1,1,74,87,0.0431216,"\int \cos ^7(c+d x) (a+a \sin (c+d x)) \, dx","Integrate[Cos[c + d*x]^7*(a + a*Sin[c + d*x]),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{a \cos ^8(c+d x)}{8 d}","-\frac{(a \sin (c+d x)+a)^8}{8 a^7 d}+\frac{6 (a \sin (c+d x)+a)^7}{7 a^6 d}-\frac{2 (a \sin (c+d x)+a)^6}{a^5 d}+\frac{8 (a \sin (c+d x)+a)^5}{5 a^4 d}",1,"-1/8*(a*Cos[c + d*x]^8)/d + (a*Sin[c + d*x])/d - (a*Sin[c + d*x]^3)/d + (3*a*Sin[c + d*x]^5)/(5*d) - (a*Sin[c + d*x]^7)/(7*d)","A",1
2,1,57,87,0.1583692,"\int \cos ^6(c+d x) (a+a \sin (c+d x)) \, dx","Integrate[Cos[c + d*x]^6*(a + a*Sin[c + d*x]),x]","\frac{a \left(7 (45 \sin (2 (c+d x))+9 \sin (4 (c+d x))+\sin (6 (c+d x))+60 c+60 d x)-192 \cos ^7(c+d x)\right)}{1344 d}","-\frac{a \cos ^7(c+d x)}{7 d}+\frac{a \sin (c+d x) \cos ^5(c+d x)}{6 d}+\frac{5 a \sin (c+d x) \cos ^3(c+d x)}{24 d}+\frac{5 a \sin (c+d x) \cos (c+d x)}{16 d}+\frac{5 a x}{16}",1,"(a*(-192*Cos[c + d*x]^7 + 7*(60*c + 60*d*x + 45*Sin[2*(c + d*x)] + 9*Sin[4*(c + d*x)] + Sin[6*(c + d*x)])))/(1344*d)","A",1
3,1,60,64,0.0227116,"\int \cos ^5(c+d x) (a+a \sin (c+d x)) \, dx","Integrate[Cos[c + d*x]^5*(a + a*Sin[c + d*x]),x]","\frac{a \sin ^5(c+d x)}{5 d}-\frac{2 a \sin ^3(c+d x)}{3 d}+\frac{a \sin (c+d x)}{d}-\frac{a \cos ^6(c+d x)}{6 d}","\frac{(a \sin (c+d x)+a)^6}{6 a^5 d}-\frac{4 (a \sin (c+d x)+a)^5}{5 a^4 d}+\frac{(a \sin (c+d x)+a)^4}{a^3 d}",1,"-1/6*(a*Cos[c + d*x]^6)/d + (a*Sin[c + d*x])/d - (2*a*Sin[c + d*x]^3)/(3*d) + (a*Sin[c + d*x]^5)/(5*d)","A",1
4,1,62,65,0.0835239,"\int \cos ^4(c+d x) (a+a \sin (c+d x)) \, dx","Integrate[Cos[c + d*x]^4*(a + a*Sin[c + d*x]),x]","\frac{3 a (c+d x)}{8 d}+\frac{a \sin (2 (c+d x))}{4 d}+\frac{a \sin (4 (c+d x))}{32 d}-\frac{a \cos ^5(c+d x)}{5 d}","-\frac{a \cos ^5(c+d x)}{5 d}+\frac{a \sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac{3 a \sin (c+d x) \cos (c+d x)}{8 d}+\frac{3 a x}{8}",1,"(3*a*(c + d*x))/(8*d) - (a*Cos[c + d*x]^5)/(5*d) + (a*Sin[2*(c + d*x)])/(4*d) + (a*Sin[4*(c + d*x)])/(32*d)","A",1
5,1,44,45,0.0150237,"\int \cos ^3(c+d x) (a+a \sin (c+d x)) \, dx","Integrate[Cos[c + d*x]^3*(a + a*Sin[c + d*x]),x]","-\frac{a \sin ^3(c+d x)}{3 d}+\frac{a \sin (c+d x)}{d}-\frac{a \cos ^4(c+d x)}{4 d}","\frac{2 (a \sin (c+d x)+a)^3}{3 a^2 d}-\frac{(a \sin (c+d x)+a)^4}{4 a^3 d}",1,"-1/4*(a*Cos[c + d*x]^4)/d + (a*Sin[c + d*x])/d - (a*Sin[c + d*x]^3)/(3*d)","A",1
6,1,46,43,0.0482328,"\int \cos ^2(c+d x) (a+a \sin (c+d x)) \, dx","Integrate[Cos[c + d*x]^2*(a + a*Sin[c + d*x]),x]","\frac{a (c+d x)}{2 d}+\frac{a \sin (2 (c+d x))}{4 d}-\frac{a \cos ^3(c+d x)}{3 d}","-\frac{a \cos ^3(c+d x)}{3 d}+\frac{a \sin (c+d x) \cos (c+d x)}{2 d}+\frac{a x}{2}",1,"(a*(c + d*x))/(2*d) - (a*Cos[c + d*x]^3)/(3*d) + (a*Sin[2*(c + d*x)])/(4*d)","A",1
7,1,39,22,0.0127846,"\int \cos (c+d x) (a+a \sin (c+d x)) \, dx","Integrate[Cos[c + d*x]*(a + a*Sin[c + d*x]),x]","-\frac{a \cos ^2(c+d x)}{2 d}+\frac{a \sin (c) \cos (d x)}{d}+\frac{a \cos (c) \sin (d x)}{d}","\frac{(a \sin (c+d x)+a)^2}{2 a d}",1,"-1/2*(a*Cos[c + d*x]^2)/d + (a*Cos[d*x]*Sin[c])/d + (a*Cos[c]*Sin[d*x])/d","A",1
8,1,26,17,0.0136989,"\int \sec (c+d x) (a+a \sin (c+d x)) \, dx","Integrate[Sec[c + d*x]*(a + a*Sin[c + d*x]),x]","\frac{a \tanh ^{-1}(\sin (c+d x))}{d}-\frac{a \log (\cos (c+d x))}{d}","-\frac{a \log (1-\sin (c+d x))}{d}",1,"(a*ArcTanh[Sin[c + d*x]])/d - (a*Log[Cos[c + d*x]])/d","A",1
9,1,23,23,0.0136049,"\int \sec ^2(c+d x) (a+a \sin (c+d x)) \, dx","Integrate[Sec[c + d*x]^2*(a + a*Sin[c + d*x]),x]","\frac{a \tan (c+d x)}{d}+\frac{a \sec (c+d x)}{d}","\frac{a \tan (c+d x)}{d}+\frac{a \sec (c+d x)}{d}",1,"(a*Sec[c + d*x])/d + (a*Tan[c + d*x])/d","A",1
10,1,52,39,0.017986,"\int \sec ^3(c+d x) (a+a \sin (c+d x)) \, dx","Integrate[Sec[c + d*x]^3*(a + a*Sin[c + d*x]),x]","\frac{a \sec ^2(c+d x)}{2 d}+\frac{a \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{a \tan (c+d x) \sec (c+d x)}{2 d}","\frac{a^2}{2 d (a-a \sin (c+d x))}+\frac{a \tanh ^{-1}(\sin (c+d x))}{2 d}",1,"(a*ArcTanh[Sin[c + d*x]])/(2*d) + (a*Sec[c + d*x]^2)/(2*d) + (a*Sec[c + d*x]*Tan[c + d*x])/(2*d)","A",1
11,1,41,44,0.0608557,"\int \sec ^4(c+d x) (a+a \sin (c+d x)) \, dx","Integrate[Sec[c + d*x]^4*(a + a*Sin[c + d*x]),x]","\frac{a \left(\frac{1}{3} \tan ^3(c+d x)+\tan (c+d x)\right)}{d}+\frac{a \sec ^3(c+d x)}{3 d}","\frac{a \tan ^3(c+d x)}{3 d}+\frac{a \tan (c+d x)}{d}+\frac{a \sec ^3(c+d x)}{3 d}",1,"(a*Sec[c + d*x]^3)/(3*d) + (a*(Tan[c + d*x] + Tan[c + d*x]^3/3))/d","A",1
12,1,68,84,0.0794815,"\int \sec ^5(c+d x) (a+a \sin (c+d x)) \, dx","Integrate[Sec[c + d*x]^5*(a + a*Sin[c + d*x]),x]","\frac{a \sec ^4(c+d x)}{4 d}+\frac{a \tan (c+d x) \sec ^3(c+d x)}{4 d}+\frac{3 a \left(\tanh ^{-1}(\sin (c+d x))+\tan (c+d x) \sec (c+d x)\right)}{8 d}","\frac{a^3}{8 d (a-a \sin (c+d x))^2}+\frac{a^2}{4 d (a-a \sin (c+d x))}-\frac{a^2}{8 d (a \sin (c+d x)+a)}+\frac{3 a \tanh ^{-1}(\sin (c+d x))}{8 d}",1,"(a*Sec[c + d*x]^4)/(4*d) + (a*Sec[c + d*x]^3*Tan[c + d*x])/(4*d) + (3*a*(ArcTanh[Sin[c + d*x]] + Sec[c + d*x]*Tan[c + d*x]))/(8*d)","A",1
13,1,171,126,1.5195381,"\int \cos ^6(c+d x) (a+a \sin (c+d x))^2 \, dx","Integrate[Cos[c + d*x]^6*(a + a*Sin[c + d*x])^2,x]","-\frac{a^2 \left(630 \sqrt{1-\sin (c+d x)} \sin ^{-1}\left(\frac{\sqrt{1-\sin (c+d x)}}{\sqrt{2}}\right)+\sqrt{\sin (c+d x)+1} \left(112 \sin ^8(c+d x)+144 \sin ^7(c+d x)-424 \sin ^6(c+d x)-600 \sin ^5(c+d x)+558 \sin ^4(c+d x)+978 \sin ^3(c+d x)-187 \sin ^2(c+d x)-837 \sin (c+d x)+256\right)\right) \cos ^7(c+d x)}{896 d (\sin (c+d x)-1)^4 (\sin (c+d x)+1)^{7/2}}","-\frac{9 a^2 \cos ^7(c+d x)}{56 d}-\frac{\cos ^7(c+d x) \left(a^2 \sin (c+d x)+a^2\right)}{8 d}+\frac{3 a^2 \sin (c+d x) \cos ^5(c+d x)}{16 d}+\frac{15 a^2 \sin (c+d x) \cos ^3(c+d x)}{64 d}+\frac{45 a^2 \sin (c+d x) \cos (c+d x)}{128 d}+\frac{45 a^2 x}{128}",1,"-1/896*(a^2*Cos[c + d*x]^7*(630*ArcSin[Sqrt[1 - Sin[c + d*x]]/Sqrt[2]]*Sqrt[1 - Sin[c + d*x]] + Sqrt[1 + Sin[c + d*x]]*(256 - 837*Sin[c + d*x] - 187*Sin[c + d*x]^2 + 978*Sin[c + d*x]^3 + 558*Sin[c + d*x]^4 - 600*Sin[c + d*x]^5 - 424*Sin[c + d*x]^6 + 144*Sin[c + d*x]^7 + 112*Sin[c + d*x]^8)))/(d*(-1 + Sin[c + d*x])^4*(1 + Sin[c + d*x])^(7/2))","A",1
14,1,58,67,0.0739204,"\int \cos ^5(c+d x) (a+a \sin (c+d x))^2 \, dx","Integrate[Cos[c + d*x]^5*(a + a*Sin[c + d*x])^2,x]","-\frac{a^2 (\sin (c+d x)+1)^2 \left(15 \sin ^2(c+d x)-40 \sin (c+d x)+29\right) \cos ^6(c+d x)}{105 d (\sin (c+d x)-1)^3}","\frac{(a \sin (c+d x)+a)^7}{7 a^5 d}-\frac{2 (a \sin (c+d x)+a)^6}{3 a^4 d}+\frac{4 (a \sin (c+d x)+a)^5}{5 a^3 d}",1,"-1/105*(a^2*Cos[c + d*x]^6*(1 + Sin[c + d*x])^2*(29 - 40*Sin[c + d*x] + 15*Sin[c + d*x]^2))/(d*(-1 + Sin[c + d*x])^3)","A",1
15,1,151,102,0.5435132,"\int \cos ^4(c+d x) (a+a \sin (c+d x))^2 \, dx","Integrate[Cos[c + d*x]^4*(a + a*Sin[c + d*x])^2,x]","-\frac{a^2 \left(\sqrt{\sin (c+d x)+1} \left(40 \sin ^6(c+d x)+56 \sin ^5(c+d x)-106 \sin ^4(c+d x)-182 \sin ^3(c+d x)+57 \sin ^2(c+d x)+231 \sin (c+d x)-96\right)-210 \sin ^{-1}\left(\frac{\sqrt{1-\sin (c+d x)}}{\sqrt{2}}\right) \sqrt{1-\sin (c+d x)}\right) \cos ^5(c+d x)}{240 d (\sin (c+d x)-1)^3 (\sin (c+d x)+1)^{5/2}}","-\frac{7 a^2 \cos ^5(c+d x)}{30 d}-\frac{\cos ^5(c+d x) \left(a^2 \sin (c+d x)+a^2\right)}{6 d}+\frac{7 a^2 \sin (c+d x) \cos ^3(c+d x)}{24 d}+\frac{7 a^2 \sin (c+d x) \cos (c+d x)}{16 d}+\frac{7 a^2 x}{16}",1,"-1/240*(a^2*Cos[c + d*x]^5*(-210*ArcSin[Sqrt[1 - Sin[c + d*x]]/Sqrt[2]]*Sqrt[1 - Sin[c + d*x]] + Sqrt[1 + Sin[c + d*x]]*(-96 + 231*Sin[c + d*x] + 57*Sin[c + d*x]^2 - 182*Sin[c + d*x]^3 - 106*Sin[c + d*x]^4 + 56*Sin[c + d*x]^5 + 40*Sin[c + d*x]^6)))/(d*(-1 + Sin[c + d*x])^3*(1 + Sin[c + d*x])^(5/2))","A",1
16,1,46,45,0.0843826,"\int \cos ^3(c+d x) (a+a \sin (c+d x))^2 \, dx","Integrate[Cos[c + d*x]^3*(a + a*Sin[c + d*x])^2,x]","-\frac{a^2 \sin (c+d x) \left(2 \sin ^4(c+d x)+5 \sin ^3(c+d x)-10 \sin (c+d x)-10\right)}{10 d}","\frac{(a \sin (c+d x)+a)^4}{2 a^2 d}-\frac{(a \sin (c+d x)+a)^5}{5 a^3 d}",1,"-1/10*(a^2*Sin[c + d*x]*(-10 - 10*Sin[c + d*x] + 5*Sin[c + d*x]^3 + 2*Sin[c + d*x]^4))/d","A",1
17,1,131,78,0.2960625,"\int \cos ^2(c+d x) (a+a \sin (c+d x))^2 \, dx","Integrate[Cos[c + d*x]^2*(a + a*Sin[c + d*x])^2,x]","-\frac{a^2 \left(30 \sqrt{1-\sin (c+d x)} \sin ^{-1}\left(\frac{\sqrt{1-\sin (c+d x)}}{\sqrt{2}}\right)+\sqrt{\sin (c+d x)+1} \left(6 \sin ^4(c+d x)+10 \sin ^3(c+d x)-7 \sin ^2(c+d x)-25 \sin (c+d x)+16\right)\right) \cos ^3(c+d x)}{24 d (\sin (c+d x)-1)^2 (\sin (c+d x)+1)^{3/2}}","-\frac{5 a^2 \cos ^3(c+d x)}{12 d}-\frac{\cos ^3(c+d x) \left(a^2 \sin (c+d x)+a^2\right)}{4 d}+\frac{5 a^2 \sin (c+d x) \cos (c+d x)}{8 d}+\frac{5 a^2 x}{8}",1,"-1/24*(a^2*Cos[c + d*x]^3*(30*ArcSin[Sqrt[1 - Sin[c + d*x]]/Sqrt[2]]*Sqrt[1 - Sin[c + d*x]] + Sqrt[1 + Sin[c + d*x]]*(16 - 25*Sin[c + d*x] - 7*Sin[c + d*x]^2 + 10*Sin[c + d*x]^3 + 6*Sin[c + d*x]^4)))/(d*(-1 + Sin[c + d*x])^2*(1 + Sin[c + d*x])^(3/2))","A",1
18,1,47,22,0.0210755,"\int \cos (c+d x) (a+a \sin (c+d x))^2 \, dx","Integrate[Cos[c + d*x]*(a + a*Sin[c + d*x])^2,x]","\frac{a^2 \sin ^3(c+d x)}{3 d}+\frac{a^2 \sin ^2(c+d x)}{d}+\frac{a^2 \sin (c+d x)}{d}","\frac{(a \sin (c+d x)+a)^3}{3 a d}",1,"(a^2*Sin[c + d*x])/d + (a^2*Sin[c + d*x]^2)/d + (a^2*Sin[c + d*x]^3)/(3*d)","B",1
19,1,29,34,0.0197758,"\int \sec (c+d x) (a+a \sin (c+d x))^2 \, dx","Integrate[Sec[c + d*x]*(a + a*Sin[c + d*x])^2,x]","\frac{a^2 (-\sin (c+d x)-2 \log (1-\sin (c+d x)))}{d}","-\frac{a^2 \sin (c+d x)}{d}-\frac{2 a^2 \log (1-\sin (c+d x))}{d}",1,"(a^2*(-2*Log[1 - Sin[c + d*x]] - Sin[c + d*x]))/d","A",1
20,1,75,38,0.0553211,"\int \sec ^2(c+d x) (a+a \sin (c+d x))^2 \, dx","Integrate[Sec[c + d*x]^2*(a + a*Sin[c + d*x])^2,x]","\frac{2 a^2 \sqrt{\sin (c+d x)+1} \left(\sqrt{1-\sin (c+d x)} \sin ^{-1}\left(\frac{\sqrt{1-\sin (c+d x)}}{\sqrt{2}}\right)+\sqrt{\sin (c+d x)+1}\right) \sec (c+d x)}{d}","\frac{2 a^4 \cos (c+d x)}{d \left(a^2-a^2 \sin (c+d x)\right)}-a^2 x",1,"(2*a^2*Sec[c + d*x]*Sqrt[1 + Sin[c + d*x]]*(ArcSin[Sqrt[1 - Sin[c + d*x]]/Sqrt[2]]*Sqrt[1 - Sin[c + d*x]] + Sqrt[1 + Sin[c + d*x]]))/d","A",1
21,1,32,20,0.1363662,"\int \sec ^3(c+d x) (a+a \sin (c+d x))^2 \, dx","Integrate[Sec[c + d*x]^3*(a + a*Sin[c + d*x])^2,x]","\frac{a^2}{d \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^2}","\frac{a^3}{d (a-a \sin (c+d x))}",1,"a^2/(d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2)","A",1
22,1,58,63,0.0089894,"\int \sec ^4(c+d x) (a+a \sin (c+d x))^2 \, dx","Integrate[Sec[c + d*x]^4*(a + a*Sin[c + d*x])^2,x]","-\frac{a^2 \tan ^3(c+d x)}{3 d}+\frac{2 a^2 \sec ^3(c+d x)}{3 d}+\frac{a^2 \tan (c+d x) \sec ^2(c+d x)}{d}","\frac{a^4 \cos (c+d x)}{3 d (a-a \sin (c+d x))^2}+\frac{a^4 \cos (c+d x)}{3 d \left(a^2-a^2 \sin (c+d x)\right)}",1,"(2*a^2*Sec[c + d*x]^3)/(3*d) + (a^2*Sec[c + d*x]^2*Tan[c + d*x])/d - (a^2*Tan[c + d*x]^3)/(3*d)","A",1
23,1,56,64,0.0675366,"\int \sec ^5(c+d x) (a+a \sin (c+d x))^2 \, dx","Integrate[Sec[c + d*x]^5*(a + a*Sin[c + d*x])^2,x]","\frac{a^2 (\sin (c+d x)+1)^2 \sec ^4(c+d x) \left(-\sin (c+d x)+(\sin (c+d x)-1)^2 \tanh ^{-1}(\sin (c+d x))+2\right)}{4 d}","\frac{a^4}{4 d (a-a \sin (c+d x))^2}+\frac{a^3}{4 d (a-a \sin (c+d x))}+\frac{a^2 \tanh ^{-1}(\sin (c+d x))}{4 d}",1,"(a^2*Sec[c + d*x]^4*(2 + ArcTanh[Sin[c + d*x]]*(-1 + Sin[c + d*x])^2 - Sin[c + d*x])*(1 + Sin[c + d*x])^2)/(4*d)","A",1
24,1,82,64,0.0105417,"\int \sec ^6(c+d x) (a+a \sin (c+d x))^2 \, dx","Integrate[Sec[c + d*x]^6*(a + a*Sin[c + d*x])^2,x]","\frac{2 a^2 \tan ^5(c+d x)}{5 d}+\frac{2 a^2 \sec ^5(c+d x)}{5 d}+\frac{a^2 \tan (c+d x) \sec ^4(c+d x)}{d}-\frac{a^2 \tan ^3(c+d x) \sec ^2(c+d x)}{d}","\frac{a^2 \tan ^3(c+d x)}{5 d}+\frac{3 a^2 \tan (c+d x)}{5 d}+\frac{2 \sec ^5(c+d x) \left(a^2 \sin (c+d x)+a^2\right)}{5 d}",1,"(2*a^2*Sec[c + d*x]^5)/(5*d) + (a^2*Sec[c + d*x]^4*Tan[c + d*x])/d - (a^2*Sec[c + d*x]^2*Tan[c + d*x]^3)/d + (2*a^2*Tan[c + d*x]^5)/(5*d)","A",1
25,1,85,109,0.1450566,"\int \sec ^7(c+d x) (a+a \sin (c+d x))^2 \, dx","Integrate[Sec[c + d*x]^7*(a + a*Sin[c + d*x])^2,x]","-\frac{a^2 (\sin (c+d x)+1)^2 \sec ^6(c+d x) \left(-3 \sin ^3(c+d x)+6 \sin ^2(c+d x)-\sin (c+d x)+3 (\sin (c+d x)+1) (\sin (c+d x)-1)^3 \tanh ^{-1}(\sin (c+d x))-4\right)}{12 d}","\frac{a^5}{12 d (a-a \sin (c+d x))^3}+\frac{a^4}{8 d (a-a \sin (c+d x))^2}+\frac{3 a^3}{16 d (a-a \sin (c+d x))}-\frac{a^3}{16 d (a \sin (c+d x)+a)}+\frac{a^2 \tanh ^{-1}(\sin (c+d x))}{4 d}",1,"-1/12*(a^2*Sec[c + d*x]^6*(1 + Sin[c + d*x])^2*(-4 - Sin[c + d*x] + 6*Sin[c + d*x]^2 - 3*Sin[c + d*x]^3 + 3*ArcTanh[Sin[c + d*x]]*(-1 + Sin[c + d*x])^3*(1 + Sin[c + d*x])))/d","A",1
26,1,110,82,0.0237138,"\int \sec ^8(c+d x) (a+a \sin (c+d x))^2 \, dx","Integrate[Sec[c + d*x]^8*(a + a*Sin[c + d*x])^2,x]","-\frac{8 a^2 \tan ^7(c+d x)}{21 d}+\frac{2 a^2 \sec ^7(c+d x)}{7 d}+\frac{a^2 \tan (c+d x) \sec ^6(c+d x)}{d}-\frac{5 a^2 \tan ^3(c+d x) \sec ^4(c+d x)}{3 d}+\frac{4 a^2 \tan ^5(c+d x) \sec ^2(c+d x)}{3 d}","\frac{a^2 \tan ^5(c+d x)}{7 d}+\frac{10 a^2 \tan ^3(c+d x)}{21 d}+\frac{5 a^2 \tan (c+d x)}{7 d}+\frac{2 \sec ^7(c+d x) \left(a^2 \sin (c+d x)+a^2\right)}{7 d}",1,"(2*a^2*Sec[c + d*x]^7)/(7*d) + (a^2*Sec[c + d*x]^6*Tan[c + d*x])/d - (5*a^2*Sec[c + d*x]^4*Tan[c + d*x]^3)/(3*d) + (4*a^2*Sec[c + d*x]^2*Tan[c + d*x]^5)/(3*d) - (8*a^2*Tan[c + d*x]^7)/(21*d)","A",1
27,1,181,154,1.9950122,"\int \cos ^6(c+d x) (a+a \sin (c+d x))^3 \, dx","Integrate[Cos[c + d*x]^6*(a + a*Sin[c + d*x])^3,x]","-\frac{a^3 \left(6930 \sqrt{1-\sin (c+d x)} \sin ^{-1}\left(\frac{\sqrt{1-\sin (c+d x)}}{\sqrt{2}}\right)+\sqrt{\sin (c+d x)+1} \left(896 \sin ^9(c+d x)+2128 \sin ^8(c+d x)-2000 \sin ^7(c+d x)-8248 \sin ^6(c+d x)-1224 \sin ^5(c+d x)+11514 \sin ^4(c+d x)+7174 \sin ^3(c+d x)-5641 \sin ^2(c+d x)-8311 \sin (c+d x)+3712\right)\right) \cos ^7(c+d x)}{8064 d (\sin (c+d x)-1)^4 (\sin (c+d x)+1)^{7/2}}","-\frac{11 a^3 \cos ^7(c+d x)}{56 d}-\frac{11 \cos ^7(c+d x) \left(a^3 \sin (c+d x)+a^3\right)}{72 d}+\frac{11 a^3 \sin (c+d x) \cos ^5(c+d x)}{48 d}+\frac{55 a^3 \sin (c+d x) \cos ^3(c+d x)}{192 d}+\frac{55 a^3 \sin (c+d x) \cos (c+d x)}{128 d}+\frac{55 a^3 x}{128}-\frac{a \cos ^7(c+d x) (a \sin (c+d x)+a)^2}{9 d}",1,"-1/8064*(a^3*Cos[c + d*x]^7*(6930*ArcSin[Sqrt[1 - Sin[c + d*x]]/Sqrt[2]]*Sqrt[1 - Sin[c + d*x]] + Sqrt[1 + Sin[c + d*x]]*(3712 - 8311*Sin[c + d*x] - 5641*Sin[c + d*x]^2 + 7174*Sin[c + d*x]^3 + 11514*Sin[c + d*x]^4 - 1224*Sin[c + d*x]^5 - 8248*Sin[c + d*x]^6 - 2000*Sin[c + d*x]^7 + 2128*Sin[c + d*x]^8 + 896*Sin[c + d*x]^9)))/(d*(-1 + Sin[c + d*x])^4*(1 + Sin[c + d*x])^(7/2))","A",1
28,1,58,67,0.0928796,"\int \cos ^5(c+d x) (a+a \sin (c+d x))^3 \, dx","Integrate[Cos[c + d*x]^5*(a + a*Sin[c + d*x])^3,x]","-\frac{a^3 (\sin (c+d x)+1)^3 \left(21 \sin ^2(c+d x)-54 \sin (c+d x)+37\right) \cos ^6(c+d x)}{168 d (\sin (c+d x)-1)^3}","\frac{(a \sin (c+d x)+a)^8}{8 a^5 d}-\frac{4 (a \sin (c+d x)+a)^7}{7 a^4 d}+\frac{2 (a \sin (c+d x)+a)^6}{3 a^3 d}",1,"-1/168*(a^3*Cos[c + d*x]^6*(1 + Sin[c + d*x])^3*(37 - 54*Sin[c + d*x] + 21*Sin[c + d*x]^2))/(d*(-1 + Sin[c + d*x])^3)","A",1
29,1,161,130,0.7785899,"\int \cos ^4(c+d x) (a+a \sin (c+d x))^3 \, dx","Integrate[Cos[c + d*x]^4*(a + a*Sin[c + d*x])^3,x]","-\frac{a^3 \left(\sqrt{\sin (c+d x)+1} \left(80 \sin ^7(c+d x)+200 \sin ^6(c+d x)-72 \sin ^5(c+d x)-558 \sin ^4(c+d x)-306 \sin ^3(c+d x)+411 \sin ^2(c+d x)+613 \sin (c+d x)-368\right)-630 \sin ^{-1}\left(\frac{\sqrt{1-\sin (c+d x)}}{\sqrt{2}}\right) \sqrt{1-\sin (c+d x)}\right) \cos ^5(c+d x)}{560 d (\sin (c+d x)-1)^3 (\sin (c+d x)+1)^{5/2}}","-\frac{3 a^3 \cos ^5(c+d x)}{10 d}-\frac{3 \cos ^5(c+d x) \left(a^3 \sin (c+d x)+a^3\right)}{14 d}+\frac{3 a^3 \sin (c+d x) \cos ^3(c+d x)}{8 d}+\frac{9 a^3 \sin (c+d x) \cos (c+d x)}{16 d}+\frac{9 a^3 x}{16}-\frac{a \cos ^5(c+d x) (a \sin (c+d x)+a)^2}{7 d}",1,"-1/560*(a^3*Cos[c + d*x]^5*(-630*ArcSin[Sqrt[1 - Sin[c + d*x]]/Sqrt[2]]*Sqrt[1 - Sin[c + d*x]] + Sqrt[1 + Sin[c + d*x]]*(-368 + 613*Sin[c + d*x] + 411*Sin[c + d*x]^2 - 306*Sin[c + d*x]^3 - 558*Sin[c + d*x]^4 - 72*Sin[c + d*x]^5 + 200*Sin[c + d*x]^6 + 80*Sin[c + d*x]^7)))/(d*(-1 + Sin[c + d*x])^3*(1 + Sin[c + d*x])^(5/2))","A",1
30,1,43,45,0.1505698,"\int \cos ^3(c+d x) (a+a \sin (c+d x))^3 \, dx","Integrate[Cos[c + d*x]^3*(a + a*Sin[c + d*x])^3,x]","-\frac{a^3 (5 \sin (c+d x)-7) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^{10}}{30 d}","\frac{2 (a \sin (c+d x)+a)^5}{5 a^2 d}-\frac{(a \sin (c+d x)+a)^6}{6 a^3 d}",1,"-1/30*(a^3*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^10*(-7 + 5*Sin[c + d*x]))/d","A",1
31,1,141,106,0.4081956,"\int \cos ^2(c+d x) (a+a \sin (c+d x))^3 \, dx","Integrate[Cos[c + d*x]^2*(a + a*Sin[c + d*x])^3,x]","-\frac{a^3 \left(210 \sqrt{1-\sin (c+d x)} \sin ^{-1}\left(\frac{\sqrt{1-\sin (c+d x)}}{\sqrt{2}}\right)+\sqrt{\sin (c+d x)+1} \left(24 \sin ^5(c+d x)+66 \sin ^4(c+d x)+22 \sin ^3(c+d x)-97 \sin ^2(c+d x)-151 \sin (c+d x)+136\right)\right) \cos ^3(c+d x)}{120 d (\sin (c+d x)-1)^2 (\sin (c+d x)+1)^{3/2}}","-\frac{7 a^3 \cos ^3(c+d x)}{12 d}-\frac{7 \cos ^3(c+d x) \left(a^3 \sin (c+d x)+a^3\right)}{20 d}+\frac{7 a^3 \sin (c+d x) \cos (c+d x)}{8 d}+\frac{7 a^3 x}{8}-\frac{a \cos ^3(c+d x) (a \sin (c+d x)+a)^2}{5 d}",1,"-1/120*(a^3*Cos[c + d*x]^3*(210*ArcSin[Sqrt[1 - Sin[c + d*x]]/Sqrt[2]]*Sqrt[1 - Sin[c + d*x]] + Sqrt[1 + Sin[c + d*x]]*(136 - 151*Sin[c + d*x] - 97*Sin[c + d*x]^2 + 22*Sin[c + d*x]^3 + 66*Sin[c + d*x]^4 + 24*Sin[c + d*x]^5)))/(d*(-1 + Sin[c + d*x])^2*(1 + Sin[c + d*x])^(3/2))","A",1
32,1,65,22,0.0267655,"\int \cos (c+d x) (a+a \sin (c+d x))^3 \, dx","Integrate[Cos[c + d*x]*(a + a*Sin[c + d*x])^3,x]","\frac{a^3 \sin ^4(c+d x)}{4 d}+\frac{a^3 \sin ^3(c+d x)}{d}+\frac{3 a^3 \sin ^2(c+d x)}{2 d}+\frac{a^3 \sin (c+d x)}{d}","\frac{(a \sin (c+d x)+a)^4}{4 a d}",1,"(a^3*Sin[c + d*x])/d + (3*a^3*Sin[c + d*x]^2)/(2*d) + (a^3*Sin[c + d*x]^3)/d + (a^3*Sin[c + d*x]^4)/(4*d)","B",1
33,1,41,52,0.0293713,"\int \sec (c+d x) (a+a \sin (c+d x))^3 \, dx","Integrate[Sec[c + d*x]*(a + a*Sin[c + d*x])^3,x]","\frac{a^3 \left(-\frac{1}{2} \sin ^2(c+d x)-3 \sin (c+d x)-4 \log (1-\sin (c+d x))\right)}{d}","-\frac{a^3 \sin ^2(c+d x)}{2 d}-\frac{3 a^3 \sin (c+d x)}{d}-\frac{4 a^3 \log (1-\sin (c+d x))}{d}",1,"(a^3*(-4*Log[1 - Sin[c + d*x]] - 3*Sin[c + d*x] - Sin[c + d*x]^2/2))/d","A",1
34,1,55,50,0.0340295,"\int \sec ^2(c+d x) (a+a \sin (c+d x))^3 \, dx","Integrate[Sec[c + d*x]^2*(a + a*Sin[c + d*x])^3,x]","\frac{4 \sqrt{2} a^3 \sqrt{\sin (c+d x)+1} \sec (c+d x) \, _2F_1\left(-\frac{3}{2},-\frac{1}{2};\frac{1}{2};\frac{1}{2} (1-\sin (c+d x))\right)}{d}","\frac{2 a^5 \cos ^3(c+d x)}{d (a-a \sin (c+d x))^2}+\frac{3 a^3 \cos (c+d x)}{d}-3 a^3 x",1,"(4*Sqrt[2]*a^3*Hypergeometric2F1[-3/2, -1/2, 1/2, (1 - Sin[c + d*x])/2]*Sec[c + d*x]*Sqrt[1 + Sin[c + d*x]])/d","C",1
35,1,59,40,0.0420163,"\int \sec ^3(c+d x) (a+a \sin (c+d x))^3 \, dx","Integrate[Sec[c + d*x]^3*(a + a*Sin[c + d*x])^3,x]","\frac{a^3 (1-\sin (c+d x)) (\sin (c+d x)+1) \sec ^2(c+d x) \left(\frac{2}{1-\sin (c+d x)}+\log (1-\sin (c+d x))\right)}{d}","\frac{2 a^4}{d (a-a \sin (c+d x))}+\frac{a^3 \log (1-\sin (c+d x))}{d}",1,"(a^3*Sec[c + d*x]^2*(Log[1 - Sin[c + d*x]] + 2/(1 - Sin[c + d*x]))*(1 - Sin[c + d*x])*(1 + Sin[c + d*x]))/d","A",1
36,1,28,31,0.0254152,"\int \sec ^4(c+d x) (a+a \sin (c+d x))^3 \, dx","Integrate[Sec[c + d*x]^4*(a + a*Sin[c + d*x])^3,x]","\frac{a^3 (\sin (c+d x)+1)^3 \sec ^3(c+d x)}{3 d}","\frac{a^6 \cos ^3(c+d x)}{3 d (a-a \sin (c+d x))^3}",1,"(a^3*Sec[c + d*x]^3*(1 + Sin[c + d*x])^3)/(3*d)","A",1
37,1,35,23,0.2321503,"\int \sec ^5(c+d x) (a+a \sin (c+d x))^3 \, dx","Integrate[Sec[c + d*x]^5*(a + a*Sin[c + d*x])^3,x]","\frac{a^3}{2 d \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^4}","\frac{a^5}{2 d (a-a \sin (c+d x))^2}",1,"a^3/(2*d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^4)","A",1
38,1,110,92,0.0182246,"\int \sec ^6(c+d x) (a+a \sin (c+d x))^3 \, dx","Integrate[Sec[c + d*x]^6*(a + a*Sin[c + d*x])^3,x]","\frac{2 a^3 \tan ^5(c+d x)}{15 d}+\frac{7 a^3 \sec ^5(c+d x)}{15 d}+\frac{a^3 \tan (c+d x) \sec ^4(c+d x)}{d}+\frac{a^3 \tan ^2(c+d x) \sec ^3(c+d x)}{3 d}-\frac{a^3 \tan ^3(c+d x) \sec ^2(c+d x)}{3 d}","\frac{a^6 \cos (c+d x)}{5 d (a-a \sin (c+d x))^3}+\frac{2 a^5 \cos (c+d x)}{15 d (a-a \sin (c+d x))^2}+\frac{2 a^6 \cos (c+d x)}{15 d \left(a^3-a^3 \sin (c+d x)\right)}",1,"(7*a^3*Sec[c + d*x]^5)/(15*d) + (a^3*Sec[c + d*x]^4*Tan[c + d*x])/d + (a^3*Sec[c + d*x]^3*Tan[c + d*x]^2)/(3*d) - (a^3*Sec[c + d*x]^2*Tan[c + d*x]^3)/(3*d) + (2*a^3*Tan[c + d*x]^5)/(15*d)","A",1
39,1,67,87,0.1028228,"\int \sec ^7(c+d x) (a+a \sin (c+d x))^3 \, dx","Integrate[Sec[c + d*x]^7*(a + a*Sin[c + d*x])^3,x]","-\frac{a^3 (\sin (c+d x)+1)^3 \sec ^6(c+d x) \left(-3 \sin ^2(c+d x)+9 \sin (c+d x)+3 (\sin (c+d x)-1)^3 \tanh ^{-1}(\sin (c+d x))-10\right)}{24 d}","\frac{a^6}{6 d (a-a \sin (c+d x))^3}+\frac{a^5}{8 d (a-a \sin (c+d x))^2}+\frac{a^4}{8 d (a-a \sin (c+d x))}+\frac{a^3 \tanh ^{-1}(\sin (c+d x))}{8 d}",1,"-1/24*(a^3*Sec[c + d*x]^6*(1 + Sin[c + d*x])^3*(-10 + 3*ArcTanh[Sin[c + d*x]]*(-1 + Sin[c + d*x])^3 + 9*Sin[c + d*x] - 3*Sin[c + d*x]^2))/d","A",1
40,1,134,99,0.0118107,"\int \sec ^8(c+d x) (a+a \sin (c+d x))^3 \, dx","Integrate[Sec[c + d*x]^8*(a + a*Sin[c + d*x])^3,x]","-\frac{8 a^3 \tan ^7(c+d x)}{35 d}+\frac{13 a^3 \sec ^7(c+d x)}{35 d}+\frac{a^3 \tan (c+d x) \sec ^6(c+d x)}{d}+\frac{a^3 \tan ^2(c+d x) \sec ^5(c+d x)}{5 d}-\frac{a^3 \tan ^3(c+d x) \sec ^4(c+d x)}{d}+\frac{4 a^3 \tan ^5(c+d x) \sec ^2(c+d x)}{5 d}","\frac{3 a^3 \tan ^5(c+d x)}{35 d}+\frac{2 a^3 \tan ^3(c+d x)}{7 d}+\frac{3 a^3 \tan (c+d x)}{7 d}+\frac{3 a^3 \sec ^5(c+d x)}{35 d}+\frac{2 a \sec ^7(c+d x) (a \sin (c+d x)+a)^2}{7 d}",1,"(13*a^3*Sec[c + d*x]^7)/(35*d) + (a^3*Sec[c + d*x]^6*Tan[c + d*x])/d + (a^3*Sec[c + d*x]^5*Tan[c + d*x]^2)/(5*d) - (a^3*Sec[c + d*x]^4*Tan[c + d*x]^3)/d + (4*a^3*Sec[c + d*x]^2*Tan[c + d*x]^5)/(5*d) - (8*a^3*Tan[c + d*x]^7)/(35*d)","A",1
41,1,58,67,0.4161859,"\int \cos ^5(c+d x) (a+a \sin (c+d x))^8 \, dx","Integrate[Cos[c + d*x]^5*(a + a*Sin[c + d*x])^8,x]","-\frac{a^8 (\sin (c+d x)+1)^8 \left(33 \sin ^2(c+d x)-77 \sin (c+d x)+46\right) \cos ^6(c+d x)}{429 d (\sin (c+d x)-1)^3}","\frac{(a \sin (c+d x)+a)^{13}}{13 a^5 d}-\frac{(a \sin (c+d x)+a)^{12}}{3 a^4 d}+\frac{4 (a \sin (c+d x)+a)^{11}}{11 a^3 d}",1,"-1/429*(a^8*Cos[c + d*x]^6*(1 + Sin[c + d*x])^8*(46 - 77*Sin[c + d*x] + 33*Sin[c + d*x]^2))/(d*(-1 + Sin[c + d*x])^3)","A",1
42,1,211,286,3.202612,"\int \cos ^4(c+d x) (a+a \sin (c+d x))^8 \, dx","Integrate[Cos[c + d*x]^4*(a + a*Sin[c + d*x])^8,x]","-\frac{a^8 \left(\sqrt{\sin (c+d x)+1} \left(295680 \sin ^{12}(c+d x)+2284800 \sin ^{11}(c+d x)+6969984 \sin ^{10}(c+d x)+9086336 \sin ^9(c+d x)-1239728 \sin ^8(c+d x)-20428112 \sin ^7(c+d x)-26346616 \sin ^6(c+d x)-8321928 \sin ^5(c+d x)+14283114 \sin ^4(c+d x)+20459158 \sin ^3(c+d x)+13958687 \sin ^2(c+d x)+11469281 \sin (c+d x)-22470656\right)-29099070 \sin ^{-1}\left(\frac{\sqrt{1-\sin (c+d x)}}{\sqrt{2}}\right) \sqrt{1-\sin (c+d x)}\right) \cos ^5(c+d x)}{3548160 d (\sin (c+d x)-1)^3 (\sin (c+d x)+1)^{5/2}}","-\frac{4199 a^8 \cos ^5(c+d x)}{1920 d}-\frac{4199 \cos ^5(c+d x) \left(a^8 \sin (c+d x)+a^8\right)}{2688 d}+\frac{4199 a^8 \sin (c+d x) \cos ^3(c+d x)}{1536 d}+\frac{4199 a^8 \sin (c+d x) \cos (c+d x)}{1024 d}+\frac{4199 a^8 x}{1024}-\frac{4199 \cos ^5(c+d x) \left(a^4 \sin (c+d x)+a^4\right)^2}{4032 d}-\frac{323 a^3 \cos ^5(c+d x) (a \sin (c+d x)+a)^5}{1320 d}-\frac{19 a^2 \cos ^5(c+d x) (a \sin (c+d x)+a)^6}{132 d}-\frac{4199 a^2 \cos ^5(c+d x) \left(a^2 \sin (c+d x)+a^2\right)^3}{6336 d}-\frac{323 \cos ^5(c+d x) \left(a^2 \sin (c+d x)+a^2\right)^4}{792 d}-\frac{a \cos ^5(c+d x) (a \sin (c+d x)+a)^7}{12 d}",1,"-1/3548160*(a^8*Cos[c + d*x]^5*(-29099070*ArcSin[Sqrt[1 - Sin[c + d*x]]/Sqrt[2]]*Sqrt[1 - Sin[c + d*x]] + Sqrt[1 + Sin[c + d*x]]*(-22470656 + 11469281*Sin[c + d*x] + 13958687*Sin[c + d*x]^2 + 20459158*Sin[c + d*x]^3 + 14283114*Sin[c + d*x]^4 - 8321928*Sin[c + d*x]^5 - 26346616*Sin[c + d*x]^6 - 20428112*Sin[c + d*x]^7 - 1239728*Sin[c + d*x]^8 + 9086336*Sin[c + d*x]^9 + 6969984*Sin[c + d*x]^10 + 2284800*Sin[c + d*x]^11 + 295680*Sin[c + d*x]^12)))/(d*(-1 + Sin[c + d*x])^3*(1 + Sin[c + d*x])^(5/2))","A",1
43,1,43,45,1.0891795,"\int \cos ^3(c+d x) (a+a \sin (c+d x))^8 \, dx","Integrate[Cos[c + d*x]^3*(a + a*Sin[c + d*x])^8,x]","-\frac{a^8 (5 \sin (c+d x)-6) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^{20}}{55 d}","\frac{(a \sin (c+d x)+a)^{10}}{5 a^2 d}-\frac{(a \sin (c+d x)+a)^{11}}{11 a^3 d}",1,"-1/55*(a^8*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^20*(-6 + 5*Sin[c + d*x]))/d","A",1
44,1,191,262,1.4939716,"\int \cos ^2(c+d x) (a+a \sin (c+d x))^8 \, dx","Integrate[Cos[c + d*x]^2*(a + a*Sin[c + d*x])^8,x]","-\frac{a^8 \left(1531530 \sqrt{1-\sin (c+d x)} \sin ^{-1}\left(\frac{\sqrt{1-\sin (c+d x)}}{\sqrt{2}}\right)+\sqrt{\sin (c+d x)+1} \left(8064 \sin ^{10}(c+d x)+63616 \sin ^9(c+d x)+209552 \sin ^8(c+d x)+353648 \sin ^7(c+d x)+257704 \sin ^6(c+d x)-130728 \sin ^5(c+d x)-492846 \sin ^4(c+d x)-543442 \sin ^3(c+d x)-410693 \sin ^2(c+d x)-508859 \sin (c+d x)+1193984\right)\right) \cos ^3(c+d x)}{80640 d (\sin (c+d x)-1)^2 (\sin (c+d x)+1)^{3/2}}","-\frac{2431 a^8 \cos ^3(c+d x)}{384 d}-\frac{2431 \cos ^3(c+d x) \left(a^8 \sin (c+d x)+a^8\right)}{640 d}+\frac{2431 a^8 \sin (c+d x) \cos (c+d x)}{256 d}+\frac{2431 a^8 x}{256}-\frac{2431 \cos ^3(c+d x) \left(a^4 \sin (c+d x)+a^4\right)^2}{1120 d}-\frac{17 a^3 \cos ^3(c+d x) (a \sin (c+d x)+a)^5}{48 d}-\frac{17 a^2 \cos ^3(c+d x) (a \sin (c+d x)+a)^6}{90 d}-\frac{2431 a^2 \cos ^3(c+d x) \left(a^2 \sin (c+d x)+a^2\right)^3}{2016 d}-\frac{221 \cos ^3(c+d x) \left(a^2 \sin (c+d x)+a^2\right)^4}{336 d}-\frac{a \cos ^3(c+d x) (a \sin (c+d x)+a)^7}{10 d}",1,"-1/80640*(a^8*Cos[c + d*x]^3*(1531530*ArcSin[Sqrt[1 - Sin[c + d*x]]/Sqrt[2]]*Sqrt[1 - Sin[c + d*x]] + Sqrt[1 + Sin[c + d*x]]*(1193984 - 508859*Sin[c + d*x] - 410693*Sin[c + d*x]^2 - 543442*Sin[c + d*x]^3 - 492846*Sin[c + d*x]^4 - 130728*Sin[c + d*x]^5 + 257704*Sin[c + d*x]^6 + 353648*Sin[c + d*x]^7 + 209552*Sin[c + d*x]^8 + 63616*Sin[c + d*x]^9 + 8064*Sin[c + d*x]^10)))/(d*(-1 + Sin[c + d*x])^2*(1 + Sin[c + d*x])^(3/2))","A",1
45,1,147,22,0.0902877,"\int \cos (c+d x) (a+a \sin (c+d x))^8 \, dx","Integrate[Cos[c + d*x]*(a + a*Sin[c + d*x])^8,x]","\frac{a^8 \sin ^9(c+d x)}{9 d}+\frac{a^8 \sin ^8(c+d x)}{d}+\frac{4 a^8 \sin ^7(c+d x)}{d}+\frac{28 a^8 \sin ^6(c+d x)}{3 d}+\frac{14 a^8 \sin ^5(c+d x)}{d}+\frac{14 a^8 \sin ^4(c+d x)}{d}+\frac{28 a^8 \sin ^3(c+d x)}{3 d}+\frac{4 a^8 \sin ^2(c+d x)}{d}+\frac{a^8 \sin (c+d x)}{d}","\frac{(a \sin (c+d x)+a)^9}{9 a d}",1,"(a^8*Sin[c + d*x])/d + (4*a^8*Sin[c + d*x]^2)/d + (28*a^8*Sin[c + d*x]^3)/(3*d) + (14*a^8*Sin[c + d*x]^4)/d + (14*a^8*Sin[c + d*x]^5)/d + (28*a^8*Sin[c + d*x]^6)/(3*d) + (4*a^8*Sin[c + d*x]^7)/d + (a^8*Sin[c + d*x]^8)/d + (a^8*Sin[c + d*x]^9)/(9*d)","B",1
46,1,95,162,0.1700664,"\int \sec (c+d x) (a+a \sin (c+d x))^8 \, dx","Integrate[Sec[c + d*x]*(a + a*Sin[c + d*x])^8,x]","\frac{a^8 \left(-\frac{1}{7} \sin ^7(c+d x)-\frac{4}{3} \sin ^6(c+d x)-\frac{29}{5} \sin ^5(c+d x)-16 \sin ^4(c+d x)-33 \sin ^3(c+d x)-60 \sin ^2(c+d x)-127 \sin (c+d x)-128 \log (1-\sin (c+d x))\right)}{d}","-\frac{64 a^8 \sin (c+d x)}{d}-\frac{128 a^8 \log (1-\sin (c+d x))}{d}-\frac{16 a^5 (a \sin (c+d x)+a)^3}{3 d}-\frac{16 \left(a^4 \sin (c+d x)+a^4\right)^2}{d}-\frac{4 a^3 (a \sin (c+d x)+a)^5}{5 d}-\frac{a^2 (a \sin (c+d x)+a)^6}{3 d}-\frac{2 \left(a^2 \sin (c+d x)+a^2\right)^4}{d}-\frac{a (a \sin (c+d x)+a)^7}{7 d}",1,"(a^8*(-128*Log[1 - Sin[c + d*x]] - 127*Sin[c + d*x] - 60*Sin[c + d*x]^2 - 33*Sin[c + d*x]^3 - 16*Sin[c + d*x]^4 - (29*Sin[c + d*x]^5)/5 - (4*Sin[c + d*x]^6)/3 - Sin[c + d*x]^7/7))/d","A",1
47,1,55,201,0.0620831,"\int \sec ^2(c+d x) (a+a \sin (c+d x))^8 \, dx","Integrate[Sec[c + d*x]^2*(a + a*Sin[c + d*x])^8,x]","\frac{128 \sqrt{2} a^8 \sqrt{\sin (c+d x)+1} \sec (c+d x) \, _2F_1\left(-\frac{13}{2},-\frac{1}{2};\frac{1}{2};\frac{1}{2} (1-\sin (c+d x))\right)}{d}","\frac{2 a^{15} \cos ^{13}(c+d x)}{d (a-a \sin (c+d x))^7}+\frac{26 a^{13} \cos ^{11}(c+d x)}{d (a-a \sin (c+d x))^5}+\frac{1001 a^8 \cos ^5(c+d x)}{10 d}-\frac{1001 a^8 \sin (c+d x) \cos ^3(c+d x)}{8 d}-\frac{3003 a^8 \sin (c+d x) \cos (c+d x)}{16 d}-\frac{3003 a^8 x}{16}+\frac{143 a^{16} \cos ^7(c+d x)}{2 d \left(a^8-a^8 \sin (c+d x)\right)}+\frac{286 a^{14} \cos ^9(c+d x)}{3 d \left(a^2-a^2 \sin (c+d x)\right)^3}",1,"(128*Sqrt[2]*a^8*Hypergeometric2F1[-13/2, -1/2, 1/2, (1 - Sin[c + d*x])/2]*Sec[c + d*x]*Sqrt[1 + Sin[c + d*x]])/d","C",1
48,1,111,121,0.2700314,"\int \sec ^3(c+d x) (a+a \sin (c+d x))^8 \, dx","Integrate[Sec[c + d*x]^3*(a + a*Sin[c + d*x])^8,x]","\frac{a^8 (1-\sin (c+d x)) (\sin (c+d x)+1) \sec ^2(c+d x) \left(\frac{1}{5} \sin ^5(c+d x)+2 \sin ^4(c+d x)+10 \sin ^3(c+d x)+36 \sin ^2(c+d x)+129 \sin (c+d x)+\frac{64}{1-\sin (c+d x)}+192 \log (1-\sin (c+d x))\right)}{d}","\frac{64 a^9}{d (a-a \sin (c+d x))}+\frac{a^8 \sin ^5(c+d x)}{5 d}+\frac{2 a^8 \sin ^4(c+d x)}{d}+\frac{10 a^8 \sin ^3(c+d x)}{d}+\frac{36 a^8 \sin ^2(c+d x)}{d}+\frac{129 a^8 \sin (c+d x)}{d}+\frac{192 a^8 \log (1-\sin (c+d x))}{d}",1,"(a^8*Sec[c + d*x]^2*(1 - Sin[c + d*x])*(1 + Sin[c + d*x])*(192*Log[1 - Sin[c + d*x]] + 64/(1 - Sin[c + d*x]) + 129*Sin[c + d*x] + 36*Sin[c + d*x]^2 + 10*Sin[c + d*x]^3 + 2*Sin[c + d*x]^4 + Sin[c + d*x]^5/5))/d","A",1
49,1,59,179,0.052014,"\int \sec ^4(c+d x) (a+a \sin (c+d x))^8 \, dx","Integrate[Sec[c + d*x]^4*(a + a*Sin[c + d*x])^8,x]","\frac{64 \sqrt{2} a^8 (\sin (c+d x)+1)^{3/2} \sec ^3(c+d x) \, _2F_1\left(-\frac{11}{2},-\frac{3}{2};-\frac{1}{2};\frac{1}{2} (1-\sin (c+d x))\right)}{3 d}","\frac{2 a^{15} \cos ^{11}(c+d x)}{3 d (a-a \sin (c+d x))^7}-\frac{22 a^{13} \cos ^9(c+d x)}{3 d (a-a \sin (c+d x))^5}-\frac{385 a^8 \cos ^3(c+d x)}{4 d}+\frac{1155 a^8 \sin (c+d x) \cos (c+d x)}{8 d}+\frac{1155 a^8 x}{8}-\frac{231 a^{16} \cos ^5(c+d x)}{4 d \left(a^8-a^8 \sin (c+d x)\right)}-\frac{66 a^{14} \cos ^7(c+d x)}{d \left(a^2-a^2 \sin (c+d x)\right)^3}",1,"(64*Sqrt[2]*a^8*Hypergeometric2F1[-11/2, -3/2, -1/2, (1 - Sin[c + d*x])/2]*Sec[c + d*x]^3*(1 + Sin[c + d*x])^(3/2))/(3*d)","C",1
50,1,73,110,0.4440652,"\int \sec ^5(c+d x) (a+a \sin (c+d x))^8 \, dx","Integrate[Sec[c + d*x]^5*(a + a*Sin[c + d*x])^8,x]","\frac{a^8 \left(-\frac{1}{3} \sin ^3(c+d x)-4 \sin ^2(c+d x)-31 \sin (c+d x)+\frac{16 (5 \sin (c+d x)-4)}{(\sin (c+d x)-1)^2}-80 \log (1-\sin (c+d x))\right)}{d}","\frac{16 a^{10}}{d (a-a \sin (c+d x))^2}-\frac{80 a^9}{d (a-a \sin (c+d x))}-\frac{a^8 \sin ^3(c+d x)}{3 d}-\frac{4 a^8 \sin ^2(c+d x)}{d}-\frac{31 a^8 \sin (c+d x)}{d}-\frac{80 a^8 \log (1-\sin (c+d x))}{d}",1,"(a^8*(-80*Log[1 - Sin[c + d*x]] - 31*Sin[c + d*x] - 4*Sin[c + d*x]^2 - Sin[c + d*x]^3/3 + (16*(-4 + 5*Sin[c + d*x]))/(-1 + Sin[c + d*x])^2))/d","A",1
51,1,141,73,0.7992179,"\int \frac{\cos ^6(c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[Cos[c + d*x]^6/(a + a*Sin[c + d*x]),x]","-\frac{\left(30 \sqrt{1-\sin (c+d x)} \sin ^{-1}\left(\frac{\sqrt{1-\sin (c+d x)}}{\sqrt{2}}\right)+\sqrt{\sin (c+d x)+1} \left(8 \sin ^5(c+d x)-18 \sin ^4(c+d x)-6 \sin ^3(c+d x)+41 \sin ^2(c+d x)-17 \sin (c+d x)-8\right)\right) \cos ^7(c+d x)}{40 a d (\sin (c+d x)-1)^4 (\sin (c+d x)+1)^{7/2}}","\frac{\cos ^5(c+d x)}{5 a d}+\frac{\sin (c+d x) \cos ^3(c+d x)}{4 a d}+\frac{3 \sin (c+d x) \cos (c+d x)}{8 a d}+\frac{3 x}{8 a}",1,"-1/40*(Cos[c + d*x]^7*(30*ArcSin[Sqrt[1 - Sin[c + d*x]]/Sqrt[2]]*Sqrt[1 - Sin[c + d*x]] + Sqrt[1 + Sin[c + d*x]]*(-8 - 17*Sin[c + d*x] + 41*Sin[c + d*x]^2 - 6*Sin[c + d*x]^3 - 18*Sin[c + d*x]^4 + 8*Sin[c + d*x]^5)))/(a*d*(-1 + Sin[c + d*x])^4*(1 + Sin[c + d*x])^(7/2))","A",1
52,1,46,47,0.091109,"\int \frac{\cos ^5(c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[Cos[c + d*x]^5/(a + a*Sin[c + d*x]),x]","\frac{\sin (c+d x) \left(3 \sin ^3(c+d x)-4 \sin ^2(c+d x)-6 \sin (c+d x)+12\right)}{12 a d}","\frac{(a-a \sin (c+d x))^4}{4 a^5 d}-\frac{2 (a-a \sin (c+d x))^3}{3 a^4 d}",1,"(Sin[c + d*x]*(12 - 6*Sin[c + d*x] - 4*Sin[c + d*x]^2 + 3*Sin[c + d*x]^3))/(12*a*d)","A",1
53,1,119,49,0.3199404,"\int \frac{\cos ^4(c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[Cos[c + d*x]^4/(a + a*Sin[c + d*x]),x]","-\frac{\left(\sqrt{\sin (c+d x)+1} \left(2 \sin ^3(c+d x)-5 \sin ^2(c+d x)+\sin (c+d x)+2\right)-6 \sin ^{-1}\left(\frac{\sqrt{1-\sin (c+d x)}}{\sqrt{2}}\right) \sqrt{1-\sin (c+d x)}\right) \cos ^5(c+d x)}{6 a d (\sin (c+d x)-1)^3 (\sin (c+d x)+1)^{5/2}}","\frac{\cos ^3(c+d x)}{3 a d}+\frac{\sin (c+d x) \cos (c+d x)}{2 a d}+\frac{x}{2 a}",1,"-1/6*(Cos[c + d*x]^5*(-6*ArcSin[Sqrt[1 - Sin[c + d*x]]/Sqrt[2]]*Sqrt[1 - Sin[c + d*x]] + Sqrt[1 + Sin[c + d*x]]*(2 + Sin[c + d*x] - 5*Sin[c + d*x]^2 + 2*Sin[c + d*x]^3)))/(a*d*(-1 + Sin[c + d*x])^3*(1 + Sin[c + d*x])^(5/2))","B",1
54,1,24,32,0.0376925,"\int \frac{\cos ^3(c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[Cos[c + d*x]^3/(a + a*Sin[c + d*x]),x]","-\frac{(\sin (c+d x)-2) \sin (c+d x)}{2 a d}","\frac{\sin (c+d x)}{a d}-\frac{\sin ^2(c+d x)}{2 a d}",1,"-1/2*((-2 + Sin[c + d*x])*Sin[c + d*x])/(a*d)","A",1
55,1,97,19,0.121635,"\int \frac{\cos ^2(c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[Cos[c + d*x]^2/(a + a*Sin[c + d*x]),x]","-\frac{\left(2 \sqrt{1-\sin (c+d x)} \sin ^{-1}\left(\frac{\sqrt{1-\sin (c+d x)}}{\sqrt{2}}\right)+(\sin (c+d x)-1) \sqrt{\sin (c+d x)+1}\right) \cos ^3(c+d x)}{a d (\sin (c+d x)-1)^2 (\sin (c+d x)+1)^{3/2}}","\frac{\cos (c+d x)}{a d}+\frac{x}{a}",1,"-((Cos[c + d*x]^3*(2*ArcSin[Sqrt[1 - Sin[c + d*x]]/Sqrt[2]]*Sqrt[1 - Sin[c + d*x]] + (-1 + Sin[c + d*x])*Sqrt[1 + Sin[c + d*x]]))/(a*d*(-1 + Sin[c + d*x])^2*(1 + Sin[c + d*x])^(3/2)))","B",1
56,1,16,16,0.0106491,"\int \frac{\cos (c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[Cos[c + d*x]/(a + a*Sin[c + d*x]),x]","\frac{\log (\sin (c+d x)+1)}{a d}","\frac{\log (\sin (c+d x)+1)}{a d}",1,"Log[1 + Sin[c + d*x]]/(a*d)","A",1
57,1,30,37,0.0397508,"\int \frac{\sec (c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[Sec[c + d*x]/(a + a*Sin[c + d*x]),x]","\frac{\tanh ^{-1}(\sin (c+d x))-\frac{1}{\sin (c+d x)+1}}{2 a d}","\frac{\tanh ^{-1}(\sin (c+d x))}{2 a d}-\frac{1}{2 d (a \sin (c+d x)+a)}",1,"(ArcTanh[Sin[c + d*x]] - (1 + Sin[c + d*x])^(-1))/(2*a*d)","A",1
58,1,45,42,0.0564776,"\int \frac{\sec ^2(c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[Sec[c + d*x]^2/(a + a*Sin[c + d*x]),x]","\frac{2 \tan (c+d x)-\cos (2 (c+d x)) \sec (c+d x)}{3 a d (\sin (c+d x)+1)}","\frac{2 \tan (c+d x)}{3 a d}-\frac{\sec (c+d x)}{3 d (a \sin (c+d x)+a)}",1,"(-(Cos[2*(c + d*x)]*Sec[c + d*x]) + 2*Tan[c + d*x])/(3*a*d*(1 + Sin[c + d*x]))","A",1
59,1,75,77,0.0994742,"\int \frac{\sec ^3(c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[Sec[c + d*x]^3/(a + a*Sin[c + d*x]),x]","-\frac{\sec ^2(c+d x) \left(-3 \sin ^2(c+d x)-3 \sin (c+d x)+3 (\sin (c+d x)-1) (\sin (c+d x)+1)^2 \tanh ^{-1}(\sin (c+d x))+2\right)}{8 a d (\sin (c+d x)+1)}","-\frac{a}{8 d (a \sin (c+d x)+a)^2}+\frac{1}{8 d (a-a \sin (c+d x))}-\frac{1}{4 d (a \sin (c+d x)+a)}+\frac{3 \tanh ^{-1}(\sin (c+d x))}{8 a d}",1,"-1/8*(Sec[c + d*x]^2*(2 - 3*Sin[c + d*x] - 3*Sin[c + d*x]^2 + 3*ArcTanh[Sin[c + d*x]]*(-1 + Sin[c + d*x])*(1 + Sin[c + d*x])^2))/(a*d*(1 + Sin[c + d*x]))","A",1
60,1,66,62,0.0982977,"\int \frac{\sec ^4(c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[Sec[c + d*x]^4/(a + a*Sin[c + d*x]),x]","-\frac{\sec ^3(c+d x) (-2 (3 \sin (c+d x)+\sin (3 (c+d x)))+2 \cos (2 (c+d x))+\cos (4 (c+d x)))}{15 a d (\sin (c+d x)+1)}","\frac{4 \tan ^3(c+d x)}{15 a d}+\frac{4 \tan (c+d x)}{5 a d}-\frac{\sec ^3(c+d x)}{5 d (a \sin (c+d x)+a)}",1,"-1/15*(Sec[c + d*x]^3*(2*Cos[2*(c + d*x)] + Cos[4*(c + d*x)] - 2*(3*Sin[c + d*x] + Sin[3*(c + d*x)])))/(a*d*(1 + Sin[c + d*x]))","A",1
61,1,97,120,0.1538211,"\int \frac{\sec ^5(c+d x)}{a+a \sin (c+d x)} \, dx","Integrate[Sec[c + d*x]^5/(a + a*Sin[c + d*x]),x]","\frac{\sec ^4(c+d x) \left(-15 \sin ^4(c+d x)-15 \sin ^3(c+d x)+25 \sin ^2(c+d x)+25 \sin (c+d x)+15 (\sin (c+d x)-1)^2 (\sin (c+d x)+1)^3 \tanh ^{-1}(\sin (c+d x))-8\right)}{48 a d (\sin (c+d x)+1)}","-\frac{a^2}{24 d (a \sin (c+d x)+a)^3}+\frac{a}{32 d (a-a \sin (c+d x))^2}-\frac{3 a}{32 d (a \sin (c+d x)+a)^2}+\frac{1}{8 d (a-a \sin (c+d x))}-\frac{3}{16 d (a \sin (c+d x)+a)}+\frac{5 \tanh ^{-1}(\sin (c+d x))}{16 a d}",1,"(Sec[c + d*x]^4*(-8 + 25*Sin[c + d*x] + 25*Sin[c + d*x]^2 - 15*Sin[c + d*x]^3 - 15*Sin[c + d*x]^4 + 15*ArcTanh[Sin[c + d*x]]*(-1 + Sin[c + d*x])^2*(1 + Sin[c + d*x])^3))/(48*a*d*(1 + Sin[c + d*x]))","A",1
62,1,151,104,1.1162208,"\int \frac{\cos ^8(c+d x)}{(a+a \sin (c+d x))^2} \, dx","Integrate[Cos[c + d*x]^8/(a + a*Sin[c + d*x])^2,x]","-\frac{\left(\sqrt{\sin (c+d x)+1} \left(40 \sin ^6(c+d x)-136 \sin ^5(c+d x)+86 \sin ^4(c+d x)+202 \sin ^3(c+d x)-327 \sin ^2(c+d x)+39 \sin (c+d x)+96\right)-210 \sin ^{-1}\left(\frac{\sqrt{1-\sin (c+d x)}}{\sqrt{2}}\right) \sqrt{1-\sin (c+d x)}\right) \cos ^9(c+d x)}{240 a^2 d (\sin (c+d x)-1)^5 (\sin (c+d x)+1)^{9/2}}","\frac{7 \cos ^5(c+d x)}{30 a^2 d}+\frac{\cos ^7(c+d x)}{6 d \left(a^2 \sin (c+d x)+a^2\right)}+\frac{7 \sin (c+d x) \cos ^3(c+d x)}{24 a^2 d}+\frac{7 \sin (c+d x) \cos (c+d x)}{16 a^2 d}+\frac{7 x}{16 a^2}",1,"-1/240*(Cos[c + d*x]^9*(-210*ArcSin[Sqrt[1 - Sin[c + d*x]]/Sqrt[2]]*Sqrt[1 - Sin[c + d*x]] + Sqrt[1 + Sin[c + d*x]]*(96 + 39*Sin[c + d*x] - 327*Sin[c + d*x]^2 + 202*Sin[c + d*x]^3 + 86*Sin[c + d*x]^4 - 136*Sin[c + d*x]^5 + 40*Sin[c + d*x]^6)))/(a^2*d*(-1 + Sin[c + d*x])^5*(1 + Sin[c + d*x])^(9/2))","A",1
63,1,46,47,0.1619316,"\int \frac{\cos ^7(c+d x)}{(a+a \sin (c+d x))^2} \, dx","Integrate[Cos[c + d*x]^7/(a + a*Sin[c + d*x])^2,x]","-\frac{\sin (c+d x) \left(2 \sin ^4(c+d x)-5 \sin ^3(c+d x)+10 \sin (c+d x)-10\right)}{10 a^2 d}","\frac{(a-a \sin (c+d x))^5}{5 a^7 d}-\frac{(a-a \sin (c+d x))^4}{2 a^6 d}",1,"-1/10*(Sin[c + d*x]*(-10 + 10*Sin[c + d*x] - 5*Sin[c + d*x]^3 + 2*Sin[c + d*x]^4))/(a^2*d)","A",1
64,1,131,80,0.5088314,"\int \frac{\cos ^6(c+d x)}{(a+a \sin (c+d x))^2} \, dx","Integrate[Cos[c + d*x]^6/(a + a*Sin[c + d*x])^2,x]","-\frac{\left(30 \sqrt{1-\sin (c+d x)} \sin ^{-1}\left(\frac{\sqrt{1-\sin (c+d x)}}{\sqrt{2}}\right)+\sqrt{\sin (c+d x)+1} \left(6 \sin ^4(c+d x)-22 \sin ^3(c+d x)+25 \sin ^2(c+d x)+7 \sin (c+d x)-16\right)\right) \cos ^7(c+d x)}{24 a^2 d (\sin (c+d x)-1)^4 (\sin (c+d x)+1)^{7/2}}","\frac{5 \cos ^3(c+d x)}{12 a^2 d}+\frac{\cos ^5(c+d x)}{4 d \left(a^2 \sin (c+d x)+a^2\right)}+\frac{5 \sin (c+d x) \cos (c+d x)}{8 a^2 d}+\frac{5 x}{8 a^2}",1,"-1/24*(Cos[c + d*x]^7*(30*ArcSin[Sqrt[1 - Sin[c + d*x]]/Sqrt[2]]*Sqrt[1 - Sin[c + d*x]] + Sqrt[1 + Sin[c + d*x]]*(-16 + 7*Sin[c + d*x] + 25*Sin[c + d*x]^2 - 22*Sin[c + d*x]^3 + 6*Sin[c + d*x]^4)))/(a^2*d*(-1 + Sin[c + d*x])^4*(1 + Sin[c + d*x])^(7/2))","A",1
65,1,34,23,0.0627641,"\int \frac{\cos ^5(c+d x)}{(a+a \sin (c+d x))^2} \, dx","Integrate[Cos[c + d*x]^5/(a + a*Sin[c + d*x])^2,x]","\frac{\sin (c+d x) \left(\sin ^2(c+d x)-3 \sin (c+d x)+3\right)}{3 a^2 d}","-\frac{(a-a \sin (c+d x))^3}{3 a^5 d}",1,"(Sin[c + d*x]*(3 - 3*Sin[c + d*x] + Sin[c + d*x]^2))/(3*a^2*d)","A",1
66,1,109,56,0.1757791,"\int \frac{\cos ^4(c+d x)}{(a+a \sin (c+d x))^2} \, dx","Integrate[Cos[c + d*x]^4/(a + a*Sin[c + d*x])^2,x]","-\frac{\left(\sqrt{\sin (c+d x)+1} \left(\sin ^2(c+d x)-5 \sin (c+d x)+4\right)-6 \sin ^{-1}\left(\frac{\sqrt{1-\sin (c+d x)}}{\sqrt{2}}\right) \sqrt{1-\sin (c+d x)}\right) \cos ^5(c+d x)}{2 a^2 d (\sin (c+d x)-1)^3 (\sin (c+d x)+1)^{5/2}}","\frac{3 \cos (c+d x)}{2 a^2 d}+\frac{\cos ^3(c+d x)}{2 d \left(a^2 \sin (c+d x)+a^2\right)}+\frac{3 x}{2 a^2}",1,"-1/2*(Cos[c + d*x]^5*(-6*ArcSin[Sqrt[1 - Sin[c + d*x]]/Sqrt[2]]*Sqrt[1 - Sin[c + d*x]] + Sqrt[1 + Sin[c + d*x]]*(4 - 5*Sin[c + d*x] + Sin[c + d*x]^2)))/(a^2*d*(-1 + Sin[c + d*x])^3*(1 + Sin[c + d*x])^(5/2))","A",1
67,1,26,32,0.0326225,"\int \frac{\cos ^3(c+d x)}{(a+a \sin (c+d x))^2} \, dx","Integrate[Cos[c + d*x]^3/(a + a*Sin[c + d*x])^2,x]","-\frac{\sin (c+d x)-2 \log (\sin (c+d x)+1)}{a^2 d}","\frac{2 \log (\sin (c+d x)+1)}{a^2 d}-\frac{\sin (c+d x)}{a^2 d}",1,"-((-2*Log[1 + Sin[c + d*x]] + Sin[c + d*x])/(a^2*d))","A",1
68,1,104,34,0.1838556,"\int \frac{\cos ^2(c+d x)}{(a+a \sin (c+d x))^2} \, dx","Integrate[Cos[c + d*x]^2/(a + a*Sin[c + d*x])^2,x]","\frac{2 \left(\sin ^{-1}\left(\frac{\sqrt{1-\sin (c+d x)}}{\sqrt{2}}\right) \sqrt{1-\sin (c+d x)} (\sin (c+d x)+1)+\sqrt{\sin (c+d x)+1} (\sin (c+d x)-1)\right) \cos ^3(c+d x)}{a^2 d (\sin (c+d x)-1)^2 (\sin (c+d x)+1)^{5/2}}","-\frac{2 \cos (c+d x)}{d \left(a^2 \sin (c+d x)+a^2\right)}-\frac{x}{a^2}",1,"(2*Cos[c + d*x]^3*((-1 + Sin[c + d*x])*Sqrt[1 + Sin[c + d*x]] + ArcSin[Sqrt[1 - Sin[c + d*x]]/Sqrt[2]]*Sqrt[1 - Sin[c + d*x]]*(1 + Sin[c + d*x])))/(a^2*d*(-1 + Sin[c + d*x])^2*(1 + Sin[c + d*x])^(5/2))","B",1
69,1,31,21,0.0658426,"\int \frac{\cos (c+d x)}{(a+a \sin (c+d x))^2} \, dx","Integrate[Cos[c + d*x]/(a + a*Sin[c + d*x])^2,x]","-\frac{1}{a^2 d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2}","-\frac{1}{d \left(a^2 \sin (c+d x)+a^2\right)}",1,"-(1/(a^2*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2))","A",1
70,1,38,60,0.0972084,"\int \frac{\sec (c+d x)}{(a+a \sin (c+d x))^2} \, dx","Integrate[Sec[c + d*x]/(a + a*Sin[c + d*x])^2,x]","\frac{\tanh ^{-1}(\sin (c+d x))-\frac{\sin (c+d x)+2}{(\sin (c+d x)+1)^2}}{4 a^2 d}","-\frac{1}{4 d \left(a^2 \sin (c+d x)+a^2\right)}+\frac{\tanh ^{-1}(\sin (c+d x))}{4 a^2 d}-\frac{1}{4 d (a \sin (c+d x)+a)^2}",1,"(ArcTanh[Sin[c + d*x]] - (2 + Sin[c + d*x])/(1 + Sin[c + d*x])^2)/(4*a^2*d)","A",1
71,1,53,71,0.075243,"\int \frac{\sec ^2(c+d x)}{(a+a \sin (c+d x))^2} \, dx","Integrate[Sec[c + d*x]^2/(a + a*Sin[c + d*x])^2,x]","-\frac{\sec (c+d x) (-5 \sin (c+d x)+\sin (3 (c+d x))+4 \cos (2 (c+d x)))}{10 a^2 d (\sin (c+d x)+1)^2}","\frac{2 \tan (c+d x)}{5 a^2 d}-\frac{\sec (c+d x)}{5 d \left(a^2 \sin (c+d x)+a^2\right)}-\frac{\sec (c+d x)}{5 d (a \sin (c+d x)+a)^2}",1,"-1/10*(Sec[c + d*x]*(4*Cos[2*(c + d*x)] - 5*Sin[c + d*x] + Sin[3*(c + d*x)]))/(a^2*d*(1 + Sin[c + d*x])^2)","A",1
72,1,85,104,0.1153768,"\int \frac{\sec ^3(c+d x)}{(a+a \sin (c+d x))^2} \, dx","Integrate[Sec[c + d*x]^3/(a + a*Sin[c + d*x])^2,x]","-\frac{\sec ^2(c+d x) \left(-3 \sin ^3(c+d x)-6 \sin ^2(c+d x)-\sin (c+d x)+3 (\sin (c+d x)-1) (\sin (c+d x)+1)^3 \tanh ^{-1}(\sin (c+d x))+4\right)}{12 a^2 d (\sin (c+d x)+1)^2}","\frac{1}{16 d \left(a^2-a^2 \sin (c+d x)\right)}-\frac{3}{16 d \left(a^2 \sin (c+d x)+a^2\right)}+\frac{\tanh ^{-1}(\sin (c+d x))}{4 a^2 d}-\frac{a}{12 d (a \sin (c+d x)+a)^3}-\frac{1}{8 d (a \sin (c+d x)+a)^2}",1,"-1/12*(Sec[c + d*x]^2*(4 - Sin[c + d*x] - 6*Sin[c + d*x]^2 - 3*Sin[c + d*x]^3 + 3*ArcTanh[Sin[c + d*x]]*(-1 + Sin[c + d*x])*(1 + Sin[c + d*x])^3))/(a^2*d*(1 + Sin[c + d*x])^2)","A",1
73,1,78,93,0.0571132,"\int \frac{\sec ^4(c+d x)}{(a+a \sin (c+d x))^2} \, dx","Integrate[Sec[c + d*x]^4/(a + a*Sin[c + d*x])^2,x]","-\frac{\left(8 \sin ^5(c+d x)+16 \sin ^4(c+d x)-4 \sin ^3(c+d x)-24 \sin ^2(c+d x)-9 \sin (c+d x)+6\right) \sec ^3(c+d x)}{21 a^2 d (\sin (c+d x)+1)^2}","\frac{4 \tan ^3(c+d x)}{21 a^2 d}+\frac{4 \tan (c+d x)}{7 a^2 d}-\frac{\sec ^3(c+d x)}{7 d \left(a^2 \sin (c+d x)+a^2\right)}-\frac{\sec ^3(c+d x)}{7 d (a \sin (c+d x)+a)^2}",1,"-1/21*(Sec[c + d*x]^3*(6 - 9*Sin[c + d*x] - 24*Sin[c + d*x]^2 - 4*Sin[c + d*x]^3 + 16*Sin[c + d*x]^4 + 8*Sin[c + d*x]^5))/(a^2*d*(1 + Sin[c + d*x])^2)","A",1
74,1,137,146,0.3325416,"\int \frac{\sec ^5(c+d x)}{(a+a \sin (c+d x))^2} \, dx","Integrate[Sec[c + d*x]^5/(a + a*Sin[c + d*x])^2,x]","\frac{(1-\sin (c+d x))^2 (\sin (c+d x)+1)^2 \sec ^4(c+d x) \left(\frac{5}{64 (1-\sin (c+d x))}-\frac{5}{32 (\sin (c+d x)+1)}+\frac{1}{64 (1-\sin (c+d x))^2}-\frac{3}{32 (\sin (c+d x)+1)^2}-\frac{1}{16 (\sin (c+d x)+1)^3}-\frac{1}{32 (\sin (c+d x)+1)^4}+\frac{15}{64} \tanh ^{-1}(\sin (c+d x))\right)}{a^2 d}","-\frac{a^2}{32 d (a \sin (c+d x)+a)^4}+\frac{5}{64 d \left(a^2-a^2 \sin (c+d x)\right)}-\frac{5}{32 d \left(a^2 \sin (c+d x)+a^2\right)}+\frac{15 \tanh ^{-1}(\sin (c+d x))}{64 a^2 d}-\frac{a}{16 d (a \sin (c+d x)+a)^3}+\frac{1}{64 d (a-a \sin (c+d x))^2}-\frac{3}{32 d (a \sin (c+d x)+a)^2}",1,"(Sec[c + d*x]^4*(1 - Sin[c + d*x])^2*(1 + Sin[c + d*x])^2*((15*ArcTanh[Sin[c + d*x]])/64 + 1/(64*(1 - Sin[c + d*x])^2) + 5/(64*(1 - Sin[c + d*x])) - 1/(32*(1 + Sin[c + d*x])^4) - 1/(16*(1 + Sin[c + d*x])^3) - 3/(32*(1 + Sin[c + d*x])^2) - 5/(32*(1 + Sin[c + d*x]))))/(a^2*d)","A",1
75,1,141,103,1.0760687,"\int \frac{\cos ^8(c+d x)}{(a+a \sin (c+d x))^3} \, dx","Integrate[Cos[c + d*x]^8/(a + a*Sin[c + d*x])^3,x]","-\frac{\left(\sqrt{\sin (c+d x)+1} \left(24 \sin ^5(c+d x)-114 \sin ^4(c+d x)+202 \sin ^3(c+d x)-127 \sin ^2(c+d x)-121 \sin (c+d x)+136\right)-210 \sin ^{-1}\left(\frac{\sqrt{1-\sin (c+d x)}}{\sqrt{2}}\right) \sqrt{1-\sin (c+d x)}\right) \cos ^9(c+d x)}{120 a^3 d (\sin (c+d x)-1)^5 (\sin (c+d x)+1)^{9/2}}","\frac{7 \cos ^5(c+d x)}{15 a^3 d}+\frac{7 \sin (c+d x) \cos ^3(c+d x)}{12 a^3 d}+\frac{7 \sin (c+d x) \cos (c+d x)}{8 a^3 d}+\frac{7 x}{8 a^3}+\frac{2 \cos ^7(c+d x)}{3 a d (a \sin (c+d x)+a)^2}",1,"-1/120*(Cos[c + d*x]^9*(-210*ArcSin[Sqrt[1 - Sin[c + d*x]]/Sqrt[2]]*Sqrt[1 - Sin[c + d*x]] + Sqrt[1 + Sin[c + d*x]]*(136 - 121*Sin[c + d*x] - 127*Sin[c + d*x]^2 + 202*Sin[c + d*x]^3 - 114*Sin[c + d*x]^4 + 24*Sin[c + d*x]^5)))/(a^3*d*(-1 + Sin[c + d*x])^5*(1 + Sin[c + d*x])^(9/2))","A",1
76,1,44,23,0.1596789,"\int \frac{\cos ^7(c+d x)}{(a+a \sin (c+d x))^3} \, dx","Integrate[Cos[c + d*x]^7/(a + a*Sin[c + d*x])^3,x]","-\frac{\sin (c+d x) \left(\sin ^3(c+d x)-4 \sin ^2(c+d x)+6 \sin (c+d x)-4\right)}{4 a^3 d}","-\frac{(a-a \sin (c+d x))^4}{4 a^7 d}",1,"-1/4*(Sin[c + d*x]*(-4 + 6*Sin[c + d*x] - 4*Sin[c + d*x]^2 + Sin[c + d*x]^3))/(a^3*d)","A",1
77,1,121,77,0.465066,"\int \frac{\cos ^6(c+d x)}{(a+a \sin (c+d x))^3} \, dx","Integrate[Cos[c + d*x]^6/(a + a*Sin[c + d*x])^3,x]","-\frac{\left(30 \sqrt{1-\sin (c+d x)} \sin ^{-1}\left(\frac{\sqrt{1-\sin (c+d x)}}{\sqrt{2}}\right)+\sqrt{\sin (c+d x)+1} \left(2 \sin ^3(c+d x)-11 \sin ^2(c+d x)+31 \sin (c+d x)-22\right)\right) \cos ^7(c+d x)}{6 a^3 d (\sin (c+d x)-1)^4 (\sin (c+d x)+1)^{7/2}}","\frac{5 \cos ^3(c+d x)}{3 a^3 d}+\frac{5 \sin (c+d x) \cos (c+d x)}{2 a^3 d}+\frac{5 x}{2 a^3}+\frac{2 \cos ^5(c+d x)}{a d (a \sin (c+d x)+a)^2}",1,"-1/6*(Cos[c + d*x]^7*(30*ArcSin[Sqrt[1 - Sin[c + d*x]]/Sqrt[2]]*Sqrt[1 - Sin[c + d*x]] + Sqrt[1 + Sin[c + d*x]]*(-22 + 31*Sin[c + d*x] - 11*Sin[c + d*x]^2 + 2*Sin[c + d*x]^3)))/(a^3*d*(-1 + Sin[c + d*x])^4*(1 + Sin[c + d*x])^(7/2))","A",1
78,1,38,50,0.0547211,"\int \frac{\cos ^5(c+d x)}{(a+a \sin (c+d x))^3} \, dx","Integrate[Cos[c + d*x]^5/(a + a*Sin[c + d*x])^3,x]","\frac{\sin ^2(c+d x)-6 \sin (c+d x)+8 \log (\sin (c+d x)+1)}{2 a^3 d}","\frac{\sin ^2(c+d x)}{2 a^3 d}-\frac{3 \sin (c+d x)}{a^3 d}+\frac{4 \log (\sin (c+d x)+1)}{a^3 d}",1,"(8*Log[1 + Sin[c + d*x]] - 6*Sin[c + d*x] + Sin[c + d*x]^2)/(2*a^3*d)","A",1
79,1,59,49,0.0438827,"\int \frac{\cos ^4(c+d x)}{(a+a \sin (c+d x))^3} \, dx","Integrate[Cos[c + d*x]^4/(a + a*Sin[c + d*x])^3,x]","-\frac{\cos ^5(c+d x) \, _2F_1\left(\frac{3}{2},\frac{5}{2};\frac{7}{2};\frac{1}{2} (1-\sin (c+d x))\right)}{5 \sqrt{2} a^3 d (\sin (c+d x)+1)^{5/2}}","-\frac{3 \cos (c+d x)}{a^3 d}-\frac{3 x}{a^3}-\frac{2 \cos ^3(c+d x)}{a d (a \sin (c+d x)+a)^2}",1,"-1/5*(Cos[c + d*x]^5*Hypergeometric2F1[3/2, 5/2, 7/2, (1 - Sin[c + d*x])/2])/(Sqrt[2]*a^3*d*(1 + Sin[c + d*x])^(5/2))","C",1
80,1,58,39,0.0591141,"\int \frac{\cos ^3(c+d x)}{(a+a \sin (c+d x))^3} \, dx","Integrate[Cos[c + d*x]^3/(a + a*Sin[c + d*x])^3,x]","-\frac{\sin (c+d x) \log (\sin (c+d x)+1)+\log (\sin (c+d x)+1)+2}{a^3 d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2}","-\frac{2}{d \left(a^3 \sin (c+d x)+a^3\right)}-\frac{\log (\sin (c+d x)+1)}{a^3 d}",1,"-((2 + Log[1 + Sin[c + d*x]] + Log[1 + Sin[c + d*x]]*Sin[c + d*x])/(a^3*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2))","A",1
81,1,28,27,0.0194373,"\int \frac{\cos ^2(c+d x)}{(a+a \sin (c+d x))^3} \, dx","Integrate[Cos[c + d*x]^2/(a + a*Sin[c + d*x])^3,x]","-\frac{\cos ^3(c+d x)}{3 a^3 d (\sin (c+d x)+1)^3}","-\frac{\cos ^3(c+d x)}{3 d (a \sin (c+d x)+a)^3}",1,"-1/3*Cos[c + d*x]^3/(a^3*d*(1 + Sin[c + d*x])^3)","A",1
82,1,33,22,0.0589885,"\int \frac{\cos (c+d x)}{(a+a \sin (c+d x))^3} \, dx","Integrate[Cos[c + d*x]/(a + a*Sin[c + d*x])^3,x]","-\frac{1}{2 a^3 d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^4}","-\frac{1}{2 a d (a \sin (c+d x)+a)^2}",1,"-1/2*1/(a^3*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^4)","A",1
83,1,61,82,0.0876907,"\int \frac{\sec (c+d x)}{(a+a \sin (c+d x))^3} \, dx","Integrate[Sec[c + d*x]/(a + a*Sin[c + d*x])^3,x]","\frac{-\frac{1}{8 (\sin (c+d x)+1)}-\frac{1}{8 (\sin (c+d x)+1)^2}-\frac{1}{6 (\sin (c+d x)+1)^3}+\frac{1}{8} \tanh ^{-1}(\sin (c+d x))}{a^3 d}","-\frac{1}{8 d \left(a^3 \sin (c+d x)+a^3\right)}+\frac{\tanh ^{-1}(\sin (c+d x))}{8 a^3 d}-\frac{1}{8 a d (a \sin (c+d x)+a)^2}-\frac{1}{6 d (a \sin (c+d x)+a)^3}",1,"(ArcTanh[Sin[c + d*x]]/8 - 1/(6*(1 + Sin[c + d*x])^3) - 1/(8*(1 + Sin[c + d*x])^2) - 1/(8*(1 + Sin[c + d*x])))/(a^3*d)","A",1
84,1,63,99,0.1092601,"\int \frac{\sec ^2(c+d x)}{(a+a \sin (c+d x))^3} \, dx","Integrate[Sec[c + d*x]^2/(a + a*Sin[c + d*x])^3,x]","\frac{\sec (c+d x) (14 \sin (c+d x)-6 \sin (3 (c+d x))-14 \cos (2 (c+d x))+\cos (4 (c+d x)))}{35 a^3 d (\sin (c+d x)+1)^3}","\frac{8 \tan (c+d x)}{35 a^3 d}-\frac{4 \sec (c+d x)}{35 d \left(a^3 \sin (c+d x)+a^3\right)}-\frac{4 \sec (c+d x)}{35 a d (a \sin (c+d x)+a)^2}-\frac{\sec (c+d x)}{7 d (a \sin (c+d x)+a)^3}",1,"(Sec[c + d*x]*(-14*Cos[2*(c + d*x)] + Cos[4*(c + d*x)] + 14*Sin[c + d*x] - 6*Sin[3*(c + d*x)]))/(35*a^3*d*(1 + Sin[c + d*x])^3)","A",1
85,1,95,126,0.1662762,"\int \frac{\sec ^3(c+d x)}{(a+a \sin (c+d x))^3} \, dx","Integrate[Sec[c + d*x]^3/(a + a*Sin[c + d*x])^3,x]","-\frac{\sec ^2(c+d x) \left(-15 \sin ^4(c+d x)-45 \sin ^3(c+d x)-35 \sin ^2(c+d x)+15 \sin (c+d x)+15 (\sin (c+d x)-1) (\sin (c+d x)+1)^4 \tanh ^{-1}(\sin (c+d x))+32\right)}{96 a^3 d (\sin (c+d x)+1)^3}","\frac{1}{32 d \left(a^3-a^3 \sin (c+d x)\right)}-\frac{1}{8 d \left(a^3 \sin (c+d x)+a^3\right)}+\frac{5 \tanh ^{-1}(\sin (c+d x))}{32 a^3 d}-\frac{a}{16 d (a \sin (c+d x)+a)^4}-\frac{1}{12 d (a \sin (c+d x)+a)^3}-\frac{3}{32 a d (a \sin (c+d x)+a)^2}",1,"-1/96*(Sec[c + d*x]^2*(32 + 15*Sin[c + d*x] - 35*Sin[c + d*x]^2 - 45*Sin[c + d*x]^3 - 15*Sin[c + d*x]^4 + 15*ArcTanh[Sin[c + d*x]]*(-1 + Sin[c + d*x])*(1 + Sin[c + d*x])^4))/(a^3*d*(1 + Sin[c + d*x])^3)","A",1
86,1,85,123,0.1140944,"\int \frac{\sec ^4(c+d x)}{(a+a \sin (c+d x))^3} \, dx","Integrate[Sec[c + d*x]^4/(a + a*Sin[c + d*x])^3,x]","\frac{\sec ^3(c+d x) (36 \sin (c+d x)+2 \sin (3 (c+d x))-6 \sin (5 (c+d x))-27 \cos (2 (c+d x))-12 \cos (4 (c+d x))+\cos (6 (c+d x)))}{126 a^3 d (\sin (c+d x)+1)^3}","\frac{8 \tan ^3(c+d x)}{63 a^3 d}+\frac{8 \tan (c+d x)}{21 a^3 d}-\frac{2 \sec ^3(c+d x)}{21 d \left(a^3 \sin (c+d x)+a^3\right)}-\frac{2 \sec ^3(c+d x)}{21 a d (a \sin (c+d x)+a)^2}-\frac{\sec ^3(c+d x)}{9 d (a \sin (c+d x)+a)^3}",1,"(Sec[c + d*x]^3*(-27*Cos[2*(c + d*x)] - 12*Cos[4*(c + d*x)] + Cos[6*(c + d*x)] + 36*Sin[c + d*x] + 2*Sin[3*(c + d*x)] - 6*Sin[5*(c + d*x)]))/(126*a^3*d*(1 + Sin[c + d*x])^3)","A",1
87,1,145,171,0.519252,"\int \frac{\sec ^5(c+d x)}{(a+a \sin (c+d x))^3} \, dx","Integrate[Sec[c + d*x]^5/(a + a*Sin[c + d*x])^3,x]","\frac{\sec ^4(c+d x) \left(-105 \sin ^6(c+d x)-315 \sin ^5(c+d x)-140 \sin ^4(c+d x)+420 \sin ^3(c+d x)+469 \sin ^2(c+d x)+7 \sin (c+d x)+105 \tanh ^{-1}(\sin (c+d x)) \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^4 \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^{10}-176\right)}{640 a^3 d (\sin (c+d x)+1)^3}","\frac{3}{64 d \left(a^3-a^3 \sin (c+d x)\right)}-\frac{15}{128 d \left(a^3 \sin (c+d x)+a^3\right)}+\frac{21 \tanh ^{-1}(\sin (c+d x))}{128 a^3 d}-\frac{a^2}{40 d (a \sin (c+d x)+a)^5}-\frac{3 a}{64 d (a \sin (c+d x)+a)^4}-\frac{1}{16 d (a \sin (c+d x)+a)^3}+\frac{1}{128 a d (a-a \sin (c+d x))^2}-\frac{5}{64 a d (a \sin (c+d x)+a)^2}",1,"(Sec[c + d*x]^4*(-176 + 105*ArcTanh[Sin[c + d*x]]*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^4*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^10 + 7*Sin[c + d*x] + 469*Sin[c + d*x]^2 + 420*Sin[c + d*x]^3 - 140*Sin[c + d*x]^4 - 315*Sin[c + d*x]^5 - 105*Sin[c + d*x]^6))/(640*a^3*d*(1 + Sin[c + d*x])^3)","A",1
88,1,275,127,6.0809901,"\int \frac{\cos ^8(c+d x)}{(a+a \sin (c+d x))^8} \, dx","Integrate[Cos[c + d*x]^8/(a + a*Sin[c + d*x])^8,x]","-\frac{2 \sqrt{2} \left(\frac{1}{2} (1-\sin (c+d x))-1\right)^4 \left(\frac{\sin ^{-1}\left(\frac{\sqrt{1-\sin (c+d x)}}{\sqrt{2}}\right) \sqrt{1-\sin (c+d x)}}{\sqrt{2} \sqrt{\frac{1}{2} (\sin (c+d x)-1)+1}}+\frac{(1-\sin (c+d x))^4}{112 \left(\frac{1}{2} (1-\sin (c+d x))-1\right)^4}+\frac{(1-\sin (c+d x))^3}{40 \left(\frac{1}{2} (1-\sin (c+d x))-1\right)^3}+\frac{(1-\sin (c+d x))^2}{12 \left(\frac{1}{2} (1-\sin (c+d x))-1\right)^2}+\frac{1-\sin (c+d x)}{2 \left(\frac{1}{2} (1-\sin (c+d x))-1\right)}\right) \cos ^9(c+d x)}{a^8 d \left(\frac{1}{2} (\sin (c+d x)-1)+1\right)^{7/2} (1-\sin (c+d x))^5 (\sin (c+d x)+1)^{9/2}}","\frac{2 \cos (c+d x)}{d \left(a^8 \sin (c+d x)+a^8\right)}+\frac{x}{a^8}+\frac{2 \cos ^5(c+d x)}{5 a^3 d (a \sin (c+d x)+a)^5}-\frac{2 \cos ^3(c+d x)}{3 a^2 d \left(a^2 \sin (c+d x)+a^2\right)^3}-\frac{2 \cos ^7(c+d x)}{7 a d (a \sin (c+d x)+a)^7}",1,"(-2*Sqrt[2]*Cos[c + d*x]^9*(-1 + (1 - Sin[c + d*x])/2)^4*((ArcSin[Sqrt[1 - Sin[c + d*x]]/Sqrt[2]]*Sqrt[1 - Sin[c + d*x]])/(Sqrt[2]*Sqrt[1 + (-1 + Sin[c + d*x])/2]) + (1 - Sin[c + d*x])/(2*(-1 + (1 - Sin[c + d*x])/2)) + (1 - Sin[c + d*x])^2/(12*(-1 + (1 - Sin[c + d*x])/2)^2) + (1 - Sin[c + d*x])^3/(40*(-1 + (1 - Sin[c + d*x])/2)^3) + (1 - Sin[c + d*x])^4/(112*(-1 + (1 - Sin[c + d*x])/2)^4)))/(a^8*d*(1 + (-1 + Sin[c + d*x])/2)^(7/2)*(1 - Sin[c + d*x])^5*(1 + Sin[c + d*x])^(9/2))","B",1
89,1,28,36,0.0797588,"\int \frac{\cos ^7(c+d x)}{(a+a \sin (c+d x))^8} \, dx","Integrate[Cos[c + d*x]^7/(a + a*Sin[c + d*x])^8,x]","-\frac{\cos ^8(c+d x)}{8 a^8 d (\sin (c+d x)+1)^8}","-\frac{(a-a \sin (c+d x))^4}{8 d \left(a^3 \sin (c+d x)+a^3\right)^4}",1,"-1/8*Cos[c + d*x]^8/(a^8*d*(1 + Sin[c + d*x])^8)","A",1
90,1,36,58,0.1065207,"\int \frac{\cos ^6(c+d x)}{(a+a \sin (c+d x))^8} \, dx","Integrate[Cos[c + d*x]^6/(a + a*Sin[c + d*x])^8,x]","-\frac{(\sin (c+d x)+8) \cos ^7(c+d x)}{63 a^8 d (\sin (c+d x)+1)^8}","-\frac{\cos ^7(c+d x)}{63 a d (a \sin (c+d x)+a)^7}-\frac{\cos ^7(c+d x)}{9 d (a \sin (c+d x)+a)^8}",1,"-1/63*(Cos[c + d*x]^7*(8 + Sin[c + d*x]))/(a^8*d*(1 + Sin[c + d*x])^8)","A",1
91,1,58,65,0.1285064,"\int \frac{\cos ^5(c+d x)}{(a+a \sin (c+d x))^8} \, dx","Integrate[Cos[c + d*x]^5/(a + a*Sin[c + d*x])^8,x]","\frac{\left(5 \sin ^2(c+d x)-5 \sin (c+d x)+2\right) \cos ^6(c+d x)}{15 a^8 d (\sin (c+d x)-1)^3 (\sin (c+d x)+1)^8}","-\frac{1}{3 a^5 d (a \sin (c+d x)+a)^3}-\frac{4}{5 a^3 d (a \sin (c+d x)+a)^5}+\frac{1}{d \left(a^2 \sin (c+d x)+a^2\right)^4}",1,"(Cos[c + d*x]^6*(2 - 5*Sin[c + d*x] + 5*Sin[c + d*x]^2))/(15*a^8*d*(-1 + Sin[c + d*x])^3*(1 + Sin[c + d*x])^8)","A",1
92,1,58,118,0.0899466,"\int \frac{\cos ^4(c+d x)}{(a+a \sin (c+d x))^8} \, dx","Integrate[Cos[c + d*x]^4/(a + a*Sin[c + d*x])^8,x]","-\frac{\left(2 \sin ^3(c+d x)+16 \sin ^2(c+d x)+61 \sin (c+d x)+152\right) \cos ^5(c+d x)}{1155 a^8 d (\sin (c+d x)+1)^8}","-\frac{2 \cos ^5(c+d x)}{1155 a^3 d (a \sin (c+d x)+a)^5}-\frac{2 \cos ^5(c+d x)}{231 a^2 d (a \sin (c+d x)+a)^6}-\frac{\cos ^5(c+d x)}{33 a d (a \sin (c+d x)+a)^7}-\frac{\cos ^5(c+d x)}{11 d (a \sin (c+d x)+a)^8}",1,"-1/1155*(Cos[c + d*x]^5*(152 + 61*Sin[c + d*x] + 16*Sin[c + d*x]^2 + 2*Sin[c + d*x]^3))/(a^8*d*(1 + Sin[c + d*x])^8)","A",1
93,1,43,45,0.1732409,"\int \frac{\cos ^3(c+d x)}{(a+a \sin (c+d x))^8} \, dx","Integrate[Cos[c + d*x]^3/(a + a*Sin[c + d*x])^8,x]","\frac{3 \sin (c+d x)-2}{15 a^8 d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^{12}}","\frac{1}{5 a^3 d (a \sin (c+d x)+a)^5}-\frac{1}{3 a^2 d (a \sin (c+d x)+a)^6}",1,"(-2 + 3*Sin[c + d*x])/(15*a^8*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^12)","A",1
94,1,78,183,0.1303355,"\int \frac{\cos ^2(c+d x)}{(a+a \sin (c+d x))^8} \, dx","Integrate[Cos[c + d*x]^2/(a + a*Sin[c + d*x])^8,x]","-\frac{\left(8 \sin ^5(c+d x)+64 \sin ^4(c+d x)+236 \sin ^3(c+d x)+544 \sin ^2(c+d x)+911 \sin (c+d x)+1240\right) \cos ^3(c+d x)}{9009 a^8 d (\sin (c+d x)+1)^8}","-\frac{20 \cos ^3(c+d x)}{3003 a^3 d (a \sin (c+d x)+a)^5}-\frac{8 \cos ^3(c+d x)}{9009 a^2 d \left(a^2 \sin (c+d x)+a^2\right)^3}-\frac{8 \cos ^3(c+d x)}{3003 d \left(a^2 \sin (c+d x)+a^2\right)^4}-\frac{20 \cos ^3(c+d x)}{1287 a^2 d (a \sin (c+d x)+a)^6}-\frac{5 \cos ^3(c+d x)}{143 a d (a \sin (c+d x)+a)^7}-\frac{\cos ^3(c+d x)}{13 d (a \sin (c+d x)+a)^8}",1,"-1/9009*(Cos[c + d*x]^3*(1240 + 911*Sin[c + d*x] + 544*Sin[c + d*x]^2 + 236*Sin[c + d*x]^3 + 64*Sin[c + d*x]^4 + 8*Sin[c + d*x]^5))/(a^8*d*(1 + Sin[c + d*x])^8)","A",1
95,1,33,22,0.2287278,"\int \frac{\cos (c+d x)}{(a+a \sin (c+d x))^8} \, dx","Integrate[Cos[c + d*x]/(a + a*Sin[c + d*x])^8,x]","-\frac{1}{7 a^8 d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^{14}}","-\frac{1}{7 a d (a \sin (c+d x)+a)^7}",1,"-1/7*1/(a^8*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^14)","A",1
96,1,122,194,0.8327052,"\int \frac{\sec (c+d x)}{(a+a \sin (c+d x))^8} \, dx","Integrate[Sec[c + d*x]/(a + a*Sin[c + d*x])^8,x]","-\frac{105 \sin ^7(c+d x)+840 \sin ^6(c+d x)+2975 \sin ^5(c+d x)+6160 \sin ^4(c+d x)+8351 \sin ^3(c+d x)+8008 \sin ^2(c+d x)+5993 \sin (c+d x)-105 \tanh ^{-1}(\sin (c+d x)) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^{16}+4096}{26880 a^8 d (\sin (c+d x)+1)^8}","-\frac{1}{256 d \left(a^8 \sin (c+d x)+a^8\right)}+\frac{\tanh ^{-1}(\sin (c+d x))}{256 a^8 d}-\frac{1}{192 a^5 d (a \sin (c+d x)+a)^3}-\frac{1}{256 d \left(a^4 \sin (c+d x)+a^4\right)^2}-\frac{1}{80 a^3 d (a \sin (c+d x)+a)^5}-\frac{1}{128 d \left(a^2 \sin (c+d x)+a^2\right)^4}-\frac{1}{48 a^2 d (a \sin (c+d x)+a)^6}-\frac{1}{28 a d (a \sin (c+d x)+a)^7}-\frac{1}{16 d (a \sin (c+d x)+a)^8}",1,"-1/26880*(4096 - 105*ArcTanh[Sin[c + d*x]]*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^16 + 5993*Sin[c + d*x] + 8008*Sin[c + d*x]^2 + 8351*Sin[c + d*x]^3 + 6160*Sin[c + d*x]^4 + 2975*Sin[c + d*x]^5 + 840*Sin[c + d*x]^6 + 105*Sin[c + d*x]^7)/(a^8*d*(1 + Sin[c + d*x])^8)","A",1
97,1,113,245,0.4019285,"\int \frac{\sec ^2(c+d x)}{(a+a \sin (c+d x))^8} \, dx","Integrate[Sec[c + d*x]^2/(a + a*Sin[c + d*x])^8,x]","\frac{\sec (c+d x) (4862 \sin (c+d x)-6188 \sin (3 (c+d x))+1700 \sin (5 (c+d x))-119 \sin (7 (c+d x))+\sin (9 (c+d x))-7072 \cos (2 (c+d x))+3808 \cos (4 (c+d x))-544 \cos (6 (c+d x))+16 \cos (8 (c+d x)))}{24310 a^8 d (\sin (c+d x)+1)^8}","\frac{128 \tan (c+d x)}{12155 a^8 d}-\frac{64 \sec (c+d x)}{12155 d \left(a^8 \sin (c+d x)+a^8\right)}-\frac{64 \sec (c+d x)}{12155 d \left(a^4 \sin (c+d x)+a^4\right)^2}-\frac{168 \sec (c+d x)}{12155 a^3 d (a \sin (c+d x)+a)^5}-\frac{16 \sec (c+d x)}{2431 a^2 d \left(a^2 \sin (c+d x)+a^2\right)^3}-\frac{112 \sec (c+d x)}{12155 d \left(a^2 \sin (c+d x)+a^2\right)^4}-\frac{24 \sec (c+d x)}{1105 a^2 d (a \sin (c+d x)+a)^6}-\frac{3 \sec (c+d x)}{85 a d (a \sin (c+d x)+a)^7}-\frac{\sec (c+d x)}{17 d (a \sin (c+d x)+a)^8}",1,"(Sec[c + d*x]*(-7072*Cos[2*(c + d*x)] + 3808*Cos[4*(c + d*x)] - 544*Cos[6*(c + d*x)] + 16*Cos[8*(c + d*x)] + 4862*Sin[c + d*x] - 6188*Sin[3*(c + d*x)] + 1700*Sin[5*(c + d*x)] - 119*Sin[7*(c + d*x)] + Sin[9*(c + d*x)]))/(24310*a^8*d*(1 + Sin[c + d*x])^8)","A",1
98,1,175,238,1.8303757,"\int \frac{\sec ^3(c+d x)}{(a+a \sin (c+d x))^8} \, dx","Integrate[Sec[c + d*x]^3/(a + a*Sin[c + d*x])^8,x]","-\frac{\sec ^2(c+d x) \left(-315 \sin ^9(c+d x)-2520 \sin ^8(c+d x)-8610 \sin ^7(c+d x)-15960 \sin ^6(c+d x)-16128 \sin ^5(c+d x)-5544 \sin ^4(c+d x)+7074 \sin ^3(c+d x)+11736 \sin ^2(c+d x)+9019 \sin (c+d x)-315 \tanh ^{-1}(\sin (c+d x)) \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^2 \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^{18}+5120\right)}{32256 a^8 d (\sin (c+d x)+1)^8}","\frac{1}{1024 d \left(a^8-a^8 \sin (c+d x)\right)}-\frac{9}{1024 d \left(a^8 \sin (c+d x)+a^8\right)}+\frac{5 \tanh ^{-1}(\sin (c+d x))}{512 a^8 d}-\frac{7}{768 a^5 d (a \sin (c+d x)+a)^3}-\frac{1}{128 d \left(a^4 \sin (c+d x)+a^4\right)^2}-\frac{1}{64 a^3 d (a \sin (c+d x)+a)^5}-\frac{3}{256 d \left(a^2 \sin (c+d x)+a^2\right)^4}-\frac{1}{48 a^2 d (a \sin (c+d x)+a)^6}-\frac{a}{36 d (a \sin (c+d x)+a)^9}-\frac{1}{32 d (a \sin (c+d x)+a)^8}-\frac{3}{112 a d (a \sin (c+d x)+a)^7}",1,"-1/32256*(Sec[c + d*x]^2*(5120 - 315*ArcTanh[Sin[c + d*x]]*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^18 + 9019*Sin[c + d*x] + 11736*Sin[c + d*x]^2 + 7074*Sin[c + d*x]^3 - 5544*Sin[c + d*x]^4 - 16128*Sin[c + d*x]^5 - 15960*Sin[c + d*x]^6 - 8610*Sin[c + d*x]^7 - 2520*Sin[c + d*x]^8 - 315*Sin[c + d*x]^9))/(a^8*d*(1 + Sin[c + d*x])^8)","A",1
99,1,125,279,0.4334377,"\int \frac{\sec ^4(c+d x)}{(a+a \sin (c+d x))^8} \, dx","Integrate[Sec[c + d*x]^4/(a + a*Sin[c + d*x])^8,x]","\frac{\sec ^3(c+d x) (8398 \sin (c+d x)-5814 \sin (3 (c+d x))-2907 \sin (5 (c+d x))+1463 \sin (7 (c+d x))-117 \sin (9 (c+d x))+\sin (11 (c+d x))-10336 \cos (2 (c+d x))+2736 \cos (6 (c+d x))-512 \cos (8 (c+d x))+16 \cos (10 (c+d x)))}{50388 a^8 d (\sin (c+d x)+1)^8}","\frac{128 \tan ^3(c+d x)}{12597 a^8 d}+\frac{128 \tan (c+d x)}{4199 a^8 d}-\frac{32 \sec ^3(c+d x)}{4199 d \left(a^8 \sin (c+d x)+a^8\right)}-\frac{32 \sec ^3(c+d x)}{4199 d \left(a^4 \sin (c+d x)+a^4\right)^2}-\frac{66 \sec ^3(c+d x)}{4199 a^3 d (a \sin (c+d x)+a)^5}-\frac{112 \sec ^3(c+d x)}{12597 a^2 d \left(a^2 \sin (c+d x)+a^2\right)^3}-\frac{48 \sec ^3(c+d x)}{4199 d \left(a^2 \sin (c+d x)+a^2\right)^4}-\frac{22 \sec ^3(c+d x)}{969 a^2 d (a \sin (c+d x)+a)^6}-\frac{11 \sec ^3(c+d x)}{323 a d (a \sin (c+d x)+a)^7}-\frac{\sec ^3(c+d x)}{19 d (a \sin (c+d x)+a)^8}",1,"(Sec[c + d*x]^3*(-10336*Cos[2*(c + d*x)] + 2736*Cos[6*(c + d*x)] - 512*Cos[8*(c + d*x)] + 16*Cos[10*(c + d*x)] + 8398*Sin[c + d*x] - 5814*Sin[3*(c + d*x)] - 2907*Sin[5*(c + d*x)] + 1463*Sin[7*(c + d*x)] - 117*Sin[9*(c + d*x)] + Sin[11*(c + d*x)]))/(50388*a^8*d*(1 + Sin[c + d*x])^8)","A",1
100,1,195,284,2.6099124,"\int \frac{\sec ^5(c+d x)}{(a+a \sin (c+d x))^8} \, dx","Integrate[Sec[c + d*x]^5/(a + a*Sin[c + d*x])^8,x]","\frac{\sec ^4(c+d x) \left(-3465 \sin ^{11}(c+d x)-27720 \sin ^{10}(c+d x)-91245 \sin ^9(c+d x)-147840 \sin ^8(c+d x)-82698 \sin ^7(c+d x)+114576 \sin ^6(c+d x)+255222 \sin ^5(c+d x)+190080 \sin ^4(c+d x)+21395 \sin ^3(c+d x)-72776 \sin ^2(c+d x)-66953 \sin (c+d x)+3465 \tanh ^{-1}(\sin (c+d x)) \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^4 \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^{20}-34816\right)}{215040 a^8 d (\sin (c+d x)+1)^8}","\frac{11}{4096 d \left(a^8-a^8 \sin (c+d x)\right)}-\frac{55}{4096 d \left(a^8 \sin (c+d x)+a^8\right)}+\frac{33 \tanh ^{-1}(\sin (c+d x))}{2048 a^8 d}-\frac{3}{256 a^5 d (a \sin (c+d x)+a)^3}+\frac{1}{4096 d \left(a^4-a^4 \sin (c+d x)\right)^2}-\frac{45}{4096 d \left(a^4 \sin (c+d x)+a^4\right)^2}-\frac{21}{1280 a^3 d (a \sin (c+d x)+a)^5}-\frac{a^2}{80 d (a \sin (c+d x)+a)^{10}}-\frac{7}{512 d \left(a^2 \sin (c+d x)+a^2\right)^4}-\frac{5}{256 a^2 d (a \sin (c+d x)+a)^6}-\frac{a}{48 d (a \sin (c+d x)+a)^9}-\frac{3}{128 d (a \sin (c+d x)+a)^8}-\frac{5}{224 a d (a \sin (c+d x)+a)^7}",1,"(Sec[c + d*x]^4*(-34816 + 3465*ArcTanh[Sin[c + d*x]]*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^4*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^20 - 66953*Sin[c + d*x] - 72776*Sin[c + d*x]^2 + 21395*Sin[c + d*x]^3 + 190080*Sin[c + d*x]^4 + 255222*Sin[c + d*x]^5 + 114576*Sin[c + d*x]^6 - 82698*Sin[c + d*x]^7 - 147840*Sin[c + d*x]^8 - 91245*Sin[c + d*x]^9 - 27720*Sin[c + d*x]^10 - 3465*Sin[c + d*x]^11))/(215040*a^8*d*(1 + Sin[c + d*x])^8)","A",1
101,1,74,97,4.323725,"\int \cos ^7(c+d x) \sqrt{a+a \sin (c+d x)} \, dx","Integrate[Cos[c + d*x]^7*Sqrt[a + a*Sin[c + d*x]],x]","\frac{\sqrt{a (\sin (c+d x)+1)} \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^8 (-10755 \sin (c+d x)+429 \sin (3 (c+d x))-3366 \cos (2 (c+d x))+8330)}{12870 d}","-\frac{2 (a \sin (c+d x)+a)^{15/2}}{15 a^7 d}+\frac{12 (a \sin (c+d x)+a)^{13/2}}{13 a^6 d}-\frac{24 (a \sin (c+d x)+a)^{11/2}}{11 a^5 d}+\frac{16 (a \sin (c+d x)+a)^{9/2}}{9 a^4 d}",1,"((Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^8*Sqrt[a*(1 + Sin[c + d*x])]*(8330 - 3366*Cos[2*(c + d*x)] - 10755*Sin[c + d*x] + 429*Sin[3*(c + d*x)]))/(12870*d)","A",1
102,1,99,127,3.9428319,"\int \cos ^6(c+d x) \sqrt{a+a \sin (c+d x)} \, dx","Integrate[Cos[c + d*x]^6*Sqrt[a + a*Sin[c + d*x]],x]","\frac{\sqrt{a (\sin (c+d x)+1)} \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^7 (-6377 \sin (c+d x)+231 \sin (3 (c+d x))+1890 \cos (2 (c+d x))-5230)}{6006 d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}","-\frac{256 a^4 \cos ^7(c+d x)}{3003 d (a \sin (c+d x)+a)^{7/2}}-\frac{64 a^3 \cos ^7(c+d x)}{429 d (a \sin (c+d x)+a)^{5/2}}-\frac{24 a^2 \cos ^7(c+d x)}{143 d (a \sin (c+d x)+a)^{3/2}}-\frac{2 a \cos ^7(c+d x)}{13 d \sqrt{a \sin (c+d x)+a}}",1,"((Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^7*Sqrt[a*(1 + Sin[c + d*x])]*(-5230 + 1890*Cos[2*(c + d*x)] - 6377*Sin[c + d*x] + 231*Sin[3*(c + d*x)]))/(6006*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))","A",1
103,1,64,73,1.024734,"\int \cos ^5(c+d x) \sqrt{a+a \sin (c+d x)} \, dx","Integrate[Cos[c + d*x]^5*Sqrt[a + a*Sin[c + d*x]],x]","-\frac{\sqrt{a (\sin (c+d x)+1)} \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^6 (364 \sin (c+d x)+63 \cos (2 (c+d x))-365)}{693 d}","\frac{2 (a \sin (c+d x)+a)^{11/2}}{11 a^5 d}-\frac{8 (a \sin (c+d x)+a)^{9/2}}{9 a^4 d}+\frac{8 (a \sin (c+d x)+a)^{7/2}}{7 a^3 d}",1,"-1/693*((Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^6*Sqrt[a*(1 + Sin[c + d*x])]*(-365 + 63*Cos[2*(c + d*x)] + 364*Sin[c + d*x]))/d","A",1
104,1,89,95,0.7676776,"\int \cos ^4(c+d x) \sqrt{a+a \sin (c+d x)} \, dx","Integrate[Cos[c + d*x]^4*Sqrt[a + a*Sin[c + d*x]],x]","-\frac{\sqrt{a (\sin (c+d x)+1)} \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^5 (220 \sin (c+d x)-35 \cos (2 (c+d x))+249)}{315 d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}","-\frac{64 a^3 \cos ^5(c+d x)}{315 d (a \sin (c+d x)+a)^{5/2}}-\frac{16 a^2 \cos ^5(c+d x)}{63 d (a \sin (c+d x)+a)^{3/2}}-\frac{2 a \cos ^5(c+d x)}{9 d \sqrt{a \sin (c+d x)+a}}",1,"-1/315*((Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^5*Sqrt[a*(1 + Sin[c + d*x])]*(249 - 35*Cos[2*(c + d*x)] + 220*Sin[c + d*x]))/(d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))","A",1
105,1,54,49,0.2310034,"\int \cos ^3(c+d x) \sqrt{a+a \sin (c+d x)} \, dx","Integrate[Cos[c + d*x]^3*Sqrt[a + a*Sin[c + d*x]],x]","-\frac{2 (5 \sin (c+d x)-9) \sqrt{a (\sin (c+d x)+1)} \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^4}{35 d}","\frac{4 (a \sin (c+d x)+a)^{5/2}}{5 a^2 d}-\frac{2 (a \sin (c+d x)+a)^{7/2}}{7 a^3 d}",1,"(-2*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^4*Sqrt[a*(1 + Sin[c + d*x])]*(-9 + 5*Sin[c + d*x]))/(35*d)","A",1
106,1,79,63,0.1697519,"\int \cos ^2(c+d x) \sqrt{a+a \sin (c+d x)} \, dx","Integrate[Cos[c + d*x]^2*Sqrt[a + a*Sin[c + d*x]],x]","-\frac{2 (3 \sin (c+d x)+7) \sqrt{a (\sin (c+d x)+1)} \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^3}{15 d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}","-\frac{8 a^2 \cos ^3(c+d x)}{15 d (a \sin (c+d x)+a)^{3/2}}-\frac{2 a \cos ^3(c+d x)}{5 d \sqrt{a \sin (c+d x)+a}}",1,"(-2*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^3*Sqrt[a*(1 + Sin[c + d*x])]*(7 + 3*Sin[c + d*x]))/(15*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))","A",1
107,1,44,24,0.0825987,"\int \cos (c+d x) \sqrt{a+a \sin (c+d x)} \, dx","Integrate[Cos[c + d*x]*Sqrt[a + a*Sin[c + d*x]],x]","\frac{2 \sqrt{a (\sin (c+d x)+1)} \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2}{3 d}","\frac{2 (a \sin (c+d x)+a)^{3/2}}{3 a d}",1,"(2*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2*Sqrt[a*(1 + Sin[c + d*x])])/(3*d)","A",1
108,1,95,40,0.1011309,"\int \sec (c+d x) \sqrt{a+a \sin (c+d x)} \, dx","Integrate[Sec[c + d*x]*Sqrt[a + a*Sin[c + d*x]],x]","-\frac{(2-2 i) \sqrt[4]{-1} \sqrt{a (\sin (c+d x)+1)} \tanh ^{-1}\left(\left(\frac{1}{2}+\frac{i}{2}\right) (-1)^{3/4} \sec \left(\frac{d x}{4}\right) \left(\sin \left(\frac{1}{4} (2 c+d x)\right)+\cos \left(\frac{1}{4} (2 c+d x)\right)\right)\right)}{d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}","\frac{\sqrt{2} \sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a \sin (c+d x)+a}}{\sqrt{2} \sqrt{a}}\right)}{d}",1,"((-2 + 2*I)*(-1)^(1/4)*ArcTanh[(1/2 + I/2)*(-1)^(3/4)*Sec[(d*x)/4]*(Cos[(2*c + d*x)/4] + Sin[(2*c + d*x)/4])]*Sqrt[a*(1 + Sin[c + d*x])])/(d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))","C",1
109,1,106,72,0.2310658,"\int \sec ^2(c+d x) \sqrt{a+a \sin (c+d x)} \, dx","Integrate[Sec[c + d*x]^2*Sqrt[a + a*Sin[c + d*x]],x]","\frac{\sec (c+d x) \sqrt{a (\sin (c+d x)+1)} \left(1-(1+i) (-1)^{3/4} \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right) \tanh ^{-1}\left(\left(\frac{1}{2}+\frac{i}{2}\right) (-1)^{3/4} \sec \left(\frac{d x}{4}\right) \left(\cos \left(\frac{1}{4} (2 c+d x)\right)-\sin \left(\frac{1}{4} (2 c+d x)\right)\right)\right)\right)}{d}","\frac{\sec (c+d x) \sqrt{a \sin (c+d x)+a}}{d}-\frac{\sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a \sin (c+d x)+a}}\right)}{\sqrt{2} d}",1,"(Sec[c + d*x]*(1 - (1 + I)*(-1)^(3/4)*ArcTanh[(1/2 + I/2)*(-1)^(3/4)*Sec[(d*x)/4]*(Cos[(2*c + d*x)/4] - Sin[(2*c + d*x)/4])]*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2]))*Sqrt[a*(1 + Sin[c + d*x])])/d","C",1
110,1,271,95,0.3775756,"\int \sec ^3(c+d x) \sqrt{a+a \sin (c+d x)} \, dx","Integrate[Sec[c + d*x]^3*Sqrt[a + a*Sin[c + d*x]],x]","\frac{\sqrt{a (\sin (c+d x)+1)} \left(\frac{\left(\sin \left(\frac{c}{2}\right)+\cos \left(\frac{c}{2}\right)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}{\left(\cos \left(\frac{c}{2}\right)-\sin \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}+\frac{2 \sin \left(\frac{d x}{2}\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}{\left(\cos \left(\frac{c}{2}\right)-\sin \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^2}+(-3+3 i) \sqrt[4]{-1} \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right) \tanh ^{-1}\left(\left(\frac{1}{2}+\frac{i}{2}\right) (-1)^{3/4} \sec \left(\frac{d x}{4}\right) \left(\sin \left(\frac{1}{4} (2 c+d x)\right)+\cos \left(\frac{1}{4} (2 c+d x)\right)\right)\right)-2\right)}{4 d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2}","-\frac{3 a}{4 d \sqrt{a \sin (c+d x)+a}}+\frac{3 \sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a \sin (c+d x)+a}}{\sqrt{2} \sqrt{a}}\right)}{4 \sqrt{2} d}+\frac{\sec ^2(c+d x) \sqrt{a \sin (c+d x)+a}}{2 d}",1,"((-2 - (3 - 3*I)*(-1)^(1/4)*ArcTanh[(1/2 + I/2)*(-1)^(3/4)*Sec[(d*x)/4]*(Cos[(2*c + d*x)/4] + Sin[(2*c + d*x)/4])]*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]) + (2*Sin[(d*x)/2]*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))/((Cos[c/2] - Sin[c/2])*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2) + ((Cos[c/2] + Sin[c/2])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))/((Cos[c/2] - Sin[c/2])*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])))*Sqrt[a*(1 + Sin[c + d*x])])/(4*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2)","C",1
111,1,302,137,0.4178542,"\int \sec ^4(c+d x) \sqrt{a+a \sin (c+d x)} \, dx","Integrate[Sec[c + d*x]^4*Sqrt[a + a*Sin[c + d*x]],x]","\frac{\sqrt{a (\sin (c+d x)+1)} \left(\frac{12 \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2}{\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)}+\frac{4 \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2}{\left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^3}-\frac{3 \left(\cos \left(\frac{c}{2}\right)-\sin \left(\frac{c}{2}\right)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}{\sin \left(\frac{c}{2}\right)+\cos \left(\frac{c}{2}\right)}+\frac{6 \sin \left(\frac{d x}{2}\right)}{\sin \left(\frac{c}{2}\right)+\cos \left(\frac{c}{2}\right)}+(-15-15 i) (-1)^{3/4} \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2 \tanh ^{-1}\left(\left(\frac{1}{2}+\frac{i}{2}\right) (-1)^{3/4} \sec \left(\frac{d x}{4}\right) \left(\cos \left(\frac{1}{4} (2 c+d x)\right)-\sin \left(\frac{1}{4} (2 c+d x)\right)\right)\right)\right)}{24 d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^3}","-\frac{5 a^2 \cos (c+d x)}{8 d (a \sin (c+d x)+a)^{3/2}}+\frac{\sec ^3(c+d x) \sqrt{a \sin (c+d x)+a}}{3 d}+\frac{5 a \sec (c+d x)}{6 d \sqrt{a \sin (c+d x)+a}}-\frac{5 \sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a \sin (c+d x)+a}}\right)}{8 \sqrt{2} d}",1,"(((6*Sin[(d*x)/2])/(Cos[c/2] + Sin[c/2]) - (3*(Cos[c/2] - Sin[c/2])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))/(Cos[c/2] + Sin[c/2]) - (15 + 15*I)*(-1)^(3/4)*ArcTanh[(1/2 + I/2)*(-1)^(3/4)*Sec[(d*x)/4]*(Cos[(2*c + d*x)/4] - Sin[(2*c + d*x)/4])]*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2 + (4*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2)/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^3 + (12*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2)/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2]))*Sqrt[a*(1 + Sin[c + d*x])])/(24*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3)","C",1
112,1,179,149,0.4885999,"\int \sec ^5(c+d x) \sqrt{a+a \sin (c+d x)} \, dx","Integrate[Sec[c + d*x]^5*Sqrt[a + a*Sin[c + d*x]],x]","\frac{\sqrt{a (\sin (c+d x)+1)} \left(\frac{329 \sin (c+d x)+105 \sin (3 (c+d x))-70 \cos (2 (c+d x))-102}{\left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^4}-(420-420 i) \sqrt[4]{-1} \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^3 \tanh ^{-1}\left(\left(\frac{1}{2}+\frac{i}{2}\right) (-1)^{3/4} \sec \left(\frac{d x}{4}\right) \left(\sin \left(\frac{1}{4} (2 c+d x)\right)+\cos \left(\frac{1}{4} (2 c+d x)\right)\right)\right)\right)}{768 d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^4}","-\frac{35 a^2}{96 d (a \sin (c+d x)+a)^{3/2}}-\frac{35 a}{64 d \sqrt{a \sin (c+d x)+a}}+\frac{35 \sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a \sin (c+d x)+a}}{\sqrt{2} \sqrt{a}}\right)}{64 \sqrt{2} d}+\frac{\sec ^4(c+d x) \sqrt{a \sin (c+d x)+a}}{4 d}+\frac{7 a \sec ^2(c+d x)}{16 d \sqrt{a \sin (c+d x)+a}}",1,"(Sqrt[a*(1 + Sin[c + d*x])]*((-420 + 420*I)*(-1)^(1/4)*ArcTanh[(1/2 + I/2)*(-1)^(3/4)*Sec[(d*x)/4]*(Cos[(2*c + d*x)/4] + Sin[(2*c + d*x)/4])]*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3 + (-102 - 70*Cos[2*(c + d*x)] + 329*Sin[c + d*x] + 105*Sin[3*(c + d*x)])/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^4))/(768*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^4)","C",0
113,1,191,197,0.6454245,"\int \sec ^6(c+d x) \sqrt{a+a \sin (c+d x)} \, dx","Integrate[Sec[c + d*x]^6*Sqrt[a + a*Sin[c + d*x]],x]","\frac{\sqrt{a (\sin (c+d x)+1)} \left(\frac{1572 \sin (c+d x)+420 \sin (3 (c+d x))+1092 \cos (2 (c+d x))+315 \cos (4 (c+d x))+649}{\left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^5}-(2520+2520 i) (-1)^{3/4} \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^4 \tanh ^{-1}\left(\left(\frac{1}{2}+\frac{i}{2}\right) (-1)^{3/4} \sec \left(\frac{d x}{4}\right) \left(\cos \left(\frac{1}{4} (2 c+d x)\right)-\sin \left(\frac{1}{4} (2 c+d x)\right)\right)\right)\right)}{5120 d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^5}","-\frac{63 a^2 \cos (c+d x)}{128 d (a \sin (c+d x)+a)^{3/2}}-\frac{21 a^2 \sec (c+d x)}{80 d (a \sin (c+d x)+a)^{3/2}}+\frac{\sec ^5(c+d x) \sqrt{a \sin (c+d x)+a}}{5 d}+\frac{3 a \sec ^3(c+d x)}{10 d \sqrt{a \sin (c+d x)+a}}+\frac{21 a \sec (c+d x)}{32 d \sqrt{a \sin (c+d x)+a}}-\frac{63 \sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a \sin (c+d x)+a}}\right)}{128 \sqrt{2} d}",1,"(Sqrt[a*(1 + Sin[c + d*x])]*((-2520 - 2520*I)*(-1)^(3/4)*ArcTanh[(1/2 + I/2)*(-1)^(3/4)*Sec[(d*x)/4]*(Cos[(2*c + d*x)/4] - Sin[(2*c + d*x)/4])]*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^4 + (649 + 1092*Cos[2*(c + d*x)] + 315*Cos[4*(c + d*x)] + 1572*Sin[c + d*x] + 420*Sin[3*(c + d*x)])/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^5))/(5120*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^5)","C",1
114,1,61,97,0.4504342,"\int \cos ^7(c+d x) (a+a \sin (c+d x))^{3/2} \, dx","Integrate[Cos[c + d*x]^7*(a + a*Sin[c + d*x])^(3/2),x]","-\frac{2 (\sin (c+d x)+1)^4 \left(715 \sin ^3(c+d x)-2717 \sin ^2(c+d x)+3641 \sin (c+d x)-1767\right) (a (\sin (c+d x)+1))^{3/2}}{12155 d}","-\frac{2 (a \sin (c+d x)+a)^{17/2}}{17 a^7 d}+\frac{4 (a \sin (c+d x)+a)^{15/2}}{5 a^6 d}-\frac{24 (a \sin (c+d x)+a)^{13/2}}{13 a^5 d}+\frac{16 (a \sin (c+d x)+a)^{11/2}}{11 a^4 d}",1,"(-2*(1 + Sin[c + d*x])^4*(a*(1 + Sin[c + d*x]))^(3/2)*(-1767 + 3641*Sin[c + d*x] - 2717*Sin[c + d*x]^2 + 715*Sin[c + d*x]^3))/(12155*d)","A",1
115,1,79,159,0.655623,"\int \cos ^6(c+d x) (a+a \sin (c+d x))^{3/2} \, dx","Integrate[Cos[c + d*x]^6*(a + a*Sin[c + d*x])^(3/2),x]","-\frac{2 \left(3003 \sin ^4(c+d x)+15708 \sin ^3(c+d x)+33138 \sin ^2(c+d x)+34748 \sin (c+d x)+16363\right) \cos ^7(c+d x) (a (\sin (c+d x)+1))^{3/2}}{45045 d (\sin (c+d x)+1)^5}","-\frac{4096 a^5 \cos ^7(c+d x)}{45045 d (a \sin (c+d x)+a)^{7/2}}-\frac{1024 a^4 \cos ^7(c+d x)}{6435 d (a \sin (c+d x)+a)^{5/2}}-\frac{128 a^3 \cos ^7(c+d x)}{715 d (a \sin (c+d x)+a)^{3/2}}-\frac{32 a^2 \cos ^7(c+d x)}{195 d \sqrt{a \sin (c+d x)+a}}-\frac{2 a \cos ^7(c+d x) \sqrt{a \sin (c+d x)+a}}{15 d}",1,"(-2*Cos[c + d*x]^7*(a*(1 + Sin[c + d*x]))^(3/2)*(16363 + 34748*Sin[c + d*x] + 33138*Sin[c + d*x]^2 + 15708*Sin[c + d*x]^3 + 3003*Sin[c + d*x]^4))/(45045*d*(1 + Sin[c + d*x])^5)","A",1
116,1,51,73,0.1698652,"\int \cos ^5(c+d x) (a+a \sin (c+d x))^{3/2} \, dx","Integrate[Cos[c + d*x]^5*(a + a*Sin[c + d*x])^(3/2),x]","\frac{2 (\sin (c+d x)+1)^3 \left(99 \sin ^2(c+d x)-270 \sin (c+d x)+203\right) (a (\sin (c+d x)+1))^{3/2}}{1287 d}","\frac{2 (a \sin (c+d x)+a)^{13/2}}{13 a^5 d}-\frac{8 (a \sin (c+d x)+a)^{11/2}}{11 a^4 d}+\frac{8 (a \sin (c+d x)+a)^{9/2}}{9 a^3 d}",1,"(2*(1 + Sin[c + d*x])^3*(a*(1 + Sin[c + d*x]))^(3/2)*(203 - 270*Sin[c + d*x] + 99*Sin[c + d*x]^2))/(1287*d)","A",1
117,1,69,127,0.1605434,"\int \cos ^4(c+d x) (a+a \sin (c+d x))^{3/2} \, dx","Integrate[Cos[c + d*x]^4*(a + a*Sin[c + d*x])^(3/2),x]","-\frac{2 \left(105 \sin ^3(c+d x)+455 \sin ^2(c+d x)+755 \sin (c+d x)+533\right) \cos ^5(c+d x) (a (\sin (c+d x)+1))^{3/2}}{1155 d (\sin (c+d x)+1)^4}","-\frac{256 a^4 \cos ^5(c+d x)}{1155 d (a \sin (c+d x)+a)^{5/2}}-\frac{64 a^3 \cos ^5(c+d x)}{231 d (a \sin (c+d x)+a)^{3/2}}-\frac{8 a^2 \cos ^5(c+d x)}{33 d \sqrt{a \sin (c+d x)+a}}-\frac{2 a \cos ^5(c+d x) \sqrt{a \sin (c+d x)+a}}{11 d}",1,"(-2*Cos[c + d*x]^5*(a*(1 + Sin[c + d*x]))^(3/2)*(533 + 755*Sin[c + d*x] + 455*Sin[c + d*x]^2 + 105*Sin[c + d*x]^3))/(1155*d*(1 + Sin[c + d*x])^4)","A",1
118,1,41,49,0.0866407,"\int \cos ^3(c+d x) (a+a \sin (c+d x))^{3/2} \, dx","Integrate[Cos[c + d*x]^3*(a + a*Sin[c + d*x])^(3/2),x]","-\frac{2 (\sin (c+d x)+1)^2 (7 \sin (c+d x)-11) (a (\sin (c+d x)+1))^{3/2}}{63 d}","\frac{4 (a \sin (c+d x)+a)^{7/2}}{7 a^2 d}-\frac{2 (a \sin (c+d x)+a)^{9/2}}{9 a^3 d}",1,"(-2*(1 + Sin[c + d*x])^2*(a*(1 + Sin[c + d*x]))^(3/2)*(-11 + 7*Sin[c + d*x]))/(63*d)","A",1
119,1,59,95,0.1495364,"\int \cos ^2(c+d x) (a+a \sin (c+d x))^{3/2} \, dx","Integrate[Cos[c + d*x]^2*(a + a*Sin[c + d*x])^(3/2),x]","-\frac{2 \left(15 \sin ^2(c+d x)+54 \sin (c+d x)+71\right) \cos ^3(c+d x) (a (\sin (c+d x)+1))^{3/2}}{105 d (\sin (c+d x)+1)^3}","-\frac{64 a^3 \cos ^3(c+d x)}{105 d (a \sin (c+d x)+a)^{3/2}}-\frac{16 a^2 \cos ^3(c+d x)}{35 d \sqrt{a \sin (c+d x)+a}}-\frac{2 a \cos ^3(c+d x) \sqrt{a \sin (c+d x)+a}}{7 d}",1,"(-2*Cos[c + d*x]^3*(a*(1 + Sin[c + d*x]))^(3/2)*(71 + 54*Sin[c + d*x] + 15*Sin[c + d*x]^2))/(105*d*(1 + Sin[c + d*x])^3)","A",1
120,1,24,24,0.0422388,"\int \cos (c+d x) (a+a \sin (c+d x))^{3/2} \, dx","Integrate[Cos[c + d*x]*(a + a*Sin[c + d*x])^(3/2),x]","\frac{2 (a \sin (c+d x)+a)^{5/2}}{5 a d}","\frac{2 (a \sin (c+d x)+a)^{5/2}}{5 a d}",1,"(2*(a + a*Sin[c + d*x])^(5/2))/(5*a*d)","A",1
121,1,60,62,0.1046623,"\int \sec (c+d x) (a+a \sin (c+d x))^{3/2} \, dx","Integrate[Sec[c + d*x]*(a + a*Sin[c + d*x])^(3/2),x]","\frac{2 a \left(\sqrt{2} \sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a \sin (c+d x)+a}}{\sqrt{2} \sqrt{a}}\right)-\sqrt{a \sin (c+d x)+a}\right)}{d}","\frac{2 \sqrt{2} a^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a \sin (c+d x)+a}}{\sqrt{2} \sqrt{a}}\right)}{d}-\frac{2 a \sqrt{a \sin (c+d x)+a}}{d}",1,"(2*a*(Sqrt[2]*Sqrt[a]*ArcTanh[Sqrt[a + a*Sin[c + d*x]]/(Sqrt[2]*Sqrt[a])] - Sqrt[a + a*Sin[c + d*x]]))/d","A",1
122,1,67,26,0.1589054,"\int \sec ^2(c+d x) (a+a \sin (c+d x))^{3/2} \, dx","Integrate[Sec[c + d*x]^2*(a + a*Sin[c + d*x])^(3/2),x]","\frac{2 (a (\sin (c+d x)+1))^{3/2}}{d \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^3}","\frac{2 a \sec (c+d x) \sqrt{a \sin (c+d x)+a}}{d}",1,"(2*(a*(1 + Sin[c + d*x]))^(3/2))/(d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3)","B",1
123,1,72,73,0.2589391,"\int \sec ^3(c+d x) (a+a \sin (c+d x))^{3/2} \, dx","Integrate[Sec[c + d*x]^3*(a + a*Sin[c + d*x])^(3/2),x]","\frac{a \left(\sqrt{2} \sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a (\sin (c+d x)+1)}}{\sqrt{2} \sqrt{a}}\right)-\frac{2 \sqrt{a (\sin (c+d x)+1)}}{\sin (c+d x)-1}\right)}{4 d}","\frac{a^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a \sin (c+d x)+a}}{\sqrt{2} \sqrt{a}}\right)}{2 \sqrt{2} d}+\frac{\sec ^2(c+d x) (a \sin (c+d x)+a)^{3/2}}{2 d}",1,"(a*(Sqrt[2]*Sqrt[a]*ArcTanh[Sqrt[a*(1 + Sin[c + d*x])]/(Sqrt[2]*Sqrt[a])] - (2*Sqrt[a*(1 + Sin[c + d*x])])/(-1 + Sin[c + d*x])))/(4*d)","A",1
124,1,130,107,0.3953482,"\int \sec ^4(c+d x) (a+a \sin (c+d x))^{3/2} \, dx","Integrate[Sec[c + d*x]^4*(a + a*Sin[c + d*x])^(3/2),x]","\frac{\left(\frac{1}{12}+\frac{i}{12}\right) a \sec ^3(c+d x) \sqrt{a (\sin (c+d x)+1)} \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2 \left(6 (-1)^{3/4} \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^3 \tanh ^{-1}\left(\left(\frac{1}{2}+\frac{i}{2}\right) (-1)^{3/4} \left(\tan \left(\frac{1}{4} (c+d x)\right)-1\right)\right)-(1-i) (3 \sin (c+d x)-5)\right)}{d}","-\frac{a^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a \sin (c+d x)+a}}\right)}{2 \sqrt{2} d}+\frac{\sec ^3(c+d x) (a \sin (c+d x)+a)^{3/2}}{3 d}+\frac{a \sec (c+d x) \sqrt{a \sin (c+d x)+a}}{2 d}",1,"((1/12 + I/12)*a*Sec[c + d*x]^3*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2*Sqrt[a*(1 + Sin[c + d*x])]*(6*(-1)^(3/4)*ArcTanh[(1/2 + I/2)*(-1)^(3/4)*(-1 + Tan[(c + d*x)/4])]*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^3 - (1 - I)*(-5 + 3*Sin[c + d*x])))/d","C",1
125,1,44,127,0.0736576,"\int \sec ^5(c+d x) (a+a \sin (c+d x))^{3/2} \, dx","Integrate[Sec[c + d*x]^5*(a + a*Sin[c + d*x])^(3/2),x]","-\frac{a^2 \, _2F_1\left(-\frac{1}{2},3;\frac{1}{2};\frac{1}{2} (\sin (c+d x)+1)\right)}{4 d \sqrt{a \sin (c+d x)+a}}","\frac{15 a^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a \sin (c+d x)+a}}{\sqrt{2} \sqrt{a}}\right)}{32 \sqrt{2} d}-\frac{15 a^2}{32 d \sqrt{a \sin (c+d x)+a}}+\frac{\sec ^4(c+d x) (a \sin (c+d x)+a)^{3/2}}{4 d}+\frac{5 a \sec ^2(c+d x) \sqrt{a \sin (c+d x)+a}}{16 d}",1,"-1/4*(a^2*Hypergeometric2F1[-1/2, 3, 1/2, (1 + Sin[c + d*x])/2])/(d*Sqrt[a + a*Sin[c + d*x]])","C",1
126,1,288,169,0.4283262,"\int \sec ^6(c+d x) (a+a \sin (c+d x))^{3/2} \, dx","Integrate[Sec[c + d*x]^6*(a + a*Sin[c + d*x])^(3/2),x]","\frac{(a (\sin (c+d x)+1))^{3/2} \left(30 \sin \left(\frac{1}{2} (c+d x)\right)+\frac{90 \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2}{\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)}+\frac{40 \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2}{\left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^3}+\frac{24 \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2}{\left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^5}-15 \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+(105+105 i) (-1)^{3/4} \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2 \tanh ^{-1}\left(\left(\frac{1}{2}+\frac{i}{2}\right) (-1)^{3/4} \left(\tan \left(\frac{1}{4} (c+d x)\right)-1\right)\right)\right)}{240 d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^5}","-\frac{7 a^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a \sin (c+d x)+a}}\right)}{16 \sqrt{2} d}-\frac{7 a^3 \cos (c+d x)}{16 d (a \sin (c+d x)+a)^{3/2}}+\frac{7 a^2 \sec (c+d x)}{12 d \sqrt{a \sin (c+d x)+a}}+\frac{\sec ^5(c+d x) (a \sin (c+d x)+a)^{3/2}}{5 d}+\frac{7 a \sec ^3(c+d x) \sqrt{a \sin (c+d x)+a}}{30 d}",1,"((30*Sin[(c + d*x)/2] - 15*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]) + (105 + 105*I)*(-1)^(3/4)*ArcTanh[(1/2 + I/2)*(-1)^(3/4)*(-1 + Tan[(c + d*x)/4])]*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2 + (24*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2)/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^5 + (40*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2)/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^3 + (90*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2)/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2]))*(a*(1 + Sin[c + d*x]))^(3/2))/(240*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^5)","C",1
127,1,51,73,0.2240504,"\int \cos ^5(c+d x) (a+a \sin (c+d x))^{5/2} \, dx","Integrate[Cos[c + d*x]^5*(a + a*Sin[c + d*x])^(5/2),x]","\frac{2 (\sin (c+d x)+1)^3 \left(143 \sin ^2(c+d x)-374 \sin (c+d x)+263\right) (a (\sin (c+d x)+1))^{5/2}}{2145 d}","\frac{2 (a \sin (c+d x)+a)^{15/2}}{15 a^5 d}-\frac{8 (a \sin (c+d x)+a)^{13/2}}{13 a^4 d}+\frac{8 (a \sin (c+d x)+a)^{11/2}}{11 a^3 d}",1,"(2*(1 + Sin[c + d*x])^3*(a*(1 + Sin[c + d*x]))^(5/2)*(263 - 374*Sin[c + d*x] + 143*Sin[c + d*x]^2))/(2145*d)","A",1
128,1,79,159,0.3316839,"\int \cos ^4(c+d x) (a+a \sin (c+d x))^{5/2} \, dx","Integrate[Cos[c + d*x]^4*(a + a*Sin[c + d*x])^(5/2),x]","-\frac{2 \left(1155 \sin ^4(c+d x)+6300 \sin ^3(c+d x)+14210 \sin ^2(c+d x)+16700 \sin (c+d x)+9683\right) \cos ^5(c+d x) (a (\sin (c+d x)+1))^{5/2}}{15015 d (\sin (c+d x)+1)^5}","-\frac{4096 a^5 \cos ^5(c+d x)}{15015 d (a \sin (c+d x)+a)^{5/2}}-\frac{1024 a^4 \cos ^5(c+d x)}{3003 d (a \sin (c+d x)+a)^{3/2}}-\frac{128 a^3 \cos ^5(c+d x)}{429 d \sqrt{a \sin (c+d x)+a}}-\frac{32 a^2 \cos ^5(c+d x) \sqrt{a \sin (c+d x)+a}}{143 d}-\frac{2 a \cos ^5(c+d x) (a \sin (c+d x)+a)^{3/2}}{13 d}",1,"(-2*Cos[c + d*x]^5*(a*(1 + Sin[c + d*x]))^(5/2)*(9683 + 16700*Sin[c + d*x] + 14210*Sin[c + d*x]^2 + 6300*Sin[c + d*x]^3 + 1155*Sin[c + d*x]^4))/(15015*d*(1 + Sin[c + d*x])^5)","A",1
129,1,41,49,0.1407475,"\int \cos ^3(c+d x) (a+a \sin (c+d x))^{5/2} \, dx","Integrate[Cos[c + d*x]^3*(a + a*Sin[c + d*x])^(5/2),x]","-\frac{2 (\sin (c+d x)+1)^2 (9 \sin (c+d x)-13) (a (\sin (c+d x)+1))^{5/2}}{99 d}","\frac{4 (a \sin (c+d x)+a)^{9/2}}{9 a^2 d}-\frac{2 (a \sin (c+d x)+a)^{11/2}}{11 a^3 d}",1,"(-2*(1 + Sin[c + d*x])^2*(a*(1 + Sin[c + d*x]))^(5/2)*(-13 + 9*Sin[c + d*x]))/(99*d)","A",1
130,1,69,127,0.3088432,"\int \cos ^2(c+d x) (a+a \sin (c+d x))^{5/2} \, dx","Integrate[Cos[c + d*x]^2*(a + a*Sin[c + d*x])^(5/2),x]","-\frac{2 \left(35 \sin ^3(c+d x)+165 \sin ^2(c+d x)+321 \sin (c+d x)+319\right) \cos ^3(c+d x) (a (\sin (c+d x)+1))^{5/2}}{315 d (\sin (c+d x)+1)^4}","-\frac{256 a^4 \cos ^3(c+d x)}{315 d (a \sin (c+d x)+a)^{3/2}}-\frac{64 a^3 \cos ^3(c+d x)}{105 d \sqrt{a \sin (c+d x)+a}}-\frac{8 a^2 \cos ^3(c+d x) \sqrt{a \sin (c+d x)+a}}{21 d}-\frac{2 a \cos ^3(c+d x) (a \sin (c+d x)+a)^{3/2}}{9 d}",1,"(-2*Cos[c + d*x]^3*(a*(1 + Sin[c + d*x]))^(5/2)*(319 + 321*Sin[c + d*x] + 165*Sin[c + d*x]^2 + 35*Sin[c + d*x]^3))/(315*d*(1 + Sin[c + d*x])^4)","A",1
131,1,24,24,0.0664663,"\int \cos (c+d x) (a+a \sin (c+d x))^{5/2} \, dx","Integrate[Cos[c + d*x]*(a + a*Sin[c + d*x])^(5/2),x]","\frac{2 (a \sin (c+d x)+a)^{7/2}}{7 a d}","\frac{2 (a \sin (c+d x)+a)^{7/2}}{7 a d}",1,"(2*(a + a*Sin[c + d*x])^(7/2))/(7*a*d)","A",1
132,1,73,86,0.2043541,"\int \sec (c+d x) (a+a \sin (c+d x))^{5/2} \, dx","Integrate[Sec[c + d*x]*(a + a*Sin[c + d*x])^(5/2),x]","\frac{12 \sqrt{2} a^{5/2} \tanh ^{-1}\left(\frac{\sqrt{a (\sin (c+d x)+1)}}{\sqrt{2} \sqrt{a}}\right)-2 a^2 (\sin (c+d x)+7) \sqrt{a (\sin (c+d x)+1)}}{3 d}","\frac{4 \sqrt{2} a^{5/2} \tanh ^{-1}\left(\frac{\sqrt{a \sin (c+d x)+a}}{\sqrt{2} \sqrt{a}}\right)}{d}-\frac{4 a^2 \sqrt{a \sin (c+d x)+a}}{d}-\frac{2 a (a \sin (c+d x)+a)^{3/2}}{3 d}",1,"(12*Sqrt[2]*a^(5/2)*ArcTanh[Sqrt[a*(1 + Sin[c + d*x])]/(Sqrt[2]*Sqrt[a])] - 2*a^2*Sqrt[a*(1 + Sin[c + d*x])]*(7 + Sin[c + d*x]))/(3*d)","A",1
133,1,36,55,4.6167573,"\int \sec ^2(c+d x) (a+a \sin (c+d x))^{5/2} \, dx","Integrate[Sec[c + d*x]^2*(a + a*Sin[c + d*x])^(5/2),x]","-\frac{2 a^2 (\sin (c+d x)-3) \sec (c+d x) \sqrt{a (\sin (c+d x)+1)}}{d}","\frac{8 a^2 \sec (c+d x) \sqrt{a \sin (c+d x)+a}}{d}-\frac{2 a \sec (c+d x) (a \sin (c+d x)+a)^{3/2}}{d}",1,"(-2*a^2*Sec[c + d*x]*(-3 + Sin[c + d*x])*Sqrt[a*(1 + Sin[c + d*x])])/d","A",1
134,1,75,69,0.4587134,"\int \sec ^3(c+d x) (a+a \sin (c+d x))^{5/2} \, dx","Integrate[Sec[c + d*x]^3*(a + a*Sin[c + d*x])^(5/2),x]","\frac{a^2 \left(-\frac{2 \sqrt{a (\sin (c+d x)+1)}}{\sin (c+d x)-1}-\sqrt{2} \sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a (\sin (c+d x)+1)}}{\sqrt{2} \sqrt{a}}\right)\right)}{2 d}","\frac{a \sec ^2(c+d x) (a \sin (c+d x)+a)^{3/2}}{d}-\frac{a^{5/2} \tanh ^{-1}\left(\frac{\sqrt{a \sin (c+d x)+a}}{\sqrt{2} \sqrt{a}}\right)}{\sqrt{2} d}",1,"(a^2*(-(Sqrt[2]*Sqrt[a]*ArcTanh[Sqrt[a*(1 + Sin[c + d*x])]/(Sqrt[2]*Sqrt[a])]) - (2*Sqrt[a*(1 + Sin[c + d*x])])/(-1 + Sin[c + d*x])))/(2*d)","A",1
135,1,69,30,5.1586288,"\int \sec ^4(c+d x) (a+a \sin (c+d x))^{5/2} \, dx","Integrate[Sec[c + d*x]^4*(a + a*Sin[c + d*x])^(5/2),x]","\frac{2 (a (\sin (c+d x)+1))^{5/2}}{3 d \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^3 \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^5}","\frac{2 a \sec ^3(c+d x) (a \sin (c+d x)+a)^{3/2}}{3 d}",1,"(2*(a*(1 + Sin[c + d*x]))^(5/2))/(3*d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^3*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^5)","B",1
136,1,110,103,0.2857209,"\int \sec ^5(c+d x) (a+a \sin (c+d x))^{5/2} \, dx","Integrate[Sec[c + d*x]^5*(a + a*Sin[c + d*x])^(5/2),x]","\frac{3 \sqrt{2} a^{5/2} \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^4 \tanh ^{-1}\left(\frac{\sqrt{a (\sin (c+d x)+1)}}{\sqrt{2} \sqrt{a}}\right)+2 a^2 (7-3 \sin (c+d x)) \sqrt{a (\sin (c+d x)+1)}}{32 d (\sin (c+d x)-1)^2}","\frac{3 a^{5/2} \tanh ^{-1}\left(\frac{\sqrt{a \sin (c+d x)+a}}{\sqrt{2} \sqrt{a}}\right)}{16 \sqrt{2} d}+\frac{\sec ^4(c+d x) (a \sin (c+d x)+a)^{5/2}}{4 d}+\frac{3 a \sec ^2(c+d x) (a \sin (c+d x)+a)^{3/2}}{16 d}",1,"(3*Sqrt[2]*a^(5/2)*ArcTanh[Sqrt[a*(1 + Sin[c + d*x])]/(Sqrt[2]*Sqrt[a])]*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^4 + 2*a^2*(7 - 3*Sin[c + d*x])*Sqrt[a*(1 + Sin[c + d*x])])/(32*d*(-1 + Sin[c + d*x])^2)","A",1
137,1,129,139,5.3067661,"\int \sec ^6(c+d x) (a+a \sin (c+d x))^{5/2} \, dx","Integrate[Sec[c + d*x]^6*(a + a*Sin[c + d*x])^(5/2),x]","\frac{(a (\sin (c+d x)+1))^{5/2} \left(\frac{-80 \sin (c+d x)-15 \cos (2 (c+d x))+89}{2 \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^5}+(15+15 i) (-1)^{3/4} \tanh ^{-1}\left(\left(\frac{1}{2}+\frac{i}{2}\right) (-1)^{3/4} \left(\tan \left(\frac{1}{4} (c+d x)\right)-1\right)\right)\right)}{60 d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^5}","-\frac{a^{5/2} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a \sin (c+d x)+a}}\right)}{4 \sqrt{2} d}+\frac{a^2 \sec (c+d x) \sqrt{a \sin (c+d x)+a}}{4 d}+\frac{\sec ^5(c+d x) (a \sin (c+d x)+a)^{5/2}}{5 d}+\frac{a \sec ^3(c+d x) (a \sin (c+d x)+a)^{3/2}}{6 d}",1,"(((15 + 15*I)*(-1)^(3/4)*ArcTanh[(1/2 + I/2)*(-1)^(3/4)*(-1 + Tan[(c + d*x)/4])] + (89 - 15*Cos[2*(c + d*x)] - 80*Sin[c + d*x])/(2*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^5))*(a*(1 + Sin[c + d*x]))^(5/2))/(60*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^5)","C",1
138,1,44,159,0.1090957,"\int \sec ^7(c+d x) (a+a \sin (c+d x))^{5/2} \, dx","Integrate[Sec[c + d*x]^7*(a + a*Sin[c + d*x])^(5/2),x]","-\frac{a^3 \, _2F_1\left(-\frac{1}{2},4;\frac{1}{2};\frac{1}{2} (\sin (c+d x)+1)\right)}{8 d \sqrt{a \sin (c+d x)+a}}","\frac{35 a^{5/2} \tanh ^{-1}\left(\frac{\sqrt{a \sin (c+d x)+a}}{\sqrt{2} \sqrt{a}}\right)}{128 \sqrt{2} d}-\frac{35 a^3}{128 d \sqrt{a \sin (c+d x)+a}}+\frac{35 a^2 \sec ^2(c+d x) \sqrt{a \sin (c+d x)+a}}{192 d}+\frac{\sec ^6(c+d x) (a \sin (c+d x)+a)^{5/2}}{6 d}+\frac{7 a \sec ^4(c+d x) (a \sin (c+d x)+a)^{3/2}}{48 d}",1,"-1/8*(a^3*Hypergeometric2F1[-1/2, 4, 1/2, (1 + Sin[c + d*x])/2])/(d*Sqrt[a + a*Sin[c + d*x]])","C",1
139,1,64,97,0.6335767,"\int \cos ^7(c+d x) (a+a \sin (c+d x))^{7/2} \, dx","Integrate[Cos[c + d*x]^7*(a + a*Sin[c + d*x])^(7/2),x]","-\frac{2 a^3 (\sin (c+d x)+1)^7 \left(1615 \sin ^3(c+d x)-5865 \sin ^2(c+d x)+7365 \sin (c+d x)-3243\right) \sqrt{a (\sin (c+d x)+1)}}{33915 d}","-\frac{2 (a \sin (c+d x)+a)^{21/2}}{21 a^7 d}+\frac{12 (a \sin (c+d x)+a)^{19/2}}{19 a^6 d}-\frac{24 (a \sin (c+d x)+a)^{17/2}}{17 a^5 d}+\frac{16 (a \sin (c+d x)+a)^{15/2}}{15 a^4 d}",1,"(-2*a^3*(1 + Sin[c + d*x])^7*Sqrt[a*(1 + Sin[c + d*x])]*(-3243 + 7365*Sin[c + d*x] - 5865*Sin[c + d*x]^2 + 1615*Sin[c + d*x]^3))/(33915*d)","A",1
140,1,102,223,0.5473209,"\int \cos ^6(c+d x) (a+a \sin (c+d x))^{7/2} \, dx","Integrate[Cos[c + d*x]^6*(a + a*Sin[c + d*x])^(7/2),x]","-\frac{2 a^3 \left(51051 \sin ^6(c+d x)+378378 \sin ^5(c+d x)+1222221 \sin ^4(c+d x)+2244396 \sin ^3(c+d x)+2546901 \sin ^2(c+d x)+1778602 \sin (c+d x)+646739\right) \cos ^7(c+d x) \sqrt{a (\sin (c+d x)+1)}}{969969 d (\sin (c+d x)+1)^4}","-\frac{131072 a^7 \cos ^7(c+d x)}{969969 d (a \sin (c+d x)+a)^{7/2}}-\frac{32768 a^6 \cos ^7(c+d x)}{138567 d (a \sin (c+d x)+a)^{5/2}}-\frac{12288 a^5 \cos ^7(c+d x)}{46189 d (a \sin (c+d x)+a)^{3/2}}-\frac{1024 a^4 \cos ^7(c+d x)}{4199 d \sqrt{a \sin (c+d x)+a}}-\frac{64 a^3 \cos ^7(c+d x) \sqrt{a \sin (c+d x)+a}}{323 d}-\frac{48 a^2 \cos ^7(c+d x) (a \sin (c+d x)+a)^{3/2}}{323 d}-\frac{2 a \cos ^7(c+d x) (a \sin (c+d x)+a)^{5/2}}{19 d}",1,"(-2*a^3*Cos[c + d*x]^7*Sqrt[a*(1 + Sin[c + d*x])]*(646739 + 1778602*Sin[c + d*x] + 2546901*Sin[c + d*x]^2 + 2244396*Sin[c + d*x]^3 + 1222221*Sin[c + d*x]^4 + 378378*Sin[c + d*x]^5 + 51051*Sin[c + d*x]^6))/(969969*d*(1 + Sin[c + d*x])^4)","A",1
141,1,54,73,0.3048654,"\int \cos ^5(c+d x) (a+a \sin (c+d x))^{7/2} \, dx","Integrate[Cos[c + d*x]^5*(a + a*Sin[c + d*x])^(7/2),x]","\frac{2 a^3 (\sin (c+d x)+1)^6 \left(195 \sin ^2(c+d x)-494 \sin (c+d x)+331\right) \sqrt{a (\sin (c+d x)+1)}}{3315 d}","\frac{2 (a \sin (c+d x)+a)^{17/2}}{17 a^5 d}-\frac{8 (a \sin (c+d x)+a)^{15/2}}{15 a^4 d}+\frac{8 (a \sin (c+d x)+a)^{13/2}}{13 a^3 d}",1,"(2*a^3*(1 + Sin[c + d*x])^6*Sqrt[a*(1 + Sin[c + d*x])]*(331 - 494*Sin[c + d*x] + 195*Sin[c + d*x]^2))/(3315*d)","A",1
142,1,92,191,0.2592476,"\int \cos ^4(c+d x) (a+a \sin (c+d x))^{7/2} \, dx","Integrate[Cos[c + d*x]^4*(a + a*Sin[c + d*x])^(7/2),x]","-\frac{2 a^3 \left(3003 \sin ^5(c+d x)+19635 \sin ^4(c+d x)+55230 \sin ^3(c+d x)+86870 \sin ^2(c+d x)+81815 \sin (c+d x)+41735\right) \cos ^5(c+d x) \sqrt{a (\sin (c+d x)+1)}}{45045 d (\sin (c+d x)+1)^3}","-\frac{16384 a^6 \cos ^5(c+d x)}{45045 d (a \sin (c+d x)+a)^{5/2}}-\frac{4096 a^5 \cos ^5(c+d x)}{9009 d (a \sin (c+d x)+a)^{3/2}}-\frac{512 a^4 \cos ^5(c+d x)}{1287 d \sqrt{a \sin (c+d x)+a}}-\frac{128 a^3 \cos ^5(c+d x) \sqrt{a \sin (c+d x)+a}}{429 d}-\frac{8 a^2 \cos ^5(c+d x) (a \sin (c+d x)+a)^{3/2}}{39 d}-\frac{2 a \cos ^5(c+d x) (a \sin (c+d x)+a)^{5/2}}{15 d}",1,"(-2*a^3*Cos[c + d*x]^5*Sqrt[a*(1 + Sin[c + d*x])]*(41735 + 81815*Sin[c + d*x] + 86870*Sin[c + d*x]^2 + 55230*Sin[c + d*x]^3 + 19635*Sin[c + d*x]^4 + 3003*Sin[c + d*x]^5))/(45045*d*(1 + Sin[c + d*x])^3)","A",1
143,1,44,49,0.1469459,"\int \cos ^3(c+d x) (a+a \sin (c+d x))^{7/2} \, dx","Integrate[Cos[c + d*x]^3*(a + a*Sin[c + d*x])^(7/2),x]","\frac{2 \left(26 a (a \sin (c+d x)+a)^{11/2}-11 (a \sin (c+d x)+a)^{13/2}\right)}{143 a^3 d}","\frac{4 (a \sin (c+d x)+a)^{11/2}}{11 a^2 d}-\frac{2 (a \sin (c+d x)+a)^{13/2}}{13 a^3 d}",1,"(2*(26*a*(a + a*Sin[c + d*x])^(11/2) - 11*(a + a*Sin[c + d*x])^(13/2)))/(143*a^3*d)","A",1
144,1,82,159,0.2289569,"\int \cos ^2(c+d x) (a+a \sin (c+d x))^{7/2} \, dx","Integrate[Cos[c + d*x]^2*(a + a*Sin[c + d*x])^(7/2),x]","-\frac{2 a^3 \left(315 \sin ^4(c+d x)+1820 \sin ^3(c+d x)+4530 \sin ^2(c+d x)+6396 \sin (c+d x)+5419\right) \cos ^3(c+d x) \sqrt{a (\sin (c+d x)+1)}}{3465 d (\sin (c+d x)+1)^2}","-\frac{4096 a^5 \cos ^3(c+d x)}{3465 d (a \sin (c+d x)+a)^{3/2}}-\frac{1024 a^4 \cos ^3(c+d x)}{1155 d \sqrt{a \sin (c+d x)+a}}-\frac{128 a^3 \cos ^3(c+d x) \sqrt{a \sin (c+d x)+a}}{231 d}-\frac{32 a^2 \cos ^3(c+d x) (a \sin (c+d x)+a)^{3/2}}{99 d}-\frac{2 a \cos ^3(c+d x) (a \sin (c+d x)+a)^{5/2}}{11 d}",1,"(-2*a^3*Cos[c + d*x]^3*Sqrt[a*(1 + Sin[c + d*x])]*(5419 + 6396*Sin[c + d*x] + 4530*Sin[c + d*x]^2 + 1820*Sin[c + d*x]^3 + 315*Sin[c + d*x]^4))/(3465*d*(1 + Sin[c + d*x])^2)","A",1
145,1,24,24,0.0858894,"\int \cos (c+d x) (a+a \sin (c+d x))^{7/2} \, dx","Integrate[Cos[c + d*x]*(a + a*Sin[c + d*x])^(7/2),x]","\frac{2 (a \sin (c+d x)+a)^{9/2}}{9 a d}","\frac{2 (a \sin (c+d x)+a)^{9/2}}{9 a d}",1,"(2*(a + a*Sin[c + d*x])^(9/2))/(9*a*d)","A",1
146,1,85,110,0.3454182,"\int \sec (c+d x) (a+a \sin (c+d x))^{7/2} \, dx","Integrate[Sec[c + d*x]*(a + a*Sin[c + d*x])^(7/2),x]","\frac{120 \sqrt{2} a^{7/2} \tanh ^{-1}\left(\frac{\sqrt{a (\sin (c+d x)+1)}}{\sqrt{2} \sqrt{a}}\right)-2 a^3 \left(3 \sin ^2(c+d x)+16 \sin (c+d x)+73\right) \sqrt{a (\sin (c+d x)+1)}}{15 d}","\frac{8 \sqrt{2} a^{7/2} \tanh ^{-1}\left(\frac{\sqrt{a \sin (c+d x)+a}}{\sqrt{2} \sqrt{a}}\right)}{d}-\frac{8 a^3 \sqrt{a \sin (c+d x)+a}}{d}-\frac{4 a^2 (a \sin (c+d x)+a)^{3/2}}{3 d}-\frac{2 a (a \sin (c+d x)+a)^{5/2}}{5 d}",1,"(120*Sqrt[2]*a^(7/2)*ArcTanh[Sqrt[a*(1 + Sin[c + d*x])]/(Sqrt[2]*Sqrt[a])] - 2*a^3*Sqrt[a*(1 + Sin[c + d*x])]*(73 + 16*Sin[c + d*x] + 3*Sin[c + d*x]^2))/(15*d)","A",1
147,1,48,89,5.4846912,"\int \sec ^2(c+d x) (a+a \sin (c+d x))^{7/2} \, dx","Integrate[Sec[c + d*x]^2*(a + a*Sin[c + d*x])^(7/2),x]","\frac{a^3 \sec (c+d x) \sqrt{a (\sin (c+d x)+1)} (-20 \sin (c+d x)+\cos (2 (c+d x))+45)}{3 d}","\frac{64 a^3 \sec (c+d x) \sqrt{a \sin (c+d x)+a}}{3 d}-\frac{16 a^2 \sec (c+d x) (a \sin (c+d x)+a)^{3/2}}{3 d}-\frac{2 a \sec (c+d x) (a \sin (c+d x)+a)^{5/2}}{3 d}",1,"(a^3*Sec[c + d*x]*(45 + Cos[2*(c + d*x)] - 20*Sin[c + d*x])*Sqrt[a*(1 + Sin[c + d*x])])/(3*d)","A",1
148,1,42,91,0.0967148,"\int \sec ^3(c+d x) (a+a \sin (c+d x))^{7/2} \, dx","Integrate[Sec[c + d*x]^3*(a + a*Sin[c + d*x])^(7/2),x]","\frac{a (a \sin (c+d x)+a)^{5/2} \, _2F_1\left(2,\frac{5}{2};\frac{7}{2};\frac{1}{2} (\sin (c+d x)+1)\right)}{10 d}","-\frac{3 \sqrt{2} a^{7/2} \tanh ^{-1}\left(\frac{\sqrt{a \sin (c+d x)+a}}{\sqrt{2} \sqrt{a}}\right)}{d}+\frac{3 a^3 \sqrt{a \sin (c+d x)+a}}{d}+\frac{a \sec ^2(c+d x) (a \sin (c+d x)+a)^{5/2}}{d}",1,"(a*Hypergeometric2F1[2, 5/2, 7/2, (1 + Sin[c + d*x])/2]*(a + a*Sin[c + d*x])^(5/2))/(10*d)","C",1
149,1,82,61,5.2992089,"\int \sec ^4(c+d x) (a+a \sin (c+d x))^{7/2} \, dx","Integrate[Sec[c + d*x]^4*(a + a*Sin[c + d*x])^(7/2),x]","\frac{2 a^3 (3 \sin (c+d x)-1) \sqrt{a (\sin (c+d x)+1)}}{3 d \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^3 \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}","\frac{2 a \sec ^3(c+d x) (a \sin (c+d x)+a)^{5/2}}{d}-\frac{8 a^2 \sec ^3(c+d x) (a \sin (c+d x)+a)^{3/2}}{3 d}",1,"(2*a^3*Sqrt[a*(1 + Sin[c + d*x])]*(-1 + 3*Sin[c + d*x]))/(3*d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^3*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))","A",1
150,1,108,106,0.293852,"\int \sec ^5(c+d x) (a+a \sin (c+d x))^{7/2} \, dx","Integrate[Sec[c + d*x]^5*(a + a*Sin[c + d*x])^(7/2),x]","\frac{2 a^3 (\sin (c+d x)+3) \sqrt{a (\sin (c+d x)+1)}-\sqrt{2} a^{7/2} \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^4 \tanh ^{-1}\left(\frac{\sqrt{a (\sin (c+d x)+1)}}{\sqrt{2} \sqrt{a}}\right)}{16 d (\sin (c+d x)-1)^2}","-\frac{a^{7/2} \tanh ^{-1}\left(\frac{\sqrt{a \sin (c+d x)+a}}{\sqrt{2} \sqrt{a}}\right)}{8 \sqrt{2} d}-\frac{a^2 \sec ^2(c+d x) (a \sin (c+d x)+a)^{3/2}}{8 d}+\frac{a \sec ^4(c+d x) (a \sin (c+d x)+a)^{5/2}}{2 d}",1,"(-(Sqrt[2]*a^(7/2)*ArcTanh[Sqrt[a*(1 + Sin[c + d*x])]/(Sqrt[2]*Sqrt[a])]*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^4) + 2*a^3*Sqrt[a*(1 + Sin[c + d*x])]*(3 + Sin[c + d*x]))/(16*d*(-1 + Sin[c + d*x])^2)","A",1
151,1,69,30,5.271583,"\int \sec ^6(c+d x) (a+a \sin (c+d x))^{7/2} \, dx","Integrate[Sec[c + d*x]^6*(a + a*Sin[c + d*x])^(7/2),x]","\frac{2 (a (\sin (c+d x)+1))^{7/2}}{5 d \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^5 \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^7}","\frac{2 a \sec ^5(c+d x) (a \sin (c+d x)+a)^{5/2}}{5 d}",1,"(2*(a*(1 + Sin[c + d*x]))^(7/2))/(5*d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^5*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^7)","B",1
152,1,120,135,0.5550086,"\int \sec ^7(c+d x) (a+a \sin (c+d x))^{7/2} \, dx","Integrate[Sec[c + d*x]^7*(a + a*Sin[c + d*x])^(7/2),x]","-\frac{15 \sqrt{2} a^{7/2} \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^6 \tanh ^{-1}\left(\frac{\sqrt{a (\sin (c+d x)+1)}}{\sqrt{2} \sqrt{a}}\right)+2 a^3 \left(15 \sin ^2(c+d x)-50 \sin (c+d x)+67\right) \sqrt{a (\sin (c+d x)+1)}}{384 d (\sin (c+d x)-1)^3}","\frac{5 a^{7/2} \tanh ^{-1}\left(\frac{\sqrt{a \sin (c+d x)+a}}{\sqrt{2} \sqrt{a}}\right)}{64 \sqrt{2} d}+\frac{5 a^2 \sec ^2(c+d x) (a \sin (c+d x)+a)^{3/2}}{64 d}+\frac{\sec ^6(c+d x) (a \sin (c+d x)+a)^{7/2}}{6 d}+\frac{5 a \sec ^4(c+d x) (a \sin (c+d x)+a)^{5/2}}{48 d}",1,"-1/384*(15*Sqrt[2]*a^(7/2)*ArcTanh[Sqrt[a*(1 + Sin[c + d*x])]/(Sqrt[2]*Sqrt[a])]*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^6 + 2*a^3*Sqrt[a*(1 + Sin[c + d*x])]*(67 - 50*Sin[c + d*x] + 15*Sin[c + d*x]^2))/(d*(-1 + Sin[c + d*x])^3)","A",1
153,1,139,171,5.4966205,"\int \sec ^8(c+d x) (a+a \sin (c+d x))^{7/2} \, dx","Integrate[Sec[c + d*x]^8*(a + a*Sin[c + d*x])^(7/2),x]","\frac{(a (\sin (c+d x)+1))^{7/2} \left(\frac{-2471 \sin (c+d x)+105 \sin (3 (c+d x))-770 \cos (2 (c+d x))+2286}{4 \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^7}+(105+105 i) (-1)^{3/4} \tanh ^{-1}\left(\left(\frac{1}{2}+\frac{i}{2}\right) (-1)^{3/4} \left(\tan \left(\frac{1}{4} (c+d x)\right)-1\right)\right)\right)}{840 d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^7}","-\frac{a^{7/2} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a \sin (c+d x)+a}}\right)}{8 \sqrt{2} d}+\frac{a^3 \sec (c+d x) \sqrt{a \sin (c+d x)+a}}{8 d}+\frac{a^2 \sec ^3(c+d x) (a \sin (c+d x)+a)^{3/2}}{12 d}+\frac{\sec ^7(c+d x) (a \sin (c+d x)+a)^{7/2}}{7 d}+\frac{a \sec ^5(c+d x) (a \sin (c+d x)+a)^{5/2}}{10 d}",1,"((a*(1 + Sin[c + d*x]))^(7/2)*((105 + 105*I)*(-1)^(3/4)*ArcTanh[(1/2 + I/2)*(-1)^(3/4)*(-1 + Tan[(c + d*x)/4])] + (2286 - 770*Cos[2*(c + d*x)] - 2471*Sin[c + d*x] + 105*Sin[3*(c + d*x)])/(4*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^7)))/(840*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^7)","C",1
154,1,44,191,0.1079783,"\int \sec ^9(c+d x) (a+a \sin (c+d x))^{7/2} \, dx","Integrate[Sec[c + d*x]^9*(a + a*Sin[c + d*x])^(7/2),x]","-\frac{a^4 \, _2F_1\left(-\frac{1}{2},5;\frac{1}{2};\frac{1}{2} (\sin (c+d x)+1)\right)}{16 d \sqrt{a \sin (c+d x)+a}}","\frac{315 a^{7/2} \tanh ^{-1}\left(\frac{\sqrt{a \sin (c+d x)+a}}{\sqrt{2} \sqrt{a}}\right)}{2048 \sqrt{2} d}-\frac{315 a^4}{2048 d \sqrt{a \sin (c+d x)+a}}+\frac{105 a^3 \sec ^2(c+d x) \sqrt{a \sin (c+d x)+a}}{1024 d}+\frac{21 a^2 \sec ^4(c+d x) (a \sin (c+d x)+a)^{3/2}}{256 d}+\frac{\sec ^8(c+d x) (a \sin (c+d x)+a)^{7/2}}{8 d}+\frac{3 a \sec ^6(c+d x) (a \sin (c+d x)+a)^{5/2}}{32 d}",1,"-1/16*(a^4*Hypergeometric2F1[-1/2, 5, 1/2, (1 + Sin[c + d*x])/2])/(d*Sqrt[a + a*Sin[c + d*x]])","C",1
155,1,388,233,5.678347,"\int \sec ^{10}(c+d x) (a+a \sin (c+d x))^{7/2} \, dx","Integrate[Sec[c + d*x]^10*(a + a*Sin[c + d*x])^(7/2),x]","\frac{(a (\sin (c+d x)+1))^{7/2} \left(630 \sin \left(\frac{1}{2} (c+d x)\right)+\frac{3150 \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2}{\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)}+\frac{1680 \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2}{\left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^3}+\frac{1512 \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2}{\left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^5}+\frac{1440 \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2}{\left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^7}+\frac{1120 \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2}{\left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^9}-315 \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+(3465+3465 i) (-1)^{3/4} \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2 \tanh ^{-1}\left(\left(\frac{1}{2}+\frac{i}{2}\right) (-1)^{3/4} \left(\tan \left(\frac{1}{4} (c+d x)\right)-1\right)\right)\right)}{20160 d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^9}","-\frac{11 a^{7/2} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a \sin (c+d x)+a}}\right)}{64 \sqrt{2} d}-\frac{11 a^5 \cos (c+d x)}{64 d (a \sin (c+d x)+a)^{3/2}}+\frac{11 a^4 \sec (c+d x)}{48 d \sqrt{a \sin (c+d x)+a}}+\frac{11 a^3 \sec ^3(c+d x) \sqrt{a \sin (c+d x)+a}}{120 d}+\frac{11 a^2 \sec ^5(c+d x) (a \sin (c+d x)+a)^{3/2}}{140 d}+\frac{\sec ^9(c+d x) (a \sin (c+d x)+a)^{7/2}}{9 d}+\frac{11 a \sec ^7(c+d x) (a \sin (c+d x)+a)^{5/2}}{126 d}",1,"((630*Sin[(c + d*x)/2] - 315*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]) + (3465 + 3465*I)*(-1)^(3/4)*ArcTanh[(1/2 + I/2)*(-1)^(3/4)*(-1 + Tan[(c + d*x)/4])]*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2 + (1120*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2)/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^9 + (1440*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2)/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^7 + (1512*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2)/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^5 + (1680*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2)/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^3 + (3150*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2)/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2]))*(a*(1 + Sin[c + d*x]))^(7/2))/(20160*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^9)","C",1
156,1,61,97,0.2980942,"\int \frac{\cos ^7(c+d x)}{\sqrt{a+a \sin (c+d x)}} \, dx","Integrate[Cos[c + d*x]^7/Sqrt[a + a*Sin[c + d*x]],x]","-\frac{2 (\sin (c+d x)+1)^4 \left(231 \sin ^3(c+d x)-945 \sin ^2(c+d x)+1421 \sin (c+d x)-835\right)}{3003 d \sqrt{a (\sin (c+d x)+1)}}","-\frac{2 (a \sin (c+d x)+a)^{13/2}}{13 a^7 d}+\frac{12 (a \sin (c+d x)+a)^{11/2}}{11 a^6 d}-\frac{8 (a \sin (c+d x)+a)^{9/2}}{3 a^5 d}+\frac{16 (a \sin (c+d x)+a)^{7/2}}{7 a^4 d}",1,"(-2*(1 + Sin[c + d*x])^4*(-835 + 1421*Sin[c + d*x] - 945*Sin[c + d*x]^2 + 231*Sin[c + d*x]^3))/(3003*d*Sqrt[a*(1 + Sin[c + d*x])])","A",1
157,1,59,95,0.2584471,"\int \frac{\cos ^6(c+d x)}{\sqrt{a+a \sin (c+d x)}} \, dx","Integrate[Cos[c + d*x]^6/Sqrt[a + a*Sin[c + d*x]],x]","-\frac{2 \left(63 \sin ^2(c+d x)+182 \sin (c+d x)+151\right) \cos ^7(c+d x)}{693 d (\sin (c+d x)+1)^3 \sqrt{a (\sin (c+d x)+1)}}","-\frac{64 a^3 \cos ^7(c+d x)}{693 d (a \sin (c+d x)+a)^{7/2}}-\frac{16 a^2 \cos ^7(c+d x)}{99 d (a \sin (c+d x)+a)^{5/2}}-\frac{2 a \cos ^7(c+d x)}{11 d (a \sin (c+d x)+a)^{3/2}}",1,"(-2*Cos[c + d*x]^7*(151 + 182*Sin[c + d*x] + 63*Sin[c + d*x]^2))/(693*d*(1 + Sin[c + d*x])^3*Sqrt[a*(1 + Sin[c + d*x])])","A",1
158,1,51,73,0.1420771,"\int \frac{\cos ^5(c+d x)}{\sqrt{a+a \sin (c+d x)}} \, dx","Integrate[Cos[c + d*x]^5/Sqrt[a + a*Sin[c + d*x]],x]","\frac{2 (\sin (c+d x)+1)^3 \left(35 \sin ^2(c+d x)-110 \sin (c+d x)+107\right)}{315 d \sqrt{a (\sin (c+d x)+1)}}","\frac{2 (a \sin (c+d x)+a)^{9/2}}{9 a^5 d}-\frac{8 (a \sin (c+d x)+a)^{7/2}}{7 a^4 d}+\frac{8 (a \sin (c+d x)+a)^{5/2}}{5 a^3 d}",1,"(2*(1 + Sin[c + d*x])^3*(107 - 110*Sin[c + d*x] + 35*Sin[c + d*x]^2))/(315*d*Sqrt[a*(1 + Sin[c + d*x])])","A",1
159,1,49,63,0.0706719,"\int \frac{\cos ^4(c+d x)}{\sqrt{a+a \sin (c+d x)}} \, dx","Integrate[Cos[c + d*x]^4/Sqrt[a + a*Sin[c + d*x]],x]","-\frac{2 (5 \sin (c+d x)+9) \cos ^5(c+d x)}{35 d (\sin (c+d x)+1)^2 \sqrt{a (\sin (c+d x)+1)}}","-\frac{8 a^2 \cos ^5(c+d x)}{35 d (a \sin (c+d x)+a)^{5/2}}-\frac{2 a \cos ^5(c+d x)}{7 d (a \sin (c+d x)+a)^{3/2}}",1,"(-2*Cos[c + d*x]^5*(9 + 5*Sin[c + d*x]))/(35*d*(1 + Sin[c + d*x])^2*Sqrt[a*(1 + Sin[c + d*x])])","A",1
160,1,34,49,0.0620448,"\int \frac{\cos ^3(c+d x)}{\sqrt{a+a \sin (c+d x)}} \, dx","Integrate[Cos[c + d*x]^3/Sqrt[a + a*Sin[c + d*x]],x]","-\frac{2 (3 \sin (c+d x)-7) (a (\sin (c+d x)+1))^{3/2}}{15 a^2 d}","\frac{4 (a \sin (c+d x)+a)^{3/2}}{3 a^2 d}-\frac{2 (a \sin (c+d x)+a)^{5/2}}{5 a^3 d}",1,"(-2*(a*(1 + Sin[c + d*x]))^(3/2)*(-7 + 3*Sin[c + d*x]))/(15*a^2*d)","A",1
161,1,30,30,0.0682845,"\int \frac{\cos ^2(c+d x)}{\sqrt{a+a \sin (c+d x)}} \, dx","Integrate[Cos[c + d*x]^2/Sqrt[a + a*Sin[c + d*x]],x]","-\frac{2 a \cos ^3(c+d x)}{3 d (a (\sin (c+d x)+1))^{3/2}}","-\frac{2 a \cos ^3(c+d x)}{3 d (a \sin (c+d x)+a)^{3/2}}",1,"(-2*a*Cos[c + d*x]^3)/(3*d*(a*(1 + Sin[c + d*x]))^(3/2))","A",1
162,1,22,22,0.0255377,"\int \frac{\cos (c+d x)}{\sqrt{a+a \sin (c+d x)}} \, dx","Integrate[Cos[c + d*x]/Sqrt[a + a*Sin[c + d*x]],x]","\frac{2 \sqrt{a \sin (c+d x)+a}}{a d}","\frac{2 \sqrt{a \sin (c+d x)+a}}{a d}",1,"(2*Sqrt[a + a*Sin[c + d*x]])/(a*d)","A",1
163,1,39,60,0.0499208,"\int \frac{\sec (c+d x)}{\sqrt{a+a \sin (c+d x)}} \, dx","Integrate[Sec[c + d*x]/Sqrt[a + a*Sin[c + d*x]],x]","-\frac{\, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\frac{1}{2} (\sin (c+d x)+1)\right)}{d \sqrt{a \sin (c+d x)+a}}","\frac{\tanh ^{-1}\left(\frac{\sqrt{a \sin (c+d x)+a}}{\sqrt{2} \sqrt{a}}\right)}{\sqrt{2} \sqrt{a} d}-\frac{1}{d \sqrt{a \sin (c+d x)+a}}",1,"-(Hypergeometric2F1[-1/2, 1, 1/2, (1 + Sin[c + d*x])/2]/(d*Sqrt[a + a*Sin[c + d*x]]))","C",1
164,1,118,102,0.2632474,"\int \frac{\sec ^2(c+d x)}{\sqrt{a+a \sin (c+d x)}} \, dx","Integrate[Sec[c + d*x]^2/Sqrt[a + a*Sin[c + d*x]],x]","-\frac{\sec (c+d x) \left(-3 \sin (c+d x)+(-3-3 i) (-1)^{3/4} \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2 \tanh ^{-1}\left(\left(\frac{1}{2}+\frac{i}{2}\right) (-1)^{3/4} \left(\tan \left(\frac{1}{4} (c+d x)\right)-1\right)\right)-1\right)}{4 d \sqrt{a (\sin (c+d x)+1)}}","-\frac{3 a \cos (c+d x)}{4 d (a \sin (c+d x)+a)^{3/2}}+\frac{\sec (c+d x)}{d \sqrt{a \sin (c+d x)+a}}-\frac{3 \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a \sin (c+d x)+a}}\right)}{4 \sqrt{2} \sqrt{a} d}",1,"-1/4*(Sec[c + d*x]*(-1 - (3 + 3*I)*(-1)^(3/4)*ArcTanh[(1/2 + I/2)*(-1)^(3/4)*(-1 + Tan[(c + d*x)/4])]*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2 - 3*Sin[c + d*x]))/(d*Sqrt[a*(1 + Sin[c + d*x])])","C",1
165,1,42,116,0.0722225,"\int \frac{\sec ^3(c+d x)}{\sqrt{a+a \sin (c+d x)}} \, dx","Integrate[Sec[c + d*x]^3/Sqrt[a + a*Sin[c + d*x]],x]","-\frac{a \, _2F_1\left(-\frac{3}{2},2;-\frac{1}{2};\frac{1}{2} (\sin (c+d x)+1)\right)}{6 d (a \sin (c+d x)+a)^{3/2}}","-\frac{5}{8 d \sqrt{a \sin (c+d x)+a}}-\frac{5 a}{12 d (a \sin (c+d x)+a)^{3/2}}+\frac{5 \tanh ^{-1}\left(\frac{\sqrt{a \sin (c+d x)+a}}{\sqrt{2} \sqrt{a}}\right)}{8 \sqrt{2} \sqrt{a} d}+\frac{\sec ^2(c+d x)}{2 d \sqrt{a \sin (c+d x)+a}}",1,"-1/6*(a*Hypergeometric2F1[-3/2, 2, -1/2, (1 + Sin[c + d*x])/2])/(d*(a + a*Sin[c + d*x])^(3/2))","C",1
166,1,117,162,0.6345739,"\int \frac{\sec ^4(c+d x)}{\sqrt{a+a \sin (c+d x)}} \, dx","Integrate[Sec[c + d*x]^4/Sqrt[a + a*Sin[c + d*x]],x]","\frac{\sec ^3(c+d x) (329 \sin (c+d x)+105 \sin (3 (c+d x))+70 \cos (2 (c+d x))+102)+(420+420 i) (-1)^{3/4} \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right) \tanh ^{-1}\left(\left(\frac{1}{2}+\frac{i}{2}\right) (-1)^{3/4} \left(\tan \left(\frac{1}{4} (c+d x)\right)-1\right)\right)}{768 d \sqrt{a (\sin (c+d x)+1)}}","-\frac{35 a \cos (c+d x)}{64 d (a \sin (c+d x)+a)^{3/2}}+\frac{\sec ^3(c+d x)}{3 d \sqrt{a \sin (c+d x)+a}}+\frac{35 \sec (c+d x)}{48 d \sqrt{a \sin (c+d x)+a}}-\frac{7 a \sec (c+d x)}{24 d (a \sin (c+d x)+a)^{3/2}}-\frac{35 \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a \sin (c+d x)+a}}\right)}{64 \sqrt{2} \sqrt{a} d}",1,"((420 + 420*I)*(-1)^(3/4)*ArcTanh[(1/2 + I/2)*(-1)^(3/4)*(-1 + Tan[(c + d*x)/4])]*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]) + Sec[c + d*x]^3*(102 + 70*Cos[2*(c + d*x)] + 329*Sin[c + d*x] + 105*Sin[3*(c + d*x)]))/(768*d*Sqrt[a*(1 + Sin[c + d*x])])","C",1
167,1,44,175,0.0815387,"\int \frac{\sec ^5(c+d x)}{\sqrt{a+a \sin (c+d x)}} \, dx","Integrate[Sec[c + d*x]^5/Sqrt[a + a*Sin[c + d*x]],x]","-\frac{a^2 \, _2F_1\left(-\frac{5}{2},3;-\frac{3}{2};\frac{1}{2} (\sin (c+d x)+1)\right)}{20 d (a \sin (c+d x)+a)^{5/2}}","-\frac{63}{128 d \sqrt{a \sin (c+d x)+a}}-\frac{21 a}{64 d (a \sin (c+d x)+a)^{3/2}}+\frac{63 \tanh ^{-1}\left(\frac{\sqrt{a \sin (c+d x)+a}}{\sqrt{2} \sqrt{a}}\right)}{128 \sqrt{2} \sqrt{a} d}+\frac{\sec ^4(c+d x)}{4 d \sqrt{a \sin (c+d x)+a}}+\frac{63 \sec ^2(c+d x)}{160 d \sqrt{a \sin (c+d x)+a}}-\frac{9 a \sec ^2(c+d x)}{40 d (a \sin (c+d x)+a)^{3/2}}",1,"-1/20*(a^2*Hypergeometric2F1[-5/2, 3, -3/2, (1 + Sin[c + d*x])/2])/(d*(a + a*Sin[c + d*x])^(5/2))","C",1
168,1,140,221,0.7176337,"\int \frac{\sec ^6(c+d x)}{\sqrt{a+a \sin (c+d x)}} \, dx","Integrate[Sec[c + d*x]^6/Sqrt[a + a*Sin[c + d*x]],x]","\frac{\frac{1}{16} \sec ^5(c+d x) (36850 \sin (c+d x)+17787 \sin (3 (c+d x))+3465 \sin (5 (c+d x))+11352 \cos (2 (c+d x))+2310 \cos (4 (c+d x))+11090)+(3465+3465 i) (-1)^{3/4} \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right) \tanh ^{-1}\left(\left(\frac{1}{2}+\frac{i}{2}\right) (-1)^{3/4} \left(\tan \left(\frac{1}{4} (c+d x)\right)-1\right)\right)}{7680 d \sqrt{a (\sin (c+d x)+1)}}","-\frac{231 a \cos (c+d x)}{512 d (a \sin (c+d x)+a)^{3/2}}+\frac{\sec ^5(c+d x)}{5 d \sqrt{a \sin (c+d x)+a}}+\frac{11 \sec ^3(c+d x)}{40 d \sqrt{a \sin (c+d x)+a}}-\frac{11 a \sec ^3(c+d x)}{60 d (a \sin (c+d x)+a)^{3/2}}+\frac{77 \sec (c+d x)}{128 d \sqrt{a \sin (c+d x)+a}}-\frac{77 a \sec (c+d x)}{320 d (a \sin (c+d x)+a)^{3/2}}-\frac{231 \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a \sin (c+d x)+a}}\right)}{512 \sqrt{2} \sqrt{a} d}",1,"((3465 + 3465*I)*(-1)^(3/4)*ArcTanh[(1/2 + I/2)*(-1)^(3/4)*(-1 + Tan[(c + d*x)/4])]*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]) + (Sec[c + d*x]^5*(11090 + 11352*Cos[2*(c + d*x)] + 2310*Cos[4*(c + d*x)] + 36850*Sin[c + d*x] + 17787*Sin[3*(c + d*x)] + 3465*Sin[5*(c + d*x)]))/16)/(7680*d*Sqrt[a*(1 + Sin[c + d*x])])","C",1
169,1,54,97,0.2389708,"\int \frac{\cos ^7(c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx","Integrate[Cos[c + d*x]^7/(a + a*Sin[c + d*x])^(3/2),x]","-\frac{2 \left(105 \sin ^3(c+d x)-455 \sin ^2(c+d x)+755 \sin (c+d x)-533\right) (a (\sin (c+d x)+1))^{5/2}}{1155 a^4 d}","-\frac{2 (a \sin (c+d x)+a)^{11/2}}{11 a^7 d}+\frac{4 (a \sin (c+d x)+a)^{9/2}}{3 a^6 d}-\frac{24 (a \sin (c+d x)+a)^{7/2}}{7 a^5 d}+\frac{16 (a \sin (c+d x)+a)^{5/2}}{5 a^4 d}",1,"(-2*(a*(1 + Sin[c + d*x]))^(5/2)*(-533 + 755*Sin[c + d*x] - 455*Sin[c + d*x]^2 + 105*Sin[c + d*x]^3))/(1155*a^4*d)","A",1
170,1,49,63,0.198142,"\int \frac{\cos ^6(c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx","Integrate[Cos[c + d*x]^6/(a + a*Sin[c + d*x])^(3/2),x]","-\frac{2 (7 \sin (c+d x)+11) \cos ^7(c+d x)}{63 d (\sin (c+d x)+1)^2 (a (\sin (c+d x)+1))^{3/2}}","-\frac{8 a^2 \cos ^7(c+d x)}{63 d (a \sin (c+d x)+a)^{7/2}}-\frac{2 a \cos ^7(c+d x)}{9 d (a \sin (c+d x)+a)^{5/2}}",1,"(-2*Cos[c + d*x]^7*(11 + 7*Sin[c + d*x]))/(63*d*(1 + Sin[c + d*x])^2*(a*(1 + Sin[c + d*x]))^(3/2))","A",1
171,1,44,73,0.0984457,"\int \frac{\cos ^5(c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx","Integrate[Cos[c + d*x]^5/(a + a*Sin[c + d*x])^(3/2),x]","\frac{2 \left(15 \sin ^2(c+d x)-54 \sin (c+d x)+71\right) (a (\sin (c+d x)+1))^{3/2}}{105 a^3 d}","\frac{2 (a \sin (c+d x)+a)^{7/2}}{7 a^5 d}-\frac{8 (a \sin (c+d x)+a)^{5/2}}{5 a^4 d}+\frac{8 (a \sin (c+d x)+a)^{3/2}}{3 a^3 d}",1,"(2*(a*(1 + Sin[c + d*x]))^(3/2)*(71 - 54*Sin[c + d*x] + 15*Sin[c + d*x]^2))/(105*a^3*d)","A",1
172,1,42,30,0.0613152,"\int \frac{\cos ^4(c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx","Integrate[Cos[c + d*x]^4/(a + a*Sin[c + d*x])^(3/2),x]","-\frac{2 \cos ^5(c+d x) \sqrt{a (\sin (c+d x)+1)}}{5 a^2 d (\sin (c+d x)+1)^3}","-\frac{2 a \cos ^5(c+d x)}{5 d (a \sin (c+d x)+a)^{5/2}}",1,"(-2*Cos[c + d*x]^5*Sqrt[a*(1 + Sin[c + d*x])])/(5*a^2*d*(1 + Sin[c + d*x])^3)","A",1
173,1,32,47,0.053019,"\int \frac{\cos ^3(c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx","Integrate[Cos[c + d*x]^3/(a + a*Sin[c + d*x])^(3/2),x]","-\frac{2 (\sin (c+d x)-5) \sqrt{a (\sin (c+d x)+1)}}{3 a^2 d}","\frac{4 \sqrt{a \sin (c+d x)+a}}{a^2 d}-\frac{2 (a \sin (c+d x)+a)^{3/2}}{3 a^3 d}",1,"(-2*(-5 + Sin[c + d*x])*Sqrt[a*(1 + Sin[c + d*x])])/(3*a^2*d)","A",1
174,1,84,76,0.1496337,"\int \frac{\cos ^2(c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx","Integrate[Cos[c + d*x]^2/(a + a*Sin[c + d*x])^(3/2),x]","\frac{2 \cos ^3(c+d x) \left(\sqrt{1-\sin (c+d x)}-\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{1-\sin (c+d x)}}{\sqrt{2}}\right)\right)}{d (1-\sin (c+d x))^{3/2} (a (\sin (c+d x)+1))^{3/2}}","\frac{2 \cos (c+d x)}{a d \sqrt{a \sin (c+d x)+a}}-\frac{2 \sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a \sin (c+d x)+a}}\right)}{a^{3/2} d}",1,"(2*Cos[c + d*x]^3*(-(Sqrt[2]*ArcTanh[Sqrt[1 - Sin[c + d*x]]/Sqrt[2]]) + Sqrt[1 - Sin[c + d*x]]))/(d*(1 - Sin[c + d*x])^(3/2)*(a*(1 + Sin[c + d*x]))^(3/2))","A",1
175,1,22,22,0.0315674,"\int \frac{\cos (c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx","Integrate[Cos[c + d*x]/(a + a*Sin[c + d*x])^(3/2),x]","-\frac{2}{a d \sqrt{a \sin (c+d x)+a}}","-\frac{2}{a d \sqrt{a \sin (c+d x)+a}}",1,"-2/(a*d*Sqrt[a + a*Sin[c + d*x]])","A",1
176,1,41,89,0.0701615,"\int \frac{\sec (c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx","Integrate[Sec[c + d*x]/(a + a*Sin[c + d*x])^(3/2),x]","-\frac{\, _2F_1\left(-\frac{3}{2},1;-\frac{1}{2};\frac{1}{2} (\sin (c+d x)+1)\right)}{3 d (a \sin (c+d x)+a)^{3/2}}","\frac{\tanh ^{-1}\left(\frac{\sqrt{a \sin (c+d x)+a}}{\sqrt{2} \sqrt{a}}\right)}{2 \sqrt{2} a^{3/2} d}-\frac{1}{2 a d \sqrt{a \sin (c+d x)+a}}-\frac{1}{3 d (a \sin (c+d x)+a)^{3/2}}",1,"-1/3*Hypergeometric2F1[-3/2, 1, -1/2, (1 + Sin[c + d*x])/2]/(d*(a + a*Sin[c + d*x])^(3/2))","C",1
177,1,224,134,0.315457,"\int \frac{\sec ^2(c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx","Integrate[Sec[c + d*x]^2/(a + a*Sin[c + d*x])^(3/2),x]","\frac{\frac{8 \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^3}{\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)}-7 \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2+14 \sin \left(\frac{1}{2} (c+d x)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+\frac{8 \sin \left(\frac{1}{2} (c+d x)\right)}{\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)}+(15+15 i) (-1)^{3/4} \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^3 \tanh ^{-1}\left(\left(\frac{1}{2}+\frac{i}{2}\right) (-1)^{3/4} \left(\tan \left(\frac{1}{4} (c+d x)\right)-1\right)\right)-4}{32 d (a (\sin (c+d x)+1))^{3/2}}","-\frac{15 \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a \sin (c+d x)+a}}\right)}{32 \sqrt{2} a^{3/2} d}-\frac{15 \cos (c+d x)}{32 d (a \sin (c+d x)+a)^{3/2}}+\frac{5 \sec (c+d x)}{8 a d \sqrt{a \sin (c+d x)+a}}-\frac{\sec (c+d x)}{4 d (a \sin (c+d x)+a)^{3/2}}",1,"(-4 + (8*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]) + 14*Sin[(c + d*x)/2]*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]) - 7*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2 + (15 + 15*I)*(-1)^(3/4)*ArcTanh[(1/2 + I/2)*(-1)^(3/4)*(-1 + Tan[(c + d*x)/4])]*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3 + (8*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3)/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2]))/(32*d*(a*(1 + Sin[c + d*x]))^(3/2))","C",1
178,1,42,150,0.0771286,"\int \frac{\sec ^3(c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx","Integrate[Sec[c + d*x]^3/(a + a*Sin[c + d*x])^(3/2),x]","-\frac{a \, _2F_1\left(-\frac{5}{2},2;-\frac{3}{2};\frac{1}{2} (\sin (c+d x)+1)\right)}{10 d (a \sin (c+d x)+a)^{5/2}}","\frac{7 \tanh ^{-1}\left(\frac{\sqrt{a \sin (c+d x)+a}}{\sqrt{2} \sqrt{a}}\right)}{16 \sqrt{2} a^{3/2} d}-\frac{7}{16 a d \sqrt{a \sin (c+d x)+a}}-\frac{7}{24 d (a \sin (c+d x)+a)^{3/2}}+\frac{7 \sec ^2(c+d x)}{20 a d \sqrt{a \sin (c+d x)+a}}-\frac{\sec ^2(c+d x)}{5 d (a \sin (c+d x)+a)^{3/2}}",1,"-1/10*(a*Hypergeometric2F1[-5/2, 2, -3/2, (1 + Sin[c + d*x])/2])/(d*(a + a*Sin[c + d*x])^(5/2))","C",1
179,1,334,195,0.3464312,"\int \frac{\sec ^4(c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx","Integrate[Sec[c + d*x]^4/(a + a*Sin[c + d*x])^(3/2),x]","\frac{\frac{192 \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^3}{\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)}+\frac{32 \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^3}{\left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^3}-123 \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2+246 \sin \left(\frac{1}{2} (c+d x)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+\frac{136 \sin \left(\frac{1}{2} (c+d x)\right)}{\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)}-\frac{32}{\left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2}+\frac{64 \sin \left(\frac{1}{2} (c+d x)\right)}{\left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^3}+(315+315 i) (-1)^{3/4} \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^3 \tanh ^{-1}\left(\left(\frac{1}{2}+\frac{i}{2}\right) (-1)^{3/4} \left(\tan \left(\frac{1}{4} (c+d x)\right)-1\right)\right)-68}{768 d (a (\sin (c+d x)+1))^{3/2}}","-\frac{105 \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a \sin (c+d x)+a}}\right)}{256 \sqrt{2} a^{3/2} d}-\frac{105 \cos (c+d x)}{256 d (a \sin (c+d x)+a)^{3/2}}+\frac{\sec ^3(c+d x)}{4 a d \sqrt{a \sin (c+d x)+a}}-\frac{\sec ^3(c+d x)}{6 d (a \sin (c+d x)+a)^{3/2}}+\frac{35 \sec (c+d x)}{64 a d \sqrt{a \sin (c+d x)+a}}-\frac{7 \sec (c+d x)}{32 d (a \sin (c+d x)+a)^{3/2}}",1,"(-68 + (64*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3 - 32/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2 + (136*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]) + 246*Sin[(c + d*x)/2]*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]) - 123*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2 + (315 + 315*I)*(-1)^(3/4)*ArcTanh[(1/2 + I/2)*(-1)^(3/4)*(-1 + Tan[(c + d*x)/4])]*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3 + (32*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3)/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^3 + (192*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3)/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2]))/(768*d*(a*(1 + Sin[c + d*x]))^(3/2))","C",1
180,1,44,211,0.0916956,"\int \frac{\sec ^5(c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx","Integrate[Sec[c + d*x]^5/(a + a*Sin[c + d*x])^(3/2),x]","-\frac{a^2 \, _2F_1\left(-\frac{7}{2},3;-\frac{5}{2};\frac{1}{2} (\sin (c+d x)+1)\right)}{28 d (a \sin (c+d x)+a)^{7/2}}","\frac{99 \tanh ^{-1}\left(\frac{\sqrt{a \sin (c+d x)+a}}{\sqrt{2} \sqrt{a}}\right)}{256 \sqrt{2} a^{3/2} d}-\frac{99}{256 a d \sqrt{a \sin (c+d x)+a}}-\frac{33}{128 d (a \sin (c+d x)+a)^{3/2}}+\frac{11 \sec ^4(c+d x)}{56 a d \sqrt{a \sin (c+d x)+a}}-\frac{\sec ^4(c+d x)}{7 d (a \sin (c+d x)+a)^{3/2}}+\frac{99 \sec ^2(c+d x)}{320 a d \sqrt{a \sin (c+d x)+a}}-\frac{99 \sec ^2(c+d x)}{560 d (a \sin (c+d x)+a)^{3/2}}",1,"-1/28*(a^2*Hypergeometric2F1[-7/2, 3, -5/2, (1 + Sin[c + d*x])/2])/(d*(a + a*Sin[c + d*x])^(7/2))","C",1
181,1,444,256,1.5085631,"\int \frac{\sec ^6(c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx","Integrate[Sec[c + d*x]^6/(a + a*Sin[c + d*x])^(3/2),x]","\frac{\frac{28800 \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^3}{\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)}+\frac{6400 \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^3}{\left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^3}+\frac{1536 \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^3}{\left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^5}-16245 \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2+32490 \sin \left(\frac{1}{2} (c+d x)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+\frac{17720 \sin \left(\frac{1}{2} (c+d x)\right)}{\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)}-\frac{4960}{\left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2}+\frac{9920 \sin \left(\frac{1}{2} (c+d x)\right)}{\left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^3}-\frac{1920}{\left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^4}+\frac{3840 \sin \left(\frac{1}{2} (c+d x)\right)}{\left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^5}+(45045+45045 i) (-1)^{3/4} \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^3 \tanh ^{-1}\left(\left(\frac{1}{2}+\frac{i}{2}\right) (-1)^{3/4} \left(\tan \left(\frac{1}{4} (c+d x)\right)-1\right)\right)-8860}{122880 d (a (\sin (c+d x)+1))^{3/2}}","-\frac{3003 \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a \sin (c+d x)+a}}\right)}{8192 \sqrt{2} a^{3/2} d}-\frac{3003 \cos (c+d x)}{8192 d (a \sin (c+d x)+a)^{3/2}}+\frac{13 \sec ^5(c+d x)}{80 a d \sqrt{a \sin (c+d x)+a}}-\frac{\sec ^5(c+d x)}{8 d (a \sin (c+d x)+a)^{3/2}}+\frac{143 \sec ^3(c+d x)}{640 a d \sqrt{a \sin (c+d x)+a}}-\frac{143 \sec ^3(c+d x)}{960 d (a \sin (c+d x)+a)^{3/2}}+\frac{1001 \sec (c+d x)}{2048 a d \sqrt{a \sin (c+d x)+a}}-\frac{1001 \sec (c+d x)}{5120 d (a \sin (c+d x)+a)^{3/2}}",1,"(-8860 + (3840*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^5 - 1920/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^4 + (9920*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3 - 4960/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2 + (17720*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]) + 32490*Sin[(c + d*x)/2]*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]) - 16245*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2 + (45045 + 45045*I)*(-1)^(3/4)*ArcTanh[(1/2 + I/2)*(-1)^(3/4)*(-1 + Tan[(c + d*x)/4])]*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3 + (1536*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3)/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^5 + (6400*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3)/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^3 + (28800*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3)/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2]))/(122880*d*(a*(1 + Sin[c + d*x]))^(3/2))","C",1
182,1,59,95,0.7697122,"\int \frac{\cos ^{10}(c+d x)}{(a+a \sin (c+d x))^{5/2}} \, dx","Integrate[Cos[c + d*x]^10/(a + a*Sin[c + d*x])^(5/2),x]","-\frac{2 \left(143 \sin ^2(c+d x)+374 \sin (c+d x)+263\right) \cos ^{11}(c+d x)}{2145 d (\sin (c+d x)+1)^3 (a (\sin (c+d x)+1))^{5/2}}","-\frac{64 a^3 \cos ^{11}(c+d x)}{2145 d (a \sin (c+d x)+a)^{11/2}}-\frac{16 a^2 \cos ^{11}(c+d x)}{195 d (a \sin (c+d x)+a)^{9/2}}-\frac{2 a \cos ^{11}(c+d x)}{15 d (a \sin (c+d x)+a)^{7/2}}",1,"(-2*Cos[c + d*x]^11*(263 + 374*Sin[c + d*x] + 143*Sin[c + d*x]^2))/(2145*d*(1 + Sin[c + d*x])^3*(a*(1 + Sin[c + d*x]))^(5/2))","A",1
183,1,64,121,0.2901167,"\int \frac{\cos ^9(c+d x)}{(a+a \sin (c+d x))^{5/2}} \, dx","Integrate[Cos[c + d*x]^9/(a + a*Sin[c + d*x])^(5/2),x]","\frac{2 \left(1155 \sin ^4(c+d x)-6300 \sin ^3(c+d x)+14210 \sin ^2(c+d x)-16700 \sin (c+d x)+9683\right) (a (\sin (c+d x)+1))^{5/2}}{15015 a^5 d}","\frac{2 (a \sin (c+d x)+a)^{13/2}}{13 a^9 d}-\frac{16 (a \sin (c+d x)+a)^{11/2}}{11 a^8 d}+\frac{16 (a \sin (c+d x)+a)^{9/2}}{3 a^7 d}-\frac{64 (a \sin (c+d x)+a)^{7/2}}{7 a^6 d}+\frac{32 (a \sin (c+d x)+a)^{5/2}}{5 a^5 d}",1,"(2*(a*(1 + Sin[c + d*x]))^(5/2)*(9683 - 16700*Sin[c + d*x] + 14210*Sin[c + d*x]^2 - 6300*Sin[c + d*x]^3 + 1155*Sin[c + d*x]^4))/(15015*a^5*d)","A",1
184,1,49,63,0.3536027,"\int \frac{\cos ^8(c+d x)}{(a+a \sin (c+d x))^{5/2}} \, dx","Integrate[Cos[c + d*x]^8/(a + a*Sin[c + d*x])^(5/2),x]","-\frac{2 (9 \sin (c+d x)+13) \cos ^9(c+d x)}{99 d (\sin (c+d x)+1)^2 (a (\sin (c+d x)+1))^{5/2}}","-\frac{8 a^2 \cos ^9(c+d x)}{99 d (a \sin (c+d x)+a)^{9/2}}-\frac{2 a \cos ^9(c+d x)}{11 d (a \sin (c+d x)+a)^{7/2}}",1,"(-2*Cos[c + d*x]^9*(13 + 9*Sin[c + d*x]))/(99*d*(1 + Sin[c + d*x])^2*(a*(1 + Sin[c + d*x]))^(5/2))","A",1
185,1,54,97,0.1884725,"\int \frac{\cos ^7(c+d x)}{(a+a \sin (c+d x))^{5/2}} \, dx","Integrate[Cos[c + d*x]^7/(a + a*Sin[c + d*x])^(5/2),x]","-\frac{2 \left(35 \sin ^3(c+d x)-165 \sin ^2(c+d x)+321 \sin (c+d x)-319\right) (a (\sin (c+d x)+1))^{3/2}}{315 a^4 d}","-\frac{2 (a \sin (c+d x)+a)^{9/2}}{9 a^7 d}+\frac{12 (a \sin (c+d x)+a)^{7/2}}{7 a^6 d}-\frac{24 (a \sin (c+d x)+a)^{5/2}}{5 a^5 d}+\frac{16 (a \sin (c+d x)+a)^{3/2}}{3 a^4 d}",1,"(-2*(a*(1 + Sin[c + d*x]))^(3/2)*(-319 + 321*Sin[c + d*x] - 165*Sin[c + d*x]^2 + 35*Sin[c + d*x]^3))/(315*a^4*d)","A",1
186,1,42,30,0.1093552,"\int \frac{\cos ^6(c+d x)}{(a+a \sin (c+d x))^{5/2}} \, dx","Integrate[Cos[c + d*x]^6/(a + a*Sin[c + d*x])^(5/2),x]","-\frac{2 \cos ^7(c+d x) \sqrt{a (\sin (c+d x)+1)}}{7 a^3 d (\sin (c+d x)+1)^4}","-\frac{2 a \cos ^7(c+d x)}{7 d (a \sin (c+d x)+a)^{7/2}}",1,"(-2*Cos[c + d*x]^7*Sqrt[a*(1 + Sin[c + d*x])])/(7*a^3*d*(1 + Sin[c + d*x])^4)","A",1
187,1,44,71,0.0721654,"\int \frac{\cos ^5(c+d x)}{(a+a \sin (c+d x))^{5/2}} \, dx","Integrate[Cos[c + d*x]^5/(a + a*Sin[c + d*x])^(5/2),x]","\frac{2 \left(3 \sin ^2(c+d x)-14 \sin (c+d x)+43\right) \sqrt{a (\sin (c+d x)+1)}}{15 a^3 d}","\frac{2 (a \sin (c+d x)+a)^{5/2}}{5 a^5 d}-\frac{8 (a \sin (c+d x)+a)^{3/2}}{3 a^4 d}+\frac{8 \sqrt{a \sin (c+d x)+a}}{a^3 d}",1,"(2*Sqrt[a*(1 + Sin[c + d*x])]*(43 - 14*Sin[c + d*x] + 3*Sin[c + d*x]^2))/(15*a^3*d)","A",1
188,1,96,108,0.1842739,"\int \frac{\cos ^4(c+d x)}{(a+a \sin (c+d x))^{5/2}} \, dx","Integrate[Cos[c + d*x]^4/(a + a*Sin[c + d*x])^(5/2),x]","-\frac{2 \cos (c+d x) \left(\sqrt{1-\sin (c+d x)} (\sin (c+d x)-7)+6 \sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{1-\sin (c+d x)}}{\sqrt{2}}\right)\right)}{3 a^2 d \sqrt{1-\sin (c+d x)} \sqrt{a (\sin (c+d x)+1)}}","-\frac{4 \sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a \sin (c+d x)+a}}\right)}{a^{5/2} d}+\frac{4 \cos (c+d x)}{a^2 d \sqrt{a \sin (c+d x)+a}}+\frac{2 \cos ^3(c+d x)}{3 a d (a \sin (c+d x)+a)^{3/2}}",1,"(-2*Cos[c + d*x]*(6*Sqrt[2]*ArcTanh[Sqrt[1 - Sin[c + d*x]]/Sqrt[2]] + Sqrt[1 - Sin[c + d*x]]*(-7 + Sin[c + d*x])))/(3*a^2*d*Sqrt[1 - Sin[c + d*x]]*Sqrt[a*(1 + Sin[c + d*x])])","A",1
189,1,30,45,0.0613832,"\int \frac{\cos ^3(c+d x)}{(a+a \sin (c+d x))^{5/2}} \, dx","Integrate[Cos[c + d*x]^3/(a + a*Sin[c + d*x])^(5/2),x]","-\frac{2 (\sin (c+d x)+3)}{a^2 d \sqrt{a (\sin (c+d x)+1)}}","-\frac{2 \sqrt{a \sin (c+d x)+a}}{a^3 d}-\frac{4}{a^2 d \sqrt{a \sin (c+d x)+a}}",1,"(-2*(3 + Sin[c + d*x]))/(a^2*d*Sqrt[a*(1 + Sin[c + d*x])])","A",1
190,1,100,75,0.2458987,"\int \frac{\cos ^2(c+d x)}{(a+a \sin (c+d x))^{5/2}} \, dx","Integrate[Cos[c + d*x]^2/(a + a*Sin[c + d*x])^(5/2),x]","\frac{\sec (c+d x) \left(2 (\sin (c+d x)-1)+\sqrt{2-2 \sin (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{1-\sin (c+d x)}}{\sqrt{2}}\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2\right)}{2 a^2 d \sqrt{a (\sin (c+d x)+1)}}","\frac{\tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a \sin (c+d x)+a}}\right)}{\sqrt{2} a^{5/2} d}-\frac{\cos (c+d x)}{a d (a \sin (c+d x)+a)^{3/2}}",1,"(Sec[c + d*x]*(ArcTanh[Sqrt[1 - Sin[c + d*x]]/Sqrt[2]]*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2*Sqrt[2 - 2*Sin[c + d*x]] + 2*(-1 + Sin[c + d*x])))/(2*a^2*d*Sqrt[a*(1 + Sin[c + d*x])])","A",1
191,1,24,24,0.0474063,"\int \frac{\cos (c+d x)}{(a+a \sin (c+d x))^{5/2}} \, dx","Integrate[Cos[c + d*x]/(a + a*Sin[c + d*x])^(5/2),x]","-\frac{2}{3 a d (a \sin (c+d x)+a)^{3/2}}","-\frac{2}{3 a d (a \sin (c+d x)+a)^{3/2}}",1,"-2/(3*a*d*(a + a*Sin[c + d*x])^(3/2))","A",1
192,1,41,113,0.0900666,"\int \frac{\sec (c+d x)}{(a+a \sin (c+d x))^{5/2}} \, dx","Integrate[Sec[c + d*x]/(a + a*Sin[c + d*x])^(5/2),x]","-\frac{\, _2F_1\left(-\frac{5}{2},1;-\frac{3}{2};\frac{1}{2} (\sin (c+d x)+1)\right)}{5 d (a \sin (c+d x)+a)^{5/2}}","\frac{\tanh ^{-1}\left(\frac{\sqrt{a \sin (c+d x)+a}}{\sqrt{2} \sqrt{a}}\right)}{4 \sqrt{2} a^{5/2} d}-\frac{1}{4 a^2 d \sqrt{a \sin (c+d x)+a}}-\frac{1}{6 a d (a \sin (c+d x)+a)^{3/2}}-\frac{1}{5 d (a \sin (c+d x)+a)^{5/2}}",1,"-1/5*Hypergeometric2F1[-5/2, 1, -3/2, (1 + Sin[c + d*x])/2]/(d*(a + a*Sin[c + d*x])^(5/2))","C",1
193,1,284,167,0.4573941,"\int \frac{\sec ^2(c+d x)}{(a+a \sin (c+d x))^{5/2}} \, dx","Integrate[Sec[c + d*x]^2/(a + a*Sin[c + d*x])^(5/2),x]","\frac{\frac{48 \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^5}{\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)}-57 \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^4+114 \sin \left(\frac{1}{2} (c+d x)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^3-44 \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2+88 \sin \left(\frac{1}{2} (c+d x)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+\frac{64 \sin \left(\frac{1}{2} (c+d x)\right)}{\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)}+(105+105 i) (-1)^{3/4} \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^5 \tanh ^{-1}\left(\left(\frac{1}{2}+\frac{i}{2}\right) (-1)^{3/4} \left(\tan \left(\frac{1}{4} (c+d x)\right)-1\right)\right)-32}{384 d (a (\sin (c+d x)+1))^{5/2}}","-\frac{35 \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a \sin (c+d x)+a}}\right)}{128 \sqrt{2} a^{5/2} d}+\frac{35 \sec (c+d x)}{96 a^2 d \sqrt{a \sin (c+d x)+a}}-\frac{35 \cos (c+d x)}{128 a d (a \sin (c+d x)+a)^{3/2}}-\frac{7 \sec (c+d x)}{48 a d (a \sin (c+d x)+a)^{3/2}}-\frac{\sec (c+d x)}{6 d (a \sin (c+d x)+a)^{5/2}}",1,"(-32 + (64*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]) + 88*Sin[(c + d*x)/2]*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]) - 44*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2 + 114*Sin[(c + d*x)/2]*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3 - 57*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^4 + (105 + 105*I)*(-1)^(3/4)*ArcTanh[(1/2 + I/2)*(-1)^(3/4)*(-1 + Tan[(c + d*x)/4])]*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^5 + (48*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^5)/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2]))/(384*d*(a*(1 + Sin[c + d*x]))^(5/2))","C",1
194,1,42,185,0.0931822,"\int \frac{\sec ^3(c+d x)}{(a+a \sin (c+d x))^{5/2}} \, dx","Integrate[Sec[c + d*x]^3/(a + a*Sin[c + d*x])^(5/2),x]","-\frac{a \, _2F_1\left(-\frac{7}{2},2;-\frac{5}{2};\frac{1}{2} (\sin (c+d x)+1)\right)}{14 d (a \sin (c+d x)+a)^{7/2}}","\frac{9 \tanh ^{-1}\left(\frac{\sqrt{a \sin (c+d x)+a}}{\sqrt{2} \sqrt{a}}\right)}{32 \sqrt{2} a^{5/2} d}-\frac{9}{32 a^2 d \sqrt{a \sin (c+d x)+a}}+\frac{9 \sec ^2(c+d x)}{40 a^2 d \sqrt{a \sin (c+d x)+a}}-\frac{3}{16 a d (a \sin (c+d x)+a)^{3/2}}-\frac{9 \sec ^2(c+d x)}{70 a d (a \sin (c+d x)+a)^{3/2}}-\frac{\sec ^2(c+d x)}{7 d (a \sin (c+d x)+a)^{5/2}}",1,"-1/14*(a*Hypergeometric2F1[-7/2, 2, -5/2, (1 + Sin[c + d*x])/2])/(d*(a + a*Sin[c + d*x])^(7/2))","C",1
195,1,394,233,0.5516764,"\int \frac{\sec ^4(c+d x)}{(a+a \sin (c+d x))^{5/2}} \, dx","Integrate[Sec[c + d*x]^4/(a + a*Sin[c + d*x])^(5/2),x]","\frac{\frac{1920 \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^5}{\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)}+\frac{256 \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^5}{\left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^3}-1545 \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^4+3090 \sin \left(\frac{1}{2} (c+d x)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^3-1036 \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2+2072 \sin \left(\frac{1}{2} (c+d x)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+\frac{1472 \sin \left(\frac{1}{2} (c+d x)\right)}{\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)}-\frac{384}{\left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2}+\frac{768 \sin \left(\frac{1}{2} (c+d x)\right)}{\left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^3}+(3465+3465 i) (-1)^{3/4} \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^5 \tanh ^{-1}\left(\left(\frac{1}{2}+\frac{i}{2}\right) (-1)^{3/4} \left(\tan \left(\frac{1}{4} (c+d x)\right)-1\right)\right)-736}{12288 d (a (\sin (c+d x)+1))^{5/2}}","-\frac{1155 \tanh ^{-1}\left(\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a \sin (c+d x)+a}}\right)}{4096 \sqrt{2} a^{5/2} d}+\frac{11 \sec ^3(c+d x)}{64 a^2 d \sqrt{a \sin (c+d x)+a}}+\frac{385 \sec (c+d x)}{1024 a^2 d \sqrt{a \sin (c+d x)+a}}-\frac{1155 \cos (c+d x)}{4096 a d (a \sin (c+d x)+a)^{3/2}}-\frac{11 \sec ^3(c+d x)}{96 a d (a \sin (c+d x)+a)^{3/2}}-\frac{\sec ^3(c+d x)}{8 d (a \sin (c+d x)+a)^{5/2}}-\frac{77 \sec (c+d x)}{512 a d (a \sin (c+d x)+a)^{3/2}}",1,"(-736 + (768*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3 - 384/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2 + (1472*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]) + 2072*Sin[(c + d*x)/2]*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]) - 1036*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2 + 3090*Sin[(c + d*x)/2]*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3 - 1545*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^4 + (3465 + 3465*I)*(-1)^(3/4)*ArcTanh[(1/2 + I/2)*(-1)^(3/4)*(-1 + Tan[(c + d*x)/4])]*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^5 + (256*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^5)/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^3 + (1920*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^5)/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2]))/(12288*d*(a*(1 + Sin[c + d*x]))^(5/2))","C",1
196,1,98,124,0.6299734,"\int (e \cos (c+d x))^{7/2} (a+a \sin (c+d x)) \, dx","Integrate[(e*Cos[c + d*x])^(7/2)*(a + a*Sin[c + d*x]),x]","-\frac{a e^3 \sqrt{e \cos (c+d x)} \left(\sqrt{\cos (c+d x)} (-138 \sin (c+d x)-18 \sin (3 (c+d x))+28 \cos (2 (c+d x))+7 \cos (4 (c+d x))+21)-120 F\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{252 d \sqrt{\cos (c+d x)}}","\frac{10 a e^4 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d \sqrt{e \cos (c+d x)}}+\frac{10 a e^3 \sin (c+d x) \sqrt{e \cos (c+d x)}}{21 d}-\frac{2 a (e \cos (c+d x))^{9/2}}{9 d e}+\frac{2 a e \sin (c+d x) (e \cos (c+d x))^{5/2}}{7 d}",1,"-1/252*(a*e^3*Sqrt[e*Cos[c + d*x]]*(-120*EllipticF[(c + d*x)/2, 2] + Sqrt[Cos[c + d*x]]*(21 + 28*Cos[2*(c + d*x)] + 7*Cos[4*(c + d*x)] - 138*Sin[c + d*x] - 18*Sin[3*(c + d*x)])))/(d*Sqrt[Cos[c + d*x]])","A",1
197,1,264,95,2.838284,"\int (e \cos (c+d x))^{5/2} (a+a \sin (c+d x)) \, dx","Integrate[(e*Cos[c + d*x])^(5/2)*(a + a*Sin[c + d*x]),x]","\frac{a e^3 \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \left(168 (\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)+56 (\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)+20 \sin (c+2 d x)-20 \sin (3 c+2 d x)+5 \sin (3 c+4 d x)-5 \sin (5 c+4 d x)-182 \cos (2 c+d x)+14 \cos (2 c+3 d x)-14 \cos (4 c+3 d x)-30 \sin (c)-154 \cos (d x)\right)}{560 d \sqrt{e \cos (c+d x)}}","\frac{6 a e^2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{5 d \sqrt{\cos (c+d x)}}-\frac{2 a (e \cos (c+d x))^{7/2}}{7 d e}+\frac{2 a e \sin (c+d x) (e \cos (c+d x))^{3/2}}{5 d}",1,"(a*e^3*Csc[c/2]*Sec[c/2]*(-154*Cos[d*x] - 182*Cos[2*c + d*x] + 14*Cos[2*c + 3*d*x] - 14*Cos[4*c + 3*d*x] - 30*Sin[c] + 168*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)]] + 56*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)]] + 20*Sin[c + 2*d*x] - 20*Sin[3*c + 2*d*x] + 5*Sin[3*c + 4*d*x] - 5*Sin[5*c + 4*d*x]))/(560*d*Sqrt[e*Cos[c + d*x]])","C",1
198,1,75,95,0.3769991,"\int (e \cos (c+d x))^{3/2} (a+a \sin (c+d x)) \, dx","Integrate[(e*Cos[c + d*x])^(3/2)*(a + a*Sin[c + d*x]),x]","-\frac{a (e \cos (c+d x))^{3/2} \left(\sqrt{\cos (c+d x)} (-10 \sin (c+d x)+3 \cos (2 (c+d x))+3)-10 F\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{15 d \cos ^{\frac{3}{2}}(c+d x)}","\frac{2 a e^2 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d \sqrt{e \cos (c+d x)}}-\frac{2 a (e \cos (c+d x))^{5/2}}{5 d e}+\frac{2 a e \sin (c+d x) \sqrt{e \cos (c+d x)}}{3 d}",1,"-1/15*(a*(e*Cos[c + d*x])^(3/2)*(-10*EllipticF[(c + d*x)/2, 2] + Sqrt[Cos[c + d*x]]*(3 + 3*Cos[2*(c + d*x)] - 10*Sin[c + d*x])))/(d*Cos[c + d*x]^(3/2))","A",1
199,1,260,63,1.0613468,"\int \sqrt{e \cos (c+d x)} (a+a \sin (c+d x)) \, dx","Integrate[Sqrt[e*Cos[c + d*x]]*(a + a*Sin[c + d*x]),x]","\frac{a \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) (\cos (d x)+i \sin (d x)) \sqrt{e \cos (c+d x)} \left(6 (\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)+2 (\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)+\sin (c+2 d x)-\sin (3 c+2 d x)-6 \cos (2 c+d x)-2 \sin (c)-6 \cos (d x)\right)}{6 d \left(i \sin (c) \left(-1+e^{2 i d x}\right)+\cos (c) \left(1+e^{2 i d x}\right)\right)}","\frac{2 a E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{d \sqrt{\cos (c+d x)}}-\frac{2 a (e \cos (c+d x))^{3/2}}{3 d e}",1,"(a*Sqrt[e*Cos[c + d*x]]*Csc[c/2]*Sec[c/2]*(Cos[d*x] + I*Sin[d*x])*(-6*Cos[d*x] - 6*Cos[2*c + d*x] - 2*Sin[c] + 6*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)]] + 2*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)]] + Sin[c + 2*d*x] - Sin[3*c + 2*d*x]))/(6*d*((1 + E^((2*I)*d*x))*Cos[c] + I*(-1 + E^((2*I)*d*x))*Sin[c]))","C",1
200,1,48,61,22.3163846,"\int \frac{a+a \sin (c+d x)}{\sqrt{e \cos (c+d x)}} \, dx","Integrate[(a + a*Sin[c + d*x])/Sqrt[e*Cos[c + d*x]],x]","-\frac{2 a \left(\cos (c+d x)-\sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{d \sqrt{e \cos (c+d x)}}","\frac{2 a \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d \sqrt{e \cos (c+d x)}}-\frac{2 a \sqrt{e \cos (c+d x)}}{d e}",1,"(-2*a*(Cos[c + d*x] - Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]))/(d*Sqrt[e*Cos[c + d*x]])","A",1
201,1,188,91,0.9792774,"\int \frac{a+a \sin (c+d x)}{(e \cos (c+d x))^{3/2}} \, dx","Integrate[(a + a*Sin[c + d*x])/(e*Cos[c + d*x])^(3/2),x]","-\frac{a \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \left(3 (\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)+(\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 (\sin (c)+\cos (d x))\right)}{6 d e \sqrt{e \cos (c+d x)}}","-\frac{2 a E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{d e^2 \sqrt{\cos (c+d x)}}+\frac{2 a}{d e \sqrt{e \cos (c+d x)}}+\frac{2 a \sin (c+d x)}{d e \sqrt{e \cos (c+d x)}}",1,"-1/6*(a*Csc[c/2]*Sec[c/2]*(-6*(Cos[d*x] + Sin[c]) + 3*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)]] + 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*e*Sqrt[e*Cos[c + d*x]])","C",1
202,1,86,97,0.4608196,"\int \frac{a+a \sin (c+d x)}{(e \cos (c+d x))^{5/2}} \, dx","Integrate[(a + a*Sin[c + d*x])/(e*Cos[c + d*x])^(5/2),x]","\frac{2 a \left(\cos (c+d x)-(\sin (c+d x)-1) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{3 d e^2 \sqrt{e \cos (c+d x)} \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^2}","\frac{2 a \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d e^2 \sqrt{e \cos (c+d x)}}+\frac{2 a}{3 d e (e \cos (c+d x))^{3/2}}+\frac{2 a \sin (c+d x)}{3 d e (e \cos (c+d x))^{3/2}}",1,"(2*a*(Cos[c + d*x] - Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*(-1 + Sin[c + d*x])))/(3*d*e^2*Sqrt[e*Cos[c + d*x]]*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2)","A",1
203,1,144,126,1.2745863,"\int \frac{a+a \sin (c+d x)}{(e \cos (c+d x))^{7/2}} \, dx","Integrate[(a + a*Sin[c + d*x])/(e*Cos[c + d*x])^(7/2),x]","\frac{2 a e^{i (c+d x)} \left(i \sqrt{1+e^{2 i (c+d x)}} \left(e^{i (c+d x)}-i\right)^2 \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-6 e^{i (c+d x)}-3 i e^{2 i (c+d x)}+i\right)}{5 d e^3 \left(e^{i (c+d x)}-i\right)^2 \sqrt{e \cos (c+d x)}}","-\frac{6 a E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{5 d e^4 \sqrt{\cos (c+d x)}}+\frac{6 a \sin (c+d x)}{5 d e^3 \sqrt{e \cos (c+d x)}}+\frac{2 a}{5 d e (e \cos (c+d x))^{5/2}}+\frac{2 a \sin (c+d x)}{5 d e (e \cos (c+d x))^{5/2}}",1,"(2*a*E^(I*(c + d*x))*(I - 6*E^(I*(c + d*x)) - (3*I)*E^((2*I)*(c + d*x)) + I*(-I + E^(I*(c + d*x)))^2*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))]))/(5*d*e^3*(-I + E^(I*(c + d*x)))^2*Sqrt[e*Cos[c + d*x]])","C",1
204,1,66,168,0.109177,"\int (e \cos (c+d x))^{7/2} (a+a \sin (c+d x))^2 \, dx","Integrate[(e*Cos[c + d*x])^(7/2)*(a + a*Sin[c + d*x])^2,x]","-\frac{32 \sqrt[4]{2} a^2 (e \cos (c+d x))^{9/2} \, _2F_1\left(-\frac{13}{4},\frac{9}{4};\frac{13}{4};\frac{1}{2} (1-\sin (c+d x))\right)}{9 d e (\sin (c+d x)+1)^{9/4}}","\frac{130 a^2 e^4 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{231 d \sqrt{e \cos (c+d x)}}+\frac{130 a^2 e^3 \sin (c+d x) \sqrt{e \cos (c+d x)}}{231 d}-\frac{26 a^2 (e \cos (c+d x))^{9/2}}{99 d e}-\frac{2 \left(a^2 \sin (c+d x)+a^2\right) (e \cos (c+d x))^{9/2}}{11 d e}+\frac{26 a^2 e \sin (c+d x) (e \cos (c+d x))^{5/2}}{77 d}",1,"(-32*2^(1/4)*a^2*(e*Cos[c + d*x])^(9/2)*Hypergeometric2F1[-13/4, 9/4, 13/4, (1 - Sin[c + d*x])/2])/(9*d*e*(1 + Sin[c + d*x])^(9/4))","C",1
205,1,66,137,0.1179215,"\int (e \cos (c+d x))^{5/2} (a+a \sin (c+d x))^2 \, dx","Integrate[(e*Cos[c + d*x])^(5/2)*(a + a*Sin[c + d*x])^2,x]","-\frac{16\ 2^{3/4} a^2 (e \cos (c+d x))^{7/2} \, _2F_1\left(-\frac{11}{4},\frac{7}{4};\frac{11}{4};\frac{1}{2} (1-\sin (c+d x))\right)}{7 d e (\sin (c+d x)+1)^{7/4}}","\frac{22 a^2 e^2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{15 d \sqrt{\cos (c+d x)}}-\frac{22 a^2 (e \cos (c+d x))^{7/2}}{63 d e}-\frac{2 \left(a^2 \sin (c+d x)+a^2\right) (e \cos (c+d x))^{7/2}}{9 d e}+\frac{22 a^2 e \sin (c+d x) (e \cos (c+d x))^{3/2}}{45 d}",1,"(-16*2^(3/4)*a^2*(e*Cos[c + d*x])^(7/2)*Hypergeometric2F1[-11/4, 7/4, 11/4, (1 - Sin[c + d*x])/2])/(7*d*e*(1 + Sin[c + d*x])^(7/4))","C",1
206,1,66,137,0.0786957,"\int (e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^2 \, dx","Integrate[(e*Cos[c + d*x])^(3/2)*(a + a*Sin[c + d*x])^2,x]","-\frac{16 \sqrt[4]{2} a^2 (e \cos (c+d x))^{5/2} \, _2F_1\left(-\frac{9}{4},\frac{5}{4};\frac{9}{4};\frac{1}{2} (1-\sin (c+d x))\right)}{5 d e (\sin (c+d x)+1)^{5/4}}","\frac{6 a^2 e^2 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{7 d \sqrt{e \cos (c+d x)}}-\frac{18 a^2 (e \cos (c+d x))^{5/2}}{35 d e}-\frac{2 \left(a^2 \sin (c+d x)+a^2\right) (e \cos (c+d x))^{5/2}}{7 d e}+\frac{6 a^2 e \sin (c+d x) \sqrt{e \cos (c+d x)}}{7 d}",1,"(-16*2^(1/4)*a^2*(e*Cos[c + d*x])^(5/2)*Hypergeometric2F1[-9/4, 5/4, 9/4, (1 - Sin[c + d*x])/2])/(5*d*e*(1 + Sin[c + d*x])^(5/4))","C",1
207,1,66,105,0.0458701,"\int \sqrt{e \cos (c+d x)} (a+a \sin (c+d x))^2 \, dx","Integrate[Sqrt[e*Cos[c + d*x]]*(a + a*Sin[c + d*x])^2,x]","-\frac{8\ 2^{3/4} a^2 (e \cos (c+d x))^{3/2} \, _2F_1\left(-\frac{7}{4},\frac{3}{4};\frac{7}{4};\frac{1}{2} (1-\sin (c+d x))\right)}{3 d e (\sin (c+d x)+1)^{3/4}}","-\frac{14 a^2 (e \cos (c+d x))^{3/2}}{15 d e}-\frac{2 \left(a^2 \sin (c+d x)+a^2\right) (e \cos (c+d x))^{3/2}}{5 d e}+\frac{14 a^2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{5 d \sqrt{\cos (c+d x)}}",1,"(-8*2^(3/4)*a^2*(e*Cos[c + d*x])^(3/2)*Hypergeometric2F1[-7/4, 3/4, 7/4, (1 - Sin[c + d*x])/2])/(3*d*e*(1 + Sin[c + d*x])^(3/4))","C",1
208,1,64,105,0.034964,"\int \frac{(a+a \sin (c+d x))^2}{\sqrt{e \cos (c+d x)}} \, dx","Integrate[(a + a*Sin[c + d*x])^2/Sqrt[e*Cos[c + d*x]],x]","-\frac{8 \sqrt[4]{2} a^2 \sqrt{e \cos (c+d x)} \, _2F_1\left(-\frac{5}{4},\frac{1}{4};\frac{5}{4};\frac{1}{2} (1-\sin (c+d x))\right)}{d e \sqrt[4]{\sin (c+d x)+1}}","-\frac{10 a^2 \sqrt{e \cos (c+d x)}}{3 d e}-\frac{2 \left(a^2 \sin (c+d x)+a^2\right) \sqrt{e \cos (c+d x)}}{3 d e}+\frac{10 a^2 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d \sqrt{e \cos (c+d x)}}",1,"(-8*2^(1/4)*a^2*Sqrt[e*Cos[c + d*x]]*Hypergeometric2F1[-5/4, 1/4, 5/4, (1 - Sin[c + d*x])/2])/(d*e*(1 + Sin[c + d*x])^(1/4))","C",1
209,1,64,85,0.0645742,"\int \frac{(a+a \sin (c+d x))^2}{(e \cos (c+d x))^{3/2}} \, dx","Integrate[(a + a*Sin[c + d*x])^2/(e*Cos[c + d*x])^(3/2),x]","\frac{4\ 2^{3/4} a^2 \sqrt[4]{\sin (c+d x)+1} \, _2F_1\left(-\frac{3}{4},-\frac{1}{4};\frac{3}{4};\frac{1}{2} (1-\sin (c+d x))\right)}{d e \sqrt{e \cos (c+d x)}}","\frac{4 a^4 (e \cos (c+d x))^{3/2}}{d e^3 \left(a^2-a^2 \sin (c+d x)\right)}-\frac{6 a^2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{d e^2 \sqrt{\cos (c+d x)}}",1,"(4*2^(3/4)*a^2*Hypergeometric2F1[-3/4, -1/4, 3/4, (1 - Sin[c + d*x])/2]*(1 + Sin[c + d*x])^(1/4))/(d*e*Sqrt[e*Cos[c + d*x]])","C",1
210,1,66,89,0.0675886,"\int \frac{(a+a \sin (c+d x))^2}{(e \cos (c+d x))^{5/2}} \, dx","Integrate[(a + a*Sin[c + d*x])^2/(e*Cos[c + d*x])^(5/2),x]","\frac{4 \sqrt[4]{2} a^2 (\sin (c+d x)+1)^{3/4} \, _2F_1\left(-\frac{3}{4},-\frac{1}{4};\frac{1}{4};\frac{1}{2} (1-\sin (c+d x))\right)}{3 d e (e \cos (c+d x))^{3/2}}","\frac{4 a^4 \sqrt{e \cos (c+d x)}}{3 d e^3 \left(a^2-a^2 \sin (c+d x)\right)}-\frac{2 a^2 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d e^2 \sqrt{e \cos (c+d x)}}",1,"(4*2^(1/4)*a^2*Hypergeometric2F1[-3/4, -1/4, 1/4, (1 - Sin[c + d*x])/2]*(1 + Sin[c + d*x])^(3/4))/(3*d*e*(e*Cos[c + d*x])^(3/2))","C",1
211,1,66,127,0.0862165,"\int \frac{(a+a \sin (c+d x))^2}{(e \cos (c+d x))^{7/2}} \, dx","Integrate[(a + a*Sin[c + d*x])^2/(e*Cos[c + d*x])^(7/2),x]","\frac{2\ 2^{3/4} a^2 (\sin (c+d x)+1)^{5/4} \, _2F_1\left(-\frac{5}{4},\frac{1}{4};-\frac{1}{4};\frac{1}{2} (1-\sin (c+d x))\right)}{5 d e (e \cos (c+d x))^{5/2}}","\frac{2 a^4 (e \cos (c+d x))^{3/2}}{5 d e^5 (a-a \sin (c+d x))^2}-\frac{2 a^2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{5 d e^4 \sqrt{\cos (c+d x)}}+\frac{2 a^4 (e \cos (c+d x))^{3/2}}{5 d e^5 \left(a^2-a^2 \sin (c+d x)\right)}",1,"(2*2^(3/4)*a^2*Hypergeometric2F1[-5/4, 1/4, -1/4, (1 - Sin[c + d*x])/2]*(1 + Sin[c + d*x])^(5/4))/(5*d*e*(e*Cos[c + d*x])^(5/2))","C",1
212,1,66,114,0.1151041,"\int \frac{(a+a \sin (c+d x))^2}{(e \cos (c+d x))^{9/2}} \, dx","Integrate[(a + a*Sin[c + d*x])^2/(e*Cos[c + d*x])^(9/2),x]","\frac{2 \sqrt[4]{2} a^2 (\sin (c+d x)+1)^{7/4} \, _2F_1\left(-\frac{7}{4},\frac{3}{4};-\frac{3}{4};\frac{1}{2} (1-\sin (c+d x))\right)}{7 d e (e \cos (c+d x))^{7/2}}","\frac{2 a^2 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{7 d e^4 \sqrt{e \cos (c+d x)}}+\frac{2 a^2 \sin (c+d x)}{7 d e^3 (e \cos (c+d x))^{3/2}}+\frac{4 \left(a^2 \sin (c+d x)+a^2\right)}{7 d e (e \cos (c+d x))^{7/2}}",1,"(2*2^(1/4)*a^2*Hypergeometric2F1[-7/4, 3/4, -3/4, (1 - Sin[c + d*x])/2]*(1 + Sin[c + d*x])^(7/4))/(7*d*e*(e*Cos[c + d*x])^(7/2))","C",1
213,1,66,145,0.1558154,"\int \frac{(a+a \sin (c+d x))^2}{(e \cos (c+d x))^{11/2}} \, dx","Integrate[(a + a*Sin[c + d*x])^2/(e*Cos[c + d*x])^(11/2),x]","\frac{2^{3/4} a^2 (\sin (c+d x)+1)^{9/4} \, _2F_1\left(-\frac{9}{4},\frac{5}{4};-\frac{5}{4};\frac{1}{2} (1-\sin (c+d x))\right)}{9 d e (e \cos (c+d x))^{9/2}}","-\frac{2 a^2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{3 d e^6 \sqrt{\cos (c+d x)}}+\frac{2 a^2 \sin (c+d x)}{3 d e^5 \sqrt{e \cos (c+d x)}}+\frac{2 a^2 \sin (c+d x)}{9 d e^3 (e \cos (c+d x))^{5/2}}+\frac{4 \left(a^2 \sin (c+d x)+a^2\right)}{9 d e (e \cos (c+d x))^{9/2}}",1,"(2^(3/4)*a^2*Hypergeometric2F1[-9/4, 5/4, -5/4, (1 - Sin[c + d*x])/2]*(1 + Sin[c + d*x])^(9/4))/(9*d*e*(e*Cos[c + d*x])^(9/2))","C",1
214,1,66,203,0.077944,"\int (e \cos (c+d x))^{7/2} (a+a \sin (c+d x))^3 \, dx","Integrate[(e*Cos[c + d*x])^(7/2)*(a + a*Sin[c + d*x])^3,x]","-\frac{64 \sqrt[4]{2} a^3 (e \cos (c+d x))^{9/2} \, _2F_1\left(-\frac{17}{4},\frac{9}{4};\frac{13}{4};\frac{1}{2} (1-\sin (c+d x))\right)}{9 d e (\sin (c+d x)+1)^{9/4}}","\frac{170 a^3 e^4 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{231 d \sqrt{e \cos (c+d x)}}+\frac{170 a^3 e^3 \sin (c+d x) \sqrt{e \cos (c+d x)}}{231 d}-\frac{34 a^3 (e \cos (c+d x))^{9/2}}{99 d e}-\frac{34 \left(a^3 \sin (c+d x)+a^3\right) (e \cos (c+d x))^{9/2}}{143 d e}+\frac{34 a^3 e \sin (c+d x) (e \cos (c+d x))^{5/2}}{77 d}-\frac{2 a (a \sin (c+d x)+a)^2 (e \cos (c+d x))^{9/2}}{13 d e}",1,"(-64*2^(1/4)*a^3*(e*Cos[c + d*x])^(9/2)*Hypergeometric2F1[-17/4, 9/4, 13/4, (1 - Sin[c + d*x])/2])/(9*d*e*(1 + Sin[c + d*x])^(9/4))","C",1
215,1,66,170,0.1065417,"\int (e \cos (c+d x))^{5/2} (a+a \sin (c+d x))^3 \, dx","Integrate[(e*Cos[c + d*x])^(5/2)*(a + a*Sin[c + d*x])^3,x]","-\frac{32\ 2^{3/4} a^3 (e \cos (c+d x))^{7/2} \, _2F_1\left(-\frac{15}{4},\frac{7}{4};\frac{11}{4};\frac{1}{2} (1-\sin (c+d x))\right)}{7 d e (\sin (c+d x)+1)^{7/4}}","\frac{2 a^3 e^2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{d \sqrt{\cos (c+d x)}}-\frac{10 a^3 (e \cos (c+d x))^{7/2}}{21 d e}-\frac{10 \left(a^3 \sin (c+d x)+a^3\right) (e \cos (c+d x))^{7/2}}{33 d e}+\frac{2 a^3 e \sin (c+d x) (e \cos (c+d x))^{3/2}}{3 d}-\frac{2 a (a \sin (c+d x)+a)^2 (e \cos (c+d x))^{7/2}}{11 d e}",1,"(-32*2^(3/4)*a^3*(e*Cos[c + d*x])^(7/2)*Hypergeometric2F1[-15/4, 7/4, 11/4, (1 - Sin[c + d*x])/2])/(7*d*e*(1 + Sin[c + d*x])^(7/4))","C",1
216,1,66,172,0.0729176,"\int (e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^3 \, dx","Integrate[(e*Cos[c + d*x])^(3/2)*(a + a*Sin[c + d*x])^3,x]","-\frac{32 \sqrt[4]{2} a^3 (e \cos (c+d x))^{5/2} \, _2F_1\left(-\frac{13}{4},\frac{5}{4};\frac{9}{4};\frac{1}{2} (1-\sin (c+d x))\right)}{5 d e (\sin (c+d x)+1)^{5/4}}","\frac{26 a^3 e^2 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d \sqrt{e \cos (c+d x)}}-\frac{26 a^3 (e \cos (c+d x))^{5/2}}{35 d e}+\frac{26 a^3 e \sin (c+d x) \sqrt{e \cos (c+d x)}}{21 d}-\frac{26 \left(a^3 \sin (c+d x)+a^3\right) (e \cos (c+d x))^{5/2}}{63 d e}-\frac{2 a (a \sin (c+d x)+a)^2 (e \cos (c+d x))^{5/2}}{9 d e}",1,"(-32*2^(1/4)*a^3*(e*Cos[c + d*x])^(5/2)*Hypergeometric2F1[-13/4, 5/4, 9/4, (1 - Sin[c + d*x])/2])/(5*d*e*(1 + Sin[c + d*x])^(5/4))","C",1
217,1,66,140,0.0462984,"\int \sqrt{e \cos (c+d x)} (a+a \sin (c+d x))^3 \, dx","Integrate[Sqrt[e*Cos[c + d*x]]*(a + a*Sin[c + d*x])^3,x]","-\frac{16\ 2^{3/4} a^3 (e \cos (c+d x))^{3/2} \, _2F_1\left(-\frac{11}{4},\frac{3}{4};\frac{7}{4};\frac{1}{2} (1-\sin (c+d x))\right)}{3 d e (\sin (c+d x)+1)^{3/4}}","-\frac{22 a^3 (e \cos (c+d x))^{3/2}}{15 d e}-\frac{22 \left(a^3 \sin (c+d x)+a^3\right) (e \cos (c+d x))^{3/2}}{35 d e}+\frac{22 a^3 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{5 d \sqrt{\cos (c+d x)}}-\frac{2 a (a \sin (c+d x)+a)^2 (e \cos (c+d x))^{3/2}}{7 d e}",1,"(-16*2^(3/4)*a^3*(e*Cos[c + d*x])^(3/2)*Hypergeometric2F1[-11/4, 3/4, 7/4, (1 - Sin[c + d*x])/2])/(3*d*e*(1 + Sin[c + d*x])^(3/4))","C",1
218,1,64,136,0.0359755,"\int \frac{(a+a \sin (c+d x))^3}{\sqrt{e \cos (c+d x)}} \, dx","Integrate[(a + a*Sin[c + d*x])^3/Sqrt[e*Cos[c + d*x]],x]","-\frac{16 \sqrt[4]{2} a^3 \sqrt{e \cos (c+d x)} \, _2F_1\left(-\frac{9}{4},\frac{1}{4};\frac{5}{4};\frac{1}{2} (1-\sin (c+d x))\right)}{d e \sqrt[4]{\sin (c+d x)+1}}","-\frac{6 a^3 \sqrt{e \cos (c+d x)}}{d e}-\frac{6 \left(a^3 \sin (c+d x)+a^3\right) \sqrt{e \cos (c+d x)}}{5 d e}+\frac{6 a^3 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d \sqrt{e \cos (c+d x)}}-\frac{2 a (a \sin (c+d x)+a)^2 \sqrt{e \cos (c+d x)}}{5 d e}",1,"(-16*2^(1/4)*a^3*Sqrt[e*Cos[c + d*x]]*Hypergeometric2F1[-9/4, 1/4, 5/4, (1 - Sin[c + d*x])/2])/(d*e*(1 + Sin[c + d*x])^(1/4))","C",1
219,1,64,106,0.057455,"\int \frac{(a+a \sin (c+d x))^3}{(e \cos (c+d x))^{3/2}} \, dx","Integrate[(a + a*Sin[c + d*x])^3/(e*Cos[c + d*x])^(3/2),x]","\frac{8\ 2^{3/4} a^3 \sqrt[4]{\sin (c+d x)+1} \, _2F_1\left(-\frac{7}{4},-\frac{1}{4};\frac{3}{4};\frac{1}{2} (1-\sin (c+d x))\right)}{d e \sqrt{e \cos (c+d x)}}","\frac{4 a^5 (e \cos (c+d x))^{7/2}}{d e^5 (a-a \sin (c+d x))^2}+\frac{14 a^3 (e \cos (c+d x))^{3/2}}{3 d e^3}-\frac{14 a^3 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{d e^2 \sqrt{\cos (c+d x)}}",1,"(8*2^(3/4)*a^3*Hypergeometric2F1[-7/4, -1/4, 3/4, (1 - Sin[c + d*x])/2]*(1 + Sin[c + d*x])^(1/4))/(d*e*Sqrt[e*Cos[c + d*x]])","C",1
220,1,66,110,0.0539164,"\int \frac{(a+a \sin (c+d x))^3}{(e \cos (c+d x))^{5/2}} \, dx","Integrate[(a + a*Sin[c + d*x])^3/(e*Cos[c + d*x])^(5/2),x]","\frac{8 \sqrt[4]{2} a^3 (\sin (c+d x)+1)^{3/4} \, _2F_1\left(-\frac{5}{4},-\frac{3}{4};\frac{1}{4};\frac{1}{2} (1-\sin (c+d x))\right)}{3 d e (e \cos (c+d x))^{3/2}}","\frac{4 a^5 (e \cos (c+d x))^{5/2}}{3 d e^5 (a-a \sin (c+d x))^2}+\frac{10 a^3 \sqrt{e \cos (c+d x)}}{3 d e^3}-\frac{10 a^3 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d e^2 \sqrt{e \cos (c+d x)}}",1,"(8*2^(1/4)*a^3*Hypergeometric2F1[-5/4, -3/4, 1/4, (1 - Sin[c + d*x])/2]*(1 + Sin[c + d*x])^(3/4))/(3*d*e*(e*Cos[c + d*x])^(3/2))","C",1
221,1,66,127,0.0883108,"\int \frac{(a+a \sin (c+d x))^3}{(e \cos (c+d x))^{7/2}} \, dx","Integrate[(a + a*Sin[c + d*x])^3/(e*Cos[c + d*x])^(7/2),x]","\frac{4\ 2^{3/4} a^3 (\sin (c+d x)+1)^{5/4} \, _2F_1\left(-\frac{5}{4},-\frac{3}{4};-\frac{1}{4};\frac{1}{2} (1-\sin (c+d x))\right)}{5 d e (e \cos (c+d x))^{5/2}}","\frac{4 a^5 (e \cos (c+d x))^{3/2}}{5 d e^5 (a-a \sin (c+d x))^2}+\frac{6 a^3 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{5 d e^4 \sqrt{\cos (c+d x)}}-\frac{6 a^6 (e \cos (c+d x))^{3/2}}{5 d e^5 \left(a^3-a^3 \sin (c+d x)\right)}",1,"(4*2^(3/4)*a^3*Hypergeometric2F1[-5/4, -3/4, -1/4, (1 - Sin[c + d*x])/2]*(1 + Sin[c + d*x])^(5/4))/(5*d*e*(e*Cos[c + d*x])^(5/2))","C",1
222,1,66,127,0.0869154,"\int \frac{(a+a \sin (c+d x))^3}{(e \cos (c+d x))^{9/2}} \, dx","Integrate[(a + a*Sin[c + d*x])^3/(e*Cos[c + d*x])^(9/2),x]","\frac{4 \sqrt[4]{2} a^3 (\sin (c+d x)+1)^{7/4} \, _2F_1\left(-\frac{7}{4},-\frac{1}{4};-\frac{3}{4};\frac{1}{2} (1-\sin (c+d x))\right)}{7 d e (e \cos (c+d x))^{7/2}}","\frac{4 a^5 \sqrt{e \cos (c+d x)}}{7 d e^5 (a-a \sin (c+d x))^2}-\frac{2 a^3 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d e^4 \sqrt{e \cos (c+d x)}}-\frac{2 a^6 \sqrt{e \cos (c+d x)}}{21 d e^5 \left(a^3-a^3 \sin (c+d x)\right)}",1,"(4*2^(1/4)*a^3*Hypergeometric2F1[-7/4, -1/4, -3/4, (1 - Sin[c + d*x])/2]*(1 + Sin[c + d*x])^(7/4))/(7*d*e*(e*Cos[c + d*x])^(7/2))","C",1
223,1,66,165,0.1265299,"\int \frac{(a+a \sin (c+d x))^3}{(e \cos (c+d x))^{11/2}} \, dx","Integrate[(a + a*Sin[c + d*x])^3/(e*Cos[c + d*x])^(11/2),x]","\frac{2\ 2^{3/4} a^3 (\sin (c+d x)+1)^{9/4} \, _2F_1\left(-\frac{9}{4},\frac{1}{4};-\frac{5}{4};\frac{1}{2} (1-\sin (c+d x))\right)}{9 d e (e \cos (c+d x))^{9/2}}","\frac{2 a^6 (e \cos (c+d x))^{3/2}}{9 d e^7 (a-a \sin (c+d x))^3}+\frac{2 a^5 (e \cos (c+d x))^{3/2}}{15 d e^7 (a-a \sin (c+d x))^2}-\frac{2 a^3 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{15 d e^6 \sqrt{\cos (c+d x)}}+\frac{2 a^6 (e \cos (c+d x))^{3/2}}{15 d e^7 \left(a^3-a^3 \sin (c+d x)\right)}",1,"(2*2^(3/4)*a^3*Hypergeometric2F1[-9/4, 1/4, -5/4, (1 - Sin[c + d*x])/2]*(1 + Sin[c + d*x])^(9/4))/(9*d*e*(e*Cos[c + d*x])^(9/2))","C",1
224,1,66,210,0.113288,"\int (e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^4 \, dx","Integrate[(e*Cos[c + d*x])^(3/2)*(a + a*Sin[c + d*x])^4,x]","-\frac{64 \sqrt[4]{2} a^4 (e \cos (c+d x))^{5/2} \, _2F_1\left(-\frac{17}{4},\frac{5}{4};\frac{9}{4};\frac{1}{2} (1-\sin (c+d x))\right)}{5 d e (\sin (c+d x)+1)^{5/4}}","\frac{442 a^4 e^2 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{231 d \sqrt{e \cos (c+d x)}}-\frac{442 a^4 (e \cos (c+d x))^{5/2}}{385 d e}+\frac{442 a^4 e \sin (c+d x) \sqrt{e \cos (c+d x)}}{231 d}-\frac{442 \left(a^4 \sin (c+d x)+a^4\right) (e \cos (c+d x))^{5/2}}{693 d e}-\frac{34 \left(a^2 \sin (c+d x)+a^2\right)^2 (e \cos (c+d x))^{5/2}}{99 d e}-\frac{2 a (a \sin (c+d x)+a)^3 (e \cos (c+d x))^{5/2}}{11 d e}",1,"(-64*2^(1/4)*a^4*(e*Cos[c + d*x])^(5/2)*Hypergeometric2F1[-17/4, 5/4, 9/4, (1 - Sin[c + d*x])/2])/(5*d*e*(1 + Sin[c + d*x])^(5/4))","C",1
225,1,66,178,0.0645731,"\int \sqrt{e \cos (c+d x)} (a+a \sin (c+d x))^4 \, dx","Integrate[Sqrt[e*Cos[c + d*x]]*(a + a*Sin[c + d*x])^4,x]","-\frac{32\ 2^{3/4} a^4 (e \cos (c+d x))^{3/2} \, _2F_1\left(-\frac{15}{4},\frac{3}{4};\frac{7}{4};\frac{1}{2} (1-\sin (c+d x))\right)}{3 d e (\sin (c+d x)+1)^{3/4}}","-\frac{22 a^4 (e \cos (c+d x))^{3/2}}{9 d e}-\frac{22 \left(a^4 \sin (c+d x)+a^4\right) (e \cos (c+d x))^{3/2}}{21 d e}+\frac{22 a^4 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{3 d \sqrt{\cos (c+d x)}}-\frac{10 \left(a^2 \sin (c+d x)+a^2\right)^2 (e \cos (c+d x))^{3/2}}{21 d e}-\frac{2 a (a \sin (c+d x)+a)^3 (e \cos (c+d x))^{3/2}}{9 d e}",1,"(-32*2^(3/4)*a^4*(e*Cos[c + d*x])^(3/2)*Hypergeometric2F1[-15/4, 3/4, 7/4, (1 - Sin[c + d*x])/2])/(3*d*e*(1 + Sin[c + d*x])^(3/4))","C",1
226,1,64,178,0.0584857,"\int \frac{(a+a \sin (c+d x))^4}{\sqrt{e \cos (c+d x)}} \, dx","Integrate[(a + a*Sin[c + d*x])^4/Sqrt[e*Cos[c + d*x]],x]","-\frac{32 \sqrt[4]{2} a^4 \sqrt{e \cos (c+d x)} \, _2F_1\left(-\frac{13}{4},\frac{1}{4};\frac{5}{4};\frac{1}{2} (1-\sin (c+d x))\right)}{d e \sqrt[4]{\sin (c+d x)+1}}","-\frac{78 a^4 \sqrt{e \cos (c+d x)}}{7 d e}-\frac{78 \left(a^4 \sin (c+d x)+a^4\right) \sqrt{e \cos (c+d x)}}{35 d e}+\frac{78 a^4 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{7 d \sqrt{e \cos (c+d x)}}-\frac{26 \left(a^2 \sin (c+d x)+a^2\right)^2 \sqrt{e \cos (c+d x)}}{35 d e}-\frac{2 a (a \sin (c+d x)+a)^3 \sqrt{e \cos (c+d x)}}{7 d e}",1,"(-32*2^(1/4)*a^4*Sqrt[e*Cos[c + d*x]]*Hypergeometric2F1[-13/4, 1/4, 5/4, (1 - Sin[c + d*x])/2])/(d*e*(1 + Sin[c + d*x])^(1/4))","C",1
227,1,64,156,0.0726974,"\int \frac{(a+a \sin (c+d x))^4}{(e \cos (c+d x))^{3/2}} \, dx","Integrate[(a + a*Sin[c + d*x])^4/(e*Cos[c + d*x])^(3/2),x]","\frac{16\ 2^{3/4} a^4 \sqrt[4]{\sin (c+d x)+1} \, _2F_1\left(-\frac{11}{4},-\frac{1}{4};\frac{3}{4};\frac{1}{2} (1-\sin (c+d x))\right)}{d e \sqrt{e \cos (c+d x)}}","\frac{4 a^7 (e \cos (c+d x))^{11/2}}{d e^7 (a-a \sin (c+d x))^3}-\frac{154 a^4 \sin (c+d x) (e \cos (c+d x))^{3/2}}{15 d e^3}-\frac{154 a^4 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{5 d e^2 \sqrt{\cos (c+d x)}}+\frac{44 a^8 (e \cos (c+d x))^{7/2}}{3 d e^5 \left(a^4-a^4 \sin (c+d x)\right)}",1,"(16*2^(3/4)*a^4*Hypergeometric2F1[-11/4, -1/4, 3/4, (1 - Sin[c + d*x])/2]*(1 + Sin[c + d*x])^(1/4))/(d*e*Sqrt[e*Cos[c + d*x]])","C",1
228,1,66,152,0.0783818,"\int \frac{(a+a \sin (c+d x))^4}{(e \cos (c+d x))^{5/2}} \, dx","Integrate[(a + a*Sin[c + d*x])^4/(e*Cos[c + d*x])^(5/2),x]","\frac{16 \sqrt[4]{2} a^4 (\sin (c+d x)+1)^{3/4} \, _2F_1\left(-\frac{9}{4},-\frac{3}{4};\frac{1}{4};\frac{1}{2} (1-\sin (c+d x))\right)}{3 d e (e \cos (c+d x))^{3/2}}","\frac{4 a^7 (e \cos (c+d x))^{9/2}}{3 d e^7 (a-a \sin (c+d x))^3}-\frac{10 a^4 \sin (c+d x) \sqrt{e \cos (c+d x)}}{d e^3}-\frac{10 a^4 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d e^2 \sqrt{e \cos (c+d x)}}+\frac{12 a^8 (e \cos (c+d x))^{5/2}}{d e^5 \left(a^4-a^4 \sin (c+d x)\right)}",1,"(16*2^(1/4)*a^4*Hypergeometric2F1[-9/4, -3/4, 1/4, (1 - Sin[c + d*x])/2]*(1 + Sin[c + d*x])^(3/4))/(3*d*e*(e*Cos[c + d*x])^(3/2))","C",1
229,1,66,127,0.1008669,"\int \frac{(a+a \sin (c+d x))^4}{(e \cos (c+d x))^{7/2}} \, dx","Integrate[(a + a*Sin[c + d*x])^4/(e*Cos[c + d*x])^(7/2),x]","\frac{8\ 2^{3/4} a^4 (\sin (c+d x)+1)^{5/4} \, _2F_1\left(-\frac{7}{4},-\frac{5}{4};-\frac{1}{4};\frac{1}{2} (1-\sin (c+d x))\right)}{5 d e (e \cos (c+d x))^{5/2}}","\frac{4 a^7 (e \cos (c+d x))^{7/2}}{5 d e^7 (a-a \sin (c+d x))^3}+\frac{42 a^4 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{5 d e^4 \sqrt{\cos (c+d x)}}-\frac{28 a^8 (e \cos (c+d x))^{3/2}}{5 d e^5 \left(a^4-a^4 \sin (c+d x)\right)}",1,"(8*2^(3/4)*a^4*Hypergeometric2F1[-7/4, -5/4, -1/4, (1 - Sin[c + d*x])/2]*(1 + Sin[c + d*x])^(5/4))/(5*d*e*(e*Cos[c + d*x])^(5/2))","C",1
230,1,66,127,0.1100697,"\int \frac{(a+a \sin (c+d x))^4}{(e \cos (c+d x))^{9/2}} \, dx","Integrate[(a + a*Sin[c + d*x])^4/(e*Cos[c + d*x])^(9/2),x]","\frac{8 \sqrt[4]{2} a^4 (\sin (c+d x)+1)^{7/4} \, _2F_1\left(-\frac{7}{4},-\frac{5}{4};-\frac{3}{4};\frac{1}{2} (1-\sin (c+d x))\right)}{7 d e (e \cos (c+d x))^{7/2}}","\frac{4 a^7 (e \cos (c+d x))^{5/2}}{7 d e^7 (a-a \sin (c+d x))^3}+\frac{10 a^4 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d e^4 \sqrt{e \cos (c+d x)}}-\frac{20 a^8 \sqrt{e \cos (c+d x)}}{21 d e^5 \left(a^4-a^4 \sin (c+d x)\right)}",1,"(8*2^(1/4)*a^4*Hypergeometric2F1[-7/4, -5/4, -3/4, (1 - Sin[c + d*x])/2]*(1 + Sin[c + d*x])^(7/4))/(7*d*e*(e*Cos[c + d*x])^(7/2))","C",1
231,1,66,169,0.139793,"\int \frac{(a+a \sin (c+d x))^4}{(e \cos (c+d x))^{11/2}} \, dx","Integrate[(a + a*Sin[c + d*x])^4/(e*Cos[c + d*x])^(11/2),x]","\frac{4\ 2^{3/4} a^4 (\sin (c+d x)+1)^{9/4} \, _2F_1\left(-\frac{9}{4},-\frac{3}{4};-\frac{5}{4};\frac{1}{2} (1-\sin (c+d x))\right)}{9 d e (e \cos (c+d x))^{9/2}}","\frac{4 a^7 (e \cos (c+d x))^{3/2}}{9 d e^7 (a-a \sin (c+d x))^3}+\frac{2 a^4 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{15 d e^6 \sqrt{\cos (c+d x)}}-\frac{2 a^8 (e \cos (c+d x))^{3/2}}{15 d e^7 \left(a^4-a^4 \sin (c+d x)\right)}-\frac{2 a^8 (e \cos (c+d x))^{3/2}}{15 d e^7 \left(a^2-a^2 \sin (c+d x)\right)^2}",1,"(4*2^(3/4)*a^4*Hypergeometric2F1[-9/4, -3/4, -5/4, (1 - Sin[c + d*x])/2]*(1 + Sin[c + d*x])^(9/4))/(9*d*e*(e*Cos[c + d*x])^(9/2))","C",1
232,1,66,169,0.2444393,"\int \frac{(a+a \sin (c+d x))^4}{(e \cos (c+d x))^{13/2}} \, dx","Integrate[(a + a*Sin[c + d*x])^4/(e*Cos[c + d*x])^(13/2),x]","\frac{4 \sqrt[4]{2} a^4 (\sin (c+d x)+1)^{11/4} \, _2F_1\left(-\frac{11}{4},-\frac{1}{4};-\frac{7}{4};\frac{1}{2} (1-\sin (c+d x))\right)}{11 d e (e \cos (c+d x))^{11/2}}","\frac{4 a^7 \sqrt{e \cos (c+d x)}}{11 d e^7 (a-a \sin (c+d x))^3}-\frac{2 a^4 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{77 d e^6 \sqrt{e \cos (c+d x)}}-\frac{2 a^8 \sqrt{e \cos (c+d x)}}{77 d e^7 \left(a^4-a^4 \sin (c+d x)\right)}-\frac{2 a^8 \sqrt{e \cos (c+d x)}}{77 d e^7 \left(a^2-a^2 \sin (c+d x)\right)^2}",1,"(4*2^(1/4)*a^4*Hypergeometric2F1[-11/4, -1/4, -7/4, (1 - Sin[c + d*x])/2]*(1 + Sin[c + d*x])^(11/4))/(11*d*e*(e*Cos[c + d*x])^(11/2))","C",1
233,1,66,132,0.1429957,"\int \frac{(e \cos (c+d x))^{11/2}}{a+a \sin (c+d x)} \, dx","Integrate[(e*Cos[c + d*x])^(11/2)/(a + a*Sin[c + d*x]),x]","-\frac{8 \sqrt[4]{2} (e \cos (c+d x))^{13/2} \, _2F_1\left(-\frac{5}{4},\frac{13}{4};\frac{17}{4};\frac{1}{2} (1-\sin (c+d x))\right)}{13 a d e (\sin (c+d x)+1)^{13/4}}","\frac{10 e^6 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 a d \sqrt{e \cos (c+d x)}}+\frac{10 e^5 \sin (c+d x) \sqrt{e \cos (c+d x)}}{21 a d}+\frac{2 e^3 \sin (c+d x) (e \cos (c+d x))^{5/2}}{7 a d}+\frac{2 e (e \cos (c+d x))^{9/2}}{9 a d}",1,"(-8*2^(1/4)*(e*Cos[c + d*x])^(13/2)*Hypergeometric2F1[-5/4, 13/4, 17/4, (1 - Sin[c + d*x])/2])/(13*a*d*e*(1 + Sin[c + d*x])^(13/4))","C",1
234,1,66,101,0.0981168,"\int \frac{(e \cos (c+d x))^{9/2}}{a+a \sin (c+d x)} \, dx","Integrate[(e*Cos[c + d*x])^(9/2)/(a + a*Sin[c + d*x]),x]","-\frac{4\ 2^{3/4} (e \cos (c+d x))^{11/2} \, _2F_1\left(-\frac{3}{4},\frac{11}{4};\frac{15}{4};\frac{1}{2} (1-\sin (c+d x))\right)}{11 a d e (\sin (c+d x)+1)^{11/4}}","\frac{6 e^4 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{5 a d \sqrt{\cos (c+d x)}}+\frac{2 e^3 \sin (c+d x) (e \cos (c+d x))^{3/2}}{5 a d}+\frac{2 e (e \cos (c+d x))^{7/2}}{7 a d}",1,"(-4*2^(3/4)*(e*Cos[c + d*x])^(11/2)*Hypergeometric2F1[-3/4, 11/4, 15/4, (1 - Sin[c + d*x])/2])/(11*a*d*e*(1 + Sin[c + d*x])^(11/4))","C",1
235,1,66,101,0.0729198,"\int \frac{(e \cos (c+d x))^{7/2}}{a+a \sin (c+d x)} \, dx","Integrate[(e*Cos[c + d*x])^(7/2)/(a + a*Sin[c + d*x]),x]","-\frac{4 \sqrt[4]{2} (e \cos (c+d x))^{9/2} \, _2F_1\left(-\frac{1}{4},\frac{9}{4};\frac{13}{4};\frac{1}{2} (1-\sin (c+d x))\right)}{9 a d e (\sin (c+d x)+1)^{9/4}}","\frac{2 e^4 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a d \sqrt{e \cos (c+d x)}}+\frac{2 e^3 \sin (c+d x) \sqrt{e \cos (c+d x)}}{3 a d}+\frac{2 e (e \cos (c+d x))^{5/2}}{5 a d}",1,"(-4*2^(1/4)*(e*Cos[c + d*x])^(9/2)*Hypergeometric2F1[-1/4, 9/4, 13/4, (1 - Sin[c + d*x])/2])/(9*a*d*e*(1 + Sin[c + d*x])^(9/4))","C",1
236,1,66,68,0.0985319,"\int \frac{(e \cos (c+d x))^{5/2}}{a+a \sin (c+d x)} \, dx","Integrate[(e*Cos[c + d*x])^(5/2)/(a + a*Sin[c + d*x]),x]","-\frac{2\ 2^{3/4} (e \cos (c+d x))^{7/2} \, _2F_1\left(\frac{1}{4},\frac{7}{4};\frac{11}{4};\frac{1}{2} (1-\sin (c+d x))\right)}{7 a d e (\sin (c+d x)+1)^{7/4}}","\frac{2 e^2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{a d \sqrt{\cos (c+d x)}}+\frac{2 e (e \cos (c+d x))^{3/2}}{3 a d}",1,"(-2*2^(3/4)*(e*Cos[c + d*x])^(7/2)*Hypergeometric2F1[1/4, 7/4, 11/4, (1 - Sin[c + d*x])/2])/(7*a*d*e*(1 + Sin[c + d*x])^(7/4))","C",1
237,1,66,66,0.0760971,"\int \frac{(e \cos (c+d x))^{3/2}}{a+a \sin (c+d x)} \, dx","Integrate[(e*Cos[c + d*x])^(3/2)/(a + a*Sin[c + d*x]),x]","-\frac{2 \sqrt[4]{2} (e \cos (c+d x))^{5/2} \, _2F_1\left(\frac{3}{4},\frac{5}{4};\frac{9}{4};\frac{1}{2} (1-\sin (c+d x))\right)}{5 a d e (\sin (c+d x)+1)^{5/4}}","\frac{2 e^2 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d \sqrt{e \cos (c+d x)}}+\frac{2 e \sqrt{e \cos (c+d x)}}{a d}",1,"(-2*2^(1/4)*(e*Cos[c + d*x])^(5/2)*Hypergeometric2F1[3/4, 5/4, 9/4, (1 - Sin[c + d*x])/2])/(5*a*d*e*(1 + Sin[c + d*x])^(5/4))","C",1
238,1,66,74,0.0415033,"\int \frac{\sqrt{e \cos (c+d x)}}{a+a \sin (c+d x)} \, dx","Integrate[Sqrt[e*Cos[c + d*x]]/(a + a*Sin[c + d*x]),x]","-\frac{2^{3/4} (e \cos (c+d x))^{3/2} \, _2F_1\left(\frac{3}{4},\frac{5}{4};\frac{7}{4};\frac{1}{2} (1-\sin (c+d x))\right)}{3 a d e (\sin (c+d x)+1)^{3/4}}","-\frac{2 (e \cos (c+d x))^{3/2}}{d e (a \sin (c+d x)+a)}-\frac{2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{a d \sqrt{\cos (c+d x)}}",1,"-1/3*(2^(3/4)*(e*Cos[c + d*x])^(3/2)*Hypergeometric2F1[3/4, 5/4, 7/4, (1 - Sin[c + d*x])/2])/(a*d*e*(1 + Sin[c + d*x])^(3/4))","C",1
239,1,64,78,0.0388518,"\int \frac{1}{\sqrt{e \cos (c+d x)} (a+a \sin (c+d x))} \, dx","Integrate[1/(Sqrt[e*Cos[c + d*x]]*(a + a*Sin[c + d*x])),x]","-\frac{\sqrt[4]{2} \sqrt{e \cos (c+d x)} \, _2F_1\left(\frac{1}{4},\frac{7}{4};\frac{5}{4};\frac{1}{2} (1-\sin (c+d x))\right)}{a d e \sqrt[4]{\sin (c+d x)+1}}","\frac{2 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a d \sqrt{e \cos (c+d x)}}-\frac{2 \sqrt{e \cos (c+d x)}}{3 d e (a \sin (c+d x)+a)}",1,"-((2^(1/4)*Sqrt[e*Cos[c + d*x]]*Hypergeometric2F1[1/4, 7/4, 5/4, (1 - Sin[c + d*x])/2])/(a*d*e*(1 + Sin[c + d*x])^(1/4)))","C",1
240,1,63,112,0.0584435,"\int \frac{1}{(e \cos (c+d x))^{3/2} (a+a \sin (c+d x))} \, dx","Integrate[1/((e*Cos[c + d*x])^(3/2)*(a + a*Sin[c + d*x])),x]","\frac{\sqrt[4]{\sin (c+d x)+1} \, _2F_1\left(-\frac{1}{4},\frac{9}{4};\frac{3}{4};\frac{1}{2} (1-\sin (c+d x))\right)}{\sqrt[4]{2} a d e \sqrt{e \cos (c+d x)}}","-\frac{6 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{5 a d e^2 \sqrt{\cos (c+d x)}}+\frac{6 \sin (c+d x)}{5 a d e \sqrt{e \cos (c+d x)}}-\frac{2}{5 d e (a \sin (c+d x)+a) \sqrt{e \cos (c+d x)}}",1,"(Hypergeometric2F1[-1/4, 9/4, 3/4, (1 - Sin[c + d*x])/2]*(1 + Sin[c + d*x])^(1/4))/(2^(1/4)*a*d*e*Sqrt[e*Cos[c + d*x]])","C",1
241,1,66,112,0.076127,"\int \frac{1}{(e \cos (c+d x))^{5/2} (a+a \sin (c+d x))} \, dx","Integrate[1/((e*Cos[c + d*x])^(5/2)*(a + a*Sin[c + d*x])),x]","\frac{(\sin (c+d x)+1)^{3/4} \, _2F_1\left(-\frac{3}{4},\frac{11}{4};\frac{1}{4};\frac{1}{2} (1-\sin (c+d x))\right)}{3\ 2^{3/4} a d e (e \cos (c+d x))^{3/2}}","\frac{10 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 a d e^2 \sqrt{e \cos (c+d x)}}+\frac{10 \sin (c+d x)}{21 a d e (e \cos (c+d x))^{3/2}}-\frac{2}{7 d e (a \sin (c+d x)+a) (e \cos (c+d x))^{3/2}}",1,"(Hypergeometric2F1[-3/4, 11/4, 1/4, (1 - Sin[c + d*x])/2]*(1 + Sin[c + d*x])^(3/4))/(3*2^(3/4)*a*d*e*(e*Cos[c + d*x])^(3/2))","C",1
242,1,66,143,0.1019155,"\int \frac{1}{(e \cos (c+d x))^{7/2} (a+a \sin (c+d x))} \, dx","Integrate[1/((e*Cos[c + d*x])^(7/2)*(a + a*Sin[c + d*x])),x]","\frac{(\sin (c+d x)+1)^{5/4} \, _2F_1\left(-\frac{5}{4},\frac{13}{4};-\frac{1}{4};\frac{1}{2} (1-\sin (c+d x))\right)}{10 \sqrt[4]{2} a d e (e \cos (c+d x))^{5/2}}","-\frac{14 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{15 a d e^4 \sqrt{\cos (c+d x)}}+\frac{14 \sin (c+d x)}{15 a d e^3 \sqrt{e \cos (c+d x)}}+\frac{14 \sin (c+d x)}{45 a d e (e \cos (c+d x))^{5/2}}-\frac{2}{9 d e (a \sin (c+d x)+a) (e \cos (c+d x))^{5/2}}",1,"(Hypergeometric2F1[-5/4, 13/4, -1/4, (1 - Sin[c + d*x])/2]*(1 + Sin[c + d*x])^(5/4))/(10*2^(1/4)*a*d*e*(e*Cos[c + d*x])^(5/2))","C",1
243,1,66,145,0.1779805,"\int \frac{(e \cos (c+d x))^{11/2}}{(a+a \sin (c+d x))^2} \, dx","Integrate[(e*Cos[c + d*x])^(11/2)/(a + a*Sin[c + d*x])^2,x]","-\frac{4 \sqrt[4]{2} (e \cos (c+d x))^{13/2} \, _2F_1\left(-\frac{1}{4},\frac{13}{4};\frac{17}{4};\frac{1}{2} (1-\sin (c+d x))\right)}{13 a^2 d e (\sin (c+d x)+1)^{13/4}}","\frac{6 e^6 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{7 a^2 d \sqrt{e \cos (c+d x)}}+\frac{6 e^5 \sin (c+d x) \sqrt{e \cos (c+d x)}}{7 a^2 d}+\frac{18 e^3 \sin (c+d x) (e \cos (c+d x))^{5/2}}{35 a^2 d}+\frac{4 e (e \cos (c+d x))^{9/2}}{5 d \left(a^2 \sin (c+d x)+a^2\right)}",1,"(-4*2^(1/4)*(e*Cos[c + d*x])^(13/2)*Hypergeometric2F1[-1/4, 13/4, 17/4, (1 - Sin[c + d*x])/2])/(13*a^2*d*e*(1 + Sin[c + d*x])^(13/4))","C",1
244,1,66,114,0.1010401,"\int \frac{(e \cos (c+d x))^{9/2}}{(a+a \sin (c+d x))^2} \, dx","Integrate[(e*Cos[c + d*x])^(9/2)/(a + a*Sin[c + d*x])^2,x]","-\frac{2\ 2^{3/4} (e \cos (c+d x))^{11/2} \, _2F_1\left(\frac{1}{4},\frac{11}{4};\frac{15}{4};\frac{1}{2} (1-\sin (c+d x))\right)}{11 a^2 d e (\sin (c+d x)+1)^{11/4}}","\frac{14 e^4 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{5 a^2 d \sqrt{\cos (c+d x)}}+\frac{14 e^3 \sin (c+d x) (e \cos (c+d x))^{3/2}}{15 a^2 d}+\frac{4 e (e \cos (c+d x))^{7/2}}{3 d \left(a^2 \sin (c+d x)+a^2\right)}",1,"(-2*2^(3/4)*(e*Cos[c + d*x])^(11/2)*Hypergeometric2F1[1/4, 11/4, 15/4, (1 - Sin[c + d*x])/2])/(11*a^2*d*e*(1 + Sin[c + d*x])^(11/4))","C",1
245,1,66,112,0.074569,"\int \frac{(e \cos (c+d x))^{7/2}}{(a+a \sin (c+d x))^2} \, dx","Integrate[(e*Cos[c + d*x])^(7/2)/(a + a*Sin[c + d*x])^2,x]","-\frac{2 \sqrt[4]{2} (e \cos (c+d x))^{9/2} \, _2F_1\left(\frac{3}{4},\frac{9}{4};\frac{13}{4};\frac{1}{2} (1-\sin (c+d x))\right)}{9 a^2 d e (\sin (c+d x)+1)^{9/4}}","\frac{10 e^4 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^2 d \sqrt{e \cos (c+d x)}}+\frac{10 e^3 \sin (c+d x) \sqrt{e \cos (c+d x)}}{3 a^2 d}+\frac{4 e (e \cos (c+d x))^{5/2}}{d \left(a^2 \sin (c+d x)+a^2\right)}",1,"(-2*2^(1/4)*(e*Cos[c + d*x])^(9/2)*Hypergeometric2F1[3/4, 9/4, 13/4, (1 - Sin[c + d*x])/2])/(9*a^2*d*e*(1 + Sin[c + d*x])^(9/4))","C",1
246,1,66,79,0.094976,"\int \frac{(e \cos (c+d x))^{5/2}}{(a+a \sin (c+d x))^2} \, dx","Integrate[(e*Cos[c + d*x])^(5/2)/(a + a*Sin[c + d*x])^2,x]","-\frac{2^{3/4} (e \cos (c+d x))^{7/2} \, _2F_1\left(\frac{5}{4},\frac{7}{4};\frac{11}{4};\frac{1}{2} (1-\sin (c+d x))\right)}{7 a^2 d e (\sin (c+d x)+1)^{7/4}}","-\frac{6 e^2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{a^2 d \sqrt{\cos (c+d x)}}-\frac{4 e (e \cos (c+d x))^{3/2}}{d \left(a^2 \sin (c+d x)+a^2\right)}",1,"-1/7*(2^(3/4)*(e*Cos[c + d*x])^(7/2)*Hypergeometric2F1[5/4, 7/4, 11/4, (1 - Sin[c + d*x])/2])/(a^2*d*e*(1 + Sin[c + d*x])^(7/4))","C",1
247,1,66,83,0.0765502,"\int \frac{(e \cos (c+d x))^{3/2}}{(a+a \sin (c+d x))^2} \, dx","Integrate[(e*Cos[c + d*x])^(3/2)/(a + a*Sin[c + d*x])^2,x]","-\frac{\sqrt[4]{2} (e \cos (c+d x))^{5/2} \, _2F_1\left(\frac{5}{4},\frac{7}{4};\frac{9}{4};\frac{1}{2} (1-\sin (c+d x))\right)}{5 a^2 d e (\sin (c+d x)+1)^{5/4}}","-\frac{2 e^2 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^2 d \sqrt{e \cos (c+d x)}}-\frac{4 e \sqrt{e \cos (c+d x)}}{3 d \left(a^2 \sin (c+d x)+a^2\right)}",1,"-1/5*(2^(1/4)*(e*Cos[c + d*x])^(5/2)*Hypergeometric2F1[5/4, 7/4, 9/4, (1 - Sin[c + d*x])/2])/(a^2*d*e*(1 + Sin[c + d*x])^(5/4))","C",1
248,1,66,116,0.0417457,"\int \frac{\sqrt{e \cos (c+d x)}}{(a+a \sin (c+d x))^2} \, dx","Integrate[Sqrt[e*Cos[c + d*x]]/(a + a*Sin[c + d*x])^2,x]","-\frac{(e \cos (c+d x))^{3/2} \, _2F_1\left(\frac{3}{4},\frac{9}{4};\frac{7}{4};\frac{1}{2} (1-\sin (c+d x))\right)}{3 \sqrt[4]{2} a^2 d e (\sin (c+d x)+1)^{3/4}}","-\frac{2 (e \cos (c+d x))^{3/2}}{5 d e \left(a^2 \sin (c+d x)+a^2\right)}-\frac{2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{5 a^2 d \sqrt{\cos (c+d x)}}-\frac{2 (e \cos (c+d x))^{3/2}}{5 d e (a \sin (c+d x)+a)^2}",1,"-1/3*((e*Cos[c + d*x])^(3/2)*Hypergeometric2F1[3/4, 9/4, 7/4, (1 - Sin[c + d*x])/2])/(2^(1/4)*a^2*d*e*(1 + Sin[c + d*x])^(3/4))","C",1
249,1,64,116,0.0433616,"\int \frac{1}{\sqrt{e \cos (c+d x)} (a+a \sin (c+d x))^2} \, dx","Integrate[1/(Sqrt[e*Cos[c + d*x]]*(a + a*Sin[c + d*x])^2),x]","-\frac{\sqrt{e \cos (c+d x)} \, _2F_1\left(\frac{1}{4},\frac{11}{4};\frac{5}{4};\frac{1}{2} (1-\sin (c+d x))\right)}{2^{3/4} a^2 d e \sqrt[4]{\sin (c+d x)+1}}","-\frac{2 \sqrt{e \cos (c+d x)}}{7 d e \left(a^2 \sin (c+d x)+a^2\right)}+\frac{2 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{7 a^2 d \sqrt{e \cos (c+d x)}}-\frac{2 \sqrt{e \cos (c+d x)}}{7 d e (a \sin (c+d x)+a)^2}",1,"-((Sqrt[e*Cos[c + d*x]]*Hypergeometric2F1[1/4, 11/4, 5/4, (1 - Sin[c + d*x])/2])/(2^(3/4)*a^2*d*e*(1 + Sin[c + d*x])^(1/4)))","C",1
250,1,66,150,0.0729754,"\int \frac{1}{(e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^2} \, dx","Integrate[1/((e*Cos[c + d*x])^(3/2)*(a + a*Sin[c + d*x])^2),x]","\frac{\sqrt[4]{\sin (c+d x)+1} \, _2F_1\left(-\frac{1}{4},\frac{13}{4};\frac{3}{4};\frac{1}{2} (1-\sin (c+d x))\right)}{2 \sqrt[4]{2} a^2 d e \sqrt{e \cos (c+d x)}}","-\frac{2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{3 a^2 d e^2 \sqrt{\cos (c+d x)}}+\frac{2 \sin (c+d x)}{3 a^2 d e \sqrt{e \cos (c+d x)}}-\frac{2}{9 d e \left(a^2 \sin (c+d x)+a^2\right) \sqrt{e \cos (c+d x)}}-\frac{2}{9 d e (a \sin (c+d x)+a)^2 \sqrt{e \cos (c+d x)}}",1,"(Hypergeometric2F1[-1/4, 13/4, 3/4, (1 - Sin[c + d*x])/2]*(1 + Sin[c + d*x])^(1/4))/(2*2^(1/4)*a^2*d*e*Sqrt[e*Cos[c + d*x]])","C",1
251,1,66,150,0.0758868,"\int \frac{1}{(e \cos (c+d x))^{5/2} (a+a \sin (c+d x))^2} \, dx","Integrate[1/((e*Cos[c + d*x])^(5/2)*(a + a*Sin[c + d*x])^2),x]","\frac{(\sin (c+d x)+1)^{3/4} \, _2F_1\left(-\frac{3}{4},\frac{15}{4};\frac{1}{4};\frac{1}{2} (1-\sin (c+d x))\right)}{6\ 2^{3/4} a^2 d e (e \cos (c+d x))^{3/2}}","\frac{10 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{33 a^2 d e^2 \sqrt{e \cos (c+d x)}}+\frac{10 \sin (c+d x)}{33 a^2 d e (e \cos (c+d x))^{3/2}}-\frac{2}{11 d e \left(a^2 \sin (c+d x)+a^2\right) (e \cos (c+d x))^{3/2}}-\frac{2}{11 d e (a \sin (c+d x)+a)^2 (e \cos (c+d x))^{3/2}}",1,"(Hypergeometric2F1[-3/4, 15/4, 1/4, (1 - Sin[c + d*x])/2]*(1 + Sin[c + d*x])^(3/4))/(6*2^(3/4)*a^2*d*e*(e*Cos[c + d*x])^(3/2))","C",1
252,1,66,181,0.1124845,"\int \frac{1}{(e \cos (c+d x))^{7/2} (a+a \sin (c+d x))^2} \, dx","Integrate[1/((e*Cos[c + d*x])^(7/2)*(a + a*Sin[c + d*x])^2),x]","\frac{(\sin (c+d x)+1)^{5/4} \, _2F_1\left(-\frac{5}{4},\frac{17}{4};-\frac{1}{4};\frac{1}{2} (1-\sin (c+d x))\right)}{20 \sqrt[4]{2} a^2 d e (e \cos (c+d x))^{5/2}}","-\frac{42 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{65 a^2 d e^4 \sqrt{\cos (c+d x)}}+\frac{42 \sin (c+d x)}{65 a^2 d e^3 \sqrt{e \cos (c+d x)}}+\frac{14 \sin (c+d x)}{65 a^2 d e (e \cos (c+d x))^{5/2}}-\frac{2}{13 d e \left(a^2 \sin (c+d x)+a^2\right) (e \cos (c+d x))^{5/2}}-\frac{2}{13 d e (a \sin (c+d x)+a)^2 (e \cos (c+d x))^{5/2}}",1,"(Hypergeometric2F1[-5/4, 17/4, -1/4, (1 - Sin[c + d*x])/2]*(1 + Sin[c + d*x])^(5/4))/(20*2^(1/4)*a^2*d*e*(e*Cos[c + d*x])^(5/2))","C",1
253,1,66,169,0.3949033,"\int \frac{(e \cos (c+d x))^{15/2}}{(a+a \sin (c+d x))^3} \, dx","Integrate[(e*Cos[c + d*x])^(15/2)/(a + a*Sin[c + d*x])^3,x]","-\frac{4 \sqrt[4]{2} (e \cos (c+d x))^{17/2} \, _2F_1\left(-\frac{1}{4},\frac{17}{4};\frac{21}{4};\frac{1}{2} (1-\sin (c+d x))\right)}{17 a^3 d e (\sin (c+d x)+1)^{17/4}}","\frac{26 e^8 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 a^3 d \sqrt{e \cos (c+d x)}}+\frac{26 e^7 \sin (c+d x) \sqrt{e \cos (c+d x)}}{21 a^3 d}+\frac{26 e^5 \sin (c+d x) (e \cos (c+d x))^{5/2}}{35 a^3 d}+\frac{26 e^3 (e \cos (c+d x))^{9/2}}{45 a^3 d}+\frac{4 e (e \cos (c+d x))^{13/2}}{5 a d (a \sin (c+d x)+a)^2}",1,"(-4*2^(1/4)*(e*Cos[c + d*x])^(17/2)*Hypergeometric2F1[-1/4, 17/4, 21/4, (1 - Sin[c + d*x])/2])/(17*a^3*d*e*(1 + Sin[c + d*x])^(17/4))","C",1
254,1,66,138,0.2367408,"\int \frac{(e \cos (c+d x))^{13/2}}{(a+a \sin (c+d x))^3} \, dx","Integrate[(e*Cos[c + d*x])^(13/2)/(a + a*Sin[c + d*x])^3,x]","-\frac{2\ 2^{3/4} (e \cos (c+d x))^{15/2} \, _2F_1\left(\frac{1}{4},\frac{15}{4};\frac{19}{4};\frac{1}{2} (1-\sin (c+d x))\right)}{15 a^3 d e (\sin (c+d x)+1)^{15/4}}","\frac{22 e^6 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{5 a^3 d \sqrt{\cos (c+d x)}}+\frac{22 e^5 \sin (c+d x) (e \cos (c+d x))^{3/2}}{15 a^3 d}+\frac{22 e^3 (e \cos (c+d x))^{7/2}}{21 a^3 d}+\frac{4 e (e \cos (c+d x))^{11/2}}{3 a d (a \sin (c+d x)+a)^2}",1,"(-2*2^(3/4)*(e*Cos[c + d*x])^(15/2)*Hypergeometric2F1[1/4, 15/4, 19/4, (1 - Sin[c + d*x])/2])/(15*a^3*d*e*(1 + Sin[c + d*x])^(15/4))","C",1
255,1,66,132,0.163944,"\int \frac{(e \cos (c+d x))^{11/2}}{(a+a \sin (c+d x))^3} \, dx","Integrate[(e*Cos[c + d*x])^(11/2)/(a + a*Sin[c + d*x])^3,x]","-\frac{2 \sqrt[4]{2} (e \cos (c+d x))^{13/2} \, _2F_1\left(\frac{3}{4},\frac{13}{4};\frac{17}{4};\frac{1}{2} (1-\sin (c+d x))\right)}{13 a^3 d e (\sin (c+d x)+1)^{13/4}}","\frac{6 e^6 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^3 d \sqrt{e \cos (c+d x)}}+\frac{6 e^5 \sin (c+d x) \sqrt{e \cos (c+d x)}}{a^3 d}+\frac{18 e^3 (e \cos (c+d x))^{5/2}}{5 a^3 d}+\frac{4 e (e \cos (c+d x))^{9/2}}{a d (a \sin (c+d x)+a)^2}",1,"(-2*2^(1/4)*(e*Cos[c + d*x])^(13/2)*Hypergeometric2F1[3/4, 13/4, 17/4, (1 - Sin[c + d*x])/2])/(13*a^3*d*e*(1 + Sin[c + d*x])^(13/4))","C",1
256,1,66,103,0.1043215,"\int \frac{(e \cos (c+d x))^{9/2}}{(a+a \sin (c+d x))^3} \, dx","Integrate[(e*Cos[c + d*x])^(9/2)/(a + a*Sin[c + d*x])^3,x]","-\frac{2^{3/4} (e \cos (c+d x))^{11/2} \, _2F_1\left(\frac{5}{4},\frac{11}{4};\frac{15}{4};\frac{1}{2} (1-\sin (c+d x))\right)}{11 a^3 d e (\sin (c+d x)+1)^{11/4}}","-\frac{14 e^4 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{a^3 d \sqrt{\cos (c+d x)}}-\frac{14 e^3 (e \cos (c+d x))^{3/2}}{3 a^3 d}-\frac{4 e (e \cos (c+d x))^{7/2}}{a d (a \sin (c+d x)+a)^2}",1,"-1/11*(2^(3/4)*(e*Cos[c + d*x])^(11/2)*Hypergeometric2F1[5/4, 11/4, 15/4, (1 - Sin[c + d*x])/2])/(a^3*d*e*(1 + Sin[c + d*x])^(11/4))","C",1
257,1,66,107,0.0806966,"\int \frac{(e \cos (c+d x))^{7/2}}{(a+a \sin (c+d x))^3} \, dx","Integrate[(e*Cos[c + d*x])^(7/2)/(a + a*Sin[c + d*x])^3,x]","-\frac{\sqrt[4]{2} (e \cos (c+d x))^{9/2} \, _2F_1\left(\frac{7}{4},\frac{9}{4};\frac{13}{4};\frac{1}{2} (1-\sin (c+d x))\right)}{9 a^3 d e (\sin (c+d x)+1)^{9/4}}","-\frac{10 e^4 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^3 d \sqrt{e \cos (c+d x)}}-\frac{10 e^3 \sqrt{e \cos (c+d x)}}{3 a^3 d}-\frac{4 e (e \cos (c+d x))^{5/2}}{3 a d (a \sin (c+d x)+a)^2}",1,"-1/9*(2^(1/4)*(e*Cos[c + d*x])^(9/2)*Hypergeometric2F1[7/4, 9/4, 13/4, (1 - Sin[c + d*x])/2])/(a^3*d*e*(1 + Sin[c + d*x])^(9/4))","C",1
258,1,66,118,0.0813469,"\int \frac{(e \cos (c+d x))^{5/2}}{(a+a \sin (c+d x))^3} \, dx","Integrate[(e*Cos[c + d*x])^(5/2)/(a + a*Sin[c + d*x])^3,x]","-\frac{(e \cos (c+d x))^{7/2} \, _2F_1\left(\frac{7}{4},\frac{9}{4};\frac{11}{4};\frac{1}{2} (1-\sin (c+d x))\right)}{7 \sqrt[4]{2} a^3 d e (\sin (c+d x)+1)^{7/4}}","\frac{6 e^2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{5 a^3 d \sqrt{\cos (c+d x)}}+\frac{6 e (e \cos (c+d x))^{3/2}}{5 d \left(a^3 \sin (c+d x)+a^3\right)}-\frac{4 e (e \cos (c+d x))^{3/2}}{5 a d (a \sin (c+d x)+a)^2}",1,"-1/7*((e*Cos[c + d*x])^(7/2)*Hypergeometric2F1[7/4, 9/4, 11/4, (1 - Sin[c + d*x])/2])/(2^(1/4)*a^3*d*e*(1 + Sin[c + d*x])^(7/4))","C",1
259,1,66,118,0.0762769,"\int \frac{(e \cos (c+d x))^{3/2}}{(a+a \sin (c+d x))^3} \, dx","Integrate[(e*Cos[c + d*x])^(3/2)/(a + a*Sin[c + d*x])^3,x]","-\frac{(e \cos (c+d x))^{5/2} \, _2F_1\left(\frac{5}{4},\frac{11}{4};\frac{9}{4};\frac{1}{2} (1-\sin (c+d x))\right)}{5\ 2^{3/4} a^3 d e (\sin (c+d x)+1)^{5/4}}","-\frac{2 e^2 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 a^3 d \sqrt{e \cos (c+d x)}}+\frac{2 e \sqrt{e \cos (c+d x)}}{21 d \left(a^3 \sin (c+d x)+a^3\right)}-\frac{4 e \sqrt{e \cos (c+d x)}}{7 a d (a \sin (c+d x)+a)^2}",1,"-1/5*((e*Cos[c + d*x])^(5/2)*Hypergeometric2F1[5/4, 11/4, 9/4, (1 - Sin[c + d*x])/2])/(2^(3/4)*a^3*d*e*(1 + Sin[c + d*x])^(5/4))","C",1
260,1,66,153,0.0426435,"\int \frac{\sqrt{e \cos (c+d x)}}{(a+a \sin (c+d x))^3} \, dx","Integrate[Sqrt[e*Cos[c + d*x]]/(a + a*Sin[c + d*x])^3,x]","-\frac{(e \cos (c+d x))^{3/2} \, _2F_1\left(\frac{3}{4},\frac{13}{4};\frac{7}{4};\frac{1}{2} (1-\sin (c+d x))\right)}{6 \sqrt[4]{2} a^3 d e (\sin (c+d x)+1)^{3/4}}","-\frac{2 (e \cos (c+d x))^{3/2}}{15 d e \left(a^3 \sin (c+d x)+a^3\right)}-\frac{2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{15 a^3 d \sqrt{\cos (c+d x)}}-\frac{2 (e \cos (c+d x))^{3/2}}{15 a d e (a \sin (c+d x)+a)^2}-\frac{2 (e \cos (c+d x))^{3/2}}{9 d e (a \sin (c+d x)+a)^3}",1,"-1/6*((e*Cos[c + d*x])^(3/2)*Hypergeometric2F1[3/4, 13/4, 7/4, (1 - Sin[c + d*x])/2])/(2^(1/4)*a^3*d*e*(1 + Sin[c + d*x])^(3/4))","C",1
261,1,66,153,0.0487861,"\int \frac{1}{\sqrt{e \cos (c+d x)} (a+a \sin (c+d x))^3} \, dx","Integrate[1/(Sqrt[e*Cos[c + d*x]]*(a + a*Sin[c + d*x])^3),x]","-\frac{\sqrt{e \cos (c+d x)} \, _2F_1\left(\frac{1}{4},\frac{15}{4};\frac{5}{4};\frac{1}{2} (1-\sin (c+d x))\right)}{2\ 2^{3/4} a^3 d e \sqrt[4]{\sin (c+d x)+1}}","-\frac{10 \sqrt{e \cos (c+d x)}}{77 d e \left(a^3 \sin (c+d x)+a^3\right)}+\frac{10 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{77 a^3 d \sqrt{e \cos (c+d x)}}-\frac{10 \sqrt{e \cos (c+d x)}}{77 a d e (a \sin (c+d x)+a)^2}-\frac{2 \sqrt{e \cos (c+d x)}}{11 d e (a \sin (c+d x)+a)^3}",1,"-1/2*(Sqrt[e*Cos[c + d*x]]*Hypergeometric2F1[1/4, 15/4, 5/4, (1 - Sin[c + d*x])/2])/(2^(3/4)*a^3*d*e*(1 + Sin[c + d*x])^(1/4))","C",1
262,1,66,187,0.0639664,"\int \frac{1}{(e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^3} \, dx","Integrate[1/((e*Cos[c + d*x])^(3/2)*(a + a*Sin[c + d*x])^3),x]","\frac{\sqrt[4]{\sin (c+d x)+1} \, _2F_1\left(-\frac{1}{4},\frac{17}{4};\frac{3}{4};\frac{1}{2} (1-\sin (c+d x))\right)}{4 \sqrt[4]{2} a^3 d e \sqrt{e \cos (c+d x)}}","-\frac{14 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{39 a^3 d e^2 \sqrt{\cos (c+d x)}}+\frac{14 \sin (c+d x)}{39 a^3 d e \sqrt{e \cos (c+d x)}}-\frac{14}{117 d e \left(a^3 \sin (c+d x)+a^3\right) \sqrt{e \cos (c+d x)}}-\frac{14}{117 a d e (a \sin (c+d x)+a)^2 \sqrt{e \cos (c+d x)}}-\frac{2}{13 d e (a \sin (c+d x)+a)^3 \sqrt{e \cos (c+d x)}}",1,"(Hypergeometric2F1[-1/4, 17/4, 3/4, (1 - Sin[c + d*x])/2]*(1 + Sin[c + d*x])^(1/4))/(4*2^(1/4)*a^3*d*e*Sqrt[e*Cos[c + d*x]])","C",1
263,1,66,180,0.3222298,"\int \frac{(e \cos (c+d x))^{15/2}}{(a+a \sin (c+d x))^4} \, dx","Integrate[(e*Cos[c + d*x])^(15/2)/(a + a*Sin[c + d*x])^4,x]","-\frac{2 \sqrt[4]{2} (e \cos (c+d x))^{17/2} \, _2F_1\left(\frac{3}{4},\frac{17}{4};\frac{21}{4};\frac{1}{2} (1-\sin (c+d x))\right)}{17 a^4 d e (\sin (c+d x)+1)^{17/4}}","\frac{78 e^8 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{7 a^4 d \sqrt{e \cos (c+d x)}}+\frac{78 e^7 \sin (c+d x) \sqrt{e \cos (c+d x)}}{7 a^4 d}+\frac{234 e^5 \sin (c+d x) (e \cos (c+d x))^{5/2}}{35 a^4 d}+\frac{52 e^3 (e \cos (c+d x))^{9/2}}{5 d \left(a^4 \sin (c+d x)+a^4\right)}+\frac{4 e (e \cos (c+d x))^{13/2}}{a d (a \sin (c+d x)+a)^3}",1,"(-2*2^(1/4)*(e*Cos[c + d*x])^(17/2)*Hypergeometric2F1[3/4, 17/4, 21/4, (1 - Sin[c + d*x])/2])/(17*a^4*d*e*(1 + Sin[c + d*x])^(17/4))","C",1
264,1,66,149,0.1891153,"\int \frac{(e \cos (c+d x))^{13/2}}{(a+a \sin (c+d x))^4} \, dx","Integrate[(e*Cos[c + d*x])^(13/2)/(a + a*Sin[c + d*x])^4,x]","-\frac{2^{3/4} (e \cos (c+d x))^{15/2} \, _2F_1\left(\frac{5}{4},\frac{15}{4};\frac{19}{4};\frac{1}{2} (1-\sin (c+d x))\right)}{15 a^4 d e (\sin (c+d x)+1)^{15/4}}","-\frac{154 e^6 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{5 a^4 d \sqrt{\cos (c+d x)}}-\frac{154 e^5 \sin (c+d x) (e \cos (c+d x))^{3/2}}{15 a^4 d}-\frac{44 e^3 (e \cos (c+d x))^{7/2}}{3 d \left(a^4 \sin (c+d x)+a^4\right)}-\frac{4 e (e \cos (c+d x))^{11/2}}{a d (a \sin (c+d x)+a)^3}",1,"-1/15*(2^(3/4)*(e*Cos[c + d*x])^(15/2)*Hypergeometric2F1[5/4, 15/4, 19/4, (1 - Sin[c + d*x])/2])/(a^4*d*e*(1 + Sin[c + d*x])^(15/4))","C",1
265,1,66,145,0.1654316,"\int \frac{(e \cos (c+d x))^{11/2}}{(a+a \sin (c+d x))^4} \, dx","Integrate[(e*Cos[c + d*x])^(11/2)/(a + a*Sin[c + d*x])^4,x]","-\frac{\sqrt[4]{2} (e \cos (c+d x))^{13/2} \, _2F_1\left(\frac{7}{4},\frac{13}{4};\frac{17}{4};\frac{1}{2} (1-\sin (c+d x))\right)}{13 a^4 d e (\sin (c+d x)+1)^{13/4}}","-\frac{10 e^6 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^4 d \sqrt{e \cos (c+d x)}}-\frac{10 e^5 \sin (c+d x) \sqrt{e \cos (c+d x)}}{a^4 d}-\frac{12 e^3 (e \cos (c+d x))^{5/2}}{d \left(a^4 \sin (c+d x)+a^4\right)}-\frac{4 e (e \cos (c+d x))^{9/2}}{3 a d (a \sin (c+d x)+a)^3}",1,"-1/13*(2^(1/4)*(e*Cos[c + d*x])^(13/2)*Hypergeometric2F1[7/4, 13/4, 17/4, (1 - Sin[c + d*x])/2])/(a^4*d*e*(1 + Sin[c + d*x])^(13/4))","C",1
266,1,66,120,0.0942851,"\int \frac{(e \cos (c+d x))^{9/2}}{(a+a \sin (c+d x))^4} \, dx","Integrate[(e*Cos[c + d*x])^(9/2)/(a + a*Sin[c + d*x])^4,x]","-\frac{(e \cos (c+d x))^{11/2} \, _2F_1\left(\frac{9}{4},\frac{11}{4};\frac{15}{4};\frac{1}{2} (1-\sin (c+d x))\right)}{11 \sqrt[4]{2} a^4 d e (\sin (c+d x)+1)^{11/4}}","\frac{42 e^4 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{5 a^4 d \sqrt{\cos (c+d x)}}+\frac{28 e^3 (e \cos (c+d x))^{3/2}}{5 d \left(a^4 \sin (c+d x)+a^4\right)}-\frac{4 e (e \cos (c+d x))^{7/2}}{5 a d (a \sin (c+d x)+a)^3}",1,"-1/11*((e*Cos[c + d*x])^(11/2)*Hypergeometric2F1[9/4, 11/4, 15/4, (1 - Sin[c + d*x])/2])/(2^(1/4)*a^4*d*e*(1 + Sin[c + d*x])^(11/4))","C",1
267,1,66,120,0.0808947,"\int \frac{(e \cos (c+d x))^{7/2}}{(a+a \sin (c+d x))^4} \, dx","Integrate[(e*Cos[c + d*x])^(7/2)/(a + a*Sin[c + d*x])^4,x]","-\frac{(e \cos (c+d x))^{9/2} \, _2F_1\left(\frac{9}{4},\frac{11}{4};\frac{13}{4};\frac{1}{2} (1-\sin (c+d x))\right)}{9\ 2^{3/4} a^4 d e (\sin (c+d x)+1)^{9/4}}","\frac{10 e^4 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 a^4 d \sqrt{e \cos (c+d x)}}+\frac{20 e^3 \sqrt{e \cos (c+d x)}}{21 d \left(a^4 \sin (c+d x)+a^4\right)}-\frac{4 e (e \cos (c+d x))^{5/2}}{7 a d (a \sin (c+d x)+a)^3}",1,"-1/9*((e*Cos[c + d*x])^(9/2)*Hypergeometric2F1[9/4, 11/4, 13/4, (1 - Sin[c + d*x])/2])/(2^(3/4)*a^4*d*e*(1 + Sin[c + d*x])^(9/4))","C",1
268,1,66,154,0.0876255,"\int \frac{(e \cos (c+d x))^{5/2}}{(a+a \sin (c+d x))^4} \, dx","Integrate[(e*Cos[c + d*x])^(5/2)/(a + a*Sin[c + d*x])^4,x]","-\frac{(e \cos (c+d x))^{7/2} \, _2F_1\left(\frac{7}{4},\frac{13}{4};\frac{11}{4};\frac{1}{2} (1-\sin (c+d x))\right)}{14 \sqrt[4]{2} a^4 d e (\sin (c+d x)+1)^{7/4}}","\frac{2 e^2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{15 a^4 d \sqrt{\cos (c+d x)}}+\frac{2 e (e \cos (c+d x))^{3/2}}{15 d \left(a^4 \sin (c+d x)+a^4\right)}+\frac{2 e (e \cos (c+d x))^{3/2}}{15 d \left(a^2 \sin (c+d x)+a^2\right)^2}-\frac{4 e (e \cos (c+d x))^{3/2}}{9 a d (a \sin (c+d x)+a)^3}",1,"-1/14*((e*Cos[c + d*x])^(7/2)*Hypergeometric2F1[7/4, 13/4, 11/4, (1 - Sin[c + d*x])/2])/(2^(1/4)*a^4*d*e*(1 + Sin[c + d*x])^(7/4))","C",1
269,1,66,154,0.0712352,"\int \frac{(e \cos (c+d x))^{3/2}}{(a+a \sin (c+d x))^4} \, dx","Integrate[(e*Cos[c + d*x])^(3/2)/(a + a*Sin[c + d*x])^4,x]","-\frac{(e \cos (c+d x))^{5/2} \, _2F_1\left(\frac{5}{4},\frac{15}{4};\frac{9}{4};\frac{1}{2} (1-\sin (c+d x))\right)}{10\ 2^{3/4} a^4 d e (\sin (c+d x)+1)^{5/4}}","-\frac{2 e^2 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{77 a^4 d \sqrt{e \cos (c+d x)}}+\frac{2 e \sqrt{e \cos (c+d x)}}{77 d \left(a^4 \sin (c+d x)+a^4\right)}+\frac{2 e \sqrt{e \cos (c+d x)}}{77 d \left(a^2 \sin (c+d x)+a^2\right)^2}-\frac{4 e \sqrt{e \cos (c+d x)}}{11 a d (a \sin (c+d x)+a)^3}",1,"-1/10*((e*Cos[c + d*x])^(5/2)*Hypergeometric2F1[5/4, 15/4, 9/4, (1 - Sin[c + d*x])/2])/(2^(3/4)*a^4*d*e*(1 + Sin[c + d*x])^(5/4))","C",1
270,1,66,191,0.0457843,"\int \frac{\sqrt{e \cos (c+d x)}}{(a+a \sin (c+d x))^4} \, dx","Integrate[Sqrt[e*Cos[c + d*x]]/(a + a*Sin[c + d*x])^4,x]","-\frac{(e \cos (c+d x))^{3/2} \, _2F_1\left(\frac{3}{4},\frac{17}{4};\frac{7}{4};\frac{1}{2} (1-\sin (c+d x))\right)}{12 \sqrt[4]{2} a^4 d e (\sin (c+d x)+1)^{3/4}}","-\frac{2 (e \cos (c+d x))^{3/2}}{39 d e \left(a^4 \sin (c+d x)+a^4\right)}-\frac{2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{39 a^4 d \sqrt{\cos (c+d x)}}-\frac{2 (e \cos (c+d x))^{3/2}}{39 d e \left(a^2 \sin (c+d x)+a^2\right)^2}-\frac{10 (e \cos (c+d x))^{3/2}}{117 a d e (a \sin (c+d x)+a)^3}-\frac{2 (e \cos (c+d x))^{3/2}}{13 d e (a \sin (c+d x)+a)^4}",1,"-1/12*((e*Cos[c + d*x])^(3/2)*Hypergeometric2F1[3/4, 17/4, 7/4, (1 - Sin[c + d*x])/2])/(2^(1/4)*a^4*d*e*(1 + Sin[c + d*x])^(3/4))","C",1
271,1,66,191,0.0583017,"\int \frac{1}{\sqrt{e \cos (c+d x)} (a+a \sin (c+d x))^4} \, dx","Integrate[1/(Sqrt[e*Cos[c + d*x]]*(a + a*Sin[c + d*x])^4),x]","-\frac{\sqrt{e \cos (c+d x)} \, _2F_1\left(\frac{1}{4},\frac{19}{4};\frac{5}{4};\frac{1}{2} (1-\sin (c+d x))\right)}{4\ 2^{3/4} a^4 d e \sqrt[4]{\sin (c+d x)+1}}","-\frac{2 \sqrt{e \cos (c+d x)}}{33 d e \left(a^4 \sin (c+d x)+a^4\right)}+\frac{2 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{33 a^4 d \sqrt{e \cos (c+d x)}}-\frac{2 \sqrt{e \cos (c+d x)}}{33 d e \left(a^2 \sin (c+d x)+a^2\right)^2}-\frac{14 \sqrt{e \cos (c+d x)}}{165 a d e (a \sin (c+d x)+a)^3}-\frac{2 \sqrt{e \cos (c+d x)}}{15 d e (a \sin (c+d x)+a)^4}",1,"-1/4*(Sqrt[e*Cos[c + d*x]]*Hypergeometric2F1[1/4, 19/4, 5/4, (1 - Sin[c + d*x])/2])/(2^(3/4)*a^4*d*e*(1 + Sin[c + d*x])^(1/4))","C",1
272,1,66,225,0.088007,"\int \frac{1}{(e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^4} \, dx","Integrate[1/((e*Cos[c + d*x])^(3/2)*(a + a*Sin[c + d*x])^4),x]","\frac{\sqrt[4]{\sin (c+d x)+1} \, _2F_1\left(-\frac{1}{4},\frac{21}{4};\frac{3}{4};\frac{1}{2} (1-\sin (c+d x))\right)}{8 \sqrt[4]{2} a^4 d e \sqrt{e \cos (c+d x)}}","-\frac{42 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{221 a^4 d e^2 \sqrt{\cos (c+d x)}}+\frac{42 \sin (c+d x)}{221 a^4 d e \sqrt{e \cos (c+d x)}}-\frac{14}{221 d e \left(a^4 \sin (c+d x)+a^4\right) \sqrt{e \cos (c+d x)}}-\frac{14}{221 d e \left(a^2 \sin (c+d x)+a^2\right)^2 \sqrt{e \cos (c+d x)}}-\frac{18}{221 a d e (a \sin (c+d x)+a)^3 \sqrt{e \cos (c+d x)}}-\frac{2}{17 d e (a \sin (c+d x)+a)^4 \sqrt{e \cos (c+d x)}}",1,"(Hypergeometric2F1[-1/4, 21/4, 3/4, (1 - Sin[c + d*x])/2]*(1 + Sin[c + d*x])^(1/4))/(8*2^(1/4)*a^4*d*e*Sqrt[e*Cos[c + d*x]])","C",1
273,1,269,236,0.9549185,"\int (e \cos (c+d x))^{3/2} \sqrt{a+a \sin (c+d x)} \, dx","Integrate[(e*Cos[c + d*x])^(3/2)*Sqrt[a + a*Sin[c + d*x]],x]","-\frac{i e e^{-i (c+d x)} \sqrt{a (\sin (c+d x)+1)} \sqrt{e \cos (c+d x)} \left(-3 d x e^{2 i (c+d x)}-2 e^{i (c+d x)} \sqrt{1+e^{2 i (c+d x)}}+2 i e^{2 i (c+d x)} \sqrt{1+e^{2 i (c+d x)}}+e^{3 i (c+d x)} \sqrt{1+e^{2 i (c+d x)}}-i \sqrt{1+e^{2 i (c+d x)}}-3 i e^{2 i (c+d x)} \log \left(1+\sqrt{1+e^{2 i (c+d x)}}\right)+3 e^{2 i (c+d x)} \sinh ^{-1}\left(e^{i (c+d x)}\right)\right)}{4 d \left(e^{i (c+d x)}+i\right) \sqrt{1+e^{2 i (c+d x)}}}","\frac{3 e^{3/2} \sqrt{\cos (c+d x)+1} \sqrt{a \sin (c+d x)+a} \tan ^{-1}\left(\frac{\sqrt{e} \sin (c+d x)}{\sqrt{\cos (c+d x)+1} \sqrt{e \cos (c+d x)}}\right)}{4 d (\sin (c+d x)+\cos (c+d x)+1)}-\frac{3 e^{3/2} \sqrt{\cos (c+d x)+1} \sqrt{a \sin (c+d x)+a} \sinh ^{-1}\left(\frac{\sqrt{e \cos (c+d x)}}{\sqrt{e}}\right)}{4 d (\sin (c+d x)+\cos (c+d x)+1)}-\frac{a (e \cos (c+d x))^{5/2}}{2 d e \sqrt{a \sin (c+d x)+a}}+\frac{3 e \sqrt{a \sin (c+d x)+a} \sqrt{e \cos (c+d x)}}{4 d}",1,"((-1/4*I)*e*Sqrt[e*Cos[c + d*x]]*((-I)*Sqrt[1 + E^((2*I)*(c + d*x))] - 2*E^(I*(c + d*x))*Sqrt[1 + E^((2*I)*(c + d*x))] + (2*I)*E^((2*I)*(c + d*x))*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((3*I)*(c + d*x))*Sqrt[1 + E^((2*I)*(c + d*x))] - 3*d*E^((2*I)*(c + d*x))*x + 3*E^((2*I)*(c + d*x))*ArcSinh[E^(I*(c + d*x))] - (3*I)*E^((2*I)*(c + d*x))*Log[1 + Sqrt[1 + E^((2*I)*(c + d*x))]])*Sqrt[a*(1 + Sin[c + d*x])])/(d*E^(I*(c + d*x))*(I + E^(I*(c + d*x)))*Sqrt[1 + E^((2*I)*(c + d*x))])","C",1
274,1,195,194,0.7398849,"\int \sqrt{e \cos (c+d x)} \sqrt{a+a \sin (c+d x)} \, dx","Integrate[Sqrt[e*Cos[c + d*x]]*Sqrt[a + a*Sin[c + d*x]],x]","-\frac{i \sqrt{a (\sin (c+d x)+1)} \sqrt{e \cos (c+d x)} \left(i d x e^{i (c+d x)}+e^{i (c+d x)} \sqrt{1+e^{2 i (c+d x)}}-i \sqrt{1+e^{2 i (c+d x)}}-e^{i (c+d x)} \log \left(1+\sqrt{1+e^{2 i (c+d x)}}\right)+i e^{i (c+d x)} \sinh ^{-1}\left(e^{i (c+d x)}\right)\right)}{d \left(e^{i (c+d x)}+i\right) \sqrt{1+e^{2 i (c+d x)}}}","-\frac{a (e \cos (c+d x))^{3/2}}{d e \sqrt{a \sin (c+d x)+a}}+\frac{\sqrt{e} \sqrt{\cos (c+d x)+1} \sqrt{a \sin (c+d x)+a} \tan ^{-1}\left(\frac{\sqrt{e} \sin (c+d x)}{\sqrt{\cos (c+d x)+1} \sqrt{e \cos (c+d x)}}\right)}{d (\sin (c+d x)+\cos (c+d x)+1)}+\frac{\sqrt{e} \sqrt{\cos (c+d x)+1} \sqrt{a \sin (c+d x)+a} \sinh ^{-1}\left(\frac{\sqrt{e \cos (c+d x)}}{\sqrt{e}}\right)}{d (\sin (c+d x)+\cos (c+d x)+1)}",1,"((-I)*Sqrt[e*Cos[c + d*x]]*((-I)*Sqrt[1 + E^((2*I)*(c + d*x))] + E^(I*(c + d*x))*Sqrt[1 + E^((2*I)*(c + d*x))] + I*d*E^(I*(c + d*x))*x + I*E^(I*(c + d*x))*ArcSinh[E^(I*(c + d*x))] - E^(I*(c + d*x))*Log[1 + Sqrt[1 + E^((2*I)*(c + d*x))]])*Sqrt[a*(1 + Sin[c + d*x])])/(d*(I + E^(I*(c + d*x)))*Sqrt[1 + E^((2*I)*(c + d*x))])","C",1
275,1,108,161,0.4720717,"\int \frac{\sqrt{a+a \sin (c+d x)}}{\sqrt{e \cos (c+d x)}} \, dx","Integrate[Sqrt[a + a*Sin[c + d*x]]/Sqrt[e*Cos[c + d*x]],x]","\frac{\sqrt{1+e^{2 i (c+d x)}} \sqrt{a (\sin (c+d x)+1)} \left(i \log \left(1+\sqrt{1+e^{2 i (c+d x)}}\right)-\sinh ^{-1}\left(e^{i (c+d x)}\right)+d x\right)}{d \left(1-i e^{i (c+d x)}\right) \sqrt{e \cos (c+d x)}}","\frac{2 \sqrt{\cos (c+d x)+1} \sqrt{a \sin (c+d x)+a} \tan ^{-1}\left(\frac{\sqrt{e} \sin (c+d x)}{\sqrt{\cos (c+d x)+1} \sqrt{e \cos (c+d x)}}\right)}{d \sqrt{e} (\sin (c+d x)+\cos (c+d x)+1)}-\frac{2 \sqrt{\cos (c+d x)+1} \sqrt{a \sin (c+d x)+a} \sinh ^{-1}\left(\frac{\sqrt{e \cos (c+d x)}}{\sqrt{e}}\right)}{d \sqrt{e} (\sin (c+d x)+\cos (c+d x)+1)}",1,"(Sqrt[1 + E^((2*I)*(c + d*x))]*(d*x - ArcSinh[E^(I*(c + d*x))] + I*Log[1 + Sqrt[1 + E^((2*I)*(c + d*x))]])*Sqrt[a*(1 + Sin[c + d*x])])/(d*(1 - I*E^(I*(c + d*x)))*Sqrt[e*Cos[c + d*x]])","C",1
276,1,34,34,0.1125776,"\int \frac{\sqrt{a+a \sin (c+d x)}}{(e \cos (c+d x))^{3/2}} \, dx","Integrate[Sqrt[a + a*Sin[c + d*x]]/(e*Cos[c + d*x])^(3/2),x]","\frac{2 \sqrt{a (\sin (c+d x)+1)}}{d e \sqrt{e \cos (c+d x)}}","\frac{2 \sqrt{a \sin (c+d x)+a}}{d e \sqrt{e \cos (c+d x)}}",1,"(2*Sqrt[a*(1 + Sin[c + d*x])])/(d*e*Sqrt[e*Cos[c + d*x]])","A",1
277,1,46,74,0.2496909,"\int \frac{\sqrt{a+a \sin (c+d x)}}{(e \cos (c+d x))^{5/2}} \, dx","Integrate[Sqrt[a + a*Sin[c + d*x]]/(e*Cos[c + d*x])^(5/2),x]","\frac{2 (2 \sin (c+d x)-1) \sqrt{a (\sin (c+d x)+1)}}{3 d e (e \cos (c+d x))^{3/2}}","\frac{4 (a \sin (c+d x)+a)^{3/2}}{3 a d e (e \cos (c+d x))^{3/2}}-\frac{2 \sqrt{a \sin (c+d x)+a}}{d e (e \cos (c+d x))^{3/2}}",1,"(2*Sqrt[a*(1 + Sin[c + d*x])]*(-1 + 2*Sin[c + d*x]))/(3*d*e*(e*Cos[c + d*x])^(3/2))","A",1
278,1,56,115,0.3440466,"\int \frac{\sqrt{a+a \sin (c+d x)}}{(e \cos (c+d x))^{7/2}} \, dx","Integrate[Sqrt[a + a*Sin[c + d*x]]/(e*Cos[c + d*x])^(7/2),x]","\frac{2 \sqrt{a (\sin (c+d x)+1)} (4 \sin (c+d x)+4 \cos (2 (c+d x))+3)}{15 d e (e \cos (c+d x))^{5/2}}","-\frac{16 (a \sin (c+d x)+a)^{5/2}}{15 a^2 d e (e \cos (c+d x))^{5/2}}+\frac{8 (a \sin (c+d x)+a)^{3/2}}{3 a d e (e \cos (c+d x))^{5/2}}-\frac{2 \sqrt{a \sin (c+d x)+a}}{3 d e (e \cos (c+d x))^{5/2}}",1,"(2*Sqrt[a*(1 + Sin[c + d*x])]*(3 + 4*Cos[2*(c + d*x)] + 4*Sin[c + d*x]))/(15*d*e*(e*Cos[c + d*x])^(5/2))","A",1
279,1,74,154,0.7958588,"\int \frac{\sqrt{a+a \sin (c+d x)}}{(e \cos (c+d x))^{9/2}} \, dx","Integrate[Sqrt[a + a*Sin[c + d*x]]/(e*Cos[c + d*x])^(9/2),x]","\frac{2 \sec ^4(c+d x) \sqrt{a (\sin (c+d x)+1)} \sqrt{e \cos (c+d x)} (10 \sin (c+d x)+4 \sin (3 (c+d x))-4 \cos (2 (c+d x))-5)}{35 d e^5}","-\frac{32 (a \sin (c+d x)+a)^{7/2}}{35 a^3 d e (e \cos (c+d x))^{7/2}}+\frac{16 (a \sin (c+d x)+a)^{5/2}}{5 a^2 d e (e \cos (c+d x))^{7/2}}-\frac{12 (a \sin (c+d x)+a)^{3/2}}{5 a d e (e \cos (c+d x))^{7/2}}-\frac{2 \sqrt{a \sin (c+d x)+a}}{5 d e (e \cos (c+d x))^{7/2}}",1,"(2*Sqrt[e*Cos[c + d*x]]*Sec[c + d*x]^4*Sqrt[a*(1 + Sin[c + d*x])]*(-5 - 4*Cos[2*(c + d*x)] + 10*Sin[c + d*x] + 4*Sin[3*(c + d*x)]))/(35*d*e^5)","A",1
280,1,78,319,0.1681881,"\int (e \cos (c+d x))^{5/2} (a+a \sin (c+d x))^{3/2} \, dx","Integrate[(e*Cos[c + d*x])^(5/2)*(a + a*Sin[c + d*x])^(3/2),x]","-\frac{16 \sqrt[4]{2} a \sqrt{a (\sin (c+d x)+1)} (e \cos (c+d x))^{7/2} \, _2F_1\left(-\frac{9}{4},\frac{7}{4};\frac{11}{4};\frac{1}{2} (1-\sin (c+d x))\right)}{7 d e (\sin (c+d x)+1)^{9/4}}","-\frac{15 a^3 (e \cos (c+d x))^{7/2}}{32 d e (a \sin (c+d x)+a)^{3/2}}-\frac{3 a^2 (e \cos (c+d x))^{7/2}}{8 d e \sqrt{a \sin (c+d x)+a}}+\frac{15 a^2 e (e \cos (c+d x))^{3/2}}{64 d \sqrt{a \sin (c+d x)+a}}+\frac{45 a e^{5/2} \sqrt{\cos (c+d x)+1} \sqrt{a \sin (c+d x)+a} \tan ^{-1}\left(\frac{\sqrt{e} \sin (c+d x)}{\sqrt{\cos (c+d x)+1} \sqrt{e \cos (c+d x)}}\right)}{64 d (\sin (c+d x)+\cos (c+d x)+1)}+\frac{45 a e^{5/2} \sqrt{\cos (c+d x)+1} \sqrt{a \sin (c+d x)+a} \sinh ^{-1}\left(\frac{\sqrt{e \cos (c+d x)}}{\sqrt{e}}\right)}{64 d (\sin (c+d x)+\cos (c+d x)+1)}-\frac{a \sqrt{a \sin (c+d x)+a} (e \cos (c+d x))^{7/2}}{4 d e}",1,"(-16*2^(1/4)*a*(e*Cos[c + d*x])^(7/2)*Hypergeometric2F1[-9/4, 7/4, 11/4, (1 - Sin[c + d*x])/2]*Sqrt[a*(1 + Sin[c + d*x])])/(7*d*e*(1 + Sin[c + d*x])^(9/4))","C",1
281,1,78,278,0.1315133,"\int (e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^{3/2} \, dx","Integrate[(e*Cos[c + d*x])^(3/2)*(a + a*Sin[c + d*x])^(3/2),x]","-\frac{8\ 2^{3/4} a \sqrt{a (\sin (c+d x)+1)} (e \cos (c+d x))^{5/2} \, _2F_1\left(-\frac{7}{4},\frac{5}{4};\frac{9}{4};\frac{1}{2} (1-\sin (c+d x))\right)}{5 d e (\sin (c+d x)+1)^{7/4}}","-\frac{7 a^2 (e \cos (c+d x))^{5/2}}{12 d e \sqrt{a \sin (c+d x)+a}}+\frac{7 a e^{3/2} \sqrt{\cos (c+d x)+1} \sqrt{a \sin (c+d x)+a} \tan ^{-1}\left(\frac{\sqrt{e} \sin (c+d x)}{\sqrt{\cos (c+d x)+1} \sqrt{e \cos (c+d x)}}\right)}{8 d (\sin (c+d x)+\cos (c+d x)+1)}-\frac{7 a e^{3/2} \sqrt{\cos (c+d x)+1} \sqrt{a \sin (c+d x)+a} \sinh ^{-1}\left(\frac{\sqrt{e \cos (c+d x)}}{\sqrt{e}}\right)}{8 d (\sin (c+d x)+\cos (c+d x)+1)}-\frac{a \sqrt{a \sin (c+d x)+a} (e \cos (c+d x))^{5/2}}{3 d e}+\frac{7 a e \sqrt{a \sin (c+d x)+a} \sqrt{e \cos (c+d x)}}{8 d}",1,"(-8*2^(3/4)*a*(e*Cos[c + d*x])^(5/2)*Hypergeometric2F1[-7/4, 5/4, 9/4, (1 - Sin[c + d*x])/2]*Sqrt[a*(1 + Sin[c + d*x])])/(5*d*e*(1 + Sin[c + d*x])^(7/4))","C",1
282,1,77,243,0.1207678,"\int \sqrt{e \cos (c+d x)} (a+a \sin (c+d x))^{3/2} \, dx","Integrate[Sqrt[e*Cos[c + d*x]]*(a + a*Sin[c + d*x])^(3/2),x]","-\frac{8 \sqrt[4]{2} (a (\sin (c+d x)+1))^{3/2} (e \cos (c+d x))^{3/2} \, _2F_1\left(-\frac{5}{4},\frac{3}{4};\frac{7}{4};\frac{1}{2} (1-\sin (c+d x))\right)}{3 d e (\sin (c+d x)+1)^{9/4}}","-\frac{5 a^2 (e \cos (c+d x))^{3/2}}{4 d e \sqrt{a \sin (c+d x)+a}}-\frac{a \sqrt{a \sin (c+d x)+a} (e \cos (c+d x))^{3/2}}{2 d e}+\frac{5 a \sqrt{e} \sqrt{\cos (c+d x)+1} \sqrt{a \sin (c+d x)+a} \tan ^{-1}\left(\frac{\sqrt{e} \sin (c+d x)}{\sqrt{\cos (c+d x)+1} \sqrt{e \cos (c+d x)}}\right)}{4 d (\sin (c+d x)+\cos (c+d x)+1)}+\frac{5 a \sqrt{e} \sqrt{\cos (c+d x)+1} \sqrt{a \sin (c+d x)+a} \sinh ^{-1}\left(\frac{\sqrt{e \cos (c+d x)}}{\sqrt{e}}\right)}{4 d (\sin (c+d x)+\cos (c+d x)+1)}",1,"(-8*2^(1/4)*(e*Cos[c + d*x])^(3/2)*Hypergeometric2F1[-5/4, 3/4, 7/4, (1 - Sin[c + d*x])/2]*(a*(1 + Sin[c + d*x]))^(3/2))/(3*d*e*(1 + Sin[c + d*x])^(9/4))","C",1
283,1,75,198,0.1063889,"\int \frac{(a+a \sin (c+d x))^{3/2}}{\sqrt{e \cos (c+d x)}} \, dx","Integrate[(a + a*Sin[c + d*x])^(3/2)/Sqrt[e*Cos[c + d*x]],x]","-\frac{4\ 2^{3/4} (a (\sin (c+d x)+1))^{3/2} \sqrt{e \cos (c+d x)} \, _2F_1\left(-\frac{3}{4},\frac{1}{4};\frac{5}{4};\frac{1}{2} (1-\sin (c+d x))\right)}{d e (\sin (c+d x)+1)^{7/4}}","-\frac{a \sqrt{a \sin (c+d x)+a} \sqrt{e \cos (c+d x)}}{d e}+\frac{3 a \sqrt{\cos (c+d x)+1} \sqrt{a \sin (c+d x)+a} \tan ^{-1}\left(\frac{\sqrt{e} \sin (c+d x)}{\sqrt{\cos (c+d x)+1} \sqrt{e \cos (c+d x)}}\right)}{d \sqrt{e} (\sin (c+d x)+\cos (c+d x)+1)}-\frac{3 a \sqrt{\cos (c+d x)+1} \sqrt{a \sin (c+d x)+a} \sinh ^{-1}\left(\frac{\sqrt{e \cos (c+d x)}}{\sqrt{e}}\right)}{d \sqrt{e} (\sin (c+d x)+\cos (c+d x)+1)}",1,"(-4*2^(3/4)*Sqrt[e*Cos[c + d*x]]*Hypergeometric2F1[-3/4, 1/4, 5/4, (1 - Sin[c + d*x])/2]*(a*(1 + Sin[c + d*x]))^(3/2))/(d*e*(1 + Sin[c + d*x])^(7/4))","C",1
284,1,75,210,0.1126255,"\int \frac{(a+a \sin (c+d x))^{3/2}}{(e \cos (c+d x))^{3/2}} \, dx","Integrate[(a + a*Sin[c + d*x])^(3/2)/(e*Cos[c + d*x])^(3/2),x]","\frac{4 \sqrt[4]{2} (a (\sin (c+d x)+1))^{3/2} \, _2F_1\left(-\frac{1}{4},-\frac{1}{4};\frac{3}{4};\frac{1}{2} (1-\sin (c+d x))\right)}{d e (\sin (c+d x)+1)^{5/4} \sqrt{e \cos (c+d x)}}","-\frac{2 a^2 \sqrt{\cos (c+d x)+1} \sqrt{a \sin (c+d x)+a} \tan ^{-1}\left(\frac{\sqrt{e} \sin (c+d x)}{\sqrt{\cos (c+d x)+1} \sqrt{e \cos (c+d x)}}\right)}{d e^{3/2} (a \sin (c+d x)+a \cos (c+d x)+a)}-\frac{2 a^2 \sqrt{\cos (c+d x)+1} \sqrt{a \sin (c+d x)+a} \sinh ^{-1}\left(\frac{\sqrt{e \cos (c+d x)}}{\sqrt{e}}\right)}{d e^{3/2} (a \sin (c+d x)+a \cos (c+d x)+a)}+\frac{4 a \sqrt{a \sin (c+d x)+a}}{d e \sqrt{e \cos (c+d x)}}",1,"(4*2^(1/4)*Hypergeometric2F1[-1/4, -1/4, 3/4, (1 - Sin[c + d*x])/2]*(a*(1 + Sin[c + d*x]))^(3/2))/(d*e*Sqrt[e*Cos[c + d*x]]*(1 + Sin[c + d*x])^(5/4))","C",1
285,1,36,36,0.09632,"\int \frac{(a+a \sin (c+d x))^{3/2}}{(e \cos (c+d x))^{5/2}} \, dx","Integrate[(a + a*Sin[c + d*x])^(3/2)/(e*Cos[c + d*x])^(5/2),x]","\frac{2 (a (\sin (c+d x)+1))^{3/2}}{3 d e (e \cos (c+d x))^{3/2}}","\frac{2 (a \sin (c+d x)+a)^{3/2}}{3 d e (e \cos (c+d x))^{3/2}}",1,"(2*(a*(1 + Sin[c + d*x]))^(3/2))/(3*d*e*(e*Cos[c + d*x])^(3/2))","A",1
286,1,72,74,0.1345135,"\int \frac{(a+a \sin (c+d x))^{3/2}}{(e \cos (c+d x))^{7/2}} \, dx","Integrate[(a + a*Sin[c + d*x])^(3/2)/(e*Cos[c + d*x])^(7/2),x]","-\frac{2 a (2 \sin (c+d x)-3) \sqrt{a (\sin (c+d x)+1)}}{5 d e^3 \sqrt{e \cos (c+d x)} \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^2}","\frac{2 (a \sin (c+d x)+a)^{3/2}}{d e (e \cos (c+d x))^{5/2}}-\frac{4 (a \sin (c+d x)+a)^{5/2}}{5 a d e (e \cos (c+d x))^{5/2}}",1,"(-2*a*Sqrt[a*(1 + Sin[c + d*x])]*(-3 + 2*Sin[c + d*x]))/(5*d*e^3*Sqrt[e*Cos[c + d*x]]*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2)","A",1
287,1,105,113,0.2024683,"\int \frac{(a+a \sin (c+d x))^{3/2}}{(e \cos (c+d x))^{9/2}} \, dx","Integrate[(a + a*Sin[c + d*x])^(3/2)/(e*Cos[c + d*x])^(9/2),x]","\frac{2 a \sqrt{a (\sin (c+d x)+1)} (12 \sin (c+d x)+4 \cos (2 (c+d x))-5)}{21 d e^4 \sqrt{e \cos (c+d x)} \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^3 \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}","-\frac{16 (a \sin (c+d x)+a)^{7/2}}{21 a^2 d e (e \cos (c+d x))^{7/2}}+\frac{8 (a \sin (c+d x)+a)^{5/2}}{3 a d e (e \cos (c+d x))^{7/2}}-\frac{2 (a \sin (c+d x)+a)^{3/2}}{d e (e \cos (c+d x))^{7/2}}",1,"(2*a*Sqrt[a*(1 + Sin[c + d*x])]*(-5 + 4*Cos[2*(c + d*x)] + 12*Sin[c + d*x]))/(21*d*e^4*Sqrt[e*Cos[c + d*x]]*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^3*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))","A",1
288,1,74,152,0.2366146,"\int \frac{(a+a \sin (c+d x))^{3/2}}{(e \cos (c+d x))^{11/2}} \, dx","Integrate[(a + a*Sin[c + d*x])^(3/2)/(e*Cos[c + d*x])^(11/2),x]","\frac{2 \sec ^5(c+d x) (a (\sin (c+d x)+1))^{3/2} \sqrt{e \cos (c+d x)} (6 \sin (c+d x)-4 \sin (3 (c+d x))+12 \cos (2 (c+d x))+7)}{45 d e^6}","\frac{32 (a \sin (c+d x)+a)^{9/2}}{45 a^3 d e (e \cos (c+d x))^{9/2}}-\frac{16 (a \sin (c+d x)+a)^{7/2}}{5 a^2 d e (e \cos (c+d x))^{9/2}}+\frac{4 (a \sin (c+d x)+a)^{5/2}}{a d e (e \cos (c+d x))^{9/2}}-\frac{2 (a \sin (c+d x)+a)^{3/2}}{3 d e (e \cos (c+d x))^{9/2}}",1,"(2*Sqrt[e*Cos[c + d*x]]*Sec[c + d*x]^5*(a*(1 + Sin[c + d*x]))^(3/2)*(7 + 12*Cos[2*(c + d*x)] + 6*Sin[c + d*x] - 4*Sin[3*(c + d*x)]))/(45*d*e^6)","A",1
289,1,77,323,0.2967172,"\int (e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^{5/2} \, dx","Integrate[(e*Cos[c + d*x])^(3/2)*(a + a*Sin[c + d*x])^(5/2),x]","-\frac{16\ 2^{3/4} (a (\sin (c+d x)+1))^{5/2} (e \cos (c+d x))^{5/2} \, _2F_1\left(-\frac{11}{4},\frac{5}{4};\frac{9}{4};\frac{1}{2} (1-\sin (c+d x))\right)}{5 d e (\sin (c+d x)+1)^{15/4}}","-\frac{77 a^3 (e \cos (c+d x))^{5/2}}{96 d e \sqrt{a \sin (c+d x)+a}}+\frac{77 a^2 e^{3/2} \sqrt{\cos (c+d x)+1} \sqrt{a \sin (c+d x)+a} \tan ^{-1}\left(\frac{\sqrt{e} \sin (c+d x)}{\sqrt{\cos (c+d x)+1} \sqrt{e \cos (c+d x)}}\right)}{64 d (\sin (c+d x)+\cos (c+d x)+1)}-\frac{77 a^2 e^{3/2} \sqrt{\cos (c+d x)+1} \sqrt{a \sin (c+d x)+a} \sinh ^{-1}\left(\frac{\sqrt{e \cos (c+d x)}}{\sqrt{e}}\right)}{64 d (\sin (c+d x)+\cos (c+d x)+1)}-\frac{11 a^2 \sqrt{a \sin (c+d x)+a} (e \cos (c+d x))^{5/2}}{24 d e}+\frac{77 a^2 e \sqrt{a \sin (c+d x)+a} \sqrt{e \cos (c+d x)}}{64 d}-\frac{a (a \sin (c+d x)+a)^{3/2} (e \cos (c+d x))^{5/2}}{4 d e}",1,"(-16*2^(3/4)*(e*Cos[c + d*x])^(5/2)*Hypergeometric2F1[-11/4, 5/4, 9/4, (1 - Sin[c + d*x])/2]*(a*(1 + Sin[c + d*x]))^(5/2))/(5*d*e*(1 + Sin[c + d*x])^(15/4))","C",1
290,1,78,286,0.1192488,"\int \sqrt{e \cos (c+d x)} (a+a \sin (c+d x))^{5/2} \, dx","Integrate[Sqrt[e*Cos[c + d*x]]*(a + a*Sin[c + d*x])^(5/2),x]","-\frac{16 \sqrt[4]{2} a (a (\sin (c+d x)+1))^{3/2} (e \cos (c+d x))^{3/2} \, _2F_1\left(-\frac{9}{4},\frac{3}{4};\frac{7}{4};\frac{1}{2} (1-\sin (c+d x))\right)}{3 d e (\sin (c+d x)+1)^{9/4}}","-\frac{15 a^3 (e \cos (c+d x))^{3/2}}{8 d e \sqrt{a \sin (c+d x)+a}}-\frac{3 a^2 \sqrt{a \sin (c+d x)+a} (e \cos (c+d x))^{3/2}}{4 d e}+\frac{15 a^2 \sqrt{e} \sqrt{\cos (c+d x)+1} \sqrt{a \sin (c+d x)+a} \tan ^{-1}\left(\frac{\sqrt{e} \sin (c+d x)}{\sqrt{\cos (c+d x)+1} \sqrt{e \cos (c+d x)}}\right)}{8 d (\sin (c+d x)+\cos (c+d x)+1)}+\frac{15 a^2 \sqrt{e} \sqrt{\cos (c+d x)+1} \sqrt{a \sin (c+d x)+a} \sinh ^{-1}\left(\frac{\sqrt{e \cos (c+d x)}}{\sqrt{e}}\right)}{8 d (\sin (c+d x)+\cos (c+d x)+1)}-\frac{a (a \sin (c+d x)+a)^{3/2} (e \cos (c+d x))^{3/2}}{3 d e}",1,"(-16*2^(1/4)*a*(e*Cos[c + d*x])^(3/2)*Hypergeometric2F1[-9/4, 3/4, 7/4, (1 - Sin[c + d*x])/2]*(a*(1 + Sin[c + d*x]))^(3/2))/(3*d*e*(1 + Sin[c + d*x])^(9/4))","C",1
291,1,76,247,0.1017215,"\int \frac{(a+a \sin (c+d x))^{5/2}}{\sqrt{e \cos (c+d x)}} \, dx","Integrate[(a + a*Sin[c + d*x])^(5/2)/Sqrt[e*Cos[c + d*x]],x]","-\frac{8\ 2^{3/4} a (a (\sin (c+d x)+1))^{3/2} \sqrt{e \cos (c+d x)} \, _2F_1\left(-\frac{7}{4},\frac{1}{4};\frac{5}{4};\frac{1}{2} (1-\sin (c+d x))\right)}{d e (\sin (c+d x)+1)^{7/4}}","-\frac{7 a^2 \sqrt{a \sin (c+d x)+a} \sqrt{e \cos (c+d x)}}{4 d e}+\frac{21 a^2 \sqrt{\cos (c+d x)+1} \sqrt{a \sin (c+d x)+a} \tan ^{-1}\left(\frac{\sqrt{e} \sin (c+d x)}{\sqrt{\cos (c+d x)+1} \sqrt{e \cos (c+d x)}}\right)}{4 d \sqrt{e} (\sin (c+d x)+\cos (c+d x)+1)}-\frac{21 a^2 \sqrt{\cos (c+d x)+1} \sqrt{a \sin (c+d x)+a} \sinh ^{-1}\left(\frac{\sqrt{e \cos (c+d x)}}{\sqrt{e}}\right)}{4 d \sqrt{e} (\sin (c+d x)+\cos (c+d x)+1)}-\frac{a (a \sin (c+d x)+a)^{3/2} \sqrt{e \cos (c+d x)}}{2 d e}",1,"(-8*2^(3/4)*a*Sqrt[e*Cos[c + d*x]]*Hypergeometric2F1[-7/4, 1/4, 5/4, (1 - Sin[c + d*x])/2]*(a*(1 + Sin[c + d*x]))^(3/2))/(d*e*(1 + Sin[c + d*x])^(7/4))","C",1
292,1,75,239,0.1647872,"\int \frac{(a+a \sin (c+d x))^{5/2}}{(e \cos (c+d x))^{3/2}} \, dx","Integrate[(a + a*Sin[c + d*x])^(5/2)/(e*Cos[c + d*x])^(3/2),x]","\frac{8 \sqrt[4]{2} (a (\sin (c+d x)+1))^{5/2} \, _2F_1\left(-\frac{5}{4},-\frac{1}{4};\frac{3}{4};\frac{1}{2} (1-\sin (c+d x))\right)}{d e (\sin (c+d x)+1)^{9/4} \sqrt{e \cos (c+d x)}}","\frac{5 a^3 (e \cos (c+d x))^{3/2}}{d e^3 \sqrt{a \sin (c+d x)+a}}-\frac{5 a^2 \sqrt{\cos (c+d x)+1} \sqrt{a \sin (c+d x)+a} \tan ^{-1}\left(\frac{\sqrt{e} \sin (c+d x)}{\sqrt{\cos (c+d x)+1} \sqrt{e \cos (c+d x)}}\right)}{d e^{3/2} (\sin (c+d x)+\cos (c+d x)+1)}-\frac{5 a^2 \sqrt{\cos (c+d x)+1} \sqrt{a \sin (c+d x)+a} \sinh ^{-1}\left(\frac{\sqrt{e \cos (c+d x)}}{\sqrt{e}}\right)}{d e^{3/2} (\sin (c+d x)+\cos (c+d x)+1)}+\frac{4 a (a \sin (c+d x)+a)^{3/2}}{d e \sqrt{e \cos (c+d x)}}",1,"(8*2^(1/4)*Hypergeometric2F1[-5/4, -1/4, 3/4, (1 - Sin[c + d*x])/2]*(a*(1 + Sin[c + d*x]))^(5/2))/(d*e*Sqrt[e*Cos[c + d*x]]*(1 + Sin[c + d*x])^(9/4))","C",1
293,1,77,204,0.1895629,"\int \frac{(a+a \sin (c+d x))^{5/2}}{(e \cos (c+d x))^{5/2}} \, dx","Integrate[(a + a*Sin[c + d*x])^(5/2)/(e*Cos[c + d*x])^(5/2),x]","\frac{4\ 2^{3/4} (a (\sin (c+d x)+1))^{5/2} \, _2F_1\left(-\frac{3}{4},-\frac{3}{4};\frac{1}{4};\frac{1}{2} (1-\sin (c+d x))\right)}{3 d e (\sin (c+d x)+1)^{7/4} (e \cos (c+d x))^{3/2}}","-\frac{2 a^2 \sqrt{\cos (c+d x)+1} \sqrt{a \sin (c+d x)+a} \tan ^{-1}\left(\frac{\sqrt{e} \sin (c+d x)}{\sqrt{\cos (c+d x)+1} \sqrt{e \cos (c+d x)}}\right)}{d e^{5/2} (\sin (c+d x)+\cos (c+d x)+1)}+\frac{2 a^2 \sqrt{\cos (c+d x)+1} \sqrt{a \sin (c+d x)+a} \sinh ^{-1}\left(\frac{\sqrt{e \cos (c+d x)}}{\sqrt{e}}\right)}{d e^{5/2} (\sin (c+d x)+\cos (c+d x)+1)}+\frac{4 a (a \sin (c+d x)+a)^{3/2}}{3 d e (e \cos (c+d x))^{3/2}}",1,"(4*2^(3/4)*Hypergeometric2F1[-3/4, -3/4, 1/4, (1 - Sin[c + d*x])/2]*(a*(1 + Sin[c + d*x]))^(5/2))/(3*d*e*(e*Cos[c + d*x])^(3/2)*(1 + Sin[c + d*x])^(7/4))","C",1
294,1,36,36,0.1435122,"\int \frac{(a+a \sin (c+d x))^{5/2}}{(e \cos (c+d x))^{7/2}} \, dx","Integrate[(a + a*Sin[c + d*x])^(5/2)/(e*Cos[c + d*x])^(7/2),x]","\frac{2 (a (\sin (c+d x)+1))^{5/2}}{5 d e (e \cos (c+d x))^{5/2}}","\frac{2 (a \sin (c+d x)+a)^{5/2}}{5 d e (e \cos (c+d x))^{5/2}}",1,"(2*(a*(1 + Sin[c + d*x]))^(5/2))/(5*d*e*(e*Cos[c + d*x])^(5/2))","A",1
295,1,54,76,0.1723829,"\int \frac{(a+a \sin (c+d x))^{5/2}}{(e \cos (c+d x))^{9/2}} \, dx","Integrate[(a + a*Sin[c + d*x])^(5/2)/(e*Cos[c + d*x])^(9/2),x]","-\frac{2 (2 \sin (c+d x)-5) \sec ^4(c+d x) (a (\sin (c+d x)+1))^{5/2} \sqrt{e \cos (c+d x)}}{21 d e^5}","\frac{2 (a \sin (c+d x)+a)^{5/2}}{3 d e (e \cos (c+d x))^{7/2}}-\frac{4 (a \sin (c+d x)+a)^{7/2}}{21 a d e (e \cos (c+d x))^{7/2}}",1,"(-2*Sqrt[e*Cos[c + d*x]]*Sec[c + d*x]^4*(a*(1 + Sin[c + d*x]))^(5/2)*(-5 + 2*Sin[c + d*x]))/(21*d*e^5)","A",1
296,1,64,113,0.2317269,"\int \frac{(a+a \sin (c+d x))^{5/2}}{(e \cos (c+d x))^{11/2}} \, dx","Integrate[(a + a*Sin[c + d*x])^(5/2)/(e*Cos[c + d*x])^(11/2),x]","\frac{2 \left(8 \sin ^2(c+d x)-20 \sin (c+d x)+17\right) \sec ^5(c+d x) (a (\sin (c+d x)+1))^{5/2} \sqrt{e \cos (c+d x)}}{45 d e^6}","\frac{16 (a \sin (c+d x)+a)^{9/2}}{45 a^2 d e (e \cos (c+d x))^{9/2}}-\frac{8 (a \sin (c+d x)+a)^{7/2}}{5 a d e (e \cos (c+d x))^{9/2}}+\frac{2 (a \sin (c+d x)+a)^{5/2}}{d e (e \cos (c+d x))^{9/2}}",1,"(2*Sqrt[e*Cos[c + d*x]]*Sec[c + d*x]^5*(a*(1 + Sin[c + d*x]))^(5/2)*(17 - 20*Sin[c + d*x] + 8*Sin[c + d*x]^2))/(45*d*e^6)","A",1
297,1,74,150,0.2358211,"\int \frac{(a+a \sin (c+d x))^{5/2}}{(e \cos (c+d x))^{13/2}} \, dx","Integrate[(a + a*Sin[c + d*x])^(5/2)/(e*Cos[c + d*x])^(13/2),x]","\frac{2 \left(16 \sin ^3(c+d x)-40 \sin ^2(c+d x)+26 \sin (c+d x)+5\right) \sec ^6(c+d x) (a (\sin (c+d x)+1))^{5/2} \sqrt{e \cos (c+d x)}}{77 d e^7}","\frac{32 (a \sin (c+d x)+a)^{11/2}}{77 a^3 d e (e \cos (c+d x))^{11/2}}-\frac{16 (a \sin (c+d x)+a)^{9/2}}{7 a^2 d e (e \cos (c+d x))^{11/2}}+\frac{4 (a \sin (c+d x)+a)^{7/2}}{a d e (e \cos (c+d x))^{11/2}}-\frac{2 (a \sin (c+d x)+a)^{5/2}}{d e (e \cos (c+d x))^{11/2}}",1,"(2*Sqrt[e*Cos[c + d*x]]*Sec[c + d*x]^6*(a*(1 + Sin[c + d*x]))^(5/2)*(5 + 26*Sin[c + d*x] - 40*Sin[c + d*x]^2 + 16*Sin[c + d*x]^3))/(77*d*e^7)","A",1
298,1,77,244,0.1630652,"\int \frac{(e \cos (c+d x))^{5/2}}{\sqrt{a+a \sin (c+d x)}} \, dx","Integrate[(e*Cos[c + d*x])^(5/2)/Sqrt[a + a*Sin[c + d*x]],x]","-\frac{4 \sqrt[4]{2} (e \cos (c+d x))^{7/2} \, _2F_1\left(-\frac{1}{4},\frac{7}{4};\frac{11}{4};\frac{1}{2} (1-\sin (c+d x))\right)}{7 d e (\sin (c+d x)+1)^{5/4} \sqrt{a (\sin (c+d x)+1)}}","\frac{3 e^{5/2} \sqrt{\cos (c+d x)+1} \sqrt{a \sin (c+d x)+a} \tan ^{-1}\left(\frac{\sqrt{e} \sin (c+d x)}{\sqrt{\cos (c+d x)+1} \sqrt{e \cos (c+d x)}}\right)}{4 d (a \sin (c+d x)+a \cos (c+d x)+a)}+\frac{3 e^{5/2} \sqrt{\cos (c+d x)+1} \sqrt{a \sin (c+d x)+a} \sinh ^{-1}\left(\frac{\sqrt{e \cos (c+d x)}}{\sqrt{e}}\right)}{4 d (a \sin (c+d x)+a \cos (c+d x)+a)}-\frac{a (e \cos (c+d x))^{7/2}}{2 d e (a \sin (c+d x)+a)^{3/2}}+\frac{e (e \cos (c+d x))^{3/2}}{4 d \sqrt{a \sin (c+d x)+a}}",1,"(-4*2^(1/4)*(e*Cos[c + d*x])^(7/2)*Hypergeometric2F1[-1/4, 7/4, 11/4, (1 - Sin[c + d*x])/2])/(7*d*e*(1 + Sin[c + d*x])^(5/4)*Sqrt[a*(1 + Sin[c + d*x])])","C",1
299,1,77,200,0.1072516,"\int \frac{(e \cos (c+d x))^{3/2}}{\sqrt{a+a \sin (c+d x)}} \, dx","Integrate[(e*Cos[c + d*x])^(3/2)/Sqrt[a + a*Sin[c + d*x]],x]","-\frac{2\ 2^{3/4} (e \cos (c+d x))^{5/2} \, _2F_1\left(\frac{1}{4},\frac{5}{4};\frac{9}{4};\frac{1}{2} (1-\sin (c+d x))\right)}{5 d e (\sin (c+d x)+1)^{3/4} \sqrt{a (\sin (c+d x)+1)}}","\frac{e^{3/2} \sqrt{\cos (c+d x)+1} \sqrt{a \sin (c+d x)+a} \tan ^{-1}\left(\frac{\sqrt{e} \sin (c+d x)}{\sqrt{\cos (c+d x)+1} \sqrt{e \cos (c+d x)}}\right)}{a d (\sin (c+d x)+\cos (c+d x)+1)}-\frac{e^{3/2} \sqrt{\cos (c+d x)+1} \sqrt{a \sin (c+d x)+a} \sinh ^{-1}\left(\frac{\sqrt{e \cos (c+d x)}}{\sqrt{e}}\right)}{a d (\sin (c+d x)+\cos (c+d x)+1)}+\frac{e \sqrt{a \sin (c+d x)+a} \sqrt{e \cos (c+d x)}}{a d}",1,"(-2*2^(3/4)*(e*Cos[c + d*x])^(5/2)*Hypergeometric2F1[1/4, 5/4, 9/4, (1 - Sin[c + d*x])/2])/(5*d*e*(1 + Sin[c + d*x])^(3/4)*Sqrt[a*(1 + Sin[c + d*x])])","C",1
300,1,77,169,0.0785961,"\int \frac{\sqrt{e \cos (c+d x)}}{\sqrt{a+a \sin (c+d x)}} \, dx","Integrate[Sqrt[e*Cos[c + d*x]]/Sqrt[a + a*Sin[c + d*x]],x]","-\frac{2 \sqrt[4]{2} (e \cos (c+d x))^{3/2} \, _2F_1\left(\frac{3}{4},\frac{3}{4};\frac{7}{4};\frac{1}{2} (1-\sin (c+d x))\right)}{3 d e \sqrt[4]{\sin (c+d x)+1} \sqrt{a (\sin (c+d x)+1)}}","\frac{2 \sqrt{e} \sqrt{\cos (c+d x)+1} \sqrt{a \sin (c+d x)+a} \tan ^{-1}\left(\frac{\sqrt{e} \sin (c+d x)}{\sqrt{\cos (c+d x)+1} \sqrt{e \cos (c+d x)}}\right)}{d (a \sin (c+d x)+a \cos (c+d x)+a)}+\frac{2 \sqrt{e} \sqrt{\cos (c+d x)+1} \sqrt{a \sin (c+d x)+a} \sinh ^{-1}\left(\frac{\sqrt{e \cos (c+d x)}}{\sqrt{e}}\right)}{d (a \sin (c+d x)+a \cos (c+d x)+a)}",1,"(-2*2^(1/4)*(e*Cos[c + d*x])^(3/2)*Hypergeometric2F1[3/4, 3/4, 7/4, (1 - Sin[c + d*x])/2])/(3*d*e*(1 + Sin[c + d*x])^(1/4)*Sqrt[a*(1 + Sin[c + d*x])])","C",1
301,1,34,34,0.0659649,"\int \frac{1}{\sqrt{e \cos (c+d x)} \sqrt{a+a \sin (c+d x)}} \, dx","Integrate[1/(Sqrt[e*Cos[c + d*x]]*Sqrt[a + a*Sin[c + d*x]]),x]","-\frac{2 \sqrt{e \cos (c+d x)}}{d e \sqrt{a (\sin (c+d x)+1)}}","-\frac{2 \sqrt{e \cos (c+d x)}}{d e \sqrt{a \sin (c+d x)+a}}",1,"(-2*Sqrt[e*Cos[c + d*x]])/(d*e*Sqrt[a*(1 + Sin[c + d*x])])","A",1
302,1,46,76,0.1060606,"\int \frac{1}{(e \cos (c+d x))^{3/2} \sqrt{a+a \sin (c+d x)}} \, dx","Integrate[1/((e*Cos[c + d*x])^(3/2)*Sqrt[a + a*Sin[c + d*x]]),x]","\frac{2 (2 \sin (c+d x)+1)}{3 d e \sqrt{a (\sin (c+d x)+1)} \sqrt{e \cos (c+d x)}}","\frac{4 \sqrt{a \sin (c+d x)+a}}{3 a d e \sqrt{e \cos (c+d x)}}-\frac{2}{3 d e \sqrt{a \sin (c+d x)+a} \sqrt{e \cos (c+d x)}}",1,"(2*(1 + 2*Sin[c + d*x]))/(3*d*e*Sqrt[e*Cos[c + d*x]]*Sqrt[a*(1 + Sin[c + d*x])])","A",1
303,1,56,115,0.0895672,"\int \frac{1}{(e \cos (c+d x))^{5/2} \sqrt{a+a \sin (c+d x)}} \, dx","Integrate[1/((e*Cos[c + d*x])^(5/2)*Sqrt[a + a*Sin[c + d*x]]),x]","\frac{2 \left(8 \sin ^2(c+d x)+4 \sin (c+d x)-7\right)}{15 d e \sqrt{a (\sin (c+d x)+1)} (e \cos (c+d x))^{3/2}}","\frac{16 (a \sin (c+d x)+a)^{3/2}}{15 a^2 d e (e \cos (c+d x))^{3/2}}-\frac{8 \sqrt{a \sin (c+d x)+a}}{5 a d e (e \cos (c+d x))^{3/2}}-\frac{2}{5 d e \sqrt{a \sin (c+d x)+a} (e \cos (c+d x))^{3/2}}",1,"(2*(-7 + 4*Sin[c + d*x] + 8*Sin[c + d*x]^2))/(15*d*e*(e*Cos[c + d*x])^(3/2)*Sqrt[a*(1 + Sin[c + d*x])])","A",1
304,1,66,154,0.1697674,"\int \frac{1}{(e \cos (c+d x))^{7/2} \sqrt{a+a \sin (c+d x)}} \, dx","Integrate[1/((e*Cos[c + d*x])^(7/2)*Sqrt[a + a*Sin[c + d*x]]),x]","\frac{2 (10 \sin (c+d x)+4 \sin (3 (c+d x))+4 \cos (2 (c+d x))+5)}{35 d e \sqrt{a (\sin (c+d x)+1)} (e \cos (c+d x))^{5/2}}","-\frac{32 (a \sin (c+d x)+a)^{5/2}}{35 a^3 d e (e \cos (c+d x))^{5/2}}+\frac{16 (a \sin (c+d x)+a)^{3/2}}{7 a^2 d e (e \cos (c+d x))^{5/2}}-\frac{4 \sqrt{a \sin (c+d x)+a}}{7 a d e (e \cos (c+d x))^{5/2}}-\frac{2}{7 d e \sqrt{a \sin (c+d x)+a} (e \cos (c+d x))^{5/2}}",1,"(2*(5 + 4*Cos[2*(c + d*x)] + 10*Sin[c + d*x] + 4*Sin[3*(c + d*x)]))/(35*d*e*(e*Cos[c + d*x])^(5/2)*Sqrt[a*(1 + Sin[c + d*x])])","A",1
305,1,80,247,0.1297865,"\int \frac{(e \cos (c+d x))^{7/2}}{(a+a \sin (c+d x))^{3/2}} \, dx","Integrate[(e*Cos[c + d*x])^(7/2)/(a + a*Sin[c + d*x])^(3/2),x]","-\frac{2\ 2^{3/4} \sqrt{a (\sin (c+d x)+1)} (e \cos (c+d x))^{9/2} \, _2F_1\left(\frac{1}{4},\frac{9}{4};\frac{13}{4};\frac{1}{2} (1-\sin (c+d x))\right)}{9 a^2 d e (\sin (c+d x)+1)^{11/4}}","\frac{5 e^{7/2} \sqrt{\cos (c+d x)+1} \sqrt{a \sin (c+d x)+a} \tan ^{-1}\left(\frac{\sqrt{e} \sin (c+d x)}{\sqrt{\cos (c+d x)+1} \sqrt{e \cos (c+d x)}}\right)}{4 a^2 d (\sin (c+d x)+\cos (c+d x)+1)}-\frac{5 e^{7/2} \sqrt{\cos (c+d x)+1} \sqrt{a \sin (c+d x)+a} \sinh ^{-1}\left(\frac{\sqrt{e \cos (c+d x)}}{\sqrt{e}}\right)}{4 a^2 d (\sin (c+d x)+\cos (c+d x)+1)}+\frac{5 e^3 \sqrt{a \sin (c+d x)+a} \sqrt{e \cos (c+d x)}}{4 a^2 d}+\frac{e (e \cos (c+d x))^{5/2}}{2 a d \sqrt{a \sin (c+d x)+a}}",1,"(-2*2^(3/4)*(e*Cos[c + d*x])^(9/2)*Hypergeometric2F1[1/4, 9/4, 13/4, (1 - Sin[c + d*x])/2]*Sqrt[a*(1 + Sin[c + d*x])])/(9*a^2*d*e*(1 + Sin[c + d*x])^(11/4))","C",1
306,1,80,215,0.1472953,"\int \frac{(e \cos (c+d x))^{5/2}}{(a+a \sin (c+d x))^{3/2}} \, dx","Integrate[(e*Cos[c + d*x])^(5/2)/(a + a*Sin[c + d*x])^(3/2),x]","-\frac{2 \sqrt[4]{2} \sqrt{a (\sin (c+d x)+1)} (e \cos (c+d x))^{7/2} \, _2F_1\left(\frac{3}{4},\frac{7}{4};\frac{11}{4};\frac{1}{2} (1-\sin (c+d x))\right)}{7 a^2 d e (\sin (c+d x)+1)^{9/4}}","\frac{3 e^{5/2} \sqrt{\cos (c+d x)+1} \sqrt{a \sin (c+d x)+a} \tan ^{-1}\left(\frac{\sqrt{e} \sin (c+d x)}{\sqrt{\cos (c+d x)+1} \sqrt{e \cos (c+d x)}}\right)}{d \left(a^2 \sin (c+d x)+a^2 \cos (c+d x)+a^2\right)}+\frac{3 e^{5/2} \sqrt{\cos (c+d x)+1} \sqrt{a \sin (c+d x)+a} \sinh ^{-1}\left(\frac{\sqrt{e \cos (c+d x)}}{\sqrt{e}}\right)}{d \left(a^2 \sin (c+d x)+a^2 \cos (c+d x)+a^2\right)}+\frac{e (e \cos (c+d x))^{3/2}}{a d \sqrt{a \sin (c+d x)+a}}",1,"(-2*2^(1/4)*(e*Cos[c + d*x])^(7/2)*Hypergeometric2F1[3/4, 7/4, 11/4, (1 - Sin[c + d*x])/2]*Sqrt[a*(1 + Sin[c + d*x])])/(7*a^2*d*e*(1 + Sin[c + d*x])^(9/4))","C",1
307,1,80,236,0.1245328,"\int \frac{(e \cos (c+d x))^{3/2}}{(a+a \sin (c+d x))^{3/2}} \, dx","Integrate[(e*Cos[c + d*x])^(3/2)/(a + a*Sin[c + d*x])^(3/2),x]","-\frac{2^{3/4} \sqrt{a (\sin (c+d x)+1)} (e \cos (c+d x))^{5/2} \, _2F_1\left(\frac{5}{4},\frac{5}{4};\frac{9}{4};\frac{1}{2} (1-\sin (c+d x))\right)}{5 a^2 d e (\sin (c+d x)+1)^{7/4}}","-\frac{2 e^{3/2} \sqrt{\cos (c+d x)+1} \sqrt{a \sin (c+d x)+a} \tan ^{-1}\left(\frac{\sqrt{e} \sin (c+d x)}{\sqrt{\cos (c+d x)+1} \sqrt{e \cos (c+d x)}}\right)}{a^2 d (\sin (c+d x)+\cos (c+d x)+1)}+\frac{2 e^{3/2} \sqrt{\cos (c+d x)+1} \sqrt{a \sin (c+d x)+a} \sinh ^{-1}\left(\frac{\sqrt{e \cos (c+d x)}}{\sqrt{e}}\right)}{a^2 d (\sin (c+d x)+\cos (c+d x)+1)}-\frac{2 e \sqrt{a \sin (c+d x)+a} \sqrt{e \cos (c+d x)}}{a^2 d}-\frac{2 (e \cos (c+d x))^{5/2}}{d e (a \sin (c+d x)+a)^{3/2}}",1,"-1/5*(2^(3/4)*(e*Cos[c + d*x])^(5/2)*Hypergeometric2F1[5/4, 5/4, 9/4, (1 - Sin[c + d*x])/2]*Sqrt[a*(1 + Sin[c + d*x])])/(a^2*d*e*(1 + Sin[c + d*x])^(7/4))","C",1
308,1,49,36,0.0689817,"\int \frac{\sqrt{e \cos (c+d x)}}{(a+a \sin (c+d x))^{3/2}} \, dx","Integrate[Sqrt[e*Cos[c + d*x]]/(a + a*Sin[c + d*x])^(3/2),x]","-\frac{2 \sqrt{a (\sin (c+d x)+1)} (e \cos (c+d x))^{3/2}}{3 a^2 d e (\sin (c+d x)+1)^2}","-\frac{2 (e \cos (c+d x))^{3/2}}{3 d e (a \sin (c+d x)+a)^{3/2}}",1,"(-2*(e*Cos[c + d*x])^(3/2)*Sqrt[a*(1 + Sin[c + d*x])])/(3*a^2*d*e*(1 + Sin[c + d*x])^2)","A",1
309,1,59,76,0.1123357,"\int \frac{1}{\sqrt{e \cos (c+d x)} (a+a \sin (c+d x))^{3/2}} \, dx","Integrate[1/(Sqrt[e*Cos[c + d*x]]*(a + a*Sin[c + d*x])^(3/2)),x]","-\frac{2 (2 \sin (c+d x)+3) \sqrt{a (\sin (c+d x)+1)} \sqrt{e \cos (c+d x)}}{5 a^2 d e (\sin (c+d x)+1)^2}","-\frac{4 \sqrt{e \cos (c+d x)}}{5 a d e \sqrt{a \sin (c+d x)+a}}-\frac{2 \sqrt{e \cos (c+d x)}}{5 d e (a \sin (c+d x)+a)^{3/2}}",1,"(-2*Sqrt[e*Cos[c + d*x]]*Sqrt[a*(1 + Sin[c + d*x])]*(3 + 2*Sin[c + d*x]))/(5*a^2*d*e*(1 + Sin[c + d*x])^2)","A",1
310,1,56,115,0.1016129,"\int \frac{1}{(e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^{3/2}} \, dx","Integrate[1/((e*Cos[c + d*x])^(3/2)*(a + a*Sin[c + d*x])^(3/2)),x]","\frac{16 \sin ^2(c+d x)+24 \sin (c+d x)+2}{21 d e (a (\sin (c+d x)+1))^{3/2} \sqrt{e \cos (c+d x)}}","\frac{16 \sqrt{a \sin (c+d x)+a}}{21 a^2 d e \sqrt{e \cos (c+d x)}}-\frac{8}{21 a d e \sqrt{a \sin (c+d x)+a} \sqrt{e \cos (c+d x)}}-\frac{2}{7 d e (a \sin (c+d x)+a)^{3/2} \sqrt{e \cos (c+d x)}}",1,"(2 + 24*Sin[c + d*x] + 16*Sin[c + d*x]^2)/(21*d*e*Sqrt[e*Cos[c + d*x]]*(a*(1 + Sin[c + d*x]))^(3/2))","A",1
311,1,66,154,0.1705908,"\int \frac{1}{(e \cos (c+d x))^{5/2} (a+a \sin (c+d x))^{3/2}} \, dx","Integrate[1/((e*Cos[c + d*x])^(5/2)*(a + a*Sin[c + d*x])^(3/2)),x]","-\frac{2 (-6 \sin (c+d x)+4 \sin (3 (c+d x))+12 \cos (2 (c+d x))+7)}{45 d e (a (\sin (c+d x)+1))^{3/2} (e \cos (c+d x))^{3/2}}","\frac{32 (a \sin (c+d x)+a)^{3/2}}{45 a^3 d e (e \cos (c+d x))^{3/2}}-\frac{16 \sqrt{a \sin (c+d x)+a}}{15 a^2 d e (e \cos (c+d x))^{3/2}}-\frac{4}{15 a d e \sqrt{a \sin (c+d x)+a} (e \cos (c+d x))^{3/2}}-\frac{2}{9 d e (a \sin (c+d x)+a)^{3/2} (e \cos (c+d x))^{3/2}}",1,"(-2*(7 + 12*Cos[2*(c + d*x)] - 6*Sin[c + d*x] + 4*Sin[3*(c + d*x)]))/(45*d*e*(e*Cos[c + d*x])^(3/2)*(a*(1 + Sin[c + d*x]))^(3/2))","A",1
312,1,76,193,0.299928,"\int \frac{1}{(e \cos (c+d x))^{7/2} (a+a \sin (c+d x))^{3/2}} \, dx","Integrate[1/((e*Cos[c + d*x])^(7/2)*(a + a*Sin[c + d*x])^(3/2)),x]","\frac{2 (104 \sin (c+d x)+48 \sin (3 (c+d x))+8 \cos (2 (c+d x))-16 \cos (4 (c+d x))+45)}{385 d e (a (\sin (c+d x)+1))^{3/2} (e \cos (c+d x))^{5/2}}","-\frac{256 (a \sin (c+d x)+a)^{5/2}}{385 a^4 d e (e \cos (c+d x))^{5/2}}+\frac{128 (a \sin (c+d x)+a)^{3/2}}{77 a^3 d e (e \cos (c+d x))^{5/2}}-\frac{32 \sqrt{a \sin (c+d x)+a}}{77 a^2 d e (e \cos (c+d x))^{5/2}}-\frac{16}{77 a d e \sqrt{a \sin (c+d x)+a} (e \cos (c+d x))^{5/2}}-\frac{2}{11 d e (a \sin (c+d x)+a)^{3/2} (e \cos (c+d x))^{5/2}}",1,"(2*(45 + 8*Cos[2*(c + d*x)] - 16*Cos[4*(c + d*x)] + 104*Sin[c + d*x] + 48*Sin[3*(c + d*x)]))/(385*d*e*(e*Cos[c + d*x])^(5/2)*(a*(1 + Sin[c + d*x]))^(3/2))","A",1
313,1,80,261,0.1759375,"\int \frac{(e \cos (c+d x))^{9/2}}{(a+a \sin (c+d x))^{5/2}} \, dx","Integrate[(e*Cos[c + d*x])^(9/2)/(a + a*Sin[c + d*x])^(5/2),x]","-\frac{2 \sqrt[4]{2} \sqrt{a (\sin (c+d x)+1)} (e \cos (c+d x))^{11/2} \, _2F_1\left(\frac{3}{4},\frac{11}{4};\frac{15}{4};\frac{1}{2} (1-\sin (c+d x))\right)}{11 a^3 d e (\sin (c+d x)+1)^{13/4}}","\frac{21 e^{9/2} \sqrt{\cos (c+d x)+1} \sqrt{a \sin (c+d x)+a} \tan ^{-1}\left(\frac{\sqrt{e} \sin (c+d x)}{\sqrt{\cos (c+d x)+1} \sqrt{e \cos (c+d x)}}\right)}{4 d \left(a^3 \sin (c+d x)+a^3 \cos (c+d x)+a^3\right)}+\frac{21 e^{9/2} \sqrt{\cos (c+d x)+1} \sqrt{a \sin (c+d x)+a} \sinh ^{-1}\left(\frac{\sqrt{e \cos (c+d x)}}{\sqrt{e}}\right)}{4 d \left(a^3 \sin (c+d x)+a^3 \cos (c+d x)+a^3\right)}+\frac{7 e^3 (e \cos (c+d x))^{3/2}}{4 a^2 d \sqrt{a \sin (c+d x)+a}}+\frac{e (e \cos (c+d x))^{7/2}}{2 a d (a \sin (c+d x)+a)^{3/2}}",1,"(-2*2^(1/4)*(e*Cos[c + d*x])^(11/2)*Hypergeometric2F1[3/4, 11/4, 15/4, (1 - Sin[c + d*x])/2]*Sqrt[a*(1 + Sin[c + d*x])])/(11*a^3*d*e*(1 + Sin[c + d*x])^(13/4))","C",1
314,1,80,239,0.122048,"\int \frac{(e \cos (c+d x))^{7/2}}{(a+a \sin (c+d x))^{5/2}} \, dx","Integrate[(e*Cos[c + d*x])^(7/2)/(a + a*Sin[c + d*x])^(5/2),x]","-\frac{2^{3/4} \sqrt{a (\sin (c+d x)+1)} (e \cos (c+d x))^{9/2} \, _2F_1\left(\frac{5}{4},\frac{9}{4};\frac{13}{4};\frac{1}{2} (1-\sin (c+d x))\right)}{9 a^3 d e (\sin (c+d x)+1)^{11/4}}","-\frac{5 e^{7/2} \sqrt{\cos (c+d x)+1} \sqrt{a \sin (c+d x)+a} \tan ^{-1}\left(\frac{\sqrt{e} \sin (c+d x)}{\sqrt{\cos (c+d x)+1} \sqrt{e \cos (c+d x)}}\right)}{a^3 d (\sin (c+d x)+\cos (c+d x)+1)}+\frac{5 e^{7/2} \sqrt{\cos (c+d x)+1} \sqrt{a \sin (c+d x)+a} \sinh ^{-1}\left(\frac{\sqrt{e \cos (c+d x)}}{\sqrt{e}}\right)}{a^3 d (\sin (c+d x)+\cos (c+d x)+1)}-\frac{5 e^3 \sqrt{a \sin (c+d x)+a} \sqrt{e \cos (c+d x)}}{a^3 d}-\frac{4 e (e \cos (c+d x))^{5/2}}{a d (a \sin (c+d x)+a)^{3/2}}",1,"-1/9*(2^(3/4)*(e*Cos[c + d*x])^(9/2)*Hypergeometric2F1[5/4, 9/4, 13/4, (1 - Sin[c + d*x])/2]*Sqrt[a*(1 + Sin[c + d*x])])/(a^3*d*e*(1 + Sin[c + d*x])^(11/4))","C",1
315,1,80,218,0.1379094,"\int \frac{(e \cos (c+d x))^{5/2}}{(a+a \sin (c+d x))^{5/2}} \, dx","Integrate[(e*Cos[c + d*x])^(5/2)/(a + a*Sin[c + d*x])^(5/2),x]","-\frac{\sqrt[4]{2} \sqrt{a (\sin (c+d x)+1)} (e \cos (c+d x))^{7/2} \, _2F_1\left(\frac{7}{4},\frac{7}{4};\frac{11}{4};\frac{1}{2} (1-\sin (c+d x))\right)}{7 a^3 d e (\sin (c+d x)+1)^{9/4}}","-\frac{2 e^{5/2} \sqrt{\cos (c+d x)+1} \sqrt{a \sin (c+d x)+a} \tan ^{-1}\left(\frac{\sqrt{e} \sin (c+d x)}{\sqrt{\cos (c+d x)+1} \sqrt{e \cos (c+d x)}}\right)}{d \left(a^3 \sin (c+d x)+a^3 \cos (c+d x)+a^3\right)}-\frac{2 e^{5/2} \sqrt{\cos (c+d x)+1} \sqrt{a \sin (c+d x)+a} \sinh ^{-1}\left(\frac{\sqrt{e \cos (c+d x)}}{\sqrt{e}}\right)}{d \left(a^3 \sin (c+d x)+a^3 \cos (c+d x)+a^3\right)}-\frac{4 e (e \cos (c+d x))^{3/2}}{3 a d (a \sin (c+d x)+a)^{3/2}}",1,"-1/7*(2^(1/4)*(e*Cos[c + d*x])^(7/2)*Hypergeometric2F1[7/4, 7/4, 11/4, (1 - Sin[c + d*x])/2]*Sqrt[a*(1 + Sin[c + d*x])])/(a^3*d*e*(1 + Sin[c + d*x])^(9/4))","C",1
316,1,49,36,0.118965,"\int \frac{(e \cos (c+d x))^{3/2}}{(a+a \sin (c+d x))^{5/2}} \, dx","Integrate[(e*Cos[c + d*x])^(3/2)/(a + a*Sin[c + d*x])^(5/2),x]","-\frac{2 \sqrt{a (\sin (c+d x)+1)} (e \cos (c+d x))^{5/2}}{5 a^3 d e (\sin (c+d x)+1)^3}","-\frac{2 (e \cos (c+d x))^{5/2}}{5 d e (a \sin (c+d x)+a)^{5/2}}",1,"(-2*(e*Cos[c + d*x])^(5/2)*Sqrt[a*(1 + Sin[c + d*x])])/(5*a^3*d*e*(1 + Sin[c + d*x])^3)","A",1
317,1,59,76,0.0965879,"\int \frac{\sqrt{e \cos (c+d x)}}{(a+a \sin (c+d x))^{5/2}} \, dx","Integrate[Sqrt[e*Cos[c + d*x]]/(a + a*Sin[c + d*x])^(5/2),x]","-\frac{2 (2 \sin (c+d x)+5) \sqrt{a (\sin (c+d x)+1)} (e \cos (c+d x))^{3/2}}{21 a^3 d e (\sin (c+d x)+1)^3}","-\frac{4 (e \cos (c+d x))^{3/2}}{21 a d e (a \sin (c+d x)+a)^{3/2}}-\frac{2 (e \cos (c+d x))^{3/2}}{7 d e (a \sin (c+d x)+a)^{5/2}}",1,"(-2*(e*Cos[c + d*x])^(3/2)*Sqrt[a*(1 + Sin[c + d*x])]*(5 + 2*Sin[c + d*x]))/(21*a^3*d*e*(1 + Sin[c + d*x])^3)","A",1
318,1,69,115,0.1545965,"\int \frac{1}{\sqrt{e \cos (c+d x)} (a+a \sin (c+d x))^{5/2}} \, dx","Integrate[1/(Sqrt[e*Cos[c + d*x]]*(a + a*Sin[c + d*x])^(5/2)),x]","-\frac{2 \left(8 \sin ^2(c+d x)+20 \sin (c+d x)+17\right) \sqrt{a (\sin (c+d x)+1)} \sqrt{e \cos (c+d x)}}{45 a^3 d e (\sin (c+d x)+1)^3}","-\frac{16 \sqrt{e \cos (c+d x)}}{45 a^2 d e \sqrt{a \sin (c+d x)+a}}-\frac{8 \sqrt{e \cos (c+d x)}}{45 a d e (a \sin (c+d x)+a)^{3/2}}-\frac{2 \sqrt{e \cos (c+d x)}}{9 d e (a \sin (c+d x)+a)^{5/2}}",1,"(-2*Sqrt[e*Cos[c + d*x]]*Sqrt[a*(1 + Sin[c + d*x])]*(17 + 20*Sin[c + d*x] + 8*Sin[c + d*x]^2))/(45*a^3*d*e*(1 + Sin[c + d*x])^3)","A",1
319,1,66,154,0.1518222,"\int \frac{1}{(e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^{5/2}} \, dx","Integrate[1/((e*Cos[c + d*x])^(3/2)*(a + a*Sin[c + d*x])^(5/2)),x]","\frac{32 \sin ^3(c+d x)+80 \sin ^2(c+d x)+52 \sin (c+d x)-10}{77 d e (a (\sin (c+d x)+1))^{5/2} \sqrt{e \cos (c+d x)}}","\frac{32 \sqrt{a \sin (c+d x)+a}}{77 a^3 d e \sqrt{e \cos (c+d x)}}-\frac{16}{77 a^2 d e \sqrt{a \sin (c+d x)+a} \sqrt{e \cos (c+d x)}}-\frac{12}{77 a d e (a \sin (c+d x)+a)^{3/2} \sqrt{e \cos (c+d x)}}-\frac{2}{11 d e (a \sin (c+d x)+a)^{5/2} \sqrt{e \cos (c+d x)}}",1,"(-10 + 52*Sin[c + d*x] + 80*Sin[c + d*x]^2 + 32*Sin[c + d*x]^3)/(77*d*e*Sqrt[e*Cos[c + d*x]]*(a*(1 + Sin[c + d*x]))^(5/2))","A",1
320,1,76,193,0.2751741,"\int \frac{1}{(e \cos (c+d x))^{5/2} (a+a \sin (c+d x))^{5/2}} \, dx","Integrate[1/((e*Cos[c + d*x])^(5/2)*(a + a*Sin[c + d*x])^(5/2)),x]","-\frac{2 (-40 \sin (c+d x)+80 \sin (3 (c+d x))+136 \cos (2 (c+d x))-16 \cos (4 (c+d x))+77)}{585 d e (a (\sin (c+d x)+1))^{5/2} (e \cos (c+d x))^{3/2}}","\frac{256 (a \sin (c+d x)+a)^{3/2}}{585 a^4 d e (e \cos (c+d x))^{3/2}}-\frac{128 \sqrt{a \sin (c+d x)+a}}{195 a^3 d e (e \cos (c+d x))^{3/2}}-\frac{32}{195 a^2 d e \sqrt{a \sin (c+d x)+a} (e \cos (c+d x))^{3/2}}-\frac{16}{117 a d e (a \sin (c+d x)+a)^{3/2} (e \cos (c+d x))^{3/2}}-\frac{2}{13 d e (a \sin (c+d x)+a)^{5/2} (e \cos (c+d x))^{3/2}}",1,"(-2*(77 + 136*Cos[2*(c + d*x)] - 16*Cos[4*(c + d*x)] - 40*Sin[c + d*x] + 80*Sin[3*(c + d*x)]))/(585*d*e*(e*Cos[c + d*x])^(3/2)*(a*(1 + Sin[c + d*x]))^(5/2))","A",1
321,1,77,78,0.1567913,"\int \frac{(e \cos (c+d x))^{7/3}}{\sqrt{a+a \sin (c+d x)}} \, dx","Integrate[(e*Cos[c + d*x])^(7/3)/Sqrt[a + a*Sin[c + d*x]],x]","-\frac{3 \sqrt[6]{2} (e \cos (c+d x))^{10/3} \, _2F_1\left(-\frac{1}{6},\frac{5}{3};\frac{8}{3};\frac{1}{2} (1-\sin (c+d x))\right)}{5 d e (\sin (c+d x)+1)^{7/6} \sqrt{a (\sin (c+d x)+1)}}","-\frac{3 \sqrt[6]{2} a (e \cos (c+d x))^{10/3} \, _2F_1\left(-\frac{1}{6},\frac{5}{3};\frac{8}{3};\frac{1}{2} (1-\sin (c+d x))\right)}{5 d e \sqrt[6]{\sin (c+d x)+1} (a \sin (c+d x)+a)^{3/2}}",1,"(-3*2^(1/6)*(e*Cos[c + d*x])^(10/3)*Hypergeometric2F1[-1/6, 5/3, 8/3, (1 - Sin[c + d*x])/2])/(5*d*e*(1 + Sin[c + d*x])^(7/6)*Sqrt[a*(1 + Sin[c + d*x])])","A",1
322,1,77,78,0.1125369,"\int \frac{(e \cos (c+d x))^{5/3}}{\sqrt{a+a \sin (c+d x)}} \, dx","Integrate[(e*Cos[c + d*x])^(5/3)/Sqrt[a + a*Sin[c + d*x]],x]","-\frac{3 (e \cos (c+d x))^{8/3} \, _2F_1\left(\frac{1}{6},\frac{4}{3};\frac{7}{3};\frac{1}{2} (1-\sin (c+d x))\right)}{4 \sqrt[6]{2} d e (\sin (c+d x)+1)^{5/6} \sqrt{a (\sin (c+d x)+1)}}","-\frac{3 a \sqrt[6]{\sin (c+d x)+1} (e \cos (c+d x))^{8/3} \, _2F_1\left(\frac{1}{6},\frac{4}{3};\frac{7}{3};\frac{1}{2} (1-\sin (c+d x))\right)}{4 \sqrt[6]{2} d e (a \sin (c+d x)+a)^{3/2}}",1,"(-3*(e*Cos[c + d*x])^(8/3)*Hypergeometric2F1[1/6, 4/3, 7/3, (1 - Sin[c + d*x])/2])/(4*2^(1/6)*d*e*(1 + Sin[c + d*x])^(5/6)*Sqrt[a*(1 + Sin[c + d*x])])","A",1
323,1,77,78,0.078219,"\int \frac{(e \cos (c+d x))^{2/3}}{\sqrt{a+a \sin (c+d x)}} \, dx","Integrate[(e*Cos[c + d*x])^(2/3)/Sqrt[a + a*Sin[c + d*x]],x]","-\frac{3 \sqrt[3]{2} (e \cos (c+d x))^{5/3} \, _2F_1\left(\frac{2}{3},\frac{5}{6};\frac{11}{6};\frac{1}{2} (1-\sin (c+d x))\right)}{5 d e \sqrt[3]{\sin (c+d x)+1} \sqrt{a (\sin (c+d x)+1)}}","-\frac{3 \sqrt[3]{2} a (\sin (c+d x)+1)^{2/3} (e \cos (c+d x))^{5/3} \, _2F_1\left(\frac{2}{3},\frac{5}{6};\frac{11}{6};\frac{1}{2} (1-\sin (c+d x))\right)}{5 d e (a \sin (c+d x)+a)^{3/2}}",1,"(-3*2^(1/3)*(e*Cos[c + d*x])^(5/3)*Hypergeometric2F1[2/3, 5/6, 11/6, (1 - Sin[c + d*x])/2])/(5*d*e*(1 + Sin[c + d*x])^(1/3)*Sqrt[a*(1 + Sin[c + d*x])])","A",1
324,1,77,78,0.0846917,"\int \frac{\sqrt[3]{e \cos (c+d x)}}{\sqrt{a+a \sin (c+d x)}} \, dx","Integrate[(e*Cos[c + d*x])^(1/3)/Sqrt[a + a*Sin[c + d*x]],x]","-\frac{3 (e \cos (c+d x))^{4/3} \, _2F_1\left(\frac{2}{3},\frac{5}{6};\frac{5}{3};\frac{1}{2} (1-\sin (c+d x))\right)}{2\ 2^{5/6} d e \sqrt[6]{\sin (c+d x)+1} \sqrt{a (\sin (c+d x)+1)}}","-\frac{3 a (\sin (c+d x)+1)^{5/6} (e \cos (c+d x))^{4/3} \, _2F_1\left(\frac{2}{3},\frac{5}{6};\frac{5}{3};\frac{1}{2} (1-\sin (c+d x))\right)}{2\ 2^{5/6} d e (a \sin (c+d x)+a)^{3/2}}",1,"(-3*(e*Cos[c + d*x])^(4/3)*Hypergeometric2F1[2/3, 5/6, 5/3, (1 - Sin[c + d*x])/2])/(2*2^(5/6)*d*e*(1 + Sin[c + d*x])^(1/6)*Sqrt[a*(1 + Sin[c + d*x])])","A",1
325,1,77,77,0.080474,"\int \frac{1}{\sqrt[3]{e \cos (c+d x)} \sqrt{a+a \sin (c+d x)}} \, dx","Integrate[1/((e*Cos[c + d*x])^(1/3)*Sqrt[a + a*Sin[c + d*x]]),x]","-\frac{3 \sqrt[6]{\sin (c+d x)+1} (e \cos (c+d x))^{2/3} \, _2F_1\left(\frac{1}{3},\frac{7}{6};\frac{4}{3};\frac{1}{2} (1-\sin (c+d x))\right)}{2 \sqrt[6]{2} d e \sqrt{a (\sin (c+d x)+1)}}","-\frac{3 \sqrt[6]{\sin (c+d x)+1} (e \cos (c+d x))^{2/3} \, _2F_1\left(\frac{1}{3},\frac{7}{6};\frac{4}{3};\frac{1}{2} (1-\sin (c+d x))\right)}{2 \sqrt[6]{2} d e \sqrt{a \sin (c+d x)+a}}",1,"(-3*(e*Cos[c + d*x])^(2/3)*Hypergeometric2F1[1/3, 7/6, 4/3, (1 - Sin[c + d*x])/2]*(1 + Sin[c + d*x])^(1/6))/(2*2^(1/6)*d*e*Sqrt[a*(1 + Sin[c + d*x])])","A",1
326,1,75,75,0.0928606,"\int \frac{1}{(e \cos (c+d x))^{4/3} \sqrt{a+a \sin (c+d x)}} \, dx","Integrate[1/((e*Cos[c + d*x])^(4/3)*Sqrt[a + a*Sin[c + d*x]]),x]","\frac{3 (\sin (c+d x)+1)^{2/3} \, _2F_1\left(-\frac{1}{6},\frac{5}{3};\frac{5}{6};\frac{1}{2} (1-\sin (c+d x))\right)}{2^{2/3} d e \sqrt{a (\sin (c+d x)+1)} \sqrt[3]{e \cos (c+d x)}}","\frac{3 (\sin (c+d x)+1)^{2/3} \, _2F_1\left(-\frac{1}{6},\frac{5}{3};\frac{5}{6};\frac{1}{2} (1-\sin (c+d x))\right)}{2^{2/3} d e \sqrt{a \sin (c+d x)+a} \sqrt[3]{e \cos (c+d x)}}",1,"(3*Hypergeometric2F1[-1/6, 5/3, 5/6, (1 - Sin[c + d*x])/2]*(1 + Sin[c + d*x])^(2/3))/(2^(2/3)*d*e*(e*Cos[c + d*x])^(1/3)*Sqrt[a*(1 + Sin[c + d*x])])","A",1
327,1,94,95,0.1980783,"\int (e \cos (c+d x))^p (a+a \sin (c+d x))^8 \, dx","Integrate[(e*Cos[c + d*x])^p*(a + a*Sin[c + d*x])^8,x]","-\frac{a^8 2^{\frac{p+17}{2}} \cos (c+d x) (\sin (c+d x)+1)^{\frac{1}{2} (-p-1)} (e \cos (c+d x))^p \, _2F_1\left(\frac{1}{2} (-p-15),\frac{p+1}{2};\frac{p+3}{2};\frac{1}{2} (1-\sin (c+d x))\right)}{d (p+1)}","-\frac{a^8 2^{\frac{p}{2}+\frac{17}{2}} (\sin (c+d x)+1)^{\frac{1}{2} (-p-1)} (e \cos (c+d x))^{p+1} \, _2F_1\left(\frac{1}{2} (-p-15),\frac{p+1}{2};\frac{p+3}{2};\frac{1}{2} (1-\sin (c+d x))\right)}{d e (p+1)}",1,"-((2^((17 + p)/2)*a^8*Cos[c + d*x]*(e*Cos[c + d*x])^p*Hypergeometric2F1[(-15 - p)/2, (1 + p)/2, (3 + p)/2, (1 - Sin[c + d*x])/2]*(1 + Sin[c + d*x])^((-1 - p)/2))/(d*(1 + p)))","A",1
328,1,94,95,0.1104186,"\int (e \cos (c+d x))^p (a+a \sin (c+d x))^3 \, dx","Integrate[(e*Cos[c + d*x])^p*(a + a*Sin[c + d*x])^3,x]","-\frac{a^3 2^{\frac{p+7}{2}} \cos (c+d x) (\sin (c+d x)+1)^{\frac{1}{2} (-p-1)} (e \cos (c+d x))^p \, _2F_1\left(\frac{1}{2} (-p-5),\frac{p+1}{2};\frac{p+3}{2};\frac{1}{2} (1-\sin (c+d x))\right)}{d (p+1)}","-\frac{a^3 2^{\frac{p}{2}+\frac{7}{2}} (\sin (c+d x)+1)^{\frac{1}{2} (-p-1)} (e \cos (c+d x))^{p+1} \, _2F_1\left(\frac{1}{2} (-p-5),\frac{p+1}{2};\frac{p+3}{2};\frac{1}{2} (1-\sin (c+d x))\right)}{d e (p+1)}",1,"-((2^((7 + p)/2)*a^3*Cos[c + d*x]*(e*Cos[c + d*x])^p*Hypergeometric2F1[(-5 - p)/2, (1 + p)/2, (3 + p)/2, (1 - Sin[c + d*x])/2]*(1 + Sin[c + d*x])^((-1 - p)/2))/(d*(1 + p)))","A",1
329,1,94,95,0.1055234,"\int (e \cos (c+d x))^p (a+a \sin (c+d x))^2 \, dx","Integrate[(e*Cos[c + d*x])^p*(a + a*Sin[c + d*x])^2,x]","-\frac{a^2 2^{\frac{p+5}{2}} \cos (c+d x) (\sin (c+d x)+1)^{\frac{1}{2} (-p-1)} (e \cos (c+d x))^p \, _2F_1\left(\frac{1}{2} (-p-3),\frac{p+1}{2};\frac{p+3}{2};\frac{1}{2} (1-\sin (c+d x))\right)}{d (p+1)}","-\frac{a^2 2^{\frac{p}{2}+\frac{5}{2}} (\sin (c+d x)+1)^{\frac{1}{2} (-p-1)} (e \cos (c+d x))^{p+1} \, _2F_1\left(\frac{1}{2} (-p-3),\frac{p+1}{2};\frac{p+3}{2};\frac{1}{2} (1-\sin (c+d x))\right)}{d e (p+1)}",1,"-((2^((5 + p)/2)*a^2*Cos[c + d*x]*(e*Cos[c + d*x])^p*Hypergeometric2F1[(-3 - p)/2, (1 + p)/2, (3 + p)/2, (1 - Sin[c + d*x])/2]*(1 + Sin[c + d*x])^((-1 - p)/2))/(d*(1 + p)))","A",1
330,1,245,93,1.3966276,"\int (e \cos (c+d x))^p (a+a \sin (c+d x)) \, dx","Integrate[(e*Cos[c + d*x])^p*(a + a*Sin[c + d*x]),x]","-\frac{i a 2^{-p-1} \left(e^{-i (c+d x)} \left(1+e^{2 i (c+d x)}\right)\right)^{p+1} (\sin (c+d x)+1) \left((p+1) e^{i (c+d x)} \left(i p e^{i (c+d x)} \, _2F_1\left(1,\frac{p+3}{2};\frac{3-p}{2};-e^{2 i (c+d x)}\right)-2 (p-1) \, _2F_1\left(1,\frac{p+2}{2};1-\frac{p}{2};-e^{2 i (c+d x)}\right)\right)-i (p-1) p \, _2F_1\left(1,\frac{p+1}{2};\frac{1-p}{2};-e^{2 i (c+d x)}\right)\right) \cos ^{-p}(c+d x) (e \cos (c+d x))^p}{d (p-1) p (p+1) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2}","-\frac{a 2^{\frac{p}{2}+\frac{3}{2}} (\sin (c+d x)+1)^{\frac{1}{2} (-p-1)} (e \cos (c+d x))^{p+1} \, _2F_1\left(\frac{1}{2} (-p-1),\frac{p+1}{2};\frac{p+3}{2};\frac{1}{2} (1-\sin (c+d x))\right)}{d e (p+1)}",1,"((-I)*2^(-1 - p)*a*((1 + E^((2*I)*(c + d*x)))/E^(I*(c + d*x)))^(1 + p)*(e*Cos[c + d*x])^p*((-I)*(-1 + p)*p*Hypergeometric2F1[1, (1 + p)/2, (1 - p)/2, -E^((2*I)*(c + d*x))] + E^(I*(c + d*x))*(1 + p)*(-2*(-1 + p)*Hypergeometric2F1[1, (2 + p)/2, 1 - p/2, -E^((2*I)*(c + d*x))] + I*E^(I*(c + d*x))*p*Hypergeometric2F1[1, (3 + p)/2, (3 - p)/2, -E^((2*I)*(c + d*x))]))*(1 + Sin[c + d*x]))/(d*(-1 + p)*p*(1 + p)*Cos[c + d*x]^p*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2)","C",0
331,1,94,95,0.1614013,"\int \frac{(e \cos (c+d x))^p}{a+a \sin (c+d x)} \, dx","Integrate[(e*Cos[c + d*x])^p/(a + a*Sin[c + d*x]),x]","-\frac{2^{\frac{p-1}{2}} \cos (c+d x) (\sin (c+d x)+1)^{\frac{1}{2} (-p-1)} (e \cos (c+d x))^p \, _2F_1\left(\frac{3-p}{2},\frac{p+1}{2};\frac{p+3}{2};\frac{1}{2} (1-\sin (c+d x))\right)}{a d (p+1)}","-\frac{2^{\frac{p}{2}-\frac{1}{2}} (\sin (c+d x)+1)^{\frac{1}{2} (-p-1)} (e \cos (c+d x))^{p+1} \, _2F_1\left(\frac{3-p}{2},\frac{p+1}{2};\frac{p+3}{2};\frac{1}{2} (1-\sin (c+d x))\right)}{a d e (p+1)}",1,"-((2^((-1 + p)/2)*Cos[c + d*x]*(e*Cos[c + d*x])^p*Hypergeometric2F1[(3 - p)/2, (1 + p)/2, (3 + p)/2, (1 - Sin[c + d*x])/2]*(1 + Sin[c + d*x])^((-1 - p)/2))/(a*d*(1 + p)))","A",1
332,1,94,93,0.1748708,"\int \frac{(e \cos (c+d x))^p}{(a+a \sin (c+d x))^2} \, dx","Integrate[(e*Cos[c + d*x])^p/(a + a*Sin[c + d*x])^2,x]","-\frac{2^{\frac{p-3}{2}} \cos (c+d x) (\sin (c+d x)+1)^{\frac{1}{2} (-p-1)} (e \cos (c+d x))^p \, _2F_1\left(\frac{5-p}{2},\frac{p+1}{2};\frac{p+3}{2};\frac{1}{2} (1-\sin (c+d x))\right)}{a^2 d (p+1)}","-\frac{2^{\frac{p-3}{2}} (\sin (c+d x)+1)^{\frac{1}{2} (-p-1)} (e \cos (c+d x))^{p+1} \, _2F_1\left(\frac{5-p}{2},\frac{p+1}{2};\frac{p+3}{2};\frac{1}{2} (1-\sin (c+d x))\right)}{a^2 d e (p+1)}",1,"-((2^((-3 + p)/2)*Cos[c + d*x]*(e*Cos[c + d*x])^p*Hypergeometric2F1[(5 - p)/2, (1 + p)/2, (3 + p)/2, (1 - Sin[c + d*x])/2]*(1 + Sin[c + d*x])^((-1 - p)/2))/(a^2*d*(1 + p)))","A",1
333,1,94,93,0.1643913,"\int \frac{(e \cos (c+d x))^p}{(a+a \sin (c+d x))^3} \, dx","Integrate[(e*Cos[c + d*x])^p/(a + a*Sin[c + d*x])^3,x]","-\frac{2^{\frac{p-5}{2}} \cos (c+d x) (\sin (c+d x)+1)^{\frac{1}{2} (-p-1)} (e \cos (c+d x))^p \, _2F_1\left(\frac{7-p}{2},\frac{p+1}{2};\frac{p+3}{2};\frac{1}{2} (1-\sin (c+d x))\right)}{a^3 d (p+1)}","-\frac{2^{\frac{p-5}{2}} (\sin (c+d x)+1)^{\frac{1}{2} (-p-1)} (e \cos (c+d x))^{p+1} \, _2F_1\left(\frac{7-p}{2},\frac{p+1}{2};\frac{p+3}{2};\frac{1}{2} (1-\sin (c+d x))\right)}{a^3 d e (p+1)}",1,"-((2^((-5 + p)/2)*Cos[c + d*x]*(e*Cos[c + d*x])^p*Hypergeometric2F1[(7 - p)/2, (1 + p)/2, (3 + p)/2, (1 - Sin[c + d*x])/2]*(1 + Sin[c + d*x])^((-1 - p)/2))/(a^3*d*(1 + p)))","A",1
334,1,94,93,0.206862,"\int \frac{(e \cos (c+d x))^p}{(a+a \sin (c+d x))^8} \, dx","Integrate[(e*Cos[c + d*x])^p/(a + a*Sin[c + d*x])^8,x]","-\frac{2^{\frac{p-15}{2}} \cos (c+d x) (\sin (c+d x)+1)^{\frac{1}{2} (-p-1)} (e \cos (c+d x))^p \, _2F_1\left(\frac{17-p}{2},\frac{p+1}{2};\frac{p+3}{2};\frac{1}{2} (1-\sin (c+d x))\right)}{a^8 d (p+1)}","-\frac{2^{\frac{p-15}{2}} (\sin (c+d x)+1)^{\frac{1}{2} (-p-1)} (e \cos (c+d x))^{p+1} \, _2F_1\left(\frac{17-p}{2},\frac{p+1}{2};\frac{p+3}{2};\frac{1}{2} (1-\sin (c+d x))\right)}{a^8 d e (p+1)}",1,"-((2^((-15 + p)/2)*Cos[c + d*x]*(e*Cos[c + d*x])^p*Hypergeometric2F1[(17 - p)/2, (1 + p)/2, (3 + p)/2, (1 - Sin[c + d*x])/2]*(1 + Sin[c + d*x])^((-1 - p)/2))/(a^8*d*(1 + p)))","A",1
335,1,102,103,0.2240098,"\int (e \cos (c+d x))^p (a+a \sin (c+d x))^{7/2} \, dx","Integrate[(e*Cos[c + d*x])^p*(a + a*Sin[c + d*x])^(7/2),x]","-\frac{a^4 2^{\frac{p}{2}+4} \cos (c+d x) (\sin (c+d x)+1)^{-p/2} (e \cos (c+d x))^p \, _2F_1\left(-\frac{p}{2}-3,\frac{p+1}{2};\frac{p+3}{2};\frac{1}{2} (1-\sin (c+d x))\right)}{d (p+1) \sqrt{a (\sin (c+d x)+1)}}","-\frac{a^4 2^{\frac{p}{2}+4} (\sin (c+d x)+1)^{-p/2} (e \cos (c+d x))^{p+1} \, _2F_1\left(\frac{1}{2} (-p-6),\frac{p+1}{2};\frac{p+3}{2};\frac{1}{2} (1-\sin (c+d x))\right)}{d e (p+1) \sqrt{a \sin (c+d x)+a}}",1,"-((2^(4 + p/2)*a^4*Cos[c + d*x]*(e*Cos[c + d*x])^p*Hypergeometric2F1[-3 - p/2, (1 + p)/2, (3 + p)/2, (1 - Sin[c + d*x])/2])/(d*(1 + p)*(1 + Sin[c + d*x])^(p/2)*Sqrt[a*(1 + Sin[c + d*x])]))","A",1
336,1,102,103,0.184354,"\int (e \cos (c+d x))^p (a+a \sin (c+d x))^{5/2} \, dx","Integrate[(e*Cos[c + d*x])^p*(a + a*Sin[c + d*x])^(5/2),x]","-\frac{a^3 2^{\frac{p}{2}+3} \cos (c+d x) (\sin (c+d x)+1)^{-p/2} (e \cos (c+d x))^p \, _2F_1\left(-\frac{p}{2}-2,\frac{p+1}{2};\frac{p+3}{2};\frac{1}{2} (1-\sin (c+d x))\right)}{d (p+1) \sqrt{a (\sin (c+d x)+1)}}","-\frac{a^3 2^{\frac{p}{2}+3} (\sin (c+d x)+1)^{-p/2} (e \cos (c+d x))^{p+1} \, _2F_1\left(\frac{1}{2} (-p-4),\frac{p+1}{2};\frac{p+3}{2};\frac{1}{2} (1-\sin (c+d x))\right)}{d e (p+1) \sqrt{a \sin (c+d x)+a}}",1,"-((2^(3 + p/2)*a^3*Cos[c + d*x]*(e*Cos[c + d*x])^p*Hypergeometric2F1[-2 - p/2, (1 + p)/2, (3 + p)/2, (1 - Sin[c + d*x])/2])/(d*(1 + p)*(1 + Sin[c + d*x])^(p/2)*Sqrt[a*(1 + Sin[c + d*x])]))","A",1
337,1,101,103,0.1872578,"\int (e \cos (c+d x))^p (a+a \sin (c+d x))^{3/2} \, dx","Integrate[(e*Cos[c + d*x])^p*(a + a*Sin[c + d*x])^(3/2),x]","-\frac{2^{\frac{p}{2}+2} \cos (c+d x) (a (\sin (c+d x)+1))^{3/2} (\sin (c+d x)+1)^{-\frac{p}{2}-2} (e \cos (c+d x))^p \, _2F_1\left(-\frac{p}{2}-1,\frac{p+1}{2};\frac{p+3}{2};\frac{1}{2} (1-\sin (c+d x))\right)}{d (p+1)}","-\frac{a^2 2^{\frac{p}{2}+2} (\sin (c+d x)+1)^{-p/2} (e \cos (c+d x))^{p+1} \, _2F_1\left(\frac{1}{2} (-p-2),\frac{p+1}{2};\frac{p+3}{2};\frac{1}{2} (1-\sin (c+d x))\right)}{d e (p+1) \sqrt{a \sin (c+d x)+a}}",1,"-((2^(2 + p/2)*Cos[c + d*x]*(e*Cos[c + d*x])^p*Hypergeometric2F1[-1 - p/2, (1 + p)/2, (3 + p)/2, (1 - Sin[c + d*x])/2]*(1 + Sin[c + d*x])^(-2 - p/2)*(a*(1 + Sin[c + d*x]))^(3/2))/(d*(1 + p)))","A",1
338,1,310,97,3.885962,"\int (e \cos (c+d x))^p \sqrt{a+a \sin (c+d x)} \, dx","Integrate[(e*Cos[c + d*x])^p*Sqrt[a + a*Sin[c + d*x]],x]","\frac{(1+i) 2^{-p} e^{-\frac{1}{2} i d x} \sqrt{a (\sin (c+d x)+1)} \cos ^{-p}(c+d x) (e \cos (c+d x))^p \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)^p \left(i \sin (2 c) e^{2 i d x}+\cos (2 c) e^{2 i d x}+1\right)^{-p} \left((2 p+1) e^{i d x} \left(\cos \left(\frac{c}{2}\right)+i \sin \left(\frac{c}{2}\right)\right) \, _2F_1\left(\frac{1}{4} (1-2 p),-p;\frac{1}{4} (5-2 p);-e^{2 i d x} (\cos (c)+i \sin (c))^2\right)+(2 p-1) \left(\sin \left(\frac{c}{2}\right)+i \cos \left(\frac{c}{2}\right)\right) \, _2F_1\left(\frac{1}{4} (-2 p-1),-p;\frac{1}{4} (3-2 p);-e^{2 i d x} (\cos (c)+i \sin (c))^2\right)\right)}{d (2 p-1) (2 p+1) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}","-\frac{a 2^{\frac{p}{2}+1} (\sin (c+d x)+1)^{-p/2} (e \cos (c+d x))^{p+1} \, _2F_1\left(-\frac{p}{2},\frac{p+1}{2};\frac{p+3}{2};\frac{1}{2} (1-\sin (c+d x))\right)}{d e (p+1) \sqrt{a \sin (c+d x)+a}}",1,"((1 + I)*(e*Cos[c + d*x])^p*(E^(I*d*x)*(1 + 2*p)*Hypergeometric2F1[(1 - 2*p)/4, -p, (5 - 2*p)/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*(Cos[c/2] + I*Sin[c/2]) + (-1 + 2*p)*Hypergeometric2F1[(-1 - 2*p)/4, -p, (3 - 2*p)/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*(I*Cos[c/2] + Sin[c/2]))*(((1 + E^((2*I)*d*x))*Cos[c] + I*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x))^p*Sqrt[a*(1 + Sin[c + d*x])])/(2^p*d*E^((I/2)*d*x)*(-1 + 2*p)*(1 + 2*p)*Cos[c + d*x]^p*(1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c])^p*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))","C",1
339,1,97,101,0.1032587,"\int \frac{(e \cos (c+d x))^p}{\sqrt{a+a \sin (c+d x)}} \, dx","Integrate[(e*Cos[c + d*x])^p/Sqrt[a + a*Sin[c + d*x]],x]","-\frac{2^{p/2} \cos (c+d x) (\sin (c+d x)+1)^{-p/2} (e \cos (c+d x))^p \, _2F_1\left(1-\frac{p}{2},\frac{p+1}{2};\frac{p+3}{2};\frac{1}{2} (1-\sin (c+d x))\right)}{d (p+1) \sqrt{a (\sin (c+d x)+1)}}","-\frac{a 2^{p/2} (\sin (c+d x)+1)^{1-\frac{p}{2}} (e \cos (c+d x))^{p+1} \, _2F_1\left(\frac{2-p}{2},\frac{p+1}{2};\frac{p+3}{2};\frac{1}{2} (1-\sin (c+d x))\right)}{d e (p+1) (a \sin (c+d x)+a)^{3/2}}",1,"-((2^(p/2)*Cos[c + d*x]*(e*Cos[c + d*x])^p*Hypergeometric2F1[1 - p/2, (1 + p)/2, (3 + p)/2, (1 - Sin[c + d*x])/2])/(d*(1 + p)*(1 + Sin[c + d*x])^(p/2)*Sqrt[a*(1 + Sin[c + d*x])]))","A",1
340,1,101,102,0.1700595,"\int \frac{(e \cos (c+d x))^p}{(a+a \sin (c+d x))^{3/2}} \, dx","Integrate[(e*Cos[c + d*x])^p/(a + a*Sin[c + d*x])^(3/2),x]","-\frac{2^{\frac{p}{2}-1} \cos (c+d x) (\sin (c+d x)+1)^{1-\frac{p}{2}} (e \cos (c+d x))^p \, _2F_1\left(2-\frac{p}{2},\frac{p+1}{2};\frac{p+3}{2};\frac{1}{2} (1-\sin (c+d x))\right)}{d (p+1) (a (\sin (c+d x)+1))^{3/2}}","-\frac{2^{\frac{p}{2}-1} (\sin (c+d x)+1)^{1-\frac{p}{2}} (e \cos (c+d x))^{p+1} \, _2F_1\left(\frac{4-p}{2},\frac{p+1}{2};\frac{p+3}{2};\frac{1}{2} (1-\sin (c+d x))\right)}{d e (p+1) (a \sin (c+d x)+a)^{3/2}}",1,"-((2^(-1 + p/2)*Cos[c + d*x]*(e*Cos[c + d*x])^p*Hypergeometric2F1[2 - p/2, (1 + p)/2, (3 + p)/2, (1 - Sin[c + d*x])/2]*(1 + Sin[c + d*x])^(1 - p/2))/(d*(1 + p)*(a*(1 + Sin[c + d*x]))^(3/2)))","A",1
341,1,102,105,0.1362526,"\int \frac{(e \cos (c+d x))^p}{(a+a \sin (c+d x))^{5/2}} \, dx","Integrate[(e*Cos[c + d*x])^p/(a + a*Sin[c + d*x])^(5/2),x]","-\frac{2^{\frac{p}{2}-2} \cos (c+d x) (\sin (c+d x)+1)^{-p/2} (e \cos (c+d x))^p \, _2F_1\left(3-\frac{p}{2},\frac{p+1}{2};\frac{p+3}{2};\frac{1}{2} (1-\sin (c+d x))\right)}{a^2 d (p+1) \sqrt{a (\sin (c+d x)+1)}}","-\frac{2^{\frac{p}{2}-2} (\sin (c+d x)+1)^{1-\frac{p}{2}} (e \cos (c+d x))^{p+1} \, _2F_1\left(\frac{6-p}{2},\frac{p+1}{2};\frac{p+3}{2};\frac{1}{2} (1-\sin (c+d x))\right)}{a d e (p+1) (a \sin (c+d x)+a)^{3/2}}",1,"-((2^(-2 + p/2)*Cos[c + d*x]*(e*Cos[c + d*x])^p*Hypergeometric2F1[3 - p/2, (1 + p)/2, (3 + p)/2, (1 - Sin[c + d*x])/2])/(a^2*d*(1 + p)*(1 + Sin[c + d*x])^(p/2)*Sqrt[a*(1 + Sin[c + d*x])]))","A",1
342,1,112,114,0.1981897,"\int (e \cos (c+d x))^p (a+a \sin (c+d x))^m \, dx","Integrate[(e*Cos[c + d*x])^p*(a + a*Sin[c + d*x])^m,x]","-\frac{2^{\frac{1}{2} (2 m+p+1)} \cos (c+d x) (a (\sin (c+d x)+1))^m (e \cos (c+d x))^p (\sin (c+d x)+1)^{\frac{1}{2} (-2 m-p-1)} \, _2F_1\left(\frac{1}{2} (-2 m-p+1),\frac{p+1}{2};\frac{p+3}{2};\frac{1}{2} (1-\sin (c+d x))\right)}{d (p+1)}","-\frac{a 2^{m+\frac{p}{2}+\frac{1}{2}} (a \sin (c+d x)+a)^{m-1} (e \cos (c+d x))^{p+1} (\sin (c+d x)+1)^{\frac{1}{2} (-2 m-p+1)} \, _2F_1\left(\frac{1}{2} (-2 m-p+1),\frac{p+1}{2};\frac{p+3}{2};\frac{1}{2} (1-\sin (c+d x))\right)}{d e (p+1)}",1,"-((2^((1 + 2*m + p)/2)*Cos[c + d*x]*(e*Cos[c + d*x])^p*Hypergeometric2F1[(1 - 2*m - p)/2, (1 + p)/2, (3 + p)/2, (1 - Sin[c + d*x])/2]*(1 + Sin[c + d*x])^((-1 - 2*m - p)/2)*(a*(1 + Sin[c + d*x]))^m)/(d*(1 + p)))","A",1
343,1,89,109,0.7020325,"\int \cos ^7(c+d x) (a+a \sin (c+d x))^m \, dx","Integrate[Cos[c + d*x]^7*(a + a*Sin[c + d*x])^m,x]","\frac{(a (\sin (c+d x)+1))^{m+4} \left(\frac{6 a^3 (\sin (c+d x)+1)^2}{m+6}-\frac{12 a^3 (\sin (c+d x)+1)}{m+5}+\frac{8 a^3}{m+4}-\frac{(a \sin (c+d x)+a)^3}{m+7}\right)}{a^7 d}","-\frac{(a \sin (c+d x)+a)^{m+7}}{a^7 d (m+7)}+\frac{6 (a \sin (c+d x)+a)^{m+6}}{a^6 d (m+6)}-\frac{12 (a \sin (c+d x)+a)^{m+5}}{a^5 d (m+5)}+\frac{8 (a \sin (c+d x)+a)^{m+4}}{a^4 d (m+4)}",1,"((a*(1 + Sin[c + d*x]))^(4 + m)*((8*a^3)/(4 + m) - (12*a^3*(1 + Sin[c + d*x]))/(5 + m) + (6*a^3*(1 + Sin[c + d*x])^2)/(6 + m) - (a + a*Sin[c + d*x])^3/(7 + m)))/(a^7*d)","A",1
344,1,68,81,0.3197891,"\int \cos ^5(c+d x) (a+a \sin (c+d x))^m \, dx","Integrate[Cos[c + d*x]^5*(a + a*Sin[c + d*x])^m,x]","\frac{(a (\sin (c+d x)+1))^{m+3} \left(-\frac{4 a^2 (\sin (c+d x)+1)}{m+4}+\frac{4 a^2}{m+3}+\frac{(a \sin (c+d x)+a)^2}{m+5}\right)}{a^5 d}","\frac{(a \sin (c+d x)+a)^{m+5}}{a^5 d (m+5)}-\frac{4 (a \sin (c+d x)+a)^{m+4}}{a^4 d (m+4)}+\frac{4 (a \sin (c+d x)+a)^{m+3}}{a^3 d (m+3)}",1,"((a*(1 + Sin[c + d*x]))^(3 + m)*((4*a^2)/(3 + m) - (4*a^2*(1 + Sin[c + d*x]))/(4 + m) + (a + a*Sin[c + d*x])^2/(5 + m)))/(a^5*d)","A",1
345,1,52,55,0.1179646,"\int \cos ^3(c+d x) (a+a \sin (c+d x))^m \, dx","Integrate[Cos[c + d*x]^3*(a + a*Sin[c + d*x])^m,x]","-\frac{(\sin (c+d x)+1)^2 ((m+2) \sin (c+d x)-m-4) (a (\sin (c+d x)+1))^m}{d (m+2) (m+3)}","\frac{2 (a \sin (c+d x)+a)^{m+2}}{a^2 d (m+2)}-\frac{(a \sin (c+d x)+a)^{m+3}}{a^3 d (m+3)}",1,"-(((1 + Sin[c + d*x])^2*(a*(1 + Sin[c + d*x]))^m*(-4 - m + (2 + m)*Sin[c + d*x]))/(d*(2 + m)*(3 + m)))","A",1
346,1,26,26,0.0371892,"\int \cos (c+d x) (a+a \sin (c+d x))^m \, dx","Integrate[Cos[c + d*x]*(a + a*Sin[c + d*x])^m,x]","\frac{(a (\sin (c+d x)+1))^{m+1}}{a d (m+1)}","\frac{(a \sin (c+d x)+a)^{m+1}}{a d (m+1)}",1,"(a*(1 + Sin[c + d*x]))^(1 + m)/(a*d*(1 + m))","A",1
347,1,63,40,0.1007434,"\int \sec (c+d x) (a+a \sin (c+d x))^m \, dx","Integrate[Sec[c + d*x]*(a + a*Sin[c + d*x])^m,x]","\frac{(a (\sin (c+d x)+1))^m \left(m (\sin (c+d x)+1) \, _2F_1\left(1,m+1;m+2;\frac{1}{2} (\sin (c+d x)+1)\right)+2 (m+1)\right)}{4 d m (m+1)}","\frac{(a \sin (c+d x)+a)^m \, _2F_1\left(1,m;m+1;\frac{1}{2} (\sin (c+d x)+1)\right)}{2 d m}",1,"((a*(1 + Sin[c + d*x]))^m*(2*(1 + m) + m*Hypergeometric2F1[1, 1 + m, 2 + m, (1 + Sin[c + d*x])/2]*(1 + Sin[c + d*x])))/(4*d*m*(1 + m))","A",1
348,1,111,47,0.3558756,"\int \sec ^3(c+d x) (a+a \sin (c+d x))^m \, dx","Integrate[Sec[c + d*x]^3*(a + a*Sin[c + d*x])^m,x]","\frac{(a (\sin (c+d x)+1))^m \left(\frac{2 (\sin (c+d x)+1) \, _2F_1\left(1,m+1;m+2;\frac{1}{2} (\sin (c+d x)+1)\right)}{m+1}+\frac{(\sin (c+d x)+1) \, _2F_1\left(2,m+1;m+2;\frac{1}{2} (\sin (c+d x)+1)\right)}{m+1}+4 \left(\frac{1}{(m-1) (\sin (c+d x)+1)}+\frac{1}{m}\right)\right)}{16 d}","-\frac{a (a \sin (c+d x)+a)^{m-1} \, _2F_1\left(2,m-1;m;\frac{1}{2} (\sin (c+d x)+1)\right)}{4 d (1-m)}",1,"((a*(1 + Sin[c + d*x]))^m*((2*Hypergeometric2F1[1, 1 + m, 2 + m, (1 + Sin[c + d*x])/2]*(1 + Sin[c + d*x]))/(1 + m) + (Hypergeometric2F1[2, 1 + m, 2 + m, (1 + Sin[c + d*x])/2]*(1 + Sin[c + d*x]))/(1 + m) + 4*(m^(-1) + 1/((-1 + m)*(1 + Sin[c + d*x])))))/(16*d)","B",1
349,1,163,51,0.7358498,"\int \sec ^5(c+d x) (a+a \sin (c+d x))^m \, dx","Integrate[Sec[c + d*x]^5*(a + a*Sin[c + d*x])^m,x]","\frac{(a (\sin (c+d x)+1))^m \left(\frac{6 (\sin (c+d x)+1) \, _2F_1\left(1,m+1;m+2;\frac{1}{2} (\sin (c+d x)+1)\right)}{m+1}+\frac{3 (\sin (c+d x)+1) \, _2F_1\left(2,m+1;m+2;\frac{1}{2} (\sin (c+d x)+1)\right)}{m+1}+\frac{(\sin (c+d x)+1) \, _2F_1\left(3,m+1;m+2;\frac{1}{2} (\sin (c+d x)+1)\right)}{m+1}+\frac{12}{(m-1) (\sin (c+d x)+1)}+\frac{8}{(m-2) (\sin (c+d x)+1)^2}+\frac{12}{m}\right)}{64 d}","-\frac{a^2 (a \sin (c+d x)+a)^{m-2} \, _2F_1\left(3,m-2;m-1;\frac{1}{2} (\sin (c+d x)+1)\right)}{8 d (2-m)}",1,"((a*(1 + Sin[c + d*x]))^m*(12/m + 8/((-2 + m)*(1 + Sin[c + d*x])^2) + 12/((-1 + m)*(1 + Sin[c + d*x])) + (6*Hypergeometric2F1[1, 1 + m, 2 + m, (1 + Sin[c + d*x])/2]*(1 + Sin[c + d*x]))/(1 + m) + (3*Hypergeometric2F1[2, 1 + m, 2 + m, (1 + Sin[c + d*x])/2]*(1 + Sin[c + d*x]))/(1 + m) + (Hypergeometric2F1[3, 1 + m, 2 + m, (1 + Sin[c + d*x])/2]*(1 + Sin[c + d*x]))/(1 + m)))/(64*d)","B",1
350,1,78,83,0.1245578,"\int \cos ^4(c+d x) (a+a \sin (c+d x))^m \, dx","Integrate[Cos[c + d*x]^4*(a + a*Sin[c + d*x])^m,x]","-\frac{2^{m+\frac{5}{2}} \cos ^5(c+d x) (\sin (c+d x)+1)^{-m-\frac{5}{2}} (a (\sin (c+d x)+1))^m \, _2F_1\left(\frac{5}{2},-m-\frac{3}{2};\frac{7}{2};\frac{1}{2} (1-\sin (c+d x))\right)}{5 d}","-\frac{a^2 2^{m+\frac{5}{2}} \cos ^5(c+d x) (\sin (c+d x)+1)^{-m-\frac{1}{2}} (a \sin (c+d x)+a)^{m-2} \, _2F_1\left(\frac{5}{2},-m-\frac{3}{2};\frac{7}{2};\frac{1}{2} (1-\sin (c+d x))\right)}{5 d}",1,"-1/5*(2^(5/2 + m)*Cos[c + d*x]^5*Hypergeometric2F1[5/2, -3/2 - m, 7/2, (1 - Sin[c + d*x])/2]*(1 + Sin[c + d*x])^(-5/2 - m)*(a*(1 + Sin[c + d*x]))^m)/d","A",1
351,1,78,81,0.1001582,"\int \cos ^2(c+d x) (a+a \sin (c+d x))^m \, dx","Integrate[Cos[c + d*x]^2*(a + a*Sin[c + d*x])^m,x]","-\frac{2^{m+\frac{3}{2}} \cos ^3(c+d x) (\sin (c+d x)+1)^{-m-\frac{3}{2}} (a (\sin (c+d x)+1))^m \, _2F_1\left(\frac{3}{2},-m-\frac{1}{2};\frac{5}{2};\frac{1}{2} (1-\sin (c+d x))\right)}{3 d}","-\frac{a 2^{m+\frac{3}{2}} \cos ^3(c+d x) (\sin (c+d x)+1)^{-m-\frac{1}{2}} (a \sin (c+d x)+a)^{m-1} \, _2F_1\left(\frac{3}{2},-m-\frac{1}{2};\frac{5}{2};\frac{1}{2} (1-\sin (c+d x))\right)}{3 d}",1,"-1/3*(2^(3/2 + m)*Cos[c + d*x]^3*Hypergeometric2F1[3/2, -1/2 - m, 5/2, (1 - Sin[c + d*x])/2]*(1 + Sin[c + d*x])^(-3/2 - m)*(a*(1 + Sin[c + d*x]))^m)/d","A",1
352,1,3917,73,16.1163249,"\int \sec ^2(c+d x) (a+a \sin (c+d x))^m \, dx","Integrate[Sec[c + d*x]^2*(a + a*Sin[c + d*x])^m,x]","\text{Result too large to show}","\frac{2^{m-\frac{1}{2}} \sec (c+d x) (\sin (c+d x)+1)^{\frac{1}{2}-m} (a \sin (c+d x)+a)^m \, _2F_1\left(-\frac{1}{2},\frac{3}{2}-m;\frac{1}{2};\frac{1}{2} (1-\sin (c+d x))\right)}{d}",1,"-1/4*((Cos[(-c + Pi/2 - d*x)/4]^2)^(2*m)*Cot[(-c + Pi/2 - d*x)/4]*(a + a*Sin[c + d*x])^m*(-(AppellF1[-1/2, -2*m, 2*m, 1/2, Tan[(-c + Pi/2 - d*x)/4]^2, -Tan[(-c + Pi/2 - d*x)/4]^2]*(Sec[(-c + Pi/2 - d*x)/4]^2)^(2*m)) + (3*AppellF1[1/2, -2*m, 2*m, 3/2, Tan[(-c + Pi/2 - d*x)/4]^2, -Tan[(-c + Pi/2 - d*x)/4]^2]*Tan[(-c + Pi/2 - d*x)/4]^2*(1 - Tan[(-c + Pi/2 - d*x)/4]^2)^(2*m))/(3*AppellF1[1/2, -2*m, 2*m, 3/2, Tan[(-c + Pi/2 - d*x)/4]^2, -Tan[(-c + Pi/2 - d*x)/4]^2] - 4*m*(AppellF1[3/2, 1 - 2*m, 2*m, 5/2, Tan[(-c + Pi/2 - d*x)/4]^2, -Tan[(-c + Pi/2 - d*x)/4]^2] + AppellF1[3/2, -2*m, 1 + 2*m, 5/2, Tan[(-c + Pi/2 - d*x)/4]^2, -Tan[(-c + Pi/2 - d*x)/4]^2])*Tan[(-c + Pi/2 - d*x)/4]^2)))/(d*(Cos[Pi/4 + (c - Pi/2 + d*x)/2] - Sin[Pi/4 + (c - Pi/2 + d*x)/2])^2*(-1/2*(m*(Cos[(-c + Pi/2 - d*x)/4]^2)^(2*m)*(-(AppellF1[-1/2, -2*m, 2*m, 1/2, Tan[(-c + Pi/2 - d*x)/4]^2, -Tan[(-c + Pi/2 - d*x)/4]^2]*(Sec[(-c + Pi/2 - d*x)/4]^2)^(2*m)) + (3*AppellF1[1/2, -2*m, 2*m, 3/2, Tan[(-c + Pi/2 - d*x)/4]^2, -Tan[(-c + Pi/2 - d*x)/4]^2]*Tan[(-c + Pi/2 - d*x)/4]^2*(1 - Tan[(-c + Pi/2 - d*x)/4]^2)^(2*m))/(3*AppellF1[1/2, -2*m, 2*m, 3/2, Tan[(-c + Pi/2 - d*x)/4]^2, -Tan[(-c + Pi/2 - d*x)/4]^2] - 4*m*(AppellF1[3/2, 1 - 2*m, 2*m, 5/2, Tan[(-c + Pi/2 - d*x)/4]^2, -Tan[(-c + Pi/2 - d*x)/4]^2] + AppellF1[3/2, -2*m, 1 + 2*m, 5/2, Tan[(-c + Pi/2 - d*x)/4]^2, -Tan[(-c + Pi/2 - d*x)/4]^2])*Tan[(-c + Pi/2 - d*x)/4]^2))) - ((Cos[(-c + Pi/2 - d*x)/4]^2)^(2*m)*Csc[(-c + Pi/2 - d*x)/4]^2*(-(AppellF1[-1/2, -2*m, 2*m, 1/2, Tan[(-c + Pi/2 - d*x)/4]^2, -Tan[(-c + Pi/2 - d*x)/4]^2]*(Sec[(-c + Pi/2 - d*x)/4]^2)^(2*m)) + (3*AppellF1[1/2, -2*m, 2*m, 3/2, Tan[(-c + Pi/2 - d*x)/4]^2, -Tan[(-c + Pi/2 - d*x)/4]^2]*Tan[(-c + Pi/2 - d*x)/4]^2*(1 - Tan[(-c + Pi/2 - d*x)/4]^2)^(2*m))/(3*AppellF1[1/2, -2*m, 2*m, 3/2, Tan[(-c + Pi/2 - d*x)/4]^2, -Tan[(-c + Pi/2 - d*x)/4]^2] - 4*m*(AppellF1[3/2, 1 - 2*m, 2*m, 5/2, Tan[(-c + Pi/2 - d*x)/4]^2, -Tan[(-c + Pi/2 - d*x)/4]^2] + AppellF1[3/2, -2*m, 1 + 2*m, 5/2, Tan[(-c + Pi/2 - d*x)/4]^2, -Tan[(-c + Pi/2 - d*x)/4]^2])*Tan[(-c + Pi/2 - d*x)/4]^2)))/8 + ((Cos[(-c + Pi/2 - d*x)/4]^2)^(2*m)*Cot[(-c + Pi/2 - d*x)/4]*(-(m*AppellF1[-1/2, -2*m, 2*m, 1/2, Tan[(-c + Pi/2 - d*x)/4]^2, -Tan[(-c + Pi/2 - d*x)/4]^2]*(Sec[(-c + Pi/2 - d*x)/4]^2)^(2*m)*Tan[(-c + Pi/2 - d*x)/4]) - (Sec[(-c + Pi/2 - d*x)/4]^2)^(2*m)*(m*AppellF1[1/2, 1 - 2*m, 2*m, 3/2, Tan[(-c + Pi/2 - d*x)/4]^2, -Tan[(-c + Pi/2 - d*x)/4]^2]*Sec[(-c + Pi/2 - d*x)/4]^2*Tan[(-c + Pi/2 - d*x)/4] + m*AppellF1[1/2, -2*m, 1 + 2*m, 3/2, Tan[(-c + Pi/2 - d*x)/4]^2, -Tan[(-c + Pi/2 - d*x)/4]^2]*Sec[(-c + Pi/2 - d*x)/4]^2*Tan[(-c + Pi/2 - d*x)/4]) + (3*AppellF1[1/2, -2*m, 2*m, 3/2, Tan[(-c + Pi/2 - d*x)/4]^2, -Tan[(-c + Pi/2 - d*x)/4]^2]*Sec[(-c + Pi/2 - d*x)/4]^2*Tan[(-c + Pi/2 - d*x)/4]*(1 - Tan[(-c + Pi/2 - d*x)/4]^2)^(2*m))/(2*(3*AppellF1[1/2, -2*m, 2*m, 3/2, Tan[(-c + Pi/2 - d*x)/4]^2, -Tan[(-c + Pi/2 - d*x)/4]^2] - 4*m*(AppellF1[3/2, 1 - 2*m, 2*m, 5/2, Tan[(-c + Pi/2 - d*x)/4]^2, -Tan[(-c + Pi/2 - d*x)/4]^2] + AppellF1[3/2, -2*m, 1 + 2*m, 5/2, Tan[(-c + Pi/2 - d*x)/4]^2, -Tan[(-c + Pi/2 - d*x)/4]^2])*Tan[(-c + Pi/2 - d*x)/4]^2)) + (3*Tan[(-c + Pi/2 - d*x)/4]^2*(-1/3*(m*AppellF1[3/2, 1 - 2*m, 2*m, 5/2, Tan[(-c + Pi/2 - d*x)/4]^2, -Tan[(-c + Pi/2 - d*x)/4]^2]*Sec[(-c + Pi/2 - d*x)/4]^2*Tan[(-c + Pi/2 - d*x)/4]) - (m*AppellF1[3/2, -2*m, 1 + 2*m, 5/2, Tan[(-c + Pi/2 - d*x)/4]^2, -Tan[(-c + Pi/2 - d*x)/4]^2]*Sec[(-c + Pi/2 - d*x)/4]^2*Tan[(-c + Pi/2 - d*x)/4])/3)*(1 - Tan[(-c + Pi/2 - d*x)/4]^2)^(2*m))/(3*AppellF1[1/2, -2*m, 2*m, 3/2, Tan[(-c + Pi/2 - d*x)/4]^2, -Tan[(-c + Pi/2 - d*x)/4]^2] - 4*m*(AppellF1[3/2, 1 - 2*m, 2*m, 5/2, Tan[(-c + Pi/2 - d*x)/4]^2, -Tan[(-c + Pi/2 - d*x)/4]^2] + AppellF1[3/2, -2*m, 1 + 2*m, 5/2, Tan[(-c + Pi/2 - d*x)/4]^2, -Tan[(-c + Pi/2 - d*x)/4]^2])*Tan[(-c + Pi/2 - d*x)/4]^2) - (3*m*AppellF1[1/2, -2*m, 2*m, 3/2, Tan[(-c + Pi/2 - d*x)/4]^2, -Tan[(-c + Pi/2 - d*x)/4]^2]*Sec[(-c + Pi/2 - d*x)/4]^2*Tan[(-c + Pi/2 - d*x)/4]^3*(1 - Tan[(-c + Pi/2 - d*x)/4]^2)^(-1 + 2*m))/(3*AppellF1[1/2, -2*m, 2*m, 3/2, Tan[(-c + Pi/2 - d*x)/4]^2, -Tan[(-c + Pi/2 - d*x)/4]^2] - 4*m*(AppellF1[3/2, 1 - 2*m, 2*m, 5/2, Tan[(-c + Pi/2 - d*x)/4]^2, -Tan[(-c + Pi/2 - d*x)/4]^2] + AppellF1[3/2, -2*m, 1 + 2*m, 5/2, Tan[(-c + Pi/2 - d*x)/4]^2, -Tan[(-c + Pi/2 - d*x)/4]^2])*Tan[(-c + Pi/2 - d*x)/4]^2) - (3*AppellF1[1/2, -2*m, 2*m, 3/2, Tan[(-c + Pi/2 - d*x)/4]^2, -Tan[(-c + Pi/2 - d*x)/4]^2]*Tan[(-c + Pi/2 - d*x)/4]^2*(1 - Tan[(-c + Pi/2 - d*x)/4]^2)^(2*m)*(-2*m*(AppellF1[3/2, 1 - 2*m, 2*m, 5/2, Tan[(-c + Pi/2 - d*x)/4]^2, -Tan[(-c + Pi/2 - d*x)/4]^2] + AppellF1[3/2, -2*m, 1 + 2*m, 5/2, Tan[(-c + Pi/2 - d*x)/4]^2, -Tan[(-c + Pi/2 - d*x)/4]^2])*Sec[(-c + Pi/2 - d*x)/4]^2*Tan[(-c + Pi/2 - d*x)/4] + 3*(-1/3*(m*AppellF1[3/2, 1 - 2*m, 2*m, 5/2, Tan[(-c + Pi/2 - d*x)/4]^2, -Tan[(-c + Pi/2 - d*x)/4]^2]*Sec[(-c + Pi/2 - d*x)/4]^2*Tan[(-c + Pi/2 - d*x)/4]) - (m*AppellF1[3/2, -2*m, 1 + 2*m, 5/2, Tan[(-c + Pi/2 - d*x)/4]^2, -Tan[(-c + Pi/2 - d*x)/4]^2]*Sec[(-c + Pi/2 - d*x)/4]^2*Tan[(-c + Pi/2 - d*x)/4])/3) - 4*m*Tan[(-c + Pi/2 - d*x)/4]^2*((-6*m*AppellF1[5/2, 1 - 2*m, 1 + 2*m, 7/2, Tan[(-c + Pi/2 - d*x)/4]^2, -Tan[(-c + Pi/2 - d*x)/4]^2]*Sec[(-c + Pi/2 - d*x)/4]^2*Tan[(-c + Pi/2 - d*x)/4])/5 + (3*(1 - 2*m)*AppellF1[5/2, 2 - 2*m, 2*m, 7/2, Tan[(-c + Pi/2 - d*x)/4]^2, -Tan[(-c + Pi/2 - d*x)/4]^2]*Sec[(-c + Pi/2 - d*x)/4]^2*Tan[(-c + Pi/2 - d*x)/4])/10 - (3*(1 + 2*m)*AppellF1[5/2, -2*m, 2 + 2*m, 7/2, Tan[(-c + Pi/2 - d*x)/4]^2, -Tan[(-c + Pi/2 - d*x)/4]^2]*Sec[(-c + Pi/2 - d*x)/4]^2*Tan[(-c + Pi/2 - d*x)/4])/10)))/(3*AppellF1[1/2, -2*m, 2*m, 3/2, Tan[(-c + Pi/2 - d*x)/4]^2, -Tan[(-c + Pi/2 - d*x)/4]^2] - 4*m*(AppellF1[3/2, 1 - 2*m, 2*m, 5/2, Tan[(-c + Pi/2 - d*x)/4]^2, -Tan[(-c + Pi/2 - d*x)/4]^2] + AppellF1[3/2, -2*m, 1 + 2*m, 5/2, Tan[(-c + Pi/2 - d*x)/4]^2, -Tan[(-c + Pi/2 - d*x)/4]^2])*Tan[(-c + Pi/2 - d*x)/4]^2)^2))/2)) + (Hypergeometric2F1[1/2, (-1 + 2*m)/2, (1 + 2*m)/2, Cos[(-c + Pi/2 - d*x)/2]^2]*(a + a*Sin[c + d*x])^m*Tan[(-c + Pi/2 - d*x)/2])/(2*d*(-1 + 2*m)*Sqrt[Sin[(-c + Pi/2 - d*x)/2]^2])","C",0
353,1,9400,83,21.2767768,"\int \sec ^4(c+d x) (a+a \sin (c+d x))^m \, dx","Integrate[Sec[c + d*x]^4*(a + a*Sin[c + d*x])^m,x]","\text{Result too large to show}","\frac{2^{m-\frac{3}{2}} \sec ^3(c+d x) (\sin (c+d x)+1)^{\frac{1}{2}-m} (a \sin (c+d x)+a)^{m+1} \, _2F_1\left(-\frac{3}{2},\frac{5}{2}-m;-\frac{1}{2};\frac{1}{2} (1-\sin (c+d x))\right)}{3 a d}",1,"Result too large to show","C",0
354,1,85,88,0.2054783,"\int (e \cos (c+d x))^{5/2} (a+a \sin (c+d x))^m \, dx","Integrate[(e*Cos[c + d*x])^(5/2)*(a + a*Sin[c + d*x])^m,x]","-\frac{2^{m+\frac{11}{4}} (e \cos (c+d x))^{7/2} (\sin (c+d x)+1)^{-m-\frac{7}{4}} (a (\sin (c+d x)+1))^m \, _2F_1\left(\frac{7}{4},-m-\frac{3}{4};\frac{11}{4};\frac{1}{2} (1-\sin (c+d x))\right)}{7 d e}","-\frac{a 2^{m+\frac{11}{4}} (e \cos (c+d x))^{7/2} (\sin (c+d x)+1)^{-m-\frac{3}{4}} (a \sin (c+d x)+a)^{m-1} \, _2F_1\left(\frac{7}{4},-m-\frac{3}{4};\frac{11}{4};\frac{1}{2} (1-\sin (c+d x))\right)}{7 d e}",1,"-1/7*(2^(11/4 + m)*(e*Cos[c + d*x])^(7/2)*Hypergeometric2F1[7/4, -3/4 - m, 11/4, (1 - Sin[c + d*x])/2]*(1 + Sin[c + d*x])^(-7/4 - m)*(a*(1 + Sin[c + d*x]))^m)/(d*e)","A",1
355,1,85,88,0.1535328,"\int (e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^m \, dx","Integrate[(e*Cos[c + d*x])^(3/2)*(a + a*Sin[c + d*x])^m,x]","-\frac{2^{m+\frac{9}{4}} (e \cos (c+d x))^{5/2} (\sin (c+d x)+1)^{-m-\frac{5}{4}} (a (\sin (c+d x)+1))^m \, _2F_1\left(\frac{5}{4},-m-\frac{1}{4};\frac{9}{4};\frac{1}{2} (1-\sin (c+d x))\right)}{5 d e}","-\frac{a 2^{m+\frac{9}{4}} (e \cos (c+d x))^{5/2} (\sin (c+d x)+1)^{-m-\frac{1}{4}} (a \sin (c+d x)+a)^{m-1} \, _2F_1\left(\frac{5}{4},-m-\frac{1}{4};\frac{9}{4};\frac{1}{2} (1-\sin (c+d x))\right)}{5 d e}",1,"-1/5*(2^(9/4 + m)*(e*Cos[c + d*x])^(5/2)*Hypergeometric2F1[5/4, -1/4 - m, 9/4, (1 - Sin[c + d*x])/2]*(1 + Sin[c + d*x])^(-5/4 - m)*(a*(1 + Sin[c + d*x]))^m)/(d*e)","A",1
356,1,85,88,0.0841687,"\int \sqrt{e \cos (c+d x)} (a+a \sin (c+d x))^m \, dx","Integrate[Sqrt[e*Cos[c + d*x]]*(a + a*Sin[c + d*x])^m,x]","-\frac{2^{m+\frac{7}{4}} (e \cos (c+d x))^{3/2} (\sin (c+d x)+1)^{-m-\frac{3}{4}} (a (\sin (c+d x)+1))^m \, _2F_1\left(\frac{3}{4},\frac{1}{4}-m;\frac{7}{4};\frac{1}{2} (1-\sin (c+d x))\right)}{3 d e}","-\frac{a 2^{m+\frac{7}{4}} (e \cos (c+d x))^{3/2} (\sin (c+d x)+1)^{\frac{1}{4}-m} (a \sin (c+d x)+a)^{m-1} \, _2F_1\left(\frac{3}{4},\frac{1}{4}-m;\frac{7}{4};\frac{1}{2} (1-\sin (c+d x))\right)}{3 d e}",1,"-1/3*(2^(7/4 + m)*(e*Cos[c + d*x])^(3/2)*Hypergeometric2F1[3/4, 1/4 - m, 7/4, (1 - Sin[c + d*x])/2]*(1 + Sin[c + d*x])^(-3/4 - m)*(a*(1 + Sin[c + d*x]))^m)/(d*e)","A",1
357,1,83,86,0.0776136,"\int \frac{(a+a \sin (c+d x))^m}{\sqrt{e \cos (c+d x)}} \, dx","Integrate[(a + a*Sin[c + d*x])^m/Sqrt[e*Cos[c + d*x]],x]","-\frac{2^{m+\frac{5}{4}} \sqrt{e \cos (c+d x)} (\sin (c+d x)+1)^{-m-\frac{1}{4}} (a (\sin (c+d x)+1))^m \, _2F_1\left(\frac{1}{4},\frac{3}{4}-m;\frac{5}{4};\frac{1}{2} (1-\sin (c+d x))\right)}{d e}","-\frac{a 2^{m+\frac{5}{4}} \sqrt{e \cos (c+d x)} (\sin (c+d x)+1)^{\frac{3}{4}-m} (a \sin (c+d x)+a)^{m-1} \, _2F_1\left(\frac{1}{4},\frac{3}{4}-m;\frac{5}{4};\frac{1}{2} (1-\sin (c+d x))\right)}{d e}",1,"-((2^(5/4 + m)*Sqrt[e*Cos[c + d*x]]*Hypergeometric2F1[1/4, 3/4 - m, 5/4, (1 - Sin[c + d*x])/2]*(1 + Sin[c + d*x])^(-1/4 - m)*(a*(1 + Sin[c + d*x]))^m)/(d*e))","A",1
358,1,82,82,0.0991366,"\int \frac{(a+a \sin (c+d x))^m}{(e \cos (c+d x))^{3/2}} \, dx","Integrate[(a + a*Sin[c + d*x])^m/(e*Cos[c + d*x])^(3/2),x]","\frac{2^{m+\frac{3}{4}} (\sin (c+d x)+1)^{\frac{1}{4}-m} (a (\sin (c+d x)+1))^m \, _2F_1\left(-\frac{1}{4},\frac{5}{4}-m;\frac{3}{4};\frac{1}{2} (1-\sin (c+d x))\right)}{d e \sqrt{e \cos (c+d x)}}","\frac{2^{m+\frac{3}{4}} (\sin (c+d x)+1)^{\frac{1}{4}-m} (a \sin (c+d x)+a)^m \, _2F_1\left(-\frac{1}{4},\frac{5}{4}-m;\frac{3}{4};\frac{1}{2} (1-\sin (c+d x))\right)}{d e \sqrt{e \cos (c+d x)}}",1,"(2^(3/4 + m)*Hypergeometric2F1[-1/4, 5/4 - m, 3/4, (1 - Sin[c + d*x])/2]*(1 + Sin[c + d*x])^(1/4 - m)*(a*(1 + Sin[c + d*x]))^m)/(d*e*Sqrt[e*Cos[c + d*x]])","A",1
359,1,85,85,0.10135,"\int \frac{(a+a \sin (c+d x))^m}{(e \cos (c+d x))^{5/2}} \, dx","Integrate[(a + a*Sin[c + d*x])^m/(e*Cos[c + d*x])^(5/2),x]","\frac{2^{m+\frac{1}{4}} (\sin (c+d x)+1)^{\frac{3}{4}-m} (a (\sin (c+d x)+1))^m \, _2F_1\left(-\frac{3}{4},\frac{7}{4}-m;\frac{1}{4};\frac{1}{2} (1-\sin (c+d x))\right)}{3 d e (e \cos (c+d x))^{3/2}}","\frac{2^{m+\frac{1}{4}} (\sin (c+d x)+1)^{\frac{3}{4}-m} (a \sin (c+d x)+a)^m \, _2F_1\left(-\frac{3}{4},\frac{7}{4}-m;\frac{1}{4};\frac{1}{2} (1-\sin (c+d x))\right)}{3 d e (e \cos (c+d x))^{3/2}}",1,"(2^(1/4 + m)*Hypergeometric2F1[-3/4, 7/4 - m, 1/4, (1 - Sin[c + d*x])/2]*(1 + Sin[c + d*x])^(3/4 - m)*(a*(1 + Sin[c + d*x]))^m)/(3*d*e*(e*Cos[c + d*x])^(3/2))","A",1
360,1,101,201,0.1926942,"\int (e \cos (c+d x))^{-4-m} (a+a \sin (c+d x))^m \, dx","Integrate[(e*Cos[c + d*x])^(-4 - m)*(a + a*Sin[c + d*x])^m,x]","\frac{\sec ^3(c+d x) \left(-3 \left(m^2-3\right) \sin (c+d x)+6 m \sin ^2(c+d x)-6 \sin ^3(c+d x)+m \left(m^2-7\right)\right) (a (\sin (c+d x)+1))^m (e \cos (c+d x))^{-m}}{d e^4 (m-3) (m-1) (m+1) (m+3)}","-\frac{6 (a \sin (c+d x)+a)^{m+3} (e \cos (c+d x))^{-m-3}}{a^3 d e \left(m^4-10 m^2+9\right)}+\frac{6 (a \sin (c+d x)+a)^{m+2} (e \cos (c+d x))^{-m-3}}{a^2 d e (3-m) \left(1-m^2\right)}-\frac{(a \sin (c+d x)+a)^m (e \cos (c+d x))^{-m-3}}{d e (3-m)}-\frac{3 (a \sin (c+d x)+a)^{m+1} (e \cos (c+d x))^{-m-3}}{a d e (1-m) (3-m)}",1,"(Sec[c + d*x]^3*(a*(1 + Sin[c + d*x]))^m*(m*(-7 + m^2) - 3*(-3 + m^2)*Sin[c + d*x] + 6*m*Sin[c + d*x]^2 - 6*Sin[c + d*x]^3))/(d*e^4*(-3 + m)*(-1 + m)*(1 + m)*(3 + m)*(e*Cos[c + d*x])^m)","A",1
361,1,76,142,0.1269881,"\int (e \cos (c+d x))^{-3-m} (a+a \sin (c+d x))^m \, dx","Integrate[(e*Cos[c + d*x])^(-3 - m)*(a + a*Sin[c + d*x])^m,x]","\frac{\sec ^2(c+d x) \left(-2 m \sin (c+d x)+2 \sin ^2(c+d x)+m^2-2\right) (a (\sin (c+d x)+1))^m (e \cos (c+d x))^{-m}}{d e^3 (m-2) m (m+2)}","-\frac{2 (a \sin (c+d x)+a)^{m+2} (e \cos (c+d x))^{-m-2}}{a^2 d e m \left(4-m^2\right)}-\frac{(a \sin (c+d x)+a)^m (e \cos (c+d x))^{-m-2}}{d e (2-m)}+\frac{2 (a \sin (c+d x)+a)^{m+1} (e \cos (c+d x))^{-m-2}}{a d e (2-m) m}",1,"(Sec[c + d*x]^2*(a*(1 + Sin[c + d*x]))^m*(-2 + m^2 - 2*m*Sin[c + d*x] + 2*Sin[c + d*x]^2))/(d*e^3*(-2 + m)*m*(2 + m)*(e*Cos[c + d*x])^m)","A",1
362,1,53,89,0.116483,"\int (e \cos (c+d x))^{-2-m} (a+a \sin (c+d x))^m \, dx","Integrate[(e*Cos[c + d*x])^(-2 - m)*(a + a*Sin[c + d*x])^m,x]","\frac{(m-\sin (c+d x)) (a (\sin (c+d x)+1))^m (e \cos (c+d x))^{-m-1}}{d e (m-1) (m+1)}","\frac{(a \sin (c+d x)+a)^{m+1} (e \cos (c+d x))^{-m-1}}{a d e \left(1-m^2\right)}-\frac{(a \sin (c+d x)+a)^m (e \cos (c+d x))^{-m-1}}{d e (1-m)}",1,"((e*Cos[c + d*x])^(-1 - m)*(m - Sin[c + d*x])*(a*(1 + Sin[c + d*x]))^m)/(d*e*(-1 + m)*(1 + m))","A",1
363,1,34,34,0.0507501,"\int (e \cos (c+d x))^{-1-m} (a+a \sin (c+d x))^m \, dx","Integrate[(e*Cos[c + d*x])^(-1 - m)*(a + a*Sin[c + d*x])^m,x]","\frac{(a (\sin (c+d x)+1))^m (e \cos (c+d x))^{-m}}{d e m}","\frac{(a \sin (c+d x)+a)^m (e \cos (c+d x))^{-m}}{d e m}",1,"(a*(1 + Sin[c + d*x]))^m/(d*e*m*(e*Cos[c + d*x])^m)","A",1
364,1,108,115,0.1201144,"\int (e \cos (c+d x))^{-m} (a+a \sin (c+d x))^m \, dx","Integrate[(a + a*Sin[c + d*x])^m/(e*Cos[c + d*x])^m,x]","\frac{2^{\frac{m+1}{2}} \cos (c+d x) (\sin (c+d x)+1)^{\frac{1}{2} (-m-1)} (a (\sin (c+d x)+1))^m (e \cos (c+d x))^{-m} \, _2F_1\left(\frac{1-m}{2},\frac{1-m}{2};\frac{3-m}{2};\frac{1}{2} (1-\sin (c+d x))\right)}{d (m-1)}","-\frac{a 2^{\frac{m}{2}+\frac{1}{2}} (\sin (c+d x)+1)^{\frac{1-m}{2}} (a \sin (c+d x)+a)^{m-1} (e \cos (c+d x))^{1-m} \, _2F_1\left(\frac{1-m}{2},\frac{1-m}{2};\frac{3-m}{2};\frac{1}{2} (1-\sin (c+d x))\right)}{d e (1-m)}",1,"(2^((1 + m)/2)*Cos[c + d*x]*Hypergeometric2F1[(1 - m)/2, (1 - m)/2, (3 - m)/2, (1 - Sin[c + d*x])/2]*(1 + Sin[c + d*x])^((-1 - m)/2)*(a*(1 + Sin[c + d*x]))^m)/(d*(-1 + m)*(e*Cos[c + d*x])^m)","A",1
365,1,97,97,0.2558872,"\int (e \cos (c+d x))^{1-m} (a+a \sin (c+d x))^m \, dx","Integrate[(e*Cos[c + d*x])^(1 - m)*(a + a*Sin[c + d*x])^m,x]","\frac{2^{\frac{m}{2}+1} (\sin (c+d x)+1)^{-\frac{m}{2}-1} (a (\sin (c+d x)+1))^m (e \cos (c+d x))^{2-m} \, _2F_1\left(1-\frac{m}{2},-\frac{m}{2};2-\frac{m}{2};\frac{1}{2} (1-\sin (c+d x))\right)}{d e (m-2)}","\frac{2^{1-\frac{m}{2}} (1-\sin (c+d x))^{\frac{m}{2}-1} (a \sin (c+d x)+a)^m (e \cos (c+d x))^{2-m} \, _2F_1\left(\frac{m}{2},\frac{m+2}{2};\frac{m+4}{2};\frac{1}{2} (\sin (c+d x)+1)\right)}{d e (m+2)}",1,"(2^(1 + m/2)*(e*Cos[c + d*x])^(2 - m)*Hypergeometric2F1[1 - m/2, -1/2*m, 2 - m/2, (1 - Sin[c + d*x])/2]*(1 + Sin[c + d*x])^(-1 - m/2)*(a*(1 + Sin[c + d*x]))^m)/(d*e*(-2 + m))","A",1
366,1,113,115,0.2388055,"\int (e \cos (c+d x))^{2-m} (a+a \sin (c+d x))^m \, dx","Integrate[(e*Cos[c + d*x])^(2 - m)*(a + a*Sin[c + d*x])^m,x]","\frac{e^2 2^{\frac{m+3}{2}} \cos ^3(c+d x) (\sin (c+d x)+1)^{\frac{1}{2} (-m-3)} (a (\sin (c+d x)+1))^m (e \cos (c+d x))^{-m} \, _2F_1\left(\frac{1}{2} (-m-1),\frac{3-m}{2};\frac{5-m}{2};\frac{1}{2} (1-\sin (c+d x))\right)}{d (m-3)}","-\frac{a 2^{\frac{m}{2}+\frac{3}{2}} (\sin (c+d x)+1)^{\frac{1}{2} (-m-1)} (a \sin (c+d x)+a)^{m-1} (e \cos (c+d x))^{3-m} \, _2F_1\left(\frac{1}{2} (-m-1),\frac{3-m}{2};\frac{5-m}{2};\frac{1}{2} (1-\sin (c+d x))\right)}{d e (3-m)}",1,"(2^((3 + m)/2)*e^2*Cos[c + d*x]^3*Hypergeometric2F1[(-1 - m)/2, (3 - m)/2, (5 - m)/2, (1 - Sin[c + d*x])/2]*(1 + Sin[c + d*x])^((-3 - m)/2)*(a*(1 + Sin[c + d*x]))^m)/(d*(-3 + m)*(e*Cos[c + d*x])^m)","A",1
367,1,105,150,0.3869366,"\int (e \cos (c+d x))^{5-2 m} (a+a \sin (c+d x))^m \, dx","Integrate[(e*Cos[c + d*x])^(5 - 2*m)*(a + a*Sin[c + d*x])^m,x]","\frac{e^5 \cos ^6(c+d x) \left(\left(m^2-7 m+12\right) \sin ^2(c+d x)+2 \left(m^2-9 m+18\right) \sin (c+d x)+m^2-11 m+32\right) (a (\sin (c+d x)+1))^m (e \cos (c+d x))^{-2 m}}{d (m-5) (m-4) (m-3) (\sin (c+d x)+1)^3}","-\frac{8 a^3 (a \sin (c+d x)+a)^{m-3} (e \cos (c+d x))^{6-2 m}}{d e (5-m) \left(m^2-7 m+12\right)}-\frac{4 a^2 (a \sin (c+d x)+a)^{m-2} (e \cos (c+d x))^{6-2 m}}{d e \left(m^2-9 m+20\right)}-\frac{a (a \sin (c+d x)+a)^{m-1} (e \cos (c+d x))^{6-2 m}}{d e (5-m)}",1,"(e^5*Cos[c + d*x]^6*(a*(1 + Sin[c + d*x]))^m*(32 - 11*m + m^2 + 2*(18 - 9*m + m^2)*Sin[c + d*x] + (12 - 7*m + m^2)*Sin[c + d*x]^2))/(d*(-5 + m)*(-4 + m)*(-3 + m)*(e*Cos[c + d*x])^(2*m)*(1 + Sin[c + d*x])^3)","A",1
368,1,72,94,0.2207001,"\int (e \cos (c+d x))^{3-2 m} (a+a \sin (c+d x))^m \, dx","Integrate[(e*Cos[c + d*x])^(3 - 2*m)*(a + a*Sin[c + d*x])^m,x]","\frac{e^3 \cos ^4(c+d x) ((m-2) \sin (c+d x)+m-4) (a (\sin (c+d x)+1))^m (e \cos (c+d x))^{-2 m}}{d (m-3) (m-2) (\sin (c+d x)+1)^2}","-\frac{2 a^2 (a \sin (c+d x)+a)^{m-2} (e \cos (c+d x))^{4-2 m}}{d e \left(m^2-5 m+6\right)}-\frac{a (a \sin (c+d x)+a)^{m-1} (e \cos (c+d x))^{4-2 m}}{d e (3-m)}",1,"(e^3*Cos[c + d*x]^4*(a*(1 + Sin[c + d*x]))^m*(-4 + m + (-2 + m)*Sin[c + d*x]))/(d*(-3 + m)*(-2 + m)*(e*Cos[c + d*x])^(2*m)*(1 + Sin[c + d*x])^2)","A",1
369,1,43,44,0.1497146,"\int (e \cos (c+d x))^{1-2 m} (a+a \sin (c+d x))^m \, dx","Integrate[(e*Cos[c + d*x])^(1 - 2*m)*(a + a*Sin[c + d*x])^m,x]","-\frac{e (\sin (c+d x)-1) (a (\sin (c+d x)+1))^m (e \cos (c+d x))^{-2 m}}{d (m-1)}","-\frac{a (a \sin (c+d x)+a)^{m-1} (e \cos (c+d x))^{2-2 m}}{d e (1-m)}",1,"-((e*(-1 + Sin[c + d*x])*(a*(1 + Sin[c + d*x]))^m)/(d*(-1 + m)*(e*Cos[c + d*x])^(2*m)))","A",1
370,1,61,61,0.0619054,"\int (e \cos (c+d x))^{-1-2 m} (a+a \sin (c+d x))^m \, dx","Integrate[(e*Cos[c + d*x])^(-1 - 2*m)*(a + a*Sin[c + d*x])^m,x]","\frac{(a (\sin (c+d x)+1))^m (e \cos (c+d x))^{-2 m} \, _2F_1\left(1,-m;1-m;\frac{1}{2} (1-\sin (c+d x))\right)}{2 d e m}","\frac{(a \sin (c+d x)+a)^m (e \cos (c+d x))^{-2 m} \, _2F_1\left(1,-m;1-m;\frac{1}{2} (1-\sin (c+d x))\right)}{2 d e m}",1,"(Hypergeometric2F1[1, -m, 1 - m, (1 - Sin[c + d*x])/2]*(a*(1 + Sin[c + d*x]))^m)/(2*d*e*m*(e*Cos[c + d*x])^(2*m))","A",1
371,1,76,70,0.1322443,"\int (e \cos (c+d x))^{-3-2 m} (a+a \sin (c+d x))^m \, dx","Integrate[(e*Cos[c + d*x])^(-3 - 2*m)*(a + a*Sin[c + d*x])^m,x]","\frac{\sec ^2(c+d x) (a (\sin (c+d x)+1))^{m+1} (e \cos (c+d x))^{-2 m} \, _2F_1\left(2,-m-1;-m;\frac{1}{2} (1-\sin (c+d x))\right)}{4 a d e^3 (m+1)}","\frac{(a \sin (c+d x)+a)^{m+1} (e \cos (c+d x))^{-2 (m+1)} \, _2F_1\left(2,-m-1;-m;\frac{1}{2} (1-\sin (c+d x))\right)}{4 a d e (m+1)}",1,"(Hypergeometric2F1[2, -1 - m, -m, (1 - Sin[c + d*x])/2]*Sec[c + d*x]^2*(a*(1 + Sin[c + d*x]))^(1 + m))/(4*a*d*e^3*(1 + m)*(e*Cos[c + d*x])^(2*m))","A",1
372,1,96,89,0.204237,"\int (e \cos (c+d x))^{4-2 m} (a+a \sin (c+d x))^m \, dx","Integrate[(e*Cos[c + d*x])^(4 - 2*m)*(a + a*Sin[c + d*x])^m,x]","\frac{4 \sqrt{2} e^4 \cos ^5(c+d x) (a (\sin (c+d x)+1))^m (e \cos (c+d x))^{-2 m} \, _2F_1\left(-\frac{3}{2},\frac{5}{2}-m;\frac{7}{2}-m;\frac{1}{2} (1-\sin (c+d x))\right)}{d (2 m-5) (\sin (c+d x)+1)^{5/2}}","\frac{2^{\frac{5}{2}-m} (1-\sin (c+d x))^{m-\frac{5}{2}} (a \sin (c+d x)+a)^m (e \cos (c+d x))^{5-2 m} \, _2F_1\left(\frac{5}{2},\frac{1}{2} (2 m-3);\frac{7}{2};\frac{1}{2} (\sin (c+d x)+1)\right)}{5 d e}",1,"(4*Sqrt[2]*e^4*Cos[c + d*x]^5*Hypergeometric2F1[-3/2, 5/2 - m, 7/2 - m, (1 - Sin[c + d*x])/2]*(a*(1 + Sin[c + d*x]))^m)/(d*(-5 + 2*m)*(e*Cos[c + d*x])^(2*m)*(1 + Sin[c + d*x])^(5/2))","A",1
373,1,96,89,0.1945707,"\int (e \cos (c+d x))^{2-2 m} (a+a \sin (c+d x))^m \, dx","Integrate[(e*Cos[c + d*x])^(2 - 2*m)*(a + a*Sin[c + d*x])^m,x]","\frac{2 \sqrt{2} e^2 \cos ^3(c+d x) (a (\sin (c+d x)+1))^m (e \cos (c+d x))^{-2 m} \, _2F_1\left(-\frac{1}{2},\frac{3}{2}-m;\frac{5}{2}-m;\frac{1}{2} (1-\sin (c+d x))\right)}{d (2 m-3) (\sin (c+d x)+1)^{3/2}}","\frac{2^{\frac{3}{2}-m} (1-\sin (c+d x))^{m-\frac{3}{2}} (a \sin (c+d x)+a)^m (e \cos (c+d x))^{3-2 m} \, _2F_1\left(\frac{3}{2},\frac{1}{2} (2 m-1);\frac{5}{2};\frac{1}{2} (\sin (c+d x)+1)\right)}{3 d e}",1,"(2*Sqrt[2]*e^2*Cos[c + d*x]^3*Hypergeometric2F1[-1/2, 3/2 - m, 5/2 - m, (1 - Sin[c + d*x])/2]*(a*(1 + Sin[c + d*x]))^m)/(d*(-3 + 2*m)*(e*Cos[c + d*x])^(2*m)*(1 + Sin[c + d*x])^(3/2))","A",1
374,1,90,86,0.0827343,"\int (e \cos (c+d x))^{-2 m} (a+a \sin (c+d x))^m \, dx","Integrate[(a + a*Sin[c + d*x])^m/(e*Cos[c + d*x])^(2*m),x]","\frac{\sqrt{2} \cos (c+d x) (a (\sin (c+d x)+1))^m (e \cos (c+d x))^{-2 m} \, _2F_1\left(\frac{1}{2},\frac{1}{2}-m;\frac{3}{2}-m;\frac{1}{2} (1-\sin (c+d x))\right)}{d (2 m-1) \sqrt{\sin (c+d x)+1}}","\frac{2^{\frac{1}{2}-m} (1-\sin (c+d x))^{m-\frac{1}{2}} (a \sin (c+d x)+a)^m (e \cos (c+d x))^{1-2 m} \, _2F_1\left(\frac{1}{2},\frac{1}{2} (2 m+1);\frac{3}{2};\frac{1}{2} (\sin (c+d x)+1)\right)}{d e}",1,"(Sqrt[2]*Cos[c + d*x]*Hypergeometric2F1[1/2, 1/2 - m, 3/2 - m, (1 - Sin[c + d*x])/2]*(a*(1 + Sin[c + d*x]))^m)/(d*(-1 + 2*m)*(e*Cos[c + d*x])^(2*m)*Sqrt[1 + Sin[c + d*x]])","A",1
375,1,87,87,0.2180318,"\int (e \cos (c+d x))^{-2-2 m} (a+a \sin (c+d x))^m \, dx","Integrate[(e*Cos[c + d*x])^(-2 - 2*m)*(a + a*Sin[c + d*x])^m,x]","\frac{\sqrt{\sin (c+d x)+1} (a (\sin (c+d x)+1))^m (e \cos (c+d x))^{-2 m-1} \, _2F_1\left(\frac{3}{2},-m-\frac{1}{2};\frac{1}{2}-m;\frac{1}{2} (1-\sin (c+d x))\right)}{\sqrt{2} e (2 d m+d)}","-\frac{2^{-m-\frac{1}{2}} (1-\sin (c+d x))^{m+\frac{1}{2}} (a \sin (c+d x)+a)^m (e \cos (c+d x))^{-2 m-1} \, _2F_1\left(-\frac{1}{2},\frac{1}{2} (2 m+3);\frac{1}{2};\frac{1}{2} (\sin (c+d x)+1)\right)}{d e}",1,"((e*Cos[c + d*x])^(-1 - 2*m)*Hypergeometric2F1[3/2, -1/2 - m, 1/2 - m, (1 - Sin[c + d*x])/2]*Sqrt[1 + Sin[c + d*x]]*(a*(1 + Sin[c + d*x]))^m)/(Sqrt[2]*e*(d + 2*d*m))","A",1
376,1,60,60,0.0209383,"\int \cos ^5(c+d x) (a+b \sin (c+d x)) \, dx","Integrate[Cos[c + d*x]^5*(a + b*Sin[c + d*x]),x]","\frac{a \sin ^5(c+d x)}{5 d}-\frac{2 a \sin ^3(c+d x)}{3 d}+\frac{a \sin (c+d x)}{d}-\frac{b \cos ^6(c+d x)}{6 d}","\frac{a \sin ^5(c+d x)}{5 d}-\frac{2 a \sin ^3(c+d x)}{3 d}+\frac{a \sin (c+d x)}{d}-\frac{b \cos ^6(c+d x)}{6 d}",1,"-1/6*(b*Cos[c + d*x]^6)/d + (a*Sin[c + d*x])/d - (2*a*Sin[c + d*x]^3)/(3*d) + (a*Sin[c + d*x]^5)/(5*d)","A",1
377,1,44,44,0.0130627,"\int \cos ^3(c+d x) (a+b \sin (c+d x)) \, dx","Integrate[Cos[c + d*x]^3*(a + b*Sin[c + d*x]),x]","-\frac{a \sin ^3(c+d x)}{3 d}+\frac{a \sin (c+d x)}{d}-\frac{b \cos ^4(c+d x)}{4 d}","-\frac{a \sin ^3(c+d x)}{3 d}+\frac{a \sin (c+d x)}{d}-\frac{b \cos ^4(c+d x)}{4 d}",1,"-1/4*(b*Cos[c + d*x]^4)/d + (a*Sin[c + d*x])/d - (a*Sin[c + d*x]^3)/(3*d)","A",1
378,1,39,22,0.0111007,"\int \cos (c+d x) (a+b \sin (c+d x)) \, dx","Integrate[Cos[c + d*x]*(a + b*Sin[c + d*x]),x]","\frac{a \sin (c) \cos (d x)}{d}+\frac{a \cos (c) \sin (d x)}{d}-\frac{b \cos ^2(c+d x)}{2 d}","\frac{(a+b \sin (c+d x))^2}{2 b d}",1,"-1/2*(b*Cos[c + d*x]^2)/d + (a*Cos[d*x]*Sin[c])/d + (a*Cos[c]*Sin[d*x])/d","A",1
379,1,26,43,0.0109833,"\int \sec (c+d x) (a+b \sin (c+d x)) \, dx","Integrate[Sec[c + d*x]*(a + b*Sin[c + d*x]),x]","\frac{a \tanh ^{-1}(\sin (c+d x))}{d}-\frac{b \log (\cos (c+d x))}{d}","\frac{(a-b) \log (\sin (c+d x)+1)}{2 d}-\frac{(a+b) \log (1-\sin (c+d x))}{2 d}",1,"(a*ArcTanh[Sin[c + d*x]])/d - (b*Log[Cos[c + d*x]])/d","A",1
380,1,52,41,0.020385,"\int \sec ^3(c+d x) (a+b \sin (c+d x)) \, dx","Integrate[Sec[c + d*x]^3*(a + b*Sin[c + d*x]),x]","\frac{a \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{a \tan (c+d x) \sec (c+d x)}{2 d}+\frac{b \sec ^2(c+d x)}{2 d}","\frac{\sec ^2(c+d x) (a \sin (c+d x)+b)}{2 d}+\frac{a \tanh ^{-1}(\sin (c+d x))}{2 d}",1,"(a*ArcTanh[Sin[c + d*x]])/(2*d) + (b*Sec[c + d*x]^2)/(2*d) + (a*Sec[c + d*x]*Tan[c + d*x])/(2*d)","A",1
381,1,68,61,0.1487311,"\int \sec ^5(c+d x) (a+b \sin (c+d x)) \, dx","Integrate[Sec[c + d*x]^5*(a + b*Sin[c + d*x]),x]","\frac{a \tan (c+d x) \sec ^3(c+d x)}{4 d}+\frac{3 a \left(\tanh ^{-1}(\sin (c+d x))+\tan (c+d x) \sec (c+d x)\right)}{8 d}+\frac{b \sec ^4(c+d x)}{4 d}","\frac{\sec ^4(c+d x) (a \sin (c+d x)+b)}{4 d}+\frac{3 a \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{3 a \tan (c+d x) \sec (c+d x)}{8 d}",1,"(b*Sec[c + d*x]^4)/(4*d) + (a*Sec[c + d*x]^3*Tan[c + d*x])/(4*d) + (3*a*(ArcTanh[Sin[c + d*x]] + Sec[c + d*x]*Tan[c + d*x]))/(8*d)","A",1
382,1,62,65,0.1063886,"\int \cos ^4(c+d x) (a+b \sin (c+d x)) \, dx","Integrate[Cos[c + d*x]^4*(a + b*Sin[c + d*x]),x]","\frac{3 a (c+d x)}{8 d}+\frac{a \sin (2 (c+d x))}{4 d}+\frac{a \sin (4 (c+d x))}{32 d}-\frac{b \cos ^5(c+d x)}{5 d}","\frac{a \sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac{3 a \sin (c+d x) \cos (c+d x)}{8 d}+\frac{3 a x}{8}-\frac{b \cos ^5(c+d x)}{5 d}",1,"(3*a*(c + d*x))/(8*d) - (b*Cos[c + d*x]^5)/(5*d) + (a*Sin[2*(c + d*x)])/(4*d) + (a*Sin[4*(c + d*x)])/(32*d)","A",1
383,1,46,43,0.0610411,"\int \cos ^2(c+d x) (a+b \sin (c+d x)) \, dx","Integrate[Cos[c + d*x]^2*(a + b*Sin[c + d*x]),x]","\frac{a (c+d x)}{2 d}+\frac{a \sin (2 (c+d x))}{4 d}-\frac{b \cos ^3(c+d x)}{3 d}","\frac{a \sin (c+d x) \cos (c+d x)}{2 d}+\frac{a x}{2}-\frac{b \cos ^3(c+d x)}{3 d}",1,"(a*(c + d*x))/(2*d) - (b*Cos[c + d*x]^3)/(3*d) + (a*Sin[2*(c + d*x)])/(4*d)","A",1
384,1,23,23,0.0108356,"\int \sec ^2(c+d x) (a+b \sin (c+d x)) \, dx","Integrate[Sec[c + d*x]^2*(a + b*Sin[c + d*x]),x]","\frac{a \tan (c+d x)}{d}+\frac{b \sec (c+d x)}{d}","\frac{a \tan (c+d x)}{d}+\frac{b \sec (c+d x)}{d}",1,"(b*Sec[c + d*x])/d + (a*Tan[c + d*x])/d","A",1
385,1,41,44,0.0732003,"\int \sec ^4(c+d x) (a+b \sin (c+d x)) \, dx","Integrate[Sec[c + d*x]^4*(a + b*Sin[c + d*x]),x]","\frac{a \left(\frac{1}{3} \tan ^3(c+d x)+\tan (c+d x)\right)}{d}+\frac{b \sec ^3(c+d x)}{3 d}","\frac{a \tan ^3(c+d x)}{3 d}+\frac{a \tan (c+d x)}{d}+\frac{b \sec ^3(c+d x)}{3 d}",1,"(b*Sec[c + d*x]^3)/(3*d) + (a*(Tan[c + d*x] + Tan[c + d*x]^3/3))/d","A",1
386,1,53,60,0.1752025,"\int \sec ^6(c+d x) (a+b \sin (c+d x)) \, dx","Integrate[Sec[c + d*x]^6*(a + b*Sin[c + d*x]),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{b \sec ^5(c+d x)}{5 d}","\frac{a \tan ^5(c+d x)}{5 d}+\frac{2 a \tan ^3(c+d x)}{3 d}+\frac{a \tan (c+d x)}{d}+\frac{b \sec ^5(c+d x)}{5 d}",1,"(b*Sec[c + d*x]^5)/(5*d) + (a*(Tan[c + d*x] + (2*Tan[c + d*x]^3)/3 + Tan[c + d*x]^5/5))/d","A",1
387,1,104,99,0.2049347,"\int \cos ^5(c+d x) (a+b \sin (c+d x))^2 \, dx","Integrate[Cos[c + d*x]^5*(a + b*Sin[c + d*x])^2,x]","\frac{\sin (c+d x) \left(21 \left(a^2-2 b^2\right) \sin ^4(c+d x)+35 \left(b^2-2 a^2\right) \sin ^2(c+d x)+105 a^2+35 a b \sin ^5(c+d x)-105 a b \sin ^3(c+d x)+105 a b \sin (c+d x)+15 b^2 \sin ^6(c+d x)\right)}{105 d}","\frac{\left(a^2-2 b^2\right) \sin ^5(c+d x)}{5 d}-\frac{\left(2 a^2-b^2\right) \sin ^3(c+d x)}{3 d}+\frac{a^2 \sin (c+d x)}{d}-\frac{a b \cos ^6(c+d x)}{3 d}+\frac{b^2 \sin ^7(c+d x)}{7 d}",1,"(Sin[c + d*x]*(105*a^2 + 105*a*b*Sin[c + d*x] + 35*(-2*a^2 + b^2)*Sin[c + d*x]^2 - 105*a*b*Sin[c + d*x]^3 + 21*(a^2 - 2*b^2)*Sin[c + d*x]^4 + 35*a*b*Sin[c + d*x]^5 + 15*b^2*Sin[c + d*x]^6))/(105*d)","A",1
388,1,56,77,0.1191457,"\int \cos ^3(c+d x) (a+b \sin (c+d x))^2 \, dx","Integrate[Cos[c + d*x]^3*(a + b*Sin[c + d*x])^2,x]","\frac{(a+b \sin (c+d x))^3 \left(-a^2+3 a b \sin (c+d x)+3 b^2 \cos (2 (c+d x))+7 b^2\right)}{30 b^3 d}","-\frac{\left(a^2-b^2\right) (a+b \sin (c+d x))^3}{3 b^3 d}-\frac{(a+b \sin (c+d x))^5}{5 b^3 d}+\frac{a (a+b \sin (c+d x))^4}{2 b^3 d}",1,"((a + b*Sin[c + d*x])^3*(-a^2 + 7*b^2 + 3*b^2*Cos[2*(c + d*x)] + 3*a*b*Sin[c + d*x]))/(30*b^3*d)","A",1
389,1,46,22,0.0126756,"\int \cos (c+d x) (a+b \sin (c+d x))^2 \, dx","Integrate[Cos[c + d*x]*(a + b*Sin[c + d*x])^2,x]","\frac{a^2 \sin (c+d x)}{d}+\frac{a b \sin ^2(c+d x)}{d}+\frac{b^2 \sin ^3(c+d x)}{3 d}","\frac{(a+b \sin (c+d x))^3}{3 b d}",1,"(a^2*Sin[c + d*x])/d + (a*b*Sin[c + d*x]^2)/d + (b^2*Sin[c + d*x]^3)/(3*d)","B",1
390,1,54,61,0.0632909,"\int \sec (c+d x) (a+b \sin (c+d x))^2 \, dx","Integrate[Sec[c + d*x]*(a + b*Sin[c + d*x])^2,x]","\frac{(a-b)^2 \log (\sin (c+d x)+1)-(a+b)^2 \log (1-\sin (c+d x))-2 b^2 \sin (c+d x)}{2 d}","\frac{(a-b)^2 \log (\sin (c+d x)+1)}{2 d}-\frac{(a+b)^2 \log (1-\sin (c+d x))}{2 d}-\frac{b^2 \sin (c+d x)}{d}",1,"(-((a + b)^2*Log[1 - Sin[c + d*x]]) + (a - b)^2*Log[1 + Sin[c + d*x]] - 2*b^2*Sin[c + d*x])/(2*d)","A",1
391,1,113,59,0.9228568,"\int \sec ^3(c+d x) (a+b \sin (c+d x))^2 \, dx","Integrate[Sec[c + d*x]^3*(a + b*Sin[c + d*x])^2,x]","\frac{-2 \left(a^4-b^4\right) \tan (c+d x) \sec (c+d x)+\left(4 a b^3-6 a^3 b\right) \tan ^2(c+d x)+2 a^3 b \sec ^2(c+d x)+\left(a^2-b^2\right)^2 (\log (1-\sin (c+d x))-\log (\sin (c+d x)+1))}{4 d \left(b^2-a^2\right)}","\frac{\left(a^2-b^2\right) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{\sec ^2(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))}{2 d}",1,"((a^2 - b^2)^2*(Log[1 - Sin[c + d*x]] - Log[1 + Sin[c + d*x]]) + 2*a^3*b*Sec[c + d*x]^2 - 2*(a^4 - b^4)*Sec[c + d*x]*Tan[c + d*x] + (-6*a^3*b + 4*a*b^3)*Tan[c + d*x]^2)/(4*(-a^2 + b^2)*d)","A",1
392,1,166,99,0.7367575,"\int \sec ^5(c+d x) (a+b \sin (c+d x))^2 \, dx","Integrate[Sec[c + d*x]^5*(a + b*Sin[c + d*x])^2,x]","\frac{4 \left(b^2-a^2\right) \sec ^4(c+d x) (b-a \sin (c+d x)) (a+b \sin (c+d x))^3+\left(b^2-3 a^2\right) \left(-2 \left(a^4-b^4\right) \tan (c+d x) \sec (c+d x)+\left(4 a b^3-6 a^3 b\right) \tan ^2(c+d x)+2 a^3 b \sec ^2(c+d x)+\left(a^2-b^2\right)^2 (\log (1-\sin (c+d x))-\log (\sin (c+d x)+1))\right)}{16 d \left(a^2-b^2\right)^2}","\frac{\left(3 a^2-b^2\right) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{\sec ^2(c+d x) \left(\left(3 a^2-b^2\right) \sin (c+d x)+2 a b\right)}{8 d}+\frac{\sec ^4(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))}{4 d}",1,"(4*(-a^2 + b^2)*Sec[c + d*x]^4*(b - a*Sin[c + d*x])*(a + b*Sin[c + d*x])^3 + (-3*a^2 + b^2)*((a^2 - b^2)^2*(Log[1 - Sin[c + d*x]] - Log[1 + Sin[c + d*x]]) + 2*a^3*b*Sec[c + d*x]^2 - 2*(a^4 - b^4)*Sec[c + d*x]*Tan[c + d*x] + (-6*a^3*b + 4*a*b^3)*Tan[c + d*x]^2))/(16*(a^2 - b^2)^2*d)","A",1
393,1,141,146,0.3453178,"\int \cos ^6(c+d x) (a+b \sin (c+d x))^2 \, dx","Integrate[Cos[c + d*x]^6*(a + b*Sin[c + d*x])^2,x]","\frac{840 \left(8 a^2+b^2\right) (c+d x)+336 \left(15 a^2+b^2\right) \sin (2 (c+d x))+168 \left(6 a^2-b^2\right) \sin (4 (c+d x))+112 (a-b) (a+b) \sin (6 (c+d x))-3360 a b \cos (c+d x)-2016 a b \cos (3 (c+d x))-672 a b \cos (5 (c+d x))-96 a b \cos (7 (c+d x))-21 b^2 \sin (8 (c+d x))}{21504 d}","\frac{\left(8 a^2+b^2\right) \sin (c+d x) \cos ^5(c+d x)}{48 d}+\frac{5 \left(8 a^2+b^2\right) \sin (c+d x) \cos ^3(c+d x)}{192 d}+\frac{5 \left(8 a^2+b^2\right) \sin (c+d x) \cos (c+d x)}{128 d}+\frac{5}{128} x \left(8 a^2+b^2\right)-\frac{9 a b \cos ^7(c+d x)}{56 d}-\frac{b \cos ^7(c+d x) (a+b \sin (c+d x))}{8 d}",1,"(840*(8*a^2 + b^2)*(c + d*x) - 3360*a*b*Cos[c + d*x] - 2016*a*b*Cos[3*(c + d*x)] - 672*a*b*Cos[5*(c + d*x)] - 96*a*b*Cos[7*(c + d*x)] + 336*(15*a^2 + b^2)*Sin[2*(c + d*x)] + 168*(6*a^2 - b^2)*Sin[4*(c + d*x)] + 112*(a - b)*(a + b)*Sin[6*(c + d*x)] - 21*b^2*Sin[8*(c + d*x)])/(21504*d)","A",1
394,1,133,116,0.1945412,"\int \cos ^4(c+d x) (a+b \sin (c+d x))^2 \, dx","Integrate[Cos[c + d*x]^4*(a + b*Sin[c + d*x])^2,x]","\frac{240 a^2 \sin (2 (c+d x))+30 a^2 \sin (4 (c+d x))+360 a^2 c+360 a^2 d x-240 a b \cos (c+d x)-120 a b \cos (3 (c+d x))-24 a b \cos (5 (c+d x))+15 b^2 \sin (2 (c+d x))-15 b^2 \sin (4 (c+d x))-5 b^2 \sin (6 (c+d x))+60 b^2 c+60 b^2 d x}{960 d}","\frac{\left(6 a^2+b^2\right) \sin (c+d x) \cos ^3(c+d x)}{24 d}+\frac{\left(6 a^2+b^2\right) \sin (c+d x) \cos (c+d x)}{16 d}+\frac{1}{16} x \left(6 a^2+b^2\right)-\frac{7 a b \cos ^5(c+d x)}{30 d}-\frac{b \cos ^5(c+d x) (a+b \sin (c+d x))}{6 d}",1,"(360*a^2*c + 60*b^2*c + 360*a^2*d*x + 60*b^2*d*x - 240*a*b*Cos[c + d*x] - 120*a*b*Cos[3*(c + d*x)] - 24*a*b*Cos[5*(c + d*x)] + 240*a^2*Sin[2*(c + d*x)] + 15*b^2*Sin[2*(c + d*x)] + 30*a^2*Sin[4*(c + d*x)] - 15*b^2*Sin[4*(c + d*x)] - 5*b^2*Sin[6*(c + d*x)])/(960*d)","A",1
395,1,85,86,0.2308296,"\int \cos ^2(c+d x) (a+b \sin (c+d x))^2 \, dx","Integrate[Cos[c + d*x]^2*(a + b*Sin[c + d*x])^2,x]","\frac{3 \left(8 a^2 \sin (2 (c+d x))+16 a^2 c+16 a^2 d x-b^2 \sin (4 (c+d x))+4 b^2 c+4 b^2 d x\right)-48 a b \cos (c+d x)-16 a b \cos (3 (c+d x))}{96 d}","\frac{\left(4 a^2+b^2\right) \sin (c+d x) \cos (c+d x)}{8 d}+\frac{1}{8} x \left(4 a^2+b^2\right)-\frac{5 a b \cos ^3(c+d x)}{12 d}-\frac{b \cos ^3(c+d x) (a+b \sin (c+d x))}{4 d}",1,"(-48*a*b*Cos[c + d*x] - 16*a*b*Cos[3*(c + d*x)] + 3*(16*a^2*c + 4*b^2*c + 16*a^2*d*x + 4*b^2*d*x + 8*a^2*Sin[2*(c + d*x)] - b^2*Sin[4*(c + d*x)]))/(96*d)","A",1
396,1,55,49,0.0581656,"\int \sec ^2(c+d x) (a+b \sin (c+d x))^2 \, dx","Integrate[Sec[c + d*x]^2*(a + b*Sin[c + d*x])^2,x]","\frac{a^2 \tan (c+d x)}{d}+\frac{2 a b \sec (c+d x)}{d}-\frac{b^2 \tan ^{-1}(\tan (c+d x))}{d}+\frac{b^2 \tan (c+d x)}{d}","\frac{a b \cos (c+d x)}{d}+\frac{\sec (c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))}{d}+b^2 (-x)",1,"-((b^2*ArcTan[Tan[c + d*x]])/d) + (2*a*b*Sec[c + d*x])/d + (a^2*Tan[c + d*x])/d + (b^2*Tan[c + d*x])/d","A",1
397,1,105,75,0.3274174,"\int \sec ^4(c+d x) (a+b \sin (c+d x))^2 \, dx","Integrate[Sec[c + d*x]^4*(a + b*Sin[c + d*x])^2,x]","\frac{\left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right) \left(3 \left(2 a^2+b^2\right) \sin (c+d x)+\left(2 a^2-b^2\right) \sin (3 (c+d x))+8 a b\right)}{12 d (\sin (c+d x)-1)^2 \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^3}","\frac{\left(2 a^2-b^2\right) \tan (c+d x)}{3 d}+\frac{a b \sec (c+d x)}{3 d}+\frac{\sec ^3(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))}{3 d}",1,"((Cos[(c + d*x)/2] - Sin[(c + d*x)/2])*(8*a*b + 3*(2*a^2 + b^2)*Sin[c + d*x] + (2*a^2 - b^2)*Sin[3*(c + d*x)]))/(12*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3*(-1 + Sin[c + d*x])^2)","A",1
398,1,84,103,0.434781,"\int \sec ^6(c+d x) (a+b \sin (c+d x))^2 \, dx","Integrate[Sec[c + d*x]^6*(a + b*Sin[c + d*x])^2,x]","\frac{\sec ^5(c+d x) \left(20 \left(2 a^2+b^2\right) \sin (c+d x)+5 \left(4 a^2-b^2\right) \sin (3 (c+d x))+4 a^2 \sin (5 (c+d x))+48 a b-b^2 \sin (5 (c+d x))\right)}{120 d}","\frac{\left(4 a^2-b^2\right) \tan ^3(c+d x)}{15 d}+\frac{\left(4 a^2-b^2\right) \tan (c+d x)}{5 d}+\frac{a b \sec ^3(c+d x)}{5 d}+\frac{\sec ^5(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))}{5 d}",1,"(Sec[c + d*x]^5*(48*a*b + 20*(2*a^2 + b^2)*Sin[c + d*x] + 5*(4*a^2 - b^2)*Sin[3*(c + d*x)] + 4*a^2*Sin[5*(c + d*x)] - b^2*Sin[5*(c + d*x)]))/(120*d)","A",1
399,1,110,129,0.8106047,"\int \sec ^8(c+d x) (a+b \sin (c+d x))^2 \, dx","Integrate[Sec[c + d*x]^8*(a + b*Sin[c + d*x])^2,x]","\frac{\sec ^7(c+d x) \left(105 \left(2 a^2+b^2\right) \sin (c+d x)+21 \left(6 a^2-b^2\right) \sin (3 (c+d x))+42 a^2 \sin (5 (c+d x))+6 a^2 \sin (7 (c+d x))+240 a b-7 b^2 \sin (5 (c+d x))-b^2 \sin (7 (c+d x))\right)}{840 d}","\frac{\left(6 a^2-b^2\right) \tan ^5(c+d x)}{35 d}+\frac{2 \left(6 a^2-b^2\right) \tan ^3(c+d x)}{21 d}+\frac{\left(6 a^2-b^2\right) \tan (c+d x)}{7 d}+\frac{a b \sec ^5(c+d x)}{7 d}+\frac{\sec ^7(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))}{7 d}",1,"(Sec[c + d*x]^7*(240*a*b + 105*(2*a^2 + b^2)*Sin[c + d*x] + 21*(6*a^2 - b^2)*Sin[3*(c + d*x)] + 42*a^2*Sin[5*(c + d*x)] - 7*b^2*Sin[5*(c + d*x)] + 6*a^2*Sin[7*(c + d*x)] - b^2*Sin[7*(c + d*x)]))/(840*d)","A",1
400,1,120,144,0.5163787,"\int \cos ^5(c+d x) (a+b \sin (c+d x))^3 \, dx","Integrate[Cos[c + d*x]^5*(a + b*Sin[c + d*x])^3,x]","\frac{\frac{1}{3} \left(3 a^2-b^2\right) (a+b \sin (c+d x))^6+\frac{1}{4} \left(a^2-b^2\right)^2 (a+b \sin (c+d x))^4+\frac{1}{8} (a+b \sin (c+d x))^8-\frac{4}{7} a (a+b \sin (c+d x))^7-\frac{4}{5} a (a-b) (a+b) (a+b \sin (c+d x))^5}{b^5 d}","\frac{\left(3 a^2-b^2\right) (a+b \sin (c+d x))^6}{3 b^5 d}-\frac{4 a \left(a^2-b^2\right) (a+b \sin (c+d x))^5}{5 b^5 d}+\frac{\left(a^2-b^2\right)^2 (a+b \sin (c+d x))^4}{4 b^5 d}+\frac{(a+b \sin (c+d x))^8}{8 b^5 d}-\frac{4 a (a+b \sin (c+d x))^7}{7 b^5 d}",1,"(((a^2 - b^2)^2*(a + b*Sin[c + d*x])^4)/4 - (4*a*(a - b)*(a + b)*(a + b*Sin[c + d*x])^5)/5 + ((3*a^2 - b^2)*(a + b*Sin[c + d*x])^6)/3 - (4*a*(a + b*Sin[c + d*x])^7)/7 + (a + b*Sin[c + d*x])^8/8)/(b^5*d)","A",1
401,1,56,77,0.1405907,"\int \cos ^3(c+d x) (a+b \sin (c+d x))^3 \, dx","Integrate[Cos[c + d*x]^3*(a + b*Sin[c + d*x])^3,x]","\frac{(a+b \sin (c+d x))^4 \left(-a^2+4 a b \sin (c+d x)+5 b^2 \cos (2 (c+d x))+10 b^2\right)}{60 b^3 d}","-\frac{\left(a^2-b^2\right) (a+b \sin (c+d x))^4}{4 b^3 d}-\frac{(a+b \sin (c+d x))^6}{6 b^3 d}+\frac{2 a (a+b \sin (c+d x))^5}{5 b^3 d}",1,"((a + b*Sin[c + d*x])^4*(-a^2 + 10*b^2 + 5*b^2*Cos[2*(c + d*x)] + 4*a*b*Sin[c + d*x]))/(60*b^3*d)","A",1
402,1,57,22,0.0639886,"\int \cos (c+d x) (a+b \sin (c+d x))^3 \, dx","Integrate[Cos[c + d*x]*(a + b*Sin[c + d*x])^3,x]","\frac{\sin (c+d x) \left(4 a^3+6 a^2 b \sin (c+d x)+4 a b^2 \sin ^2(c+d x)+b^3 \sin ^3(c+d x)\right)}{4 d}","\frac{(a+b \sin (c+d x))^4}{4 b d}",1,"(Sin[c + d*x]*(4*a^3 + 6*a^2*b*Sin[c + d*x] + 4*a*b^2*Sin[c + d*x]^2 + b^3*Sin[c + d*x]^3))/(4*d)","B",1
403,1,67,80,0.1177125,"\int \sec (c+d x) (a+b \sin (c+d x))^3 \, dx","Integrate[Sec[c + d*x]*(a + b*Sin[c + d*x])^3,x]","-\frac{6 a b^2 \sin (c+d x)+(a-b)^3 (-\log (\sin (c+d x)+1))+(a+b)^3 \log (1-\sin (c+d x))+b^3 \sin ^2(c+d x)}{2 d}","-\frac{3 a b^2 \sin (c+d x)}{d}+\frac{(a-b)^3 \log (\sin (c+d x)+1)}{2 d}-\frac{(a+b)^3 \log (1-\sin (c+d x))}{2 d}-\frac{b^3 \sin ^2(c+d x)}{2 d}",1,"-1/2*((a + b)^3*Log[1 - Sin[c + d*x]] - (a - b)^3*Log[1 + Sin[c + d*x]] + 6*a*b^2*Sin[c + d*x] + b^3*Sin[c + d*x]^2)/d","A",1
404,1,176,111,1.335554,"\int \sec ^3(c+d x) (a+b \sin (c+d x))^3 \, dx","Integrate[Sec[c + d*x]^3*(a + b*Sin[c + d*x])^3,x]","\frac{2 a^4 b \sec ^2(c+d x)+\left(a^2-b^2\right) \left((a-2 b) (a+b)^2 \log (1-\sin (c+d x))-(a-b)^2 (a+2 b) \log (\sin (c+d x)+1)\right)-a \tan (c+d x) \sec (c+d x) \left(2 a^4+4 a^2 b^2+b^4 \cos (2 (c+d x))-7 b^4\right)+\tan ^2(c+d x) \left(-8 a^4 b+4 a^2 b^3-2 a b^4 \sin (c+d x)+2 b^5\right)}{4 d \left(b^2-a^2\right)}","\frac{a b^2 \sin (c+d x)}{2 d}+\frac{(a+2 b) (a-b)^2 \log (\sin (c+d x)+1)}{4 d}-\frac{(a-2 b) (a+b)^2 \log (1-\sin (c+d x))}{4 d}+\frac{\sec ^2(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^2}{2 d}",1,"((a^2 - b^2)*((a - 2*b)*(a + b)^2*Log[1 - Sin[c + d*x]] - (a - b)^2*(a + 2*b)*Log[1 + Sin[c + d*x]]) + 2*a^4*b*Sec[c + d*x]^2 - a*(2*a^4 + 4*a^2*b^2 - 7*b^4 + b^4*Cos[2*(c + d*x)])*Sec[c + d*x]*Tan[c + d*x] + (-8*a^4*b + 4*a^2*b^3 + 2*b^5 - 2*a*b^4*Sin[c + d*x])*Tan[c + d*x]^2)/(4*(-a^2 + b^2)*d)","A",1
405,1,318,94,4.0371452,"\int \sec ^5(c+d x) (a+b \sin (c+d x))^3 \, dx","Integrate[Sec[c + d*x]^5*(a + b*Sin[c + d*x])^3,x]","\frac{-6 a \left(a^2-b^2\right)^3 (\log (1-\sin (c+d x))-\log (\sin (c+d x)+1))+16 a^4 b \left(3 a^2-2 b^2\right) \tan ^2(c+d x)+16 a^2 b \sec ^2(c+d x) \left(\left(2 a^4-5 a^2 b^2+3 b^4\right) \tan ^2(c+d x)-a^4\right)+8 b^3 \left(4 a^4-5 a^2 b^2+b^4\right) \tan ^4(c+d x)+4 a \tan (c+d x) \sec (c+d x) \left(3 \left(a^6-5 a^4 b^2\right)+4 b^2 \left(3 a^4-5 a^2 b^2+2 b^4\right) \tan ^2(c+d x)\right)+a \left(8 a^6-22 a^4 b^2+29 a^2 b^4-3 b^6\right) \tan (c+d x) \sec ^3(c+d x)+a b \sec ^4(c+d x) \left(-8 a^5+8 a^3 b^2+\left(18 a^4 b-11 a^2 b^3+5 b^5\right) \sin (3 (c+d x))\right)}{32 d \left(a^2-b^2\right)^2}","\frac{3 a \left(a^2-b^2\right) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{3 a \sec ^2(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))}{8 d}+\frac{\tan (c+d x) \sec ^3(c+d x) (a+b \sin (c+d x))^3}{4 d}",1,"(-6*a*(a^2 - b^2)^3*(Log[1 - Sin[c + d*x]] - Log[1 + Sin[c + d*x]]) + a*b*Sec[c + d*x]^4*(-8*a^5 + 8*a^3*b^2 + (18*a^4*b - 11*a^2*b^3 + 5*b^5)*Sin[3*(c + d*x)]) + a*(8*a^6 - 22*a^4*b^2 + 29*a^2*b^4 - 3*b^6)*Sec[c + d*x]^3*Tan[c + d*x] + 16*a^4*b*(3*a^2 - 2*b^2)*Tan[c + d*x]^2 + 8*b^3*(4*a^4 - 5*a^2*b^2 + b^4)*Tan[c + d*x]^4 + 4*a*Sec[c + d*x]*Tan[c + d*x]*(3*(a^6 - 5*a^4*b^2) + 4*b^2*(3*a^4 - 5*a^2*b^2 + 2*b^4)*Tan[c + d*x]^2) + 16*a^2*b*Sec[c + d*x]^2*(-a^4 + (2*a^4 - 5*a^2*b^2 + 3*b^4)*Tan[c + d*x]^2))/(32*(a^2 - b^2)^2*d)","B",1
406,1,182,158,0.3787415,"\int \cos ^4(c+d x) (a+b \sin (c+d x))^3 \, dx","Integrate[Cos[c + d*x]^4*(a + b*Sin[c + d*x])^3,x]","\frac{560 a^3 \sin (2 (c+d x))+70 a^3 \sin (4 (c+d x))+840 a^3 c+840 a^3 d x-35 \left(12 a^2 b+b^3\right) \cos (3 (c+d x))-105 b \left(8 a^2+b^2\right) \cos (c+d x)-84 a^2 b \cos (5 (c+d x))+105 a b^2 \sin (2 (c+d x))-105 a b^2 \sin (4 (c+d x))-35 a b^2 \sin (6 (c+d x))+420 a b^2 c+420 a b^2 d x+7 b^3 \cos (5 (c+d x))+5 b^3 \cos (7 (c+d x))}{2240 d}","-\frac{b \left(17 a^2+4 b^2\right) \cos ^5(c+d x)}{70 d}+\frac{a \left(2 a^2+b^2\right) \sin (c+d x) \cos ^3(c+d x)}{8 d}+\frac{3 a \left(2 a^2+b^2\right) \sin (c+d x) \cos (c+d x)}{16 d}+\frac{3}{16} a x \left(2 a^2+b^2\right)-\frac{b \cos ^5(c+d x) (a+b \sin (c+d x))^2}{7 d}-\frac{3 a b \cos ^5(c+d x) (a+b \sin (c+d x))}{14 d}",1,"(840*a^3*c + 420*a*b^2*c + 840*a^3*d*x + 420*a*b^2*d*x - 105*b*(8*a^2 + b^2)*Cos[c + d*x] - 35*(12*a^2*b + b^3)*Cos[3*(c + d*x)] - 84*a^2*b*Cos[5*(c + d*x)] + 7*b^3*Cos[5*(c + d*x)] + 5*b^3*Cos[7*(c + d*x)] + 560*a^3*Sin[2*(c + d*x)] + 105*a*b^2*Sin[2*(c + d*x)] + 70*a^3*Sin[4*(c + d*x)] - 105*a*b^2*Sin[4*(c + d*x)] - 35*a*b^2*Sin[6*(c + d*x)])/(2240*d)","A",1
407,1,107,131,0.5139162,"\int \cos ^2(c+d x) (a+b \sin (c+d x))^3 \, dx","Integrate[Cos[c + d*x]^2*(a + b*Sin[c + d*x])^3,x]","\frac{-10 \left(12 a^2 b+b^3\right) \cos (3 (c+d x))+15 a \left(4 \left(4 a^2+3 b^2\right) (c+d x)+8 a^2 \sin (2 (c+d x))-3 b^2 \sin (4 (c+d x))\right)-60 b \left(6 a^2+b^2\right) \cos (c+d x)+6 b^3 \cos (5 (c+d x))}{480 d}","-\frac{b \left(27 a^2+8 b^2\right) \cos ^3(c+d x)}{60 d}+\frac{a \left(4 a^2+3 b^2\right) \sin (c+d x) \cos (c+d x)}{8 d}+\frac{1}{8} a x \left(4 a^2+3 b^2\right)-\frac{b \cos ^3(c+d x) (a+b \sin (c+d x))^2}{5 d}-\frac{7 a b \cos ^3(c+d x) (a+b \sin (c+d x))}{20 d}",1,"(-60*b*(6*a^2 + b^2)*Cos[c + d*x] - 10*(12*a^2*b + b^3)*Cos[3*(c + d*x)] + 6*b^3*Cos[5*(c + d*x)] + 15*a*(4*(4*a^2 + 3*b^2)*(c + d*x) + 8*a^2*Sin[2*(c + d*x)] - 3*b^2*Sin[4*(c + d*x)]))/(480*d)","A",1
408,1,68,79,0.3138091,"\int \sec ^2(c+d x) (a+b \sin (c+d x))^3 \, dx","Integrate[Sec[c + d*x]^2*(a + b*Sin[c + d*x])^3,x]","\frac{\sec (c+d x) \left(6 a^2 b+b^3 \cos (2 (c+d x))+3 b^3\right)+2 a \left(a^2+3 b^2\right) \tan (c+d x)-6 a b^2 (c+d x)}{2 d}","\frac{2 b \left(a^2+b^2\right) \cos (c+d x)}{d}+\frac{a b^2 \sin (c+d x) \cos (c+d x)}{d}-3 a b^2 x+\frac{\sec (c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^2}{d}",1,"(-6*a*b^2*(c + d*x) + (6*a^2*b + 3*b^3 + b^3*Cos[2*(c + d*x)])*Sec[c + d*x] + 2*a*(a^2 + 3*b^2)*Tan[c + d*x])/(2*d)","A",1
409,1,136,84,0.4222813,"\int \sec ^4(c+d x) (a+b \sin (c+d x))^3 \, dx","Integrate[Sec[c + d*x]^4*(a + b*Sin[c + d*x])^3,x]","\frac{\sec ^3(c+d x) \left(12 a^3 \sin (c+d x)+4 a^3 \sin (3 (c+d x))+\left(15 b^3-9 a^2 b\right) \cos (c+d x)-3 a^2 b \cos (3 (c+d x))+24 a^2 b+18 a b^2 \sin (c+d x)-6 a b^2 \sin (3 (c+d x))-12 b^3 \cos (2 (c+d x))+5 b^3 \cos (3 (c+d x))-4 b^3\right)}{24 d}","\frac{2 a \left(a^2-b^2\right) \tan (c+d x)}{3 d}+\frac{2 b \left(a^2-b^2\right) \sec (c+d x)}{3 d}+\frac{\sec ^3(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^2}{3 d}",1,"(Sec[c + d*x]^3*(24*a^2*b - 4*b^3 + (-9*a^2*b + 15*b^3)*Cos[c + d*x] - 12*b^3*Cos[2*(c + d*x)] - 3*a^2*b*Cos[3*(c + d*x)] + 5*b^3*Cos[3*(c + d*x)] + 12*a^3*Sin[c + d*x] + 18*a*b^2*Sin[c + d*x] + 4*a^3*Sin[3*(c + d*x)] - 6*a*b^2*Sin[3*(c + d*x)]))/(24*d)","A",1
410,1,190,135,0.5395127,"\int \sec ^6(c+d x) (a+b \sin (c+d x))^3 \, dx","Integrate[Sec[c + d*x]^6*(a + b*Sin[c + d*x])^3,x]","\frac{\sec ^5(c+d x) \left(640 a^3 \sin (c+d x)+320 a^3 \sin (3 (c+d x))+64 a^3 \sin (5 (c+d x))+\left(110 b^3-270 a^2 b\right) \cos (c+d x)-135 a^2 b \cos (3 (c+d x))-27 a^2 b \cos (5 (c+d x))+1152 a^2 b+960 a b^2 \sin (c+d x)-240 a b^2 \sin (3 (c+d x))-48 a b^2 \sin (5 (c+d x))-320 b^3 \cos (2 (c+d x))+55 b^3 \cos (3 (c+d x))+11 b^3 \cos (5 (c+d x))+64 b^3\right)}{1920 d}","\frac{2 a \left(4 a^2-3 b^2\right) \tan (c+d x)}{15 d}+\frac{2 b \left(2 a^2-b^2\right) \sec (c+d x)}{15 d}+\frac{2 \sec ^3(c+d x) (a+b \sin (c+d x)) \left(\left(2 a^2-b^2\right) \sin (c+d x)+a b\right)}{15 d}+\frac{\sec ^5(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^2}{5 d}",1,"(Sec[c + d*x]^5*(1152*a^2*b + 64*b^3 + (-270*a^2*b + 110*b^3)*Cos[c + d*x] - 320*b^3*Cos[2*(c + d*x)] - 135*a^2*b*Cos[3*(c + d*x)] + 55*b^3*Cos[3*(c + d*x)] - 27*a^2*b*Cos[5*(c + d*x)] + 11*b^3*Cos[5*(c + d*x)] + 640*a^3*Sin[c + d*x] + 960*a*b^2*Sin[c + d*x] + 320*a^3*Sin[3*(c + d*x)] - 240*a*b^2*Sin[3*(c + d*x)] + 64*a^3*Sin[5*(c + d*x)] - 48*a*b^2*Sin[5*(c + d*x)]))/(1920*d)","A",1
411,1,245,165,0.8967829,"\int \sec ^8(c+d x) (a+b \sin (c+d x))^3 \, dx","Integrate[Sec[c + d*x]^8*(a + b*Sin[c + d*x])^3,x]","\frac{\sec ^7(c+d x) \left(8960 a^3 \sin (c+d x)+5376 a^3 \sin (3 (c+d x))+1792 a^3 \sin (5 (c+d x))+256 a^3 \sin (7 (c+d x))+35 b \left(17 b^2-75 a^2\right) \cos (c+d x)-1575 a^2 b \cos (3 (c+d x))-525 a^2 b \cos (5 (c+d x))-75 a^2 b \cos (7 (c+d x))+15360 a^2 b+13440 a b^2 \sin (c+d x)-2688 a b^2 \sin (3 (c+d x))-896 a b^2 \sin (5 (c+d x))-128 a b^2 \sin (7 (c+d x))-3584 b^3 \cos (2 (c+d x))+357 b^3 \cos (3 (c+d x))+119 b^3 \cos (5 (c+d x))+17 b^3 \cos (7 (c+d x))+1536 b^3\right)}{35840 d}","\frac{4 a \left(2 a^2-b^2\right) \tan ^3(c+d x)}{35 d}+\frac{12 a \left(2 a^2-b^2\right) \tan (c+d x)}{35 d}+\frac{2 b \left(3 a^2-b^2\right) \sec ^3(c+d x)}{35 d}+\frac{2 \sec ^5(c+d x) (a+b \sin (c+d x)) \left(\left(3 a^2-b^2\right) \sin (c+d x)+2 a b\right)}{35 d}+\frac{\sec ^7(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^2}{7 d}",1,"(Sec[c + d*x]^7*(15360*a^2*b + 1536*b^3 + 35*b*(-75*a^2 + 17*b^2)*Cos[c + d*x] - 3584*b^3*Cos[2*(c + d*x)] - 1575*a^2*b*Cos[3*(c + d*x)] + 357*b^3*Cos[3*(c + d*x)] - 525*a^2*b*Cos[5*(c + d*x)] + 119*b^3*Cos[5*(c + d*x)] - 75*a^2*b*Cos[7*(c + d*x)] + 17*b^3*Cos[7*(c + d*x)] + 8960*a^3*Sin[c + d*x] + 13440*a*b^2*Sin[c + d*x] + 5376*a^3*Sin[3*(c + d*x)] - 2688*a*b^2*Sin[3*(c + d*x)] + 1792*a^3*Sin[5*(c + d*x)] - 896*a*b^2*Sin[5*(c + d*x)] + 256*a^3*Sin[7*(c + d*x)] - 128*a*b^2*Sin[7*(c + d*x)]))/(35840*d)","A",1
412,1,299,192,1.5195904,"\int \sec ^{10}(c+d x) (a+b \sin (c+d x))^3 \, dx","Integrate[Sec[c + d*x]^10*(a + b*Sin[c + d*x])^3,x]","\frac{\sec ^9(c+d x) \left(2064384 a^3 \sin (c+d x)+1376256 a^3 \sin (3 (c+d x))+589824 a^3 \sin (5 (c+d x))+147456 a^3 \sin (7 (c+d x))+16384 a^3 \sin (9 (c+d x))+3150 b \left(23 b^2-147 a^2\right) \cos (c+d x)-308700 a^2 b \cos (3 (c+d x))-132300 a^2 b \cos (5 (c+d x))-33075 a^2 b \cos (7 (c+d x))-3675 a^2 b \cos (9 (c+d x))+3440640 a^2 b+3096576 a b^2 \sin (c+d x)-516096 a b^2 \sin (3 (c+d x))-221184 a b^2 \sin (5 (c+d x))-55296 a b^2 \sin (7 (c+d x))-6144 a b^2 \sin (9 (c+d x))-737280 b^3 \cos (2 (c+d x))+48300 b^3 \cos (3 (c+d x))+20700 b^3 \cos (5 (c+d x))+5175 b^3 \cos (7 (c+d x))+575 b^3 \cos (9 (c+d x))+409600 b^3\right)}{10321920 d}","\frac{2 a \left(8 a^2-3 b^2\right) \tan ^5(c+d x)}{105 d}+\frac{4 a \left(8 a^2-3 b^2\right) \tan ^3(c+d x)}{63 d}+\frac{2 a \left(8 a^2-3 b^2\right) \tan (c+d x)}{21 d}+\frac{2 b \left(4 a^2-b^2\right) \sec ^5(c+d x)}{63 d}+\frac{2 \sec ^7(c+d x) (a+b \sin (c+d x)) \left(\left(4 a^2-b^2\right) \sin (c+d x)+3 a b\right)}{63 d}+\frac{\sec ^9(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^2}{9 d}",1,"(Sec[c + d*x]^9*(3440640*a^2*b + 409600*b^3 + 3150*b*(-147*a^2 + 23*b^2)*Cos[c + d*x] - 737280*b^3*Cos[2*(c + d*x)] - 308700*a^2*b*Cos[3*(c + d*x)] + 48300*b^3*Cos[3*(c + d*x)] - 132300*a^2*b*Cos[5*(c + d*x)] + 20700*b^3*Cos[5*(c + d*x)] - 33075*a^2*b*Cos[7*(c + d*x)] + 5175*b^3*Cos[7*(c + d*x)] - 3675*a^2*b*Cos[9*(c + d*x)] + 575*b^3*Cos[9*(c + d*x)] + 2064384*a^3*Sin[c + d*x] + 3096576*a*b^2*Sin[c + d*x] + 1376256*a^3*Sin[3*(c + d*x)] - 516096*a*b^2*Sin[3*(c + d*x)] + 589824*a^3*Sin[5*(c + d*x)] - 221184*a*b^2*Sin[5*(c + d*x)] + 147456*a^3*Sin[7*(c + d*x)] - 55296*a*b^2*Sin[7*(c + d*x)] + 16384*a^3*Sin[9*(c + d*x)] - 6144*a*b^2*Sin[9*(c + d*x)]))/(10321920*d)","A",1
413,1,120,144,1.9905121,"\int \cos ^5(c+d x) (a+b \sin (c+d x))^8 \, dx","Integrate[Cos[c + d*x]^5*(a + b*Sin[c + d*x])^8,x]","\frac{\frac{2}{11} \left(3 a^2-b^2\right) (a+b \sin (c+d x))^{11}+\frac{1}{9} \left(a^2-b^2\right)^2 (a+b \sin (c+d x))^9+\frac{1}{13} (a+b \sin (c+d x))^{13}-\frac{1}{3} a (a+b \sin (c+d x))^{12}-\frac{2}{5} a (a-b) (a+b) (a+b \sin (c+d x))^{10}}{b^5 d}","\frac{2 \left(3 a^2-b^2\right) (a+b \sin (c+d x))^{11}}{11 b^5 d}-\frac{2 a \left(a^2-b^2\right) (a+b \sin (c+d x))^{10}}{5 b^5 d}+\frac{\left(a^2-b^2\right)^2 (a+b \sin (c+d x))^9}{9 b^5 d}+\frac{(a+b \sin (c+d x))^{13}}{13 b^5 d}-\frac{a (a+b \sin (c+d x))^{12}}{3 b^5 d}",1,"(((a^2 - b^2)^2*(a + b*Sin[c + d*x])^9)/9 - (2*a*(a - b)*(a + b)*(a + b*Sin[c + d*x])^10)/5 + (2*(3*a^2 - b^2)*(a + b*Sin[c + d*x])^11)/11 - (a*(a + b*Sin[c + d*x])^12)/3 + (a + b*Sin[c + d*x])^13/13)/(b^5*d)","A",1
414,1,56,77,0.8474215,"\int \cos ^3(c+d x) (a+b \sin (c+d x))^8 \, dx","Integrate[Cos[c + d*x]^3*(a + b*Sin[c + d*x])^8,x]","\frac{(a+b \sin (c+d x))^9 \left(-2 a^2+18 a b \sin (c+d x)+45 b^2 \cos (2 (c+d x))+65 b^2\right)}{990 b^3 d}","-\frac{\left(a^2-b^2\right) (a+b \sin (c+d x))^9}{9 b^3 d}-\frac{(a+b \sin (c+d x))^{11}}{11 b^3 d}+\frac{a (a+b \sin (c+d x))^{10}}{5 b^3 d}",1,"((a + b*Sin[c + d*x])^9*(-2*a^2 + 65*b^2 + 45*b^2*Cos[2*(c + d*x)] + 18*a*b*Sin[c + d*x]))/(990*b^3*d)","A",1
415,1,137,22,0.3606574,"\int \cos (c+d x) (a+b \sin (c+d x))^8 \, dx","Integrate[Cos[c + d*x]*(a + b*Sin[c + d*x])^8,x]","\frac{\sin (c+d x) \left(9 a^8+36 a^7 b \sin (c+d x)+84 a^6 b^2 \sin ^2(c+d x)+126 a^5 b^3 \sin ^3(c+d x)+126 a^4 b^4 \sin ^4(c+d x)+84 a^3 b^5 \sin ^5(c+d x)+36 a^2 b^6 \sin ^6(c+d x)+9 a b^7 \sin ^7(c+d x)+b^8 \sin ^8(c+d x)\right)}{9 d}","\frac{(a+b \sin (c+d x))^9}{9 b d}",1,"(Sin[c + d*x]*(9*a^8 + 36*a^7*b*Sin[c + d*x] + 84*a^6*b^2*Sin[c + d*x]^2 + 126*a^5*b^3*Sin[c + d*x]^3 + 126*a^4*b^4*Sin[c + d*x]^4 + 84*a^3*b^5*Sin[c + d*x]^5 + 36*a^2*b^6*Sin[c + d*x]^6 + 9*a*b^7*Sin[c + d*x]^7 + b^8*Sin[c + d*x]^8))/(9*d)","B",1
416,1,227,245,0.2136217,"\int \sec (c+d x) (a+b \sin (c+d x))^8 \, dx","Integrate[Sec[c + d*x]*(a + b*Sin[c + d*x])^8,x]","\frac{b \left(-\frac{1}{5} b^5 \left(28 a^2+b^2\right) \sin ^5(c+d x)-2 a b^4 \left(7 a^2+b^2\right) \sin ^4(c+d x)-4 a b^2 \left(7 a^4+7 a^2 b^2+b^4\right) \sin ^2(c+d x)-\frac{1}{3} b^3 \left(70 a^4+28 a^2 b^2+b^4\right) \sin ^3(c+d x)-b \left(28 a^6+70 a^4 b^2+28 a^2 b^4+b^6\right) \sin (c+d x)-\frac{4}{3} a b^6 \sin ^6(c+d x)+\frac{(a-b)^8 \log (\sin (c+d x)+1)}{2 b}-\frac{(a+b)^8 \log (1-\sin (c+d x))}{2 b}-\frac{1}{7} b^7 \sin ^7(c+d x)\right)}{d}","-\frac{b^6 \left(28 a^2+b^2\right) \sin ^5(c+d x)}{5 d}-\frac{2 a b^5 \left(7 a^2+b^2\right) \sin ^4(c+d x)}{d}-\frac{b^4 \left(70 a^4+28 a^2 b^2+b^4\right) \sin ^3(c+d x)}{3 d}-\frac{4 a b^3 \left(7 a^4+7 a^2 b^2+b^4\right) \sin ^2(c+d x)}{d}-\frac{b^2 \left(28 a^6+70 a^4 b^2+28 a^2 b^4+b^6\right) \sin (c+d x)}{d}-\frac{4 a b^7 \sin ^6(c+d x)}{3 d}+\frac{(a-b)^8 \log (\sin (c+d x)+1)}{2 d}-\frac{(a+b)^8 \log (1-\sin (c+d x))}{2 d}-\frac{b^8 \sin ^7(c+d x)}{7 d}",1,"(b*(-1/2*((a + b)^8*Log[1 - Sin[c + d*x]])/b + ((a - b)^8*Log[1 + Sin[c + d*x]])/(2*b) - b*(28*a^6 + 70*a^4*b^2 + 28*a^2*b^4 + b^6)*Sin[c + d*x] - 4*a*b^2*(7*a^4 + 7*a^2*b^2 + b^4)*Sin[c + d*x]^2 - (b^3*(70*a^4 + 28*a^2*b^2 + b^4)*Sin[c + d*x]^3)/3 - 2*a*b^4*(7*a^2 + b^2)*Sin[c + d*x]^4 - (b^5*(28*a^2 + b^2)*Sin[c + d*x]^5)/5 - (4*a*b^6*Sin[c + d*x]^6)/3 - (b^7*Sin[c + d*x]^7)/7))/d","A",1
417,1,366,284,2.3468084,"\int \sec ^3(c+d x) (a+b \sin (c+d x))^8 \, dx","Integrate[Sec[c + d*x]^3*(a + b*Sin[c + d*x])^8,x]","\frac{\frac{1}{2} b \left(a^2-b^2\right) \left((a-7 b) (a+b)^7 \log (1-\sin (c+d x))-(a-b)^7 (a+7 b) \log (\sin (c+d x)+1)\right)+b^9 \left(b^2-9 a^2\right) \sin ^7(c+d x)-4 a b^8 \left(9 a^2-2 b^2\right) \sin ^6(c+d x)+\frac{7}{5} b^7 \left(-60 a^4+19 a^2 b^2+b^4\right) \sin ^5(c+d x)-2 a b^6 \left(63 a^4-22 a^2 b^2-6 b^4\right) \sin ^4(c+d x)-4 a b^4 \left(21 a^6+14 a^4 b^2-22 a^2 b^4-6 b^6\right) \sin ^2(c+d x)+\frac{7}{3} b^5 \left(-54 a^6+10 a^4 b^2+19 a^2 b^4+b^6\right) \sin ^3(c+d x)+b^3 \left(-36 a^8-182 a^6 b^2+70 a^4 b^4+133 a^2 b^6+7 b^8\right) \sin (c+d x)-a b^{10} \sin ^8(c+d x)+b \sec ^2(c+d x) (b-a \sin (c+d x)) (a+b \sin (c+d x))^9}{2 b d \left(b^2-a^2\right)}","\frac{7 b^6 \left(5 a^2+b^2\right) \sin ^5(c+d x)}{10 d}+\frac{3 a b^5 \left(7 a^2+4 b^2\right) \sin ^4(c+d x)}{2 d}+\frac{7 b^4 \left(15 a^4+20 a^2 b^2+b^4\right) \sin ^3(c+d x)}{6 d}+\frac{a b^3 \left(35 a^4+112 a^2 b^2+24 b^4\right) \sin ^2(c+d x)}{2 d}+\frac{7 b^2 \left(3 a^6+30 a^4 b^2+20 a^2 b^4+b^6\right) \sin (c+d x)}{2 d}+\frac{a b^7 \sin ^6(c+d x)}{2 d}+\frac{(a+7 b) (a-b)^7 \log (\sin (c+d x)+1)}{4 d}-\frac{(a-7 b) (a+b)^7 \log (1-\sin (c+d x))}{4 d}+\frac{\sec ^2(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^7}{2 d}",1,"((b*(a^2 - b^2)*((a - 7*b)*(a + b)^7*Log[1 - Sin[c + d*x]] - (a - b)^7*(a + 7*b)*Log[1 + Sin[c + d*x]]))/2 + b^3*(-36*a^8 - 182*a^6*b^2 + 70*a^4*b^4 + 133*a^2*b^6 + 7*b^8)*Sin[c + d*x] - 4*a*b^4*(21*a^6 + 14*a^4*b^2 - 22*a^2*b^4 - 6*b^6)*Sin[c + d*x]^2 + (7*b^5*(-54*a^6 + 10*a^4*b^2 + 19*a^2*b^4 + b^6)*Sin[c + d*x]^3)/3 - 2*a*b^6*(63*a^4 - 22*a^2*b^2 - 6*b^4)*Sin[c + d*x]^4 + (7*b^7*(-60*a^4 + 19*a^2*b^2 + b^4)*Sin[c + d*x]^5)/5 - 4*a*b^8*(9*a^2 - 2*b^2)*Sin[c + d*x]^6 + b^9*(-9*a^2 + b^2)*Sin[c + d*x]^7 - a*b^10*Sin[c + d*x]^8 + b*Sec[c + d*x]^2*(b - a*Sin[c + d*x])*(a + b*Sin[c + d*x])^9)/(2*b*(-a^2 + b^2)*d)","A",1
418,1,514,320,4.1179728,"\int \sec ^5(c+d x) (a+b \sin (c+d x))^8 \, dx","Integrate[Sec[c + d*x]^5*(a + b*Sin[c + d*x])^8,x]","-\frac{3 \left(a^2-b^2\right)^2 \left((a+b)^6 \left(3 a^2-18 a b+35 b^2\right) \log (1-\sin (c+d x))-(a-b)^6 \left(3 a^2+18 a b+35 b^2\right) \log (\sin (c+d x)+1)\right)+12 \left(a^2-b^2\right) \sec ^4(c+d x) (b-a \sin (c+d x)) (a+b \sin (c+d x))^9-6 a b^9 \left(3 a^2+11 b^2\right) \sin ^8(c+d x)+6 \sec ^2(c+d x) (a+b \sin (c+d x))^9 \left(-a \left(3 a^2+11 b^2\right) \sin (c+d x)+9 a^2 b+5 b^3\right)+6 b^8 \left(-27 a^4-90 a^2 b^2+5 b^4\right) \sin ^7(c+d x)-24 a b^7 \left(27 a^4+79 a^2 b^2-8 b^4\right) \sin ^6(c+d x)+42 b^6 \left(-36 a^6-87 a^4 b^2+10 a^2 b^4+b^6\right) \sin ^5(c+d x)-12 a b^5 \left(189 a^6+333 a^4 b^2-8 a^2 b^4-24 b^6\right) \sin ^4(c+d x)+14 b^4 \left(-162 a^8-144 a^6 b^2-85 a^4 b^4+50 a^2 b^6+5 b^8\right) \sin ^3(c+d x)-24 a b^3 \left(63 a^8-21 a^6 b^2+88 a^4 b^4-8 a^2 b^6-24 b^8\right) \sin ^2(c+d x)+6 b^2 \left(-108 a^{10}+234 a^8 b^2-28 a^6 b^4-595 a^4 b^6+350 a^2 b^8+35 b^{10}\right) \sin (c+d x)}{48 d \left(a^2-b^2\right)^2}","-\frac{(a+b)^6 \left(3 a^2-18 a b+35 b^2\right) \log (1-\sin (c+d x))}{16 d}+\frac{(a-b)^6 \left(3 a^2+18 a b+35 b^2\right) \log (\sin (c+d x)+1)}{16 d}-\frac{\sec ^2(c+d x) (a+b \sin (c+d x))^5 \left(b \left(a^2+7 b^2\right)-a \left(3 a^2-11 b^2\right) \sin (c+d x)\right)}{8 d}-\frac{a b^7 \left(13-\frac{3 a^2}{b^2}\right) \sin ^4(c+d x)}{8 d}+\frac{5 b^4 \left(9 a^4-42 a^2 b^2-7 b^4\right) \sin ^3(c+d x)}{24 d}+\frac{a b^3 \left(15 a^4-77 a^2 b^2-48 b^4\right) \sin ^2(c+d x)}{4 d}+\frac{5 b^2 \left(6 a^6-35 a^4 b^2-84 a^2 b^4-7 b^6\right) \sin (c+d x)}{8 d}+\frac{\sec ^4(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^7}{4 d}",1,"-1/48*(3*(a^2 - b^2)^2*((a + b)^6*(3*a^2 - 18*a*b + 35*b^2)*Log[1 - Sin[c + d*x]] - (a - b)^6*(3*a^2 + 18*a*b + 35*b^2)*Log[1 + Sin[c + d*x]]) + 6*b^2*(-108*a^10 + 234*a^8*b^2 - 28*a^6*b^4 - 595*a^4*b^6 + 350*a^2*b^8 + 35*b^10)*Sin[c + d*x] - 24*a*b^3*(63*a^8 - 21*a^6*b^2 + 88*a^4*b^4 - 8*a^2*b^6 - 24*b^8)*Sin[c + d*x]^2 + 14*b^4*(-162*a^8 - 144*a^6*b^2 - 85*a^4*b^4 + 50*a^2*b^6 + 5*b^8)*Sin[c + d*x]^3 - 12*a*b^5*(189*a^6 + 333*a^4*b^2 - 8*a^2*b^4 - 24*b^6)*Sin[c + d*x]^4 + 42*b^6*(-36*a^6 - 87*a^4*b^2 + 10*a^2*b^4 + b^6)*Sin[c + d*x]^5 - 24*a*b^7*(27*a^4 + 79*a^2*b^2 - 8*b^4)*Sin[c + d*x]^6 + 6*b^8*(-27*a^4 - 90*a^2*b^2 + 5*b^4)*Sin[c + d*x]^7 - 6*a*b^9*(3*a^2 + 11*b^2)*Sin[c + d*x]^8 + 12*(a^2 - b^2)*Sec[c + d*x]^4*(b - a*Sin[c + d*x])*(a + b*Sin[c + d*x])^9 + 6*Sec[c + d*x]^2*(a + b*Sin[c + d*x])^9*(9*a^2*b + 5*b^3 - a*(3*a^2 + 11*b^2)*Sin[c + d*x]))/((a^2 - b^2)^2*d)","A",1
419,1,457,423,1.0118635,"\int \cos ^2(c+d x) (a+b \sin (c+d x))^8 \, dx","Integrate[Cos[c + d*x]^2*(a + b*Sin[c + d*x])^8,x]","\frac{161280 a^8 \sin (2 (c+d x))+322560 a^8 c+322560 a^8 d x-564480 a^6 b^2 \sin (4 (c+d x))+2257920 a^6 b^2 c+2257920 a^6 b^2 d x+451584 a^5 b^3 \cos (5 (c+d x))-705600 a^4 b^4 \sin (2 (c+d x))-705600 a^4 b^4 \sin (4 (c+d x))+235200 a^4 b^4 \sin (6 (c+d x))+2822400 a^4 b^4 c+2822400 a^4 b^4 d x+338688 a^3 b^5 \cos (5 (c+d x))-80640 a^3 b^5 \cos (7 (c+d x))-282240 a^2 b^6 \sin (2 (c+d x))-141120 a^2 b^6 \sin (4 (c+d x))+94080 a^2 b^6 \sin (6 (c+d x))-17640 a^2 b^6 \sin (8 (c+d x))+705600 a^2 b^6 c+705600 a^2 b^6 d x-26880 \left(16 a^7 b+28 a^5 b^3+7 a^3 b^5\right) \cos (3 (c+d x))-40320 a b \left(32 a^6+112 a^4 b^2+70 a^2 b^4+7 b^6\right) \cos (c+d x)+32256 a b^7 \cos (5 (c+d x))-14400 a b^7 \cos (7 (c+d x))+2240 a b^7 \cos (9 (c+d x))-8820 b^8 \sin (2 (c+d x))-2520 b^8 \sin (4 (c+d x))+2730 b^8 \sin (6 (c+d x))-945 b^8 \sin (8 (c+d x))+126 b^8 \sin (10 (c+d x))+17640 b^8 c+17640 b^8 d x}{645120 d}","-\frac{b \left(64 a^2+21 b^2\right) \cos ^3(c+d x) (a+b \sin (c+d x))^5}{240 d}-\frac{a b \left(112 a^2+109 b^2\right) \cos ^3(c+d x) (a+b \sin (c+d x))^4}{336 d}-\frac{b \left(784 a^4+1500 a^2 b^2+147 b^4\right) \cos ^3(c+d x) (a+b \sin (c+d x))^3}{2016 d}-\frac{13 a b \left(112 a^4+348 a^2 b^2+101 b^4\right) \cos ^3(c+d x) (a+b \sin (c+d x))^2}{3360 d}-\frac{11 a b \left(1792 a^6+10536 a^4 b^2+9588 a^2 b^4+1289 b^6\right) \cos ^3(c+d x)}{40320 d}-\frac{b \left(6272 a^6+28088 a^4 b^2+15956 a^2 b^4+735 b^6\right) \cos ^3(c+d x) (a+b \sin (c+d x))}{13440 d}+\frac{\left(128 a^8+896 a^6 b^2+1120 a^4 b^4+280 a^2 b^6+7 b^8\right) \sin (c+d x) \cos (c+d x)}{256 d}+\frac{1}{256} x \left(128 a^8+896 a^6 b^2+1120 a^4 b^4+280 a^2 b^6+7 b^8\right)-\frac{b \cos ^3(c+d x) (a+b \sin (c+d x))^7}{10 d}-\frac{17 a b \cos ^3(c+d x) (a+b \sin (c+d x))^6}{90 d}",1,"(322560*a^8*c + 2257920*a^6*b^2*c + 2822400*a^4*b^4*c + 705600*a^2*b^6*c + 17640*b^8*c + 322560*a^8*d*x + 2257920*a^6*b^2*d*x + 2822400*a^4*b^4*d*x + 705600*a^2*b^6*d*x + 17640*b^8*d*x - 40320*a*b*(32*a^6 + 112*a^4*b^2 + 70*a^2*b^4 + 7*b^6)*Cos[c + d*x] - 26880*(16*a^7*b + 28*a^5*b^3 + 7*a^3*b^5)*Cos[3*(c + d*x)] + 451584*a^5*b^3*Cos[5*(c + d*x)] + 338688*a^3*b^5*Cos[5*(c + d*x)] + 32256*a*b^7*Cos[5*(c + d*x)] - 80640*a^3*b^5*Cos[7*(c + d*x)] - 14400*a*b^7*Cos[7*(c + d*x)] + 2240*a*b^7*Cos[9*(c + d*x)] + 161280*a^8*Sin[2*(c + d*x)] - 705600*a^4*b^4*Sin[2*(c + d*x)] - 282240*a^2*b^6*Sin[2*(c + d*x)] - 8820*b^8*Sin[2*(c + d*x)] - 564480*a^6*b^2*Sin[4*(c + d*x)] - 705600*a^4*b^4*Sin[4*(c + d*x)] - 141120*a^2*b^6*Sin[4*(c + d*x)] - 2520*b^8*Sin[4*(c + d*x)] + 235200*a^4*b^4*Sin[6*(c + d*x)] + 94080*a^2*b^6*Sin[6*(c + d*x)] + 2730*b^8*Sin[6*(c + d*x)] - 17640*a^2*b^6*Sin[8*(c + d*x)] - 945*b^8*Sin[8*(c + d*x)] + 126*b^8*Sin[10*(c + d*x)])/(645120*d)","A",1
420,1,313,349,1.0890325,"\int \sec ^2(c+d x) (a+b \sin (c+d x))^8 \, dx","Integrate[Sec[c + d*x]^2*(a + b*Sin[c + d*x])^8,x]","\frac{\sec (c+d x) \left(1920 a^8 \sin (c+d x)+15360 a^7 b+53760 a^6 b^2 \sin (c+d x)+161280 a^5 b^3+151200 a^4 b^4 \sin (c+d x)+16800 a^4 b^4 \sin (3 (c+d x))-4480 a^3 b^5 \cos (4 (c+d x))+201600 a^3 b^5+67200 a^2 b^6 \sin (c+d x)+12600 a^2 b^6 \sin (3 (c+d x))-840 a^2 b^6 \sin (5 (c+d x))+1120 \left(48 a^5 b^3+80 a^3 b^5+15 a b^7\right) \cos (2 (c+d x))-840 b^2 \left(64 a^6+240 a^4 b^2+120 a^2 b^4+5 b^6\right) (c+d x) \cos (c+d x)-1344 a b^7 \cos (4 (c+d x))+96 a b^7 \cos (6 (c+d x))+33600 a b^7+2625 b^8 \sin (c+d x)+630 b^8 \sin (3 (c+d x))-70 b^8 \sin (5 (c+d x))+5 b^8 \sin (7 (c+d x))\right)}{1920 d}","\frac{b \left(6 a^2+7 b^2\right) \cos (c+d x) (a+b \sin (c+d x))^5}{6 d}+\frac{a b \left(30 a^2+113 b^2\right) \cos (c+d x) (a+b \sin (c+d x))^4}{30 d}+\frac{b \left(120 a^4+992 a^2 b^2+175 b^4\right) \cos (c+d x) (a+b \sin (c+d x))^3}{120 d}+\frac{a b \left(40 a^4+624 a^2 b^2+337 b^4\right) \cos (c+d x) (a+b \sin (c+d x))^2}{40 d}+\frac{a b \left(40 a^6+1664 a^4 b^2+2789 a^2 b^4+512 b^6\right) \cos (c+d x)}{20 d}+\frac{b^2 \left(80 a^6+2248 a^4 b^2+2502 a^2 b^4+175 b^6\right) \sin (c+d x) \cos (c+d x)}{80 d}-\frac{7}{16} b^2 x \left(64 a^6+240 a^4 b^2+120 a^2 b^4+5 b^6\right)+\frac{a b \cos (c+d x) (a+b \sin (c+d x))^6}{d}+\frac{\sec (c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^7}{d}",1,"(Sec[c + d*x]*(15360*a^7*b + 161280*a^5*b^3 + 201600*a^3*b^5 + 33600*a*b^7 - 840*b^2*(64*a^6 + 240*a^4*b^2 + 120*a^2*b^4 + 5*b^6)*(c + d*x)*Cos[c + d*x] + 1120*(48*a^5*b^3 + 80*a^3*b^5 + 15*a*b^7)*Cos[2*(c + d*x)] - 4480*a^3*b^5*Cos[4*(c + d*x)] - 1344*a*b^7*Cos[4*(c + d*x)] + 96*a*b^7*Cos[6*(c + d*x)] + 1920*a^8*Sin[c + d*x] + 53760*a^6*b^2*Sin[c + d*x] + 151200*a^4*b^4*Sin[c + d*x] + 67200*a^2*b^6*Sin[c + d*x] + 2625*b^8*Sin[c + d*x] + 16800*a^4*b^4*Sin[3*(c + d*x)] + 12600*a^2*b^6*Sin[3*(c + d*x)] + 630*b^8*Sin[3*(c + d*x)] - 840*a^2*b^6*Sin[5*(c + d*x)] - 70*b^8*Sin[5*(c + d*x)] + 5*b^8*Sin[7*(c + d*x)]))/(1920*d)","A",1
421,1,414,369,1.1111261,"\int \sec ^4(c+d x) (a+b \sin (c+d x))^8 \, dx","Integrate[Sec[c + d*x]^4*(a + b*Sin[c + d*x])^8,x]","\frac{\sec ^3(c+d x) \left(384 a^8 \sin (c+d x)+128 a^8 \sin (3 (c+d x))+2048 a^7 b+5376 a^6 b^2 \sin (c+d x)-1792 a^6 b^2 \sin (3 (c+d x))-21504 a^5 b^3 \cos (2 (c+d x))-7168 a^5 b^3-17920 a^4 b^4 \sin (3 (c+d x))+40320 a^4 b^4 (c+d x) \cos (c+d x)+13440 a^4 b^4 (c+d x) \cos (3 (c+d x))-64512 a^3 b^5 \cos (2 (c+d x))-5376 a^3 b^5 \cos (4 (c+d x))-44800 a^3 b^5-6720 a^2 b^6 \sin (c+d x)-14560 a^2 b^6 \sin (3 (c+d x))-672 a^2 b^6 \sin (5 (c+d x))+40320 a^2 b^6 (c+d x) \cos (c+d x)+13440 a^2 b^6 (c+d x) \cos (3 (c+d x))-17472 a b^7 \cos (2 (c+d x))-1920 a b^7 \cos (4 (c+d x))+64 a b^7 \cos (6 (c+d x))-13440 a b^7-525 b^8 \sin (c+d x)-847 b^8 \sin (3 (c+d x))-63 b^8 \sin (5 (c+d x))+3 b^8 \sin (7 (c+d x))+2520 b^8 (c+d x) \cos (c+d x)+840 b^8 (c+d x) \cos (3 (c+d x))\right)}{768 d}","\frac{b \left(2 a^2-7 b^2\right) \cos (c+d x) (a+b \sin (c+d x))^5}{3 d}+\frac{a b \left(2 a^2-13 b^2\right) \cos (c+d x) (a+b \sin (c+d x))^4}{3 d}-\frac{\sec (c+d x) \left(5 a b-\left(2 a^2-7 b^2\right) \sin (c+d x)\right) (a+b \sin (c+d x))^6}{3 d}+\frac{b \left(8 a^4-72 a^2 b^2-35 b^4\right) \cos (c+d x) (a+b \sin (c+d x))^3}{12 d}+\frac{a b \left(8 a^4-88 a^2 b^2-151 b^4\right) \cos (c+d x) (a+b \sin (c+d x))^2}{12 d}+\frac{35}{8} b^4 x \left(16 a^4+16 a^2 b^2+b^4\right)+\frac{a b \left(8 a^6-104 a^4 b^2-803 a^2 b^4-256 b^6\right) \cos (c+d x)}{6 d}+\frac{b^2 \left(16 a^6-200 a^4 b^2-866 a^2 b^4-105 b^6\right) \sin (c+d x) \cos (c+d x)}{24 d}+\frac{\sec ^3(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^7}{3 d}",1,"(Sec[c + d*x]^3*(2048*a^7*b - 7168*a^5*b^3 - 44800*a^3*b^5 - 13440*a*b^7 + 40320*a^4*b^4*(c + d*x)*Cos[c + d*x] + 40320*a^2*b^6*(c + d*x)*Cos[c + d*x] + 2520*b^8*(c + d*x)*Cos[c + d*x] - 21504*a^5*b^3*Cos[2*(c + d*x)] - 64512*a^3*b^5*Cos[2*(c + d*x)] - 17472*a*b^7*Cos[2*(c + d*x)] + 13440*a^4*b^4*(c + d*x)*Cos[3*(c + d*x)] + 13440*a^2*b^6*(c + d*x)*Cos[3*(c + d*x)] + 840*b^8*(c + d*x)*Cos[3*(c + d*x)] - 5376*a^3*b^5*Cos[4*(c + d*x)] - 1920*a*b^7*Cos[4*(c + d*x)] + 64*a*b^7*Cos[6*(c + d*x)] + 384*a^8*Sin[c + d*x] + 5376*a^6*b^2*Sin[c + d*x] - 6720*a^2*b^6*Sin[c + d*x] - 525*b^8*Sin[c + d*x] + 128*a^8*Sin[3*(c + d*x)] - 1792*a^6*b^2*Sin[3*(c + d*x)] - 17920*a^4*b^4*Sin[3*(c + d*x)] - 14560*a^2*b^6*Sin[3*(c + d*x)] - 847*b^8*Sin[3*(c + d*x)] - 672*a^2*b^6*Sin[5*(c + d*x)] - 63*b^8*Sin[5*(c + d*x)] + 3*b^8*Sin[7*(c + d*x)]))/(768*d)","A",1
422,1,472,381,1.333658,"\int \sec ^6(c+d x) (a+b \sin (c+d x))^8 \, dx","Integrate[Sec[c + d*x]^6*(a + b*Sin[c + d*x])^8,x]","\frac{\sec ^5(c+d x) \left(640 a^8 \sin (c+d x)+320 a^8 \sin (3 (c+d x))+64 a^8 \sin (5 (c+d x))+3072 a^7 b+8960 a^6 b^2 \sin (c+d x)-2240 a^6 b^2 \sin (3 (c+d x))-448 a^6 b^2 \sin (5 (c+d x))-17920 a^5 b^3 \cos (2 (c+d x))+3584 a^5 b^3+16800 a^4 b^4 \sin (c+d x)-8400 a^4 b^4 \sin (3 (c+d x))+1680 a^4 b^4 \sin (5 (c+d x))+17920 a^3 b^5 \cos (2 (c+d x))+13440 a^3 b^5 \cos (4 (c+d x))+25984 a^3 b^5+11200 a^2 b^6 \sin (c+d x)+5600 a^2 b^6 \sin (3 (c+d x))+5152 a^2 b^6 \sin (5 (c+d x))-33600 a^2 b^6 (c+d x) \cos (c+d x)-16800 a^2 b^6 (c+d x) \cos (3 (c+d x))-3360 a^2 b^6 (c+d x) \cos (5 (c+d x))+22560 a b^7 \cos (2 (c+d x))+8640 a b^7 \cos (4 (c+d x))+480 a b^7 \cos (6 (c+d x))+17472 a b^7+875 b^8 \sin (c+d x)+1015 b^8 \sin (3 (c+d x))+539 b^8 \sin (5 (c+d x))+15 b^8 \sin (7 (c+d x))-4200 b^8 (c+d x) \cos (c+d x)-2100 b^8 (c+d x) \cos (3 (c+d x))-420 b^8 (c+d x) \cos (5 (c+d x))\right)}{1920 d}","\frac{4 a b \left(2 a^2+b^2\right) \cos (c+d x) (a+b \sin (c+d x))^4}{15 d}-\frac{\sec ^3(c+d x) \left(3 a b-\left(4 a^2-7 b^2\right) \sin (c+d x)\right) (a+b \sin (c+d x))^6}{15 d}-\frac{4 \sec (c+d x) \left(b \left(4 a^2-7 b^2\right)-a \left(2 a^2+b^2\right) \sin (c+d x)\right) (a+b \sin (c+d x))^5}{15 d}-\frac{7}{2} b^6 x \left(8 a^2+b^2\right)+\frac{b \left(8 a^4-16 a^2 b^2+35 b^4\right) \cos (c+d x) (a+b \sin (c+d x))^3}{15 d}+\frac{a b \left(8 a^4-32 a^2 b^2+87 b^4\right) \cos (c+d x) (a+b \sin (c+d x))^2}{15 d}+\frac{2 a b \left(8 a^6-48 a^4 b^2+163 a^2 b^4+192 b^6\right) \cos (c+d x)}{15 d}+\frac{b^2 \left(16 a^6-88 a^4 b^2+282 a^2 b^4+105 b^6\right) \sin (c+d x) \cos (c+d x)}{30 d}+\frac{\sec ^5(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^7}{5 d}",1,"(Sec[c + d*x]^5*(3072*a^7*b + 3584*a^5*b^3 + 25984*a^3*b^5 + 17472*a*b^7 - 33600*a^2*b^6*(c + d*x)*Cos[c + d*x] - 4200*b^8*(c + d*x)*Cos[c + d*x] - 17920*a^5*b^3*Cos[2*(c + d*x)] + 17920*a^3*b^5*Cos[2*(c + d*x)] + 22560*a*b^7*Cos[2*(c + d*x)] - 16800*a^2*b^6*(c + d*x)*Cos[3*(c + d*x)] - 2100*b^8*(c + d*x)*Cos[3*(c + d*x)] + 13440*a^3*b^5*Cos[4*(c + d*x)] + 8640*a*b^7*Cos[4*(c + d*x)] - 3360*a^2*b^6*(c + d*x)*Cos[5*(c + d*x)] - 420*b^8*(c + d*x)*Cos[5*(c + d*x)] + 480*a*b^7*Cos[6*(c + d*x)] + 640*a^8*Sin[c + d*x] + 8960*a^6*b^2*Sin[c + d*x] + 16800*a^4*b^4*Sin[c + d*x] + 11200*a^2*b^6*Sin[c + d*x] + 875*b^8*Sin[c + d*x] + 320*a^8*Sin[3*(c + d*x)] - 2240*a^6*b^2*Sin[3*(c + d*x)] - 8400*a^4*b^4*Sin[3*(c + d*x)] + 5600*a^2*b^6*Sin[3*(c + d*x)] + 1015*b^8*Sin[3*(c + d*x)] + 64*a^8*Sin[5*(c + d*x)] - 448*a^6*b^2*Sin[5*(c + d*x)] + 1680*a^4*b^4*Sin[5*(c + d*x)] + 5152*a^2*b^6*Sin[5*(c + d*x)] + 539*b^8*Sin[5*(c + d*x)] + 15*b^8*Sin[7*(c + d*x)]))/(1920*d)","A",1
423,1,479,404,1.4348744,"\int \sec ^8(c+d x) (a+b \sin (c+d x))^8 \, dx","Integrate[Sec[c + d*x]^8*(a + b*Sin[c + d*x])^8,x]","\frac{\sec ^7(c+d x) \left(1680 a^8 \sin (c+d x)+1008 a^8 \sin (3 (c+d x))+336 a^8 \sin (5 (c+d x))+48 a^8 \sin (7 (c+d x))+7680 a^7 b+23520 a^6 b^2 \sin (c+d x)-4704 a^6 b^2 \sin (3 (c+d x))-1568 a^6 b^2 \sin (5 (c+d x))-224 a^6 b^2 \sin (7 (c+d x))-37632 a^5 b^3 \cos (2 (c+d x))+16128 a^5 b^3+44100 a^4 b^4 \sin (c+d x)-20580 a^4 b^4 \sin (3 (c+d x))+2940 a^4 b^4 \sin (5 (c+d x))+420 a^4 b^4 \sin (7 (c+d x))-12544 a^3 b^5 \cos (2 (c+d x))+15680 a^3 b^5 \cos (4 (c+d x))+25536 a^3 b^5+14700 a^2 b^6 \sin (c+d x)-8820 a^2 b^6 \sin (3 (c+d x))+2940 a^2 b^6 \sin (5 (c+d x))-420 a^2 b^6 \sin (7 (c+d x))-14448 a b^7 \cos (2 (c+d x))-3360 a b^7 \cos (4 (c+d x))-1680 a b^7 \cos (6 (c+d x))-5088 a b^7-1176 b^8 \sin (3 (c+d x))-392 b^8 \sin (5 (c+d x))-176 b^8 \sin (7 (c+d x))+3675 b^8 (c+d x) \cos (c+d x)+2205 b^8 (c+d x) \cos (3 (c+d x))+735 b^8 (c+d x) \cos (5 (c+d x))+105 b^8 (c+d x) \cos (7 (c+d x))\right)}{6720 d}","-\frac{\sec ^5(c+d x) (a+b \sin (c+d x))^6 \left(a b-\left(6 a^2-7 b^2\right) \sin (c+d x)\right)}{35 d}-\frac{2 \sec ^3(c+d x) (a+b \sin (c+d x))^5 \left(b \left(6 a^2-7 b^2\right)-a \left(12 a^2-11 b^2\right) \sin (c+d x)\right)}{105 d}+\frac{2 b \left(24 a^4+8 a^2 b^2-35 b^4\right) \cos (c+d x) (a+b \sin (c+d x))^3}{105 d}+\frac{2 a b \left(24 a^4-40 a^2 b^2+9 b^4\right) \cos (c+d x) (a+b \sin (c+d x))^2}{105 d}-\frac{2 \sec (c+d x) (a+b \sin (c+d x))^4 \left(3 a b \left(12 a^2-11 b^2\right)-\left(24 a^4+8 a^2 b^2-35 b^4\right) \sin (c+d x)\right)}{105 d}+\frac{4 a b \left(24 a^6-88 a^4 b^2+125 a^2 b^4-96 b^6\right) \cos (c+d x)}{105 d}+\frac{b^2 \left(48 a^6-152 a^4 b^2+174 a^2 b^4-105 b^6\right) \sin (c+d x) \cos (c+d x)}{105 d}+\frac{\sec ^7(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^7}{7 d}+b^8 x",1,"(Sec[c + d*x]^7*(7680*a^7*b + 16128*a^5*b^3 + 25536*a^3*b^5 - 5088*a*b^7 + 3675*b^8*(c + d*x)*Cos[c + d*x] - 37632*a^5*b^3*Cos[2*(c + d*x)] - 12544*a^3*b^5*Cos[2*(c + d*x)] - 14448*a*b^7*Cos[2*(c + d*x)] + 2205*b^8*(c + d*x)*Cos[3*(c + d*x)] + 15680*a^3*b^5*Cos[4*(c + d*x)] - 3360*a*b^7*Cos[4*(c + d*x)] + 735*b^8*(c + d*x)*Cos[5*(c + d*x)] - 1680*a*b^7*Cos[6*(c + d*x)] + 105*b^8*(c + d*x)*Cos[7*(c + d*x)] + 1680*a^8*Sin[c + d*x] + 23520*a^6*b^2*Sin[c + d*x] + 44100*a^4*b^4*Sin[c + d*x] + 14700*a^2*b^6*Sin[c + d*x] + 1008*a^8*Sin[3*(c + d*x)] - 4704*a^6*b^2*Sin[3*(c + d*x)] - 20580*a^4*b^4*Sin[3*(c + d*x)] - 8820*a^2*b^6*Sin[3*(c + d*x)] - 1176*b^8*Sin[3*(c + d*x)] + 336*a^8*Sin[5*(c + d*x)] - 1568*a^6*b^2*Sin[5*(c + d*x)] + 2940*a^4*b^4*Sin[5*(c + d*x)] + 2940*a^2*b^6*Sin[5*(c + d*x)] - 392*b^8*Sin[5*(c + d*x)] + 48*a^8*Sin[7*(c + d*x)] - 224*a^6*b^2*Sin[7*(c + d*x)] + 420*a^4*b^4*Sin[7*(c + d*x)] - 420*a^2*b^6*Sin[7*(c + d*x)] - 176*b^8*Sin[7*(c + d*x)]))/(6720*d)","A",1
424,1,313,236,4.4602218,"\int \sec ^{10}(c+d x) (a+b \sin (c+d x))^8 \, dx","Integrate[Sec[c + d*x]^10*(a + b*Sin[c + d*x])^8,x]","\frac{\cos (c+d x) \left(\frac{a \left(8 (a-b) (1-\sin (c+d x)) \left((a-b) (1-\sin (c+d x)) \left(2 (a-b) (1-\sin (c+d x)) \left((a-b) (1-\sin (c+d x)) \left(35 (a+b \sin (c+d x))^4-4 (a-b) (1-\sin (c+d x)) \left((a+b) (\sin (c+d x)+1) \left(\left(2 a^2+6 a b+7 b^2\right) \sin ^2(c+d x)+6 \left(a^2+3 a b+b^2\right) \sin (c+d x)+7 a^2+6 a b+2 b^2\right)+5 (a+b \sin (c+d x))^3\right)\right)+7 (a+b \sin (c+d x))^5\right)+7 (a+b \sin (c+d x))^6\right)+5 (a+b \sin (c+d x))^7\right)+35 (a+b \sin (c+d x))^8\right)}{35 (1-\sin (c+d x))^5 (\sin (c+d x)+1)^4}-\sec ^{10}(c+d x) (a+b \sin (c+d x))^9\right)}{9 d (a-b)}","\frac{128 a^2 \left(a^2-b^2\right)^3 \tan (c+d x)}{315 d}+\frac{128 a b \left(a^2-b^2\right)^3 \sec (c+d x)}{315 d}+\frac{\sec ^7(c+d x) (a+b \sin (c+d x))^6 \left(\left(8 a^2-7 b^2\right) \sin (c+d x)+a b\right)}{63 d}+\frac{16 a \left(a^2-b^2\right) \sec ^5(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^4}{105 d}+\frac{64 a \left(a^2-b^2\right)^2 \sec ^3(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^2}{315 d}+\frac{\sec ^9(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^7}{9 d}",1,"(Cos[c + d*x]*(-(Sec[c + d*x]^10*(a + b*Sin[c + d*x])^9) + (a*(35*(a + b*Sin[c + d*x])^8 + 8*(a - b)*(1 - Sin[c + d*x])*(5*(a + b*Sin[c + d*x])^7 + (a - b)*(1 - Sin[c + d*x])*(7*(a + b*Sin[c + d*x])^6 + 2*(a - b)*(1 - Sin[c + d*x])*(7*(a + b*Sin[c + d*x])^5 + (a - b)*(1 - Sin[c + d*x])*(35*(a + b*Sin[c + d*x])^4 - 4*(a - b)*(1 - Sin[c + d*x])*(5*(a + b*Sin[c + d*x])^3 + (a + b)*(1 + Sin[c + d*x])*(7*a^2 + 6*a*b + 2*b^2 + 6*(a^2 + 3*a*b + b^2)*Sin[c + d*x] + (2*a^2 + 6*a*b + 7*b^2)*Sin[c + d*x]^2))))))))/(35*(1 - Sin[c + d*x])^5*(1 + Sin[c + d*x])^4)))/(9*(a - b)*d)","A",1
425,1,103,118,0.2264422,"\int \frac{\cos ^5(c+d x)}{a+b \sin (c+d x)} \, dx","Integrate[Cos[c + d*x]^5/(a + b*Sin[c + d*x]),x]","\frac{6 b^2 \left(a^2-b^2\right) \sin ^2(c+d x)-12 a b \left(a^2-2 b^2\right) \sin (c+d x)+12 \left(a^2-b^2\right)^2 \log (a+b \sin (c+d x))-4 a b^3 \sin ^3(c+d x)+3 b^4 \cos ^4(c+d x)}{12 b^5 d}","\frac{\left(a^2-b^2\right)^2 \log (a+b \sin (c+d x))}{b^5 d}-\frac{a \left(a^2-2 b^2\right) \sin (c+d x)}{b^4 d}+\frac{\left(a^2-2 b^2\right) \sin ^2(c+d x)}{2 b^3 d}-\frac{a \sin ^3(c+d x)}{3 b^2 d}+\frac{\sin ^4(c+d x)}{4 b d}",1,"(3*b^4*Cos[c + d*x]^4 + 12*(a^2 - b^2)^2*Log[a + b*Sin[c + d*x]] - 12*a*b*(a^2 - 2*b^2)*Sin[c + d*x] + 6*b^2*(a^2 - b^2)*Sin[c + d*x]^2 - 4*a*b^3*Sin[c + d*x]^3)/(12*b^5*d)","A",1
426,1,54,61,0.0647271,"\int \frac{\cos ^3(c+d x)}{a+b \sin (c+d x)} \, dx","Integrate[Cos[c + d*x]^3/(a + b*Sin[c + d*x]),x]","\frac{-\left(a^2-b^2\right) \log (a+b \sin (c+d x))+a b \sin (c+d x)-\frac{1}{2} b^2 \sin ^2(c+d x)}{b^3 d}","-\frac{\left(a^2-b^2\right) \log (a+b \sin (c+d x))}{b^3 d}+\frac{a \sin (c+d x)}{b^2 d}-\frac{\sin ^2(c+d x)}{2 b d}",1,"(-((a^2 - b^2)*Log[a + b*Sin[c + d*x]]) + a*b*Sin[c + d*x] - (b^2*Sin[c + d*x]^2)/2)/(b^3*d)","A",1
427,1,18,18,0.006324,"\int \frac{\cos (c+d x)}{a+b \sin (c+d x)} \, dx","Integrate[Cos[c + d*x]/(a + b*Sin[c + d*x]),x]","\frac{\log (a+b \sin (c+d x))}{b d}","\frac{\log (a+b \sin (c+d x))}{b d}",1,"Log[a + b*Sin[c + d*x]]/(b*d)","A",1
428,1,64,75,0.0563436,"\int \frac{\sec (c+d x)}{a+b \sin (c+d x)} \, dx","Integrate[Sec[c + d*x]/(a + b*Sin[c + d*x]),x]","\frac{(b-a) \log (1-\sin (c+d x))+(a+b) \log (\sin (c+d x)+1)-2 b \log (a+b \sin (c+d x))}{2 d (a-b) (a+b)}","-\frac{b \log (a+b \sin (c+d x))}{d \left(a^2-b^2\right)}-\frac{\log (1-\sin (c+d x))}{2 d (a+b)}+\frac{\log (\sin (c+d x)+1)}{2 d (a-b)}",1,"((-a + b)*Log[1 - Sin[c + d*x]] + (a + b)*Log[1 + Sin[c + d*x]] - 2*b*Log[a + b*Sin[c + d*x]])/(2*(a - b)*(a + b)*d)","A",1
429,1,170,123,0.5825598,"\int \frac{\sec ^3(c+d x)}{a+b \sin (c+d x)} \, dx","Integrate[Sec[c + d*x]^3/(a + b*Sin[c + d*x]),x]","\frac{\frac{4 b^3 \log (a+b \sin (c+d x))}{\left(a^2-b^2\right)^2}+\frac{1}{(a+b) \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^2}-\frac{1}{(a-b) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2}-\frac{2 (a+2 b) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}{(a+b)^2}+\frac{2 (a-2 b) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}{(a-b)^2}}{4 d}","-\frac{\sec ^2(c+d x) (b-a \sin (c+d x))}{2 d \left(a^2-b^2\right)}+\frac{b^3 \log (a+b \sin (c+d x))}{d \left(a^2-b^2\right)^2}-\frac{(a+2 b) \log (1-\sin (c+d x))}{4 d (a+b)^2}+\frac{(a-2 b) \log (\sin (c+d x)+1)}{4 d (a-b)^2}",1,"((-2*(a + 2*b)*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]])/(a + b)^2 + (2*(a - 2*b)*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]])/(a - b)^2 + (4*b^3*Log[a + b*Sin[c + d*x]])/(a^2 - b^2)^2 + 1/((a + b)*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2) - 1/((a - b)*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2))/(4*d)","A",1
430,1,266,195,0.9484145,"\int \frac{\sec ^5(c+d x)}{a+b \sin (c+d x)} \, dx","Integrate[Sec[c + d*x]^5/(a + b*Sin[c + d*x]),x]","\frac{-\frac{2 \left(3 a^2+9 a b+8 b^2\right) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}{(a+b)^3}+\frac{2 \left(3 a^2-9 a b+8 b^2\right) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}{(a-b)^3}+\frac{16 b^5 \log (a+b \sin (c+d x))}{\left(b^2-a^2\right)^3}+\frac{3 a+5 b}{(a+b)^2 \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^2}+\frac{5 b-3 a}{(a-b)^2 \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2}+\frac{1}{(a+b) \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^4}-\frac{1}{(a-b) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^4}}{16 d}","-\frac{\left(3 a^2+9 a b+8 b^2\right) \log (1-\sin (c+d x))}{16 d (a+b)^3}+\frac{\left(3 a^2-9 a b+8 b^2\right) \log (\sin (c+d x)+1)}{16 d (a-b)^3}-\frac{\sec ^4(c+d x) (b-a \sin (c+d x))}{4 d \left(a^2-b^2\right)}-\frac{b^5 \log (a+b \sin (c+d x))}{d \left(a^2-b^2\right)^3}+\frac{\sec ^2(c+d x) \left(a \left(3 a^2-7 b^2\right) \sin (c+d x)+4 b^3\right)}{8 d \left(a^2-b^2\right)^2}",1,"((-2*(3*a^2 + 9*a*b + 8*b^2)*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]])/(a + b)^3 + (2*(3*a^2 - 9*a*b + 8*b^2)*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]])/(a - b)^3 + (16*b^5*Log[a + b*Sin[c + d*x]])/(-a^2 + b^2)^3 + 1/((a + b)*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^4) + (3*a + 5*b)/((a + b)^2*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2) - 1/((a - b)*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^4) + (-3*a + 5*b)/((a - b)^2*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2))/(16*d)","A",1
431,1,2827,188,6.3058359,"\int \frac{\cos ^6(c+d x)}{a+b \sin (c+d x)} \, dx","Integrate[Cos[c + d*x]^6/(a + b*Sin[c + d*x]),x]","\text{Result too large to show}","-\frac{2 \left(a^2-b^2\right)^{5/2} \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{b^6 d}+\frac{\cos (c+d x) \left(8 \left(a^2-b^2\right)^2-a b \left(4 a^2-7 b^2\right) \sin (c+d x)\right)}{8 b^5 d}-\frac{\cos ^3(c+d x) \left(4 \left(a^2-b^2\right)-3 a b \sin (c+d x)\right)}{12 b^3 d}+\frac{a x \left(8 a^4-20 a^2 b^2+15 b^4\right)}{8 b^6}+\frac{\cos ^5(c+d x)}{5 b d}",1,"(Cos[c + d*x]^5*((8*Sqrt[2]*b*(-(b/(a - b)) - (b*Sin[c + d*x])/(a - b))^(5/2)*Sqrt[b/(a + b) - (b*Sin[c + d*x])/(a + b)]*(1 + ((a - b)*(-(b/(a - b)) - (b*Sin[c + d*x])/(a - b)))/(2*b))^(7/2)*((5/(16*(1 + ((a - b)*(-(b/(a - b)) - (b*Sin[c + d*x])/(a - b)))/(2*b))^3) + 5/(8*(1 + ((a - b)*(-(b/(a - b)) - (b*Sin[c + d*x])/(a - b)))/(2*b))^2) + (1 + ((a - b)*(-(b/(a - b)) - (b*Sin[c + d*x])/(a - b)))/(2*b))^(-1))/2 - (15*b^3*(((a - b)*(-(b/(a - b)) - (b*Sin[c + d*x])/(a - b)))/b - ((a - b)^2*(-(b/(a - b)) - (b*Sin[c + d*x])/(a - b))^2)/(3*b^2) - (Sqrt[2]*Sqrt[a - b]*ArcSinh[(Sqrt[a - b]*Sqrt[-(b/(a - b)) - (b*Sin[c + d*x])/(a - b)])/(Sqrt[2]*Sqrt[b])]*Sqrt[-(b/(a - b)) - (b*Sin[c + d*x])/(a - b)])/(Sqrt[b]*Sqrt[1 + ((a - b)*(-(b/(a - b)) - (b*Sin[c + d*x])/(a - b)))/(2*b)])))/(64*(a - b)^3*(-(b/(a - b)) - (b*Sin[c + d*x])/(a - b))^3*(1 + ((a - b)*(-(b/(a - b)) - (b*Sin[c + d*x])/(a - b)))/(2*b))^3)))/(5*(a + b)^2*Sqrt[((a + b)*(b/(a + b) - (b*Sin[c + d*x])/(a + b)))/b]) - ((-((a*b)/(a - b)) + b^2/(a - b))*((8*Sqrt[2]*b*(-(b/(a - b)) - (b*Sin[c + d*x])/(a - b))^(3/2)*Sqrt[b/(a + b) - (b*Sin[c + d*x])/(a + b)]*(1 + ((a - b)*(-(b/(a - b)) - (b*Sin[c + d*x])/(a - b)))/(2*b))^(7/2)*((3*(5/(8*(1 + ((a - b)*(-(b/(a - b)) - (b*Sin[c + d*x])/(a - b)))/(2*b))^3) + 5/(6*(1 + ((a - b)*(-(b/(a - b)) - (b*Sin[c + d*x])/(a - b)))/(2*b))^2) + (1 + ((a - b)*(-(b/(a - b)) - (b*Sin[c + d*x])/(a - b)))/(2*b))^(-1)))/8 + (15*b^2*(((a - b)*(-(b/(a - b)) - (b*Sin[c + d*x])/(a - b)))/b - (Sqrt[2]*Sqrt[a - b]*ArcSinh[(Sqrt[a - b]*Sqrt[-(b/(a - b)) - (b*Sin[c + d*x])/(a - b)])/(Sqrt[2]*Sqrt[b])]*Sqrt[-(b/(a - b)) - (b*Sin[c + d*x])/(a - b)])/(Sqrt[b]*Sqrt[1 + ((a - b)*(-(b/(a - b)) - (b*Sin[c + d*x])/(a - b)))/(2*b)])))/(64*(a - b)^2*(-(b/(a - b)) - (b*Sin[c + d*x])/(a - b))^2*(1 + ((a - b)*(-(b/(a - b)) - (b*Sin[c + d*x])/(a - b)))/(2*b))^3)))/(3*(a + b)^2*Sqrt[((a + b)*(b/(a + b) - (b*Sin[c + d*x])/(a + b)))/b]) - ((-((a*b)/(a - b)) + b^2/(a - b))*((8*Sqrt[2]*b*Sqrt[-(b/(a - b)) - (b*Sin[c + d*x])/(a - b)]*Sqrt[b/(a + b) - (b*Sin[c + d*x])/(a + b)]*(1 + ((a - b)*(-(b/(a - b)) - (b*Sin[c + d*x])/(a - b)))/(2*b))^(7/2)*((5*Sqrt[b]*ArcSinh[(Sqrt[a - b]*Sqrt[-(b/(a - b)) - (b*Sin[c + d*x])/(a - b)])/(Sqrt[2]*Sqrt[b])])/(8*Sqrt[2]*Sqrt[a - b]*Sqrt[-(b/(a - b)) - (b*Sin[c + d*x])/(a - b)]*(1 + ((a - b)*(-(b/(a - b)) - (b*Sin[c + d*x])/(a - b)))/(2*b))^(7/2)) + (15/(8*(1 + ((a - b)*(-(b/(a - b)) - (b*Sin[c + d*x])/(a - b)))/(2*b))^3) + 5/(4*(1 + ((a - b)*(-(b/(a - b)) - (b*Sin[c + d*x])/(a - b)))/(2*b))^2) + (1 + ((a - b)*(-(b/(a - b)) - (b*Sin[c + d*x])/(a - b)))/(2*b))^(-1))/6))/((a + b)^2*Sqrt[((a + b)*(b/(a + b) - (b*Sin[c + d*x])/(a + b)))/b]) - ((-((a*b)/(a - b)) + b^2/(a - b))*(-(((-((a*b)/(a + b)) - b^2/(a + b))*(-(((-((a*b)/(a + b)) - b^2/(a + b))*((2*Sqrt[a - b]*ArcTanh[(Sqrt[a - b]*Sqrt[-(b/(a - b)) - (b*Sin[c + d*x])/(a - b)])/(Sqrt[a + b]*Sqrt[b/(a + b) - (b*Sin[c + d*x])/(a + b)])])/(b*Sqrt[a + b]) - (2*Sqrt[-((a*b)/(a + b)) - b^2/(a + b)]*ArcTanh[(Sqrt[-((a*b)/(a + b)) - b^2/(a + b)]*Sqrt[-(b/(a - b)) - (b*Sin[c + d*x])/(a - b)])/(Sqrt[-((a*b)/(a - b)) + b^2/(a - b)]*Sqrt[b/(a + b) - (b*Sin[c + d*x])/(a + b)])])/(b*Sqrt[-((a*b)/(a - b)) + b^2/(a - b)])))/b) + (2*Sqrt[2]*(a - b)*Sqrt[-(b/(a - b)) - (b*Sin[c + d*x])/(a - b)]*Sqrt[b/(a + b) - (b*Sin[c + d*x])/(a + b)]*(1 + ((a - b)*(-(b/(a - b)) - (b*Sin[c + d*x])/(a - b)))/(2*b))^(3/2)*((Sqrt[b]*ArcSinh[(Sqrt[a - b]*Sqrt[-(b/(a - b)) - (b*Sin[c + d*x])/(a - b)])/(Sqrt[2]*Sqrt[b])])/(Sqrt[2]*Sqrt[a - b]*Sqrt[-(b/(a - b)) - (b*Sin[c + d*x])/(a - b)]*(1 + ((a - b)*(-(b/(a - b)) - (b*Sin[c + d*x])/(a - b)))/(2*b))^(3/2)) + 1/(2*(1 + ((a - b)*(-(b/(a - b)) - (b*Sin[c + d*x])/(a - b)))/(2*b)))))/(b*(a + b)*Sqrt[((a + b)*(b/(a + b) - (b*Sin[c + d*x])/(a + b)))/b])))/b) + (4*Sqrt[2]*(a - b)*Sqrt[-(b/(a - b)) - (b*Sin[c + d*x])/(a - b)]*Sqrt[b/(a + b) - (b*Sin[c + d*x])/(a + b)]*(1 + ((a - b)*(-(b/(a - b)) - (b*Sin[c + d*x])/(a - b)))/(2*b))^(5/2)*((3*Sqrt[b]*ArcSinh[(Sqrt[a - b]*Sqrt[-(b/(a - b)) - (b*Sin[c + d*x])/(a - b)])/(Sqrt[2]*Sqrt[b])])/(4*Sqrt[2]*Sqrt[a - b]*Sqrt[-(b/(a - b)) - (b*Sin[c + d*x])/(a - b)]*(1 + ((a - b)*(-(b/(a - b)) - (b*Sin[c + d*x])/(a - b)))/(2*b))^(5/2)) + (3/(2*(1 + ((a - b)*(-(b/(a - b)) - (b*Sin[c + d*x])/(a - b)))/(2*b))^2) + (1 + ((a - b)*(-(b/(a - b)) - (b*Sin[c + d*x])/(a - b)))/(2*b))^(-1))/4))/((a + b)^2*Sqrt[((a + b)*(b/(a + b) - (b*Sin[c + d*x])/(a + b)))/b])))/b))/b))/b))/(d*(1 - (a + b*Sin[c + d*x])/(a - b))^(5/2)*(1 - (a + b*Sin[c + d*x])/(a + b))^(5/2))","B",1
432,1,428,127,4.5102255,"\int \frac{\cos ^4(c+d x)}{a+b \sin (c+d x)} \, dx","Integrate[Cos[c + d*x]^4/(a + b*Sin[c + d*x]),x]","\frac{\cos ^3(c+d x) \left(\sqrt{a+b} \left(\sqrt{-\frac{b (\sin (c+d x)-1)}{a+b}} \left(\sqrt{a-b} \sqrt{1-\sin (c+d x)} \sqrt{\frac{b (\sin (c+d x)+1)}{b-a}} \left(6 a^2-3 a b \sin (c+d x)+2 b^2 \sin ^2(c+d x)-8 b^2\right)-6 \sqrt{b} \left(-2 a^2+a b+2 b^2\right) \sinh ^{-1}\left(\frac{\sqrt{a-b} \sqrt{-\frac{b (\sin (c+d x)+1)}{a-b}}}{\sqrt{2} \sqrt{b}}\right)\right)-12 \sqrt{a-b} \left(a^2-b^2\right) \sqrt{1-\sin (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{\frac{b (\sin (c+d x)+1)}{b-a}}}{\sqrt{-\frac{b (\sin (c+d x)-1)}{a+b}}}\right)\right)+12 (a+b) (a-b)^2 \sqrt{1-\sin (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{a-b} \sqrt{-\frac{b (\sin (c+d x)+1)}{a-b}}}{\sqrt{a+b} \sqrt{-\frac{b (\sin (c+d x)-1)}{a+b}}}\right)\right)}{6 b^2 d (a-b)^{3/2} \sqrt{a+b} (1-\sin (c+d x))^{3/2} \sqrt{-\frac{b (\sin (c+d x)-1)}{a+b}} \left(-\frac{b (\sin (c+d x)+1)}{a-b}\right)^{3/2}}","\frac{2 \left(a^2-b^2\right)^{3/2} \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{b^4 d}-\frac{a x \left(2 a^2-3 b^2\right)}{2 b^4}-\frac{\cos (c+d x) \left(2 \left(a^2-b^2\right)-a b \sin (c+d x)\right)}{2 b^3 d}+\frac{\cos ^3(c+d x)}{3 b d}",1,"(Cos[c + d*x]^3*(12*(a - b)^2*(a + b)*ArcTanh[(Sqrt[a - b]*Sqrt[-((b*(1 + Sin[c + d*x]))/(a - b))])/(Sqrt[a + b]*Sqrt[-((b*(-1 + Sin[c + d*x]))/(a + b))])]*Sqrt[1 - Sin[c + d*x]] + Sqrt[a + b]*(-12*Sqrt[a - b]*(a^2 - b^2)*ArcTanh[Sqrt[(b*(1 + Sin[c + d*x]))/(-a + b)]/Sqrt[-((b*(-1 + Sin[c + d*x]))/(a + b))]]*Sqrt[1 - Sin[c + d*x]] + Sqrt[-((b*(-1 + Sin[c + d*x]))/(a + b))]*(-6*Sqrt[b]*(-2*a^2 + a*b + 2*b^2)*ArcSinh[(Sqrt[a - b]*Sqrt[-((b*(1 + Sin[c + d*x]))/(a - b))])/(Sqrt[2]*Sqrt[b])] + Sqrt[a - b]*Sqrt[1 - Sin[c + d*x]]*Sqrt[(b*(1 + Sin[c + d*x]))/(-a + b)]*(6*a^2 - 8*b^2 - 3*a*b*Sin[c + d*x] + 2*b^2*Sin[c + d*x]^2)))))/(6*(a - b)^(3/2)*b^2*Sqrt[a + b]*d*(1 - Sin[c + d*x])^(3/2)*Sqrt[-((b*(-1 + Sin[c + d*x]))/(a + b))]*(-((b*(1 + Sin[c + d*x]))/(a - b)))^(3/2))","B",1
433,1,361,70,1.4108752,"\int \frac{\cos ^2(c+d x)}{a+b \sin (c+d x)} \, dx","Integrate[Cos[c + d*x]^2/(a + b*Sin[c + d*x]),x]","\frac{\cos (c+d x) \left(2 (a-b) \sqrt{1-\sin (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{a-b} \sqrt{-\frac{b (\sin (c+d x)+1)}{a-b}}}{\sqrt{a+b} \sqrt{-\frac{b (\sin (c+d x)-1)}{a+b}}}\right)+\sqrt{a+b} \left(\sqrt{-\frac{b (\sin (c+d x)-1)}{a+b}} \left(\sqrt{a-b} \sqrt{1-\sin (c+d x)} \sqrt{\frac{b (\sin (c+d x)+1)}{b-a}}+2 \sqrt{b} \sinh ^{-1}\left(\frac{\sqrt{a-b} \sqrt{-\frac{b (\sin (c+d x)+1)}{a-b}}}{\sqrt{2} \sqrt{b}}\right)\right)-2 \sqrt{a-b} \sqrt{1-\sin (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{\frac{b (\sin (c+d x)+1)}{b-a}}}{\sqrt{-\frac{b (\sin (c+d x)-1)}{a+b}}}\right)\right)\right)}{b d \sqrt{a-b} \sqrt{a+b} \sqrt{1-\sin (c+d x)} \sqrt{-\frac{b (\sin (c+d x)-1)}{a+b}} \sqrt{-\frac{b (\sin (c+d x)+1)}{a-b}}}","-\frac{2 \sqrt{a^2-b^2} \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{b^2 d}+\frac{a x}{b^2}+\frac{\cos (c+d x)}{b d}",1,"(Cos[c + d*x]*(2*(a - b)*ArcTanh[(Sqrt[a - b]*Sqrt[-((b*(1 + Sin[c + d*x]))/(a - b))])/(Sqrt[a + b]*Sqrt[-((b*(-1 + Sin[c + d*x]))/(a + b))])]*Sqrt[1 - Sin[c + d*x]] + Sqrt[a + b]*(-2*Sqrt[a - b]*ArcTanh[Sqrt[(b*(1 + Sin[c + d*x]))/(-a + b)]/Sqrt[-((b*(-1 + Sin[c + d*x]))/(a + b))]]*Sqrt[1 - Sin[c + d*x]] + Sqrt[-((b*(-1 + Sin[c + d*x]))/(a + b))]*(2*Sqrt[b]*ArcSinh[(Sqrt[a - b]*Sqrt[-((b*(1 + Sin[c + d*x]))/(a - b))])/(Sqrt[2]*Sqrt[b])] + Sqrt[a - b]*Sqrt[1 - Sin[c + d*x]]*Sqrt[(b*(1 + Sin[c + d*x]))/(-a + b)]))))/(Sqrt[a - b]*b*Sqrt[a + b]*d*Sqrt[1 - Sin[c + d*x]]*Sqrt[-((b*(-1 + Sin[c + d*x]))/(a + b))]*Sqrt[-((b*(1 + Sin[c + d*x]))/(a - b))])","B",1
434,1,152,84,0.3234805,"\int \frac{\sec ^2(c+d x)}{a+b \sin (c+d x)} \, dx","Integrate[Sec[c + d*x]^2/(a + b*Sin[c + d*x]),x]","\frac{\sqrt{a^2-b^2} (-a \sin (c+d x)+b (-\cos (c+d x))+b)+2 b^2 \cos (c+d x) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{d (b-a) (a+b) \sqrt{a^2-b^2} \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}","-\frac{2 b^2 \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{d \left(a^2-b^2\right)^{3/2}}-\frac{\sec (c+d x) (b-a \sin (c+d x))}{d \left(a^2-b^2\right)}",1,"(2*b^2*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]]*Cos[c + d*x] + Sqrt[a^2 - b^2]*(b - b*Cos[c + d*x] - a*Sin[c + d*x]))/((-a + b)*(a + b)*Sqrt[a^2 - b^2]*d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))","A",1
435,1,202,137,1.3395674,"\int \frac{\sec ^4(c+d x)}{a+b \sin (c+d x)} \, dx","Integrate[Sec[c + d*x]^4/(a + b*Sin[c + d*x]),x]","\frac{\frac{24 b^4 \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{\left(a^2-b^2\right)^{5/2}}+\frac{\sec ^3(c+d x) \left(6 a^3 \sin (c+d x)+2 a^3 \sin (3 (c+d x))+\frac{3}{2} b \left(a^2-7 b^2\right) \cos (c+d x)+\frac{1}{2} a^2 b \cos (3 (c+d x))-4 a^2 b-9 a b^2 \sin (c+d x)-5 a b^2 \sin (3 (c+d x))+6 b^3 \cos (2 (c+d x))-\frac{7}{2} b^3 \cos (3 (c+d x))+10 b^3\right)}{(a-b)^2 (a+b)^2}}{12 d}","-\frac{\sec ^3(c+d x) (b-a \sin (c+d x))}{3 d \left(a^2-b^2\right)}+\frac{2 b^4 \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{d \left(a^2-b^2\right)^{5/2}}+\frac{\sec (c+d x) \left(a \left(2 a^2-5 b^2\right) \sin (c+d x)+3 b^3\right)}{3 d \left(a^2-b^2\right)^2}",1,"((24*b^4*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(5/2) + (Sec[c + d*x]^3*(-4*a^2*b + 10*b^3 + (3*b*(a^2 - 7*b^2)*Cos[c + d*x])/2 + 6*b^3*Cos[2*(c + d*x)] + (a^2*b*Cos[3*(c + d*x)])/2 - (7*b^3*Cos[3*(c + d*x)])/2 + 6*a^3*Sin[c + d*x] - 9*a*b^2*Sin[c + d*x] + 2*a^3*Sin[3*(c + d*x)] - 5*a*b^2*Sin[3*(c + d*x)]))/((a - b)^2*(a + b)^2))/(12*d)","A",1
436,1,370,197,2.526814,"\int \frac{\sec ^6(c+d x)}{a+b \sin (c+d x)} \, dx","Integrate[Sec[c + d*x]^6/(a + b*Sin[c + d*x]),x]","\frac{\sec ^5(c+d x) \left(640 a^5 \sin (c+d x)+320 a^5 \sin (3 (c+d x))+64 a^5 \sin (5 (c+d x))+45 a^4 b \cos (3 (c+d x))+9 a^4 b \cos (5 (c+d x))-384 a^4 b-1600 a^3 b^2 \sin (c+d x)-1040 a^3 b^2 \sin (3 (c+d x))-208 a^3 b^2 \sin (5 (c+d x))-190 a^2 b^3 \cos (3 (c+d x))-38 a^2 b^3 \cos (5 (c+d x))+1088 a^2 b^3+320 b^3 \left(a^2-4 b^2\right) \cos (2 (c+d x))+10 b \left(9 a^4-38 a^2 b^2+149 b^4\right) \cos (c+d x)+1200 a b^4 \sin (c+d x)+1080 a b^4 \sin (3 (c+d x))+264 a b^4 \sin (5 (c+d x))+745 b^5 \cos (3 (c+d x))-240 b^5 \cos (4 (c+d x))+149 b^5 \cos (5 (c+d x))-1424 b^5\right)}{1920 d (a-b)^3 (a+b)^3}-\frac{2 b^6 \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{d \left(a^2-b^2\right)^{7/2}}","-\frac{\sec ^5(c+d x) (b-a \sin (c+d x))}{5 d \left(a^2-b^2\right)}-\frac{2 b^6 \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{d \left(a^2-b^2\right)^{7/2}}+\frac{\sec ^3(c+d x) \left(a \left(4 a^2-9 b^2\right) \sin (c+d x)+5 b^3\right)}{15 d \left(a^2-b^2\right)^2}-\frac{\sec (c+d x) \left(15 b^5-a \left(8 a^4-26 a^2 b^2+33 b^4\right) \sin (c+d x)\right)}{15 d \left(a^2-b^2\right)^3}",1,"(-2*b^6*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/((a^2 - b^2)^(7/2)*d) + (Sec[c + d*x]^5*(-384*a^4*b + 1088*a^2*b^3 - 1424*b^5 + 10*b*(9*a^4 - 38*a^2*b^2 + 149*b^4)*Cos[c + d*x] + 320*b^3*(a^2 - 4*b^2)*Cos[2*(c + d*x)] + 45*a^4*b*Cos[3*(c + d*x)] - 190*a^2*b^3*Cos[3*(c + d*x)] + 745*b^5*Cos[3*(c + d*x)] - 240*b^5*Cos[4*(c + d*x)] + 9*a^4*b*Cos[5*(c + d*x)] - 38*a^2*b^3*Cos[5*(c + d*x)] + 149*b^5*Cos[5*(c + d*x)] + 640*a^5*Sin[c + d*x] - 1600*a^3*b^2*Sin[c + d*x] + 1200*a*b^4*Sin[c + d*x] + 320*a^5*Sin[3*(c + d*x)] - 1040*a^3*b^2*Sin[3*(c + d*x)] + 1080*a*b^4*Sin[3*(c + d*x)] + 64*a^5*Sin[5*(c + d*x)] - 208*a^3*b^2*Sin[5*(c + d*x)] + 264*a*b^4*Sin[5*(c + d*x)]))/(1920*(a - b)^3*(a + b)^3*d)","A",1
437,1,235,184,0.4758258,"\int \frac{\cos ^7(c+d x)}{(a+b \sin (c+d x))^2} \, dx","Integrate[Cos[c + d*x]^7/(a + b*Sin[c + d*x])^2,x]","\frac{-4 a^2 b^4 \sin ^4(c+d x)+4 \left(a^2-b^2\right)^2 \left(15 a^2 \log (a+b \sin (c+d x))+4 a^2-4 b^2\right)+b^4 \cos ^4(c+d x) \left(-a^2+3 a b \sin (c+d x)+4 b^2\right)+2 a b^3 \left(5 a^2-7 b^2\right) \sin ^3(c+d x)-2 b^2 \left(15 a^4-29 a^2 b^2+8 b^4\right) \sin ^2(c+d x)+4 a b \sin (c+d x) \left(-11 a^4+15 \left(a^2-b^2\right)^2 \log (a+b \sin (c+d x))+18 a^2 b^2-4 b^4\right)+2 b^6 \cos ^6(c+d x)}{10 b^7 d (a+b \sin (c+d x))}","\frac{\left(a^2-b^2\right)^3}{b^7 d (a+b \sin (c+d x))}+\frac{6 a \left(a^2-b^2\right)^2 \log (a+b \sin (c+d x))}{b^7 d}+\frac{a \left(2 a^2-3 b^2\right) \sin ^2(c+d x)}{b^5 d}-\frac{\left(a^2-b^2\right) \sin ^3(c+d x)}{b^4 d}-\frac{\left(5 a^4-9 a^2 b^2+3 b^4\right) \sin (c+d x)}{b^6 d}+\frac{a \sin ^4(c+d x)}{2 b^3 d}-\frac{\sin ^5(c+d x)}{5 b^2 d}",1,"(2*b^6*Cos[c + d*x]^6 + 4*(a^2 - b^2)^2*(4*a^2 - 4*b^2 + 15*a^2*Log[a + b*Sin[c + d*x]]) + 4*a*b*(-11*a^4 + 18*a^2*b^2 - 4*b^4 + 15*(a^2 - b^2)^2*Log[a + b*Sin[c + d*x]])*Sin[c + d*x] - 2*b^2*(15*a^4 - 29*a^2*b^2 + 8*b^4)*Sin[c + d*x]^2 + 2*a*b^3*(5*a^2 - 7*b^2)*Sin[c + d*x]^3 - 4*a^2*b^4*Sin[c + d*x]^4 + b^4*Cos[c + d*x]^4*(-a^2 + 4*b^2 + 3*a*b*Sin[c + d*x]))/(10*b^7*d*(a + b*Sin[c + d*x]))","A",1
438,1,127,120,0.6310148,"\int \frac{\cos ^5(c+d x)}{(a+b \sin (c+d x))^2} \, dx","Integrate[Cos[c + d*x]^5/(a + b*Sin[c + d*x])^2,x]","\frac{\left(8 a^2 b-4 b^3\right) \sin (c+d x)+\frac{b^4 \cos ^4(c+d x)-4 \left(a^2-b^2\right) \left(3 a^2 \log (a+b \sin (c+d x))+a^2+3 a b \sin (c+d x) \log (a+b \sin (c+d x))-b^2\right)}{a+b \sin (c+d x)}-2 a b^2 \sin ^2(c+d x)}{3 b^5 d}","-\frac{\left(a^2-b^2\right)^2}{b^5 d (a+b \sin (c+d x))}-\frac{4 a \left(a^2-b^2\right) \log (a+b \sin (c+d x))}{b^5 d}+\frac{\left(3 a^2-2 b^2\right) \sin (c+d x)}{b^4 d}-\frac{a \sin ^2(c+d x)}{b^3 d}+\frac{\sin ^3(c+d x)}{3 b^2 d}",1,"((8*a^2*b - 4*b^3)*Sin[c + d*x] - 2*a*b^2*Sin[c + d*x]^2 + (b^4*Cos[c + d*x]^4 - 4*(a^2 - b^2)*(a^2 - b^2 + 3*a^2*Log[a + b*Sin[c + d*x]] + 3*a*b*Log[a + b*Sin[c + d*x]]*Sin[c + d*x]))/(a + b*Sin[c + d*x]))/(3*b^5*d)","A",1
439,1,52,63,0.0353163,"\int \frac{\cos ^3(c+d x)}{(a+b \sin (c+d x))^2} \, dx","Integrate[Cos[c + d*x]^3/(a + b*Sin[c + d*x])^2,x]","\frac{\frac{(a-b) (a+b)}{a+b \sin (c+d x)}+2 a \log (a+b \sin (c+d x))-b \sin (c+d x)}{b^3 d}","\frac{a^2-b^2}{b^3 d (a+b \sin (c+d x))}+\frac{2 a \log (a+b \sin (c+d x))}{b^3 d}-\frac{\sin (c+d x)}{b^2 d}",1,"(2*a*Log[a + b*Sin[c + d*x]] - b*Sin[c + d*x] + ((a - b)*(a + b))/(a + b*Sin[c + d*x]))/(b^3*d)","A",1
440,1,20,20,0.0242878,"\int \frac{\cos (c+d x)}{(a+b \sin (c+d x))^2} \, dx","Integrate[Cos[c + d*x]/(a + b*Sin[c + d*x])^2,x]","-\frac{1}{b d (a+b \sin (c+d x))}","-\frac{1}{b d (a+b \sin (c+d x))}",1,"-(1/(b*d*(a + b*Sin[c + d*x])))","A",1
441,1,102,104,0.2110961,"\int \frac{\sec (c+d x)}{(a+b \sin (c+d x))^2} \, dx","Integrate[Sec[c + d*x]/(a + b*Sin[c + d*x])^2,x]","\frac{b \left(\frac{1}{\left(a^2-b^2\right) (a+b \sin (c+d x))}-\frac{\log (1-\sin (c+d x))}{2 b (a+b)^2}+\frac{\log (\sin (c+d x)+1)}{2 b (a-b)^2}-\frac{2 a \log (a+b \sin (c+d x))}{(a-b)^2 (a+b)^2}\right)}{d}","\frac{b}{d \left(a^2-b^2\right) (a+b \sin (c+d x))}-\frac{2 a b \log (a+b \sin (c+d x))}{d \left(a^2-b^2\right)^2}-\frac{\log (1-\sin (c+d x))}{2 d (a+b)^2}+\frac{\log (\sin (c+d x)+1)}{2 d (a-b)^2}",1,"(b*(-1/2*Log[1 - Sin[c + d*x]]/(b*(a + b)^2) + Log[1 + Sin[c + d*x]]/(2*(a - b)^2*b) - (2*a*Log[a + b*Sin[c + d*x]])/((a - b)^2*(a + b)^2) + 1/((a^2 - b^2)*(a + b*Sin[c + d*x]))))/d","A",1
442,1,222,177,1.7545192,"\int \frac{\sec ^3(c+d x)}{(a+b \sin (c+d x))^2} \, dx","Integrate[Sec[c + d*x]^3/(a + b*Sin[c + d*x])^2,x]","\frac{-b \left(-a^2-3 b^2\right) \left(\frac{1}{\left(a^2-b^2\right) (a+b \sin (c+d x))}-\frac{\log (1-\sin (c+d x))}{2 b (a+b)^2}+\frac{\log (\sin (c+d x)+1)}{2 b (a-b)^2}-\frac{2 a \log (a+b \sin (c+d x))}{(a-b)^2 (a+b)^2}\right)+\frac{a ((a-b) \log (1-\sin (c+d x))-(a+b) \log (\sin (c+d x)+1)+2 b \log (a+b \sin (c+d x)))}{(a-b) (a+b)}+\frac{\sec ^2(c+d x) (b-a \sin (c+d x))}{a+b \sin (c+d x)}}{2 d \left(b^2-a^2\right)}","-\frac{b \left(a^2+3 b^2\right)}{2 d \left(a^2-b^2\right)^2 (a+b \sin (c+d x))}-\frac{\sec ^2(c+d x) (b-a \sin (c+d x))}{2 d \left(a^2-b^2\right) (a+b \sin (c+d x))}+\frac{4 a b^3 \log (a+b \sin (c+d x))}{d \left(a^2-b^2\right)^3}-\frac{(a+3 b) \log (1-\sin (c+d x))}{4 d (a+b)^3}+\frac{(a-3 b) \log (\sin (c+d x)+1)}{4 d (a-b)^3}",1,"((a*((a - b)*Log[1 - Sin[c + d*x]] - (a + b)*Log[1 + Sin[c + d*x]] + 2*b*Log[a + b*Sin[c + d*x]]))/((a - b)*(a + b)) + (Sec[c + d*x]^2*(b - a*Sin[c + d*x]))/(a + b*Sin[c + d*x]) - b*(-a^2 - 3*b^2)*(-1/2*Log[1 - Sin[c + d*x]]/(b*(a + b)^2) + Log[1 + Sin[c + d*x]]/(2*(a - b)^2*b) - (2*a*Log[a + b*Sin[c + d*x]])/((a - b)^2*(a + b)^2) + 1/((a^2 - b^2)*(a + b*Sin[c + d*x]))))/(2*(-a^2 + b^2)*d)","A",1
443,1,406,269,6.1092395,"\int \frac{\sec ^5(c+d x)}{(a+b \sin (c+d x))^2} \, dx","Integrate[Sec[c + d*x]^5/(a + b*Sin[c + d*x])^2,x]","\frac{b^5 \left(\frac{\sec ^4(c+d x) \left(b^2-a b \sin (c+d x)\right)}{4 b^6 \left(b^2-a^2\right) (a+b \sin (c+d x))}-\frac{\frac{\left(6 a^2 \left(a^2-3 b^2\right)-3 \left(a^4-2 a^2 b^2+5 b^4\right)\right) \left(\frac{1}{\left(a^2-b^2\right) (a+b \sin (c+d x))}-\frac{\log (1-\sin (c+d x))}{2 b (a+b)^2}+\frac{\log (\sin (c+d x)+1)}{2 b (a-b)^2}-\frac{2 a \log (a+b \sin (c+d x))}{(a-b)^2 (a+b)^2}\right)-6 a \left(a^2-3 b^2\right) \left(-\frac{\log (a+b \sin (c+d x))}{a^2-b^2}-\frac{\log (1-\sin (c+d x))}{2 b (a+b)}+\frac{\log (\sin (c+d x)+1)}{2 b (a-b)}\right)}{2 b^2 \left(b^2-a^2\right)}-\frac{\sec ^2(c+d x) \left(-b \left(4 a b^2-a \left(3 a^2-5 b^2\right)\right) \sin (c+d x)+4 a^2 b^2-b^2 \left(3 a^2-5 b^2\right)\right)}{2 b^4 \left(b^2-a^2\right) (a+b \sin (c+d x))}}{4 b^2 \left(b^2-a^2\right)}\right)}{d}","-\frac{3 \left(a^2+4 a b+5 b^2\right) \log (1-\sin (c+d x))}{16 d (a+b)^4}+\frac{3 \left(a^2-4 a b+5 b^2\right) \log (\sin (c+d x)+1)}{16 d (a-b)^4}-\frac{\sec ^4(c+d x) (b-a \sin (c+d x))}{4 d \left(a^2-b^2\right) (a+b \sin (c+d x))}+\frac{\sec ^2(c+d x) \left(3 a \left(a^2-3 b^2\right) \sin (c+d x)+b \left(a^2+5 b^2\right)\right)}{8 d \left(a^2-b^2\right)^2 (a+b \sin (c+d x))}-\frac{6 a b^5 \log (a+b \sin (c+d x))}{d \left(a^2-b^2\right)^4}-\frac{3 b \left(a^4-4 a^2 b^2-5 b^4\right)}{8 d \left(a^2-b^2\right)^3 (a+b \sin (c+d x))}",1,"(b^5*((Sec[c + d*x]^4*(b^2 - a*b*Sin[c + d*x]))/(4*b^6*(-a^2 + b^2)*(a + b*Sin[c + d*x])) - (-1/2*(Sec[c + d*x]^2*(4*a^2*b^2 - b^2*(3*a^2 - 5*b^2) - b*(4*a*b^2 - a*(3*a^2 - 5*b^2))*Sin[c + d*x]))/(b^4*(-a^2 + b^2)*(a + b*Sin[c + d*x])) + (-6*a*(a^2 - 3*b^2)*(-1/2*Log[1 - Sin[c + d*x]]/(b*(a + b)) + Log[1 + Sin[c + d*x]]/(2*(a - b)*b) - Log[a + b*Sin[c + d*x]]/(a^2 - b^2)) + (6*a^2*(a^2 - 3*b^2) - 3*(a^4 - 2*a^2*b^2 + 5*b^4))*(-1/2*Log[1 - Sin[c + d*x]]/(b*(a + b)^2) + Log[1 + Sin[c + d*x]]/(2*(a - b)^2*b) - (2*a*Log[a + b*Sin[c + d*x]])/((a - b)^2*(a + b)^2) + 1/((a^2 - b^2)*(a + b*Sin[c + d*x]))))/(2*b^2*(-a^2 + b^2)))/(4*b^2*(-a^2 + b^2))))/d","A",1
444,1,3679,187,6.5231917,"\int \frac{\cos ^6(c+d x)}{(a+b \sin (c+d x))^2} \, dx","Integrate[Cos[c + d*x]^6/(a + b*Sin[c + d*x])^2,x]","\text{Result too large to show}","\frac{10 a \left(a^2-b^2\right)^{3/2} \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{b^6 d}-\frac{5 \cos (c+d x) \left(8 a \left(a^2-b^2\right)-b \left(4 a^2-3 b^2\right) \sin (c+d x)\right)}{8 b^5 d}-\frac{5 x \left(8 a^4-12 a^2 b^2+3 b^4\right)}{8 b^6}+\frac{5 \cos ^3(c+d x) (4 a-3 b \sin (c+d x))}{12 b^3 d}-\frac{\cos ^5(c+d x)}{b d (a+b \sin (c+d x))}",1,"(Cos[c + d*x]^5*(-((b*(-(b/(a - b)) - (b*Sin[c + d*x])/(a - b))^(7/2)*(b/(a + b) - (b*Sin[c + d*x])/(a + b))^(7/2))/(((a*b)/(a - b) - b^2/(a - b))*((a*b)/(a + b) + b^2/(a + b))*(a + b*Sin[c + d*x]))) - ((48*Sqrt[2]*(a - b)*b^3*(-(b/(a - b)) - (b*Sin[c + d*x])/(a - b))^(7/2)*Sqrt[b/(a + b) - (b*Sin[c + d*x])/(a + b)]*(1 + ((a - b)*(-(b/(a - b)) - (b*Sin[c + d*x])/(a - b)))/(2*b))^(7/2)*((7*(3/(16*(1 + ((a - b)*(-(b/(a - b)) - (b*Sin[c + d*x])/(a - b)))/(2*b))^3) + 1/(2*(1 + ((a - b)*(-(b/(a - b)) - (b*Sin[c + d*x])/(a - b)))/(2*b))^2) + (1 + ((a - b)*(-(b/(a - b)) - (b*Sin[c + d*x])/(a - b)))/(2*b))^(-1)))/12 + (35*b^4*(((a - b)*(-(b/(a - b)) - (b*Sin[c + d*x])/(a - b)))/b - ((a - b)^2*(-(b/(a - b)) - (b*Sin[c + d*x])/(a - b))^2)/(3*b^2) + (2*(a - b)^3*(-(b/(a - b)) - (b*Sin[c + d*x])/(a - b))^3)/(15*b^3) - (Sqrt[2]*Sqrt[a - b]*ArcSinh[(Sqrt[a - b]*Sqrt[-(b/(a - b)) - (b*Sin[c + d*x])/(a - b)])/(Sqrt[2]*Sqrt[b])]*Sqrt[-(b/(a - b)) - (b*Sin[c + d*x])/(a - b)])/(Sqrt[b]*Sqrt[1 + ((a - b)*(-(b/(a - b)) - (b*Sin[c + d*x])/(a - b)))/(2*b)])))/(128*(a - b)^4*(-(b/(a - b)) - (b*Sin[c + d*x])/(a - b))^4*(1 + ((a - b)*(-(b/(a - b)) - (b*Sin[c + d*x])/(a - b)))/(2*b))^3)))/(7*(a + b)^2*(a^2 - b^2)*Sqrt[((a + b)*(b/(a + b) - (b*Sin[c + d*x])/(a + b)))/b]) + (5*a*b^2*((8*Sqrt[2]*b*(-(b/(a - b)) - (b*Sin[c + d*x])/(a - b))^(5/2)*Sqrt[b/(a + b) - (b*Sin[c + d*x])/(a + b)]*(1 + ((a - b)*(-(b/(a - b)) - (b*Sin[c + d*x])/(a - b)))/(2*b))^(7/2)*((5/(16*(1 + ((a - b)*(-(b/(a - b)) - (b*Sin[c + d*x])/(a - b)))/(2*b))^3) + 5/(8*(1 + ((a - b)*(-(b/(a - b)) - (b*Sin[c + d*x])/(a - b)))/(2*b))^2) + (1 + ((a - b)*(-(b/(a - b)) - (b*Sin[c + d*x])/(a - b)))/(2*b))^(-1))/2 - (15*b^3*(((a - b)*(-(b/(a - b)) - (b*Sin[c + d*x])/(a - b)))/b - ((a - b)^2*(-(b/(a - b)) - (b*Sin[c + d*x])/(a - b))^2)/(3*b^2) - (Sqrt[2]*Sqrt[a - b]*ArcSinh[(Sqrt[a - b]*Sqrt[-(b/(a - b)) - (b*Sin[c + d*x])/(a - b)])/(Sqrt[2]*Sqrt[b])]*Sqrt[-(b/(a - b)) - (b*Sin[c + d*x])/(a - b)])/(Sqrt[b]*Sqrt[1 + ((a - b)*(-(b/(a - b)) - (b*Sin[c + d*x])/(a - b)))/(2*b)])))/(64*(a - b)^3*(-(b/(a - b)) - (b*Sin[c + d*x])/(a - b))^3*(1 + ((a - b)*(-(b/(a - b)) - (b*Sin[c + d*x])/(a - b)))/(2*b))^3)))/(5*(a + b)^2*Sqrt[((a + b)*(b/(a + b) - (b*Sin[c + d*x])/(a + b)))/b]) - ((-((a*b)/(a - b)) + b^2/(a - b))*((8*Sqrt[2]*b*(-(b/(a - b)) - (b*Sin[c + d*x])/(a - b))^(3/2)*Sqrt[b/(a + b) - (b*Sin[c + d*x])/(a + b)]*(1 + ((a - b)*(-(b/(a - b)) - (b*Sin[c + d*x])/(a - b)))/(2*b))^(7/2)*((3*(5/(8*(1 + ((a - b)*(-(b/(a - b)) - (b*Sin[c + d*x])/(a - b)))/(2*b))^3) + 5/(6*(1 + ((a - b)*(-(b/(a - b)) - (b*Sin[c + d*x])/(a - b)))/(2*b))^2) + (1 + ((a - b)*(-(b/(a - b)) - (b*Sin[c + d*x])/(a - b)))/(2*b))^(-1)))/8 + (15*b^2*(((a - b)*(-(b/(a - b)) - (b*Sin[c + d*x])/(a - b)))/b - (Sqrt[2]*Sqrt[a - b]*ArcSinh[(Sqrt[a - b]*Sqrt[-(b/(a - b)) - (b*Sin[c + d*x])/(a - b)])/(Sqrt[2]*Sqrt[b])]*Sqrt[-(b/(a - b)) - (b*Sin[c + d*x])/(a - b)])/(Sqrt[b]*Sqrt[1 + ((a - b)*(-(b/(a - b)) - (b*Sin[c + d*x])/(a - b)))/(2*b)])))/(64*(a - b)^2*(-(b/(a - b)) - (b*Sin[c + d*x])/(a - b))^2*(1 + ((a - b)*(-(b/(a - b)) - (b*Sin[c + d*x])/(a - b)))/(2*b))^3)))/(3*(a + b)^2*Sqrt[((a + b)*(b/(a + b) - (b*Sin[c + d*x])/(a + b)))/b]) - ((-((a*b)/(a - b)) + b^2/(a - b))*((8*Sqrt[2]*b*Sqrt[-(b/(a - b)) - (b*Sin[c + d*x])/(a - b)]*Sqrt[b/(a + b) - (b*Sin[c + d*x])/(a + b)]*(1 + ((a - b)*(-(b/(a - b)) - (b*Sin[c + d*x])/(a - b)))/(2*b))^(7/2)*((5*Sqrt[b]*ArcSinh[(Sqrt[a - b]*Sqrt[-(b/(a - b)) - (b*Sin[c + d*x])/(a - b)])/(Sqrt[2]*Sqrt[b])])/(8*Sqrt[2]*Sqrt[a - b]*Sqrt[-(b/(a - b)) - (b*Sin[c + d*x])/(a - b)]*(1 + ((a - b)*(-(b/(a - b)) - (b*Sin[c + d*x])/(a - b)))/(2*b))^(7/2)) + (15/(8*(1 + ((a - b)*(-(b/(a - b)) - (b*Sin[c + d*x])/(a - b)))/(2*b))^3) + 5/(4*(1 + ((a - b)*(-(b/(a - b)) - (b*Sin[c + d*x])/(a - b)))/(2*b))^2) + (1 + ((a - b)*(-(b/(a - b)) - (b*Sin[c + d*x])/(a - b)))/(2*b))^(-1))/6))/((a + b)^2*Sqrt[((a + b)*(b/(a + b) - (b*Sin[c + d*x])/(a + b)))/b]) - ((-((a*b)/(a - b)) + b^2/(a - b))*(-(((-((a*b)/(a + b)) - b^2/(a + b))*(-(((-((a*b)/(a + b)) - b^2/(a + b))*((2*Sqrt[a - b]*ArcTanh[(Sqrt[a - b]*Sqrt[-(b/(a - b)) - (b*Sin[c + d*x])/(a - b)])/(Sqrt[a + b]*Sqrt[b/(a + b) - (b*Sin[c + d*x])/(a + b)])])/(b*Sqrt[a + b]) - (2*Sqrt[-((a*b)/(a + b)) - b^2/(a + b)]*ArcTanh[(Sqrt[-((a*b)/(a + b)) - b^2/(a + b)]*Sqrt[-(b/(a - b)) - (b*Sin[c + d*x])/(a - b)])/(Sqrt[-((a*b)/(a - b)) + b^2/(a - b)]*Sqrt[b/(a + b) - (b*Sin[c + d*x])/(a + b)])])/(b*Sqrt[-((a*b)/(a - b)) + b^2/(a - b)])))/b) + (2*Sqrt[2]*(a - b)*Sqrt[-(b/(a - b)) - (b*Sin[c + d*x])/(a - b)]*Sqrt[b/(a + b) - (b*Sin[c + d*x])/(a + b)]*(1 + ((a - b)*(-(b/(a - b)) - (b*Sin[c + d*x])/(a - b)))/(2*b))^(3/2)*((Sqrt[b]*ArcSinh[(Sqrt[a - b]*Sqrt[-(b/(a - b)) - (b*Sin[c + d*x])/(a - b)])/(Sqrt[2]*Sqrt[b])])/(Sqrt[2]*Sqrt[a - b]*Sqrt[-(b/(a - b)) - (b*Sin[c + d*x])/(a - b)]*(1 + ((a - b)*(-(b/(a - b)) - (b*Sin[c + d*x])/(a - b)))/(2*b))^(3/2)) + 1/(2*(1 + ((a - b)*(-(b/(a - b)) - (b*Sin[c + d*x])/(a - b)))/(2*b)))))/(b*(a + b)*Sqrt[((a + b)*(b/(a + b) - (b*Sin[c + d*x])/(a + b)))/b])))/b) + (4*Sqrt[2]*(a - b)*Sqrt[-(b/(a - b)) - (b*Sin[c + d*x])/(a - b)]*Sqrt[b/(a + b) - (b*Sin[c + d*x])/(a + b)]*(1 + ((a - b)*(-(b/(a - b)) - (b*Sin[c + d*x])/(a - b)))/(2*b))^(5/2)*((3*Sqrt[b]*ArcSinh[(Sqrt[a - b]*Sqrt[-(b/(a - b)) - (b*Sin[c + d*x])/(a - b)])/(Sqrt[2]*Sqrt[b])])/(4*Sqrt[2]*Sqrt[a - b]*Sqrt[-(b/(a - b)) - (b*Sin[c + d*x])/(a - b)]*(1 + ((a - b)*(-(b/(a - b)) - (b*Sin[c + d*x])/(a - b)))/(2*b))^(5/2)) + (3/(2*(1 + ((a - b)*(-(b/(a - b)) - (b*Sin[c + d*x])/(a - b)))/(2*b))^2) + (1 + ((a - b)*(-(b/(a - b)) - (b*Sin[c + d*x])/(a - b)))/(2*b))^(-1))/4))/((a + b)^2*Sqrt[((a + b)*(b/(a + b) - (b*Sin[c + d*x])/(a + b)))/b])))/b))/b))/b))/(a^2 - b^2))/(((a*b)/(a - b) - b^2/(a - b))*((a*b)/(a + b) + b^2/(a + b)))))/(d*(1 - (a + b*Sin[c + d*x])/(a - b))^(5/2)*(1 - (a + b*Sin[c + d*x])/(a + b))^(5/2))","B",1
445,1,448,128,5.1118442,"\int \frac{\cos ^4(c+d x)}{(a+b \sin (c+d x))^2} \, dx","Integrate[Cos[c + d*x]^4/(a + b*Sin[c + d*x])^2,x]","\frac{\cos ^3(c+d x) \left(\sqrt{a+b} \left(\sqrt{-\frac{b (\sin (c+d x)-1)}{a+b}} \left(\sqrt{a-b} \sqrt{1-\sin (c+d x)} \sqrt{\frac{b (\sin (c+d x)+1)}{b-a}} \left(-6 a^2-3 a b \sin (c+d x)+b^2 \sin ^2(c+d x)+2 b^2\right)+6 \sqrt{b} (b-2 a) (a+b \sin (c+d x)) \sinh ^{-1}\left(\frac{\sqrt{a-b} \sqrt{-\frac{b (\sin (c+d x)+1)}{a-b}}}{\sqrt{2} \sqrt{b}}\right)\right)+12 a \sqrt{a-b} \sqrt{1-\sin (c+d x)} (a+b \sin (c+d x)) \tanh ^{-1}\left(\frac{\sqrt{\frac{b (\sin (c+d x)+1)}{b-a}}}{\sqrt{-\frac{b (\sin (c+d x)-1)}{a+b}}}\right)\right)-12 a (a-b) \sqrt{1-\sin (c+d x)} (a+b \sin (c+d x)) \tanh ^{-1}\left(\frac{\sqrt{a-b} \sqrt{-\frac{b (\sin (c+d x)+1)}{a-b}}}{\sqrt{a+b} \sqrt{-\frac{b (\sin (c+d x)-1)}{a+b}}}\right)\right)}{2 b^2 d (a-b)^{3/2} \sqrt{a+b} (1-\sin (c+d x))^{3/2} \sqrt{-\frac{b (\sin (c+d x)-1)}{a+b}} \left(-\frac{b (\sin (c+d x)+1)}{a-b}\right)^{3/2} (a+b \sin (c+d x))}","-\frac{6 a \sqrt{a^2-b^2} \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{b^4 d}+\frac{3 x \left(2 a^2-b^2\right)}{2 b^4}+\frac{3 \cos (c+d x) (2 a-b \sin (c+d x))}{2 b^3 d}-\frac{\cos ^3(c+d x)}{b d (a+b \sin (c+d x))}",1,"(Cos[c + d*x]^3*(-12*a*(a - b)*ArcTanh[(Sqrt[a - b]*Sqrt[-((b*(1 + Sin[c + d*x]))/(a - b))])/(Sqrt[a + b]*Sqrt[-((b*(-1 + Sin[c + d*x]))/(a + b))])]*Sqrt[1 - Sin[c + d*x]]*(a + b*Sin[c + d*x]) + Sqrt[a + b]*(12*a*Sqrt[a - b]*ArcTanh[Sqrt[(b*(1 + Sin[c + d*x]))/(-a + b)]/Sqrt[-((b*(-1 + Sin[c + d*x]))/(a + b))]]*Sqrt[1 - Sin[c + d*x]]*(a + b*Sin[c + d*x]) + Sqrt[-((b*(-1 + Sin[c + d*x]))/(a + b))]*(6*Sqrt[b]*(-2*a + b)*ArcSinh[(Sqrt[a - b]*Sqrt[-((b*(1 + Sin[c + d*x]))/(a - b))])/(Sqrt[2]*Sqrt[b])]*(a + b*Sin[c + d*x]) + Sqrt[a - b]*Sqrt[1 - Sin[c + d*x]]*Sqrt[(b*(1 + Sin[c + d*x]))/(-a + b)]*(-6*a^2 + 2*b^2 - 3*a*b*Sin[c + d*x] + b^2*Sin[c + d*x]^2)))))/(2*(a - b)^(3/2)*b^2*Sqrt[a + b]*d*(1 - Sin[c + d*x])^(3/2)*Sqrt[-((b*(-1 + Sin[c + d*x]))/(a + b))]*(-((b*(1 + Sin[c + d*x]))/(a - b)))^(3/2)*(a + b*Sin[c + d*x]))","B",1
446,1,414,84,2.6922245,"\int \frac{\cos ^2(c+d x)}{(a+b \sin (c+d x))^2} \, dx","Integrate[Cos[c + d*x]^2/(a + b*Sin[c + d*x])^2,x]","\frac{\cos (c+d x) \left(\sqrt{a+b} \left((b-a) \sqrt{-\frac{b (\sin (c+d x)-1)}{a+b}} \left(\sqrt{a-b} (a+b) \sqrt{1-\sin (c+d x)} \sqrt{-\frac{b (\sin (c+d x)+1)}{a-b}}+2 \sqrt{b} (a+b \sin (c+d x)) \sinh ^{-1}\left(\frac{\sqrt{a-b} \sqrt{-\frac{b (\sin (c+d x)+1)}{a-b}}}{\sqrt{2} \sqrt{b}}\right)\right)+2 a \sqrt{a-b} \sqrt{1-\sin (c+d x)} (a+b \sin (c+d x)) \tanh ^{-1}\left(\frac{\sqrt{\frac{b (\sin (c+d x)+1)}{b-a}}}{\sqrt{-\frac{b (\sin (c+d x)-1)}{a+b}}}\right)\right)-2 a (a-b) \sqrt{1-\sin (c+d x)} (a+b \sin (c+d x)) \tanh ^{-1}\left(\frac{\sqrt{a-b} \sqrt{-\frac{b (\sin (c+d x)+1)}{a-b}}}{\sqrt{a+b} \sqrt{-\frac{b (\sin (c+d x)-1)}{a+b}}}\right)\right)}{b d (a-b)^{3/2} (a+b)^{3/2} \sqrt{1-\sin (c+d x)} \sqrt{-\frac{b (\sin (c+d x)-1)}{a+b}} \sqrt{-\frac{b (\sin (c+d x)+1)}{a-b}} (a+b \sin (c+d x))}","\frac{2 a \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{b^2 d \sqrt{a^2-b^2}}-\frac{\cos (c+d x)}{b d (a+b \sin (c+d x))}-\frac{x}{b^2}",1,"(Cos[c + d*x]*(-2*a*(a - b)*ArcTanh[(Sqrt[a - b]*Sqrt[-((b*(1 + Sin[c + d*x]))/(a - b))])/(Sqrt[a + b]*Sqrt[-((b*(-1 + Sin[c + d*x]))/(a + b))])]*Sqrt[1 - Sin[c + d*x]]*(a + b*Sin[c + d*x]) + Sqrt[a + b]*(2*a*Sqrt[a - b]*ArcTanh[Sqrt[(b*(1 + Sin[c + d*x]))/(-a + b)]/Sqrt[-((b*(-1 + Sin[c + d*x]))/(a + b))]]*Sqrt[1 - Sin[c + d*x]]*(a + b*Sin[c + d*x]) + (-a + b)*Sqrt[-((b*(-1 + Sin[c + d*x]))/(a + b))]*(Sqrt[a - b]*(a + b)*Sqrt[1 - Sin[c + d*x]]*Sqrt[-((b*(1 + Sin[c + d*x]))/(a - b))] + 2*Sqrt[b]*ArcSinh[(Sqrt[a - b]*Sqrt[-((b*(1 + Sin[c + d*x]))/(a - b))])/(Sqrt[2]*Sqrt[b])]*(a + b*Sin[c + d*x])))))/((a - b)^(3/2)*b*(a + b)^(3/2)*d*Sqrt[1 - Sin[c + d*x]]*Sqrt[-((b*(-1 + Sin[c + d*x]))/(a + b))]*Sqrt[-((b*(1 + Sin[c + d*x]))/(a - b))]*(a + b*Sin[c + d*x]))","B",1
447,1,162,130,1.1503594,"\int \frac{\sec ^2(c+d x)}{(a+b \sin (c+d x))^2} \, dx","Integrate[Sec[c + d*x]^2/(a + b*Sin[c + d*x])^2,x]","\frac{-\frac{6 a b^2 \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{\left(a^2-b^2\right)^{5/2}}-\frac{b^3 \cos (c+d x)}{(a-b)^2 (a+b)^2 (a+b \sin (c+d x))}+\sin \left(\frac{1}{2} (c+d x)\right) \left(\frac{1}{(a-b)^2 \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}+\frac{1}{(a+b)^2 \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}\right)}{d}","-\frac{6 a b^2 \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{d \left(a^2-b^2\right)^{5/2}}+\frac{b \sec (c+d x)}{d \left(a^2-b^2\right) (a+b \sin (c+d x))}-\frac{\sec (c+d x) \left(3 a b-\left(a^2+2 b^2\right) \sin (c+d x)\right)}{d \left(a^2-b^2\right)^2}",1,"((-6*a*b^2*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(5/2) + Sin[(c + d*x)/2]*(1/((a + b)^2*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])) + 1/((a - b)^2*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))) - (b^3*Cos[c + d*x])/((a - b)^2*(a + b)^2*(a + b*Sin[c + d*x])))/d","A",1
448,1,336,193,1.8627324,"\int \frac{\sec ^4(c+d x)}{(a+b \sin (c+d x))^2} \, dx","Integrate[Sec[c + d*x]^4/(a + b*Sin[c + d*x])^2,x]","\frac{\frac{120 a b^4 \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{\left(a^2-b^2\right)^{7/2}}+\frac{12 b^5 \cos (c+d x)}{(a-b)^3 (a+b)^3 (a+b \sin (c+d x))}+\frac{4 (2 a+5 b) \sin \left(\frac{1}{2} (c+d x)\right)}{(a+b)^3 \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}+\frac{2 \sin \left(\frac{1}{2} (c+d x)\right)}{(a+b)^2 \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^3}+\frac{4 (2 a-5 b) \sin \left(\frac{1}{2} (c+d x)\right)}{(a-b)^3 \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}+\frac{1}{(a+b)^2 \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^2}-\frac{1}{(a-b)^2 \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2}+\frac{2 \sin \left(\frac{1}{2} (c+d x)\right)}{(a-b)^2 \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^3}}{12 d}","\frac{b \sec ^3(c+d x)}{d \left(a^2-b^2\right) (a+b \sin (c+d x))}-\frac{\sec ^3(c+d x) \left(5 a b-\left(a^2+4 b^2\right) \sin (c+d x)\right)}{3 d \left(a^2-b^2\right)^2}+\frac{10 a b^4 \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{d \left(a^2-b^2\right)^{7/2}}+\frac{\sec (c+d x) \left(\left(2 a^4-9 a^2 b^2-8 b^4\right) \sin (c+d x)+15 a b^3\right)}{3 d \left(a^2-b^2\right)^3}",1,"((120*a*b^4*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(7/2) + 1/((a + b)^2*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2) + (2*Sin[(c + d*x)/2])/((a + b)^2*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^3) + (4*(2*a + 5*b)*Sin[(c + d*x)/2])/((a + b)^3*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])) + (2*Sin[(c + d*x)/2])/((a - b)^2*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3) - 1/((a - b)^2*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2) + (4*(2*a - 5*b)*Sin[(c + d*x)/2])/((a - b)^3*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])) + (12*b^5*Cos[c + d*x])/((a - b)^3*(a + b)^3*(a + b*Sin[c + d*x])))/(12*d)","A",1
449,1,282,190,0.6395953,"\int \frac{\cos ^7(c+d x)}{(a+b \sin (c+d x))^3} \, dx","Integrate[Cos[c + d*x]^7/(a + b*Sin[c + d*x])^3,x]","\frac{b^4 \cos ^4(c+d x) \left(-a^2+2 a b \sin (c+d x)+3 b^2\right)-2 \left(2 a^2 b^4 \sin ^4(c+d x)-10 a b^3 \left(a^2-b^2\right) \sin ^3(c+d x)+2 b^2 \sin ^2(c+d x) \left(-13 a^4+10 a^2 b^2+3 \left(5 a^4-6 a^2 b^2+b^4\right) \log (a+b \sin (c+d x))\right)+2 a b \sin (c+d x) \left(4 a^4-17 a^2 b^2+6 \left(5 a^4-6 a^2 b^2+b^4\right) \log (a+b \sin (c+d x))+11 b^4\right)+\left(a^2-b^2\right) \left(19 a^4+6 a^2 \left(5 a^2-b^2\right) \log (a+b \sin (c+d x))-16 a^2 b^2-3 b^4\right)\right)+b^6 \cos ^6(c+d x)}{4 b^7 d (a+b \sin (c+d x))^2}","-\frac{6 a \left(a^2-b^2\right)^2}{b^7 d (a+b \sin (c+d x))}+\frac{\left(a^2-b^2\right)^3}{2 b^7 d (a+b \sin (c+d x))^2}+\frac{a \left(10 a^2-9 b^2\right) \sin (c+d x)}{b^6 d}-\frac{3 \left(2 a^2-b^2\right) \sin ^2(c+d x)}{2 b^5 d}-\frac{3 \left(5 a^4-6 a^2 b^2+b^4\right) \log (a+b \sin (c+d x))}{b^7 d}+\frac{a \sin ^3(c+d x)}{b^4 d}-\frac{\sin ^4(c+d x)}{4 b^3 d}",1,"(b^6*Cos[c + d*x]^6 + b^4*Cos[c + d*x]^4*(-a^2 + 3*b^2 + 2*a*b*Sin[c + d*x]) - 2*((a^2 - b^2)*(19*a^4 - 16*a^2*b^2 - 3*b^4 + 6*a^2*(5*a^2 - b^2)*Log[a + b*Sin[c + d*x]]) + 2*a*b*(4*a^4 - 17*a^2*b^2 + 11*b^4 + 6*(5*a^4 - 6*a^2*b^2 + b^4)*Log[a + b*Sin[c + d*x]])*Sin[c + d*x] + 2*b^2*(-13*a^4 + 10*a^2*b^2 + 3*(5*a^4 - 6*a^2*b^2 + b^4)*Log[a + b*Sin[c + d*x]])*Sin[c + d*x]^2 - 10*a*b^3*(a^2 - b^2)*Sin[c + d*x]^3 + 2*a^2*b^4*Sin[c + d*x]^4))/(4*b^7*d*(a + b*Sin[c + d*x])^2)","A",1
450,1,143,127,0.9841057,"\int \frac{\cos ^5(c+d x)}{(a+b \sin (c+d x))^3} \, dx","Integrate[Cos[c + d*x]^5/(a + b*Sin[c + d*x])^3,x]","\frac{2 \left(b^2-a^2\right) \left(-\frac{3 a^2+4 a b \sin (c+d x)+b^2}{2 (a+b \sin (c+d x))^2}-\log (a+b \sin (c+d x))\right)+\frac{b^4 \cos ^4(c+d x)}{2 (a+b \sin (c+d x))^2}+2 a \left(\frac{(a-b) (a+b)}{a+b \sin (c+d x)}+2 a \log (a+b \sin (c+d x))-b \sin (c+d x)\right)}{b^5 d}","-\frac{\left(a^2-b^2\right)^2}{2 b^5 d (a+b \sin (c+d x))^2}+\frac{4 a \left(a^2-b^2\right)}{b^5 d (a+b \sin (c+d x))}+\frac{2 \left(3 a^2-b^2\right) \log (a+b \sin (c+d x))}{b^5 d}-\frac{3 a \sin (c+d x)}{b^4 d}+\frac{\sin ^2(c+d x)}{2 b^3 d}",1,"((b^4*Cos[c + d*x]^4)/(2*(a + b*Sin[c + d*x])^2) + 2*a*(2*a*Log[a + b*Sin[c + d*x]] - b*Sin[c + d*x] + ((a - b)*(a + b))/(a + b*Sin[c + d*x])) + 2*(-a^2 + b^2)*(-Log[a + b*Sin[c + d*x]] - (3*a^2 + b^2 + 4*a*b*Sin[c + d*x])/(2*(a + b*Sin[c + d*x])^2)))/(b^5*d)","A",1
451,1,55,72,0.1182919,"\int \frac{\cos ^3(c+d x)}{(a+b \sin (c+d x))^3} \, dx","Integrate[Cos[c + d*x]^3/(a + b*Sin[c + d*x])^3,x]","-\frac{\frac{3 a^2+4 a b \sin (c+d x)+b^2}{2 (a+b \sin (c+d x))^2}+\log (a+b \sin (c+d x))}{b^3 d}","\frac{a^2-b^2}{2 b^3 d (a+b \sin (c+d x))^2}-\frac{2 a}{b^3 d (a+b \sin (c+d x))}-\frac{\log (a+b \sin (c+d x))}{b^3 d}",1,"-((Log[a + b*Sin[c + d*x]] + (3*a^2 + b^2 + 4*a*b*Sin[c + d*x])/(2*(a + b*Sin[c + d*x])^2))/(b^3*d))","A",1
452,1,22,22,0.0272506,"\int \frac{\cos (c+d x)}{(a+b \sin (c+d x))^3} \, dx","Integrate[Cos[c + d*x]/(a + b*Sin[c + d*x])^3,x]","-\frac{1}{2 b d (a+b \sin (c+d x))^2}","-\frac{1}{2 b d (a+b \sin (c+d x))^2}",1,"-1/2*1/(b*d*(a + b*Sin[c + d*x])^2)","A",1
453,1,135,145,0.5893737,"\int \frac{\sec (c+d x)}{(a+b \sin (c+d x))^3} \, dx","Integrate[Sec[c + d*x]/(a + b*Sin[c + d*x])^3,x]","\frac{b \left(\frac{1}{\left(a^2-b^2\right) (a+b \sin (c+d x))^2}-\frac{2 \left(3 a^2+b^2\right) \log (a+b \sin (c+d x))}{(a-b)^3 (a+b)^3}+\frac{4 a}{(a-b)^2 (a+b)^2 (a+b \sin (c+d x))}-\frac{\log (1-\sin (c+d x))}{b (a+b)^3}+\frac{\log (\sin (c+d x)+1)}{b (a-b)^3}\right)}{2 d}","\frac{2 a b}{d \left(a^2-b^2\right)^2 (a+b \sin (c+d x))}+\frac{b}{2 d \left(a^2-b^2\right) (a+b \sin (c+d x))^2}-\frac{b \left(3 a^2+b^2\right) \log (a+b \sin (c+d x))}{d \left(a^2-b^2\right)^3}-\frac{\log (1-\sin (c+d x))}{2 d (a+b)^3}+\frac{\log (\sin (c+d x)+1)}{2 d (a-b)^3}",1,"(b*(-(Log[1 - Sin[c + d*x]]/(b*(a + b)^3)) + Log[1 + Sin[c + d*x]]/((a - b)^3*b) - (2*(3*a^2 + b^2)*Log[a + b*Sin[c + d*x]])/((a - b)^3*(a + b)^3) + 1/((a^2 - b^2)*(a + b*Sin[c + d*x])^2) + (4*a)/((a - b)^2*(a + b)^2*(a + b*Sin[c + d*x]))))/(2*d)","A",1
454,1,283,226,3.9520174,"\int \frac{\sec ^3(c+d x)}{(a+b \sin (c+d x))^3} \, dx","Integrate[Sec[c + d*x]^3/(a + b*Sin[c + d*x])^3,x]","\frac{b \left(a^2+2 b^2\right) \left(\frac{1}{\left(a^2-b^2\right) (a+b \sin (c+d x))^2}-\frac{2 \left(3 a^2+b^2\right) \log (a+b \sin (c+d x))}{(a-b)^3 (a+b)^3}+\frac{4 a}{(a-b)^2 (a+b)^2 (a+b \sin (c+d x))}-\frac{\log (1-\sin (c+d x))}{b (a+b)^3}+\frac{\log (\sin (c+d x)+1)}{b (a-b)^3}\right)+\frac{3}{2} a \left(\frac{2 b}{\left(b^2-a^2\right) (a+b \sin (c+d x))}+\frac{4 a b \log (a+b \sin (c+d x))}{(a-b)^2 (a+b)^2}+\frac{\log (1-\sin (c+d x))}{(a+b)^2}-\frac{\log (\sin (c+d x)+1)}{(a-b)^2}\right)+\frac{\sec ^2(c+d x) (b-a \sin (c+d x))}{(a+b \sin (c+d x))^2}}{2 d \left(b^2-a^2\right)}","-\frac{a b \left(a^2+11 b^2\right)}{2 d \left(a^2-b^2\right)^3 (a+b \sin (c+d x))}-\frac{b \left(a^2+2 b^2\right)}{2 d \left(a^2-b^2\right)^2 (a+b \sin (c+d x))^2}-\frac{\sec ^2(c+d x) (b-a \sin (c+d x))}{2 d \left(a^2-b^2\right) (a+b \sin (c+d x))^2}+\frac{2 b^3 \left(5 a^2+b^2\right) \log (a+b \sin (c+d x))}{d \left(a^2-b^2\right)^4}-\frac{(a+4 b) \log (1-\sin (c+d x))}{4 d (a+b)^4}+\frac{(a-4 b) \log (\sin (c+d x)+1)}{4 d (a-b)^4}",1,"((Sec[c + d*x]^2*(b - a*Sin[c + d*x]))/(a + b*Sin[c + d*x])^2 + b*(a^2 + 2*b^2)*(-(Log[1 - Sin[c + d*x]]/(b*(a + b)^3)) + Log[1 + Sin[c + d*x]]/((a - b)^3*b) - (2*(3*a^2 + b^2)*Log[a + b*Sin[c + d*x]])/((a - b)^3*(a + b)^3) + 1/((a^2 - b^2)*(a + b*Sin[c + d*x])^2) + (4*a)/((a - b)^2*(a + b)^2*(a + b*Sin[c + d*x]))) + (3*a*(Log[1 - Sin[c + d*x]]/(a + b)^2 - Log[1 + Sin[c + d*x]]/(a - b)^2 + (4*a*b*Log[a + b*Sin[c + d*x]])/((a - b)^2*(a + b)^2) + (2*b)/((-a^2 + b^2)*(a + b*Sin[c + d*x]))))/2)/(2*(-a^2 + b^2)*d)","A",1
455,1,388,328,2.7350145,"\int \frac{\sec ^5(c+d x)}{(a+b \sin (c+d x))^3} \, dx","Integrate[Sec[c + d*x]^5/(a + b*Sin[c + d*x])^3,x]","\frac{\frac{\sec ^2(c+d x) \left(a \left(3 a^2-11 b^2\right) \sin (c+d x)+2 b \left(a^2+3 b^2\right)\right)}{\left(b^2-a^2\right) (a+b \sin (c+d x))^2}-\frac{b \left(3 \left(a^4-5 a^2 b^2-4 b^4\right) \left(\frac{1}{\left(a^2-b^2\right) (a+b \sin (c+d x))^2}-\frac{2 \left(3 a^2+b^2\right) \log (a+b \sin (c+d x))}{(a-b)^3 (a+b)^3}+\frac{4 a}{(a-b)^2 (a+b)^2 (a+b \sin (c+d x))}-\frac{\log (1-\sin (c+d x))}{b (a+b)^3}+\frac{\log (\sin (c+d x)+1)}{b (a-b)^3}\right)-3 a \left(3 a^2-11 b^2\right) \left(\frac{1}{\left(a^2-b^2\right) (a+b \sin (c+d x))}-\frac{\log (1-\sin (c+d x))}{2 b (a+b)^2}+\frac{\log (\sin (c+d x)+1)}{2 b (a-b)^2}-\frac{2 a \log (a+b \sin (c+d x))}{(a-b)^2 (a+b)^2}\right)\right)}{b^2-a^2}+\frac{2 \sec ^4(c+d x) (b-a \sin (c+d x))}{(a+b \sin (c+d x))^2}}{8 d \left(b^2-a^2\right)}","-\frac{3 \left(a^2+5 a b+8 b^2\right) \log (1-\sin (c+d x))}{16 d (a+b)^5}+\frac{3 \left(a^2-5 a b+8 b^2\right) \log (\sin (c+d x)+1)}{16 d (a-b)^5}-\frac{\sec ^4(c+d x) (b-a \sin (c+d x))}{4 d \left(a^2-b^2\right) (a+b \sin (c+d x))^2}+\frac{\sec ^2(c+d x) \left(a \left(3 a^2-11 b^2\right) \sin (c+d x)+2 b \left(a^2+3 b^2\right)\right)}{8 d \left(a^2-b^2\right)^2 (a+b \sin (c+d x))^2}-\frac{3 b^5 \left(7 a^2+b^2\right) \log (a+b \sin (c+d x))}{d \left(a^2-b^2\right)^5}-\frac{3 a b \left(a^4-6 a^2 b^2-27 b^4\right)}{8 d \left(a^2-b^2\right)^4 (a+b \sin (c+d x))}-\frac{3 b \left(a^4-5 a^2 b^2-4 b^4\right)}{8 d \left(a^2-b^2\right)^3 (a+b \sin (c+d x))^2}",1,"((2*Sec[c + d*x]^4*(b - a*Sin[c + d*x]))/(a + b*Sin[c + d*x])^2 + (Sec[c + d*x]^2*(2*b*(a^2 + 3*b^2) + a*(3*a^2 - 11*b^2)*Sin[c + d*x]))/((-a^2 + b^2)*(a + b*Sin[c + d*x])^2) - (b*(3*(a^4 - 5*a^2*b^2 - 4*b^4)*(-(Log[1 - Sin[c + d*x]]/(b*(a + b)^3)) + Log[1 + Sin[c + d*x]]/((a - b)^3*b) - (2*(3*a^2 + b^2)*Log[a + b*Sin[c + d*x]])/((a - b)^3*(a + b)^3) + 1/((a^2 - b^2)*(a + b*Sin[c + d*x])^2) + (4*a)/((a - b)^2*(a + b)^2*(a + b*Sin[c + d*x]))) - 3*a*(3*a^2 - 11*b^2)*(-1/2*Log[1 - Sin[c + d*x]]/(b*(a + b)^2) + Log[1 + Sin[c + d*x]]/(2*(a - b)^2*b) - (2*a*Log[a + b*Sin[c + d*x]])/((a - b)^2*(a + b)^2) + 1/((a^2 - b^2)*(a + b*Sin[c + d*x])))))/(-a^2 + b^2))/(8*(-a^2 + b^2)*d)","A",1
456,1,3889,197,6.5717254,"\int \frac{\cos ^6(c+d x)}{(a+b \sin (c+d x))^3} \, dx","Integrate[Cos[c + d*x]^6/(a + b*Sin[c + d*x])^3,x]","\text{Result too large to show}","\frac{5 a x \left(4 a^2-3 b^2\right)}{2 b^6}+\frac{5 \cos (c+d x) \left(4 a^2-2 a b \sin (c+d x)-b^2\right)}{2 b^5 d}-\frac{5 \left(4 a^4-5 a^2 b^2+b^4\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{b^6 d \sqrt{a^2-b^2}}-\frac{5 \cos ^3(c+d x) (4 a+b \sin (c+d x))}{6 b^3 d (a+b \sin (c+d x))}-\frac{\cos ^5(c+d x)}{2 b d (a+b \sin (c+d x))^2}",1,"(Cos[c + d*x]^5*(-1/2*(b*(-(b/(a - b)) - (b*Sin[c + d*x])/(a - b))^(7/2)*(b/(a + b) - (b*Sin[c + d*x])/(a + b))^(7/2))/(((a*b)/(a - b) - b^2/(a - b))*((a*b)/(a + b) + b^2/(a + b))*(a + b*Sin[c + d*x])^2) - ((-3*a*b^3*(-(b/(a - b)) - (b*Sin[c + d*x])/(a - b))^(7/2)*(b/(a + b) - (b*Sin[c + d*x])/(a + b))^(7/2))/((a^2 - b^2)*((a*b)/(a - b) - b^2/(a - b))*((a*b)/(a + b) + b^2/(a + b))*(a + b*Sin[c + d*x])) - ((144*Sqrt[2]*a*b^5*(-(b/(a - b)) - (b*Sin[c + d*x])/(a - b))^(7/2)*Sqrt[b/(a + b) - (b*Sin[c + d*x])/(a + b)]*(1 + ((a - b)*(-(b/(a - b)) - (b*Sin[c + d*x])/(a - b)))/(2*b))^(7/2)*((7*(3/(16*(1 + ((a - b)*(-(b/(a - b)) - (b*Sin[c + d*x])/(a - b)))/(2*b))^3) + 1/(2*(1 + ((a - b)*(-(b/(a - b)) - (b*Sin[c + d*x])/(a - b)))/(2*b))^2) + (1 + ((a - b)*(-(b/(a - b)) - (b*Sin[c + d*x])/(a - b)))/(2*b))^(-1)))/12 + (35*b^4*(((a - b)*(-(b/(a - b)) - (b*Sin[c + d*x])/(a - b)))/b - ((a - b)^2*(-(b/(a - b)) - (b*Sin[c + d*x])/(a - b))^2)/(3*b^2) + (2*(a - b)^3*(-(b/(a - b)) - (b*Sin[c + d*x])/(a - b))^3)/(15*b^3) - (Sqrt[2]*Sqrt[a - b]*ArcSinh[(Sqrt[a - b]*Sqrt[-(b/(a - b)) - (b*Sin[c + d*x])/(a - b)])/(Sqrt[2]*Sqrt[b])]*Sqrt[-(b/(a - b)) - (b*Sin[c + d*x])/(a - b)])/(Sqrt[b]*Sqrt[1 + ((a - b)*(-(b/(a - b)) - (b*Sin[c + d*x])/(a - b)))/(2*b)])))/(128*(a - b)^4*(-(b/(a - b)) - (b*Sin[c + d*x])/(a - b))^4*(1 + ((a - b)*(-(b/(a - b)) - (b*Sin[c + d*x])/(a - b)))/(2*b))^3)))/(7*(a - b)*(a + b)^4*Sqrt[((a + b)*(b/(a + b) - (b*Sin[c + d*x])/(a + b)))/b]) + (((18*a^2*b^5)/((a - b)^2*(a + b)^2) + (b^5*(2*a^2 - 5*b^2))/((a - b)^2*(a + b)^2))*((8*Sqrt[2]*b*(-(b/(a - b)) - (b*Sin[c + d*x])/(a - b))^(5/2)*Sqrt[b/(a + b) - (b*Sin[c + d*x])/(a + b)]*(1 + ((a - b)*(-(b/(a - b)) - (b*Sin[c + d*x])/(a - b)))/(2*b))^(7/2)*((5/(16*(1 + ((a - b)*(-(b/(a - b)) - (b*Sin[c + d*x])/(a - b)))/(2*b))^3) + 5/(8*(1 + ((a - b)*(-(b/(a - b)) - (b*Sin[c + d*x])/(a - b)))/(2*b))^2) + (1 + ((a - b)*(-(b/(a - b)) - (b*Sin[c + d*x])/(a - b)))/(2*b))^(-1))/2 - (15*b^3*(((a - b)*(-(b/(a - b)) - (b*Sin[c + d*x])/(a - b)))/b - ((a - b)^2*(-(b/(a - b)) - (b*Sin[c + d*x])/(a - b))^2)/(3*b^2) - (Sqrt[2]*Sqrt[a - b]*ArcSinh[(Sqrt[a - b]*Sqrt[-(b/(a - b)) - (b*Sin[c + d*x])/(a - b)])/(Sqrt[2]*Sqrt[b])]*Sqrt[-(b/(a - b)) - (b*Sin[c + d*x])/(a - b)])/(Sqrt[b]*Sqrt[1 + ((a - b)*(-(b/(a - b)) - (b*Sin[c + d*x])/(a - b)))/(2*b)])))/(64*(a - b)^3*(-(b/(a - b)) - (b*Sin[c + d*x])/(a - b))^3*(1 + ((a - b)*(-(b/(a - b)) - (b*Sin[c + d*x])/(a - b)))/(2*b))^3)))/(5*(a + b)^2*Sqrt[((a + b)*(b/(a + b) - (b*Sin[c + d*x])/(a + b)))/b]) - ((-((a*b)/(a - b)) + b^2/(a - b))*((8*Sqrt[2]*b*(-(b/(a - b)) - (b*Sin[c + d*x])/(a - b))^(3/2)*Sqrt[b/(a + b) - (b*Sin[c + d*x])/(a + b)]*(1 + ((a - b)*(-(b/(a - b)) - (b*Sin[c + d*x])/(a - b)))/(2*b))^(7/2)*((3*(5/(8*(1 + ((a - b)*(-(b/(a - b)) - (b*Sin[c + d*x])/(a - b)))/(2*b))^3) + 5/(6*(1 + ((a - b)*(-(b/(a - b)) - (b*Sin[c + d*x])/(a - b)))/(2*b))^2) + (1 + ((a - b)*(-(b/(a - b)) - (b*Sin[c + d*x])/(a - b)))/(2*b))^(-1)))/8 + (15*b^2*(((a - b)*(-(b/(a - b)) - (b*Sin[c + d*x])/(a - b)))/b - (Sqrt[2]*Sqrt[a - b]*ArcSinh[(Sqrt[a - b]*Sqrt[-(b/(a - b)) - (b*Sin[c + d*x])/(a - b)])/(Sqrt[2]*Sqrt[b])]*Sqrt[-(b/(a - b)) - (b*Sin[c + d*x])/(a - b)])/(Sqrt[b]*Sqrt[1 + ((a - b)*(-(b/(a - b)) - (b*Sin[c + d*x])/(a - b)))/(2*b)])))/(64*(a - b)^2*(-(b/(a - b)) - (b*Sin[c + d*x])/(a - b))^2*(1 + ((a - b)*(-(b/(a - b)) - (b*Sin[c + d*x])/(a - b)))/(2*b))^3)))/(3*(a + b)^2*Sqrt[((a + b)*(b/(a + b) - (b*Sin[c + d*x])/(a + b)))/b]) - ((-((a*b)/(a - b)) + b^2/(a - b))*((8*Sqrt[2]*b*Sqrt[-(b/(a - b)) - (b*Sin[c + d*x])/(a - b)]*Sqrt[b/(a + b) - (b*Sin[c + d*x])/(a + b)]*(1 + ((a - b)*(-(b/(a - b)) - (b*Sin[c + d*x])/(a - b)))/(2*b))^(7/2)*((5*Sqrt[b]*ArcSinh[(Sqrt[a - b]*Sqrt[-(b/(a - b)) - (b*Sin[c + d*x])/(a - b)])/(Sqrt[2]*Sqrt[b])])/(8*Sqrt[2]*Sqrt[a - b]*Sqrt[-(b/(a - b)) - (b*Sin[c + d*x])/(a - b)]*(1 + ((a - b)*(-(b/(a - b)) - (b*Sin[c + d*x])/(a - b)))/(2*b))^(7/2)) + (15/(8*(1 + ((a - b)*(-(b/(a - b)) - (b*Sin[c + d*x])/(a - b)))/(2*b))^3) + 5/(4*(1 + ((a - b)*(-(b/(a - b)) - (b*Sin[c + d*x])/(a - b)))/(2*b))^2) + (1 + ((a - b)*(-(b/(a - b)) - (b*Sin[c + d*x])/(a - b)))/(2*b))^(-1))/6))/((a + b)^2*Sqrt[((a + b)*(b/(a + b) - (b*Sin[c + d*x])/(a + b)))/b]) - ((-((a*b)/(a - b)) + b^2/(a - b))*(-(((-((a*b)/(a + b)) - b^2/(a + b))*(-(((-((a*b)/(a + b)) - b^2/(a + b))*((2*Sqrt[a - b]*ArcTanh[(Sqrt[a - b]*Sqrt[-(b/(a - b)) - (b*Sin[c + d*x])/(a - b)])/(Sqrt[a + b]*Sqrt[b/(a + b) - (b*Sin[c + d*x])/(a + b)])])/(b*Sqrt[a + b]) - (2*Sqrt[-((a*b)/(a + b)) - b^2/(a + b)]*ArcTanh[(Sqrt[-((a*b)/(a + b)) - b^2/(a + b)]*Sqrt[-(b/(a - b)) - (b*Sin[c + d*x])/(a - b)])/(Sqrt[-((a*b)/(a - b)) + b^2/(a - b)]*Sqrt[b/(a + b) - (b*Sin[c + d*x])/(a + b)])])/(b*Sqrt[-((a*b)/(a - b)) + b^2/(a - b)])))/b) + (2*Sqrt[2]*(a - b)*Sqrt[-(b/(a - b)) - (b*Sin[c + d*x])/(a - b)]*Sqrt[b/(a + b) - (b*Sin[c + d*x])/(a + b)]*(1 + ((a - b)*(-(b/(a - b)) - (b*Sin[c + d*x])/(a - b)))/(2*b))^(3/2)*((Sqrt[b]*ArcSinh[(Sqrt[a - b]*Sqrt[-(b/(a - b)) - (b*Sin[c + d*x])/(a - b)])/(Sqrt[2]*Sqrt[b])])/(Sqrt[2]*Sqrt[a - b]*Sqrt[-(b/(a - b)) - (b*Sin[c + d*x])/(a - b)]*(1 + ((a - b)*(-(b/(a - b)) - (b*Sin[c + d*x])/(a - b)))/(2*b))^(3/2)) + 1/(2*(1 + ((a - b)*(-(b/(a - b)) - (b*Sin[c + d*x])/(a - b)))/(2*b)))))/(b*(a + b)*Sqrt[((a + b)*(b/(a + b) - (b*Sin[c + d*x])/(a + b)))/b])))/b) + (4*Sqrt[2]*(a - b)*Sqrt[-(b/(a - b)) - (b*Sin[c + d*x])/(a - b)]*Sqrt[b/(a + b) - (b*Sin[c + d*x])/(a + b)]*(1 + ((a - b)*(-(b/(a - b)) - (b*Sin[c + d*x])/(a - b)))/(2*b))^(5/2)*((3*Sqrt[b]*ArcSinh[(Sqrt[a - b]*Sqrt[-(b/(a - b)) - (b*Sin[c + d*x])/(a - b)])/(Sqrt[2]*Sqrt[b])])/(4*Sqrt[2]*Sqrt[a - b]*Sqrt[-(b/(a - b)) - (b*Sin[c + d*x])/(a - b)]*(1 + ((a - b)*(-(b/(a - b)) - (b*Sin[c + d*x])/(a - b)))/(2*b))^(5/2)) + (3/(2*(1 + ((a - b)*(-(b/(a - b)) - (b*Sin[c + d*x])/(a - b)))/(2*b))^2) + (1 + ((a - b)*(-(b/(a - b)) - (b*Sin[c + d*x])/(a - b)))/(2*b))^(-1))/4))/((a + b)^2*Sqrt[((a + b)*(b/(a + b) - (b*Sin[c + d*x])/(a + b)))/b])))/b))/b))/b))/b)/(((a*b)/(a - b) - b^2/(a - b))*((a*b)/(a + b) + b^2/(a + b))))/(2*((a*b)/(a - b) - b^2/(a - b))*((a*b)/(a + b) + b^2/(a + b)))))/(d*(1 - (a + b*Sin[c + d*x])/(a - b))^(5/2)*(1 - (a + b*Sin[c + d*x])/(a + b))^(5/2))","B",1
457,1,2641,139,6.2752734,"\int \frac{\cos ^4(c+d x)}{(a+b \sin (c+d x))^3} \, dx","Integrate[Cos[c + d*x]^4/(a + b*Sin[c + d*x])^3,x]","\text{Result too large to show}","\frac{3 \left(2 a^2-b^2\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{b^4 d \sqrt{a^2-b^2}}-\frac{3 a x}{b^4}-\frac{3 \cos (c+d x) (2 a+b \sin (c+d x))}{2 b^3 d (a+b \sin (c+d x))}-\frac{\cos ^3(c+d x)}{2 b d (a+b \sin (c+d x))^2}",1,"(Cos[c + d*x]^3*(-1/2*(b*(-(b/(a - b)) - (b*Sin[c + d*x])/(a - b))^(5/2)*(b/(a + b) - (b*Sin[c + d*x])/(a + b))^(5/2))/(((a*b)/(a - b) - b^2/(a - b))*((a*b)/(a + b) + b^2/(a + b))*(a + b*Sin[c + d*x])^2) - (-((a*b^3*(-(b/(a - b)) - (b*Sin[c + d*x])/(a - b))^(5/2)*(b/(a + b) - (b*Sin[c + d*x])/(a + b))^(5/2))/((a^2 - b^2)*((a*b)/(a - b) - b^2/(a - b))*((a*b)/(a + b) + b^2/(a + b))*(a + b*Sin[c + d*x]))) - ((16*Sqrt[2]*a*b^4*(-(b/(a - b)) - (b*Sin[c + d*x])/(a - b))^(5/2)*Sqrt[b/(a + b) - (b*Sin[c + d*x])/(a + b)]*(1 + ((a - b)*(-(b/(a - b)) - (b*Sin[c + d*x])/(a - b)))/(2*b))^(5/2)*((5*(1/(2*(1 + ((a - b)*(-(b/(a - b)) - (b*Sin[c + d*x])/(a - b)))/(2*b))^2) + (1 + ((a - b)*(-(b/(a - b)) - (b*Sin[c + d*x])/(a - b)))/(2*b))^(-1)))/8 - (15*b^3*(((a - b)*(-(b/(a - b)) - (b*Sin[c + d*x])/(a - b)))/b - ((a - b)^2*(-(b/(a - b)) - (b*Sin[c + d*x])/(a - b))^2)/(3*b^2) - (Sqrt[2]*Sqrt[a - b]*ArcSinh[(Sqrt[a - b]*Sqrt[-(b/(a - b)) - (b*Sin[c + d*x])/(a - b)])/(Sqrt[2]*Sqrt[b])]*Sqrt[-(b/(a - b)) - (b*Sin[c + d*x])/(a - b)])/(Sqrt[b]*Sqrt[1 + ((a - b)*(-(b/(a - b)) - (b*Sin[c + d*x])/(a - b)))/(2*b)])))/(32*(a - b)^3*(-(b/(a - b)) - (b*Sin[c + d*x])/(a - b))^3*(1 + ((a - b)*(-(b/(a - b)) - (b*Sin[c + d*x])/(a - b)))/(2*b))^2)))/(5*(a - b)*(a + b)^3*Sqrt[((a + b)*(b/(a + b) - (b*Sin[c + d*x])/(a + b)))/b]) + (((4*a^2*b^5)/((a - b)^2*(a + b)^2) + (b^5*(2*a^2 - 3*b^2))/((a - b)^2*(a + b)^2))*((4*Sqrt[2]*(-(b/(a - b)) - (b*Sin[c + d*x])/(a - b))^(3/2)*Sqrt[b/(a + b) - (b*Sin[c + d*x])/(a + b)]*(1 + ((a - b)*(-(b/(a - b)) - (b*Sin[c + d*x])/(a - b)))/(2*b))^(5/2)*((3/(4*(1 + ((a - b)*(-(b/(a - b)) - (b*Sin[c + d*x])/(a - b)))/(2*b))^2) + (1 + ((a - b)*(-(b/(a - b)) - (b*Sin[c + d*x])/(a - b)))/(2*b))^(-1))/2 + (3*b^2*(((a - b)*(-(b/(a - b)) - (b*Sin[c + d*x])/(a - b)))/b - (Sqrt[2]*Sqrt[a - b]*ArcSinh[(Sqrt[a - b]*Sqrt[-(b/(a - b)) - (b*Sin[c + d*x])/(a - b)])/(Sqrt[2]*Sqrt[b])]*Sqrt[-(b/(a - b)) - (b*Sin[c + d*x])/(a - b)])/(Sqrt[b]*Sqrt[1 + ((a - b)*(-(b/(a - b)) - (b*Sin[c + d*x])/(a - b)))/(2*b)])))/(8*(a - b)^2*(-(b/(a - b)) - (b*Sin[c + d*x])/(a - b))^2*(1 + ((a - b)*(-(b/(a - b)) - (b*Sin[c + d*x])/(a - b)))/(2*b))^2)))/(3*(a + b)*Sqrt[((a + b)*(b/(a + b) - (b*Sin[c + d*x])/(a + b)))/b]) - ((-((a*b)/(a - b)) + b^2/(a - b))*(-(((-((a*b)/(a - b)) + b^2/(a - b))*(-(((-((a*b)/(a + b)) - b^2/(a + b))*((2*Sqrt[a - b]*ArcTanh[(Sqrt[a - b]*Sqrt[-(b/(a - b)) - (b*Sin[c + d*x])/(a - b)])/(Sqrt[a + b]*Sqrt[b/(a + b) - (b*Sin[c + d*x])/(a + b)])])/(b*Sqrt[a + b]) - (2*Sqrt[-((a*b)/(a + b)) - b^2/(a + b)]*ArcTanh[(Sqrt[-((a*b)/(a + b)) - b^2/(a + b)]*Sqrt[-(b/(a - b)) - (b*Sin[c + d*x])/(a - b)])/(Sqrt[-((a*b)/(a - b)) + b^2/(a - b)]*Sqrt[b/(a + b) - (b*Sin[c + d*x])/(a + b)])])/(b*Sqrt[-((a*b)/(a - b)) + b^2/(a - b)])))/b) + (2*Sqrt[2]*(a - b)*Sqrt[-(b/(a - b)) - (b*Sin[c + d*x])/(a - b)]*Sqrt[b/(a + b) - (b*Sin[c + d*x])/(a + b)]*(1 + ((a - b)*(-(b/(a - b)) - (b*Sin[c + d*x])/(a - b)))/(2*b))^(3/2)*((Sqrt[b]*ArcSinh[(Sqrt[a - b]*Sqrt[-(b/(a - b)) - (b*Sin[c + d*x])/(a - b)])/(Sqrt[2]*Sqrt[b])])/(Sqrt[2]*Sqrt[a - b]*Sqrt[-(b/(a - b)) - (b*Sin[c + d*x])/(a - b)]*(1 + ((a - b)*(-(b/(a - b)) - (b*Sin[c + d*x])/(a - b)))/(2*b))^(3/2)) + 1/(2*(1 + ((a - b)*(-(b/(a - b)) - (b*Sin[c + d*x])/(a - b)))/(2*b)))))/(b*(a + b)*Sqrt[((a + b)*(b/(a + b) - (b*Sin[c + d*x])/(a + b)))/b])))/b) + (4*Sqrt[2]*Sqrt[-(b/(a - b)) - (b*Sin[c + d*x])/(a - b)]*Sqrt[b/(a + b) - (b*Sin[c + d*x])/(a + b)]*(1 + ((a - b)*(-(b/(a - b)) - (b*Sin[c + d*x])/(a - b)))/(2*b))^(5/2)*((3*Sqrt[b]*ArcSinh[(Sqrt[a - b]*Sqrt[-(b/(a - b)) - (b*Sin[c + d*x])/(a - b)])/(Sqrt[2]*Sqrt[b])])/(4*Sqrt[2]*Sqrt[a - b]*Sqrt[-(b/(a - b)) - (b*Sin[c + d*x])/(a - b)]*(1 + ((a - b)*(-(b/(a - b)) - (b*Sin[c + d*x])/(a - b)))/(2*b))^(5/2)) + (3/(2*(1 + ((a - b)*(-(b/(a - b)) - (b*Sin[c + d*x])/(a - b)))/(2*b))^2) + (1 + ((a - b)*(-(b/(a - b)) - (b*Sin[c + d*x])/(a - b)))/(2*b))^(-1))/4))/((a + b)*Sqrt[((a + b)*(b/(a + b) - (b*Sin[c + d*x])/(a + b)))/b])))/b))/b)/(((a*b)/(a - b) - b^2/(a - b))*((a*b)/(a + b) + b^2/(a + b))))/(2*((a*b)/(a - b) - b^2/(a - b))*((a*b)/(a + b) + b^2/(a + b)))))/(d*(1 - (a + b*Sin[c + d*x])/(a - b))^(3/2)*(1 - (a + b*Sin[c + d*x])/(a + b))^(3/2))","B",1
458,1,93,115,0.2541001,"\int \frac{\cos ^2(c+d x)}{(a+b \sin (c+d x))^3} \, dx","Integrate[Cos[c + d*x]^2/(a + b*Sin[c + d*x])^3,x]","\frac{\frac{2 \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{\sqrt{a^2-b^2}}+\frac{\cos (c+d x) (a \sin (c+d x)+b)}{(a+b \sin (c+d x))^2}}{2 d (a-b) (a+b)}","\frac{\tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{d \left(a^2-b^2\right)^{3/2}}+\frac{a \cos (c+d x)}{2 b d \left(a^2-b^2\right) (a+b \sin (c+d x))}-\frac{\cos (c+d x)}{2 b d (a+b \sin (c+d x))^2}",1,"((2*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/Sqrt[a^2 - b^2] + (Cos[c + d*x]*(b + a*Sin[c + d*x]))/(a + b*Sin[c + d*x])^2)/(2*(a - b)*(a + b)*d)","A",1
459,1,193,192,3.0602927,"\int \frac{\sec ^2(c+d x)}{(a+b \sin (c+d x))^3} \, dx","Integrate[Sec[c + d*x]^2/(a + b*Sin[c + d*x])^3,x]","\frac{-\frac{6 b^2 \left(4 a^2+b^2\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{\left(a^2-b^2\right)^{7/2}}+\frac{b^3 \cos (c+d x) \left(-8 a^2-7 a b \sin (c+d x)+b^2\right)}{(a-b)^3 (a+b)^3 (a+b \sin (c+d x))^2}+\sin \left(\frac{1}{2} (c+d x)\right) \left(\frac{2}{(a-b)^3 \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}+\frac{2}{(a+b)^3 \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}\right)}{2 d}","-\frac{3 b^2 \left(4 a^2+b^2\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{d \left(a^2-b^2\right)^{7/2}}+\frac{5 a b \sec (c+d x)}{2 d \left(a^2-b^2\right)^2 (a+b \sin (c+d x))}+\frac{b \sec (c+d x)}{2 d \left(a^2-b^2\right) (a+b \sin (c+d x))^2}-\frac{\sec (c+d x) \left(3 b \left(4 a^2+b^2\right)-a \left(2 a^2+13 b^2\right) \sin (c+d x)\right)}{2 d \left(a^2-b^2\right)^3}",1,"((-6*b^2*(4*a^2 + b^2)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(7/2) + Sin[(c + d*x)/2]*(2/((a + b)^3*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])) + 2/((a - b)^3*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))) + (b^3*Cos[c + d*x]*(-8*a^2 + b^2 - 7*a*b*Sin[c + d*x]))/((a - b)^3*(a + b)^3*(a + b*Sin[c + d*x])^2))/(2*d)","A",1
460,1,380,264,2.808512,"\int \frac{\sec ^4(c+d x)}{(a+b \sin (c+d x))^3} \, dx","Integrate[Sec[c + d*x]^4/(a + b*Sin[c + d*x])^3,x]","\frac{\frac{60 b^4 \left(6 a^2+b^2\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{\left(a^2-b^2\right)^{9/2}}+\frac{66 a b^5 \cos (c+d x)}{(a-b)^4 (a+b)^4 (a+b \sin (c+d x))}+\frac{6 b^5 \cos (c+d x)}{(a-b)^3 (a+b)^3 (a+b \sin (c+d x))^2}+\frac{2 (4 a+13 b) \sin \left(\frac{1}{2} (c+d x)\right)}{(a+b)^4 \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}+\frac{2 \sin \left(\frac{1}{2} (c+d x)\right)}{(a+b)^3 \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^3}+\frac{2 (4 a-13 b) \sin \left(\frac{1}{2} (c+d x)\right)}{(a-b)^4 \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}+\frac{1}{(a+b)^3 \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^2}-\frac{1}{(a-b)^3 \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2}+\frac{2 \sin \left(\frac{1}{2} (c+d x)\right)}{(a-b)^3 \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^3}}{12 d}","\frac{7 a b \sec ^3(c+d x)}{2 d \left(a^2-b^2\right)^2 (a+b \sin (c+d x))}+\frac{b \sec ^3(c+d x)}{2 d \left(a^2-b^2\right) (a+b \sin (c+d x))^2}-\frac{\sec ^3(c+d x) \left(5 b \left(6 a^2+b^2\right)-a \left(2 a^2+33 b^2\right) \sin (c+d x)\right)}{6 d \left(a^2-b^2\right)^3}+\frac{5 b^4 \left(6 a^2+b^2\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{d \left(a^2-b^2\right)^{9/2}}+\frac{\sec (c+d x) \left(15 b^3 \left(6 a^2+b^2\right)+a \left(4 a^4-28 a^2 b^2-81 b^4\right) \sin (c+d x)\right)}{6 d \left(a^2-b^2\right)^4}",1,"((60*b^4*(6*a^2 + b^2)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(9/2) + 1/((a + b)^3*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2) + (2*Sin[(c + d*x)/2])/((a + b)^3*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^3) + (2*(4*a + 13*b)*Sin[(c + d*x)/2])/((a + b)^4*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])) + (2*Sin[(c + d*x)/2])/((a - b)^3*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3) - 1/((a - b)^3*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2) + (2*(4*a - 13*b)*Sin[(c + d*x)/2])/((a - b)^4*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])) + (6*b^5*Cos[c + d*x])/((a - b)^3*(a + b)^3*(a + b*Sin[c + d*x])^2) + (66*a*b^5*Cos[c + d*x])/((a - b)^4*(a + b)^4*(a + b*Sin[c + d*x])))/(12*d)","A",1
461,1,171,207,1.1438226,"\int \frac{\cos ^7(c+d x)}{(a+b \sin (c+d x))^8} \, dx","Integrate[Cos[c + d*x]^7/(a + b*Sin[c + d*x])^8,x]","\frac{\frac{\left(a^2-b^2\right)^3}{7 (a+b \sin (c+d x))^7}-\frac{a \left(a^2-b^2\right)^2}{(a+b \sin (c+d x))^6}+\frac{5 a^2-b^2}{(a+b \sin (c+d x))^3}-\frac{a \left(5 a^2-3 b^2\right)}{(a+b \sin (c+d x))^4}+\frac{3 \left(5 a^4-6 a^2 b^2+b^4\right)}{5 (a+b \sin (c+d x))^5}+\frac{1}{a+b \sin (c+d x)}-\frac{3 a}{(a+b \sin (c+d x))^2}}{b^7 d}","\frac{\left(a^2-b^2\right)^3}{7 b^7 d (a+b \sin (c+d x))^7}-\frac{a \left(a^2-b^2\right)^2}{b^7 d (a+b \sin (c+d x))^6}+\frac{5 a^2-b^2}{b^7 d (a+b \sin (c+d x))^3}-\frac{a \left(5 a^2-3 b^2\right)}{b^7 d (a+b \sin (c+d x))^4}+\frac{3 \left(5 a^4-6 a^2 b^2+b^4\right)}{5 b^7 d (a+b \sin (c+d x))^5}+\frac{1}{b^7 d (a+b \sin (c+d x))}-\frac{3 a}{b^7 d (a+b \sin (c+d x))^2}",1,"((a^2 - b^2)^3/(7*(a + b*Sin[c + d*x])^7) - (a*(a^2 - b^2)^2)/(a + b*Sin[c + d*x])^6 + (3*(5*a^4 - 6*a^2*b^2 + b^4))/(5*(a + b*Sin[c + d*x])^5) - (a*(5*a^2 - 3*b^2))/(a + b*Sin[c + d*x])^4 + (5*a^2 - b^2)/(a + b*Sin[c + d*x])^3 - (3*a)/(a + b*Sin[c + d*x])^2 + (a + b*Sin[c + d*x])^(-1))/(b^7*d)","A",1
462,1,107,141,0.2722926,"\int \frac{\cos ^5(c+d x)}{(a+b \sin (c+d x))^8} \, dx","Integrate[Cos[c + d*x]^5/(a + b*Sin[c + d*x])^8,x]","-\frac{a^4+21 b^2 \left(a^2-2 b^2\right) \sin ^2(c+d x)+7 a b \left(a^2-2 b^2\right) \sin (c+d x)-2 a^2 b^2+35 a b^3 \sin ^3(c+d x)+35 b^4 \sin ^4(c+d x)+15 b^4}{105 b^5 d (a+b \sin (c+d x))^7}","-\frac{\left(a^2-b^2\right)^2}{7 b^5 d (a+b \sin (c+d x))^7}+\frac{2 a \left(a^2-b^2\right)}{3 b^5 d (a+b \sin (c+d x))^6}-\frac{2 \left(3 a^2-b^2\right)}{5 b^5 d (a+b \sin (c+d x))^5}-\frac{1}{3 b^5 d (a+b \sin (c+d x))^3}+\frac{a}{b^5 d (a+b \sin (c+d x))^4}",1,"-1/105*(a^4 - 2*a^2*b^2 + 15*b^4 + 7*a*b*(a^2 - 2*b^2)*Sin[c + d*x] + 21*b^2*(a^2 - 2*b^2)*Sin[c + d*x]^2 + 35*a*b^3*Sin[c + d*x]^3 + 35*b^4*Sin[c + d*x]^4)/(b^5*d*(a + b*Sin[c + d*x])^7)","A",1
463,1,54,77,0.1904429,"\int \frac{\cos ^3(c+d x)}{(a+b \sin (c+d x))^8} \, dx","Integrate[Cos[c + d*x]^3/(a + b*Sin[c + d*x])^8,x]","\frac{a^2+7 a b \sin (c+d x)+21 b^2 \sin ^2(c+d x)-15 b^2}{105 b^3 d (a+b \sin (c+d x))^7}","\frac{a^2-b^2}{7 b^3 d (a+b \sin (c+d x))^7}-\frac{a}{3 b^3 d (a+b \sin (c+d x))^6}+\frac{1}{5 b^3 d (a+b \sin (c+d x))^5}",1,"(a^2 - 15*b^2 + 7*a*b*Sin[c + d*x] + 21*b^2*Sin[c + d*x]^2)/(105*b^3*d*(a + b*Sin[c + d*x])^7)","A",1
464,1,22,22,0.0802375,"\int \frac{\cos (c+d x)}{(a+b \sin (c+d x))^8} \, dx","Integrate[Cos[c + d*x]/(a + b*Sin[c + d*x])^8,x]","-\frac{1}{7 b d (a+b \sin (c+d x))^7}","-\frac{1}{7 b d (a+b \sin (c+d x))^7}",1,"-1/7*1/(b*d*(a + b*Sin[c + d*x])^7)","A",1
465,1,365,385,2.6877794,"\int \frac{\sec (c+d x)}{(a+b \sin (c+d x))^8} \, dx","Integrate[Sec[c + d*x]/(a + b*Sin[c + d*x])^8,x]","\frac{b \left(\frac{a \left(3 a^2+b^2\right) \left(a^2+3 b^2\right)}{(a-b)^6 (a+b)^6 (a+b \sin (c+d x))^2}+\frac{a \left(a^2+b^2\right)}{(a-b)^4 (a+b)^4 (a+b \sin (c+d x))^4}+\frac{3 a^2+b^2}{5 (a-b)^3 (a+b)^3 (a+b \sin (c+d x))^5}+\frac{1}{7 \left(a^2-b^2\right) (a+b \sin (c+d x))^7}+\frac{5 a^4+10 a^2 b^2+b^4}{3 (a-b)^5 (a+b)^5 (a+b \sin (c+d x))^3}-\frac{8 a \left(a^2+b^2\right) \left(a^4+6 a^2 b^2+b^4\right) \log (a+b \sin (c+d x))}{(a-b)^8 (a+b)^8}+\frac{7 a^6+35 a^4 b^2+21 a^2 b^4+b^6}{(a-b)^7 (a+b)^7 (a+b \sin (c+d x))}+\frac{a}{3 (a-b)^2 (a+b)^2 (a+b \sin (c+d x))^6}-\frac{\log (1-\sin (c+d x))}{2 b (a+b)^8}+\frac{\log (\sin (c+d x)+1)}{2 b (a-b)^8}\right)}{d}","\frac{a b \left(3 a^2+b^2\right) \left(a^2+3 b^2\right)}{d \left(a^2-b^2\right)^6 (a+b \sin (c+d x))^2}+\frac{a b \left(a^2+b^2\right)}{d \left(a^2-b^2\right)^4 (a+b \sin (c+d x))^4}+\frac{b \left(3 a^2+b^2\right)}{5 d \left(a^2-b^2\right)^3 (a+b \sin (c+d x))^5}+\frac{a b}{3 d \left(a^2-b^2\right)^2 (a+b \sin (c+d x))^6}+\frac{b}{7 d \left(a^2-b^2\right) (a+b \sin (c+d x))^7}+\frac{b \left(5 a^4+10 a^2 b^2+b^4\right)}{3 d \left(a^2-b^2\right)^5 (a+b \sin (c+d x))^3}-\frac{8 a b \left(a^2+b^2\right) \left(a^4+6 a^2 b^2+b^4\right) \log (a+b \sin (c+d x))}{d \left(a^2-b^2\right)^8}+\frac{b \left(7 a^6+35 a^4 b^2+21 a^2 b^4+b^6\right)}{d \left(a^2-b^2\right)^7 (a+b \sin (c+d x))}-\frac{\log (1-\sin (c+d x))}{2 d (a+b)^8}+\frac{\log (\sin (c+d x)+1)}{2 d (a-b)^8}",1,"(b*(-1/2*Log[1 - Sin[c + d*x]]/(b*(a + b)^8) + Log[1 + Sin[c + d*x]]/(2*(a - b)^8*b) - (8*a*(a^2 + b^2)*(a^4 + 6*a^2*b^2 + b^4)*Log[a + b*Sin[c + d*x]])/((a - b)^8*(a + b)^8) + 1/(7*(a^2 - b^2)*(a + b*Sin[c + d*x])^7) + a/(3*(a - b)^2*(a + b)^2*(a + b*Sin[c + d*x])^6) + (3*a^2 + b^2)/(5*(a - b)^3*(a + b)^3*(a + b*Sin[c + d*x])^5) + (a*(a^2 + b^2))/((a - b)^4*(a + b)^4*(a + b*Sin[c + d*x])^4) + (5*a^4 + 10*a^2*b^2 + b^4)/(3*(a - b)^5*(a + b)^5*(a + b*Sin[c + d*x])^3) + (a*(3*a^2 + b^2)*(a^2 + 3*b^2))/((a - b)^6*(a + b)^6*(a + b*Sin[c + d*x])^2) + (7*a^6 + 35*a^4*b^2 + 21*a^2*b^4 + b^6)/((a - b)^7*(a + b)^7*(a + b*Sin[c + d*x]))))/d","A",1
466,1,770,527,6.7481642,"\int \frac{\sec ^3(c+d x)}{(a+b \sin (c+d x))^8} \, dx","Integrate[Sec[c + d*x]^3/(a + b*Sin[c + d*x])^8,x]","\frac{b^3 \left(\frac{\sec ^2(c+d x) \left(b^2-a b \sin (c+d x)\right)}{2 b^4 \left(b^2-a^2\right) (a+b \sin (c+d x))^7}-\frac{8 a \left(\frac{2 a \left(3 a^2+b^2\right) \left(a^2+3 b^2\right)}{(a-b)^6 (a+b)^6 (a+b \sin (c+d x))}+\frac{4 a \left(a^2+b^2\right)}{3 (a-b)^4 (a+b)^4 (a+b \sin (c+d x))^3}+\frac{3 a^2+b^2}{4 (a-b)^3 (a+b)^3 (a+b \sin (c+d x))^4}+\frac{1}{6 \left(a^2-b^2\right) (a+b \sin (c+d x))^6}+\frac{5 a^4+10 a^2 b^2+b^4}{2 (a-b)^5 (a+b)^5 (a+b \sin (c+d x))^2}-\frac{\left(7 a^6+35 a^4 b^2+21 a^2 b^4+b^6\right) \log (a+b \sin (c+d x))}{(a-b)^7 (a+b)^7}+\frac{2 a}{5 (a-b)^2 (a+b)^2 (a+b \sin (c+d x))^5}-\frac{\log (1-\sin (c+d x))}{2 b (a+b)^7}+\frac{\log (\sin (c+d x)+1)}{2 b (a-b)^7}\right)+\left(-7 a^2-9 b^2\right) \left(\frac{a \left(3 a^2+b^2\right) \left(a^2+3 b^2\right)}{(a-b)^6 (a+b)^6 (a+b \sin (c+d x))^2}+\frac{a \left(a^2+b^2\right)}{(a-b)^4 (a+b)^4 (a+b \sin (c+d x))^4}+\frac{3 a^2+b^2}{5 (a-b)^3 (a+b)^3 (a+b \sin (c+d x))^5}+\frac{1}{7 \left(a^2-b^2\right) (a+b \sin (c+d x))^7}+\frac{5 a^4+10 a^2 b^2+b^4}{3 (a-b)^5 (a+b)^5 (a+b \sin (c+d x))^3}-\frac{8 a \left(a^2+b^2\right) \left(a^4+6 a^2 b^2+b^4\right) \log (a+b \sin (c+d x))}{(a-b)^8 (a+b)^8}+\frac{7 a^6+35 a^4 b^2+21 a^2 b^4+b^6}{(a-b)^7 (a+b)^7 (a+b \sin (c+d x))}+\frac{a}{3 (a-b)^2 (a+b)^2 (a+b \sin (c+d x))^6}-\frac{\log (1-\sin (c+d x))}{2 b (a+b)^8}+\frac{\log (\sin (c+d x)+1)}{2 b (a-b)^8}\right)}{2 b^2 \left(b^2-a^2\right)}\right)}{d}","-\frac{a b \left(3 a^2+13 b^2\right)}{6 d \left(a^2-b^2\right)^3 (a+b \sin (c+d x))^6}-\frac{b \left(7 a^2+9 b^2\right)}{14 d \left(a^2-b^2\right)^2 (a+b \sin (c+d x))^7}-\frac{\sec ^2(c+d x) (b-a \sin (c+d x))}{2 d \left(a^2-b^2\right) (a+b \sin (c+d x))^7}-\frac{a b \left(a^4+20 a^2 b^2+11 b^4\right)}{2 d \left(a^2-b^2\right)^5 (a+b \sin (c+d x))^4}-\frac{b \left(5 a^4+50 a^2 b^2+9 b^4\right)}{10 d \left(a^2-b^2\right)^4 (a+b \sin (c+d x))^5}-\frac{a b \left(a^6+77 a^4 b^2+147 a^2 b^4+31 b^6\right)}{2 d \left(a^2-b^2\right)^7 (a+b \sin (c+d x))^2}-\frac{b \left(3 a^6+115 a^4 b^2+129 a^2 b^4+9 b^6\right)}{6 d \left(a^2-b^2\right)^6 (a+b \sin (c+d x))^3}+\frac{8 a b^3 \left(15 a^6+63 a^4 b^2+45 a^2 b^4+5 b^6\right) \log (a+b \sin (c+d x))}{d \left(a^2-b^2\right)^9}-\frac{b \left(a^8+196 a^6 b^2+574 a^4 b^4+244 a^2 b^6+9 b^8\right)}{2 d \left(a^2-b^2\right)^8 (a+b \sin (c+d x))}-\frac{(a+9 b) \log (1-\sin (c+d x))}{4 d (a+b)^9}+\frac{(a-9 b) \log (\sin (c+d x)+1)}{4 d (a-b)^9}",1,"(b^3*((Sec[c + d*x]^2*(b^2 - a*b*Sin[c + d*x]))/(2*b^4*(-a^2 + b^2)*(a + b*Sin[c + d*x])^7) - (8*a*(-1/2*Log[1 - Sin[c + d*x]]/(b*(a + b)^7) + Log[1 + Sin[c + d*x]]/(2*(a - b)^7*b) - ((7*a^6 + 35*a^4*b^2 + 21*a^2*b^4 + b^6)*Log[a + b*Sin[c + d*x]])/((a - b)^7*(a + b)^7) + 1/(6*(a^2 - b^2)*(a + b*Sin[c + d*x])^6) + (2*a)/(5*(a - b)^2*(a + b)^2*(a + b*Sin[c + d*x])^5) + (3*a^2 + b^2)/(4*(a - b)^3*(a + b)^3*(a + b*Sin[c + d*x])^4) + (4*a*(a^2 + b^2))/(3*(a - b)^4*(a + b)^4*(a + b*Sin[c + d*x])^3) + (5*a^4 + 10*a^2*b^2 + b^4)/(2*(a - b)^5*(a + b)^5*(a + b*Sin[c + d*x])^2) + (2*a*(3*a^2 + b^2)*(a^2 + 3*b^2))/((a - b)^6*(a + b)^6*(a + b*Sin[c + d*x]))) + (-7*a^2 - 9*b^2)*(-1/2*Log[1 - Sin[c + d*x]]/(b*(a + b)^8) + Log[1 + Sin[c + d*x]]/(2*(a - b)^8*b) - (8*a*(a^2 + b^2)*(a^4 + 6*a^2*b^2 + b^4)*Log[a + b*Sin[c + d*x]])/((a - b)^8*(a + b)^8) + 1/(7*(a^2 - b^2)*(a + b*Sin[c + d*x])^7) + a/(3*(a - b)^2*(a + b)^2*(a + b*Sin[c + d*x])^6) + (3*a^2 + b^2)/(5*(a - b)^3*(a + b)^3*(a + b*Sin[c + d*x])^5) + (a*(a^2 + b^2))/((a - b)^4*(a + b)^4*(a + b*Sin[c + d*x])^4) + (5*a^4 + 10*a^2*b^2 + b^4)/(3*(a - b)^5*(a + b)^5*(a + b*Sin[c + d*x])^3) + (a*(3*a^2 + b^2)*(a^2 + 3*b^2))/((a - b)^6*(a + b)^6*(a + b*Sin[c + d*x])^2) + (7*a^6 + 35*a^4*b^2 + 21*a^2*b^4 + b^6)/((a - b)^7*(a + b)^7*(a + b*Sin[c + d*x]))))/(2*b^2*(-a^2 + b^2))))/d","A",1
467,1,6570,491,8.5047993,"\int \frac{\cos ^8(c+d x)}{(a+b \sin (c+d x))^8} \, dx","Integrate[Cos[c + d*x]^8/(a + b*Sin[c + d*x])^8,x]","\text{Result too large to show}","\frac{a \cos ^7(c+d x)}{6 b d \left(a^2-b^2\right) (a+b \sin (c+d x))^6}-\frac{\cos ^3(c+d x) \left(a b \left(6 a^2-11 b^2\right) \sin (c+d x)+8 \left(a^2-b^2\right)^2\right)}{24 b^5 d \left(a^2-b^2\right)^2 (a+b \sin (c+d x))^3}+\frac{\cos ^5(c+d x) \left(6 \left(a^2-b^2\right)+5 a b \sin (c+d x)\right)}{30 b^3 d \left(a^2-b^2\right) (a+b \sin (c+d x))^5}-\frac{a \left(6 a^2-11 b^2\right) \cos ^5(c+d x)}{24 b^3 d \left(a^2-b^2\right)^2 (a+b \sin (c+d x))^4}+\frac{\cos (c+d x) \left(16 \left(a^2-b^2\right)^3+a b \left(8 a^4-22 a^2 b^2+19 b^4\right) \sin (c+d x)\right)}{16 b^7 d \left(a^2-b^2\right)^3 (a+b \sin (c+d x))}+\frac{a \left(8 a^4-22 a^2 b^2+19 b^4\right) \cos ^3(c+d x)}{16 b^5 d \left(a^2-b^2\right)^3 (a+b \sin (c+d x))^2}-\frac{a \left(16 a^6-56 a^4 b^2+70 a^2 b^4-35 b^6\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{8 b^8 d \left(a^2-b^2\right)^{7/2}}-\frac{\cos ^7(c+d x)}{7 b d (a+b \sin (c+d x))^7}+\frac{x}{b^8}",1,"Result too large to show","B",0
468,1,386,407,6.0008653,"\int \frac{\cos ^6(c+d x)}{(a+b \sin (c+d x))^8} \, dx","Integrate[Cos[c + d*x]^6/(a + b*Sin[c + d*x])^8,x]","\frac{\cos (c+d x) \left(\frac{35 a (\sin (c+d x)+1)}{8 (b-a)^3 (a+b)^3 (a+b \sin (c+d x))^2}+\frac{7 a (\sin (c+d x)+1)^2}{4 (b-a)^3 (a+b)^2 (a+b \sin (c+d x))^3}+\frac{3 a (\sin (c+d x)+1)^3}{4 (b-a)^3 (a+b) (a+b \sin (c+d x))^4}+\frac{3 a (\sin (c+d x)+1)^4}{(a-b)^3 (a+b \sin (c+d x))^5}-\frac{5 a (\sin (c+d x)-1) (\sin (c+d x)+1)^4}{(a-b)^2 (a+b \sin (c+d x))^6}+\frac{6 a (\sin (c+d x)-1)^2 (\sin (c+d x)+1)^4}{(a-b) (a+b \sin (c+d x))^7}+\frac{6 \cos ^6(c+d x)}{(a+b \sin (c+d x))^7}+\frac{105}{8} a \left(\frac{2 \tanh ^{-1}\left(\frac{\sqrt{a-b} \sqrt{1-\sin (c+d x)}}{\sqrt{-a-b} \sqrt{\sin (c+d x)+1}}\right)}{(-a-b)^{9/2} (a-b)^{7/2} \sqrt{\cos ^2(c+d x)}}-\frac{1}{(a-b)^3 (a+b)^4 (a+b \sin (c+d x))}\right)\right)}{42 d (a-b)}","\frac{5 a \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{8 d \left(a^2-b^2\right)^{9/2}}-\frac{\cos (c+d x) \left(4 a^2+10 a b \sin (c+d x)+9 b^2\right)}{42 b^5 d (a+b \sin (c+d x))^5}+\frac{a \left(4 a^2-b^2\right) \cos (c+d x)}{168 b^5 d \left(a^2-b^2\right) (a+b \sin (c+d x))^4}+\frac{a \left(8 a^4-30 a^2 b^2+57 b^4\right) \cos (c+d x)}{336 b^5 d \left(a^2-b^2\right)^3 (a+b \sin (c+d x))^2}+\frac{\left(4 a^4-9 a^2 b^2+12 b^4\right) \cos (c+d x)}{168 b^5 d \left(a^2-b^2\right)^2 (a+b \sin (c+d x))^3}+\frac{\left(8 a^6-38 a^4 b^2+87 a^2 b^4+48 b^6\right) \cos (c+d x)}{336 b^5 d \left(a^2-b^2\right)^4 (a+b \sin (c+d x))}+\frac{5 \cos ^3(c+d x) (2 a+3 b \sin (c+d x))}{42 b^3 d (a+b \sin (c+d x))^6}-\frac{\cos ^5(c+d x)}{7 b d (a+b \sin (c+d x))^7}",1,"(Cos[c + d*x]*((6*Cos[c + d*x]^6)/(a + b*Sin[c + d*x])^7 + (6*a*(-1 + Sin[c + d*x])^2*(1 + Sin[c + d*x])^4)/((a - b)*(a + b*Sin[c + d*x])^7) - (5*a*(-1 + Sin[c + d*x])*(1 + Sin[c + d*x])^4)/((a - b)^2*(a + b*Sin[c + d*x])^6) + (3*a*(1 + Sin[c + d*x])^4)/((a - b)^3*(a + b*Sin[c + d*x])^5) + (3*a*(1 + Sin[c + d*x])^3)/(4*(-a + b)^3*(a + b)*(a + b*Sin[c + d*x])^4) + (7*a*(1 + Sin[c + d*x])^2)/(4*(-a + b)^3*(a + b)^2*(a + b*Sin[c + d*x])^3) + (35*a*(1 + Sin[c + d*x]))/(8*(-a + b)^3*(a + b)^3*(a + b*Sin[c + d*x])^2) + (105*a*((2*ArcTanh[(Sqrt[a - b]*Sqrt[1 - Sin[c + d*x]])/(Sqrt[-a - b]*Sqrt[1 + Sin[c + d*x]])])/((-a - b)^(9/2)*(a - b)^(7/2)*Sqrt[Cos[c + d*x]^2]) - 1/((a - b)^3*(a + b)^4*(a + b*Sin[c + d*x]))))/8))/(42*(a - b)*d)","A",1
469,1,1167,411,6.077375,"\int \frac{\cos ^4(c+d x)}{(a+b \sin (c+d x))^8} \, dx","Integrate[Cos[c + d*x]^4/(a + b*Sin[c + d*x])^8,x]","\frac{\cos ^5(c+d x)}{5 (a-b) d (a+b \sin (c+d x))^7}+\frac{a \left(-\frac{b (1-\sin (c+d x))^{5/2} (\sin (c+d x)+1)^{7/2}}{7 (b-a) (a+b) (a+b \sin (c+d x))^7}-\frac{-\frac{(a b+(7 a-b) b) (1-\sin (c+d x))^{5/2} (\sin (c+d x)+1)^{7/2}}{6 (b-a) (a+b) (a+b \sin (c+d x))^6}-\frac{7 \left(6 a^2-2 b a+b^2\right) \left(-\frac{(1-\sin (c+d x))^{3/2} (\sin (c+d x)+1)^{7/2}}{5 (b-a) (a+b \sin (c+d x))^5}-\frac{3 \left(-\frac{\sqrt{1-\sin (c+d x)} (\sin (c+d x)+1)^{7/2}}{4 (b-a) (a+b \sin (c+d x))^4}-\frac{\frac{5 \left(\frac{3 \left(\frac{\sqrt{1-\sin (c+d x)} \sqrt{\sin (c+d x)+1}}{(-a-b) (a+b \sin (c+d x))}-\frac{2 \tanh ^{-1}\left(\frac{\sqrt{a-b} \sqrt{1-\sin (c+d x)}}{\sqrt{-a-b} \sqrt{\sin (c+d x)+1}}\right)}{(-a-b)^{3/2} \sqrt{a-b}}\right)}{2 (a+b)}-\frac{\sqrt{1-\sin (c+d x)} (\sin (c+d x)+1)^{3/2}}{2 (a+b) (a+b \sin (c+d x))^2}\right)}{3 (a+b)}-\frac{\sqrt{1-\sin (c+d x)} (\sin (c+d x)+1)^{5/2}}{3 (a+b) (a+b \sin (c+d x))^3}}{4 (b-a)}\right)}{5 (b-a)}\right)}{6 (b-a) (a+b)}}{7 (b-a) (a+b)}\right) \cos (c+d x)}{(a-b) d \sqrt{1-\sin (c+d x)} \sqrt{\sin (c+d x)+1}}+\frac{2 b \left(\frac{\cos ^7(c+d x)}{7 (a-b) d (a+b \sin (c+d x))^7}+\frac{a \left(-\frac{(1-\sin (c+d x))^{5/2} (\sin (c+d x)+1)^{9/2}}{7 (b-a) (a+b \sin (c+d x))^7}-\frac{5 \left(-\frac{(1-\sin (c+d x))^{3/2} (\sin (c+d x)+1)^{9/2}}{6 (b-a) (a+b \sin (c+d x))^6}-\frac{-\frac{\sqrt{1-\sin (c+d x)} (\sin (c+d x)+1)^{9/2}}{5 (b-a) (a+b \sin (c+d x))^5}-\frac{\frac{7 \left(\frac{5 \left(\frac{3 \left(\frac{\sqrt{1-\sin (c+d x)} \sqrt{\sin (c+d x)+1}}{(-a-b) (a+b \sin (c+d x))}-\frac{2 \tanh ^{-1}\left(\frac{\sqrt{a-b} \sqrt{1-\sin (c+d x)}}{\sqrt{-a-b} \sqrt{\sin (c+d x)+1}}\right)}{(-a-b)^{3/2} \sqrt{a-b}}\right)}{2 (a+b)}-\frac{\sqrt{1-\sin (c+d x)} (\sin (c+d x)+1)^{3/2}}{2 (a+b) (a+b \sin (c+d x))^2}\right)}{3 (a+b)}-\frac{\sqrt{1-\sin (c+d x)} (\sin (c+d x)+1)^{5/2}}{3 (a+b) (a+b \sin (c+d x))^3}\right)}{4 (a+b)}-\frac{\sqrt{1-\sin (c+d x)} (\sin (c+d x)+1)^{7/2}}{4 (a+b) (a+b \sin (c+d x))^4}}{5 (b-a)}}{2 (b-a)}\right)}{7 (b-a)}\right) \cos (c+d x)}{(a-b) d \sqrt{1-\sin (c+d x)} \sqrt{\sin (c+d x)+1}}\right)}{5 (a-b)}","\frac{3 a \left(2 a^2+b^2\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{8 d \left(a^2-b^2\right)^{11/2}}-\frac{a \left(2 a^2-11 b^2\right) \cos (c+d x)}{280 b^3 d \left(a^2-b^2\right)^2 (a+b \sin (c+d x))^4}-\frac{\left(a^2-3 b^2\right) \cos (c+d x)}{140 b^3 d \left(a^2-b^2\right) (a+b \sin (c+d x))^5}-\frac{a \left(4 a^4-36 a^2 b^2-73 b^4\right) \cos (c+d x)}{560 b^3 d \left(a^2-b^2\right)^4 (a+b \sin (c+d x))^2}-\frac{\left(2 a^4-15 a^2 b^2-8 b^4\right) \cos (c+d x)}{280 b^3 d \left(a^2-b^2\right)^3 (a+b \sin (c+d x))^3}-\frac{\left(4 a^6-40 a^4 b^2-247 a^2 b^4-32 b^6\right) \cos (c+d x)}{560 b^3 d \left(a^2-b^2\right)^5 (a+b \sin (c+d x))}+\frac{\cos (c+d x) (a+3 b \sin (c+d x))}{28 b^3 d (a+b \sin (c+d x))^6}-\frac{\cos ^3(c+d x)}{7 b d (a+b \sin (c+d x))^7}",1,"Cos[c + d*x]^5/(5*(a - b)*d*(a + b*Sin[c + d*x])^7) + (a*Cos[c + d*x]*(-1/7*(b*(1 - Sin[c + d*x])^(5/2)*(1 + Sin[c + d*x])^(7/2))/((-a + b)*(a + b)*(a + b*Sin[c + d*x])^7) - (-1/6*((a*b + (7*a - b)*b)*(1 - Sin[c + d*x])^(5/2)*(1 + Sin[c + d*x])^(7/2))/((-a + b)*(a + b)*(a + b*Sin[c + d*x])^6) - (7*(6*a^2 - 2*a*b + b^2)*(-1/5*((1 - Sin[c + d*x])^(3/2)*(1 + Sin[c + d*x])^(7/2))/((-a + b)*(a + b*Sin[c + d*x])^5) - (3*(-1/4*(Sqrt[1 - Sin[c + d*x]]*(1 + Sin[c + d*x])^(7/2))/((-a + b)*(a + b*Sin[c + d*x])^4) - (-1/3*(Sqrt[1 - Sin[c + d*x]]*(1 + Sin[c + d*x])^(5/2))/((a + b)*(a + b*Sin[c + d*x])^3) + (5*(-1/2*(Sqrt[1 - Sin[c + d*x]]*(1 + Sin[c + d*x])^(3/2))/((a + b)*(a + b*Sin[c + d*x])^2) + (3*((-2*ArcTanh[(Sqrt[a - b]*Sqrt[1 - Sin[c + d*x]])/(Sqrt[-a - b]*Sqrt[1 + Sin[c + d*x]])])/((-a - b)^(3/2)*Sqrt[a - b]) + (Sqrt[1 - Sin[c + d*x]]*Sqrt[1 + Sin[c + d*x]])/((-a - b)*(a + b*Sin[c + d*x]))))/(2*(a + b))))/(3*(a + b)))/(4*(-a + b))))/(5*(-a + b))))/(6*(-a + b)*(a + b)))/(7*(-a + b)*(a + b))))/((a - b)*d*Sqrt[1 - Sin[c + d*x]]*Sqrt[1 + Sin[c + d*x]]) + (2*b*(Cos[c + d*x]^7/(7*(a - b)*d*(a + b*Sin[c + d*x])^7) + (a*Cos[c + d*x]*(-1/7*((1 - Sin[c + d*x])^(5/2)*(1 + Sin[c + d*x])^(9/2))/((-a + b)*(a + b*Sin[c + d*x])^7) - (5*(-1/6*((1 - Sin[c + d*x])^(3/2)*(1 + Sin[c + d*x])^(9/2))/((-a + b)*(a + b*Sin[c + d*x])^6) - (-1/5*(Sqrt[1 - Sin[c + d*x]]*(1 + Sin[c + d*x])^(9/2))/((-a + b)*(a + b*Sin[c + d*x])^5) - (-1/4*(Sqrt[1 - Sin[c + d*x]]*(1 + Sin[c + d*x])^(7/2))/((a + b)*(a + b*Sin[c + d*x])^4) + (7*(-1/3*(Sqrt[1 - Sin[c + d*x]]*(1 + Sin[c + d*x])^(5/2))/((a + b)*(a + b*Sin[c + d*x])^3) + (5*(-1/2*(Sqrt[1 - Sin[c + d*x]]*(1 + Sin[c + d*x])^(3/2))/((a + b)*(a + b*Sin[c + d*x])^2) + (3*((-2*ArcTanh[(Sqrt[a - b]*Sqrt[1 - Sin[c + d*x]])/(Sqrt[-a - b]*Sqrt[1 + Sin[c + d*x]])])/((-a - b)^(3/2)*Sqrt[a - b]) + (Sqrt[1 - Sin[c + d*x]]*Sqrt[1 + Sin[c + d*x]])/((-a - b)*(a + b*Sin[c + d*x]))))/(2*(a + b))))/(3*(a + b))))/(4*(a + b)))/(5*(-a + b)))/(2*(-a + b))))/(7*(-a + b))))/((a - b)*d*Sqrt[1 - Sin[c + d*x]]*Sqrt[1 + Sin[c + d*x]])))/(5*(a - b))","B",0
470,1,1896,422,6.2041421,"\int \frac{\cos ^2(c+d x)}{(a+b \sin (c+d x))^8} \, dx","Integrate[Cos[c + d*x]^2/(a + b*Sin[c + d*x])^8,x]","\frac{\cos ^3(c+d x)}{3 (a-b) d (a+b \sin (c+d x))^7}+\frac{a \left(-\frac{b (1-\sin (c+d x))^{3/2} (\sin (c+d x)+1)^{5/2}}{7 (b-a) (a+b) (a+b \sin (c+d x))^7}-\frac{-\frac{(3 a b+(7 a-b) b) (1-\sin (c+d x))^{3/2} (\sin (c+d x)+1)^{5/2}}{6 (b-a) (a+b) (a+b \sin (c+d x))^6}-\frac{-\frac{\left(2 a (10 a-b) b+\left(42 a^2-16 b a+19 b^2\right) b\right) (1-\sin (c+d x))^{3/2} (\sin (c+d x)+1)^{5/2}}{5 (b-a) (a+b) (a+b \sin (c+d x))^5}-\frac{-\frac{\left(a b \left(62 a^2-18 b a+19 b^2\right)+b \left(210 a^3-142 b a^2+213 b^2 a-29 b^3\right)\right) (1-\sin (c+d x))^{3/2} (\sin (c+d x)+1)^{5/2}}{4 (b-a) (a+b) (a+b \sin (c+d x))^4}-\frac{105 \left(8 a^4-8 b a^3+12 b^2 a^2-4 b^3 a+b^4\right) \left(-\frac{\sqrt{1-\sin (c+d x)} (\sin (c+d x)+1)^{5/2}}{3 (b-a) (a+b \sin (c+d x))^3}-\frac{\frac{3 \left(\frac{\sqrt{1-\sin (c+d x)} \sqrt{\sin (c+d x)+1}}{(-a-b) (a+b \sin (c+d x))}-\frac{2 \tanh ^{-1}\left(\frac{\sqrt{a-b} \sqrt{1-\sin (c+d x)}}{\sqrt{-a-b} \sqrt{\sin (c+d x)+1}}\right)}{(-a-b)^{3/2} \sqrt{a-b}}\right)}{2 (a+b)}-\frac{\sqrt{1-\sin (c+d x)} (\sin (c+d x)+1)^{3/2}}{2 (a+b) (a+b \sin (c+d x))^2}}{3 (b-a)}\right)}{4 (b-a) (a+b)}}{5 (b-a) (a+b)}}{6 (b-a) (a+b)}}{7 (b-a) (a+b)}\right) \cos (c+d x)}{(a-b) d \sqrt{1-\sin (c+d x)} \sqrt{\sin (c+d x)+1}}+\frac{4 b \left(\frac{\cos ^5(c+d x)}{5 (a-b) d (a+b \sin (c+d x))^7}+\frac{a \left(-\frac{b (1-\sin (c+d x))^{5/2} (\sin (c+d x)+1)^{7/2}}{7 (b-a) (a+b) (a+b \sin (c+d x))^7}-\frac{-\frac{(a b+(7 a-b) b) (1-\sin (c+d x))^{5/2} (\sin (c+d x)+1)^{7/2}}{6 (b-a) (a+b) (a+b \sin (c+d x))^6}-\frac{7 \left(6 a^2-2 b a+b^2\right) \left(-\frac{(1-\sin (c+d x))^{3/2} (\sin (c+d x)+1)^{7/2}}{5 (b-a) (a+b \sin (c+d x))^5}-\frac{3 \left(-\frac{\sqrt{1-\sin (c+d x)} (\sin (c+d x)+1)^{7/2}}{4 (b-a) (a+b \sin (c+d x))^4}-\frac{\frac{5 \left(\frac{3 \left(\frac{\sqrt{1-\sin (c+d x)} \sqrt{\sin (c+d x)+1}}{(-a-b) (a+b \sin (c+d x))}-\frac{2 \tanh ^{-1}\left(\frac{\sqrt{a-b} \sqrt{1-\sin (c+d x)}}{\sqrt{-a-b} \sqrt{\sin (c+d x)+1}}\right)}{(-a-b)^{3/2} \sqrt{a-b}}\right)}{2 (a+b)}-\frac{\sqrt{1-\sin (c+d x)} (\sin (c+d x)+1)^{3/2}}{2 (a+b) (a+b \sin (c+d x))^2}\right)}{3 (a+b)}-\frac{\sqrt{1-\sin (c+d x)} (\sin (c+d x)+1)^{5/2}}{3 (a+b) (a+b \sin (c+d x))^3}}{4 (b-a)}\right)}{5 (b-a)}\right)}{6 (b-a) (a+b)}}{7 (b-a) (a+b)}\right) \cos (c+d x)}{(a-b) d \sqrt{1-\sin (c+d x)} \sqrt{\sin (c+d x)+1}}+\frac{2 b \left(\frac{\cos ^7(c+d x)}{7 (a-b) d (a+b \sin (c+d x))^7}+\frac{a \left(-\frac{(1-\sin (c+d x))^{5/2} (\sin (c+d x)+1)^{9/2}}{7 (b-a) (a+b \sin (c+d x))^7}-\frac{5 \left(-\frac{(1-\sin (c+d x))^{3/2} (\sin (c+d x)+1)^{9/2}}{6 (b-a) (a+b \sin (c+d x))^6}-\frac{-\frac{\sqrt{1-\sin (c+d x)} (\sin (c+d x)+1)^{9/2}}{5 (b-a) (a+b \sin (c+d x))^5}-\frac{\frac{7 \left(\frac{5 \left(\frac{3 \left(\frac{\sqrt{1-\sin (c+d x)} \sqrt{\sin (c+d x)+1}}{(-a-b) (a+b \sin (c+d x))}-\frac{2 \tanh ^{-1}\left(\frac{\sqrt{a-b} \sqrt{1-\sin (c+d x)}}{\sqrt{-a-b} \sqrt{\sin (c+d x)+1}}\right)}{(-a-b)^{3/2} \sqrt{a-b}}\right)}{2 (a+b)}-\frac{\sqrt{1-\sin (c+d x)} (\sin (c+d x)+1)^{3/2}}{2 (a+b) (a+b \sin (c+d x))^2}\right)}{3 (a+b)}-\frac{\sqrt{1-\sin (c+d x)} (\sin (c+d x)+1)^{5/2}}{3 (a+b) (a+b \sin (c+d x))^3}\right)}{4 (a+b)}-\frac{\sqrt{1-\sin (c+d x)} (\sin (c+d x)+1)^{7/2}}{4 (a+b) (a+b \sin (c+d x))^4}}{5 (b-a)}}{2 (b-a)}\right)}{7 (b-a)}\right) \cos (c+d x)}{(a-b) d \sqrt{1-\sin (c+d x)} \sqrt{\sin (c+d x)+1}}\right)}{5 (a-b)}\right)}{3 (a-b)}","\frac{a \left(20 a^2+79 b^2\right) \cos (c+d x)}{840 b d \left(a^2-b^2\right)^3 (a+b \sin (c+d x))^4}+\frac{\left(5 a^2+6 b^2\right) \cos (c+d x)}{210 b d \left(a^2-b^2\right)^2 (a+b \sin (c+d x))^5}+\frac{a \cos (c+d x)}{42 b d \left(a^2-b^2\right) (a+b \sin (c+d x))^6}+\frac{a \left(8 a^4+20 a^2 b^2+5 b^4\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{8 d \left(a^2-b^2\right)^{13/2}}+\frac{a \left(40 a^4+718 a^2 b^2+397 b^4\right) \cos (c+d x)}{1680 b d \left(a^2-b^2\right)^5 (a+b \sin (c+d x))^2}+\frac{\left(20 a^4+179 a^2 b^2+32 b^4\right) \cos (c+d x)}{840 b d \left(a^2-b^2\right)^4 (a+b \sin (c+d x))^3}+\frac{\left(40 a^6+1518 a^4 b^2+1779 a^2 b^4+128 b^6\right) \cos (c+d x)}{1680 b d \left(a^2-b^2\right)^6 (a+b \sin (c+d x))}-\frac{\cos (c+d x)}{7 b d (a+b \sin (c+d x))^7}",1,"Cos[c + d*x]^3/(3*(a - b)*d*(a + b*Sin[c + d*x])^7) + (a*Cos[c + d*x]*(-1/7*(b*(1 - Sin[c + d*x])^(3/2)*(1 + Sin[c + d*x])^(5/2))/((-a + b)*(a + b)*(a + b*Sin[c + d*x])^7) - (-1/6*((3*a*b + (7*a - b)*b)*(1 - Sin[c + d*x])^(3/2)*(1 + Sin[c + d*x])^(5/2))/((-a + b)*(a + b)*(a + b*Sin[c + d*x])^6) - (-1/5*((2*a*(10*a - b)*b + b*(42*a^2 - 16*a*b + 19*b^2))*(1 - Sin[c + d*x])^(3/2)*(1 + Sin[c + d*x])^(5/2))/((-a + b)*(a + b)*(a + b*Sin[c + d*x])^5) - (-1/4*((a*b*(62*a^2 - 18*a*b + 19*b^2) + b*(210*a^3 - 142*a^2*b + 213*a*b^2 - 29*b^3))*(1 - Sin[c + d*x])^(3/2)*(1 + Sin[c + d*x])^(5/2))/((-a + b)*(a + b)*(a + b*Sin[c + d*x])^4) - (105*(8*a^4 - 8*a^3*b + 12*a^2*b^2 - 4*a*b^3 + b^4)*(-1/3*(Sqrt[1 - Sin[c + d*x]]*(1 + Sin[c + d*x])^(5/2))/((-a + b)*(a + b*Sin[c + d*x])^3) - (-1/2*(Sqrt[1 - Sin[c + d*x]]*(1 + Sin[c + d*x])^(3/2))/((a + b)*(a + b*Sin[c + d*x])^2) + (3*((-2*ArcTanh[(Sqrt[a - b]*Sqrt[1 - Sin[c + d*x]])/(Sqrt[-a - b]*Sqrt[1 + Sin[c + d*x]])])/((-a - b)^(3/2)*Sqrt[a - b]) + (Sqrt[1 - Sin[c + d*x]]*Sqrt[1 + Sin[c + d*x]])/((-a - b)*(a + b*Sin[c + d*x]))))/(2*(a + b)))/(3*(-a + b))))/(4*(-a + b)*(a + b)))/(5*(-a + b)*(a + b)))/(6*(-a + b)*(a + b)))/(7*(-a + b)*(a + b))))/((a - b)*d*Sqrt[1 - Sin[c + d*x]]*Sqrt[1 + Sin[c + d*x]]) + (4*b*(Cos[c + d*x]^5/(5*(a - b)*d*(a + b*Sin[c + d*x])^7) + (a*Cos[c + d*x]*(-1/7*(b*(1 - Sin[c + d*x])^(5/2)*(1 + Sin[c + d*x])^(7/2))/((-a + b)*(a + b)*(a + b*Sin[c + d*x])^7) - (-1/6*((a*b + (7*a - b)*b)*(1 - Sin[c + d*x])^(5/2)*(1 + Sin[c + d*x])^(7/2))/((-a + b)*(a + b)*(a + b*Sin[c + d*x])^6) - (7*(6*a^2 - 2*a*b + b^2)*(-1/5*((1 - Sin[c + d*x])^(3/2)*(1 + Sin[c + d*x])^(7/2))/((-a + b)*(a + b*Sin[c + d*x])^5) - (3*(-1/4*(Sqrt[1 - Sin[c + d*x]]*(1 + Sin[c + d*x])^(7/2))/((-a + b)*(a + b*Sin[c + d*x])^4) - (-1/3*(Sqrt[1 - Sin[c + d*x]]*(1 + Sin[c + d*x])^(5/2))/((a + b)*(a + b*Sin[c + d*x])^3) + (5*(-1/2*(Sqrt[1 - Sin[c + d*x]]*(1 + Sin[c + d*x])^(3/2))/((a + b)*(a + b*Sin[c + d*x])^2) + (3*((-2*ArcTanh[(Sqrt[a - b]*Sqrt[1 - Sin[c + d*x]])/(Sqrt[-a - b]*Sqrt[1 + Sin[c + d*x]])])/((-a - b)^(3/2)*Sqrt[a - b]) + (Sqrt[1 - Sin[c + d*x]]*Sqrt[1 + Sin[c + d*x]])/((-a - b)*(a + b*Sin[c + d*x]))))/(2*(a + b))))/(3*(a + b)))/(4*(-a + b))))/(5*(-a + b))))/(6*(-a + b)*(a + b)))/(7*(-a + b)*(a + b))))/((a - b)*d*Sqrt[1 - Sin[c + d*x]]*Sqrt[1 + Sin[c + d*x]]) + (2*b*(Cos[c + d*x]^7/(7*(a - b)*d*(a + b*Sin[c + d*x])^7) + (a*Cos[c + d*x]*(-1/7*((1 - Sin[c + d*x])^(5/2)*(1 + Sin[c + d*x])^(9/2))/((-a + b)*(a + b*Sin[c + d*x])^7) - (5*(-1/6*((1 - Sin[c + d*x])^(3/2)*(1 + Sin[c + d*x])^(9/2))/((-a + b)*(a + b*Sin[c + d*x])^6) - (-1/5*(Sqrt[1 - Sin[c + d*x]]*(1 + Sin[c + d*x])^(9/2))/((-a + b)*(a + b*Sin[c + d*x])^5) - (-1/4*(Sqrt[1 - Sin[c + d*x]]*(1 + Sin[c + d*x])^(7/2))/((a + b)*(a + b*Sin[c + d*x])^4) + (7*(-1/3*(Sqrt[1 - Sin[c + d*x]]*(1 + Sin[c + d*x])^(5/2))/((a + b)*(a + b*Sin[c + d*x])^3) + (5*(-1/2*(Sqrt[1 - Sin[c + d*x]]*(1 + Sin[c + d*x])^(3/2))/((a + b)*(a + b*Sin[c + d*x])^2) + (3*((-2*ArcTanh[(Sqrt[a - b]*Sqrt[1 - Sin[c + d*x]])/(Sqrt[-a - b]*Sqrt[1 + Sin[c + d*x]])])/((-a - b)^(3/2)*Sqrt[a - b]) + (Sqrt[1 - Sin[c + d*x]]*Sqrt[1 + Sin[c + d*x]])/((-a - b)*(a + b*Sin[c + d*x]))))/(2*(a + b))))/(3*(a + b))))/(4*(a + b)))/(5*(-a + b)))/(2*(-a + b))))/(7*(-a + b))))/((a - b)*d*Sqrt[1 - Sin[c + d*x]]*Sqrt[1 + Sin[c + d*x]])))/(5*(a - b))))/(3*(a - b))","B",0
471,1,494,529,5.2484576,"\int \frac{\sec ^2(c+d x)}{(a+b \sin (c+d x))^8} \, dx","Integrate[Sec[c + d*x]^2/(a + b*Sin[c + d*x])^8,x]","-\frac{\frac{2 a b^3 \left(1216 a^2+739 b^2\right) \cos (c+d x)}{\left(a^2-b^2\right)^5 (a+b \sin (c+d x))^4}+\frac{8 b^3 \left(129 a^2+26 b^2\right) \cos (c+d x)}{\left(a^2-b^2\right)^4 (a+b \sin (c+d x))^5}+\frac{360 a b^3 \cos (c+d x)}{\left(a^2-b^2\right)^3 (a+b \sin (c+d x))^6}+\frac{80 b^3 \cos (c+d x)}{\left(a^2-b^2\right)^2 (a+b \sin (c+d x))^7}+\frac{a b^3 \left(11112 a^4+23066 a^2 b^2+5057 b^4\right) \cos (c+d x)}{\left(a^2-b^2\right)^7 (a+b \sin (c+d x))^2}+\frac{2 b^3 \left(2616 a^4+3207 a^2 b^2+232 b^4\right) \cos (c+d x)}{\left(a^2-b^2\right)^6 (a+b \sin (c+d x))^3}+\frac{630 a b^2 \left(64 a^6+336 a^4 b^2+280 a^2 b^4+35 b^6\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{\left(a^2-b^2\right)^{17/2}}+\frac{b^3 \left(26792 a^6+86434 a^4 b^2+38831 a^2 b^4+1488 b^6\right) \cos (c+d x)}{\left(a^2-b^2\right)^8 (a+b \sin (c+d x))}-\frac{560 \sec (c+d x) \left(\left(a^8+28 a^6 b^2+70 a^4 b^4+28 a^2 b^6+b^8\right) \sin (c+d x)-8 a b \left(a^6+7 a^4 b^2+7 a^2 b^4+b^6\right)\right)}{\left(a^2-b^2\right)^8}}{560 d}","\frac{13 a b \left(28 a^2+27 b^2\right) \sec (c+d x)}{280 d \left(a^2-b^2\right)^4 (a+b \sin (c+d x))^4}+\frac{b \left(49 a^2+16 b^2\right) \sec (c+d x)}{70 d \left(a^2-b^2\right)^3 (a+b \sin (c+d x))^5}+\frac{5 a b \sec (c+d x)}{14 d \left(a^2-b^2\right)^2 (a+b \sin (c+d x))^6}+\frac{b \sec (c+d x)}{7 d \left(a^2-b^2\right) (a+b \sin (c+d x))^7}+\frac{11 a b \left(280 a^4+844 a^2 b^2+241 b^4\right) \sec (c+d x)}{560 d \left(a^2-b^2\right)^6 (a+b \sin (c+d x))^2}+\frac{b \left(700 a^4+1317 a^2 b^2+128 b^4\right) \sec (c+d x)}{280 d \left(a^2-b^2\right)^5 (a+b \sin (c+d x))^3}-\frac{9 a b^2 \left(64 a^6+336 a^4 b^2+280 a^2 b^4+35 b^6\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{8 d \left(a^2-b^2\right)^{17/2}}+\frac{b \left(9800 a^6+41484 a^4 b^2+22767 a^2 b^4+1024 b^6\right) \sec (c+d x)}{560 d \left(a^2-b^2\right)^7 (a+b \sin (c+d x))}-\frac{\sec (c+d x) \left(315 a b \left(64 a^6+336 a^4 b^2+280 a^2 b^4+35 b^6\right)-\left(560 a^8+42472 a^6 b^2+125634 a^4 b^4+54511 a^2 b^6+2048 b^8\right) \sin (c+d x)\right)}{560 d \left(a^2-b^2\right)^8}",1,"-1/560*((630*a*b^2*(64*a^6 + 336*a^4*b^2 + 280*a^2*b^4 + 35*b^6)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(17/2) + (80*b^3*Cos[c + d*x])/((a^2 - b^2)^2*(a + b*Sin[c + d*x])^7) + (360*a*b^3*Cos[c + d*x])/((a^2 - b^2)^3*(a + b*Sin[c + d*x])^6) + (8*b^3*(129*a^2 + 26*b^2)*Cos[c + d*x])/((a^2 - b^2)^4*(a + b*Sin[c + d*x])^5) + (2*a*b^3*(1216*a^2 + 739*b^2)*Cos[c + d*x])/((a^2 - b^2)^5*(a + b*Sin[c + d*x])^4) + (2*b^3*(2616*a^4 + 3207*a^2*b^2 + 232*b^4)*Cos[c + d*x])/((a^2 - b^2)^6*(a + b*Sin[c + d*x])^3) + (a*b^3*(11112*a^4 + 23066*a^2*b^2 + 5057*b^4)*Cos[c + d*x])/((a^2 - b^2)^7*(a + b*Sin[c + d*x])^2) + (b^3*(26792*a^6 + 86434*a^4*b^2 + 38831*a^2*b^4 + 1488*b^6)*Cos[c + d*x])/((a^2 - b^2)^8*(a + b*Sin[c + d*x])) - (560*Sec[c + d*x]*(-8*a*b*(a^6 + 7*a^4*b^2 + 7*a^2*b^4 + b^6) + (a^8 + 28*a^6*b^2 + 70*a^4*b^4 + 28*a^2*b^6 + b^8)*Sin[c + d*x]))/(a^2 - b^2)^8)/d","A",1
472,1,597,653,5.9951183,"\int \frac{\sec ^4(c+d x)}{(a+b \sin (c+d x))^8} \, dx","Integrate[Sec[c + d*x]^4/(a + b*Sin[c + d*x])^8,x]","\frac{\frac{2 a b^5 \left(2138 a^2+925 b^2\right) \cos (c+d x)}{\left(a^2-b^2\right)^6 (a+b \sin (c+d x))^4}+\frac{8 b^5 \left(167 a^2+24 b^2\right) \cos (c+d x)}{\left(a^2-b^2\right)^5 (a+b \sin (c+d x))^5}+\frac{328 a b^5 \cos (c+d x)}{\left(a^2-b^2\right)^4 (a+b \sin (c+d x))^6}+\frac{48 b^5 \cos (c+d x)}{\left(a^2-b^2\right)^3 (a+b \sin (c+d x))^7}+\frac{a b^5 \left(33284 a^4+48820 a^2 b^2+8287 b^4\right) \cos (c+d x)}{\left(a^2-b^2\right)^8 (a+b \sin (c+d x))^2}+\frac{2 b^5 \left(6058 a^4+5273 a^2 b^2+296 b^4\right) \cos (c+d x)}{\left(a^2-b^2\right)^7 (a+b \sin (c+d x))^3}+\frac{6930 a b^4 \left(32 a^6+112 a^4 b^2+70 a^2 b^4+7 b^6\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{\left(a^2-b^2\right)^{19/2}}+\frac{b^5 \left(103844 a^6+234272 a^4 b^2+81057 a^2 b^4+2528 b^6\right) \cos (c+d x)}{\left(a^2-b^2\right)^9 (a+b \sin (c+d x))}+\frac{112 \sec ^3(c+d x) \left(\left(a^8+28 a^6 b^2+70 a^4 b^4+28 a^2 b^6+b^8\right) \sin (c+d x)-8 a b \left(a^6+7 a^4 b^2+7 a^2 b^4+b^6\right)\right)}{\left(a^2-b^2\right)^8}+\frac{224 \sec (c+d x) \left(12 \left(15 a^7 b^3+63 a^5 b^5+45 a^3 b^7+5 a b^9\right)+\left(a^{10}-27 a^8 b^2-462 a^6 b^4-798 a^4 b^6-243 a^2 b^8-7 b^{10}\right) \sin (c+d x)\right)}{\left(a^2-b^2\right)^9}}{336 d}","\frac{a b \left(118 a^2+103 b^2\right) \sec ^3(c+d x)}{56 d \left(a^2-b^2\right)^4 (a+b \sin (c+d x))^4}+\frac{b \left(13 a^2+4 b^2\right) \sec ^3(c+d x)}{14 d \left(a^2-b^2\right)^3 (a+b \sin (c+d x))^5}+\frac{17 a b \sec ^3(c+d x)}{42 d \left(a^2-b^2\right)^2 (a+b \sin (c+d x))^6}+\frac{b \sec ^3(c+d x)}{7 d \left(a^2-b^2\right) (a+b \sin (c+d x))^7}+\frac{13 a b \left(140 a^4+336 a^2 b^2+85 b^4\right) \sec ^3(c+d x)}{112 d \left(a^2-b^2\right)^6 (a+b \sin (c+d x))^2}+\frac{b \left(882 a^4+1421 a^2 b^2+128 b^4\right) \sec ^3(c+d x)}{168 d \left(a^2-b^2\right)^5 (a+b \sin (c+d x))^3}+\frac{165 a b^4 \left(32 a^6+112 a^4 b^2+70 a^2 b^4+7 b^6\right) \tan ^{-1}\left(\frac{a \tan \left(\frac{1}{2} (c+d x)\right)+b}{\sqrt{a^2-b^2}}\right)}{8 d \left(a^2-b^2\right)^{19/2}}+\frac{b \left(9212 a^6+28420 a^4 b^2+12907 a^2 b^4+512 b^6\right) \sec ^3(c+d x)}{112 d \left(a^2-b^2\right)^7 (a+b \sin (c+d x))}-\frac{\sec ^3(c+d x) \left(1155 a b \left(32 a^6+112 a^4 b^2+70 a^2 b^4+7 b^6\right)-\left(112 a^8+52528 a^6 b^2+142902 a^4 b^4+57665 a^2 b^6+2048 b^8\right) \sin (c+d x)\right)}{336 d \left(a^2-b^2\right)^8}+\frac{\sec (c+d x) \left(3465 a b^3 \left(32 a^6+112 a^4 b^2+70 a^2 b^4+7 b^6\right)+\left(224 a^{10}-6048 a^8 b^2-207332 a^6 b^4-413024 a^4 b^6-135489 a^2 b^8-4096 b^{10}\right) \sin (c+d x)\right)}{336 d \left(a^2-b^2\right)^9}",1,"((6930*a*b^4*(32*a^6 + 112*a^4*b^2 + 70*a^2*b^4 + 7*b^6)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(19/2) + (48*b^5*Cos[c + d*x])/((a^2 - b^2)^3*(a + b*Sin[c + d*x])^7) + (328*a*b^5*Cos[c + d*x])/((a^2 - b^2)^4*(a + b*Sin[c + d*x])^6) + (8*b^5*(167*a^2 + 24*b^2)*Cos[c + d*x])/((a^2 - b^2)^5*(a + b*Sin[c + d*x])^5) + (2*a*b^5*(2138*a^2 + 925*b^2)*Cos[c + d*x])/((a^2 - b^2)^6*(a + b*Sin[c + d*x])^4) + (2*b^5*(6058*a^4 + 5273*a^2*b^2 + 296*b^4)*Cos[c + d*x])/((a^2 - b^2)^7*(a + b*Sin[c + d*x])^3) + (a*b^5*(33284*a^4 + 48820*a^2*b^2 + 8287*b^4)*Cos[c + d*x])/((a^2 - b^2)^8*(a + b*Sin[c + d*x])^2) + (b^5*(103844*a^6 + 234272*a^4*b^2 + 81057*a^2*b^4 + 2528*b^6)*Cos[c + d*x])/((a^2 - b^2)^9*(a + b*Sin[c + d*x])) + (112*Sec[c + d*x]^3*(-8*a*b*(a^6 + 7*a^4*b^2 + 7*a^2*b^4 + b^6) + (a^8 + 28*a^6*b^2 + 70*a^4*b^4 + 28*a^2*b^6 + b^8)*Sin[c + d*x]))/(a^2 - b^2)^8 + (224*Sec[c + d*x]*(12*(15*a^7*b^3 + 63*a^5*b^5 + 45*a^3*b^7 + 5*a*b^9) + (a^10 - 27*a^8*b^2 - 462*a^6*b^4 - 798*a^4*b^6 - 243*a^2*b^8 - 7*b^10)*Sin[c + d*x]))/(a^2 - b^2)^9)/(336*d)","A",1
473,1,117,154,0.3440198,"\int \cos ^5(c+d x) \sqrt{a+b \sin (c+d x)} \, dx","Integrate[Cos[c + d*x]^5*Sqrt[a + b*Sin[c + d*x]],x]","\frac{2 (a+b \sin (c+d x))^{3/2} \left(8 \left(16 a^4+\left(99 a b^3-24 a^3 b\right) \sin (c+d x)+15 b^2 \left(2 a^2-3 b^2\right) \sin ^2(c+d x)-66 a^2 b^2-35 a b^3 \sin ^3(c+d x)+105 b^4\right)+315 b^4 \cos ^4(c+d x)\right)}{3465 b^5 d}","\frac{4 \left(3 a^2-b^2\right) (a+b \sin (c+d x))^{7/2}}{7 b^5 d}-\frac{8 a \left(a^2-b^2\right) (a+b \sin (c+d x))^{5/2}}{5 b^5 d}+\frac{2 \left(a^2-b^2\right)^2 (a+b \sin (c+d x))^{3/2}}{3 b^5 d}+\frac{2 (a+b \sin (c+d x))^{11/2}}{11 b^5 d}-\frac{8 a (a+b \sin (c+d x))^{9/2}}{9 b^5 d}",1,"(2*(a + b*Sin[c + d*x])^(3/2)*(315*b^4*Cos[c + d*x]^4 + 8*(16*a^4 - 66*a^2*b^2 + 105*b^4 + (-24*a^3*b + 99*a*b^3)*Sin[c + d*x] + 15*b^2*(2*a^2 - 3*b^2)*Sin[c + d*x]^2 - 35*a*b^3*Sin[c + d*x]^3)))/(3465*b^5*d)","A",1
474,1,58,83,0.1275283,"\int \cos ^3(c+d x) \sqrt{a+b \sin (c+d x)} \, dx","Integrate[Cos[c + d*x]^3*Sqrt[a + b*Sin[c + d*x]],x]","\frac{(a+b \sin (c+d x))^{3/2} \left(-16 a^2+24 a b \sin (c+d x)+15 b^2 \cos (2 (c+d x))+55 b^2\right)}{105 b^3 d}","-\frac{2 \left(a^2-b^2\right) (a+b \sin (c+d x))^{3/2}}{3 b^3 d}-\frac{2 (a+b \sin (c+d x))^{7/2}}{7 b^3 d}+\frac{4 a (a+b \sin (c+d x))^{5/2}}{5 b^3 d}",1,"((a + b*Sin[c + d*x])^(3/2)*(-16*a^2 + 55*b^2 + 15*b^2*Cos[2*(c + d*x)] + 24*a*b*Sin[c + d*x]))/(105*b^3*d)","A",1
475,1,24,24,0.0162215,"\int \cos (c+d x) \sqrt{a+b \sin (c+d x)} \, dx","Integrate[Cos[c + d*x]*Sqrt[a + b*Sin[c + d*x]],x]","\frac{2 (a+b \sin (c+d x))^{3/2}}{3 b d}","\frac{2 (a+b \sin (c+d x))^{3/2}}{3 b d}",1,"(2*(a + b*Sin[c + d*x])^(3/2))/(3*b*d)","A",1
476,1,74,74,0.0515781,"\int \sec (c+d x) \sqrt{a+b \sin (c+d x)} \, dx","Integrate[Sec[c + d*x]*Sqrt[a + b*Sin[c + d*x]],x]","\frac{\sqrt{a+b} \tanh ^{-1}\left(\frac{\sqrt{a+b \sin (c+d x)}}{\sqrt{a+b}}\right)}{d}-\frac{\sqrt{a-b} \tanh ^{-1}\left(\frac{\sqrt{a+b \sin (c+d x)}}{\sqrt{a-b}}\right)}{d}","\frac{\sqrt{a+b} \tanh ^{-1}\left(\frac{\sqrt{a+b \sin (c+d x)}}{\sqrt{a+b}}\right)}{d}-\frac{\sqrt{a-b} \tanh ^{-1}\left(\frac{\sqrt{a+b \sin (c+d x)}}{\sqrt{a-b}}\right)}{d}",1,"-((Sqrt[a - b]*ArcTanh[Sqrt[a + b*Sin[c + d*x]]/Sqrt[a - b]])/d) + (Sqrt[a + b]*ArcTanh[Sqrt[a + b*Sin[c + d*x]]/Sqrt[a + b]])/d","A",1
477,1,143,124,0.7062117,"\int \sec ^3(c+d x) \sqrt{a+b \sin (c+d x)} \, dx","Integrate[Sec[c + d*x]^3*Sqrt[a + b*Sin[c + d*x]],x]","\frac{(a-b) \left(\sqrt{a+b} (2 a+b) \tanh ^{-1}\left(\frac{\sqrt{a+b \sin (c+d x)}}{\sqrt{a+b}}\right)+2 (a+b) \tan (c+d x) \sec (c+d x) \sqrt{a+b \sin (c+d x)}\right)-\sqrt{a-b} \left(2 a^2+a b-b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a+b \sin (c+d x)}}{\sqrt{a-b}}\right)}{4 d \left(a^2-b^2\right)}","-\frac{(2 a-b) \tanh ^{-1}\left(\frac{\sqrt{a+b \sin (c+d x)}}{\sqrt{a-b}}\right)}{4 d \sqrt{a-b}}+\frac{(2 a+b) \tanh ^{-1}\left(\frac{\sqrt{a+b \sin (c+d x)}}{\sqrt{a+b}}\right)}{4 d \sqrt{a+b}}+\frac{\tan (c+d x) \sec (c+d x) \sqrt{a+b \sin (c+d x)}}{2 d}",1,"(-(Sqrt[a - b]*(2*a^2 + a*b - b^2)*ArcTanh[Sqrt[a + b*Sin[c + d*x]]/Sqrt[a - b]]) + (a - b)*(Sqrt[a + b]*(2*a + b)*ArcTanh[Sqrt[a + b*Sin[c + d*x]]/Sqrt[a + b]] + 2*(a + b)*Sec[c + d*x]*Sqrt[a + b*Sin[c + d*x]]*Tan[c + d*x]))/(4*(a^2 - b^2)*d)","A",1
478,1,224,207,1.52489,"\int \sec ^5(c+d x) \sqrt{a+b \sin (c+d x)} \, dx","Integrate[Sec[c + d*x]^5*Sqrt[a + b*Sin[c + d*x]],x]","\frac{-\sqrt{a-b} (a+b)^2 \left(12 a^2-18 a b+5 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a+b \sin (c+d x)}}{\sqrt{a-b}}\right)+(a-b)^2 \sqrt{a+b} \left(12 a^2+18 a b+5 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a+b \sin (c+d x)}}{\sqrt{a+b}}\right)+\frac{1}{2} \left(a^2-b^2\right) \sec ^4(c+d x) \sqrt{a+b \sin (c+d x)} \left(\left(22 a^2-21 b^2\right) \sin (c+d x)+6 a^2 \sin (3 (c+d x))-2 a b \cos (2 (c+d x))-2 a b-5 b^2 \sin (3 (c+d x))\right)}{32 d \left(a^2-b^2\right)^2}","-\frac{\left(12 a^2-18 a b+5 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a+b \sin (c+d x)}}{\sqrt{a-b}}\right)}{32 d (a-b)^{3/2}}+\frac{\left(12 a^2+18 a b+5 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a+b \sin (c+d x)}}{\sqrt{a+b}}\right)}{32 d (a+b)^{3/2}}-\frac{\sec ^2(c+d x) \sqrt{a+b \sin (c+d x)} \left(a b-\left(6 a^2-5 b^2\right) \sin (c+d x)\right)}{16 d \left(a^2-b^2\right)}+\frac{\tan (c+d x) \sec ^3(c+d x) \sqrt{a+b \sin (c+d x)}}{4 d}",1,"(-(Sqrt[a - b]*(a + b)^2*(12*a^2 - 18*a*b + 5*b^2)*ArcTanh[Sqrt[a + b*Sin[c + d*x]]/Sqrt[a - b]]) + (a - b)^2*Sqrt[a + b]*(12*a^2 + 18*a*b + 5*b^2)*ArcTanh[Sqrt[a + b*Sin[c + d*x]]/Sqrt[a + b]] + ((a^2 - b^2)*Sec[c + d*x]^4*Sqrt[a + b*Sin[c + d*x]]*(-2*a*b - 2*a*b*Cos[2*(c + d*x)] + (22*a^2 - 21*b^2)*Sin[c + d*x] + 6*a^2*Sin[3*(c + d*x)] - 5*b^2*Sin[3*(c + d*x)]))/2)/(32*(a^2 - b^2)^2*d)","A",1
479,1,233,298,0.9106429,"\int \cos ^4(c+d x) \sqrt{a+b \sin (c+d x)} \, dx","Integrate[Cos[c + d*x]^4*Sqrt[a + b*Sin[c + d*x]],x]","\frac{32 \sqrt{\frac{a+b \sin (c+d x)}{a+b}} \left(a b^2 \left(a^2-33 b^2\right) F\left(\frac{1}{4} (-2 c-2 d x+\pi )|\frac{2 b}{a+b}\right)+\left(4 a^4-15 a^2 b^2-21 b^4\right) \left((a+b) E\left(\frac{1}{4} (-2 c-2 d x+\pi )|\frac{2 b}{a+b}\right)-a F\left(\frac{1}{4} (-2 c-2 d x+\pi )|\frac{2 b}{a+b}\right)\right)\right)+2 b \cos (c+d x) (a+b \sin (c+d x)) \left(-32 a^3+b \left(24 a^2+203 b^2\right) \sin (c+d x)+10 a b^2 \cos (2 (c+d x))+106 a b^2+35 b^3 \sin (3 (c+d x))\right)}{1260 b^4 d \sqrt{a+b \sin (c+d x)}}","-\frac{4 \cos (c+d x) \sqrt{a+b \sin (c+d x)} \left(4 a \left(a^2-3 b^2\right)-3 b \left(a^2+7 b^2\right) \sin (c+d x)\right)}{315 b^3 d}+\frac{32 a \left(a^4-4 a^2 b^2+3 b^4\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{315 b^4 d \sqrt{a+b \sin (c+d x)}}-\frac{8 \left(4 a^4-15 a^2 b^2-21 b^4\right) \sqrt{a+b \sin (c+d x)} E\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{315 b^4 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}+\frac{2 \cos ^3(c+d x) (a+b \sin (c+d x))^{3/2}}{9 b d}-\frac{4 a \cos ^3(c+d x) \sqrt{a+b \sin (c+d x)}}{21 b d}",1,"(32*(a*b^2*(a^2 - 33*b^2)*EllipticF[(-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)] + (4*a^4 - 15*a^2*b^2 - 21*b^4)*((a + b)*EllipticE[(-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)] - a*EllipticF[(-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)]))*Sqrt[(a + b*Sin[c + d*x])/(a + b)] + 2*b*Cos[c + d*x]*(a + b*Sin[c + d*x])*(-32*a^3 + 106*a*b^2 + 10*a*b^2*Cos[2*(c + d*x)] + b*(24*a^2 + 203*b^2)*Sin[c + d*x] + 35*b^3*Sin[3*(c + d*x)]))/(1260*b^4*d*Sqrt[a + b*Sin[c + d*x]])","A",1
480,1,185,215,0.8283215,"\int \cos ^2(c+d x) \sqrt{a+b \sin (c+d x)} \, dx","Integrate[Cos[c + d*x]^2*Sqrt[a + b*Sin[c + d*x]],x]","\frac{b \cos (c+d x) \left(2 a^2+8 a b \sin (c+d x)-3 b^2 \cos (2 (c+d x))+3 b^2\right)+4 a \left(a^2-b^2\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{4} (-2 c-2 d x+\pi )|\frac{2 b}{a+b}\right)-4 \left(a^3+a^2 b+3 a b^2+3 b^3\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} E\left(\frac{1}{4} (-2 c-2 d x+\pi )|\frac{2 b}{a+b}\right)}{15 b^2 d \sqrt{a+b \sin (c+d x)}}","-\frac{4 a \left(a^2-b^2\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{15 b^2 d \sqrt{a+b \sin (c+d x)}}+\frac{4 \left(a^2+3 b^2\right) \sqrt{a+b \sin (c+d x)} E\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{15 b^2 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}+\frac{2 \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{5 b d}-\frac{4 a \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{15 b d}",1,"(-4*(a^3 + a^2*b + 3*a*b^2 + 3*b^3)*EllipticE[(-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)] + 4*a*(a^2 - b^2)*EllipticF[(-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)] + b*Cos[c + d*x]*(2*a^2 + 3*b^2 - 3*b^2*Cos[2*(c + d*x)] + 8*a*b*Sin[c + d*x]))/(15*b^2*d*Sqrt[a + b*Sin[c + d*x]])","A",1
481,1,127,149,2.8124964,"\int \sec ^2(c+d x) \sqrt{a+b \sin (c+d x)} \, dx","Integrate[Sec[c + d*x]^2*Sqrt[a + b*Sin[c + d*x]],x]","\frac{\tan (c+d x) (a+b \sin (c+d x))-a \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{4} (-2 c-2 d x+\pi )|\frac{2 b}{a+b}\right)+(a+b) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} E\left(\frac{1}{4} (-2 c-2 d x+\pi )|\frac{2 b}{a+b}\right)}{d \sqrt{a+b \sin (c+d x)}}","\frac{\tan (c+d x) \sqrt{a+b \sin (c+d x)}}{d}+\frac{a \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{d \sqrt{a+b \sin (c+d x)}}-\frac{\sqrt{a+b \sin (c+d x)} E\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}",1,"((a + b)*EllipticE[(-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)] - a*EllipticF[(-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)] + (a + b*Sin[c + d*x])*Tan[c + d*x])/(d*Sqrt[a + b*Sin[c + d*x]])","A",1
482,1,270,248,3.3623195,"\int \sec ^4(c+d x) \sqrt{a+b \sin (c+d x)} \, dx","Integrate[Sec[c + d*x]^4*Sqrt[a + b*Sin[c + d*x]],x]","\frac{-4 a \left(a^2-b^2\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{4} (-2 c-2 d x+\pi )|\frac{2 b}{a+b}\right)+\left(4 a^3+4 a^2 b-3 a b^2-3 b^3\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} E\left(\frac{1}{4} (-2 c-2 d x+\pi )|\frac{2 b}{a+b}\right)+\frac{1}{8} \sec ^3(c+d x) \left(24 a^3 \sin (c+d x)+8 a^3 \sin (3 (c+d x))+\left(8 b^3-12 a^2 b\right) \cos (2 (c+d x))+\left(3 b^3-4 a^2 b\right) \cos (4 (c+d x))+8 a^2 b-24 a b^2 \sin (c+d x)-8 a b^2 \sin (3 (c+d x))-11 b^3\right)}{6 d (a-b) (a+b) \sqrt{a+b \sin (c+d x)}}","-\frac{\sec (c+d x) \sqrt{a+b \sin (c+d x)} \left(a b-\left(4 a^2-3 b^2\right) \sin (c+d x)\right)}{6 d \left(a^2-b^2\right)}-\frac{\left(4 a^2-3 b^2\right) \sqrt{a+b \sin (c+d x)} E\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{6 d \left(a^2-b^2\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}+\frac{2 a \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{3 d \sqrt{a+b \sin (c+d x)}}+\frac{\tan (c+d x) \sec ^2(c+d x) \sqrt{a+b \sin (c+d x)}}{3 d}",1,"((4*a^3 + 4*a^2*b - 3*a*b^2 - 3*b^3)*EllipticE[(-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)] - 4*a*(a^2 - b^2)*EllipticF[(-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)] + (Sec[c + d*x]^3*(8*a^2*b - 11*b^3 + (-12*a^2*b + 8*b^3)*Cos[2*(c + d*x)] + (-4*a^2*b + 3*b^3)*Cos[4*(c + d*x)] + 24*a^3*Sin[c + d*x] - 24*a*b^2*Sin[c + d*x] + 8*a^3*Sin[3*(c + d*x)] - 8*a*b^2*Sin[3*(c + d*x)]))/8)/(6*(a - b)*(a + b)*d*Sqrt[a + b*Sin[c + d*x]])","A",1
483,1,131,154,0.7178735,"\int \cos ^5(c+d x) (a+b \sin (c+d x))^{3/2} \, dx","Integrate[Cos[c + d*x]^5*(a + b*Sin[c + d*x])^(3/2),x]","\frac{2 \left(\frac{2}{9} \left(3 a^2-b^2\right) (a+b \sin (c+d x))^{9/2}+\frac{1}{5} \left(a^2-b^2\right)^2 (a+b \sin (c+d x))^{5/2}+\frac{1}{13} (a+b \sin (c+d x))^{13/2}-\frac{4}{11} a (a+b \sin (c+d x))^{11/2}-\frac{4}{7} a (a-b) (a+b) (a+b \sin (c+d x))^{7/2}\right)}{b^5 d}","\frac{4 \left(3 a^2-b^2\right) (a+b \sin (c+d x))^{9/2}}{9 b^5 d}-\frac{8 a \left(a^2-b^2\right) (a+b \sin (c+d x))^{7/2}}{7 b^5 d}+\frac{2 \left(a^2-b^2\right)^2 (a+b \sin (c+d x))^{5/2}}{5 b^5 d}+\frac{2 (a+b \sin (c+d x))^{13/2}}{13 b^5 d}-\frac{8 a (a+b \sin (c+d x))^{11/2}}{11 b^5 d}",1,"(2*(((a^2 - b^2)^2*(a + b*Sin[c + d*x])^(5/2))/5 - (4*a*(a - b)*(a + b)*(a + b*Sin[c + d*x])^(7/2))/7 + (2*(3*a^2 - b^2)*(a + b*Sin[c + d*x])^(9/2))/9 - (4*a*(a + b*Sin[c + d*x])^(11/2))/11 + (a + b*Sin[c + d*x])^(13/2)/13))/(b^5*d)","A",1
484,1,58,83,0.1803586,"\int \cos ^3(c+d x) (a+b \sin (c+d x))^{3/2} \, dx","Integrate[Cos[c + d*x]^3*(a + b*Sin[c + d*x])^(3/2),x]","\frac{(a+b \sin (c+d x))^{5/2} \left(-16 a^2+40 a b \sin (c+d x)+35 b^2 \cos (2 (c+d x))+91 b^2\right)}{315 b^3 d}","-\frac{2 \left(a^2-b^2\right) (a+b \sin (c+d x))^{5/2}}{5 b^3 d}-\frac{2 (a+b \sin (c+d x))^{9/2}}{9 b^3 d}+\frac{4 a (a+b \sin (c+d x))^{7/2}}{7 b^3 d}",1,"((a + b*Sin[c + d*x])^(5/2)*(-16*a^2 + 91*b^2 + 35*b^2*Cos[2*(c + d*x)] + 40*a*b*Sin[c + d*x]))/(315*b^3*d)","A",1
485,1,24,24,0.0215569,"\int \cos (c+d x) (a+b \sin (c+d x))^{3/2} \, dx","Integrate[Cos[c + d*x]*(a + b*Sin[c + d*x])^(3/2),x]","\frac{2 (a+b \sin (c+d x))^{5/2}}{5 b d}","\frac{2 (a+b \sin (c+d x))^{5/2}}{5 b d}",1,"(2*(a + b*Sin[c + d*x])^(5/2))/(5*b*d)","A",1
486,1,89,94,0.0953151,"\int \sec (c+d x) (a+b \sin (c+d x))^{3/2} \, dx","Integrate[Sec[c + d*x]*(a + b*Sin[c + d*x])^(3/2),x]","\frac{-2 b \sqrt{a+b \sin (c+d x)}+(a-b)^{3/2} \left(-\tanh ^{-1}\left(\frac{\sqrt{a+b \sin (c+d x)}}{\sqrt{a-b}}\right)\right)+(a+b)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a+b \sin (c+d x)}}{\sqrt{a+b}}\right)}{d}","-\frac{2 b \sqrt{a+b \sin (c+d x)}}{d}-\frac{(a-b)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a+b \sin (c+d x)}}{\sqrt{a-b}}\right)}{d}+\frac{(a+b)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a+b \sin (c+d x)}}{\sqrt{a+b}}\right)}{d}",1,"(-((a - b)^(3/2)*ArcTanh[Sqrt[a + b*Sin[c + d*x]]/Sqrt[a - b]]) + (a + b)^(3/2)*ArcTanh[Sqrt[a + b*Sin[c + d*x]]/Sqrt[a + b]] - 2*b*Sqrt[a + b*Sin[c + d*x]])/d","A",1
487,1,121,130,0.6996805,"\int \sec ^3(c+d x) (a+b \sin (c+d x))^{3/2} \, dx","Integrate[Sec[c + d*x]^3*(a + b*Sin[c + d*x])^(3/2),x]","\frac{-\sqrt{a-b} (2 a+b) \tanh ^{-1}\left(\frac{\sqrt{a+b \sin (c+d x)}}{\sqrt{a-b}}\right)+(2 a-b) \sqrt{a+b} \tanh ^{-1}\left(\frac{\sqrt{a+b \sin (c+d x)}}{\sqrt{a+b}}\right)+2 \sec ^2(c+d x) (a \sin (c+d x)+b) \sqrt{a+b \sin (c+d x)}}{4 d}","-\frac{\sqrt{a-b} (2 a+b) \tanh ^{-1}\left(\frac{\sqrt{a+b \sin (c+d x)}}{\sqrt{a-b}}\right)}{4 d}+\frac{(2 a-b) \sqrt{a+b} \tanh ^{-1}\left(\frac{\sqrt{a+b \sin (c+d x)}}{\sqrt{a+b}}\right)}{4 d}+\frac{\sec ^2(c+d x) (a \sin (c+d x)+b) \sqrt{a+b \sin (c+d x)}}{2 d}",1,"(-(Sqrt[a - b]*(2*a + b)*ArcTanh[Sqrt[a + b*Sin[c + d*x]]/Sqrt[a - b]]) + (2*a - b)*Sqrt[a + b]*ArcTanh[Sqrt[a + b*Sin[c + d*x]]/Sqrt[a + b]] + 2*Sec[c + d*x]^2*(b + a*Sin[c + d*x])*Sqrt[a + b*Sin[c + d*x]])/(4*d)","A",1
488,1,297,188,2.6038823,"\int \sec ^5(c+d x) (a+b \sin (c+d x))^{3/2} \, dx","Integrate[Sec[c + d*x]^5*(a + b*Sin[c + d*x])^(3/2),x]","-\frac{8 \left(b^2-a^2\right) \sec ^4(c+d x) (a \sin (c+d x)-b) (a+b \sin (c+d x))^{5/2}+3 \sqrt{a-b} (a+b)^2 \left(4 a^3-6 a^2 b+a b^2+b^3\right) \tanh ^{-1}\left(\frac{\sqrt{a+b \sin (c+d x)}}{\sqrt{a-b}}\right)-3 (a-b)^2 \sqrt{a+b} \left(4 a^3+6 a^2 b+a b^2-b^3\right) \tanh ^{-1}\left(\frac{\sqrt{a+b \sin (c+d x)}}{\sqrt{a+b}}\right)+2 \sec ^2(c+d x) (a+b \sin (c+d x))^{5/2} \left(\left(4 a b^2-6 a^3\right) \sin (c+d x)+5 a^2 b-3 b^3\right)-2 b \sqrt{a+b \sin (c+d x)} \left(12 a^4+\left(6 a^3 b-4 a b^3\right) \sin (c+d x)-13 a^2 b^2+3 b^4\right)}{32 d \left(a^2-b^2\right)^2}","-\frac{3 \left(4 a^2-2 a b-b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a+b \sin (c+d x)}}{\sqrt{a-b}}\right)}{32 d \sqrt{a-b}}+\frac{3 \left(4 a^2+2 a b-b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a+b \sin (c+d x)}}{\sqrt{a+b}}\right)}{32 d \sqrt{a+b}}+\frac{\sec ^4(c+d x) (a \sin (c+d x)+b) \sqrt{a+b \sin (c+d x)}}{4 d}-\frac{\sec ^2(c+d x) (b-6 a \sin (c+d x)) \sqrt{a+b \sin (c+d x)}}{16 d}",1,"-1/32*(3*Sqrt[a - b]*(a + b)^2*(4*a^3 - 6*a^2*b + a*b^2 + b^3)*ArcTanh[Sqrt[a + b*Sin[c + d*x]]/Sqrt[a - b]] - 3*(a - b)^2*Sqrt[a + b]*(4*a^3 + 6*a^2*b + a*b^2 - b^3)*ArcTanh[Sqrt[a + b*Sin[c + d*x]]/Sqrt[a + b]] + 8*(-a^2 + b^2)*Sec[c + d*x]^4*(-b + a*Sin[c + d*x])*(a + b*Sin[c + d*x])^(5/2) + 2*Sec[c + d*x]^2*(a + b*Sin[c + d*x])^(5/2)*(5*a^2*b - 3*b^3 + (-6*a^3 + 4*a*b^2)*Sin[c + d*x]) - 2*b*Sqrt[a + b*Sin[c + d*x]]*(12*a^4 - 13*a^2*b^2 + 3*b^4 + (6*a^3*b - 4*a*b^3)*Sin[c + d*x]))/((a^2 - b^2)^2*d)","A",1
489,1,278,329,1.0903513,"\int \cos ^4(c+d x) (a+b \sin (c+d x))^{3/2} \, dx","Integrate[Cos[c + d*x]^4*(a + b*Sin[c + d*x])^(3/2),x]","\frac{64 \sqrt{\frac{a+b \sin (c+d x)}{a+b}} \left(4 \left(a^5-6 a^3 b^2-27 a b^4\right) \left((a+b) E\left(\frac{1}{4} (-2 c-2 d x+\pi )|\frac{2 b}{a+b}\right)-a F\left(\frac{1}{4} (-2 c-2 d x+\pi )|\frac{2 b}{a+b}\right)\right)+b^2 \left(a^4-114 a^2 b^2-15 b^4\right) F\left(\frac{1}{4} (-2 c-2 d x+\pi )|\frac{2 b}{a+b}\right)\right)-b (a+b \sin (c+d x)) \left(-16 a b \left(3 a^2+128 b^2\right) \sin (2 (c+d x))+5 b^2 \left(93 b^2-4 a^2\right) \cos (3 (c+d x))+2 \left(64 a^4-366 a^2 b^2+195 b^4\right) \cos (c+d x)-280 a b^3 \sin (4 (c+d x))+105 b^4 \cos (5 (c+d x))\right)}{9240 b^4 d \sqrt{a+b \sin (c+d x)}}","\frac{2 \cos ^3(c+d x) \sqrt{a+b \sin (c+d x)} \left(a^2+28 a b \sin (c+d x)+3 b^2\right)}{231 b d}-\frac{32 a \left(a^4-6 a^2 b^2-27 b^4\right) \sqrt{a+b \sin (c+d x)} E\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{1155 b^4 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}-\frac{4 \cos (c+d x) \sqrt{a+b \sin (c+d x)} \left(4 a^4-3 a b \left(a^2+31 b^2\right) \sin (c+d x)-21 a^2 b^2-15 b^4\right)}{1155 b^3 d}+\frac{8 \left(4 a^6-25 a^4 b^2+6 a^2 b^4+15 b^6\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{1155 b^4 d \sqrt{a+b \sin (c+d x)}}-\frac{2 b \cos ^5(c+d x) \sqrt{a+b \sin (c+d x)}}{11 d}",1,"(64*(b^2*(a^4 - 114*a^2*b^2 - 15*b^4)*EllipticF[(-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)] + 4*(a^5 - 6*a^3*b^2 - 27*a*b^4)*((a + b)*EllipticE[(-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)] - a*EllipticF[(-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)]))*Sqrt[(a + b*Sin[c + d*x])/(a + b)] - b*(a + b*Sin[c + d*x])*(2*(64*a^4 - 366*a^2*b^2 + 195*b^4)*Cos[c + d*x] + 5*b^2*(-4*a^2 + 93*b^2)*Cos[3*(c + d*x)] + 105*b^4*Cos[5*(c + d*x)] - 16*a*b*(3*a^2 + 128*b^2)*Sin[2*(c + d*x)] - 280*a*b^3*Sin[4*(c + d*x)]))/(9240*b^4*d*Sqrt[a + b*Sin[c + d*x]])","A",1
490,1,222,247,1.0412764,"\int \cos ^2(c+d x) (a+b \sin (c+d x))^{3/2} \, dx","Integrate[Cos[c + d*x]^2*(a + b*Sin[c + d*x])^(3/2),x]","\frac{8 \left(3 a^4+2 a^2 b^2-5 b^4\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{4} (-2 c-2 d x+\pi )|\frac{2 b}{a+b}\right)+b \cos (c+d x) \left(12 a^3+b \left(108 a^2+5 b^2\right) \sin (c+d x)-78 a b^2 \cos (2 (c+d x))+38 a b^2-15 b^3 \sin (3 (c+d x))\right)-8 a \left(3 a^3+3 a^2 b+29 a b^2+29 b^3\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} E\left(\frac{1}{4} (-2 c-2 d x+\pi )|\frac{2 b}{a+b}\right)}{210 b^2 d \sqrt{a+b \sin (c+d x)}}","\frac{2 \cos (c+d x) \sqrt{a+b \sin (c+d x)} \left(3 a^2+24 a b \sin (c+d x)+5 b^2\right)}{105 b d}+\frac{4 a \left(3 a^2+29 b^2\right) \sqrt{a+b \sin (c+d x)} E\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{105 b^2 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}-\frac{4 \left(3 a^4+2 a^2 b^2-5 b^4\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{105 b^2 d \sqrt{a+b \sin (c+d x)}}-\frac{2 b \cos ^3(c+d x) \sqrt{a+b \sin (c+d x)}}{7 d}",1,"(-8*a*(3*a^3 + 3*a^2*b + 29*a*b^2 + 29*b^3)*EllipticE[(-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)] + 8*(3*a^4 + 2*a^2*b^2 - 5*b^4)*EllipticF[(-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)] + b*Cos[c + d*x]*(12*a^3 + 38*a*b^2 - 78*a*b^2*Cos[2*(c + d*x)] + b*(108*a^2 + 5*b^2)*Sin[c + d*x] - 15*b^3*Sin[3*(c + d*x)]))/(210*b^2*d*Sqrt[a + b*Sin[c + d*x]])","A",1
491,1,163,168,0.6569112,"\int \sec ^2(c+d x) (a+b \sin (c+d x))^{3/2} \, dx","Integrate[Sec[c + d*x]^2*(a + b*Sin[c + d*x])^(3/2),x]","\frac{-\left(a^2-b^2\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{4} (-2 c-2 d x+\pi )|\frac{2 b}{a+b}\right)+a^2 \tan (c+d x)+a b \sec (c+d x)+a b \sin (c+d x) \tan (c+d x)+a (a+b) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} E\left(\frac{1}{4} (-2 c-2 d x+\pi )|\frac{2 b}{a+b}\right)+b^2 \tan (c+d x)}{d \sqrt{a+b \sin (c+d x)}}","\frac{\left(a^2-b^2\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{d \sqrt{a+b \sin (c+d x)}}+\frac{\sec (c+d x) (a \sin (c+d x)+b) \sqrt{a+b \sin (c+d x)}}{d}-\frac{a \sqrt{a+b \sin (c+d x)} E\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}",1,"(a*b*Sec[c + d*x] + a*(a + b)*EllipticE[(-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)] - (a^2 - b^2)*EllipticF[(-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)] + a^2*Tan[c + d*x] + b^2*Tan[c + d*x] + a*b*Sin[c + d*x]*Tan[c + d*x])/(d*Sqrt[a + b*Sin[c + d*x]])","A",1
492,1,211,218,2.5525269,"\int \sec ^4(c+d x) (a+b \sin (c+d x))^{3/2} \, dx","Integrate[Sec[c + d*x]^4*(a + b*Sin[c + d*x])^(3/2),x]","\frac{-4 \left(4 a^2-b^2\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{4} (-2 c-2 d x+\pi )|\frac{2 b}{a+b}\right)+\sec ^3(c+d x) \left(12 a^2 \sin (c+d x)+4 a^2 \sin (3 (c+d x))-6 a b \cos (2 (c+d x))-2 a b \cos (4 (c+d x))+12 a b+7 b^2 \sin (c+d x)-b^2 \sin (3 (c+d x))\right)+16 a (a+b) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} E\left(\frac{1}{4} (-2 c-2 d x+\pi )|\frac{2 b}{a+b}\right)}{24 d \sqrt{a+b \sin (c+d x)}}","\frac{\left(4 a^2-b^2\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{6 d \sqrt{a+b \sin (c+d x)}}+\frac{\sec ^3(c+d x) (a \sin (c+d x)+b) \sqrt{a+b \sin (c+d x)}}{3 d}-\frac{\sec (c+d x) (b-4 a \sin (c+d x)) \sqrt{a+b \sin (c+d x)}}{6 d}-\frac{2 a \sqrt{a+b \sin (c+d x)} E\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{3 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}",1,"(16*a*(a + b)*EllipticE[(-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)] - 4*(4*a^2 - b^2)*EllipticF[(-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)] + Sec[c + d*x]^3*(12*a*b - 6*a*b*Cos[2*(c + d*x)] - 2*a*b*Cos[4*(c + d*x)] + 12*a^2*Sin[c + d*x] + 7*b^2*Sin[c + d*x] + 4*a^2*Sin[3*(c + d*x)] - b^2*Sin[3*(c + d*x)]))/(24*d*Sqrt[a + b*Sin[c + d*x]])","A",1
493,1,364,330,6.2778162,"\int \sec ^6(c+d x) (a+b \sin (c+d x))^{3/2} \, dx","Integrate[Sec[c + d*x]^6*(a + b*Sin[c + d*x])^(3/2),x]","\frac{\sqrt{a+b \sin (c+d x)} \left(\frac{\sec (c+d x) \left(32 a^3 \sin (c+d x)-8 a^2 b-29 a b^2 \sin (c+d x)+5 b^3\right)}{60 \left(a^2-b^2\right)}+\frac{1}{5} \sec ^5(c+d x) (a \sin (c+d x)+b)+\frac{1}{30} \sec ^3(c+d x) (8 a \sin (c+d x)-b)\right)}{d}-\frac{b \left(-\frac{\left(32 a^3-29 a b^2\right) \left(\frac{2 (a+b) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} E\left(\frac{1}{2} \left(-c-d x+\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{\sqrt{a+b \sin (c+d x)}}-\frac{2 a \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{2} \left(-c-d x+\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{\sqrt{a+b \sin (c+d x)}}\right)}{b}-\frac{2 \left(8 a^2 b-5 b^3\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{2} \left(-c-d x+\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{\sqrt{a+b \sin (c+d x)}}\right)}{120 d (a-b) (a+b)}","\frac{\left(32 a^2-5 b^2\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{60 d \sqrt{a+b \sin (c+d x)}}-\frac{a \left(32 a^2-29 b^2\right) \sqrt{a+b \sin (c+d x)} E\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{60 d \left(a^2-b^2\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}-\frac{\sec (c+d x) \sqrt{a+b \sin (c+d x)} \left(b \left(8 a^4-13 a^2 b^2+5 b^4\right)-a \left(32 a^4-61 a^2 b^2+29 b^4\right) \sin (c+d x)\right)}{60 d \left(a^2-b^2\right)^2}+\frac{\sec ^5(c+d x) (a \sin (c+d x)+b) \sqrt{a+b \sin (c+d x)}}{5 d}-\frac{\sec ^3(c+d x) (b-8 a \sin (c+d x)) \sqrt{a+b \sin (c+d x)}}{30 d}",1,"(Sqrt[a + b*Sin[c + d*x]]*((Sec[c + d*x]^5*(b + a*Sin[c + d*x]))/5 + (Sec[c + d*x]^3*(-b + 8*a*Sin[c + d*x]))/30 + (Sec[c + d*x]*(-8*a^2*b + 5*b^3 + 32*a^3*Sin[c + d*x] - 29*a*b^2*Sin[c + d*x]))/(60*(a^2 - b^2))))/d - (b*((-2*(8*a^2*b - 5*b^3)*EllipticF[(-c + Pi/2 - d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/Sqrt[a + b*Sin[c + d*x]] - ((32*a^3 - 29*a*b^2)*((2*(a + b)*EllipticE[(-c + Pi/2 - d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/Sqrt[a + b*Sin[c + d*x]] - (2*a*EllipticF[(-c + Pi/2 - d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/Sqrt[a + b*Sin[c + d*x]]))/b))/(120*(a - b)*(a + b)*d)","A",1
494,1,113,154,0.5762,"\int \cos ^5(c+d x) (a+b \sin (c+d x))^{5/2} \, dx","Integrate[Cos[c + d*x]^5*(a + b*Sin[c + d*x])^(5/2),x]","\frac{2 (a+b \sin (c+d x))^{7/2} \left(8190 \left(3 a^2-b^2\right) (a+b \sin (c+d x))^2+6435 \left(a^2-b^2\right)^2+3003 (a+b \sin (c+d x))^4-13860 a (a+b \sin (c+d x))^3-20020 a (a-b) (a+b) (a+b \sin (c+d x))\right)}{45045 b^5 d}","\frac{4 \left(3 a^2-b^2\right) (a+b \sin (c+d x))^{11/2}}{11 b^5 d}-\frac{8 a \left(a^2-b^2\right) (a+b \sin (c+d x))^{9/2}}{9 b^5 d}+\frac{2 \left(a^2-b^2\right)^2 (a+b \sin (c+d x))^{7/2}}{7 b^5 d}+\frac{2 (a+b \sin (c+d x))^{15/2}}{15 b^5 d}-\frac{8 a (a+b \sin (c+d x))^{13/2}}{13 b^5 d}",1,"(2*(a + b*Sin[c + d*x])^(7/2)*(6435*(a^2 - b^2)^2 - 20020*a*(a - b)*(a + b)*(a + b*Sin[c + d*x]) + 8190*(3*a^2 - b^2)*(a + b*Sin[c + d*x])^2 - 13860*a*(a + b*Sin[c + d*x])^3 + 3003*(a + b*Sin[c + d*x])^4))/(45045*b^5*d)","A",1
495,1,58,83,0.0977875,"\int \cos ^3(c+d x) (a+b \sin (c+d x))^{5/2} \, dx","Integrate[Cos[c + d*x]^3*(a + b*Sin[c + d*x])^(5/2),x]","-\frac{2 (a+b \sin (c+d x))^{7/2} \left(8 a^2-28 a b \sin (c+d x)+63 b^2 \sin ^2(c+d x)-99 b^2\right)}{693 b^3 d}","-\frac{2 \left(a^2-b^2\right) (a+b \sin (c+d x))^{7/2}}{7 b^3 d}-\frac{2 (a+b \sin (c+d x))^{11/2}}{11 b^3 d}+\frac{4 a (a+b \sin (c+d x))^{9/2}}{9 b^3 d}",1,"(-2*(a + b*Sin[c + d*x])^(7/2)*(8*a^2 - 99*b^2 - 28*a*b*Sin[c + d*x] + 63*b^2*Sin[c + d*x]^2))/(693*b^3*d)","A",1
496,1,24,24,0.033843,"\int \cos (c+d x) (a+b \sin (c+d x))^{5/2} \, dx","Integrate[Cos[c + d*x]*(a + b*Sin[c + d*x])^(5/2),x]","\frac{2 (a+b \sin (c+d x))^{7/2}}{7 b d}","\frac{2 (a+b \sin (c+d x))^{7/2}}{7 b d}",1,"(2*(a + b*Sin[c + d*x])^(7/2))/(7*b*d)","A",1
497,1,105,117,0.1613762,"\int \sec (c+d x) (a+b \sin (c+d x))^{5/2} \, dx","Integrate[Sec[c + d*x]*(a + b*Sin[c + d*x])^(5/2),x]","\frac{-2 b \sqrt{a+b \sin (c+d x)} (7 a+b \sin (c+d x))-3 (a-b)^{5/2} \tanh ^{-1}\left(\frac{\sqrt{a+b \sin (c+d x)}}{\sqrt{a-b}}\right)+3 (a+b)^{5/2} \tanh ^{-1}\left(\frac{\sqrt{a+b \sin (c+d x)}}{\sqrt{a+b}}\right)}{3 d}","-\frac{2 b (a+b \sin (c+d x))^{3/2}}{3 d}-\frac{4 a b \sqrt{a+b \sin (c+d x)}}{d}-\frac{(a-b)^{5/2} \tanh ^{-1}\left(\frac{\sqrt{a+b \sin (c+d x)}}{\sqrt{a-b}}\right)}{d}+\frac{(a+b)^{5/2} \tanh ^{-1}\left(\frac{\sqrt{a+b \sin (c+d x)}}{\sqrt{a+b}}\right)}{d}",1,"(-3*(a - b)^(5/2)*ArcTanh[Sqrt[a + b*Sin[c + d*x]]/Sqrt[a - b]] + 3*(a + b)^(5/2)*ArcTanh[Sqrt[a + b*Sin[c + d*x]]/Sqrt[a + b]] - 2*b*Sqrt[a + b*Sin[c + d*x]]*(7*a + b*Sin[c + d*x]))/(3*d)","A",1
498,1,147,155,0.8897514,"\int \sec ^3(c+d x) (a+b \sin (c+d x))^{5/2} \, dx","Integrate[Sec[c + d*x]^3*(a + b*Sin[c + d*x])^(5/2),x]","\frac{-\sqrt{a-b} \left(2 a^2+a b-3 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a+b \sin (c+d x)}}{\sqrt{a-b}}\right)+\sqrt{a+b} \left(2 a^2-a b-3 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a+b \sin (c+d x)}}{\sqrt{a+b}}\right)+2 \sec ^2(c+d x) \sqrt{a+b \sin (c+d x)} \left(\left(a^2+b^2\right) \sin (c+d x)+2 a b\right)}{4 d}","\frac{a b \sqrt{a+b \sin (c+d x)}}{2 d}-\frac{(a-b)^{3/2} (2 a+3 b) \tanh ^{-1}\left(\frac{\sqrt{a+b \sin (c+d x)}}{\sqrt{a-b}}\right)}{4 d}+\frac{(2 a-3 b) (a+b)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a+b \sin (c+d x)}}{\sqrt{a+b}}\right)}{4 d}+\frac{\sec ^2(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^{3/2}}{2 d}",1,"(-(Sqrt[a - b]*(2*a^2 + a*b - 3*b^2)*ArcTanh[Sqrt[a + b*Sin[c + d*x]]/Sqrt[a - b]]) + Sqrt[a + b]*(2*a^2 - a*b - 3*b^2)*ArcTanh[Sqrt[a + b*Sin[c + d*x]]/Sqrt[a + b]] + 2*Sec[c + d*x]^2*Sqrt[a + b*Sin[c + d*x]]*(2*a*b + (a^2 + b^2)*Sin[c + d*x]))/(4*d)","A",1
499,1,307,199,3.3840394,"\int \sec ^5(c+d x) (a+b \sin (c+d x))^{5/2} \, dx","Integrate[Sec[c + d*x]^5*(a + b*Sin[c + d*x])^(5/2),x]","-\frac{3 \sqrt{a-b} \left(a^2-b^2\right)^2 \left(4 a^2+2 a b-b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a+b \sin (c+d x)}}{\sqrt{a-b}}\right)-3 \sqrt{a+b} \left(a^2-b^2\right)^2 \left(4 a^2-2 a b-b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a+b \sin (c+d x)}}{\sqrt{a+b}}\right)+8 \left(b^2-a^2\right) \sec ^4(c+d x) (a \sin (c+d x)-b) (a+b \sin (c+d x))^{7/2}-2 \sec ^2(c+d x) \left(6 a^3 \sin (c+d x)-7 a^2 b+b^3\right) (a+b \sin (c+d x))^{7/2}-2 b \sqrt{a+b \sin (c+d x)} \left(18 a^5-3 a^3 b^2 \cos (2 (c+d x))-16 a^3 b^2+b \left(18 a^4-7 a^2 b^2+b^4\right) \sin (c+d x)+7 a b^4\right)}{32 d \left(a^2-b^2\right)^2}","-\frac{3 \sqrt{a-b} \left(4 a^2+2 a b-b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a+b \sin (c+d x)}}{\sqrt{a-b}}\right)}{32 d}+\frac{3 \sqrt{a+b} \left(4 a^2-2 a b-b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a+b \sin (c+d x)}}{\sqrt{a+b}}\right)}{32 d}+\frac{3 \sec ^2(c+d x) \sqrt{a+b \sin (c+d x)} \left(\left(2 a^2-b^2\right) \sin (c+d x)+a b\right)}{16 d}+\frac{\sec ^4(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^{3/2}}{4 d}",1,"-1/32*(3*Sqrt[a - b]*(a^2 - b^2)^2*(4*a^2 + 2*a*b - b^2)*ArcTanh[Sqrt[a + b*Sin[c + d*x]]/Sqrt[a - b]] - 3*Sqrt[a + b]*(a^2 - b^2)^2*(4*a^2 - 2*a*b - b^2)*ArcTanh[Sqrt[a + b*Sin[c + d*x]]/Sqrt[a + b]] + 8*(-a^2 + b^2)*Sec[c + d*x]^4*(-b + a*Sin[c + d*x])*(a + b*Sin[c + d*x])^(7/2) - 2*Sec[c + d*x]^2*(-7*a^2*b + b^3 + 6*a^3*Sin[c + d*x])*(a + b*Sin[c + d*x])^(7/2) - 2*b*Sqrt[a + b*Sin[c + d*x]]*(18*a^5 - 16*a^3*b^2 + 7*a*b^4 - 3*a^3*b^2*Cos[2*(c + d*x)] + b*(18*a^4 - 7*a^2*b^2 + b^4)*Sin[c + d*x]))/((a^2 - b^2)^2*d)","A",1
500,1,321,398,1.2961281,"\int \cos ^4(c+d x) (a+b \sin (c+d x))^{5/2} \, dx","Integrate[Cos[c + d*x]^4*(a + b*Sin[c + d*x])^(5/2),x]","\frac{128 \sqrt{\frac{a+b \sin (c+d x)}{a+b}} \left(b \left(5 a^5 b-1450 a^3 b^3-603 a b^5\right) F\left(\frac{1}{4} (-2 c-2 d x+\pi )|\frac{2 b}{a+b}\right)+\left(20 a^6-175 a^4 b^2-1662 a^2 b^4-231 b^6\right) \left((a+b) E\left(\frac{1}{4} (-2 c-2 d x+\pi )|\frac{2 b}{a+b}\right)-a F\left(\frac{1}{4} (-2 c-2 d x+\pi )|\frac{2 b}{a+b}\right)\right)\right)-b (a+b \sin (c+d x)) \left(-10 a b^2 \left(20 a^2-2599 b^2\right) \cos (3 (c+d x))+140 b^3 \left(22 b^2-53 a^2\right) \sin (4 (c+d x))-b \left(480 a^4+56120 a^2 b^2+4697 b^4\right) \sin (2 (c+d x))+4 a \left(320 a^4-2710 a^2 b^2+6453 b^4\right) \cos (c+d x)+5670 a b^4 \cos (5 (c+d x))+1155 b^5 \sin (6 (c+d x))\right)}{240240 b^4 d \sqrt{a+b \sin (c+d x)}}","\frac{2 \cos ^3(c+d x) \sqrt{a+b \sin (c+d x)} \left(7 b \left(53 a^2+11 b^2\right) \sin (c+d x)+a \left(5 a^2+59 b^2\right)\right)}{3003 b d}-\frac{4 \cos (c+d x) \sqrt{a+b \sin (c+d x)} \left(4 a \left(5 a^4-40 a^2 b^2-93 b^4\right)-3 b \left(5 a^4+430 a^2 b^2+77 b^4\right) \sin (c+d x)\right)}{15015 b^3 d}+\frac{32 a \left(5 a^6-45 a^4 b^2-53 a^2 b^4+93 b^6\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{15015 b^4 d \sqrt{a+b \sin (c+d x)}}-\frac{8 \left(20 a^6-175 a^4 b^2-1662 a^2 b^4-231 b^6\right) \sqrt{a+b \sin (c+d x)} E\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{15015 b^4 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}-\frac{2 b \cos ^5(c+d x) (a+b \sin (c+d x))^{3/2}}{13 d}-\frac{32 a b \cos ^5(c+d x) \sqrt{a+b \sin (c+d x)}}{143 d}",1,"(128*(b*(5*a^5*b - 1450*a^3*b^3 - 603*a*b^5)*EllipticF[(-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)] + (20*a^6 - 175*a^4*b^2 - 1662*a^2*b^4 - 231*b^6)*((a + b)*EllipticE[(-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)] - a*EllipticF[(-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)]))*Sqrt[(a + b*Sin[c + d*x])/(a + b)] - b*(a + b*Sin[c + d*x])*(4*a*(320*a^4 - 2710*a^2*b^2 + 6453*b^4)*Cos[c + d*x] - 10*a*b^2*(20*a^2 - 2599*b^2)*Cos[3*(c + d*x)] + 5670*a*b^4*Cos[5*(c + d*x)] - b*(480*a^4 + 56120*a^2*b^2 + 4697*b^4)*Sin[2*(c + d*x)] + 140*b^3*(-53*a^2 + 22*b^2)*Sin[4*(c + d*x)] + 1155*b^5*Sin[6*(c + d*x)]))/(240240*b^4*d*Sqrt[a + b*Sin[c + d*x]])","A",1
501,1,239,299,1.0371517,"\int \cos ^2(c+d x) (a+b \sin (c+d x))^{5/2} \, dx","Integrate[Cos[c + d*x]^2*(a + b*Sin[c + d*x])^(5/2),x]","\frac{b (a+b \sin (c+d x)) \left(\left(40 a^3-354 a b^2\right) \cos (c+d x)+2 b \left(\sin (2 (c+d x)) \left(150 a^2-35 b^2 \cos (2 (c+d x))+7 b^2\right)-95 a b \cos (3 (c+d x))\right)\right)-16 \sqrt{\frac{a+b \sin (c+d x)}{a+b}} \left(16 b \left(5 a^3 b+3 a b^3\right) F\left(\frac{1}{4} (-2 c-2 d x+\pi )|\frac{2 b}{a+b}\right)+\left(5 a^4+102 a^2 b^2+21 b^4\right) \left((a+b) E\left(\frac{1}{4} (-2 c-2 d x+\pi )|\frac{2 b}{a+b}\right)-a F\left(\frac{1}{4} (-2 c-2 d x+\pi )|\frac{2 b}{a+b}\right)\right)\right)}{1260 b^2 d \sqrt{a+b \sin (c+d x)}}","\frac{2 \cos (c+d x) \sqrt{a+b \sin (c+d x)} \left(3 b \left(25 a^2+7 b^2\right) \sin (c+d x)+a \left(5 a^2+27 b^2\right)\right)}{315 b d}-\frac{4 a \left(5 a^4+22 a^2 b^2-27 b^4\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{315 b^2 d \sqrt{a+b \sin (c+d x)}}+\frac{4 \left(5 a^4+102 a^2 b^2+21 b^4\right) \sqrt{a+b \sin (c+d x)} E\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{315 b^2 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}-\frac{2 b \cos ^3(c+d x) (a+b \sin (c+d x))^{3/2}}{9 d}-\frac{8 a b \cos ^3(c+d x) \sqrt{a+b \sin (c+d x)}}{21 d}",1,"(-16*(16*b*(5*a^3*b + 3*a*b^3)*EllipticF[(-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)] + (5*a^4 + 102*a^2*b^2 + 21*b^4)*((a + b)*EllipticE[(-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)] - a*EllipticF[(-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)]))*Sqrt[(a + b*Sin[c + d*x])/(a + b)] + b*(a + b*Sin[c + d*x])*((40*a^3 - 354*a*b^2)*Cos[c + d*x] + 2*b*(-95*a*b*Cos[3*(c + d*x)] + (150*a^2 + 7*b^2 - 35*b^2*Cos[2*(c + d*x)])*Sin[2*(c + d*x)])))/(1260*b^2*d*Sqrt[a + b*Sin[c + d*x]])","A",1
502,1,203,203,0.895837,"\int \sec ^2(c+d x) (a+b \sin (c+d x))^{5/2} \, dx","Integrate[Sec[c + d*x]^2*(a + b*Sin[c + d*x])^(5/2),x]","\frac{a^3 \tan (c+d x)-a \left(a^2-b^2\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{4} (-2 c-2 d x+\pi )|\frac{2 b}{a+b}\right)+2 a^2 b \sec (c+d x)+a^2 b \sin (c+d x) \tan (c+d x)+\left(a^3+a^2 b+3 a b^2+3 b^3\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} E\left(\frac{1}{4} (-2 c-2 d x+\pi )|\frac{2 b}{a+b}\right)+3 a b^2 \tan (c+d x)+b^3 \sin (c+d x) \tan (c+d x)}{d \sqrt{a+b \sin (c+d x)}}","\frac{a \left(a^2-b^2\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{d \sqrt{a+b \sin (c+d x)}}-\frac{\left(a^2+3 b^2\right) \sqrt{a+b \sin (c+d x)} E\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}+\frac{a b \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{d}+\frac{\sec (c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^{3/2}}{d}",1,"(2*a^2*b*Sec[c + d*x] + (a^3 + a^2*b + 3*a*b^2 + 3*b^3)*EllipticE[(-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)] - a*(a^2 - b^2)*EllipticF[(-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)] + a^3*Tan[c + d*x] + 3*a*b^2*Tan[c + d*x] + a^2*b*Sin[c + d*x]*Tan[c + d*x] + b^3*Sin[c + d*x]*Tan[c + d*x])/(d*Sqrt[a + b*Sin[c + d*x]])","A",1
503,1,259,238,3.5200862,"\int \sec ^4(c+d x) (a+b \sin (c+d x))^{5/2} \, dx","Integrate[Sec[c + d*x]^4*(a + b*Sin[c + d*x])^(5/2),x]","\frac{-4 a \left(a^2-b^2\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{4} (-2 c-2 d x+\pi )|\frac{2 b}{a+b}\right)+\left(4 a^3+4 a^2 b-3 a b^2-3 b^3\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} E\left(\frac{1}{4} (-2 c-2 d x+\pi )|\frac{2 b}{a+b}\right)+\frac{1}{8} \sec ^3(c+d x) \left(24 a^3 \sin (c+d x)+8 a^3 \sin (3 (c+d x))-4 \left(3 a^2 b+2 b^3\right) \cos (2 (c+d x))+\left(3 b^3-4 a^2 b\right) \cos (4 (c+d x))+40 a^2 b+40 a b^2 \sin (c+d x)-8 a b^2 \sin (3 (c+d x))+5 b^3\right)}{6 d \sqrt{a+b \sin (c+d x)}}","\frac{\sec (c+d x) \sqrt{a+b \sin (c+d x)} \left(\left(4 a^2-3 b^2\right) \sin (c+d x)+a b\right)}{6 d}+\frac{2 a \left(a^2-b^2\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{3 d \sqrt{a+b \sin (c+d x)}}-\frac{\left(4 a^2-3 b^2\right) \sqrt{a+b \sin (c+d x)} E\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{6 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}+\frac{\sec ^3(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^{3/2}}{3 d}",1,"((4*a^3 + 4*a^2*b - 3*a*b^2 - 3*b^3)*EllipticE[(-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)] - 4*a*(a^2 - b^2)*EllipticF[(-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)] + (Sec[c + d*x]^3*(40*a^2*b + 5*b^3 - 4*(3*a^2*b + 2*b^3)*Cos[2*(c + d*x)] + (-4*a^2*b + 3*b^3)*Cos[4*(c + d*x)] + 24*a^3*Sin[c + d*x] + 40*a*b^2*Sin[c + d*x] + 8*a^3*Sin[3*(c + d*x)] - 8*a*b^2*Sin[3*(c + d*x)]))/8)/(6*d*Sqrt[a + b*Sin[c + d*x]])","A",1
504,1,351,322,6.2735193,"\int \sec ^6(c+d x) (a+b \sin (c+d x))^{5/2} \, dx","Integrate[Sec[c + d*x]^6*(a + b*Sin[c + d*x])^(5/2),x]","\frac{\sqrt{a+b \sin (c+d x)} \left(\frac{1}{5} \sec ^5(c+d x) \left(a^2 \sin (c+d x)+2 a b+b^2 \sin (c+d x)\right)+\frac{1}{30} \sec ^3(c+d x) \left(8 a^2 \sin (c+d x)-a b-3 b^2 \sin (c+d x)\right)+\frac{1}{60} \sec (c+d x) \left(32 a^2 \sin (c+d x)-8 a b-9 b^2 \sin (c+d x)\right)\right)}{d}-\frac{b \left(-\frac{\left(32 a^2-9 b^2\right) \left(\frac{2 (a+b) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} E\left(\frac{1}{2} \left(-c-d x+\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{\sqrt{a+b \sin (c+d x)}}-\frac{2 a \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{2} \left(-c-d x+\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{\sqrt{a+b \sin (c+d x)}}\right)}{b}-\frac{16 a b \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{2} \left(-c-d x+\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{\sqrt{a+b \sin (c+d x)}}\right)}{120 d}","\frac{\sec ^3(c+d x) \sqrt{a+b \sin (c+d x)} \left(\left(8 a^2-3 b^2\right) \sin (c+d x)+5 a b\right)}{30 d}+\frac{a \left(32 a^2-17 b^2\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{60 d \sqrt{a+b \sin (c+d x)}}-\frac{\left(32 a^2-9 b^2\right) \sqrt{a+b \sin (c+d x)} E\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{60 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}-\frac{\sec (c+d x) \sqrt{a+b \sin (c+d x)} \left(8 a b \left(a^2-b^2\right)-\left(32 a^4-41 a^2 b^2+9 b^4\right) \sin (c+d x)\right)}{60 d \left(a^2-b^2\right)}+\frac{\sec ^5(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^{3/2}}{5 d}",1,"(Sqrt[a + b*Sin[c + d*x]]*((Sec[c + d*x]*(-8*a*b + 32*a^2*Sin[c + d*x] - 9*b^2*Sin[c + d*x]))/60 + (Sec[c + d*x]^3*(-(a*b) + 8*a^2*Sin[c + d*x] - 3*b^2*Sin[c + d*x]))/30 + (Sec[c + d*x]^5*(2*a*b + a^2*Sin[c + d*x] + b^2*Sin[c + d*x]))/5))/d - (b*((-16*a*b*EllipticF[(-c + Pi/2 - d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/Sqrt[a + b*Sin[c + d*x]] - ((32*a^2 - 9*b^2)*((2*(a + b)*EllipticE[(-c + Pi/2 - d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/Sqrt[a + b*Sin[c + d*x]] - (2*a*EllipticF[(-c + Pi/2 - d*x)/2, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/Sqrt[a + b*Sin[c + d*x]]))/b))/(120*d)","A",1
505,1,338,439,4.4405292,"\int \sec ^8(c+d x) (a+b \sin (c+d x))^{5/2} \, dx","Integrate[Sec[c + d*x]^8*(a + b*Sin[c + d*x])^(5/2),x]","\frac{\frac{\sec (c+d x) (a+b \sin (c+d x)) \left(128 a^4 \sin (c+d x)-32 a^3 b-144 a^2 b^2 \sin (c+d x)+40 \left(a^2-b^2\right) \sec ^6(c+d x) \left(\left(a^2+b^2\right) \sin (c+d x)+2 a b\right)-4 \left(a^2-b^2\right) \sec ^4(c+d x) \left(3 \left(b^2-4 a^2\right) \sin (c+d x)+a b\right)+2 \left(a^2-b^2\right) \sec ^2(c+d x) \left(\left(32 a^2-7 b^2\right) \sin (c+d x)-4 a b\right)+27 a b^3+21 b^4 \sin (c+d x)\right)}{a^2-b^2}+\frac{\sqrt{\frac{a+b \sin (c+d x)}{a+b}} \left(\left(128 a^4-144 a^2 b^2+21 b^4\right) E\left(\frac{1}{4} (-2 c-2 d x+\pi )|\frac{2 b}{a+b}\right)-16 a \left(8 a^3-8 a^2 b-3 a b^2+3 b^3\right) F\left(\frac{1}{4} (-2 c-2 d x+\pi )|\frac{2 b}{a+b}\right)\right)}{a-b}}{280 d \sqrt{a+b \sin (c+d x)}}","\frac{3 \sec ^5(c+d x) \sqrt{a+b \sin (c+d x)} \left(\left(4 a^2-b^2\right) \sin (c+d x)+3 a b\right)}{70 d}+\frac{2 a \left(8 a^2-3 b^2\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{35 d \sqrt{a+b \sin (c+d x)}}-\frac{\sec ^3(c+d x) \sqrt{a+b \sin (c+d x)} \left(4 a b \left(a^2-b^2\right)-\left(32 a^4-39 a^2 b^2+7 b^4\right) \sin (c+d x)\right)}{140 d \left(a^2-b^2\right)}-\frac{\left(128 a^4-144 a^2 b^2+21 b^4\right) \sqrt{a+b \sin (c+d x)} E\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{280 d \left(a^2-b^2\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}-\frac{\sec (c+d x) \sqrt{a+b \sin (c+d x)} \left(a b \left(32 a^4-59 a^2 b^2+27 b^4\right)-\left(128 a^6-272 a^4 b^2+165 a^2 b^4-21 b^6\right) \sin (c+d x)\right)}{280 d \left(a^2-b^2\right)^2}+\frac{\sec ^7(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^{3/2}}{7 d}",1,"((((128*a^4 - 144*a^2*b^2 + 21*b^4)*EllipticE[(-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)] - 16*a*(8*a^3 - 8*a^2*b - 3*a*b^2 + 3*b^3)*EllipticF[(-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)])*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(a - b) + (Sec[c + d*x]*(a + b*Sin[c + d*x])*(-32*a^3*b + 27*a*b^3 + 128*a^4*Sin[c + d*x] - 144*a^2*b^2*Sin[c + d*x] + 21*b^4*Sin[c + d*x] + 2*(a^2 - b^2)*Sec[c + d*x]^2*(-4*a*b + (32*a^2 - 7*b^2)*Sin[c + d*x]) - 4*(a^2 - b^2)*Sec[c + d*x]^4*(a*b + 3*(-4*a^2 + b^2)*Sin[c + d*x]) + 40*(a^2 - b^2)*Sec[c + d*x]^6*(2*a*b + (a^2 + b^2)*Sin[c + d*x])))/(a^2 - b^2))/(280*d*Sqrt[a + b*Sin[c + d*x]])","A",1
506,1,118,152,0.3145215,"\int \frac{\cos ^5(c+d x)}{\sqrt{a+b \sin (c+d x)}} \, dx","Integrate[Cos[c + d*x]^5/Sqrt[a + b*Sin[c + d*x]],x]","\frac{\sqrt{a+b \sin (c+d x)} \left(1024 a^4-512 a^3 b \sin (c+d x)-2496 a^2 b^2-4 \left(48 a^2 b^2-91 b^4\right) \cos (2 (c+d x))+1104 a b^3 \sin (c+d x)+80 a b^3 \sin (3 (c+d x))+35 b^4 \cos (4 (c+d x))+2121 b^4\right)}{1260 b^5 d}","\frac{4 \left(3 a^2-b^2\right) (a+b \sin (c+d x))^{5/2}}{5 b^5 d}-\frac{8 a \left(a^2-b^2\right) (a+b \sin (c+d x))^{3/2}}{3 b^5 d}+\frac{2 \left(a^2-b^2\right)^2 \sqrt{a+b \sin (c+d x)}}{b^5 d}+\frac{2 (a+b \sin (c+d x))^{9/2}}{9 b^5 d}-\frac{8 a (a+b \sin (c+d x))^{7/2}}{7 b^5 d}",1,"(Sqrt[a + b*Sin[c + d*x]]*(1024*a^4 - 2496*a^2*b^2 + 2121*b^4 - 4*(48*a^2*b^2 - 91*b^4)*Cos[2*(c + d*x)] + 35*b^4*Cos[4*(c + d*x)] - 512*a^3*b*Sin[c + d*x] + 1104*a*b^3*Sin[c + d*x] + 80*a*b^3*Sin[3*(c + d*x)]))/(1260*b^5*d)","A",1
507,1,58,81,0.0976764,"\int \frac{\cos ^3(c+d x)}{\sqrt{a+b \sin (c+d x)}} \, dx","Integrate[Cos[c + d*x]^3/Sqrt[a + b*Sin[c + d*x]],x]","\frac{2 \sqrt{a+b \sin (c+d x)} \left(-8 a^2+4 a b \sin (c+d x)-3 b^2 \sin ^2(c+d x)+15 b^2\right)}{15 b^3 d}","-\frac{2 \left(a^2-b^2\right) \sqrt{a+b \sin (c+d x)}}{b^3 d}-\frac{2 (a+b \sin (c+d x))^{5/2}}{5 b^3 d}+\frac{4 a (a+b \sin (c+d x))^{3/2}}{3 b^3 d}",1,"(2*Sqrt[a + b*Sin[c + d*x]]*(-8*a^2 + 15*b^2 + 4*a*b*Sin[c + d*x] - 3*b^2*Sin[c + d*x]^2))/(15*b^3*d)","A",1
508,1,22,22,0.0130325,"\int \frac{\cos (c+d x)}{\sqrt{a+b \sin (c+d x)}} \, dx","Integrate[Cos[c + d*x]/Sqrt[a + b*Sin[c + d*x]],x]","\frac{2 \sqrt{a+b \sin (c+d x)}}{b d}","\frac{2 \sqrt{a+b \sin (c+d x)}}{b d}",1,"(2*Sqrt[a + b*Sin[c + d*x]])/(b*d)","A",1
509,1,74,74,0.054422,"\int \frac{\sec (c+d x)}{\sqrt{a+b \sin (c+d x)}} \, dx","Integrate[Sec[c + d*x]/Sqrt[a + b*Sin[c + d*x]],x]","\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \sin (c+d x)}}{\sqrt{a+b}}\right)}{d \sqrt{a+b}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \sin (c+d x)}}{\sqrt{a-b}}\right)}{d \sqrt{a-b}}","\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \sin (c+d x)}}{\sqrt{a+b}}\right)}{d \sqrt{a+b}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \sin (c+d x)}}{\sqrt{a-b}}\right)}{d \sqrt{a-b}}",1,"-(ArcTanh[Sqrt[a + b*Sin[c + d*x]]/Sqrt[a - b]]/(Sqrt[a - b]*d)) + ArcTanh[Sqrt[a + b*Sin[c + d*x]]/Sqrt[a + b]]/(Sqrt[a + b]*d)","A",1
510,1,176,144,0.5123687,"\int \frac{\sec ^3(c+d x)}{\sqrt{a+b \sin (c+d x)}} \, dx","Integrate[Sec[c + d*x]^3/Sqrt[a + b*Sin[c + d*x]],x]","\frac{\sqrt{a+b} \left(2 a^2-a b-3 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a+b \sin (c+d x)}}{\sqrt{a-b}}\right)-\sqrt{a-b} \left(\left(2 a^2+a b-3 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a+b \sin (c+d x)}}{\sqrt{a+b}}\right)+2 \sqrt{a+b} \sec ^2(c+d x) (a \sin (c+d x)-b) \sqrt{a+b \sin (c+d x)}\right)}{4 d \sqrt{a-b} \sqrt{a+b} \left(b^2-a^2\right)}","-\frac{\sec ^2(c+d x) (b-a \sin (c+d x)) \sqrt{a+b \sin (c+d x)}}{2 d \left(a^2-b^2\right)}-\frac{(2 a-3 b) \tanh ^{-1}\left(\frac{\sqrt{a+b \sin (c+d x)}}{\sqrt{a-b}}\right)}{4 d (a-b)^{3/2}}+\frac{(2 a+3 b) \tanh ^{-1}\left(\frac{\sqrt{a+b \sin (c+d x)}}{\sqrt{a+b}}\right)}{4 d (a+b)^{3/2}}",1,"(Sqrt[a + b]*(2*a^2 - a*b - 3*b^2)*ArcTanh[Sqrt[a + b*Sin[c + d*x]]/Sqrt[a - b]] - Sqrt[a - b]*((2*a^2 + a*b - 3*b^2)*ArcTanh[Sqrt[a + b*Sin[c + d*x]]/Sqrt[a + b]] + 2*Sqrt[a + b]*Sec[c + d*x]^2*(-b + a*Sin[c + d*x])*Sqrt[a + b*Sin[c + d*x]]))/(4*Sqrt[a - b]*Sqrt[a + b]*(-a^2 + b^2)*d)","A",1
511,1,244,230,1.9025149,"\int \frac{\sec ^5(c+d x)}{\sqrt{a+b \sin (c+d x)}} \, dx","Integrate[Sec[c + d*x]^5/Sqrt[a + b*Sin[c + d*x]],x]","\frac{\sqrt{a-b} \left(3 (a-b)^2 \left(4 a^2+10 a b+7 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a+b \sin (c+d x)}}{\sqrt{a+b}}\right)+\sqrt{a+b} \sec ^4(c+d x) \sqrt{a+b \sin (c+d x)} \left(3 \left(a^3-2 a b^2\right) \sin (3 (c+d x))+\left(7 b^3-a^2 b\right) \cos (2 (c+d x))+a \left(11 a^2-14 b^2\right) \sin (c+d x)-9 a^2 b+15 b^3\right)\right)-3 (a+b)^{5/2} \left(4 a^2-10 a b+7 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a+b \sin (c+d x)}}{\sqrt{a-b}}\right)}{32 d \sqrt{a-b} \sqrt{a+b} \left(a^2-b^2\right)^2}","-\frac{3 \left(4 a^2-10 a b+7 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a+b \sin (c+d x)}}{\sqrt{a-b}}\right)}{32 d (a-b)^{5/2}}+\frac{3 \left(4 a^2+10 a b+7 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a+b \sin (c+d x)}}{\sqrt{a+b}}\right)}{32 d (a+b)^{5/2}}-\frac{\sec ^4(c+d x) (b-a \sin (c+d x)) \sqrt{a+b \sin (c+d x)}}{4 d \left(a^2-b^2\right)}-\frac{\sec ^2(c+d x) \sqrt{a+b \sin (c+d x)} \left(b \left(a^2-7 b^2\right)-6 a \left(a^2-2 b^2\right) \sin (c+d x)\right)}{16 d \left(a^2-b^2\right)^2}",1,"(-3*(a + b)^(5/2)*(4*a^2 - 10*a*b + 7*b^2)*ArcTanh[Sqrt[a + b*Sin[c + d*x]]/Sqrt[a - b]] + Sqrt[a - b]*(3*(a - b)^2*(4*a^2 + 10*a*b + 7*b^2)*ArcTanh[Sqrt[a + b*Sin[c + d*x]]/Sqrt[a + b]] + Sqrt[a + b]*Sec[c + d*x]^4*Sqrt[a + b*Sin[c + d*x]]*(-9*a^2*b + 15*b^3 + (-(a^2*b) + 7*b^3)*Cos[2*(c + d*x)] + a*(11*a^2 - 14*b^2)*Sin[c + d*x] + 3*(a^3 - 2*a*b^2)*Sin[3*(c + d*x)])))/(32*Sqrt[a - b]*Sqrt[a + b]*(a^2 - b^2)^2*d)","A",1
512,1,219,247,1.073232,"\int \frac{\cos ^4(c+d x)}{\sqrt{a+b \sin (c+d x)}} \, dx","Integrate[Cos[c + d*x]^4/Sqrt[a + b*Sin[c + d*x]],x]","\frac{-16 \left(4 a^4-9 a^2 b^2+5 b^4\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{4} (-2 c-2 d x+\pi )|\frac{2 b}{a+b}\right)+b \cos (c+d x) \left(-32 a^3+\left(45 b^3-8 a^2 b\right) \sin (c+d x)-2 a b^2 \cos (2 (c+d x))+62 a b^2+5 b^3 \sin (3 (c+d x))\right)+64 a \left(a^3+a^2 b-2 a b^2-2 b^3\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} E\left(\frac{1}{4} (-2 c-2 d x+\pi )|\frac{2 b}{a+b}\right)}{70 b^4 d \sqrt{a+b \sin (c+d x)}}","-\frac{32 a \left(a^2-2 b^2\right) \sqrt{a+b \sin (c+d x)} E\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{35 b^4 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}-\frac{4 \cos (c+d x) \sqrt{a+b \sin (c+d x)} \left(4 a^2-3 a b \sin (c+d x)-5 b^2\right)}{35 b^3 d}+\frac{8 \left(4 a^4-9 a^2 b^2+5 b^4\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{35 b^4 d \sqrt{a+b \sin (c+d x)}}+\frac{2 \cos ^3(c+d x) \sqrt{a+b \sin (c+d x)}}{7 b d}",1,"(64*a*(a^3 + a^2*b - 2*a*b^2 - 2*b^3)*EllipticE[(-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)] - 16*(4*a^4 - 9*a^2*b^2 + 5*b^4)*EllipticF[(-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)] + b*Cos[c + d*x]*(-32*a^3 + 62*a*b^2 - 2*a*b^2*Cos[2*(c + d*x)] + (-8*a^2*b + 45*b^3)*Sin[c + d*x] + 5*b^3*Sin[3*(c + d*x)]))/(70*b^4*d*Sqrt[a + b*Sin[c + d*x]])","A",1
513,1,145,175,0.8160047,"\int \frac{\cos ^2(c+d x)}{\sqrt{a+b \sin (c+d x)}} \, dx","Integrate[Cos[c + d*x]^2/Sqrt[a + b*Sin[c + d*x]],x]","\frac{4 \left(a^2-b^2\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{4} (-2 c-2 d x+\pi )|\frac{2 b}{a+b}\right)+2 b \cos (c+d x) (a+b \sin (c+d x))-4 a (a+b) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} E\left(\frac{1}{4} (-2 c-2 d x+\pi )|\frac{2 b}{a+b}\right)}{3 b^2 d \sqrt{a+b \sin (c+d x)}}","-\frac{4 \left(a^2-b^2\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{3 b^2 d \sqrt{a+b \sin (c+d x)}}+\frac{4 a \sqrt{a+b \sin (c+d x)} E\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{3 b^2 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}+\frac{2 \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{3 b d}",1,"(2*b*Cos[c + d*x]*(a + b*Sin[c + d*x]) - 4*a*(a + b)*EllipticE[(-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)] + 4*(a^2 - b^2)*EllipticF[(-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/(3*b^2*d*Sqrt[a + b*Sin[c + d*x]])","A",1
514,1,177,183,0.6341692,"\int \frac{\sec ^2(c+d x)}{\sqrt{a+b \sin (c+d x)}} \, dx","Integrate[Sec[c + d*x]^2/Sqrt[a + b*Sin[c + d*x]],x]","\frac{-\left(a^2-b^2\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{4} (-2 c-2 d x+\pi )|\frac{2 b}{a+b}\right)+a^2 \tan (c+d x)-a b \sec (c+d x)+a b \sin (c+d x) \tan (c+d x)+a (a+b) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} E\left(\frac{1}{4} (-2 c-2 d x+\pi )|\frac{2 b}{a+b}\right)-b^2 \tan (c+d x)}{d (a-b) (a+b) \sqrt{a+b \sin (c+d x)}}","-\frac{\sec (c+d x) (b-a \sin (c+d x)) \sqrt{a+b \sin (c+d x)}}{d \left(a^2-b^2\right)}-\frac{a \sqrt{a+b \sin (c+d x)} E\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{d \left(a^2-b^2\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}+\frac{\sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{d \sqrt{a+b \sin (c+d x)}}",1,"(-(a*b*Sec[c + d*x]) + a*(a + b)*EllipticE[(-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)] - (a^2 - b^2)*EllipticF[(-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)] + a^2*Tan[c + d*x] - b^2*Tan[c + d*x] + a*b*Sin[c + d*x]*Tan[c + d*x])/((a - b)*(a + b)*d*Sqrt[a + b*Sin[c + d*x]])","A",1
515,1,306,291,4.1428207,"\int \frac{\sec ^4(c+d x)}{\sqrt{a+b \sin (c+d x)}} \, dx","Integrate[Sec[c + d*x]^4/Sqrt[a + b*Sin[c + d*x]],x]","\frac{-4 \left(4 a^4-9 a^2 b^2+5 b^4\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{4} (-2 c-2 d x+\pi )|\frac{2 b}{a+b}\right)+16 a \left(a^3+a^2 b-2 a b^2-2 b^3\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} E\left(\frac{1}{4} (-2 c-2 d x+\pi )|\frac{2 b}{a+b}\right)+\sec ^3(c+d x) \left(12 a^4 \sin (c+d x)+4 a^4 \sin (3 (c+d x))+\left(14 a b^3-6 a^3 b\right) \cos (2 (c+d x))+\left(4 a b^3-2 a^3 b\right) \cos (4 (c+d x))-4 a^3 b-25 a^2 b^2 \sin (c+d x)-9 a^2 b^2 \sin (3 (c+d x))+10 a b^3+13 b^4 \sin (c+d x)+5 b^4 \sin (3 (c+d x))\right)}{24 d (a-b)^2 (a+b)^2 \sqrt{a+b \sin (c+d x)}}","-\frac{\sec ^3(c+d x) (b-a \sin (c+d x)) \sqrt{a+b \sin (c+d x)}}{3 d \left(a^2-b^2\right)}-\frac{\sec (c+d x) \sqrt{a+b \sin (c+d x)} \left(b \left(a^2-5 b^2\right)-4 a \left(a^2-2 b^2\right) \sin (c+d x)\right)}{6 d \left(a^2-b^2\right)^2}+\frac{\left(4 a^2-5 b^2\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{6 d \left(a^2-b^2\right) \sqrt{a+b \sin (c+d x)}}-\frac{2 a \left(a^2-2 b^2\right) \sqrt{a+b \sin (c+d x)} E\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{3 d \left(a^2-b^2\right)^2 \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}",1,"(16*a*(a^3 + a^2*b - 2*a*b^2 - 2*b^3)*EllipticE[(-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)] - 4*(4*a^4 - 9*a^2*b^2 + 5*b^4)*EllipticF[(-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)] + Sec[c + d*x]^3*(-4*a^3*b + 10*a*b^3 + (-6*a^3*b + 14*a*b^3)*Cos[2*(c + d*x)] + (-2*a^3*b + 4*a*b^3)*Cos[4*(c + d*x)] + 12*a^4*Sin[c + d*x] - 25*a^2*b^2*Sin[c + d*x] + 13*b^4*Sin[c + d*x] + 4*a^4*Sin[3*(c + d*x)] - 9*a^2*b^2*Sin[3*(c + d*x)] + 5*b^4*Sin[3*(c + d*x)]))/(24*(a - b)^2*(a + b)^2*d*Sqrt[a + b*Sin[c + d*x]])","A",1
516,1,116,150,0.2559262,"\int \frac{\cos ^5(c+d x)}{(a+b \sin (c+d x))^{3/2}} \, dx","Integrate[Cos[c + d*x]^5/(a + b*Sin[c + d*x])^(3/2),x]","\frac{30 b^4 \cos ^4(c+d x)-16 \left(48 a^4+a b \left(24 a^2-35 b^2\right) \sin (c+d x)-70 a^2 b^2+\left(5 b^4-6 a^2 b^2\right) \sin ^2(c+d x)+3 a b^3 \sin ^3(c+d x)+15 b^4\right)}{105 b^5 d \sqrt{a+b \sin (c+d x)}}","\frac{4 \left(3 a^2-b^2\right) (a+b \sin (c+d x))^{3/2}}{3 b^5 d}-\frac{8 a \left(a^2-b^2\right) \sqrt{a+b \sin (c+d x)}}{b^5 d}-\frac{2 \left(a^2-b^2\right)^2}{b^5 d \sqrt{a+b \sin (c+d x)}}+\frac{2 (a+b \sin (c+d x))^{7/2}}{7 b^5 d}-\frac{8 a (a+b \sin (c+d x))^{5/2}}{5 b^5 d}",1,"(30*b^4*Cos[c + d*x]^4 - 16*(48*a^4 - 70*a^2*b^2 + 15*b^4 + a*b*(24*a^2 - 35*b^2)*Sin[c + d*x] + (-6*a^2*b^2 + 5*b^4)*Sin[c + d*x]^2 + 3*a*b^3*Sin[c + d*x]^3))/(105*b^5*d*Sqrt[a + b*Sin[c + d*x]])","A",1
517,1,57,79,0.0640468,"\int \frac{\cos ^3(c+d x)}{(a+b \sin (c+d x))^{3/2}} \, dx","Integrate[Cos[c + d*x]^3/(a + b*Sin[c + d*x])^(3/2),x]","\frac{16 a^2+8 a b \sin (c+d x)+b^2 \cos (2 (c+d x))-7 b^2}{3 b^3 d \sqrt{a+b \sin (c+d x)}}","\frac{2 \left(a^2-b^2\right)}{b^3 d \sqrt{a+b \sin (c+d x)}}-\frac{2 (a+b \sin (c+d x))^{3/2}}{3 b^3 d}+\frac{4 a \sqrt{a+b \sin (c+d x)}}{b^3 d}",1,"(16*a^2 - 7*b^2 + b^2*Cos[2*(c + d*x)] + 8*a*b*Sin[c + d*x])/(3*b^3*d*Sqrt[a + b*Sin[c + d*x]])","A",1
518,1,22,22,0.0153099,"\int \frac{\cos (c+d x)}{(a+b \sin (c+d x))^{3/2}} \, dx","Integrate[Cos[c + d*x]/(a + b*Sin[c + d*x])^(3/2),x]","-\frac{2}{b d \sqrt{a+b \sin (c+d x)}}","-\frac{2}{b d \sqrt{a+b \sin (c+d x)}}",1,"-2/(b*d*Sqrt[a + b*Sin[c + d*x]])","A",1
519,1,91,105,0.0860011,"\int \frac{\sec (c+d x)}{(a+b \sin (c+d x))^{3/2}} \, dx","Integrate[Sec[c + d*x]/(a + b*Sin[c + d*x])^(3/2),x]","\frac{(a+b) \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\frac{a+b \sin (c+d x)}{a-b}\right)+(b-a) \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\frac{a+b \sin (c+d x)}{a+b}\right)}{d (a-b) (a+b) \sqrt{a+b \sin (c+d x)}}","\frac{2 b}{d \left(a^2-b^2\right) \sqrt{a+b \sin (c+d x)}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \sin (c+d x)}}{\sqrt{a-b}}\right)}{d (a-b)^{3/2}}+\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \sin (c+d x)}}{\sqrt{a+b}}\right)}{d (a+b)^{3/2}}",1,"((a + b)*Hypergeometric2F1[-1/2, 1, 1/2, (a + b*Sin[c + d*x])/(a - b)] + (-a + b)*Hypergeometric2F1[-1/2, 1, 1/2, (a + b*Sin[c + d*x])/(a + b)])/((a - b)*(a + b)*d*Sqrt[a + b*Sin[c + d*x]])","C",1
520,1,221,186,1.1985755,"\int \frac{\sec ^3(c+d x)}{(a+b \sin (c+d x))^{3/2}} \, dx","Integrate[Sec[c + d*x]^3/(a + b*Sin[c + d*x])^(3/2),x]","\frac{\frac{\left(a^2+5 b^2\right) \left((a+b) \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\frac{a+b \sin (c+d x)}{a-b}\right)+(b-a) \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\frac{a+b \sin (c+d x)}{a+b}\right)\right)}{(a-b) (a+b) \sqrt{a+b \sin (c+d x)}}+\frac{3 a \tanh ^{-1}\left(\frac{\sqrt{a+b \sin (c+d x)}}{\sqrt{a-b}}\right)}{\sqrt{a-b}}-\frac{3 a \tanh ^{-1}\left(\frac{\sqrt{a+b \sin (c+d x)}}{\sqrt{a+b}}\right)}{\sqrt{a+b}}+\frac{2 \sec ^2(c+d x) (b-a \sin (c+d x))}{\sqrt{a+b \sin (c+d x)}}}{4 d \left(b^2-a^2\right)}","-\frac{b \left(a^2+5 b^2\right)}{2 d \left(a^2-b^2\right)^2 \sqrt{a+b \sin (c+d x)}}-\frac{\sec ^2(c+d x) (b-a \sin (c+d x))}{2 d \left(a^2-b^2\right) \sqrt{a+b \sin (c+d x)}}-\frac{(2 a-5 b) \tanh ^{-1}\left(\frac{\sqrt{a+b \sin (c+d x)}}{\sqrt{a-b}}\right)}{4 d (a-b)^{5/2}}+\frac{(2 a+5 b) \tanh ^{-1}\left(\frac{\sqrt{a+b \sin (c+d x)}}{\sqrt{a+b}}\right)}{4 d (a+b)^{5/2}}",1,"((3*a*ArcTanh[Sqrt[a + b*Sin[c + d*x]]/Sqrt[a - b]])/Sqrt[a - b] - (3*a*ArcTanh[Sqrt[a + b*Sin[c + d*x]]/Sqrt[a + b]])/Sqrt[a + b] + ((a^2 + 5*b^2)*((a + b)*Hypergeometric2F1[-1/2, 1, 1/2, (a + b*Sin[c + d*x])/(a - b)] + (-a + b)*Hypergeometric2F1[-1/2, 1, 1/2, (a + b*Sin[c + d*x])/(a + b)]))/((a - b)*(a + b)*Sqrt[a + b*Sin[c + d*x]]) + (2*Sec[c + d*x]^2*(b - a*Sin[c + d*x]))/Sqrt[a + b*Sin[c + d*x]])/(4*(-a^2 + b^2)*d)","C",1
521,1,324,284,2.2396957,"\int \frac{\sec ^5(c+d x)}{(a+b \sin (c+d x))^{3/2}} \, dx","Integrate[Sec[c + d*x]^5/(a + b*Sin[c + d*x])^(3/2),x]","\frac{3 a \sqrt{a-b} \sqrt{a+b} \left(3 a^2-8 b^2\right) \sqrt{a+b \sin (c+d x)} \left(\sqrt{a+b} \tanh ^{-1}\left(\frac{\sqrt{a+b \sin (c+d x)}}{\sqrt{a-b}}\right)-\sqrt{a-b} \tanh ^{-1}\left(\frac{\sqrt{a+b \sin (c+d x)}}{\sqrt{a+b}}\right)\right)-(a-b) (a+b) \sec ^2(c+d x) \left(2 a \left(3 a^2-8 b^2\right) \sin (c+d x)+b \left(a^2+9 b^2\right)\right)+\frac{3}{2} \left(2 a^4-7 a^2 b^2-15 b^4\right) \left((a+b) \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\frac{a+b \sin (c+d x)}{a-b}\right)+(b-a) \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\frac{a+b \sin (c+d x)}{a+b}\right)\right)-4 (a-b)^2 (a+b)^2 \sec ^4(c+d x) (a \sin (c+d x)-b)}{16 d \left(a^2-b^2\right)^2 \left(b^2-a^2\right) \sqrt{a+b \sin (c+d x)}}","-\frac{3 \left(4 a^2-14 a b+15 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a+b \sin (c+d x)}}{\sqrt{a-b}}\right)}{32 d (a-b)^{7/2}}+\frac{3 \left(4 a^2+14 a b+15 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a+b \sin (c+d x)}}{\sqrt{a+b}}\right)}{32 d (a+b)^{7/2}}-\frac{\sec ^4(c+d x) (b-a \sin (c+d x))}{4 d \left(a^2-b^2\right) \sqrt{a+b \sin (c+d x)}}+\frac{\sec ^2(c+d x) \left(2 a \left(3 a^2-8 b^2\right) \sin (c+d x)+b \left(a^2+9 b^2\right)\right)}{16 d \left(a^2-b^2\right)^2 \sqrt{a+b \sin (c+d x)}}-\frac{3 b \left(2 a^4-7 a^2 b^2-15 b^4\right)}{16 d \left(a^2-b^2\right)^3 \sqrt{a+b \sin (c+d x)}}",1,"((3*(2*a^4 - 7*a^2*b^2 - 15*b^4)*((a + b)*Hypergeometric2F1[-1/2, 1, 1/2, (a + b*Sin[c + d*x])/(a - b)] + (-a + b)*Hypergeometric2F1[-1/2, 1, 1/2, (a + b*Sin[c + d*x])/(a + b)]))/2 - 4*(a - b)^2*(a + b)^2*Sec[c + d*x]^4*(-b + a*Sin[c + d*x]) + 3*a*Sqrt[a - b]*Sqrt[a + b]*(3*a^2 - 8*b^2)*(Sqrt[a + b]*ArcTanh[Sqrt[a + b*Sin[c + d*x]]/Sqrt[a - b]] - Sqrt[a - b]*ArcTanh[Sqrt[a + b*Sin[c + d*x]]/Sqrt[a + b]])*Sqrt[a + b*Sin[c + d*x]] - (a - b)*(a + b)*Sec[c + d*x]^2*(b*(a^2 + 9*b^2) + 2*a*(3*a^2 - 8*b^2)*Sin[c + d*x]))/(16*(a^2 - b^2)^2*(-a^2 + b^2)*d*Sqrt[a + b*Sin[c + d*x]])","C",1
522,1,273,313,1.5122356,"\int \frac{\cos ^6(c+d x)}{(a+b \sin (c+d x))^{3/2}} \, dx","Integrate[Cos[c + d*x]^6/(a + b*Sin[c + d*x])^(3/2),x]","\frac{-64 a \left(32 a^4-65 a^2 b^2+33 b^4\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{4} (-2 c-2 d x+\pi )|\frac{2 b}{a+b}\right)+b \cos (c+d x) \left(-1024 a^4-256 a^3 b \sin (c+d x)+1760 a^2 b^2+\left(84 b^4-64 a^2 b^2\right) \cos (2 (c+d x))+404 a b^3 \sin (c+d x)+20 a b^3 \sin (3 (c+d x))+7 b^4 \cos (4 (c+d x))-595 b^4\right)+64 \left(32 a^5+32 a^4 b-57 a^3 b^2-57 a^2 b^3+21 a b^4+21 b^5\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} E\left(\frac{1}{4} (-2 c-2 d x+\pi )|\frac{2 b}{a+b}\right)}{252 b^6 d \sqrt{a+b \sin (c+d x)}}","-\frac{8 \cos (c+d x) \sqrt{a+b \sin (c+d x)} \left(a \left(32 a^2-33 b^2\right)-3 b \left(8 a^2-7 b^2\right) \sin (c+d x)\right)}{63 b^5 d}+\frac{16 a \left(32 a^4-65 a^2 b^2+33 b^4\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{63 b^6 d \sqrt{a+b \sin (c+d x)}}-\frac{16 \left(32 a^4-57 a^2 b^2+21 b^4\right) \sqrt{a+b \sin (c+d x)} E\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{63 b^6 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}+\frac{20 \cos ^3(c+d x) (8 a-7 b \sin (c+d x)) \sqrt{a+b \sin (c+d x)}}{63 b^3 d}-\frac{2 \cos ^5(c+d x)}{b d \sqrt{a+b \sin (c+d x)}}",1,"(64*(32*a^5 + 32*a^4*b - 57*a^3*b^2 - 57*a^2*b^3 + 21*a*b^4 + 21*b^5)*EllipticE[(-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)] - 64*a*(32*a^4 - 65*a^2*b^2 + 33*b^4)*EllipticF[(-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)] + b*Cos[c + d*x]*(-1024*a^4 + 1760*a^2*b^2 - 595*b^4 + (-64*a^2*b^2 + 84*b^4)*Cos[2*(c + d*x)] + 7*b^4*Cos[4*(c + d*x)] - 256*a^3*b*Sin[c + d*x] + 404*a*b^3*Sin[c + d*x] + 20*a*b^3*Sin[3*(c + d*x)]))/(252*b^6*d*Sqrt[a + b*Sin[c + d*x]])","A",1
523,1,187,229,1.1209438,"\int \frac{\cos ^4(c+d x)}{(a+b \sin (c+d x))^{3/2}} \, dx","Integrate[Cos[c + d*x]^4/(a + b*Sin[c + d*x])^(3/2),x]","\frac{b \cos (c+d x) \left(16 a^2+4 a b \sin (c+d x)+b^2 \cos (2 (c+d x))-11 b^2\right)+32 a \left(a^2-b^2\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{4} (-2 c-2 d x+\pi )|\frac{2 b}{a+b}\right)-8 \left(4 a^3+4 a^2 b-3 a b^2-3 b^3\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} E\left(\frac{1}{4} (-2 c-2 d x+\pi )|\frac{2 b}{a+b}\right)}{5 b^4 d \sqrt{a+b \sin (c+d x)}}","-\frac{32 a \left(a^2-b^2\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{5 b^4 d \sqrt{a+b \sin (c+d x)}}+\frac{8 \left(4 a^2-3 b^2\right) \sqrt{a+b \sin (c+d x)} E\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{5 b^4 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}+\frac{4 \cos (c+d x) (4 a-3 b \sin (c+d x)) \sqrt{a+b \sin (c+d x)}}{5 b^3 d}-\frac{2 \cos ^3(c+d x)}{b d \sqrt{a+b \sin (c+d x)}}",1,"(-8*(4*a^3 + 4*a^2*b - 3*a*b^2 - 3*b^3)*EllipticE[(-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)] + 32*a*(a^2 - b^2)*EllipticF[(-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)] + b*Cos[c + d*x]*(16*a^2 - 11*b^2 + b^2*Cos[2*(c + d*x)] + 4*a*b*Sin[c + d*x]))/(5*b^4*d*Sqrt[a + b*Sin[c + d*x]])","A",1
524,1,125,160,2.7695324,"\int \frac{\cos ^2(c+d x)}{(a+b \sin (c+d x))^{3/2}} \, dx","Integrate[Cos[c + d*x]^2/(a + b*Sin[c + d*x])^(3/2),x]","\frac{4 (a+b) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} E\left(\frac{1}{4} (-2 c-2 d x+\pi )|\frac{2 b}{a+b}\right)-2 \left(2 a \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{4} (-2 c-2 d x+\pi )|\frac{2 b}{a+b}\right)+b \cos (c+d x)\right)}{b^2 d \sqrt{a+b \sin (c+d x)}}","\frac{4 a \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{b^2 d \sqrt{a+b \sin (c+d x)}}-\frac{4 \sqrt{a+b \sin (c+d x)} E\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{b^2 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}-\frac{2 \cos (c+d x)}{b d \sqrt{a+b \sin (c+d x)}}",1,"(4*(a + b)*EllipticE[(-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)] - 2*(b*Cos[c + d*x] + 2*a*EllipticF[(-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)]))/(b^2*d*Sqrt[a + b*Sin[c + d*x]])","A",1
525,1,205,251,1.7484555,"\int \frac{\sec ^2(c+d x)}{(a+b \sin (c+d x))^{3/2}} \, dx","Integrate[Sec[c + d*x]^2/(a + b*Sin[c + d*x])^(3/2),x]","\frac{-a \left(a^2-b^2\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{4} (-2 c-2 d x+\pi )|\frac{2 b}{a+b}\right)-\frac{1}{2} \sec (c+d x) \left(-2 a \left(a^2-b^2\right) \sin (c+d x)+b \left(a^2+3 b^2\right) \cos (2 (c+d x))+3 a^2 b+b^3\right)+\left(a^3+a^2 b+3 a b^2+3 b^3\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} E\left(\frac{1}{4} (-2 c-2 d x+\pi )|\frac{2 b}{a+b}\right)}{d (a-b)^2 (a+b)^2 \sqrt{a+b \sin (c+d x)}}","-\frac{\sec (c+d x) \sqrt{a+b \sin (c+d x)} \left(4 a b-\left(a^2+3 b^2\right) \sin (c+d x)\right)}{d \left(a^2-b^2\right)^2}+\frac{2 b \sec (c+d x)}{d \left(a^2-b^2\right) \sqrt{a+b \sin (c+d x)}}+\frac{a \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{d \left(a^2-b^2\right) \sqrt{a+b \sin (c+d x)}}-\frac{\left(a^2+3 b^2\right) \sqrt{a+b \sin (c+d x)} E\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{d \left(a^2-b^2\right)^2 \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}",1,"((a^3 + a^2*b + 3*a*b^2 + 3*b^3)*EllipticE[(-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)] - a*(a^2 - b^2)*EllipticF[(-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)] - (Sec[c + d*x]*(3*a^2*b + b^3 + b*(a^2 + 3*b^2)*Cos[2*(c + d*x)] - 2*a*(a^2 - b^2)*Sin[c + d*x]))/2)/((a - b)^2*(a + b)^2*d*Sqrt[a + b*Sin[c + d*x]])","A",1
526,1,348,359,3.1096876,"\int \frac{\sec ^4(c+d x)}{(a+b \sin (c+d x))^{3/2}} \, dx","Integrate[Sec[c + d*x]^4/(a + b*Sin[c + d*x])^(3/2),x]","\frac{-4 a \left(a^4-4 a^2 b^2+3 b^4\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{4} (-2 c-2 d x+\pi )|\frac{2 b}{a+b}\right)+\left(4 a^5+4 a^4 b-15 a^3 b^2-15 a^2 b^3-21 a b^4-21 b^5\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} E\left(\frac{1}{4} (-2 c-2 d x+\pi )|\frac{2 b}{a+b}\right)+\frac{1}{8} \sec ^3(c+d x) \left(24 a^5 \sin (c+d x)+8 a^5 \sin (3 (c+d x))-24 a^4 b-64 a^3 b^2 \sin (c+d x)-32 a^3 b^2 \sin (3 (c+d x))+101 a^2 b^3+\left(-12 a^4 b+84 a^2 b^3+56 b^5\right) \cos (2 (c+d x))+\left(-4 a^4 b+15 a^2 b^3+21 b^5\right) \cos (4 (c+d x))+40 a b^4 \sin (c+d x)+24 a b^4 \sin (3 (c+d x))+19 b^5\right)}{6 d (a-b)^3 (a+b)^3 \sqrt{a+b \sin (c+d x)}}","-\frac{\sec ^3(c+d x) \sqrt{a+b \sin (c+d x)} \left(8 a b-\left(a^2+7 b^2\right) \sin (c+d x)\right)}{3 d \left(a^2-b^2\right)^2}+\frac{2 b \sec ^3(c+d x)}{d \left(a^2-b^2\right) \sqrt{a+b \sin (c+d x)}}+\frac{2 a \left(a^2-3 b^2\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{3 d \left(a^2-b^2\right)^2 \sqrt{a+b \sin (c+d x)}}-\frac{\sec (c+d x) \sqrt{a+b \sin (c+d x)} \left(a b \left(a^2-33 b^2\right)-\left(4 a^4-15 a^2 b^2-21 b^4\right) \sin (c+d x)\right)}{6 d \left(a^2-b^2\right)^3}-\frac{\left(4 a^4-15 a^2 b^2-21 b^4\right) \sqrt{a+b \sin (c+d x)} E\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{6 d \left(a^2-b^2\right)^3 \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}",1,"((4*a^5 + 4*a^4*b - 15*a^3*b^2 - 15*a^2*b^3 - 21*a*b^4 - 21*b^5)*EllipticE[(-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)] - 4*a*(a^4 - 4*a^2*b^2 + 3*b^4)*EllipticF[(-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)] + (Sec[c + d*x]^3*(-24*a^4*b + 101*a^2*b^3 + 19*b^5 + (-12*a^4*b + 84*a^2*b^3 + 56*b^5)*Cos[2*(c + d*x)] + (-4*a^4*b + 15*a^2*b^3 + 21*b^5)*Cos[4*(c + d*x)] + 24*a^5*Sin[c + d*x] - 64*a^3*b^2*Sin[c + d*x] + 40*a*b^4*Sin[c + d*x] + 8*a^5*Sin[3*(c + d*x)] - 32*a^3*b^2*Sin[3*(c + d*x)] + 24*a*b^4*Sin[3*(c + d*x)]))/8)/(6*(a - b)^3*(a + b)^3*d*Sqrt[a + b*Sin[c + d*x]])","A",1
527,1,117,150,0.3033693,"\int \frac{\cos ^5(c+d x)}{(a+b \sin (c+d x))^{5/2}} \, dx","Integrate[Cos[c + d*x]^5/(a + b*Sin[c + d*x])^(5/2),x]","\frac{16 \left(16 a^4+3 a b \left(8 a^2-5 b^2\right) \sin (c+d x)-10 a^2 b^2+\left(6 a^2 b^2-3 b^4\right) \sin ^2(c+d x)-a b^3 \sin ^3(c+d x)-b^4\right)+6 b^4 \cos ^4(c+d x)}{15 b^5 d (a+b \sin (c+d x))^{3/2}}","\frac{4 \left(3 a^2-b^2\right) \sqrt{a+b \sin (c+d x)}}{b^5 d}+\frac{8 a \left(a^2-b^2\right)}{b^5 d \sqrt{a+b \sin (c+d x)}}-\frac{2 \left(a^2-b^2\right)^2}{3 b^5 d (a+b \sin (c+d x))^{3/2}}+\frac{2 (a+b \sin (c+d x))^{5/2}}{5 b^5 d}-\frac{8 a (a+b \sin (c+d x))^{3/2}}{3 b^5 d}",1,"(6*b^4*Cos[c + d*x]^4 + 16*(16*a^4 - 10*a^2*b^2 - b^4 + 3*a*b*(8*a^2 - 5*b^2)*Sin[c + d*x] + (6*a^2*b^2 - 3*b^4)*Sin[c + d*x]^2 - a*b^3*Sin[c + d*x]^3))/(15*b^5*d*(a + b*Sin[c + d*x])^(3/2))","A",1
528,1,56,79,0.0561388,"\int \frac{\cos ^3(c+d x)}{(a+b \sin (c+d x))^{5/2}} \, dx","Integrate[Cos[c + d*x]^3/(a + b*Sin[c + d*x])^(5/2),x]","-\frac{2 \left(8 a^2+12 a b \sin (c+d x)+3 b^2 \sin ^2(c+d x)+b^2\right)}{3 b^3 d (a+b \sin (c+d x))^{3/2}}","\frac{2 \left(a^2-b^2\right)}{3 b^3 d (a+b \sin (c+d x))^{3/2}}-\frac{4 a}{b^3 d \sqrt{a+b \sin (c+d x)}}-\frac{2 \sqrt{a+b \sin (c+d x)}}{b^3 d}",1,"(-2*(8*a^2 + b^2 + 12*a*b*Sin[c + d*x] + 3*b^2*Sin[c + d*x]^2))/(3*b^3*d*(a + b*Sin[c + d*x])^(3/2))","A",1
529,1,24,24,0.0194217,"\int \frac{\cos (c+d x)}{(a+b \sin (c+d x))^{5/2}} \, dx","Integrate[Cos[c + d*x]/(a + b*Sin[c + d*x])^(5/2),x]","-\frac{2}{3 b d (a+b \sin (c+d x))^{3/2}}","-\frac{2}{3 b d (a+b \sin (c+d x))^{3/2}}",1,"-2/(3*b*d*(a + b*Sin[c + d*x])^(3/2))","A",1
530,1,94,139,0.081033,"\int \frac{\sec (c+d x)}{(a+b \sin (c+d x))^{5/2}} \, dx","Integrate[Sec[c + d*x]/(a + b*Sin[c + d*x])^(5/2),x]","\frac{(a+b) \, _2F_1\left(-\frac{3}{2},1;-\frac{1}{2};\frac{a+b \sin (c+d x)}{a-b}\right)+(b-a) \, _2F_1\left(-\frac{3}{2},1;-\frac{1}{2};\frac{a+b \sin (c+d x)}{a+b}\right)}{3 d (a-b) (a+b) (a+b \sin (c+d x))^{3/2}}","\frac{4 a b}{d \left(a^2-b^2\right)^2 \sqrt{a+b \sin (c+d x)}}+\frac{2 b}{3 d \left(a^2-b^2\right) (a+b \sin (c+d x))^{3/2}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \sin (c+d x)}}{\sqrt{a-b}}\right)}{d (a-b)^{5/2}}+\frac{\tanh ^{-1}\left(\frac{\sqrt{a+b \sin (c+d x)}}{\sqrt{a+b}}\right)}{d (a+b)^{5/2}}",1,"((a + b)*Hypergeometric2F1[-3/2, 1, -1/2, (a + b*Sin[c + d*x])/(a - b)] + (-a + b)*Hypergeometric2F1[-3/2, 1, -1/2, (a + b*Sin[c + d*x])/(a + b)])/(3*(a - b)*(a + b)*d*(a + b*Sin[c + d*x])^(3/2))","C",1
531,1,245,231,0.8807985,"\int \frac{\sec ^3(c+d x)}{(a+b \sin (c+d x))^{5/2}} \, dx","Integrate[Sec[c + d*x]^3/(a + b*Sin[c + d*x])^(5/2),x]","\frac{-\left(\left(3 a^3+3 a^2 b+7 a b^2+7 b^3\right) \, _2F_1\left(-\frac{3}{2},1;-\frac{1}{2};\frac{a+b \sin (c+d x)}{a-b}\right)\right)+\left(3 a^3-3 a^2 b+7 a b^2-7 b^3\right) \, _2F_1\left(-\frac{3}{2},1;-\frac{1}{2};\frac{a+b \sin (c+d x)}{a+b}\right)+15 a (a+b) (a+b \sin (c+d x)) \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\frac{a+b \sin (c+d x)}{a-b}\right)-3 (a-b) \left(5 a (a+b \sin (c+d x)) \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\frac{a+b \sin (c+d x)}{a+b}\right)-2 (a+b) \sec ^2(c+d x) (a \sin (c+d x)-b)\right)}{12 d (a-b)^2 (a+b)^2 (a+b \sin (c+d x))^{3/2}}","-\frac{a b \left(a^2+19 b^2\right)}{2 d \left(a^2-b^2\right)^3 \sqrt{a+b \sin (c+d x)}}-\frac{b \left(3 a^2+7 b^2\right)}{6 d \left(a^2-b^2\right)^2 (a+b \sin (c+d x))^{3/2}}-\frac{\sec ^2(c+d x) (b-a \sin (c+d x))}{2 d \left(a^2-b^2\right) (a+b \sin (c+d x))^{3/2}}-\frac{(2 a-7 b) \tanh ^{-1}\left(\frac{\sqrt{a+b \sin (c+d x)}}{\sqrt{a-b}}\right)}{4 d (a-b)^{7/2}}+\frac{(2 a+7 b) \tanh ^{-1}\left(\frac{\sqrt{a+b \sin (c+d x)}}{\sqrt{a+b}}\right)}{4 d (a+b)^{7/2}}",1,"(-((3*a^3 + 3*a^2*b + 7*a*b^2 + 7*b^3)*Hypergeometric2F1[-3/2, 1, -1/2, (a + b*Sin[c + d*x])/(a - b)]) + (3*a^3 - 3*a^2*b + 7*a*b^2 - 7*b^3)*Hypergeometric2F1[-3/2, 1, -1/2, (a + b*Sin[c + d*x])/(a + b)] + 15*a*(a + b)*Hypergeometric2F1[-1/2, 1, 1/2, (a + b*Sin[c + d*x])/(a - b)]*(a + b*Sin[c + d*x]) - 3*(a - b)*(-2*(a + b)*Sec[c + d*x]^2*(-b + a*Sin[c + d*x]) + 5*a*Hypergeometric2F1[-1/2, 1, 1/2, (a + b*Sin[c + d*x])/(a + b)]*(a + b*Sin[c + d*x])))/(12*(a - b)^2*(a + b)^2*d*(a + b*Sin[c + d*x])^(3/2))","C",1
532,1,296,339,3.4323155,"\int \frac{\sec ^5(c+d x)}{(a+b \sin (c+d x))^{5/2}} \, dx","Integrate[Sec[c + d*x]^5/(a + b*Sin[c + d*x])^(5/2),x]","\frac{-15 a \left(3 a^2-10 b^2\right) (a+b \sin (c+d x)) \left((a+b) \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\frac{a+b \sin (c+d x)}{a-b}\right)+(b-a) \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\frac{a+b \sin (c+d x)}{a+b}\right)\right)+\frac{1}{2} \left(18 a^4-81 a^2 b^2-77 b^4\right) \left((a+b) \, _2F_1\left(-\frac{3}{2},1;-\frac{1}{2};\frac{a+b \sin (c+d x)}{a-b}\right)+(b-a) \, _2F_1\left(-\frac{3}{2},1;-\frac{1}{2};\frac{a+b \sin (c+d x)}{a+b}\right)\right)-3 (a-b) (a+b) \sec ^2(c+d x) \left(\left(6 a^3-20 a b^2\right) \sin (c+d x)+3 a^2 b+11 b^3\right)-12 (a-b)^2 (a+b)^2 \sec ^4(c+d x) (a \sin (c+d x)-b)}{48 d \left(a^2-b^2\right)^2 \left(b^2-a^2\right) (a+b \sin (c+d x))^{3/2}}","-\frac{\left(12 a^2-54 a b+77 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a+b \sin (c+d x)}}{\sqrt{a-b}}\right)}{32 d (a-b)^{9/2}}+\frac{\left(12 a^2+54 a b+77 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a+b \sin (c+d x)}}{\sqrt{a+b}}\right)}{32 d (a+b)^{9/2}}-\frac{\sec ^4(c+d x) (b-a \sin (c+d x))}{4 d \left(a^2-b^2\right) (a+b \sin (c+d x))^{3/2}}+\frac{\sec ^2(c+d x) \left(2 a \left(3 a^2-10 b^2\right) \sin (c+d x)+b \left(3 a^2+11 b^2\right)\right)}{16 d \left(a^2-b^2\right)^2 (a+b \sin (c+d x))^{3/2}}-\frac{a b \left(3 a^4-16 a^2 b^2-127 b^4\right)}{8 d \left(a^2-b^2\right)^4 \sqrt{a+b \sin (c+d x)}}-\frac{b \left(18 a^4-81 a^2 b^2-77 b^4\right)}{48 d \left(a^2-b^2\right)^3 (a+b \sin (c+d x))^{3/2}}",1,"(((18*a^4 - 81*a^2*b^2 - 77*b^4)*((a + b)*Hypergeometric2F1[-3/2, 1, -1/2, (a + b*Sin[c + d*x])/(a - b)] + (-a + b)*Hypergeometric2F1[-3/2, 1, -1/2, (a + b*Sin[c + d*x])/(a + b)]))/2 - 12*(a - b)^2*(a + b)^2*Sec[c + d*x]^4*(-b + a*Sin[c + d*x]) - 15*a*(3*a^2 - 10*b^2)*((a + b)*Hypergeometric2F1[-1/2, 1, 1/2, (a + b*Sin[c + d*x])/(a - b)] + (-a + b)*Hypergeometric2F1[-1/2, 1, 1/2, (a + b*Sin[c + d*x])/(a + b)])*(a + b*Sin[c + d*x]) - 3*(a - b)*(a + b)*Sec[c + d*x]^2*(3*a^2*b + 11*b^3 + (6*a^3 - 20*a*b^2)*Sin[c + d*x]))/(48*(a^2 - b^2)^2*(-a^2 + b^2)*d*(a + b*Sin[c + d*x])^(3/2))","C",1
533,1,356,384,1.7870103,"\int \frac{\cos ^8(c+d x)}{(a+b \sin (c+d x))^{5/2}} \, dx","Integrate[Cos[c + d*x]^8/(a + b*Sin[c + d*x])^(5/2),x]","\frac{256 (a+b) \left(\frac{a+b \sin (c+d x)}{a+b}\right)^{3/2} \left(4 \left(32 a^5-60 a^3 b^2+27 a b^4\right) \left((a+b) E\left(\frac{1}{4} (-2 c-2 d x+\pi )|\frac{2 b}{a+b}\right)-a F\left(\frac{1}{4} (-2 c-2 d x+\pi )|\frac{2 b}{a+b}\right)\right)+b \left(32 a^4 b-51 a^2 b^3+15 b^5\right) F\left(\frac{1}{4} (-2 c-2 d x+\pi )|\frac{2 b}{a+b}\right)\right)+\frac{1}{2} b \cos (c+d x) \left(-32768 a^6-40960 a^5 b \sin (c+d x)+55296 a^4 b^2+74112 a^3 b^3 \sin (c+d x)-384 a^3 b^3 \sin (3 (c+d x))-18144 a^2 b^4+\left(126 b^6-96 a^2 b^4\right) \cos (4 (c+d x))+\left(2048 a^4 b^2-3648 a^2 b^4+1383 b^6\right) \cos (2 (c+d x))-30920 a b^5 \sin (c+d x)+596 a b^5 \sin (3 (c+d x))+28 a b^5 \sin (5 (c+d x))+9 b^6 \cos (6 (c+d x))-2574 b^6\right)}{792 b^8 d (a+b \sin (c+d x))^{3/2}}","-\frac{128 a \left(8 a^2-9 b^2\right) \left(4 a^2-3 b^2\right) \sqrt{a+b \sin (c+d x)} E\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{99 b^8 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}+\frac{40 \cos ^3(c+d x) \sqrt{a+b \sin (c+d x)} \left(32 a^2-28 a b \sin (c+d x)-3 b^2\right)}{99 b^5 d}-\frac{16 \cos (c+d x) \sqrt{a+b \sin (c+d x)} \left(128 a^4-3 a b \left(32 a^2-31 b^2\right) \sin (c+d x)-144 a^2 b^2+15 b^4\right)}{99 b^7 d}+\frac{32 \left(128 a^6-272 a^4 b^2+159 a^2 b^4-15 b^6\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{99 b^8 d \sqrt{a+b \sin (c+d x)}}-\frac{28 \cos ^5(c+d x) (12 a+b \sin (c+d x))}{33 b^3 d \sqrt{a+b \sin (c+d x)}}-\frac{2 \cos ^7(c+d x)}{3 b d (a+b \sin (c+d x))^{3/2}}",1,"(256*(a + b)*(b*(32*a^4*b - 51*a^2*b^3 + 15*b^5)*EllipticF[(-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)] + 4*(32*a^5 - 60*a^3*b^2 + 27*a*b^4)*((a + b)*EllipticE[(-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)] - a*EllipticF[(-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)]))*((a + b*Sin[c + d*x])/(a + b))^(3/2) + (b*Cos[c + d*x]*(-32768*a^6 + 55296*a^4*b^2 - 18144*a^2*b^4 - 2574*b^6 + (2048*a^4*b^2 - 3648*a^2*b^4 + 1383*b^6)*Cos[2*(c + d*x)] + (-96*a^2*b^4 + 126*b^6)*Cos[4*(c + d*x)] + 9*b^6*Cos[6*(c + d*x)] - 40960*a^5*b*Sin[c + d*x] + 74112*a^3*b^3*Sin[c + d*x] - 30920*a*b^5*Sin[c + d*x] - 384*a^3*b^3*Sin[3*(c + d*x)] + 596*a*b^5*Sin[3*(c + d*x)] + 28*a*b^5*Sin[5*(c + d*x)]))/2)/(792*b^8*d*(a + b*Sin[c + d*x])^(3/2))","A",1
534,1,244,293,1.2209053,"\int \frac{\cos ^6(c+d x)}{(a+b \sin (c+d x))^{5/2}} \, dx","Integrate[Cos[c + d*x]^6/(a + b*Sin[c + d*x])^(5/2),x]","\frac{\frac{1}{2} b \cos (c+d x) \left(1024 a^4+1280 a^3 b \sin (c+d x)-736 a^2 b^2+\left(52 b^4-64 a^2 b^2\right) \cos (2 (c+d x))-1076 a b^3 \sin (c+d x)+12 a b^3 \sin (3 (c+d x))+3 b^4 \cos (4 (c+d x))-111 b^4\right)-32 (a+b) \left(\frac{a+b \sin (c+d x)}{a+b}\right)^{3/2} \left(\left(-32 a^4+37 a^2 b^2-5 b^4\right) F\left(\frac{1}{4} (-2 c-2 d x+\pi )|\frac{2 b}{a+b}\right)+a \left(32 a^3+32 a^2 b-29 a b^2-29 b^3\right) E\left(\frac{1}{4} (-2 c-2 d x+\pi )|\frac{2 b}{a+b}\right)\right)}{42 b^6 d (a+b \sin (c+d x))^{3/2}}","\frac{16 a \left(32 a^2-29 b^2\right) \sqrt{a+b \sin (c+d x)} E\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{21 b^6 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}+\frac{8 \cos (c+d x) \sqrt{a+b \sin (c+d x)} \left(32 a^2-24 a b \sin (c+d x)-5 b^2\right)}{21 b^5 d}-\frac{16 \left(32 a^4-37 a^2 b^2+5 b^4\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{21 b^6 d \sqrt{a+b \sin (c+d x)}}-\frac{20 \cos ^3(c+d x) (8 a+b \sin (c+d x))}{21 b^3 d \sqrt{a+b \sin (c+d x)}}-\frac{2 \cos ^5(c+d x)}{3 b d (a+b \sin (c+d x))^{3/2}}",1,"(-32*(a + b)*(a*(32*a^3 + 32*a^2*b - 29*a*b^2 - 29*b^3)*EllipticE[(-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)] + (-32*a^4 + 37*a^2*b^2 - 5*b^4)*EllipticF[(-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)])*((a + b*Sin[c + d*x])/(a + b))^(3/2) + (b*Cos[c + d*x]*(1024*a^4 - 736*a^2*b^2 - 111*b^4 + (-64*a^2*b^2 + 52*b^4)*Cos[2*(c + d*x)] + 3*b^4*Cos[4*(c + d*x)] + 1280*a^3*b*Sin[c + d*x] - 1076*a*b^3*Sin[c + d*x] + 12*a*b^3*Sin[3*(c + d*x)]))/2)/(42*b^6*d*(a + b*Sin[c + d*x])^(3/2))","A",1
535,1,174,221,1.0259866,"\int \frac{\cos ^4(c+d x)}{(a+b \sin (c+d x))^{5/2}} \, dx","Integrate[Cos[c + d*x]^4/(a + b*Sin[c + d*x])^(5/2),x]","\frac{b \cos (c+d x) \left(-16 a^2-20 a b \sin (c+d x)+b^2 \cos (2 (c+d x))-3 b^2\right)-8 \left(4 a^2-b^2\right) (a+b) \left(\frac{a+b \sin (c+d x)}{a+b}\right)^{3/2} F\left(\frac{1}{4} (-2 c-2 d x+\pi )|\frac{2 b}{a+b}\right)+32 a (a+b)^2 \left(\frac{a+b \sin (c+d x)}{a+b}\right)^{3/2} E\left(\frac{1}{4} (-2 c-2 d x+\pi )|\frac{2 b}{a+b}\right)}{3 b^4 d (a+b \sin (c+d x))^{3/2}}","\frac{8 \left(4 a^2-b^2\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{3 b^4 d \sqrt{a+b \sin (c+d x)}}-\frac{32 a \sqrt{a+b \sin (c+d x)} E\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{3 b^4 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}-\frac{4 \cos (c+d x) (4 a+b \sin (c+d x))}{3 b^3 d \sqrt{a+b \sin (c+d x)}}-\frac{2 \cos ^3(c+d x)}{3 b d (a+b \sin (c+d x))^{3/2}}",1,"(32*a*(a + b)^2*EllipticE[(-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)]*((a + b*Sin[c + d*x])/(a + b))^(3/2) - 8*(a + b)*(4*a^2 - b^2)*EllipticF[(-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)]*((a + b*Sin[c + d*x])/(a + b))^(3/2) + b*Cos[c + d*x]*(-16*a^2 - 3*b^2 + b^2*Cos[2*(c + d*x)] - 20*a*b*Sin[c + d*x]))/(3*b^4*d*(a + b*Sin[c + d*x])^(3/2))","A",1
536,1,167,219,1.0294178,"\int \frac{\cos ^2(c+d x)}{(a+b \sin (c+d x))^{5/2}} \, dx","Integrate[Cos[c + d*x]^2/(a + b*Sin[c + d*x])^(5/2),x]","\frac{2 b \cos (c+d x) \left(a^2+2 a b \sin (c+d x)+b^2\right)+4 (a-b) (a+b)^2 \left(\frac{a+b \sin (c+d x)}{a+b}\right)^{3/2} F\left(\frac{1}{4} (-2 c-2 d x+\pi )|\frac{2 b}{a+b}\right)-4 a (a+b)^2 \left(\frac{a+b \sin (c+d x)}{a+b}\right)^{3/2} E\left(\frac{1}{4} (-2 c-2 d x+\pi )|\frac{2 b}{a+b}\right)}{3 b^2 d (a-b) (a+b) (a+b \sin (c+d x))^{3/2}}","\frac{4 a \cos (c+d x)}{3 b d \left(a^2-b^2\right) \sqrt{a+b \sin (c+d x)}}+\frac{4 a \sqrt{a+b \sin (c+d x)} E\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{3 b^2 d \left(a^2-b^2\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}-\frac{4 \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{3 b^2 d \sqrt{a+b \sin (c+d x)}}-\frac{2 \cos (c+d x)}{3 b d (a+b \sin (c+d x))^{3/2}}",1,"(-4*a*(a + b)^2*EllipticE[(-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)]*((a + b*Sin[c + d*x])/(a + b))^(3/2) + 4*(a - b)*(a + b)^2*EllipticF[(-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)]*((a + b*Sin[c + d*x])/(a + b))^(3/2) + 2*b*Cos[c + d*x]*(a^2 + b^2 + 2*a*b*Sin[c + d*x]))/(3*(a - b)*b^2*(a + b)*d*(a + b*Sin[c + d*x])^(3/2))","A",1
537,1,241,325,1.8926407,"\int \frac{\sec ^2(c+d x)}{(a+b \sin (c+d x))^{5/2}} \, dx","Integrate[Sec[c + d*x]^2/(a + b*Sin[c + d*x])^(5/2),x]","\frac{\frac{\left(\frac{a+b \sin (c+d x)}{a+b}\right)^{3/2} \left(\left(3 a^3+29 a b^2\right) E\left(\frac{1}{4} (-2 c-2 d x+\pi )|\frac{2 b}{a+b}\right)+\left(-3 a^3+3 a^2 b-5 a b^2+5 b^3\right) F\left(\frac{1}{4} (-2 c-2 d x+\pi )|\frac{2 b}{a+b}\right)\right)}{(a-b)^3 (a+b)}-\frac{2 b^3 \left(a^2-b^2\right) \cos (c+d x)+3 \sec (c+d x) (a+b \sin (c+d x))^2 \left(-a \left(a^2+3 b^2\right) \sin (c+d x)+3 a^2 b+b^3\right)+20 a b^3 \cos (c+d x) (a+b \sin (c+d x))}{\left(a^2-b^2\right)^3}}{3 d (a+b \sin (c+d x))^{3/2}}","-\frac{\sec (c+d x) \sqrt{a+b \sin (c+d x)} \left(b \left(27 a^2+5 b^2\right)-a \left(3 a^2+29 b^2\right) \sin (c+d x)\right)}{3 d \left(a^2-b^2\right)^3}+\frac{16 a b \sec (c+d x)}{3 d \left(a^2-b^2\right)^2 \sqrt{a+b \sin (c+d x)}}+\frac{2 b \sec (c+d x)}{3 d \left(a^2-b^2\right) (a+b \sin (c+d x))^{3/2}}+\frac{\left(3 a^2+5 b^2\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{3 d \left(a^2-b^2\right)^2 \sqrt{a+b \sin (c+d x)}}-\frac{a \left(3 a^2+29 b^2\right) \sqrt{a+b \sin (c+d x)} E\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{3 d \left(a^2-b^2\right)^3 \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}",1,"((((3*a^3 + 29*a*b^2)*EllipticE[(-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)] + (-3*a^3 + 3*a^2*b - 5*a*b^2 + 5*b^3)*EllipticF[(-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)])*((a + b*Sin[c + d*x])/(a + b))^(3/2))/((a - b)^3*(a + b)) - (2*b^3*(a^2 - b^2)*Cos[c + d*x] + 20*a*b^3*Cos[c + d*x]*(a + b*Sin[c + d*x]) + 3*Sec[c + d*x]*(a + b*Sin[c + d*x])^2*(3*a^2*b + b^3 - a*(a^2 + 3*b^2)*Sin[c + d*x]))/(a^2 - b^2)^3)/(3*d*(a + b*Sin[c + d*x])^(3/2))","A",1
538,1,341,425,2.5025775,"\int \frac{\sec ^4(c+d x)}{(a+b \sin (c+d x))^{5/2}} \, dx","Integrate[Sec[c + d*x]^4/(a + b*Sin[c + d*x])^(5/2),x]","\frac{\frac{2 \left(a^2-b^2\right) \sec ^3(c+d x) (a+b \sin (c+d x))^2 \left(a \left(a^2+3 b^2\right) \sin (c+d x)-b \left(3 a^2+b^2\right)\right)+4 b^5 \left(a^2-b^2\right) \cos (c+d x)+\sec (c+d x) (a+b \sin (c+d x))^2 \left(-a^4 b+54 a^2 b^3+4 a \left(a^4-6 a^2 b^2-11 b^4\right) \sin (c+d x)+11 b^5\right)+64 a b^5 \cos (c+d x) (a+b \sin (c+d x))}{\left(a^2-b^2\right)^4}+\frac{\left(\frac{a+b \sin (c+d x)}{a+b}\right)^{3/2} \left(4 \left(a^5-6 a^3 b^2-27 a b^4\right) E\left(\frac{1}{4} (-2 c-2 d x+\pi )|\frac{2 b}{a+b}\right)+\left(-4 a^5+4 a^4 b+21 a^3 b^2-21 a^2 b^3+15 a b^4-15 b^5\right) F\left(\frac{1}{4} (-2 c-2 d x+\pi )|\frac{2 b}{a+b}\right)\right)}{(a-b)^4 (a+b)^2}}{6 d (a+b \sin (c+d x))^{3/2}}","-\frac{\sec ^3(c+d x) \sqrt{a+b \sin (c+d x)} \left(b \left(29 a^2+3 b^2\right)-a \left(a^2+31 b^2\right) \sin (c+d x)\right)}{3 d \left(a^2-b^2\right)^3}+\frac{8 a b \sec ^3(c+d x)}{d \left(a^2-b^2\right)^2 \sqrt{a+b \sin (c+d x)}}+\frac{2 b \sec ^3(c+d x)}{3 d \left(a^2-b^2\right) (a+b \sin (c+d x))^{3/2}}-\frac{\sec (c+d x) \sqrt{a+b \sin (c+d x)} \left(b \left(a^4-114 a^2 b^2-15 b^4\right)-4 a \left(a^4-6 a^2 b^2-27 b^4\right) \sin (c+d x)\right)}{6 d \left(a^2-b^2\right)^4}+\frac{\left(4 a^4-21 a^2 b^2-15 b^4\right) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{6 d \left(a^2-b^2\right)^3 \sqrt{a+b \sin (c+d x)}}-\frac{2 a \left(a^4-6 a^2 b^2-27 b^4\right) \sqrt{a+b \sin (c+d x)} E\left(\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)|\frac{2 b}{a+b}\right)}{3 d \left(a^2-b^2\right)^4 \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}",1,"(((4*(a^5 - 6*a^3*b^2 - 27*a*b^4)*EllipticE[(-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)] + (-4*a^5 + 4*a^4*b + 21*a^3*b^2 - 21*a^2*b^3 + 15*a*b^4 - 15*b^5)*EllipticF[(-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)])*((a + b*Sin[c + d*x])/(a + b))^(3/2))/((a - b)^4*(a + b)^2) + (4*b^5*(a^2 - b^2)*Cos[c + d*x] + 64*a*b^5*Cos[c + d*x]*(a + b*Sin[c + d*x]) + 2*(a^2 - b^2)*Sec[c + d*x]^3*(a + b*Sin[c + d*x])^2*(-(b*(3*a^2 + b^2)) + a*(a^2 + 3*b^2)*Sin[c + d*x]) + Sec[c + d*x]*(a + b*Sin[c + d*x])^2*(-(a^4*b) + 54*a^2*b^3 + 11*b^5 + 4*a*(a^4 - 6*a^2*b^2 - 11*b^4)*Sin[c + d*x]))/(a^2 - b^2)^4)/(6*d*(a + b*Sin[c + d*x])^(3/2))","A",1
539,1,104,124,0.9012098,"\int (e \cos (c+d x))^{7/2} (a+b \sin (c+d x)) \, dx","Integrate[(e*Cos[c + d*x])^(7/2)*(a + b*Sin[c + d*x]),x]","\frac{e^3 \sqrt{e \cos (c+d x)} \left(\sqrt{\cos (c+d x)} (138 a \sin (c+d x)+18 a \sin (3 (c+d x))-28 b \cos (2 (c+d x))-7 b \cos (4 (c+d x))-21 b)+120 a F\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{252 d \sqrt{\cos (c+d x)}}","\frac{10 a e^4 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d \sqrt{e \cos (c+d x)}}+\frac{10 a e^3 \sin (c+d x) \sqrt{e \cos (c+d x)}}{21 d}+\frac{2 a e \sin (c+d x) (e \cos (c+d x))^{5/2}}{7 d}-\frac{2 b (e \cos (c+d x))^{9/2}}{9 d e}",1,"(e^3*Sqrt[e*Cos[c + d*x]]*(120*a*EllipticF[(c + d*x)/2, 2] + Sqrt[Cos[c + d*x]]*(-21*b - 28*b*Cos[2*(c + d*x)] - 7*b*Cos[4*(c + d*x)] + 138*a*Sin[c + d*x] + 18*a*Sin[3*(c + d*x)])))/(252*d*Sqrt[Cos[c + d*x]])","A",1
540,1,79,95,0.4929173,"\int (e \cos (c+d x))^{5/2} (a+b \sin (c+d x)) \, dx","Integrate[(e*Cos[c + d*x])^(5/2)*(a + b*Sin[c + d*x]),x]","\frac{(e \cos (c+d x))^{5/2} \left(\cos ^{\frac{3}{2}}(c+d x) (14 a \sin (c+d x)-5 b \cos (2 (c+d x))-5 b)+42 a E\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{35 d \cos ^{\frac{5}{2}}(c+d x)}","\frac{6 a e^2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{5 d \sqrt{\cos (c+d x)}}+\frac{2 a e \sin (c+d x) (e \cos (c+d x))^{3/2}}{5 d}-\frac{2 b (e \cos (c+d x))^{7/2}}{7 d e}",1,"((e*Cos[c + d*x])^(5/2)*(42*a*EllipticE[(c + d*x)/2, 2] + Cos[c + d*x]^(3/2)*(-5*b - 5*b*Cos[2*(c + d*x)] + 14*a*Sin[c + d*x])))/(35*d*Cos[c + d*x]^(5/2))","A",1
541,1,79,95,0.4825641,"\int (e \cos (c+d x))^{3/2} (a+b \sin (c+d x)) \, dx","Integrate[(e*Cos[c + d*x])^(3/2)*(a + b*Sin[c + d*x]),x]","\frac{(e \cos (c+d x))^{3/2} \left(\sqrt{\cos (c+d x)} (10 a \sin (c+d x)-3 b \cos (2 (c+d x))-3 b)+10 a F\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{15 d \cos ^{\frac{3}{2}}(c+d x)}","\frac{2 a e^2 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d \sqrt{e \cos (c+d x)}}+\frac{2 a e \sin (c+d x) \sqrt{e \cos (c+d x)}}{3 d}-\frac{2 b (e \cos (c+d x))^{5/2}}{5 d e}",1,"((e*Cos[c + d*x])^(3/2)*(10*a*EllipticF[(c + d*x)/2, 2] + Sqrt[Cos[c + d*x]]*(-3*b - 3*b*Cos[2*(c + d*x)] + 10*a*Sin[c + d*x])))/(15*d*Cos[c + d*x]^(3/2))","A",1
542,1,56,63,0.1000321,"\int \sqrt{e \cos (c+d x)} (a+b \sin (c+d x)) \, dx","Integrate[Sqrt[e*Cos[c + d*x]]*(a + b*Sin[c + d*x]),x]","-\frac{2 \sqrt{e \cos (c+d x)} \left(b \cos ^{\frac{3}{2}}(c+d x)-3 a E\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{3 d \sqrt{\cos (c+d x)}}","\frac{2 a E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{d \sqrt{\cos (c+d x)}}-\frac{2 b (e \cos (c+d x))^{3/2}}{3 d e}",1,"(-2*Sqrt[e*Cos[c + d*x]]*(b*Cos[c + d*x]^(3/2) - 3*a*EllipticE[(c + d*x)/2, 2]))/(3*d*Sqrt[Cos[c + d*x]])","A",1
543,1,50,61,0.1991757,"\int \frac{a+b \sin (c+d x)}{\sqrt{e \cos (c+d x)}} \, dx","Integrate[(a + b*Sin[c + d*x])/Sqrt[e*Cos[c + d*x]],x]","\frac{2 a \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-2 b \cos (c+d x)}{d \sqrt{e \cos (c+d x)}}","\frac{2 a \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d \sqrt{e \cos (c+d x)}}-\frac{2 b \sqrt{e \cos (c+d x)}}{d e}",1,"(-2*b*Cos[c + d*x] + 2*a*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])/(d*Sqrt[e*Cos[c + d*x]])","A",1
544,1,54,91,0.1092002,"\int \frac{a+b \sin (c+d x)}{(e \cos (c+d x))^{3/2}} \, dx","Integrate[(a + b*Sin[c + d*x])/(e*Cos[c + d*x])^(3/2),x]","\frac{2 \left(a \sin (c+d x)-a \sqrt{\cos (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+b\right)}{d e \sqrt{e \cos (c+d x)}}","-\frac{2 a E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{d e^2 \sqrt{\cos (c+d x)}}+\frac{2 a \sin (c+d x)}{d e \sqrt{e \cos (c+d x)}}+\frac{2 b}{d e \sqrt{e \cos (c+d x)}}",1,"(2*(b - a*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2] + a*Sin[c + d*x]))/(d*e*Sqrt[e*Cos[c + d*x]])","A",1
545,1,55,97,0.1446822,"\int \frac{a+b \sin (c+d x)}{(e \cos (c+d x))^{5/2}} \, dx","Integrate[(a + b*Sin[c + d*x])/(e*Cos[c + d*x])^(5/2),x]","\frac{2 \left(a \sin (c+d x)+a \cos ^{\frac{3}{2}}(c+d x) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+b\right)}{3 d e (e \cos (c+d x))^{3/2}}","\frac{2 a \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d e^2 \sqrt{e \cos (c+d x)}}+\frac{2 a \sin (c+d x)}{3 d e (e \cos (c+d x))^{3/2}}+\frac{2 b}{3 d e (e \cos (c+d x))^{3/2}}",1,"(2*(b + a*Cos[c + d*x]^(3/2)*EllipticF[(c + d*x)/2, 2] + a*Sin[c + d*x]))/(3*d*e*(e*Cos[c + d*x])^(3/2))","A",1
546,1,70,126,0.344724,"\int \frac{a+b \sin (c+d x)}{(e \cos (c+d x))^{7/2}} \, dx","Integrate[(a + b*Sin[c + d*x])/(e*Cos[c + d*x])^(7/2),x]","\frac{7 a \sin (c+d x)+3 a \sin (3 (c+d x))-12 a \cos ^{\frac{5}{2}}(c+d x) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+4 b}{10 d e (e \cos (c+d x))^{5/2}}","-\frac{6 a E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{5 d e^4 \sqrt{\cos (c+d x)}}+\frac{6 a \sin (c+d x)}{5 d e^3 \sqrt{e \cos (c+d x)}}+\frac{2 a \sin (c+d x)}{5 d e (e \cos (c+d x))^{5/2}}+\frac{2 b}{5 d e (e \cos (c+d x))^{5/2}}",1,"(4*b - 12*a*Cos[c + d*x]^(5/2)*EllipticE[(c + d*x)/2, 2] + 7*a*Sin[c + d*x] + 3*a*Sin[3*(c + d*x)])/(10*d*e*(e*Cos[c + d*x])^(5/2))","A",1
547,1,160,188,1.978463,"\int (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))^2 \, dx","Integrate[(e*Cos[c + d*x])^(7/2)*(a + b*Sin[c + d*x])^2,x]","\frac{(e \cos (c+d x))^{7/2} \left(40 \left(11 a^2+2 b^2\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+\frac{1}{6} \sqrt{\cos (c+d x)} \left(6 \left(572 a^2+41 b^2\right) \sin (c+d x)+8 \cos (2 (c+d x)) \left(9 \left(11 a^2-5 b^2\right) \sin (c+d x)-154 a b\right)-14 b \cos (4 (c+d x)) (22 a+9 b \sin (c+d x))\right)-154 a b \sqrt{\cos (c+d x)}\right)}{924 d \cos ^{\frac{7}{2}}(c+d x)}","\frac{10 e^4 \left(11 a^2+2 b^2\right) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{231 d \sqrt{e \cos (c+d x)}}+\frac{10 e^3 \left(11 a^2+2 b^2\right) \sin (c+d x) \sqrt{e \cos (c+d x)}}{231 d}+\frac{2 e \left(11 a^2+2 b^2\right) \sin (c+d x) (e \cos (c+d x))^{5/2}}{77 d}-\frac{26 a b (e \cos (c+d x))^{9/2}}{99 d e}-\frac{2 b (e \cos (c+d x))^{9/2} (a+b \sin (c+d x))}{11 d e}",1,"((e*Cos[c + d*x])^(7/2)*(-154*a*b*Sqrt[Cos[c + d*x]] + 40*(11*a^2 + 2*b^2)*EllipticF[(c + d*x)/2, 2] + (Sqrt[Cos[c + d*x]]*(6*(572*a^2 + 41*b^2)*Sin[c + d*x] - 14*b*Cos[4*(c + d*x)]*(22*a + 9*b*Sin[c + d*x]) + 8*Cos[2*(c + d*x)]*(-154*a*b + 9*(11*a^2 - 5*b^2)*Sin[c + d*x])))/6))/(924*d*Cos[c + d*x]^(7/2))","A",1
548,1,113,149,0.9176722,"\int (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))^2 \, dx","Integrate[(e*Cos[c + d*x])^(5/2)*(a + b*Sin[c + d*x])^2,x]","\frac{(e \cos (c+d x))^{5/2} \left(84 \left(9 a^2+2 b^2\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+\cos ^{\frac{3}{2}}(c+d x) \left(21 \left(12 a^2+b^2\right) \sin (c+d x)-5 b (36 a+7 b \sin (3 (c+d x)))-180 a b \cos (2 (c+d x))\right)\right)}{630 d \cos ^{\frac{5}{2}}(c+d x)}","\frac{2 e^2 \left(9 a^2+2 b^2\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{15 d \sqrt{\cos (c+d x)}}+\frac{2 e \left(9 a^2+2 b^2\right) \sin (c+d x) (e \cos (c+d x))^{3/2}}{45 d}-\frac{22 a b (e \cos (c+d x))^{7/2}}{63 d e}-\frac{2 b (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))}{9 d e}",1,"((e*Cos[c + d*x])^(5/2)*(84*(9*a^2 + 2*b^2)*EllipticE[(c + d*x)/2, 2] + Cos[c + d*x]^(3/2)*(-180*a*b*Cos[2*(c + d*x)] + 21*(12*a^2 + b^2)*Sin[c + d*x] - 5*b*(36*a + 7*b*Sin[3*(c + d*x)]))))/(630*d*Cos[c + d*x]^(5/2))","A",1
549,1,115,149,1.178695,"\int (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^2 \, dx","Integrate[(e*Cos[c + d*x])^(3/2)*(a + b*Sin[c + d*x])^2,x]","\frac{(e \cos (c+d x))^{3/2} \left(20 \left(7 a^2+2 b^2\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+\sqrt{\cos (c+d x)} \left(5 \left(28 a^2+5 b^2\right) \sin (c+d x)-3 b (28 a+5 b \sin (3 (c+d x)))-84 a b \cos (2 (c+d x))\right)\right)}{210 d \cos ^{\frac{3}{2}}(c+d x)}","\frac{2 e^2 \left(7 a^2+2 b^2\right) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d \sqrt{e \cos (c+d x)}}+\frac{2 e \left(7 a^2+2 b^2\right) \sin (c+d x) \sqrt{e \cos (c+d x)}}{21 d}-\frac{18 a b (e \cos (c+d x))^{5/2}}{35 d e}-\frac{2 b (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))}{7 d e}",1,"((e*Cos[c + d*x])^(3/2)*(20*(7*a^2 + 2*b^2)*EllipticF[(c + d*x)/2, 2] + Sqrt[Cos[c + d*x]]*(-84*a*b*Cos[2*(c + d*x)] + 5*(28*a^2 + 5*b^2)*Sin[c + d*x] - 3*b*(28*a + 5*b*Sin[3*(c + d*x)]))))/(210*d*Cos[c + d*x]^(3/2))","A",1
550,1,80,109,0.3349291,"\int \sqrt{e \cos (c+d x)} (a+b \sin (c+d x))^2 \, dx","Integrate[Sqrt[e*Cos[c + d*x]]*(a + b*Sin[c + d*x])^2,x]","\frac{\sqrt{e \cos (c+d x)} \left(6 \left(5 a^2+2 b^2\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)-2 b \cos ^{\frac{3}{2}}(c+d x) (10 a+3 b \sin (c+d x))\right)}{15 d \sqrt{\cos (c+d x)}}","\frac{2 \left(5 a^2+2 b^2\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{5 d \sqrt{\cos (c+d x)}}-\frac{14 a b (e \cos (c+d x))^{3/2}}{15 d e}-\frac{2 b (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))}{5 d e}",1,"(Sqrt[e*Cos[c + d*x]]*(6*(5*a^2 + 2*b^2)*EllipticE[(c + d*x)/2, 2] - 2*b*Cos[c + d*x]^(3/2)*(10*a + 3*b*Sin[c + d*x])))/(15*d*Sqrt[Cos[c + d*x]])","A",1
551,1,75,109,0.4276457,"\int \frac{(a+b \sin (c+d x))^2}{\sqrt{e \cos (c+d x)}} \, dx","Integrate[(a + b*Sin[c + d*x])^2/Sqrt[e*Cos[c + d*x]],x]","\frac{2 \left(3 a^2+2 b^2\right) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-2 b \cos (c+d x) (6 a+b \sin (c+d x))}{3 d \sqrt{e \cos (c+d x)}}","\frac{2 \left(3 a^2+2 b^2\right) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d \sqrt{e \cos (c+d x)}}-\frac{10 a b \sqrt{e \cos (c+d x)}}{3 d e}-\frac{2 b \sqrt{e \cos (c+d x)} (a+b \sin (c+d x))}{3 d e}",1,"(2*(3*a^2 + 2*b^2)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] - 2*b*Cos[c + d*x]*(6*a + b*Sin[c + d*x]))/(3*d*Sqrt[e*Cos[c + d*x]])","A",1
552,1,71,113,0.2414866,"\int \frac{(a+b \sin (c+d x))^2}{(e \cos (c+d x))^{3/2}} \, dx","Integrate[(a + b*Sin[c + d*x])^2/(e*Cos[c + d*x])^(3/2),x]","\frac{2 \left(a^2+b^2\right) \sin (c+d x)-2 \left(a^2+2 b^2\right) \sqrt{\cos (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+4 a b}{d e \sqrt{e \cos (c+d x)}}","-\frac{2 \left(a^2+2 b^2\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{d e^2 \sqrt{\cos (c+d x)}}+\frac{2 a b (e \cos (c+d x))^{3/2}}{d e^3}+\frac{2 (a \sin (c+d x)+b) (a+b \sin (c+d x))}{d e \sqrt{e \cos (c+d x)}}",1,"(4*a*b - 2*(a^2 + 2*b^2)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2] + 2*(a^2 + b^2)*Sin[c + d*x])/(d*e*Sqrt[e*Cos[c + d*x]])","A",1
553,1,72,119,0.2558119,"\int \frac{(a+b \sin (c+d x))^2}{(e \cos (c+d x))^{5/2}} \, dx","Integrate[(a + b*Sin[c + d*x])^2/(e*Cos[c + d*x])^(5/2),x]","\frac{2 \left(\left(a^2+b^2\right) \sin (c+d x)+\left(a^2-2 b^2\right) \cos ^{\frac{3}{2}}(c+d x) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+2 a b\right)}{3 d e (e \cos (c+d x))^{3/2}}","\frac{2 \left(a^2-2 b^2\right) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d e^2 \sqrt{e \cos (c+d x)}}+\frac{2 a b \sqrt{e \cos (c+d x)}}{3 d e^3}+\frac{2 (a \sin (c+d x)+b) (a+b \sin (c+d x))}{3 d e (e \cos (c+d x))^{3/2}}",1,"(2*(2*a*b + (a^2 - 2*b^2)*Cos[c + d*x]^(3/2)*EllipticF[(c + d*x)/2, 2] + (a^2 + b^2)*Sin[c + d*x]))/(3*d*e*(e*Cos[c + d*x])^(3/2))","A",1
554,1,105,160,0.5598878,"\int \frac{(a+b \sin (c+d x))^2}{(e \cos (c+d x))^{7/2}} \, dx","Integrate[(a + b*Sin[c + d*x])^2/(e*Cos[c + d*x])^(7/2),x]","\frac{\left(7 a^2+2 b^2\right) \sin (c+d x)-4 \left(3 a^2-2 b^2\right) \cos ^{\frac{5}{2}}(c+d x) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+3 a^2 \sin (3 (c+d x))+8 a b-2 b^2 \sin (3 (c+d x))}{10 d e (e \cos (c+d x))^{5/2}}","-\frac{2 \left(3 a^2-2 b^2\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{5 d e^4 \sqrt{\cos (c+d x)}}+\frac{2 \left(3 a^2-2 b^2\right) \sin (c+d x)}{5 d e^3 \sqrt{e \cos (c+d x)}}+\frac{2 a b}{5 d e^3 \sqrt{e \cos (c+d x)}}+\frac{2 (a \sin (c+d x)+b) (a+b \sin (c+d x))}{5 d e (e \cos (c+d x))^{5/2}}",1,"(8*a*b - 4*(3*a^2 - 2*b^2)*Cos[c + d*x]^(5/2)*EllipticE[(c + d*x)/2, 2] + (7*a^2 + 2*b^2)*Sin[c + d*x] + 3*a^2*Sin[3*(c + d*x)] - 2*b^2*Sin[3*(c + d*x)])/(10*d*e*(e*Cos[c + d*x])^(5/2))","A",1
555,1,205,237,2.087402,"\int (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))^3 \, dx","Integrate[(e*Cos[c + d*x])^(7/2)*(a + b*Sin[c + d*x])^3,x]","\frac{(e \cos (c+d x))^{7/2} \left(2080 \left(11 a^3+6 a b^2\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-154 b \left(78 a^2+11 b^2\right) \sqrt{\cos (c+d x)}+\frac{1}{3} \sqrt{\cos (c+d x)} \left(156 a \left(506 a^2+213 b^2\right) \sin (c+d x)+234 a \left(44 a^2-39 b^2\right) \sin (3 (c+d x))-77 b \left(624 a^2+73 b^2\right) \cos (2 (c+d x))+154 b \left(b^2-78 a^2\right) \cos (4 (c+d x))-4914 a b^2 \sin (5 (c+d x))+693 b^3 \cos (6 (c+d x))\right)\right)}{48048 d \cos ^{\frac{7}{2}}(c+d x)}","\frac{10 a e^4 \left(11 a^2+6 b^2\right) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{231 d \sqrt{e \cos (c+d x)}}+\frac{10 a e^3 \left(11 a^2+6 b^2\right) \sin (c+d x) \sqrt{e \cos (c+d x)}}{231 d}-\frac{2 b \left(177 a^2+44 b^2\right) (e \cos (c+d x))^{9/2}}{1287 d e}+\frac{2 a e \left(11 a^2+6 b^2\right) \sin (c+d x) (e \cos (c+d x))^{5/2}}{77 d}-\frac{2 b (e \cos (c+d x))^{9/2} (a+b \sin (c+d x))^2}{13 d e}-\frac{34 a b (e \cos (c+d x))^{9/2} (a+b \sin (c+d x))}{143 d e}",1,"((e*Cos[c + d*x])^(7/2)*(-154*b*(78*a^2 + 11*b^2)*Sqrt[Cos[c + d*x]] + 2080*(11*a^3 + 6*a*b^2)*EllipticF[(c + d*x)/2, 2] + (Sqrt[Cos[c + d*x]]*(-77*b*(624*a^2 + 73*b^2)*Cos[2*(c + d*x)] + 154*b*(-78*a^2 + b^2)*Cos[4*(c + d*x)] + 693*b^3*Cos[6*(c + d*x)] + 156*a*(506*a^2 + 213*b^2)*Sin[c + d*x] + 234*a*(44*a^2 - 39*b^2)*Sin[3*(c + d*x)] - 4914*a*b^2*Sin[5*(c + d*x)]))/3))/(48048*d*Cos[c + d*x]^(7/2))","A",1
556,1,150,197,1.412929,"\int (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))^3 \, dx","Integrate[(e*Cos[c + d*x])^(5/2)*(a + b*Sin[c + d*x])^3,x]","\frac{(e \cos (c+d x))^{5/2} \left(1848 \left(3 a^3+2 a b^2\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+\cos ^{\frac{3}{2}}(c+d x) \left(1848 a^3 \sin (c+d x)-60 \left(33 a^2 b+4 b^3\right) \cos (2 (c+d x))-1980 a^2 b+462 a b^2 \sin (c+d x)-770 a b^2 \sin (3 (c+d x))+105 b^3 \cos (4 (c+d x))-345 b^3\right)\right)}{4620 d \cos ^{\frac{5}{2}}(c+d x)}","\frac{2 a e^2 \left(3 a^2+2 b^2\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{5 d \sqrt{\cos (c+d x)}}-\frac{2 b \left(43 a^2+12 b^2\right) (e \cos (c+d x))^{7/2}}{231 d e}+\frac{2 a e \left(3 a^2+2 b^2\right) \sin (c+d x) (e \cos (c+d x))^{3/2}}{15 d}-\frac{2 b (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))^2}{11 d e}-\frac{10 a b (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))}{33 d e}",1,"((e*Cos[c + d*x])^(5/2)*(1848*(3*a^3 + 2*a*b^2)*EllipticE[(c + d*x)/2, 2] + Cos[c + d*x]^(3/2)*(-1980*a^2*b - 345*b^3 - 60*(33*a^2*b + 4*b^3)*Cos[2*(c + d*x)] + 105*b^3*Cos[4*(c + d*x)] + 1848*a^3*Sin[c + d*x] + 462*a*b^2*Sin[c + d*x] - 770*a*b^2*Sin[3*(c + d*x)])))/(4620*d*Cos[c + d*x]^(5/2))","A",1
557,1,153,197,1.4477826,"\int (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^3 \, dx","Integrate[(e*Cos[c + d*x])^(3/2)*(a + b*Sin[c + d*x])^3,x]","\frac{(e \cos (c+d x))^{3/2} \left(80 \left(7 a^3+6 a b^2\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+\frac{2}{3} \sqrt{\cos (c+d x)} \left(840 a^3 \sin (c+d x)-28 \left(27 a^2 b+4 b^3\right) \cos (2 (c+d x))-756 a^2 b+450 a b^2 \sin (c+d x)-270 a b^2 \sin (3 (c+d x))+35 b^3 \cos (4 (c+d x))-147 b^3\right)\right)}{840 d \cos ^{\frac{3}{2}}(c+d x)}","\frac{2 a e^2 \left(7 a^2+6 b^2\right) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d \sqrt{e \cos (c+d x)}}-\frac{2 b \left(89 a^2+28 b^2\right) (e \cos (c+d x))^{5/2}}{315 d e}+\frac{2 a e \left(7 a^2+6 b^2\right) \sin (c+d x) \sqrt{e \cos (c+d x)}}{21 d}-\frac{2 b (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))^2}{9 d e}-\frac{26 a b (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))}{63 d e}",1,"((e*Cos[c + d*x])^(3/2)*(80*(7*a^3 + 6*a*b^2)*EllipticF[(c + d*x)/2, 2] + (2*Sqrt[Cos[c + d*x]]*(-756*a^2*b - 147*b^3 - 28*(27*a^2*b + 4*b^3)*Cos[2*(c + d*x)] + 35*b^3*Cos[4*(c + d*x)] + 840*a^3*Sin[c + d*x] + 450*a*b^2*Sin[c + d*x] - 270*a*b^2*Sin[3*(c + d*x)]))/3))/(840*d*Cos[c + d*x]^(3/2))","A",1
558,1,101,156,0.6304848,"\int \sqrt{e \cos (c+d x)} (a+b \sin (c+d x))^3 \, dx","Integrate[Sqrt[e*Cos[c + d*x]]*(a + b*Sin[c + d*x])^3,x]","\frac{\sqrt{e \cos (c+d x)} \left(42 \left(5 a^3+6 a b^2\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+b \cos ^{\frac{3}{2}}(c+d x) \left(-210 a^2-126 a b \sin (c+d x)+15 b^2 \cos (2 (c+d x))-55 b^2\right)\right)}{105 d \sqrt{\cos (c+d x)}}","-\frac{2 b \left(57 a^2+20 b^2\right) (e \cos (c+d x))^{3/2}}{105 d e}+\frac{2 a \left(5 a^2+6 b^2\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{5 d \sqrt{\cos (c+d x)}}-\frac{2 b (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^2}{7 d e}-\frac{22 a b (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))}{35 d e}",1,"(Sqrt[e*Cos[c + d*x]]*(42*(5*a^3 + 6*a*b^2)*EllipticE[(c + d*x)/2, 2] + b*Cos[c + d*x]^(3/2)*(-210*a^2 - 55*b^2 + 15*b^2*Cos[2*(c + d*x)] - 126*a*b*Sin[c + d*x])))/(105*d*Sqrt[Cos[c + d*x]])","A",1
559,1,94,152,0.7897901,"\int \frac{(a+b \sin (c+d x))^3}{\sqrt{e \cos (c+d x)}} \, dx","Integrate[(a + b*Sin[c + d*x])^3/Sqrt[e*Cos[c + d*x]],x]","\frac{b \cos (c+d x) \left(-30 a^2-10 a b \sin (c+d x)+b^2 \cos (2 (c+d x))-9 b^2\right)+10 a \left(a^2+2 b^2\right) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d \sqrt{e \cos (c+d x)}}","-\frac{2 b \left(11 a^2+4 b^2\right) \sqrt{e \cos (c+d x)}}{5 d e}+\frac{2 a \left(a^2+2 b^2\right) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d \sqrt{e \cos (c+d x)}}-\frac{2 b \sqrt{e \cos (c+d x)} (a+b \sin (c+d x))^2}{5 d e}-\frac{6 a b \sqrt{e \cos (c+d x)} (a+b \sin (c+d x))}{5 d e}",1,"(10*a*(a^2 + 2*b^2)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + b*Cos[c + d*x]*(-30*a^2 - 9*b^2 + b^2*Cos[2*(c + d*x)] - 10*a*b*Sin[c + d*x]))/(5*d*Sqrt[e*Cos[c + d*x]])","A",1
560,1,98,160,0.4122104,"\int \frac{(a+b \sin (c+d x))^3}{(e \cos (c+d x))^{3/2}} \, dx","Integrate[(a + b*Sin[c + d*x])^3/(e*Cos[c + d*x])^(3/2),x]","\frac{-6 a \left(a^2+6 b^2\right) \sqrt{\cos (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+6 \left(a \left(a^2+3 b^2\right) \sin (c+d x)+3 a^2 b+b^3\right)+2 b^3 \cos ^2(c+d x)}{3 d e \sqrt{e \cos (c+d x)}}","\frac{2 b \left(3 a^2+4 b^2\right) (e \cos (c+d x))^{3/2}}{3 d e^3}-\frac{2 a \left(a^2+6 b^2\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{d e^2 \sqrt{\cos (c+d x)}}+\frac{2 a b (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))}{d e^3}+\frac{2 (a \sin (c+d x)+b) (a+b \sin (c+d x))^2}{d e \sqrt{e \cos (c+d x)}}",1,"(2*b^3*Cos[c + d*x]^2 - 6*a*(a^2 + 6*b^2)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2] + 6*(3*a^2*b + b^3 + a*(a^2 + 3*b^2)*Sin[c + d*x]))/(3*d*e*Sqrt[e*Cos[c + d*x]])","A",1
561,1,103,164,0.7087891,"\int \frac{(a+b \sin (c+d x))^3}{(e \cos (c+d x))^{5/2}} \, dx","Integrate[(a + b*Sin[c + d*x])^3/(e*Cos[c + d*x])^(5/2),x]","\frac{2 a^3 \sin (c+d x)+2 a \left(a^2-6 b^2\right) \cos ^{\frac{3}{2}}(c+d x) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+6 a^2 b+6 a b^2 \sin (c+d x)+3 b^3 \cos (2 (c+d x))+5 b^3}{3 d e (e \cos (c+d x))^{3/2}}","\frac{2 b \left(a^2+4 b^2\right) \sqrt{e \cos (c+d x)}}{3 d e^3}+\frac{2 a \left(a^2-6 b^2\right) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d e^2 \sqrt{e \cos (c+d x)}}+\frac{2 a b \sqrt{e \cos (c+d x)} (a+b \sin (c+d x))}{3 d e^3}+\frac{2 (a \sin (c+d x)+b) (a+b \sin (c+d x))^2}{3 d e (e \cos (c+d x))^{3/2}}",1,"(6*a^2*b + 5*b^3 + 3*b^3*Cos[2*(c + d*x)] + 2*a*(a^2 - 6*b^2)*Cos[c + d*x]^(3/2)*EllipticF[(c + d*x)/2, 2] + 2*a^3*Sin[c + d*x] + 6*a*b^2*Sin[c + d*x])/(3*d*e*(e*Cos[c + d*x])^(3/2))","A",1
562,1,126,187,0.7351892,"\int \frac{(a+b \sin (c+d x))^3}{(e \cos (c+d x))^{7/2}} \, dx","Integrate[(a + b*Sin[c + d*x])^3/(e*Cos[c + d*x])^(7/2),x]","\frac{2 \left(3 a^3 \sin (c+d x)+b \left(3 a^2+b^2\right) \sec ^2(c+d x)-3 a \left(a^2-2 b^2\right) \sqrt{\cos (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+a \left(a^2+3 b^2\right) \tan (c+d x) \sec (c+d x)-6 a b^2 \sin (c+d x)-5 b^3\right)}{5 d e^3 \sqrt{e \cos (c+d x)}}","\frac{2 b \left(3 a^2-4 b^2\right) (e \cos (c+d x))^{3/2}}{5 d e^5}-\frac{6 a \left(a^2-2 b^2\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{5 d e^4 \sqrt{\cos (c+d x)}}-\frac{2 \left(a b-\left(3 a^2-4 b^2\right) \sin (c+d x)\right) (a+b \sin (c+d x))}{5 d e^3 \sqrt{e \cos (c+d x)}}+\frac{2 (a \sin (c+d x)+b) (a+b \sin (c+d x))^2}{5 d e (e \cos (c+d x))^{5/2}}",1,"(2*(-5*b^3 - 3*a*(a^2 - 2*b^2)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2] + b*(3*a^2 + b^2)*Sec[c + d*x]^2 + 3*a^3*Sin[c + d*x] - 6*a*b^2*Sin[c + d*x] + a*(a^2 + 3*b^2)*Sec[c + d*x]*Tan[c + d*x]))/(5*d*e^3*Sqrt[e*Cos[c + d*x]])","A",1
563,1,140,188,0.653227,"\int \frac{(a+b \sin (c+d x))^3}{(e \cos (c+d x))^{9/2}} \, dx","Integrate[(a + b*Sin[c + d*x])^3/(e*Cos[c + d*x])^(9/2),x]","\frac{\sec ^4(c+d x) \sqrt{e \cos (c+d x)} \left(17 a^3 \sin (c+d x)+5 a^3 \sin (3 (c+d x))+4 a \left(5 a^2-6 b^2\right) \cos ^{\frac{7}{2}}(c+d x) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+36 a^2 b+30 a b^2 \sin (c+d x)-6 a b^2 \sin (3 (c+d x))-14 b^3 \cos (2 (c+d x))-2 b^3\right)}{42 d e^5}","\frac{2 b \left(5 a^2-4 b^2\right) \sqrt{e \cos (c+d x)}}{21 d e^5}+\frac{2 a \left(5 a^2-6 b^2\right) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d e^4 \sqrt{e \cos (c+d x)}}+\frac{2 \left(\left(5 a^2-4 b^2\right) \sin (c+d x)+a b\right) (a+b \sin (c+d x))}{21 d e^3 (e \cos (c+d x))^{3/2}}+\frac{2 (a \sin (c+d x)+b) (a+b \sin (c+d x))^2}{7 d e (e \cos (c+d x))^{7/2}}",1,"(Sqrt[e*Cos[c + d*x]]*Sec[c + d*x]^4*(36*a^2*b - 2*b^3 - 14*b^3*Cos[2*(c + d*x)] + 4*a*(5*a^2 - 6*b^2)*Cos[c + d*x]^(7/2)*EllipticF[(c + d*x)/2, 2] + 17*a^3*Sin[c + d*x] + 30*a*b^2*Sin[c + d*x] + 5*a^3*Sin[3*(c + d*x)] - 6*a*b^2*Sin[3*(c + d*x)]))/(42*d*e^5)","A",1
564,1,251,305,4.8054119,"\int (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))^4 \, dx","Integrate[(e*Cos[c + d*x])^(7/2)*(a + b*Sin[c + d*x])^4,x]","\frac{(e \cos (c+d x))^{7/2} \left(-154 a b \left(26 a^2+11 b^2\right) \sqrt{\cos (c+d x)}+104 \left(55 a^4+60 a^2 b^2+4 b^4\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+\frac{1}{120} \sqrt{\cos (c+d x)} \left(-28 b \cos (4 (c+d x)) \left(39 b \left(180 a^2+b^2\right) \sin (c+d x)+220 a \left(26 a^2-b^2\right)\right)+156 \left(5720 a^4+2460 a^2 b^2+87 b^4\right) \sin (c+d x)+\cos (2 (c+d x)) \left(78 \left(2640 a^4-7200 a^2 b^2-557 b^4\right) \sin (c+d x)-3080 \left(208 a^3 b+73 a b^3\right)\right)+462 b^3 \cos (6 (c+d x)) (60 a+13 b \sin (c+d x))\right)\right)}{12012 d \cos ^{\frac{7}{2}}(c+d x)}","-\frac{34 a b \left(53 a^2+38 b^2\right) (e \cos (c+d x))^{9/2}}{6435 d e}-\frac{2 b \left(93 a^2+26 b^2\right) (e \cos (c+d x))^{9/2} (a+b \sin (c+d x))}{715 d e}+\frac{2 e^4 \left(55 a^4+60 a^2 b^2+4 b^4\right) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{231 d \sqrt{e \cos (c+d x)}}+\frac{2 e^3 \left(55 a^4+60 a^2 b^2+4 b^4\right) \sin (c+d x) \sqrt{e \cos (c+d x)}}{231 d}+\frac{2 e \left(55 a^4+60 a^2 b^2+4 b^4\right) \sin (c+d x) (e \cos (c+d x))^{5/2}}{385 d}-\frac{2 b (e \cos (c+d x))^{9/2} (a+b \sin (c+d x))^3}{15 d e}-\frac{14 a b (e \cos (c+d x))^{9/2} (a+b \sin (c+d x))^2}{65 d e}",1,"((e*Cos[c + d*x])^(7/2)*(-154*a*b*(26*a^2 + 11*b^2)*Sqrt[Cos[c + d*x]] + 104*(55*a^4 + 60*a^2*b^2 + 4*b^4)*EllipticF[(c + d*x)/2, 2] + (Sqrt[Cos[c + d*x]]*(156*(5720*a^4 + 2460*a^2*b^2 + 87*b^4)*Sin[c + d*x] + 462*b^3*Cos[6*(c + d*x)]*(60*a + 13*b*Sin[c + d*x]) - 28*b*Cos[4*(c + d*x)]*(220*a*(26*a^2 - b^2) + 39*b*(180*a^2 + b^2)*Sin[c + d*x]) + Cos[2*(c + d*x)]*(-3080*(208*a^3*b + 73*a*b^3) + 78*(2640*a^4 - 7200*a^2*b^2 - 557*b^4)*Sin[c + d*x])))/120))/(12012*d*Cos[c + d*x]^(7/2))","A",1
565,1,209,258,2.1509789,"\int (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))^4 \, dx","Integrate[(e*Cos[c + d*x])^(5/2)*(a + b*Sin[c + d*x])^4,x]","\frac{(e \cos (c+d x))^{5/2} \left(2 \left(39 a^4+52 a^2 b^2+4 b^4\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+65 \sqrt{\cos (c+d x)} \left(-\frac{1}{78} b^2 \left(13 a^2+b^2\right) \sin (4 (c+d x))-\frac{1}{77} a b \left(66 a^2+31 b^2\right) \cos (c+d x)-\frac{1}{154} a b \left(44 a^2+9 b^2\right) \cos (3 (c+d x))+\frac{\left(624 a^4-208 a^2 b^2-61 b^4\right) \sin (2 (c+d x))}{3120}+\frac{1}{22} a b^3 \cos (5 (c+d x))+\frac{1}{208} b^4 \sin (6 (c+d x))\right)\right)}{65 d \cos ^{\frac{5}{2}}(c+d x)}","-\frac{10 a b \left(115 a^2+94 b^2\right) (e \cos (c+d x))^{7/2}}{3003 d e}-\frac{2 b \left(73 a^2+22 b^2\right) (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))}{429 d e}+\frac{2 e^2 \left(39 a^4+52 a^2 b^2+4 b^4\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{65 d \sqrt{\cos (c+d x)}}+\frac{2 e \left(39 a^4+52 a^2 b^2+4 b^4\right) \sin (c+d x) (e \cos (c+d x))^{3/2}}{195 d}-\frac{2 b (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))^3}{13 d e}-\frac{38 a b (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))^2}{143 d e}",1,"((e*Cos[c + d*x])^(5/2)*(2*(39*a^4 + 52*a^2*b^2 + 4*b^4)*EllipticE[(c + d*x)/2, 2] + 65*Sqrt[Cos[c + d*x]]*(-1/77*(a*b*(66*a^2 + 31*b^2)*Cos[c + d*x]) - (a*b*(44*a^2 + 9*b^2)*Cos[3*(c + d*x)])/154 + (a*b^3*Cos[5*(c + d*x)])/22 + ((624*a^4 - 208*a^2*b^2 - 61*b^4)*Sin[2*(c + d*x)])/3120 - (b^2*(13*a^2 + b^2)*Sin[4*(c + d*x)])/78 + (b^4*Sin[6*(c + d*x)])/208)))/(65*d*Cos[c + d*x]^(5/2))","A",1
566,1,189,258,2.8099899,"\int (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^4 \, dx","Integrate[(e*Cos[c + d*x])^(3/2)*(a + b*Sin[c + d*x])^4,x]","\frac{(e \cos (c+d x))^{3/2} \left(240 \left(77 a^4+132 a^2 b^2+12 b^4\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+\sqrt{\cos (c+d x)} \left(-2464 \left(9 a^3 b+4 a b^3\right) \cos (2 (c+d x))-1848 b \left(12 a^3+7 a b^2\right)-45 b \left(264 a^2 b+31 b^3\right) \sin (3 (c+d x))+30 \left(616 a^4+660 a^2 b^2+39 b^4\right) \sin (c+d x)+3080 a b^3 \cos (4 (c+d x))+315 b^4 \sin (5 (c+d x))\right)\right)}{27720 d \cos ^{\frac{3}{2}}(c+d x)}","-\frac{26 a b \left(79 a^2+74 b^2\right) (e \cos (c+d x))^{5/2}}{3465 d e}-\frac{2 b \left(167 a^2+54 b^2\right) (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))}{693 d e}+\frac{2 e^2 \left(77 a^4+132 a^2 b^2+12 b^4\right) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{231 d \sqrt{e \cos (c+d x)}}+\frac{2 e \left(77 a^4+132 a^2 b^2+12 b^4\right) \sin (c+d x) \sqrt{e \cos (c+d x)}}{231 d}-\frac{2 b (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))^3}{11 d e}-\frac{34 a b (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))^2}{99 d e}",1,"((e*Cos[c + d*x])^(3/2)*(240*(77*a^4 + 132*a^2*b^2 + 12*b^4)*EllipticF[(c + d*x)/2, 2] + Sqrt[Cos[c + d*x]]*(-1848*b*(12*a^3 + 7*a*b^2) - 2464*(9*a^3*b + 4*a*b^3)*Cos[2*(c + d*x)] + 3080*a*b^3*Cos[4*(c + d*x)] + 30*(616*a^4 + 660*a^2*b^2 + 39*b^4)*Sin[c + d*x] - 45*b*(264*a^2*b + 31*b^3)*Sin[3*(c + d*x)] + 315*b^4*Sin[5*(c + d*x)])))/(27720*d*Cos[c + d*x]^(3/2))","A",1
567,1,137,210,1.0981271,"\int \sqrt{e \cos (c+d x)} (a+b \sin (c+d x))^4 \, dx","Integrate[Sqrt[e*Cos[c + d*x]]*(a + b*Sin[c + d*x])^4,x]","\frac{\sqrt{e \cos (c+d x)} \left(84 \left(15 a^4+36 a^2 b^2+4 b^4\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)-b \cos ^{\frac{3}{2}}(c+d x) \left(5 \left(336 a^3+264 a b^2-7 b^3 \sin (3 (c+d x))\right)+21 b \left(72 a^2+13 b^2\right) \sin (c+d x)-360 a b^2 \cos (2 (c+d x))\right)\right)}{630 d \sqrt{\cos (c+d x)}}","-\frac{22 a b \left(17 a^2+18 b^2\right) (e \cos (c+d x))^{3/2}}{315 d e}-\frac{2 b \left(41 a^2+14 b^2\right) (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))}{105 d e}+\frac{2 \left(15 a^4+36 a^2 b^2+4 b^4\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{15 d \sqrt{\cos (c+d x)}}-\frac{2 b (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^3}{9 d e}-\frac{10 a b (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^2}{21 d e}",1,"(Sqrt[e*Cos[c + d*x]]*(84*(15*a^4 + 36*a^2*b^2 + 4*b^4)*EllipticE[(c + d*x)/2, 2] - b*Cos[c + d*x]^(3/2)*(-360*a*b^2*Cos[2*(c + d*x)] + 21*b*(72*a^2 + 13*b^2)*Sin[c + d*x] + 5*(336*a^3 + 264*a*b^2 - 7*b^3*Sin[3*(c + d*x)]))))/(630*d*Sqrt[Cos[c + d*x]])","A",1
568,1,130,210,1.1290301,"\int \frac{(a+b \sin (c+d x))^4}{\sqrt{e \cos (c+d x)}} \, dx","Integrate[(a + b*Sin[c + d*x])^4/Sqrt[e*Cos[c + d*x]],x]","\frac{20 \left(7 a^4+28 a^2 b^2+4 b^4\right) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-b \cos (c+d x) \left(560 a^3+5 b \left(56 a^2+11 b^2\right) \sin (c+d x)-56 a b^2 \cos (2 (c+d x))+504 a b^2-5 b^3 \sin (3 (c+d x))\right)}{70 d \sqrt{e \cos (c+d x)}}","-\frac{6 a b \left(31 a^2+34 b^2\right) \sqrt{e \cos (c+d x)}}{35 d e}-\frac{2 b \left(29 a^2+10 b^2\right) \sqrt{e \cos (c+d x)} (a+b \sin (c+d x))}{35 d e}+\frac{2 \left(7 a^4+28 a^2 b^2+4 b^4\right) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{7 d \sqrt{e \cos (c+d x)}}-\frac{2 b \sqrt{e \cos (c+d x)} (a+b \sin (c+d x))^3}{7 d e}-\frac{26 a b \sqrt{e \cos (c+d x)} (a+b \sin (c+d x))^2}{35 d e}",1,"(20*(7*a^4 + 28*a^2*b^2 + 4*b^4)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] - b*Cos[c + d*x]*(560*a^3 + 504*a*b^2 - 56*a*b^2*Cos[2*(c + d*x)] + 5*b*(56*a^2 + 11*b^2)*Sin[c + d*x] - 5*b^3*Sin[3*(c + d*x)]))/(70*d*Sqrt[e*Cos[c + d*x]])","A",1
569,1,135,218,0.5990873,"\int \frac{(a+b \sin (c+d x))^4}{(e \cos (c+d x))^{3/2}} \, dx","Integrate[(a + b*Sin[c + d*x])^4/(e*Cos[c + d*x])^(3/2),x]","\frac{\frac{1}{2} \left(240 a^3 b+\left(60 a^4+360 a^2 b^2+63 b^4\right) \sin (c+d x)+40 a b^3 \cos (2 (c+d x))+280 a b^3+3 b^4 \sin (3 (c+d x))\right)-6 \left(5 a^4+60 a^2 b^2+12 b^4\right) \sqrt{\cos (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d e \sqrt{e \cos (c+d x)}}","\frac{2 a b \left(15 a^2+62 b^2\right) (e \cos (c+d x))^{3/2}}{15 d e^3}+\frac{2 b \left(5 a^2+6 b^2\right) (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))}{5 d e^3}-\frac{2 \left(5 a^4+60 a^2 b^2+12 b^4\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{5 d e^2 \sqrt{\cos (c+d x)}}+\frac{2 a b (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^2}{d e^3}+\frac{2 (a \sin (c+d x)+b) (a+b \sin (c+d x))^3}{d e \sqrt{e \cos (c+d x)}}",1,"(-6*(5*a^4 + 60*a^2*b^2 + 12*b^4)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2] + (240*a^3*b + 280*a*b^3 + 40*a*b^3*Cos[2*(c + d*x)] + (60*a^4 + 360*a^2*b^2 + 63*b^4)*Sin[c + d*x] + 3*b^4*Sin[3*(c + d*x)])/2)/(15*d*e*Sqrt[e*Cos[c + d*x]])","A",1
570,1,137,216,1.1889577,"\int \frac{(a+b \sin (c+d x))^4}{(e \cos (c+d x))^{5/2}} \, dx","Integrate[(a + b*Sin[c + d*x])^4/(e*Cos[c + d*x])^(5/2),x]","\frac{4 a^4 \sin (c+d x)+16 a^3 b+24 a^2 b^2 \sin (c+d x)+4 \left(a^4-12 a^2 b^2-4 b^4\right) \cos ^{\frac{3}{2}}(c+d x) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+24 a b^3 \cos (2 (c+d x))+40 a b^3+5 b^4 \sin (c+d x)+b^4 \sin (3 (c+d x))}{6 d e (e \cos (c+d x))^{3/2}}","\frac{2 a b \left(a^2+14 b^2\right) \sqrt{e \cos (c+d x)}}{3 d e^3}+\frac{2 b \left(a^2+2 b^2\right) \sqrt{e \cos (c+d x)} (a+b \sin (c+d x))}{3 d e^3}+\frac{2 \left(a^4-12 a^2 b^2-4 b^4\right) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d e^2 \sqrt{e \cos (c+d x)}}+\frac{2 a b \sqrt{e \cos (c+d x)} (a+b \sin (c+d x))^2}{3 d e^3}+\frac{2 (a \sin (c+d x)+b) (a+b \sin (c+d x))^3}{3 d e (e \cos (c+d x))^{3/2}}",1,"(16*a^3*b + 40*a*b^3 + 24*a*b^3*Cos[2*(c + d*x)] + 4*(a^4 - 12*a^2*b^2 - 4*b^4)*Cos[c + d*x]^(3/2)*EllipticF[(c + d*x)/2, 2] + 4*a^4*Sin[c + d*x] + 24*a^2*b^2*Sin[c + d*x] + 5*b^4*Sin[c + d*x] + b^4*Sin[3*(c + d*x)])/(6*d*e*(e*Cos[c + d*x])^(3/2))","A",1
571,1,152,237,0.6282977,"\int \frac{(a+b \sin (c+d x))^4}{(e \cos (c+d x))^{7/2}} \, dx","Integrate[(a + b*Sin[c + d*x])^4/(e*Cos[c + d*x])^(7/2),x]","\frac{2 \left(3 a^4 \sin (c+d x)-12 a^2 b^2 \sin (c+d x)+4 a b \left(a^2+b^2\right) \sec ^2(c+d x)-3 \left(a^4-4 a^2 b^2-4 b^4\right) \sqrt{\cos (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+\left(a^4+6 a^2 b^2+b^4\right) \tan (c+d x) \sec (c+d x)-20 a b^3-7 b^4 \sin (c+d x)\right)}{5 d e^3 \sqrt{e \cos (c+d x)}}","\frac{2 a b \left(3 a^2-10 b^2\right) (e \cos (c+d x))^{3/2}}{5 d e^5}+\frac{6 b \left(a^2-2 b^2\right) (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))}{5 d e^5}-\frac{6 \left(a b-\left(a^2-2 b^2\right) \sin (c+d x)\right) (a+b \sin (c+d x))^2}{5 d e^3 \sqrt{e \cos (c+d x)}}-\frac{6 \left(a^4-4 a^2 b^2-4 b^4\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{5 d e^4 \sqrt{\cos (c+d x)}}+\frac{2 (a \sin (c+d x)+b) (a+b \sin (c+d x))^3}{5 d e (e \cos (c+d x))^{5/2}}",1,"(2*(-20*a*b^3 - 3*(a^4 - 4*a^2*b^2 - 4*b^4)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2] + 4*a*b*(a^2 + b^2)*Sec[c + d*x]^2 + 3*a^4*Sin[c + d*x] - 12*a^2*b^2*Sin[c + d*x] - 7*b^4*Sin[c + d*x] + (a^4 + 6*a^2*b^2 + b^4)*Sec[c + d*x]*Tan[c + d*x]))/(5*d*e^3*Sqrt[e*Cos[c + d*x]])","A",1
572,1,177,241,0.9323499,"\int \frac{(a+b \sin (c+d x))^4}{(e \cos (c+d x))^{9/2}} \, dx","Integrate[(a + b*Sin[c + d*x])^4/(e*Cos[c + d*x])^(9/2),x]","\frac{\sec ^4(c+d x) \sqrt{e \cos (c+d x)} \left(17 a^4 \sin (c+d x)+5 a^4 \sin (3 (c+d x))+48 a^3 b+60 a^2 b^2 \sin (c+d x)-12 a^2 b^2 \sin (3 (c+d x))+4 \left(5 a^4-12 a^2 b^2+12 b^4\right) \cos ^{\frac{7}{2}}(c+d x) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-56 a b^3 \cos (2 (c+d x))-8 a b^3+3 b^4 \sin (c+d x)-9 b^4 \sin (3 (c+d x))\right)}{42 d e^5}","\frac{10 a b \left(a^2-2 b^2\right) \sqrt{e \cos (c+d x)}}{21 d e^5}+\frac{2 b \left(5 a^2-6 b^2\right) \sqrt{e \cos (c+d x)} (a+b \sin (c+d x))}{21 d e^5}-\frac{2 \left(a b-\left(5 a^2-6 b^2\right) \sin (c+d x)\right) (a+b \sin (c+d x))^2}{21 d e^3 (e \cos (c+d x))^{3/2}}+\frac{2 \left(5 a^4-12 a^2 b^2+12 b^4\right) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d e^4 \sqrt{e \cos (c+d x)}}+\frac{2 (a \sin (c+d x)+b) (a+b \sin (c+d x))^3}{7 d e (e \cos (c+d x))^{7/2}}",1,"(Sqrt[e*Cos[c + d*x]]*Sec[c + d*x]^4*(48*a^3*b - 8*a*b^3 - 56*a*b^3*Cos[2*(c + d*x)] + 4*(5*a^4 - 12*a^2*b^2 + 12*b^4)*Cos[c + d*x]^(7/2)*EllipticF[(c + d*x)/2, 2] + 17*a^4*Sin[c + d*x] + 60*a^2*b^2*Sin[c + d*x] + 3*b^4*Sin[c + d*x] + 5*a^4*Sin[3*(c + d*x)] - 12*a^2*b^2*Sin[3*(c + d*x)] - 9*b^4*Sin[3*(c + d*x)]))/(42*d*e^5)","A",1
573,1,219,264,1.6232734,"\int \frac{(a+b \sin (c+d x))^4}{(e \cos (c+d x))^{11/2}} \, dx","Integrate[(a + b*Sin[c + d*x])^4/(e*Cos[c + d*x])^(11/2),x]","\frac{\sec ^5(c+d x) \sqrt{e \cos (c+d x)} \left(150 a^4 \sin (c+d x)+91 a^4 \sin (3 (c+d x))+21 a^4 \sin (5 (c+d x))+320 a^3 b+360 a^2 b^2 \sin (c+d x)-156 a^2 b^2 \sin (3 (c+d x))-36 a^2 b^2 \sin (5 (c+d x))-48 \left(7 a^4-12 a^2 b^2+4 b^4\right) \cos ^{\frac{9}{2}}(c+d x) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)-288 a b^3 \cos (2 (c+d x))+32 a b^3+60 b^4 \sin (c+d x)-8 b^4 \sin (3 (c+d x))+12 b^4 \sin (5 (c+d x))\right)}{360 d e^6}","\frac{2 a b \left(21 a^2-22 b^2\right) (e \cos (c+d x))^{3/2}}{45 d e^7}-\frac{2 \left(b \left(7 a^2-6 b^2\right)-a \left(21 a^2-22 b^2\right) \sin (c+d x)\right) (a+b \sin (c+d x))}{45 d e^5 \sqrt{e \cos (c+d x)}}+\frac{2 \left(\left(7 a^2-6 b^2\right) \sin (c+d x)+a b\right) (a+b \sin (c+d x))^2}{45 d e^3 (e \cos (c+d x))^{5/2}}-\frac{2 \left(7 a^4-12 a^2 b^2+4 b^4\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{15 d e^6 \sqrt{\cos (c+d x)}}+\frac{2 (a \sin (c+d x)+b) (a+b \sin (c+d x))^3}{9 d e (e \cos (c+d x))^{9/2}}",1,"(Sqrt[e*Cos[c + d*x]]*Sec[c + d*x]^5*(320*a^3*b + 32*a*b^3 - 288*a*b^3*Cos[2*(c + d*x)] - 48*(7*a^4 - 12*a^2*b^2 + 4*b^4)*Cos[c + d*x]^(9/2)*EllipticE[(c + d*x)/2, 2] + 150*a^4*Sin[c + d*x] + 360*a^2*b^2*Sin[c + d*x] + 60*b^4*Sin[c + d*x] + 91*a^4*Sin[3*(c + d*x)] - 156*a^2*b^2*Sin[3*(c + d*x)] - 8*b^4*Sin[3*(c + d*x)] + 21*a^4*Sin[5*(c + d*x)] - 36*a^2*b^2*Sin[5*(c + d*x)] + 12*b^4*Sin[5*(c + d*x)]))/(360*d*e^6)","A",1
574,1,2035,531,28.3046908,"\int \frac{(e \cos (c+d x))^{11/2}}{a+b \sin (c+d x)} \, dx","Integrate[(e*Cos[c + d*x])^(11/2)/(a + b*Sin[c + d*x]),x]","\text{Result too large to show}","-\frac{e^{11/2} \left(b^2-a^2\right)^{9/4} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{b^{11/2} d}-\frac{e^{11/2} \left(b^2-a^2\right)^{9/4} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{b^{11/2} d}-\frac{a e^6 \left(a^2-b^2\right)^3 \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{b^6 d \left(a^2-b \left(b-\sqrt{b^2-a^2}\right)\right) \sqrt{e \cos (c+d x)}}-\frac{a e^6 \left(a^2-b^2\right)^3 \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{b^6 d \left(a^2-b \left(\sqrt{b^2-a^2}+b\right)\right) \sqrt{e \cos (c+d x)}}+\frac{2 e^5 \sqrt{e \cos (c+d x)} \left(21 \left(a^2-b^2\right)^2-a b \left(7 a^2-12 b^2\right) \sin (c+d x)\right)}{21 b^5 d}-\frac{2 e^3 (e \cos (c+d x))^{5/2} \left(7 \left(a^2-b^2\right)-5 a b \sin (c+d x)\right)}{35 b^3 d}+\frac{2 a e^6 \left(21 a^4-49 a^2 b^2+33 b^4\right) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 b^6 d \sqrt{e \cos (c+d x)}}+\frac{2 e (e \cos (c+d x))^{9/2}}{9 b d}",1,"((e*Cos[c + d*x])^(11/2)*((-2*(280*a^4 - 636*a^2*b^2 + 721*b^4)*(a + b*Sqrt[1 - Cos[c + d*x]^2])*((5*a*(a^2 - b^2)*AppellF1[1/4, 1/2, 1, 5/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)]*Sqrt[Cos[c + d*x]])/(Sqrt[1 - Cos[c + d*x]^2]*(5*(a^2 - b^2)*AppellF1[1/4, 1/2, 1, 5/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)] - 2*(2*b^2*AppellF1[5/4, 1/2, 2, 9/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)] + (-a^2 + b^2)*AppellF1[5/4, 3/2, 1, 9/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)])*Cos[c + d*x]^2)*(a^2 + b^2*(-1 + Cos[c + d*x]^2))) - ((1/8 - I/8)*Sqrt[b]*(2*ArcTan[1 - ((1 + I)*Sqrt[b]*Sqrt[Cos[c + d*x]])/(-a^2 + b^2)^(1/4)] - 2*ArcTan[1 + ((1 + I)*Sqrt[b]*Sqrt[Cos[c + d*x]])/(-a^2 + b^2)^(1/4)] + Log[Sqrt[-a^2 + b^2] - (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Cos[c + d*x]] + I*b*Cos[c + d*x]] - Log[Sqrt[-a^2 + b^2] + (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Cos[c + d*x]] + I*b*Cos[c + d*x]]))/(-a^2 + b^2)^(3/4))*Sin[c + d*x])/(Sqrt[1 - Cos[c + d*x]^2]*(a + b*Sin[c + d*x])) + ((840*a^4 - 1764*a^2*b^2 + 959*b^4)*(a + b*Sqrt[1 - Cos[c + d*x]^2])*Cos[2*(c + d*x)]*(((1/2 - I/2)*(-2*a^2 + b^2)*ArcTan[1 - ((1 + I)*Sqrt[b]*Sqrt[Cos[c + d*x]])/(-a^2 + b^2)^(1/4)])/(b^(3/2)*(-a^2 + b^2)^(3/4)) - ((1/2 - I/2)*(-2*a^2 + b^2)*ArcTan[1 + ((1 + I)*Sqrt[b]*Sqrt[Cos[c + d*x]])/(-a^2 + b^2)^(1/4)])/(b^(3/2)*(-a^2 + b^2)^(3/4)) + (4*Sqrt[Cos[c + d*x]])/b - (4*a*AppellF1[5/4, 1/2, 1, 9/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)]*Cos[c + d*x]^(5/2))/(5*(a^2 - b^2)) + (10*a*(a^2 - b^2)*AppellF1[1/4, 1/2, 1, 5/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)]*Sqrt[Cos[c + d*x]])/(Sqrt[1 - Cos[c + d*x]^2]*(5*(a^2 - b^2)*AppellF1[1/4, 1/2, 1, 5/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)] - 2*(2*b^2*AppellF1[5/4, 1/2, 2, 9/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)] + (-a^2 + b^2)*AppellF1[5/4, 3/2, 1, 9/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)])*Cos[c + d*x]^2)*(a^2 + b^2*(-1 + Cos[c + d*x]^2))) + ((1/4 - I/4)*(-2*a^2 + b^2)*Log[Sqrt[-a^2 + b^2] - (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Cos[c + d*x]] + I*b*Cos[c + d*x]])/(b^(3/2)*(-a^2 + b^2)^(3/4)) - ((1/4 - I/4)*(-2*a^2 + b^2)*Log[Sqrt[-a^2 + b^2] + (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Cos[c + d*x]] + I*b*Cos[c + d*x]])/(b^(3/2)*(-a^2 + b^2)^(3/4)))*Sin[c + d*x])/(Sqrt[1 - Cos[c + d*x]^2]*(-1 + 2*Cos[c + d*x]^2)*(a + b*Sin[c + d*x])) - (2*(-392*a^3*b + 722*a*b^3)*(a + b*Sqrt[1 - Cos[c + d*x]^2])*((5*b*(a^2 - b^2)*AppellF1[1/4, -1/2, 1, 5/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)]*Sqrt[Cos[c + d*x]]*Sqrt[1 - Cos[c + d*x]^2])/((-5*(a^2 - b^2)*AppellF1[1/4, -1/2, 1, 5/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)] + 2*(2*b^2*AppellF1[5/4, -1/2, 2, 9/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)] + (a^2 - b^2)*AppellF1[5/4, 1/2, 1, 9/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)])*Cos[c + d*x]^2)*(a^2 + b^2*(-1 + Cos[c + d*x]^2))) + (a*(-2*ArcTan[1 - (Sqrt[2]*Sqrt[b]*Sqrt[Cos[c + d*x]])/(a^2 - b^2)^(1/4)] + 2*ArcTan[1 + (Sqrt[2]*Sqrt[b]*Sqrt[Cos[c + d*x]])/(a^2 - b^2)^(1/4)] - Log[Sqrt[a^2 - b^2] - Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Cos[c + d*x]] + b*Cos[c + d*x]] + Log[Sqrt[a^2 - b^2] + Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Cos[c + d*x]] + b*Cos[c + d*x]]))/(4*Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(3/4)))*Sin[c + d*x]^2)/((1 - Cos[c + d*x]^2)*(a + b*Sin[c + d*x]))))/(1680*b^4*d*Cos[c + d*x]^(11/2)) + ((e*Cos[c + d*x])^(11/2)*Sec[c + d*x]^5*(((-9*a^2 + 14*b^2)*Cos[2*(c + d*x)])/(45*b^3) + Cos[4*(c + d*x)]/(36*b) - (a*(28*a^2 - 51*b^2)*Sin[c + d*x])/(42*b^4) + (a*Sin[3*(c + d*x)])/(14*b^2)))/d","C",0
575,1,834,446,27.1715607,"\int \frac{(e \cos (c+d x))^{9/2}}{a+b \sin (c+d x)} \, dx","Integrate[(e*Cos[c + d*x])^(9/2)/(a + b*Sin[c + d*x]),x]","\frac{(e \cos (c+d x))^{9/2} \sec ^4(c+d x) \left(\frac{\left(37 b^2-28 a^2\right) \cos (c+d x)}{42 b^3}+\frac{\cos (3 (c+d x))}{14 b}+\frac{a \sin (2 (c+d x))}{5 b^2}\right)}{d}-\frac{(e \cos (c+d x))^{9/2} \left(-\frac{\left(5 a^3-8 a b^2\right) \left(a+b \sqrt{1-\cos ^2(c+d x)}\right) \left(8 F_1\left(\frac{3}{4};-\frac{1}{2},1;\frac{7}{4};\cos ^2(c+d x),\frac{b^2 \cos ^2(c+d x)}{b^2-a^2}\right) \cos ^{\frac{3}{2}}(c+d x) b^{5/2}+3 \sqrt{2} a \left(a^2-b^2\right)^{3/4} \left(2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{b} \sqrt{\cos (c+d x)}}{\sqrt[4]{a^2-b^2}}\right)-2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{b} \sqrt{\cos (c+d x)}}{\sqrt[4]{a^2-b^2}}+1\right)-\log \left(b \cos (c+d x)-\sqrt{2} \sqrt{b} \sqrt[4]{a^2-b^2} \sqrt{\cos (c+d x)}+\sqrt{a^2-b^2}\right)+\log \left(b \cos (c+d x)+\sqrt{2} \sqrt{b} \sqrt[4]{a^2-b^2} \sqrt{\cos (c+d x)}+\sqrt{a^2-b^2}\right)\right)\right) \sin ^2(c+d x)}{12 b^{3/2} \left(b^2-a^2\right) \left(1-\cos ^2(c+d x)\right) (a+b \sin (c+d x))}-\frac{2 \left(2 a^2 b-5 b^3\right) \left(a+b \sqrt{1-\cos ^2(c+d x)}\right) \left(\frac{a F_1\left(\frac{3}{4};\frac{1}{2},1;\frac{7}{4};\cos ^2(c+d x),\frac{b^2 \cos ^2(c+d x)}{b^2-a^2}\right) \cos ^{\frac{3}{2}}(c+d x)}{3 \left(a^2-b^2\right)}+\frac{\left(\frac{1}{8}+\frac{i}{8}\right) \left(2 \tan ^{-1}\left(1-\frac{(1+i) \sqrt{b} \sqrt{\cos (c+d x)}}{\sqrt[4]{b^2-a^2}}\right)-2 \tan ^{-1}\left(\frac{(1+i) \sqrt{b} \sqrt{\cos (c+d x)}}{\sqrt[4]{b^2-a^2}}+1\right)-\log \left(i b \cos (c+d x)-(1+i) \sqrt{b} \sqrt[4]{b^2-a^2} \sqrt{\cos (c+d x)}+\sqrt{b^2-a^2}\right)+\log \left(i b \cos (c+d x)+(1+i) \sqrt{b} \sqrt[4]{b^2-a^2} \sqrt{\cos (c+d x)}+\sqrt{b^2-a^2}\right)\right)}{\sqrt{b} \sqrt[4]{b^2-a^2}}\right) \sin (c+d x)}{\sqrt{1-\cos ^2(c+d x)} (a+b \sin (c+d x))}\right)}{5 b^3 d \cos ^{\frac{9}{2}}(c+d x)}","\frac{e^{9/2} \left(b^2-a^2\right)^{7/4} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{b^{9/2} d}-\frac{e^{9/2} \left(b^2-a^2\right)^{7/4} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{b^{9/2} d}+\frac{a e^5 \left(a^2-b^2\right)^2 \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{b^5 d \left(b-\sqrt{b^2-a^2}\right) \sqrt{e \cos (c+d x)}}+\frac{a e^5 \left(a^2-b^2\right)^2 \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{b^5 d \left(\sqrt{b^2-a^2}+b\right) \sqrt{e \cos (c+d x)}}-\frac{2 a e^4 \left(5 a^2-8 b^2\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{5 b^4 d \sqrt{\cos (c+d x)}}-\frac{2 e^3 (e \cos (c+d x))^{3/2} \left(5 \left(a^2-b^2\right)-3 a b \sin (c+d x)\right)}{15 b^3 d}+\frac{2 e (e \cos (c+d x))^{7/2}}{7 b d}",1,"-1/5*((e*Cos[c + d*x])^(9/2)*((-2*(2*a^2*b - 5*b^3)*(a + b*Sqrt[1 - Cos[c + d*x]^2])*((a*AppellF1[3/4, 1/2, 1, 7/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)]*Cos[c + d*x]^(3/2))/(3*(a^2 - b^2)) + ((1/8 + I/8)*(2*ArcTan[1 - ((1 + I)*Sqrt[b]*Sqrt[Cos[c + d*x]])/(-a^2 + b^2)^(1/4)] - 2*ArcTan[1 + ((1 + I)*Sqrt[b]*Sqrt[Cos[c + d*x]])/(-a^2 + b^2)^(1/4)] - Log[Sqrt[-a^2 + b^2] - (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Cos[c + d*x]] + I*b*Cos[c + d*x]] + Log[Sqrt[-a^2 + b^2] + (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Cos[c + d*x]] + I*b*Cos[c + d*x]]))/(Sqrt[b]*(-a^2 + b^2)^(1/4)))*Sin[c + d*x])/(Sqrt[1 - Cos[c + d*x]^2]*(a + b*Sin[c + d*x])) - ((5*a^3 - 8*a*b^2)*(a + b*Sqrt[1 - Cos[c + d*x]^2])*(8*b^(5/2)*AppellF1[3/4, -1/2, 1, 7/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)]*Cos[c + d*x]^(3/2) + 3*Sqrt[2]*a*(a^2 - b^2)^(3/4)*(2*ArcTan[1 - (Sqrt[2]*Sqrt[b]*Sqrt[Cos[c + d*x]])/(a^2 - b^2)^(1/4)] - 2*ArcTan[1 + (Sqrt[2]*Sqrt[b]*Sqrt[Cos[c + d*x]])/(a^2 - b^2)^(1/4)] - Log[Sqrt[a^2 - b^2] - Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Cos[c + d*x]] + b*Cos[c + d*x]] + Log[Sqrt[a^2 - b^2] + Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Cos[c + d*x]] + b*Cos[c + d*x]]))*Sin[c + d*x]^2)/(12*b^(3/2)*(-a^2 + b^2)*(1 - Cos[c + d*x]^2)*(a + b*Sin[c + d*x]))))/(b^3*d*Cos[c + d*x]^(9/2)) + ((e*Cos[c + d*x])^(9/2)*Sec[c + d*x]^4*(((-28*a^2 + 37*b^2)*Cos[c + d*x])/(42*b^3) + Cos[3*(c + d*x)]/(14*b) + (a*Sin[2*(c + d*x)])/(5*b^2)))/d","C",0
576,1,1955,461,29.0414019,"\int \frac{(e \cos (c+d x))^{7/2}}{a+b \sin (c+d x)} \, dx","Integrate[(e*Cos[c + d*x])^(7/2)/(a + b*Sin[c + d*x]),x]","\frac{(e \cos (c+d x))^{7/2} \sec ^3(c+d x) \left(\frac{\cos (2 (c+d x))}{5 b}+\frac{2 a \sin (c+d x)}{3 b^2}\right)}{d}-\frac{(e \cos (c+d x))^{7/2} \left(\frac{28 a b \left(a+b \sqrt{1-\cos ^2(c+d x)}\right) \left(\frac{5 b \left(a^2-b^2\right) \sqrt{\cos (c+d x)} \sqrt{1-\cos ^2(c+d x)} F_1\left(\frac{1}{4};-\frac{1}{2},1;\frac{5}{4};\cos ^2(c+d x),\frac{b^2 \cos ^2(c+d x)}{b^2-a^2}\right)}{\left(2 \left(2 F_1\left(\frac{5}{4};-\frac{1}{2},2;\frac{9}{4};\cos ^2(c+d x),\frac{b^2 \cos ^2(c+d x)}{b^2-a^2}\right) b^2+\left(a^2-b^2\right) F_1\left(\frac{5}{4};\frac{1}{2},1;\frac{9}{4};\cos ^2(c+d x),\frac{b^2 \cos ^2(c+d x)}{b^2-a^2}\right)\right) \cos ^2(c+d x)-5 \left(a^2-b^2\right) F_1\left(\frac{1}{4};-\frac{1}{2},1;\frac{5}{4};\cos ^2(c+d x),\frac{b^2 \cos ^2(c+d x)}{b^2-a^2}\right)\right) \left(a^2+b^2 \left(\cos ^2(c+d x)-1\right)\right)}+\frac{a \left(-2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{b} \sqrt{\cos (c+d x)}}{\sqrt[4]{a^2-b^2}}\right)+2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{b} \sqrt{\cos (c+d x)}}{\sqrt[4]{a^2-b^2}}+1\right)-\log \left(b \cos (c+d x)-\sqrt{2} \sqrt{b} \sqrt[4]{a^2-b^2} \sqrt{\cos (c+d x)}+\sqrt{a^2-b^2}\right)+\log \left(b \cos (c+d x)+\sqrt{2} \sqrt{b} \sqrt[4]{a^2-b^2} \sqrt{\cos (c+d x)}+\sqrt{a^2-b^2}\right)\right)}{4 \sqrt{2} \sqrt{b} \left(a^2-b^2\right)^{3/4}}\right) \sin ^2(c+d x)}{\left(1-\cos ^2(c+d x)\right) (a+b \sin (c+d x))}-\frac{2 \left(10 a^2-27 b^2\right) \left(a+b \sqrt{1-\cos ^2(c+d x)}\right) \left(\frac{5 a \left(a^2-b^2\right) F_1\left(\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\cos ^2(c+d x),\frac{b^2 \cos ^2(c+d x)}{b^2-a^2}\right) \sqrt{\cos (c+d x)}}{\sqrt{1-\cos ^2(c+d x)} \left(5 \left(a^2-b^2\right) F_1\left(\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\cos ^2(c+d x),\frac{b^2 \cos ^2(c+d x)}{b^2-a^2}\right)-2 \left(2 F_1\left(\frac{5}{4};\frac{1}{2},2;\frac{9}{4};\cos ^2(c+d x),\frac{b^2 \cos ^2(c+d x)}{b^2-a^2}\right) b^2+\left(b^2-a^2\right) F_1\left(\frac{5}{4};\frac{3}{2},1;\frac{9}{4};\cos ^2(c+d x),\frac{b^2 \cos ^2(c+d x)}{b^2-a^2}\right)\right) \cos ^2(c+d x)\right) \left(a^2+b^2 \left(\cos ^2(c+d x)-1\right)\right)}-\frac{\left(\frac{1}{8}-\frac{i}{8}\right) \sqrt{b} \left(2 \tan ^{-1}\left(1-\frac{(1+i) \sqrt{b} \sqrt{\cos (c+d x)}}{\sqrt[4]{b^2-a^2}}\right)-2 \tan ^{-1}\left(\frac{(1+i) \sqrt{b} \sqrt{\cos (c+d x)}}{\sqrt[4]{b^2-a^2}}+1\right)+\log \left(i b \cos (c+d x)-(1+i) \sqrt{b} \sqrt[4]{b^2-a^2} \sqrt{\cos (c+d x)}+\sqrt{b^2-a^2}\right)-\log \left(i b \cos (c+d x)+(1+i) \sqrt{b} \sqrt[4]{b^2-a^2} \sqrt{\cos (c+d x)}+\sqrt{b^2-a^2}\right)\right)}{\left(b^2-a^2\right)^{3/4}}\right) \sin (c+d x)}{\sqrt{1-\cos ^2(c+d x)} (a+b \sin (c+d x))}+\frac{\left(30 a^2-33 b^2\right) \left(a+b \sqrt{1-\cos ^2(c+d x)}\right) \cos (2 (c+d x)) \left(-\frac{4 a F_1\left(\frac{5}{4};\frac{1}{2},1;\frac{9}{4};\cos ^2(c+d x),\frac{b^2 \cos ^2(c+d x)}{b^2-a^2}\right) \cos ^{\frac{5}{2}}(c+d x)}{5 \left(a^2-b^2\right)}+\frac{4 \sqrt{\cos (c+d x)}}{b}+\frac{10 a \left(a^2-b^2\right) F_1\left(\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\cos ^2(c+d x),\frac{b^2 \cos ^2(c+d x)}{b^2-a^2}\right) \sqrt{\cos (c+d x)}}{\sqrt{1-\cos ^2(c+d x)} \left(5 \left(a^2-b^2\right) F_1\left(\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\cos ^2(c+d x),\frac{b^2 \cos ^2(c+d x)}{b^2-a^2}\right)-2 \left(2 F_1\left(\frac{5}{4};\frac{1}{2},2;\frac{9}{4};\cos ^2(c+d x),\frac{b^2 \cos ^2(c+d x)}{b^2-a^2}\right) b^2+\left(b^2-a^2\right) F_1\left(\frac{5}{4};\frac{3}{2},1;\frac{9}{4};\cos ^2(c+d x),\frac{b^2 \cos ^2(c+d x)}{b^2-a^2}\right)\right) \cos ^2(c+d x)\right) \left(a^2+b^2 \left(\cos ^2(c+d x)-1\right)\right)}+\frac{\left(\frac{1}{2}-\frac{i}{2}\right) \left(b^2-2 a^2\right) \tan ^{-1}\left(1-\frac{(1+i) \sqrt{b} \sqrt{\cos (c+d x)}}{\sqrt[4]{b^2-a^2}}\right)}{b^{3/2} \left(b^2-a^2\right)^{3/4}}-\frac{\left(\frac{1}{2}-\frac{i}{2}\right) \left(b^2-2 a^2\right) \tan ^{-1}\left(\frac{(1+i) \sqrt{b} \sqrt{\cos (c+d x)}}{\sqrt[4]{b^2-a^2}}+1\right)}{b^{3/2} \left(b^2-a^2\right)^{3/4}}+\frac{\left(\frac{1}{4}-\frac{i}{4}\right) \left(b^2-2 a^2\right) \log \left(i b \cos (c+d x)-(1+i) \sqrt{b} \sqrt[4]{b^2-a^2} \sqrt{\cos (c+d x)}+\sqrt{b^2-a^2}\right)}{b^{3/2} \left(b^2-a^2\right)^{3/4}}-\frac{\left(\frac{1}{4}-\frac{i}{4}\right) \left(b^2-2 a^2\right) \log \left(i b \cos (c+d x)+(1+i) \sqrt{b} \sqrt[4]{b^2-a^2} \sqrt{\cos (c+d x)}+\sqrt{b^2-a^2}\right)}{b^{3/2} \left(b^2-a^2\right)^{3/4}}\right) \sin (c+d x)}{\sqrt{1-\cos ^2(c+d x)} \left(2 \cos ^2(c+d x)-1\right) (a+b \sin (c+d x))}\right)}{60 b^2 d \cos ^{\frac{7}{2}}(c+d x)}","-\frac{e^{7/2} \left(b^2-a^2\right)^{5/4} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{b^{7/2} d}-\frac{e^{7/2} \left(b^2-a^2\right)^{5/4} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{b^{7/2} d}-\frac{2 a e^4 \left(3 a^2-4 b^2\right) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 b^4 d \sqrt{e \cos (c+d x)}}+\frac{a e^4 \left(a^2-b^2\right)^2 \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{b^4 d \left(a^2-b \left(b-\sqrt{b^2-a^2}\right)\right) \sqrt{e \cos (c+d x)}}+\frac{a e^4 \left(a^2-b^2\right)^2 \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{b^4 d \left(a^2-b \left(\sqrt{b^2-a^2}+b\right)\right) \sqrt{e \cos (c+d x)}}-\frac{2 e^3 \sqrt{e \cos (c+d x)} \left(3 \left(a^2-b^2\right)-a b \sin (c+d x)\right)}{3 b^3 d}+\frac{2 e (e \cos (c+d x))^{5/2}}{5 b d}",1,"((e*Cos[c + d*x])^(7/2)*Sec[c + d*x]^3*(Cos[2*(c + d*x)]/(5*b) + (2*a*Sin[c + d*x])/(3*b^2)))/d - ((e*Cos[c + d*x])^(7/2)*((-2*(10*a^2 - 27*b^2)*(a + b*Sqrt[1 - Cos[c + d*x]^2])*((5*a*(a^2 - b^2)*AppellF1[1/4, 1/2, 1, 5/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)]*Sqrt[Cos[c + d*x]])/(Sqrt[1 - Cos[c + d*x]^2]*(5*(a^2 - b^2)*AppellF1[1/4, 1/2, 1, 5/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)] - 2*(2*b^2*AppellF1[5/4, 1/2, 2, 9/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)] + (-a^2 + b^2)*AppellF1[5/4, 3/2, 1, 9/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)])*Cos[c + d*x]^2)*(a^2 + b^2*(-1 + Cos[c + d*x]^2))) - ((1/8 - I/8)*Sqrt[b]*(2*ArcTan[1 - ((1 + I)*Sqrt[b]*Sqrt[Cos[c + d*x]])/(-a^2 + b^2)^(1/4)] - 2*ArcTan[1 + ((1 + I)*Sqrt[b]*Sqrt[Cos[c + d*x]])/(-a^2 + b^2)^(1/4)] + Log[Sqrt[-a^2 + b^2] - (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Cos[c + d*x]] + I*b*Cos[c + d*x]] - Log[Sqrt[-a^2 + b^2] + (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Cos[c + d*x]] + I*b*Cos[c + d*x]]))/(-a^2 + b^2)^(3/4))*Sin[c + d*x])/(Sqrt[1 - Cos[c + d*x]^2]*(a + b*Sin[c + d*x])) + ((30*a^2 - 33*b^2)*(a + b*Sqrt[1 - Cos[c + d*x]^2])*Cos[2*(c + d*x)]*(((1/2 - I/2)*(-2*a^2 + b^2)*ArcTan[1 - ((1 + I)*Sqrt[b]*Sqrt[Cos[c + d*x]])/(-a^2 + b^2)^(1/4)])/(b^(3/2)*(-a^2 + b^2)^(3/4)) - ((1/2 - I/2)*(-2*a^2 + b^2)*ArcTan[1 + ((1 + I)*Sqrt[b]*Sqrt[Cos[c + d*x]])/(-a^2 + b^2)^(1/4)])/(b^(3/2)*(-a^2 + b^2)^(3/4)) + (4*Sqrt[Cos[c + d*x]])/b - (4*a*AppellF1[5/4, 1/2, 1, 9/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)]*Cos[c + d*x]^(5/2))/(5*(a^2 - b^2)) + (10*a*(a^2 - b^2)*AppellF1[1/4, 1/2, 1, 5/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)]*Sqrt[Cos[c + d*x]])/(Sqrt[1 - Cos[c + d*x]^2]*(5*(a^2 - b^2)*AppellF1[1/4, 1/2, 1, 5/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)] - 2*(2*b^2*AppellF1[5/4, 1/2, 2, 9/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)] + (-a^2 + b^2)*AppellF1[5/4, 3/2, 1, 9/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)])*Cos[c + d*x]^2)*(a^2 + b^2*(-1 + Cos[c + d*x]^2))) + ((1/4 - I/4)*(-2*a^2 + b^2)*Log[Sqrt[-a^2 + b^2] - (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Cos[c + d*x]] + I*b*Cos[c + d*x]])/(b^(3/2)*(-a^2 + b^2)^(3/4)) - ((1/4 - I/4)*(-2*a^2 + b^2)*Log[Sqrt[-a^2 + b^2] + (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Cos[c + d*x]] + I*b*Cos[c + d*x]])/(b^(3/2)*(-a^2 + b^2)^(3/4)))*Sin[c + d*x])/(Sqrt[1 - Cos[c + d*x]^2]*(-1 + 2*Cos[c + d*x]^2)*(a + b*Sin[c + d*x])) + (28*a*b*(a + b*Sqrt[1 - Cos[c + d*x]^2])*((5*b*(a^2 - b^2)*AppellF1[1/4, -1/2, 1, 5/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)]*Sqrt[Cos[c + d*x]]*Sqrt[1 - Cos[c + d*x]^2])/((-5*(a^2 - b^2)*AppellF1[1/4, -1/2, 1, 5/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)] + 2*(2*b^2*AppellF1[5/4, -1/2, 2, 9/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)] + (a^2 - b^2)*AppellF1[5/4, 1/2, 1, 9/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)])*Cos[c + d*x]^2)*(a^2 + b^2*(-1 + Cos[c + d*x]^2))) + (a*(-2*ArcTan[1 - (Sqrt[2]*Sqrt[b]*Sqrt[Cos[c + d*x]])/(a^2 - b^2)^(1/4)] + 2*ArcTan[1 + (Sqrt[2]*Sqrt[b]*Sqrt[Cos[c + d*x]])/(a^2 - b^2)^(1/4)] - Log[Sqrt[a^2 - b^2] - Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Cos[c + d*x]] + b*Cos[c + d*x]] + Log[Sqrt[a^2 - b^2] + Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Cos[c + d*x]] + b*Cos[c + d*x]]))/(4*Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(3/4)))*Sin[c + d*x]^2)/((1 - Cos[c + d*x]^2)*(a + b*Sin[c + d*x]))))/(60*b^2*d*Cos[c + d*x]^(7/2))","C",0
577,1,709,384,21.8064914,"\int \frac{(e \cos (c+d x))^{5/2}}{a+b \sin (c+d x)} \, dx","Integrate[(e*Cos[c + d*x])^(5/2)/(a + b*Sin[c + d*x]),x]","\frac{(e \cos (c+d x))^{5/2} \left(-\frac{6 b \sin (c+d x) \left(a+b \sqrt{\sin ^2(c+d x)}\right) \left(\frac{a \cos ^{\frac{3}{2}}(c+d x) F_1\left(\frac{3}{4};\frac{1}{2},1;\frac{7}{4};\cos ^2(c+d x),\frac{b^2 \cos ^2(c+d x)}{b^2-a^2}\right)}{3 \left(a^2-b^2\right)}+\frac{\left(\frac{1}{8}+\frac{i}{8}\right) \left(-\log \left(-(1+i) \sqrt{b} \sqrt[4]{b^2-a^2} \sqrt{\cos (c+d x)}+\sqrt{b^2-a^2}+i b \cos (c+d x)\right)+\log \left((1+i) \sqrt{b} \sqrt[4]{b^2-a^2} \sqrt{\cos (c+d x)}+\sqrt{b^2-a^2}+i b \cos (c+d x)\right)+2 \tan ^{-1}\left(1-\frac{(1+i) \sqrt{b} \sqrt{\cos (c+d x)}}{\sqrt[4]{b^2-a^2}}\right)-2 \tan ^{-1}\left(1+\frac{(1+i) \sqrt{b} \sqrt{\cos (c+d x)}}{\sqrt[4]{b^2-a^2}}\right)\right)}{\sqrt{b} \sqrt[4]{b^2-a^2}}\right)}{\sqrt{\sin ^2(c+d x)} (a+b \sin (c+d x))}-\frac{a \left(a+b \sqrt{\sin ^2(c+d x)}\right) \left(8 b^{5/2} \cos ^{\frac{3}{2}}(c+d x) F_1\left(\frac{3}{4};-\frac{1}{2},1;\frac{7}{4};\cos ^2(c+d x),\frac{b^2 \cos ^2(c+d x)}{b^2-a^2}\right)+3 \sqrt{2} a \left(a^2-b^2\right)^{3/4} \left(-\log \left(-\sqrt{2} \sqrt{b} \sqrt[4]{a^2-b^2} \sqrt{\cos (c+d x)}+\sqrt{a^2-b^2}+b \cos (c+d x)\right)+\log \left(\sqrt{2} \sqrt{b} \sqrt[4]{a^2-b^2} \sqrt{\cos (c+d x)}+\sqrt{a^2-b^2}+b \cos (c+d x)\right)+2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{b} \sqrt{\cos (c+d x)}}{\sqrt[4]{a^2-b^2}}\right)-2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{b} \sqrt{\cos (c+d x)}}{\sqrt[4]{a^2-b^2}}+1\right)\right)\right)}{4 b^{3/2} \left(b^2-a^2\right) (a+b \sin (c+d x))}+2 \cos ^{\frac{3}{2}}(c+d x)\right)}{3 b d \cos ^{\frac{5}{2}}(c+d x)}","\frac{e^{5/2} \left(b^2-a^2\right)^{3/4} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{b^{5/2} d}-\frac{e^{5/2} \left(b^2-a^2\right)^{3/4} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{b^{5/2} d}-\frac{a e^3 \left(a^2-b^2\right) \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{b^3 d \left(b-\sqrt{b^2-a^2}\right) \sqrt{e \cos (c+d x)}}-\frac{a e^3 \left(a^2-b^2\right) \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{b^3 d \left(\sqrt{b^2-a^2}+b\right) \sqrt{e \cos (c+d x)}}+\frac{2 a e^2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{b^2 d \sqrt{\cos (c+d x)}}+\frac{2 e (e \cos (c+d x))^{3/2}}{3 b d}",1,"((e*Cos[c + d*x])^(5/2)*(2*Cos[c + d*x]^(3/2) - (a*(8*b^(5/2)*AppellF1[3/4, -1/2, 1, 7/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)]*Cos[c + d*x]^(3/2) + 3*Sqrt[2]*a*(a^2 - b^2)^(3/4)*(2*ArcTan[1 - (Sqrt[2]*Sqrt[b]*Sqrt[Cos[c + d*x]])/(a^2 - b^2)^(1/4)] - 2*ArcTan[1 + (Sqrt[2]*Sqrt[b]*Sqrt[Cos[c + d*x]])/(a^2 - b^2)^(1/4)] - Log[Sqrt[a^2 - b^2] - Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Cos[c + d*x]] + b*Cos[c + d*x]] + Log[Sqrt[a^2 - b^2] + Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Cos[c + d*x]] + b*Cos[c + d*x]]))*(a + b*Sqrt[Sin[c + d*x]^2]))/(4*b^(3/2)*(-a^2 + b^2)*(a + b*Sin[c + d*x])) - (6*b*((a*AppellF1[3/4, 1/2, 1, 7/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)]*Cos[c + d*x]^(3/2))/(3*(a^2 - b^2)) + ((1/8 + I/8)*(2*ArcTan[1 - ((1 + I)*Sqrt[b]*Sqrt[Cos[c + d*x]])/(-a^2 + b^2)^(1/4)] - 2*ArcTan[1 + ((1 + I)*Sqrt[b]*Sqrt[Cos[c + d*x]])/(-a^2 + b^2)^(1/4)] - Log[Sqrt[-a^2 + b^2] - (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Cos[c + d*x]] + I*b*Cos[c + d*x]] + Log[Sqrt[-a^2 + b^2] + (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Cos[c + d*x]] + I*b*Cos[c + d*x]]))/(Sqrt[b]*(-a^2 + b^2)^(1/4)))*Sin[c + d*x]*(a + b*Sqrt[Sin[c + d*x]^2]))/(Sqrt[Sin[c + d*x]^2]*(a + b*Sin[c + d*x]))))/(3*b*d*Cos[c + d*x]^(5/2))","C",0
578,1,219,397,4.6706756,"\int \frac{(e \cos (c+d x))^{3/2}}{a+b \sin (c+d x)} \, dx","Integrate[(e*Cos[c + d*x])^(3/2)/(a + b*Sin[c + d*x]),x]","-\frac{e \csc ^2(c+d x) \sqrt{e \cos (c+d x)} \left(a^2+b^2 \cos ^2(c+d x)-b^2\right) \left(2 b \tan (c+d x) \, _2F_1\left(-\frac{1}{4},1;\frac{3}{4};\left(1-\frac{a^2}{b^2}\right) \sec ^2(c+d x)\right)+a \sqrt{-\tan ^2(c+d x)} \sqrt[4]{\sec ^2(c+d x)} \left(\Pi \left(-\frac{\sqrt{b^2-a^2}}{b};\left.\sin ^{-1}\left(\sqrt[4]{\sec ^2(c+d x)}\right)\right|-1\right)+\Pi \left(\frac{\sqrt{b^2-a^2}}{b};\left.\sin ^{-1}\left(\sqrt[4]{\sec ^2(c+d x)}\right)\right|-1\right)\right)\right)}{b^2 d (a \csc (c+d x)+b) \left(b \tan (c+d x)-a \sqrt{\sec ^2(c+d x)}\right)}","-\frac{a e^2 \left(a^2-b^2\right) \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{b^2 d \left(a^2-b \left(b-\sqrt{b^2-a^2}\right)\right) \sqrt{e \cos (c+d x)}}-\frac{a e^2 \left(a^2-b^2\right) \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{b^2 d \left(a^2-b \left(\sqrt{b^2-a^2}+b\right)\right) \sqrt{e \cos (c+d x)}}-\frac{e^{3/2} \sqrt[4]{b^2-a^2} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{b^{3/2} d}-\frac{e^{3/2} \sqrt[4]{b^2-a^2} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{b^{3/2} d}+\frac{2 a e^2 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b^2 d \sqrt{e \cos (c+d x)}}+\frac{2 e \sqrt{e \cos (c+d x)}}{b d}",1,"-((e*Sqrt[e*Cos[c + d*x]]*(a^2 - b^2 + b^2*Cos[c + d*x]^2)*Csc[c + d*x]^2*(2*b*Hypergeometric2F1[-1/4, 1, 3/4, (1 - a^2/b^2)*Sec[c + d*x]^2]*Tan[c + d*x] + a*(EllipticPi[-(Sqrt[-a^2 + b^2]/b), ArcSin[(Sec[c + d*x]^2)^(1/4)], -1] + EllipticPi[Sqrt[-a^2 + b^2]/b, ArcSin[(Sec[c + d*x]^2)^(1/4)], -1])*(Sec[c + d*x]^2)^(1/4)*Sqrt[-Tan[c + d*x]^2]))/(b^2*d*(b + a*Csc[c + d*x])*(-(a*Sqrt[Sec[c + d*x]^2]) + b*Tan[c + d*x])))","C",0
579,1,361,292,16.0340101,"\int \frac{\sqrt{e \cos (c+d x)}}{a+b \sin (c+d x)} \, dx","Integrate[Sqrt[e*Cos[c + d*x]]/(a + b*Sin[c + d*x]),x]","-\frac{2 \sin (c+d x) \sqrt{e \cos (c+d x)} \left(a+b \sqrt{\sin ^2(c+d x)}\right) \left(\frac{a \cos ^{\frac{3}{2}}(c+d x) F_1\left(\frac{3}{4};\frac{1}{2},1;\frac{7}{4};\cos ^2(c+d x),\frac{b^2 \cos ^2(c+d x)}{b^2-a^2}\right)}{3 \left(a^2-b^2\right)}+\frac{\left(\frac{1}{8}+\frac{i}{8}\right) \left(-\log \left(-(1+i) \sqrt{b} \sqrt[4]{b^2-a^2} \sqrt{\cos (c+d x)}+\sqrt{b^2-a^2}+i b \cos (c+d x)\right)+\log \left((1+i) \sqrt{b} \sqrt[4]{b^2-a^2} \sqrt{\cos (c+d x)}+\sqrt{b^2-a^2}+i b \cos (c+d x)\right)+2 \tan ^{-1}\left(1-\frac{(1+i) \sqrt{b} \sqrt{\cos (c+d x)}}{\sqrt[4]{b^2-a^2}}\right)-2 \tan ^{-1}\left(1+\frac{(1+i) \sqrt{b} \sqrt{\cos (c+d x)}}{\sqrt[4]{b^2-a^2}}\right)\right)}{\sqrt{b} \sqrt[4]{b^2-a^2}}\right)}{d \sqrt{\sin ^2(c+d x)} \sqrt{\cos (c+d x)} (a+b \sin (c+d x))}","\frac{\sqrt{e} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{\sqrt{b} d \sqrt[4]{b^2-a^2}}-\frac{\sqrt{e} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{\sqrt{b} d \sqrt[4]{b^2-a^2}}+\frac{a e \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{b d \left(b-\sqrt{b^2-a^2}\right) \sqrt{e \cos (c+d x)}}+\frac{a e \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{b d \left(\sqrt{b^2-a^2}+b\right) \sqrt{e \cos (c+d x)}}",1,"(-2*Sqrt[e*Cos[c + d*x]]*((a*AppellF1[3/4, 1/2, 1, 7/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)]*Cos[c + d*x]^(3/2))/(3*(a^2 - b^2)) + ((1/8 + I/8)*(2*ArcTan[1 - ((1 + I)*Sqrt[b]*Sqrt[Cos[c + d*x]])/(-a^2 + b^2)^(1/4)] - 2*ArcTan[1 + ((1 + I)*Sqrt[b]*Sqrt[Cos[c + d*x]])/(-a^2 + b^2)^(1/4)] - Log[Sqrt[-a^2 + b^2] - (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Cos[c + d*x]] + I*b*Cos[c + d*x]] + Log[Sqrt[-a^2 + b^2] + (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Cos[c + d*x]] + I*b*Cos[c + d*x]]))/(Sqrt[b]*(-a^2 + b^2)^(1/4)))*Sin[c + d*x]*(a + b*Sqrt[Sin[c + d*x]^2]))/(d*Sqrt[Cos[c + d*x]]*Sqrt[Sin[c + d*x]^2]*(a + b*Sin[c + d*x]))","C",0
580,1,558,299,16.1255624,"\int \frac{1}{\sqrt{e \cos (c+d x)} (a+b \sin (c+d x))} \, dx","Integrate[1/(Sqrt[e*Cos[c + d*x]]*(a + b*Sin[c + d*x])),x]","-\frac{2 \sin (c+d x) \sqrt{\cos (c+d x)} \left(a+b \sqrt{\sin ^2(c+d x)}\right) \left(\frac{5 a \left(a^2-b^2\right) \sqrt{\cos (c+d x)} F_1\left(\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\cos ^2(c+d x),\frac{b^2 \cos ^2(c+d x)}{b^2-a^2}\right)}{\sqrt{\sin ^2(c+d x)} \left(a^2+b^2 \cos ^2(c+d x)-b^2\right) \left(5 \left(a^2-b^2\right) F_1\left(\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\cos ^2(c+d x),\frac{b^2 \cos ^2(c+d x)}{b^2-a^2}\right)-2 \cos ^2(c+d x) \left(2 b^2 F_1\left(\frac{5}{4};\frac{1}{2},2;\frac{9}{4};\cos ^2(c+d x),\frac{b^2 \cos ^2(c+d x)}{b^2-a^2}\right)+\left(b^2-a^2\right) F_1\left(\frac{5}{4};\frac{3}{2},1;\frac{9}{4};\cos ^2(c+d x),\frac{b^2 \cos ^2(c+d x)}{b^2-a^2}\right)\right)\right)}-\frac{\left(\frac{1}{8}-\frac{i}{8}\right) \sqrt{b} \left(\log \left(-(1+i) \sqrt{b} \sqrt[4]{b^2-a^2} \sqrt{\cos (c+d x)}+\sqrt{b^2-a^2}+i b \cos (c+d x)\right)-\log \left((1+i) \sqrt{b} \sqrt[4]{b^2-a^2} \sqrt{\cos (c+d x)}+\sqrt{b^2-a^2}+i b \cos (c+d x)\right)+2 \tan ^{-1}\left(1-\frac{(1+i) \sqrt{b} \sqrt{\cos (c+d x)}}{\sqrt[4]{b^2-a^2}}\right)-2 \tan ^{-1}\left(1+\frac{(1+i) \sqrt{b} \sqrt{\cos (c+d x)}}{\sqrt[4]{b^2-a^2}}\right)\right)}{\left(b^2-a^2\right)^{3/4}}\right)}{d \sqrt{\sin ^2(c+d x)} \sqrt{e \cos (c+d x)} (a+b \sin (c+d x))}","-\frac{\sqrt{b} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{d \sqrt{e} \left(b^2-a^2\right)^{3/4}}-\frac{\sqrt{b} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{d \sqrt{e} \left(b^2-a^2\right)^{3/4}}+\frac{a \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{d \left(a^2-b \left(b-\sqrt{b^2-a^2}\right)\right) \sqrt{e \cos (c+d x)}}+\frac{a \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{d \left(a^2-b \left(\sqrt{b^2-a^2}+b\right)\right) \sqrt{e \cos (c+d x)}}",1,"(-2*Sqrt[Cos[c + d*x]]*Sin[c + d*x]*(((-1/8 + I/8)*Sqrt[b]*(2*ArcTan[1 - ((1 + I)*Sqrt[b]*Sqrt[Cos[c + d*x]])/(-a^2 + b^2)^(1/4)] - 2*ArcTan[1 + ((1 + I)*Sqrt[b]*Sqrt[Cos[c + d*x]])/(-a^2 + b^2)^(1/4)] + Log[Sqrt[-a^2 + b^2] - (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Cos[c + d*x]] + I*b*Cos[c + d*x]] - Log[Sqrt[-a^2 + b^2] + (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Cos[c + d*x]] + I*b*Cos[c + d*x]]))/(-a^2 + b^2)^(3/4) + (5*a*(a^2 - b^2)*AppellF1[1/4, 1/2, 1, 5/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)]*Sqrt[Cos[c + d*x]])/((a^2 - b^2 + b^2*Cos[c + d*x]^2)*(5*(a^2 - b^2)*AppellF1[1/4, 1/2, 1, 5/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)] - 2*(2*b^2*AppellF1[5/4, 1/2, 2, 9/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)] + (-a^2 + b^2)*AppellF1[5/4, 3/2, 1, 9/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)])*Cos[c + d*x]^2)*Sqrt[Sin[c + d*x]^2]))*(a + b*Sqrt[Sin[c + d*x]^2]))/(d*Sqrt[e*Cos[c + d*x]]*Sqrt[Sin[c + d*x]^2]*(a + b*Sin[c + d*x]))","C",0
581,1,791,411,22.8539829,"\int \frac{1}{(e \cos (c+d x))^{3/2} (a+b \sin (c+d x))} \, dx","Integrate[1/((e*Cos[c + d*x])^(3/2)*(a + b*Sin[c + d*x])),x]","\frac{2 \cos (c+d x) (a \sin (c+d x)-b)}{d \left(a^2-b^2\right) (e \cos (c+d x))^{3/2}}-\frac{\cos ^{\frac{3}{2}}(c+d x) \left(-\frac{a \sin ^2(c+d x) \left(a+b \sqrt{1-\cos ^2(c+d x)}\right) \left(8 b^{5/2} \cos ^{\frac{3}{2}}(c+d x) F_1\left(\frac{3}{4};-\frac{1}{2},1;\frac{7}{4};\cos ^2(c+d x),\frac{b^2 \cos ^2(c+d x)}{b^2-a^2}\right)+3 \sqrt{2} a \left(a^2-b^2\right)^{3/4} \left(-\log \left(-\sqrt{2} \sqrt{b} \sqrt[4]{a^2-b^2} \sqrt{\cos (c+d x)}+\sqrt{a^2-b^2}+b \cos (c+d x)\right)+\log \left(\sqrt{2} \sqrt{b} \sqrt[4]{a^2-b^2} \sqrt{\cos (c+d x)}+\sqrt{a^2-b^2}+b \cos (c+d x)\right)+2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{b} \sqrt{\cos (c+d x)}}{\sqrt[4]{a^2-b^2}}\right)-2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{b} \sqrt{\cos (c+d x)}}{\sqrt[4]{a^2-b^2}}+1\right)\right)\right)}{12 \sqrt{b} \left(b^2-a^2\right) \left(1-\cos ^2(c+d x)\right) (a+b \sin (c+d x))}-\frac{2 \left(a^2+b^2\right) \sin (c+d x) \left(a+b \sqrt{1-\cos ^2(c+d x)}\right) \left(\frac{a \cos ^{\frac{3}{2}}(c+d x) F_1\left(\frac{3}{4};\frac{1}{2},1;\frac{7}{4};\cos ^2(c+d x),\frac{b^2 \cos ^2(c+d x)}{b^2-a^2}\right)}{3 \left(a^2-b^2\right)}+\frac{\left(\frac{1}{8}+\frac{i}{8}\right) \left(-\log \left(-(1+i) \sqrt{b} \sqrt[4]{b^2-a^2} \sqrt{\cos (c+d x)}+\sqrt{b^2-a^2}+i b \cos (c+d x)\right)+\log \left((1+i) \sqrt{b} \sqrt[4]{b^2-a^2} \sqrt{\cos (c+d x)}+\sqrt{b^2-a^2}+i b \cos (c+d x)\right)+2 \tan ^{-1}\left(1-\frac{(1+i) \sqrt{b} \sqrt{\cos (c+d x)}}{\sqrt[4]{b^2-a^2}}\right)-2 \tan ^{-1}\left(1+\frac{(1+i) \sqrt{b} \sqrt{\cos (c+d x)}}{\sqrt[4]{b^2-a^2}}\right)\right)}{\sqrt{b} \sqrt[4]{b^2-a^2}}\right)}{\sqrt{1-\cos ^2(c+d x)} (a+b \sin (c+d x))}\right)}{d (a-b) (a+b) (e \cos (c+d x))^{3/2}}","-\frac{2 a E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{d e^2 \left(a^2-b^2\right) \sqrt{\cos (c+d x)}}-\frac{2 (b-a \sin (c+d x))}{d e \left(a^2-b^2\right) \sqrt{e \cos (c+d x)}}-\frac{a b \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{d e \left(a^2-b^2\right) \left(b-\sqrt{b^2-a^2}\right) \sqrt{e \cos (c+d x)}}-\frac{a b \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{d e \left(a^2-b^2\right) \left(\sqrt{b^2-a^2}+b\right) \sqrt{e \cos (c+d x)}}+\frac{b^{3/2} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{d e^{3/2} \left(b^2-a^2\right)^{5/4}}-\frac{b^{3/2} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{d e^{3/2} \left(b^2-a^2\right)^{5/4}}",1,"(2*Cos[c + d*x]*(-b + a*Sin[c + d*x]))/((a^2 - b^2)*d*(e*Cos[c + d*x])^(3/2)) - (Cos[c + d*x]^(3/2)*((-2*(a^2 + b^2)*(a + b*Sqrt[1 - Cos[c + d*x]^2])*((a*AppellF1[3/4, 1/2, 1, 7/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)]*Cos[c + d*x]^(3/2))/(3*(a^2 - b^2)) + ((1/8 + I/8)*(2*ArcTan[1 - ((1 + I)*Sqrt[b]*Sqrt[Cos[c + d*x]])/(-a^2 + b^2)^(1/4)] - 2*ArcTan[1 + ((1 + I)*Sqrt[b]*Sqrt[Cos[c + d*x]])/(-a^2 + b^2)^(1/4)] - Log[Sqrt[-a^2 + b^2] - (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Cos[c + d*x]] + I*b*Cos[c + d*x]] + Log[Sqrt[-a^2 + b^2] + (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Cos[c + d*x]] + I*b*Cos[c + d*x]]))/(Sqrt[b]*(-a^2 + b^2)^(1/4)))*Sin[c + d*x])/(Sqrt[1 - Cos[c + d*x]^2]*(a + b*Sin[c + d*x])) - (a*(a + b*Sqrt[1 - Cos[c + d*x]^2])*(8*b^(5/2)*AppellF1[3/4, -1/2, 1, 7/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)]*Cos[c + d*x]^(3/2) + 3*Sqrt[2]*a*(a^2 - b^2)^(3/4)*(2*ArcTan[1 - (Sqrt[2]*Sqrt[b]*Sqrt[Cos[c + d*x]])/(a^2 - b^2)^(1/4)] - 2*ArcTan[1 + (Sqrt[2]*Sqrt[b]*Sqrt[Cos[c + d*x]])/(a^2 - b^2)^(1/4)] - Log[Sqrt[a^2 - b^2] - Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Cos[c + d*x]] + b*Cos[c + d*x]] + Log[Sqrt[a^2 - b^2] + Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Cos[c + d*x]] + b*Cos[c + d*x]]))*Sin[c + d*x]^2)/(12*Sqrt[b]*(-a^2 + b^2)*(1 - Cos[c + d*x]^2)*(a + b*Sin[c + d*x]))))/((a - b)*(a + b)*d*(e*Cos[c + d*x])^(3/2))","C",0
582,1,1192,434,24.5968627,"\int \frac{1}{(e \cos (c+d x))^{5/2} (a+b \sin (c+d x))} \, dx","Integrate[1/((e*Cos[c + d*x])^(5/2)*(a + b*Sin[c + d*x])),x]","\frac{\left(-\frac{2 a b \left(a+b \sqrt{1-\cos ^2(c+d x)}\right) \left(\frac{5 b \left(a^2-b^2\right) \sqrt{\cos (c+d x)} \sqrt{1-\cos ^2(c+d x)} F_1\left(\frac{1}{4};-\frac{1}{2},1;\frac{5}{4};\cos ^2(c+d x),\frac{b^2 \cos ^2(c+d x)}{b^2-a^2}\right)}{\left(2 \left(2 F_1\left(\frac{5}{4};-\frac{1}{2},2;\frac{9}{4};\cos ^2(c+d x),\frac{b^2 \cos ^2(c+d x)}{b^2-a^2}\right) b^2+\left(a^2-b^2\right) F_1\left(\frac{5}{4};\frac{1}{2},1;\frac{9}{4};\cos ^2(c+d x),\frac{b^2 \cos ^2(c+d x)}{b^2-a^2}\right)\right) \cos ^2(c+d x)-5 \left(a^2-b^2\right) F_1\left(\frac{1}{4};-\frac{1}{2},1;\frac{5}{4};\cos ^2(c+d x),\frac{b^2 \cos ^2(c+d x)}{b^2-a^2}\right)\right) \left(a^2+b^2 \left(\cos ^2(c+d x)-1\right)\right)}+\frac{a \left(-2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{b} \sqrt{\cos (c+d x)}}{\sqrt[4]{a^2-b^2}}\right)+2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{b} \sqrt{\cos (c+d x)}}{\sqrt[4]{a^2-b^2}}+1\right)-\log \left(b \cos (c+d x)-\sqrt{2} \sqrt{b} \sqrt[4]{a^2-b^2} \sqrt{\cos (c+d x)}+\sqrt{a^2-b^2}\right)+\log \left(b \cos (c+d x)+\sqrt{2} \sqrt{b} \sqrt[4]{a^2-b^2} \sqrt{\cos (c+d x)}+\sqrt{a^2-b^2}\right)\right)}{4 \sqrt{2} \sqrt{b} \left(a^2-b^2\right)^{3/4}}\right) \sin ^2(c+d x)}{\left(1-\cos ^2(c+d x)\right) (a+b \sin (c+d x))}-\frac{2 \left(a^2-3 b^2\right) \left(a+b \sqrt{1-\cos ^2(c+d x)}\right) \left(\frac{5 a \left(a^2-b^2\right) F_1\left(\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\cos ^2(c+d x),\frac{b^2 \cos ^2(c+d x)}{b^2-a^2}\right) \sqrt{\cos (c+d x)}}{\sqrt{1-\cos ^2(c+d x)} \left(5 \left(a^2-b^2\right) F_1\left(\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\cos ^2(c+d x),\frac{b^2 \cos ^2(c+d x)}{b^2-a^2}\right)-2 \left(2 F_1\left(\frac{5}{4};\frac{1}{2},2;\frac{9}{4};\cos ^2(c+d x),\frac{b^2 \cos ^2(c+d x)}{b^2-a^2}\right) b^2+\left(b^2-a^2\right) F_1\left(\frac{5}{4};\frac{3}{2},1;\frac{9}{4};\cos ^2(c+d x),\frac{b^2 \cos ^2(c+d x)}{b^2-a^2}\right)\right) \cos ^2(c+d x)\right) \left(a^2+b^2 \left(\cos ^2(c+d x)-1\right)\right)}-\frac{\left(\frac{1}{8}-\frac{i}{8}\right) \sqrt{b} \left(2 \tan ^{-1}\left(1-\frac{(1+i) \sqrt{b} \sqrt{\cos (c+d x)}}{\sqrt[4]{b^2-a^2}}\right)-2 \tan ^{-1}\left(\frac{(1+i) \sqrt{b} \sqrt{\cos (c+d x)}}{\sqrt[4]{b^2-a^2}}+1\right)+\log \left(i b \cos (c+d x)-(1+i) \sqrt{b} \sqrt[4]{b^2-a^2} \sqrt{\cos (c+d x)}+\sqrt{b^2-a^2}\right)-\log \left(i b \cos (c+d x)+(1+i) \sqrt{b} \sqrt[4]{b^2-a^2} \sqrt{\cos (c+d x)}+\sqrt{b^2-a^2}\right)\right)}{\left(b^2-a^2\right)^{3/4}}\right) \sin (c+d x)}{\sqrt{1-\cos ^2(c+d x)} (a+b \sin (c+d x))}\right) \cos ^{\frac{5}{2}}(c+d x)}{3 (a-b) (a+b) d (e \cos (c+d x))^{5/2}}+\frac{2 (a \sin (c+d x)-b) \cos (c+d x)}{3 \left(a^2-b^2\right) d (e \cos (c+d x))^{5/2}}","\frac{2 a \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d e^2 \left(a^2-b^2\right) \sqrt{e \cos (c+d x)}}-\frac{a b^2 \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{d e^2 \left(a^2-b^2\right) \left(a^2-b \left(b-\sqrt{b^2-a^2}\right)\right) \sqrt{e \cos (c+d x)}}-\frac{a b^2 \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{d e^2 \left(a^2-b^2\right) \left(a^2-b \left(\sqrt{b^2-a^2}+b\right)\right) \sqrt{e \cos (c+d x)}}-\frac{2 (b-a \sin (c+d x))}{3 d e \left(a^2-b^2\right) (e \cos (c+d x))^{3/2}}-\frac{b^{5/2} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{d e^{5/2} \left(b^2-a^2\right)^{7/4}}-\frac{b^{5/2} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{d e^{5/2} \left(b^2-a^2\right)^{7/4}}",1,"(2*Cos[c + d*x]*(-b + a*Sin[c + d*x]))/(3*(a^2 - b^2)*d*(e*Cos[c + d*x])^(5/2)) + (Cos[c + d*x]^(5/2)*((-2*(a^2 - 3*b^2)*(a + b*Sqrt[1 - Cos[c + d*x]^2])*((5*a*(a^2 - b^2)*AppellF1[1/4, 1/2, 1, 5/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)]*Sqrt[Cos[c + d*x]])/(Sqrt[1 - Cos[c + d*x]^2]*(5*(a^2 - b^2)*AppellF1[1/4, 1/2, 1, 5/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)] - 2*(2*b^2*AppellF1[5/4, 1/2, 2, 9/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)] + (-a^2 + b^2)*AppellF1[5/4, 3/2, 1, 9/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)])*Cos[c + d*x]^2)*(a^2 + b^2*(-1 + Cos[c + d*x]^2))) - ((1/8 - I/8)*Sqrt[b]*(2*ArcTan[1 - ((1 + I)*Sqrt[b]*Sqrt[Cos[c + d*x]])/(-a^2 + b^2)^(1/4)] - 2*ArcTan[1 + ((1 + I)*Sqrt[b]*Sqrt[Cos[c + d*x]])/(-a^2 + b^2)^(1/4)] + Log[Sqrt[-a^2 + b^2] - (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Cos[c + d*x]] + I*b*Cos[c + d*x]] - Log[Sqrt[-a^2 + b^2] + (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Cos[c + d*x]] + I*b*Cos[c + d*x]]))/(-a^2 + b^2)^(3/4))*Sin[c + d*x])/(Sqrt[1 - Cos[c + d*x]^2]*(a + b*Sin[c + d*x])) - (2*a*b*(a + b*Sqrt[1 - Cos[c + d*x]^2])*((5*b*(a^2 - b^2)*AppellF1[1/4, -1/2, 1, 5/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)]*Sqrt[Cos[c + d*x]]*Sqrt[1 - Cos[c + d*x]^2])/((-5*(a^2 - b^2)*AppellF1[1/4, -1/2, 1, 5/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)] + 2*(2*b^2*AppellF1[5/4, -1/2, 2, 9/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)] + (a^2 - b^2)*AppellF1[5/4, 1/2, 1, 9/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)])*Cos[c + d*x]^2)*(a^2 + b^2*(-1 + Cos[c + d*x]^2))) + (a*(-2*ArcTan[1 - (Sqrt[2]*Sqrt[b]*Sqrt[Cos[c + d*x]])/(a^2 - b^2)^(1/4)] + 2*ArcTan[1 + (Sqrt[2]*Sqrt[b]*Sqrt[Cos[c + d*x]])/(a^2 - b^2)^(1/4)] - Log[Sqrt[a^2 - b^2] - Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Cos[c + d*x]] + b*Cos[c + d*x]] + Log[Sqrt[a^2 - b^2] + Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Cos[c + d*x]] + b*Cos[c + d*x]]))/(4*Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(3/4)))*Sin[c + d*x]^2)/((1 - Cos[c + d*x]^2)*(a + b*Sin[c + d*x]))))/(3*(a - b)*(a + b)*d*(e*Cos[c + d*x])^(5/2))","C",0
583,1,881,486,6.7585199,"\int \frac{1}{(e \cos (c+d x))^{7/2} (a+b \sin (c+d x))} \, dx","Integrate[1/((e*Cos[c + d*x])^(7/2)*(a + b*Sin[c + d*x])),x]","\frac{\cos ^4(c+d x) \left(\frac{2 (a \sin (c+d x)-b) \sec ^3(c+d x)}{5 \left(a^2-b^2\right)}+\frac{2 \left(3 \sin (c+d x) a^3-8 b^2 \sin (c+d x) a+5 b^3\right) \sec (c+d x)}{5 \left(a^2-b^2\right)^2}\right)}{d (e \cos (c+d x))^{7/2}}-\frac{\cos ^{\frac{7}{2}}(c+d x) \left(-\frac{\left(3 a^3 b-8 a b^3\right) \left(a+b \sqrt{1-\cos ^2(c+d x)}\right) \left(8 F_1\left(\frac{3}{4};-\frac{1}{2},1;\frac{7}{4};\cos ^2(c+d x),\frac{b^2 \cos ^2(c+d x)}{b^2-a^2}\right) \cos ^{\frac{3}{2}}(c+d x) b^{5/2}+3 \sqrt{2} a \left(a^2-b^2\right)^{3/4} \left(2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{b} \sqrt{\cos (c+d x)}}{\sqrt[4]{a^2-b^2}}\right)-2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{b} \sqrt{\cos (c+d x)}}{\sqrt[4]{a^2-b^2}}+1\right)-\log \left(b \cos (c+d x)-\sqrt{2} \sqrt{b} \sqrt[4]{a^2-b^2} \sqrt{\cos (c+d x)}+\sqrt{a^2-b^2}\right)+\log \left(b \cos (c+d x)+\sqrt{2} \sqrt{b} \sqrt[4]{a^2-b^2} \sqrt{\cos (c+d x)}+\sqrt{a^2-b^2}\right)\right)\right) \sin ^2(c+d x)}{12 b^{3/2} \left(b^2-a^2\right) \left(1-\cos ^2(c+d x)\right) (a+b \sin (c+d x))}-\frac{2 \left(3 a^4-8 b^2 a^2-5 b^4\right) \left(a+b \sqrt{1-\cos ^2(c+d x)}\right) \left(\frac{a F_1\left(\frac{3}{4};\frac{1}{2},1;\frac{7}{4};\cos ^2(c+d x),\frac{b^2 \cos ^2(c+d x)}{b^2-a^2}\right) \cos ^{\frac{3}{2}}(c+d x)}{3 \left(a^2-b^2\right)}+\frac{\left(\frac{1}{8}+\frac{i}{8}\right) \left(2 \tan ^{-1}\left(1-\frac{(1+i) \sqrt{b} \sqrt{\cos (c+d x)}}{\sqrt[4]{b^2-a^2}}\right)-2 \tan ^{-1}\left(\frac{(1+i) \sqrt{b} \sqrt{\cos (c+d x)}}{\sqrt[4]{b^2-a^2}}+1\right)-\log \left(i b \cos (c+d x)-(1+i) \sqrt{b} \sqrt[4]{b^2-a^2} \sqrt{\cos (c+d x)}+\sqrt{b^2-a^2}\right)+\log \left(i b \cos (c+d x)+(1+i) \sqrt{b} \sqrt[4]{b^2-a^2} \sqrt{\cos (c+d x)}+\sqrt{b^2-a^2}\right)\right)}{\sqrt{b} \sqrt[4]{b^2-a^2}}\right) \sin (c+d x)}{\sqrt{1-\cos ^2(c+d x)} (a+b \sin (c+d x))}\right)}{5 (a-b)^2 (a+b)^2 d (e \cos (c+d x))^{7/2}}","-\frac{2 a \left(3 a^2-8 b^2\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{5 d e^4 \left(a^2-b^2\right)^2 \sqrt{\cos (c+d x)}}-\frac{2 (b-a \sin (c+d x))}{5 d e \left(a^2-b^2\right) (e \cos (c+d x))^{5/2}}+\frac{b^{7/2} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{d e^{7/2} \left(b^2-a^2\right)^{9/4}}-\frac{b^{7/2} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{d e^{7/2} \left(b^2-a^2\right)^{9/4}}+\frac{2 \left(a \left(3 a^2-8 b^2\right) \sin (c+d x)+5 b^3\right)}{5 d e^3 \left(a^2-b^2\right)^2 \sqrt{e \cos (c+d x)}}+\frac{a b^3 \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{d e^3 \left(a^2-b^2\right)^2 \left(b-\sqrt{b^2-a^2}\right) \sqrt{e \cos (c+d x)}}+\frac{a b^3 \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{d e^3 \left(a^2-b^2\right)^2 \left(\sqrt{b^2-a^2}+b\right) \sqrt{e \cos (c+d x)}}",1,"-1/5*(Cos[c + d*x]^(7/2)*((-2*(3*a^4 - 8*a^2*b^2 - 5*b^4)*(a + b*Sqrt[1 - Cos[c + d*x]^2])*((a*AppellF1[3/4, 1/2, 1, 7/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)]*Cos[c + d*x]^(3/2))/(3*(a^2 - b^2)) + ((1/8 + I/8)*(2*ArcTan[1 - ((1 + I)*Sqrt[b]*Sqrt[Cos[c + d*x]])/(-a^2 + b^2)^(1/4)] - 2*ArcTan[1 + ((1 + I)*Sqrt[b]*Sqrt[Cos[c + d*x]])/(-a^2 + b^2)^(1/4)] - Log[Sqrt[-a^2 + b^2] - (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Cos[c + d*x]] + I*b*Cos[c + d*x]] + Log[Sqrt[-a^2 + b^2] + (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Cos[c + d*x]] + I*b*Cos[c + d*x]]))/(Sqrt[b]*(-a^2 + b^2)^(1/4)))*Sin[c + d*x])/(Sqrt[1 - Cos[c + d*x]^2]*(a + b*Sin[c + d*x])) - ((3*a^3*b - 8*a*b^3)*(a + b*Sqrt[1 - Cos[c + d*x]^2])*(8*b^(5/2)*AppellF1[3/4, -1/2, 1, 7/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)]*Cos[c + d*x]^(3/2) + 3*Sqrt[2]*a*(a^2 - b^2)^(3/4)*(2*ArcTan[1 - (Sqrt[2]*Sqrt[b]*Sqrt[Cos[c + d*x]])/(a^2 - b^2)^(1/4)] - 2*ArcTan[1 + (Sqrt[2]*Sqrt[b]*Sqrt[Cos[c + d*x]])/(a^2 - b^2)^(1/4)] - Log[Sqrt[a^2 - b^2] - Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Cos[c + d*x]] + b*Cos[c + d*x]] + Log[Sqrt[a^2 - b^2] + Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Cos[c + d*x]] + b*Cos[c + d*x]]))*Sin[c + d*x]^2)/(12*b^(3/2)*(-a^2 + b^2)*(1 - Cos[c + d*x]^2)*(a + b*Sin[c + d*x]))))/((a - b)^2*(a + b)^2*d*(e*Cos[c + d*x])^(7/2)) + (Cos[c + d*x]^4*((2*Sec[c + d*x]^3*(-b + a*Sin[c + d*x]))/(5*(a^2 - b^2)) + (2*Sec[c + d*x]*(5*b^3 + 3*a^3*Sin[c + d*x] - 8*a*b^2*Sin[c + d*x]))/(5*(a^2 - b^2)^2)))/(d*(e*Cos[c + d*x])^(7/2))","C",0
584,1,2030,543,27.8020252,"\int \frac{(e \cos (c+d x))^{11/2}}{(a+b \sin (c+d x))^2} \, dx","Integrate[(e*Cos[c + d*x])^(11/2)/(a + b*Sin[c + d*x])^2,x]","\text{Result too large to show}","-\frac{9 a e^{11/2} \left(b^2-a^2\right)^{5/4} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{2 b^{11/2} d}-\frac{9 a e^{11/2} \left(b^2-a^2\right)^{5/4} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{2 b^{11/2} d}+\frac{9 a^2 e^6 \left(a^2-b^2\right)^2 \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{2 b^6 d \left(a^2-b \left(b-\sqrt{b^2-a^2}\right)\right) \sqrt{e \cos (c+d x)}}+\frac{9 a^2 e^6 \left(a^2-b^2\right)^2 \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{2 b^6 d \left(a^2-b \left(\sqrt{b^2-a^2}+b\right)\right) \sqrt{e \cos (c+d x)}}-\frac{3 e^5 \sqrt{e \cos (c+d x)} \left(21 a \left(a^2-b^2\right)-b \left(7 a^2-5 b^2\right) \sin (c+d x)\right)}{7 b^5 d}-\frac{3 e^6 \left(21 a^4-28 a^2 b^2+5 b^4\right) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{7 b^6 d \sqrt{e \cos (c+d x)}}+\frac{9 e^3 (e \cos (c+d x))^{5/2} (7 a-5 b \sin (c+d x))}{35 b^3 d}-\frac{e (e \cos (c+d x))^{9/2}}{b d (a+b \sin (c+d x))}",1,"-1/70*((e*Cos[c + d*x])^(11/2)*((-2*(70*a^3*b - 93*a*b^3)*(a + b*Sqrt[1 - Cos[c + d*x]^2])*((5*a*(a^2 - b^2)*AppellF1[1/4, 1/2, 1, 5/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)]*Sqrt[Cos[c + d*x]])/(Sqrt[1 - Cos[c + d*x]^2]*(5*(a^2 - b^2)*AppellF1[1/4, 1/2, 1, 5/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)] - 2*(2*b^2*AppellF1[5/4, 1/2, 2, 9/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)] + (-a^2 + b^2)*AppellF1[5/4, 3/2, 1, 9/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)])*Cos[c + d*x]^2)*(a^2 + b^2*(-1 + Cos[c + d*x]^2))) - ((1/8 - I/8)*Sqrt[b]*(2*ArcTan[1 - ((1 + I)*Sqrt[b]*Sqrt[Cos[c + d*x]])/(-a^2 + b^2)^(1/4)] - 2*ArcTan[1 + ((1 + I)*Sqrt[b]*Sqrt[Cos[c + d*x]])/(-a^2 + b^2)^(1/4)] + Log[Sqrt[-a^2 + b^2] - (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Cos[c + d*x]] + I*b*Cos[c + d*x]] - Log[Sqrt[-a^2 + b^2] + (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Cos[c + d*x]] + I*b*Cos[c + d*x]]))/(-a^2 + b^2)^(3/4))*Sin[c + d*x])/(Sqrt[1 - Cos[c + d*x]^2]*(a + b*Sin[c + d*x])) + ((140*a^3*b - 147*a*b^3)*(a + b*Sqrt[1 - Cos[c + d*x]^2])*Cos[2*(c + d*x)]*(((1/2 - I/2)*(-2*a^2 + b^2)*ArcTan[1 - ((1 + I)*Sqrt[b]*Sqrt[Cos[c + d*x]])/(-a^2 + b^2)^(1/4)])/(b^(3/2)*(-a^2 + b^2)^(3/4)) - ((1/2 - I/2)*(-2*a^2 + b^2)*ArcTan[1 + ((1 + I)*Sqrt[b]*Sqrt[Cos[c + d*x]])/(-a^2 + b^2)^(1/4)])/(b^(3/2)*(-a^2 + b^2)^(3/4)) + (4*Sqrt[Cos[c + d*x]])/b - (4*a*AppellF1[5/4, 1/2, 1, 9/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)]*Cos[c + d*x]^(5/2))/(5*(a^2 - b^2)) + (10*a*(a^2 - b^2)*AppellF1[1/4, 1/2, 1, 5/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)]*Sqrt[Cos[c + d*x]])/(Sqrt[1 - Cos[c + d*x]^2]*(5*(a^2 - b^2)*AppellF1[1/4, 1/2, 1, 5/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)] - 2*(2*b^2*AppellF1[5/4, 1/2, 2, 9/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)] + (-a^2 + b^2)*AppellF1[5/4, 3/2, 1, 9/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)])*Cos[c + d*x]^2)*(a^2 + b^2*(-1 + Cos[c + d*x]^2))) + ((1/4 - I/4)*(-2*a^2 + b^2)*Log[Sqrt[-a^2 + b^2] - (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Cos[c + d*x]] + I*b*Cos[c + d*x]])/(b^(3/2)*(-a^2 + b^2)^(3/4)) - ((1/4 - I/4)*(-2*a^2 + b^2)*Log[Sqrt[-a^2 + b^2] + (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Cos[c + d*x]] + I*b*Cos[c + d*x]])/(b^(3/2)*(-a^2 + b^2)^(3/4)))*Sin[c + d*x])/(Sqrt[1 - Cos[c + d*x]^2]*(-1 + 2*Cos[c + d*x]^2)*(a + b*Sin[c + d*x])) - (2*(35*a^4 - 126*a^2*b^2 + 75*b^4)*(a + b*Sqrt[1 - Cos[c + d*x]^2])*((5*b*(a^2 - b^2)*AppellF1[1/4, -1/2, 1, 5/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)]*Sqrt[Cos[c + d*x]]*Sqrt[1 - Cos[c + d*x]^2])/((-5*(a^2 - b^2)*AppellF1[1/4, -1/2, 1, 5/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)] + 2*(2*b^2*AppellF1[5/4, -1/2, 2, 9/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)] + (a^2 - b^2)*AppellF1[5/4, 1/2, 1, 9/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)])*Cos[c + d*x]^2)*(a^2 + b^2*(-1 + Cos[c + d*x]^2))) + (a*(-2*ArcTan[1 - (Sqrt[2]*Sqrt[b]*Sqrt[Cos[c + d*x]])/(a^2 - b^2)^(1/4)] + 2*ArcTan[1 + (Sqrt[2]*Sqrt[b]*Sqrt[Cos[c + d*x]])/(a^2 - b^2)^(1/4)] - Log[Sqrt[a^2 - b^2] - Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Cos[c + d*x]] + b*Cos[c + d*x]] + Log[Sqrt[a^2 - b^2] + Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Cos[c + d*x]] + b*Cos[c + d*x]]))/(4*Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(3/4)))*Sin[c + d*x]^2)/((1 - Cos[c + d*x]^2)*(a + b*Sin[c + d*x]))))/(b^5*d*Cos[c + d*x]^(11/2)) + ((e*Cos[c + d*x])^(11/2)*Sec[c + d*x]^5*((2*a*Cos[2*(c + d*x)])/(5*b^3) - ((-28*a^2 + 17*b^2)*Sin[c + d*x])/(14*b^4) - (-a^2 + b^2)^2/(b^5*(a + b*Sin[c + d*x])) - Sin[3*(c + d*x)]/(14*b^2)))/d","C",0
585,1,835,459,26.8174512,"\int \frac{(e \cos (c+d x))^{9/2}}{(a+b \sin (c+d x))^2} \, dx","Integrate[(e*Cos[c + d*x])^(9/2)/(a + b*Sin[c + d*x])^2,x]","\frac{7 \left(-\frac{\left(5 a^2-3 b^2\right) \left(a+b \sqrt{1-\cos ^2(c+d x)}\right) \left(8 F_1\left(\frac{3}{4};-\frac{1}{2},1;\frac{7}{4};\cos ^2(c+d x),\frac{b^2 \cos ^2(c+d x)}{b^2-a^2}\right) \cos ^{\frac{3}{2}}(c+d x) b^{5/2}+3 \sqrt{2} a \left(a^2-b^2\right)^{3/4} \left(2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{b} \sqrt{\cos (c+d x)}}{\sqrt[4]{a^2-b^2}}\right)-2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{b} \sqrt{\cos (c+d x)}}{\sqrt[4]{a^2-b^2}}+1\right)-\log \left(b \cos (c+d x)-\sqrt{2} \sqrt{b} \sqrt[4]{a^2-b^2} \sqrt{\cos (c+d x)}+\sqrt{a^2-b^2}\right)+\log \left(b \cos (c+d x)+\sqrt{2} \sqrt{b} \sqrt[4]{a^2-b^2} \sqrt{\cos (c+d x)}+\sqrt{a^2-b^2}\right)\right)\right) \sin ^2(c+d x)}{12 b^{3/2} \left(b^2-a^2\right) \left(1-\cos ^2(c+d x)\right) (a+b \sin (c+d x))}-\frac{4 a b \left(a+b \sqrt{1-\cos ^2(c+d x)}\right) \left(\frac{a F_1\left(\frac{3}{4};\frac{1}{2},1;\frac{7}{4};\cos ^2(c+d x),\frac{b^2 \cos ^2(c+d x)}{b^2-a^2}\right) \cos ^{\frac{3}{2}}(c+d x)}{3 \left(a^2-b^2\right)}+\frac{\left(\frac{1}{8}+\frac{i}{8}\right) \left(2 \tan ^{-1}\left(1-\frac{(1+i) \sqrt{b} \sqrt{\cos (c+d x)}}{\sqrt[4]{b^2-a^2}}\right)-2 \tan ^{-1}\left(\frac{(1+i) \sqrt{b} \sqrt{\cos (c+d x)}}{\sqrt[4]{b^2-a^2}}+1\right)-\log \left(i b \cos (c+d x)-(1+i) \sqrt{b} \sqrt[4]{b^2-a^2} \sqrt{\cos (c+d x)}+\sqrt{b^2-a^2}\right)+\log \left(i b \cos (c+d x)+(1+i) \sqrt{b} \sqrt[4]{b^2-a^2} \sqrt{\cos (c+d x)}+\sqrt{b^2-a^2}\right)\right)}{\sqrt{b} \sqrt[4]{b^2-a^2}}\right) \sin (c+d x)}{\sqrt{1-\cos ^2(c+d x)} (a+b \sin (c+d x))}\right) (e \cos (c+d x))^{9/2}}{10 b^3 d \cos ^{\frac{9}{2}}(c+d x)}+\frac{\sec ^4(c+d x) \left(\frac{4 a \cos (c+d x)}{3 b^3}-\frac{\sin (2 (c+d x))}{5 b^2}+\frac{a^2 \cos (c+d x)-b^2 \cos (c+d x)}{b^3 (a+b \sin (c+d x))}\right) (e \cos (c+d x))^{9/2}}{d}","\frac{7 a e^{9/2} \left(b^2-a^2\right)^{3/4} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{2 b^{9/2} d}-\frac{7 a e^{9/2} \left(b^2-a^2\right)^{3/4} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{2 b^{9/2} d}-\frac{7 a^2 e^5 \left(a^2-b^2\right) \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{2 b^5 d \left(b-\sqrt{b^2-a^2}\right) \sqrt{e \cos (c+d x)}}-\frac{7 a^2 e^5 \left(a^2-b^2\right) \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{2 b^5 d \left(\sqrt{b^2-a^2}+b\right) \sqrt{e \cos (c+d x)}}+\frac{7 e^4 \left(5 a^2-3 b^2\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{5 b^4 d \sqrt{\cos (c+d x)}}+\frac{7 e^3 (e \cos (c+d x))^{3/2} (5 a-3 b \sin (c+d x))}{15 b^3 d}-\frac{e (e \cos (c+d x))^{7/2}}{b d (a+b \sin (c+d x))}",1,"(7*(e*Cos[c + d*x])^(9/2)*((-4*a*b*(a + b*Sqrt[1 - Cos[c + d*x]^2])*((a*AppellF1[3/4, 1/2, 1, 7/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)]*Cos[c + d*x]^(3/2))/(3*(a^2 - b^2)) + ((1/8 + I/8)*(2*ArcTan[1 - ((1 + I)*Sqrt[b]*Sqrt[Cos[c + d*x]])/(-a^2 + b^2)^(1/4)] - 2*ArcTan[1 + ((1 + I)*Sqrt[b]*Sqrt[Cos[c + d*x]])/(-a^2 + b^2)^(1/4)] - Log[Sqrt[-a^2 + b^2] - (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Cos[c + d*x]] + I*b*Cos[c + d*x]] + Log[Sqrt[-a^2 + b^2] + (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Cos[c + d*x]] + I*b*Cos[c + d*x]]))/(Sqrt[b]*(-a^2 + b^2)^(1/4)))*Sin[c + d*x])/(Sqrt[1 - Cos[c + d*x]^2]*(a + b*Sin[c + d*x])) - ((5*a^2 - 3*b^2)*(a + b*Sqrt[1 - Cos[c + d*x]^2])*(8*b^(5/2)*AppellF1[3/4, -1/2, 1, 7/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)]*Cos[c + d*x]^(3/2) + 3*Sqrt[2]*a*(a^2 - b^2)^(3/4)*(2*ArcTan[1 - (Sqrt[2]*Sqrt[b]*Sqrt[Cos[c + d*x]])/(a^2 - b^2)^(1/4)] - 2*ArcTan[1 + (Sqrt[2]*Sqrt[b]*Sqrt[Cos[c + d*x]])/(a^2 - b^2)^(1/4)] - Log[Sqrt[a^2 - b^2] - Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Cos[c + d*x]] + b*Cos[c + d*x]] + Log[Sqrt[a^2 - b^2] + Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Cos[c + d*x]] + b*Cos[c + d*x]]))*Sin[c + d*x]^2)/(12*b^(3/2)*(-a^2 + b^2)*(1 - Cos[c + d*x]^2)*(a + b*Sin[c + d*x]))))/(10*b^3*d*Cos[c + d*x]^(9/2)) + ((e*Cos[c + d*x])^(9/2)*Sec[c + d*x]^4*((4*a*Cos[c + d*x])/(3*b^3) + (a^2*Cos[c + d*x] - b^2*Cos[c + d*x])/(b^3*(a + b*Sin[c + d*x])) - Sin[2*(c + d*x)]/(5*b^2)))/d","C",0
586,1,1956,473,27.0393301,"\int \frac{(e \cos (c+d x))^{7/2}}{(a+b \sin (c+d x))^2} \, dx","Integrate[(e*Cos[c + d*x])^(7/2)/(a + b*Sin[c + d*x])^2,x]","\frac{\sec ^3(c+d x) \left(\frac{a^2-b^2}{b^3 (a+b \sin (c+d x))}-\frac{2 \sin (c+d x)}{3 b^2}\right) (e \cos (c+d x))^{7/2}}{d}+\frac{\left(-\frac{2 \left(3 a^2-5 b^2\right) \left(a+b \sqrt{1-\cos ^2(c+d x)}\right) \left(\frac{5 b \left(a^2-b^2\right) \sqrt{\cos (c+d x)} \sqrt{1-\cos ^2(c+d x)} F_1\left(\frac{1}{4};-\frac{1}{2},1;\frac{5}{4};\cos ^2(c+d x),\frac{b^2 \cos ^2(c+d x)}{b^2-a^2}\right)}{\left(2 \left(2 F_1\left(\frac{5}{4};-\frac{1}{2},2;\frac{9}{4};\cos ^2(c+d x),\frac{b^2 \cos ^2(c+d x)}{b^2-a^2}\right) b^2+\left(a^2-b^2\right) F_1\left(\frac{5}{4};\frac{1}{2},1;\frac{9}{4};\cos ^2(c+d x),\frac{b^2 \cos ^2(c+d x)}{b^2-a^2}\right)\right) \cos ^2(c+d x)-5 \left(a^2-b^2\right) F_1\left(\frac{1}{4};-\frac{1}{2},1;\frac{5}{4};\cos ^2(c+d x),\frac{b^2 \cos ^2(c+d x)}{b^2-a^2}\right)\right) \left(a^2+b^2 \left(\cos ^2(c+d x)-1\right)\right)}+\frac{a \left(-2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{b} \sqrt{\cos (c+d x)}}{\sqrt[4]{a^2-b^2}}\right)+2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{b} \sqrt{\cos (c+d x)}}{\sqrt[4]{a^2-b^2}}+1\right)-\log \left(b \cos (c+d x)-\sqrt{2} \sqrt{b} \sqrt[4]{a^2-b^2} \sqrt{\cos (c+d x)}+\sqrt{a^2-b^2}\right)+\log \left(b \cos (c+d x)+\sqrt{2} \sqrt{b} \sqrt[4]{a^2-b^2} \sqrt{\cos (c+d x)}+\sqrt{a^2-b^2}\right)\right)}{4 \sqrt{2} \sqrt{b} \left(a^2-b^2\right)^{3/4}}\right) \sin ^2(c+d x)}{\left(1-\cos ^2(c+d x)\right) (a+b \sin (c+d x))}-\frac{8 a b \left(a+b \sqrt{1-\cos ^2(c+d x)}\right) \left(\frac{5 a \left(a^2-b^2\right) F_1\left(\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\cos ^2(c+d x),\frac{b^2 \cos ^2(c+d x)}{b^2-a^2}\right) \sqrt{\cos (c+d x)}}{\sqrt{1-\cos ^2(c+d x)} \left(5 \left(a^2-b^2\right) F_1\left(\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\cos ^2(c+d x),\frac{b^2 \cos ^2(c+d x)}{b^2-a^2}\right)-2 \left(2 F_1\left(\frac{5}{4};\frac{1}{2},2;\frac{9}{4};\cos ^2(c+d x),\frac{b^2 \cos ^2(c+d x)}{b^2-a^2}\right) b^2+\left(b^2-a^2\right) F_1\left(\frac{5}{4};\frac{3}{2},1;\frac{9}{4};\cos ^2(c+d x),\frac{b^2 \cos ^2(c+d x)}{b^2-a^2}\right)\right) \cos ^2(c+d x)\right) \left(a^2+b^2 \left(\cos ^2(c+d x)-1\right)\right)}-\frac{\left(\frac{1}{8}-\frac{i}{8}\right) \sqrt{b} \left(2 \tan ^{-1}\left(1-\frac{(1+i) \sqrt{b} \sqrt{\cos (c+d x)}}{\sqrt[4]{b^2-a^2}}\right)-2 \tan ^{-1}\left(\frac{(1+i) \sqrt{b} \sqrt{\cos (c+d x)}}{\sqrt[4]{b^2-a^2}}+1\right)+\log \left(i b \cos (c+d x)-(1+i) \sqrt{b} \sqrt[4]{b^2-a^2} \sqrt{\cos (c+d x)}+\sqrt{b^2-a^2}\right)-\log \left(i b \cos (c+d x)+(1+i) \sqrt{b} \sqrt[4]{b^2-a^2} \sqrt{\cos (c+d x)}+\sqrt{b^2-a^2}\right)\right)}{\left(b^2-a^2\right)^{3/4}}\right) \sin (c+d x)}{\sqrt{1-\cos ^2(c+d x)} (a+b \sin (c+d x))}+\frac{6 a b \left(a+b \sqrt{1-\cos ^2(c+d x)}\right) \cos (2 (c+d x)) \left(-\frac{4 a F_1\left(\frac{5}{4};\frac{1}{2},1;\frac{9}{4};\cos ^2(c+d x),\frac{b^2 \cos ^2(c+d x)}{b^2-a^2}\right) \cos ^{\frac{5}{2}}(c+d x)}{5 \left(a^2-b^2\right)}+\frac{4 \sqrt{\cos (c+d x)}}{b}+\frac{10 a \left(a^2-b^2\right) F_1\left(\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\cos ^2(c+d x),\frac{b^2 \cos ^2(c+d x)}{b^2-a^2}\right) \sqrt{\cos (c+d x)}}{\sqrt{1-\cos ^2(c+d x)} \left(5 \left(a^2-b^2\right) F_1\left(\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\cos ^2(c+d x),\frac{b^2 \cos ^2(c+d x)}{b^2-a^2}\right)-2 \left(2 F_1\left(\frac{5}{4};\frac{1}{2},2;\frac{9}{4};\cos ^2(c+d x),\frac{b^2 \cos ^2(c+d x)}{b^2-a^2}\right) b^2+\left(b^2-a^2\right) F_1\left(\frac{5}{4};\frac{3}{2},1;\frac{9}{4};\cos ^2(c+d x),\frac{b^2 \cos ^2(c+d x)}{b^2-a^2}\right)\right) \cos ^2(c+d x)\right) \left(a^2+b^2 \left(\cos ^2(c+d x)-1\right)\right)}+\frac{\left(\frac{1}{2}-\frac{i}{2}\right) \left(b^2-2 a^2\right) \tan ^{-1}\left(1-\frac{(1+i) \sqrt{b} \sqrt{\cos (c+d x)}}{\sqrt[4]{b^2-a^2}}\right)}{b^{3/2} \left(b^2-a^2\right)^{3/4}}-\frac{\left(\frac{1}{2}-\frac{i}{2}\right) \left(b^2-2 a^2\right) \tan ^{-1}\left(\frac{(1+i) \sqrt{b} \sqrt{\cos (c+d x)}}{\sqrt[4]{b^2-a^2}}+1\right)}{b^{3/2} \left(b^2-a^2\right)^{3/4}}+\frac{\left(\frac{1}{4}-\frac{i}{4}\right) \left(b^2-2 a^2\right) \log \left(i b \cos (c+d x)-(1+i) \sqrt{b} \sqrt[4]{b^2-a^2} \sqrt{\cos (c+d x)}+\sqrt{b^2-a^2}\right)}{b^{3/2} \left(b^2-a^2\right)^{3/4}}-\frac{\left(\frac{1}{4}-\frac{i}{4}\right) \left(b^2-2 a^2\right) \log \left(i b \cos (c+d x)+(1+i) \sqrt{b} \sqrt[4]{b^2-a^2} \sqrt{\cos (c+d x)}+\sqrt{b^2-a^2}\right)}{b^{3/2} \left(b^2-a^2\right)^{3/4}}\right) \sin (c+d x)}{\sqrt{1-\cos ^2(c+d x)} \left(2 \cos ^2(c+d x)-1\right) (a+b \sin (c+d x))}\right) (e \cos (c+d x))^{7/2}}{6 b^3 d \cos ^{\frac{7}{2}}(c+d x)}","-\frac{5 a e^{7/2} \sqrt[4]{b^2-a^2} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{2 b^{7/2} d}-\frac{5 a e^{7/2} \sqrt[4]{b^2-a^2} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{2 b^{7/2} d}+\frac{5 e^4 \left(3 a^2-b^2\right) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 b^4 d \sqrt{e \cos (c+d x)}}-\frac{5 a^2 e^4 \left(a^2-b^2\right) \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{2 b^4 d \left(a^2-b \left(b-\sqrt{b^2-a^2}\right)\right) \sqrt{e \cos (c+d x)}}-\frac{5 a^2 e^4 \left(a^2-b^2\right) \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{2 b^4 d \left(a^2-b \left(\sqrt{b^2-a^2}+b\right)\right) \sqrt{e \cos (c+d x)}}+\frac{5 e^3 \sqrt{e \cos (c+d x)} (3 a-b \sin (c+d x))}{3 b^3 d}-\frac{e (e \cos (c+d x))^{5/2}}{b d (a+b \sin (c+d x))}",1,"((e*Cos[c + d*x])^(7/2)*Sec[c + d*x]^3*((-2*Sin[c + d*x])/(3*b^2) + (a^2 - b^2)/(b^3*(a + b*Sin[c + d*x]))))/d + ((e*Cos[c + d*x])^(7/2)*((-8*a*b*(a + b*Sqrt[1 - Cos[c + d*x]^2])*((5*a*(a^2 - b^2)*AppellF1[1/4, 1/2, 1, 5/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)]*Sqrt[Cos[c + d*x]])/(Sqrt[1 - Cos[c + d*x]^2]*(5*(a^2 - b^2)*AppellF1[1/4, 1/2, 1, 5/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)] - 2*(2*b^2*AppellF1[5/4, 1/2, 2, 9/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)] + (-a^2 + b^2)*AppellF1[5/4, 3/2, 1, 9/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)])*Cos[c + d*x]^2)*(a^2 + b^2*(-1 + Cos[c + d*x]^2))) - ((1/8 - I/8)*Sqrt[b]*(2*ArcTan[1 - ((1 + I)*Sqrt[b]*Sqrt[Cos[c + d*x]])/(-a^2 + b^2)^(1/4)] - 2*ArcTan[1 + ((1 + I)*Sqrt[b]*Sqrt[Cos[c + d*x]])/(-a^2 + b^2)^(1/4)] + Log[Sqrt[-a^2 + b^2] - (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Cos[c + d*x]] + I*b*Cos[c + d*x]] - Log[Sqrt[-a^2 + b^2] + (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Cos[c + d*x]] + I*b*Cos[c + d*x]]))/(-a^2 + b^2)^(3/4))*Sin[c + d*x])/(Sqrt[1 - Cos[c + d*x]^2]*(a + b*Sin[c + d*x])) + (6*a*b*(a + b*Sqrt[1 - Cos[c + d*x]^2])*Cos[2*(c + d*x)]*(((1/2 - I/2)*(-2*a^2 + b^2)*ArcTan[1 - ((1 + I)*Sqrt[b]*Sqrt[Cos[c + d*x]])/(-a^2 + b^2)^(1/4)])/(b^(3/2)*(-a^2 + b^2)^(3/4)) - ((1/2 - I/2)*(-2*a^2 + b^2)*ArcTan[1 + ((1 + I)*Sqrt[b]*Sqrt[Cos[c + d*x]])/(-a^2 + b^2)^(1/4)])/(b^(3/2)*(-a^2 + b^2)^(3/4)) + (4*Sqrt[Cos[c + d*x]])/b - (4*a*AppellF1[5/4, 1/2, 1, 9/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)]*Cos[c + d*x]^(5/2))/(5*(a^2 - b^2)) + (10*a*(a^2 - b^2)*AppellF1[1/4, 1/2, 1, 5/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)]*Sqrt[Cos[c + d*x]])/(Sqrt[1 - Cos[c + d*x]^2]*(5*(a^2 - b^2)*AppellF1[1/4, 1/2, 1, 5/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)] - 2*(2*b^2*AppellF1[5/4, 1/2, 2, 9/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)] + (-a^2 + b^2)*AppellF1[5/4, 3/2, 1, 9/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)])*Cos[c + d*x]^2)*(a^2 + b^2*(-1 + Cos[c + d*x]^2))) + ((1/4 - I/4)*(-2*a^2 + b^2)*Log[Sqrt[-a^2 + b^2] - (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Cos[c + d*x]] + I*b*Cos[c + d*x]])/(b^(3/2)*(-a^2 + b^2)^(3/4)) - ((1/4 - I/4)*(-2*a^2 + b^2)*Log[Sqrt[-a^2 + b^2] + (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Cos[c + d*x]] + I*b*Cos[c + d*x]])/(b^(3/2)*(-a^2 + b^2)^(3/4)))*Sin[c + d*x])/(Sqrt[1 - Cos[c + d*x]^2]*(-1 + 2*Cos[c + d*x]^2)*(a + b*Sin[c + d*x])) - (2*(3*a^2 - 5*b^2)*(a + b*Sqrt[1 - Cos[c + d*x]^2])*((5*b*(a^2 - b^2)*AppellF1[1/4, -1/2, 1, 5/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)]*Sqrt[Cos[c + d*x]]*Sqrt[1 - Cos[c + d*x]^2])/((-5*(a^2 - b^2)*AppellF1[1/4, -1/2, 1, 5/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)] + 2*(2*b^2*AppellF1[5/4, -1/2, 2, 9/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)] + (a^2 - b^2)*AppellF1[5/4, 1/2, 1, 9/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)])*Cos[c + d*x]^2)*(a^2 + b^2*(-1 + Cos[c + d*x]^2))) + (a*(-2*ArcTan[1 - (Sqrt[2]*Sqrt[b]*Sqrt[Cos[c + d*x]])/(a^2 - b^2)^(1/4)] + 2*ArcTan[1 + (Sqrt[2]*Sqrt[b]*Sqrt[Cos[c + d*x]])/(a^2 - b^2)^(1/4)] - Log[Sqrt[a^2 - b^2] - Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Cos[c + d*x]] + b*Cos[c + d*x]] + Log[Sqrt[a^2 - b^2] + Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Cos[c + d*x]] + b*Cos[c + d*x]]))/(4*Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(3/4)))*Sin[c + d*x]^2)/((1 - Cos[c + d*x]^2)*(a + b*Sin[c + d*x]))))/(6*b^3*d*Cos[c + d*x]^(7/2))","C",0
587,1,371,390,37.9460844,"\int \frac{(e \cos (c+d x))^{5/2}}{(a+b \sin (c+d x))^2} \, dx","Integrate[(e*Cos[c + d*x])^(5/2)/(a + b*Sin[c + d*x])^2,x]","\frac{(e \cos (c+d x))^{5/2} \left(-\frac{\left(a+b \sqrt{\sin ^2(c+d x)}\right) \left(8 b^{5/2} \cos ^{\frac{3}{2}}(c+d x) F_1\left(\frac{3}{4};-\frac{1}{2},1;\frac{7}{4};\cos ^2(c+d x),\frac{b^2 \cos ^2(c+d x)}{b^2-a^2}\right)+3 \sqrt{2} a \left(a^2-b^2\right)^{3/4} \left(-\log \left(-\sqrt{2} \sqrt{b} \sqrt[4]{a^2-b^2} \sqrt{\cos (c+d x)}+\sqrt{a^2-b^2}+b \cos (c+d x)\right)+\log \left(\sqrt{2} \sqrt{b} \sqrt[4]{a^2-b^2} \sqrt{\cos (c+d x)}+\sqrt{a^2-b^2}+b \cos (c+d x)\right)+2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{b} \sqrt{\cos (c+d x)}}{\sqrt[4]{a^2-b^2}}\right)-2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{b} \sqrt{\cos (c+d x)}}{\sqrt[4]{a^2-b^2}}+1\right)\right)\right)}{a^2-b^2}-8 b^{3/2} \cos ^{\frac{3}{2}}(c+d x)\right)}{8 b^{5/2} d \cos ^{\frac{5}{2}}(c+d x) (a+b \sin (c+d x))}","\frac{3 a e^{5/2} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{2 b^{5/2} d \sqrt[4]{b^2-a^2}}-\frac{3 a e^{5/2} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{2 b^{5/2} d \sqrt[4]{b^2-a^2}}+\frac{3 a^2 e^3 \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{2 b^3 d \left(b-\sqrt{b^2-a^2}\right) \sqrt{e \cos (c+d x)}}+\frac{3 a^2 e^3 \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{2 b^3 d \left(\sqrt{b^2-a^2}+b\right) \sqrt{e \cos (c+d x)}}-\frac{e (e \cos (c+d x))^{3/2}}{b d (a+b \sin (c+d x))}-\frac{3 e^2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{b^2 d \sqrt{\cos (c+d x)}}",1,"((e*Cos[c + d*x])^(5/2)*(-8*b^(3/2)*Cos[c + d*x]^(3/2) - ((8*b^(5/2)*AppellF1[3/4, -1/2, 1, 7/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)]*Cos[c + d*x]^(3/2) + 3*Sqrt[2]*a*(a^2 - b^2)^(3/4)*(2*ArcTan[1 - (Sqrt[2]*Sqrt[b]*Sqrt[Cos[c + d*x]])/(a^2 - b^2)^(1/4)] - 2*ArcTan[1 + (Sqrt[2]*Sqrt[b]*Sqrt[Cos[c + d*x]])/(a^2 - b^2)^(1/4)] - Log[Sqrt[a^2 - b^2] - Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Cos[c + d*x]] + b*Cos[c + d*x]] + Log[Sqrt[a^2 - b^2] + Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Cos[c + d*x]] + b*Cos[c + d*x]]))*(a + b*Sqrt[Sin[c + d*x]^2]))/(a^2 - b^2)))/(8*b^(5/2)*d*Cos[c + d*x]^(5/2)*(a + b*Sin[c + d*x]))","C",0
588,1,614,404,12.529892,"\int \frac{(e \cos (c+d x))^{3/2}}{(a+b \sin (c+d x))^2} \, dx","Integrate[(e*Cos[c + d*x])^(3/2)/(a + b*Sin[c + d*x])^2,x]","\frac{\sin ^2(c+d x) (e \cos (c+d x))^{3/2} \left(a+b \sqrt{1-\cos ^2(c+d x)}\right) \left(\frac{5 b \left(a^2-b^2\right) \sqrt{\cos (c+d x)} \sqrt{1-\cos ^2(c+d x)} F_1\left(\frac{1}{4};-\frac{1}{2},1;\frac{5}{4};\cos ^2(c+d x),\frac{b^2 \cos ^2(c+d x)}{b^2-a^2}\right)}{\left(a^2+b^2 \left(\cos ^2(c+d x)-1\right)\right) \left(2 \cos ^2(c+d x) \left(2 b^2 F_1\left(\frac{5}{4};-\frac{1}{2},2;\frac{9}{4};\cos ^2(c+d x),\frac{b^2 \cos ^2(c+d x)}{b^2-a^2}\right)+\left(a^2-b^2\right) F_1\left(\frac{5}{4};\frac{1}{2},1;\frac{9}{4};\cos ^2(c+d x),\frac{b^2 \cos ^2(c+d x)}{b^2-a^2}\right)\right)-5 \left(a^2-b^2\right) F_1\left(\frac{1}{4};-\frac{1}{2},1;\frac{5}{4};\cos ^2(c+d x),\frac{b^2 \cos ^2(c+d x)}{b^2-a^2}\right)\right)}+\frac{a \left(-\log \left(-\sqrt{2} \sqrt{b} \sqrt[4]{a^2-b^2} \sqrt{\cos (c+d x)}+\sqrt{a^2-b^2}+b \cos (c+d x)\right)+\log \left(\sqrt{2} \sqrt{b} \sqrt[4]{a^2-b^2} \sqrt{\cos (c+d x)}+\sqrt{a^2-b^2}+b \cos (c+d x)\right)-2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{b} \sqrt{\cos (c+d x)}}{\sqrt[4]{a^2-b^2}}\right)+2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{b} \sqrt{\cos (c+d x)}}{\sqrt[4]{a^2-b^2}}+1\right)\right)}{4 \sqrt{2} \sqrt{b} \left(a^2-b^2\right)^{3/4}}\right)}{b d \cos ^{\frac{3}{2}}(c+d x) \left(1-\cos ^2(c+d x)\right) (a+b \sin (c+d x))}-\frac{\sec (c+d x) (e \cos (c+d x))^{3/2}}{b d (a+b \sin (c+d x))}","\frac{a^2 e^2 \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{2 b^2 d \left(a^2-b \left(b-\sqrt{b^2-a^2}\right)\right) \sqrt{e \cos (c+d x)}}+\frac{a^2 e^2 \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{2 b^2 d \left(a^2-b \left(\sqrt{b^2-a^2}+b\right)\right) \sqrt{e \cos (c+d x)}}-\frac{a e^{3/2} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{2 b^{3/2} d \left(b^2-a^2\right)^{3/4}}-\frac{a e^{3/2} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{2 b^{3/2} d \left(b^2-a^2\right)^{3/4}}-\frac{e \sqrt{e \cos (c+d x)}}{b d (a+b \sin (c+d x))}-\frac{e^2 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b^2 d \sqrt{e \cos (c+d x)}}",1,"-(((e*Cos[c + d*x])^(3/2)*Sec[c + d*x])/(b*d*(a + b*Sin[c + d*x]))) + ((e*Cos[c + d*x])^(3/2)*(a + b*Sqrt[1 - Cos[c + d*x]^2])*((5*b*(a^2 - b^2)*AppellF1[1/4, -1/2, 1, 5/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)]*Sqrt[Cos[c + d*x]]*Sqrt[1 - Cos[c + d*x]^2])/((-5*(a^2 - b^2)*AppellF1[1/4, -1/2, 1, 5/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)] + 2*(2*b^2*AppellF1[5/4, -1/2, 2, 9/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)] + (a^2 - b^2)*AppellF1[5/4, 1/2, 1, 9/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)])*Cos[c + d*x]^2)*(a^2 + b^2*(-1 + Cos[c + d*x]^2))) + (a*(-2*ArcTan[1 - (Sqrt[2]*Sqrt[b]*Sqrt[Cos[c + d*x]])/(a^2 - b^2)^(1/4)] + 2*ArcTan[1 + (Sqrt[2]*Sqrt[b]*Sqrt[Cos[c + d*x]])/(a^2 - b^2)^(1/4)] - Log[Sqrt[a^2 - b^2] - Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Cos[c + d*x]] + b*Cos[c + d*x]] + Log[Sqrt[a^2 - b^2] + Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Cos[c + d*x]] + b*Cos[c + d*x]]))/(4*Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(3/4)))*Sin[c + d*x]^2)/(b*d*Cos[c + d*x]^(3/2)*(1 - Cos[c + d*x]^2)*(a + b*Sin[c + d*x]))","C",0
589,1,787,422,16.3823183,"\int \frac{\sqrt{e \cos (c+d x)}}{(a+b \sin (c+d x))^2} \, dx","Integrate[Sqrt[e*Cos[c + d*x]]/(a + b*Sin[c + d*x])^2,x]","-\frac{b \cos (c+d x) \sqrt{e \cos (c+d x)}}{d \left(b^2-a^2\right) (a+b \sin (c+d x))}+\frac{\sqrt{e \cos (c+d x)} \left(-\frac{\sin ^2(c+d x) \left(a+b \sqrt{1-\cos ^2(c+d x)}\right) \left(8 b^{5/2} \cos ^{\frac{3}{2}}(c+d x) F_1\left(\frac{3}{4};-\frac{1}{2},1;\frac{7}{4};\cos ^2(c+d x),\frac{b^2 \cos ^2(c+d x)}{b^2-a^2}\right)+3 \sqrt{2} a \left(a^2-b^2\right)^{3/4} \left(-\log \left(-\sqrt{2} \sqrt{b} \sqrt[4]{a^2-b^2} \sqrt{\cos (c+d x)}+\sqrt{a^2-b^2}+b \cos (c+d x)\right)+\log \left(\sqrt{2} \sqrt{b} \sqrt[4]{a^2-b^2} \sqrt{\cos (c+d x)}+\sqrt{a^2-b^2}+b \cos (c+d x)\right)+2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{b} \sqrt{\cos (c+d x)}}{\sqrt[4]{a^2-b^2}}\right)-2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{b} \sqrt{\cos (c+d x)}}{\sqrt[4]{a^2-b^2}}+1\right)\right)\right)}{12 \sqrt{b} \left(b^2-a^2\right) \left(1-\cos ^2(c+d x)\right) (a+b \sin (c+d x))}-\frac{4 a \sin (c+d x) \left(a+b \sqrt{1-\cos ^2(c+d x)}\right) \left(\frac{a \cos ^{\frac{3}{2}}(c+d x) F_1\left(\frac{3}{4};\frac{1}{2},1;\frac{7}{4};\cos ^2(c+d x),\frac{b^2 \cos ^2(c+d x)}{b^2-a^2}\right)}{3 \left(a^2-b^2\right)}+\frac{\left(\frac{1}{8}+\frac{i}{8}\right) \left(-\log \left(-(1+i) \sqrt{b} \sqrt[4]{b^2-a^2} \sqrt{\cos (c+d x)}+\sqrt{b^2-a^2}+i b \cos (c+d x)\right)+\log \left((1+i) \sqrt{b} \sqrt[4]{b^2-a^2} \sqrt{\cos (c+d x)}+\sqrt{b^2-a^2}+i b \cos (c+d x)\right)+2 \tan ^{-1}\left(1-\frac{(1+i) \sqrt{b} \sqrt{\cos (c+d x)}}{\sqrt[4]{b^2-a^2}}\right)-2 \tan ^{-1}\left(1+\frac{(1+i) \sqrt{b} \sqrt{\cos (c+d x)}}{\sqrt[4]{b^2-a^2}}\right)\right)}{\sqrt{b} \sqrt[4]{b^2-a^2}}\right)}{\sqrt{1-\cos ^2(c+d x)} (a+b \sin (c+d x))}\right)}{2 d (a-b) (a+b) \sqrt{\cos (c+d x)}}","\frac{b (e \cos (c+d x))^{3/2}}{d e \left(a^2-b^2\right) (a+b \sin (c+d x))}-\frac{a \sqrt{e} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{2 \sqrt{b} d \left(b^2-a^2\right)^{5/4}}+\frac{a \sqrt{e} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{2 \sqrt{b} d \left(b^2-a^2\right)^{5/4}}+\frac{E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{d \left(a^2-b^2\right) \sqrt{\cos (c+d x)}}+\frac{a^2 e \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{2 b d \left(a^2-b^2\right) \left(b-\sqrt{b^2-a^2}\right) \sqrt{e \cos (c+d x)}}+\frac{a^2 e \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{2 b d \left(a^2-b^2\right) \left(\sqrt{b^2-a^2}+b\right) \sqrt{e \cos (c+d x)}}",1,"-((b*Cos[c + d*x]*Sqrt[e*Cos[c + d*x]])/((-a^2 + b^2)*d*(a + b*Sin[c + d*x]))) + (Sqrt[e*Cos[c + d*x]]*((-4*a*(a + b*Sqrt[1 - Cos[c + d*x]^2])*((a*AppellF1[3/4, 1/2, 1, 7/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)]*Cos[c + d*x]^(3/2))/(3*(a^2 - b^2)) + ((1/8 + I/8)*(2*ArcTan[1 - ((1 + I)*Sqrt[b]*Sqrt[Cos[c + d*x]])/(-a^2 + b^2)^(1/4)] - 2*ArcTan[1 + ((1 + I)*Sqrt[b]*Sqrt[Cos[c + d*x]])/(-a^2 + b^2)^(1/4)] - Log[Sqrt[-a^2 + b^2] - (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Cos[c + d*x]] + I*b*Cos[c + d*x]] + Log[Sqrt[-a^2 + b^2] + (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Cos[c + d*x]] + I*b*Cos[c + d*x]]))/(Sqrt[b]*(-a^2 + b^2)^(1/4)))*Sin[c + d*x])/(Sqrt[1 - Cos[c + d*x]^2]*(a + b*Sin[c + d*x])) - ((a + b*Sqrt[1 - Cos[c + d*x]^2])*(8*b^(5/2)*AppellF1[3/4, -1/2, 1, 7/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)]*Cos[c + d*x]^(3/2) + 3*Sqrt[2]*a*(a^2 - b^2)^(3/4)*(2*ArcTan[1 - (Sqrt[2]*Sqrt[b]*Sqrt[Cos[c + d*x]])/(a^2 - b^2)^(1/4)] - 2*ArcTan[1 + (Sqrt[2]*Sqrt[b]*Sqrt[Cos[c + d*x]])/(a^2 - b^2)^(1/4)] - Log[Sqrt[a^2 - b^2] - Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Cos[c + d*x]] + b*Cos[c + d*x]] + Log[Sqrt[a^2 - b^2] + Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Cos[c + d*x]] + b*Cos[c + d*x]]))*Sin[c + d*x]^2)/(12*Sqrt[b]*(-a^2 + b^2)*(1 - Cos[c + d*x]^2)*(a + b*Sin[c + d*x]))))/(2*(a - b)*(a + b)*d*Sqrt[Cos[c + d*x]])","C",0
590,1,1181,429,23.7617659,"\int \frac{1}{\sqrt{e \cos (c+d x)} (a+b \sin (c+d x))^2} \, dx","Integrate[1/(Sqrt[e*Cos[c + d*x]]*(a + b*Sin[c + d*x])^2),x]","\frac{b \cos (c+d x)}{\left(a^2-b^2\right) d \sqrt{e \cos (c+d x)} (a+b \sin (c+d x))}+\frac{\left(\frac{2 b \left(a+b \sqrt{1-\cos ^2(c+d x)}\right) \left(\frac{5 b \left(a^2-b^2\right) \sqrt{\cos (c+d x)} \sqrt{1-\cos ^2(c+d x)} F_1\left(\frac{1}{4};-\frac{1}{2},1;\frac{5}{4};\cos ^2(c+d x),\frac{b^2 \cos ^2(c+d x)}{b^2-a^2}\right)}{\left(2 \left(2 F_1\left(\frac{5}{4};-\frac{1}{2},2;\frac{9}{4};\cos ^2(c+d x),\frac{b^2 \cos ^2(c+d x)}{b^2-a^2}\right) b^2+\left(a^2-b^2\right) F_1\left(\frac{5}{4};\frac{1}{2},1;\frac{9}{4};\cos ^2(c+d x),\frac{b^2 \cos ^2(c+d x)}{b^2-a^2}\right)\right) \cos ^2(c+d x)-5 \left(a^2-b^2\right) F_1\left(\frac{1}{4};-\frac{1}{2},1;\frac{5}{4};\cos ^2(c+d x),\frac{b^2 \cos ^2(c+d x)}{b^2-a^2}\right)\right) \left(a^2+b^2 \left(\cos ^2(c+d x)-1\right)\right)}+\frac{a \left(-2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{b} \sqrt{\cos (c+d x)}}{\sqrt[4]{a^2-b^2}}\right)+2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{b} \sqrt{\cos (c+d x)}}{\sqrt[4]{a^2-b^2}}+1\right)-\log \left(b \cos (c+d x)-\sqrt{2} \sqrt{b} \sqrt[4]{a^2-b^2} \sqrt{\cos (c+d x)}+\sqrt{a^2-b^2}\right)+\log \left(b \cos (c+d x)+\sqrt{2} \sqrt{b} \sqrt[4]{a^2-b^2} \sqrt{\cos (c+d x)}+\sqrt{a^2-b^2}\right)\right)}{4 \sqrt{2} \sqrt{b} \left(a^2-b^2\right)^{3/4}}\right) \sin ^2(c+d x)}{\left(1-\cos ^2(c+d x)\right) (a+b \sin (c+d x))}-\frac{4 a \left(a+b \sqrt{1-\cos ^2(c+d x)}\right) \left(\frac{5 a \left(a^2-b^2\right) F_1\left(\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\cos ^2(c+d x),\frac{b^2 \cos ^2(c+d x)}{b^2-a^2}\right) \sqrt{\cos (c+d x)}}{\sqrt{1-\cos ^2(c+d x)} \left(5 \left(a^2-b^2\right) F_1\left(\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\cos ^2(c+d x),\frac{b^2 \cos ^2(c+d x)}{b^2-a^2}\right)-2 \left(2 F_1\left(\frac{5}{4};\frac{1}{2},2;\frac{9}{4};\cos ^2(c+d x),\frac{b^2 \cos ^2(c+d x)}{b^2-a^2}\right) b^2+\left(b^2-a^2\right) F_1\left(\frac{5}{4};\frac{3}{2},1;\frac{9}{4};\cos ^2(c+d x),\frac{b^2 \cos ^2(c+d x)}{b^2-a^2}\right)\right) \cos ^2(c+d x)\right) \left(a^2+b^2 \left(\cos ^2(c+d x)-1\right)\right)}-\frac{\left(\frac{1}{8}-\frac{i}{8}\right) \sqrt{b} \left(2 \tan ^{-1}\left(1-\frac{(1+i) \sqrt{b} \sqrt{\cos (c+d x)}}{\sqrt[4]{b^2-a^2}}\right)-2 \tan ^{-1}\left(\frac{(1+i) \sqrt{b} \sqrt{\cos (c+d x)}}{\sqrt[4]{b^2-a^2}}+1\right)+\log \left(i b \cos (c+d x)-(1+i) \sqrt{b} \sqrt[4]{b^2-a^2} \sqrt{\cos (c+d x)}+\sqrt{b^2-a^2}\right)-\log \left(i b \cos (c+d x)+(1+i) \sqrt{b} \sqrt[4]{b^2-a^2} \sqrt{\cos (c+d x)}+\sqrt{b^2-a^2}\right)\right)}{\left(b^2-a^2\right)^{3/4}}\right) \sin (c+d x)}{\sqrt{1-\cos ^2(c+d x)} (a+b \sin (c+d x))}\right) \sqrt{\cos (c+d x)}}{2 (a-b) (a+b) d \sqrt{e \cos (c+d x)}}","\frac{b \sqrt{e \cos (c+d x)}}{d e \left(a^2-b^2\right) (a+b \sin (c+d x))}+\frac{3 a \sqrt{b} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{2 d \sqrt{e} \left(b^2-a^2\right)^{7/4}}+\frac{3 a \sqrt{b} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{2 d \sqrt{e} \left(b^2-a^2\right)^{7/4}}-\frac{\sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d \left(a^2-b^2\right) \sqrt{e \cos (c+d x)}}+\frac{3 a^2 \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{2 d \left(a^2-b^2\right) \left(a^2-b \left(b-\sqrt{b^2-a^2}\right)\right) \sqrt{e \cos (c+d x)}}+\frac{3 a^2 \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{2 d \left(a^2-b^2\right) \left(a^2-b \left(\sqrt{b^2-a^2}+b\right)\right) \sqrt{e \cos (c+d x)}}",1,"(b*Cos[c + d*x])/((a^2 - b^2)*d*Sqrt[e*Cos[c + d*x]]*(a + b*Sin[c + d*x])) + (Sqrt[Cos[c + d*x]]*((-4*a*(a + b*Sqrt[1 - Cos[c + d*x]^2])*((5*a*(a^2 - b^2)*AppellF1[1/4, 1/2, 1, 5/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)]*Sqrt[Cos[c + d*x]])/(Sqrt[1 - Cos[c + d*x]^2]*(5*(a^2 - b^2)*AppellF1[1/4, 1/2, 1, 5/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)] - 2*(2*b^2*AppellF1[5/4, 1/2, 2, 9/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)] + (-a^2 + b^2)*AppellF1[5/4, 3/2, 1, 9/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)])*Cos[c + d*x]^2)*(a^2 + b^2*(-1 + Cos[c + d*x]^2))) - ((1/8 - I/8)*Sqrt[b]*(2*ArcTan[1 - ((1 + I)*Sqrt[b]*Sqrt[Cos[c + d*x]])/(-a^2 + b^2)^(1/4)] - 2*ArcTan[1 + ((1 + I)*Sqrt[b]*Sqrt[Cos[c + d*x]])/(-a^2 + b^2)^(1/4)] + Log[Sqrt[-a^2 + b^2] - (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Cos[c + d*x]] + I*b*Cos[c + d*x]] - Log[Sqrt[-a^2 + b^2] + (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Cos[c + d*x]] + I*b*Cos[c + d*x]]))/(-a^2 + b^2)^(3/4))*Sin[c + d*x])/(Sqrt[1 - Cos[c + d*x]^2]*(a + b*Sin[c + d*x])) + (2*b*(a + b*Sqrt[1 - Cos[c + d*x]^2])*((5*b*(a^2 - b^2)*AppellF1[1/4, -1/2, 1, 5/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)]*Sqrt[Cos[c + d*x]]*Sqrt[1 - Cos[c + d*x]^2])/((-5*(a^2 - b^2)*AppellF1[1/4, -1/2, 1, 5/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)] + 2*(2*b^2*AppellF1[5/4, -1/2, 2, 9/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)] + (a^2 - b^2)*AppellF1[5/4, 1/2, 1, 9/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)])*Cos[c + d*x]^2)*(a^2 + b^2*(-1 + Cos[c + d*x]^2))) + (a*(-2*ArcTan[1 - (Sqrt[2]*Sqrt[b]*Sqrt[Cos[c + d*x]])/(a^2 - b^2)^(1/4)] + 2*ArcTan[1 + (Sqrt[2]*Sqrt[b]*Sqrt[Cos[c + d*x]])/(a^2 - b^2)^(1/4)] - Log[Sqrt[a^2 - b^2] - Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Cos[c + d*x]] + b*Cos[c + d*x]] + Log[Sqrt[a^2 - b^2] + Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Cos[c + d*x]] + b*Cos[c + d*x]]))/(4*Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(3/4)))*Sin[c + d*x]^2)/((1 - Cos[c + d*x]^2)*(a + b*Sin[c + d*x]))))/(2*(a - b)*(a + b)*d*Sqrt[e*Cos[c + d*x]])","C",0
591,1,777,492,6.4111473,"\int \frac{1}{(e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^2} \, dx","Integrate[1/((e*Cos[c + d*x])^(3/2)*(a + b*Sin[c + d*x])^2),x]","\frac{\cos ^{\frac{3}{2}}(c+d x) \left(\frac{12 \left(-\left(2 a^2 b+3 b^3\right) \cos (2 (c+d x))+4 a \left(a^2-b^2\right) \sin (c+d x)-6 a^2 b+b^3\right)}{\left(a^2-b^2\right)^2 \sqrt{\cos (c+d x)}}-\frac{\sin (c+d x) \left(a+b \sqrt{\sin ^2(c+d x)}\right) \left(-\frac{\left(2 a^2+3 b^2\right) \csc (c+d x) \left(8 b^{5/2} \cos ^{\frac{3}{2}}(c+d x) F_1\left(\frac{3}{4};-\frac{1}{2},1;\frac{7}{4};\cos ^2(c+d x),\frac{b^2 \cos ^2(c+d x)}{b^2-a^2}\right)+3 \sqrt{2} a \left(a^2-b^2\right)^{3/4} \left(-\log \left(-\sqrt{2} \sqrt{b} \sqrt[4]{a^2-b^2} \sqrt{\cos (c+d x)}+\sqrt{a^2-b^2}+b \cos (c+d x)\right)+\log \left(\sqrt{2} \sqrt{b} \sqrt[4]{a^2-b^2} \sqrt{\cos (c+d x)}+\sqrt{a^2-b^2}+b \cos (c+d x)\right)+2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{b} \sqrt{\cos (c+d x)}}{\sqrt[4]{a^2-b^2}}\right)-2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{b} \sqrt{\cos (c+d x)}}{\sqrt[4]{a^2-b^2}}+1\right)\right)\right)}{\sqrt{b} \left(b^2-a^2\right)}-\frac{48 a \left(a^2+4 b^2\right) \left(\frac{a \cos ^{\frac{3}{2}}(c+d x) F_1\left(\frac{3}{4};\frac{1}{2},1;\frac{7}{4};\cos ^2(c+d x),\frac{b^2 \cos ^2(c+d x)}{b^2-a^2}\right)}{3 \left(a^2-b^2\right)}+\frac{\left(\frac{1}{8}+\frac{i}{8}\right) \left(-\log \left(-(1+i) \sqrt{b} \sqrt[4]{b^2-a^2} \sqrt{\cos (c+d x)}+\sqrt{b^2-a^2}+i b \cos (c+d x)\right)+\log \left((1+i) \sqrt{b} \sqrt[4]{b^2-a^2} \sqrt{\cos (c+d x)}+\sqrt{b^2-a^2}+i b \cos (c+d x)\right)+2 \tan ^{-1}\left(1-\frac{(1+i) \sqrt{b} \sqrt{\cos (c+d x)}}{\sqrt[4]{b^2-a^2}}\right)-2 \tan ^{-1}\left(1+\frac{(1+i) \sqrt{b} \sqrt{\cos (c+d x)}}{\sqrt[4]{b^2-a^2}}\right)\right)}{\sqrt{b} \sqrt[4]{b^2-a^2}}\right)}{\sqrt{\sin ^2(c+d x)}}\right)}{(a-b)^2 (a+b)^2}\right)}{24 d (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))}","-\frac{\left(2 a^2+3 b^2\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{d e^2 \left(a^2-b^2\right)^2 \sqrt{\cos (c+d x)}}-\frac{5 a b-\left(2 a^2+3 b^2\right) \sin (c+d x)}{d e \left(a^2-b^2\right)^2 \sqrt{e \cos (c+d x)}}+\frac{b}{d e \left(a^2-b^2\right) \sqrt{e \cos (c+d x)} (a+b \sin (c+d x))}-\frac{5 a^2 b \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{2 d e \left(a^2-b^2\right)^2 \left(b-\sqrt{b^2-a^2}\right) \sqrt{e \cos (c+d x)}}-\frac{5 a^2 b \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{2 d e \left(a^2-b^2\right)^2 \left(\sqrt{b^2-a^2}+b\right) \sqrt{e \cos (c+d x)}}-\frac{5 a b^{3/2} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{2 d e^{3/2} \left(b^2-a^2\right)^{9/4}}+\frac{5 a b^{3/2} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{2 d e^{3/2} \left(b^2-a^2\right)^{9/4}}",1,"(Cos[c + d*x]^(3/2)*((12*(-6*a^2*b + b^3 - (2*a^2*b + 3*b^3)*Cos[2*(c + d*x)] + 4*a*(a^2 - b^2)*Sin[c + d*x]))/((a^2 - b^2)^2*Sqrt[Cos[c + d*x]]) - (Sin[c + d*x]*(-(((2*a^2 + 3*b^2)*Csc[c + d*x]*(8*b^(5/2)*AppellF1[3/4, -1/2, 1, 7/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)]*Cos[c + d*x]^(3/2) + 3*Sqrt[2]*a*(a^2 - b^2)^(3/4)*(2*ArcTan[1 - (Sqrt[2]*Sqrt[b]*Sqrt[Cos[c + d*x]])/(a^2 - b^2)^(1/4)] - 2*ArcTan[1 + (Sqrt[2]*Sqrt[b]*Sqrt[Cos[c + d*x]])/(a^2 - b^2)^(1/4)] - Log[Sqrt[a^2 - b^2] - Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Cos[c + d*x]] + b*Cos[c + d*x]] + Log[Sqrt[a^2 - b^2] + Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Cos[c + d*x]] + b*Cos[c + d*x]])))/(Sqrt[b]*(-a^2 + b^2))) - (48*a*(a^2 + 4*b^2)*((a*AppellF1[3/4, 1/2, 1, 7/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)]*Cos[c + d*x]^(3/2))/(3*(a^2 - b^2)) + ((1/8 + I/8)*(2*ArcTan[1 - ((1 + I)*Sqrt[b]*Sqrt[Cos[c + d*x]])/(-a^2 + b^2)^(1/4)] - 2*ArcTan[1 + ((1 + I)*Sqrt[b]*Sqrt[Cos[c + d*x]])/(-a^2 + b^2)^(1/4)] - Log[Sqrt[-a^2 + b^2] - (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Cos[c + d*x]] + I*b*Cos[c + d*x]] + Log[Sqrt[-a^2 + b^2] + (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Cos[c + d*x]] + I*b*Cos[c + d*x]]))/(Sqrt[b]*(-a^2 + b^2)^(1/4))))/Sqrt[Sin[c + d*x]^2])*(a + b*Sqrt[Sin[c + d*x]^2]))/((a - b)^2*(a + b)^2)))/(24*d*(e*Cos[c + d*x])^(3/2)*(a + b*Sin[c + d*x]))","C",0
592,1,1258,514,24.3444224,"\int \frac{1}{(e \cos (c+d x))^{5/2} (a+b \sin (c+d x))^2} \, dx","Integrate[1/((e*Cos[c + d*x])^(5/2)*(a + b*Sin[c + d*x])^2),x]","\frac{\left(\frac{2 \sec ^2(c+d x) \left(\sin (c+d x) a^2-2 b a+b^2 \sin (c+d x)\right)}{3 \left(a^2-b^2\right)^2}-\frac{b^3}{\left(a^2-b^2\right)^2 (a+b \sin (c+d x))}\right) \cos ^3(c+d x)}{d (e \cos (c+d x))^{5/2}}+\frac{\left(-\frac{2 \left(5 b^3+2 a^2 b\right) \left(a+b \sqrt{1-\cos ^2(c+d x)}\right) \left(\frac{5 b \left(a^2-b^2\right) \sqrt{\cos (c+d x)} \sqrt{1-\cos ^2(c+d x)} F_1\left(\frac{1}{4};-\frac{1}{2},1;\frac{5}{4};\cos ^2(c+d x),\frac{b^2 \cos ^2(c+d x)}{b^2-a^2}\right)}{\left(2 \left(2 F_1\left(\frac{5}{4};-\frac{1}{2},2;\frac{9}{4};\cos ^2(c+d x),\frac{b^2 \cos ^2(c+d x)}{b^2-a^2}\right) b^2+\left(a^2-b^2\right) F_1\left(\frac{5}{4};\frac{1}{2},1;\frac{9}{4};\cos ^2(c+d x),\frac{b^2 \cos ^2(c+d x)}{b^2-a^2}\right)\right) \cos ^2(c+d x)-5 \left(a^2-b^2\right) F_1\left(\frac{1}{4};-\frac{1}{2},1;\frac{5}{4};\cos ^2(c+d x),\frac{b^2 \cos ^2(c+d x)}{b^2-a^2}\right)\right) \left(a^2+b^2 \left(\cos ^2(c+d x)-1\right)\right)}+\frac{a \left(-2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{b} \sqrt{\cos (c+d x)}}{\sqrt[4]{a^2-b^2}}\right)+2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{b} \sqrt{\cos (c+d x)}}{\sqrt[4]{a^2-b^2}}+1\right)-\log \left(b \cos (c+d x)-\sqrt{2} \sqrt{b} \sqrt[4]{a^2-b^2} \sqrt{\cos (c+d x)}+\sqrt{a^2-b^2}\right)+\log \left(b \cos (c+d x)+\sqrt{2} \sqrt{b} \sqrt[4]{a^2-b^2} \sqrt{\cos (c+d x)}+\sqrt{a^2-b^2}\right)\right)}{4 \sqrt{2} \sqrt{b} \left(a^2-b^2\right)^{3/4}}\right) \sin ^2(c+d x)}{\left(1-\cos ^2(c+d x)\right) (a+b \sin (c+d x))}-\frac{2 \left(2 a^3-16 a b^2\right) \left(a+b \sqrt{1-\cos ^2(c+d x)}\right) \left(\frac{5 a \left(a^2-b^2\right) F_1\left(\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\cos ^2(c+d x),\frac{b^2 \cos ^2(c+d x)}{b^2-a^2}\right) \sqrt{\cos (c+d x)}}{\sqrt{1-\cos ^2(c+d x)} \left(5 \left(a^2-b^2\right) F_1\left(\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\cos ^2(c+d x),\frac{b^2 \cos ^2(c+d x)}{b^2-a^2}\right)-2 \left(2 F_1\left(\frac{5}{4};\frac{1}{2},2;\frac{9}{4};\cos ^2(c+d x),\frac{b^2 \cos ^2(c+d x)}{b^2-a^2}\right) b^2+\left(b^2-a^2\right) F_1\left(\frac{5}{4};\frac{3}{2},1;\frac{9}{4};\cos ^2(c+d x),\frac{b^2 \cos ^2(c+d x)}{b^2-a^2}\right)\right) \cos ^2(c+d x)\right) \left(a^2+b^2 \left(\cos ^2(c+d x)-1\right)\right)}-\frac{\left(\frac{1}{8}-\frac{i}{8}\right) \sqrt{b} \left(2 \tan ^{-1}\left(1-\frac{(1+i) \sqrt{b} \sqrt{\cos (c+d x)}}{\sqrt[4]{b^2-a^2}}\right)-2 \tan ^{-1}\left(\frac{(1+i) \sqrt{b} \sqrt{\cos (c+d x)}}{\sqrt[4]{b^2-a^2}}+1\right)+\log \left(i b \cos (c+d x)-(1+i) \sqrt{b} \sqrt[4]{b^2-a^2} \sqrt{\cos (c+d x)}+\sqrt{b^2-a^2}\right)-\log \left(i b \cos (c+d x)+(1+i) \sqrt{b} \sqrt[4]{b^2-a^2} \sqrt{\cos (c+d x)}+\sqrt{b^2-a^2}\right)\right)}{\left(b^2-a^2\right)^{3/4}}\right) \sin (c+d x)}{\sqrt{1-\cos ^2(c+d x)} (a+b \sin (c+d x))}\right) \cos ^{\frac{5}{2}}(c+d x)}{6 (a-b)^2 (a+b)^2 d (e \cos (c+d x))^{5/2}}","\frac{\left(2 a^2+5 b^2\right) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d e^2 \left(a^2-b^2\right)^2 \sqrt{e \cos (c+d x)}}-\frac{7 a^2 b^2 \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{2 d e^2 \left(a^2-b^2\right)^2 \left(a^2-b \left(b-\sqrt{b^2-a^2}\right)\right) \sqrt{e \cos (c+d x)}}-\frac{7 a^2 b^2 \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{2 d e^2 \left(a^2-b^2\right)^2 \left(a^2-b \left(\sqrt{b^2-a^2}+b\right)\right) \sqrt{e \cos (c+d x)}}+\frac{b}{d e \left(a^2-b^2\right) (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))}-\frac{7 a b-\left(2 a^2+5 b^2\right) \sin (c+d x)}{3 d e \left(a^2-b^2\right)^2 (e \cos (c+d x))^{3/2}}+\frac{7 a b^{5/2} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{2 d e^{5/2} \left(b^2-a^2\right)^{11/4}}+\frac{7 a b^{5/2} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{2 d e^{5/2} \left(b^2-a^2\right)^{11/4}}",1,"(Cos[c + d*x]^(5/2)*((-2*(2*a^3 - 16*a*b^2)*(a + b*Sqrt[1 - Cos[c + d*x]^2])*((5*a*(a^2 - b^2)*AppellF1[1/4, 1/2, 1, 5/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)]*Sqrt[Cos[c + d*x]])/(Sqrt[1 - Cos[c + d*x]^2]*(5*(a^2 - b^2)*AppellF1[1/4, 1/2, 1, 5/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)] - 2*(2*b^2*AppellF1[5/4, 1/2, 2, 9/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)] + (-a^2 + b^2)*AppellF1[5/4, 3/2, 1, 9/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)])*Cos[c + d*x]^2)*(a^2 + b^2*(-1 + Cos[c + d*x]^2))) - ((1/8 - I/8)*Sqrt[b]*(2*ArcTan[1 - ((1 + I)*Sqrt[b]*Sqrt[Cos[c + d*x]])/(-a^2 + b^2)^(1/4)] - 2*ArcTan[1 + ((1 + I)*Sqrt[b]*Sqrt[Cos[c + d*x]])/(-a^2 + b^2)^(1/4)] + Log[Sqrt[-a^2 + b^2] - (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Cos[c + d*x]] + I*b*Cos[c + d*x]] - Log[Sqrt[-a^2 + b^2] + (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Cos[c + d*x]] + I*b*Cos[c + d*x]]))/(-a^2 + b^2)^(3/4))*Sin[c + d*x])/(Sqrt[1 - Cos[c + d*x]^2]*(a + b*Sin[c + d*x])) - (2*(2*a^2*b + 5*b^3)*(a + b*Sqrt[1 - Cos[c + d*x]^2])*((5*b*(a^2 - b^2)*AppellF1[1/4, -1/2, 1, 5/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)]*Sqrt[Cos[c + d*x]]*Sqrt[1 - Cos[c + d*x]^2])/((-5*(a^2 - b^2)*AppellF1[1/4, -1/2, 1, 5/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)] + 2*(2*b^2*AppellF1[5/4, -1/2, 2, 9/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)] + (a^2 - b^2)*AppellF1[5/4, 1/2, 1, 9/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)])*Cos[c + d*x]^2)*(a^2 + b^2*(-1 + Cos[c + d*x]^2))) + (a*(-2*ArcTan[1 - (Sqrt[2]*Sqrt[b]*Sqrt[Cos[c + d*x]])/(a^2 - b^2)^(1/4)] + 2*ArcTan[1 + (Sqrt[2]*Sqrt[b]*Sqrt[Cos[c + d*x]])/(a^2 - b^2)^(1/4)] - Log[Sqrt[a^2 - b^2] - Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Cos[c + d*x]] + b*Cos[c + d*x]] + Log[Sqrt[a^2 - b^2] + Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Cos[c + d*x]] + b*Cos[c + d*x]]))/(4*Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(3/4)))*Sin[c + d*x]^2)/((1 - Cos[c + d*x]^2)*(a + b*Sin[c + d*x]))))/(6*(a - b)^2*(a + b)^2*d*(e*Cos[c + d*x])^(5/2)) + (Cos[c + d*x]^3*(-(b^3/((a^2 - b^2)^2*(a + b*Sin[c + d*x]))) + (2*Sec[c + d*x]^2*(-2*a*b + a^2*Sin[c + d*x] + b^2*Sin[c + d*x]))/(3*(a^2 - b^2)^2)))/(d*(e*Cos[c + d*x])^(5/2))","C",0
593,1,949,574,6.7719271,"\int \frac{1}{(e \cos (c+d x))^{7/2} (a+b \sin (c+d x))^2} \, dx","Integrate[1/((e*Cos[c + d*x])^(7/2)*(a + b*Sin[c + d*x])^2),x]","\frac{\cos ^4(c+d x) \left(\frac{\cos (c+d x) b^5}{\left(a^2-b^2\right)^3 (a+b \sin (c+d x))}+\frac{2 \sec ^3(c+d x) \left(\sin (c+d x) a^2-2 b a+b^2 \sin (c+d x)\right)}{5 \left(a^2-b^2\right)^2}+\frac{2 \sec (c+d x) \left(3 \sin (c+d x) a^4-15 b^2 \sin (c+d x) a^2+20 b^3 a-8 b^4 \sin (c+d x)\right)}{5 \left(a^2-b^2\right)^3}\right)}{d (e \cos (c+d x))^{7/2}}-\frac{3 \cos ^{\frac{7}{2}}(c+d x) \left(-\frac{\left(-7 b^5-10 a^2 b^3+2 a^4 b\right) \left(a+b \sqrt{1-\cos ^2(c+d x)}\right) \left(8 F_1\left(\frac{3}{4};-\frac{1}{2},1;\frac{7}{4};\cos ^2(c+d x),\frac{b^2 \cos ^2(c+d x)}{b^2-a^2}\right) \cos ^{\frac{3}{2}}(c+d x) b^{5/2}+3 \sqrt{2} a \left(a^2-b^2\right)^{3/4} \left(2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{b} \sqrt{\cos (c+d x)}}{\sqrt[4]{a^2-b^2}}\right)-2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{b} \sqrt{\cos (c+d x)}}{\sqrt[4]{a^2-b^2}}+1\right)-\log \left(b \cos (c+d x)-\sqrt{2} \sqrt{b} \sqrt[4]{a^2-b^2} \sqrt{\cos (c+d x)}+\sqrt{a^2-b^2}\right)+\log \left(b \cos (c+d x)+\sqrt{2} \sqrt{b} \sqrt[4]{a^2-b^2} \sqrt{\cos (c+d x)}+\sqrt{a^2-b^2}\right)\right)\right) \sin ^2(c+d x)}{12 b^{3/2} \left(b^2-a^2\right) \left(1-\cos ^2(c+d x)\right) (a+b \sin (c+d x))}-\frac{2 \left(2 a^5-10 b^2 a^3-22 b^4 a\right) \left(a+b \sqrt{1-\cos ^2(c+d x)}\right) \left(\frac{a F_1\left(\frac{3}{4};\frac{1}{2},1;\frac{7}{4};\cos ^2(c+d x),\frac{b^2 \cos ^2(c+d x)}{b^2-a^2}\right) \cos ^{\frac{3}{2}}(c+d x)}{3 \left(a^2-b^2\right)}+\frac{\left(\frac{1}{8}+\frac{i}{8}\right) \left(2 \tan ^{-1}\left(1-\frac{(1+i) \sqrt{b} \sqrt{\cos (c+d x)}}{\sqrt[4]{b^2-a^2}}\right)-2 \tan ^{-1}\left(\frac{(1+i) \sqrt{b} \sqrt{\cos (c+d x)}}{\sqrt[4]{b^2-a^2}}+1\right)-\log \left(i b \cos (c+d x)-(1+i) \sqrt{b} \sqrt[4]{b^2-a^2} \sqrt{\cos (c+d x)}+\sqrt{b^2-a^2}\right)+\log \left(i b \cos (c+d x)+(1+i) \sqrt{b} \sqrt[4]{b^2-a^2} \sqrt{\cos (c+d x)}+\sqrt{b^2-a^2}\right)\right)}{\sqrt{b} \sqrt[4]{b^2-a^2}}\right) \sin (c+d x)}{\sqrt{1-\cos ^2(c+d x)} (a+b \sin (c+d x))}\right)}{10 (a-b)^3 (a+b)^3 d (e \cos (c+d x))^{7/2}}","\frac{b}{d e \left(a^2-b^2\right) (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))}-\frac{9 a b-\left(2 a^2+7 b^2\right) \sin (c+d x)}{5 d e \left(a^2-b^2\right)^2 (e \cos (c+d x))^{5/2}}-\frac{9 a b^{7/2} \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{2 d e^{7/2} \left(b^2-a^2\right)^{13/4}}+\frac{9 a b^{7/2} \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{2 d e^{7/2} \left(b^2-a^2\right)^{13/4}}+\frac{9 a^2 b^3 \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{2 d e^3 \left(a^2-b^2\right)^3 \left(b-\sqrt{b^2-a^2}\right) \sqrt{e \cos (c+d x)}}+\frac{9 a^2 b^3 \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{2 d e^3 \left(a^2-b^2\right)^3 \left(\sqrt{b^2-a^2}+b\right) \sqrt{e \cos (c+d x)}}-\frac{3 \left(2 a^4-10 a^2 b^2-7 b^4\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{5 d e^4 \left(a^2-b^2\right)^3 \sqrt{\cos (c+d x)}}+\frac{3 \left(\left(2 a^4-10 a^2 b^2-7 b^4\right) \sin (c+d x)+15 a b^3\right)}{5 d e^3 \left(a^2-b^2\right)^3 \sqrt{e \cos (c+d x)}}",1,"(-3*Cos[c + d*x]^(7/2)*((-2*(2*a^5 - 10*a^3*b^2 - 22*a*b^4)*(a + b*Sqrt[1 - Cos[c + d*x]^2])*((a*AppellF1[3/4, 1/2, 1, 7/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)]*Cos[c + d*x]^(3/2))/(3*(a^2 - b^2)) + ((1/8 + I/8)*(2*ArcTan[1 - ((1 + I)*Sqrt[b]*Sqrt[Cos[c + d*x]])/(-a^2 + b^2)^(1/4)] - 2*ArcTan[1 + ((1 + I)*Sqrt[b]*Sqrt[Cos[c + d*x]])/(-a^2 + b^2)^(1/4)] - Log[Sqrt[-a^2 + b^2] - (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Cos[c + d*x]] + I*b*Cos[c + d*x]] + Log[Sqrt[-a^2 + b^2] + (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Cos[c + d*x]] + I*b*Cos[c + d*x]]))/(Sqrt[b]*(-a^2 + b^2)^(1/4)))*Sin[c + d*x])/(Sqrt[1 - Cos[c + d*x]^2]*(a + b*Sin[c + d*x])) - ((2*a^4*b - 10*a^2*b^3 - 7*b^5)*(a + b*Sqrt[1 - Cos[c + d*x]^2])*(8*b^(5/2)*AppellF1[3/4, -1/2, 1, 7/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)]*Cos[c + d*x]^(3/2) + 3*Sqrt[2]*a*(a^2 - b^2)^(3/4)*(2*ArcTan[1 - (Sqrt[2]*Sqrt[b]*Sqrt[Cos[c + d*x]])/(a^2 - b^2)^(1/4)] - 2*ArcTan[1 + (Sqrt[2]*Sqrt[b]*Sqrt[Cos[c + d*x]])/(a^2 - b^2)^(1/4)] - Log[Sqrt[a^2 - b^2] - Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Cos[c + d*x]] + b*Cos[c + d*x]] + Log[Sqrt[a^2 - b^2] + Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Cos[c + d*x]] + b*Cos[c + d*x]]))*Sin[c + d*x]^2)/(12*b^(3/2)*(-a^2 + b^2)*(1 - Cos[c + d*x]^2)*(a + b*Sin[c + d*x]))))/(10*(a - b)^3*(a + b)^3*d*(e*Cos[c + d*x])^(7/2)) + (Cos[c + d*x]^4*((b^5*Cos[c + d*x])/((a^2 - b^2)^3*(a + b*Sin[c + d*x])) + (2*Sec[c + d*x]^3*(-2*a*b + a^2*Sin[c + d*x] + b^2*Sin[c + d*x]))/(5*(a^2 - b^2)^2) + (2*Sec[c + d*x]*(20*a*b^3 + 3*a^4*Sin[c + d*x] - 15*a^2*b^2*Sin[c + d*x] - 8*b^4*Sin[c + d*x]))/(5*(a^2 - b^2)^3)))/(d*(e*Cos[c + d*x])^(7/2))","C",0
594,1,932,575,27.051823,"\int \frac{(e \cos (c+d x))^{13/2}}{(a+b \sin (c+d x))^3} \, dx","Integrate[(e*Cos[c + d*x])^(13/2)/(a + b*Sin[c + d*x])^3,x]","\frac{11 \left(-\frac{\left(45 a^3-37 a b^2\right) \left(a+b \sqrt{1-\cos ^2(c+d x)}\right) \left(8 F_1\left(\frac{3}{4};-\frac{1}{2},1;\frac{7}{4};\cos ^2(c+d x),\frac{b^2 \cos ^2(c+d x)}{b^2-a^2}\right) \cos ^{\frac{3}{2}}(c+d x) b^{5/2}+3 \sqrt{2} a \left(a^2-b^2\right)^{3/4} \left(2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{b} \sqrt{\cos (c+d x)}}{\sqrt[4]{a^2-b^2}}\right)-2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{b} \sqrt{\cos (c+d x)}}{\sqrt[4]{a^2-b^2}}+1\right)-\log \left(b \cos (c+d x)-\sqrt{2} \sqrt{b} \sqrt[4]{a^2-b^2} \sqrt{\cos (c+d x)}+\sqrt{a^2-b^2}\right)+\log \left(b \cos (c+d x)+\sqrt{2} \sqrt{b} \sqrt[4]{a^2-b^2} \sqrt{\cos (c+d x)}+\sqrt{a^2-b^2}\right)\right)\right) \sin ^2(c+d x)}{12 b^{3/2} \left(b^2-a^2\right) \left(1-\cos ^2(c+d x)\right) (a+b \sin (c+d x))}-\frac{2 \left(18 a^2 b-10 b^3\right) \left(a+b \sqrt{1-\cos ^2(c+d x)}\right) \left(\frac{a F_1\left(\frac{3}{4};\frac{1}{2},1;\frac{7}{4};\cos ^2(c+d x),\frac{b^2 \cos ^2(c+d x)}{b^2-a^2}\right) \cos ^{\frac{3}{2}}(c+d x)}{3 \left(a^2-b^2\right)}+\frac{\left(\frac{1}{8}+\frac{i}{8}\right) \left(2 \tan ^{-1}\left(1-\frac{(1+i) \sqrt{b} \sqrt{\cos (c+d x)}}{\sqrt[4]{b^2-a^2}}\right)-2 \tan ^{-1}\left(\frac{(1+i) \sqrt{b} \sqrt{\cos (c+d x)}}{\sqrt[4]{b^2-a^2}}+1\right)-\log \left(i b \cos (c+d x)-(1+i) \sqrt{b} \sqrt[4]{b^2-a^2} \sqrt{\cos (c+d x)}+\sqrt{b^2-a^2}\right)+\log \left(i b \cos (c+d x)+(1+i) \sqrt{b} \sqrt[4]{b^2-a^2} \sqrt{\cos (c+d x)}+\sqrt{b^2-a^2}\right)\right)}{\sqrt{b} \sqrt[4]{b^2-a^2}}\right) \sin (c+d x)}{\sqrt{1-\cos ^2(c+d x)} (a+b \sin (c+d x))}\right) (e \cos (c+d x))^{13/2}}{40 b^5 d \cos ^{\frac{13}{2}}(c+d x)}+\frac{\sec ^6(c+d x) \left(-\frac{\left(65 b^2-168 a^2\right) \cos (c+d x)}{42 b^5}-\frac{\cos (3 (c+d x))}{14 b^3}-\frac{3 a \sin (2 (c+d x))}{5 b^4}+\frac{19 \left(a^3 \cos (c+d x)-a b^2 \cos (c+d x)\right)}{4 b^5 (a+b \sin (c+d x))}+\frac{-\cos (c+d x) a^4+2 b^2 \cos (c+d x) a^2-b^4 \cos (c+d x)}{2 b^5 (a+b \sin (c+d x))^2}\right) (e \cos (c+d x))^{13/2}}{d}","\frac{11 a e^6 \left(45 a^2-37 b^2\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{20 b^6 d \sqrt{\cos (c+d x)}}+\frac{11 e^5 (e \cos (c+d x))^{3/2} \left(5 \left(9 a^2-2 b^2\right)-27 a b \sin (c+d x)\right)}{60 b^5 d}-\frac{11 e^{13/2} \left(9 a^4-11 a^2 b^2+2 b^4\right) \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{8 b^{13/2} d \sqrt[4]{b^2-a^2}}+\frac{11 e^{13/2} \left(9 a^4-11 a^2 b^2+2 b^4\right) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{8 b^{13/2} d \sqrt[4]{b^2-a^2}}-\frac{11 a e^7 \left(9 a^4-11 a^2 b^2+2 b^4\right) \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{8 b^7 d \left(b-\sqrt{b^2-a^2}\right) \sqrt{e \cos (c+d x)}}-\frac{11 a e^7 \left(9 a^4-11 a^2 b^2+2 b^4\right) \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{8 b^7 d \left(\sqrt{b^2-a^2}+b\right) \sqrt{e \cos (c+d x)}}-\frac{11 e^3 (e \cos (c+d x))^{7/2} (9 a+2 b \sin (c+d x))}{28 b^3 d (a+b \sin (c+d x))}-\frac{e (e \cos (c+d x))^{11/2}}{2 b d (a+b \sin (c+d x))^2}",1,"(11*(e*Cos[c + d*x])^(13/2)*((-2*(18*a^2*b - 10*b^3)*(a + b*Sqrt[1 - Cos[c + d*x]^2])*((a*AppellF1[3/4, 1/2, 1, 7/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)]*Cos[c + d*x]^(3/2))/(3*(a^2 - b^2)) + ((1/8 + I/8)*(2*ArcTan[1 - ((1 + I)*Sqrt[b]*Sqrt[Cos[c + d*x]])/(-a^2 + b^2)^(1/4)] - 2*ArcTan[1 + ((1 + I)*Sqrt[b]*Sqrt[Cos[c + d*x]])/(-a^2 + b^2)^(1/4)] - Log[Sqrt[-a^2 + b^2] - (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Cos[c + d*x]] + I*b*Cos[c + d*x]] + Log[Sqrt[-a^2 + b^2] + (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Cos[c + d*x]] + I*b*Cos[c + d*x]]))/(Sqrt[b]*(-a^2 + b^2)^(1/4)))*Sin[c + d*x])/(Sqrt[1 - Cos[c + d*x]^2]*(a + b*Sin[c + d*x])) - ((45*a^3 - 37*a*b^2)*(a + b*Sqrt[1 - Cos[c + d*x]^2])*(8*b^(5/2)*AppellF1[3/4, -1/2, 1, 7/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)]*Cos[c + d*x]^(3/2) + 3*Sqrt[2]*a*(a^2 - b^2)^(3/4)*(2*ArcTan[1 - (Sqrt[2]*Sqrt[b]*Sqrt[Cos[c + d*x]])/(a^2 - b^2)^(1/4)] - 2*ArcTan[1 + (Sqrt[2]*Sqrt[b]*Sqrt[Cos[c + d*x]])/(a^2 - b^2)^(1/4)] - Log[Sqrt[a^2 - b^2] - Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Cos[c + d*x]] + b*Cos[c + d*x]] + Log[Sqrt[a^2 - b^2] + Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Cos[c + d*x]] + b*Cos[c + d*x]]))*Sin[c + d*x]^2)/(12*b^(3/2)*(-a^2 + b^2)*(1 - Cos[c + d*x]^2)*(a + b*Sin[c + d*x]))))/(40*b^5*d*Cos[c + d*x]^(13/2)) + ((e*Cos[c + d*x])^(13/2)*Sec[c + d*x]^6*(-1/42*((-168*a^2 + 65*b^2)*Cos[c + d*x])/b^5 - Cos[3*(c + d*x)]/(14*b^3) + (-(a^4*Cos[c + d*x]) + 2*a^2*b^2*Cos[c + d*x] - b^4*Cos[c + d*x])/(2*b^5*(a + b*Sin[c + d*x])^2) + (19*(a^3*Cos[c + d*x] - a*b^2*Cos[c + d*x]))/(4*b^5*(a + b*Sin[c + d*x])) - (3*a*Sin[2*(c + d*x)])/(5*b^4)))/d","C",0
595,1,2024,589,27.6213584,"\int \frac{(e \cos (c+d x))^{11/2}}{(a+b \sin (c+d x))^3} \, dx","Integrate[(e*Cos[c + d*x])^(11/2)/(a + b*Sin[c + d*x])^3,x]","\text{Result too large to show}","\frac{3 a e^6 \left(21 a^2-13 b^2\right) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{4 b^6 d \sqrt{e \cos (c+d x)}}+\frac{3 e^5 \sqrt{e \cos (c+d x)} \left(3 \left(7 a^2-2 b^2\right)-7 a b \sin (c+d x)\right)}{4 b^5 d}+\frac{9 e^{11/2} \left(7 a^4-9 a^2 b^2+2 b^4\right) \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{8 b^{11/2} d \left(b^2-a^2\right)^{3/4}}+\frac{9 e^{11/2} \left(7 a^4-9 a^2 b^2+2 b^4\right) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{8 b^{11/2} d \left(b^2-a^2\right)^{3/4}}-\frac{9 a e^6 \left(7 a^4-9 a^2 b^2+2 b^4\right) \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{8 b^6 d \left(a^2-b \left(b-\sqrt{b^2-a^2}\right)\right) \sqrt{e \cos (c+d x)}}-\frac{9 a e^6 \left(7 a^4-9 a^2 b^2+2 b^4\right) \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{8 b^6 d \left(a^2-b \left(\sqrt{b^2-a^2}+b\right)\right) \sqrt{e \cos (c+d x)}}-\frac{9 e^3 (e \cos (c+d x))^{5/2} (7 a+2 b \sin (c+d x))}{20 b^3 d (a+b \sin (c+d x))}-\frac{e (e \cos (c+d x))^{9/2}}{2 b d (a+b \sin (c+d x))^2}",1,"((e*Cos[c + d*x])^(11/2)*Sec[c + d*x]^5*(-1/5*Cos[2*(c + d*x)]/b^3 - (2*a*Sin[c + d*x])/b^4 - (-a^2 + b^2)^2/(2*b^5*(a + b*Sin[c + d*x])^2) + (17*a*(a^2 - b^2))/(4*b^5*(a + b*Sin[c + d*x]))))/d + (3*(e*Cos[c + d*x])^(11/2)*((-2*(30*a^2*b - 16*b^3)*(a + b*Sqrt[1 - Cos[c + d*x]^2])*((5*a*(a^2 - b^2)*AppellF1[1/4, 1/2, 1, 5/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)]*Sqrt[Cos[c + d*x]])/(Sqrt[1 - Cos[c + d*x]^2]*(5*(a^2 - b^2)*AppellF1[1/4, 1/2, 1, 5/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)] - 2*(2*b^2*AppellF1[5/4, 1/2, 2, 9/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)] + (-a^2 + b^2)*AppellF1[5/4, 3/2, 1, 9/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)])*Cos[c + d*x]^2)*(a^2 + b^2*(-1 + Cos[c + d*x]^2))) - ((1/8 - I/8)*Sqrt[b]*(2*ArcTan[1 - ((1 + I)*Sqrt[b]*Sqrt[Cos[c + d*x]])/(-a^2 + b^2)^(1/4)] - 2*ArcTan[1 + ((1 + I)*Sqrt[b]*Sqrt[Cos[c + d*x]])/(-a^2 + b^2)^(1/4)] + Log[Sqrt[-a^2 + b^2] - (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Cos[c + d*x]] + I*b*Cos[c + d*x]] - Log[Sqrt[-a^2 + b^2] + (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Cos[c + d*x]] + I*b*Cos[c + d*x]]))/(-a^2 + b^2)^(3/4))*Sin[c + d*x])/(Sqrt[1 - Cos[c + d*x]^2]*(a + b*Sin[c + d*x])) + ((40*a^2*b - 14*b^3)*(a + b*Sqrt[1 - Cos[c + d*x]^2])*Cos[2*(c + d*x)]*(((1/2 - I/2)*(-2*a^2 + b^2)*ArcTan[1 - ((1 + I)*Sqrt[b]*Sqrt[Cos[c + d*x]])/(-a^2 + b^2)^(1/4)])/(b^(3/2)*(-a^2 + b^2)^(3/4)) - ((1/2 - I/2)*(-2*a^2 + b^2)*ArcTan[1 + ((1 + I)*Sqrt[b]*Sqrt[Cos[c + d*x]])/(-a^2 + b^2)^(1/4)])/(b^(3/2)*(-a^2 + b^2)^(3/4)) + (4*Sqrt[Cos[c + d*x]])/b - (4*a*AppellF1[5/4, 1/2, 1, 9/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)]*Cos[c + d*x]^(5/2))/(5*(a^2 - b^2)) + (10*a*(a^2 - b^2)*AppellF1[1/4, 1/2, 1, 5/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)]*Sqrt[Cos[c + d*x]])/(Sqrt[1 - Cos[c + d*x]^2]*(5*(a^2 - b^2)*AppellF1[1/4, 1/2, 1, 5/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)] - 2*(2*b^2*AppellF1[5/4, 1/2, 2, 9/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)] + (-a^2 + b^2)*AppellF1[5/4, 3/2, 1, 9/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)])*Cos[c + d*x]^2)*(a^2 + b^2*(-1 + Cos[c + d*x]^2))) + ((1/4 - I/4)*(-2*a^2 + b^2)*Log[Sqrt[-a^2 + b^2] - (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Cos[c + d*x]] + I*b*Cos[c + d*x]])/(b^(3/2)*(-a^2 + b^2)^(3/4)) - ((1/4 - I/4)*(-2*a^2 + b^2)*Log[Sqrt[-a^2 + b^2] + (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Cos[c + d*x]] + I*b*Cos[c + d*x]])/(b^(3/2)*(-a^2 + b^2)^(3/4)))*Sin[c + d*x])/(Sqrt[1 - Cos[c + d*x]^2]*(-1 + 2*Cos[c + d*x]^2)*(a + b*Sin[c + d*x])) - (2*(25*a^3 - 37*a*b^2)*(a + b*Sqrt[1 - Cos[c + d*x]^2])*((5*b*(a^2 - b^2)*AppellF1[1/4, -1/2, 1, 5/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)]*Sqrt[Cos[c + d*x]]*Sqrt[1 - Cos[c + d*x]^2])/((-5*(a^2 - b^2)*AppellF1[1/4, -1/2, 1, 5/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)] + 2*(2*b^2*AppellF1[5/4, -1/2, 2, 9/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)] + (a^2 - b^2)*AppellF1[5/4, 1/2, 1, 9/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)])*Cos[c + d*x]^2)*(a^2 + b^2*(-1 + Cos[c + d*x]^2))) + (a*(-2*ArcTan[1 - (Sqrt[2]*Sqrt[b]*Sqrt[Cos[c + d*x]])/(a^2 - b^2)^(1/4)] + 2*ArcTan[1 + (Sqrt[2]*Sqrt[b]*Sqrt[Cos[c + d*x]])/(a^2 - b^2)^(1/4)] - Log[Sqrt[a^2 - b^2] - Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Cos[c + d*x]] + b*Cos[c + d*x]] + Log[Sqrt[a^2 - b^2] + Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Cos[c + d*x]] + b*Cos[c + d*x]]))/(4*Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(3/4)))*Sin[c + d*x]^2)/((1 - Cos[c + d*x]^2)*(a + b*Sin[c + d*x]))))/(40*b^5*d*Cos[c + d*x]^(11/2))","C",0
596,1,777,483,25.9901928,"\int \frac{(e \cos (c+d x))^{9/2}}{(a+b \sin (c+d x))^3} \, dx","Integrate[(e*Cos[c + d*x])^(9/2)/(a + b*Sin[c + d*x])^3,x]","\frac{(e \cos (c+d x))^{9/2} \left(\frac{28 \sin (c+d x) \left(a+b \sqrt{\sin ^2(c+d x)}\right) \left(\frac{a \cos ^{\frac{3}{2}}(c+d x) F_1\left(\frac{3}{4};\frac{1}{2},1;\frac{7}{4};\cos ^2(c+d x),\frac{b^2 \cos ^2(c+d x)}{b^2-a^2}\right)}{3 \left(a^2-b^2\right)}+\frac{\left(\frac{1}{8}+\frac{i}{8}\right) \left(-\log \left(-(1+i) \sqrt{b} \sqrt[4]{b^2-a^2} \sqrt{\cos (c+d x)}+\sqrt{b^2-a^2}+i b \cos (c+d x)\right)+\log \left((1+i) \sqrt{b} \sqrt[4]{b^2-a^2} \sqrt{\cos (c+d x)}+\sqrt{b^2-a^2}+i b \cos (c+d x)\right)+2 \tan ^{-1}\left(1-\frac{(1+i) \sqrt{b} \sqrt{\cos (c+d x)}}{\sqrt[4]{b^2-a^2}}\right)-2 \tan ^{-1}\left(1+\frac{(1+i) \sqrt{b} \sqrt{\cos (c+d x)}}{\sqrt[4]{b^2-a^2}}\right)\right)}{\sqrt{b} \sqrt[4]{b^2-a^2}}\right)}{b^2 \sqrt{\sin ^2(c+d x)} (a+b \sin (c+d x))}+\frac{35 a \left(a+b \sqrt{\sin ^2(c+d x)}\right) \left(8 b^{5/2} \cos ^{\frac{3}{2}}(c+d x) F_1\left(\frac{3}{4};-\frac{1}{2},1;\frac{7}{4};\cos ^2(c+d x),\frac{b^2 \cos ^2(c+d x)}{b^2-a^2}\right)+3 \sqrt{2} a \left(a^2-b^2\right)^{3/4} \left(-\log \left(-\sqrt{2} \sqrt{b} \sqrt[4]{a^2-b^2} \sqrt{\cos (c+d x)}+\sqrt{a^2-b^2}+b \cos (c+d x)\right)+\log \left(\sqrt{2} \sqrt{b} \sqrt[4]{a^2-b^2} \sqrt{\cos (c+d x)}+\sqrt{a^2-b^2}+b \cos (c+d x)\right)+2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{b} \sqrt{\cos (c+d x)}}{\sqrt[4]{a^2-b^2}}\right)-2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{b} \sqrt{\cos (c+d x)}}{\sqrt[4]{a^2-b^2}}+1\right)\right)\right)}{12 b^{9/2} \left(b^2-a^2\right) (a+b \sin (c+d x))}+\frac{4 \left(a^2-b^2\right) \cos ^{\frac{3}{2}}(c+d x)}{b^3 (a+b \sin (c+d x))^2}-\frac{22 a \cos ^{\frac{3}{2}}(c+d x)}{b^3 (a+b \sin (c+d x))}-\frac{16 \cos ^{\frac{3}{2}}(c+d x)}{3 b^3}\right)}{8 d \cos ^{\frac{9}{2}}(c+d x)}","\frac{7 e^{9/2} \left(5 a^2-2 b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{8 b^{9/2} d \sqrt[4]{b^2-a^2}}-\frac{7 e^{9/2} \left(5 a^2-2 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{8 b^{9/2} d \sqrt[4]{b^2-a^2}}+\frac{7 a e^5 \left(5 a^2-2 b^2\right) \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{8 b^5 d \left(b-\sqrt{b^2-a^2}\right) \sqrt{e \cos (c+d x)}}+\frac{7 a e^5 \left(5 a^2-2 b^2\right) \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{8 b^5 d \left(\sqrt{b^2-a^2}+b\right) \sqrt{e \cos (c+d x)}}-\frac{35 a e^4 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{4 b^4 d \sqrt{\cos (c+d x)}}-\frac{7 e^3 (e \cos (c+d x))^{3/2} (5 a+2 b \sin (c+d x))}{12 b^3 d (a+b \sin (c+d x))}-\frac{e (e \cos (c+d x))^{7/2}}{2 b d (a+b \sin (c+d x))^2}",1,"((e*Cos[c + d*x])^(9/2)*((-16*Cos[c + d*x]^(3/2))/(3*b^3) + (4*(a^2 - b^2)*Cos[c + d*x]^(3/2))/(b^3*(a + b*Sin[c + d*x])^2) - (22*a*Cos[c + d*x]^(3/2))/(b^3*(a + b*Sin[c + d*x])) + (35*a*(8*b^(5/2)*AppellF1[3/4, -1/2, 1, 7/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)]*Cos[c + d*x]^(3/2) + 3*Sqrt[2]*a*(a^2 - b^2)^(3/4)*(2*ArcTan[1 - (Sqrt[2]*Sqrt[b]*Sqrt[Cos[c + d*x]])/(a^2 - b^2)^(1/4)] - 2*ArcTan[1 + (Sqrt[2]*Sqrt[b]*Sqrt[Cos[c + d*x]])/(a^2 - b^2)^(1/4)] - Log[Sqrt[a^2 - b^2] - Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Cos[c + d*x]] + b*Cos[c + d*x]] + Log[Sqrt[a^2 - b^2] + Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Cos[c + d*x]] + b*Cos[c + d*x]]))*(a + b*Sqrt[Sin[c + d*x]^2]))/(12*b^(9/2)*(-a^2 + b^2)*(a + b*Sin[c + d*x])) + (28*((a*AppellF1[3/4, 1/2, 1, 7/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)]*Cos[c + d*x]^(3/2))/(3*(a^2 - b^2)) + ((1/8 + I/8)*(2*ArcTan[1 - ((1 + I)*Sqrt[b]*Sqrt[Cos[c + d*x]])/(-a^2 + b^2)^(1/4)] - 2*ArcTan[1 + ((1 + I)*Sqrt[b]*Sqrt[Cos[c + d*x]])/(-a^2 + b^2)^(1/4)] - Log[Sqrt[-a^2 + b^2] - (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Cos[c + d*x]] + I*b*Cos[c + d*x]] + Log[Sqrt[-a^2 + b^2] + (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Cos[c + d*x]] + I*b*Cos[c + d*x]]))/(Sqrt[b]*(-a^2 + b^2)^(1/4)))*Sin[c + d*x]*(a + b*Sqrt[Sin[c + d*x]^2]))/(b^2*Sqrt[Sin[c + d*x]^2]*(a + b*Sin[c + d*x]))))/(8*d*Cos[c + d*x]^(9/2))","C",0
597,1,1954,497,26.3711502,"\int \frac{(e \cos (c+d x))^{7/2}}{(a+b \sin (c+d x))^3} \, dx","Integrate[(e*Cos[c + d*x])^(7/2)/(a + b*Sin[c + d*x])^3,x]","\frac{(e \cos (c+d x))^{7/2} \sec ^3(c+d x) \left(\frac{a^2-b^2}{2 b^3 (a+b \sin (c+d x))^2}-\frac{9 a}{4 b^3 (a+b \sin (c+d x))}\right)}{d}-\frac{(e \cos (c+d x))^{7/2} \left(-\frac{14 a \left(a+b \sqrt{1-\cos ^2(c+d x)}\right) \left(\frac{5 b \left(a^2-b^2\right) \sqrt{\cos (c+d x)} \sqrt{1-\cos ^2(c+d x)} F_1\left(\frac{1}{4};-\frac{1}{2},1;\frac{5}{4};\cos ^2(c+d x),\frac{b^2 \cos ^2(c+d x)}{b^2-a^2}\right)}{\left(2 \left(2 F_1\left(\frac{5}{4};-\frac{1}{2},2;\frac{9}{4};\cos ^2(c+d x),\frac{b^2 \cos ^2(c+d x)}{b^2-a^2}\right) b^2+\left(a^2-b^2\right) F_1\left(\frac{5}{4};\frac{1}{2},1;\frac{9}{4};\cos ^2(c+d x),\frac{b^2 \cos ^2(c+d x)}{b^2-a^2}\right)\right) \cos ^2(c+d x)-5 \left(a^2-b^2\right) F_1\left(\frac{1}{4};-\frac{1}{2},1;\frac{5}{4};\cos ^2(c+d x),\frac{b^2 \cos ^2(c+d x)}{b^2-a^2}\right)\right) \left(a^2+b^2 \left(\cos ^2(c+d x)-1\right)\right)}+\frac{a \left(-2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{b} \sqrt{\cos (c+d x)}}{\sqrt[4]{a^2-b^2}}\right)+2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{b} \sqrt{\cos (c+d x)}}{\sqrt[4]{a^2-b^2}}+1\right)-\log \left(b \cos (c+d x)-\sqrt{2} \sqrt{b} \sqrt[4]{a^2-b^2} \sqrt{\cos (c+d x)}+\sqrt{a^2-b^2}\right)+\log \left(b \cos (c+d x)+\sqrt{2} \sqrt{b} \sqrt[4]{a^2-b^2} \sqrt{\cos (c+d x)}+\sqrt{a^2-b^2}\right)\right)}{4 \sqrt{2} \sqrt{b} \left(a^2-b^2\right)^{3/4}}\right) \sin ^2(c+d x)}{\left(1-\cos ^2(c+d x)\right) (a+b \sin (c+d x))}-\frac{12 b \left(a+b \sqrt{1-\cos ^2(c+d x)}\right) \left(\frac{5 a \left(a^2-b^2\right) F_1\left(\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\cos ^2(c+d x),\frac{b^2 \cos ^2(c+d x)}{b^2-a^2}\right) \sqrt{\cos (c+d x)}}{\sqrt{1-\cos ^2(c+d x)} \left(5 \left(a^2-b^2\right) F_1\left(\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\cos ^2(c+d x),\frac{b^2 \cos ^2(c+d x)}{b^2-a^2}\right)-2 \left(2 F_1\left(\frac{5}{4};\frac{1}{2},2;\frac{9}{4};\cos ^2(c+d x),\frac{b^2 \cos ^2(c+d x)}{b^2-a^2}\right) b^2+\left(b^2-a^2\right) F_1\left(\frac{5}{4};\frac{3}{2},1;\frac{9}{4};\cos ^2(c+d x),\frac{b^2 \cos ^2(c+d x)}{b^2-a^2}\right)\right) \cos ^2(c+d x)\right) \left(a^2+b^2 \left(\cos ^2(c+d x)-1\right)\right)}-\frac{\left(\frac{1}{8}-\frac{i}{8}\right) \sqrt{b} \left(2 \tan ^{-1}\left(1-\frac{(1+i) \sqrt{b} \sqrt{\cos (c+d x)}}{\sqrt[4]{b^2-a^2}}\right)-2 \tan ^{-1}\left(\frac{(1+i) \sqrt{b} \sqrt{\cos (c+d x)}}{\sqrt[4]{b^2-a^2}}+1\right)+\log \left(i b \cos (c+d x)-(1+i) \sqrt{b} \sqrt[4]{b^2-a^2} \sqrt{\cos (c+d x)}+\sqrt{b^2-a^2}\right)-\log \left(i b \cos (c+d x)+(1+i) \sqrt{b} \sqrt[4]{b^2-a^2} \sqrt{\cos (c+d x)}+\sqrt{b^2-a^2}\right)\right)}{\left(b^2-a^2\right)^{3/4}}\right) \sin (c+d x)}{\sqrt{1-\cos ^2(c+d x)} (a+b \sin (c+d x))}+\frac{4 b \left(a+b \sqrt{1-\cos ^2(c+d x)}\right) \cos (2 (c+d x)) \left(-\frac{4 a F_1\left(\frac{5}{4};\frac{1}{2},1;\frac{9}{4};\cos ^2(c+d x),\frac{b^2 \cos ^2(c+d x)}{b^2-a^2}\right) \cos ^{\frac{5}{2}}(c+d x)}{5 \left(a^2-b^2\right)}+\frac{4 \sqrt{\cos (c+d x)}}{b}+\frac{10 a \left(a^2-b^2\right) F_1\left(\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\cos ^2(c+d x),\frac{b^2 \cos ^2(c+d x)}{b^2-a^2}\right) \sqrt{\cos (c+d x)}}{\sqrt{1-\cos ^2(c+d x)} \left(5 \left(a^2-b^2\right) F_1\left(\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\cos ^2(c+d x),\frac{b^2 \cos ^2(c+d x)}{b^2-a^2}\right)-2 \left(2 F_1\left(\frac{5}{4};\frac{1}{2},2;\frac{9}{4};\cos ^2(c+d x),\frac{b^2 \cos ^2(c+d x)}{b^2-a^2}\right) b^2+\left(b^2-a^2\right) F_1\left(\frac{5}{4};\frac{3}{2},1;\frac{9}{4};\cos ^2(c+d x),\frac{b^2 \cos ^2(c+d x)}{b^2-a^2}\right)\right) \cos ^2(c+d x)\right) \left(a^2+b^2 \left(\cos ^2(c+d x)-1\right)\right)}+\frac{\left(\frac{1}{2}-\frac{i}{2}\right) \left(b^2-2 a^2\right) \tan ^{-1}\left(1-\frac{(1+i) \sqrt{b} \sqrt{\cos (c+d x)}}{\sqrt[4]{b^2-a^2}}\right)}{b^{3/2} \left(b^2-a^2\right)^{3/4}}-\frac{\left(\frac{1}{2}-\frac{i}{2}\right) \left(b^2-2 a^2\right) \tan ^{-1}\left(\frac{(1+i) \sqrt{b} \sqrt{\cos (c+d x)}}{\sqrt[4]{b^2-a^2}}+1\right)}{b^{3/2} \left(b^2-a^2\right)^{3/4}}+\frac{\left(\frac{1}{4}-\frac{i}{4}\right) \left(b^2-2 a^2\right) \log \left(i b \cos (c+d x)-(1+i) \sqrt{b} \sqrt[4]{b^2-a^2} \sqrt{\cos (c+d x)}+\sqrt{b^2-a^2}\right)}{b^{3/2} \left(b^2-a^2\right)^{3/4}}-\frac{\left(\frac{1}{4}-\frac{i}{4}\right) \left(b^2-2 a^2\right) \log \left(i b \cos (c+d x)+(1+i) \sqrt{b} \sqrt[4]{b^2-a^2} \sqrt{\cos (c+d x)}+\sqrt{b^2-a^2}\right)}{b^{3/2} \left(b^2-a^2\right)^{3/4}}\right) \sin (c+d x)}{\sqrt{1-\cos ^2(c+d x)} \left(2 \cos ^2(c+d x)-1\right) (a+b \sin (c+d x))}\right)}{8 b^3 d \cos ^{\frac{7}{2}}(c+d x)}","-\frac{5 e^{7/2} \left(3 a^2-2 b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{8 b^{7/2} d \left(b^2-a^2\right)^{3/4}}-\frac{5 e^{7/2} \left(3 a^2-2 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{8 b^{7/2} d \left(b^2-a^2\right)^{3/4}}+\frac{5 a e^4 \left(3 a^2-2 b^2\right) \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{8 b^4 d \left(a^2-b \left(b-\sqrt{b^2-a^2}\right)\right) \sqrt{e \cos (c+d x)}}+\frac{5 a e^4 \left(3 a^2-2 b^2\right) \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{8 b^4 d \left(a^2-b \left(\sqrt{b^2-a^2}+b\right)\right) \sqrt{e \cos (c+d x)}}-\frac{15 a e^4 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{4 b^4 d \sqrt{e \cos (c+d x)}}-\frac{5 e^3 \sqrt{e \cos (c+d x)} (3 a+2 b \sin (c+d x))}{4 b^3 d (a+b \sin (c+d x))}-\frac{e (e \cos (c+d x))^{5/2}}{2 b d (a+b \sin (c+d x))^2}",1,"((e*Cos[c + d*x])^(7/2)*Sec[c + d*x]^3*((a^2 - b^2)/(2*b^3*(a + b*Sin[c + d*x])^2) - (9*a)/(4*b^3*(a + b*Sin[c + d*x]))))/d - ((e*Cos[c + d*x])^(7/2)*((-12*b*(a + b*Sqrt[1 - Cos[c + d*x]^2])*((5*a*(a^2 - b^2)*AppellF1[1/4, 1/2, 1, 5/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)]*Sqrt[Cos[c + d*x]])/(Sqrt[1 - Cos[c + d*x]^2]*(5*(a^2 - b^2)*AppellF1[1/4, 1/2, 1, 5/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)] - 2*(2*b^2*AppellF1[5/4, 1/2, 2, 9/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)] + (-a^2 + b^2)*AppellF1[5/4, 3/2, 1, 9/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)])*Cos[c + d*x]^2)*(a^2 + b^2*(-1 + Cos[c + d*x]^2))) - ((1/8 - I/8)*Sqrt[b]*(2*ArcTan[1 - ((1 + I)*Sqrt[b]*Sqrt[Cos[c + d*x]])/(-a^2 + b^2)^(1/4)] - 2*ArcTan[1 + ((1 + I)*Sqrt[b]*Sqrt[Cos[c + d*x]])/(-a^2 + b^2)^(1/4)] + Log[Sqrt[-a^2 + b^2] - (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Cos[c + d*x]] + I*b*Cos[c + d*x]] - Log[Sqrt[-a^2 + b^2] + (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Cos[c + d*x]] + I*b*Cos[c + d*x]]))/(-a^2 + b^2)^(3/4))*Sin[c + d*x])/(Sqrt[1 - Cos[c + d*x]^2]*(a + b*Sin[c + d*x])) + (4*b*(a + b*Sqrt[1 - Cos[c + d*x]^2])*Cos[2*(c + d*x)]*(((1/2 - I/2)*(-2*a^2 + b^2)*ArcTan[1 - ((1 + I)*Sqrt[b]*Sqrt[Cos[c + d*x]])/(-a^2 + b^2)^(1/4)])/(b^(3/2)*(-a^2 + b^2)^(3/4)) - ((1/2 - I/2)*(-2*a^2 + b^2)*ArcTan[1 + ((1 + I)*Sqrt[b]*Sqrt[Cos[c + d*x]])/(-a^2 + b^2)^(1/4)])/(b^(3/2)*(-a^2 + b^2)^(3/4)) + (4*Sqrt[Cos[c + d*x]])/b - (4*a*AppellF1[5/4, 1/2, 1, 9/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)]*Cos[c + d*x]^(5/2))/(5*(a^2 - b^2)) + (10*a*(a^2 - b^2)*AppellF1[1/4, 1/2, 1, 5/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)]*Sqrt[Cos[c + d*x]])/(Sqrt[1 - Cos[c + d*x]^2]*(5*(a^2 - b^2)*AppellF1[1/4, 1/2, 1, 5/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)] - 2*(2*b^2*AppellF1[5/4, 1/2, 2, 9/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)] + (-a^2 + b^2)*AppellF1[5/4, 3/2, 1, 9/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)])*Cos[c + d*x]^2)*(a^2 + b^2*(-1 + Cos[c + d*x]^2))) + ((1/4 - I/4)*(-2*a^2 + b^2)*Log[Sqrt[-a^2 + b^2] - (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Cos[c + d*x]] + I*b*Cos[c + d*x]])/(b^(3/2)*(-a^2 + b^2)^(3/4)) - ((1/4 - I/4)*(-2*a^2 + b^2)*Log[Sqrt[-a^2 + b^2] + (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Cos[c + d*x]] + I*b*Cos[c + d*x]])/(b^(3/2)*(-a^2 + b^2)^(3/4)))*Sin[c + d*x])/(Sqrt[1 - Cos[c + d*x]^2]*(-1 + 2*Cos[c + d*x]^2)*(a + b*Sin[c + d*x])) - (14*a*(a + b*Sqrt[1 - Cos[c + d*x]^2])*((5*b*(a^2 - b^2)*AppellF1[1/4, -1/2, 1, 5/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)]*Sqrt[Cos[c + d*x]]*Sqrt[1 - Cos[c + d*x]^2])/((-5*(a^2 - b^2)*AppellF1[1/4, -1/2, 1, 5/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)] + 2*(2*b^2*AppellF1[5/4, -1/2, 2, 9/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)] + (a^2 - b^2)*AppellF1[5/4, 1/2, 1, 9/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)])*Cos[c + d*x]^2)*(a^2 + b^2*(-1 + Cos[c + d*x]^2))) + (a*(-2*ArcTan[1 - (Sqrt[2]*Sqrt[b]*Sqrt[Cos[c + d*x]])/(a^2 - b^2)^(1/4)] + 2*ArcTan[1 + (Sqrt[2]*Sqrt[b]*Sqrt[Cos[c + d*x]])/(a^2 - b^2)^(1/4)] - Log[Sqrt[a^2 - b^2] - Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Cos[c + d*x]] + b*Cos[c + d*x]] + Log[Sqrt[a^2 - b^2] + Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Cos[c + d*x]] + b*Cos[c + d*x]]))/(4*Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(3/4)))*Sin[c + d*x]^2)/((1 - Cos[c + d*x]^2)*(a + b*Sin[c + d*x]))))/(8*b^3*d*Cos[c + d*x]^(7/2))","C",0
598,1,831,505,24.058419,"\int \frac{(e \cos (c+d x))^{5/2}}{(a+b \sin (c+d x))^3} \, dx","Integrate[(e*Cos[c + d*x])^(5/2)/(a + b*Sin[c + d*x])^3,x]","\frac{\sec ^2(c+d x) \left(-\frac{3 a \cos (c+d x)}{4 b \left(b^2-a^2\right) (a+b \sin (c+d x))}-\frac{\cos (c+d x)}{2 b (a+b \sin (c+d x))^2}\right) (e \cos (c+d x))^{5/2}}{d}+\frac{3 \left(-\frac{a \left(a+b \sqrt{1-\cos ^2(c+d x)}\right) \left(8 F_1\left(\frac{3}{4};-\frac{1}{2},1;\frac{7}{4};\cos ^2(c+d x),\frac{b^2 \cos ^2(c+d x)}{b^2-a^2}\right) \cos ^{\frac{3}{2}}(c+d x) b^{5/2}+3 \sqrt{2} a \left(a^2-b^2\right)^{3/4} \left(2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{b} \sqrt{\cos (c+d x)}}{\sqrt[4]{a^2-b^2}}\right)-2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{b} \sqrt{\cos (c+d x)}}{\sqrt[4]{a^2-b^2}}+1\right)-\log \left(b \cos (c+d x)-\sqrt{2} \sqrt{b} \sqrt[4]{a^2-b^2} \sqrt{\cos (c+d x)}+\sqrt{a^2-b^2}\right)+\log \left(b \cos (c+d x)+\sqrt{2} \sqrt{b} \sqrt[4]{a^2-b^2} \sqrt{\cos (c+d x)}+\sqrt{a^2-b^2}\right)\right)\right) \sin ^2(c+d x)}{12 b^{3/2} \left(b^2-a^2\right) \left(1-\cos ^2(c+d x)\right) (a+b \sin (c+d x))}-\frac{4 b \left(a+b \sqrt{1-\cos ^2(c+d x)}\right) \left(\frac{a F_1\left(\frac{3}{4};\frac{1}{2},1;\frac{7}{4};\cos ^2(c+d x),\frac{b^2 \cos ^2(c+d x)}{b^2-a^2}\right) \cos ^{\frac{3}{2}}(c+d x)}{3 \left(a^2-b^2\right)}+\frac{\left(\frac{1}{8}+\frac{i}{8}\right) \left(2 \tan ^{-1}\left(1-\frac{(1+i) \sqrt{b} \sqrt{\cos (c+d x)}}{\sqrt[4]{b^2-a^2}}\right)-2 \tan ^{-1}\left(\frac{(1+i) \sqrt{b} \sqrt{\cos (c+d x)}}{\sqrt[4]{b^2-a^2}}+1\right)-\log \left(i b \cos (c+d x)-(1+i) \sqrt{b} \sqrt[4]{b^2-a^2} \sqrt{\cos (c+d x)}+\sqrt{b^2-a^2}\right)+\log \left(i b \cos (c+d x)+(1+i) \sqrt{b} \sqrt[4]{b^2-a^2} \sqrt{\cos (c+d x)}+\sqrt{b^2-a^2}\right)\right)}{\sqrt{b} \sqrt[4]{b^2-a^2}}\right) \sin (c+d x)}{\sqrt{1-\cos ^2(c+d x)} (a+b \sin (c+d x))}\right) (e \cos (c+d x))^{5/2}}{8 (a-b) b (a+b) d \cos ^{\frac{5}{2}}(c+d x)}","\frac{3 a e^2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{4 b^2 d \left(a^2-b^2\right) \sqrt{\cos (c+d x)}}+\frac{3 a e (e \cos (c+d x))^{3/2}}{4 b d \left(a^2-b^2\right) (a+b \sin (c+d x))}+\frac{3 e^{5/2} \left(a^2-2 b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{8 b^{5/2} d \left(b^2-a^2\right)^{5/4}}-\frac{3 e^{5/2} \left(a^2-2 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{8 b^{5/2} d \left(b^2-a^2\right)^{5/4}}-\frac{3 a e^3 \left(a^2-2 b^2\right) \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{8 b^3 d \left(a^2-b^2\right) \left(b-\sqrt{b^2-a^2}\right) \sqrt{e \cos (c+d x)}}-\frac{3 a e^3 \left(a^2-2 b^2\right) \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{8 b^3 d \left(a^2-b^2\right) \left(\sqrt{b^2-a^2}+b\right) \sqrt{e \cos (c+d x)}}-\frac{e (e \cos (c+d x))^{3/2}}{2 b d (a+b \sin (c+d x))^2}",1,"((e*Cos[c + d*x])^(5/2)*Sec[c + d*x]^2*(-1/2*Cos[c + d*x]/(b*(a + b*Sin[c + d*x])^2) - (3*a*Cos[c + d*x])/(4*b*(-a^2 + b^2)*(a + b*Sin[c + d*x]))))/d + (3*(e*Cos[c + d*x])^(5/2)*((-4*b*(a + b*Sqrt[1 - Cos[c + d*x]^2])*((a*AppellF1[3/4, 1/2, 1, 7/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)]*Cos[c + d*x]^(3/2))/(3*(a^2 - b^2)) + ((1/8 + I/8)*(2*ArcTan[1 - ((1 + I)*Sqrt[b]*Sqrt[Cos[c + d*x]])/(-a^2 + b^2)^(1/4)] - 2*ArcTan[1 + ((1 + I)*Sqrt[b]*Sqrt[Cos[c + d*x]])/(-a^2 + b^2)^(1/4)] - Log[Sqrt[-a^2 + b^2] - (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Cos[c + d*x]] + I*b*Cos[c + d*x]] + Log[Sqrt[-a^2 + b^2] + (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Cos[c + d*x]] + I*b*Cos[c + d*x]]))/(Sqrt[b]*(-a^2 + b^2)^(1/4)))*Sin[c + d*x])/(Sqrt[1 - Cos[c + d*x]^2]*(a + b*Sin[c + d*x])) - (a*(a + b*Sqrt[1 - Cos[c + d*x]^2])*(8*b^(5/2)*AppellF1[3/4, -1/2, 1, 7/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)]*Cos[c + d*x]^(3/2) + 3*Sqrt[2]*a*(a^2 - b^2)^(3/4)*(2*ArcTan[1 - (Sqrt[2]*Sqrt[b]*Sqrt[Cos[c + d*x]])/(a^2 - b^2)^(1/4)] - 2*ArcTan[1 + (Sqrt[2]*Sqrt[b]*Sqrt[Cos[c + d*x]])/(a^2 - b^2)^(1/4)] - Log[Sqrt[a^2 - b^2] - Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Cos[c + d*x]] + b*Cos[c + d*x]] + Log[Sqrt[a^2 - b^2] + Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Cos[c + d*x]] + b*Cos[c + d*x]]))*Sin[c + d*x]^2)/(12*b^(3/2)*(-a^2 + b^2)*(1 - Cos[c + d*x]^2)*(a + b*Sin[c + d*x]))))/(8*(a - b)*b*(a + b)*d*Cos[c + d*x]^(5/2))","C",0
599,1,1211,519,23.7483909,"\int \frac{(e \cos (c+d x))^{3/2}}{(a+b \sin (c+d x))^3} \, dx","Integrate[(e*Cos[c + d*x])^(3/2)/(a + b*Sin[c + d*x])^3,x]","\frac{(e \cos (c+d x))^{3/2} \sec (c+d x) \left(-\frac{a}{4 b \left(b^2-a^2\right) (a+b \sin (c+d x))}-\frac{1}{2 b (a+b \sin (c+d x))^2}\right)}{d}-\frac{(e \cos (c+d x))^{3/2} \left(\frac{4 b \left(a+b \sqrt{1-\cos ^2(c+d x)}\right) \left(\frac{5 a \left(a^2-b^2\right) F_1\left(\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\cos ^2(c+d x),\frac{b^2 \cos ^2(c+d x)}{b^2-a^2}\right) \sqrt{\cos (c+d x)}}{\sqrt{1-\cos ^2(c+d x)} \left(5 \left(a^2-b^2\right) F_1\left(\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\cos ^2(c+d x),\frac{b^2 \cos ^2(c+d x)}{b^2-a^2}\right)-2 \left(2 F_1\left(\frac{5}{4};\frac{1}{2},2;\frac{9}{4};\cos ^2(c+d x),\frac{b^2 \cos ^2(c+d x)}{b^2-a^2}\right) b^2+\left(b^2-a^2\right) F_1\left(\frac{5}{4};\frac{3}{2},1;\frac{9}{4};\cos ^2(c+d x),\frac{b^2 \cos ^2(c+d x)}{b^2-a^2}\right)\right) \cos ^2(c+d x)\right) \left(a^2+b^2 \left(\cos ^2(c+d x)-1\right)\right)}-\frac{\left(\frac{1}{8}-\frac{i}{8}\right) \sqrt{b} \left(2 \tan ^{-1}\left(1-\frac{(1+i) \sqrt{b} \sqrt{\cos (c+d x)}}{\sqrt[4]{b^2-a^2}}\right)-2 \tan ^{-1}\left(\frac{(1+i) \sqrt{b} \sqrt{\cos (c+d x)}}{\sqrt[4]{b^2-a^2}}+1\right)+\log \left(i b \cos (c+d x)-(1+i) \sqrt{b} \sqrt[4]{b^2-a^2} \sqrt{\cos (c+d x)}+\sqrt{b^2-a^2}\right)-\log \left(i b \cos (c+d x)+(1+i) \sqrt{b} \sqrt[4]{b^2-a^2} \sqrt{\cos (c+d x)}+\sqrt{b^2-a^2}\right)\right)}{\left(b^2-a^2\right)^{3/4}}\right) \sin (c+d x)}{\sqrt{1-\cos ^2(c+d x)} (a+b \sin (c+d x))}-\frac{2 a \left(a+b \sqrt{1-\cos ^2(c+d x)}\right) \left(\frac{5 b \left(a^2-b^2\right) \sqrt{\cos (c+d x)} \sqrt{1-\cos ^2(c+d x)} F_1\left(\frac{1}{4};-\frac{1}{2},1;\frac{5}{4};\cos ^2(c+d x),\frac{b^2 \cos ^2(c+d x)}{b^2-a^2}\right)}{\left(2 \left(2 F_1\left(\frac{5}{4};-\frac{1}{2},2;\frac{9}{4};\cos ^2(c+d x),\frac{b^2 \cos ^2(c+d x)}{b^2-a^2}\right) b^2+\left(a^2-b^2\right) F_1\left(\frac{5}{4};\frac{1}{2},1;\frac{9}{4};\cos ^2(c+d x),\frac{b^2 \cos ^2(c+d x)}{b^2-a^2}\right)\right) \cos ^2(c+d x)-5 \left(a^2-b^2\right) F_1\left(\frac{1}{4};-\frac{1}{2},1;\frac{5}{4};\cos ^2(c+d x),\frac{b^2 \cos ^2(c+d x)}{b^2-a^2}\right)\right) \left(a^2+b^2 \left(\cos ^2(c+d x)-1\right)\right)}+\frac{a \left(-2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{b} \sqrt{\cos (c+d x)}}{\sqrt[4]{a^2-b^2}}\right)+2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{b} \sqrt{\cos (c+d x)}}{\sqrt[4]{a^2-b^2}}+1\right)-\log \left(b \cos (c+d x)-\sqrt{2} \sqrt{b} \sqrt[4]{a^2-b^2} \sqrt{\cos (c+d x)}+\sqrt{a^2-b^2}\right)+\log \left(b \cos (c+d x)+\sqrt{2} \sqrt{b} \sqrt[4]{a^2-b^2} \sqrt{\cos (c+d x)}+\sqrt{a^2-b^2}\right)\right)}{4 \sqrt{2} \sqrt{b} \left(a^2-b^2\right)^{3/4}}\right) \sin ^2(c+d x)}{\left(1-\cos ^2(c+d x)\right) (a+b \sin (c+d x))}\right)}{8 (a-b) b (a+b) d \cos ^{\frac{3}{2}}(c+d x)}","-\frac{a e^2 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{4 b^2 d \left(a^2-b^2\right) \sqrt{e \cos (c+d x)}}+\frac{a e^2 \left(a^2+2 b^2\right) \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{8 b^2 d \left(a^2-b^2\right) \left(a^2-b \left(b-\sqrt{b^2-a^2}\right)\right) \sqrt{e \cos (c+d x)}}+\frac{a e^2 \left(a^2+2 b^2\right) \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{8 b^2 d \left(a^2-b^2\right) \left(a^2-b \left(\sqrt{b^2-a^2}+b\right)\right) \sqrt{e \cos (c+d x)}}+\frac{a e \sqrt{e \cos (c+d x)}}{4 b d \left(a^2-b^2\right) (a+b \sin (c+d x))}+\frac{e^{3/2} \left(a^2+2 b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{8 b^{3/2} d \left(b^2-a^2\right)^{7/4}}+\frac{e^{3/2} \left(a^2+2 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{8 b^{3/2} d \left(b^2-a^2\right)^{7/4}}-\frac{e \sqrt{e \cos (c+d x)}}{2 b d (a+b \sin (c+d x))^2}",1,"((e*Cos[c + d*x])^(3/2)*Sec[c + d*x]*(-1/2*1/(b*(a + b*Sin[c + d*x])^2) - a/(4*b*(-a^2 + b^2)*(a + b*Sin[c + d*x]))))/d - ((e*Cos[c + d*x])^(3/2)*((4*b*(a + b*Sqrt[1 - Cos[c + d*x]^2])*((5*a*(a^2 - b^2)*AppellF1[1/4, 1/2, 1, 5/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)]*Sqrt[Cos[c + d*x]])/(Sqrt[1 - Cos[c + d*x]^2]*(5*(a^2 - b^2)*AppellF1[1/4, 1/2, 1, 5/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)] - 2*(2*b^2*AppellF1[5/4, 1/2, 2, 9/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)] + (-a^2 + b^2)*AppellF1[5/4, 3/2, 1, 9/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)])*Cos[c + d*x]^2)*(a^2 + b^2*(-1 + Cos[c + d*x]^2))) - ((1/8 - I/8)*Sqrt[b]*(2*ArcTan[1 - ((1 + I)*Sqrt[b]*Sqrt[Cos[c + d*x]])/(-a^2 + b^2)^(1/4)] - 2*ArcTan[1 + ((1 + I)*Sqrt[b]*Sqrt[Cos[c + d*x]])/(-a^2 + b^2)^(1/4)] + Log[Sqrt[-a^2 + b^2] - (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Cos[c + d*x]] + I*b*Cos[c + d*x]] - Log[Sqrt[-a^2 + b^2] + (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Cos[c + d*x]] + I*b*Cos[c + d*x]]))/(-a^2 + b^2)^(3/4))*Sin[c + d*x])/(Sqrt[1 - Cos[c + d*x]^2]*(a + b*Sin[c + d*x])) - (2*a*(a + b*Sqrt[1 - Cos[c + d*x]^2])*((5*b*(a^2 - b^2)*AppellF1[1/4, -1/2, 1, 5/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)]*Sqrt[Cos[c + d*x]]*Sqrt[1 - Cos[c + d*x]^2])/((-5*(a^2 - b^2)*AppellF1[1/4, -1/2, 1, 5/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)] + 2*(2*b^2*AppellF1[5/4, -1/2, 2, 9/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)] + (a^2 - b^2)*AppellF1[5/4, 1/2, 1, 9/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)])*Cos[c + d*x]^2)*(a^2 + b^2*(-1 + Cos[c + d*x]^2))) + (a*(-2*ArcTan[1 - (Sqrt[2]*Sqrt[b]*Sqrt[Cos[c + d*x]])/(a^2 - b^2)^(1/4)] + 2*ArcTan[1 + (Sqrt[2]*Sqrt[b]*Sqrt[Cos[c + d*x]])/(a^2 - b^2)^(1/4)] - Log[Sqrt[a^2 - b^2] - Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Cos[c + d*x]] + b*Cos[c + d*x]] + Log[Sqrt[a^2 - b^2] + Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Cos[c + d*x]] + b*Cos[c + d*x]]))/(4*Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(3/4)))*Sin[c + d*x]^2)/((1 - Cos[c + d*x]^2)*(a + b*Sin[c + d*x]))))/(8*(a - b)*b*(a + b)*d*Cos[c + d*x]^(3/2))","C",0
600,1,837,514,24.3338892,"\int \frac{\sqrt{e \cos (c+d x)}}{(a+b \sin (c+d x))^3} \, dx","Integrate[Sqrt[e*Cos[c + d*x]]/(a + b*Sin[c + d*x])^3,x]","\frac{\sqrt{e \cos (c+d x)} \left(\frac{5 a b \cos (c+d x)}{4 \left(a^2-b^2\right)^2 (a+b \sin (c+d x))}+\frac{b \cos (c+d x)}{2 \left(a^2-b^2\right) (a+b \sin (c+d x))^2}\right)}{d}+\frac{\sqrt{e \cos (c+d x)} \left(-\frac{5 a \left(a+b \sqrt{1-\cos ^2(c+d x)}\right) \left(8 F_1\left(\frac{3}{4};-\frac{1}{2},1;\frac{7}{4};\cos ^2(c+d x),\frac{b^2 \cos ^2(c+d x)}{b^2-a^2}\right) \cos ^{\frac{3}{2}}(c+d x) b^{5/2}+3 \sqrt{2} a \left(a^2-b^2\right)^{3/4} \left(2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{b} \sqrt{\cos (c+d x)}}{\sqrt[4]{a^2-b^2}}\right)-2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{b} \sqrt{\cos (c+d x)}}{\sqrt[4]{a^2-b^2}}+1\right)-\log \left(b \cos (c+d x)-\sqrt{2} \sqrt{b} \sqrt[4]{a^2-b^2} \sqrt{\cos (c+d x)}+\sqrt{a^2-b^2}\right)+\log \left(b \cos (c+d x)+\sqrt{2} \sqrt{b} \sqrt[4]{a^2-b^2} \sqrt{\cos (c+d x)}+\sqrt{a^2-b^2}\right)\right)\right) \sin ^2(c+d x)}{12 \sqrt{b} \left(b^2-a^2\right) \left(1-\cos ^2(c+d x)\right) (a+b \sin (c+d x))}-\frac{2 \left(8 a^2+2 b^2\right) \left(a+b \sqrt{1-\cos ^2(c+d x)}\right) \left(\frac{a F_1\left(\frac{3}{4};\frac{1}{2},1;\frac{7}{4};\cos ^2(c+d x),\frac{b^2 \cos ^2(c+d x)}{b^2-a^2}\right) \cos ^{\frac{3}{2}}(c+d x)}{3 \left(a^2-b^2\right)}+\frac{\left(\frac{1}{8}+\frac{i}{8}\right) \left(2 \tan ^{-1}\left(1-\frac{(1+i) \sqrt{b} \sqrt{\cos (c+d x)}}{\sqrt[4]{b^2-a^2}}\right)-2 \tan ^{-1}\left(\frac{(1+i) \sqrt{b} \sqrt{\cos (c+d x)}}{\sqrt[4]{b^2-a^2}}+1\right)-\log \left(i b \cos (c+d x)-(1+i) \sqrt{b} \sqrt[4]{b^2-a^2} \sqrt{\cos (c+d x)}+\sqrt{b^2-a^2}\right)+\log \left(i b \cos (c+d x)+(1+i) \sqrt{b} \sqrt[4]{b^2-a^2} \sqrt{\cos (c+d x)}+\sqrt{b^2-a^2}\right)\right)}{\sqrt{b} \sqrt[4]{b^2-a^2}}\right) \sin (c+d x)}{\sqrt{1-\cos ^2(c+d x)} (a+b \sin (c+d x))}\right)}{8 (a-b)^2 (a+b)^2 d \sqrt{\cos (c+d x)}}","\frac{5 a b (e \cos (c+d x))^{3/2}}{4 d e \left(a^2-b^2\right)^2 (a+b \sin (c+d x))}+\frac{b (e \cos (c+d x))^{3/2}}{2 d e \left(a^2-b^2\right) (a+b \sin (c+d x))^2}+\frac{\sqrt{e} \left(3 a^2+2 b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{8 \sqrt{b} d \left(b^2-a^2\right)^{9/4}}-\frac{\sqrt{e} \left(3 a^2+2 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{8 \sqrt{b} d \left(b^2-a^2\right)^{9/4}}+\frac{5 a E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{4 d \left(a^2-b^2\right)^2 \sqrt{\cos (c+d x)}}+\frac{a e \left(3 a^2+2 b^2\right) \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{8 b d \left(a^2-b^2\right)^2 \left(b-\sqrt{b^2-a^2}\right) \sqrt{e \cos (c+d x)}}+\frac{a e \left(3 a^2+2 b^2\right) \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{8 b d \left(a^2-b^2\right)^2 \left(\sqrt{b^2-a^2}+b\right) \sqrt{e \cos (c+d x)}}",1,"(Sqrt[e*Cos[c + d*x]]*((b*Cos[c + d*x])/(2*(a^2 - b^2)*(a + b*Sin[c + d*x])^2) + (5*a*b*Cos[c + d*x])/(4*(a^2 - b^2)^2*(a + b*Sin[c + d*x]))))/d + (Sqrt[e*Cos[c + d*x]]*((-2*(8*a^2 + 2*b^2)*(a + b*Sqrt[1 - Cos[c + d*x]^2])*((a*AppellF1[3/4, 1/2, 1, 7/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)]*Cos[c + d*x]^(3/2))/(3*(a^2 - b^2)) + ((1/8 + I/8)*(2*ArcTan[1 - ((1 + I)*Sqrt[b]*Sqrt[Cos[c + d*x]])/(-a^2 + b^2)^(1/4)] - 2*ArcTan[1 + ((1 + I)*Sqrt[b]*Sqrt[Cos[c + d*x]])/(-a^2 + b^2)^(1/4)] - Log[Sqrt[-a^2 + b^2] - (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Cos[c + d*x]] + I*b*Cos[c + d*x]] + Log[Sqrt[-a^2 + b^2] + (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Cos[c + d*x]] + I*b*Cos[c + d*x]]))/(Sqrt[b]*(-a^2 + b^2)^(1/4)))*Sin[c + d*x])/(Sqrt[1 - Cos[c + d*x]^2]*(a + b*Sin[c + d*x])) - (5*a*(a + b*Sqrt[1 - Cos[c + d*x]^2])*(8*b^(5/2)*AppellF1[3/4, -1/2, 1, 7/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)]*Cos[c + d*x]^(3/2) + 3*Sqrt[2]*a*(a^2 - b^2)^(3/4)*(2*ArcTan[1 - (Sqrt[2]*Sqrt[b]*Sqrt[Cos[c + d*x]])/(a^2 - b^2)^(1/4)] - 2*ArcTan[1 + (Sqrt[2]*Sqrt[b]*Sqrt[Cos[c + d*x]])/(a^2 - b^2)^(1/4)] - Log[Sqrt[a^2 - b^2] - Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Cos[c + d*x]] + b*Cos[c + d*x]] + Log[Sqrt[a^2 - b^2] + Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Cos[c + d*x]] + b*Cos[c + d*x]]))*Sin[c + d*x]^2)/(12*Sqrt[b]*(-a^2 + b^2)*(1 - Cos[c + d*x]^2)*(a + b*Sin[c + d*x]))))/(8*(a - b)^2*(a + b)^2*d*Sqrt[Cos[c + d*x]])","C",0
601,1,1226,520,24.5476894,"\int \frac{1}{\sqrt{e \cos (c+d x)} (a+b \sin (c+d x))^3} \, dx","Integrate[1/(Sqrt[e*Cos[c + d*x]]*(a + b*Sin[c + d*x])^3),x]","\frac{\cos (c+d x) \left(\frac{7 a b}{4 \left(a^2-b^2\right)^2 (a+b \sin (c+d x))}+\frac{b}{2 \left(a^2-b^2\right) (a+b \sin (c+d x))^2}\right)}{d \sqrt{e \cos (c+d x)}}+\frac{\sqrt{\cos (c+d x)} \left(\frac{14 a b \left(a+b \sqrt{1-\cos ^2(c+d x)}\right) \left(\frac{5 b \left(a^2-b^2\right) \sqrt{\cos (c+d x)} \sqrt{1-\cos ^2(c+d x)} F_1\left(\frac{1}{4};-\frac{1}{2},1;\frac{5}{4};\cos ^2(c+d x),\frac{b^2 \cos ^2(c+d x)}{b^2-a^2}\right)}{\left(2 \left(2 F_1\left(\frac{5}{4};-\frac{1}{2},2;\frac{9}{4};\cos ^2(c+d x),\frac{b^2 \cos ^2(c+d x)}{b^2-a^2}\right) b^2+\left(a^2-b^2\right) F_1\left(\frac{5}{4};\frac{1}{2},1;\frac{9}{4};\cos ^2(c+d x),\frac{b^2 \cos ^2(c+d x)}{b^2-a^2}\right)\right) \cos ^2(c+d x)-5 \left(a^2-b^2\right) F_1\left(\frac{1}{4};-\frac{1}{2},1;\frac{5}{4};\cos ^2(c+d x),\frac{b^2 \cos ^2(c+d x)}{b^2-a^2}\right)\right) \left(a^2+b^2 \left(\cos ^2(c+d x)-1\right)\right)}+\frac{a \left(-2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{b} \sqrt{\cos (c+d x)}}{\sqrt[4]{a^2-b^2}}\right)+2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{b} \sqrt{\cos (c+d x)}}{\sqrt[4]{a^2-b^2}}+1\right)-\log \left(b \cos (c+d x)-\sqrt{2} \sqrt{b} \sqrt[4]{a^2-b^2} \sqrt{\cos (c+d x)}+\sqrt{a^2-b^2}\right)+\log \left(b \cos (c+d x)+\sqrt{2} \sqrt{b} \sqrt[4]{a^2-b^2} \sqrt{\cos (c+d x)}+\sqrt{a^2-b^2}\right)\right)}{4 \sqrt{2} \sqrt{b} \left(a^2-b^2\right)^{3/4}}\right) \sin ^2(c+d x)}{\left(1-\cos ^2(c+d x)\right) (a+b \sin (c+d x))}-\frac{2 \left(8 a^2+6 b^2\right) \left(a+b \sqrt{1-\cos ^2(c+d x)}\right) \left(\frac{5 a \left(a^2-b^2\right) F_1\left(\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\cos ^2(c+d x),\frac{b^2 \cos ^2(c+d x)}{b^2-a^2}\right) \sqrt{\cos (c+d x)}}{\sqrt{1-\cos ^2(c+d x)} \left(5 \left(a^2-b^2\right) F_1\left(\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\cos ^2(c+d x),\frac{b^2 \cos ^2(c+d x)}{b^2-a^2}\right)-2 \left(2 F_1\left(\frac{5}{4};\frac{1}{2},2;\frac{9}{4};\cos ^2(c+d x),\frac{b^2 \cos ^2(c+d x)}{b^2-a^2}\right) b^2+\left(b^2-a^2\right) F_1\left(\frac{5}{4};\frac{3}{2},1;\frac{9}{4};\cos ^2(c+d x),\frac{b^2 \cos ^2(c+d x)}{b^2-a^2}\right)\right) \cos ^2(c+d x)\right) \left(a^2+b^2 \left(\cos ^2(c+d x)-1\right)\right)}-\frac{\left(\frac{1}{8}-\frac{i}{8}\right) \sqrt{b} \left(2 \tan ^{-1}\left(1-\frac{(1+i) \sqrt{b} \sqrt{\cos (c+d x)}}{\sqrt[4]{b^2-a^2}}\right)-2 \tan ^{-1}\left(\frac{(1+i) \sqrt{b} \sqrt{\cos (c+d x)}}{\sqrt[4]{b^2-a^2}}+1\right)+\log \left(i b \cos (c+d x)-(1+i) \sqrt{b} \sqrt[4]{b^2-a^2} \sqrt{\cos (c+d x)}+\sqrt{b^2-a^2}\right)-\log \left(i b \cos (c+d x)+(1+i) \sqrt{b} \sqrt[4]{b^2-a^2} \sqrt{\cos (c+d x)}+\sqrt{b^2-a^2}\right)\right)}{\left(b^2-a^2\right)^{3/4}}\right) \sin (c+d x)}{\sqrt{1-\cos ^2(c+d x)} (a+b \sin (c+d x))}\right)}{8 (a-b)^2 (a+b)^2 d \sqrt{e \cos (c+d x)}}","\frac{7 a b \sqrt{e \cos (c+d x)}}{4 d e \left(a^2-b^2\right)^2 (a+b \sin (c+d x))}+\frac{b \sqrt{e \cos (c+d x)}}{2 d e \left(a^2-b^2\right) (a+b \sin (c+d x))^2}-\frac{3 \sqrt{b} \left(5 a^2+2 b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{8 d \sqrt{e} \left(b^2-a^2\right)^{11/4}}-\frac{3 \sqrt{b} \left(5 a^2+2 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{8 d \sqrt{e} \left(b^2-a^2\right)^{11/4}}-\frac{7 a \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{4 d \left(a^2-b^2\right)^2 \sqrt{e \cos (c+d x)}}+\frac{3 a \left(5 a^2+2 b^2\right) \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{8 d \left(a^2-b^2\right)^2 \left(a^2-b \left(b-\sqrt{b^2-a^2}\right)\right) \sqrt{e \cos (c+d x)}}+\frac{3 a \left(5 a^2+2 b^2\right) \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{8 d \left(a^2-b^2\right)^2 \left(a^2-b \left(\sqrt{b^2-a^2}+b\right)\right) \sqrt{e \cos (c+d x)}}",1,"(Cos[c + d*x]*(b/(2*(a^2 - b^2)*(a + b*Sin[c + d*x])^2) + (7*a*b)/(4*(a^2 - b^2)^2*(a + b*Sin[c + d*x]))))/(d*Sqrt[e*Cos[c + d*x]]) + (Sqrt[Cos[c + d*x]]*((-2*(8*a^2 + 6*b^2)*(a + b*Sqrt[1 - Cos[c + d*x]^2])*((5*a*(a^2 - b^2)*AppellF1[1/4, 1/2, 1, 5/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)]*Sqrt[Cos[c + d*x]])/(Sqrt[1 - Cos[c + d*x]^2]*(5*(a^2 - b^2)*AppellF1[1/4, 1/2, 1, 5/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)] - 2*(2*b^2*AppellF1[5/4, 1/2, 2, 9/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)] + (-a^2 + b^2)*AppellF1[5/4, 3/2, 1, 9/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)])*Cos[c + d*x]^2)*(a^2 + b^2*(-1 + Cos[c + d*x]^2))) - ((1/8 - I/8)*Sqrt[b]*(2*ArcTan[1 - ((1 + I)*Sqrt[b]*Sqrt[Cos[c + d*x]])/(-a^2 + b^2)^(1/4)] - 2*ArcTan[1 + ((1 + I)*Sqrt[b]*Sqrt[Cos[c + d*x]])/(-a^2 + b^2)^(1/4)] + Log[Sqrt[-a^2 + b^2] - (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Cos[c + d*x]] + I*b*Cos[c + d*x]] - Log[Sqrt[-a^2 + b^2] + (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Cos[c + d*x]] + I*b*Cos[c + d*x]]))/(-a^2 + b^2)^(3/4))*Sin[c + d*x])/(Sqrt[1 - Cos[c + d*x]^2]*(a + b*Sin[c + d*x])) + (14*a*b*(a + b*Sqrt[1 - Cos[c + d*x]^2])*((5*b*(a^2 - b^2)*AppellF1[1/4, -1/2, 1, 5/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)]*Sqrt[Cos[c + d*x]]*Sqrt[1 - Cos[c + d*x]^2])/((-5*(a^2 - b^2)*AppellF1[1/4, -1/2, 1, 5/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)] + 2*(2*b^2*AppellF1[5/4, -1/2, 2, 9/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)] + (a^2 - b^2)*AppellF1[5/4, 1/2, 1, 9/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)])*Cos[c + d*x]^2)*(a^2 + b^2*(-1 + Cos[c + d*x]^2))) + (a*(-2*ArcTan[1 - (Sqrt[2]*Sqrt[b]*Sqrt[Cos[c + d*x]])/(a^2 - b^2)^(1/4)] + 2*ArcTan[1 + (Sqrt[2]*Sqrt[b]*Sqrt[Cos[c + d*x]])/(a^2 - b^2)^(1/4)] - Log[Sqrt[a^2 - b^2] - Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Cos[c + d*x]] + b*Cos[c + d*x]] + Log[Sqrt[a^2 - b^2] + Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Cos[c + d*x]] + b*Cos[c + d*x]]))/(4*Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(3/4)))*Sin[c + d*x]^2)/((1 - Cos[c + d*x]^2)*(a + b*Sin[c + d*x]))))/(8*(a - b)^2*(a + b)^2*d*Sqrt[e*Cos[c + d*x]])","C",0
602,1,922,596,6.727009,"\int \frac{1}{(e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^3} \, dx","Integrate[1/((e*Cos[c + d*x])^(3/2)*(a + b*Sin[c + d*x])^3),x]","\frac{\cos ^2(c+d x) \left(-\frac{13 a \cos (c+d x) b^3}{4 \left(a^2-b^2\right)^3 (a+b \sin (c+d x))}-\frac{\cos (c+d x) b^3}{2 \left(a^2-b^2\right)^2 (a+b \sin (c+d x))^2}+\frac{2 \sec (c+d x) \left(\sin (c+d x) a^3-3 b a^2+3 b^2 \sin (c+d x) a-b^3\right)}{\left(a^2-b^2\right)^3}\right)}{d (e \cos (c+d x))^{3/2}}-\frac{\cos ^{\frac{3}{2}}(c+d x) \left(-\frac{\left(8 b a^3+37 b^3 a\right) \left(a+b \sqrt{1-\cos ^2(c+d x)}\right) \left(8 F_1\left(\frac{3}{4};-\frac{1}{2},1;\frac{7}{4};\cos ^2(c+d x),\frac{b^2 \cos ^2(c+d x)}{b^2-a^2}\right) \cos ^{\frac{3}{2}}(c+d x) b^{5/2}+3 \sqrt{2} a \left(a^2-b^2\right)^{3/4} \left(2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{b} \sqrt{\cos (c+d x)}}{\sqrt[4]{a^2-b^2}}\right)-2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{b} \sqrt{\cos (c+d x)}}{\sqrt[4]{a^2-b^2}}+1\right)-\log \left(b \cos (c+d x)-\sqrt{2} \sqrt{b} \sqrt[4]{a^2-b^2} \sqrt{\cos (c+d x)}+\sqrt{a^2-b^2}\right)+\log \left(b \cos (c+d x)+\sqrt{2} \sqrt{b} \sqrt[4]{a^2-b^2} \sqrt{\cos (c+d x)}+\sqrt{a^2-b^2}\right)\right)\right) \sin ^2(c+d x)}{12 b^{3/2} \left(b^2-a^2\right) \left(1-\cos ^2(c+d x)\right) (a+b \sin (c+d x))}-\frac{2 \left(8 a^4+72 b^2 a^2+10 b^4\right) \left(a+b \sqrt{1-\cos ^2(c+d x)}\right) \left(\frac{a F_1\left(\frac{3}{4};\frac{1}{2},1;\frac{7}{4};\cos ^2(c+d x),\frac{b^2 \cos ^2(c+d x)}{b^2-a^2}\right) \cos ^{\frac{3}{2}}(c+d x)}{3 \left(a^2-b^2\right)}+\frac{\left(\frac{1}{8}+\frac{i}{8}\right) \left(2 \tan ^{-1}\left(1-\frac{(1+i) \sqrt{b} \sqrt{\cos (c+d x)}}{\sqrt[4]{b^2-a^2}}\right)-2 \tan ^{-1}\left(\frac{(1+i) \sqrt{b} \sqrt{\cos (c+d x)}}{\sqrt[4]{b^2-a^2}}+1\right)-\log \left(i b \cos (c+d x)-(1+i) \sqrt{b} \sqrt[4]{b^2-a^2} \sqrt{\cos (c+d x)}+\sqrt{b^2-a^2}\right)+\log \left(i b \cos (c+d x)+(1+i) \sqrt{b} \sqrt[4]{b^2-a^2} \sqrt{\cos (c+d x)}+\sqrt{b^2-a^2}\right)\right)}{\sqrt{b} \sqrt[4]{b^2-a^2}}\right) \sin (c+d x)}{\sqrt{1-\cos ^2(c+d x)} (a+b \sin (c+d x))}\right)}{8 (a-b)^3 (a+b)^3 d (e \cos (c+d x))^{3/2}}","-\frac{a \left(8 a^2+37 b^2\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{4 d e^2 \left(a^2-b^2\right)^3 \sqrt{\cos (c+d x)}}+\frac{9 a b}{4 d e \left(a^2-b^2\right)^2 \sqrt{e \cos (c+d x)} (a+b \sin (c+d x))}+\frac{b}{2 d e \left(a^2-b^2\right) \sqrt{e \cos (c+d x)} (a+b \sin (c+d x))^2}-\frac{5 b \left(7 a^2+2 b^2\right)-a \left(8 a^2+37 b^2\right) \sin (c+d x)}{4 d e \left(a^2-b^2\right)^3 \sqrt{e \cos (c+d x)}}-\frac{5 a b \left(7 a^2+2 b^2\right) \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{8 d e \left(a^2-b^2\right)^3 \left(b-\sqrt{b^2-a^2}\right) \sqrt{e \cos (c+d x)}}-\frac{5 a b \left(7 a^2+2 b^2\right) \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{8 d e \left(a^2-b^2\right)^3 \left(\sqrt{b^2-a^2}+b\right) \sqrt{e \cos (c+d x)}}+\frac{5 b^{3/2} \left(7 a^2+2 b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{8 d e^{3/2} \left(b^2-a^2\right)^{13/4}}-\frac{5 b^{3/2} \left(7 a^2+2 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{8 d e^{3/2} \left(b^2-a^2\right)^{13/4}}",1,"-1/8*(Cos[c + d*x]^(3/2)*((-2*(8*a^4 + 72*a^2*b^2 + 10*b^4)*(a + b*Sqrt[1 - Cos[c + d*x]^2])*((a*AppellF1[3/4, 1/2, 1, 7/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)]*Cos[c + d*x]^(3/2))/(3*(a^2 - b^2)) + ((1/8 + I/8)*(2*ArcTan[1 - ((1 + I)*Sqrt[b]*Sqrt[Cos[c + d*x]])/(-a^2 + b^2)^(1/4)] - 2*ArcTan[1 + ((1 + I)*Sqrt[b]*Sqrt[Cos[c + d*x]])/(-a^2 + b^2)^(1/4)] - Log[Sqrt[-a^2 + b^2] - (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Cos[c + d*x]] + I*b*Cos[c + d*x]] + Log[Sqrt[-a^2 + b^2] + (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Cos[c + d*x]] + I*b*Cos[c + d*x]]))/(Sqrt[b]*(-a^2 + b^2)^(1/4)))*Sin[c + d*x])/(Sqrt[1 - Cos[c + d*x]^2]*(a + b*Sin[c + d*x])) - ((8*a^3*b + 37*a*b^3)*(a + b*Sqrt[1 - Cos[c + d*x]^2])*(8*b^(5/2)*AppellF1[3/4, -1/2, 1, 7/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)]*Cos[c + d*x]^(3/2) + 3*Sqrt[2]*a*(a^2 - b^2)^(3/4)*(2*ArcTan[1 - (Sqrt[2]*Sqrt[b]*Sqrt[Cos[c + d*x]])/(a^2 - b^2)^(1/4)] - 2*ArcTan[1 + (Sqrt[2]*Sqrt[b]*Sqrt[Cos[c + d*x]])/(a^2 - b^2)^(1/4)] - Log[Sqrt[a^2 - b^2] - Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Cos[c + d*x]] + b*Cos[c + d*x]] + Log[Sqrt[a^2 - b^2] + Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Cos[c + d*x]] + b*Cos[c + d*x]]))*Sin[c + d*x]^2)/(12*b^(3/2)*(-a^2 + b^2)*(1 - Cos[c + d*x]^2)*(a + b*Sin[c + d*x]))))/((a - b)^3*(a + b)^3*d*(e*Cos[c + d*x])^(3/2)) + (Cos[c + d*x]^2*(-1/2*(b^3*Cos[c + d*x])/((a^2 - b^2)^2*(a + b*Sin[c + d*x])^2) - (13*a*b^3*Cos[c + d*x])/(4*(a^2 - b^2)^3*(a + b*Sin[c + d*x])) + (2*Sec[c + d*x]*(-3*a^2*b - b^3 + a^3*Sin[c + d*x] + 3*a*b^2*Sin[c + d*x]))/(a^2 - b^2)^3))/(d*(e*Cos[c + d*x])^(3/2))","C",0
603,1,1308,614,23.4895947,"\int \frac{1}{(e \cos (c+d x))^{5/2} (a+b \sin (c+d x))^3} \, dx","Integrate[1/((e*Cos[c + d*x])^(5/2)*(a + b*Sin[c + d*x])^3),x]","\frac{\left(-\frac{15 a b^3}{4 \left(a^2-b^2\right)^3 (a+b \sin (c+d x))}-\frac{b^3}{2 \left(a^2-b^2\right)^2 (a+b \sin (c+d x))^2}+\frac{2 \sec ^2(c+d x) \left(\sin (c+d x) a^3-3 b a^2+3 b^2 \sin (c+d x) a-b^3\right)}{3 \left(a^2-b^2\right)^3}\right) \cos ^3(c+d x)}{d (e \cos (c+d x))^{5/2}}+\frac{\left(-\frac{2 \left(8 b a^3+69 b^3 a\right) \left(a+b \sqrt{1-\cos ^2(c+d x)}\right) \left(\frac{5 b \left(a^2-b^2\right) \sqrt{\cos (c+d x)} \sqrt{1-\cos ^2(c+d x)} F_1\left(\frac{1}{4};-\frac{1}{2},1;\frac{5}{4};\cos ^2(c+d x),\frac{b^2 \cos ^2(c+d x)}{b^2-a^2}\right)}{\left(2 \left(2 F_1\left(\frac{5}{4};-\frac{1}{2},2;\frac{9}{4};\cos ^2(c+d x),\frac{b^2 \cos ^2(c+d x)}{b^2-a^2}\right) b^2+\left(a^2-b^2\right) F_1\left(\frac{5}{4};\frac{1}{2},1;\frac{9}{4};\cos ^2(c+d x),\frac{b^2 \cos ^2(c+d x)}{b^2-a^2}\right)\right) \cos ^2(c+d x)-5 \left(a^2-b^2\right) F_1\left(\frac{1}{4};-\frac{1}{2},1;\frac{5}{4};\cos ^2(c+d x),\frac{b^2 \cos ^2(c+d x)}{b^2-a^2}\right)\right) \left(a^2+b^2 \left(\cos ^2(c+d x)-1\right)\right)}+\frac{a \left(-2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{b} \sqrt{\cos (c+d x)}}{\sqrt[4]{a^2-b^2}}\right)+2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{b} \sqrt{\cos (c+d x)}}{\sqrt[4]{a^2-b^2}}+1\right)-\log \left(b \cos (c+d x)-\sqrt{2} \sqrt{b} \sqrt[4]{a^2-b^2} \sqrt{\cos (c+d x)}+\sqrt{a^2-b^2}\right)+\log \left(b \cos (c+d x)+\sqrt{2} \sqrt{b} \sqrt[4]{a^2-b^2} \sqrt{\cos (c+d x)}+\sqrt{a^2-b^2}\right)\right)}{4 \sqrt{2} \sqrt{b} \left(a^2-b^2\right)^{3/4}}\right) \sin ^2(c+d x)}{\left(1-\cos ^2(c+d x)\right) (a+b \sin (c+d x))}-\frac{2 \left(8 a^4-120 b^2 a^2-42 b^4\right) \left(a+b \sqrt{1-\cos ^2(c+d x)}\right) \left(\frac{5 a \left(a^2-b^2\right) F_1\left(\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\cos ^2(c+d x),\frac{b^2 \cos ^2(c+d x)}{b^2-a^2}\right) \sqrt{\cos (c+d x)}}{\sqrt{1-\cos ^2(c+d x)} \left(5 \left(a^2-b^2\right) F_1\left(\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\cos ^2(c+d x),\frac{b^2 \cos ^2(c+d x)}{b^2-a^2}\right)-2 \left(2 F_1\left(\frac{5}{4};\frac{1}{2},2;\frac{9}{4};\cos ^2(c+d x),\frac{b^2 \cos ^2(c+d x)}{b^2-a^2}\right) b^2+\left(b^2-a^2\right) F_1\left(\frac{5}{4};\frac{3}{2},1;\frac{9}{4};\cos ^2(c+d x),\frac{b^2 \cos ^2(c+d x)}{b^2-a^2}\right)\right) \cos ^2(c+d x)\right) \left(a^2+b^2 \left(\cos ^2(c+d x)-1\right)\right)}-\frac{\left(\frac{1}{8}-\frac{i}{8}\right) \sqrt{b} \left(2 \tan ^{-1}\left(1-\frac{(1+i) \sqrt{b} \sqrt{\cos (c+d x)}}{\sqrt[4]{b^2-a^2}}\right)-2 \tan ^{-1}\left(\frac{(1+i) \sqrt{b} \sqrt{\cos (c+d x)}}{\sqrt[4]{b^2-a^2}}+1\right)+\log \left(i b \cos (c+d x)-(1+i) \sqrt{b} \sqrt[4]{b^2-a^2} \sqrt{\cos (c+d x)}+\sqrt{b^2-a^2}\right)-\log \left(i b \cos (c+d x)+(1+i) \sqrt{b} \sqrt[4]{b^2-a^2} \sqrt{\cos (c+d x)}+\sqrt{b^2-a^2}\right)\right)}{\left(b^2-a^2\right)^{3/4}}\right) \sin (c+d x)}{\sqrt{1-\cos ^2(c+d x)} (a+b \sin (c+d x))}\right) \cos ^{\frac{5}{2}}(c+d x)}{24 (a-b)^3 (a+b)^3 d (e \cos (c+d x))^{5/2}}","\frac{a \left(8 a^2+69 b^2\right) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{12 d e^2 \left(a^2-b^2\right)^3 \sqrt{e \cos (c+d x)}}-\frac{7 a b^2 \left(9 a^2+2 b^2\right) \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{8 d e^2 \left(a^2-b^2\right)^3 \left(a^2-b \left(b-\sqrt{b^2-a^2}\right)\right) \sqrt{e \cos (c+d x)}}-\frac{7 a b^2 \left(9 a^2+2 b^2\right) \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{8 d e^2 \left(a^2-b^2\right)^3 \left(a^2-b \left(\sqrt{b^2-a^2}+b\right)\right) \sqrt{e \cos (c+d x)}}+\frac{11 a b}{4 d e \left(a^2-b^2\right)^2 (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))}+\frac{b}{2 d e \left(a^2-b^2\right) (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^2}-\frac{7 b \left(9 a^2+2 b^2\right)-a \left(8 a^2+69 b^2\right) \sin (c+d x)}{12 d e \left(a^2-b^2\right)^3 (e \cos (c+d x))^{3/2}}-\frac{7 b^{5/2} \left(9 a^2+2 b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{8 d e^{5/2} \left(b^2-a^2\right)^{15/4}}-\frac{7 b^{5/2} \left(9 a^2+2 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{8 d e^{5/2} \left(b^2-a^2\right)^{15/4}}",1,"(Cos[c + d*x]^(5/2)*((-2*(8*a^4 - 120*a^2*b^2 - 42*b^4)*(a + b*Sqrt[1 - Cos[c + d*x]^2])*((5*a*(a^2 - b^2)*AppellF1[1/4, 1/2, 1, 5/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)]*Sqrt[Cos[c + d*x]])/(Sqrt[1 - Cos[c + d*x]^2]*(5*(a^2 - b^2)*AppellF1[1/4, 1/2, 1, 5/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)] - 2*(2*b^2*AppellF1[5/4, 1/2, 2, 9/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)] + (-a^2 + b^2)*AppellF1[5/4, 3/2, 1, 9/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)])*Cos[c + d*x]^2)*(a^2 + b^2*(-1 + Cos[c + d*x]^2))) - ((1/8 - I/8)*Sqrt[b]*(2*ArcTan[1 - ((1 + I)*Sqrt[b]*Sqrt[Cos[c + d*x]])/(-a^2 + b^2)^(1/4)] - 2*ArcTan[1 + ((1 + I)*Sqrt[b]*Sqrt[Cos[c + d*x]])/(-a^2 + b^2)^(1/4)] + Log[Sqrt[-a^2 + b^2] - (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Cos[c + d*x]] + I*b*Cos[c + d*x]] - Log[Sqrt[-a^2 + b^2] + (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Cos[c + d*x]] + I*b*Cos[c + d*x]]))/(-a^2 + b^2)^(3/4))*Sin[c + d*x])/(Sqrt[1 - Cos[c + d*x]^2]*(a + b*Sin[c + d*x])) - (2*(8*a^3*b + 69*a*b^3)*(a + b*Sqrt[1 - Cos[c + d*x]^2])*((5*b*(a^2 - b^2)*AppellF1[1/4, -1/2, 1, 5/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)]*Sqrt[Cos[c + d*x]]*Sqrt[1 - Cos[c + d*x]^2])/((-5*(a^2 - b^2)*AppellF1[1/4, -1/2, 1, 5/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)] + 2*(2*b^2*AppellF1[5/4, -1/2, 2, 9/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)] + (a^2 - b^2)*AppellF1[5/4, 1/2, 1, 9/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)])*Cos[c + d*x]^2)*(a^2 + b^2*(-1 + Cos[c + d*x]^2))) + (a*(-2*ArcTan[1 - (Sqrt[2]*Sqrt[b]*Sqrt[Cos[c + d*x]])/(a^2 - b^2)^(1/4)] + 2*ArcTan[1 + (Sqrt[2]*Sqrt[b]*Sqrt[Cos[c + d*x]])/(a^2 - b^2)^(1/4)] - Log[Sqrt[a^2 - b^2] - Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Cos[c + d*x]] + b*Cos[c + d*x]] + Log[Sqrt[a^2 - b^2] + Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Cos[c + d*x]] + b*Cos[c + d*x]]))/(4*Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(3/4)))*Sin[c + d*x]^2)/((1 - Cos[c + d*x]^2)*(a + b*Sin[c + d*x]))))/(24*(a - b)^3*(a + b)^3*d*(e*Cos[c + d*x])^(5/2)) + (Cos[c + d*x]^3*(-1/2*b^3/((a^2 - b^2)^2*(a + b*Sin[c + d*x])^2) - (15*a*b^3)/(4*(a^2 - b^2)^3*(a + b*Sin[c + d*x])) + (2*Sec[c + d*x]^2*(-3*a^2*b - b^3 + a^3*Sin[c + d*x] + 3*a*b^2*Sin[c + d*x]))/(3*(a^2 - b^2)^3)))/(d*(e*Cos[c + d*x])^(5/2))","C",0
604,1,1014,685,6.90425,"\int \frac{1}{(e \cos (c+d x))^{7/2} (a+b \sin (c+d x))^3} \, dx","Integrate[1/((e*Cos[c + d*x])^(7/2)*(a + b*Sin[c + d*x])^3),x]","\frac{\cos ^4(c+d x) \left(\frac{21 a \cos (c+d x) b^5}{4 \left(a^2-b^2\right)^4 (a+b \sin (c+d x))}+\frac{\cos (c+d x) b^5}{2 \left(a^2-b^2\right)^3 (a+b \sin (c+d x))^2}+\frac{2 \sec ^3(c+d x) \left(\sin (c+d x) a^3-3 b a^2+3 b^2 \sin (c+d x) a-b^3\right)}{5 \left(a^2-b^2\right)^3}+\frac{2 \sec (c+d x) \left(3 \sin (c+d x) a^5-24 b^2 \sin (c+d x) a^3+50 b^3 a^2-39 b^4 \sin (c+d x) a+10 b^5\right)}{5 \left(a^2-b^2\right)^4}\right)}{d (e \cos (c+d x))^{7/2}}-\frac{3 \cos ^{\frac{7}{2}}(c+d x) \left(-\frac{\left(8 b a^5-64 b^3 a^3-139 b^5 a\right) \left(a+b \sqrt{1-\cos ^2(c+d x)}\right) \left(8 F_1\left(\frac{3}{4};-\frac{1}{2},1;\frac{7}{4};\cos ^2(c+d x),\frac{b^2 \cos ^2(c+d x)}{b^2-a^2}\right) \cos ^{\frac{3}{2}}(c+d x) b^{5/2}+3 \sqrt{2} a \left(a^2-b^2\right)^{3/4} \left(2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{b} \sqrt{\cos (c+d x)}}{\sqrt[4]{a^2-b^2}}\right)-2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{b} \sqrt{\cos (c+d x)}}{\sqrt[4]{a^2-b^2}}+1\right)-\log \left(b \cos (c+d x)-\sqrt{2} \sqrt{b} \sqrt[4]{a^2-b^2} \sqrt{\cos (c+d x)}+\sqrt{a^2-b^2}\right)+\log \left(b \cos (c+d x)+\sqrt{2} \sqrt{b} \sqrt[4]{a^2-b^2} \sqrt{\cos (c+d x)}+\sqrt{a^2-b^2}\right)\right)\right) \sin ^2(c+d x)}{12 b^{3/2} \left(b^2-a^2\right) \left(1-\cos ^2(c+d x)\right) (a+b \sin (c+d x))}-\frac{2 \left(8 a^6-64 b^2 a^4-304 b^4 a^2-30 b^6\right) \left(a+b \sqrt{1-\cos ^2(c+d x)}\right) \left(\frac{a F_1\left(\frac{3}{4};\frac{1}{2},1;\frac{7}{4};\cos ^2(c+d x),\frac{b^2 \cos ^2(c+d x)}{b^2-a^2}\right) \cos ^{\frac{3}{2}}(c+d x)}{3 \left(a^2-b^2\right)}+\frac{\left(\frac{1}{8}+\frac{i}{8}\right) \left(2 \tan ^{-1}\left(1-\frac{(1+i) \sqrt{b} \sqrt{\cos (c+d x)}}{\sqrt[4]{b^2-a^2}}\right)-2 \tan ^{-1}\left(\frac{(1+i) \sqrt{b} \sqrt{\cos (c+d x)}}{\sqrt[4]{b^2-a^2}}+1\right)-\log \left(i b \cos (c+d x)-(1+i) \sqrt{b} \sqrt[4]{b^2-a^2} \sqrt{\cos (c+d x)}+\sqrt{b^2-a^2}\right)+\log \left(i b \cos (c+d x)+(1+i) \sqrt{b} \sqrt[4]{b^2-a^2} \sqrt{\cos (c+d x)}+\sqrt{b^2-a^2}\right)\right)}{\sqrt{b} \sqrt[4]{b^2-a^2}}\right) \sin (c+d x)}{\sqrt{1-\cos ^2(c+d x)} (a+b \sin (c+d x))}\right)}{40 (a-b)^4 (a+b)^4 d (e \cos (c+d x))^{7/2}}","\frac{13 a b}{4 d e \left(a^2-b^2\right)^2 (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))}+\frac{b}{2 d e \left(a^2-b^2\right) (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))^2}-\frac{9 b \left(11 a^2+2 b^2\right)-a \left(8 a^2+109 b^2\right) \sin (c+d x)}{20 d e \left(a^2-b^2\right)^3 (e \cos (c+d x))^{5/2}}+\frac{9 b^{7/2} \left(11 a^2+2 b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{8 d e^{7/2} \left(b^2-a^2\right)^{17/4}}-\frac{9 b^{7/2} \left(11 a^2+2 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{8 d e^{7/2} \left(b^2-a^2\right)^{17/4}}+\frac{9 a b^3 \left(11 a^2+2 b^2\right) \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{8 d e^3 \left(a^2-b^2\right)^4 \left(b-\sqrt{b^2-a^2}\right) \sqrt{e \cos (c+d x)}}+\frac{9 a b^3 \left(11 a^2+2 b^2\right) \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{8 d e^3 \left(a^2-b^2\right)^4 \left(\sqrt{b^2-a^2}+b\right) \sqrt{e \cos (c+d x)}}-\frac{3 a \left(8 a^4-64 a^2 b^2-139 b^4\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{20 d e^4 \left(a^2-b^2\right)^4 \sqrt{\cos (c+d x)}}+\frac{3 \left(15 b^3 \left(11 a^2+2 b^2\right)+a \left(8 a^4-64 a^2 b^2-139 b^4\right) \sin (c+d x)\right)}{20 d e^3 \left(a^2-b^2\right)^4 \sqrt{e \cos (c+d x)}}",1,"(-3*Cos[c + d*x]^(7/2)*((-2*(8*a^6 - 64*a^4*b^2 - 304*a^2*b^4 - 30*b^6)*(a + b*Sqrt[1 - Cos[c + d*x]^2])*((a*AppellF1[3/4, 1/2, 1, 7/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)]*Cos[c + d*x]^(3/2))/(3*(a^2 - b^2)) + ((1/8 + I/8)*(2*ArcTan[1 - ((1 + I)*Sqrt[b]*Sqrt[Cos[c + d*x]])/(-a^2 + b^2)^(1/4)] - 2*ArcTan[1 + ((1 + I)*Sqrt[b]*Sqrt[Cos[c + d*x]])/(-a^2 + b^2)^(1/4)] - Log[Sqrt[-a^2 + b^2] - (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Cos[c + d*x]] + I*b*Cos[c + d*x]] + Log[Sqrt[-a^2 + b^2] + (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Cos[c + d*x]] + I*b*Cos[c + d*x]]))/(Sqrt[b]*(-a^2 + b^2)^(1/4)))*Sin[c + d*x])/(Sqrt[1 - Cos[c + d*x]^2]*(a + b*Sin[c + d*x])) - ((8*a^5*b - 64*a^3*b^3 - 139*a*b^5)*(a + b*Sqrt[1 - Cos[c + d*x]^2])*(8*b^(5/2)*AppellF1[3/4, -1/2, 1, 7/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)]*Cos[c + d*x]^(3/2) + 3*Sqrt[2]*a*(a^2 - b^2)^(3/4)*(2*ArcTan[1 - (Sqrt[2]*Sqrt[b]*Sqrt[Cos[c + d*x]])/(a^2 - b^2)^(1/4)] - 2*ArcTan[1 + (Sqrt[2]*Sqrt[b]*Sqrt[Cos[c + d*x]])/(a^2 - b^2)^(1/4)] - Log[Sqrt[a^2 - b^2] - Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Cos[c + d*x]] + b*Cos[c + d*x]] + Log[Sqrt[a^2 - b^2] + Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Cos[c + d*x]] + b*Cos[c + d*x]]))*Sin[c + d*x]^2)/(12*b^(3/2)*(-a^2 + b^2)*(1 - Cos[c + d*x]^2)*(a + b*Sin[c + d*x]))))/(40*(a - b)^4*(a + b)^4*d*(e*Cos[c + d*x])^(7/2)) + (Cos[c + d*x]^4*((b^5*Cos[c + d*x])/(2*(a^2 - b^2)^3*(a + b*Sin[c + d*x])^2) + (21*a*b^5*Cos[c + d*x])/(4*(a^2 - b^2)^4*(a + b*Sin[c + d*x])) + (2*Sec[c + d*x]^3*(-3*a^2*b - b^3 + a^3*Sin[c + d*x] + 3*a*b^2*Sin[c + d*x]))/(5*(a^2 - b^2)^3) + (2*Sec[c + d*x]*(50*a^2*b^3 + 10*b^5 + 3*a^5*Sin[c + d*x] - 24*a^3*b^2*Sin[c + d*x] - 39*a*b^4*Sin[c + d*x]))/(5*(a^2 - b^2)^4)))/(d*(e*Cos[c + d*x])^(7/2))","C",0
605,1,2102,671,27.8138847,"\int \frac{(e \cos (c+d x))^{15/2}}{(a+b \sin (c+d x))^4} \, dx","Integrate[(e*Cos[c + d*x])^(15/2)/(a + b*Sin[c + d*x])^4,x]","\text{Result too large to show}","\frac{13 e^7 \sqrt{e \cos (c+d x)} \left(21 a \left(11 a^2-6 b^2\right)-b \left(77 a^2-20 b^2\right) \sin (c+d x)\right)}{56 b^7 d}-\frac{39 e^5 (e \cos (c+d x))^{5/2} \left(77 a^2+22 a b \sin (c+d x)-20 b^2\right)}{280 b^5 d (a+b \sin (c+d x))}+\frac{39 a e^{15/2} \left(11 a^4-17 a^2 b^2+6 b^4\right) \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{16 b^{15/2} d \left(b^2-a^2\right)^{3/4}}+\frac{39 a e^{15/2} \left(11 a^4-17 a^2 b^2+6 b^4\right) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{16 b^{15/2} d \left(b^2-a^2\right)^{3/4}}+\frac{13 e^8 \left(231 a^4-203 a^2 b^2+20 b^4\right) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{56 b^8 d \sqrt{e \cos (c+d x)}}-\frac{39 a^2 e^8 \left(11 a^4-17 a^2 b^2+6 b^4\right) \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{16 b^8 d \left(a^2-b \left(b-\sqrt{b^2-a^2}\right)\right) \sqrt{e \cos (c+d x)}}-\frac{39 a^2 e^8 \left(11 a^4-17 a^2 b^2+6 b^4\right) \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{16 b^8 d \left(a^2-b \left(\sqrt{b^2-a^2}+b\right)\right) \sqrt{e \cos (c+d x)}}-\frac{13 e^3 (e \cos (c+d x))^{9/2} (11 a+4 b \sin (c+d x))}{84 b^3 d (a+b \sin (c+d x))^2}-\frac{e (e \cos (c+d x))^{13/2}}{3 b d (a+b \sin (c+d x))^3}",1,"((e*Cos[c + d*x])^(15/2)*((-2*(4410*a^3*b - 3418*a*b^3)*(a + b*Sqrt[1 - Cos[c + d*x]^2])*((5*a*(a^2 - b^2)*AppellF1[1/4, 1/2, 1, 5/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)]*Sqrt[Cos[c + d*x]])/(Sqrt[1 - Cos[c + d*x]^2]*(5*(a^2 - b^2)*AppellF1[1/4, 1/2, 1, 5/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)] - 2*(2*b^2*AppellF1[5/4, 1/2, 2, 9/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)] + (-a^2 + b^2)*AppellF1[5/4, 3/2, 1, 9/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)])*Cos[c + d*x]^2)*(a^2 + b^2*(-1 + Cos[c + d*x]^2))) - ((1/8 - I/8)*Sqrt[b]*(2*ArcTan[1 - ((1 + I)*Sqrt[b]*Sqrt[Cos[c + d*x]])/(-a^2 + b^2)^(1/4)] - 2*ArcTan[1 + ((1 + I)*Sqrt[b]*Sqrt[Cos[c + d*x]])/(-a^2 + b^2)^(1/4)] + Log[Sqrt[-a^2 + b^2] - (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Cos[c + d*x]] + I*b*Cos[c + d*x]] - Log[Sqrt[-a^2 + b^2] + (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Cos[c + d*x]] + I*b*Cos[c + d*x]]))/(-a^2 + b^2)^(3/4))*Sin[c + d*x])/(Sqrt[1 - Cos[c + d*x]^2]*(a + b*Sin[c + d*x])) + ((5600*a^3*b - 3472*a*b^3)*(a + b*Sqrt[1 - Cos[c + d*x]^2])*Cos[2*(c + d*x)]*(((1/2 - I/2)*(-2*a^2 + b^2)*ArcTan[1 - ((1 + I)*Sqrt[b]*Sqrt[Cos[c + d*x]])/(-a^2 + b^2)^(1/4)])/(b^(3/2)*(-a^2 + b^2)^(3/4)) - ((1/2 - I/2)*(-2*a^2 + b^2)*ArcTan[1 + ((1 + I)*Sqrt[b]*Sqrt[Cos[c + d*x]])/(-a^2 + b^2)^(1/4)])/(b^(3/2)*(-a^2 + b^2)^(3/4)) + (4*Sqrt[Cos[c + d*x]])/b - (4*a*AppellF1[5/4, 1/2, 1, 9/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)]*Cos[c + d*x]^(5/2))/(5*(a^2 - b^2)) + (10*a*(a^2 - b^2)*AppellF1[1/4, 1/2, 1, 5/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)]*Sqrt[Cos[c + d*x]])/(Sqrt[1 - Cos[c + d*x]^2]*(5*(a^2 - b^2)*AppellF1[1/4, 1/2, 1, 5/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)] - 2*(2*b^2*AppellF1[5/4, 1/2, 2, 9/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)] + (-a^2 + b^2)*AppellF1[5/4, 3/2, 1, 9/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)])*Cos[c + d*x]^2)*(a^2 + b^2*(-1 + Cos[c + d*x]^2))) + ((1/4 - I/4)*(-2*a^2 + b^2)*Log[Sqrt[-a^2 + b^2] - (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Cos[c + d*x]] + I*b*Cos[c + d*x]])/(b^(3/2)*(-a^2 + b^2)^(3/4)) - ((1/4 - I/4)*(-2*a^2 + b^2)*Log[Sqrt[-a^2 + b^2] + (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Cos[c + d*x]] + I*b*Cos[c + d*x]])/(b^(3/2)*(-a^2 + b^2)^(3/4)))*Sin[c + d*x])/(Sqrt[1 - Cos[c + d*x]^2]*(-1 + 2*Cos[c + d*x]^2)*(a + b*Sin[c + d*x])) - (2*(3815*a^4 - 6251*a^2*b^2 + 1300*b^4)*(a + b*Sqrt[1 - Cos[c + d*x]^2])*((5*b*(a^2 - b^2)*AppellF1[1/4, -1/2, 1, 5/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)]*Sqrt[Cos[c + d*x]]*Sqrt[1 - Cos[c + d*x]^2])/((-5*(a^2 - b^2)*AppellF1[1/4, -1/2, 1, 5/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)] + 2*(2*b^2*AppellF1[5/4, -1/2, 2, 9/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)] + (a^2 - b^2)*AppellF1[5/4, 1/2, 1, 9/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)])*Cos[c + d*x]^2)*(a^2 + b^2*(-1 + Cos[c + d*x]^2))) + (a*(-2*ArcTan[1 - (Sqrt[2]*Sqrt[b]*Sqrt[Cos[c + d*x]])/(a^2 - b^2)^(1/4)] + 2*ArcTan[1 + (Sqrt[2]*Sqrt[b]*Sqrt[Cos[c + d*x]])/(a^2 - b^2)^(1/4)] - Log[Sqrt[a^2 - b^2] - Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Cos[c + d*x]] + b*Cos[c + d*x]] + Log[Sqrt[a^2 - b^2] + Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Cos[c + d*x]] + b*Cos[c + d*x]]))/(4*Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(3/4)))*Sin[c + d*x]^2)/((1 - Cos[c + d*x]^2)*(a + b*Sin[c + d*x]))))/(560*b^7*d*Cos[c + d*x]^(15/2)) + ((e*Cos[c + d*x])^(15/2)*Sec[c + d*x]^7*((-4*a*Cos[2*(c + d*x)])/(5*b^5) + ((-280*a^2 + 79*b^2)*Sin[c + d*x])/(42*b^6) - (-a^2 + b^2)^3/(3*b^7*(a + b*Sin[c + d*x])^3) - (37*a*(a^2 - b^2)^2)/(12*b^7*(a + b*Sin[c + d*x])^2) + ((-a^2 + b^2)*(-393*a^2 + 76*b^2))/(24*b^7*(a + b*Sin[c + d*x])) + Sin[3*(c + d*x)]/(14*b^4)))/d","C",0
606,1,937,557,26.8463938,"\int \frac{(e \cos (c+d x))^{13/2}}{(a+b \sin (c+d x))^4} \, dx","Integrate[(e*Cos[c + d*x])^(13/2)/(a + b*Sin[c + d*x])^4,x]","\frac{(e \cos (c+d x))^{13/2} \sec ^6(c+d x) \left(-\frac{8 a \cos (c+d x)}{3 b^5}+\frac{\sin (2 (c+d x))}{5 b^4}+\frac{20 b^2 \cos (c+d x)-71 a^2 \cos (c+d x)}{8 b^5 (a+b \sin (c+d x))}+\frac{9 \left(a^3 \cos (c+d x)-a b^2 \cos (c+d x)\right)}{4 b^5 (a+b \sin (c+d x))^2}+\frac{-\cos (c+d x) a^4+2 b^2 \cos (c+d x) a^2-b^4 \cos (c+d x)}{3 b^5 (a+b \sin (c+d x))^3}\right)}{d}-\frac{77 (e \cos (c+d x))^{13/2} \left(-\frac{\left(15 a^2-4 b^2\right) \left(a+b \sqrt{1-\cos ^2(c+d x)}\right) \left(8 F_1\left(\frac{3}{4};-\frac{1}{2},1;\frac{7}{4};\cos ^2(c+d x),\frac{b^2 \cos ^2(c+d x)}{b^2-a^2}\right) \cos ^{\frac{3}{2}}(c+d x) b^{5/2}+3 \sqrt{2} a \left(a^2-b^2\right)^{3/4} \left(2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{b} \sqrt{\cos (c+d x)}}{\sqrt[4]{a^2-b^2}}\right)-2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{b} \sqrt{\cos (c+d x)}}{\sqrt[4]{a^2-b^2}}+1\right)-\log \left(b \cos (c+d x)-\sqrt{2} \sqrt{b} \sqrt[4]{a^2-b^2} \sqrt{\cos (c+d x)}+\sqrt{a^2-b^2}\right)+\log \left(b \cos (c+d x)+\sqrt{2} \sqrt{b} \sqrt[4]{a^2-b^2} \sqrt{\cos (c+d x)}+\sqrt{a^2-b^2}\right)\right)\right) \sin ^2(c+d x)}{12 b^{3/2} \left(b^2-a^2\right) \left(1-\cos ^2(c+d x)\right) (a+b \sin (c+d x))}-\frac{12 a b \left(a+b \sqrt{1-\cos ^2(c+d x)}\right) \left(\frac{a F_1\left(\frac{3}{4};\frac{1}{2},1;\frac{7}{4};\cos ^2(c+d x),\frac{b^2 \cos ^2(c+d x)}{b^2-a^2}\right) \cos ^{\frac{3}{2}}(c+d x)}{3 \left(a^2-b^2\right)}+\frac{\left(\frac{1}{8}+\frac{i}{8}\right) \left(2 \tan ^{-1}\left(1-\frac{(1+i) \sqrt{b} \sqrt{\cos (c+d x)}}{\sqrt[4]{b^2-a^2}}\right)-2 \tan ^{-1}\left(\frac{(1+i) \sqrt{b} \sqrt{\cos (c+d x)}}{\sqrt[4]{b^2-a^2}}+1\right)-\log \left(i b \cos (c+d x)-(1+i) \sqrt{b} \sqrt[4]{b^2-a^2} \sqrt{\cos (c+d x)}+\sqrt{b^2-a^2}\right)+\log \left(i b \cos (c+d x)+(1+i) \sqrt{b} \sqrt[4]{b^2-a^2} \sqrt{\cos (c+d x)}+\sqrt{b^2-a^2}\right)\right)}{\sqrt{b} \sqrt[4]{b^2-a^2}}\right) \sin (c+d x)}{\sqrt{1-\cos ^2(c+d x)} (a+b \sin (c+d x))}\right)}{80 b^5 d \cos ^{\frac{13}{2}}(c+d x)}","\frac{77 a e^{13/2} \left(3 a^2-2 b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{16 b^{13/2} d \sqrt[4]{b^2-a^2}}-\frac{77 a e^{13/2} \left(3 a^2-2 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{16 b^{13/2} d \sqrt[4]{b^2-a^2}}+\frac{77 a^2 e^7 \left(3 a^2-2 b^2\right) \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{16 b^7 d \left(b-\sqrt{b^2-a^2}\right) \sqrt{e \cos (c+d x)}}+\frac{77 a^2 e^7 \left(3 a^2-2 b^2\right) \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{16 b^7 d \left(\sqrt{b^2-a^2}+b\right) \sqrt{e \cos (c+d x)}}-\frac{77 e^6 \left(15 a^2-4 b^2\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{40 b^6 d \sqrt{\cos (c+d x)}}-\frac{77 e^5 (e \cos (c+d x))^{3/2} \left(15 a^2+6 a b \sin (c+d x)-4 b^2\right)}{120 b^5 d (a+b \sin (c+d x))}-\frac{11 e^3 (e \cos (c+d x))^{7/2} (9 a+4 b \sin (c+d x))}{60 b^3 d (a+b \sin (c+d x))^2}-\frac{e (e \cos (c+d x))^{11/2}}{3 b d (a+b \sin (c+d x))^3}",1,"(-77*(e*Cos[c + d*x])^(13/2)*((-12*a*b*(a + b*Sqrt[1 - Cos[c + d*x]^2])*((a*AppellF1[3/4, 1/2, 1, 7/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)]*Cos[c + d*x]^(3/2))/(3*(a^2 - b^2)) + ((1/8 + I/8)*(2*ArcTan[1 - ((1 + I)*Sqrt[b]*Sqrt[Cos[c + d*x]])/(-a^2 + b^2)^(1/4)] - 2*ArcTan[1 + ((1 + I)*Sqrt[b]*Sqrt[Cos[c + d*x]])/(-a^2 + b^2)^(1/4)] - Log[Sqrt[-a^2 + b^2] - (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Cos[c + d*x]] + I*b*Cos[c + d*x]] + Log[Sqrt[-a^2 + b^2] + (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Cos[c + d*x]] + I*b*Cos[c + d*x]]))/(Sqrt[b]*(-a^2 + b^2)^(1/4)))*Sin[c + d*x])/(Sqrt[1 - Cos[c + d*x]^2]*(a + b*Sin[c + d*x])) - ((15*a^2 - 4*b^2)*(a + b*Sqrt[1 - Cos[c + d*x]^2])*(8*b^(5/2)*AppellF1[3/4, -1/2, 1, 7/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)]*Cos[c + d*x]^(3/2) + 3*Sqrt[2]*a*(a^2 - b^2)^(3/4)*(2*ArcTan[1 - (Sqrt[2]*Sqrt[b]*Sqrt[Cos[c + d*x]])/(a^2 - b^2)^(1/4)] - 2*ArcTan[1 + (Sqrt[2]*Sqrt[b]*Sqrt[Cos[c + d*x]])/(a^2 - b^2)^(1/4)] - Log[Sqrt[a^2 - b^2] - Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Cos[c + d*x]] + b*Cos[c + d*x]] + Log[Sqrt[a^2 - b^2] + Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Cos[c + d*x]] + b*Cos[c + d*x]]))*Sin[c + d*x]^2)/(12*b^(3/2)*(-a^2 + b^2)*(1 - Cos[c + d*x]^2)*(a + b*Sin[c + d*x]))))/(80*b^5*d*Cos[c + d*x]^(13/2)) + ((e*Cos[c + d*x])^(13/2)*Sec[c + d*x]^6*((-8*a*Cos[c + d*x])/(3*b^5) + (-(a^4*Cos[c + d*x]) + 2*a^2*b^2*Cos[c + d*x] - b^4*Cos[c + d*x])/(3*b^5*(a + b*Sin[c + d*x])^3) + (9*(a^3*Cos[c + d*x] - a*b^2*Cos[c + d*x]))/(4*b^5*(a + b*Sin[c + d*x])^2) + (-71*a^2*Cos[c + d*x] + 20*b^2*Cos[c + d*x])/(8*b^5*(a + b*Sin[c + d*x])) + Sin[2*(c + d*x)]/(5*b^4)))/d","C",0
607,1,2020,571,26.4966554,"\int \frac{(e \cos (c+d x))^{11/2}}{(a+b \sin (c+d x))^4} \, dx","Integrate[(e*Cos[c + d*x])^(11/2)/(a + b*Sin[c + d*x])^4,x]","\text{Result too large to show}","-\frac{15 a e^{11/2} \left(7 a^2-6 b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{16 b^{11/2} d \left(b^2-a^2\right)^{3/4}}-\frac{15 a e^{11/2} \left(7 a^2-6 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{16 b^{11/2} d \left(b^2-a^2\right)^{3/4}}-\frac{5 e^6 \left(21 a^2-4 b^2\right) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{8 b^6 d \sqrt{e \cos (c+d x)}}+\frac{15 a^2 e^6 \left(7 a^2-6 b^2\right) \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{16 b^6 d \left(a^2-b \left(b-\sqrt{b^2-a^2}\right)\right) \sqrt{e \cos (c+d x)}}+\frac{15 a^2 e^6 \left(7 a^2-6 b^2\right) \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{16 b^6 d \left(a^2-b \left(\sqrt{b^2-a^2}+b\right)\right) \sqrt{e \cos (c+d x)}}-\frac{5 e^5 \sqrt{e \cos (c+d x)} \left(21 a^2+14 a b \sin (c+d x)-4 b^2\right)}{8 b^5 d (a+b \sin (c+d x))}-\frac{e^3 (e \cos (c+d x))^{5/2} (7 a+4 b \sin (c+d x))}{4 b^3 d (a+b \sin (c+d x))^2}-\frac{e (e \cos (c+d x))^{9/2}}{3 b d (a+b \sin (c+d x))^3}",1,"((e*Cos[c + d*x])^(11/2)*Sec[c + d*x]^5*((2*Sin[c + d*x])/(3*b^4) - (-a^2 + b^2)^2/(3*b^5*(a + b*Sin[c + d*x])^3) + (25*a*(a^2 - b^2))/(12*b^5*(a + b*Sin[c + d*x])^2) + (-165*a^2 + 52*b^2)/(24*b^5*(a + b*Sin[c + d*x]))))/d - ((e*Cos[c + d*x])^(11/2)*((-76*a*b*(a + b*Sqrt[1 - Cos[c + d*x]^2])*((5*a*(a^2 - b^2)*AppellF1[1/4, 1/2, 1, 5/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)]*Sqrt[Cos[c + d*x]])/(Sqrt[1 - Cos[c + d*x]^2]*(5*(a^2 - b^2)*AppellF1[1/4, 1/2, 1, 5/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)] - 2*(2*b^2*AppellF1[5/4, 1/2, 2, 9/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)] + (-a^2 + b^2)*AppellF1[5/4, 3/2, 1, 9/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)])*Cos[c + d*x]^2)*(a^2 + b^2*(-1 + Cos[c + d*x]^2))) - ((1/8 - I/8)*Sqrt[b]*(2*ArcTan[1 - ((1 + I)*Sqrt[b]*Sqrt[Cos[c + d*x]])/(-a^2 + b^2)^(1/4)] - 2*ArcTan[1 + ((1 + I)*Sqrt[b]*Sqrt[Cos[c + d*x]])/(-a^2 + b^2)^(1/4)] + Log[Sqrt[-a^2 + b^2] - (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Cos[c + d*x]] + I*b*Cos[c + d*x]] - Log[Sqrt[-a^2 + b^2] + (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Cos[c + d*x]] + I*b*Cos[c + d*x]]))/(-a^2 + b^2)^(3/4))*Sin[c + d*x])/(Sqrt[1 - Cos[c + d*x]^2]*(a + b*Sin[c + d*x])) + (32*a*b*(a + b*Sqrt[1 - Cos[c + d*x]^2])*Cos[2*(c + d*x)]*(((1/2 - I/2)*(-2*a^2 + b^2)*ArcTan[1 - ((1 + I)*Sqrt[b]*Sqrt[Cos[c + d*x]])/(-a^2 + b^2)^(1/4)])/(b^(3/2)*(-a^2 + b^2)^(3/4)) - ((1/2 - I/2)*(-2*a^2 + b^2)*ArcTan[1 + ((1 + I)*Sqrt[b]*Sqrt[Cos[c + d*x]])/(-a^2 + b^2)^(1/4)])/(b^(3/2)*(-a^2 + b^2)^(3/4)) + (4*Sqrt[Cos[c + d*x]])/b - (4*a*AppellF1[5/4, 1/2, 1, 9/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)]*Cos[c + d*x]^(5/2))/(5*(a^2 - b^2)) + (10*a*(a^2 - b^2)*AppellF1[1/4, 1/2, 1, 5/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)]*Sqrt[Cos[c + d*x]])/(Sqrt[1 - Cos[c + d*x]^2]*(5*(a^2 - b^2)*AppellF1[1/4, 1/2, 1, 5/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)] - 2*(2*b^2*AppellF1[5/4, 1/2, 2, 9/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)] + (-a^2 + b^2)*AppellF1[5/4, 3/2, 1, 9/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)])*Cos[c + d*x]^2)*(a^2 + b^2*(-1 + Cos[c + d*x]^2))) + ((1/4 - I/4)*(-2*a^2 + b^2)*Log[Sqrt[-a^2 + b^2] - (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Cos[c + d*x]] + I*b*Cos[c + d*x]])/(b^(3/2)*(-a^2 + b^2)^(3/4)) - ((1/4 - I/4)*(-2*a^2 + b^2)*Log[Sqrt[-a^2 + b^2] + (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Cos[c + d*x]] + I*b*Cos[c + d*x]])/(b^(3/2)*(-a^2 + b^2)^(3/4)))*Sin[c + d*x])/(Sqrt[1 - Cos[c + d*x]^2]*(-1 + 2*Cos[c + d*x]^2)*(a + b*Sin[c + d*x])) - (2*(41*a^2 - 20*b^2)*(a + b*Sqrt[1 - Cos[c + d*x]^2])*((5*b*(a^2 - b^2)*AppellF1[1/4, -1/2, 1, 5/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)]*Sqrt[Cos[c + d*x]]*Sqrt[1 - Cos[c + d*x]^2])/((-5*(a^2 - b^2)*AppellF1[1/4, -1/2, 1, 5/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)] + 2*(2*b^2*AppellF1[5/4, -1/2, 2, 9/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)] + (a^2 - b^2)*AppellF1[5/4, 1/2, 1, 9/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)])*Cos[c + d*x]^2)*(a^2 + b^2*(-1 + Cos[c + d*x]^2))) + (a*(-2*ArcTan[1 - (Sqrt[2]*Sqrt[b]*Sqrt[Cos[c + d*x]])/(a^2 - b^2)^(1/4)] + 2*ArcTan[1 + (Sqrt[2]*Sqrt[b]*Sqrt[Cos[c + d*x]])/(a^2 - b^2)^(1/4)] - Log[Sqrt[a^2 - b^2] - Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Cos[c + d*x]] + b*Cos[c + d*x]] + Log[Sqrt[a^2 - b^2] + Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Cos[c + d*x]] + b*Cos[c + d*x]]))/(4*Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(3/4)))*Sin[c + d*x]^2)/((1 - Cos[c + d*x]^2)*(a + b*Sin[c + d*x]))))/(16*b^5*d*Cos[c + d*x]^(11/2))","C",0
608,1,900,591,26.825871,"\int \frac{(e \cos (c+d x))^{9/2}}{(a+b \sin (c+d x))^4} \, dx","Integrate[(e*Cos[c + d*x])^(9/2)/(a + b*Sin[c + d*x])^4,x]","\frac{\sec ^4(c+d x) \left(-\frac{5 a \cos (c+d x)}{4 b^3 (a+b \sin (c+d x))^2}+\frac{12 b^2 \cos (c+d x)-19 a^2 \cos (c+d x)}{8 b^3 \left(b^2-a^2\right) (a+b \sin (c+d x))}+\frac{a^2 \cos (c+d x)-b^2 \cos (c+d x)}{3 b^3 (a+b \sin (c+d x))^3}\right) (e \cos (c+d x))^{9/2}}{d}+\frac{7 \left(-\frac{\left(5 a^2-4 b^2\right) \left(a+b \sqrt{1-\cos ^2(c+d x)}\right) \left(8 F_1\left(\frac{3}{4};-\frac{1}{2},1;\frac{7}{4};\cos ^2(c+d x),\frac{b^2 \cos ^2(c+d x)}{b^2-a^2}\right) \cos ^{\frac{3}{2}}(c+d x) b^{5/2}+3 \sqrt{2} a \left(a^2-b^2\right)^{3/4} \left(2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{b} \sqrt{\cos (c+d x)}}{\sqrt[4]{a^2-b^2}}\right)-2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{b} \sqrt{\cos (c+d x)}}{\sqrt[4]{a^2-b^2}}+1\right)-\log \left(b \cos (c+d x)-\sqrt{2} \sqrt{b} \sqrt[4]{a^2-b^2} \sqrt{\cos (c+d x)}+\sqrt{a^2-b^2}\right)+\log \left(b \cos (c+d x)+\sqrt{2} \sqrt{b} \sqrt[4]{a^2-b^2} \sqrt{\cos (c+d x)}+\sqrt{a^2-b^2}\right)\right)\right) \sin ^2(c+d x)}{12 b^{3/2} \left(b^2-a^2\right) \left(1-\cos ^2(c+d x)\right) (a+b \sin (c+d x))}-\frac{4 a b \left(a+b \sqrt{1-\cos ^2(c+d x)}\right) \left(\frac{a F_1\left(\frac{3}{4};\frac{1}{2},1;\frac{7}{4};\cos ^2(c+d x),\frac{b^2 \cos ^2(c+d x)}{b^2-a^2}\right) \cos ^{\frac{3}{2}}(c+d x)}{3 \left(a^2-b^2\right)}+\frac{\left(\frac{1}{8}+\frac{i}{8}\right) \left(2 \tan ^{-1}\left(1-\frac{(1+i) \sqrt{b} \sqrt{\cos (c+d x)}}{\sqrt[4]{b^2-a^2}}\right)-2 \tan ^{-1}\left(\frac{(1+i) \sqrt{b} \sqrt{\cos (c+d x)}}{\sqrt[4]{b^2-a^2}}+1\right)-\log \left(i b \cos (c+d x)-(1+i) \sqrt{b} \sqrt[4]{b^2-a^2} \sqrt{\cos (c+d x)}+\sqrt{b^2-a^2}\right)+\log \left(i b \cos (c+d x)+(1+i) \sqrt{b} \sqrt[4]{b^2-a^2} \sqrt{\cos (c+d x)}+\sqrt{b^2-a^2}\right)\right)}{\sqrt{b} \sqrt[4]{b^2-a^2}}\right) \sin (c+d x)}{\sqrt{1-\cos ^2(c+d x)} (a+b \sin (c+d x))}\right) (e \cos (c+d x))^{9/2}}{16 (a-b) b^3 (a+b) d \cos ^{\frac{9}{2}}(c+d x)}","\frac{7 a e^{9/2} \left(5 a^2-6 b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{16 b^{9/2} d \left(b^2-a^2\right)^{5/4}}-\frac{7 a e^{9/2} \left(5 a^2-6 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{16 b^{9/2} d \left(b^2-a^2\right)^{5/4}}-\frac{7 a^2 e^5 \left(5 a^2-6 b^2\right) \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{16 b^5 d \left(a^2-b^2\right) \left(b-\sqrt{b^2-a^2}\right) \sqrt{e \cos (c+d x)}}-\frac{7 a^2 e^5 \left(5 a^2-6 b^2\right) \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{16 b^5 d \left(a^2-b^2\right) \left(\sqrt{b^2-a^2}+b\right) \sqrt{e \cos (c+d x)}}+\frac{7 e^4 \left(5 a^2-4 b^2\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{8 b^4 d \left(a^2-b^2\right) \sqrt{\cos (c+d x)}}+\frac{7 e^3 \left(5 a^2-4 b^2\right) (e \cos (c+d x))^{3/2}}{8 b^3 d \left(a^2-b^2\right) (a+b \sin (c+d x))}-\frac{7 e^3 (e \cos (c+d x))^{3/2} (5 a+4 b \sin (c+d x))}{12 b^3 d (a+b \sin (c+d x))^2}-\frac{e (e \cos (c+d x))^{7/2}}{3 b d (a+b \sin (c+d x))^3}",1,"((e*Cos[c + d*x])^(9/2)*Sec[c + d*x]^4*((a^2*Cos[c + d*x] - b^2*Cos[c + d*x])/(3*b^3*(a + b*Sin[c + d*x])^3) - (5*a*Cos[c + d*x])/(4*b^3*(a + b*Sin[c + d*x])^2) + (-19*a^2*Cos[c + d*x] + 12*b^2*Cos[c + d*x])/(8*b^3*(-a^2 + b^2)*(a + b*Sin[c + d*x]))))/d + (7*(e*Cos[c + d*x])^(9/2)*((-4*a*b*(a + b*Sqrt[1 - Cos[c + d*x]^2])*((a*AppellF1[3/4, 1/2, 1, 7/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)]*Cos[c + d*x]^(3/2))/(3*(a^2 - b^2)) + ((1/8 + I/8)*(2*ArcTan[1 - ((1 + I)*Sqrt[b]*Sqrt[Cos[c + d*x]])/(-a^2 + b^2)^(1/4)] - 2*ArcTan[1 + ((1 + I)*Sqrt[b]*Sqrt[Cos[c + d*x]])/(-a^2 + b^2)^(1/4)] - Log[Sqrt[-a^2 + b^2] - (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Cos[c + d*x]] + I*b*Cos[c + d*x]] + Log[Sqrt[-a^2 + b^2] + (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Cos[c + d*x]] + I*b*Cos[c + d*x]]))/(Sqrt[b]*(-a^2 + b^2)^(1/4)))*Sin[c + d*x])/(Sqrt[1 - Cos[c + d*x]^2]*(a + b*Sin[c + d*x])) - ((5*a^2 - 4*b^2)*(a + b*Sqrt[1 - Cos[c + d*x]^2])*(8*b^(5/2)*AppellF1[3/4, -1/2, 1, 7/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)]*Cos[c + d*x]^(3/2) + 3*Sqrt[2]*a*(a^2 - b^2)^(3/4)*(2*ArcTan[1 - (Sqrt[2]*Sqrt[b]*Sqrt[Cos[c + d*x]])/(a^2 - b^2)^(1/4)] - 2*ArcTan[1 + (Sqrt[2]*Sqrt[b]*Sqrt[Cos[c + d*x]])/(a^2 - b^2)^(1/4)] - Log[Sqrt[a^2 - b^2] - Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Cos[c + d*x]] + b*Cos[c + d*x]] + Log[Sqrt[a^2 - b^2] + Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Cos[c + d*x]] + b*Cos[c + d*x]]))*Sin[c + d*x]^2)/(12*b^(3/2)*(-a^2 + b^2)*(1 - Cos[c + d*x]^2)*(a + b*Sin[c + d*x]))))/(16*(a - b)*b^3*(a + b)*d*Cos[c + d*x]^(9/2))","C",0
609,1,1263,597,24.0628762,"\int \frac{(e \cos (c+d x))^{7/2}}{(a+b \sin (c+d x))^4} \, dx","Integrate[(e*Cos[c + d*x])^(7/2)/(a + b*Sin[c + d*x])^4,x]","\frac{\sec ^3(c+d x) \left(-\frac{13 a}{12 b^3 (a+b \sin (c+d x))^2}+\frac{28 b^2-33 a^2}{24 b^3 \left(b^2-a^2\right) (a+b \sin (c+d x))}+\frac{a^2-b^2}{3 b^3 (a+b \sin (c+d x))^3}\right) (e \cos (c+d x))^{7/2}}{d}+\frac{5 \left(-\frac{2 \left(3 a^2-4 b^2\right) \left(a+b \sqrt{1-\cos ^2(c+d x)}\right) \left(\frac{5 b \left(a^2-b^2\right) \sqrt{\cos (c+d x)} \sqrt{1-\cos ^2(c+d x)} F_1\left(\frac{1}{4};-\frac{1}{2},1;\frac{5}{4};\cos ^2(c+d x),\frac{b^2 \cos ^2(c+d x)}{b^2-a^2}\right)}{\left(2 \left(2 F_1\left(\frac{5}{4};-\frac{1}{2},2;\frac{9}{4};\cos ^2(c+d x),\frac{b^2 \cos ^2(c+d x)}{b^2-a^2}\right) b^2+\left(a^2-b^2\right) F_1\left(\frac{5}{4};\frac{1}{2},1;\frac{9}{4};\cos ^2(c+d x),\frac{b^2 \cos ^2(c+d x)}{b^2-a^2}\right)\right) \cos ^2(c+d x)-5 \left(a^2-b^2\right) F_1\left(\frac{1}{4};-\frac{1}{2},1;\frac{5}{4};\cos ^2(c+d x),\frac{b^2 \cos ^2(c+d x)}{b^2-a^2}\right)\right) \left(a^2+b^2 \left(\cos ^2(c+d x)-1\right)\right)}+\frac{a \left(-2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{b} \sqrt{\cos (c+d x)}}{\sqrt[4]{a^2-b^2}}\right)+2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{b} \sqrt{\cos (c+d x)}}{\sqrt[4]{a^2-b^2}}+1\right)-\log \left(b \cos (c+d x)-\sqrt{2} \sqrt{b} \sqrt[4]{a^2-b^2} \sqrt{\cos (c+d x)}+\sqrt{a^2-b^2}\right)+\log \left(b \cos (c+d x)+\sqrt{2} \sqrt{b} \sqrt[4]{a^2-b^2} \sqrt{\cos (c+d x)}+\sqrt{a^2-b^2}\right)\right)}{4 \sqrt{2} \sqrt{b} \left(a^2-b^2\right)^{3/4}}\right) \sin ^2(c+d x)}{\left(1-\cos ^2(c+d x)\right) (a+b \sin (c+d x))}-\frac{4 a b \left(a+b \sqrt{1-\cos ^2(c+d x)}\right) \left(\frac{5 a \left(a^2-b^2\right) F_1\left(\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\cos ^2(c+d x),\frac{b^2 \cos ^2(c+d x)}{b^2-a^2}\right) \sqrt{\cos (c+d x)}}{\sqrt{1-\cos ^2(c+d x)} \left(5 \left(a^2-b^2\right) F_1\left(\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\cos ^2(c+d x),\frac{b^2 \cos ^2(c+d x)}{b^2-a^2}\right)-2 \left(2 F_1\left(\frac{5}{4};\frac{1}{2},2;\frac{9}{4};\cos ^2(c+d x),\frac{b^2 \cos ^2(c+d x)}{b^2-a^2}\right) b^2+\left(b^2-a^2\right) F_1\left(\frac{5}{4};\frac{3}{2},1;\frac{9}{4};\cos ^2(c+d x),\frac{b^2 \cos ^2(c+d x)}{b^2-a^2}\right)\right) \cos ^2(c+d x)\right) \left(a^2+b^2 \left(\cos ^2(c+d x)-1\right)\right)}-\frac{\left(\frac{1}{8}-\frac{i}{8}\right) \sqrt{b} \left(2 \tan ^{-1}\left(1-\frac{(1+i) \sqrt{b} \sqrt{\cos (c+d x)}}{\sqrt[4]{b^2-a^2}}\right)-2 \tan ^{-1}\left(\frac{(1+i) \sqrt{b} \sqrt{\cos (c+d x)}}{\sqrt[4]{b^2-a^2}}+1\right)+\log \left(i b \cos (c+d x)-(1+i) \sqrt{b} \sqrt[4]{b^2-a^2} \sqrt{\cos (c+d x)}+\sqrt{b^2-a^2}\right)-\log \left(i b \cos (c+d x)+(1+i) \sqrt{b} \sqrt[4]{b^2-a^2} \sqrt{\cos (c+d x)}+\sqrt{b^2-a^2}\right)\right)}{\left(b^2-a^2\right)^{3/4}}\right) \sin (c+d x)}{\sqrt{1-\cos ^2(c+d x)} (a+b \sin (c+d x))}\right) (e \cos (c+d x))^{7/2}}{48 (a-b) b^3 (a+b) d \cos ^{\frac{7}{2}}(c+d x)}","-\frac{5 a e^{7/2} \left(a^2-2 b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{16 b^{7/2} d \left(b^2-a^2\right)^{7/4}}-\frac{5 a e^{7/2} \left(a^2-2 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{16 b^{7/2} d \left(b^2-a^2\right)^{7/4}}+\frac{5 e^4 \left(3 a^2-4 b^2\right) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{24 b^4 d \left(a^2-b^2\right) \sqrt{e \cos (c+d x)}}-\frac{5 a^2 e^4 \left(a^2-2 b^2\right) \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{16 b^4 d \left(a^2-b^2\right) \left(a^2-b \left(b-\sqrt{b^2-a^2}\right)\right) \sqrt{e \cos (c+d x)}}-\frac{5 a^2 e^4 \left(a^2-2 b^2\right) \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{16 b^4 d \left(a^2-b^2\right) \left(a^2-b \left(\sqrt{b^2-a^2}+b\right)\right) \sqrt{e \cos (c+d x)}}-\frac{5 e^3 \left(3 a^2-4 b^2\right) \sqrt{e \cos (c+d x)}}{24 b^3 d \left(a^2-b^2\right) (a+b \sin (c+d x))}+\frac{5 e^3 \sqrt{e \cos (c+d x)} (3 a+4 b \sin (c+d x))}{12 b^3 d (a+b \sin (c+d x))^2}-\frac{e (e \cos (c+d x))^{5/2}}{3 b d (a+b \sin (c+d x))^3}",1,"((e*Cos[c + d*x])^(7/2)*Sec[c + d*x]^3*((a^2 - b^2)/(3*b^3*(a + b*Sin[c + d*x])^3) - (13*a)/(12*b^3*(a + b*Sin[c + d*x])^2) + (-33*a^2 + 28*b^2)/(24*b^3*(-a^2 + b^2)*(a + b*Sin[c + d*x]))))/d + (5*(e*Cos[c + d*x])^(7/2)*((-4*a*b*(a + b*Sqrt[1 - Cos[c + d*x]^2])*((5*a*(a^2 - b^2)*AppellF1[1/4, 1/2, 1, 5/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)]*Sqrt[Cos[c + d*x]])/(Sqrt[1 - Cos[c + d*x]^2]*(5*(a^2 - b^2)*AppellF1[1/4, 1/2, 1, 5/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)] - 2*(2*b^2*AppellF1[5/4, 1/2, 2, 9/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)] + (-a^2 + b^2)*AppellF1[5/4, 3/2, 1, 9/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)])*Cos[c + d*x]^2)*(a^2 + b^2*(-1 + Cos[c + d*x]^2))) - ((1/8 - I/8)*Sqrt[b]*(2*ArcTan[1 - ((1 + I)*Sqrt[b]*Sqrt[Cos[c + d*x]])/(-a^2 + b^2)^(1/4)] - 2*ArcTan[1 + ((1 + I)*Sqrt[b]*Sqrt[Cos[c + d*x]])/(-a^2 + b^2)^(1/4)] + Log[Sqrt[-a^2 + b^2] - (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Cos[c + d*x]] + I*b*Cos[c + d*x]] - Log[Sqrt[-a^2 + b^2] + (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Cos[c + d*x]] + I*b*Cos[c + d*x]]))/(-a^2 + b^2)^(3/4))*Sin[c + d*x])/(Sqrt[1 - Cos[c + d*x]^2]*(a + b*Sin[c + d*x])) - (2*(3*a^2 - 4*b^2)*(a + b*Sqrt[1 - Cos[c + d*x]^2])*((5*b*(a^2 - b^2)*AppellF1[1/4, -1/2, 1, 5/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)]*Sqrt[Cos[c + d*x]]*Sqrt[1 - Cos[c + d*x]^2])/((-5*(a^2 - b^2)*AppellF1[1/4, -1/2, 1, 5/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)] + 2*(2*b^2*AppellF1[5/4, -1/2, 2, 9/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)] + (a^2 - b^2)*AppellF1[5/4, 1/2, 1, 9/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)])*Cos[c + d*x]^2)*(a^2 + b^2*(-1 + Cos[c + d*x]^2))) + (a*(-2*ArcTan[1 - (Sqrt[2]*Sqrt[b]*Sqrt[Cos[c + d*x]])/(a^2 - b^2)^(1/4)] + 2*ArcTan[1 + (Sqrt[2]*Sqrt[b]*Sqrt[Cos[c + d*x]])/(a^2 - b^2)^(1/4)] - Log[Sqrt[a^2 - b^2] - Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Cos[c + d*x]] + b*Cos[c + d*x]] + Log[Sqrt[a^2 - b^2] + Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Cos[c + d*x]] + b*Cos[c + d*x]]))/(4*Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(3/4)))*Sin[c + d*x]^2)/((1 - Cos[c + d*x]^2)*(a + b*Sin[c + d*x]))))/(48*(a - b)*b^3*(a + b)*d*Cos[c + d*x]^(7/2))","C",0
610,1,892,574,26.7182452,"\int \frac{(e \cos (c+d x))^{5/2}}{(a+b \sin (c+d x))^4} \, dx","Integrate[(e*Cos[c + d*x])^(5/2)/(a + b*Sin[c + d*x])^4,x]","\frac{\sec ^2(c+d x) \left(-\frac{a \cos (c+d x)}{4 b \left(b^2-a^2\right) (a+b \sin (c+d x))^2}-\frac{\cos (c+d x)}{3 b (a+b \sin (c+d x))^3}+\frac{\cos (c+d x) a^2+4 b^2 \cos (c+d x)}{8 b \left(b^2-a^2\right)^2 (a+b \sin (c+d x))}\right) (e \cos (c+d x))^{5/2}}{d}+\frac{\left(-\frac{\left(a^2+4 b^2\right) \left(a+b \sqrt{1-\cos ^2(c+d x)}\right) \left(8 F_1\left(\frac{3}{4};-\frac{1}{2},1;\frac{7}{4};\cos ^2(c+d x),\frac{b^2 \cos ^2(c+d x)}{b^2-a^2}\right) \cos ^{\frac{3}{2}}(c+d x) b^{5/2}+3 \sqrt{2} a \left(a^2-b^2\right)^{3/4} \left(2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{b} \sqrt{\cos (c+d x)}}{\sqrt[4]{a^2-b^2}}\right)-2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{b} \sqrt{\cos (c+d x)}}{\sqrt[4]{a^2-b^2}}+1\right)-\log \left(b \cos (c+d x)-\sqrt{2} \sqrt{b} \sqrt[4]{a^2-b^2} \sqrt{\cos (c+d x)}+\sqrt{a^2-b^2}\right)+\log \left(b \cos (c+d x)+\sqrt{2} \sqrt{b} \sqrt[4]{a^2-b^2} \sqrt{\cos (c+d x)}+\sqrt{a^2-b^2}\right)\right)\right) \sin ^2(c+d x)}{12 b^{3/2} \left(b^2-a^2\right) \left(1-\cos ^2(c+d x)\right) (a+b \sin (c+d x))}-\frac{20 a b \left(a+b \sqrt{1-\cos ^2(c+d x)}\right) \left(\frac{a F_1\left(\frac{3}{4};\frac{1}{2},1;\frac{7}{4};\cos ^2(c+d x),\frac{b^2 \cos ^2(c+d x)}{b^2-a^2}\right) \cos ^{\frac{3}{2}}(c+d x)}{3 \left(a^2-b^2\right)}+\frac{\left(\frac{1}{8}+\frac{i}{8}\right) \left(2 \tan ^{-1}\left(1-\frac{(1+i) \sqrt{b} \sqrt{\cos (c+d x)}}{\sqrt[4]{b^2-a^2}}\right)-2 \tan ^{-1}\left(\frac{(1+i) \sqrt{b} \sqrt{\cos (c+d x)}}{\sqrt[4]{b^2-a^2}}+1\right)-\log \left(i b \cos (c+d x)-(1+i) \sqrt{b} \sqrt[4]{b^2-a^2} \sqrt{\cos (c+d x)}+\sqrt{b^2-a^2}\right)+\log \left(i b \cos (c+d x)+(1+i) \sqrt{b} \sqrt[4]{b^2-a^2} \sqrt{\cos (c+d x)}+\sqrt{b^2-a^2}\right)\right)}{\sqrt{b} \sqrt[4]{b^2-a^2}}\right) \sin (c+d x)}{\sqrt{1-\cos ^2(c+d x)} (a+b \sin (c+d x))}\right) (e \cos (c+d x))^{5/2}}{16 (a-b)^2 b (a+b)^2 d \cos ^{\frac{5}{2}}(c+d x)}","\frac{e^2 \left(a^2+4 b^2\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{8 b^2 d \left(a^2-b^2\right)^2 \sqrt{\cos (c+d x)}}+\frac{e \left(a^2+4 b^2\right) (e \cos (c+d x))^{3/2}}{8 b d \left(a^2-b^2\right)^2 (a+b \sin (c+d x))}+\frac{a e (e \cos (c+d x))^{3/2}}{4 b d \left(a^2-b^2\right) (a+b \sin (c+d x))^2}-\frac{a e^{5/2} \left(a^2-6 b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{16 b^{5/2} d \left(b^2-a^2\right)^{9/4}}+\frac{a e^{5/2} \left(a^2-6 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{16 b^{5/2} d \left(b^2-a^2\right)^{9/4}}-\frac{a^2 e^3 \left(a^2-6 b^2\right) \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{16 b^3 d \left(a^2-b^2\right)^2 \left(b-\sqrt{b^2-a^2}\right) \sqrt{e \cos (c+d x)}}-\frac{a^2 e^3 \left(a^2-6 b^2\right) \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{16 b^3 d \left(a^2-b^2\right)^2 \left(\sqrt{b^2-a^2}+b\right) \sqrt{e \cos (c+d x)}}-\frac{e (e \cos (c+d x))^{3/2}}{3 b d (a+b \sin (c+d x))^3}",1,"((e*Cos[c + d*x])^(5/2)*Sec[c + d*x]^2*(-1/3*Cos[c + d*x]/(b*(a + b*Sin[c + d*x])^3) - (a*Cos[c + d*x])/(4*b*(-a^2 + b^2)*(a + b*Sin[c + d*x])^2) + (a^2*Cos[c + d*x] + 4*b^2*Cos[c + d*x])/(8*b*(-a^2 + b^2)^2*(a + b*Sin[c + d*x]))))/d + ((e*Cos[c + d*x])^(5/2)*((-20*a*b*(a + b*Sqrt[1 - Cos[c + d*x]^2])*((a*AppellF1[3/4, 1/2, 1, 7/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)]*Cos[c + d*x]^(3/2))/(3*(a^2 - b^2)) + ((1/8 + I/8)*(2*ArcTan[1 - ((1 + I)*Sqrt[b]*Sqrt[Cos[c + d*x]])/(-a^2 + b^2)^(1/4)] - 2*ArcTan[1 + ((1 + I)*Sqrt[b]*Sqrt[Cos[c + d*x]])/(-a^2 + b^2)^(1/4)] - Log[Sqrt[-a^2 + b^2] - (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Cos[c + d*x]] + I*b*Cos[c + d*x]] + Log[Sqrt[-a^2 + b^2] + (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Cos[c + d*x]] + I*b*Cos[c + d*x]]))/(Sqrt[b]*(-a^2 + b^2)^(1/4)))*Sin[c + d*x])/(Sqrt[1 - Cos[c + d*x]^2]*(a + b*Sin[c + d*x])) - ((a^2 + 4*b^2)*(a + b*Sqrt[1 - Cos[c + d*x]^2])*(8*b^(5/2)*AppellF1[3/4, -1/2, 1, 7/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)]*Cos[c + d*x]^(3/2) + 3*Sqrt[2]*a*(a^2 - b^2)^(3/4)*(2*ArcTan[1 - (Sqrt[2]*Sqrt[b]*Sqrt[Cos[c + d*x]])/(a^2 - b^2)^(1/4)] - 2*ArcTan[1 + (Sqrt[2]*Sqrt[b]*Sqrt[Cos[c + d*x]])/(a^2 - b^2)^(1/4)] - Log[Sqrt[a^2 - b^2] - Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Cos[c + d*x]] + b*Cos[c + d*x]] + Log[Sqrt[a^2 - b^2] + Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Cos[c + d*x]] + b*Cos[c + d*x]]))*Sin[c + d*x]^2)/(12*b^(3/2)*(-a^2 + b^2)*(1 - Cos[c + d*x]^2)*(a + b*Sin[c + d*x]))))/(16*(a - b)^2*b*(a + b)^2*d*Cos[c + d*x]^(5/2))","C",0
611,1,1263,592,23.8473054,"\int \frac{(e \cos (c+d x))^{3/2}}{(a+b \sin (c+d x))^4} \, dx","Integrate[(e*Cos[c + d*x])^(3/2)/(a + b*Sin[c + d*x])^4,x]","\frac{(e \cos (c+d x))^{3/2} \sec (c+d x) \left(-\frac{a}{12 b \left(b^2-a^2\right) (a+b \sin (c+d x))^2}+\frac{3 a^2+4 b^2}{24 b \left(b^2-a^2\right)^2 (a+b \sin (c+d x))}-\frac{1}{3 b (a+b \sin (c+d x))^3}\right)}{d}-\frac{(e \cos (c+d x))^{3/2} \left(\frac{28 a b \left(a+b \sqrt{1-\cos ^2(c+d x)}\right) \left(\frac{5 a \left(a^2-b^2\right) F_1\left(\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\cos ^2(c+d x),\frac{b^2 \cos ^2(c+d x)}{b^2-a^2}\right) \sqrt{\cos (c+d x)}}{\sqrt{1-\cos ^2(c+d x)} \left(5 \left(a^2-b^2\right) F_1\left(\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\cos ^2(c+d x),\frac{b^2 \cos ^2(c+d x)}{b^2-a^2}\right)-2 \left(2 F_1\left(\frac{5}{4};\frac{1}{2},2;\frac{9}{4};\cos ^2(c+d x),\frac{b^2 \cos ^2(c+d x)}{b^2-a^2}\right) b^2+\left(b^2-a^2\right) F_1\left(\frac{5}{4};\frac{3}{2},1;\frac{9}{4};\cos ^2(c+d x),\frac{b^2 \cos ^2(c+d x)}{b^2-a^2}\right)\right) \cos ^2(c+d x)\right) \left(a^2+b^2 \left(\cos ^2(c+d x)-1\right)\right)}-\frac{\left(\frac{1}{8}-\frac{i}{8}\right) \sqrt{b} \left(2 \tan ^{-1}\left(1-\frac{(1+i) \sqrt{b} \sqrt{\cos (c+d x)}}{\sqrt[4]{b^2-a^2}}\right)-2 \tan ^{-1}\left(\frac{(1+i) \sqrt{b} \sqrt{\cos (c+d x)}}{\sqrt[4]{b^2-a^2}}+1\right)+\log \left(i b \cos (c+d x)-(1+i) \sqrt{b} \sqrt[4]{b^2-a^2} \sqrt{\cos (c+d x)}+\sqrt{b^2-a^2}\right)-\log \left(i b \cos (c+d x)+(1+i) \sqrt{b} \sqrt[4]{b^2-a^2} \sqrt{\cos (c+d x)}+\sqrt{b^2-a^2}\right)\right)}{\left(b^2-a^2\right)^{3/4}}\right) \sin (c+d x)}{\sqrt{1-\cos ^2(c+d x)} (a+b \sin (c+d x))}-\frac{2 \left(3 a^2+4 b^2\right) \left(a+b \sqrt{1-\cos ^2(c+d x)}\right) \left(\frac{5 b \left(a^2-b^2\right) \sqrt{\cos (c+d x)} \sqrt{1-\cos ^2(c+d x)} F_1\left(\frac{1}{4};-\frac{1}{2},1;\frac{5}{4};\cos ^2(c+d x),\frac{b^2 \cos ^2(c+d x)}{b^2-a^2}\right)}{\left(2 \left(2 F_1\left(\frac{5}{4};-\frac{1}{2},2;\frac{9}{4};\cos ^2(c+d x),\frac{b^2 \cos ^2(c+d x)}{b^2-a^2}\right) b^2+\left(a^2-b^2\right) F_1\left(\frac{5}{4};\frac{1}{2},1;\frac{9}{4};\cos ^2(c+d x),\frac{b^2 \cos ^2(c+d x)}{b^2-a^2}\right)\right) \cos ^2(c+d x)-5 \left(a^2-b^2\right) F_1\left(\frac{1}{4};-\frac{1}{2},1;\frac{5}{4};\cos ^2(c+d x),\frac{b^2 \cos ^2(c+d x)}{b^2-a^2}\right)\right) \left(a^2+b^2 \left(\cos ^2(c+d x)-1\right)\right)}+\frac{a \left(-2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{b} \sqrt{\cos (c+d x)}}{\sqrt[4]{a^2-b^2}}\right)+2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{b} \sqrt{\cos (c+d x)}}{\sqrt[4]{a^2-b^2}}+1\right)-\log \left(b \cos (c+d x)-\sqrt{2} \sqrt{b} \sqrt[4]{a^2-b^2} \sqrt{\cos (c+d x)}+\sqrt{a^2-b^2}\right)+\log \left(b \cos (c+d x)+\sqrt{2} \sqrt{b} \sqrt[4]{a^2-b^2} \sqrt{\cos (c+d x)}+\sqrt{a^2-b^2}\right)\right)}{4 \sqrt{2} \sqrt{b} \left(a^2-b^2\right)^{3/4}}\right) \sin ^2(c+d x)}{\left(1-\cos ^2(c+d x)\right) (a+b \sin (c+d x))}\right)}{48 (a-b)^2 b (a+b)^2 d \cos ^{\frac{3}{2}}(c+d x)}","-\frac{e^2 \left(3 a^2+4 b^2\right) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{24 b^2 d \left(a^2-b^2\right)^2 \sqrt{e \cos (c+d x)}}+\frac{a^2 e^2 \left(a^2+6 b^2\right) \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{16 b^2 d \left(a^2-b^2\right)^2 \left(a^2-b \left(b-\sqrt{b^2-a^2}\right)\right) \sqrt{e \cos (c+d x)}}+\frac{a^2 e^2 \left(a^2+6 b^2\right) \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{16 b^2 d \left(a^2-b^2\right)^2 \left(a^2-b \left(\sqrt{b^2-a^2}+b\right)\right) \sqrt{e \cos (c+d x)}}+\frac{e \left(3 a^2+4 b^2\right) \sqrt{e \cos (c+d x)}}{24 b d \left(a^2-b^2\right)^2 (a+b \sin (c+d x))}+\frac{a e \sqrt{e \cos (c+d x)}}{12 b d \left(a^2-b^2\right) (a+b \sin (c+d x))^2}-\frac{a e^{3/2} \left(a^2+6 b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{16 b^{3/2} d \left(b^2-a^2\right)^{11/4}}-\frac{a e^{3/2} \left(a^2+6 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{16 b^{3/2} d \left(b^2-a^2\right)^{11/4}}-\frac{e \sqrt{e \cos (c+d x)}}{3 b d (a+b \sin (c+d x))^3}",1,"((e*Cos[c + d*x])^(3/2)*Sec[c + d*x]*(-1/3*1/(b*(a + b*Sin[c + d*x])^3) - a/(12*b*(-a^2 + b^2)*(a + b*Sin[c + d*x])^2) + (3*a^2 + 4*b^2)/(24*b*(-a^2 + b^2)^2*(a + b*Sin[c + d*x]))))/d - ((e*Cos[c + d*x])^(3/2)*((28*a*b*(a + b*Sqrt[1 - Cos[c + d*x]^2])*((5*a*(a^2 - b^2)*AppellF1[1/4, 1/2, 1, 5/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)]*Sqrt[Cos[c + d*x]])/(Sqrt[1 - Cos[c + d*x]^2]*(5*(a^2 - b^2)*AppellF1[1/4, 1/2, 1, 5/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)] - 2*(2*b^2*AppellF1[5/4, 1/2, 2, 9/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)] + (-a^2 + b^2)*AppellF1[5/4, 3/2, 1, 9/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)])*Cos[c + d*x]^2)*(a^2 + b^2*(-1 + Cos[c + d*x]^2))) - ((1/8 - I/8)*Sqrt[b]*(2*ArcTan[1 - ((1 + I)*Sqrt[b]*Sqrt[Cos[c + d*x]])/(-a^2 + b^2)^(1/4)] - 2*ArcTan[1 + ((1 + I)*Sqrt[b]*Sqrt[Cos[c + d*x]])/(-a^2 + b^2)^(1/4)] + Log[Sqrt[-a^2 + b^2] - (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Cos[c + d*x]] + I*b*Cos[c + d*x]] - Log[Sqrt[-a^2 + b^2] + (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Cos[c + d*x]] + I*b*Cos[c + d*x]]))/(-a^2 + b^2)^(3/4))*Sin[c + d*x])/(Sqrt[1 - Cos[c + d*x]^2]*(a + b*Sin[c + d*x])) - (2*(3*a^2 + 4*b^2)*(a + b*Sqrt[1 - Cos[c + d*x]^2])*((5*b*(a^2 - b^2)*AppellF1[1/4, -1/2, 1, 5/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)]*Sqrt[Cos[c + d*x]]*Sqrt[1 - Cos[c + d*x]^2])/((-5*(a^2 - b^2)*AppellF1[1/4, -1/2, 1, 5/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)] + 2*(2*b^2*AppellF1[5/4, -1/2, 2, 9/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)] + (a^2 - b^2)*AppellF1[5/4, 1/2, 1, 9/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)])*Cos[c + d*x]^2)*(a^2 + b^2*(-1 + Cos[c + d*x]^2))) + (a*(-2*ArcTan[1 - (Sqrt[2]*Sqrt[b]*Sqrt[Cos[c + d*x]])/(a^2 - b^2)^(1/4)] + 2*ArcTan[1 + (Sqrt[2]*Sqrt[b]*Sqrt[Cos[c + d*x]])/(a^2 - b^2)^(1/4)] - Log[Sqrt[a^2 - b^2] - Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Cos[c + d*x]] + b*Cos[c + d*x]] + Log[Sqrt[a^2 - b^2] + Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Cos[c + d*x]] + b*Cos[c + d*x]]))/(4*Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(3/4)))*Sin[c + d*x]^2)/((1 - Cos[c + d*x]^2)*(a + b*Sin[c + d*x]))))/(48*(a - b)^2*b*(a + b)^2*d*Cos[c + d*x]^(3/2))","C",0
612,1,900,579,6.630102,"\int \frac{\sqrt{e \cos (c+d x)}}{(a+b \sin (c+d x))^4} \, dx","Integrate[Sqrt[e*Cos[c + d*x]]/(a + b*Sin[c + d*x])^4,x]","\frac{\sqrt{e \cos (c+d x)} \left(\frac{3 a b \cos (c+d x)}{4 \left(a^2-b^2\right)^2 (a+b \sin (c+d x))^2}+\frac{b \cos (c+d x)}{3 \left(a^2-b^2\right) (a+b \sin (c+d x))^3}-\frac{-4 \cos (c+d x) b^3-11 a^2 \cos (c+d x) b}{8 \left(a^2-b^2\right)^3 (a+b \sin (c+d x))}\right)}{d}+\frac{\sqrt{e \cos (c+d x)} \left(-\frac{\left(4 b^3+11 a^2 b\right) \left(a+b \sqrt{1-\cos ^2(c+d x)}\right) \left(8 F_1\left(\frac{3}{4};-\frac{1}{2},1;\frac{7}{4};\cos ^2(c+d x),\frac{b^2 \cos ^2(c+d x)}{b^2-a^2}\right) \cos ^{\frac{3}{2}}(c+d x) b^{5/2}+3 \sqrt{2} a \left(a^2-b^2\right)^{3/4} \left(2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{b} \sqrt{\cos (c+d x)}}{\sqrt[4]{a^2-b^2}}\right)-2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{b} \sqrt{\cos (c+d x)}}{\sqrt[4]{a^2-b^2}}+1\right)-\log \left(b \cos (c+d x)-\sqrt{2} \sqrt{b} \sqrt[4]{a^2-b^2} \sqrt{\cos (c+d x)}+\sqrt{a^2-b^2}\right)+\log \left(b \cos (c+d x)+\sqrt{2} \sqrt{b} \sqrt[4]{a^2-b^2} \sqrt{\cos (c+d x)}+\sqrt{a^2-b^2}\right)\right)\right) \sin ^2(c+d x)}{12 b^{3/2} \left(b^2-a^2\right) \left(1-\cos ^2(c+d x)\right) (a+b \sin (c+d x))}-\frac{2 \left(16 a^3+14 b^2 a\right) \left(a+b \sqrt{1-\cos ^2(c+d x)}\right) \left(\frac{a F_1\left(\frac{3}{4};\frac{1}{2},1;\frac{7}{4};\cos ^2(c+d x),\frac{b^2 \cos ^2(c+d x)}{b^2-a^2}\right) \cos ^{\frac{3}{2}}(c+d x)}{3 \left(a^2-b^2\right)}+\frac{\left(\frac{1}{8}+\frac{i}{8}\right) \left(2 \tan ^{-1}\left(1-\frac{(1+i) \sqrt{b} \sqrt{\cos (c+d x)}}{\sqrt[4]{b^2-a^2}}\right)-2 \tan ^{-1}\left(\frac{(1+i) \sqrt{b} \sqrt{\cos (c+d x)}}{\sqrt[4]{b^2-a^2}}+1\right)-\log \left(i b \cos (c+d x)-(1+i) \sqrt{b} \sqrt[4]{b^2-a^2} \sqrt{\cos (c+d x)}+\sqrt{b^2-a^2}\right)+\log \left(i b \cos (c+d x)+(1+i) \sqrt{b} \sqrt[4]{b^2-a^2} \sqrt{\cos (c+d x)}+\sqrt{b^2-a^2}\right)\right)}{\sqrt{b} \sqrt[4]{b^2-a^2}}\right) \sin (c+d x)}{\sqrt{1-\cos ^2(c+d x)} (a+b \sin (c+d x))}\right)}{16 (a-b)^3 (a+b)^3 d \sqrt{\cos (c+d x)}}","\frac{3 a b (e \cos (c+d x))^{3/2}}{4 d e \left(a^2-b^2\right)^2 (a+b \sin (c+d x))^2}+\frac{b \left(11 a^2+4 b^2\right) (e \cos (c+d x))^{3/2}}{8 d e \left(a^2-b^2\right)^3 (a+b \sin (c+d x))}+\frac{b (e \cos (c+d x))^{3/2}}{3 d e \left(a^2-b^2\right) (a+b \sin (c+d x))^3}-\frac{5 a \sqrt{e} \left(a^2+2 b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{16 \sqrt{b} d \left(b^2-a^2\right)^{13/4}}+\frac{5 a \sqrt{e} \left(a^2+2 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{16 \sqrt{b} d \left(b^2-a^2\right)^{13/4}}+\frac{\left(11 a^2+4 b^2\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{8 d \left(a^2-b^2\right)^3 \sqrt{\cos (c+d x)}}+\frac{5 a^2 e \left(a^2+2 b^2\right) \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{16 b d \left(a^2-b^2\right)^3 \left(b-\sqrt{b^2-a^2}\right) \sqrt{e \cos (c+d x)}}+\frac{5 a^2 e \left(a^2+2 b^2\right) \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{16 b d \left(a^2-b^2\right)^3 \left(\sqrt{b^2-a^2}+b\right) \sqrt{e \cos (c+d x)}}",1,"(Sqrt[e*Cos[c + d*x]]*((b*Cos[c + d*x])/(3*(a^2 - b^2)*(a + b*Sin[c + d*x])^3) + (3*a*b*Cos[c + d*x])/(4*(a^2 - b^2)^2*(a + b*Sin[c + d*x])^2) - (-11*a^2*b*Cos[c + d*x] - 4*b^3*Cos[c + d*x])/(8*(a^2 - b^2)^3*(a + b*Sin[c + d*x]))))/d + (Sqrt[e*Cos[c + d*x]]*((-2*(16*a^3 + 14*a*b^2)*(a + b*Sqrt[1 - Cos[c + d*x]^2])*((a*AppellF1[3/4, 1/2, 1, 7/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)]*Cos[c + d*x]^(3/2))/(3*(a^2 - b^2)) + ((1/8 + I/8)*(2*ArcTan[1 - ((1 + I)*Sqrt[b]*Sqrt[Cos[c + d*x]])/(-a^2 + b^2)^(1/4)] - 2*ArcTan[1 + ((1 + I)*Sqrt[b]*Sqrt[Cos[c + d*x]])/(-a^2 + b^2)^(1/4)] - Log[Sqrt[-a^2 + b^2] - (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Cos[c + d*x]] + I*b*Cos[c + d*x]] + Log[Sqrt[-a^2 + b^2] + (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Cos[c + d*x]] + I*b*Cos[c + d*x]]))/(Sqrt[b]*(-a^2 + b^2)^(1/4)))*Sin[c + d*x])/(Sqrt[1 - Cos[c + d*x]^2]*(a + b*Sin[c + d*x])) - ((11*a^2*b + 4*b^3)*(a + b*Sqrt[1 - Cos[c + d*x]^2])*(8*b^(5/2)*AppellF1[3/4, -1/2, 1, 7/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)]*Cos[c + d*x]^(3/2) + 3*Sqrt[2]*a*(a^2 - b^2)^(3/4)*(2*ArcTan[1 - (Sqrt[2]*Sqrt[b]*Sqrt[Cos[c + d*x]])/(a^2 - b^2)^(1/4)] - 2*ArcTan[1 + (Sqrt[2]*Sqrt[b]*Sqrt[Cos[c + d*x]])/(a^2 - b^2)^(1/4)] - Log[Sqrt[a^2 - b^2] - Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Cos[c + d*x]] + b*Cos[c + d*x]] + Log[Sqrt[a^2 - b^2] + Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Cos[c + d*x]] + b*Cos[c + d*x]]))*Sin[c + d*x]^2)/(12*b^(3/2)*(-a^2 + b^2)*(1 - Cos[c + d*x]^2)*(a + b*Sin[c + d*x]))))/(16*(a - b)^3*(a + b)^3*d*Sqrt[Cos[c + d*x]])","C",0
613,1,1276,593,24.4571635,"\int \frac{1}{\sqrt{e \cos (c+d x)} (a+b \sin (c+d x))^4} \, dx","Integrate[1/(Sqrt[e*Cos[c + d*x]]*(a + b*Sin[c + d*x])^4),x]","\frac{\cos (c+d x) \left(\frac{\left(57 a^2+20 b^2\right) b}{24 \left(a^2-b^2\right)^3 (a+b \sin (c+d x))}+\frac{11 a b}{12 \left(a^2-b^2\right)^2 (a+b \sin (c+d x))^2}+\frac{b}{3 \left(a^2-b^2\right) (a+b \sin (c+d x))^3}\right)}{d \sqrt{e \cos (c+d x)}}+\frac{\sqrt{\cos (c+d x)} \left(-\frac{2 \left(-20 b^3-57 a^2 b\right) \left(a+b \sqrt{1-\cos ^2(c+d x)}\right) \left(\frac{5 b \left(a^2-b^2\right) \sqrt{\cos (c+d x)} \sqrt{1-\cos ^2(c+d x)} F_1\left(\frac{1}{4};-\frac{1}{2},1;\frac{5}{4};\cos ^2(c+d x),\frac{b^2 \cos ^2(c+d x)}{b^2-a^2}\right)}{\left(2 \left(2 F_1\left(\frac{5}{4};-\frac{1}{2},2;\frac{9}{4};\cos ^2(c+d x),\frac{b^2 \cos ^2(c+d x)}{b^2-a^2}\right) b^2+\left(a^2-b^2\right) F_1\left(\frac{5}{4};\frac{1}{2},1;\frac{9}{4};\cos ^2(c+d x),\frac{b^2 \cos ^2(c+d x)}{b^2-a^2}\right)\right) \cos ^2(c+d x)-5 \left(a^2-b^2\right) F_1\left(\frac{1}{4};-\frac{1}{2},1;\frac{5}{4};\cos ^2(c+d x),\frac{b^2 \cos ^2(c+d x)}{b^2-a^2}\right)\right) \left(a^2+b^2 \left(\cos ^2(c+d x)-1\right)\right)}+\frac{a \left(-2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{b} \sqrt{\cos (c+d x)}}{\sqrt[4]{a^2-b^2}}\right)+2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{b} \sqrt{\cos (c+d x)}}{\sqrt[4]{a^2-b^2}}+1\right)-\log \left(b \cos (c+d x)-\sqrt{2} \sqrt{b} \sqrt[4]{a^2-b^2} \sqrt{\cos (c+d x)}+\sqrt{a^2-b^2}\right)+\log \left(b \cos (c+d x)+\sqrt{2} \sqrt{b} \sqrt[4]{a^2-b^2} \sqrt{\cos (c+d x)}+\sqrt{a^2-b^2}\right)\right)}{4 \sqrt{2} \sqrt{b} \left(a^2-b^2\right)^{3/4}}\right) \sin ^2(c+d x)}{\left(1-\cos ^2(c+d x)\right) (a+b \sin (c+d x))}-\frac{2 \left(48 a^3+106 b^2 a\right) \left(a+b \sqrt{1-\cos ^2(c+d x)}\right) \left(\frac{5 a \left(a^2-b^2\right) F_1\left(\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\cos ^2(c+d x),\frac{b^2 \cos ^2(c+d x)}{b^2-a^2}\right) \sqrt{\cos (c+d x)}}{\sqrt{1-\cos ^2(c+d x)} \left(5 \left(a^2-b^2\right) F_1\left(\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\cos ^2(c+d x),\frac{b^2 \cos ^2(c+d x)}{b^2-a^2}\right)-2 \left(2 F_1\left(\frac{5}{4};\frac{1}{2},2;\frac{9}{4};\cos ^2(c+d x),\frac{b^2 \cos ^2(c+d x)}{b^2-a^2}\right) b^2+\left(b^2-a^2\right) F_1\left(\frac{5}{4};\frac{3}{2},1;\frac{9}{4};\cos ^2(c+d x),\frac{b^2 \cos ^2(c+d x)}{b^2-a^2}\right)\right) \cos ^2(c+d x)\right) \left(a^2+b^2 \left(\cos ^2(c+d x)-1\right)\right)}-\frac{\left(\frac{1}{8}-\frac{i}{8}\right) \sqrt{b} \left(2 \tan ^{-1}\left(1-\frac{(1+i) \sqrt{b} \sqrt{\cos (c+d x)}}{\sqrt[4]{b^2-a^2}}\right)-2 \tan ^{-1}\left(\frac{(1+i) \sqrt{b} \sqrt{\cos (c+d x)}}{\sqrt[4]{b^2-a^2}}+1\right)+\log \left(i b \cos (c+d x)-(1+i) \sqrt{b} \sqrt[4]{b^2-a^2} \sqrt{\cos (c+d x)}+\sqrt{b^2-a^2}\right)-\log \left(i b \cos (c+d x)+(1+i) \sqrt{b} \sqrt[4]{b^2-a^2} \sqrt{\cos (c+d x)}+\sqrt{b^2-a^2}\right)\right)}{\left(b^2-a^2\right)^{3/4}}\right) \sin (c+d x)}{\sqrt{1-\cos ^2(c+d x)} (a+b \sin (c+d x))}\right)}{48 (a-b)^3 (a+b)^3 d \sqrt{e \cos (c+d x)}}","\frac{11 a b \sqrt{e \cos (c+d x)}}{12 d e \left(a^2-b^2\right)^2 (a+b \sin (c+d x))^2}+\frac{b \left(57 a^2+20 b^2\right) \sqrt{e \cos (c+d x)}}{24 d e \left(a^2-b^2\right)^3 (a+b \sin (c+d x))}+\frac{b \sqrt{e \cos (c+d x)}}{3 d e \left(a^2-b^2\right) (a+b \sin (c+d x))^3}+\frac{7 a \sqrt{b} \left(5 a^2+6 b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{16 d \sqrt{e} \left(b^2-a^2\right)^{15/4}}+\frac{7 a \sqrt{b} \left(5 a^2+6 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{16 d \sqrt{e} \left(b^2-a^2\right)^{15/4}}-\frac{\left(57 a^2+20 b^2\right) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{24 d \left(a^2-b^2\right)^3 \sqrt{e \cos (c+d x)}}+\frac{7 a^2 \left(5 a^2+6 b^2\right) \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{16 d \left(a^2-b^2\right)^3 \left(a^2-b \left(b-\sqrt{b^2-a^2}\right)\right) \sqrt{e \cos (c+d x)}}+\frac{7 a^2 \left(5 a^2+6 b^2\right) \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{16 d \left(a^2-b^2\right)^3 \left(a^2-b \left(\sqrt{b^2-a^2}+b\right)\right) \sqrt{e \cos (c+d x)}}",1,"(Cos[c + d*x]*(b/(3*(a^2 - b^2)*(a + b*Sin[c + d*x])^3) + (11*a*b)/(12*(a^2 - b^2)^2*(a + b*Sin[c + d*x])^2) + (b*(57*a^2 + 20*b^2))/(24*(a^2 - b^2)^3*(a + b*Sin[c + d*x]))))/(d*Sqrt[e*Cos[c + d*x]]) + (Sqrt[Cos[c + d*x]]*((-2*(48*a^3 + 106*a*b^2)*(a + b*Sqrt[1 - Cos[c + d*x]^2])*((5*a*(a^2 - b^2)*AppellF1[1/4, 1/2, 1, 5/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)]*Sqrt[Cos[c + d*x]])/(Sqrt[1 - Cos[c + d*x]^2]*(5*(a^2 - b^2)*AppellF1[1/4, 1/2, 1, 5/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)] - 2*(2*b^2*AppellF1[5/4, 1/2, 2, 9/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)] + (-a^2 + b^2)*AppellF1[5/4, 3/2, 1, 9/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)])*Cos[c + d*x]^2)*(a^2 + b^2*(-1 + Cos[c + d*x]^2))) - ((1/8 - I/8)*Sqrt[b]*(2*ArcTan[1 - ((1 + I)*Sqrt[b]*Sqrt[Cos[c + d*x]])/(-a^2 + b^2)^(1/4)] - 2*ArcTan[1 + ((1 + I)*Sqrt[b]*Sqrt[Cos[c + d*x]])/(-a^2 + b^2)^(1/4)] + Log[Sqrt[-a^2 + b^2] - (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Cos[c + d*x]] + I*b*Cos[c + d*x]] - Log[Sqrt[-a^2 + b^2] + (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Cos[c + d*x]] + I*b*Cos[c + d*x]]))/(-a^2 + b^2)^(3/4))*Sin[c + d*x])/(Sqrt[1 - Cos[c + d*x]^2]*(a + b*Sin[c + d*x])) - (2*(-57*a^2*b - 20*b^3)*(a + b*Sqrt[1 - Cos[c + d*x]^2])*((5*b*(a^2 - b^2)*AppellF1[1/4, -1/2, 1, 5/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)]*Sqrt[Cos[c + d*x]]*Sqrt[1 - Cos[c + d*x]^2])/((-5*(a^2 - b^2)*AppellF1[1/4, -1/2, 1, 5/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)] + 2*(2*b^2*AppellF1[5/4, -1/2, 2, 9/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)] + (a^2 - b^2)*AppellF1[5/4, 1/2, 1, 9/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)])*Cos[c + d*x]^2)*(a^2 + b^2*(-1 + Cos[c + d*x]^2))) + (a*(-2*ArcTan[1 - (Sqrt[2]*Sqrt[b]*Sqrt[Cos[c + d*x]])/(a^2 - b^2)^(1/4)] + 2*ArcTan[1 + (Sqrt[2]*Sqrt[b]*Sqrt[Cos[c + d*x]])/(a^2 - b^2)^(1/4)] - Log[Sqrt[a^2 - b^2] - Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Cos[c + d*x]] + b*Cos[c + d*x]] + Log[Sqrt[a^2 - b^2] + Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Cos[c + d*x]] + b*Cos[c + d*x]]))/(4*Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(3/4)))*Sin[c + d*x]^2)/((1 - Cos[c + d*x]^2)*(a + b*Sin[c + d*x]))))/(48*(a - b)^3*(a + b)^3*d*Sqrt[e*Cos[c + d*x]])","C",0
614,1,996,674,6.8186212,"\int \frac{1}{(e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^4} \, dx","Integrate[1/((e*Cos[c + d*x])^(3/2)*(a + b*Sin[c + d*x])^4),x]","\frac{\cos ^2(c+d x) \left(-\frac{7 a \cos (c+d x) b^3}{4 \left(a^2-b^2\right)^3 (a+b \sin (c+d x))^2}-\frac{\cos (c+d x) b^3}{3 \left(a^2-b^2\right)^2 (a+b \sin (c+d x))^3}+\frac{2 \sec (c+d x) \left(\sin (c+d x) a^4-4 b a^3+6 b^2 \sin (c+d x) a^2-4 b^3 a+b^4 \sin (c+d x)\right)}{\left(a^2-b^2\right)^4}+\frac{-12 \cos (c+d x) b^5-55 a^2 \cos (c+d x) b^3}{8 \left(a^2-b^2\right)^4 (a+b \sin (c+d x))}\right)}{d (e \cos (c+d x))^{3/2}}-\frac{\cos ^{\frac{3}{2}}(c+d x) \left(-\frac{\left(28 b^5+151 a^2 b^3+16 a^4 b\right) \left(a+b \sqrt{1-\cos ^2(c+d x)}\right) \left(8 F_1\left(\frac{3}{4};-\frac{1}{2},1;\frac{7}{4};\cos ^2(c+d x),\frac{b^2 \cos ^2(c+d x)}{b^2-a^2}\right) \cos ^{\frac{3}{2}}(c+d x) b^{5/2}+3 \sqrt{2} a \left(a^2-b^2\right)^{3/4} \left(2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{b} \sqrt{\cos (c+d x)}}{\sqrt[4]{a^2-b^2}}\right)-2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{b} \sqrt{\cos (c+d x)}}{\sqrt[4]{a^2-b^2}}+1\right)-\log \left(b \cos (c+d x)-\sqrt{2} \sqrt{b} \sqrt[4]{a^2-b^2} \sqrt{\cos (c+d x)}+\sqrt{a^2-b^2}\right)+\log \left(b \cos (c+d x)+\sqrt{2} \sqrt{b} \sqrt[4]{a^2-b^2} \sqrt{\cos (c+d x)}+\sqrt{a^2-b^2}\right)\right)\right) \sin ^2(c+d x)}{12 b^{3/2} \left(b^2-a^2\right) \left(1-\cos ^2(c+d x)\right) (a+b \sin (c+d x))}-\frac{2 \left(16 a^5+256 b^2 a^3+118 b^4 a\right) \left(a+b \sqrt{1-\cos ^2(c+d x)}\right) \left(\frac{a F_1\left(\frac{3}{4};\frac{1}{2},1;\frac{7}{4};\cos ^2(c+d x),\frac{b^2 \cos ^2(c+d x)}{b^2-a^2}\right) \cos ^{\frac{3}{2}}(c+d x)}{3 \left(a^2-b^2\right)}+\frac{\left(\frac{1}{8}+\frac{i}{8}\right) \left(2 \tan ^{-1}\left(1-\frac{(1+i) \sqrt{b} \sqrt{\cos (c+d x)}}{\sqrt[4]{b^2-a^2}}\right)-2 \tan ^{-1}\left(\frac{(1+i) \sqrt{b} \sqrt{\cos (c+d x)}}{\sqrt[4]{b^2-a^2}}+1\right)-\log \left(i b \cos (c+d x)-(1+i) \sqrt{b} \sqrt[4]{b^2-a^2} \sqrt{\cos (c+d x)}+\sqrt{b^2-a^2}\right)+\log \left(i b \cos (c+d x)+(1+i) \sqrt{b} \sqrt[4]{b^2-a^2} \sqrt{\cos (c+d x)}+\sqrt{b^2-a^2}\right)\right)}{\sqrt{b} \sqrt[4]{b^2-a^2}}\right) \sin (c+d x)}{\sqrt{1-\cos ^2(c+d x)} (a+b \sin (c+d x))}\right)}{16 (a-b)^4 (a+b)^4 d (e \cos (c+d x))^{3/2}}","\frac{13 a b}{12 d e \left(a^2-b^2\right)^2 \sqrt{e \cos (c+d x)} (a+b \sin (c+d x))^2}+\frac{b \left(89 a^2+28 b^2\right)}{24 d e \left(a^2-b^2\right)^3 \sqrt{e \cos (c+d x)} (a+b \sin (c+d x))}+\frac{b}{3 d e \left(a^2-b^2\right) \sqrt{e \cos (c+d x)} (a+b \sin (c+d x))^3}-\frac{15 a^2 b \left(7 a^2+6 b^2\right) \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b-\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{16 d e \left(a^2-b^2\right)^4 \left(b-\sqrt{b^2-a^2}\right) \sqrt{e \cos (c+d x)}}-\frac{15 a^2 b \left(7 a^2+6 b^2\right) \sqrt{\cos (c+d x)} \Pi \left(\frac{2 b}{b+\sqrt{b^2-a^2}};\left.\frac{1}{2} (c+d x)\right|2\right)}{16 d e \left(a^2-b^2\right)^4 \left(\sqrt{b^2-a^2}+b\right) \sqrt{e \cos (c+d x)}}-\frac{15 a b^{3/2} \left(7 a^2+6 b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{16 d e^{3/2} \left(b^2-a^2\right)^{17/4}}+\frac{15 a b^{3/2} \left(7 a^2+6 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt[4]{b^2-a^2}}\right)}{16 d e^{3/2} \left(b^2-a^2\right)^{17/4}}-\frac{\left(16 a^4+151 a^2 b^2+28 b^4\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{e \cos (c+d x)}}{8 d e^2 \left(a^2-b^2\right)^4 \sqrt{\cos (c+d x)}}-\frac{15 a b \left(7 a^2+6 b^2\right)-\left(16 a^4+151 a^2 b^2+28 b^4\right) \sin (c+d x)}{8 d e \left(a^2-b^2\right)^4 \sqrt{e \cos (c+d x)}}",1,"-1/16*(Cos[c + d*x]^(3/2)*((-2*(16*a^5 + 256*a^3*b^2 + 118*a*b^4)*(a + b*Sqrt[1 - Cos[c + d*x]^2])*((a*AppellF1[3/4, 1/2, 1, 7/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)]*Cos[c + d*x]^(3/2))/(3*(a^2 - b^2)) + ((1/8 + I/8)*(2*ArcTan[1 - ((1 + I)*Sqrt[b]*Sqrt[Cos[c + d*x]])/(-a^2 + b^2)^(1/4)] - 2*ArcTan[1 + ((1 + I)*Sqrt[b]*Sqrt[Cos[c + d*x]])/(-a^2 + b^2)^(1/4)] - Log[Sqrt[-a^2 + b^2] - (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Cos[c + d*x]] + I*b*Cos[c + d*x]] + Log[Sqrt[-a^2 + b^2] + (1 + I)*Sqrt[b]*(-a^2 + b^2)^(1/4)*Sqrt[Cos[c + d*x]] + I*b*Cos[c + d*x]]))/(Sqrt[b]*(-a^2 + b^2)^(1/4)))*Sin[c + d*x])/(Sqrt[1 - Cos[c + d*x]^2]*(a + b*Sin[c + d*x])) - ((16*a^4*b + 151*a^2*b^3 + 28*b^5)*(a + b*Sqrt[1 - Cos[c + d*x]^2])*(8*b^(5/2)*AppellF1[3/4, -1/2, 1, 7/4, Cos[c + d*x]^2, (b^2*Cos[c + d*x]^2)/(-a^2 + b^2)]*Cos[c + d*x]^(3/2) + 3*Sqrt[2]*a*(a^2 - b^2)^(3/4)*(2*ArcTan[1 - (Sqrt[2]*Sqrt[b]*Sqrt[Cos[c + d*x]])/(a^2 - b^2)^(1/4)] - 2*ArcTan[1 + (Sqrt[2]*Sqrt[b]*Sqrt[Cos[c + d*x]])/(a^2 - b^2)^(1/4)] - Log[Sqrt[a^2 - b^2] - Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Cos[c + d*x]] + b*Cos[c + d*x]] + Log[Sqrt[a^2 - b^2] + Sqrt[2]*Sqrt[b]*(a^2 - b^2)^(1/4)*Sqrt[Cos[c + d*x]] + b*Cos[c + d*x]]))*Sin[c + d*x]^2)/(12*b^(3/2)*(-a^2 + b^2)*(1 - Cos[c + d*x]^2)*(a + b*Sin[c + d*x]))))/((a - b)^4*(a + b)^4*d*(e*Cos[c + d*x])^(3/2)) + (Cos[c + d*x]^2*(-1/3*(b^3*Cos[c + d*x])/((a^2 - b^2)^2*(a + b*Sin[c + d*x])^3) - (7*a*b^3*Cos[c + d*x])/(4*(a^2 - b^2)^3*(a + b*Sin[c + d*x])^2) + (-55*a^2*b^3*Cos[c + d*x] - 12*b^5*Cos[c + d*x])/(8*(a^2 - b^2)^4*(a + b*Sin[c + d*x])) + (2*Sec[c + d*x]*(-4*a^3*b - 4*a*b^3 + a^4*Sin[c + d*x] + 6*a^2*b^2*Sin[c + d*x] + b^4*Sin[c + d*x]))/(a^2 - b^2)^4))/(d*(e*Cos[c + d*x])^(3/2))","C",0
615,1,117,183,0.3195252,"\int \frac{1}{\sqrt{c \cos (e+f x)} \sqrt{a+b \sin (e+f x)}} \, dx","Integrate[1/(Sqrt[c*Cos[e + f*x]]*Sqrt[a + b*Sin[e + f*x]]),x]","-\frac{2 c (\sin (e+f x)-1) \left(\frac{(a+b) (\sin (e+f x)+1)}{(a-b) (\sin (e+f x)-1)}\right)^{3/4} \sqrt{a+b \sin (e+f x)} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{3}{2};-\frac{2 (a+b \sin (e+f x))}{(a-b) (\sin (e+f x)-1)}\right)}{f (a+b) (c \cos (e+f x))^{3/2}}","\frac{2 \sqrt{2} \sqrt[4]{b-a} \sqrt{c \cos (e+f x)} \sqrt{\frac{a+b \sin (e+f x)}{(a-b) (1-\sin (e+f x))}} F\left(\left.\sin ^{-1}\left(\frac{\sqrt[4]{a+b} \sqrt{\frac{\cos (e+f x)+\sin (e+f x)+1}{\cos (e+f x)-\sin (e+f x)+1}}}{\sqrt[4]{b-a}}\right)\right|-1\right)}{c f \sqrt[4]{a+b} \sqrt{\frac{\sin (e+f x)+\cos (e+f x)+1}{-\sin (e+f x)+\cos (e+f x)+1}} \sqrt{a+b \sin (e+f x)}}",1,"(-2*c*Hypergeometric2F1[1/2, 3/4, 3/2, (-2*(a + b*Sin[e + f*x]))/((a - b)*(-1 + Sin[e + f*x]))]*(-1 + Sin[e + f*x])*(((a + b)*(1 + Sin[e + f*x]))/((a - b)*(-1 + Sin[e + f*x])))^(3/4)*Sqrt[a + b*Sin[e + f*x]])/((a + b)*f*(c*Cos[e + f*x])^(3/2))","C",1
616,1,290,229,54.8841081,"\int (e \cos (c+d x))^p (a+b \sin (c+d x))^3 \, dx","Integrate[(e*Cos[c + d*x])^p*(a + b*Sin[c + d*x])^3,x]","\frac{8 \sec ^2(c+d x)^{p/2} (a+b \sin (c+d x))^3 (e \cos (c+d x))^p \left(a^3 \tan (c+d x) \, _2F_1\left(\frac{1}{2},\frac{p+4}{2};\frac{3}{2};-\tan ^2(c+d x)\right)+\frac{1}{3} a \left(a^2+3 b^2\right) \tan ^3(c+d x) \, _2F_1\left(\frac{3}{2},\frac{p+4}{2};\frac{5}{2};-\tan ^2(c+d x)\right)-\frac{b \left(3 a^2+b^2\right) \left((p+3) \tan ^2(c+d x)+2\right) \sec ^2(c+d x)^{-\frac{p}{2}-\frac{3}{2}}}{(p+1) (p+3)}-\frac{3 a^2 b \sec ^2(c+d x)^{-\frac{p}{2}-\frac{3}{2}}}{p+3}\right)}{d \left(8 a^3+2 b \left(6 a^2+b^2\right) \sin (2 (c+d x)) \sqrt{\sec ^2(c+d x)}-12 a b^2 \cos (2 (c+d x))+12 a b^2-b^3 \sin (4 (c+d x)) \sqrt{\sec ^2(c+d x)}\right)}","-\frac{a \left(a^2 (p+2)+3 b^2\right) \sin (c+d x) (e \cos (c+d x))^{p+1} \, _2F_1\left(\frac{1}{2},\frac{p+1}{2};\frac{p+3}{2};\cos ^2(c+d x)\right)}{d e (p+1) (p+2) \sqrt{\sin ^2(c+d x)}}-\frac{b \left(a^2 \left(p^2+6 p+11\right)+2 b^2 (p+2)\right) (e \cos (c+d x))^{p+1}}{d e (p+1) (p+2) (p+3)}-\frac{b (a+b \sin (c+d x))^2 (e \cos (c+d x))^{p+1}}{d e (p+3)}-\frac{a b (p+5) (a+b \sin (c+d x)) (e \cos (c+d x))^{p+1}}{d e (p+2) (p+3)}",1,"(8*(e*Cos[c + d*x])^p*(Sec[c + d*x]^2)^(p/2)*(a + b*Sin[c + d*x])^3*((-3*a^2*b*(Sec[c + d*x]^2)^(-3/2 - p/2))/(3 + p) + a^3*Hypergeometric2F1[1/2, (4 + p)/2, 3/2, -Tan[c + d*x]^2]*Tan[c + d*x] + (a*(a^2 + 3*b^2)*Hypergeometric2F1[3/2, (4 + p)/2, 5/2, -Tan[c + d*x]^2]*Tan[c + d*x]^3)/3 - (b*(3*a^2 + b^2)*(Sec[c + d*x]^2)^(-3/2 - p/2)*(2 + (3 + p)*Tan[c + d*x]^2))/((1 + p)*(3 + p))))/(d*(8*a^3 + 12*a*b^2 - 12*a*b^2*Cos[2*(c + d*x)] + 2*b*(6*a^2 + b^2)*Sqrt[Sec[c + d*x]^2]*Sin[2*(c + d*x)] - b^3*Sqrt[Sec[c + d*x]^2]*Sin[4*(c + d*x)]))","A",0
617,1,285,157,1.0496928,"\int (e \cos (c+d x))^p (a+b \sin (c+d x))^2 \, dx","Integrate[(e*Cos[c + d*x])^p*(a + b*Sin[c + d*x])^2,x]","-\frac{(e \cos (c+d x))^p \left(-\frac{1}{2} a^2 (p-1) \sin (2 (c+d x)) \, _2F_1\left(\frac{1}{2},\frac{p+1}{2};\frac{p+3}{2};\cos ^2(c+d x)\right)+a b 2^{-p} \left(1+e^{2 i (c+d x)}\right) \left(e^{-i (c+d x)}+e^{i (c+d x)}\right)^p \sqrt{\sin ^2(c+d x)} \left((p+1) e^{i (c+d x)} \, _2F_1\left(1,\frac{p+3}{2};\frac{3-p}{2};-e^{2 i (c+d x)}\right)-(p-1) e^{-i (c+d x)} \, _2F_1\left(1,\frac{p+1}{2};\frac{1-p}{2};-e^{2 i (c+d x)}\right)\right) \cos ^{-p}(c+d x)-\frac{1}{2} b^2 (p-1) \sin (2 (c+d x)) \, _2F_1\left(-\frac{1}{2},\frac{p+1}{2};\frac{p+3}{2};\cos ^2(c+d x)\right)\right)}{\left(d-d p^2\right) \sqrt{\sin ^2(c+d x)}}","-\frac{\left(a^2 (p+2)+b^2\right) \sin (c+d x) (e \cos (c+d x))^{p+1} \, _2F_1\left(\frac{1}{2},\frac{p+1}{2};\frac{p+3}{2};\cos ^2(c+d x)\right)}{d e (p+1) (p+2) \sqrt{\sin ^2(c+d x)}}-\frac{a b (p+3) (e \cos (c+d x))^{p+1}}{d e (p+1) (p+2)}-\frac{b (a+b \sin (c+d x)) (e \cos (c+d x))^{p+1}}{d e (p+2)}",1,"-(((e*Cos[c + d*x])^p*((a*b*(E^((-I)*(c + d*x)) + E^(I*(c + d*x)))^p*(1 + E^((2*I)*(c + d*x)))*(-(((-1 + p)*Hypergeometric2F1[1, (1 + p)/2, (1 - p)/2, -E^((2*I)*(c + d*x))])/E^(I*(c + d*x))) + E^(I*(c + d*x))*(1 + p)*Hypergeometric2F1[1, (3 + p)/2, (3 - p)/2, -E^((2*I)*(c + d*x))])*Sqrt[Sin[c + d*x]^2])/(2^p*Cos[c + d*x]^p) - (b^2*(-1 + p)*Hypergeometric2F1[-1/2, (1 + p)/2, (3 + p)/2, Cos[c + d*x]^2]*Sin[2*(c + d*x)])/2 - (a^2*(-1 + p)*Hypergeometric2F1[1/2, (1 + p)/2, (3 + p)/2, Cos[c + d*x]^2]*Sin[2*(c + d*x)])/2))/((d - d*p^2)*Sqrt[Sin[c + d*x]^2]))","C",0
618,1,240,97,0.9934295,"\int (e \cos (c+d x))^p (a+b \sin (c+d x)) \, dx","Integrate[(e*Cos[c + d*x])^p*(a + b*Sin[c + d*x]),x]","-\frac{(e \cos (c+d x))^p \left(-\frac{1}{2} a (p-1) \sin (2 (c+d x)) \, _2F_1\left(\frac{1}{2},\frac{p+1}{2};\frac{p+3}{2};\cos ^2(c+d x)\right)+b 2^{-p-1} \left(1+e^{2 i (c+d x)}\right) \left(e^{-i (c+d x)}+e^{i (c+d x)}\right)^p \sqrt{\sin ^2(c+d x)} \left((p+1) e^{i (c+d x)} \, _2F_1\left(1,\frac{p+3}{2};\frac{3-p}{2};-e^{2 i (c+d x)}\right)-(p-1) e^{-i (c+d x)} \, _2F_1\left(1,\frac{p+1}{2};\frac{1-p}{2};-e^{2 i (c+d x)}\right)\right) \cos ^{-p}(c+d x)\right)}{\left(d-d p^2\right) \sqrt{\sin ^2(c+d x)}}","-\frac{a \sin (c+d x) (e \cos (c+d x))^{p+1} \, _2F_1\left(\frac{1}{2},\frac{p+1}{2};\frac{p+3}{2};\cos ^2(c+d x)\right)}{d e (p+1) \sqrt{\sin ^2(c+d x)}}-\frac{b (e \cos (c+d x))^{p+1}}{d e (p+1)}",1,"-(((e*Cos[c + d*x])^p*((2^(-1 - p)*b*(E^((-I)*(c + d*x)) + E^(I*(c + d*x)))^p*(1 + E^((2*I)*(c + d*x)))*(-(((-1 + p)*Hypergeometric2F1[1, (1 + p)/2, (1 - p)/2, -E^((2*I)*(c + d*x))])/E^(I*(c + d*x))) + E^(I*(c + d*x))*(1 + p)*Hypergeometric2F1[1, (3 + p)/2, (3 - p)/2, -E^((2*I)*(c + d*x))])*Sqrt[Sin[c + d*x]^2])/Cos[c + d*x]^p - (a*(-1 + p)*Hypergeometric2F1[1/2, (1 + p)/2, (3 + p)/2, Cos[c + d*x]^2]*Sin[2*(c + d*x)])/2))/((d - d*p^2)*Sqrt[Sin[c + d*x]^2]))","C",0
619,1,3815,158,20.2674823,"\int \frac{(e \cos (c+d x))^p}{a+b \sin (c+d x)} \, dx","Integrate[(e*Cos[c + d*x])^p/(a + b*Sin[c + d*x]),x]","\text{Result too large to show}","-\frac{e (e \cos (c+d x))^{p-1} \left(-\frac{b (1-\sin (c+d x))}{a+b \sin (c+d x)}\right)^{\frac{1-p}{2}} \left(\frac{b (\sin (c+d x)+1)}{a+b \sin (c+d x)}\right)^{\frac{1-p}{2}} F_1\left(1-p;\frac{1-p}{2},\frac{1-p}{2};2-p;\frac{a+b}{a+b \sin (c+d x)},\frac{a-b}{a+b \sin (c+d x)}\right)}{b d (1-p)}",1,"((e*Cos[c + d*x])^p*Tan[c + d*x]*(a*Sqrt[Sec[c + d*x]^2] + b*Tan[c + d*x])*(-(b*AppellF1[1, (1 + p)/2, 1, 2, -Tan[c + d*x]^2, (-1 + b^2/a^2)*Tan[c + d*x]^2]*Tan[c + d*x]) - (6*a^5*AppellF1[1/2, p/2, 1, 3/2, -Tan[c + d*x]^2, ((-a^2 + b^2)*Tan[c + d*x]^2)/a^2])/((Sec[c + d*x]^2)^(p/2)*(a^2 + (a^2 - b^2)*Tan[c + d*x]^2)*(-3*a^2*AppellF1[1/2, p/2, 1, 3/2, -Tan[c + d*x]^2, (-1 + b^2/a^2)*Tan[c + d*x]^2] + (2*(a^2 - b^2)*AppellF1[3/2, p/2, 2, 5/2, -Tan[c + d*x]^2, (-1 + b^2/a^2)*Tan[c + d*x]^2] + a^2*p*AppellF1[3/2, (2 + p)/2, 1, 5/2, -Tan[c + d*x]^2, (-1 + b^2/a^2)*Tan[c + d*x]^2])*Tan[c + d*x]^2))))/(2*a^2*d*Sqrt[Sec[c + d*x]^2]*(a + b*Sin[c + d*x])*(a + (b*Tan[c + d*x])/Sqrt[Sec[c + d*x]^2])*((Sqrt[Sec[c + d*x]^2]*(a*Sqrt[Sec[c + d*x]^2] + b*Tan[c + d*x])*(-(b*AppellF1[1, (1 + p)/2, 1, 2, -Tan[c + d*x]^2, (-1 + b^2/a^2)*Tan[c + d*x]^2]*Tan[c + d*x]) - (6*a^5*AppellF1[1/2, p/2, 1, 3/2, -Tan[c + d*x]^2, ((-a^2 + b^2)*Tan[c + d*x]^2)/a^2])/((Sec[c + d*x]^2)^(p/2)*(a^2 + (a^2 - b^2)*Tan[c + d*x]^2)*(-3*a^2*AppellF1[1/2, p/2, 1, 3/2, -Tan[c + d*x]^2, (-1 + b^2/a^2)*Tan[c + d*x]^2] + (2*(a^2 - b^2)*AppellF1[3/2, p/2, 2, 5/2, -Tan[c + d*x]^2, (-1 + b^2/a^2)*Tan[c + d*x]^2] + a^2*p*AppellF1[3/2, (2 + p)/2, 1, 5/2, -Tan[c + d*x]^2, (-1 + b^2/a^2)*Tan[c + d*x]^2])*Tan[c + d*x]^2))))/(2*a^2*(a + (b*Tan[c + d*x])/Sqrt[Sec[c + d*x]^2])) - (Tan[c + d*x]^2*(a*Sqrt[Sec[c + d*x]^2] + b*Tan[c + d*x])*(-(b*AppellF1[1, (1 + p)/2, 1, 2, -Tan[c + d*x]^2, (-1 + b^2/a^2)*Tan[c + d*x]^2]*Tan[c + d*x]) - (6*a^5*AppellF1[1/2, p/2, 1, 3/2, -Tan[c + d*x]^2, ((-a^2 + b^2)*Tan[c + d*x]^2)/a^2])/((Sec[c + d*x]^2)^(p/2)*(a^2 + (a^2 - b^2)*Tan[c + d*x]^2)*(-3*a^2*AppellF1[1/2, p/2, 1, 3/2, -Tan[c + d*x]^2, (-1 + b^2/a^2)*Tan[c + d*x]^2] + (2*(a^2 - b^2)*AppellF1[3/2, p/2, 2, 5/2, -Tan[c + d*x]^2, (-1 + b^2/a^2)*Tan[c + d*x]^2] + a^2*p*AppellF1[3/2, (2 + p)/2, 1, 5/2, -Tan[c + d*x]^2, (-1 + b^2/a^2)*Tan[c + d*x]^2])*Tan[c + d*x]^2))))/(2*a^2*Sqrt[Sec[c + d*x]^2]*(a + (b*Tan[c + d*x])/Sqrt[Sec[c + d*x]^2])) + (Tan[c + d*x]*(b*Sec[c + d*x]^2 + a*Sqrt[Sec[c + d*x]^2]*Tan[c + d*x])*(-(b*AppellF1[1, (1 + p)/2, 1, 2, -Tan[c + d*x]^2, (-1 + b^2/a^2)*Tan[c + d*x]^2]*Tan[c + d*x]) - (6*a^5*AppellF1[1/2, p/2, 1, 3/2, -Tan[c + d*x]^2, ((-a^2 + b^2)*Tan[c + d*x]^2)/a^2])/((Sec[c + d*x]^2)^(p/2)*(a^2 + (a^2 - b^2)*Tan[c + d*x]^2)*(-3*a^2*AppellF1[1/2, p/2, 1, 3/2, -Tan[c + d*x]^2, (-1 + b^2/a^2)*Tan[c + d*x]^2] + (2*(a^2 - b^2)*AppellF1[3/2, p/2, 2, 5/2, -Tan[c + d*x]^2, (-1 + b^2/a^2)*Tan[c + d*x]^2] + a^2*p*AppellF1[3/2, (2 + p)/2, 1, 5/2, -Tan[c + d*x]^2, (-1 + b^2/a^2)*Tan[c + d*x]^2])*Tan[c + d*x]^2))))/(2*a^2*Sqrt[Sec[c + d*x]^2]*(a + (b*Tan[c + d*x])/Sqrt[Sec[c + d*x]^2])) - (Tan[c + d*x]*(a*Sqrt[Sec[c + d*x]^2] + b*Tan[c + d*x])*(b*Sqrt[Sec[c + d*x]^2] - (b*Tan[c + d*x]^2)/Sqrt[Sec[c + d*x]^2])*(-(b*AppellF1[1, (1 + p)/2, 1, 2, -Tan[c + d*x]^2, (-1 + b^2/a^2)*Tan[c + d*x]^2]*Tan[c + d*x]) - (6*a^5*AppellF1[1/2, p/2, 1, 3/2, -Tan[c + d*x]^2, ((-a^2 + b^2)*Tan[c + d*x]^2)/a^2])/((Sec[c + d*x]^2)^(p/2)*(a^2 + (a^2 - b^2)*Tan[c + d*x]^2)*(-3*a^2*AppellF1[1/2, p/2, 1, 3/2, -Tan[c + d*x]^2, (-1 + b^2/a^2)*Tan[c + d*x]^2] + (2*(a^2 - b^2)*AppellF1[3/2, p/2, 2, 5/2, -Tan[c + d*x]^2, (-1 + b^2/a^2)*Tan[c + d*x]^2] + a^2*p*AppellF1[3/2, (2 + p)/2, 1, 5/2, -Tan[c + d*x]^2, (-1 + b^2/a^2)*Tan[c + d*x]^2])*Tan[c + d*x]^2))))/(2*a^2*Sqrt[Sec[c + d*x]^2]*(a + (b*Tan[c + d*x])/Sqrt[Sec[c + d*x]^2])^2) + (Tan[c + d*x]*(a*Sqrt[Sec[c + d*x]^2] + b*Tan[c + d*x])*(-(b*AppellF1[1, (1 + p)/2, 1, 2, -Tan[c + d*x]^2, (-1 + b^2/a^2)*Tan[c + d*x]^2]*Sec[c + d*x]^2) - b*Tan[c + d*x]*((-1 + b^2/a^2)*AppellF1[2, (1 + p)/2, 2, 3, -Tan[c + d*x]^2, (-1 + b^2/a^2)*Tan[c + d*x]^2]*Sec[c + d*x]^2*Tan[c + d*x] - ((1 + p)*AppellF1[2, 1 + (1 + p)/2, 1, 3, -Tan[c + d*x]^2, (-1 + b^2/a^2)*Tan[c + d*x]^2]*Sec[c + d*x]^2*Tan[c + d*x])/2) + (12*a^5*(a^2 - b^2)*AppellF1[1/2, p/2, 1, 3/2, -Tan[c + d*x]^2, ((-a^2 + b^2)*Tan[c + d*x]^2)/a^2]*(Sec[c + d*x]^2)^(1 - p/2)*Tan[c + d*x])/((a^2 + (a^2 - b^2)*Tan[c + d*x]^2)^2*(-3*a^2*AppellF1[1/2, p/2, 1, 3/2, -Tan[c + d*x]^2, (-1 + b^2/a^2)*Tan[c + d*x]^2] + (2*(a^2 - b^2)*AppellF1[3/2, p/2, 2, 5/2, -Tan[c + d*x]^2, (-1 + b^2/a^2)*Tan[c + d*x]^2] + a^2*p*AppellF1[3/2, (2 + p)/2, 1, 5/2, -Tan[c + d*x]^2, (-1 + b^2/a^2)*Tan[c + d*x]^2])*Tan[c + d*x]^2)) + (6*a^5*p*AppellF1[1/2, p/2, 1, 3/2, -Tan[c + d*x]^2, ((-a^2 + b^2)*Tan[c + d*x]^2)/a^2]*Tan[c + d*x])/((Sec[c + d*x]^2)^(p/2)*(a^2 + (a^2 - b^2)*Tan[c + d*x]^2)*(-3*a^2*AppellF1[1/2, p/2, 1, 3/2, -Tan[c + d*x]^2, (-1 + b^2/a^2)*Tan[c + d*x]^2] + (2*(a^2 - b^2)*AppellF1[3/2, p/2, 2, 5/2, -Tan[c + d*x]^2, (-1 + b^2/a^2)*Tan[c + d*x]^2] + a^2*p*AppellF1[3/2, (2 + p)/2, 1, 5/2, -Tan[c + d*x]^2, (-1 + b^2/a^2)*Tan[c + d*x]^2])*Tan[c + d*x]^2)) - (6*a^5*(-1/3*(p*AppellF1[3/2, 1 + p/2, 1, 5/2, -Tan[c + d*x]^2, ((-a^2 + b^2)*Tan[c + d*x]^2)/a^2]*Sec[c + d*x]^2*Tan[c + d*x]) + (2*(-a^2 + b^2)*AppellF1[3/2, p/2, 2, 5/2, -Tan[c + d*x]^2, ((-a^2 + b^2)*Tan[c + d*x]^2)/a^2]*Sec[c + d*x]^2*Tan[c + d*x])/(3*a^2)))/((Sec[c + d*x]^2)^(p/2)*(a^2 + (a^2 - b^2)*Tan[c + d*x]^2)*(-3*a^2*AppellF1[1/2, p/2, 1, 3/2, -Tan[c + d*x]^2, (-1 + b^2/a^2)*Tan[c + d*x]^2] + (2*(a^2 - b^2)*AppellF1[3/2, p/2, 2, 5/2, -Tan[c + d*x]^2, (-1 + b^2/a^2)*Tan[c + d*x]^2] + a^2*p*AppellF1[3/2, (2 + p)/2, 1, 5/2, -Tan[c + d*x]^2, (-1 + b^2/a^2)*Tan[c + d*x]^2])*Tan[c + d*x]^2)) + (6*a^5*AppellF1[1/2, p/2, 1, 3/2, -Tan[c + d*x]^2, ((-a^2 + b^2)*Tan[c + d*x]^2)/a^2]*(2*(2*(a^2 - b^2)*AppellF1[3/2, p/2, 2, 5/2, -Tan[c + d*x]^2, (-1 + b^2/a^2)*Tan[c + d*x]^2] + a^2*p*AppellF1[3/2, (2 + p)/2, 1, 5/2, -Tan[c + d*x]^2, (-1 + b^2/a^2)*Tan[c + d*x]^2])*Sec[c + d*x]^2*Tan[c + d*x] - 3*a^2*(-1/3*(p*AppellF1[3/2, 1 + p/2, 1, 5/2, -Tan[c + d*x]^2, (-1 + b^2/a^2)*Tan[c + d*x]^2]*Sec[c + d*x]^2*Tan[c + d*x]) + (2*(-1 + b^2/a^2)*AppellF1[3/2, p/2, 2, 5/2, -Tan[c + d*x]^2, (-1 + b^2/a^2)*Tan[c + d*x]^2]*Sec[c + d*x]^2*Tan[c + d*x])/3) + Tan[c + d*x]^2*(2*(a^2 - b^2)*((-3*p*AppellF1[5/2, 1 + p/2, 2, 7/2, -Tan[c + d*x]^2, (-1 + b^2/a^2)*Tan[c + d*x]^2]*Sec[c + d*x]^2*Tan[c + d*x])/5 + (12*(-1 + b^2/a^2)*AppellF1[5/2, p/2, 3, 7/2, -Tan[c + d*x]^2, (-1 + b^2/a^2)*Tan[c + d*x]^2]*Sec[c + d*x]^2*Tan[c + d*x])/5) + a^2*p*((6*(-1 + b^2/a^2)*AppellF1[5/2, (2 + p)/2, 2, 7/2, -Tan[c + d*x]^2, (-1 + b^2/a^2)*Tan[c + d*x]^2]*Sec[c + d*x]^2*Tan[c + d*x])/5 - (3*(2 + p)*AppellF1[5/2, 1 + (2 + p)/2, 1, 7/2, -Tan[c + d*x]^2, (-1 + b^2/a^2)*Tan[c + d*x]^2]*Sec[c + d*x]^2*Tan[c + d*x])/5))))/((Sec[c + d*x]^2)^(p/2)*(a^2 + (a^2 - b^2)*Tan[c + d*x]^2)*(-3*a^2*AppellF1[1/2, p/2, 1, 3/2, -Tan[c + d*x]^2, (-1 + b^2/a^2)*Tan[c + d*x]^2] + (2*(a^2 - b^2)*AppellF1[3/2, p/2, 2, 5/2, -Tan[c + d*x]^2, (-1 + b^2/a^2)*Tan[c + d*x]^2] + a^2*p*AppellF1[3/2, (2 + p)/2, 1, 5/2, -Tan[c + d*x]^2, (-1 + b^2/a^2)*Tan[c + d*x]^2])*Tan[c + d*x]^2)^2)))/(2*a^2*Sqrt[Sec[c + d*x]^2]*(a + (b*Tan[c + d*x])/Sqrt[Sec[c + d*x]^2]))))","B",0
620,1,4727,170,25.5405214,"\int \frac{(e \cos (c+d x))^p}{(a+b \sin (c+d x))^2} \, dx","Integrate[(e*Cos[c + d*x])^p/(a + b*Sin[c + d*x])^2,x]","\text{Result too large to show}","-\frac{e (e \cos (c+d x))^{p-1} \left(-\frac{b (1-\sin (c+d x))}{a+b \sin (c+d x)}\right)^{\frac{1-p}{2}} \left(\frac{b (\sin (c+d x)+1)}{a+b \sin (c+d x)}\right)^{\frac{1-p}{2}} F_1\left(2-p;\frac{1-p}{2},\frac{1-p}{2};3-p;\frac{a+b}{a+b \sin (c+d x)},\frac{a-b}{a+b \sin (c+d x)}\right)}{b d (2-p) (a+b \sin (c+d x))}",1,"((e*Cos[c + d*x])^p*Tan[c + d*x]*(b*(a^2 - b^2)*AppellF1[1, (-1 + p)/2, 2, 2, -Tan[c + d*x]^2, (-1 + b^2/a^2)*Tan[c + d*x]^2]*Tan[c + d*x] + (3*a^5*((-2*a^2*b^2*AppellF1[1/2, p/2, 2, 3/2, -Tan[c + d*x]^2, (-1 + b^2/a^2)*Tan[c + d*x]^2])/((-3*a^2*AppellF1[1/2, p/2, 2, 3/2, -Tan[c + d*x]^2, (-1 + b^2/a^2)*Tan[c + d*x]^2] + (4*(a^2 - b^2)*AppellF1[3/2, p/2, 3, 5/2, -Tan[c + d*x]^2, (-1 + b^2/a^2)*Tan[c + d*x]^2] + a^2*p*AppellF1[3/2, (2 + p)/2, 2, 5/2, -Tan[c + d*x]^2, (-1 + b^2/a^2)*Tan[c + d*x]^2])*Tan[c + d*x]^2)*(b^2*Tan[c + d*x]^2 - a^2*(1 + Tan[c + d*x]^2))^2) + ((a^2 + b^2)*AppellF1[1/2, p/2, 1, 3/2, -Tan[c + d*x]^2, (-1 + b^2/a^2)*Tan[c + d*x]^2])/((-3*a^2*AppellF1[1/2, p/2, 1, 3/2, -Tan[c + d*x]^2, (-1 + b^2/a^2)*Tan[c + d*x]^2] + (2*(a^2 - b^2)*AppellF1[3/2, p/2, 2, 5/2, -Tan[c + d*x]^2, (-1 + b^2/a^2)*Tan[c + d*x]^2] + a^2*p*AppellF1[3/2, (2 + p)/2, 1, 5/2, -Tan[c + d*x]^2, (-1 + b^2/a^2)*Tan[c + d*x]^2])*Tan[c + d*x]^2)*(-(b^2*Tan[c + d*x]^2) + a^2*(1 + Tan[c + d*x]^2)))))/(1 + Tan[c + d*x]^2)^(p/2)))/(a^3*(-a^2 + b^2)*d*(a + b*Sin[c + d*x])^2*((Sec[c + d*x]^2*(b*(a^2 - b^2)*AppellF1[1, (-1 + p)/2, 2, 2, -Tan[c + d*x]^2, (-1 + b^2/a^2)*Tan[c + d*x]^2]*Tan[c + d*x] + (3*a^5*((-2*a^2*b^2*AppellF1[1/2, p/2, 2, 3/2, -Tan[c + d*x]^2, (-1 + b^2/a^2)*Tan[c + d*x]^2])/((-3*a^2*AppellF1[1/2, p/2, 2, 3/2, -Tan[c + d*x]^2, (-1 + b^2/a^2)*Tan[c + d*x]^2] + (4*(a^2 - b^2)*AppellF1[3/2, p/2, 3, 5/2, -Tan[c + d*x]^2, (-1 + b^2/a^2)*Tan[c + d*x]^2] + a^2*p*AppellF1[3/2, (2 + p)/2, 2, 5/2, -Tan[c + d*x]^2, (-1 + b^2/a^2)*Tan[c + d*x]^2])*Tan[c + d*x]^2)*(b^2*Tan[c + d*x]^2 - a^2*(1 + Tan[c + d*x]^2))^2) + ((a^2 + b^2)*AppellF1[1/2, p/2, 1, 3/2, -Tan[c + d*x]^2, (-1 + b^2/a^2)*Tan[c + d*x]^2])/((-3*a^2*AppellF1[1/2, p/2, 1, 3/2, -Tan[c + d*x]^2, (-1 + b^2/a^2)*Tan[c + d*x]^2] + (2*(a^2 - b^2)*AppellF1[3/2, p/2, 2, 5/2, -Tan[c + d*x]^2, (-1 + b^2/a^2)*Tan[c + d*x]^2] + a^2*p*AppellF1[3/2, (2 + p)/2, 1, 5/2, -Tan[c + d*x]^2, (-1 + b^2/a^2)*Tan[c + d*x]^2])*Tan[c + d*x]^2)*(-(b^2*Tan[c + d*x]^2) + a^2*(1 + Tan[c + d*x]^2)))))/(1 + Tan[c + d*x]^2)^(p/2)))/(a^3*(-a^2 + b^2)) + (Tan[c + d*x]*(b*(a^2 - b^2)*AppellF1[1, (-1 + p)/2, 2, 2, -Tan[c + d*x]^2, (-1 + b^2/a^2)*Tan[c + d*x]^2]*Sec[c + d*x]^2 + b*(a^2 - b^2)*Tan[c + d*x]*(-1/2*((-1 + p)*AppellF1[2, 1 + (-1 + p)/2, 2, 3, -Tan[c + d*x]^2, (-1 + b^2/a^2)*Tan[c + d*x]^2]*Sec[c + d*x]^2*Tan[c + d*x]) + 2*(-1 + b^2/a^2)*AppellF1[2, (-1 + p)/2, 3, 3, -Tan[c + d*x]^2, (-1 + b^2/a^2)*Tan[c + d*x]^2]*Sec[c + d*x]^2*Tan[c + d*x]) - 3*a^5*p*Sec[c + d*x]^2*Tan[c + d*x]*(1 + Tan[c + d*x]^2)^(-1 - p/2)*((-2*a^2*b^2*AppellF1[1/2, p/2, 2, 3/2, -Tan[c + d*x]^2, (-1 + b^2/a^2)*Tan[c + d*x]^2])/((-3*a^2*AppellF1[1/2, p/2, 2, 3/2, -Tan[c + d*x]^2, (-1 + b^2/a^2)*Tan[c + d*x]^2] + (4*(a^2 - b^2)*AppellF1[3/2, p/2, 3, 5/2, -Tan[c + d*x]^2, (-1 + b^2/a^2)*Tan[c + d*x]^2] + a^2*p*AppellF1[3/2, (2 + p)/2, 2, 5/2, -Tan[c + d*x]^2, (-1 + b^2/a^2)*Tan[c + d*x]^2])*Tan[c + d*x]^2)*(b^2*Tan[c + d*x]^2 - a^2*(1 + Tan[c + d*x]^2))^2) + ((a^2 + b^2)*AppellF1[1/2, p/2, 1, 3/2, -Tan[c + d*x]^2, (-1 + b^2/a^2)*Tan[c + d*x]^2])/((-3*a^2*AppellF1[1/2, p/2, 1, 3/2, -Tan[c + d*x]^2, (-1 + b^2/a^2)*Tan[c + d*x]^2] + (2*(a^2 - b^2)*AppellF1[3/2, p/2, 2, 5/2, -Tan[c + d*x]^2, (-1 + b^2/a^2)*Tan[c + d*x]^2] + a^2*p*AppellF1[3/2, (2 + p)/2, 1, 5/2, -Tan[c + d*x]^2, (-1 + b^2/a^2)*Tan[c + d*x]^2])*Tan[c + d*x]^2)*(-(b^2*Tan[c + d*x]^2) + a^2*(1 + Tan[c + d*x]^2)))) + (3*a^5*((4*a^2*b^2*AppellF1[1/2, p/2, 2, 3/2, -Tan[c + d*x]^2, (-1 + b^2/a^2)*Tan[c + d*x]^2]*(-2*a^2*Sec[c + d*x]^2*Tan[c + d*x] + 2*b^2*Sec[c + d*x]^2*Tan[c + d*x]))/((-3*a^2*AppellF1[1/2, p/2, 2, 3/2, -Tan[c + d*x]^2, (-1 + b^2/a^2)*Tan[c + d*x]^2] + (4*(a^2 - b^2)*AppellF1[3/2, p/2, 3, 5/2, -Tan[c + d*x]^2, (-1 + b^2/a^2)*Tan[c + d*x]^2] + a^2*p*AppellF1[3/2, (2 + p)/2, 2, 5/2, -Tan[c + d*x]^2, (-1 + b^2/a^2)*Tan[c + d*x]^2])*Tan[c + d*x]^2)*(b^2*Tan[c + d*x]^2 - a^2*(1 + Tan[c + d*x]^2))^3) - (2*a^2*b^2*(-1/3*(p*AppellF1[3/2, 1 + p/2, 2, 5/2, -Tan[c + d*x]^2, (-1 + b^2/a^2)*Tan[c + d*x]^2]*Sec[c + d*x]^2*Tan[c + d*x]) + (4*(-1 + b^2/a^2)*AppellF1[3/2, p/2, 3, 5/2, -Tan[c + d*x]^2, (-1 + b^2/a^2)*Tan[c + d*x]^2]*Sec[c + d*x]^2*Tan[c + d*x])/3))/((-3*a^2*AppellF1[1/2, p/2, 2, 3/2, -Tan[c + d*x]^2, (-1 + b^2/a^2)*Tan[c + d*x]^2] + (4*(a^2 - b^2)*AppellF1[3/2, p/2, 3, 5/2, -Tan[c + d*x]^2, (-1 + b^2/a^2)*Tan[c + d*x]^2] + a^2*p*AppellF1[3/2, (2 + p)/2, 2, 5/2, -Tan[c + d*x]^2, (-1 + b^2/a^2)*Tan[c + d*x]^2])*Tan[c + d*x]^2)*(b^2*Tan[c + d*x]^2 - a^2*(1 + Tan[c + d*x]^2))^2) - ((a^2 + b^2)*AppellF1[1/2, p/2, 1, 3/2, -Tan[c + d*x]^2, (-1 + b^2/a^2)*Tan[c + d*x]^2]*(2*a^2*Sec[c + d*x]^2*Tan[c + d*x] - 2*b^2*Sec[c + d*x]^2*Tan[c + d*x]))/((-3*a^2*AppellF1[1/2, p/2, 1, 3/2, -Tan[c + d*x]^2, (-1 + b^2/a^2)*Tan[c + d*x]^2] + (2*(a^2 - b^2)*AppellF1[3/2, p/2, 2, 5/2, -Tan[c + d*x]^2, (-1 + b^2/a^2)*Tan[c + d*x]^2] + a^2*p*AppellF1[3/2, (2 + p)/2, 1, 5/2, -Tan[c + d*x]^2, (-1 + b^2/a^2)*Tan[c + d*x]^2])*Tan[c + d*x]^2)*(-(b^2*Tan[c + d*x]^2) + a^2*(1 + Tan[c + d*x]^2))^2) + ((a^2 + b^2)*(-1/3*(p*AppellF1[3/2, 1 + p/2, 1, 5/2, -Tan[c + d*x]^2, (-1 + b^2/a^2)*Tan[c + d*x]^2]*Sec[c + d*x]^2*Tan[c + d*x]) + (2*(-1 + b^2/a^2)*AppellF1[3/2, p/2, 2, 5/2, -Tan[c + d*x]^2, (-1 + b^2/a^2)*Tan[c + d*x]^2]*Sec[c + d*x]^2*Tan[c + d*x])/3))/((-3*a^2*AppellF1[1/2, p/2, 1, 3/2, -Tan[c + d*x]^2, (-1 + b^2/a^2)*Tan[c + d*x]^2] + (2*(a^2 - b^2)*AppellF1[3/2, p/2, 2, 5/2, -Tan[c + d*x]^2, (-1 + b^2/a^2)*Tan[c + d*x]^2] + a^2*p*AppellF1[3/2, (2 + p)/2, 1, 5/2, -Tan[c + d*x]^2, (-1 + b^2/a^2)*Tan[c + d*x]^2])*Tan[c + d*x]^2)*(-(b^2*Tan[c + d*x]^2) + a^2*(1 + Tan[c + d*x]^2))) - ((a^2 + b^2)*AppellF1[1/2, p/2, 1, 3/2, -Tan[c + d*x]^2, (-1 + b^2/a^2)*Tan[c + d*x]^2]*(2*(2*(a^2 - b^2)*AppellF1[3/2, p/2, 2, 5/2, -Tan[c + d*x]^2, (-1 + b^2/a^2)*Tan[c + d*x]^2] + a^2*p*AppellF1[3/2, (2 + p)/2, 1, 5/2, -Tan[c + d*x]^2, (-1 + b^2/a^2)*Tan[c + d*x]^2])*Sec[c + d*x]^2*Tan[c + d*x] - 3*a^2*(-1/3*(p*AppellF1[3/2, 1 + p/2, 1, 5/2, -Tan[c + d*x]^2, (-1 + b^2/a^2)*Tan[c + d*x]^2]*Sec[c + d*x]^2*Tan[c + d*x]) + (2*(-1 + b^2/a^2)*AppellF1[3/2, p/2, 2, 5/2, -Tan[c + d*x]^2, (-1 + b^2/a^2)*Tan[c + d*x]^2]*Sec[c + d*x]^2*Tan[c + d*x])/3) + Tan[c + d*x]^2*(2*(a^2 - b^2)*((-3*p*AppellF1[5/2, 1 + p/2, 2, 7/2, -Tan[c + d*x]^2, (-1 + b^2/a^2)*Tan[c + d*x]^2]*Sec[c + d*x]^2*Tan[c + d*x])/5 + (12*(-1 + b^2/a^2)*AppellF1[5/2, p/2, 3, 7/2, -Tan[c + d*x]^2, (-1 + b^2/a^2)*Tan[c + d*x]^2]*Sec[c + d*x]^2*Tan[c + d*x])/5) + a^2*p*((6*(-1 + b^2/a^2)*AppellF1[5/2, (2 + p)/2, 2, 7/2, -Tan[c + d*x]^2, (-1 + b^2/a^2)*Tan[c + d*x]^2]*Sec[c + d*x]^2*Tan[c + d*x])/5 - (3*(2 + p)*AppellF1[5/2, 1 + (2 + p)/2, 1, 7/2, -Tan[c + d*x]^2, (-1 + b^2/a^2)*Tan[c + d*x]^2]*Sec[c + d*x]^2*Tan[c + d*x])/5))))/((-3*a^2*AppellF1[1/2, p/2, 1, 3/2, -Tan[c + d*x]^2, (-1 + b^2/a^2)*Tan[c + d*x]^2] + (2*(a^2 - b^2)*AppellF1[3/2, p/2, 2, 5/2, -Tan[c + d*x]^2, (-1 + b^2/a^2)*Tan[c + d*x]^2] + a^2*p*AppellF1[3/2, (2 + p)/2, 1, 5/2, -Tan[c + d*x]^2, (-1 + b^2/a^2)*Tan[c + d*x]^2])*Tan[c + d*x]^2)^2*(-(b^2*Tan[c + d*x]^2) + a^2*(1 + Tan[c + d*x]^2))) + (2*a^2*b^2*AppellF1[1/2, p/2, 2, 3/2, -Tan[c + d*x]^2, (-1 + b^2/a^2)*Tan[c + d*x]^2]*(2*(4*(a^2 - b^2)*AppellF1[3/2, p/2, 3, 5/2, -Tan[c + d*x]^2, (-1 + b^2/a^2)*Tan[c + d*x]^2] + a^2*p*AppellF1[3/2, (2 + p)/2, 2, 5/2, -Tan[c + d*x]^2, (-1 + b^2/a^2)*Tan[c + d*x]^2])*Sec[c + d*x]^2*Tan[c + d*x] - 3*a^2*(-1/3*(p*AppellF1[3/2, 1 + p/2, 2, 5/2, -Tan[c + d*x]^2, (-1 + b^2/a^2)*Tan[c + d*x]^2]*Sec[c + d*x]^2*Tan[c + d*x]) + (4*(-1 + b^2/a^2)*AppellF1[3/2, p/2, 3, 5/2, -Tan[c + d*x]^2, (-1 + b^2/a^2)*Tan[c + d*x]^2]*Sec[c + d*x]^2*Tan[c + d*x])/3) + Tan[c + d*x]^2*(4*(a^2 - b^2)*((-3*p*AppellF1[5/2, 1 + p/2, 3, 7/2, -Tan[c + d*x]^2, (-1 + b^2/a^2)*Tan[c + d*x]^2]*Sec[c + d*x]^2*Tan[c + d*x])/5 + (18*(-1 + b^2/a^2)*AppellF1[5/2, p/2, 4, 7/2, -Tan[c + d*x]^2, (-1 + b^2/a^2)*Tan[c + d*x]^2]*Sec[c + d*x]^2*Tan[c + d*x])/5) + a^2*p*((12*(-1 + b^2/a^2)*AppellF1[5/2, (2 + p)/2, 3, 7/2, -Tan[c + d*x]^2, (-1 + b^2/a^2)*Tan[c + d*x]^2]*Sec[c + d*x]^2*Tan[c + d*x])/5 - (3*(2 + p)*AppellF1[5/2, 1 + (2 + p)/2, 2, 7/2, -Tan[c + d*x]^2, (-1 + b^2/a^2)*Tan[c + d*x]^2]*Sec[c + d*x]^2*Tan[c + d*x])/5))))/((-3*a^2*AppellF1[1/2, p/2, 2, 3/2, -Tan[c + d*x]^2, (-1 + b^2/a^2)*Tan[c + d*x]^2] + (4*(a^2 - b^2)*AppellF1[3/2, p/2, 3, 5/2, -Tan[c + d*x]^2, (-1 + b^2/a^2)*Tan[c + d*x]^2] + a^2*p*AppellF1[3/2, (2 + p)/2, 2, 5/2, -Tan[c + d*x]^2, (-1 + b^2/a^2)*Tan[c + d*x]^2])*Tan[c + d*x]^2)^2*(b^2*Tan[c + d*x]^2 - a^2*(1 + Tan[c + d*x]^2))^2)))/(1 + Tan[c + d*x]^2)^(p/2)))/(a^3*(-a^2 + b^2))))","B",0
621,1,7781,170,28.2561838,"\int \frac{(e \cos (c+d x))^p}{(a+b \sin (c+d x))^3} \, dx","Integrate[(e*Cos[c + d*x])^p/(a + b*Sin[c + d*x])^3,x]","\text{Result too large to show}","-\frac{e (e \cos (c+d x))^{p-1} \left(-\frac{b (1-\sin (c+d x))}{a+b \sin (c+d x)}\right)^{\frac{1-p}{2}} \left(\frac{b (\sin (c+d x)+1)}{a+b \sin (c+d x)}\right)^{\frac{1-p}{2}} F_1\left(3-p;\frac{1-p}{2},\frac{1-p}{2};4-p;\frac{a+b}{a+b \sin (c+d x)},\frac{a-b}{a+b \sin (c+d x)}\right)}{b d (3-p) (a+b \sin (c+d x))^2}",1,"Result too large to show","B",0
622,0,0,170,67.6808327,"\int \frac{(e \cos (c+d x))^p}{(a+b \sin (c+d x))^8} \, dx","Integrate[(e*Cos[c + d*x])^p/(a + b*Sin[c + d*x])^8,x]","\int \frac{(e \cos (c+d x))^p}{(a+b \sin (c+d x))^8} \, dx","-\frac{e (e \cos (c+d x))^{p-1} \left(-\frac{b (1-\sin (c+d x))}{a+b \sin (c+d x)}\right)^{\frac{1-p}{2}} \left(\frac{b (\sin (c+d x)+1)}{a+b \sin (c+d x)}\right)^{\frac{1-p}{2}} F_1\left(8-p;\frac{1-p}{2},\frac{1-p}{2};9-p;\frac{a+b}{a+b \sin (c+d x)},\frac{a-b}{a+b \sin (c+d x)}\right)}{b d (8-p) (a+b \sin (c+d x))^7}",1,"Integrate[(e*Cos[c + d*x])^p/(a + b*Sin[c + d*x])^8, x]","F",-1
623,1,187,156,7.4640387,"\int (e \cos (c+d x))^p (a+b \sin (c+d x))^{5/2} \, dx","Integrate[(e*Cos[c + d*x])^p*(a + b*Sin[c + d*x])^(5/2),x]","\frac{2 e (a+b \sin (c+d x))^{7/2} (e \cos (c+d x))^{p-1} \left(\frac{\sqrt{b^2}-b \sin (c+d x)}{a+\sqrt{b^2}}\right)^{\frac{1-p}{2}} \left(\frac{\sqrt{b^2}+b \sin (c+d x)}{\sqrt{b^2}-a}\right)^{\frac{1-p}{2}} F_1\left(\frac{7}{2};\frac{1-p}{2},\frac{1-p}{2};\frac{9}{2};\frac{a+b \sin (c+d x)}{a-\sqrt{b^2}},\frac{a+b \sin (c+d x)}{a+\sqrt{b^2}}\right)}{7 b d}","\frac{2 e (a+b \sin (c+d x))^{7/2} (e \cos (c+d x))^{p-1} \left(1-\frac{a+b \sin (c+d x)}{a-b}\right)^{\frac{1-p}{2}} \left(1-\frac{a+b \sin (c+d x)}{a+b}\right)^{\frac{1-p}{2}} F_1\left(\frac{7}{2};\frac{1-p}{2},\frac{1-p}{2};\frac{9}{2};\frac{a+b \sin (c+d x)}{a-b},\frac{a+b \sin (c+d x)}{a+b}\right)}{7 b d}",1,"(2*e*AppellF1[7/2, (1 - p)/2, (1 - p)/2, 9/2, (a + b*Sin[c + d*x])/(a - Sqrt[b^2]), (a + b*Sin[c + d*x])/(a + Sqrt[b^2])]*(e*Cos[c + d*x])^(-1 + p)*((Sqrt[b^2] - b*Sin[c + d*x])/(a + Sqrt[b^2]))^((1 - p)/2)*(a + b*Sin[c + d*x])^(7/2)*((Sqrt[b^2] + b*Sin[c + d*x])/(-a + Sqrt[b^2]))^((1 - p)/2))/(7*b*d)","A",0
624,1,187,156,0.7908703,"\int (e \cos (c+d x))^p (a+b \sin (c+d x))^{3/2} \, dx","Integrate[(e*Cos[c + d*x])^p*(a + b*Sin[c + d*x])^(3/2),x]","\frac{2 e (a+b \sin (c+d x))^{5/2} (e \cos (c+d x))^{p-1} \left(\frac{\sqrt{b^2}-b \sin (c+d x)}{a+\sqrt{b^2}}\right)^{\frac{1-p}{2}} \left(\frac{\sqrt{b^2}+b \sin (c+d x)}{\sqrt{b^2}-a}\right)^{\frac{1-p}{2}} F_1\left(\frac{5}{2};\frac{1-p}{2},\frac{1-p}{2};\frac{7}{2};\frac{a+b \sin (c+d x)}{a-\sqrt{b^2}},\frac{a+b \sin (c+d x)}{a+\sqrt{b^2}}\right)}{5 b d}","\frac{2 e (a+b \sin (c+d x))^{5/2} (e \cos (c+d x))^{p-1} \left(1-\frac{a+b \sin (c+d x)}{a-b}\right)^{\frac{1-p}{2}} \left(1-\frac{a+b \sin (c+d x)}{a+b}\right)^{\frac{1-p}{2}} F_1\left(\frac{5}{2};\frac{1-p}{2},\frac{1-p}{2};\frac{7}{2};\frac{a+b \sin (c+d x)}{a-b},\frac{a+b \sin (c+d x)}{a+b}\right)}{5 b d}",1,"(2*e*AppellF1[5/2, (1 - p)/2, (1 - p)/2, 7/2, (a + b*Sin[c + d*x])/(a - Sqrt[b^2]), (a + b*Sin[c + d*x])/(a + Sqrt[b^2])]*(e*Cos[c + d*x])^(-1 + p)*((Sqrt[b^2] - b*Sin[c + d*x])/(a + Sqrt[b^2]))^((1 - p)/2)*(a + b*Sin[c + d*x])^(5/2)*((Sqrt[b^2] + b*Sin[c + d*x])/(-a + Sqrt[b^2]))^((1 - p)/2))/(5*b*d)","A",0
625,1,187,156,0.9618431,"\int (e \cos (c+d x))^p \sqrt{a+b \sin (c+d x)} \, dx","Integrate[(e*Cos[c + d*x])^p*Sqrt[a + b*Sin[c + d*x]],x]","\frac{2 e (a+b \sin (c+d x))^{3/2} (e \cos (c+d x))^{p-1} \left(\frac{\sqrt{b^2}-b \sin (c+d x)}{a+\sqrt{b^2}}\right)^{\frac{1-p}{2}} \left(\frac{\sqrt{b^2}+b \sin (c+d x)}{\sqrt{b^2}-a}\right)^{\frac{1-p}{2}} F_1\left(\frac{3}{2};\frac{1-p}{2},\frac{1-p}{2};\frac{5}{2};\frac{a+b \sin (c+d x)}{a-\sqrt{b^2}},\frac{a+b \sin (c+d x)}{a+\sqrt{b^2}}\right)}{3 b d}","\frac{2 e (a+b \sin (c+d x))^{3/2} (e \cos (c+d x))^{p-1} \left(1-\frac{a+b \sin (c+d x)}{a-b}\right)^{\frac{1-p}{2}} \left(1-\frac{a+b \sin (c+d x)}{a+b}\right)^{\frac{1-p}{2}} F_1\left(\frac{3}{2};\frac{1-p}{2},\frac{1-p}{2};\frac{5}{2};\frac{a+b \sin (c+d x)}{a-b},\frac{a+b \sin (c+d x)}{a+b}\right)}{3 b d}",1,"(2*e*AppellF1[3/2, (1 - p)/2, (1 - p)/2, 5/2, (a + b*Sin[c + d*x])/(a - Sqrt[b^2]), (a + b*Sin[c + d*x])/(a + Sqrt[b^2])]*(e*Cos[c + d*x])^(-1 + p)*((Sqrt[b^2] - b*Sin[c + d*x])/(a + Sqrt[b^2]))^((1 - p)/2)*(a + b*Sin[c + d*x])^(3/2)*((Sqrt[b^2] + b*Sin[c + d*x])/(-a + Sqrt[b^2]))^((1 - p)/2))/(3*b*d)","A",0
626,1,185,154,1.0755695,"\int \frac{(e \cos (c+d x))^p}{\sqrt{a+b \sin (c+d x)}} \, dx","Integrate[(e*Cos[c + d*x])^p/Sqrt[a + b*Sin[c + d*x]],x]","\frac{2 e \sqrt{a+b \sin (c+d x)} (e \cos (c+d x))^{p-1} \left(\frac{\sqrt{b^2}-b \sin (c+d x)}{a+\sqrt{b^2}}\right)^{\frac{1-p}{2}} \left(\frac{\sqrt{b^2}+b \sin (c+d x)}{\sqrt{b^2}-a}\right)^{\frac{1-p}{2}} F_1\left(\frac{1}{2};\frac{1-p}{2},\frac{1-p}{2};\frac{3}{2};\frac{a+b \sin (c+d x)}{a-\sqrt{b^2}},\frac{a+b \sin (c+d x)}{a+\sqrt{b^2}}\right)}{b d}","\frac{2 e \sqrt{a+b \sin (c+d x)} (e \cos (c+d x))^{p-1} \left(1-\frac{a+b \sin (c+d x)}{a-b}\right)^{\frac{1-p}{2}} \left(1-\frac{a+b \sin (c+d x)}{a+b}\right)^{\frac{1-p}{2}} F_1\left(\frac{1}{2};\frac{1-p}{2},\frac{1-p}{2};\frac{3}{2};\frac{a+b \sin (c+d x)}{a-b},\frac{a+b \sin (c+d x)}{a+b}\right)}{b d}",1,"(2*e*AppellF1[1/2, (1 - p)/2, (1 - p)/2, 3/2, (a + b*Sin[c + d*x])/(a - Sqrt[b^2]), (a + b*Sin[c + d*x])/(a + Sqrt[b^2])]*(e*Cos[c + d*x])^(-1 + p)*((Sqrt[b^2] - b*Sin[c + d*x])/(a + Sqrt[b^2]))^((1 - p)/2)*Sqrt[a + b*Sin[c + d*x]]*((Sqrt[b^2] + b*Sin[c + d*x])/(-a + Sqrt[b^2]))^((1 - p)/2))/(b*d)","A",0
627,1,185,154,2.7484794,"\int \frac{(e \cos (c+d x))^p}{(a+b \sin (c+d x))^{3/2}} \, dx","Integrate[(e*Cos[c + d*x])^p/(a + b*Sin[c + d*x])^(3/2),x]","-\frac{2 e (e \cos (c+d x))^{p-1} \left(\frac{\sqrt{b^2}-b \sin (c+d x)}{a+\sqrt{b^2}}\right)^{\frac{1-p}{2}} \left(\frac{\sqrt{b^2}+b \sin (c+d x)}{\sqrt{b^2}-a}\right)^{\frac{1-p}{2}} F_1\left(-\frac{1}{2};\frac{1-p}{2},\frac{1-p}{2};\frac{1}{2};\frac{a+b \sin (c+d x)}{a-\sqrt{b^2}},\frac{a+b \sin (c+d x)}{a+\sqrt{b^2}}\right)}{b d \sqrt{a+b \sin (c+d x)}}","-\frac{2 e (e \cos (c+d x))^{p-1} \left(1-\frac{a+b \sin (c+d x)}{a-b}\right)^{\frac{1-p}{2}} \left(1-\frac{a+b \sin (c+d x)}{a+b}\right)^{\frac{1-p}{2}} F_1\left(-\frac{1}{2};\frac{1-p}{2},\frac{1-p}{2};\frac{1}{2};\frac{a+b \sin (c+d x)}{a-b},\frac{a+b \sin (c+d x)}{a+b}\right)}{b d \sqrt{a+b \sin (c+d x)}}",1,"(-2*e*AppellF1[-1/2, (1 - p)/2, (1 - p)/2, 1/2, (a + b*Sin[c + d*x])/(a - Sqrt[b^2]), (a + b*Sin[c + d*x])/(a + Sqrt[b^2])]*(e*Cos[c + d*x])^(-1 + p)*((Sqrt[b^2] - b*Sin[c + d*x])/(a + Sqrt[b^2]))^((1 - p)/2)*((Sqrt[b^2] + b*Sin[c + d*x])/(-a + Sqrt[b^2]))^((1 - p)/2))/(b*d*Sqrt[a + b*Sin[c + d*x]])","A",0
628,1,187,156,3.0378196,"\int \frac{(e \cos (c+d x))^p}{(a+b \sin (c+d x))^{5/2}} \, dx","Integrate[(e*Cos[c + d*x])^p/(a + b*Sin[c + d*x])^(5/2),x]","-\frac{2 e (e \cos (c+d x))^{p-1} \left(\frac{\sqrt{b^2}-b \sin (c+d x)}{a+\sqrt{b^2}}\right)^{\frac{1-p}{2}} \left(\frac{\sqrt{b^2}+b \sin (c+d x)}{\sqrt{b^2}-a}\right)^{\frac{1-p}{2}} F_1\left(-\frac{3}{2};\frac{1-p}{2},\frac{1-p}{2};-\frac{1}{2};\frac{a+b \sin (c+d x)}{a-\sqrt{b^2}},\frac{a+b \sin (c+d x)}{a+\sqrt{b^2}}\right)}{3 b d (a+b \sin (c+d x))^{3/2}}","-\frac{2 e (e \cos (c+d x))^{p-1} \left(1-\frac{a+b \sin (c+d x)}{a-b}\right)^{\frac{1-p}{2}} \left(1-\frac{a+b \sin (c+d x)}{a+b}\right)^{\frac{1-p}{2}} F_1\left(-\frac{3}{2};\frac{1-p}{2},\frac{1-p}{2};-\frac{1}{2};\frac{a+b \sin (c+d x)}{a-b},\frac{a+b \sin (c+d x)}{a+b}\right)}{3 b d (a+b \sin (c+d x))^{3/2}}",1,"(-2*e*AppellF1[-3/2, (1 - p)/2, (1 - p)/2, -1/2, (a + b*Sin[c + d*x])/(a - Sqrt[b^2]), (a + b*Sin[c + d*x])/(a + Sqrt[b^2])]*(e*Cos[c + d*x])^(-1 + p)*((Sqrt[b^2] - b*Sin[c + d*x])/(a + Sqrt[b^2]))^((1 - p)/2)*((Sqrt[b^2] + b*Sin[c + d*x])/(-a + Sqrt[b^2]))^((1 - p)/2))/(3*b*d*(a + b*Sin[c + d*x])^(3/2))","A",0
629,0,0,158,2.4122267,"\int (e \cos (c+d x))^p (a+b \sin (c+d x))^m \, dx","Integrate[(e*Cos[c + d*x])^p*(a + b*Sin[c + d*x])^m,x]","\int (e \cos (c+d x))^p (a+b \sin (c+d x))^m \, dx","\frac{e (e \cos (c+d x))^{p-1} (a+b \sin (c+d x))^{m+1} \left(1-\frac{a+b \sin (c+d x)}{a-b}\right)^{\frac{1-p}{2}} \left(1-\frac{a+b \sin (c+d x)}{a+b}\right)^{\frac{1-p}{2}} F_1\left(m+1;\frac{1-p}{2},\frac{1-p}{2};m+2;\frac{a+b \sin (c+d x)}{a-b},\frac{a+b \sin (c+d x)}{a+b}\right)}{b d (m+1)}",1,"Integrate[(e*Cos[c + d*x])^p*(a + b*Sin[c + d*x])^m, x]","F",-1
630,1,459,254,6.1141292,"\int \cos ^7(c+d x) (a+b \sin (c+d x))^m \, dx","Integrate[Cos[c + d*x]^7*(a + b*Sin[c + d*x])^m,x]","\frac{\frac{6 \left(\left(b^2-a^2\right) \left(\frac{4 \left(\left(b^2-a^2\right) \left(-\frac{\left(a^2-b^2\right) (a+b \sin (c+d x))^{m+1}}{m+1}+\frac{2 a (a+b \sin (c+d x))^{m+2}}{m+2}-\frac{(a+b \sin (c+d x))^{m+3}}{m+3}\right)+a \left(-\frac{\left(a^2-b^2\right) (a+b \sin (c+d x))^{m+2}}{m+2}+\frac{2 a (a+b \sin (c+d x))^{m+3}}{m+3}-\frac{(a+b \sin (c+d x))^{m+4}}{m+4}\right)\right)}{m+5}+\frac{b^4 \cos ^4(c+d x) (a+b \sin (c+d x))^{m+1}}{m+5}\right)+a \left(\frac{4 \left(\left(b^2-a^2\right) \left(-\frac{\left(a^2-b^2\right) (a+b \sin (c+d x))^{m+2}}{m+2}+\frac{2 a (a+b \sin (c+d x))^{m+3}}{m+3}-\frac{(a+b \sin (c+d x))^{m+4}}{m+4}\right)+a \left(-\frac{\left(a^2-b^2\right) (a+b \sin (c+d x))^{m+3}}{m+3}+\frac{2 a (a+b \sin (c+d x))^{m+4}}{m+4}-\frac{(a+b \sin (c+d x))^{m+5}}{m+5}\right)\right)}{m+6}+\frac{b^4 \cos ^4(c+d x) (a+b \sin (c+d x))^{m+2}}{m+6}\right)\right)}{m+7}+\frac{b^6 \cos ^6(c+d x) (a+b \sin (c+d x))^{m+1}}{m+7}}{b^7 d}","-\frac{\left(a^2-b^2\right)^3 (a+b \sin (c+d x))^{m+1}}{b^7 d (m+1)}+\frac{6 a \left(a^2-b^2\right)^2 (a+b \sin (c+d x))^{m+2}}{b^7 d (m+2)}+\frac{4 a \left(5 a^2-3 b^2\right) (a+b \sin (c+d x))^{m+4}}{b^7 d (m+4)}-\frac{3 \left(5 a^2-b^2\right) (a+b \sin (c+d x))^{m+5}}{b^7 d (m+5)}-\frac{3 \left(5 a^4-6 a^2 b^2+b^4\right) (a+b \sin (c+d x))^{m+3}}{b^7 d (m+3)}+\frac{6 a (a+b \sin (c+d x))^{m+6}}{b^7 d (m+6)}-\frac{(a+b \sin (c+d x))^{m+7}}{b^7 d (m+7)}",1,"((b^6*Cos[c + d*x]^6*(a + b*Sin[c + d*x])^(1 + m))/(7 + m) + (6*((-a^2 + b^2)*((b^4*Cos[c + d*x]^4*(a + b*Sin[c + d*x])^(1 + m))/(5 + m) + (4*((-a^2 + b^2)*(-(((a^2 - b^2)*(a + b*Sin[c + d*x])^(1 + m))/(1 + m)) + (2*a*(a + b*Sin[c + d*x])^(2 + m))/(2 + m) - (a + b*Sin[c + d*x])^(3 + m)/(3 + m)) + a*(-(((a^2 - b^2)*(a + b*Sin[c + d*x])^(2 + m))/(2 + m)) + (2*a*(a + b*Sin[c + d*x])^(3 + m))/(3 + m) - (a + b*Sin[c + d*x])^(4 + m)/(4 + m))))/(5 + m)) + a*((b^4*Cos[c + d*x]^4*(a + b*Sin[c + d*x])^(2 + m))/(6 + m) + (4*((-a^2 + b^2)*(-(((a^2 - b^2)*(a + b*Sin[c + d*x])^(2 + m))/(2 + m)) + (2*a*(a + b*Sin[c + d*x])^(3 + m))/(3 + m) - (a + b*Sin[c + d*x])^(4 + m)/(4 + m)) + a*(-(((a^2 - b^2)*(a + b*Sin[c + d*x])^(3 + m))/(3 + m)) + (2*a*(a + b*Sin[c + d*x])^(4 + m))/(4 + m) - (a + b*Sin[c + d*x])^(5 + m)/(5 + m))))/(6 + m))))/(7 + m))/(b^7*d)","A",1
631,1,169,167,0.90182,"\int \cos ^5(c+d x) (a+b \sin (c+d x))^m \, dx","Integrate[Cos[c + d*x]^5*(a + b*Sin[c + d*x])^m,x]","\frac{(a+b \sin (c+d x))^{m+1} \left(4 \left(b^2-a^2\right) \left(\frac{b^2-a^2}{m+1}-\frac{(a+b \sin (c+d x))^2}{m+3}+\frac{2 a (a+b \sin (c+d x))}{m+2}\right)+4 a (a+b \sin (c+d x)) \left(\frac{b^2-a^2}{m+2}-\frac{(a+b \sin (c+d x))^2}{m+4}+\frac{2 a (a+b \sin (c+d x))}{m+3}\right)+b^4 \cos ^4(c+d x)\right)}{b^5 d (m+5)}","\frac{\left(a^2-b^2\right)^2 (a+b \sin (c+d x))^{m+1}}{b^5 d (m+1)}-\frac{4 a \left(a^2-b^2\right) (a+b \sin (c+d x))^{m+2}}{b^5 d (m+2)}+\frac{2 \left(3 a^2-b^2\right) (a+b \sin (c+d x))^{m+3}}{b^5 d (m+3)}-\frac{4 a (a+b \sin (c+d x))^{m+4}}{b^5 d (m+4)}+\frac{(a+b \sin (c+d x))^{m+5}}{b^5 d (m+5)}",1,"((a + b*Sin[c + d*x])^(1 + m)*(b^4*Cos[c + d*x]^4 + 4*(-a^2 + b^2)*((-a^2 + b^2)/(1 + m) + (2*a*(a + b*Sin[c + d*x]))/(2 + m) - (a + b*Sin[c + d*x])^2/(3 + m)) + 4*a*(a + b*Sin[c + d*x])*((-a^2 + b^2)/(2 + m) + (2*a*(a + b*Sin[c + d*x]))/(3 + m) - (a + b*Sin[c + d*x])^2/(4 + m))))/(b^5*d*(5 + m))","A",1
632,1,74,92,0.1774891,"\int \cos ^3(c+d x) (a+b \sin (c+d x))^m \, dx","Integrate[Cos[c + d*x]^3*(a + b*Sin[c + d*x])^m,x]","\frac{(a+b \sin (c+d x))^{m+1} \left(\frac{b^2-a^2}{m+1}-\frac{(a+b \sin (c+d x))^2}{m+3}+\frac{2 a (a+b \sin (c+d x))}{m+2}\right)}{b^3 d}","-\frac{\left(a^2-b^2\right) (a+b \sin (c+d x))^{m+1}}{b^3 d (m+1)}+\frac{2 a (a+b \sin (c+d x))^{m+2}}{b^3 d (m+2)}-\frac{(a+b \sin (c+d x))^{m+3}}{b^3 d (m+3)}",1,"((a + b*Sin[c + d*x])^(1 + m)*((-a^2 + b^2)/(1 + m) + (2*a*(a + b*Sin[c + d*x]))/(2 + m) - (a + b*Sin[c + d*x])^2/(3 + m)))/(b^3*d)","A",1
633,1,26,26,0.0242412,"\int \cos (c+d x) (a+b \sin (c+d x))^m \, dx","Integrate[Cos[c + d*x]*(a + b*Sin[c + d*x])^m,x]","\frac{(a+b \sin (c+d x))^{m+1}}{b d (m+1)}","\frac{(a+b \sin (c+d x))^{m+1}}{b d (m+1)}",1,"(a + b*Sin[c + d*x])^(1 + m)/(b*d*(1 + m))","A",1
634,1,99,115,0.1176047,"\int \sec (c+d x) (a+b \sin (c+d x))^m \, dx","Integrate[Sec[c + d*x]*(a + b*Sin[c + d*x])^m,x]","-\frac{(a+b \sin (c+d x))^{m+1} \left((a+b) \, _2F_1\left(1,m+1;m+2;\frac{a+b \sin (c+d x)}{a-b}\right)+(b-a) \, _2F_1\left(1,m+1;m+2;\frac{a+b \sin (c+d x)}{a+b}\right)\right)}{2 d (m+1) (a-b) (a+b)}","\frac{(a+b \sin (c+d x))^{m+1} \, _2F_1\left(1,m+1;m+2;\frac{a+b \sin (c+d x)}{a+b}\right)}{2 d (m+1) (a+b)}-\frac{(a+b \sin (c+d x))^{m+1} \, _2F_1\left(1,m+1;m+2;\frac{a+b \sin (c+d x)}{a-b}\right)}{2 d (m+1) (a-b)}",1,"-1/2*(((a + b)*Hypergeometric2F1[1, 1 + m, 2 + m, (a + b*Sin[c + d*x])/(a - b)] + (-a + b)*Hypergeometric2F1[1, 1 + m, 2 + m, (a + b*Sin[c + d*x])/(a + b)])*(a + b*Sin[c + d*x])^(1 + m))/((a - b)*(a + b)*d*(1 + m))","A",1
635,1,157,183,0.620596,"\int \sec ^3(c+d x) (a+b \sin (c+d x))^m \, dx","Integrate[Sec[c + d*x]^3*(a + b*Sin[c + d*x])^m,x]","\frac{(a+b \sin (c+d x))^{m+1} \left(\frac{b \left((a+b)^2 (a+b (m-1)) \, _2F_1\left(1,m+1;m+2;\frac{a+b \sin (c+d x)}{a-b}\right)-(a-b)^2 (a-b m+b) \, _2F_1\left(1,m+1;m+2;\frac{a+b \sin (c+d x)}{a+b}\right)\right)}{(m+1) (a-b) (a+b)}+2 b \sec ^2(c+d x) (b-a \sin (c+d x))\right)}{4 b d \left(b^2-a^2\right)}","-\frac{\sec ^2(c+d x) (b-a \sin (c+d x)) (a+b \sin (c+d x))^{m+1}}{2 d \left(a^2-b^2\right)}-\frac{(a-b (1-m)) (a+b \sin (c+d x))^{m+1} \, _2F_1\left(1,m+1;m+2;\frac{a+b \sin (c+d x)}{a-b}\right)}{4 d (m+1) (a-b)^2}+\frac{(a-b m+b) (a+b \sin (c+d x))^{m+1} \, _2F_1\left(1,m+1;m+2;\frac{a+b \sin (c+d x)}{a+b}\right)}{4 d (m+1) (a+b)^2}",1,"((a + b*Sin[c + d*x])^(1 + m)*((b*((a + b)^2*(a + b*(-1 + m))*Hypergeometric2F1[1, 1 + m, 2 + m, (a + b*Sin[c + d*x])/(a - b)] - (a - b)^2*(a + b - b*m)*Hypergeometric2F1[1, 1 + m, 2 + m, (a + b*Sin[c + d*x])/(a + b)]))/((a - b)*(a + b)*(1 + m)) + 2*b*Sec[c + d*x]^2*(b - a*Sin[c + d*x])))/(4*b*(-a^2 + b^2)*d)","A",1
636,1,260,305,3.9964236,"\int \sec ^5(c+d x) (a+b \sin (c+d x))^m \, dx","Integrate[Sec[c + d*x]^5*(a + b*Sin[c + d*x])^m,x]","\frac{(a+b \sin (c+d x))^{m+1} \left(\frac{(a+b)^3 \left(3 a^2+3 a b (m-2)+b^2 \left(m^2-4 m+3\right)\right) \, _2F_1\left(1,m+1;m+2;\frac{a+b \sin (c+d x)}{a-b}\right)-(a-b)^3 \left(3 a^2-3 a b (m-2)+b^2 \left(m^2-4 m+3\right)\right) \, _2F_1\left(1,m+1;m+2;\frac{a+b \sin (c+d x)}{a+b}\right)}{(m+1) (a-b) (a+b) \left(a^2-b^2\right)}+\frac{2 \sec ^2(c+d x) \left(-a \left(3 a^2+b^2 (2 m-5)\right) \sin (c+d x)+a^2 b (m+1)+b^3 (m-3)\right)}{a^2-b^2}+4 \sec ^4(c+d x) (b-a \sin (c+d x))\right)}{16 d \left(b^2-a^2\right)}","-\frac{\left(3 a^2-3 a b (2-m)+b^2 \left(m^2-4 m+3\right)\right) (a+b \sin (c+d x))^{m+1} \, _2F_1\left(1,m+1;m+2;\frac{a+b \sin (c+d x)}{a-b}\right)}{16 d (m+1) (a-b)^3}+\frac{\left(3 a^2+3 a b (2-m)+b^2 \left(m^2-4 m+3\right)\right) (a+b \sin (c+d x))^{m+1} \, _2F_1\left(1,m+1;m+2;\frac{a+b \sin (c+d x)}{a+b}\right)}{16 d (m+1) (a+b)^3}-\frac{\sec ^4(c+d x) (b-a \sin (c+d x)) (a+b \sin (c+d x))^{m+1}}{4 d \left(a^2-b^2\right)}+\frac{\sec ^2(c+d x) \left(a \left(3 a^2-b^2 (5-2 m)\right) \sin (c+d x)+b \left(b^2 (3-m)-a^2 (m+1)\right)\right) (a+b \sin (c+d x))^{m+1}}{8 d \left(a^2-b^2\right)^2}",1,"((a + b*Sin[c + d*x])^(1 + m)*(((a + b)^3*(3*a^2 + 3*a*b*(-2 + m) + b^2*(3 - 4*m + m^2))*Hypergeometric2F1[1, 1 + m, 2 + m, (a + b*Sin[c + d*x])/(a - b)] - (a - b)^3*(3*a^2 - 3*a*b*(-2 + m) + b^2*(3 - 4*m + m^2))*Hypergeometric2F1[1, 1 + m, 2 + m, (a + b*Sin[c + d*x])/(a + b)])/((a - b)*(a + b)*(a^2 - b^2)*(1 + m)) + 4*Sec[c + d*x]^4*(b - a*Sin[c + d*x]) + (2*Sec[c + d*x]^2*(b^3*(-3 + m) + a^2*b*(1 + m) - a*(3*a^2 + b^2*(-5 + 2*m))*Sin[c + d*x]))/(a^2 - b^2)))/(16*(-a^2 + b^2)*d)","A",1
637,0,0,129,4.2178974,"\int \cos ^4(c+d x) (a+b \sin (c+d x))^m \, dx","Integrate[Cos[c + d*x]^4*(a + b*Sin[c + d*x])^m,x]","\int \cos ^4(c+d x) (a+b \sin (c+d x))^m \, dx","\frac{\cos ^3(c+d x) (a+b \sin (c+d x))^{m+1} F_1\left(m+1;-\frac{3}{2},-\frac{3}{2};m+2;\frac{a+b \sin (c+d x)}{a-b},\frac{a+b \sin (c+d x)}{a+b}\right)}{b d (m+1) \left(1-\frac{a+b \sin (c+d x)}{a-b}\right)^{3/2} \left(1-\frac{a+b \sin (c+d x)}{a+b}\right)^{3/2}}",1,"Integrate[Cos[c + d*x]^4*(a + b*Sin[c + d*x])^m, x]","F",-1
638,0,0,127,5.8834076,"\int \cos ^2(c+d x) (a+b \sin (c+d x))^m \, dx","Integrate[Cos[c + d*x]^2*(a + b*Sin[c + d*x])^m,x]","\int \cos ^2(c+d x) (a+b \sin (c+d x))^m \, dx","\frac{\cos (c+d x) (a+b \sin (c+d x))^{m+1} F_1\left(m+1;-\frac{1}{2},-\frac{1}{2};m+2;\frac{a+b \sin (c+d x)}{a-b},\frac{a+b \sin (c+d x)}{a+b}\right)}{b d (m+1) \sqrt{1-\frac{a+b \sin (c+d x)}{a-b}} \sqrt{1-\frac{a+b \sin (c+d x)}{a+b}}}",1,"Integrate[Cos[c + d*x]^2*(a + b*Sin[c + d*x])^m, x]","F",-1
639,0,0,129,2.2288548,"\int \sec ^2(c+d x) (a+b \sin (c+d x))^m \, dx","Integrate[Sec[c + d*x]^2*(a + b*Sin[c + d*x])^m,x]","\int \sec ^2(c+d x) (a+b \sin (c+d x))^m \, dx","\frac{\sec ^3(c+d x) \left(1-\frac{a+b \sin (c+d x)}{a-b}\right)^{3/2} \left(1-\frac{a+b \sin (c+d x)}{a+b}\right)^{3/2} (a+b \sin (c+d x))^{m+1} F_1\left(m+1;\frac{3}{2},\frac{3}{2};m+2;\frac{a+b \sin (c+d x)}{a-b},\frac{a+b \sin (c+d x)}{a+b}\right)}{b d (m+1)}",1,"Integrate[Sec[c + d*x]^2*(a + b*Sin[c + d*x])^m, x]","F",-1
640,0,0,129,4.9164971,"\int \sec ^4(c+d x) (a+b \sin (c+d x))^m \, dx","Integrate[Sec[c + d*x]^4*(a + b*Sin[c + d*x])^m,x]","\int \sec ^4(c+d x) (a+b \sin (c+d x))^m \, dx","\frac{\sec ^5(c+d x) \left(1-\frac{a+b \sin (c+d x)}{a-b}\right)^{5/2} \left(1-\frac{a+b \sin (c+d x)}{a+b}\right)^{5/2} (a+b \sin (c+d x))^{m+1} F_1\left(m+1;\frac{5}{2},\frac{5}{2};m+2;\frac{a+b \sin (c+d x)}{a-b},\frac{a+b \sin (c+d x)}{a+b}\right)}{b d (m+1)}",1,"Integrate[Sec[c + d*x]^4*(a + b*Sin[c + d*x])^m, x]","F",-1
641,0,0,134,55.4538213,"\int (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))^m \, dx","Integrate[(e*Cos[c + d*x])^(5/2)*(a + b*Sin[c + d*x])^m,x]","\int (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))^m \, dx","\frac{e (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^{m+1} F_1\left(m+1;-\frac{3}{4},-\frac{3}{4};m+2;\frac{a+b \sin (c+d x)}{a-b},\frac{a+b \sin (c+d x)}{a+b}\right)}{b d (m+1) \left(1-\frac{a+b \sin (c+d x)}{a-b}\right)^{3/4} \left(1-\frac{a+b \sin (c+d x)}{a+b}\right)^{3/4}}",1,"Integrate[(e*Cos[c + d*x])^(5/2)*(a + b*Sin[c + d*x])^m, x]","F",-1
642,0,0,134,5.697418,"\int (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^m \, dx","Integrate[(e*Cos[c + d*x])^(3/2)*(a + b*Sin[c + d*x])^m,x]","\int (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^m \, dx","\frac{e \sqrt{e \cos (c+d x)} (a+b \sin (c+d x))^{m+1} F_1\left(m+1;-\frac{1}{4},-\frac{1}{4};m+2;\frac{a+b \sin (c+d x)}{a-b},\frac{a+b \sin (c+d x)}{a+b}\right)}{b d (m+1) \sqrt[4]{1-\frac{a+b \sin (c+d x)}{a-b}} \sqrt[4]{1-\frac{a+b \sin (c+d x)}{a+b}}}",1,"Integrate[(e*Cos[c + d*x])^(3/2)*(a + b*Sin[c + d*x])^m, x]","F",-1
643,0,0,134,1.9306654,"\int \sqrt{e \cos (c+d x)} (a+b \sin (c+d x))^m \, dx","Integrate[Sqrt[e*Cos[c + d*x]]*(a + b*Sin[c + d*x])^m,x]","\int \sqrt{e \cos (c+d x)} (a+b \sin (c+d x))^m \, dx","\frac{e \sqrt[4]{1-\frac{a+b \sin (c+d x)}{a-b}} \sqrt[4]{1-\frac{a+b \sin (c+d x)}{a+b}} (a+b \sin (c+d x))^{m+1} F_1\left(m+1;\frac{1}{4},\frac{1}{4};m+2;\frac{a+b \sin (c+d x)}{a-b},\frac{a+b \sin (c+d x)}{a+b}\right)}{b d (m+1) \sqrt{e \cos (c+d x)}}",1,"Integrate[Sqrt[e*Cos[c + d*x]]*(a + b*Sin[c + d*x])^m, x]","F",-1
644,0,0,134,1.7714472,"\int \frac{(a+b \sin (c+d x))^m}{\sqrt{e \cos (c+d x)}} \, dx","Integrate[(a + b*Sin[c + d*x])^m/Sqrt[e*Cos[c + d*x]],x]","\int \frac{(a+b \sin (c+d x))^m}{\sqrt{e \cos (c+d x)}} \, dx","\frac{e \left(1-\frac{a+b \sin (c+d x)}{a-b}\right)^{3/4} \left(1-\frac{a+b \sin (c+d x)}{a+b}\right)^{3/4} (a+b \sin (c+d x))^{m+1} F_1\left(m+1;\frac{3}{4},\frac{3}{4};m+2;\frac{a+b \sin (c+d x)}{a-b},\frac{a+b \sin (c+d x)}{a+b}\right)}{b d (m+1) (e \cos (c+d x))^{3/2}}",1,"Integrate[(a + b*Sin[c + d*x])^m/Sqrt[e*Cos[c + d*x]], x]","F",-1
645,0,0,134,1.9236294,"\int \frac{(a+b \sin (c+d x))^m}{(e \cos (c+d x))^{3/2}} \, dx","Integrate[(a + b*Sin[c + d*x])^m/(e*Cos[c + d*x])^(3/2),x]","\int \frac{(a+b \sin (c+d x))^m}{(e \cos (c+d x))^{3/2}} \, dx","\frac{e \left(1-\frac{a+b \sin (c+d x)}{a-b}\right)^{5/4} \left(1-\frac{a+b \sin (c+d x)}{a+b}\right)^{5/4} (a+b \sin (c+d x))^{m+1} F_1\left(m+1;\frac{5}{4},\frac{5}{4};m+2;\frac{a+b \sin (c+d x)}{a-b},\frac{a+b \sin (c+d x)}{a+b}\right)}{b d (m+1) (e \cos (c+d x))^{5/2}}",1,"Integrate[(a + b*Sin[c + d*x])^m/(e*Cos[c + d*x])^(3/2), x]","F",-1
646,0,0,134,2.1040226,"\int \frac{(a+b \sin (c+d x))^m}{(e \cos (c+d x))^{5/2}} \, dx","Integrate[(a + b*Sin[c + d*x])^m/(e*Cos[c + d*x])^(5/2),x]","\int \frac{(a+b \sin (c+d x))^m}{(e \cos (c+d x))^{5/2}} \, dx","\frac{e \left(1-\frac{a+b \sin (c+d x)}{a-b}\right)^{7/4} \left(1-\frac{a+b \sin (c+d x)}{a+b}\right)^{7/4} (a+b \sin (c+d x))^{m+1} F_1\left(m+1;\frac{7}{4},\frac{7}{4};m+2;\frac{a+b \sin (c+d x)}{a-b},\frac{a+b \sin (c+d x)}{a+b}\right)}{b d (m+1) (e \cos (c+d x))^{7/2}}",1,"Integrate[(a + b*Sin[c + d*x])^m/(e*Cos[c + d*x])^(5/2), x]","F",-1
647,1,826,598,6.0954683,"\int (e \cos (c+d x))^{-4-m} (a+b \sin (c+d x))^m \, dx","Integrate[(e*Cos[c + d*x])^(-4 - m)*(a + b*Sin[c + d*x])^m,x]","\frac{\cos (c+d x) (a+b \sin (c+d x))^{m+1} (e \cos (c+d x))^{-m-4}}{(a-b) d (-m-3)}+\frac{2 b \cos ^{m+4}(c+d x) \left(\frac{2^{\frac{1}{2} (-m-1)+1} a \, _2F_1\left(\frac{1}{2} (-m-1),\frac{m+1}{2};\frac{1}{2} (-m-1)+1;-\frac{(b-a) (1-\sin (c+d x))}{2 (a+b \sin (c+d x))}\right) (1-\sin (c+d x))^{\frac{1}{2} (-m-1)+\frac{m+1}{2}} (\sin (c+d x)+1)^{\frac{1}{2} (-m-1)+\frac{m+1}{2}} \left(-\frac{(-a-b) (\sin (c+d x)+1)}{a+b \sin (c+d x)}\right)^{\frac{m+1}{2}} (a+b \sin (c+d x))^{m+1} \cos ^{-m-1}(c+d x)}{(-a-b) (a-b) d (-m-1)}+\frac{(a+b \sin (c+d x))^{m+1} \cos ^{-m-1}(c+d x)}{(a-b) d (-m-1)}\right) (e \cos (c+d x))^{-m-4}}{(a-b) (-m-3)}+\frac{a \cos (c+d x) (1-\sin (c+d x))^{\frac{m+3}{2}} (\sin (c+d x)+1)^{\frac{m+3}{2}} \left(\frac{(1-\sin (c+d x))^{\frac{1}{2} (-m-3)} (\sin (c+d x)+1)^{\frac{1}{2} (-m-3)+1} (a+b \sin (c+d x))^{m+1}}{(-a-b) (-m-3)}-\frac{-\frac{(3 b+a (m+2)) (1-\sin (c+d x))^{\frac{1}{2} (-m-3)+1} (a+b \sin (c+d x))^{m+1} (\sin (c+d x)+1)^{\frac{1}{2} (-m-3)+1}}{2 (-a-b) \left(\frac{1}{2} (-m-3)+1\right)}-\frac{2^{\frac{1}{2} (-m-3)-1} (m+1) \left((m+2) a^2+2 b a-b^2\right) \, _2F_1\left(\frac{1}{2} (-m-3)+2,\frac{m+3}{2};\frac{1}{2} (-m-3)+3;-\frac{(b-a) (1-\sin (c+d x))}{2 (a+b \sin (c+d x))}\right) (1-\sin (c+d x))^{\frac{1}{2} (-m-3)+2} \left(-\frac{(-a-b) (\sin (c+d x)+1)}{a+b \sin (c+d x)}\right)^{\frac{m+3}{2}} (a+b \sin (c+d x))^{m+1} (\sin (c+d x)+1)^{\frac{1}{2} (-m-3)}}{(-a-b)^2 \left(\frac{1}{2} (-m-3)+1\right) \left(\frac{1}{2} (-m-3)+2\right)}}{(-a-b) (-m-3)}\right) (e \cos (c+d x))^{-m-4}}{(a-b) d}","-\frac{a 2^{-\frac{m}{2}-\frac{1}{2}} \left(a^2 (m+2)+2 a b-b^2\right) (1-\sin (c+d x))^2 (e \cos (c+d x))^{-m-3} \left(\frac{(a+b) (\sin (c+d x)+1)}{a+b \sin (c+d x)}\right)^{\frac{m+3}{2}} (a+b \sin (c+d x))^{m+1} \, _2F_1\left(\frac{1-m}{2},\frac{m+3}{2};\frac{3-m}{2};\frac{(a-b) (1-\sin (c+d x))}{2 (a+b \sin (c+d x))}\right)}{d e (1-m) (m+3) (a-b) (a+b)^3}+\frac{a (\sin (c+d x)+1) (e \cos (c+d x))^{-m-3} (a+b \sin (c+d x))^{m+1}}{d e (m+3) \left(a^2-b^2\right)}-\frac{a b 2^{\frac{3}{2}-\frac{m}{2}} (e \cos (c+d x))^{-m-1} \left(\frac{(a+b) (\sin (c+d x)+1)}{a+b \sin (c+d x)}\right)^{\frac{m+1}{2}} (a+b \sin (c+d x))^{m+1} \, _2F_1\left(\frac{1}{2} (-m-1),\frac{m+1}{2};\frac{1-m}{2};\frac{(a-b) (1-\sin (c+d x))}{2 (a+b \sin (c+d x))}\right)}{d e^3 (m+1) (m+3) (a-b)^2 (a+b)}+\frac{2 b (e \cos (c+d x))^{-m-1} (a+b \sin (c+d x))^{m+1}}{d e^3 (m+1) (m+3) (a-b)^2}+\frac{a (a (m+2)+3 b) (1-\sin (c+d x)) (\sin (c+d x)+1) (e \cos (c+d x))^{-m-3} (a+b \sin (c+d x))^{m+1}}{d e (m+1) (m+3) (a-b) (a+b)^2}-\frac{(e \cos (c+d x))^{-m-3} (a+b \sin (c+d x))^{m+1}}{d e (m+3) (a-b)}",1,"(Cos[c + d*x]*(e*Cos[c + d*x])^(-4 - m)*(a + b*Sin[c + d*x])^(1 + m))/((a - b)*d*(-3 - m)) + (2*b*Cos[c + d*x]^(4 + m)*(e*Cos[c + d*x])^(-4 - m)*((Cos[c + d*x]^(-1 - m)*(a + b*Sin[c + d*x])^(1 + m))/((a - b)*d*(-1 - m)) + (2^(1 + (-1 - m)/2)*a*Cos[c + d*x]^(-1 - m)*Hypergeometric2F1[(-1 - m)/2, (1 + m)/2, 1 + (-1 - m)/2, -1/2*((-a + b)*(1 - Sin[c + d*x]))/(a + b*Sin[c + d*x])]*(1 - Sin[c + d*x])^((-1 - m)/2 + (1 + m)/2)*(1 + Sin[c + d*x])^((-1 - m)/2 + (1 + m)/2)*(-(((-a - b)*(1 + Sin[c + d*x]))/(a + b*Sin[c + d*x])))^((1 + m)/2)*(a + b*Sin[c + d*x])^(1 + m))/((-a - b)*(a - b)*d*(-1 - m))))/((a - b)*(-3 - m)) + (a*Cos[c + d*x]*(e*Cos[c + d*x])^(-4 - m)*(1 - Sin[c + d*x])^((3 + m)/2)*(1 + Sin[c + d*x])^((3 + m)/2)*(((1 - Sin[c + d*x])^((-3 - m)/2)*(1 + Sin[c + d*x])^(1 + (-3 - m)/2)*(a + b*Sin[c + d*x])^(1 + m))/((-a - b)*(-3 - m)) - (-1/2*((3*b + a*(2 + m))*(1 - Sin[c + d*x])^(1 + (-3 - m)/2)*(1 + Sin[c + d*x])^(1 + (-3 - m)/2)*(a + b*Sin[c + d*x])^(1 + m))/((-a - b)*(1 + (-3 - m)/2)) - (2^(-1 + (-3 - m)/2)*(1 + m)*(2*a*b - b^2 + a^2*(2 + m))*Hypergeometric2F1[2 + (-3 - m)/2, (3 + m)/2, 3 + (-3 - m)/2, -1/2*((-a + b)*(1 - Sin[c + d*x]))/(a + b*Sin[c + d*x])]*(1 - Sin[c + d*x])^(2 + (-3 - m)/2)*(1 + Sin[c + d*x])^((-3 - m)/2)*(-(((-a - b)*(1 + Sin[c + d*x]))/(a + b*Sin[c + d*x])))^((3 + m)/2)*(a + b*Sin[c + d*x])^(1 + m))/((-a - b)^2*(1 + (-3 - m)/2)*(2 + (-3 - m)/2)))/((-a - b)*(-3 - m))))/((a - b)*d)","A",1
648,1,319,311,5.0469349,"\int (e \cos (c+d x))^{-3-m} (a+b \sin (c+d x))^m \, dx","Integrate[(e*Cos[c + d*x])^(-3 - m)*(a + b*Sin[c + d*x])^m,x]","\frac{\sec ^2(c+d x) (e \cos (c+d x))^{-m} (a+b \sin (c+d x))^m \left(\frac{b (\sin (c+d x)+1) (a+b \sin (c+d x)) \left(\frac{(a+b) (\sin (c+d x)+1)}{(a-b) (\sin (c+d x)-1)}\right)^{m/2} \, _2F_1\left(m+1,\frac{m+2}{2};m+2;-\frac{2 (a+b \sin (c+d x))}{(a-b) (\sin (c+d x)-1)}\right)}{(m+1) (a-b)}+\frac{a (1-\sin (c+d x)) (\sin (c+d x)+1) \left(2^{-m/2} (a m+a+b) \left(\frac{(a+b) (\sin (c+d x)+1)}{a+b \sin (c+d x)}\right)^{m/2} \, _2F_1\left(-\frac{m}{2},\frac{m+2}{2};1-\frac{m}{2};-\frac{(a-b) (\sin (c+d x)-1)}{2 (a+b \sin (c+d x))}\right)-\frac{m (a+b \sin (c+d x))}{\sin (c+d x)-1}\right)}{m (a+b)}-a-b \sin (c+d x)\right)}{d e^3 (m+2) (a-b)}","-\frac{\left(a^2 (m+1)-b^2\right) (\sin (c+d x)+1)^3 \sec ^4(c+d x) (e \cos (c+d x))^{-m} (a+b \sin (c+d x))^{m+1} \left(\frac{(a+b) (\sin (c+d x)+1)}{(a-b) (\sin (c+d x)-1)}\right)^{\frac{m-2}{2}} \, _2F_1\left(\frac{m}{2},m+1;m+2;-\frac{2 (a+b \sin (c+d x))}{(a-b) (\sin (c+d x)-1)}\right)}{d e^3 m (m+1) (a-b)^3}+\frac{(a (m+2)-2 b) (\sin (c+d x)-1) (\sin (c+d x)+1)^2 \sec ^4(c+d x) (e \cos (c+d x))^{-m} (a+b \sin (c+d x))^{m+1}}{d e^3 m (m+2) (a-b)^2}+\frac{(\sin (c+d x)-1) (\sin (c+d x)+1) \sec ^4(c+d x) (e \cos (c+d x))^{-m} (a+b \sin (c+d x))^{m+1}}{d e^3 (m+2) (a-b)}",1,"(Sec[c + d*x]^2*(a + b*Sin[c + d*x])^m*(-a - b*Sin[c + d*x] + (b*Hypergeometric2F1[1 + m, (2 + m)/2, 2 + m, (-2*(a + b*Sin[c + d*x]))/((a - b)*(-1 + Sin[c + d*x]))]*(1 + Sin[c + d*x])*(((a + b)*(1 + Sin[c + d*x]))/((a - b)*(-1 + Sin[c + d*x])))^(m/2)*(a + b*Sin[c + d*x]))/((a - b)*(1 + m)) + (a*(1 - Sin[c + d*x])*(1 + Sin[c + d*x])*(((a + b + a*m)*Hypergeometric2F1[-1/2*m, (2 + m)/2, 1 - m/2, -1/2*((a - b)*(-1 + Sin[c + d*x]))/(a + b*Sin[c + d*x])]*(((a + b)*(1 + Sin[c + d*x]))/(a + b*Sin[c + d*x]))^(m/2))/2^(m/2) - (m*(a + b*Sin[c + d*x]))/(-1 + Sin[c + d*x])))/((a + b)*m)))/((a - b)*d*e^3*(2 + m)*(e*Cos[c + d*x])^m)","A",1
649,1,168,201,0.9592535,"\int (e \cos (c+d x))^{-2-m} (a+b \sin (c+d x))^m \, dx","Integrate[(e*Cos[c + d*x])^(-2 - m)*(a + b*Sin[c + d*x])^m,x]","-\frac{2^{\frac{1}{2} (-m-1)} (e \cos (c+d x))^{-m-1} (a+b \sin (c+d x))^{m+1} \left(2^{\frac{m+1}{2}} (a+b)-2 a \left(\frac{(a+b) (\sin (c+d x)+1)}{a+b \sin (c+d x)}\right)^{\frac{m+1}{2}} \, _2F_1\left(\frac{1}{2} (-m-1),\frac{m+1}{2};\frac{1-m}{2};-\frac{(a-b) (\sin (c+d x)-1)}{2 (a+b \sin (c+d x))}\right)\right)}{d e (m+1) (a-b) (a+b)}","\frac{a 2^{\frac{1}{2}-\frac{m}{2}} (e \cos (c+d x))^{-m-1} \left(\frac{(a+b) (\sin (c+d x)+1)}{a+b \sin (c+d x)}\right)^{\frac{m+1}{2}} (a+b \sin (c+d x))^{m+1} \, _2F_1\left(\frac{1}{2} (-m-1),\frac{m+1}{2};\frac{1-m}{2};\frac{(a-b) (1-\sin (c+d x))}{2 (a+b \sin (c+d x))}\right)}{d e (m+1) \left(a^2-b^2\right)}-\frac{(e \cos (c+d x))^{-m-1} (a+b \sin (c+d x))^{m+1}}{d e (m+1) (a-b)}",1,"-((2^((-1 - m)/2)*(e*Cos[c + d*x])^(-1 - m)*(a + b*Sin[c + d*x])^(1 + m)*(2^((1 + m)/2)*(a + b) - 2*a*Hypergeometric2F1[(-1 - m)/2, (1 + m)/2, (1 - m)/2, -1/2*((a - b)*(-1 + Sin[c + d*x]))/(a + b*Sin[c + d*x])]*(((a + b)*(1 + Sin[c + d*x]))/(a + b*Sin[c + d*x]))^((1 + m)/2)))/((a - b)*(a + b)*d*e*(1 + m)))","A",1
650,1,132,132,0.3881986,"\int (e \cos (c+d x))^{-1-m} (a+b \sin (c+d x))^m \, dx","Integrate[(e*Cos[c + d*x])^(-1 - m)*(a + b*Sin[c + d*x])^m,x]","-\frac{e (\sin (c+d x)+1) (e \cos (c+d x))^{-m-2} \left(\frac{(a+b) (\sin (c+d x)+1)}{(a-b) (\sin (c+d x)-1)}\right)^{m/2} (a+b \sin (c+d x))^{m+1} \, _2F_1\left(m+1,\frac{m+2}{2};m+2;-\frac{2 (a+b \sin (c+d x))}{(a-b) (\sin (c+d x)-1)}\right)}{d (m+1) (a-b)}","\frac{e (1-\sin (c+d x)) (e \cos (c+d x))^{-m-2} \left(-\frac{(a-b) (1-\sin (c+d x))}{(a+b) (\sin (c+d x)+1)}\right)^{m/2} (a+b \sin (c+d x))^{m+1} \, _2F_1\left(m+1,\frac{m+2}{2};m+2;\frac{2 (a+b \sin (c+d x))}{(a+b) (\sin (c+d x)+1)}\right)}{d (m+1) (a+b)}",1,"-((e*(e*Cos[c + d*x])^(-2 - m)*Hypergeometric2F1[1 + m, (2 + m)/2, 2 + m, (-2*(a + b*Sin[c + d*x]))/((a - b)*(-1 + Sin[c + d*x]))]*(1 + Sin[c + d*x])*(((a + b)*(1 + Sin[c + d*x]))/((a - b)*(-1 + Sin[c + d*x])))^(m/2)*(a + b*Sin[c + d*x])^(1 + m))/((a - b)*d*(1 + m)))","A",1
651,0,0,152,1.6688252,"\int (e \cos (c+d x))^{-m} (a+b \sin (c+d x))^m \, dx","Integrate[(a + b*Sin[c + d*x])^m/(e*Cos[c + d*x])^m,x]","\int (e \cos (c+d x))^{-m} (a+b \sin (c+d x))^m \, dx","\frac{e (e \cos (c+d x))^{-m-1} (a+b \sin (c+d x))^{m+1} \left(1-\frac{a+b \sin (c+d x)}{a-b}\right)^{\frac{m+1}{2}} \left(1-\frac{a+b \sin (c+d x)}{a+b}\right)^{\frac{m+1}{2}} F_1\left(m+1;\frac{m+1}{2},\frac{m+1}{2};m+2;\frac{a+b \sin (c+d x)}{a-b},\frac{a+b \sin (c+d x)}{a+b}\right)}{b d (m+1)}",1,"Integrate[(a + b*Sin[c + d*x])^m/(e*Cos[c + d*x])^m, x]","F",-1
652,0,0,142,5.6322091,"\int (e \cos (c+d x))^{1-m} (a+b \sin (c+d x))^m \, dx","Integrate[(e*Cos[c + d*x])^(1 - m)*(a + b*Sin[c + d*x])^m,x]","\int (e \cos (c+d x))^{1-m} (a+b \sin (c+d x))^m \, dx","\frac{e (e \cos (c+d x))^{-m} (a+b \sin (c+d x))^{m+1} \left(1-\frac{a+b \sin (c+d x)}{a-b}\right)^{m/2} \left(1-\frac{a+b \sin (c+d x)}{a+b}\right)^{m/2} F_1\left(m+1;\frac{m}{2},\frac{m}{2};m+2;\frac{a+b \sin (c+d x)}{a-b},\frac{a+b \sin (c+d x)}{a+b}\right)}{b d (m+1)}",1,"Integrate[(e*Cos[c + d*x])^(1 - m)*(a + b*Sin[c + d*x])^m, x]","F",-1
653,0,0,152,4.7268348,"\int (e \cos (c+d x))^{2-m} (a+b \sin (c+d x))^m \, dx","Integrate[(e*Cos[c + d*x])^(2 - m)*(a + b*Sin[c + d*x])^m,x]","\int (e \cos (c+d x))^{2-m} (a+b \sin (c+d x))^m \, dx","\frac{e (e \cos (c+d x))^{1-m} (a+b \sin (c+d x))^{m+1} \left(1-\frac{a+b \sin (c+d x)}{a-b}\right)^{\frac{m-1}{2}} \left(1-\frac{a+b \sin (c+d x)}{a+b}\right)^{\frac{m-1}{2}} F_1\left(m+1;\frac{m-1}{2},\frac{m-1}{2};m+2;\frac{a+b \sin (c+d x)}{a-b},\frac{a+b \sin (c+d x)}{a+b}\right)}{b d (m+1)}",1,"Integrate[(e*Cos[c + d*x])^(2 - m)*(a + b*Sin[c + d*x])^m, x]","F",-1