3*C*Cos[2*(c + d*x)])*Sin[c + d*x])/(42*b*d*Sqrt[b*Sec[c + d*x]])","A",1
33,1,81,115,0.6774919,"\int \frac{A+C \cos ^2(c+d x)}{(b \sec (c+d x))^{5/2}} \, dx","Integrate[(A + C*Cos[c + d*x]^2)/(b*Sec[c + d*x])^(5/2),x]","\frac{4 \sin (2 (c+d x)) (18 A+5 C \cos (2 (c+d x))+19 C)+\frac{48 (9 A+7 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{\sqrt{\cos (c+d x)}}}{360 b^2 d \sqrt{b \sec (c+d x)}}","\frac{2 (9 A+7 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 b^2 d \sqrt{\cos (c+d x)} \sqrt{b \sec (c+d x)}}+\frac{2 (9 A+7 C) \sin (c+d x)}{45 b d (b \sec (c+d x))^{3/2}}+\frac{2 b^2 C \tan (c+d x)}{9 d (b \sec (c+d x))^{9/2}}",1,"((48*(9*A + 7*C)*EllipticE[(c + d*x)/2, 2])/Sqrt[Cos[c + d*x]] + 4*(18*A + 19*C + 5*C*Cos[2*(c + d*x)])*Sin[2*(c + d*x)])/(360*b^2*d*Sqrt[b*Sec[c + d*x]])","A",1
34,1,114,117,0.1949249,"\int (b \cos (c+d x))^m \left(A+C \cos ^2(c+d x)\right) \, dx","Integrate[(b*Cos[c + d*x])^m*(A + C*Cos[c + d*x]^2),x]","-\frac{\sqrt{\sin ^2(c+d x)} \cot (c+d x) (b \cos (c+d x))^m \left(A (m+3) \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};\cos ^2(c+d x)\right)+C (m+1) \cos ^2(c+d x) \, _2F_1\left(\frac{1}{2},\frac{m+3}{2};\frac{m+5}{2};\cos ^2(c+d x)\right)\right)}{d (m+1) (m+3)}","\frac{C \sin (c+d x) (b \cos (c+d x))^{m+1}}{b d (m+2)}-\frac{(A (m+2)+C (m+1)) \sin (c+d x) (b \cos (c+d x))^{m+1} \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};\cos ^2(c+d x)\right)}{b d (m+1) (m+2) \sqrt{\sin ^2(c+d x)}}",1,"-(((b*Cos[c + d*x])^m*Cot[c + d*x]*(A*(3 + m)*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, Cos[c + d*x]^2] + C*(1 + m)*Cos[c + d*x]^2*Hypergeometric2F1[1/2, (3 + m)/2, (5 + m)/2, Cos[c + d*x]^2])*Sqrt[Sin[c + d*x]^2])/(d*(1 + m)*(3 + m)))","A",1
35,1,113,31,0.2230079,"\int (b \cos (c+d x))^m \left(-\frac{C (1+m)}{2+m}+C \cos ^2(c+d x)\right) \, dx","Integrate[(b*Cos[c + d*x])^m*(-((C*(1 + m))/(2 + m)) + C*Cos[c + d*x]^2),x]","\frac{C \sqrt{\sin ^2(c+d x)} \cot (c+d x) (b \cos (c+d x))^m \left((m+3) \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};\cos ^2(c+d x)\right)-(m+2) \cos ^2(c+d x) \, _2F_1\left(\frac{1}{2},\frac{m+3}{2};\frac{m+5}{2};\cos ^2(c+d x)\right)\right)}{d (m+2) (m+3)}","\frac{C \sin (c+d x) (b \cos (c+d x))^{m+1}}{b d (m+2)}",1,"(C*(b*Cos[c + d*x])^m*Cot[c + d*x]*((3 + m)*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, Cos[c + d*x]^2] - (2 + m)*Cos[c + d*x]^2*Hypergeometric2F1[1/2, (3 + m)/2, (5 + m)/2, Cos[c + d*x]^2])*Sqrt[Sin[c + d*x]^2])/(d*(2 + m)*(3 + m))","C",1
36,1,119,32,0.2172991,"\int (b \cos (c+d x))^m \left(A-\frac{A (2+m) \cos ^2(c+d x)}{1+m}\right) \, dx","Integrate[(b*Cos[c + d*x])^m*(A - (A*(2 + m)*Cos[c + d*x]^2)/(1 + m)),x]","\frac{A \sin (c+d x) \cos (c+d x) (b \cos (c+d x))^m \left((m+2) \cos ^2(c+d x) \, _2F_1\left(\frac{1}{2},\frac{m+3}{2};\frac{m+5}{2};\cos ^2(c+d x)\right)-(m+3) \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};\cos ^2(c+d x)\right)\right)}{d (m+1) (m+3) \sqrt{\sin ^2(c+d x)}}","-\frac{A \sin (c+d x) (b \cos (c+d x))^{m+1}}{b d (m+1)}",1,"(A*Cos[c + d*x]*(b*Cos[c + d*x])^m*(-((3 + m)*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, Cos[c + d*x]^2]) + (2 + m)*Cos[c + d*x]^2*Hypergeometric2F1[1/2, (3 + m)/2, (5 + m)/2, Cos[c + d*x]^2])*Sin[c + d*x])/(d*(1 + m)*(3 + m)*Sqrt[Sin[c + d*x]^2])","C",1
37,1,88,112,0.371122,"\int \cos ^2(c+d x) \sqrt{b \cos (c+d x)} \left(A+C \cos ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^2*Sqrt[b*Cos[c + d*x]]*(A + C*Cos[c + d*x]^2),x]","\frac{\sqrt{b \cos (c+d x)} \left(24 (9 A+7 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+2 \sin (2 (c+d x)) \sqrt{\cos (c+d x)} (18 A+5 C \cos (2 (c+d x))+19 C)\right)}{180 d \sqrt{\cos (c+d x)}}","\frac{2 (9 A+7 C) \sin (c+d x) (b \cos (c+d x))^{3/2}}{45 b d}+\frac{2 (9 A+7 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \cos (c+d x)}}{15 d \sqrt{\cos (c+d x)}}+\frac{2 C \sin (c+d x) (b \cos (c+d x))^{7/2}}{9 b^3 d}",1,"(Sqrt[b*Cos[c + d*x]]*(24*(9*A + 7*C)*EllipticE[(c + d*x)/2, 2] + 2*Sqrt[Cos[c + d*x]]*(18*A + 19*C + 5*C*Cos[2*(c + d*x)])*Sin[2*(c + d*x)]))/(180*d*Sqrt[Cos[c + d*x]])","A",1
38,1,89,110,0.3624721,"\int \cos (c+d x) \sqrt{b \cos (c+d x)} \left(A+C \cos ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]*Sqrt[b*Cos[c + d*x]]*(A + C*Cos[c + d*x]^2),x]","\frac{(b \cos (c+d x))^{3/2} \left(4 (7 A+5 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+2 \sin (c+d x) \sqrt{\cos (c+d x)} (14 A+3 C \cos (2 (c+d x))+13 C)\right)}{42 b d \cos ^{\frac{3}{2}}(c+d x)}","\frac{2 (7 A+5 C) \sin (c+d x) \sqrt{b \cos (c+d x)}}{21 d}+\frac{2 b (7 A+5 C) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d \sqrt{b \cos (c+d x)}}+\frac{2 C \sin (c+d x) (b \cos (c+d x))^{5/2}}{7 b^2 d}",1,"((b*Cos[c + d*x])^(3/2)*(4*(7*A + 5*C)*EllipticF[(c + d*x)/2, 2] + 2*Sqrt[Cos[c + d*x]]*(14*A + 13*C + 3*C*Cos[2*(c + d*x)])*Sin[c + d*x]))/(42*b*d*Cos[c + d*x]^(3/2))","A",1
39,1,70,77,0.0872766,"\int \sqrt{b \cos (c+d x)} \left(A+C \cos ^2(c+d x)\right) \, dx","Integrate[Sqrt[b*Cos[c + d*x]]*(A + C*Cos[c + d*x]^2),x]","\frac{\sqrt{b \cos (c+d x)} \left(2 (5 A+3 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+C \sin (2 (c+d x)) \sqrt{\cos (c+d x)}\right)}{5 d \sqrt{\cos (c+d x)}}","\frac{2 (5 A+3 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \cos (c+d x)}}{5 d \sqrt{\cos (c+d x)}}+\frac{2 C \sin (c+d x) (b \cos (c+d x))^{3/2}}{5 b d}",1,"(Sqrt[b*Cos[c + d*x]]*(2*(5*A + 3*C)*EllipticE[(c + d*x)/2, 2] + C*Sqrt[Cos[c + d*x]]*Sin[2*(c + d*x)]))/(5*d*Sqrt[Cos[c + d*x]])","A",1
40,1,59,73,0.1436877,"\int \sqrt{b \cos (c+d x)} \left(A+C \cos ^2(c+d x)\right) \sec (c+d x) \, dx","Integrate[Sqrt[b*Cos[c + d*x]]*(A + C*Cos[c + d*x]^2)*Sec[c + d*x],x]","\frac{b \left(2 (3 A+C) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+C \sin (2 (c+d x))\right)}{3 d \sqrt{b \cos (c+d x)}}","\frac{2 b (3 A+C) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d \sqrt{b \cos (c+d x)}}+\frac{2 C \sin (c+d x) \sqrt{b \cos (c+d x)}}{3 d}",1,"(b*(2*(3*A + C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + C*Sin[2*(c + d*x)]))/(3*d*Sqrt[b*Cos[c + d*x]])","A",1
41,1,55,69,0.2230232,"\int \sqrt{b \cos (c+d x)} \left(A+C \cos ^2(c+d x)\right) \sec ^2(c+d x) \, dx","Integrate[Sqrt[b*Cos[c + d*x]]*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^2,x]","\frac{2 b \left(A \sin (c+d x)-(A-C) \sqrt{\cos (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{d \sqrt{b \cos (c+d x)}}","\frac{2 A b \sin (c+d x)}{d \sqrt{b \cos (c+d x)}}-\frac{2 (A-C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \cos (c+d x)}}{d \sqrt{\cos (c+d x)}}",1,"(2*b*(-((A - C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]) + A*Sin[c + d*x]))/(d*Sqrt[b*Cos[c + d*x]])","A",1
42,1,56,76,0.2144869,"\int \sqrt{b \cos (c+d x)} \left(A+C \cos ^2(c+d x)\right) \sec ^3(c+d x) \, dx","Integrate[Sqrt[b*Cos[c + d*x]]*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^3,x]","\frac{2 b \left((A+3 C) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+A \tan (c+d x)\right)}{3 d \sqrt{b \cos (c+d x)}}","\frac{2 A b^2 \sin (c+d x)}{3 d (b \cos (c+d x))^{3/2}}+\frac{2 b (A+3 C) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d \sqrt{b \cos (c+d x)}}",1,"(2*b*((A + 3*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + A*Tan[c + d*x]))/(3*d*Sqrt[b*Cos[c + d*x]])","A",1
43,1,84,110,0.2809306,"\int \sqrt{b \cos (c+d x)} \left(A+C \cos ^2(c+d x)\right) \sec ^4(c+d x) \, dx","Integrate[Sqrt[b*Cos[c + d*x]]*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^4,x]","-\frac{\sec ^2(c+d x) \sqrt{b \cos (c+d x)} \left(-(3 A+5 C) \sin (2 (c+d x))+2 (3 A+5 C) \cos ^{\frac{3}{2}}(c+d x) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)-2 A \tan (c+d x)\right)}{5 d}","\frac{2 A b^3 \sin (c+d x)}{5 d (b \cos (c+d x))^{5/2}}+\frac{2 b (3 A+5 C) \sin (c+d x)}{5 d \sqrt{b \cos (c+d x)}}-\frac{2 (3 A+5 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \cos (c+d x)}}{5 d \sqrt{\cos (c+d x)}}",1,"-1/5*(Sqrt[b*Cos[c + d*x]]*Sec[c + d*x]^2*(2*(3*A + 5*C)*Cos[c + d*x]^(3/2)*EllipticE[(c + d*x)/2, 2] - (3*A + 5*C)*Sin[2*(c + d*x)] - 2*A*Tan[c + d*x]))/d","A",1
44,1,83,113,0.5261515,"\int \sqrt{b \cos (c+d x)} \left(A+C \cos ^2(c+d x)\right) \sec ^5(c+d x) \, dx","Integrate[Sqrt[b*Cos[c + d*x]]*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^5,x]","\frac{\sec ^3(c+d x) \sqrt{b \cos (c+d x)} \left((5 A+7 C) \sin (2 (c+d x))+2 (5 A+7 C) \cos ^{\frac{5}{2}}(c+d x) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+6 A \tan (c+d x)\right)}{21 d}","\frac{2 A b^4 \sin (c+d x)}{7 d (b \cos (c+d x))^{7/2}}+\frac{2 b^2 (5 A+7 C) \sin (c+d x)}{21 d (b \cos (c+d x))^{3/2}}+\frac{2 b (5 A+7 C) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d \sqrt{b \cos (c+d x)}}",1,"(Sqrt[b*Cos[c + d*x]]*Sec[c + d*x]^3*(2*(5*A + 7*C)*Cos[c + d*x]^(5/2)*EllipticF[(c + d*x)/2, 2] + (5*A + 7*C)*Sin[2*(c + d*x)] + 6*A*Tan[c + d*x]))/(21*d)","A",1
45,1,91,110,0.2678289,"\int \cos (c+d x) (b \cos (c+d x))^{3/2} \left(A+C \cos ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]*(b*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2),x]","\frac{(b \cos (c+d x))^{5/2} \left(24 (9 A+7 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+2 \sin (2 (c+d x)) \sqrt{\cos (c+d x)} (18 A+5 C \cos (2 (c+d x))+19 C)\right)}{180 b d \cos ^{\frac{5}{2}}(c+d x)}","\frac{2 (9 A+7 C) \sin (c+d x) (b \cos (c+d x))^{3/2}}{45 d}+\frac{2 b (9 A+7 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \cos (c+d x)}}{15 d \sqrt{\cos (c+d x)}}+\frac{2 C \sin (c+d x) (b \cos (c+d x))^{7/2}}{9 b^2 d}",1,"((b*Cos[c + d*x])^(5/2)*(24*(9*A + 7*C)*EllipticE[(c + d*x)/2, 2] + 2*Sqrt[Cos[c + d*x]]*(18*A + 19*C + 5*C*Cos[2*(c + d*x)])*Sin[2*(c + d*x)]))/(180*b*d*Cos[c + d*x]^(5/2))","A",1
46,1,86,113,0.0789379,"\int (b \cos (c+d x))^{3/2} \left(A+C \cos ^2(c+d x)\right) \, dx","Integrate[(b*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2),x]","\frac{(b \cos (c+d x))^{3/2} \left(4 (7 A+5 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+2 \sin (c+d x) \sqrt{\cos (c+d x)} (14 A+3 C \cos (2 (c+d x))+13 C)\right)}{42 d \cos ^{\frac{3}{2}}(c+d x)}","\frac{2 b^2 (7 A+5 C) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d \sqrt{b \cos (c+d x)}}+\frac{2 b (7 A+5 C) \sin (c+d x) \sqrt{b \cos (c+d x)}}{21 d}+\frac{2 C \sin (c+d x) (b \cos (c+d x))^{5/2}}{7 b d}",1,"((b*Cos[c + d*x])^(3/2)*(4*(7*A + 5*C)*EllipticF[(c + d*x)/2, 2] + 2*Sqrt[Cos[c + d*x]]*(14*A + 13*C + 3*C*Cos[2*(c + d*x)])*Sin[c + d*x]))/(42*d*Cos[c + d*x]^(3/2))","A",1
47,1,71,75,0.0712084,"\int (b \cos (c+d x))^{3/2} \left(A+C \cos ^2(c+d x)\right) \sec (c+d x) \, dx","Integrate[(b*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x],x]","\frac{b \sqrt{b \cos (c+d x)} \left(2 (5 A+3 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+C \sin (2 (c+d x)) \sqrt{\cos (c+d x)}\right)}{5 d \sqrt{\cos (c+d x)}}","\frac{2 b (5 A+3 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \cos (c+d x)}}{5 d \sqrt{\cos (c+d x)}}+\frac{2 C \sin (c+d x) (b \cos (c+d x))^{3/2}}{5 d}",1,"(b*Sqrt[b*Cos[c + d*x]]*(2*(5*A + 3*C)*EllipticE[(c + d*x)/2, 2] + C*Sqrt[Cos[c + d*x]]*Sin[2*(c + d*x)]))/(5*d*Sqrt[Cos[c + d*x]])","A",1
48,1,61,76,0.090499,"\int (b \cos (c+d x))^{3/2} \left(A+C \cos ^2(c+d x)\right) \sec ^2(c+d x) \, dx","Integrate[(b*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^2,x]","\frac{b^2 \left(2 (3 A+C) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+C \sin (2 (c+d x))\right)}{3 d \sqrt{b \cos (c+d x)}}","\frac{2 b^2 (3 A+C) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d \sqrt{b \cos (c+d x)}}+\frac{2 b C \sin (c+d x) \sqrt{b \cos (c+d x)}}{3 d}",1,"(b^2*(2*(3*A + C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + C*Sin[2*(c + d*x)]))/(3*d*Sqrt[b*Cos[c + d*x]])","A",1
49,1,57,72,0.1489541,"\int (b \cos (c+d x))^{3/2} \left(A+C \cos ^2(c+d x)\right) \sec ^3(c+d x) \, dx","Integrate[(b*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^3,x]","\frac{2 b^2 \left(A \sin (c+d x)-(A-C) \sqrt{\cos (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{d \sqrt{b \cos (c+d x)}}","\frac{2 A b^2 \sin (c+d x)}{d \sqrt{b \cos (c+d x)}}-\frac{2 b (A-C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \cos (c+d x)}}{d \sqrt{\cos (c+d x)}}",1,"(2*b^2*(-((A - C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]) + A*Sin[c + d*x]))/(d*Sqrt[b*Cos[c + d*x]])","A",1
50,1,58,78,0.1905042,"\int (b \cos (c+d x))^{3/2} \left(A+C \cos ^2(c+d x)\right) \sec ^4(c+d x) \, dx","Integrate[(b*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^4,x]","\frac{2 b^2 \left((A+3 C) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+A \tan (c+d x)\right)}{3 d \sqrt{b \cos (c+d x)}}","\frac{2 A b^3 \sin (c+d x)}{3 d (b \cos (c+d x))^{3/2}}+\frac{2 b^2 (A+3 C) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d \sqrt{b \cos (c+d x)}}",1,"(2*b^2*((A + 3*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + A*Tan[c + d*x]))/(3*d*Sqrt[b*Cos[c + d*x]])","A",1
51,1,84,113,0.2081131,"\int (b \cos (c+d x))^{3/2} \left(A+C \cos ^2(c+d x)\right) \sec ^5(c+d x) \, dx","Integrate[(b*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^5,x]","-\frac{\sec ^3(c+d x) (b \cos (c+d x))^{3/2} \left(-(3 A+5 C) \sin (2 (c+d x))+2 (3 A+5 C) \cos ^{\frac{3}{2}}(c+d x) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)-2 A \tan (c+d x)\right)}{5 d}","\frac{2 A b^4 \sin (c+d x)}{5 d (b \cos (c+d x))^{5/2}}+\frac{2 b^2 (3 A+5 C) \sin (c+d x)}{5 d \sqrt{b \cos (c+d x)}}-\frac{2 b (3 A+5 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \cos (c+d x)}}{5 d \sqrt{\cos (c+d x)}}",1,"-1/5*((b*Cos[c + d*x])^(3/2)*Sec[c + d*x]^3*(2*(3*A + 5*C)*Cos[c + d*x]^(3/2)*EllipticE[(c + d*x)/2, 2] - (3*A + 5*C)*Sin[2*(c + d*x)] - 2*A*Tan[c + d*x]))/d","A",1
52,1,83,115,0.4082858,"\int (b \cos (c+d x))^{3/2} \left(A+C \cos ^2(c+d x)\right) \sec ^6(c+d x) \, dx","Integrate[(b*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^6,x]","\frac{\sec ^4(c+d x) (b \cos (c+d x))^{3/2} \left((5 A+7 C) \sin (2 (c+d x))+2 (5 A+7 C) \cos ^{\frac{5}{2}}(c+d x) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+6 A \tan (c+d x)\right)}{21 d}","\frac{2 A b^5 \sin (c+d x)}{7 d (b \cos (c+d x))^{7/2}}+\frac{2 b^3 (5 A+7 C) \sin (c+d x)}{21 d (b \cos (c+d x))^{3/2}}+\frac{2 b^2 (5 A+7 C) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d \sqrt{b \cos (c+d x)}}",1,"((b*Cos[c + d*x])^(3/2)*Sec[c + d*x]^4*(2*(5*A + 7*C)*Cos[c + d*x]^(5/2)*EllipticF[(c + d*x)/2, 2] + (5*A + 7*C)*Sin[2*(c + d*x)] + 6*A*Tan[c + d*x]))/(21*d)","A",1
53,1,88,113,0.0713772,"\int (b \cos (c+d x))^{5/2} \left(A+C \cos ^2(c+d x)\right) \, dx","Integrate[(b*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2),x]","\frac{(b \cos (c+d x))^{5/2} \left(24 (9 A+7 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+2 \sin (2 (c+d x)) \sqrt{\cos (c+d x)} (18 A+5 C \cos (2 (c+d x))+19 C)\right)}{180 d \cos ^{\frac{5}{2}}(c+d x)}","\frac{2 b^2 (9 A+7 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \cos (c+d x)}}{15 d \sqrt{\cos (c+d x)}}+\frac{2 b (9 A+7 C) \sin (c+d x) (b \cos (c+d x))^{3/2}}{45 d}+\frac{2 C \sin (c+d x) (b \cos (c+d x))^{7/2}}{9 b d}",1,"((b*Cos[c + d*x])^(5/2)*(24*(9*A + 7*C)*EllipticE[(c + d*x)/2, 2] + 2*Sqrt[Cos[c + d*x]]*(18*A + 19*C + 5*C*Cos[2*(c + d*x)])*Sin[2*(c + d*x)]))/(180*d*Cos[c + d*x]^(5/2))","A",1
54,1,87,112,0.0867974,"\int (b \cos (c+d x))^{5/2} \left(A+C \cos ^2(c+d x)\right) \sec (c+d x) \, dx","Integrate[(b*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x],x]","\frac{b (b \cos (c+d x))^{3/2} \left(4 (7 A+5 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+2 \sin (c+d x) \sqrt{\cos (c+d x)} (14 A+3 C \cos (2 (c+d x))+13 C)\right)}{42 d \cos ^{\frac{3}{2}}(c+d x)}","\frac{2 b^3 (7 A+5 C) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d \sqrt{b \cos (c+d x)}}+\frac{2 b^2 (7 A+5 C) \sin (c+d x) \sqrt{b \cos (c+d x)}}{21 d}+\frac{2 C \sin (c+d x) (b \cos (c+d x))^{5/2}}{7 d}",1,"(b*(b*Cos[c + d*x])^(3/2)*(4*(7*A + 5*C)*EllipticF[(c + d*x)/2, 2] + 2*Sqrt[Cos[c + d*x]]*(14*A + 13*C + 3*C*Cos[2*(c + d*x)])*Sin[c + d*x]))/(42*d*Cos[c + d*x]^(3/2))","A",1
55,1,73,78,0.0740349,"\int (b \cos (c+d x))^{5/2} \left(A+C \cos ^2(c+d x)\right) \sec ^2(c+d x) \, dx","Integrate[(b*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^2,x]","\frac{b^2 \sqrt{b \cos (c+d x)} \left(2 (5 A+3 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+C \sin (2 (c+d x)) \sqrt{\cos (c+d x)}\right)}{5 d \sqrt{\cos (c+d x)}}","\frac{2 b^2 (5 A+3 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \cos (c+d x)}}{5 d \sqrt{\cos (c+d x)}}+\frac{2 b C \sin (c+d x) (b \cos (c+d x))^{3/2}}{5 d}",1,"(b^2*Sqrt[b*Cos[c + d*x]]*(2*(5*A + 3*C)*EllipticE[(c + d*x)/2, 2] + C*Sqrt[Cos[c + d*x]]*Sin[2*(c + d*x)]))/(5*d*Sqrt[Cos[c + d*x]])","A",1
56,1,65,78,0.1646949,"\int (b \cos (c+d x))^{5/2} \left(A+C \cos ^2(c+d x)\right) \sec ^3(c+d x) \, dx","Integrate[(b*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^3,x]","\frac{2 (b \cos (c+d x))^{5/2} \left((3 A+C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+C \sin (c+d x) \sqrt{\cos (c+d x)}\right)}{3 d \cos ^{\frac{5}{2}}(c+d x)}","\frac{2 b^3 (3 A+C) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d \sqrt{b \cos (c+d x)}}+\frac{2 b^2 C \sin (c+d x) \sqrt{b \cos (c+d x)}}{3 d}",1,"(2*(b*Cos[c + d*x])^(5/2)*((3*A + C)*EllipticF[(c + d*x)/2, 2] + C*Sqrt[Cos[c + d*x]]*Sin[c + d*x]))/(3*d*Cos[c + d*x]^(5/2))","A",1
57,1,57,74,0.1450782,"\int (b \cos (c+d x))^{5/2} \left(A+C \cos ^2(c+d x)\right) \sec ^4(c+d x) \, dx","Integrate[(b*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^4,x]","\frac{2 b^3 \left(A \sin (c+d x)-(A-C) \sqrt{\cos (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{d \sqrt{b \cos (c+d x)}}","\frac{2 A b^3 \sin (c+d x)}{d \sqrt{b \cos (c+d x)}}-\frac{2 b^2 (A-C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \cos (c+d x)}}{d \sqrt{\cos (c+d x)}}",1,"(2*b^3*(-((A - C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]) + A*Sin[c + d*x]))/(d*Sqrt[b*Cos[c + d*x]])","A",1
58,1,58,78,0.1873384,"\int (b \cos (c+d x))^{5/2} \left(A+C \cos ^2(c+d x)\right) \sec ^5(c+d x) \, dx","Integrate[(b*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^5,x]","\frac{2 b^3 \left((A+3 C) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+A \tan (c+d x)\right)}{3 d \sqrt{b \cos (c+d x)}}","\frac{2 A b^4 \sin (c+d x)}{3 d (b \cos (c+d x))^{3/2}}+\frac{2 b^3 (A+3 C) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d \sqrt{b \cos (c+d x)}}",1,"(2*b^3*((A + 3*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + A*Tan[c + d*x]))/(3*d*Sqrt[b*Cos[c + d*x]])","A",1
59,1,80,115,0.2203659,"\int (b \cos (c+d x))^{5/2} \left(A+C \cos ^2(c+d x)\right) \sec ^6(c+d x) \, dx","Integrate[(b*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^6,x]","-\frac{2 b^4 \left(-\frac{1}{2} (3 A+5 C) \sin (2 (c+d x))+(3 A+5 C) \cos ^{\frac{3}{2}}(c+d x) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)-A \tan (c+d x)\right)}{5 d (b \cos (c+d x))^{3/2}}","\frac{2 A b^5 \sin (c+d x)}{5 d (b \cos (c+d x))^{5/2}}+\frac{2 b^3 (3 A+5 C) \sin (c+d x)}{5 d \sqrt{b \cos (c+d x)}}-\frac{2 b^2 (3 A+5 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \cos (c+d x)}}{5 d \sqrt{\cos (c+d x)}}",1,"(-2*b^4*((3*A + 5*C)*Cos[c + d*x]^(3/2)*EllipticE[(c + d*x)/2, 2] - ((3*A + 5*C)*Sin[2*(c + d*x)])/2 - A*Tan[c + d*x]))/(5*d*(b*Cos[c + d*x])^(3/2))","A",1
60,1,83,115,0.4161705,"\int (b \cos (c+d x))^{5/2} \left(A+C \cos ^2(c+d x)\right) \sec ^7(c+d x) \, dx","Integrate[(b*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^7,x]","\frac{\sec ^5(c+d x) (b \cos (c+d x))^{5/2} \left((5 A+7 C) \sin (2 (c+d x))+2 (5 A+7 C) \cos ^{\frac{5}{2}}(c+d x) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+6 A \tan (c+d x)\right)}{21 d}","\frac{2 A b^6 \sin (c+d x)}{7 d (b \cos (c+d x))^{7/2}}+\frac{2 b^4 (5 A+7 C) \sin (c+d x)}{21 d (b \cos (c+d x))^{3/2}}+\frac{2 b^3 (5 A+7 C) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d \sqrt{b \cos (c+d x)}}",1,"((b*Cos[c + d*x])^(5/2)*Sec[c + d*x]^5*(2*(5*A + 7*C)*Cos[c + d*x]^(5/2)*EllipticF[(c + d*x)/2, 2] + (5*A + 7*C)*Sin[2*(c + d*x)] + 6*A*Tan[c + d*x]))/(21*d)","A",1
61,1,94,147,0.4078345,"\int \frac{\cos ^4(c+d x) \left(A+C \cos ^2(c+d x)\right)}{\sqrt{b \cos (c+d x)}} \, dx","Integrate[(Cos[c + d*x]^4*(A + C*Cos[c + d*x]^2))/Sqrt[b*Cos[c + d*x]],x]","\frac{\sin (2 (c+d x)) (12 (11 A+16 C) \cos (2 (c+d x))+572 A+21 C \cos (4 (c+d x))+531 C)+80 (11 A+9 C) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{1848 d \sqrt{b \cos (c+d x)}}","\frac{2 (11 A+9 C) \sin (c+d x) (b \cos (c+d x))^{5/2}}{77 b^3 d}+\frac{10 (11 A+9 C) \sin (c+d x) \sqrt{b \cos (c+d x)}}{231 b d}+\frac{10 (11 A+9 C) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{231 d \sqrt{b \cos (c+d x)}}+\frac{2 C \sin (c+d x) (b \cos (c+d x))^{9/2}}{11 b^5 d}",1,"(80*(11*A + 9*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + (572*A + 531*C + 12*(11*A + 16*C)*Cos[2*(c + d*x)] + 21*C*Cos[4*(c + d*x)])*Sin[2*(c + d*x)])/(1848*d*Sqrt[b*Cos[c + d*x]])","A",1
62,1,83,115,0.4361801,"\int \frac{\cos ^3(c+d x) \left(A+C \cos ^2(c+d x)\right)}{\sqrt{b \cos (c+d x)}} \, dx","Integrate[(Cos[c + d*x]^3*(A + C*Cos[c + d*x]^2))/Sqrt[b*Cos[c + d*x]],x]","\frac{\sin (c+d x) \cos ^2(c+d x) (18 A+5 C \cos (2 (c+d x))+19 C)+6 (9 A+7 C) \sqrt{\cos (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{45 d \sqrt{b \cos (c+d x)}}","\frac{2 (9 A+7 C) \sin (c+d x) (b \cos (c+d x))^{3/2}}{45 b^2 d}+\frac{2 (9 A+7 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \cos (c+d x)}}{15 b d \sqrt{\cos (c+d x)}}+\frac{2 C \sin (c+d x) (b \cos (c+d x))^{7/2}}{9 b^4 d}",1,"(6*(9*A + 7*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2] + Cos[c + d*x]^2*(18*A + 19*C + 5*C*Cos[2*(c + d*x)])*Sin[c + d*x])/(45*d*Sqrt[b*Cos[c + d*x]])","A",1
63,1,77,112,0.2561825,"\int \frac{\cos ^2(c+d x) \left(A+C \cos ^2(c+d x)\right)}{\sqrt{b \cos (c+d x)}} \, dx","Integrate[(Cos[c + d*x]^2*(A + C*Cos[c + d*x]^2))/Sqrt[b*Cos[c + d*x]],x]","\frac{\sin (2 (c+d x)) (14 A+3 C \cos (2 (c+d x))+13 C)+4 (7 A+5 C) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{42 d \sqrt{b \cos (c+d x)}}","\frac{2 (7 A+5 C) \sin (c+d x) \sqrt{b \cos (c+d x)}}{21 b d}+\frac{2 (7 A+5 C) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d \sqrt{b \cos (c+d x)}}+\frac{2 C \sin (c+d x) (b \cos (c+d x))^{5/2}}{7 b^3 d}",1,"(4*(7*A + 5*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + (14*A + 13*C + 3*C*Cos[2*(c + d*x)])*Sin[2*(c + d*x)])/(42*d*Sqrt[b*Cos[c + d*x]])","A",1
64,1,73,80,0.0751245,"\int \frac{\cos (c+d x) \left(A+C \cos ^2(c+d x)\right)}{\sqrt{b \cos (c+d x)}} \, dx","Integrate[(Cos[c + d*x]*(A + C*Cos[c + d*x]^2))/Sqrt[b*Cos[c + d*x]],x]","\frac{\sqrt{b \cos (c+d x)} \left(2 (5 A+3 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+C \sin (2 (c+d x)) \sqrt{\cos (c+d x)}\right)}{5 b d \sqrt{\cos (c+d x)}}","\frac{2 (5 A+3 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \cos (c+d x)}}{5 b d \sqrt{\cos (c+d x)}}+\frac{2 C \sin (c+d x) (b \cos (c+d x))^{3/2}}{5 b^2 d}",1,"(Sqrt[b*Cos[c + d*x]]*(2*(5*A + 3*C)*EllipticE[(c + d*x)/2, 2] + C*Sqrt[Cos[c + d*x]]*Sin[2*(c + d*x)]))/(5*b*d*Sqrt[Cos[c + d*x]])","A",1
65,1,58,75,0.1324879,"\int \frac{A+C \cos ^2(c+d x)}{\sqrt{b \cos (c+d x)}} \, dx","Integrate[(A + C*Cos[c + d*x]^2)/Sqrt[b*Cos[c + d*x]],x]","\frac{2 (3 A+C) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+C \sin (2 (c+d x))}{3 d \sqrt{b \cos (c+d x)}}","\frac{2 (3 A+C) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d \sqrt{b \cos (c+d x)}}+\frac{2 C \sin (c+d x) \sqrt{b \cos (c+d x)}}{3 b d}",1,"(2*(3*A + C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + C*Sin[2*(c + d*x)])/(3*d*Sqrt[b*Cos[c + d*x]])","A",1
66,1,200,71,1.3718658,"\int \frac{\left(A+C \cos ^2(c+d x)\right) \sec (c+d x)}{\sqrt{b \cos (c+d x)}} \, dx","Integrate[((A + C*Cos[c + d*x]^2)*Sec[c + d*x])/Sqrt[b*Cos[c + d*x]],x]","-\frac{\csc (c) \left(3 (A-C) (\cos (d x)-i \sin (d x)) \sqrt{i \sin (2 (c+d x))+\cos (2 (c+d x))+1} \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right)+(A-C) (\cos (d x)+i \sin (d x)) \sqrt{i \sin (2 (c+d x))+\cos (2 (c+d x))+1} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right)-6 A \cos (d x)+3 C \cos (2 c+d x)+3 C \cos (d x)\right)}{3 d \sqrt{b \cos (c+d x)}}","\frac{2 A \sin (c+d x)}{d \sqrt{b \cos (c+d x)}}-\frac{2 (A-C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \cos (c+d x)}}{b d \sqrt{\cos (c+d x)}}",1,"-1/3*(Csc[c]*(-6*A*Cos[d*x] + 3*C*Cos[d*x] + 3*C*Cos[2*c + d*x] + 3*(A - C)*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*(Cos[d*x] - I*Sin[d*x])*Sqrt[1 + Cos[2*(c + d*x)] + I*Sin[2*(c + d*x)]] + (A - C)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*(Cos[d*x] + I*Sin[d*x])*Sqrt[1 + Cos[2*(c + d*x)] + I*Sin[2*(c + d*x)]]))/(d*Sqrt[b*Cos[c + d*x]])","C",1
67,1,141,73,1.5482699,"\int \frac{\left(A+C \cos ^2(c+d x)\right) \sec ^2(c+d x)}{\sqrt{b \cos (c+d x)}} \, dx","Integrate[((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^2)/Sqrt[b*Cos[c + d*x]],x]","-\frac{4 b \left(A+C \cos ^2(c+d x)\right) \left((A+3 C) \csc (c) \cos ^2(c+d x) \sqrt{\cos ^2\left(d x-\tan ^{-1}(\cot (c))\right)} \sec \left(d x-\tan ^{-1}(\cot (c))\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right)-A \sqrt{\csc ^2(c)} \sin (c+d x)\right)}{3 d \sqrt{\csc ^2(c)} (b \cos (c+d x))^{3/2} (2 A+C \cos (2 (c+d x))+C)}","\frac{2 (A+3 C) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d \sqrt{b \cos (c+d x)}}+\frac{2 A b \sin (c+d x)}{3 d (b \cos (c+d x))^{3/2}}",1,"(-4*b*(A + C*Cos[c + d*x]^2)*((A + 3*C)*Cos[c + d*x]^2*Sqrt[Cos[d*x - ArcTan[Cot[c]]]^2]*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[d*x - ArcTan[Cot[c]]] - A*Sqrt[Csc[c]^2]*Sin[c + d*x]))/(3*d*(b*Cos[c + d*x])^(3/2)*(2*A + C + C*Cos[2*(c + d*x)])*Sqrt[Csc[c]^2])","C",0
68,1,522,112,6.3207378,"\int \frac{\left(A+C \cos ^2(c+d x)\right) \sec ^3(c+d x)}{\sqrt{b \cos (c+d x)}} \, dx","Integrate[((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^3)/Sqrt[b*Cos[c + d*x]],x]","b \left(\frac{\cos ^4(c+d x) \left(A \sec ^2(c+d x)+C\right) \left(\frac{4 \sec (c) \sec (c+d x) (3 A \sin (d x)+5 C \sin (d x))}{5 d}+\frac{4 (3 A+5 C) \csc (c) \sec (c)}{5 d}+\frac{4 A \sec (c) \sin (d x) \sec ^3(c+d x)}{5 d}+\frac{4 A \tan (c) \sec ^2(c+d x)}{5 d}\right)}{(b \cos (c+d x))^{3/2} (2 A+C \cos (2 c+2 d x)+C)}-\frac{i (3 A+5 C) \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \cos ^{\frac{7}{2}}(c+d x) \left(A \sec ^2(c+d x)+C\right) \left(\frac{2 e^{2 i d x} \sqrt{e^{-i d x} \left(2 i \sin (c) \left(-1+e^{2 i d x}\right)+2 \cos (c) \left(1+e^{2 i d x}\right)\right)} \sqrt{i \sin (2 c) e^{2 i d x}+\cos (2 c) e^{2 i d x}+1} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right)}{3 i d \cos (c) \left(1+e^{2 i d x}\right)-3 d \sin (c) \left(-1+e^{2 i d x}\right)}-\frac{2 \sqrt{e^{-i d x} \left(2 i \sin (c) \left(-1+e^{2 i d x}\right)+2 \cos (c) \left(1+e^{2 i d x}\right)\right)} \sqrt{i \sin (2 c) e^{2 i d x}+\cos (2 c) e^{2 i d x}+1} \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right)}{d \sin (c) \left(-1+e^{2 i d x}\right)-i d \cos (c) \left(1+e^{2 i d x}\right)}\right)}{10 (b \cos (c+d x))^{3/2} (2 A+C \cos (2 c+2 d x)+C)}\right)","\frac{2 A b^2 \sin (c+d x)}{5 d (b \cos (c+d x))^{5/2}}+\frac{2 (3 A+5 C) \sin (c+d x)}{5 d \sqrt{b \cos (c+d x)}}-\frac{2 (3 A+5 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \cos (c+d x)}}{5 b d \sqrt{\cos (c+d x)}}",1,"b*(((-1/10*I)*(3*A + 5*C)*Cos[c + d*x]^(7/2)*Csc[c/2]*Sec[c/2]*(C + A*Sec[c + d*x]^2)*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((b*Cos[c + d*x])^(3/2)*(2*A + C + C*Cos[2*c + 2*d*x])) + (Cos[c + d*x]^4*(C + A*Sec[c + d*x]^2)*((4*(3*A + 5*C)*Csc[c]*Sec[c])/(5*d) + (4*A*Sec[c]*Sec[c + d*x]^3*Sin[d*x])/(5*d) + (4*Sec[c]*Sec[c + d*x]*(3*A*Sin[d*x] + 5*C*Sin[d*x]))/(5*d) + (4*A*Sec[c + d*x]^2*Tan[c])/(5*d)))/((b*Cos[c + d*x])^(3/2)*(2*A + C + C*Cos[2*c + 2*d*x])))","C",1
69,1,74,110,0.3803289,"\int \frac{\left(A+C \cos ^2(c+d x)\right) \sec ^4(c+d x)}{\sqrt{b \cos (c+d x)}} \, dx","Integrate[((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^4)/Sqrt[b*Cos[c + d*x]],x]","\frac{2 \left((5 A+7 C) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+\tan (c+d x) \left(3 A \sec ^2(c+d x)+5 A+7 C\right)\right)}{21 d \sqrt{b \cos (c+d x)}}","\frac{2 A b^3 \sin (c+d x)}{7 d (b \cos (c+d x))^{7/2}}+\frac{2 b (5 A+7 C) \sin (c+d x)}{21 d (b \cos (c+d x))^{3/2}}+\frac{2 (5 A+7 C) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d \sqrt{b \cos (c+d x)}}",1,"(2*((5*A + 7*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + (5*A + 7*C + 3*A*Sec[c + d*x]^2)*Tan[c + d*x]))/(21*d*Sqrt[b*Cos[c + d*x]])","A",1
70,1,97,147,0.8305489,"\int \frac{\left(A+C \cos ^2(c+d x)\right) \sec ^5(c+d x)}{\sqrt{b \cos (c+d x)}} \, dx","Integrate[((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^5)/Sqrt[b*Cos[c + d*x]],x]","\frac{6 (7 A+9 C) \sin (c+d x)-6 (7 A+9 C) \sqrt{\cos (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+2 \tan (c+d x) \sec (c+d x) \left(5 A \sec ^2(c+d x)+7 A+9 C\right)}{45 d \sqrt{b \cos (c+d x)}}","\frac{2 A b^4 \sin (c+d x)}{9 d (b \cos (c+d x))^{9/2}}+\frac{2 b^2 (7 A+9 C) \sin (c+d x)}{45 d (b \cos (c+d x))^{5/2}}+\frac{2 (7 A+9 C) \sin (c+d x)}{15 d \sqrt{b \cos (c+d x)}}-\frac{2 (7 A+9 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \cos (c+d x)}}{15 b d \sqrt{\cos (c+d x)}}",1,"(-6*(7*A + 9*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2] + 6*(7*A + 9*C)*Sin[c + d*x] + 2*Sec[c + d*x]*(7*A + 9*C + 5*A*Sec[c + d*x]^2)*Tan[c + d*x])/(45*d*Sqrt[b*Cos[c + d*x]])","A",1
71,1,86,115,0.4233323,"\int \frac{\cos ^4(c+d x) \left(A+C \cos ^2(c+d x)\right)}{(b \cos (c+d x))^{3/2}} \, dx","Integrate[(Cos[c + d*x]^4*(A + C*Cos[c + d*x]^2))/(b*Cos[c + d*x])^(3/2),x]","\frac{\sin (c+d x) \cos ^2(c+d x) (18 A+5 C \cos (2 (c+d x))+19 C)+6 (9 A+7 C) \sqrt{\cos (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{45 b d \sqrt{b \cos (c+d x)}}","\frac{2 (9 A+7 C) \sin (c+d x) (b \cos (c+d x))^{3/2}}{45 b^3 d}+\frac{2 (9 A+7 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \cos (c+d x)}}{15 b^2 d \sqrt{\cos (c+d x)}}+\frac{2 C \sin (c+d x) (b \cos (c+d x))^{7/2}}{9 b^5 d}",1,"(6*(9*A + 7*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2] + Cos[c + d*x]^2*(18*A + 19*C + 5*C*Cos[2*(c + d*x)])*Sin[c + d*x])/(45*b*d*Sqrt[b*Cos[c + d*x]])","A",1
72,1,80,115,0.2686765,"\int \frac{\cos ^3(c+d x) \left(A+C \cos ^2(c+d x)\right)}{(b \cos (c+d x))^{3/2}} \, dx","Integrate[(Cos[c + d*x]^3*(A + C*Cos[c + d*x]^2))/(b*Cos[c + d*x])^(3/2),x]","\frac{\sin (2 (c+d x)) (14 A+3 C \cos (2 (c+d x))+13 C)+4 (7 A+5 C) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{42 b d \sqrt{b \cos (c+d x)}}","\frac{2 (7 A+5 C) \sin (c+d x) \sqrt{b \cos (c+d x)}}{21 b^2 d}+\frac{2 (7 A+5 C) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 b d \sqrt{b \cos (c+d x)}}+\frac{2 C \sin (c+d x) (b \cos (c+d x))^{5/2}}{7 b^4 d}",1,"(4*(7*A + 5*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + (14*A + 13*C + 3*C*Cos[2*(c + d*x)])*Sin[2*(c + d*x)])/(42*b*d*Sqrt[b*Cos[c + d*x]])","A",1
73,1,69,80,0.1701876,"\int \frac{\cos ^2(c+d x) \left(A+C \cos ^2(c+d x)\right)}{(b \cos (c+d x))^{3/2}} \, dx","Integrate[(Cos[c + d*x]^2*(A + C*Cos[c + d*x]^2))/(b*Cos[c + d*x])^(3/2),x]","\frac{2 (5 A+3 C) \sqrt{\cos (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+C \sin (2 (c+d x)) \cos (c+d x)}{5 b d \sqrt{b \cos (c+d x)}}","\frac{2 (5 A+3 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \cos (c+d x)}}{5 b^2 d \sqrt{\cos (c+d x)}}+\frac{2 C \sin (c+d x) (b \cos (c+d x))^{3/2}}{5 b^3 d}",1,"(2*(5*A + 3*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2] + C*Cos[c + d*x]*Sin[2*(c + d*x)])/(5*b*d*Sqrt[b*Cos[c + d*x]])","A",1
74,1,61,78,0.1405295,"\int \frac{\cos (c+d x) \left(A+C \cos ^2(c+d x)\right)}{(b \cos (c+d x))^{3/2}} \, dx","Integrate[(Cos[c + d*x]*(A + C*Cos[c + d*x]^2))/(b*Cos[c + d*x])^(3/2),x]","\frac{2 (3 A+C) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+C \sin (2 (c+d x))}{3 b d \sqrt{b \cos (c+d x)}}","\frac{2 (3 A+C) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 b d \sqrt{b \cos (c+d x)}}+\frac{2 C \sin (c+d x) \sqrt{b \cos (c+d x)}}{3 b^2 d}",1,"(2*(3*A + C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + C*Sin[2*(c + d*x)])/(3*b*d*Sqrt[b*Cos[c + d*x]])","A",1
75,1,57,74,0.1430661,"\int \frac{A+C \cos ^2(c+d x)}{(b \cos (c+d x))^{3/2}} \, dx","Integrate[(A + C*Cos[c + d*x]^2)/(b*Cos[c + d*x])^(3/2),x]","\frac{2 A \sin (c+d x)-2 (A-C) \sqrt{\cos (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b d \sqrt{b \cos (c+d x)}}","\frac{2 A \sin (c+d x)}{b d \sqrt{b \cos (c+d x)}}-\frac{2 (A-C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \cos (c+d x)}}{b^2 d \sqrt{\cos (c+d x)}}",1,"(-2*(A - C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2] + 2*A*Sin[c + d*x])/(b*d*Sqrt[b*Cos[c + d*x]])","A",1
76,1,140,75,1.3654004,"\int \frac{\left(A+C \cos ^2(c+d x)\right) \sec (c+d x)}{(b \cos (c+d x))^{3/2}} \, dx","Integrate[((A + C*Cos[c + d*x]^2)*Sec[c + d*x])/(b*Cos[c + d*x])^(3/2),x]","-\frac{4 \left(A+C \cos ^2(c+d x)\right) \left((A+3 C) \csc (c) \cos ^2(c+d x) \sqrt{\cos ^2\left(d x-\tan ^{-1}(\cot (c))\right)} \sec \left(d x-\tan ^{-1}(\cot (c))\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right)-A \sqrt{\csc ^2(c)} \sin (c+d x)\right)}{3 d \sqrt{\csc ^2(c)} (b \cos (c+d x))^{3/2} (2 A+C \cos (2 (c+d x))+C)}","\frac{2 (A+3 C) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 b d \sqrt{b \cos (c+d x)}}+\frac{2 A \sin (c+d x)}{3 d (b \cos (c+d x))^{3/2}}",1,"(-4*(A + C*Cos[c + d*x]^2)*((A + 3*C)*Cos[c + d*x]^2*Sqrt[Cos[d*x - ArcTan[Cot[c]]]^2]*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[d*x - ArcTan[Cot[c]]] - A*Sqrt[Csc[c]^2]*Sin[c + d*x]))/(3*d*(b*Cos[c + d*x])^(3/2)*(2*A + C + C*Cos[2*(c + d*x)])*Sqrt[Csc[c]^2])","C",0
77,1,81,113,0.3062079,"\int \frac{\left(A+C \cos ^2(c+d x)\right) \sec ^2(c+d x)}{(b \cos (c+d x))^{3/2}} \, dx","Integrate[((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^2)/(b*Cos[c + d*x])^(3/2),x]","\frac{2 \left((3 A+5 C) \sin (c+d x)-\left((3 A+5 C) \sqrt{\cos (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)+A \tan (c+d x) \sec (c+d x)\right)}{5 b d \sqrt{b \cos (c+d x)}}","-\frac{2 (3 A+5 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \cos (c+d x)}}{5 b^2 d \sqrt{\cos (c+d x)}}+\frac{2 (3 A+5 C) \sin (c+d x)}{5 b d \sqrt{b \cos (c+d x)}}+\frac{2 A b \sin (c+d x)}{5 d (b \cos (c+d x))^{5/2}}",1,"(2*(-((3*A + 5*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]) + (3*A + 5*C)*Sin[c + d*x] + A*Sec[c + d*x]*Tan[c + d*x]))/(5*b*d*Sqrt[b*Cos[c + d*x]])","A",1
78,1,77,112,0.3949881,"\int \frac{\left(A+C \cos ^2(c+d x)\right) \sec ^3(c+d x)}{(b \cos (c+d x))^{3/2}} \, dx","Integrate[((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^3)/(b*Cos[c + d*x])^(3/2),x]","\frac{2 \left((5 A+7 C) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+\tan (c+d x) \left(3 A \sec ^2(c+d x)+5 A+7 C\right)\right)}{21 b d \sqrt{b \cos (c+d x)}}","\frac{2 A b^2 \sin (c+d x)}{7 d (b \cos (c+d x))^{7/2}}+\frac{2 (5 A+7 C) \sin (c+d x)}{21 d (b \cos (c+d x))^{3/2}}+\frac{2 (5 A+7 C) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 b d \sqrt{b \cos (c+d x)}}",1,"(2*((5*A + 7*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + (5*A + 7*C + 3*A*Sec[c + d*x]^2)*Tan[c + d*x]))/(21*b*d*Sqrt[b*Cos[c + d*x]])","A",1
79,1,86,115,0.4336254,"\int \frac{\cos ^5(c+d x) \left(A+C \cos ^2(c+d x)\right)}{(b \cos (c+d x))^{5/2}} \, dx","Integrate[(Cos[c + d*x]^5*(A + C*Cos[c + d*x]^2))/(b*Cos[c + d*x])^(5/2),x]","\frac{\sin (c+d x) \cos ^2(c+d x) (18 A+5 C \cos (2 (c+d x))+19 C)+6 (9 A+7 C) \sqrt{\cos (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{45 b^2 d \sqrt{b \cos (c+d x)}}","\frac{2 (9 A+7 C) \sin (c+d x) (b \cos (c+d x))^{3/2}}{45 b^4 d}+\frac{2 (9 A+7 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \cos (c+d x)}}{15 b^3 d \sqrt{\cos (c+d x)}}+\frac{2 C \sin (c+d x) (b \cos (c+d x))^{7/2}}{9 b^6 d}",1,"(6*(9*A + 7*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2] + Cos[c + d*x]^2*(18*A + 19*C + 5*C*Cos[2*(c + d*x)])*Sin[c + d*x])/(45*b^2*d*Sqrt[b*Cos[c + d*x]])","A",1
80,1,80,115,0.2598427,"\int \frac{\cos ^4(c+d x) \left(A+C \cos ^2(c+d x)\right)}{(b \cos (c+d x))^{5/2}} \, dx","Integrate[(Cos[c + d*x]^4*(A + C*Cos[c + d*x]^2))/(b*Cos[c + d*x])^(5/2),x]","\frac{\sin (2 (c+d x)) (14 A+3 C \cos (2 (c+d x))+13 C)+4 (7 A+5 C) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{42 b^2 d \sqrt{b \cos (c+d x)}}","\frac{2 (7 A+5 C) \sin (c+d x) \sqrt{b \cos (c+d x)}}{21 b^3 d}+\frac{2 (7 A+5 C) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 b^2 d \sqrt{b \cos (c+d x)}}+\frac{2 C \sin (c+d x) (b \cos (c+d x))^{5/2}}{7 b^5 d}",1,"(4*(7*A + 5*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + (14*A + 13*C + 3*C*Cos[2*(c + d*x)])*Sin[2*(c + d*x)])/(42*b^2*d*Sqrt[b*Cos[c + d*x]])","A",1
81,1,69,80,0.1709352,"\int \frac{\cos ^3(c+d x) \left(A+C \cos ^2(c+d x)\right)}{(b \cos (c+d x))^{5/2}} \, dx","Integrate[(Cos[c + d*x]^3*(A + C*Cos[c + d*x]^2))/(b*Cos[c + d*x])^(5/2),x]","\frac{2 (5 A+3 C) \sqrt{\cos (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+C \sin (2 (c+d x)) \cos (c+d x)}{5 b^2 d \sqrt{b \cos (c+d x)}}","\frac{2 (5 A+3 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \cos (c+d x)}}{5 b^3 d \sqrt{\cos (c+d x)}}+\frac{2 C \sin (c+d x) (b \cos (c+d x))^{3/2}}{5 b^4 d}",1,"(2*(5*A + 3*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2] + C*Cos[c + d*x]*Sin[2*(c + d*x)])/(5*b^2*d*Sqrt[b*Cos[c + d*x]])","A",1
82,1,61,78,0.1404918,"\int \frac{\cos ^2(c+d x) \left(A+C \cos ^2(c+d x)\right)}{(b \cos (c+d x))^{5/2}} \, dx","Integrate[(Cos[c + d*x]^2*(A + C*Cos[c + d*x]^2))/(b*Cos[c + d*x])^(5/2),x]","\frac{2 (3 A+C) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+C \sin (2 (c+d x))}{3 b^2 d \sqrt{b \cos (c+d x)}}","\frac{2 (3 A+C) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 b^2 d \sqrt{b \cos (c+d x)}}+\frac{2 C \sin (c+d x) \sqrt{b \cos (c+d x)}}{3 b^3 d}",1,"(2*(3*A + C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + C*Sin[2*(c + d*x)])/(3*b^2*d*Sqrt[b*Cos[c + d*x]])","A",1
83,1,57,74,0.1500318,"\int \frac{\cos (c+d x) \left(A+C \cos ^2(c+d x)\right)}{(b \cos (c+d x))^{5/2}} \, dx","Integrate[(Cos[c + d*x]*(A + C*Cos[c + d*x]^2))/(b*Cos[c + d*x])^(5/2),x]","\frac{2 A \sin (c+d x)-2 (A-C) \sqrt{\cos (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b^2 d \sqrt{b \cos (c+d x)}}","\frac{2 A \sin (c+d x)}{b^2 d \sqrt{b \cos (c+d x)}}-\frac{2 (A-C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \cos (c+d x)}}{b^3 d \sqrt{\cos (c+d x)}}",1,"(-2*(A - C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2] + 2*A*Sin[c + d*x])/(b^2*d*Sqrt[b*Cos[c + d*x]])","A",1
84,1,58,78,0.2140733,"\int \frac{A+C \cos ^2(c+d x)}{(b \cos (c+d x))^{5/2}} \, dx","Integrate[(A + C*Cos[c + d*x]^2)/(b*Cos[c + d*x])^(5/2),x]","\frac{2 \left((A+3 C) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+A \tan (c+d x)\right)}{3 b^2 d \sqrt{b \cos (c+d x)}}","\frac{2 (A+3 C) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 b^2 d \sqrt{b \cos (c+d x)}}+\frac{2 A \sin (c+d x)}{3 b d (b \cos (c+d x))^{3/2}}",1,"(2*((A + 3*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + A*Tan[c + d*x]))/(3*b^2*d*Sqrt[b*Cos[c + d*x]])","A",1
85,1,81,112,0.0997935,"\int \frac{\left(A+C \cos ^2(c+d x)\right) \sec (c+d x)}{(b \cos (c+d x))^{5/2}} \, dx","Integrate[((A + C*Cos[c + d*x]^2)*Sec[c + d*x])/(b*Cos[c + d*x])^(5/2),x]","\frac{2 \left((3 A+5 C) \sin (c+d x)-\left((3 A+5 C) \sqrt{\cos (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)+A \tan (c+d x) \sec (c+d x)\right)}{5 b^2 d \sqrt{b \cos (c+d x)}}","-\frac{2 (3 A+5 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \cos (c+d x)}}{5 b^3 d \sqrt{\cos (c+d x)}}+\frac{2 (3 A+5 C) \sin (c+d x)}{5 b^2 d \sqrt{b \cos (c+d x)}}+\frac{2 A \sin (c+d x)}{5 d (b \cos (c+d x))^{5/2}}",1,"(2*(-((3*A + 5*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]) + (3*A + 5*C)*Sin[c + d*x] + A*Sec[c + d*x]*Tan[c + d*x]))/(5*b^2*d*Sqrt[b*Cos[c + d*x]])","A",1
86,1,77,113,0.3207926,"\int \frac{\left(A+C \cos ^2(c+d x)\right) \sec ^2(c+d x)}{(b \cos (c+d x))^{5/2}} \, dx","Integrate[((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^2)/(b*Cos[c + d*x])^(5/2),x]","\frac{2 \left((5 A+7 C) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+\tan (c+d x) \left(3 A \sec ^2(c+d x)+5 A+7 C\right)\right)}{21 b^2 d \sqrt{b \cos (c+d x)}}","\frac{2 (5 A+7 C) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 b^2 d \sqrt{b \cos (c+d x)}}+\frac{2 (5 A+7 C) \sin (c+d x)}{21 b d (b \cos (c+d x))^{3/2}}+\frac{2 A b \sin (c+d x)}{7 d (b \cos (c+d x))^{7/2}}",1,"(2*((5*A + 7*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + (5*A + 7*C + 3*A*Sec[c + d*x]^2)*Tan[c + d*x]))/(21*b^2*d*Sqrt[b*Cos[c + d*x]])","A",1
87,1,81,115,0.0919684,"\int \frac{A+C \cos ^2(c+d x)}{(b \cos (c+d x))^{7/2}} \, dx","Integrate[(A + C*Cos[c + d*x]^2)/(b*Cos[c + d*x])^(7/2),x]","\frac{2 \left((3 A+5 C) \sin (c+d x)-\left((3 A+5 C) \sqrt{\cos (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)+A \tan (c+d x) \sec (c+d x)\right)}{5 b^3 d \sqrt{b \cos (c+d x)}}","-\frac{2 (3 A+5 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \cos (c+d x)}}{5 b^4 d \sqrt{\cos (c+d x)}}+\frac{2 (3 A+5 C) \sin (c+d x)}{5 b^3 d \sqrt{b \cos (c+d x)}}+\frac{2 A \sin (c+d x)}{5 b d (b \cos (c+d x))^{5/2}}",1,"(2*(-((3*A + 5*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]) + (3*A + 5*C)*Sin[c + d*x] + A*Sec[c + d*x]*Tan[c + d*x]))/(5*b^3*d*Sqrt[b*Cos[c + d*x]])","A",1
88,1,77,115,0.095616,"\int \frac{A+C \cos ^2(c+d x)}{(b \cos (c+d x))^{9/2}} \, dx","Integrate[(A + C*Cos[c + d*x]^2)/(b*Cos[c + d*x])^(9/2),x]","\frac{2 \left((5 A+7 C) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+\tan (c+d x) \left(3 A \sec ^2(c+d x)+5 A+7 C\right)\right)}{21 b^4 d \sqrt{b \cos (c+d x)}}","\frac{2 (5 A+7 C) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 b^4 d \sqrt{b \cos (c+d x)}}+\frac{2 (5 A+7 C) \sin (c+d x)}{21 b^3 d (b \cos (c+d x))^{3/2}}+\frac{2 A \sin (c+d x)}{7 b d (b \cos (c+d x))^{7/2}}",1,"(2*((5*A + 7*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + (5*A + 7*C + 3*A*Sec[c + d*x]^2)*Tan[c + d*x]))/(21*b^4*d*Sqrt[b*Cos[c + d*x]])","A",1
89,1,70,116,0.2389918,"\int \cos ^{\frac{5}{2}}(c+d x) \sqrt{b \cos (c+d x)} \left(A+C \cos ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^(5/2)*Sqrt[b*Cos[c + d*x]]*(A + C*Cos[c + d*x]^2),x]","\frac{\sin (c+d x) \sqrt{b \cos (c+d x)} (4 (5 A+7 C) \cos (2 (c+d x))+100 A+3 C \cos (4 (c+d x))+89 C)}{120 d \sqrt{\cos (c+d x)}}","-\frac{(A+2 C) \sin ^3(c+d x) \sqrt{b \cos (c+d x)}}{3 d \sqrt{\cos (c+d x)}}+\frac{(A+C) \sin (c+d x) \sqrt{b \cos (c+d x)}}{d \sqrt{\cos (c+d x)}}+\frac{C \sin ^5(c+d x) \sqrt{b \cos (c+d x)}}{5 d \sqrt{\cos (c+d x)}}",1,"(Sqrt[b*Cos[c + d*x]]*(100*A + 89*C + 4*(5*A + 7*C)*Cos[2*(c + d*x)] + 3*C*Cos[4*(c + d*x)])*Sin[c + d*x])/(120*d*Sqrt[Cos[c + d*x]])","A",1
90,1,67,113,0.1846953,"\int \cos ^{\frac{3}{2}}(c+d x) \sqrt{b \cos (c+d x)} \left(A+C \cos ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^(3/2)*Sqrt[b*Cos[c + d*x]]*(A + C*Cos[c + d*x]^2),x]","\frac{\sqrt{b \cos (c+d x)} (4 (4 A+3 C) (c+d x)+8 (A+C) \sin (2 (c+d x))+C \sin (4 (c+d x)))}{32 d \sqrt{\cos (c+d x)}}","\frac{x (4 A+3 C) \sqrt{b \cos (c+d x)}}{8 \sqrt{\cos (c+d x)}}+\frac{(4 A+3 C) \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{b \cos (c+d x)}}{8 d}+\frac{C \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) \sqrt{b \cos (c+d x)}}{4 d}",1,"(Sqrt[b*Cos[c + d*x]]*(4*(4*A + 3*C)*(c + d*x) + 8*(A + C)*Sin[2*(c + d*x)] + C*Sin[4*(c + d*x)]))/(32*d*Sqrt[Cos[c + d*x]])","A",1
91,1,52,74,0.0936507,"\int \sqrt{\cos (c+d x)} \sqrt{b \cos (c+d x)} \left(A+C \cos ^2(c+d x)\right) \, dx","Integrate[Sqrt[Cos[c + d*x]]*Sqrt[b*Cos[c + d*x]]*(A + C*Cos[c + d*x]^2),x]","\frac{\sin (c+d x) \sqrt{b \cos (c+d x)} (6 A+C \cos (2 (c+d x))+5 C)}{6 d \sqrt{\cos (c+d x)}}","\frac{(A+C) \sin (c+d x) \sqrt{b \cos (c+d x)}}{d \sqrt{\cos (c+d x)}}-\frac{C \sin ^3(c+d x) \sqrt{b \cos (c+d x)}}{3 d \sqrt{\cos (c+d x)}}",1,"(Sqrt[b*Cos[c + d*x]]*(6*A + 5*C + C*Cos[2*(c + d*x)])*Sin[c + d*x])/(6*d*Sqrt[Cos[c + d*x]])","A",1
92,1,52,90,0.0851688,"\int \frac{\sqrt{b \cos (c+d x)} \left(A+C \cos ^2(c+d x)\right)}{\sqrt{\cos (c+d x)}} \, dx","Integrate[(Sqrt[b*Cos[c + d*x]]*(A + C*Cos[c + d*x]^2))/Sqrt[Cos[c + d*x]],x]","\frac{\sqrt{b \cos (c+d x)} (2 (2 A+C) (c+d x)+C \sin (2 (c+d x)))}{4 d \sqrt{\cos (c+d x)}}","\frac{A x \sqrt{b \cos (c+d x)}}{\sqrt{\cos (c+d x)}}+\frac{C x \sqrt{b \cos (c+d x)}}{2 \sqrt{\cos (c+d x)}}+\frac{C \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{b \cos (c+d x)}}{2 d}",1,"(Sqrt[b*Cos[c + d*x]]*(2*(2*A + C)*(c + d*x) + C*Sin[2*(c + d*x)]))/(4*d*Sqrt[Cos[c + d*x]])","A",1
93,1,44,68,0.0503218,"\int \frac{\sqrt{b \cos (c+d x)} \left(A+C \cos ^2(c+d x)\right)}{\cos ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[(Sqrt[b*Cos[c + d*x]]*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(3/2),x]","\frac{\sqrt{b \cos (c+d x)} \left(A \tanh ^{-1}(\sin (c+d x))+C \sin (c+d x)\right)}{d \sqrt{\cos (c+d x)}}","\frac{A \sqrt{b \cos (c+d x)} \tanh ^{-1}(\sin (c+d x))}{d \sqrt{\cos (c+d x)}}+\frac{C \sin (c+d x) \sqrt{b \cos (c+d x)}}{d \sqrt{\cos (c+d x)}}",1,"(Sqrt[b*Cos[c + d*x]]*(A*ArcTanh[Sin[c + d*x]] + C*Sin[c + d*x]))/(d*Sqrt[Cos[c + d*x]])","A",1
94,1,45,59,0.0682982,"\int \frac{\sqrt{b \cos (c+d x)} \left(A+C \cos ^2(c+d x)\right)}{\cos ^{\frac{5}{2}}(c+d x)} \, dx","Integrate[(Sqrt[b*Cos[c + d*x]]*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(5/2),x]","\frac{\sqrt{b \cos (c+d x)} (A \sin (c+d x)+C d x \cos (c+d x))}{d \cos ^{\frac{3}{2}}(c+d x)}","\frac{A \sin (c+d x) \sqrt{b \cos (c+d x)}}{d \cos ^{\frac{3}{2}}(c+d x)}+\frac{C x \sqrt{b \cos (c+d x)}}{\sqrt{\cos (c+d x)}}",1,"(Sqrt[b*Cos[c + d*x]]*(C*d*x*Cos[c + d*x] + A*Sin[c + d*x]))/(d*Cos[c + d*x]^(3/2))","A",1
95,1,59,78,0.101474,"\int \frac{\sqrt{b \cos (c+d x)} \left(A+C \cos ^2(c+d x)\right)}{\cos ^{\frac{7}{2}}(c+d x)} \, dx","Integrate[(Sqrt[b*Cos[c + d*x]]*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(7/2),x]","\frac{\sqrt{b \cos (c+d x)} \left((A+2 C) \cos ^2(c+d x) \tanh ^{-1}(\sin (c+d x))+A \sin (c+d x)\right)}{2 d \cos ^{\frac{5}{2}}(c+d x)}","\frac{(A+2 C) \sqrt{b \cos (c+d x)} \tanh ^{-1}(\sin (c+d x))}{2 d \sqrt{\cos (c+d x)}}+\frac{A \sin (c+d x) \sqrt{b \cos (c+d x)}}{2 d \cos ^{\frac{5}{2}}(c+d x)}",1,"(Sqrt[b*Cos[c + d*x]]*((A + 2*C)*ArcTanh[Sin[c + d*x]]*Cos[c + d*x]^2 + A*Sin[c + d*x]))/(2*d*Cos[c + d*x]^(5/2))","A",1
96,1,51,79,0.2211827,"\int \frac{\sqrt{b \cos (c+d x)} \left(A+C \cos ^2(c+d x)\right)}{\cos ^{\frac{9}{2}}(c+d x)} \, dx","Integrate[(Sqrt[b*Cos[c + d*x]]*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(9/2),x]","\frac{\sin (c+d x) \sqrt{b \cos (c+d x)} \left(A \tan ^2(c+d x)+3 (A+C)\right)}{3 d \cos ^{\frac{3}{2}}(c+d x)}","\frac{(2 A+3 C) \sin (c+d x) \sqrt{b \cos (c+d x)}}{3 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{A \sin (c+d x) \sqrt{b \cos (c+d x)}}{3 d \cos ^{\frac{7}{2}}(c+d x)}",1,"(Sqrt[b*Cos[c + d*x]]*Sin[c + d*x]*(3*(A + C) + A*Tan[c + d*x]^2))/(3*d*Cos[c + d*x]^(3/2))","A",1
97,1,80,122,0.2560634,"\int \frac{\sqrt{b \cos (c+d x)} \left(A+C \cos ^2(c+d x)\right)}{\cos ^{\frac{11}{2}}(c+d x)} \, dx","Integrate[(Sqrt[b*Cos[c + d*x]]*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(11/2),x]","\frac{\sqrt{b \cos (c+d x)} \left(\sin (c+d x) \left((3 A+4 C) \cos ^2(c+d x)+2 A\right)+(3 A+4 C) \cos ^4(c+d x) \tanh ^{-1}(\sin (c+d x))\right)}{8 d \cos ^{\frac{9}{2}}(c+d x)}","\frac{(3 A+4 C) \sin (c+d x) \sqrt{b \cos (c+d x)}}{8 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{(3 A+4 C) \sqrt{b \cos (c+d x)} \tanh ^{-1}(\sin (c+d x))}{8 d \sqrt{\cos (c+d x)}}+\frac{A \sin (c+d x) \sqrt{b \cos (c+d x)}}{4 d \cos ^{\frac{9}{2}}(c+d x)}",1,"(Sqrt[b*Cos[c + d*x]]*((3*A + 4*C)*ArcTanh[Sin[c + d*x]]*Cos[c + d*x]^4 + (2*A + (3*A + 4*C)*Cos[c + d*x]^2)*Sin[c + d*x]))/(8*d*Cos[c + d*x]^(9/2))","A",1
98,1,70,119,0.2567299,"\int \cos ^{\frac{3}{2}}(c+d x) (b \cos (c+d x))^{3/2} \left(A+C \cos ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^(3/2)*(b*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2),x]","\frac{\sin (c+d x) (b \cos (c+d x))^{3/2} (4 (5 A+7 C) \cos (2 (c+d x))+100 A+3 C \cos (4 (c+d x))+89 C)}{120 d \cos ^{\frac{3}{2}}(c+d x)}","-\frac{b (A+2 C) \sin ^3(c+d x) \sqrt{b \cos (c+d x)}}{3 d \sqrt{\cos (c+d x)}}+\frac{b (A+C) \sin (c+d x) \sqrt{b \cos (c+d x)}}{d \sqrt{\cos (c+d x)}}+\frac{b C \sin ^5(c+d x) \sqrt{b \cos (c+d x)}}{5 d \sqrt{\cos (c+d x)}}",1,"((b*Cos[c + d*x])^(3/2)*(100*A + 89*C + 4*(5*A + 7*C)*Cos[2*(c + d*x)] + 3*C*Cos[4*(c + d*x)])*Sin[c + d*x])/(120*d*Cos[c + d*x]^(3/2))","A",1
99,1,67,116,0.2021833,"\int \sqrt{\cos (c+d x)} (b \cos (c+d x))^{3/2} \left(A+C \cos ^2(c+d x)\right) \, dx","Integrate[Sqrt[Cos[c + d*x]]*(b*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2),x]","\frac{(b \cos (c+d x))^{3/2} (4 (4 A+3 C) (c+d x)+8 (A+C) \sin (2 (c+d x))+C \sin (4 (c+d x)))}{32 d \cos ^{\frac{3}{2}}(c+d x)}","\frac{b x (4 A+3 C) \sqrt{b \cos (c+d x)}}{8 \sqrt{\cos (c+d x)}}+\frac{b (4 A+3 C) \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{b \cos (c+d x)}}{8 d}+\frac{b C \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) \sqrt{b \cos (c+d x)}}{4 d}",1,"((b*Cos[c + d*x])^(3/2)*(4*(4*A + 3*C)*(c + d*x) + 8*(A + C)*Sin[2*(c + d*x)] + C*Sin[4*(c + d*x)]))/(32*d*Cos[c + d*x]^(3/2))","A",1
100,1,53,76,0.0318306,"\int \frac{(b \cos (c+d x))^{3/2} \left(A+C \cos ^2(c+d x)\right)}{\sqrt{\cos (c+d x)}} \, dx","Integrate[((b*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2))/Sqrt[Cos[c + d*x]],x]","\frac{b \sin (c+d x) \sqrt{b \cos (c+d x)} (6 A+C \cos (2 (c+d x))+5 C)}{6 d \sqrt{\cos (c+d x)}}","\frac{b (A+C) \sin (c+d x) \sqrt{b \cos (c+d x)}}{d \sqrt{\cos (c+d x)}}-\frac{b C \sin ^3(c+d x) \sqrt{b \cos (c+d x)}}{3 d \sqrt{\cos (c+d x)}}",1,"(b*Sqrt[b*Cos[c + d*x]]*(6*A + 5*C + C*Cos[2*(c + d*x)])*Sin[c + d*x])/(6*d*Sqrt[Cos[c + d*x]])","A",1
101,1,52,93,0.1021003,"\int \frac{(b \cos (c+d x))^{3/2} \left(A+C \cos ^2(c+d x)\right)}{\cos ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[((b*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(3/2),x]","\frac{(b \cos (c+d x))^{3/2} (2 (2 A+C) (c+d x)+C \sin (2 (c+d x)))}{4 d \cos ^{\frac{3}{2}}(c+d x)}","\frac{A b x \sqrt{b \cos (c+d x)}}{\sqrt{\cos (c+d x)}}+\frac{b C x \sqrt{b \cos (c+d x)}}{2 \sqrt{\cos (c+d x)}}+\frac{b C \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{b \cos (c+d x)}}{2 d}",1,"((b*Cos[c + d*x])^(3/2)*(2*(2*A + C)*(c + d*x) + C*Sin[2*(c + d*x)]))/(4*d*Cos[c + d*x]^(3/2))","A",1
102,1,44,70,0.0593003,"\int \frac{(b \cos (c+d x))^{3/2} \left(A+C \cos ^2(c+d x)\right)}{\cos ^{\frac{5}{2}}(c+d x)} \, dx","Integrate[((b*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(5/2),x]","\frac{(b \cos (c+d x))^{3/2} \left(A \tanh ^{-1}(\sin (c+d x))+C \sin (c+d x)\right)}{d \cos ^{\frac{3}{2}}(c+d x)}","\frac{A b \sqrt{b \cos (c+d x)} \tanh ^{-1}(\sin (c+d x))}{d \sqrt{\cos (c+d x)}}+\frac{b C \sin (c+d x) \sqrt{b \cos (c+d x)}}{d \sqrt{\cos (c+d x)}}",1,"((b*Cos[c + d*x])^(3/2)*(A*ArcTanh[Sin[c + d*x]] + C*Sin[c + d*x]))/(d*Cos[c + d*x]^(3/2))","A",1
103,1,45,61,0.0624672,"\int \frac{(b \cos (c+d x))^{3/2} \left(A+C \cos ^2(c+d x)\right)}{\cos ^{\frac{7}{2}}(c+d x)} \, dx","Integrate[((b*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(7/2),x]","\frac{(b \cos (c+d x))^{3/2} (A \sin (c+d x)+C d x \cos (c+d x))}{d \cos ^{\frac{5}{2}}(c+d x)}","\frac{A b \sin (c+d x) \sqrt{b \cos (c+d x)}}{d \cos ^{\frac{3}{2}}(c+d x)}+\frac{b C x \sqrt{b \cos (c+d x)}}{\sqrt{\cos (c+d x)}}",1,"((b*Cos[c + d*x])^(3/2)*(C*d*x*Cos[c + d*x] + A*Sin[c + d*x]))/(d*Cos[c + d*x]^(5/2))","A",1
104,1,59,80,0.1148209,"\int \frac{(b \cos (c+d x))^{3/2} \left(A+C \cos ^2(c+d x)\right)}{\cos ^{\frac{9}{2}}(c+d x)} \, dx","Integrate[((b*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(9/2),x]","\frac{(b \cos (c+d x))^{3/2} \left((A+2 C) \cos ^2(c+d x) \tanh ^{-1}(\sin (c+d x))+A \sin (c+d x)\right)}{2 d \cos ^{\frac{7}{2}}(c+d x)}","\frac{b (A+2 C) \sqrt{b \cos (c+d x)} \tanh ^{-1}(\sin (c+d x))}{2 d \sqrt{\cos (c+d x)}}+\frac{A b \sin (c+d x) \sqrt{b \cos (c+d x)}}{2 d \cos ^{\frac{5}{2}}(c+d x)}",1,"((b*Cos[c + d*x])^(3/2)*((A + 2*C)*ArcTanh[Sin[c + d*x]]*Cos[c + d*x]^2 + A*Sin[c + d*x]))/(2*d*Cos[c + d*x]^(7/2))","A",1
105,1,52,81,0.1500233,"\int \frac{(b \cos (c+d x))^{3/2} \left(A+C \cos ^2(c+d x)\right)}{\cos ^{\frac{11}{2}}(c+d x)} \, dx","Integrate[((b*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(11/2),x]","\frac{b \sin (c+d x) \sqrt{b \cos (c+d x)} \left(A \tan ^2(c+d x)+3 (A+C)\right)}{3 d \cos ^{\frac{3}{2}}(c+d x)}","\frac{b (2 A+3 C) \sin (c+d x) \sqrt{b \cos (c+d x)}}{3 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{A b \sin (c+d x) \sqrt{b \cos (c+d x)}}{3 d \cos ^{\frac{7}{2}}(c+d x)}",1,"(b*Sqrt[b*Cos[c + d*x]]*Sin[c + d*x]*(3*(A + C) + A*Tan[c + d*x]^2))/(3*d*Cos[c + d*x]^(3/2))","A",1
106,1,81,125,0.2096314,"\int \frac{(b \cos (c+d x))^{3/2} \left(A+C \cos ^2(c+d x)\right)}{\cos ^{\frac{13}{2}}(c+d x)} \, dx","Integrate[((b*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(13/2),x]","\frac{b \sqrt{b \cos (c+d x)} \left(\sin (c+d x) \left((3 A+4 C) \cos ^2(c+d x)+2 A\right)+(3 A+4 C) \cos ^4(c+d x) \tanh ^{-1}(\sin (c+d x))\right)}{8 d \cos ^{\frac{9}{2}}(c+d x)}","\frac{b (3 A+4 C) \sin (c+d x) \sqrt{b \cos (c+d x)}}{8 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{b (3 A+4 C) \sqrt{b \cos (c+d x)} \tanh ^{-1}(\sin (c+d x))}{8 d \sqrt{\cos (c+d x)}}+\frac{A b \sin (c+d x) \sqrt{b \cos (c+d x)}}{4 d \cos ^{\frac{9}{2}}(c+d x)}",1,"(b*Sqrt[b*Cos[c + d*x]]*((3*A + 4*C)*ArcTanh[Sin[c + d*x]]*Cos[c + d*x]^4 + (2*A + (3*A + 4*C)*Cos[c + d*x]^2)*Sin[c + d*x]))/(8*d*Cos[c + d*x]^(9/2))","A",1
107,1,70,125,0.2888269,"\int \sqrt{\cos (c+d x)} (b \cos (c+d x))^{5/2} \left(A+C \cos ^2(c+d x)\right) \, dx","Integrate[Sqrt[Cos[c + d*x]]*(b*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2),x]","\frac{\sin (c+d x) (b \cos (c+d x))^{5/2} (4 (5 A+7 C) \cos (2 (c+d x))+100 A+3 C \cos (4 (c+d x))+89 C)}{120 d \cos ^{\frac{5}{2}}(c+d x)}","-\frac{b^2 (A+2 C) \sin ^3(c+d x) \sqrt{b \cos (c+d x)}}{3 d \sqrt{\cos (c+d x)}}+\frac{b^2 (A+C) \sin (c+d x) \sqrt{b \cos (c+d x)}}{d \sqrt{\cos (c+d x)}}+\frac{b^2 C \sin ^5(c+d x) \sqrt{b \cos (c+d x)}}{5 d \sqrt{\cos (c+d x)}}",1,"((b*Cos[c + d*x])^(5/2)*(100*A + 89*C + 4*(5*A + 7*C)*Cos[2*(c + d*x)] + 3*C*Cos[4*(c + d*x)])*Sin[c + d*x])/(120*d*Cos[c + d*x]^(5/2))","A",1
108,1,67,122,0.1884798,"\int \frac{(b \cos (c+d x))^{5/2} \left(A+C \cos ^2(c+d x)\right)}{\sqrt{\cos (c+d x)}} \, dx","Integrate[((b*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2))/Sqrt[Cos[c + d*x]],x]","\frac{(b \cos (c+d x))^{5/2} (4 (4 A+3 C) (c+d x)+8 (A+C) \sin (2 (c+d x))+C \sin (4 (c+d x)))}{32 d \cos ^{\frac{5}{2}}(c+d x)}","\frac{b^2 x (4 A+3 C) \sqrt{b \cos (c+d x)}}{8 \sqrt{\cos (c+d x)}}+\frac{b^2 (4 A+3 C) \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{b \cos (c+d x)}}{8 d}+\frac{b^2 C \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) \sqrt{b \cos (c+d x)}}{4 d}",1,"((b*Cos[c + d*x])^(5/2)*(4*(4*A + 3*C)*(c + d*x) + 8*(A + C)*Sin[2*(c + d*x)] + C*Sin[4*(c + d*x)]))/(32*d*Cos[c + d*x]^(5/2))","A",1
109,1,52,80,0.1429609,"\int \frac{(b \cos (c+d x))^{5/2} \left(A+C \cos ^2(c+d x)\right)}{\cos ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[((b*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(3/2),x]","\frac{\sin (c+d x) (b \cos (c+d x))^{5/2} (6 A+C \cos (2 (c+d x))+5 C)}{6 d \cos ^{\frac{5}{2}}(c+d x)}","\frac{b^2 (A+C) \sin (c+d x) \sqrt{b \cos (c+d x)}}{d \sqrt{\cos (c+d x)}}-\frac{b^2 C \sin ^3(c+d x) \sqrt{b \cos (c+d x)}}{3 d \sqrt{\cos (c+d x)}}",1,"((b*Cos[c + d*x])^(5/2)*(6*A + 5*C + C*Cos[2*(c + d*x)])*Sin[c + d*x])/(6*d*Cos[c + d*x]^(5/2))","A",1
110,1,52,99,0.1119696,"\int \frac{(b \cos (c+d x))^{5/2} \left(A+C \cos ^2(c+d x)\right)}{\cos ^{\frac{5}{2}}(c+d x)} \, dx","Integrate[((b*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(5/2),x]","\frac{(b \cos (c+d x))^{5/2} (2 (2 A+C) (c+d x)+C \sin (2 (c+d x)))}{4 d \cos ^{\frac{5}{2}}(c+d x)}","\frac{A b^2 x \sqrt{b \cos (c+d x)}}{\sqrt{\cos (c+d x)}}+\frac{b^2 C x \sqrt{b \cos (c+d x)}}{2 \sqrt{\cos (c+d x)}}+\frac{b^2 C \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{b \cos (c+d x)}}{2 d}",1,"((b*Cos[c + d*x])^(5/2)*(2*(2*A + C)*(c + d*x) + C*Sin[2*(c + d*x)]))/(4*d*Cos[c + d*x]^(5/2))","A",1
111,1,44,74,0.0859238,"\int \frac{(b \cos (c+d x))^{5/2} \left(A+C \cos ^2(c+d x)\right)}{\cos ^{\frac{7}{2}}(c+d x)} \, dx","Integrate[((b*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(7/2),x]","\frac{(b \cos (c+d x))^{5/2} \left(A \tanh ^{-1}(\sin (c+d x))+C \sin (c+d x)\right)}{d \cos ^{\frac{5}{2}}(c+d x)}","\frac{A b^2 \sqrt{b \cos (c+d x)} \tanh ^{-1}(\sin (c+d x))}{d \sqrt{\cos (c+d x)}}+\frac{b^2 C \sin (c+d x) \sqrt{b \cos (c+d x)}}{d \sqrt{\cos (c+d x)}}",1,"((b*Cos[c + d*x])^(5/2)*(A*ArcTanh[Sin[c + d*x]] + C*Sin[c + d*x]))/(d*Cos[c + d*x]^(5/2))","A",1
112,1,45,65,0.0801929,"\int \frac{(b \cos (c+d x))^{5/2} \left(A+C \cos ^2(c+d x)\right)}{\cos ^{\frac{9}{2}}(c+d x)} \, dx","Integrate[((b*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(9/2),x]","\frac{(b \cos (c+d x))^{5/2} (A \sin (c+d x)+C d x \cos (c+d x))}{d \cos ^{\frac{7}{2}}(c+d x)}","\frac{A b^2 \sin (c+d x) \sqrt{b \cos (c+d x)}}{d \cos ^{\frac{3}{2}}(c+d x)}+\frac{b^2 C x \sqrt{b \cos (c+d x)}}{\sqrt{\cos (c+d x)}}",1,"((b*Cos[c + d*x])^(5/2)*(C*d*x*Cos[c + d*x] + A*Sin[c + d*x]))/(d*Cos[c + d*x]^(7/2))","A",1
113,1,59,84,0.1062864,"\int \frac{(b \cos (c+d x))^{5/2} \left(A+C \cos ^2(c+d x)\right)}{\cos ^{\frac{11}{2}}(c+d x)} \, dx","Integrate[((b*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(11/2),x]","\frac{(b \cos (c+d x))^{5/2} \left((A+2 C) \cos ^2(c+d x) \tanh ^{-1}(\sin (c+d x))+A \sin (c+d x)\right)}{2 d \cos ^{\frac{9}{2}}(c+d x)}","\frac{b^2 (A+2 C) \sqrt{b \cos (c+d x)} \tanh ^{-1}(\sin (c+d x))}{2 d \sqrt{\cos (c+d x)}}+\frac{A b^2 \sin (c+d x) \sqrt{b \cos (c+d x)}}{2 d \cos ^{\frac{5}{2}}(c+d x)}",1,"((b*Cos[c + d*x])^(5/2)*((A + 2*C)*ArcTanh[Sin[c + d*x]]*Cos[c + d*x]^2 + A*Sin[c + d*x]))/(2*d*Cos[c + d*x]^(9/2))","A",1
114,1,51,85,0.2518572,"\int \frac{(b \cos (c+d x))^{5/2} \left(A+C \cos ^2(c+d x)\right)}{\cos ^{\frac{13}{2}}(c+d x)} \, dx","Integrate[((b*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(13/2),x]","\frac{\sin (c+d x) (b \cos (c+d x))^{5/2} \left(A \tan ^2(c+d x)+3 (A+C)\right)}{3 d \cos ^{\frac{7}{2}}(c+d x)}","\frac{b^2 (2 A+3 C) \sin (c+d x) \sqrt{b \cos (c+d x)}}{3 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{A b^2 \sin (c+d x) \sqrt{b \cos (c+d x)}}{3 d \cos ^{\frac{7}{2}}(c+d x)}",1,"((b*Cos[c + d*x])^(5/2)*Sin[c + d*x]*(3*(A + C) + A*Tan[c + d*x]^2))/(3*d*Cos[c + d*x]^(7/2))","A",1
115,1,80,131,0.1889842,"\int \frac{(b \cos (c+d x))^{5/2} \left(A+C \cos ^2(c+d x)\right)}{\cos ^{\frac{15}{2}}(c+d x)} \, dx","Integrate[((b*Cos[c + d*x])^(5/2)*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(15/2),x]","\frac{(b \cos (c+d x))^{5/2} \left(\sin (c+d x) \left((3 A+4 C) \cos ^2(c+d x)+2 A\right)+(3 A+4 C) \cos ^4(c+d x) \tanh ^{-1}(\sin (c+d x))\right)}{8 d \cos ^{\frac{13}{2}}(c+d x)}","\frac{b^2 (3 A+4 C) \sin (c+d x) \sqrt{b \cos (c+d x)}}{8 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{b^2 (3 A+4 C) \sqrt{b \cos (c+d x)} \tanh ^{-1}(\sin (c+d x))}{8 d \sqrt{\cos (c+d x)}}+\frac{A b^2 \sin (c+d x) \sqrt{b \cos (c+d x)}}{4 d \cos ^{\frac{9}{2}}(c+d x)}",1,"((b*Cos[c + d*x])^(5/2)*((3*A + 4*C)*ArcTanh[Sin[c + d*x]]*Cos[c + d*x]^4 + (2*A + (3*A + 4*C)*Cos[c + d*x]^2)*Sin[c + d*x]))/(8*d*Cos[c + d*x]^(13/2))","A",1
116,1,67,113,0.1399114,"\int \frac{\cos ^{\frac{5}{2}}(c+d x) \left(A+C \cos ^2(c+d x)\right)}{\sqrt{b \cos (c+d x)}} \, dx","Integrate[(Cos[c + d*x]^(5/2)*(A + C*Cos[c + d*x]^2))/Sqrt[b*Cos[c + d*x]],x]","\frac{\sqrt{\cos (c+d x)} (4 (4 A+3 C) (c+d x)+8 (A+C) \sin (2 (c+d x))+C \sin (4 (c+d x)))}{32 d \sqrt{b \cos (c+d x)}}","\frac{x (4 A+3 C) \sqrt{\cos (c+d x)}}{8 \sqrt{b \cos (c+d x)}}+\frac{(4 A+3 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{8 d \sqrt{b \cos (c+d x)}}+\frac{C \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x)}{4 d \sqrt{b \cos (c+d x)}}",1,"(Sqrt[Cos[c + d*x]]*(4*(4*A + 3*C)*(c + d*x) + 8*(A + C)*Sin[2*(c + d*x)] + C*Sin[4*(c + d*x)]))/(32*d*Sqrt[b*Cos[c + d*x]])","A",1
117,1,52,74,0.090577,"\int \frac{\cos ^{\frac{3}{2}}(c+d x) \left(A+C \cos ^2(c+d x)\right)}{\sqrt{b \cos (c+d x)}} \, dx","Integrate[(Cos[c + d*x]^(3/2)*(A + C*Cos[c + d*x]^2))/Sqrt[b*Cos[c + d*x]],x]","\frac{\sin (c+d x) \sqrt{\cos (c+d x)} (6 A+C \cos (2 (c+d x))+5 C)}{6 d \sqrt{b \cos (c+d x)}}","\frac{(A+C) \sin (c+d x) \sqrt{\cos (c+d x)}}{d \sqrt{b \cos (c+d x)}}-\frac{C \sin ^3(c+d x) \sqrt{\cos (c+d x)}}{3 d \sqrt{b \cos (c+d x)}}",1,"(Sqrt[Cos[c + d*x]]*(6*A + 5*C + C*Cos[2*(c + d*x)])*Sin[c + d*x])/(6*d*Sqrt[b*Cos[c + d*x]])","A",1
118,1,52,90,0.070008,"\int \frac{\sqrt{\cos (c+d x)} \left(A+C \cos ^2(c+d x)\right)}{\sqrt{b \cos (c+d x)}} \, dx","Integrate[(Sqrt[Cos[c + d*x]]*(A + C*Cos[c + d*x]^2))/Sqrt[b*Cos[c + d*x]],x]","\frac{\sqrt{\cos (c+d x)} (2 (2 A+C) (c+d x)+C \sin (2 (c+d x)))}{4 d \sqrt{b \cos (c+d x)}}","\frac{A x \sqrt{\cos (c+d x)}}{\sqrt{b \cos (c+d x)}}+\frac{C x \sqrt{\cos (c+d x)}}{2 \sqrt{b \cos (c+d x)}}+\frac{C \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{2 d \sqrt{b \cos (c+d x)}}",1,"(Sqrt[Cos[c + d*x]]*(2*(2*A + C)*(c + d*x) + C*Sin[2*(c + d*x)]))/(4*d*Sqrt[b*Cos[c + d*x]])","A",1
119,1,44,68,0.0480089,"\int \frac{A+C \cos ^2(c+d x)}{\sqrt{\cos (c+d x)} \sqrt{b \cos (c+d x)}} \, dx","Integrate[(A + C*Cos[c + d*x]^2)/(Sqrt[Cos[c + d*x]]*Sqrt[b*Cos[c + d*x]]),x]","\frac{\sqrt{\cos (c+d x)} \left(A \tanh ^{-1}(\sin (c+d x))+C \sin (c+d x)\right)}{d \sqrt{b \cos (c+d x)}}","\frac{A \sqrt{\cos (c+d x)} \tanh ^{-1}(\sin (c+d x))}{d \sqrt{b \cos (c+d x)}}+\frac{C \sin (c+d x) \sqrt{\cos (c+d x)}}{d \sqrt{b \cos (c+d x)}}",1,"(Sqrt[Cos[c + d*x]]*(A*ArcTanh[Sin[c + d*x]] + C*Sin[c + d*x]))/(d*Sqrt[b*Cos[c + d*x]])","A",1
120,1,45,59,0.0501298,"\int \frac{A+C \cos ^2(c+d x)}{\cos ^{\frac{3}{2}}(c+d x) \sqrt{b \cos (c+d x)}} \, dx","Integrate[(A + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(3/2)*Sqrt[b*Cos[c + d*x]]),x]","\frac{A \sin (c+d x)+C d x \cos (c+d x)}{d \sqrt{\cos (c+d x)} \sqrt{b \cos (c+d x)}}","\frac{A \sin (c+d x)}{d \sqrt{\cos (c+d x)} \sqrt{b \cos (c+d x)}}+\frac{C x \sqrt{\cos (c+d x)}}{\sqrt{b \cos (c+d x)}}",1,"(C*d*x*Cos[c + d*x] + A*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]*Sqrt[b*Cos[c + d*x]])","A",1
121,1,59,78,0.0825036,"\int \frac{A+C \cos ^2(c+d x)}{\cos ^{\frac{5}{2}}(c+d x) \sqrt{b \cos (c+d x)}} \, dx","Integrate[(A + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(5/2)*Sqrt[b*Cos[c + d*x]]),x]","\frac{(A+2 C) \cos ^2(c+d x) \tanh ^{-1}(\sin (c+d x))+A \sin (c+d x)}{2 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{b \cos (c+d x)}}","\frac{(A+2 C) \sqrt{\cos (c+d x)} \tanh ^{-1}(\sin (c+d x))}{2 d \sqrt{b \cos (c+d x)}}+\frac{A \sin (c+d x)}{2 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{b \cos (c+d x)}}",1,"((A + 2*C)*ArcTanh[Sin[c + d*x]]*Cos[c + d*x]^2 + A*Sin[c + d*x])/(2*d*Cos[c + d*x]^(3/2)*Sqrt[b*Cos[c + d*x]])","A",1
122,1,51,79,0.1345884,"\int \frac{A+C \cos ^2(c+d x)}{\cos ^{\frac{7}{2}}(c+d x) \sqrt{b \cos (c+d x)}} \, dx","Integrate[(A + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(7/2)*Sqrt[b*Cos[c + d*x]]),x]","\frac{\sin (c+d x) \left(A \tan ^2(c+d x)+3 (A+C)\right)}{3 d \sqrt{\cos (c+d x)} \sqrt{b \cos (c+d x)}}","\frac{(2 A+3 C) \sin (c+d x)}{3 d \sqrt{\cos (c+d x)} \sqrt{b \cos (c+d x)}}+\frac{A \sin (c+d x)}{3 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{b \cos (c+d x)}}",1,"(Sin[c + d*x]*(3*(A + C) + A*Tan[c + d*x]^2))/(3*d*Sqrt[Cos[c + d*x]]*Sqrt[b*Cos[c + d*x]])","A",1
123,1,80,122,0.1832909,"\int \frac{A+C \cos ^2(c+d x)}{\cos ^{\frac{9}{2}}(c+d x) \sqrt{b \cos (c+d x)}} \, dx","Integrate[(A + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(9/2)*Sqrt[b*Cos[c + d*x]]),x]","\frac{\sin (c+d x) \left((3 A+4 C) \cos ^2(c+d x)+2 A\right)+(3 A+4 C) \cos ^4(c+d x) \tanh ^{-1}(\sin (c+d x))}{8 d \cos ^{\frac{7}{2}}(c+d x) \sqrt{b \cos (c+d x)}}","\frac{(3 A+4 C) \sin (c+d x)}{8 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{b \cos (c+d x)}}+\frac{(3 A+4 C) \sqrt{\cos (c+d x)} \tanh ^{-1}(\sin (c+d x))}{8 d \sqrt{b \cos (c+d x)}}+\frac{A \sin (c+d x)}{4 d \cos ^{\frac{7}{2}}(c+d x) \sqrt{b \cos (c+d x)}}",1,"((3*A + 4*C)*ArcTanh[Sin[c + d*x]]*Cos[c + d*x]^4 + (2*A + (3*A + 4*C)*Cos[c + d*x]^2)*Sin[c + d*x])/(8*d*Cos[c + d*x]^(7/2)*Sqrt[b*Cos[c + d*x]])","A",1
124,1,67,122,0.13689,"\int \frac{\cos ^{\frac{7}{2}}(c+d x) \left(A+C \cos ^2(c+d x)\right)}{(b \cos (c+d x))^{3/2}} \, dx","Integrate[(Cos[c + d*x]^(7/2)*(A + C*Cos[c + d*x]^2))/(b*Cos[c + d*x])^(3/2),x]","\frac{\cos ^{\frac{3}{2}}(c+d x) (4 (4 A+3 C) (c+d x)+8 (A+C) \sin (2 (c+d x))+C \sin (4 (c+d x)))}{32 d (b \cos (c+d x))^{3/2}}","\frac{x (4 A+3 C) \sqrt{\cos (c+d x)}}{8 b \sqrt{b \cos (c+d x)}}+\frac{(4 A+3 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{8 b d \sqrt{b \cos (c+d x)}}+\frac{C \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x)}{4 b d \sqrt{b \cos (c+d x)}}",1,"(Cos[c + d*x]^(3/2)*(4*(4*A + 3*C)*(c + d*x) + 8*(A + C)*Sin[2*(c + d*x)] + C*Sin[4*(c + d*x)]))/(32*d*(b*Cos[c + d*x])^(3/2))","A",1
125,1,52,80,0.1128336,"\int \frac{\cos ^{\frac{5}{2}}(c+d x) \left(A+C \cos ^2(c+d x)\right)}{(b \cos (c+d x))^{3/2}} \, dx","Integrate[(Cos[c + d*x]^(5/2)*(A + C*Cos[c + d*x]^2))/(b*Cos[c + d*x])^(3/2),x]","\frac{\sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) (6 A+C \cos (2 (c+d x))+5 C)}{6 d (b \cos (c+d x))^{3/2}}","\frac{(A+C) \sin (c+d x) \sqrt{\cos (c+d x)}}{b d \sqrt{b \cos (c+d x)}}-\frac{C \sin ^3(c+d x) \sqrt{\cos (c+d x)}}{3 b d \sqrt{b \cos (c+d x)}}",1,"(Cos[c + d*x]^(3/2)*(6*A + 5*C + C*Cos[2*(c + d*x)])*Sin[c + d*x])/(6*d*(b*Cos[c + d*x])^(3/2))","A",1
126,1,52,99,0.0808463,"\int \frac{\cos ^{\frac{3}{2}}(c+d x) \left(A+C \cos ^2(c+d x)\right)}{(b \cos (c+d x))^{3/2}} \, dx","Integrate[(Cos[c + d*x]^(3/2)*(A + C*Cos[c + d*x]^2))/(b*Cos[c + d*x])^(3/2),x]","\frac{\cos ^{\frac{3}{2}}(c+d x) (2 (2 A+C) (c+d x)+C \sin (2 (c+d x)))}{4 d (b \cos (c+d x))^{3/2}}","\frac{A x \sqrt{\cos (c+d x)}}{b \sqrt{b \cos (c+d x)}}+\frac{C x \sqrt{\cos (c+d x)}}{2 b \sqrt{b \cos (c+d x)}}+\frac{C \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{2 b d \sqrt{b \cos (c+d x)}}",1,"(Cos[c + d*x]^(3/2)*(2*(2*A + C)*(c + d*x) + C*Sin[2*(c + d*x)]))/(4*d*(b*Cos[c + d*x])^(3/2))","A",1
127,1,44,74,0.0484792,"\int \frac{\sqrt{\cos (c+d x)} \left(A+C \cos ^2(c+d x)\right)}{(b \cos (c+d x))^{3/2}} \, dx","Integrate[(Sqrt[Cos[c + d*x]]*(A + C*Cos[c + d*x]^2))/(b*Cos[c + d*x])^(3/2),x]","\frac{\cos ^{\frac{3}{2}}(c+d x) \left(A \tanh ^{-1}(\sin (c+d x))+C \sin (c+d x)\right)}{d (b \cos (c+d x))^{3/2}}","\frac{A \sqrt{\cos (c+d x)} \tanh ^{-1}(\sin (c+d x))}{b d \sqrt{b \cos (c+d x)}}+\frac{C \sin (c+d x) \sqrt{\cos (c+d x)}}{b d \sqrt{b \cos (c+d x)}}",1,"(Cos[c + d*x]^(3/2)*(A*ArcTanh[Sin[c + d*x]] + C*Sin[c + d*x]))/(d*(b*Cos[c + d*x])^(3/2))","A",1
128,1,45,65,0.0587874,"\int \frac{A+C \cos ^2(c+d x)}{\sqrt{\cos (c+d x)} (b \cos (c+d x))^{3/2}} \, dx","Integrate[(A + C*Cos[c + d*x]^2)/(Sqrt[Cos[c + d*x]]*(b*Cos[c + d*x])^(3/2)),x]","\frac{\sqrt{\cos (c+d x)} (A \sin (c+d x)+C d x \cos (c+d x))}{d (b \cos (c+d x))^{3/2}}","\frac{A \sin (c+d x)}{b d \sqrt{\cos (c+d x)} \sqrt{b \cos (c+d x)}}+\frac{C x \sqrt{\cos (c+d x)}}{b \sqrt{b \cos (c+d x)}}",1,"(Sqrt[Cos[c + d*x]]*(C*d*x*Cos[c + d*x] + A*Sin[c + d*x]))/(d*(b*Cos[c + d*x])^(3/2))","A",1
129,1,59,84,0.0754806,"\int \frac{A+C \cos ^2(c+d x)}{\cos ^{\frac{3}{2}}(c+d x) (b \cos (c+d x))^{3/2}} \, dx","Integrate[(A + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(3/2)*(b*Cos[c + d*x])^(3/2)),x]","\frac{(A+2 C) \cos ^2(c+d x) \tanh ^{-1}(\sin (c+d x))+A \sin (c+d x)}{2 d \sqrt{\cos (c+d x)} (b \cos (c+d x))^{3/2}}","\frac{(A+2 C) \sqrt{\cos (c+d x)} \tanh ^{-1}(\sin (c+d x))}{2 b d \sqrt{b \cos (c+d x)}}+\frac{A \sin (c+d x)}{2 b d \cos ^{\frac{3}{2}}(c+d x) \sqrt{b \cos (c+d x)}}",1,"((A + 2*C)*ArcTanh[Sin[c + d*x]]*Cos[c + d*x]^2 + A*Sin[c + d*x])/(2*d*Sqrt[Cos[c + d*x]]*(b*Cos[c + d*x])^(3/2))","A",1
130,1,51,85,0.1449393,"\int \frac{A+C \cos ^2(c+d x)}{\cos ^{\frac{5}{2}}(c+d x) (b \cos (c+d x))^{3/2}} \, dx","Integrate[(A + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(5/2)*(b*Cos[c + d*x])^(3/2)),x]","\frac{\sin (c+d x) \sqrt{\cos (c+d x)} \left(A \tan ^2(c+d x)+3 (A+C)\right)}{3 d (b \cos (c+d x))^{3/2}}","\frac{(2 A+3 C) \sin (c+d x)}{3 b d \sqrt{\cos (c+d x)} \sqrt{b \cos (c+d x)}}+\frac{A \sin (c+d x)}{3 b d \cos ^{\frac{5}{2}}(c+d x) \sqrt{b \cos (c+d x)}}",1,"(Sqrt[Cos[c + d*x]]*Sin[c + d*x]*(3*(A + C) + A*Tan[c + d*x]^2))/(3*d*(b*Cos[c + d*x])^(3/2))","A",1
131,1,80,131,0.1236009,"\int \frac{A+C \cos ^2(c+d x)}{\cos ^{\frac{7}{2}}(c+d x) (b \cos (c+d x))^{3/2}} \, dx","Integrate[(A + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(7/2)*(b*Cos[c + d*x])^(3/2)),x]","\frac{\sin (c+d x) \left((3 A+4 C) \cos ^2(c+d x)+2 A\right)+(3 A+4 C) \cos ^4(c+d x) \tanh ^{-1}(\sin (c+d x))}{8 d \cos ^{\frac{5}{2}}(c+d x) (b \cos (c+d x))^{3/2}}","\frac{(3 A+4 C) \sin (c+d x)}{8 b d \cos ^{\frac{3}{2}}(c+d x) \sqrt{b \cos (c+d x)}}+\frac{(3 A+4 C) \sqrt{\cos (c+d x)} \tanh ^{-1}(\sin (c+d x))}{8 b d \sqrt{b \cos (c+d x)}}+\frac{A \sin (c+d x)}{4 b d \cos ^{\frac{7}{2}}(c+d x) \sqrt{b \cos (c+d x)}}",1,"((3*A + 4*C)*ArcTanh[Sin[c + d*x]]*Cos[c + d*x]^4 + (2*A + (3*A + 4*C)*Cos[c + d*x]^2)*Sin[c + d*x])/(8*d*Cos[c + d*x]^(5/2)*(b*Cos[c + d*x])^(3/2))","A",1
132,1,70,122,0.133525,"\int \frac{\cos ^{\frac{9}{2}}(c+d x) \left(A+C \cos ^2(c+d x)\right)}{(b \cos (c+d x))^{5/2}} \, dx","Integrate[(Cos[c + d*x]^(9/2)*(A + C*Cos[c + d*x]^2))/(b*Cos[c + d*x])^(5/2),x]","\frac{\sqrt{\cos (c+d x)} (4 (4 A+3 C) (c+d x)+8 (A+C) \sin (2 (c+d x))+C \sin (4 (c+d x)))}{32 b^2 d \sqrt{b \cos (c+d x)}}","\frac{x (4 A+3 C) \sqrt{\cos (c+d x)}}{8 b^2 \sqrt{b \cos (c+d x)}}+\frac{(4 A+3 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{8 b^2 d \sqrt{b \cos (c+d x)}}+\frac{C \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x)}{4 b^2 d \sqrt{b \cos (c+d x)}}",1,"(Sqrt[Cos[c + d*x]]*(4*(4*A + 3*C)*(c + d*x) + 8*(A + C)*Sin[2*(c + d*x)] + C*Sin[4*(c + d*x)]))/(32*b^2*d*Sqrt[b*Cos[c + d*x]])","A",1
133,1,55,80,0.0635705,"\int \frac{\cos ^{\frac{7}{2}}(c+d x) \left(A+C \cos ^2(c+d x)\right)}{(b \cos (c+d x))^{5/2}} \, dx","Integrate[(Cos[c + d*x]^(7/2)*(A + C*Cos[c + d*x]^2))/(b*Cos[c + d*x])^(5/2),x]","\frac{\sin (c+d x) \sqrt{\cos (c+d x)} (6 A+C \cos (2 (c+d x))+5 C)}{6 b^2 d \sqrt{b \cos (c+d x)}}","\frac{(A+C) \sin (c+d x) \sqrt{\cos (c+d x)}}{b^2 d \sqrt{b \cos (c+d x)}}-\frac{C \sin ^3(c+d x) \sqrt{\cos (c+d x)}}{3 b^2 d \sqrt{b \cos (c+d x)}}",1,"(Sqrt[Cos[c + d*x]]*(6*A + 5*C + C*Cos[2*(c + d*x)])*Sin[c + d*x])/(6*b^2*d*Sqrt[b*Cos[c + d*x]])","A",1
134,1,55,99,0.0612059,"\int \frac{\cos ^{\frac{5}{2}}(c+d x) \left(A+C \cos ^2(c+d x)\right)}{(b \cos (c+d x))^{5/2}} \, dx","Integrate[(Cos[c + d*x]^(5/2)*(A + C*Cos[c + d*x]^2))/(b*Cos[c + d*x])^(5/2),x]","\frac{\sqrt{\cos (c+d x)} (2 (2 A+C) (c+d x)+C \sin (2 (c+d x)))}{4 b^2 d \sqrt{b \cos (c+d x)}}","\frac{A x \sqrt{\cos (c+d x)}}{b^2 \sqrt{b \cos (c+d x)}}+\frac{C x \sqrt{\cos (c+d x)}}{2 b^2 \sqrt{b \cos (c+d x)}}+\frac{C \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{2 b^2 d \sqrt{b \cos (c+d x)}}",1,"(Sqrt[Cos[c + d*x]]*(2*(2*A + C)*(c + d*x) + C*Sin[2*(c + d*x)]))/(4*b^2*d*Sqrt[b*Cos[c + d*x]])","A",1
135,1,47,74,0.0488709,"\int \frac{\cos ^{\frac{3}{2}}(c+d x) \left(A+C \cos ^2(c+d x)\right)}{(b \cos (c+d x))^{5/2}} \, dx","Integrate[(Cos[c + d*x]^(3/2)*(A + C*Cos[c + d*x]^2))/(b*Cos[c + d*x])^(5/2),x]","\frac{\sqrt{\cos (c+d x)} \left(A \tanh ^{-1}(\sin (c+d x))+C \sin (c+d x)\right)}{b^2 d \sqrt{b \cos (c+d x)}}","\frac{A \sqrt{\cos (c+d x)} \tanh ^{-1}(\sin (c+d x))}{b^2 d \sqrt{b \cos (c+d x)}}+\frac{C \sin (c+d x) \sqrt{\cos (c+d x)}}{b^2 d \sqrt{b \cos (c+d x)}}",1,"(Sqrt[Cos[c + d*x]]*(A*ArcTanh[Sin[c + d*x]] + C*Sin[c + d*x]))/(b^2*d*Sqrt[b*Cos[c + d*x]])","A",1
136,1,45,65,0.0553514,"\int \frac{\sqrt{\cos (c+d x)} \left(A+C \cos ^2(c+d x)\right)}{(b \cos (c+d x))^{5/2}} \, dx","Integrate[(Sqrt[Cos[c + d*x]]*(A + C*Cos[c + d*x]^2))/(b*Cos[c + d*x])^(5/2),x]","\frac{\cos ^{\frac{3}{2}}(c+d x) (A \sin (c+d x)+C d x \cos (c+d x))}{d (b \cos (c+d x))^{5/2}}","\frac{A \sin (c+d x)}{b^2 d \sqrt{\cos (c+d x)} \sqrt{b \cos (c+d x)}}+\frac{C x \sqrt{\cos (c+d x)}}{b^2 \sqrt{b \cos (c+d x)}}",1,"(Cos[c + d*x]^(3/2)*(C*d*x*Cos[c + d*x] + A*Sin[c + d*x]))/(d*(b*Cos[c + d*x])^(5/2))","A",1
137,1,59,84,0.0768425,"\int \frac{A+C \cos ^2(c+d x)}{\sqrt{\cos (c+d x)} (b \cos (c+d x))^{5/2}} \, dx","Integrate[(A + C*Cos[c + d*x]^2)/(Sqrt[Cos[c + d*x]]*(b*Cos[c + d*x])^(5/2)),x]","\frac{\sqrt{\cos (c+d x)} \left((A+2 C) \cos ^2(c+d x) \tanh ^{-1}(\sin (c+d x))+A \sin (c+d x)\right)}{2 d (b \cos (c+d x))^{5/2}}","\frac{(A+2 C) \sqrt{\cos (c+d x)} \tanh ^{-1}(\sin (c+d x))}{2 b^2 d \sqrt{b \cos (c+d x)}}+\frac{A \sin (c+d x)}{2 b^2 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{b \cos (c+d x)}}",1,"(Sqrt[Cos[c + d*x]]*((A + 2*C)*ArcTanh[Sin[c + d*x]]*Cos[c + d*x]^2 + A*Sin[c + d*x]))/(2*d*(b*Cos[c + d*x])^(5/2))","A",1
138,1,51,85,0.1303363,"\int \frac{A+C \cos ^2(c+d x)}{\cos ^{\frac{3}{2}}(c+d x) (b \cos (c+d x))^{5/2}} \, dx","Integrate[(A + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(3/2)*(b*Cos[c + d*x])^(5/2)),x]","\frac{\sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \left(A \tan ^2(c+d x)+3 (A+C)\right)}{3 d (b \cos (c+d x))^{5/2}}","\frac{(2 A+3 C) \sin (c+d x)}{3 b^2 d \sqrt{\cos (c+d x)} \sqrt{b \cos (c+d x)}}+\frac{A \sin (c+d x)}{3 b^2 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{b \cos (c+d x)}}",1,"(Cos[c + d*x]^(3/2)*Sin[c + d*x]*(3*(A + C) + A*Tan[c + d*x]^2))/(3*d*(b*Cos[c + d*x])^(5/2))","A",1
139,1,80,131,0.1542746,"\int \frac{A+C \cos ^2(c+d x)}{\cos ^{\frac{5}{2}}(c+d x) (b \cos (c+d x))^{5/2}} \, dx","Integrate[(A + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(5/2)*(b*Cos[c + d*x])^(5/2)),x]","\frac{\sin (c+d x) \left((3 A+4 C) \cos ^2(c+d x)+2 A\right)+(3 A+4 C) \cos ^4(c+d x) \tanh ^{-1}(\sin (c+d x))}{8 d \cos ^{\frac{3}{2}}(c+d x) (b \cos (c+d x))^{5/2}}","\frac{(3 A+4 C) \sin (c+d x)}{8 b^2 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{b \cos (c+d x)}}+\frac{(3 A+4 C) \sqrt{\cos (c+d x)} \tanh ^{-1}(\sin (c+d x))}{8 b^2 d \sqrt{b \cos (c+d x)}}+\frac{A \sin (c+d x)}{4 b^2 d \cos ^{\frac{7}{2}}(c+d x) \sqrt{b \cos (c+d x)}}",1,"((3*A + 4*C)*ArcTanh[Sin[c + d*x]]*Cos[c + d*x]^4 + (2*A + (3*A + 4*C)*Cos[c + d*x]^2)*Sin[c + d*x])/(8*d*Cos[c + d*x]^(3/2)*(b*Cos[c + d*x])^(5/2))","A",1
140,1,96,95,0.1132338,"\int \cos ^2(c+d x) \sqrt[3]{b \cos (c+d x)} \left(A+C \cos ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^2*(b*Cos[c + d*x])^(1/3)*(A + C*Cos[c + d*x]^2),x]","-\frac{3 \sqrt{\sin ^2(c+d x)} \cot (c+d x) \sqrt[3]{b \cos (c+d x)} \left(8 A \cos ^2(c+d x) \, _2F_1\left(\frac{1}{2},\frac{5}{3};\frac{8}{3};\cos ^2(c+d x)\right)+5 C \cos ^4(c+d x) \, _2F_1\left(\frac{1}{2},\frac{8}{3};\frac{11}{3};\cos ^2(c+d x)\right)\right)}{80 d}","\frac{3 C \sin (c+d x) (b \cos (c+d x))^{10/3}}{13 b^3 d}-\frac{3 (13 A+10 C) \sin (c+d x) (b \cos (c+d x))^{10/3} \, _2F_1\left(\frac{1}{2},\frac{5}{3};\frac{8}{3};\cos ^2(c+d x)\right)}{130 b^3 d \sqrt{\sin ^2(c+d x)}}",1,"(-3*(b*Cos[c + d*x])^(1/3)*Cot[c + d*x]*(8*A*Cos[c + d*x]^2*Hypergeometric2F1[1/2, 5/3, 8/3, Cos[c + d*x]^2] + 5*C*Cos[c + d*x]^4*Hypergeometric2F1[1/2, 8/3, 11/3, Cos[c + d*x]^2])*Sqrt[Sin[c + d*x]^2])/(80*d)","A",1
141,1,91,95,0.1763358,"\int \cos (c+d x) \sqrt[3]{b \cos (c+d x)} \left(A+C \cos ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]*(b*Cos[c + d*x])^(1/3)*(A + C*Cos[c + d*x]^2),x]","-\frac{3 \sqrt{\sin ^2(c+d x)} \cot (c+d x) (b \cos (c+d x))^{4/3} \left(13 A \, _2F_1\left(\frac{1}{2},\frac{7}{6};\frac{13}{6};\cos ^2(c+d x)\right)+7 C \cos ^2(c+d x) \, _2F_1\left(\frac{1}{2},\frac{13}{6};\frac{19}{6};\cos ^2(c+d x)\right)\right)}{91 b d}","\frac{3 C \sin (c+d x) (b \cos (c+d x))^{7/3}}{10 b^2 d}-\frac{3 (10 A+7 C) \sin (c+d x) (b \cos (c+d x))^{7/3} \, _2F_1\left(\frac{1}{2},\frac{7}{6};\frac{13}{6};\cos ^2(c+d x)\right)}{70 b^2 d \sqrt{\sin ^2(c+d x)}}",1,"(-3*(b*Cos[c + d*x])^(4/3)*Cot[c + d*x]*(13*A*Hypergeometric2F1[1/2, 7/6, 13/6, Cos[c + d*x]^2] + 7*C*Cos[c + d*x]^2*Hypergeometric2F1[1/2, 13/6, 19/6, Cos[c + d*x]^2])*Sqrt[Sin[c + d*x]^2])/(91*b*d)","A",1
142,1,88,95,0.1186772,"\int \sqrt[3]{b \cos (c+d x)} \left(A+C \cos ^2(c+d x)\right) \, dx","Integrate[(b*Cos[c + d*x])^(1/3)*(A + C*Cos[c + d*x]^2),x]","-\frac{3 \sqrt{\sin ^2(c+d x)} \cot (c+d x) \sqrt[3]{b \cos (c+d x)} \left(5 A \, _2F_1\left(\frac{1}{2},\frac{2}{3};\frac{5}{3};\cos ^2(c+d x)\right)+2 C \cos ^2(c+d x) \, _2F_1\left(\frac{1}{2},\frac{5}{3};\frac{8}{3};\cos ^2(c+d x)\right)\right)}{20 d}","\frac{3 C \sin (c+d x) (b \cos (c+d x))^{4/3}}{7 b d}-\frac{3 (7 A+4 C) \sin (c+d x) (b \cos (c+d x))^{4/3} \, _2F_1\left(\frac{1}{2},\frac{2}{3};\frac{5}{3};\cos ^2(c+d x)\right)}{28 b d \sqrt{\sin ^2(c+d x)}}",1,"(-3*(b*Cos[c + d*x])^(1/3)*Cot[c + d*x]*(5*A*Hypergeometric2F1[1/2, 2/3, 5/3, Cos[c + d*x]^2] + 2*C*Cos[c + d*x]^2*Hypergeometric2F1[1/2, 5/3, 8/3, Cos[c + d*x]^2])*Sqrt[Sin[c + d*x]^2])/(20*d)","A",1
143,1,88,87,0.1283945,"\int \sqrt[3]{b \cos (c+d x)} \left(A+C \cos ^2(c+d x)\right) \sec (c+d x) \, dx","Integrate[(b*Cos[c + d*x])^(1/3)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x],x]","-\frac{3 b \sqrt{\sin ^2(c+d x)} \cot (c+d x) \left(7 A \, _2F_1\left(\frac{1}{6},\frac{1}{2};\frac{7}{6};\cos ^2(c+d x)\right)+C \cos ^2(c+d x) \, _2F_1\left(\frac{1}{2},\frac{7}{6};\frac{13}{6};\cos ^2(c+d x)\right)\right)}{7 d (b \cos (c+d x))^{2/3}}","\frac{3 C \sin (c+d x) \sqrt[3]{b \cos (c+d x)}}{4 d}-\frac{3 (4 A+C) \sin (c+d x) \sqrt[3]{b \cos (c+d x)} \, _2F_1\left(\frac{1}{6},\frac{1}{2};\frac{7}{6};\cos ^2(c+d x)\right)}{4 d \sqrt{\sin ^2(c+d x)}}",1,"(-3*b*Cot[c + d*x]*(7*A*Hypergeometric2F1[1/6, 1/2, 7/6, Cos[c + d*x]^2] + C*Cos[c + d*x]^2*Hypergeometric2F1[1/2, 7/6, 13/6, Cos[c + d*x]^2])*Sqrt[Sin[c + d*x]^2])/(7*d*(b*Cos[c + d*x])^(2/3))","A",1
144,1,88,91,0.1722042,"\int \sqrt[3]{b \cos (c+d x)} \left(A+C \cos ^2(c+d x)\right) \sec ^2(c+d x) \, dx","Integrate[(b*Cos[c + d*x])^(1/3)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^2,x]","-\frac{3 b \sqrt{\sin ^2(c+d x)} \csc (c+d x) \left(C \cos ^2(c+d x) \, _2F_1\left(\frac{1}{2},\frac{2}{3};\frac{5}{3};\cos ^2(c+d x)\right)-2 A \, _2F_1\left(-\frac{1}{3},\frac{1}{2};\frac{2}{3};\cos ^2(c+d x)\right)\right)}{4 d (b \cos (c+d x))^{2/3}}","\frac{3 (A-2 C) \sin (c+d x) (b \cos (c+d x))^{4/3} \, _2F_1\left(\frac{1}{2},\frac{2}{3};\frac{5}{3};\cos ^2(c+d x)\right)}{8 b d \sqrt{\sin ^2(c+d x)}}+\frac{3 A b \sin (c+d x)}{2 d (b \cos (c+d x))^{2/3}}",1,"(-3*b*Csc[c + d*x]*(-2*A*Hypergeometric2F1[-1/3, 1/2, 2/3, Cos[c + d*x]^2] + C*Cos[c + d*x]^2*Hypergeometric2F1[1/2, 2/3, 5/3, Cos[c + d*x]^2])*Sqrt[Sin[c + d*x]^2])/(4*d*(b*Cos[c + d*x])^(2/3))","A",1
145,1,96,92,0.1193515,"\int \sqrt[3]{b \cos (c+d x)} \left(A+C \cos ^2(c+d x)\right) \sec ^3(c+d x) \, dx","Integrate[(b*Cos[c + d*x])^(1/3)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^3,x]","-\frac{3 \sqrt{\sin ^2(c+d x)} \csc (c+d x) \sec ^2(c+d x) \sqrt[3]{b \cos (c+d x)} \left(5 C \cos ^2(c+d x) \, _2F_1\left(\frac{1}{6},\frac{1}{2};\frac{7}{6};\cos ^2(c+d x)\right)-A \, _2F_1\left(-\frac{5}{6},\frac{1}{2};\frac{1}{6};\cos ^2(c+d x)\right)\right)}{5 d}","\frac{3 A b^2 \sin (c+d x)}{5 d (b \cos (c+d x))^{5/3}}-\frac{3 (2 A+5 C) \sin (c+d x) \sqrt[3]{b \cos (c+d x)} \, _2F_1\left(\frac{1}{6},\frac{1}{2};\frac{7}{6};\cos ^2(c+d x)\right)}{5 d \sqrt{\sin ^2(c+d x)}}",1,"(-3*(b*Cos[c + d*x])^(1/3)*Csc[c + d*x]*(-(A*Hypergeometric2F1[-5/6, 1/2, 1/6, Cos[c + d*x]^2]) + 5*C*Cos[c + d*x]^2*Hypergeometric2F1[1/6, 1/2, 7/6, Cos[c + d*x]^2])*Sec[c + d*x]^2*Sqrt[Sin[c + d*x]^2])/(5*d)","A",1
146,1,96,95,0.1156118,"\int \cos ^2(c+d x) (b \cos (c+d x))^{2/3} \left(A+C \cos ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^2*(b*Cos[c + d*x])^(2/3)*(A + C*Cos[c + d*x]^2),x]","-\frac{3 \sqrt{\sin ^2(c+d x)} \cot (c+d x) (b \cos (c+d x))^{2/3} \left(17 A \cos ^2(c+d x) \, _2F_1\left(\frac{1}{2},\frac{11}{6};\frac{17}{6};\cos ^2(c+d x)\right)+11 C \cos ^4(c+d x) \, _2F_1\left(\frac{1}{2},\frac{17}{6};\frac{23}{6};\cos ^2(c+d x)\right)\right)}{187 d}","\frac{3 C \sin (c+d x) (b \cos (c+d x))^{11/3}}{14 b^3 d}-\frac{3 (14 A+11 C) \sin (c+d x) (b \cos (c+d x))^{11/3} \, _2F_1\left(\frac{1}{2},\frac{11}{6};\frac{17}{6};\cos ^2(c+d x)\right)}{154 b^3 d \sqrt{\sin ^2(c+d x)}}",1,"(-3*(b*Cos[c + d*x])^(2/3)*Cot[c + d*x]*(17*A*Cos[c + d*x]^2*Hypergeometric2F1[1/2, 11/6, 17/6, Cos[c + d*x]^2] + 11*C*Cos[c + d*x]^4*Hypergeometric2F1[1/2, 17/6, 23/6, Cos[c + d*x]^2])*Sqrt[Sin[c + d*x]^2])/(187*d)","A",1
147,1,91,95,0.1748468,"\int \cos (c+d x) (b \cos (c+d x))^{2/3} \left(A+C \cos ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]*(b*Cos[c + d*x])^(2/3)*(A + C*Cos[c + d*x]^2),x]","-\frac{3 \sqrt{\sin ^2(c+d x)} \cot (c+d x) (b \cos (c+d x))^{5/3} \left(7 A \, _2F_1\left(\frac{1}{2},\frac{4}{3};\frac{7}{3};\cos ^2(c+d x)\right)+4 C \cos ^2(c+d x) \, _2F_1\left(\frac{1}{2},\frac{7}{3};\frac{10}{3};\cos ^2(c+d x)\right)\right)}{56 b d}","\frac{3 C \sin (c+d x) (b \cos (c+d x))^{8/3}}{11 b^2 d}-\frac{3 (11 A+8 C) \sin (c+d x) (b \cos (c+d x))^{8/3} \, _2F_1\left(\frac{1}{2},\frac{4}{3};\frac{7}{3};\cos ^2(c+d x)\right)}{88 b^2 d \sqrt{\sin ^2(c+d x)}}",1,"(-3*(b*Cos[c + d*x])^(5/3)*Cot[c + d*x]*(7*A*Hypergeometric2F1[1/2, 4/3, 7/3, Cos[c + d*x]^2] + 4*C*Cos[c + d*x]^2*Hypergeometric2F1[1/2, 7/3, 10/3, Cos[c + d*x]^2])*Sqrt[Sin[c + d*x]^2])/(56*b*d)","A",1
148,1,88,95,0.1208608,"\int (b \cos (c+d x))^{2/3} \left(A+C \cos ^2(c+d x)\right) \, dx","Integrate[(b*Cos[c + d*x])^(2/3)*(A + C*Cos[c + d*x]^2),x]","-\frac{3 \sqrt{\sin ^2(c+d x)} \cot (c+d x) (b \cos (c+d x))^{2/3} \left(11 A \, _2F_1\left(\frac{1}{2},\frac{5}{6};\frac{11}{6};\cos ^2(c+d x)\right)+5 C \cos ^2(c+d x) \, _2F_1\left(\frac{1}{2},\frac{11}{6};\frac{17}{6};\cos ^2(c+d x)\right)\right)}{55 d}","\frac{3 C \sin (c+d x) (b \cos (c+d x))^{5/3}}{8 b d}-\frac{3 (8 A+5 C) \sin (c+d x) (b \cos (c+d x))^{5/3} \, _2F_1\left(\frac{1}{2},\frac{5}{6};\frac{11}{6};\cos ^2(c+d x)\right)}{40 b d \sqrt{\sin ^2(c+d x)}}",1,"(-3*(b*Cos[c + d*x])^(2/3)*Cot[c + d*x]*(11*A*Hypergeometric2F1[1/2, 5/6, 11/6, Cos[c + d*x]^2] + 5*C*Cos[c + d*x]^2*Hypergeometric2F1[1/2, 11/6, 17/6, Cos[c + d*x]^2])*Sqrt[Sin[c + d*x]^2])/(55*d)","A",1
149,1,88,89,0.1241231,"\int (b \cos (c+d x))^{2/3} \left(A+C \cos ^2(c+d x)\right) \sec (c+d x) \, dx","Integrate[(b*Cos[c + d*x])^(2/3)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x],x]","-\frac{3 b \sqrt{\sin ^2(c+d x)} \cot (c+d x) \left(4 A \, _2F_1\left(\frac{1}{3},\frac{1}{2};\frac{4}{3};\cos ^2(c+d x)\right)+C \cos ^2(c+d x) \, _2F_1\left(\frac{1}{2},\frac{4}{3};\frac{7}{3};\cos ^2(c+d x)\right)\right)}{8 d \sqrt[3]{b \cos (c+d x)}}","\frac{3 C \sin (c+d x) (b \cos (c+d x))^{2/3}}{5 d}-\frac{3 (5 A+2 C) \sin (c+d x) (b \cos (c+d x))^{2/3} \, _2F_1\left(\frac{1}{3},\frac{1}{2};\frac{4}{3};\cos ^2(c+d x)\right)}{10 d \sqrt{\sin ^2(c+d x)}}",1,"(-3*b*Cot[c + d*x]*(4*A*Hypergeometric2F1[1/3, 1/2, 4/3, Cos[c + d*x]^2] + C*Cos[c + d*x]^2*Hypergeometric2F1[1/2, 4/3, 7/3, Cos[c + d*x]^2])*Sqrt[Sin[c + d*x]^2])/(8*d*(b*Cos[c + d*x])^(1/3))","A",1
150,1,88,91,0.174831,"\int (b \cos (c+d x))^{2/3} \left(A+C \cos ^2(c+d x)\right) \sec ^2(c+d x) \, dx","Integrate[(b*Cos[c + d*x])^(2/3)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^2,x]","-\frac{3 b \sqrt{\sin ^2(c+d x)} \csc (c+d x) \left(C \cos ^2(c+d x) \, _2F_1\left(\frac{1}{2},\frac{5}{6};\frac{11}{6};\cos ^2(c+d x)\right)-5 A \, _2F_1\left(-\frac{1}{6},\frac{1}{2};\frac{5}{6};\cos ^2(c+d x)\right)\right)}{5 d \sqrt[3]{b \cos (c+d x)}}","\frac{3 (2 A-C) \sin (c+d x) (b \cos (c+d x))^{5/3} \, _2F_1\left(\frac{1}{2},\frac{5}{6};\frac{11}{6};\cos ^2(c+d x)\right)}{5 b d \sqrt{\sin ^2(c+d x)}}+\frac{3 A b \sin (c+d x)}{d \sqrt[3]{b \cos (c+d x)}}",1,"(-3*b*Csc[c + d*x]*(-5*A*Hypergeometric2F1[-1/6, 1/2, 5/6, Cos[c + d*x]^2] + C*Cos[c + d*x]^2*Hypergeometric2F1[1/2, 5/6, 11/6, Cos[c + d*x]^2])*Sqrt[Sin[c + d*x]^2])/(5*d*(b*Cos[c + d*x])^(1/3))","A",1
151,1,96,90,0.1312644,"\int (b \cos (c+d x))^{2/3} \left(A+C \cos ^2(c+d x)\right) \sec ^3(c+d x) \, dx","Integrate[(b*Cos[c + d*x])^(2/3)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^3,x]","-\frac{3 \sqrt{\sin ^2(c+d x)} \csc (c+d x) \sec ^2(c+d x) (b \cos (c+d x))^{2/3} \left(2 C \cos ^2(c+d x) \, _2F_1\left(\frac{1}{3},\frac{1}{2};\frac{4}{3};\cos ^2(c+d x)\right)-A \, _2F_1\left(-\frac{2}{3},\frac{1}{2};\frac{1}{3};\cos ^2(c+d x)\right)\right)}{4 d}","\frac{3 A b^2 \sin (c+d x)}{4 d (b \cos (c+d x))^{4/3}}-\frac{3 (A+4 C) \sin (c+d x) (b \cos (c+d x))^{2/3} \, _2F_1\left(\frac{1}{3},\frac{1}{2};\frac{4}{3};\cos ^2(c+d x)\right)}{8 d \sqrt{\sin ^2(c+d x)}}",1,"(-3*(b*Cos[c + d*x])^(2/3)*Csc[c + d*x]*(-(A*Hypergeometric2F1[-2/3, 1/2, 1/3, Cos[c + d*x]^2]) + 2*C*Cos[c + d*x]^2*Hypergeometric2F1[1/3, 1/2, 4/3, Cos[c + d*x]^2])*Sec[c + d*x]^2*Sqrt[Sin[c + d*x]^2])/(4*d)","A",1
152,1,96,95,0.2774516,"\int \cos ^2(c+d x) (b \cos (c+d x))^{4/3} \left(A+C \cos ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^2*(b*Cos[c + d*x])^(4/3)*(A + C*Cos[c + d*x]^2),x]","-\frac{3 \sqrt{\sin ^2(c+d x)} \cos ^2(c+d x) \cot (c+d x) (b \cos (c+d x))^{4/3} \left(19 A \, _2F_1\left(\frac{1}{2},\frac{13}{6};\frac{19}{6};\cos ^2(c+d x)\right)+13 C \cos ^2(c+d x) \, _2F_1\left(\frac{1}{2},\frac{19}{6};\frac{25}{6};\cos ^2(c+d x)\right)\right)}{247 d}","\frac{3 C \sin (c+d x) (b \cos (c+d x))^{13/3}}{16 b^3 d}-\frac{3 (16 A+13 C) \sin (c+d x) (b \cos (c+d x))^{13/3} \, _2F_1\left(\frac{1}{2},\frac{13}{6};\frac{19}{6};\cos ^2(c+d x)\right)}{208 b^3 d \sqrt{\sin ^2(c+d x)}}",1,"(-3*Cos[c + d*x]^2*(b*Cos[c + d*x])^(4/3)*Cot[c + d*x]*(19*A*Hypergeometric2F1[1/2, 13/6, 19/6, Cos[c + d*x]^2] + 13*C*Cos[c + d*x]^2*Hypergeometric2F1[1/2, 19/6, 25/6, Cos[c + d*x]^2])*Sqrt[Sin[c + d*x]^2])/(247*d)","A",1
153,1,91,95,0.1776992,"\int \cos (c+d x) (b \cos (c+d x))^{4/3} \left(A+C \cos ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]*(b*Cos[c + d*x])^(4/3)*(A + C*Cos[c + d*x]^2),x]","-\frac{3 \sqrt{\sin ^2(c+d x)} \cot (c+d x) (b \cos (c+d x))^{7/3} \left(8 A \, _2F_1\left(\frac{1}{2},\frac{5}{3};\frac{8}{3};\cos ^2(c+d x)\right)+5 C \cos ^2(c+d x) \, _2F_1\left(\frac{1}{2},\frac{8}{3};\frac{11}{3};\cos ^2(c+d x)\right)\right)}{80 b d}","\frac{3 C \sin (c+d x) (b \cos (c+d x))^{10/3}}{13 b^2 d}-\frac{3 (13 A+10 C) \sin (c+d x) (b \cos (c+d x))^{10/3} \, _2F_1\left(\frac{1}{2},\frac{5}{3};\frac{8}{3};\cos ^2(c+d x)\right)}{130 b^2 d \sqrt{\sin ^2(c+d x)}}",1,"(-3*(b*Cos[c + d*x])^(7/3)*Cot[c + d*x]*(8*A*Hypergeometric2F1[1/2, 5/3, 8/3, Cos[c + d*x]^2] + 5*C*Cos[c + d*x]^2*Hypergeometric2F1[1/2, 8/3, 11/3, Cos[c + d*x]^2])*Sqrt[Sin[c + d*x]^2])/(80*b*d)","A",1
154,1,88,95,0.155846,"\int (b \cos (c+d x))^{4/3} \left(A+C \cos ^2(c+d x)\right) \, dx","Integrate[(b*Cos[c + d*x])^(4/3)*(A + C*Cos[c + d*x]^2),x]","-\frac{3 \sqrt{\sin ^2(c+d x)} \cot (c+d x) (b \cos (c+d x))^{4/3} \left(13 A \, _2F_1\left(\frac{1}{2},\frac{7}{6};\frac{13}{6};\cos ^2(c+d x)\right)+7 C \cos ^2(c+d x) \, _2F_1\left(\frac{1}{2},\frac{13}{6};\frac{19}{6};\cos ^2(c+d x)\right)\right)}{91 d}","\frac{3 C \sin (c+d x) (b \cos (c+d x))^{7/3}}{10 b d}-\frac{3 (10 A+7 C) \sin (c+d x) (b \cos (c+d x))^{7/3} \, _2F_1\left(\frac{1}{2},\frac{7}{6};\frac{13}{6};\cos ^2(c+d x)\right)}{70 b d \sqrt{\sin ^2(c+d x)}}",1,"(-3*(b*Cos[c + d*x])^(4/3)*Cot[c + d*x]*(13*A*Hypergeometric2F1[1/2, 7/6, 13/6, Cos[c + d*x]^2] + 7*C*Cos[c + d*x]^2*Hypergeometric2F1[1/2, 13/6, 19/6, Cos[c + d*x]^2])*Sqrt[Sin[c + d*x]^2])/(91*d)","A",1
155,1,89,89,0.0884709,"\int (b \cos (c+d x))^{4/3} \left(A+C \cos ^2(c+d x)\right) \sec (c+d x) \, dx","Integrate[(b*Cos[c + d*x])^(4/3)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x],x]","-\frac{3 b \sqrt{\sin ^2(c+d x)} \cot (c+d x) \sqrt[3]{b \cos (c+d x)} \left(5 A \, _2F_1\left(\frac{1}{2},\frac{2}{3};\frac{5}{3};\cos ^2(c+d x)\right)+2 C \cos ^2(c+d x) \, _2F_1\left(\frac{1}{2},\frac{5}{3};\frac{8}{3};\cos ^2(c+d x)\right)\right)}{20 d}","\frac{3 C \sin (c+d x) (b \cos (c+d x))^{4/3}}{7 d}-\frac{3 (7 A+4 C) \sin (c+d x) (b \cos (c+d x))^{4/3} \, _2F_1\left(\frac{1}{2},\frac{2}{3};\frac{5}{3};\cos ^2(c+d x)\right)}{28 d \sqrt{\sin ^2(c+d x)}}",1,"(-3*b*(b*Cos[c + d*x])^(1/3)*Cot[c + d*x]*(5*A*Hypergeometric2F1[1/2, 2/3, 5/3, Cos[c + d*x]^2] + 2*C*Cos[c + d*x]^2*Hypergeometric2F1[1/2, 5/3, 8/3, Cos[c + d*x]^2])*Sqrt[Sin[c + d*x]^2])/(20*d)","A",1
156,1,90,89,0.1093528,"\int (b \cos (c+d x))^{4/3} \left(A+C \cos ^2(c+d x)\right) \sec ^2(c+d x) \, dx","Integrate[(b*Cos[c + d*x])^(4/3)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^2,x]","-\frac{3 b^2 \sqrt{\sin ^2(c+d x)} \cot (c+d x) \left(7 A \, _2F_1\left(\frac{1}{6},\frac{1}{2};\frac{7}{6};\cos ^2(c+d x)\right)+C \cos ^2(c+d x) \, _2F_1\left(\frac{1}{2},\frac{7}{6};\frac{13}{6};\cos ^2(c+d x)\right)\right)}{7 d (b \cos (c+d x))^{2/3}}","\frac{3 b C \sin (c+d x) \sqrt[3]{b \cos (c+d x)}}{4 d}-\frac{3 b (4 A+C) \sin (c+d x) \sqrt[3]{b \cos (c+d x)} \, _2F_1\left(\frac{1}{6},\frac{1}{2};\frac{7}{6};\cos ^2(c+d x)\right)}{4 d \sqrt{\sin ^2(c+d x)}}",1,"(-3*b^2*Cot[c + d*x]*(7*A*Hypergeometric2F1[1/6, 1/2, 7/6, Cos[c + d*x]^2] + C*Cos[c + d*x]^2*Hypergeometric2F1[1/2, 7/6, 13/6, Cos[c + d*x]^2])*Sqrt[Sin[c + d*x]^2])/(7*d*(b*Cos[c + d*x])^(2/3))","A",1
157,1,90,90,0.1766072,"\int (b \cos (c+d x))^{4/3} \left(A+C \cos ^2(c+d x)\right) \sec ^3(c+d x) \, dx","Integrate[(b*Cos[c + d*x])^(4/3)*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^3,x]","-\frac{3 b^2 \sqrt{\sin ^2(c+d x)} \csc (c+d x) \left(C \cos ^2(c+d x) \, _2F_1\left(\frac{1}{2},\frac{2}{3};\frac{5}{3};\cos ^2(c+d x)\right)-2 A \, _2F_1\left(-\frac{1}{3},\frac{1}{2};\frac{2}{3};\cos ^2(c+d x)\right)\right)}{4 d (b \cos (c+d x))^{2/3}}","\frac{3 A b^2 \sin (c+d x)}{2 d (b \cos (c+d x))^{2/3}}+\frac{3 (A-2 C) \sin (c+d x) (b \cos (c+d x))^{4/3} \, _2F_1\left(\frac{1}{2},\frac{2}{3};\frac{5}{3};\cos ^2(c+d x)\right)}{8 d \sqrt{\sin ^2(c+d x)}}",1,"(-3*b^2*Csc[c + d*x]*(-2*A*Hypergeometric2F1[-1/3, 1/2, 2/3, Cos[c + d*x]^2] + C*Cos[c + d*x]^2*Hypergeometric2F1[1/2, 2/3, 5/3, Cos[c + d*x]^2])*Sqrt[Sin[c + d*x]^2])/(4*d*(b*Cos[c + d*x])^(2/3))","A",1
158,1,96,95,0.1202166,"\int \frac{\cos ^2(c+d x) \left(A+C \cos ^2(c+d x)\right)}{\sqrt[3]{b \cos (c+d x)}} \, dx","Integrate[(Cos[c + d*x]^2*(A + C*Cos[c + d*x]^2))/(b*Cos[c + d*x])^(1/3),x]","-\frac{3 \sqrt{\sin ^2(c+d x)} \cot (c+d x) \left(7 A \cos ^2(c+d x) \, _2F_1\left(\frac{1}{2},\frac{4}{3};\frac{7}{3};\cos ^2(c+d x)\right)+4 C \cos ^4(c+d x) \, _2F_1\left(\frac{1}{2},\frac{7}{3};\frac{10}{3};\cos ^2(c+d x)\right)\right)}{56 d \sqrt[3]{b \cos (c+d x)}}","\frac{3 C \sin (c+d x) (b \cos (c+d x))^{8/3}}{11 b^3 d}-\frac{3 (11 A+8 C) \sin (c+d x) (b \cos (c+d x))^{8/3} \, _2F_1\left(\frac{1}{2},\frac{4}{3};\frac{7}{3};\cos ^2(c+d x)\right)}{88 b^3 d \sqrt{\sin ^2(c+d x)}}",1,"(-3*Cot[c + d*x]*(7*A*Cos[c + d*x]^2*Hypergeometric2F1[1/2, 4/3, 7/3, Cos[c + d*x]^2] + 4*C*Cos[c + d*x]^4*Hypergeometric2F1[1/2, 7/3, 10/3, Cos[c + d*x]^2])*Sqrt[Sin[c + d*x]^2])/(56*d*(b*Cos[c + d*x])^(1/3))","A",1
159,1,91,95,0.1121544,"\int \frac{\cos (c+d x) \left(A+C \cos ^2(c+d x)\right)}{\sqrt[3]{b \cos (c+d x)}} \, dx","Integrate[(Cos[c + d*x]*(A + C*Cos[c + d*x]^2))/(b*Cos[c + d*x])^(1/3),x]","-\frac{3 \sqrt{\sin ^2(c+d x)} \cot (c+d x) (b \cos (c+d x))^{2/3} \left(11 A \, _2F_1\left(\frac{1}{2},\frac{5}{6};\frac{11}{6};\cos ^2(c+d x)\right)+5 C \cos ^2(c+d x) \, _2F_1\left(\frac{1}{2},\frac{11}{6};\frac{17}{6};\cos ^2(c+d x)\right)\right)}{55 b d}","\frac{3 C \sin (c+d x) (b \cos (c+d x))^{5/3}}{8 b^2 d}-\frac{3 (8 A+5 C) \sin (c+d x) (b \cos (c+d x))^{5/3} \, _2F_1\left(\frac{1}{2},\frac{5}{6};\frac{11}{6};\cos ^2(c+d x)\right)}{40 b^2 d \sqrt{\sin ^2(c+d x)}}",1,"(-3*(b*Cos[c + d*x])^(2/3)*Cot[c + d*x]*(11*A*Hypergeometric2F1[1/2, 5/6, 11/6, Cos[c + d*x]^2] + 5*C*Cos[c + d*x]^2*Hypergeometric2F1[1/2, 11/6, 17/6, Cos[c + d*x]^2])*Sqrt[Sin[c + d*x]^2])/(55*b*d)","A",1
160,1,87,95,0.0929435,"\int \frac{A+C \cos ^2(c+d x)}{\sqrt[3]{b \cos (c+d x)}} \, dx","Integrate[(A + C*Cos[c + d*x]^2)/(b*Cos[c + d*x])^(1/3),x]","-\frac{3 \sqrt{\sin ^2(c+d x)} \cot (c+d x) \left(4 A \, _2F_1\left(\frac{1}{3},\frac{1}{2};\frac{4}{3};\cos ^2(c+d x)\right)+C \cos ^2(c+d x) \, _2F_1\left(\frac{1}{2},\frac{4}{3};\frac{7}{3};\cos ^2(c+d x)\right)\right)}{8 d \sqrt[3]{b \cos (c+d x)}}","\frac{3 C \sin (c+d x) (b \cos (c+d x))^{2/3}}{5 b d}-\frac{3 (5 A+2 C) \sin (c+d x) (b \cos (c+d x))^{2/3} \, _2F_1\left(\frac{1}{3},\frac{1}{2};\frac{4}{3};\cos ^2(c+d x)\right)}{10 b d \sqrt{\sin ^2(c+d x)}}",1,"(-3*Cot[c + d*x]*(4*A*Hypergeometric2F1[1/3, 1/2, 4/3, Cos[c + d*x]^2] + C*Cos[c + d*x]^2*Hypergeometric2F1[1/2, 4/3, 7/3, Cos[c + d*x]^2])*Sqrt[Sin[c + d*x]^2])/(8*d*(b*Cos[c + d*x])^(1/3))","A",1
161,1,283,90,3.8723377,"\int \frac{\left(A+C \cos ^2(c+d x)\right) \sec (c+d x)}{\sqrt[3]{b \cos (c+d x)}} \, dx","Integrate[((A + C*Cos[c + d*x]^2)*Sec[c + d*x])/(b*Cos[c + d*x])^(1/3),x]","-\frac{3 \csc (c) e^{-i d x} \sqrt[3]{\cos (c+d x)} (\cos (d x)+i \sin (d x)) \left(2 (2 A-C) (\cos (d x)-i \sin (d x)) \sqrt[3]{i \sin (2 (c+d x))+\cos (2 (c+d x))+1} \, _2F_1\left(-\frac{1}{3},\frac{1}{3};\frac{2}{3};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right)+(2 A-C) (\cos (d x)+i \sin (d x)) \sqrt[3]{i \sin (2 (c+d x))+\cos (2 (c+d x))+1} \, _2F_1\left(\frac{1}{3},\frac{2}{3};\frac{5}{3};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right)-8 A \cos (d x)+2 C \cos (2 c+d x)+2 C \cos (d x)\right)}{4\ 2^{2/3} d \sqrt[3]{b \cos (c+d x)} \sqrt[3]{e^{-i d x} \left(i \sin (c) \left(-1+e^{2 i d x}\right)+\cos (c) \left(1+e^{2 i d x}\right)\right)}}","\frac{3 (2 A-C) \sin (c+d x) (b \cos (c+d x))^{5/3} \, _2F_1\left(\frac{1}{2},\frac{5}{6};\frac{11}{6};\cos ^2(c+d x)\right)}{5 b^2 d \sqrt{\sin ^2(c+d x)}}+\frac{3 A \sin (c+d x)}{d \sqrt[3]{b \cos (c+d x)}}",1,"(-3*Cos[c + d*x]^(1/3)*Csc[c]*(Cos[d*x] + I*Sin[d*x])*(-8*A*Cos[d*x] + 2*C*Cos[d*x] + 2*C*Cos[2*c + d*x] + 2*(2*A - C)*Hypergeometric2F1[-1/3, 1/3, 2/3, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*(Cos[d*x] - I*Sin[d*x])*(1 + Cos[2*(c + d*x)] + I*Sin[2*(c + d*x)])^(1/3) + (2*A - C)*Hypergeometric2F1[1/3, 2/3, 5/3, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*(Cos[d*x] + I*Sin[d*x])*(1 + Cos[2*(c + d*x)] + I*Sin[2*(c + d*x)])^(1/3)))/(4*2^(2/3)*d*E^(I*d*x)*(b*Cos[c + d*x])^(1/3)*(((1 + E^((2*I)*d*x))*Cos[c] + I*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x))^(1/3))","C",1
162,1,101,91,0.8204861,"\int \frac{\left(A+C \cos ^2(c+d x)\right) \sec ^2(c+d x)}{\sqrt[3]{b \cos (c+d x)}} \, dx","Integrate[((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^2)/(b*Cos[c + d*x])^(1/3),x]","\frac{6 A \tan (c+d x) \sqrt[3]{\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)}-(A+4 C) \sin \left(2 d x-2 \tan ^{-1}(\cot (c))\right) \, _2F_1\left(\frac{1}{2},\frac{2}{3};\frac{3}{2};\cos ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right)}{8 d \sqrt[3]{b \cos (c+d x)} \sqrt[3]{\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)}}","\frac{3 A b \sin (c+d x)}{4 d (b \cos (c+d x))^{4/3}}-\frac{3 (A+4 C) \sin (c+d x) (b \cos (c+d x))^{2/3} \, _2F_1\left(\frac{1}{3},\frac{1}{2};\frac{4}{3};\cos ^2(c+d x)\right)}{8 b d \sqrt{\sin ^2(c+d x)}}",1,"(-((A + 4*C)*Hypergeometric2F1[1/2, 2/3, 3/2, Cos[d*x - ArcTan[Cot[c]]]^2]*Sin[2*d*x - 2*ArcTan[Cot[c]]]) + 6*A*(Sin[d*x - ArcTan[Cot[c]]]^2)^(1/3)*Tan[c + d*x])/(8*d*(b*Cos[c + d*x])^(1/3)*(Sin[d*x - ArcTan[Cot[c]]]^2)^(1/3))","A",1
163,1,481,92,6.2949044,"\int \frac{\left(A+C \cos ^2(c+d x)\right) \sec ^3(c+d x)}{\sqrt[3]{b \cos (c+d x)}} \, dx","Integrate[((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^3)/(b*Cos[c + d*x])^(1/3),x]","b \left(\frac{\cos ^4(c+d x) \left(A \sec ^2(c+d x)+C\right) \left(\frac{6 \sec (c) \sec (c+d x) (4 A \sin (d x)+7 C \sin (d x))}{7 d}+\frac{6 (4 A+7 C) \csc (c) \sec (c)}{7 d}+\frac{6 A \sec (c) \sin (d x) \sec ^3(c+d x)}{7 d}+\frac{6 A \tan (c) \sec ^2(c+d x)}{7 d}\right)}{(b \cos (c+d x))^{4/3} (2 A+C \cos (2 c+2 d x)+C)}-\frac{i (4 A+7 C) \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \cos ^{\frac{10}{3}}(c+d x) \left(A \sec ^2(c+d x)+C\right) \left(-\frac{3 i e^{-i d x} \sqrt[3]{i \sin (2 c) e^{2 i d x}+\cos (2 c) e^{2 i d x}+1} \, _2F_1\left(-\frac{1}{3},\frac{1}{3};\frac{2}{3};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right)}{2^{2/3} d \sqrt[3]{e^{-i d x} \left(i \sin (c) \left(-1+e^{2 i d x}\right)+\cos (c) \left(1+e^{2 i d x}\right)\right)}}-\frac{3 i e^{i d x} \sqrt[3]{i \sin (2 c) e^{2 i d x}+\cos (2 c) e^{2 i d x}+1} \, _2F_1\left(\frac{1}{3},\frac{2}{3};\frac{5}{3};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right)}{2\ 2^{2/3} d \sqrt[3]{e^{-i d x} \left(i \sin (c) \left(-1+e^{2 i d x}\right)+\cos (c) \left(1+e^{2 i d x}\right)\right)}}\right)}{7 (b \cos (c+d x))^{4/3} (2 A+C \cos (2 c+2 d x)+C)}\right)","\frac{3 A b^2 \sin (c+d x)}{7 d (b \cos (c+d x))^{7/3}}+\frac{3 (4 A+7 C) \sin (c+d x) \, _2F_1\left(-\frac{1}{6},\frac{1}{2};\frac{5}{6};\cos ^2(c+d x)\right)}{7 d \sqrt{\sin ^2(c+d x)} \sqrt[3]{b \cos (c+d x)}}",1,"b*(((-1/7*I)*(4*A + 7*C)*Cos[c + d*x]^(10/3)*Csc[c/2]*Sec[c/2]*(C + A*Sec[c + d*x]^2)*(((-3*I)*Hypergeometric2F1[-1/3, 1/3, 2/3, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*(1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c])^(1/3))/(2^(2/3)*d*E^(I*d*x)*(((1 + E^((2*I)*d*x))*Cos[c] + I*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x))^(1/3)) - (((3*I)/2)*E^(I*d*x)*Hypergeometric2F1[1/3, 2/3, 5/3, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*(1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c])^(1/3))/(2^(2/3)*d*(((1 + E^((2*I)*d*x))*Cos[c] + I*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x))^(1/3))))/((b*Cos[c + d*x])^(4/3)*(2*A + C + C*Cos[2*c + 2*d*x])) + (Cos[c + d*x]^4*(C + A*Sec[c + d*x]^2)*((6*(4*A + 7*C)*Csc[c]*Sec[c])/(7*d) + (6*A*Sec[c]*Sec[c + d*x]^3*Sin[d*x])/(7*d) + (6*Sec[c]*Sec[c + d*x]*(4*A*Sin[d*x] + 7*C*Sin[d*x]))/(7*d) + (6*A*Sec[c + d*x]^2*Tan[c])/(7*d)))/((b*Cos[c + d*x])^(4/3)*(2*A + C + C*Cos[2*c + 2*d*x])))","C",1
164,1,96,95,0.132829,"\int \frac{\cos ^2(c+d x) \left(A+C \cos ^2(c+d x)\right)}{(b \cos (c+d x))^{2/3}} \, dx","Integrate[(Cos[c + d*x]^2*(A + C*Cos[c + d*x]^2))/(b*Cos[c + d*x])^(2/3),x]","-\frac{3 \sqrt{\sin ^2(c+d x)} \cot (c+d x) \left(13 A \cos ^2(c+d x) \, _2F_1\left(\frac{1}{2},\frac{7}{6};\frac{13}{6};\cos ^2(c+d x)\right)+7 C \cos ^4(c+d x) \, _2F_1\left(\frac{1}{2},\frac{13}{6};\frac{19}{6};\cos ^2(c+d x)\right)\right)}{91 d (b \cos (c+d x))^{2/3}}","\frac{3 C \sin (c+d x) (b \cos (c+d x))^{7/3}}{10 b^3 d}-\frac{3 (10 A+7 C) \sin (c+d x) (b \cos (c+d x))^{7/3} \, _2F_1\left(\frac{1}{2},\frac{7}{6};\frac{13}{6};\cos ^2(c+d x)\right)}{70 b^3 d \sqrt{\sin ^2(c+d x)}}",1,"(-3*Cot[c + d*x]*(13*A*Cos[c + d*x]^2*Hypergeometric2F1[1/2, 7/6, 13/6, Cos[c + d*x]^2] + 7*C*Cos[c + d*x]^4*Hypergeometric2F1[1/2, 13/6, 19/6, Cos[c + d*x]^2])*Sqrt[Sin[c + d*x]^2])/(91*d*(b*Cos[c + d*x])^(2/3))","A",1
165,1,91,95,0.1408269,"\int \frac{\cos (c+d x) \left(A+C \cos ^2(c+d x)\right)}{(b \cos (c+d x))^{2/3}} \, dx","Integrate[(Cos[c + d*x]*(A + C*Cos[c + d*x]^2))/(b*Cos[c + d*x])^(2/3),x]","-\frac{3 \sqrt{\sin ^2(c+d x)} \cot (c+d x) \sqrt[3]{b \cos (c+d x)} \left(5 A \, _2F_1\left(\frac{1}{2},\frac{2}{3};\frac{5}{3};\cos ^2(c+d x)\right)+2 C \cos ^2(c+d x) \, _2F_1\left(\frac{1}{2},\frac{5}{3};\frac{8}{3};\cos ^2(c+d x)\right)\right)}{20 b d}","\frac{3 C \sin (c+d x) (b \cos (c+d x))^{4/3}}{7 b^2 d}-\frac{3 (7 A+4 C) \sin (c+d x) (b \cos (c+d x))^{4/3} \, _2F_1\left(\frac{1}{2},\frac{2}{3};\frac{5}{3};\cos ^2(c+d x)\right)}{28 b^2 d \sqrt{\sin ^2(c+d x)}}",1,"(-3*(b*Cos[c + d*x])^(1/3)*Cot[c + d*x]*(5*A*Hypergeometric2F1[1/2, 2/3, 5/3, Cos[c + d*x]^2] + 2*C*Cos[c + d*x]^2*Hypergeometric2F1[1/2, 5/3, 8/3, Cos[c + d*x]^2])*Sqrt[Sin[c + d*x]^2])/(20*b*d)","A",1
166,1,87,93,0.1148368,"\int \frac{A+C \cos ^2(c+d x)}{(b \cos (c+d x))^{2/3}} \, dx","Integrate[(A + C*Cos[c + d*x]^2)/(b*Cos[c + d*x])^(2/3),x]","-\frac{3 \sqrt{\sin ^2(c+d x)} \cot (c+d x) \left(7 A \, _2F_1\left(\frac{1}{6},\frac{1}{2};\frac{7}{6};\cos ^2(c+d x)\right)+C \cos ^2(c+d x) \, _2F_1\left(\frac{1}{2},\frac{7}{6};\frac{13}{6};\cos ^2(c+d x)\right)\right)}{7 d (b \cos (c+d x))^{2/3}}","\frac{3 C \sin (c+d x) \sqrt[3]{b \cos (c+d x)}}{4 b d}-\frac{3 (4 A+C) \sin (c+d x) \sqrt[3]{b \cos (c+d x)} \, _2F_1\left(\frac{1}{6},\frac{1}{2};\frac{7}{6};\cos ^2(c+d x)\right)}{4 b d \sqrt{\sin ^2(c+d x)}}",1,"(-3*Cot[c + d*x]*(7*A*Hypergeometric2F1[1/6, 1/2, 7/6, Cos[c + d*x]^2] + C*Cos[c + d*x]^2*Hypergeometric2F1[1/2, 7/6, 13/6, Cos[c + d*x]^2])*Sqrt[Sin[c + d*x]^2])/(7*d*(b*Cos[c + d*x])^(2/3))","A",1
167,1,277,90,3.926708,"\int \frac{\left(A+C \cos ^2(c+d x)\right) \sec (c+d x)}{(b \cos (c+d x))^{2/3}} \, dx","Integrate[((A + C*Cos[c + d*x]^2)*Sec[c + d*x])/(b*Cos[c + d*x])^(2/3),x]","-\frac{3 \csc (c) e^{-i d x} \cos ^{\frac{2}{3}}(c+d x) (\cos (d x)+i \sin (d x)) \left(5 (A-2 C) (\cos (d x)-i \sin (d x)) (i \sin (2 (c+d x))+\cos (2 (c+d x))+1)^{2/3} \, _2F_1\left(-\frac{1}{6},\frac{2}{3};\frac{5}{6};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right)+(A-2 C) (\cos (d x)+i \sin (d x)) (i \sin (2 (c+d x))+\cos (2 (c+d x))+1)^{2/3} \, _2F_1\left(\frac{2}{3},\frac{5}{6};\frac{11}{6};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right)+10 ((C-A) \cos (d x)+C \cos (2 c+d x))\right)}{10 \sqrt[3]{2} d (b \cos (c+d x))^{2/3} \left(e^{-i d x} \left(i \sin (c) \left(-1+e^{2 i d x}\right)+\cos (c) \left(1+e^{2 i d x}\right)\right)\right)^{2/3}}","\frac{3 (A-2 C) \sin (c+d x) (b \cos (c+d x))^{4/3} \, _2F_1\left(\frac{1}{2},\frac{2}{3};\frac{5}{3};\cos ^2(c+d x)\right)}{8 b^2 d \sqrt{\sin ^2(c+d x)}}+\frac{3 A \sin (c+d x)}{2 d (b \cos (c+d x))^{2/3}}",1,"(-3*Cos[c + d*x]^(2/3)*Csc[c]*(Cos[d*x] + I*Sin[d*x])*(10*((-A + C)*Cos[d*x] + C*Cos[2*c + d*x]) + 5*(A - 2*C)*Hypergeometric2F1[-1/6, 2/3, 5/6, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*(Cos[d*x] - I*Sin[d*x])*(1 + Cos[2*(c + d*x)] + I*Sin[2*(c + d*x)])^(2/3) + (A - 2*C)*Hypergeometric2F1[2/3, 5/6, 11/6, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*(Cos[d*x] + I*Sin[d*x])*(1 + Cos[2*(c + d*x)] + I*Sin[2*(c + d*x)])^(2/3)))/(10*2^(1/3)*d*E^(I*d*x)*(b*Cos[c + d*x])^(2/3)*(((1 + E^((2*I)*d*x))*Cos[c] + I*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x))^(2/3))","C",1
168,1,103,93,0.7296246,"\int \frac{\left(A+C \cos ^2(c+d x)\right) \sec ^2(c+d x)}{(b \cos (c+d x))^{2/3}} \, dx","Integrate[((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^2)/(b*Cos[c + d*x])^(2/3),x]","\frac{6 A \tan (c+d x) \sqrt[6]{\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)}-(2 A+5 C) \sin \left(2 d x-2 \tan ^{-1}(\cot (c))\right) \, _2F_1\left(\frac{1}{2},\frac{5}{6};\frac{3}{2};\cos ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right)}{10 d (b \cos (c+d x))^{2/3} \sqrt[6]{\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)}}","\frac{3 A b \sin (c+d x)}{5 d (b \cos (c+d x))^{5/3}}-\frac{3 (2 A+5 C) \sin (c+d x) \sqrt[3]{b \cos (c+d x)} \, _2F_1\left(\frac{1}{6},\frac{1}{2};\frac{7}{6};\cos ^2(c+d x)\right)}{5 b d \sqrt{\sin ^2(c+d x)}}",1,"(-((2*A + 5*C)*Hypergeometric2F1[1/2, 5/6, 3/2, Cos[d*x - ArcTan[Cot[c]]]^2]*Sin[2*d*x - 2*ArcTan[Cot[c]]]) + 6*A*(Sin[d*x - ArcTan[Cot[c]]]^2)^(1/6)*Tan[c + d*x])/(10*d*(b*Cos[c + d*x])^(2/3)*(Sin[d*x - ArcTan[Cot[c]]]^2)^(1/6))","A",1
169,1,473,92,6.289821,"\int \frac{\left(A+C \cos ^2(c+d x)\right) \sec ^3(c+d x)}{(b \cos (c+d x))^{2/3}} \, dx","Integrate[((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^3)/(b*Cos[c + d*x])^(2/3),x]","b \left(\frac{\cos ^4(c+d x) \left(A \sec ^2(c+d x)+C\right) \left(\frac{3 \sec (c) \sec (c+d x) (5 A \sin (d x)+8 C \sin (d x))}{8 d}+\frac{3 (5 A+8 C) \csc (c) \sec (c)}{8 d}+\frac{3 A \sec (c) \sin (d x) \sec ^3(c+d x)}{4 d}+\frac{3 A \tan (c) \sec ^2(c+d x)}{4 d}\right)}{(b \cos (c+d x))^{5/3} (2 A+C \cos (2 c+2 d x)+C)}-\frac{i (5 A+8 C) \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \cos ^{\frac{11}{3}}(c+d x) \left(A \sec ^2(c+d x)+C\right) \left(-\frac{3 i e^{-i d x} \left(2 i \sin (2 c) e^{2 i d x}+2 \cos (2 c) e^{2 i d x}+2\right)^{2/3} \, _2F_1\left(-\frac{1}{6},\frac{2}{3};\frac{5}{6};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right)}{d \left(e^{-i d x} \left(i \sin (c) \left(-1+e^{2 i d x}\right)+\cos (c) \left(1+e^{2 i d x}\right)\right)\right)^{2/3}}-\frac{3 i e^{i d x} \left(2 i \sin (2 c) e^{2 i d x}+2 \cos (2 c) e^{2 i d x}+2\right)^{2/3} \, _2F_1\left(\frac{2}{3},\frac{5}{6};\frac{11}{6};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right)}{5 d \left(e^{-i d x} \left(i \sin (c) \left(-1+e^{2 i d x}\right)+\cos (c) \left(1+e^{2 i d x}\right)\right)\right)^{2/3}}\right)}{32 (b \cos (c+d x))^{5/3} (2 A+C \cos (2 c+2 d x)+C)}\right)","\frac{3 A b^2 \sin (c+d x)}{8 d (b \cos (c+d x))^{8/3}}+\frac{3 (5 A+8 C) \sin (c+d x) \, _2F_1\left(-\frac{1}{3},\frac{1}{2};\frac{2}{3};\cos ^2(c+d x)\right)}{16 d \sqrt{\sin ^2(c+d x)} (b \cos (c+d x))^{2/3}}",1,"b*(((-1/32*I)*(5*A + 8*C)*Cos[c + d*x]^(11/3)*Csc[c/2]*Sec[c/2]*(C + A*Sec[c + d*x]^2)*(((-3*I)*Hypergeometric2F1[-1/6, 2/3, 5/6, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*(2 + 2*E^((2*I)*d*x)*Cos[2*c] + (2*I)*E^((2*I)*d*x)*Sin[2*c])^(2/3))/(d*E^(I*d*x)*(((1 + E^((2*I)*d*x))*Cos[c] + I*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x))^(2/3)) - (((3*I)/5)*E^(I*d*x)*Hypergeometric2F1[2/3, 5/6, 11/6, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*(2 + 2*E^((2*I)*d*x)*Cos[2*c] + (2*I)*E^((2*I)*d*x)*Sin[2*c])^(2/3))/(d*(((1 + E^((2*I)*d*x))*Cos[c] + I*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x))^(2/3))))/((b*Cos[c + d*x])^(5/3)*(2*A + C + C*Cos[2*c + 2*d*x])) + (Cos[c + d*x]^4*(C + A*Sec[c + d*x]^2)*((3*(5*A + 8*C)*Csc[c]*Sec[c])/(8*d) + (3*A*Sec[c]*Sec[c + d*x]^3*Sin[d*x])/(4*d) + (3*Sec[c]*Sec[c + d*x]*(5*A*Sin[d*x] + 8*C*Sin[d*x]))/(8*d) + (3*A*Sec[c + d*x]^2*Tan[c])/(4*d)))/((b*Cos[c + d*x])^(5/3)*(2*A + C + C*Cos[2*c + 2*d*x])))","C",1
170,1,96,95,0.2022024,"\int \frac{\cos ^2(c+d x) \left(A+C \cos ^2(c+d x)\right)}{(b \cos (c+d x))^{4/3}} \, dx","Integrate[(Cos[c + d*x]^2*(A + C*Cos[c + d*x]^2))/(b*Cos[c + d*x])^(4/3),x]","-\frac{3 \sqrt{\sin ^2(c+d x)} \cos ^2(c+d x) \cot (c+d x) \left(11 A \, _2F_1\left(\frac{1}{2},\frac{5}{6};\frac{11}{6};\cos ^2(c+d x)\right)+5 C \cos ^2(c+d x) \, _2F_1\left(\frac{1}{2},\frac{11}{6};\frac{17}{6};\cos ^2(c+d x)\right)\right)}{55 d (b \cos (c+d x))^{4/3}}","\frac{3 C \sin (c+d x) (b \cos (c+d x))^{5/3}}{8 b^3 d}-\frac{3 (8 A+5 C) \sin (c+d x) (b \cos (c+d x))^{5/3} \, _2F_1\left(\frac{1}{2},\frac{5}{6};\frac{11}{6};\cos ^2(c+d x)\right)}{40 b^3 d \sqrt{\sin ^2(c+d x)}}",1,"(-3*Cos[c + d*x]^2*Cot[c + d*x]*(11*A*Hypergeometric2F1[1/2, 5/6, 11/6, Cos[c + d*x]^2] + 5*C*Cos[c + d*x]^2*Hypergeometric2F1[1/2, 11/6, 17/6, Cos[c + d*x]^2])*Sqrt[Sin[c + d*x]^2])/(55*d*(b*Cos[c + d*x])^(4/3))","A",1
171,1,90,95,0.0996795,"\int \frac{\cos (c+d x) \left(A+C \cos ^2(c+d x)\right)}{(b \cos (c+d x))^{4/3}} \, dx","Integrate[(Cos[c + d*x]*(A + C*Cos[c + d*x]^2))/(b*Cos[c + d*x])^(4/3),x]","-\frac{3 \sqrt{\sin ^2(c+d x)} \cot (c+d x) \left(4 A \, _2F_1\left(\frac{1}{3},\frac{1}{2};\frac{4}{3};\cos ^2(c+d x)\right)+C \cos ^2(c+d x) \, _2F_1\left(\frac{1}{2},\frac{4}{3};\frac{7}{3};\cos ^2(c+d x)\right)\right)}{8 b d \sqrt[3]{b \cos (c+d x)}}","\frac{3 C \sin (c+d x) (b \cos (c+d x))^{2/3}}{5 b^2 d}-\frac{3 (5 A+2 C) \sin (c+d x) (b \cos (c+d x))^{2/3} \, _2F_1\left(\frac{1}{3},\frac{1}{2};\frac{4}{3};\cos ^2(c+d x)\right)}{10 b^2 d \sqrt{\sin ^2(c+d x)}}",1,"(-3*Cot[c + d*x]*(4*A*Hypergeometric2F1[1/3, 1/2, 4/3, Cos[c + d*x]^2] + C*Cos[c + d*x]^2*Hypergeometric2F1[1/2, 4/3, 7/3, Cos[c + d*x]^2])*Sqrt[Sin[c + d*x]^2])/(8*b*d*(b*Cos[c + d*x])^(1/3))","A",1
172,1,87,93,0.1533263,"\int \frac{A+C \cos ^2(c+d x)}{(b \cos (c+d x))^{4/3}} \, dx","Integrate[(A + C*Cos[c + d*x]^2)/(b*Cos[c + d*x])^(4/3),x]","-\frac{3 \sqrt{\sin ^2(c+d x)} \cot (c+d x) \left(C \cos ^2(c+d x) \, _2F_1\left(\frac{1}{2},\frac{5}{6};\frac{11}{6};\cos ^2(c+d x)\right)-5 A \, _2F_1\left(-\frac{1}{6},\frac{1}{2};\frac{5}{6};\cos ^2(c+d x)\right)\right)}{5 d (b \cos (c+d x))^{4/3}}","\frac{3 (2 A-C) \sin (c+d x) (b \cos (c+d x))^{5/3} \, _2F_1\left(\frac{1}{2},\frac{5}{6};\frac{11}{6};\cos ^2(c+d x)\right)}{5 b^3 d \sqrt{\sin ^2(c+d x)}}+\frac{3 A \sin (c+d x)}{b d \sqrt[3]{b \cos (c+d x)}}",1,"(-3*Cot[c + d*x]*(-5*A*Hypergeometric2F1[-1/6, 1/2, 5/6, Cos[c + d*x]^2] + C*Cos[c + d*x]^2*Hypergeometric2F1[1/2, 5/6, 11/6, Cos[c + d*x]^2])*Sqrt[Sin[c + d*x]^2])/(5*d*(b*Cos[c + d*x])^(4/3))","A",1
173,1,104,90,0.7033741,"\int \frac{\left(A+C \cos ^2(c+d x)\right) \sec (c+d x)}{(b \cos (c+d x))^{4/3}} \, dx","Integrate[((A + C*Cos[c + d*x]^2)*Sec[c + d*x])/(b*Cos[c + d*x])^(4/3),x]","\frac{6 A \tan (c+d x) \sqrt[3]{\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)}-(A+4 C) \sin \left(2 d x-2 \tan ^{-1}(\cot (c))\right) \, _2F_1\left(\frac{1}{2},\frac{2}{3};\frac{3}{2};\cos ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right)}{8 b d \sqrt[3]{b \cos (c+d x)} \sqrt[3]{\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)}}","\frac{3 A \sin (c+d x)}{4 d (b \cos (c+d x))^{4/3}}-\frac{3 (A+4 C) \sin (c+d x) (b \cos (c+d x))^{2/3} \, _2F_1\left(\frac{1}{3},\frac{1}{2};\frac{4}{3};\cos ^2(c+d x)\right)}{8 b^2 d \sqrt{\sin ^2(c+d x)}}",1,"(-((A + 4*C)*Hypergeometric2F1[1/2, 2/3, 3/2, Cos[d*x - ArcTan[Cot[c]]]^2]*Sin[2*d*x - 2*ArcTan[Cot[c]]]) + 6*A*(Sin[d*x - ArcTan[Cot[c]]]^2)^(1/3)*Tan[c + d*x])/(8*b*d*(b*Cos[c + d*x])^(1/3)*(Sin[d*x - ArcTan[Cot[c]]]^2)^(1/3))","A",1
174,1,90,93,0.2261512,"\int \frac{\left(A+C \cos ^2(c+d x)\right) \sec ^2(c+d x)}{(b \cos (c+d x))^{4/3}} \, dx","Integrate[((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^2)/(b*Cos[c + d*x])^(4/3),x]","\frac{3 b^2 \sqrt{\sin ^2(c+d x)} \cot (c+d x) \left(A \, _2F_1\left(-\frac{7}{6},\frac{1}{2};-\frac{1}{6};\cos ^2(c+d x)\right)+7 C \cos ^2(c+d x) \, _2F_1\left(-\frac{1}{6},\frac{1}{2};\frac{5}{6};\cos ^2(c+d x)\right)\right)}{7 d (b \cos (c+d x))^{10/3}}","\frac{3 (4 A+7 C) \sin (c+d x) \, _2F_1\left(-\frac{1}{6},\frac{1}{2};\frac{5}{6};\cos ^2(c+d x)\right)}{7 b d \sqrt{\sin ^2(c+d x)} \sqrt[3]{b \cos (c+d x)}}+\frac{3 A b \sin (c+d x)}{7 d (b \cos (c+d x))^{7/3}}",1,"(3*b^2*Cot[c + d*x]*(A*Hypergeometric2F1[-7/6, 1/2, -1/6, Cos[c + d*x]^2] + 7*C*Cos[c + d*x]^2*Hypergeometric2F1[-1/6, 1/2, 5/6, Cos[c + d*x]^2])*Sqrt[Sin[c + d*x]^2])/(7*d*(b*Cos[c + d*x])^(10/3))","A",1
175,1,91,92,0.2223572,"\int \frac{\left(A+C \cos ^2(c+d x)\right) \sec ^3(c+d x)}{(b \cos (c+d x))^{4/3}} \, dx","Integrate[((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^3)/(b*Cos[c + d*x])^(4/3),x]","\frac{3 b^2 \sqrt{\sin ^2(c+d x)} \csc (c+d x) \left(2 A \, _2F_1\left(-\frac{5}{3},\frac{1}{2};-\frac{2}{3};\cos ^2(c+d x)\right)+5 C \cos ^2(c+d x) \, _2F_1\left(-\frac{2}{3},\frac{1}{2};\frac{1}{3};\cos ^2(c+d x)\right)\right)}{20 d (b \cos (c+d x))^{10/3}}","\frac{3 A b^2 \sin (c+d x)}{10 d (b \cos (c+d x))^{10/3}}+\frac{3 (7 A+10 C) \sin (c+d x) \, _2F_1\left(-\frac{2}{3},\frac{1}{2};\frac{1}{3};\cos ^2(c+d x)\right)}{40 d \sqrt{\sin ^2(c+d x)} (b \cos (c+d x))^{4/3}}",1,"(3*b^2*Csc[c + d*x]*(2*A*Hypergeometric2F1[-5/3, 1/2, -2/3, Cos[c + d*x]^2] + 5*C*Cos[c + d*x]^2*Hypergeometric2F1[-2/3, 1/2, 1/3, Cos[c + d*x]^2])*Sqrt[Sin[c + d*x]^2])/(20*d*(b*Cos[c + d*x])^(10/3))","A",1
176,1,142,148,0.303626,"\int \cos ^m(c+d x) (b \cos (c+d x))^{4/3} \left(A+C \cos ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^m*(b*Cos[c + d*x])^(4/3)*(A + C*Cos[c + d*x]^2),x]","-\frac{3 \sqrt{\sin ^2(c+d x)} \csc (c+d x) (b \cos (c+d x))^{4/3} \cos ^{m+1}(c+d x) \left(A (3 m+13) \, _2F_1\left(\frac{1}{2},\frac{1}{6} (3 m+7);\frac{1}{6} (3 m+13);\cos ^2(c+d x)\right)+C (3 m+7) \cos ^2(c+d x) \, _2F_1\left(\frac{1}{2},\frac{1}{6} (3 m+13);\frac{1}{6} (3 m+19);\cos ^2(c+d x)\right)\right)}{d (3 m+7) (3 m+13)}","\frac{3 b C \sin (c+d x) \sqrt[3]{b \cos (c+d x)} \cos ^{m+2}(c+d x)}{d (3 m+10)}-\frac{3 b (A (3 m+10)+C (3 m+7)) \sin (c+d x) \sqrt[3]{b \cos (c+d x)} \cos ^{m+2}(c+d x) \, _2F_1\left(\frac{1}{2},\frac{1}{6} (3 m+7);\frac{1}{6} (3 m+13);\cos ^2(c+d x)\right)}{d (3 m+7) (3 m+10) \sqrt{\sin ^2(c+d x)}}",1,"(-3*Cos[c + d*x]^(1 + m)*(b*Cos[c + d*x])^(4/3)*Csc[c + d*x]*(A*(13 + 3*m)*Hypergeometric2F1[1/2, (7 + 3*m)/6, (13 + 3*m)/6, Cos[c + d*x]^2] + C*(7 + 3*m)*Cos[c + d*x]^2*Hypergeometric2F1[1/2, (13 + 3*m)/6, (19 + 3*m)/6, Cos[c + d*x]^2])*Sqrt[Sin[c + d*x]^2])/(d*(7 + 3*m)*(13 + 3*m))","A",1
177,1,142,146,0.2309826,"\int \cos ^m(c+d x) (b \cos (c+d x))^{2/3} \left(A+C \cos ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^m*(b*Cos[c + d*x])^(2/3)*(A + C*Cos[c + d*x]^2),x]","-\frac{3 \sqrt{\sin ^2(c+d x)} \csc (c+d x) (b \cos (c+d x))^{2/3} \cos ^{m+1}(c+d x) \left(A (3 m+11) \, _2F_1\left(\frac{1}{2},\frac{1}{6} (3 m+5);\frac{1}{6} (3 m+11);\cos ^2(c+d x)\right)+C (3 m+5) \cos ^2(c+d x) \, _2F_1\left(\frac{1}{2},\frac{1}{6} (3 m+11);\frac{1}{6} (3 m+17);\cos ^2(c+d x)\right)\right)}{d (3 m+5) (3 m+11)}","\frac{3 C \sin (c+d x) (b \cos (c+d x))^{2/3} \cos ^{m+1}(c+d x)}{d (3 m+8)}-\frac{3 (A (3 m+8)+C (3 m+5)) \sin (c+d x) (b \cos (c+d x))^{2/3} \cos ^{m+1}(c+d x) \, _2F_1\left(\frac{1}{2},\frac{1}{6} (3 m+5);\frac{1}{6} (3 m+11);\cos ^2(c+d x)\right)}{d (3 m+5) (3 m+8) \sqrt{\sin ^2(c+d x)}}",1,"(-3*Cos[c + d*x]^(1 + m)*(b*Cos[c + d*x])^(2/3)*Csc[c + d*x]*(A*(11 + 3*m)*Hypergeometric2F1[1/2, (5 + 3*m)/6, (11 + 3*m)/6, Cos[c + d*x]^2] + C*(5 + 3*m)*Cos[c + d*x]^2*Hypergeometric2F1[1/2, (11 + 3*m)/6, (17 + 3*m)/6, Cos[c + d*x]^2])*Sqrt[Sin[c + d*x]^2])/(d*(5 + 3*m)*(11 + 3*m))","A",1
178,1,142,146,0.3244512,"\int \cos ^m(c+d x) \sqrt[3]{b \cos (c+d x)} \left(A+C \cos ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^m*(b*Cos[c + d*x])^(1/3)*(A + C*Cos[c + d*x]^2),x]","-\frac{3 \sqrt{\sin ^2(c+d x)} \csc (c+d x) \sqrt[3]{b \cos (c+d x)} \cos ^{m+1}(c+d x) \left(A (3 m+10) \, _2F_1\left(\frac{1}{2},\frac{1}{6} (3 m+4);\frac{m}{2}+\frac{5}{3};\cos ^2(c+d x)\right)+C (3 m+4) \cos ^2(c+d x) \, _2F_1\left(\frac{1}{2},\frac{m}{2}+\frac{5}{3};\frac{m}{2}+\frac{8}{3};\cos ^2(c+d x)\right)\right)}{d (3 m+4) (3 m+10)}","\frac{3 C \sin (c+d x) \sqrt[3]{b \cos (c+d x)} \cos ^{m+1}(c+d x)}{d (3 m+7)}-\frac{3 (A (3 m+7)+C (3 m+4)) \sin (c+d x) \sqrt[3]{b \cos (c+d x)} \cos ^{m+1}(c+d x) \, _2F_1\left(\frac{1}{2},\frac{1}{6} (3 m+4);\frac{1}{6} (3 m+10);\cos ^2(c+d x)\right)}{d (3 m+4) (3 m+7) \sqrt{\sin ^2(c+d x)}}",1,"(-3*Cos[c + d*x]^(1 + m)*(b*Cos[c + d*x])^(1/3)*Csc[c + d*x]*(C*(4 + 3*m)*Cos[c + d*x]^2*Hypergeometric2F1[1/2, 5/3 + m/2, 8/3 + m/2, Cos[c + d*x]^2] + A*(10 + 3*m)*Hypergeometric2F1[1/2, (4 + 3*m)/6, 5/3 + m/2, Cos[c + d*x]^2])*Sqrt[Sin[c + d*x]^2])/(d*(4 + 3*m)*(10 + 3*m))","A",1
179,1,142,146,0.2973258,"\int \frac{\cos ^m(c+d x) \left(A+C \cos ^2(c+d x)\right)}{\sqrt[3]{b \cos (c+d x)}} \, dx","Integrate[(Cos[c + d*x]^m*(A + C*Cos[c + d*x]^2))/(b*Cos[c + d*x])^(1/3),x]","-\frac{3 \sqrt{\sin ^2(c+d x)} \csc (c+d x) \cos ^{m+1}(c+d x) \left(A (3 m+8) \, _2F_1\left(\frac{1}{2},\frac{1}{6} (3 m+2);\frac{1}{6} (3 m+8);\cos ^2(c+d x)\right)+C (3 m+2) \cos ^2(c+d x) \, _2F_1\left(\frac{1}{2},\frac{1}{6} (3 m+8);\frac{m}{2}+\frac{7}{3};\cos ^2(c+d x)\right)\right)}{d (3 m+2) (3 m+8) \sqrt[3]{b \cos (c+d x)}}","\frac{3 C \sin (c+d x) \cos ^{m+1}(c+d x)}{d (3 m+5) \sqrt[3]{b \cos (c+d x)}}-\frac{3 (A (3 m+5)+C (3 m+2)) \sin (c+d x) \cos ^{m+1}(c+d x) \, _2F_1\left(\frac{1}{2},\frac{1}{6} (3 m+2);\frac{1}{6} (3 m+8);\cos ^2(c+d x)\right)}{d (3 m+2) (3 m+5) \sqrt{\sin ^2(c+d x)} \sqrt[3]{b \cos (c+d x)}}",1,"(-3*Cos[c + d*x]^(1 + m)*Csc[c + d*x]*(A*(8 + 3*m)*Hypergeometric2F1[1/2, (2 + 3*m)/6, (8 + 3*m)/6, Cos[c + d*x]^2] + C*(2 + 3*m)*Cos[c + d*x]^2*Hypergeometric2F1[1/2, (8 + 3*m)/6, 7/3 + m/2, Cos[c + d*x]^2])*Sqrt[Sin[c + d*x]^2])/(d*(2 + 3*m)*(8 + 3*m)*(b*Cos[c + d*x])^(1/3))","A",1
180,1,142,144,0.2653421,"\int \frac{\cos ^m(c+d x) \left(A+C \cos ^2(c+d x)\right)}{(b \cos (c+d x))^{2/3}} \, dx","Integrate[(Cos[c + d*x]^m*(A + C*Cos[c + d*x]^2))/(b*Cos[c + d*x])^(2/3),x]","-\frac{3 \sqrt{\sin ^2(c+d x)} \csc (c+d x) \cos ^{m+1}(c+d x) \left(A (3 m+7) \, _2F_1\left(\frac{1}{2},\frac{1}{6} (3 m+1);\frac{1}{6} (3 m+7);\cos ^2(c+d x)\right)+C (3 m+1) \cos ^2(c+d x) \, _2F_1\left(\frac{1}{2},\frac{1}{6} (3 m+7);\frac{1}{6} (3 m+13);\cos ^2(c+d x)\right)\right)}{d (3 m+1) (3 m+7) (b \cos (c+d x))^{2/3}}","\frac{3 C \sin (c+d x) \cos ^{m+1}(c+d x)}{d (3 m+4) (b \cos (c+d x))^{2/3}}-\frac{3 (A (3 m+4)+3 C m+C) \sin (c+d x) \cos ^{m+1}(c+d x) \, _2F_1\left(\frac{1}{2},\frac{1}{6} (3 m+1);\frac{1}{6} (3 m+7);\cos ^2(c+d x)\right)}{d (3 m+1) (3 m+4) \sqrt{\sin ^2(c+d x)} (b \cos (c+d x))^{2/3}}",1,"(-3*Cos[c + d*x]^(1 + m)*Csc[c + d*x]*(A*(7 + 3*m)*Hypergeometric2F1[1/2, (1 + 3*m)/6, (7 + 3*m)/6, Cos[c + d*x]^2] + C*(1 + 3*m)*Cos[c + d*x]^2*Hypergeometric2F1[1/2, (7 + 3*m)/6, (13 + 3*m)/6, Cos[c + d*x]^2])*Sqrt[Sin[c + d*x]^2])/(d*(1 + 3*m)*(7 + 3*m)*(b*Cos[c + d*x])^(2/3))","A",1
181,1,142,149,0.2738685,"\int \frac{\cos ^m(c+d x) \left(A+C \cos ^2(c+d x)\right)}{(b \cos (c+d x))^{4/3}} \, dx","Integrate[(Cos[c + d*x]^m*(A + C*Cos[c + d*x]^2))/(b*Cos[c + d*x])^(4/3),x]","-\frac{3 \sqrt{\sin ^2(c+d x)} \csc (c+d x) \cos ^{m+1}(c+d x) \left(A (3 m+5) \, _2F_1\left(\frac{1}{2},\frac{1}{6} (3 m-1);\frac{1}{6} (3 m+5);\cos ^2(c+d x)\right)+C (3 m-1) \cos ^2(c+d x) \, _2F_1\left(\frac{1}{2},\frac{1}{6} (3 m+5);\frac{1}{6} (3 m+11);\cos ^2(c+d x)\right)\right)}{d (3 m-1) (3 m+5) (b \cos (c+d x))^{4/3}}","\frac{3 C \sin (c+d x) \cos ^m(c+d x)}{b d (3 m+2) \sqrt[3]{b \cos (c+d x)}}-\frac{3 (C (1-3 m)-A (3 m+2)) \sin (c+d x) \cos ^m(c+d x) \, _2F_1\left(\frac{1}{2},\frac{1}{6} (3 m-1);\frac{1}{6} (3 m+5);\cos ^2(c+d x)\right)}{b d (1-3 m) (3 m+2) \sqrt{\sin ^2(c+d x)} \sqrt[3]{b \cos (c+d x)}}",1,"(-3*Cos[c + d*x]^(1 + m)*Csc[c + d*x]*(A*(5 + 3*m)*Hypergeometric2F1[1/2, (-1 + 3*m)/6, (5 + 3*m)/6, Cos[c + d*x]^2] + C*(-1 + 3*m)*Cos[c + d*x]^2*Hypergeometric2F1[1/2, (5 + 3*m)/6, (11 + 3*m)/6, Cos[c + d*x]^2])*Sqrt[Sin[c + d*x]^2])/(d*(-1 + 3*m)*(5 + 3*m)*(b*Cos[c + d*x])^(4/3))","A",1
182,1,132,144,0.2608626,"\int (a \cos (c+d x))^m (b \cos (c+d x))^n \left(A+C \cos ^2(c+d x)\right) \, dx","Integrate[(a*Cos[c + d*x])^m*(b*Cos[c + d*x])^n*(A + C*Cos[c + d*x]^2),x]","-\frac{\sqrt{\sin ^2(c+d x)} \cot (c+d x) (a \cos (c+d x))^m (b \cos (c+d x))^n \left(A (m+n+3) \, _2F_1\left(\frac{1}{2},\frac{1}{2} (m+n+1);\frac{1}{2} (m+n+3);\cos ^2(c+d x)\right)+C (m+n+1) \cos ^2(c+d x) \, _2F_1\left(\frac{1}{2},\frac{1}{2} (m+n+3);\frac{1}{2} (m+n+5);\cos ^2(c+d x)\right)\right)}{d (m+n+1) (m+n+3)}","\frac{C \sin (c+d x) (a \cos (c+d x))^{m+1} (b \cos (c+d x))^n}{a d (m+n+2)}-\frac{(A (m+n+2)+C (m+n+1)) \sin (c+d x) (a \cos (c+d x))^{m+1} (b \cos (c+d x))^n \, _2F_1\left(\frac{1}{2},\frac{1}{2} (m+n+1);\frac{1}{2} (m+n+3);\cos ^2(c+d x)\right)}{a d (m+n+1) (m+n+2) \sqrt{\sin ^2(c+d x)}}",1,"-(((a*Cos[c + d*x])^m*(b*Cos[c + d*x])^n*Cot[c + d*x]*(A*(3 + m + n)*Hypergeometric2F1[1/2, (1 + m + n)/2, (3 + m + n)/2, Cos[c + d*x]^2] + C*(1 + m + n)*Cos[c + d*x]^2*Hypergeometric2F1[1/2, (3 + m + n)/2, (5 + m + n)/2, Cos[c + d*x]^2])*Sqrt[Sin[c + d*x]^2])/(d*(1 + m + n)*(3 + m + n)))","A",1
183,1,122,117,0.2187416,"\int \cos ^2(c+d x) (b \cos (c+d x))^n \left(A+C \cos ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^2*(b*Cos[c + d*x])^n*(A + C*Cos[c + d*x]^2),x]","-\frac{\sqrt{\sin ^2(c+d x)} \cos ^2(c+d x) \cot (c+d x) (b \cos (c+d x))^n \left(A (n+5) \, _2F_1\left(\frac{1}{2},\frac{n+3}{2};\frac{n+5}{2};\cos ^2(c+d x)\right)+C (n+3) \cos ^2(c+d x) \, _2F_1\left(\frac{1}{2},\frac{n+5}{2};\frac{n+7}{2};\cos ^2(c+d x)\right)\right)}{d (n+3) (n+5)}","\frac{C \sin (c+d x) (b \cos (c+d x))^{n+3}}{b^3 d (n+4)}-\frac{(A (n+4)+C (n+3)) \sin (c+d x) (b \cos (c+d x))^{n+3} \, _2F_1\left(\frac{1}{2},\frac{n+3}{2};\frac{n+5}{2};\cos ^2(c+d x)\right)}{b^3 d (n+3) (n+4) \sqrt{\sin ^2(c+d x)}}",1,"-((Cos[c + d*x]^2*(b*Cos[c + d*x])^n*Cot[c + d*x]*(A*(5 + n)*Hypergeometric2F1[1/2, (3 + n)/2, (5 + n)/2, Cos[c + d*x]^2] + C*(3 + n)*Cos[c + d*x]^2*Hypergeometric2F1[1/2, (5 + n)/2, (7 + n)/2, Cos[c + d*x]^2])*Sqrt[Sin[c + d*x]^2])/(d*(3 + n)*(5 + n)))","A",1
184,1,120,117,0.1930113,"\int \cos (c+d x) (b \cos (c+d x))^n \left(A+C \cos ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]*(b*Cos[c + d*x])^n*(A + C*Cos[c + d*x]^2),x]","-\frac{\sqrt{\sin ^2(c+d x)} \cos (c+d x) \cot (c+d x) (b \cos (c+d x))^n \left(A (n+4) \, _2F_1\left(\frac{1}{2},\frac{n+2}{2};\frac{n+4}{2};\cos ^2(c+d x)\right)+C (n+2) \cos ^2(c+d x) \, _2F_1\left(\frac{1}{2},\frac{n+4}{2};\frac{n+6}{2};\cos ^2(c+d x)\right)\right)}{d (n+2) (n+4)}","\frac{C \sin (c+d x) (b \cos (c+d x))^{n+2}}{b^2 d (n+3)}-\frac{(A (n+3)+C (n+2)) \sin (c+d x) (b \cos (c+d x))^{n+2} \, _2F_1\left(\frac{1}{2},\frac{n+2}{2};\frac{n+4}{2};\cos ^2(c+d x)\right)}{b^2 d (n+2) (n+3) \sqrt{\sin ^2(c+d x)}}",1,"-((Cos[c + d*x]*(b*Cos[c + d*x])^n*Cot[c + d*x]*(A*(4 + n)*Hypergeometric2F1[1/2, (2 + n)/2, (4 + n)/2, Cos[c + d*x]^2] + C*(2 + n)*Cos[c + d*x]^2*Hypergeometric2F1[1/2, (4 + n)/2, (6 + n)/2, Cos[c + d*x]^2])*Sqrt[Sin[c + d*x]^2])/(d*(2 + n)*(4 + n)))","A",1
185,1,114,117,0.1764025,"\int (b \cos (c+d x))^n \left(A+C \cos ^2(c+d x)\right) \, dx","Integrate[(b*Cos[c + d*x])^n*(A + C*Cos[c + d*x]^2),x]","-\frac{\sqrt{\sin ^2(c+d x)} \cot (c+d x) (b \cos (c+d x))^n \left(A (n+3) \, _2F_1\left(\frac{1}{2},\frac{n+1}{2};\frac{n+3}{2};\cos ^2(c+d x)\right)+C (n+1) \cos ^2(c+d x) \, _2F_1\left(\frac{1}{2},\frac{n+3}{2};\frac{n+5}{2};\cos ^2(c+d x)\right)\right)}{d (n+1) (n+3)}","\frac{C \sin (c+d x) (b \cos (c+d x))^{n+1}}{b d (n+2)}-\frac{(A (n+2)+C (n+1)) \sin (c+d x) (b \cos (c+d x))^{n+1} \, _2F_1\left(\frac{1}{2},\frac{n+1}{2};\frac{n+3}{2};\cos ^2(c+d x)\right)}{b d (n+1) (n+2) \sqrt{\sin ^2(c+d x)}}",1,"-(((b*Cos[c + d*x])^n*Cot[c + d*x]*(A*(3 + n)*Hypergeometric2F1[1/2, (1 + n)/2, (3 + n)/2, Cos[c + d*x]^2] + C*(1 + n)*Cos[c + d*x]^2*Hypergeometric2F1[1/2, (3 + n)/2, (5 + n)/2, Cos[c + d*x]^2])*Sqrt[Sin[c + d*x]^2])/(d*(1 + n)*(3 + n)))","A",1
186,1,111,100,0.243885,"\int (b \cos (c+d x))^n \left(A+C \cos ^2(c+d x)\right) \sec (c+d x) \, dx","Integrate[(b*Cos[c + d*x])^n*(A + C*Cos[c + d*x]^2)*Sec[c + d*x],x]","-\frac{b \sqrt{\sin ^2(c+d x)} \cot (c+d x) (b \cos (c+d x))^{n-1} \left(A (n+2) \, _2F_1\left(\frac{1}{2},\frac{n}{2};\frac{n+2}{2};\cos ^2(c+d x)\right)+C n \cos ^2(c+d x) \, _2F_1\left(\frac{1}{2},\frac{n+2}{2};\frac{n+4}{2};\cos ^2(c+d x)\right)\right)}{d n (n+2)}","\frac{C \sin (c+d x) (b \cos (c+d x))^n}{d (n+1)}-\frac{(A n+A+C n) \sin (c+d x) (b \cos (c+d x))^n \, _2F_1\left(\frac{1}{2},\frac{n}{2};\frac{n+2}{2};\cos ^2(c+d x)\right)}{d n (n+1) \sqrt{\sin ^2(c+d x)}}",1,"-((b*(b*Cos[c + d*x])^(-1 + n)*Cot[c + d*x]*(A*(2 + n)*Hypergeometric2F1[1/2, n/2, (2 + n)/2, Cos[c + d*x]^2] + C*n*Cos[c + d*x]^2*Hypergeometric2F1[1/2, (2 + n)/2, (4 + n)/2, Cos[c + d*x]^2])*Sqrt[Sin[c + d*x]^2])/(d*n*(2 + n)))","A",1
187,1,117,112,0.2186743,"\int (b \cos (c+d x))^n \left(A+C \cos ^2(c+d x)\right) \sec ^2(c+d x) \, dx","Integrate[(b*Cos[c + d*x])^n*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^2,x]","-\frac{b \sqrt{\sin ^2(c+d x)} \csc (c+d x) (b \cos (c+d x))^{n-1} \left(A (n+1) \, _2F_1\left(\frac{1}{2},\frac{n-1}{2};\frac{n+1}{2};\cos ^2(c+d x)\right)+C (n-1) \cos ^2(c+d x) \, _2F_1\left(\frac{1}{2},\frac{n+1}{2};\frac{n+3}{2};\cos ^2(c+d x)\right)\right)}{d (n-1) (n+1)}","\frac{b C \sin (c+d x) (b \cos (c+d x))^{n-1}}{d n}-\frac{b (C (1-n)-A n) \sin (c+d x) (b \cos (c+d x))^{n-1} \, _2F_1\left(\frac{1}{2},\frac{n-1}{2};\frac{n+1}{2};\cos ^2(c+d x)\right)}{d (1-n) n \sqrt{\sin ^2(c+d x)}}",1,"-((b*(b*Cos[c + d*x])^(-1 + n)*Csc[c + d*x]*(A*(1 + n)*Hypergeometric2F1[1/2, (-1 + n)/2, (1 + n)/2, Cos[c + d*x]^2] + C*(-1 + n)*Cos[c + d*x]^2*Hypergeometric2F1[1/2, (1 + n)/2, (3 + n)/2, Cos[c + d*x]^2])*Sqrt[Sin[c + d*x]^2])/(d*(-1 + n)*(1 + n)))","A",1
188,1,114,125,0.1675245,"\int (b \cos (c+d x))^n \left(A+C \cos ^2(c+d x)\right) \sec ^3(c+d x) \, dx","Integrate[(b*Cos[c + d*x])^n*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^3,x]","-\frac{\sqrt{\sin ^2(c+d x)} \csc (c+d x) \sec ^2(c+d x) (b \cos (c+d x))^n \left(A n \, _2F_1\left(\frac{1}{2},\frac{n-2}{2};\frac{n}{2};\cos ^2(c+d x)\right)+C (n-2) \cos ^2(c+d x) \, _2F_1\left(\frac{1}{2},\frac{n}{2};\frac{n+2}{2};\cos ^2(c+d x)\right)\right)}{d (n-2) n}","\frac{b^2 (A (1-n)+C (2-n)) \sin (c+d x) (b \cos (c+d x))^{n-2} \, _2F_1\left(\frac{1}{2},\frac{n-2}{2};\frac{n}{2};\cos ^2(c+d x)\right)}{d (1-n) (2-n) \sqrt{\sin ^2(c+d x)}}-\frac{b^2 C \sin (c+d x) (b \cos (c+d x))^{n-2}}{d (1-n)}",1,"-(((b*Cos[c + d*x])^n*Csc[c + d*x]*(A*n*Hypergeometric2F1[1/2, (-2 + n)/2, n/2, Cos[c + d*x]^2] + C*(-2 + n)*Cos[c + d*x]^2*Hypergeometric2F1[1/2, n/2, (2 + n)/2, Cos[c + d*x]^2])*Sec[c + d*x]^2*Sqrt[Sin[c + d*x]^2])/(d*(-2 + n)*n))","A",1
189,1,122,127,0.1727933,"\int (b \cos (c+d x))^n \left(A+C \cos ^2(c+d x)\right) \sec ^4(c+d x) \, dx","Integrate[(b*Cos[c + d*x])^n*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^4,x]","-\frac{\sqrt{\sin ^2(c+d x)} \csc (c+d x) \sec ^3(c+d x) (b \cos (c+d x))^n \left(A (n-1) \, _2F_1\left(\frac{1}{2},\frac{n-3}{2};\frac{n-1}{2};\cos ^2(c+d x)\right)+C (n-3) \cos ^2(c+d x) \, _2F_1\left(\frac{1}{2},\frac{n-1}{2};\frac{n+1}{2};\cos ^2(c+d x)\right)\right)}{d (n-3) (n-1)}","\frac{b^3 (A (2-n)+C (3-n)) \sin (c+d x) (b \cos (c+d x))^{n-3} \, _2F_1\left(\frac{1}{2},\frac{n-3}{2};\frac{n-1}{2};\cos ^2(c+d x)\right)}{d (2-n) (3-n) \sqrt{\sin ^2(c+d x)}}-\frac{b^3 C \sin (c+d x) (b \cos (c+d x))^{n-3}}{d (2-n)}",1,"-(((b*Cos[c + d*x])^n*Csc[c + d*x]*(A*(-1 + n)*Hypergeometric2F1[1/2, (-3 + n)/2, (-1 + n)/2, Cos[c + d*x]^2] + C*(-3 + n)*Cos[c + d*x]^2*Hypergeometric2F1[1/2, (-1 + n)/2, (1 + n)/2, Cos[c + d*x]^2])*Sec[c + d*x]^3*Sqrt[Sin[c + d*x]^2])/(d*(-3 + n)*(-1 + n)))","A",1
190,1,140,142,0.2460271,"\int \cos ^{\frac{5}{2}}(c+d x) (b \cos (c+d x))^n \left(A+C \cos ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^(5/2)*(b*Cos[c + d*x])^n*(A + C*Cos[c + d*x]^2),x]","-\frac{2 \sqrt{\sin ^2(c+d x)} \cos ^{\frac{7}{2}}(c+d x) \csc (c+d x) (b \cos (c+d x))^n \left(A (2 n+11) \, _2F_1\left(\frac{1}{2},\frac{1}{4} (2 n+7);\frac{1}{4} (2 n+11);\cos ^2(c+d x)\right)+C (2 n+7) \cos ^2(c+d x) \, _2F_1\left(\frac{1}{2},\frac{1}{4} (2 n+11);\frac{1}{4} (2 n+15);\cos ^2(c+d x)\right)\right)}{d (2 n+7) (2 n+11)}","\frac{2 C \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x) (b \cos (c+d x))^n}{d (2 n+9)}-\frac{2 (A (2 n+9)+C (2 n+7)) \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x) (b \cos (c+d x))^n \, _2F_1\left(\frac{1}{2},\frac{1}{4} (2 n+7);\frac{1}{4} (2 n+11);\cos ^2(c+d x)\right)}{d (2 n+7) (2 n+9) \sqrt{\sin ^2(c+d x)}}",1,"(-2*Cos[c + d*x]^(7/2)*(b*Cos[c + d*x])^n*Csc[c + d*x]*(A*(11 + 2*n)*Hypergeometric2F1[1/2, (7 + 2*n)/4, (11 + 2*n)/4, Cos[c + d*x]^2] + C*(7 + 2*n)*Cos[c + d*x]^2*Hypergeometric2F1[1/2, (11 + 2*n)/4, (15 + 2*n)/4, Cos[c + d*x]^2])*Sqrt[Sin[c + d*x]^2])/(d*(7 + 2*n)*(11 + 2*n))","A",1
191,1,140,142,0.21364,"\int \cos ^{\frac{3}{2}}(c+d x) (b \cos (c+d x))^n \left(A+C \cos ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^(3/2)*(b*Cos[c + d*x])^n*(A + C*Cos[c + d*x]^2),x]","-\frac{2 \sqrt{\sin ^2(c+d x)} \cos ^{\frac{5}{2}}(c+d x) \csc (c+d x) (b \cos (c+d x))^n \left(A (2 n+9) \, _2F_1\left(\frac{1}{2},\frac{1}{4} (2 n+5);\frac{1}{4} (2 n+9);\cos ^2(c+d x)\right)+C (2 n+5) \cos ^2(c+d x) \, _2F_1\left(\frac{1}{2},\frac{1}{4} (2 n+9);\frac{1}{4} (2 n+13);\cos ^2(c+d x)\right)\right)}{d (2 n+5) (2 n+9)}","\frac{2 C \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) (b \cos (c+d x))^n}{d (2 n+7)}-\frac{2 (A (2 n+7)+C (2 n+5)) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) (b \cos (c+d x))^n \, _2F_1\left(\frac{1}{2},\frac{1}{4} (2 n+5);\frac{1}{4} (2 n+9);\cos ^2(c+d x)\right)}{d (2 n+5) (2 n+7) \sqrt{\sin ^2(c+d x)}}",1,"(-2*Cos[c + d*x]^(5/2)*(b*Cos[c + d*x])^n*Csc[c + d*x]*(A*(9 + 2*n)*Hypergeometric2F1[1/2, (5 + 2*n)/4, (9 + 2*n)/4, Cos[c + d*x]^2] + C*(5 + 2*n)*Cos[c + d*x]^2*Hypergeometric2F1[1/2, (9 + 2*n)/4, (13 + 2*n)/4, Cos[c + d*x]^2])*Sqrt[Sin[c + d*x]^2])/(d*(5 + 2*n)*(9 + 2*n))","A",1
192,1,140,142,0.1912785,"\int \sqrt{\cos (c+d x)} (b \cos (c+d x))^n \left(A+C \cos ^2(c+d x)\right) \, dx","Integrate[Sqrt[Cos[c + d*x]]*(b*Cos[c + d*x])^n*(A + C*Cos[c + d*x]^2),x]","-\frac{2 \sqrt{\sin ^2(c+d x)} \cos ^{\frac{3}{2}}(c+d x) \csc (c+d x) (b \cos (c+d x))^n \left(A (2 n+7) \, _2F_1\left(\frac{1}{2},\frac{1}{4} (2 n+3);\frac{1}{4} (2 n+7);\cos ^2(c+d x)\right)+C (2 n+3) \cos ^2(c+d x) \, _2F_1\left(\frac{1}{2},\frac{1}{4} (2 n+7);\frac{1}{4} (2 n+11);\cos ^2(c+d x)\right)\right)}{d (2 n+3) (2 n+7)}","\frac{2 C \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) (b \cos (c+d x))^n}{d (2 n+5)}-\frac{2 (A (2 n+5)+C (2 n+3)) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) (b \cos (c+d x))^n \, _2F_1\left(\frac{1}{2},\frac{1}{4} (2 n+3);\frac{1}{4} (2 n+7);\cos ^2(c+d x)\right)}{d (2 n+3) (2 n+5) \sqrt{\sin ^2(c+d x)}}",1,"(-2*Cos[c + d*x]^(3/2)*(b*Cos[c + d*x])^n*Csc[c + d*x]*(A*(7 + 2*n)*Hypergeometric2F1[1/2, (3 + 2*n)/4, (7 + 2*n)/4, Cos[c + d*x]^2] + C*(3 + 2*n)*Cos[c + d*x]^2*Hypergeometric2F1[1/2, (7 + 2*n)/4, (11 + 2*n)/4, Cos[c + d*x]^2])*Sqrt[Sin[c + d*x]^2])/(d*(3 + 2*n)*(7 + 2*n))","A",1
193,1,140,140,0.1882841,"\int \frac{(b \cos (c+d x))^n \left(A+C \cos ^2(c+d x)\right)}{\sqrt{\cos (c+d x)}} \, dx","Integrate[((b*Cos[c + d*x])^n*(A + C*Cos[c + d*x]^2))/Sqrt[Cos[c + d*x]],x]","-\frac{2 \sqrt{\sin ^2(c+d x)} \sqrt{\cos (c+d x)} \csc (c+d x) (b \cos (c+d x))^n \left(A (2 n+5) \, _2F_1\left(\frac{1}{2},\frac{1}{4} (2 n+1);\frac{1}{4} (2 n+5);\cos ^2(c+d x)\right)+C (2 n+1) \cos ^2(c+d x) \, _2F_1\left(\frac{1}{2},\frac{1}{4} (2 n+5);\frac{1}{4} (2 n+9);\cos ^2(c+d x)\right)\right)}{d (2 n+1) (2 n+5)}","\frac{2 C \sin (c+d x) \sqrt{\cos (c+d x)} (b \cos (c+d x))^n}{d (2 n+3)}-\frac{2 (A (2 n+3)+2 C n+C) \sin (c+d x) \sqrt{\cos (c+d x)} (b \cos (c+d x))^n \, _2F_1\left(\frac{1}{2},\frac{1}{4} (2 n+1);\frac{1}{4} (2 n+5);\cos ^2(c+d x)\right)}{d (2 n+1) (2 n+3) \sqrt{\sin ^2(c+d x)}}",1,"(-2*Sqrt[Cos[c + d*x]]*(b*Cos[c + d*x])^n*Csc[c + d*x]*(A*(5 + 2*n)*Hypergeometric2F1[1/2, (1 + 2*n)/4, (5 + 2*n)/4, Cos[c + d*x]^2] + C*(1 + 2*n)*Cos[c + d*x]^2*Hypergeometric2F1[1/2, (5 + 2*n)/4, (9 + 2*n)/4, Cos[c + d*x]^2])*Sqrt[Sin[c + d*x]^2])/(d*(1 + 2*n)*(5 + 2*n))","A",1
194,1,140,136,0.1827089,"\int \frac{(b \cos (c+d x))^n \left(A+C \cos ^2(c+d x)\right)}{\cos ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[((b*Cos[c + d*x])^n*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(3/2),x]","-\frac{2 \sqrt{\sin ^2(c+d x)} \csc (c+d x) (b \cos (c+d x))^n \left(A (2 n+3) \, _2F_1\left(\frac{1}{2},\frac{1}{4} (2 n-1);\frac{1}{4} (2 n+3);\cos ^2(c+d x)\right)+C (2 n-1) \cos ^2(c+d x) \, _2F_1\left(\frac{1}{2},\frac{1}{4} (2 n+3);\frac{1}{4} (2 n+7);\cos ^2(c+d x)\right)\right)}{d (2 n-1) (2 n+3) \sqrt{\cos (c+d x)}}","\frac{2 (2 A n+A-C (1-2 n)) \sin (c+d x) (b \cos (c+d x))^n \, _2F_1\left(\frac{1}{2},\frac{1}{4} (2 n-1);\frac{1}{4} (2 n+3);\cos ^2(c+d x)\right)}{d \left(1-4 n^2\right) \sqrt{\sin ^2(c+d x)} \sqrt{\cos (c+d x)}}+\frac{2 C \sin (c+d x) (b \cos (c+d x))^n}{d (2 n+1) \sqrt{\cos (c+d x)}}",1,"(-2*(b*Cos[c + d*x])^n*Csc[c + d*x]*(A*(3 + 2*n)*Hypergeometric2F1[1/2, (-1 + 2*n)/4, (3 + 2*n)/4, Cos[c + d*x]^2] + C*(-1 + 2*n)*Cos[c + d*x]^2*Hypergeometric2F1[1/2, (3 + 2*n)/4, (7 + 2*n)/4, Cos[c + d*x]^2])*Sqrt[Sin[c + d*x]^2])/(d*(-1 + 2*n)*(3 + 2*n)*Sqrt[Cos[c + d*x]])","A",1
195,1,140,140,0.1824855,"\int \frac{(b \cos (c+d x))^n \left(A+C \cos ^2(c+d x)\right)}{\cos ^{\frac{5}{2}}(c+d x)} \, dx","Integrate[((b*Cos[c + d*x])^n*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(5/2),x]","-\frac{2 \sqrt{\sin ^2(c+d x)} \csc (c+d x) (b \cos (c+d x))^n \left(A (2 n+1) \, _2F_1\left(\frac{1}{2},\frac{1}{4} (2 n-3);\frac{1}{4} (2 n+1);\cos ^2(c+d x)\right)+C (2 n-3) \cos ^2(c+d x) \, _2F_1\left(\frac{1}{2},\frac{1}{4} (2 n+1);\frac{1}{4} (2 n+5);\cos ^2(c+d x)\right)\right)}{d (2 n-3) (2 n+1) \cos ^{\frac{3}{2}}(c+d x)}","\frac{2 (-2 A n+A+C (3-2 n)) \sin (c+d x) (b \cos (c+d x))^n \, _2F_1\left(\frac{1}{2},\frac{1}{4} (2 n-3);\frac{1}{4} (2 n+1);\cos ^2(c+d x)\right)}{d (1-2 n) (3-2 n) \sqrt{\sin ^2(c+d x)} \cos ^{\frac{3}{2}}(c+d x)}-\frac{2 C \sin (c+d x) (b \cos (c+d x))^n}{d (1-2 n) \cos ^{\frac{3}{2}}(c+d x)}",1,"(-2*(b*Cos[c + d*x])^n*Csc[c + d*x]*(A*(1 + 2*n)*Hypergeometric2F1[1/2, (-3 + 2*n)/4, (1 + 2*n)/4, Cos[c + d*x]^2] + C*(-3 + 2*n)*Cos[c + d*x]^2*Hypergeometric2F1[1/2, (1 + 2*n)/4, (5 + 2*n)/4, Cos[c + d*x]^2])*Sqrt[Sin[c + d*x]^2])/(d*(-3 + 2*n)*(1 + 2*n)*Cos[c + d*x]^(3/2))","A",1
196,1,140,142,0.1925594,"\int \frac{(b \cos (c+d x))^n \left(A+C \cos ^2(c+d x)\right)}{\cos ^{\frac{7}{2}}(c+d x)} \, dx","Integrate[((b*Cos[c + d*x])^n*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(7/2),x]","-\frac{2 \sqrt{\sin ^2(c+d x)} \csc (c+d x) (b \cos (c+d x))^n \left(A (2 n-1) \, _2F_1\left(\frac{1}{2},\frac{1}{4} (2 n-5);\frac{1}{4} (2 n-1);\cos ^2(c+d x)\right)+C (2 n-5) \cos ^2(c+d x) \, _2F_1\left(\frac{1}{2},\frac{1}{4} (2 n-1);\frac{1}{4} (2 n+3);\cos ^2(c+d x)\right)\right)}{d (2 n-5) (2 n-1) \cos ^{\frac{5}{2}}(c+d x)}","\frac{2 (A (3-2 n)+C (5-2 n)) \sin (c+d x) (b \cos (c+d x))^n \, _2F_1\left(\frac{1}{2},\frac{1}{4} (2 n-5);\frac{1}{4} (2 n-1);\cos ^2(c+d x)\right)}{d (3-2 n) (5-2 n) \sqrt{\sin ^2(c+d x)} \cos ^{\frac{5}{2}}(c+d x)}-\frac{2 C \sin (c+d x) (b \cos (c+d x))^n}{d (3-2 n) \cos ^{\frac{5}{2}}(c+d x)}",1,"(-2*(b*Cos[c + d*x])^n*Csc[c + d*x]*(A*(-1 + 2*n)*Hypergeometric2F1[1/2, (-5 + 2*n)/4, (-1 + 2*n)/4, Cos[c + d*x]^2] + C*(-5 + 2*n)*Cos[c + d*x]^2*Hypergeometric2F1[1/2, (-1 + 2*n)/4, (3 + 2*n)/4, Cos[c + d*x]^2])*Sqrt[Sin[c + d*x]^2])/(d*(-5 + 2*n)*(-1 + 2*n)*Cos[c + d*x]^(5/2))","A",1
197,1,140,142,0.1891969,"\int \frac{(b \cos (c+d x))^n \left(A+C \cos ^2(c+d x)\right)}{\cos ^{\frac{9}{2}}(c+d x)} \, dx","Integrate[((b*Cos[c + d*x])^n*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(9/2),x]","-\frac{2 \sqrt{\sin ^2(c+d x)} \csc (c+d x) (b \cos (c+d x))^n \left(A (2 n-3) \, _2F_1\left(\frac{1}{2},\frac{1}{4} (2 n-7);\frac{1}{4} (2 n-3);\cos ^2(c+d x)\right)+C (2 n-7) \cos ^2(c+d x) \, _2F_1\left(\frac{1}{2},\frac{1}{4} (2 n-3);\frac{1}{4} (2 n+1);\cos ^2(c+d x)\right)\right)}{d (2 n-7) (2 n-3) \cos ^{\frac{7}{2}}(c+d x)}","\frac{2 (A (5-2 n)+C (7-2 n)) \sin (c+d x) (b \cos (c+d x))^n \, _2F_1\left(\frac{1}{2},\frac{1}{4} (2 n-7);\frac{1}{4} (2 n-3);\cos ^2(c+d x)\right)}{d (5-2 n) (7-2 n) \sqrt{\sin ^2(c+d x)} \cos ^{\frac{7}{2}}(c+d x)}-\frac{2 C \sin (c+d x) (b \cos (c+d x))^n}{d (5-2 n) \cos ^{\frac{7}{2}}(c+d x)}",1,"(-2*(b*Cos[c + d*x])^n*Csc[c + d*x]*(A*(-3 + 2*n)*Hypergeometric2F1[1/2, (-7 + 2*n)/4, (-3 + 2*n)/4, Cos[c + d*x]^2] + C*(-7 + 2*n)*Cos[c + d*x]^2*Hypergeometric2F1[1/2, (-3 + 2*n)/4, (1 + 2*n)/4, Cos[c + d*x]^2])*Sqrt[Sin[c + d*x]^2])/(d*(-7 + 2*n)*(-3 + 2*n)*Cos[c + d*x]^(7/2))","A",1
198,1,242,170,1.5365508,"\int (a+a \cos (e+f x))^m \left(A+C \cos ^2(e+f x)\right) \, dx","Integrate[(a + a*Cos[e + f*x])^m*(A + C*Cos[e + f*x]^2),x]","\frac{i 4^{-m-1} \left(1+e^{i (e+f x)}\right) e^{-i (m+2) (e+f x)} \left(e^{-\frac{1}{2} i (e+f x)} \left(1+e^{i (e+f x)}\right)\right)^{2 m} \cos ^{-2 m}\left(\frac{1}{2} (e+f x)\right) (a (\cos (e+f x)+1))^m \left((m+2) e^{i (m+2) (e+f x)} \left(2 (m-2) (2 A+C) \, _2F_1\left(1,m+1;1-m;-e^{i (e+f x)}\right)+C m e^{2 i (e+f x)} \, _2F_1\left(1,m+3;3-m;-e^{i (e+f x)}\right)\right)+C (m-2) m e^{i m (e+f x)} \, _2F_1\left(1,m-1;-m-1;-e^{i (e+f x)}\right)\right)}{f (m-2) m (m+2)}","\frac{2^{m+\frac{1}{2}} \left(A \left(m^2+3 m+2\right)+C \left(m^2+m+1\right)\right) \sin (e+f x) (\cos (e+f x)+1)^{-m-\frac{1}{2}} (a \cos (e+f x)+a)^m \, _2F_1\left(\frac{1}{2},\frac{1}{2}-m;\frac{3}{2};\frac{1}{2} (1-\cos (e+f x))\right)}{f (m+1) (m+2)}-\frac{C \sin (e+f x) (a \cos (e+f x)+a)^m}{f \left(m^2+3 m+2\right)}+\frac{C \sin (e+f x) (a \cos (e+f x)+a)^{m+1}}{a f (m+2)}",1,"(I*4^(-1 - m)*(1 + E^(I*(e + f*x)))*((1 + E^(I*(e + f*x)))/E^((I/2)*(e + f*x)))^(2*m)*(a*(1 + Cos[e + f*x]))^m*(C*E^(I*m*(e + f*x))*(-2 + m)*m*Hypergeometric2F1[1, -1 + m, -1 - m, -E^(I*(e + f*x))] + E^(I*(2 + m)*(e + f*x))*(2 + m)*(2*(2*A + C)*(-2 + m)*Hypergeometric2F1[1, 1 + m, 1 - m, -E^(I*(e + f*x))] + C*E^((2*I)*(e + f*x))*m*Hypergeometric2F1[1, 3 + m, 3 - m, -E^(I*(e + f*x))])))/(E^(I*(2 + m)*(e + f*x))*f*(-2 + m)*m*(2 + m)*Cos[(e + f*x)/2]^(2*m))","C",0
199,1,175,135,0.7952256,"\int (a+a \cos (c+d x))^{2/3} \left(A+C \cos ^2(c+d x)\right) \, dx","Integrate[(a + a*Cos[c + d*x])^(2/3)*(A + C*Cos[c + d*x]^2),x]","\frac{\sec ^2\left(\frac{1}{2} (c+d x)\right) (a (\cos (c+d x)+1))^{2/3} \left(6\ 2^{5/6} \sin (c+d x) (40 A+14 C \cos (c+d x)+5 C \cos (2 (c+d x))+28 C) \sqrt[6]{1-\cos \left(d x-2 \tan ^{-1}\left(\cot \left(\frac{c}{2}\right)\right)\right)}-4 (40 A+19 C) \sin \left(d x-2 \tan ^{-1}\left(\cot \left(\frac{c}{2}\right)\right)\right) \, _2F_1\left(\frac{1}{2},\frac{5}{6};\frac{3}{2};\cos ^2\left(\frac{d x}{2}-\tan ^{-1}\left(\cot \left(\frac{c}{2}\right)\right)\right)\right)\right)}{320\ 2^{5/6} d \sqrt[6]{1-\cos \left(d x-2 \tan ^{-1}\left(\cot \left(\frac{c}{2}\right)\right)\right)}}","\frac{(40 A+19 C) \sin (c+d x) (a \cos (c+d x)+a)^{2/3} \, _2F_1\left(-\frac{1}{6},\frac{1}{2};\frac{3}{2};\frac{1}{2} (1-\cos (c+d x))\right)}{10\ 2^{5/6} d (\cos (c+d x)+1)^{7/6}}+\frac{3 C \sin (c+d x) (a \cos (c+d x)+a)^{5/3}}{8 a d}-\frac{9 C \sin (c+d x) (a \cos (c+d x)+a)^{2/3}}{40 d}",1,"((a*(1 + Cos[c + d*x]))^(2/3)*Sec[(c + d*x)/2]^2*(6*2^(5/6)*(40*A + 28*C + 14*C*Cos[c + d*x] + 5*C*Cos[2*(c + d*x)])*(1 - Cos[d*x - 2*ArcTan[Cot[c/2]]])^(1/6)*Sin[c + d*x] - 4*(40*A + 19*C)*Hypergeometric2F1[1/2, 5/6, 3/2, Cos[(d*x)/2 - ArcTan[Cot[c/2]]]^2]*Sin[d*x - 2*ArcTan[Cot[c/2]]]))/(320*2^(5/6)*d*(1 - Cos[d*x - 2*ArcTan[Cot[c/2]]])^(1/6))","A",1
200,1,240,135,0.9024516,"\int \sqrt[3]{a+a \cos (c+d x)} \left(A+C \cos ^2(c+d x)\right) \, dx","Integrate[(a + a*Cos[c + d*x])^(1/3)*(A + C*Cos[c + d*x]^2),x]","\frac{3 \sqrt[3]{a (\cos (c+d x)+1)} \left(\frac{(28 A+13 C) \csc \left(\frac{c}{4}\right) \sec \left(\frac{c}{4}\right) \sqrt[3]{i \sin (c) e^{i d x}+\cos (c) e^{i d x}+1} \left(2 \, _2F_1\left(-\frac{1}{3},\frac{1}{3};\frac{2}{3};-e^{i d x} (\cos (c)+i \sin (c))\right)+e^{i d x} \, _2F_1\left(\frac{1}{3},\frac{2}{3};\frac{5}{3};-e^{i d x} (\cos (c)+i \sin (c))\right)\right)}{i \sin \left(\frac{c}{2}\right) \left(-1+e^{i d x}\right)+\cos \left(\frac{c}{2}\right) \left(1+e^{i d x}\right)}-4 (28 A+13 C) \cot \left(\frac{c}{2}\right)+4 C \sin (c) \cos (d x)+8 C \sin (2 c) \cos (2 d x)+4 C \cos (c) \sin (d x)+8 C \cos (2 c) \sin (2 d x)\right)}{112 d}","\frac{(28 A+13 C) \sin (c+d x) \sqrt[3]{a \cos (c+d x)+a} \, _2F_1\left(\frac{1}{6},\frac{1}{2};\frac{3}{2};\frac{1}{2} (1-\cos (c+d x))\right)}{14 \sqrt[6]{2} d (\cos (c+d x)+1)^{5/6}}+\frac{3 C \sin (c+d x) (a \cos (c+d x)+a)^{4/3}}{7 a d}-\frac{9 C \sin (c+d x) \sqrt[3]{a \cos (c+d x)+a}}{28 d}",1,"(3*(a*(1 + Cos[c + d*x]))^(1/3)*(-4*(28*A + 13*C)*Cot[c/2] + 4*C*Cos[d*x]*Sin[c] + ((28*A + 13*C)*Csc[c/4]*(2*Hypergeometric2F1[-1/3, 1/3, 2/3, -(E^(I*d*x)*(Cos[c] + I*Sin[c]))] + E^(I*d*x)*Hypergeometric2F1[1/3, 2/3, 5/3, -(E^(I*d*x)*(Cos[c] + I*Sin[c]))])*Sec[c/4]*(1 + E^(I*d*x)*Cos[c] + I*E^(I*d*x)*Sin[c])^(1/3))/((1 + E^(I*d*x))*Cos[c/2] + I*(-1 + E^(I*d*x))*Sin[c/2]) + 8*C*Cos[2*d*x]*Sin[2*c] + 4*C*Cos[c]*Sin[d*x] + 8*C*Cos[2*c]*Sin[2*d*x]))/(112*d)","C",1
201,1,144,135,0.466319,"\int \frac{A+C \cos ^2(c+d x)}{\sqrt[3]{a+a \cos (c+d x)}} \, dx","Integrate[(A + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^(1/3),x]","-\frac{2 (10 A+7 C) \sin \left(d x-2 \tan ^{-1}\left(\cot \left(\frac{c}{2}\right)\right)\right) \, _2F_1\left(\frac{1}{2},\frac{5}{6};\frac{3}{2};\cos ^2\left(\frac{d x}{2}-\tan ^{-1}\left(\cot \left(\frac{c}{2}\right)\right)\right)\right)+3\ 2^{5/6} C (\sin (c+d x)-\sin (2 (c+d x))) \sqrt[6]{1-\cos \left(d x-2 \tan ^{-1}\left(\cot \left(\frac{c}{2}\right)\right)\right)}}{20 d \sqrt[3]{a (\cos (c+d x)+1)} \sqrt[6]{\sin ^2\left(\frac{d x}{2}-\tan ^{-1}\left(\cot \left(\frac{c}{2}\right)\right)\right)}}","\frac{(10 A+7 C) \sin (c+d x) \, _2F_1\left(\frac{1}{2},\frac{5}{6};\frac{3}{2};\frac{1}{2} (1-\cos (c+d x))\right)}{5\ 2^{5/6} d \sqrt[6]{\cos (c+d x)+1} \sqrt[3]{a \cos (c+d x)+a}}+\frac{3 C \sin (c+d x) (a \cos (c+d x)+a)^{2/3}}{5 a d}-\frac{9 C \sin (c+d x)}{10 d \sqrt[3]{a \cos (c+d x)+a}}",1,"-1/20*(3*2^(5/6)*C*(1 - Cos[d*x - 2*ArcTan[Cot[c/2]]])^(1/6)*(Sin[c + d*x] - Sin[2*(c + d*x)]) + 2*(10*A + 7*C)*Hypergeometric2F1[1/2, 5/6, 3/2, Cos[(d*x)/2 - ArcTan[Cot[c/2]]]^2]*Sin[d*x - 2*ArcTan[Cot[c/2]]])/(d*(a*(1 + Cos[c + d*x]))^(1/3)*(Sin[(d*x)/2 - ArcTan[Cot[c/2]]]^2)^(1/6))","A",1
202,0,0,138,0.1521987,"\int \frac{A+C \cos ^2(c+d x)}{(a+a \cos (c+d x))^{2/3}} \, dx","Integrate[(A + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^(2/3),x]","\int \frac{A+C \cos ^2(c+d x)}{(a+a \cos (c+d x))^{2/3}} \, dx","-\frac{(4 A+7 C) \sin (c+d x) \sqrt[3]{a \cos (c+d x)+a} \, _2F_1\left(\frac{1}{6},\frac{1}{2};\frac{3}{2};\frac{1}{2} (1-\cos (c+d x))\right)}{2 \sqrt[6]{2} a d (\cos (c+d x)+1)^{5/6}}+\frac{3 (A+C) \sin (c+d x)}{d (a \cos (c+d x)+a)^{2/3}}+\frac{3 C \sin (c+d x) \sqrt[3]{a \cos (c+d x)+a}}{4 a d}",1,"Integrate[(A + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^(2/3), x]","F",-1
203,1,279,277,2.779073,"\int (a+b \cos (c+d x))^{2/3} \left(A+C \cos ^2(c+d x)\right) \, dx","Integrate[(a + b*Cos[c + d*x])^(2/3)*(A + C*Cos[c + d*x]^2),x]","-\frac{3 \csc (c+d x) (a+b \cos (c+d x))^{2/3} \left(4 \left(-6 a^2 C+40 A b^2+25 b^2 C\right) \sqrt{-\frac{b (\cos (c+d x)-1)}{a+b}} \sqrt{\frac{b (\cos (c+d x)+1)}{b-a}} (a+b \cos (c+d x)) F_1\left(\frac{5}{3};\frac{1}{2},\frac{1}{2};\frac{8}{3};\frac{a+b \cos (c+d x)}{a-b},\frac{a+b \cos (c+d x)}{a+b}\right)+60 a C \left(a^2-b^2\right) \sqrt{-\frac{b (\cos (c+d x)-1)}{a+b}} \sqrt{-\frac{b (\cos (c+d x)+1)}{a-b}} F_1\left(\frac{2}{3};\frac{1}{2},\frac{1}{2};\frac{5}{3};\frac{a+b \cos (c+d x)}{a-b},\frac{a+b \cos (c+d x)}{a+b}\right)-20 b^2 C \sin ^2(c+d x) (2 a+5 b \cos (c+d x))\right)}{800 b^3 d}","\frac{\left(3 a^2 C+b^2 (8 A+5 C)\right) \sin (c+d x) (a+b \cos (c+d x))^{2/3} F_1\left(\frac{1}{2};\frac{1}{2},-\frac{2}{3};\frac{3}{2};\frac{1}{2} (1-\cos (c+d x)),\frac{b (1-\cos (c+d x))}{a+b}\right)}{4 \sqrt{2} b^2 d \sqrt{\cos (c+d x)+1} \left(\frac{a+b \cos (c+d x)}{a+b}\right)^{2/3}}-\frac{3 a C (a+b) \sin (c+d x) (a+b \cos (c+d x))^{2/3} F_1\left(\frac{1}{2};\frac{1}{2},-\frac{5}{3};\frac{3}{2};\frac{1}{2} (1-\cos (c+d x)),\frac{b (1-\cos (c+d x))}{a+b}\right)}{4 \sqrt{2} b^2 d \sqrt{\cos (c+d x)+1} \left(\frac{a+b \cos (c+d x)}{a+b}\right)^{2/3}}+\frac{3 C \sin (c+d x) (a+b \cos (c+d x))^{5/3}}{8 b d}",1,"(-3*(a + b*Cos[c + d*x])^(2/3)*Csc[c + d*x]*(60*a*(a^2 - b^2)*C*AppellF1[2/3, 1/2, 1/2, 5/3, (a + b*Cos[c + d*x])/(a - b), (a + b*Cos[c + d*x])/(a + b)]*Sqrt[-((b*(-1 + Cos[c + d*x]))/(a + b))]*Sqrt[-((b*(1 + Cos[c + d*x]))/(a - b))] + 4*(40*A*b^2 - 6*a^2*C + 25*b^2*C)*AppellF1[5/3, 1/2, 1/2, 8/3, (a + b*Cos[c + d*x])/(a - b), (a + b*Cos[c + d*x])/(a + b)]*Sqrt[-((b*(-1 + Cos[c + d*x]))/(a + b))]*Sqrt[(b*(1 + Cos[c + d*x]))/(-a + b)]*(a + b*Cos[c + d*x]) - 20*b^2*C*(2*a + 5*b*Cos[c + d*x])*Sin[c + d*x]^2))/(800*b^3*d)","A",0
204,1,276,277,2.7255756,"\int \sqrt[3]{a+b \cos (c+d x)} \left(A+C \cos ^2(c+d x)\right) \, dx","Integrate[(a + b*Cos[c + d*x])^(1/3)*(A + C*Cos[c + d*x]^2),x]","-\frac{3 \csc (c+d x) \sqrt[3]{a+b \cos (c+d x)} \left(\left(-3 a^2 C+28 A b^2+16 b^2 C\right) \sqrt{-\frac{b (\cos (c+d x)-1)}{a+b}} \sqrt{\frac{b (\cos (c+d x)+1)}{b-a}} (a+b \cos (c+d x)) F_1\left(\frac{4}{3};\frac{1}{2},\frac{1}{2};\frac{7}{3};\frac{a+b \cos (c+d x)}{a-b},\frac{a+b \cos (c+d x)}{a+b}\right)+12 a C \left(a^2-b^2\right) \sqrt{-\frac{b (\cos (c+d x)-1)}{a+b}} \sqrt{-\frac{b (\cos (c+d x)+1)}{a-b}} F_1\left(\frac{1}{3};\frac{1}{2},\frac{1}{2};\frac{4}{3};\frac{a+b \cos (c+d x)}{a-b},\frac{a+b \cos (c+d x)}{a+b}\right)-4 b^2 C \sin ^2(c+d x) (a+4 b \cos (c+d x))\right)}{112 b^3 d}","\frac{\sqrt{2} \left(3 a^2 C+b^2 (7 A+4 C)\right) \sin (c+d x) \sqrt[3]{a+b \cos (c+d x)} F_1\left(\frac{1}{2};\frac{1}{2},-\frac{1}{3};\frac{3}{2};\frac{1}{2} (1-\cos (c+d x)),\frac{b (1-\cos (c+d x))}{a+b}\right)}{7 b^2 d \sqrt{\cos (c+d x)+1} \sqrt[3]{\frac{a+b \cos (c+d x)}{a+b}}}-\frac{3 \sqrt{2} a C (a+b) \sin (c+d x) \sqrt[3]{a+b \cos (c+d x)} F_1\left(\frac{1}{2};\frac{1}{2},-\frac{4}{3};\frac{3}{2};\frac{1}{2} (1-\cos (c+d x)),\frac{b (1-\cos (c+d x))}{a+b}\right)}{7 b^2 d \sqrt{\cos (c+d x)+1} \sqrt[3]{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{3 C \sin (c+d x) (a+b \cos (c+d x))^{4/3}}{7 b d}",1,"(-3*(a + b*Cos[c + d*x])^(1/3)*Csc[c + d*x]*(12*a*(a^2 - b^2)*C*AppellF1[1/3, 1/2, 1/2, 4/3, (a + b*Cos[c + d*x])/(a - b), (a + b*Cos[c + d*x])/(a + b)]*Sqrt[-((b*(-1 + Cos[c + d*x]))/(a + b))]*Sqrt[-((b*(1 + Cos[c + d*x]))/(a - b))] + (28*A*b^2 - 3*a^2*C + 16*b^2*C)*AppellF1[4/3, 1/2, 1/2, 7/3, (a + b*Cos[c + d*x])/(a - b), (a + b*Cos[c + d*x])/(a + b)]*Sqrt[-((b*(-1 + Cos[c + d*x]))/(a + b))]*Sqrt[(b*(1 + Cos[c + d*x]))/(-a + b)]*(a + b*Cos[c + d*x]) - 4*b^2*C*(a + 4*b*Cos[c + d*x])*Sin[c + d*x]^2))/(112*b^3*d)","A",0
205,1,256,274,1.8085512,"\int \frac{A+C \cos ^2(c+d x)}{\sqrt[3]{a+b \cos (c+d x)}} \, dx","Integrate[(A + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^(1/3),x]","-\frac{3 \csc (c+d x) (a+b \cos (c+d x))^{2/3} \left(5 \left(3 a^2 C+5 A b^2+2 b^2 C\right) \sqrt{-\frac{b (\cos (c+d x)-1)}{a+b}} \sqrt{\frac{b (\cos (c+d x)+1)}{b-a}} F_1\left(\frac{2}{3};\frac{1}{2},\frac{1}{2};\frac{5}{3};\frac{a+b \cos (c+d x)}{a-b},\frac{a+b \cos (c+d x)}{a+b}\right)-6 a C \sqrt{-\frac{b (\cos (c+d x)-1)}{a+b}} \sqrt{\frac{b (\cos (c+d x)+1)}{b-a}} (a+b \cos (c+d x)) F_1\left(\frac{5}{3};\frac{1}{2},\frac{1}{2};\frac{8}{3};\frac{a+b \cos (c+d x)}{a-b},\frac{a+b \cos (c+d x)}{a+b}\right)-10 b^2 C \sin ^2(c+d x)\right)}{50 b^3 d}","\frac{\sqrt{2} \left(3 a^2 C+b^2 (5 A+2 C)\right) \sin (c+d x) \sqrt[3]{\frac{a+b \cos (c+d x)}{a+b}} F_1\left(\frac{1}{2};\frac{1}{2},\frac{1}{3};\frac{3}{2};\frac{1}{2} (1-\cos (c+d x)),\frac{b (1-\cos (c+d x))}{a+b}\right)}{5 b^2 d \sqrt{\cos (c+d x)+1} \sqrt[3]{a+b \cos (c+d x)}}-\frac{3 \sqrt{2} a C \sin (c+d x) (a+b \cos (c+d x))^{2/3} F_1\left(\frac{1}{2};\frac{1}{2},-\frac{2}{3};\frac{3}{2};\frac{1}{2} (1-\cos (c+d x)),\frac{b (1-\cos (c+d x))}{a+b}\right)}{5 b^2 d \sqrt{\cos (c+d x)+1} \left(\frac{a+b \cos (c+d x)}{a+b}\right)^{2/3}}+\frac{3 C \sin (c+d x) (a+b \cos (c+d x))^{2/3}}{5 b d}",1,"(-3*(a + b*Cos[c + d*x])^(2/3)*Csc[c + d*x]*(5*(5*A*b^2 + 3*a^2*C + 2*b^2*C)*AppellF1[2/3, 1/2, 1/2, 5/3, (a + b*Cos[c + d*x])/(a - b), (a + b*Cos[c + d*x])/(a + b)]*Sqrt[-((b*(-1 + Cos[c + d*x]))/(a + b))]*Sqrt[(b*(1 + Cos[c + d*x]))/(-a + b)] - 6*a*C*AppellF1[5/3, 1/2, 1/2, 8/3, (a + b*Cos[c + d*x])/(a - b), (a + b*Cos[c + d*x])/(a + b)]*Sqrt[-((b*(-1 + Cos[c + d*x]))/(a + b))]*Sqrt[(b*(1 + Cos[c + d*x]))/(-a + b)]*(a + b*Cos[c + d*x]) - 10*b^2*C*Sin[c + d*x]^2))/(50*b^3*d)","A",0
206,1,256,272,1.9366574,"\int \frac{A+C \cos ^2(c+d x)}{(a+b \cos (c+d x))^{2/3}} \, dx","Integrate[(A + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^(2/3),x]","-\frac{3 \csc (c+d x) \sqrt[3]{a+b \cos (c+d x)} \left(4 \left(C \left(3 a^2+b^2\right)+4 A b^2\right) \sqrt{-\frac{b (\cos (c+d x)-1)}{a+b}} \sqrt{\frac{b (\cos (c+d x)+1)}{b-a}} F_1\left(\frac{1}{3};\frac{1}{2},\frac{1}{2};\frac{4}{3};\frac{a+b \cos (c+d x)}{a-b},\frac{a+b \cos (c+d x)}{a+b}\right)+C \left(-3 a \sqrt{-\frac{b (\cos (c+d x)-1)}{a+b}} \sqrt{\frac{b (\cos (c+d x)+1)}{b-a}} (a+b \cos (c+d x)) F_1\left(\frac{4}{3};\frac{1}{2},\frac{1}{2};\frac{7}{3};\frac{a+b \cos (c+d x)}{a-b},\frac{a+b \cos (c+d x)}{a+b}\right)-4 b^2 \sin ^2(c+d x)\right)\right)}{16 b^3 d}","\frac{\left(3 a^2 C+b^2 (4 A+C)\right) \sin (c+d x) \left(\frac{a+b \cos (c+d x)}{a+b}\right)^{2/3} F_1\left(\frac{1}{2};\frac{1}{2},\frac{2}{3};\frac{3}{2};\frac{1}{2} (1-\cos (c+d x)),\frac{b (1-\cos (c+d x))}{a+b}\right)}{2 \sqrt{2} b^2 d \sqrt{\cos (c+d x)+1} (a+b \cos (c+d x))^{2/3}}-\frac{3 a C \sin (c+d x) \sqrt[3]{a+b \cos (c+d x)} F_1\left(\frac{1}{2};\frac{1}{2},-\frac{1}{3};\frac{3}{2};\frac{1}{2} (1-\cos (c+d x)),\frac{b (1-\cos (c+d x))}{a+b}\right)}{2 \sqrt{2} b^2 d \sqrt{\cos (c+d x)+1} \sqrt[3]{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{3 C \sin (c+d x) \sqrt[3]{a+b \cos (c+d x)}}{4 b d}",1,"(-3*(a + b*Cos[c + d*x])^(1/3)*Csc[c + d*x]*(4*(4*A*b^2 + (3*a^2 + b^2)*C)*AppellF1[1/3, 1/2, 1/2, 4/3, (a + b*Cos[c + d*x])/(a - b), (a + b*Cos[c + d*x])/(a + b)]*Sqrt[-((b*(-1 + Cos[c + d*x]))/(a + b))]*Sqrt[(b*(1 + Cos[c + d*x]))/(-a + b)] + C*(-3*a*AppellF1[4/3, 1/2, 1/2, 7/3, (a + b*Cos[c + d*x])/(a - b), (a + b*Cos[c + d*x])/(a + b)]*Sqrt[-((b*(-1 + Cos[c + d*x]))/(a + b))]*Sqrt[(b*(1 + Cos[c + d*x]))/(-a + b)]*(a + b*Cos[c + d*x]) - 4*b^2*Sin[c + d*x]^2)))/(16*b^3*d)","A",0
207,1,119,211,0.3704513,"\int (a+b \cos (e+f x))^m \left(A-A \cos ^2(e+f x)\right) \, dx","Integrate[(a + b*Cos[e + f*x])^m*(A - A*Cos[e + f*x]^2),x]","\frac{4 A \sin (e+f x) \sqrt{\cos ^2\left(\frac{1}{2} (e+f x)\right)} \tan ^2\left(\frac{1}{2} (e+f x)\right) (a+b \cos (e+f x))^m \left(\frac{a+b \cos (e+f x)}{a+b}\right)^{-m} F_1\left(\frac{3}{2};-\frac{1}{2},-m;\frac{5}{2};\sin ^2\left(\frac{1}{2} (e+f x)\right),\frac{2 b \sin ^2\left(\frac{1}{2} (e+f x)\right)}{a+b}\right)}{3 f}","\frac{4 \sqrt{2} A \sin (e+f x) (a+b \cos (e+f x))^m \left(\frac{a+b \cos (e+f x)}{a+b}\right)^{-m} F_1\left(\frac{1}{2};-\frac{1}{2},-m;\frac{3}{2};\frac{1}{2} (1-\cos (e+f x)),\frac{b (1-\cos (e+f x))}{a+b}\right)}{f \sqrt{\cos (e+f x)+1}}-\frac{4 \sqrt{2} A \sin (e+f x) (a+b \cos (e+f x))^m \left(\frac{a+b \cos (e+f x)}{a+b}\right)^{-m} F_1\left(\frac{1}{2};-\frac{3}{2},-m;\frac{3}{2};\frac{1}{2} (1-\cos (e+f x)),\frac{b (1-\cos (e+f x))}{a+b}\right)}{f \sqrt{\cos (e+f x)+1}}",1,"(4*A*AppellF1[3/2, -1/2, -m, 5/2, Sin[(e + f*x)/2]^2, (2*b*Sin[(e + f*x)/2]^2)/(a + b)]*Sqrt[Cos[(e + f*x)/2]^2]*(a + b*Cos[e + f*x])^m*Sin[e + f*x]*Tan[(e + f*x)/2]^2)/(3*f*((a + b*Cos[e + f*x])/(a + b))^m)","A",0
208,1,10805,285,26.4941706,"\int (a+b \cos (e+f x))^m \left(A+C \cos ^2(e+f x)\right) \, dx","Integrate[(a + b*Cos[e + f*x])^m*(A + C*Cos[e + f*x]^2),x]","\text{Result too large to show}","\frac{\sqrt{2} \sin (e+f x) \left(a^2 C+b^2 (A (m+2)+C (m+1))\right) (a+b \cos (e+f x))^m \left(\frac{a+b \cos (e+f x)}{a+b}\right)^{-m} F_1\left(\frac{1}{2};\frac{1}{2},-m;\frac{3}{2};\frac{1}{2} (1-\cos (e+f x)),\frac{b (1-\cos (e+f x))}{a+b}\right)}{b^2 f (m+2) \sqrt{\cos (e+f x)+1}}-\frac{\sqrt{2} a C (a+b) \sin (e+f x) (a+b \cos (e+f x))^m \left(\frac{a+b \cos (e+f x)}{a+b}\right)^{-m} F_1\left(\frac{1}{2};\frac{1}{2},-m-1;\frac{3}{2};\frac{1}{2} (1-\cos (e+f x)),\frac{b (1-\cos (e+f x))}{a+b}\right)}{b^2 f (m+2) \sqrt{\cos (e+f x)+1}}+\frac{C \sin (e+f x) (a+b \cos (e+f x))^{m+1}}{b f (m+2)}",1,"Result too large to show","B",0
209,1,118,141,0.2635038,"\int (a \cos (e+f x))^m \left(B \cos (e+f x)+C \cos ^2(e+f x)\right) \, dx","Integrate[(a*Cos[e + f*x])^m*(B*Cos[e + f*x] + C*Cos[e + f*x]^2),x]","-\frac{\sqrt{\sin ^2(e+f x)} \cos (e+f x) \cot (e+f x) (a \cos (e+f x))^m \left(B (m+3) \, _2F_1\left(\frac{1}{2},\frac{m+2}{2};\frac{m+4}{2};\cos ^2(e+f x)\right)+C (m+2) \cos (e+f x) \, _2F_1\left(\frac{1}{2},\frac{m+3}{2};\frac{m+5}{2};\cos ^2(e+f x)\right)\right)}{f (m+2) (m+3)}","-\frac{C \sin (e+f x) (a \cos (e+f x))^{m+3} \, _2F_1\left(\frac{1}{2},\frac{m+3}{2};\frac{m+5}{2};\cos ^2(e+f x)\right)}{a^3 f (m+3) \sqrt{\sin ^2(e+f x)}}-\frac{B \sin (e+f x) (a \cos (e+f x))^{m+2} \, _2F_1\left(\frac{1}{2},\frac{m+2}{2};\frac{m+4}{2};\cos ^2(e+f x)\right)}{a^2 f (m+2) \sqrt{\sin ^2(e+f x)}}",1,"-((Cos[e + f*x]*(a*Cos[e + f*x])^m*Cot[e + f*x]*(B*(3 + m)*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, Cos[e + f*x]^2] + C*(2 + m)*Cos[e + f*x]*Hypergeometric2F1[1/2, (3 + m)/2, (5 + m)/2, Cos[e + f*x]^2])*Sqrt[Sin[e + f*x]^2])/(f*(2 + m)*(3 + m)))","A",1
210,1,140,167,0.4000207,"\int \cos ^m(c+d x) \sqrt[3]{b \cos (c+d x)} \left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^m*(b*Cos[c + d*x])^(1/3)*(B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","-\frac{3 \sqrt{\sin ^2(c+d x)} \csc (c+d x) \sqrt[3]{b \cos (c+d x)} \cos ^{m+2}(c+d x) \left(B (3 m+10) \, _2F_1\left(\frac{1}{2},\frac{1}{6} (3 m+7);\frac{1}{6} (3 m+13);\cos ^2(c+d x)\right)+C (3 m+7) \cos (c+d x) \, _2F_1\left(\frac{1}{2},\frac{m}{2}+\frac{5}{3};\frac{m}{2}+\frac{8}{3};\cos ^2(c+d x)\right)\right)}{d (3 m+7) (3 m+10)}","-\frac{3 B \sin (c+d x) \sqrt[3]{b \cos (c+d x)} \cos ^{m+2}(c+d x) \, _2F_1\left(\frac{1}{2},\frac{1}{6} (3 m+7);\frac{1}{6} (3 m+13);\cos ^2(c+d x)\right)}{d (3 m+7) \sqrt{\sin ^2(c+d x)}}-\frac{3 C \sin (c+d x) \sqrt[3]{b \cos (c+d x)} \cos ^{m+3}(c+d x) \, _2F_1\left(\frac{1}{2},\frac{1}{6} (3 m+10);\frac{1}{6} (3 m+16);\cos ^2(c+d x)\right)}{d (3 m+10) \sqrt{\sin ^2(c+d x)}}",1,"(-3*Cos[c + d*x]^(2 + m)*(b*Cos[c + d*x])^(1/3)*Csc[c + d*x]*(C*(7 + 3*m)*Cos[c + d*x]*Hypergeometric2F1[1/2, 5/3 + m/2, 8/3 + m/2, Cos[c + d*x]^2] + B*(10 + 3*m)*Hypergeometric2F1[1/2, (7 + 3*m)/6, (13 + 3*m)/6, Cos[c + d*x]^2])*Sqrt[Sin[c + d*x]^2])/(d*(7 + 3*m)*(10 + 3*m))","A",1
211,1,140,167,0.4013942,"\int \cos ^m(c+d x) (b \cos (c+d x))^{2/3} \left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^m*(b*Cos[c + d*x])^(2/3)*(B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","-\frac{3 \sqrt{\sin ^2(c+d x)} \csc (c+d x) (b \cos (c+d x))^{2/3} \cos ^{m+2}(c+d x) \left(B (3 m+11) \, _2F_1\left(\frac{1}{2},\frac{1}{6} (3 m+8);\frac{m}{2}+\frac{7}{3};\cos ^2(c+d x)\right)+C (3 m+8) \cos (c+d x) \, _2F_1\left(\frac{1}{2},\frac{1}{6} (3 m+11);\frac{1}{6} (3 m+17);\cos ^2(c+d x)\right)\right)}{d (3 m+8) (3 m+11)}","-\frac{3 B \sin (c+d x) (b \cos (c+d x))^{2/3} \cos ^{m+2}(c+d x) \, _2F_1\left(\frac{1}{2},\frac{1}{6} (3 m+8);\frac{1}{6} (3 m+14);\cos ^2(c+d x)\right)}{d (3 m+8) \sqrt{\sin ^2(c+d x)}}-\frac{3 C \sin (c+d x) (b \cos (c+d x))^{2/3} \cos ^{m+3}(c+d x) \, _2F_1\left(\frac{1}{2},\frac{1}{6} (3 m+11);\frac{1}{6} (3 m+17);\cos ^2(c+d x)\right)}{d (3 m+11) \sqrt{\sin ^2(c+d x)}}",1,"(-3*Cos[c + d*x]^(2 + m)*(b*Cos[c + d*x])^(2/3)*Csc[c + d*x]*(B*(11 + 3*m)*Hypergeometric2F1[1/2, (8 + 3*m)/6, 7/3 + m/2, Cos[c + d*x]^2] + C*(8 + 3*m)*Cos[c + d*x]*Hypergeometric2F1[1/2, (11 + 3*m)/6, (17 + 3*m)/6, Cos[c + d*x]^2])*Sqrt[Sin[c + d*x]^2])/(d*(8 + 3*m)*(11 + 3*m))","A",1
212,1,140,169,0.5795759,"\int \cos ^m(c+d x) (b \cos (c+d x))^{4/3} \left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^m*(b*Cos[c + d*x])^(4/3)*(B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","-\frac{3 \sqrt{\sin ^2(c+d x)} \csc (c+d x) (b \cos (c+d x))^{4/3} \cos ^{m+2}(c+d x) \left(B (3 m+13) \, _2F_1\left(\frac{1}{2},\frac{m}{2}+\frac{5}{3};\frac{m}{2}+\frac{8}{3};\cos ^2(c+d x)\right)+C (3 m+10) \cos (c+d x) \, _2F_1\left(\frac{1}{2},\frac{1}{6} (3 m+13);\frac{1}{6} (3 m+19);\cos ^2(c+d x)\right)\right)}{d (3 m+10) (3 m+13)}","-\frac{3 b B \sin (c+d x) \sqrt[3]{b \cos (c+d x)} \cos ^{m+3}(c+d x) \, _2F_1\left(\frac{1}{2},\frac{1}{6} (3 m+10);\frac{1}{6} (3 m+16);\cos ^2(c+d x)\right)}{d (3 m+10) \sqrt{\sin ^2(c+d x)}}-\frac{3 b C \sin (c+d x) \sqrt[3]{b \cos (c+d x)} \cos ^{m+4}(c+d x) \, _2F_1\left(\frac{1}{2},\frac{1}{6} (3 m+13);\frac{1}{6} (3 m+19);\cos ^2(c+d x)\right)}{d (3 m+13) \sqrt{\sin ^2(c+d x)}}",1,"(-3*Cos[c + d*x]^(2 + m)*(b*Cos[c + d*x])^(4/3)*Csc[c + d*x]*(B*(13 + 3*m)*Hypergeometric2F1[1/2, 5/3 + m/2, 8/3 + m/2, Cos[c + d*x]^2] + C*(10 + 3*m)*Cos[c + d*x]*Hypergeometric2F1[1/2, (13 + 3*m)/6, (19 + 3*m)/6, Cos[c + d*x]^2])*Sqrt[Sin[c + d*x]^2])/(d*(10 + 3*m)*(13 + 3*m))","A",1
213,1,140,167,0.4545802,"\int \frac{\cos ^m(c+d x) \left(B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\sqrt[3]{b \cos (c+d x)}} \, dx","Integrate[(Cos[c + d*x]^m*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(b*Cos[c + d*x])^(1/3),x]","-\frac{3 \sqrt{\sin ^2(c+d x)} \csc (c+d x) \cos ^{m+2}(c+d x) \left(B (3 m+8) \, _2F_1\left(\frac{1}{2},\frac{1}{6} (3 m+5);\frac{1}{6} (3 m+11);\cos ^2(c+d x)\right)+C (3 m+5) \cos (c+d x) \, _2F_1\left(\frac{1}{2},\frac{1}{6} (3 m+8);\frac{m}{2}+\frac{7}{3};\cos ^2(c+d x)\right)\right)}{d (3 m+5) (3 m+8) \sqrt[3]{b \cos (c+d x)}}","-\frac{3 B \sin (c+d x) \cos ^{m+2}(c+d x) \, _2F_1\left(\frac{1}{2},\frac{1}{6} (3 m+5);\frac{1}{6} (3 m+11);\cos ^2(c+d x)\right)}{d (3 m+5) \sqrt{\sin ^2(c+d x)} \sqrt[3]{b \cos (c+d x)}}-\frac{3 C \sin (c+d x) \cos ^{m+3}(c+d x) \, _2F_1\left(\frac{1}{2},\frac{1}{6} (3 m+8);\frac{1}{6} (3 m+14);\cos ^2(c+d x)\right)}{d (3 m+8) \sqrt{\sin ^2(c+d x)} \sqrt[3]{b \cos (c+d x)}}",1,"(-3*Cos[c + d*x]^(2 + m)*Csc[c + d*x]*(B*(8 + 3*m)*Hypergeometric2F1[1/2, (5 + 3*m)/6, (11 + 3*m)/6, Cos[c + d*x]^2] + C*(5 + 3*m)*Cos[c + d*x]*Hypergeometric2F1[1/2, (8 + 3*m)/6, 7/3 + m/2, Cos[c + d*x]^2])*Sqrt[Sin[c + d*x]^2])/(d*(5 + 3*m)*(8 + 3*m)*(b*Cos[c + d*x])^(1/3))","A",1
214,1,140,167,0.4497897,"\int \frac{\cos ^m(c+d x) \left(B \cos (c+d x)+C \cos ^2(c+d x)\right)}{(b \cos (c+d x))^{2/3}} \, dx","Integrate[(Cos[c + d*x]^m*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(b*Cos[c + d*x])^(2/3),x]","-\frac{3 \sqrt{\sin ^2(c+d x)} \csc (c+d x) \cos ^{m+2}(c+d x) \left(B (3 m+7) \, _2F_1\left(\frac{1}{2},\frac{1}{6} (3 m+4);\frac{m}{2}+\frac{5}{3};\cos ^2(c+d x)\right)+C (3 m+4) \cos (c+d x) \, _2F_1\left(\frac{1}{2},\frac{1}{6} (3 m+7);\frac{1}{6} (3 m+13);\cos ^2(c+d x)\right)\right)}{d (3 m+4) (3 m+7) (b \cos (c+d x))^{2/3}}","-\frac{3 B \sin (c+d x) \cos ^{m+2}(c+d x) \, _2F_1\left(\frac{1}{2},\frac{1}{6} (3 m+4);\frac{1}{6} (3 m+10);\cos ^2(c+d x)\right)}{d (3 m+4) \sqrt{\sin ^2(c+d x)} (b \cos (c+d x))^{2/3}}-\frac{3 C \sin (c+d x) \cos ^{m+3}(c+d x) \, _2F_1\left(\frac{1}{2},\frac{1}{6} (3 m+7);\frac{1}{6} (3 m+13);\cos ^2(c+d x)\right)}{d (3 m+7) \sqrt{\sin ^2(c+d x)} (b \cos (c+d x))^{2/3}}",1,"(-3*Cos[c + d*x]^(2 + m)*Csc[c + d*x]*(B*(7 + 3*m)*Hypergeometric2F1[1/2, (4 + 3*m)/6, 5/3 + m/2, Cos[c + d*x]^2] + C*(4 + 3*m)*Cos[c + d*x]*Hypergeometric2F1[1/2, (7 + 3*m)/6, (13 + 3*m)/6, Cos[c + d*x]^2])*Sqrt[Sin[c + d*x]^2])/(d*(4 + 3*m)*(7 + 3*m)*(b*Cos[c + d*x])^(2/3))","A",1
215,1,140,173,0.3769798,"\int \frac{\cos ^m(c+d x) \left(B \cos (c+d x)+C \cos ^2(c+d x)\right)}{(b \cos (c+d x))^{4/3}} \, dx","Integrate[(Cos[c + d*x]^m*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(b*Cos[c + d*x])^(4/3),x]","-\frac{3 \sqrt{\sin ^2(c+d x)} \csc (c+d x) \cos ^{m+2}(c+d x) \left(B (3 m+5) \, _2F_1\left(\frac{1}{2},\frac{1}{6} (3 m+2);\frac{1}{6} (3 m+8);\cos ^2(c+d x)\right)+C (3 m+2) \cos (c+d x) \, _2F_1\left(\frac{1}{2},\frac{1}{6} (3 m+5);\frac{1}{6} (3 m+11);\cos ^2(c+d x)\right)\right)}{d (3 m+2) (3 m+5) (b \cos (c+d x))^{4/3}}","-\frac{3 B \sin (c+d x) \cos ^{m+1}(c+d x) \, _2F_1\left(\frac{1}{2},\frac{1}{6} (3 m+2);\frac{1}{6} (3 m+8);\cos ^2(c+d x)\right)}{b d (3 m+2) \sqrt{\sin ^2(c+d x)} \sqrt[3]{b \cos (c+d x)}}-\frac{3 C \sin (c+d x) \cos ^{m+2}(c+d x) \, _2F_1\left(\frac{1}{2},\frac{1}{6} (3 m+5);\frac{1}{6} (3 m+11);\cos ^2(c+d x)\right)}{b d (3 m+5) \sqrt{\sin ^2(c+d x)} \sqrt[3]{b \cos (c+d x)}}",1,"(-3*Cos[c + d*x]^(2 + m)*Csc[c + d*x]*(B*(5 + 3*m)*Hypergeometric2F1[1/2, (2 + 3*m)/6, (8 + 3*m)/6, Cos[c + d*x]^2] + C*(2 + 3*m)*Cos[c + d*x]*Hypergeometric2F1[1/2, (5 + 3*m)/6, (11 + 3*m)/6, Cos[c + d*x]^2])*Sqrt[Sin[c + d*x]^2])/(d*(2 + 3*m)*(5 + 3*m)*(b*Cos[c + d*x])^(4/3))","A",1
216,1,136,167,0.3685028,"\int (a \cos (c+d x))^m (b \cos (c+d x))^n \left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Integrate[(a*Cos[c + d*x])^m*(b*Cos[c + d*x])^n*(B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","-\frac{\sqrt{\sin ^2(c+d x)} \cos (c+d x) \cot (c+d x) (a \cos (c+d x))^m (b \cos (c+d x))^n \left(B (m+n+3) \, _2F_1\left(\frac{1}{2},\frac{1}{2} (m+n+2);\frac{1}{2} (m+n+4);\cos ^2(c+d x)\right)+C (m+n+2) \cos (c+d x) \, _2F_1\left(\frac{1}{2},\frac{1}{2} (m+n+3);\frac{1}{2} (m+n+5);\cos ^2(c+d x)\right)\right)}{d (m+n+2) (m+n+3)}","-\frac{C \sin (c+d x) (a \cos (c+d x))^{m+3} (b \cos (c+d x))^n \, _2F_1\left(\frac{1}{2},\frac{1}{2} (m+n+3);\frac{1}{2} (m+n+5);\cos ^2(c+d x)\right)}{a^3 d (m+n+3) \sqrt{\sin ^2(c+d x)}}-\frac{B \sin (c+d x) (a \cos (c+d x))^{m+2} (b \cos (c+d x))^n \, _2F_1\left(\frac{1}{2},\frac{1}{2} (m+n+2);\frac{1}{2} (m+n+4);\cos ^2(c+d x)\right)}{a^2 d (m+n+2) \sqrt{\sin ^2(c+d x)}}",1,"-((Cos[c + d*x]*(a*Cos[c + d*x])^m*(b*Cos[c + d*x])^n*Cot[c + d*x]*(B*(3 + m + n)*Hypergeometric2F1[1/2, (2 + m + n)/2, (4 + m + n)/2, Cos[c + d*x]^2] + C*(2 + m + n)*Cos[c + d*x]*Hypergeometric2F1[1/2, (3 + m + n)/2, (5 + m + n)/2, Cos[c + d*x]^2])*Sqrt[Sin[c + d*x]^2])/(d*(2 + m + n)*(3 + m + n)))","A",1
217,1,120,141,0.4850185,"\int \cos ^2(c+d x) (b \cos (c+d x))^n \left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^2*(b*Cos[c + d*x])^n*(B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","-\frac{\sqrt{\sin ^2(c+d x)} \cos ^3(c+d x) \cot (c+d x) (b \cos (c+d x))^n \left(B (n+5) \, _2F_1\left(\frac{1}{2},\frac{n+4}{2};\frac{n+6}{2};\cos ^2(c+d x)\right)+C (n+4) \cos (c+d x) \, _2F_1\left(\frac{1}{2},\frac{n+5}{2};\frac{n+7}{2};\cos ^2(c+d x)\right)\right)}{d (n+4) (n+5)}","-\frac{C \sin (c+d x) (b \cos (c+d x))^{n+5} \, _2F_1\left(\frac{1}{2},\frac{n+5}{2};\frac{n+7}{2};\cos ^2(c+d x)\right)}{b^5 d (n+5) \sqrt{\sin ^2(c+d x)}}-\frac{B \sin (c+d x) (b \cos (c+d x))^{n+4} \, _2F_1\left(\frac{1}{2},\frac{n+4}{2};\frac{n+6}{2};\cos ^2(c+d x)\right)}{b^4 d (n+4) \sqrt{\sin ^2(c+d x)}}",1,"-((Cos[c + d*x]^3*(b*Cos[c + d*x])^n*Cot[c + d*x]*(B*(5 + n)*Hypergeometric2F1[1/2, (4 + n)/2, (6 + n)/2, Cos[c + d*x]^2] + C*(4 + n)*Cos[c + d*x]*Hypergeometric2F1[1/2, (5 + n)/2, (7 + n)/2, Cos[c + d*x]^2])*Sqrt[Sin[c + d*x]^2])/(d*(4 + n)*(5 + n)))","A",1
218,1,120,141,0.2961894,"\int \cos (c+d x) (b \cos (c+d x))^n \left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]*(b*Cos[c + d*x])^n*(B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","-\frac{\sqrt{\sin ^2(c+d x)} \cos ^2(c+d x) \cot (c+d x) (b \cos (c+d x))^n \left(B (n+4) \, _2F_1\left(\frac{1}{2},\frac{n+3}{2};\frac{n+5}{2};\cos ^2(c+d x)\right)+C (n+3) \cos (c+d x) \, _2F_1\left(\frac{1}{2},\frac{n+4}{2};\frac{n+6}{2};\cos ^2(c+d x)\right)\right)}{d (n+3) (n+4)}","-\frac{C \sin (c+d x) (b \cos (c+d x))^{n+4} \, _2F_1\left(\frac{1}{2},\frac{n+4}{2};\frac{n+6}{2};\cos ^2(c+d x)\right)}{b^4 d (n+4) \sqrt{\sin ^2(c+d x)}}-\frac{B \sin (c+d x) (b \cos (c+d x))^{n+3} \, _2F_1\left(\frac{1}{2},\frac{n+3}{2};\frac{n+5}{2};\cos ^2(c+d x)\right)}{b^3 d (n+3) \sqrt{\sin ^2(c+d x)}}",1,"-((Cos[c + d*x]^2*(b*Cos[c + d*x])^n*Cot[c + d*x]*(B*(4 + n)*Hypergeometric2F1[1/2, (3 + n)/2, (5 + n)/2, Cos[c + d*x]^2] + C*(3 + n)*Cos[c + d*x]*Hypergeometric2F1[1/2, (4 + n)/2, (6 + n)/2, Cos[c + d*x]^2])*Sqrt[Sin[c + d*x]^2])/(d*(3 + n)*(4 + n)))","A",1
219,1,118,141,0.241252,"\int (b \cos (c+d x))^n \left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Integrate[(b*Cos[c + d*x])^n*(B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","-\frac{\sqrt{\sin ^2(c+d x)} \cos (c+d x) \cot (c+d x) (b \cos (c+d x))^n \left(B (n+3) \, _2F_1\left(\frac{1}{2},\frac{n+2}{2};\frac{n+4}{2};\cos ^2(c+d x)\right)+C (n+2) \cos (c+d x) \, _2F_1\left(\frac{1}{2},\frac{n+3}{2};\frac{n+5}{2};\cos ^2(c+d x)\right)\right)}{d (n+2) (n+3)}","-\frac{C \sin (c+d x) (b \cos (c+d x))^{n+3} \, _2F_1\left(\frac{1}{2},\frac{n+3}{2};\frac{n+5}{2};\cos ^2(c+d x)\right)}{b^3 d (n+3) \sqrt{\sin ^2(c+d x)}}-\frac{B \sin (c+d x) (b \cos (c+d x))^{n+2} \, _2F_1\left(\frac{1}{2},\frac{n+2}{2};\frac{n+4}{2};\cos ^2(c+d x)\right)}{b^2 d (n+2) \sqrt{\sin ^2(c+d x)}}",1,"-((Cos[c + d*x]*(b*Cos[c + d*x])^n*Cot[c + d*x]*(B*(3 + n)*Hypergeometric2F1[1/2, (2 + n)/2, (4 + n)/2, Cos[c + d*x]^2] + C*(2 + n)*Cos[c + d*x]*Hypergeometric2F1[1/2, (3 + n)/2, (5 + n)/2, Cos[c + d*x]^2])*Sqrt[Sin[c + d*x]^2])/(d*(2 + n)*(3 + n)))","A",1
220,1,112,141,0.1732178,"\int (b \cos (c+d x))^n \left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec (c+d x) \, dx","Integrate[(b*Cos[c + d*x])^n*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x],x]","-\frac{\sqrt{\sin ^2(c+d x)} \cot (c+d x) (b \cos (c+d x))^n \left(B (n+2) \, _2F_1\left(\frac{1}{2},\frac{n+1}{2};\frac{n+3}{2};\cos ^2(c+d x)\right)+C (n+1) \cos (c+d x) \, _2F_1\left(\frac{1}{2},\frac{n+2}{2};\frac{n+4}{2};\cos ^2(c+d x)\right)\right)}{d (n+1) (n+2)}","-\frac{C \sin (c+d x) (b \cos (c+d x))^{n+2} \, _2F_1\left(\frac{1}{2},\frac{n+2}{2};\frac{n+4}{2};\cos ^2(c+d x)\right)}{b^2 d (n+2) \sqrt{\sin ^2(c+d x)}}-\frac{B \sin (c+d x) (b \cos (c+d x))^{n+1} \, _2F_1\left(\frac{1}{2},\frac{n+1}{2};\frac{n+3}{2};\cos ^2(c+d x)\right)}{b d (n+1) \sqrt{\sin ^2(c+d x)}}",1,"-(((b*Cos[c + d*x])^n*Cot[c + d*x]*(B*(2 + n)*Hypergeometric2F1[1/2, (1 + n)/2, (3 + n)/2, Cos[c + d*x]^2] + C*(1 + n)*Cos[c + d*x]*Hypergeometric2F1[1/2, (2 + n)/2, (4 + n)/2, Cos[c + d*x]^2])*Sqrt[Sin[c + d*x]^2])/(d*(1 + n)*(2 + n)))","A",1
221,1,109,132,0.5257475,"\int (b \cos (c+d x))^n \left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^2(c+d x) \, dx","Integrate[(b*Cos[c + d*x])^n*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^2,x]","-\frac{b \sqrt{\sin ^2(c+d x)} \cot (c+d x) (b \cos (c+d x))^{n-1} \left(B (n+1) \, _2F_1\left(\frac{1}{2},\frac{n}{2};\frac{n+2}{2};\cos ^2(c+d x)\right)+C n \cos (c+d x) \, _2F_1\left(\frac{1}{2},\frac{n+1}{2};\frac{n+3}{2};\cos ^2(c+d x)\right)\right)}{d n (n+1)}","-\frac{B \sin (c+d x) (b \cos (c+d x))^n \, _2F_1\left(\frac{1}{2},\frac{n}{2};\frac{n+2}{2};\cos ^2(c+d x)\right)}{d n \sqrt{\sin ^2(c+d x)}}-\frac{C \sin (c+d x) (b \cos (c+d x))^{n+1} \, _2F_1\left(\frac{1}{2},\frac{n+1}{2};\frac{n+3}{2};\cos ^2(c+d x)\right)}{b d (n+1) \sqrt{\sin ^2(c+d x)}}",1,"-((b*(b*Cos[c + d*x])^(-1 + n)*Cot[c + d*x]*(B*(1 + n)*Hypergeometric2F1[1/2, n/2, (2 + n)/2, Cos[c + d*x]^2] + C*n*Cos[c + d*x]*Hypergeometric2F1[1/2, (1 + n)/2, (3 + n)/2, Cos[c + d*x]^2])*Sqrt[Sin[c + d*x]^2])/(d*n*(1 + n)))","A",1
222,1,109,131,0.1851985,"\int (b \cos (c+d x))^n \left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^3(c+d x) \, dx","Integrate[(b*Cos[c + d*x])^n*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^3,x]","-\frac{b \sqrt{\sin ^2(c+d x)} \csc (c+d x) (b \cos (c+d x))^{n-1} \left(B n \, _2F_1\left(\frac{1}{2},\frac{n-1}{2};\frac{n+1}{2};\cos ^2(c+d x)\right)+C (n-1) \cos (c+d x) \, _2F_1\left(\frac{1}{2},\frac{n}{2};\frac{n+2}{2};\cos ^2(c+d x)\right)\right)}{d (n-1) n}","\frac{b B \sin (c+d x) (b \cos (c+d x))^{n-1} \, _2F_1\left(\frac{1}{2},\frac{n-1}{2};\frac{n+1}{2};\cos ^2(c+d x)\right)}{d (1-n) \sqrt{\sin ^2(c+d x)}}-\frac{C \sin (c+d x) (b \cos (c+d x))^n \, _2F_1\left(\frac{1}{2},\frac{n}{2};\frac{n+2}{2};\cos ^2(c+d x)\right)}{d n \sqrt{\sin ^2(c+d x)}}",1,"-((b*(b*Cos[c + d*x])^(-1 + n)*Csc[c + d*x]*(B*n*Hypergeometric2F1[1/2, (-1 + n)/2, (1 + n)/2, Cos[c + d*x]^2] + C*(-1 + n)*Cos[c + d*x]*Hypergeometric2F1[1/2, n/2, (2 + n)/2, Cos[c + d*x]^2])*Sqrt[Sin[c + d*x]^2])/(d*(-1 + n)*n))","A",1
223,1,118,139,0.1661166,"\int (b \cos (c+d x))^n \left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^4(c+d x) \, dx","Integrate[(b*Cos[c + d*x])^n*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^4,x]","-\frac{\sqrt{\sin ^2(c+d x)} \csc (c+d x) \sec ^2(c+d x) (b \cos (c+d x))^n \left(B (n-1) \, _2F_1\left(\frac{1}{2},\frac{n-2}{2};\frac{n}{2};\cos ^2(c+d x)\right)+C (n-2) \cos (c+d x) \, _2F_1\left(\frac{1}{2},\frac{n-1}{2};\frac{n+1}{2};\cos ^2(c+d x)\right)\right)}{d (n-2) (n-1)}","\frac{b^2 B \sin (c+d x) (b \cos (c+d x))^{n-2} \, _2F_1\left(\frac{1}{2},\frac{n-2}{2};\frac{n}{2};\cos ^2(c+d x)\right)}{d (2-n) \sqrt{\sin ^2(c+d x)}}+\frac{b C \sin (c+d x) (b \cos (c+d x))^{n-1} \, _2F_1\left(\frac{1}{2},\frac{n-1}{2};\frac{n+1}{2};\cos ^2(c+d x)\right)}{d (1-n) \sqrt{\sin ^2(c+d x)}}",1,"-(((b*Cos[c + d*x])^n*Csc[c + d*x]*(B*(-1 + n)*Hypergeometric2F1[1/2, (-2 + n)/2, n/2, Cos[c + d*x]^2] + C*(-2 + n)*Cos[c + d*x]*Hypergeometric2F1[1/2, (-1 + n)/2, (1 + n)/2, Cos[c + d*x]^2])*Sec[c + d*x]^2*Sqrt[Sin[c + d*x]^2])/(d*(-2 + n)*(-1 + n)))","A",1
224,1,138,163,0.282926,"\int \cos ^{\frac{5}{2}}(c+d x) (b \cos (c+d x))^n \left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^(5/2)*(b*Cos[c + d*x])^n*(B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","-\frac{2 \sqrt{\sin ^2(c+d x)} \cos ^{\frac{9}{2}}(c+d x) \csc (c+d x) (b \cos (c+d x))^n \left(B (2 n+11) \, _2F_1\left(\frac{1}{2},\frac{1}{4} (2 n+9);\frac{1}{4} (2 n+13);\cos ^2(c+d x)\right)+C (2 n+9) \cos (c+d x) \, _2F_1\left(\frac{1}{2},\frac{1}{4} (2 n+11);\frac{1}{4} (2 n+15);\cos ^2(c+d x)\right)\right)}{d (2 n+9) (2 n+11)}","-\frac{2 B \sin (c+d x) \cos ^{\frac{9}{2}}(c+d x) (b \cos (c+d x))^n \, _2F_1\left(\frac{1}{2},\frac{1}{4} (2 n+9);\frac{1}{4} (2 n+13);\cos ^2(c+d x)\right)}{d (2 n+9) \sqrt{\sin ^2(c+d x)}}-\frac{2 C \sin (c+d x) \cos ^{\frac{11}{2}}(c+d x) (b \cos (c+d x))^n \, _2F_1\left(\frac{1}{2},\frac{1}{4} (2 n+11);\frac{1}{4} (2 n+15);\cos ^2(c+d x)\right)}{d (2 n+11) \sqrt{\sin ^2(c+d x)}}",1,"(-2*Cos[c + d*x]^(9/2)*(b*Cos[c + d*x])^n*Csc[c + d*x]*(B*(11 + 2*n)*Hypergeometric2F1[1/2, (9 + 2*n)/4, (13 + 2*n)/4, Cos[c + d*x]^2] + C*(9 + 2*n)*Cos[c + d*x]*Hypergeometric2F1[1/2, (11 + 2*n)/4, (15 + 2*n)/4, Cos[c + d*x]^2])*Sqrt[Sin[c + d*x]^2])/(d*(9 + 2*n)*(11 + 2*n))","A",1
225,1,138,163,0.4173548,"\int \cos ^{\frac{3}{2}}(c+d x) (b \cos (c+d x))^n \left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^(3/2)*(b*Cos[c + d*x])^n*(B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","-\frac{2 \sqrt{\sin ^2(c+d x)} \cos ^{\frac{7}{2}}(c+d x) \csc (c+d x) (b \cos (c+d x))^n \left(B (2 n+9) \, _2F_1\left(\frac{1}{2},\frac{1}{4} (2 n+7);\frac{1}{4} (2 n+11);\cos ^2(c+d x)\right)+C (2 n+7) \cos (c+d x) \, _2F_1\left(\frac{1}{2},\frac{1}{4} (2 n+9);\frac{1}{4} (2 n+13);\cos ^2(c+d x)\right)\right)}{d (2 n+7) (2 n+9)}","-\frac{2 B \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x) (b \cos (c+d x))^n \, _2F_1\left(\frac{1}{2},\frac{1}{4} (2 n+7);\frac{1}{4} (2 n+11);\cos ^2(c+d x)\right)}{d (2 n+7) \sqrt{\sin ^2(c+d x)}}-\frac{2 C \sin (c+d x) \cos ^{\frac{9}{2}}(c+d x) (b \cos (c+d x))^n \, _2F_1\left(\frac{1}{2},\frac{1}{4} (2 n+9);\frac{1}{4} (2 n+13);\cos ^2(c+d x)\right)}{d (2 n+9) \sqrt{\sin ^2(c+d x)}}",1,"(-2*Cos[c + d*x]^(7/2)*(b*Cos[c + d*x])^n*Csc[c + d*x]*(B*(9 + 2*n)*Hypergeometric2F1[1/2, (7 + 2*n)/4, (11 + 2*n)/4, Cos[c + d*x]^2] + C*(7 + 2*n)*Cos[c + d*x]*Hypergeometric2F1[1/2, (9 + 2*n)/4, (13 + 2*n)/4, Cos[c + d*x]^2])*Sqrt[Sin[c + d*x]^2])/(d*(7 + 2*n)*(9 + 2*n))","A",1
226,1,138,163,0.343703,"\int \sqrt{\cos (c+d x)} (b \cos (c+d x))^n \left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Integrate[Sqrt[Cos[c + d*x]]*(b*Cos[c + d*x])^n*(B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","-\frac{2 \sqrt{\sin ^2(c+d x)} \cos ^{\frac{5}{2}}(c+d x) \csc (c+d x) (b \cos (c+d x))^n \left(B (2 n+7) \, _2F_1\left(\frac{1}{2},\frac{1}{4} (2 n+5);\frac{1}{4} (2 n+9);\cos ^2(c+d x)\right)+C (2 n+5) \cos (c+d x) \, _2F_1\left(\frac{1}{2},\frac{1}{4} (2 n+7);\frac{1}{4} (2 n+11);\cos ^2(c+d x)\right)\right)}{d (2 n+5) (2 n+7)}","-\frac{2 B \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) (b \cos (c+d x))^n \, _2F_1\left(\frac{1}{2},\frac{1}{4} (2 n+5);\frac{1}{4} (2 n+9);\cos ^2(c+d x)\right)}{d (2 n+5) \sqrt{\sin ^2(c+d x)}}-\frac{2 C \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x) (b \cos (c+d x))^n \, _2F_1\left(\frac{1}{2},\frac{1}{4} (2 n+7);\frac{1}{4} (2 n+11);\cos ^2(c+d x)\right)}{d (2 n+7) \sqrt{\sin ^2(c+d x)}}",1,"(-2*Cos[c + d*x]^(5/2)*(b*Cos[c + d*x])^n*Csc[c + d*x]*(B*(7 + 2*n)*Hypergeometric2F1[1/2, (5 + 2*n)/4, (9 + 2*n)/4, Cos[c + d*x]^2] + C*(5 + 2*n)*Cos[c + d*x]*Hypergeometric2F1[1/2, (7 + 2*n)/4, (11 + 2*n)/4, Cos[c + d*x]^2])*Sqrt[Sin[c + d*x]^2])/(d*(5 + 2*n)*(7 + 2*n))","A",1
227,1,138,163,0.2497005,"\int \frac{(b \cos (c+d x))^n \left(B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\sqrt{\cos (c+d x)}} \, dx","Integrate[((b*Cos[c + d*x])^n*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Sqrt[Cos[c + d*x]],x]","-\frac{2 \sqrt{\sin ^2(c+d x)} \cos ^{\frac{3}{2}}(c+d x) \csc (c+d x) (b \cos (c+d x))^n \left(B (2 n+5) \, _2F_1\left(\frac{1}{2},\frac{1}{4} (2 n+3);\frac{1}{4} (2 n+7);\cos ^2(c+d x)\right)+C (2 n+3) \cos (c+d x) \, _2F_1\left(\frac{1}{2},\frac{1}{4} (2 n+5);\frac{1}{4} (2 n+9);\cos ^2(c+d x)\right)\right)}{d (2 n+3) (2 n+5)}","-\frac{2 B \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) (b \cos (c+d x))^n \, _2F_1\left(\frac{1}{2},\frac{1}{4} (2 n+3);\frac{1}{4} (2 n+7);\cos ^2(c+d x)\right)}{d (2 n+3) \sqrt{\sin ^2(c+d x)}}-\frac{2 C \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) (b \cos (c+d x))^n \, _2F_1\left(\frac{1}{2},\frac{1}{4} (2 n+5);\frac{1}{4} (2 n+9);\cos ^2(c+d x)\right)}{d (2 n+5) \sqrt{\sin ^2(c+d x)}}",1,"(-2*Cos[c + d*x]^(3/2)*(b*Cos[c + d*x])^n*Csc[c + d*x]*(B*(5 + 2*n)*Hypergeometric2F1[1/2, (3 + 2*n)/4, (7 + 2*n)/4, Cos[c + d*x]^2] + C*(3 + 2*n)*Cos[c + d*x]*Hypergeometric2F1[1/2, (5 + 2*n)/4, (9 + 2*n)/4, Cos[c + d*x]^2])*Sqrt[Sin[c + d*x]^2])/(d*(3 + 2*n)*(5 + 2*n))","A",1
228,1,138,163,0.241107,"\int \frac{(b \cos (c+d x))^n \left(B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[((b*Cos[c + d*x])^n*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(3/2),x]","-\frac{2 \sqrt{\sin ^2(c+d x)} \sqrt{\cos (c+d x)} \csc (c+d x) (b \cos (c+d x))^n \left(B (2 n+3) \, _2F_1\left(\frac{1}{2},\frac{1}{4} (2 n+1);\frac{1}{4} (2 n+5);\cos ^2(c+d x)\right)+C (2 n+1) \cos (c+d x) \, _2F_1\left(\frac{1}{2},\frac{1}{4} (2 n+3);\frac{1}{4} (2 n+7);\cos ^2(c+d x)\right)\right)}{d (2 n+1) (2 n+3)}","-\frac{2 B \sin (c+d x) \sqrt{\cos (c+d x)} (b \cos (c+d x))^n \, _2F_1\left(\frac{1}{2},\frac{1}{4} (2 n+1);\frac{1}{4} (2 n+5);\cos ^2(c+d x)\right)}{d (2 n+1) \sqrt{\sin ^2(c+d x)}}-\frac{2 C \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) (b \cos (c+d x))^n \, _2F_1\left(\frac{1}{2},\frac{1}{4} (2 n+3);\frac{1}{4} (2 n+7);\cos ^2(c+d x)\right)}{d (2 n+3) \sqrt{\sin ^2(c+d x)}}",1,"(-2*Sqrt[Cos[c + d*x]]*(b*Cos[c + d*x])^n*Csc[c + d*x]*(B*(3 + 2*n)*Hypergeometric2F1[1/2, (1 + 2*n)/4, (5 + 2*n)/4, Cos[c + d*x]^2] + C*(1 + 2*n)*Cos[c + d*x]*Hypergeometric2F1[1/2, (3 + 2*n)/4, (7 + 2*n)/4, Cos[c + d*x]^2])*Sqrt[Sin[c + d*x]^2])/(d*(1 + 2*n)*(3 + 2*n))","A",1
229,1,133,163,0.2668711,"\int \frac{(b \cos (c+d x))^n \left(B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{5}{2}}(c+d x)} \, dx","Integrate[((b*Cos[c + d*x])^n*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(5/2),x]","-\frac{2 \sqrt{\sin ^2(c+d x)} \csc (c+d x) (b \cos (c+d x))^n \left(B (2 n+1) \, _2F_1\left(\frac{1}{2},\frac{1}{4} (2 n-1);\frac{1}{4} (2 n+3);\cos ^2(c+d x)\right)+C (2 n-1) \cos (c+d x) \, _2F_1\left(\frac{1}{2},\frac{1}{4} (2 n+1);\frac{1}{4} (2 n+5);\cos ^2(c+d x)\right)\right)}{d \left(4 n^2-1\right) \sqrt{\cos (c+d x)}}","\frac{2 B \sin (c+d x) (b \cos (c+d x))^n \, _2F_1\left(\frac{1}{2},\frac{1}{4} (2 n-1);\frac{1}{4} (2 n+3);\cos ^2(c+d x)\right)}{d (1-2 n) \sqrt{\sin ^2(c+d x)} \sqrt{\cos (c+d x)}}-\frac{2 C \sin (c+d x) \sqrt{\cos (c+d x)} (b \cos (c+d x))^n \, _2F_1\left(\frac{1}{2},\frac{1}{4} (2 n+1);\frac{1}{4} (2 n+5);\cos ^2(c+d x)\right)}{d (2 n+1) \sqrt{\sin ^2(c+d x)}}",1,"(-2*(b*Cos[c + d*x])^n*Csc[c + d*x]*(B*(1 + 2*n)*Hypergeometric2F1[1/2, (-1 + 2*n)/4, (3 + 2*n)/4, Cos[c + d*x]^2] + C*(-1 + 2*n)*Cos[c + d*x]*Hypergeometric2F1[1/2, (1 + 2*n)/4, (5 + 2*n)/4, Cos[c + d*x]^2])*Sqrt[Sin[c + d*x]^2])/(d*(-1 + 4*n^2)*Sqrt[Cos[c + d*x]])","A",1
230,1,138,163,0.2291897,"\int \frac{(b \cos (c+d x))^n \left(B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{7}{2}}(c+d x)} \, dx","Integrate[((b*Cos[c + d*x])^n*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(7/2),x]","-\frac{2 \sqrt{\sin ^2(c+d x)} \csc (c+d x) (b \cos (c+d x))^n \left(B (2 n-1) \, _2F_1\left(\frac{1}{2},\frac{1}{4} (2 n-3);\frac{1}{4} (2 n+1);\cos ^2(c+d x)\right)+C (2 n-3) \cos (c+d x) \, _2F_1\left(\frac{1}{2},\frac{1}{4} (2 n-1);\frac{1}{4} (2 n+3);\cos ^2(c+d x)\right)\right)}{d (2 n-3) (2 n-1) \cos ^{\frac{3}{2}}(c+d x)}","\frac{2 B \sin (c+d x) (b \cos (c+d x))^n \, _2F_1\left(\frac{1}{2},\frac{1}{4} (2 n-3);\frac{1}{4} (2 n+1);\cos ^2(c+d x)\right)}{d (3-2 n) \sqrt{\sin ^2(c+d x)} \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 C \sin (c+d x) (b \cos (c+d x))^n \, _2F_1\left(\frac{1}{2},\frac{1}{4} (2 n-1);\frac{1}{4} (2 n+3);\cos ^2(c+d x)\right)}{d (1-2 n) \sqrt{\sin ^2(c+d x)} \sqrt{\cos (c+d x)}}",1,"(-2*(b*Cos[c + d*x])^n*Csc[c + d*x]*(B*(-1 + 2*n)*Hypergeometric2F1[1/2, (-3 + 2*n)/4, (1 + 2*n)/4, Cos[c + d*x]^2] + C*(-3 + 2*n)*Cos[c + d*x]*Hypergeometric2F1[1/2, (-1 + 2*n)/4, (3 + 2*n)/4, Cos[c + d*x]^2])*Sqrt[Sin[c + d*x]^2])/(d*(-3 + 2*n)*(-1 + 2*n)*Cos[c + d*x]^(3/2))","A",1
231,1,138,163,0.2312979,"\int \frac{(b \cos (c+d x))^n \left(B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{9}{2}}(c+d x)} \, dx","Integrate[((b*Cos[c + d*x])^n*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(9/2),x]","-\frac{2 \sqrt{\sin ^2(c+d x)} \csc (c+d x) (b \cos (c+d x))^n \left(B (2 n-3) \, _2F_1\left(\frac{1}{2},\frac{1}{4} (2 n-5);\frac{1}{4} (2 n-1);\cos ^2(c+d x)\right)+C (2 n-5) \cos (c+d x) \, _2F_1\left(\frac{1}{2},\frac{1}{4} (2 n-3);\frac{1}{4} (2 n+1);\cos ^2(c+d x)\right)\right)}{d (2 n-5) (2 n-3) \cos ^{\frac{5}{2}}(c+d x)}","\frac{2 B \sin (c+d x) (b \cos (c+d x))^n \, _2F_1\left(\frac{1}{2},\frac{1}{4} (2 n-5);\frac{1}{4} (2 n-1);\cos ^2(c+d x)\right)}{d (5-2 n) \sqrt{\sin ^2(c+d x)} \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 C \sin (c+d x) (b \cos (c+d x))^n \, _2F_1\left(\frac{1}{2},\frac{1}{4} (2 n-3);\frac{1}{4} (2 n+1);\cos ^2(c+d x)\right)}{d (3-2 n) \sqrt{\sin ^2(c+d x)} \cos ^{\frac{3}{2}}(c+d x)}",1,"(-2*(b*Cos[c + d*x])^n*Csc[c + d*x]*(B*(-3 + 2*n)*Hypergeometric2F1[1/2, (-5 + 2*n)/4, (-1 + 2*n)/4, Cos[c + d*x]^2] + C*(-5 + 2*n)*Cos[c + d*x]*Hypergeometric2F1[1/2, (-3 + 2*n)/4, (1 + 2*n)/4, Cos[c + d*x]^2])*Sqrt[Sin[c + d*x]^2])/(d*(-5 + 2*n)*(-3 + 2*n)*Cos[c + d*x]^(5/2))","A",1
232,1,356,173,50.2908297,"\int (a+a \cos (e+f x))^m \left(B \cos (e+f x)+C \cos ^2(e+f x)\right) \, dx","Integrate[(a + a*Cos[e + f*x])^m*(B*Cos[e + f*x] + C*Cos[e + f*x]^2),x]","\frac{i 4^{-m-1} e^{-2 i (e+f x)} \left(1+e^{i (e+f x)}\right)^{-2 m} \left(e^{-\frac{1}{2} i (e+f x)} \left(1+e^{i (e+f x)}\right)\right)^{2 m} \cos ^{-2 m}\left(\frac{1}{2} (e+f x)\right) (a (\cos (e+f x)+1))^m \left((m+2) e^{i (e+f x)} \left((m+1) e^{i (e+f x)} \left(2 B (m-2) m e^{i (e+f x)} \, _2F_1\left(1-m,-2 m;2-m;-e^{i (e+f x)}\right)+C (m-1) \left(m e^{2 i (e+f x)} \, _2F_1\left(2-m,-2 m;3-m;-e^{i (e+f x)}\right)+2 (m-2) \, _2F_1\left(-2 m,-m;1-m;-e^{i (e+f x)}\right)\right)\right)+2 B m \left(m^2-3 m+2\right) \, _2F_1\left(-m-1,-2 m;-m;-e^{i (e+f x)}\right)\right)+C m \left(m^3-2 m^2-m+2\right) \, _2F_1\left(-m-2,-2 m;-m-1;-e^{i (e+f x)}\right)\right)}{f (m-2) (m-1) m (m+1) (m+2)}","\frac{2^{m+\frac{1}{2}} \left(B m (m+2)+C \left(m^2+m+1\right)\right) \sin (e+f x) (\cos (e+f x)+1)^{-m-\frac{1}{2}} (a \cos (e+f x)+a)^m \, _2F_1\left(\frac{1}{2},\frac{1}{2}-m;\frac{3}{2};\frac{1}{2} (1-\cos (e+f x))\right)}{f (m+1) (m+2)}-\frac{(C-B (m+2)) \sin (e+f x) (a \cos (e+f x)+a)^m}{f (m+1) (m+2)}+\frac{C \sin (e+f x) (a \cos (e+f x)+a)^{m+1}}{a f (m+2)}",1,"(I*4^(-1 - m)*((1 + E^(I*(e + f*x)))/E^((I/2)*(e + f*x)))^(2*m)*(a*(1 + Cos[e + f*x]))^m*(C*m*(2 - m - 2*m^2 + m^3)*Hypergeometric2F1[-2 - m, -2*m, -1 - m, -E^(I*(e + f*x))] + E^(I*(e + f*x))*(2 + m)*(2*B*m*(2 - 3*m + m^2)*Hypergeometric2F1[-1 - m, -2*m, -m, -E^(I*(e + f*x))] + E^(I*(e + f*x))*(1 + m)*(2*B*E^(I*(e + f*x))*(-2 + m)*m*Hypergeometric2F1[1 - m, -2*m, 2 - m, -E^(I*(e + f*x))] + C*(-1 + m)*(E^((2*I)*(e + f*x))*m*Hypergeometric2F1[2 - m, -2*m, 3 - m, -E^(I*(e + f*x))] + 2*(-2 + m)*Hypergeometric2F1[-2*m, -m, 1 - m, -E^(I*(e + f*x))])))))/(E^((2*I)*(e + f*x))*(1 + E^(I*(e + f*x)))^(2*m)*f*(-2 + m)*(-1 + m)*m*(1 + m)*(2 + m)*Cos[(e + f*x)/2]^(2*m))","C",1
233,1,13441,295,26.7550192,"\int (a+b \cos (e+f x))^m \left(B \cos (e+f x)+C \cos ^2(e+f x)\right) \, dx","Integrate[(a + b*Cos[e + f*x])^m*(B*Cos[e + f*x] + C*Cos[e + f*x]^2),x]","\text{Result too large to show}","\frac{\sqrt{2} \sin (e+f x) \left(a^2 C-a b B (m+2)+b^2 C (m+1)\right) (a+b \cos (e+f x))^m \left(\frac{a+b \cos (e+f x)}{a+b}\right)^{-m} F_1\left(\frac{1}{2};\frac{1}{2},-m;\frac{3}{2};\frac{1}{2} (1-\cos (e+f x)),\frac{b (1-\cos (e+f x))}{a+b}\right)}{b^2 f (m+2) \sqrt{\cos (e+f x)+1}}-\frac{\sqrt{2} (a+b) \sin (e+f x) (a C-b B (m+2)) (a+b \cos (e+f x))^m \left(\frac{a+b \cos (e+f x)}{a+b}\right)^{-m} F_1\left(\frac{1}{2};\frac{1}{2},-m-1;\frac{3}{2};\frac{1}{2} (1-\cos (e+f x)),\frac{b (1-\cos (e+f x))}{a+b}\right)}{b^2 f (m+2) \sqrt{\cos (e+f x)+1}}+\frac{C \sin (e+f x) (a+b \cos (e+f x))^{m+1}}{b f (m+2)}",1,"Result too large to show","B",0
234,1,290,284,3.0331851,"\int (a+b \cos (c+d x))^{2/3} \left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Integrate[(a + b*Cos[c + d*x])^(2/3)*(B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","-\frac{3 \csc (c+d x) (a+b \cos (c+d x))^{2/3} \left(\left(-6 a^2 C+16 a b B+25 b^2 C\right) \sqrt{-\frac{b (\cos (c+d x)-1)}{a+b}} \sqrt{-\frac{b (\cos (c+d x)+1)}{a-b}} (a+b \cos (c+d x)) F_1\left(\frac{5}{3};\frac{1}{2},\frac{1}{2};\frac{8}{3};\frac{a+b \cos (c+d x)}{a-b},\frac{a+b \cos (c+d x)}{a+b}\right)+5 \left(b^2-a^2\right) (8 b B-3 a C) \sqrt{-\frac{b (\cos (c+d x)-1)}{a+b}} \sqrt{-\frac{b (\cos (c+d x)+1)}{a-b}} F_1\left(\frac{2}{3};\frac{1}{2},\frac{1}{2};\frac{5}{3};\frac{a+b \cos (c+d x)}{a-b},\frac{a+b \cos (c+d x)}{a+b}\right)-5 b^2 \sin ^2(c+d x) (2 a C+8 b B+5 b C \cos (c+d x))\right)}{200 b^3 d}","-\frac{\left(-3 a^2 C+8 a b B-5 b^2 C\right) \sin (c+d x) (a+b \cos (c+d x))^{2/3} F_1\left(\frac{1}{2};\frac{1}{2},-\frac{2}{3};\frac{3}{2};\frac{1}{2} (1-\cos (c+d x)),\frac{b (1-\cos (c+d x))}{a+b}\right)}{4 \sqrt{2} b^2 d \sqrt{\cos (c+d x)+1} \left(\frac{a+b \cos (c+d x)}{a+b}\right)^{2/3}}+\frac{(a+b) (8 b B-3 a C) \sin (c+d x) (a+b \cos (c+d x))^{2/3} F_1\left(\frac{1}{2};\frac{1}{2},-\frac{5}{3};\frac{3}{2};\frac{1}{2} (1-\cos (c+d x)),\frac{b (1-\cos (c+d x))}{a+b}\right)}{4 \sqrt{2} b^2 d \sqrt{\cos (c+d x)+1} \left(\frac{a+b \cos (c+d x)}{a+b}\right)^{2/3}}+\frac{3 C \sin (c+d x) (a+b \cos (c+d x))^{5/3}}{8 b d}",1,"(-3*(a + b*Cos[c + d*x])^(2/3)*Csc[c + d*x]*(5*(-a^2 + b^2)*(8*b*B - 3*a*C)*AppellF1[2/3, 1/2, 1/2, 5/3, (a + b*Cos[c + d*x])/(a - b), (a + b*Cos[c + d*x])/(a + b)]*Sqrt[-((b*(-1 + Cos[c + d*x]))/(a + b))]*Sqrt[-((b*(1 + Cos[c + d*x]))/(a - b))] + (16*a*b*B - 6*a^2*C + 25*b^2*C)*AppellF1[5/3, 1/2, 1/2, 8/3, (a + b*Cos[c + d*x])/(a - b), (a + b*Cos[c + d*x])/(a + b)]*Sqrt[-((b*(-1 + Cos[c + d*x]))/(a + b))]*Sqrt[-((b*(1 + Cos[c + d*x]))/(a - b))]*(a + b*Cos[c + d*x]) - 5*b^2*(8*b*B + 2*a*C + 5*b*C*Cos[c + d*x])*Sin[c + d*x]^2))/(200*b^3*d)","A",0
235,1,289,284,3.0136292,"\int \sqrt[3]{a+b \cos (c+d x)} \left(B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Integrate[(a + b*Cos[c + d*x])^(1/3)*(B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","-\frac{3 \csc (c+d x) \sqrt[3]{a+b \cos (c+d x)} \left(\left(-3 a^2 C+7 a b B+16 b^2 C\right) \sqrt{-\frac{b (\cos (c+d x)-1)}{a+b}} \sqrt{-\frac{b (\cos (c+d x)+1)}{a-b}} (a+b \cos (c+d x)) F_1\left(\frac{4}{3};\frac{1}{2},\frac{1}{2};\frac{7}{3};\frac{a+b \cos (c+d x)}{a-b},\frac{a+b \cos (c+d x)}{a+b}\right)+4 \left(b^2-a^2\right) (7 b B-3 a C) \sqrt{-\frac{b (\cos (c+d x)-1)}{a+b}} \sqrt{-\frac{b (\cos (c+d x)+1)}{a-b}} F_1\left(\frac{1}{3};\frac{1}{2},\frac{1}{2};\frac{4}{3};\frac{a+b \cos (c+d x)}{a-b},\frac{a+b \cos (c+d x)}{a+b}\right)-4 b^2 \sin ^2(c+d x) (a C+7 b B+4 b C \cos (c+d x))\right)}{112 b^3 d}","-\frac{\sqrt{2} \left(-3 a^2 C+7 a b B-4 b^2 C\right) \sin (c+d x) \sqrt[3]{a+b \cos (c+d x)} F_1\left(\frac{1}{2};\frac{1}{2},-\frac{1}{3};\frac{3}{2};\frac{1}{2} (1-\cos (c+d x)),\frac{b (1-\cos (c+d x))}{a+b}\right)}{7 b^2 d \sqrt{\cos (c+d x)+1} \sqrt[3]{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{\sqrt{2} (a+b) (7 b B-3 a C) \sin (c+d x) \sqrt[3]{a+b \cos (c+d x)} F_1\left(\frac{1}{2};\frac{1}{2},-\frac{4}{3};\frac{3}{2};\frac{1}{2} (1-\cos (c+d x)),\frac{b (1-\cos (c+d x))}{a+b}\right)}{7 b^2 d \sqrt{\cos (c+d x)+1} \sqrt[3]{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{3 C \sin (c+d x) (a+b \cos (c+d x))^{4/3}}{7 b d}",1,"(-3*(a + b*Cos[c + d*x])^(1/3)*Csc[c + d*x]*(4*(-a^2 + b^2)*(7*b*B - 3*a*C)*AppellF1[1/3, 1/2, 1/2, 4/3, (a + b*Cos[c + d*x])/(a - b), (a + b*Cos[c + d*x])/(a + b)]*Sqrt[-((b*(-1 + Cos[c + d*x]))/(a + b))]*Sqrt[-((b*(1 + Cos[c + d*x]))/(a - b))] + (7*a*b*B - 3*a^2*C + 16*b^2*C)*AppellF1[4/3, 1/2, 1/2, 7/3, (a + b*Cos[c + d*x])/(a - b), (a + b*Cos[c + d*x])/(a + b)]*Sqrt[-((b*(-1 + Cos[c + d*x]))/(a + b))]*Sqrt[-((b*(1 + Cos[c + d*x]))/(a - b))]*(a + b*Cos[c + d*x]) - 4*b^2*(7*b*B + a*C + 4*b*C*Cos[c + d*x])*Sin[c + d*x]^2))/(112*b^3*d)","A",0
236,1,263,281,2.1674691,"\int \frac{B \cos (c+d x)+C \cos ^2(c+d x)}{\sqrt[3]{a+b \cos (c+d x)}} \, dx","Integrate[(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^(1/3),x]","-\frac{3 \csc (c+d x) (a+b \cos (c+d x))^{2/3} \left(5 \left(3 a^2 C-5 a b B+2 b^2 C\right) \sqrt{-\frac{b (\cos (c+d x)-1)}{a+b}} \sqrt{-\frac{b (\cos (c+d x)+1)}{a-b}} F_1\left(\frac{2}{3};\frac{1}{2},\frac{1}{2};\frac{5}{3};\frac{a+b \cos (c+d x)}{a-b},\frac{a+b \cos (c+d x)}{a+b}\right)+2 (5 b B-3 a C) \sqrt{-\frac{b (\cos (c+d x)-1)}{a+b}} \sqrt{\frac{b (\cos (c+d x)+1)}{b-a}} (a+b \cos (c+d x)) F_1\left(\frac{5}{3};\frac{1}{2},\frac{1}{2};\frac{8}{3};\frac{a+b \cos (c+d x)}{a-b},\frac{a+b \cos (c+d x)}{a+b}\right)-10 b^2 C \sin ^2(c+d x)\right)}{50 b^3 d}","-\frac{\sqrt{2} \left(-3 a^2 C+5 a b B-2 b^2 C\right) \sin (c+d x) \sqrt[3]{\frac{a+b \cos (c+d x)}{a+b}} F_1\left(\frac{1}{2};\frac{1}{2},\frac{1}{3};\frac{3}{2};\frac{1}{2} (1-\cos (c+d x)),\frac{b (1-\cos (c+d x))}{a+b}\right)}{5 b^2 d \sqrt{\cos (c+d x)+1} \sqrt[3]{a+b \cos (c+d x)}}+\frac{\sqrt{2} (5 b B-3 a C) \sin (c+d x) (a+b \cos (c+d x))^{2/3} F_1\left(\frac{1}{2};\frac{1}{2},-\frac{2}{3};\frac{3}{2};\frac{1}{2} (1-\cos (c+d x)),\frac{b (1-\cos (c+d x))}{a+b}\right)}{5 b^2 d \sqrt{\cos (c+d x)+1} \left(\frac{a+b \cos (c+d x)}{a+b}\right)^{2/3}}+\frac{3 C \sin (c+d x) (a+b \cos (c+d x))^{2/3}}{5 b d}",1,"(-3*(a + b*Cos[c + d*x])^(2/3)*Csc[c + d*x]*(5*(-5*a*b*B + 3*a^2*C + 2*b^2*C)*AppellF1[2/3, 1/2, 1/2, 5/3, (a + b*Cos[c + d*x])/(a - b), (a + b*Cos[c + d*x])/(a + b)]*Sqrt[-((b*(-1 + Cos[c + d*x]))/(a + b))]*Sqrt[-((b*(1 + Cos[c + d*x]))/(a - b))] + 2*(5*b*B - 3*a*C)*AppellF1[5/3, 1/2, 1/2, 8/3, (a + b*Cos[c + d*x])/(a - b), (a + b*Cos[c + d*x])/(a + b)]*Sqrt[-((b*(-1 + Cos[c + d*x]))/(a + b))]*Sqrt[(b*(1 + Cos[c + d*x]))/(-a + b)]*(a + b*Cos[c + d*x]) - 10*b^2*C*Sin[c + d*x]^2))/(50*b^3*d)","A",0
237,1,261,281,2.2057318,"\int \frac{B \cos (c+d x)+C \cos ^2(c+d x)}{(a+b \cos (c+d x))^{2/3}} \, dx","Integrate[(B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^(2/3),x]","-\frac{3 \csc (c+d x) \sqrt[3]{a+b \cos (c+d x)} \left(4 \left(3 a^2 C-4 a b B+b^2 C\right) \sqrt{-\frac{b (\cos (c+d x)-1)}{a+b}} \sqrt{-\frac{b (\cos (c+d x)+1)}{a-b}} F_1\left(\frac{1}{3};\frac{1}{2},\frac{1}{2};\frac{4}{3};\frac{a+b \cos (c+d x)}{a-b},\frac{a+b \cos (c+d x)}{a+b}\right)+(4 b B-3 a C) \sqrt{-\frac{b (\cos (c+d x)-1)}{a+b}} \sqrt{\frac{b (\cos (c+d x)+1)}{b-a}} (a+b \cos (c+d x)) F_1\left(\frac{4}{3};\frac{1}{2},\frac{1}{2};\frac{7}{3};\frac{a+b \cos (c+d x)}{a-b},\frac{a+b \cos (c+d x)}{a+b}\right)-4 b^2 C \sin ^2(c+d x)\right)}{16 b^3 d}","-\frac{\left(-3 a^2 C+4 a b B-b^2 C\right) \sin (c+d x) \left(\frac{a+b \cos (c+d x)}{a+b}\right)^{2/3} F_1\left(\frac{1}{2};\frac{1}{2},\frac{2}{3};\frac{3}{2};\frac{1}{2} (1-\cos (c+d x)),\frac{b (1-\cos (c+d x))}{a+b}\right)}{2 \sqrt{2} b^2 d \sqrt{\cos (c+d x)+1} (a+b \cos (c+d x))^{2/3}}+\frac{(4 b B-3 a C) \sin (c+d x) \sqrt[3]{a+b \cos (c+d x)} F_1\left(\frac{1}{2};\frac{1}{2},-\frac{1}{3};\frac{3}{2};\frac{1}{2} (1-\cos (c+d x)),\frac{b (1-\cos (c+d x))}{a+b}\right)}{2 \sqrt{2} b^2 d \sqrt{\cos (c+d x)+1} \sqrt[3]{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{3 C \sin (c+d x) \sqrt[3]{a+b \cos (c+d x)}}{4 b d}",1,"(-3*(a + b*Cos[c + d*x])^(1/3)*Csc[c + d*x]*(4*(-4*a*b*B + 3*a^2*C + b^2*C)*AppellF1[1/3, 1/2, 1/2, 4/3, (a + b*Cos[c + d*x])/(a - b), (a + b*Cos[c + d*x])/(a + b)]*Sqrt[-((b*(-1 + Cos[c + d*x]))/(a + b))]*Sqrt[-((b*(1 + Cos[c + d*x]))/(a - b))] + (4*b*B - 3*a*C)*AppellF1[4/3, 1/2, 1/2, 7/3, (a + b*Cos[c + d*x])/(a - b), (a + b*Cos[c + d*x])/(a + b)]*Sqrt[-((b*(-1 + Cos[c + d*x]))/(a + b))]*Sqrt[(b*(1 + Cos[c + d*x]))/(-a + b)]*(a + b*Cos[c + d*x]) - 4*b^2*C*Sin[c + d*x]^2))/(16*b^3*d)","A",0
238,1,142,187,0.295128,"\int (a \cos (e+f x))^m \left(A+B \cos (e+f x)+C \cos ^2(e+f x)\right) \, dx","Integrate[(a*Cos[e + f*x])^m*(A + B*Cos[e + f*x] + C*Cos[e + f*x]^2),x]","-\frac{\sin (e+f x) \cos (e+f x) (a \cos (e+f x))^m \left((A (m+2)+C (m+1)) \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};\cos ^2(e+f x)\right)+(m+1) \left(B \cos (e+f x) \, _2F_1\left(\frac{1}{2},\frac{m+2}{2};\frac{m+4}{2};\cos ^2(e+f x)\right)-C \sqrt{\sin ^2(e+f x)}\right)\right)}{f (m+1) (m+2) \sqrt{\sin ^2(e+f x)}}","-\frac{B \sin (e+f x) (a \cos (e+f x))^{m+2} \, _2F_1\left(\frac{1}{2},\frac{m+2}{2};\frac{m+4}{2};\cos ^2(e+f x)\right)}{a^2 f (m+2) \sqrt{\sin ^2(e+f x)}}-\frac{(A (m+2)+C (m+1)) \sin (e+f x) (a \cos (e+f x))^{m+1} \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};\cos ^2(e+f x)\right)}{a f (m+1) (m+2) \sqrt{\sin ^2(e+f x)}}+\frac{C \sin (e+f x) (a \cos (e+f x))^{m+1}}{a f (m+2)}",1,"-((Cos[e + f*x]*(a*Cos[e + f*x])^m*Sin[e + f*x]*((C*(1 + m) + A*(2 + m))*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, Cos[e + f*x]^2] + (1 + m)*(B*Cos[e + f*x]*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, Cos[e + f*x]^2] - C*Sqrt[Sin[e + f*x]^2])))/(f*(1 + m)*(2 + m)*Sqrt[Sin[e + f*x]^2]))","A",1
239,1,125,209,1.0783453,"\int \cos ^2(c+d x) \sqrt{b \cos (c+d x)} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^2*Sqrt[b*Cos[c + d*x]]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","\frac{\sqrt{b \cos (c+d x)} \left(\sin (c+d x) \sqrt{\cos (c+d x)} (7 (36 A+43 C) \cos (c+d x)+5 (18 B \cos (2 (c+d x))+78 B+7 C \cos (3 (c+d x))))+84 (9 A+7 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+300 B F\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{630 d \sqrt{\cos (c+d x)}}","\frac{2 (9 A+7 C) \sin (c+d x) (b \cos (c+d x))^{3/2}}{45 b d}+\frac{2 (9 A+7 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \cos (c+d x)}}{15 d \sqrt{\cos (c+d x)}}+\frac{2 C \sin (c+d x) (b \cos (c+d x))^{7/2}}{9 b^3 d}+\frac{2 B \sin (c+d x) (b \cos (c+d x))^{5/2}}{7 b^2 d}+\frac{10 B \sin (c+d x) \sqrt{b \cos (c+d x)}}{21 d}+\frac{10 b B \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d \sqrt{b \cos (c+d x)}}",1,"(Sqrt[b*Cos[c + d*x]]*(84*(9*A + 7*C)*EllipticE[(c + d*x)/2, 2] + 300*B*EllipticF[(c + d*x)/2, 2] + Sqrt[Cos[c + d*x]]*(7*(36*A + 43*C)*Cos[c + d*x] + 5*(78*B + 18*B*Cos[2*(c + d*x)] + 7*C*Cos[3*(c + d*x)]))*Sin[c + d*x]))/(630*d*Sqrt[Cos[c + d*x]])","A",1
240,1,111,180,0.8017908,"\int \cos (c+d x) \sqrt{b \cos (c+d x)} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]*Sqrt[b*Cos[c + d*x]]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","\frac{(b \cos (c+d x))^{3/2} \left(\sin (c+d x) \sqrt{\cos (c+d x)} (70 A+42 B \cos (c+d x)+15 C \cos (2 (c+d x))+65 C)+10 (7 A+5 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+126 B E\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{105 b d \cos ^{\frac{3}{2}}(c+d x)}","\frac{2 (7 A+5 C) \sin (c+d x) \sqrt{b \cos (c+d x)}}{21 d}+\frac{2 b (7 A+5 C) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d \sqrt{b \cos (c+d x)}}+\frac{2 C \sin (c+d x) (b \cos (c+d x))^{5/2}}{7 b^2 d}+\frac{2 B \sin (c+d x) (b \cos (c+d x))^{3/2}}{5 b d}+\frac{6 B E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \cos (c+d x)}}{5 d \sqrt{\cos (c+d x)}}",1,"((b*Cos[c + d*x])^(3/2)*(126*B*EllipticE[(c + d*x)/2, 2] + 10*(7*A + 5*C)*EllipticF[(c + d*x)/2, 2] + Sqrt[Cos[c + d*x]]*(70*A + 65*C + 42*B*Cos[c + d*x] + 15*C*Cos[2*(c + d*x)])*Sin[c + d*x]))/(105*b*d*Cos[c + d*x]^(3/2))","A",1
241,1,94,145,0.3541097,"\int \sqrt{b \cos (c+d x)} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Integrate[Sqrt[b*Cos[c + d*x]]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","\frac{2 \sqrt{b \cos (c+d x)} \left(3 (5 A+3 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+\sin (c+d x) \sqrt{\cos (c+d x)} (5 B+3 C \cos (c+d x))+5 B F\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{15 d \sqrt{\cos (c+d x)}}","\frac{2 (5 A+3 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \cos (c+d x)}}{5 d \sqrt{\cos (c+d x)}}+\frac{2 B \sin (c+d x) \sqrt{b \cos (c+d x)}}{3 d}+\frac{2 b B \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d \sqrt{b \cos (c+d x)}}+\frac{2 C \sin (c+d x) (b \cos (c+d x))^{3/2}}{5 b d}",1,"(2*Sqrt[b*Cos[c + d*x]]*(3*(5*A + 3*C)*EllipticE[(c + d*x)/2, 2] + 5*B*EllipticF[(c + d*x)/2, 2] + Sqrt[Cos[c + d*x]]*(5*B + 3*C*Cos[c + d*x])*Sin[c + d*x]))/(15*d*Sqrt[Cos[c + d*x]])","A",1
242,1,83,112,0.2235107,"\int \sqrt{b \cos (c+d x)} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec (c+d x) \, dx","Integrate[Sqrt[b*Cos[c + d*x]]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x],x]","\frac{b \left(2 (3 A+C) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+6 B \sqrt{\cos (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+C \sin (2 (c+d x))\right)}{3 d \sqrt{b \cos (c+d x)}}","\frac{2 b (3 A+C) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d \sqrt{b \cos (c+d x)}}+\frac{2 B E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \cos (c+d x)}}{d \sqrt{\cos (c+d x)}}+\frac{2 C \sin (c+d x) \sqrt{b \cos (c+d x)}}{3 d}",1,"(b*(6*B*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2] + 2*(3*A + C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + C*Sin[2*(c + d*x)]))/(3*d*Sqrt[b*Cos[c + d*x]])","A",1
243,1,78,109,0.2911518,"\int \sqrt{b \cos (c+d x)} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^2(c+d x) \, dx","Integrate[Sqrt[b*Cos[c + d*x]]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^2,x]","\frac{2 b \left(-\left((A-C) \sqrt{\cos (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)+A \sin (c+d x)+B \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{d \sqrt{b \cos (c+d x)}}","-\frac{2 (A-C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \cos (c+d x)}}{d \sqrt{\cos (c+d x)}}+\frac{2 A b \sin (c+d x)}{d \sqrt{b \cos (c+d x)}}+\frac{2 b B \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d \sqrt{b \cos (c+d x)}}",1,"(2*b*(-((A - C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]) + B*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + A*Sin[c + d*x]))/(d*Sqrt[b*Cos[c + d*x]])","A",1
244,1,90,140,0.3975011,"\int \sqrt{b \cos (c+d x)} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^3(c+d x) \, dx","Integrate[Sqrt[b*Cos[c + d*x]]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^3,x]","\frac{2 b \left(\tan (c+d x) (A+3 B \cos (c+d x))+(A+3 C) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-3 B \sqrt{\cos (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{3 d \sqrt{b \cos (c+d x)}}","\frac{2 A b^2 \sin (c+d x)}{3 d (b \cos (c+d x))^{3/2}}+\frac{2 b (A+3 C) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d \sqrt{b \cos (c+d x)}}+\frac{2 b B \sin (c+d x)}{d \sqrt{b \cos (c+d x)}}-\frac{2 B E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \cos (c+d x)}}{d \sqrt{\cos (c+d x)}}",1,"(2*b*(-3*B*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2] + (A + 3*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + (A + 3*B*Cos[c + d*x])*Tan[c + d*x]))/(3*d*Sqrt[b*Cos[c + d*x]])","A",1
245,1,122,181,0.5997595,"\int \sqrt{b \cos (c+d x)} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^4(c+d x) \, dx","Integrate[Sqrt[b*Cos[c + d*x]]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^4,x]","-\frac{\sec ^2(c+d x) \sqrt{b \cos (c+d x)} \left(6 (3 A+5 C) \cos ^{\frac{3}{2}}(c+d x) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)-9 A \sin (2 (c+d x))-6 A \tan (c+d x)-10 B \sin (c+d x)-10 B \cos ^{\frac{3}{2}}(c+d x) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-15 C \sin (2 (c+d x))\right)}{15 d}","\frac{2 A b^3 \sin (c+d x)}{5 d (b \cos (c+d x))^{5/2}}+\frac{2 b (3 A+5 C) \sin (c+d x)}{5 d \sqrt{b \cos (c+d x)}}-\frac{2 (3 A+5 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \cos (c+d x)}}{5 d \sqrt{\cos (c+d x)}}+\frac{2 b^2 B \sin (c+d x)}{3 d (b \cos (c+d x))^{3/2}}+\frac{2 b B \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d \sqrt{b \cos (c+d x)}}",1,"-1/15*(Sqrt[b*Cos[c + d*x]]*Sec[c + d*x]^2*(6*(3*A + 5*C)*Cos[c + d*x]^(3/2)*EllipticE[(c + d*x)/2, 2] - 10*B*Cos[c + d*x]^(3/2)*EllipticF[(c + d*x)/2, 2] - 10*B*Sin[c + d*x] - 9*A*Sin[2*(c + d*x)] - 15*C*Sin[2*(c + d*x)] - 6*A*Tan[c + d*x]))/d","A",1
246,1,143,210,0.9896368,"\int \sqrt{b \cos (c+d x)} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^5(c+d x) \, dx","Integrate[Sqrt[b*Cos[c + d*x]]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^5,x]","\frac{2 \sec ^3(c+d x) \sqrt{b \cos (c+d x)} \left(5 (5 A+7 C) \cos ^{\frac{5}{2}}(c+d x) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+\frac{25}{2} A \sin (2 (c+d x))+15 A \tan (c+d x)+21 B \sin (c+d x)-63 B \cos ^{\frac{5}{2}}(c+d x) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+63 B \sin (c+d x) \cos ^2(c+d x)+\frac{35}{2} C \sin (2 (c+d x))\right)}{105 d}","\frac{2 A b^4 \sin (c+d x)}{7 d (b \cos (c+d x))^{7/2}}+\frac{2 b^2 (5 A+7 C) \sin (c+d x)}{21 d (b \cos (c+d x))^{3/2}}+\frac{2 b (5 A+7 C) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d \sqrt{b \cos (c+d x)}}+\frac{2 b^3 B \sin (c+d x)}{5 d (b \cos (c+d x))^{5/2}}+\frac{6 b B \sin (c+d x)}{5 d \sqrt{b \cos (c+d x)}}-\frac{6 B E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \cos (c+d x)}}{5 d \sqrt{\cos (c+d x)}}",1,"(2*Sqrt[b*Cos[c + d*x]]*Sec[c + d*x]^3*(-63*B*Cos[c + d*x]^(5/2)*EllipticE[(c + d*x)/2, 2] + 5*(5*A + 7*C)*Cos[c + d*x]^(5/2)*EllipticF[(c + d*x)/2, 2] + 21*B*Sin[c + d*x] + 63*B*Cos[c + d*x]^2*Sin[c + d*x] + (25*A*Sin[2*(c + d*x)])/2 + (35*C*Sin[2*(c + d*x)])/2 + 15*A*Tan[c + d*x]))/(105*d)","A",1
247,1,128,210,0.941068,"\int \cos (c+d x) (b \cos (c+d x))^{3/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]*(b*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","\frac{(b \cos (c+d x))^{5/2} \left(\sin (c+d x) \sqrt{\cos (c+d x)} (7 (36 A+43 C) \cos (c+d x)+5 (18 B \cos (2 (c+d x))+78 B+7 C \cos (3 (c+d x))))+84 (9 A+7 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+300 B F\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{630 b d \cos ^{\frac{5}{2}}(c+d x)}","\frac{2 (9 A+7 C) \sin (c+d x) (b \cos (c+d x))^{3/2}}{45 d}+\frac{2 b (9 A+7 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \cos (c+d x)}}{15 d \sqrt{\cos (c+d x)}}+\frac{10 b^2 B \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d \sqrt{b \cos (c+d x)}}+\frac{2 C \sin (c+d x) (b \cos (c+d x))^{7/2}}{9 b^2 d}+\frac{2 B \sin (c+d x) (b \cos (c+d x))^{5/2}}{7 b d}+\frac{10 b B \sin (c+d x) \sqrt{b \cos (c+d x)}}{21 d}",1,"((b*Cos[c + d*x])^(5/2)*(84*(9*A + 7*C)*EllipticE[(c + d*x)/2, 2] + 300*B*EllipticF[(c + d*x)/2, 2] + Sqrt[Cos[c + d*x]]*(7*(36*A + 43*C)*Cos[c + d*x] + 5*(78*B + 18*B*Cos[2*(c + d*x)] + 7*C*Cos[3*(c + d*x)]))*Sin[c + d*x]))/(630*b*d*Cos[c + d*x]^(5/2))","A",1
248,1,108,181,0.0961628,"\int (b \cos (c+d x))^{3/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Integrate[(b*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","\frac{(b \cos (c+d x))^{3/2} \left(\sin (c+d x) \sqrt{\cos (c+d x)} (70 A+42 B \cos (c+d x)+15 C \cos (2 (c+d x))+65 C)+10 (7 A+5 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+126 B E\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{105 d \cos ^{\frac{3}{2}}(c+d x)}","\frac{2 b^2 (7 A+5 C) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d \sqrt{b \cos (c+d x)}}+\frac{2 b (7 A+5 C) \sin (c+d x) \sqrt{b \cos (c+d x)}}{21 d}+\frac{2 B \sin (c+d x) (b \cos (c+d x))^{3/2}}{5 d}+\frac{6 b B E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \cos (c+d x)}}{5 d \sqrt{\cos (c+d x)}}+\frac{2 C \sin (c+d x) (b \cos (c+d x))^{5/2}}{7 b d}",1,"((b*Cos[c + d*x])^(3/2)*(126*B*EllipticE[(c + d*x)/2, 2] + 10*(7*A + 5*C)*EllipticF[(c + d*x)/2, 2] + Sqrt[Cos[c + d*x]]*(70*A + 65*C + 42*B*Cos[c + d*x] + 15*C*Cos[2*(c + d*x)])*Sin[c + d*x]))/(105*d*Cos[c + d*x]^(3/2))","A",1
249,1,95,146,0.2498929,"\int (b \cos (c+d x))^{3/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec (c+d x) \, dx","Integrate[(b*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x],x]","\frac{2 b \sqrt{b \cos (c+d x)} \left(3 (5 A+3 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+\sin (c+d x) \sqrt{\cos (c+d x)} (5 B+3 C \cos (c+d x))+5 B F\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{15 d \sqrt{\cos (c+d x)}}","\frac{2 b (5 A+3 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \cos (c+d x)}}{5 d \sqrt{\cos (c+d x)}}+\frac{2 b^2 B \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d \sqrt{b \cos (c+d x)}}+\frac{2 b B \sin (c+d x) \sqrt{b \cos (c+d x)}}{3 d}+\frac{2 C \sin (c+d x) (b \cos (c+d x))^{3/2}}{5 d}",1,"(2*b*Sqrt[b*Cos[c + d*x]]*(3*(5*A + 3*C)*EllipticE[(c + d*x)/2, 2] + 5*B*EllipticF[(c + d*x)/2, 2] + Sqrt[Cos[c + d*x]]*(5*B + 3*C*Cos[c + d*x])*Sin[c + d*x]))/(15*d*Sqrt[Cos[c + d*x]])","A",1
250,1,85,116,0.167411,"\int (b \cos (c+d x))^{3/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^2(c+d x) \, dx","Integrate[(b*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^2,x]","\frac{b^2 \left(2 (3 A+C) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+6 B \sqrt{\cos (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+C \sin (2 (c+d x))\right)}{3 d \sqrt{b \cos (c+d x)}}","\frac{2 b^2 (3 A+C) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d \sqrt{b \cos (c+d x)}}+\frac{2 b B E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \cos (c+d x)}}{d \sqrt{\cos (c+d x)}}+\frac{2 b C \sin (c+d x) \sqrt{b \cos (c+d x)}}{3 d}",1,"(b^2*(6*B*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2] + 2*(3*A + C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + C*Sin[2*(c + d*x)]))/(3*d*Sqrt[b*Cos[c + d*x]])","A",1
251,1,80,114,0.2653278,"\int (b \cos (c+d x))^{3/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^3(c+d x) \, dx","Integrate[(b*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^3,x]","\frac{2 b^2 \left(-\left((A-C) \sqrt{\cos (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)+A \sin (c+d x)+B \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{d \sqrt{b \cos (c+d x)}}","\frac{2 A b^2 \sin (c+d x)}{d \sqrt{b \cos (c+d x)}}-\frac{2 b (A-C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \cos (c+d x)}}{d \sqrt{\cos (c+d x)}}+\frac{2 b^2 B \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d \sqrt{b \cos (c+d x)}}",1,"(2*b^2*(-((A - C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]) + B*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + A*Sin[c + d*x]))/(d*Sqrt[b*Cos[c + d*x]])","A",1
252,1,92,145,0.3224529,"\int (b \cos (c+d x))^{3/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^4(c+d x) \, dx","Integrate[(b*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^4,x]","\frac{2 b^2 \left(\tan (c+d x) (A+3 B \cos (c+d x))+(A+3 C) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-3 B \sqrt{\cos (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{3 d \sqrt{b \cos (c+d x)}}","\frac{2 A b^3 \sin (c+d x)}{3 d (b \cos (c+d x))^{3/2}}+\frac{2 b^2 (A+3 C) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d \sqrt{b \cos (c+d x)}}+\frac{2 b^2 B \sin (c+d x)}{d \sqrt{b \cos (c+d x)}}-\frac{2 b B E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \cos (c+d x)}}{d \sqrt{\cos (c+d x)}}",1,"(2*b^2*(-3*B*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2] + (A + 3*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + (A + 3*B*Cos[c + d*x])*Tan[c + d*x]))/(3*d*Sqrt[b*Cos[c + d*x]])","A",1
253,1,122,186,0.512317,"\int (b \cos (c+d x))^{3/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^5(c+d x) \, dx","Integrate[(b*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^5,x]","-\frac{\sec ^3(c+d x) (b \cos (c+d x))^{3/2} \left(6 (3 A+5 C) \cos ^{\frac{3}{2}}(c+d x) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)-9 A \sin (2 (c+d x))-6 A \tan (c+d x)-10 B \sin (c+d x)-10 B \cos ^{\frac{3}{2}}(c+d x) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-15 C \sin (2 (c+d x))\right)}{15 d}","\frac{2 A b^4 \sin (c+d x)}{5 d (b \cos (c+d x))^{5/2}}+\frac{2 b^2 (3 A+5 C) \sin (c+d x)}{5 d \sqrt{b \cos (c+d x)}}-\frac{2 b (3 A+5 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \cos (c+d x)}}{5 d \sqrt{\cos (c+d x)}}+\frac{2 b^3 B \sin (c+d x)}{3 d (b \cos (c+d x))^{3/2}}+\frac{2 b^2 B \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d \sqrt{b \cos (c+d x)}}",1,"-1/15*((b*Cos[c + d*x])^(3/2)*Sec[c + d*x]^3*(6*(3*A + 5*C)*Cos[c + d*x]^(3/2)*EllipticE[(c + d*x)/2, 2] - 10*B*Cos[c + d*x]^(3/2)*EllipticF[(c + d*x)/2, 2] - 10*B*Sin[c + d*x] - 9*A*Sin[2*(c + d*x)] - 15*C*Sin[2*(c + d*x)] - 6*A*Tan[c + d*x]))/d","A",1
254,1,134,215,1.7565008,"\int (b \cos (c+d x))^{3/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^6(c+d x) \, dx","Integrate[(b*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^6,x]","\frac{\sec ^5(c+d x) (b \cos (c+d x))^{3/2} \left(2 \sin (c+d x) (10 (5 A+7 C) \cos (2 (c+d x))+110 A+273 B \cos (c+d x)+63 B \cos (3 (c+d x))+70 C)+40 (5 A+7 C) \cos ^{\frac{7}{2}}(c+d x) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-504 B \cos ^{\frac{7}{2}}(c+d x) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{420 d}","\frac{2 A b^5 \sin (c+d x)}{7 d (b \cos (c+d x))^{7/2}}+\frac{2 b^3 (5 A+7 C) \sin (c+d x)}{21 d (b \cos (c+d x))^{3/2}}+\frac{2 b^2 (5 A+7 C) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d \sqrt{b \cos (c+d x)}}+\frac{2 b^4 B \sin (c+d x)}{5 d (b \cos (c+d x))^{5/2}}+\frac{6 b^2 B \sin (c+d x)}{5 d \sqrt{b \cos (c+d x)}}-\frac{6 b B E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \cos (c+d x)}}{5 d \sqrt{\cos (c+d x)}}",1,"((b*Cos[c + d*x])^(3/2)*Sec[c + d*x]^5*(-504*B*Cos[c + d*x]^(7/2)*EllipticE[(c + d*x)/2, 2] + 40*(5*A + 7*C)*Cos[c + d*x]^(7/2)*EllipticF[(c + d*x)/2, 2] + 2*(110*A + 70*C + 273*B*Cos[c + d*x] + 10*(5*A + 7*C)*Cos[2*(c + d*x)] + 63*B*Cos[3*(c + d*x)])*Sin[c + d*x]))/(420*d)","A",1
255,1,125,212,0.2562914,"\int (b \cos (c+d x))^{5/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Integrate[(b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","\frac{(b \cos (c+d x))^{5/2} \left(\sin (c+d x) \sqrt{\cos (c+d x)} (7 (36 A+43 C) \cos (c+d x)+5 (18 B \cos (2 (c+d x))+78 B+7 C \cos (3 (c+d x))))+84 (9 A+7 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+300 B F\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{630 d \cos ^{\frac{5}{2}}(c+d x)}","\frac{2 b^2 (9 A+7 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \cos (c+d x)}}{15 d \sqrt{\cos (c+d x)}}+\frac{2 b (9 A+7 C) \sin (c+d x) (b \cos (c+d x))^{3/2}}{45 d}+\frac{10 b^3 B \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d \sqrt{b \cos (c+d x)}}+\frac{10 b^2 B \sin (c+d x) \sqrt{b \cos (c+d x)}}{21 d}+\frac{2 B \sin (c+d x) (b \cos (c+d x))^{5/2}}{7 d}+\frac{2 C \sin (c+d x) (b \cos (c+d x))^{7/2}}{9 b d}",1,"((b*Cos[c + d*x])^(5/2)*(84*(9*A + 7*C)*EllipticE[(c + d*x)/2, 2] + 300*B*EllipticF[(c + d*x)/2, 2] + Sqrt[Cos[c + d*x]]*(7*(36*A + 43*C)*Cos[c + d*x] + 5*(78*B + 18*B*Cos[2*(c + d*x)] + 7*C*Cos[3*(c + d*x)]))*Sin[c + d*x]))/(630*d*Cos[c + d*x]^(5/2))","A",1
256,1,109,183,0.1061526,"\int (b \cos (c+d x))^{5/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec (c+d x) \, dx","Integrate[(b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x],x]","\frac{b (b \cos (c+d x))^{3/2} \left(\sin (c+d x) \sqrt{\cos (c+d x)} (70 A+42 B \cos (c+d x)+15 C \cos (2 (c+d x))+65 C)+10 (7 A+5 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+126 B E\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{105 d \cos ^{\frac{3}{2}}(c+d x)}","\frac{2 b^3 (7 A+5 C) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d \sqrt{b \cos (c+d x)}}+\frac{2 b^2 (7 A+5 C) \sin (c+d x) \sqrt{b \cos (c+d x)}}{21 d}+\frac{6 b^2 B E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \cos (c+d x)}}{5 d \sqrt{\cos (c+d x)}}+\frac{2 b B \sin (c+d x) (b \cos (c+d x))^{3/2}}{5 d}+\frac{2 C \sin (c+d x) (b \cos (c+d x))^{5/2}}{7 d}",1,"(b*(b*Cos[c + d*x])^(3/2)*(126*B*EllipticE[(c + d*x)/2, 2] + 10*(7*A + 5*C)*EllipticF[(c + d*x)/2, 2] + Sqrt[Cos[c + d*x]]*(70*A + 65*C + 42*B*Cos[c + d*x] + 15*C*Cos[2*(c + d*x)])*Sin[c + d*x]))/(105*d*Cos[c + d*x]^(3/2))","A",1
257,1,97,151,0.2273613,"\int (b \cos (c+d x))^{5/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^2(c+d x) \, dx","Integrate[(b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^2,x]","\frac{2 b^2 \sqrt{b \cos (c+d x)} \left(3 (5 A+3 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+\sin (c+d x) \sqrt{\cos (c+d x)} (5 B+3 C \cos (c+d x))+5 B F\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{15 d \sqrt{\cos (c+d x)}}","\frac{2 b^2 (5 A+3 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \cos (c+d x)}}{5 d \sqrt{\cos (c+d x)}}+\frac{2 b^3 B \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d \sqrt{b \cos (c+d x)}}+\frac{2 b^2 B \sin (c+d x) \sqrt{b \cos (c+d x)}}{3 d}+\frac{2 b C \sin (c+d x) (b \cos (c+d x))^{3/2}}{5 d}",1,"(2*b^2*Sqrt[b*Cos[c + d*x]]*(3*(5*A + 3*C)*EllipticE[(c + d*x)/2, 2] + 5*B*EllipticF[(c + d*x)/2, 2] + Sqrt[Cos[c + d*x]]*(5*B + 3*C*Cos[c + d*x])*Sin[c + d*x]))/(15*d*Sqrt[Cos[c + d*x]])","A",1
258,1,79,120,0.2145452,"\int (b \cos (c+d x))^{5/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^3(c+d x) \, dx","Integrate[(b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^3,x]","\frac{2 (b \cos (c+d x))^{5/2} \left((3 A+C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+3 B E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+C \sin (c+d x) \sqrt{\cos (c+d x)}\right)}{3 d \cos ^{\frac{5}{2}}(c+d x)}","\frac{2 b^3 (3 A+C) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d \sqrt{b \cos (c+d x)}}+\frac{2 b^2 B E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \cos (c+d x)}}{d \sqrt{\cos (c+d x)}}+\frac{2 b^2 C \sin (c+d x) \sqrt{b \cos (c+d x)}}{3 d}",1,"(2*(b*Cos[c + d*x])^(5/2)*(3*B*EllipticE[(c + d*x)/2, 2] + (3*A + C)*EllipticF[(c + d*x)/2, 2] + C*Sqrt[Cos[c + d*x]]*Sin[c + d*x]))/(3*d*Cos[c + d*x]^(5/2))","A",1
259,1,80,116,0.2919678,"\int (b \cos (c+d x))^{5/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^4(c+d x) \, dx","Integrate[(b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^4,x]","\frac{2 b^3 \left(-\left((A-C) \sqrt{\cos (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)+A \sin (c+d x)+B \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{d \sqrt{b \cos (c+d x)}}","\frac{2 A b^3 \sin (c+d x)}{d \sqrt{b \cos (c+d x)}}-\frac{2 b^2 (A-C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \cos (c+d x)}}{d \sqrt{\cos (c+d x)}}+\frac{2 b^3 B \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d \sqrt{b \cos (c+d x)}}",1,"(2*b^3*(-((A - C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]) + B*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + A*Sin[c + d*x]))/(d*Sqrt[b*Cos[c + d*x]])","A",1
260,1,92,147,0.3104873,"\int (b \cos (c+d x))^{5/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^5(c+d x) \, dx","Integrate[(b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^5,x]","\frac{2 b^3 \left(\tan (c+d x) (A+3 B \cos (c+d x))+(A+3 C) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-3 B \sqrt{\cos (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{3 d \sqrt{b \cos (c+d x)}}","\frac{2 A b^4 \sin (c+d x)}{3 d (b \cos (c+d x))^{3/2}}+\frac{2 b^3 (A+3 C) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d \sqrt{b \cos (c+d x)}}+\frac{2 b^3 B \sin (c+d x)}{d \sqrt{b \cos (c+d x)}}-\frac{2 b^2 B E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \cos (c+d x)}}{d \sqrt{\cos (c+d x)}}",1,"(2*b^3*(-3*B*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2] + (A + 3*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + (A + 3*B*Cos[c + d*x])*Tan[c + d*x]))/(3*d*Sqrt[b*Cos[c + d*x]])","A",1
261,1,121,188,0.3852661,"\int (b \cos (c+d x))^{5/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^6(c+d x) \, dx","Integrate[(b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^6,x]","-\frac{2 b^4 \left(3 (3 A+5 C) \cos ^{\frac{3}{2}}(c+d x) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)-\frac{9}{2} A \sin (2 (c+d x))-3 A \tan (c+d x)-5 B \sin (c+d x)-5 B \cos ^{\frac{3}{2}}(c+d x) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-\frac{15}{2} C \sin (2 (c+d x))\right)}{15 d (b \cos (c+d x))^{3/2}}","\frac{2 A b^5 \sin (c+d x)}{5 d (b \cos (c+d x))^{5/2}}+\frac{2 b^3 (3 A+5 C) \sin (c+d x)}{5 d \sqrt{b \cos (c+d x)}}-\frac{2 b^2 (3 A+5 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \cos (c+d x)}}{5 d \sqrt{\cos (c+d x)}}+\frac{2 b^4 B \sin (c+d x)}{3 d (b \cos (c+d x))^{3/2}}+\frac{2 b^3 B \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d \sqrt{b \cos (c+d x)}}",1,"(-2*b^4*(3*(3*A + 5*C)*Cos[c + d*x]^(3/2)*EllipticE[(c + d*x)/2, 2] - 5*B*Cos[c + d*x]^(3/2)*EllipticF[(c + d*x)/2, 2] - 5*B*Sin[c + d*x] - (9*A*Sin[2*(c + d*x)])/2 - (15*C*Sin[2*(c + d*x)])/2 - 3*A*Tan[c + d*x]))/(15*d*(b*Cos[c + d*x])^(3/2))","A",1
262,1,134,217,0.7874031,"\int (b \cos (c+d x))^{5/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^7(c+d x) \, dx","Integrate[(b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^7,x]","\frac{\sec ^6(c+d x) (b \cos (c+d x))^{5/2} \left(2 \sin (c+d x) (10 (5 A+7 C) \cos (2 (c+d x))+110 A+273 B \cos (c+d x)+63 B \cos (3 (c+d x))+70 C)+40 (5 A+7 C) \cos ^{\frac{7}{2}}(c+d x) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-504 B \cos ^{\frac{7}{2}}(c+d x) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{420 d}","\frac{2 A b^6 \sin (c+d x)}{7 d (b \cos (c+d x))^{7/2}}+\frac{2 b^4 (5 A+7 C) \sin (c+d x)}{21 d (b \cos (c+d x))^{3/2}}+\frac{2 b^3 (5 A+7 C) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d \sqrt{b \cos (c+d x)}}+\frac{2 b^5 B \sin (c+d x)}{5 d (b \cos (c+d x))^{5/2}}+\frac{6 b^3 B \sin (c+d x)}{5 d \sqrt{b \cos (c+d x)}}-\frac{6 b^2 B E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \cos (c+d x)}}{5 d \sqrt{\cos (c+d x)}}",1,"((b*Cos[c + d*x])^(5/2)*Sec[c + d*x]^6*(-504*B*Cos[c + d*x]^(7/2)*EllipticE[(c + d*x)/2, 2] + 40*(5*A + 7*C)*Cos[c + d*x]^(7/2)*EllipticF[(c + d*x)/2, 2] + 2*(110*A + 70*C + 273*B*Cos[c + d*x] + 10*(5*A + 7*C)*Cos[2*(c + d*x)] + 63*B*Cos[3*(c + d*x)])*Sin[c + d*x]))/(420*d)","A",1
263,1,127,214,0.7197299,"\int \frac{\cos ^3(c+d x) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\sqrt{b \cos (c+d x)}} \, dx","Integrate[(Cos[c + d*x]^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Sqrt[b*Cos[c + d*x]],x]","\frac{\sin (2 (c+d x)) (7 (36 A+43 C) \cos (c+d x)+5 (18 B \cos (2 (c+d x))+78 B+7 C \cos (3 (c+d x))))+168 (9 A+7 C) \sqrt{\cos (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+600 B \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{1260 d \sqrt{b \cos (c+d x)}}","\frac{2 (9 A+7 C) \sin (c+d x) (b \cos (c+d x))^{3/2}}{45 b^2 d}+\frac{2 (9 A+7 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \cos (c+d x)}}{15 b d \sqrt{\cos (c+d x)}}+\frac{2 C \sin (c+d x) (b \cos (c+d x))^{7/2}}{9 b^4 d}+\frac{2 B \sin (c+d x) (b \cos (c+d x))^{5/2}}{7 b^3 d}+\frac{10 B \sin (c+d x) \sqrt{b \cos (c+d x)}}{21 b d}+\frac{10 B \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d \sqrt{b \cos (c+d x)}}",1,"(168*(9*A + 7*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2] + 600*B*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + (7*(36*A + 43*C)*Cos[c + d*x] + 5*(78*B + 18*B*Cos[2*(c + d*x)] + 7*C*Cos[3*(c + d*x)]))*Sin[2*(c + d*x)])/(1260*d*Sqrt[b*Cos[c + d*x]])","A",1
264,1,108,185,0.67648,"\int \frac{\cos ^2(c+d x) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\sqrt{b \cos (c+d x)}} \, dx","Integrate[(Cos[c + d*x]^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Sqrt[b*Cos[c + d*x]],x]","\frac{\sqrt{\cos (c+d x)} \left(\sin (c+d x) \sqrt{\cos (c+d x)} (70 A+42 B \cos (c+d x)+15 C \cos (2 (c+d x))+65 C)+10 (7 A+5 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+126 B E\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{105 d \sqrt{b \cos (c+d x)}}","\frac{2 (7 A+5 C) \sin (c+d x) \sqrt{b \cos (c+d x)}}{21 b d}+\frac{2 (7 A+5 C) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d \sqrt{b \cos (c+d x)}}+\frac{2 C \sin (c+d x) (b \cos (c+d x))^{5/2}}{7 b^3 d}+\frac{2 B \sin (c+d x) (b \cos (c+d x))^{3/2}}{5 b^2 d}+\frac{6 B E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \cos (c+d x)}}{5 b d \sqrt{\cos (c+d x)}}",1,"(Sqrt[Cos[c + d*x]]*(126*B*EllipticE[(c + d*x)/2, 2] + 10*(7*A + 5*C)*EllipticF[(c + d*x)/2, 2] + Sqrt[Cos[c + d*x]]*(70*A + 65*C + 42*B*Cos[c + d*x] + 15*C*Cos[2*(c + d*x)])*Sin[c + d*x]))/(105*d*Sqrt[b*Cos[c + d*x]])","A",1
265,1,97,150,0.1994052,"\int \frac{\cos (c+d x) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\sqrt{b \cos (c+d x)}} \, dx","Integrate[(Cos[c + d*x]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Sqrt[b*Cos[c + d*x]],x]","\frac{2 \sqrt{b \cos (c+d x)} \left(3 (5 A+3 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+\sin (c+d x) \sqrt{\cos (c+d x)} (5 B+3 C \cos (c+d x))+5 B F\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{15 b d \sqrt{\cos (c+d x)}}","\frac{2 (5 A+3 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \cos (c+d x)}}{5 b d \sqrt{\cos (c+d x)}}+\frac{2 C \sin (c+d x) (b \cos (c+d x))^{3/2}}{5 b^2 d}+\frac{2 B \sin (c+d x) \sqrt{b \cos (c+d x)}}{3 b d}+\frac{2 B \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d \sqrt{b \cos (c+d x)}}",1,"(2*Sqrt[b*Cos[c + d*x]]*(3*(5*A + 3*C)*EllipticE[(c + d*x)/2, 2] + 5*B*EllipticF[(c + d*x)/2, 2] + Sqrt[Cos[c + d*x]]*(5*B + 3*C*Cos[c + d*x])*Sin[c + d*x]))/(15*b*d*Sqrt[Cos[c + d*x]])","A",1
266,1,82,117,0.0911974,"\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{\sqrt{b \cos (c+d x)}} \, dx","Integrate[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/Sqrt[b*Cos[c + d*x]],x]","\frac{2 (3 A+C) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+6 B \sqrt{\cos (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+C \sin (2 (c+d x))}{3 d \sqrt{b \cos (c+d x)}}","\frac{2 (3 A+C) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d \sqrt{b \cos (c+d x)}}+\frac{2 B E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \cos (c+d x)}}{b d \sqrt{\cos (c+d x)}}+\frac{2 C \sin (c+d x) \sqrt{b \cos (c+d x)}}{3 b d}",1,"(6*B*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2] + 2*(3*A + C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + C*Sin[2*(c + d*x)])/(3*d*Sqrt[b*Cos[c + d*x]])","A",1
267,1,803,110,6.3203086,"\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec (c+d x)}{\sqrt{b \cos (c+d x)}} \, dx","Integrate[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x])/Sqrt[b*Cos[c + d*x]],x]","\frac{(B+C \cos (c+d x)+A \sec (c+d x)) \left(\frac{4 A \sec (c) \sec (c+d x) \sin (d x)}{d}-\frac{2 (-2 A+C+C \cos (2 c)) \csc (c) \sec (c)}{d}\right) \cos ^2(c+d x)}{\sqrt{b \cos (c+d x)} (2 A+C+2 B \cos (c+d x)+C \cos (2 c+2 d x))}+\frac{2 A \csc (c) (B+C \cos (c+d x)+A \sec (c+d x)) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \cos ^{\frac{3}{2}}(c+d x)}{d \sqrt{b \cos (c+d x)} (2 A+C+2 B \cos (c+d x)+C \cos (2 c+2 d x))}-\frac{2 C \csc (c) (B+C \cos (c+d x)+A \sec (c+d x)) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \cos ^{\frac{3}{2}}(c+d x)}{d \sqrt{b \cos (c+d x)} (2 A+C+2 B \cos (c+d x)+C \cos (2 c+2 d x))}-\frac{4 B \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) (B+C \cos (c+d x)+A \sec (c+d x)) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^{\frac{3}{2}}(c+d x)}{d \sqrt{b \cos (c+d x)} (2 A+C+2 B \cos (c+d x)+C \cos (2 c+2 d x)) \sqrt{\cot ^2(c)+1}}","-\frac{2 (A-C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \cos (c+d x)}}{b d \sqrt{\cos (c+d x)}}+\frac{2 A \sin (c+d x)}{d \sqrt{b \cos (c+d x)}}+\frac{2 B \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d \sqrt{b \cos (c+d x)}}",1,"(Cos[c + d*x]^2*(B + C*Cos[c + d*x] + A*Sec[c + d*x])*((-2*(-2*A + C + C*Cos[2*c])*Csc[c]*Sec[c])/d + (4*A*Sec[c]*Sec[c + d*x]*Sin[d*x])/d))/(Sqrt[b*Cos[c + d*x]]*(2*A + C + 2*B*Cos[c + d*x] + C*Cos[2*c + 2*d*x])) - (4*B*Cos[c + d*x]^(3/2)*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*(B + C*Cos[c + d*x] + A*Sec[c + d*x])*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(d*Sqrt[b*Cos[c + d*x]]*(2*A + C + 2*B*Cos[c + d*x] + C*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]) + (2*A*Cos[c + d*x]^(3/2)*Csc[c]*(B + C*Cos[c + d*x] + A*Sec[c + d*x])*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(d*Sqrt[b*Cos[c + d*x]]*(2*A + C + 2*B*Cos[c + d*x] + C*Cos[2*c + 2*d*x])) - (2*C*Cos[c + d*x]^(3/2)*Csc[c]*(B + C*Cos[c + d*x] + A*Sec[c + d*x])*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(d*Sqrt[b*Cos[c + d*x]]*(2*A + C + 2*B*Cos[c + d*x] + C*Cos[2*c + 2*d*x]))","C",0
268,1,757,139,6.3319638,"\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^2(c+d x)}{\sqrt{b \cos (c+d x)}} \, dx","Integrate[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^2)/Sqrt[b*Cos[c + d*x]],x]","\frac{2 B \csc (c) \cos ^{\frac{5}{2}}(c+d x) \left(A \sec ^2(c+d x)+B \sec (c+d x)+C\right) \left(\frac{\tan (c) \sin \left(\tan ^{-1}(\tan (c))+d x\right) \, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right)}{\sqrt{\tan ^2(c)+1} \sqrt{1-\cos \left(\tan ^{-1}(\tan (c))+d x\right)} \sqrt{\cos \left(\tan ^{-1}(\tan (c))+d x\right)+1} \sqrt{\cos (c) \sqrt{\tan ^2(c)+1} \cos \left(\tan ^{-1}(\tan (c))+d x\right)}}-\frac{\frac{\tan (c) \sin \left(\tan ^{-1}(\tan (c))+d x\right)}{\sqrt{\tan ^2(c)+1}}+\frac{2 \cos ^2(c) \sqrt{\tan ^2(c)+1} \cos \left(\tan ^{-1}(\tan (c))+d x\right)}{\sin ^2(c)+\cos ^2(c)}}{\sqrt{\cos (c) \sqrt{\tan ^2(c)+1} \cos \left(\tan ^{-1}(\tan (c))+d x\right)}}\right)}{d \sqrt{b \cos (c+d x)} (2 A+2 B \cos (c+d x)+C \cos (2 c+2 d x)+C)}-\frac{4 A \csc (c) \cos ^{\frac{5}{2}}(c+d x) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin (c) \left(-\sqrt{\cot ^2(c)+1}\right) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec \left(d x-\tan ^{-1}(\cot (c))\right) \left(A \sec ^2(c+d x)+B \sec (c+d x)+C\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right)}{3 d \sqrt{\cot ^2(c)+1} \sqrt{b \cos (c+d x)} (2 A+2 B \cos (c+d x)+C \cos (2 c+2 d x)+C)}-\frac{4 C \csc (c) \cos ^{\frac{5}{2}}(c+d x) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin (c) \left(-\sqrt{\cot ^2(c)+1}\right) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec \left(d x-\tan ^{-1}(\cot (c))\right) \left(A \sec ^2(c+d x)+B \sec (c+d x)+C\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right)}{d \sqrt{\cot ^2(c)+1} \sqrt{b \cos (c+d x)} (2 A+2 B \cos (c+d x)+C \cos (2 c+2 d x)+C)}+\frac{\cos ^3(c+d x) \left(A \sec ^2(c+d x)+B \sec (c+d x)+C\right) \left(\frac{4 \sec (c) \sec (c+d x) (A \sin (c)+3 B \sin (d x))}{3 d}+\frac{4 A \sec (c) \sin (d x) \sec ^2(c+d x)}{3 d}+\frac{4 B \csc (c) \sec (c)}{d}\right)}{\sqrt{b \cos (c+d x)} (2 A+2 B \cos (c+d x)+C \cos (2 c+2 d x)+C)}","\frac{2 (A+3 C) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d \sqrt{b \cos (c+d x)}}+\frac{2 A b \sin (c+d x)}{3 d (b \cos (c+d x))^{3/2}}+\frac{2 B \sin (c+d x)}{d \sqrt{b \cos (c+d x)}}-\frac{2 B E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \cos (c+d x)}}{b d \sqrt{\cos (c+d x)}}",1,"(Cos[c + d*x]^3*(C + B*Sec[c + d*x] + A*Sec[c + d*x]^2)*((4*B*Csc[c]*Sec[c])/d + (4*A*Sec[c]*Sec[c + d*x]^2*Sin[d*x])/(3*d) + (4*Sec[c]*Sec[c + d*x]*(A*Sin[c] + 3*B*Sin[d*x]))/(3*d)))/(Sqrt[b*Cos[c + d*x]]*(2*A + C + 2*B*Cos[c + d*x] + C*Cos[2*c + 2*d*x])) - (4*A*Cos[c + d*x]^(5/2)*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*(C + B*Sec[c + d*x] + A*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*Sqrt[b*Cos[c + d*x]]*(2*A + C + 2*B*Cos[c + d*x] + C*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]) - (4*C*Cos[c + d*x]^(5/2)*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*(C + B*Sec[c + d*x] + A*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(d*Sqrt[b*Cos[c + d*x]]*(2*A + C + 2*B*Cos[c + d*x] + C*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]) + (2*B*Cos[c + d*x]^(5/2)*Csc[c]*(C + B*Sec[c + d*x] + A*Sec[c + d*x]^2)*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(d*Sqrt[b*Cos[c + d*x]]*(2*A + C + 2*B*Cos[c + d*x] + C*Cos[2*c + 2*d*x]))","C",0
269,1,116,180,0.4481008,"\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^3(c+d x)}{\sqrt{b \cos (c+d x)}} \, dx","Integrate[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^3)/Sqrt[b*Cos[c + d*x]],x]","\frac{2 \left(-3 (3 A+5 C) \sqrt{\cos (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+9 A \sin (c+d x)+3 A \tan (c+d x) \sec (c+d x)+5 B \tan (c+d x)+5 B \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+15 C \sin (c+d x)\right)}{15 d \sqrt{b \cos (c+d x)}}","\frac{2 A b^2 \sin (c+d x)}{5 d (b \cos (c+d x))^{5/2}}+\frac{2 (3 A+5 C) \sin (c+d x)}{5 d \sqrt{b \cos (c+d x)}}-\frac{2 (3 A+5 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \cos (c+d x)}}{5 b d \sqrt{\cos (c+d x)}}+\frac{2 b B \sin (c+d x)}{3 d (b \cos (c+d x))^{3/2}}+\frac{2 B \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d \sqrt{b \cos (c+d x)}}",1,"(2*(-3*(3*A + 5*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2] + 5*B*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + 9*A*Sin[c + d*x] + 15*C*Sin[c + d*x] + 5*B*Tan[c + d*x] + 3*A*Sec[c + d*x]*Tan[c + d*x]))/(15*d*Sqrt[b*Cos[c + d*x]])","A",1
270,1,133,209,0.690361,"\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^4(c+d x)}{\sqrt{b \cos (c+d x)}} \, dx","Integrate[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^4)/Sqrt[b*Cos[c + d*x]],x]","\frac{2 \left(5 (5 A+7 C) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+25 A \tan (c+d x)+15 A \tan (c+d x) \sec ^2(c+d x)+63 B \sin (c+d x)-63 B \sqrt{\cos (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+21 B \tan (c+d x) \sec (c+d x)+35 C \tan (c+d x)\right)}{105 d \sqrt{b \cos (c+d x)}}","\frac{2 A b^3 \sin (c+d x)}{7 d (b \cos (c+d x))^{7/2}}+\frac{2 b (5 A+7 C) \sin (c+d x)}{21 d (b \cos (c+d x))^{3/2}}+\frac{2 (5 A+7 C) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d \sqrt{b \cos (c+d x)}}+\frac{2 b^2 B \sin (c+d x)}{5 d (b \cos (c+d x))^{5/2}}+\frac{6 B \sin (c+d x)}{5 d \sqrt{b \cos (c+d x)}}-\frac{6 B E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \cos (c+d x)}}{5 b d \sqrt{\cos (c+d x)}}",1,"(2*(-63*B*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2] + 5*(5*A + 7*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + 63*B*Sin[c + d*x] + 25*A*Tan[c + d*x] + 35*C*Tan[c + d*x] + 21*B*Sec[c + d*x]*Tan[c + d*x] + 15*A*Sec[c + d*x]^2*Tan[c + d*x]))/(105*d*Sqrt[b*Cos[c + d*x]])","A",1
271,1,130,217,0.7171382,"\int \frac{\cos ^4(c+d x) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{(b \cos (c+d x))^{3/2}} \, dx","Integrate[(Cos[c + d*x]^4*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(b*Cos[c + d*x])^(3/2),x]","\frac{\sin (2 (c+d x)) (7 (36 A+43 C) \cos (c+d x)+5 (18 B \cos (2 (c+d x))+78 B+7 C \cos (3 (c+d x))))+168 (9 A+7 C) \sqrt{\cos (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+600 B \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{1260 b d \sqrt{b \cos (c+d x)}}","\frac{2 (9 A+7 C) \sin (c+d x) (b \cos (c+d x))^{3/2}}{45 b^3 d}+\frac{2 (9 A+7 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \cos (c+d x)}}{15 b^2 d \sqrt{\cos (c+d x)}}+\frac{2 C \sin (c+d x) (b \cos (c+d x))^{7/2}}{9 b^5 d}+\frac{2 B \sin (c+d x) (b \cos (c+d x))^{5/2}}{7 b^4 d}+\frac{10 B \sin (c+d x) \sqrt{b \cos (c+d x)}}{21 b^2 d}+\frac{10 B \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 b d \sqrt{b \cos (c+d x)}}",1,"(168*(9*A + 7*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2] + 600*B*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + (7*(36*A + 43*C)*Cos[c + d*x] + 5*(78*B + 18*B*Cos[2*(c + d*x)] + 7*C*Cos[3*(c + d*x)]))*Sin[2*(c + d*x)])/(1260*b*d*Sqrt[b*Cos[c + d*x]])","A",1
272,1,108,188,0.6470512,"\int \frac{\cos ^3(c+d x) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{(b \cos (c+d x))^{3/2}} \, dx","Integrate[(Cos[c + d*x]^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(b*Cos[c + d*x])^(3/2),x]","\frac{\cos ^{\frac{3}{2}}(c+d x) \left(\sin (c+d x) \sqrt{\cos (c+d x)} (70 A+42 B \cos (c+d x)+15 C \cos (2 (c+d x))+65 C)+10 (7 A+5 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+126 B E\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{105 d (b \cos (c+d x))^{3/2}}","\frac{2 (7 A+5 C) \sin (c+d x) \sqrt{b \cos (c+d x)}}{21 b^2 d}+\frac{2 (7 A+5 C) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 b d \sqrt{b \cos (c+d x)}}+\frac{2 C \sin (c+d x) (b \cos (c+d x))^{5/2}}{7 b^4 d}+\frac{2 B \sin (c+d x) (b \cos (c+d x))^{3/2}}{5 b^3 d}+\frac{6 B E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \cos (c+d x)}}{5 b^2 d \sqrt{\cos (c+d x)}}",1,"(Cos[c + d*x]^(3/2)*(126*B*EllipticE[(c + d*x)/2, 2] + 10*(7*A + 5*C)*EllipticF[(c + d*x)/2, 2] + Sqrt[Cos[c + d*x]]*(70*A + 65*C + 42*B*Cos[c + d*x] + 15*C*Cos[2*(c + d*x)])*Sin[c + d*x]))/(105*d*(b*Cos[c + d*x])^(3/2))","A",1
273,1,94,153,0.3721485,"\int \frac{\cos ^2(c+d x) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{(b \cos (c+d x))^{3/2}} \, dx","Integrate[(Cos[c + d*x]^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(b*Cos[c + d*x])^(3/2),x]","\frac{2 \cos ^{\frac{3}{2}}(c+d x) \left(3 (5 A+3 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+\sin (c+d x) \sqrt{\cos (c+d x)} (5 B+3 C \cos (c+d x))+5 B F\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{15 d (b \cos (c+d x))^{3/2}}","\frac{2 (5 A+3 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \cos (c+d x)}}{5 b^2 d \sqrt{\cos (c+d x)}}+\frac{2 C \sin (c+d x) (b \cos (c+d x))^{3/2}}{5 b^3 d}+\frac{2 B \sin (c+d x) \sqrt{b \cos (c+d x)}}{3 b^2 d}+\frac{2 B \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 b d \sqrt{b \cos (c+d x)}}",1,"(2*Cos[c + d*x]^(3/2)*(3*(5*A + 3*C)*EllipticE[(c + d*x)/2, 2] + 5*B*EllipticF[(c + d*x)/2, 2] + Sqrt[Cos[c + d*x]]*(5*B + 3*C*Cos[c + d*x])*Sin[c + d*x]))/(15*d*(b*Cos[c + d*x])^(3/2))","A",1
274,1,85,120,0.1770387,"\int \frac{\cos (c+d x) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{(b \cos (c+d x))^{3/2}} \, dx","Integrate[(Cos[c + d*x]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(b*Cos[c + d*x])^(3/2),x]","\frac{2 (3 A+C) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+6 B \sqrt{\cos (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+C \sin (2 (c+d x))}{3 b d \sqrt{b \cos (c+d x)}}","\frac{2 (3 A+C) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 b d \sqrt{b \cos (c+d x)}}+\frac{2 B E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \cos (c+d x)}}{b^2 d \sqrt{\cos (c+d x)}}+\frac{2 C \sin (c+d x) \sqrt{b \cos (c+d x)}}{3 b^2 d}",1,"(6*B*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2] + 2*(3*A + C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + C*Sin[2*(c + d*x)])/(3*b*d*Sqrt[b*Cos[c + d*x]])","A",1
275,1,80,116,0.177797,"\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{(b \cos (c+d x))^{3/2}} \, dx","Integrate[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(b*Cos[c + d*x])^(3/2),x]","\frac{2 \left(-\left((A-C) \sqrt{\cos (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)+A \sin (c+d x)+B \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{b d \sqrt{b \cos (c+d x)}}","-\frac{2 (A-C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \cos (c+d x)}}{b^2 d \sqrt{\cos (c+d x)}}+\frac{2 A \sin (c+d x)}{b d \sqrt{b \cos (c+d x)}}+\frac{2 B \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b d \sqrt{b \cos (c+d x)}}",1,"(2*(-((A - C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]) + B*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + A*Sin[c + d*x]))/(b*d*Sqrt[b*Cos[c + d*x]])","A",1
276,1,761,144,6.2816437,"\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec (c+d x)}{(b \cos (c+d x))^{3/2}} \, dx","Integrate[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x])/(b*Cos[c + d*x])^(3/2),x]","\frac{\frac{2 B \csc (c) \cos ^{\frac{5}{2}}(c+d x) \left(A \sec ^2(c+d x)+B \sec (c+d x)+C\right) \left(\frac{\tan (c) \sin \left(\tan ^{-1}(\tan (c))+d x\right) \, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right)}{\sqrt{\tan ^2(c)+1} \sqrt{1-\cos \left(\tan ^{-1}(\tan (c))+d x\right)} \sqrt{\cos \left(\tan ^{-1}(\tan (c))+d x\right)+1} \sqrt{\cos (c) \sqrt{\tan ^2(c)+1} \cos \left(\tan ^{-1}(\tan (c))+d x\right)}}-\frac{\frac{\tan (c) \sin \left(\tan ^{-1}(\tan (c))+d x\right)}{\sqrt{\tan ^2(c)+1}}+\frac{2 \cos ^2(c) \sqrt{\tan ^2(c)+1} \cos \left(\tan ^{-1}(\tan (c))+d x\right)}{\sin ^2(c)+\cos ^2(c)}}{\sqrt{\cos (c) \sqrt{\tan ^2(c)+1} \cos \left(\tan ^{-1}(\tan (c))+d x\right)}}\right)}{d \sqrt{b \cos (c+d x)} (2 A+2 B \cos (c+d x)+C \cos (2 c+2 d x)+C)}-\frac{4 A \csc (c) \cos ^{\frac{5}{2}}(c+d x) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin (c) \left(-\sqrt{\cot ^2(c)+1}\right) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec \left(d x-\tan ^{-1}(\cot (c))\right) \left(A \sec ^2(c+d x)+B \sec (c+d x)+C\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right)}{3 d \sqrt{\cot ^2(c)+1} \sqrt{b \cos (c+d x)} (2 A+2 B \cos (c+d x)+C \cos (2 c+2 d x)+C)}-\frac{4 C \csc (c) \cos ^{\frac{5}{2}}(c+d x) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin (c) \left(-\sqrt{\cot ^2(c)+1}\right) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec \left(d x-\tan ^{-1}(\cot (c))\right) \left(A \sec ^2(c+d x)+B \sec (c+d x)+C\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right)}{d \sqrt{\cot ^2(c)+1} \sqrt{b \cos (c+d x)} (2 A+2 B \cos (c+d x)+C \cos (2 c+2 d x)+C)}+\frac{\cos ^3(c+d x) \left(A \sec ^2(c+d x)+B \sec (c+d x)+C\right) \left(\frac{4 \sec (c) \sec (c+d x) (A \sin (c)+3 B \sin (d x))}{3 d}+\frac{4 A \sec (c) \sin (d x) \sec ^2(c+d x)}{3 d}+\frac{4 B \csc (c) \sec (c)}{d}\right)}{\sqrt{b \cos (c+d x)} (2 A+2 B \cos (c+d x)+C \cos (2 c+2 d x)+C)}}{b}","\frac{2 (A+3 C) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 b d \sqrt{b \cos (c+d x)}}+\frac{2 A \sin (c+d x)}{3 d (b \cos (c+d x))^{3/2}}-\frac{2 B E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \cos (c+d x)}}{b^2 d \sqrt{\cos (c+d x)}}+\frac{2 B \sin (c+d x)}{b d \sqrt{b \cos (c+d x)}}",1,"((Cos[c + d*x]^3*(C + B*Sec[c + d*x] + A*Sec[c + d*x]^2)*((4*B*Csc[c]*Sec[c])/d + (4*A*Sec[c]*Sec[c + d*x]^2*Sin[d*x])/(3*d) + (4*Sec[c]*Sec[c + d*x]*(A*Sin[c] + 3*B*Sin[d*x]))/(3*d)))/(Sqrt[b*Cos[c + d*x]]*(2*A + C + 2*B*Cos[c + d*x] + C*Cos[2*c + 2*d*x])) - (4*A*Cos[c + d*x]^(5/2)*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*(C + B*Sec[c + d*x] + A*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*Sqrt[b*Cos[c + d*x]]*(2*A + C + 2*B*Cos[c + d*x] + C*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]) - (4*C*Cos[c + d*x]^(5/2)*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*(C + B*Sec[c + d*x] + A*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(d*Sqrt[b*Cos[c + d*x]]*(2*A + C + 2*B*Cos[c + d*x] + C*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]) + (2*B*Cos[c + d*x]^(5/2)*Csc[c]*(C + B*Sec[c + d*x] + A*Sec[c + d*x]^2)*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(d*Sqrt[b*Cos[c + d*x]]*(2*A + C + 2*B*Cos[c + d*x] + C*Cos[2*c + 2*d*x])))/b","C",0
277,1,119,183,0.3640912,"\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^2(c+d x)}{(b \cos (c+d x))^{3/2}} \, dx","Integrate[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^2)/(b*Cos[c + d*x])^(3/2),x]","\frac{2 \left(-3 (3 A+5 C) \sqrt{\cos (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+9 A \sin (c+d x)+3 A \tan (c+d x) \sec (c+d x)+5 B \tan (c+d x)+5 B \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+15 C \sin (c+d x)\right)}{15 b d \sqrt{b \cos (c+d x)}}","-\frac{2 (3 A+5 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \cos (c+d x)}}{5 b^2 d \sqrt{\cos (c+d x)}}+\frac{2 (3 A+5 C) \sin (c+d x)}{5 b d \sqrt{b \cos (c+d x)}}+\frac{2 A b \sin (c+d x)}{5 d (b \cos (c+d x))^{5/2}}+\frac{2 B \sin (c+d x)}{3 d (b \cos (c+d x))^{3/2}}+\frac{2 B \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 b d \sqrt{b \cos (c+d x)}}",1,"(2*(-3*(3*A + 5*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2] + 5*B*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + 9*A*Sin[c + d*x] + 15*C*Sin[c + d*x] + 5*B*Tan[c + d*x] + 3*A*Sec[c + d*x]*Tan[c + d*x]))/(15*b*d*Sqrt[b*Cos[c + d*x]])","A",1
278,1,136,212,0.7479365,"\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^3(c+d x)}{(b \cos (c+d x))^{3/2}} \, dx","Integrate[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^3)/(b*Cos[c + d*x])^(3/2),x]","\frac{2 \left(5 (5 A+7 C) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+25 A \tan (c+d x)+15 A \tan (c+d x) \sec ^2(c+d x)+63 B \sin (c+d x)-63 B \sqrt{\cos (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+21 B \tan (c+d x) \sec (c+d x)+35 C \tan (c+d x)\right)}{105 b d \sqrt{b \cos (c+d x)}}","\frac{2 A b^2 \sin (c+d x)}{7 d (b \cos (c+d x))^{7/2}}+\frac{2 (5 A+7 C) \sin (c+d x)}{21 d (b \cos (c+d x))^{3/2}}+\frac{2 (5 A+7 C) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 b d \sqrt{b \cos (c+d x)}}-\frac{6 B E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \cos (c+d x)}}{5 b^2 d \sqrt{\cos (c+d x)}}+\frac{2 b B \sin (c+d x)}{5 d (b \cos (c+d x))^{5/2}}+\frac{6 B \sin (c+d x)}{5 b d \sqrt{b \cos (c+d x)}}",1,"(2*(-63*B*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2] + 5*(5*A + 7*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + 63*B*Sin[c + d*x] + 25*A*Tan[c + d*x] + 35*C*Tan[c + d*x] + 21*B*Sec[c + d*x]*Tan[c + d*x] + 15*A*Sec[c + d*x]^2*Tan[c + d*x]))/(105*b*d*Sqrt[b*Cos[c + d*x]])","A",1
279,1,130,217,0.7030799,"\int \frac{\cos ^5(c+d x) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{(b \cos (c+d x))^{5/2}} \, dx","Integrate[(Cos[c + d*x]^5*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(b*Cos[c + d*x])^(5/2),x]","\frac{\sin (2 (c+d x)) (7 (36 A+43 C) \cos (c+d x)+5 (18 B \cos (2 (c+d x))+78 B+7 C \cos (3 (c+d x))))+168 (9 A+7 C) \sqrt{\cos (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+600 B \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{1260 b^2 d \sqrt{b \cos (c+d x)}}","\frac{2 (9 A+7 C) \sin (c+d x) (b \cos (c+d x))^{3/2}}{45 b^4 d}+\frac{2 (9 A+7 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \cos (c+d x)}}{15 b^3 d \sqrt{\cos (c+d x)}}+\frac{2 C \sin (c+d x) (b \cos (c+d x))^{7/2}}{9 b^6 d}+\frac{2 B \sin (c+d x) (b \cos (c+d x))^{5/2}}{7 b^5 d}+\frac{10 B \sin (c+d x) \sqrt{b \cos (c+d x)}}{21 b^3 d}+\frac{10 B \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 b^2 d \sqrt{b \cos (c+d x)}}",1,"(168*(9*A + 7*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2] + 600*B*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + (7*(36*A + 43*C)*Cos[c + d*x] + 5*(78*B + 18*B*Cos[2*(c + d*x)] + 7*C*Cos[3*(c + d*x)]))*Sin[2*(c + d*x)])/(1260*b^2*d*Sqrt[b*Cos[c + d*x]])","A",1
280,1,111,188,0.5935136,"\int \frac{\cos ^4(c+d x) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{(b \cos (c+d x))^{5/2}} \, dx","Integrate[(Cos[c + d*x]^4*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(b*Cos[c + d*x])^(5/2),x]","\frac{\sqrt{\cos (c+d x)} \left(\sin (c+d x) \sqrt{\cos (c+d x)} (70 A+42 B \cos (c+d x)+15 C \cos (2 (c+d x))+65 C)+10 (7 A+5 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+126 B E\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{105 b^2 d \sqrt{b \cos (c+d x)}}","\frac{2 (7 A+5 C) \sin (c+d x) \sqrt{b \cos (c+d x)}}{21 b^3 d}+\frac{2 (7 A+5 C) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 b^2 d \sqrt{b \cos (c+d x)}}+\frac{2 C \sin (c+d x) (b \cos (c+d x))^{5/2}}{7 b^5 d}+\frac{2 B \sin (c+d x) (b \cos (c+d x))^{3/2}}{5 b^4 d}+\frac{6 B E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \cos (c+d x)}}{5 b^3 d \sqrt{\cos (c+d x)}}",1,"(Sqrt[Cos[c + d*x]]*(126*B*EllipticE[(c + d*x)/2, 2] + 10*(7*A + 5*C)*EllipticF[(c + d*x)/2, 2] + Sqrt[Cos[c + d*x]]*(70*A + 65*C + 42*B*Cos[c + d*x] + 15*C*Cos[2*(c + d*x)])*Sin[c + d*x]))/(105*b^2*d*Sqrt[b*Cos[c + d*x]])","A",1
281,1,97,153,0.3172429,"\int \frac{\cos ^3(c+d x) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{(b \cos (c+d x))^{5/2}} \, dx","Integrate[(Cos[c + d*x]^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(b*Cos[c + d*x])^(5/2),x]","\frac{2 \sqrt{\cos (c+d x)} \left(3 (5 A+3 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+\sin (c+d x) \sqrt{\cos (c+d x)} (5 B+3 C \cos (c+d x))+5 B F\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{15 b^2 d \sqrt{b \cos (c+d x)}}","\frac{2 (5 A+3 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \cos (c+d x)}}{5 b^3 d \sqrt{\cos (c+d x)}}+\frac{2 C \sin (c+d x) (b \cos (c+d x))^{3/2}}{5 b^4 d}+\frac{2 B \sin (c+d x) \sqrt{b \cos (c+d x)}}{3 b^3 d}+\frac{2 B \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 b^2 d \sqrt{b \cos (c+d x)}}",1,"(2*Sqrt[Cos[c + d*x]]*(3*(5*A + 3*C)*EllipticE[(c + d*x)/2, 2] + 5*B*EllipticF[(c + d*x)/2, 2] + Sqrt[Cos[c + d*x]]*(5*B + 3*C*Cos[c + d*x])*Sin[c + d*x]))/(15*b^2*d*Sqrt[b*Cos[c + d*x]])","A",1
282,1,85,120,0.1787546,"\int \frac{\cos ^2(c+d x) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{(b \cos (c+d x))^{5/2}} \, dx","Integrate[(Cos[c + d*x]^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(b*Cos[c + d*x])^(5/2),x]","\frac{2 (3 A+C) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+6 B \sqrt{\cos (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+C \sin (2 (c+d x))}{3 b^2 d \sqrt{b \cos (c+d x)}}","\frac{2 (3 A+C) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 b^2 d \sqrt{b \cos (c+d x)}}+\frac{2 B E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \cos (c+d x)}}{b^3 d \sqrt{\cos (c+d x)}}+\frac{2 C \sin (c+d x) \sqrt{b \cos (c+d x)}}{3 b^3 d}",1,"(6*B*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2] + 2*(3*A + C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + C*Sin[2*(c + d*x)])/(3*b^2*d*Sqrt[b*Cos[c + d*x]])","A",1
283,1,807,116,6.2259928,"\int \frac{\cos (c+d x) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{(b \cos (c+d x))^{5/2}} \, dx","Integrate[(Cos[c + d*x]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(b*Cos[c + d*x])^(5/2),x]","\frac{\frac{(B+C \cos (c+d x)+A \sec (c+d x)) \left(\frac{4 A \sec (c) \sec (c+d x) \sin (d x)}{d}-\frac{2 (-2 A+C+C \cos (2 c)) \csc (c) \sec (c)}{d}\right) \cos ^2(c+d x)}{\sqrt{b \cos (c+d x)} (2 A+C+2 B \cos (c+d x)+C \cos (2 c+2 d x))}+\frac{2 A \csc (c) (B+C \cos (c+d x)+A \sec (c+d x)) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \cos ^{\frac{3}{2}}(c+d x)}{d \sqrt{b \cos (c+d x)} (2 A+C+2 B \cos (c+d x)+C \cos (2 c+2 d x))}-\frac{2 C \csc (c) (B+C \cos (c+d x)+A \sec (c+d x)) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \cos ^{\frac{3}{2}}(c+d x)}{d \sqrt{b \cos (c+d x)} (2 A+C+2 B \cos (c+d x)+C \cos (2 c+2 d x))}-\frac{4 B \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) (B+C \cos (c+d x)+A \sec (c+d x)) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^{\frac{3}{2}}(c+d x)}{d \sqrt{b \cos (c+d x)} (2 A+C+2 B \cos (c+d x)+C \cos (2 c+2 d x)) \sqrt{\cot ^2(c)+1}}}{b^2}","-\frac{2 (A-C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \cos (c+d x)}}{b^3 d \sqrt{\cos (c+d x)}}+\frac{2 A \sin (c+d x)}{b^2 d \sqrt{b \cos (c+d x)}}+\frac{2 B \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b^2 d \sqrt{b \cos (c+d x)}}",1,"((Cos[c + d*x]^2*(B + C*Cos[c + d*x] + A*Sec[c + d*x])*((-2*(-2*A + C + C*Cos[2*c])*Csc[c]*Sec[c])/d + (4*A*Sec[c]*Sec[c + d*x]*Sin[d*x])/d))/(Sqrt[b*Cos[c + d*x]]*(2*A + C + 2*B*Cos[c + d*x] + C*Cos[2*c + 2*d*x])) - (4*B*Cos[c + d*x]^(3/2)*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*(B + C*Cos[c + d*x] + A*Sec[c + d*x])*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(d*Sqrt[b*Cos[c + d*x]]*(2*A + C + 2*B*Cos[c + d*x] + C*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]) + (2*A*Cos[c + d*x]^(3/2)*Csc[c]*(B + C*Cos[c + d*x] + A*Sec[c + d*x])*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(d*Sqrt[b*Cos[c + d*x]]*(2*A + C + 2*B*Cos[c + d*x] + C*Cos[2*c + 2*d*x])) - (2*C*Cos[c + d*x]^(3/2)*Csc[c]*(B + C*Cos[c + d*x] + A*Sec[c + d*x])*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(d*Sqrt[b*Cos[c + d*x]]*(2*A + C + 2*B*Cos[c + d*x] + C*Cos[2*c + 2*d*x])))/b^2","C",1
284,1,92,147,0.3863453,"\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{(b \cos (c+d x))^{5/2}} \, dx","Integrate[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(b*Cos[c + d*x])^(5/2),x]","\frac{2 \left(\tan (c+d x) (A+3 B \cos (c+d x))+(A+3 C) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-3 B \sqrt{\cos (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{3 b^2 d \sqrt{b \cos (c+d x)}}","\frac{2 (A+3 C) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 b^2 d \sqrt{b \cos (c+d x)}}+\frac{2 A \sin (c+d x)}{3 b d (b \cos (c+d x))^{3/2}}-\frac{2 B E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \cos (c+d x)}}{b^3 d \sqrt{\cos (c+d x)}}+\frac{2 B \sin (c+d x)}{b^2 d \sqrt{b \cos (c+d x)}}",1,"(2*(-3*B*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2] + (A + 3*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + (A + 3*B*Cos[c + d*x])*Tan[c + d*x]))/(3*b^2*d*Sqrt[b*Cos[c + d*x]])","A",1
285,1,119,185,0.3426497,"\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec (c+d x)}{(b \cos (c+d x))^{5/2}} \, dx","Integrate[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x])/(b*Cos[c + d*x])^(5/2),x]","\frac{2 \left(-3 (3 A+5 C) \sqrt{\cos (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+9 A \sin (c+d x)+3 A \tan (c+d x) \sec (c+d x)+5 B \tan (c+d x)+5 B \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+15 C \sin (c+d x)\right)}{15 b^2 d \sqrt{b \cos (c+d x)}}","-\frac{2 (3 A+5 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \cos (c+d x)}}{5 b^3 d \sqrt{\cos (c+d x)}}+\frac{2 (3 A+5 C) \sin (c+d x)}{5 b^2 d \sqrt{b \cos (c+d x)}}+\frac{2 A \sin (c+d x)}{5 d (b \cos (c+d x))^{5/2}}+\frac{2 B \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 b^2 d \sqrt{b \cos (c+d x)}}+\frac{2 B \sin (c+d x)}{3 b d (b \cos (c+d x))^{3/2}}",1,"(2*(-3*(3*A + 5*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2] + 5*B*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + 9*A*Sin[c + d*x] + 15*C*Sin[c + d*x] + 5*B*Tan[c + d*x] + 3*A*Sec[c + d*x]*Tan[c + d*x]))/(15*b^2*d*Sqrt[b*Cos[c + d*x]])","A",1
286,1,136,212,0.4648881,"\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^2(c+d x)}{(b \cos (c+d x))^{5/2}} \, dx","Integrate[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^2)/(b*Cos[c + d*x])^(5/2),x]","\frac{2 \left(5 (5 A+7 C) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+25 A \tan (c+d x)+15 A \tan (c+d x) \sec ^2(c+d x)+63 B \sin (c+d x)-63 B \sqrt{\cos (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+21 B \tan (c+d x) \sec (c+d x)+35 C \tan (c+d x)\right)}{105 b^2 d \sqrt{b \cos (c+d x)}}","\frac{2 (5 A+7 C) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 b^2 d \sqrt{b \cos (c+d x)}}+\frac{2 (5 A+7 C) \sin (c+d x)}{21 b d (b \cos (c+d x))^{3/2}}+\frac{2 A b \sin (c+d x)}{7 d (b \cos (c+d x))^{7/2}}-\frac{6 B E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \cos (c+d x)}}{5 b^3 d \sqrt{\cos (c+d x)}}+\frac{6 B \sin (c+d x)}{5 b^2 d \sqrt{b \cos (c+d x)}}+\frac{2 B \sin (c+d x)}{5 d (b \cos (c+d x))^{5/2}}",1,"(2*(-63*B*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2] + 5*(5*A + 7*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + 63*B*Sin[c + d*x] + 25*A*Tan[c + d*x] + 35*C*Tan[c + d*x] + 21*B*Sec[c + d*x]*Tan[c + d*x] + 15*A*Sec[c + d*x]^2*Tan[c + d*x]))/(105*b^2*d*Sqrt[b*Cos[c + d*x]])","A",1
287,1,119,188,0.1239227,"\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{(b \cos (c+d x))^{7/2}} \, dx","Integrate[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(b*Cos[c + d*x])^(7/2),x]","\frac{2 \left(-3 (3 A+5 C) \sqrt{\cos (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+9 A \sin (c+d x)+3 A \tan (c+d x) \sec (c+d x)+5 B \tan (c+d x)+5 B \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+15 C \sin (c+d x)\right)}{15 b^3 d \sqrt{b \cos (c+d x)}}","-\frac{2 (3 A+5 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \cos (c+d x)}}{5 b^4 d \sqrt{\cos (c+d x)}}+\frac{2 (3 A+5 C) \sin (c+d x)}{5 b^3 d \sqrt{b \cos (c+d x)}}+\frac{2 A \sin (c+d x)}{5 b d (b \cos (c+d x))^{5/2}}+\frac{2 B \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 b^3 d \sqrt{b \cos (c+d x)}}+\frac{2 B \sin (c+d x)}{3 b^2 d (b \cos (c+d x))^{3/2}}",1,"(2*(-3*(3*A + 5*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2] + 5*B*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + 9*A*Sin[c + d*x] + 15*C*Sin[c + d*x] + 5*B*Tan[c + d*x] + 3*A*Sec[c + d*x]*Tan[c + d*x]))/(15*b^3*d*Sqrt[b*Cos[c + d*x]])","A",1
288,1,109,223,0.2933534,"\int \cos ^{\frac{5}{2}}(c+d x) \sqrt{b \cos (c+d x)} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^(5/2)*Sqrt[b*Cos[c + d*x]]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","\frac{\sqrt{b \cos (c+d x)} (60 (6 A+5 C) \sin (c+d x)+40 A \sin (3 (c+d x))+120 B \sin (2 (c+d x))+15 B \sin (4 (c+d x))+180 B c+180 B d x+50 C \sin (3 (c+d x))+6 C \sin (5 (c+d x)))}{480 d \sqrt{\cos (c+d x)}}","-\frac{(5 A+4 C) \sin ^3(c+d x) \sqrt{b \cos (c+d x)}}{15 d \sqrt{\cos (c+d x)}}+\frac{(5 A+4 C) \sin (c+d x) \sqrt{b \cos (c+d x)}}{5 d \sqrt{\cos (c+d x)}}+\frac{3 B x \sqrt{b \cos (c+d x)}}{8 \sqrt{\cos (c+d x)}}+\frac{B \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) \sqrt{b \cos (c+d x)}}{4 d}+\frac{3 B \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{b \cos (c+d x)}}{8 d}+\frac{C \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x) \sqrt{b \cos (c+d x)}}{5 d}",1,"(Sqrt[b*Cos[c + d*x]]*(180*B*c + 180*B*d*x + 60*(6*A + 5*C)*Sin[c + d*x] + 120*B*Sin[2*(c + d*x)] + 40*A*Sin[3*(c + d*x)] + 50*C*Sin[3*(c + d*x)] + 15*B*Sin[4*(c + d*x)] + 6*C*Sin[5*(c + d*x)]))/(480*d*Sqrt[Cos[c + d*x]])","A",1
289,1,92,184,0.2706208,"\int \cos ^{\frac{3}{2}}(c+d x) \sqrt{b \cos (c+d x)} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^(3/2)*Sqrt[b*Cos[c + d*x]]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","\frac{\sqrt{b \cos (c+d x)} (24 (A+C) \sin (2 (c+d x))+48 A c+48 A d x+72 B \sin (c+d x)+8 B \sin (3 (c+d x))+3 C \sin (4 (c+d x))+36 c C+36 C d x)}{96 d \sqrt{\cos (c+d x)}}","\frac{x (4 A+3 C) \sqrt{b \cos (c+d x)}}{8 \sqrt{\cos (c+d x)}}+\frac{(4 A+3 C) \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{b \cos (c+d x)}}{8 d}-\frac{B \sin ^3(c+d x) \sqrt{b \cos (c+d x)}}{3 d \sqrt{\cos (c+d x)}}+\frac{B \sin (c+d x) \sqrt{b \cos (c+d x)}}{d \sqrt{\cos (c+d x)}}+\frac{C \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) \sqrt{b \cos (c+d x)}}{4 d}",1,"(Sqrt[b*Cos[c + d*x]]*(48*A*c + 36*c*C + 48*A*d*x + 36*C*d*x + 72*B*Sin[c + d*x] + 24*(A + C)*Sin[2*(c + d*x)] + 8*B*Sin[3*(c + d*x)] + 3*C*Sin[4*(c + d*x)]))/(96*d*Sqrt[Cos[c + d*x]])","A",1
290,1,75,143,0.1932513,"\int \sqrt{\cos (c+d x)} \sqrt{b \cos (c+d x)} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Integrate[Sqrt[Cos[c + d*x]]*Sqrt[b*Cos[c + d*x]]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","\frac{\sqrt{b \cos (c+d x)} (3 (4 A+3 C) \sin (c+d x)+3 B \sin (2 (c+d x))+6 B c+6 B d x+C \sin (3 (c+d x)))}{12 d \sqrt{\cos (c+d x)}}","\frac{(3 A+2 C) \sin (c+d x) \sqrt{b \cos (c+d x)}}{3 d \sqrt{\cos (c+d x)}}+\frac{B x \sqrt{b \cos (c+d x)}}{2 \sqrt{\cos (c+d x)}}+\frac{B \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{b \cos (c+d x)}}{2 d}+\frac{C \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \sqrt{b \cos (c+d x)}}{3 d}",1,"(Sqrt[b*Cos[c + d*x]]*(6*B*c + 6*B*d*x + 3*(4*A + 3*C)*Sin[c + d*x] + 3*B*Sin[2*(c + d*x)] + C*Sin[3*(c + d*x)]))/(12*d*Sqrt[Cos[c + d*x]])","A",1
291,1,61,123,0.1093621,"\int \frac{\sqrt{b \cos (c+d x)} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\sqrt{\cos (c+d x)}} \, dx","Integrate[(Sqrt[b*Cos[c + d*x]]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Sqrt[Cos[c + d*x]],x]","\frac{\sqrt{b \cos (c+d x)} (2 (2 A+C) (c+d x)+4 B \sin (c+d x)+C \sin (2 (c+d x)))}{4 d \sqrt{\cos (c+d x)}}","\frac{A x \sqrt{b \cos (c+d x)}}{\sqrt{\cos (c+d x)}}+\frac{B \sin (c+d x) \sqrt{b \cos (c+d x)}}{d \sqrt{\cos (c+d x)}}+\frac{C x \sqrt{b \cos (c+d x)}}{2 \sqrt{\cos (c+d x)}}+\frac{C \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{b \cos (c+d x)}}{2 d}",1,"(Sqrt[b*Cos[c + d*x]]*(2*(2*A + C)*(c + d*x) + 4*B*Sin[c + d*x] + C*Sin[2*(c + d*x)]))/(4*d*Sqrt[Cos[c + d*x]])","A",1
292,1,93,93,0.1025999,"\int \frac{\sqrt{b \cos (c+d x)} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[(Sqrt[b*Cos[c + d*x]]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(3/2),x]","\frac{\sqrt{b \cos (c+d x)} \left(-A \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+A \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+B c+B d x+C \sin (c+d x)\right)}{d \sqrt{\cos (c+d x)}}","\frac{A \sqrt{b \cos (c+d x)} \tanh ^{-1}(\sin (c+d x))}{d \sqrt{\cos (c+d x)}}+\frac{B x \sqrt{b \cos (c+d x)}}{\sqrt{\cos (c+d x)}}+\frac{C \sin (c+d x) \sqrt{b \cos (c+d x)}}{d \sqrt{\cos (c+d x)}}",1,"(Sqrt[b*Cos[c + d*x]]*(B*c + B*d*x - A*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + A*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + C*Sin[c + d*x]))/(d*Sqrt[Cos[c + d*x]])","A",1
293,1,60,93,0.0731529,"\int \frac{\sqrt{b \cos (c+d x)} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{5}{2}}(c+d x)} \, dx","Integrate[(Sqrt[b*Cos[c + d*x]]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(5/2),x]","\frac{\sqrt{b \cos (c+d x)} \left(A \sin (c+d x)+B \cos (c+d x) \tanh ^{-1}(\sin (c+d x))+C d x \cos (c+d x)\right)}{d \cos ^{\frac{3}{2}}(c+d x)}","\frac{A \sin (c+d x) \sqrt{b \cos (c+d x)}}{d \cos ^{\frac{3}{2}}(c+d x)}+\frac{B \sqrt{b \cos (c+d x)} \tanh ^{-1}(\sin (c+d x))}{d \sqrt{\cos (c+d x)}}+\frac{C x \sqrt{b \cos (c+d x)}}{\sqrt{\cos (c+d x)}}",1,"(Sqrt[b*Cos[c + d*x]]*(C*d*x*Cos[c + d*x] + B*ArcTanh[Sin[c + d*x]]*Cos[c + d*x] + A*Sin[c + d*x]))/(d*Cos[c + d*x]^(3/2))","A",1
294,1,69,111,0.1451053,"\int \frac{\sqrt{b \cos (c+d x)} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{7}{2}}(c+d x)} \, dx","Integrate[(Sqrt[b*Cos[c + d*x]]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(7/2),x]","\frac{\sqrt{b \cos (c+d x)} \left(\sin (c+d x) (A+2 B \cos (c+d x))+(A+2 C) \cos ^2(c+d x) \tanh ^{-1}(\sin (c+d x))\right)}{2 d \cos ^{\frac{5}{2}}(c+d x)}","\frac{(A+2 C) \sqrt{b \cos (c+d x)} \tanh ^{-1}(\sin (c+d x))}{2 d \sqrt{\cos (c+d x)}}+\frac{A \sin (c+d x) \sqrt{b \cos (c+d x)}}{2 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{B \sin (c+d x) \sqrt{b \cos (c+d x)}}{d \cos ^{\frac{3}{2}}(c+d x)}",1,"(Sqrt[b*Cos[c + d*x]]*((A + 2*C)*ArcTanh[Sin[c + d*x]]*Cos[c + d*x]^2 + (A + 2*B*Cos[c + d*x])*Sin[c + d*x]))/(2*d*Cos[c + d*x]^(5/2))","A",1
295,1,87,152,0.4309302,"\int \frac{\sqrt{b \cos (c+d x)} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{9}{2}}(c+d x)} \, dx","Integrate[(Sqrt[b*Cos[c + d*x]]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(9/2),x]","\frac{\sqrt{b \cos (c+d x)} \left(\tan (c+d x) ((2 A+3 C) \cos (2 (c+d x))+4 A+3 B \cos (c+d x)+3 C)+3 B \cos ^2(c+d x) \tanh ^{-1}(\sin (c+d x))\right)}{6 d \cos ^{\frac{5}{2}}(c+d x)}","\frac{(2 A+3 C) \sin (c+d x) \sqrt{b \cos (c+d x)}}{3 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{A \sin (c+d x) \sqrt{b \cos (c+d x)}}{3 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{B \sin (c+d x) \sqrt{b \cos (c+d x)}}{2 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{B \sqrt{b \cos (c+d x)} \tanh ^{-1}(\sin (c+d x))}{2 d \sqrt{\cos (c+d x)}}",1,"(Sqrt[b*Cos[c + d*x]]*(3*B*ArcTanh[Sin[c + d*x]]*Cos[c + d*x]^2 + (4*A + 3*C + 3*B*Cos[c + d*x] + (2*A + 3*C)*Cos[2*(c + d*x)])*Tan[c + d*x]))/(6*d*Cos[c + d*x]^(5/2))","A",1
296,1,110,193,0.3537454,"\int \frac{\sqrt{b \cos (c+d x)} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{11}{2}}(c+d x)} \, dx","Integrate[(Sqrt[b*Cos[c + d*x]]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(11/2),x]","\frac{\sqrt{b \cos (c+d x)} \left(\sin (c+d x) \left(3 (3 A+4 C) \cos ^2(c+d x)+6 A+24 B \cos ^3(c+d x)+8 B \sin ^2(c+d x) \cos (c+d x)\right)+3 (3 A+4 C) \cos ^4(c+d x) \tanh ^{-1}(\sin (c+d x))\right)}{24 d \cos ^{\frac{9}{2}}(c+d x)}","\frac{(3 A+4 C) \sin (c+d x) \sqrt{b \cos (c+d x)}}{8 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{(3 A+4 C) \sqrt{b \cos (c+d x)} \tanh ^{-1}(\sin (c+d x))}{8 d \sqrt{\cos (c+d x)}}+\frac{A \sin (c+d x) \sqrt{b \cos (c+d x)}}{4 d \cos ^{\frac{9}{2}}(c+d x)}+\frac{B \sin ^3(c+d x) \sqrt{b \cos (c+d x)}}{3 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{B \sin (c+d x) \sqrt{b \cos (c+d x)}}{d \cos ^{\frac{3}{2}}(c+d x)}",1,"(Sqrt[b*Cos[c + d*x]]*(3*(3*A + 4*C)*ArcTanh[Sin[c + d*x]]*Cos[c + d*x]^4 + Sin[c + d*x]*(6*A + 3*(3*A + 4*C)*Cos[c + d*x]^2 + 24*B*Cos[c + d*x]^3 + 8*B*Cos[c + d*x]*Sin[c + d*x]^2)))/(24*d*Cos[c + d*x]^(9/2))","A",1
297,1,109,229,0.3246343,"\int \cos ^{\frac{3}{2}}(c+d x) (b \cos (c+d x))^{3/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^(3/2)*(b*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","\frac{(b \cos (c+d x))^{3/2} (60 (6 A+5 C) \sin (c+d x)+40 A \sin (3 (c+d x))+120 B \sin (2 (c+d x))+15 B \sin (4 (c+d x))+180 B c+180 B d x+50 C \sin (3 (c+d x))+6 C \sin (5 (c+d x)))}{480 d \cos ^{\frac{3}{2}}(c+d x)}","-\frac{b (5 A+4 C) \sin ^3(c+d x) \sqrt{b \cos (c+d x)}}{15 d \sqrt{\cos (c+d x)}}+\frac{b (5 A+4 C) \sin (c+d x) \sqrt{b \cos (c+d x)}}{5 d \sqrt{\cos (c+d x)}}+\frac{3 b B x \sqrt{b \cos (c+d x)}}{8 \sqrt{\cos (c+d x)}}+\frac{b B \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) \sqrt{b \cos (c+d x)}}{4 d}+\frac{3 b B \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{b \cos (c+d x)}}{8 d}+\frac{b C \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x) \sqrt{b \cos (c+d x)}}{5 d}",1,"((b*Cos[c + d*x])^(3/2)*(180*B*c + 180*B*d*x + 60*(6*A + 5*C)*Sin[c + d*x] + 120*B*Sin[2*(c + d*x)] + 40*A*Sin[3*(c + d*x)] + 50*C*Sin[3*(c + d*x)] + 15*B*Sin[4*(c + d*x)] + 6*C*Sin[5*(c + d*x)]))/(480*d*Cos[c + d*x]^(3/2))","A",1
298,1,92,189,0.2217003,"\int \sqrt{\cos (c+d x)} (b \cos (c+d x))^{3/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Integrate[Sqrt[Cos[c + d*x]]*(b*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","\frac{(b \cos (c+d x))^{3/2} (24 (A+C) \sin (2 (c+d x))+48 A c+48 A d x+72 B \sin (c+d x)+8 B \sin (3 (c+d x))+3 C \sin (4 (c+d x))+36 c C+36 C d x)}{96 d \cos ^{\frac{3}{2}}(c+d x)}","\frac{b x (4 A+3 C) \sqrt{b \cos (c+d x)}}{8 \sqrt{\cos (c+d x)}}+\frac{b (4 A+3 C) \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{b \cos (c+d x)}}{8 d}-\frac{b B \sin ^3(c+d x) \sqrt{b \cos (c+d x)}}{3 d \sqrt{\cos (c+d x)}}+\frac{b B \sin (c+d x) \sqrt{b \cos (c+d x)}}{d \sqrt{\cos (c+d x)}}+\frac{b C \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) \sqrt{b \cos (c+d x)}}{4 d}",1,"((b*Cos[c + d*x])^(3/2)*(48*A*c + 36*c*C + 48*A*d*x + 36*C*d*x + 72*B*Sin[c + d*x] + 24*(A + C)*Sin[2*(c + d*x)] + 8*B*Sin[3*(c + d*x)] + 3*C*Sin[4*(c + d*x)]))/(96*d*Cos[c + d*x]^(3/2))","A",1
299,1,76,147,0.0736273,"\int \frac{(b \cos (c+d x))^{3/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\sqrt{\cos (c+d x)}} \, dx","Integrate[((b*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Sqrt[Cos[c + d*x]],x]","\frac{b \sqrt{b \cos (c+d x)} (3 (4 A+3 C) \sin (c+d x)+3 B \sin (2 (c+d x))+6 B c+6 B d x+C \sin (3 (c+d x)))}{12 d \sqrt{\cos (c+d x)}}","\frac{b (3 A+2 C) \sin (c+d x) \sqrt{b \cos (c+d x)}}{3 d \sqrt{\cos (c+d x)}}+\frac{b B x \sqrt{b \cos (c+d x)}}{2 \sqrt{\cos (c+d x)}}+\frac{b B \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{b \cos (c+d x)}}{2 d}+\frac{b C \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \sqrt{b \cos (c+d x)}}{3 d}",1,"(b*Sqrt[b*Cos[c + d*x]]*(6*B*c + 6*B*d*x + 3*(4*A + 3*C)*Sin[c + d*x] + 3*B*Sin[2*(c + d*x)] + C*Sin[3*(c + d*x)]))/(12*d*Sqrt[Cos[c + d*x]])","A",1
300,1,61,127,0.1337695,"\int \frac{(b \cos (c+d x))^{3/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[((b*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(3/2),x]","\frac{(b \cos (c+d x))^{3/2} (2 (2 A+C) (c+d x)+4 B \sin (c+d x)+C \sin (2 (c+d x)))}{4 d \cos ^{\frac{3}{2}}(c+d x)}","\frac{A b x \sqrt{b \cos (c+d x)}}{\sqrt{\cos (c+d x)}}+\frac{b B \sin (c+d x) \sqrt{b \cos (c+d x)}}{d \sqrt{\cos (c+d x)}}+\frac{b C x \sqrt{b \cos (c+d x)}}{2 \sqrt{\cos (c+d x)}}+\frac{b C \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{b \cos (c+d x)}}{2 d}",1,"((b*Cos[c + d*x])^(3/2)*(2*(2*A + C)*(c + d*x) + 4*B*Sin[c + d*x] + C*Sin[2*(c + d*x)]))/(4*d*Cos[c + d*x]^(3/2))","A",1
301,1,93,96,0.1305204,"\int \frac{(b \cos (c+d x))^{3/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{5}{2}}(c+d x)} \, dx","Integrate[((b*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(5/2),x]","\frac{(b \cos (c+d x))^{3/2} \left(-A \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+A \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+B c+B d x+C \sin (c+d x)\right)}{d \cos ^{\frac{3}{2}}(c+d x)}","\frac{A b \sqrt{b \cos (c+d x)} \tanh ^{-1}(\sin (c+d x))}{d \sqrt{\cos (c+d x)}}+\frac{b B x \sqrt{b \cos (c+d x)}}{\sqrt{\cos (c+d x)}}+\frac{b C \sin (c+d x) \sqrt{b \cos (c+d x)}}{d \sqrt{\cos (c+d x)}}",1,"((b*Cos[c + d*x])^(3/2)*(B*c + B*d*x - A*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + A*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + C*Sin[c + d*x]))/(d*Cos[c + d*x]^(3/2))","A",1
302,1,60,96,0.0751076,"\int \frac{(b \cos (c+d x))^{3/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{7}{2}}(c+d x)} \, dx","Integrate[((b*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(7/2),x]","\frac{(b \cos (c+d x))^{3/2} \left(A \sin (c+d x)+B \cos (c+d x) \tanh ^{-1}(\sin (c+d x))+C d x \cos (c+d x)\right)}{d \cos ^{\frac{5}{2}}(c+d x)}","\frac{A b \sin (c+d x) \sqrt{b \cos (c+d x)}}{d \cos ^{\frac{3}{2}}(c+d x)}+\frac{b B \sqrt{b \cos (c+d x)} \tanh ^{-1}(\sin (c+d x))}{d \sqrt{\cos (c+d x)}}+\frac{b C x \sqrt{b \cos (c+d x)}}{\sqrt{\cos (c+d x)}}",1,"((b*Cos[c + d*x])^(3/2)*(C*d*x*Cos[c + d*x] + B*ArcTanh[Sin[c + d*x]]*Cos[c + d*x] + A*Sin[c + d*x]))/(d*Cos[c + d*x]^(5/2))","A",1
303,1,69,114,0.1203015,"\int \frac{(b \cos (c+d x))^{3/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{9}{2}}(c+d x)} \, dx","Integrate[((b*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(9/2),x]","\frac{(b \cos (c+d x))^{3/2} \left(\sin (c+d x) (A+2 B \cos (c+d x))+(A+2 C) \cos ^2(c+d x) \tanh ^{-1}(\sin (c+d x))\right)}{2 d \cos ^{\frac{7}{2}}(c+d x)}","\frac{b (A+2 C) \sqrt{b \cos (c+d x)} \tanh ^{-1}(\sin (c+d x))}{2 d \sqrt{\cos (c+d x)}}+\frac{A b \sin (c+d x) \sqrt{b \cos (c+d x)}}{2 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{b B \sin (c+d x) \sqrt{b \cos (c+d x)}}{d \cos ^{\frac{3}{2}}(c+d x)}",1,"((b*Cos[c + d*x])^(3/2)*((A + 2*C)*ArcTanh[Sin[c + d*x]]*Cos[c + d*x]^2 + (A + 2*B*Cos[c + d*x])*Sin[c + d*x]))/(2*d*Cos[c + d*x]^(7/2))","A",1
304,1,88,156,0.0581052,"\int \frac{(b \cos (c+d x))^{3/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{11}{2}}(c+d x)} \, dx","Integrate[((b*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(11/2),x]","\frac{b \sqrt{b \cos (c+d x)} \left(\tan (c+d x) ((2 A+3 C) \cos (2 (c+d x))+4 A+3 B \cos (c+d x)+3 C)+3 B \cos ^2(c+d x) \tanh ^{-1}(\sin (c+d x))\right)}{6 d \cos ^{\frac{5}{2}}(c+d x)}","\frac{b (2 A+3 C) \sin (c+d x) \sqrt{b \cos (c+d x)}}{3 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{A b \sin (c+d x) \sqrt{b \cos (c+d x)}}{3 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{b B \sin (c+d x) \sqrt{b \cos (c+d x)}}{2 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{b B \sqrt{b \cos (c+d x)} \tanh ^{-1}(\sin (c+d x))}{2 d \sqrt{\cos (c+d x)}}",1,"(b*Sqrt[b*Cos[c + d*x]]*(3*B*ArcTanh[Sin[c + d*x]]*Cos[c + d*x]^2 + (4*A + 3*C + 3*B*Cos[c + d*x] + (2*A + 3*C)*Cos[2*(c + d*x)])*Tan[c + d*x]))/(6*d*Cos[c + d*x]^(5/2))","A",1
305,1,111,198,0.2684711,"\int \frac{(b \cos (c+d x))^{3/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{13}{2}}(c+d x)} \, dx","Integrate[((b*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(13/2),x]","\frac{b \sqrt{b \cos (c+d x)} \left(\sin (c+d x) \left(3 (3 A+4 C) \cos ^2(c+d x)+6 A+24 B \cos ^3(c+d x)+8 B \sin ^2(c+d x) \cos (c+d x)\right)+3 (3 A+4 C) \cos ^4(c+d x) \tanh ^{-1}(\sin (c+d x))\right)}{24 d \cos ^{\frac{9}{2}}(c+d x)}","\frac{b (3 A+4 C) \sin (c+d x) \sqrt{b \cos (c+d x)}}{8 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{b (3 A+4 C) \sqrt{b \cos (c+d x)} \tanh ^{-1}(\sin (c+d x))}{8 d \sqrt{\cos (c+d x)}}+\frac{A b \sin (c+d x) \sqrt{b \cos (c+d x)}}{4 d \cos ^{\frac{9}{2}}(c+d x)}+\frac{b B \sin ^3(c+d x) \sqrt{b \cos (c+d x)}}{3 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{b B \sin (c+d x) \sqrt{b \cos (c+d x)}}{d \cos ^{\frac{3}{2}}(c+d x)}",1,"(b*Sqrt[b*Cos[c + d*x]]*(3*(3*A + 4*C)*ArcTanh[Sin[c + d*x]]*Cos[c + d*x]^4 + Sin[c + d*x]*(6*A + 3*(3*A + 4*C)*Cos[c + d*x]^2 + 24*B*Cos[c + d*x]^3 + 8*B*Cos[c + d*x]*Sin[c + d*x]^2)))/(24*d*Cos[c + d*x]^(9/2))","A",1
306,1,109,241,0.3236596,"\int \sqrt{\cos (c+d x)} (b \cos (c+d x))^{5/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Integrate[Sqrt[Cos[c + d*x]]*(b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","\frac{(b \cos (c+d x))^{5/2} (60 (6 A+5 C) \sin (c+d x)+40 A \sin (3 (c+d x))+120 B \sin (2 (c+d x))+15 B \sin (4 (c+d x))+180 B c+180 B d x+50 C \sin (3 (c+d x))+6 C \sin (5 (c+d x)))}{480 d \cos ^{\frac{5}{2}}(c+d x)}","-\frac{b^2 (5 A+4 C) \sin ^3(c+d x) \sqrt{b \cos (c+d x)}}{15 d \sqrt{\cos (c+d x)}}+\frac{b^2 (5 A+4 C) \sin (c+d x) \sqrt{b \cos (c+d x)}}{5 d \sqrt{\cos (c+d x)}}+\frac{3 b^2 B x \sqrt{b \cos (c+d x)}}{8 \sqrt{\cos (c+d x)}}+\frac{b^2 B \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) \sqrt{b \cos (c+d x)}}{4 d}+\frac{3 b^2 B \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{b \cos (c+d x)}}{8 d}+\frac{b^2 C \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x) \sqrt{b \cos (c+d x)}}{5 d}",1,"((b*Cos[c + d*x])^(5/2)*(180*B*c + 180*B*d*x + 60*(6*A + 5*C)*Sin[c + d*x] + 120*B*Sin[2*(c + d*x)] + 40*A*Sin[3*(c + d*x)] + 50*C*Sin[3*(c + d*x)] + 15*B*Sin[4*(c + d*x)] + 6*C*Sin[5*(c + d*x)]))/(480*d*Cos[c + d*x]^(5/2))","A",1
307,1,92,199,0.3024513,"\int \frac{(b \cos (c+d x))^{5/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\sqrt{\cos (c+d x)}} \, dx","Integrate[((b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Sqrt[Cos[c + d*x]],x]","\frac{(b \cos (c+d x))^{5/2} (24 (A+C) \sin (2 (c+d x))+48 A c+48 A d x+72 B \sin (c+d x)+8 B \sin (3 (c+d x))+3 C \sin (4 (c+d x))+36 c C+36 C d x)}{96 d \cos ^{\frac{5}{2}}(c+d x)}","\frac{b^2 x (4 A+3 C) \sqrt{b \cos (c+d x)}}{8 \sqrt{\cos (c+d x)}}+\frac{b^2 (4 A+3 C) \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{b \cos (c+d x)}}{8 d}-\frac{b^2 B \sin ^3(c+d x) \sqrt{b \cos (c+d x)}}{3 d \sqrt{\cos (c+d x)}}+\frac{b^2 B \sin (c+d x) \sqrt{b \cos (c+d x)}}{d \sqrt{\cos (c+d x)}}+\frac{b^2 C \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) \sqrt{b \cos (c+d x)}}{4 d}",1,"((b*Cos[c + d*x])^(5/2)*(48*A*c + 36*c*C + 48*A*d*x + 36*C*d*x + 72*B*Sin[c + d*x] + 24*(A + C)*Sin[2*(c + d*x)] + 8*B*Sin[3*(c + d*x)] + 3*C*Sin[4*(c + d*x)]))/(96*d*Cos[c + d*x]^(5/2))","A",1
308,1,75,155,0.2572446,"\int \frac{(b \cos (c+d x))^{5/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[((b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(3/2),x]","\frac{(b \cos (c+d x))^{5/2} (3 (4 A+3 C) \sin (c+d x)+3 B \sin (2 (c+d x))+6 B c+6 B d x+C \sin (3 (c+d x)))}{12 d \cos ^{\frac{5}{2}}(c+d x)}","\frac{b^2 (3 A+2 C) \sin (c+d x) \sqrt{b \cos (c+d x)}}{3 d \sqrt{\cos (c+d x)}}+\frac{b^2 B x \sqrt{b \cos (c+d x)}}{2 \sqrt{\cos (c+d x)}}+\frac{b^2 B \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{b \cos (c+d x)}}{2 d}+\frac{b^2 C \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \sqrt{b \cos (c+d x)}}{3 d}",1,"((b*Cos[c + d*x])^(5/2)*(6*B*c + 6*B*d*x + 3*(4*A + 3*C)*Sin[c + d*x] + 3*B*Sin[2*(c + d*x)] + C*Sin[3*(c + d*x)]))/(12*d*Cos[c + d*x]^(5/2))","A",1
309,1,61,135,0.1521653,"\int \frac{(b \cos (c+d x))^{5/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{5}{2}}(c+d x)} \, dx","Integrate[((b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(5/2),x]","\frac{(b \cos (c+d x))^{5/2} (2 (2 A+C) (c+d x)+4 B \sin (c+d x)+C \sin (2 (c+d x)))}{4 d \cos ^{\frac{5}{2}}(c+d x)}","\frac{A b^2 x \sqrt{b \cos (c+d x)}}{\sqrt{\cos (c+d x)}}+\frac{b^2 B \sin (c+d x) \sqrt{b \cos (c+d x)}}{d \sqrt{\cos (c+d x)}}+\frac{b^2 C x \sqrt{b \cos (c+d x)}}{2 \sqrt{\cos (c+d x)}}+\frac{b^2 C \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{b \cos (c+d x)}}{2 d}",1,"((b*Cos[c + d*x])^(5/2)*(2*(2*A + C)*(c + d*x) + 4*B*Sin[c + d*x] + C*Sin[2*(c + d*x)]))/(4*d*Cos[c + d*x]^(5/2))","A",1
310,1,93,102,0.1524476,"\int \frac{(b \cos (c+d x))^{5/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{7}{2}}(c+d x)} \, dx","Integrate[((b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(7/2),x]","\frac{(b \cos (c+d x))^{5/2} \left(-A \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+A \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+B c+B d x+C \sin (c+d x)\right)}{d \cos ^{\frac{5}{2}}(c+d x)}","\frac{A b^2 \sqrt{b \cos (c+d x)} \tanh ^{-1}(\sin (c+d x))}{d \sqrt{\cos (c+d x)}}+\frac{b^2 B x \sqrt{b \cos (c+d x)}}{\sqrt{\cos (c+d x)}}+\frac{b^2 C \sin (c+d x) \sqrt{b \cos (c+d x)}}{d \sqrt{\cos (c+d x)}}",1,"((b*Cos[c + d*x])^(5/2)*(B*c + B*d*x - A*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + A*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + C*Sin[c + d*x]))/(d*Cos[c + d*x]^(5/2))","A",1
311,1,60,102,0.1243293,"\int \frac{(b \cos (c+d x))^{5/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{9}{2}}(c+d x)} \, dx","Integrate[((b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(9/2),x]","\frac{(b \cos (c+d x))^{5/2} \left(A \sin (c+d x)+B \cos (c+d x) \tanh ^{-1}(\sin (c+d x))+C d x \cos (c+d x)\right)}{d \cos ^{\frac{7}{2}}(c+d x)}","\frac{A b^2 \sin (c+d x) \sqrt{b \cos (c+d x)}}{d \cos ^{\frac{3}{2}}(c+d x)}+\frac{b^2 B \sqrt{b \cos (c+d x)} \tanh ^{-1}(\sin (c+d x))}{d \sqrt{\cos (c+d x)}}+\frac{b^2 C x \sqrt{b \cos (c+d x)}}{\sqrt{\cos (c+d x)}}",1,"((b*Cos[c + d*x])^(5/2)*(C*d*x*Cos[c + d*x] + B*ArcTanh[Sin[c + d*x]]*Cos[c + d*x] + A*Sin[c + d*x]))/(d*Cos[c + d*x]^(7/2))","A",1
312,1,69,120,0.133504,"\int \frac{(b \cos (c+d x))^{5/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{11}{2}}(c+d x)} \, dx","Integrate[((b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(11/2),x]","\frac{(b \cos (c+d x))^{5/2} \left(\sin (c+d x) (A+2 B \cos (c+d x))+(A+2 C) \cos ^2(c+d x) \tanh ^{-1}(\sin (c+d x))\right)}{2 d \cos ^{\frac{9}{2}}(c+d x)}","\frac{b^2 (A+2 C) \sqrt{b \cos (c+d x)} \tanh ^{-1}(\sin (c+d x))}{2 d \sqrt{\cos (c+d x)}}+\frac{A b^2 \sin (c+d x) \sqrt{b \cos (c+d x)}}{2 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{b^2 B \sin (c+d x) \sqrt{b \cos (c+d x)}}{d \cos ^{\frac{3}{2}}(c+d x)}",1,"((b*Cos[c + d*x])^(5/2)*((A + 2*C)*ArcTanh[Sin[c + d*x]]*Cos[c + d*x]^2 + (A + 2*B*Cos[c + d*x])*Sin[c + d*x]))/(2*d*Cos[c + d*x]^(9/2))","A",1
313,1,87,164,0.4589769,"\int \frac{(b \cos (c+d x))^{5/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{13}{2}}(c+d x)} \, dx","Integrate[((b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(13/2),x]","\frac{(b \cos (c+d x))^{5/2} \left(\tan (c+d x) ((2 A+3 C) \cos (2 (c+d x))+4 A+3 B \cos (c+d x)+3 C)+3 B \cos ^2(c+d x) \tanh ^{-1}(\sin (c+d x))\right)}{6 d \cos ^{\frac{9}{2}}(c+d x)}","\frac{b^2 (2 A+3 C) \sin (c+d x) \sqrt{b \cos (c+d x)}}{3 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{A b^2 \sin (c+d x) \sqrt{b \cos (c+d x)}}{3 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{b^2 B \sin (c+d x) \sqrt{b \cos (c+d x)}}{2 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{b^2 B \sqrt{b \cos (c+d x)} \tanh ^{-1}(\sin (c+d x))}{2 d \sqrt{\cos (c+d x)}}",1,"((b*Cos[c + d*x])^(5/2)*(3*B*ArcTanh[Sin[c + d*x]]*Cos[c + d*x]^2 + (4*A + 3*C + 3*B*Cos[c + d*x] + (2*A + 3*C)*Cos[2*(c + d*x)])*Tan[c + d*x]))/(6*d*Cos[c + d*x]^(9/2))","A",1
314,1,110,208,0.3767007,"\int \frac{(b \cos (c+d x))^{5/2} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{15}{2}}(c+d x)} \, dx","Integrate[((b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(15/2),x]","\frac{(b \cos (c+d x))^{5/2} \left(\sin (c+d x) \left(3 (3 A+4 C) \cos ^2(c+d x)+6 A+24 B \cos ^3(c+d x)+8 B \sin ^2(c+d x) \cos (c+d x)\right)+3 (3 A+4 C) \cos ^4(c+d x) \tanh ^{-1}(\sin (c+d x))\right)}{24 d \cos ^{\frac{13}{2}}(c+d x)}","\frac{b^2 (3 A+4 C) \sin (c+d x) \sqrt{b \cos (c+d x)}}{8 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{b^2 (3 A+4 C) \sqrt{b \cos (c+d x)} \tanh ^{-1}(\sin (c+d x))}{8 d \sqrt{\cos (c+d x)}}+\frac{A b^2 \sin (c+d x) \sqrt{b \cos (c+d x)}}{4 d \cos ^{\frac{9}{2}}(c+d x)}+\frac{b^2 B \sin ^3(c+d x) \sqrt{b \cos (c+d x)}}{3 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{b^2 B \sin (c+d x) \sqrt{b \cos (c+d x)}}{d \cos ^{\frac{3}{2}}(c+d x)}",1,"((b*Cos[c + d*x])^(5/2)*(3*(3*A + 4*C)*ArcTanh[Sin[c + d*x]]*Cos[c + d*x]^4 + Sin[c + d*x]*(6*A + 3*(3*A + 4*C)*Cos[c + d*x]^2 + 24*B*Cos[c + d*x]^3 + 8*B*Cos[c + d*x]*Sin[c + d*x]^2)))/(24*d*Cos[c + d*x]^(13/2))","A",1
315,1,92,184,0.2378871,"\int \frac{\cos ^{\frac{5}{2}}(c+d x) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\sqrt{b \cos (c+d x)}} \, dx","Integrate[(Cos[c + d*x]^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Sqrt[b*Cos[c + d*x]],x]","\frac{\sqrt{\cos (c+d x)} (24 (A+C) \sin (2 (c+d x))+48 A c+48 A d x+72 B \sin (c+d x)+8 B \sin (3 (c+d x))+3 C \sin (4 (c+d x))+36 c C+36 C d x)}{96 d \sqrt{b \cos (c+d x)}}","\frac{x (4 A+3 C) \sqrt{\cos (c+d x)}}{8 \sqrt{b \cos (c+d x)}}+\frac{(4 A+3 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{8 d \sqrt{b \cos (c+d x)}}-\frac{B \sin ^3(c+d x) \sqrt{\cos (c+d x)}}{3 d \sqrt{b \cos (c+d x)}}+\frac{B \sin (c+d x) \sqrt{\cos (c+d x)}}{d \sqrt{b \cos (c+d x)}}+\frac{C \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x)}{4 d \sqrt{b \cos (c+d x)}}",1,"(Sqrt[Cos[c + d*x]]*(48*A*c + 36*c*C + 48*A*d*x + 36*C*d*x + 72*B*Sin[c + d*x] + 24*(A + C)*Sin[2*(c + d*x)] + 8*B*Sin[3*(c + d*x)] + 3*C*Sin[4*(c + d*x)]))/(96*d*Sqrt[b*Cos[c + d*x]])","A",1
316,1,75,143,0.2020529,"\int \frac{\cos ^{\frac{3}{2}}(c+d x) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\sqrt{b \cos (c+d x)}} \, dx","Integrate[(Cos[c + d*x]^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Sqrt[b*Cos[c + d*x]],x]","\frac{\sqrt{\cos (c+d x)} (3 (4 A+3 C) \sin (c+d x)+3 B \sin (2 (c+d x))+6 B c+6 B d x+C \sin (3 (c+d x)))}{12 d \sqrt{b \cos (c+d x)}}","\frac{(3 A+2 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{3 d \sqrt{b \cos (c+d x)}}+\frac{B x \sqrt{\cos (c+d x)}}{2 \sqrt{b \cos (c+d x)}}+\frac{B \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{2 d \sqrt{b \cos (c+d x)}}+\frac{C \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{3 d \sqrt{b \cos (c+d x)}}",1,"(Sqrt[Cos[c + d*x]]*(6*B*c + 6*B*d*x + 3*(4*A + 3*C)*Sin[c + d*x] + 3*B*Sin[2*(c + d*x)] + C*Sin[3*(c + d*x)]))/(12*d*Sqrt[b*Cos[c + d*x]])","A",1
317,1,61,123,0.0975641,"\int \frac{\sqrt{\cos (c+d x)} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\sqrt{b \cos (c+d x)}} \, dx","Integrate[(Sqrt[Cos[c + d*x]]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Sqrt[b*Cos[c + d*x]],x]","\frac{\sqrt{\cos (c+d x)} (2 (2 A+C) (c+d x)+4 B \sin (c+d x)+C \sin (2 (c+d x)))}{4 d \sqrt{b \cos (c+d x)}}","\frac{A x \sqrt{\cos (c+d x)}}{\sqrt{b \cos (c+d x)}}+\frac{B \sin (c+d x) \sqrt{\cos (c+d x)}}{d \sqrt{b \cos (c+d x)}}+\frac{C x \sqrt{\cos (c+d x)}}{2 \sqrt{b \cos (c+d x)}}+\frac{C \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{2 d \sqrt{b \cos (c+d x)}}",1,"(Sqrt[Cos[c + d*x]]*(2*(2*A + C)*(c + d*x) + 4*B*Sin[c + d*x] + C*Sin[2*(c + d*x)]))/(4*d*Sqrt[b*Cos[c + d*x]])","A",1
318,1,93,93,0.110295,"\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{\sqrt{\cos (c+d x)} \sqrt{b \cos (c+d x)}} \, dx","Integrate[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Sqrt[Cos[c + d*x]]*Sqrt[b*Cos[c + d*x]]),x]","\frac{\sqrt{\cos (c+d x)} \left(-A \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+A \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+B c+B d x+C \sin (c+d x)\right)}{d \sqrt{b \cos (c+d x)}}","\frac{A \sqrt{\cos (c+d x)} \tanh ^{-1}(\sin (c+d x))}{d \sqrt{b \cos (c+d x)}}+\frac{B x \sqrt{\cos (c+d x)}}{\sqrt{b \cos (c+d x)}}+\frac{C \sin (c+d x) \sqrt{\cos (c+d x)}}{d \sqrt{b \cos (c+d x)}}",1,"(Sqrt[Cos[c + d*x]]*(B*c + B*d*x - A*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + A*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + C*Sin[c + d*x]))/(d*Sqrt[b*Cos[c + d*x]])","A",1
319,1,60,93,0.071501,"\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{\cos ^{\frac{3}{2}}(c+d x) \sqrt{b \cos (c+d x)}} \, dx","Integrate[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(3/2)*Sqrt[b*Cos[c + d*x]]),x]","\frac{A \sin (c+d x)+B \cos (c+d x) \tanh ^{-1}(\sin (c+d x))+C d x \cos (c+d x)}{d \sqrt{\cos (c+d x)} \sqrt{b \cos (c+d x)}}","\frac{A \sin (c+d x)}{d \sqrt{\cos (c+d x)} \sqrt{b \cos (c+d x)}}+\frac{B \sqrt{\cos (c+d x)} \tanh ^{-1}(\sin (c+d x))}{d \sqrt{b \cos (c+d x)}}+\frac{C x \sqrt{\cos (c+d x)}}{\sqrt{b \cos (c+d x)}}",1,"(C*d*x*Cos[c + d*x] + B*ArcTanh[Sin[c + d*x]]*Cos[c + d*x] + A*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]*Sqrt[b*Cos[c + d*x]])","A",1
320,1,69,111,0.0920009,"\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{\cos ^{\frac{5}{2}}(c+d x) \sqrt{b \cos (c+d x)}} \, dx","Integrate[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(5/2)*Sqrt[b*Cos[c + d*x]]),x]","\frac{\sin (c+d x) (A+2 B \cos (c+d x))+(A+2 C) \cos ^2(c+d x) \tanh ^{-1}(\sin (c+d x))}{2 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{b \cos (c+d x)}}","\frac{(A+2 C) \sqrt{\cos (c+d x)} \tanh ^{-1}(\sin (c+d x))}{2 d \sqrt{b \cos (c+d x)}}+\frac{A \sin (c+d x)}{2 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{b \cos (c+d x)}}+\frac{B \sin (c+d x)}{d \sqrt{\cos (c+d x)} \sqrt{b \cos (c+d x)}}",1,"((A + 2*C)*ArcTanh[Sin[c + d*x]]*Cos[c + d*x]^2 + (A + 2*B*Cos[c + d*x])*Sin[c + d*x])/(2*d*Cos[c + d*x]^(3/2)*Sqrt[b*Cos[c + d*x]])","A",1
321,1,87,152,0.1938319,"\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{\cos ^{\frac{7}{2}}(c+d x) \sqrt{b \cos (c+d x)}} \, dx","Integrate[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(7/2)*Sqrt[b*Cos[c + d*x]]),x]","\frac{\tan (c+d x) ((2 A+3 C) \cos (2 (c+d x))+4 A+3 B \cos (c+d x)+3 C)+3 B \cos ^2(c+d x) \tanh ^{-1}(\sin (c+d x))}{6 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{b \cos (c+d x)}}","\frac{(2 A+3 C) \sin (c+d x)}{3 d \sqrt{\cos (c+d x)} \sqrt{b \cos (c+d x)}}+\frac{A \sin (c+d x)}{3 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{b \cos (c+d x)}}+\frac{B \sin (c+d x)}{2 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{b \cos (c+d x)}}+\frac{B \sqrt{\cos (c+d x)} \tanh ^{-1}(\sin (c+d x))}{2 d \sqrt{b \cos (c+d x)}}",1,"(3*B*ArcTanh[Sin[c + d*x]]*Cos[c + d*x]^2 + (4*A + 3*C + 3*B*Cos[c + d*x] + (2*A + 3*C)*Cos[2*(c + d*x)])*Tan[c + d*x])/(6*d*Cos[c + d*x]^(3/2)*Sqrt[b*Cos[c + d*x]])","A",1
322,1,110,193,0.2411265,"\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{\cos ^{\frac{9}{2}}(c+d x) \sqrt{b \cos (c+d x)}} \, dx","Integrate[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(9/2)*Sqrt[b*Cos[c + d*x]]),x]","\frac{\sin (c+d x) \left(3 (3 A+4 C) \cos ^2(c+d x)+6 A+24 B \cos ^3(c+d x)+8 B \sin ^2(c+d x) \cos (c+d x)\right)+3 (3 A+4 C) \cos ^4(c+d x) \tanh ^{-1}(\sin (c+d x))}{24 d \cos ^{\frac{7}{2}}(c+d x) \sqrt{b \cos (c+d x)}}","\frac{(3 A+4 C) \sin (c+d x)}{8 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{b \cos (c+d x)}}+\frac{(3 A+4 C) \sqrt{\cos (c+d x)} \tanh ^{-1}(\sin (c+d x))}{8 d \sqrt{b \cos (c+d x)}}+\frac{A \sin (c+d x)}{4 d \cos ^{\frac{7}{2}}(c+d x) \sqrt{b \cos (c+d x)}}+\frac{B \sin ^3(c+d x)}{3 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{b \cos (c+d x)}}+\frac{B \sin (c+d x)}{d \sqrt{\cos (c+d x)} \sqrt{b \cos (c+d x)}}",1,"(3*(3*A + 4*C)*ArcTanh[Sin[c + d*x]]*Cos[c + d*x]^4 + Sin[c + d*x]*(6*A + 3*(3*A + 4*C)*Cos[c + d*x]^2 + 24*B*Cos[c + d*x]^3 + 8*B*Cos[c + d*x]*Sin[c + d*x]^2))/(24*d*Cos[c + d*x]^(7/2)*Sqrt[b*Cos[c + d*x]])","A",1
323,1,92,199,0.1719085,"\int \frac{\cos ^{\frac{7}{2}}(c+d x) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{(b \cos (c+d x))^{3/2}} \, dx","Integrate[(Cos[c + d*x]^(7/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(b*Cos[c + d*x])^(3/2),x]","\frac{\cos ^{\frac{3}{2}}(c+d x) (24 (A+C) \sin (2 (c+d x))+48 A c+48 A d x+72 B \sin (c+d x)+8 B \sin (3 (c+d x))+3 C \sin (4 (c+d x))+36 c C+36 C d x)}{96 d (b \cos (c+d x))^{3/2}}","\frac{x (4 A+3 C) \sqrt{\cos (c+d x)}}{8 b \sqrt{b \cos (c+d x)}}+\frac{(4 A+3 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{8 b d \sqrt{b \cos (c+d x)}}-\frac{B \sin ^3(c+d x) \sqrt{\cos (c+d x)}}{3 b d \sqrt{b \cos (c+d x)}}+\frac{B \sin (c+d x) \sqrt{\cos (c+d x)}}{b d \sqrt{b \cos (c+d x)}}+\frac{C \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x)}{4 b d \sqrt{b \cos (c+d x)}}",1,"(Cos[c + d*x]^(3/2)*(48*A*c + 36*c*C + 48*A*d*x + 36*C*d*x + 72*B*Sin[c + d*x] + 24*(A + C)*Sin[2*(c + d*x)] + 8*B*Sin[3*(c + d*x)] + 3*C*Sin[4*(c + d*x)]))/(96*d*(b*Cos[c + d*x])^(3/2))","A",1
324,1,75,155,0.1794502,"\int \frac{\cos ^{\frac{5}{2}}(c+d x) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{(b \cos (c+d x))^{3/2}} \, dx","Integrate[(Cos[c + d*x]^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(b*Cos[c + d*x])^(3/2),x]","\frac{\cos ^{\frac{3}{2}}(c+d x) (3 (4 A+3 C) \sin (c+d x)+3 B \sin (2 (c+d x))+6 B c+6 B d x+C \sin (3 (c+d x)))}{12 d (b \cos (c+d x))^{3/2}}","\frac{(3 A+2 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{3 b d \sqrt{b \cos (c+d x)}}+\frac{B x \sqrt{\cos (c+d x)}}{2 b \sqrt{b \cos (c+d x)}}+\frac{B \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{2 b d \sqrt{b \cos (c+d x)}}+\frac{C \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{3 b d \sqrt{b \cos (c+d x)}}",1,"(Cos[c + d*x]^(3/2)*(6*B*c + 6*B*d*x + 3*(4*A + 3*C)*Sin[c + d*x] + 3*B*Sin[2*(c + d*x)] + C*Sin[3*(c + d*x)]))/(12*d*(b*Cos[c + d*x])^(3/2))","A",1
325,1,61,135,0.1131916,"\int \frac{\cos ^{\frac{3}{2}}(c+d x) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{(b \cos (c+d x))^{3/2}} \, dx","Integrate[(Cos[c + d*x]^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(b*Cos[c + d*x])^(3/2),x]","\frac{\cos ^{\frac{3}{2}}(c+d x) (2 (2 A+C) (c+d x)+4 B \sin (c+d x)+C \sin (2 (c+d x)))}{4 d (b \cos (c+d x))^{3/2}}","\frac{A x \sqrt{\cos (c+d x)}}{b \sqrt{b \cos (c+d x)}}+\frac{B \sin (c+d x) \sqrt{\cos (c+d x)}}{b d \sqrt{b \cos (c+d x)}}+\frac{C x \sqrt{\cos (c+d x)}}{2 b \sqrt{b \cos (c+d x)}}+\frac{C \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{2 b d \sqrt{b \cos (c+d x)}}",1,"(Cos[c + d*x]^(3/2)*(2*(2*A + C)*(c + d*x) + 4*B*Sin[c + d*x] + C*Sin[2*(c + d*x)]))/(4*d*(b*Cos[c + d*x])^(3/2))","A",1
326,1,93,102,0.1117058,"\int \frac{\sqrt{\cos (c+d x)} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{(b \cos (c+d x))^{3/2}} \, dx","Integrate[(Sqrt[Cos[c + d*x]]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(b*Cos[c + d*x])^(3/2),x]","\frac{\cos ^{\frac{3}{2}}(c+d x) \left(-A \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+A \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+B c+B d x+C \sin (c+d x)\right)}{d (b \cos (c+d x))^{3/2}}","\frac{A \sqrt{\cos (c+d x)} \tanh ^{-1}(\sin (c+d x))}{b d \sqrt{b \cos (c+d x)}}+\frac{B x \sqrt{\cos (c+d x)}}{b \sqrt{b \cos (c+d x)}}+\frac{C \sin (c+d x) \sqrt{\cos (c+d x)}}{b d \sqrt{b \cos (c+d x)}}",1,"(Cos[c + d*x]^(3/2)*(B*c + B*d*x - A*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + A*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + C*Sin[c + d*x]))/(d*(b*Cos[c + d*x])^(3/2))","A",1
327,1,60,102,0.0751512,"\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{\sqrt{\cos (c+d x)} (b \cos (c+d x))^{3/2}} \, dx","Integrate[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Sqrt[Cos[c + d*x]]*(b*Cos[c + d*x])^(3/2)),x]","\frac{\sqrt{\cos (c+d x)} \left(A \sin (c+d x)+B \cos (c+d x) \tanh ^{-1}(\sin (c+d x))+C d x \cos (c+d x)\right)}{d (b \cos (c+d x))^{3/2}}","\frac{A \sin (c+d x)}{b d \sqrt{\cos (c+d x)} \sqrt{b \cos (c+d x)}}+\frac{B \sqrt{\cos (c+d x)} \tanh ^{-1}(\sin (c+d x))}{b d \sqrt{b \cos (c+d x)}}+\frac{C x \sqrt{\cos (c+d x)}}{b \sqrt{b \cos (c+d x)}}",1,"(Sqrt[Cos[c + d*x]]*(C*d*x*Cos[c + d*x] + B*ArcTanh[Sin[c + d*x]]*Cos[c + d*x] + A*Sin[c + d*x]))/(d*(b*Cos[c + d*x])^(3/2))","A",1
328,1,69,120,0.097747,"\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{\cos ^{\frac{3}{2}}(c+d x) (b \cos (c+d x))^{3/2}} \, dx","Integrate[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(3/2)*(b*Cos[c + d*x])^(3/2)),x]","\frac{\sin (c+d x) (A+2 B \cos (c+d x))+(A+2 C) \cos ^2(c+d x) \tanh ^{-1}(\sin (c+d x))}{2 d \sqrt{\cos (c+d x)} (b \cos (c+d x))^{3/2}}","\frac{(A+2 C) \sqrt{\cos (c+d x)} \tanh ^{-1}(\sin (c+d x))}{2 b d \sqrt{b \cos (c+d x)}}+\frac{A \sin (c+d x)}{2 b d \cos ^{\frac{3}{2}}(c+d x) \sqrt{b \cos (c+d x)}}+\frac{B \sin (c+d x)}{b d \sqrt{\cos (c+d x)} \sqrt{b \cos (c+d x)}}",1,"((A + 2*C)*ArcTanh[Sin[c + d*x]]*Cos[c + d*x]^2 + (A + 2*B*Cos[c + d*x])*Sin[c + d*x])/(2*d*Sqrt[Cos[c + d*x]]*(b*Cos[c + d*x])^(3/2))","A",1
329,1,87,164,0.1665439,"\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{\cos ^{\frac{5}{2}}(c+d x) (b \cos (c+d x))^{3/2}} \, dx","Integrate[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(5/2)*(b*Cos[c + d*x])^(3/2)),x]","\frac{\tan (c+d x) ((2 A+3 C) \cos (2 (c+d x))+4 A+3 B \cos (c+d x)+3 C)+3 B \cos ^2(c+d x) \tanh ^{-1}(\sin (c+d x))}{6 d \sqrt{\cos (c+d x)} (b \cos (c+d x))^{3/2}}","\frac{(2 A+3 C) \sin (c+d x)}{3 b d \sqrt{\cos (c+d x)} \sqrt{b \cos (c+d x)}}+\frac{A \sin (c+d x)}{3 b d \cos ^{\frac{5}{2}}(c+d x) \sqrt{b \cos (c+d x)}}+\frac{B \sin (c+d x)}{2 b d \cos ^{\frac{3}{2}}(c+d x) \sqrt{b \cos (c+d x)}}+\frac{B \sqrt{\cos (c+d x)} \tanh ^{-1}(\sin (c+d x))}{2 b d \sqrt{b \cos (c+d x)}}",1,"(3*B*ArcTanh[Sin[c + d*x]]*Cos[c + d*x]^2 + (4*A + 3*C + 3*B*Cos[c + d*x] + (2*A + 3*C)*Cos[2*(c + d*x)])*Tan[c + d*x])/(6*d*Sqrt[Cos[c + d*x]]*(b*Cos[c + d*x])^(3/2))","A",1
330,1,110,208,0.169449,"\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{\cos ^{\frac{7}{2}}(c+d x) (b \cos (c+d x))^{3/2}} \, dx","Integrate[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(7/2)*(b*Cos[c + d*x])^(3/2)),x]","\frac{\sin (c+d x) \left(3 (3 A+4 C) \cos ^2(c+d x)+6 A+24 B \cos ^3(c+d x)+8 B \sin ^2(c+d x) \cos (c+d x)\right)+3 (3 A+4 C) \cos ^4(c+d x) \tanh ^{-1}(\sin (c+d x))}{24 d \cos ^{\frac{5}{2}}(c+d x) (b \cos (c+d x))^{3/2}}","\frac{(3 A+4 C) \sin (c+d x)}{8 b d \cos ^{\frac{3}{2}}(c+d x) \sqrt{b \cos (c+d x)}}+\frac{(3 A+4 C) \sqrt{\cos (c+d x)} \tanh ^{-1}(\sin (c+d x))}{8 b d \sqrt{b \cos (c+d x)}}+\frac{A \sin (c+d x)}{4 b d \cos ^{\frac{7}{2}}(c+d x) \sqrt{b \cos (c+d x)}}+\frac{B \sin ^3(c+d x)}{3 b d \cos ^{\frac{5}{2}}(c+d x) \sqrt{b \cos (c+d x)}}+\frac{B \sin (c+d x)}{b d \sqrt{\cos (c+d x)} \sqrt{b \cos (c+d x)}}",1,"(3*(3*A + 4*C)*ArcTanh[Sin[c + d*x]]*Cos[c + d*x]^4 + Sin[c + d*x]*(6*A + 3*(3*A + 4*C)*Cos[c + d*x]^2 + 24*B*Cos[c + d*x]^3 + 8*B*Cos[c + d*x]*Sin[c + d*x]^2))/(24*d*Cos[c + d*x]^(5/2)*(b*Cos[c + d*x])^(3/2))","A",1
331,1,95,199,0.1327932,"\int \frac{\cos ^{\frac{9}{2}}(c+d x) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{(b \cos (c+d x))^{5/2}} \, dx","Integrate[(Cos[c + d*x]^(9/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(b*Cos[c + d*x])^(5/2),x]","\frac{\sqrt{\cos (c+d x)} (24 (A+C) \sin (2 (c+d x))+48 A c+48 A d x+72 B \sin (c+d x)+8 B \sin (3 (c+d x))+3 C \sin (4 (c+d x))+36 c C+36 C d x)}{96 b^2 d \sqrt{b \cos (c+d x)}}","\frac{x (4 A+3 C) \sqrt{\cos (c+d x)}}{8 b^2 \sqrt{b \cos (c+d x)}}+\frac{(4 A+3 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{8 b^2 d \sqrt{b \cos (c+d x)}}-\frac{B \sin ^3(c+d x) \sqrt{\cos (c+d x)}}{3 b^2 d \sqrt{b \cos (c+d x)}}+\frac{B \sin (c+d x) \sqrt{\cos (c+d x)}}{b^2 d \sqrt{b \cos (c+d x)}}+\frac{C \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x)}{4 b^2 d \sqrt{b \cos (c+d x)}}",1,"(Sqrt[Cos[c + d*x]]*(48*A*c + 36*c*C + 48*A*d*x + 36*C*d*x + 72*B*Sin[c + d*x] + 24*(A + C)*Sin[2*(c + d*x)] + 8*B*Sin[3*(c + d*x)] + 3*C*Sin[4*(c + d*x)]))/(96*b^2*d*Sqrt[b*Cos[c + d*x]])","A",1
332,1,78,155,0.1134821,"\int \frac{\cos ^{\frac{7}{2}}(c+d x) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{(b \cos (c+d x))^{5/2}} \, dx","Integrate[(Cos[c + d*x]^(7/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(b*Cos[c + d*x])^(5/2),x]","\frac{\sqrt{\cos (c+d x)} (3 (4 A+3 C) \sin (c+d x)+3 B \sin (2 (c+d x))+6 B c+6 B d x+C \sin (3 (c+d x)))}{12 b^2 d \sqrt{b \cos (c+d x)}}","\frac{(3 A+2 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{3 b^2 d \sqrt{b \cos (c+d x)}}+\frac{B x \sqrt{\cos (c+d x)}}{2 b^2 \sqrt{b \cos (c+d x)}}+\frac{B \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{2 b^2 d \sqrt{b \cos (c+d x)}}+\frac{C \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{3 b^2 d \sqrt{b \cos (c+d x)}}",1,"(Sqrt[Cos[c + d*x]]*(6*B*c + 6*B*d*x + 3*(4*A + 3*C)*Sin[c + d*x] + 3*B*Sin[2*(c + d*x)] + C*Sin[3*(c + d*x)]))/(12*b^2*d*Sqrt[b*Cos[c + d*x]])","A",1
333,1,64,135,0.0821213,"\int \frac{\cos ^{\frac{5}{2}}(c+d x) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{(b \cos (c+d x))^{5/2}} \, dx","Integrate[(Cos[c + d*x]^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(b*Cos[c + d*x])^(5/2),x]","\frac{\sqrt{\cos (c+d x)} (2 (2 A+C) (c+d x)+4 B \sin (c+d x)+C \sin (2 (c+d x)))}{4 b^2 d \sqrt{b \cos (c+d x)}}","\frac{A x \sqrt{\cos (c+d x)}}{b^2 \sqrt{b \cos (c+d x)}}+\frac{B \sin (c+d x) \sqrt{\cos (c+d x)}}{b^2 d \sqrt{b \cos (c+d x)}}+\frac{C x \sqrt{\cos (c+d x)}}{2 b^2 \sqrt{b \cos (c+d x)}}+\frac{C \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{2 b^2 d \sqrt{b \cos (c+d x)}}",1,"(Sqrt[Cos[c + d*x]]*(2*(2*A + C)*(c + d*x) + 4*B*Sin[c + d*x] + C*Sin[2*(c + d*x)]))/(4*b^2*d*Sqrt[b*Cos[c + d*x]])","A",1
334,1,96,102,0.0958325,"\int \frac{\cos ^{\frac{3}{2}}(c+d x) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{(b \cos (c+d x))^{5/2}} \, dx","Integrate[(Cos[c + d*x]^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(b*Cos[c + d*x])^(5/2),x]","\frac{\sqrt{\cos (c+d x)} \left(-A \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+A \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+B c+B d x+C \sin (c+d x)\right)}{b^2 d \sqrt{b \cos (c+d x)}}","\frac{A \sqrt{\cos (c+d x)} \tanh ^{-1}(\sin (c+d x))}{b^2 d \sqrt{b \cos (c+d x)}}+\frac{B x \sqrt{\cos (c+d x)}}{b^2 \sqrt{b \cos (c+d x)}}+\frac{C \sin (c+d x) \sqrt{\cos (c+d x)}}{b^2 d \sqrt{b \cos (c+d x)}}",1,"(Sqrt[Cos[c + d*x]]*(B*c + B*d*x - A*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + A*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + C*Sin[c + d*x]))/(b^2*d*Sqrt[b*Cos[c + d*x]])","A",1
335,1,60,102,0.0655955,"\int \frac{\sqrt{\cos (c+d x)} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{(b \cos (c+d x))^{5/2}} \, dx","Integrate[(Sqrt[Cos[c + d*x]]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(b*Cos[c + d*x])^(5/2),x]","\frac{\cos ^{\frac{3}{2}}(c+d x) \left(A \sin (c+d x)+B \cos (c+d x) \tanh ^{-1}(\sin (c+d x))+C d x \cos (c+d x)\right)}{d (b \cos (c+d x))^{5/2}}","\frac{A \sin (c+d x)}{b^2 d \sqrt{\cos (c+d x)} \sqrt{b \cos (c+d x)}}+\frac{B \sqrt{\cos (c+d x)} \tanh ^{-1}(\sin (c+d x))}{b^2 d \sqrt{b \cos (c+d x)}}+\frac{C x \sqrt{\cos (c+d x)}}{b^2 \sqrt{b \cos (c+d x)}}",1,"(Cos[c + d*x]^(3/2)*(C*d*x*Cos[c + d*x] + B*ArcTanh[Sin[c + d*x]]*Cos[c + d*x] + A*Sin[c + d*x]))/(d*(b*Cos[c + d*x])^(5/2))","A",1
336,1,69,120,0.0963124,"\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{\sqrt{\cos (c+d x)} (b \cos (c+d x))^{5/2}} \, dx","Integrate[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Sqrt[Cos[c + d*x]]*(b*Cos[c + d*x])^(5/2)),x]","\frac{\sqrt{\cos (c+d x)} \left(\sin (c+d x) (A+2 B \cos (c+d x))+(A+2 C) \cos ^2(c+d x) \tanh ^{-1}(\sin (c+d x))\right)}{2 d (b \cos (c+d x))^{5/2}}","\frac{(A+2 C) \sqrt{\cos (c+d x)} \tanh ^{-1}(\sin (c+d x))}{2 b^2 d \sqrt{b \cos (c+d x)}}+\frac{A \sin (c+d x)}{2 b^2 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{b \cos (c+d x)}}+\frac{B \sin (c+d x)}{b^2 d \sqrt{\cos (c+d x)} \sqrt{b \cos (c+d x)}}",1,"(Sqrt[Cos[c + d*x]]*((A + 2*C)*ArcTanh[Sin[c + d*x]]*Cos[c + d*x]^2 + (A + 2*B*Cos[c + d*x])*Sin[c + d*x]))/(2*d*(b*Cos[c + d*x])^(5/2))","A",1
337,1,87,164,0.3019373,"\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{\cos ^{\frac{3}{2}}(c+d x) (b \cos (c+d x))^{5/2}} \, dx","Integrate[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(3/2)*(b*Cos[c + d*x])^(5/2)),x]","\frac{\sqrt{\cos (c+d x)} \left(\tan (c+d x) ((2 A+3 C) \cos (2 (c+d x))+4 A+3 B \cos (c+d x)+3 C)+3 B \cos ^2(c+d x) \tanh ^{-1}(\sin (c+d x))\right)}{6 d (b \cos (c+d x))^{5/2}}","\frac{(2 A+3 C) \sin (c+d x)}{3 b^2 d \sqrt{\cos (c+d x)} \sqrt{b \cos (c+d x)}}+\frac{A \sin (c+d x)}{3 b^2 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{b \cos (c+d x)}}+\frac{B \sin (c+d x)}{2 b^2 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{b \cos (c+d x)}}+\frac{B \sqrt{\cos (c+d x)} \tanh ^{-1}(\sin (c+d x))}{2 b^2 d \sqrt{b \cos (c+d x)}}",1,"(Sqrt[Cos[c + d*x]]*(3*B*ArcTanh[Sin[c + d*x]]*Cos[c + d*x]^2 + (4*A + 3*C + 3*B*Cos[c + d*x] + (2*A + 3*C)*Cos[2*(c + d*x)])*Tan[c + d*x]))/(6*d*(b*Cos[c + d*x])^(5/2))","A",1
338,1,110,208,0.2246401,"\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{\cos ^{\frac{5}{2}}(c+d x) (b \cos (c+d x))^{5/2}} \, dx","Integrate[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(Cos[c + d*x]^(5/2)*(b*Cos[c + d*x])^(5/2)),x]","\frac{\sin (c+d x) \left(3 (3 A+4 C) \cos ^2(c+d x)+6 A+24 B \cos ^3(c+d x)+8 B \sin ^2(c+d x) \cos (c+d x)\right)+3 (3 A+4 C) \cos ^4(c+d x) \tanh ^{-1}(\sin (c+d x))}{24 d \cos ^{\frac{3}{2}}(c+d x) (b \cos (c+d x))^{5/2}}","\frac{(3 A+4 C) \sin (c+d x)}{8 b^2 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{b \cos (c+d x)}}+\frac{(3 A+4 C) \sqrt{\cos (c+d x)} \tanh ^{-1}(\sin (c+d x))}{8 b^2 d \sqrt{b \cos (c+d x)}}+\frac{A \sin (c+d x)}{4 b^2 d \cos ^{\frac{7}{2}}(c+d x) \sqrt{b \cos (c+d x)}}+\frac{B \sin ^3(c+d x)}{3 b^2 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{b \cos (c+d x)}}+\frac{B \sin (c+d x)}{b^2 d \sqrt{\cos (c+d x)} \sqrt{b \cos (c+d x)}}",1,"(3*(3*A + 4*C)*ArcTanh[Sin[c + d*x]]*Cos[c + d*x]^4 + Sin[c + d*x]*(6*A + 3*(3*A + 4*C)*Cos[c + d*x]^2 + 24*B*Cos[c + d*x]^3 + 8*B*Cos[c + d*x]*Sin[c + d*x]^2))/(24*d*Cos[c + d*x]^(3/2)*(b*Cos[c + d*x])^(5/2))","A",1
339,1,109,154,0.2538772,"\int \cos (c+d x) (b \cos (c+d x))^{2/3} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]*(b*Cos[c + d*x])^(2/3)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","-\frac{3 \sin (c+d x) (b \cos (c+d x))^{8/3} \left((11 A+8 C) \, _2F_1\left(\frac{1}{2},\frac{4}{3};\frac{7}{3};\cos ^2(c+d x)\right)+8 B \cos (c+d x) \, _2F_1\left(\frac{1}{2},\frac{11}{6};\frac{17}{6};\cos ^2(c+d x)\right)-8 C \sqrt{\sin ^2(c+d x)}\right)}{88 b^2 d \sqrt{\sin ^2(c+d x)}}","-\frac{3 (11 A+8 C) \sin (c+d x) (b \cos (c+d x))^{8/3} \, _2F_1\left(\frac{1}{2},\frac{4}{3};\frac{7}{3};\cos ^2(c+d x)\right)}{88 b^2 d \sqrt{\sin ^2(c+d x)}}-\frac{3 B \sin (c+d x) (b \cos (c+d x))^{11/3} \, _2F_1\left(\frac{1}{2},\frac{11}{6};\frac{17}{6};\cos ^2(c+d x)\right)}{11 b^3 d \sqrt{\sin ^2(c+d x)}}+\frac{3 C \sin (c+d x) (b \cos (c+d x))^{8/3}}{11 b^2 d}",1,"(-3*(b*Cos[c + d*x])^(8/3)*Sin[c + d*x]*((11*A + 8*C)*Hypergeometric2F1[1/2, 4/3, 7/3, Cos[c + d*x]^2] + 8*B*Cos[c + d*x]*Hypergeometric2F1[1/2, 11/6, 17/6, Cos[c + d*x]^2] - 8*C*Sqrt[Sin[c + d*x]^2]))/(88*b^2*d*Sqrt[Sin[c + d*x]^2])","A",1
340,1,109,154,0.2276649,"\int (b \cos (c+d x))^{2/3} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Integrate[(b*Cos[c + d*x])^(2/3)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","-\frac{3 \sin (c+d x) (b \cos (c+d x))^{5/3} \left((8 A+5 C) \, _2F_1\left(\frac{1}{2},\frac{5}{6};\frac{11}{6};\cos ^2(c+d x)\right)+5 B \cos (c+d x) \, _2F_1\left(\frac{1}{2},\frac{4}{3};\frac{7}{3};\cos ^2(c+d x)\right)-5 C \sqrt{\sin ^2(c+d x)}\right)}{40 b d \sqrt{\sin ^2(c+d x)}}","-\frac{3 (8 A+5 C) \sin (c+d x) (b \cos (c+d x))^{5/3} \, _2F_1\left(\frac{1}{2},\frac{5}{6};\frac{11}{6};\cos ^2(c+d x)\right)}{40 b d \sqrt{\sin ^2(c+d x)}}-\frac{3 B \sin (c+d x) (b \cos (c+d x))^{8/3} \, _2F_1\left(\frac{1}{2},\frac{4}{3};\frac{7}{3};\cos ^2(c+d x)\right)}{8 b^2 d \sqrt{\sin ^2(c+d x)}}+\frac{3 C \sin (c+d x) (b \cos (c+d x))^{5/3}}{8 b d}",1,"(-3*(b*Cos[c + d*x])^(5/3)*Sin[c + d*x]*((8*A + 5*C)*Hypergeometric2F1[1/2, 5/6, 11/6, Cos[c + d*x]^2] + 5*B*Cos[c + d*x]*Hypergeometric2F1[1/2, 4/3, 7/3, Cos[c + d*x]^2] - 5*C*Sqrt[Sin[c + d*x]^2]))/(40*b*d*Sqrt[Sin[c + d*x]^2])","A",1
341,1,109,148,0.2271375,"\int (b \cos (c+d x))^{2/3} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec (c+d x) \, dx","Integrate[(b*Cos[c + d*x])^(2/3)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x],x]","-\frac{3 b \sin (2 (c+d x)) \left((5 A+2 C) \, _2F_1\left(\frac{1}{3},\frac{1}{2};\frac{4}{3};\cos ^2(c+d x)\right)+2 B \cos (c+d x) \, _2F_1\left(\frac{1}{2},\frac{5}{6};\frac{11}{6};\cos ^2(c+d x)\right)-2 C \sqrt{\sin ^2(c+d x)}\right)}{20 d \sqrt{\sin ^2(c+d x)} \sqrt[3]{b \cos (c+d x)}}","-\frac{3 (5 A+2 C) \sin (c+d x) (b \cos (c+d x))^{2/3} \, _2F_1\left(\frac{1}{3},\frac{1}{2};\frac{4}{3};\cos ^2(c+d x)\right)}{10 d \sqrt{\sin ^2(c+d x)}}-\frac{3 B \sin (c+d x) (b \cos (c+d x))^{5/3} \, _2F_1\left(\frac{1}{2},\frac{5}{6};\frac{11}{6};\cos ^2(c+d x)\right)}{5 b d \sqrt{\sin ^2(c+d x)}}+\frac{3 C \sin (c+d x) (b \cos (c+d x))^{2/3}}{5 d}",1,"(-3*b*((5*A + 2*C)*Hypergeometric2F1[1/3, 1/2, 4/3, Cos[c + d*x]^2] + 2*B*Cos[c + d*x]*Hypergeometric2F1[1/2, 5/6, 11/6, Cos[c + d*x]^2] - 2*C*Sqrt[Sin[c + d*x]^2])*Sin[2*(c + d*x)])/(20*d*(b*Cos[c + d*x])^(1/3)*Sqrt[Sin[c + d*x]^2])","A",1
342,1,116,147,0.3573286,"\int (b \cos (c+d x))^{2/3} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^2(c+d x) \, dx","Integrate[(b*Cos[c + d*x])^(2/3)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^2,x]","-\frac{3 b \sqrt{\sin ^2(c+d x)} \left(\cot (c+d x) \left(5 B \, _2F_1\left(\frac{1}{3},\frac{1}{2};\frac{4}{3};\cos ^2(c+d x)\right)+2 C \cos (c+d x) \, _2F_1\left(\frac{1}{2},\frac{5}{6};\frac{11}{6};\cos ^2(c+d x)\right)\right)-10 A \csc (c+d x) \, _2F_1\left(-\frac{1}{6},\frac{1}{2};\frac{5}{6};\cos ^2(c+d x)\right)\right)}{10 d \sqrt[3]{b \cos (c+d x)}}","\frac{3 (2 A-C) \sin (c+d x) (b \cos (c+d x))^{5/3} \, _2F_1\left(\frac{1}{2},\frac{5}{6};\frac{11}{6};\cos ^2(c+d x)\right)}{5 b d \sqrt{\sin ^2(c+d x)}}+\frac{3 A b \sin (c+d x)}{d \sqrt[3]{b \cos (c+d x)}}-\frac{3 B \sin (c+d x) (b \cos (c+d x))^{2/3} \, _2F_1\left(\frac{1}{3},\frac{1}{2};\frac{4}{3};\cos ^2(c+d x)\right)}{2 d \sqrt{\sin ^2(c+d x)}}",1,"(-3*b*(-10*A*Csc[c + d*x]*Hypergeometric2F1[-1/6, 1/2, 5/6, Cos[c + d*x]^2] + Cot[c + d*x]*(5*B*Hypergeometric2F1[1/3, 1/2, 4/3, Cos[c + d*x]^2] + 2*C*Cos[c + d*x]*Hypergeometric2F1[1/2, 5/6, 11/6, Cos[c + d*x]^2]))*Sqrt[Sin[c + d*x]^2])/(10*d*(b*Cos[c + d*x])^(1/3))","A",1
343,1,123,145,0.2357218,"\int (b \cos (c+d x))^{2/3} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^3(c+d x) \, dx","Integrate[(b*Cos[c + d*x])^(2/3)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^3,x]","-\frac{3 \sqrt{\sin ^2(c+d x)} \csc (c+d x) \sec ^2(c+d x) (b \cos (c+d x))^{2/3} \left(2 \cos (c+d x) \left(C \cos (c+d x) \, _2F_1\left(\frac{1}{3},\frac{1}{2};\frac{4}{3};\cos ^2(c+d x)\right)-2 B \, _2F_1\left(-\frac{1}{6},\frac{1}{2};\frac{5}{6};\cos ^2(c+d x)\right)\right)-A \, _2F_1\left(-\frac{2}{3},\frac{1}{2};\frac{1}{3};\cos ^2(c+d x)\right)\right)}{4 d}","\frac{3 A b^2 \sin (c+d x)}{4 d (b \cos (c+d x))^{4/3}}-\frac{3 (A+4 C) \sin (c+d x) (b \cos (c+d x))^{2/3} \, _2F_1\left(\frac{1}{3},\frac{1}{2};\frac{4}{3};\cos ^2(c+d x)\right)}{8 d \sqrt{\sin ^2(c+d x)}}+\frac{3 b B \sin (c+d x) \, _2F_1\left(-\frac{1}{6},\frac{1}{2};\frac{5}{6};\cos ^2(c+d x)\right)}{d \sqrt{\sin ^2(c+d x)} \sqrt[3]{b \cos (c+d x)}}",1,"(-3*(b*Cos[c + d*x])^(2/3)*Csc[c + d*x]*(-(A*Hypergeometric2F1[-2/3, 1/2, 1/3, Cos[c + d*x]^2]) + 2*Cos[c + d*x]*(-2*B*Hypergeometric2F1[-1/6, 1/2, 5/6, Cos[c + d*x]^2] + C*Cos[c + d*x]*Hypergeometric2F1[1/3, 1/2, 4/3, Cos[c + d*x]^2]))*Sec[c + d*x]^2*Sqrt[Sin[c + d*x]^2])/(4*d)","A",1
344,1,123,152,0.2116157,"\int (b \cos (c+d x))^{2/3} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^4(c+d x) \, dx","Integrate[(b*Cos[c + d*x])^(2/3)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^4,x]","\frac{3 \sqrt{\sin ^2(c+d x)} \csc (c+d x) \sec ^3(c+d x) (b \cos (c+d x))^{2/3} \left(4 A \, _2F_1\left(-\frac{7}{6},\frac{1}{2};-\frac{1}{6};\cos ^2(c+d x)\right)+7 \cos (c+d x) \left(B \, _2F_1\left(-\frac{2}{3},\frac{1}{2};\frac{1}{3};\cos ^2(c+d x)\right)+4 C \cos (c+d x) \, _2F_1\left(-\frac{1}{6},\frac{1}{2};\frac{5}{6};\cos ^2(c+d x)\right)\right)\right)}{28 d}","\frac{3 A b^3 \sin (c+d x)}{7 d (b \cos (c+d x))^{7/3}}+\frac{3 b (4 A+7 C) \sin (c+d x) \, _2F_1\left(-\frac{1}{6},\frac{1}{2};\frac{5}{6};\cos ^2(c+d x)\right)}{7 d \sqrt{\sin ^2(c+d x)} \sqrt[3]{b \cos (c+d x)}}+\frac{3 b^2 B \sin (c+d x) \, _2F_1\left(-\frac{2}{3},\frac{1}{2};\frac{1}{3};\cos ^2(c+d x)\right)}{4 d \sqrt{\sin ^2(c+d x)} (b \cos (c+d x))^{4/3}}",1,"(3*(b*Cos[c + d*x])^(2/3)*Csc[c + d*x]*(4*A*Hypergeometric2F1[-7/6, 1/2, -1/6, Cos[c + d*x]^2] + 7*Cos[c + d*x]*(B*Hypergeometric2F1[-2/3, 1/2, 1/3, Cos[c + d*x]^2] + 4*C*Cos[c + d*x]*Hypergeometric2F1[-1/6, 1/2, 5/6, Cos[c + d*x]^2]))*Sec[c + d*x]^3*Sqrt[Sin[c + d*x]^2])/(28*d)","A",1
345,1,111,154,0.3665811,"\int \cos (c+d x) (b \cos (c+d x))^{4/3} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]*(b*Cos[c + d*x])^(4/3)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","-\frac{3 \sin (c+d x) (b \cos (c+d x))^{10/3} \left((13 A+10 C) \, _2F_1\left(\frac{1}{2},\frac{5}{3};\frac{8}{3};\cos ^2(c+d x)\right)+10 \left(B \cos (c+d x) \, _2F_1\left(\frac{1}{2},\frac{13}{6};\frac{19}{6};\cos ^2(c+d x)\right)-C \sqrt{\sin ^2(c+d x)}\right)\right)}{130 b^2 d \sqrt{\sin ^2(c+d x)}}","-\frac{3 (13 A+10 C) \sin (c+d x) (b \cos (c+d x))^{10/3} \, _2F_1\left(\frac{1}{2},\frac{5}{3};\frac{8}{3};\cos ^2(c+d x)\right)}{130 b^2 d \sqrt{\sin ^2(c+d x)}}-\frac{3 B \sin (c+d x) (b \cos (c+d x))^{13/3} \, _2F_1\left(\frac{1}{2},\frac{13}{6};\frac{19}{6};\cos ^2(c+d x)\right)}{13 b^3 d \sqrt{\sin ^2(c+d x)}}+\frac{3 C \sin (c+d x) (b \cos (c+d x))^{10/3}}{13 b^2 d}",1,"(-3*(b*Cos[c + d*x])^(10/3)*Sin[c + d*x]*((13*A + 10*C)*Hypergeometric2F1[1/2, 5/3, 8/3, Cos[c + d*x]^2] + 10*(B*Cos[c + d*x]*Hypergeometric2F1[1/2, 13/6, 19/6, Cos[c + d*x]^2] - C*Sqrt[Sin[c + d*x]^2])))/(130*b^2*d*Sqrt[Sin[c + d*x]^2])","A",1
346,1,109,154,0.2287268,"\int (b \cos (c+d x))^{4/3} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Integrate[(b*Cos[c + d*x])^(4/3)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","-\frac{3 \sin (c+d x) (b \cos (c+d x))^{7/3} \left((10 A+7 C) \, _2F_1\left(\frac{1}{2},\frac{7}{6};\frac{13}{6};\cos ^2(c+d x)\right)+7 B \cos (c+d x) \, _2F_1\left(\frac{1}{2},\frac{5}{3};\frac{8}{3};\cos ^2(c+d x)\right)-7 C \sqrt{\sin ^2(c+d x)}\right)}{70 b d \sqrt{\sin ^2(c+d x)}}","-\frac{3 (10 A+7 C) \sin (c+d x) (b \cos (c+d x))^{7/3} \, _2F_1\left(\frac{1}{2},\frac{7}{6};\frac{13}{6};\cos ^2(c+d x)\right)}{70 b d \sqrt{\sin ^2(c+d x)}}-\frac{3 B \sin (c+d x) (b \cos (c+d x))^{10/3} \, _2F_1\left(\frac{1}{2},\frac{5}{3};\frac{8}{3};\cos ^2(c+d x)\right)}{10 b^2 d \sqrt{\sin ^2(c+d x)}}+\frac{3 C \sin (c+d x) (b \cos (c+d x))^{7/3}}{10 b d}",1,"(-3*(b*Cos[c + d*x])^(7/3)*Sin[c + d*x]*((10*A + 7*C)*Hypergeometric2F1[1/2, 7/6, 13/6, Cos[c + d*x]^2] + 7*B*Cos[c + d*x]*Hypergeometric2F1[1/2, 5/3, 8/3, Cos[c + d*x]^2] - 7*C*Sqrt[Sin[c + d*x]^2]))/(70*b*d*Sqrt[Sin[c + d*x]^2])","A",1
347,1,109,148,0.2350819,"\int (b \cos (c+d x))^{4/3} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec (c+d x) \, dx","Integrate[(b*Cos[c + d*x])^(4/3)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x],x]","-\frac{3 b \sin (2 (c+d x)) \sqrt[3]{b \cos (c+d x)} \left((7 A+4 C) \, _2F_1\left(\frac{1}{2},\frac{2}{3};\frac{5}{3};\cos ^2(c+d x)\right)+4 B \cos (c+d x) \, _2F_1\left(\frac{1}{2},\frac{7}{6};\frac{13}{6};\cos ^2(c+d x)\right)-4 C \sqrt{\sin ^2(c+d x)}\right)}{56 d \sqrt{\sin ^2(c+d x)}}","-\frac{3 (7 A+4 C) \sin (c+d x) (b \cos (c+d x))^{4/3} \, _2F_1\left(\frac{1}{2},\frac{2}{3};\frac{5}{3};\cos ^2(c+d x)\right)}{28 d \sqrt{\sin ^2(c+d x)}}-\frac{3 B \sin (c+d x) (b \cos (c+d x))^{7/3} \, _2F_1\left(\frac{1}{2},\frac{7}{6};\frac{13}{6};\cos ^2(c+d x)\right)}{7 b d \sqrt{\sin ^2(c+d x)}}+\frac{3 C \sin (c+d x) (b \cos (c+d x))^{4/3}}{7 d}",1,"(-3*b*(b*Cos[c + d*x])^(1/3)*((7*A + 4*C)*Hypergeometric2F1[1/2, 2/3, 5/3, Cos[c + d*x]^2] + 4*B*Cos[c + d*x]*Hypergeometric2F1[1/2, 7/6, 13/6, Cos[c + d*x]^2] - 4*C*Sqrt[Sin[c + d*x]^2])*Sin[2*(c + d*x)])/(56*d*Sqrt[Sin[c + d*x]^2])","A",1
348,1,108,145,0.2106795,"\int (b \cos (c+d x))^{4/3} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^2(c+d x) \, dx","Integrate[(b*Cos[c + d*x])^(4/3)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^2,x]","-\frac{3 b^2 \sin (2 (c+d x)) \left((4 A+C) \, _2F_1\left(\frac{1}{6},\frac{1}{2};\frac{7}{6};\cos ^2(c+d x)\right)+B \cos (c+d x) \, _2F_1\left(\frac{1}{2},\frac{2}{3};\frac{5}{3};\cos ^2(c+d x)\right)+C \left(-\sqrt{\sin ^2(c+d x)}\right)\right)}{8 d \sqrt{\sin ^2(c+d x)} (b \cos (c+d x))^{2/3}}","-\frac{3 b (4 A+C) \sin (c+d x) \sqrt[3]{b \cos (c+d x)} \, _2F_1\left(\frac{1}{6},\frac{1}{2};\frac{7}{6};\cos ^2(c+d x)\right)}{4 d \sqrt{\sin ^2(c+d x)}}-\frac{3 B \sin (c+d x) (b \cos (c+d x))^{4/3} \, _2F_1\left(\frac{1}{2},\frac{2}{3};\frac{5}{3};\cos ^2(c+d x)\right)}{4 d \sqrt{\sin ^2(c+d x)}}+\frac{3 b C \sin (c+d x) \sqrt[3]{b \cos (c+d x)}}{4 d}",1,"(-3*b^2*((4*A + C)*Hypergeometric2F1[1/6, 1/2, 7/6, Cos[c + d*x]^2] + B*Cos[c + d*x]*Hypergeometric2F1[1/2, 2/3, 5/3, Cos[c + d*x]^2] - C*Sqrt[Sin[c + d*x]^2])*Sin[2*(c + d*x)])/(8*d*(b*Cos[c + d*x])^(2/3)*Sqrt[Sin[c + d*x]^2])","A",1
349,1,117,145,0.2380046,"\int (b \cos (c+d x))^{4/3} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^3(c+d x) \, dx","Integrate[(b*Cos[c + d*x])^(4/3)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^3,x]","-\frac{3 b^2 \sqrt{\sin ^2(c+d x)} \csc (c+d x) \left(\cos (c+d x) \left(4 B \, _2F_1\left(\frac{1}{6},\frac{1}{2};\frac{7}{6};\cos ^2(c+d x)\right)+C \cos (c+d x) \, _2F_1\left(\frac{1}{2},\frac{2}{3};\frac{5}{3};\cos ^2(c+d x)\right)\right)-2 A \, _2F_1\left(-\frac{1}{3},\frac{1}{2};\frac{2}{3};\cos ^2(c+d x)\right)\right)}{4 d (b \cos (c+d x))^{2/3}}","\frac{3 A b^2 \sin (c+d x)}{2 d (b \cos (c+d x))^{2/3}}+\frac{3 (A-2 C) \sin (c+d x) (b \cos (c+d x))^{4/3} \, _2F_1\left(\frac{1}{2},\frac{2}{3};\frac{5}{3};\cos ^2(c+d x)\right)}{8 d \sqrt{\sin ^2(c+d x)}}-\frac{3 b B \sin (c+d x) \sqrt[3]{b \cos (c+d x)} \, _2F_1\left(\frac{1}{6},\frac{1}{2};\frac{7}{6};\cos ^2(c+d x)\right)}{d \sqrt{\sin ^2(c+d x)}}",1,"(-3*b^2*Csc[c + d*x]*(-2*A*Hypergeometric2F1[-1/3, 1/2, 2/3, Cos[c + d*x]^2] + Cos[c + d*x]*(4*B*Hypergeometric2F1[1/6, 1/2, 7/6, Cos[c + d*x]^2] + C*Cos[c + d*x]*Hypergeometric2F1[1/2, 2/3, 5/3, Cos[c + d*x]^2]))*Sqrt[Sin[c + d*x]^2])/(4*d*(b*Cos[c + d*x])^(2/3))","A",1
350,1,124,152,0.2107502,"\int (b \cos (c+d x))^{4/3} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^4(c+d x) \, dx","Integrate[(b*Cos[c + d*x])^(4/3)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^4,x]","-\frac{3 \sqrt{\sin ^2(c+d x)} \csc (c+d x) \sec ^3(c+d x) (b \cos (c+d x))^{4/3} \left(5 \cos (c+d x) \left(2 C \cos (c+d x) \, _2F_1\left(\frac{1}{6},\frac{1}{2};\frac{7}{6};\cos ^2(c+d x)\right)-B \, _2F_1\left(-\frac{1}{3},\frac{1}{2};\frac{2}{3};\cos ^2(c+d x)\right)\right)-2 A \, _2F_1\left(-\frac{5}{6},\frac{1}{2};\frac{1}{6};\cos ^2(c+d x)\right)\right)}{10 d}","\frac{3 A b^3 \sin (c+d x)}{5 d (b \cos (c+d x))^{5/3}}-\frac{3 b (2 A+5 C) \sin (c+d x) \sqrt[3]{b \cos (c+d x)} \, _2F_1\left(\frac{1}{6},\frac{1}{2};\frac{7}{6};\cos ^2(c+d x)\right)}{5 d \sqrt{\sin ^2(c+d x)}}+\frac{3 b^2 B \sin (c+d x) \, _2F_1\left(-\frac{1}{3},\frac{1}{2};\frac{2}{3};\cos ^2(c+d x)\right)}{2 d \sqrt{\sin ^2(c+d x)} (b \cos (c+d x))^{2/3}}",1,"(-3*(b*Cos[c + d*x])^(4/3)*Csc[c + d*x]*(-2*A*Hypergeometric2F1[-5/6, 1/2, 1/6, Cos[c + d*x]^2] + 5*Cos[c + d*x]*(-(B*Hypergeometric2F1[-1/3, 1/2, 2/3, Cos[c + d*x]^2]) + 2*C*Cos[c + d*x]*Hypergeometric2F1[1/6, 1/2, 7/6, Cos[c + d*x]^2]))*Sec[c + d*x]^3*Sqrt[Sin[c + d*x]^2])/(10*d)","A",1
351,1,114,154,0.2788468,"\int \frac{\cos ^2(c+d x) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\sqrt[3]{b \cos (c+d x)}} \, dx","Integrate[(Cos[c + d*x]^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(b*Cos[c + d*x])^(1/3),x]","-\frac{3 \sin (c+d x) \cos ^3(c+d x) \left((11 A+8 C) \, _2F_1\left(\frac{1}{2},\frac{4}{3};\frac{7}{3};\cos ^2(c+d x)\right)+8 B \cos (c+d x) \, _2F_1\left(\frac{1}{2},\frac{11}{6};\frac{17}{6};\cos ^2(c+d x)\right)-8 C \sqrt{\sin ^2(c+d x)}\right)}{88 d \sqrt{\sin ^2(c+d x)} \sqrt[3]{b \cos (c+d x)}}","-\frac{3 (11 A+8 C) \sin (c+d x) (b \cos (c+d x))^{8/3} \, _2F_1\left(\frac{1}{2},\frac{4}{3};\frac{7}{3};\cos ^2(c+d x)\right)}{88 b^3 d \sqrt{\sin ^2(c+d x)}}-\frac{3 B \sin (c+d x) (b \cos (c+d x))^{11/3} \, _2F_1\left(\frac{1}{2},\frac{11}{6};\frac{17}{6};\cos ^2(c+d x)\right)}{11 b^4 d \sqrt{\sin ^2(c+d x)}}+\frac{3 C \sin (c+d x) (b \cos (c+d x))^{8/3}}{11 b^3 d}",1,"(-3*Cos[c + d*x]^3*Sin[c + d*x]*((11*A + 8*C)*Hypergeometric2F1[1/2, 4/3, 7/3, Cos[c + d*x]^2] + 8*B*Cos[c + d*x]*Hypergeometric2F1[1/2, 11/6, 17/6, Cos[c + d*x]^2] - 8*C*Sqrt[Sin[c + d*x]^2]))/(88*d*(b*Cos[c + d*x])^(1/3)*Sqrt[Sin[c + d*x]^2])","A",1
352,1,109,154,0.2249519,"\int \frac{\cos (c+d x) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\sqrt[3]{b \cos (c+d x)}} \, dx","Integrate[(Cos[c + d*x]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(b*Cos[c + d*x])^(1/3),x]","-\frac{3 \sin (c+d x) (b \cos (c+d x))^{5/3} \left((8 A+5 C) \, _2F_1\left(\frac{1}{2},\frac{5}{6};\frac{11}{6};\cos ^2(c+d x)\right)+5 B \cos (c+d x) \, _2F_1\left(\frac{1}{2},\frac{4}{3};\frac{7}{3};\cos ^2(c+d x)\right)-5 C \sqrt{\sin ^2(c+d x)}\right)}{40 b^2 d \sqrt{\sin ^2(c+d x)}}","-\frac{3 (8 A+5 C) \sin (c+d x) (b \cos (c+d x))^{5/3} \, _2F_1\left(\frac{1}{2},\frac{5}{6};\frac{11}{6};\cos ^2(c+d x)\right)}{40 b^2 d \sqrt{\sin ^2(c+d x)}}-\frac{3 B \sin (c+d x) (b \cos (c+d x))^{8/3} \, _2F_1\left(\frac{1}{2},\frac{4}{3};\frac{7}{3};\cos ^2(c+d x)\right)}{8 b^3 d \sqrt{\sin ^2(c+d x)}}+\frac{3 C \sin (c+d x) (b \cos (c+d x))^{5/3}}{8 b^2 d}",1,"(-3*(b*Cos[c + d*x])^(5/3)*Sin[c + d*x]*((8*A + 5*C)*Hypergeometric2F1[1/2, 5/6, 11/6, Cos[c + d*x]^2] + 5*B*Cos[c + d*x]*Hypergeometric2F1[1/2, 4/3, 7/3, Cos[c + d*x]^2] - 5*C*Sqrt[Sin[c + d*x]^2]))/(40*b^2*d*Sqrt[Sin[c + d*x]^2])","A",1
353,1,108,154,0.1429619,"\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{\sqrt[3]{b \cos (c+d x)}} \, dx","Integrate[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(b*Cos[c + d*x])^(1/3),x]","-\frac{3 \sin (2 (c+d x)) \left((5 A+2 C) \, _2F_1\left(\frac{1}{3},\frac{1}{2};\frac{4}{3};\cos ^2(c+d x)\right)+2 B \cos (c+d x) \, _2F_1\left(\frac{1}{2},\frac{5}{6};\frac{11}{6};\cos ^2(c+d x)\right)-2 C \sqrt{\sin ^2(c+d x)}\right)}{20 d \sqrt{\sin ^2(c+d x)} \sqrt[3]{b \cos (c+d x)}}","-\frac{3 (5 A+2 C) \sin (c+d x) (b \cos (c+d x))^{2/3} \, _2F_1\left(\frac{1}{3},\frac{1}{2};\frac{4}{3};\cos ^2(c+d x)\right)}{10 b d \sqrt{\sin ^2(c+d x)}}-\frac{3 B \sin (c+d x) (b \cos (c+d x))^{5/3} \, _2F_1\left(\frac{1}{2},\frac{5}{6};\frac{11}{6};\cos ^2(c+d x)\right)}{5 b^2 d \sqrt{\sin ^2(c+d x)}}+\frac{3 C \sin (c+d x) (b \cos (c+d x))^{2/3}}{5 b d}",1,"(-3*((5*A + 2*C)*Hypergeometric2F1[1/3, 1/2, 4/3, Cos[c + d*x]^2] + 2*B*Cos[c + d*x]*Hypergeometric2F1[1/2, 5/6, 11/6, Cos[c + d*x]^2] - 2*C*Sqrt[Sin[c + d*x]^2])*Sin[2*(c + d*x)])/(20*d*(b*Cos[c + d*x])^(1/3)*Sqrt[Sin[c + d*x]^2])","A",1
354,1,779,149,6.2893218,"\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec (c+d x)}{\sqrt[3]{b \cos (c+d x)}} \, dx","Integrate[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x])/(b*Cos[c + d*x])^(1/3),x]","\frac{4 A \csc (c) \cos ^{\frac{4}{3}}(c+d x) (A \sec (c+d x)+B+C \cos (c+d x)) \left(\frac{\tan (c) \sin \left(\tan ^{-1}(\tan (c))+d x\right) \, _2F_1\left(-\frac{1}{2},-\frac{1}{6};\frac{5}{6};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right)}{\sqrt{\tan ^2(c)+1} \sqrt{1-\cos \left(\tan ^{-1}(\tan (c))+d x\right)} \sqrt{\cos \left(\tan ^{-1}(\tan (c))+d x\right)+1} \sqrt[3]{\cos (c) \sqrt{\tan ^2(c)+1} \cos \left(\tan ^{-1}(\tan (c))+d x\right)}}-\frac{\frac{\tan (c) \sin \left(\tan ^{-1}(\tan (c))+d x\right)}{\sqrt{\tan ^2(c)+1}}+\frac{3 \cos ^2(c) \sqrt{\tan ^2(c)+1} \cos \left(\tan ^{-1}(\tan (c))+d x\right)}{2 \left(\sin ^2(c)+\cos ^2(c)\right)}}{\sqrt[3]{\cos (c) \sqrt{\tan ^2(c)+1} \cos \left(\tan ^{-1}(\tan (c))+d x\right)}}\right)}{d \sqrt[3]{b \cos (c+d x)} (2 A+2 B \cos (c+d x)+C \cos (2 c+2 d x)+C)}-\frac{2 C \csc (c) \cos ^{\frac{4}{3}}(c+d x) (A \sec (c+d x)+B+C \cos (c+d x)) \left(\frac{\tan (c) \sin \left(\tan ^{-1}(\tan (c))+d x\right) \, _2F_1\left(-\frac{1}{2},-\frac{1}{6};\frac{5}{6};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right)}{\sqrt{\tan ^2(c)+1} \sqrt{1-\cos \left(\tan ^{-1}(\tan (c))+d x\right)} \sqrt{\cos \left(\tan ^{-1}(\tan (c))+d x\right)+1} \sqrt[3]{\cos (c) \sqrt{\tan ^2(c)+1} \cos \left(\tan ^{-1}(\tan (c))+d x\right)}}-\frac{\frac{\tan (c) \sin \left(\tan ^{-1}(\tan (c))+d x\right)}{\sqrt{\tan ^2(c)+1}}+\frac{3 \cos ^2(c) \sqrt{\tan ^2(c)+1} \cos \left(\tan ^{-1}(\tan (c))+d x\right)}{2 \left(\sin ^2(c)+\cos ^2(c)\right)}}{\sqrt[3]{\cos (c) \sqrt{\tan ^2(c)+1} \cos \left(\tan ^{-1}(\tan (c))+d x\right)}}\right)}{d \sqrt[3]{b \cos (c+d x)} (2 A+2 B \cos (c+d x)+C \cos (2 c+2 d x)+C)}-\frac{2 B \cos ^{\frac{4}{3}}(c+d x) \cos \left(d x-\tan ^{-1}(\cot (c))\right) \sin \left(d x-\tan ^{-1}(\cot (c))\right) \, _2F_1\left(\frac{1}{2},\frac{2}{3};\frac{3}{2};\cos ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) (A \sec (c+d x)+B+C \cos (c+d x))}{d \sqrt[3]{b \cos (c+d x)} \sqrt[3]{\cos (c) \cos (d x)-\sin (c) \sin (d x)} \sqrt[3]{\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)} (2 A+2 B \cos (c+d x)+C \cos (2 c+2 d x)+C)}+\frac{\cos ^2(c+d x) (A \sec (c+d x)+B+C \cos (c+d x)) \left(\frac{6 A \sec (c) \sin (d x) \sec (c+d x)}{d}-\frac{3 \csc (c) \sec (c) (-4 A+C \cos (2 c)+C)}{2 d}\right)}{\sqrt[3]{b \cos (c+d x)} (2 A+2 B \cos (c+d x)+C \cos (2 c+2 d x)+C)}","\frac{3 (2 A-C) \sin (c+d x) (b \cos (c+d x))^{5/3} \, _2F_1\left(\frac{1}{2},\frac{5}{6};\frac{11}{6};\cos ^2(c+d x)\right)}{5 b^2 d \sqrt{\sin ^2(c+d x)}}+\frac{3 A \sin (c+d x)}{d \sqrt[3]{b \cos (c+d x)}}-\frac{3 B \sin (c+d x) (b \cos (c+d x))^{2/3} \, _2F_1\left(\frac{1}{3},\frac{1}{2};\frac{4}{3};\cos ^2(c+d x)\right)}{2 b d \sqrt{\sin ^2(c+d x)}}",1,"(Cos[c + d*x]^2*(B + C*Cos[c + d*x] + A*Sec[c + d*x])*((-3*(-4*A + C + C*Cos[2*c])*Csc[c]*Sec[c])/(2*d) + (6*A*Sec[c]*Sec[c + d*x]*Sin[d*x])/d))/((b*Cos[c + d*x])^(1/3)*(2*A + C + 2*B*Cos[c + d*x] + C*Cos[2*c + 2*d*x])) - (2*B*Cos[c + d*x]^(4/3)*Cos[d*x - ArcTan[Cot[c]]]*Hypergeometric2F1[1/2, 2/3, 3/2, Cos[d*x - ArcTan[Cot[c]]]^2]*(B + C*Cos[c + d*x] + A*Sec[c + d*x])*Sin[d*x - ArcTan[Cot[c]]])/(d*(b*Cos[c + d*x])^(1/3)*(2*A + C + 2*B*Cos[c + d*x] + C*Cos[2*c + 2*d*x])*(Cos[c]*Cos[d*x] - Sin[c]*Sin[d*x])^(1/3)*(Sin[d*x - ArcTan[Cot[c]]]^2)^(1/3)) + (4*A*Cos[c + d*x]^(4/3)*Csc[c]*(B + C*Cos[c + d*x] + A*Sec[c + d*x])*((HypergeometricPFQ[{-1/2, -1/6}, {5/6}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*(Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])^(1/3)*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (3*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(2*(Cos[c]^2 + Sin[c]^2)))/(Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])^(1/3)))/(d*(b*Cos[c + d*x])^(1/3)*(2*A + C + 2*B*Cos[c + d*x] + C*Cos[2*c + 2*d*x])) - (2*C*Cos[c + d*x]^(4/3)*Csc[c]*(B + C*Cos[c + d*x] + A*Sec[c + d*x])*((HypergeometricPFQ[{-1/2, -1/6}, {5/6}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*(Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])^(1/3)*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (3*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(2*(Cos[c]^2 + Sin[c]^2)))/(Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])^(1/3)))/(d*(b*Cos[c + d*x])^(1/3)*(2*A + C + 2*B*Cos[c + d*x] + C*Cos[2*c + 2*d*x]))","B",0
355,1,699,145,6.3220564,"\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^2(c+d x)}{\sqrt[3]{b \cos (c+d x)}} \, dx","Integrate[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^2)/(b*Cos[c + d*x])^(1/3),x]","\frac{4 B \csc (c) \cos ^{\frac{7}{3}}(c+d x) \left(A \sec ^2(c+d x)+B \sec (c+d x)+C\right) \left(\frac{\tan (c) \sin \left(\tan ^{-1}(\tan (c))+d x\right) \, _2F_1\left(-\frac{1}{2},-\frac{1}{6};\frac{5}{6};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right)}{\sqrt{\tan ^2(c)+1} \sqrt{1-\cos \left(\tan ^{-1}(\tan (c))+d x\right)} \sqrt{\cos \left(\tan ^{-1}(\tan (c))+d x\right)+1} \sqrt[3]{\cos (c) \sqrt{\tan ^2(c)+1} \cos \left(\tan ^{-1}(\tan (c))+d x\right)}}-\frac{\frac{\tan (c) \sin \left(\tan ^{-1}(\tan (c))+d x\right)}{\sqrt{\tan ^2(c)+1}}+\frac{3 \cos ^2(c) \sqrt{\tan ^2(c)+1} \cos \left(\tan ^{-1}(\tan (c))+d x\right)}{2 \left(\sin ^2(c)+\cos ^2(c)\right)}}{\sqrt[3]{\cos (c) \sqrt{\tan ^2(c)+1} \cos \left(\tan ^{-1}(\tan (c))+d x\right)}}\right)}{d \sqrt[3]{b \cos (c+d x)} (2 A+2 B \cos (c+d x)+C \cos (2 c+2 d x)+C)}-\frac{A \cos ^{\frac{7}{3}}(c+d x) \cos \left(d x-\tan ^{-1}(\cot (c))\right) \sin \left(d x-\tan ^{-1}(\cot (c))\right) \left(A \sec ^2(c+d x)+B \sec (c+d x)+C\right) \, _2F_1\left(\frac{1}{2},\frac{2}{3};\frac{3}{2};\cos ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right)}{2 d \sqrt[3]{b \cos (c+d x)} \sqrt[3]{\cos (c) \cos (d x)-\sin (c) \sin (d x)} \sqrt[3]{\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)} (2 A+2 B \cos (c+d x)+C \cos (2 c+2 d x)+C)}-\frac{2 C \cos ^{\frac{7}{3}}(c+d x) \cos \left(d x-\tan ^{-1}(\cot (c))\right) \sin \left(d x-\tan ^{-1}(\cot (c))\right) \left(A \sec ^2(c+d x)+B \sec (c+d x)+C\right) \, _2F_1\left(\frac{1}{2},\frac{2}{3};\frac{3}{2};\cos ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right)}{d \sqrt[3]{b \cos (c+d x)} \sqrt[3]{\cos (c) \cos (d x)-\sin (c) \sin (d x)} \sqrt[3]{\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)} (2 A+2 B \cos (c+d x)+C \cos (2 c+2 d x)+C)}+\frac{\cos ^3(c+d x) \left(A \sec ^2(c+d x)+B \sec (c+d x)+C\right) \left(\frac{3 \sec (c) \sec (c+d x) (A \sin (c)+4 B \sin (d x))}{2 d}+\frac{3 A \sec (c) \sin (d x) \sec ^2(c+d x)}{2 d}+\frac{6 B \csc (c) \sec (c)}{d}\right)}{\sqrt[3]{b \cos (c+d x)} (2 A+2 B \cos (c+d x)+C \cos (2 c+2 d x)+C)}","-\frac{3 (A+4 C) \sin (c+d x) (b \cos (c+d x))^{2/3} \, _2F_1\left(\frac{1}{3},\frac{1}{2};\frac{4}{3};\cos ^2(c+d x)\right)}{8 b d \sqrt{\sin ^2(c+d x)}}+\frac{3 A b \sin (c+d x)}{4 d (b \cos (c+d x))^{4/3}}+\frac{3 B \sin (c+d x) \, _2F_1\left(-\frac{1}{6},\frac{1}{2};\frac{5}{6};\cos ^2(c+d x)\right)}{d \sqrt{\sin ^2(c+d x)} \sqrt[3]{b \cos (c+d x)}}",1,"(Cos[c + d*x]^3*(C + B*Sec[c + d*x] + A*Sec[c + d*x]^2)*((6*B*Csc[c]*Sec[c])/d + (3*A*Sec[c]*Sec[c + d*x]^2*Sin[d*x])/(2*d) + (3*Sec[c]*Sec[c + d*x]*(A*Sin[c] + 4*B*Sin[d*x]))/(2*d)))/((b*Cos[c + d*x])^(1/3)*(2*A + C + 2*B*Cos[c + d*x] + C*Cos[2*c + 2*d*x])) - (A*Cos[c + d*x]^(7/3)*Cos[d*x - ArcTan[Cot[c]]]*Hypergeometric2F1[1/2, 2/3, 3/2, Cos[d*x - ArcTan[Cot[c]]]^2]*(C + B*Sec[c + d*x] + A*Sec[c + d*x]^2)*Sin[d*x - ArcTan[Cot[c]]])/(2*d*(b*Cos[c + d*x])^(1/3)*(2*A + C + 2*B*Cos[c + d*x] + C*Cos[2*c + 2*d*x])*(Cos[c]*Cos[d*x] - Sin[c]*Sin[d*x])^(1/3)*(Sin[d*x - ArcTan[Cot[c]]]^2)^(1/3)) - (2*C*Cos[c + d*x]^(7/3)*Cos[d*x - ArcTan[Cot[c]]]*Hypergeometric2F1[1/2, 2/3, 3/2, Cos[d*x - ArcTan[Cot[c]]]^2]*(C + B*Sec[c + d*x] + A*Sec[c + d*x]^2)*Sin[d*x - ArcTan[Cot[c]]])/(d*(b*Cos[c + d*x])^(1/3)*(2*A + C + 2*B*Cos[c + d*x] + C*Cos[2*c + 2*d*x])*(Cos[c]*Cos[d*x] - Sin[c]*Sin[d*x])^(1/3)*(Sin[d*x - ArcTan[Cot[c]]]^2)^(1/3)) + (4*B*Cos[c + d*x]^(7/3)*Csc[c]*(C + B*Sec[c + d*x] + A*Sec[c + d*x]^2)*((HypergeometricPFQ[{-1/2, -1/6}, {5/6}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*(Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])^(1/3)*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (3*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(2*(Cos[c]^2 + Sin[c]^2)))/(Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])^(1/3)))/(d*(b*Cos[c + d*x])^(1/3)*(2*A + C + 2*B*Cos[c + d*x] + C*Cos[2*c + 2*d*x]))","B",0
356,1,118,149,0.2653673,"\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^3(c+d x)}{\sqrt[3]{b \cos (c+d x)}} \, dx","Integrate[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^3)/(b*Cos[c + d*x])^(1/3),x]","\frac{3 b^2 \sqrt{\sin ^2(c+d x)} \csc (c+d x) \left(4 A \, _2F_1\left(-\frac{7}{6},\frac{1}{2};-\frac{1}{6};\cos ^2(c+d x)\right)+7 \cos (c+d x) \left(B \, _2F_1\left(-\frac{2}{3},\frac{1}{2};\frac{1}{3};\cos ^2(c+d x)\right)+4 C \cos (c+d x) \, _2F_1\left(-\frac{1}{6},\frac{1}{2};\frac{5}{6};\cos ^2(c+d x)\right)\right)\right)}{28 d (b \cos (c+d x))^{7/3}}","\frac{3 A b^2 \sin (c+d x)}{7 d (b \cos (c+d x))^{7/3}}+\frac{3 (4 A+7 C) \sin (c+d x) \, _2F_1\left(-\frac{1}{6},\frac{1}{2};\frac{5}{6};\cos ^2(c+d x)\right)}{7 d \sqrt{\sin ^2(c+d x)} \sqrt[3]{b \cos (c+d x)}}+\frac{3 b B \sin (c+d x) \, _2F_1\left(-\frac{2}{3},\frac{1}{2};\frac{1}{3};\cos ^2(c+d x)\right)}{4 d \sqrt{\sin ^2(c+d x)} (b \cos (c+d x))^{4/3}}",1,"(3*b^2*Csc[c + d*x]*(4*A*Hypergeometric2F1[-7/6, 1/2, -1/6, Cos[c + d*x]^2] + 7*Cos[c + d*x]*(B*Hypergeometric2F1[-2/3, 1/2, 1/3, Cos[c + d*x]^2] + 4*C*Cos[c + d*x]*Hypergeometric2F1[-1/6, 1/2, 5/6, Cos[c + d*x]^2]))*Sqrt[Sin[c + d*x]^2])/(28*d*(b*Cos[c + d*x])^(7/3))","A",1
357,1,114,154,0.3471848,"\int \frac{\cos ^3(c+d x) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{(b \cos (c+d x))^{4/3}} \, dx","Integrate[(Cos[c + d*x]^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(b*Cos[c + d*x])^(4/3),x]","-\frac{3 \sin (c+d x) \cos ^4(c+d x) \left((11 A+8 C) \, _2F_1\left(\frac{1}{2},\frac{4}{3};\frac{7}{3};\cos ^2(c+d x)\right)+8 B \cos (c+d x) \, _2F_1\left(\frac{1}{2},\frac{11}{6};\frac{17}{6};\cos ^2(c+d x)\right)-8 C \sqrt{\sin ^2(c+d x)}\right)}{88 d \sqrt{\sin ^2(c+d x)} (b \cos (c+d x))^{4/3}}","-\frac{3 (11 A+8 C) \sin (c+d x) (b \cos (c+d x))^{8/3} \, _2F_1\left(\frac{1}{2},\frac{4}{3};\frac{7}{3};\cos ^2(c+d x)\right)}{88 b^4 d \sqrt{\sin ^2(c+d x)}}-\frac{3 B \sin (c+d x) (b \cos (c+d x))^{11/3} \, _2F_1\left(\frac{1}{2},\frac{11}{6};\frac{17}{6};\cos ^2(c+d x)\right)}{11 b^5 d \sqrt{\sin ^2(c+d x)}}+\frac{3 C \sin (c+d x) (b \cos (c+d x))^{8/3}}{11 b^4 d}",1,"(-3*Cos[c + d*x]^4*Sin[c + d*x]*((11*A + 8*C)*Hypergeometric2F1[1/2, 4/3, 7/3, Cos[c + d*x]^2] + 8*B*Cos[c + d*x]*Hypergeometric2F1[1/2, 11/6, 17/6, Cos[c + d*x]^2] - 8*C*Sqrt[Sin[c + d*x]^2]))/(88*d*(b*Cos[c + d*x])^(4/3)*Sqrt[Sin[c + d*x]^2])","A",1
358,1,114,154,0.2409442,"\int \frac{\cos ^2(c+d x) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{(b \cos (c+d x))^{4/3}} \, dx","Integrate[(Cos[c + d*x]^2*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(b*Cos[c + d*x])^(4/3),x]","-\frac{3 \sin (c+d x) \cos ^3(c+d x) \left((8 A+5 C) \, _2F_1\left(\frac{1}{2},\frac{5}{6};\frac{11}{6};\cos ^2(c+d x)\right)+5 B \cos (c+d x) \, _2F_1\left(\frac{1}{2},\frac{4}{3};\frac{7}{3};\cos ^2(c+d x)\right)-5 C \sqrt{\sin ^2(c+d x)}\right)}{40 d \sqrt{\sin ^2(c+d x)} (b \cos (c+d x))^{4/3}}","-\frac{3 (8 A+5 C) \sin (c+d x) (b \cos (c+d x))^{5/3} \, _2F_1\left(\frac{1}{2},\frac{5}{6};\frac{11}{6};\cos ^2(c+d x)\right)}{40 b^3 d \sqrt{\sin ^2(c+d x)}}-\frac{3 B \sin (c+d x) (b \cos (c+d x))^{8/3} \, _2F_1\left(\frac{1}{2},\frac{4}{3};\frac{7}{3};\cos ^2(c+d x)\right)}{8 b^4 d \sqrt{\sin ^2(c+d x)}}+\frac{3 C \sin (c+d x) (b \cos (c+d x))^{5/3}}{8 b^3 d}",1,"(-3*Cos[c + d*x]^3*Sin[c + d*x]*((8*A + 5*C)*Hypergeometric2F1[1/2, 5/6, 11/6, Cos[c + d*x]^2] + 5*B*Cos[c + d*x]*Hypergeometric2F1[1/2, 4/3, 7/3, Cos[c + d*x]^2] - 5*C*Sqrt[Sin[c + d*x]^2]))/(40*d*(b*Cos[c + d*x])^(4/3)*Sqrt[Sin[c + d*x]^2])","A",1
359,1,111,154,0.211914,"\int \frac{\cos (c+d x) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{(b \cos (c+d x))^{4/3}} \, dx","Integrate[(Cos[c + d*x]*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(b*Cos[c + d*x])^(4/3),x]","-\frac{3 \sin (2 (c+d x)) \left((5 A+2 C) \, _2F_1\left(\frac{1}{3},\frac{1}{2};\frac{4}{3};\cos ^2(c+d x)\right)+2 B \cos (c+d x) \, _2F_1\left(\frac{1}{2},\frac{5}{6};\frac{11}{6};\cos ^2(c+d x)\right)-2 C \sqrt{\sin ^2(c+d x)}\right)}{20 b d \sqrt{\sin ^2(c+d x)} \sqrt[3]{b \cos (c+d x)}}","-\frac{3 (5 A+2 C) \sin (c+d x) (b \cos (c+d x))^{2/3} \, _2F_1\left(\frac{1}{3},\frac{1}{2};\frac{4}{3};\cos ^2(c+d x)\right)}{10 b^2 d \sqrt{\sin ^2(c+d x)}}-\frac{3 B \sin (c+d x) (b \cos (c+d x))^{5/3} \, _2F_1\left(\frac{1}{2},\frac{5}{6};\frac{11}{6};\cos ^2(c+d x)\right)}{5 b^3 d \sqrt{\sin ^2(c+d x)}}+\frac{3 C \sin (c+d x) (b \cos (c+d x))^{2/3}}{5 b^2 d}",1,"(-3*((5*A + 2*C)*Hypergeometric2F1[1/3, 1/2, 4/3, Cos[c + d*x]^2] + 2*B*Cos[c + d*x]*Hypergeometric2F1[1/2, 5/6, 11/6, Cos[c + d*x]^2] - 2*C*Sqrt[Sin[c + d*x]^2])*Sin[2*(c + d*x)])/(20*b*d*(b*Cos[c + d*x])^(1/3)*Sqrt[Sin[c + d*x]^2])","A",1
360,1,115,152,0.2283408,"\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{(b \cos (c+d x))^{4/3}} \, dx","Integrate[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(b*Cos[c + d*x])^(4/3),x]","-\frac{3 \sqrt{\sin ^2(c+d x)} \cot (c+d x) \left(\cos (c+d x) \left(5 B \, _2F_1\left(\frac{1}{3},\frac{1}{2};\frac{4}{3};\cos ^2(c+d x)\right)+2 C \cos (c+d x) \, _2F_1\left(\frac{1}{2},\frac{5}{6};\frac{11}{6};\cos ^2(c+d x)\right)\right)-10 A \, _2F_1\left(-\frac{1}{6},\frac{1}{2};\frac{5}{6};\cos ^2(c+d x)\right)\right)}{10 d (b \cos (c+d x))^{4/3}}","\frac{3 (2 A-C) \sin (c+d x) (b \cos (c+d x))^{5/3} \, _2F_1\left(\frac{1}{2},\frac{5}{6};\frac{11}{6};\cos ^2(c+d x)\right)}{5 b^3 d \sqrt{\sin ^2(c+d x)}}+\frac{3 A \sin (c+d x)}{b d \sqrt[3]{b \cos (c+d x)}}-\frac{3 B \sin (c+d x) (b \cos (c+d x))^{2/3} \, _2F_1\left(\frac{1}{3},\frac{1}{2};\frac{4}{3};\cos ^2(c+d x)\right)}{2 b^2 d \sqrt{\sin ^2(c+d x)}}",1,"(-3*Cot[c + d*x]*(-10*A*Hypergeometric2F1[-1/6, 1/2, 5/6, Cos[c + d*x]^2] + Cos[c + d*x]*(5*B*Hypergeometric2F1[1/3, 1/2, 4/3, Cos[c + d*x]^2] + 2*C*Cos[c + d*x]*Hypergeometric2F1[1/2, 5/6, 11/6, Cos[c + d*x]^2]))*Sqrt[Sin[c + d*x]^2])/(10*d*(b*Cos[c + d*x])^(4/3))","A",1
361,1,703,147,6.2940205,"\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec (c+d x)}{(b \cos (c+d x))^{4/3}} \, dx","Integrate[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x])/(b*Cos[c + d*x])^(4/3),x]","\frac{\frac{4 B \csc (c) \cos ^{\frac{7}{3}}(c+d x) \left(A \sec ^2(c+d x)+B \sec (c+d x)+C\right) \left(\frac{\tan (c) \sin \left(\tan ^{-1}(\tan (c))+d x\right) \, _2F_1\left(-\frac{1}{2},-\frac{1}{6};\frac{5}{6};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right)}{\sqrt{\tan ^2(c)+1} \sqrt{1-\cos \left(\tan ^{-1}(\tan (c))+d x\right)} \sqrt{\cos \left(\tan ^{-1}(\tan (c))+d x\right)+1} \sqrt[3]{\cos (c) \sqrt{\tan ^2(c)+1} \cos \left(\tan ^{-1}(\tan (c))+d x\right)}}-\frac{\frac{\tan (c) \sin \left(\tan ^{-1}(\tan (c))+d x\right)}{\sqrt{\tan ^2(c)+1}}+\frac{3 \cos ^2(c) \sqrt{\tan ^2(c)+1} \cos \left(\tan ^{-1}(\tan (c))+d x\right)}{2 \left(\sin ^2(c)+\cos ^2(c)\right)}}{\sqrt[3]{\cos (c) \sqrt{\tan ^2(c)+1} \cos \left(\tan ^{-1}(\tan (c))+d x\right)}}\right)}{d \sqrt[3]{b \cos (c+d x)} (2 A+2 B \cos (c+d x)+C \cos (2 c+2 d x)+C)}-\frac{A \cos ^{\frac{7}{3}}(c+d x) \cos \left(d x-\tan ^{-1}(\cot (c))\right) \sin \left(d x-\tan ^{-1}(\cot (c))\right) \left(A \sec ^2(c+d x)+B \sec (c+d x)+C\right) \, _2F_1\left(\frac{1}{2},\frac{2}{3};\frac{3}{2};\cos ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right)}{2 d \sqrt[3]{b \cos (c+d x)} \sqrt[3]{\cos (c) \cos (d x)-\sin (c) \sin (d x)} \sqrt[3]{\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)} (2 A+2 B \cos (c+d x)+C \cos (2 c+2 d x)+C)}-\frac{2 C \cos ^{\frac{7}{3}}(c+d x) \cos \left(d x-\tan ^{-1}(\cot (c))\right) \sin \left(d x-\tan ^{-1}(\cot (c))\right) \left(A \sec ^2(c+d x)+B \sec (c+d x)+C\right) \, _2F_1\left(\frac{1}{2},\frac{2}{3};\frac{3}{2};\cos ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right)}{d \sqrt[3]{b \cos (c+d x)} \sqrt[3]{\cos (c) \cos (d x)-\sin (c) \sin (d x)} \sqrt[3]{\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)} (2 A+2 B \cos (c+d x)+C \cos (2 c+2 d x)+C)}+\frac{\cos ^3(c+d x) \left(A \sec ^2(c+d x)+B \sec (c+d x)+C\right) \left(\frac{3 \sec (c) \sec (c+d x) (A \sin (c)+4 B \sin (d x))}{2 d}+\frac{3 A \sec (c) \sin (d x) \sec ^2(c+d x)}{2 d}+\frac{6 B \csc (c) \sec (c)}{d}\right)}{\sqrt[3]{b \cos (c+d x)} (2 A+2 B \cos (c+d x)+C \cos (2 c+2 d x)+C)}}{b}","-\frac{3 (A+4 C) \sin (c+d x) (b \cos (c+d x))^{2/3} \, _2F_1\left(\frac{1}{3},\frac{1}{2};\frac{4}{3};\cos ^2(c+d x)\right)}{8 b^2 d \sqrt{\sin ^2(c+d x)}}+\frac{3 A \sin (c+d x)}{4 d (b \cos (c+d x))^{4/3}}+\frac{3 B \sin (c+d x) \, _2F_1\left(-\frac{1}{6},\frac{1}{2};\frac{5}{6};\cos ^2(c+d x)\right)}{b d \sqrt{\sin ^2(c+d x)} \sqrt[3]{b \cos (c+d x)}}",1,"((Cos[c + d*x]^3*(C + B*Sec[c + d*x] + A*Sec[c + d*x]^2)*((6*B*Csc[c]*Sec[c])/d + (3*A*Sec[c]*Sec[c + d*x]^2*Sin[d*x])/(2*d) + (3*Sec[c]*Sec[c + d*x]*(A*Sin[c] + 4*B*Sin[d*x]))/(2*d)))/((b*Cos[c + d*x])^(1/3)*(2*A + C + 2*B*Cos[c + d*x] + C*Cos[2*c + 2*d*x])) - (A*Cos[c + d*x]^(7/3)*Cos[d*x - ArcTan[Cot[c]]]*Hypergeometric2F1[1/2, 2/3, 3/2, Cos[d*x - ArcTan[Cot[c]]]^2]*(C + B*Sec[c + d*x] + A*Sec[c + d*x]^2)*Sin[d*x - ArcTan[Cot[c]]])/(2*d*(b*Cos[c + d*x])^(1/3)*(2*A + C + 2*B*Cos[c + d*x] + C*Cos[2*c + 2*d*x])*(Cos[c]*Cos[d*x] - Sin[c]*Sin[d*x])^(1/3)*(Sin[d*x - ArcTan[Cot[c]]]^2)^(1/3)) - (2*C*Cos[c + d*x]^(7/3)*Cos[d*x - ArcTan[Cot[c]]]*Hypergeometric2F1[1/2, 2/3, 3/2, Cos[d*x - ArcTan[Cot[c]]]^2]*(C + B*Sec[c + d*x] + A*Sec[c + d*x]^2)*Sin[d*x - ArcTan[Cot[c]]])/(d*(b*Cos[c + d*x])^(1/3)*(2*A + C + 2*B*Cos[c + d*x] + C*Cos[2*c + 2*d*x])*(Cos[c]*Cos[d*x] - Sin[c]*Sin[d*x])^(1/3)*(Sin[d*x - ArcTan[Cot[c]]]^2)^(1/3)) + (4*B*Cos[c + d*x]^(7/3)*Csc[c]*(C + B*Sec[c + d*x] + A*Sec[c + d*x]^2)*((HypergeometricPFQ[{-1/2, -1/6}, {5/6}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*(Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])^(1/3)*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (3*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(2*(Cos[c]^2 + Sin[c]^2)))/(Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])^(1/3)))/(d*(b*Cos[c + d*x])^(1/3)*(2*A + C + 2*B*Cos[c + d*x] + C*Cos[2*c + 2*d*x])))/b","B",1
362,1,118,149,0.331026,"\int \frac{\left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^2(c+d x)}{(b \cos (c+d x))^{4/3}} \, dx","Integrate[((A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^2)/(b*Cos[c + d*x])^(4/3),x]","\frac{3 b^2 \sqrt{\sin ^2(c+d x)} \cot (c+d x) \left(4 A \, _2F_1\left(-\frac{7}{6},\frac{1}{2};-\frac{1}{6};\cos ^2(c+d x)\right)+7 \cos (c+d x) \left(B \, _2F_1\left(-\frac{2}{3},\frac{1}{2};\frac{1}{3};\cos ^2(c+d x)\right)+4 C \cos (c+d x) \, _2F_1\left(-\frac{1}{6},\frac{1}{2};\frac{5}{6};\cos ^2(c+d x)\right)\right)\right)}{28 d (b \cos (c+d x))^{10/3}}","\frac{3 (4 A+7 C) \sin (c+d x) \, _2F_1\left(-\frac{1}{6},\frac{1}{2};\frac{5}{6};\cos ^2(c+d x)\right)}{7 b d \sqrt{\sin ^2(c+d x)} \sqrt[3]{b \cos (c+d x)}}+\frac{3 A b \sin (c+d x)}{7 d (b \cos (c+d x))^{7/3}}+\frac{3 B \sin (c+d x) \, _2F_1\left(-\frac{2}{3},\frac{1}{2};\frac{1}{3};\cos ^2(c+d x)\right)}{4 d \sqrt{\sin ^2(c+d x)} (b \cos (c+d x))^{4/3}}",1,"(3*b^2*Cot[c + d*x]*(4*A*Hypergeometric2F1[-7/6, 1/2, -1/6, Cos[c + d*x]^2] + 7*Cos[c + d*x]*(B*Hypergeometric2F1[-2/3, 1/2, 1/3, Cos[c + d*x]^2] + 4*C*Cos[c + d*x]*Hypergeometric2F1[-1/6, 1/2, 5/6, Cos[c + d*x]^2]))*Sqrt[Sin[c + d*x]^2])/(28*d*(b*Cos[c + d*x])^(10/3))","A",1
363,1,169,232,0.6747977,"\int \cos ^m(c+d x) (b \cos (c+d x))^{4/3} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^m*(b*Cos[c + d*x])^(4/3)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","-\frac{3 \sin (c+d x) (b \cos (c+d x))^{4/3} \cos ^{m+1}(c+d x) \left((A (3 m+10)+C (3 m+7)) \, _2F_1\left(\frac{1}{2},\frac{1}{6} (3 m+7);\frac{1}{6} (3 m+13);\cos ^2(c+d x)\right)+B (3 m+7) \cos (c+d x) \, _2F_1\left(\frac{1}{2},\frac{m}{2}+\frac{5}{3};\frac{m}{2}+\frac{8}{3};\cos ^2(c+d x)\right)-C (3 m+7) \sqrt{\sin ^2(c+d x)}\right)}{d (3 m+7) (3 m+10) \sqrt{\sin ^2(c+d x)}}","-\frac{3 b (A (3 m+10)+C (3 m+7)) \sin (c+d x) \sqrt[3]{b \cos (c+d x)} \cos ^{m+2}(c+d x) \, _2F_1\left(\frac{1}{2},\frac{1}{6} (3 m+7);\frac{1}{6} (3 m+13);\cos ^2(c+d x)\right)}{d (3 m+7) (3 m+10) \sqrt{\sin ^2(c+d x)}}-\frac{3 b B \sin (c+d x) \sqrt[3]{b \cos (c+d x)} \cos ^{m+3}(c+d x) \, _2F_1\left(\frac{1}{2},\frac{1}{6} (3 m+10);\frac{1}{6} (3 m+16);\cos ^2(c+d x)\right)}{d (3 m+10) \sqrt{\sin ^2(c+d x)}}+\frac{3 b C \sin (c+d x) \sqrt[3]{b \cos (c+d x)} \cos ^{m+2}(c+d x)}{d (3 m+10)}",1,"(-3*Cos[c + d*x]^(1 + m)*(b*Cos[c + d*x])^(4/3)*Sin[c + d*x]*(B*(7 + 3*m)*Cos[c + d*x]*Hypergeometric2F1[1/2, 5/3 + m/2, 8/3 + m/2, Cos[c + d*x]^2] + (C*(7 + 3*m) + A*(10 + 3*m))*Hypergeometric2F1[1/2, (7 + 3*m)/6, (13 + 3*m)/6, Cos[c + d*x]^2] - C*(7 + 3*m)*Sqrt[Sin[c + d*x]^2]))/(d*(7 + 3*m)*(10 + 3*m)*Sqrt[Sin[c + d*x]^2])","A",1
364,1,166,229,0.4354533,"\int \cos ^m(c+d x) (b \cos (c+d x))^{2/3} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^m*(b*Cos[c + d*x])^(2/3)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","-\frac{3 \sin (c+d x) (b \cos (c+d x))^{2/3} \cos ^{m+1}(c+d x) \left((A (3 m+8)+C (3 m+5)) \, _2F_1\left(\frac{1}{2},\frac{1}{6} (3 m+5);\frac{1}{6} (3 m+11);\cos ^2(c+d x)\right)+(3 m+5) \left(B \cos (c+d x) \, _2F_1\left(\frac{1}{2},\frac{1}{6} (3 m+8);\frac{m}{2}+\frac{7}{3};\cos ^2(c+d x)\right)-C \sqrt{\sin ^2(c+d x)}\right)\right)}{d (3 m+5) (3 m+8) \sqrt{\sin ^2(c+d x)}}","-\frac{3 (A (3 m+8)+C (3 m+5)) \sin (c+d x) (b \cos (c+d x))^{2/3} \cos ^{m+1}(c+d x) \, _2F_1\left(\frac{1}{2},\frac{1}{6} (3 m+5);\frac{1}{6} (3 m+11);\cos ^2(c+d x)\right)}{d (3 m+5) (3 m+8) \sqrt{\sin ^2(c+d x)}}-\frac{3 B \sin (c+d x) (b \cos (c+d x))^{2/3} \cos ^{m+2}(c+d x) \, _2F_1\left(\frac{1}{2},\frac{1}{6} (3 m+8);\frac{1}{6} (3 m+14);\cos ^2(c+d x)\right)}{d (3 m+8) \sqrt{\sin ^2(c+d x)}}+\frac{3 C \sin (c+d x) (b \cos (c+d x))^{2/3} \cos ^{m+1}(c+d x)}{d (3 m+8)}",1,"(-3*Cos[c + d*x]^(1 + m)*(b*Cos[c + d*x])^(2/3)*Sin[c + d*x]*((C*(5 + 3*m) + A*(8 + 3*m))*Hypergeometric2F1[1/2, (5 + 3*m)/6, (11 + 3*m)/6, Cos[c + d*x]^2] + (5 + 3*m)*(B*Cos[c + d*x]*Hypergeometric2F1[1/2, (8 + 3*m)/6, 7/3 + m/2, Cos[c + d*x]^2] - C*Sqrt[Sin[c + d*x]^2])))/(d*(5 + 3*m)*(8 + 3*m)*Sqrt[Sin[c + d*x]^2])","A",1
365,1,166,229,0.405914,"\int \cos ^m(c+d x) \sqrt[3]{b \cos (c+d x)} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^m*(b*Cos[c + d*x])^(1/3)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","-\frac{3 \sin (c+d x) \sqrt[3]{b \cos (c+d x)} \cos ^{m+1}(c+d x) \left((A (3 m+7)+C (3 m+4)) \, _2F_1\left(\frac{1}{2},\frac{1}{6} (3 m+4);\frac{m}{2}+\frac{5}{3};\cos ^2(c+d x)\right)+(3 m+4) \left(B \cos (c+d x) \, _2F_1\left(\frac{1}{2},\frac{1}{6} (3 m+7);\frac{1}{6} (3 m+13);\cos ^2(c+d x)\right)-C \sqrt{\sin ^2(c+d x)}\right)\right)}{d (3 m+4) (3 m+7) \sqrt{\sin ^2(c+d x)}}","-\frac{3 (A (3 m+7)+C (3 m+4)) \sin (c+d x) \sqrt[3]{b \cos (c+d x)} \cos ^{m+1}(c+d x) \, _2F_1\left(\frac{1}{2},\frac{1}{6} (3 m+4);\frac{1}{6} (3 m+10);\cos ^2(c+d x)\right)}{d (3 m+4) (3 m+7) \sqrt{\sin ^2(c+d x)}}-\frac{3 B \sin (c+d x) \sqrt[3]{b \cos (c+d x)} \cos ^{m+2}(c+d x) \, _2F_1\left(\frac{1}{2},\frac{1}{6} (3 m+7);\frac{1}{6} (3 m+13);\cos ^2(c+d x)\right)}{d (3 m+7) \sqrt{\sin ^2(c+d x)}}+\frac{3 C \sin (c+d x) \sqrt[3]{b \cos (c+d x)} \cos ^{m+1}(c+d x)}{d (3 m+7)}",1,"(-3*Cos[c + d*x]^(1 + m)*(b*Cos[c + d*x])^(1/3)*Sin[c + d*x]*((C*(4 + 3*m) + A*(7 + 3*m))*Hypergeometric2F1[1/2, (4 + 3*m)/6, 5/3 + m/2, Cos[c + d*x]^2] + (4 + 3*m)*(B*Cos[c + d*x]*Hypergeometric2F1[1/2, (7 + 3*m)/6, (13 + 3*m)/6, Cos[c + d*x]^2] - C*Sqrt[Sin[c + d*x]^2])))/(d*(4 + 3*m)*(7 + 3*m)*Sqrt[Sin[c + d*x]^2])","A",1
366,1,166,229,0.4255085,"\int \frac{\cos ^m(c+d x) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\sqrt[3]{b \cos (c+d x)}} \, dx","Integrate[(Cos[c + d*x]^m*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(b*Cos[c + d*x])^(1/3),x]","-\frac{3 \sin (c+d x) \cos ^{m+1}(c+d x) \left((A (3 m+5)+C (3 m+2)) \, _2F_1\left(\frac{1}{2},\frac{1}{6} (3 m+2);\frac{1}{6} (3 m+8);\cos ^2(c+d x)\right)+(3 m+2) \left(B \cos (c+d x) \, _2F_1\left(\frac{1}{2},\frac{1}{6} (3 m+5);\frac{1}{6} (3 m+11);\cos ^2(c+d x)\right)-C \sqrt{\sin ^2(c+d x)}\right)\right)}{d (3 m+2) (3 m+5) \sqrt{\sin ^2(c+d x)} \sqrt[3]{b \cos (c+d x)}}","-\frac{3 (A (3 m+5)+C (3 m+2)) \sin (c+d x) \cos ^{m+1}(c+d x) \, _2F_1\left(\frac{1}{2},\frac{1}{6} (3 m+2);\frac{1}{6} (3 m+8);\cos ^2(c+d x)\right)}{d (3 m+2) (3 m+5) \sqrt{\sin ^2(c+d x)} \sqrt[3]{b \cos (c+d x)}}-\frac{3 B \sin (c+d x) \cos ^{m+2}(c+d x) \, _2F_1\left(\frac{1}{2},\frac{1}{6} (3 m+5);\frac{1}{6} (3 m+11);\cos ^2(c+d x)\right)}{d (3 m+5) \sqrt{\sin ^2(c+d x)} \sqrt[3]{b \cos (c+d x)}}+\frac{3 C \sin (c+d x) \cos ^{m+1}(c+d x)}{d (3 m+5) \sqrt[3]{b \cos (c+d x)}}",1,"(-3*Cos[c + d*x]^(1 + m)*Sin[c + d*x]*((C*(2 + 3*m) + A*(5 + 3*m))*Hypergeometric2F1[1/2, (2 + 3*m)/6, (8 + 3*m)/6, Cos[c + d*x]^2] + (2 + 3*m)*(B*Cos[c + d*x]*Hypergeometric2F1[1/2, (5 + 3*m)/6, (11 + 3*m)/6, Cos[c + d*x]^2] - C*Sqrt[Sin[c + d*x]^2])))/(d*(2 + 3*m)*(5 + 3*m)*(b*Cos[c + d*x])^(1/3)*Sqrt[Sin[c + d*x]^2])","A",1
367,1,164,227,0.4056318,"\int \frac{\cos ^m(c+d x) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{(b \cos (c+d x))^{2/3}} \, dx","Integrate[(Cos[c + d*x]^m*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(b*Cos[c + d*x])^(2/3),x]","-\frac{3 \sin (c+d x) \cos ^{m+1}(c+d x) \left((A (3 m+4)+3 C m+C) \, _2F_1\left(\frac{1}{2},\frac{1}{6} (3 m+1);\frac{1}{6} (3 m+7);\cos ^2(c+d x)\right)+(3 m+1) \left(B \cos (c+d x) \, _2F_1\left(\frac{1}{2},\frac{1}{6} (3 m+4);\frac{m}{2}+\frac{5}{3};\cos ^2(c+d x)\right)-C \sqrt{\sin ^2(c+d x)}\right)\right)}{d (3 m+1) (3 m+4) \sqrt{\sin ^2(c+d x)} (b \cos (c+d x))^{2/3}}","-\frac{3 (A (3 m+4)+3 C m+C) \sin (c+d x) \cos ^{m+1}(c+d x) \, _2F_1\left(\frac{1}{2},\frac{1}{6} (3 m+1);\frac{1}{6} (3 m+7);\cos ^2(c+d x)\right)}{d (3 m+1) (3 m+4) \sqrt{\sin ^2(c+d x)} (b \cos (c+d x))^{2/3}}-\frac{3 B \sin (c+d x) \cos ^{m+2}(c+d x) \, _2F_1\left(\frac{1}{2},\frac{1}{6} (3 m+4);\frac{1}{6} (3 m+10);\cos ^2(c+d x)\right)}{d (3 m+4) \sqrt{\sin ^2(c+d x)} (b \cos (c+d x))^{2/3}}+\frac{3 C \sin (c+d x) \cos ^{m+1}(c+d x)}{d (3 m+4) (b \cos (c+d x))^{2/3}}",1,"(-3*Cos[c + d*x]^(1 + m)*Sin[c + d*x]*((C + 3*C*m + A*(4 + 3*m))*Hypergeometric2F1[1/2, (1 + 3*m)/6, (7 + 3*m)/6, Cos[c + d*x]^2] + (1 + 3*m)*(B*Cos[c + d*x]*Hypergeometric2F1[1/2, (4 + 3*m)/6, 5/3 + m/2, Cos[c + d*x]^2] - C*Sqrt[Sin[c + d*x]^2])))/(d*(1 + 3*m)*(4 + 3*m)*(b*Cos[c + d*x])^(2/3)*Sqrt[Sin[c + d*x]^2])","A",1
368,1,166,235,0.5525908,"\int \frac{\cos ^m(c+d x) \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{(b \cos (c+d x))^{4/3}} \, dx","Integrate[(Cos[c + d*x]^m*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(b*Cos[c + d*x])^(4/3),x]","-\frac{3 \sin (c+d x) \cos ^{m+1}(c+d x) \left((A (3 m+2)+C (3 m-1)) \, _2F_1\left(\frac{1}{2},\frac{1}{6} (3 m-1);\frac{1}{6} (3 m+5);\cos ^2(c+d x)\right)+(3 m-1) \left(B \cos (c+d x) \, _2F_1\left(\frac{1}{2},\frac{1}{6} (3 m+2);\frac{1}{6} (3 m+8);\cos ^2(c+d x)\right)-C \sqrt{\sin ^2(c+d x)}\right)\right)}{d (3 m-1) (3 m+2) \sqrt{\sin ^2(c+d x)} (b \cos (c+d x))^{4/3}}","-\frac{3 (C (1-3 m)-A (3 m+2)) \sin (c+d x) \cos ^m(c+d x) \, _2F_1\left(\frac{1}{2},\frac{1}{6} (3 m-1);\frac{1}{6} (3 m+5);\cos ^2(c+d x)\right)}{b d (1-3 m) (3 m+2) \sqrt{\sin ^2(c+d x)} \sqrt[3]{b \cos (c+d x)}}-\frac{3 B \sin (c+d x) \cos ^{m+1}(c+d x) \, _2F_1\left(\frac{1}{2},\frac{1}{6} (3 m+2);\frac{1}{6} (3 m+8);\cos ^2(c+d x)\right)}{b d (3 m+2) \sqrt{\sin ^2(c+d x)} \sqrt[3]{b \cos (c+d x)}}+\frac{3 C \sin (c+d x) \cos ^m(c+d x)}{b d (3 m+2) \sqrt[3]{b \cos (c+d x)}}",1,"(-3*Cos[c + d*x]^(1 + m)*Sin[c + d*x]*((C*(-1 + 3*m) + A*(2 + 3*m))*Hypergeometric2F1[1/2, (-1 + 3*m)/6, (5 + 3*m)/6, Cos[c + d*x]^2] + (-1 + 3*m)*(B*Cos[c + d*x]*Hypergeometric2F1[1/2, (2 + 3*m)/6, (8 + 3*m)/6, Cos[c + d*x]^2] - C*Sqrt[Sin[c + d*x]^2])))/(d*(-1 + 3*m)*(2 + 3*m)*(b*Cos[c + d*x])^(4/3)*Sqrt[Sin[c + d*x]^2])","A",1
369,1,161,227,0.2592676,"\int (a \cos (c+d x))^m (b \cos (c+d x))^n \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Integrate[(a*Cos[c + d*x])^m*(b*Cos[c + d*x])^n*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","-\frac{\sin (c+d x) \cos (c+d x) (a \cos (c+d x))^m (b \cos (c+d x))^n \left((A (m+n+2)+C (m+n+1)) \, _2F_1\left(\frac{1}{2},\frac{1}{2} (m+n+1);\frac{1}{2} (m+n+3);\cos ^2(c+d x)\right)+(m+n+1) \left(B \cos (c+d x) \, _2F_1\left(\frac{1}{2},\frac{1}{2} (m+n+2);\frac{1}{2} (m+n+4);\cos ^2(c+d x)\right)-C \sqrt{\sin ^2(c+d x)}\right)\right)}{d (m+n+1) (m+n+2) \sqrt{\sin ^2(c+d x)}}","-\frac{B \sin (c+d x) (a \cos (c+d x))^{m+2} (b \cos (c+d x))^n \, _2F_1\left(\frac{1}{2},\frac{1}{2} (m+n+2);\frac{1}{2} (m+n+4);\cos ^2(c+d x)\right)}{a^2 d (m+n+2) \sqrt{\sin ^2(c+d x)}}-\frac{(A (m+n+2)+C (m+n+1)) \sin (c+d x) (a \cos (c+d x))^{m+1} (b \cos (c+d x))^n \, _2F_1\left(\frac{1}{2},\frac{1}{2} (m+n+1);\frac{1}{2} (m+n+3);\cos ^2(c+d x)\right)}{a d (m+n+1) (m+n+2) \sqrt{\sin ^2(c+d x)}}+\frac{C \sin (c+d x) (a \cos (c+d x))^{m+1} (b \cos (c+d x))^n}{a d (m+n+2)}",1,"-((Cos[c + d*x]*(a*Cos[c + d*x])^m*(b*Cos[c + d*x])^n*Sin[c + d*x]*((C*(1 + m + n) + A*(2 + m + n))*Hypergeometric2F1[1/2, (1 + m + n)/2, (3 + m + n)/2, Cos[c + d*x]^2] + (1 + m + n)*(B*Cos[c + d*x]*Hypergeometric2F1[1/2, (2 + m + n)/2, (4 + m + n)/2, Cos[c + d*x]^2] - C*Sqrt[Sin[c + d*x]^2])))/(d*(1 + m + n)*(2 + m + n)*Sqrt[Sin[c + d*x]^2]))","A",1
370,1,144,187,0.4959754,"\int \cos ^2(c+d x) (b \cos (c+d x))^n \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^2*(b*Cos[c + d*x])^n*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","-\frac{\sin (c+d x) \cos ^3(c+d x) (b \cos (c+d x))^n \left((A (n+4)+C (n+3)) \, _2F_1\left(\frac{1}{2},\frac{n+3}{2};\frac{n+5}{2};\cos ^2(c+d x)\right)+(n+3) \left(B \cos (c+d x) \, _2F_1\left(\frac{1}{2},\frac{n+4}{2};\frac{n+6}{2};\cos ^2(c+d x)\right)-C \sqrt{\sin ^2(c+d x)}\right)\right)}{d (n+3) (n+4) \sqrt{\sin ^2(c+d x)}}","-\frac{(A (n+4)+C (n+3)) \sin (c+d x) (b \cos (c+d x))^{n+3} \, _2F_1\left(\frac{1}{2},\frac{n+3}{2};\frac{n+5}{2};\cos ^2(c+d x)\right)}{b^3 d (n+3) (n+4) \sqrt{\sin ^2(c+d x)}}-\frac{B \sin (c+d x) (b \cos (c+d x))^{n+4} \, _2F_1\left(\frac{1}{2},\frac{n+4}{2};\frac{n+6}{2};\cos ^2(c+d x)\right)}{b^4 d (n+4) \sqrt{\sin ^2(c+d x)}}+\frac{C \sin (c+d x) (b \cos (c+d x))^{n+3}}{b^3 d (n+4)}",1,"-((Cos[c + d*x]^3*(b*Cos[c + d*x])^n*Sin[c + d*x]*((C*(3 + n) + A*(4 + n))*Hypergeometric2F1[1/2, (3 + n)/2, (5 + n)/2, Cos[c + d*x]^2] + (3 + n)*(B*Cos[c + d*x]*Hypergeometric2F1[1/2, (4 + n)/2, (6 + n)/2, Cos[c + d*x]^2] - C*Sqrt[Sin[c + d*x]^2])))/(d*(3 + n)*(4 + n)*Sqrt[Sin[c + d*x]^2]))","A",1
371,1,144,187,0.3312861,"\int \cos (c+d x) (b \cos (c+d x))^n \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]*(b*Cos[c + d*x])^n*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","-\frac{\sin (c+d x) \cos ^2(c+d x) (b \cos (c+d x))^n \left((A (n+3)+C (n+2)) \, _2F_1\left(\frac{1}{2},\frac{n+2}{2};\frac{n+4}{2};\cos ^2(c+d x)\right)+(n+2) \left(B \cos (c+d x) \, _2F_1\left(\frac{1}{2},\frac{n+3}{2};\frac{n+5}{2};\cos ^2(c+d x)\right)-C \sqrt{\sin ^2(c+d x)}\right)\right)}{d (n+2) (n+3) \sqrt{\sin ^2(c+d x)}}","-\frac{(A (n+3)+C (n+2)) \sin (c+d x) (b \cos (c+d x))^{n+2} \, _2F_1\left(\frac{1}{2},\frac{n+2}{2};\frac{n+4}{2};\cos ^2(c+d x)\right)}{b^2 d (n+2) (n+3) \sqrt{\sin ^2(c+d x)}}-\frac{B \sin (c+d x) (b \cos (c+d x))^{n+3} \, _2F_1\left(\frac{1}{2},\frac{n+3}{2};\frac{n+5}{2};\cos ^2(c+d x)\right)}{b^3 d (n+3) \sqrt{\sin ^2(c+d x)}}+\frac{C \sin (c+d x) (b \cos (c+d x))^{n+2}}{b^2 d (n+3)}",1,"-((Cos[c + d*x]^2*(b*Cos[c + d*x])^n*Sin[c + d*x]*((C*(2 + n) + A*(3 + n))*Hypergeometric2F1[1/2, (2 + n)/2, (4 + n)/2, Cos[c + d*x]^2] + (2 + n)*(B*Cos[c + d*x]*Hypergeometric2F1[1/2, (3 + n)/2, (5 + n)/2, Cos[c + d*x]^2] - C*Sqrt[Sin[c + d*x]^2])))/(d*(2 + n)*(3 + n)*Sqrt[Sin[c + d*x]^2]))","A",1
372,1,142,187,0.2378259,"\int (b \cos (c+d x))^n \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Integrate[(b*Cos[c + d*x])^n*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","-\frac{\sin (c+d x) \cos (c+d x) (b \cos (c+d x))^n \left((A (n+2)+C (n+1)) \, _2F_1\left(\frac{1}{2},\frac{n+1}{2};\frac{n+3}{2};\cos ^2(c+d x)\right)+(n+1) \left(B \cos (c+d x) \, _2F_1\left(\frac{1}{2},\frac{n+2}{2};\frac{n+4}{2};\cos ^2(c+d x)\right)-C \sqrt{\sin ^2(c+d x)}\right)\right)}{d (n+1) (n+2) \sqrt{\sin ^2(c+d x)}}","-\frac{(A (n+2)+C (n+1)) \sin (c+d x) (b \cos (c+d x))^{n+1} \, _2F_1\left(\frac{1}{2},\frac{n+1}{2};\frac{n+3}{2};\cos ^2(c+d x)\right)}{b d (n+1) (n+2) \sqrt{\sin ^2(c+d x)}}-\frac{B \sin (c+d x) (b \cos (c+d x))^{n+2} \, _2F_1\left(\frac{1}{2},\frac{n+2}{2};\frac{n+4}{2};\cos ^2(c+d x)\right)}{b^2 d (n+2) \sqrt{\sin ^2(c+d x)}}+\frac{C \sin (c+d x) (b \cos (c+d x))^{n+1}}{b d (n+2)}",1,"-((Cos[c + d*x]*(b*Cos[c + d*x])^n*Sin[c + d*x]*((C*(1 + n) + A*(2 + n))*Hypergeometric2F1[1/2, (1 + n)/2, (3 + n)/2, Cos[c + d*x]^2] + (1 + n)*(B*Cos[c + d*x]*Hypergeometric2F1[1/2, (2 + n)/2, (4 + n)/2, Cos[c + d*x]^2] - C*Sqrt[Sin[c + d*x]^2])))/(d*(1 + n)*(2 + n)*Sqrt[Sin[c + d*x]^2]))","A",1
373,1,127,170,0.210035,"\int (b \cos (c+d x))^n \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec (c+d x) \, dx","Integrate[(b*Cos[c + d*x])^n*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x],x]","-\frac{\sin (c+d x) (b \cos (c+d x))^n \left((A n+A+C n) \, _2F_1\left(\frac{1}{2},\frac{n}{2};\frac{n+2}{2};\cos ^2(c+d x)\right)+n \left(B \cos (c+d x) \, _2F_1\left(\frac{1}{2},\frac{n+1}{2};\frac{n+3}{2};\cos ^2(c+d x)\right)-C \sqrt{\sin ^2(c+d x)}\right)\right)}{d n (n+1) \sqrt{\sin ^2(c+d x)}}","-\frac{(A n+A+C n) \sin (c+d x) (b \cos (c+d x))^n \, _2F_1\left(\frac{1}{2},\frac{n}{2};\frac{n+2}{2};\cos ^2(c+d x)\right)}{d n (n+1) \sqrt{\sin ^2(c+d x)}}-\frac{B \sin (c+d x) (b \cos (c+d x))^{n+1} \, _2F_1\left(\frac{1}{2},\frac{n+1}{2};\frac{n+3}{2};\cos ^2(c+d x)\right)}{b d (n+1) \sqrt{\sin ^2(c+d x)}}+\frac{C \sin (c+d x) (b \cos (c+d x))^n}{d (n+1)}",1,"-(((b*Cos[c + d*x])^n*Sin[c + d*x]*((A + A*n + C*n)*Hypergeometric2F1[1/2, n/2, (2 + n)/2, Cos[c + d*x]^2] + n*(B*Cos[c + d*x]*Hypergeometric2F1[1/2, (1 + n)/2, (3 + n)/2, Cos[c + d*x]^2] - C*Sqrt[Sin[c + d*x]^2])))/(d*n*(1 + n)*Sqrt[Sin[c + d*x]^2]))","A",1
374,1,131,173,0.2939231,"\int (b \cos (c+d x))^n \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^2(c+d x) \, dx","Integrate[(b*Cos[c + d*x])^n*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^2,x]","\frac{\tan (c+d x) (b \cos (c+d x))^n \left((-A n+C (-n)+C) \, _2F_1\left(\frac{1}{2},\frac{n-1}{2};\frac{n+1}{2};\cos ^2(c+d x)\right)-(n-1) \left(B \cos (c+d x) \, _2F_1\left(\frac{1}{2},\frac{n}{2};\frac{n+2}{2};\cos ^2(c+d x)\right)-C \sqrt{\sin ^2(c+d x)}\right)\right)}{d (n-1) n \sqrt{\sin ^2(c+d x)}}","-\frac{b (C (1-n)-A n) \sin (c+d x) (b \cos (c+d x))^{n-1} \, _2F_1\left(\frac{1}{2},\frac{n-1}{2};\frac{n+1}{2};\cos ^2(c+d x)\right)}{d (1-n) n \sqrt{\sin ^2(c+d x)}}-\frac{B \sin (c+d x) (b \cos (c+d x))^n \, _2F_1\left(\frac{1}{2},\frac{n}{2};\frac{n+2}{2};\cos ^2(c+d x)\right)}{d n \sqrt{\sin ^2(c+d x)}}+\frac{b C \sin (c+d x) (b \cos (c+d x))^{n-1}}{d n}",1,"((b*Cos[c + d*x])^n*((C - A*n - C*n)*Hypergeometric2F1[1/2, (-1 + n)/2, (1 + n)/2, Cos[c + d*x]^2] - (-1 + n)*(B*Cos[c + d*x]*Hypergeometric2F1[1/2, n/2, (2 + n)/2, Cos[c + d*x]^2] - C*Sqrt[Sin[c + d*x]^2]))*Tan[c + d*x])/(d*(-1 + n)*n*Sqrt[Sin[c + d*x]^2])","A",1
375,1,137,194,0.4654026,"\int (b \cos (c+d x))^n \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^3(c+d x) \, dx","Integrate[(b*Cos[c + d*x])^n*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^3,x]","-\frac{b \tan (c+d x) (b \cos (c+d x))^{n-1} \left((A (n-1)+C (n-2)) \, _2F_1\left(\frac{1}{2},\frac{n-2}{2};\frac{n}{2};\cos ^2(c+d x)\right)+(n-2) \left(B \cos (c+d x) \, _2F_1\left(\frac{1}{2},\frac{n-1}{2};\frac{n+1}{2};\cos ^2(c+d x)\right)-C \sqrt{\sin ^2(c+d x)}\right)\right)}{d (n-2) (n-1) \sqrt{\sin ^2(c+d x)}}","\frac{b^2 (A (1-n)+C (2-n)) \sin (c+d x) (b \cos (c+d x))^{n-2} \, _2F_1\left(\frac{1}{2},\frac{n-2}{2};\frac{n}{2};\cos ^2(c+d x)\right)}{d (1-n) (2-n) \sqrt{\sin ^2(c+d x)}}-\frac{b^2 C \sin (c+d x) (b \cos (c+d x))^{n-2}}{d (1-n)}+\frac{b B \sin (c+d x) (b \cos (c+d x))^{n-1} \, _2F_1\left(\frac{1}{2},\frac{n-1}{2};\frac{n+1}{2};\cos ^2(c+d x)\right)}{d (1-n) \sqrt{\sin ^2(c+d x)}}",1,"-((b*(b*Cos[c + d*x])^(-1 + n)*((C*(-2 + n) + A*(-1 + n))*Hypergeometric2F1[1/2, (-2 + n)/2, n/2, Cos[c + d*x]^2] + (-2 + n)*(B*Cos[c + d*x]*Hypergeometric2F1[1/2, (-1 + n)/2, (1 + n)/2, Cos[c + d*x]^2] - C*Sqrt[Sin[c + d*x]^2]))*Tan[c + d*x])/(d*(-2 + n)*(-1 + n)*Sqrt[Sin[c + d*x]^2]))","A",1
376,1,142,196,0.3310557,"\int (b \cos (c+d x))^n \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \sec ^4(c+d x) \, dx","Integrate[(b*Cos[c + d*x])^n*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^4,x]","-\frac{\tan (c+d x) \sec ^2(c+d x) (b \cos (c+d x))^n \left((A (n-2)+C (n-3)) \, _2F_1\left(\frac{1}{2},\frac{n-3}{2};\frac{n-1}{2};\cos ^2(c+d x)\right)+(n-3) \left(B \cos (c+d x) \, _2F_1\left(\frac{1}{2},\frac{n-2}{2};\frac{n}{2};\cos ^2(c+d x)\right)-C \sqrt{\sin ^2(c+d x)}\right)\right)}{d (n-3) (n-2) \sqrt{\sin ^2(c+d x)}}","\frac{b^3 (A (2-n)+C (3-n)) \sin (c+d x) (b \cos (c+d x))^{n-3} \, _2F_1\left(\frac{1}{2},\frac{n-3}{2};\frac{n-1}{2};\cos ^2(c+d x)\right)}{d (2-n) (3-n) \sqrt{\sin ^2(c+d x)}}-\frac{b^3 C \sin (c+d x) (b \cos (c+d x))^{n-3}}{d (2-n)}+\frac{b^2 B \sin (c+d x) (b \cos (c+d x))^{n-2} \, _2F_1\left(\frac{1}{2},\frac{n-2}{2};\frac{n}{2};\cos ^2(c+d x)\right)}{d (2-n) \sqrt{\sin ^2(c+d x)}}",1,"-(((b*Cos[c + d*x])^n*Sec[c + d*x]^2*((C*(-3 + n) + A*(-2 + n))*Hypergeometric2F1[1/2, (-3 + n)/2, (-1 + n)/2, Cos[c + d*x]^2] + (-3 + n)*(B*Cos[c + d*x]*Hypergeometric2F1[1/2, (-2 + n)/2, n/2, Cos[c + d*x]^2] - C*Sqrt[Sin[c + d*x]^2]))*Tan[c + d*x])/(d*(-3 + n)*(-2 + n)*Sqrt[Sin[c + d*x]^2]))","A",1
377,1,164,223,0.5320864,"\int \cos ^{\frac{3}{2}}(c+d x) (b \cos (c+d x))^n \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^(3/2)*(b*Cos[c + d*x])^n*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","-\frac{2 \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) (b \cos (c+d x))^n \left((A (2 n+7)+C (2 n+5)) \, _2F_1\left(\frac{1}{2},\frac{1}{4} (2 n+5);\frac{1}{4} (2 n+9);\cos ^2(c+d x)\right)+(2 n+5) \left(B \cos (c+d x) \, _2F_1\left(\frac{1}{2},\frac{1}{4} (2 n+7);\frac{1}{4} (2 n+11);\cos ^2(c+d x)\right)-C \sqrt{\sin ^2(c+d x)}\right)\right)}{d (2 n+5) (2 n+7) \sqrt{\sin ^2(c+d x)}}","-\frac{2 (A (2 n+7)+C (2 n+5)) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) (b \cos (c+d x))^n \, _2F_1\left(\frac{1}{2},\frac{1}{4} (2 n+5);\frac{1}{4} (2 n+9);\cos ^2(c+d x)\right)}{d (2 n+5) (2 n+7) \sqrt{\sin ^2(c+d x)}}-\frac{2 B \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x) (b \cos (c+d x))^n \, _2F_1\left(\frac{1}{2},\frac{1}{4} (2 n+7);\frac{1}{4} (2 n+11);\cos ^2(c+d x)\right)}{d (2 n+7) \sqrt{\sin ^2(c+d x)}}+\frac{2 C \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) (b \cos (c+d x))^n}{d (2 n+7)}",1,"(-2*Cos[c + d*x]^(5/2)*(b*Cos[c + d*x])^n*Sin[c + d*x]*((C*(5 + 2*n) + A*(7 + 2*n))*Hypergeometric2F1[1/2, (5 + 2*n)/4, (9 + 2*n)/4, Cos[c + d*x]^2] + (5 + 2*n)*(B*Cos[c + d*x]*Hypergeometric2F1[1/2, (7 + 2*n)/4, (11 + 2*n)/4, Cos[c + d*x]^2] - C*Sqrt[Sin[c + d*x]^2])))/(d*(5 + 2*n)*(7 + 2*n)*Sqrt[Sin[c + d*x]^2])","A",1
378,1,164,223,0.4368161,"\int \sqrt{\cos (c+d x)} (b \cos (c+d x))^n \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Integrate[Sqrt[Cos[c + d*x]]*(b*Cos[c + d*x])^n*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","-\frac{2 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) (b \cos (c+d x))^n \left((A (2 n+5)+C (2 n+3)) \, _2F_1\left(\frac{1}{2},\frac{1}{4} (2 n+3);\frac{1}{4} (2 n+7);\cos ^2(c+d x)\right)+(2 n+3) \left(B \cos (c+d x) \, _2F_1\left(\frac{1}{2},\frac{1}{4} (2 n+5);\frac{1}{4} (2 n+9);\cos ^2(c+d x)\right)-C \sqrt{\sin ^2(c+d x)}\right)\right)}{d (2 n+3) (2 n+5) \sqrt{\sin ^2(c+d x)}}","-\frac{2 (A (2 n+5)+C (2 n+3)) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) (b \cos (c+d x))^n \, _2F_1\left(\frac{1}{2},\frac{1}{4} (2 n+3);\frac{1}{4} (2 n+7);\cos ^2(c+d x)\right)}{d (2 n+3) (2 n+5) \sqrt{\sin ^2(c+d x)}}-\frac{2 B \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) (b \cos (c+d x))^n \, _2F_1\left(\frac{1}{2},\frac{1}{4} (2 n+5);\frac{1}{4} (2 n+9);\cos ^2(c+d x)\right)}{d (2 n+5) \sqrt{\sin ^2(c+d x)}}+\frac{2 C \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) (b \cos (c+d x))^n}{d (2 n+5)}",1,"(-2*Cos[c + d*x]^(3/2)*(b*Cos[c + d*x])^n*Sin[c + d*x]*((C*(3 + 2*n) + A*(5 + 2*n))*Hypergeometric2F1[1/2, (3 + 2*n)/4, (7 + 2*n)/4, Cos[c + d*x]^2] + (3 + 2*n)*(B*Cos[c + d*x]*Hypergeometric2F1[1/2, (5 + 2*n)/4, (9 + 2*n)/4, Cos[c + d*x]^2] - C*Sqrt[Sin[c + d*x]^2])))/(d*(3 + 2*n)*(5 + 2*n)*Sqrt[Sin[c + d*x]^2])","A",1
379,1,162,221,0.3834758,"\int \frac{(b \cos (c+d x))^n \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\sqrt{\cos (c+d x)}} \, dx","Integrate[((b*Cos[c + d*x])^n*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Sqrt[Cos[c + d*x]],x]","-\frac{2 \sin (c+d x) \sqrt{\cos (c+d x)} (b \cos (c+d x))^n \left((A (2 n+3)+2 C n+C) \, _2F_1\left(\frac{1}{2},\frac{1}{4} (2 n+1);\frac{1}{4} (2 n+5);\cos ^2(c+d x)\right)+(2 n+1) \left(B \cos (c+d x) \, _2F_1\left(\frac{1}{2},\frac{1}{4} (2 n+3);\frac{1}{4} (2 n+7);\cos ^2(c+d x)\right)-C \sqrt{\sin ^2(c+d x)}\right)\right)}{d (2 n+1) (2 n+3) \sqrt{\sin ^2(c+d x)}}","-\frac{2 (A (2 n+3)+2 C n+C) \sin (c+d x) \sqrt{\cos (c+d x)} (b \cos (c+d x))^n \, _2F_1\left(\frac{1}{2},\frac{1}{4} (2 n+1);\frac{1}{4} (2 n+5);\cos ^2(c+d x)\right)}{d (2 n+1) (2 n+3) \sqrt{\sin ^2(c+d x)}}-\frac{2 B \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) (b \cos (c+d x))^n \, _2F_1\left(\frac{1}{2},\frac{1}{4} (2 n+3);\frac{1}{4} (2 n+7);\cos ^2(c+d x)\right)}{d (2 n+3) \sqrt{\sin ^2(c+d x)}}+\frac{2 C \sin (c+d x) \sqrt{\cos (c+d x)} (b \cos (c+d x))^n}{d (2 n+3)}",1,"(-2*Sqrt[Cos[c + d*x]]*(b*Cos[c + d*x])^n*Sin[c + d*x]*((C + 2*C*n + A*(3 + 2*n))*Hypergeometric2F1[1/2, (1 + 2*n)/4, (5 + 2*n)/4, Cos[c + d*x]^2] + (1 + 2*n)*(B*Cos[c + d*x]*Hypergeometric2F1[1/2, (3 + 2*n)/4, (7 + 2*n)/4, Cos[c + d*x]^2] - C*Sqrt[Sin[c + d*x]^2])))/(d*(1 + 2*n)*(3 + 2*n)*Sqrt[Sin[c + d*x]^2])","A",1
380,1,157,217,0.4501666,"\int \frac{(b \cos (c+d x))^n \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[((b*Cos[c + d*x])^n*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(3/2),x]","-\frac{2 \sin (c+d x) (b \cos (c+d x))^n \left((2 A n+A+C (2 n-1)) \, _2F_1\left(\frac{1}{2},\frac{1}{4} (2 n-1);\frac{1}{4} (2 n+3);\cos ^2(c+d x)\right)+(2 n-1) \left(B \cos (c+d x) \, _2F_1\left(\frac{1}{2},\frac{1}{4} (2 n+1);\frac{1}{4} (2 n+5);\cos ^2(c+d x)\right)-C \sqrt{\sin ^2(c+d x)}\right)\right)}{d \left(4 n^2-1\right) \sqrt{\sin ^2(c+d x)} \sqrt{\cos (c+d x)}}","\frac{2 (2 A n+A-C (1-2 n)) \sin (c+d x) (b \cos (c+d x))^n \, _2F_1\left(\frac{1}{2},\frac{1}{4} (2 n-1);\frac{1}{4} (2 n+3);\cos ^2(c+d x)\right)}{d \left(1-4 n^2\right) \sqrt{\sin ^2(c+d x)} \sqrt{\cos (c+d x)}}-\frac{2 B \sin (c+d x) \sqrt{\cos (c+d x)} (b \cos (c+d x))^n \, _2F_1\left(\frac{1}{2},\frac{1}{4} (2 n+1);\frac{1}{4} (2 n+5);\cos ^2(c+d x)\right)}{d (2 n+1) \sqrt{\sin ^2(c+d x)}}+\frac{2 C \sin (c+d x) (b \cos (c+d x))^n}{d (2 n+1) \sqrt{\cos (c+d x)}}",1,"(-2*(b*Cos[c + d*x])^n*Sin[c + d*x]*((A + 2*A*n + C*(-1 + 2*n))*Hypergeometric2F1[1/2, (-1 + 2*n)/4, (3 + 2*n)/4, Cos[c + d*x]^2] + (-1 + 2*n)*(B*Cos[c + d*x]*Hypergeometric2F1[1/2, (1 + 2*n)/4, (5 + 2*n)/4, Cos[c + d*x]^2] - C*Sqrt[Sin[c + d*x]^2])))/(d*(-1 + 4*n^2)*Sqrt[Cos[c + d*x]]*Sqrt[Sin[c + d*x]^2])","A",1
381,1,163,221,0.4115926,"\int \frac{(b \cos (c+d x))^n \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{5}{2}}(c+d x)} \, dx","Integrate[((b*Cos[c + d*x])^n*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(5/2),x]","\frac{2 \sin (c+d x) (b \cos (c+d x))^n \left((-2 A n+A+C (3-2 n)) \, _2F_1\left(\frac{1}{2},\frac{1}{4} (2 n-3);\frac{1}{4} (2 n+1);\cos ^2(c+d x)\right)-(2 n-3) \left(B \cos (c+d x) \, _2F_1\left(\frac{1}{2},\frac{1}{4} (2 n-1);\frac{1}{4} (2 n+3);\cos ^2(c+d x)\right)-C \sqrt{\sin ^2(c+d x)}\right)\right)}{d (2 n-3) (2 n-1) \sqrt{\sin ^2(c+d x)} \cos ^{\frac{3}{2}}(c+d x)}","\frac{2 (-2 A n+A+C (3-2 n)) \sin (c+d x) (b \cos (c+d x))^n \, _2F_1\left(\frac{1}{2},\frac{1}{4} (2 n-3);\frac{1}{4} (2 n+1);\cos ^2(c+d x)\right)}{d (1-2 n) (3-2 n) \sqrt{\sin ^2(c+d x)} \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 B \sin (c+d x) (b \cos (c+d x))^n \, _2F_1\left(\frac{1}{2},\frac{1}{4} (2 n-1);\frac{1}{4} (2 n+3);\cos ^2(c+d x)\right)}{d (1-2 n) \sqrt{\sin ^2(c+d x)} \sqrt{\cos (c+d x)}}-\frac{2 C \sin (c+d x) (b \cos (c+d x))^n}{d (1-2 n) \cos ^{\frac{3}{2}}(c+d x)}",1,"(2*(b*Cos[c + d*x])^n*Sin[c + d*x]*((A + C*(3 - 2*n) - 2*A*n)*Hypergeometric2F1[1/2, (-3 + 2*n)/4, (1 + 2*n)/4, Cos[c + d*x]^2] - (-3 + 2*n)*(B*Cos[c + d*x]*Hypergeometric2F1[1/2, (-1 + 2*n)/4, (3 + 2*n)/4, Cos[c + d*x]^2] - C*Sqrt[Sin[c + d*x]^2])))/(d*(-3 + 2*n)*(-1 + 2*n)*Cos[c + d*x]^(3/2)*Sqrt[Sin[c + d*x]^2])","A",1
382,1,164,223,0.4138342,"\int \frac{(b \cos (c+d x))^n \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right)}{\cos ^{\frac{7}{2}}(c+d x)} \, dx","Integrate[((b*Cos[c + d*x])^n*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(7/2),x]","-\frac{2 \sin (c+d x) (b \cos (c+d x))^n \left((A (2 n-3)+C (2 n-5)) \, _2F_1\left(\frac{1}{2},\frac{1}{4} (2 n-5);\frac{1}{4} (2 n-1);\cos ^2(c+d x)\right)+(2 n-5) \left(B \cos (c+d x) \, _2F_1\left(\frac{1}{2},\frac{1}{4} (2 n-3);\frac{1}{4} (2 n+1);\cos ^2(c+d x)\right)-C \sqrt{\sin ^2(c+d x)}\right)\right)}{d (2 n-5) (2 n-3) \sqrt{\sin ^2(c+d x)} \cos ^{\frac{5}{2}}(c+d x)}","\frac{2 (A (3-2 n)+C (5-2 n)) \sin (c+d x) (b \cos (c+d x))^n \, _2F_1\left(\frac{1}{2},\frac{1}{4} (2 n-5);\frac{1}{4} (2 n-1);\cos ^2(c+d x)\right)}{d (3-2 n) (5-2 n) \sqrt{\sin ^2(c+d x)} \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 B \sin (c+d x) (b \cos (c+d x))^n \, _2F_1\left(\frac{1}{2},\frac{1}{4} (2 n-3);\frac{1}{4} (2 n+1);\cos ^2(c+d x)\right)}{d (3-2 n) \sqrt{\sin ^2(c+d x)} \cos ^{\frac{3}{2}}(c+d x)}-\frac{2 C \sin (c+d x) (b \cos (c+d x))^n}{d (3-2 n) \cos ^{\frac{5}{2}}(c+d x)}",1,"(-2*(b*Cos[c + d*x])^n*Sin[c + d*x]*((C*(-5 + 2*n) + A*(-3 + 2*n))*Hypergeometric2F1[1/2, (-5 + 2*n)/4, (-1 + 2*n)/4, Cos[c + d*x]^2] + (-5 + 2*n)*(B*Cos[c + d*x]*Hypergeometric2F1[1/2, (-3 + 2*n)/4, (1 + 2*n)/4, Cos[c + d*x]^2] - C*Sqrt[Sin[c + d*x]^2])))/(d*(-5 + 2*n)*(-3 + 2*n)*Cos[c + d*x]^(5/2)*Sqrt[Sin[c + d*x]^2])","A",1
383,1,557,183,3.8158654,"\int (a+a \cos (e+f x))^m \left(A+B \cos (e+f x)+C \cos ^2(e+f x)\right) \, dx","Integrate[(a + a*Cos[e + f*x])^m*(A + B*Cos[e + f*x] + C*Cos[e + f*x]^2),x]","\frac{\cos ^{-2 m}\left(\frac{1}{2} (e+f x)\right) (a (\cos (e+f x)+1))^m \left(\frac{i A 4^{1-m} \left(1+e^{i (e+f x)}\right) \left(e^{-\frac{1}{2} i (e+f x)} \left(1+e^{i (e+f x)}\right)\right)^{2 m} \, _2F_1\left(1,m+1;1-m;-e^{i (e+f x)}\right)}{m}+\frac{2 i B e^{-i f x} (\cos (f x)+i \sin (f x)) \cos ^{2 m}\left(\frac{1}{2} (e+f x)\right) \left(i \sin (e) e^{i f x}+\cos (e) e^{i f x}+1\right)^{-2 m} \left((m-1) (\cos (e+f x)-i \sin (e+f x)) \, _2F_1\left(-m-1,-2 m;-m;-e^{i f x} (\cos (e)+i \sin (e))\right)+(m+1) (\cos (e+f x)+i \sin (e+f x)) \, _2F_1\left(1-m,-2 m;2-m;-e^{i f x} (\cos (e)+i \sin (e))\right)\right)}{m^2-1}+\frac{C e^{-2 i f x} \cos ^{2 m}\left(\frac{1}{2} (e+f x)\right) \left(i \sin (e) e^{i f x}+\cos (e) e^{i f x}+1\right)^{-2 m} \left(i (m+2) e^{4 i f x} (\cos (2 e)+i \sin (2 e)) \, _2F_1\left(2-m,-2 m;3-m;-e^{i f x} (\cos (e)+i \sin (e))\right)+(m-2) (\sin (2 e)+i \cos (2 e)) \, _2F_1\left(-m-2,-2 m;-m-1;-e^{i f x} (\cos (e)+i \sin (e))\right)\right)}{m^2-4}+\frac{i C 2^{1-2 m} \left(1+e^{i (e+f x)}\right) \left(e^{-\frac{1}{2} i (e+f x)} \left(1+e^{i (e+f x)}\right)\right)^{2 m} \, _2F_1\left(1,m+1;1-m;-e^{i (e+f x)}\right)}{m}\right)}{4 f}","\frac{2^{m+\frac{1}{2}} \left(A \left(m^2+3 m+2\right)+B m (m+2)+C \left(m^2+m+1\right)\right) \sin (e+f x) (\cos (e+f x)+1)^{-m-\frac{1}{2}} (a \cos (e+f x)+a)^m \, _2F_1\left(\frac{1}{2},\frac{1}{2}-m;\frac{3}{2};\frac{1}{2} (1-\cos (e+f x))\right)}{f (m+1) (m+2)}-\frac{(C-B (m+2)) \sin (e+f x) (a \cos (e+f x)+a)^m}{f (m+1) (m+2)}+\frac{C \sin (e+f x) (a \cos (e+f x)+a)^{m+1}}{a f (m+2)}",1,"((a*(1 + Cos[e + f*x]))^m*((I*4^(1 - m)*A*(1 + E^(I*(e + f*x)))*((1 + E^(I*(e + f*x)))/E^((I/2)*(e + f*x)))^(2*m)*Hypergeometric2F1[1, 1 + m, 1 - m, -E^(I*(e + f*x))])/m + (I*2^(1 - 2*m)*C*(1 + E^(I*(e + f*x)))*((1 + E^(I*(e + f*x)))/E^((I/2)*(e + f*x)))^(2*m)*Hypergeometric2F1[1, 1 + m, 1 - m, -E^(I*(e + f*x))])/m + (C*Cos[(e + f*x)/2]^(2*m)*(I*E^((4*I)*f*x)*(2 + m)*Hypergeometric2F1[2 - m, -2*m, 3 - m, -(E^(I*f*x)*(Cos[e] + I*Sin[e]))]*(Cos[2*e] + I*Sin[2*e]) + (-2 + m)*Hypergeometric2F1[-2 - m, -2*m, -1 - m, -(E^(I*f*x)*(Cos[e] + I*Sin[e]))]*(I*Cos[2*e] + Sin[2*e])))/(E^((2*I)*f*x)*(-4 + m^2)*(1 + E^(I*f*x)*Cos[e] + I*E^(I*f*x)*Sin[e])^(2*m)) + ((2*I)*B*Cos[(e + f*x)/2]^(2*m)*(Cos[f*x] + I*Sin[f*x])*((-1 + m)*Hypergeometric2F1[-1 - m, -2*m, -m, -(E^(I*f*x)*(Cos[e] + I*Sin[e]))]*(Cos[e + f*x] - I*Sin[e + f*x]) + (1 + m)*Hypergeometric2F1[1 - m, -2*m, 2 - m, -(E^(I*f*x)*(Cos[e] + I*Sin[e]))]*(Cos[e + f*x] + I*Sin[e + f*x])))/(E^(I*f*x)*(-1 + m^2)*(1 + E^(I*f*x)*Cos[e] + I*E^(I*f*x)*Sin[e])^(2*m))))/(4*f*Cos[(e + f*x)/2]^(2*m))","C",0
384,1,137,144,0.8597237,"\int (a+a \cos (c+d x))^{2/3} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Integrate[(a + a*Cos[c + d*x])^(2/3)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","\frac{3 \sec ^2\left(\frac{1}{2} (c+d x)\right) (a (\cos (c+d x)+1))^{2/3} \left(2 \sin (c+d x) (40 A+2 (8 B+7 C) \cos (c+d x)+32 B+5 C \cos (2 (c+d x))+28 C)-2 i (40 A+16 B+19 C) \, _2F_1\left(\frac{1}{3},\frac{2}{3};\frac{4}{3};-e^{i (c+d x)}\right) (i \sin (c+d x)+\cos (c+d x)+1)^{2/3}\right)}{320 d}","\frac{(40 A+16 B+19 C) \sin (c+d x) (a \cos (c+d x)+a)^{2/3} \, _2F_1\left(-\frac{1}{6},\frac{1}{2};\frac{3}{2};\frac{1}{2} (1-\cos (c+d x))\right)}{10\ 2^{5/6} d (\cos (c+d x)+1)^{7/6}}+\frac{3 (8 B-3 C) \sin (c+d x) (a \cos (c+d x)+a)^{2/3}}{40 d}+\frac{3 C \sin (c+d x) (a \cos (c+d x)+a)^{5/3}}{8 a d}",1,"(3*(a*(1 + Cos[c + d*x]))^(2/3)*Sec[(c + d*x)/2]^2*((-2*I)*(40*A + 16*B + 19*C)*Hypergeometric2F1[1/3, 2/3, 4/3, -E^(I*(c + d*x))]*(1 + Cos[c + d*x] + I*Sin[c + d*x])^(2/3) + 2*(40*A + 32*B + 28*C + 2*(8*B + 7*C)*Cos[c + d*x] + 5*C*Cos[2*(c + d*x)])*Sin[c + d*x]))/(320*d)","C",1
385,0,0,144,0.1214603,"\int \sqrt[3]{a+a \cos (c+d x)} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Integrate[(a + a*Cos[c + d*x])^(1/3)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","\int \sqrt[3]{a+a \cos (c+d x)} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","\frac{(28 A+7 B+13 C) \sin (c+d x) \sqrt[3]{a \cos (c+d x)+a} \, _2F_1\left(\frac{1}{6},\frac{1}{2};\frac{3}{2};\frac{1}{2} (1-\cos (c+d x))\right)}{14 \sqrt[6]{2} d (\cos (c+d x)+1)^{5/6}}+\frac{3 (7 B-3 C) \sin (c+d x) \sqrt[3]{a \cos (c+d x)+a}}{28 d}+\frac{3 C \sin (c+d x) (a \cos (c+d x)+a)^{4/3}}{7 a d}",1,"Integrate[(a + a*Cos[c + d*x])^(1/3)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2), x]","F",-1
386,1,105,144,0.5956655,"\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{\sqrt[3]{a+a \cos (c+d x)}} \, dx","Integrate[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^(1/3),x]","\frac{3 \sin (c+d x) (5 B+2 C \cos (c+d x)-C)-3 i (10 A-5 B+7 C) \, _2F_1\left(\frac{1}{3},\frac{2}{3};\frac{4}{3};-e^{i (c+d x)}\right) (i \sin (c+d x)+\cos (c+d x)+1)^{2/3}}{10 d \sqrt[3]{a (\cos (c+d x)+1)}}","\frac{(10 A-5 B+7 C) \sin (c+d x) \, _2F_1\left(\frac{1}{2},\frac{5}{6};\frac{3}{2};\frac{1}{2} (1-\cos (c+d x))\right)}{5\ 2^{5/6} d \sqrt[6]{\cos (c+d x)+1} \sqrt[3]{a \cos (c+d x)+a}}+\frac{3 (5 B-3 C) \sin (c+d x)}{10 d \sqrt[3]{a \cos (c+d x)+a}}+\frac{3 C \sin (c+d x) (a \cos (c+d x)+a)^{2/3}}{5 a d}",1,"((-3*I)*(10*A - 5*B + 7*C)*Hypergeometric2F1[1/3, 2/3, 4/3, -E^(I*(c + d*x))]*(1 + Cos[c + d*x] + I*Sin[c + d*x])^(2/3) + 3*(5*B - C + 2*C*Cos[c + d*x])*Sin[c + d*x])/(10*d*(a*(1 + Cos[c + d*x]))^(1/3))","C",1
387,0,0,144,0.1942673,"\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{(a+a \cos (c+d x))^{2/3}} \, dx","Integrate[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^(2/3),x]","\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{(a+a \cos (c+d x))^{2/3}} \, dx","-\frac{(4 A-8 B+7 C) \sin (c+d x) \sqrt[3]{a \cos (c+d x)+a} \, _2F_1\left(\frac{1}{6},\frac{1}{2};\frac{3}{2};\frac{1}{2} (1-\cos (c+d x))\right)}{2 \sqrt[6]{2} a d (\cos (c+d x)+1)^{5/6}}+\frac{3 (A-B+C) \sin (c+d x)}{d (a \cos (c+d x)+a)^{2/3}}+\frac{3 C \sin (c+d x) \sqrt[3]{a \cos (c+d x)+a}}{4 a d}",1,"Integrate[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + a*Cos[c + d*x])^(2/3), x]","F",-1
388,1,296,290,3.4395174,"\int (a+b \cos (c+d x))^{2/3} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Integrate[(a + b*Cos[c + d*x])^(2/3)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","-\frac{3 \csc (c+d x) (a+b \cos (c+d x))^{2/3} \left(4 \left(-6 a^2 C+16 a b B+40 A b^2+25 b^2 C\right) \sqrt{-\frac{b (\cos (c+d x)-1)}{a+b}} \sqrt{\frac{b (\cos (c+d x)+1)}{b-a}} (a+b \cos (c+d x)) F_1\left(\frac{5}{3};\frac{1}{2},\frac{1}{2};\frac{8}{3};\frac{a+b \cos (c+d x)}{a-b},\frac{a+b \cos (c+d x)}{a+b}\right)+20 \left(b^2-a^2\right) (8 b B-3 a C) \sqrt{-\frac{b (\cos (c+d x)-1)}{a+b}} \sqrt{-\frac{b (\cos (c+d x)+1)}{a-b}} F_1\left(\frac{2}{3};\frac{1}{2},\frac{1}{2};\frac{5}{3};\frac{a+b \cos (c+d x)}{a-b},\frac{a+b \cos (c+d x)}{a+b}\right)-20 b^2 \sin ^2(c+d x) (2 a C+8 b B+5 b C \cos (c+d x))\right)}{800 b^3 d}","\frac{\sin (c+d x) \left(3 a^2 C-8 a b B+8 A b^2+5 b^2 C\right) (a+b \cos (c+d x))^{2/3} F_1\left(\frac{1}{2};\frac{1}{2},-\frac{2}{3};\frac{3}{2};\frac{1}{2} (1-\cos (c+d x)),\frac{b (1-\cos (c+d x))}{a+b}\right)}{4 \sqrt{2} b^2 d \sqrt{\cos (c+d x)+1} \left(\frac{a+b \cos (c+d x)}{a+b}\right)^{2/3}}+\frac{(a+b) (8 b B-3 a C) \sin (c+d x) (a+b \cos (c+d x))^{2/3} F_1\left(\frac{1}{2};\frac{1}{2},-\frac{5}{3};\frac{3}{2};\frac{1}{2} (1-\cos (c+d x)),\frac{b (1-\cos (c+d x))}{a+b}\right)}{4 \sqrt{2} b^2 d \sqrt{\cos (c+d x)+1} \left(\frac{a+b \cos (c+d x)}{a+b}\right)^{2/3}}+\frac{3 C \sin (c+d x) (a+b \cos (c+d x))^{5/3}}{8 b d}",1,"(-3*(a + b*Cos[c + d*x])^(2/3)*Csc[c + d*x]*(20*(-a^2 + b^2)*(8*b*B - 3*a*C)*AppellF1[2/3, 1/2, 1/2, 5/3, (a + b*Cos[c + d*x])/(a - b), (a + b*Cos[c + d*x])/(a + b)]*Sqrt[-((b*(-1 + Cos[c + d*x]))/(a + b))]*Sqrt[-((b*(1 + Cos[c + d*x]))/(a - b))] + 4*(40*A*b^2 + 16*a*b*B - 6*a^2*C + 25*b^2*C)*AppellF1[5/3, 1/2, 1/2, 8/3, (a + b*Cos[c + d*x])/(a - b), (a + b*Cos[c + d*x])/(a + b)]*Sqrt[-((b*(-1 + Cos[c + d*x]))/(a + b))]*Sqrt[(b*(1 + Cos[c + d*x]))/(-a + b)]*(a + b*Cos[c + d*x]) - 20*b^2*(8*b*B + 2*a*C + 5*b*C*Cos[c + d*x])*Sin[c + d*x]^2))/(800*b^3*d)","A",0
389,1,294,290,3.5200051,"\int \sqrt[3]{a+b \cos (c+d x)} \left(A+B \cos (c+d x)+C \cos ^2(c+d x)\right) \, dx","Integrate[(a + b*Cos[c + d*x])^(1/3)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]","-\frac{3 \csc (c+d x) \sqrt[3]{a+b \cos (c+d x)} \left(\left(-3 a^2 C+7 a b B+28 A b^2+16 b^2 C\right) \sqrt{-\frac{b (\cos (c+d x)-1)}{a+b}} \sqrt{\frac{b (\cos (c+d x)+1)}{b-a}} (a+b \cos (c+d x)) F_1\left(\frac{4}{3};\frac{1}{2},\frac{1}{2};\frac{7}{3};\frac{a+b \cos (c+d x)}{a-b},\frac{a+b \cos (c+d x)}{a+b}\right)+4 \left(b^2-a^2\right) (7 b B-3 a C) \sqrt{-\frac{b (\cos (c+d x)-1)}{a+b}} \sqrt{-\frac{b (\cos (c+d x)+1)}{a-b}} F_1\left(\frac{1}{3};\frac{1}{2},\frac{1}{2};\frac{4}{3};\frac{a+b \cos (c+d x)}{a-b},\frac{a+b \cos (c+d x)}{a+b}\right)-4 b^2 \sin ^2(c+d x) (a C+7 b B+4 b C \cos (c+d x))\right)}{112 b^3 d}","\frac{\sqrt{2} \sin (c+d x) \left(3 a^2 C-7 a b B+7 A b^2+4 b^2 C\right) \sqrt[3]{a+b \cos (c+d x)} F_1\left(\frac{1}{2};\frac{1}{2},-\frac{1}{3};\frac{3}{2};\frac{1}{2} (1-\cos (c+d x)),\frac{b (1-\cos (c+d x))}{a+b}\right)}{7 b^2 d \sqrt{\cos (c+d x)+1} \sqrt[3]{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{\sqrt{2} (a+b) (7 b B-3 a C) \sin (c+d x) \sqrt[3]{a+b \cos (c+d x)} F_1\left(\frac{1}{2};\frac{1}{2},-\frac{4}{3};\frac{3}{2};\frac{1}{2} (1-\cos (c+d x)),\frac{b (1-\cos (c+d x))}{a+b}\right)}{7 b^2 d \sqrt{\cos (c+d x)+1} \sqrt[3]{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{3 C \sin (c+d x) (a+b \cos (c+d x))^{4/3}}{7 b d}",1,"(-3*(a + b*Cos[c + d*x])^(1/3)*Csc[c + d*x]*(4*(-a^2 + b^2)*(7*b*B - 3*a*C)*AppellF1[1/3, 1/2, 1/2, 4/3, (a + b*Cos[c + d*x])/(a - b), (a + b*Cos[c + d*x])/(a + b)]*Sqrt[-((b*(-1 + Cos[c + d*x]))/(a + b))]*Sqrt[-((b*(1 + Cos[c + d*x]))/(a - b))] + (28*A*b^2 + 7*a*b*B - 3*a^2*C + 16*b^2*C)*AppellF1[4/3, 1/2, 1/2, 7/3, (a + b*Cos[c + d*x])/(a - b), (a + b*Cos[c + d*x])/(a + b)]*Sqrt[-((b*(-1 + Cos[c + d*x]))/(a + b))]*Sqrt[(b*(1 + Cos[c + d*x]))/(-a + b)]*(a + b*Cos[c + d*x]) - 4*b^2*(7*b*B + a*C + 4*b*C*Cos[c + d*x])*Sin[c + d*x]^2))/(112*b^3*d)","A",0
390,1,268,287,2.3978256,"\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{\sqrt[3]{a+b \cos (c+d x)}} \, dx","Integrate[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^(1/3),x]","-\frac{3 \csc (c+d x) (a+b \cos (c+d x))^{2/3} \left(5 \left(3 a^2 C-5 a b B+5 A b^2+2 b^2 C\right) \sqrt{-\frac{b (\cos (c+d x)-1)}{a+b}} \sqrt{\frac{b (\cos (c+d x)+1)}{b-a}} F_1\left(\frac{2}{3};\frac{1}{2},\frac{1}{2};\frac{5}{3};\frac{a+b \cos (c+d x)}{a-b},\frac{a+b \cos (c+d x)}{a+b}\right)+2 (5 b B-3 a C) \sqrt{-\frac{b (\cos (c+d x)-1)}{a+b}} \sqrt{\frac{b (\cos (c+d x)+1)}{b-a}} (a+b \cos (c+d x)) F_1\left(\frac{5}{3};\frac{1}{2},\frac{1}{2};\frac{8}{3};\frac{a+b \cos (c+d x)}{a-b},\frac{a+b \cos (c+d x)}{a+b}\right)-10 b^2 C \sin ^2(c+d x)\right)}{50 b^3 d}","\frac{\sqrt{2} \sin (c+d x) \left(3 a^2 C-5 a b B+5 A b^2+2 b^2 C\right) \sqrt[3]{\frac{a+b \cos (c+d x)}{a+b}} F_1\left(\frac{1}{2};\frac{1}{2},\frac{1}{3};\frac{3}{2};\frac{1}{2} (1-\cos (c+d x)),\frac{b (1-\cos (c+d x))}{a+b}\right)}{5 b^2 d \sqrt{\cos (c+d x)+1} \sqrt[3]{a+b \cos (c+d x)}}+\frac{\sqrt{2} (5 b B-3 a C) \sin (c+d x) (a+b \cos (c+d x))^{2/3} F_1\left(\frac{1}{2};\frac{1}{2},-\frac{2}{3};\frac{3}{2};\frac{1}{2} (1-\cos (c+d x)),\frac{b (1-\cos (c+d x))}{a+b}\right)}{5 b^2 d \sqrt{\cos (c+d x)+1} \left(\frac{a+b \cos (c+d x)}{a+b}\right)^{2/3}}+\frac{3 C \sin (c+d x) (a+b \cos (c+d x))^{2/3}}{5 b d}",1,"(-3*(a + b*Cos[c + d*x])^(2/3)*Csc[c + d*x]*(5*(5*A*b^2 - 5*a*b*B + 3*a^2*C + 2*b^2*C)*AppellF1[2/3, 1/2, 1/2, 5/3, (a + b*Cos[c + d*x])/(a - b), (a + b*Cos[c + d*x])/(a + b)]*Sqrt[-((b*(-1 + Cos[c + d*x]))/(a + b))]*Sqrt[(b*(1 + Cos[c + d*x]))/(-a + b)] + 2*(5*b*B - 3*a*C)*AppellF1[5/3, 1/2, 1/2, 8/3, (a + b*Cos[c + d*x])/(a - b), (a + b*Cos[c + d*x])/(a + b)]*Sqrt[-((b*(-1 + Cos[c + d*x]))/(a + b))]*Sqrt[(b*(1 + Cos[c + d*x]))/(-a + b)]*(a + b*Cos[c + d*x]) - 10*b^2*C*Sin[c + d*x]^2))/(50*b^3*d)","A",0
391,1,266,286,2.4469563,"\int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{(a+b \cos (c+d x))^{2/3}} \, dx","Integrate[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/(a + b*Cos[c + d*x])^(2/3),x]","-\frac{3 \csc (c+d x) \sqrt[3]{a+b \cos (c+d x)} \left(4 \left(3 a^2 C-4 a b B+4 A b^2+b^2 C\right) \sqrt{-\frac{b (\cos (c+d x)-1)}{a+b}} \sqrt{\frac{b (\cos (c+d x)+1)}{b-a}} F_1\left(\frac{1}{3};\frac{1}{2},\frac{1}{2};\frac{4}{3};\frac{a+b \cos (c+d x)}{a-b},\frac{a+b \cos (c+d x)}{a+b}\right)+(4 b B-3 a C) \sqrt{-\frac{b (\cos (c+d x)-1)}{a+b}} \sqrt{\frac{b (\cos (c+d x)+1)}{b-a}} (a+b \cos (c+d x)) F_1\left(\frac{4}{3};\frac{1}{2},\frac{1}{2};\frac{7}{3};\frac{a+b \cos (c+d x)}{a-b},\frac{a+b \cos (c+d x)}{a+b}\right)-4 b^2 C \sin ^2(c+d x)\right)}{16 b^3 d}","\frac{\sin (c+d x) \left(3 a^2 C-4 a b B+4 A b^2+b^2 C\right) \left(\frac{a+b \cos (c+d x)}{a+b}\right)^{2/3} F_1\left(\frac{1}{2};\frac{1}{2},\frac{2}{3};\frac{3}{2};\frac{1}{2} (1-\cos (c+d x)),\frac{b (1-\cos (c+d x))}{a+b}\right)}{2 \sqrt{2} b^2 d \sqrt{\cos (c+d x)+1} (a+b \cos (c+d x))^{2/3}}+\frac{(4 b B-3 a C) \sin (c+d x) \sqrt[3]{a+b \cos (c+d x)} F_1\left(\frac{1}{2};\frac{1}{2},-\frac{1}{3};\frac{3}{2};\frac{1}{2} (1-\cos (c+d x)),\frac{b (1-\cos (c+d x))}{a+b}\right)}{2 \sqrt{2} b^2 d \sqrt{\cos (c+d x)+1} \sqrt[3]{\frac{a+b \cos (c+d x)}{a+b}}}+\frac{3 C \sin (c+d x) \sqrt[3]{a+b \cos (c+d x)}}{4 b d}",1,"(-3*(a + b*Cos[c + d*x])^(1/3)*Csc[c + d*x]*(4*(4*A*b^2 - 4*a*b*B + 3*a^2*C + b^2*C)*AppellF1[1/3, 1/2, 1/2, 4/3, (a + b*Cos[c + d*x])/(a - b), (a + b*Cos[c + d*x])/(a + b)]*Sqrt[-((b*(-1 + Cos[c + d*x]))/(a + b))]*Sqrt[(b*(1 + Cos[c + d*x]))/(-a + b)] + (4*b*B - 3*a*C)*AppellF1[4/3, 1/2, 1/2, 7/3, (a + b*Cos[c + d*x])/(a - b), (a + b*Cos[c + d*x])/(a + b)]*Sqrt[-((b*(-1 + Cos[c + d*x]))/(a + b))]*Sqrt[(b*(1 + Cos[c + d*x]))/(-a + b)]*(a + b*Cos[c + d*x]) - 4*b^2*C*Sin[c + d*x]^2))/(16*b^3*d)","A",0
392,0,0,215,4.0016191,"\int (a+b \cos (e+f x))^m \left(A+(A+C) \cos (e+f x)+C \cos ^2(e+f x)\right) \, dx","Integrate[(a + b*Cos[e + f*x])^m*(A + (A + C)*Cos[e + f*x] + C*Cos[e + f*x]^2),x]","\int (a+b \cos (e+f x))^m \left(A+(A+C) \cos (e+f x)+C \cos ^2(e+f x)\right) \, dx","\frac{2 \sqrt{2} (A-C) \sin (e+f x) (a+b \cos (e+f x))^m \left(\frac{a+b \cos (e+f x)}{a+b}\right)^{-m} F_1\left(\frac{1}{2};-\frac{1}{2},-m;\frac{3}{2};\frac{1}{2} (1-\cos (e+f x)),\frac{b (1-\cos (e+f x))}{a+b}\right)}{f \sqrt{\cos (e+f x)+1}}+\frac{4 \sqrt{2} C \sin (e+f x) (a+b \cos (e+f x))^m \left(\frac{a+b \cos (e+f x)}{a+b}\right)^{-m} F_1\left(\frac{1}{2};-\frac{3}{2},-m;\frac{3}{2};\frac{1}{2} (1-\cos (e+f x)),\frac{b (1-\cos (e+f x))}{a+b}\right)}{f \sqrt{\cos (e+f x)+1}}",1,"Integrate[(a + b*Cos[e + f*x])^m*(A + (A + C)*Cos[e + f*x] + C*Cos[e + f*x]^2), x]","F",-1
393,1,16142,303,27.1121149,"\int (a+b \cos (e+f x))^m \left(A+B \cos (e+f x)+C \cos ^2(e+f x)\right) \, dx","Integrate[(a + b*Cos[e + f*x])^m*(A + B*Cos[e + f*x] + C*Cos[e + f*x]^2),x]","\text{Result too large to show}","\frac{\sqrt{2} \sin (e+f x) \left(a^2 C-a b B (m+2)+A b^2 (m+2)+b^2 C (m+1)\right) (a+b \cos (e+f x))^m \left(\frac{a+b \cos (e+f x)}{a+b}\right)^{-m} F_1\left(\frac{1}{2};\frac{1}{2},-m;\frac{3}{2};\frac{1}{2} (1-\cos (e+f x)),\frac{b (1-\cos (e+f x))}{a+b}\right)}{b^2 f (m+2) \sqrt{\cos (e+f x)+1}}-\frac{\sqrt{2} (a+b) \sin (e+f x) (a C-b B (m+2)) (a+b \cos (e+f x))^m \left(\frac{a+b \cos (e+f x)}{a+b}\right)^{-m} F_1\left(\frac{1}{2};\frac{1}{2},-m-1;\frac{3}{2};\frac{1}{2} (1-\cos (e+f x)),\frac{b (1-\cos (e+f x))}{a+b}\right)}{b^2 f (m+2) \sqrt{\cos (e+f x)+1}}+\frac{C \sin (e+f x) (a+b \cos (e+f x))^{m+1}}{b f (m+2)}",1,"Result too large to show","B",0