1,1,76,85,0.1824211,"\int \sec ^4(c+d x) (a+a \sec (c+d x)) \, dx","Integrate[Sec[c + d*x]^4*(a + a*Sec[c + d*x]),x]","\frac{a \left(\frac{1}{3} \tan ^3(c+d x)+\tan (c+d x)\right)}{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 \tan ^3(c+d x)}{3 d}+\frac{a \tan (c+d x)}{d}+\frac{3 a \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a \tan (c+d x) \sec ^3(c+d x)}{4 d}+\frac{3 a \tan (c+d x) \sec (c+d x)}{8 d}",1,"(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*(Tan[c + d*x] + Tan[c + d*x]^3/3))/d","A",1
2,1,60,63,0.1551784,"\int \sec ^3(c+d x) (a+a \sec (c+d x)) \, dx","Integrate[Sec[c + d*x]^3*(a + a*Sec[c + d*x]),x]","\frac{a \left(\frac{1}{3} \tan ^3(c+d x)+\tan (c+d x)\right)}{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 \tan ^3(c+d x)}{3 d}+\frac{a \tan (c+d x)}{d}+\frac{a \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{a \tan (c+d x) \sec (c+d x)}{2 d}",1,"(a*ArcTanh[Sin[c + d*x]])/(2*d) + (a*Sec[c + d*x]*Tan[c + d*x])/(2*d) + (a*(Tan[c + d*x] + Tan[c + d*x]^3/3))/d","A",1
3,1,47,47,0.0216569,"\int \sec ^2(c+d x) (a+a \sec (c+d x)) \, dx","Integrate[Sec[c + d*x]^2*(a + a*Sec[c + d*x]),x]","\frac{a \tan (c+d x)}{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 \tan (c+d x)}{d}+\frac{a \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{a \tan (c+d x) \sec (c+d x)}{2 d}",1,"(a*ArcTanh[Sin[c + d*x]])/(2*d) + (a*Tan[c + d*x])/d + (a*Sec[c + d*x]*Tan[c + d*x])/(2*d)","A",1
4,1,24,24,0.0116287,"\int \sec (c+d x) (a+a \sec (c+d x)) \, dx","Integrate[Sec[c + d*x]*(a + a*Sec[c + d*x]),x]","\frac{a \tan (c+d x)}{d}+\frac{a \tanh ^{-1}(\sin (c+d x))}{d}","\frac{a \tan (c+d x)}{d}+\frac{a \tanh ^{-1}(\sin (c+d x))}{d}",1,"(a*ArcTanh[Sin[c + d*x]])/d + (a*Tan[c + d*x])/d","A",1
5,1,16,16,0.0017567,"\int (a+a \sec (c+d x)) \, dx","Integrate[a + a*Sec[c + d*x],x]","\frac{a \tanh ^{-1}(\sin (c+d x))}{d}+a x","\frac{a \tanh ^{-1}(\sin (c+d x))}{d}+a x",1,"a*x + (a*ArcTanh[Sin[c + d*x]])/d","A",1
6,1,26,15,0.0090725,"\int \cos (c+d x) (a+a \sec (c+d x)) \, dx","Integrate[Cos[c + d*x]*(a + a*Sec[c + d*x]),x]","\frac{a \sin (c) \cos (d x)}{d}+\frac{a \cos (c) \sin (d x)}{d}+a x","\frac{a \sin (c+d x)}{d}+a x",1,"a*x + (a*Cos[d*x]*Sin[c])/d + (a*Cos[c]*Sin[d*x])/d","A",1
7,1,32,38,0.0557892,"\int \cos ^2(c+d x) (a+a \sec (c+d x)) \, dx","Integrate[Cos[c + d*x]^2*(a + a*Sec[c + d*x]),x]","\frac{a (2 (c+d x)+4 \sin (c+d x)+\sin (2 (c+d x)))}{4 d}","\frac{a \sin (c+d x)}{d}+\frac{a \sin (c+d x) \cos (c+d x)}{2 d}+\frac{a x}{2}",1,"(a*(2*(c + d*x) + 4*Sin[c + d*x] + Sin[2*(c + d*x)]))/(4*d)","A",1
8,1,57,54,0.0702216,"\int \cos ^3(c+d x) (a+a \sec (c+d x)) \, dx","Integrate[Cos[c + d*x]^3*(a + a*Sec[c + d*x]),x]","\frac{a (c+d x)}{2 d}-\frac{a \sin ^3(c+d x)}{3 d}+\frac{a \sin (c+d x)}{d}+\frac{a \sin (2 (c+d x))}{4 d}","-\frac{a \sin ^3(c+d x)}{3 d}+\frac{a \sin (c+d x)}{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*Sin[c + d*x])/d - (a*Sin[c + d*x]^3)/(3*d) + (a*Sin[2*(c + d*x)])/(4*d)","A",1
9,1,73,76,0.1254167,"\int \cos ^4(c+d x) (a+a \sec (c+d x)) \, dx","Integrate[Cos[c + d*x]^4*(a + a*Sec[c + d*x]),x]","\frac{3 a (c+d x)}{8 d}-\frac{a \sin ^3(c+d x)}{3 d}+\frac{a \sin (c+d x)}{d}+\frac{a \sin (2 (c+d x))}{4 d}+\frac{a \sin (4 (c+d x))}{32 d}","-\frac{a \sin ^3(c+d x)}{3 d}+\frac{a \sin (c+d x)}{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*Sin[c + d*x])/d - (a*Sin[c + d*x]^3)/(3*d) + (a*Sin[2*(c + d*x)])/(4*d) + (a*Sin[4*(c + d*x)])/(32*d)","A",1
10,1,487,122,1.7238821,"\int \sec ^4(c+d x) (a+a \sec (c+d x))^2 \, dx","Integrate[Sec[c + d*x]^4*(a + a*Sec[c + d*x])^2,x]","-\frac{a^2 \sec (c) \sec ^5(c+d x) \left(80 \sin (2 c+d x)-140 \sin (c+2 d x)-140 \sin (3 c+2 d x)-240 \sin (2 c+3 d x)-30 \sin (3 c+4 d x)-30 \sin (5 c+4 d x)-48 \sin (4 c+5 d x)+75 \cos (2 c+3 d x) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+75 \cos (4 c+3 d x) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+15 \cos (4 c+5 d x) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+15 \cos (6 c+5 d x) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+150 \cos (d x) \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)+150 \cos (2 c+d x) \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)-75 \cos (2 c+3 d x) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)-75 \cos (4 c+3 d x) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)-15 \cos (4 c+5 d x) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)-15 \cos (6 c+5 d x) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)-400 \sin (d x)\right)}{640 d}","\frac{3 a^2 \tan ^3(c+d x)}{5 d}+\frac{9 a^2 \tan (c+d x)}{5 d}+\frac{3 a^2 \tanh ^{-1}(\sin (c+d x))}{4 d}+\frac{a^2 \tan (c+d x) \sec ^4(c+d x)}{5 d}+\frac{a^2 \tan (c+d x) \sec ^3(c+d x)}{2 d}+\frac{3 a^2 \tan (c+d x) \sec (c+d x)}{4 d}",1,"-1/640*(a^2*Sec[c]*Sec[c + d*x]^5*(75*Cos[2*c + 3*d*x]*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 75*Cos[4*c + 3*d*x]*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 15*Cos[4*c + 5*d*x]*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 15*Cos[6*c + 5*d*x]*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 150*Cos[d*x]*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) + 150*Cos[2*c + d*x]*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) - 75*Cos[2*c + 3*d*x]*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] - 75*Cos[4*c + 3*d*x]*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] - 15*Cos[4*c + 5*d*x]*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] - 15*Cos[6*c + 5*d*x]*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] - 400*Sin[d*x] + 80*Sin[2*c + d*x] - 140*Sin[c + 2*d*x] - 140*Sin[3*c + 2*d*x] - 240*Sin[2*c + 3*d*x] - 30*Sin[3*c + 4*d*x] - 30*Sin[5*c + 4*d*x] - 48*Sin[4*c + 5*d*x]))/d","B",1
11,1,877,96,6.4628258,"\int \sec ^3(c+d x) (a+a \sec (c+d x))^2 \, dx","Integrate[Sec[c + d*x]^3*(a + a*Sec[c + d*x])^2,x]","-\frac{7 \cos ^2(c+d x) \log \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)-\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right) (\sec (c+d x) a+a)^2 \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{32 d}+\frac{7 \cos ^2(c+d x) \log \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)+\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right) (\sec (c+d x) a+a)^2 \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{32 d}+\frac{\cos ^2(c+d x) (\sec (c+d x) a+a)^2 \sin \left(\frac{d x}{2}\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d \left(\cos \left(\frac{c}{2}\right)-\sin \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)-\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)}+\frac{\cos ^2(c+d x) (\sec (c+d x) a+a)^2 \sin \left(\frac{d x}{2}\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d \left(\cos \left(\frac{c}{2}\right)+\sin \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)+\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)}+\frac{\cos ^2(c+d x) (\sec (c+d x) a+a)^2 \left(29 \cos \left(\frac{c}{2}\right)-13 \sin \left(\frac{c}{2}\right)\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{192 d \left(\cos \left(\frac{c}{2}\right)-\sin \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)-\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)^2}+\frac{\cos ^2(c+d x) (\sec (c+d x) a+a)^2 \left(-29 \cos \left(\frac{c}{2}\right)-13 \sin \left(\frac{c}{2}\right)\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{192 d \left(\cos \left(\frac{c}{2}\right)+\sin \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)+\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)^2}+\frac{\cos ^2(c+d x) (\sec (c+d x) a+a)^2 \sin \left(\frac{d x}{2}\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{12 d \left(\cos \left(\frac{c}{2}\right)-\sin \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)-\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)^3}+\frac{\cos ^2(c+d x) (\sec (c+d x) a+a)^2 \sin \left(\frac{d x}{2}\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{12 d \left(\cos \left(\frac{c}{2}\right)+\sin \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)+\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)^3}+\frac{\cos ^2(c+d x) (\sec (c+d x) a+a)^2 \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{64 d \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)-\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)^4}-\frac{\cos ^2(c+d x) (\sec (c+d x) a+a)^2 \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{64 d \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)+\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)^4}","\frac{2 a^2 \tan ^3(c+d x)}{3 d}+\frac{2 a^2 \tan (c+d x)}{d}+\frac{7 a^2 \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a^2 \tan (c+d x) \sec ^3(c+d x)}{4 d}+\frac{7 a^2 \tan (c+d x) \sec (c+d x)}{8 d}",1,"(-7*Cos[c + d*x]^2*Log[Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2]]*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2)/(32*d) + (7*Cos[c + d*x]^2*Log[Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2]]*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2)/(32*d) + (Cos[c + d*x]^2*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2)/(64*d*(Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2])^4) + (Cos[c + d*x]^2*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2*Sin[(d*x)/2])/(12*d*(Cos[c/2] - Sin[c/2])*(Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2])^3) + (Cos[c + d*x]^2*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2*(29*Cos[c/2] - 13*Sin[c/2]))/(192*d*(Cos[c/2] - Sin[c/2])*(Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2])^2) + (Cos[c + d*x]^2*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2*Sin[(d*x)/2])/(3*d*(Cos[c/2] - Sin[c/2])*(Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2])) - (Cos[c + d*x]^2*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2)/(64*d*(Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2])^4) + (Cos[c + d*x]^2*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2*Sin[(d*x)/2])/(12*d*(Cos[c/2] + Sin[c/2])*(Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2])^3) + (Cos[c + d*x]^2*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2*(-29*Cos[c/2] - 13*Sin[c/2]))/(192*d*(Cos[c/2] + Sin[c/2])*(Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2])^2) + (Cos[c + d*x]^2*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2*Sin[(d*x)/2])/(3*d*(Cos[c/2] + Sin[c/2])*(Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2]))","B",0
12,1,318,74,0.6501258,"\int \sec ^2(c+d x) (a+a \sec (c+d x))^2 \, dx","Integrate[Sec[c + d*x]^2*(a + a*Sec[c + d*x])^2,x]","-\frac{a^2 \sec (c) \sec ^3(c+d x) \left(6 \sin (2 c+d x)-6 \sin (c+2 d x)-6 \sin (3 c+2 d x)-10 \sin (2 c+3 d x)+3 \cos (2 c+3 d x) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+3 \cos (4 c+3 d x) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+9 \cos (d x) \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)+9 \cos (2 c+d x) \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)-3 \cos (2 c+3 d x) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)-3 \cos (4 c+3 d x) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)-24 \sin (d x)\right)}{24 d}","\frac{5 a^2 \tan (c+d x)}{3 d}+\frac{a^2 \tanh ^{-1}(\sin (c+d x))}{d}+\frac{a^2 \tan (c+d x) \sec ^2(c+d x)}{3 d}+\frac{a^2 \tan (c+d x) \sec (c+d x)}{d}",1,"-1/24*(a^2*Sec[c]*Sec[c + d*x]^3*(3*Cos[2*c + 3*d*x]*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 3*Cos[4*c + 3*d*x]*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 9*Cos[d*x]*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) + 9*Cos[2*c + d*x]*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) - 3*Cos[2*c + 3*d*x]*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] - 3*Cos[4*c + 3*d*x]*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] - 24*Sin[d*x] + 6*Sin[2*c + d*x] - 6*Sin[c + 2*d*x] - 6*Sin[3*c + 2*d*x] - 10*Sin[2*c + 3*d*x]))/d","B",1
13,1,219,54,0.6096727,"\int \sec (c+d x) (a+a \sec (c+d x))^2 \, dx","Integrate[Sec[c + d*x]*(a + a*Sec[c + d*x])^2,x]","\frac{a^2 (\cos (c+d x)+1)^2 \sec ^4\left(\frac{1}{2} (c+d x)\right) \left(\frac{8 \sin (d x)}{\left(\cos \left(\frac{c}{2}\right)-\sin \left(\frac{c}{2}\right)\right) \left(\sin \left(\frac{c}{2}\right)+\cos \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) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}+\frac{1}{\left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^2}-\frac{1}{\left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2}-6 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+6 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)}{16 d}","\frac{2 a^2 \tan (c+d x)}{d}+\frac{3 a^2 \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{a^2 \tan (c+d x) \sec (c+d x)}{2 d}",1,"(a^2*(1 + Cos[c + d*x])^2*Sec[(c + d*x)/2]^4*(-6*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 6*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + (Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^(-2) - (Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^(-2) + (8*Sin[d*x])/((Cos[c/2] - Sin[c/2])*(Cos[c/2] + Sin[c/2])*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))))/(16*d)","B",1
14,1,171,34,0.4678692,"\int (a+a \sec (c+d x))^2 \, dx","Integrate[(a + a*Sec[c + d*x])^2,x]","\frac{a^2 (\cos (c+d x)+1)^2 \sec ^4\left(\frac{1}{2} (c+d x)\right) \left(\frac{\sin (d x)}{\left(\cos \left(\frac{c}{2}\right)-\sin \left(\frac{c}{2}\right)\right) \left(\sin \left(\frac{c}{2}\right)+\cos \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) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}-2 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+2 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+d x\right)}{4 d}","\frac{a^2 \tan (c+d x)}{d}+\frac{2 a^2 \tanh ^{-1}(\sin (c+d x))}{d}+a^2 x",1,"(a^2*(1 + Cos[c + d*x])^2*Sec[(c + d*x)/2]^4*(d*x - 2*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 2*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + Sin[d*x]/((Cos[c/2] - Sin[c/2])*(Cos[c/2] + Sin[c/2])*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))))/(4*d)","B",1
15,1,47,34,0.0119719,"\int \cos (c+d x) (a+a \sec (c+d x))^2 \, dx","Integrate[Cos[c + d*x]*(a + a*Sec[c + d*x])^2,x]","\frac{a^2 \tanh ^{-1}(\sin (c+d x))}{d}+\frac{a^2 \sin (c) \cos (d x)}{d}+\frac{a^2 \cos (c) \sin (d x)}{d}+2 a^2 x","\frac{a^2 \sin (c+d x)}{d}+\frac{a^2 \tanh ^{-1}(\sin (c+d x))}{d}+2 a^2 x",1,"2*a^2*x + (a^2*ArcTanh[Sin[c + d*x]])/d + (a^2*Cos[d*x]*Sin[c])/d + (a^2*Cos[c]*Sin[d*x])/d","A",1
16,1,34,45,0.0383356,"\int \cos ^2(c+d x) (a+a \sec (c+d x))^2 \, dx","Integrate[Cos[c + d*x]^2*(a + a*Sec[c + d*x])^2,x]","\frac{a^2 (6 (c+d x)+8 \sin (c+d x)+\sin (2 (c+d x)))}{4 d}","\frac{2 a^2 \sin (c+d x)}{d}+\frac{a^2 \sin (c+d x) \cos (c+d x)}{2 d}+\frac{3 a^2 x}{2}",1,"(a^2*(6*(c + d*x) + 8*Sin[c + d*x] + Sin[2*(c + d*x)]))/(4*d)","A",1
17,1,41,57,0.0853174,"\int \cos ^3(c+d x) (a+a \sec (c+d x))^2 \, dx","Integrate[Cos[c + d*x]^3*(a + a*Sec[c + d*x])^2,x]","\frac{a^2 (21 \sin (c+d x)+6 \sin (2 (c+d x))+\sin (3 (c+d x))+12 d x)}{12 d}","-\frac{a^2 \sin ^3(c+d x)}{3 d}+\frac{2 a^2 \sin (c+d x)}{d}+\frac{a^2 \sin (c+d x) \cos (c+d x)}{d}+a^2 x",1,"(a^2*(12*d*x + 21*Sin[c + d*x] + 6*Sin[2*(c + d*x)] + Sin[3*(c + d*x)]))/(12*d)","A",1
18,1,53,87,0.1397174,"\int \cos ^4(c+d x) (a+a \sec (c+d x))^2 \, dx","Integrate[Cos[c + d*x]^4*(a + a*Sec[c + d*x])^2,x]","\frac{a^2 (144 \sin (c+d x)+48 \sin (2 (c+d x))+16 \sin (3 (c+d x))+3 \sin (4 (c+d x))+84 d x)}{96 d}","-\frac{2 a^2 \sin ^3(c+d x)}{3 d}+\frac{2 a^2 \sin (c+d x)}{d}+\frac{a^2 \sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac{7 a^2 \sin (c+d x) \cos (c+d x)}{8 d}+\frac{7 a^2 x}{8}",1,"(a^2*(84*d*x + 144*Sin[c + d*x] + 48*Sin[2*(c + d*x)] + 16*Sin[3*(c + d*x)] + 3*Sin[4*(c + d*x)]))/(96*d)","A",1
19,1,61,103,0.1362913,"\int \cos ^5(c+d x) (a+a \sec (c+d x))^2 \, dx","Integrate[Cos[c + d*x]^5*(a + a*Sec[c + d*x])^2,x]","\frac{a^2 (110 \sin (c+d x)+40 \sin (2 (c+d x))+15 \sin (3 (c+d x))+5 \sin (4 (c+d x))+\sin (5 (c+d x))+60 d x)}{80 d}","\frac{a^2 \sin ^5(c+d x)}{5 d}-\frac{a^2 \sin ^3(c+d x)}{d}+\frac{2 a^2 \sin (c+d x)}{d}+\frac{a^2 \sin (c+d x) \cos ^3(c+d x)}{2 d}+\frac{3 a^2 \sin (c+d x) \cos (c+d x)}{4 d}+\frac{3 a^2 x}{4}",1,"(a^2*(60*d*x + 110*Sin[c + d*x] + 40*Sin[2*(c + d*x)] + 15*Sin[3*(c + d*x)] + 5*Sin[4*(c + d*x)] + Sin[5*(c + d*x)]))/(80*d)","A",1
20,1,487,114,1.5073764,"\int \sec ^3(c+d x) (a+a \sec (c+d x))^3 \, dx","Integrate[Sec[c + d*x]^3*(a + a*Sec[c + d*x])^3,x]","-\frac{a^3 \sec (c) \sec ^5(c+d x) \left(1440 \sin (2 c+d x)-1500 \sin (c+2 d x)-1500 \sin (3 c+2 d x)-3040 \sin (2 c+3 d x)-390 \sin (3 c+4 d x)-390 \sin (5 c+4 d x)-608 \sin (4 c+5 d x)+975 \cos (2 c+3 d x) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+975 \cos (4 c+3 d x) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+195 \cos (4 c+5 d x) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+195 \cos (6 c+5 d x) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+1950 \cos (d x) \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)+1950 \cos (2 c+d x) \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)-975 \cos (2 c+3 d x) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)-975 \cos (4 c+3 d x) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)-195 \cos (4 c+5 d x) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)-195 \cos (6 c+5 d x) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)-4640 \sin (d x)\right)}{3840 d}","\frac{a^3 \tan ^5(c+d x)}{5 d}+\frac{5 a^3 \tan ^3(c+d x)}{3 d}+\frac{4 a^3 \tan (c+d x)}{d}+\frac{13 a^3 \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{3 a^3 \tan (c+d x) \sec ^3(c+d x)}{4 d}+\frac{13 a^3 \tan (c+d x) \sec (c+d x)}{8 d}",1,"-1/3840*(a^3*Sec[c]*Sec[c + d*x]^5*(975*Cos[2*c + 3*d*x]*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 975*Cos[4*c + 3*d*x]*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 195*Cos[4*c + 5*d*x]*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 195*Cos[6*c + 5*d*x]*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 1950*Cos[d*x]*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) + 1950*Cos[2*c + d*x]*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) - 975*Cos[2*c + 3*d*x]*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] - 975*Cos[4*c + 3*d*x]*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] - 195*Cos[4*c + 5*d*x]*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] - 195*Cos[6*c + 5*d*x]*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] - 4640*Sin[d*x] + 1440*Sin[2*c + d*x] - 1500*Sin[c + 2*d*x] - 1500*Sin[3*c + 2*d*x] - 3040*Sin[2*c + 3*d*x] - 390*Sin[3*c + 4*d*x] - 390*Sin[5*c + 4*d*x] - 608*Sin[4*c + 5*d*x]))/d","B",1
21,1,877,93,6.4217803,"\int \sec ^2(c+d x) (a+a \sec (c+d x))^3 \, dx","Integrate[Sec[c + d*x]^2*(a + a*Sec[c + d*x])^3,x]","-\frac{15 \cos ^3(c+d x) \log \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)-\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right) (\sec (c+d x) a+a)^3 \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{64 d}+\frac{15 \cos ^3(c+d x) \log \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)+\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right) (\sec (c+d x) a+a)^3 \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{64 d}+\frac{3 \cos ^3(c+d x) (\sec (c+d x) a+a)^3 \sin \left(\frac{d x}{2}\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{8 d \left(\cos \left(\frac{c}{2}\right)-\sin \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)-\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)}+\frac{3 \cos ^3(c+d x) (\sec (c+d x) a+a)^3 \sin \left(\frac{d x}{2}\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{8 d \left(\cos \left(\frac{c}{2}\right)+\sin \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)+\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)}+\frac{\cos ^3(c+d x) (\sec (c+d x) a+a)^3 \left(19 \cos \left(\frac{c}{2}\right)-11 \sin \left(\frac{c}{2}\right)\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{128 d \left(\cos \left(\frac{c}{2}\right)-\sin \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)-\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)^2}+\frac{\cos ^3(c+d x) (\sec (c+d x) a+a)^3 \left(-19 \cos \left(\frac{c}{2}\right)-11 \sin \left(\frac{c}{2}\right)\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{128 d \left(\cos \left(\frac{c}{2}\right)+\sin \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)+\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)^2}+\frac{\cos ^3(c+d x) (\sec (c+d x) a+a)^3 \sin \left(\frac{d x}{2}\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{16 d \left(\cos \left(\frac{c}{2}\right)-\sin \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)-\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)^3}+\frac{\cos ^3(c+d x) (\sec (c+d x) a+a)^3 \sin \left(\frac{d x}{2}\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{16 d \left(\cos \left(\frac{c}{2}\right)+\sin \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)+\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)^3}+\frac{\cos ^3(c+d x) (\sec (c+d x) a+a)^3 \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{128 d \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)-\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)^4}-\frac{\cos ^3(c+d x) (\sec (c+d x) a+a)^3 \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{128 d \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)+\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)^4}","\frac{a^3 \tan ^3(c+d x)}{d}+\frac{4 a^3 \tan (c+d x)}{d}+\frac{15 a^3 \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a^3 \tan (c+d x) \sec ^3(c+d x)}{4 d}+\frac{15 a^3 \tan (c+d x) \sec (c+d x)}{8 d}",1,"(-15*Cos[c + d*x]^3*Log[Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2]]*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3)/(64*d) + (15*Cos[c + d*x]^3*Log[Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2]]*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3)/(64*d) + (Cos[c + d*x]^3*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3)/(128*d*(Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2])^4) + (Cos[c + d*x]^3*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3*Sin[(d*x)/2])/(16*d*(Cos[c/2] - Sin[c/2])*(Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2])^3) + (Cos[c + d*x]^3*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3*(19*Cos[c/2] - 11*Sin[c/2]))/(128*d*(Cos[c/2] - Sin[c/2])*(Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2])^2) + (3*Cos[c + d*x]^3*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3*Sin[(d*x)/2])/(8*d*(Cos[c/2] - Sin[c/2])*(Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2])) - (Cos[c + d*x]^3*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3)/(128*d*(Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2])^4) + (Cos[c + d*x]^3*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3*Sin[(d*x)/2])/(16*d*(Cos[c/2] + Sin[c/2])*(Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2])^3) + (Cos[c + d*x]^3*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3*(-19*Cos[c/2] - 11*Sin[c/2]))/(128*d*(Cos[c/2] + Sin[c/2])*(Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2])^2) + (3*Cos[c + d*x]^3*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3*Sin[(d*x)/2])/(8*d*(Cos[c/2] + Sin[c/2])*(Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2]))","B",0
22,1,154,72,5.669352,"\int \sec (c+d x) (a+a \sec (c+d x))^3 \, dx","Integrate[Sec[c + d*x]*(a + a*Sec[c + d*x])^3,x]","-\frac{a^3 \sec ^6\left(\frac{1}{2} (c+d x)\right) (\sec (c+d x)+1)^3 \left(-4 \tan (c) \cos (c+d x)-\sec (c) (-20 \sin (2 c+d x)+9 \sin (c+2 d x)+9 \sin (3 c+2 d x)+22 \sin (2 c+3 d x)+50 \sin (d x))+60 \cos ^3(c+d x) \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)\right)}{192 d}","\frac{a^3 \tan ^3(c+d x)}{3 d}+\frac{4 a^3 \tan (c+d x)}{d}+\frac{5 a^3 \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{3 a^3 \tan (c+d x) \sec (c+d x)}{2 d}",1,"-1/192*(a^3*Sec[(c + d*x)/2]^6*(1 + Sec[c + d*x])^3*(60*Cos[c + d*x]^3*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) - Sec[c]*(50*Sin[d*x] - 20*Sin[2*c + d*x] + 9*Sin[c + 2*d*x] + 9*Sin[3*c + 2*d*x] + 22*Sin[2*c + 3*d*x]) - 4*Cos[c + d*x]*Tan[c]))/d","B",1
23,1,235,66,0.9258011,"\int (a+a \sec (c+d x))^3 \, dx","Integrate[(a + a*Sec[c + d*x])^3,x]","\frac{1}{32} a^3 (\cos (c+d x)+1)^3 \sec ^6\left(\frac{1}{2} (c+d x)\right) \left(\frac{12 \sin (d x)}{d \left(\cos \left(\frac{c}{2}\right)-\sin \left(\frac{c}{2}\right)\right) \left(\sin \left(\frac{c}{2}\right)+\cos \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) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}+\frac{1}{d \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^2}-\frac{1}{d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2}-\frac{14 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}{d}+\frac{14 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}{d}+4 x\right)","\frac{5 a^3 \tan (c+d x)}{2 d}+\frac{7 a^3 \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{\tan (c+d x) \left(a^3 \sec (c+d x)+a^3\right)}{2 d}+a^3 x",1,"(a^3*(1 + Cos[c + d*x])^3*Sec[(c + d*x)/2]^6*(4*x - (14*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]])/d + (14*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]])/d + 1/(d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2) - 1/(d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2) + (12*Sin[d*x])/(d*(Cos[c/2] - Sin[c/2])*(Cos[c/2] + Sin[c/2])*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))))/32","B",1
24,1,211,48,0.8647409,"\int \cos (c+d x) (a+a \sec (c+d x))^3 \, dx","Integrate[Cos[c + d*x]*(a + a*Sec[c + d*x])^3,x]","\frac{1}{8} a^3 (\cos (c+d x)+1)^3 \sec ^6\left(\frac{1}{2} (c+d x)\right) \left(\frac{\sin (c) \cos (d x)}{d}+\frac{\cos (c) \sin (d x)}{d}+\frac{\sin \left(\frac{d x}{2}\right)}{d \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{\sin \left(\frac{d x}{2}\right)}{d \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)}-\frac{3 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}{d}+\frac{3 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}{d}+3 x\right)","\frac{a^3 \sin (c+d x)}{d}+\frac{a^3 \tan (c+d x)}{d}+\frac{3 a^3 \tanh ^{-1}(\sin (c+d x))}{d}+3 a^3 x",1,"(a^3*(1 + Cos[c + d*x])^3*Sec[(c + d*x)/2]^6*(3*x - (3*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]])/d + (3*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]])/d + (Cos[d*x]*Sin[c])/d + (Cos[c]*Sin[d*x])/d + Sin[(d*x)/2]/(d*(Cos[c/2] - Sin[c/2])*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])) + Sin[(d*x)/2]/(d*(Cos[c/2] + Sin[c/2])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))))/8","B",1
25,1,81,59,0.0709952,"\int \cos ^2(c+d x) (a+a \sec (c+d x))^3 \, dx","Integrate[Cos[c + d*x]^2*(a + a*Sec[c + d*x])^3,x]","\frac{a^3 \left(12 \sin (c+d x)+\sin (2 (c+d x))-4 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+4 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+14 d x\right)}{4 d}","\frac{3 a^3 \sin (c+d x)}{d}+\frac{a^3 \tanh ^{-1}(\sin (c+d x))}{d}+\frac{a^3 \sin (c+d x) \cos (c+d x)}{2 d}+\frac{7 a^3 x}{2}",1,"(a^3*(14*d*x - 4*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 4*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + 12*Sin[c + d*x] + Sin[2*(c + d*x)]))/(4*d)","A",1
26,1,44,63,0.06128,"\int \cos ^3(c+d x) (a+a \sec (c+d x))^3 \, dx","Integrate[Cos[c + d*x]^3*(a + a*Sec[c + d*x])^3,x]","\frac{a^3 (45 \sin (c+d x)+9 \sin (2 (c+d x))+\sin (3 (c+d x))+30 c+30 d x)}{12 d}","-\frac{a^3 \sin ^3(c+d x)}{3 d}+\frac{4 a^3 \sin (c+d x)}{d}+\frac{3 a^3 \sin (c+d x) \cos (c+d x)}{2 d}+\frac{5 a^3 x}{2}",1,"(a^3*(30*c + 30*d*x + 45*Sin[c + d*x] + 9*Sin[2*(c + d*x)] + Sin[3*(c + d*x)]))/(12*d)","A",1
27,1,51,85,0.1250424,"\int \cos ^4(c+d x) (a+a \sec (c+d x))^3 \, dx","Integrate[Cos[c + d*x]^4*(a + a*Sec[c + d*x])^3,x]","\frac{a^3 (104 \sin (c+d x)+32 \sin (2 (c+d x))+8 \sin (3 (c+d x))+\sin (4 (c+d x))+60 d x)}{32 d}","-\frac{a^3 \sin ^3(c+d x)}{d}+\frac{4 a^3 \sin (c+d x)}{d}+\frac{a^3 \sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac{15 a^3 \sin (c+d x) \cos (c+d x)}{8 d}+\frac{15 a^3 x}{8}",1,"(a^3*(60*d*x + 104*Sin[c + d*x] + 32*Sin[2*(c + d*x)] + 8*Sin[3*(c + d*x)] + Sin[4*(c + d*x)]))/(32*d)","A",1
28,1,63,105,0.1487747,"\int \cos ^5(c+d x) (a+a \sec (c+d x))^3 \, dx","Integrate[Cos[c + d*x]^5*(a + a*Sec[c + d*x])^3,x]","\frac{a^3 (1380 \sin (c+d x)+480 \sin (2 (c+d x))+170 \sin (3 (c+d x))+45 \sin (4 (c+d x))+6 \sin (5 (c+d x))+780 d x)}{480 d}","\frac{a^3 \sin ^5(c+d x)}{5 d}-\frac{5 a^3 \sin ^3(c+d x)}{3 d}+\frac{4 a^3 \sin (c+d x)}{d}+\frac{3 a^3 \sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac{13 a^3 \sin (c+d x) \cos (c+d x)}{8 d}+\frac{13 a^3 x}{8}",1,"(a^3*(780*d*x + 1380*Sin[c + d*x] + 480*Sin[2*(c + d*x)] + 170*Sin[3*(c + d*x)] + 45*Sin[4*(c + d*x)] + 6*Sin[5*(c + d*x)]))/(480*d)","A",1
29,1,73,129,0.2092496,"\int \cos ^6(c+d x) (a+a \sec (c+d x))^3 \, dx","Integrate[Cos[c + d*x]^6*(a + a*Sec[c + d*x])^3,x]","\frac{a^3 (2520 \sin (c+d x)+945 \sin (2 (c+d x))+380 \sin (3 (c+d x))+135 \sin (4 (c+d x))+36 \sin (5 (c+d x))+5 \sin (6 (c+d x))+1380 d x)}{960 d}","\frac{3 a^3 \sin ^5(c+d x)}{5 d}-\frac{7 a^3 \sin ^3(c+d x)}{3 d}+\frac{4 a^3 \sin (c+d x)}{d}+\frac{a^3 \sin (c+d x) \cos ^5(c+d x)}{6 d}+\frac{23 a^3 \sin (c+d x) \cos ^3(c+d x)}{24 d}+\frac{23 a^3 \sin (c+d x) \cos (c+d x)}{16 d}+\frac{23 a^3 x}{16}",1,"(a^3*(1380*d*x + 2520*Sin[c + d*x] + 945*Sin[2*(c + d*x)] + 380*Sin[3*(c + d*x)] + 135*Sin[4*(c + d*x)] + 36*Sin[5*(c + d*x)] + 5*Sin[6*(c + d*x)]))/(960*d)","A",1
30,1,211,136,0.8189273,"\int \sec ^3(c+d x) (a+a \sec (c+d x))^4 \, dx","Integrate[Sec[c + d*x]^3*(a + a*Sec[c + d*x])^4,x]","-\frac{a^4 (\cos (c+d x)+1)^4 \sec ^8\left(\frac{1}{2} (c+d x)\right) \sec ^6(c+d x) \left(23520 \cos ^6(c+d x) \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)-\sec (c) (3750 \sin (2 c+d x)+15360 \sin (c+2 d x)-1920 \sin (3 c+2 d x)+3845 \sin (2 c+3 d x)+3845 \sin (4 c+3 d x)+6912 \sin (3 c+4 d x)+735 \sin (4 c+5 d x)+735 \sin (6 c+5 d x)+1152 \sin (5 c+6 d x)-11520 \sin (c)+3750 \sin (d x))\right)}{122880 d}","\frac{4 a^4 \tan ^5(c+d x)}{5 d}+\frac{4 a^4 \tan ^3(c+d x)}{d}+\frac{8 a^4 \tan (c+d x)}{d}+\frac{49 a^4 \tanh ^{-1}(\sin (c+d x))}{16 d}+\frac{a^4 \tan (c+d x) \sec ^5(c+d x)}{6 d}+\frac{41 a^4 \tan (c+d x) \sec ^3(c+d x)}{24 d}+\frac{49 a^4 \tan (c+d x) \sec (c+d x)}{16 d}",1,"-1/122880*(a^4*(1 + Cos[c + d*x])^4*Sec[(c + d*x)/2]^8*Sec[c + d*x]^6*(23520*Cos[c + d*x]^6*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) - Sec[c]*(-11520*Sin[c] + 3750*Sin[d*x] + 3750*Sin[2*c + d*x] + 15360*Sin[c + 2*d*x] - 1920*Sin[3*c + 2*d*x] + 3845*Sin[2*c + 3*d*x] + 3845*Sin[4*c + 3*d*x] + 6912*Sin[3*c + 4*d*x] + 735*Sin[4*c + 5*d*x] + 735*Sin[6*c + 5*d*x] + 1152*Sin[5*c + 6*d*x])))/d","A",1
31,1,498,111,1.5233682,"\int \sec ^2(c+d x) (a+a \sec (c+d x))^4 \, dx","Integrate[Sec[c + d*x]^2*(a + a*Sec[c + d*x])^4,x]","-\frac{a^4 \sec (c) \sec ^5(c+d x) \left(960 \sin (2 c+d x)-660 \sin (c+2 d x)-660 \sin (3 c+2 d x)-1600 \sin (2 c+3 d x)+60 \sin (4 c+3 d x)-210 \sin (3 c+4 d x)-210 \sin (5 c+4 d x)-332 \sin (4 c+5 d x)+525 \cos (2 c+3 d x) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+525 \cos (4 c+3 d x) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+105 \cos (4 c+5 d x) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+105 \cos (6 c+5 d x) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+1050 \cos (d x) \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)+1050 \cos (2 c+d x) \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)-525 \cos (2 c+3 d x) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)-525 \cos (4 c+3 d x) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)-105 \cos (4 c+5 d x) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)-105 \cos (6 c+5 d x) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)-2360 \sin (d x)\right)}{960 d}","\frac{a^4 \tan ^5(c+d x)}{5 d}+\frac{8 a^4 \tan ^3(c+d x)}{3 d}+\frac{8 a^4 \tan (c+d x)}{d}+\frac{7 a^4 \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{a^4 \tan (c+d x) \sec ^3(c+d x)}{d}+\frac{7 a^4 \tan (c+d x) \sec (c+d x)}{2 d}",1,"-1/960*(a^4*Sec[c]*Sec[c + d*x]^5*(525*Cos[2*c + 3*d*x]*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 525*Cos[4*c + 3*d*x]*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 105*Cos[4*c + 5*d*x]*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 105*Cos[6*c + 5*d*x]*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 1050*Cos[d*x]*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) + 1050*Cos[2*c + d*x]*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) - 525*Cos[2*c + 3*d*x]*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] - 525*Cos[4*c + 3*d*x]*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] - 105*Cos[4*c + 5*d*x]*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] - 105*Cos[6*c + 5*d*x]*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] - 2360*Sin[d*x] + 960*Sin[2*c + d*x] - 660*Sin[c + 2*d*x] - 660*Sin[3*c + 2*d*x] - 1600*Sin[2*c + 3*d*x] + 60*Sin[4*c + 3*d*x] - 210*Sin[3*c + 4*d*x] - 210*Sin[5*c + 4*d*x] - 332*Sin[4*c + 5*d*x]))/d","B",1
32,1,877,96,6.4218285,"\int \sec (c+d x) (a+a \sec (c+d x))^4 \, dx","Integrate[Sec[c + d*x]*(a + a*Sec[c + d*x])^4,x]","-\frac{35 \cos ^4(c+d x) \log \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)-\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right) (\sec (c+d x) a+a)^4 \sec ^8\left(\frac{c}{2}+\frac{d x}{2}\right)}{128 d}+\frac{35 \cos ^4(c+d x) \log \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)+\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right) (\sec (c+d x) a+a)^4 \sec ^8\left(\frac{c}{2}+\frac{d x}{2}\right)}{128 d}+\frac{5 \cos ^4(c+d x) (\sec (c+d x) a+a)^4 \sin \left(\frac{d x}{2}\right) \sec ^8\left(\frac{c}{2}+\frac{d x}{2}\right)}{12 d \left(\cos \left(\frac{c}{2}\right)-\sin \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)-\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)}+\frac{5 \cos ^4(c+d x) (\sec (c+d x) a+a)^4 \sin \left(\frac{d x}{2}\right) \sec ^8\left(\frac{c}{2}+\frac{d x}{2}\right)}{12 d \left(\cos \left(\frac{c}{2}\right)+\sin \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)+\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)}+\frac{\cos ^4(c+d x) (\sec (c+d x) a+a)^4 \left(97 \cos \left(\frac{c}{2}\right)-65 \sin \left(\frac{c}{2}\right)\right) \sec ^8\left(\frac{c}{2}+\frac{d x}{2}\right)}{768 d \left(\cos \left(\frac{c}{2}\right)-\sin \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)-\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)^2}+\frac{\cos ^4(c+d x) (\sec (c+d x) a+a)^4 \left(-97 \cos \left(\frac{c}{2}\right)-65 \sin \left(\frac{c}{2}\right)\right) \sec ^8\left(\frac{c}{2}+\frac{d x}{2}\right)}{768 d \left(\cos \left(\frac{c}{2}\right)+\sin \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)+\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)^2}+\frac{\cos ^4(c+d x) (\sec (c+d x) a+a)^4 \sin \left(\frac{d x}{2}\right) \sec ^8\left(\frac{c}{2}+\frac{d x}{2}\right)}{24 d \left(\cos \left(\frac{c}{2}\right)-\sin \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)-\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)^3}+\frac{\cos ^4(c+d x) (\sec (c+d x) a+a)^4 \sin \left(\frac{d x}{2}\right) \sec ^8\left(\frac{c}{2}+\frac{d x}{2}\right)}{24 d \left(\cos \left(\frac{c}{2}\right)+\sin \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)+\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)^3}+\frac{\cos ^4(c+d x) (\sec (c+d x) a+a)^4 \sec ^8\left(\frac{c}{2}+\frac{d x}{2}\right)}{256 d \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)-\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)^4}-\frac{\cos ^4(c+d x) (\sec (c+d x) a+a)^4 \sec ^8\left(\frac{c}{2}+\frac{d x}{2}\right)}{256 d \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)+\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)^4}","\frac{4 a^4 \tan ^3(c+d x)}{3 d}+\frac{8 a^4 \tan (c+d x)}{d}+\frac{35 a^4 \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a^4 \tan (c+d x) \sec ^3(c+d x)}{4 d}+\frac{27 a^4 \tan (c+d x) \sec (c+d x)}{8 d}",1,"(-35*Cos[c + d*x]^4*Log[Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2]]*Sec[c/2 + (d*x)/2]^8*(a + a*Sec[c + d*x])^4)/(128*d) + (35*Cos[c + d*x]^4*Log[Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2]]*Sec[c/2 + (d*x)/2]^8*(a + a*Sec[c + d*x])^4)/(128*d) + (Cos[c + d*x]^4*Sec[c/2 + (d*x)/2]^8*(a + a*Sec[c + d*x])^4)/(256*d*(Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2])^4) + (Cos[c + d*x]^4*Sec[c/2 + (d*x)/2]^8*(a + a*Sec[c + d*x])^4*Sin[(d*x)/2])/(24*d*(Cos[c/2] - Sin[c/2])*(Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2])^3) + (Cos[c + d*x]^4*Sec[c/2 + (d*x)/2]^8*(a + a*Sec[c + d*x])^4*(97*Cos[c/2] - 65*Sin[c/2]))/(768*d*(Cos[c/2] - Sin[c/2])*(Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2])^2) + (5*Cos[c + d*x]^4*Sec[c/2 + (d*x)/2]^8*(a + a*Sec[c + d*x])^4*Sin[(d*x)/2])/(12*d*(Cos[c/2] - Sin[c/2])*(Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2])) - (Cos[c + d*x]^4*Sec[c/2 + (d*x)/2]^8*(a + a*Sec[c + d*x])^4)/(256*d*(Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2])^4) + (Cos[c + d*x]^4*Sec[c/2 + (d*x)/2]^8*(a + a*Sec[c + d*x])^4*Sin[(d*x)/2])/(24*d*(Cos[c/2] + Sin[c/2])*(Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2])^3) + (Cos[c + d*x]^4*Sec[c/2 + (d*x)/2]^8*(a + a*Sec[c + d*x])^4*(-97*Cos[c/2] - 65*Sin[c/2]))/(768*d*(Cos[c/2] + Sin[c/2])*(Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2])^2) + (5*Cos[c + d*x]^4*Sec[c/2 + (d*x)/2]^8*(a + a*Sec[c + d*x])^4*Sin[(d*x)/2])/(12*d*(Cos[c/2] + Sin[c/2])*(Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2]))","B",1
33,1,773,91,6.271532,"\int (a+a \sec (c+d x))^4 \, dx","Integrate[(a + a*Sec[c + d*x])^4,x]","\frac{1}{16} x \cos ^4(c+d x) \sec ^8\left(\frac{c}{2}+\frac{d x}{2}\right) (a \sec (c+d x)+a)^4+\frac{5 \sin \left(\frac{d x}{2}\right) \cos ^4(c+d x) \sec ^8\left(\frac{c}{2}+\frac{d x}{2}\right) (a \sec (c+d x)+a)^4}{12 d \left(\cos \left(\frac{c}{2}\right)-\sin \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)-\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)}+\frac{5 \sin \left(\frac{d x}{2}\right) \cos ^4(c+d x) \sec ^8\left(\frac{c}{2}+\frac{d x}{2}\right) (a \sec (c+d x)+a)^4}{12 d \left(\sin \left(\frac{c}{2}\right)+\cos \left(\frac{c}{2}\right)\right) \left(\sin \left(\frac{c}{2}+\frac{d x}{2}\right)+\cos \left(\frac{c}{2}+\frac{d x}{2}\right)\right)}+\frac{\left(13 \cos \left(\frac{c}{2}\right)-11 \sin \left(\frac{c}{2}\right)\right) \cos ^4(c+d x) \sec ^8\left(\frac{c}{2}+\frac{d x}{2}\right) (a \sec (c+d x)+a)^4}{192 d \left(\cos \left(\frac{c}{2}\right)-\sin \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)-\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)^2}+\frac{\left(-11 \sin \left(\frac{c}{2}\right)-13 \cos \left(\frac{c}{2}\right)\right) \cos ^4(c+d x) \sec ^8\left(\frac{c}{2}+\frac{d x}{2}\right) (a \sec (c+d x)+a)^4}{192 d \left(\sin \left(\frac{c}{2}\right)+\cos \left(\frac{c}{2}\right)\right) \left(\sin \left(\frac{c}{2}+\frac{d x}{2}\right)+\cos \left(\frac{c}{2}+\frac{d x}{2}\right)\right)^2}+\frac{\sin \left(\frac{d x}{2}\right) \cos ^4(c+d x) \sec ^8\left(\frac{c}{2}+\frac{d x}{2}\right) (a \sec (c+d x)+a)^4}{96 d \left(\cos \left(\frac{c}{2}\right)-\sin \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)-\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)^3}+\frac{\sin \left(\frac{d x}{2}\right) \cos ^4(c+d x) \sec ^8\left(\frac{c}{2}+\frac{d x}{2}\right) (a \sec (c+d x)+a)^4}{96 d \left(\sin \left(\frac{c}{2}\right)+\cos \left(\frac{c}{2}\right)\right) \left(\sin \left(\frac{c}{2}+\frac{d x}{2}\right)+\cos \left(\frac{c}{2}+\frac{d x}{2}\right)\right)^3}-\frac{3 \cos ^4(c+d x) \sec ^8\left(\frac{c}{2}+\frac{d x}{2}\right) (a \sec (c+d x)+a)^4 \log \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)-\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)}{8 d}+\frac{3 \cos ^4(c+d x) \sec ^8\left(\frac{c}{2}+\frac{d x}{2}\right) (a \sec (c+d x)+a)^4 \log \left(\sin \left(\frac{c}{2}+\frac{d x}{2}\right)+\cos \left(\frac{c}{2}+\frac{d x}{2}\right)\right)}{8 d}","\frac{5 a^4 \tan (c+d x)}{d}+\frac{6 a^4 \tanh ^{-1}(\sin (c+d x))}{d}+\frac{4 \tan (c+d x) \left(a^4 \sec (c+d x)+a^4\right)}{3 d}+a^4 x+\frac{\tan (c+d x) \left(a^2 \sec (c+d x)+a^2\right)^2}{3 d}",1,"(x*Cos[c + d*x]^4*Sec[c/2 + (d*x)/2]^8*(a + a*Sec[c + d*x])^4)/16 - (3*Cos[c + d*x]^4*Log[Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2]]*Sec[c/2 + (d*x)/2]^8*(a + a*Sec[c + d*x])^4)/(8*d) + (3*Cos[c + d*x]^4*Log[Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2]]*Sec[c/2 + (d*x)/2]^8*(a + a*Sec[c + d*x])^4)/(8*d) + (Cos[c + d*x]^4*Sec[c/2 + (d*x)/2]^8*(a + a*Sec[c + d*x])^4*Sin[(d*x)/2])/(96*d*(Cos[c/2] - Sin[c/2])*(Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2])^3) + (Cos[c + d*x]^4*Sec[c/2 + (d*x)/2]^8*(a + a*Sec[c + d*x])^4*(13*Cos[c/2] - 11*Sin[c/2]))/(192*d*(Cos[c/2] - Sin[c/2])*(Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2])^2) + (5*Cos[c + d*x]^4*Sec[c/2 + (d*x)/2]^8*(a + a*Sec[c + d*x])^4*Sin[(d*x)/2])/(12*d*(Cos[c/2] - Sin[c/2])*(Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2])) + (Cos[c + d*x]^4*Sec[c/2 + (d*x)/2]^8*(a + a*Sec[c + d*x])^4*Sin[(d*x)/2])/(96*d*(Cos[c/2] + Sin[c/2])*(Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2])^3) + (Cos[c + d*x]^4*Sec[c/2 + (d*x)/2]^8*(a + a*Sec[c + d*x])^4*(-13*Cos[c/2] - 11*Sin[c/2]))/(192*d*(Cos[c/2] + Sin[c/2])*(Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2])^2) + (5*Cos[c + d*x]^4*Sec[c/2 + (d*x)/2]^8*(a + a*Sec[c + d*x])^4*Sin[(d*x)/2])/(12*d*(Cos[c/2] + Sin[c/2])*(Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2]))","B",1
34,1,272,73,1.3827452,"\int \cos (c+d x) (a+a \sec (c+d x))^4 \, dx","Integrate[Cos[c + d*x]*(a + a*Sec[c + d*x])^4,x]","\frac{1}{64} a^4 (\cos (c+d x)+1)^4 \sec ^8\left(\frac{1}{2} (c+d x)\right) \left(\frac{4 \sin (c) \cos (d x)}{d}+\frac{4 \cos (c) \sin (d x)}{d}+\frac{16 \sin \left(\frac{d x}{2}\right)}{d \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{16 \sin \left(\frac{d x}{2}\right)}{d \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)}+\frac{1}{d \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^2}-\frac{1}{d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2}-\frac{26 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}{d}+\frac{26 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}{d}+16 x\right)","\frac{a^4 \sin (c+d x)}{d}+\frac{4 a^4 \tan (c+d x)}{d}+\frac{13 a^4 \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{a^4 \tan (c+d x) \sec (c+d x)}{2 d}+4 a^4 x",1,"(a^4*(1 + Cos[c + d*x])^4*Sec[(c + d*x)/2]^8*(16*x - (26*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]])/d + (26*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]])/d + (4*Cos[d*x]*Sin[c])/d + (4*Cos[c]*Sin[d*x])/d + 1/(d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2) + (16*Sin[(d*x)/2])/(d*(Cos[c/2] - Sin[c/2])*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])) - 1/(d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2) + (16*Sin[(d*x)/2])/(d*(Cos[c/2] + Sin[c/2])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))))/64","B",1
35,1,241,73,1.8040114,"\int \cos ^2(c+d x) (a+a \sec (c+d x))^4 \, dx","Integrate[Cos[c + d*x]^2*(a + a*Sec[c + d*x])^4,x]","\frac{1}{64} a^4 (\cos (c+d x)+1)^4 \sec ^8\left(\frac{1}{2} (c+d x)\right) \left(\frac{16 \sin (c) \cos (d x)}{d}+\frac{\sin (2 c) \cos (2 d x)}{d}+\frac{16 \cos (c) \sin (d x)}{d}+\frac{\cos (2 c) \sin (2 d x)}{d}+\frac{4 \sin \left(\frac{d x}{2}\right)}{d \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{4 \sin \left(\frac{d x}{2}\right)}{d \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)}-\frac{16 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}{d}+\frac{16 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}{d}+26 x\right)","\frac{4 a^4 \sin (c+d x)}{d}+\frac{a^4 \tan (c+d x)}{d}+\frac{4 a^4 \tanh ^{-1}(\sin (c+d x))}{d}+\frac{a^4 \sin (c+d x) \cos (c+d x)}{2 d}+\frac{13 a^4 x}{2}",1,"(a^4*(1 + Cos[c + d*x])^4*Sec[(c + d*x)/2]^8*(26*x - (16*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]])/d + (16*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]])/d + (16*Cos[d*x]*Sin[c])/d + (Cos[2*d*x]*Sin[2*c])/d + (16*Cos[c]*Sin[d*x])/d + (Cos[2*c]*Sin[2*d*x])/d + (4*Sin[(d*x)/2])/(d*(Cos[c/2] - Sin[c/2])*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])) + (4*Sin[(d*x)/2])/(d*(Cos[c/2] + Sin[c/2])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))))/64","B",1
36,1,91,73,0.1064789,"\int \cos ^3(c+d x) (a+a \sec (c+d x))^4 \, dx","Integrate[Cos[c + d*x]^3*(a + a*Sec[c + d*x])^4,x]","\frac{a^4 \left(81 \sin (c+d x)+12 \sin (2 (c+d x))+\sin (3 (c+d x))-12 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+12 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+72 d x\right)}{12 d}","-\frac{a^4 \sin ^3(c+d x)}{3 d}+\frac{7 a^4 \sin (c+d x)}{d}+\frac{a^4 \tanh ^{-1}(\sin (c+d x))}{d}+\frac{2 a^4 \sin (c+d x) \cos (c+d x)}{d}+6 a^4 x",1,"(a^4*(72*d*x - 12*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 12*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + 81*Sin[c + d*x] + 12*Sin[2*(c + d*x)] + Sin[3*(c + d*x)]))/(12*d)","A",1
37,1,56,87,0.1034333,"\int \cos ^4(c+d x) (a+a \sec (c+d x))^4 \, dx","Integrate[Cos[c + d*x]^4*(a + a*Sec[c + d*x])^4,x]","\frac{a^4 (672 \sin (c+d x)+168 \sin (2 (c+d x))+32 \sin (3 (c+d x))+3 \sin (4 (c+d x))+420 c+420 d x)}{96 d}","-\frac{4 a^4 \sin ^3(c+d x)}{3 d}+\frac{8 a^4 \sin (c+d x)}{d}+\frac{a^4 \sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac{27 a^4 \sin (c+d x) \cos (c+d x)}{8 d}+\frac{35 a^4 x}{8}",1,"(a^4*(420*c + 420*d*x + 672*Sin[c + d*x] + 168*Sin[2*(c + d*x)] + 32*Sin[3*(c + d*x)] + 3*Sin[4*(c + d*x)]))/(96*d)","A",1
38,1,63,102,0.1534895,"\int \cos ^5(c+d x) (a+a \sec (c+d x))^4 \, dx","Integrate[Cos[c + d*x]^5*(a + a*Sec[c + d*x])^4,x]","\frac{a^4 (1470 \sin (c+d x)+480 \sin (2 (c+d x))+145 \sin (3 (c+d x))+30 \sin (4 (c+d x))+3 \sin (5 (c+d x))+840 d x)}{240 d}","\frac{a^4 \sin ^5(c+d x)}{5 d}-\frac{8 a^4 \sin ^3(c+d x)}{3 d}+\frac{8 a^4 \sin (c+d x)}{d}+\frac{a^4 \sin (c+d x) \cos ^3(c+d x)}{d}+\frac{7 a^4 \sin (c+d x) \cos (c+d x)}{2 d}+\frac{7 a^4 x}{2}",1,"(a^4*(840*d*x + 1470*Sin[c + d*x] + 480*Sin[2*(c + d*x)] + 145*Sin[3*(c + d*x)] + 30*Sin[4*(c + d*x)] + 3*Sin[5*(c + d*x)]))/(240*d)","A",1
39,1,73,127,0.2199924,"\int \cos ^6(c+d x) (a+a \sec (c+d x))^4 \, dx","Integrate[Cos[c + d*x]^6*(a + a*Sec[c + d*x])^4,x]","\frac{a^4 (5280 \sin (c+d x)+1905 \sin (2 (c+d x))+720 \sin (3 (c+d x))+225 \sin (4 (c+d x))+48 \sin (5 (c+d x))+5 \sin (6 (c+d x))+2940 d x)}{960 d}","\frac{4 a^4 \sin ^5(c+d x)}{5 d}-\frac{4 a^4 \sin ^3(c+d x)}{d}+\frac{8 a^4 \sin (c+d x)}{d}+\frac{a^4 \sin (c+d x) \cos ^5(c+d x)}{6 d}+\frac{41 a^4 \sin (c+d x) \cos ^3(c+d x)}{24 d}+\frac{49 a^4 \sin (c+d x) \cos (c+d x)}{16 d}+\frac{49 a^4 x}{16}",1,"(a^4*(2940*d*x + 5280*Sin[c + d*x] + 1905*Sin[2*(c + d*x)] + 720*Sin[3*(c + d*x)] + 225*Sin[4*(c + d*x)] + 48*Sin[5*(c + d*x)] + 5*Sin[6*(c + d*x)]))/(960*d)","A",1
40,1,83,147,0.2673187,"\int \cos ^7(c+d x) (a+a \sec (c+d x))^4 \, dx","Integrate[Cos[c + d*x]^7*(a + a*Sec[c + d*x])^4,x]","\frac{a^4 (33915 \sin (c+d x)+13020 \sin (2 (c+d x))+5495 \sin (3 (c+d x))+2100 \sin (4 (c+d x))+651 \sin (5 (c+d x))+140 \sin (6 (c+d x))+15 \sin (7 (c+d x))+18480 d x)}{6720 d}","-\frac{a^4 \sin ^7(c+d x)}{7 d}+\frac{9 a^4 \sin ^5(c+d x)}{5 d}-\frac{16 a^4 \sin ^3(c+d x)}{3 d}+\frac{8 a^4 \sin (c+d x)}{d}+\frac{2 a^4 \sin (c+d x) \cos ^5(c+d x)}{3 d}+\frac{11 a^4 \sin (c+d x) \cos ^3(c+d x)}{6 d}+\frac{11 a^4 \sin (c+d x) \cos (c+d x)}{4 d}+\frac{11 a^4 x}{4}",1,"(a^4*(18480*d*x + 33915*Sin[c + d*x] + 13020*Sin[2*(c + d*x)] + 5495*Sin[3*(c + d*x)] + 2100*Sin[4*(c + d*x)] + 651*Sin[5*(c + d*x)] + 140*Sin[6*(c + d*x)] + 15*Sin[7*(c + d*x)]))/(6720*d)","A",1
41,1,229,156,1.3936442,"\int \sec ^3(c+d x) (a+a \sec (c+d x))^5 \, dx","Integrate[Sec[c + d*x]^3*(a + a*Sec[c + d*x])^5,x]","-\frac{a^5 (\cos (c+d x)+1)^5 \sec ^{10}\left(\frac{1}{2} (c+d x)\right) \sec ^7(c+d x) \left(624960 \cos ^7(c+d x) \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)-\sec (c) (-162400 \sin (2 c+d x)+118825 \sin (c+2 d x)+118825 \sin (3 c+2 d x)+305088 \sin (2 c+3 d x)-16800 \sin (4 c+3 d x)+62860 \sin (3 c+4 d x)+62860 \sin (5 c+4 d x)+107296 \sin (4 c+5 d x)+9765 \sin (5 c+6 d x)+9765 \sin (7 c+6 d x)+15328 \sin (6 c+7 d x)+374080 \sin (d x))\right)}{3440640 d}","\frac{a^5 \tan ^7(c+d x)}{7 d}+\frac{13 a^5 \tan ^5(c+d x)}{5 d}+\frac{28 a^5 \tan ^3(c+d x)}{3 d}+\frac{16 a^5 \tan (c+d x)}{d}+\frac{93 a^5 \tanh ^{-1}(\sin (c+d x))}{16 d}+\frac{5 a^5 \tan (c+d x) \sec ^5(c+d x)}{6 d}+\frac{85 a^5 \tan (c+d x) \sec ^3(c+d x)}{24 d}+\frac{93 a^5 \tan (c+d x) \sec (c+d x)}{16 d}",1,"-1/3440640*(a^5*(1 + Cos[c + d*x])^5*Sec[(c + d*x)/2]^10*Sec[c + d*x]^7*(624960*Cos[c + d*x]^7*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) - Sec[c]*(374080*Sin[d*x] - 162400*Sin[2*c + d*x] + 118825*Sin[c + 2*d*x] + 118825*Sin[3*c + 2*d*x] + 305088*Sin[2*c + 3*d*x] - 16800*Sin[4*c + 3*d*x] + 62860*Sin[3*c + 4*d*x] + 62860*Sin[5*c + 4*d*x] + 107296*Sin[4*c + 5*d*x] + 9765*Sin[5*c + 6*d*x] + 9765*Sin[7*c + 6*d*x] + 15328*Sin[6*c + 7*d*x])))/d","A",1
42,1,374,103,3.4848301,"\int \frac{\sec ^5(c+d x)}{a+a \sec (c+d x)} \, dx","Integrate[Sec[c + d*x]^5/(a + a*Sec[c + d*x]),x]","\frac{\cos \left(\frac{1}{2} (c+d x)\right) \sec (c+d x) \left(6 \sec \left(\frac{c}{2}\right) \sin \left(\frac{d x}{2}\right)+\frac{1}{8} \sec (c) \cos \left(\frac{1}{2} (c+d x)\right) \sec ^3(c+d x) \left(-12 \sin (2 c+d x)-6 \sin (c+2 d x)-6 \sin (3 c+2 d x)+20 \sin (2 c+3 d x)+9 \cos (2 c+3 d x) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+9 \cos (4 c+3 d x) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+27 \cos (d x) \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)+27 \cos (2 c+d x) \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)-9 \cos (2 c+3 d x) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)-9 \cos (4 c+3 d x) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+48 \sin (d x)\right)\right)}{3 a d (\sec (c+d x)+1)}","\frac{4 \tan ^3(c+d x)}{3 a d}+\frac{4 \tan (c+d x)}{a d}-\frac{3 \tanh ^{-1}(\sin (c+d x))}{2 a d}-\frac{\tan (c+d x) \sec ^3(c+d x)}{d (a \sec (c+d x)+a)}-\frac{3 \tan (c+d x) \sec (c+d x)}{2 a d}",1,"(Cos[(c + d*x)/2]*Sec[c + d*x]*(6*Sec[c/2]*Sin[(d*x)/2] + (Cos[(c + d*x)/2]*Sec[c]*Sec[c + d*x]^3*(9*Cos[2*c + 3*d*x]*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 9*Cos[4*c + 3*d*x]*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 27*Cos[d*x]*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) + 27*Cos[2*c + d*x]*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) - 9*Cos[2*c + 3*d*x]*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] - 9*Cos[4*c + 3*d*x]*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + 48*Sin[d*x] - 12*Sin[2*c + d*x] - 6*Sin[c + 2*d*x] - 6*Sin[3*c + 2*d*x] + 20*Sin[2*c + 3*d*x]))/8))/(3*a*d*(1 + Sec[c + d*x]))","B",1
43,1,250,85,1.4570768,"\int \frac{\sec ^4(c+d x)}{a+a \sec (c+d x)} \, dx","Integrate[Sec[c + d*x]^4/(a + a*Sec[c + d*x]),x]","\frac{\cos \left(\frac{1}{2} (c+d x)\right) \sec (c+d x) \left(\cos \left(\frac{1}{2} (c+d x)\right) \left(-\frac{4 \sin (d x)}{\left(\cos \left(\frac{c}{2}\right)-\sin \left(\frac{c}{2}\right)\right) \left(\sin \left(\frac{c}{2}\right)+\cos \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) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}+\frac{1}{\left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^2}-\frac{1}{\left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2}-6 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+6 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)-4 \sec \left(\frac{c}{2}\right) \sin \left(\frac{d x}{2}\right)\right)}{2 a d (\sec (c+d x)+1)}","-\frac{2 \tan (c+d x)}{a d}+\frac{3 \tanh ^{-1}(\sin (c+d x))}{2 a d}-\frac{\tan (c+d x) \sec ^2(c+d x)}{d (a \sec (c+d x)+a)}+\frac{3 \tan (c+d x) \sec (c+d x)}{2 a d}",1,"(Cos[(c + d*x)/2]*Sec[c + d*x]*(-4*Sec[c/2]*Sin[(d*x)/2] + Cos[(c + d*x)/2]*(-6*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 6*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + (Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^(-2) - (Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^(-2) - (4*Sin[d*x])/((Cos[c/2] - Sin[c/2])*(Cos[c/2] + Sin[c/2])*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])))))/(2*a*d*(1 + Sec[c + d*x]))","B",1
44,1,194,51,0.7979856,"\int \frac{\sec ^3(c+d x)}{a+a \sec (c+d x)} \, dx","Integrate[Sec[c + d*x]^3/(a + a*Sec[c + d*x]),x]","\frac{2 \cos \left(\frac{1}{2} (c+d x)\right) \sec (c+d x) \left(\sec \left(\frac{c}{2}\right) \sin \left(\frac{d x}{2}\right)+\cos \left(\frac{1}{2} (c+d x)\right) \left(\frac{\sin (d x)}{\left(\cos \left(\frac{c}{2}\right)-\sin \left(\frac{c}{2}\right)\right) \left(\sin \left(\frac{c}{2}\right)+\cos \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) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}+\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)\right)}{a d (\sec (c+d x)+1)}","\frac{\tan (c+d x)}{a d}-\frac{\tanh ^{-1}(\sin (c+d x))}{a d}+\frac{\tan (c+d x)}{d (a \sec (c+d x)+a)}",1,"(2*Cos[(c + d*x)/2]*Sec[c + d*x]*(Sec[c/2]*Sin[(d*x)/2] + Cos[(c + d*x)/2]*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + Sin[d*x]/((Cos[c/2] - Sin[c/2])*(Cos[c/2] + Sin[c/2])*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])))))/(a*d*(1 + Sec[c + d*x]))","B",1
45,1,109,38,0.1803562,"\int \frac{\sec ^2(c+d x)}{a+a \sec (c+d x)} \, dx","Integrate[Sec[c + d*x]^2/(a + a*Sec[c + d*x]),x]","-\frac{2 \cos \left(\frac{1}{2} (c+d x)\right) \sec (c+d x) \left(\sec \left(\frac{c}{2}\right) \sin \left(\frac{d x}{2}\right)+\cos \left(\frac{1}{2} (c+d x)\right) \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)\right)}{a d (\sec (c+d x)+1)}","\frac{\tanh ^{-1}(\sin (c+d x))}{a d}-\frac{\tan (c+d x)}{d (a \sec (c+d x)+a)}",1,"(-2*Cos[(c + d*x)/2]*Sec[c + d*x]*(Cos[(c + d*x)/2]*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) + Sec[c/2]*Sin[(d*x)/2]))/(a*d*(1 + Sec[c + d*x]))","B",1
46,1,17,22,0.0245559,"\int \frac{\sec (c+d x)}{a+a \sec (c+d x)} \, dx","Integrate[Sec[c + d*x]/(a + a*Sec[c + d*x]),x]","\frac{\tan \left(\frac{1}{2} (c+d x)\right)}{a d}","\frac{\tan (c+d x)}{d (a \sec (c+d x)+a)}",1,"Tan[(c + d*x)/2]/(a*d)","A",1
47,1,58,29,0.1282118,"\int \frac{1}{a+a \sec (c+d x)} \, dx","Integrate[(a + a*Sec[c + d*x])^(-1),x]","\frac{\sec \left(\frac{c}{2}\right) \sec \left(\frac{1}{2} (c+d x)\right) \left(d x \cos \left(c+\frac{d x}{2}\right)-2 \sin \left(\frac{d x}{2}\right)+d x \cos \left(\frac{d x}{2}\right)\right)}{2 a d}","\frac{x}{a}-\frac{\tan (c+d x)}{d (a \sec (c+d x)+a)}",1,"(Sec[c/2]*Sec[(c + d*x)/2]*(d*x*Cos[(d*x)/2] + d*x*Cos[c + (d*x)/2] - 2*Sin[(d*x)/2]))/(2*a*d)","A",1
48,1,89,44,0.2329365,"\int \frac{\cos (c+d x)}{a+a \sec (c+d x)} \, dx","Integrate[Cos[c + d*x]/(a + a*Sec[c + d*x]),x]","\frac{\sec \left(\frac{c}{2}\right) \sec \left(\frac{1}{2} (c+d x)\right) \left(\sin \left(c+\frac{d x}{2}\right)+\sin \left(c+\frac{3 d x}{2}\right)+\sin \left(2 c+\frac{3 d x}{2}\right)-2 d x \cos \left(c+\frac{d x}{2}\right)+5 \sin \left(\frac{d x}{2}\right)-2 d x \cos \left(\frac{d x}{2}\right)\right)}{4 a d}","\frac{2 \sin (c+d x)}{a d}-\frac{\sin (c+d x)}{d (a \sec (c+d x)+a)}-\frac{x}{a}",1,"(Sec[c/2]*Sec[(c + d*x)/2]*(-2*d*x*Cos[(d*x)/2] - 2*d*x*Cos[c + (d*x)/2] + 5*Sin[(d*x)/2] + Sin[c + (d*x)/2] + Sin[c + (3*d*x)/2] + Sin[2*c + (3*d*x)/2]))/(4*a*d)","B",1
49,1,117,74,0.24575,"\int \frac{\cos ^2(c+d x)}{a+a \sec (c+d x)} \, dx","Integrate[Cos[c + d*x]^2/(a + a*Sec[c + d*x]),x]","\frac{\sec \left(\frac{c}{2}\right) \sec \left(\frac{1}{2} (c+d x)\right) \left(-4 \sin \left(c+\frac{d x}{2}\right)-3 \sin \left(c+\frac{3 d x}{2}\right)-3 \sin \left(2 c+\frac{3 d x}{2}\right)+\sin \left(2 c+\frac{5 d x}{2}\right)+\sin \left(3 c+\frac{5 d x}{2}\right)+12 d x \cos \left(c+\frac{d x}{2}\right)-20 \sin \left(\frac{d x}{2}\right)+12 d x \cos \left(\frac{d x}{2}\right)\right)}{16 a d}","-\frac{2 \sin (c+d x)}{a d}+\frac{3 \sin (c+d x) \cos (c+d x)}{2 a d}-\frac{\sin (c+d x) \cos (c+d x)}{d (a \sec (c+d x)+a)}+\frac{3 x}{2 a}",1,"(Sec[c/2]*Sec[(c + d*x)/2]*(12*d*x*Cos[(d*x)/2] + 12*d*x*Cos[c + (d*x)/2] - 20*Sin[(d*x)/2] - 4*Sin[c + (d*x)/2] - 3*Sin[c + (3*d*x)/2] - 3*Sin[2*c + (3*d*x)/2] + Sin[2*c + (5*d*x)/2] + Sin[3*c + (5*d*x)/2]))/(16*a*d)","A",1
50,1,143,94,0.3227625,"\int \frac{\cos ^3(c+d x)}{a+a \sec (c+d x)} \, dx","Integrate[Cos[c + d*x]^3/(a + a*Sec[c + d*x]),x]","\frac{\sec \left(\frac{c}{2}\right) \sec \left(\frac{1}{2} (c+d x)\right) \left(21 \sin \left(c+\frac{d x}{2}\right)+18 \sin \left(c+\frac{3 d x}{2}\right)+18 \sin \left(2 c+\frac{3 d x}{2}\right)-2 \sin \left(2 c+\frac{5 d x}{2}\right)-2 \sin \left(3 c+\frac{5 d x}{2}\right)+\sin \left(3 c+\frac{7 d x}{2}\right)+\sin \left(4 c+\frac{7 d x}{2}\right)-36 d x \cos \left(c+\frac{d x}{2}\right)+69 \sin \left(\frac{d x}{2}\right)-36 d x \cos \left(\frac{d x}{2}\right)\right)}{48 a d}","-\frac{4 \sin ^3(c+d x)}{3 a d}+\frac{4 \sin (c+d x)}{a d}-\frac{3 \sin (c+d x) \cos (c+d x)}{2 a d}-\frac{\sin (c+d x) \cos ^2(c+d x)}{d (a \sec (c+d x)+a)}-\frac{3 x}{2 a}",1,"(Sec[c/2]*Sec[(c + d*x)/2]*(-36*d*x*Cos[(d*x)/2] - 36*d*x*Cos[c + (d*x)/2] + 69*Sin[(d*x)/2] + 21*Sin[c + (d*x)/2] + 18*Sin[c + (3*d*x)/2] + 18*Sin[2*c + (3*d*x)/2] - 2*Sin[2*c + (5*d*x)/2] - 2*Sin[3*c + (5*d*x)/2] + Sin[3*c + (7*d*x)/2] + Sin[4*c + (7*d*x)/2]))/(48*a*d)","A",1
51,1,173,118,0.316391,"\int \frac{\cos ^4(c+d x)}{a+a \sec (c+d x)} \, dx","Integrate[Cos[c + d*x]^4/(a + a*Sec[c + d*x]),x]","\frac{\sec \left(\frac{c}{2}\right) \sec \left(\frac{1}{2} (c+d x)\right) \left(-168 \sin \left(c+\frac{d x}{2}\right)-120 \sin \left(c+\frac{3 d x}{2}\right)-120 \sin \left(2 c+\frac{3 d x}{2}\right)+40 \sin \left(2 c+\frac{5 d x}{2}\right)+40 \sin \left(3 c+\frac{5 d x}{2}\right)-5 \sin \left(3 c+\frac{7 d x}{2}\right)-5 \sin \left(4 c+\frac{7 d x}{2}\right)+3 \sin \left(4 c+\frac{9 d x}{2}\right)+3 \sin \left(5 c+\frac{9 d x}{2}\right)+360 d x \cos \left(c+\frac{d x}{2}\right)-552 \sin \left(\frac{d x}{2}\right)+360 d x \cos \left(\frac{d x}{2}\right)\right)}{384 a d}","\frac{4 \sin ^3(c+d x)}{3 a d}-\frac{4 \sin (c+d x)}{a d}+\frac{5 \sin (c+d x) \cos ^3(c+d x)}{4 a d}+\frac{15 \sin (c+d x) \cos (c+d x)}{8 a d}-\frac{\sin (c+d x) \cos ^3(c+d x)}{d (a \sec (c+d x)+a)}+\frac{15 x}{8 a}",1,"(Sec[c/2]*Sec[(c + d*x)/2]*(360*d*x*Cos[(d*x)/2] + 360*d*x*Cos[c + (d*x)/2] - 552*Sin[(d*x)/2] - 168*Sin[c + (d*x)/2] - 120*Sin[c + (3*d*x)/2] - 120*Sin[2*c + (3*d*x)/2] + 40*Sin[2*c + (5*d*x)/2] + 40*Sin[3*c + (5*d*x)/2] - 5*Sin[3*c + (7*d*x)/2] - 5*Sin[4*c + (7*d*x)/2] + 3*Sin[4*c + (9*d*x)/2] + 3*Sin[5*c + (9*d*x)/2]))/(384*a*d)","A",1
52,1,300,123,1.9045683,"\int \frac{\sec ^5(c+d x)}{(a+a \sec (c+d x))^2} \, dx","Integrate[Sec[c + d*x]^5/(a + a*Sec[c + d*x])^2,x]","\frac{\cos \left(\frac{1}{2} (c+d x)\right) \sec ^2(c+d x) \left(-2 \tan \left(\frac{c}{2}\right) \cos \left(\frac{1}{2} (c+d x)\right)-2 \sec \left(\frac{c}{2}\right) \sin \left(\frac{d x}{2}\right)+3 \cos ^3\left(\frac{1}{2} (c+d x)\right) \left(-\frac{8 \sin (d x)}{\left(\cos \left(\frac{c}{2}\right)-\sin \left(\frac{c}{2}\right)\right) \left(\sin \left(\frac{c}{2}\right)+\cos \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) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}+\frac{1}{\left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^2}-\frac{1}{\left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2}-14 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+14 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)-40 \sec \left(\frac{c}{2}\right) \sin \left(\frac{d x}{2}\right) \cos ^2\left(\frac{1}{2} (c+d x)\right)\right)}{3 a^2 d (\sec (c+d x)+1)^2}","-\frac{16 \tan (c+d x)}{3 a^2 d}+\frac{7 \tanh ^{-1}(\sin (c+d x))}{2 a^2 d}-\frac{8 \tan (c+d x) \sec ^2(c+d x)}{3 a^2 d (\sec (c+d x)+1)}+\frac{7 \tan (c+d x) \sec (c+d x)}{2 a^2 d}-\frac{\tan (c+d x) \sec ^3(c+d x)}{3 d (a \sec (c+d x)+a)^2}",1,"(Cos[(c + d*x)/2]*Sec[c + d*x]^2*(-2*Sec[c/2]*Sin[(d*x)/2] - 40*Cos[(c + d*x)/2]^2*Sec[c/2]*Sin[(d*x)/2] + 3*Cos[(c + d*x)/2]^3*(-14*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 14*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + (Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^(-2) - (Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^(-2) - (8*Sin[d*x])/((Cos[c/2] - Sin[c/2])*(Cos[c/2] + Sin[c/2])*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))) - 2*Cos[(c + d*x)/2]*Tan[c/2]))/(3*a^2*d*(1 + Sec[c + d*x])^2)","B",1
53,1,247,89,1.2624332,"\int \frac{\sec ^4(c+d x)}{(a+a \sec (c+d x))^2} \, dx","Integrate[Sec[c + d*x]^4/(a + a*Sec[c + d*x])^2,x]","\frac{2 \cos \left(\frac{1}{2} (c+d x)\right) \sec ^2(c+d x) \left(\tan \left(\frac{c}{2}\right) \cos \left(\frac{1}{2} (c+d x)\right)+\sec \left(\frac{c}{2}\right) \sin \left(\frac{d x}{2}\right)+6 \cos ^3\left(\frac{1}{2} (c+d x)\right) \left(\frac{\sin (d x)}{\left(\cos \left(\frac{c}{2}\right)-\sin \left(\frac{c}{2}\right)\right) \left(\sin \left(\frac{c}{2}\right)+\cos \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) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}+2 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-2 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)+14 \sec \left(\frac{c}{2}\right) \sin \left(\frac{d x}{2}\right) \cos ^2\left(\frac{1}{2} (c+d x)\right)\right)}{3 a^2 d (\sec (c+d x)+1)^2}","\frac{4 \tan (c+d x)}{3 a^2 d}-\frac{2 \tanh ^{-1}(\sin (c+d x))}{a^2 d}+\frac{2 \tan (c+d x)}{a^2 d (\sec (c+d x)+1)}-\frac{\tan (c+d x) \sec ^2(c+d x)}{3 d (a \sec (c+d x)+a)^2}",1,"(2*Cos[(c + d*x)/2]*Sec[c + d*x]^2*(Sec[c/2]*Sin[(d*x)/2] + 14*Cos[(c + d*x)/2]^2*Sec[c/2]*Sin[(d*x)/2] + 6*Cos[(c + d*x)/2]^3*(2*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - 2*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + Sin[d*x]/((Cos[c/2] - Sin[c/2])*(Cos[c/2] + Sin[c/2])*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))) + Cos[(c + d*x)/2]*Tan[c/2]))/(3*a^2*d*(1 + Sec[c + d*x])^2)","B",1
54,1,160,66,0.367746,"\int \frac{\sec ^3(c+d x)}{(a+a \sec (c+d x))^2} \, dx","Integrate[Sec[c + d*x]^3/(a + a*Sec[c + d*x])^2,x]","-\frac{2 \cos \left(\frac{1}{2} (c+d x)\right) \sec ^2(c+d x) \left(\tan \left(\frac{c}{2}\right) \cos \left(\frac{1}{2} (c+d x)\right)+\sec \left(\frac{c}{2}\right) \sin \left(\frac{d x}{2}\right)+6 \cos ^3\left(\frac{1}{2} (c+d x)\right) \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)+8 \sec \left(\frac{c}{2}\right) \sin \left(\frac{d x}{2}\right) \cos ^2\left(\frac{1}{2} (c+d x)\right)\right)}{3 a^2 d (\sec (c+d x)+1)^2}","\frac{\tanh ^{-1}(\sin (c+d x))}{a^2 d}-\frac{5 \tan (c+d x)}{3 a^2 d (\sec (c+d x)+1)}+\frac{\tan (c+d x)}{3 d (a \sec (c+d x)+a)^2}",1,"(-2*Cos[(c + d*x)/2]*Sec[c + d*x]^2*(6*Cos[(c + d*x)/2]^3*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) + Sec[c/2]*Sin[(d*x)/2] + 8*Cos[(c + d*x)/2]^2*Sec[c/2]*Sin[(d*x)/2] + Cos[(c + d*x)/2]*Tan[c/2]))/(3*a^2*d*(1 + Sec[c + d*x])^2)","B",1
55,1,45,55,0.0689301,"\int \frac{\sec ^2(c+d x)}{(a+a \sec (c+d x))^2} \, dx","Integrate[Sec[c + d*x]^2/(a + a*Sec[c + d*x])^2,x]","\frac{\left(3 \sin \left(\frac{1}{2} (c+d x)\right)+\sin \left(\frac{3}{2} (c+d x)\right)\right) \sec ^3\left(\frac{1}{2} (c+d x)\right)}{12 a^2 d}","\frac{2 \tan (c+d x)}{3 d \left(a^2 \sec (c+d x)+a^2\right)}-\frac{\tan (c+d x)}{3 d (a \sec (c+d x)+a)^2}",1,"(Sec[(c + d*x)/2]^3*(3*Sin[(c + d*x)/2] + Sin[(3*(c + d*x))/2]))/(12*a^2*d)","A",1
56,1,60,55,0.1433582,"\int \frac{\sec (c+d x)}{(a+a \sec (c+d x))^2} \, dx","Integrate[Sec[c + d*x]/(a + a*Sec[c + d*x])^2,x]","\frac{\sec \left(\frac{c}{2}\right) \left(-3 \sin \left(c+\frac{d x}{2}\right)+2 \sin \left(c+\frac{3 d x}{2}\right)+3 \sin \left(\frac{d x}{2}\right)\right) \sec ^3\left(\frac{1}{2} (c+d x)\right)}{12 a^2 d}","\frac{\tan (c+d x)}{3 d \left(a^2 \sec (c+d x)+a^2\right)}+\frac{\tan (c+d x)}{3 d (a \sec (c+d x)+a)^2}",1,"(Sec[c/2]*Sec[(c + d*x)/2]^3*(3*Sin[(d*x)/2] - 3*Sin[c + (d*x)/2] + 2*Sin[c + (3*d*x)/2]))/(12*a^2*d)","A",1
57,1,112,57,0.2829844,"\int \frac{1}{(a+a \sec (c+d x))^2} \, dx","Integrate[(a + a*Sec[c + d*x])^(-2),x]","\frac{\sec \left(\frac{c}{2}\right) \sec ^3\left(\frac{1}{2} (c+d x)\right) \left(12 \sin \left(c+\frac{d x}{2}\right)-10 \sin \left(c+\frac{3 d x}{2}\right)+9 d x \cos \left(c+\frac{d x}{2}\right)+3 d x \cos \left(c+\frac{3 d x}{2}\right)+3 d x \cos \left(2 c+\frac{3 d x}{2}\right)-18 \sin \left(\frac{d x}{2}\right)+9 d x \cos \left(\frac{d x}{2}\right)\right)}{24 a^2 d}","-\frac{4 \tan (c+d x)}{3 a^2 d (\sec (c+d x)+1)}+\frac{x}{a^2}-\frac{\tan (c+d x)}{3 d (a \sec (c+d x)+a)^2}",1,"(Sec[c/2]*Sec[(c + d*x)/2]^3*(9*d*x*Cos[(d*x)/2] + 9*d*x*Cos[c + (d*x)/2] + 3*d*x*Cos[c + (3*d*x)/2] + 3*d*x*Cos[2*c + (3*d*x)/2] - 18*Sin[(d*x)/2] + 12*Sin[c + (d*x)/2] - 10*Sin[c + (3*d*x)/2]))/(24*a^2*d)","A",1
58,1,151,72,0.5251975,"\int \frac{\cos (c+d x)}{(a+a \sec (c+d x))^2} \, dx","Integrate[Cos[c + d*x]/(a + a*Sec[c + d*x])^2,x]","\frac{\sec \left(\frac{c}{2}\right) \sec ^3\left(\frac{1}{2} (c+d x)\right) \left(-30 \sin \left(c+\frac{d x}{2}\right)+41 \sin \left(c+\frac{3 d x}{2}\right)+9 \sin \left(2 c+\frac{3 d x}{2}\right)+3 \sin \left(2 c+\frac{5 d x}{2}\right)+3 \sin \left(3 c+\frac{5 d x}{2}\right)-36 d x \cos \left(c+\frac{d x}{2}\right)-12 d x \cos \left(c+\frac{3 d x}{2}\right)-12 d x \cos \left(2 c+\frac{3 d x}{2}\right)+66 \sin \left(\frac{d x}{2}\right)-36 d x \cos \left(\frac{d x}{2}\right)\right)}{48 a^2 d}","\frac{10 \sin (c+d x)}{3 a^2 d}-\frac{2 \sin (c+d x)}{a^2 d (\sec (c+d x)+1)}-\frac{2 x}{a^2}-\frac{\sin (c+d x)}{3 d (a \sec (c+d x)+a)^2}",1,"(Sec[c/2]*Sec[(c + d*x)/2]^3*(-36*d*x*Cos[(d*x)/2] - 36*d*x*Cos[c + (d*x)/2] - 12*d*x*Cos[c + (3*d*x)/2] - 12*d*x*Cos[2*c + (3*d*x)/2] + 66*Sin[(d*x)/2] - 30*Sin[c + (d*x)/2] + 41*Sin[c + (3*d*x)/2] + 9*Sin[2*c + (3*d*x)/2] + 3*Sin[2*c + (5*d*x)/2] + 3*Sin[3*c + (5*d*x)/2]))/(48*a^2*d)","B",1
59,1,177,110,0.3903033,"\int \frac{\cos ^2(c+d x)}{(a+a \sec (c+d x))^2} \, dx","Integrate[Cos[c + d*x]^2/(a + a*Sec[c + d*x])^2,x]","\frac{\sec \left(\frac{c}{2}\right) \sec ^3\left(\frac{1}{2} (c+d x)\right) \left(147 \sin \left(c+\frac{d x}{2}\right)-239 \sin \left(c+\frac{3 d x}{2}\right)-63 \sin \left(2 c+\frac{3 d x}{2}\right)-15 \sin \left(2 c+\frac{5 d x}{2}\right)-15 \sin \left(3 c+\frac{5 d x}{2}\right)+3 \sin \left(3 c+\frac{7 d x}{2}\right)+3 \sin \left(4 c+\frac{7 d x}{2}\right)+252 d x \cos \left(c+\frac{d x}{2}\right)+84 d x \cos \left(c+\frac{3 d x}{2}\right)+84 d x \cos \left(2 c+\frac{3 d x}{2}\right)-381 \sin \left(\frac{d x}{2}\right)+252 d x \cos \left(\frac{d x}{2}\right)\right)}{192 a^2 d}","-\frac{16 \sin (c+d x)}{3 a^2 d}+\frac{7 \sin (c+d x) \cos (c+d x)}{2 a^2 d}-\frac{8 \sin (c+d x) \cos (c+d x)}{3 a^2 d (\sec (c+d x)+1)}+\frac{7 x}{2 a^2}-\frac{\sin (c+d x) \cos (c+d x)}{3 d (a \sec (c+d x)+a)^2}",1,"(Sec[c/2]*Sec[(c + d*x)/2]^3*(252*d*x*Cos[(d*x)/2] + 252*d*x*Cos[c + (d*x)/2] + 84*d*x*Cos[c + (3*d*x)/2] + 84*d*x*Cos[2*c + (3*d*x)/2] - 381*Sin[(d*x)/2] + 147*Sin[c + (d*x)/2] - 239*Sin[c + (3*d*x)/2] - 63*Sin[2*c + (3*d*x)/2] - 15*Sin[2*c + (5*d*x)/2] - 15*Sin[3*c + (5*d*x)/2] + 3*Sin[3*c + (7*d*x)/2] + 3*Sin[4*c + (7*d*x)/2]))/(192*a^2*d)","A",1
60,1,199,124,0.4514649,"\int \frac{\cos ^3(c+d x)}{(a+a \sec (c+d x))^2} \, dx","Integrate[Cos[c + d*x]^3/(a + a*Sec[c + d*x])^2,x]","\frac{\sec \left(\frac{c}{2}\right) \sec ^3\left(\frac{1}{2} (c+d x)\right) \left(-156 \sin \left(c+\frac{d x}{2}\right)+342 \sin \left(c+\frac{3 d x}{2}\right)+118 \sin \left(2 c+\frac{3 d x}{2}\right)+30 \sin \left(2 c+\frac{5 d x}{2}\right)+30 \sin \left(3 c+\frac{5 d x}{2}\right)-3 \sin \left(3 c+\frac{7 d x}{2}\right)-3 \sin \left(4 c+\frac{7 d x}{2}\right)+\sin \left(4 c+\frac{9 d x}{2}\right)+\sin \left(5 c+\frac{9 d x}{2}\right)-360 d x \cos \left(c+\frac{d x}{2}\right)-120 d x \cos \left(c+\frac{3 d x}{2}\right)-120 d x \cos \left(2 c+\frac{3 d x}{2}\right)+516 \sin \left(\frac{d x}{2}\right)-360 d x \cos \left(\frac{d x}{2}\right)\right)}{192 a^2 d}","-\frac{4 \sin ^3(c+d x)}{a^2 d}+\frac{12 \sin (c+d x)}{a^2 d}-\frac{5 \sin (c+d x) \cos (c+d x)}{a^2 d}-\frac{10 \sin (c+d x) \cos ^2(c+d x)}{3 a^2 d (\sec (c+d x)+1)}-\frac{5 x}{a^2}-\frac{\sin (c+d x) \cos ^2(c+d x)}{3 d (a \sec (c+d x)+a)^2}",1,"(Sec[c/2]*Sec[(c + d*x)/2]^3*(-360*d*x*Cos[(d*x)/2] - 360*d*x*Cos[c + (d*x)/2] - 120*d*x*Cos[c + (3*d*x)/2] - 120*d*x*Cos[2*c + (3*d*x)/2] + 516*Sin[(d*x)/2] - 156*Sin[c + (d*x)/2] + 342*Sin[c + (3*d*x)/2] + 118*Sin[2*c + (3*d*x)/2] + 30*Sin[2*c + (5*d*x)/2] + 30*Sin[3*c + (5*d*x)/2] - 3*Sin[3*c + (7*d*x)/2] - 3*Sin[4*c + (7*d*x)/2] + Sin[4*c + (9*d*x)/2] + Sin[5*c + (9*d*x)/2]))/(192*a^2*d)","A",1
61,1,351,162,1.0514528,"\int \frac{\sec ^6(c+d x)}{(a+a \sec (c+d x))^3} \, dx","Integrate[Sec[c + d*x]^6/(a + a*Sec[c + d*x])^3,x]","-\frac{\cos \left(\frac{1}{2} (c+d x)\right) \sec ^3(c+d x) \left(\sec \left(\frac{c}{2}\right) \sec (c) \left(-4329 \sin \left(c-\frac{d x}{2}\right)+1989 \sin \left(c+\frac{d x}{2}\right)-3575 \sin \left(2 c+\frac{d x}{2}\right)-475 \sin \left(c+\frac{3 d x}{2}\right)+2005 \sin \left(2 c+\frac{3 d x}{2}\right)-2275 \sin \left(3 c+\frac{3 d x}{2}\right)+2673 \sin \left(c+\frac{5 d x}{2}\right)+105 \sin \left(2 c+\frac{5 d x}{2}\right)+1593 \sin \left(3 c+\frac{5 d x}{2}\right)-975 \sin \left(4 c+\frac{5 d x}{2}\right)+1325 \sin \left(2 c+\frac{7 d x}{2}\right)+255 \sin \left(3 c+\frac{7 d x}{2}\right)+875 \sin \left(4 c+\frac{7 d x}{2}\right)-195 \sin \left(5 c+\frac{7 d x}{2}\right)+304 \sin \left(3 c+\frac{9 d x}{2}\right)+90 \sin \left(4 c+\frac{9 d x}{2}\right)+214 \sin \left(5 c+\frac{9 d x}{2}\right)-1235 \sin \left(\frac{d x}{2}\right)+3805 \sin \left(\frac{3 d x}{2}\right)\right) \sec ^2(c+d x)+24960 \cos ^5\left(\frac{1}{2} (c+d x)\right) \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)\right)}{480 a^3 d (\sec (c+d x)+1)^3}","-\frac{152 \tan (c+d x)}{15 a^3 d}+\frac{13 \tanh ^{-1}(\sin (c+d x))}{2 a^3 d}-\frac{76 \tan (c+d x) \sec ^2(c+d x)}{15 d \left(a^3 \sec (c+d x)+a^3\right)}+\frac{13 \tan (c+d x) \sec (c+d x)}{2 a^3 d}-\frac{\tan (c+d x) \sec ^4(c+d x)}{5 d (a \sec (c+d x)+a)^3}-\frac{11 \tan (c+d x) \sec ^3(c+d x)}{15 a d (a \sec (c+d x)+a)^2}",1,"-1/480*(Cos[(c + d*x)/2]*Sec[c + d*x]^3*(24960*Cos[(c + d*x)/2]^5*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) + Sec[c/2]*Sec[c]*Sec[c + d*x]^2*(-1235*Sin[(d*x)/2] + 3805*Sin[(3*d*x)/2] - 4329*Sin[c - (d*x)/2] + 1989*Sin[c + (d*x)/2] - 3575*Sin[2*c + (d*x)/2] - 475*Sin[c + (3*d*x)/2] + 2005*Sin[2*c + (3*d*x)/2] - 2275*Sin[3*c + (3*d*x)/2] + 2673*Sin[c + (5*d*x)/2] + 105*Sin[2*c + (5*d*x)/2] + 1593*Sin[3*c + (5*d*x)/2] - 975*Sin[4*c + (5*d*x)/2] + 1325*Sin[2*c + (7*d*x)/2] + 255*Sin[3*c + (7*d*x)/2] + 875*Sin[4*c + (7*d*x)/2] - 195*Sin[5*c + (7*d*x)/2] + 304*Sin[3*c + (9*d*x)/2] + 90*Sin[4*c + (9*d*x)/2] + 214*Sin[5*c + (9*d*x)/2])))/(a^3*d*(1 + Sec[c + d*x])^3)","B",1
62,1,294,128,1.3379042,"\int \frac{\sec ^5(c+d x)}{(a+a \sec (c+d x))^3} \, dx","Integrate[Sec[c + d*x]^5/(a + a*Sec[c + d*x])^3,x]","\frac{2 \cos \left(\frac{1}{2} (c+d x)\right) \sec ^3(c+d x) \left(8 \tan \left(\frac{c}{2}\right) \cos ^3\left(\frac{1}{2} (c+d x)\right)+\tan \left(\frac{c}{2}\right) \cos \left(\frac{1}{2} (c+d x)\right)+\sec \left(\frac{c}{2}\right) \sin \left(\frac{d x}{2}\right)+20 \cos ^5\left(\frac{1}{2} (c+d x)\right) \left(\frac{\sin (d x)}{\left(\cos \left(\frac{c}{2}\right)-\sin \left(\frac{c}{2}\right)\right) \left(\sin \left(\frac{c}{2}\right)+\cos \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) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}+3 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-3 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)+76 \sec \left(\frac{c}{2}\right) \sin \left(\frac{d x}{2}\right) \cos ^4\left(\frac{1}{2} (c+d x)\right)+8 \sec \left(\frac{c}{2}\right) \sin \left(\frac{d x}{2}\right) \cos ^2\left(\frac{1}{2} (c+d x)\right)\right)}{5 a^3 d (\sec (c+d x)+1)^3}","\frac{9 \tan (c+d x)}{5 a^3 d}-\frac{3 \tanh ^{-1}(\sin (c+d x))}{a^3 d}+\frac{3 \tan (c+d x)}{d \left(a^3 \sec (c+d x)+a^3\right)}-\frac{\tan (c+d x) \sec ^3(c+d x)}{5 d (a \sec (c+d x)+a)^3}-\frac{3 \tan (c+d x) \sec ^2(c+d x)}{5 a d (a \sec (c+d x)+a)^2}",1,"(2*Cos[(c + d*x)/2]*Sec[c + d*x]^3*(Sec[c/2]*Sin[(d*x)/2] + 8*Cos[(c + d*x)/2]^2*Sec[c/2]*Sin[(d*x)/2] + 76*Cos[(c + d*x)/2]^4*Sec[c/2]*Sin[(d*x)/2] + 20*Cos[(c + d*x)/2]^5*(3*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - 3*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + Sin[d*x]/((Cos[c/2] - Sin[c/2])*(Cos[c/2] + Sin[c/2])*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))) + Cos[(c + d*x)/2]*Tan[c/2] + 8*Cos[(c + d*x)/2]^3*Tan[c/2]))/(5*a^3*d*(1 + Sec[c + d*x])^3)","B",1
63,1,209,105,0.5123381,"\int \frac{\sec ^4(c+d x)}{(a+a \sec (c+d x))^3} \, dx","Integrate[Sec[c + d*x]^4/(a + a*Sec[c + d*x])^3,x]","-\frac{2 \cos \left(\frac{1}{2} (c+d x)\right) \sec ^3(c+d x) \left(14 \tan \left(\frac{c}{2}\right) \cos ^3\left(\frac{1}{2} (c+d x)\right)+3 \tan \left(\frac{c}{2}\right) \cos \left(\frac{1}{2} (c+d x)\right)+3 \sec \left(\frac{c}{2}\right) \sin \left(\frac{d x}{2}\right)+60 \cos ^5\left(\frac{1}{2} (c+d x)\right) \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)+88 \sec \left(\frac{c}{2}\right) \sin \left(\frac{d x}{2}\right) \cos ^4\left(\frac{1}{2} (c+d x)\right)+14 \sec \left(\frac{c}{2}\right) \sin \left(\frac{d x}{2}\right) \cos ^2\left(\frac{1}{2} (c+d x)\right)\right)}{15 a^3 d (\sec (c+d x)+1)^3}","\frac{\tanh ^{-1}(\sin (c+d x))}{a^3 d}-\frac{29 \tan (c+d x)}{15 d \left(a^3 \sec (c+d x)+a^3\right)}-\frac{\tan (c+d x) \sec ^2(c+d x)}{5 d (a \sec (c+d x)+a)^3}+\frac{7 \tan (c+d x)}{15 a d (a \sec (c+d x)+a)^2}",1,"(-2*Cos[(c + d*x)/2]*Sec[c + d*x]^3*(60*Cos[(c + d*x)/2]^5*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) + 3*Sec[c/2]*Sin[(d*x)/2] + 14*Cos[(c + d*x)/2]^2*Sec[c/2]*Sin[(d*x)/2] + 88*Cos[(c + d*x)/2]^4*Sec[c/2]*Sin[(d*x)/2] + 3*Cos[(c + d*x)/2]*Tan[c/2] + 14*Cos[(c + d*x)/2]^3*Tan[c/2]))/(15*a^3*d*(1 + Sec[c + d*x])^3)","A",1
64,1,57,83,0.1103184,"\int \frac{\sec ^3(c+d x)}{(a+a \sec (c+d x))^3} \, dx","Integrate[Sec[c + d*x]^3/(a + a*Sec[c + d*x])^3,x]","\frac{\left(10 \sin \left(\frac{1}{2} (c+d x)\right)+5 \sin \left(\frac{3}{2} (c+d x)\right)+\sin \left(\frac{5}{2} (c+d x)\right)\right) \sec ^5\left(\frac{1}{2} (c+d x)\right)}{120 a^3 d}","\frac{7 \tan (c+d x)}{15 d \left(a^3 \sec (c+d x)+a^3\right)}-\frac{8 \tan (c+d x)}{15 a d (a \sec (c+d x)+a)^2}+\frac{\tan (c+d x)}{5 d (a \sec (c+d x)+a)^3}",1,"(Sec[(c + d*x)/2]^5*(10*Sin[(c + d*x)/2] + 5*Sin[(3*(c + d*x))/2] + Sin[(5*(c + d*x))/2]))/(120*a^3*d)","A",1
65,1,71,83,0.1641301,"\int \frac{\sec ^2(c+d x)}{(a+a \sec (c+d x))^3} \, dx","Integrate[Sec[c + d*x]^2/(a + a*Sec[c + d*x])^3,x]","\frac{\sec \left(\frac{c}{2}\right) \left(-5 \sin \left(c+\frac{d x}{2}\right)+5 \sin \left(c+\frac{3 d x}{2}\right)+\sin \left(2 c+\frac{5 d x}{2}\right)+5 \sin \left(\frac{d x}{2}\right)\right) \sec ^5\left(\frac{1}{2} (c+d x)\right)}{80 a^3 d}","\frac{\tan (c+d x)}{5 d \left(a^3 \sec (c+d x)+a^3\right)}+\frac{\tan (c+d x)}{5 a d (a \sec (c+d x)+a)^2}-\frac{\tan (c+d x)}{5 d (a \sec (c+d x)+a)^3}",1,"(Sec[c/2]*Sec[(c + d*x)/2]^5*(5*Sin[(d*x)/2] - 5*Sin[c + (d*x)/2] + 5*Sin[c + (3*d*x)/2] + Sin[2*c + (5*d*x)/2]))/(80*a^3*d)","A",1
66,1,86,83,0.2240217,"\int \frac{\sec (c+d x)}{(a+a \sec (c+d x))^3} \, dx","Integrate[Sec[c + d*x]/(a + a*Sec[c + d*x])^3,x]","\frac{\sec \left(\frac{c}{2}\right) \left(-30 \sin \left(c+\frac{d x}{2}\right)+20 \sin \left(c+\frac{3 d x}{2}\right)-15 \sin \left(2 c+\frac{3 d x}{2}\right)+7 \sin \left(2 c+\frac{5 d x}{2}\right)+40 \sin \left(\frac{d x}{2}\right)\right) \sec ^5\left(\frac{1}{2} (c+d x)\right)}{240 a^3 d}","\frac{2 \tan (c+d x)}{15 d \left(a^3 \sec (c+d x)+a^3\right)}+\frac{2 \tan (c+d x)}{15 a d (a \sec (c+d x)+a)^2}+\frac{\tan (c+d x)}{5 d (a \sec (c+d x)+a)^3}",1,"(Sec[c/2]*Sec[(c + d*x)/2]^5*(40*Sin[(d*x)/2] - 30*Sin[c + (d*x)/2] + 20*Sin[c + (3*d*x)/2] - 15*Sin[2*c + (3*d*x)/2] + 7*Sin[2*c + (5*d*x)/2]))/(240*a^3*d)","A",1
67,1,162,88,0.2777296,"\int \frac{1}{(a+a \sec (c+d x))^3} \, dx","Integrate[(a + a*Sec[c + d*x])^(-3),x]","\frac{2 \cos \left(\frac{1}{2} (c+d x)\right) \sec ^3(c+d x) \left(60 d x \cos ^5\left(\frac{1}{2} (c+d x)\right)+26 \tan \left(\frac{c}{2}\right) \cos ^3\left(\frac{1}{2} (c+d x)\right)-3 \tan \left(\frac{c}{2}\right) \cos \left(\frac{1}{2} (c+d x)\right)-3 \sec \left(\frac{c}{2}\right) \sin \left(\frac{d x}{2}\right)-128 \sec \left(\frac{c}{2}\right) \sin \left(\frac{d x}{2}\right) \cos ^4\left(\frac{1}{2} (c+d x)\right)+26 \sec \left(\frac{c}{2}\right) \sin \left(\frac{d x}{2}\right) \cos ^2\left(\frac{1}{2} (c+d x)\right)\right)}{15 a^3 d (\sec (c+d x)+1)^3}","-\frac{22 \tan (c+d x)}{15 d \left(a^3 \sec (c+d x)+a^3\right)}+\frac{x}{a^3}-\frac{7 \tan (c+d x)}{15 a d (a \sec (c+d x)+a)^2}-\frac{\tan (c+d x)}{5 d (a \sec (c+d x)+a)^3}",1,"(2*Cos[(c + d*x)/2]*Sec[c + d*x]^3*(60*d*x*Cos[(c + d*x)/2]^5 - 3*Sec[c/2]*Sin[(d*x)/2] + 26*Cos[(c + d*x)/2]^2*Sec[c/2]*Sin[(d*x)/2] - 128*Cos[(c + d*x)/2]^4*Sec[c/2]*Sin[(d*x)/2] - 3*Cos[(c + d*x)/2]*Tan[c/2] + 26*Cos[(c + d*x)/2]^3*Tan[c/2]))/(15*a^3*d*(1 + Sec[c + d*x])^3)","A",1
68,1,169,103,0.5807163,"\int \frac{\cos (c+d x)}{(a+a \sec (c+d x))^3} \, dx","Integrate[Cos[c + d*x]/(a + a*Sec[c + d*x])^3,x]","\frac{2 \cos \left(\frac{1}{2} (c+d x)\right) \sec ^3(c+d x) \left(20 (\sin (c+d x)-3 d x) \cos ^5\left(\frac{1}{2} (c+d x)\right)-12 \tan \left(\frac{c}{2}\right) \cos ^3\left(\frac{1}{2} (c+d x)\right)+\tan \left(\frac{c}{2}\right) \cos \left(\frac{1}{2} (c+d x)\right)+\sec \left(\frac{c}{2}\right) \sin \left(\frac{d x}{2}\right)+96 \sec \left(\frac{c}{2}\right) \sin \left(\frac{d x}{2}\right) \cos ^4\left(\frac{1}{2} (c+d x)\right)-12 \sec \left(\frac{c}{2}\right) \sin \left(\frac{d x}{2}\right) \cos ^2\left(\frac{1}{2} (c+d x)\right)\right)}{5 a^3 d (\sec (c+d x)+1)^3}","\frac{24 \sin (c+d x)}{5 a^3 d}-\frac{3 \sin (c+d x)}{d \left(a^3 \sec (c+d x)+a^3\right)}-\frac{3 x}{a^3}-\frac{3 \sin (c+d x)}{5 a d (a \sec (c+d x)+a)^2}-\frac{\sin (c+d x)}{5 d (a \sec (c+d x)+a)^3}",1,"(2*Cos[(c + d*x)/2]*Sec[c + d*x]^3*(Sec[c/2]*Sin[(d*x)/2] - 12*Cos[(c + d*x)/2]^2*Sec[c/2]*Sin[(d*x)/2] + 96*Cos[(c + d*x)/2]^4*Sec[c/2]*Sin[(d*x)/2] + 20*Cos[(c + d*x)/2]^5*(-3*d*x + Sin[c + d*x]) + Cos[(c + d*x)/2]*Tan[c/2] - 12*Cos[(c + d*x)/2]^3*Tan[c/2]))/(5*a^3*d*(1 + Sec[c + d*x])^3)","A",1
69,1,181,147,0.5770724,"\int \frac{\cos ^2(c+d x)}{(a+a \sec (c+d x))^3} \, dx","Integrate[Cos[c + d*x]^2/(a + a*Sec[c + d*x])^3,x]","\frac{2 \cos \left(\frac{1}{2} (c+d x)\right) \sec ^3(c+d x) \left(15 (-12 \sin (c+d x)+\sin (2 (c+d x))+26 d x) \cos ^5\left(\frac{1}{2} (c+d x)\right)+46 \tan \left(\frac{c}{2}\right) \cos ^3\left(\frac{1}{2} (c+d x)\right)-3 \tan \left(\frac{c}{2}\right) \cos \left(\frac{1}{2} (c+d x)\right)-3 \sec \left(\frac{c}{2}\right) \sin \left(\frac{d x}{2}\right)-508 \sec \left(\frac{c}{2}\right) \sin \left(\frac{d x}{2}\right) \cos ^4\left(\frac{1}{2} (c+d x)\right)+46 \sec \left(\frac{c}{2}\right) \sin \left(\frac{d x}{2}\right) \cos ^2\left(\frac{1}{2} (c+d x)\right)\right)}{15 a^3 d (\sec (c+d x)+1)^3}","-\frac{152 \sin (c+d x)}{15 a^3 d}+\frac{13 \sin (c+d x) \cos (c+d x)}{2 a^3 d}-\frac{76 \sin (c+d x) \cos (c+d x)}{15 d \left(a^3 \sec (c+d x)+a^3\right)}+\frac{13 x}{2 a^3}-\frac{11 \sin (c+d x) \cos (c+d x)}{15 a d (a \sec (c+d x)+a)^2}-\frac{\sin (c+d x) \cos (c+d x)}{5 d (a \sec (c+d x)+a)^3}",1,"(2*Cos[(c + d*x)/2]*Sec[c + d*x]^3*(-3*Sec[c/2]*Sin[(d*x)/2] + 46*Cos[(c + d*x)/2]^2*Sec[c/2]*Sin[(d*x)/2] - 508*Cos[(c + d*x)/2]^4*Sec[c/2]*Sin[(d*x)/2] + 15*Cos[(c + d*x)/2]^5*(26*d*x - 12*Sin[c + d*x] + Sin[2*(c + d*x)]) - 3*Cos[(c + d*x)/2]*Tan[c/2] + 46*Cos[(c + d*x)/2]^3*Tan[c/2]))/(15*a^3*d*(1 + Sec[c + d*x])^3)","A",1
70,1,403,193,1.5575608,"\int \frac{\sec ^7(c+d x)}{(a+a \sec (c+d x))^4} \, dx","Integrate[Sec[c + d*x]^7/(a + a*Sec[c + d*x])^4,x]","-\frac{\cos \left(\frac{1}{2} (c+d x)\right) \sec ^4(c+d x) \left(\sec \left(\frac{c}{2}\right) \sec (c) \left(-61054 \sin \left(c-\frac{d x}{2}\right)+33614 \sin \left(c+\frac{d x}{2}\right)-51842 \sin \left(2 c+\frac{d x}{2}\right)-12460 \sin \left(c+\frac{3 d x}{2}\right)+33716 \sin \left(2 c+\frac{3 d x}{2}\right)-34300 \sin \left(3 c+\frac{3 d x}{2}\right)+39788 \sin \left(c+\frac{5 d x}{2}\right)-2940 \sin \left(2 c+\frac{5 d x}{2}\right)+26068 \sin \left(3 c+\frac{5 d x}{2}\right)-16660 \sin \left(4 c+\frac{5 d x}{2}\right)+21351 \sin \left(2 c+\frac{7 d x}{2}\right)+1295 \sin \left(3 c+\frac{7 d x}{2}\right)+14911 \sin \left(4 c+\frac{7 d x}{2}\right)-5145 \sin \left(5 c+\frac{7 d x}{2}\right)+7329 \sin \left(3 c+\frac{9 d x}{2}\right)+1225 \sin \left(4 c+\frac{9 d x}{2}\right)+5369 \sin \left(5 c+\frac{9 d x}{2}\right)-735 \sin \left(6 c+\frac{9 d x}{2}\right)+1152 \sin \left(4 c+\frac{11 d x}{2}\right)+280 \sin \left(5 c+\frac{11 d x}{2}\right)+872 \sin \left(6 c+\frac{11 d x}{2}\right)-24402 \sin \left(\frac{d x}{2}\right)+55556 \sin \left(\frac{3 d x}{2}\right)\right) \sec ^2(c+d x)+376320 \cos ^7\left(\frac{1}{2} (c+d x)\right) \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)\right)}{2240 a^4 d (\sec (c+d x)+1)^4}","-\frac{576 \tan (c+d x)}{35 a^4 d}+\frac{21 \tanh ^{-1}(\sin (c+d x))}{2 a^4 d}-\frac{43 \tan (c+d x) \sec ^3(c+d x)}{35 a^4 d (\sec (c+d x)+1)^2}-\frac{288 \tan (c+d x) \sec ^2(c+d x)}{35 a^4 d (\sec (c+d x)+1)}+\frac{21 \tan (c+d x) \sec (c+d x)}{2 a^4 d}-\frac{\tan (c+d x) \sec ^5(c+d x)}{7 d (a \sec (c+d x)+a)^4}-\frac{2 \tan (c+d x) \sec ^4(c+d x)}{5 a d (a \sec (c+d x)+a)^3}",1,"-1/2240*(Cos[(c + d*x)/2]*Sec[c + d*x]^4*(376320*Cos[(c + d*x)/2]^7*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) + Sec[c/2]*Sec[c]*Sec[c + d*x]^2*(-24402*Sin[(d*x)/2] + 55556*Sin[(3*d*x)/2] - 61054*Sin[c - (d*x)/2] + 33614*Sin[c + (d*x)/2] - 51842*Sin[2*c + (d*x)/2] - 12460*Sin[c + (3*d*x)/2] + 33716*Sin[2*c + (3*d*x)/2] - 34300*Sin[3*c + (3*d*x)/2] + 39788*Sin[c + (5*d*x)/2] - 2940*Sin[2*c + (5*d*x)/2] + 26068*Sin[3*c + (5*d*x)/2] - 16660*Sin[4*c + (5*d*x)/2] + 21351*Sin[2*c + (7*d*x)/2] + 1295*Sin[3*c + (7*d*x)/2] + 14911*Sin[4*c + (7*d*x)/2] - 5145*Sin[5*c + (7*d*x)/2] + 7329*Sin[3*c + (9*d*x)/2] + 1225*Sin[4*c + (9*d*x)/2] + 5369*Sin[5*c + (9*d*x)/2] - 735*Sin[6*c + (9*d*x)/2] + 1152*Sin[4*c + (11*d*x)/2] + 280*Sin[5*c + (11*d*x)/2] + 872*Sin[6*c + (11*d*x)/2])))/(a^4*d*(1 + Sec[c + d*x])^4)","B",1
71,1,349,159,1.2868296,"\int \frac{\sec ^6(c+d x)}{(a+a \sec (c+d x))^4} \, dx","Integrate[Sec[c + d*x]^6/(a + a*Sec[c + d*x])^4,x]","\frac{\cos \left(\frac{1}{2} (c+d x)\right) \sec ^4(c+d x) \left(\sec \left(\frac{c}{2}\right) \sec (c) \left(-20524 \sin \left(c-\frac{d x}{2}\right)+14644 \sin \left(c+\frac{d x}{2}\right)-16660 \sin \left(2 c+\frac{d x}{2}\right)-4690 \sin \left(c+\frac{3 d x}{2}\right)+14378 \sin \left(2 c+\frac{3 d x}{2}\right)-9100 \sin \left(3 c+\frac{3 d x}{2}\right)+11668 \sin \left(c+\frac{5 d x}{2}\right)-630 \sin \left(2 c+\frac{5 d x}{2}\right)+9358 \sin \left(3 c+\frac{5 d x}{2}\right)-2940 \sin \left(4 c+\frac{5 d x}{2}\right)+4228 \sin \left(2 c+\frac{7 d x}{2}\right)+315 \sin \left(3 c+\frac{7 d x}{2}\right)+3493 \sin \left(4 c+\frac{7 d x}{2}\right)-420 \sin \left(5 c+\frac{7 d x}{2}\right)+664 \sin \left(3 c+\frac{9 d x}{2}\right)+105 \sin \left(4 c+\frac{9 d x}{2}\right)+559 \sin \left(5 c+\frac{9 d x}{2}\right)-10780 \sin \left(\frac{d x}{2}\right)+18788 \sin \left(\frac{3 d x}{2}\right)\right) \sec (c+d x)+107520 \cos ^7\left(\frac{1}{2} (c+d x)\right) \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)\right)}{1680 a^4 d (\sec (c+d x)+1)^4}","\frac{244 \tan (c+d x)}{105 a^4 d}-\frac{4 \tanh ^{-1}(\sin (c+d x))}{a^4 d}-\frac{88 \tan (c+d x) \sec ^2(c+d x)}{105 a^4 d (\sec (c+d x)+1)^2}+\frac{4 \tan (c+d x)}{a^4 d (\sec (c+d x)+1)}-\frac{\tan (c+d x) \sec ^4(c+d x)}{7 d (a \sec (c+d x)+a)^4}-\frac{12 \tan (c+d x) \sec ^3(c+d x)}{35 a d (a \sec (c+d x)+a)^3}",1,"(Cos[(c + d*x)/2]*Sec[c + d*x]^4*(107520*Cos[(c + d*x)/2]^7*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) + Sec[c/2]*Sec[c]*Sec[c + d*x]*(-10780*Sin[(d*x)/2] + 18788*Sin[(3*d*x)/2] - 20524*Sin[c - (d*x)/2] + 14644*Sin[c + (d*x)/2] - 16660*Sin[2*c + (d*x)/2] - 4690*Sin[c + (3*d*x)/2] + 14378*Sin[2*c + (3*d*x)/2] - 9100*Sin[3*c + (3*d*x)/2] + 11668*Sin[c + (5*d*x)/2] - 630*Sin[2*c + (5*d*x)/2] + 9358*Sin[3*c + (5*d*x)/2] - 2940*Sin[4*c + (5*d*x)/2] + 4228*Sin[2*c + (7*d*x)/2] + 315*Sin[3*c + (7*d*x)/2] + 3493*Sin[4*c + (7*d*x)/2] - 420*Sin[5*c + (7*d*x)/2] + 664*Sin[3*c + (9*d*x)/2] + 105*Sin[4*c + (9*d*x)/2] + 559*Sin[5*c + (9*d*x)/2])))/(1680*a^4*d*(1 + Sec[c + d*x])^4)","B",1
72,1,193,136,0.9355437,"\int \frac{\sec ^5(c+d x)}{(a+a \sec (c+d x))^4} \, dx","Integrate[Sec[c + d*x]^5/(a + a*Sec[c + d*x])^4,x]","-\frac{\cos \left(\frac{1}{2} (c+d x)\right) \sec ^4(c+d x) \left(\sec \left(\frac{c}{2}\right) \left(-434 \sin \left(c+\frac{d x}{2}\right)+525 \sin \left(c+\frac{3 d x}{2}\right)-147 \sin \left(2 c+\frac{3 d x}{2}\right)+203 \sin \left(2 c+\frac{5 d x}{2}\right)-21 \sin \left(3 c+\frac{5 d x}{2}\right)+32 \sin \left(3 c+\frac{7 d x}{2}\right)+686 \sin \left(\frac{d x}{2}\right)\right)+1344 \cos ^7\left(\frac{1}{2} (c+d x)\right) \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)\right)}{84 a^4 d (\sec (c+d x)+1)^4}","\frac{\tanh ^{-1}(\sin (c+d x))}{a^4 d}-\frac{43 \tan (c+d x)}{21 a^4 d (\sec (c+d x)+1)}+\frac{11 \tan (c+d x)}{21 a^4 d (\sec (c+d x)+1)^2}-\frac{\tan (c+d x) \sec ^3(c+d x)}{7 d (a \sec (c+d x)+a)^4}-\frac{2 \tan (c+d x) \sec ^2(c+d x)}{7 a d (a \sec (c+d x)+a)^3}",1,"-1/84*(Cos[(c + d*x)/2]*Sec[c + d*x]^4*(1344*Cos[(c + d*x)/2]^7*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) + Sec[c/2]*(686*Sin[(d*x)/2] - 434*Sin[c + (d*x)/2] + 525*Sin[c + (3*d*x)/2] - 147*Sin[2*c + (3*d*x)/2] + 203*Sin[2*c + (5*d*x)/2] - 21*Sin[3*c + (5*d*x)/2] + 32*Sin[3*c + (7*d*x)/2])))/(a^4*d*(1 + Sec[c + d*x])^4)","A",1
73,1,69,120,0.2001571,"\int \frac{\sec ^4(c+d x)}{(a+a \sec (c+d x))^4} \, dx","Integrate[Sec[c + d*x]^4/(a + a*Sec[c + d*x])^4,x]","\frac{\left(35 \sin \left(\frac{1}{2} (c+d x)\right)+21 \sin \left(\frac{3}{2} (c+d x)\right)+7 \sin \left(\frac{5}{2} (c+d x)\right)+\sin \left(\frac{7}{2} (c+d x)\right)\right) \sec ^7\left(\frac{1}{2} (c+d x)\right)}{1120 a^4 d}","\frac{\tan (c+d x)}{5 d \left(a^4 \sec (c+d x)+a^4\right)}-\frac{8 \tan (c+d x)}{35 d \left(a^2 \sec (c+d x)+a^2\right)^2}+\frac{\tan (c+d x) \sec ^3(c+d x)}{7 d (a \sec (c+d x)+a)^4}+\frac{3 \tan (c+d x)}{35 a d (a \sec (c+d x)+a)^3}",1,"(Sec[(c + d*x)/2]^7*(35*Sin[(c + d*x)/2] + 21*Sin[(3*(c + d*x))/2] + 7*Sin[(5*(c + d*x))/2] + Sin[(7*(c + d*x))/2]))/(1120*a^4*d)","A",1
74,1,87,112,0.2434639,"\int \frac{\sec ^3(c+d x)}{(a+a \sec (c+d x))^4} \, dx","Integrate[Sec[c + d*x]^3/(a + a*Sec[c + d*x])^4,x]","\frac{\sec \left(\frac{c}{2}\right) \left(-35 \sin \left(c+\frac{d x}{2}\right)+2 \left(21 \sin \left(c+\frac{3 d x}{2}\right)+7 \sin \left(2 c+\frac{5 d x}{2}\right)+\sin \left(3 c+\frac{7 d x}{2}\right)\right)+35 \sin \left(\frac{d x}{2}\right)\right) \sec ^7\left(\frac{1}{2} (c+d x)\right)}{1680 a^4 d}","\frac{13 \tan (c+d x)}{105 d \left(a^4 \sec (c+d x)+a^4\right)}+\frac{13 \tan (c+d x)}{105 d \left(a^2 \sec (c+d x)+a^2\right)^2}-\frac{11 \tan (c+d x)}{35 a d (a \sec (c+d x)+a)^3}+\frac{\tan (c+d x)}{7 d (a \sec (c+d x)+a)^4}",1,"(Sec[c/2]*Sec[(c + d*x)/2]^7*(35*Sin[(d*x)/2] - 35*Sin[c + (d*x)/2] + 2*(21*Sin[c + (3*d*x)/2] + 7*Sin[2*c + (5*d*x)/2] + Sin[3*c + (7*d*x)/2])))/(1680*a^4*d)","A",1
75,1,99,112,0.2678877,"\int \frac{\sec ^2(c+d x)}{(a+a \sec (c+d x))^4} \, dx","Integrate[Sec[c + d*x]^2/(a + a*Sec[c + d*x])^4,x]","\frac{\sec \left(\frac{c}{2}\right) \left(-175 \sin \left(c+\frac{d x}{2}\right)+168 \sin \left(c+\frac{3 d x}{2}\right)-105 \sin \left(2 c+\frac{3 d x}{2}\right)+91 \sin \left(2 c+\frac{5 d x}{2}\right)+13 \sin \left(3 c+\frac{7 d x}{2}\right)+280 \sin \left(\frac{d x}{2}\right)\right) \sec ^7\left(\frac{1}{2} (c+d x)\right)}{6720 a^4 d}","\frac{8 \tan (c+d x)}{105 d \left(a^4 \sec (c+d x)+a^4\right)}+\frac{8 \tan (c+d x)}{105 d \left(a^2 \sec (c+d x)+a^2\right)^2}+\frac{4 \tan (c+d x)}{35 a d (a \sec (c+d x)+a)^3}-\frac{\tan (c+d x)}{7 d (a \sec (c+d x)+a)^4}",1,"(Sec[c/2]*Sec[(c + d*x)/2]^7*(280*Sin[(d*x)/2] - 175*Sin[c + (d*x)/2] + 168*Sin[c + (3*d*x)/2] - 105*Sin[2*c + (3*d*x)/2] + 91*Sin[2*c + (5*d*x)/2] + 13*Sin[3*c + (7*d*x)/2]))/(6720*a^4*d)","A",1
76,1,112,112,0.2572382,"\int \frac{\sec (c+d x)}{(a+a \sec (c+d x))^4} \, dx","Integrate[Sec[c + d*x]/(a + a*Sec[c + d*x])^4,x]","\frac{\sec \left(\frac{c}{2}\right) \left(-210 \sin \left(c+\frac{d x}{2}\right)+147 \sin \left(c+\frac{3 d x}{2}\right)-105 \sin \left(2 c+\frac{3 d x}{2}\right)+49 \sin \left(2 c+\frac{5 d x}{2}\right)-35 \sin \left(3 c+\frac{5 d x}{2}\right)+12 \sin \left(3 c+\frac{7 d x}{2}\right)+210 \sin \left(\frac{d x}{2}\right)\right) \sec ^7\left(\frac{1}{2} (c+d x)\right)}{2240 a^4 d}","\frac{2 \tan (c+d x)}{35 d \left(a^4 \sec (c+d x)+a^4\right)}+\frac{2 \tan (c+d x)}{35 d \left(a^2 \sec (c+d x)+a^2\right)^2}+\frac{3 \tan (c+d x)}{35 a d (a \sec (c+d x)+a)^3}+\frac{\tan (c+d x)}{7 d (a \sec (c+d x)+a)^4}",1,"(Sec[c/2]*Sec[(c + d*x)/2]^7*(210*Sin[(d*x)/2] - 210*Sin[c + (d*x)/2] + 147*Sin[c + (3*d*x)/2] - 105*Sin[2*c + (3*d*x)/2] + 49*Sin[2*c + (5*d*x)/2] - 35*Sin[3*c + (5*d*x)/2] + 12*Sin[3*c + (7*d*x)/2]))/(2240*a^4*d)","A",1
77,1,224,111,0.3946055,"\int \frac{1}{(a+a \sec (c+d x))^4} \, dx","Integrate[(a + a*Sec[c + d*x])^(-4),x]","\frac{\sec \left(\frac{c}{2}\right) \sec ^7\left(\frac{1}{2} (c+d x)\right) \left(1652 \sin \left(c+\frac{d x}{2}\right)-1428 \sin \left(c+\frac{3 d x}{2}\right)+756 \sin \left(2 c+\frac{3 d x}{2}\right)-560 \sin \left(2 c+\frac{5 d x}{2}\right)+168 \sin \left(3 c+\frac{5 d x}{2}\right)-104 \sin \left(3 c+\frac{7 d x}{2}\right)+735 d x \cos \left(c+\frac{d x}{2}\right)+441 d x \cos \left(c+\frac{3 d x}{2}\right)+441 d x \cos \left(2 c+\frac{3 d x}{2}\right)+147 d x \cos \left(2 c+\frac{5 d x}{2}\right)+147 d x \cos \left(3 c+\frac{5 d x}{2}\right)+21 d x \cos \left(3 c+\frac{7 d x}{2}\right)+21 d x \cos \left(4 c+\frac{7 d x}{2}\right)-1988 \sin \left(\frac{d x}{2}\right)+735 d x \cos \left(\frac{d x}{2}\right)\right)}{2688 a^4 d}","-\frac{32 \tan (c+d x)}{21 a^4 d (\sec (c+d x)+1)}-\frac{11 \tan (c+d x)}{21 a^4 d (\sec (c+d x)+1)^2}+\frac{x}{a^4}-\frac{2 \tan (c+d x)}{7 a d (a \sec (c+d x)+a)^3}-\frac{\tan (c+d x)}{7 d (a \sec (c+d x)+a)^4}",1,"(Sec[c/2]*Sec[(c + d*x)/2]^7*(735*d*x*Cos[(d*x)/2] + 735*d*x*Cos[c + (d*x)/2] + 441*d*x*Cos[c + (3*d*x)/2] + 441*d*x*Cos[2*c + (3*d*x)/2] + 147*d*x*Cos[2*c + (5*d*x)/2] + 147*d*x*Cos[3*c + (5*d*x)/2] + 21*d*x*Cos[3*c + (7*d*x)/2] + 21*d*x*Cos[4*c + (7*d*x)/2] - 1988*Sin[(d*x)/2] + 1652*Sin[c + (d*x)/2] - 1428*Sin[c + (3*d*x)/2] + 756*Sin[2*c + (3*d*x)/2] - 560*Sin[2*c + (5*d*x)/2] + 168*Sin[3*c + (5*d*x)/2] - 104*Sin[3*c + (7*d*x)/2]))/(2688*a^4*d)","B",1
78,1,263,126,0.4643124,"\int \frac{\cos (c+d x)}{(a+a \sec (c+d x))^4} \, dx","Integrate[Cos[c + d*x]/(a + a*Sec[c + d*x])^4,x]","-\frac{\sec \left(\frac{c}{2}\right) \sec ^7\left(\frac{1}{2} (c+d x)\right) \left(46130 \sin \left(c+\frac{d x}{2}\right)-46116 \sin \left(c+\frac{3 d x}{2}\right)+18060 \sin \left(2 c+\frac{3 d x}{2}\right)-19292 \sin \left(2 c+\frac{5 d x}{2}\right)+2100 \sin \left(3 c+\frac{5 d x}{2}\right)-3791 \sin \left(3 c+\frac{7 d x}{2}\right)-735 \sin \left(4 c+\frac{7 d x}{2}\right)-105 \sin \left(4 c+\frac{9 d x}{2}\right)-105 \sin \left(5 c+\frac{9 d x}{2}\right)+29400 d x \cos \left(c+\frac{d x}{2}\right)+17640 d x \cos \left(c+\frac{3 d x}{2}\right)+17640 d x \cos \left(2 c+\frac{3 d x}{2}\right)+5880 d x \cos \left(2 c+\frac{5 d x}{2}\right)+5880 d x \cos \left(3 c+\frac{5 d x}{2}\right)+840 d x \cos \left(3 c+\frac{7 d x}{2}\right)+840 d x \cos \left(4 c+\frac{7 d x}{2}\right)-60830 \sin \left(\frac{d x}{2}\right)+29400 d x \cos \left(\frac{d x}{2}\right)\right)}{26880 a^4 d}","\frac{664 \sin (c+d x)}{105 a^4 d}-\frac{4 \sin (c+d x)}{a^4 d (\sec (c+d x)+1)}-\frac{88 \sin (c+d x)}{105 a^4 d (\sec (c+d x)+1)^2}-\frac{4 x}{a^4}-\frac{12 \sin (c+d x)}{35 a d (a \sec (c+d x)+a)^3}-\frac{\sin (c+d x)}{7 d (a \sec (c+d x)+a)^4}",1,"-1/26880*(Sec[c/2]*Sec[(c + d*x)/2]^7*(29400*d*x*Cos[(d*x)/2] + 29400*d*x*Cos[c + (d*x)/2] + 17640*d*x*Cos[c + (3*d*x)/2] + 17640*d*x*Cos[2*c + (3*d*x)/2] + 5880*d*x*Cos[2*c + (5*d*x)/2] + 5880*d*x*Cos[3*c + (5*d*x)/2] + 840*d*x*Cos[3*c + (7*d*x)/2] + 840*d*x*Cos[4*c + (7*d*x)/2] - 60830*Sin[(d*x)/2] + 46130*Sin[c + (d*x)/2] - 46116*Sin[c + (3*d*x)/2] + 18060*Sin[2*c + (3*d*x)/2] - 19292*Sin[2*c + (5*d*x)/2] + 2100*Sin[3*c + (5*d*x)/2] - 3791*Sin[3*c + (7*d*x)/2] - 735*Sin[4*c + (7*d*x)/2] - 105*Sin[4*c + (9*d*x)/2] - 105*Sin[5*c + (9*d*x)/2]))/(a^4*d)","B",1
79,1,289,176,0.5849822,"\int \frac{\cos ^2(c+d x)}{(a+a \sec (c+d x))^4} \, dx","Integrate[Cos[c + d*x]^2/(a + a*Sec[c + d*x])^4,x]","\frac{\sec \left(\frac{c}{2}\right) \sec ^7\left(\frac{1}{2} (c+d x)\right) \left(128730 \sin \left(c+\frac{d x}{2}\right)-140826 \sin \left(c+\frac{3 d x}{2}\right)+44310 \sin \left(2 c+\frac{3 d x}{2}\right)-60487 \sin \left(2 c+\frac{5 d x}{2}\right)+1225 \sin \left(3 c+\frac{5 d x}{2}\right)-12001 \sin \left(3 c+\frac{7 d x}{2}\right)-3185 \sin \left(4 c+\frac{7 d x}{2}\right)-315 \sin \left(4 c+\frac{9 d x}{2}\right)-315 \sin \left(5 c+\frac{9 d x}{2}\right)+35 \sin \left(5 c+\frac{11 d x}{2}\right)+35 \sin \left(6 c+\frac{11 d x}{2}\right)+102900 d x \cos \left(c+\frac{d x}{2}\right)+61740 d x \cos \left(c+\frac{3 d x}{2}\right)+61740 d x \cos \left(2 c+\frac{3 d x}{2}\right)+20580 d x \cos \left(2 c+\frac{5 d x}{2}\right)+20580 d x \cos \left(3 c+\frac{5 d x}{2}\right)+2940 d x \cos \left(3 c+\frac{7 d x}{2}\right)+2940 d x \cos \left(4 c+\frac{7 d x}{2}\right)-179830 \sin \left(\frac{d x}{2}\right)+102900 d x \cos \left(\frac{d x}{2}\right)\right)}{35840 a^4 d}","-\frac{576 \sin (c+d x)}{35 a^4 d}+\frac{21 \sin (c+d x) \cos (c+d x)}{2 a^4 d}-\frac{288 \sin (c+d x) \cos (c+d x)}{35 a^4 d (\sec (c+d x)+1)}-\frac{43 \sin (c+d x) \cos (c+d x)}{35 a^4 d (\sec (c+d x)+1)^2}+\frac{21 x}{2 a^4}-\frac{2 \sin (c+d x) \cos (c+d x)}{5 a d (a \sec (c+d x)+a)^3}-\frac{\sin (c+d x) \cos (c+d x)}{7 d (a \sec (c+d x)+a)^4}",1,"(Sec[c/2]*Sec[(c + d*x)/2]^7*(102900*d*x*Cos[(d*x)/2] + 102900*d*x*Cos[c + (d*x)/2] + 61740*d*x*Cos[c + (3*d*x)/2] + 61740*d*x*Cos[2*c + (3*d*x)/2] + 20580*d*x*Cos[2*c + (5*d*x)/2] + 20580*d*x*Cos[3*c + (5*d*x)/2] + 2940*d*x*Cos[3*c + (7*d*x)/2] + 2940*d*x*Cos[4*c + (7*d*x)/2] - 179830*Sin[(d*x)/2] + 128730*Sin[c + (d*x)/2] - 140826*Sin[c + (3*d*x)/2] + 44310*Sin[2*c + (3*d*x)/2] - 60487*Sin[2*c + (5*d*x)/2] + 1225*Sin[3*c + (5*d*x)/2] - 12001*Sin[3*c + (7*d*x)/2] - 3185*Sin[4*c + (7*d*x)/2] - 315*Sin[4*c + (9*d*x)/2] - 315*Sin[5*c + (9*d*x)/2] + 35*Sin[5*c + (11*d*x)/2] + 35*Sin[6*c + (11*d*x)/2]))/(35840*a^4*d)","A",1
80,1,401,200,1.9711643,"\int \frac{\sec ^7(c+d x)}{(a+a \sec (c+d x))^5} \, dx","Integrate[Sec[c + d*x]^7/(a + a*Sec[c + d*x])^5,x]","\frac{\cos \left(\frac{1}{2} (c+d x)\right) \sec ^5(c+d x) \left(\sec \left(\frac{c}{2}\right) \sec (c) \left(-56952 \sin \left(c-\frac{d x}{2}\right)+43722 \sin \left(c+\frac{d x}{2}\right)-47208 \sin \left(2 c+\frac{d x}{2}\right)-18144 \sin \left(c+\frac{3 d x}{2}\right)+41796 \sin \left(2 c+\frac{3 d x}{2}\right)-28350 \sin \left(3 c+\frac{3 d x}{2}\right)+34578 \sin \left(c+\frac{5 d x}{2}\right)-5691 \sin \left(2 c+\frac{5 d x}{2}\right)+28719 \sin \left(3 c+\frac{5 d x}{2}\right)-11550 \sin \left(4 c+\frac{5 d x}{2}\right)+15517 \sin \left(2 c+\frac{7 d x}{2}\right)-504 \sin \left(3 c+\frac{7 d x}{2}\right)+13186 \sin \left(4 c+\frac{7 d x}{2}\right)-2835 \sin \left(5 c+\frac{7 d x}{2}\right)+4149 \sin \left(3 c+\frac{9 d x}{2}\right)+252 \sin \left(4 c+\frac{9 d x}{2}\right)+3582 \sin \left(5 c+\frac{9 d x}{2}\right)-315 \sin \left(6 c+\frac{9 d x}{2}\right)+496 \sin \left(4 c+\frac{11 d x}{2}\right)+63 \sin \left(5 c+\frac{11 d x}{2}\right)+433 \sin \left(6 c+\frac{11 d x}{2}\right)-33978 \sin \left(\frac{d x}{2}\right)+52002 \sin \left(\frac{3 d x}{2}\right)\right) \sec (c+d x)+322560 \cos ^9\left(\frac{1}{2} (c+d x)\right) \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)\right)}{2016 a^5 d (\sec (c+d x)+1)^5}","\frac{181 \tan (c+d x)}{63 a^5 d}-\frac{5 \tanh ^{-1}(\sin (c+d x))}{a^5 d}+\frac{5 \tan (c+d x)}{d \left(a^5 \sec (c+d x)+a^5\right)}-\frac{67 \tan (c+d x) \sec ^2(c+d x)}{63 a^3 d (a \sec (c+d x)+a)^2}-\frac{29 \tan (c+d x) \sec ^3(c+d x)}{63 a^2 d (a \sec (c+d x)+a)^3}-\frac{\tan (c+d x) \sec ^5(c+d x)}{9 d (a \sec (c+d x)+a)^5}-\frac{5 \tan (c+d x) \sec ^4(c+d x)}{21 a d (a \sec (c+d x)+a)^4}",1,"(Cos[(c + d*x)/2]*Sec[c + d*x]^5*(322560*Cos[(c + d*x)/2]^9*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) + Sec[c/2]*Sec[c]*Sec[c + d*x]*(-33978*Sin[(d*x)/2] + 52002*Sin[(3*d*x)/2] - 56952*Sin[c - (d*x)/2] + 43722*Sin[c + (d*x)/2] - 47208*Sin[2*c + (d*x)/2] - 18144*Sin[c + (3*d*x)/2] + 41796*Sin[2*c + (3*d*x)/2] - 28350*Sin[3*c + (3*d*x)/2] + 34578*Sin[c + (5*d*x)/2] - 5691*Sin[2*c + (5*d*x)/2] + 28719*Sin[3*c + (5*d*x)/2] - 11550*Sin[4*c + (5*d*x)/2] + 15517*Sin[2*c + (7*d*x)/2] - 504*Sin[3*c + (7*d*x)/2] + 13186*Sin[4*c + (7*d*x)/2] - 2835*Sin[5*c + (7*d*x)/2] + 4149*Sin[3*c + (9*d*x)/2] + 252*Sin[4*c + (9*d*x)/2] + 3582*Sin[5*c + (9*d*x)/2] - 315*Sin[6*c + (9*d*x)/2] + 496*Sin[4*c + (11*d*x)/2] + 63*Sin[5*c + (11*d*x)/2] + 433*Sin[6*c + (11*d*x)/2])))/(2016*a^5*d*(1 + Sec[c + d*x])^5)","B",1
81,1,219,177,1.9596756,"\int \frac{\sec ^6(c+d x)}{(a+a \sec (c+d x))^5} \, dx","Integrate[Sec[c + d*x]^6/(a + a*Sec[c + d*x])^5,x]","-\frac{\cos \left(\frac{1}{2} (c+d x)\right) \sec ^5(c+d x) \left(\sec \left(\frac{c}{2}\right) \left(-25515 \sin \left(c+\frac{d x}{2}\right)+29757 \sin \left(c+\frac{3 d x}{2}\right)-11235 \sin \left(2 c+\frac{3 d x}{2}\right)+14733 \sin \left(2 c+\frac{5 d x}{2}\right)-2835 \sin \left(3 c+\frac{5 d x}{2}\right)+4077 \sin \left(3 c+\frac{7 d x}{2}\right)-315 \sin \left(4 c+\frac{7 d x}{2}\right)+488 \sin \left(4 c+\frac{9 d x}{2}\right)+35973 \sin \left(\frac{d x}{2}\right)\right)+80640 \cos ^9\left(\frac{1}{2} (c+d x)\right) \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)\right)}{2520 a^5 d (\sec (c+d x)+1)^5}","\frac{\tanh ^{-1}(\sin (c+d x))}{a^5 d}-\frac{661 \tan (c+d x)}{315 d \left(a^5 \sec (c+d x)+a^5\right)}+\frac{173 \tan (c+d x)}{315 a^3 d (a \sec (c+d x)+a)^2}-\frac{34 \tan (c+d x) \sec ^2(c+d x)}{105 a^2 d (a \sec (c+d x)+a)^3}-\frac{\tan (c+d x) \sec ^4(c+d x)}{9 d (a \sec (c+d x)+a)^5}-\frac{13 \tan (c+d x) \sec ^3(c+d x)}{63 a d (a \sec (c+d x)+a)^4}",1,"-1/2520*(Cos[(c + d*x)/2]*Sec[c + d*x]^5*(80640*Cos[(c + d*x)/2]^9*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) + Sec[c/2]*(35973*Sin[(d*x)/2] - 25515*Sin[c + (d*x)/2] + 29757*Sin[c + (3*d*x)/2] - 11235*Sin[2*c + (3*d*x)/2] + 14733*Sin[2*c + (5*d*x)/2] - 2835*Sin[3*c + (5*d*x)/2] + 4077*Sin[3*c + (7*d*x)/2] - 315*Sin[4*c + (7*d*x)/2] + 488*Sin[4*c + (9*d*x)/2])))/(a^5*d*(1 + Sec[c + d*x])^5)","A",1
82,1,97,159,0.1871492,"\int \frac{\sec ^5(c+d x)}{(a+a \sec (c+d x))^5} \, dx","Integrate[Sec[c + d*x]^5/(a + a*Sec[c + d*x])^5,x]","\frac{\left(126 \sin \left(\frac{1}{2} (c+d x)\right)+84 \sin \left(\frac{3}{2} (c+d x)\right)+36 \sin \left(\frac{5}{2} (c+d x)\right)+9 \sin \left(\frac{7}{2} (c+d x)\right)+\sin \left(\frac{9}{2} (c+d x)\right)\right) \cos \left(\frac{1}{2} (c+d x)\right) \sec ^5(c+d x)}{315 a^5 d (\sec (c+d x)+1)^5}","\frac{4 \tan (c+d x)}{45 d \left(a^5 \sec (c+d x)+a^5\right)}-\frac{32 \tan (c+d x)}{315 a d \left(a^2 \sec (c+d x)+a^2\right)^2}+\frac{4 \tan (c+d x)}{105 a^2 d (a \sec (c+d x)+a)^3}+\frac{\tan (c+d x) \sec ^4(c+d x)}{9 d (a \sec (c+d x)+a)^5}+\frac{4 \tan (c+d x) \sec ^3(c+d x)}{63 a d (a \sec (c+d x)+a)^4}",1,"(Cos[(c + d*x)/2]*Sec[c + d*x]^5*(126*Sin[(c + d*x)/2] + 84*Sin[(3*(c + d*x))/2] + 36*Sin[(5*(c + d*x))/2] + 9*Sin[(7*(c + d*x))/2] + Sin[(9*(c + d*x))/2]))/(315*a^5*d*(1 + Sec[c + d*x])^5)","A",1
83,1,97,159,0.2068491,"\int \frac{\sec ^4(c+d x)}{(a+a \sec (c+d x))^5} \, dx","Integrate[Sec[c + d*x]^4/(a + a*Sec[c + d*x])^5,x]","\frac{\sec \left(\frac{c}{2}\right) \left(-63 \sin \left(c+\frac{d x}{2}\right)+84 \sin \left(c+\frac{3 d x}{2}\right)+36 \sin \left(2 c+\frac{5 d x}{2}\right)+9 \sin \left(3 c+\frac{7 d x}{2}\right)+\sin \left(4 c+\frac{9 d x}{2}\right)+63 \sin \left(\frac{d x}{2}\right)\right) \sec ^9\left(\frac{1}{2} (c+d x)\right)}{8064 a^5 d}","\frac{\tan (c+d x)}{9 d \left(a^5 \sec (c+d x)+a^5\right)}-\frac{8 \tan (c+d x)}{63 a d \left(a^2 \sec (c+d x)+a^2\right)^2}+\frac{\tan (c+d x)}{21 a^2 d (a \sec (c+d x)+a)^3}-\frac{\tan (c+d x) \sec ^4(c+d x)}{9 d (a \sec (c+d x)+a)^5}+\frac{5 \tan (c+d x) \sec ^3(c+d x)}{63 a d (a \sec (c+d x)+a)^4}",1,"(Sec[c/2]*Sec[(c + d*x)/2]^9*(63*Sin[(d*x)/2] - 63*Sin[c + (d*x)/2] + 84*Sin[c + (3*d*x)/2] + 36*Sin[2*c + (5*d*x)/2] + 9*Sin[3*c + (7*d*x)/2] + Sin[4*c + (9*d*x)/2]))/(8064*a^5*d)","A",1
84,1,110,139,0.2461752,"\int \frac{\sec ^3(c+d x)}{(a+a \sec (c+d x))^5} \, dx","Integrate[Sec[c + d*x]^3/(a + a*Sec[c + d*x])^5,x]","\frac{\sec \left(\frac{c}{2}\right) \left(-45 \sin \left(c+\frac{d x}{2}\right)+54 \sin \left(c+\frac{3 d x}{2}\right)-30 \sin \left(2 c+\frac{3 d x}{2}\right)+36 \sin \left(2 c+\frac{5 d x}{2}\right)+9 \sin \left(3 c+\frac{7 d x}{2}\right)+\sin \left(4 c+\frac{9 d x}{2}\right)+81 \sin \left(\frac{d x}{2}\right)\right) \sec ^9\left(\frac{1}{2} (c+d x)\right)}{5760 a^5 d}","\frac{2 \tan (c+d x)}{45 d \left(a^5 \sec (c+d x)+a^5\right)}+\frac{2 \tan (c+d x)}{45 a^3 d (a \sec (c+d x)+a)^2}+\frac{\tan (c+d x)}{15 a^2 d (a \sec (c+d x)+a)^3}-\frac{2 \tan (c+d x)}{9 a d (a \sec (c+d x)+a)^4}+\frac{\tan (c+d x)}{9 d (a \sec (c+d x)+a)^5}",1,"(Sec[c/2]*Sec[(c + d*x)/2]^9*(81*Sin[(d*x)/2] - 45*Sin[c + (d*x)/2] + 54*Sin[c + (3*d*x)/2] - 30*Sin[2*c + (3*d*x)/2] + 36*Sin[2*c + (5*d*x)/2] + 9*Sin[3*c + (7*d*x)/2] + Sin[4*c + (9*d*x)/2]))/(5760*a^5*d)","A",1
85,1,125,143,0.2501534,"\int \frac{\sec ^2(c+d x)}{(a+a \sec (c+d x))^5} \, dx","Integrate[Sec[c + d*x]^2/(a + a*Sec[c + d*x])^5,x]","\frac{\sec \left(\frac{c}{2}\right) \left(-315 \sin \left(c+\frac{d x}{2}\right)+273 \sin \left(c+\frac{3 d x}{2}\right)-147 \sin \left(2 c+\frac{3 d x}{2}\right)+117 \sin \left(2 c+\frac{5 d x}{2}\right)-63 \sin \left(3 c+\frac{5 d x}{2}\right)+45 \sin \left(3 c+\frac{7 d x}{2}\right)+5 \sin \left(4 c+\frac{9 d x}{2}\right)+315 \sin \left(\frac{d x}{2}\right)\right) \sec ^9\left(\frac{1}{2} (c+d x)\right)}{16128 a^5 d}","\frac{2 \tan (c+d x)}{63 d \left(a^5 \sec (c+d x)+a^5\right)}+\frac{2 \tan (c+d x)}{63 a d \left(a^2 \sec (c+d x)+a^2\right)^2}+\frac{\tan (c+d x)}{21 a^2 d (a \sec (c+d x)+a)^3}+\frac{5 \tan (c+d x)}{63 a d (a \sec (c+d x)+a)^4}-\frac{\tan (c+d x)}{9 d (a \sec (c+d x)+a)^5}",1,"(Sec[c/2]*Sec[(c + d*x)/2]^9*(315*Sin[(d*x)/2] - 315*Sin[c + (d*x)/2] + 273*Sin[c + (3*d*x)/2] - 147*Sin[2*c + (3*d*x)/2] + 117*Sin[2*c + (5*d*x)/2] - 63*Sin[3*c + (5*d*x)/2] + 45*Sin[3*c + (7*d*x)/2] + 5*Sin[4*c + (9*d*x)/2]))/(16128*a^5*d)","A",1
86,1,138,143,0.2885524,"\int \frac{\sec (c+d x)}{(a+a \sec (c+d x))^5} \, dx","Integrate[Sec[c + d*x]/(a + a*Sec[c + d*x])^5,x]","\frac{\sec \left(\frac{c}{2}\right) \left(-5040 \sin \left(c+\frac{d x}{2}\right)+3612 \sin \left(c+\frac{3 d x}{2}\right)-3360 \sin \left(2 c+\frac{3 d x}{2}\right)+1728 \sin \left(2 c+\frac{5 d x}{2}\right)-1260 \sin \left(3 c+\frac{5 d x}{2}\right)+432 \sin \left(3 c+\frac{7 d x}{2}\right)-315 \sin \left(4 c+\frac{7 d x}{2}\right)+83 \sin \left(4 c+\frac{9 d x}{2}\right)+5418 \sin \left(\frac{d x}{2}\right)\right) \sec ^9\left(\frac{1}{2} (c+d x)\right)}{80640 a^5 d}","\frac{8 \tan (c+d x)}{315 d \left(a^5 \sec (c+d x)+a^5\right)}+\frac{8 \tan (c+d x)}{315 a d \left(a^2 \sec (c+d x)+a^2\right)^2}+\frac{4 \tan (c+d x)}{105 a^2 d (a \sec (c+d x)+a)^3}+\frac{4 \tan (c+d x)}{63 a d (a \sec (c+d x)+a)^4}+\frac{\tan (c+d x)}{9 d (a \sec (c+d x)+a)^5}",1,"(Sec[c/2]*Sec[(c + d*x)/2]^9*(5418*Sin[(d*x)/2] - 5040*Sin[c + (d*x)/2] + 3612*Sin[c + (3*d*x)/2] - 3360*Sin[2*c + (3*d*x)/2] + 1728*Sin[2*c + (5*d*x)/2] - 1260*Sin[3*c + (5*d*x)/2] + 432*Sin[3*c + (7*d*x)/2] - 315*Sin[4*c + (7*d*x)/2] + 83*Sin[4*c + (9*d*x)/2]))/(80640*a^5*d)","A",1
87,1,280,144,0.5578029,"\int \frac{1}{(a+a \sec (c+d x))^5} \, dx","Integrate[(a + a*Sec[c + d*x])^(-5),x]","\frac{\sec \left(\frac{c}{2}\right) \sec ^9\left(\frac{1}{2} (c+d x)\right) \left(100800 \sin \left(c+\frac{d x}{2}\right)-88284 \sin \left(c+\frac{3 d x}{2}\right)+56700 \sin \left(2 c+\frac{3 d x}{2}\right)-43236 \sin \left(2 c+\frac{5 d x}{2}\right)+18900 \sin \left(3 c+\frac{5 d x}{2}\right)-12384 \sin \left(3 c+\frac{7 d x}{2}\right)+3150 \sin \left(4 c+\frac{7 d x}{2}\right)-1726 \sin \left(4 c+\frac{9 d x}{2}\right)+39690 d x \cos \left(c+\frac{d x}{2}\right)+26460 d x \cos \left(c+\frac{3 d x}{2}\right)+26460 d x \cos \left(2 c+\frac{3 d x}{2}\right)+11340 d x \cos \left(2 c+\frac{5 d x}{2}\right)+11340 d x \cos \left(3 c+\frac{5 d x}{2}\right)+2835 d x \cos \left(3 c+\frac{7 d x}{2}\right)+2835 d x \cos \left(4 c+\frac{7 d x}{2}\right)+315 d x \cos \left(4 c+\frac{9 d x}{2}\right)+315 d x \cos \left(5 c+\frac{9 d x}{2}\right)-116676 \sin \left(\frac{d x}{2}\right)+39690 d x \cos \left(\frac{d x}{2}\right)\right)}{161280 a^5 d}","-\frac{488 \tan (c+d x)}{315 d \left(a^5 \sec (c+d x)+a^5\right)}+\frac{x}{a^5}-\frac{173 \tan (c+d x)}{315 a^3 d (a \sec (c+d x)+a)^2}-\frac{34 \tan (c+d x)}{105 a^2 d (a \sec (c+d x)+a)^3}-\frac{13 \tan (c+d x)}{63 a d (a \sec (c+d x)+a)^4}-\frac{\tan (c+d x)}{9 d (a \sec (c+d x)+a)^5}",1,"(Sec[c/2]*Sec[(c + d*x)/2]^9*(39690*d*x*Cos[(d*x)/2] + 39690*d*x*Cos[c + (d*x)/2] + 26460*d*x*Cos[c + (3*d*x)/2] + 26460*d*x*Cos[2*c + (3*d*x)/2] + 11340*d*x*Cos[2*c + (5*d*x)/2] + 11340*d*x*Cos[3*c + (5*d*x)/2] + 2835*d*x*Cos[3*c + (7*d*x)/2] + 2835*d*x*Cos[4*c + (7*d*x)/2] + 315*d*x*Cos[4*c + (9*d*x)/2] + 315*d*x*Cos[5*c + (9*d*x)/2] - 116676*Sin[(d*x)/2] + 100800*Sin[c + (d*x)/2] - 88284*Sin[c + (3*d*x)/2] + 56700*Sin[2*c + (3*d*x)/2] - 43236*Sin[2*c + (5*d*x)/2] + 18900*Sin[3*c + (5*d*x)/2] - 12384*Sin[3*c + (7*d*x)/2] + 3150*Sin[4*c + (7*d*x)/2] - 1726*Sin[4*c + (9*d*x)/2]))/(161280*a^5*d)","A",1
88,1,319,159,0.6954545,"\int \frac{\cos (c+d x)}{(a+a \sec (c+d x))^5} \, dx","Integrate[Cos[c + d*x]/(a + a*Sec[c + d*x])^5,x]","-\frac{\sec \left(\frac{c}{2}\right) \sec ^9\left(\frac{1}{2} (c+d x)\right) \left(143010 \sin \left(c+\frac{d x}{2}\right)-138726 \sin \left(c+\frac{3 d x}{2}\right)+73290 \sin \left(2 c+\frac{3 d x}{2}\right)-70389 \sin \left(2 c+\frac{5 d x}{2}\right)+20475 \sin \left(3 c+\frac{5 d x}{2}\right)-21141 \sin \left(3 c+\frac{7 d x}{2}\right)+1575 \sin \left(4 c+\frac{7 d x}{2}\right)-3091 \sin \left(4 c+\frac{9 d x}{2}\right)-567 \sin \left(5 c+\frac{9 d x}{2}\right)-63 \sin \left(5 c+\frac{11 d x}{2}\right)-63 \sin \left(6 c+\frac{11 d x}{2}\right)+79380 d x \cos \left(c+\frac{d x}{2}\right)+52920 d x \cos \left(c+\frac{3 d x}{2}\right)+52920 d x \cos \left(2 c+\frac{3 d x}{2}\right)+22680 d x \cos \left(2 c+\frac{5 d x}{2}\right)+22680 d x \cos \left(3 c+\frac{5 d x}{2}\right)+5670 d x \cos \left(3 c+\frac{7 d x}{2}\right)+5670 d x \cos \left(4 c+\frac{7 d x}{2}\right)+630 d x \cos \left(4 c+\frac{9 d x}{2}\right)+630 d x \cos \left(5 c+\frac{9 d x}{2}\right)-175014 \sin \left(\frac{d x}{2}\right)+79380 d x \cos \left(\frac{d x}{2}\right)\right)}{64512 a^5 d}","\frac{496 \sin (c+d x)}{63 a^5 d}-\frac{5 \sin (c+d x)}{d \left(a^5 \sec (c+d x)+a^5\right)}-\frac{5 x}{a^5}-\frac{67 \sin (c+d x)}{63 a^3 d (a \sec (c+d x)+a)^2}-\frac{29 \sin (c+d x)}{63 a^2 d (a \sec (c+d x)+a)^3}-\frac{5 \sin (c+d x)}{21 a d (a \sec (c+d x)+a)^4}-\frac{\sin (c+d x)}{9 d (a \sec (c+d x)+a)^5}",1,"-1/64512*(Sec[c/2]*Sec[(c + d*x)/2]^9*(79380*d*x*Cos[(d*x)/2] + 79380*d*x*Cos[c + (d*x)/2] + 52920*d*x*Cos[c + (3*d*x)/2] + 52920*d*x*Cos[2*c + (3*d*x)/2] + 22680*d*x*Cos[2*c + (5*d*x)/2] + 22680*d*x*Cos[3*c + (5*d*x)/2] + 5670*d*x*Cos[3*c + (7*d*x)/2] + 5670*d*x*Cos[4*c + (7*d*x)/2] + 630*d*x*Cos[4*c + (9*d*x)/2] + 630*d*x*Cos[5*c + (9*d*x)/2] - 175014*Sin[(d*x)/2] + 143010*Sin[c + (d*x)/2] - 138726*Sin[c + (3*d*x)/2] + 73290*Sin[2*c + (3*d*x)/2] - 70389*Sin[2*c + (5*d*x)/2] + 20475*Sin[3*c + (5*d*x)/2] - 21141*Sin[3*c + (7*d*x)/2] + 1575*Sin[4*c + (7*d*x)/2] - 3091*Sin[4*c + (9*d*x)/2] - 567*Sin[5*c + (9*d*x)/2] - 63*Sin[5*c + (11*d*x)/2] - 63*Sin[6*c + (11*d*x)/2]))/(a^5*d)","B",1
89,1,345,215,0.778486,"\int \frac{\cos ^2(c+d x)}{(a+a \sec (c+d x))^5} \, dx","Integrate[Cos[c + d*x]^2/(a + a*Sec[c + d*x])^5,x]","\frac{\sec \left(\frac{c}{2}\right) \sec ^9\left(\frac{1}{2} (c+d x)\right) \left(7194600 \sin \left(c+\frac{d x}{2}\right)-7472241 \sin \left(c+\frac{3 d x}{2}\right)+3432975 \sin \left(2 c+\frac{3 d x}{2}\right)-3871989 \sin \left(2 c+\frac{5 d x}{2}\right)+801675 \sin \left(3 c+\frac{5 d x}{2}\right)-1186056 \sin \left(3 c+\frac{7 d x}{2}\right)-17640 \sin \left(4 c+\frac{7 d x}{2}\right)-175184 \sin \left(4 c+\frac{9 d x}{2}\right)-45360 \sin \left(5 c+\frac{9 d x}{2}\right)-3465 \sin \left(5 c+\frac{11 d x}{2}\right)-3465 \sin \left(6 c+\frac{11 d x}{2}\right)+315 \sin \left(6 c+\frac{13 d x}{2}\right)+315 \sin \left(7 c+\frac{13 d x}{2}\right)+4921560 d x \cos \left(c+\frac{d x}{2}\right)+3281040 d x \cos \left(c+\frac{3 d x}{2}\right)+3281040 d x \cos \left(2 c+\frac{3 d x}{2}\right)+1406160 d x \cos \left(2 c+\frac{5 d x}{2}\right)+1406160 d x \cos \left(3 c+\frac{5 d x}{2}\right)+351540 d x \cos \left(3 c+\frac{7 d x}{2}\right)+351540 d x \cos \left(4 c+\frac{7 d x}{2}\right)+39060 d x \cos \left(4 c+\frac{9 d x}{2}\right)+39060 d x \cos \left(5 c+\frac{9 d x}{2}\right)-9163224 \sin \left(\frac{d x}{2}\right)+4921560 d x \cos \left(\frac{d x}{2}\right)\right)}{1290240 a^5 d}","-\frac{7664 \sin (c+d x)}{315 a^5 d}+\frac{31 \sin (c+d x) \cos (c+d x)}{2 a^5 d}-\frac{3832 \sin (c+d x) \cos (c+d x)}{315 d \left(a^5 \sec (c+d x)+a^5\right)}+\frac{31 x}{2 a^5}-\frac{577 \sin (c+d x) \cos (c+d x)}{315 a^3 d (a \sec (c+d x)+a)^2}-\frac{28 \sin (c+d x) \cos (c+d x)}{45 a^2 d (a \sec (c+d x)+a)^3}-\frac{17 \sin (c+d x) \cos (c+d x)}{63 a d (a \sec (c+d x)+a)^4}-\frac{\sin (c+d x) \cos (c+d x)}{9 d (a \sec (c+d x)+a)^5}",1,"(Sec[c/2]*Sec[(c + d*x)/2]^9*(4921560*d*x*Cos[(d*x)/2] + 4921560*d*x*Cos[c + (d*x)/2] + 3281040*d*x*Cos[c + (3*d*x)/2] + 3281040*d*x*Cos[2*c + (3*d*x)/2] + 1406160*d*x*Cos[2*c + (5*d*x)/2] + 1406160*d*x*Cos[3*c + (5*d*x)/2] + 351540*d*x*Cos[3*c + (7*d*x)/2] + 351540*d*x*Cos[4*c + (7*d*x)/2] + 39060*d*x*Cos[4*c + (9*d*x)/2] + 39060*d*x*Cos[5*c + (9*d*x)/2] - 9163224*Sin[(d*x)/2] + 7194600*Sin[c + (d*x)/2] - 7472241*Sin[c + (3*d*x)/2] + 3432975*Sin[2*c + (3*d*x)/2] - 3871989*Sin[2*c + (5*d*x)/2] + 801675*Sin[3*c + (5*d*x)/2] - 1186056*Sin[3*c + (7*d*x)/2] - 17640*Sin[4*c + (7*d*x)/2] - 175184*Sin[4*c + (9*d*x)/2] - 45360*Sin[5*c + (9*d*x)/2] - 3465*Sin[5*c + (11*d*x)/2] - 3465*Sin[6*c + (11*d*x)/2] + 315*Sin[6*c + (13*d*x)/2] + 315*Sin[7*c + (13*d*x)/2]))/(1290240*a^5*d)","A",1
90,1,58,122,0.138057,"\int \sec ^4(c+d x) \sqrt{a+a \sec (c+d x)} \, dx","Integrate[Sec[c + d*x]^4*Sqrt[a + a*Sec[c + d*x]],x]","\frac{2 a \tan (c+d x) \left(5 \sec ^3(c+d x)+6 \sec ^2(c+d x)+8 \sec (c+d x)+16\right)}{35 d \sqrt{a (\sec (c+d x)+1)}}","\frac{2 a \tan (c+d x) \sec ^3(c+d x)}{7 d \sqrt{a \sec (c+d x)+a}}+\frac{12 \tan (c+d x) (a \sec (c+d x)+a)^{3/2}}{35 a d}-\frac{8 \tan (c+d x) \sqrt{a \sec (c+d x)+a}}{35 d}+\frac{4 a \tan (c+d x)}{5 d \sqrt{a \sec (c+d x)+a}}",1,"(2*a*(16 + 8*Sec[c + d*x] + 6*Sec[c + d*x]^2 + 5*Sec[c + d*x]^3)*Tan[c + d*x])/(35*d*Sqrt[a*(1 + Sec[c + d*x])])","A",1
91,1,48,86,0.1119249,"\int \sec ^3(c+d x) \sqrt{a+a \sec (c+d x)} \, dx","Integrate[Sec[c + d*x]^3*Sqrt[a + a*Sec[c + d*x]],x]","\frac{2 a \tan (c+d x) \left(3 \sec ^2(c+d x)+4 \sec (c+d x)+8\right)}{15 d \sqrt{a (\sec (c+d x)+1)}}","\frac{2 \tan (c+d x) (a \sec (c+d x)+a)^{3/2}}{5 a d}-\frac{4 \tan (c+d x) \sqrt{a \sec (c+d x)+a}}{15 d}+\frac{14 a \tan (c+d x)}{15 d \sqrt{a \sec (c+d x)+a}}",1,"(2*a*(8 + 4*Sec[c + d*x] + 3*Sec[c + d*x]^2)*Tan[c + d*x])/(15*d*Sqrt[a*(1 + Sec[c + d*x])])","A",1
92,1,36,56,0.0998996,"\int \sec ^2(c+d x) \sqrt{a+a \sec (c+d x)} \, dx","Integrate[Sec[c + d*x]^2*Sqrt[a + a*Sec[c + d*x]],x]","\frac{2 a \tan (c+d x) (\sec (c+d x)+2)}{3 d \sqrt{a (\sec (c+d x)+1)}}","\frac{2 \tan (c+d x) \sqrt{a \sec (c+d x)+a}}{3 d}+\frac{2 a \tan (c+d x)}{3 d \sqrt{a \sec (c+d x)+a}}",1,"(2*a*(2 + Sec[c + d*x])*Tan[c + d*x])/(3*d*Sqrt[a*(1 + Sec[c + d*x])])","A",1
93,1,29,26,0.070022,"\int \sec (c+d x) \sqrt{a+a \sec (c+d x)} \, dx","Integrate[Sec[c + d*x]*Sqrt[a + a*Sec[c + d*x]],x]","\frac{2 \tan \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\sec (c+d x)+1)}}{d}","\frac{2 a \tan (c+d x)}{d \sqrt{a \sec (c+d x)+a}}",1,"(2*Sqrt[a*(1 + Sec[c + d*x])]*Tan[(c + d*x)/2])/d","A",1
94,1,60,37,0.0961701,"\int \sqrt{a+a \sec (c+d x)} \, dx","Integrate[Sqrt[a + a*Sec[c + d*x]],x]","\frac{\sqrt{2} \sin ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right) \sqrt{\cos (c+d x)} \sec \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\sec (c+d x)+1)}}{d}","\frac{2 \sqrt{a} \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{d}",1,"(Sqrt[2]*ArcSin[Sqrt[2]*Sin[(c + d*x)/2]]*Sqrt[Cos[c + d*x]]*Sec[(c + d*x)/2]*Sqrt[a*(1 + Sec[c + d*x])])/d","A",1
95,1,62,62,0.2155915,"\int \cos (c+d x) \sqrt{a+a \sec (c+d x)} \, dx","Integrate[Cos[c + d*x]*Sqrt[a + a*Sec[c + d*x]],x]","\frac{a \tan (c+d x) \left(\cos (c+d x)+\frac{\tanh ^{-1}\left(\sqrt{1-\sec (c+d x)}\right)}{\sqrt{1-\sec (c+d x)}}\right)}{d \sqrt{a (\sec (c+d x)+1)}}","\frac{a \sin (c+d x)}{d \sqrt{a \sec (c+d x)+a}}+\frac{\sqrt{a} \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{d}",1,"(a*(Cos[c + d*x] + ArcTanh[Sqrt[1 - Sec[c + d*x]]]/Sqrt[1 - Sec[c + d*x]])*Tan[c + d*x])/(d*Sqrt[a*(1 + Sec[c + d*x])])","A",1
96,1,47,102,0.1062345,"\int \cos ^2(c+d x) \sqrt{a+a \sec (c+d x)} \, dx","Integrate[Cos[c + d*x]^2*Sqrt[a + a*Sec[c + d*x]],x]","\frac{2 \tan \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\sec (c+d x)+1)} \, _2F_1\left(\frac{1}{2},3;\frac{3}{2};1-\sec (c+d x)\right)}{d}","\frac{3 a \sin (c+d x)}{4 d \sqrt{a \sec (c+d x)+a}}+\frac{3 \sqrt{a} \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{4 d}+\frac{a \sin (c+d x) \cos (c+d x)}{2 d \sqrt{a \sec (c+d x)+a}}",1,"(2*Hypergeometric2F1[1/2, 3, 3/2, 1 - Sec[c + d*x]]*Sqrt[a*(1 + Sec[c + d*x])]*Tan[(c + d*x)/2])/d","C",1
97,1,47,138,0.0990979,"\int \cos ^3(c+d x) \sqrt{a+a \sec (c+d x)} \, dx","Integrate[Cos[c + d*x]^3*Sqrt[a + a*Sec[c + d*x]],x]","\frac{2 \tan \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\sec (c+d x)+1)} \, _2F_1\left(\frac{1}{2},4;\frac{3}{2};1-\sec (c+d x)\right)}{d}","\frac{5 a \sin (c+d x)}{8 d \sqrt{a \sec (c+d x)+a}}+\frac{5 \sqrt{a} \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{8 d}+\frac{a \sin (c+d x) \cos ^2(c+d x)}{3 d \sqrt{a \sec (c+d x)+a}}+\frac{5 a \sin (c+d x) \cos (c+d x)}{12 d \sqrt{a \sec (c+d x)+a}}",1,"(2*Hypergeometric2F1[1/2, 4, 3/2, 1 - Sec[c + d*x]]*Sqrt[a*(1 + Sec[c + d*x])]*Tan[(c + d*x)/2])/d","C",1
98,1,47,174,0.0991851,"\int \cos ^4(c+d x) \sqrt{a+a \sec (c+d x)} \, dx","Integrate[Cos[c + d*x]^4*Sqrt[a + a*Sec[c + d*x]],x]","\frac{2 \tan \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\sec (c+d x)+1)} \, _2F_1\left(\frac{1}{2},5;\frac{3}{2};1-\sec (c+d x)\right)}{d}","\frac{35 a \sin (c+d x)}{64 d \sqrt{a \sec (c+d x)+a}}+\frac{35 \sqrt{a} \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{64 d}+\frac{a \sin (c+d x) \cos ^3(c+d x)}{4 d \sqrt{a \sec (c+d x)+a}}+\frac{7 a \sin (c+d x) \cos ^2(c+d x)}{24 d \sqrt{a \sec (c+d x)+a}}+\frac{35 a \sin (c+d x) \cos (c+d x)}{96 d \sqrt{a \sec (c+d x)+a}}",1,"(2*Hypergeometric2F1[1/2, 5, 3/2, 1 - Sec[c + d*x]]*Sqrt[a*(1 + Sec[c + d*x])]*Tan[(c + d*x)/2])/d","C",1
99,1,70,162,0.553817,"\int \sec ^4(c+d x) (a+a \sec (c+d x))^{3/2} \, dx","Integrate[Sec[c + d*x]^4*(a + a*Sec[c + d*x])^(3/2),x]","\frac{2 a^2 \tan (c+d x) \left(35 \sec ^4(c+d x)+85 \sec ^3(c+d x)+102 \sec ^2(c+d x)+136 \sec (c+d x)+272\right)}{315 d \sqrt{a (\sec (c+d x)+1)}}","\frac{2 a^2 \tan (c+d x) \sec ^4(c+d x)}{9 d \sqrt{a \sec (c+d x)+a}}+\frac{34 a^2 \tan (c+d x) \sec ^3(c+d x)}{63 d \sqrt{a \sec (c+d x)+a}}+\frac{68 a^2 \tan (c+d x)}{45 d \sqrt{a \sec (c+d x)+a}}+\frac{68 \tan (c+d x) (a \sec (c+d x)+a)^{3/2}}{105 d}-\frac{136 a \tan (c+d x) \sqrt{a \sec (c+d x)+a}}{315 d}",1,"(2*a^2*(272 + 136*Sec[c + d*x] + 102*Sec[c + d*x]^2 + 85*Sec[c + d*x]^3 + 35*Sec[c + d*x]^4)*Tan[c + d*x])/(315*d*Sqrt[a*(1 + Sec[c + d*x])])","A",1
100,1,60,116,0.1795253,"\int \sec ^3(c+d x) (a+a \sec (c+d x))^{3/2} \, dx","Integrate[Sec[c + d*x]^3*(a + a*Sec[c + d*x])^(3/2),x]","\frac{2 a^2 \tan (c+d x) \left(15 \sec ^3(c+d x)+39 \sec ^2(c+d x)+52 \sec (c+d x)+104\right)}{105 d \sqrt{a (\sec (c+d x)+1)}}","\frac{152 a^2 \tan (c+d x)}{105 d \sqrt{a \sec (c+d x)+a}}+\frac{2 \tan (c+d x) (a \sec (c+d x)+a)^{5/2}}{7 a d}-\frac{4 \tan (c+d x) (a \sec (c+d x)+a)^{3/2}}{35 d}+\frac{38 a \tan (c+d x) \sqrt{a \sec (c+d x)+a}}{105 d}",1,"(2*a^2*(104 + 52*Sec[c + d*x] + 39*Sec[c + d*x]^2 + 15*Sec[c + d*x]^3)*Tan[c + d*x])/(105*d*Sqrt[a*(1 + Sec[c + d*x])])","A",1
101,1,48,86,0.1341619,"\int \sec ^2(c+d x) (a+a \sec (c+d x))^{3/2} \, dx","Integrate[Sec[c + d*x]^2*(a + a*Sec[c + d*x])^(3/2),x]","\frac{2 a^2 \tan (c+d x) \left(\sec ^2(c+d x)+3 \sec (c+d x)+6\right)}{5 d \sqrt{a (\sec (c+d x)+1)}}","\frac{8 a^2 \tan (c+d x)}{5 d \sqrt{a \sec (c+d x)+a}}+\frac{2 a \tan (c+d x) \sqrt{a \sec (c+d x)+a}}{5 d}+\frac{2 \tan (c+d x) (a \sec (c+d x)+a)^{3/2}}{5 d}",1,"(2*a^2*(6 + 3*Sec[c + d*x] + Sec[c + d*x]^2)*Tan[c + d*x])/(5*d*Sqrt[a*(1 + Sec[c + d*x])])","A",1
102,1,38,59,0.091499,"\int \sec (c+d x) (a+a \sec (c+d x))^{3/2} \, dx","Integrate[Sec[c + d*x]*(a + a*Sec[c + d*x])^(3/2),x]","\frac{2 a^2 \tan (c+d x) (\sec (c+d x)+5)}{3 d \sqrt{a (\sec (c+d x)+1)}}","\frac{8 a^2 \tan (c+d x)}{3 d \sqrt{a \sec (c+d x)+a}}+\frac{2 a \tan (c+d x) \sqrt{a \sec (c+d x)+a}}{3 d}",1,"(2*a^2*(5 + Sec[c + d*x])*Tan[c + d*x])/(3*d*Sqrt[a*(1 + Sec[c + d*x])])","A",1
103,1,75,66,0.212659,"\int (a+a \sec (c+d x))^{3/2} \, dx","Integrate[(a + a*Sec[c + d*x])^(3/2),x]","\frac{a \sec \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\sec (c+d x)+1)} \left(2 \sin \left(\frac{1}{2} (c+d x)\right)+\sqrt{2} \sin ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right) \sqrt{\cos (c+d x)}\right)}{d}","\frac{2 a^{3/2} \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{d}+\frac{2 a^2 \tan (c+d x)}{d \sqrt{a \sec (c+d x)+a}}",1,"(a*Sec[(c + d*x)/2]*Sqrt[a*(1 + Sec[c + d*x])]*(Sqrt[2]*ArcSin[Sqrt[2]*Sin[(c + d*x)/2]]*Sqrt[Cos[c + d*x]] + 2*Sin[(c + d*x)/2]))/d","A",1
104,1,89,65,0.2060818,"\int \cos (c+d x) (a+a \sec (c+d x))^{3/2} \, dx","Integrate[Cos[c + d*x]*(a + a*Sec[c + d*x])^(3/2),x]","\frac{a \sqrt{\cos (c+d x)} \sec \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\sec (c+d x)+1)} \left(3 \sqrt{2} \sin ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)+2 \sin \left(\frac{1}{2} (c+d x)\right) \sqrt{\cos (c+d x)}\right)}{2 d}","\frac{3 a^{3/2} \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{d}+\frac{a^2 \sin (c+d x)}{d \sqrt{a \sec (c+d x)+a}}",1,"(a*Sqrt[Cos[c + d*x]]*Sec[(c + d*x)/2]*Sqrt[a*(1 + Sec[c + d*x])]*(3*Sqrt[2]*ArcSin[Sqrt[2]*Sin[(c + d*x)/2]] + 2*Sqrt[Cos[c + d*x]]*Sin[(c + d*x)/2]))/(2*d)","A",1
105,1,108,106,0.3768128,"\int \cos ^2(c+d x) (a+a \sec (c+d x))^{3/2} \, dx","Integrate[Cos[c + d*x]^2*(a + a*Sec[c + d*x])^(3/2),x]","\frac{a \cos (c+d x) \sqrt{a (\sec (c+d x)+1)} \left((7 \sin (c+d x)+\sin (2 (c+d x))) \sqrt{1-\sec (c+d x)}+7 \tan (c+d x) \tanh ^{-1}\left(\sqrt{1-\sec (c+d x)}\right)\right)}{4 d (\cos (c+d x)+1) \sqrt{1-\sec (c+d x)}}","\frac{7 a^{3/2} \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{4 d}+\frac{7 a^2 \sin (c+d x)}{4 d \sqrt{a \sec (c+d x)+a}}+\frac{a^2 \sin (c+d x) \cos (c+d x)}{2 d \sqrt{a \sec (c+d x)+a}}",1,"(a*Cos[c + d*x]*Sqrt[a*(1 + Sec[c + d*x])]*(Sqrt[1 - Sec[c + d*x]]*(7*Sin[c + d*x] + Sin[2*(c + d*x)]) + 7*ArcTanh[Sqrt[1 - Sec[c + d*x]]]*Tan[c + d*x]))/(4*d*(1 + Cos[c + d*x])*Sqrt[1 - Sec[c + d*x]])","A",1
106,1,120,144,0.5501969,"\int \cos ^3(c+d x) (a+a \sec (c+d x))^{3/2} \, dx","Integrate[Cos[c + d*x]^3*(a + a*Sec[c + d*x])^(3/2),x]","\frac{a \cos (c+d x) \sqrt{a (\sec (c+d x)+1)} \left((35 \sin (c+d x)+11 \sin (2 (c+d x))+2 \sin (3 (c+d x))) \sqrt{1-\sec (c+d x)}+33 \tan (c+d x) \tanh ^{-1}\left(\sqrt{1-\sec (c+d x)}\right)\right)}{24 d (\cos (c+d x)+1) \sqrt{1-\sec (c+d x)}}","\frac{11 a^{3/2} \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{8 d}+\frac{11 a^2 \sin (c+d x)}{8 d \sqrt{a \sec (c+d x)+a}}+\frac{a^2 \sin (c+d x) \cos ^2(c+d x)}{3 d \sqrt{a \sec (c+d x)+a}}+\frac{11 a^2 \sin (c+d x) \cos (c+d x)}{12 d \sqrt{a \sec (c+d x)+a}}",1,"(a*Cos[c + d*x]*Sqrt[a*(1 + Sec[c + d*x])]*(Sqrt[1 - Sec[c + d*x]]*(35*Sin[c + d*x] + 11*Sin[2*(c + d*x)] + 2*Sin[3*(c + d*x)]) + 33*ArcTanh[Sqrt[1 - Sec[c + d*x]]]*Tan[c + d*x]))/(24*d*(1 + Cos[c + d*x])*Sqrt[1 - Sec[c + d*x]])","A",1
107,1,80,203,0.2135691,"\int \sec ^4(c+d x) (a+a \sec (c+d x))^{5/2} \, dx","Integrate[Sec[c + d*x]^4*(a + a*Sec[c + d*x])^(5/2),x]","\frac{2 a^3 \tan (c+d x) \left(63 \sec ^5(c+d x)+224 \sec ^4(c+d x)+355 \sec ^3(c+d x)+426 \sec ^2(c+d x)+568 \sec (c+d x)+1136\right)}{693 d \sqrt{a (\sec (c+d x)+1)}}","\frac{46 a^3 \tan (c+d x) \sec ^4(c+d x)}{99 d \sqrt{a \sec (c+d x)+a}}+\frac{710 a^3 \tan (c+d x) \sec ^3(c+d x)}{693 d \sqrt{a \sec (c+d x)+a}}+\frac{284 a^3 \tan (c+d x)}{99 d \sqrt{a \sec (c+d x)+a}}+\frac{2 a^2 \tan (c+d x) \sec ^4(c+d x) \sqrt{a \sec (c+d x)+a}}{11 d}-\frac{568 a^2 \tan (c+d x) \sqrt{a \sec (c+d x)+a}}{693 d}+\frac{284 a \tan (c+d x) (a \sec (c+d x)+a)^{3/2}}{231 d}",1,"(2*a^3*(1136 + 568*Sec[c + d*x] + 426*Sec[c + d*x]^2 + 355*Sec[c + d*x]^3 + 224*Sec[c + d*x]^4 + 63*Sec[c + d*x]^5)*Tan[c + d*x])/(693*d*Sqrt[a*(1 + Sec[c + d*x])])","A",1
108,1,70,146,0.5337812,"\int \sec ^3(c+d x) (a+a \sec (c+d x))^{5/2} \, dx","Integrate[Sec[c + d*x]^3*(a + a*Sec[c + d*x])^(5/2),x]","\frac{2 a^3 \tan (c+d x) \left(35 \sec ^4(c+d x)+130 \sec ^3(c+d x)+219 \sec ^2(c+d x)+292 \sec (c+d x)+584\right)}{315 d \sqrt{a (\sec (c+d x)+1)}}","\frac{832 a^3 \tan (c+d x)}{315 d \sqrt{a \sec (c+d x)+a}}+\frac{208 a^2 \tan (c+d x) \sqrt{a \sec (c+d x)+a}}{315 d}+\frac{2 \tan (c+d x) (a \sec (c+d x)+a)^{7/2}}{9 a d}-\frac{4 \tan (c+d x) (a \sec (c+d x)+a)^{5/2}}{63 d}+\frac{26 a \tan (c+d x) (a \sec (c+d x)+a)^{3/2}}{105 d}",1,"(2*a^3*(584 + 292*Sec[c + d*x] + 219*Sec[c + d*x]^2 + 130*Sec[c + d*x]^3 + 35*Sec[c + d*x]^4)*Tan[c + d*x])/(315*d*Sqrt[a*(1 + Sec[c + d*x])])","A",1
109,1,60,116,0.1822289,"\int \sec ^2(c+d x) (a+a \sec (c+d x))^{5/2} \, dx","Integrate[Sec[c + d*x]^2*(a + a*Sec[c + d*x])^(5/2),x]","\frac{2 a^3 \tan (c+d x) \left(3 \sec ^3(c+d x)+12 \sec ^2(c+d x)+23 \sec (c+d x)+46\right)}{21 d \sqrt{a (\sec (c+d x)+1)}}","\frac{64 a^3 \tan (c+d x)}{21 d \sqrt{a \sec (c+d x)+a}}+\frac{16 a^2 \tan (c+d x) \sqrt{a \sec (c+d x)+a}}{21 d}+\frac{2 a \tan (c+d x) (a \sec (c+d x)+a)^{3/2}}{7 d}+\frac{2 \tan (c+d x) (a \sec (c+d x)+a)^{5/2}}{7 d}",1,"(2*a^3*(46 + 23*Sec[c + d*x] + 12*Sec[c + d*x]^2 + 3*Sec[c + d*x]^3)*Tan[c + d*x])/(21*d*Sqrt[a*(1 + Sec[c + d*x])])","A",1
110,1,50,89,0.0991733,"\int \sec (c+d x) (a+a \sec (c+d x))^{5/2} \, dx","Integrate[Sec[c + d*x]*(a + a*Sec[c + d*x])^(5/2),x]","\frac{2 a^3 \tan (c+d x) \left(3 \sec ^2(c+d x)+14 \sec (c+d x)+43\right)}{15 d \sqrt{a (\sec (c+d x)+1)}}","\frac{64 a^3 \tan (c+d x)}{15 d \sqrt{a \sec (c+d x)+a}}+\frac{16 a^2 \tan (c+d x) \sqrt{a \sec (c+d x)+a}}{15 d}+\frac{2 a \tan (c+d x) (a \sec (c+d x)+a)^{3/2}}{5 d}",1,"(2*a^3*(43 + 14*Sec[c + d*x] + 3*Sec[c + d*x]^2)*Tan[c + d*x])/(15*d*Sqrt[a*(1 + Sec[c + d*x])])","A",1
111,1,360,98,6.3563072,"\int (a+a \sec (c+d x))^{5/2} \, dx","Integrate[(a + a*Sec[c + d*x])^(5/2),x]","\frac{\sqrt{\frac{1}{1-2 \sin ^2\left(\frac{1}{2} (c+d x)\right)}} \sqrt{1-2 \sin ^2\left(\frac{1}{2} (c+d x)\right)} \csc ^3\left(\frac{1}{2} (c+d x)\right) \sec ^5\left(\frac{1}{2} (c+d x)\right) (a (\sec (c+d x)+1))^{5/2} \left(256 \sin ^6\left(\frac{1}{2} (c+d x)\right) \cos ^4\left(\frac{1}{2} (c+d x)\right) \, _3F_2\left(\frac{3}{2},2,\frac{7}{2};1,\frac{9}{2};2 \sin ^2\left(\frac{1}{2} (c+d x)\right)\right)+512 \left(\sin ^4\left(\frac{1}{2} (c+d x)\right)-3 \sin ^2\left(\frac{1}{2} (c+d x)\right)+2\right) \sin ^6\left(\frac{1}{2} (c+d x)\right) \, _2F_1\left(\frac{3}{2},\frac{7}{2};\frac{9}{2};2 \sin ^2\left(\frac{1}{2} (c+d x)\right)\right)+\frac{21 \sqrt{2} \sin ^{-1}\left(\sqrt{2} \sqrt{\sin ^2\left(\frac{1}{2} (c+d x)\right)}\right) \left(3 \sin ^4\left(\frac{1}{2} (c+d x)\right)-10 \sin ^2\left(\frac{1}{2} (c+d x)\right)+15\right)}{\sqrt{\sin ^2\left(\frac{1}{2} (c+d x)\right)}}-14 \sqrt{1-2 \sin ^2\left(\frac{1}{2} (c+d x)\right)} \left(12 \sin ^6\left(\frac{1}{2} (c+d x)\right)-31 \sin ^4\left(\frac{1}{2} (c+d x)\right)+30 \sin ^2\left(\frac{1}{2} (c+d x)\right)+45\right)\right)}{672 d \sec ^{\frac{5}{2}}(c+d x)}","\frac{2 a^{5/2} \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{d}+\frac{14 a^3 \tan (c+d x)}{3 d \sqrt{a \sec (c+d x)+a}}+\frac{2 a^2 \tan (c+d x) \sqrt{a \sec (c+d x)+a}}{3 d}",1,"(Csc[(c + d*x)/2]^3*Sec[(c + d*x)/2]^5*(a*(1 + Sec[c + d*x]))^(5/2)*Sqrt[(1 - 2*Sin[(c + d*x)/2]^2)^(-1)]*Sqrt[1 - 2*Sin[(c + d*x)/2]^2]*(256*Cos[(c + d*x)/2]^4*HypergeometricPFQ[{3/2, 2, 7/2}, {1, 9/2}, 2*Sin[(c + d*x)/2]^2]*Sin[(c + d*x)/2]^6 + 512*Hypergeometric2F1[3/2, 7/2, 9/2, 2*Sin[(c + d*x)/2]^2]*Sin[(c + d*x)/2]^6*(2 - 3*Sin[(c + d*x)/2]^2 + Sin[(c + d*x)/2]^4) + (21*Sqrt[2]*ArcSin[Sqrt[2]*Sqrt[Sin[(c + d*x)/2]^2]]*(15 - 10*Sin[(c + d*x)/2]^2 + 3*Sin[(c + d*x)/2]^4))/Sqrt[Sin[(c + d*x)/2]^2] - 14*Sqrt[1 - 2*Sin[(c + d*x)/2]^2]*(45 + 30*Sin[(c + d*x)/2]^2 - 31*Sin[(c + d*x)/2]^4 + 12*Sin[(c + d*x)/2]^6)))/(672*d*Sec[c + d*x]^(5/2))","C",0
112,1,189,94,2.5127425,"\int \cos (c+d x) (a+a \sec (c+d x))^{5/2} \, dx","Integrate[Cos[c + d*x]*(a + a*Sec[c + d*x])^(5/2),x]","\frac{2 \cos ^{\frac{5}{2}}(c+d x) \tan \left(\frac{1}{2} (c+d x)\right) (a (\sec (c+d x)+1))^{5/2} \left(12 \sin ^2\left(\frac{1}{2} (c+d x)\right) \, _3F_2\left(\frac{3}{2},2,\frac{5}{2};1,\frac{9}{2};2 \sin ^2\left(\frac{1}{2} (c+d x)\right)\right)+\frac{1}{8} \sec ^4\left(\frac{1}{2} (c+d x)\right) \left(24 \sin ^2(c+d x) (\cos (c+d x)+3) \, _2F_1\left(\frac{3}{2},\frac{5}{2};\frac{9}{2};2 \sin ^2\left(\frac{1}{2} (c+d x)\right)\right)+7 (28 \cos (c+d x)+3 \cos (2 (c+d x))+89) \, _2F_1\left(\frac{1}{2},\frac{3}{2};\frac{7}{2};2 \sin ^2\left(\frac{1}{2} (c+d x)\right)\right)\right)\right)}{105 d}","\frac{5 a^{5/2} \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{d}-\frac{a^3 \sin (c+d x)}{d \sqrt{a \sec (c+d x)+a}}+\frac{2 a^2 \sin (c+d x) \sqrt{a \sec (c+d x)+a}}{d}",1,"(2*Cos[c + d*x]^(5/2)*(a*(1 + Sec[c + d*x]))^(5/2)*(12*HypergeometricPFQ[{3/2, 2, 5/2}, {1, 9/2}, 2*Sin[(c + d*x)/2]^2]*Sin[(c + d*x)/2]^2 + (Sec[(c + d*x)/2]^4*(7*(89 + 28*Cos[c + d*x] + 3*Cos[2*(c + d*x)])*Hypergeometric2F1[1/2, 3/2, 7/2, 2*Sin[(c + d*x)/2]^2] + 24*(3 + Cos[c + d*x])*Hypergeometric2F1[3/2, 5/2, 9/2, 2*Sin[(c + d*x)/2]^2]*Sin[c + d*x]^2))/8)*Tan[(c + d*x)/2])/(105*d)","C",0
113,1,150,106,0.5624855,"\int \cos ^2(c+d x) (a+a \sec (c+d x))^{5/2} \, dx","Integrate[Cos[c + d*x]^2*(a + a*Sec[c + d*x])^(5/2),x]","-\frac{a^2 \cos (c+d x) \sqrt{a (\sec (c+d x)+1)} \left(-32 \tan (c+d x) \sqrt{1-\sec (c+d x)} \, _2F_1\left(\frac{1}{2},3;\frac{3}{2};1-\sec (c+d x)\right)+(\sin (c+d x)+3 \sin (2 (c+d x))) \sqrt{1-\sec (c+d x)}-7 \tan (c+d x) \tanh ^{-1}\left(\sqrt{1-\sec (c+d x)}\right)\right)}{4 d (\cos (c+d x)+1) \sqrt{1-\sec (c+d x)}}","\frac{19 a^{5/2} \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{4 d}+\frac{9 a^3 \sin (c+d x)}{4 d \sqrt{a \sec (c+d x)+a}}+\frac{a^2 \sin (c+d x) \cos (c+d x) \sqrt{a \sec (c+d x)+a}}{2 d}",1,"-1/4*(a^2*Cos[c + d*x]*Sqrt[a*(1 + Sec[c + d*x])]*(Sqrt[1 - Sec[c + d*x]]*(Sin[c + d*x] + 3*Sin[2*(c + d*x)]) - 7*ArcTanh[Sqrt[1 - Sec[c + d*x]]]*Tan[c + d*x] - 32*Hypergeometric2F1[1/2, 3, 3/2, 1 - Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]]*Tan[c + d*x]))/(d*(1 + Cos[c + d*x])*Sqrt[1 - Sec[c + d*x]])","C",1
114,1,151,144,0.8170729,"\int \cos ^3(c+d x) (a+a \sec (c+d x))^{5/2} \, dx","Integrate[Cos[c + d*x]^3*(a + a*Sec[c + d*x])^(5/2),x]","\frac{a^2 \sin (c+d x) \sqrt{a (\sec (c+d x)+1)} \left(192 \sqrt{1-\sec (c+d x)} \, _2F_1\left(\frac{1}{2},4;\frac{3}{2};1-\sec (c+d x)\right)+(159 \cos (c+d x)+31 \cos (2 (c+d x))-2 \cos (3 (c+d x))+31) \sqrt{1-\sec (c+d x)}+165 \tanh ^{-1}\left(\sqrt{1-\sec (c+d x)}\right)\right)}{72 d (\cos (c+d x)+1) \sqrt{1-\sec (c+d x)}}","\frac{25 a^{5/2} \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{8 d}+\frac{25 a^3 \sin (c+d x)}{8 d \sqrt{a \sec (c+d x)+a}}+\frac{13 a^3 \sin (c+d x) \cos (c+d x)}{12 d \sqrt{a \sec (c+d x)+a}}+\frac{a^2 \sin (c+d x) \cos ^2(c+d x) \sqrt{a \sec (c+d x)+a}}{3 d}",1,"(a^2*(165*ArcTanh[Sqrt[1 - Sec[c + d*x]]] + (31 + 159*Cos[c + d*x] + 31*Cos[2*(c + d*x)] - 2*Cos[3*(c + d*x)])*Sqrt[1 - Sec[c + d*x]] + 192*Hypergeometric2F1[1/2, 4, 3/2, 1 - Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]])*Sqrt[a*(1 + Sec[c + d*x])]*Sin[c + d*x])/(72*d*(1 + Cos[c + d*x])*Sqrt[1 - Sec[c + d*x]])","C",1
115,1,161,182,0.8192239,"\int \cos ^4(c+d x) (a+a \sec (c+d x))^{5/2} \, dx","Integrate[Cos[c + d*x]^4*(a + a*Sec[c + d*x])^(5/2),x]","\frac{a^2 \sin (c+d x) \sqrt{a (\sec (c+d x)+1)} \left(512 \sqrt{1-\sec (c+d x)} \, _2F_1\left(\frac{1}{2},5;\frac{3}{2};1-\sec (c+d x)\right)+(849 \cos (c+d x)+233 \cos (2 (c+d x))+58 \cos (3 (c+d x))+2 \cos (4 (c+d x))+231) \sqrt{1-\sec (c+d x)}+675 \tanh ^{-1}\left(\sqrt{1-\sec (c+d x)}\right)\right)}{320 d (\cos (c+d x)+1) \sqrt{1-\sec (c+d x)}}","\frac{163 a^{5/2} \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{64 d}+\frac{163 a^3 \sin (c+d x)}{64 d \sqrt{a \sec (c+d x)+a}}+\frac{17 a^3 \sin (c+d x) \cos ^2(c+d x)}{24 d \sqrt{a \sec (c+d x)+a}}+\frac{163 a^3 \sin (c+d x) \cos (c+d x)}{96 d \sqrt{a \sec (c+d x)+a}}+\frac{a^2 \sin (c+d x) \cos ^3(c+d x) \sqrt{a \sec (c+d x)+a}}{4 d}",1,"(a^2*(675*ArcTanh[Sqrt[1 - Sec[c + d*x]]] + (231 + 849*Cos[c + d*x] + 233*Cos[2*(c + d*x)] + 58*Cos[3*(c + d*x)] + 2*Cos[4*(c + d*x)])*Sqrt[1 - Sec[c + d*x]] + 512*Hypergeometric2F1[1/2, 5, 3/2, 1 - Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]])*Sqrt[a*(1 + Sec[c + d*x])]*Sin[c + d*x])/(320*d*(1 + Cos[c + d*x])*Sqrt[1 - Sec[c + d*x]])","C",1
116,1,30,27,0.1174036,"\int \sec (c+d x) \sqrt{a-a \sec (c+d x)} \, dx","Integrate[Sec[c + d*x]*Sqrt[a - a*Sec[c + d*x]],x]","\frac{2 \cot \left(\frac{1}{2} (c+d x)\right) \sqrt{a-a \sec (c+d x)}}{d}","-\frac{2 a \tan (c+d x)}{d \sqrt{a-a \sec (c+d x)}}",1,"(2*Cot[(c + d*x)/2]*Sqrt[a - a*Sec[c + d*x]])/d","A",1
117,1,188,38,0.6211804,"\int \sqrt{a-a \sec (c+d x)} \, dx","Integrate[Sqrt[a - a*Sec[c + d*x]],x]","-\frac{\sqrt{\cos (c)-i \sin (c)} \cos (c+d x) \left(\cot \left(\frac{1}{2} (c+d x)\right)+i\right) \sqrt{a-a \sec (c+d x)} \left(\tanh ^{-1}\left(\frac{e^{i d x}}{\sqrt{\cos (c)-i \sin (c)} \sqrt{e^{2 i d x} (\cos (c)+i \sin (c))-i \sin (c)+\cos (c)}}\right)+\tanh ^{-1}\left(\frac{\sqrt{e^{2 i d x} (\cos (c)+i \sin (c))-i \sin (c)+\cos (c)}}{\sqrt{\cos (c)-i \sin (c)}}\right)\right)}{d \sqrt{i \sin (c) \left(-1+e^{2 i d x}\right)+\cos (c) \left(1+e^{2 i d x}\right)}}","\frac{2 \sqrt{a} \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a-a \sec (c+d x)}}\right)}{d}",1,"-(((ArcTanh[E^(I*d*x)/(Sqrt[Cos[c] - I*Sin[c]]*Sqrt[Cos[c] + E^((2*I)*d*x)*(Cos[c] + I*Sin[c]) - I*Sin[c]])] + ArcTanh[Sqrt[Cos[c] + E^((2*I)*d*x)*(Cos[c] + I*Sin[c]) - I*Sin[c]]/Sqrt[Cos[c] - I*Sin[c]]])*Cos[c + d*x]*(I + Cot[(c + d*x)/2])*Sqrt[a - a*Sec[c + d*x]]*Sqrt[Cos[c] - I*Sin[c]])/(d*Sqrt[(1 + E^((2*I)*d*x))*Cos[c] + I*(-1 + E^((2*I)*d*x))*Sin[c]]))","C",1
118,1,260,65,0.9359594,"\int \cos (c+d x) \sqrt{a-a \sec (c+d x)} \, dx","Integrate[Cos[c + d*x]*Sqrt[a - a*Sec[c + d*x]],x]","\frac{\cos (c+d x) \sqrt{a-a \sec (c+d x)} \left(-2 \sqrt{2} \cot \left(\frac{1}{2} (c+d x)\right) \sqrt{\cos (c+d x) (\cos (d x)+i \sin (d x))}+\sqrt{\cos (c)-i \sin (c)} \left(\cot \left(\frac{1}{2} (c+d x)\right)+i\right) \tanh ^{-1}\left(\frac{e^{i d x}}{\sqrt{\cos (c)-i \sin (c)} \sqrt{e^{2 i d x} (\cos (c)+i \sin (c))-i \sin (c)+\cos (c)}}\right)+\sqrt{\cos (c)-i \sin (c)} \left(\cot \left(\frac{1}{2} (c+d x)\right)+i\right) \tanh ^{-1}\left(\frac{\sqrt{e^{2 i d x} (\cos (c)+i \sin (c))-i \sin (c)+\cos (c)}}{\sqrt{\cos (c)-i \sin (c)}}\right)\right)}{2 d \sqrt{i \sin (c) \left(-1+e^{2 i d x}\right)+\cos (c) \left(1+e^{2 i d x}\right)}}","\frac{a \sin (c+d x)}{d \sqrt{a-a \sec (c+d x)}}-\frac{\sqrt{a} \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a-a \sec (c+d x)}}\right)}{d}",1,"(Cos[c + d*x]*Sqrt[a - a*Sec[c + d*x]]*(ArcTanh[E^(I*d*x)/(Sqrt[Cos[c] - I*Sin[c]]*Sqrt[Cos[c] + E^((2*I)*d*x)*(Cos[c] + I*Sin[c]) - I*Sin[c]])]*(I + Cot[(c + d*x)/2])*Sqrt[Cos[c] - I*Sin[c]] + ArcTanh[Sqrt[Cos[c] + E^((2*I)*d*x)*(Cos[c] + I*Sin[c]) - I*Sin[c]]/Sqrt[Cos[c] - I*Sin[c]]]*(I + Cot[(c + d*x)/2])*Sqrt[Cos[c] - I*Sin[c]] - 2*Sqrt[2]*Cot[(c + d*x)/2]*Sqrt[Cos[c + d*x]*(Cos[d*x] + I*Sin[d*x])]))/(2*d*Sqrt[(1 + E^((2*I)*d*x))*Cos[c] + I*(-1 + E^((2*I)*d*x))*Sin[c]])","C",1
119,1,106,140,0.2235634,"\int \frac{\sec ^4(c+d x)}{\sqrt{a+a \sec (c+d x)}} \, dx","Integrate[Sec[c + d*x]^4/Sqrt[a + a*Sec[c + d*x]],x]","\frac{\tan (c+d x) \left(2 \sqrt{1-\sec (c+d x)} \left(3 \sec ^2(c+d x)-\sec (c+d x)+13\right)-15 \sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{1-\sec (c+d x)}}{\sqrt{2}}\right)\right)}{15 d \sqrt{1-\sec (c+d x)} \sqrt{a (\sec (c+d x)+1)}}","\frac{2 \tan (c+d x) \sec ^2(c+d x)}{5 d \sqrt{a \sec (c+d x)+a}}-\frac{\sqrt{2} \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{\sqrt{a} d}-\frac{2 \tan (c+d x) \sqrt{a \sec (c+d x)+a}}{15 a d}+\frac{28 \tan (c+d x)}{15 d \sqrt{a \sec (c+d x)+a}}",1,"((-15*Sqrt[2]*ArcTanh[Sqrt[1 - Sec[c + d*x]]/Sqrt[2]] + 2*Sqrt[1 - Sec[c + d*x]]*(13 - Sec[c + d*x] + 3*Sec[c + d*x]^2))*Tan[c + d*x])/(15*d*Sqrt[1 - Sec[c + d*x]]*Sqrt[a*(1 + Sec[c + d*x])])","A",1
120,1,86,104,0.1577209,"\int \frac{\sec ^3(c+d x)}{\sqrt{a+a \sec (c+d x)}} \, dx","Integrate[Sec[c + d*x]^3/Sqrt[a + a*Sec[c + d*x]],x]","-\frac{\tan (c+d x) \left(\frac{2}{3} (1-\sec (c+d x))^{3/2}-\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{1-\sec (c+d x)}}{\sqrt{2}}\right)\right)}{d \sqrt{1-\sec (c+d x)} \sqrt{a (\sec (c+d x)+1)}}","\frac{\sqrt{2} \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{\sqrt{a} d}+\frac{2 \tan (c+d x) \sqrt{a \sec (c+d x)+a}}{3 a d}-\frac{4 \tan (c+d x)}{3 d \sqrt{a \sec (c+d x)+a}}",1,"-(((-(Sqrt[2]*ArcTanh[Sqrt[1 - Sec[c + d*x]]/Sqrt[2]]) + (2*(1 - Sec[c + d*x])^(3/2))/3)*Tan[c + d*x])/(d*Sqrt[1 - Sec[c + d*x]]*Sqrt[a*(1 + Sec[c + d*x])]))","A",1
121,1,83,73,0.082894,"\int \frac{\sec ^2(c+d x)}{\sqrt{a+a \sec (c+d x)}} \, dx","Integrate[Sec[c + d*x]^2/Sqrt[a + a*Sec[c + d*x]],x]","-\frac{\tan (c+d x) \left(\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{1-\sec (c+d x)}}{\sqrt{2}}\right)-2 \sqrt{1-\sec (c+d x)}\right)}{d \sqrt{1-\sec (c+d x)} \sqrt{a (\sec (c+d x)+1)}}","\frac{2 \tan (c+d x)}{d \sqrt{a \sec (c+d x)+a}}-\frac{\sqrt{2} \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{\sqrt{a} d}",1,"-(((Sqrt[2]*ArcTanh[Sqrt[1 - Sec[c + d*x]]/Sqrt[2]] - 2*Sqrt[1 - Sec[c + d*x]])*Tan[c + d*x])/(d*Sqrt[1 - Sec[c + d*x]]*Sqrt[a*(1 + Sec[c + d*x])]))","A",1
122,1,64,46,0.0499083,"\int \frac{\sec (c+d x)}{\sqrt{a+a \sec (c+d x)}} \, dx","Integrate[Sec[c + d*x]/Sqrt[a + a*Sec[c + d*x]],x]","\frac{\sqrt{2} \tan (c+d x) \tanh ^{-1}\left(\frac{\sqrt{1-\sec (c+d x)}}{\sqrt{2}}\right)}{d \sqrt{1-\sec (c+d x)} \sqrt{a (\sec (c+d x)+1)}}","\frac{\sqrt{2} \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{\sqrt{a} d}",1,"(Sqrt[2]*ArcTanh[Sqrt[1 - Sec[c + d*x]]/Sqrt[2]]*Tan[c + d*x])/(d*Sqrt[1 - Sec[c + d*x]]*Sqrt[a*(1 + Sec[c + d*x])])","A",1
123,1,5402,85,24.0713072,"\int \frac{1}{\sqrt{a+a \sec (c+d x)}} \, dx","Integrate[1/Sqrt[a + a*Sec[c + d*x]],x]","\text{Result too large to show}","\frac{2 \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{\sqrt{a} d}-\frac{\sqrt{2} \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{\sqrt{a} d}",1,"Result too large to show","C",0
124,1,105,108,0.1643805,"\int \frac{\cos (c+d x)}{\sqrt{a+a \sec (c+d x)}} \, dx","Integrate[Cos[c + d*x]/Sqrt[a + a*Sec[c + d*x]],x]","-\frac{\tan (c+d x) \left(-\cos (c+d x) \sqrt{1-\sec (c+d x)}+\tanh ^{-1}\left(\sqrt{1-\sec (c+d x)}\right)-\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{1-\sec (c+d x)}}{\sqrt{2}}\right)\right)}{d \sqrt{1-\sec (c+d x)} \sqrt{a (\sec (c+d x)+1)}}","\frac{\sin (c+d x)}{d \sqrt{a \sec (c+d x)+a}}-\frac{\tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{\sqrt{a} d}+\frac{\sqrt{2} \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{\sqrt{a} d}",1,"-(((ArcTanh[Sqrt[1 - Sec[c + d*x]]] - Sqrt[2]*ArcTanh[Sqrt[1 - Sec[c + d*x]]/Sqrt[2]] - Cos[c + d*x]*Sqrt[1 - Sec[c + d*x]])*Tan[c + d*x])/(d*Sqrt[1 - Sec[c + d*x]]*Sqrt[a*(1 + Sec[c + d*x])]))","A",1
125,1,118,147,0.3189186,"\int \frac{\cos ^2(c+d x)}{\sqrt{a+a \sec (c+d x)}} \, dx","Integrate[Cos[c + d*x]^2/Sqrt[a + a*Sec[c + d*x]],x]","\frac{\tan (c+d x) \left(\cos (c+d x) (2 \cos (c+d x)-1) \sqrt{1-\sec (c+d x)}+7 \tanh ^{-1}\left(\sqrt{1-\sec (c+d x)}\right)-4 \sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{1-\sec (c+d x)}}{\sqrt{2}}\right)\right)}{4 d \sqrt{1-\sec (c+d x)} \sqrt{a (\sec (c+d x)+1)}}","-\frac{\sin (c+d x)}{4 d \sqrt{a \sec (c+d x)+a}}+\frac{7 \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{4 \sqrt{a} d}-\frac{\sqrt{2} \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{\sqrt{a} d}+\frac{\sin (c+d x) \cos (c+d x)}{2 d \sqrt{a \sec (c+d x)+a}}",1,"((7*ArcTanh[Sqrt[1 - Sec[c + d*x]]] - 4*Sqrt[2]*ArcTanh[Sqrt[1 - Sec[c + d*x]]/Sqrt[2]] + Cos[c + d*x]*(-1 + 2*Cos[c + d*x])*Sqrt[1 - Sec[c + d*x]])*Tan[c + d*x])/(4*d*Sqrt[1 - Sec[c + d*x]]*Sqrt[a*(1 + Sec[c + d*x])])","A",1
126,1,124,183,0.5763178,"\int \frac{\sec ^5(c+d x)}{(a+a \sec (c+d x))^{3/2}} \, dx","Integrate[Sec[c + d*x]^5/(a + a*Sec[c + d*x])^(3/2),x]","\frac{\tan (c+d x) \left(2 \sqrt{1-\sec (c+d x)} \left(4 \sec ^3(c+d x)-4 \sec ^2(c+d x)+36 \sec (c+d x)+49\right)-75 \sqrt{2} (\sec (c+d x)+1) \tanh ^{-1}\left(\frac{\sqrt{1-\sec (c+d x)}}{\sqrt{2}}\right)\right)}{20 d \sqrt{1-\sec (c+d x)} (a (\sec (c+d x)+1))^{3/2}}","-\frac{15 \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}-\frac{13 \tan (c+d x) \sqrt{a \sec (c+d x)+a}}{10 a^2 d}-\frac{\tan (c+d x) \sec ^3(c+d x)}{2 d (a \sec (c+d x)+a)^{3/2}}+\frac{9 \tan (c+d x) \sec ^2(c+d x)}{10 a d \sqrt{a \sec (c+d x)+a}}+\frac{31 \tan (c+d x)}{5 a d \sqrt{a \sec (c+d x)+a}}",1,"((-75*Sqrt[2]*ArcTanh[Sqrt[1 - Sec[c + d*x]]/Sqrt[2]]*(1 + Sec[c + d*x]) + 2*Sqrt[1 - Sec[c + d*x]]*(49 + 36*Sec[c + d*x] - 4*Sec[c + d*x]^2 + 4*Sec[c + d*x]^3))*Tan[c + d*x])/(20*d*Sqrt[1 - Sec[c + d*x]]*(a*(1 + Sec[c + d*x]))^(3/2))","A",1
127,1,114,145,0.3619997,"\int \frac{\sec ^4(c+d x)}{(a+a \sec (c+d x))^{3/2}} \, dx","Integrate[Sec[c + d*x]^4/(a + a*Sec[c + d*x])^(3/2),x]","\frac{\tan (c+d x) \left(2 \sqrt{1-\sec (c+d x)} \left(4 \sec ^2(c+d x)-12 \sec (c+d x)-19\right)+33 \sqrt{2} (\sec (c+d x)+1) \tanh ^{-1}\left(\frac{\sqrt{1-\sec (c+d x)}}{\sqrt{2}}\right)\right)}{12 d \sqrt{1-\sec (c+d x)} (a (\sec (c+d x)+1))^{3/2}}","\frac{11 \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}+\frac{7 \tan (c+d x) \sqrt{a \sec (c+d x)+a}}{6 a^2 d}-\frac{\tan (c+d x) \sec ^2(c+d x)}{2 d (a \sec (c+d x)+a)^{3/2}}-\frac{13 \tan (c+d x)}{3 a d \sqrt{a \sec (c+d x)+a}}",1,"((33*Sqrt[2]*ArcTanh[Sqrt[1 - Sec[c + d*x]]/Sqrt[2]]*(1 + Sec[c + d*x]) + 2*Sqrt[1 - Sec[c + d*x]]*(-19 - 12*Sec[c + d*x] + 4*Sec[c + d*x]^2))*Tan[c + d*x])/(12*d*Sqrt[1 - Sec[c + d*x]]*(a*(1 + Sec[c + d*x]))^(3/2))","A",1
128,1,104,105,0.3942987,"\int \frac{\sec ^3(c+d x)}{(a+a \sec (c+d x))^{3/2}} \, dx","Integrate[Sec[c + d*x]^3/(a + a*Sec[c + d*x])^(3/2),x]","\frac{\tan (c+d x) \left(2 \sqrt{1-\sec (c+d x)} (4 \sec (c+d x)+5)-7 \sqrt{2} (\sec (c+d x)+1) \tanh ^{-1}\left(\frac{\sqrt{1-\sec (c+d x)}}{\sqrt{2}}\right)\right)}{4 d \sqrt{1-\sec (c+d x)} (a (\sec (c+d x)+1))^{3/2}}","-\frac{7 \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}+\frac{2 \tan (c+d x)}{a d \sqrt{a \sec (c+d x)+a}}+\frac{\tan (c+d x)}{2 d (a \sec (c+d x)+a)^{3/2}}",1,"((-7*Sqrt[2]*ArcTanh[Sqrt[1 - Sec[c + d*x]]/Sqrt[2]]*(1 + Sec[c + d*x]) + 2*Sqrt[1 - Sec[c + d*x]]*(5 + 4*Sec[c + d*x]))*Tan[c + d*x])/(4*d*Sqrt[1 - Sec[c + d*x]]*(a*(1 + Sec[c + d*x]))^(3/2))","A",1
129,1,94,77,0.2620804,"\int \frac{\sec ^2(c+d x)}{(a+a \sec (c+d x))^{3/2}} \, dx","Integrate[Sec[c + d*x]^2/(a + a*Sec[c + d*x])^(3/2),x]","\frac{\tan (c+d x) \left(3 \sqrt{2} (\sec (c+d x)+1) \tanh ^{-1}\left(\frac{\sqrt{1-\sec (c+d x)}}{\sqrt{2}}\right)-2 \sqrt{1-\sec (c+d x)}\right)}{4 d \sqrt{1-\sec (c+d x)} (a (\sec (c+d x)+1))^{3/2}}","\frac{3 \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}-\frac{\tan (c+d x)}{2 d (a \sec (c+d x)+a)^{3/2}}",1,"((-2*Sqrt[1 - Sec[c + d*x]] + 3*Sqrt[2]*ArcTanh[Sqrt[1 - Sec[c + d*x]]/Sqrt[2]]*(1 + Sec[c + d*x]))*Tan[c + d*x])/(4*d*Sqrt[1 - Sec[c + d*x]]*(a*(1 + Sec[c + d*x]))^(3/2))","A",1
130,1,93,77,0.1438314,"\int \frac{\sec (c+d x)}{(a+a \sec (c+d x))^{3/2}} \, dx","Integrate[Sec[c + d*x]/(a + a*Sec[c + d*x])^(3/2),x]","\frac{\tan (c+d x) \left(2 \sqrt{1-\sec (c+d x)}+\sqrt{2} (\sec (c+d x)+1) \tanh ^{-1}\left(\frac{\sqrt{1-\sec (c+d x)}}{\sqrt{2}}\right)\right)}{4 d \sqrt{1-\sec (c+d x)} (a (\sec (c+d x)+1))^{3/2}}","\frac{\tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}+\frac{\tan (c+d x)}{2 d (a \sec (c+d x)+a)^{3/2}}",1,"((2*Sqrt[1 - Sec[c + d*x]] + Sqrt[2]*ArcTanh[Sqrt[1 - Sec[c + d*x]]/Sqrt[2]]*(1 + Sec[c + d*x]))*Tan[c + d*x])/(4*d*Sqrt[1 - Sec[c + d*x]]*(a*(1 + Sec[c + d*x]))^(3/2))","A",1
131,1,5524,114,24.8444133,"\int \frac{1}{(a+a \sec (c+d x))^{3/2}} \, dx","Integrate[(a + a*Sec[c + d*x])^(-3/2),x]","\text{Result too large to show}","\frac{2 \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{a^{3/2} d}-\frac{5 \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}-\frac{\tan (c+d x)}{2 d (a \sec (c+d x)+a)^{3/2}}",1,"Result too large to show","C",0
132,1,129,144,0.9807442,"\int \frac{\cos (c+d x)}{(a+a \sec (c+d x))^{3/2}} \, dx","Integrate[Cos[c + d*x]/(a + a*Sec[c + d*x])^(3/2),x]","\frac{\tan (c+d x) \left(2 (2 \cos (c+d x)+3) \sqrt{1-\sec (c+d x)}-12 (\sec (c+d x)+1) \tanh ^{-1}\left(\sqrt{1-\sec (c+d x)}\right)+9 \sqrt{2} (\sec (c+d x)+1) \tanh ^{-1}\left(\frac{\sqrt{1-\sec (c+d x)}}{\sqrt{2}}\right)\right)}{4 d \sqrt{1-\sec (c+d x)} (a (\sec (c+d x)+1))^{3/2}}","-\frac{3 \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{a^{3/2} d}+\frac{9 \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}+\frac{3 \sin (c+d x)}{2 a d \sqrt{a \sec (c+d x)+a}}-\frac{\sin (c+d x)}{2 d (a \sec (c+d x)+a)^{3/2}}",1,"((2*(3 + 2*Cos[c + d*x])*Sqrt[1 - Sec[c + d*x]] - 12*ArcTanh[Sqrt[1 - Sec[c + d*x]]]*(1 + Sec[c + d*x]) + 9*Sqrt[2]*ArcTanh[Sqrt[1 - Sec[c + d*x]]/Sqrt[2]]*(1 + Sec[c + d*x]))*Tan[c + d*x])/(4*d*Sqrt[1 - Sec[c + d*x]]*(a*(1 + Sec[c + d*x]))^(3/2))","A",1
133,1,197,185,3.3730383,"\int \frac{\cos ^2(c+d x)}{(a+a \sec (c+d x))^{3/2}} \, dx","Integrate[Cos[c + d*x]^2/(a + a*Sec[c + d*x])^(3/2),x]","-\frac{\sin (2 (c+d x))-\frac{(\cos (c+d x)+1) \tan (c+d x) \sec (c+d x) \left(13 \left(2 \cos ^2(c+d x) \sqrt{1-\sec (c+d x)}-\cos (c+d x) \sqrt{1-\sec (c+d x)}+7 \tanh ^{-1}\left(\sqrt{1-\sec (c+d x)}\right)-4 \sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{1-\sec (c+d x)}}{\sqrt{2}}\right)\right)-40 \sqrt{1-\sec (c+d x)} \, _2F_1\left(\frac{1}{2},3;\frac{3}{2};1-\sec (c+d x)\right)\right)}{4 \sqrt{1-\sec (c+d x)}}}{4 d (a (\sec (c+d x)+1))^{3/2}}","\frac{19 \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{4 a^{3/2} d}-\frac{13 \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}-\frac{7 \sin (c+d x)}{4 a d \sqrt{a \sec (c+d x)+a}}+\frac{\sin (c+d x) \cos (c+d x)}{a d \sqrt{a \sec (c+d x)+a}}-\frac{\sin (c+d x) \cos (c+d x)}{2 d (a \sec (c+d x)+a)^{3/2}}",1,"-1/4*(Sin[2*(c + d*x)] - ((1 + Cos[c + d*x])*(13*(7*ArcTanh[Sqrt[1 - Sec[c + d*x]]] - 4*Sqrt[2]*ArcTanh[Sqrt[1 - Sec[c + d*x]]/Sqrt[2]] - Cos[c + d*x]*Sqrt[1 - Sec[c + d*x]] + 2*Cos[c + d*x]^2*Sqrt[1 - Sec[c + d*x]]) - 40*Hypergeometric2F1[1/2, 3, 3/2, 1 - Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]])*Sec[c + d*x]*Tan[c + d*x])/(4*Sqrt[1 - Sec[c + d*x]]))/(d*(a*(1 + Sec[c + d*x]))^(3/2))","C",1
134,1,135,183,1.5462047,"\int \frac{\sec ^5(c+d x)}{(a+a \sec (c+d x))^{5/2}} \, dx","Integrate[Sec[c + d*x]^5/(a + a*Sec[c + d*x])^(5/2),x]","\frac{\tan (c+d x) \left(\sqrt{1-\sec (c+d x)} \left(32 \sec ^3(c+d x)-160 \sec ^2(c+d x)-503 \sec (c+d x)-299\right)+978 \sqrt{2} \cos ^4\left(\frac{1}{2} (c+d x)\right) \sec ^2(c+d x) \tanh ^{-1}\left(\frac{\sqrt{1-\sec (c+d x)}}{\sqrt{2}}\right)\right)}{48 d \sqrt{1-\sec (c+d x)} (a (\sec (c+d x)+1))^{5/2}}","\frac{163 \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}+\frac{95 \tan (c+d x) \sqrt{a \sec (c+d x)+a}}{48 a^3 d}-\frac{197 \tan (c+d x)}{24 a^2 d \sqrt{a \sec (c+d x)+a}}-\frac{\tan (c+d x) \sec ^3(c+d x)}{4 d (a \sec (c+d x)+a)^{5/2}}-\frac{17 \tan (c+d x) \sec ^2(c+d x)}{16 a d (a \sec (c+d x)+a)^{3/2}}",1,"((978*Sqrt[2]*ArcTanh[Sqrt[1 - Sec[c + d*x]]/Sqrt[2]]*Cos[(c + d*x)/2]^4*Sec[c + d*x]^2 + Sqrt[1 - Sec[c + d*x]]*(-299 - 503*Sec[c + d*x] - 160*Sec[c + d*x]^2 + 32*Sec[c + d*x]^3))*Tan[c + d*x])/(48*d*Sqrt[1 - Sec[c + d*x]]*(a*(1 + Sec[c + d*x]))^(5/2))","A",1
135,1,125,145,0.8483794,"\int \frac{\sec ^4(c+d x)}{(a+a \sec (c+d x))^{5/2}} \, dx","Integrate[Sec[c + d*x]^4/(a + a*Sec[c + d*x])^(5/2),x]","\frac{\tan (c+d x) \left(\sqrt{1-\sec (c+d x)} \left(32 \sec ^2(c+d x)+85 \sec (c+d x)+49\right)-150 \sqrt{2} \cos ^4\left(\frac{1}{2} (c+d x)\right) \sec ^2(c+d x) \tanh ^{-1}\left(\frac{\sqrt{1-\sec (c+d x)}}{\sqrt{2}}\right)\right)}{16 d \sqrt{1-\sec (c+d x)} (a (\sec (c+d x)+1))^{5/2}}","-\frac{75 \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}+\frac{9 \tan (c+d x)}{4 a^2 d \sqrt{a \sec (c+d x)+a}}-\frac{\tan (c+d x) \sec ^2(c+d x)}{4 d (a \sec (c+d x)+a)^{5/2}}+\frac{13 \tan (c+d x)}{16 a d (a \sec (c+d x)+a)^{3/2}}",1,"((-150*Sqrt[2]*ArcTanh[Sqrt[1 - Sec[c + d*x]]/Sqrt[2]]*Cos[(c + d*x)/2]^4*Sec[c + d*x]^2 + Sqrt[1 - Sec[c + d*x]]*(49 + 85*Sec[c + d*x] + 32*Sec[c + d*x]^2))*Tan[c + d*x])/(16*d*Sqrt[1 - Sec[c + d*x]]*(a*(1 + Sec[c + d*x]))^(5/2))","A",1
136,1,116,107,0.7767712,"\int \frac{\sec ^3(c+d x)}{(a+a \sec (c+d x))^{5/2}} \, dx","Integrate[Sec[c + d*x]^3/(a + a*Sec[c + d*x])^(5/2),x]","\frac{\tan (c+d x) \left(76 \sqrt{2} \cos ^4\left(\frac{1}{2} (c+d x)\right) \sec ^2(c+d x) \tanh ^{-1}\left(\frac{\sqrt{1-\sec (c+d x)}}{\sqrt{2}}\right)-2 \sqrt{1-\sec (c+d x)} (13 \sec (c+d x)+9)\right)}{32 d \sqrt{1-\sec (c+d x)} (a (\sec (c+d x)+1))^{5/2}}","\frac{19 \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}-\frac{13 \tan (c+d x)}{16 a d (a \sec (c+d x)+a)^{3/2}}+\frac{\tan (c+d x)}{4 d (a \sec (c+d x)+a)^{5/2}}",1,"((76*Sqrt[2]*ArcTanh[Sqrt[1 - Sec[c + d*x]]/Sqrt[2]]*Cos[(c + d*x)/2]^4*Sec[c + d*x]^2 - 2*Sqrt[1 - Sec[c + d*x]]*(9 + 13*Sec[c + d*x]))*Tan[c + d*x])/(32*d*Sqrt[1 - Sec[c + d*x]]*(a*(1 + Sec[c + d*x]))^(5/2))","A",1
137,1,115,107,0.7422797,"\int \frac{\sec ^2(c+d x)}{(a+a \sec (c+d x))^{5/2}} \, dx","Integrate[Sec[c + d*x]^2/(a + a*Sec[c + d*x])^(5/2),x]","\frac{\tan (c+d x) \left(\sqrt{1-\sec (c+d x)} (5 \sec (c+d x)+1)+10 \sqrt{2} \cos ^4\left(\frac{1}{2} (c+d x)\right) \sec ^2(c+d x) \tanh ^{-1}\left(\frac{\sqrt{1-\sec (c+d x)}}{\sqrt{2}}\right)\right)}{16 d \sqrt{1-\sec (c+d x)} (a (\sec (c+d x)+1))^{5/2}}","\frac{5 \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}+\frac{5 \tan (c+d x)}{16 a d (a \sec (c+d x)+a)^{3/2}}-\frac{\tan (c+d x)}{4 d (a \sec (c+d x)+a)^{5/2}}",1,"((10*Sqrt[2]*ArcTanh[Sqrt[1 - Sec[c + d*x]]/Sqrt[2]]*Cos[(c + d*x)/2]^4*Sec[c + d*x]^2 + Sqrt[1 - Sec[c + d*x]]*(1 + 5*Sec[c + d*x]))*Tan[c + d*x])/(16*d*Sqrt[1 - Sec[c + d*x]]*(a*(1 + Sec[c + d*x]))^(5/2))","A",1
138,1,52,107,0.0650687,"\int \frac{\sec (c+d x)}{(a+a \sec (c+d x))^{5/2}} \, dx","Integrate[Sec[c + d*x]/(a + a*Sec[c + d*x])^(5/2),x]","\frac{\tan (c+d x) \, _2F_1\left(\frac{1}{2},3;\frac{3}{2};\frac{1}{2} (1-\sec (c+d x))\right)}{4 a^2 d \sqrt{a (\sec (c+d x)+1)}}","\frac{3 \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}+\frac{3 \tan (c+d x)}{16 a d (a \sec (c+d x)+a)^{3/2}}+\frac{\tan (c+d x)}{4 d (a \sec (c+d x)+a)^{5/2}}",1,"(Hypergeometric2F1[1/2, 3, 3/2, (1 - Sec[c + d*x])/2]*Tan[c + d*x])/(4*a^2*d*Sqrt[a*(1 + Sec[c + d*x])])","C",1
139,1,5564,144,25.2019109,"\int \frac{1}{(a+a \sec (c+d x))^{5/2}} \, dx","Integrate[(a + a*Sec[c + d*x])^(-5/2),x]","\text{Result too large to show}","\frac{2 \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{a^{5/2} d}-\frac{43 \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}-\frac{11 \tan (c+d x)}{16 a d (a \sec (c+d x)+a)^{3/2}}-\frac{\tan (c+d x)}{4 d (a \sec (c+d x)+a)^{5/2}}",1,"Result too large to show","C",0
140,1,169,174,2.2308181,"\int \frac{\cos (c+d x)}{(a+a \sec (c+d x))^{5/2}} \, dx","Integrate[Cos[c + d*x]/(a + a*Sec[c + d*x])^(5/2),x]","\frac{\sqrt{1-\sec (c+d x)} (16 \sin (c+d x)+5 \tan (c+d x) (7 \sec (c+d x)+11))-80 \tan (c+d x) (\sec (c+d x)+1)^2 \tanh ^{-1}\left(\sqrt{1-\sec (c+d x)}\right)+460 \sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right) \cos ^5\left(\frac{1}{2} (c+d x)\right) \sec ^3(c+d x) \tanh ^{-1}\left(\frac{\sqrt{1-\sec (c+d x)}}{\sqrt{2}}\right)}{16 d \sqrt{1-\sec (c+d x)} (a (\sec (c+d x)+1))^{5/2}}","-\frac{5 \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{a^{5/2} d}+\frac{115 \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}+\frac{35 \sin (c+d x)}{16 a^2 d \sqrt{a \sec (c+d x)+a}}-\frac{15 \sin (c+d x)}{16 a d (a \sec (c+d x)+a)^{3/2}}-\frac{\sin (c+d x)}{4 d (a \sec (c+d x)+a)^{5/2}}",1,"(460*Sqrt[2]*ArcTanh[Sqrt[1 - Sec[c + d*x]]/Sqrt[2]]*Cos[(c + d*x)/2]^5*Sec[c + d*x]^3*Sin[(c + d*x)/2] - 80*ArcTanh[Sqrt[1 - Sec[c + d*x]]]*(1 + Sec[c + d*x])^2*Tan[c + d*x] + Sqrt[1 - Sec[c + d*x]]*(16*Sin[c + d*x] + 5*(11 + 7*Sec[c + d*x])*Tan[c + d*x]))/(16*d*Sqrt[1 - Sec[c + d*x]]*(a*(1 + Sec[c + d*x]))^(5/2))","A",1
141,1,94,48,0.4608275,"\int \frac{\sec (c+d x)}{\sqrt{a-a \sec (c+d x)}} \, dx","Integrate[Sec[c + d*x]/Sqrt[a - a*Sec[c + d*x]],x]","\frac{i \sqrt{2} \left(-1+e^{i (c+d x)}\right) \tanh ^{-1}\left(\frac{1+e^{i (c+d x)}}{\sqrt{2} \sqrt{1+e^{2 i (c+d x)}}}\right)}{d \sqrt{1+e^{2 i (c+d x)}} \sqrt{a-a \sec (c+d x)}}","-\frac{\sqrt{2} \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a-a \sec (c+d x)}}\right)}{\sqrt{a} d}",1,"(I*Sqrt[2]*(-1 + E^(I*(c + d*x)))*ArcTanh[(1 + E^(I*(c + d*x)))/(Sqrt[2]*Sqrt[1 + E^((2*I)*(c + d*x))])])/(d*Sqrt[1 + E^((2*I)*(c + d*x))]*Sqrt[a - a*Sec[c + d*x]])","C",1
142,1,127,87,0.5398773,"\int \frac{1}{\sqrt{a-a \sec (c+d x)}} \, dx","Integrate[1/Sqrt[a - a*Sec[c + d*x]],x]","-\frac{i \left(-1+e^{i (c+d x)}\right) \left(\sinh ^{-1}\left(e^{i (c+d x)}\right)-\sqrt{2} \tanh ^{-1}\left(\frac{1+e^{i (c+d x)}}{\sqrt{2} \sqrt{1+e^{2 i (c+d x)}}}\right)+\tanh ^{-1}\left(\sqrt{1+e^{2 i (c+d x)}}\right)\right)}{d \sqrt{1+e^{2 i (c+d x)}} \sqrt{a-a \sec (c+d x)}}","\frac{2 \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a-a \sec (c+d x)}}\right)}{\sqrt{a} d}-\frac{\sqrt{2} \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a-a \sec (c+d x)}}\right)}{\sqrt{a} d}",1,"((-I)*(-1 + E^(I*(c + d*x)))*(ArcSinh[E^(I*(c + d*x))] - Sqrt[2]*ArcTanh[(1 + E^(I*(c + d*x)))/(Sqrt[2]*Sqrt[1 + E^((2*I)*(c + d*x))])] + ArcTanh[Sqrt[1 + E^((2*I)*(c + d*x))]]))/(d*Sqrt[1 + E^((2*I)*(c + d*x))]*Sqrt[a - a*Sec[c + d*x]])","C",1
143,1,105,383,0.3398884,"\int \sec ^3(c+d x) (a+a \sec (c+d x))^{2/3} \, dx","Integrate[Sec[c + d*x]^3*(a + a*Sec[c + d*x])^(2/3),x]","\frac{\tan (c+d x) (a (\sec (c+d x)+1))^{2/3} \left(38 \sqrt[6]{2} \, _2F_1\left(-\frac{1}{6},\frac{1}{2};\frac{3}{2};\frac{1}{2} (1-\sec (c+d x))\right)+3 \sqrt[6]{\sec (c+d x)+1} \left(5 \sec ^2(c+d x)+7 \sec (c+d x)+2\right)\right)}{40 d (\sec (c+d x)+1)^{7/6}}","\frac{3 \tan (c+d x) (a \sec (c+d x)+a)^{5/3}}{8 a d}-\frac{9 \tan (c+d x) (a \sec (c+d x)+a)^{2/3}}{40 d}+\frac{57 \tan (c+d x) (a \sec (c+d x)+a)^{2/3}}{80 d (\sec (c+d x)+1)}-\frac{19\ 3^{3/4} \tan (c+d x) \left(\sqrt[3]{2}-\sqrt[3]{\sec (c+d x)+1}\right) \sqrt{\frac{(\sec (c+d x)+1)^{2/3}+\sqrt[3]{2} \sqrt[3]{\sec (c+d x)+1}+2^{2/3}}{\left(\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}\right)^2}} (a \sec (c+d x)+a)^{2/3} F\left(\cos ^{-1}\left(\frac{\sqrt[3]{2}-\left(1-\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}}{\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}}\right)|\frac{1}{4} \left(2+\sqrt{3}\right)\right)}{80 \sqrt[3]{2} d (1-\sec (c+d x)) (\sec (c+d x)+1) \sqrt{-\frac{\sqrt[3]{\sec (c+d x)+1} \left(\sqrt[3]{2}-\sqrt[3]{\sec (c+d x)+1}\right)}{\left(\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}\right)^2}}}",1,"((a*(1 + Sec[c + d*x]))^(2/3)*(38*2^(1/6)*Hypergeometric2F1[-1/6, 1/2, 3/2, (1 - Sec[c + d*x])/2] + 3*(1 + Sec[c + d*x])^(1/6)*(2 + 7*Sec[c + d*x] + 5*Sec[c + d*x]^2))*Tan[c + d*x])/(40*d*(1 + Sec[c + d*x])^(7/6))","C",1
144,1,85,353,0.1497319,"\int \sec ^2(c+d x) (a+a \sec (c+d x))^{2/3} \, dx","Integrate[Sec[c + d*x]^2*(a + a*Sec[c + d*x])^(2/3),x]","\frac{\tan (c+d x) (a (\sec (c+d x)+1))^{2/3} \left(4 \sqrt[6]{2} \, _2F_1\left(-\frac{1}{6},\frac{1}{2};\frac{3}{2};\frac{1}{2} (1-\sec (c+d x))\right)+3 (\sec (c+d x)+1)^{7/6}\right)}{5 d (\sec (c+d x)+1)^{7/6}}","\frac{3 \tan (c+d x) (a \sec (c+d x)+a)^{2/3}}{5 d}+\frac{3 \tan (c+d x) (a \sec (c+d x)+a)^{2/3}}{5 d (\sec (c+d x)+1)}-\frac{3^{3/4} \tan (c+d x) \left(\sqrt[3]{2}-\sqrt[3]{\sec (c+d x)+1}\right) \sqrt{\frac{(\sec (c+d x)+1)^{2/3}+\sqrt[3]{2} \sqrt[3]{\sec (c+d x)+1}+2^{2/3}}{\left(\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}\right)^2}} (a \sec (c+d x)+a)^{2/3} F\left(\cos ^{-1}\left(\frac{\sqrt[3]{2}-\left(1-\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}}{\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}}\right)|\frac{1}{4} \left(2+\sqrt{3}\right)\right)}{5 \sqrt[3]{2} d (1-\sec (c+d x)) (\sec (c+d x)+1) \sqrt{-\frac{\sqrt[3]{\sec (c+d x)+1} \left(\sqrt[3]{2}-\sqrt[3]{\sec (c+d x)+1}\right)}{\left(\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}\right)^2}}}",1,"((a*(1 + Sec[c + d*x]))^(2/3)*(4*2^(1/6)*Hypergeometric2F1[-1/6, 1/2, 3/2, (1 - Sec[c + d*x])/2] + 3*(1 + Sec[c + d*x])^(7/6))*Tan[c + d*x])/(5*d*(1 + Sec[c + d*x])^(7/6))","C",1
145,1,66,326,0.0613127,"\int \sec (c+d x) (a+a \sec (c+d x))^{2/3} \, dx","Integrate[Sec[c + d*x]*(a + a*Sec[c + d*x])^(2/3),x]","\frac{2 \sqrt[6]{2} \tan (c+d x) (a (\sec (c+d x)+1))^{2/3} \, _2F_1\left(-\frac{1}{6},\frac{1}{2};\frac{3}{2};\frac{1}{2} (1-\sec (c+d x))\right)}{d (\sec (c+d x)+1)^{7/6}}","\frac{3 \tan (c+d x) (a \sec (c+d x)+a)^{2/3}}{2 d (\sec (c+d x)+1)}-\frac{3^{3/4} \tan (c+d x) \left(\sqrt[3]{2}-\sqrt[3]{\sec (c+d x)+1}\right) \sqrt{\frac{(\sec (c+d x)+1)^{2/3}+\sqrt[3]{2} \sqrt[3]{\sec (c+d x)+1}+2^{2/3}}{\left(\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}\right)^2}} (a \sec (c+d x)+a)^{2/3} F\left(\cos ^{-1}\left(\frac{\sqrt[3]{2}-\left(1-\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}}{\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}}\right)|\frac{1}{4} \left(2+\sqrt{3}\right)\right)}{2 \sqrt[3]{2} d (1-\sec (c+d x)) (\sec (c+d x)+1) \sqrt{-\frac{\sqrt[3]{\sec (c+d x)+1} \left(\sqrt[3]{2}-\sqrt[3]{\sec (c+d x)+1}\right)}{\left(\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}\right)^2}}}",1,"(2*2^(1/6)*Hypergeometric2F1[-1/6, 1/2, 3/2, (1 - Sec[c + d*x])/2]*(a*(1 + Sec[c + d*x]))^(2/3)*Tan[c + d*x])/(d*(1 + Sec[c + d*x])^(7/6))","C",1
146,1,694,77,8.9756195,"\int (a+a \sec (c+d x))^{2/3} \, dx","Integrate[(a + a*Sec[c + d*x])^(2/3),x]","\frac{45 \tan (c+d x) (a (\sec (c+d x)+1))^{5/3} F_1\left(\frac{1}{2};\frac{2}{3},1;\frac{3}{2};\tan ^2\left(\frac{1}{2} (c+d x)\right),-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right) \left(2 \tan ^2\left(\frac{1}{2} (c+d x)\right) \left(2 F_1\left(\frac{3}{2};\frac{5}{3},1;\frac{5}{2};\tan ^2\left(\frac{1}{2} (c+d x)\right),-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right)-3 F_1\left(\frac{3}{2};\frac{2}{3},2;\frac{5}{2};\tan ^2\left(\frac{1}{2} (c+d x)\right),-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right)\right)+9 F_1\left(\frac{1}{2};\frac{2}{3},1;\frac{3}{2};\tan ^2\left(\frac{1}{2} (c+d x)\right),-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right)\right)}{a d \left(135 F_1\left(\frac{1}{2};\frac{2}{3},1;\frac{3}{2};\tan ^2\left(\frac{1}{2} (c+d x)\right),-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right){}^2 \left(\left(2 \tan ^2(c+d x)+3\right) \sec (c+d x)+3\right)+40 \sin ^2\left(\frac{1}{2} (c+d x)\right) \tan ^2\left(\frac{1}{2} (c+d x)\right) \sec ^2(c+d x) \left(3 F_1\left(\frac{3}{2};\frac{2}{3},2;\frac{5}{2};\tan ^2\left(\frac{1}{2} (c+d x)\right),-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right)-2 F_1\left(\frac{3}{2};\frac{5}{3},1;\frac{5}{2};\tan ^2\left(\frac{1}{2} (c+d x)\right),-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right)\right){}^2-6 \sin ^2\left(\frac{1}{2} (c+d x)\right) \sec ^3(c+d x) F_1\left(\frac{1}{2};\frac{2}{3},1;\frac{3}{2};\tan ^2\left(\frac{1}{2} (c+d x)\right),-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right) \left(24 \cos (c+d x) \tan ^2\left(\frac{1}{2} (c+d x)\right) \left(9 F_1\left(\frac{5}{2};\frac{2}{3},3;\frac{7}{2};\tan ^2\left(\frac{1}{2} (c+d x)\right),-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right)-6 F_1\left(\frac{5}{2};\frac{5}{3},2;\frac{7}{2};\tan ^2\left(\frac{1}{2} (c+d x)\right),-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right)+5 F_1\left(\frac{5}{2};\frac{8}{3},1;\frac{7}{2};\tan ^2\left(\frac{1}{2} (c+d x)\right),-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right)\right)+10 (16 \cos (c+d x)-3 \cos (2 (c+d x))-7) F_1\left(\frac{3}{2};\frac{5}{3},1;\frac{5}{2};\tan ^2\left(\frac{1}{2} (c+d x)\right),-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right)+15 (-16 \cos (c+d x)+3 \cos (2 (c+d x))+7) F_1\left(\frac{3}{2};\frac{2}{3},2;\frac{5}{2};\tan ^2\left(\frac{1}{2} (c+d x)\right),-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right)\right)\right)}","\frac{3 \sqrt{2} \tan (c+d x) (a \sec (c+d x)+a)^{2/3} F_1\left(\frac{7}{6};\frac{1}{2},1;\frac{13}{6};\frac{1}{2} (\sec (c+d x)+1),\sec (c+d x)+1\right)}{7 d \sqrt{1-\sec (c+d x)}}",1,"(45*AppellF1[1/2, 2/3, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*(a*(1 + Sec[c + d*x]))^(5/3)*(9*AppellF1[1/2, 2/3, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + 2*(-3*AppellF1[3/2, 2/3, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + 2*AppellF1[3/2, 5/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Tan[(c + d*x)/2]^2)*Tan[c + d*x])/(a*d*(40*(3*AppellF1[3/2, 2/3, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] - 2*AppellF1[3/2, 5/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])^2*Sec[c + d*x]^2*Sin[(c + d*x)/2]^2*Tan[(c + d*x)/2]^2 - 6*AppellF1[1/2, 2/3, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[c + d*x]^3*Sin[(c + d*x)/2]^2*(10*AppellF1[3/2, 5/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*(-7 + 16*Cos[c + d*x] - 3*Cos[2*(c + d*x)]) + 15*AppellF1[3/2, 2/3, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*(7 - 16*Cos[c + d*x] + 3*Cos[2*(c + d*x)]) + 24*(9*AppellF1[5/2, 2/3, 3, 7/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] - 6*AppellF1[5/2, 5/3, 2, 7/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + 5*AppellF1[5/2, 8/3, 1, 7/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Cos[c + d*x]*Tan[(c + d*x)/2]^2) + 135*AppellF1[1/2, 2/3, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]^2*(3 + Sec[c + d*x]*(3 + 2*Tan[c + d*x]^2))))","B",0
147,1,2700,77,16.3629471,"\int \cos (c+d x) (a+a \sec (c+d x))^{2/3} \, dx","Integrate[Cos[c + d*x]*(a + a*Sec[c + d*x])^(2/3),x]","\text{Result too large to show}","-\frac{3 \sqrt{2} \tan (c+d x) (a \sec (c+d x)+a)^{2/3} F_1\left(\frac{7}{6};\frac{1}{2},2;\frac{13}{6};\frac{1}{2} (\sec (c+d x)+1),\sec (c+d x)+1\right)}{7 d \sqrt{1-\sec (c+d x)}}",1,"(((1 + Cos[c + d*x])*Sec[c + d*x])^(2/3)*(a*(1 + Sec[c + d*x]))^(2/3)*(Sin[c + d*x] - Tan[(c + d*x)/2]))/(d*(1 + Sec[c + d*x])^(2/3)) - (2^(2/3)*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^(2/3)*(a*(1 + Sec[c + d*x]))^(2/3)*((Sec[(c + d*x)/2]^2*(1 + Sec[c + d*x])^(2/3))/6 + (Cos[c + d*x]*Sec[(c + d*x)/2]^2*(1 + Sec[c + d*x])^(2/3))/3)*Tan[(c + d*x)/2]*(AppellF1[3/2, 2/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(2/3)*Tan[(c + d*x)/2]^2 + (81*AppellF1[1/2, 2/3, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Cos[(c + d*x)/2]^2)/(-9*AppellF1[1/2, 2/3, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + 2*(3*AppellF1[3/2, 2/3, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] - 2*AppellF1[3/2, 5/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Tan[(c + d*x)/2]^2)))/(9*d*(1 + Sec[c + d*x])^(2/3)*(-1/9*(Sec[(c + d*x)/2]^2*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^(2/3)*(AppellF1[3/2, 2/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(2/3)*Tan[(c + d*x)/2]^2 + (81*AppellF1[1/2, 2/3, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Cos[(c + d*x)/2]^2)/(-9*AppellF1[1/2, 2/3, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + 2*(3*AppellF1[3/2, 2/3, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] - 2*AppellF1[3/2, 5/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Tan[(c + d*x)/2]^2)))/2^(1/3) - (2^(2/3)*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^(2/3)*Tan[(c + d*x)/2]*(AppellF1[3/2, 2/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(2/3)*Tan[(c + d*x)/2] + (Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(2/3)*Tan[(c + d*x)/2]^2*((-3*AppellF1[5/2, 2/3, 2, 7/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/5 + (2*AppellF1[5/2, 5/3, 1, 7/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/5) + (2*AppellF1[3/2, 2/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Tan[(c + d*x)/2]^2*(-(Sec[(c + d*x)/2]^2*Sin[c + d*x]) + Cos[c + d*x]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(3*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(1/3)) - (81*AppellF1[1/2, 2/3, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Cos[(c + d*x)/2]*Sin[(c + d*x)/2])/(-9*AppellF1[1/2, 2/3, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + 2*(3*AppellF1[3/2, 2/3, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] - 2*AppellF1[3/2, 5/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Tan[(c + d*x)/2]^2) + (81*Cos[(c + d*x)/2]^2*(-1/3*(AppellF1[3/2, 2/3, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]) + (2*AppellF1[3/2, 5/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/9))/(-9*AppellF1[1/2, 2/3, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + 2*(3*AppellF1[3/2, 2/3, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] - 2*AppellF1[3/2, 5/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Tan[(c + d*x)/2]^2) - (81*AppellF1[1/2, 2/3, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Cos[(c + d*x)/2]^2*(2*(3*AppellF1[3/2, 2/3, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] - 2*AppellF1[3/2, 5/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2] - 9*(-1/3*(AppellF1[3/2, 2/3, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]) + (2*AppellF1[3/2, 5/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/9) + 2*Tan[(c + d*x)/2]^2*(3*((-6*AppellF1[5/2, 2/3, 3, 7/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/5 + (2*AppellF1[5/2, 5/3, 2, 7/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/5) - 2*((-3*AppellF1[5/2, 5/3, 2, 7/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/5 + AppellF1[5/2, 8/3, 1, 7/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))))/(-9*AppellF1[1/2, 2/3, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + 2*(3*AppellF1[3/2, 2/3, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] - 2*AppellF1[3/2, 5/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Tan[(c + d*x)/2]^2)^2))/9 - (2*2^(2/3)*Tan[(c + d*x)/2]*(AppellF1[3/2, 2/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(2/3)*Tan[(c + d*x)/2]^2 + (81*AppellF1[1/2, 2/3, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Cos[(c + d*x)/2]^2)/(-9*AppellF1[1/2, 2/3, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + 2*(3*AppellF1[3/2, 2/3, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] - 2*AppellF1[3/2, 5/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Tan[(c + d*x)/2]^2))*(-(Cos[(c + d*x)/2]*Sec[c + d*x]*Sin[(c + d*x)/2]) + Cos[(c + d*x)/2]^2*Sec[c + d*x]*Tan[c + d*x]))/(27*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^(1/3))))","B",0
148,1,96,413,0.3351058,"\int \sec ^3(c+d x) (a+a \sec (c+d x))^{5/3} \, dx","Integrate[Sec[c + d*x]^3*(a + a*Sec[c + d*x])^(5/3),x]","\frac{a \tan (c+d x) (a (\sec (c+d x)+1))^{2/3} \left(196 \sqrt[6]{2} \, _2F_1\left(-\frac{7}{6},\frac{1}{2};\frac{3}{2};\frac{1}{2} (1-\sec (c+d x))\right)+3 (8 \sec (c+d x)+5) (\sec (c+d x)+1)^{13/6}\right)}{88 d (\sec (c+d x)+1)^{7/6}}","\frac{3 \tan (c+d x) (a \sec (c+d x)+a)^{8/3}}{11 a d}-\frac{9 \tan (c+d x) (a \sec (c+d x)+a)^{5/3}}{88 d}+\frac{147 a \tan (c+d x) (a \sec (c+d x)+a)^{2/3}}{440 d}+\frac{1029 a \tan (c+d x) (a \sec (c+d x)+a)^{2/3}}{880 d (\sec (c+d x)+1)}-\frac{343\ 3^{3/4} a \tan (c+d x) \left(\sqrt[3]{2}-\sqrt[3]{\sec (c+d x)+1}\right) \sqrt{\frac{(\sec (c+d x)+1)^{2/3}+\sqrt[3]{2} \sqrt[3]{\sec (c+d x)+1}+2^{2/3}}{\left(\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}\right)^2}} (a \sec (c+d x)+a)^{2/3} F\left(\cos ^{-1}\left(\frac{\sqrt[3]{2}-\left(1-\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}}{\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}}\right)|\frac{1}{4} \left(2+\sqrt{3}\right)\right)}{880 \sqrt[3]{2} d (1-\sec (c+d x)) (\sec (c+d x)+1) \sqrt{-\frac{\sqrt[3]{\sec (c+d x)+1} \left(\sqrt[3]{2}-\sqrt[3]{\sec (c+d x)+1}\right)}{\left(\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}\right)^2}}}",1,"(a*(a*(1 + Sec[c + d*x]))^(2/3)*(196*2^(1/6)*Hypergeometric2F1[-7/6, 1/2, 3/2, (1 - Sec[c + d*x])/2] + 3*(1 + Sec[c + d*x])^(13/6)*(5 + 8*Sec[c + d*x]))*Tan[c + d*x])/(88*d*(1 + Sec[c + d*x])^(7/6))","C",1
149,1,106,383,0.4505954,"\int \sec ^2(c+d x) (a+a \sec (c+d x))^{5/3} \, dx","Integrate[Sec[c + d*x]^2*(a + a*Sec[c + d*x])^(5/3),x]","\frac{a \tan (c+d x) (a (\sec (c+d x)+1))^{2/3} \left(5 \sqrt[6]{2} \, _2F_1\left(-\frac{7}{6},\frac{1}{2};\frac{3}{2};\frac{1}{2} (1-\sec (c+d x))\right)+3 \cos ^4\left(\frac{1}{2} (c+d x)\right) \sec ^2(c+d x) \sqrt[6]{\sec (c+d x)+1}\right)}{2 d (\sec (c+d x)+1)^{7/6}}","\frac{3 \tan (c+d x) (a \sec (c+d x)+a)^{5/3}}{8 d}+\frac{3 a \tan (c+d x) (a \sec (c+d x)+a)^{2/3}}{8 d}+\frac{21 a \tan (c+d x) (a \sec (c+d x)+a)^{2/3}}{16 d (\sec (c+d x)+1)}-\frac{7\ 3^{3/4} a \tan (c+d x) \left(\sqrt[3]{2}-\sqrt[3]{\sec (c+d x)+1}\right) \sqrt{\frac{(\sec (c+d x)+1)^{2/3}+\sqrt[3]{2} \sqrt[3]{\sec (c+d x)+1}+2^{2/3}}{\left(\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}\right)^2}} (a \sec (c+d x)+a)^{2/3} F\left(\cos ^{-1}\left(\frac{\sqrt[3]{2}-\left(1-\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}}{\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}}\right)|\frac{1}{4} \left(2+\sqrt{3}\right)\right)}{16 \sqrt[3]{2} d (1-\sec (c+d x)) (\sec (c+d x)+1) \sqrt{-\frac{\sqrt[3]{\sec (c+d x)+1} \left(\sqrt[3]{2}-\sqrt[3]{\sec (c+d x)+1}\right)}{\left(\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}\right)^2}}}",1,"(a*(a*(1 + Sec[c + d*x]))^(2/3)*(5*2^(1/6)*Hypergeometric2F1[-7/6, 1/2, 3/2, (1 - Sec[c + d*x])/2] + 3*Cos[(c + d*x)/2]^4*Sec[c + d*x]^2*(1 + Sec[c + d*x])^(1/6))*Tan[c + d*x])/(2*d*(1 + Sec[c + d*x])^(7/6))","C",1
150,1,66,356,0.0853966,"\int \sec (c+d x) (a+a \sec (c+d x))^{5/3} \, dx","Integrate[Sec[c + d*x]*(a + a*Sec[c + d*x])^(5/3),x]","\frac{4 \sqrt[6]{2} \tan (c+d x) (a (\sec (c+d x)+1))^{5/3} \, _2F_1\left(-\frac{7}{6},\frac{1}{2};\frac{3}{2};\frac{1}{2} (1-\sec (c+d x))\right)}{d (\sec (c+d x)+1)^{13/6}}","\frac{3 a \tan (c+d x) (a \sec (c+d x)+a)^{2/3}}{5 d}+\frac{21 a \tan (c+d x) (a \sec (c+d x)+a)^{2/3}}{10 d (\sec (c+d x)+1)}-\frac{7\ 3^{3/4} a \tan (c+d x) \left(\sqrt[3]{2}-\sqrt[3]{\sec (c+d x)+1}\right) \sqrt{\frac{(\sec (c+d x)+1)^{2/3}+\sqrt[3]{2} \sqrt[3]{\sec (c+d x)+1}+2^{2/3}}{\left(\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}\right)^2}} (a \sec (c+d x)+a)^{2/3} F\left(\cos ^{-1}\left(\frac{\sqrt[3]{2}-\left(1-\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}}{\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}}\right)|\frac{1}{4} \left(2+\sqrt{3}\right)\right)}{10 \sqrt[3]{2} d (1-\sec (c+d x)) (\sec (c+d x)+1) \sqrt{-\frac{\sqrt[3]{\sec (c+d x)+1} \left(\sqrt[3]{2}-\sqrt[3]{\sec (c+d x)+1}\right)}{\left(\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}\right)^2}}}",1,"(4*2^(1/6)*Hypergeometric2F1[-7/6, 1/2, 3/2, (1 - Sec[c + d*x])/2]*(a*(1 + Sec[c + d*x]))^(5/3)*Tan[c + d*x])/(d*(1 + Sec[c + d*x])^(13/6))","C",1
151,1,2694,86,16.0888856,"\int (a+a \sec (c+d x))^{5/3} \, dx","Integrate[(a + a*Sec[c + d*x])^(5/3),x]","\text{Result too large to show}","\frac{3 \sqrt{2} a \tan (c+d x) (\sec (c+d x)+1) (a \sec (c+d x)+a)^{2/3} F_1\left(\frac{13}{6};\frac{1}{2},1;\frac{19}{6};\frac{1}{2} (\sec (c+d x)+1),\sec (c+d x)+1\right)}{13 d \sqrt{1-\sec (c+d x)}}",1,"(3*((1 + Cos[c + d*x])*Sec[c + d*x])^(2/3)*(a*(1 + Sec[c + d*x]))^(5/3)*Tan[(c + d*x)/2])/(2*d*(1 + Sec[c + d*x])^(5/3)) + ((Cos[(c + d*x)/2]^2*Sec[c + d*x])^(2/3)*(a*(1 + Sec[c + d*x]))^(5/3)*((3*Sec[(c + d*x)/2]^2*(1 + Sec[c + d*x])^(2/3))/4 + (Cos[c + d*x]*Sec[(c + d*x)/2]^2*(1 + Sec[c + d*x])^(2/3))/2)*Tan[(c + d*x)/2]*(AppellF1[3/2, 2/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(2/3)*Tan[(c + d*x)/2]^2 + (135*AppellF1[1/2, 2/3, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Cos[(c + d*x)/2]^2)/(9*AppellF1[1/2, 2/3, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + 2*(-3*AppellF1[3/2, 2/3, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + 2*AppellF1[3/2, 5/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Tan[(c + d*x)/2]^2)))/(3*2^(1/3)*d*(1 + Sec[c + d*x])^(5/3)*((Sec[(c + d*x)/2]^2*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^(2/3)*(AppellF1[3/2, 2/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(2/3)*Tan[(c + d*x)/2]^2 + (135*AppellF1[1/2, 2/3, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Cos[(c + d*x)/2]^2)/(9*AppellF1[1/2, 2/3, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + 2*(-3*AppellF1[3/2, 2/3, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + 2*AppellF1[3/2, 5/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Tan[(c + d*x)/2]^2)))/(6*2^(1/3)) + ((Cos[(c + d*x)/2]^2*Sec[c + d*x])^(2/3)*Tan[(c + d*x)/2]*(AppellF1[3/2, 2/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(2/3)*Tan[(c + d*x)/2] + (Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(2/3)*Tan[(c + d*x)/2]^2*((-3*AppellF1[5/2, 2/3, 2, 7/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/5 + (2*AppellF1[5/2, 5/3, 1, 7/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/5) + (2*AppellF1[3/2, 2/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Tan[(c + d*x)/2]^2*(-(Sec[(c + d*x)/2]^2*Sin[c + d*x]) + Cos[c + d*x]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(3*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(1/3)) - (135*AppellF1[1/2, 2/3, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Cos[(c + d*x)/2]*Sin[(c + d*x)/2])/(9*AppellF1[1/2, 2/3, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + 2*(-3*AppellF1[3/2, 2/3, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + 2*AppellF1[3/2, 5/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Tan[(c + d*x)/2]^2) + (135*Cos[(c + d*x)/2]^2*(-1/3*(AppellF1[3/2, 2/3, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]) + (2*AppellF1[3/2, 5/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/9))/(9*AppellF1[1/2, 2/3, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + 2*(-3*AppellF1[3/2, 2/3, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + 2*AppellF1[3/2, 5/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Tan[(c + d*x)/2]^2) - (135*AppellF1[1/2, 2/3, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Cos[(c + d*x)/2]^2*(2*(-3*AppellF1[3/2, 2/3, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + 2*AppellF1[3/2, 5/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2] + 9*(-1/3*(AppellF1[3/2, 2/3, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]) + (2*AppellF1[3/2, 5/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/9) + 2*Tan[(c + d*x)/2]^2*(-3*((-6*AppellF1[5/2, 2/3, 3, 7/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/5 + (2*AppellF1[5/2, 5/3, 2, 7/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/5) + 2*((-3*AppellF1[5/2, 5/3, 2, 7/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/5 + AppellF1[5/2, 8/3, 1, 7/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))))/(9*AppellF1[1/2, 2/3, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + 2*(-3*AppellF1[3/2, 2/3, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + 2*AppellF1[3/2, 5/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Tan[(c + d*x)/2]^2)^2))/(3*2^(1/3)) + (2^(2/3)*Tan[(c + d*x)/2]*(AppellF1[3/2, 2/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(2/3)*Tan[(c + d*x)/2]^2 + (135*AppellF1[1/2, 2/3, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Cos[(c + d*x)/2]^2)/(9*AppellF1[1/2, 2/3, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + 2*(-3*AppellF1[3/2, 2/3, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + 2*AppellF1[3/2, 5/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Tan[(c + d*x)/2]^2))*(-(Cos[(c + d*x)/2]*Sec[c + d*x]*Sin[(c + d*x)/2]) + Cos[(c + d*x)/2]^2*Sec[c + d*x]*Tan[c + d*x]))/(9*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^(1/3))))","B",0
152,1,2700,86,16.3338463,"\int \cos (c+d x) (a+a \sec (c+d x))^{5/3} \, dx","Integrate[Cos[c + d*x]*(a + a*Sec[c + d*x])^(5/3),x]","\text{Result too large to show}","-\frac{3 \sqrt{2} a \tan (c+d x) (\sec (c+d x)+1) (a \sec (c+d x)+a)^{2/3} F_1\left(\frac{13}{6};\frac{1}{2},2;\frac{19}{6};\frac{1}{2} (\sec (c+d x)+1),\sec (c+d x)+1\right)}{13 d \sqrt{1-\sec (c+d x)}}",1,"(((1 + Cos[c + d*x])*Sec[c + d*x])^(2/3)*(a*(1 + Sec[c + d*x]))^(5/3)*(Sin[c + d*x] - Tan[(c + d*x)/2]))/(d*(1 + Sec[c + d*x])^(5/3)) - (2^(2/3)*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^(2/3)*(a*(1 + Sec[c + d*x]))^(5/3)*((2*Sec[(c + d*x)/2]^2*(1 + Sec[c + d*x])^(2/3))/3 + (5*Cos[c + d*x]*Sec[(c + d*x)/2]^2*(1 + Sec[c + d*x])^(2/3))/6)*Tan[(c + d*x)/2]*(AppellF1[3/2, 2/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(2/3)*Tan[(c + d*x)/2]^2 + (243*AppellF1[1/2, 2/3, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Cos[(c + d*x)/2]^2)/(-9*AppellF1[1/2, 2/3, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + 2*(3*AppellF1[3/2, 2/3, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] - 2*AppellF1[3/2, 5/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Tan[(c + d*x)/2]^2)))/(9*d*(1 + Sec[c + d*x])^(5/3)*(-1/9*(Sec[(c + d*x)/2]^2*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^(2/3)*(AppellF1[3/2, 2/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(2/3)*Tan[(c + d*x)/2]^2 + (243*AppellF1[1/2, 2/3, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Cos[(c + d*x)/2]^2)/(-9*AppellF1[1/2, 2/3, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + 2*(3*AppellF1[3/2, 2/3, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] - 2*AppellF1[3/2, 5/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Tan[(c + d*x)/2]^2)))/2^(1/3) - (2^(2/3)*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^(2/3)*Tan[(c + d*x)/2]*(AppellF1[3/2, 2/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(2/3)*Tan[(c + d*x)/2] + (Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(2/3)*Tan[(c + d*x)/2]^2*((-3*AppellF1[5/2, 2/3, 2, 7/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/5 + (2*AppellF1[5/2, 5/3, 1, 7/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/5) + (2*AppellF1[3/2, 2/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Tan[(c + d*x)/2]^2*(-(Sec[(c + d*x)/2]^2*Sin[c + d*x]) + Cos[c + d*x]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(3*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(1/3)) - (243*AppellF1[1/2, 2/3, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Cos[(c + d*x)/2]*Sin[(c + d*x)/2])/(-9*AppellF1[1/2, 2/3, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + 2*(3*AppellF1[3/2, 2/3, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] - 2*AppellF1[3/2, 5/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Tan[(c + d*x)/2]^2) + (243*Cos[(c + d*x)/2]^2*(-1/3*(AppellF1[3/2, 2/3, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]) + (2*AppellF1[3/2, 5/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/9))/(-9*AppellF1[1/2, 2/3, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + 2*(3*AppellF1[3/2, 2/3, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] - 2*AppellF1[3/2, 5/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Tan[(c + d*x)/2]^2) - (243*AppellF1[1/2, 2/3, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Cos[(c + d*x)/2]^2*(2*(3*AppellF1[3/2, 2/3, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] - 2*AppellF1[3/2, 5/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2] - 9*(-1/3*(AppellF1[3/2, 2/3, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]) + (2*AppellF1[3/2, 5/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/9) + 2*Tan[(c + d*x)/2]^2*(3*((-6*AppellF1[5/2, 2/3, 3, 7/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/5 + (2*AppellF1[5/2, 5/3, 2, 7/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/5) - 2*((-3*AppellF1[5/2, 5/3, 2, 7/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/5 + AppellF1[5/2, 8/3, 1, 7/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))))/(-9*AppellF1[1/2, 2/3, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + 2*(3*AppellF1[3/2, 2/3, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] - 2*AppellF1[3/2, 5/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Tan[(c + d*x)/2]^2)^2))/9 - (2*2^(2/3)*Tan[(c + d*x)/2]*(AppellF1[3/2, 2/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(2/3)*Tan[(c + d*x)/2]^2 + (243*AppellF1[1/2, 2/3, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Cos[(c + d*x)/2]^2)/(-9*AppellF1[1/2, 2/3, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + 2*(3*AppellF1[3/2, 2/3, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] - 2*AppellF1[3/2, 5/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Tan[(c + d*x)/2]^2))*(-(Cos[(c + d*x)/2]*Sec[c + d*x]*Sin[(c + d*x)/2]) + Cos[(c + d*x)/2]^2*Sec[c + d*x]*Tan[c + d*x]))/(27*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^(1/3))))","B",0
153,1,155,371,0.4067777,"\int \frac{\sec ^4(c+d x)}{\sqrt[3]{a+a \sec (c+d x)}} \, dx","Integrate[Sec[c + d*x]^4/(a + a*Sec[c + d*x])^(1/3),x]","\frac{\tan (c+d x) \left(-4 \sqrt[6]{2} \, _2F_1\left(-\frac{7}{6},\frac{1}{2};\frac{3}{2};\frac{1}{2} (1-\sec (c+d x))\right)+16 \sqrt[6]{2} \, _2F_1\left(-\frac{1}{6},\frac{1}{2};\frac{3}{2};\frac{1}{2} (1-\sec (c+d x))\right)-7 \sqrt[6]{2} \, _2F_1\left(\frac{1}{2},\frac{5}{6};\frac{3}{2};\frac{1}{2} (1-\sec (c+d x))\right)+3 \sqrt[6]{\sec (c+d x)+1} \sec ^2(c+d x)\right)}{8 d \sqrt[6]{\sec (c+d x)+1} \sqrt[3]{a (\sec (c+d x)+1)}}","\frac{3 \tan (c+d x) \sec ^2(c+d x)}{8 d \sqrt[3]{a \sec (c+d x)+a}}-\frac{3 \tan (c+d x) (a \sec (c+d x)+a)^{2/3}}{40 a d}+\frac{99 \tan (c+d x)}{80 d \sqrt[3]{a \sec (c+d x)+a}}+\frac{37\ 3^{3/4} \tan (c+d x) \left(\sqrt[3]{2}-\sqrt[3]{\sec (c+d x)+1}\right) \sqrt{\frac{(\sec (c+d x)+1)^{2/3}+\sqrt[3]{2} \sqrt[3]{\sec (c+d x)+1}+2^{2/3}}{\left(\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}\right)^2}} F\left(\cos ^{-1}\left(\frac{\sqrt[3]{2}-\left(1-\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}}{\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}}\right)|\frac{1}{4} \left(2+\sqrt{3}\right)\right)}{80 \sqrt[3]{2} d (1-\sec (c+d x)) \sqrt{-\frac{\sqrt[3]{\sec (c+d x)+1} \left(\sqrt[3]{2}-\sqrt[3]{\sec (c+d x)+1}\right)}{\left(\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}\right)^2}} \sqrt[3]{a \sec (c+d x)+a}}",1,"((-4*2^(1/6)*Hypergeometric2F1[-7/6, 1/2, 3/2, (1 - Sec[c + d*x])/2] + 16*2^(1/6)*Hypergeometric2F1[-1/6, 1/2, 3/2, (1 - Sec[c + d*x])/2] - 7*2^(1/6)*Hypergeometric2F1[1/2, 5/6, 3/2, (1 - Sec[c + d*x])/2] + 3*Sec[c + d*x]^2*(1 + Sec[c + d*x])^(1/6))*Tan[c + d*x])/(8*d*(1 + Sec[c + d*x])^(1/6)*(a*(1 + Sec[c + d*x]))^(1/3))","C",1
154,1,95,336,0.1827265,"\int \frac{\sec ^3(c+d x)}{\sqrt[3]{a+a \sec (c+d x)}} \, dx","Integrate[Sec[c + d*x]^3/(a + a*Sec[c + d*x])^(1/3),x]","\frac{\tan (c+d x) \left(7 \sqrt[6]{2} \, _2F_1\left(\frac{1}{2},\frac{5}{6};\frac{3}{2};\frac{1}{2} (1-\sec (c+d x))\right)+3 \sqrt[6]{\sec (c+d x)+1} (2 \sec (c+d x)-1)\right)}{10 d \sqrt[6]{\sec (c+d x)+1} \sqrt[3]{a (\sec (c+d x)+1)}}","\frac{3 \tan (c+d x) (a \sec (c+d x)+a)^{2/3}}{5 a d}-\frac{9 \tan (c+d x)}{10 d \sqrt[3]{a \sec (c+d x)+a}}-\frac{7\ 3^{3/4} \tan (c+d x) \left(\sqrt[3]{2}-\sqrt[3]{\sec (c+d x)+1}\right) \sqrt{\frac{(\sec (c+d x)+1)^{2/3}+\sqrt[3]{2} \sqrt[3]{\sec (c+d x)+1}+2^{2/3}}{\left(\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}\right)^2}} F\left(\cos ^{-1}\left(\frac{\sqrt[3]{2}-\left(1-\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}}{\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}}\right)|\frac{1}{4} \left(2+\sqrt{3}\right)\right)}{10 \sqrt[3]{2} d (1-\sec (c+d x)) \sqrt{-\frac{\sqrt[3]{\sec (c+d x)+1} \left(\sqrt[3]{2}-\sqrt[3]{\sec (c+d x)+1}\right)}{\left(\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}\right)^2}} \sqrt[3]{a \sec (c+d x)+a}}",1,"((7*2^(1/6)*Hypergeometric2F1[1/2, 5/6, 3/2, (1 - Sec[c + d*x])/2] + 3*(1 + Sec[c + d*x])^(1/6)*(-1 + 2*Sec[c + d*x]))*Tan[c + d*x])/(10*d*(1 + Sec[c + d*x])^(1/6)*(a*(1 + Sec[c + d*x]))^(1/3))","C",1
155,1,85,306,0.1261389,"\int \frac{\sec ^2(c+d x)}{\sqrt[3]{a+a \sec (c+d x)}} \, dx","Integrate[Sec[c + d*x]^2/(a + a*Sec[c + d*x])^(1/3),x]","\frac{\tan (c+d x) \left(3 \sqrt[6]{\sec (c+d x)+1}-\sqrt[6]{2} \, _2F_1\left(\frac{1}{2},\frac{5}{6};\frac{3}{2};\frac{1}{2} (1-\sec (c+d x))\right)\right)}{2 d \sqrt[6]{\sec (c+d x)+1} \sqrt[3]{a (\sec (c+d x)+1)}}","\frac{3 \tan (c+d x)}{2 d \sqrt[3]{a \sec (c+d x)+a}}+\frac{3^{3/4} \tan (c+d x) \left(\sqrt[3]{2}-\sqrt[3]{\sec (c+d x)+1}\right) \sqrt{\frac{(\sec (c+d x)+1)^{2/3}+\sqrt[3]{2} \sqrt[3]{\sec (c+d x)+1}+2^{2/3}}{\left(\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}\right)^2}} F\left(\cos ^{-1}\left(\frac{\sqrt[3]{2}-\left(1-\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}}{\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}}\right)|\frac{1}{4} \left(2+\sqrt{3}\right)\right)}{2 \sqrt[3]{2} d (1-\sec (c+d x)) \sqrt{-\frac{\sqrt[3]{\sec (c+d x)+1} \left(\sqrt[3]{2}-\sqrt[3]{\sec (c+d x)+1}\right)}{\left(\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}\right)^2}} \sqrt[3]{a \sec (c+d x)+a}}",1,"((-(2^(1/6)*Hypergeometric2F1[1/2, 5/6, 3/2, (1 - Sec[c + d*x])/2]) + 3*(1 + Sec[c + d*x])^(1/6))*Tan[c + d*x])/(2*d*(1 + Sec[c + d*x])^(1/6)*(a*(1 + Sec[c + d*x]))^(1/3))","C",1
156,1,65,276,0.0826911,"\int \frac{\sec (c+d x)}{\sqrt[3]{a+a \sec (c+d x)}} \, dx","Integrate[Sec[c + d*x]/(a + a*Sec[c + d*x])^(1/3),x]","\frac{\sqrt[6]{2} \tan (c+d x) \, _2F_1\left(\frac{1}{2},\frac{5}{6};\frac{3}{2};\frac{1}{2} (1-\sec (c+d x))\right)}{d \sqrt[6]{\sec (c+d x)+1} \sqrt[3]{a (\sec (c+d x)+1)}}","-\frac{3^{3/4} \tan (c+d x) \left(\sqrt[3]{2}-\sqrt[3]{\sec (c+d x)+1}\right) \sqrt{\frac{(\sec (c+d x)+1)^{2/3}+\sqrt[3]{2} \sqrt[3]{\sec (c+d x)+1}+2^{2/3}}{\left(\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}\right)^2}} F\left(\cos ^{-1}\left(\frac{\sqrt[3]{2}-\left(1-\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}}{\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}}\right)|\frac{1}{4} \left(2+\sqrt{3}\right)\right)}{\sqrt[3]{2} d (1-\sec (c+d x)) \sqrt{-\frac{\sqrt[3]{\sec (c+d x)+1} \left(\sqrt[3]{2}-\sqrt[3]{\sec (c+d x)+1}\right)}{\left(\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}\right)^2}} \sqrt[3]{a \sec (c+d x)+a}}",1,"(2^(1/6)*Hypergeometric2F1[1/2, 5/6, 3/2, (1 - Sec[c + d*x])/2]*Tan[c + d*x])/(d*(1 + Sec[c + d*x])^(1/6)*(a*(1 + Sec[c + d*x]))^(1/3))","C",1
157,1,718,75,5.3307265,"\int \frac{1}{\sqrt[3]{a+a \sec (c+d x)}} \, dx","Integrate[(a + a*Sec[c + d*x])^(-1/3),x]","\frac{45 \cos (c+d x) \tan \left(\frac{1}{2} (c+d x)\right) (\sec (c+d x)+1)^2 F_1\left(\frac{1}{2};-\frac{1}{3},1;\frac{3}{2};\tan ^2\left(\frac{1}{2} (c+d x)\right),-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right) \left(9 F_1\left(\frac{1}{2};-\frac{1}{3},1;\frac{3}{2};\tan ^2\left(\frac{1}{2} (c+d x)\right),-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right)-2 \tan ^2\left(\frac{1}{2} (c+d x)\right) \left(3 F_1\left(\frac{3}{2};-\frac{1}{3},2;\frac{5}{2};\tan ^2\left(\frac{1}{2} (c+d x)\right),-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right)+F_1\left(\frac{3}{2};\frac{2}{3},1;\frac{5}{2};\tan ^2\left(\frac{1}{2} (c+d x)\right),-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right)\right)\right)}{d \sqrt[3]{a (\sec (c+d x)+1)} \left(40 \sin ^2\left(\frac{1}{2} (c+d x)\right) \tan ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x) \left(3 F_1\left(\frac{3}{2};-\frac{1}{3},2;\frac{5}{2};\tan ^2\left(\frac{1}{2} (c+d x)\right),-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right)+F_1\left(\frac{3}{2};\frac{2}{3},1;\frac{5}{2};\tan ^2\left(\frac{1}{2} (c+d x)\right),-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right)\right){}^2+135 F_1\left(\frac{1}{2};-\frac{1}{3},1;\frac{3}{2};\tan ^2\left(\frac{1}{2} (c+d x)\right),-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right){}^2 \left(-\tan ^2(c+d x)+3 \sec (c+d x)-3 \sin (c+d x) \tan (c+d x)+3\right)+6 \sin ^2\left(\frac{1}{2} (c+d x)\right) \sec ^2(c+d x) F_1\left(\frac{1}{2};-\frac{1}{3},1;\frac{3}{2};\tan ^2\left(\frac{1}{2} (c+d x)\right),-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right) \left(-24 \cos (c+d x) \tan ^2\left(\frac{1}{2} (c+d x)\right) \left(9 F_1\left(\frac{5}{2};-\frac{1}{3},3;\frac{7}{2};\tan ^2\left(\frac{1}{2} (c+d x)\right),-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right)+3 F_1\left(\frac{5}{2};\frac{2}{3},2;\frac{7}{2};\tan ^2\left(\frac{1}{2} (c+d x)\right),-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right)-F_1\left(\frac{5}{2};\frac{5}{3},1;\frac{7}{2};\tan ^2\left(\frac{1}{2} (c+d x)\right),-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right)\right)-15 (-10 \cos (c+d x)+3 \cos (2 (c+d x))+1) F_1\left(\frac{3}{2};-\frac{1}{3},2;\frac{5}{2};\tan ^2\left(\frac{1}{2} (c+d x)\right),-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right)-5 (-10 \cos (c+d x)+3 \cos (2 (c+d x))+1) F_1\left(\frac{3}{2};\frac{2}{3},1;\frac{5}{2};\tan ^2\left(\frac{1}{2} (c+d x)\right),-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right)\right)\right)}","\frac{3 \sqrt{2} \tan (c+d x) F_1\left(\frac{1}{6};\frac{1}{2},1;\frac{7}{6};\frac{1}{2} (\sec (c+d x)+1),\sec (c+d x)+1\right)}{d \sqrt{1-\sec (c+d x)} \sqrt[3]{a \sec (c+d x)+a}}",1,"(45*AppellF1[1/2, -1/3, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Cos[c + d*x]*(1 + Sec[c + d*x])^2*Tan[(c + d*x)/2]*(9*AppellF1[1/2, -1/3, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] - 2*(3*AppellF1[3/2, -1/3, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + AppellF1[3/2, 2/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Tan[(c + d*x)/2]^2))/(d*(a*(1 + Sec[c + d*x]))^(1/3)*(40*(3*AppellF1[3/2, -1/3, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + AppellF1[3/2, 2/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])^2*Sec[c + d*x]*Sin[(c + d*x)/2]^2*Tan[(c + d*x)/2]^2 + 6*AppellF1[1/2, -1/3, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[c + d*x]^2*Sin[(c + d*x)/2]^2*(-15*AppellF1[3/2, -1/3, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*(1 - 10*Cos[c + d*x] + 3*Cos[2*(c + d*x)]) - 5*AppellF1[3/2, 2/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*(1 - 10*Cos[c + d*x] + 3*Cos[2*(c + d*x)]) - 24*(9*AppellF1[5/2, -1/3, 3, 7/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + 3*AppellF1[5/2, 2/3, 2, 7/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] - AppellF1[5/2, 5/3, 1, 7/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Cos[c + d*x]*Tan[(c + d*x)/2]^2) + 135*AppellF1[1/2, -1/3, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]^2*(3 + 3*Sec[c + d*x] - 3*Sin[c + d*x]*Tan[c + d*x] - Tan[c + d*x]^2)))","B",0
158,1,240,75,2.0410791,"\int \frac{\cos (c+d x)}{\sqrt[3]{a+a \sec (c+d x)}} \, dx","Integrate[Cos[c + d*x]/(a + a*Sec[c + d*x])^(1/3),x]","\frac{(a (\sec (c+d x)+1))^{2/3} \left(\frac{20 \sin ^3\left(\frac{1}{2} (c+d x)\right) \cos \left(\frac{1}{2} (c+d x)\right) F_1\left(\frac{3}{2};\frac{2}{3},1;\frac{5}{2};\tan ^2\left(\frac{1}{2} (c+d x)\right),-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right)}{6 (\cos (c+d x)-1) \left(3 F_1\left(\frac{5}{2};\frac{2}{3},2;\frac{7}{2};\tan ^2\left(\frac{1}{2} (c+d x)\right),-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right)-2 F_1\left(\frac{5}{2};\frac{5}{3},1;\frac{7}{2};\tan ^2\left(\frac{1}{2} (c+d x)\right),-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right)\right)+45 (\cos (c+d x)+1) F_1\left(\frac{3}{2};\frac{2}{3},1;\frac{5}{2};\tan ^2\left(\frac{1}{2} (c+d x)\right),-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right)}+\sin (c+d x)-\tan \left(\frac{1}{2} (c+d x)\right)\right)}{a d}","-\frac{3 \sqrt{2} \tan (c+d x) F_1\left(\frac{1}{6};\frac{1}{2},2;\frac{7}{6};\frac{1}{2} (\sec (c+d x)+1),\sec (c+d x)+1\right)}{d \sqrt{1-\sec (c+d x)} \sqrt[3]{a \sec (c+d x)+a}}",1,"((a*(1 + Sec[c + d*x]))^(2/3)*((20*AppellF1[3/2, 2/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Cos[(c + d*x)/2]*Sin[(c + d*x)/2]^3)/(6*(3*AppellF1[5/2, 2/3, 2, 7/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] - 2*AppellF1[5/2, 5/3, 1, 7/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*(-1 + Cos[c + d*x]) + 45*AppellF1[3/2, 2/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*(1 + Cos[c + d*x])) + Sin[c + d*x] - Tan[(c + d*x)/2]))/(a*d)","B",0
159,1,111,766,0.5606226,"\int \frac{\sec ^4(c+d x)}{(a+a \sec (c+d x))^{5/3}} \, dx","Integrate[Sec[c + d*x]^4/(a + a*Sec[c + d*x])^(5/3),x]","\frac{\tan (c+d x) \left(3 \left(7 \sec ^2(c+d x)+90 \sec (c+d x)+79\right)-250\ 2^{5/6} \cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x) \sqrt[6]{\sec (c+d x)+1} \, _2F_1\left(\frac{1}{6},\frac{1}{2};\frac{3}{2};\frac{1}{2} (1-\sec (c+d x))\right)\right)}{28 d (a (\sec (c+d x)+1))^{5/3}}","\frac{375 \left(1+\sqrt{3}\right) \tan (c+d x) \sqrt[3]{a \sec (c+d x)+a}}{28 a^2 d (\sec (c+d x)+1)^{2/3} \left(\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}\right)}-\frac{125\ 3^{3/4} \left(1-\sqrt{3}\right) \tan (c+d x) \left(\sqrt[3]{2}-\sqrt[3]{\sec (c+d x)+1}\right) \sqrt{\frac{(\sec (c+d x)+1)^{2/3}+\sqrt[3]{2} \sqrt[3]{\sec (c+d x)+1}+2^{2/3}}{\left(\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}\right)^2}} \sqrt[3]{a \sec (c+d x)+a} F\left(\cos ^{-1}\left(\frac{\sqrt[3]{2}-\left(1-\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}}{\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}}\right)|\frac{1}{4} \left(2+\sqrt{3}\right)\right)}{28\ 2^{2/3} a^2 d (1-\sec (c+d x)) (\sec (c+d x)+1)^{2/3} \sqrt{-\frac{\sqrt[3]{\sec (c+d x)+1} \left(\sqrt[3]{2}-\sqrt[3]{\sec (c+d x)+1}\right)}{\left(\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}\right)^2}}}-\frac{375 \sqrt[4]{3} \tan (c+d x) \left(\sqrt[3]{2}-\sqrt[3]{\sec (c+d x)+1}\right) \sqrt{\frac{(\sec (c+d x)+1)^{2/3}+\sqrt[3]{2} \sqrt[3]{\sec (c+d x)+1}+2^{2/3}}{\left(\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}\right)^2}} \sqrt[3]{a \sec (c+d x)+a} E\left(\cos ^{-1}\left(\frac{\sqrt[3]{2}-\left(1-\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}}{\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}}\right)|\frac{1}{4} \left(2+\sqrt{3}\right)\right)}{14\ 2^{2/3} a^2 d (1-\sec (c+d x)) (\sec (c+d x)+1)^{2/3} \sqrt{-\frac{\sqrt[3]{\sec (c+d x)+1} \left(\sqrt[3]{2}-\sqrt[3]{\sec (c+d x)+1}\right)}{\left(\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}\right)^2}}}+\frac{3 \tan (c+d x) \sec ^2(c+d x)}{4 d (a \sec (c+d x)+a)^{5/3}}+\frac{135 \tan (c+d x)}{14 a d (a \sec (c+d x)+a)^{2/3}}-\frac{33 \tan (c+d x)}{28 d (a \sec (c+d x)+a)^{5/3}}",1,"((-250*2^(5/6)*Cos[(c + d*x)/2]^2*Hypergeometric2F1[1/6, 1/2, 3/2, (1 - Sec[c + d*x])/2]*Sec[c + d*x]*(1 + Sec[c + d*x])^(1/6) + 3*(79 + 90*Sec[c + d*x] + 7*Sec[c + d*x]^2))*Tan[c + d*x])/(28*d*(a*(1 + Sec[c + d*x]))^(5/3))","C",1
160,1,98,731,0.2819603,"\int \frac{\sec ^3(c+d x)}{(a+a \sec (c+d x))^{5/3}} \, dx","Integrate[Sec[c + d*x]^3/(a + a*Sec[c + d*x])^(5/3),x]","\frac{\tan (c+d x) \left(38\ 2^{5/6} \cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x) \sqrt[6]{\sec (c+d x)+1} \, _2F_1\left(\frac{1}{6},\frac{1}{2};\frac{3}{2};\frac{1}{2} (1-\sec (c+d x))\right)-36 \sec (c+d x)-33\right)}{7 d (a (\sec (c+d x)+1))^{5/3}}","-\frac{57 \left(1+\sqrt{3}\right) \tan (c+d x) \sqrt[3]{a \sec (c+d x)+a}}{7 a^2 d (\sec (c+d x)+1)^{2/3} \left(\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}\right)}+\frac{19\ 3^{3/4} \left(1-\sqrt{3}\right) \tan (c+d x) \left(\sqrt[3]{2}-\sqrt[3]{\sec (c+d x)+1}\right) \sqrt{\frac{(\sec (c+d x)+1)^{2/3}+\sqrt[3]{2} \sqrt[3]{\sec (c+d x)+1}+2^{2/3}}{\left(\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}\right)^2}} \sqrt[3]{a \sec (c+d x)+a} F\left(\cos ^{-1}\left(\frac{\sqrt[3]{2}-\left(1-\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}}{\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}}\right)|\frac{1}{4} \left(2+\sqrt{3}\right)\right)}{7\ 2^{2/3} a^2 d (1-\sec (c+d x)) (\sec (c+d x)+1)^{2/3} \sqrt{-\frac{\sqrt[3]{\sec (c+d x)+1} \left(\sqrt[3]{2}-\sqrt[3]{\sec (c+d x)+1}\right)}{\left(\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}\right)^2}}}+\frac{57 \sqrt[3]{2} \sqrt[4]{3} \tan (c+d x) \left(\sqrt[3]{2}-\sqrt[3]{\sec (c+d x)+1}\right) \sqrt{\frac{(\sec (c+d x)+1)^{2/3}+\sqrt[3]{2} \sqrt[3]{\sec (c+d x)+1}+2^{2/3}}{\left(\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}\right)^2}} \sqrt[3]{a \sec (c+d x)+a} E\left(\cos ^{-1}\left(\frac{\sqrt[3]{2}-\left(1-\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}}{\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}}\right)|\frac{1}{4} \left(2+\sqrt{3}\right)\right)}{7 a^2 d (1-\sec (c+d x)) (\sec (c+d x)+1)^{2/3} \sqrt{-\frac{\sqrt[3]{\sec (c+d x)+1} \left(\sqrt[3]{2}-\sqrt[3]{\sec (c+d x)+1}\right)}{\left(\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}\right)^2}}}-\frac{36 \tan (c+d x)}{7 a d (a \sec (c+d x)+a)^{2/3}}+\frac{3 \tan (c+d x)}{7 d (a \sec (c+d x)+a)^{5/3}}",1,"((-33 - 36*Sec[c + d*x] + 38*2^(5/6)*Cos[(c + d*x)/2]^2*Hypergeometric2F1[1/6, 1/2, 3/2, (1 - Sec[c + d*x])/2]*Sec[c + d*x]*(1 + Sec[c + d*x])^(1/6))*Tan[c + d*x])/(7*d*(a*(1 + Sec[c + d*x]))^(5/3))","C",1
161,1,90,731,0.358372,"\int \frac{\sec ^2(c+d x)}{(a+a \sec (c+d x))^{5/3}} \, dx","Integrate[Sec[c + d*x]^2/(a + a*Sec[c + d*x])^(5/3),x]","\frac{\tan (c+d x) \left(5\ 2^{5/6} \cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x) \sqrt[6]{\sec (c+d x)+1} \, _2F_1\left(\frac{1}{2},\frac{7}{6};\frac{3}{2};\frac{1}{2} (1-\sec (c+d x))\right)-3\right)}{7 d (a (\sec (c+d x)+1))^{5/3}}","\frac{15 \tan (c+d x)}{7 a d (a \sec (c+d x)+a)^{2/3}}+\frac{15 \left(1+\sqrt{3}\right) \tan (c+d x) \sqrt[3]{\sec (c+d x)+1}}{7 a d \left(\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}\right) (a \sec (c+d x)+a)^{2/3}}-\frac{3 \tan (c+d x)}{7 d (a \sec (c+d x)+a)^{5/3}}-\frac{5\ 3^{3/4} \left(1-\sqrt{3}\right) \tan (c+d x) \sqrt[3]{\sec (c+d x)+1} \left(\sqrt[3]{2}-\sqrt[3]{\sec (c+d x)+1}\right) \sqrt{\frac{(\sec (c+d x)+1)^{2/3}+\sqrt[3]{2} \sqrt[3]{\sec (c+d x)+1}+2^{2/3}}{\left(\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}\right)^2}} F\left(\cos ^{-1}\left(\frac{\sqrt[3]{2}-\left(1-\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}}{\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}}\right)|\frac{1}{4} \left(2+\sqrt{3}\right)\right)}{7\ 2^{2/3} a d (1-\sec (c+d x)) \sqrt{-\frac{\sqrt[3]{\sec (c+d x)+1} \left(\sqrt[3]{2}-\sqrt[3]{\sec (c+d x)+1}\right)}{\left(\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}\right)^2}} (a \sec (c+d x)+a)^{2/3}}-\frac{15 \sqrt[3]{2} \sqrt[4]{3} \tan (c+d x) \sqrt[3]{\sec (c+d x)+1} \left(\sqrt[3]{2}-\sqrt[3]{\sec (c+d x)+1}\right) \sqrt{\frac{(\sec (c+d x)+1)^{2/3}+\sqrt[3]{2} \sqrt[3]{\sec (c+d x)+1}+2^{2/3}}{\left(\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}\right)^2}} E\left(\cos ^{-1}\left(\frac{\sqrt[3]{2}-\left(1-\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}}{\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}}\right)|\frac{1}{4} \left(2+\sqrt{3}\right)\right)}{7 a d (1-\sec (c+d x)) \sqrt{-\frac{\sqrt[3]{\sec (c+d x)+1} \left(\sqrt[3]{2}-\sqrt[3]{\sec (c+d x)+1}\right)}{\left(\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}\right)^2}} (a \sec (c+d x)+a)^{2/3}}",1,"((-3 + 5*2^(5/6)*Cos[(c + d*x)/2]^2*Hypergeometric2F1[1/2, 7/6, 3/2, (1 - Sec[c + d*x])/2]*Sec[c + d*x]*(1 + Sec[c + d*x])^(1/6))*Tan[c + d*x])/(7*d*(a*(1 + Sec[c + d*x]))^(5/3))","C",1
162,1,68,744,0.0755971,"\int \frac{\sec (c+d x)}{(a+a \sec (c+d x))^{5/3}} \, dx","Integrate[Sec[c + d*x]/(a + a*Sec[c + d*x])^(5/3),x]","\frac{\tan (c+d x) (\sec (c+d x)+1)^{7/6} \, _2F_1\left(\frac{1}{2},\frac{13}{6};\frac{3}{2};\frac{1}{2} (1-\sec (c+d x))\right)}{2 \sqrt[6]{2} d (a (\sec (c+d x)+1))^{5/3}}","\frac{6 \tan (c+d x)}{7 a d (a \sec (c+d x)+a)^{2/3}}+\frac{3 \tan (c+d x)}{7 a d (\sec (c+d x)+1) (a \sec (c+d x)+a)^{2/3}}+\frac{6 \left(1+\sqrt{3}\right) \tan (c+d x) \sqrt[3]{\sec (c+d x)+1}}{7 a d \left(\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}\right) (a \sec (c+d x)+a)^{2/3}}-\frac{\sqrt[3]{2} 3^{3/4} \left(1-\sqrt{3}\right) \tan (c+d x) \sqrt[3]{\sec (c+d x)+1} \left(\sqrt[3]{2}-\sqrt[3]{\sec (c+d x)+1}\right) \sqrt{\frac{(\sec (c+d x)+1)^{2/3}+\sqrt[3]{2} \sqrt[3]{\sec (c+d x)+1}+2^{2/3}}{\left(\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}\right)^2}} F\left(\cos ^{-1}\left(\frac{\sqrt[3]{2}-\left(1-\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}}{\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}}\right)|\frac{1}{4} \left(2+\sqrt{3}\right)\right)}{7 a d (1-\sec (c+d x)) \sqrt{-\frac{\sqrt[3]{\sec (c+d x)+1} \left(\sqrt[3]{2}-\sqrt[3]{\sec (c+d x)+1}\right)}{\left(\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}\right)^2}} (a \sec (c+d x)+a)^{2/3}}-\frac{6 \sqrt[3]{2} \sqrt[4]{3} \tan (c+d x) \sqrt[3]{\sec (c+d x)+1} \left(\sqrt[3]{2}-\sqrt[3]{\sec (c+d x)+1}\right) \sqrt{\frac{(\sec (c+d x)+1)^{2/3}+\sqrt[3]{2} \sqrt[3]{\sec (c+d x)+1}+2^{2/3}}{\left(\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}\right)^2}} E\left(\cos ^{-1}\left(\frac{\sqrt[3]{2}-\left(1-\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}}{\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}}\right)|\frac{1}{4} \left(2+\sqrt{3}\right)\right)}{7 a d (1-\sec (c+d x)) \sqrt{-\frac{\sqrt[3]{\sec (c+d x)+1} \left(\sqrt[3]{2}-\sqrt[3]{\sec (c+d x)+1}\right)}{\left(\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}\right)^2}} (a \sec (c+d x)+a)^{2/3}}",1,"(Hypergeometric2F1[1/2, 13/6, 3/2, (1 - Sec[c + d*x])/2]*(1 + Sec[c + d*x])^(7/6)*Tan[c + d*x])/(2*2^(1/6)*d*(a*(1 + Sec[c + d*x]))^(5/3))","C",1
163,1,3007,90,16.7188002,"\int \frac{1}{(a+a \sec (c+d x))^{5/3}} \, dx","Integrate[(a + a*Sec[c + d*x])^(-5/3),x]","\text{Result too large to show}","-\frac{3 \sqrt{2} \tan (c+d x) F_1\left(-\frac{7}{6};\frac{1}{2},1;-\frac{1}{6};\frac{1}{2} (\sec (c+d x)+1),\sec (c+d x)+1\right)}{7 a d \sqrt{1-\sec (c+d x)} (\sec (c+d x)+1) (a \sec (c+d x)+a)^{2/3}}",1,"(((1 + Cos[c + d*x])*Sec[c + d*x])^(1/3)*(1 + Sec[c + d*x])^(5/3)*((27*Sin[c + d*x])/7 - (30*Tan[(c + d*x)/2])/7 + (3*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/14))/(d*(a*(1 + Sec[c + d*x]))^(5/3)) + (2^(1/3)*(1 + Sec[c + d*x])^(5/3)*((16*(1 + Sec[c + d*x])^(1/3))/7 - (27*Cos[c + d*x]*(1 + Sec[c + d*x])^(1/3))/7)*Tan[(c + d*x)/2]*((-3*AppellF1[3/2, 1/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Tan[(c + d*x)/2]^2)/(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(2/3) + Cos[(c + d*x)/2]^2*(-27 - (5*AppellF1[1/2, 1/3, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])/((-1 + Tan[(c + d*x)/2]^2)*(AppellF1[1/2, 1/3, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + (2*(-3*AppellF1[3/2, 1/3, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + AppellF1[3/2, 4/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Tan[(c + d*x)/2]^2)/9)))))/(7*d*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^(2/3)*(a*(1 + Sec[c + d*x]))^(5/3)*((Sec[(c + d*x)/2]^2*((-3*AppellF1[3/2, 1/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Tan[(c + d*x)/2]^2)/(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(2/3) + Cos[(c + d*x)/2]^2*(-27 - (5*AppellF1[1/2, 1/3, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])/((-1 + Tan[(c + d*x)/2]^2)*(AppellF1[1/2, 1/3, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + (2*(-3*AppellF1[3/2, 1/3, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + AppellF1[3/2, 4/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Tan[(c + d*x)/2]^2)/9)))))/(7*2^(2/3)*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^(2/3)) + (2^(1/3)*Tan[(c + d*x)/2]*((-3*AppellF1[3/2, 1/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(2/3) - (3*Tan[(c + d*x)/2]^2*((-3*AppellF1[5/2, 1/3, 2, 7/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/5 + (AppellF1[5/2, 4/3, 1, 7/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/5))/(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(2/3) + (2*AppellF1[3/2, 1/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Tan[(c + d*x)/2]^2*(-(Sec[(c + d*x)/2]^2*Sin[c + d*x]) + Cos[c + d*x]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(5/3) - Cos[(c + d*x)/2]*Sin[(c + d*x)/2]*(-27 - (5*AppellF1[1/2, 1/3, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])/((-1 + Tan[(c + d*x)/2]^2)*(AppellF1[1/2, 1/3, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + (2*(-3*AppellF1[3/2, 1/3, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + AppellF1[3/2, 4/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Tan[(c + d*x)/2]^2)/9))) + Cos[(c + d*x)/2]^2*((5*AppellF1[1/2, 1/3, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/((-1 + Tan[(c + d*x)/2]^2)^2*(AppellF1[1/2, 1/3, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + (2*(-3*AppellF1[3/2, 1/3, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + AppellF1[3/2, 4/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Tan[(c + d*x)/2]^2)/9)) - (5*(-1/3*(AppellF1[3/2, 1/3, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]) + (AppellF1[3/2, 4/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/9))/((-1 + Tan[(c + d*x)/2]^2)*(AppellF1[1/2, 1/3, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + (2*(-3*AppellF1[3/2, 1/3, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + AppellF1[3/2, 4/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Tan[(c + d*x)/2]^2)/9)) + (5*AppellF1[1/2, 1/3, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*(-1/3*(AppellF1[3/2, 1/3, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]) + (AppellF1[3/2, 4/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/9 + (2*(-3*AppellF1[3/2, 1/3, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + AppellF1[3/2, 4/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/9 + (2*Tan[(c + d*x)/2]^2*((-3*AppellF1[5/2, 4/3, 2, 7/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/5 + (4*AppellF1[5/2, 7/3, 1, 7/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/5 - 3*((-6*AppellF1[5/2, 1/3, 3, 7/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/5 + (AppellF1[5/2, 4/3, 2, 7/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/5)))/9))/((-1 + Tan[(c + d*x)/2]^2)*(AppellF1[1/2, 1/3, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + (2*(-3*AppellF1[3/2, 1/3, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + AppellF1[3/2, 4/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Tan[(c + d*x)/2]^2)/9)^2))))/(7*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^(2/3)) - (2*2^(1/3)*Tan[(c + d*x)/2]*((-3*AppellF1[3/2, 1/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Tan[(c + d*x)/2]^2)/(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(2/3) + Cos[(c + d*x)/2]^2*(-27 - (5*AppellF1[1/2, 1/3, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])/((-1 + Tan[(c + d*x)/2]^2)*(AppellF1[1/2, 1/3, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + (2*(-3*AppellF1[3/2, 1/3, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + AppellF1[3/2, 4/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Tan[(c + d*x)/2]^2)/9))))*(-(Cos[(c + d*x)/2]*Sec[c + d*x]*Sin[(c + d*x)/2]) + Cos[(c + d*x)/2]^2*Sec[c + d*x]*Tan[c + d*x]))/(21*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^(5/3))))","B",0
164,1,3011,90,16.5838626,"\int \frac{\cos (c+d x)}{(a+a \sec (c+d x))^{5/3}} \, dx","Integrate[Cos[c + d*x]/(a + a*Sec[c + d*x])^(5/3),x]","\text{Result too large to show}","\frac{3 \sqrt{2} \tan (c+d x) F_1\left(-\frac{7}{6};\frac{1}{2},2;-\frac{1}{6};\frac{1}{2} (\sec (c+d x)+1),\sec (c+d x)+1\right)}{7 a d \sqrt{1-\sec (c+d x)} (\sec (c+d x)+1) (a \sec (c+d x)+a)^{2/3}}",1,"(((1 + Cos[c + d*x])*Sec[c + d*x])^(1/3)*(1 + Sec[c + d*x])^(5/3)*((-48*Sin[c + d*x])/7 + (51*Tan[(c + d*x)/2])/7 - (3*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/14))/(d*(a*(1 + Sec[c + d*x]))^(5/3)) - (5*2^(1/3)*(1 + Sec[c + d*x])^(5/3)*((-30*(1 + Sec[c + d*x])^(1/3))/7 + (55*Cos[c + d*x]*(1 + Sec[c + d*x])^(1/3))/7)*Tan[(c + d*x)/2]*((-11*AppellF1[3/2, 1/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Tan[(c + d*x)/2]^2)/(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(2/3) + 9*Cos[(c + d*x)/2]^2*(-11 - AppellF1[1/2, 1/3, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]/((-1 + Tan[(c + d*x)/2]^2)*(AppellF1[1/2, 1/3, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + (2*(-3*AppellF1[3/2, 1/3, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + AppellF1[3/2, 4/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Tan[(c + d*x)/2]^2)/9)))))/(63*d*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^(2/3)*(a*(1 + Sec[c + d*x]))^(5/3)*((-5*Sec[(c + d*x)/2]^2*((-11*AppellF1[3/2, 1/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Tan[(c + d*x)/2]^2)/(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(2/3) + 9*Cos[(c + d*x)/2]^2*(-11 - AppellF1[1/2, 1/3, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]/((-1 + Tan[(c + d*x)/2]^2)*(AppellF1[1/2, 1/3, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + (2*(-3*AppellF1[3/2, 1/3, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + AppellF1[3/2, 4/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Tan[(c + d*x)/2]^2)/9)))))/(63*2^(2/3)*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^(2/3)) - (5*2^(1/3)*Tan[(c + d*x)/2]*((-11*AppellF1[3/2, 1/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(2/3) - (11*Tan[(c + d*x)/2]^2*((-3*AppellF1[5/2, 1/3, 2, 7/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/5 + (AppellF1[5/2, 4/3, 1, 7/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/5))/(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(2/3) + (22*AppellF1[3/2, 1/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Tan[(c + d*x)/2]^2*(-(Sec[(c + d*x)/2]^2*Sin[c + d*x]) + Cos[c + d*x]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(3*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(5/3)) - 9*Cos[(c + d*x)/2]*Sin[(c + d*x)/2]*(-11 - AppellF1[1/2, 1/3, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]/((-1 + Tan[(c + d*x)/2]^2)*(AppellF1[1/2, 1/3, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + (2*(-3*AppellF1[3/2, 1/3, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + AppellF1[3/2, 4/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Tan[(c + d*x)/2]^2)/9))) + 9*Cos[(c + d*x)/2]^2*((AppellF1[1/2, 1/3, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/((-1 + Tan[(c + d*x)/2]^2)^2*(AppellF1[1/2, 1/3, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + (2*(-3*AppellF1[3/2, 1/3, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + AppellF1[3/2, 4/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Tan[(c + d*x)/2]^2)/9)) - (-1/3*(AppellF1[3/2, 1/3, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]) + (AppellF1[3/2, 4/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/9)/((-1 + Tan[(c + d*x)/2]^2)*(AppellF1[1/2, 1/3, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + (2*(-3*AppellF1[3/2, 1/3, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + AppellF1[3/2, 4/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Tan[(c + d*x)/2]^2)/9)) + (AppellF1[1/2, 1/3, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*(-1/3*(AppellF1[3/2, 1/3, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]) + (AppellF1[3/2, 4/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/9 + (2*(-3*AppellF1[3/2, 1/3, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + AppellF1[3/2, 4/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/9 + (2*Tan[(c + d*x)/2]^2*((-3*AppellF1[5/2, 4/3, 2, 7/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/5 + (4*AppellF1[5/2, 7/3, 1, 7/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/5 - 3*((-6*AppellF1[5/2, 1/3, 3, 7/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/5 + (AppellF1[5/2, 4/3, 2, 7/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/5)))/9))/((-1 + Tan[(c + d*x)/2]^2)*(AppellF1[1/2, 1/3, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + (2*(-3*AppellF1[3/2, 1/3, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + AppellF1[3/2, 4/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Tan[(c + d*x)/2]^2)/9)^2))))/(63*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^(2/3)) + (10*2^(1/3)*Tan[(c + d*x)/2]*((-11*AppellF1[3/2, 1/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Tan[(c + d*x)/2]^2)/(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(2/3) + 9*Cos[(c + d*x)/2]^2*(-11 - AppellF1[1/2, 1/3, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]/((-1 + Tan[(c + d*x)/2]^2)*(AppellF1[1/2, 1/3, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + (2*(-3*AppellF1[3/2, 1/3, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + AppellF1[3/2, 4/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Tan[(c + d*x)/2]^2)/9))))*(-(Cos[(c + d*x)/2]*Sec[c + d*x]*Sin[(c + d*x)/2]) + Cos[(c + d*x)/2]^2*Sec[c + d*x]*Tan[c + d*x]))/(189*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^(5/3))))","B",0
165,1,115,151,0.2462174,"\int \sec ^{\frac{5}{2}}(c+d x) (a+a \sec (c+d x)) \, dx","Integrate[Sec[c + d*x]^(5/2)*(a + a*Sec[c + d*x]),x]","\frac{a \sec ^2\left(\frac{1}{2} (c+d x)\right) (\sec (c+d x)+1) \left(9 \sin (c+d x)+5 \tan (c+d x)+5 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-9 \sqrt{\cos (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+3 \tan (c+d x) \sec (c+d x)\right)}{15 d \sqrt{\sec (c+d x)}}","\frac{2 a \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{5 d}+\frac{2 a \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 d}+\frac{6 a \sin (c+d x) \sqrt{\sec (c+d x)}}{5 d}+\frac{2 a \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{6 a \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}",1,"(a*Sec[(c + d*x)/2]^2*(1 + Sec[c + d*x])*(-9*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2] + 5*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + 9*Sin[c + d*x] + 5*Tan[c + d*x] + 3*Sec[c + d*x]*Tan[c + d*x]))/(15*d*Sqrt[Sec[c + d*x]])","A",1
166,1,83,123,0.2209202,"\int \sec ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x)) \, dx","Integrate[Sec[c + d*x]^(3/2)*(a + a*Sec[c + d*x]),x]","\frac{a \sec ^{\frac{3}{2}}(c+d x) \left(2 \sin (c+d x)+3 \sin (2 (c+d x))+2 \cos ^{\frac{3}{2}}(c+d x) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-6 \cos ^{\frac{3}{2}}(c+d x) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{3 d}","\frac{2 a \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 d}+\frac{2 a \sin (c+d x) \sqrt{\sec (c+d x)}}{d}+\frac{2 a \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{2 a \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}",1,"(a*Sec[c + d*x]^(3/2)*(-6*Cos[c + d*x]^(3/2)*EllipticE[(c + d*x)/2, 2] + 2*Cos[c + d*x]^(3/2)*EllipticF[(c + d*x)/2, 2] + 2*Sin[c + d*x] + 3*Sin[2*(c + d*x)]))/(3*d)","A",1
167,1,68,97,0.1665793,"\int \sqrt{\sec (c+d x)} (a+a \sec (c+d x)) \, dx","Integrate[Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x]),x]","\frac{2 a \sqrt{\sec (c+d x)} \left(\sin (c+d x)+\sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-\sqrt{\cos (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{d}","\frac{2 a \sin (c+d x) \sqrt{\sec (c+d x)}}{d}+\frac{2 a \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}-\frac{2 a \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}",1,"(2*a*Sqrt[Sec[c + d*x]]*(-(Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]) + Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + Sin[c + d*x]))/d","A",1
168,1,49,75,0.0981085,"\int \frac{a+a \sec (c+d x)}{\sqrt{\sec (c+d x)}} \, dx","Integrate[(a + a*Sec[c + d*x])/Sqrt[Sec[c + d*x]],x]","\frac{2 a \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \left(F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+E\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{d}","\frac{2 a \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 a \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}",1,"(2*a*Sqrt[Cos[c + d*x]]*(EllipticE[(c + d*x)/2, 2] + EllipticF[(c + d*x)/2, 2])*Sqrt[Sec[c + d*x]])/d","A",1
169,1,73,101,0.152926,"\int \frac{a+a \sec (c+d x)}{\sec ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[(a + a*Sec[c + d*x])/Sec[c + d*x]^(3/2),x]","\frac{a \sqrt{\sec (c+d x)} \left(\sin (2 (c+d x))+2 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+6 \sqrt{\cos (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{3 d}","\frac{2 a \sin (c+d x)}{3 d \sqrt{\sec (c+d x)}}+\frac{2 a \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{2 a \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}",1,"(a*Sqrt[Sec[c + d*x]]*(6*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2] + 2*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + Sin[2*(c + d*x)]))/(3*d)","A",1
170,1,93,127,0.2224108,"\int \frac{a+a \sec (c+d x)}{\sec ^{\frac{5}{2}}(c+d x)} \, dx","Integrate[(a + a*Sec[c + d*x])/Sec[c + d*x]^(5/2),x]","\frac{a \sqrt{\sec (c+d x)} \left(3 \sin (c+d x)+10 \sin (2 (c+d x))+3 \sin (3 (c+d x))+20 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+36 \sqrt{\cos (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{30 d}","\frac{2 a \sin (c+d x)}{5 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 a \sin (c+d x)}{3 d \sqrt{\sec (c+d x)}}+\frac{2 a \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{6 a \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}",1,"(a*Sqrt[Sec[c + d*x]]*(36*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2] + 20*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + 3*Sin[c + d*x] + 10*Sin[2*(c + d*x)] + 3*Sin[3*(c + d*x)]))/(30*d)","A",1
171,1,103,151,0.2944512,"\int \frac{a+a \sec (c+d x)}{\sec ^{\frac{7}{2}}(c+d x)} \, dx","Integrate[(a + a*Sec[c + d*x])/Sec[c + d*x]^(7/2),x]","\frac{a \sqrt{\sec (c+d x)} \left(42 \sin (c+d x)+130 \sin (2 (c+d x))+42 \sin (3 (c+d x))+15 \sin (4 (c+d x))+200 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+504 \sqrt{\cos (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{420 d}","\frac{2 a \sin (c+d x)}{5 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 a \sin (c+d x)}{7 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{10 a \sin (c+d x)}{21 d \sqrt{\sec (c+d x)}}+\frac{10 a \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{6 a \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}",1,"(a*Sqrt[Sec[c + d*x]]*(504*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2] + 200*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + 42*Sin[c + d*x] + 130*Sin[2*(c + d*x)] + 42*Sin[3*(c + d*x)] + 15*Sin[4*(c + d*x)]))/(420*d)","A",1
172,1,269,187,2.6459708,"\int \sec ^{\frac{5}{2}}(c+d x) (a+a \sec (c+d x))^2 \, dx","Integrate[Sec[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^2,x]","\frac{1}{35} a^2 \sec ^4\left(\frac{1}{2} (c+d x)\right) (\sec (c+d x)+1)^2 \left(\frac{42 \csc (c) \cos (d x)+(14 \cos (c+d x)+10 \cos (2 (c+d x))+15) \tan (c+d x) \sec ^2(c+d x)}{2 d \sec ^{\frac{3}{2}}(c+d x)}-\frac{i \sqrt{2} \left(21 \left(-1+e^{2 i c}\right) \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i (c+d x)}\right)+10 \left(-1+e^{2 i c}\right) e^{i (c+d x)} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-e^{2 i (c+d x)}\right)+21 \sqrt{1+e^{2 i (c+d x)}}\right) \cos ^2(c+d x)}{\left(-1+e^{2 i c}\right) d \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}}}\right)","\frac{2 a^2 \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x)}{7 d}+\frac{4 a^2 \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{5 d}+\frac{8 a^2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{7 d}+\frac{12 a^2 \sin (c+d x) \sqrt{\sec (c+d x)}}{5 d}+\frac{8 a^2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{7 d}-\frac{12 a^2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}",1,"(a^2*Sec[(c + d*x)/2]^4*(1 + Sec[c + d*x])^2*(((-I)*Sqrt[2]*Cos[c + d*x]^2*(21*Sqrt[1 + E^((2*I)*(c + d*x))] + 21*(-1 + E^((2*I)*c))*Hypergeometric2F1[-1/4, 1/2, 3/4, -E^((2*I)*(c + d*x))] + 10*E^(I*(c + d*x))*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/4, 1/2, 5/4, -E^((2*I)*(c + d*x))]))/(d*(-1 + E^((2*I)*c))*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]) + (42*Cos[d*x]*Csc[c] + (15 + 14*Cos[c + d*x] + 10*Cos[2*(c + d*x)])*Sec[c + d*x]^2*Tan[c + d*x])/(2*d*Sec[c + d*x]^(3/2))))/35","C",1
173,1,269,161,1.5799628,"\int \sec ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x))^2 \, dx","Integrate[Sec[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^2,x]","\frac{a^2 \sec ^4\left(\frac{1}{2} (c+d x)\right) (\sec (c+d x)+1)^2 \left(\frac{24 \csc (c) \cos (d x)+\tan (c+d x) (3 \sec (c+d x)+10)}{\sec ^{\frac{3}{2}}(c+d x)}-\frac{2 i \sqrt{2} e^{-i (c+d x)} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \left(12 \left(-1+e^{2 i c}\right) \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i (c+d x)}\right)+5 \left(-1+e^{2 i c}\right) e^{i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-e^{2 i (c+d x)}\right)+12 \left(1+e^{2 i (c+d x)}\right)\right) \cos ^2(c+d x)}{-1+e^{2 i c}}\right)}{30 d}","\frac{2 a^2 \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{5 d}+\frac{4 a^2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 d}+\frac{16 a^2 \sin (c+d x) \sqrt{\sec (c+d x)}}{5 d}+\frac{4 a^2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{16 a^2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}",1,"(a^2*Sec[(c + d*x)/2]^4*(1 + Sec[c + d*x])^2*(((-2*I)*Sqrt[2]*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Cos[c + d*x]^2*(12*(1 + E^((2*I)*(c + d*x))) + 12*(-1 + E^((2*I)*c))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[-1/4, 1/2, 3/4, -E^((2*I)*(c + d*x))] + 5*E^(I*(c + d*x))*(-1 + E^((2*I)*c))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/4, 1/2, 5/4, -E^((2*I)*(c + d*x))]))/(E^(I*(c + d*x))*(-1 + E^((2*I)*c))) + (24*Cos[d*x]*Csc[c] + (10 + 3*Sec[c + d*x])*Tan[c + d*x])/Sec[c + d*x]^(3/2)))/(30*d)","C",1
174,1,264,131,1.2539342,"\int \sqrt{\sec (c+d x)} (a+a \sec (c+d x))^2 \, dx","Integrate[Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x])^2,x]","\frac{1}{3} a^2 \sec ^4\left(\frac{1}{2} (c+d x)\right) (\sec (c+d x)+1)^2 \left(\frac{\tan (c+d x)+6 \csc (c) \cos (d x)}{2 d \sec ^{\frac{3}{2}}(c+d x)}-\frac{i \sqrt{2} e^{-i (c+d x)} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \left(3 \left(-1+e^{2 i c}\right) \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i (c+d x)}\right)+2 \left(-1+e^{2 i c}\right) e^{i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-e^{2 i (c+d x)}\right)+3 \left(1+e^{2 i (c+d x)}\right)\right) \cos ^2(c+d x)}{\left(-1+e^{2 i c}\right) d}\right)","\frac{2 a^2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 d}+\frac{4 a^2 \sin (c+d x) \sqrt{\sec (c+d x)}}{d}+\frac{8 a^2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{4 a^2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}",1,"(a^2*Sec[(c + d*x)/2]^4*(1 + Sec[c + d*x])^2*(((-I)*Sqrt[2]*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Cos[c + d*x]^2*(3*(1 + E^((2*I)*(c + d*x))) + 3*(-1 + E^((2*I)*c))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[-1/4, 1/2, 3/4, -E^((2*I)*(c + d*x))] + 2*E^(I*(c + d*x))*(-1 + E^((2*I)*c))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/4, 1/2, 5/4, -E^((2*I)*(c + d*x))]))/(d*E^(I*(c + d*x))*(-1 + E^((2*I)*c))) + (6*Cos[d*x]*Csc[c] + Tan[c + d*x])/(2*d*Sec[c + d*x]^(3/2))))/3","C",1
175,1,48,64,0.1583548,"\int \frac{(a+a \sec (c+d x))^2}{\sqrt{\sec (c+d x)}} \, dx","Integrate[(a + a*Sec[c + d*x])^2/Sqrt[Sec[c + d*x]],x]","\frac{2 a^2 \sqrt{\sec (c+d x)} \left(\sin (c+d x)+2 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{d}","\frac{2 a^2 \sin (c+d x) \sqrt{\sec (c+d x)}}{d}+\frac{4 a^2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}",1,"(2*a^2*Sqrt[Sec[c + d*x]]*(2*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + Sin[c + d*x]))/d","A",1
176,1,156,107,0.8541323,"\int \frac{(a+a \sec (c+d x))^2}{\sec ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[(a + a*Sec[c + d*x])^2/Sec[c + d*x]^(3/2),x]","\frac{a^2 \left(\cos \left(\frac{c}{2}\right)-i \sin \left(\frac{c}{2}\right)\right) \left(\sin \left(\frac{c}{2}\right)-i \cos \left(\frac{c}{2}\right)\right) \left(-\frac{24 \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i (c+d x)}\right)}{\sqrt{1+e^{2 i (c+d x)}}}+8 \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-e^{2 i (c+d x)}\right) \sec (c+d x)+2 i \sin (c+d x)+12\right)}{3 d \sqrt{\sec (c+d x)}}","\frac{2 a^2 \sin (c+d x)}{3 d \sqrt{\sec (c+d x)}}+\frac{8 a^2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{4 a^2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}",1,"(a^2*(Cos[c/2] - I*Sin[c/2])*((-I)*Cos[c/2] + Sin[c/2])*(12 - (24*Hypergeometric2F1[-1/4, 1/2, 3/4, -E^((2*I)*(c + d*x))])/Sqrt[1 + E^((2*I)*(c + d*x))] + 8*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/4, 1/2, 5/4, -E^((2*I)*(c + d*x))]*Sec[c + d*x] + (2*I)*Sin[c + d*x]))/(3*d*Sqrt[Sec[c + d*x]])","C",1
177,1,136,135,1.1420519,"\int \frac{(a+a \sec (c+d x))^2}{\sec ^{\frac{5}{2}}(c+d x)} \, dx","Integrate[(a + a*Sec[c + d*x])^2/Sec[c + d*x]^(5/2),x]","\frac{a^2 \left(\frac{192 i \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i (c+d x)}\right)}{\sqrt{1+e^{2 i (c+d x)}}}-40 i \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-e^{2 i (c+d x)}\right) \sec (c+d x)+40 \sin (c+d x)+6 \sin (2 (c+d x))-96 i\right)}{30 d \sqrt{\sec (c+d x)}}","\frac{2 a^2 \sin (c+d x)}{5 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{4 a^2 \sin (c+d x)}{3 d \sqrt{\sec (c+d x)}}+\frac{4 a^2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{16 a^2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}",1,"(a^2*(-96*I + ((192*I)*Hypergeometric2F1[-1/4, 1/2, 3/4, -E^((2*I)*(c + d*x))])/Sqrt[1 + E^((2*I)*(c + d*x))] - (40*I)*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/4, 1/2, 5/4, -E^((2*I)*(c + d*x))]*Sec[c + d*x] + 40*Sin[c + d*x] + 6*Sin[2*(c + d*x)]))/(30*d*Sqrt[Sec[c + d*x]])","C",1
178,1,149,161,1.3764353,"\int \frac{(a+a \sec (c+d x))^2}{\sec ^{\frac{7}{2}}(c+d x)} \, dx","Integrate[(a + a*Sec[c + d*x])^2/Sec[c + d*x]^(7/2),x]","\frac{a^2 \left(\frac{672 i \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i (c+d x)}\right)}{\sqrt{1+e^{2 i (c+d x)}}}+2 \left(-80 i \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-e^{2 i (c+d x)}\right) \sec (c+d x)+85 \sin (c+d x)+28 \sin (2 (c+d x))+5 \sin (3 (c+d x))-168 i\right)\right)}{140 d \sqrt{\sec (c+d x)}}","\frac{4 a^2 \sin (c+d x)}{5 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 a^2 \sin (c+d x)}{7 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{8 a^2 \sin (c+d x)}{7 d \sqrt{\sec (c+d x)}}+\frac{8 a^2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{7 d}+\frac{12 a^2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}",1,"(a^2*(((672*I)*Hypergeometric2F1[-1/4, 1/2, 3/4, -E^((2*I)*(c + d*x))])/Sqrt[1 + E^((2*I)*(c + d*x))] + 2*(-168*I - (80*I)*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/4, 1/2, 5/4, -E^((2*I)*(c + d*x))]*Sec[c + d*x] + 85*Sin[c + d*x] + 28*Sin[2*(c + d*x)] + 5*Sin[3*(c + d*x)])))/(140*d*Sqrt[Sec[c + d*x]])","C",1
179,1,287,187,2.2613564,"\int \sec ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x))^3 \, dx","Integrate[Sec[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^3,x]","\frac{a^3 \sec ^6\left(\frac{1}{2} (c+d x)\right) (\sec (c+d x)+1)^3 \left(\frac{294 \csc (c) \cos (d x)+(63 \cos (c+d x)+65 \cos (2 (c+d x))+80) \tan (c+d x) \sec ^2(c+d x)}{\sec ^{\frac{5}{2}}(c+d x)}-\frac{2 i \sqrt{2} e^{-i (c+d x)} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \left(147 \left(-1+e^{2 i c}\right) \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i (c+d x)}\right)+65 \left(-1+e^{2 i c}\right) e^{i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-e^{2 i (c+d x)}\right)+147 \left(1+e^{2 i (c+d x)}\right)\right) \cos ^3(c+d x)}{-1+e^{2 i c}}\right)}{420 d}","\frac{2 a^3 \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x)}{7 d}+\frac{6 a^3 \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{5 d}+\frac{52 a^3 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{21 d}+\frac{28 a^3 \sin (c+d x) \sqrt{\sec (c+d x)}}{5 d}+\frac{52 a^3 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}-\frac{28 a^3 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}",1,"(a^3*Sec[(c + d*x)/2]^6*(1 + Sec[c + d*x])^3*(((-2*I)*Sqrt[2]*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Cos[c + d*x]^3*(147*(1 + E^((2*I)*(c + d*x))) + 147*(-1 + E^((2*I)*c))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[-1/4, 1/2, 3/4, -E^((2*I)*(c + d*x))] + 65*E^(I*(c + d*x))*(-1 + E^((2*I)*c))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/4, 1/2, 5/4, -E^((2*I)*(c + d*x))]))/(E^(I*(c + d*x))*(-1 + E^((2*I)*c))) + (294*Cos[d*x]*Csc[c] + (80 + 63*Cos[c + d*x] + 65*Cos[2*(c + d*x)])*Sec[c + d*x]^2*Tan[c + d*x])/Sec[c + d*x]^(5/2)))/(420*d)","C",1
180,1,267,157,1.91006,"\int \sqrt{\sec (c+d x)} (a+a \sec (c+d x))^3 \, dx","Integrate[Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x])^3,x]","\frac{a^3 \sec ^6\left(\frac{1}{2} (c+d x)\right) (\sec (c+d x)+1)^3 \left(\frac{18 \csc (c) \cos (d x)+\tan (c+d x) (\sec (c+d x)+5)}{\sec ^{\frac{5}{2}}(c+d x)}-\frac{2 i \sqrt{2} e^{-i (c+d x)} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \left(9 \left(-1+e^{2 i c}\right) \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i (c+d x)}\right)+5 \left(-1+e^{2 i c}\right) e^{i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-e^{2 i (c+d x)}\right)+9 \left(1+e^{2 i (c+d x)}\right)\right) \cos ^3(c+d x)}{-1+e^{2 i c}}\right)}{20 d}","\frac{2 a^3 \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{5 d}+\frac{2 a^3 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{d}+\frac{36 a^3 \sin (c+d x) \sqrt{\sec (c+d x)}}{5 d}+\frac{4 a^3 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}-\frac{36 a^3 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}",1,"(a^3*Sec[(c + d*x)/2]^6*(1 + Sec[c + d*x])^3*(((-2*I)*Sqrt[2]*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Cos[c + d*x]^3*(9*(1 + E^((2*I)*(c + d*x))) + 9*(-1 + E^((2*I)*c))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[-1/4, 1/2, 3/4, -E^((2*I)*(c + d*x))] + 5*E^(I*(c + d*x))*(-1 + E^((2*I)*c))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/4, 1/2, 5/4, -E^((2*I)*(c + d*x))]))/(E^(I*(c + d*x))*(-1 + E^((2*I)*c))) + (18*Cos[d*x]*Csc[c] + (5 + Sec[c + d*x])*Tan[c + d*x])/Sec[c + d*x]^(5/2)))/(20*d)","C",1
181,1,187,131,1.4131427,"\int \frac{(a+a \sec (c+d x))^3}{\sqrt{\sec (c+d x)}} \, dx","Integrate[(a + a*Sec[c + d*x])^3/Sqrt[Sec[c + d*x]],x]","\frac{a^3 e^{-2 i (c+d x)} \sec ^{\frac{3}{2}}(c+d x) (\sin (2 (c+d x))-i \cos (2 (c+d x))) \left(6 e^{-2 i (c+d x)} \left(1+e^{2 i (c+d x)}\right)^{3/2} \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i (c+d x)}\right)+20 \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-e^{2 i (c+d x)}\right) \cos (c+d x)+2 i \sin (c+d x)+9 i \sin (2 (c+d x))-6 \cos (2 (c+d x))-6\right)}{3 d}","\frac{2 a^3 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 d}+\frac{6 a^3 \sin (c+d x) \sqrt{\sec (c+d x)}}{d}+\frac{20 a^3 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{4 a^3 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}",1,"(a^3*Sec[c + d*x]^(3/2)*(-6 - 6*Cos[2*(c + d*x)] + (6*(1 + E^((2*I)*(c + d*x)))^(3/2)*Hypergeometric2F1[-1/4, 1/2, 3/4, -E^((2*I)*(c + d*x))])/E^((2*I)*(c + d*x)) + 20*Sqrt[1 + E^((2*I)*(c + d*x))]*Cos[c + d*x]*Hypergeometric2F1[1/4, 1/2, 5/4, -E^((2*I)*(c + d*x))] + (2*I)*Sin[c + d*x] + (9*I)*Sin[2*(c + d*x)])*((-I)*Cos[2*(c + d*x)] + Sin[2*(c + d*x)]))/(3*d*E^((2*I)*(c + d*x)))","C",1
182,1,169,131,1.0925226,"\int \frac{(a+a \sec (c+d x))^3}{\sec ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[(a + a*Sec[c + d*x])^3/Sec[c + d*x]^(3/2),x]","\frac{a^3 \left(\cos \left(\frac{c}{2}\right)-i \sin \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{c}{2}\right)+i \sin \left(\frac{c}{2}\right)\right) \left(\frac{24 i \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i (c+d x)}\right)}{\sqrt{1+e^{2 i (c+d x)}}}+2 \left(-10 i \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-e^{2 i (c+d x)}\right) \sec (c+d x)+\sin (c+d x)+3 \tan (c+d x)-6 i\right)\right)}{3 d \sqrt{\sec (c+d x)}}","\frac{2 a^3 \sin (c+d x) \sqrt{\sec (c+d x)}}{d}+\frac{2 a^3 \sin (c+d x)}{3 d \sqrt{\sec (c+d x)}}+\frac{20 a^3 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{4 a^3 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}",1,"(a^3*(Cos[c/2] - I*Sin[c/2])*(Cos[c/2] + I*Sin[c/2])*(((24*I)*Hypergeometric2F1[-1/4, 1/2, 3/4, -E^((2*I)*(c + d*x))])/Sqrt[1 + E^((2*I)*(c + d*x))] + 2*(-6*I - (10*I)*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/4, 1/2, 5/4, -E^((2*I)*(c + d*x))]*Sec[c + d*x] + Sin[c + d*x] + 3*Tan[c + d*x])))/(3*d*Sqrt[Sec[c + d*x]])","C",1
183,1,171,131,1.0681098,"\int \frac{(a+a \sec (c+d x))^3}{\sec ^{\frac{5}{2}}(c+d x)} \, dx","Integrate[(a + a*Sec[c + d*x])^3/Sec[c + d*x]^(5/2),x]","\frac{a^3 \left(\cos \left(\frac{c}{2}\right)-i \sin \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{c}{2}\right)+i \sin \left(\frac{c}{2}\right)\right) \left(\frac{144 i \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i (c+d x)}\right)}{\sqrt{1+e^{2 i (c+d x)}}}+2 \left(-20 i \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-e^{2 i (c+d x)}\right) \sec (c+d x)+10 \sin (c+d x)+\sin (2 (c+d x))-36 i\right)\right)}{10 d \sqrt{\sec (c+d x)}}","\frac{2 a^3 \sin (c+d x)}{5 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 a^3 \sin (c+d x)}{d \sqrt{\sec (c+d x)}}+\frac{4 a^3 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{36 a^3 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}",1,"(a^3*(Cos[c/2] - I*Sin[c/2])*(Cos[c/2] + I*Sin[c/2])*(((144*I)*Hypergeometric2F1[-1/4, 1/2, 3/4, -E^((2*I)*(c + d*x))])/Sqrt[1 + E^((2*I)*(c + d*x))] + 2*(-36*I - (20*I)*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/4, 1/2, 5/4, -E^((2*I)*(c + d*x))]*Sec[c + d*x] + 10*Sin[c + d*x] + Sin[2*(c + d*x)])))/(10*d*Sqrt[Sec[c + d*x]])","C",1
184,1,146,161,1.5188474,"\int \frac{(a+a \sec (c+d x))^3}{\sec ^{\frac{7}{2}}(c+d x)} \, dx","Integrate[(a + a*Sec[c + d*x])^3/Sec[c + d*x]^(7/2),x]","\frac{a^3 \left(\frac{4704 i \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i (c+d x)}\right)}{\sqrt{1+e^{2 i (c+d x)}}}-1040 i \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-e^{2 i (c+d x)}\right) \sec (c+d x)+1070 \sin (c+d x)+252 \sin (2 (c+d x))+30 \sin (3 (c+d x))-2352 i\right)}{420 d \sqrt{\sec (c+d x)}}","\frac{6 a^3 \sin (c+d x)}{5 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 a^3 \sin (c+d x)}{7 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{52 a^3 \sin (c+d x)}{21 d \sqrt{\sec (c+d x)}}+\frac{52 a^3 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{28 a^3 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}",1,"(a^3*(-2352*I + ((4704*I)*Hypergeometric2F1[-1/4, 1/2, 3/4, -E^((2*I)*(c + d*x))])/Sqrt[1 + E^((2*I)*(c + d*x))] - (1040*I)*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/4, 1/2, 5/4, -E^((2*I)*(c + d*x))]*Sec[c + d*x] + 1070*Sin[c + d*x] + 252*Sin[2*(c + d*x)] + 30*Sin[3*(c + d*x)]))/(420*d*Sqrt[Sec[c + d*x]])","C",1
185,1,156,187,1.9839197,"\int \frac{(a+a \sec (c+d x))^3}{\sec ^{\frac{9}{2}}(c+d x)} \, dx","Integrate[(a + a*Sec[c + d*x])^3/Sec[c + d*x]^(9/2),x]","\frac{a^3 \left(\frac{22848 i \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i (c+d x)}\right)}{\sqrt{1+e^{2 i (c+d x)}}}-5280 i \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-e^{2 i (c+d x)}\right) \sec (c+d x)+5820 \sin (c+d x)+2044 \sin (2 (c+d x))+540 \sin (3 (c+d x))+70 \sin (4 (c+d x))-11424 i\right)}{2520 d \sqrt{\sec (c+d x)}}","\frac{68 a^3 \sin (c+d x)}{45 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{6 a^3 \sin (c+d x)}{7 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 a^3 \sin (c+d x)}{9 d \sec ^{\frac{7}{2}}(c+d x)}+\frac{44 a^3 \sin (c+d x)}{21 d \sqrt{\sec (c+d x)}}+\frac{44 a^3 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{68 a^3 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}",1,"(a^3*(-11424*I + ((22848*I)*Hypergeometric2F1[-1/4, 1/2, 3/4, -E^((2*I)*(c + d*x))])/Sqrt[1 + E^((2*I)*(c + d*x))] - (5280*I)*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/4, 1/2, 5/4, -E^((2*I)*(c + d*x))]*Sec[c + d*x] + 5820*Sin[c + d*x] + 2044*Sin[2*(c + d*x)] + 540*Sin[3*(c + d*x)] + 70*Sin[4*(c + d*x)]))/(2520*d*Sqrt[Sec[c + d*x]])","C",1
186,1,271,213,5.088923,"\int \sec ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x))^4 \, dx","Integrate[Sec[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^4,x]","\frac{1}{210} a^4 \sec ^8\left(\frac{1}{2} (c+d x)\right) (\sec (c+d x)+1)^4 \left(\frac{1596 \csc (c) \cos (d x)+\tan (c+d x) \left(35 \sec ^3(c+d x)+180 \sec ^2(c+d x)+427 \sec (c+d x)+720\right)}{12 d \sec ^{\frac{7}{2}}(c+d x)}-\frac{i \sqrt{2} \left(133 \left(-1+e^{2 i c}\right) \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i (c+d x)}\right)+60 \left(-1+e^{2 i c}\right) e^{i (c+d x)} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-e^{2 i (c+d x)}\right)+133 \sqrt{1+e^{2 i (c+d x)}}\right) \cos ^4(c+d x)}{\left(-1+e^{2 i c}\right) d \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}}}\right)","\frac{2 a^4 \sin (c+d x) \sec ^{\frac{9}{2}}(c+d x)}{9 d}+\frac{8 a^4 \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x)}{7 d}+\frac{122 a^4 \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{45 d}+\frac{32 a^4 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{7 d}+\frac{152 a^4 \sin (c+d x) \sqrt{\sec (c+d x)}}{15 d}+\frac{32 a^4 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{7 d}-\frac{152 a^4 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}",1,"(a^4*Sec[(c + d*x)/2]^8*(1 + Sec[c + d*x])^4*(((-I)*Sqrt[2]*Cos[c + d*x]^4*(133*Sqrt[1 + E^((2*I)*(c + d*x))] + 133*(-1 + E^((2*I)*c))*Hypergeometric2F1[-1/4, 1/2, 3/4, -E^((2*I)*(c + d*x))] + 60*E^(I*(c + d*x))*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/4, 1/2, 5/4, -E^((2*I)*(c + d*x))]))/(d*(-1 + E^((2*I)*c))*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]) + (1596*Cos[d*x]*Csc[c] + (720 + 427*Sec[c + d*x] + 180*Sec[c + d*x]^2 + 35*Sec[c + d*x]^3)*Tan[c + d*x])/(12*d*Sec[c + d*x]^(7/2))))/210","C",0
187,1,279,187,2.7248454,"\int \sqrt{\sec (c+d x)} (a+a \sec (c+d x))^4 \, dx","Integrate[Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x])^4,x]","\frac{a^4 \sec ^8\left(\frac{1}{2} (c+d x)\right) (\sec (c+d x)+1)^4 \left(\frac{672 \csc (c) \cos (d x)+\tan (c+d x) \left(15 \sec ^2(c+d x)+84 \sec (c+d x)+235\right)}{\sec ^{\frac{7}{2}}(c+d x)}-\frac{4 i \sqrt{2} e^{-i (c+d x)} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \left(168 \left(-1+e^{2 i c}\right) \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i (c+d x)}\right)+85 \left(-1+e^{2 i c}\right) e^{i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-e^{2 i (c+d x)}\right)+168 \left(1+e^{2 i (c+d x)}\right)\right) \cos ^4(c+d x)}{-1+e^{2 i c}}\right)}{840 d}","\frac{2 a^4 \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x)}{7 d}+\frac{8 a^4 \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{5 d}+\frac{94 a^4 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{21 d}+\frac{64 a^4 \sin (c+d x) \sqrt{\sec (c+d x)}}{5 d}+\frac{136 a^4 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}-\frac{64 a^4 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}",1,"(a^4*Sec[(c + d*x)/2]^8*(1 + Sec[c + d*x])^4*(((-4*I)*Sqrt[2]*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Cos[c + d*x]^4*(168*(1 + E^((2*I)*(c + d*x))) + 168*(-1 + E^((2*I)*c))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[-1/4, 1/2, 3/4, -E^((2*I)*(c + d*x))] + 85*E^(I*(c + d*x))*(-1 + E^((2*I)*c))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/4, 1/2, 5/4, -E^((2*I)*(c + d*x))]))/(E^(I*(c + d*x))*(-1 + E^((2*I)*c))) + (672*Cos[d*x]*Csc[c] + (235 + 84*Sec[c + d*x] + 15*Sec[c + d*x]^2)*Tan[c + d*x])/Sec[c + d*x]^(7/2)))/(840*d)","C",1
188,1,286,161,3.2683747,"\int \frac{(a+a \sec (c+d x))^4}{\sqrt{\sec (c+d x)}} \, dx","Integrate[(a + a*Sec[c + d*x])^4/Sqrt[Sec[c + d*x]],x]","\frac{a^4 \sec ^8\left(\frac{1}{2} (c+d x)\right) (\sec (c+d x)+1)^4 \left(\frac{30 \cos (c) \sin (d x)-3 (5 \cos (2 c)-61) \csc (c) \cos (d x)+2 \tan (c+d x) (3 \sec (c+d x)+20)}{\sec ^{\frac{7}{2}}(c+d x)}-\frac{8 i \sqrt{2} e^{-i (c+d x)} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \left(21 \left(-1+e^{2 i c}\right) \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i (c+d x)}\right)+20 \left(-1+e^{2 i c}\right) e^{i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-e^{2 i (c+d x)}\right)+21 \left(1+e^{2 i (c+d x)}\right)\right) \cos ^4(c+d x)}{-1+e^{2 i c}}\right)}{240 d}","\frac{2 a^4 \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{5 d}+\frac{8 a^4 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 d}+\frac{66 a^4 \sin (c+d x) \sqrt{\sec (c+d x)}}{5 d}+\frac{32 a^4 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{56 a^4 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}",1,"(a^4*Sec[(c + d*x)/2]^8*(1 + Sec[c + d*x])^4*(((-8*I)*Sqrt[2]*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Cos[c + d*x]^4*(21*(1 + E^((2*I)*(c + d*x))) + 21*(-1 + E^((2*I)*c))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[-1/4, 1/2, 3/4, -E^((2*I)*(c + d*x))] + 20*E^(I*(c + d*x))*(-1 + E^((2*I)*c))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/4, 1/2, 5/4, -E^((2*I)*(c + d*x))]))/(E^(I*(c + d*x))*(-1 + E^((2*I)*c))) + (-3*(-61 + 5*Cos[2*c])*Cos[d*x]*Csc[c] + 30*Cos[c]*Sin[d*x] + 2*(20 + 3*Sec[c + d*x])*Tan[c + d*x])/Sec[c + d*x]^(7/2)))/(240*d)","C",1
189,1,70,118,0.3222889,"\int \frac{(a+a \sec (c+d x))^4}{\sec ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[(a + a*Sec[c + d*x])^4/Sec[c + d*x]^(3/2),x]","\frac{a^4 \sec ^{\frac{3}{2}}(c+d x) \left(5 \sin (c+d x)+24 \sin (2 (c+d x))+\sin (3 (c+d x))+80 \cos ^{\frac{3}{2}}(c+d x) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{6 d}","\frac{2 a^4 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 d}+\frac{8 a^4 \sin (c+d x) \sqrt{\sec (c+d x)}}{d}+\frac{2 a^4 \sin (c+d x)}{3 d \sqrt{\sec (c+d x)}}+\frac{40 a^4 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}",1,"(a^4*Sec[c + d*x]^(3/2)*(80*Cos[c + d*x]^(3/2)*EllipticF[(c + d*x)/2, 2] + 5*Sin[c + d*x] + 24*Sin[2*(c + d*x)] + Sin[3*(c + d*x)]))/(6*d)","A",1
190,1,184,159,1.242803,"\int \frac{(a+a \sec (c+d x))^4}{\sec ^{\frac{5}{2}}(c+d x)} \, dx","Integrate[(a + a*Sec[c + d*x])^4/Sec[c + d*x]^(5/2),x]","\frac{a^4 \left(\cos \left(\frac{c}{2}\right)-i \sin \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{c}{2}\right)+i \sin \left(\frac{c}{2}\right)\right) \left(\frac{672 i \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i (c+d x)}\right)}{\sqrt{1+e^{2 i (c+d x)}}}-320 i \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-e^{2 i (c+d x)}\right) \sec (c+d x)+80 \sin (c+d x)+63 \tan (c+d x)+3 \sin (3 (c+d x)) \sec (c+d x)-336 i\right)}{30 d \sqrt{\sec (c+d x)}}","\frac{2 a^4 \sin (c+d x)}{5 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 a^4 \sin (c+d x) \sqrt{\sec (c+d x)}}{d}+\frac{8 a^4 \sin (c+d x)}{3 d \sqrt{\sec (c+d x)}}+\frac{32 a^4 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{56 a^4 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}",1,"(a^4*(Cos[c/2] - I*Sin[c/2])*(Cos[c/2] + I*Sin[c/2])*(-336*I + ((672*I)*Hypergeometric2F1[-1/4, 1/2, 3/4, -E^((2*I)*(c + d*x))])/Sqrt[1 + E^((2*I)*(c + d*x))] - (320*I)*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/4, 1/2, 5/4, -E^((2*I)*(c + d*x))]*Sec[c + d*x] + 80*Sin[c + d*x] + 3*Sec[c + d*x]*Sin[3*(c + d*x)] + 63*Tan[c + d*x]))/(30*d*Sqrt[Sec[c + d*x]])","C",1
191,1,180,161,1.3233536,"\int \frac{(a+a \sec (c+d x))^4}{\sec ^{\frac{7}{2}}(c+d x)} \, dx","Integrate[(a + a*Sec[c + d*x])^4/Sec[c + d*x]^(7/2),x]","\frac{a^4 \left(\cos \left(\frac{c}{2}\right)-i \sin \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{c}{2}\right)+i \sin \left(\frac{c}{2}\right)\right) \left(\frac{10752 i \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i (c+d x)}\right)}{\sqrt{1+e^{2 i (c+d x)}}}-2720 i \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-e^{2 i (c+d x)}\right) \sec (c+d x)+1910 \sin (c+d x)+336 \sin (2 (c+d x))+30 \sin (3 (c+d x))-5376 i\right)}{420 d \sqrt{\sec (c+d x)}}","\frac{8 a^4 \sin (c+d x)}{5 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 a^4 \sin (c+d x)}{7 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{94 a^4 \sin (c+d x)}{21 d \sqrt{\sec (c+d x)}}+\frac{136 a^4 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{64 a^4 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}",1,"(a^4*(Cos[c/2] - I*Sin[c/2])*(Cos[c/2] + I*Sin[c/2])*(-5376*I + ((10752*I)*Hypergeometric2F1[-1/4, 1/2, 3/4, -E^((2*I)*(c + d*x))])/Sqrt[1 + E^((2*I)*(c + d*x))] - (2720*I)*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/4, 1/2, 5/4, -E^((2*I)*(c + d*x))]*Sec[c + d*x] + 1910*Sin[c + d*x] + 336*Sin[2*(c + d*x)] + 30*Sin[3*(c + d*x)]))/(420*d*Sqrt[Sec[c + d*x]])","C",1
192,1,156,187,1.8540401,"\int \frac{(a+a \sec (c+d x))^4}{\sec ^{\frac{9}{2}}(c+d x)} \, dx","Integrate[(a + a*Sec[c + d*x])^4/Sec[c + d*x]^(9/2),x]","\frac{a^4 \left(\frac{51072 i \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i (c+d x)}\right)}{\sqrt{1+e^{2 i (c+d x)}}}-11520 i \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-e^{2 i (c+d x)}\right) \sec (c+d x)+12240 \sin (c+d x)+3556 \sin (2 (c+d x))+720 \sin (3 (c+d x))+70 \sin (4 (c+d x))-25536 i\right)}{2520 d \sqrt{\sec (c+d x)}}","\frac{122 a^4 \sin (c+d x)}{45 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{8 a^4 \sin (c+d x)}{7 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 a^4 \sin (c+d x)}{9 d \sec ^{\frac{7}{2}}(c+d x)}+\frac{32 a^4 \sin (c+d x)}{7 d \sqrt{\sec (c+d x)}}+\frac{32 a^4 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{7 d}+\frac{152 a^4 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}",1,"(a^4*(-25536*I + ((51072*I)*Hypergeometric2F1[-1/4, 1/2, 3/4, -E^((2*I)*(c + d*x))])/Sqrt[1 + E^((2*I)*(c + d*x))] - (11520*I)*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/4, 1/2, 5/4, -E^((2*I)*(c + d*x))]*Sec[c + d*x] + 12240*Sin[c + d*x] + 3556*Sin[2*(c + d*x)] + 720*Sin[3*(c + d*x)] + 70*Sin[4*(c + d*x)]))/(2520*d*Sqrt[Sec[c + d*x]])","C",1
193,1,296,213,3.3754893,"\int \frac{(a+a \sec (c+d x))^4}{\sec ^{\frac{11}{2}}(c+d x)} \, dx","Integrate[(a + a*Sec[c + d*x])^4/Sec[c + d*x]^(11/2),x]","\frac{a^4 \sec ^8\left(\frac{1}{2} (c+d x)\right) (\sec (c+d x)+1)^4 \left(\frac{137055 \sin (2 (c+d x))+48664 \sin (3 (c+d x))+14760 \sin (4 (c+d x))+3080 \sin (5 (c+d x))+315 \sin (6 (c+d x))-213752 \csc (c) \cos (d x)-259336 \csc (c) \cos (2 c+d x)}{384 d \sec ^{\frac{7}{2}}(c+d x)}+\frac{i \sqrt{2} \left(1232 \left(-1+e^{2 i c}\right) \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i (c+d x)}\right)-565 \left(-1+e^{2 i c}\right) e^{i (c+d x)} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-e^{2 i (c+d x)}\right)+1232 \sqrt{1+e^{2 i (c+d x)}}\right) \cos ^4(c+d x)}{\left(-1+e^{2 i c}\right) d \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}}}\right)}{2310}","\frac{128 a^4 \sin (c+d x)}{45 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{150 a^4 \sin (c+d x)}{77 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{8 a^4 \sin (c+d x)}{9 d \sec ^{\frac{7}{2}}(c+d x)}+\frac{2 a^4 \sin (c+d x)}{11 d \sec ^{\frac{9}{2}}(c+d x)}+\frac{904 a^4 \sin (c+d x)}{231 d \sqrt{\sec (c+d x)}}+\frac{904 a^4 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{231 d}+\frac{128 a^4 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}",1,"(a^4*Sec[(c + d*x)/2]^8*(1 + Sec[c + d*x])^4*((I*Sqrt[2]*Cos[c + d*x]^4*(1232*Sqrt[1 + E^((2*I)*(c + d*x))] + 1232*(-1 + E^((2*I)*c))*Hypergeometric2F1[-1/4, 1/2, 3/4, -E^((2*I)*(c + d*x))] - 565*E^(I*(c + d*x))*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/4, 1/2, 5/4, -E^((2*I)*(c + d*x))]))/(d*(-1 + E^((2*I)*c))*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]) + (-213752*Cos[d*x]*Csc[c] - 259336*Cos[2*c + d*x]*Csc[c] + 137055*Sin[2*(c + d*x)] + 48664*Sin[3*(c + d*x)] + 14760*Sin[4*(c + d*x)] + 3080*Sin[5*(c + d*x)] + 315*Sin[6*(c + d*x)])/(384*d*Sec[c + d*x]^(7/2))))/2310","C",1
194,1,291,164,3.2087495,"\int \frac{\sec ^{\frac{7}{2}}(c+d x)}{a+a \sec (c+d x)} \, dx","Integrate[Sec[c + d*x]^(7/2)/(a + a*Sec[c + d*x]),x]","\frac{\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x) \left(-\sqrt{\sec (c+d x)} \left(18 \csc (c) \cos (d x)+\sec (c+d x) \left(\tan \left(\frac{1}{2} (c+d x)\right)-5 \sin \left(\frac{3}{2} (c+d x)\right) \sec \left(\frac{1}{2} (c+d x)\right)\right)\right)+\frac{2 i \sqrt{2} e^{-i (c+d x)} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \left(9 \left(-1+e^{2 i c}\right) \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i (c+d x)}\right)-5 \left(-1+e^{2 i c}\right) e^{i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-e^{2 i (c+d x)}\right)+9 \left(1+e^{2 i (c+d x)}\right)\right)}{-1+e^{2 i c}}\right)}{3 a d (\sec (c+d x)+1)}","-\frac{\sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{d (a \sec (c+d x)+a)}+\frac{5 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 a d}-\frac{3 \sin (c+d x) \sqrt{\sec (c+d x)}}{a d}+\frac{5 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a d}+\frac{3 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}",1,"(Cos[(c + d*x)/2]^2*Sec[c + d*x]*(((2*I)*Sqrt[2]*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*(9*(1 + E^((2*I)*(c + d*x))) + 9*(-1 + E^((2*I)*c))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[-1/4, 1/2, 3/4, -E^((2*I)*(c + d*x))] - 5*E^(I*(c + d*x))*(-1 + E^((2*I)*c))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/4, 1/2, 5/4, -E^((2*I)*(c + d*x))]))/(E^(I*(c + d*x))*(-1 + E^((2*I)*c))) - Sqrt[Sec[c + d*x]]*(18*Cos[d*x]*Csc[c] + Sec[c + d*x]*(-5*Sec[(c + d*x)/2]*Sin[(3*(c + d*x))/2] + Tan[(c + d*x)/2]))))/(3*a*d*(1 + Sec[c + d*x]))","C",1
195,1,262,136,1.7804006,"\int \frac{\sec ^{\frac{5}{2}}(c+d x)}{a+a \sec (c+d x)} \, dx","Integrate[Sec[c + d*x]^(5/2)/(a + a*Sec[c + d*x]),x]","\frac{\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x) \left(\frac{\sqrt{\sec (c+d x)} \left(6 \csc (c) \cos (d x)-2 \tan \left(\frac{1}{2} (c+d x)\right)\right)}{d}-\frac{2 i \sqrt{2} e^{-i (c+d x)} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \left(3 \left(-1+e^{2 i c}\right) \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i (c+d x)}\right)-\left(-1+e^{2 i c}\right) e^{i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-e^{2 i (c+d x)}\right)+3 \left(1+e^{2 i (c+d x)}\right)\right)}{\left(-1+e^{2 i c}\right) d}\right)}{a (\sec (c+d x)+1)}","-\frac{\sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{d (a \sec (c+d x)+a)}+\frac{3 \sin (c+d x) \sqrt{\sec (c+d x)}}{a d}-\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}-\frac{3 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}",1,"(Cos[(c + d*x)/2]^2*Sec[c + d*x]*(((-2*I)*Sqrt[2]*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*(3*(1 + E^((2*I)*(c + d*x))) + 3*(-1 + E^((2*I)*c))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[-1/4, 1/2, 3/4, -E^((2*I)*(c + d*x))] - E^(I*(c + d*x))*(-1 + E^((2*I)*c))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/4, 1/2, 5/4, -E^((2*I)*(c + d*x))]))/(d*E^(I*(c + d*x))*(-1 + E^((2*I)*c))) + (Sqrt[Sec[c + d*x]]*(6*Cos[d*x]*Csc[c] - 2*Tan[(c + d*x)/2]))/d))/(a*(1 + Sec[c + d*x]))","C",1
196,1,201,110,0.5915469,"\int \frac{\sec ^{\frac{3}{2}}(c+d x)}{a+a \sec (c+d x)} \, dx","Integrate[Sec[c + d*x]^(3/2)/(a + a*Sec[c + d*x]),x]","-\frac{2 i e^{-i (c+d x)} \left(-\left(\left(1+e^{i (c+d x)}\right) \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i (c+d x)}\right)\right)+e^{i (c+d x)} \left(1+e^{i (c+d x)}\right) \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-e^{2 i (c+d x)}\right)+e^{2 i (c+d x)}+1\right) \cos ^2\left(\frac{1}{2} (c+d x)\right) \sec ^{\frac{3}{2}}(c+d x)}{a d \left(1+e^{i (c+d x)}\right) (\sec (c+d x)+1)}","-\frac{\sin (c+d x) \sqrt{\sec (c+d x)}}{d (a \sec (c+d x)+a)}+\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}+\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}",1,"((-2*I)*Cos[(c + d*x)/2]^2*(1 + E^((2*I)*(c + d*x)) - (1 + E^(I*(c + d*x)))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[-1/4, 1/2, 3/4, -E^((2*I)*(c + d*x))] + E^(I*(c + d*x))*(1 + E^(I*(c + d*x)))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/4, 1/2, 5/4, -E^((2*I)*(c + d*x))])*Sec[c + d*x]^(3/2))/(a*d*E^(I*(c + d*x))*(1 + E^(I*(c + d*x)))*(1 + Sec[c + d*x]))","C",1
197,1,202,110,0.5620399,"\int \frac{\sqrt{\sec (c+d x)}}{a+a \sec (c+d x)} \, dx","Integrate[Sqrt[Sec[c + d*x]]/(a + a*Sec[c + d*x]),x]","-\frac{2 i e^{-i (c+d x)} \left(\left(1+e^{i (c+d x)}\right) \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i (c+d x)}\right)+e^{i (c+d x)} \left(1+e^{i (c+d x)}\right) \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-e^{2 i (c+d x)}\right)-e^{2 i (c+d x)}-1\right) \cos ^2\left(\frac{1}{2} (c+d x)\right) \sec ^{\frac{3}{2}}(c+d x)}{a d \left(1+e^{i (c+d x)}\right) (\sec (c+d x)+1)}","\frac{\sin (c+d x) \sqrt{\sec (c+d x)}}{d (a \sec (c+d x)+a)}+\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}-\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}",1,"((-2*I)*Cos[(c + d*x)/2]^2*(-1 - E^((2*I)*(c + d*x)) + (1 + E^(I*(c + d*x)))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[-1/4, 1/2, 3/4, -E^((2*I)*(c + d*x))] + E^(I*(c + d*x))*(1 + E^(I*(c + d*x)))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/4, 1/2, 5/4, -E^((2*I)*(c + d*x))])*Sec[c + d*x]^(3/2))/(a*d*E^(I*(c + d*x))*(1 + E^(I*(c + d*x)))*(1 + Sec[c + d*x]))","C",1
198,1,317,112,1.5698341,"\int \frac{1}{\sqrt{\sec (c+d x)} (a+a \sec (c+d x))} \, dx","Integrate[1/(Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x])),x]","\frac{\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x) \left(-\frac{\csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \left(\cos \left(\frac{1}{2} (c-d x)\right)+2 \cos \left(\frac{1}{2} (3 c+d x)\right)+2 \cos \left(\frac{1}{2} (c+3 d x)\right)+\cos \left(\frac{1}{2} (5 c+3 d x)\right)\right) \sec \left(\frac{1}{2} (c+d x)\right) \sqrt{\sec (c+d x)}}{2 d}+\frac{2 i \sqrt{2} e^{-i (c+d x)} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \left(3 \left(-1+e^{2 i c}\right) \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i (c+d x)}\right)+\left(-1+e^{2 i c}\right) e^{i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-e^{2 i (c+d x)}\right)+3 \left(1+e^{2 i (c+d x)}\right)\right)}{\left(-1+e^{2 i c}\right) d}\right)}{a (\sec (c+d x)+1)}","-\frac{\sin (c+d x) \sqrt{\sec (c+d x)}}{d (a \sec (c+d x)+a)}-\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}+\frac{3 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}",1,"(Cos[(c + d*x)/2]^2*(((2*I)*Sqrt[2]*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*(3*(1 + E^((2*I)*(c + d*x))) + 3*(-1 + E^((2*I)*c))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[-1/4, 1/2, 3/4, -E^((2*I)*(c + d*x))] + E^(I*(c + d*x))*(-1 + E^((2*I)*c))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/4, 1/2, 5/4, -E^((2*I)*(c + d*x))]))/(d*E^(I*(c + d*x))*(-1 + E^((2*I)*c))) - ((Cos[(c - d*x)/2] + 2*Cos[(3*c + d*x)/2] + 2*Cos[(c + 3*d*x)/2] + Cos[(5*c + 3*d*x)/2])*Csc[c/2]*Sec[c/2]*Sec[(c + d*x)/2]*Sqrt[Sec[c + d*x]])/(2*d))*Sec[c + d*x])/(a*(1 + Sec[c + d*x]))","C",1
199,1,318,140,4.4360546,"\int \frac{1}{\sec ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x))} \, dx","Integrate[1/(Sec[c + d*x]^(3/2)*(a + a*Sec[c + d*x])),x]","\frac{\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x) \left(2 \sqrt{\sec (c+d x)} \left(\sin (2 c) \cos (2 d x)-6 \cos (c) \sin (d x)+\cos (2 c) \sin (2 d x)+3 (\cos (2 c)+2) \csc (c) \cos (d x)-3 \sec \left(\frac{c}{2}\right) \sin \left(\frac{d x}{2}\right) \sec \left(\frac{1}{2} (c+d x)\right)-3 \tan \left(\frac{c}{2}\right)\right)-\frac{2 i \sqrt{2} e^{-i (c+d x)} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \left(9 \left(-1+e^{2 i c}\right) \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i (c+d x)}\right)+5 \left(-1+e^{2 i c}\right) e^{i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-e^{2 i (c+d x)}\right)+9 \left(1+e^{2 i (c+d x)}\right)\right)}{-1+e^{2 i c}}\right)}{3 a d (\sec (c+d x)+1)}","\frac{5 \sin (c+d x)}{3 a d \sqrt{\sec (c+d x)}}-\frac{\sin (c+d x)}{d \sqrt{\sec (c+d x)} (a \sec (c+d x)+a)}+\frac{5 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a d}-\frac{3 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}",1,"(Cos[(c + d*x)/2]^2*Sec[c + d*x]*(((-2*I)*Sqrt[2]*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*(9*(1 + E^((2*I)*(c + d*x))) + 9*(-1 + E^((2*I)*c))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[-1/4, 1/2, 3/4, -E^((2*I)*(c + d*x))] + 5*E^(I*(c + d*x))*(-1 + E^((2*I)*c))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/4, 1/2, 5/4, -E^((2*I)*(c + d*x))]))/(E^(I*(c + d*x))*(-1 + E^((2*I)*c))) + 2*Sqrt[Sec[c + d*x]]*(3*(2 + Cos[2*c])*Cos[d*x]*Csc[c] + Cos[2*d*x]*Sin[2*c] - 3*Sec[c/2]*Sec[(c + d*x)/2]*Sin[(d*x)/2] - 6*Cos[c]*Sin[d*x] + Cos[2*c]*Sin[2*d*x] - 3*Tan[c/2])))/(3*a*d*(1 + Sec[c + d*x]))","C",1
200,1,347,168,2.5825327,"\int \frac{1}{\sec ^{\frac{5}{2}}(c+d x) (a+a \sec (c+d x))} \, dx","Integrate[1/(Sec[c + d*x]^(5/2)*(a + a*Sec[c + d*x])),x]","\frac{\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x) \left(-\sqrt{\sec (c+d x)} \left(18 (11 \cos (2 c)+17) \csc (c) \cos (d x)+4 \left(10 \sin (2 c) \cos (2 d x)-3 \sin (3 c) \cos (3 d x)-99 \cos (c) \sin (d x)+10 \cos (2 c) \sin (2 d x)-3 \cos (3 c) \sin (3 d x)-30 \sec \left(\frac{c}{2}\right) \sin \left(\frac{d x}{2}\right) \sec \left(\frac{1}{2} (c+d x)\right)-30 \tan \left(\frac{c}{2}\right)\right)\right)+\frac{8 i \sqrt{2} e^{-i (c+d x)} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \left(63 \left(-1+e^{2 i c}\right) \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i (c+d x)}\right)+25 \left(-1+e^{2 i c}\right) e^{i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-e^{2 i (c+d x)}\right)+63 \left(1+e^{2 i (c+d x)}\right)\right)}{-1+e^{2 i c}}\right)}{60 a d (\sec (c+d x)+1)}","-\frac{\sin (c+d x)}{d \sec ^{\frac{3}{2}}(c+d x) (a \sec (c+d x)+a)}+\frac{7 \sin (c+d x)}{5 a d \sec ^{\frac{3}{2}}(c+d x)}-\frac{5 \sin (c+d x)}{3 a d \sqrt{\sec (c+d x)}}-\frac{5 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a d}+\frac{21 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 a d}",1,"(Cos[(c + d*x)/2]^2*Sec[c + d*x]*(((8*I)*Sqrt[2]*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*(63*(1 + E^((2*I)*(c + d*x))) + 63*(-1 + E^((2*I)*c))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[-1/4, 1/2, 3/4, -E^((2*I)*(c + d*x))] + 25*E^(I*(c + d*x))*(-1 + E^((2*I)*c))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/4, 1/2, 5/4, -E^((2*I)*(c + d*x))]))/(E^(I*(c + d*x))*(-1 + E^((2*I)*c))) - Sqrt[Sec[c + d*x]]*(18*(17 + 11*Cos[2*c])*Cos[d*x]*Csc[c] + 4*(10*Cos[2*d*x]*Sin[2*c] - 3*Cos[3*d*x]*Sin[3*c] - 30*Sec[c/2]*Sec[(c + d*x)/2]*Sin[(d*x)/2] - 99*Cos[c]*Sin[d*x] + 10*Cos[2*c]*Sin[2*d*x] - 3*Cos[3*c]*Sin[3*d*x] - 30*Tan[c/2]))))/(60*a*d*(1 + Sec[c + d*x]))","C",1
201,1,287,202,3.8410084,"\int \frac{\sec ^{\frac{9}{2}}(c+d x)}{(a+a \sec (c+d x))^2} \, dx","Integrate[Sec[c + d*x]^(9/2)/(a + a*Sec[c + d*x])^2,x]","-\frac{\left(-1+e^{i c}\right) \csc \left(\frac{c}{2}\right) e^{-\frac{1}{2} i (4 c+3 d x)} \cos \left(\frac{1}{2} (c+d x)\right) \sec ^{\frac{5}{2}}(c+d x) \left(7 e^{i (c+d x)} \left(1+e^{2 i (c+d x)}\right)^{3/2} \left(1+e^{i (c+d x)}\right)^3 \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-37 e^{i (c+d x)}-65 e^{2 i (c+d x)}-82 e^{3 i (c+d x)}-68 e^{4 i (c+d x)}-53 e^{5 i (c+d x)}-21 e^{6 i (c+d x)}+10 i \left(1+e^{2 i (c+d x)}\right) \left(1+e^{i (c+d x)}\right)^3 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-10\right)}{12 a^2 d \left(1+e^{2 i (c+d x)}\right) (\sec (c+d x)+1)^2}","-\frac{7 \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{3 a^2 d (\sec (c+d x)+1)}+\frac{10 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 a^2 d}-\frac{7 \sin (c+d x) \sqrt{\sec (c+d x)}}{a^2 d}+\frac{10 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^2 d}+\frac{7 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d}-\frac{\sin (c+d x) \sec ^{\frac{7}{2}}(c+d x)}{3 d (a \sec (c+d x)+a)^2}",1,"-1/12*((-1 + E^(I*c))*Cos[(c + d*x)/2]*Csc[c/2]*(-10 - 37*E^(I*(c + d*x)) - 65*E^((2*I)*(c + d*x)) - 82*E^((3*I)*(c + d*x)) - 68*E^((4*I)*(c + d*x)) - 53*E^((5*I)*(c + d*x)) - 21*E^((6*I)*(c + d*x)) + (10*I)*(1 + E^(I*(c + d*x)))^3*(1 + E^((2*I)*(c + d*x)))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + 7*E^(I*(c + d*x))*(1 + E^(I*(c + d*x)))^3*(1 + E^((2*I)*(c + d*x)))^(3/2)*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])*Sec[c + d*x]^(5/2))/(a^2*d*E^((I/2)*(4*c + 3*d*x))*(1 + E^((2*I)*(c + d*x)))*(1 + Sec[c + d*x])^2)","C",1
202,1,252,176,1.4075448,"\int \frac{\sec ^{\frac{7}{2}}(c+d x)}{(a+a \sec (c+d x))^2} \, dx","Integrate[Sec[c + d*x]^(7/2)/(a + a*Sec[c + d*x])^2,x]","-\frac{e^{-i d x} \cos \left(\frac{1}{2} (c+d x)\right) \sec ^{\frac{5}{2}}(c+d x) \left(\cos \left(\frac{1}{2} (c+3 d x)\right)+i \sin \left(\frac{1}{2} (c+3 d x)\right)\right) \left(-4 i e^{-i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \left(1+e^{i (c+d x)}\right)^3 \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)+i (12 i \sin (c+d x)+7 i \sin (2 (c+d x))+50 \cos (c+d x)+17 \cos (2 (c+d x))+29)+40 \cos ^3\left(\frac{1}{2} (c+d x)\right) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(\cos \left(\frac{1}{2} (c+d x)\right)-i \sin \left(\frac{1}{2} (c+d x)\right)\right)\right)}{6 a^2 d (\sec (c+d x)+1)^2}","-\frac{5 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 a^2 d (\sec (c+d x)+1)}+\frac{4 \sin (c+d x) \sqrt{\sec (c+d x)}}{a^2 d}-\frac{5 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^2 d}-\frac{4 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d}-\frac{\sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{3 d (a \sec (c+d x)+a)^2}",1,"-1/6*(Cos[(c + d*x)/2]*Sec[c + d*x]^(5/2)*(((-4*I)*(1 + E^(I*(c + d*x)))^3*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])/E^(I*(c + d*x)) + 40*Cos[(c + d*x)/2]^3*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*(Cos[(c + d*x)/2] - I*Sin[(c + d*x)/2]) + I*(29 + 50*Cos[c + d*x] + 17*Cos[2*(c + d*x)] + (12*I)*Sin[c + d*x] + (7*I)*Sin[2*(c + d*x)]))*(Cos[(c + 3*d*x)/2] + I*Sin[(c + 3*d*x)/2]))/(a^2*d*E^(I*d*x)*(1 + Sec[c + d*x])^2)","C",1
203,1,242,149,1.2665774,"\int \frac{\sec ^{\frac{5}{2}}(c+d x)}{(a+a \sec (c+d x))^2} \, dx","Integrate[Sec[c + d*x]^(5/2)/(a + a*Sec[c + d*x])^2,x]","\frac{e^{-i d x} \cos \left(\frac{1}{2} (c+d x)\right) \sec ^{\frac{5}{2}}(c+d x) \left(\cos \left(\frac{1}{2} (c+3 d x)\right)+i \sin \left(\frac{1}{2} (c+3 d x)\right)\right) \left(-i e^{-i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \left(1+e^{i (c+d x)}\right)^3 \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)+i (i \sin (2 (c+d x))+14 \cos (c+d x)+5 \cos (2 (c+d x))+5)+16 \cos ^3\left(\frac{1}{2} (c+d x)\right) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(\cos \left(\frac{1}{2} (c+d x)\right)-i \sin \left(\frac{1}{2} (c+d x)\right)\right)\right)}{6 a^2 d (\sec (c+d x)+1)^2}","-\frac{\sin (c+d x) \sqrt{\sec (c+d x)}}{a^2 d (\sec (c+d x)+1)}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^2 d}+\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d}-\frac{\sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 d (a \sec (c+d x)+a)^2}",1,"(Cos[(c + d*x)/2]*Sec[c + d*x]^(5/2)*(((-I)*(1 + E^(I*(c + d*x)))^3*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])/E^(I*(c + d*x)) + 16*Cos[(c + d*x)/2]^3*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*(Cos[(c + d*x)/2] - I*Sin[(c + d*x)/2]) + I*(5 + 14*Cos[c + d*x] + 5*Cos[2*(c + d*x)] + I*Sin[2*(c + d*x)]))*(Cos[(c + 3*d*x)/2] + I*Sin[(c + 3*d*x)/2]))/(6*a^2*d*E^(I*d*x)*(1 + Sec[c + d*x])^2)","C",1
204,1,98,77,0.4256283,"\int \frac{\sec ^{\frac{3}{2}}(c+d x)}{(a+a \sec (c+d x))^2} \, dx","Integrate[Sec[c + d*x]^(3/2)/(a + a*Sec[c + d*x])^2,x]","\frac{\cos \left(\frac{1}{2} (c+d x)\right) \sec ^{\frac{5}{2}}(c+d x) \left(-\sin \left(\frac{1}{2} (c+d x)\right)+\sin \left(\frac{3}{2} (c+d x)\right)+4 \sqrt{\cos (c+d x)} \cos ^3\left(\frac{1}{2} (c+d x)\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{3 a^2 d (\sec (c+d x)+1)^2}","\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^2 d}+\frac{\sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 d (a \sec (c+d x)+a)^2}",1,"(Cos[(c + d*x)/2]*Sec[c + d*x]^(5/2)*(4*Cos[(c + d*x)/2]^3*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] - Sin[(c + d*x)/2] + Sin[(3*(c + d*x))/2]))/(3*a^2*d*(1 + Sec[c + d*x])^2)","A",1
205,1,239,149,1.4987394,"\int \frac{\sqrt{\sec (c+d x)}}{(a+a \sec (c+d x))^2} \, dx","Integrate[Sqrt[Sec[c + d*x]]/(a + a*Sec[c + d*x])^2,x]","\frac{e^{-i d x} \cos \left(\frac{1}{2} (c+d x)\right) \sec ^{\frac{5}{2}}(c+d x) \left(\cos \left(\frac{1}{2} (c+3 d x)\right)+i \sin \left(\frac{1}{2} (c+3 d x)\right)\right) \left(i \left(e^{-i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \left(1+e^{i (c+d x)}\right)^3 \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)+i \sin (2 (c+d x))-10 \cos (c+d x)-7 \cos (2 (c+d x))-7\right)+16 \sqrt{\cos (c+d x)} \cos ^3\left(\frac{1}{2} (c+d x)\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(\cos \left(\frac{1}{2} (c+d x)\right)-i \sin \left(\frac{1}{2} (c+d x)\right)\right)\right)}{6 a^2 d (\sec (c+d x)+1)^2}","\frac{\sin (c+d x) \sqrt{\sec (c+d x)}}{a^2 d (\sec (c+d x)+1)}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^2 d}-\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d}-\frac{\sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 d (a \sec (c+d x)+a)^2}",1,"(Cos[(c + d*x)/2]*Sec[c + d*x]^(5/2)*(16*Cos[(c + d*x)/2]^3*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*(Cos[(c + d*x)/2] - I*Sin[(c + d*x)/2]) + I*(-7 - 10*Cos[c + d*x] - 7*Cos[2*(c + d*x)] + ((1 + E^(I*(c + d*x)))^3*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])/E^(I*(c + d*x)) + I*Sin[2*(c + d*x)]))*(Cos[(c + 3*d*x)/2] + I*Sin[(c + 3*d*x)/2]))/(6*a^2*d*E^(I*d*x)*(1 + Sec[c + d*x])^2)","C",1
206,1,260,152,0.9518733,"\int \frac{1}{\sqrt{\sec (c+d x)} (a+a \sec (c+d x))^2} \, dx","Integrate[1/(Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x])^2),x]","-\frac{i \sqrt{2} e^{-i (c+d x)} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \left(4 e^{2 i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \left(1+e^{i (c+d x)}\right)^3 \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-16 e^{i (c+d x)}-23 e^{2 i (c+d x)}-25 e^{3 i (c+d x)}-20 e^{4 i (c+d x)}-9 e^{5 i (c+d x)}-5 i e^{i (c+d x)} \left(1+e^{i (c+d x)}\right)^3 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-3\right)}{3 a^2 d \left(1+e^{i (c+d x)}\right)^3}","-\frac{5 \sin (c+d x) \sqrt{\sec (c+d x)}}{3 a^2 d (\sec (c+d x)+1)}-\frac{5 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^2 d}+\frac{4 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d}-\frac{\sin (c+d x) \sqrt{\sec (c+d x)}}{3 d (a \sec (c+d x)+a)^2}",1,"((-1/3*I)*Sqrt[2]*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*(-3 - 16*E^(I*(c + d*x)) - 23*E^((2*I)*(c + d*x)) - 25*E^((3*I)*(c + d*x)) - 20*E^((4*I)*(c + d*x)) - 9*E^((5*I)*(c + d*x)) - (5*I)*E^(I*(c + d*x))*(1 + E^(I*(c + d*x)))^3*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + 4*E^((2*I)*(c + d*x))*(1 + E^(I*(c + d*x)))^3*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))]))/(a^2*d*E^(I*(c + d*x))*(1 + E^(I*(c + d*x)))^3)","C",1
207,1,257,178,1.9916799,"\int \frac{1}{\sec ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x))^2} \, dx","Integrate[1/(Sec[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^2),x]","\frac{e^{-i d x} \cos \left(\frac{1}{2} (c+d x)\right) \sec ^{\frac{5}{2}}(c+d x) (\cos (d x)+i \sin (d x)) \left(7 i e^{-\frac{1}{2} i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \left(1+e^{i (c+d x)}\right)^3 \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)+3 \sin \left(\frac{1}{2} (c+d x)\right)+10 \sin \left(\frac{3}{2} (c+d x)\right)+12 \sin \left(\frac{5}{2} (c+d x)\right)+\sin \left(\frac{7}{2} (c+d x)\right)-84 i \cos \left(\frac{1}{2} (c+d x)\right)-63 i \cos \left(\frac{3}{2} (c+d x)\right)-21 i \cos \left(\frac{5}{2} (c+d x)\right)+80 \cos ^3\left(\frac{1}{2} (c+d x)\right) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{6 a^2 d (\sec (c+d x)+1)^2}","\frac{10 \sin (c+d x)}{3 a^2 d \sqrt{\sec (c+d x)}}-\frac{7 \sin (c+d x)}{3 a^2 d \sqrt{\sec (c+d x)} (\sec (c+d x)+1)}+\frac{10 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^2 d}-\frac{7 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d}-\frac{\sin (c+d x)}{3 d \sqrt{\sec (c+d x)} (a \sec (c+d x)+a)^2}",1,"(Cos[(c + d*x)/2]*Sec[c + d*x]^(5/2)*(Cos[d*x] + I*Sin[d*x])*((-84*I)*Cos[(c + d*x)/2] - (63*I)*Cos[(3*(c + d*x))/2] - (21*I)*Cos[(5*(c + d*x))/2] + 80*Cos[(c + d*x)/2]^3*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + ((7*I)*(1 + E^(I*(c + d*x)))^3*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])/E^((I/2)*(c + d*x)) + 3*Sin[(c + d*x)/2] + 10*Sin[(3*(c + d*x))/2] + 12*Sin[(5*(c + d*x))/2] + Sin[(7*(c + d*x))/2]))/(6*a^2*d*E^(I*d*x)*(1 + Sec[c + d*x])^2)","C",1
208,1,271,200,1.9753542,"\int \frac{1}{\sec ^{\frac{5}{2}}(c+d x) (a+a \sec (c+d x))^2} \, dx","Integrate[1/(Sec[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^2),x]","\frac{e^{-i d x} \cos \left(\frac{1}{2} (c+d x)\right) \sec ^{\frac{5}{2}}(c+d x) (\cos (d x)+i \sin (d x)) \left(-112 i e^{-\frac{1}{2} i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \left(1+e^{i (c+d x)}\right)^3 \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-34 \sin \left(\frac{1}{2} (c+d x)\right)-148 \sin \left(\frac{3}{2} (c+d x)\right)-168 \sin \left(\frac{5}{2} (c+d x)\right)-11 \sin \left(\frac{7}{2} (c+d x)\right)+3 \sin \left(\frac{9}{2} (c+d x)\right)+1344 i \cos \left(\frac{1}{2} (c+d x)\right)+1008 i \cos \left(\frac{3}{2} (c+d x)\right)+336 i \cos \left(\frac{5}{2} (c+d x)\right)-1200 \cos ^3\left(\frac{1}{2} (c+d x)\right) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{60 a^2 d (\sec (c+d x)+1)^2}","-\frac{3 \sin (c+d x)}{a^2 d \sec ^{\frac{3}{2}}(c+d x) (\sec (c+d x)+1)}+\frac{56 \sin (c+d x)}{15 a^2 d \sec ^{\frac{3}{2}}(c+d x)}-\frac{5 \sin (c+d x)}{a^2 d \sqrt{\sec (c+d x)}}-\frac{5 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d}+\frac{56 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 a^2 d}-\frac{\sin (c+d x)}{3 d \sec ^{\frac{3}{2}}(c+d x) (a \sec (c+d x)+a)^2}",1,"(Cos[(c + d*x)/2]*Sec[c + d*x]^(5/2)*(Cos[d*x] + I*Sin[d*x])*((1344*I)*Cos[(c + d*x)/2] + (1008*I)*Cos[(3*(c + d*x))/2] + (336*I)*Cos[(5*(c + d*x))/2] - 1200*Cos[(c + d*x)/2]^3*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] - ((112*I)*(1 + E^(I*(c + d*x)))^3*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])/E^((I/2)*(c + d*x)) - 34*Sin[(c + d*x)/2] - 148*Sin[(3*(c + d*x))/2] - 168*Sin[(5*(c + d*x))/2] - 11*Sin[(7*(c + d*x))/2] + 3*Sin[(9*(c + d*x))/2]))/(60*a^2*d*E^(I*d*x)*(1 + Sec[c + d*x])^2)","C",1
209,1,378,247,5.3366362,"\int \frac{\sec ^{\frac{11}{2}}(c+d x)}{(a+a \sec (c+d x))^3} \, dx","Integrate[Sec[c + d*x]^(11/2)/(a + a*Sec[c + d*x])^3,x]","\frac{\csc \left(\frac{c}{2}\right) e^{-i d x} \left(\frac{\left(-1+e^{i c}\right) e^{-\frac{3}{2} i (2 c+d x)} \cos \left(\frac{1}{2} (c+d x)\right) \sec ^{\frac{7}{2}}(c+d x) \left(944 e^{i (c+d x)}+2476 e^{2 i (c+d x)}+4148 e^{3 i (c+d x)}+5134 e^{4 i (c+d x)}+4664 e^{5 i (c+d x)}+3340 e^{6 i (c+d x)}+1620 e^{7 i (c+d x)}+357 e^{8 i (c+d x)}-165 i \left(1+e^{2 i (c+d x)}\right) \left(1+e^{i (c+d x)}\right)^5 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+165\right)}{16 \left(1+e^{2 i (c+d x)}\right)}-119 \sqrt{2} \left(-1+e^{2 i c}\right) \sec \left(\frac{c}{2}\right) e^{2 i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right) \cos ^6\left(\frac{1}{2} (c+d x)\right) \sec ^3(c+d x)\right)}{15 a^3 d (\sec (c+d x)+1)^3}","-\frac{119 \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{30 d \left(a^3 \sec (c+d x)+a^3\right)}+\frac{11 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{2 a^3 d}-\frac{119 \sin (c+d x) \sqrt{\sec (c+d x)}}{10 a^3 d}+\frac{11 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{2 a^3 d}+\frac{119 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{10 a^3 d}-\frac{\sin (c+d x) \sec ^{\frac{9}{2}}(c+d x)}{5 d (a \sec (c+d x)+a)^3}-\frac{2 \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x)}{3 a d (a \sec (c+d x)+a)^2}",1,"(Csc[c/2]*(-119*Sqrt[2]*E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*Cos[(c + d*x)/2]^6*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))]*Sec[c/2]*Sec[c + d*x]^3 + ((-1 + E^(I*c))*Cos[(c + d*x)/2]*(165 + 944*E^(I*(c + d*x)) + 2476*E^((2*I)*(c + d*x)) + 4148*E^((3*I)*(c + d*x)) + 5134*E^((4*I)*(c + d*x)) + 4664*E^((5*I)*(c + d*x)) + 3340*E^((6*I)*(c + d*x)) + 1620*E^((7*I)*(c + d*x)) + 357*E^((8*I)*(c + d*x)) - (165*I)*(1 + E^(I*(c + d*x)))^5*(1 + E^((2*I)*(c + d*x)))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])*Sec[c + d*x]^(7/2))/(16*E^(((3*I)/2)*(2*c + d*x))*(1 + E^((2*I)*(c + d*x))))))/(15*a^3*d*E^(I*d*x)*(1 + Sec[c + d*x])^3)","C",1
210,1,371,221,2.1202653,"\int \frac{\sec ^{\frac{9}{2}}(c+d x)}{(a+a \sec (c+d x))^3} \, dx","Integrate[Sec[c + d*x]^(9/2)/(a + a*Sec[c + d*x])^3,x]","\frac{2 \cos ^6\left(\frac{1}{2} (c+d x)\right) \sec ^3(c+d x) \left(\frac{1}{32} \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \left(1284 \cos \left(\frac{1}{2} (c-d x)\right)+921 \cos \left(\frac{1}{2} (3 c+d x)\right)+1243 \cos \left(\frac{1}{2} (c+3 d x)\right)+374 \cos \left(\frac{1}{2} (5 c+3 d x)\right)+670 \cos \left(\frac{1}{2} (3 c+5 d x)\right)+65 \cos \left(\frac{1}{2} (7 c+5 d x)\right)+147 \cos \left(\frac{1}{2} (5 c+7 d x)\right)\right) \sec ^5\left(\frac{1}{2} (c+d x)\right) \sqrt{\sec (c+d x)}-\frac{2 i \sqrt{2} e^{-i (c+d x)} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \left(147 \left(-1+e^{2 i c}\right) \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i (c+d x)}\right)-65 \left(-1+e^{2 i c}\right) e^{i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-e^{2 i (c+d x)}\right)+147 \left(1+e^{2 i (c+d x)}\right)\right)}{-1+e^{2 i c}}\right)}{15 a^3 d (\sec (c+d x)+1)^3}","-\frac{13 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{6 d \left(a^3 \sec (c+d x)+a^3\right)}+\frac{49 \sin (c+d x) \sqrt{\sec (c+d x)}}{10 a^3 d}-\frac{13 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{6 a^3 d}-\frac{49 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{10 a^3 d}-\frac{\sin (c+d x) \sec ^{\frac{7}{2}}(c+d x)}{5 d (a \sec (c+d x)+a)^3}-\frac{8 \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{15 a d (a \sec (c+d x)+a)^2}",1,"(2*Cos[(c + d*x)/2]^6*(((-2*I)*Sqrt[2]*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*(147*(1 + E^((2*I)*(c + d*x))) + 147*(-1 + E^((2*I)*c))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[-1/4, 1/2, 3/4, -E^((2*I)*(c + d*x))] - 65*E^(I*(c + d*x))*(-1 + E^((2*I)*c))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/4, 1/2, 5/4, -E^((2*I)*(c + d*x))]))/(E^(I*(c + d*x))*(-1 + E^((2*I)*c))) + ((1284*Cos[(c - d*x)/2] + 921*Cos[(3*c + d*x)/2] + 1243*Cos[(c + 3*d*x)/2] + 374*Cos[(5*c + 3*d*x)/2] + 670*Cos[(3*c + 5*d*x)/2] + 65*Cos[(7*c + 5*d*x)/2] + 147*Cos[(5*c + 7*d*x)/2])*Csc[c/2]*Sec[c/2]*Sec[(c + d*x)/2]^5*Sqrt[Sec[c + d*x]])/32)*Sec[c + d*x]^3)/(15*a^3*d*(1 + Sec[c + d*x])^3)","C",1
211,1,274,195,4.8399374,"\int \frac{\sec ^{\frac{7}{2}}(c+d x)}{(a+a \sec (c+d x))^3} \, dx","Integrate[Sec[c + d*x]^(7/2)/(a + a*Sec[c + d*x])^3,x]","\frac{e^{-i d x} \cos \left(\frac{1}{2} (c+d x)\right) \sec ^{\frac{7}{2}}(c+d x) \left(\cos \left(\frac{1}{2} (c+3 d x)\right)+i \sin \left(\frac{1}{2} (c+3 d x)\right)\right) \left(-3 i e^{-2 i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \left(1+e^{i (c+d x)}\right)^5 \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)+2 i (2 i \sin (c+d x)+6 i \sin (2 (c+d x))+2 i \sin (3 (c+d x))+69 \cos (c+d x)+34 \cos (2 (c+d x))+7 \cos (3 (c+d x))+34)+160 \cos ^5\left(\frac{1}{2} (c+d x)\right) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(\cos \left(\frac{1}{2} (c+d x)\right)-i \sin \left(\frac{1}{2} (c+d x)\right)\right)\right)}{40 a^3 d (\sec (c+d x)+1)^3}","-\frac{9 \sin (c+d x) \sqrt{\sec (c+d x)}}{10 d \left(a^3 \sec (c+d x)+a^3\right)}+\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{2 a^3 d}+\frac{9 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{10 a^3 d}-\frac{\sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{5 d (a \sec (c+d x)+a)^3}-\frac{2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{5 a d (a \sec (c+d x)+a)^2}",1,"(Cos[(c + d*x)/2]*Sec[c + d*x]^(7/2)*(((-3*I)*(1 + E^(I*(c + d*x)))^5*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])/E^((2*I)*(c + d*x)) + 160*Cos[(c + d*x)/2]^5*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*(Cos[(c + d*x)/2] - I*Sin[(c + d*x)/2]) + (2*I)*(34 + 69*Cos[c + d*x] + 34*Cos[2*(c + d*x)] + 7*Cos[3*(c + d*x)] + (2*I)*Sin[c + d*x] + (6*I)*Sin[2*(c + d*x)] + (2*I)*Sin[3*(c + d*x)]))*(Cos[(c + 3*d*x)/2] + I*Sin[(c + 3*d*x)/2]))/(40*a^3*d*E^(I*d*x)*(1 + Sec[c + d*x])^3)","C",1
212,1,371,195,1.8150013,"\int \frac{\sec ^{\frac{5}{2}}(c+d x)}{(a+a \sec (c+d x))^3} \, dx","Integrate[Sec[c + d*x]^(5/2)/(a + a*Sec[c + d*x])^3,x]","\frac{2 \cos ^6\left(\frac{1}{2} (c+d x)\right) \sec ^3(c+d x) \left(-\frac{1}{32} \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \left(36 \cos \left(\frac{1}{2} (c-d x)\right)+9 \cos \left(\frac{1}{2} (3 c+d x)\right)+7 \cos \left(\frac{1}{2} (c+3 d x)\right)+26 \cos \left(\frac{1}{2} (5 c+3 d x)\right)+10 \cos \left(\frac{1}{2} (3 c+5 d x)\right)+5 \cos \left(\frac{1}{2} (7 c+5 d x)\right)+3 \cos \left(\frac{1}{2} (5 c+7 d x)\right)\right) \sec ^5\left(\frac{1}{2} (c+d x)\right) \sqrt{\sec (c+d x)}+\frac{2 i \sqrt{2} e^{-i (c+d x)} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \left(3 \left(-1+e^{2 i c}\right) \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i (c+d x)}\right)-5 \left(-1+e^{2 i c}\right) e^{i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-e^{2 i (c+d x)}\right)+3 \left(1+e^{2 i (c+d x)}\right)\right)}{-1+e^{2 i c}}\right)}{15 a^3 d (\sec (c+d x)+1)^3}","\frac{\sin (c+d x) \sqrt{\sec (c+d x)}}{6 d \left(a^3 \sec (c+d x)+a^3\right)}+\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{6 a^3 d}+\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{10 a^3 d}-\frac{\sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{5 d (a \sec (c+d x)+a)^3}-\frac{4 \sin (c+d x) \sqrt{\sec (c+d x)}}{15 a d (a \sec (c+d x)+a)^2}",1,"(2*Cos[(c + d*x)/2]^6*(((2*I)*Sqrt[2]*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*(3*(1 + E^((2*I)*(c + d*x))) + 3*(-1 + E^((2*I)*c))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[-1/4, 1/2, 3/4, -E^((2*I)*(c + d*x))] - 5*E^(I*(c + d*x))*(-1 + E^((2*I)*c))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/4, 1/2, 5/4, -E^((2*I)*(c + d*x))]))/(E^(I*(c + d*x))*(-1 + E^((2*I)*c))) - ((36*Cos[(c - d*x)/2] + 9*Cos[(3*c + d*x)/2] + 7*Cos[(c + 3*d*x)/2] + 26*Cos[(5*c + 3*d*x)/2] + 10*Cos[(3*c + 5*d*x)/2] + 5*Cos[(7*c + 5*d*x)/2] + 3*Cos[(5*c + 7*d*x)/2])*Csc[c/2]*Sec[c/2]*Sec[(c + d*x)/2]^5*Sqrt[Sec[c + d*x]])/32)*Sec[c + d*x]^3)/(15*a^3*d*(1 + Sec[c + d*x])^3)","C",1
213,1,371,195,1.9658452,"\int \frac{\sec ^{\frac{3}{2}}(c+d x)}{(a+a \sec (c+d x))^3} \, dx","Integrate[Sec[c + d*x]^(3/2)/(a + a*Sec[c + d*x])^3,x]","\frac{2 \cos ^6\left(\frac{1}{2} (c+d x)\right) \sec ^3(c+d x) \left(\frac{1}{32} \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \left(36 \cos \left(\frac{1}{2} (c-d x)\right)+9 \cos \left(\frac{1}{2} (3 c+d x)\right)+17 \cos \left(\frac{1}{2} (c+3 d x)\right)+16 \cos \left(\frac{1}{2} (5 c+3 d x)\right)+20 \cos \left(\frac{1}{2} (3 c+5 d x)\right)-5 \cos \left(\frac{1}{2} (7 c+5 d x)\right)+3 \cos \left(\frac{1}{2} (5 c+7 d x)\right)\right) \sec ^5\left(\frac{1}{2} (c+d x)\right) \sqrt{\sec (c+d x)}-\frac{2 i \sqrt{2} e^{-i (c+d x)} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \left(3 \left(-1+e^{2 i c}\right) \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i (c+d x)}\right)+5 \left(-1+e^{2 i c}\right) e^{i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-e^{2 i (c+d x)}\right)+3 \left(1+e^{2 i (c+d x)}\right)\right)}{-1+e^{2 i c}}\right)}{15 a^3 d (\sec (c+d x)+1)^3}","\frac{\sin (c+d x) \sqrt{\sec (c+d x)}}{6 d \left(a^3 \sec (c+d x)+a^3\right)}+\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{6 a^3 d}-\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{10 a^3 d}+\frac{\sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{5 d (a \sec (c+d x)+a)^3}-\frac{\sin (c+d x) \sqrt{\sec (c+d x)}}{15 a d (a \sec (c+d x)+a)^2}",1,"(2*Cos[(c + d*x)/2]^6*(((-2*I)*Sqrt[2]*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*(3*(1 + E^((2*I)*(c + d*x))) + 3*(-1 + E^((2*I)*c))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[-1/4, 1/2, 3/4, -E^((2*I)*(c + d*x))] + 5*E^(I*(c + d*x))*(-1 + E^((2*I)*c))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/4, 1/2, 5/4, -E^((2*I)*(c + d*x))]))/(E^(I*(c + d*x))*(-1 + E^((2*I)*c))) + ((36*Cos[(c - d*x)/2] + 9*Cos[(3*c + d*x)/2] + 17*Cos[(c + 3*d*x)/2] + 16*Cos[(5*c + 3*d*x)/2] + 20*Cos[(3*c + 5*d*x)/2] - 5*Cos[(7*c + 5*d*x)/2] + 3*Cos[(5*c + 7*d*x)/2])*Csc[c/2]*Sec[c/2]*Sec[(c + d*x)/2]^5*Sqrt[Sec[c + d*x]])/32)*Sec[c + d*x]^3)/(15*a^3*d*(1 + Sec[c + d*x])^3)","C",1
214,1,272,195,5.6229503,"\int \frac{\sqrt{\sec (c+d x)}}{(a+a \sec (c+d x))^3} \, dx","Integrate[Sqrt[Sec[c + d*x]]/(a + a*Sec[c + d*x])^3,x]","\frac{e^{-i d x} \cos \left(\frac{1}{2} (c+d x)\right) \sec ^{\frac{7}{2}}(c+d x) \left(\cos \left(\frac{1}{2} (c+3 d x)\right)+i \sin \left(\frac{1}{2} (c+3 d x)\right)\right) \left(i \left(3 e^{-2 i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \left(1+e^{i (c+d x)}\right)^5 \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)+6 i \sin (c+d x)+8 i \sin (2 (c+d x))+6 i \sin (3 (c+d x))-128 \cos (c+d x)-68 \cos (2 (c+d x))-24 \cos (3 (c+d x))-68\right)+160 \sqrt{\cos (c+d x)} \cos ^5\left(\frac{1}{2} (c+d x)\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(\cos \left(\frac{1}{2} (c+d x)\right)-i \sin \left(\frac{1}{2} (c+d x)\right)\right)\right)}{40 a^3 d (\sec (c+d x)+1)^3}","\frac{\sin (c+d x) \sqrt{\sec (c+d x)}}{2 d \left(a^3 \sec (c+d x)+a^3\right)}+\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{2 a^3 d}-\frac{9 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{10 a^3 d}-\frac{\sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{5 d (a \sec (c+d x)+a)^3}+\frac{2 \sin (c+d x) \sqrt{\sec (c+d x)}}{5 a d (a \sec (c+d x)+a)^2}",1,"(Cos[(c + d*x)/2]*Sec[c + d*x]^(7/2)*(160*Cos[(c + d*x)/2]^5*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*(Cos[(c + d*x)/2] - I*Sin[(c + d*x)/2]) + I*(-68 - 128*Cos[c + d*x] - 68*Cos[2*(c + d*x)] - 24*Cos[3*(c + d*x)] + (3*(1 + E^(I*(c + d*x)))^5*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])/E^((2*I)*(c + d*x)) + (6*I)*Sin[c + d*x] + (8*I)*Sin[2*(c + d*x)] + (6*I)*Sin[3*(c + d*x)]))*(Cos[(c + 3*d*x)/2] + I*Sin[(c + 3*d*x)/2]))/(40*a^3*d*E^(I*d*x)*(1 + Sec[c + d*x])^3)","C",1
215,1,386,195,2.0669334,"\int \frac{1}{\sqrt{\sec (c+d x)} (a+a \sec (c+d x))^3} \, dx","Integrate[1/(Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x])^3),x]","\frac{2 \cos ^6\left(\frac{1}{2} (c+d x)\right) \sec ^3(c+d x) \left(-\frac{1}{32} \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \left(1134 \cos \left(\frac{1}{2} (c-d x)\right)+1071 \cos \left(\frac{1}{2} (3 c+d x)\right)+923 \cos \left(\frac{1}{2} (c+3 d x)\right)+694 \cos \left(\frac{1}{2} (5 c+3 d x)\right)+470 \cos \left(\frac{1}{2} (3 c+5 d x)\right)+265 \cos \left(\frac{1}{2} (7 c+5 d x)\right)+117 \cos \left(\frac{1}{2} (5 c+7 d x)\right)+30 \cos \left(\frac{1}{2} (9 c+7 d x)\right)\right) \sec ^5\left(\frac{1}{2} (c+d x)\right) \sqrt{\sec (c+d x)}+\frac{2 i \sqrt{2} e^{-i (c+d x)} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \left(147 \left(-1+e^{2 i c}\right) \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i (c+d x)}\right)+65 \left(-1+e^{2 i c}\right) e^{i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-e^{2 i (c+d x)}\right)+147 \left(1+e^{2 i (c+d x)}\right)\right)}{-1+e^{2 i c}}\right)}{15 a^3 d (\sec (c+d x)+1)^3}","-\frac{13 \sin (c+d x) \sqrt{\sec (c+d x)}}{6 d \left(a^3 \sec (c+d x)+a^3\right)}-\frac{13 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{6 a^3 d}+\frac{49 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{10 a^3 d}-\frac{8 \sin (c+d x) \sqrt{\sec (c+d x)}}{15 a d (a \sec (c+d x)+a)^2}-\frac{\sin (c+d x) \sqrt{\sec (c+d x)}}{5 d (a \sec (c+d x)+a)^3}",1,"(2*Cos[(c + d*x)/2]^6*(((2*I)*Sqrt[2]*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*(147*(1 + E^((2*I)*(c + d*x))) + 147*(-1 + E^((2*I)*c))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[-1/4, 1/2, 3/4, -E^((2*I)*(c + d*x))] + 65*E^(I*(c + d*x))*(-1 + E^((2*I)*c))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/4, 1/2, 5/4, -E^((2*I)*(c + d*x))]))/(E^(I*(c + d*x))*(-1 + E^((2*I)*c))) - ((1134*Cos[(c - d*x)/2] + 1071*Cos[(3*c + d*x)/2] + 923*Cos[(c + 3*d*x)/2] + 694*Cos[(5*c + 3*d*x)/2] + 470*Cos[(3*c + 5*d*x)/2] + 265*Cos[(7*c + 5*d*x)/2] + 117*Cos[(5*c + 7*d*x)/2] + 30*Cos[(9*c + 7*d*x)/2])*Csc[c/2]*Sec[c/2]*Sec[(c + d*x)/2]^5*Sqrt[Sec[c + d*x]])/32)*Sec[c + d*x]^3)/(15*a^3*d*(1 + Sec[c + d*x])^3)","C",1
216,1,285,221,2.4662591,"\int \frac{1}{\sec ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x))^3} \, dx","Integrate[1/(Sec[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^3),x]","\frac{e^{-i d x} \cos \left(\frac{1}{2} (c+d x)\right) \sec ^{\frac{7}{2}}(c+d x) (\cos (d x)+i \sin (d x)) \left(119 i e^{-\frac{3}{2} i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \left(1+e^{i (c+d x)}\right)^5 \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)+193 \sin \left(\frac{1}{2} (c+d x)\right)+579 \sin \left(\frac{3}{2} (c+d x)\right)+555 \sin \left(\frac{5}{2} (c+d x)\right)+227 \sin \left(\frac{7}{2} (c+d x)\right)+10 \sin \left(\frac{9}{2} (c+d x)\right)-5355 i \cos \left(\frac{1}{2} (c+d x)\right)-3927 i \cos \left(\frac{3}{2} (c+d x)\right)-1785 i \cos \left(\frac{5}{2} (c+d x)\right)-357 i \cos \left(\frac{7}{2} (c+d x)\right)+5280 \cos ^5\left(\frac{1}{2} (c+d x)\right) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{120 a^3 d (\sec (c+d x)+1)^3}","\frac{11 \sin (c+d x)}{2 a^3 d \sqrt{\sec (c+d x)}}-\frac{119 \sin (c+d x)}{30 d \sqrt{\sec (c+d x)} \left(a^3 \sec (c+d x)+a^3\right)}+\frac{11 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{2 a^3 d}-\frac{119 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{10 a^3 d}-\frac{2 \sin (c+d x)}{3 a d \sqrt{\sec (c+d x)} (a \sec (c+d x)+a)^2}-\frac{\sin (c+d x)}{5 d \sqrt{\sec (c+d x)} (a \sec (c+d x)+a)^3}",1,"(Cos[(c + d*x)/2]*Sec[c + d*x]^(7/2)*(Cos[d*x] + I*Sin[d*x])*((-5355*I)*Cos[(c + d*x)/2] - (3927*I)*Cos[(3*(c + d*x))/2] - (1785*I)*Cos[(5*(c + d*x))/2] - (357*I)*Cos[(7*(c + d*x))/2] + 5280*Cos[(c + d*x)/2]^5*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + ((119*I)*(1 + E^(I*(c + d*x)))^5*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])/E^(((3*I)/2)*(c + d*x)) + 193*Sin[(c + d*x)/2] + 579*Sin[(3*(c + d*x))/2] + 555*Sin[(5*(c + d*x))/2] + 227*Sin[(7*(c + d*x))/2] + 10*Sin[(9*(c + d*x))/2]))/(120*a^3*d*E^(I*d*x)*(1 + Sec[c + d*x])^3)","C",1
217,1,297,247,2.8592418,"\int \frac{1}{\sec ^{\frac{5}{2}}(c+d x) (a+a \sec (c+d x))^3} \, dx","Integrate[1/(Sec[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^3),x]","-\frac{e^{-i d x} \cos \left(\frac{1}{2} (c+d x)\right) \sec ^{\frac{7}{2}}(c+d x) (\cos (d x)+i \sin (d x)) \left(77 i e^{-\frac{3}{2} i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \left(1+e^{i (c+d x)}\right)^5 \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)+125 \sin \left(\frac{1}{2} (c+d x)\right)+359 \sin \left(\frac{3}{2} (c+d x)\right)+350 \sin \left(\frac{5}{2} (c+d x)\right)+138 \sin \left(\frac{7}{2} (c+d x)\right)+5 \sin \left(\frac{9}{2} (c+d x)\right)-\sin \left(\frac{11}{2} (c+d x)\right)-3465 i \cos \left(\frac{1}{2} (c+d x)\right)-2541 i \cos \left(\frac{3}{2} (c+d x)\right)-1155 i \cos \left(\frac{5}{2} (c+d x)\right)-231 i \cos \left(\frac{7}{2} (c+d x)\right)+3360 \cos ^5\left(\frac{1}{2} (c+d x)\right) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{40 a^3 d (\sec (c+d x)+1)^3}","-\frac{63 \sin (c+d x)}{10 d \sec ^{\frac{3}{2}}(c+d x) \left(a^3 \sec (c+d x)+a^3\right)}+\frac{77 \sin (c+d x)}{10 a^3 d \sec ^{\frac{3}{2}}(c+d x)}-\frac{21 \sin (c+d x)}{2 a^3 d \sqrt{\sec (c+d x)}}-\frac{21 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{2 a^3 d}+\frac{231 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{10 a^3 d}-\frac{4 \sin (c+d x)}{5 a d \sec ^{\frac{3}{2}}(c+d x) (a \sec (c+d x)+a)^2}-\frac{\sin (c+d x)}{5 d \sec ^{\frac{3}{2}}(c+d x) (a \sec (c+d x)+a)^3}",1,"-1/40*(Cos[(c + d*x)/2]*Sec[c + d*x]^(7/2)*(Cos[d*x] + I*Sin[d*x])*((-3465*I)*Cos[(c + d*x)/2] - (2541*I)*Cos[(3*(c + d*x))/2] - (1155*I)*Cos[(5*(c + d*x))/2] - (231*I)*Cos[(7*(c + d*x))/2] + 3360*Cos[(c + d*x)/2]^5*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + ((77*I)*(1 + E^(I*(c + d*x)))^5*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])/E^(((3*I)/2)*(c + d*x)) + 125*Sin[(c + d*x)/2] + 359*Sin[(3*(c + d*x))/2] + 350*Sin[(5*(c + d*x))/2] + 138*Sin[(7*(c + d*x))/2] + 5*Sin[(9*(c + d*x))/2] - Sin[(11*(c + d*x))/2]))/(a^3*d*E^(I*d*x)*(1 + Sec[c + d*x])^3)","C",1
218,1,100,116,0.5572619,"\int \sec ^{\frac{5}{2}}(c+d x) \sqrt{a+a \sec (c+d x)} \, dx","Integrate[Sec[c + d*x]^(5/2)*Sqrt[a + a*Sec[c + d*x]],x]","\frac{2 a \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) \left(\frac{1}{8} \cos (c+d x) (3 \cos (c+d x)+2)+\frac{3 \sin ^{-1}\left(\sqrt{1-\sec (c+d x)}\right)}{8 \sqrt{1-\sec (c+d x)} \sec ^{\frac{5}{2}}(c+d x)}\right)}{d \sqrt{a (\sec (c+d x)+1)}}","\frac{a \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{2 d \sqrt{a \sec (c+d x)+a}}+\frac{3 a \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{4 d \sqrt{a \sec (c+d x)+a}}+\frac{3 \sqrt{a} \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{4 d}",1,"(2*a*((Cos[c + d*x]*(2 + 3*Cos[c + d*x]))/8 + (3*ArcSin[Sqrt[1 - Sec[c + d*x]]])/(8*Sqrt[1 - Sec[c + d*x]]*Sec[c + d*x]^(5/2)))*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(d*Sqrt[a*(1 + Sec[c + d*x])])","A",1
219,1,75,72,0.2499258,"\int \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+a \sec (c+d x)} \, dx","Integrate[Sec[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]],x]","\frac{a \tan (c+d x) \left(\sqrt{-((\sec (c+d x)-1) \sec (c+d x))}+\sin ^{-1}\left(\sqrt{1-\sec (c+d x)}\right)\right)}{d \sqrt{1-\sec (c+d x)} \sqrt{a (\sec (c+d x)+1)}}","\frac{a \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{d \sqrt{a \sec (c+d x)+a}}+\frac{\sqrt{a} \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{d}",1,"(a*(ArcSin[Sqrt[1 - Sec[c + d*x]]] + Sqrt[-((-1 + Sec[c + d*x])*Sec[c + d*x])])*Tan[c + d*x])/(d*Sqrt[1 - Sec[c + d*x]]*Sqrt[a*(1 + Sec[c + d*x])])","A",0
220,1,54,37,0.1116865,"\int \sqrt{\sec (c+d x)} \sqrt{a+a \sec (c+d x)} \, dx","Integrate[Sqrt[Sec[c + d*x]]*Sqrt[a + a*Sec[c + d*x]],x]","-\frac{2 \tan \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\sec (c+d x)+1)} \sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)}{d \sqrt{1-\sec (c+d x)}}","\frac{2 \sqrt{a} \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{d}",1,"(-2*ArcSin[Sqrt[Sec[c + d*x]]]*Sqrt[a*(1 + Sec[c + d*x])]*Tan[(c + d*x)/2])/(d*Sqrt[1 - Sec[c + d*x]])","A",1
221,1,39,36,0.0943847,"\int \frac{\sqrt{a+a \sec (c+d x)}}{\sqrt{\sec (c+d x)}} \, dx","Integrate[Sqrt[a + a*Sec[c + d*x]]/Sqrt[Sec[c + d*x]],x]","\frac{2 \tan \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\sec (c+d x)+1)}}{d \sqrt{\sec (c+d x)}}","\frac{2 a \sin (c+d x) \sqrt{\sec (c+d x)}}{d \sqrt{a \sec (c+d x)+a}}",1,"(2*Sqrt[a*(1 + Sec[c + d*x])]*Tan[(c + d*x)/2])/(d*Sqrt[Sec[c + d*x]])","A",1
222,1,49,77,0.1638382,"\int \frac{\sqrt{a+a \sec (c+d x)}}{\sec ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[Sqrt[a + a*Sec[c + d*x]]/Sec[c + d*x]^(3/2),x]","\frac{2 (\cos (c+d x)+2) \tan \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\sec (c+d x)+1)}}{3 d \sqrt{\sec (c+d x)}}","\frac{4 a \sin (c+d x) \sqrt{\sec (c+d x)}}{3 d \sqrt{a \sec (c+d x)+a}}+\frac{2 a \sin (c+d x)}{3 d \sqrt{\sec (c+d x)} \sqrt{a \sec (c+d x)+a}}",1,"(2*(2 + Cos[c + d*x])*Sqrt[a*(1 + Sec[c + d*x])]*Tan[(c + d*x)/2])/(3*d*Sqrt[Sec[c + d*x]])","A",1
223,1,61,115,0.1942836,"\int \frac{\sqrt{a+a \sec (c+d x)}}{\sec ^{\frac{5}{2}}(c+d x)} \, dx","Integrate[Sqrt[a + a*Sec[c + d*x]]/Sec[c + d*x]^(5/2),x]","\frac{(8 \cos (c+d x)+3 \cos (2 (c+d x))+19) \tan \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\sec (c+d x)+1)}}{15 d \sqrt{\sec (c+d x)}}","\frac{2 a \sin (c+d x)}{5 d \sec ^{\frac{3}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{16 a \sin (c+d x) \sqrt{\sec (c+d x)}}{15 d \sqrt{a \sec (c+d x)+a}}+\frac{8 a \sin (c+d x)}{15 d \sqrt{\sec (c+d x)} \sqrt{a \sec (c+d x)+a}}",1,"((19 + 8*Cos[c + d*x] + 3*Cos[2*(c + d*x)])*Sqrt[a*(1 + Sec[c + d*x])]*Tan[(c + d*x)/2])/(15*d*Sqrt[Sec[c + d*x]])","A",1
224,1,71,153,0.2708635,"\int \frac{\sqrt{a+a \sec (c+d x)}}{\sec ^{\frac{7}{2}}(c+d x)} \, dx","Integrate[Sqrt[a + a*Sec[c + d*x]]/Sec[c + d*x]^(7/2),x]","\frac{(47 \cos (c+d x)+12 \cos (2 (c+d x))+5 \cos (3 (c+d x))+76) \tan \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\sec (c+d x)+1)}}{70 d \sqrt{\sec (c+d x)}}","\frac{12 a \sin (c+d x)}{35 d \sec ^{\frac{3}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{2 a \sin (c+d x)}{7 d \sec ^{\frac{5}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{32 a \sin (c+d x) \sqrt{\sec (c+d x)}}{35 d \sqrt{a \sec (c+d x)+a}}+\frac{16 a \sin (c+d x)}{35 d \sqrt{\sec (c+d x)} \sqrt{a \sec (c+d x)+a}}",1,"((76 + 47*Cos[c + d*x] + 12*Cos[2*(c + d*x)] + 5*Cos[3*(c + d*x)])*Sqrt[a*(1 + Sec[c + d*x])]*Tan[(c + d*x)/2])/(70*d*Sqrt[Sec[c + d*x]])","A",1
225,1,112,160,0.5911649,"\int \sec ^{\frac{5}{2}}(c+d x) (a+a \sec (c+d x))^{3/2} \, dx","Integrate[Sec[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^(3/2),x]","\frac{a \sec \left(\frac{1}{2} (c+d x)\right) \sec ^{\frac{5}{2}}(c+d x) \sqrt{a (\sec (c+d x)+1)} \left(54 \sin \left(\frac{1}{2} (c+d x)\right)+11 \left(\sin \left(\frac{3}{2} (c+d x)\right)+3 \sin \left(\frac{5}{2} (c+d x)\right)\right)+66 \sqrt{2} \cos ^3(c+d x) \tanh ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)\right)}{96 d}","\frac{11 a^{3/2} \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{8 d}+\frac{a^2 \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x)}{3 d \sqrt{a \sec (c+d x)+a}}+\frac{11 a^2 \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{12 d \sqrt{a \sec (c+d x)+a}}+\frac{11 a^2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{8 d \sqrt{a \sec (c+d x)+a}}",1,"(a*Sec[(c + d*x)/2]*Sec[c + d*x]^(5/2)*Sqrt[a*(1 + Sec[c + d*x])]*(66*Sqrt[2]*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[c + d*x]^3 + 54*Sin[(c + d*x)/2] + 11*(Sin[(3*(c + d*x))/2] + 3*Sin[(5*(c + d*x))/2])))/(96*d)","A",1
226,1,99,120,0.4534981,"\int \sec ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x))^{3/2} \, dx","Integrate[Sec[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^(3/2),x]","\frac{a \sec \left(\frac{1}{2} (c+d x)\right) \sec ^{\frac{3}{2}}(c+d x) \sqrt{a (\sec (c+d x)+1)} \left(-3 \sin \left(\frac{1}{2} (c+d x)\right)+7 \sin \left(\frac{3}{2} (c+d x)\right)+7 \sqrt{2} \cos ^2(c+d x) \tanh ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)\right)}{8 d}","\frac{7 a^{3/2} \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{4 d}+\frac{a^2 \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{2 d \sqrt{a \sec (c+d x)+a}}+\frac{7 a^2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{4 d \sqrt{a \sec (c+d x)+a}}",1,"(a*Sec[(c + d*x)/2]*Sec[c + d*x]^(3/2)*Sqrt[a*(1 + Sec[c + d*x])]*(7*Sqrt[2]*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[c + d*x]^2 - 3*Sin[(c + d*x)/2] + 7*Sin[(3*(c + d*x))/2]))/(8*d)","A",1
227,1,75,75,0.2941342,"\int \sqrt{\sec (c+d x)} (a+a \sec (c+d x))^{3/2} \, dx","Integrate[Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x])^(3/2),x]","\frac{a^2 \tan (c+d x) \left(\sqrt{-((\sec (c+d x)-1) \sec (c+d x))}-3 \sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right)}{d \sqrt{1-\sec (c+d x)} \sqrt{a (\sec (c+d x)+1)}}","\frac{3 a^{3/2} \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{d}+\frac{a^2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{d \sqrt{a \sec (c+d x)+a}}",1,"(a^2*(-3*ArcSin[Sqrt[Sec[c + d*x]]] + Sqrt[-((-1 + Sec[c + d*x])*Sec[c + d*x])])*Tan[c + d*x])/(d*Sqrt[1 - Sec[c + d*x]]*Sqrt[a*(1 + Sec[c + d*x])])","A",0
228,1,86,76,0.3812041,"\int \frac{(a+a \sec (c+d x))^{3/2}}{\sqrt{\sec (c+d x)}} \, dx","Integrate[(a + a*Sec[c + d*x])^(3/2)/Sqrt[Sec[c + d*x]],x]","\frac{2 a^2 \left(\sin (c+d x) \sqrt{-((\sec (c+d x)-1) \sec (c+d x))}+\tan (c+d x) \sin ^{-1}\left(\sqrt{1-\sec (c+d x)}\right)\right)}{d \sqrt{1-\sec (c+d x)} \sqrt{a (\sec (c+d x)+1)}}","\frac{2 a^{3/2} \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{d}+\frac{2 a^2 \sin (c+d x) \sqrt{\sec (c+d x)}}{d \sqrt{a \sec (c+d x)+a}}",1,"(2*a^2*(Sqrt[-((-1 + Sec[c + d*x])*Sec[c + d*x])]*Sin[c + d*x] + ArcSin[Sqrt[1 - Sec[c + d*x]]]*Tan[c + d*x]))/(d*Sqrt[1 - Sec[c + d*x]]*Sqrt[a*(1 + Sec[c + d*x])])","A",0
229,1,50,79,0.2189831,"\int \frac{(a+a \sec (c+d x))^{3/2}}{\sec ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[(a + a*Sec[c + d*x])^(3/2)/Sec[c + d*x]^(3/2),x]","\frac{2 a (\cos (c+d x)+5) \tan \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\sec (c+d x)+1)}}{3 d \sqrt{\sec (c+d x)}}","\frac{8 a^2 \sin (c+d x) \sqrt{\sec (c+d x)}}{3 d \sqrt{a \sec (c+d x)+a}}+\frac{2 a \sin (c+d x) \sqrt{a \sec (c+d x)+a}}{3 d \sqrt{\sec (c+d x)}}",1,"(2*a*(5 + Cos[c + d*x])*Sqrt[a*(1 + Sec[c + d*x])]*Tan[(c + d*x)/2])/(3*d*Sqrt[Sec[c + d*x]])","A",1
230,1,60,116,0.2904826,"\int \frac{(a+a \sec (c+d x))^{3/2}}{\sec ^{\frac{5}{2}}(c+d x)} \, dx","Integrate[(a + a*Sec[c + d*x])^(3/2)/Sec[c + d*x]^(5/2),x]","\frac{a (6 \cos (c+d x)+\cos (2 (c+d x))+13) \tan \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\sec (c+d x)+1)}}{5 d \sqrt{\sec (c+d x)}}","\frac{8 a^2 \sin (c+d x) \sqrt{\sec (c+d x)}}{5 d \sqrt{a \sec (c+d x)+a}}+\frac{2 \sin (c+d x) (a \sec (c+d x)+a)^{3/2}}{5 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 a \sin (c+d x) \sqrt{a \sec (c+d x)+a}}{5 d \sqrt{\sec (c+d x)}}",1,"(a*(13 + 6*Cos[c + d*x] + Cos[2*(c + d*x)])*Sqrt[a*(1 + Sec[c + d*x])]*Tan[(c + d*x)/2])/(5*d*Sqrt[Sec[c + d*x]])","A",1
231,1,72,161,0.371298,"\int \frac{(a+a \sec (c+d x))^{3/2}}{\sec ^{\frac{7}{2}}(c+d x)} \, dx","Integrate[(a + a*Sec[c + d*x])^(3/2)/Sec[c + d*x]^(7/2),x]","\frac{a (253 \cos (c+d x)+78 \cos (2 (c+d x))+15 \cos (3 (c+d x))+494) \tan \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\sec (c+d x)+1)}}{210 d \sqrt{\sec (c+d x)}}","\frac{26 a^2 \sin (c+d x)}{35 d \sec ^{\frac{3}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{2 a^2 \sin (c+d x)}{7 d \sec ^{\frac{5}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{208 a^2 \sin (c+d x) \sqrt{\sec (c+d x)}}{105 d \sqrt{a \sec (c+d x)+a}}+\frac{104 a^2 \sin (c+d x)}{105 d \sqrt{\sec (c+d x)} \sqrt{a \sec (c+d x)+a}}",1,"(a*(494 + 253*Cos[c + d*x] + 78*Cos[2*(c + d*x)] + 15*Cos[3*(c + d*x)])*Sqrt[a*(1 + Sec[c + d*x])]*Tan[(c + d*x)/2])/(210*d*Sqrt[Sec[c + d*x]])","A",1
232,1,80,201,0.5654112,"\int \frac{(a+a \sec (c+d x))^{3/2}}{\sec ^{\frac{9}{2}}(c+d x)} \, dx","Integrate[(a + a*Sec[c + d*x])^(3/2)/Sec[c + d*x]^(9/2),x]","\frac{2 a^2 \sin (c+d x) \left(272 \sec ^4(c+d x)+136 \sec ^3(c+d x)+102 \sec ^2(c+d x)+85 \sec (c+d x)+35\right)}{315 d \sec ^{\frac{7}{2}}(c+d x) \sqrt{a (\sec (c+d x)+1)}}","\frac{68 a^2 \sin (c+d x)}{105 d \sec ^{\frac{3}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{34 a^2 \sin (c+d x)}{63 d \sec ^{\frac{5}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{2 a^2 \sin (c+d x)}{9 d \sec ^{\frac{7}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{544 a^2 \sin (c+d x) \sqrt{\sec (c+d x)}}{315 d \sqrt{a \sec (c+d x)+a}}+\frac{272 a^2 \sin (c+d x)}{315 d \sqrt{\sec (c+d x)} \sqrt{a \sec (c+d x)+a}}",1,"(2*a^2*(35 + 85*Sec[c + d*x] + 102*Sec[c + d*x]^2 + 136*Sec[c + d*x]^3 + 272*Sec[c + d*x]^4)*Sin[c + d*x])/(315*d*Sec[c + d*x]^(7/2)*Sqrt[a*(1 + Sec[c + d*x])])","A",1
233,1,582,200,8.5019658,"\int \sec ^{\frac{5}{2}}(c+d x) (a+a \sec (c+d x))^{5/2} \, dx","Integrate[Sec[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^(5/2),x]","-\frac{a^2 \sec \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\sec (c+d x)+1)} \left(7824 i \tan ^{-1}\left(\frac{\cos \left(\frac{1}{4} (c+d x)\right)-\left(\sqrt{2}-1\right) \sin \left(\frac{1}{4} (c+d x)\right)}{\left(1+\sqrt{2}\right) \cos \left(\frac{1}{4} (c+d x)\right)-\sin \left(\frac{1}{4} (c+d x)\right)}\right)+7824 i \tan ^{-1}\left(\frac{\cos \left(\frac{1}{4} (c+d x)\right)-\left(1+\sqrt{2}\right) \sin \left(\frac{1}{4} (c+d x)\right)}{\left(\sqrt{2}-1\right) \cos \left(\frac{1}{4} (c+d x)\right)-\sin \left(\frac{1}{4} (c+d x)\right)}\right)+\sec ^4(c+d x) \left(2060 \sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)-6204 \sqrt{2} \sin \left(\frac{3}{2} (c+d x)\right)-652 \sqrt{2} \sin \left(\frac{5}{2} (c+d x)\right)-1956 \sqrt{2} \sin \left(\frac{7}{2} (c+d x)\right)-2934 \log \left(2 \sin \left(\frac{1}{2} (c+d x)\right)+\sqrt{2}\right)+1467 \log \left(-\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)-\sqrt{2} \cos \left(\frac{1}{2} (c+d x)\right)+2\right)-1956 \cos (2 (c+d x)) \left(2 \log \left(2 \sin \left(\frac{1}{2} (c+d x)\right)+\sqrt{2}\right)-\log \left(-\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)-\sqrt{2} \cos \left(\frac{1}{2} (c+d x)\right)+2\right)-\log \left(-\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)+\sqrt{2} \cos \left(\frac{1}{2} (c+d x)\right)+2\right)\right)-489 \cos (4 (c+d x)) \left(2 \log \left(2 \sin \left(\frac{1}{2} (c+d x)\right)+\sqrt{2}\right)-\log \left(-\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)-\sqrt{2} \cos \left(\frac{1}{2} (c+d x)\right)+2\right)-\log \left(-\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)+\sqrt{2} \cos \left(\frac{1}{2} (c+d x)\right)+2\right)\right)+1467 \log \left(-\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)+\sqrt{2} \cos \left(\frac{1}{2} (c+d x)\right)+2\right)\right)\right)}{6144 \sqrt{2} d \sqrt{\sec (c+d x)}}","\frac{163 a^{5/2} \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{64 d}+\frac{17 a^3 \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x)}{24 d \sqrt{a \sec (c+d x)+a}}+\frac{163 a^3 \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{96 d \sqrt{a \sec (c+d x)+a}}+\frac{163 a^3 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{64 d \sqrt{a \sec (c+d x)+a}}+\frac{a^2 \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}{4 d}",1,"-1/6144*(a^2*Sec[(c + d*x)/2]*Sqrt[a*(1 + Sec[c + d*x])]*((7824*I)*ArcTan[(Cos[(c + d*x)/4] - (-1 + Sqrt[2])*Sin[(c + d*x)/4])/((1 + Sqrt[2])*Cos[(c + d*x)/4] - Sin[(c + d*x)/4])] + (7824*I)*ArcTan[(Cos[(c + d*x)/4] - (1 + Sqrt[2])*Sin[(c + d*x)/4])/((-1 + Sqrt[2])*Cos[(c + d*x)/4] - Sin[(c + d*x)/4])] + Sec[c + d*x]^4*(-2934*Log[Sqrt[2] + 2*Sin[(c + d*x)/2]] + 1467*Log[2 - Sqrt[2]*Cos[(c + d*x)/2] - Sqrt[2]*Sin[(c + d*x)/2]] - 1956*Cos[2*(c + d*x)]*(2*Log[Sqrt[2] + 2*Sin[(c + d*x)/2]] - Log[2 - Sqrt[2]*Cos[(c + d*x)/2] - Sqrt[2]*Sin[(c + d*x)/2]] - Log[2 + Sqrt[2]*Cos[(c + d*x)/2] - Sqrt[2]*Sin[(c + d*x)/2]]) - 489*Cos[4*(c + d*x)]*(2*Log[Sqrt[2] + 2*Sin[(c + d*x)/2]] - Log[2 - Sqrt[2]*Cos[(c + d*x)/2] - Sqrt[2]*Sin[(c + d*x)/2]] - Log[2 + Sqrt[2]*Cos[(c + d*x)/2] - Sqrt[2]*Sin[(c + d*x)/2]]) + 1467*Log[2 + Sqrt[2]*Cos[(c + d*x)/2] - Sqrt[2]*Sin[(c + d*x)/2]] + 2060*Sqrt[2]*Sin[(c + d*x)/2] - 6204*Sqrt[2]*Sin[(3*(c + d*x))/2] - 652*Sqrt[2]*Sin[(5*(c + d*x))/2] - 1956*Sqrt[2]*Sin[(7*(c + d*x))/2])))/(Sqrt[2]*d*Sqrt[Sec[c + d*x]])","C",1
234,1,458,160,8.0583857,"\int \sec ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x))^{5/2} \, dx","Integrate[Sec[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^(5/2),x]","\frac{a^2 \sec \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\sec (c+d x)+1)} \left(-600 i \tan ^{-1}\left(\frac{\cos \left(\frac{1}{4} (c+d x)\right)-\left(\sqrt{2}-1\right) \sin \left(\frac{1}{4} (c+d x)\right)}{\left(1+\sqrt{2}\right) \cos \left(\frac{1}{4} (c+d x)\right)-\sin \left(\frac{1}{4} (c+d x)\right)}\right)-600 i \tan ^{-1}\left(\frac{\cos \left(\frac{1}{4} (c+d x)\right)-\left(1+\sqrt{2}\right) \sin \left(\frac{1}{4} (c+d x)\right)}{\left(\sqrt{2}-1\right) \cos \left(\frac{1}{4} (c+d x)\right)-\sin \left(\frac{1}{4} (c+d x)\right)}\right)+\sec ^3(c+d x) \left(4 \sqrt{2} \left(114 \sin \left(\frac{1}{2} (c+d x)\right)-7 \sin \left(\frac{3}{2} (c+d x)\right)+75 \sin \left(\frac{5}{2} (c+d x)\right)\right)+225 \cos (c+d x) \left(2 \log \left(2 \sin \left(\frac{1}{2} (c+d x)\right)+\sqrt{2}\right)-\log \left(-\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)-\sqrt{2} \cos \left(\frac{1}{2} (c+d x)\right)+2\right)-\log \left(-\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)+\sqrt{2} \cos \left(\frac{1}{2} (c+d x)\right)+2\right)\right)+75 \cos (3 (c+d x)) \left(2 \log \left(2 \sin \left(\frac{1}{2} (c+d x)\right)+\sqrt{2}\right)-\log \left(-\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)-\sqrt{2} \cos \left(\frac{1}{2} (c+d x)\right)+2\right)-\log \left(-\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)+\sqrt{2} \cos \left(\frac{1}{2} (c+d x)\right)+2\right)\right)\right)\right)}{384 \sqrt{2} d \sqrt{\sec (c+d x)}}","\frac{25 a^{5/2} \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{8 d}+\frac{13 a^3 \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{12 d \sqrt{a \sec (c+d x)+a}}+\frac{25 a^3 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{8 d \sqrt{a \sec (c+d x)+a}}+\frac{a^2 \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}{3 d}",1,"(a^2*Sec[(c + d*x)/2]*Sqrt[a*(1 + Sec[c + d*x])]*((-600*I)*ArcTan[(Cos[(c + d*x)/4] - (-1 + Sqrt[2])*Sin[(c + d*x)/4])/((1 + Sqrt[2])*Cos[(c + d*x)/4] - Sin[(c + d*x)/4])] - (600*I)*ArcTan[(Cos[(c + d*x)/4] - (1 + Sqrt[2])*Sin[(c + d*x)/4])/((-1 + Sqrt[2])*Cos[(c + d*x)/4] - Sin[(c + d*x)/4])] + Sec[c + d*x]^3*(225*Cos[c + d*x]*(2*Log[Sqrt[2] + 2*Sin[(c + d*x)/2]] - Log[2 - Sqrt[2]*Cos[(c + d*x)/2] - Sqrt[2]*Sin[(c + d*x)/2]] - Log[2 + Sqrt[2]*Cos[(c + d*x)/2] - Sqrt[2]*Sin[(c + d*x)/2]]) + 75*Cos[3*(c + d*x)]*(2*Log[Sqrt[2] + 2*Sin[(c + d*x)/2]] - Log[2 - Sqrt[2]*Cos[(c + d*x)/2] - Sqrt[2]*Sin[(c + d*x)/2]] - Log[2 + Sqrt[2]*Cos[(c + d*x)/2] - Sqrt[2]*Sin[(c + d*x)/2]]) + 4*Sqrt[2]*(114*Sin[(c + d*x)/2] - 7*Sin[(3*(c + d*x))/2] + 75*Sin[(5*(c + d*x))/2]))))/(384*Sqrt[2]*d*Sqrt[Sec[c + d*x]])","C",1
235,1,106,120,0.5092028,"\int \sqrt{\sec (c+d x)} (a+a \sec (c+d x))^{5/2} \, dx","Integrate[Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x])^(5/2),x]","\frac{a^3 \tan (c+d x) \left(2 \sqrt{1-\sec (c+d x)} \sec ^{\frac{3}{2}}(c+d x)+11 \sqrt{-((\sec (c+d x)-1) \sec (c+d x))}-19 \sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right)}{4 d \sqrt{1-\sec (c+d x)} \sqrt{a (\sec (c+d x)+1)}}","\frac{19 a^{5/2} \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{4 d}+\frac{9 a^3 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{4 d \sqrt{a \sec (c+d x)+a}}+\frac{a^2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}{2 d}",1,"(a^3*(-19*ArcSin[Sqrt[Sec[c + d*x]]] + 2*Sqrt[1 - Sec[c + d*x]]*Sec[c + d*x]^(3/2) + 11*Sqrt[-((-1 + Sec[c + d*x])*Sec[c + d*x])])*Tan[c + d*x])/(4*d*Sqrt[1 - Sec[c + d*x]]*Sqrt[a*(1 + Sec[c + d*x])])","A",0
236,1,91,112,0.7485061,"\int \frac{(a+a \sec (c+d x))^{5/2}}{\sqrt{\sec (c+d x)}} \, dx","Integrate[(a + a*Sec[c + d*x])^(5/2)/Sqrt[Sec[c + d*x]],x]","\frac{a^3 \tan (c+d x) \left((2 \cos (c+d x)+1) \sqrt{(\cos (c+d x)-1) \sec ^2(c+d x)}+5 \sin ^{-1}\left(\sqrt{1-\sec (c+d x)}\right)\right)}{d \sqrt{1-\sec (c+d x)} \sqrt{a (\sec (c+d x)+1)}}","\frac{5 a^{5/2} \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{d}+\frac{a^3 \sin (c+d x) \sqrt{\sec (c+d x)}}{d \sqrt{a \sec (c+d x)+a}}+\frac{a^2 \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a \sec (c+d x)+a}}{d}",1,"(a^3*(5*ArcSin[Sqrt[1 - Sec[c + d*x]]] + (1 + 2*Cos[c + d*x])*Sqrt[(-1 + Cos[c + d*x])*Sec[c + d*x]^2])*Tan[c + d*x])/(d*Sqrt[1 - Sec[c + d*x]]*Sqrt[a*(1 + Sec[c + d*x])])","A",0
237,1,103,118,0.45478,"\int \frac{(a+a \sec (c+d x))^{5/2}}{\sec ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[(a + a*Sec[c + d*x])^(5/2)/Sec[c + d*x]^(3/2),x]","\frac{2 a^3 \sin (c+d x) \left(\sqrt{1-\sec (c+d x)} (8 \sec (c+d x)+1)+3 \sec ^{\frac{3}{2}}(c+d x) \sin ^{-1}\left(\sqrt{1-\sec (c+d x)}\right)\right)}{3 d \sqrt{-((\sec (c+d x)-1) \sec (c+d x))} \sqrt{a (\sec (c+d x)+1)}}","\frac{2 a^{5/2} \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{d}+\frac{14 a^3 \sin (c+d x) \sqrt{\sec (c+d x)}}{3 d \sqrt{a \sec (c+d x)+a}}+\frac{2 a^2 \sin (c+d x) \sqrt{a \sec (c+d x)+a}}{3 d \sqrt{\sec (c+d x)}}",1,"(2*a^3*(3*ArcSin[Sqrt[1 - Sec[c + d*x]]]*Sec[c + d*x]^(3/2) + Sqrt[1 - Sec[c + d*x]]*(1 + 8*Sec[c + d*x]))*Sin[c + d*x])/(3*d*Sqrt[-((-1 + Sec[c + d*x])*Sec[c + d*x])]*Sqrt[a*(1 + Sec[c + d*x])])","A",0
238,1,64,119,0.3296219,"\int \frac{(a+a \sec (c+d x))^{5/2}}{\sec ^{\frac{5}{2}}(c+d x)} \, dx","Integrate[(a + a*Sec[c + d*x])^(5/2)/Sec[c + d*x]^(5/2),x]","\frac{a^2 (28 \cos (c+d x)+3 \cos (2 (c+d x))+89) \tan \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\sec (c+d x)+1)}}{15 d \sqrt{\sec (c+d x)}}","\frac{64 a^3 \sin (c+d x) \sqrt{\sec (c+d x)}}{15 d \sqrt{a \sec (c+d x)+a}}+\frac{16 a^2 \sin (c+d x) \sqrt{a \sec (c+d x)+a}}{15 d \sqrt{\sec (c+d x)}}+\frac{2 a \sin (c+d x) (a \sec (c+d x)+a)^{3/2}}{5 d \sec ^{\frac{3}{2}}(c+d x)}",1,"(a^2*(89 + 28*Cos[c + d*x] + 3*Cos[2*(c + d*x)])*Sqrt[a*(1 + Sec[c + d*x])]*Tan[(c + d*x)/2])/(15*d*Sqrt[Sec[c + d*x]])","A",1
239,1,74,156,0.3707585,"\int \frac{(a+a \sec (c+d x))^{5/2}}{\sec ^{\frac{7}{2}}(c+d x)} \, dx","Integrate[(a + a*Sec[c + d*x])^(5/2)/Sec[c + d*x]^(7/2),x]","\frac{a^2 (101 \cos (c+d x)+24 \cos (2 (c+d x))+3 \cos (3 (c+d x))+208) \tan \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\sec (c+d x)+1)}}{42 d \sqrt{\sec (c+d x)}}","\frac{64 a^3 \sin (c+d x) \sqrt{\sec (c+d x)}}{21 d \sqrt{a \sec (c+d x)+a}}+\frac{16 a^2 \sin (c+d x) \sqrt{a \sec (c+d x)+a}}{21 d \sqrt{\sec (c+d x)}}+\frac{2 a \sin (c+d x) (a \sec (c+d x)+a)^{3/2}}{7 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 \sin (c+d x) (a \sec (c+d x)+a)^{5/2}}{7 d \sec ^{\frac{5}{2}}(c+d x)}",1,"(a^2*(208 + 101*Cos[c + d*x] + 24*Cos[2*(c + d*x)] + 3*Cos[3*(c + d*x)])*Sqrt[a*(1 + Sec[c + d*x])]*Tan[(c + d*x)/2])/(42*d*Sqrt[Sec[c + d*x]])","A",1
240,1,80,201,0.603259,"\int \frac{(a+a \sec (c+d x))^{5/2}}{\sec ^{\frac{9}{2}}(c+d x)} \, dx","Integrate[(a + a*Sec[c + d*x])^(5/2)/Sec[c + d*x]^(9/2),x]","\frac{2 a^3 \sin (c+d x) \left(584 \sec ^4(c+d x)+292 \sec ^3(c+d x)+219 \sec ^2(c+d x)+130 \sec (c+d x)+35\right)}{315 d \sec ^{\frac{7}{2}}(c+d x) \sqrt{a (\sec (c+d x)+1)}}","\frac{146 a^3 \sin (c+d x)}{105 d \sec ^{\frac{3}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{38 a^3 \sin (c+d x)}{63 d \sec ^{\frac{5}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{1168 a^3 \sin (c+d x) \sqrt{\sec (c+d x)}}{315 d \sqrt{a \sec (c+d x)+a}}+\frac{584 a^3 \sin (c+d x)}{315 d \sqrt{\sec (c+d x)} \sqrt{a \sec (c+d x)+a}}+\frac{2 a^2 \sin (c+d x) \sqrt{a \sec (c+d x)+a}}{9 d \sec ^{\frac{7}{2}}(c+d x)}",1,"(2*a^3*(35 + 130*Sec[c + d*x] + 219*Sec[c + d*x]^2 + 292*Sec[c + d*x]^3 + 584*Sec[c + d*x]^4)*Sin[c + d*x])/(315*d*Sec[c + d*x]^(7/2)*Sqrt[a*(1 + Sec[c + d*x])])","A",1
241,1,90,241,0.4016644,"\int \frac{(a+a \sec (c+d x))^{5/2}}{\sec ^{\frac{11}{2}}(c+d x)} \, dx","Integrate[(a + a*Sec[c + d*x])^(5/2)/Sec[c + d*x]^(11/2),x]","\frac{2 a^3 \sin (c+d x) \left(1136 \sec ^5(c+d x)+568 \sec ^4(c+d x)+426 \sec ^3(c+d x)+355 \sec ^2(c+d x)+224 \sec (c+d x)+63\right)}{693 d \sec ^{\frac{9}{2}}(c+d x) \sqrt{a (\sec (c+d x)+1)}}","\frac{284 a^3 \sin (c+d x)}{231 d \sec ^{\frac{3}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{710 a^3 \sin (c+d x)}{693 d \sec ^{\frac{5}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{46 a^3 \sin (c+d x)}{99 d \sec ^{\frac{7}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{2272 a^3 \sin (c+d x) \sqrt{\sec (c+d x)}}{693 d \sqrt{a \sec (c+d x)+a}}+\frac{1136 a^3 \sin (c+d x)}{693 d \sqrt{\sec (c+d x)} \sqrt{a \sec (c+d x)+a}}+\frac{2 a^2 \sin (c+d x) \sqrt{a \sec (c+d x)+a}}{11 d \sec ^{\frac{9}{2}}(c+d x)}",1,"(2*a^3*(63 + 224*Sec[c + d*x] + 355*Sec[c + d*x]^2 + 426*Sec[c + d*x]^3 + 568*Sec[c + d*x]^4 + 1136*Sec[c + d*x]^5)*Sin[c + d*x])/(693*d*Sec[c + d*x]^(9/2)*Sqrt[a*(1 + Sec[c + d*x])])","A",1
242,1,45,38,0.0928484,"\int \frac{(a+a \sec (c+d x))^{3/2}}{\sqrt[4]{\sec (c+d x)}} \, dx","Integrate[(a + a*Sec[c + d*x])^(3/2)/Sec[c + d*x]^(1/4),x]","\frac{4 \sin (c+d x) \sec ^{\frac{3}{4}}(c+d x) (a (\sec (c+d x)+1))^{3/2}}{d (\sec (c+d x)+1)^2}","\frac{4 a^2 \sin (c+d x) \sec ^{\frac{3}{4}}(c+d x)}{d \sqrt{a \sec (c+d x)+a}}",1,"(4*Sec[c + d*x]^(3/4)*(a*(1 + Sec[c + d*x]))^(3/2)*Sin[c + d*x])/(d*(1 + Sec[c + d*x])^2)","A",1
243,1,54,37,0.1496759,"\int \sqrt{\sec (e+f x)} \sqrt{a+a \sec (e+f x)} \, dx","Integrate[Sqrt[Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]],x]","-\frac{2 \tan \left(\frac{1}{2} (e+f x)\right) \sqrt{a (\sec (e+f x)+1)} \sin ^{-1}\left(\sqrt{\sec (e+f x)}\right)}{f \sqrt{1-\sec (e+f x)}}","\frac{2 \sqrt{a} \sinh ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a \sec (e+f x)+a}}\right)}{f}",1,"(-2*ArcSin[Sqrt[Sec[e + f*x]]]*Sqrt[a*(1 + Sec[e + f*x])]*Tan[(e + f*x)/2])/(f*Sqrt[1 - Sec[e + f*x]])","A",1
244,1,299,38,2.0606528,"\int \sqrt{-\sec (e+f x)} \sqrt{a-a \sec (e+f x)} \, dx","Integrate[Sqrt[-Sec[e + f*x]]*Sqrt[a - a*Sec[e + f*x]],x]","\frac{\csc \left(\frac{1}{2} (e+f x)\right) \sqrt{a-a \sec (e+f x)} \left(\log \left(-\sqrt{2} \sin \left(\frac{1}{2} (e+f x)\right)-\sqrt{2} \cos \left(\frac{1}{2} (e+f x)\right)+2\right)-\log \left(-\sqrt{2} \sin \left(\frac{1}{2} (e+f x)\right)+\sqrt{2} \cos \left(\frac{1}{2} (e+f x)\right)+2\right)-2 i \tan ^{-1}\left(\frac{\cos \left(\frac{1}{4} (e+f x)\right)-\left(\sqrt{2}-1\right) \sin \left(\frac{1}{4} (e+f x)\right)}{\left(1+\sqrt{2}\right) \cos \left(\frac{1}{4} (e+f x)\right)-\sin \left(\frac{1}{4} (e+f x)\right)}\right)+2 i \tan ^{-1}\left(\frac{\cos \left(\frac{1}{4} (e+f x)\right)-\left(1+\sqrt{2}\right) \sin \left(\frac{1}{4} (e+f x)\right)}{\left(\sqrt{2}-1\right) \cos \left(\frac{1}{4} (e+f x)\right)-\sin \left(\frac{1}{4} (e+f x)\right)}\right)-4 \tanh ^{-1}\left(\sqrt{2} \sec \left(\frac{f x}{4}\right) \cos \left(\frac{1}{4} (2 e+f x)\right)+\tan \left(\frac{f x}{4}\right)\right)\right)}{2 \sqrt{2} f \sqrt{-\sec (e+f x)}}","\frac{2 \sqrt{a} \sinh ^{-1}\left(\frac{\sqrt{a} \tan (e+f x)}{\sqrt{a-a \sec (e+f x)}}\right)}{f}",1,"(Csc[(e + f*x)/2]*((-2*I)*ArcTan[(Cos[(e + f*x)/4] - (-1 + Sqrt[2])*Sin[(e + f*x)/4])/((1 + Sqrt[2])*Cos[(e + f*x)/4] - Sin[(e + f*x)/4])] + (2*I)*ArcTan[(Cos[(e + f*x)/4] - (1 + Sqrt[2])*Sin[(e + f*x)/4])/((-1 + Sqrt[2])*Cos[(e + f*x)/4] - Sin[(e + f*x)/4])] - 4*ArcTanh[Sqrt[2]*Cos[(2*e + f*x)/4]*Sec[(f*x)/4] + Tan[(f*x)/4]] + Log[2 - Sqrt[2]*Cos[(e + f*x)/2] - Sqrt[2]*Sin[(e + f*x)/2]] - Log[2 + Sqrt[2]*Cos[(e + f*x)/2] - Sqrt[2]*Sin[(e + f*x)/2]])*Sqrt[a - a*Sec[e + f*x]])/(2*Sqrt[2]*f*Sqrt[-Sec[e + f*x]])","C",1
245,1,125,128,0.2611207,"\int \frac{\sec ^{\frac{5}{2}}(c+d x)}{\sqrt{a+a \sec (c+d x)}} \, dx","Integrate[Sec[c + d*x]^(5/2)/Sqrt[a + a*Sec[c + d*x]],x]","\frac{\tan (c+d x) \left(\sqrt{-((\sec (c+d x)-1) \sec (c+d x))}+\sin ^{-1}\left(\sqrt{1-\sec (c+d x)}\right)+2 \sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)-\sqrt{2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{\sec (c+d x)}}{\sqrt{1-\sec (c+d x)}}\right)\right)}{d \sqrt{1-\sec (c+d x)} \sqrt{a (\sec (c+d x)+1)}}","\frac{\sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{d \sqrt{a \sec (c+d x)+a}}+\frac{\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{\sqrt{a} d}-\frac{\sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{\sqrt{a} d}",1,"((ArcSin[Sqrt[1 - Sec[c + d*x]]] + 2*ArcSin[Sqrt[Sec[c + d*x]]] - Sqrt[2]*ArcTan[(Sqrt[2]*Sqrt[Sec[c + d*x]])/Sqrt[1 - Sec[c + d*x]]] + Sqrt[-((-1 + Sec[c + d*x])*Sec[c + d*x])])*Tan[c + d*x])/(d*Sqrt[1 - Sec[c + d*x]]*Sqrt[a*(1 + Sec[c + d*x])])","A",0
246,1,89,95,0.1043289,"\int \frac{\sec ^{\frac{3}{2}}(c+d x)}{\sqrt{a+a \sec (c+d x)}} \, dx","Integrate[Sec[c + d*x]^(3/2)/Sqrt[a + a*Sec[c + d*x]],x]","\frac{\tan (c+d x) \left(\sqrt{2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{\sec (c+d x)}}{\sqrt{1-\sec (c+d x)}}\right)-2 \sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right)}{d \sqrt{1-\sec (c+d x)} \sqrt{a (\sec (c+d x)+1)}}","\frac{2 \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{\sqrt{a} d}-\frac{\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{\sqrt{a} d}",1,"((-2*ArcSin[Sqrt[Sec[c + d*x]]] + Sqrt[2]*ArcTan[(Sqrt[2]*Sqrt[Sec[c + d*x]])/Sqrt[1 - Sec[c + d*x]]])*Tan[c + d*x])/(d*Sqrt[1 - Sec[c + d*x]]*Sqrt[a*(1 + Sec[c + d*x])])","A",1
247,1,75,56,0.0736531,"\int \frac{\sqrt{\sec (c+d x)}}{\sqrt{a+a \sec (c+d x)}} \, dx","Integrate[Sqrt[Sec[c + d*x]]/Sqrt[a + a*Sec[c + d*x]],x]","-\frac{\sqrt{2} \tan (c+d x) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{\sec (c+d x)}}{\sqrt{1-\sec (c+d x)}}\right)}{d \sqrt{1-\sec (c+d x)} \sqrt{a (\sec (c+d x)+1)}}","\frac{\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{\sqrt{a} d}",1,"-((Sqrt[2]*ArcTan[(Sqrt[2]*Sqrt[Sec[c + d*x]])/Sqrt[1 - Sec[c + d*x]]]*Tan[c + d*x])/(d*Sqrt[1 - Sec[c + d*x]]*Sqrt[a*(1 + Sec[c + d*x])]))","A",1
248,1,102,93,0.2349271,"\int \frac{1}{\sqrt{\sec (c+d x)} \sqrt{a+a \sec (c+d x)}} \, dx","Integrate[1/(Sqrt[Sec[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]),x]","\frac{\tan (c+d x) \left(\frac{2 \sqrt{1-\sec (c+d x)}}{\sqrt{\sec (c+d x)}}+\sqrt{2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{\sec (c+d x)}}{\sqrt{1-\sec (c+d x)}}\right)\right)}{d \sqrt{1-\sec (c+d x)} \sqrt{a (\sec (c+d x)+1)}}","\frac{2 \sin (c+d x) \sqrt{\sec (c+d x)}}{d \sqrt{a \sec (c+d x)+a}}-\frac{\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{\sqrt{a} d}",1,"((Sqrt[2]*ArcTan[(Sqrt[2]*Sqrt[Sec[c + d*x]])/Sqrt[1 - Sec[c + d*x]]] + (2*Sqrt[1 - Sec[c + d*x]])/Sqrt[Sec[c + d*x]])*Tan[c + d*x])/(d*Sqrt[1 - Sec[c + d*x]]*Sqrt[a*(1 + Sec[c + d*x])])","A",1
249,1,120,131,0.2648763,"\int \frac{1}{\sec ^{\frac{3}{2}}(c+d x) \sqrt{a+a \sec (c+d x)}} \, dx","Integrate[1/(Sec[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]]),x]","\frac{\tan (c+d x) \left(2 (\cos (c+d x)-1) \sqrt{1-\sec (c+d x)}-3 \sqrt{2} \sqrt{\sec (c+d x)} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{\sec (c+d x)}}{\sqrt{1-\sec (c+d x)}}\right)\right)}{3 d \sqrt{-((\sec (c+d x)-1) \sec (c+d x))} \sqrt{a (\sec (c+d x)+1)}}","-\frac{2 \sin (c+d x) \sqrt{\sec (c+d x)}}{3 d \sqrt{a \sec (c+d x)+a}}+\frac{2 \sin (c+d x)}{3 d \sqrt{\sec (c+d x)} \sqrt{a \sec (c+d x)+a}}+\frac{\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{\sqrt{a} d}",1,"((2*(-1 + Cos[c + d*x])*Sqrt[1 - Sec[c + d*x]] - 3*Sqrt[2]*ArcTan[(Sqrt[2]*Sqrt[Sec[c + d*x]])/Sqrt[1 - Sec[c + d*x]]]*Sqrt[Sec[c + d*x]])*Tan[c + d*x])/(3*d*Sqrt[-((-1 + Sec[c + d*x])*Sec[c + d*x])]*Sqrt[a*(1 + Sec[c + d*x])])","A",0
250,1,117,169,1.2091751,"\int \frac{1}{\sec ^{\frac{5}{2}}(c+d x) \sqrt{a+a \sec (c+d x)}} \, dx","Integrate[1/(Sec[c + d*x]^(5/2)*Sqrt[a + a*Sec[c + d*x]]),x]","\frac{\frac{15 \sqrt{2} \tan (c+d x) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{\sec (c+d x)}}{\sqrt{1-\sec (c+d x)}}\right)}{\sqrt{1-\sec (c+d x)}}+\sin (c+d x) (-2 \cos (c+d x)+3 \cos (2 (c+d x))+29) \sqrt{\sec (c+d x)}}{15 d \sqrt{a (\sec (c+d x)+1)}}","\frac{2 \sin (c+d x)}{5 d \sec ^{\frac{3}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{26 \sin (c+d x) \sqrt{\sec (c+d x)}}{15 d \sqrt{a \sec (c+d x)+a}}-\frac{2 \sin (c+d x)}{15 d \sqrt{\sec (c+d x)} \sqrt{a \sec (c+d x)+a}}-\frac{\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{\sqrt{a} d}",1,"((29 - 2*Cos[c + d*x] + 3*Cos[2*(c + d*x)])*Sqrt[Sec[c + d*x]]*Sin[c + d*x] + (15*Sqrt[2]*ArcTan[(Sqrt[2]*Sqrt[Sec[c + d*x]])/Sqrt[1 - Sec[c + d*x]]]*Tan[c + d*x])/Sqrt[1 - Sec[c + d*x]])/(15*d*Sqrt[a*(1 + Sec[c + d*x])])","A",1
251,1,252,174,0.6761553,"\int \frac{\sec ^{\frac{7}{2}}(c+d x)}{(a+a \sec (c+d x))^{3/2}} \, dx","Integrate[Sec[c + d*x]^(7/2)/(a + a*Sec[c + d*x])^(3/2),x]","\frac{4 \sin (c+d x) \sqrt{1-\sec (c+d x)} \sec ^{\frac{5}{2}}(c+d x)+6 \sin (c+d x) \sqrt{1-\sec (c+d x)} \sec ^{\frac{3}{2}}(c+d x)-9 \sqrt{2} \tan (c+d x) \sec (c+d x) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{\sec (c+d x)}}{\sqrt{1-\sec (c+d x)}}\right)-9 \sqrt{2} \tan (c+d x) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{\sec (c+d x)}}{\sqrt{1-\sec (c+d x)}}\right)+6 \tan (c+d x) (\sec (c+d x)+1) \sin ^{-1}\left(\sqrt{1-\sec (c+d x)}\right)+18 \tan (c+d x) (\sec (c+d x)+1) \sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)}{4 d \sqrt{1-\sec (c+d x)} (a (\sec (c+d x)+1))^{3/2}}","\frac{9 \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}-\frac{3 \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{a^{3/2} d}-\frac{\sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{2 d (a \sec (c+d x)+a)^{3/2}}+\frac{3 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{2 a d \sqrt{a \sec (c+d x)+a}}",1,"(6*Sqrt[1 - Sec[c + d*x]]*Sec[c + d*x]^(3/2)*Sin[c + d*x] + 4*Sqrt[1 - Sec[c + d*x]]*Sec[c + d*x]^(5/2)*Sin[c + d*x] - 9*Sqrt[2]*ArcTan[(Sqrt[2]*Sqrt[Sec[c + d*x]])/Sqrt[1 - Sec[c + d*x]]]*Tan[c + d*x] - 9*Sqrt[2]*ArcTan[(Sqrt[2]*Sqrt[Sec[c + d*x]])/Sqrt[1 - Sec[c + d*x]]]*Sec[c + d*x]*Tan[c + d*x] + 6*ArcSin[Sqrt[1 - Sec[c + d*x]]]*(1 + Sec[c + d*x])*Tan[c + d*x] + 18*ArcSin[Sqrt[Sec[c + d*x]]]*(1 + Sec[c + d*x])*Tan[c + d*x])/(4*d*Sqrt[1 - Sec[c + d*x]]*(a*(1 + Sec[c + d*x]))^(3/2))","A",1
252,1,220,134,0.6185467,"\int \frac{\sec ^{\frac{5}{2}}(c+d x)}{(a+a \sec (c+d x))^{3/2}} \, dx","Integrate[Sec[c + d*x]^(5/2)/(a + a*Sec[c + d*x])^(3/2),x]","\frac{-2 \sin (c+d x) \sqrt{1-\sec (c+d x)} \sec ^{\frac{3}{2}}(c+d x)+5 \sqrt{2} \tan (c+d x) \sec (c+d x) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{\sec (c+d x)}}{\sqrt{1-\sec (c+d x)}}\right)+5 \sqrt{2} \tan (c+d x) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{\sec (c+d x)}}{\sqrt{1-\sec (c+d x)}}\right)-2 \tan (c+d x) (\sec (c+d x)+1) \sin ^{-1}\left(\sqrt{1-\sec (c+d x)}\right)-10 \tan (c+d x) (\sec (c+d x)+1) \sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)}{4 d \sqrt{1-\sec (c+d x)} (a (\sec (c+d x)+1))^{3/2}}","-\frac{5 \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}+\frac{2 \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{a^{3/2} d}-\frac{\sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{2 d (a \sec (c+d x)+a)^{3/2}}",1,"(-2*Sqrt[1 - Sec[c + d*x]]*Sec[c + d*x]^(3/2)*Sin[c + d*x] + 5*Sqrt[2]*ArcTan[(Sqrt[2]*Sqrt[Sec[c + d*x]])/Sqrt[1 - Sec[c + d*x]]]*Tan[c + d*x] + 5*Sqrt[2]*ArcTan[(Sqrt[2]*Sqrt[Sec[c + d*x]])/Sqrt[1 - Sec[c + d*x]]]*Sec[c + d*x]*Tan[c + d*x] - 2*ArcSin[Sqrt[1 - Sec[c + d*x]]]*(1 + Sec[c + d*x])*Tan[c + d*x] - 10*ArcSin[Sqrt[Sec[c + d*x]]]*(1 + Sec[c + d*x])*Tan[c + d*x])/(4*d*Sqrt[1 - Sec[c + d*x]]*(a*(1 + Sec[c + d*x]))^(3/2))","A",1
253,1,220,97,0.6435967,"\int \frac{\sec ^{\frac{3}{2}}(c+d x)}{(a+a \sec (c+d x))^{3/2}} \, dx","Integrate[Sec[c + d*x]^(3/2)/(a + a*Sec[c + d*x])^(3/2),x]","\frac{2 \sin (c+d x) \sqrt{1-\sec (c+d x)} \sec ^{\frac{3}{2}}(c+d x)-\sqrt{2} \tan (c+d x) \sec (c+d x) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{\sec (c+d x)}}{\sqrt{1-\sec (c+d x)}}\right)-\sqrt{2} \tan (c+d x) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{\sec (c+d x)}}{\sqrt{1-\sec (c+d x)}}\right)+2 \tan (c+d x) (\sec (c+d x)+1) \sin ^{-1}\left(\sqrt{1-\sec (c+d x)}\right)+2 \tan (c+d x) (\sec (c+d x)+1) \sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)}{4 d \sqrt{1-\sec (c+d x)} (a (\sec (c+d x)+1))^{3/2}}","\frac{\tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}+\frac{\sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{2 d (a \sec (c+d x)+a)^{3/2}}",1,"(2*Sqrt[1 - Sec[c + d*x]]*Sec[c + d*x]^(3/2)*Sin[c + d*x] - Sqrt[2]*ArcTan[(Sqrt[2]*Sqrt[Sec[c + d*x]])/Sqrt[1 - Sec[c + d*x]]]*Tan[c + d*x] - Sqrt[2]*ArcTan[(Sqrt[2]*Sqrt[Sec[c + d*x]])/Sqrt[1 - Sec[c + d*x]]]*Sec[c + d*x]*Tan[c + d*x] + 2*ArcSin[Sqrt[1 - Sec[c + d*x]]]*(1 + Sec[c + d*x])*Tan[c + d*x] + 2*ArcSin[Sqrt[Sec[c + d*x]]]*(1 + Sec[c + d*x])*Tan[c + d*x])/(4*d*Sqrt[1 - Sec[c + d*x]]*(a*(1 + Sec[c + d*x]))^(3/2))","B",1
254,1,120,97,0.1893853,"\int \frac{\sqrt{\sec (c+d x)}}{(a+a \sec (c+d x))^{3/2}} \, dx","Integrate[Sqrt[Sec[c + d*x]]/(a + a*Sec[c + d*x])^(3/2),x]","\frac{-2 \sin (c+d x) \sqrt{1-\sec (c+d x)} \sec ^{\frac{3}{2}}(c+d x)-3 \sqrt{2} \tan (c+d x) (\sec (c+d x)+1) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{\sec (c+d x)}}{\sqrt{1-\sec (c+d x)}}\right)}{4 d \sqrt{1-\sec (c+d x)} (a (\sec (c+d x)+1))^{3/2}}","\frac{3 \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}-\frac{\sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{2 d (a \sec (c+d x)+a)^{3/2}}",1,"(-2*Sqrt[1 - Sec[c + d*x]]*Sec[c + d*x]^(3/2)*Sin[c + d*x] - 3*Sqrt[2]*ArcTan[(Sqrt[2]*Sqrt[Sec[c + d*x]])/Sqrt[1 - Sec[c + d*x]]]*(1 + Sec[c + d*x])*Tan[c + d*x])/(4*d*Sqrt[1 - Sec[c + d*x]]*(a*(1 + Sec[c + d*x]))^(3/2))","A",1
255,1,145,137,0.5550949,"\int \frac{1}{\sqrt{\sec (c+d x)} (a+a \sec (c+d x))^{3/2}} \, dx","Integrate[1/(Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x])^(3/2)),x]","\frac{2 \sin (c+d x) \left(5 \sqrt{1-\sec (c+d x)} \sec ^{\frac{3}{2}}(c+d x)+4 \sqrt{-((\sec (c+d x)-1) \sec (c+d x))}\right)+7 \sqrt{2} \tan (c+d x) (\sec (c+d x)+1) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{\sec (c+d x)}}{\sqrt{1-\sec (c+d x)}}\right)}{4 d \sqrt{1-\sec (c+d x)} (a (\sec (c+d x)+1))^{3/2}}","-\frac{7 \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}+\frac{5 \sin (c+d x) \sqrt{\sec (c+d x)}}{2 a d \sqrt{a \sec (c+d x)+a}}-\frac{\sin (c+d x) \sqrt{\sec (c+d x)}}{2 d (a \sec (c+d x)+a)^{3/2}}",1,"(2*(5*Sqrt[1 - Sec[c + d*x]]*Sec[c + d*x]^(3/2) + 4*Sqrt[-((-1 + Sec[c + d*x])*Sec[c + d*x])])*Sin[c + d*x] + 7*Sqrt[2]*ArcTan[(Sqrt[2]*Sqrt[Sec[c + d*x]])/Sqrt[1 - Sec[c + d*x]]]*(1 + Sec[c + d*x])*Tan[c + d*x])/(4*d*Sqrt[1 - Sec[c + d*x]]*(a*(1 + Sec[c + d*x]))^(3/2))","A",0
256,1,150,177,0.9844875,"\int \frac{1}{\sec ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x))^{3/2}} \, dx","Integrate[1/(Sec[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^(3/2)),x]","\frac{\sqrt{1-\sec (c+d x)} (4 \sin (c+d x)-\tan (c+d x) (19 \sec (c+d x)+12))-33 \sqrt{2} \sin (c+d x) \cos ^2\left(\frac{1}{2} (c+d x)\right) \sec ^{\frac{5}{2}}(c+d x) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{\sec (c+d x)}}{\sqrt{1-\sec (c+d x)}}\right)}{6 d \sqrt{-((\sec (c+d x)-1) \sec (c+d x))} (a (\sec (c+d x)+1))^{3/2}}","\frac{11 \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}-\frac{19 \sin (c+d x) \sqrt{\sec (c+d x)}}{6 a d \sqrt{a \sec (c+d x)+a}}+\frac{7 \sin (c+d x)}{6 a d \sqrt{\sec (c+d x)} \sqrt{a \sec (c+d x)+a}}-\frac{\sin (c+d x)}{2 d \sqrt{\sec (c+d x)} (a \sec (c+d x)+a)^{3/2}}",1,"(-33*Sqrt[2]*ArcTan[(Sqrt[2]*Sqrt[Sec[c + d*x]])/Sqrt[1 - Sec[c + d*x]]]*Cos[(c + d*x)/2]^2*Sec[c + d*x]^(5/2)*Sin[c + d*x] + Sqrt[1 - Sec[c + d*x]]*(4*Sin[c + d*x] - (12 + 19*Sec[c + d*x])*Tan[c + d*x]))/(6*d*Sqrt[-((-1 + Sec[c + d*x])*Sec[c + d*x])]*(a*(1 + Sec[c + d*x]))^(3/2))","A",0
257,1,163,217,1.3833718,"\int \frac{1}{\sec ^{\frac{5}{2}}(c+d x) (a+a \sec (c+d x))^{3/2}} \, dx","Integrate[1/(Sec[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^(3/2)),x]","\frac{(39 \cos (c+d x)-2 \cos (2 (c+d x))+\cos (3 (c+d x))+47) \tan (c+d x) \sqrt{1-\sec (c+d x)} \sec (c+d x)+75 \sqrt{2} \sin (c+d x) \cos ^2\left(\frac{1}{2} (c+d x)\right) \sec ^{\frac{5}{2}}(c+d x) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{\sec (c+d x)}}{\sqrt{1-\sec (c+d x)}}\right)}{10 d \sqrt{-((\sec (c+d x)-1) \sec (c+d x))} (a (\sec (c+d x)+1))^{3/2}}","-\frac{15 \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}+\frac{9 \sin (c+d x)}{10 a d \sec ^{\frac{3}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}-\frac{\sin (c+d x)}{2 d \sec ^{\frac{3}{2}}(c+d x) (a \sec (c+d x)+a)^{3/2}}+\frac{49 \sin (c+d x) \sqrt{\sec (c+d x)}}{10 a d \sqrt{a \sec (c+d x)+a}}-\frac{13 \sin (c+d x)}{10 a d \sqrt{\sec (c+d x)} \sqrt{a \sec (c+d x)+a}}",1,"(75*Sqrt[2]*ArcTan[(Sqrt[2]*Sqrt[Sec[c + d*x]])/Sqrt[1 - Sec[c + d*x]]]*Cos[(c + d*x)/2]^2*Sec[c + d*x]^(5/2)*Sin[c + d*x] + (47 + 39*Cos[c + d*x] - 2*Cos[2*(c + d*x)] + Cos[3*(c + d*x)])*Sqrt[1 - Sec[c + d*x]]*Sec[c + d*x]*Tan[c + d*x])/(10*d*Sqrt[-((-1 + Sec[c + d*x])*Sec[c + d*x])]*(a*(1 + Sec[c + d*x]))^(3/2))","A",0
258,1,340,214,1.3450082,"\int \frac{\sec ^{\frac{9}{2}}(c+d x)}{(a+a \sec (c+d x))^{5/2}} \, dx","Integrate[Sec[c + d*x]^(9/2)/(a + a*Sec[c + d*x])^(5/2),x]","\frac{32 \sin (c+d x) \sqrt{1-\sec (c+d x)} \sec ^{\frac{7}{2}}(c+d x)+110 \sin (c+d x) \sqrt{1-\sec (c+d x)} \sec ^{\frac{5}{2}}(c+d x)+70 \sin (c+d x) \sqrt{1-\sec (c+d x)} \sec ^{\frac{3}{2}}(c+d x)-115 \sqrt{2} \tan (c+d x) \sec ^2(c+d x) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{\sec (c+d x)}}{\sqrt{1-\sec (c+d x)}}\right)-230 \sqrt{2} \tan (c+d x) \sec (c+d x) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{\sec (c+d x)}}{\sqrt{1-\sec (c+d x)}}\right)-115 \sqrt{2} \tan (c+d x) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{\sec (c+d x)}}{\sqrt{1-\sec (c+d x)}}\right)+70 \tan (c+d x) (\sec (c+d x)+1)^2 \sin ^{-1}\left(\sqrt{1-\sec (c+d x)}\right)+230 \tan (c+d x) (\sec (c+d x)+1)^2 \sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)}{32 d \sqrt{1-\sec (c+d x)} (a (\sec (c+d x)+1))^{5/2}}","\frac{115 \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}-\frac{5 \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{a^{5/2} d}+\frac{35 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{16 a^2 d \sqrt{a \sec (c+d x)+a}}-\frac{\sin (c+d x) \sec ^{\frac{7}{2}}(c+d x)}{4 d (a \sec (c+d x)+a)^{5/2}}-\frac{15 \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{16 a d (a \sec (c+d x)+a)^{3/2}}",1,"(70*Sqrt[1 - Sec[c + d*x]]*Sec[c + d*x]^(3/2)*Sin[c + d*x] + 110*Sqrt[1 - Sec[c + d*x]]*Sec[c + d*x]^(5/2)*Sin[c + d*x] + 32*Sqrt[1 - Sec[c + d*x]]*Sec[c + d*x]^(7/2)*Sin[c + d*x] - 115*Sqrt[2]*ArcTan[(Sqrt[2]*Sqrt[Sec[c + d*x]])/Sqrt[1 - Sec[c + d*x]]]*Tan[c + d*x] - 230*Sqrt[2]*ArcTan[(Sqrt[2]*Sqrt[Sec[c + d*x]])/Sqrt[1 - Sec[c + d*x]]]*Sec[c + d*x]*Tan[c + d*x] - 115*Sqrt[2]*ArcTan[(Sqrt[2]*Sqrt[Sec[c + d*x]])/Sqrt[1 - Sec[c + d*x]]]*Sec[c + d*x]^2*Tan[c + d*x] + 70*ArcSin[Sqrt[1 - Sec[c + d*x]]]*(1 + Sec[c + d*x])^2*Tan[c + d*x] + 230*ArcSin[Sqrt[Sec[c + d*x]]]*(1 + Sec[c + d*x])^2*Tan[c + d*x])/(32*d*Sqrt[1 - Sec[c + d*x]]*(a*(1 + Sec[c + d*x]))^(5/2))","A",1
259,1,308,174,0.7336548,"\int \frac{\sec ^{\frac{7}{2}}(c+d x)}{(a+a \sec (c+d x))^{5/2}} \, dx","Integrate[Sec[c + d*x]^(7/2)/(a + a*Sec[c + d*x])^(5/2),x]","\frac{-30 \sin (c+d x) \sqrt{1-\sec (c+d x)} \sec ^{\frac{5}{2}}(c+d x)-22 \sin (c+d x) \sqrt{1-\sec (c+d x)} \sec ^{\frac{3}{2}}(c+d x)+43 \sqrt{2} \tan (c+d x) \sec ^2(c+d x) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{\sec (c+d x)}}{\sqrt{1-\sec (c+d x)}}\right)+86 \sqrt{2} \tan (c+d x) \sec (c+d x) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{\sec (c+d x)}}{\sqrt{1-\sec (c+d x)}}\right)+43 \sqrt{2} \tan (c+d x) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{\sec (c+d x)}}{\sqrt{1-\sec (c+d x)}}\right)-22 \tan (c+d x) (\sec (c+d x)+1)^2 \sin ^{-1}\left(\sqrt{1-\sec (c+d x)}\right)-86 \tan (c+d x) (\sec (c+d x)+1)^2 \sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)}{32 d \sqrt{1-\sec (c+d x)} (a (\sec (c+d x)+1))^{5/2}}","-\frac{43 \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}+\frac{2 \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{a^{5/2} d}-\frac{\sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{4 d (a \sec (c+d x)+a)^{5/2}}-\frac{11 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{16 a d (a \sec (c+d x)+a)^{3/2}}",1,"(-22*Sqrt[1 - Sec[c + d*x]]*Sec[c + d*x]^(3/2)*Sin[c + d*x] - 30*Sqrt[1 - Sec[c + d*x]]*Sec[c + d*x]^(5/2)*Sin[c + d*x] + 43*Sqrt[2]*ArcTan[(Sqrt[2]*Sqrt[Sec[c + d*x]])/Sqrt[1 - Sec[c + d*x]]]*Tan[c + d*x] + 86*Sqrt[2]*ArcTan[(Sqrt[2]*Sqrt[Sec[c + d*x]])/Sqrt[1 - Sec[c + d*x]]]*Sec[c + d*x]*Tan[c + d*x] + 43*Sqrt[2]*ArcTan[(Sqrt[2]*Sqrt[Sec[c + d*x]])/Sqrt[1 - Sec[c + d*x]]]*Sec[c + d*x]^2*Tan[c + d*x] - 22*ArcSin[Sqrt[1 - Sec[c + d*x]]]*(1 + Sec[c + d*x])^2*Tan[c + d*x] - 86*ArcSin[Sqrt[Sec[c + d*x]]]*(1 + Sec[c + d*x])^2*Tan[c + d*x])/(32*d*Sqrt[1 - Sec[c + d*x]]*(a*(1 + Sec[c + d*x]))^(5/2))","A",1
260,1,308,137,0.803798,"\int \frac{\sec ^{\frac{5}{2}}(c+d x)}{(a+a \sec (c+d x))^{5/2}} \, dx","Integrate[Sec[c + d*x]^(5/2)/(a + a*Sec[c + d*x])^(5/2),x]","\frac{14 \sin (c+d x) \sqrt{1-\sec (c+d x)} \sec ^{\frac{5}{2}}(c+d x)+6 \sin (c+d x) \sqrt{1-\sec (c+d x)} \sec ^{\frac{3}{2}}(c+d x)-3 \sqrt{2} \tan (c+d x) \sec ^2(c+d x) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{\sec (c+d x)}}{\sqrt{1-\sec (c+d x)}}\right)-6 \sqrt{2} \tan (c+d x) \sec (c+d x) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{\sec (c+d x)}}{\sqrt{1-\sec (c+d x)}}\right)-3 \sqrt{2} \tan (c+d x) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{\sec (c+d x)}}{\sqrt{1-\sec (c+d x)}}\right)+6 \tan (c+d x) (\sec (c+d x)+1)^2 \sin ^{-1}\left(\sqrt{1-\sec (c+d x)}\right)+6 \tan (c+d x) (\sec (c+d x)+1)^2 \sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)}{32 d \sqrt{1-\sec (c+d x)} (a (\sec (c+d x)+1))^{5/2}}","\frac{3 \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}+\frac{\sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{4 d (a \sec (c+d x)+a)^{5/2}}+\frac{3 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{16 a d (a \sec (c+d x)+a)^{3/2}}",1,"(6*Sqrt[1 - Sec[c + d*x]]*Sec[c + d*x]^(3/2)*Sin[c + d*x] + 14*Sqrt[1 - Sec[c + d*x]]*Sec[c + d*x]^(5/2)*Sin[c + d*x] - 3*Sqrt[2]*ArcTan[(Sqrt[2]*Sqrt[Sec[c + d*x]])/Sqrt[1 - Sec[c + d*x]]]*Tan[c + d*x] - 6*Sqrt[2]*ArcTan[(Sqrt[2]*Sqrt[Sec[c + d*x]])/Sqrt[1 - Sec[c + d*x]]]*Sec[c + d*x]*Tan[c + d*x] - 3*Sqrt[2]*ArcTan[(Sqrt[2]*Sqrt[Sec[c + d*x]])/Sqrt[1 - Sec[c + d*x]]]*Sec[c + d*x]^2*Tan[c + d*x] + 6*ArcSin[Sqrt[1 - Sec[c + d*x]]]*(1 + Sec[c + d*x])^2*Tan[c + d*x] + 6*ArcSin[Sqrt[Sec[c + d*x]]]*(1 + Sec[c + d*x])^2*Tan[c + d*x])/(32*d*Sqrt[1 - Sec[c + d*x]]*(a*(1 + Sec[c + d*x]))^(5/2))","B",1
261,1,266,137,1.9804507,"\int \frac{\sec ^{\frac{3}{2}}(c+d x)}{(a+a \sec (c+d x))^{5/2}} \, dx","Integrate[Sec[c + d*x]^(3/2)/(a + a*Sec[c + d*x])^(5/2),x]","-\frac{10 \sin (c+d x) \sqrt{1-\sec (c+d x)} (\sec (c+d x)+1) \sec ^{\frac{5}{2}}(c+d x)+8 \sin (c+d x) \sqrt{1-\sec (c+d x)} \sec ^{\frac{5}{2}}(c+d x)-10 \sin (c+d x) \sqrt{1-\sec (c+d x)} (\sec (c+d x)+1)^2 \sec ^{\frac{3}{2}}(c+d x)+5 \sqrt{2} \tan (c+d x) (\sec (c+d x)+1)^2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{\sec (c+d x)}}{\sqrt{1-\sec (c+d x)}}\right)-10 \tan (c+d x) (\sec (c+d x)+1)^2 \sin ^{-1}\left(\sqrt{1-\sec (c+d x)}\right)-10 \tan (c+d x) (\sec (c+d x)+1)^2 \sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)}{32 d \sqrt{1-\sec (c+d x)} (a (\sec (c+d x)+1))^{5/2}}","\frac{5 \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}-\frac{\sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{4 d (a \sec (c+d x)+a)^{5/2}}+\frac{5 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{16 a d (a \sec (c+d x)+a)^{3/2}}",1,"-1/32*(8*Sqrt[1 - Sec[c + d*x]]*Sec[c + d*x]^(5/2)*Sin[c + d*x] + 10*Sqrt[1 - Sec[c + d*x]]*Sec[c + d*x]^(5/2)*(1 + Sec[c + d*x])*Sin[c + d*x] - 10*Sqrt[1 - Sec[c + d*x]]*Sec[c + d*x]^(3/2)*(1 + Sec[c + d*x])^2*Sin[c + d*x] - 10*ArcSin[Sqrt[1 - Sec[c + d*x]]]*(1 + Sec[c + d*x])^2*Tan[c + d*x] - 10*ArcSin[Sqrt[Sec[c + d*x]]]*(1 + Sec[c + d*x])^2*Tan[c + d*x] + 5*Sqrt[2]*ArcTan[(Sqrt[2]*Sqrt[Sec[c + d*x]])/Sqrt[1 - Sec[c + d*x]]]*(1 + Sec[c + d*x])^2*Tan[c + d*x])/(d*Sqrt[1 - Sec[c + d*x]]*(a*(1 + Sec[c + d*x]))^(5/2))","A",1
262,1,146,137,1.0018017,"\int \frac{\sqrt{\sec (c+d x)}}{(a+a \sec (c+d x))^{5/2}} \, dx","Integrate[Sqrt[Sec[c + d*x]]/(a + a*Sec[c + d*x])^(5/2),x]","\frac{-\sin (c+d x) (13 \cos (c+d x)+9) \sqrt{1-\sec (c+d x)} \sec ^{\frac{5}{2}}(c+d x)-76 \sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right) \cos ^5\left(\frac{1}{2} (c+d x)\right) \sec ^3(c+d x) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{\sec (c+d x)}}{\sqrt{1-\sec (c+d x)}}\right)}{16 d \sqrt{1-\sec (c+d x)} (a (\sec (c+d x)+1))^{5/2}}","\frac{19 \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}-\frac{9 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{16 a d (a \sec (c+d x)+a)^{3/2}}-\frac{\sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{4 d (a \sec (c+d x)+a)^{5/2}}",1,"(-76*Sqrt[2]*ArcTan[(Sqrt[2]*Sqrt[Sec[c + d*x]])/Sqrt[1 - Sec[c + d*x]]]*Cos[(c + d*x)/2]^5*Sec[c + d*x]^3*Sin[(c + d*x)/2] - (9 + 13*Cos[c + d*x])*Sqrt[1 - Sec[c + d*x]]*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(16*d*Sqrt[1 - Sec[c + d*x]]*(a*(1 + Sec[c + d*x]))^(5/2))","A",1
263,1,186,177,1.3918161,"\int \frac{1}{\sqrt{\sec (c+d x)} (a+a \sec (c+d x))^{5/2}} \, dx","Integrate[1/(Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x])^(5/2)),x]","\frac{\sin (c+d x) \left(49 \sqrt{1-\sec (c+d x)} \sec ^{\frac{5}{2}}(c+d x)+85 \sqrt{1-\sec (c+d x)} \sec ^{\frac{3}{2}}(c+d x)+32 \sqrt{-((\sec (c+d x)-1) \sec (c+d x))}\right)+300 \sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right) \cos ^5\left(\frac{1}{2} (c+d x)\right) \sec ^3(c+d x) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{\sec (c+d x)}}{\sqrt{1-\sec (c+d x)}}\right)}{16 d \sqrt{1-\sec (c+d x)} (a (\sec (c+d x)+1))^{5/2}}","-\frac{75 \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}+\frac{49 \sin (c+d x) \sqrt{\sec (c+d x)}}{16 a^2 d \sqrt{a \sec (c+d x)+a}}-\frac{13 \sin (c+d x) \sqrt{\sec (c+d x)}}{16 a d (a \sec (c+d x)+a)^{3/2}}-\frac{\sin (c+d x) \sqrt{\sec (c+d x)}}{4 d (a \sec (c+d x)+a)^{5/2}}",1,"(300*Sqrt[2]*ArcTan[(Sqrt[2]*Sqrt[Sec[c + d*x]])/Sqrt[1 - Sec[c + d*x]]]*Cos[(c + d*x)/2]^5*Sec[c + d*x]^3*Sin[(c + d*x)/2] + (85*Sqrt[1 - Sec[c + d*x]]*Sec[c + d*x]^(3/2) + 49*Sqrt[1 - Sec[c + d*x]]*Sec[c + d*x]^(5/2) + 32*Sqrt[-((-1 + Sec[c + d*x])*Sec[c + d*x])])*Sin[c + d*x])/(16*d*Sqrt[1 - Sec[c + d*x]]*(a*(1 + Sec[c + d*x]))^(5/2))","A",1
264,1,165,217,2.492237,"\int \frac{1}{\sec ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x))^{5/2}} \, dx","Integrate[1/(Sec[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^(5/2)),x]","-\frac{\sec (c+d x) \left(\tan (c+d x) \sqrt{1-\sec (c+d x)} (379 \sec (c+d x)+16 \cos (2 (c+d x)) (5 \sec (c+d x)-1)+487)+978 \sqrt{2} \sin (c+d x) \cos ^4\left(\frac{1}{2} (c+d x)\right) \sec ^{\frac{5}{2}}(c+d x) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{\sec (c+d x)}}{\sqrt{1-\sec (c+d x)}}\right)\right)}{48 d \sqrt{-((\sec (c+d x)-1) \sec (c+d x))} (a (\sec (c+d x)+1))^{5/2}}","\frac{163 \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}-\frac{299 \sin (c+d x) \sqrt{\sec (c+d x)}}{48 a^2 d \sqrt{a \sec (c+d x)+a}}+\frac{95 \sin (c+d x)}{48 a^2 d \sqrt{\sec (c+d x)} \sqrt{a \sec (c+d x)+a}}-\frac{17 \sin (c+d x)}{16 a d \sqrt{\sec (c+d x)} (a \sec (c+d x)+a)^{3/2}}-\frac{\sin (c+d x)}{4 d \sqrt{\sec (c+d x)} (a \sec (c+d x)+a)^{5/2}}",1,"-1/48*(Sec[c + d*x]*(978*Sqrt[2]*ArcTan[(Sqrt[2]*Sqrt[Sec[c + d*x]])/Sqrt[1 - Sec[c + d*x]]]*Cos[(c + d*x)/2]^4*Sec[c + d*x]^(5/2)*Sin[c + d*x] + Sqrt[1 - Sec[c + d*x]]*(487 + 379*Sec[c + d*x] + 16*Cos[2*(c + d*x)]*(-1 + 5*Sec[c + d*x]))*Tan[c + d*x]))/(d*Sqrt[-((-1 + Sec[c + d*x])*Sec[c + d*x])]*(a*(1 + Sec[c + d*x]))^(5/2))","A",0
265,1,140,126,0.4575108,"\int \frac{\sec ^{\frac{7}{2}}(c+d x)}{\sqrt{1+\sec (c+d x)}} \, dx","Integrate[Sec[c + d*x]^(7/2)/Sqrt[1 + Sec[c + d*x]],x]","\frac{\sqrt{-\tan ^2(c+d x)} \cot (c+d x) \left(-2 \sqrt{1-\sec (c+d x)} \sec ^{\frac{3}{2}}(c+d x)+\sqrt{-((\sec (c+d x)-1) \sec (c+d x))}+\sin ^{-1}\left(\sqrt{1-\sec (c+d x)}\right)+8 \sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)-4 \sqrt{2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{\sec (c+d x)}}{\sqrt{1-\sec (c+d x)}}\right)\right)}{4 d}","\frac{\sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{2 d \sqrt{\sec (c+d x)+1}}-\frac{\sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{4 d \sqrt{\sec (c+d x)+1}}-\frac{\sqrt{2} \sinh ^{-1}\left(\frac{\tan (c+d x)}{\sec (c+d x)+1}\right)}{d}+\frac{7 \sinh ^{-1}\left(\frac{\tan (c+d x)}{\sqrt{\sec (c+d x)+1}}\right)}{4 d}",1,"(Cot[c + d*x]*(ArcSin[Sqrt[1 - Sec[c + d*x]]] + 8*ArcSin[Sqrt[Sec[c + d*x]]] - 4*Sqrt[2]*ArcTan[(Sqrt[2]*Sqrt[Sec[c + d*x]])/Sqrt[1 - Sec[c + d*x]]] - 2*Sqrt[1 - Sec[c + d*x]]*Sec[c + d*x]^(3/2) + Sqrt[-((-1 + Sec[c + d*x])*Sec[c + d*x])])*Sqrt[-Tan[c + d*x]^2])/(4*d)","A",0
266,1,111,85,0.3092749,"\int \frac{\sec ^{\frac{5}{2}}(c+d x)}{\sqrt{1+\sec (c+d x)}} \, dx","Integrate[Sec[c + d*x]^(5/2)/Sqrt[1 + Sec[c + d*x]],x]","\frac{\tan (c+d x) \left(\sqrt{-((\sec (c+d x)-1) \sec (c+d x))}+\sin ^{-1}\left(\sqrt{1-\sec (c+d x)}\right)+2 \sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)-\sqrt{2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{\sec (c+d x)}}{\sqrt{1-\sec (c+d x)}}\right)\right)}{d \sqrt{-\tan ^2(c+d x)}}","\frac{\sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{d \sqrt{\sec (c+d x)+1}}+\frac{\sqrt{2} \sinh ^{-1}\left(\frac{\tan (c+d x)}{\sec (c+d x)+1}\right)}{d}-\frac{\sinh ^{-1}\left(\frac{\tan (c+d x)}{\sqrt{\sec (c+d x)+1}}\right)}{d}",1,"((ArcSin[Sqrt[1 - Sec[c + d*x]]] + 2*ArcSin[Sqrt[Sec[c + d*x]]] - Sqrt[2]*ArcTan[(Sqrt[2]*Sqrt[Sec[c + d*x]])/Sqrt[1 - Sec[c + d*x]]] + Sqrt[-((-1 + Sec[c + d*x])*Sec[c + d*x])])*Tan[c + d*x])/(d*Sqrt[-Tan[c + d*x]^2])","A",0
267,1,76,54,0.0877317,"\int \frac{\sec ^{\frac{3}{2}}(c+d x)}{\sqrt{1+\sec (c+d x)}} \, dx","Integrate[Sec[c + d*x]^(3/2)/Sqrt[1 + Sec[c + d*x]],x]","\frac{\sqrt{-\tan ^2(c+d x)} \cot (c+d x) \left(2 \sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)-\sqrt{2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{\sec (c+d x)}}{\sqrt{1-\sec (c+d x)}}\right)\right)}{d}","\frac{2 \sinh ^{-1}\left(\frac{\tan (c+d x)}{\sqrt{\sec (c+d x)+1}}\right)}{d}-\frac{\sqrt{2} \sinh ^{-1}\left(\frac{\tan (c+d x)}{\sec (c+d x)+1}\right)}{d}",1,"((2*ArcSin[Sqrt[Sec[c + d*x]]] - Sqrt[2]*ArcTan[(Sqrt[2]*Sqrt[Sec[c + d*x]])/Sqrt[1 - Sec[c + d*x]]])*Cot[c + d*x]*Sqrt[-Tan[c + d*x]^2])/d","A",0
268,1,40,27,0.0310526,"\int \frac{\sqrt{\sec (c+d x)}}{\sqrt{1+\sec (c+d x)}} \, dx","Integrate[Sqrt[Sec[c + d*x]]/Sqrt[1 + Sec[c + d*x]],x]","\frac{2 \cos \left(\frac{1}{2} (c+d x)\right) \sqrt{\frac{1}{\cos (c+d x)+1}} \tanh ^{-1}\left(\sin \left(\frac{1}{2} (c+d x)\right)\right)}{d}","\frac{\sqrt{2} \sinh ^{-1}\left(\frac{\tan (c+d x)}{\sec (c+d x)+1}\right)}{d}",1,"(2*ArcTanh[Sin[(c + d*x)/2]]*Cos[(c + d*x)/2]*Sqrt[(1 + Cos[c + d*x])^(-1)])/d","A",1
269,1,90,62,0.2310397,"\int \frac{1}{\sqrt{\sec (c+d x)} \sqrt{1+\sec (c+d x)}} \, dx","Integrate[1/(Sqrt[Sec[c + d*x]]*Sqrt[1 + Sec[c + d*x]]),x]","\frac{2 \sin (c+d x) \sqrt{-((\sec (c+d x)-1) \sec (c+d x))}+\sqrt{2} \tan (c+d x) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{\sec (c+d x)}}{\sqrt{1-\sec (c+d x)}}\right)}{d \sqrt{-\tan ^2(c+d x)}}","\frac{2 \sin (c+d x) \sqrt{\sec (c+d x)}}{d \sqrt{\sec (c+d x)+1}}-\frac{\sqrt{2} \sinh ^{-1}\left(\frac{\tan (c+d x)}{\sec (c+d x)+1}\right)}{d}",1,"(2*Sqrt[-((-1 + Sec[c + d*x])*Sec[c + d*x])]*Sin[c + d*x] + Sqrt[2]*ArcTan[(Sqrt[2]*Sqrt[Sec[c + d*x]])/Sqrt[1 - Sec[c + d*x]]]*Tan[c + d*x])/(d*Sqrt[-Tan[c + d*x]^2])","A",0
270,1,118,98,0.2564316,"\int \frac{1}{\sec ^{\frac{3}{2}}(c+d x) \sqrt{1+\sec (c+d x)}} \, dx","Integrate[1/(Sec[c + d*x]^(3/2)*Sqrt[1 + Sec[c + d*x]]),x]","\frac{\tan (c+d x) \left(2 (\cos (c+d x)-1) \sqrt{1-\sec (c+d x)}-3 \sqrt{2} \sqrt{\sec (c+d x)} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{\sec (c+d x)}}{\sqrt{1-\sec (c+d x)}}\right)\right)}{3 d \sqrt{-((\sec (c+d x)-1) \sec (c+d x))} \sqrt{\sec (c+d x)+1}}","-\frac{2 \sin (c+d x) \sqrt{\sec (c+d x)}}{3 d \sqrt{\sec (c+d x)+1}}+\frac{2 \sin (c+d x)}{3 d \sqrt{\sec (c+d x)} \sqrt{\sec (c+d x)+1}}+\frac{\sqrt{2} \sinh ^{-1}\left(\frac{\tan (c+d x)}{\sec (c+d x)+1}\right)}{d}",1,"((2*(-1 + Cos[c + d*x])*Sqrt[1 - Sec[c + d*x]] - 3*Sqrt[2]*ArcTan[(Sqrt[2]*Sqrt[Sec[c + d*x]])/Sqrt[1 - Sec[c + d*x]]]*Sqrt[Sec[c + d*x]])*Tan[c + d*x])/(3*d*Sqrt[-((-1 + Sec[c + d*x])*Sec[c + d*x])]*Sqrt[1 + Sec[c + d*x]])","A",0
271,1,122,134,0.3062926,"\int \frac{1}{\sec ^{\frac{5}{2}}(c+d x) \sqrt{1+\sec (c+d x)}} \, dx","Integrate[1/(Sec[c + d*x]^(5/2)*Sqrt[1 + Sec[c + d*x]]),x]","\frac{\sin (c+d x) \left(2 \sqrt{1-\sec (c+d x)} \left(13 \sec ^2(c+d x)-\sec (c+d x)+3\right)+15 \sqrt{2} \sec ^{\frac{5}{2}}(c+d x) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{\sec (c+d x)}}{\sqrt{1-\sec (c+d x)}}\right)\right)}{15 d \sqrt{-\tan ^2(c+d x)} \sec ^{\frac{3}{2}}(c+d x)}","\frac{2 \sin (c+d x)}{5 d \sec ^{\frac{3}{2}}(c+d x) \sqrt{\sec (c+d x)+1}}+\frac{26 \sin (c+d x) \sqrt{\sec (c+d x)}}{15 d \sqrt{\sec (c+d x)+1}}-\frac{2 \sin (c+d x)}{15 d \sqrt{\sec (c+d x)} \sqrt{\sec (c+d x)+1}}-\frac{\sqrt{2} \sinh ^{-1}\left(\frac{\tan (c+d x)}{\sec (c+d x)+1}\right)}{d}",1,"((15*Sqrt[2]*ArcTan[(Sqrt[2]*Sqrt[Sec[c + d*x]])/Sqrt[1 - Sec[c + d*x]]]*Sec[c + d*x]^(5/2) + 2*Sqrt[1 - Sec[c + d*x]]*(3 - Sec[c + d*x] + 13*Sec[c + d*x]^2))*Sin[c + d*x])/(15*d*Sec[c + d*x]^(3/2)*Sqrt[-Tan[c + d*x]^2])","A",0
272,1,71,325,0.277991,"\int (e \sec (c+d x))^{4/3} \sqrt{a+a \sec (c+d x)} \, dx","Integrate[(e*Sec[c + d*x])^(4/3)*Sqrt[a + a*Sec[c + d*x]],x]","\frac{2 \tan \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\sec (c+d x)+1)} (e \sec (c+d x))^{4/3} \, _2F_1\left(-\frac{1}{3},\frac{1}{2};\frac{3}{2};1-\sec (c+d x)\right)}{d \sec ^{\frac{4}{3}}(c+d x)}","\frac{4\ 3^{3/4} \sqrt{2+\sqrt{3}} a^2 e \tan (c+d x) \left(\sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}\right) \sqrt{\frac{\sqrt[3]{e} \sqrt[3]{e \sec (c+d x)}+(e \sec (c+d x))^{2/3}+e^{2/3}}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}\right)^2}} F\left(\sin ^{-1}\left(\frac{\left(1-\sqrt{3}\right) \sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}}{\left(1+\sqrt{3}\right) \sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}}\right)|-7-4 \sqrt{3}\right)}{5 d (a-a \sec (c+d x)) \sqrt{a \sec (c+d x)+a} \sqrt{\frac{\sqrt[3]{e} \left(\sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}\right)^2}}}+\frac{6 a e \tan (c+d x) \sqrt[3]{e \sec (c+d x)}}{5 d \sqrt{a \sec (c+d x)+a}}",1,"(2*Hypergeometric2F1[-1/3, 1/2, 3/2, 1 - Sec[c + d*x]]*(e*Sec[c + d*x])^(4/3)*Sqrt[a*(1 + Sec[c + d*x])]*Tan[(c + d*x)/2])/(d*Sec[c + d*x]^(4/3))","C",1
273,1,71,280,0.1630085,"\int \sqrt[3]{e \sec (c+d x)} \sqrt{a+a \sec (c+d x)} \, dx","Integrate[(e*Sec[c + d*x])^(1/3)*Sqrt[a + a*Sec[c + d*x]],x]","\frac{2 \tan \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\sec (c+d x)+1)} \sqrt[3]{e \sec (c+d x)} \, _2F_1\left(\frac{1}{2},\frac{2}{3};\frac{3}{2};1-\sec (c+d x)\right)}{d \sqrt[3]{\sec (c+d x)}}","\frac{2\ 3^{3/4} \sqrt{2+\sqrt{3}} a^2 \tan (c+d x) \left(\sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}\right) \sqrt{\frac{\sqrt[3]{e} \sqrt[3]{e \sec (c+d x)}+(e \sec (c+d x))^{2/3}+e^{2/3}}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}\right)^2}} F\left(\sin ^{-1}\left(\frac{\left(1-\sqrt{3}\right) \sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}}{\left(1+\sqrt{3}\right) \sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}}\right)|-7-4 \sqrt{3}\right)}{d (a-a \sec (c+d x)) \sqrt{a \sec (c+d x)+a} \sqrt{\frac{\sqrt[3]{e} \left(\sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}\right)^2}}}",1,"(2*Hypergeometric2F1[1/2, 2/3, 3/2, 1 - Sec[c + d*x]]*(e*Sec[c + d*x])^(1/3)*Sqrt[a*(1 + Sec[c + d*x])]*Tan[(c + d*x)/2])/(d*Sec[c + d*x]^(1/3))","C",1
274,1,71,326,0.171215,"\int \frac{\sqrt{a+a \sec (c+d x)}}{(e \sec (c+d x))^{2/3}} \, dx","Integrate[Sqrt[a + a*Sec[c + d*x]]/(e*Sec[c + d*x])^(2/3),x]","\frac{2 \tan \left(\frac{1}{2} (c+d x)\right) \sec ^{\frac{2}{3}}(c+d x) \sqrt{a (\sec (c+d x)+1)} \, _2F_1\left(\frac{1}{2},\frac{5}{3};\frac{3}{2};1-\sec (c+d x)\right)}{d (e \sec (c+d x))^{2/3}}","\frac{3^{3/4} \sqrt{2+\sqrt{3}} a^2 \tan (c+d x) \left(\sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}\right) \sqrt{\frac{\sqrt[3]{e} \sqrt[3]{e \sec (c+d x)}+(e \sec (c+d x))^{2/3}+e^{2/3}}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}\right)^2}} F\left(\sin ^{-1}\left(\frac{\left(1-\sqrt{3}\right) \sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}}{\left(1+\sqrt{3}\right) \sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}}\right)|-7-4 \sqrt{3}\right)}{2 d e (a-a \sec (c+d x)) \sqrt{a \sec (c+d x)+a} \sqrt{\frac{\sqrt[3]{e} \left(\sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}\right)^2}}}+\frac{3 a \tan (c+d x)}{2 d \sqrt{a \sec (c+d x)+a} (e \sec (c+d x))^{2/3}}",1,"(2*Hypergeometric2F1[1/2, 5/3, 3/2, 1 - Sec[c + d*x]]*Sec[c + d*x]^(2/3)*Sqrt[a*(1 + Sec[c + d*x])]*Tan[(c + d*x)/2])/(d*(e*Sec[c + d*x])^(2/3))","C",1
275,1,71,716,0.2609964,"\int (e \sec (c+d x))^{8/3} \sqrt{a+a \sec (c+d x)} \, dx","Integrate[(e*Sec[c + d*x])^(8/3)*Sqrt[a + a*Sec[c + d*x]],x]","\frac{2 \tan \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\sec (c+d x)+1)} (e \sec (c+d x))^{8/3} \, _2F_1\left(-\frac{5}{3},\frac{1}{2};\frac{3}{2};1-\sec (c+d x)\right)}{d \sec ^{\frac{8}{3}}(c+d x)}","-\frac{80 \sqrt{2} 3^{3/4} a^2 e^{7/3} \tan (c+d x) \left(\sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}\right) \sqrt{\frac{\sqrt[3]{e} \sqrt[3]{e \sec (c+d x)}+(e \sec (c+d x))^{2/3}+e^{2/3}}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}\right)^2}} F\left(\sin ^{-1}\left(\frac{\left(1-\sqrt{3}\right) \sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}}{\left(1+\sqrt{3}\right) \sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}}\right)|-7-4 \sqrt{3}\right)}{91 d (a-a \sec (c+d x)) \sqrt{a \sec (c+d x)+a} \sqrt{\frac{\sqrt[3]{e} \left(\sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}\right)^2}}}+\frac{120 \sqrt[4]{3} \sqrt{2-\sqrt{3}} a^2 e^{7/3} \tan (c+d x) \left(\sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}\right) \sqrt{\frac{\sqrt[3]{e} \sqrt[3]{e \sec (c+d x)}+(e \sec (c+d x))^{2/3}+e^{2/3}}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}\right)^2}} E\left(\sin ^{-1}\left(\frac{\left(1-\sqrt{3}\right) \sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}}{\left(1+\sqrt{3}\right) \sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}}\right)|-7-4 \sqrt{3}\right)}{91 d (a-a \sec (c+d x)) \sqrt{a \sec (c+d x)+a} \sqrt{\frac{\sqrt[3]{e} \left(\sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}\right)^2}}}-\frac{240 a e^3 \tan (c+d x)}{91 d \sqrt{a \sec (c+d x)+a} \left(\left(1+\sqrt{3}\right) \sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}\right)}+\frac{60 a e^2 \tan (c+d x) (e \sec (c+d x))^{2/3}}{91 d \sqrt{a \sec (c+d x)+a}}+\frac{6 a e \tan (c+d x) (e \sec (c+d x))^{5/3}}{13 d \sqrt{a \sec (c+d x)+a}}",1,"(2*Hypergeometric2F1[-5/3, 1/2, 3/2, 1 - Sec[c + d*x]]*(e*Sec[c + d*x])^(8/3)*Sqrt[a*(1 + Sec[c + d*x])]*Tan[(c + d*x)/2])/(d*Sec[c + d*x]^(8/3))","C",1
276,1,71,673,0.2536057,"\int (e \sec (c+d x))^{5/3} \sqrt{a+a \sec (c+d x)} \, dx","Integrate[(e*Sec[c + d*x])^(5/3)*Sqrt[a + a*Sec[c + d*x]],x]","\frac{2 \tan \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\sec (c+d x)+1)} (e \sec (c+d x))^{5/3} \, _2F_1\left(-\frac{2}{3},\frac{1}{2};\frac{3}{2};1-\sec (c+d x)\right)}{d \sec ^{\frac{5}{3}}(c+d x)}","-\frac{8 \sqrt{2} 3^{3/4} a^2 e^{4/3} \tan (c+d x) \left(\sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}\right) \sqrt{\frac{\sqrt[3]{e} \sqrt[3]{e \sec (c+d x)}+(e \sec (c+d x))^{2/3}+e^{2/3}}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}\right)^2}} F\left(\sin ^{-1}\left(\frac{\left(1-\sqrt{3}\right) \sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}}{\left(1+\sqrt{3}\right) \sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}}\right)|-7-4 \sqrt{3}\right)}{7 d (a-a \sec (c+d x)) \sqrt{a \sec (c+d x)+a} \sqrt{\frac{\sqrt[3]{e} \left(\sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}\right)^2}}}+\frac{12 \sqrt[4]{3} \sqrt{2-\sqrt{3}} a^2 e^{4/3} \tan (c+d x) \left(\sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}\right) \sqrt{\frac{\sqrt[3]{e} \sqrt[3]{e \sec (c+d x)}+(e \sec (c+d x))^{2/3}+e^{2/3}}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}\right)^2}} E\left(\sin ^{-1}\left(\frac{\left(1-\sqrt{3}\right) \sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}}{\left(1+\sqrt{3}\right) \sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}}\right)|-7-4 \sqrt{3}\right)}{7 d (a-a \sec (c+d x)) \sqrt{a \sec (c+d x)+a} \sqrt{\frac{\sqrt[3]{e} \left(\sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}\right)^2}}}-\frac{24 a e^2 \tan (c+d x)}{7 d \sqrt{a \sec (c+d x)+a} \left(\left(1+\sqrt{3}\right) \sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}\right)}+\frac{6 a e \tan (c+d x) (e \sec (c+d x))^{2/3}}{7 d \sqrt{a \sec (c+d x)+a}}",1,"(2*Hypergeometric2F1[-2/3, 1/2, 3/2, 1 - Sec[c + d*x]]*(e*Sec[c + d*x])^(5/3)*Sqrt[a*(1 + Sec[c + d*x])]*Tan[(c + d*x)/2])/(d*Sec[c + d*x]^(5/3))","C",1
277,1,71,624,0.159083,"\int (e \sec (c+d x))^{2/3} \sqrt{a+a \sec (c+d x)} \, dx","Integrate[(e*Sec[c + d*x])^(2/3)*Sqrt[a + a*Sec[c + d*x]],x]","\frac{2 \tan \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\sec (c+d x)+1)} (e \sec (c+d x))^{2/3} \, _2F_1\left(\frac{1}{3},\frac{1}{2};\frac{3}{2};1-\sec (c+d x)\right)}{d \sec ^{\frac{2}{3}}(c+d x)}","-\frac{2 \sqrt{2} 3^{3/4} a^2 \sqrt[3]{e} \tan (c+d x) \left(\sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}\right) \sqrt{\frac{\sqrt[3]{e} \sqrt[3]{e \sec (c+d x)}+(e \sec (c+d x))^{2/3}+e^{2/3}}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}\right)^2}} F\left(\sin ^{-1}\left(\frac{\left(1-\sqrt{3}\right) \sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}}{\left(1+\sqrt{3}\right) \sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}}\right)|-7-4 \sqrt{3}\right)}{d (a-a \sec (c+d x)) \sqrt{a \sec (c+d x)+a} \sqrt{\frac{\sqrt[3]{e} \left(\sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}\right)^2}}}+\frac{3 \sqrt[4]{3} \sqrt{2-\sqrt{3}} a^2 \sqrt[3]{e} \tan (c+d x) \left(\sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}\right) \sqrt{\frac{\sqrt[3]{e} \sqrt[3]{e \sec (c+d x)}+(e \sec (c+d x))^{2/3}+e^{2/3}}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}\right)^2}} E\left(\sin ^{-1}\left(\frac{\left(1-\sqrt{3}\right) \sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}}{\left(1+\sqrt{3}\right) \sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}}\right)|-7-4 \sqrt{3}\right)}{d (a-a \sec (c+d x)) \sqrt{a \sec (c+d x)+a} \sqrt{\frac{\sqrt[3]{e} \left(\sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}\right)^2}}}-\frac{6 a e \tan (c+d x)}{d \sqrt{a \sec (c+d x)+a} \left(\left(1+\sqrt{3}\right) \sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}\right)}",1,"(2*Hypergeometric2F1[1/3, 1/2, 3/2, 1 - Sec[c + d*x]]*(e*Sec[c + d*x])^(2/3)*Sqrt[a*(1 + Sec[c + d*x])]*Tan[(c + d*x)/2])/(d*Sec[c + d*x]^(2/3))","C",1
278,1,71,662,0.1644863,"\int \frac{\sqrt{a+a \sec (c+d x)}}{\sqrt[3]{e \sec (c+d x)}} \, dx","Integrate[Sqrt[a + a*Sec[c + d*x]]/(e*Sec[c + d*x])^(1/3),x]","\frac{2 \tan \left(\frac{1}{2} (c+d x)\right) \sqrt[3]{\sec (c+d x)} \sqrt{a (\sec (c+d x)+1)} \, _2F_1\left(\frac{1}{2},\frac{4}{3};\frac{3}{2};1-\sec (c+d x)\right)}{d \sqrt[3]{e \sec (c+d x)}}","\frac{\sqrt{2} 3^{3/4} a^2 \tan (c+d x) \left(\sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}\right) \sqrt{\frac{\sqrt[3]{e} \sqrt[3]{e \sec (c+d x)}+(e \sec (c+d x))^{2/3}+e^{2/3}}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}\right)^2}} F\left(\sin ^{-1}\left(\frac{\left(1-\sqrt{3}\right) \sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}}{\left(1+\sqrt{3}\right) \sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}}\right)|-7-4 \sqrt{3}\right)}{d e^{2/3} (a-a \sec (c+d x)) \sqrt{a \sec (c+d x)+a} \sqrt{\frac{\sqrt[3]{e} \left(\sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}\right)^2}}}-\frac{3 \sqrt[4]{3} \sqrt{2-\sqrt{3}} a^2 \tan (c+d x) \left(\sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}\right) \sqrt{\frac{\sqrt[3]{e} \sqrt[3]{e \sec (c+d x)}+(e \sec (c+d x))^{2/3}+e^{2/3}}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}\right)^2}} E\left(\sin ^{-1}\left(\frac{\left(1-\sqrt{3}\right) \sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}}{\left(1+\sqrt{3}\right) \sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}}\right)|-7-4 \sqrt{3}\right)}{2 d e^{2/3} (a-a \sec (c+d x)) \sqrt{a \sec (c+d x)+a} \sqrt{\frac{\sqrt[3]{e} \left(\sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}\right)^2}}}+\frac{3 a \tan (c+d x)}{d \sqrt{a \sec (c+d x)+a} \sqrt[3]{e \sec (c+d x)}}+\frac{3 a \tan (c+d x)}{d \sqrt{a \sec (c+d x)+a} \left(\left(1+\sqrt{3}\right) \sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}\right)}",1,"(2*Hypergeometric2F1[1/2, 4/3, 3/2, 1 - Sec[c + d*x]]*Sec[c + d*x]^(1/3)*Sqrt[a*(1 + Sec[c + d*x])]*Tan[(c + d*x)/2])/(d*(e*Sec[c + d*x])^(1/3))","C",1
279,1,71,715,0.2034005,"\int \frac{\sqrt{a+a \sec (c+d x)}}{(e \sec (c+d x))^{4/3}} \, dx","Integrate[Sqrt[a + a*Sec[c + d*x]]/(e*Sec[c + d*x])^(4/3),x]","\frac{2 \tan \left(\frac{1}{2} (c+d x)\right) \sec ^{\frac{4}{3}}(c+d x) \sqrt{a (\sec (c+d x)+1)} \, _2F_1\left(\frac{1}{2},\frac{7}{3};\frac{3}{2};1-\sec (c+d x)\right)}{d (e \sec (c+d x))^{4/3}}","\frac{5\ 3^{3/4} a^2 \tan (c+d x) \left(\sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}\right) \sqrt{\frac{\sqrt[3]{e} \sqrt[3]{e \sec (c+d x)}+(e \sec (c+d x))^{2/3}+e^{2/3}}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}\right)^2}} F\left(\sin ^{-1}\left(\frac{\left(1-\sqrt{3}\right) \sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}}{\left(1+\sqrt{3}\right) \sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}}\right)|-7-4 \sqrt{3}\right)}{4 \sqrt{2} d e^{5/3} (a-a \sec (c+d x)) \sqrt{a \sec (c+d x)+a} \sqrt{\frac{\sqrt[3]{e} \left(\sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}\right)^2}}}-\frac{15 \sqrt[4]{3} \sqrt{2-\sqrt{3}} a^2 \tan (c+d x) \left(\sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}\right) \sqrt{\frac{\sqrt[3]{e} \sqrt[3]{e \sec (c+d x)}+(e \sec (c+d x))^{2/3}+e^{2/3}}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}\right)^2}} E\left(\sin ^{-1}\left(\frac{\left(1-\sqrt{3}\right) \sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}}{\left(1+\sqrt{3}\right) \sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}}\right)|-7-4 \sqrt{3}\right)}{16 d e^{5/3} (a-a \sec (c+d x)) \sqrt{a \sec (c+d x)+a} \sqrt{\frac{\sqrt[3]{e} \left(\sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}\right)}{\left(\left(1+\sqrt{3}\right) \sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}\right)^2}}}+\frac{15 a \tan (c+d x)}{8 d e \sqrt{a \sec (c+d x)+a} \sqrt[3]{e \sec (c+d x)}}+\frac{3 a \tan (c+d x)}{4 d \sqrt{a \sec (c+d x)+a} (e \sec (c+d x))^{4/3}}+\frac{15 a \tan (c+d x)}{8 d e \sqrt{a \sec (c+d x)+a} \left(\left(1+\sqrt{3}\right) \sqrt[3]{e}-\sqrt[3]{e \sec (c+d x)}\right)}",1,"(2*Hypergeometric2F1[1/2, 7/3, 3/2, 1 - Sec[c + d*x]]*Sec[c + d*x]^(4/3)*Sqrt[a*(1 + Sec[c + d*x])]*Tan[(c + d*x)/2])/(d*(e*Sec[c + d*x])^(4/3))","C",1
280,1,750,78,9.2461479,"\int \frac{(e \sec (c+d x))^{2/3}}{\sqrt{a+a \sec (c+d x)}} \, dx","Integrate[(e*Sec[c + d*x])^(2/3)/Sqrt[a + a*Sec[c + d*x]],x]","\frac{90 \cos ^2(c+d x) \tan \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\sec (c+d x)+1)} F_1\left(\frac{1}{2};\frac{1}{6},\frac{1}{3};\frac{3}{2};\tan ^2\left(\frac{1}{2} (c+d x)\right),-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right) \left(\tan ^2\left(\frac{1}{2} (c+d x)\right) \left(F_1\left(\frac{3}{2};\frac{7}{6},\frac{1}{3};\frac{5}{2};\tan ^2\left(\frac{1}{2} (c+d x)\right),-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right)-2 F_1\left(\frac{3}{2};\frac{1}{6},\frac{4}{3};\frac{5}{2};\tan ^2\left(\frac{1}{2} (c+d x)\right),-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right)\right)+9 F_1\left(\frac{1}{2};\frac{1}{6},\frac{1}{3};\frac{3}{2};\tan ^2\left(\frac{1}{2} (c+d x)\right),-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right)\right) (e \sec (c+d x))^{2/3}}{a d \left(270 \cos ^2\left(\frac{1}{2} (c+d x)\right) (2 \cos (c+d x)+1) F_1\left(\frac{1}{2};\frac{1}{6},\frac{1}{3};\frac{3}{2};\tan ^2\left(\frac{1}{2} (c+d x)\right),-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right){}^2+10 \cos (c+d x) \tan ^4\left(\frac{1}{2} (c+d x)\right) \left(F_1\left(\frac{3}{2};\frac{7}{6},\frac{1}{3};\frac{5}{2};\tan ^2\left(\frac{1}{2} (c+d x)\right),-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right)-2 F_1\left(\frac{3}{2};\frac{1}{6},\frac{4}{3};\frac{5}{2};\tan ^2\left(\frac{1}{2} (c+d x)\right),-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right)\right){}^2+3 \tan ^2\left(\frac{1}{2} (c+d x)\right) \sec ^2\left(\frac{1}{2} (c+d x)\right) F_1\left(\frac{1}{2};\frac{1}{6},\frac{1}{3};\frac{3}{2};\tan ^2\left(\frac{1}{2} (c+d x)\right),-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right) \left(-10 (-9 \cos (c+d x)+\cos (2 (c+d x))+2) \cos ^2\left(\frac{1}{2} (c+d x)\right) F_1\left(\frac{3}{2};\frac{1}{6},\frac{4}{3};\frac{5}{2};\tan ^2\left(\frac{1}{2} (c+d x)\right),-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right)+5 (-9 \cos (c+d x)+\cos (2 (c+d x))+2) \cos ^2\left(\frac{1}{2} (c+d x)\right) F_1\left(\frac{3}{2};\frac{7}{6},\frac{1}{3};\frac{5}{2};\tan ^2\left(\frac{1}{2} (c+d x)\right),-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right)-6 \sin ^2\left(\frac{1}{2} (c+d x)\right) \cos (c+d x) \left(16 F_1\left(\frac{5}{2};\frac{1}{6},\frac{7}{3};\frac{7}{2};\tan ^2\left(\frac{1}{2} (c+d x)\right),-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right)-4 F_1\left(\frac{5}{2};\frac{7}{6},\frac{4}{3};\frac{7}{2};\tan ^2\left(\frac{1}{2} (c+d x)\right),-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right)+7 F_1\left(\frac{5}{2};\frac{13}{6},\frac{1}{3};\frac{7}{2};\tan ^2\left(\frac{1}{2} (c+d x)\right),-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right)\right)\right)\right)}","-\frac{3 \tan (c+d x) F_1\left(\frac{2}{3};\frac{1}{2},1;\frac{5}{3};\sec (c+d x),-\sec (c+d x)\right) (e \sec (c+d x))^{2/3}}{2 d \sqrt{1-\sec (c+d x)} \sqrt{a \sec (c+d x)+a}}",1,"(90*AppellF1[1/2, 1/6, 1/3, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Cos[c + d*x]^2*(e*Sec[c + d*x])^(2/3)*Sqrt[a*(1 + Sec[c + d*x])]*Tan[(c + d*x)/2]*(9*AppellF1[1/2, 1/6, 1/3, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + (-2*AppellF1[3/2, 1/6, 4/3, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + AppellF1[3/2, 7/6, 1/3, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Tan[(c + d*x)/2]^2))/(a*d*(270*AppellF1[1/2, 1/6, 1/3, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]^2*Cos[(c + d*x)/2]^2*(1 + 2*Cos[c + d*x]) + 3*AppellF1[1/2, 1/6, 1/3, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*(-10*AppellF1[3/2, 1/6, 4/3, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Cos[(c + d*x)/2]^2*(2 - 9*Cos[c + d*x] + Cos[2*(c + d*x)]) + 5*AppellF1[3/2, 7/6, 1/3, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Cos[(c + d*x)/2]^2*(2 - 9*Cos[c + d*x] + Cos[2*(c + d*x)]) - 6*(16*AppellF1[5/2, 1/6, 7/3, 7/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] - 4*AppellF1[5/2, 7/6, 4/3, 7/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + 7*AppellF1[5/2, 13/6, 1/3, 7/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Cos[c + d*x]*Sin[(c + d*x)/2]^2)*Tan[(c + d*x)/2]^2 + 10*(-2*AppellF1[3/2, 1/6, 4/3, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + AppellF1[3/2, 7/6, 1/3, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])^2*Cos[c + d*x]*Tan[(c + d*x)/2]^4))","B",0
281,1,749,76,8.1238661,"\int \frac{\sqrt[3]{e \sec (c+d x)}}{\sqrt{a+a \sec (c+d x)}} \, dx","Integrate[(e*Sec[c + d*x])^(1/3)/Sqrt[a + a*Sec[c + d*x]],x]","\frac{720 e \sin \left(\frac{1}{2} (c+d x)\right) \cos \left(\frac{1}{2} (c+d x)\right) (\cos (c+d x)+1)^2 F_1\left(\frac{1}{2};-\frac{1}{6},\frac{2}{3};\frac{3}{2};\tan ^2\left(\frac{1}{2} (c+d x)\right),-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right) \left(9 F_1\left(\frac{1}{2};-\frac{1}{6},\frac{2}{3};\frac{3}{2};\tan ^2\left(\frac{1}{2} (c+d x)\right),-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right)-\tan ^2\left(\frac{1}{2} (c+d x)\right) \left(4 F_1\left(\frac{3}{2};-\frac{1}{6},\frac{5}{3};\frac{5}{2};\tan ^2\left(\frac{1}{2} (c+d x)\right),-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right)+F_1\left(\frac{3}{2};\frac{5}{6},\frac{2}{3};\frac{5}{2};\tan ^2\left(\frac{1}{2} (c+d x)\right),-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right)\right)\right)}{d \sqrt{a (\sec (c+d x)+1)} (e \sec (c+d x))^{2/3} \left(4320 (4 \cos (c+d x)-1) \cos ^6\left(\frac{1}{2} (c+d x)\right) F_1\left(\frac{1}{2};-\frac{1}{6},\frac{2}{3};\frac{3}{2};\tan ^2\left(\frac{1}{2} (c+d x)\right),-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right){}^2+160 \sin ^4\left(\frac{1}{2} (c+d x)\right) \cos (c+d x) \left(4 F_1\left(\frac{3}{2};-\frac{1}{6},\frac{5}{3};\frac{5}{2};\tan ^2\left(\frac{1}{2} (c+d x)\right),-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right)+F_1\left(\frac{3}{2};\frac{5}{6},\frac{2}{3};\frac{5}{2};\tan ^2\left(\frac{1}{2} (c+d x)\right),-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right)\right){}^2+12 \sin ^2\left(\frac{1}{2} (c+d x)\right) F_1\left(\frac{1}{2};-\frac{1}{6},\frac{2}{3};\frac{3}{2};\tan ^2\left(\frac{1}{2} (c+d x)\right),-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right) \left(20 (14 \cos (c+d x)+5 \cos (2 (c+d x))-2 \cos (3 (c+d x))+7) F_1\left(\frac{3}{2};-\frac{1}{6},\frac{5}{3};\frac{5}{2};\tan ^2\left(\frac{1}{2} (c+d x)\right),-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right)+5 (14 \cos (c+d x)+5 \cos (2 (c+d x))-2 \cos (3 (c+d x))+7) F_1\left(\frac{3}{2};\frac{5}{6},\frac{2}{3};\frac{5}{2};\tan ^2\left(\frac{1}{2} (c+d x)\right),-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right)-24 \sin ^2\left(\frac{1}{2} (c+d x)\right) \cos (c+d x) \left(40 F_1\left(\frac{5}{2};-\frac{1}{6},\frac{8}{3};\frac{7}{2};\tan ^2\left(\frac{1}{2} (c+d x)\right),-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right)+8 F_1\left(\frac{5}{2};\frac{5}{6},\frac{5}{3};\frac{7}{2};\tan ^2\left(\frac{1}{2} (c+d x)\right),-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right)-5 F_1\left(\frac{5}{2};\frac{11}{6},\frac{2}{3};\frac{7}{2};\tan ^2\left(\frac{1}{2} (c+d x)\right),-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right)\right)\right)\right)}","-\frac{3 \tan (c+d x) F_1\left(\frac{1}{3};\frac{1}{2},1;\frac{4}{3};\sec (c+d x),-\sec (c+d x)\right) \sqrt[3]{e \sec (c+d x)}}{d \sqrt{1-\sec (c+d x)} \sqrt{a \sec (c+d x)+a}}",1,"(720*e*AppellF1[1/2, -1/6, 2/3, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Cos[(c + d*x)/2]*(1 + Cos[c + d*x])^2*Sin[(c + d*x)/2]*(9*AppellF1[1/2, -1/6, 2/3, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] - (4*AppellF1[3/2, -1/6, 5/3, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + AppellF1[3/2, 5/6, 2/3, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Tan[(c + d*x)/2]^2))/(d*(e*Sec[c + d*x])^(2/3)*Sqrt[a*(1 + Sec[c + d*x])]*(4320*AppellF1[1/2, -1/6, 2/3, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]^2*Cos[(c + d*x)/2]^6*(-1 + 4*Cos[c + d*x]) + 160*(4*AppellF1[3/2, -1/6, 5/3, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + AppellF1[3/2, 5/6, 2/3, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])^2*Cos[c + d*x]*Sin[(c + d*x)/2]^4 + 12*AppellF1[1/2, -1/6, 2/3, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sin[(c + d*x)/2]^2*(20*AppellF1[3/2, -1/6, 5/3, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*(7 + 14*Cos[c + d*x] + 5*Cos[2*(c + d*x)] - 2*Cos[3*(c + d*x)]) + 5*AppellF1[3/2, 5/6, 2/3, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*(7 + 14*Cos[c + d*x] + 5*Cos[2*(c + d*x)] - 2*Cos[3*(c + d*x)]) - 24*(40*AppellF1[5/2, -1/6, 8/3, 7/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + 8*AppellF1[5/2, 5/6, 5/3, 7/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] - 5*AppellF1[5/2, 11/6, 2/3, 7/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Cos[c + d*x]*Sin[(c + d*x)/2]^2)))","B",0
282,1,3346,76,20.5881408,"\int \frac{1}{\sqrt[3]{e \sec (c+d x)} \sqrt{a+a \sec (c+d x)}} \, dx","Integrate[1/((e*Sec[c + d*x])^(1/3)*Sqrt[a + a*Sec[c + d*x]]),x]","\text{Result too large to show}","\frac{3 \tan (c+d x) F_1\left(-\frac{1}{3};\frac{1}{2},1;\frac{2}{3};\sec (c+d x),-\sec (c+d x)\right)}{d \sqrt{1-\sec (c+d x)} \sqrt{a \sec (c+d x)+a} \sqrt[3]{e \sec (c+d x)}}",1,"-(((Cos[(c + d*x)/2]^2*Sec[c + d*x])^(1/6)*Tan[(c + d*x)/2]*(-1 + Tan[(c + d*x)/2]^2)*((2*AppellF1[3/2, 1/6, 1/3, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Tan[(c + d*x)/2]^2)/(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(5/6) + (3*(1 + (3*AppellF1[1/2, 1/6, 1/3, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])/((-1 + Tan[(c + d*x)/2]^2)*(9*AppellF1[1/2, 1/6, 1/3, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + (-2*AppellF1[3/2, 1/6, 4/3, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + AppellF1[3/2, 7/6, 1/3, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Tan[(c + d*x)/2]^2))))/(Sec[(c + d*x)/2]^2)^(1/3)))/(d*(e*Sec[c + d*x])^(1/3)*Sqrt[a*(1 + Sec[c + d*x])]*(-(Sec[(c + d*x)/2]^2*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^(1/6)*Tan[(c + d*x)/2]^2*((2*AppellF1[3/2, 1/6, 1/3, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Tan[(c + d*x)/2]^2)/(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(5/6) + (3*(1 + (3*AppellF1[1/2, 1/6, 1/3, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])/((-1 + Tan[(c + d*x)/2]^2)*(9*AppellF1[1/2, 1/6, 1/3, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + (-2*AppellF1[3/2, 1/6, 4/3, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + AppellF1[3/2, 7/6, 1/3, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Tan[(c + d*x)/2]^2))))/(Sec[(c + d*x)/2]^2)^(1/3))) - (Sec[(c + d*x)/2]^2*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^(1/6)*(-1 + Tan[(c + d*x)/2]^2)*((2*AppellF1[3/2, 1/6, 1/3, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Tan[(c + d*x)/2]^2)/(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(5/6) + (3*(1 + (3*AppellF1[1/2, 1/6, 1/3, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])/((-1 + Tan[(c + d*x)/2]^2)*(9*AppellF1[1/2, 1/6, 1/3, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + (-2*AppellF1[3/2, 1/6, 4/3, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + AppellF1[3/2, 7/6, 1/3, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Tan[(c + d*x)/2]^2))))/(Sec[(c + d*x)/2]^2)^(1/3)))/2 - (Cos[(c + d*x)/2]^2*Sec[c + d*x])^(1/6)*Tan[(c + d*x)/2]*(-1 + Tan[(c + d*x)/2]^2)*((2*AppellF1[3/2, 1/6, 1/3, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(5/6) + (2*Tan[(c + d*x)/2]^2*(-1/5*(AppellF1[5/2, 1/6, 4/3, 7/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]) + (AppellF1[5/2, 7/6, 1/3, 7/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/10))/(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(5/6) - (5*AppellF1[3/2, 1/6, 1/3, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Tan[(c + d*x)/2]^2*(-(Sec[(c + d*x)/2]^2*Sin[c + d*x]) + Cos[c + d*x]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(3*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(11/6)) - (Tan[(c + d*x)/2]*(1 + (3*AppellF1[1/2, 1/6, 1/3, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])/((-1 + Tan[(c + d*x)/2]^2)*(9*AppellF1[1/2, 1/6, 1/3, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + (-2*AppellF1[3/2, 1/6, 4/3, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + AppellF1[3/2, 7/6, 1/3, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Tan[(c + d*x)/2]^2))))/(Sec[(c + d*x)/2]^2)^(1/3) + (3*((-3*AppellF1[1/2, 1/6, 1/3, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/((-1 + Tan[(c + d*x)/2]^2)^2*(9*AppellF1[1/2, 1/6, 1/3, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + (-2*AppellF1[3/2, 1/6, 4/3, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + AppellF1[3/2, 7/6, 1/3, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Tan[(c + d*x)/2]^2)) + (3*(-1/9*(AppellF1[3/2, 1/6, 4/3, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]) + (AppellF1[3/2, 7/6, 1/3, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/18))/((-1 + Tan[(c + d*x)/2]^2)*(9*AppellF1[1/2, 1/6, 1/3, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + (-2*AppellF1[3/2, 1/6, 4/3, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + AppellF1[3/2, 7/6, 1/3, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Tan[(c + d*x)/2]^2)) - (3*AppellF1[1/2, 1/6, 1/3, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*((-2*AppellF1[3/2, 1/6, 4/3, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + AppellF1[3/2, 7/6, 1/3, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2] + 9*(-1/9*(AppellF1[3/2, 1/6, 4/3, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]) + (AppellF1[3/2, 7/6, 1/3, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/18) + Tan[(c + d*x)/2]^2*(-1/5*(AppellF1[5/2, 7/6, 4/3, 7/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]) + (7*AppellF1[5/2, 13/6, 1/3, 7/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/10 - 2*((-4*AppellF1[5/2, 1/6, 7/3, 7/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/5 + (AppellF1[5/2, 7/6, 4/3, 7/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/10))))/((-1 + Tan[(c + d*x)/2]^2)*(9*AppellF1[1/2, 1/6, 1/3, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + (-2*AppellF1[3/2, 1/6, 4/3, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + AppellF1[3/2, 7/6, 1/3, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Tan[(c + d*x)/2]^2)^2)))/(Sec[(c + d*x)/2]^2)^(1/3)) - (Tan[(c + d*x)/2]*(-1 + Tan[(c + d*x)/2]^2)*((2*AppellF1[3/2, 1/6, 1/3, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Tan[(c + d*x)/2]^2)/(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(5/6) + (3*(1 + (3*AppellF1[1/2, 1/6, 1/3, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])/((-1 + Tan[(c + d*x)/2]^2)*(9*AppellF1[1/2, 1/6, 1/3, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + (-2*AppellF1[3/2, 1/6, 4/3, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + AppellF1[3/2, 7/6, 1/3, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Tan[(c + d*x)/2]^2))))/(Sec[(c + d*x)/2]^2)^(1/3))*(-(Cos[(c + d*x)/2]*Sec[c + d*x]*Sin[(c + d*x)/2]) + Cos[(c + d*x)/2]^2*Sec[c + d*x]*Tan[c + d*x]))/(6*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^(5/6)))))","B",0
283,1,585,78,7.4720683,"\int \frac{1}{(e \sec (c+d x))^{2/3} \sqrt{a+a \sec (c+d x)}} \, dx","Integrate[1/((e*Sec[c + d*x])^(2/3)*Sqrt[a + a*Sec[c + d*x]]),x]","\frac{\sec ^{\frac{7}{6}}(c+d x) \left(\frac{5 \sin \left(\frac{1}{2} (c+d x)\right) \sqrt{\frac{1}{\cos (c+d x)+1}} (3 \cos (c+d x)-1) \left(\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)\right)^{5/6} \left(2 \tan ^2\left(\frac{1}{2} (c+d x)\right) F_1\left(\frac{3}{2};\frac{5}{6},\frac{2}{3};\frac{5}{2};\tan ^2\left(\frac{1}{2} (c+d x)\right),-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right) \left(\cos (c+d x) \sec ^2\left(\frac{1}{2} (c+d x)\right)\right)^{5/6}-3 \cos ^{\frac{5}{6}}(c+d x) \sqrt[3]{\sec ^2\left(\frac{1}{2} (c+d x)\right)} \, _2F_1\left(\frac{1}{2},\frac{5}{6};\frac{3}{2};2 \sin ^2\left(\frac{1}{2} (c+d x)\right)\right)\right)}{5 \sqrt{2} \cos \left(\frac{1}{2} (c+d x)\right) \left(3-4 \sqrt{2} \left(\frac{1}{\cos (c+d x)+1}\right)^{2/3} \left(\frac{\cos (c+d x)}{\cos (c+d x)+1}\right)^{5/6} \tan ^4\left(\frac{1}{2} (c+d x)\right) F_1\left(\frac{5}{2};\frac{11}{6},\frac{2}{3};\frac{7}{2};\tan ^2\left(\frac{1}{2} (c+d x)\right),-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right)\right)-120 \sin \left(\frac{1}{2} (c+d x)\right) \left(\frac{1}{\cos (c+d x)+1}\right)^{2/3} \left(\frac{\cos (c+d x)}{\cos (c+d x)+1}\right)^{5/6} \tan \left(\frac{1}{2} (c+d x)\right) F_1\left(\frac{3}{2};\frac{5}{6},\frac{2}{3};\frac{5}{2};\tan ^2\left(\frac{1}{2} (c+d x)\right),-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right)+32 \sin \left(\frac{1}{2} (c+d x)\right) \left(\frac{1}{\cos (c+d x)+1}\right)^{2/3} \left(\frac{\cos (c+d x)}{\cos (c+d x)+1}\right)^{5/6} \tan ^3\left(\frac{1}{2} (c+d x)\right) F_1\left(\frac{5}{2};\frac{5}{6},\frac{5}{3};\frac{7}{2};\tan ^2\left(\frac{1}{2} (c+d x)\right),-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right)}-\frac{3}{2} \left(\sin \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{3}{2} (c+d x)\right)\right) \cos \left(\frac{1}{2} (c+d x)\right) \sec ^{\frac{5}{6}}(c+d x)\right)}{d \sqrt{a (\sec (c+d x)+1)} (e \sec (c+d x))^{2/3}}","\frac{3 \tan (c+d x) F_1\left(-\frac{2}{3};\frac{1}{2},1;\frac{1}{3};\sec (c+d x),-\sec (c+d x)\right)}{2 d \sqrt{1-\sec (c+d x)} \sqrt{a \sec (c+d x)+a} (e \sec (c+d x))^{2/3}}",1,"(Sec[c + d*x]^(7/6)*((-3*Cos[(c + d*x)/2]*Sec[c + d*x]^(5/6)*(Sin[(c + d*x)/2] - Sin[(3*(c + d*x))/2]))/2 + (5*Sqrt[(1 + Cos[c + d*x])^(-1)]*(-1 + 3*Cos[c + d*x])*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^(5/6)*Sin[(c + d*x)/2]*(-3*Cos[c + d*x]^(5/6)*Hypergeometric2F1[1/2, 5/6, 3/2, 2*Sin[(c + d*x)/2]^2]*(Sec[(c + d*x)/2]^2)^(1/3) + 2*AppellF1[3/2, 5/6, 2/3, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(5/6)*Tan[(c + d*x)/2]^2))/(-120*AppellF1[3/2, 5/6, 2/3, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*((1 + Cos[c + d*x])^(-1))^(2/3)*(Cos[c + d*x]/(1 + Cos[c + d*x]))^(5/6)*Sin[(c + d*x)/2]*Tan[(c + d*x)/2] + 32*AppellF1[5/2, 5/6, 5/3, 7/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*((1 + Cos[c + d*x])^(-1))^(2/3)*(Cos[c + d*x]/(1 + Cos[c + d*x]))^(5/6)*Sin[(c + d*x)/2]*Tan[(c + d*x)/2]^3 + 5*Sqrt[2]*Cos[(c + d*x)/2]*(3 - 4*Sqrt[2]*AppellF1[5/2, 11/6, 2/3, 7/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*((1 + Cos[c + d*x])^(-1))^(2/3)*(Cos[c + d*x]/(1 + Cos[c + d*x]))^(5/6)*Tan[(c + d*x)/2]^4))))/(d*(e*Sec[c + d*x])^(2/3)*Sqrt[a*(1 + Sec[c + d*x])])","B",0
284,1,1982,78,15.0055398,"\int \sec ^{\frac{4}{3}}(c+d x) \sqrt[3]{a+a \sec (c+d x)} \, dx","Integrate[Sec[c + d*x]^(4/3)*(a + a*Sec[c + d*x])^(1/3),x]","\frac{3 \sqrt[3]{\sec (c+d x)} \sqrt[3]{(\cos (c+d x)+1) \sec (c+d x)} \sqrt[3]{a (\sec (c+d x)+1)} \sin (c+d x)}{2 d \sqrt[3]{\sec (c+d x)+1}}+\frac{3 \sqrt[3]{a (\sec (c+d x)+1)} \left(\frac{1}{2} \sqrt[3]{\sec (c+d x)} \sqrt[3]{\sec (c+d x)+1}-\frac{\sqrt[3]{\sec (c+d x)+1}}{\sec ^{\frac{2}{3}}(c+d x)}\right) \tan \left(\frac{1}{2} (c+d x)\right) \left(\frac{3 F_1\left(\frac{1}{2};-\frac{1}{3},\frac{2}{3};\frac{3}{2};\tan ^2\left(\frac{1}{2} (c+d x)\right),-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right)}{9 F_1\left(\frac{1}{2};-\frac{1}{3},\frac{2}{3};\frac{3}{2};\tan ^2\left(\frac{1}{2} (c+d x)\right),-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right)-2 \left(2 F_1\left(\frac{3}{2};-\frac{1}{3},\frac{5}{3};\frac{5}{2};\tan ^2\left(\frac{1}{2} (c+d x)\right),-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right)+\, _2F_1\left(\frac{2}{3},\frac{3}{4};\frac{7}{4};\tan ^4\left(\frac{1}{2} (c+d x)\right)\right)\right) \tan ^2\left(\frac{1}{2} (c+d x)\right)}-1\right)}{2^{2/3} d \sec ^2\left(\frac{1}{2} (c+d x)\right)^{2/3} \sqrt[3]{\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)} \sqrt[3]{\sec (c+d x)+1} \left(-\frac{\sqrt[3]{2} \left(\frac{3 F_1\left(\frac{1}{2};-\frac{1}{3},\frac{2}{3};\frac{3}{2};\tan ^2\left(\frac{1}{2} (c+d x)\right),-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right)}{9 F_1\left(\frac{1}{2};-\frac{1}{3},\frac{2}{3};\frac{3}{2};\tan ^2\left(\frac{1}{2} (c+d x)\right),-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right)-2 \left(2 F_1\left(\frac{3}{2};-\frac{1}{3},\frac{5}{3};\frac{5}{2};\tan ^2\left(\frac{1}{2} (c+d x)\right),-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right)+\, _2F_1\left(\frac{2}{3},\frac{3}{4};\frac{7}{4};\tan ^4\left(\frac{1}{2} (c+d x)\right)\right)\right) \tan ^2\left(\frac{1}{2} (c+d x)\right)}-1\right) \tan ^2\left(\frac{1}{2} (c+d x)\right)}{\sec ^2\left(\frac{1}{2} (c+d x)\right)^{2/3} \sqrt[3]{\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)}}+\frac{3 \left(\frac{3 \left(-\frac{2}{9} F_1\left(\frac{3}{2};-\frac{1}{3},\frac{5}{3};\frac{5}{2};\tan ^2\left(\frac{1}{2} (c+d x)\right),-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right) \tan \left(\frac{1}{2} (c+d x)\right) \sec ^2\left(\frac{1}{2} (c+d x)\right)-\frac{1}{9} \, _2F_1\left(\frac{2}{3},\frac{3}{4};\frac{7}{4};\tan ^4\left(\frac{1}{2} (c+d x)\right)\right) \tan \left(\frac{1}{2} (c+d x)\right) \sec ^2\left(\frac{1}{2} (c+d x)\right)\right)}{9 F_1\left(\frac{1}{2};-\frac{1}{3},\frac{2}{3};\frac{3}{2};\tan ^2\left(\frac{1}{2} (c+d x)\right),-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right)-2 \left(2 F_1\left(\frac{3}{2};-\frac{1}{3},\frac{5}{3};\frac{5}{2};\tan ^2\left(\frac{1}{2} (c+d x)\right),-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right)+\, _2F_1\left(\frac{2}{3},\frac{3}{4};\frac{7}{4};\tan ^4\left(\frac{1}{2} (c+d x)\right)\right)\right) \tan ^2\left(\frac{1}{2} (c+d x)\right)}-\frac{3 F_1\left(\frac{1}{2};-\frac{1}{3},\frac{2}{3};\frac{3}{2};\tan ^2\left(\frac{1}{2} (c+d x)\right),-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right) \left(-2 \left(2 F_1\left(\frac{3}{2};-\frac{1}{3},\frac{5}{3};\frac{5}{2};\tan ^2\left(\frac{1}{2} (c+d x)\right),-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right)+\, _2F_1\left(\frac{2}{3},\frac{3}{4};\frac{7}{4};\tan ^4\left(\frac{1}{2} (c+d x)\right)\right)\right) \tan \left(\frac{1}{2} (c+d x)\right) \sec ^2\left(\frac{1}{2} (c+d x)\right)+9 \left(-\frac{2}{9} F_1\left(\frac{3}{2};-\frac{1}{3},\frac{5}{3};\frac{5}{2};\tan ^2\left(\frac{1}{2} (c+d x)\right),-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right) \tan \left(\frac{1}{2} (c+d x)\right) \sec ^2\left(\frac{1}{2} (c+d x)\right)-\frac{1}{9} \, _2F_1\left(\frac{2}{3},\frac{3}{4};\frac{7}{4};\tan ^4\left(\frac{1}{2} (c+d x)\right)\right) \tan \left(\frac{1}{2} (c+d x)\right) \sec ^2\left(\frac{1}{2} (c+d x)\right)\right)-2 \tan ^2\left(\frac{1}{2} (c+d x)\right) \left(2 \left(-F_1\left(\frac{5}{2};-\frac{1}{3},\frac{8}{3};\frac{7}{2};\tan ^2\left(\frac{1}{2} (c+d x)\right),-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right) \tan \left(\frac{1}{2} (c+d x)\right) \sec ^2\left(\frac{1}{2} (c+d x)\right)-\frac{1}{5} F_1\left(\frac{5}{2};\frac{2}{3},\frac{5}{3};\frac{7}{2};\tan ^2\left(\frac{1}{2} (c+d x)\right),-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right) \tan \left(\frac{1}{2} (c+d x)\right) \sec ^2\left(\frac{1}{2} (c+d x)\right)\right)+\frac{3}{2} \csc \left(\frac{1}{2} (c+d x)\right) \sec \left(\frac{1}{2} (c+d x)\right) \left(\frac{1}{\left(1-\tan ^4\left(\frac{1}{2} (c+d x)\right)\right)^{2/3}}-\, _2F_1\left(\frac{2}{3},\frac{3}{4};\frac{7}{4};\tan ^4\left(\frac{1}{2} (c+d x)\right)\right)\right)\right)\right)}{\left(9 F_1\left(\frac{1}{2};-\frac{1}{3},\frac{2}{3};\frac{3}{2};\tan ^2\left(\frac{1}{2} (c+d x)\right),-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right)-2 \left(2 F_1\left(\frac{3}{2};-\frac{1}{3},\frac{5}{3};\frac{5}{2};\tan ^2\left(\frac{1}{2} (c+d x)\right),-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right)+\, _2F_1\left(\frac{2}{3},\frac{3}{4};\frac{7}{4};\tan ^4\left(\frac{1}{2} (c+d x)\right)\right)\right) \tan ^2\left(\frac{1}{2} (c+d x)\right)\right){}^2}\right) \tan \left(\frac{1}{2} (c+d x)\right)}{2^{2/3} \sec ^2\left(\frac{1}{2} (c+d x)\right)^{2/3} \sqrt[3]{\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)}}-\frac{\left(\frac{3 F_1\left(\frac{1}{2};-\frac{1}{3},\frac{2}{3};\frac{3}{2};\tan ^2\left(\frac{1}{2} (c+d x)\right),-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right)}{9 F_1\left(\frac{1}{2};-\frac{1}{3},\frac{2}{3};\frac{3}{2};\tan ^2\left(\frac{1}{2} (c+d x)\right),-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right)-2 \left(2 F_1\left(\frac{3}{2};-\frac{1}{3},\frac{5}{3};\frac{5}{2};\tan ^2\left(\frac{1}{2} (c+d x)\right),-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right)+\, _2F_1\left(\frac{2}{3},\frac{3}{4};\frac{7}{4};\tan ^4\left(\frac{1}{2} (c+d x)\right)\right)\right) \tan ^2\left(\frac{1}{2} (c+d x)\right)}-1\right) \left(\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x) \tan (c+d x)-\cos \left(\frac{1}{2} (c+d x)\right) \sec (c+d x) \sin \left(\frac{1}{2} (c+d x)\right)\right) \tan \left(\frac{1}{2} (c+d x)\right)}{2^{2/3} \sec ^2\left(\frac{1}{2} (c+d x)\right)^{2/3} \left(\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)\right)^{4/3}}+\frac{3 \sqrt[3]{\sec ^2\left(\frac{1}{2} (c+d x)\right)} \left(\frac{3 F_1\left(\frac{1}{2};-\frac{1}{3},\frac{2}{3};\frac{3}{2};\tan ^2\left(\frac{1}{2} (c+d x)\right),-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right)}{9 F_1\left(\frac{1}{2};-\frac{1}{3},\frac{2}{3};\frac{3}{2};\tan ^2\left(\frac{1}{2} (c+d x)\right),-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right)-2 \left(2 F_1\left(\frac{3}{2};-\frac{1}{3},\frac{5}{3};\frac{5}{2};\tan ^2\left(\frac{1}{2} (c+d x)\right),-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right)+\, _2F_1\left(\frac{2}{3},\frac{3}{4};\frac{7}{4};\tan ^4\left(\frac{1}{2} (c+d x)\right)\right)\right) \tan ^2\left(\frac{1}{2} (c+d x)\right)}-1\right)}{2\ 2^{2/3} \sqrt[3]{\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)}}\right)}","\frac{2^{5/6} \tan (c+d x) \sqrt[3]{a \sec (c+d x)+a} F_1\left(\frac{1}{2};-\frac{1}{3},\frac{1}{6};\frac{3}{2};1-\sec (c+d x),\frac{1}{2} (1-\sec (c+d x))\right)}{d (\sec (c+d x)+1)^{5/6}}",1,"(3*Sec[c + d*x]^(1/3)*((1 + Cos[c + d*x])*Sec[c + d*x])^(1/3)*(a*(1 + Sec[c + d*x]))^(1/3)*Sin[c + d*x])/(2*d*(1 + Sec[c + d*x])^(1/3)) + (3*(a*(1 + Sec[c + d*x]))^(1/3)*(-((1 + Sec[c + d*x])^(1/3)/Sec[c + d*x]^(2/3)) + (Sec[c + d*x]^(1/3)*(1 + Sec[c + d*x])^(1/3))/2)*Tan[(c + d*x)/2]*(-1 + (3*AppellF1[1/2, -1/3, 2/3, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])/(9*AppellF1[1/2, -1/3, 2/3, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] - 2*(2*AppellF1[3/2, -1/3, 5/3, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + HypergeometricPFQ[{2/3, 3/4}, {7/4}, Tan[(c + d*x)/2]^4])*Tan[(c + d*x)/2]^2)))/(2^(2/3)*d*(Sec[(c + d*x)/2]^2)^(2/3)*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^(1/3)*(1 + Sec[c + d*x])^(1/3)*((3*(Sec[(c + d*x)/2]^2)^(1/3)*(-1 + (3*AppellF1[1/2, -1/3, 2/3, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])/(9*AppellF1[1/2, -1/3, 2/3, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] - 2*(2*AppellF1[3/2, -1/3, 5/3, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + HypergeometricPFQ[{2/3, 3/4}, {7/4}, Tan[(c + d*x)/2]^4])*Tan[(c + d*x)/2]^2)))/(2*2^(2/3)*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^(1/3)) - (2^(1/3)*Tan[(c + d*x)/2]^2*(-1 + (3*AppellF1[1/2, -1/3, 2/3, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])/(9*AppellF1[1/2, -1/3, 2/3, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] - 2*(2*AppellF1[3/2, -1/3, 5/3, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + HypergeometricPFQ[{2/3, 3/4}, {7/4}, Tan[(c + d*x)/2]^4])*Tan[(c + d*x)/2]^2)))/((Sec[(c + d*x)/2]^2)^(2/3)*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^(1/3)) + (3*Tan[(c + d*x)/2]*((3*((-2*AppellF1[3/2, -1/3, 5/3, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/9 - (HypergeometricPFQ[{2/3, 3/4}, {7/4}, Tan[(c + d*x)/2]^4]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/9))/(9*AppellF1[1/2, -1/3, 2/3, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] - 2*(2*AppellF1[3/2, -1/3, 5/3, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + HypergeometricPFQ[{2/3, 3/4}, {7/4}, Tan[(c + d*x)/2]^4])*Tan[(c + d*x)/2]^2) - (3*AppellF1[1/2, -1/3, 2/3, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*(-2*(2*AppellF1[3/2, -1/3, 5/3, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + HypergeometricPFQ[{2/3, 3/4}, {7/4}, Tan[(c + d*x)/2]^4])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2] + 9*((-2*AppellF1[3/2, -1/3, 5/3, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/9 - (HypergeometricPFQ[{2/3, 3/4}, {7/4}, Tan[(c + d*x)/2]^4]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/9) - 2*Tan[(c + d*x)/2]^2*(2*(-(AppellF1[5/2, -1/3, 8/3, 7/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]) - (AppellF1[5/2, 2/3, 5/3, 7/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/5) + (3*Csc[(c + d*x)/2]*Sec[(c + d*x)/2]*(-HypergeometricPFQ[{2/3, 3/4}, {7/4}, Tan[(c + d*x)/2]^4] + (1 - Tan[(c + d*x)/2]^4)^(-2/3)))/2)))/(9*AppellF1[1/2, -1/3, 2/3, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] - 2*(2*AppellF1[3/2, -1/3, 5/3, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + HypergeometricPFQ[{2/3, 3/4}, {7/4}, Tan[(c + d*x)/2]^4])*Tan[(c + d*x)/2]^2)^2))/(2^(2/3)*(Sec[(c + d*x)/2]^2)^(2/3)*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^(1/3)) - (Tan[(c + d*x)/2]*(-1 + (3*AppellF1[1/2, -1/3, 2/3, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])/(9*AppellF1[1/2, -1/3, 2/3, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] - 2*(2*AppellF1[3/2, -1/3, 5/3, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + HypergeometricPFQ[{2/3, 3/4}, {7/4}, Tan[(c + d*x)/2]^4])*Tan[(c + d*x)/2]^2))*(-(Cos[(c + d*x)/2]*Sec[c + d*x]*Sin[(c + d*x)/2]) + Cos[(c + d*x)/2]^2*Sec[c + d*x]*Tan[c + d*x]))/(2^(2/3)*(Sec[(c + d*x)/2]^2)^(2/3)*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^(4/3))))","B",0
285,1,2618,79,20.3907636,"\int \sec ^{\frac{4}{3}}(c+d x) (a+a \sec (c+d x))^{2/3} \, dx","Integrate[Sec[c + d*x]^(4/3)*(a + a*Sec[c + d*x])^(2/3),x]","\text{Result too large to show}","\frac{2 \sqrt[6]{2} \tan (c+d x) (a \sec (c+d x)+a)^{2/3} F_1\left(\frac{1}{2};-\frac{1}{3},-\frac{1}{6};\frac{3}{2};1-\sec (c+d x),\frac{1}{2} (1-\sec (c+d x))\right)}{d (\sec (c+d x)+1)^{7/6}}",1,"(Sec[c + d*x]^(1/3)*((1 + Cos[c + d*x])*Sec[c + d*x])^(2/3)*(a*(1 + Sec[c + d*x]))^(2/3)*Tan[(c + d*x)/2])/(d*(1 + Sec[c + d*x])^(2/3)) + (15*AppellF1[1/2, 2/3, 1/3, 3/2, Tan[(c + d*x)/4]^2, -Tan[(c + d*x)/4]^2]*(a*(1 + Sec[c + d*x]))^(2/3)*(9*AppellF1[1/2, 2/3, 1/3, 3/2, Tan[(c + d*x)/4]^2, -Tan[(c + d*x)/4]^2] - 2*(AppellF1[3/2, 2/3, 4/3, 5/2, Tan[(c + d*x)/4]^2, -Tan[(c + d*x)/4]^2] - 2*AppellF1[3/2, 5/3, 1/3, 5/2, Tan[(c + d*x)/4]^2, -Tan[(c + d*x)/4]^2])*Tan[(c + d*x)/4]^2)*Tan[(c + d*x)/2])/(d*(Sec[c + d*x]/(1 + Sec[c + d*x]))^(2/3)*(1 + Sec[c + d*x])^(2/3)*((135*AppellF1[1/2, 2/3, 1/3, 3/2, Tan[(c + d*x)/4]^2, -Tan[(c + d*x)/4]^2]^2*(5 + Cos[c + d*x]))/2 + 20*(AppellF1[3/2, 2/3, 4/3, 5/2, Tan[(c + d*x)/4]^2, -Tan[(c + d*x)/4]^2] - 2*AppellF1[3/2, 5/3, 1/3, 5/2, Tan[(c + d*x)/4]^2, -Tan[(c + d*x)/4]^2])^2*Cos[(c + d*x)/2]*Tan[(c + d*x)/4]^4 - 3*AppellF1[1/2, 2/3, 1/3, 3/2, Tan[(c + d*x)/4]^2, -Tan[(c + d*x)/4]^2]*Tan[(c + d*x)/4]^2*(5*AppellF1[3/2, 2/3, 4/3, 5/2, Tan[(c + d*x)/4]^2, -Tan[(c + d*x)/4]^2]*(5 - 12*Cos[(c + d*x)/2] + Cos[c + d*x]) - 10*AppellF1[3/2, 5/3, 1/3, 5/2, Tan[(c + d*x)/4]^2, -Tan[(c + d*x)/4]^2]*(5 - 12*Cos[(c + d*x)/2] + Cos[c + d*x]) + 24*(2*AppellF1[5/2, 2/3, 7/3, 7/2, Tan[(c + d*x)/4]^2, -Tan[(c + d*x)/4]^2] - 2*AppellF1[5/2, 5/3, 4/3, 7/2, Tan[(c + d*x)/4]^2, -Tan[(c + d*x)/4]^2] + 5*AppellF1[5/2, 8/3, 1/3, 7/2, Tan[(c + d*x)/4]^2, -Tan[(c + d*x)/4]^2])*Cos[(c + d*x)/2]*Tan[(c + d*x)/4]^2))) - ((Sec[(c + d*x)/2]^2)^(4/3)*Sec[c + d*x]^(1/3)*(a*(1 + Sec[c + d*x]))^(2/3)*(AppellF1[-2/3, -1/3, -1/3, 1/3, (-1 - I)/(-1 + Tan[(c + d*x)/2]), (-1 + I)/(-1 + Tan[(c + d*x)/2])]/(((-I + Tan[(c + d*x)/2])/(-1 + Tan[(c + d*x)/2]))^(1/3)*((I + Tan[(c + d*x)/2])/(-1 + Tan[(c + d*x)/2]))^(1/3)) - AppellF1[-2/3, -1/3, -1/3, 1/3, (1 - I)/(1 + Tan[(c + d*x)/2]), (1 + I)/(1 + Tan[(c + d*x)/2])]/(((-I + Tan[(c + d*x)/2])/(1 + Tan[(c + d*x)/2]))^(1/3)*((I + Tan[(c + d*x)/2])/(1 + Tan[(c + d*x)/2]))^(1/3))))/(2^(1/3)*d*(-(((Sec[(c + d*x)/2]^2)^(1/3)*Tan[(c + d*x)/2]*(AppellF1[-2/3, -1/3, -1/3, 1/3, (-1 - I)/(-1 + Tan[(c + d*x)/2]), (-1 + I)/(-1 + Tan[(c + d*x)/2])]/(((-I + Tan[(c + d*x)/2])/(-1 + Tan[(c + d*x)/2]))^(1/3)*((I + Tan[(c + d*x)/2])/(-1 + Tan[(c + d*x)/2]))^(1/3)) - AppellF1[-2/3, -1/3, -1/3, 1/3, (1 - I)/(1 + Tan[(c + d*x)/2]), (1 + I)/(1 + Tan[(c + d*x)/2])]/(((-I + Tan[(c + d*x)/2])/(1 + Tan[(c + d*x)/2]))^(1/3)*((I + Tan[(c + d*x)/2])/(1 + Tan[(c + d*x)/2]))^(1/3))))/2^(1/3)) - (3*(Sec[(c + d*x)/2]^2)^(1/3)*((((1/3 - I/3)*AppellF1[1/3, -1/3, 2/3, 4/3, (-1 - I)/(-1 + Tan[(c + d*x)/2]), (-1 + I)/(-1 + Tan[(c + d*x)/2])]*Sec[(c + d*x)/2]^2)/(-1 + Tan[(c + d*x)/2])^2 + ((1/3 + I/3)*AppellF1[1/3, 2/3, -1/3, 4/3, (-1 - I)/(-1 + Tan[(c + d*x)/2]), (-1 + I)/(-1 + Tan[(c + d*x)/2])]*Sec[(c + d*x)/2]^2)/(-1 + Tan[(c + d*x)/2])^2)/(((-I + Tan[(c + d*x)/2])/(-1 + Tan[(c + d*x)/2]))^(1/3)*((I + Tan[(c + d*x)/2])/(-1 + Tan[(c + d*x)/2]))^(1/3)) - (AppellF1[-2/3, -1/3, -1/3, 1/3, (-1 - I)/(-1 + Tan[(c + d*x)/2]), (-1 + I)/(-1 + Tan[(c + d*x)/2])]*(Sec[(c + d*x)/2]^2/(2*(-1 + Tan[(c + d*x)/2])) - (Sec[(c + d*x)/2]^2*(-I + Tan[(c + d*x)/2]))/(2*(-1 + Tan[(c + d*x)/2])^2)))/(3*((-I + Tan[(c + d*x)/2])/(-1 + Tan[(c + d*x)/2]))^(4/3)*((I + Tan[(c + d*x)/2])/(-1 + Tan[(c + d*x)/2]))^(1/3)) - (AppellF1[-2/3, -1/3, -1/3, 1/3, (-1 - I)/(-1 + Tan[(c + d*x)/2]), (-1 + I)/(-1 + Tan[(c + d*x)/2])]*(Sec[(c + d*x)/2]^2/(2*(-1 + Tan[(c + d*x)/2])) - (Sec[(c + d*x)/2]^2*(I + Tan[(c + d*x)/2]))/(2*(-1 + Tan[(c + d*x)/2])^2)))/(3*((-I + Tan[(c + d*x)/2])/(-1 + Tan[(c + d*x)/2]))^(1/3)*((I + Tan[(c + d*x)/2])/(-1 + Tan[(c + d*x)/2]))^(4/3)) - (((-1/3 - I/3)*AppellF1[1/3, -1/3, 2/3, 4/3, (1 - I)/(1 + Tan[(c + d*x)/2]), (1 + I)/(1 + Tan[(c + d*x)/2])]*Sec[(c + d*x)/2]^2)/(1 + Tan[(c + d*x)/2])^2 - ((1/3 - I/3)*AppellF1[1/3, 2/3, -1/3, 4/3, (1 - I)/(1 + Tan[(c + d*x)/2]), (1 + I)/(1 + Tan[(c + d*x)/2])]*Sec[(c + d*x)/2]^2)/(1 + Tan[(c + d*x)/2])^2)/(((-I + Tan[(c + d*x)/2])/(1 + Tan[(c + d*x)/2]))^(1/3)*((I + Tan[(c + d*x)/2])/(1 + Tan[(c + d*x)/2]))^(1/3)) + (AppellF1[-2/3, -1/3, -1/3, 1/3, (1 - I)/(1 + Tan[(c + d*x)/2]), (1 + I)/(1 + Tan[(c + d*x)/2])]*(-1/2*(Sec[(c + d*x)/2]^2*(-I + Tan[(c + d*x)/2]))/(1 + Tan[(c + d*x)/2])^2 + Sec[(c + d*x)/2]^2/(2*(1 + Tan[(c + d*x)/2]))))/(3*((-I + Tan[(c + d*x)/2])/(1 + Tan[(c + d*x)/2]))^(4/3)*((I + Tan[(c + d*x)/2])/(1 + Tan[(c + d*x)/2]))^(1/3)) + (AppellF1[-2/3, -1/3, -1/3, 1/3, (1 - I)/(1 + Tan[(c + d*x)/2]), (1 + I)/(1 + Tan[(c + d*x)/2])]*(-1/2*(Sec[(c + d*x)/2]^2*(I + Tan[(c + d*x)/2]))/(1 + Tan[(c + d*x)/2])^2 + Sec[(c + d*x)/2]^2/(2*(1 + Tan[(c + d*x)/2]))))/(3*((-I + Tan[(c + d*x)/2])/(1 + Tan[(c + d*x)/2]))^(1/3)*((I + Tan[(c + d*x)/2])/(1 + Tan[(c + d*x)/2]))^(4/3))))/2^(1/3)))","C",0
286,1,274,327,8.3314657,"\int \sec ^{\frac{5}{3}}(c+d x) (a+a \sec (c+d x))^{2/3} \, dx","Integrate[Sec[c + d*x]^(5/3)*(a + a*Sec[c + d*x])^(2/3),x]","\frac{(a (\sec (c+d x)+1))^{2/3} \left(\sqrt[3]{2} \tan \left(\frac{1}{2} (c+d x)\right) \, _2F_1\left(\frac{1}{4},\frac{1}{3};\frac{5}{4};\tan ^4\left(\frac{1}{2} (c+d x)\right)\right) \sqrt[3]{\cos (c+d x) \sec ^4\left(\frac{1}{2} (c+d x)\right)} \sqrt[3]{\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)}-5 \sqrt[3]{2} \tan ^3\left(\frac{1}{2} (c+d x)\right) \, _2F_1\left(\frac{1}{3},\frac{3}{4};\frac{7}{4};\tan ^4\left(\frac{1}{2} (c+d x)\right)\right) \sqrt[3]{\cos (c+d x) \sec ^4\left(\frac{1}{2} (c+d x)\right)} \sqrt[3]{\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)}-3 \left(\sin \left(\frac{1}{2} (c+d x)\right)-2 \sin \left(\frac{3}{2} (c+d x)\right)\right) \sec (c+d x) \sqrt[3]{\sec (c+d x)+1} \sec ^3\left(\frac{1}{2} (c+d x)\right)\right)}{8 d \sqrt[3]{\frac{1}{\cos (c+d x)+1}} (\sec (c+d x)+1)^{2/3}}","\frac{\tan (c+d x) (a (\sec (c+d x)+1))^{2/3} \, _2F_1\left(\frac{1}{4},\frac{1}{3};\frac{5}{4};\tan ^4\left(\frac{1}{2} (c+d x)\right)\right) \sqrt[3]{\cos (c+d x) \sec ^4\left(\frac{1}{2} (c+d x)\right)}}{8 d \sqrt[3]{\frac{1}{\cos (c+d x)+1}} (\sec (c+d x)+1)^{4/3}}-\frac{5 \tan ^3(c+d x) (a (\sec (c+d x)+1))^{2/3} \, _2F_1\left(\frac{1}{3},\frac{3}{4};\frac{7}{4};\tan ^4\left(\frac{1}{2} (c+d x)\right)\right) \sqrt[3]{\cos (c+d x) \sec ^4\left(\frac{1}{2} (c+d x)\right)}}{8 d \sqrt[3]{\frac{1}{\cos (c+d x)+1}} (\sec (c+d x)+1)^{10/3}}+\frac{9 \sin (c+d x) \sec ^{\frac{2}{3}}(c+d x) (a (\sec (c+d x)+1))^{2/3}}{4 d}-\frac{3 a \sin (c+d x) \sec ^{\frac{5}{3}}(c+d x)}{2 d \sqrt[3]{a (\sec (c+d x)+1)}}-\frac{9 \tan (c+d x) (a (\sec (c+d x)+1))^{2/3}}{4 d \sqrt[3]{\frac{1}{\cos (c+d x)+1}} (\sec (c+d x)+1)^{7/3}}",1,"((a*(1 + Sec[c + d*x]))^(2/3)*(-3*Sec[(c + d*x)/2]^3*Sec[c + d*x]*(1 + Sec[c + d*x])^(1/3)*(Sin[(c + d*x)/2] - 2*Sin[(3*(c + d*x))/2]) + 2^(1/3)*Hypergeometric2F1[1/4, 1/3, 5/4, Tan[(c + d*x)/2]^4]*(Cos[c + d*x]*Sec[(c + d*x)/2]^4)^(1/3)*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^(1/3)*Tan[(c + d*x)/2] - 5*2^(1/3)*Hypergeometric2F1[1/3, 3/4, 7/4, Tan[(c + d*x)/2]^4]*(Cos[c + d*x]*Sec[(c + d*x)/2]^4)^(1/3)*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^(1/3)*Tan[(c + d*x)/2]^3))/(8*d*((1 + Cos[c + d*x])^(-1))^(1/3)*(1 + Sec[c + d*x])^(2/3))","A",1
287,1,2325,80,15.1775772,"\int \frac{(a+a \sec (c+d x))^{4/3}}{\sqrt[3]{\sec (c+d x)}} \, dx","Integrate[(a + a*Sec[c + d*x])^(4/3)/Sec[c + d*x]^(1/3),x]","\text{Result too large to show}","\frac{2\ 2^{5/6} a \tan (c+d x) \sqrt[3]{a \sec (c+d x)+a} F_1\left(\frac{1}{2};\frac{4}{3},-\frac{5}{6};\frac{3}{2};1-\sec (c+d x),\frac{1}{2} (1-\sec (c+d x))\right)}{d (\sec (c+d x)+1)^{5/6}}",1,"(-3*(a*(1 + Sec[c + d*x]))^(4/3)*((1 + Sec[c + d*x])^(1/3)/Sec[c + d*x]^(1/3) + Sec[c + d*x]^(2/3)*(1 + Sec[c + d*x])^(1/3))*(-8*Tan[(c + d*x)/2] + (AppellF1[-4/3, -2/3, -2/3, -1/3, (-1 - I)/(-1 + Tan[(c + d*x)/2]), (-1 + I)/(-1 + Tan[(c + d*x)/2])]*Sec[(c + d*x)/2]^2)/(((-I + Tan[(c + d*x)/2])/(-1 + Tan[(c + d*x)/2]))^(2/3)*((I + Tan[(c + d*x)/2])/(-1 + Tan[(c + d*x)/2]))^(2/3)) - AppellF1[-4/3, -2/3, -2/3, -1/3, (1 - I)/(1 + Tan[(c + d*x)/2]), (1 + I)/(1 + Tan[(c + d*x)/2])]*((-I + Tan[(c + d*x)/2])/(1 + Tan[(c + d*x)/2]))^(1/3)*((I + Tan[(c + d*x)/2])/(1 + Tan[(c + d*x)/2]))^(1/3)*(1 + Tan[(c + d*x)/2])^2))/(4*2^(2/3)*d*(Sec[(c + d*x)/2]^2)^(1/3)*(1 + Sec[c + d*x])^(4/3)*((Tan[(c + d*x)/2]*(-8*Tan[(c + d*x)/2] + (AppellF1[-4/3, -2/3, -2/3, -1/3, (-1 - I)/(-1 + Tan[(c + d*x)/2]), (-1 + I)/(-1 + Tan[(c + d*x)/2])]*Sec[(c + d*x)/2]^2)/(((-I + Tan[(c + d*x)/2])/(-1 + Tan[(c + d*x)/2]))^(2/3)*((I + Tan[(c + d*x)/2])/(-1 + Tan[(c + d*x)/2]))^(2/3)) - AppellF1[-4/3, -2/3, -2/3, -1/3, (1 - I)/(1 + Tan[(c + d*x)/2]), (1 + I)/(1 + Tan[(c + d*x)/2])]*((-I + Tan[(c + d*x)/2])/(1 + Tan[(c + d*x)/2]))^(1/3)*((I + Tan[(c + d*x)/2])/(1 + Tan[(c + d*x)/2]))^(1/3)*(1 + Tan[(c + d*x)/2])^2))/(4*2^(2/3)*(Sec[(c + d*x)/2]^2)^(1/3)) - (3*(-4*Sec[(c + d*x)/2]^2 + (Sec[(c + d*x)/2]^2*(((-4/3 + (4*I)/3)*AppellF1[-1/3, -2/3, 1/3, 2/3, (-1 - I)/(-1 + Tan[(c + d*x)/2]), (-1 + I)/(-1 + Tan[(c + d*x)/2])]*Sec[(c + d*x)/2]^2)/(-1 + Tan[(c + d*x)/2])^2 - ((4/3 + (4*I)/3)*AppellF1[-1/3, 1/3, -2/3, 2/3, (-1 - I)/(-1 + Tan[(c + d*x)/2]), (-1 + I)/(-1 + Tan[(c + d*x)/2])]*Sec[(c + d*x)/2]^2)/(-1 + Tan[(c + d*x)/2])^2))/(((-I + Tan[(c + d*x)/2])/(-1 + Tan[(c + d*x)/2]))^(2/3)*((I + Tan[(c + d*x)/2])/(-1 + Tan[(c + d*x)/2]))^(2/3)) + (AppellF1[-4/3, -2/3, -2/3, -1/3, (-1 - I)/(-1 + Tan[(c + d*x)/2]), (-1 + I)/(-1 + Tan[(c + d*x)/2])]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/(((-I + Tan[(c + d*x)/2])/(-1 + Tan[(c + d*x)/2]))^(2/3)*((I + Tan[(c + d*x)/2])/(-1 + Tan[(c + d*x)/2]))^(2/3)) - AppellF1[-4/3, -2/3, -2/3, -1/3, (1 - I)/(1 + Tan[(c + d*x)/2]), (1 + I)/(1 + Tan[(c + d*x)/2])]*Sec[(c + d*x)/2]^2*((-I + Tan[(c + d*x)/2])/(1 + Tan[(c + d*x)/2]))^(1/3)*((I + Tan[(c + d*x)/2])/(1 + Tan[(c + d*x)/2]))^(1/3)*(1 + Tan[(c + d*x)/2]) - (2*AppellF1[-4/3, -2/3, -2/3, -1/3, (-1 - I)/(-1 + Tan[(c + d*x)/2]), (-1 + I)/(-1 + Tan[(c + d*x)/2])]*Sec[(c + d*x)/2]^2*(Sec[(c + d*x)/2]^2/(2*(-1 + Tan[(c + d*x)/2])) - (Sec[(c + d*x)/2]^2*(-I + Tan[(c + d*x)/2]))/(2*(-1 + Tan[(c + d*x)/2])^2)))/(3*((-I + Tan[(c + d*x)/2])/(-1 + Tan[(c + d*x)/2]))^(5/3)*((I + Tan[(c + d*x)/2])/(-1 + Tan[(c + d*x)/2]))^(2/3)) - (2*AppellF1[-4/3, -2/3, -2/3, -1/3, (-1 - I)/(-1 + Tan[(c + d*x)/2]), (-1 + I)/(-1 + Tan[(c + d*x)/2])]*Sec[(c + d*x)/2]^2*(Sec[(c + d*x)/2]^2/(2*(-1 + Tan[(c + d*x)/2])) - (Sec[(c + d*x)/2]^2*(I + Tan[(c + d*x)/2]))/(2*(-1 + Tan[(c + d*x)/2])^2)))/(3*((-I + Tan[(c + d*x)/2])/(-1 + Tan[(c + d*x)/2]))^(2/3)*((I + Tan[(c + d*x)/2])/(-1 + Tan[(c + d*x)/2]))^(5/3)) - ((-I + Tan[(c + d*x)/2])/(1 + Tan[(c + d*x)/2]))^(1/3)*((I + Tan[(c + d*x)/2])/(1 + Tan[(c + d*x)/2]))^(1/3)*(1 + Tan[(c + d*x)/2])^2*(((4/3 + (4*I)/3)*AppellF1[-1/3, -2/3, 1/3, 2/3, (1 - I)/(1 + Tan[(c + d*x)/2]), (1 + I)/(1 + Tan[(c + d*x)/2])]*Sec[(c + d*x)/2]^2)/(1 + Tan[(c + d*x)/2])^2 + ((4/3 - (4*I)/3)*AppellF1[-1/3, 1/3, -2/3, 2/3, (1 - I)/(1 + Tan[(c + d*x)/2]), (1 + I)/(1 + Tan[(c + d*x)/2])]*Sec[(c + d*x)/2]^2)/(1 + Tan[(c + d*x)/2])^2) - (AppellF1[-4/3, -2/3, -2/3, -1/3, (1 - I)/(1 + Tan[(c + d*x)/2]), (1 + I)/(1 + Tan[(c + d*x)/2])]*((I + Tan[(c + d*x)/2])/(1 + Tan[(c + d*x)/2]))^(1/3)*(1 + Tan[(c + d*x)/2])^2*(-1/2*(Sec[(c + d*x)/2]^2*(-I + Tan[(c + d*x)/2]))/(1 + Tan[(c + d*x)/2])^2 + Sec[(c + d*x)/2]^2/(2*(1 + Tan[(c + d*x)/2]))))/(3*((-I + Tan[(c + d*x)/2])/(1 + Tan[(c + d*x)/2]))^(2/3)) - (AppellF1[-4/3, -2/3, -2/3, -1/3, (1 - I)/(1 + Tan[(c + d*x)/2]), (1 + I)/(1 + Tan[(c + d*x)/2])]*((-I + Tan[(c + d*x)/2])/(1 + Tan[(c + d*x)/2]))^(1/3)*(1 + Tan[(c + d*x)/2])^2*(-1/2*(Sec[(c + d*x)/2]^2*(I + Tan[(c + d*x)/2]))/(1 + Tan[(c + d*x)/2])^2 + Sec[(c + d*x)/2]^2/(2*(1 + Tan[(c + d*x)/2]))))/(3*((I + Tan[(c + d*x)/2])/(1 + Tan[(c + d*x)/2]))^(2/3))))/(4*2^(2/3)*(Sec[(c + d*x)/2]^2)^(1/3))))","C",0
288,0,0,304,0.6597909,"\int \sec ^n(e+f x) (a+a \sec (e+f x))^4 \, dx","Integrate[Sec[e + f*x]^n*(a + a*Sec[e + f*x])^4,x]","\int \sec ^n(e+f x) (a+a \sec (e+f x))^4 \, dx","-\frac{a^4 \left(8 n^2+24 n+3\right) \sin (e+f x) \sec ^{n-1}(e+f x) \, _2F_1\left(\frac{1}{2},\frac{1-n}{2};\frac{3-n}{2};\cos ^2(e+f x)\right)}{f (1-n) (n+1) (n+3) \sqrt{\sin ^2(e+f x)}}+\frac{4 a^4 (2 n+3) \sin (e+f x) \sec ^n(e+f x) \, _2F_1\left(\frac{1}{2},-\frac{n}{2};\frac{2-n}{2};\cos ^2(e+f x)\right)}{f n (n+2) \sqrt{\sin ^2(e+f x)}}+\frac{a^4 \left(4 n^2+21 n+30\right) \sin (e+f x) \sec ^{n+1}(e+f x)}{f (n+1) (n+2) (n+3)}+\frac{2 (n+4) \sin (e+f x) \left(a^4 \sec (e+f x)+a^4\right) \sec ^{n+1}(e+f x)}{f (n+2) (n+3)}+\frac{\sin (e+f x) \left(a^2 \sec (e+f x)+a^2\right)^2 \sec ^{n+1}(e+f x)}{f (n+3)}",1,"Integrate[Sec[e + f*x]^n*(a + a*Sec[e + f*x])^4, x]","F",-1
289,1,286,230,1.5430117,"\int \sec ^n(e+f x) (a+a \sec (e+f x))^3 \, dx","Integrate[Sec[e + f*x]^n*(a + a*Sec[e + f*x])^3,x]","-\frac{i a^3 2^{n-3} \left(\frac{e^{i (e+f x)}}{1+e^{2 i (e+f x)}}\right)^n (\cos (e+f x)+1)^3 \sec ^6\left(\frac{1}{2} (e+f x)\right) \left(\frac{8 e^{3 i (e+f x)} \, _2F_1\left(1,\frac{1}{2} (-n-1);\frac{n+5}{2};-e^{2 i (e+f x)}\right)}{(n+3) \left(1+e^{2 i (e+f x)}\right)^2}+\frac{6 e^{i (e+f x)} \, _2F_1\left(1,\frac{1-n}{2};\frac{n+3}{2};-e^{2 i (e+f x)}\right)}{n+1}+\frac{\left(1+e^{2 i (e+f x)}\right) \, _2F_1\left(1,1-\frac{n}{2};\frac{n+2}{2};-e^{2 i (e+f x)}\right)}{n}+\frac{12 e^{2 i (e+f x)} \, _2F_1\left(1,-\frac{n}{2};\frac{n+4}{2};-e^{2 i (e+f x)}\right)}{(n+2) \left(1+e^{2 i (e+f x)}\right)}\right)}{f}","-\frac{a^3 (4 n+1) \sin (e+f x) \sec ^{n-1}(e+f x) \, _2F_1\left(\frac{1}{2},\frac{1-n}{2};\frac{3-n}{2};\cos ^2(e+f x)\right)}{f \left(1-n^2\right) \sqrt{\sin ^2(e+f x)}}+\frac{a^3 (4 n+7) \sin (e+f x) \sec ^n(e+f x) \, _2F_1\left(\frac{1}{2},-\frac{n}{2};\frac{2-n}{2};\cos ^2(e+f x)\right)}{f n (n+2) \sqrt{\sin ^2(e+f x)}}+\frac{a^3 (2 n+5) \sin (e+f x) \sec ^{n+1}(e+f x)}{f (n+1) (n+2)}+\frac{\sin (e+f x) \left(a^3 \sec (e+f x)+a^3\right) \sec ^{n+1}(e+f x)}{f (n+2)}",1,"((-I)*2^(-3 + n)*a^3*(E^(I*(e + f*x))/(1 + E^((2*I)*(e + f*x))))^n*(1 + Cos[e + f*x])^3*((8*E^((3*I)*(e + f*x))*Hypergeometric2F1[1, (-1 - n)/2, (5 + n)/2, -E^((2*I)*(e + f*x))])/((1 + E^((2*I)*(e + f*x)))^2*(3 + n)) + (6*E^(I*(e + f*x))*Hypergeometric2F1[1, (1 - n)/2, (3 + n)/2, -E^((2*I)*(e + f*x))])/(1 + n) + ((1 + E^((2*I)*(e + f*x)))*Hypergeometric2F1[1, 1 - n/2, (2 + n)/2, -E^((2*I)*(e + f*x))])/n + (12*E^((2*I)*(e + f*x))*Hypergeometric2F1[1, -1/2*n, (4 + n)/2, -E^((2*I)*(e + f*x))])/((1 + E^((2*I)*(e + f*x)))*(2 + n)))*Sec[(e + f*x)/2]^6)/f","C",0
290,1,222,172,1.0940393,"\int \sec ^n(e+f x) (a+a \sec (e+f x))^2 \, dx","Integrate[Sec[e + f*x]^n*(a + a*Sec[e + f*x])^2,x]","-\frac{i a^2 2^{n-2} \left(\frac{e^{i (e+f x)}}{1+e^{2 i (e+f x)}}\right)^n (\cos (e+f x)+1)^2 \sec ^4\left(\frac{1}{2} (e+f x)\right) \left(\frac{4 e^{i (e+f x)} \, _2F_1\left(1,\frac{1-n}{2};\frac{n+3}{2};-e^{2 i (e+f x)}\right)}{n+1}+\frac{\left(1+e^{2 i (e+f x)}\right) \, _2F_1\left(1,1-\frac{n}{2};\frac{n+2}{2};-e^{2 i (e+f x)}\right)}{n}+\frac{4 e^{2 i (e+f x)} \, _2F_1\left(1,-\frac{n}{2};\frac{n+4}{2};-e^{2 i (e+f x)}\right)}{(n+2) \left(1+e^{2 i (e+f x)}\right)}\right)}{f}","-\frac{a^2 (2 n+1) \sin (e+f x) \sec ^{n-1}(e+f x) \, _2F_1\left(\frac{1}{2},\frac{1-n}{2};\frac{3-n}{2};\cos ^2(e+f x)\right)}{f \left(1-n^2\right) \sqrt{\sin ^2(e+f x)}}+\frac{2 a^2 \sin (e+f x) \sec ^n(e+f x) \, _2F_1\left(\frac{1}{2},-\frac{n}{2};\frac{2-n}{2};\cos ^2(e+f x)\right)}{f n \sqrt{\sin ^2(e+f x)}}+\frac{a^2 \sin (e+f x) \sec ^{n+1}(e+f x)}{f (n+1)}",1,"((-I)*2^(-2 + n)*a^2*(E^(I*(e + f*x))/(1 + E^((2*I)*(e + f*x))))^n*(1 + Cos[e + f*x])^2*((4*E^(I*(e + f*x))*Hypergeometric2F1[1, (1 - n)/2, (3 + n)/2, -E^((2*I)*(e + f*x))])/(1 + n) + ((1 + E^((2*I)*(e + f*x)))*Hypergeometric2F1[1, 1 - n/2, (2 + n)/2, -E^((2*I)*(e + f*x))])/n + (4*E^((2*I)*(e + f*x))*Hypergeometric2F1[1, -1/2*n, (4 + n)/2, -E^((2*I)*(e + f*x))])/((1 + E^((2*I)*(e + f*x)))*(2 + n)))*Sec[(e + f*x)/2]^4)/f","C",0
291,1,106,132,0.162604,"\int \sec ^n(e+f x) (a+a \sec (e+f x)) \, dx","Integrate[Sec[e + f*x]^n*(a + a*Sec[e + f*x]),x]","\frac{a \sqrt{-\tan ^2(e+f x)} \csc (e+f x) \sec ^{n-1}(e+f x) \left((n+1) \, _2F_1\left(\frac{1}{2},\frac{n}{2};\frac{n+2}{2};\sec ^2(e+f x)\right)+n \sec (e+f x) \, _2F_1\left(\frac{1}{2},\frac{n+1}{2};\frac{n+3}{2};\sec ^2(e+f x)\right)\right)}{f n (n+1)}","\frac{a \sin (e+f x) \sec ^n(e+f x) \, _2F_1\left(\frac{1}{2},-\frac{n}{2};\frac{2-n}{2};\cos ^2(e+f x)\right)}{f n \sqrt{\sin ^2(e+f x)}}-\frac{a \sin (e+f x) \sec ^{n-1}(e+f x) \, _2F_1\left(\frac{1}{2},\frac{1-n}{2};\frac{3-n}{2};\cos ^2(e+f x)\right)}{f (1-n) \sqrt{\sin ^2(e+f x)}}",1,"(a*Csc[e + f*x]*Sec[e + f*x]^(-1 + n)*((1 + n)*Hypergeometric2F1[1/2, n/2, (2 + n)/2, Sec[e + f*x]^2] + n*Hypergeometric2F1[1/2, (1 + n)/2, (3 + n)/2, Sec[e + f*x]^2]*Sec[e + f*x])*Sqrt[-Tan[e + f*x]^2])/(f*n*(1 + n))","A",1
292,0,0,174,1.0504266,"\int \frac{\sec ^n(e+f x)}{a+a \sec (e+f x)} \, dx","Integrate[Sec[e + f*x]^n/(a + a*Sec[e + f*x]),x]","\int \frac{\sec ^n(e+f x)}{a+a \sec (e+f x)} \, dx","\frac{(1-n) \sin (e+f x) \sec ^{n-2}(e+f x) \, _2F_1\left(\frac{1}{2},\frac{2-n}{2};\frac{4-n}{2};\cos ^2(e+f x)\right)}{a f (2-n) \sqrt{\sin ^2(e+f x)}}-\frac{\sin (e+f x) \sec ^{n-1}(e+f x) \, _2F_1\left(\frac{1}{2},\frac{1-n}{2};\frac{3-n}{2};\cos ^2(e+f x)\right)}{a f \sqrt{\sin ^2(e+f x)}}+\frac{\sin (e+f x) \sec ^n(e+f x)}{f (a \sec (e+f x)+a)}",1,"Integrate[Sec[e + f*x]^n/(a + a*Sec[e + f*x]), x]","F",-1
293,0,0,217,10.1443906,"\int \frac{\sec ^n(e+f x)}{(a+a \sec (e+f x))^2} \, dx","Integrate[Sec[e + f*x]^n/(a + a*Sec[e + f*x])^2,x]","\int \frac{\sec ^n(e+f x)}{(a+a \sec (e+f x))^2} \, dx","-\frac{(3-2 n) \sin (e+f x) \sec ^{n-1}(e+f x) \, _2F_1\left(\frac{1}{2},\frac{1-n}{2};\frac{3-n}{2};\cos ^2(e+f x)\right)}{3 a^2 f \sqrt{\sin ^2(e+f x)}}+\frac{2 (2-n) \sin (e+f x) \sec ^n(e+f x) \, _2F_1\left(\frac{1}{2},-\frac{n}{2};\frac{2-n}{2};\cos ^2(e+f x)\right)}{3 a^2 f \sqrt{\sin ^2(e+f x)}}-\frac{2 (2-n) \sin (e+f x) \sec ^{n+1}(e+f x)}{3 a^2 f (\sec (e+f x)+1)}-\frac{\sin (e+f x) \sec ^{n+1}(e+f x)}{3 f (a \sec (e+f x)+a)^2}",1,"Integrate[Sec[e + f*x]^n/(a + a*Sec[e + f*x])^2, x]","F",-1
294,1,398,162,58.401714,"\int \sec ^n(e+f x) (1+\sec (e+f x))^{5/2} \, dx","Integrate[Sec[e + f*x]^n*(1 + Sec[e + f*x])^(5/2),x]","-\frac{i 2^{n-\frac{5}{2}} e^{-\frac{1}{2} i (2 n+3) (e+f x)} \left(\frac{e^{i (e+f x)}}{1+e^{2 i (e+f x)}}\right)^{n+\frac{3}{2}} \sec ^5\left(\frac{1}{2} (e+f x)\right) (\sec (e+f x)+1)^{5/2} \left(\frac{10 e^{i (n+2) (e+f x)} \, _2F_1\left(1,\frac{1}{2} (-n-1);\frac{n+4}{2};-e^{2 i (e+f x)}\right)}{n+2}+\frac{5 e^{i (n+4) (e+f x)} \, _2F_1\left(1,\frac{1-n}{2};\frac{n+6}{2};-e^{2 i (e+f x)}\right)}{n+4}+\frac{e^{i n (e+f x)} \, _2F_1\left(1,-\frac{n}{2}-\frac{3}{2};\frac{n}{2}+1;-e^{2 i (e+f x)}\right)}{n}+\frac{5 e^{i (n+1) (e+f x)} \, _2F_1\left(1,-\frac{n}{2}-1;\frac{n+3}{2};-e^{2 i (e+f x)}\right)}{n+1}+\frac{e^{i (n+5) (e+f x)} \, _2F_1\left(1,1-\frac{n}{2};\frac{n+7}{2};-e^{2 i (e+f x)}\right)}{n+5}+\frac{10 e^{i (n+3) (e+f x)} \, _2F_1\left(1,-\frac{n}{2};\frac{n+5}{2};-e^{2 i (e+f x)}\right)}{n+3}\right)}{f \sec ^{\frac{5}{2}}(e+f x)}","\frac{2 \left(16 n^2+24 n+3\right) \tan (e+f x) \, _2F_1\left(\frac{1}{2},1-n;\frac{3}{2};1-\sec (e+f x)\right)}{f (2 n+1) (2 n+3) \sqrt{\sec (e+f x)+1}}+\frac{2 \sin (e+f x) \sqrt{\sec (e+f x)+1} \sec ^{n+1}(e+f x)}{f (2 n+3)}+\frac{2 (4 n+7) \sin (e+f x) \sec ^{n+1}(e+f x)}{f (2 n+1) (2 n+3) \sqrt{\sec (e+f x)+1}}",1,"((-I)*2^(-5/2 + n)*(E^(I*(e + f*x))/(1 + E^((2*I)*(e + f*x))))^(3/2 + n)*((10*E^(I*(2 + n)*(e + f*x))*Hypergeometric2F1[1, (-1 - n)/2, (4 + n)/2, -E^((2*I)*(e + f*x))])/(2 + n) + (5*E^(I*(4 + n)*(e + f*x))*Hypergeometric2F1[1, (1 - n)/2, (6 + n)/2, -E^((2*I)*(e + f*x))])/(4 + n) + (E^(I*n*(e + f*x))*Hypergeometric2F1[1, -3/2 - n/2, 1 + n/2, -E^((2*I)*(e + f*x))])/n + (5*E^(I*(1 + n)*(e + f*x))*Hypergeometric2F1[1, -1 - n/2, (3 + n)/2, -E^((2*I)*(e + f*x))])/(1 + n) + (E^(I*(5 + n)*(e + f*x))*Hypergeometric2F1[1, 1 - n/2, (7 + n)/2, -E^((2*I)*(e + f*x))])/(5 + n) + (10*E^(I*(3 + n)*(e + f*x))*Hypergeometric2F1[1, -1/2*n, (5 + n)/2, -E^((2*I)*(e + f*x))])/(3 + n))*Sec[(e + f*x)/2]^5*(1 + Sec[e + f*x])^(5/2))/(E^((I/2)*(3 + 2*n)*(e + f*x))*f*Sec[e + f*x]^(5/2))","C",0
295,1,83,98,0.4563264,"\int \sec ^n(e+f x) (1+\sec (e+f x))^{3/2} \, dx","Integrate[Sec[e + f*x]^n*(1 + Sec[e + f*x])^(3/2),x]","\frac{\tan \left(\frac{1}{2} (e+f x)\right) \sqrt{\sec (e+f x)+1} \sec ^n(e+f x) \left((4 n+1) \cos ^{n+\frac{1}{2}}(e+f x) \, _2F_1\left(\frac{1}{2},n+\frac{3}{2};\frac{3}{2};2 \sin ^2\left(\frac{1}{2} (e+f x)\right)\right)-1\right)}{f n}","\frac{2 (4 n+1) \tan (e+f x) \, _2F_1\left(\frac{1}{2},1-n;\frac{3}{2};1-\sec (e+f x)\right)}{f (2 n+1) \sqrt{\sec (e+f x)+1}}+\frac{2 \sin (e+f x) \sec ^{n+1}(e+f x)}{f (2 n+1) \sqrt{\sec (e+f x)+1}}",1,"((-1 + (1 + 4*n)*Cos[e + f*x]^(1/2 + n)*Hypergeometric2F1[1/2, 3/2 + n, 3/2, 2*Sin[(e + f*x)/2]^2])*Sec[e + f*x]^n*Sqrt[1 + Sec[e + f*x]]*Tan[(e + f*x)/2])/(f*n)","A",1
296,1,45,45,0.0463277,"\int \sec ^n(e+f x) \sqrt{1+\sec (e+f x)} \, dx","Integrate[Sec[e + f*x]^n*Sqrt[1 + Sec[e + f*x]],x]","\frac{2 \tan (e+f x) \, _2F_1\left(\frac{1}{2},1-n;\frac{3}{2};1-\sec (e+f x)\right)}{f \sqrt{\sec (e+f x)+1}}","\frac{2 \tan (e+f x) \, _2F_1\left(\frac{1}{2},1-n;\frac{3}{2};1-\sec (e+f x)\right)}{f \sqrt{\sec (e+f x)+1}}",1,"(2*Hypergeometric2F1[1/2, 1 - n, 3/2, 1 - Sec[e + f*x]]*Tan[e + f*x])/(f*Sqrt[1 + Sec[e + f*x]])","A",1
297,1,2938,59,16.0874759,"\int \frac{\sec ^n(e+f x)}{\sqrt{1+\sec (e+f x)}} \, dx","Integrate[Sec[e + f*x]^n/Sqrt[1 + Sec[e + f*x]],x]","\text{Result too large to show}","\frac{\tan (e+f x) F_1\left(\frac{1}{2};1-n,1;\frac{3}{2};1-\sec (e+f x),\frac{1}{2} (1-\sec (e+f x))\right)}{f \sqrt{\sec (e+f x)+1}}",1,"(3*Sqrt[2]*AppellF1[1/2, -1/2 + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*(Sec[(e + f*x)/2]^2)^n*Sec[e + f*x]^(-1/2 + (-1 + 2*n)/2)*(Cos[(e + f*x)/2]^2*Sec[e + f*x])^n*Tan[(e + f*x)/2])/(f*(3*AppellF1[1/2, -1/2 + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (2*(-1 + n)*AppellF1[3/2, -1/2 + n, 2 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (-1 + 2*n)*AppellF1[3/2, 1/2 + n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2)*((3*AppellF1[1/2, -1/2 + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Cos[e + f*x]*(Sec[(e + f*x)/2]^2)^(1 + n)*(Cos[(e + f*x)/2]^2*Sec[e + f*x])^n*Sqrt[1 + Sec[e + f*x]])/(Sqrt[2]*(3*AppellF1[1/2, -1/2 + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (2*(-1 + n)*AppellF1[3/2, -1/2 + n, 2 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (-1 + 2*n)*AppellF1[3/2, 1/2 + n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2)) - (3*Sqrt[2]*AppellF1[1/2, -1/2 + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*(Sec[(e + f*x)/2]^2)^n*(Cos[(e + f*x)/2]^2*Sec[e + f*x])^n*Sqrt[1 + Sec[e + f*x]]*Sin[e + f*x]*Tan[(e + f*x)/2])/(3*AppellF1[1/2, -1/2 + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (2*(-1 + n)*AppellF1[3/2, -1/2 + n, 2 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (-1 + 2*n)*AppellF1[3/2, 1/2 + n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2) + (3*Sqrt[2]*n*AppellF1[1/2, -1/2 + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Cos[e + f*x]*(Sec[(e + f*x)/2]^2)^n*(Cos[(e + f*x)/2]^2*Sec[e + f*x])^n*Sqrt[1 + Sec[e + f*x]]*Tan[(e + f*x)/2]^2)/(3*AppellF1[1/2, -1/2 + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (2*(-1 + n)*AppellF1[3/2, -1/2 + n, 2 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (-1 + 2*n)*AppellF1[3/2, 1/2 + n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2) + (3*Sqrt[2]*Cos[e + f*x]*(Sec[(e + f*x)/2]^2)^n*(Cos[(e + f*x)/2]^2*Sec[e + f*x])^n*Sqrt[1 + Sec[e + f*x]]*Tan[(e + f*x)/2]*(-1/3*((1 - n)*AppellF1[3/2, -1/2 + n, 2 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2]) + ((-1/2 + n)*AppellF1[3/2, 1/2 + n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/3))/(3*AppellF1[1/2, -1/2 + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (2*(-1 + n)*AppellF1[3/2, -1/2 + n, 2 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (-1 + 2*n)*AppellF1[3/2, 1/2 + n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2) - (3*Sqrt[2]*AppellF1[1/2, -1/2 + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Cos[e + f*x]*(Sec[(e + f*x)/2]^2)^n*(Cos[(e + f*x)/2]^2*Sec[e + f*x])^n*Sqrt[1 + Sec[e + f*x]]*Tan[(e + f*x)/2]*((2*(-1 + n)*AppellF1[3/2, -1/2 + n, 2 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (-1 + 2*n)*AppellF1[3/2, 1/2 + n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2] + 3*(-1/3*((1 - n)*AppellF1[3/2, -1/2 + n, 2 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2]) + ((-1/2 + n)*AppellF1[3/2, 1/2 + n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/3) + Tan[(e + f*x)/2]^2*(2*(-1 + n)*((-3*(2 - n)*AppellF1[5/2, -1/2 + n, 3 - n, 7/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/5 + (3*(-1/2 + n)*AppellF1[5/2, 1/2 + n, 2 - n, 7/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/5) + (-1 + 2*n)*((-3*(1 - n)*AppellF1[5/2, 1/2 + n, 2 - n, 7/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/5 + (3*(1/2 + n)*AppellF1[5/2, 3/2 + n, 1 - n, 7/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/5))))/(3*AppellF1[1/2, -1/2 + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (2*(-1 + n)*AppellF1[3/2, -1/2 + n, 2 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (-1 + 2*n)*AppellF1[3/2, 1/2 + n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2)^2 + (3*AppellF1[1/2, -1/2 + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*(Sec[(e + f*x)/2]^2)^n*(Cos[(e + f*x)/2]^2*Sec[e + f*x])^n*Tan[(e + f*x)/2]*Tan[e + f*x])/(Sqrt[2]*Sqrt[1 + Sec[e + f*x]]*(3*AppellF1[1/2, -1/2 + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (2*(-1 + n)*AppellF1[3/2, -1/2 + n, 2 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (-1 + 2*n)*AppellF1[3/2, 1/2 + n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2)) + (3*Sqrt[2]*n*AppellF1[1/2, -1/2 + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Cos[e + f*x]*(Sec[(e + f*x)/2]^2)^n*(Cos[(e + f*x)/2]^2*Sec[e + f*x])^(-1 + n)*Sqrt[1 + Sec[e + f*x]]*Tan[(e + f*x)/2]*(-(Cos[(e + f*x)/2]*Sec[e + f*x]*Sin[(e + f*x)/2]) + Cos[(e + f*x)/2]^2*Sec[e + f*x]*Tan[e + f*x]))/(3*AppellF1[1/2, -1/2 + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (2*(-1 + n)*AppellF1[3/2, -1/2 + n, 2 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (-1 + 2*n)*AppellF1[3/2, 1/2 + n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2)))","B",0
298,1,2990,62,17.2083939,"\int \frac{\sec ^n(e+f x)}{(1+\sec (e+f x))^{3/2}} \, dx","Integrate[Sec[e + f*x]^n/(1 + Sec[e + f*x])^(3/2),x]","\text{Result too large to show}","\frac{\tan (e+f x) F_1\left(\frac{1}{2};1-n,2;\frac{3}{2};1-\sec (e+f x),\frac{1}{2} (1-\sec (e+f x))\right)}{2 f \sqrt{\sec (e+f x)+1}}",1,"(6*AppellF1[1/2, -3/2 + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*(Sec[(e + f*x)/2]^2)^n*Sec[e + f*x]^(1/2 + (-3 + 2*n)/2)*(Cos[(e + f*x)/2]^2*Sec[e + f*x])^(3/2 + n)*Tan[(e + f*x)/2]*(-1 + Tan[(e + f*x)/2]^2)^2)/(f*(1 + Sec[e + f*x])^(3/2)*(3*AppellF1[1/2, -3/2 + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (2*(-1 + n)*AppellF1[3/2, -3/2 + n, 2 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (-3 + 2*n)*AppellF1[3/2, -1/2 + n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2)*((12*AppellF1[1/2, -3/2 + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Cos[e + f*x]*(Sec[(e + f*x)/2]^2)^(1 + n)*(Cos[(e + f*x)/2]^2*Sec[e + f*x])^(3/2 + n)*Tan[(e + f*x)/2]^2*(-1 + Tan[(e + f*x)/2]^2))/(3*AppellF1[1/2, -3/2 + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (2*(-1 + n)*AppellF1[3/2, -3/2 + n, 2 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (-3 + 2*n)*AppellF1[3/2, -1/2 + n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2) + (3*AppellF1[1/2, -3/2 + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Cos[e + f*x]*(Sec[(e + f*x)/2]^2)^(1 + n)*(Cos[(e + f*x)/2]^2*Sec[e + f*x])^(3/2 + n)*(-1 + Tan[(e + f*x)/2]^2)^2)/(3*AppellF1[1/2, -3/2 + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (2*(-1 + n)*AppellF1[3/2, -3/2 + n, 2 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (-3 + 2*n)*AppellF1[3/2, -1/2 + n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2) - (6*AppellF1[1/2, -3/2 + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*(Sec[(e + f*x)/2]^2)^n*(Cos[(e + f*x)/2]^2*Sec[e + f*x])^(3/2 + n)*Sin[e + f*x]*Tan[(e + f*x)/2]*(-1 + Tan[(e + f*x)/2]^2)^2)/(3*AppellF1[1/2, -3/2 + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (2*(-1 + n)*AppellF1[3/2, -3/2 + n, 2 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (-3 + 2*n)*AppellF1[3/2, -1/2 + n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2) + (6*n*AppellF1[1/2, -3/2 + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Cos[e + f*x]*(Sec[(e + f*x)/2]^2)^n*(Cos[(e + f*x)/2]^2*Sec[e + f*x])^(3/2 + n)*Tan[(e + f*x)/2]^2*(-1 + Tan[(e + f*x)/2]^2)^2)/(3*AppellF1[1/2, -3/2 + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (2*(-1 + n)*AppellF1[3/2, -3/2 + n, 2 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (-3 + 2*n)*AppellF1[3/2, -1/2 + n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2) + (6*Cos[e + f*x]*(Sec[(e + f*x)/2]^2)^n*(Cos[(e + f*x)/2]^2*Sec[e + f*x])^(3/2 + n)*Tan[(e + f*x)/2]*(-1/3*((1 - n)*AppellF1[3/2, -3/2 + n, 2 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2]) + ((-3/2 + n)*AppellF1[3/2, -1/2 + n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/3)*(-1 + Tan[(e + f*x)/2]^2)^2)/(3*AppellF1[1/2, -3/2 + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (2*(-1 + n)*AppellF1[3/2, -3/2 + n, 2 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (-3 + 2*n)*AppellF1[3/2, -1/2 + n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2) - (6*AppellF1[1/2, -3/2 + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Cos[e + f*x]*(Sec[(e + f*x)/2]^2)^n*(Cos[(e + f*x)/2]^2*Sec[e + f*x])^(3/2 + n)*Tan[(e + f*x)/2]*(-1 + Tan[(e + f*x)/2]^2)^2*((2*(-1 + n)*AppellF1[3/2, -3/2 + n, 2 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (-3 + 2*n)*AppellF1[3/2, -1/2 + n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2] + 3*(-1/3*((1 - n)*AppellF1[3/2, -3/2 + n, 2 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2]) + ((-3/2 + n)*AppellF1[3/2, -1/2 + n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/3) + Tan[(e + f*x)/2]^2*(2*(-1 + n)*((-3*(2 - n)*AppellF1[5/2, -3/2 + n, 3 - n, 7/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/5 + (3*(-3/2 + n)*AppellF1[5/2, -1/2 + n, 2 - n, 7/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/5) + (-3 + 2*n)*((-3*(1 - n)*AppellF1[5/2, -1/2 + n, 2 - n, 7/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/5 + (3*(-1/2 + n)*AppellF1[5/2, 1/2 + n, 1 - n, 7/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/5))))/(3*AppellF1[1/2, -3/2 + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (2*(-1 + n)*AppellF1[3/2, -3/2 + n, 2 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (-3 + 2*n)*AppellF1[3/2, -1/2 + n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2)^2 + (6*(3/2 + n)*AppellF1[1/2, -3/2 + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Cos[e + f*x]*(Sec[(e + f*x)/2]^2)^n*(Cos[(e + f*x)/2]^2*Sec[e + f*x])^(1/2 + n)*Tan[(e + f*x)/2]*(-1 + Tan[(e + f*x)/2]^2)^2*(-(Cos[(e + f*x)/2]*Sec[e + f*x]*Sin[(e + f*x)/2]) + Cos[(e + f*x)/2]^2*Sec[e + f*x]*Tan[e + f*x]))/(3*AppellF1[1/2, -3/2 + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (2*(-1 + n)*AppellF1[3/2, -3/2 + n, 2 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (-3 + 2*n)*AppellF1[3/2, -1/2 + n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2)))","B",0
299,1,85,117,0.436792,"\int (-\sec (e+f x))^n (1+\sec (e+f x))^{3/2} \, dx","Integrate[(-Sec[e + f*x])^n*(1 + Sec[e + f*x])^(3/2),x]","\frac{\tan \left(\frac{1}{2} (e+f x)\right) \sqrt{\sec (e+f x)+1} (-\sec (e+f x))^n \left((4 n+1) \cos ^{n+\frac{1}{2}}(e+f x) \, _2F_1\left(\frac{1}{2},n+\frac{3}{2};\frac{3}{2};2 \sin ^2\left(\frac{1}{2} (e+f x)\right)\right)-1\right)}{f n}","\frac{2 \tan (e+f x) (-\sec (e+f x))^n}{f (2 n+1) \sqrt{\sec (e+f x)+1}}-\frac{(4 n+1) \tan (e+f x) (-\sec (e+f x))^n \, _2F_1\left(\frac{1}{2},n;n+1;\sec (e+f x)\right)}{f n (2 n+1) \sqrt{1-\sec (e+f x)} \sqrt{\sec (e+f x)+1}}",1,"((-1 + (1 + 4*n)*Cos[e + f*x]^(1/2 + n)*Hypergeometric2F1[1/2, 3/2 + n, 3/2, 2*Sin[(e + f*x)/2]^2])*(-Sec[e + f*x])^n*Sqrt[1 + Sec[e + f*x]]*Tan[(e + f*x)/2])/(f*n)","A",1
300,1,67,64,0.0582666,"\int (-\sec (e+f x))^n \sqrt{1+\sec (e+f x)} \, dx","Integrate[(-Sec[e + f*x])^n*Sqrt[1 + Sec[e + f*x]],x]","\frac{2 \sin (e+f x) (-\sec (e+f x))^n \sec ^{1-n}(e+f x) \, _2F_1\left(\frac{1}{2},1-n;\frac{3}{2};1-\sec (e+f x)\right)}{f \sqrt{\sec (e+f x)+1}}","-\frac{\tan (e+f x) (-\sec (e+f x))^n \, _2F_1\left(\frac{1}{2},n;n+1;\sec (e+f x)\right)}{f n \sqrt{1-\sec (e+f x)} \sqrt{\sec (e+f x)+1}}",1,"(2*Hypergeometric2F1[1/2, 1 - n, 3/2, 1 - Sec[e + f*x]]*(-Sec[e + f*x])^n*Sec[e + f*x]^(1 - n)*Sin[e + f*x])/(f*Sqrt[1 + Sec[e + f*x]])","A",1
301,1,2951,73,6.239929,"\int \frac{(-\sec (e+f x))^n}{\sqrt{1+\sec (e+f x)}} \, dx","Integrate[(-Sec[e + f*x])^n/Sqrt[1 + Sec[e + f*x]],x]","\text{Result too large to show}","-\frac{\tan (e+f x) (-\sec (e+f x))^n F_1\left(n;\frac{1}{2},1;n+1;\sec (e+f x),-\sec (e+f x)\right)}{f n \sqrt{1-\sec (e+f x)} \sqrt{\sec (e+f x)+1}}",1,"(3*Sqrt[2]*AppellF1[1/2, -1/2 + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*(Sec[(e + f*x)/2]^2)^n*(-Sec[e + f*x])^n*Sec[e + f*x]^(-1/2 - n + (-1 + 2*n)/2)*(Cos[(e + f*x)/2]^2*Sec[e + f*x])^n*Tan[(e + f*x)/2])/(f*(3*AppellF1[1/2, -1/2 + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (2*(-1 + n)*AppellF1[3/2, -1/2 + n, 2 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (-1 + 2*n)*AppellF1[3/2, 1/2 + n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2)*((3*AppellF1[1/2, -1/2 + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Cos[e + f*x]*(Sec[(e + f*x)/2]^2)^(1 + n)*(Cos[(e + f*x)/2]^2*Sec[e + f*x])^n*Sqrt[1 + Sec[e + f*x]])/(Sqrt[2]*(3*AppellF1[1/2, -1/2 + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (2*(-1 + n)*AppellF1[3/2, -1/2 + n, 2 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (-1 + 2*n)*AppellF1[3/2, 1/2 + n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2)) - (3*Sqrt[2]*AppellF1[1/2, -1/2 + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*(Sec[(e + f*x)/2]^2)^n*(Cos[(e + f*x)/2]^2*Sec[e + f*x])^n*Sqrt[1 + Sec[e + f*x]]*Sin[e + f*x]*Tan[(e + f*x)/2])/(3*AppellF1[1/2, -1/2 + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (2*(-1 + n)*AppellF1[3/2, -1/2 + n, 2 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (-1 + 2*n)*AppellF1[3/2, 1/2 + n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2) + (3*Sqrt[2]*n*AppellF1[1/2, -1/2 + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Cos[e + f*x]*(Sec[(e + f*x)/2]^2)^n*(Cos[(e + f*x)/2]^2*Sec[e + f*x])^n*Sqrt[1 + Sec[e + f*x]]*Tan[(e + f*x)/2]^2)/(3*AppellF1[1/2, -1/2 + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (2*(-1 + n)*AppellF1[3/2, -1/2 + n, 2 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (-1 + 2*n)*AppellF1[3/2, 1/2 + n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2) + (3*Sqrt[2]*Cos[e + f*x]*(Sec[(e + f*x)/2]^2)^n*(Cos[(e + f*x)/2]^2*Sec[e + f*x])^n*Sqrt[1 + Sec[e + f*x]]*Tan[(e + f*x)/2]*(-1/3*((1 - n)*AppellF1[3/2, -1/2 + n, 2 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2]) + ((-1/2 + n)*AppellF1[3/2, 1/2 + n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/3))/(3*AppellF1[1/2, -1/2 + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (2*(-1 + n)*AppellF1[3/2, -1/2 + n, 2 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (-1 + 2*n)*AppellF1[3/2, 1/2 + n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2) - (3*Sqrt[2]*AppellF1[1/2, -1/2 + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Cos[e + f*x]*(Sec[(e + f*x)/2]^2)^n*(Cos[(e + f*x)/2]^2*Sec[e + f*x])^n*Sqrt[1 + Sec[e + f*x]]*Tan[(e + f*x)/2]*((2*(-1 + n)*AppellF1[3/2, -1/2 + n, 2 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (-1 + 2*n)*AppellF1[3/2, 1/2 + n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2] + 3*(-1/3*((1 - n)*AppellF1[3/2, -1/2 + n, 2 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2]) + ((-1/2 + n)*AppellF1[3/2, 1/2 + n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/3) + Tan[(e + f*x)/2]^2*(2*(-1 + n)*((-3*(2 - n)*AppellF1[5/2, -1/2 + n, 3 - n, 7/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/5 + (3*(-1/2 + n)*AppellF1[5/2, 1/2 + n, 2 - n, 7/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/5) + (-1 + 2*n)*((-3*(1 - n)*AppellF1[5/2, 1/2 + n, 2 - n, 7/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/5 + (3*(1/2 + n)*AppellF1[5/2, 3/2 + n, 1 - n, 7/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/5))))/(3*AppellF1[1/2, -1/2 + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (2*(-1 + n)*AppellF1[3/2, -1/2 + n, 2 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (-1 + 2*n)*AppellF1[3/2, 1/2 + n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2)^2 + (3*AppellF1[1/2, -1/2 + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*(Sec[(e + f*x)/2]^2)^n*(Cos[(e + f*x)/2]^2*Sec[e + f*x])^n*Tan[(e + f*x)/2]*Tan[e + f*x])/(Sqrt[2]*Sqrt[1 + Sec[e + f*x]]*(3*AppellF1[1/2, -1/2 + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (2*(-1 + n)*AppellF1[3/2, -1/2 + n, 2 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (-1 + 2*n)*AppellF1[3/2, 1/2 + n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2)) + (3*Sqrt[2]*n*AppellF1[1/2, -1/2 + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Cos[e + f*x]*(Sec[(e + f*x)/2]^2)^n*(Cos[(e + f*x)/2]^2*Sec[e + f*x])^(-1 + n)*Sqrt[1 + Sec[e + f*x]]*Tan[(e + f*x)/2]*(-(Cos[(e + f*x)/2]*Sec[e + f*x]*Sin[(e + f*x)/2]) + Cos[(e + f*x)/2]^2*Sec[e + f*x]*Tan[e + f*x]))/(3*AppellF1[1/2, -1/2 + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (2*(-1 + n)*AppellF1[3/2, -1/2 + n, 2 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (-1 + 2*n)*AppellF1[3/2, 1/2 + n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2)))","B",0
302,1,3003,73,6.2815546,"\int \frac{(-\sec (e+f x))^n}{(1+\sec (e+f x))^{3/2}} \, dx","Integrate[(-Sec[e + f*x])^n/(1 + Sec[e + f*x])^(3/2),x]","\text{Result too large to show}","-\frac{\tan (e+f x) (-\sec (e+f x))^n F_1\left(n;\frac{1}{2},2;n+1;\sec (e+f x),-\sec (e+f x)\right)}{f n \sqrt{1-\sec (e+f x)} \sqrt{\sec (e+f x)+1}}",1,"(6*AppellF1[1/2, -3/2 + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*(Sec[(e + f*x)/2]^2)^n*(-Sec[e + f*x])^n*Sec[e + f*x]^(1/2 - n + (-3 + 2*n)/2)*(Cos[(e + f*x)/2]^2*Sec[e + f*x])^(3/2 + n)*Tan[(e + f*x)/2]*(-1 + Tan[(e + f*x)/2]^2)^2)/(f*(1 + Sec[e + f*x])^(3/2)*(3*AppellF1[1/2, -3/2 + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (2*(-1 + n)*AppellF1[3/2, -3/2 + n, 2 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (-3 + 2*n)*AppellF1[3/2, -1/2 + n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2)*((12*AppellF1[1/2, -3/2 + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Cos[e + f*x]*(Sec[(e + f*x)/2]^2)^(1 + n)*(Cos[(e + f*x)/2]^2*Sec[e + f*x])^(3/2 + n)*Tan[(e + f*x)/2]^2*(-1 + Tan[(e + f*x)/2]^2))/(3*AppellF1[1/2, -3/2 + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (2*(-1 + n)*AppellF1[3/2, -3/2 + n, 2 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (-3 + 2*n)*AppellF1[3/2, -1/2 + n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2) + (3*AppellF1[1/2, -3/2 + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Cos[e + f*x]*(Sec[(e + f*x)/2]^2)^(1 + n)*(Cos[(e + f*x)/2]^2*Sec[e + f*x])^(3/2 + n)*(-1 + Tan[(e + f*x)/2]^2)^2)/(3*AppellF1[1/2, -3/2 + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (2*(-1 + n)*AppellF1[3/2, -3/2 + n, 2 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (-3 + 2*n)*AppellF1[3/2, -1/2 + n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2) - (6*AppellF1[1/2, -3/2 + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*(Sec[(e + f*x)/2]^2)^n*(Cos[(e + f*x)/2]^2*Sec[e + f*x])^(3/2 + n)*Sin[e + f*x]*Tan[(e + f*x)/2]*(-1 + Tan[(e + f*x)/2]^2)^2)/(3*AppellF1[1/2, -3/2 + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (2*(-1 + n)*AppellF1[3/2, -3/2 + n, 2 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (-3 + 2*n)*AppellF1[3/2, -1/2 + n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2) + (6*n*AppellF1[1/2, -3/2 + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Cos[e + f*x]*(Sec[(e + f*x)/2]^2)^n*(Cos[(e + f*x)/2]^2*Sec[e + f*x])^(3/2 + n)*Tan[(e + f*x)/2]^2*(-1 + Tan[(e + f*x)/2]^2)^2)/(3*AppellF1[1/2, -3/2 + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (2*(-1 + n)*AppellF1[3/2, -3/2 + n, 2 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (-3 + 2*n)*AppellF1[3/2, -1/2 + n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2) + (6*Cos[e + f*x]*(Sec[(e + f*x)/2]^2)^n*(Cos[(e + f*x)/2]^2*Sec[e + f*x])^(3/2 + n)*Tan[(e + f*x)/2]*(-1/3*((1 - n)*AppellF1[3/2, -3/2 + n, 2 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2]) + ((-3/2 + n)*AppellF1[3/2, -1/2 + n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/3)*(-1 + Tan[(e + f*x)/2]^2)^2)/(3*AppellF1[1/2, -3/2 + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (2*(-1 + n)*AppellF1[3/2, -3/2 + n, 2 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (-3 + 2*n)*AppellF1[3/2, -1/2 + n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2) - (6*AppellF1[1/2, -3/2 + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Cos[e + f*x]*(Sec[(e + f*x)/2]^2)^n*(Cos[(e + f*x)/2]^2*Sec[e + f*x])^(3/2 + n)*Tan[(e + f*x)/2]*(-1 + Tan[(e + f*x)/2]^2)^2*((2*(-1 + n)*AppellF1[3/2, -3/2 + n, 2 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (-3 + 2*n)*AppellF1[3/2, -1/2 + n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2] + 3*(-1/3*((1 - n)*AppellF1[3/2, -3/2 + n, 2 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2]) + ((-3/2 + n)*AppellF1[3/2, -1/2 + n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/3) + Tan[(e + f*x)/2]^2*(2*(-1 + n)*((-3*(2 - n)*AppellF1[5/2, -3/2 + n, 3 - n, 7/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/5 + (3*(-3/2 + n)*AppellF1[5/2, -1/2 + n, 2 - n, 7/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/5) + (-3 + 2*n)*((-3*(1 - n)*AppellF1[5/2, -1/2 + n, 2 - n, 7/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/5 + (3*(-1/2 + n)*AppellF1[5/2, 1/2 + n, 1 - n, 7/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/5))))/(3*AppellF1[1/2, -3/2 + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (2*(-1 + n)*AppellF1[3/2, -3/2 + n, 2 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (-3 + 2*n)*AppellF1[3/2, -1/2 + n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2)^2 + (6*(3/2 + n)*AppellF1[1/2, -3/2 + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Cos[e + f*x]*(Sec[(e + f*x)/2]^2)^n*(Cos[(e + f*x)/2]^2*Sec[e + f*x])^(1/2 + n)*Tan[(e + f*x)/2]*(-1 + Tan[(e + f*x)/2]^2)^2*(-(Cos[(e + f*x)/2]*Sec[e + f*x]*Sin[(e + f*x)/2]) + Cos[(e + f*x)/2]^2*Sec[e + f*x]*Tan[e + f*x]))/(3*AppellF1[1/2, -3/2 + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (2*(-1 + n)*AppellF1[3/2, -3/2 + n, 2 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (-3 + 2*n)*AppellF1[3/2, -1/2 + n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2)))","B",0
303,1,85,117,0.4065734,"\int (d \sec (e+f x))^n (1+\sec (e+f x))^{3/2} \, dx","Integrate[(d*Sec[e + f*x])^n*(1 + Sec[e + f*x])^(3/2),x]","\frac{\tan \left(\frac{1}{2} (e+f x)\right) \sqrt{\sec (e+f x)+1} (d \sec (e+f x))^n \left((4 n+1) \cos ^{n+\frac{1}{2}}(e+f x) \, _2F_1\left(\frac{1}{2},n+\frac{3}{2};\frac{3}{2};2 \sin ^2\left(\frac{1}{2} (e+f x)\right)\right)-1\right)}{f n}","\frac{2 \tan (e+f x) (d \sec (e+f x))^n}{f (2 n+1) \sqrt{\sec (e+f x)+1}}-\frac{(4 n+1) \tan (e+f x) (d \sec (e+f x))^n \, _2F_1\left(\frac{1}{2},n;n+1;\sec (e+f x)\right)}{f n (2 n+1) \sqrt{1-\sec (e+f x)} \sqrt{\sec (e+f x)+1}}",1,"((-1 + (1 + 4*n)*Cos[e + f*x]^(1/2 + n)*Hypergeometric2F1[1/2, 3/2 + n, 3/2, 2*Sin[(e + f*x)/2]^2])*(d*Sec[e + f*x])^n*Sqrt[1 + Sec[e + f*x]]*Tan[(e + f*x)/2])/(f*n)","A",1
304,1,67,64,0.0450431,"\int (d \sec (e+f x))^n \sqrt{1+\sec (e+f x)} \, dx","Integrate[(d*Sec[e + f*x])^n*Sqrt[1 + Sec[e + f*x]],x]","\frac{2 \sin (e+f x) \sec ^{1-n}(e+f x) (d \sec (e+f x))^n \, _2F_1\left(\frac{1}{2},1-n;\frac{3}{2};1-\sec (e+f x)\right)}{f \sqrt{\sec (e+f x)+1}}","-\frac{\tan (e+f x) (d \sec (e+f x))^n \, _2F_1\left(\frac{1}{2},n;n+1;\sec (e+f x)\right)}{f n \sqrt{1-\sec (e+f x)} \sqrt{\sec (e+f x)+1}}",1,"(2*Hypergeometric2F1[1/2, 1 - n, 3/2, 1 - Sec[e + f*x]]*Sec[e + f*x]^(1 - n)*(d*Sec[e + f*x])^n*Sin[e + f*x])/(f*Sqrt[1 + Sec[e + f*x]])","A",1
305,1,2951,73,6.2209527,"\int \frac{(d \sec (e+f x))^n}{\sqrt{1+\sec (e+f x)}} \, dx","Integrate[(d*Sec[e + f*x])^n/Sqrt[1 + Sec[e + f*x]],x]","\text{Result too large to show}","-\frac{\tan (e+f x) F_1\left(n;\frac{1}{2},1;n+1;\sec (e+f x),-\sec (e+f x)\right) (d \sec (e+f x))^n}{f n \sqrt{1-\sec (e+f x)} \sqrt{\sec (e+f x)+1}}",1,"(3*Sqrt[2]*AppellF1[1/2, -1/2 + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*(Sec[(e + f*x)/2]^2)^n*Sec[e + f*x]^(-1/2 - n + (-1 + 2*n)/2)*(d*Sec[e + f*x])^n*(Cos[(e + f*x)/2]^2*Sec[e + f*x])^n*Tan[(e + f*x)/2])/(f*(3*AppellF1[1/2, -1/2 + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (2*(-1 + n)*AppellF1[3/2, -1/2 + n, 2 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (-1 + 2*n)*AppellF1[3/2, 1/2 + n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2)*((3*AppellF1[1/2, -1/2 + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Cos[e + f*x]*(Sec[(e + f*x)/2]^2)^(1 + n)*(Cos[(e + f*x)/2]^2*Sec[e + f*x])^n*Sqrt[1 + Sec[e + f*x]])/(Sqrt[2]*(3*AppellF1[1/2, -1/2 + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (2*(-1 + n)*AppellF1[3/2, -1/2 + n, 2 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (-1 + 2*n)*AppellF1[3/2, 1/2 + n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2)) - (3*Sqrt[2]*AppellF1[1/2, -1/2 + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*(Sec[(e + f*x)/2]^2)^n*(Cos[(e + f*x)/2]^2*Sec[e + f*x])^n*Sqrt[1 + Sec[e + f*x]]*Sin[e + f*x]*Tan[(e + f*x)/2])/(3*AppellF1[1/2, -1/2 + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (2*(-1 + n)*AppellF1[3/2, -1/2 + n, 2 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (-1 + 2*n)*AppellF1[3/2, 1/2 + n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2) + (3*Sqrt[2]*n*AppellF1[1/2, -1/2 + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Cos[e + f*x]*(Sec[(e + f*x)/2]^2)^n*(Cos[(e + f*x)/2]^2*Sec[e + f*x])^n*Sqrt[1 + Sec[e + f*x]]*Tan[(e + f*x)/2]^2)/(3*AppellF1[1/2, -1/2 + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (2*(-1 + n)*AppellF1[3/2, -1/2 + n, 2 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (-1 + 2*n)*AppellF1[3/2, 1/2 + n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2) + (3*Sqrt[2]*Cos[e + f*x]*(Sec[(e + f*x)/2]^2)^n*(Cos[(e + f*x)/2]^2*Sec[e + f*x])^n*Sqrt[1 + Sec[e + f*x]]*Tan[(e + f*x)/2]*(-1/3*((1 - n)*AppellF1[3/2, -1/2 + n, 2 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2]) + ((-1/2 + n)*AppellF1[3/2, 1/2 + n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/3))/(3*AppellF1[1/2, -1/2 + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (2*(-1 + n)*AppellF1[3/2, -1/2 + n, 2 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (-1 + 2*n)*AppellF1[3/2, 1/2 + n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2) - (3*Sqrt[2]*AppellF1[1/2, -1/2 + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Cos[e + f*x]*(Sec[(e + f*x)/2]^2)^n*(Cos[(e + f*x)/2]^2*Sec[e + f*x])^n*Sqrt[1 + Sec[e + f*x]]*Tan[(e + f*x)/2]*((2*(-1 + n)*AppellF1[3/2, -1/2 + n, 2 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (-1 + 2*n)*AppellF1[3/2, 1/2 + n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2] + 3*(-1/3*((1 - n)*AppellF1[3/2, -1/2 + n, 2 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2]) + ((-1/2 + n)*AppellF1[3/2, 1/2 + n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/3) + Tan[(e + f*x)/2]^2*(2*(-1 + n)*((-3*(2 - n)*AppellF1[5/2, -1/2 + n, 3 - n, 7/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/5 + (3*(-1/2 + n)*AppellF1[5/2, 1/2 + n, 2 - n, 7/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/5) + (-1 + 2*n)*((-3*(1 - n)*AppellF1[5/2, 1/2 + n, 2 - n, 7/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/5 + (3*(1/2 + n)*AppellF1[5/2, 3/2 + n, 1 - n, 7/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/5))))/(3*AppellF1[1/2, -1/2 + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (2*(-1 + n)*AppellF1[3/2, -1/2 + n, 2 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (-1 + 2*n)*AppellF1[3/2, 1/2 + n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2)^2 + (3*AppellF1[1/2, -1/2 + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*(Sec[(e + f*x)/2]^2)^n*(Cos[(e + f*x)/2]^2*Sec[e + f*x])^n*Tan[(e + f*x)/2]*Tan[e + f*x])/(Sqrt[2]*Sqrt[1 + Sec[e + f*x]]*(3*AppellF1[1/2, -1/2 + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (2*(-1 + n)*AppellF1[3/2, -1/2 + n, 2 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (-1 + 2*n)*AppellF1[3/2, 1/2 + n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2)) + (3*Sqrt[2]*n*AppellF1[1/2, -1/2 + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Cos[e + f*x]*(Sec[(e + f*x)/2]^2)^n*(Cos[(e + f*x)/2]^2*Sec[e + f*x])^(-1 + n)*Sqrt[1 + Sec[e + f*x]]*Tan[(e + f*x)/2]*(-(Cos[(e + f*x)/2]*Sec[e + f*x]*Sin[(e + f*x)/2]) + Cos[(e + f*x)/2]^2*Sec[e + f*x]*Tan[e + f*x]))/(3*AppellF1[1/2, -1/2 + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (2*(-1 + n)*AppellF1[3/2, -1/2 + n, 2 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (-1 + 2*n)*AppellF1[3/2, 1/2 + n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2)))","B",0
306,1,3003,73,6.236041,"\int \frac{(d \sec (e+f x))^n}{(1+\sec (e+f x))^{3/2}} \, dx","Integrate[(d*Sec[e + f*x])^n/(1 + Sec[e + f*x])^(3/2),x]","\text{Result too large to show}","-\frac{\tan (e+f x) F_1\left(n;\frac{1}{2},2;n+1;\sec (e+f x),-\sec (e+f x)\right) (d \sec (e+f x))^n}{f n \sqrt{1-\sec (e+f x)} \sqrt{\sec (e+f x)+1}}",1,"(6*AppellF1[1/2, -3/2 + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*(Sec[(e + f*x)/2]^2)^n*Sec[e + f*x]^(1/2 - n + (-3 + 2*n)/2)*(d*Sec[e + f*x])^n*(Cos[(e + f*x)/2]^2*Sec[e + f*x])^(3/2 + n)*Tan[(e + f*x)/2]*(-1 + Tan[(e + f*x)/2]^2)^2)/(f*(1 + Sec[e + f*x])^(3/2)*(3*AppellF1[1/2, -3/2 + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (2*(-1 + n)*AppellF1[3/2, -3/2 + n, 2 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (-3 + 2*n)*AppellF1[3/2, -1/2 + n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2)*((12*AppellF1[1/2, -3/2 + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Cos[e + f*x]*(Sec[(e + f*x)/2]^2)^(1 + n)*(Cos[(e + f*x)/2]^2*Sec[e + f*x])^(3/2 + n)*Tan[(e + f*x)/2]^2*(-1 + Tan[(e + f*x)/2]^2))/(3*AppellF1[1/2, -3/2 + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (2*(-1 + n)*AppellF1[3/2, -3/2 + n, 2 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (-3 + 2*n)*AppellF1[3/2, -1/2 + n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2) + (3*AppellF1[1/2, -3/2 + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Cos[e + f*x]*(Sec[(e + f*x)/2]^2)^(1 + n)*(Cos[(e + f*x)/2]^2*Sec[e + f*x])^(3/2 + n)*(-1 + Tan[(e + f*x)/2]^2)^2)/(3*AppellF1[1/2, -3/2 + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (2*(-1 + n)*AppellF1[3/2, -3/2 + n, 2 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (-3 + 2*n)*AppellF1[3/2, -1/2 + n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2) - (6*AppellF1[1/2, -3/2 + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*(Sec[(e + f*x)/2]^2)^n*(Cos[(e + f*x)/2]^2*Sec[e + f*x])^(3/2 + n)*Sin[e + f*x]*Tan[(e + f*x)/2]*(-1 + Tan[(e + f*x)/2]^2)^2)/(3*AppellF1[1/2, -3/2 + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (2*(-1 + n)*AppellF1[3/2, -3/2 + n, 2 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (-3 + 2*n)*AppellF1[3/2, -1/2 + n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2) + (6*n*AppellF1[1/2, -3/2 + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Cos[e + f*x]*(Sec[(e + f*x)/2]^2)^n*(Cos[(e + f*x)/2]^2*Sec[e + f*x])^(3/2 + n)*Tan[(e + f*x)/2]^2*(-1 + Tan[(e + f*x)/2]^2)^2)/(3*AppellF1[1/2, -3/2 + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (2*(-1 + n)*AppellF1[3/2, -3/2 + n, 2 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (-3 + 2*n)*AppellF1[3/2, -1/2 + n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2) + (6*Cos[e + f*x]*(Sec[(e + f*x)/2]^2)^n*(Cos[(e + f*x)/2]^2*Sec[e + f*x])^(3/2 + n)*Tan[(e + f*x)/2]*(-1/3*((1 - n)*AppellF1[3/2, -3/2 + n, 2 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2]) + ((-3/2 + n)*AppellF1[3/2, -1/2 + n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/3)*(-1 + Tan[(e + f*x)/2]^2)^2)/(3*AppellF1[1/2, -3/2 + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (2*(-1 + n)*AppellF1[3/2, -3/2 + n, 2 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (-3 + 2*n)*AppellF1[3/2, -1/2 + n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2) - (6*AppellF1[1/2, -3/2 + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Cos[e + f*x]*(Sec[(e + f*x)/2]^2)^n*(Cos[(e + f*x)/2]^2*Sec[e + f*x])^(3/2 + n)*Tan[(e + f*x)/2]*(-1 + Tan[(e + f*x)/2]^2)^2*((2*(-1 + n)*AppellF1[3/2, -3/2 + n, 2 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (-3 + 2*n)*AppellF1[3/2, -1/2 + n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2] + 3*(-1/3*((1 - n)*AppellF1[3/2, -3/2 + n, 2 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2]) + ((-3/2 + n)*AppellF1[3/2, -1/2 + n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/3) + Tan[(e + f*x)/2]^2*(2*(-1 + n)*((-3*(2 - n)*AppellF1[5/2, -3/2 + n, 3 - n, 7/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/5 + (3*(-3/2 + n)*AppellF1[5/2, -1/2 + n, 2 - n, 7/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/5) + (-3 + 2*n)*((-3*(1 - n)*AppellF1[5/2, -1/2 + n, 2 - n, 7/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/5 + (3*(-1/2 + n)*AppellF1[5/2, 1/2 + n, 1 - n, 7/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/5))))/(3*AppellF1[1/2, -3/2 + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (2*(-1 + n)*AppellF1[3/2, -3/2 + n, 2 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (-3 + 2*n)*AppellF1[3/2, -1/2 + n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2)^2 + (6*(3/2 + n)*AppellF1[1/2, -3/2 + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Cos[e + f*x]*(Sec[(e + f*x)/2]^2)^n*(Cos[(e + f*x)/2]^2*Sec[e + f*x])^(1/2 + n)*Tan[(e + f*x)/2]*(-1 + Tan[(e + f*x)/2]^2)^2*(-(Cos[(e + f*x)/2]*Sec[e + f*x]*Sin[(e + f*x)/2]) + Cos[(e + f*x)/2]^2*Sec[e + f*x]*Tan[e + f*x]))/(3*AppellF1[1/2, -3/2 + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (2*(-1 + n)*AppellF1[3/2, -3/2 + n, 2 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (-3 + 2*n)*AppellF1[3/2, -1/2 + n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2)))","B",0
307,1,400,177,8.1216279,"\int \sec ^n(e+f x) (a+a \sec (e+f x))^{5/2} \, dx","Integrate[Sec[e + f*x]^n*(a + a*Sec[e + f*x])^(5/2),x]","-\frac{i 2^{n-\frac{5}{2}} e^{-\frac{1}{2} i (2 n+3) (e+f x)} \left(\frac{e^{i (e+f x)}}{1+e^{2 i (e+f x)}}\right)^{n+\frac{3}{2}} \sec ^5\left(\frac{1}{2} (e+f x)\right) (a (\sec (e+f x)+1))^{5/2} \left(\frac{10 e^{i (n+2) (e+f x)} \, _2F_1\left(1,\frac{1}{2} (-n-1);\frac{n+4}{2};-e^{2 i (e+f x)}\right)}{n+2}+\frac{5 e^{i (n+4) (e+f x)} \, _2F_1\left(1,\frac{1-n}{2};\frac{n+6}{2};-e^{2 i (e+f x)}\right)}{n+4}+\frac{e^{i n (e+f x)} \, _2F_1\left(1,-\frac{n}{2}-\frac{3}{2};\frac{n}{2}+1;-e^{2 i (e+f x)}\right)}{n}+\frac{5 e^{i (n+1) (e+f x)} \, _2F_1\left(1,-\frac{n}{2}-1;\frac{n+3}{2};-e^{2 i (e+f x)}\right)}{n+1}+\frac{e^{i (n+5) (e+f x)} \, _2F_1\left(1,1-\frac{n}{2};\frac{n+7}{2};-e^{2 i (e+f x)}\right)}{n+5}+\frac{10 e^{i (n+3) (e+f x)} \, _2F_1\left(1,-\frac{n}{2};\frac{n+5}{2};-e^{2 i (e+f x)}\right)}{n+3}\right)}{f \sec ^{\frac{5}{2}}(e+f x)}","\frac{2 a^3 \left(16 n^2+24 n+3\right) \tan (e+f x) \, _2F_1\left(\frac{1}{2},1-n;\frac{3}{2};1-\sec (e+f x)\right)}{f (2 n+1) (2 n+3) \sqrt{a \sec (e+f x)+a}}+\frac{2 a^3 (4 n+7) \sin (e+f x) \sec ^{n+1}(e+f x)}{f (2 n+1) (2 n+3) \sqrt{a \sec (e+f x)+a}}+\frac{2 a^2 \sin (e+f x) \sqrt{a \sec (e+f x)+a} \sec ^{n+1}(e+f x)}{f (2 n+3)}",1,"((-I)*2^(-5/2 + n)*(E^(I*(e + f*x))/(1 + E^((2*I)*(e + f*x))))^(3/2 + n)*((10*E^(I*(2 + n)*(e + f*x))*Hypergeometric2F1[1, (-1 - n)/2, (4 + n)/2, -E^((2*I)*(e + f*x))])/(2 + n) + (5*E^(I*(4 + n)*(e + f*x))*Hypergeometric2F1[1, (1 - n)/2, (6 + n)/2, -E^((2*I)*(e + f*x))])/(4 + n) + (E^(I*n*(e + f*x))*Hypergeometric2F1[1, -3/2 - n/2, 1 + n/2, -E^((2*I)*(e + f*x))])/n + (5*E^(I*(1 + n)*(e + f*x))*Hypergeometric2F1[1, -1 - n/2, (3 + n)/2, -E^((2*I)*(e + f*x))])/(1 + n) + (E^(I*(5 + n)*(e + f*x))*Hypergeometric2F1[1, 1 - n/2, (7 + n)/2, -E^((2*I)*(e + f*x))])/(5 + n) + (10*E^(I*(3 + n)*(e + f*x))*Hypergeometric2F1[1, -1/2*n, (5 + n)/2, -E^((2*I)*(e + f*x))])/(3 + n))*Sec[(e + f*x)/2]^5*(a*(1 + Sec[e + f*x]))^(5/2))/(E^((I/2)*(3 + 2*n)*(e + f*x))*f*Sec[e + f*x]^(5/2))","C",0
308,1,86,108,0.4258564,"\int \sec ^n(e+f x) (a+a \sec (e+f x))^{3/2} \, dx","Integrate[Sec[e + f*x]^n*(a + a*Sec[e + f*x])^(3/2),x]","\frac{a \tan \left(\frac{1}{2} (e+f x)\right) \sqrt{a (\sec (e+f x)+1)} \sec ^n(e+f x) \left((4 n+1) \cos ^{n+\frac{1}{2}}(e+f x) \, _2F_1\left(\frac{1}{2},n+\frac{3}{2};\frac{3}{2};2 \sin ^2\left(\frac{1}{2} (e+f x)\right)\right)-1\right)}{f n}","\frac{2 a^2 (4 n+1) \tan (e+f x) \, _2F_1\left(\frac{1}{2},1-n;\frac{3}{2};1-\sec (e+f x)\right)}{f (2 n+1) \sqrt{a \sec (e+f x)+a}}+\frac{2 a^2 \sin (e+f x) \sec ^{n+1}(e+f x)}{f (2 n+1) \sqrt{a \sec (e+f x)+a}}",1,"(a*(-1 + (1 + 4*n)*Cos[e + f*x]^(1/2 + n)*Hypergeometric2F1[1/2, 3/2 + n, 3/2, 2*Sin[(e + f*x)/2]^2])*Sec[e + f*x]^n*Sqrt[a*(1 + Sec[e + f*x])]*Tan[(e + f*x)/2])/(f*n)","A",1
309,1,51,48,0.100957,"\int \sec ^n(e+f x) \sqrt{a+a \sec (e+f x)} \, dx","Integrate[Sec[e + f*x]^n*Sqrt[a + a*Sec[e + f*x]],x]","\frac{2 \tan \left(\frac{1}{2} (e+f x)\right) \sqrt{a (\sec (e+f x)+1)} \, _2F_1\left(\frac{1}{2},1-n;\frac{3}{2};1-\sec (e+f x)\right)}{f}","\frac{2 a \tan (e+f x) \, _2F_1\left(\frac{1}{2},1-n;\frac{3}{2};1-\sec (e+f x)\right)}{f \sqrt{a \sec (e+f x)+a}}",1,"(2*Hypergeometric2F1[1/2, 1 - n, 3/2, 1 - Sec[e + f*x]]*Sqrt[a*(1 + Sec[e + f*x])]*Tan[(e + f*x)/2])/f","A",1
310,1,2964,61,6.238406,"\int \frac{\sec ^n(e+f x)}{\sqrt{a+a \sec (e+f x)}} \, dx","Integrate[Sec[e + f*x]^n/Sqrt[a + a*Sec[e + f*x]],x]","\text{Result too large to show}","\frac{\tan (e+f x) F_1\left(\frac{1}{2};1-n,1;\frac{3}{2};1-\sec (e+f x),\frac{1}{2} (1-\sec (e+f x))\right)}{f \sqrt{a \sec (e+f x)+a}}",1,"(3*Sqrt[2]*AppellF1[1/2, -1/2 + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*(Sec[(e + f*x)/2]^2)^n*Sec[e + f*x]^(-1/2 + (-1 + 2*n)/2)*(Cos[(e + f*x)/2]^2*Sec[e + f*x])^n*Sqrt[1 + Sec[e + f*x]]*Tan[(e + f*x)/2])/(f*Sqrt[a*(1 + Sec[e + f*x])]*(3*AppellF1[1/2, -1/2 + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (2*(-1 + n)*AppellF1[3/2, -1/2 + n, 2 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (-1 + 2*n)*AppellF1[3/2, 1/2 + n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2)*((3*AppellF1[1/2, -1/2 + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Cos[e + f*x]*(Sec[(e + f*x)/2]^2)^(1 + n)*(Cos[(e + f*x)/2]^2*Sec[e + f*x])^n*Sqrt[1 + Sec[e + f*x]])/(Sqrt[2]*(3*AppellF1[1/2, -1/2 + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (2*(-1 + n)*AppellF1[3/2, -1/2 + n, 2 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (-1 + 2*n)*AppellF1[3/2, 1/2 + n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2)) - (3*Sqrt[2]*AppellF1[1/2, -1/2 + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*(Sec[(e + f*x)/2]^2)^n*(Cos[(e + f*x)/2]^2*Sec[e + f*x])^n*Sqrt[1 + Sec[e + f*x]]*Sin[e + f*x]*Tan[(e + f*x)/2])/(3*AppellF1[1/2, -1/2 + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (2*(-1 + n)*AppellF1[3/2, -1/2 + n, 2 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (-1 + 2*n)*AppellF1[3/2, 1/2 + n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2) + (3*Sqrt[2]*n*AppellF1[1/2, -1/2 + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Cos[e + f*x]*(Sec[(e + f*x)/2]^2)^n*(Cos[(e + f*x)/2]^2*Sec[e + f*x])^n*Sqrt[1 + Sec[e + f*x]]*Tan[(e + f*x)/2]^2)/(3*AppellF1[1/2, -1/2 + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (2*(-1 + n)*AppellF1[3/2, -1/2 + n, 2 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (-1 + 2*n)*AppellF1[3/2, 1/2 + n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2) + (3*Sqrt[2]*Cos[e + f*x]*(Sec[(e + f*x)/2]^2)^n*(Cos[(e + f*x)/2]^2*Sec[e + f*x])^n*Sqrt[1 + Sec[e + f*x]]*Tan[(e + f*x)/2]*(-1/3*((1 - n)*AppellF1[3/2, -1/2 + n, 2 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2]) + ((-1/2 + n)*AppellF1[3/2, 1/2 + n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/3))/(3*AppellF1[1/2, -1/2 + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (2*(-1 + n)*AppellF1[3/2, -1/2 + n, 2 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (-1 + 2*n)*AppellF1[3/2, 1/2 + n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2) - (3*Sqrt[2]*AppellF1[1/2, -1/2 + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Cos[e + f*x]*(Sec[(e + f*x)/2]^2)^n*(Cos[(e + f*x)/2]^2*Sec[e + f*x])^n*Sqrt[1 + Sec[e + f*x]]*Tan[(e + f*x)/2]*((2*(-1 + n)*AppellF1[3/2, -1/2 + n, 2 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (-1 + 2*n)*AppellF1[3/2, 1/2 + n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2] + 3*(-1/3*((1 - n)*AppellF1[3/2, -1/2 + n, 2 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2]) + ((-1/2 + n)*AppellF1[3/2, 1/2 + n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/3) + Tan[(e + f*x)/2]^2*(2*(-1 + n)*((-3*(2 - n)*AppellF1[5/2, -1/2 + n, 3 - n, 7/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/5 + (3*(-1/2 + n)*AppellF1[5/2, 1/2 + n, 2 - n, 7/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/5) + (-1 + 2*n)*((-3*(1 - n)*AppellF1[5/2, 1/2 + n, 2 - n, 7/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/5 + (3*(1/2 + n)*AppellF1[5/2, 3/2 + n, 1 - n, 7/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/5))))/(3*AppellF1[1/2, -1/2 + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (2*(-1 + n)*AppellF1[3/2, -1/2 + n, 2 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (-1 + 2*n)*AppellF1[3/2, 1/2 + n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2)^2 + (3*AppellF1[1/2, -1/2 + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*(Sec[(e + f*x)/2]^2)^n*(Cos[(e + f*x)/2]^2*Sec[e + f*x])^n*Tan[(e + f*x)/2]*Tan[e + f*x])/(Sqrt[2]*Sqrt[1 + Sec[e + f*x]]*(3*AppellF1[1/2, -1/2 + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (2*(-1 + n)*AppellF1[3/2, -1/2 + n, 2 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (-1 + 2*n)*AppellF1[3/2, 1/2 + n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2)) + (3*Sqrt[2]*n*AppellF1[1/2, -1/2 + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Cos[e + f*x]*(Sec[(e + f*x)/2]^2)^n*(Cos[(e + f*x)/2]^2*Sec[e + f*x])^(-1 + n)*Sqrt[1 + Sec[e + f*x]]*Tan[(e + f*x)/2]*(-(Cos[(e + f*x)/2]*Sec[e + f*x]*Sin[(e + f*x)/2]) + Cos[(e + f*x)/2]^2*Sec[e + f*x]*Tan[e + f*x]))/(3*AppellF1[1/2, -1/2 + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (2*(-1 + n)*AppellF1[3/2, -1/2 + n, 2 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (-1 + 2*n)*AppellF1[3/2, 1/2 + n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2)))","B",0
311,1,2992,67,6.2574528,"\int \frac{\sec ^n(e+f x)}{(a+a \sec (e+f x))^{3/2}} \, dx","Integrate[Sec[e + f*x]^n/(a + a*Sec[e + f*x])^(3/2),x]","\text{Result too large to show}","\frac{\tan (e+f x) F_1\left(\frac{1}{2};1-n,2;\frac{3}{2};1-\sec (e+f x),\frac{1}{2} (1-\sec (e+f x))\right)}{2 a f \sqrt{a \sec (e+f x)+a}}",1,"(6*AppellF1[1/2, -3/2 + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*(Sec[(e + f*x)/2]^2)^n*Sec[e + f*x]^(1/2 + (-3 + 2*n)/2)*(Cos[(e + f*x)/2]^2*Sec[e + f*x])^(3/2 + n)*Tan[(e + f*x)/2]*(-1 + Tan[(e + f*x)/2]^2)^2)/(f*(a*(1 + Sec[e + f*x]))^(3/2)*(3*AppellF1[1/2, -3/2 + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (2*(-1 + n)*AppellF1[3/2, -3/2 + n, 2 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (-3 + 2*n)*AppellF1[3/2, -1/2 + n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2)*((12*AppellF1[1/2, -3/2 + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Cos[e + f*x]*(Sec[(e + f*x)/2]^2)^(1 + n)*(Cos[(e + f*x)/2]^2*Sec[e + f*x])^(3/2 + n)*Tan[(e + f*x)/2]^2*(-1 + Tan[(e + f*x)/2]^2))/(3*AppellF1[1/2, -3/2 + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (2*(-1 + n)*AppellF1[3/2, -3/2 + n, 2 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (-3 + 2*n)*AppellF1[3/2, -1/2 + n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2) + (3*AppellF1[1/2, -3/2 + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Cos[e + f*x]*(Sec[(e + f*x)/2]^2)^(1 + n)*(Cos[(e + f*x)/2]^2*Sec[e + f*x])^(3/2 + n)*(-1 + Tan[(e + f*x)/2]^2)^2)/(3*AppellF1[1/2, -3/2 + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (2*(-1 + n)*AppellF1[3/2, -3/2 + n, 2 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (-3 + 2*n)*AppellF1[3/2, -1/2 + n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2) - (6*AppellF1[1/2, -3/2 + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*(Sec[(e + f*x)/2]^2)^n*(Cos[(e + f*x)/2]^2*Sec[e + f*x])^(3/2 + n)*Sin[e + f*x]*Tan[(e + f*x)/2]*(-1 + Tan[(e + f*x)/2]^2)^2)/(3*AppellF1[1/2, -3/2 + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (2*(-1 + n)*AppellF1[3/2, -3/2 + n, 2 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (-3 + 2*n)*AppellF1[3/2, -1/2 + n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2) + (6*n*AppellF1[1/2, -3/2 + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Cos[e + f*x]*(Sec[(e + f*x)/2]^2)^n*(Cos[(e + f*x)/2]^2*Sec[e + f*x])^(3/2 + n)*Tan[(e + f*x)/2]^2*(-1 + Tan[(e + f*x)/2]^2)^2)/(3*AppellF1[1/2, -3/2 + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (2*(-1 + n)*AppellF1[3/2, -3/2 + n, 2 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (-3 + 2*n)*AppellF1[3/2, -1/2 + n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2) + (6*Cos[e + f*x]*(Sec[(e + f*x)/2]^2)^n*(Cos[(e + f*x)/2]^2*Sec[e + f*x])^(3/2 + n)*Tan[(e + f*x)/2]*(-1/3*((1 - n)*AppellF1[3/2, -3/2 + n, 2 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2]) + ((-3/2 + n)*AppellF1[3/2, -1/2 + n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/3)*(-1 + Tan[(e + f*x)/2]^2)^2)/(3*AppellF1[1/2, -3/2 + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (2*(-1 + n)*AppellF1[3/2, -3/2 + n, 2 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (-3 + 2*n)*AppellF1[3/2, -1/2 + n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2) - (6*AppellF1[1/2, -3/2 + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Cos[e + f*x]*(Sec[(e + f*x)/2]^2)^n*(Cos[(e + f*x)/2]^2*Sec[e + f*x])^(3/2 + n)*Tan[(e + f*x)/2]*(-1 + Tan[(e + f*x)/2]^2)^2*((2*(-1 + n)*AppellF1[3/2, -3/2 + n, 2 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (-3 + 2*n)*AppellF1[3/2, -1/2 + n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2] + 3*(-1/3*((1 - n)*AppellF1[3/2, -3/2 + n, 2 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2]) + ((-3/2 + n)*AppellF1[3/2, -1/2 + n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/3) + Tan[(e + f*x)/2]^2*(2*(-1 + n)*((-3*(2 - n)*AppellF1[5/2, -3/2 + n, 3 - n, 7/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/5 + (3*(-3/2 + n)*AppellF1[5/2, -1/2 + n, 2 - n, 7/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/5) + (-3 + 2*n)*((-3*(1 - n)*AppellF1[5/2, -1/2 + n, 2 - n, 7/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/5 + (3*(-1/2 + n)*AppellF1[5/2, 1/2 + n, 1 - n, 7/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/5))))/(3*AppellF1[1/2, -3/2 + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (2*(-1 + n)*AppellF1[3/2, -3/2 + n, 2 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (-3 + 2*n)*AppellF1[3/2, -1/2 + n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2)^2 + (6*(3/2 + n)*AppellF1[1/2, -3/2 + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Cos[e + f*x]*(Sec[(e + f*x)/2]^2)^n*(Cos[(e + f*x)/2]^2*Sec[e + f*x])^(1/2 + n)*Tan[(e + f*x)/2]*(-1 + Tan[(e + f*x)/2]^2)^2*(-(Cos[(e + f*x)/2]*Sec[e + f*x]*Sin[(e + f*x)/2]) + Cos[(e + f*x)/2]^2*Sec[e + f*x]*Tan[e + f*x]))/(3*AppellF1[1/2, -3/2 + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (2*(-1 + n)*AppellF1[3/2, -3/2 + n, 2 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (-3 + 2*n)*AppellF1[3/2, -1/2 + n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2)))","B",0
312,1,88,130,0.3778068,"\int (-\sec (e+f x))^n (a+a \sec (e+f x))^{3/2} \, dx","Integrate[(-Sec[e + f*x])^n*(a + a*Sec[e + f*x])^(3/2),x]","\frac{a \tan \left(\frac{1}{2} (e+f x)\right) \sqrt{a (\sec (e+f x)+1)} (-\sec (e+f x))^n \left((4 n+1) \cos ^{n+\frac{1}{2}}(e+f x) \, _2F_1\left(\frac{1}{2},n+\frac{3}{2};\frac{3}{2};2 \sin ^2\left(\frac{1}{2} (e+f x)\right)\right)-1\right)}{f n}","\frac{2 a^2 (4 n+1) \sin (e+f x) (-\sec (e+f x))^n \sec ^{1-n}(e+f x) \, _2F_1\left(\frac{1}{2},1-n;\frac{3}{2};1-\sec (e+f x)\right)}{f (2 n+1) \sqrt{a \sec (e+f x)+a}}+\frac{2 a^2 \tan (e+f x) (-\sec (e+f x))^n}{f (2 n+1) \sqrt{a \sec (e+f x)+a}}",1,"(a*(-1 + (1 + 4*n)*Cos[e + f*x]^(1/2 + n)*Hypergeometric2F1[1/2, 3/2 + n, 3/2, 2*Sin[(e + f*x)/2]^2])*(-Sec[e + f*x])^n*Sqrt[a*(1 + Sec[e + f*x])]*Tan[(e + f*x)/2])/(f*n)","A",1
313,1,71,70,0.1399868,"\int (-\sec (e+f x))^n \sqrt{a+a \sec (e+f x)} \, dx","Integrate[(-Sec[e + f*x])^n*Sqrt[a + a*Sec[e + f*x]],x]","\frac{2 \tan \left(\frac{1}{2} (e+f x)\right) \sqrt{a (\sec (e+f x)+1)} (-\sec (e+f x))^n \sec ^{-n}(e+f x) \, _2F_1\left(\frac{1}{2},1-n;\frac{3}{2};1-\sec (e+f x)\right)}{f}","\frac{2 a \sin (e+f x) (-\sec (e+f x))^n \sec ^{1-n}(e+f x) \, _2F_1\left(\frac{1}{2},1-n;\frac{3}{2};1-\sec (e+f x)\right)}{f \sqrt{a \sec (e+f x)+a}}",1,"(2*Hypergeometric2F1[1/2, 1 - n, 3/2, 1 - Sec[e + f*x]]*(-Sec[e + f*x])^n*Sqrt[a*(1 + Sec[e + f*x])]*Tan[(e + f*x)/2])/(f*Sec[e + f*x]^n)","A",1
314,1,2977,75,6.2342364,"\int \frac{(-\sec (e+f x))^n}{\sqrt{a+a \sec (e+f x)}} \, dx","Integrate[(-Sec[e + f*x])^n/Sqrt[a + a*Sec[e + f*x]],x]","\text{Result too large to show}","-\frac{\tan (e+f x) (-\sec (e+f x))^n F_1\left(n;\frac{1}{2},1;n+1;\sec (e+f x),-\sec (e+f x)\right)}{f n \sqrt{1-\sec (e+f x)} \sqrt{a \sec (e+f x)+a}}",1,"(3*Sqrt[2]*AppellF1[1/2, -1/2 + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*(Sec[(e + f*x)/2]^2)^n*(-Sec[e + f*x])^n*Sec[e + f*x]^(-1/2 - n + (-1 + 2*n)/2)*(Cos[(e + f*x)/2]^2*Sec[e + f*x])^n*Sqrt[1 + Sec[e + f*x]]*Tan[(e + f*x)/2])/(f*Sqrt[a*(1 + Sec[e + f*x])]*(3*AppellF1[1/2, -1/2 + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (2*(-1 + n)*AppellF1[3/2, -1/2 + n, 2 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (-1 + 2*n)*AppellF1[3/2, 1/2 + n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2)*((3*AppellF1[1/2, -1/2 + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Cos[e + f*x]*(Sec[(e + f*x)/2]^2)^(1 + n)*(Cos[(e + f*x)/2]^2*Sec[e + f*x])^n*Sqrt[1 + Sec[e + f*x]])/(Sqrt[2]*(3*AppellF1[1/2, -1/2 + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (2*(-1 + n)*AppellF1[3/2, -1/2 + n, 2 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (-1 + 2*n)*AppellF1[3/2, 1/2 + n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2)) - (3*Sqrt[2]*AppellF1[1/2, -1/2 + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*(Sec[(e + f*x)/2]^2)^n*(Cos[(e + f*x)/2]^2*Sec[e + f*x])^n*Sqrt[1 + Sec[e + f*x]]*Sin[e + f*x]*Tan[(e + f*x)/2])/(3*AppellF1[1/2, -1/2 + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (2*(-1 + n)*AppellF1[3/2, -1/2 + n, 2 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (-1 + 2*n)*AppellF1[3/2, 1/2 + n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2) + (3*Sqrt[2]*n*AppellF1[1/2, -1/2 + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Cos[e + f*x]*(Sec[(e + f*x)/2]^2)^n*(Cos[(e + f*x)/2]^2*Sec[e + f*x])^n*Sqrt[1 + Sec[e + f*x]]*Tan[(e + f*x)/2]^2)/(3*AppellF1[1/2, -1/2 + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (2*(-1 + n)*AppellF1[3/2, -1/2 + n, 2 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (-1 + 2*n)*AppellF1[3/2, 1/2 + n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2) + (3*Sqrt[2]*Cos[e + f*x]*(Sec[(e + f*x)/2]^2)^n*(Cos[(e + f*x)/2]^2*Sec[e + f*x])^n*Sqrt[1 + Sec[e + f*x]]*Tan[(e + f*x)/2]*(-1/3*((1 - n)*AppellF1[3/2, -1/2 + n, 2 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2]) + ((-1/2 + n)*AppellF1[3/2, 1/2 + n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/3))/(3*AppellF1[1/2, -1/2 + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (2*(-1 + n)*AppellF1[3/2, -1/2 + n, 2 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (-1 + 2*n)*AppellF1[3/2, 1/2 + n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2) - (3*Sqrt[2]*AppellF1[1/2, -1/2 + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Cos[e + f*x]*(Sec[(e + f*x)/2]^2)^n*(Cos[(e + f*x)/2]^2*Sec[e + f*x])^n*Sqrt[1 + Sec[e + f*x]]*Tan[(e + f*x)/2]*((2*(-1 + n)*AppellF1[3/2, -1/2 + n, 2 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (-1 + 2*n)*AppellF1[3/2, 1/2 + n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2] + 3*(-1/3*((1 - n)*AppellF1[3/2, -1/2 + n, 2 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2]) + ((-1/2 + n)*AppellF1[3/2, 1/2 + n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/3) + Tan[(e + f*x)/2]^2*(2*(-1 + n)*((-3*(2 - n)*AppellF1[5/2, -1/2 + n, 3 - n, 7/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/5 + (3*(-1/2 + n)*AppellF1[5/2, 1/2 + n, 2 - n, 7/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/5) + (-1 + 2*n)*((-3*(1 - n)*AppellF1[5/2, 1/2 + n, 2 - n, 7/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/5 + (3*(1/2 + n)*AppellF1[5/2, 3/2 + n, 1 - n, 7/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/5))))/(3*AppellF1[1/2, -1/2 + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (2*(-1 + n)*AppellF1[3/2, -1/2 + n, 2 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (-1 + 2*n)*AppellF1[3/2, 1/2 + n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2)^2 + (3*AppellF1[1/2, -1/2 + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*(Sec[(e + f*x)/2]^2)^n*(Cos[(e + f*x)/2]^2*Sec[e + f*x])^n*Tan[(e + f*x)/2]*Tan[e + f*x])/(Sqrt[2]*Sqrt[1 + Sec[e + f*x]]*(3*AppellF1[1/2, -1/2 + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (2*(-1 + n)*AppellF1[3/2, -1/2 + n, 2 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (-1 + 2*n)*AppellF1[3/2, 1/2 + n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2)) + (3*Sqrt[2]*n*AppellF1[1/2, -1/2 + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Cos[e + f*x]*(Sec[(e + f*x)/2]^2)^n*(Cos[(e + f*x)/2]^2*Sec[e + f*x])^(-1 + n)*Sqrt[1 + Sec[e + f*x]]*Tan[(e + f*x)/2]*(-(Cos[(e + f*x)/2]*Sec[e + f*x]*Sin[(e + f*x)/2]) + Cos[(e + f*x)/2]^2*Sec[e + f*x]*Tan[e + f*x]))/(3*AppellF1[1/2, -1/2 + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (2*(-1 + n)*AppellF1[3/2, -1/2 + n, 2 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (-1 + 2*n)*AppellF1[3/2, 1/2 + n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2)))","B",0
315,1,3005,78,6.2230759,"\int \frac{(-\sec (e+f x))^n}{(a+a \sec (e+f x))^{3/2}} \, dx","Integrate[(-Sec[e + f*x])^n/(a + a*Sec[e + f*x])^(3/2),x]","\text{Result too large to show}","-\frac{\tan (e+f x) (-\sec (e+f x))^n F_1\left(n;\frac{1}{2},2;n+1;\sec (e+f x),-\sec (e+f x)\right)}{a f n \sqrt{1-\sec (e+f x)} \sqrt{a \sec (e+f x)+a}}",1,"(6*AppellF1[1/2, -3/2 + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*(Sec[(e + f*x)/2]^2)^n*(-Sec[e + f*x])^n*Sec[e + f*x]^(1/2 - n + (-3 + 2*n)/2)*(Cos[(e + f*x)/2]^2*Sec[e + f*x])^(3/2 + n)*Tan[(e + f*x)/2]*(-1 + Tan[(e + f*x)/2]^2)^2)/(f*(a*(1 + Sec[e + f*x]))^(3/2)*(3*AppellF1[1/2, -3/2 + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (2*(-1 + n)*AppellF1[3/2, -3/2 + n, 2 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (-3 + 2*n)*AppellF1[3/2, -1/2 + n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2)*((12*AppellF1[1/2, -3/2 + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Cos[e + f*x]*(Sec[(e + f*x)/2]^2)^(1 + n)*(Cos[(e + f*x)/2]^2*Sec[e + f*x])^(3/2 + n)*Tan[(e + f*x)/2]^2*(-1 + Tan[(e + f*x)/2]^2))/(3*AppellF1[1/2, -3/2 + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (2*(-1 + n)*AppellF1[3/2, -3/2 + n, 2 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (-3 + 2*n)*AppellF1[3/2, -1/2 + n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2) + (3*AppellF1[1/2, -3/2 + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Cos[e + f*x]*(Sec[(e + f*x)/2]^2)^(1 + n)*(Cos[(e + f*x)/2]^2*Sec[e + f*x])^(3/2 + n)*(-1 + Tan[(e + f*x)/2]^2)^2)/(3*AppellF1[1/2, -3/2 + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (2*(-1 + n)*AppellF1[3/2, -3/2 + n, 2 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (-3 + 2*n)*AppellF1[3/2, -1/2 + n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2) - (6*AppellF1[1/2, -3/2 + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*(Sec[(e + f*x)/2]^2)^n*(Cos[(e + f*x)/2]^2*Sec[e + f*x])^(3/2 + n)*Sin[e + f*x]*Tan[(e + f*x)/2]*(-1 + Tan[(e + f*x)/2]^2)^2)/(3*AppellF1[1/2, -3/2 + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (2*(-1 + n)*AppellF1[3/2, -3/2 + n, 2 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (-3 + 2*n)*AppellF1[3/2, -1/2 + n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2) + (6*n*AppellF1[1/2, -3/2 + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Cos[e + f*x]*(Sec[(e + f*x)/2]^2)^n*(Cos[(e + f*x)/2]^2*Sec[e + f*x])^(3/2 + n)*Tan[(e + f*x)/2]^2*(-1 + Tan[(e + f*x)/2]^2)^2)/(3*AppellF1[1/2, -3/2 + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (2*(-1 + n)*AppellF1[3/2, -3/2 + n, 2 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (-3 + 2*n)*AppellF1[3/2, -1/2 + n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2) + (6*Cos[e + f*x]*(Sec[(e + f*x)/2]^2)^n*(Cos[(e + f*x)/2]^2*Sec[e + f*x])^(3/2 + n)*Tan[(e + f*x)/2]*(-1/3*((1 - n)*AppellF1[3/2, -3/2 + n, 2 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2]) + ((-3/2 + n)*AppellF1[3/2, -1/2 + n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/3)*(-1 + Tan[(e + f*x)/2]^2)^2)/(3*AppellF1[1/2, -3/2 + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (2*(-1 + n)*AppellF1[3/2, -3/2 + n, 2 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (-3 + 2*n)*AppellF1[3/2, -1/2 + n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2) - (6*AppellF1[1/2, -3/2 + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Cos[e + f*x]*(Sec[(e + f*x)/2]^2)^n*(Cos[(e + f*x)/2]^2*Sec[e + f*x])^(3/2 + n)*Tan[(e + f*x)/2]*(-1 + Tan[(e + f*x)/2]^2)^2*((2*(-1 + n)*AppellF1[3/2, -3/2 + n, 2 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (-3 + 2*n)*AppellF1[3/2, -1/2 + n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2] + 3*(-1/3*((1 - n)*AppellF1[3/2, -3/2 + n, 2 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2]) + ((-3/2 + n)*AppellF1[3/2, -1/2 + n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/3) + Tan[(e + f*x)/2]^2*(2*(-1 + n)*((-3*(2 - n)*AppellF1[5/2, -3/2 + n, 3 - n, 7/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/5 + (3*(-3/2 + n)*AppellF1[5/2, -1/2 + n, 2 - n, 7/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/5) + (-3 + 2*n)*((-3*(1 - n)*AppellF1[5/2, -1/2 + n, 2 - n, 7/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/5 + (3*(-1/2 + n)*AppellF1[5/2, 1/2 + n, 1 - n, 7/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/5))))/(3*AppellF1[1/2, -3/2 + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (2*(-1 + n)*AppellF1[3/2, -3/2 + n, 2 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (-3 + 2*n)*AppellF1[3/2, -1/2 + n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2)^2 + (6*(3/2 + n)*AppellF1[1/2, -3/2 + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Cos[e + f*x]*(Sec[(e + f*x)/2]^2)^n*(Cos[(e + f*x)/2]^2*Sec[e + f*x])^(1/2 + n)*Tan[(e + f*x)/2]*(-1 + Tan[(e + f*x)/2]^2)^2*(-(Cos[(e + f*x)/2]*Sec[e + f*x]*Sin[(e + f*x)/2]) + Cos[(e + f*x)/2]^2*Sec[e + f*x]*Tan[e + f*x]))/(3*AppellF1[1/2, -3/2 + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (2*(-1 + n)*AppellF1[3/2, -3/2 + n, 2 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (-3 + 2*n)*AppellF1[3/2, -1/2 + n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2)))","B",0
316,1,88,130,0.3439904,"\int (d \sec (e+f x))^n (a+a \sec (e+f x))^{3/2} \, dx","Integrate[(d*Sec[e + f*x])^n*(a + a*Sec[e + f*x])^(3/2),x]","\frac{a \tan \left(\frac{1}{2} (e+f x)\right) \sqrt{a (\sec (e+f x)+1)} (d \sec (e+f x))^n \left((4 n+1) \cos ^{n+\frac{1}{2}}(e+f x) \, _2F_1\left(\frac{1}{2},n+\frac{3}{2};\frac{3}{2};2 \sin ^2\left(\frac{1}{2} (e+f x)\right)\right)-1\right)}{f n}","\frac{2 a^2 (4 n+1) \sin (e+f x) \sec ^{1-n}(e+f x) (d \sec (e+f x))^n \, _2F_1\left(\frac{1}{2},1-n;\frac{3}{2};1-\sec (e+f x)\right)}{f (2 n+1) \sqrt{a \sec (e+f x)+a}}+\frac{2 a^2 \tan (e+f x) (d \sec (e+f x))^n}{f (2 n+1) \sqrt{a \sec (e+f x)+a}}",1,"(a*(-1 + (1 + 4*n)*Cos[e + f*x]^(1/2 + n)*Hypergeometric2F1[1/2, 3/2 + n, 3/2, 2*Sin[(e + f*x)/2]^2])*(d*Sec[e + f*x])^n*Sqrt[a*(1 + Sec[e + f*x])]*Tan[(e + f*x)/2])/(f*n)","A",1
317,1,71,70,0.1287586,"\int (d \sec (e+f x))^n \sqrt{a+a \sec (e+f x)} \, dx","Integrate[(d*Sec[e + f*x])^n*Sqrt[a + a*Sec[e + f*x]],x]","\frac{2 \tan \left(\frac{1}{2} (e+f x)\right) \sqrt{a (\sec (e+f x)+1)} \sec ^{-n}(e+f x) (d \sec (e+f x))^n \, _2F_1\left(\frac{1}{2},1-n;\frac{3}{2};1-\sec (e+f x)\right)}{f}","\frac{2 a \sin (e+f x) \sec ^{1-n}(e+f x) (d \sec (e+f x))^n \, _2F_1\left(\frac{1}{2},1-n;\frac{3}{2};1-\sec (e+f x)\right)}{f \sqrt{a \sec (e+f x)+a}}",1,"(2*Hypergeometric2F1[1/2, 1 - n, 3/2, 1 - Sec[e + f*x]]*(d*Sec[e + f*x])^n*Sqrt[a*(1 + Sec[e + f*x])]*Tan[(e + f*x)/2])/(f*Sec[e + f*x]^n)","A",1
318,1,2977,75,6.2095004,"\int \frac{(d \sec (e+f x))^n}{\sqrt{a+a \sec (e+f x)}} \, dx","Integrate[(d*Sec[e + f*x])^n/Sqrt[a + a*Sec[e + f*x]],x]","\text{Result too large to show}","-\frac{\tan (e+f x) F_1\left(n;\frac{1}{2},1;n+1;\sec (e+f x),-\sec (e+f x)\right) (d \sec (e+f x))^n}{f n \sqrt{1-\sec (e+f x)} \sqrt{a \sec (e+f x)+a}}",1,"(3*Sqrt[2]*AppellF1[1/2, -1/2 + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*(Sec[(e + f*x)/2]^2)^n*Sec[e + f*x]^(-1/2 - n + (-1 + 2*n)/2)*(d*Sec[e + f*x])^n*(Cos[(e + f*x)/2]^2*Sec[e + f*x])^n*Sqrt[1 + Sec[e + f*x]]*Tan[(e + f*x)/2])/(f*Sqrt[a*(1 + Sec[e + f*x])]*(3*AppellF1[1/2, -1/2 + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (2*(-1 + n)*AppellF1[3/2, -1/2 + n, 2 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (-1 + 2*n)*AppellF1[3/2, 1/2 + n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2)*((3*AppellF1[1/2, -1/2 + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Cos[e + f*x]*(Sec[(e + f*x)/2]^2)^(1 + n)*(Cos[(e + f*x)/2]^2*Sec[e + f*x])^n*Sqrt[1 + Sec[e + f*x]])/(Sqrt[2]*(3*AppellF1[1/2, -1/2 + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (2*(-1 + n)*AppellF1[3/2, -1/2 + n, 2 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (-1 + 2*n)*AppellF1[3/2, 1/2 + n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2)) - (3*Sqrt[2]*AppellF1[1/2, -1/2 + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*(Sec[(e + f*x)/2]^2)^n*(Cos[(e + f*x)/2]^2*Sec[e + f*x])^n*Sqrt[1 + Sec[e + f*x]]*Sin[e + f*x]*Tan[(e + f*x)/2])/(3*AppellF1[1/2, -1/2 + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (2*(-1 + n)*AppellF1[3/2, -1/2 + n, 2 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (-1 + 2*n)*AppellF1[3/2, 1/2 + n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2) + (3*Sqrt[2]*n*AppellF1[1/2, -1/2 + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Cos[e + f*x]*(Sec[(e + f*x)/2]^2)^n*(Cos[(e + f*x)/2]^2*Sec[e + f*x])^n*Sqrt[1 + Sec[e + f*x]]*Tan[(e + f*x)/2]^2)/(3*AppellF1[1/2, -1/2 + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (2*(-1 + n)*AppellF1[3/2, -1/2 + n, 2 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (-1 + 2*n)*AppellF1[3/2, 1/2 + n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2) + (3*Sqrt[2]*Cos[e + f*x]*(Sec[(e + f*x)/2]^2)^n*(Cos[(e + f*x)/2]^2*Sec[e + f*x])^n*Sqrt[1 + Sec[e + f*x]]*Tan[(e + f*x)/2]*(-1/3*((1 - n)*AppellF1[3/2, -1/2 + n, 2 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2]) + ((-1/2 + n)*AppellF1[3/2, 1/2 + n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/3))/(3*AppellF1[1/2, -1/2 + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (2*(-1 + n)*AppellF1[3/2, -1/2 + n, 2 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (-1 + 2*n)*AppellF1[3/2, 1/2 + n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2) - (3*Sqrt[2]*AppellF1[1/2, -1/2 + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Cos[e + f*x]*(Sec[(e + f*x)/2]^2)^n*(Cos[(e + f*x)/2]^2*Sec[e + f*x])^n*Sqrt[1 + Sec[e + f*x]]*Tan[(e + f*x)/2]*((2*(-1 + n)*AppellF1[3/2, -1/2 + n, 2 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (-1 + 2*n)*AppellF1[3/2, 1/2 + n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2] + 3*(-1/3*((1 - n)*AppellF1[3/2, -1/2 + n, 2 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2]) + ((-1/2 + n)*AppellF1[3/2, 1/2 + n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/3) + Tan[(e + f*x)/2]^2*(2*(-1 + n)*((-3*(2 - n)*AppellF1[5/2, -1/2 + n, 3 - n, 7/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/5 + (3*(-1/2 + n)*AppellF1[5/2, 1/2 + n, 2 - n, 7/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/5) + (-1 + 2*n)*((-3*(1 - n)*AppellF1[5/2, 1/2 + n, 2 - n, 7/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/5 + (3*(1/2 + n)*AppellF1[5/2, 3/2 + n, 1 - n, 7/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/5))))/(3*AppellF1[1/2, -1/2 + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (2*(-1 + n)*AppellF1[3/2, -1/2 + n, 2 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (-1 + 2*n)*AppellF1[3/2, 1/2 + n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2)^2 + (3*AppellF1[1/2, -1/2 + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*(Sec[(e + f*x)/2]^2)^n*(Cos[(e + f*x)/2]^2*Sec[e + f*x])^n*Tan[(e + f*x)/2]*Tan[e + f*x])/(Sqrt[2]*Sqrt[1 + Sec[e + f*x]]*(3*AppellF1[1/2, -1/2 + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (2*(-1 + n)*AppellF1[3/2, -1/2 + n, 2 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (-1 + 2*n)*AppellF1[3/2, 1/2 + n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2)) + (3*Sqrt[2]*n*AppellF1[1/2, -1/2 + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Cos[e + f*x]*(Sec[(e + f*x)/2]^2)^n*(Cos[(e + f*x)/2]^2*Sec[e + f*x])^(-1 + n)*Sqrt[1 + Sec[e + f*x]]*Tan[(e + f*x)/2]*(-(Cos[(e + f*x)/2]*Sec[e + f*x]*Sin[(e + f*x)/2]) + Cos[(e + f*x)/2]^2*Sec[e + f*x]*Tan[e + f*x]))/(3*AppellF1[1/2, -1/2 + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (2*(-1 + n)*AppellF1[3/2, -1/2 + n, 2 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (-1 + 2*n)*AppellF1[3/2, 1/2 + n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2)))","B",0
319,1,3005,78,6.2245536,"\int \frac{(d \sec (e+f x))^n}{(a+a \sec (e+f x))^{3/2}} \, dx","Integrate[(d*Sec[e + f*x])^n/(a + a*Sec[e + f*x])^(3/2),x]","\text{Result too large to show}","-\frac{\tan (e+f x) F_1\left(n;\frac{1}{2},2;n+1;\sec (e+f x),-\sec (e+f x)\right) (d \sec (e+f x))^n}{a f n \sqrt{1-\sec (e+f x)} \sqrt{a \sec (e+f x)+a}}",1,"(6*AppellF1[1/2, -3/2 + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*(Sec[(e + f*x)/2]^2)^n*Sec[e + f*x]^(1/2 - n + (-3 + 2*n)/2)*(d*Sec[e + f*x])^n*(Cos[(e + f*x)/2]^2*Sec[e + f*x])^(3/2 + n)*Tan[(e + f*x)/2]*(-1 + Tan[(e + f*x)/2]^2)^2)/(f*(a*(1 + Sec[e + f*x]))^(3/2)*(3*AppellF1[1/2, -3/2 + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (2*(-1 + n)*AppellF1[3/2, -3/2 + n, 2 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (-3 + 2*n)*AppellF1[3/2, -1/2 + n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2)*((12*AppellF1[1/2, -3/2 + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Cos[e + f*x]*(Sec[(e + f*x)/2]^2)^(1 + n)*(Cos[(e + f*x)/2]^2*Sec[e + f*x])^(3/2 + n)*Tan[(e + f*x)/2]^2*(-1 + Tan[(e + f*x)/2]^2))/(3*AppellF1[1/2, -3/2 + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (2*(-1 + n)*AppellF1[3/2, -3/2 + n, 2 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (-3 + 2*n)*AppellF1[3/2, -1/2 + n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2) + (3*AppellF1[1/2, -3/2 + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Cos[e + f*x]*(Sec[(e + f*x)/2]^2)^(1 + n)*(Cos[(e + f*x)/2]^2*Sec[e + f*x])^(3/2 + n)*(-1 + Tan[(e + f*x)/2]^2)^2)/(3*AppellF1[1/2, -3/2 + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (2*(-1 + n)*AppellF1[3/2, -3/2 + n, 2 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (-3 + 2*n)*AppellF1[3/2, -1/2 + n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2) - (6*AppellF1[1/2, -3/2 + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*(Sec[(e + f*x)/2]^2)^n*(Cos[(e + f*x)/2]^2*Sec[e + f*x])^(3/2 + n)*Sin[e + f*x]*Tan[(e + f*x)/2]*(-1 + Tan[(e + f*x)/2]^2)^2)/(3*AppellF1[1/2, -3/2 + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (2*(-1 + n)*AppellF1[3/2, -3/2 + n, 2 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (-3 + 2*n)*AppellF1[3/2, -1/2 + n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2) + (6*n*AppellF1[1/2, -3/2 + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Cos[e + f*x]*(Sec[(e + f*x)/2]^2)^n*(Cos[(e + f*x)/2]^2*Sec[e + f*x])^(3/2 + n)*Tan[(e + f*x)/2]^2*(-1 + Tan[(e + f*x)/2]^2)^2)/(3*AppellF1[1/2, -3/2 + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (2*(-1 + n)*AppellF1[3/2, -3/2 + n, 2 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (-3 + 2*n)*AppellF1[3/2, -1/2 + n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2) + (6*Cos[e + f*x]*(Sec[(e + f*x)/2]^2)^n*(Cos[(e + f*x)/2]^2*Sec[e + f*x])^(3/2 + n)*Tan[(e + f*x)/2]*(-1/3*((1 - n)*AppellF1[3/2, -3/2 + n, 2 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2]) + ((-3/2 + n)*AppellF1[3/2, -1/2 + n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/3)*(-1 + Tan[(e + f*x)/2]^2)^2)/(3*AppellF1[1/2, -3/2 + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (2*(-1 + n)*AppellF1[3/2, -3/2 + n, 2 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (-3 + 2*n)*AppellF1[3/2, -1/2 + n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2) - (6*AppellF1[1/2, -3/2 + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Cos[e + f*x]*(Sec[(e + f*x)/2]^2)^n*(Cos[(e + f*x)/2]^2*Sec[e + f*x])^(3/2 + n)*Tan[(e + f*x)/2]*(-1 + Tan[(e + f*x)/2]^2)^2*((2*(-1 + n)*AppellF1[3/2, -3/2 + n, 2 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (-3 + 2*n)*AppellF1[3/2, -1/2 + n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2] + 3*(-1/3*((1 - n)*AppellF1[3/2, -3/2 + n, 2 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2]) + ((-3/2 + n)*AppellF1[3/2, -1/2 + n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/3) + Tan[(e + f*x)/2]^2*(2*(-1 + n)*((-3*(2 - n)*AppellF1[5/2, -3/2 + n, 3 - n, 7/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/5 + (3*(-3/2 + n)*AppellF1[5/2, -1/2 + n, 2 - n, 7/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/5) + (-3 + 2*n)*((-3*(1 - n)*AppellF1[5/2, -1/2 + n, 2 - n, 7/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/5 + (3*(-1/2 + n)*AppellF1[5/2, 1/2 + n, 1 - n, 7/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/5))))/(3*AppellF1[1/2, -3/2 + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (2*(-1 + n)*AppellF1[3/2, -3/2 + n, 2 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (-3 + 2*n)*AppellF1[3/2, -1/2 + n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2)^2 + (6*(3/2 + n)*AppellF1[1/2, -3/2 + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Cos[e + f*x]*(Sec[(e + f*x)/2]^2)^n*(Cos[(e + f*x)/2]^2*Sec[e + f*x])^(1/2 + n)*Tan[(e + f*x)/2]*(-1 + Tan[(e + f*x)/2]^2)^2*(-(Cos[(e + f*x)/2]*Sec[e + f*x]*Sin[(e + f*x)/2]) + Cos[(e + f*x)/2]^2*Sec[e + f*x]*Tan[e + f*x]))/(3*AppellF1[1/2, -3/2 + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (2*(-1 + n)*AppellF1[3/2, -3/2 + n, 2 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (-3 + 2*n)*AppellF1[3/2, -1/2 + n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2)))","B",0
320,1,429,178,24.6454572,"\int (-\sec (e+f x))^n (a-a \sec (e+f x))^{5/2} \, dx","Integrate[(-Sec[e + f*x])^n*(a - a*Sec[e + f*x])^(5/2),x]","\frac{2^{n-\frac{5}{2}} e^{-i \left(n-\frac{1}{2}\right) (e+f x)} \left(\frac{e^{i (e+f x)}}{1+e^{2 i (e+f x)}}\right)^{n-\frac{1}{2}} \csc ^5\left(\frac{e}{2}+\frac{f x}{2}\right) (a-a \sec (e+f x))^{5/2} \left(\frac{e^{i n (e+f x)} \, _2F_1\left(1,\frac{1}{2} (-n-3);\frac{n+2}{2};-e^{2 i (e+f x)}\right)}{n}+\frac{10 e^{i (n+2) (e+f x)} \, _2F_1\left(1,\frac{1}{2} (-n-1);\frac{n+4}{2};-e^{2 i (e+f x)}\right)}{n+2}+\frac{5 e^{i (n+4) (e+f x)} \, _2F_1\left(1,\frac{1-n}{2};\frac{n+6}{2};-e^{2 i (e+f x)}\right)}{n+4}-\frac{5 e^{i (n+1) (e+f x)} \, _2F_1\left(1,-\frac{n}{2}-1;\frac{n+3}{2};-e^{2 i (e+f x)}\right)}{n+1}-\frac{e^{i (n+5) (e+f x)} \, _2F_1\left(1,1-\frac{n}{2};\frac{n+7}{2};-e^{2 i (e+f x)}\right)}{n+5}-\frac{10 e^{i (n+3) (e+f x)} \, _2F_1\left(1,-\frac{n}{2};\frac{n+5}{2};-e^{2 i (e+f x)}\right)}{n+3}\right) (-\sec (e+f x))^n \sec ^{-n-\frac{5}{2}}(e+f x)}{f \left(1+e^{2 i (e+f x)}\right)^2}","\frac{2 a^3 \left(16 n^2+24 n+3\right) \tan (e+f x) \, _2F_1\left(\frac{1}{2},1-n;\frac{3}{2};\sec (e+f x)+1\right)}{f (2 n+1) (2 n+3) \sqrt{a-a \sec (e+f x)}}+\frac{2 a^3 (4 n+7) \tan (e+f x) (-\sec (e+f x))^n}{f (2 n+1) (2 n+3) \sqrt{a-a \sec (e+f x)}}+\frac{2 a^2 \tan (e+f x) \sqrt{a-a \sec (e+f x)} (-\sec (e+f x))^n}{f (2 n+3)}",1,"(2^(-5/2 + n)*(E^(I*(e + f*x))/(1 + E^((2*I)*(e + f*x))))^(-1/2 + n)*Csc[e/2 + (f*x)/2]^5*((E^(I*n*(e + f*x))*Hypergeometric2F1[1, (-3 - n)/2, (2 + n)/2, -E^((2*I)*(e + f*x))])/n + (10*E^(I*(2 + n)*(e + f*x))*Hypergeometric2F1[1, (-1 - n)/2, (4 + n)/2, -E^((2*I)*(e + f*x))])/(2 + n) + (5*E^(I*(4 + n)*(e + f*x))*Hypergeometric2F1[1, (1 - n)/2, (6 + n)/2, -E^((2*I)*(e + f*x))])/(4 + n) - (5*E^(I*(1 + n)*(e + f*x))*Hypergeometric2F1[1, -1 - n/2, (3 + n)/2, -E^((2*I)*(e + f*x))])/(1 + n) - (E^(I*(5 + n)*(e + f*x))*Hypergeometric2F1[1, 1 - n/2, (7 + n)/2, -E^((2*I)*(e + f*x))])/(5 + n) - (10*E^(I*(3 + n)*(e + f*x))*Hypergeometric2F1[1, -1/2*n, (5 + n)/2, -E^((2*I)*(e + f*x))])/(3 + n))*(-Sec[e + f*x])^n*Sec[e + f*x]^(-5/2 - n)*(a - a*Sec[e + f*x])^(5/2))/(E^(I*(-1/2 + n)*(e + f*x))*(1 + E^((2*I)*(e + f*x)))^2*f)","C",0
321,1,346,108,13.7162767,"\int (-\sec (e+f x))^n (a-a \sec (e+f x))^{3/2} \, dx","Integrate[(-Sec[e + f*x])^n*(a - a*Sec[e + f*x])^(3/2),x]","-\frac{2^{n-\frac{3}{2}} e^{-\frac{1}{2} i (2 n+1) (e+f x)} \left(\frac{e^{i (e+f x)}}{1+e^{2 i (e+f x)}}\right)^{n+\frac{1}{2}} \csc ^3\left(\frac{1}{2} (e+f x)\right) (a-a \sec (e+f x))^{3/2} \left(3 n \left(n^2+4 n+3\right) e^{i (n+2) (e+f x)} \, _2F_1\left(1,\frac{1-n}{2};\frac{n+4}{2};-e^{2 i (e+f x)}\right)+\left(n^3+6 n^2+11 n+6\right) e^{i n (e+f x)} \, _2F_1\left(1,\frac{1}{2} (-n-1);\frac{n+2}{2};-e^{2 i (e+f x)}\right)-n (n+2) \left((n+1) e^{i (n+3) (e+f x)} \, _2F_1\left(1,1-\frac{n}{2};\frac{n+5}{2};-e^{2 i (e+f x)}\right)+3 (n+3) e^{i (n+1) (e+f x)} \, _2F_1\left(1,-\frac{n}{2};\frac{n+3}{2};-e^{2 i (e+f x)}\right)\right)\right) (-\sec (e+f x))^n \sec ^{-n-\frac{3}{2}}(e+f x)}{f n (n+1) (n+2) (n+3)}","\frac{2 a^2 (4 n+1) \tan (e+f x) \, _2F_1\left(\frac{1}{2},1-n;\frac{3}{2};\sec (e+f x)+1\right)}{f (2 n+1) \sqrt{a-a \sec (e+f x)}}+\frac{2 a^2 \tan (e+f x) (-\sec (e+f x))^n}{f (2 n+1) \sqrt{a-a \sec (e+f x)}}",1,"-((2^(-3/2 + n)*(E^(I*(e + f*x))/(1 + E^((2*I)*(e + f*x))))^(1/2 + n)*Csc[(e + f*x)/2]^3*(E^(I*n*(e + f*x))*(6 + 11*n + 6*n^2 + n^3)*Hypergeometric2F1[1, (-1 - n)/2, (2 + n)/2, -E^((2*I)*(e + f*x))] + 3*E^(I*(2 + n)*(e + f*x))*n*(3 + 4*n + n^2)*Hypergeometric2F1[1, (1 - n)/2, (4 + n)/2, -E^((2*I)*(e + f*x))] - n*(2 + n)*(E^(I*(3 + n)*(e + f*x))*(1 + n)*Hypergeometric2F1[1, 1 - n/2, (5 + n)/2, -E^((2*I)*(e + f*x))] + 3*E^(I*(1 + n)*(e + f*x))*(3 + n)*Hypergeometric2F1[1, -1/2*n, (3 + n)/2, -E^((2*I)*(e + f*x))]))*(-Sec[e + f*x])^n*Sec[e + f*x]^(-3/2 - n)*(a - a*Sec[e + f*x])^(3/2))/(E^((I/2)*(1 + 2*n)*(e + f*x))*f*n*(1 + n)*(2 + n)*(3 + n)))","C",0
322,1,213,47,72.7794466,"\int (-\sec (e+f x))^n \sqrt{a-a \sec (e+f x)} \, dx","Integrate[(-Sec[e + f*x])^n*Sqrt[a - a*Sec[e + f*x]],x]","\frac{2^{n-\frac{1}{2}} e^{\frac{1}{2} i (e+f (1-2 n) x)} \left(\frac{e^{i (e+f x)}}{1+e^{2 i (e+f x)}}\right)^{n-\frac{1}{2}} \csc \left(\frac{e}{2}+\frac{f x}{2}\right) \sqrt{a-a \sec (e+f x)} \left((n+1) e^{i f n x} \, _2F_1\left(1,\frac{1-n}{2};\frac{n+2}{2};-e^{2 i (e+f x)}\right)-n e^{i (e+f (n+1) x)} \, _2F_1\left(1,1-\frac{n}{2};\frac{n+3}{2};-e^{2 i (e+f x)}\right)\right) (-\sec (e+f x))^n \sec ^{-n-\frac{1}{2}}(e+f x)}{f n (n+1)}","\frac{2 a \tan (e+f x) \, _2F_1\left(\frac{1}{2},1-n;\frac{3}{2};\sec (e+f x)+1\right)}{f \sqrt{a-a \sec (e+f x)}}",1,"(2^(-1/2 + n)*E^((I/2)*(e + f*(1 - 2*n)*x))*(E^(I*(e + f*x))/(1 + E^((2*I)*(e + f*x))))^(-1/2 + n)*Csc[e/2 + (f*x)/2]*(E^(I*f*n*x)*(1 + n)*Hypergeometric2F1[1, (1 - n)/2, (2 + n)/2, -E^((2*I)*(e + f*x))] - E^(I*(e + f*(1 + n)*x))*n*Hypergeometric2F1[1, 1 - n/2, (3 + n)/2, -E^((2*I)*(e + f*x))])*(-Sec[e + f*x])^n*Sec[e + f*x]^(-1/2 - n)*Sqrt[a - a*Sec[e + f*x]])/(f*n*(1 + n))","C",0
323,0,0,58,1.3278653,"\int \frac{(-\sec (e+f x))^n}{\sqrt{a-a \sec (e+f x)}} \, dx","Integrate[(-Sec[e + f*x])^n/Sqrt[a - a*Sec[e + f*x]],x]","\int \frac{(-\sec (e+f x))^n}{\sqrt{a-a \sec (e+f x)}} \, dx","\frac{\tan (e+f x) F_1\left(\frac{1}{2};1-n,1;\frac{3}{2};\sec (e+f x)+1,\frac{1}{2} (\sec (e+f x)+1)\right)}{f \sqrt{a-a \sec (e+f x)}}",1,"Integrate[(-Sec[e + f*x])^n/Sqrt[a - a*Sec[e + f*x]], x]","F",-1
324,0,0,64,75.5922694,"\int \frac{(-\sec (e+f x))^n}{(a-a \sec (e+f x))^{3/2}} \, dx","Integrate[(-Sec[e + f*x])^n/(a - a*Sec[e + f*x])^(3/2),x]","\int \frac{(-\sec (e+f x))^n}{(a-a \sec (e+f x))^{3/2}} \, dx","\frac{\tan (e+f x) F_1\left(\frac{1}{2};1-n,2;\frac{3}{2};\sec (e+f x)+1,\frac{1}{2} (\sec (e+f x)+1)\right)}{2 a f \sqrt{a-a \sec (e+f x)}}",1,"Integrate[(-Sec[e + f*x])^n/(a - a*Sec[e + f*x])^(3/2), x]","F",-1
325,1,332,130,2.1014283,"\int \sec ^n(e+f x) (a-a \sec (e+f x))^{3/2} \, dx","Integrate[Sec[e + f*x]^n*(a - a*Sec[e + f*x])^(3/2),x]","-\frac{2^{n-\frac{3}{2}} e^{-\frac{1}{2} i (2 n+1) (e+f x)} \left(\frac{e^{i (e+f x)}}{1+e^{2 i (e+f x)}}\right)^{n+\frac{1}{2}} \csc ^3\left(\frac{1}{2} (e+f x)\right) (a-a \sec (e+f x))^{3/2} \left(3 n \left(n^2+4 n+3\right) e^{i (n+2) (e+f x)} \, _2F_1\left(1,\frac{1-n}{2};\frac{n+4}{2};-e^{2 i (e+f x)}\right)+\left(n^3+6 n^2+11 n+6\right) e^{i n (e+f x)} \, _2F_1\left(1,\frac{1}{2} (-n-1);\frac{n+2}{2};-e^{2 i (e+f x)}\right)-n (n+2) \left((n+1) e^{i (n+3) (e+f x)} \, _2F_1\left(1,1-\frac{n}{2};\frac{n+5}{2};-e^{2 i (e+f x)}\right)+3 (n+3) e^{i (n+1) (e+f x)} \, _2F_1\left(1,-\frac{n}{2};\frac{n+3}{2};-e^{2 i (e+f x)}\right)\right)\right)}{f n (n+1) (n+2) (n+3) \sec ^{\frac{3}{2}}(e+f x)}","\frac{2 a^2 (4 n+1) \sin (e+f x) \sec ^{n+1}(e+f x) (-\sec (e+f x))^{-n} \, _2F_1\left(\frac{1}{2},1-n;\frac{3}{2};\sec (e+f x)+1\right)}{f (2 n+1) \sqrt{a-a \sec (e+f x)}}+\frac{2 a^2 \sin (e+f x) \sec ^{n+1}(e+f x)}{f (2 n+1) \sqrt{a-a \sec (e+f x)}}",1,"-((2^(-3/2 + n)*(E^(I*(e + f*x))/(1 + E^((2*I)*(e + f*x))))^(1/2 + n)*Csc[(e + f*x)/2]^3*(E^(I*n*(e + f*x))*(6 + 11*n + 6*n^2 + n^3)*Hypergeometric2F1[1, (-1 - n)/2, (2 + n)/2, -E^((2*I)*(e + f*x))] + 3*E^(I*(2 + n)*(e + f*x))*n*(3 + 4*n + n^2)*Hypergeometric2F1[1, (1 - n)/2, (4 + n)/2, -E^((2*I)*(e + f*x))] - n*(2 + n)*(E^(I*(3 + n)*(e + f*x))*(1 + n)*Hypergeometric2F1[1, 1 - n/2, (5 + n)/2, -E^((2*I)*(e + f*x))] + 3*E^(I*(1 + n)*(e + f*x))*(3 + n)*Hypergeometric2F1[1, -1/2*n, (3 + n)/2, -E^((2*I)*(e + f*x))]))*(a - a*Sec[e + f*x])^(3/2))/(E^((I/2)*(1 + 2*n)*(e + f*x))*f*n*(1 + n)*(2 + n)*(3 + n)*Sec[e + f*x]^(3/2)))","C",0
326,1,185,69,0.403731,"\int \sec ^n(e+f x) \sqrt{a-a \sec (e+f x)} \, dx","Integrate[Sec[e + f*x]^n*Sqrt[a - a*Sec[e + f*x]],x]","-\frac{2^n e^{\frac{1}{2} i (e+f (1-2 n) x)} \left(\frac{e^{i (e+f x)}}{1+e^{2 i (e+f x)}}\right)^n \cos (e+f x) \csc \left(\frac{1}{2} (e+f x)\right) \sqrt{a-a \sec (e+f x)} \left(n e^{i (e+f (n+1) x)} \, _2F_1\left(1,1-\frac{n}{2};\frac{n+3}{2};-e^{2 i (e+f x)}\right)-(n+1) e^{i f n x} \, _2F_1\left(1,\frac{1-n}{2};\frac{n+2}{2};-e^{2 i (e+f x)}\right)\right)}{f n (n+1)}","\frac{2 a \sin (e+f x) (-\sec (e+f x))^{-n} \sec ^{n+1}(e+f x) \, _2F_1\left(\frac{1}{2},1-n;\frac{3}{2};\sec (e+f x)+1\right)}{f \sqrt{a-a \sec (e+f x)}}",1,"-((2^n*E^((I/2)*(e + f*(1 - 2*n)*x))*(E^(I*(e + f*x))/(1 + E^((2*I)*(e + f*x))))^n*Cos[e + f*x]*Csc[(e + f*x)/2]*(-(E^(I*f*n*x)*(1 + n)*Hypergeometric2F1[1, (1 - n)/2, (2 + n)/2, -E^((2*I)*(e + f*x))]) + E^(I*(e + f*(1 + n)*x))*n*Hypergeometric2F1[1, 1 - n/2, (3 + n)/2, -E^((2*I)*(e + f*x))])*Sqrt[a - a*Sec[e + f*x]])/(f*n*(1 + n)))","C",0
327,1,346,130,1.1620535,"\int (d \sec (e+f x))^n (a-a \sec (e+f x))^{3/2} \, dx","Integrate[(d*Sec[e + f*x])^n*(a - a*Sec[e + f*x])^(3/2),x]","-\frac{2^{n-\frac{3}{2}} e^{-\frac{1}{2} i (2 n+1) (e+f x)} \left(\frac{e^{i (e+f x)}}{1+e^{2 i (e+f x)}}\right)^{n+\frac{1}{2}} \csc ^3\left(\frac{1}{2} (e+f x)\right) (a-a \sec (e+f x))^{3/2} \left(3 n \left(n^2+4 n+3\right) e^{i (n+2) (e+f x)} \, _2F_1\left(1,\frac{1-n}{2};\frac{n+4}{2};-e^{2 i (e+f x)}\right)+\left(n^3+6 n^2+11 n+6\right) e^{i n (e+f x)} \, _2F_1\left(1,\frac{1}{2} (-n-1);\frac{n+2}{2};-e^{2 i (e+f x)}\right)-n (n+2) \left((n+1) e^{i (n+3) (e+f x)} \, _2F_1\left(1,1-\frac{n}{2};\frac{n+5}{2};-e^{2 i (e+f x)}\right)+3 (n+3) e^{i (n+1) (e+f x)} \, _2F_1\left(1,-\frac{n}{2};\frac{n+3}{2};-e^{2 i (e+f x)}\right)\right)\right) \sec ^{-n-\frac{3}{2}}(e+f x) (d \sec (e+f x))^n}{f n (n+1) (n+2) (n+3)}","\frac{2 a^2 (4 n+1) \tan (e+f x) (-\sec (e+f x))^{-n} (d \sec (e+f x))^n \, _2F_1\left(\frac{1}{2},1-n;\frac{3}{2};\sec (e+f x)+1\right)}{f (2 n+1) \sqrt{a-a \sec (e+f x)}}+\frac{2 a^2 \tan (e+f x) (d \sec (e+f x))^n}{f (2 n+1) \sqrt{a-a \sec (e+f x)}}",1,"-((2^(-3/2 + n)*(E^(I*(e + f*x))/(1 + E^((2*I)*(e + f*x))))^(1/2 + n)*Csc[(e + f*x)/2]^3*(E^(I*n*(e + f*x))*(6 + 11*n + 6*n^2 + n^3)*Hypergeometric2F1[1, (-1 - n)/2, (2 + n)/2, -E^((2*I)*(e + f*x))] + 3*E^(I*(2 + n)*(e + f*x))*n*(3 + 4*n + n^2)*Hypergeometric2F1[1, (1 - n)/2, (4 + n)/2, -E^((2*I)*(e + f*x))] - n*(2 + n)*(E^(I*(3 + n)*(e + f*x))*(1 + n)*Hypergeometric2F1[1, 1 - n/2, (5 + n)/2, -E^((2*I)*(e + f*x))] + 3*E^(I*(1 + n)*(e + f*x))*(3 + n)*Hypergeometric2F1[1, -1/2*n, (3 + n)/2, -E^((2*I)*(e + f*x))]))*Sec[e + f*x]^(-3/2 - n)*(d*Sec[e + f*x])^n*(a - a*Sec[e + f*x])^(3/2))/(E^((I/2)*(1 + 2*n)*(e + f*x))*f*n*(1 + n)*(2 + n)*(3 + n)))","C",0
328,1,213,69,0.5807232,"\int (d \sec (e+f x))^n \sqrt{a-a \sec (e+f x)} \, dx","Integrate[(d*Sec[e + f*x])^n*Sqrt[a - a*Sec[e + f*x]],x]","\frac{2^{n-\frac{1}{2}} e^{\frac{1}{2} i (e+f (1-2 n) x)} \left(\frac{e^{i (e+f x)}}{1+e^{2 i (e+f x)}}\right)^{n-\frac{1}{2}} \csc \left(\frac{e}{2}+\frac{f x}{2}\right) \sqrt{a-a \sec (e+f x)} \left((n+1) e^{i f n x} \, _2F_1\left(1,\frac{1-n}{2};\frac{n+2}{2};-e^{2 i (e+f x)}\right)-n e^{i (e+f (n+1) x)} \, _2F_1\left(1,1-\frac{n}{2};\frac{n+3}{2};-e^{2 i (e+f x)}\right)\right) \sec ^{-n-\frac{1}{2}}(e+f x) (d \sec (e+f x))^n}{f n (n+1)}","\frac{2 a \tan (e+f x) (-\sec (e+f x))^{-n} (d \sec (e+f x))^n \, _2F_1\left(\frac{1}{2},1-n;\frac{3}{2};\sec (e+f x)+1\right)}{f \sqrt{a-a \sec (e+f x)}}",1,"(2^(-1/2 + n)*E^((I/2)*(e + f*(1 - 2*n)*x))*(E^(I*(e + f*x))/(1 + E^((2*I)*(e + f*x))))^(-1/2 + n)*Csc[e/2 + (f*x)/2]*(E^(I*f*n*x)*(1 + n)*Hypergeometric2F1[1, (1 - n)/2, (2 + n)/2, -E^((2*I)*(e + f*x))] - E^(I*(e + f*(1 + n)*x))*n*Hypergeometric2F1[1, 1 - n/2, (3 + n)/2, -E^((2*I)*(e + f*x))])*Sec[e + f*x]^(-1/2 - n)*(d*Sec[e + f*x])^n*Sqrt[a - a*Sec[e + f*x]])/(f*n*(1 + n))","C",0
329,1,2246,72,14.36745,"\int \sec ^n(e+f x) (1+\sec (e+f x))^m \, dx","Integrate[Sec[e + f*x]^n*(1 + Sec[e + f*x])^m,x]","\text{Result too large to show}","\frac{2^{m+\frac{1}{2}} \tan (e+f x) F_1\left(\frac{1}{2};1-n,\frac{1}{2}-m;\frac{3}{2};1-\sec (e+f x),\frac{1}{2} (1-\sec (e+f x))\right)}{f \sqrt{\sec (e+f x)+1}}",1,"(3*2^(1 + m)*AppellF1[1/2, m + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*(Sec[(e + f*x)/2]^2)^(-1 + n)*Sec[e + f*x]^n*(Cos[(e + f*x)/2]^2*Sec[e + f*x])^(m + n)*(1 + Sec[e + f*x])^m*Tan[(e + f*x)/2])/(f*(3*AppellF1[1/2, m + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + 2*((-1 + n)*AppellF1[3/2, m + n, 2 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (m + n)*AppellF1[3/2, 1 + m + n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2)*((3*2^m*AppellF1[1/2, m + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*(Sec[(e + f*x)/2]^2)^n*(Cos[(e + f*x)/2]^2*Sec[e + f*x])^(m + n))/(3*AppellF1[1/2, m + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + 2*((-1 + n)*AppellF1[3/2, m + n, 2 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (m + n)*AppellF1[3/2, 1 + m + n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2) + (3*2^(1 + m)*(-1 + n)*AppellF1[1/2, m + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*(Sec[(e + f*x)/2]^2)^(-1 + n)*(Cos[(e + f*x)/2]^2*Sec[e + f*x])^(m + n)*Tan[(e + f*x)/2]^2)/(3*AppellF1[1/2, m + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + 2*((-1 + n)*AppellF1[3/2, m + n, 2 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (m + n)*AppellF1[3/2, 1 + m + n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2) + (3*2^(1 + m)*(Sec[(e + f*x)/2]^2)^(-1 + n)*(Cos[(e + f*x)/2]^2*Sec[e + f*x])^(m + n)*Tan[(e + f*x)/2]*(-1/3*((1 - n)*AppellF1[3/2, m + n, 2 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2]) + ((m + n)*AppellF1[3/2, 1 + m + n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/3))/(3*AppellF1[1/2, m + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + 2*((-1 + n)*AppellF1[3/2, m + n, 2 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (m + n)*AppellF1[3/2, 1 + m + n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2) - (3*2^(1 + m)*AppellF1[1/2, m + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*(Sec[(e + f*x)/2]^2)^(-1 + n)*(Cos[(e + f*x)/2]^2*Sec[e + f*x])^(m + n)*Tan[(e + f*x)/2]*(2*((-1 + n)*AppellF1[3/2, m + n, 2 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (m + n)*AppellF1[3/2, 1 + m + n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2] + 3*(-1/3*((1 - n)*AppellF1[3/2, m + n, 2 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2]) + ((m + n)*AppellF1[3/2, 1 + m + n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/3) + 2*Tan[(e + f*x)/2]^2*((-1 + n)*((-3*(2 - n)*AppellF1[5/2, m + n, 3 - n, 7/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/5 + (3*(m + n)*AppellF1[5/2, 1 + m + n, 2 - n, 7/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/5) + (m + n)*((-3*(1 - n)*AppellF1[5/2, 1 + m + n, 2 - n, 7/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/5 + (3*(1 + m + n)*AppellF1[5/2, 2 + m + n, 1 - n, 7/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/5))))/(3*AppellF1[1/2, m + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + 2*((-1 + n)*AppellF1[3/2, m + n, 2 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (m + n)*AppellF1[3/2, 1 + m + n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2)^2 + (3*2^(1 + m)*(m + n)*AppellF1[1/2, m + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*(Sec[(e + f*x)/2]^2)^(-1 + n)*(Cos[(e + f*x)/2]^2*Sec[e + f*x])^(-1 + m + n)*Tan[(e + f*x)/2]*(-(Cos[(e + f*x)/2]*Sec[e + f*x]*Sin[(e + f*x)/2]) + Cos[(e + f*x)/2]^2*Sec[e + f*x]*Tan[e + f*x]))/(3*AppellF1[1/2, m + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + 2*((-1 + n)*AppellF1[3/2, m + n, 2 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (m + n)*AppellF1[3/2, 1 + m + n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2)))","B",0
330,1,255,89,2.3449327,"\int (1-\sec (e+f x))^m \sec ^n(e+f x) \, dx","Integrate[(1 - Sec[e + f*x])^m*Sec[e + f*x]^n,x]","\frac{(2 m+3) \sin (e+f x) (1-\sec (e+f x))^m \sec ^n(e+f x) F_1\left(m+\frac{1}{2};m+n,1-n;m+\frac{3}{2};\tan ^2\left(\frac{1}{2} (e+f x)\right),-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)}{f (2 m+1) \left(2 \tan ^2\left(\frac{1}{2} (e+f x)\right) \left((n-1) F_1\left(m+\frac{3}{2};m+n,2-n;m+\frac{5}{2};\tan ^2\left(\frac{1}{2} (e+f x)\right),-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)+(m+n) F_1\left(m+\frac{3}{2};m+n+1,1-n;m+\frac{5}{2};\tan ^2\left(\frac{1}{2} (e+f x)\right),-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)\right)+(2 m+3) F_1\left(m+\frac{1}{2};m+n,1-n;m+\frac{3}{2};\tan ^2\left(\frac{1}{2} (e+f x)\right),-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)\right)}","\frac{\sqrt{2} \tan (e+f x) (1-\sec (e+f x))^m F_1\left(m+\frac{1}{2};1-n,\frac{1}{2};m+\frac{3}{2};1-\sec (e+f x),\frac{1}{2} (1-\sec (e+f x))\right)}{f (2 m+1) \sqrt{\sec (e+f x)+1}}",1,"((3 + 2*m)*AppellF1[1/2 + m, m + n, 1 - n, 3/2 + m, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*(1 - Sec[e + f*x])^m*Sec[e + f*x]^n*Sin[e + f*x])/(f*(1 + 2*m)*((3 + 2*m)*AppellF1[1/2 + m, m + n, 1 - n, 3/2 + m, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + 2*((-1 + n)*AppellF1[3/2 + m, m + n, 2 - n, 5/2 + m, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (m + n)*AppellF1[3/2 + m, 1 + m + n, 1 - n, 5/2 + m, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2))","B",0
331,1,2248,88,6.2365667,"\int \sec ^n(e+f x) (a+a \sec (e+f x))^m \, dx","Integrate[Sec[e + f*x]^n*(a + a*Sec[e + f*x])^m,x]","\text{Result too large to show}","\frac{2^{m+\frac{1}{2}} \tan (e+f x) (\sec (e+f x)+1)^{-m-\frac{1}{2}} (a \sec (e+f x)+a)^m F_1\left(\frac{1}{2};1-n,\frac{1}{2}-m;\frac{3}{2};1-\sec (e+f x),\frac{1}{2} (1-\sec (e+f x))\right)}{f}",1,"(3*2^(1 + m)*AppellF1[1/2, m + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*(Sec[(e + f*x)/2]^2)^(-1 + n)*Sec[e + f*x]^n*(Cos[(e + f*x)/2]^2*Sec[e + f*x])^(m + n)*(a*(1 + Sec[e + f*x]))^m*Tan[(e + f*x)/2])/(f*(3*AppellF1[1/2, m + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + 2*((-1 + n)*AppellF1[3/2, m + n, 2 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (m + n)*AppellF1[3/2, 1 + m + n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2)*((3*2^m*AppellF1[1/2, m + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*(Sec[(e + f*x)/2]^2)^n*(Cos[(e + f*x)/2]^2*Sec[e + f*x])^(m + n))/(3*AppellF1[1/2, m + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + 2*((-1 + n)*AppellF1[3/2, m + n, 2 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (m + n)*AppellF1[3/2, 1 + m + n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2) + (3*2^(1 + m)*(-1 + n)*AppellF1[1/2, m + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*(Sec[(e + f*x)/2]^2)^(-1 + n)*(Cos[(e + f*x)/2]^2*Sec[e + f*x])^(m + n)*Tan[(e + f*x)/2]^2)/(3*AppellF1[1/2, m + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + 2*((-1 + n)*AppellF1[3/2, m + n, 2 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (m + n)*AppellF1[3/2, 1 + m + n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2) + (3*2^(1 + m)*(Sec[(e + f*x)/2]^2)^(-1 + n)*(Cos[(e + f*x)/2]^2*Sec[e + f*x])^(m + n)*Tan[(e + f*x)/2]*(-1/3*((1 - n)*AppellF1[3/2, m + n, 2 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2]) + ((m + n)*AppellF1[3/2, 1 + m + n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/3))/(3*AppellF1[1/2, m + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + 2*((-1 + n)*AppellF1[3/2, m + n, 2 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (m + n)*AppellF1[3/2, 1 + m + n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2) - (3*2^(1 + m)*AppellF1[1/2, m + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*(Sec[(e + f*x)/2]^2)^(-1 + n)*(Cos[(e + f*x)/2]^2*Sec[e + f*x])^(m + n)*Tan[(e + f*x)/2]*(2*((-1 + n)*AppellF1[3/2, m + n, 2 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (m + n)*AppellF1[3/2, 1 + m + n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2] + 3*(-1/3*((1 - n)*AppellF1[3/2, m + n, 2 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2]) + ((m + n)*AppellF1[3/2, 1 + m + n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/3) + 2*Tan[(e + f*x)/2]^2*((-1 + n)*((-3*(2 - n)*AppellF1[5/2, m + n, 3 - n, 7/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/5 + (3*(m + n)*AppellF1[5/2, 1 + m + n, 2 - n, 7/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/5) + (m + n)*((-3*(1 - n)*AppellF1[5/2, 1 + m + n, 2 - n, 7/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/5 + (3*(1 + m + n)*AppellF1[5/2, 2 + m + n, 1 - n, 7/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/5))))/(3*AppellF1[1/2, m + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + 2*((-1 + n)*AppellF1[3/2, m + n, 2 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (m + n)*AppellF1[3/2, 1 + m + n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2)^2 + (3*2^(1 + m)*(m + n)*AppellF1[1/2, m + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*(Sec[(e + f*x)/2]^2)^(-1 + n)*(Cos[(e + f*x)/2]^2*Sec[e + f*x])^(-1 + m + n)*Tan[(e + f*x)/2]*(-(Cos[(e + f*x)/2]*Sec[e + f*x]*Sin[(e + f*x)/2]) + Cos[(e + f*x)/2]^2*Sec[e + f*x]*Tan[e + f*x]))/(3*AppellF1[1/2, m + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + 2*((-1 + n)*AppellF1[3/2, m + n, 2 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (m + n)*AppellF1[3/2, 1 + m + n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2)))","B",0
332,0,0,90,1.1049519,"\int \sec ^n(e+f x) (a-a \sec (e+f x))^m \, dx","Integrate[Sec[e + f*x]^n*(a - a*Sec[e + f*x])^m,x]","\int \sec ^n(e+f x) (a-a \sec (e+f x))^m \, dx","\frac{\sqrt{2} \tan (e+f x) (a-a \sec (e+f x))^m F_1\left(m+\frac{1}{2};1-n,\frac{1}{2};m+\frac{3}{2};1-\sec (e+f x),\frac{1}{2} (1-\sec (e+f x))\right)}{f (2 m+1) \sqrt{\sec (e+f x)+1}}",1,"Integrate[Sec[e + f*x]^n*(a - a*Sec[e + f*x])^m, x]","F",-1
333,1,2248,85,6.2293981,"\int (-\sec (e+f x))^n (1+\sec (e+f x))^m \, dx","Integrate[(-Sec[e + f*x])^n*(1 + Sec[e + f*x])^m,x]","\text{Result too large to show}","\frac{\sqrt{2} \tan (e+f x) (\sec (e+f x)+1)^m F_1\left(m+\frac{1}{2};1-n,\frac{1}{2};m+\frac{3}{2};\sec (e+f x)+1,\frac{1}{2} (\sec (e+f x)+1)\right)}{f (2 m+1) \sqrt{1-\sec (e+f x)}}",1,"(3*2^(1 + m)*AppellF1[1/2, m + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*(Sec[(e + f*x)/2]^2)^(-1 + n)*(-Sec[e + f*x])^n*(Cos[(e + f*x)/2]^2*Sec[e + f*x])^(m + n)*(1 + Sec[e + f*x])^m*Tan[(e + f*x)/2])/(f*(3*AppellF1[1/2, m + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + 2*((-1 + n)*AppellF1[3/2, m + n, 2 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (m + n)*AppellF1[3/2, 1 + m + n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2)*((3*2^m*AppellF1[1/2, m + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*(Sec[(e + f*x)/2]^2)^n*(Cos[(e + f*x)/2]^2*Sec[e + f*x])^(m + n))/(3*AppellF1[1/2, m + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + 2*((-1 + n)*AppellF1[3/2, m + n, 2 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (m + n)*AppellF1[3/2, 1 + m + n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2) + (3*2^(1 + m)*(-1 + n)*AppellF1[1/2, m + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*(Sec[(e + f*x)/2]^2)^(-1 + n)*(Cos[(e + f*x)/2]^2*Sec[e + f*x])^(m + n)*Tan[(e + f*x)/2]^2)/(3*AppellF1[1/2, m + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + 2*((-1 + n)*AppellF1[3/2, m + n, 2 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (m + n)*AppellF1[3/2, 1 + m + n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2) + (3*2^(1 + m)*(Sec[(e + f*x)/2]^2)^(-1 + n)*(Cos[(e + f*x)/2]^2*Sec[e + f*x])^(m + n)*Tan[(e + f*x)/2]*(-1/3*((1 - n)*AppellF1[3/2, m + n, 2 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2]) + ((m + n)*AppellF1[3/2, 1 + m + n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/3))/(3*AppellF1[1/2, m + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + 2*((-1 + n)*AppellF1[3/2, m + n, 2 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (m + n)*AppellF1[3/2, 1 + m + n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2) - (3*2^(1 + m)*AppellF1[1/2, m + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*(Sec[(e + f*x)/2]^2)^(-1 + n)*(Cos[(e + f*x)/2]^2*Sec[e + f*x])^(m + n)*Tan[(e + f*x)/2]*(2*((-1 + n)*AppellF1[3/2, m + n, 2 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (m + n)*AppellF1[3/2, 1 + m + n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2] + 3*(-1/3*((1 - n)*AppellF1[3/2, m + n, 2 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2]) + ((m + n)*AppellF1[3/2, 1 + m + n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/3) + 2*Tan[(e + f*x)/2]^2*((-1 + n)*((-3*(2 - n)*AppellF1[5/2, m + n, 3 - n, 7/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/5 + (3*(m + n)*AppellF1[5/2, 1 + m + n, 2 - n, 7/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/5) + (m + n)*((-3*(1 - n)*AppellF1[5/2, 1 + m + n, 2 - n, 7/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/5 + (3*(1 + m + n)*AppellF1[5/2, 2 + m + n, 1 - n, 7/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/5))))/(3*AppellF1[1/2, m + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + 2*((-1 + n)*AppellF1[3/2, m + n, 2 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (m + n)*AppellF1[3/2, 1 + m + n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2)^2 + (3*2^(1 + m)*(m + n)*AppellF1[1/2, m + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*(Sec[(e + f*x)/2]^2)^(-1 + n)*(Cos[(e + f*x)/2]^2*Sec[e + f*x])^(-1 + m + n)*Tan[(e + f*x)/2]*(-(Cos[(e + f*x)/2]*Sec[e + f*x]*Sin[(e + f*x)/2]) + Cos[(e + f*x)/2]^2*Sec[e + f*x]*Tan[e + f*x]))/(3*AppellF1[1/2, m + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + 2*((-1 + n)*AppellF1[3/2, m + n, 2 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (m + n)*AppellF1[3/2, 1 + m + n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2)))","B",0
334,1,257,70,0.3061879,"\int (1-\sec (e+f x))^m (-\sec (e+f x))^n \, dx","Integrate[(1 - Sec[e + f*x])^m*(-Sec[e + f*x])^n,x]","\frac{(2 m+3) \sin (e+f x) (1-\sec (e+f x))^m (-\sec (e+f x))^n F_1\left(m+\frac{1}{2};m+n,1-n;m+\frac{3}{2};\tan ^2\left(\frac{1}{2} (e+f x)\right),-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)}{f (2 m+1) \left(2 \tan ^2\left(\frac{1}{2} (e+f x)\right) \left((n-1) F_1\left(m+\frac{3}{2};m+n,2-n;m+\frac{5}{2};\tan ^2\left(\frac{1}{2} (e+f x)\right),-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)+(m+n) F_1\left(m+\frac{3}{2};m+n+1,1-n;m+\frac{5}{2};\tan ^2\left(\frac{1}{2} (e+f x)\right),-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)\right)+(2 m+3) F_1\left(m+\frac{1}{2};m+n,1-n;m+\frac{3}{2};\tan ^2\left(\frac{1}{2} (e+f x)\right),-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)\right)}","\frac{2^{m+\frac{1}{2}} \tan (e+f x) F_1\left(\frac{1}{2};1-n,\frac{1}{2}-m;\frac{3}{2};\sec (e+f x)+1,\frac{1}{2} (\sec (e+f x)+1)\right)}{f \sqrt{1-\sec (e+f x)}}",1,"((3 + 2*m)*AppellF1[1/2 + m, m + n, 1 - n, 3/2 + m, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*(1 - Sec[e + f*x])^m*(-Sec[e + f*x])^n*Sin[e + f*x])/(f*(1 + 2*m)*((3 + 2*m)*AppellF1[1/2 + m, m + n, 1 - n, 3/2 + m, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + 2*((-1 + n)*AppellF1[3/2 + m, m + n, 2 - n, 5/2 + m, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (m + n)*AppellF1[3/2 + m, 1 + m + n, 1 - n, 5/2 + m, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2))","B",0
335,1,2250,87,6.2229725,"\int (-\sec (e+f x))^n (a+a \sec (e+f x))^m \, dx","Integrate[(-Sec[e + f*x])^n*(a + a*Sec[e + f*x])^m,x]","\text{Result too large to show}","\frac{\sqrt{2} \tan (e+f x) (a \sec (e+f x)+a)^m F_1\left(m+\frac{1}{2};1-n,\frac{1}{2};m+\frac{3}{2};\sec (e+f x)+1,\frac{1}{2} (\sec (e+f x)+1)\right)}{f (2 m+1) \sqrt{1-\sec (e+f x)}}",1,"(3*2^(1 + m)*AppellF1[1/2, m + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*(Sec[(e + f*x)/2]^2)^(-1 + n)*(-Sec[e + f*x])^n*(Cos[(e + f*x)/2]^2*Sec[e + f*x])^(m + n)*(a*(1 + Sec[e + f*x]))^m*Tan[(e + f*x)/2])/(f*(3*AppellF1[1/2, m + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + 2*((-1 + n)*AppellF1[3/2, m + n, 2 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (m + n)*AppellF1[3/2, 1 + m + n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2)*((3*2^m*AppellF1[1/2, m + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*(Sec[(e + f*x)/2]^2)^n*(Cos[(e + f*x)/2]^2*Sec[e + f*x])^(m + n))/(3*AppellF1[1/2, m + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + 2*((-1 + n)*AppellF1[3/2, m + n, 2 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (m + n)*AppellF1[3/2, 1 + m + n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2) + (3*2^(1 + m)*(-1 + n)*AppellF1[1/2, m + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*(Sec[(e + f*x)/2]^2)^(-1 + n)*(Cos[(e + f*x)/2]^2*Sec[e + f*x])^(m + n)*Tan[(e + f*x)/2]^2)/(3*AppellF1[1/2, m + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + 2*((-1 + n)*AppellF1[3/2, m + n, 2 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (m + n)*AppellF1[3/2, 1 + m + n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2) + (3*2^(1 + m)*(Sec[(e + f*x)/2]^2)^(-1 + n)*(Cos[(e + f*x)/2]^2*Sec[e + f*x])^(m + n)*Tan[(e + f*x)/2]*(-1/3*((1 - n)*AppellF1[3/2, m + n, 2 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2]) + ((m + n)*AppellF1[3/2, 1 + m + n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/3))/(3*AppellF1[1/2, m + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + 2*((-1 + n)*AppellF1[3/2, m + n, 2 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (m + n)*AppellF1[3/2, 1 + m + n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2) - (3*2^(1 + m)*AppellF1[1/2, m + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*(Sec[(e + f*x)/2]^2)^(-1 + n)*(Cos[(e + f*x)/2]^2*Sec[e + f*x])^(m + n)*Tan[(e + f*x)/2]*(2*((-1 + n)*AppellF1[3/2, m + n, 2 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (m + n)*AppellF1[3/2, 1 + m + n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2] + 3*(-1/3*((1 - n)*AppellF1[3/2, m + n, 2 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2]) + ((m + n)*AppellF1[3/2, 1 + m + n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/3) + 2*Tan[(e + f*x)/2]^2*((-1 + n)*((-3*(2 - n)*AppellF1[5/2, m + n, 3 - n, 7/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/5 + (3*(m + n)*AppellF1[5/2, 1 + m + n, 2 - n, 7/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/5) + (m + n)*((-3*(1 - n)*AppellF1[5/2, 1 + m + n, 2 - n, 7/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/5 + (3*(1 + m + n)*AppellF1[5/2, 2 + m + n, 1 - n, 7/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/5))))/(3*AppellF1[1/2, m + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + 2*((-1 + n)*AppellF1[3/2, m + n, 2 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (m + n)*AppellF1[3/2, 1 + m + n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2)^2 + (3*2^(1 + m)*(m + n)*AppellF1[1/2, m + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*(Sec[(e + f*x)/2]^2)^(-1 + n)*(Cos[(e + f*x)/2]^2*Sec[e + f*x])^(-1 + m + n)*Tan[(e + f*x)/2]*(-(Cos[(e + f*x)/2]*Sec[e + f*x]*Sin[(e + f*x)/2]) + Cos[(e + f*x)/2]^2*Sec[e + f*x]*Tan[e + f*x]))/(3*AppellF1[1/2, m + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + 2*((-1 + n)*AppellF1[3/2, m + n, 2 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (m + n)*AppellF1[3/2, 1 + m + n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2)))","B",0
336,0,0,87,0.2601322,"\int (-\sec (e+f x))^n (a-a \sec (e+f x))^m \, dx","Integrate[(-Sec[e + f*x])^n*(a - a*Sec[e + f*x])^m,x]","\int (-\sec (e+f x))^n (a-a \sec (e+f x))^m \, dx","\frac{2^{m+\frac{1}{2}} \tan (e+f x) (1-\sec (e+f x))^{-m-\frac{1}{2}} (a-a \sec (e+f x))^m F_1\left(\frac{1}{2};1-n,\frac{1}{2}-m;\frac{3}{2};\sec (e+f x)+1,\frac{1}{2} (\sec (e+f x)+1)\right)}{f}",1,"Integrate[(-Sec[e + f*x])^n*(a - a*Sec[e + f*x])^m, x]","F",-1
337,1,2248,79,6.2119509,"\int (d \sec (e+f x))^n (1+\sec (e+f x))^m \, dx","Integrate[(d*Sec[e + f*x])^n*(1 + Sec[e + f*x])^m,x]","\text{Result too large to show}","-\frac{\tan (e+f x) (d \sec (e+f x))^n F_1\left(n;\frac{1}{2},\frac{1}{2}-m;n+1;\sec (e+f x),-\sec (e+f x)\right)}{f n \sqrt{1-\sec (e+f x)} \sqrt{\sec (e+f x)+1}}",1,"(3*2^(1 + m)*AppellF1[1/2, m + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*(Sec[(e + f*x)/2]^2)^(-1 + n)*(d*Sec[e + f*x])^n*(Cos[(e + f*x)/2]^2*Sec[e + f*x])^(m + n)*(1 + Sec[e + f*x])^m*Tan[(e + f*x)/2])/(f*(3*AppellF1[1/2, m + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + 2*((-1 + n)*AppellF1[3/2, m + n, 2 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (m + n)*AppellF1[3/2, 1 + m + n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2)*((3*2^m*AppellF1[1/2, m + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*(Sec[(e + f*x)/2]^2)^n*(Cos[(e + f*x)/2]^2*Sec[e + f*x])^(m + n))/(3*AppellF1[1/2, m + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + 2*((-1 + n)*AppellF1[3/2, m + n, 2 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (m + n)*AppellF1[3/2, 1 + m + n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2) + (3*2^(1 + m)*(-1 + n)*AppellF1[1/2, m + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*(Sec[(e + f*x)/2]^2)^(-1 + n)*(Cos[(e + f*x)/2]^2*Sec[e + f*x])^(m + n)*Tan[(e + f*x)/2]^2)/(3*AppellF1[1/2, m + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + 2*((-1 + n)*AppellF1[3/2, m + n, 2 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (m + n)*AppellF1[3/2, 1 + m + n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2) + (3*2^(1 + m)*(Sec[(e + f*x)/2]^2)^(-1 + n)*(Cos[(e + f*x)/2]^2*Sec[e + f*x])^(m + n)*Tan[(e + f*x)/2]*(-1/3*((1 - n)*AppellF1[3/2, m + n, 2 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2]) + ((m + n)*AppellF1[3/2, 1 + m + n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/3))/(3*AppellF1[1/2, m + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + 2*((-1 + n)*AppellF1[3/2, m + n, 2 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (m + n)*AppellF1[3/2, 1 + m + n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2) - (3*2^(1 + m)*AppellF1[1/2, m + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*(Sec[(e + f*x)/2]^2)^(-1 + n)*(Cos[(e + f*x)/2]^2*Sec[e + f*x])^(m + n)*Tan[(e + f*x)/2]*(2*((-1 + n)*AppellF1[3/2, m + n, 2 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (m + n)*AppellF1[3/2, 1 + m + n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2] + 3*(-1/3*((1 - n)*AppellF1[3/2, m + n, 2 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2]) + ((m + n)*AppellF1[3/2, 1 + m + n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/3) + 2*Tan[(e + f*x)/2]^2*((-1 + n)*((-3*(2 - n)*AppellF1[5/2, m + n, 3 - n, 7/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/5 + (3*(m + n)*AppellF1[5/2, 1 + m + n, 2 - n, 7/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/5) + (m + n)*((-3*(1 - n)*AppellF1[5/2, 1 + m + n, 2 - n, 7/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/5 + (3*(1 + m + n)*AppellF1[5/2, 2 + m + n, 1 - n, 7/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/5))))/(3*AppellF1[1/2, m + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + 2*((-1 + n)*AppellF1[3/2, m + n, 2 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (m + n)*AppellF1[3/2, 1 + m + n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2)^2 + (3*2^(1 + m)*(m + n)*AppellF1[1/2, m + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*(Sec[(e + f*x)/2]^2)^(-1 + n)*(Cos[(e + f*x)/2]^2*Sec[e + f*x])^(-1 + m + n)*Tan[(e + f*x)/2]*(-(Cos[(e + f*x)/2]*Sec[e + f*x]*Sin[(e + f*x)/2]) + Cos[(e + f*x)/2]^2*Sec[e + f*x]*Tan[e + f*x]))/(3*AppellF1[1/2, m + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + 2*((-1 + n)*AppellF1[3/2, m + n, 2 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (m + n)*AppellF1[3/2, 1 + m + n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2)))","B",0
338,1,257,79,0.2781252,"\int (1-\sec (e+f x))^m (d \sec (e+f x))^n \, dx","Integrate[(1 - Sec[e + f*x])^m*(d*Sec[e + f*x])^n,x]","\frac{(2 m+3) \sin (e+f x) (1-\sec (e+f x))^m (d \sec (e+f x))^n F_1\left(m+\frac{1}{2};m+n,1-n;m+\frac{3}{2};\tan ^2\left(\frac{1}{2} (e+f x)\right),-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)}{f (2 m+1) \left(2 \tan ^2\left(\frac{1}{2} (e+f x)\right) \left((n-1) F_1\left(m+\frac{3}{2};m+n,2-n;m+\frac{5}{2};\tan ^2\left(\frac{1}{2} (e+f x)\right),-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)+(m+n) F_1\left(m+\frac{3}{2};m+n+1,1-n;m+\frac{5}{2};\tan ^2\left(\frac{1}{2} (e+f x)\right),-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)\right)+(2 m+3) F_1\left(m+\frac{1}{2};m+n,1-n;m+\frac{3}{2};\tan ^2\left(\frac{1}{2} (e+f x)\right),-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)\right)}","-\frac{\tan (e+f x) (d \sec (e+f x))^n F_1\left(n;\frac{1}{2}-m,\frac{1}{2};n+1;\sec (e+f x),-\sec (e+f x)\right)}{f n \sqrt{1-\sec (e+f x)} \sqrt{\sec (e+f x)+1}}",1,"((3 + 2*m)*AppellF1[1/2 + m, m + n, 1 - n, 3/2 + m, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*(1 - Sec[e + f*x])^m*(d*Sec[e + f*x])^n*Sin[e + f*x])/(f*(1 + 2*m)*((3 + 2*m)*AppellF1[1/2 + m, m + n, 1 - n, 3/2 + m, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + 2*((-1 + n)*AppellF1[3/2 + m, m + n, 2 - n, 5/2 + m, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (m + n)*AppellF1[3/2 + m, 1 + m + n, 1 - n, 5/2 + m, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2))","B",0
339,1,2250,95,6.2238852,"\int (d \sec (e+f x))^n (a+a \sec (e+f x))^m \, dx","Integrate[(d*Sec[e + f*x])^n*(a + a*Sec[e + f*x])^m,x]","\text{Result too large to show}","-\frac{\tan (e+f x) (\sec (e+f x)+1)^{-m-\frac{1}{2}} (a \sec (e+f x)+a)^m (d \sec (e+f x))^n F_1\left(n;\frac{1}{2},\frac{1}{2}-m;n+1;\sec (e+f x),-\sec (e+f x)\right)}{f n \sqrt{1-\sec (e+f x)}}",1,"(3*2^(1 + m)*AppellF1[1/2, m + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*(Sec[(e + f*x)/2]^2)^(-1 + n)*(d*Sec[e + f*x])^n*(Cos[(e + f*x)/2]^2*Sec[e + f*x])^(m + n)*(a*(1 + Sec[e + f*x]))^m*Tan[(e + f*x)/2])/(f*(3*AppellF1[1/2, m + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + 2*((-1 + n)*AppellF1[3/2, m + n, 2 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (m + n)*AppellF1[3/2, 1 + m + n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2)*((3*2^m*AppellF1[1/2, m + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*(Sec[(e + f*x)/2]^2)^n*(Cos[(e + f*x)/2]^2*Sec[e + f*x])^(m + n))/(3*AppellF1[1/2, m + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + 2*((-1 + n)*AppellF1[3/2, m + n, 2 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (m + n)*AppellF1[3/2, 1 + m + n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2) + (3*2^(1 + m)*(-1 + n)*AppellF1[1/2, m + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*(Sec[(e + f*x)/2]^2)^(-1 + n)*(Cos[(e + f*x)/2]^2*Sec[e + f*x])^(m + n)*Tan[(e + f*x)/2]^2)/(3*AppellF1[1/2, m + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + 2*((-1 + n)*AppellF1[3/2, m + n, 2 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (m + n)*AppellF1[3/2, 1 + m + n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2) + (3*2^(1 + m)*(Sec[(e + f*x)/2]^2)^(-1 + n)*(Cos[(e + f*x)/2]^2*Sec[e + f*x])^(m + n)*Tan[(e + f*x)/2]*(-1/3*((1 - n)*AppellF1[3/2, m + n, 2 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2]) + ((m + n)*AppellF1[3/2, 1 + m + n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/3))/(3*AppellF1[1/2, m + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + 2*((-1 + n)*AppellF1[3/2, m + n, 2 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (m + n)*AppellF1[3/2, 1 + m + n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2) - (3*2^(1 + m)*AppellF1[1/2, m + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*(Sec[(e + f*x)/2]^2)^(-1 + n)*(Cos[(e + f*x)/2]^2*Sec[e + f*x])^(m + n)*Tan[(e + f*x)/2]*(2*((-1 + n)*AppellF1[3/2, m + n, 2 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (m + n)*AppellF1[3/2, 1 + m + n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2] + 3*(-1/3*((1 - n)*AppellF1[3/2, m + n, 2 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2]) + ((m + n)*AppellF1[3/2, 1 + m + n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/3) + 2*Tan[(e + f*x)/2]^2*((-1 + n)*((-3*(2 - n)*AppellF1[5/2, m + n, 3 - n, 7/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/5 + (3*(m + n)*AppellF1[5/2, 1 + m + n, 2 - n, 7/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/5) + (m + n)*((-3*(1 - n)*AppellF1[5/2, 1 + m + n, 2 - n, 7/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/5 + (3*(1 + m + n)*AppellF1[5/2, 2 + m + n, 1 - n, 7/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/5))))/(3*AppellF1[1/2, m + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + 2*((-1 + n)*AppellF1[3/2, m + n, 2 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (m + n)*AppellF1[3/2, 1 + m + n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2)^2 + (3*2^(1 + m)*(m + n)*AppellF1[1/2, m + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*(Sec[(e + f*x)/2]^2)^(-1 + n)*(Cos[(e + f*x)/2]^2*Sec[e + f*x])^(-1 + m + n)*Tan[(e + f*x)/2]*(-(Cos[(e + f*x)/2]*Sec[e + f*x]*Sin[(e + f*x)/2]) + Cos[(e + f*x)/2]^2*Sec[e + f*x]*Tan[e + f*x]))/(3*AppellF1[1/2, m + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + 2*((-1 + n)*AppellF1[3/2, m + n, 2 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (m + n)*AppellF1[3/2, 1 + m + n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2)))","B",0
340,0,0,96,0.2406715,"\int (d \sec (e+f x))^n (a-a \sec (e+f x))^m \, dx","Integrate[(d*Sec[e + f*x])^n*(a - a*Sec[e + f*x])^m,x]","\int (d \sec (e+f x))^n (a-a \sec (e+f x))^m \, dx","-\frac{\tan (e+f x) (1-\sec (e+f x))^{-m-\frac{1}{2}} (a-a \sec (e+f x))^m (d \sec (e+f x))^n F_1\left(n;\frac{1}{2}-m,\frac{1}{2};n+1;\sec (e+f x),-\sec (e+f x)\right)}{f n \sqrt{\sec (e+f x)+1}}",1,"Integrate[(d*Sec[e + f*x])^n*(a - a*Sec[e + f*x])^m, x]","F",-1
341,1,154,211,1.3590641,"\int \sec ^4(e+f x) (a+a \sec (e+f x))^m \, dx","Integrate[Sec[e + f*x]^4*(a + a*Sec[e + f*x])^m,x]","\frac{\tan (e+f x) (\sec (e+f x)+1)^{-m-\frac{1}{2}} (a (\sec (e+f x)+1))^m \left(2^{m+\frac{3}{2}} m \left(m^2+3 m+5\right) \, _2F_1\left(\frac{1}{2},-m-\frac{1}{2};\frac{3}{2};\frac{1}{2} (1-\sec (e+f x))\right)+\left(\left(2 m^2+5 m+2\right) \sec ^2(e+f x)+(2 m+1) m \sec (e+f x)+m^2+m+4\right) (\sec (e+f x)+1)^{m+\frac{1}{2}}\right)}{f (m+2) (m+3) (2 m+1)}","\frac{2^{m+\frac{1}{2}} m \left(m^2+3 m+5\right) \tan (e+f x) (\sec (e+f x)+1)^{-m-\frac{1}{2}} (a \sec (e+f x)+a)^m \, _2F_1\left(\frac{1}{2},\frac{1}{2}-m;\frac{3}{2};\frac{1}{2} (1-\sec (e+f x))\right)}{f (m+1) (m+2) (m+3)}+\frac{m \tan (e+f x) (a \sec (e+f x)+a)^{m+1}}{a f \left(m^2+5 m+6\right)}+\frac{\tan (e+f x) \sec ^2(e+f x) (a \sec (e+f x)+a)^m}{f (m+3)}+\frac{(m+4) \tan (e+f x) (a \sec (e+f x)+a)^m}{f (m+1) (m+2) (m+3)}",1,"((1 + Sec[e + f*x])^(-1/2 - m)*(a*(1 + Sec[e + f*x]))^m*(2^(3/2 + m)*m*(5 + 3*m + m^2)*Hypergeometric2F1[1/2, -1/2 - m, 3/2, (1 - Sec[e + f*x])/2] + (1 + Sec[e + f*x])^(1/2 + m)*(4 + m + m^2 + m*(1 + 2*m)*Sec[e + f*x] + (2 + 5*m + 2*m^2)*Sec[e + f*x]^2))*Tan[e + f*x])/(f*(2 + m)*(3 + m)*(1 + 2*m))","A",1
342,1,123,155,0.6313748,"\int \sec ^3(e+f x) (a+a \sec (e+f x))^m \, dx","Integrate[Sec[e + f*x]^3*(a + a*Sec[e + f*x])^m,x]","\frac{\tan (e+f x) (\sec (e+f x)+1)^{-m-\frac{1}{2}} (a (\sec (e+f x)+1))^m \left(2^{m+\frac{3}{2}} \left(m^2+m+1\right) \, _2F_1\left(\frac{1}{2},-m-\frac{1}{2};\frac{3}{2};\frac{1}{2} (1-\sec (e+f x))\right)+((2 m+1) \sec (e+f x)+m-1) (\sec (e+f x)+1)^{m+\frac{1}{2}}\right)}{f (m+2) (2 m+1)}","\frac{2^{m+\frac{1}{2}} \left(m^2+m+1\right) \tan (e+f x) (\sec (e+f x)+1)^{-m-\frac{1}{2}} (a \sec (e+f x)+a)^m \, _2F_1\left(\frac{1}{2},\frac{1}{2}-m;\frac{3}{2};\frac{1}{2} (1-\sec (e+f x))\right)}{f (m+1) (m+2)}-\frac{\tan (e+f x) (a \sec (e+f x)+a)^m}{f \left(m^2+3 m+2\right)}+\frac{\tan (e+f x) (a \sec (e+f x)+a)^{m+1}}{a f (m+2)}",1,"((1 + Sec[e + f*x])^(-1/2 - m)*(a*(1 + Sec[e + f*x]))^m*(2^(3/2 + m)*(1 + m + m^2)*Hypergeometric2F1[1/2, -1/2 - m, 3/2, (1 - Sec[e + f*x])/2] + (1 + Sec[e + f*x])^(1/2 + m)*(-1 + m + (1 + 2*m)*Sec[e + f*x]))*Tan[e + f*x])/(f*(2 + m)*(1 + 2*m))","A",1
343,1,95,107,0.2179234,"\int \sec ^2(e+f x) (a+a \sec (e+f x))^m \, dx","Integrate[Sec[e + f*x]^2*(a + a*Sec[e + f*x])^m,x]","\frac{\tan (e+f x) (\sec (e+f x)+1)^{-m-\frac{1}{2}} (a (\sec (e+f x)+1))^m \left(2^{m+\frac{3}{2}} m \, _2F_1\left(\frac{1}{2},-m-\frac{1}{2};\frac{3}{2};\frac{1}{2} (1-\sec (e+f x))\right)+(\sec (e+f x)+1)^{m+\frac{1}{2}}\right)}{2 f m+f}","\frac{2^{m+\frac{1}{2}} m \tan (e+f x) (\sec (e+f x)+1)^{-m-\frac{1}{2}} (a \sec (e+f x)+a)^m \, _2F_1\left(\frac{1}{2},\frac{1}{2}-m;\frac{3}{2};\frac{1}{2} (1-\sec (e+f x))\right)}{f (m+1)}+\frac{\tan (e+f x) (a \sec (e+f x)+a)^m}{f (m+1)}",1,"((1 + Sec[e + f*x])^(-1/2 - m)*(a*(1 + Sec[e + f*x]))^m*(2^(3/2 + m)*m*Hypergeometric2F1[1/2, -1/2 - m, 3/2, (1 - Sec[e + f*x])/2] + (1 + Sec[e + f*x])^(1/2 + m))*Tan[e + f*x])/(f + 2*f*m)","A",1
344,1,73,73,0.1058518,"\int \sec (e+f x) (a+a \sec (e+f x))^m \, dx","Integrate[Sec[e + f*x]*(a + a*Sec[e + f*x])^m,x]","\frac{2^{m+\frac{1}{2}} \tan (e+f x) (\sec (e+f x)+1)^{-m-\frac{1}{2}} (a (\sec (e+f x)+1))^m \, _2F_1\left(\frac{1}{2},\frac{1}{2}-m;\frac{3}{2};\frac{1}{2} (1-\sec (e+f x))\right)}{f}","\frac{2^{m+\frac{1}{2}} \tan (e+f x) (\sec (e+f x)+1)^{-m-\frac{1}{2}} (a \sec (e+f x)+a)^m \, _2F_1\left(\frac{1}{2},\frac{1}{2}-m;\frac{3}{2};\frac{1}{2} (1-\sec (e+f x))\right)}{f}",1,"(2^(1/2 + m)*Hypergeometric2F1[1/2, 1/2 - m, 3/2, (1 - Sec[e + f*x])/2]*(1 + Sec[e + f*x])^(-1/2 - m)*(a*(1 + Sec[e + f*x]))^m*Tan[e + f*x])/f","A",1
345,1,711,83,6.795378,"\int (a+a \sec (e+f x))^m \, dx","Integrate[(a + a*Sec[e + f*x])^m,x]","\frac{30 \sin (e+f x) \cos ^2\left(\frac{1}{2} (e+f x)\right) \cos (e+f x) (a (\sec (e+f x)+1))^m F_1\left(\frac{1}{2};m,1;\frac{3}{2};\tan ^2\left(\frac{1}{2} (e+f x)\right),-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right) \left(3 F_1\left(\frac{1}{2};m,1;\frac{3}{2};\tan ^2\left(\frac{1}{2} (e+f x)\right),-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)-2 \tan ^2\left(\frac{1}{2} (e+f x)\right) \left(F_1\left(\frac{3}{2};m,2;\frac{5}{2};\tan ^2\left(\frac{1}{2} (e+f x)\right),-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)-m F_1\left(\frac{3}{2};m+1,1;\frac{5}{2};\tan ^2\left(\frac{1}{2} (e+f x)\right),-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)\right)\right)}{f \left(45 \cos ^2\left(\frac{1}{2} (e+f x)\right) (-2 m \cos (e+f x)+\cos (2 (e+f x))+2 m+1) F_1\left(\frac{1}{2};m,1;\frac{3}{2};\tan ^2\left(\frac{1}{2} (e+f x)\right),-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right){}^2+40 \sin ^2\left(\frac{1}{2} (e+f x)\right) \cos (e+f x) \tan ^2\left(\frac{1}{2} (e+f x)\right) \left(F_1\left(\frac{3}{2};m,2;\frac{5}{2};\tan ^2\left(\frac{1}{2} (e+f x)\right),-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)-m F_1\left(\frac{3}{2};m+1,1;\frac{5}{2};\tan ^2\left(\frac{1}{2} (e+f x)\right),-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)\right){}^2+6 \sin ^2\left(\frac{1}{2} (e+f x)\right) F_1\left(\frac{1}{2};m,1;\frac{3}{2};\tan ^2\left(\frac{1}{2} (e+f x)\right),-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right) \left(-5 (-2 (m+2) \cos (e+f x)+\cos (2 (e+f x))+2 m+1) F_1\left(\frac{3}{2};m,2;\frac{5}{2};\tan ^2\left(\frac{1}{2} (e+f x)\right),-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)+5 m (-2 (m+2) \cos (e+f x)+\cos (2 (e+f x))+2 m+1) F_1\left(\frac{3}{2};m+1,1;\frac{5}{2};\tan ^2\left(\frac{1}{2} (e+f x)\right),-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)-48 \sin ^4\left(\frac{1}{2} (e+f x)\right) \cot (e+f x) \csc (e+f x) \left(2 F_1\left(\frac{5}{2};m,3;\frac{7}{2};\tan ^2\left(\frac{1}{2} (e+f x)\right),-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)-2 m F_1\left(\frac{5}{2};m+1,2;\frac{7}{2};\tan ^2\left(\frac{1}{2} (e+f x)\right),-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)+m (m+1) F_1\left(\frac{5}{2};m+2,1;\frac{7}{2};\tan ^2\left(\frac{1}{2} (e+f x)\right),-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)\right)\right)\right)}","\frac{\sqrt{2} \tan (e+f x) (a \sec (e+f x)+a)^m F_1\left(m+\frac{1}{2};\frac{1}{2},1;m+\frac{3}{2};\frac{1}{2} (\sec (e+f x)+1),\sec (e+f x)+1\right)}{f (2 m+1) \sqrt{1-\sec (e+f x)}}",1,"(30*AppellF1[1/2, m, 1, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Cos[(e + f*x)/2]^2*Cos[e + f*x]*(a*(1 + Sec[e + f*x]))^m*Sin[e + f*x]*(3*AppellF1[1/2, m, 1, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] - 2*(AppellF1[3/2, m, 2, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] - m*AppellF1[3/2, 1 + m, 1, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2))/(f*(45*AppellF1[1/2, m, 1, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]^2*Cos[(e + f*x)/2]^2*(1 + 2*m - 2*m*Cos[e + f*x] + Cos[2*(e + f*x)]) + 6*AppellF1[1/2, m, 1, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sin[(e + f*x)/2]^2*(-5*AppellF1[3/2, m, 2, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*(1 + 2*m - 2*(2 + m)*Cos[e + f*x] + Cos[2*(e + f*x)]) + 5*m*AppellF1[3/2, 1 + m, 1, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*(1 + 2*m - 2*(2 + m)*Cos[e + f*x] + Cos[2*(e + f*x)]) - 48*(2*AppellF1[5/2, m, 3, 7/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] - 2*m*AppellF1[5/2, 1 + m, 2, 7/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + m*(1 + m)*AppellF1[5/2, 2 + m, 1, 7/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Cot[e + f*x]*Csc[e + f*x]*Sin[(e + f*x)/2]^4) + 40*(AppellF1[3/2, m, 2, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] - m*AppellF1[3/2, 1 + m, 1, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])^2*Cos[e + f*x]*Sin[(e + f*x)/2]^2*Tan[(e + f*x)/2]^2))","B",0
346,1,3781,84,16.6625548,"\int \cos (e+f x) (a+a \sec (e+f x))^m \, dx","Integrate[Cos[e + f*x]*(a + a*Sec[e + f*x])^m,x]","\text{Result too large to show}","-\frac{\sqrt{2} \tan (e+f x) (a \sec (e+f x)+a)^m F_1\left(m+\frac{1}{2};\frac{1}{2},2;m+\frac{3}{2};\frac{1}{2} (\sec (e+f x)+1),\sec (e+f x)+1\right)}{f (2 m+1) \sqrt{1-\sec (e+f x)}}",1,"(2^(1 + m)*Cos[(e + f*x)/2]^3*Cos[e + f*x]*(Cos[(e + f*x)/2]^2*Sec[e + f*x])^m*(a*(1 + Sec[e + f*x]))^m*Sin[(e + f*x)/2]*((-3*AppellF1[1/2, m, 1, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2)/(3*AppellF1[1/2, m, 1, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] - 2*(AppellF1[3/2, m, 2, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] - m*AppellF1[3/2, 1 + m, 1, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2) + (2*AppellF1[1/2, m, 2, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])/(AppellF1[1/2, m, 2, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (2*(-2*AppellF1[3/2, m, 3, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + m*AppellF1[3/2, 1 + m, 2, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2)/3)))/(f*(2^m*Cos[(e + f*x)/2]^4*(Cos[(e + f*x)/2]^2*Sec[e + f*x])^m*((-3*AppellF1[1/2, m, 1, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2)/(3*AppellF1[1/2, m, 1, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] - 2*(AppellF1[3/2, m, 2, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] - m*AppellF1[3/2, 1 + m, 1, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2) + (2*AppellF1[1/2, m, 2, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])/(AppellF1[1/2, m, 2, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (2*(-2*AppellF1[3/2, m, 3, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + m*AppellF1[3/2, 1 + m, 2, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2)/3)) - 3*2^m*Cos[(e + f*x)/2]^2*(Cos[(e + f*x)/2]^2*Sec[e + f*x])^m*Sin[(e + f*x)/2]^2*((-3*AppellF1[1/2, m, 1, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2)/(3*AppellF1[1/2, m, 1, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] - 2*(AppellF1[3/2, m, 2, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] - m*AppellF1[3/2, 1 + m, 1, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2) + (2*AppellF1[1/2, m, 2, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])/(AppellF1[1/2, m, 2, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (2*(-2*AppellF1[3/2, m, 3, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + m*AppellF1[3/2, 1 + m, 2, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2)/3)) + 2^(1 + m)*Cos[(e + f*x)/2]^3*(Cos[(e + f*x)/2]^2*Sec[e + f*x])^m*Sin[(e + f*x)/2]*((-3*AppellF1[1/2, m, 1, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/(3*AppellF1[1/2, m, 1, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] - 2*(AppellF1[3/2, m, 2, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] - m*AppellF1[3/2, 1 + m, 1, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2) - (3*Sec[(e + f*x)/2]^2*(-1/3*(AppellF1[3/2, m, 2, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2]) + (m*AppellF1[3/2, 1 + m, 1, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/3))/(3*AppellF1[1/2, m, 1, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] - 2*(AppellF1[3/2, m, 2, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] - m*AppellF1[3/2, 1 + m, 1, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2) + (2*((-2*AppellF1[3/2, m, 3, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/3 + (m*AppellF1[3/2, 1 + m, 2, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/3))/(AppellF1[1/2, m, 2, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (2*(-2*AppellF1[3/2, m, 3, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + m*AppellF1[3/2, 1 + m, 2, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2)/3) + (3*AppellF1[1/2, m, 1, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*(-2*(AppellF1[3/2, m, 2, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] - m*AppellF1[3/2, 1 + m, 1, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2] + 3*(-1/3*(AppellF1[3/2, m, 2, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2]) + (m*AppellF1[3/2, 1 + m, 1, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/3) - 2*Tan[(e + f*x)/2]^2*((-6*AppellF1[5/2, m, 3, 7/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/5 + (3*m*AppellF1[5/2, 1 + m, 2, 7/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/5 - m*((-3*AppellF1[5/2, 1 + m, 2, 7/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/5 + (3*(1 + m)*AppellF1[5/2, 2 + m, 1, 7/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/5))))/(3*AppellF1[1/2, m, 1, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] - 2*(AppellF1[3/2, m, 2, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] - m*AppellF1[3/2, 1 + m, 1, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2)^2 - (2*AppellF1[1/2, m, 2, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*((-2*AppellF1[3/2, m, 3, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/3 + (m*AppellF1[3/2, 1 + m, 2, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/3 + (2*(-2*AppellF1[3/2, m, 3, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + m*AppellF1[3/2, 1 + m, 2, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/3 + (2*Tan[(e + f*x)/2]^2*(-2*((-9*AppellF1[5/2, m, 4, 7/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/5 + (3*m*AppellF1[5/2, 1 + m, 3, 7/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/5) + m*((-6*AppellF1[5/2, 1 + m, 3, 7/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/5 + (3*(1 + m)*AppellF1[5/2, 2 + m, 2, 7/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/5)))/3))/(AppellF1[1/2, m, 2, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (2*(-2*AppellF1[3/2, m, 3, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + m*AppellF1[3/2, 1 + m, 2, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2)/3)^2) + 2^(1 + m)*m*Cos[(e + f*x)/2]^3*(Cos[(e + f*x)/2]^2*Sec[e + f*x])^(-1 + m)*Sin[(e + f*x)/2]*((-3*AppellF1[1/2, m, 1, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2)/(3*AppellF1[1/2, m, 1, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] - 2*(AppellF1[3/2, m, 2, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] - m*AppellF1[3/2, 1 + m, 1, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2) + (2*AppellF1[1/2, m, 2, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])/(AppellF1[1/2, m, 2, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (2*(-2*AppellF1[3/2, m, 3, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + m*AppellF1[3/2, 1 + m, 2, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2)/3))*(-(Cos[(e + f*x)/2]*Sec[e + f*x]*Sin[(e + f*x)/2]) + Cos[(e + f*x)/2]^2*Sec[e + f*x]*Tan[e + f*x])))","B",0
347,1,2529,98,15.0132518,"\int (d \sec (e+f x))^{3/2} (a+a \sec (e+f x))^m \, dx","Integrate[(d*Sec[e + f*x])^(3/2)*(a + a*Sec[e + f*x])^m,x]","\text{Result too large to show}","-\frac{2 \tan (e+f x) (d \sec (e+f x))^{3/2} (\sec (e+f x)+1)^{-m-\frac{1}{2}} (a \sec (e+f x)+a)^m F_1\left(\frac{3}{2};\frac{1}{2},\frac{1}{2}-m;\frac{5}{2};\sec (e+f x),-\sec (e+f x)\right)}{3 f \sqrt{1-\sec (e+f x)}}",1,"(-3*2^(1 + m)*AppellF1[1/2, 3/2 + m, -1/2, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sqrt[Sec[e + f*x]]*(d*Sec[e + f*x])^(3/2)*(Cos[(e + f*x)/2]^2*Sec[e + f*x])^m*(a*(1 + Sec[e + f*x]))^m*Tan[(e + f*x)/2])/(f*(-1 + Tan[(e + f*x)/2]^2)*(3*AppellF1[1/2, 3/2 + m, -1/2, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (AppellF1[3/2, 3/2 + m, 1/2, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (3 + 2*m)*AppellF1[3/2, 5/2 + m, -1/2, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2)*((3*2^(1 + m)*AppellF1[1/2, 3/2 + m, -1/2, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Sqrt[Sec[e + f*x]]*(Cos[(e + f*x)/2]^2*Sec[e + f*x])^m*Tan[(e + f*x)/2]^2)/((-1 + Tan[(e + f*x)/2]^2)^2*(3*AppellF1[1/2, 3/2 + m, -1/2, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (AppellF1[3/2, 3/2 + m, 1/2, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (3 + 2*m)*AppellF1[3/2, 5/2 + m, -1/2, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2)) - (3*2^m*AppellF1[1/2, 3/2 + m, -1/2, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Sqrt[Sec[e + f*x]]*(Cos[(e + f*x)/2]^2*Sec[e + f*x])^m)/((-1 + Tan[(e + f*x)/2]^2)*(3*AppellF1[1/2, 3/2 + m, -1/2, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (AppellF1[3/2, 3/2 + m, 1/2, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (3 + 2*m)*AppellF1[3/2, 5/2 + m, -1/2, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2)) - (3*2^m*AppellF1[1/2, 3/2 + m, -1/2, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[e + f*x]^(3/2)*(Cos[(e + f*x)/2]^2*Sec[e + f*x])^m*Sin[e + f*x]*Tan[(e + f*x)/2])/((-1 + Tan[(e + f*x)/2]^2)*(3*AppellF1[1/2, 3/2 + m, -1/2, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (AppellF1[3/2, 3/2 + m, 1/2, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (3 + 2*m)*AppellF1[3/2, 5/2 + m, -1/2, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2)) - (3*2^(1 + m)*Sqrt[Sec[e + f*x]]*(Cos[(e + f*x)/2]^2*Sec[e + f*x])^m*Tan[(e + f*x)/2]*((AppellF1[3/2, 3/2 + m, 1/2, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/6 + ((3/2 + m)*AppellF1[3/2, 5/2 + m, -1/2, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/3))/((-1 + Tan[(e + f*x)/2]^2)*(3*AppellF1[1/2, 3/2 + m, -1/2, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (AppellF1[3/2, 3/2 + m, 1/2, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (3 + 2*m)*AppellF1[3/2, 5/2 + m, -1/2, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2)) + (3*2^(1 + m)*AppellF1[1/2, 3/2 + m, -1/2, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sqrt[Sec[e + f*x]]*(Cos[(e + f*x)/2]^2*Sec[e + f*x])^m*Tan[(e + f*x)/2]*((AppellF1[3/2, 3/2 + m, 1/2, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (3 + 2*m)*AppellF1[3/2, 5/2 + m, -1/2, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2] + 3*((AppellF1[3/2, 3/2 + m, 1/2, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/6 + ((3/2 + m)*AppellF1[3/2, 5/2 + m, -1/2, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/3) + Tan[(e + f*x)/2]^2*((-3*AppellF1[5/2, 3/2 + m, 3/2, 7/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/10 + (3*(3/2 + m)*AppellF1[5/2, 5/2 + m, 1/2, 7/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/5 + (3 + 2*m)*((3*AppellF1[5/2, 5/2 + m, 1/2, 7/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/10 + (3*(5/2 + m)*AppellF1[5/2, 7/2 + m, -1/2, 7/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/5))))/((-1 + Tan[(e + f*x)/2]^2)*(3*AppellF1[1/2, 3/2 + m, -1/2, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (AppellF1[3/2, 3/2 + m, 1/2, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (3 + 2*m)*AppellF1[3/2, 5/2 + m, -1/2, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2)^2) - (3*2^(1 + m)*m*AppellF1[1/2, 3/2 + m, -1/2, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sqrt[Sec[e + f*x]]*(Cos[(e + f*x)/2]^2*Sec[e + f*x])^(-1 + m)*Tan[(e + f*x)/2]*(-(Cos[(e + f*x)/2]*Sec[e + f*x]*Sin[(e + f*x)/2]) + Cos[(e + f*x)/2]^2*Sec[e + f*x]*Tan[e + f*x]))/((-1 + Tan[(e + f*x)/2]^2)*(3*AppellF1[1/2, 3/2 + m, -1/2, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (AppellF1[3/2, 3/2 + m, 1/2, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (3 + 2*m)*AppellF1[3/2, 5/2 + m, -1/2, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2))))","B",0
348,1,2225,96,14.6438001,"\int \sqrt{d \sec (e+f x)} (a+a \sec (e+f x))^m \, dx","Integrate[Sqrt[d*Sec[e + f*x]]*(a + a*Sec[e + f*x])^m,x]","\text{Result too large to show}","-\frac{2 \tan (e+f x) \sqrt{d \sec (e+f x)} (\sec (e+f x)+1)^{-m-\frac{1}{2}} (a \sec (e+f x)+a)^m F_1\left(\frac{1}{2};\frac{1}{2},\frac{1}{2}-m;\frac{3}{2};\sec (e+f x),-\sec (e+f x)\right)}{f \sqrt{1-\sec (e+f x)}}",1,"(2^(1 + m)*AppellF1[1/2, 1/2 + m, 1/2, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sqrt[d*Sec[e + f*x]]*(Cos[(e + f*x)/2]^2*Sec[e + f*x])^(1/2 + m)*(a*(1 + Sec[e + f*x]))^m*Tan[(e + f*x)/2])/(f*Sqrt[Sec[(e + f*x)/2]^2]*(AppellF1[1/2, 1/2 + m, 1/2, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] - ((AppellF1[3/2, 1/2 + m, 3/2, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] - (1 + 2*m)*AppellF1[3/2, 3/2 + m, 1/2, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2)/3)*((2^m*AppellF1[1/2, 1/2 + m, 1/2, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sqrt[Sec[(e + f*x)/2]^2]*(Cos[(e + f*x)/2]^2*Sec[e + f*x])^(1/2 + m))/(AppellF1[1/2, 1/2 + m, 1/2, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] - ((AppellF1[3/2, 1/2 + m, 3/2, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] - (1 + 2*m)*AppellF1[3/2, 3/2 + m, 1/2, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2)/3) - (2^m*AppellF1[1/2, 1/2 + m, 1/2, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*(Cos[(e + f*x)/2]^2*Sec[e + f*x])^(1/2 + m)*Tan[(e + f*x)/2]^2)/(Sqrt[Sec[(e + f*x)/2]^2]*(AppellF1[1/2, 1/2 + m, 1/2, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] - ((AppellF1[3/2, 1/2 + m, 3/2, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] - (1 + 2*m)*AppellF1[3/2, 3/2 + m, 1/2, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2)/3)) + (2^(1 + m)*(Cos[(e + f*x)/2]^2*Sec[e + f*x])^(1/2 + m)*Tan[(e + f*x)/2]*(-1/6*(AppellF1[3/2, 1/2 + m, 3/2, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2]) + ((1/2 + m)*AppellF1[3/2, 3/2 + m, 1/2, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/3))/(Sqrt[Sec[(e + f*x)/2]^2]*(AppellF1[1/2, 1/2 + m, 1/2, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] - ((AppellF1[3/2, 1/2 + m, 3/2, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] - (1 + 2*m)*AppellF1[3/2, 3/2 + m, 1/2, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2)/3)) - (2^(1 + m)*AppellF1[1/2, 1/2 + m, 1/2, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*(Cos[(e + f*x)/2]^2*Sec[e + f*x])^(1/2 + m)*Tan[(e + f*x)/2]*(-1/6*(AppellF1[3/2, 1/2 + m, 3/2, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2]) + ((1/2 + m)*AppellF1[3/2, 3/2 + m, 1/2, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/3 - ((AppellF1[3/2, 1/2 + m, 3/2, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] - (1 + 2*m)*AppellF1[3/2, 3/2 + m, 1/2, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/3 - (Tan[(e + f*x)/2]^2*((-9*AppellF1[5/2, 1/2 + m, 5/2, 7/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/10 + (3*(1/2 + m)*AppellF1[5/2, 3/2 + m, 3/2, 7/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/5 - (1 + 2*m)*((-3*AppellF1[5/2, 3/2 + m, 3/2, 7/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/10 + (3*(3/2 + m)*AppellF1[5/2, 5/2 + m, 1/2, 7/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/5)))/3))/(Sqrt[Sec[(e + f*x)/2]^2]*(AppellF1[1/2, 1/2 + m, 1/2, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] - ((AppellF1[3/2, 1/2 + m, 3/2, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] - (1 + 2*m)*AppellF1[3/2, 3/2 + m, 1/2, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2)/3)^2) + (2^(1 + m)*(1/2 + m)*AppellF1[1/2, 1/2 + m, 1/2, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*(Cos[(e + f*x)/2]^2*Sec[e + f*x])^(-1/2 + m)*Tan[(e + f*x)/2]*(-(Cos[(e + f*x)/2]*Sec[e + f*x]*Sin[(e + f*x)/2]) + Cos[(e + f*x)/2]^2*Sec[e + f*x]*Tan[e + f*x]))/(Sqrt[Sec[(e + f*x)/2]^2]*(AppellF1[1/2, 1/2 + m, 1/2, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] - ((AppellF1[3/2, 1/2 + m, 3/2, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] - (1 + 2*m)*AppellF1[3/2, 3/2 + m, 1/2, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2)/3))))","B",0
349,1,2424,96,14.9475487,"\int \frac{(a+a \sec (e+f x))^m}{\sqrt{d \sec (e+f x)}} \, dx","Integrate[(a + a*Sec[e + f*x])^m/Sqrt[d*Sec[e + f*x]],x]","\text{Result too large to show}","\frac{2 \tan (e+f x) (\sec (e+f x)+1)^{-m-\frac{1}{2}} (a \sec (e+f x)+a)^m F_1\left(-\frac{1}{2};\frac{1}{2},\frac{1}{2}-m;\frac{1}{2};\sec (e+f x),-\sec (e+f x)\right)}{f \sqrt{1-\sec (e+f x)} \sqrt{d \sec (e+f x)}}",1,"(-3*2^(1 + m)*AppellF1[1/2, -1/2 + m, 3/2, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sqrt[Sec[e + f*x]]*(Cos[(e + f*x)/2]^2*Sec[e + f*x])^(-1/2 + m)*(a*(1 + Sec[e + f*x]))^m*(Cos[2*(e + f*x)]*((1 + Sec[e + f*x])^m/(2*Sqrt[Sec[e + f*x]]) - (I/2)*Sqrt[Sec[e + f*x]]*(1 + Sec[e + f*x])^m*Sin[e + f*x]) + ((1 + Sec[e + f*x])^m/2 + (I/2)*(1 + Sec[e + f*x])^m*Sin[2*(e + f*x)])/Sqrt[Sec[e + f*x]] + Sqrt[Sec[e + f*x]]*Sin[e + f*x]*((-1/2*I)*(1 + Sec[e + f*x])^m + ((1 + Sec[e + f*x])^m*Sin[2*(e + f*x)])/2))*Tan[(e + f*x)/2])/(f*(Sec[(e + f*x)/2]^2)^(3/2)*Sqrt[d*Sec[e + f*x]]*(1 + Sec[e + f*x])^m*(-3*AppellF1[1/2, -1/2 + m, 3/2, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (3*AppellF1[3/2, -1/2 + m, 5/2, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (1 - 2*m)*AppellF1[3/2, 1/2 + m, 3/2, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2)*((-3*2^m*AppellF1[1/2, -1/2 + m, 3/2, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*(Cos[(e + f*x)/2]^2*Sec[e + f*x])^(-1/2 + m))/(Sqrt[Sec[(e + f*x)/2]^2]*(-3*AppellF1[1/2, -1/2 + m, 3/2, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (3*AppellF1[3/2, -1/2 + m, 5/2, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (1 - 2*m)*AppellF1[3/2, 1/2 + m, 3/2, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2)) + (9*2^m*AppellF1[1/2, -1/2 + m, 3/2, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*(Cos[(e + f*x)/2]^2*Sec[e + f*x])^(-1/2 + m)*Tan[(e + f*x)/2]^2)/((Sec[(e + f*x)/2]^2)^(3/2)*(-3*AppellF1[1/2, -1/2 + m, 3/2, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (3*AppellF1[3/2, -1/2 + m, 5/2, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (1 - 2*m)*AppellF1[3/2, 1/2 + m, 3/2, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2)) - (3*2^(1 + m)*(Cos[(e + f*x)/2]^2*Sec[e + f*x])^(-1/2 + m)*Tan[(e + f*x)/2]*(-1/2*(AppellF1[3/2, -1/2 + m, 5/2, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2]) + ((-1/2 + m)*AppellF1[3/2, 1/2 + m, 3/2, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/3))/((Sec[(e + f*x)/2]^2)^(3/2)*(-3*AppellF1[1/2, -1/2 + m, 3/2, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (3*AppellF1[3/2, -1/2 + m, 5/2, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (1 - 2*m)*AppellF1[3/2, 1/2 + m, 3/2, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2)) + (3*2^(1 + m)*AppellF1[1/2, -1/2 + m, 3/2, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*(Cos[(e + f*x)/2]^2*Sec[e + f*x])^(-1/2 + m)*Tan[(e + f*x)/2]*((3*AppellF1[3/2, -1/2 + m, 5/2, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (1 - 2*m)*AppellF1[3/2, 1/2 + m, 3/2, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2] - 3*(-1/2*(AppellF1[3/2, -1/2 + m, 5/2, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2]) + ((-1/2 + m)*AppellF1[3/2, 1/2 + m, 3/2, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/3) + Tan[(e + f*x)/2]^2*(3*((-3*AppellF1[5/2, -1/2 + m, 7/2, 7/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/2 + (3*(-1/2 + m)*AppellF1[5/2, 1/2 + m, 5/2, 7/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/5) + (1 - 2*m)*((-9*AppellF1[5/2, 1/2 + m, 5/2, 7/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/10 + (3*(1/2 + m)*AppellF1[5/2, 3/2 + m, 3/2, 7/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/5))))/((Sec[(e + f*x)/2]^2)^(3/2)*(-3*AppellF1[1/2, -1/2 + m, 3/2, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (3*AppellF1[3/2, -1/2 + m, 5/2, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (1 - 2*m)*AppellF1[3/2, 1/2 + m, 3/2, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2)^2) - (3*2^(1 + m)*(-1/2 + m)*AppellF1[1/2, -1/2 + m, 3/2, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*(Cos[(e + f*x)/2]^2*Sec[e + f*x])^(-3/2 + m)*Tan[(e + f*x)/2]*(-(Cos[(e + f*x)/2]*Sec[e + f*x]*Sin[(e + f*x)/2]) + Cos[(e + f*x)/2]^2*Sec[e + f*x]*Tan[e + f*x]))/((Sec[(e + f*x)/2]^2)^(3/2)*(-3*AppellF1[1/2, -1/2 + m, 3/2, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (3*AppellF1[3/2, -1/2 + m, 5/2, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (1 - 2*m)*AppellF1[3/2, 1/2 + m, 3/2, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2))))","C",0
350,1,3349,98,19.4172551,"\int \frac{(a+a \sec (e+f x))^m}{(d \sec (e+f x))^{3/2}} \, dx","Integrate[(a + a*Sec[e + f*x])^m/(d*Sec[e + f*x])^(3/2),x]","\text{Result too large to show}","\frac{2 \tan (e+f x) (\sec (e+f x)+1)^{-m-\frac{1}{2}} (a \sec (e+f x)+a)^m F_1\left(-\frac{3}{2};\frac{1}{2},\frac{1}{2}-m;-\frac{1}{2};\sec (e+f x),-\sec (e+f x)\right)}{3 f \sqrt{1-\sec (e+f x)} (d \sec (e+f x))^{3/2}}",1,"(2^(1 + m)*Sec[e + f*x]^(3/2)*(Cos[(e + f*x)/2]^2*Sec[e + f*x])^(1/2 + m)*(a*(1 + Sec[e + f*x]))^m*((Cos[2*(e + f*x)]^3*Sqrt[Sec[e + f*x]]*(1 + Sec[e + f*x])^m)/4 + Cos[2*(e + f*x)]^2*Sqrt[Sec[e + f*x]]*((1 + Sec[e + f*x])^m/2 + (I/4)*(1 + Sec[e + f*x])^m*Sin[2*(e + f*x)]) + Cos[2*(e + f*x)]*Sqrt[Sec[e + f*x]]*((1 + Sec[e + f*x])^m/4 + ((1 + Sec[e + f*x])^m*Sin[2*(e + f*x)]^2)/4) + Sqrt[Sec[e + f*x]]*((-1/4*I)*(1 + Sec[e + f*x])^m*Sin[2*(e + f*x)] + ((1 + Sec[e + f*x])^m*Sin[2*(e + f*x)]^2)/2 + (I/4)*(1 + Sec[e + f*x])^m*Sin[2*(e + f*x)]^3))*Tan[(e + f*x)/2]*(-(AppellF1[3/2, -1/2 + m, 5/2, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*(Cos[e + f*x]*Sec[(e + f*x)/2]^2)^(1/2 + m)*Tan[(e + f*x)/2]^2) - (9*AppellF1[1/2, -1/2 + m, 5/2, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Cos[e + f*x])/((Sec[(e + f*x)/2]^2)^(3/2)*(-3*AppellF1[1/2, -1/2 + m, 5/2, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (5*AppellF1[3/2, -1/2 + m, 7/2, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (1 - 2*m)*AppellF1[3/2, 1/2 + m, 5/2, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2))))/(3*f*(d*Sec[e + f*x])^(3/2)*(1 + Sec[e + f*x])^m*((2^m*Sec[(e + f*x)/2]^2*(Cos[(e + f*x)/2]^2*Sec[e + f*x])^(1/2 + m)*(-(AppellF1[3/2, -1/2 + m, 5/2, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*(Cos[e + f*x]*Sec[(e + f*x)/2]^2)^(1/2 + m)*Tan[(e + f*x)/2]^2) - (9*AppellF1[1/2, -1/2 + m, 5/2, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Cos[e + f*x])/((Sec[(e + f*x)/2]^2)^(3/2)*(-3*AppellF1[1/2, -1/2 + m, 5/2, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (5*AppellF1[3/2, -1/2 + m, 7/2, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (1 - 2*m)*AppellF1[3/2, 1/2 + m, 5/2, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2))))/3 + (2^(1 + m)*(Cos[(e + f*x)/2]^2*Sec[e + f*x])^(1/2 + m)*Tan[(e + f*x)/2]*(-(AppellF1[3/2, -1/2 + m, 5/2, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*(Cos[e + f*x]*Sec[(e + f*x)/2]^2)^(1/2 + m)*Tan[(e + f*x)/2]) - (Cos[e + f*x]*Sec[(e + f*x)/2]^2)^(1/2 + m)*Tan[(e + f*x)/2]^2*((-3*AppellF1[5/2, -1/2 + m, 7/2, 7/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/2 + (3*(-1/2 + m)*AppellF1[5/2, 1/2 + m, 5/2, 7/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/5) - (1/2 + m)*AppellF1[3/2, -1/2 + m, 5/2, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*(Cos[e + f*x]*Sec[(e + f*x)/2]^2)^(-1/2 + m)*Tan[(e + f*x)/2]^2*(-(Sec[(e + f*x)/2]^2*Sin[e + f*x]) + Cos[e + f*x]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2]) + (9*AppellF1[1/2, -1/2 + m, 5/2, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sin[e + f*x])/((Sec[(e + f*x)/2]^2)^(3/2)*(-3*AppellF1[1/2, -1/2 + m, 5/2, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (5*AppellF1[3/2, -1/2 + m, 7/2, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (1 - 2*m)*AppellF1[3/2, 1/2 + m, 5/2, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2)) + (27*AppellF1[1/2, -1/2 + m, 5/2, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Cos[e + f*x]*Tan[(e + f*x)/2])/(2*(Sec[(e + f*x)/2]^2)^(3/2)*(-3*AppellF1[1/2, -1/2 + m, 5/2, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (5*AppellF1[3/2, -1/2 + m, 7/2, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (1 - 2*m)*AppellF1[3/2, 1/2 + m, 5/2, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2)) - (9*Cos[e + f*x]*((-5*AppellF1[3/2, -1/2 + m, 7/2, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/6 + ((-1/2 + m)*AppellF1[3/2, 1/2 + m, 5/2, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/3))/((Sec[(e + f*x)/2]^2)^(3/2)*(-3*AppellF1[1/2, -1/2 + m, 5/2, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (5*AppellF1[3/2, -1/2 + m, 7/2, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (1 - 2*m)*AppellF1[3/2, 1/2 + m, 5/2, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2)) + (9*AppellF1[1/2, -1/2 + m, 5/2, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Cos[e + f*x]*((5*AppellF1[3/2, -1/2 + m, 7/2, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (1 - 2*m)*AppellF1[3/2, 1/2 + m, 5/2, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2] - 3*((-5*AppellF1[3/2, -1/2 + m, 7/2, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/6 + ((-1/2 + m)*AppellF1[3/2, 1/2 + m, 5/2, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/3) + Tan[(e + f*x)/2]^2*(5*((-21*AppellF1[5/2, -1/2 + m, 9/2, 7/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/10 + (3*(-1/2 + m)*AppellF1[5/2, 1/2 + m, 7/2, 7/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/5) + (1 - 2*m)*((-3*AppellF1[5/2, 1/2 + m, 7/2, 7/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/2 + (3*(1/2 + m)*AppellF1[5/2, 3/2 + m, 5/2, 7/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/5))))/((Sec[(e + f*x)/2]^2)^(3/2)*(-3*AppellF1[1/2, -1/2 + m, 5/2, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (5*AppellF1[3/2, -1/2 + m, 7/2, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (1 - 2*m)*AppellF1[3/2, 1/2 + m, 5/2, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2)^2)))/3 + (2^(1 + m)*(1/2 + m)*(Cos[(e + f*x)/2]^2*Sec[e + f*x])^(-1/2 + m)*Tan[(e + f*x)/2]*(-(AppellF1[3/2, -1/2 + m, 5/2, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*(Cos[e + f*x]*Sec[(e + f*x)/2]^2)^(1/2 + m)*Tan[(e + f*x)/2]^2) - (9*AppellF1[1/2, -1/2 + m, 5/2, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Cos[e + f*x])/((Sec[(e + f*x)/2]^2)^(3/2)*(-3*AppellF1[1/2, -1/2 + m, 5/2, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (5*AppellF1[3/2, -1/2 + m, 7/2, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (1 - 2*m)*AppellF1[3/2, 1/2 + m, 5/2, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2)))*(-(Cos[(e + f*x)/2]*Sec[e + f*x]*Sin[(e + f*x)/2]) + Cos[(e + f*x)/2]^2*Sec[e + f*x]*Tan[e + f*x]))/3))","C",0
351,1,490,111,6.1024899,"\int \cos ^{\frac{7}{2}}(c+d x) (a+a \sec (c+d x)) \, dx","Integrate[Cos[c + d*x]^(7/2)*(a + a*Sec[c + d*x]),x]","a \left(-\frac{3 \csc (c) (\cos (c+d x)+1) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right) \left(\frac{\tan (c) \sin \left(\tan ^{-1}(\tan (c))+d x\right) \, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right)}{\sqrt{\tan ^2(c)+1} \sqrt{1-\cos \left(\tan ^{-1}(\tan (c))+d x\right)} \sqrt{\cos \left(\tan ^{-1}(\tan (c))+d x\right)+1} \sqrt{\cos (c) \sqrt{\tan ^2(c)+1} \cos \left(\tan ^{-1}(\tan (c))+d x\right)}}-\frac{\frac{\tan (c) \sin \left(\tan ^{-1}(\tan (c))+d x\right)}{\sqrt{\tan ^2(c)+1}}+\frac{2 \cos ^2(c) \sqrt{\tan ^2(c)+1} \cos \left(\tan ^{-1}(\tan (c))+d x\right)}{\sin ^2(c)+\cos ^2(c)}}{\sqrt{\cos (c) \sqrt{\tan ^2(c)+1} \cos \left(\tan ^{-1}(\tan (c))+d x\right)}}\right)}{10 d}-\frac{5 \csc (c) (\cos (c+d x)+1) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin (c) \left(-\sqrt{\cot ^2(c)+1}\right) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec \left(d x-\tan ^{-1}(\cot (c))\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right)}{21 d \sqrt{\cot ^2(c)+1}}+\sqrt{\cos (c+d x)} (\cos (c+d x)+1) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right) \left(\frac{23 \sin (c) \cos (d x)}{84 d}+\frac{\sin (2 c) \cos (2 d x)}{10 d}+\frac{\sin (3 c) \cos (3 d x)}{28 d}+\frac{23 \cos (c) \sin (d x)}{84 d}+\frac{\cos (2 c) \sin (2 d x)}{10 d}+\frac{\cos (3 c) \sin (3 d x)}{28 d}-\frac{3 \cot (c)}{5 d}\right)\right)","\frac{10 a F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{6 a E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 a \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{7 d}+\frac{2 a \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 d}+\frac{10 a \sin (c+d x) \sqrt{\cos (c+d x)}}{21 d}",1,"a*(Sqrt[Cos[c + d*x]]*(1 + Cos[c + d*x])*Sec[c/2 + (d*x)/2]^2*((-3*Cot[c])/(5*d) + (23*Cos[d*x]*Sin[c])/(84*d) + (Cos[2*d*x]*Sin[2*c])/(10*d) + (Cos[3*d*x]*Sin[3*c])/(28*d) + (23*Cos[c]*Sin[d*x])/(84*d) + (Cos[2*c]*Sin[2*d*x])/(10*d) + (Cos[3*c]*Sin[3*d*x])/(28*d)) - (5*(1 + Cos[c + d*x])*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^2*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(21*d*Sqrt[1 + Cot[c]^2]) - (3*(1 + Cos[c + d*x])*Csc[c]*Sec[c/2 + (d*x)/2]^2*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(10*d))","C",0
352,1,232,87,5.6737235,"\int \cos ^{\frac{5}{2}}(c+d x) (a+a \sec (c+d x)) \, dx","Integrate[Cos[c + d*x]^(5/2)*(a + a*Sec[c + d*x]),x]","\frac{a (\cos (c+d x)+1) \sec ^2\left(\frac{1}{2} (c+d x)\right) \left(-18 \cos (c) \sqrt{\sec ^2(c)} \sqrt{\sin ^2\left(\tan ^{-1}(\tan (c))+d x\right)} \csc \left(\tan ^{-1}(\tan (c))+d x\right) \, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right)-20 \sin (c) \sqrt{\csc ^2(c)} \cos (c+d x) \sqrt{\cos ^2\left(d x-\tan ^{-1}(\cot (c))\right)} \sec \left(d x-\tan ^{-1}(\cot (c))\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right)+2 \cos (c+d x) (10 \sin (c+d x)+3 \sin (2 (c+d x))-18 \cot (c))+\frac{9 \csc (c) \sec (c) \left(3 \cos \left(c-\tan ^{-1}(\tan (c))-d x\right)+\cos \left(c+\tan ^{-1}(\tan (c))+d x\right)\right)}{\sqrt{\sec ^2(c)}}\right)}{60 d \sqrt{\cos (c+d x)}}","\frac{2 a F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{6 a E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 a \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 d}+\frac{2 a \sin (c+d x) \sqrt{\cos (c+d x)}}{3 d}",1,"(a*(1 + Cos[c + d*x])*Sec[(c + d*x)/2]^2*((9*(3*Cos[c - d*x - ArcTan[Tan[c]]] + Cos[c + d*x + ArcTan[Tan[c]]])*Csc[c]*Sec[c])/Sqrt[Sec[c]^2] - 20*Cos[c + d*x]*Sqrt[Cos[d*x - ArcTan[Cot[c]]]^2]*Sqrt[Csc[c]^2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[d*x - ArcTan[Cot[c]]]*Sin[c] + 2*Cos[c + d*x]*(-18*Cot[c] + 10*Sin[c + d*x] + 3*Sin[2*(c + d*x)]) - 18*Cos[c]*Csc[d*x + ArcTan[Tan[c]]]*HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sqrt[Sec[c]^2]*Sqrt[Sin[d*x + ArcTan[Tan[c]]]^2]))/(60*d*Sqrt[Cos[c + d*x]])","C",0
353,1,222,61,5.2141468,"\int \cos ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x)) \, dx","Integrate[Cos[c + d*x]^(3/2)*(a + a*Sec[c + d*x]),x]","\frac{a (\cos (c+d x)+1) \sec ^2\left(\frac{1}{2} (c+d x)\right) \left(-6 \cos (c) \sqrt{\sec ^2(c)} \sqrt{\sin ^2\left(\tan ^{-1}(\tan (c))+d x\right)} \csc \left(\tan ^{-1}(\tan (c))+d x\right) \, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right)-4 \sin (c) \sqrt{\csc ^2(c)} \cos (c+d x) \sqrt{\cos ^2\left(d x-\tan ^{-1}(\cot (c))\right)} \sec \left(d x-\tan ^{-1}(\cot (c))\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right)-4 \cos (c+d x) (3 \cot (c)-\sin (c+d x))+\frac{3 \csc (c) \sec (c) \left(3 \cos \left(c-\tan ^{-1}(\tan (c))-d x\right)+\cos \left(c+\tan ^{-1}(\tan (c))+d x\right)\right)}{\sqrt{\sec ^2(c)}}\right)}{12 d \sqrt{\cos (c+d x)}}","\frac{2 a F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{2 a E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 a \sin (c+d x) \sqrt{\cos (c+d x)}}{3 d}",1,"(a*(1 + Cos[c + d*x])*Sec[(c + d*x)/2]^2*((3*(3*Cos[c - d*x - ArcTan[Tan[c]]] + Cos[c + d*x + ArcTan[Tan[c]]])*Csc[c]*Sec[c])/Sqrt[Sec[c]^2] - 4*Cos[c + d*x]*Sqrt[Cos[d*x - ArcTan[Cot[c]]]^2]*Sqrt[Csc[c]^2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[d*x - ArcTan[Cot[c]]]*Sin[c] - 4*Cos[c + d*x]*(3*Cot[c] - Sin[c + d*x]) - 6*Cos[c]*Csc[d*x + ArcTan[Tan[c]]]*HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sqrt[Sec[c]^2]*Sqrt[Sin[d*x + ArcTan[Tan[c]]]^2]))/(12*d*Sqrt[Cos[c + d*x]])","C",0
354,1,155,35,1.8549506,"\int \sqrt{\cos (c+d x)} (a+a \sec (c+d x)) \, dx","Integrate[Sqrt[Cos[c + d*x]]*(a + a*Sec[c + d*x]),x]","\frac{a \sqrt{\cos (c+d x)} (\cos (c+d x)+1) \sec ^2\left(\frac{1}{2} (c+d x)\right) \left(-\frac{\tan \left(\tan ^{-1}(\tan (c))+d x\right) \, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right)}{\sqrt{\sin ^2\left(\tan ^{-1}(\tan (c))+d x\right)}}-2 \sin (c) \sqrt{\csc ^2(c)} \sqrt{\cos ^2\left(d x-\tan ^{-1}(\cot (c))\right)} \sec \left(d x-\tan ^{-1}(\cot (c))\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right)+\tan \left(\tan ^{-1}(\tan (c))+d x\right)\right)}{2 d}","\frac{2 a F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 a E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}",1,"(a*Sqrt[Cos[c + d*x]]*(1 + Cos[c + d*x])*Sec[(c + d*x)/2]^2*(-2*Sqrt[Cos[d*x - ArcTan[Cot[c]]]^2]*Sqrt[Csc[c]^2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[d*x - ArcTan[Cot[c]]]*Sin[c] + Tan[d*x + ArcTan[Tan[c]]] - (HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Tan[d*x + ArcTan[Tan[c]]])/Sqrt[Sin[d*x + ArcTan[Tan[c]]]^2]))/(2*d)","C",0
355,1,209,57,4.9707537,"\int \frac{a+a \sec (c+d x)}{\sqrt{\cos (c+d x)}} \, dx","Integrate[(a + a*Sec[c + d*x])/Sqrt[Cos[c + d*x]],x]","\frac{a (\cos (c+d x)+1) \sec ^2\left(\frac{1}{2} (c+d x)\right) \left(2 \cos (c) \sqrt{\sec ^2(c)} \sqrt{\sin ^2\left(\tan ^{-1}(\tan (c))+d x\right)} \csc \left(\tan ^{-1}(\tan (c))+d x\right) \, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right)-4 \sin (c) \sqrt{\csc ^2(c)} \cos (c+d x) \sqrt{\cos ^2\left(d x-\tan ^{-1}(\cot (c))\right)} \sec \left(d x-\tan ^{-1}(\cot (c))\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right)+4 \csc (c) \cos (d x)-\frac{\csc (c) \sec (c) \left(3 \cos \left(c-\tan ^{-1}(\tan (c))-d x\right)+\cos \left(c+\tan ^{-1}(\tan (c))+d x\right)\right)}{\sqrt{\sec ^2(c)}}\right)}{4 d \sqrt{\cos (c+d x)}}","\frac{2 a F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}-\frac{2 a E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 a \sin (c+d x)}{d \sqrt{\cos (c+d x)}}",1,"(a*(1 + Cos[c + d*x])*Sec[(c + d*x)/2]^2*(4*Cos[d*x]*Csc[c] - ((3*Cos[c - d*x - ArcTan[Tan[c]]] + Cos[c + d*x + ArcTan[Tan[c]]])*Csc[c]*Sec[c])/Sqrt[Sec[c]^2] - 4*Cos[c + d*x]*Sqrt[Cos[d*x - ArcTan[Cot[c]]]^2]*Sqrt[Csc[c]^2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[d*x - ArcTan[Cot[c]]]*Sin[c] + 2*Cos[c]*Csc[d*x + ArcTan[Tan[c]]]*HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sqrt[Sec[c]^2]*Sqrt[Sin[d*x + ArcTan[Tan[c]]]^2]))/(4*d*Sqrt[Cos[c + d*x]])","C",0
356,1,444,83,6.1340302,"\int \frac{a+a \sec (c+d x)}{\cos ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[(a + a*Sec[c + d*x])/Cos[c + d*x]^(3/2),x]","a \left(\frac{\csc (c) (\cos (c+d x)+1) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right) \left(\frac{\tan (c) \sin \left(\tan ^{-1}(\tan (c))+d x\right) \, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right)}{\sqrt{\tan ^2(c)+1} \sqrt{1-\cos \left(\tan ^{-1}(\tan (c))+d x\right)} \sqrt{\cos \left(\tan ^{-1}(\tan (c))+d x\right)+1} \sqrt{\cos (c) \sqrt{\tan ^2(c)+1} \cos \left(\tan ^{-1}(\tan (c))+d x\right)}}-\frac{\frac{\tan (c) \sin \left(\tan ^{-1}(\tan (c))+d x\right)}{\sqrt{\tan ^2(c)+1}}+\frac{2 \cos ^2(c) \sqrt{\tan ^2(c)+1} \cos \left(\tan ^{-1}(\tan (c))+d x\right)}{\sin ^2(c)+\cos ^2(c)}}{\sqrt{\cos (c) \sqrt{\tan ^2(c)+1} \cos \left(\tan ^{-1}(\tan (c))+d x\right)}}\right)}{2 d}-\frac{\csc (c) (\cos (c+d x)+1) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin (c) \left(-\sqrt{\cot ^2(c)+1}\right) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec \left(d x-\tan ^{-1}(\cot (c))\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right)}{3 d \sqrt{\cot ^2(c)+1}}+\sqrt{\cos (c+d x)} (\cos (c+d x)+1) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right) \left(\frac{\sec (c) \sin (d x) \sec ^2(c+d x)}{3 d}+\frac{\sec (c) (\sin (c)+3 \sin (d x)) \sec (c+d x)}{3 d}+\frac{\csc (c) \sec (c)}{d}\right)\right)","\frac{2 a F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{2 a E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 a \sin (c+d x)}{3 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 a \sin (c+d x)}{d \sqrt{\cos (c+d x)}}",1,"a*(Sqrt[Cos[c + d*x]]*(1 + Cos[c + d*x])*Sec[c/2 + (d*x)/2]^2*((Csc[c]*Sec[c])/d + (Sec[c]*Sec[c + d*x]^2*Sin[d*x])/(3*d) + (Sec[c]*Sec[c + d*x]*(Sin[c] + 3*Sin[d*x]))/(3*d)) - ((1 + Cos[c + d*x])*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^2*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*Sqrt[1 + Cot[c]^2]) + ((1 + Cos[c + d*x])*Csc[c]*Sec[c/2 + (d*x)/2]^2*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(2*d))","C",0
357,1,477,111,6.1619557,"\int \frac{a+a \sec (c+d x)}{\cos ^{\frac{5}{2}}(c+d x)} \, dx","Integrate[(a + a*Sec[c + d*x])/Cos[c + d*x]^(5/2),x]","a \left(\frac{3 \csc (c) (\cos (c+d x)+1) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right) \left(\frac{\tan (c) \sin \left(\tan ^{-1}(\tan (c))+d x\right) \, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right)}{\sqrt{\tan ^2(c)+1} \sqrt{1-\cos \left(\tan ^{-1}(\tan (c))+d x\right)} \sqrt{\cos \left(\tan ^{-1}(\tan (c))+d x\right)+1} \sqrt{\cos (c) \sqrt{\tan ^2(c)+1} \cos \left(\tan ^{-1}(\tan (c))+d x\right)}}-\frac{\frac{\tan (c) \sin \left(\tan ^{-1}(\tan (c))+d x\right)}{\sqrt{\tan ^2(c)+1}}+\frac{2 \cos ^2(c) \sqrt{\tan ^2(c)+1} \cos \left(\tan ^{-1}(\tan (c))+d x\right)}{\sin ^2(c)+\cos ^2(c)}}{\sqrt{\cos (c) \sqrt{\tan ^2(c)+1} \cos \left(\tan ^{-1}(\tan (c))+d x\right)}}\right)}{10 d}-\frac{\csc (c) (\cos (c+d x)+1) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin (c) \left(-\sqrt{\cot ^2(c)+1}\right) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec \left(d x-\tan ^{-1}(\cot (c))\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right)}{3 d \sqrt{\cot ^2(c)+1}}+\sqrt{\cos (c+d x)} (\cos (c+d x)+1) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right) \left(\frac{\sec (c) \sin (d x) \sec ^3(c+d x)}{5 d}+\frac{\sec (c) (3 \sin (c)+5 \sin (d x)) \sec ^2(c+d x)}{15 d}+\frac{\sec (c) (5 \sin (c)+9 \sin (d x)) \sec (c+d x)}{15 d}+\frac{3 \csc (c) \sec (c)}{5 d}\right)\right)","\frac{2 a F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{6 a E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 a \sin (c+d x)}{3 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 a \sin (c+d x)}{5 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{6 a \sin (c+d x)}{5 d \sqrt{\cos (c+d x)}}",1,"a*(Sqrt[Cos[c + d*x]]*(1 + Cos[c + d*x])*Sec[c/2 + (d*x)/2]^2*((3*Csc[c]*Sec[c])/(5*d) + (Sec[c]*Sec[c + d*x]^3*Sin[d*x])/(5*d) + (Sec[c]*Sec[c + d*x]^2*(3*Sin[c] + 5*Sin[d*x]))/(15*d) + (Sec[c]*Sec[c + d*x]*(5*Sin[c] + 9*Sin[d*x]))/(15*d)) - ((1 + Cos[c + d*x])*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^2*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*Sqrt[1 + Cot[c]^2]) + (3*(1 + Cos[c + d*x])*Csc[c]*Sec[c/2 + (d*x)/2]^2*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(10*d))","C",0
358,1,294,135,4.6069092,"\int \frac{a+a \sec (c+d x)}{\cos ^{\frac{7}{2}}(c+d x)} \, dx","Integrate[(a + a*Sec[c + d*x])/Cos[c + d*x]^(7/2),x]","\frac{a (\cos (c+d x)+1) \sec ^2\left(\frac{1}{2} (c+d x)\right) \left(-\frac{126 \sec (c) \cos ^3(c+d x) \left(\csc (c) \sqrt{\sin ^2\left(\tan ^{-1}(\tan (c))+d x\right)} \left(3 \cos \left(c-\tan ^{-1}(\tan (c))-d x\right)+\cos \left(c+\tan ^{-1}(\tan (c))+d x\right)\right)-2 \sin \left(\tan ^{-1}(\tan (c))+d x\right) \, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right)\right)}{\sqrt{\sec ^2(c)} \sqrt{\sin ^2\left(\tan ^{-1}(\tan (c))+d x\right)}}-200 \sin (c) \sqrt{\csc ^2(c)} \cos ^4(c+d x) \sqrt{\cos ^2\left(d x-\tan ^{-1}(\cot (c))\right)} \sec \left(d x-\tan ^{-1}(\cot (c))\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right)+\csc (c) (-85 \cos (2 c+d x)+231 \cos (c+2 d x)+21 \cos (3 c+2 d x)+25 \cos (2 c+3 d x)-25 \cos (4 c+3 d x)+63 \cos (3 c+4 d x)+189 \cos (c)+85 \cos (d x))\right)}{840 d \cos ^{\frac{7}{2}}(c+d x)}","\frac{10 a F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}-\frac{6 a E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{10 a \sin (c+d x)}{21 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 a \sin (c+d x)}{5 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 a \sin (c+d x)}{7 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{6 a \sin (c+d x)}{5 d \sqrt{\cos (c+d x)}}",1,"(a*(1 + Cos[c + d*x])*Sec[(c + d*x)/2]^2*((189*Cos[c] + 85*Cos[d*x] - 85*Cos[2*c + d*x] + 231*Cos[c + 2*d*x] + 21*Cos[3*c + 2*d*x] + 25*Cos[2*c + 3*d*x] - 25*Cos[4*c + 3*d*x] + 63*Cos[3*c + 4*d*x])*Csc[c] - 200*Cos[c + d*x]^4*Sqrt[Cos[d*x - ArcTan[Cot[c]]]^2]*Sqrt[Csc[c]^2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[d*x - ArcTan[Cot[c]]]*Sin[c] - (126*Cos[c + d*x]^3*Sec[c]*(-2*HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]] + (3*Cos[c - d*x - ArcTan[Tan[c]]] + Cos[c + d*x + ArcTan[Tan[c]]])*Csc[c]*Sqrt[Sin[d*x + ArcTan[Tan[c]]]^2]))/(Sqrt[Sec[c]^2]*Sqrt[Sin[d*x + ArcTan[Tan[c]]]^2])))/(840*d*Cos[c + d*x]^(7/2))","C",0
359,1,548,147,6.1227248,"\int \cos ^{\frac{9}{2}}(c+d x) (a+a \sec (c+d x))^2 \, dx","Integrate[Cos[c + d*x]^(9/2)*(a + a*Sec[c + d*x])^2,x]","-\frac{4 \csc (c) \cos ^2(c+d x) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right) (a \sec (c+d x)+a)^2 \left(\frac{\tan (c) \sin \left(\tan ^{-1}(\tan (c))+d x\right) \, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right)}{\sqrt{\tan ^2(c)+1} \sqrt{1-\cos \left(\tan ^{-1}(\tan (c))+d x\right)} \sqrt{\cos \left(\tan ^{-1}(\tan (c))+d x\right)+1} \sqrt{\cos (c) \sqrt{\tan ^2(c)+1} \cos \left(\tan ^{-1}(\tan (c))+d x\right)}}-\frac{\frac{\tan (c) \sin \left(\tan ^{-1}(\tan (c))+d x\right)}{\sqrt{\tan ^2(c)+1}}+\frac{2 \cos ^2(c) \sqrt{\tan ^2(c)+1} \cos \left(\tan ^{-1}(\tan (c))+d x\right)}{\sin ^2(c)+\cos ^2(c)}}{\sqrt{\cos (c) \sqrt{\tan ^2(c)+1} \cos \left(\tan ^{-1}(\tan (c))+d x\right)}}\right)}{15 d}-\frac{5 \csc (c) \cos ^2(c+d x) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right) (a \sec (c+d x)+a)^2 \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin (c) \left(-\sqrt{\cot ^2(c)+1}\right) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec \left(d x-\tan ^{-1}(\cot (c))\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right)}{21 d \sqrt{\cot ^2(c)+1}}+\cos ^{\frac{5}{2}}(c+d x) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right) (a \sec (c+d x)+a)^2 \left(\frac{23 \sin (c) \cos (d x)}{84 d}+\frac{37 \sin (2 c) \cos (2 d x)}{360 d}+\frac{\sin (3 c) \cos (3 d x)}{28 d}+\frac{\sin (4 c) \cos (4 d x)}{144 d}+\frac{23 \cos (c) \sin (d x)}{84 d}+\frac{37 \cos (2 c) \sin (2 d x)}{360 d}+\frac{\cos (3 c) \sin (3 d x)}{28 d}+\frac{\cos (4 c) \sin (4 d x)}{144 d}-\frac{8 \cot (c)}{15 d}\right)","\frac{20 a^2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{32 a^2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{2 a^2 \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x)}{9 d}+\frac{4 a^2 \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{7 d}+\frac{32 a^2 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{45 d}+\frac{20 a^2 \sin (c+d x) \sqrt{\cos (c+d x)}}{21 d}",1,"Cos[c + d*x]^(5/2)*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2*((-8*Cot[c])/(15*d) + (23*Cos[d*x]*Sin[c])/(84*d) + (37*Cos[2*d*x]*Sin[2*c])/(360*d) + (Cos[3*d*x]*Sin[3*c])/(28*d) + (Cos[4*d*x]*Sin[4*c])/(144*d) + (23*Cos[c]*Sin[d*x])/(84*d) + (37*Cos[2*c]*Sin[2*d*x])/(360*d) + (Cos[3*c]*Sin[3*d*x])/(28*d) + (Cos[4*c]*Sin[4*d*x])/(144*d)) - (5*Cos[c + d*x]^2*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(21*d*Sqrt[1 + Cot[c]^2]) - (4*Cos[c + d*x]^2*Csc[c]*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(15*d)","C",0
360,1,516,121,6.1171612,"\int \cos ^{\frac{7}{2}}(c+d x) (a+a \sec (c+d x))^2 \, dx","Integrate[Cos[c + d*x]^(7/2)*(a + a*Sec[c + d*x])^2,x]","-\frac{3 \csc (c) \cos ^2(c+d x) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right) (a \sec (c+d x)+a)^2 \left(\frac{\tan (c) \sin \left(\tan ^{-1}(\tan (c))+d x\right) \, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right)}{\sqrt{\tan ^2(c)+1} \sqrt{1-\cos \left(\tan ^{-1}(\tan (c))+d x\right)} \sqrt{\cos \left(\tan ^{-1}(\tan (c))+d x\right)+1} \sqrt{\cos (c) \sqrt{\tan ^2(c)+1} \cos \left(\tan ^{-1}(\tan (c))+d x\right)}}-\frac{\frac{\tan (c) \sin \left(\tan ^{-1}(\tan (c))+d x\right)}{\sqrt{\tan ^2(c)+1}}+\frac{2 \cos ^2(c) \sqrt{\tan ^2(c)+1} \cos \left(\tan ^{-1}(\tan (c))+d x\right)}{\sin ^2(c)+\cos ^2(c)}}{\sqrt{\cos (c) \sqrt{\tan ^2(c)+1} \cos \left(\tan ^{-1}(\tan (c))+d x\right)}}\right)}{10 d}-\frac{2 \csc (c) \cos ^2(c+d x) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right) (a \sec (c+d x)+a)^2 \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin (c) \left(-\sqrt{\cot ^2(c)+1}\right) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec \left(d x-\tan ^{-1}(\cot (c))\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right)}{7 d \sqrt{\cot ^2(c)+1}}+\cos ^{\frac{5}{2}}(c+d x) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right) (a \sec (c+d x)+a)^2 \left(\frac{17 \sin (c) \cos (d x)}{56 d}+\frac{\sin (2 c) \cos (2 d x)}{10 d}+\frac{\sin (3 c) \cos (3 d x)}{56 d}+\frac{17 \cos (c) \sin (d x)}{56 d}+\frac{\cos (2 c) \sin (2 d x)}{10 d}+\frac{\cos (3 c) \sin (3 d x)}{56 d}-\frac{3 \cot (c)}{5 d}\right)","\frac{8 a^2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{7 d}+\frac{12 a^2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 a^2 \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{7 d}+\frac{4 a^2 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 d}+\frac{8 a^2 \sin (c+d x) \sqrt{\cos (c+d x)}}{7 d}",1,"Cos[c + d*x]^(5/2)*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2*((-3*Cot[c])/(5*d) + (17*Cos[d*x]*Sin[c])/(56*d) + (Cos[2*d*x]*Sin[2*c])/(10*d) + (Cos[3*d*x]*Sin[3*c])/(56*d) + (17*Cos[c]*Sin[d*x])/(56*d) + (Cos[2*c]*Sin[2*d*x])/(10*d) + (Cos[3*c]*Sin[3*d*x])/(56*d)) - (2*Cos[c + d*x]^2*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(7*d*Sqrt[1 + Cot[c]^2]) - (3*Cos[c + d*x]^2*Csc[c]*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(10*d)","C",0
361,1,235,95,5.9316101,"\int \cos ^{\frac{5}{2}}(c+d x) (a+a \sec (c+d x))^2 \, dx","Integrate[Cos[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^2,x]","\frac{a^2 (\cos (c+d x)+1)^2 \sec ^4\left(\frac{1}{2} (c+d x)\right) \left(-24 \cos (c) \sqrt{\sec ^2(c)} \sqrt{\sin ^2\left(\tan ^{-1}(\tan (c))+d x\right)} \csc \left(\tan ^{-1}(\tan (c))+d x\right) \, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right)-20 \sin (c) \sqrt{\csc ^2(c)} \cos (c+d x) \sqrt{\cos ^2\left(d x-\tan ^{-1}(\cot (c))\right)} \sec \left(d x-\tan ^{-1}(\cot (c))\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right)+\cos (c+d x) (20 \sin (c+d x)+3 \sin (2 (c+d x))-48 \cot (c))+\frac{12 \csc (c) \sec (c) \left(3 \cos \left(c-\tan ^{-1}(\tan (c))-d x\right)+\cos \left(c+\tan ^{-1}(\tan (c))+d x\right)\right)}{\sqrt{\sec ^2(c)}}\right)}{60 d \sqrt{\cos (c+d x)}}","\frac{4 a^2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{16 a^2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 a^2 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 d}+\frac{4 a^2 \sin (c+d x) \sqrt{\cos (c+d x)}}{3 d}",1,"(a^2*(1 + Cos[c + d*x])^2*Sec[(c + d*x)/2]^4*((12*(3*Cos[c - d*x - ArcTan[Tan[c]]] + Cos[c + d*x + ArcTan[Tan[c]]])*Csc[c]*Sec[c])/Sqrt[Sec[c]^2] - 20*Cos[c + d*x]*Sqrt[Cos[d*x - ArcTan[Cot[c]]]^2]*Sqrt[Csc[c]^2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[d*x - ArcTan[Cot[c]]]*Sin[c] + Cos[c + d*x]*(-48*Cot[c] + 20*Sin[c + d*x] + 3*Sin[2*(c + d*x)]) - 24*Cos[c]*Csc[d*x + ArcTan[Tan[c]]]*HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sqrt[Sec[c]^2]*Sqrt[Sin[d*x + ArcTan[Tan[c]]]^2]))/(60*d*Sqrt[Cos[c + d*x]])","C",0
362,1,224,67,5.5580336,"\int \cos ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x))^2 \, dx","Integrate[Cos[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^2,x]","\frac{a^2 (\cos (c+d x)+1)^2 \sec ^4\left(\frac{1}{2} (c+d x)\right) \left(-6 \cos (c) \sqrt{\sec ^2(c)} \sqrt{\sin ^2\left(\tan ^{-1}(\tan (c))+d x\right)} \csc \left(\tan ^{-1}(\tan (c))+d x\right) \, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right)-8 \sin (c) \sqrt{\csc ^2(c)} \cos (c+d x) \sqrt{\cos ^2\left(d x-\tan ^{-1}(\cot (c))\right)} \sec \left(d x-\tan ^{-1}(\cot (c))\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right)+2 \cos (c+d x) (\sin (c+d x)-6 \cot (c))+\frac{3 \csc (c) \sec (c) \left(3 \cos \left(c-\tan ^{-1}(\tan (c))-d x\right)+\cos \left(c+\tan ^{-1}(\tan (c))+d x\right)\right)}{\sqrt{\sec ^2(c)}}\right)}{12 d \sqrt{\cos (c+d x)}}","\frac{8 a^2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{4 a^2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 a^2 \sin (c+d x) \sqrt{\cos (c+d x)}}{3 d}",1,"(a^2*(1 + Cos[c + d*x])^2*Sec[(c + d*x)/2]^4*((3*(3*Cos[c - d*x - ArcTan[Tan[c]]] + Cos[c + d*x + ArcTan[Tan[c]]])*Csc[c]*Sec[c])/Sqrt[Sec[c]^2] - 8*Cos[c + d*x]*Sqrt[Cos[d*x - ArcTan[Cot[c]]]^2]*Sqrt[Csc[c]^2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[d*x - ArcTan[Cot[c]]]*Sin[c] + 2*Cos[c + d*x]*(-6*Cot[c] + Sin[c + d*x]) - 6*Cos[c]*Csc[d*x + ArcTan[Tan[c]]]*HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sqrt[Sec[c]^2]*Sqrt[Sin[d*x + ArcTan[Tan[c]]]^2]))/(12*d*Sqrt[Cos[c + d*x]])","C",0
363,1,39,44,0.1613387,"\int \sqrt{\cos (c+d x)} (a+a \sec (c+d x))^2 \, dx","Integrate[Sqrt[Cos[c + d*x]]*(a + a*Sec[c + d*x])^2,x]","\frac{2 a^2 \left(2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+\frac{\sin (c+d x)}{\sqrt{\cos (c+d x)}}\right)}{d}","\frac{4 a^2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 a^2 \sin (c+d x)}{d \sqrt{\cos (c+d x)}}",1,"(2*a^2*(2*EllipticF[(c + d*x)/2, 2] + Sin[c + d*x]/Sqrt[Cos[c + d*x]]))/d","A",1
364,1,470,91,6.1545297,"\int \frac{(a+a \sec (c+d x))^2}{\sqrt{\cos (c+d x)}} \, dx","Integrate[(a + a*Sec[c + d*x])^2/Sqrt[Cos[c + d*x]],x]","\frac{\csc (c) \cos ^2(c+d x) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right) (a \sec (c+d x)+a)^2 \left(\frac{\tan (c) \sin \left(\tan ^{-1}(\tan (c))+d x\right) \, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right)}{\sqrt{\tan ^2(c)+1} \sqrt{1-\cos \left(\tan ^{-1}(\tan (c))+d x\right)} \sqrt{\cos \left(\tan ^{-1}(\tan (c))+d x\right)+1} \sqrt{\cos (c) \sqrt{\tan ^2(c)+1} \cos \left(\tan ^{-1}(\tan (c))+d x\right)}}-\frac{\frac{\tan (c) \sin \left(\tan ^{-1}(\tan (c))+d x\right)}{\sqrt{\tan ^2(c)+1}}+\frac{2 \cos ^2(c) \sqrt{\tan ^2(c)+1} \cos \left(\tan ^{-1}(\tan (c))+d x\right)}{\sin ^2(c)+\cos ^2(c)}}{\sqrt{\cos (c) \sqrt{\tan ^2(c)+1} \cos \left(\tan ^{-1}(\tan (c))+d x\right)}}\right)}{2 d}-\frac{2 \csc (c) \cos ^2(c+d x) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right) (a \sec (c+d x)+a)^2 \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin (c) \left(-\sqrt{\cot ^2(c)+1}\right) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec \left(d x-\tan ^{-1}(\cot (c))\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right)}{3 d \sqrt{\cot ^2(c)+1}}+\cos ^{\frac{5}{2}}(c+d x) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right) (a \sec (c+d x)+a)^2 \left(\frac{\sec (c) \sin (d x) \sec ^2(c+d x)}{6 d}+\frac{\sec (c) (\sin (c)+6 \sin (d x)) \sec (c+d x)}{6 d}+\frac{\csc (c) \sec (c)}{d}\right)","\frac{8 a^2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{4 a^2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 a^2 \sin (c+d x)}{3 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{4 a^2 \sin (c+d x)}{d \sqrt{\cos (c+d x)}}",1,"Cos[c + d*x]^(5/2)*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2*((Csc[c]*Sec[c])/d + (Sec[c]*Sec[c + d*x]^2*Sin[d*x])/(6*d) + (Sec[c]*Sec[c + d*x]*(Sin[c] + 6*Sin[d*x]))/(6*d)) - (2*Cos[c + d*x]^2*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*Sqrt[1 + Cot[c]^2]) + (Cos[c + d*x]^2*Csc[c]*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(2*d)","C",0
365,1,503,121,6.2005228,"\int \frac{(a+a \sec (c+d x))^2}{\cos ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[(a + a*Sec[c + d*x])^2/Cos[c + d*x]^(3/2),x]","\frac{2 \csc (c) \cos ^2(c+d x) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right) (a \sec (c+d x)+a)^2 \left(\frac{\tan (c) \sin \left(\tan ^{-1}(\tan (c))+d x\right) \, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right)}{\sqrt{\tan ^2(c)+1} \sqrt{1-\cos \left(\tan ^{-1}(\tan (c))+d x\right)} \sqrt{\cos \left(\tan ^{-1}(\tan (c))+d x\right)+1} \sqrt{\cos (c) \sqrt{\tan ^2(c)+1} \cos \left(\tan ^{-1}(\tan (c))+d x\right)}}-\frac{\frac{\tan (c) \sin \left(\tan ^{-1}(\tan (c))+d x\right)}{\sqrt{\tan ^2(c)+1}}+\frac{2 \cos ^2(c) \sqrt{\tan ^2(c)+1} \cos \left(\tan ^{-1}(\tan (c))+d x\right)}{\sin ^2(c)+\cos ^2(c)}}{\sqrt{\cos (c) \sqrt{\tan ^2(c)+1} \cos \left(\tan ^{-1}(\tan (c))+d x\right)}}\right)}{5 d}-\frac{\csc (c) \cos ^2(c+d x) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right) (a \sec (c+d x)+a)^2 \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin (c) \left(-\sqrt{\cot ^2(c)+1}\right) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec \left(d x-\tan ^{-1}(\cot (c))\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right)}{3 d \sqrt{\cot ^2(c)+1}}+\cos ^{\frac{5}{2}}(c+d x) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right) (a \sec (c+d x)+a)^2 \left(\frac{\sec (c) \sin (d x) \sec ^3(c+d x)}{10 d}+\frac{\sec (c) (3 \sin (c)+10 \sin (d x)) \sec ^2(c+d x)}{30 d}+\frac{\sec (c) (5 \sin (c)+12 \sin (d x)) \sec (c+d x)}{15 d}+\frac{4 \csc (c) \sec (c)}{5 d}\right)","\frac{4 a^2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{16 a^2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{4 a^2 \sin (c+d x)}{3 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 a^2 \sin (c+d x)}{5 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{16 a^2 \sin (c+d x)}{5 d \sqrt{\cos (c+d x)}}",1,"Cos[c + d*x]^(5/2)*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2*((4*Csc[c]*Sec[c])/(5*d) + (Sec[c]*Sec[c + d*x]^3*Sin[d*x])/(10*d) + (Sec[c]*Sec[c + d*x]^2*(3*Sin[c] + 10*Sin[d*x]))/(30*d) + (Sec[c]*Sec[c + d*x]*(5*Sin[c] + 12*Sin[d*x]))/(15*d)) - (Cos[c + d*x]^2*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*Sqrt[1 + Cot[c]^2]) + (2*Cos[c + d*x]^2*Csc[c]*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(5*d)","C",0
366,1,531,147,6.243014,"\int \frac{(a+a \sec (c+d x))^2}{\cos ^{\frac{5}{2}}(c+d x)} \, dx","Integrate[(a + a*Sec[c + d*x])^2/Cos[c + d*x]^(5/2),x]","\frac{3 \csc (c) \cos ^2(c+d x) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right) (a \sec (c+d x)+a)^2 \left(\frac{\tan (c) \sin \left(\tan ^{-1}(\tan (c))+d x\right) \, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right)}{\sqrt{\tan ^2(c)+1} \sqrt{1-\cos \left(\tan ^{-1}(\tan (c))+d x\right)} \sqrt{\cos \left(\tan ^{-1}(\tan (c))+d x\right)+1} \sqrt{\cos (c) \sqrt{\tan ^2(c)+1} \cos \left(\tan ^{-1}(\tan (c))+d x\right)}}-\frac{\frac{\tan (c) \sin \left(\tan ^{-1}(\tan (c))+d x\right)}{\sqrt{\tan ^2(c)+1}}+\frac{2 \cos ^2(c) \sqrt{\tan ^2(c)+1} \cos \left(\tan ^{-1}(\tan (c))+d x\right)}{\sin ^2(c)+\cos ^2(c)}}{\sqrt{\cos (c) \sqrt{\tan ^2(c)+1} \cos \left(\tan ^{-1}(\tan (c))+d x\right)}}\right)}{10 d}-\frac{2 \csc (c) \cos ^2(c+d x) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right) (a \sec (c+d x)+a)^2 \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin (c) \left(-\sqrt{\cot ^2(c)+1}\right) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec \left(d x-\tan ^{-1}(\cot (c))\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right)}{7 d \sqrt{\cot ^2(c)+1}}+\cos ^{\frac{5}{2}}(c+d x) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right) (a \sec (c+d x)+a)^2 \left(\frac{\sec (c) \sin (d x) \sec ^4(c+d x)}{14 d}+\frac{\sec (c) (5 \sin (c)+14 \sin (d x)) \sec ^3(c+d x)}{70 d}+\frac{\sec (c) (7 \sin (c)+10 \sin (d x)) \sec ^2(c+d x)}{35 d}+\frac{\sec (c) (10 \sin (c)+21 \sin (d x)) \sec (c+d x)}{35 d}+\frac{3 \csc (c) \sec (c)}{5 d}\right)","\frac{8 a^2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{7 d}-\frac{12 a^2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{8 a^2 \sin (c+d x)}{7 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{4 a^2 \sin (c+d x)}{5 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 a^2 \sin (c+d x)}{7 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{12 a^2 \sin (c+d x)}{5 d \sqrt{\cos (c+d x)}}",1,"Cos[c + d*x]^(5/2)*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2*((3*Csc[c]*Sec[c])/(5*d) + (Sec[c]*Sec[c + d*x]^4*Sin[d*x])/(14*d) + (Sec[c]*Sec[c + d*x]^2*(7*Sin[c] + 10*Sin[d*x]))/(35*d) + (Sec[c]*Sec[c + d*x]^3*(5*Sin[c] + 14*Sin[d*x]))/(70*d) + (Sec[c]*Sec[c + d*x]*(10*Sin[c] + 21*Sin[d*x]))/(35*d)) - (2*Cos[c + d*x]^2*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(7*d*Sqrt[1 + Cot[c]^2]) + (3*Cos[c + d*x]^2*Csc[c]*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(10*d)","C",0
367,1,548,147,6.1300619,"\int \cos ^{\frac{9}{2}}(c+d x) (a+a \sec (c+d x))^3 \, dx","Integrate[Cos[c + d*x]^(9/2)*(a + a*Sec[c + d*x])^3,x]","-\frac{17 \csc (c) \cos ^3(c+d x) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right) (a \sec (c+d x)+a)^3 \left(\frac{\tan (c) \sin \left(\tan ^{-1}(\tan (c))+d x\right) \, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right)}{\sqrt{\tan ^2(c)+1} \sqrt{1-\cos \left(\tan ^{-1}(\tan (c))+d x\right)} \sqrt{\cos \left(\tan ^{-1}(\tan (c))+d x\right)+1} \sqrt{\cos (c) \sqrt{\tan ^2(c)+1} \cos \left(\tan ^{-1}(\tan (c))+d x\right)}}-\frac{\frac{\tan (c) \sin \left(\tan ^{-1}(\tan (c))+d x\right)}{\sqrt{\tan ^2(c)+1}}+\frac{2 \cos ^2(c) \sqrt{\tan ^2(c)+1} \cos \left(\tan ^{-1}(\tan (c))+d x\right)}{\sin ^2(c)+\cos ^2(c)}}{\sqrt{\cos (c) \sqrt{\tan ^2(c)+1} \cos \left(\tan ^{-1}(\tan (c))+d x\right)}}\right)}{60 d}-\frac{11 \csc (c) \cos ^3(c+d x) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right) (a \sec (c+d x)+a)^3 \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin (c) \left(-\sqrt{\cot ^2(c)+1}\right) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec \left(d x-\tan ^{-1}(\cot (c))\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right)}{42 d \sqrt{\cot ^2(c)+1}}+\cos ^{\frac{7}{2}}(c+d x) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right) (a \sec (c+d x)+a)^3 \left(\frac{97 \sin (c) \cos (d x)}{336 d}+\frac{73 \sin (2 c) \cos (2 d x)}{720 d}+\frac{3 \sin (3 c) \cos (3 d x)}{112 d}+\frac{\sin (4 c) \cos (4 d x)}{288 d}+\frac{97 \cos (c) \sin (d x)}{336 d}+\frac{73 \cos (2 c) \sin (2 d x)}{720 d}+\frac{3 \cos (3 c) \sin (3 d x)}{112 d}+\frac{\cos (4 c) \sin (4 d x)}{288 d}-\frac{17 \cot (c)}{30 d}\right)","\frac{44 a^3 F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{68 a^3 E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{2 a^3 \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x)}{9 d}+\frac{6 a^3 \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{7 d}+\frac{68 a^3 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{45 d}+\frac{44 a^3 \sin (c+d x) \sqrt{\cos (c+d x)}}{21 d}",1,"Cos[c + d*x]^(7/2)*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3*((-17*Cot[c])/(30*d) + (97*Cos[d*x]*Sin[c])/(336*d) + (73*Cos[2*d*x]*Sin[2*c])/(720*d) + (3*Cos[3*d*x]*Sin[3*c])/(112*d) + (Cos[4*d*x]*Sin[4*c])/(288*d) + (97*Cos[c]*Sin[d*x])/(336*d) + (73*Cos[2*c]*Sin[2*d*x])/(720*d) + (3*Cos[3*c]*Sin[3*d*x])/(112*d) + (Cos[4*c]*Sin[4*d*x])/(288*d)) - (11*Cos[c + d*x]^3*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(42*d*Sqrt[1 + Cot[c]^2]) - (17*Cos[c + d*x]^3*Csc[c]*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(60*d)","C",0
368,1,516,121,6.1281478,"\int \cos ^{\frac{7}{2}}(c+d x) (a+a \sec (c+d x))^3 \, dx","Integrate[Cos[c + d*x]^(7/2)*(a + a*Sec[c + d*x])^3,x]","-\frac{7 \csc (c) \cos ^3(c+d x) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right) (a \sec (c+d x)+a)^3 \left(\frac{\tan (c) \sin \left(\tan ^{-1}(\tan (c))+d x\right) \, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right)}{\sqrt{\tan ^2(c)+1} \sqrt{1-\cos \left(\tan ^{-1}(\tan (c))+d x\right)} \sqrt{\cos \left(\tan ^{-1}(\tan (c))+d x\right)+1} \sqrt{\cos (c) \sqrt{\tan ^2(c)+1} \cos \left(\tan ^{-1}(\tan (c))+d x\right)}}-\frac{\frac{\tan (c) \sin \left(\tan ^{-1}(\tan (c))+d x\right)}{\sqrt{\tan ^2(c)+1}}+\frac{2 \cos ^2(c) \sqrt{\tan ^2(c)+1} \cos \left(\tan ^{-1}(\tan (c))+d x\right)}{\sin ^2(c)+\cos ^2(c)}}{\sqrt{\cos (c) \sqrt{\tan ^2(c)+1} \cos \left(\tan ^{-1}(\tan (c))+d x\right)}}\right)}{20 d}-\frac{13 \csc (c) \cos ^3(c+d x) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right) (a \sec (c+d x)+a)^3 \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin (c) \left(-\sqrt{\cot ^2(c)+1}\right) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec \left(d x-\tan ^{-1}(\cot (c))\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right)}{42 d \sqrt{\cot ^2(c)+1}}+\cos ^{\frac{7}{2}}(c+d x) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right) (a \sec (c+d x)+a)^3 \left(\frac{107 \sin (c) \cos (d x)}{336 d}+\frac{3 \sin (2 c) \cos (2 d x)}{40 d}+\frac{\sin (3 c) \cos (3 d x)}{112 d}+\frac{107 \cos (c) \sin (d x)}{336 d}+\frac{3 \cos (2 c) \sin (2 d x)}{40 d}+\frac{\cos (3 c) \sin (3 d x)}{112 d}-\frac{7 \cot (c)}{10 d}\right)","\frac{52 a^3 F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{28 a^3 E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 a^3 \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{7 d}+\frac{6 a^3 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 d}+\frac{52 a^3 \sin (c+d x) \sqrt{\cos (c+d x)}}{21 d}",1,"Cos[c + d*x]^(7/2)*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3*((-7*Cot[c])/(10*d) + (107*Cos[d*x]*Sin[c])/(336*d) + (3*Cos[2*d*x]*Sin[2*c])/(40*d) + (Cos[3*d*x]*Sin[3*c])/(112*d) + (107*Cos[c]*Sin[d*x])/(336*d) + (3*Cos[2*c]*Sin[2*d*x])/(40*d) + (Cos[3*c]*Sin[3*d*x])/(112*d)) - (13*Cos[c + d*x]^3*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(42*d*Sqrt[1 + Cot[c]^2]) - (7*Cos[c + d*x]^3*Csc[c]*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(20*d)","C",0
369,1,233,91,6.1125394,"\int \cos ^{\frac{5}{2}}(c+d x) (a+a \sec (c+d x))^3 \, dx","Integrate[Cos[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^3,x]","\frac{a^3 (\cos (c+d x)+1)^3 \sec ^6\left(\frac{1}{2} (c+d x)\right) \left(-18 \cos (c) \sqrt{\sec ^2(c)} \sqrt{\sin ^2\left(\tan ^{-1}(\tan (c))+d x\right)} \csc \left(\tan ^{-1}(\tan (c))+d x\right) \, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right)-20 \sin (c) \sqrt{\csc ^2(c)} \cos (c+d x) \sqrt{\cos ^2\left(d x-\tan ^{-1}(\cot (c))\right)} \sec \left(d x-\tan ^{-1}(\cot (c))\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right)+\cos (c+d x) (10 \sin (c+d x)+\sin (2 (c+d x))-36 \cot (c))+\frac{9 \csc (c) \sec (c) \left(3 \cos \left(c-\tan ^{-1}(\tan (c))-d x\right)+\cos \left(c+\tan ^{-1}(\tan (c))+d x\right)\right)}{\sqrt{\sec ^2(c)}}\right)}{40 d \sqrt{\cos (c+d x)}}","\frac{4 a^3 F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{36 a^3 E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 a^3 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 d}+\frac{2 a^3 \sin (c+d x) \sqrt{\cos (c+d x)}}{d}",1,"(a^3*(1 + Cos[c + d*x])^3*Sec[(c + d*x)/2]^6*((9*(3*Cos[c - d*x - ArcTan[Tan[c]]] + Cos[c + d*x + ArcTan[Tan[c]]])*Csc[c]*Sec[c])/Sqrt[Sec[c]^2] - 20*Cos[c + d*x]*Sqrt[Cos[d*x - ArcTan[Cot[c]]]^2]*Sqrt[Csc[c]^2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[d*x - ArcTan[Cot[c]]]*Sin[c] + Cos[c + d*x]*(-36*Cot[c] + 10*Sin[c + d*x] + Sin[2*(c + d*x)]) - 18*Cos[c]*Csc[d*x + ArcTan[Tan[c]]]*HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sqrt[Sec[c]^2]*Sqrt[Sin[d*x + ArcTan[Tan[c]]]^2]))/(40*d*Sqrt[Cos[c + d*x]])","C",0
370,1,240,91,4.9143349,"\int \cos ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x))^3 \, dx","Integrate[Cos[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^3,x]","\frac{a^3 (\cos (c+d x)+1)^3 \sec ^6\left(\frac{1}{2} (c+d x)\right) \left(-6 \cos (c) \sqrt{\sec ^2(c)} \sqrt{\sin ^2\left(\tan ^{-1}(\tan (c))+d x\right)} \csc \left(\tan ^{-1}(\tan (c))+d x\right) \, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right)-20 \sin (c) \sqrt{\csc ^2(c)} \cos (c+d x) \sqrt{\cos ^2\left(d x-\tan ^{-1}(\cot (c))\right)} \sec \left(d x-\tan ^{-1}(\cot (c))\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right)+\sin (2 (c+d x))-3 \csc (c) \cos (d x)-9 \csc (c) \cos (2 c+d x)+9 \cot (c) \sqrt{\sec ^2(c)} \cos \left(c-\tan ^{-1}(\tan (c))-d x\right)+3 \cot (c) \sqrt{\sec ^2(c)} \cos \left(c+\tan ^{-1}(\tan (c))+d x\right)\right)}{24 d \sqrt{\cos (c+d x)}}","\frac{20 a^3 F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{4 a^3 E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 a^3 \sin (c+d x) \sqrt{\cos (c+d x)}}{3 d}+\frac{2 a^3 \sin (c+d x)}{d \sqrt{\cos (c+d x)}}",1,"(a^3*(1 + Cos[c + d*x])^3*Sec[(c + d*x)/2]^6*(-3*Cos[d*x]*Csc[c] - 9*Cos[2*c + d*x]*Csc[c] + 9*Cos[c - d*x - ArcTan[Tan[c]]]*Cot[c]*Sqrt[Sec[c]^2] + 3*Cos[c + d*x + ArcTan[Tan[c]]]*Cot[c]*Sqrt[Sec[c]^2] - 20*Cos[c + d*x]*Sqrt[Cos[d*x - ArcTan[Cot[c]]]^2]*Sqrt[Csc[c]^2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[d*x - ArcTan[Cot[c]]]*Sin[c] + Sin[2*(c + d*x)] - 6*Cos[c]*Csc[d*x + ArcTan[Tan[c]]]*HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sqrt[Sec[c]^2]*Sqrt[Sin[d*x + ArcTan[Tan[c]]]^2]))/(24*d*Sqrt[Cos[c + d*x]])","C",0
371,1,479,91,6.1863853,"\int \sqrt{\cos (c+d x)} (a+a \sec (c+d x))^3 \, dx","Integrate[Sqrt[Cos[c + d*x]]*(a + a*Sec[c + d*x])^3,x]","\frac{\csc (c) \cos ^3(c+d x) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right) (a \sec (c+d x)+a)^3 \left(\frac{\tan (c) \sin \left(\tan ^{-1}(\tan (c))+d x\right) \, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right)}{\sqrt{\tan ^2(c)+1} \sqrt{1-\cos \left(\tan ^{-1}(\tan (c))+d x\right)} \sqrt{\cos \left(\tan ^{-1}(\tan (c))+d x\right)+1} \sqrt{\cos (c) \sqrt{\tan ^2(c)+1} \cos \left(\tan ^{-1}(\tan (c))+d x\right)}}-\frac{\frac{\tan (c) \sin \left(\tan ^{-1}(\tan (c))+d x\right)}{\sqrt{\tan ^2(c)+1}}+\frac{2 \cos ^2(c) \sqrt{\tan ^2(c)+1} \cos \left(\tan ^{-1}(\tan (c))+d x\right)}{\sin ^2(c)+\cos ^2(c)}}{\sqrt{\cos (c) \sqrt{\tan ^2(c)+1} \cos \left(\tan ^{-1}(\tan (c))+d x\right)}}\right)}{4 d}-\frac{5 \csc (c) \cos ^3(c+d x) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right) (a \sec (c+d x)+a)^3 \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin (c) \left(-\sqrt{\cot ^2(c)+1}\right) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec \left(d x-\tan ^{-1}(\cot (c))\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right)}{6 d \sqrt{\cot ^2(c)+1}}+\cos ^{\frac{7}{2}}(c+d x) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right) (a \sec (c+d x)+a)^3 \left(\frac{\sec (c) \sin (d x) \sec ^2(c+d x)}{12 d}+\frac{\sec (c) (\sin (c)+9 \sin (d x)) \sec (c+d x)}{12 d}-\frac{(\cos (2 c)-5) \csc (c) \sec (c)}{8 d}\right)","\frac{20 a^3 F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{4 a^3 E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 a^3 \sin (c+d x)}{3 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{6 a^3 \sin (c+d x)}{d \sqrt{\cos (c+d x)}}",1,"Cos[c + d*x]^(7/2)*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3*(-1/8*((-5 + Cos[2*c])*Csc[c]*Sec[c])/d + (Sec[c]*Sec[c + d*x]^2*Sin[d*x])/(12*d) + (Sec[c]*Sec[c + d*x]*(Sin[c] + 9*Sin[d*x]))/(12*d)) - (5*Cos[c + d*x]^3*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(6*d*Sqrt[1 + Cot[c]^2]) + (Cos[c + d*x]^3*Csc[c]*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(4*d)","C",0
372,1,501,117,6.211196,"\int \frac{(a+a \sec (c+d x))^3}{\sqrt{\cos (c+d x)}} \, dx","Integrate[(a + a*Sec[c + d*x])^3/Sqrt[Cos[c + d*x]],x]","\frac{9 \csc (c) \cos ^3(c+d x) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right) (a \sec (c+d x)+a)^3 \left(\frac{\tan (c) \sin \left(\tan ^{-1}(\tan (c))+d x\right) \, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right)}{\sqrt{\tan ^2(c)+1} \sqrt{1-\cos \left(\tan ^{-1}(\tan (c))+d x\right)} \sqrt{\cos \left(\tan ^{-1}(\tan (c))+d x\right)+1} \sqrt{\cos (c) \sqrt{\tan ^2(c)+1} \cos \left(\tan ^{-1}(\tan (c))+d x\right)}}-\frac{\frac{\tan (c) \sin \left(\tan ^{-1}(\tan (c))+d x\right)}{\sqrt{\tan ^2(c)+1}}+\frac{2 \cos ^2(c) \sqrt{\tan ^2(c)+1} \cos \left(\tan ^{-1}(\tan (c))+d x\right)}{\sin ^2(c)+\cos ^2(c)}}{\sqrt{\cos (c) \sqrt{\tan ^2(c)+1} \cos \left(\tan ^{-1}(\tan (c))+d x\right)}}\right)}{20 d}-\frac{\csc (c) \cos ^3(c+d x) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right) (a \sec (c+d x)+a)^3 \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin (c) \left(-\sqrt{\cot ^2(c)+1}\right) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec \left(d x-\tan ^{-1}(\cot (c))\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right)}{2 d \sqrt{\cot ^2(c)+1}}+\cos ^{\frac{7}{2}}(c+d x) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right) (a \sec (c+d x)+a)^3 \left(\frac{\sec (c) \sin (d x) \sec ^3(c+d x)}{20 d}+\frac{\sec (c) (\sin (c)+5 \sin (d x)) \sec ^2(c+d x)}{20 d}+\frac{\sec (c) (5 \sin (c)+18 \sin (d x)) \sec (c+d x)}{20 d}+\frac{9 \csc (c) \sec (c)}{10 d}\right)","\frac{4 a^3 F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}-\frac{36 a^3 E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 a^3 \sin (c+d x)}{d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 a^3 \sin (c+d x)}{5 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{36 a^3 \sin (c+d x)}{5 d \sqrt{\cos (c+d x)}}",1,"Cos[c + d*x]^(7/2)*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3*((9*Csc[c]*Sec[c])/(10*d) + (Sec[c]*Sec[c + d*x]^3*Sin[d*x])/(20*d) + (Sec[c]*Sec[c + d*x]^2*(Sin[c] + 5*Sin[d*x]))/(20*d) + (Sec[c]*Sec[c + d*x]*(5*Sin[c] + 18*Sin[d*x]))/(20*d)) - (Cos[c + d*x]^3*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(2*d*Sqrt[1 + Cot[c]^2]) + (9*Cos[c + d*x]^3*Csc[c]*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(20*d)","C",0
373,1,531,147,6.2430307,"\int \frac{(a+a \sec (c+d x))^3}{\cos ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[(a + a*Sec[c + d*x])^3/Cos[c + d*x]^(3/2),x]","\frac{7 \csc (c) \cos ^3(c+d x) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right) (a \sec (c+d x)+a)^3 \left(\frac{\tan (c) \sin \left(\tan ^{-1}(\tan (c))+d x\right) \, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right)}{\sqrt{\tan ^2(c)+1} \sqrt{1-\cos \left(\tan ^{-1}(\tan (c))+d x\right)} \sqrt{\cos \left(\tan ^{-1}(\tan (c))+d x\right)+1} \sqrt{\cos (c) \sqrt{\tan ^2(c)+1} \cos \left(\tan ^{-1}(\tan (c))+d x\right)}}-\frac{\frac{\tan (c) \sin \left(\tan ^{-1}(\tan (c))+d x\right)}{\sqrt{\tan ^2(c)+1}}+\frac{2 \cos ^2(c) \sqrt{\tan ^2(c)+1} \cos \left(\tan ^{-1}(\tan (c))+d x\right)}{\sin ^2(c)+\cos ^2(c)}}{\sqrt{\cos (c) \sqrt{\tan ^2(c)+1} \cos \left(\tan ^{-1}(\tan (c))+d x\right)}}\right)}{20 d}-\frac{13 \csc (c) \cos ^3(c+d x) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right) (a \sec (c+d x)+a)^3 \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin (c) \left(-\sqrt{\cot ^2(c)+1}\right) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec \left(d x-\tan ^{-1}(\cot (c))\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right)}{42 d \sqrt{\cot ^2(c)+1}}+\cos ^{\frac{7}{2}}(c+d x) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right) (a \sec (c+d x)+a)^3 \left(\frac{\sec (c) \sin (d x) \sec ^4(c+d x)}{28 d}+\frac{\sec (c) (5 \sin (c)+21 \sin (d x)) \sec ^3(c+d x)}{140 d}+\frac{\sec (c) (63 \sin (c)+130 \sin (d x)) \sec ^2(c+d x)}{420 d}+\frac{\sec (c) (65 \sin (c)+147 \sin (d x)) \sec (c+d x)}{210 d}+\frac{7 \csc (c) \sec (c)}{10 d}\right)","\frac{52 a^3 F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}-\frac{28 a^3 E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{52 a^3 \sin (c+d x)}{21 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{6 a^3 \sin (c+d x)}{5 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 a^3 \sin (c+d x)}{7 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{28 a^3 \sin (c+d x)}{5 d \sqrt{\cos (c+d x)}}",1,"Cos[c + d*x]^(7/2)*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3*((7*Csc[c]*Sec[c])/(10*d) + (Sec[c]*Sec[c + d*x]^4*Sin[d*x])/(28*d) + (Sec[c]*Sec[c + d*x]^3*(5*Sin[c] + 21*Sin[d*x]))/(140*d) + (Sec[c]*Sec[c + d*x]^2*(63*Sin[c] + 130*Sin[d*x]))/(420*d) + (Sec[c]*Sec[c + d*x]*(65*Sin[c] + 147*Sin[d*x]))/(210*d)) - (13*Cos[c + d*x]^3*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(42*d*Sqrt[1 + Cot[c]^2]) + (7*Cos[c + d*x]^3*Csc[c]*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(20*d)","C",0
374,1,314,128,1.7661961,"\int \frac{\cos ^{\frac{5}{2}}(c+d x)}{a+a \sec (c+d x)} \, dx","Integrate[Cos[c + d*x]^(5/2)/(a + a*Sec[c + d*x]),x]","\frac{\cos ^2\left(\frac{1}{2} (c+d x)\right) \left(\frac{-20 \sin (c) \cos (d x)+6 \sin (2 c) \cos (2 d x)-20 \cos (c) \sin (d x)+6 \cos (2 c) \sin (2 d x)-30 \sec \left(\frac{c}{2}\right) \sin \left(\frac{d x}{2}\right) \sec \left(\frac{1}{2} (c+d x)\right)-96 \cot (c)-30 \csc (c)}{d \sqrt{\cos (c+d x)}}+\frac{2 i \sqrt{2} e^{-i (c+d x)} \left(63 \left(-1+e^{2 i c}\right) \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i (c+d x)}\right)+25 \left(-1+e^{2 i c}\right) e^{i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-e^{2 i (c+d x)}\right)+63 \left(1+e^{2 i (c+d x)}\right)\right) \sec (c+d x)}{\left(-1+e^{2 i c}\right) d \sqrt{e^{-i (c+d x)} \left(1+e^{2 i (c+d x)}\right)}}\right)}{15 a (\sec (c+d x)+1)}","-\frac{5 F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a d}+\frac{21 E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 a d}+\frac{7 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 a d}-\frac{5 \sin (c+d x) \sqrt{\cos (c+d x)}}{3 a d}-\frac{\sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{d (a \sec (c+d x)+a)}",1,"(Cos[(c + d*x)/2]^2*(((2*I)*Sqrt[2]*(63*(1 + E^((2*I)*(c + d*x))) + 63*(-1 + E^((2*I)*c))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[-1/4, 1/2, 3/4, -E^((2*I)*(c + d*x))] + 25*E^(I*(c + d*x))*(-1 + E^((2*I)*c))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/4, 1/2, 5/4, -E^((2*I)*(c + d*x))])*Sec[c + d*x])/(d*E^(I*(c + d*x))*(-1 + E^((2*I)*c))*Sqrt[(1 + E^((2*I)*(c + d*x)))/E^(I*(c + d*x))]) + (-96*Cot[c] - 30*Csc[c] - 20*Cos[d*x]*Sin[c] + 6*Cos[2*d*x]*Sin[2*c] - 30*Sec[c/2]*Sec[(c + d*x)/2]*Sin[(d*x)/2] - 20*Cos[c]*Sin[d*x] + 6*Cos[2*c]*Sin[2*d*x])/(d*Sqrt[Cos[c + d*x]])))/(15*a*(1 + Sec[c + d*x]))","C",1
375,1,292,100,2.1549814,"\int \frac{\cos ^{\frac{3}{2}}(c+d x)}{a+a \sec (c+d x)} \, dx","Integrate[Cos[c + d*x]^(3/2)/(a + a*Sec[c + d*x]),x]","\frac{\cos ^2\left(\frac{1}{2} (c+d x)\right) \left(\frac{4 \sin (c) \cos (d x)+4 \cos (c) \sin (d x)+6 \sec \left(\frac{c}{2}\right) \sin \left(\frac{d x}{2}\right) \sec \left(\frac{1}{2} (c+d x)\right)+12 \cot (c)+6 \csc (c)}{d \sqrt{\cos (c+d x)}}-\frac{2 i \sqrt{2} e^{-i (c+d x)} \left(9 \left(-1+e^{2 i c}\right) \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i (c+d x)}\right)+5 \left(-1+e^{2 i c}\right) e^{i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-e^{2 i (c+d x)}\right)+9 \left(1+e^{2 i (c+d x)}\right)\right) \sec (c+d x)}{\left(-1+e^{2 i c}\right) d \sqrt{e^{-i (c+d x)} \left(1+e^{2 i (c+d x)}\right)}}\right)}{3 a (\sec (c+d x)+1)}","\frac{5 F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a d}-\frac{3 E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}+\frac{5 \sin (c+d x) \sqrt{\cos (c+d x)}}{3 a d}-\frac{\sin (c+d x) \sqrt{\cos (c+d x)}}{d (a \sec (c+d x)+a)}",1,"(Cos[(c + d*x)/2]^2*(((-2*I)*Sqrt[2]*(9*(1 + E^((2*I)*(c + d*x))) + 9*(-1 + E^((2*I)*c))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[-1/4, 1/2, 3/4, -E^((2*I)*(c + d*x))] + 5*E^(I*(c + d*x))*(-1 + E^((2*I)*c))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/4, 1/2, 5/4, -E^((2*I)*(c + d*x))])*Sec[c + d*x])/(d*E^(I*(c + d*x))*(-1 + E^((2*I)*c))*Sqrt[(1 + E^((2*I)*(c + d*x)))/E^(I*(c + d*x))]) + (12*Cot[c] + 6*Csc[c] + 4*Cos[d*x]*Sin[c] + 6*Sec[c/2]*Sec[(c + d*x)/2]*Sin[(d*x)/2] + 4*Cos[c]*Sin[d*x])/(d*Sqrt[Cos[c + d*x]])))/(3*a*(1 + Sec[c + d*x]))","C",1
376,1,270,72,1.7445556,"\int \frac{\sqrt{\cos (c+d x)}}{a+a \sec (c+d x)} \, dx","Integrate[Sqrt[Cos[c + d*x]]/(a + a*Sec[c + d*x]),x]","\frac{\cos ^2\left(\frac{1}{2} (c+d x)\right) \left(-\frac{2 \left(\sec \left(\frac{c}{2}\right) \sin \left(\frac{d x}{2}\right) \sec \left(\frac{1}{2} (c+d x)\right)+2 \cot (c)+\csc (c)\right)}{d \sqrt{\cos (c+d x)}}+\frac{2 i \sqrt{2} e^{-i (c+d x)} \left(3 \left(-1+e^{2 i c}\right) \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i (c+d x)}\right)+\left(-1+e^{2 i c}\right) e^{i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-e^{2 i (c+d x)}\right)+3 \left(1+e^{2 i (c+d x)}\right)\right) \sec (c+d x)}{\left(-1+e^{2 i c}\right) d \sqrt{e^{-i (c+d x)} \left(1+e^{2 i (c+d x)}\right)}}\right)}{a (\sec (c+d x)+1)}","-\frac{F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}+\frac{3 E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}-\frac{\sin (c+d x)}{d \sqrt{\cos (c+d x)} (a \sec (c+d x)+a)}",1,"(Cos[(c + d*x)/2]^2*(((2*I)*Sqrt[2]*(3*(1 + E^((2*I)*(c + d*x))) + 3*(-1 + E^((2*I)*c))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[-1/4, 1/2, 3/4, -E^((2*I)*(c + d*x))] + E^(I*(c + d*x))*(-1 + E^((2*I)*c))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/4, 1/2, 5/4, -E^((2*I)*(c + d*x))])*Sec[c + d*x])/(d*E^(I*(c + d*x))*(-1 + E^((2*I)*c))*Sqrt[(1 + E^((2*I)*(c + d*x)))/E^(I*(c + d*x))]) - (2*(2*Cot[c] + Csc[c] + Sec[c/2]*Sec[(c + d*x)/2]*Sin[(d*x)/2]))/(d*Sqrt[Cos[c + d*x]])))/(a*(1 + Sec[c + d*x]))","C",1
377,1,262,70,1.089199,"\int \frac{1}{\sqrt{\cos (c+d x)} (a+a \sec (c+d x))} \, dx","Integrate[1/(Sqrt[Cos[c + d*x]]*(a + a*Sec[c + d*x])),x]","\frac{\cos ^2\left(\frac{1}{2} (c+d x)\right) \left(\frac{2 \left(\sec \left(\frac{c}{2}\right) \sin \left(\frac{d x}{2}\right) \sec \left(\frac{1}{2} (c+d x)\right)+\csc (c)\right)}{d \sqrt{\cos (c+d x)}}-\frac{2 i \sqrt{2} e^{-i (c+d x)} \left(\left(-1+e^{2 i c}\right) \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i (c+d x)}\right)+\left(-1+e^{2 i c}\right) e^{i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-e^{2 i (c+d x)}\right)+e^{2 i (c+d x)}+1\right) \sec (c+d x)}{\left(-1+e^{2 i c}\right) d \sqrt{e^{-i (c+d x)} \left(1+e^{2 i (c+d x)}\right)}}\right)}{a (\sec (c+d x)+1)}","\frac{F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}-\frac{E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}+\frac{\sin (c+d x)}{d \sqrt{\cos (c+d x)} (a \sec (c+d x)+a)}",1,"(Cos[(c + d*x)/2]^2*(((-2*I)*Sqrt[2]*(1 + E^((2*I)*(c + d*x)) + (-1 + E^((2*I)*c))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[-1/4, 1/2, 3/4, -E^((2*I)*(c + d*x))] + E^(I*(c + d*x))*(-1 + E^((2*I)*c))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/4, 1/2, 5/4, -E^((2*I)*(c + d*x))])*Sec[c + d*x])/(d*E^(I*(c + d*x))*(-1 + E^((2*I)*c))*Sqrt[(1 + E^((2*I)*(c + d*x)))/E^(I*(c + d*x))]) + (2*(Csc[c] + Sec[c/2]*Sec[(c + d*x)/2]*Sin[(d*x)/2]))/(d*Sqrt[Cos[c + d*x]])))/(a*(1 + Sec[c + d*x]))","C",1
378,1,263,70,1.106335,"\int \frac{1}{\cos ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x))} \, dx","Integrate[1/(Cos[c + d*x]^(3/2)*(a + a*Sec[c + d*x])),x]","\frac{\cos ^2\left(\frac{1}{2} (c+d x)\right) \left(-\frac{2 \left(\sec \left(\frac{c}{2}\right) \sin \left(\frac{d x}{2}\right) \sec \left(\frac{1}{2} (c+d x)\right)+\csc (c)\right)}{d \sqrt{\cos (c+d x)}}+\frac{2 i \sqrt{2} e^{-i (c+d x)} \left(\left(-1+e^{2 i c}\right) \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i (c+d x)}\right)-\left(-1+e^{2 i c}\right) e^{i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-e^{2 i (c+d x)}\right)+e^{2 i (c+d x)}+1\right) \sec (c+d x)}{\left(-1+e^{2 i c}\right) d \sqrt{e^{-i (c+d x)} \left(1+e^{2 i (c+d x)}\right)}}\right)}{a (\sec (c+d x)+1)}","\frac{F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}+\frac{E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}-\frac{\sin (c+d x)}{d \sqrt{\cos (c+d x)} (a \sec (c+d x)+a)}",1,"(Cos[(c + d*x)/2]^2*(((2*I)*Sqrt[2]*(1 + E^((2*I)*(c + d*x)) + (-1 + E^((2*I)*c))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[-1/4, 1/2, 3/4, -E^((2*I)*(c + d*x))] - E^(I*(c + d*x))*(-1 + E^((2*I)*c))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/4, 1/2, 5/4, -E^((2*I)*(c + d*x))])*Sec[c + d*x])/(d*E^(I*(c + d*x))*(-1 + E^((2*I)*c))*Sqrt[(1 + E^((2*I)*(c + d*x)))/E^(I*(c + d*x))]) - (2*(Csc[c] + Sec[c/2]*Sec[(c + d*x)/2]*Sin[(d*x)/2]))/(d*Sqrt[Cos[c + d*x]])))/(a*(1 + Sec[c + d*x]))","C",1
379,1,303,96,1.753168,"\int \frac{1}{\cos ^{\frac{5}{2}}(c+d x) (a+a \sec (c+d x))} \, dx","Integrate[1/(Cos[c + d*x]^(5/2)*(a + a*Sec[c + d*x])),x]","\frac{\cos ^2\left(\frac{1}{2} (c+d x)\right) \left(\frac{\csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \left(2 \cos \left(\frac{1}{2} (c-d x)\right)+\cos \left(\frac{1}{2} (3 c+d x)\right)+3 \cos \left(\frac{1}{2} (c+3 d x)\right)\right) \sec \left(\frac{1}{2} (c+d x)\right)}{2 d \cos ^{\frac{3}{2}}(c+d x)}-\frac{2 i \sqrt{2} e^{-i (c+d x)} \left(3 \left(-1+e^{2 i c}\right) \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i (c+d x)}\right)-\left(-1+e^{2 i c}\right) e^{i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-e^{2 i (c+d x)}\right)+3 \left(1+e^{2 i (c+d x)}\right)\right) \sec (c+d x)}{\left(-1+e^{2 i c}\right) d \sqrt{e^{-i (c+d x)} \left(1+e^{2 i (c+d x)}\right)}}\right)}{a (\sec (c+d x)+1)}","-\frac{F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}-\frac{3 E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}+\frac{3 \sin (c+d x)}{a d \sqrt{\cos (c+d x)}}-\frac{\sin (c+d x)}{d \cos ^{\frac{3}{2}}(c+d x) (a \sec (c+d x)+a)}",1,"(Cos[(c + d*x)/2]^2*(((2*Cos[(c - d*x)/2] + Cos[(3*c + d*x)/2] + 3*Cos[(c + 3*d*x)/2])*Csc[c/2]*Sec[c/2]*Sec[(c + d*x)/2])/(2*d*Cos[c + d*x]^(3/2)) - ((2*I)*Sqrt[2]*(3*(1 + E^((2*I)*(c + d*x))) + 3*(-1 + E^((2*I)*c))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[-1/4, 1/2, 3/4, -E^((2*I)*(c + d*x))] - E^(I*(c + d*x))*(-1 + E^((2*I)*c))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/4, 1/2, 5/4, -E^((2*I)*(c + d*x))])*Sec[c + d*x])/(d*E^(I*(c + d*x))*(-1 + E^((2*I)*c))*Sqrt[(1 + E^((2*I)*(c + d*x)))/E^(I*(c + d*x))])))/(a*(1 + Sec[c + d*x]))","C",1
380,1,338,124,3.7480191,"\int \frac{1}{\cos ^{\frac{7}{2}}(c+d x) (a+a \sec (c+d x))} \, dx","Integrate[1/(Cos[c + d*x]^(7/2)*(a + a*Sec[c + d*x])),x]","\frac{\cos ^2\left(\frac{1}{2} (c+d x)\right) \left(-\frac{\csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \left(10 \cos \left(\frac{1}{2} (c-d x)\right)+8 \cos \left(\frac{1}{2} (3 c+d x)\right)+4 \cos \left(\frac{1}{2} (c+3 d x)\right)+5 \cos \left(\frac{1}{2} (5 c+3 d x)\right)+9 \cos \left(\frac{1}{2} (3 c+5 d x)\right)\right) \sec \left(\frac{1}{2} (c+d x)\right)}{4 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 i \sqrt{2} e^{-i (c+d x)} \left(9 \left(-1+e^{2 i c}\right) \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i (c+d x)}\right)-5 \left(-1+e^{2 i c}\right) e^{i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-e^{2 i (c+d x)}\right)+9 \left(1+e^{2 i (c+d x)}\right)\right) \sec (c+d x)}{\left(-1+e^{2 i c}\right) d \sqrt{e^{-i (c+d x)} \left(1+e^{2 i (c+d x)}\right)}}\right)}{3 a (\sec (c+d x)+1)}","\frac{5 F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a d}+\frac{3 E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}+\frac{5 \sin (c+d x)}{3 a d \cos ^{\frac{3}{2}}(c+d x)}-\frac{3 \sin (c+d x)}{a d \sqrt{\cos (c+d x)}}-\frac{\sin (c+d x)}{d \cos ^{\frac{5}{2}}(c+d x) (a \sec (c+d x)+a)}",1,"(Cos[(c + d*x)/2]^2*(-1/4*((10*Cos[(c - d*x)/2] + 8*Cos[(3*c + d*x)/2] + 4*Cos[(c + 3*d*x)/2] + 5*Cos[(5*c + 3*d*x)/2] + 9*Cos[(3*c + 5*d*x)/2])*Csc[c/2]*Sec[c/2]*Sec[(c + d*x)/2])/(d*Cos[c + d*x]^(5/2)) + ((2*I)*Sqrt[2]*(9*(1 + E^((2*I)*(c + d*x))) + 9*(-1 + E^((2*I)*c))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[-1/4, 1/2, 3/4, -E^((2*I)*(c + d*x))] - 5*E^(I*(c + d*x))*(-1 + E^((2*I)*c))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/4, 1/2, 5/4, -E^((2*I)*(c + d*x))])*Sec[c + d*x])/(d*E^(I*(c + d*x))*(-1 + E^((2*I)*c))*Sqrt[(1 + E^((2*I)*(c + d*x)))/E^(I*(c + d*x))])))/(3*a*(1 + Sec[c + d*x]))","C",1
381,1,366,160,2.5883323,"\int \frac{\cos ^{\frac{5}{2}}(c+d x)}{(a+a \sec (c+d x))^2} \, dx","Integrate[Cos[c + d*x]^(5/2)/(a + a*Sec[c + d*x])^2,x]","\frac{\cos ^4\left(\frac{1}{2} (c+d x)\right) \left(\frac{2 \left(-40 \sin (c) \cos (d x)+6 \sin (2 c) \cos (2 d x)-40 \cos (c) \sin (d x)+6 \cos (2 c) \sin (2 d x)+5 \sec \left(\frac{c}{2}\right) \sin \left(\frac{d x}{2}\right) \sec ^3\left(\frac{1}{2} (c+d x)\right)+5 \tan \left(\frac{c}{2}\right) \sec ^2\left(\frac{1}{2} (c+d x)\right)-120 \sec \left(\frac{c}{2}\right) \sin \left(\frac{d x}{2}\right) \sec \left(\frac{1}{2} (c+d x)\right)-216 \cot (c)-120 \csc (c)\right)}{3 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{4 i \sqrt{2} e^{-i (c+d x)} \left(56 \left(-1+e^{2 i c}\right) \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i (c+d x)}\right)+25 \left(-1+e^{2 i c}\right) e^{i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-e^{2 i (c+d x)}\right)+56 \left(1+e^{2 i (c+d x)}\right)\right) \sec ^2(c+d x)}{\left(-1+e^{2 i c}\right) d \sqrt{e^{-i (c+d x)} \left(1+e^{2 i (c+d x)}\right)}}\right)}{5 a^2 (\sec (c+d x)+1)^2}","-\frac{5 F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d}+\frac{56 E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 a^2 d}+\frac{56 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{15 a^2 d}-\frac{5 \sin (c+d x) \sqrt{\cos (c+d x)}}{a^2 d}-\frac{3 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{a^2 d (\sec (c+d x)+1)}-\frac{\sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{3 d (a \sec (c+d x)+a)^2}",1,"(Cos[(c + d*x)/2]^4*(((4*I)*Sqrt[2]*(56*(1 + E^((2*I)*(c + d*x))) + 56*(-1 + E^((2*I)*c))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[-1/4, 1/2, 3/4, -E^((2*I)*(c + d*x))] + 25*E^(I*(c + d*x))*(-1 + E^((2*I)*c))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/4, 1/2, 5/4, -E^((2*I)*(c + d*x))])*Sec[c + d*x]^2)/(d*E^(I*(c + d*x))*(-1 + E^((2*I)*c))*Sqrt[(1 + E^((2*I)*(c + d*x)))/E^(I*(c + d*x))]) + (2*(-216*Cot[c] - 120*Csc[c] - 40*Cos[d*x]*Sin[c] + 6*Cos[2*d*x]*Sin[2*c] - 120*Sec[c/2]*Sec[(c + d*x)/2]*Sin[(d*x)/2] + 5*Sec[c/2]*Sec[(c + d*x)/2]^3*Sin[(d*x)/2] - 40*Cos[c]*Sin[d*x] + 6*Cos[2*c]*Sin[2*d*x] + 5*Sec[(c + d*x)/2]^2*Tan[c/2]))/(3*d*Cos[c + d*x]^(3/2))))/(5*a^2*(1 + Sec[c + d*x])^2)","C",1
382,1,341,138,1.7987922,"\int \frac{\cos ^{\frac{3}{2}}(c+d x)}{(a+a \sec (c+d x))^2} \, dx","Integrate[Cos[c + d*x]^(3/2)/(a + a*Sec[c + d*x])^2,x]","\frac{\cos ^4\left(\frac{1}{2} (c+d x)\right) \left(\frac{8 \sin (c) \cos (d x)+8 \cos (c) \sin (d x)-2 \sec \left(\frac{c}{2}\right) \sin \left(\frac{d x}{2}\right) \sec ^3\left(\frac{1}{2} (c+d x)\right)-2 \tan \left(\frac{c}{2}\right) \sec ^2\left(\frac{1}{2} (c+d x)\right)+36 \sec \left(\frac{c}{2}\right) \sin \left(\frac{d x}{2}\right) \sec \left(\frac{1}{2} (c+d x)\right)+48 \cot (c)+36 \csc (c)}{d \cos ^{\frac{3}{2}}(c+d x)}-\frac{4 i \sqrt{2} e^{-i (c+d x)} \left(21 \left(-1+e^{2 i c}\right) \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i (c+d x)}\right)+10 \left(-1+e^{2 i c}\right) e^{i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-e^{2 i (c+d x)}\right)+21 \left(1+e^{2 i (c+d x)}\right)\right) \sec ^2(c+d x)}{\left(-1+e^{2 i c}\right) d \sqrt{e^{-i (c+d x)} \left(1+e^{2 i (c+d x)}\right)}}\right)}{3 a^2 (\sec (c+d x)+1)^2}","\frac{10 F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^2 d}-\frac{7 E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d}+\frac{10 \sin (c+d x) \sqrt{\cos (c+d x)}}{3 a^2 d}-\frac{7 \sin (c+d x) \sqrt{\cos (c+d x)}}{3 a^2 d (\sec (c+d x)+1)}-\frac{\sin (c+d x) \sqrt{\cos (c+d x)}}{3 d (a \sec (c+d x)+a)^2}",1,"(Cos[(c + d*x)/2]^4*(((-4*I)*Sqrt[2]*(21*(1 + E^((2*I)*(c + d*x))) + 21*(-1 + E^((2*I)*c))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[-1/4, 1/2, 3/4, -E^((2*I)*(c + d*x))] + 10*E^(I*(c + d*x))*(-1 + E^((2*I)*c))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/4, 1/2, 5/4, -E^((2*I)*(c + d*x))])*Sec[c + d*x]^2)/(d*E^(I*(c + d*x))*(-1 + E^((2*I)*c))*Sqrt[(1 + E^((2*I)*(c + d*x)))/E^(I*(c + d*x))]) + (48*Cot[c] + 36*Csc[c] + 8*Cos[d*x]*Sin[c] + 36*Sec[c/2]*Sec[(c + d*x)/2]*Sin[(d*x)/2] - 2*Sec[c/2]*Sec[(c + d*x)/2]^3*Sin[(d*x)/2] + 8*Cos[c]*Sin[d*x] - 2*Sec[(c + d*x)/2]^2*Tan[c/2])/(d*Cos[c + d*x]^(3/2))))/(3*a^2*(1 + Sec[c + d*x])^2)","C",1
383,1,374,112,6.1908831,"\int \frac{\sqrt{\cos (c+d x)}}{(a+a \sec (c+d x))^2} \, dx","Integrate[Sqrt[Cos[c + d*x]]/(a + a*Sec[c + d*x])^2,x]","\frac{\cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right) \left(\frac{2 \sec \left(\frac{c}{2}\right) \sin \left(\frac{d x}{2}\right) \sec ^3\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d}+\frac{2 \tan \left(\frac{c}{2}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d}-\frac{8 \sec \left(\frac{c}{2}\right) \sin \left(\frac{d x}{2}\right) \sec \left(\frac{c}{2}+\frac{d x}{2}\right)}{d}-\frac{8 \cot \left(\frac{c}{2}\right)}{d}\right)}{\cos ^{\frac{3}{2}}(c+d x) (a \sec (c+d x)+a)^2}+\frac{4 i \sqrt{2} e^{-i (c+d x)} \left(12 \left(-1+e^{2 i c}\right) \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i (c+d x)}\right)+5 \left(-1+e^{2 i c}\right) e^{i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-e^{2 i (c+d x)}\right)+12 \left(1+e^{2 i (c+d x)}\right)\right) \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right) \sec ^2(c+d x)}{3 \left(-1+e^{2 i c}\right) d \sqrt{e^{-i (c+d x)} \left(1+e^{2 i (c+d x)}\right)} (a \sec (c+d x)+a)^2}","-\frac{5 F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^2 d}+\frac{4 E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d}-\frac{5 \sin (c+d x)}{3 a^2 d \sqrt{\cos (c+d x)} (\sec (c+d x)+1)}-\frac{\sin (c+d x)}{3 d \sqrt{\cos (c+d x)} (a \sec (c+d x)+a)^2}",1,"(((4*I)/3)*Sqrt[2]*Cos[c/2 + (d*x)/2]^4*(12*(1 + E^((2*I)*(c + d*x))) + 12*(-1 + E^((2*I)*c))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[-1/4, 1/2, 3/4, -E^((2*I)*(c + d*x))] + 5*E^(I*(c + d*x))*(-1 + E^((2*I)*c))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/4, 1/2, 5/4, -E^((2*I)*(c + d*x))])*Sec[c + d*x]^2)/(d*E^(I*(c + d*x))*(-1 + E^((2*I)*c))*Sqrt[(1 + E^((2*I)*(c + d*x)))/E^(I*(c + d*x))]*(a + a*Sec[c + d*x])^2) + (Cos[c/2 + (d*x)/2]^4*((-8*Cot[c/2])/d - (8*Sec[c/2]*Sec[c/2 + (d*x)/2]*Sin[(d*x)/2])/d + (2*Sec[c/2]*Sec[c/2 + (d*x)/2]^3*Sin[(d*x)/2])/(3*d) + (2*Sec[c/2 + (d*x)/2]^2*Tan[c/2])/(3*d)))/(Cos[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^2)","C",1
384,1,656,109,6.1842115,"\int \frac{1}{\sqrt{\cos (c+d x)} (a+a \sec (c+d x))^2} \, dx","Integrate[1/(Sqrt[Cos[c + d*x]]*(a + a*Sec[c + d*x])^2),x]","-\frac{4 \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right) \sec ^2(c+d x) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin (c) \left(-\sqrt{\cot ^2(c)+1}\right) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec \left(d x-\tan ^{-1}(\cot (c))\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right)}{3 d \sqrt{\cot ^2(c)+1} (a \sec (c+d x)+a)^2}-\frac{i \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right) \sec ^2(c+d x) \left(\frac{2 e^{2 i d x} \sqrt{e^{-i d x} \left(2 i \sin (c) \left(-1+e^{2 i d x}\right)+2 \cos (c) \left(1+e^{2 i d x}\right)\right)} \sqrt{i \sin (2 c) e^{2 i d x}+\cos (2 c) e^{2 i d x}+1} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right)}{3 i d \cos (c) \left(1+e^{2 i d x}\right)-3 d \sin (c) \left(-1+e^{2 i d x}\right)}-\frac{2 \sqrt{e^{-i d x} \left(2 i \sin (c) \left(-1+e^{2 i d x}\right)+2 \cos (c) \left(1+e^{2 i d x}\right)\right)} \sqrt{i \sin (2 c) e^{2 i d x}+\cos (2 c) e^{2 i d x}+1} \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right)}{d \sin (c) \left(-1+e^{2 i d x}\right)-i d \cos (c) \left(1+e^{2 i d x}\right)}\right)}{2 (a \sec (c+d x)+a)^2}+\frac{\cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right) \left(-\frac{2 \sec \left(\frac{c}{2}\right) \sin \left(\frac{d x}{2}\right) \sec ^3\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d}-\frac{2 \tan \left(\frac{c}{2}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d}+\frac{4 \sec \left(\frac{c}{2}\right) \sin \left(\frac{d x}{2}\right) \sec \left(\frac{c}{2}+\frac{d x}{2}\right)}{d}+\frac{4 \csc (c)}{d}\right)}{\cos ^{\frac{3}{2}}(c+d x) (a \sec (c+d x)+a)^2}","\frac{2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^2 d}-\frac{E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d}+\frac{\sin (c+d x)}{a^2 d \sqrt{\cos (c+d x)} (\sec (c+d x)+1)}-\frac{\sin (c+d x)}{3 d \cos ^{\frac{3}{2}}(c+d x) (a \sec (c+d x)+a)^2}",1,"((-1/2*I)*Cos[c/2 + (d*x)/2]^4*Csc[c/2]*Sec[c/2]*Sec[c + d*x]^2*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/(a + a*Sec[c + d*x])^2 - (4*Cos[c/2 + (d*x)/2]^4*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*Sec[c + d*x]^2*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*Sqrt[1 + Cot[c]^2]*(a + a*Sec[c + d*x])^2) + (Cos[c/2 + (d*x)/2]^4*((4*Csc[c])/d + (4*Sec[c/2]*Sec[c/2 + (d*x)/2]*Sin[(d*x)/2])/d - (2*Sec[c/2]*Sec[c/2 + (d*x)/2]^3*Sin[(d*x)/2])/(3*d) - (2*Sec[c/2 + (d*x)/2]^2*Tan[c/2])/(3*d)))/(Cos[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^2)","C",0
385,1,63,57,0.2373364,"\int \frac{1}{\cos ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x))^2} \, dx","Integrate[1/(Cos[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^2),x]","\frac{4 \cos ^4\left(\frac{1}{2} (c+d x)\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+\sin (c+d x) \sqrt{\cos (c+d x)}}{3 a^2 d (\cos (c+d x)+1)^2}","\frac{F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^2 d}+\frac{\sin (c+d x)}{3 d \cos ^{\frac{3}{2}}(c+d x) (a \sec (c+d x)+a)^2}",1,"(4*Cos[(c + d*x)/2]^4*EllipticF[(c + d*x)/2, 2] + Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*a^2*d*(1 + Cos[c + d*x])^2)","A",1
386,1,312,109,5.1246653,"\int \frac{1}{\cos ^{\frac{5}{2}}(c+d x) (a+a \sec (c+d x))^2} \, dx","Integrate[1/(Cos[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^2),x]","\frac{\cos ^4\left(\frac{1}{2} (c+d x)\right) \left(-\frac{\csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \left(7 \cos \left(\frac{1}{2} (c-d x)\right)+2 \cos \left(\frac{1}{2} (3 c+d x)\right)+3 \cos \left(\frac{1}{2} (c+3 d x)\right)\right) \sec ^3\left(\frac{1}{2} (c+d x)\right)}{2 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{4 i \sqrt{2} e^{-i (c+d x)} \left(3 \left(-1+e^{2 i c}\right) \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i (c+d x)}\right)-2 \left(-1+e^{2 i c}\right) e^{i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-e^{2 i (c+d x)}\right)+3 \left(1+e^{2 i (c+d x)}\right)\right) \sec ^2(c+d x)}{\left(-1+e^{2 i c}\right) d \sqrt{e^{-i (c+d x)} \left(1+e^{2 i (c+d x)}\right)}}\right)}{3 a^2 (\sec (c+d x)+1)^2}","\frac{2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^2 d}+\frac{E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d}-\frac{\sin (c+d x)}{a^2 d \sqrt{\cos (c+d x)} (\sec (c+d x)+1)}-\frac{\sin (c+d x)}{3 d \cos ^{\frac{3}{2}}(c+d x) (a \sec (c+d x)+a)^2}",1,"(Cos[(c + d*x)/2]^4*(-1/2*((7*Cos[(c - d*x)/2] + 2*Cos[(3*c + d*x)/2] + 3*Cos[(c + 3*d*x)/2])*Csc[c/2]*Sec[c/2]*Sec[(c + d*x)/2]^3)/(d*Cos[c + d*x]^(3/2)) + ((4*I)*Sqrt[2]*(3*(1 + E^((2*I)*(c + d*x))) + 3*(-1 + E^((2*I)*c))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[-1/4, 1/2, 3/4, -E^((2*I)*(c + d*x))] - 2*E^(I*(c + d*x))*(-1 + E^((2*I)*c))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/4, 1/2, 5/4, -E^((2*I)*(c + d*x))])*Sec[c + d*x]^2)/(d*E^(I*(c + d*x))*(-1 + E^((2*I)*c))*Sqrt[(1 + E^((2*I)*(c + d*x)))/E^(I*(c + d*x))])))/(3*a^2*(1 + Sec[c + d*x])^2)","C",1
387,1,393,136,6.2584195,"\int \frac{1}{\cos ^{\frac{7}{2}}(c+d x) (a+a \sec (c+d x))^2} \, dx","Integrate[1/(Cos[c + d*x]^(7/2)*(a + a*Sec[c + d*x])^2),x]","\frac{\cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right) \left(\frac{2 \sec \left(\frac{c}{2}\right) \sin \left(\frac{d x}{2}\right) \sec ^3\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d}+\frac{2 \tan \left(\frac{c}{2}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d}+\frac{8 \sec \left(\frac{c}{2}\right) \sin \left(\frac{d x}{2}\right) \sec \left(\frac{c}{2}+\frac{d x}{2}\right)}{d}+\frac{8 \sec (c) \sin (d x) \sec (c+d x)}{d}+\frac{8 \cot \left(\frac{c}{2}\right) \sec (c)}{d}\right)}{\cos ^{\frac{3}{2}}(c+d x) (a \sec (c+d x)+a)^2}-\frac{4 i \sqrt{2} e^{-i (c+d x)} \left(12 \left(-1+e^{2 i c}\right) \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i (c+d x)}\right)-5 \left(-1+e^{2 i c}\right) e^{i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-e^{2 i (c+d x)}\right)+12 \left(1+e^{2 i (c+d x)}\right)\right) \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right) \sec ^2(c+d x)}{3 \left(-1+e^{2 i c}\right) d \sqrt{e^{-i (c+d x)} \left(1+e^{2 i (c+d x)}\right)} (a \sec (c+d x)+a)^2}","-\frac{5 F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^2 d}-\frac{4 E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d}+\frac{4 \sin (c+d x)}{a^2 d \sqrt{\cos (c+d x)}}-\frac{5 \sin (c+d x)}{3 a^2 d \cos ^{\frac{3}{2}}(c+d x) (\sec (c+d x)+1)}-\frac{\sin (c+d x)}{3 d \cos ^{\frac{5}{2}}(c+d x) (a \sec (c+d x)+a)^2}",1,"(((-4*I)/3)*Sqrt[2]*Cos[c/2 + (d*x)/2]^4*(12*(1 + E^((2*I)*(c + d*x))) + 12*(-1 + E^((2*I)*c))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[-1/4, 1/2, 3/4, -E^((2*I)*(c + d*x))] - 5*E^(I*(c + d*x))*(-1 + E^((2*I)*c))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/4, 1/2, 5/4, -E^((2*I)*(c + d*x))])*Sec[c + d*x]^2)/(d*E^(I*(c + d*x))*(-1 + E^((2*I)*c))*Sqrt[(1 + E^((2*I)*(c + d*x)))/E^(I*(c + d*x))]*(a + a*Sec[c + d*x])^2) + (Cos[c/2 + (d*x)/2]^4*((8*Cot[c/2]*Sec[c])/d + (8*Sec[c/2]*Sec[c/2 + (d*x)/2]*Sin[(d*x)/2])/d + (2*Sec[c/2]*Sec[c/2 + (d*x)/2]^3*Sin[(d*x)/2])/(3*d) + (8*Sec[c]*Sec[c + d*x]*Sin[d*x])/d + (2*Sec[c/2 + (d*x)/2]^2*Tan[c/2])/(3*d)))/(Cos[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^2)","C",1
388,1,372,162,2.1360918,"\int \frac{1}{\cos ^{\frac{9}{2}}(c+d x) (a+a \sec (c+d x))^2} \, dx","Integrate[1/(Cos[c + d*x]^(9/2)*(a + a*Sec[c + d*x])^2),x]","\frac{\cos ^4\left(\frac{1}{2} (c+d x)\right) \left(-\frac{\csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \left(82 \cos \left(\frac{1}{2} (c-d x)\right)+65 \cos \left(\frac{1}{2} (3 c+d x)\right)+68 \cos \left(\frac{1}{2} (c+3 d x)\right)+37 \cos \left(\frac{1}{2} (5 c+3 d x)\right)+53 \cos \left(\frac{1}{2} (3 c+5 d x)\right)+10 \cos \left(\frac{1}{2} (7 c+5 d x)\right)+21 \cos \left(\frac{1}{2} (5 c+7 d x)\right)\right) \sec ^3\left(\frac{1}{2} (c+d x)\right)}{8 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{4 i \sqrt{2} e^{-i (c+d x)} \left(21 \left(-1+e^{2 i c}\right) \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i (c+d x)}\right)-10 \left(-1+e^{2 i c}\right) e^{i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-e^{2 i (c+d x)}\right)+21 \left(1+e^{2 i (c+d x)}\right)\right) \sec ^2(c+d x)}{\left(-1+e^{2 i c}\right) d \sqrt{e^{-i (c+d x)} \left(1+e^{2 i (c+d x)}\right)}}\right)}{3 a^2 (\sec (c+d x)+1)^2}","\frac{10 F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^2 d}+\frac{7 E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d}+\frac{10 \sin (c+d x)}{3 a^2 d \cos ^{\frac{3}{2}}(c+d x)}-\frac{7 \sin (c+d x)}{a^2 d \sqrt{\cos (c+d x)}}-\frac{7 \sin (c+d x)}{3 a^2 d \cos ^{\frac{5}{2}}(c+d x) (\sec (c+d x)+1)}-\frac{\sin (c+d x)}{3 d \cos ^{\frac{7}{2}}(c+d x) (a \sec (c+d x)+a)^2}",1,"(Cos[(c + d*x)/2]^4*(-1/8*((82*Cos[(c - d*x)/2] + 65*Cos[(3*c + d*x)/2] + 68*Cos[(c + 3*d*x)/2] + 37*Cos[(5*c + 3*d*x)/2] + 53*Cos[(3*c + 5*d*x)/2] + 10*Cos[(7*c + 5*d*x)/2] + 21*Cos[(5*c + 7*d*x)/2])*Csc[c/2]*Sec[c/2]*Sec[(c + d*x)/2]^3)/(d*Cos[c + d*x]^(7/2)) + ((4*I)*Sqrt[2]*(21*(1 + E^((2*I)*(c + d*x))) + 21*(-1 + E^((2*I)*c))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[-1/4, 1/2, 3/4, -E^((2*I)*(c + d*x))] - 10*E^(I*(c + d*x))*(-1 + E^((2*I)*c))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/4, 1/2, 5/4, -E^((2*I)*(c + d*x))])*Sec[c + d*x]^2)/(d*E^(I*(c + d*x))*(-1 + E^((2*I)*c))*Sqrt[(1 + E^((2*I)*(c + d*x)))/E^(I*(c + d*x))])))/(3*a^2*(1 + Sec[c + d*x])^2)","C",0
389,1,391,207,2.5705555,"\int \frac{\cos ^{\frac{5}{2}}(c+d x)}{(a+a \sec (c+d x))^3} \, dx","Integrate[Cos[c + d*x]^(5/2)/(a + a*Sec[c + d*x])^3,x]","\frac{2 \cos ^6\left(\frac{1}{2} (c+d x)\right) \left(\frac{\frac{1}{16} \sec \left(\frac{c}{2}\right) \left(770 \sin \left(c+\frac{d x}{2}\right)-840 \sin \left(c+\frac{3 d x}{2}\right)+150 \sin \left(2 c+\frac{3 d x}{2}\right)-238 \sin \left(2 c+\frac{5 d x}{2}\right)-40 \sin \left(3 c+\frac{5 d x}{2}\right)-5 \sin \left(3 c+\frac{7 d x}{2}\right)-5 \sin \left(4 c+\frac{7 d x}{2}\right)+\sin \left(4 c+\frac{9 d x}{2}\right)+\sin \left(5 c+\frac{9 d x}{2}\right)-1210 \sin \left(\frac{d x}{2}\right)\right) \sec ^5\left(\frac{1}{2} (c+d x)\right)-264 \cot (c)-198 \csc (c)}{\cos ^{\frac{5}{2}}(c+d x)}+\frac{42 i \sqrt{2} e^{-i (c+d x)} \left(11 \left(-1+e^{2 i c}\right) \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i (c+d x)}\right)+5 \left(-1+e^{2 i c}\right) e^{i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-e^{2 i (c+d x)}\right)+11 \left(1+e^{2 i (c+d x)}\right)\right) \sec ^3(c+d x)}{\left(-1+e^{2 i c}\right) \sqrt{e^{-i (c+d x)} \left(1+e^{2 i (c+d x)}\right)}}\right)}{5 a^3 d (\sec (c+d x)+1)^3}","-\frac{21 F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{2 a^3 d}+\frac{231 E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{10 a^3 d}+\frac{77 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{10 a^3 d}-\frac{21 \sin (c+d x) \sqrt{\cos (c+d x)}}{2 a^3 d}-\frac{63 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{10 d \left(a^3 \sec (c+d x)+a^3\right)}-\frac{4 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 a d (a \sec (c+d x)+a)^2}-\frac{\sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 d (a \sec (c+d x)+a)^3}",1,"(2*Cos[(c + d*x)/2]^6*(((42*I)*Sqrt[2]*(11*(1 + E^((2*I)*(c + d*x))) + 11*(-1 + E^((2*I)*c))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[-1/4, 1/2, 3/4, -E^((2*I)*(c + d*x))] + 5*E^(I*(c + d*x))*(-1 + E^((2*I)*c))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/4, 1/2, 5/4, -E^((2*I)*(c + d*x))])*Sec[c + d*x]^3)/(E^(I*(c + d*x))*(-1 + E^((2*I)*c))*Sqrt[(1 + E^((2*I)*(c + d*x)))/E^(I*(c + d*x))]) + (-264*Cot[c] - 198*Csc[c] + (Sec[c/2]*Sec[(c + d*x)/2]^5*(-1210*Sin[(d*x)/2] + 770*Sin[c + (d*x)/2] - 840*Sin[c + (3*d*x)/2] + 150*Sin[2*c + (3*d*x)/2] - 238*Sin[2*c + (5*d*x)/2] - 40*Sin[3*c + (5*d*x)/2] - 5*Sin[3*c + (7*d*x)/2] - 5*Sin[4*c + (7*d*x)/2] + Sin[4*c + (9*d*x)/2] + Sin[5*c + (9*d*x)/2]))/16)/Cos[c + d*x]^(5/2)))/(5*a^3*d*(1 + Sec[c + d*x])^3)","C",1
390,1,375,181,2.053637,"\int \frac{\cos ^{\frac{3}{2}}(c+d x)}{(a+a \sec (c+d x))^3} \, dx","Integrate[Cos[c + d*x]^(3/2)/(a + a*Sec[c + d*x])^3,x]","\frac{\cos ^6\left(\frac{1}{2} (c+d x)\right) \left(\frac{\frac{1}{4} \sec \left(\frac{c}{2}\right) \left(-709 \sin \left(c+\frac{d x}{2}\right)+715 \sin \left(c+\frac{3 d x}{2}\right)-170 \sin \left(2 c+\frac{3 d x}{2}\right)+202 \sin \left(2 c+\frac{5 d x}{2}\right)+25 \sin \left(3 c+\frac{5 d x}{2}\right)+5 \sin \left(3 c+\frac{7 d x}{2}\right)+5 \sin \left(4 c+\frac{7 d x}{2}\right)+1061 \sin \left(\frac{d x}{2}\right)\right) \sec ^5\left(\frac{1}{2} (c+d x)\right)+720 \cot (c)+708 \csc (c)}{3 d \cos ^{\frac{5}{2}}(c+d x)}-\frac{4 i \sqrt{2} e^{-i (c+d x)} \left(119 \left(-1+e^{2 i c}\right) \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i (c+d x)}\right)+55 \left(-1+e^{2 i c}\right) e^{i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-e^{2 i (c+d x)}\right)+119 \left(1+e^{2 i (c+d x)}\right)\right) \sec ^3(c+d x)}{\left(-1+e^{2 i c}\right) d \sqrt{e^{-i (c+d x)} \left(1+e^{2 i (c+d x)}\right)}}\right)}{5 a^3 (\sec (c+d x)+1)^3}","\frac{11 F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{2 a^3 d}-\frac{119 E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{10 a^3 d}+\frac{11 \sin (c+d x) \sqrt{\cos (c+d x)}}{2 a^3 d}-\frac{119 \sin (c+d x) \sqrt{\cos (c+d x)}}{30 d \left(a^3 \sec (c+d x)+a^3\right)}-\frac{2 \sin (c+d x) \sqrt{\cos (c+d x)}}{3 a d (a \sec (c+d x)+a)^2}-\frac{\sin (c+d x) \sqrt{\cos (c+d x)}}{5 d (a \sec (c+d x)+a)^3}",1,"(Cos[(c + d*x)/2]^6*(((-4*I)*Sqrt[2]*(119*(1 + E^((2*I)*(c + d*x))) + 119*(-1 + E^((2*I)*c))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[-1/4, 1/2, 3/4, -E^((2*I)*(c + d*x))] + 55*E^(I*(c + d*x))*(-1 + E^((2*I)*c))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/4, 1/2, 5/4, -E^((2*I)*(c + d*x))])*Sec[c + d*x]^3)/(d*E^(I*(c + d*x))*(-1 + E^((2*I)*c))*Sqrt[(1 + E^((2*I)*(c + d*x)))/E^(I*(c + d*x))]) + (720*Cot[c] + 708*Csc[c] + (Sec[c/2]*Sec[(c + d*x)/2]^5*(1061*Sin[(d*x)/2] - 709*Sin[c + (d*x)/2] + 715*Sin[c + (3*d*x)/2] - 170*Sin[2*c + (3*d*x)/2] + 202*Sin[2*c + (5*d*x)/2] + 25*Sin[3*c + (5*d*x)/2] + 5*Sin[3*c + (7*d*x)/2] + 5*Sin[4*c + (7*d*x)/2]))/4)/(3*d*Cos[c + d*x]^(5/2))))/(5*a^3*(1 + Sec[c + d*x])^3)","C",1
391,1,357,155,1.726984,"\int \frac{\sqrt{\cos (c+d x)}}{(a+a \sec (c+d x))^3} \, dx","Integrate[Sqrt[Cos[c + d*x]]/(a + a*Sec[c + d*x])^3,x]","\frac{\cos ^6\left(\frac{1}{2} (c+d x)\right) \left(-\frac{\csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \left(806 \cos \left(\frac{1}{2} (c-d x)\right)+664 \cos \left(\frac{1}{2} (3 c+d x)\right)+470 \cos \left(\frac{1}{2} (c+3 d x)\right)+265 \cos \left(\frac{1}{2} (5 c+3 d x)\right)+117 \cos \left(\frac{1}{2} (3 c+5 d x)\right)+30 \cos \left(\frac{1}{2} (7 c+5 d x)\right)\right) \sec ^5\left(\frac{1}{2} (c+d x)\right)}{8 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{4 i \sqrt{2} e^{-i (c+d x)} \left(147 \left(-1+e^{2 i c}\right) \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i (c+d x)}\right)+65 \left(-1+e^{2 i c}\right) e^{i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-e^{2 i (c+d x)}\right)+147 \left(1+e^{2 i (c+d x)}\right)\right) \sec ^3(c+d x)}{\left(-1+e^{2 i c}\right) d \sqrt{e^{-i (c+d x)} \left(1+e^{2 i (c+d x)}\right)}}\right)}{15 a^3 (\sec (c+d x)+1)^3}","-\frac{13 F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{6 a^3 d}+\frac{49 E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{10 a^3 d}-\frac{13 \sin (c+d x)}{6 d \sqrt{\cos (c+d x)} \left(a^3 \sec (c+d x)+a^3\right)}-\frac{8 \sin (c+d x)}{15 a d \sqrt{\cos (c+d x)} (a \sec (c+d x)+a)^2}-\frac{\sin (c+d x)}{5 d \sqrt{\cos (c+d x)} (a \sec (c+d x)+a)^3}",1,"(Cos[(c + d*x)/2]^6*(-1/8*((806*Cos[(c - d*x)/2] + 664*Cos[(3*c + d*x)/2] + 470*Cos[(c + 3*d*x)/2] + 265*Cos[(5*c + 3*d*x)/2] + 117*Cos[(3*c + 5*d*x)/2] + 30*Cos[(7*c + 5*d*x)/2])*Csc[c/2]*Sec[c/2]*Sec[(c + d*x)/2]^5)/(d*Cos[c + d*x]^(5/2)) + ((4*I)*Sqrt[2]*(147*(1 + E^((2*I)*(c + d*x))) + 147*(-1 + E^((2*I)*c))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[-1/4, 1/2, 3/4, -E^((2*I)*(c + d*x))] + 65*E^(I*(c + d*x))*(-1 + E^((2*I)*c))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/4, 1/2, 5/4, -E^((2*I)*(c + d*x))])*Sec[c + d*x]^3)/(d*E^(I*(c + d*x))*(-1 + E^((2*I)*c))*Sqrt[(1 + E^((2*I)*(c + d*x)))/E^(I*(c + d*x))])))/(15*a^3*(1 + Sec[c + d*x])^3)","C",0
392,1,721,155,6.2606222,"\int \frac{1}{\sqrt{\cos (c+d x)} (a+a \sec (c+d x))^3} \, dx","Integrate[1/(Sqrt[Cos[c + d*x]]*(a + a*Sec[c + d*x])^3),x]","-\frac{2 \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right) \sec ^3(c+d x) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin (c) \left(-\sqrt{\cot ^2(c)+1}\right) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec \left(d x-\tan ^{-1}(\cot (c))\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right)}{d \sqrt{\cot ^2(c)+1} (a \sec (c+d x)+a)^3}-\frac{9 i \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right) \sec ^3(c+d x) \left(\frac{2 e^{2 i d x} \sqrt{e^{-i d x} \left(2 i \sin (c) \left(-1+e^{2 i d x}\right)+2 \cos (c) \left(1+e^{2 i d x}\right)\right)} \sqrt{i \sin (2 c) e^{2 i d x}+\cos (2 c) e^{2 i d x}+1} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right)}{3 i d \cos (c) \left(1+e^{2 i d x}\right)-3 d \sin (c) \left(-1+e^{2 i d x}\right)}-\frac{2 \sqrt{e^{-i d x} \left(2 i \sin (c) \left(-1+e^{2 i d x}\right)+2 \cos (c) \left(1+e^{2 i d x}\right)\right)} \sqrt{i \sin (2 c) e^{2 i d x}+\cos (2 c) e^{2 i d x}+1} \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right)}{d \sin (c) \left(-1+e^{2 i d x}\right)-i d \cos (c) \left(1+e^{2 i d x}\right)}\right)}{10 (a \sec (c+d x)+a)^3}+\frac{\cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right) \left(\frac{2 \sec \left(\frac{c}{2}\right) \sin \left(\frac{d x}{2}\right) \sec ^5\left(\frac{c}{2}+\frac{d x}{2}\right)}{5 d}+\frac{2 \tan \left(\frac{c}{2}\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{5 d}-\frac{12 \sec \left(\frac{c}{2}\right) \sin \left(\frac{d x}{2}\right) \sec ^3\left(\frac{c}{2}+\frac{d x}{2}\right)}{5 d}-\frac{12 \tan \left(\frac{c}{2}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{5 d}+\frac{36 \sec \left(\frac{c}{2}\right) \sin \left(\frac{d x}{2}\right) \sec \left(\frac{c}{2}+\frac{d x}{2}\right)}{5 d}+\frac{36 \csc (c)}{5 d}\right)}{\cos ^{\frac{5}{2}}(c+d x) (a \sec (c+d x)+a)^3}","\frac{F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{2 a^3 d}-\frac{9 E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{10 a^3 d}+\frac{\sin (c+d x)}{2 d \sqrt{\cos (c+d x)} \left(a^3 \sec (c+d x)+a^3\right)}-\frac{\sin (c+d x)}{5 d \cos ^{\frac{3}{2}}(c+d x) (a \sec (c+d x)+a)^3}+\frac{2 \sin (c+d x)}{5 a d \sqrt{\cos (c+d x)} (a \sec (c+d x)+a)^2}",1,"(((-9*I)/10)*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*Sec[c/2]*Sec[c + d*x]^3*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/(a + a*Sec[c + d*x])^3 - (2*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*Sec[c + d*x]^3*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(d*Sqrt[1 + Cot[c]^2]*(a + a*Sec[c + d*x])^3) + (Cos[c/2 + (d*x)/2]^6*((36*Csc[c])/(5*d) + (36*Sec[c/2]*Sec[c/2 + (d*x)/2]*Sin[(d*x)/2])/(5*d) - (12*Sec[c/2]*Sec[c/2 + (d*x)/2]^3*Sin[(d*x)/2])/(5*d) + (2*Sec[c/2]*Sec[c/2 + (d*x)/2]^5*Sin[(d*x)/2])/(5*d) - (12*Sec[c/2 + (d*x)/2]^2*Tan[c/2])/(5*d) + (2*Sec[c/2 + (d*x)/2]^4*Tan[c/2])/(5*d)))/(Cos[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^3)","C",0
393,1,342,155,1.5668331,"\int \frac{1}{\cos ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x))^3} \, dx","Integrate[1/(Cos[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^3),x]","\frac{\cos ^6\left(\frac{1}{2} (c+d x)\right) \left(\frac{\csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \left(14 \cos \left(\frac{1}{2} (c-d x)\right)+16 \cos \left(\frac{1}{2} (3 c+d x)\right)+20 \cos \left(\frac{1}{2} (c+3 d x)\right)-5 \cos \left(\frac{1}{2} (5 c+3 d x)\right)+3 \cos \left(\frac{1}{2} (3 c+5 d x)\right)\right) \sec ^5\left(\frac{1}{2} (c+d x)\right)}{8 d \cos ^{\frac{5}{2}}(c+d x)}-\frac{4 i \sqrt{2} e^{-i (c+d x)} \left(3 \left(-1+e^{2 i c}\right) \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i (c+d x)}\right)+5 \left(-1+e^{2 i c}\right) e^{i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-e^{2 i (c+d x)}\right)+3 \left(1+e^{2 i (c+d x)}\right)\right) \sec ^3(c+d x)}{\left(-1+e^{2 i c}\right) d \sqrt{e^{-i (c+d x)} \left(1+e^{2 i (c+d x)}\right)}}\right)}{15 a^3 (\sec (c+d x)+1)^3}","\frac{F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{6 a^3 d}-\frac{E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{10 a^3 d}+\frac{\sin (c+d x)}{6 d \sqrt{\cos (c+d x)} \left(a^3 \sec (c+d x)+a^3\right)}+\frac{\sin (c+d x)}{5 d \cos ^{\frac{3}{2}}(c+d x) (a \sec (c+d x)+a)^3}-\frac{\sin (c+d x)}{15 a d \sqrt{\cos (c+d x)} (a \sec (c+d x)+a)^2}",1,"(Cos[(c + d*x)/2]^6*(((14*Cos[(c - d*x)/2] + 16*Cos[(3*c + d*x)/2] + 20*Cos[(c + 3*d*x)/2] - 5*Cos[(5*c + 3*d*x)/2] + 3*Cos[(3*c + 5*d*x)/2])*Csc[c/2]*Sec[c/2]*Sec[(c + d*x)/2]^5)/(8*d*Cos[c + d*x]^(5/2)) - ((4*I)*Sqrt[2]*(3*(1 + E^((2*I)*(c + d*x))) + 3*(-1 + E^((2*I)*c))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[-1/4, 1/2, 3/4, -E^((2*I)*(c + d*x))] + 5*E^(I*(c + d*x))*(-1 + E^((2*I)*c))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/4, 1/2, 5/4, -E^((2*I)*(c + d*x))])*Sec[c + d*x]^3)/(d*E^(I*(c + d*x))*(-1 + E^((2*I)*c))*Sqrt[(1 + E^((2*I)*(c + d*x)))/E^(I*(c + d*x))])))/(15*a^3*(1 + Sec[c + d*x])^3)","C",1
394,1,342,155,1.6033776,"\int \frac{1}{\cos ^{\frac{5}{2}}(c+d x) (a+a \sec (c+d x))^3} \, dx","Integrate[1/(Cos[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^3),x]","\frac{\cos ^6\left(\frac{1}{2} (c+d x)\right) \left(-\frac{\csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \left(4 \cos \left(\frac{1}{2} (c-d x)\right)+26 \cos \left(\frac{1}{2} (3 c+d x)\right)+10 \cos \left(\frac{1}{2} (c+3 d x)\right)+5 \cos \left(\frac{1}{2} (5 c+3 d x)\right)+3 \cos \left(\frac{1}{2} (3 c+5 d x)\right)\right) \sec ^5\left(\frac{1}{2} (c+d x)\right)}{8 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{4 i \sqrt{2} e^{-i (c+d x)} \left(3 \left(-1+e^{2 i c}\right) \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i (c+d x)}\right)-5 \left(-1+e^{2 i c}\right) e^{i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-e^{2 i (c+d x)}\right)+3 \left(1+e^{2 i (c+d x)}\right)\right) \sec ^3(c+d x)}{\left(-1+e^{2 i c}\right) d \sqrt{e^{-i (c+d x)} \left(1+e^{2 i (c+d x)}\right)}}\right)}{15 a^3 (\sec (c+d x)+1)^3}","\frac{F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{6 a^3 d}+\frac{E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{10 a^3 d}+\frac{\sin (c+d x)}{6 d \sqrt{\cos (c+d x)} \left(a^3 \sec (c+d x)+a^3\right)}-\frac{\sin (c+d x)}{5 d \cos ^{\frac{3}{2}}(c+d x) (a \sec (c+d x)+a)^3}-\frac{4 \sin (c+d x)}{15 a d \sqrt{\cos (c+d x)} (a \sec (c+d x)+a)^2}",1,"(Cos[(c + d*x)/2]^6*(-1/8*((4*Cos[(c - d*x)/2] + 26*Cos[(3*c + d*x)/2] + 10*Cos[(c + 3*d*x)/2] + 5*Cos[(5*c + 3*d*x)/2] + 3*Cos[(3*c + 5*d*x)/2])*Csc[c/2]*Sec[c/2]*Sec[(c + d*x)/2]^5)/(d*Cos[c + d*x]^(5/2)) + ((4*I)*Sqrt[2]*(3*(1 + E^((2*I)*(c + d*x))) + 3*(-1 + E^((2*I)*c))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[-1/4, 1/2, 3/4, -E^((2*I)*(c + d*x))] - 5*E^(I*(c + d*x))*(-1 + E^((2*I)*c))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/4, 1/2, 5/4, -E^((2*I)*(c + d*x))])*Sec[c + d*x]^3)/(d*E^(I*(c + d*x))*(-1 + E^((2*I)*c))*Sqrt[(1 + E^((2*I)*(c + d*x)))/E^(I*(c + d*x))])))/(15*a^3*(1 + Sec[c + d*x])^3)","C",1
395,1,721,155,6.2528939,"\int \frac{1}{\cos ^{\frac{7}{2}}(c+d x) (a+a \sec (c+d x))^3} \, dx","Integrate[1/(Cos[c + d*x]^(7/2)*(a + a*Sec[c + d*x])^3),x]","-\frac{2 \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right) \sec ^3(c+d x) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin (c) \left(-\sqrt{\cot ^2(c)+1}\right) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec \left(d x-\tan ^{-1}(\cot (c))\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right)}{d \sqrt{\cot ^2(c)+1} (a \sec (c+d x)+a)^3}+\frac{9 i \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right) \sec ^3(c+d x) \left(\frac{2 e^{2 i d x} \sqrt{e^{-i d x} \left(2 i \sin (c) \left(-1+e^{2 i d x}\right)+2 \cos (c) \left(1+e^{2 i d x}\right)\right)} \sqrt{i \sin (2 c) e^{2 i d x}+\cos (2 c) e^{2 i d x}+1} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right)}{3 i d \cos (c) \left(1+e^{2 i d x}\right)-3 d \sin (c) \left(-1+e^{2 i d x}\right)}-\frac{2 \sqrt{e^{-i d x} \left(2 i \sin (c) \left(-1+e^{2 i d x}\right)+2 \cos (c) \left(1+e^{2 i d x}\right)\right)} \sqrt{i \sin (2 c) e^{2 i d x}+\cos (2 c) e^{2 i d x}+1} \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right)}{d \sin (c) \left(-1+e^{2 i d x}\right)-i d \cos (c) \left(1+e^{2 i d x}\right)}\right)}{10 (a \sec (c+d x)+a)^3}+\frac{\cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right) \left(-\frac{2 \sec \left(\frac{c}{2}\right) \sin \left(\frac{d x}{2}\right) \sec ^5\left(\frac{c}{2}+\frac{d x}{2}\right)}{5 d}-\frac{2 \tan \left(\frac{c}{2}\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{5 d}-\frac{8 \sec \left(\frac{c}{2}\right) \sin \left(\frac{d x}{2}\right) \sec ^3\left(\frac{c}{2}+\frac{d x}{2}\right)}{5 d}-\frac{8 \tan \left(\frac{c}{2}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{5 d}-\frac{36 \sec \left(\frac{c}{2}\right) \sin \left(\frac{d x}{2}\right) \sec \left(\frac{c}{2}+\frac{d x}{2}\right)}{5 d}-\frac{36 \csc (c)}{5 d}\right)}{\cos ^{\frac{5}{2}}(c+d x) (a \sec (c+d x)+a)^3}","\frac{F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{2 a^3 d}+\frac{9 E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{10 a^3 d}-\frac{9 \sin (c+d x)}{10 d \sqrt{\cos (c+d x)} \left(a^3 \sec (c+d x)+a^3\right)}-\frac{2 \sin (c+d x)}{5 a d \cos ^{\frac{3}{2}}(c+d x) (a \sec (c+d x)+a)^2}-\frac{\sin (c+d x)}{5 d \cos ^{\frac{5}{2}}(c+d x) (a \sec (c+d x)+a)^3}",1,"(((9*I)/10)*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*Sec[c/2]*Sec[c + d*x]^3*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/(a + a*Sec[c + d*x])^3 - (2*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*Sec[c + d*x]^3*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(d*Sqrt[1 + Cot[c]^2]*(a + a*Sec[c + d*x])^3) + (Cos[c/2 + (d*x)/2]^6*((-36*Csc[c])/(5*d) - (36*Sec[c/2]*Sec[c/2 + (d*x)/2]*Sin[(d*x)/2])/(5*d) - (8*Sec[c/2]*Sec[c/2 + (d*x)/2]^3*Sin[(d*x)/2])/(5*d) - (2*Sec[c/2]*Sec[c/2 + (d*x)/2]^5*Sin[(d*x)/2])/(5*d) - (8*Sec[c/2 + (d*x)/2]^2*Tan[c/2])/(5*d) - (2*Sec[c/2 + (d*x)/2]^4*Tan[c/2])/(5*d)))/(Cos[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^3)","C",0
396,1,372,181,1.9118469,"\int \frac{1}{\cos ^{\frac{9}{2}}(c+d x) (a+a \sec (c+d x))^3} \, dx","Integrate[1/(Cos[c + d*x]^(9/2)*(a + a*Sec[c + d*x])^3),x]","\frac{\cos ^6\left(\frac{1}{2} (c+d x)\right) \left(\frac{\csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \left(1284 \cos \left(\frac{1}{2} (c-d x)\right)+921 \cos \left(\frac{1}{2} (3 c+d x)\right)+1243 \cos \left(\frac{1}{2} (c+3 d x)\right)+374 \cos \left(\frac{1}{2} (5 c+3 d x)\right)+670 \cos \left(\frac{1}{2} (3 c+5 d x)\right)+65 \cos \left(\frac{1}{2} (7 c+5 d x)\right)+147 \cos \left(\frac{1}{2} (5 c+7 d x)\right)\right) \sec ^5\left(\frac{1}{2} (c+d x)\right)}{16 d \cos ^{\frac{7}{2}}(c+d x)}-\frac{4 i \sqrt{2} e^{-i (c+d x)} \left(147 \left(-1+e^{2 i c}\right) \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i (c+d x)}\right)-65 \left(-1+e^{2 i c}\right) e^{i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-e^{2 i (c+d x)}\right)+147 \left(1+e^{2 i (c+d x)}\right)\right) \sec ^3(c+d x)}{\left(-1+e^{2 i c}\right) d \sqrt{e^{-i (c+d x)} \left(1+e^{2 i (c+d x)}\right)}}\right)}{15 a^3 (\sec (c+d x)+1)^3}","-\frac{13 F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{6 a^3 d}-\frac{49 E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{10 a^3 d}+\frac{49 \sin (c+d x)}{10 a^3 d \sqrt{\cos (c+d x)}}-\frac{13 \sin (c+d x)}{6 d \cos ^{\frac{3}{2}}(c+d x) \left(a^3 \sec (c+d x)+a^3\right)}-\frac{8 \sin (c+d x)}{15 a d \cos ^{\frac{5}{2}}(c+d x) (a \sec (c+d x)+a)^2}-\frac{\sin (c+d x)}{5 d \cos ^{\frac{7}{2}}(c+d x) (a \sec (c+d x)+a)^3}",1,"(Cos[(c + d*x)/2]^6*(((1284*Cos[(c - d*x)/2] + 921*Cos[(3*c + d*x)/2] + 1243*Cos[(c + 3*d*x)/2] + 374*Cos[(5*c + 3*d*x)/2] + 670*Cos[(3*c + 5*d*x)/2] + 65*Cos[(7*c + 5*d*x)/2] + 147*Cos[(5*c + 7*d*x)/2])*Csc[c/2]*Sec[c/2]*Sec[(c + d*x)/2]^5)/(16*d*Cos[c + d*x]^(7/2)) - ((4*I)*Sqrt[2]*(147*(1 + E^((2*I)*(c + d*x))) + 147*(-1 + E^((2*I)*c))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[-1/4, 1/2, 3/4, -E^((2*I)*(c + d*x))] - 65*E^(I*(c + d*x))*(-1 + E^((2*I)*c))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/4, 1/2, 5/4, -E^((2*I)*(c + d*x))])*Sec[c + d*x]^3)/(d*E^(I*(c + d*x))*(-1 + E^((2*I)*c))*Sqrt[(1 + E^((2*I)*(c + d*x)))/E^(I*(c + d*x))])))/(15*a^3*(1 + Sec[c + d*x])^3)","C",0
397,1,402,207,2.7765876,"\int \frac{1}{\cos ^{\frac{11}{2}}(c+d x) (a+a \sec (c+d x))^3} \, dx","Integrate[1/(Cos[c + d*x]^(11/2)*(a + a*Sec[c + d*x])^3),x]","\frac{\cos ^6\left(\frac{1}{2} (c+d x)\right) \left(-\frac{\csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \left(5134 \cos \left(\frac{1}{2} (c-d x)\right)+4148 \cos \left(\frac{1}{2} (3 c+d x)\right)+4664 \cos \left(\frac{1}{2} (c+3 d x)\right)+2476 \cos \left(\frac{1}{2} (5 c+3 d x)\right)+3340 \cos \left(\frac{1}{2} (3 c+5 d x)\right)+944 \cos \left(\frac{1}{2} (7 c+5 d x)\right)+1620 \cos \left(\frac{1}{2} (5 c+7 d x)\right)+165 \cos \left(\frac{1}{2} (9 c+7 d x)\right)+357 \cos \left(\frac{1}{2} (7 c+9 d x)\right)\right) \sec ^5\left(\frac{1}{2} (c+d x)\right)}{96 d \cos ^{\frac{9}{2}}(c+d x)}+\frac{4 i \sqrt{2} e^{-i (c+d x)} \left(119 \left(-1+e^{2 i c}\right) \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i (c+d x)}\right)-55 \left(-1+e^{2 i c}\right) e^{i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-e^{2 i (c+d x)}\right)+119 \left(1+e^{2 i (c+d x)}\right)\right) \sec ^3(c+d x)}{\left(-1+e^{2 i c}\right) d \sqrt{e^{-i (c+d x)} \left(1+e^{2 i (c+d x)}\right)}}\right)}{5 a^3 (\sec (c+d x)+1)^3}","\frac{11 F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{2 a^3 d}+\frac{119 E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{10 a^3 d}+\frac{11 \sin (c+d x)}{2 a^3 d \cos ^{\frac{3}{2}}(c+d x)}-\frac{119 \sin (c+d x)}{10 a^3 d \sqrt{\cos (c+d x)}}-\frac{119 \sin (c+d x)}{30 d \cos ^{\frac{5}{2}}(c+d x) \left(a^3 \sec (c+d x)+a^3\right)}-\frac{2 \sin (c+d x)}{3 a d \cos ^{\frac{7}{2}}(c+d x) (a \sec (c+d x)+a)^2}-\frac{\sin (c+d x)}{5 d \cos ^{\frac{9}{2}}(c+d x) (a \sec (c+d x)+a)^3}",1,"(Cos[(c + d*x)/2]^6*(-1/96*((5134*Cos[(c - d*x)/2] + 4148*Cos[(3*c + d*x)/2] + 4664*Cos[(c + 3*d*x)/2] + 2476*Cos[(5*c + 3*d*x)/2] + 3340*Cos[(3*c + 5*d*x)/2] + 944*Cos[(7*c + 5*d*x)/2] + 1620*Cos[(5*c + 7*d*x)/2] + 165*Cos[(9*c + 7*d*x)/2] + 357*Cos[(7*c + 9*d*x)/2])*Csc[c/2]*Sec[c/2]*Sec[(c + d*x)/2]^5)/(d*Cos[c + d*x]^(9/2)) + ((4*I)*Sqrt[2]*(119*(1 + E^((2*I)*(c + d*x))) + 119*(-1 + E^((2*I)*c))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[-1/4, 1/2, 3/4, -E^((2*I)*(c + d*x))] - 55*E^(I*(c + d*x))*(-1 + E^((2*I)*c))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/4, 1/2, 5/4, -E^((2*I)*(c + d*x))])*Sec[c + d*x]^3)/(d*E^(I*(c + d*x))*(-1 + E^((2*I)*c))*Sqrt[(1 + E^((2*I)*(c + d*x)))/E^(I*(c + d*x))])))/(5*a^3*(1 + Sec[c + d*x])^3)","C",0
398,1,80,153,0.2133287,"\int \cos ^{\frac{7}{2}}(c+d x) \sqrt{a+a \sec (c+d x)} \, dx","Integrate[Cos[c + d*x]^(7/2)*Sqrt[a + a*Sec[c + d*x]],x]","\frac{(140 \sin (c+d x)+42 \sin (2 (c+d x))+12 \sin (3 (c+d x))+5 \sin (4 (c+d x))) \sqrt{\cos (c+d x)} \sqrt{a (\sec (c+d x)+1)}}{140 d (\cos (c+d x)+1)}","\frac{2 a \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{7 d \sqrt{a \sec (c+d x)+a}}+\frac{12 a \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{35 d \sqrt{a \sec (c+d x)+a}}+\frac{16 a \sin (c+d x) \sqrt{\cos (c+d x)}}{35 d \sqrt{a \sec (c+d x)+a}}+\frac{32 a \sin (c+d x)}{35 d \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}",1,"(Sqrt[Cos[c + d*x]]*Sqrt[a*(1 + Sec[c + d*x])]*(140*Sin[c + d*x] + 42*Sin[2*(c + d*x)] + 12*Sin[3*(c + d*x)] + 5*Sin[4*(c + d*x)]))/(140*d*(1 + Cos[c + d*x]))","A",1
399,1,61,115,0.1782878,"\int \cos ^{\frac{5}{2}}(c+d x) \sqrt{a+a \sec (c+d x)} \, dx","Integrate[Cos[c + d*x]^(5/2)*Sqrt[a + a*Sec[c + d*x]],x]","\frac{\sqrt{\cos (c+d x)} (8 \cos (c+d x)+3 \cos (2 (c+d x))+19) \tan \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\sec (c+d x)+1)}}{15 d}","\frac{2 a \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 d \sqrt{a \sec (c+d x)+a}}+\frac{8 a \sin (c+d x) \sqrt{\cos (c+d x)}}{15 d \sqrt{a \sec (c+d x)+a}}+\frac{16 a \sin (c+d x)}{15 d \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}",1,"(Sqrt[Cos[c + d*x]]*(19 + 8*Cos[c + d*x] + 3*Cos[2*(c + d*x)])*Sqrt[a*(1 + Sec[c + d*x])]*Tan[(c + d*x)/2])/(15*d)","A",1
400,1,49,77,0.1154753,"\int \cos ^{\frac{3}{2}}(c+d x) \sqrt{a+a \sec (c+d x)} \, dx","Integrate[Cos[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]],x]","\frac{2 \sqrt{\cos (c+d x)} (\cos (c+d x)+2) \tan \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\sec (c+d x)+1)}}{3 d}","\frac{2 a \sin (c+d x) \sqrt{\cos (c+d x)}}{3 d \sqrt{a \sec (c+d x)+a}}+\frac{4 a \sin (c+d x)}{3 d \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}",1,"(2*Sqrt[Cos[c + d*x]]*(2 + Cos[c + d*x])*Sqrt[a*(1 + Sec[c + d*x])]*Tan[(c + d*x)/2])/(3*d)","A",1
401,1,39,36,0.1012934,"\int \sqrt{\cos (c+d x)} \sqrt{a+a \sec (c+d x)} \, dx","Integrate[Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]],x]","\frac{2 \sqrt{\cos (c+d x)} \tan \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\sec (c+d x)+1)}}{d}","\frac{2 a \sin (c+d x)}{d \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}",1,"(2*Sqrt[Cos[c + d*x]]*Sqrt[a*(1 + Sec[c + d*x])]*Tan[(c + d*x)/2])/d","A",1
402,1,74,57,0.1571373,"\int \frac{\sqrt{a+a \sec (c+d x)}}{\sqrt{\cos (c+d x)}} \, dx","Integrate[Sqrt[a + a*Sec[c + d*x]]/Sqrt[Cos[c + d*x]],x]","-\frac{2 \sqrt{\cos (c+d x)} \tan \left(\frac{1}{2} (c+d x)\right) \sqrt{\sec (c+d x)} \sqrt{a (\sec (c+d x)+1)} \sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)}{d \sqrt{1-\sec (c+d x)}}","\frac{2 \sqrt{a} \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{d}",1,"(-2*ArcSin[Sqrt[Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*Sqrt[a*(1 + Sec[c + d*x])]*Tan[(c + d*x)/2])/(d*Sqrt[1 - Sec[c + d*x]])","A",1
403,1,90,92,0.2431032,"\int \frac{\sqrt{a+a \sec (c+d x)}}{\cos ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[Sqrt[a + a*Sec[c + d*x]]/Cos[c + d*x]^(3/2),x]","\frac{2 a \sin (c+d x) \left(\frac{1}{2} \cos (c+d x)+\frac{\sin ^{-1}\left(\sqrt{1-\sec (c+d x)}\right)}{2 \sqrt{1-\sec (c+d x)} \sec ^{\frac{3}{2}}(c+d x)}\right)}{d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a (\sec (c+d x)+1)}}","\frac{a \sin (c+d x)}{d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{\sqrt{a} \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{d}",1,"(2*a*(Cos[c + d*x]/2 + ArcSin[Sqrt[1 - Sec[c + d*x]]]/(2*Sqrt[1 - Sec[c + d*x]]*Sec[c + d*x]^(3/2)))*Sin[c + d*x])/(d*Cos[c + d*x]^(5/2)*Sqrt[a*(1 + Sec[c + d*x])])","A",1
404,1,100,136,0.4858483,"\int \frac{\sqrt{a+a \sec (c+d x)}}{\cos ^{\frac{5}{2}}(c+d x)} \, dx","Integrate[Sqrt[a + a*Sec[c + d*x]]/Cos[c + d*x]^(5/2),x]","\frac{2 a \sin (c+d x) \left(\frac{1}{8} \cos (c+d x) (3 \cos (c+d x)+2)+\frac{3 \sin ^{-1}\left(\sqrt{1-\sec (c+d x)}\right)}{8 \sqrt{1-\sec (c+d x)} \sec ^{\frac{5}{2}}(c+d x)}\right)}{d \cos ^{\frac{7}{2}}(c+d x) \sqrt{a (\sec (c+d x)+1)}}","\frac{3 a \sin (c+d x)}{4 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{a \sin (c+d x)}{2 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{3 \sqrt{a} \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{4 d}",1,"(2*a*((Cos[c + d*x]*(2 + 3*Cos[c + d*x]))/8 + (3*ArcSin[Sqrt[1 - Sec[c + d*x]]])/(8*Sqrt[1 - Sec[c + d*x]]*Sec[c + d*x]^(5/2)))*Sin[c + d*x])/(d*Cos[c + d*x]^(7/2)*Sqrt[a*(1 + Sec[c + d*x])])","A",1
405,1,72,161,0.2758728,"\int \cos ^{\frac{7}{2}}(c+d x) (a+a \sec (c+d x))^{3/2} \, dx","Integrate[Cos[c + d*x]^(7/2)*(a + a*Sec[c + d*x])^(3/2),x]","\frac{a \sqrt{\cos (c+d x)} (253 \cos (c+d x)+78 \cos (2 (c+d x))+15 \cos (3 (c+d x))+494) \tan \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\sec (c+d x)+1)}}{210 d}","\frac{2 a^2 \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{7 d \sqrt{a \sec (c+d x)+a}}+\frac{26 a^2 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{35 d \sqrt{a \sec (c+d x)+a}}+\frac{104 a^2 \sin (c+d x) \sqrt{\cos (c+d x)}}{105 d \sqrt{a \sec (c+d x)+a}}+\frac{208 a^2 \sin (c+d x)}{105 d \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}",1,"(a*Sqrt[Cos[c + d*x]]*(494 + 253*Cos[c + d*x] + 78*Cos[2*(c + d*x)] + 15*Cos[3*(c + d*x)])*Sqrt[a*(1 + Sec[c + d*x])]*Tan[(c + d*x)/2])/(210*d)","A",1
406,1,60,116,0.2154589,"\int \cos ^{\frac{5}{2}}(c+d x) (a+a \sec (c+d x))^{3/2} \, dx","Integrate[Cos[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^(3/2),x]","\frac{a \sqrt{\cos (c+d x)} (6 \cos (c+d x)+\cos (2 (c+d x))+13) \tan \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\sec (c+d x)+1)}}{5 d}","\frac{8 a^2 \sin (c+d x)}{5 d \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}+\frac{2 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) (a \sec (c+d x)+a)^{3/2}}{5 d}+\frac{2 a \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}{5 d}",1,"(a*Sqrt[Cos[c + d*x]]*(13 + 6*Cos[c + d*x] + Cos[2*(c + d*x)])*Sqrt[a*(1 + Sec[c + d*x])]*Tan[(c + d*x)/2])/(5*d)","A",1
407,1,50,79,0.1645444,"\int \cos ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x))^{3/2} \, dx","Integrate[Cos[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^(3/2),x]","\frac{2 a \sqrt{\cos (c+d x)} (\cos (c+d x)+5) \tan \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\sec (c+d x)+1)}}{3 d}","\frac{8 a^2 \sin (c+d x)}{3 d \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}+\frac{2 a \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}{3 d}",1,"(2*a*Sqrt[Cos[c + d*x]]*(5 + Cos[c + d*x])*Sqrt[a*(1 + Sec[c + d*x])]*Tan[(c + d*x)/2])/(3*d)","A",1
408,1,81,96,0.1411145,"\int \sqrt{\cos (c+d x)} (a+a \sec (c+d x))^{3/2} \, dx","Integrate[Sqrt[Cos[c + d*x]]*(a + a*Sec[c + d*x])^(3/2),x]","\frac{2 a^2 \sin (c+d x) \left(\sqrt{1-\sec (c+d x)}+\sqrt{\sec (c+d x)} \sin ^{-1}\left(\sqrt{1-\sec (c+d x)}\right)\right)}{d \sqrt{\cos (c+d x)-1} \sqrt{a (\sec (c+d x)+1)}}","\frac{2 a^{3/2} \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{d}+\frac{2 a^2 \sin (c+d x)}{d \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}",1,"(2*a^2*(Sqrt[1 - Sec[c + d*x]] + ArcSin[Sqrt[1 - Sec[c + d*x]]]*Sqrt[Sec[c + d*x]])*Sin[c + d*x])/(d*Sqrt[-1 + Cos[c + d*x]]*Sqrt[a*(1 + Sec[c + d*x])])","A",0
409,1,92,95,0.3114062,"\int \frac{(a+a \sec (c+d x))^{3/2}}{\sqrt{\cos (c+d x)}} \, dx","Integrate[(a + a*Sec[c + d*x])^(3/2)/Sqrt[Cos[c + d*x]],x]","-\frac{a^2 \sin (c+d x) \left(\frac{3 \sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)}{\sqrt{\sec (c+d x)}}-\sqrt{1-\sec (c+d x)}\right)}{d \cos ^{\frac{3}{2}}(c+d x) \sqrt{1-\sec (c+d x)} \sqrt{a (\sec (c+d x)+1)}}","\frac{3 a^{3/2} \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{d}+\frac{a^2 \sin (c+d x)}{d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}",1,"-((a^2*(-Sqrt[1 - Sec[c + d*x]] + (3*ArcSin[Sqrt[Sec[c + d*x]]])/Sqrt[Sec[c + d*x]])*Sin[c + d*x])/(d*Cos[c + d*x]^(3/2)*Sqrt[1 - Sec[c + d*x]]*Sqrt[a*(1 + Sec[c + d*x])]))","A",1
410,1,99,140,0.430272,"\int \frac{(a+a \sec (c+d x))^{3/2}}{\cos ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[(a + a*Sec[c + d*x])^(3/2)/Cos[c + d*x]^(3/2),x]","\frac{a \sec \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\sec (c+d x)+1)} \left(-3 \sin \left(\frac{1}{2} (c+d x)\right)+7 \sin \left(\frac{3}{2} (c+d x)\right)+7 \sqrt{2} \cos ^2(c+d x) \tanh ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)\right)}{8 d \cos ^{\frac{3}{2}}(c+d x)}","\frac{7 a^{3/2} \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{4 d}+\frac{7 a^2 \sin (c+d x)}{4 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{a^2 \sin (c+d x)}{2 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}",1,"(a*Sec[(c + d*x)/2]*Sqrt[a*(1 + Sec[c + d*x])]*(7*Sqrt[2]*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[c + d*x]^2 - 3*Sin[(c + d*x)/2] + 7*Sin[(3*(c + d*x))/2]))/(8*d*Cos[c + d*x]^(3/2))","A",1
411,1,112,180,0.5228826,"\int \frac{(a+a \sec (c+d x))^{3/2}}{\cos ^{\frac{5}{2}}(c+d x)} \, dx","Integrate[(a + a*Sec[c + d*x])^(3/2)/Cos[c + d*x]^(5/2),x]","\frac{a \sec \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\sec (c+d x)+1)} \left(54 \sin \left(\frac{1}{2} (c+d x)\right)+11 \left(\sin \left(\frac{3}{2} (c+d x)\right)+3 \sin \left(\frac{5}{2} (c+d x)\right)\right)+66 \sqrt{2} \cos ^3(c+d x) \tanh ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)\right)}{96 d \cos ^{\frac{5}{2}}(c+d x)}","\frac{11 a^{3/2} \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{8 d}+\frac{11 a^2 \sin (c+d x)}{8 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{11 a^2 \sin (c+d x)}{12 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{a^2 \sin (c+d x)}{3 d \cos ^{\frac{7}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}",1,"(a*Sec[(c + d*x)/2]*Sqrt[a*(1 + Sec[c + d*x])]*(66*Sqrt[2]*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[c + d*x]^3 + 54*Sin[(c + d*x)/2] + 11*(Sin[(3*(c + d*x))/2] + 3*Sin[(5*(c + d*x))/2])))/(96*d*Cos[c + d*x]^(5/2))","A",1
412,1,90,201,0.2433494,"\int \cos ^{\frac{9}{2}}(c+d x) (a+a \sec (c+d x))^{5/2} \, dx","Integrate[Cos[c + d*x]^(9/2)*(a + a*Sec[c + d*x])^(5/2),x]","\frac{2 a^2 \sin (c+d x) \sqrt{\cos (c+d x)} \left(35 \cos ^4(c+d x)+130 \cos ^3(c+d x)+219 \cos ^2(c+d x)+292 \cos (c+d x)+584\right) \sqrt{a (\sec (c+d x)+1)}}{315 d (\cos (c+d x)+1)}","\frac{38 a^3 \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{63 d \sqrt{a \sec (c+d x)+a}}+\frac{146 a^3 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{105 d \sqrt{a \sec (c+d x)+a}}+\frac{584 a^3 \sin (c+d x) \sqrt{\cos (c+d x)}}{315 d \sqrt{a \sec (c+d x)+a}}+\frac{1168 a^3 \sin (c+d x)}{315 d \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}+\frac{2 a^2 \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}{9 d}",1,"(2*a^2*Sqrt[Cos[c + d*x]]*(584 + 292*Cos[c + d*x] + 219*Cos[c + d*x]^2 + 130*Cos[c + d*x]^3 + 35*Cos[c + d*x]^4)*Sqrt[a*(1 + Sec[c + d*x])]*Sin[c + d*x])/(315*d*(1 + Cos[c + d*x]))","A",1
413,1,74,156,0.2562411,"\int \cos ^{\frac{7}{2}}(c+d x) (a+a \sec (c+d x))^{5/2} \, dx","Integrate[Cos[c + d*x]^(7/2)*(a + a*Sec[c + d*x])^(5/2),x]","\frac{a^2 \sqrt{\cos (c+d x)} (101 \cos (c+d x)+24 \cos (2 (c+d x))+3 \cos (3 (c+d x))+208) \tan \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\sec (c+d x)+1)}}{42 d}","\frac{64 a^3 \sin (c+d x)}{21 d \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}+\frac{16 a^2 \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}{21 d}+\frac{2 a \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) (a \sec (c+d x)+a)^{3/2}}{7 d}+\frac{2 \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) (a \sec (c+d x)+a)^{5/2}}{7 d}",1,"(a^2*Sqrt[Cos[c + d*x]]*(208 + 101*Cos[c + d*x] + 24*Cos[2*(c + d*x)] + 3*Cos[3*(c + d*x)])*Sqrt[a*(1 + Sec[c + d*x])]*Tan[(c + d*x)/2])/(42*d)","A",1
414,1,64,119,0.2392572,"\int \cos ^{\frac{5}{2}}(c+d x) (a+a \sec (c+d x))^{5/2} \, dx","Integrate[Cos[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^(5/2),x]","\frac{a^2 \sqrt{\cos (c+d x)} (28 \cos (c+d x)+3 \cos (2 (c+d x))+89) \tan \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\sec (c+d x)+1)}}{15 d}","\frac{64 a^3 \sin (c+d x)}{15 d \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}+\frac{16 a^2 \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}{15 d}+\frac{2 a \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) (a \sec (c+d x)+a)^{3/2}}{5 d}",1,"(a^2*Sqrt[Cos[c + d*x]]*(89 + 28*Cos[c + d*x] + 3*Cos[2*(c + d*x)])*Sqrt[a*(1 + Sec[c + d*x])]*Tan[(c + d*x)/2])/(15*d)","A",1
415,1,93,138,0.244404,"\int \cos ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x))^{5/2} \, dx","Integrate[Cos[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^(5/2),x]","\frac{2 a^3 \sin (c+d x) \left((\cos (c+d x)+8) \sqrt{1-\sec (c+d x)}+3 \sqrt{\sec (c+d x)} \sin ^{-1}\left(\sqrt{1-\sec (c+d x)}\right)\right)}{3 d \sqrt{\cos (c+d x)-1} \sqrt{a (\sec (c+d x)+1)}}","\frac{2 a^{5/2} \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{d}+\frac{14 a^3 \sin (c+d x)}{3 d \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}+\frac{2 a^2 \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}{3 d}",1,"(2*a^3*((8 + Cos[c + d*x])*Sqrt[1 - Sec[c + d*x]] + 3*ArcSin[Sqrt[1 - Sec[c + d*x]]]*Sqrt[Sec[c + d*x]])*Sin[c + d*x])/(3*d*Sqrt[-1 + Cos[c + d*x]]*Sqrt[a*(1 + Sec[c + d*x])])","A",0
416,1,90,132,0.2899437,"\int \sqrt{\cos (c+d x)} (a+a \sec (c+d x))^{5/2} \, dx","Integrate[Sqrt[Cos[c + d*x]]*(a + a*Sec[c + d*x])^(5/2),x]","\frac{a^3 \sin (c+d x) \left(\sqrt{1-\sec (c+d x)} (\sec (c+d x)+2)+5 \sqrt{\sec (c+d x)} \sin ^{-1}\left(\sqrt{1-\sec (c+d x)}\right)\right)}{d \sqrt{\cos (c+d x)-1} \sqrt{a (\sec (c+d x)+1)}}","\frac{5 a^{5/2} \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{d}+\frac{a^3 \sin (c+d x)}{d \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}+\frac{a^2 \sin (c+d x) \sqrt{a \sec (c+d x)+a}}{d \sqrt{\cos (c+d x)}}",1,"(a^3*(5*ArcSin[Sqrt[1 - Sec[c + d*x]]]*Sqrt[Sec[c + d*x]] + Sqrt[1 - Sec[c + d*x]]*(2 + Sec[c + d*x]))*Sin[c + d*x])/(d*Sqrt[-1 + Cos[c + d*x]]*Sqrt[a*(1 + Sec[c + d*x])])","A",0
417,1,95,140,0.6494983,"\int \frac{(a+a \sec (c+d x))^{5/2}}{\sqrt{\cos (c+d x)}} \, dx","Integrate[(a + a*Sec[c + d*x])^(5/2)/Sqrt[Cos[c + d*x]],x]","\frac{a^2 \tan \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\sec (c+d x)+1)} \left(\sqrt{1-\sec (c+d x)} (2 \sec (c+d x)+11)-\frac{19 \sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)}{\sqrt{\sec (c+d x)}}\right)}{4 d \sqrt{\cos (c+d x)-1}}","\frac{19 a^{5/2} \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{4 d}+\frac{9 a^3 \sin (c+d x)}{4 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{a^2 \sin (c+d x) \sqrt{a \sec (c+d x)+a}}{2 d \cos ^{\frac{3}{2}}(c+d x)}",1,"(a^2*Sqrt[a*(1 + Sec[c + d*x])]*((-19*ArcSin[Sqrt[Sec[c + d*x]]])/Sqrt[Sec[c + d*x]] + Sqrt[1 - Sec[c + d*x]]*(11 + 2*Sec[c + d*x]))*Tan[(c + d*x)/2])/(4*d*Sqrt[-1 + Cos[c + d*x]])","A",0
418,1,180,180,5.4729989,"\int \frac{(a+a \sec (c+d x))^{5/2}}{\cos ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[(a + a*Sec[c + d*x])^(5/2)/Cos[c + d*x]^(3/2),x]","\frac{a^2 (\cos (c+d x)+1)^2 \sec ^5\left(\frac{1}{2} (c+d x)\right) \sqrt{a (\sec (c+d x)+1)} \left(-75 i e^{\frac{1}{2} i (c+d x)} \, _2F_1\left(\frac{1}{4},1;\frac{5}{4};-e^{2 i (c+d x)}\right) \cos ^3(c+d x)-25 i e^{\frac{3}{2} i (c+d x)} \, _2F_1\left(\frac{3}{4},1;\frac{7}{4};-e^{2 i (c+d x)}\right) \cos ^3(c+d x)+\sin \left(\frac{1}{2} (c+d x)\right) \left(75 \cos ^2(c+d x)+34 \cos (c+d x)+8\right)\right)}{96 d \cos ^{\frac{5}{2}}(c+d x)}","\frac{25 a^{5/2} \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{8 d}+\frac{25 a^3 \sin (c+d x)}{8 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{13 a^3 \sin (c+d x)}{12 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{a^2 \sin (c+d x) \sqrt{a \sec (c+d x)+a}}{3 d \cos ^{\frac{5}{2}}(c+d x)}",1,"(a^2*(1 + Cos[c + d*x])^2*Sec[(c + d*x)/2]^5*Sqrt[a*(1 + Sec[c + d*x])]*((-75*I)*E^((I/2)*(c + d*x))*Cos[c + d*x]^3*Hypergeometric2F1[1/4, 1, 5/4, -E^((2*I)*(c + d*x))] - (25*I)*E^(((3*I)/2)*(c + d*x))*Cos[c + d*x]^3*Hypergeometric2F1[3/4, 1, 7/4, -E^((2*I)*(c + d*x))] + (8 + 34*Cos[c + d*x] + 75*Cos[c + d*x]^2)*Sin[(c + d*x)/2]))/(96*d*Cos[c + d*x]^(5/2))","C",1
419,1,190,220,5.5668837,"\int \frac{(a+a \sec (c+d x))^{5/2}}{\cos ^{\frac{5}{2}}(c+d x)} \, dx","Integrate[(a + a*Sec[c + d*x])^(5/2)/Cos[c + d*x]^(5/2),x]","\frac{a^2 (\cos (c+d x)+1)^2 \sec ^5\left(\frac{1}{2} (c+d x)\right) \sqrt{a (\sec (c+d x)+1)} \left(-489 i e^{\frac{1}{2} i (c+d x)} \, _2F_1\left(\frac{1}{4},1;\frac{5}{4};-e^{2 i (c+d x)}\right) \cos ^4(c+d x)-163 i e^{\frac{3}{2} i (c+d x)} \, _2F_1\left(\frac{3}{4},1;\frac{7}{4};-e^{2 i (c+d x)}\right) \cos ^4(c+d x)+\sin \left(\frac{1}{2} (c+d x)\right) \left(489 \cos ^3(c+d x)+326 \cos ^2(c+d x)+184 \cos (c+d x)+48\right)\right)}{768 d \cos ^{\frac{7}{2}}(c+d x)}","\frac{163 a^{5/2} \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{64 d}+\frac{163 a^3 \sin (c+d x)}{64 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{163 a^3 \sin (c+d x)}{96 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{17 a^3 \sin (c+d x)}{24 d \cos ^{\frac{7}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{a^2 \sin (c+d x) \sqrt{a \sec (c+d x)+a}}{4 d \cos ^{\frac{7}{2}}(c+d x)}",1,"(a^2*(1 + Cos[c + d*x])^2*Sec[(c + d*x)/2]^5*Sqrt[a*(1 + Sec[c + d*x])]*((-489*I)*E^((I/2)*(c + d*x))*Cos[c + d*x]^4*Hypergeometric2F1[1/4, 1, 5/4, -E^((2*I)*(c + d*x))] - (163*I)*E^(((3*I)/2)*(c + d*x))*Cos[c + d*x]^4*Hypergeometric2F1[3/4, 1, 7/4, -E^((2*I)*(c + d*x))] + (48 + 184*Cos[c + d*x] + 326*Cos[c + d*x]^2 + 489*Cos[c + d*x]^3)*Sin[(c + d*x)/2]))/(768*d*Cos[c + d*x]^(7/2))","C",1
420,1,136,189,0.3081346,"\int \frac{\cos ^{\frac{5}{2}}(c+d x)}{\sqrt{a+a \sec (c+d x)}} \, dx","Integrate[Cos[c + d*x]^(5/2)/Sqrt[a + a*Sec[c + d*x]],x]","\frac{\sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \left(2 \sqrt{1-\sec (c+d x)} \left(13 \sec ^2(c+d x)-\sec (c+d x)+3\right)+15 \sqrt{2} \sec ^{\frac{5}{2}}(c+d x) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{\sec (c+d x)}}{\sqrt{1-\sec (c+d x)}}\right)\right)}{15 d \sqrt{1-\sec (c+d x)} \sqrt{a (\sec (c+d x)+1)}}","\frac{2 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 d \sqrt{a \sec (c+d x)+a}}-\frac{2 \sin (c+d x) \sqrt{\cos (c+d x)}}{15 d \sqrt{a \sec (c+d x)+a}}+\frac{26 \sin (c+d x)}{15 d \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}-\frac{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{\sqrt{a} d}",1,"(Cos[c + d*x]^(3/2)*(15*Sqrt[2]*ArcTan[(Sqrt[2]*Sqrt[Sec[c + d*x]])/Sqrt[1 - Sec[c + d*x]]]*Sec[c + d*x]^(5/2) + 2*Sqrt[1 - Sec[c + d*x]]*(3 - Sec[c + d*x] + 13*Sec[c + d*x]^2))*Sin[c + d*x])/(15*d*Sqrt[1 - Sec[c + d*x]]*Sqrt[a*(1 + Sec[c + d*x])])","A",1
421,1,116,151,0.2154997,"\int \frac{\cos ^{\frac{3}{2}}(c+d x)}{\sqrt{a+a \sec (c+d x)}} \, dx","Integrate[Cos[c + d*x]^(3/2)/Sqrt[a + a*Sec[c + d*x]],x]","\frac{\sin (c+d x) \sqrt{\cos (c+d x)} \left(2 (1-\sec (c+d x))^{3/2}-3 \sqrt{2} \sec ^{\frac{3}{2}}(c+d x) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{\sec (c+d x)}}{\sqrt{1-\sec (c+d x)}}\right)\right)}{3 d \sqrt{1-\sec (c+d x)} \sqrt{a (\sec (c+d x)+1)}}","\frac{2 \sin (c+d x) \sqrt{\cos (c+d x)}}{3 d \sqrt{a \sec (c+d x)+a}}-\frac{2 \sin (c+d x)}{3 d \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}+\frac{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{\sqrt{a} d}",1,"(Sqrt[Cos[c + d*x]]*(2*(1 - Sec[c + d*x])^(3/2) - 3*Sqrt[2]*ArcTan[(Sqrt[2]*Sqrt[Sec[c + d*x]])/Sqrt[1 - Sec[c + d*x]]]*Sec[c + d*x]^(3/2))*Sin[c + d*x])/(3*d*Sqrt[1 - Sec[c + d*x]]*Sqrt[a*(1 + Sec[c + d*x])])","A",1
422,1,100,113,0.0999774,"\int \frac{\sqrt{\cos (c+d x)}}{\sqrt{a+a \sec (c+d x)}} \, dx","Integrate[Sqrt[Cos[c + d*x]]/Sqrt[a + a*Sec[c + d*x]],x]","\frac{\sin (c+d x) \left(2 \sqrt{1-\sec (c+d x)}+\sqrt{2} \sqrt{\sec (c+d x)} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{\sec (c+d x)}}{\sqrt{1-\sec (c+d x)}}\right)\right)}{d \sqrt{\cos (c+d x)-1} \sqrt{a (\sec (c+d x)+1)}}","\frac{2 \sin (c+d x)}{d \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}-\frac{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{\sqrt{a} d}",1,"((2*Sqrt[1 - Sec[c + d*x]] + Sqrt[2]*ArcTan[(Sqrt[2]*Sqrt[Sec[c + d*x]])/Sqrt[1 - Sec[c + d*x]]]*Sqrt[Sec[c + d*x]])*Sin[c + d*x])/(d*Sqrt[-1 + Cos[c + d*x]]*Sqrt[a*(1 + Sec[c + d*x])])","A",1
423,1,95,56,0.0722243,"\int \frac{1}{\sqrt{\cos (c+d x)} \sqrt{a+a \sec (c+d x)}} \, dx","Integrate[1/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]),x]","-\frac{\sqrt{2} \sin (c+d x) \sqrt{\cos (c+d x)} \sec ^{\frac{3}{2}}(c+d x) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{\sec (c+d x)}}{\sqrt{1-\sec (c+d x)}}\right)}{d \sqrt{1-\sec (c+d x)} \sqrt{a (\sec (c+d x)+1)}}","\frac{\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}\right)}{\sqrt{a} d}",1,"-((Sqrt[2]*ArcTan[(Sqrt[2]*Sqrt[Sec[c + d*x]])/Sqrt[1 - Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(d*Sqrt[1 - Sec[c + d*x]]*Sqrt[a*(1 + Sec[c + d*x])]))","A",1
424,1,109,135,0.0922209,"\int \frac{1}{\cos ^{\frac{3}{2}}(c+d x) \sqrt{a+a \sec (c+d x)}} \, dx","Integrate[1/(Cos[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]]),x]","\frac{\sin (c+d x) \sqrt{\cos (c+d x)} \sec ^{\frac{3}{2}}(c+d x) \left(\sqrt{2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{\sec (c+d x)}}{\sqrt{1-\sec (c+d x)}}\right)-2 \sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right)}{d \sqrt{1-\sec (c+d x)} \sqrt{a (\sec (c+d x)+1)}}","\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{\sqrt{a} d}-\frac{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{\sqrt{a} d}",1,"((-2*ArcSin[Sqrt[Sec[c + d*x]]] + Sqrt[2]*ArcTan[(Sqrt[2]*Sqrt[Sec[c + d*x]])/Sqrt[1 - Sec[c + d*x]]])*Sqrt[Cos[c + d*x]]*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(d*Sqrt[1 - Sec[c + d*x]]*Sqrt[a*(1 + Sec[c + d*x])])","A",1
425,1,145,168,0.2357675,"\int \frac{1}{\cos ^{\frac{5}{2}}(c+d x) \sqrt{a+a \sec (c+d x)}} \, dx","Integrate[1/(Cos[c + d*x]^(5/2)*Sqrt[a + a*Sec[c + d*x]]),x]","\frac{\sin (c+d x) \sqrt{\cos (c+d x)} \sec ^{\frac{3}{2}}(c+d x) \left(\sqrt{-((\sec (c+d x)-1) \sec (c+d x))}+\sin ^{-1}\left(\sqrt{1-\sec (c+d x)}\right)+2 \sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)-\sqrt{2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{\sec (c+d x)}}{\sqrt{1-\sec (c+d x)}}\right)\right)}{d \sqrt{1-\sec (c+d x)} \sqrt{a (\sec (c+d x)+1)}}","\frac{\sin (c+d x)}{d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{\sqrt{a} d}-\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{\sqrt{a} d}",1,"(Sqrt[Cos[c + d*x]]*Sec[c + d*x]^(3/2)*(ArcSin[Sqrt[1 - Sec[c + d*x]]] + 2*ArcSin[Sqrt[Sec[c + d*x]]] - Sqrt[2]*ArcTan[(Sqrt[2]*Sqrt[Sec[c + d*x]])/Sqrt[1 - Sec[c + d*x]]] + Sqrt[-((-1 + Sec[c + d*x])*Sec[c + d*x])])*Sin[c + d*x])/(d*Sqrt[1 - Sec[c + d*x]]*Sqrt[a*(1 + Sec[c + d*x])])","A",0
426,1,178,211,0.3916237,"\int \frac{1}{\cos ^{\frac{7}{2}}(c+d x) \sqrt{a+a \sec (c+d x)}} \, dx","Integrate[1/(Cos[c + d*x]^(7/2)*Sqrt[a + a*Sec[c + d*x]]),x]","\frac{\sin (c+d x) \sqrt{\cos (c+d x)} \sec ^{\frac{3}{2}}(c+d x) \left(2 \sqrt{1-\sec (c+d x)} \sec ^{\frac{3}{2}}(c+d x)-\sqrt{-((\sec (c+d x)-1) \sec (c+d x))}-\sin ^{-1}\left(\sqrt{1-\sec (c+d x)}\right)-8 \sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)+4 \sqrt{2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{\sec (c+d x)}}{\sqrt{1-\sec (c+d x)}}\right)\right)}{4 d \sqrt{1-\sec (c+d x)} \sqrt{a (\sec (c+d x)+1)}}","-\frac{\sin (c+d x)}{4 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{\sin (c+d x)}{2 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}-\frac{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{\sqrt{a} d}+\frac{7 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{4 \sqrt{a} d}",1,"(Sqrt[Cos[c + d*x]]*Sec[c + d*x]^(3/2)*(-ArcSin[Sqrt[1 - Sec[c + d*x]]] - 8*ArcSin[Sqrt[Sec[c + d*x]]] + 4*Sqrt[2]*ArcTan[(Sqrt[2]*Sqrt[Sec[c + d*x]])/Sqrt[1 - Sec[c + d*x]]] + 2*Sqrt[1 - Sec[c + d*x]]*Sec[c + d*x]^(3/2) - Sqrt[-((-1 + Sec[c + d*x])*Sec[c + d*x])])*Sin[c + d*x])/(4*d*Sqrt[1 - Sec[c + d*x]]*Sqrt[a*(1 + Sec[c + d*x])])","A",0
427,1,152,237,0.9245158,"\int \frac{\cos ^{\frac{5}{2}}(c+d x)}{(a+a \sec (c+d x))^{3/2}} \, dx","Integrate[Cos[c + d*x]^(5/2)/(a + a*Sec[c + d*x])^(3/2),x]","\frac{75 \sqrt{2} \sin (c+d x) \cos ^2\left(\frac{1}{2} (c+d x)\right) \sec ^{\frac{3}{2}}(c+d x) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{\sec (c+d x)}}{\sqrt{1-\sec (c+d x)}}\right)+\sqrt{1-\sec (c+d x)} \left(-2 \sin (2 (c+d x))+49 \tan (c+d x)+4 \sin (c+d x) \left(\cos ^2(c+d x)+9\right)\right)}{10 d \sqrt{\cos (c+d x)-1} (a (\sec (c+d x)+1))^{3/2}}","-\frac{15 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}+\frac{9 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{10 a d \sqrt{a \sec (c+d x)+a}}-\frac{\sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{2 d (a \sec (c+d x)+a)^{3/2}}-\frac{13 \sin (c+d x) \sqrt{\cos (c+d x)}}{10 a d \sqrt{a \sec (c+d x)+a}}+\frac{49 \sin (c+d x)}{10 a d \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}",1,"(75*Sqrt[2]*ArcTan[(Sqrt[2]*Sqrt[Sec[c + d*x]])/Sqrt[1 - Sec[c + d*x]]]*Cos[(c + d*x)/2]^2*Sec[c + d*x]^(3/2)*Sin[c + d*x] + Sqrt[1 - Sec[c + d*x]]*(4*(9 + Cos[c + d*x]^2)*Sin[c + d*x] - 2*Sin[2*(c + d*x)] + 49*Tan[c + d*x]))/(10*d*Sqrt[-1 + Cos[c + d*x]]*(a*(1 + Sec[c + d*x]))^(3/2))","A",0
428,1,133,197,0.6481252,"\int \frac{\cos ^{\frac{3}{2}}(c+d x)}{(a+a \sec (c+d x))^{3/2}} \, dx","Integrate[Cos[c + d*x]^(3/2)/(a + a*Sec[c + d*x])^(3/2),x]","\frac{\sin (c+d x) \left(\sqrt{1-\sec (c+d x)} (4 \cos (c+d x)-19 \sec (c+d x)-12)-33 \sqrt{2} \cos ^2\left(\frac{1}{2} (c+d x)\right) \sec ^{\frac{3}{2}}(c+d x) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{\sec (c+d x)}}{\sqrt{1-\sec (c+d x)}}\right)\right)}{6 d \sqrt{\cos (c+d x)-1} (a (\sec (c+d x)+1))^{3/2}}","\frac{11 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}+\frac{7 \sin (c+d x) \sqrt{\cos (c+d x)}}{6 a d \sqrt{a \sec (c+d x)+a}}-\frac{19 \sin (c+d x)}{6 a d \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}-\frac{\sin (c+d x) \sqrt{\cos (c+d x)}}{2 d (a \sec (c+d x)+a)^{3/2}}",1,"(((-12 + 4*Cos[c + d*x] - 19*Sec[c + d*x])*Sqrt[1 - Sec[c + d*x]] - 33*Sqrt[2]*ArcTan[(Sqrt[2]*Sqrt[Sec[c + d*x]])/Sqrt[1 - Sec[c + d*x]]]*Cos[(c + d*x)/2]^2*Sec[c + d*x]^(3/2))*Sin[c + d*x])/(6*d*Sqrt[-1 + Cos[c + d*x]]*(a*(1 + Sec[c + d*x]))^(3/2))","A",0
429,1,138,157,0.9278449,"\int \frac{\sqrt{\cos (c+d x)}}{(a+a \sec (c+d x))^{3/2}} \, dx","Integrate[Sqrt[Cos[c + d*x]]/(a + a*Sec[c + d*x])^(3/2),x]","\frac{\sin (c+d x) \sqrt{\sec (c+d x)} \left((4 \cos (c+d x)+5) \sqrt{(\cos (c+d x)-1) \sec ^2(c+d x)}+7 \sqrt{2} \cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{\sec (c+d x)}}{\sqrt{1-\sec (c+d x)}}\right)\right)}{2 d \sqrt{\cos (c+d x)-1} (a (\sec (c+d x)+1))^{3/2}}","-\frac{7 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}+\frac{5 \sin (c+d x)}{2 a d \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}-\frac{\sin (c+d x)}{2 d \sqrt{\cos (c+d x)} (a \sec (c+d x)+a)^{3/2}}",1,"(Sqrt[Sec[c + d*x]]*(7*Sqrt[2]*ArcTan[(Sqrt[2]*Sqrt[Sec[c + d*x]])/Sqrt[1 - Sec[c + d*x]]]*Cos[(c + d*x)/2]^2*Sec[c + d*x] + (5 + 4*Cos[c + d*x])*Sqrt[(-1 + Cos[c + d*x])*Sec[c + d*x]^2])*Sin[c + d*x])/(2*d*Sqrt[-1 + Cos[c + d*x]]*(a*(1 + Sec[c + d*x]))^(3/2))","A",1
430,1,131,117,0.4430323,"\int \frac{1}{\sqrt{\cos (c+d x)} (a+a \sec (c+d x))^{3/2}} \, dx","Integrate[1/(Sqrt[Cos[c + d*x]]*(a + a*Sec[c + d*x])^(3/2)),x]","-\frac{\sin (c+d x) \left(2 \sqrt{-((\sec (c+d x)-1) \sec (c+d x))}+3 \sqrt{2} (\sec (c+d x)+1) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{\sec (c+d x)}}{\sqrt{1-\sec (c+d x)}}\right)\right)}{4 a d \sqrt{\cos (c+d x)-1} (\cos (c+d x)+1) \sqrt{\sec (c+d x)} \sqrt{a (\sec (c+d x)+1)}}","\frac{3 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}-\frac{\sin (c+d x)}{2 d \cos ^{\frac{3}{2}}(c+d x) (a \sec (c+d x)+a)^{3/2}}",1,"-1/4*((2*Sqrt[-((-1 + Sec[c + d*x])*Sec[c + d*x])] + 3*Sqrt[2]*ArcTan[(Sqrt[2]*Sqrt[Sec[c + d*x]])/Sqrt[1 - Sec[c + d*x]]]*(1 + Sec[c + d*x]))*Sin[c + d*x])/(a*d*Sqrt[-1 + Cos[c + d*x]]*(1 + Cos[c + d*x])*Sqrt[Sec[c + d*x]]*Sqrt[a*(1 + Sec[c + d*x])])","A",1
431,1,248,117,0.866626,"\int \frac{1}{\cos ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x))^{3/2}} \, dx","Integrate[1/(Cos[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^(3/2)),x]","\frac{\sin (c+d x) \sqrt{\cos (c+d x)} \sec ^{\frac{5}{2}}(c+d x) \left(\sqrt{1-\sec (c+d x)} \sec ^{\frac{3}{2}}(c+d x)+\cos (2 (c+d x)) \sqrt{1-\sec (c+d x)} \sec ^{\frac{3}{2}}(c+d x)-\sqrt{2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{\sec (c+d x)}}{\sqrt{1-\sec (c+d x)}}\right)+2 (\cos (c+d x)+1) \sin ^{-1}\left(\sqrt{1-\sec (c+d x)}\right)+2 (\cos (c+d x)+1) \sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)-\sqrt{2} \cos (c+d x) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{\sec (c+d x)}}{\sqrt{1-\sec (c+d x)}}\right)\right)}{4 d \sqrt{1-\sec (c+d x)} (a (\sec (c+d x)+1))^{3/2}}","\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}+\frac{\sin (c+d x)}{2 d \cos ^{\frac{3}{2}}(c+d x) (a \sec (c+d x)+a)^{3/2}}",1,"(Sqrt[Cos[c + d*x]]*Sec[c + d*x]^(5/2)*(-(Sqrt[2]*ArcTan[(Sqrt[2]*Sqrt[Sec[c + d*x]])/Sqrt[1 - Sec[c + d*x]]]) - Sqrt[2]*ArcTan[(Sqrt[2]*Sqrt[Sec[c + d*x]])/Sqrt[1 - Sec[c + d*x]]]*Cos[c + d*x] + 2*ArcSin[Sqrt[1 - Sec[c + d*x]]]*(1 + Cos[c + d*x]) + 2*ArcSin[Sqrt[Sec[c + d*x]]]*(1 + Cos[c + d*x]) + Sqrt[1 - Sec[c + d*x]]*Sec[c + d*x]^(3/2) + Cos[2*(c + d*x)]*Sqrt[1 - Sec[c + d*x]]*Sec[c + d*x]^(3/2))*Sin[c + d*x])/(4*d*Sqrt[1 - Sec[c + d*x]]*(a*(1 + Sec[c + d*x]))^(3/2))","B",1
432,1,248,174,0.8384464,"\int \frac{1}{\cos ^{\frac{5}{2}}(c+d x) (a+a \sec (c+d x))^{3/2}} \, dx","Integrate[1/(Cos[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^(3/2)),x]","-\frac{\sin (c+d x) \sqrt{\cos (c+d x)} \sec ^{\frac{5}{2}}(c+d x) \left(\sqrt{1-\sec (c+d x)} \sec ^{\frac{3}{2}}(c+d x)+\cos (2 (c+d x)) \sqrt{1-\sec (c+d x)} \sec ^{\frac{3}{2}}(c+d x)-5 \sqrt{2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{\sec (c+d x)}}{\sqrt{1-\sec (c+d x)}}\right)+2 (\cos (c+d x)+1) \sin ^{-1}\left(\sqrt{1-\sec (c+d x)}\right)+10 (\cos (c+d x)+1) \sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)-5 \sqrt{2} \cos (c+d x) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{\sec (c+d x)}}{\sqrt{1-\sec (c+d x)}}\right)\right)}{4 d \sqrt{1-\sec (c+d x)} (a (\sec (c+d x)+1))^{3/2}}","-\frac{5 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{a^{3/2} d}-\frac{\sin (c+d x)}{2 d \cos ^{\frac{3}{2}}(c+d x) (a \sec (c+d x)+a)^{3/2}}",1,"-1/4*(Sqrt[Cos[c + d*x]]*Sec[c + d*x]^(5/2)*(-5*Sqrt[2]*ArcTan[(Sqrt[2]*Sqrt[Sec[c + d*x]])/Sqrt[1 - Sec[c + d*x]]] - 5*Sqrt[2]*ArcTan[(Sqrt[2]*Sqrt[Sec[c + d*x]])/Sqrt[1 - Sec[c + d*x]]]*Cos[c + d*x] + 2*ArcSin[Sqrt[1 - Sec[c + d*x]]]*(1 + Cos[c + d*x]) + 10*ArcSin[Sqrt[Sec[c + d*x]]]*(1 + Cos[c + d*x]) + Sqrt[1 - Sec[c + d*x]]*Sec[c + d*x]^(3/2) + Cos[2*(c + d*x)]*Sqrt[1 - Sec[c + d*x]]*Sec[c + d*x]^(3/2))*Sin[c + d*x])/(d*Sqrt[1 - Sec[c + d*x]]*(a*(1 + Sec[c + d*x]))^(3/2))","A",1
433,1,242,214,0.971575,"\int \frac{1}{\cos ^{\frac{7}{2}}(c+d x) (a+a \sec (c+d x))^{3/2}} \, dx","Integrate[1/(Cos[c + d*x]^(7/2)*(a + a*Sec[c + d*x])^(3/2)),x]","\frac{\sin (c+d x) \sqrt{\cos (c+d x)} \sec ^{\frac{5}{2}}(c+d x) \left(6 \cos (c+d x) \sqrt{(\cos (c+d x)-1) \sec ^2(c+d x)}+4 \sqrt{(\cos (c+d x)-1) \sec ^2(c+d x)}-9 \sqrt{2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{\sec (c+d x)}}{\sqrt{1-\sec (c+d x)}}\right)+6 (\cos (c+d x)+1) \sin ^{-1}\left(\sqrt{1-\sec (c+d x)}\right)+18 (\cos (c+d x)+1) \sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)-9 \sqrt{2} \cos (c+d x) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{\sec (c+d x)}}{\sqrt{1-\sec (c+d x)}}\right)\right)}{4 d \sqrt{1-\sec (c+d x)} (a (\sec (c+d x)+1))^{3/2}}","\frac{9 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}-\frac{3 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{a^{3/2} d}+\frac{3 \sin (c+d x)}{2 a d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}-\frac{\sin (c+d x)}{2 d \cos ^{\frac{5}{2}}(c+d x) (a \sec (c+d x)+a)^{3/2}}",1,"(Sqrt[Cos[c + d*x]]*Sec[c + d*x]^(5/2)*(-9*Sqrt[2]*ArcTan[(Sqrt[2]*Sqrt[Sec[c + d*x]])/Sqrt[1 - Sec[c + d*x]]] - 9*Sqrt[2]*ArcTan[(Sqrt[2]*Sqrt[Sec[c + d*x]])/Sqrt[1 - Sec[c + d*x]]]*Cos[c + d*x] + 6*ArcSin[Sqrt[1 - Sec[c + d*x]]]*(1 + Cos[c + d*x]) + 18*ArcSin[Sqrt[Sec[c + d*x]]]*(1 + Cos[c + d*x]) + 4*Sqrt[(-1 + Cos[c + d*x])*Sec[c + d*x]^2] + 6*Cos[c + d*x]*Sqrt[(-1 + Cos[c + d*x])*Sec[c + d*x]^2])*Sin[c + d*x])/(4*d*Sqrt[1 - Sec[c + d*x]]*(a*(1 + Sec[c + d*x]))^(3/2))","A",0
434,1,144,237,1.1689114,"\int \frac{\cos ^{\frac{3}{2}}(c+d x)}{(a+a \sec (c+d x))^{5/2}} \, dx","Integrate[Cos[c + d*x]^(3/2)/(a + a*Sec[c + d*x])^(5/2),x]","-\frac{\sin (c+d x) \left(2 \sqrt{1-\sec (c+d x)} \left(-32 \cos (c+d x)+299 \sec ^2(c+d x)+503 \sec (c+d x)+160\right)+1956 \sqrt{2} \cos ^4\left(\frac{1}{2} (c+d x)\right) \sec ^{\frac{5}{2}}(c+d x) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{\sec (c+d x)}}{\sqrt{1-\sec (c+d x)}}\right)\right)}{96 d \sqrt{\cos (c+d x)-1} (a (\sec (c+d x)+1))^{5/2}}","\frac{163 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}+\frac{95 \sin (c+d x) \sqrt{\cos (c+d x)}}{48 a^2 d \sqrt{a \sec (c+d x)+a}}-\frac{299 \sin (c+d x)}{48 a^2 d \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}-\frac{17 \sin (c+d x) \sqrt{\cos (c+d x)}}{16 a d (a \sec (c+d x)+a)^{3/2}}-\frac{\sin (c+d x) \sqrt{\cos (c+d x)}}{4 d (a \sec (c+d x)+a)^{5/2}}",1,"-1/96*((1956*Sqrt[2]*ArcTan[(Sqrt[2]*Sqrt[Sec[c + d*x]])/Sqrt[1 - Sec[c + d*x]]]*Cos[(c + d*x)/2]^4*Sec[c + d*x]^(5/2) + 2*Sqrt[1 - Sec[c + d*x]]*(160 - 32*Cos[c + d*x] + 503*Sec[c + d*x] + 299*Sec[c + d*x]^2))*Sin[c + d*x])/(d*Sqrt[-1 + Cos[c + d*x]]*(a*(1 + Sec[c + d*x]))^(5/2))","A",0
435,1,141,197,0.9554961,"\int \frac{\sqrt{\cos (c+d x)}}{(a+a \sec (c+d x))^{5/2}} \, dx","Integrate[Sqrt[Cos[c + d*x]]/(a + a*Sec[c + d*x])^(5/2),x]","\frac{\sqrt{1-\sec (c+d x)} (32 \sin (c+d x)+\tan (c+d x) (49 \sec (c+d x)+85))+150 \sqrt{2} \sin (c+d x) \cos ^4\left(\frac{1}{2} (c+d x)\right) \sec ^{\frac{5}{2}}(c+d x) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{\sec (c+d x)}}{\sqrt{1-\sec (c+d x)}}\right)}{16 d \sqrt{\cos (c+d x)-1} (a (\sec (c+d x)+1))^{5/2}}","-\frac{75 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}+\frac{49 \sin (c+d x)}{16 a^2 d \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}-\frac{13 \sin (c+d x)}{16 a d \sqrt{\cos (c+d x)} (a \sec (c+d x)+a)^{3/2}}-\frac{\sin (c+d x)}{4 d \sqrt{\cos (c+d x)} (a \sec (c+d x)+a)^{5/2}}",1,"(150*Sqrt[2]*ArcTan[(Sqrt[2]*Sqrt[Sec[c + d*x]])/Sqrt[1 - Sec[c + d*x]]]*Cos[(c + d*x)/2]^4*Sec[c + d*x]^(5/2)*Sin[c + d*x] + Sqrt[1 - Sec[c + d*x]]*(32*Sin[c + d*x] + (85 + 49*Sec[c + d*x])*Tan[c + d*x]))/(16*d*Sqrt[-1 + Cos[c + d*x]]*(a*(1 + Sec[c + d*x]))^(5/2))","A",1
436,1,168,157,1.318469,"\int \frac{1}{\sqrt{\cos (c+d x)} (a+a \sec (c+d x))^{5/2}} \, dx","Integrate[1/(Sqrt[Cos[c + d*x]]*(a + a*Sec[c + d*x])^(5/2)),x]","-\frac{\sin (c+d x) \sqrt{\cos (c+d x)} \sec ^{\frac{3}{2}}(c+d x) \left(9 \sqrt{1-\sec (c+d x)} \sec ^{\frac{3}{2}}(c+d x)+13 \sqrt{-((\sec (c+d x)-1) \sec (c+d x))}+38 \sqrt{2} \cos ^4\left(\frac{1}{2} (c+d x)\right) \sec ^2(c+d x) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{\sec (c+d x)}}{\sqrt{1-\sec (c+d x)}}\right)\right)}{16 d \sqrt{1-\sec (c+d x)} (a (\sec (c+d x)+1))^{5/2}}","\frac{19 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}-\frac{9 \sin (c+d x)}{16 a d \cos ^{\frac{3}{2}}(c+d x) (a \sec (c+d x)+a)^{3/2}}-\frac{\sin (c+d x)}{4 d \cos ^{\frac{3}{2}}(c+d x) (a \sec (c+d x)+a)^{5/2}}",1,"-1/16*(Sqrt[Cos[c + d*x]]*Sec[c + d*x]^(3/2)*(9*Sqrt[1 - Sec[c + d*x]]*Sec[c + d*x]^(3/2) + 38*Sqrt[2]*ArcTan[(Sqrt[2]*Sqrt[Sec[c + d*x]])/Sqrt[1 - Sec[c + d*x]]]*Cos[(c + d*x)/2]^4*Sec[c + d*x]^2 + 13*Sqrt[-((-1 + Sec[c + d*x])*Sec[c + d*x])])*Sin[c + d*x])/(d*Sqrt[1 - Sec[c + d*x]]*(a*(1 + Sec[c + d*x]))^(5/2))","A",0
437,1,224,157,3.7181571,"\int \frac{1}{\cos ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x))^{5/2}} \, dx","Integrate[1/(Cos[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^(5/2)),x]","\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \left(-8 \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)-\frac{5 (\sec (c+d x)+1) \left(2 \sin (c+d x) \sqrt{1-\sec (c+d x)} \sec ^{\frac{5}{2}}(c+d x)-\tan (c+d x) (\sec (c+d x)+1) \left(2 \sqrt{-((\sec (c+d x)-1) \sec (c+d x))}+2 \sin ^{-1}\left(\sqrt{1-\sec (c+d x)}\right)+2 \sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)-\sqrt{2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{\sec (c+d x)}}{\sqrt{1-\sec (c+d x)}}\right)\right)\right)}{\sqrt{1-\sec (c+d x)}}\right)}{32 d (a (\sec (c+d x)+1))^{5/2}}","\frac{5 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}+\frac{5 \sin (c+d x)}{16 a d \cos ^{\frac{3}{2}}(c+d x) (a \sec (c+d x)+a)^{3/2}}-\frac{\sin (c+d x)}{4 d \cos ^{\frac{5}{2}}(c+d x) (a \sec (c+d x)+a)^{5/2}}",1,"(Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*(-8*Sec[c + d*x]^(5/2)*Sin[c + d*x] - (5*(1 + Sec[c + d*x])*(2*Sqrt[1 - Sec[c + d*x]]*Sec[c + d*x]^(5/2)*Sin[c + d*x] - (1 + Sec[c + d*x])*(2*ArcSin[Sqrt[1 - Sec[c + d*x]]] + 2*ArcSin[Sqrt[Sec[c + d*x]]] - Sqrt[2]*ArcTan[(Sqrt[2]*Sqrt[Sec[c + d*x]])/Sqrt[1 - Sec[c + d*x]]] + 2*Sqrt[-((-1 + Sec[c + d*x])*Sec[c + d*x])])*Tan[c + d*x]))/Sqrt[1 - Sec[c + d*x]]))/(32*d*(a*(1 + Sec[c + d*x]))^(5/2))","A",0
438,1,341,157,1.3525884,"\int \frac{1}{\cos ^{\frac{5}{2}}(c+d x) (a+a \sec (c+d x))^{5/2}} \, dx","Integrate[1/(Cos[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^(5/2)),x]","\frac{14 \sin (c+d x) \sqrt{1-\sec (c+d x)} \sec ^{\frac{3}{2}}(c+d x)+6 \sin (c+d x) \sqrt{-((\sec (c+d x)-1) \sec (c+d x))}-3 \sqrt{2} \tan (c+d x) \sec (c+d x) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{\sec (c+d x)}}{\sqrt{1-\sec (c+d x)}}\right)-6 \sqrt{2} \tan (c+d x) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{\sec (c+d x)}}{\sqrt{1-\sec (c+d x)}}\right)+6 \sin ^{-1}\left(\sqrt{1-\sec (c+d x)}\right) (\sin (c+d x)+\tan (c+d x) (\sec (c+d x)+2))+6 \sin ^{-1}\left(\sqrt{\sec (c+d x)}\right) (\sin (c+d x)+\tan (c+d x) (\sec (c+d x)+2))-3 \sqrt{2} \sin (c+d x) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{\sec (c+d x)}}{\sqrt{1-\sec (c+d x)}}\right)}{32 a^2 d \cos ^{\frac{3}{2}}(c+d x) (\cos (c+d x)+1)^2 \sqrt{1-\sec (c+d x)} \sec ^{\frac{5}{2}}(c+d x) \sqrt{a (\sec (c+d x)+1)}}","\frac{3 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}+\frac{3 \sin (c+d x)}{16 a d \cos ^{\frac{3}{2}}(c+d x) (a \sec (c+d x)+a)^{3/2}}+\frac{\sin (c+d x)}{4 d \cos ^{\frac{5}{2}}(c+d x) (a \sec (c+d x)+a)^{5/2}}",1,"(-3*Sqrt[2]*ArcTan[(Sqrt[2]*Sqrt[Sec[c + d*x]])/Sqrt[1 - Sec[c + d*x]]]*Sin[c + d*x] + 14*Sqrt[1 - Sec[c + d*x]]*Sec[c + d*x]^(3/2)*Sin[c + d*x] + 6*Sqrt[-((-1 + Sec[c + d*x])*Sec[c + d*x])]*Sin[c + d*x] - 6*Sqrt[2]*ArcTan[(Sqrt[2]*Sqrt[Sec[c + d*x]])/Sqrt[1 - Sec[c + d*x]]]*Tan[c + d*x] - 3*Sqrt[2]*ArcTan[(Sqrt[2]*Sqrt[Sec[c + d*x]])/Sqrt[1 - Sec[c + d*x]]]*Sec[c + d*x]*Tan[c + d*x] + 6*ArcSin[Sqrt[1 - Sec[c + d*x]]]*(Sin[c + d*x] + (2 + Sec[c + d*x])*Tan[c + d*x]) + 6*ArcSin[Sqrt[Sec[c + d*x]]]*(Sin[c + d*x] + (2 + Sec[c + d*x])*Tan[c + d*x]))/(32*a^2*d*Cos[c + d*x]^(3/2)*(1 + Cos[c + d*x])^2*Sqrt[1 - Sec[c + d*x]]*Sec[c + d*x]^(5/2)*Sqrt[a*(1 + Sec[c + d*x])])","B",1
439,1,328,214,1.2086546,"\int \frac{1}{\cos ^{\frac{7}{2}}(c+d x) (a+a \sec (c+d x))^{5/2}} \, dx","Integrate[1/(Cos[c + d*x]^(7/2)*(a + a*Sec[c + d*x])^(5/2)),x]","\frac{-30 \sin (c+d x) \sqrt{1-\sec (c+d x)} \sec ^{\frac{3}{2}}(c+d x)-22 \sin (c+d x) \sqrt{-((\sec (c+d x)-1) \sec (c+d x))}+43 \sqrt{2} \tan (c+d x) \sec (c+d x) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{\sec (c+d x)}}{\sqrt{1-\sec (c+d x)}}\right)+86 \sqrt{2} \tan (c+d x) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{\sec (c+d x)}}{\sqrt{1-\sec (c+d x)}}\right)-22 \sin ^{-1}\left(\sqrt{1-\sec (c+d x)}\right) (\sin (c+d x)+\tan (c+d x) (\sec (c+d x)+2))-86 \sin ^{-1}\left(\sqrt{\sec (c+d x)}\right) (\sin (c+d x)+\tan (c+d x) (\sec (c+d x)+2))+43 \sqrt{2} \sin (c+d x) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{\sec (c+d x)}}{\sqrt{1-\sec (c+d x)}}\right)}{32 d \cos ^{\frac{9}{2}}(c+d x) \sqrt{1-\sec (c+d x)} \sec ^{\frac{7}{2}}(c+d x) (a (\sec (c+d x)+1))^{5/2}}","-\frac{43 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{a^{5/2} d}-\frac{11 \sin (c+d x)}{16 a d \cos ^{\frac{3}{2}}(c+d x) (a \sec (c+d x)+a)^{3/2}}-\frac{\sin (c+d x)}{4 d \cos ^{\frac{5}{2}}(c+d x) (a \sec (c+d x)+a)^{5/2}}",1,"(43*Sqrt[2]*ArcTan[(Sqrt[2]*Sqrt[Sec[c + d*x]])/Sqrt[1 - Sec[c + d*x]]]*Sin[c + d*x] - 30*Sqrt[1 - Sec[c + d*x]]*Sec[c + d*x]^(3/2)*Sin[c + d*x] - 22*Sqrt[-((-1 + Sec[c + d*x])*Sec[c + d*x])]*Sin[c + d*x] + 86*Sqrt[2]*ArcTan[(Sqrt[2]*Sqrt[Sec[c + d*x]])/Sqrt[1 - Sec[c + d*x]]]*Tan[c + d*x] + 43*Sqrt[2]*ArcTan[(Sqrt[2]*Sqrt[Sec[c + d*x]])/Sqrt[1 - Sec[c + d*x]]]*Sec[c + d*x]*Tan[c + d*x] - 22*ArcSin[Sqrt[1 - Sec[c + d*x]]]*(Sin[c + d*x] + (2 + Sec[c + d*x])*Tan[c + d*x]) - 86*ArcSin[Sqrt[Sec[c + d*x]]]*(Sin[c + d*x] + (2 + Sec[c + d*x])*Tan[c + d*x]))/(32*d*Cos[c + d*x]^(9/2)*Sqrt[1 - Sec[c + d*x]]*Sec[c + d*x]^(7/2)*(a*(1 + Sec[c + d*x]))^(5/2))","A",1
440,1,348,254,1.5397524,"\int \frac{1}{\cos ^{\frac{9}{2}}(c+d x) (a+a \sec (c+d x))^{5/2}} \, dx","Integrate[1/(Cos[c + d*x]^(9/2)*(a + a*Sec[c + d*x])^(5/2)),x]","\frac{\sqrt{\sec (c+d x)} \left(32 \sin (c+d x) \sqrt{1-\sec (c+d x)} \sec ^{\frac{5}{2}}(c+d x)+110 \sin (c+d x) \sqrt{1-\sec (c+d x)} \sec ^{\frac{3}{2}}(c+d x)+70 \sin (c+d x) \sqrt{-((\sec (c+d x)-1) \sec (c+d x))}-115 \sqrt{2} \tan (c+d x) \sec (c+d x) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{\sec (c+d x)}}{\sqrt{1-\sec (c+d x)}}\right)-230 \sqrt{2} \tan (c+d x) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{\sec (c+d x)}}{\sqrt{1-\sec (c+d x)}}\right)+70 \sin ^{-1}\left(\sqrt{1-\sec (c+d x)}\right) (\sin (c+d x)+\tan (c+d x) (\sec (c+d x)+2))+230 \sin ^{-1}\left(\sqrt{\sec (c+d x)}\right) (\sin (c+d x)+\tan (c+d x) (\sec (c+d x)+2))-115 \sqrt{2} \sin (c+d x) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{\sec (c+d x)}}{\sqrt{1-\sec (c+d x)}}\right)\right)}{32 d \sqrt{\cos (c+d x)-1} (a (\sec (c+d x)+1))^{5/2}}","\frac{115 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}-\frac{5 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{a^{5/2} d}+\frac{35 \sin (c+d x)}{16 a^2 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}-\frac{15 \sin (c+d x)}{16 a d \cos ^{\frac{5}{2}}(c+d x) (a \sec (c+d x)+a)^{3/2}}-\frac{\sin (c+d x)}{4 d \cos ^{\frac{7}{2}}(c+d x) (a \sec (c+d x)+a)^{5/2}}",1,"(Sqrt[Sec[c + d*x]]*(-115*Sqrt[2]*ArcTan[(Sqrt[2]*Sqrt[Sec[c + d*x]])/Sqrt[1 - Sec[c + d*x]]]*Sin[c + d*x] + 110*Sqrt[1 - Sec[c + d*x]]*Sec[c + d*x]^(3/2)*Sin[c + d*x] + 32*Sqrt[1 - Sec[c + d*x]]*Sec[c + d*x]^(5/2)*Sin[c + d*x] + 70*Sqrt[-((-1 + Sec[c + d*x])*Sec[c + d*x])]*Sin[c + d*x] - 230*Sqrt[2]*ArcTan[(Sqrt[2]*Sqrt[Sec[c + d*x]])/Sqrt[1 - Sec[c + d*x]]]*Tan[c + d*x] - 115*Sqrt[2]*ArcTan[(Sqrt[2]*Sqrt[Sec[c + d*x]])/Sqrt[1 - Sec[c + d*x]]]*Sec[c + d*x]*Tan[c + d*x] + 70*ArcSin[Sqrt[1 - Sec[c + d*x]]]*(Sin[c + d*x] + (2 + Sec[c + d*x])*Tan[c + d*x]) + 230*ArcSin[Sqrt[Sec[c + d*x]]]*(Sin[c + d*x] + (2 + Sec[c + d*x])*Tan[c + d*x])))/(32*d*Sqrt[-1 + Cos[c + d*x]]*(a*(1 + Sec[c + d*x]))^(5/2))","A",0
441,1,308,244,1.5895831,"\int (d \cos (e+f x))^n (a+a \sec (e+f x))^3 \, dx","Integrate[(d*Cos[e + f*x])^n*(a + a*Sec[e + f*x])^3,x]","\frac{i a^3 2^{-n-3} \left(e^{-i (e+f x)} \left(1+e^{2 i (e+f x)}\right)\right)^n \sec ^6\left(\frac{1}{2} (e+f x)\right) (\sec (e+f x)+1)^3 \left(\frac{8 e^{3 i (e+f x)} \, _2F_1\left(1,\frac{n-1}{2};\frac{5-n}{2};-e^{2 i (e+f x)}\right)}{(n-3) \left(1+e^{2 i (e+f x)}\right)^2}+\frac{12 e^{2 i (e+f x)} \, _2F_1\left(1,\frac{n}{2};2-\frac{n}{2};-e^{2 i (e+f x)}\right)}{(n-2) \left(1+e^{2 i (e+f x)}\right)}+\frac{6 e^{i (e+f x)} \, _2F_1\left(1,\frac{n+1}{2};\frac{3-n}{2};-e^{2 i (e+f x)}\right)}{n-1}+\frac{\left(1+e^{2 i (e+f x)}\right) \, _2F_1\left(1,\frac{n+2}{2};1-\frac{n}{2};-e^{2 i (e+f x)}\right)}{n}\right) \cos ^{3-n}(e+f x) (d \cos (e+f x))^n}{f}","-\frac{a^3 (7-4 n) \sin (e+f x) (d \cos (e+f x))^n \, _2F_1\left(\frac{1}{2},\frac{n}{2};\frac{n+2}{2};\cos ^2(e+f x)\right)}{f (2-n) n \sqrt{\sin ^2(e+f x)}}-\frac{a^3 (1-4 n) \sin (e+f x) \cos (e+f x) (d \cos (e+f x))^n \, _2F_1\left(\frac{1}{2},\frac{n+1}{2};\frac{n+3}{2};\cos ^2(e+f x)\right)}{f (1-n) (n+1) \sqrt{\sin ^2(e+f x)}}+\frac{a^3 (5-2 n) \tan (e+f x) (d \cos (e+f x))^n}{f (1-n) (2-n)}+\frac{\tan (e+f x) \left(a^3 \sec (e+f x)+a^3\right) (d \cos (e+f x))^n}{f (2-n)}",1,"(I*2^(-3 - n)*a^3*((1 + E^((2*I)*(e + f*x)))/E^(I*(e + f*x)))^n*Cos[e + f*x]^(3 - n)*(d*Cos[e + f*x])^n*((8*E^((3*I)*(e + f*x))*Hypergeometric2F1[1, (-1 + n)/2, (5 - n)/2, -E^((2*I)*(e + f*x))])/((1 + E^((2*I)*(e + f*x)))^2*(-3 + n)) + (12*E^((2*I)*(e + f*x))*Hypergeometric2F1[1, n/2, 2 - n/2, -E^((2*I)*(e + f*x))])/((1 + E^((2*I)*(e + f*x)))*(-2 + n)) + (6*E^(I*(e + f*x))*Hypergeometric2F1[1, (1 + n)/2, (3 - n)/2, -E^((2*I)*(e + f*x))])/(-1 + n) + ((1 + E^((2*I)*(e + f*x)))*Hypergeometric2F1[1, (2 + n)/2, 1 - n/2, -E^((2*I)*(e + f*x))])/n)*Sec[(e + f*x)/2]^6*(1 + Sec[e + f*x])^3)/f","C",0
442,1,266,179,1.2872444,"\int (d \cos (e+f x))^n (a+a \sec (e+f x))^2 \, dx","Integrate[(d*Cos[e + f*x])^n*(a + a*Sec[e + f*x])^2,x]","\frac{i a^2 2^{-n-2} e^{-i (e+f x)} \left(e^{-i (e+f x)} \left(1+e^{2 i (e+f x)}\right)\right)^{n-1} (\cos (e+f x)+1)^2 \sec ^4\left(\frac{1}{2} (e+f x)\right) \left(4 (n-1) n e^{2 i (e+f x)} \, _2F_1\left(1,\frac{n}{2};2-\frac{n}{2};-e^{2 i (e+f x)}\right)+(n-2) \left(1+e^{2 i (e+f x)}\right) \left(4 n e^{i (e+f x)} \, _2F_1\left(1,\frac{n+1}{2};\frac{3-n}{2};-e^{2 i (e+f x)}\right)+(n-1) \left(1+e^{2 i (e+f x)}\right) \, _2F_1\left(1,\frac{n+2}{2};1-\frac{n}{2};-e^{2 i (e+f x)}\right)\right)\right) \cos ^{-n}(e+f x) (d \cos (e+f x))^n}{f (n-2) (n-1) n}","-\frac{2 a^2 \sin (e+f x) (d \cos (e+f x))^n \, _2F_1\left(\frac{1}{2},\frac{n}{2};\frac{n+2}{2};\cos ^2(e+f x)\right)}{f n \sqrt{\sin ^2(e+f x)}}-\frac{a^2 (1-2 n) \sin (e+f x) \cos (e+f x) (d \cos (e+f x))^n \, _2F_1\left(\frac{1}{2},\frac{n+1}{2};\frac{n+3}{2};\cos ^2(e+f x)\right)}{f (1-n) (n+1) \sqrt{\sin ^2(e+f x)}}+\frac{a^2 \tan (e+f x) (d \cos (e+f x))^n}{f (1-n)}",1,"(I*2^(-2 - n)*a^2*((1 + E^((2*I)*(e + f*x)))/E^(I*(e + f*x)))^(-1 + n)*(d*Cos[e + f*x])^n*(1 + Cos[e + f*x])^2*(4*E^((2*I)*(e + f*x))*(-1 + n)*n*Hypergeometric2F1[1, n/2, 2 - n/2, -E^((2*I)*(e + f*x))] + (1 + E^((2*I)*(e + f*x)))*(-2 + n)*(4*E^(I*(e + f*x))*n*Hypergeometric2F1[1, (1 + n)/2, (3 - n)/2, -E^((2*I)*(e + f*x))] + (1 + E^((2*I)*(e + f*x)))*(-1 + n)*Hypergeometric2F1[1, (2 + n)/2, 1 - n/2, -E^((2*I)*(e + f*x))]))*Sec[(e + f*x)/2]^4)/(E^(I*(e + f*x))*f*(-2 + n)*(-1 + n)*n*Cos[e + f*x]^n)","C",0
443,1,105,132,0.1421916,"\int (d \cos (e+f x))^n (a+a \sec (e+f x)) \, dx","Integrate[(d*Cos[e + f*x])^n*(a + a*Sec[e + f*x]),x]","-\frac{a \sqrt{\sin ^2(e+f x)} (d \cos (e+f x))^n \left(n \cot (e+f x) \, _2F_1\left(\frac{1}{2},\frac{n+1}{2};\frac{n+3}{2};\cos ^2(e+f x)\right)+(n+1) \csc (e+f x) \, _2F_1\left(\frac{1}{2},\frac{n}{2};\frac{n+2}{2};\cos ^2(e+f x)\right)\right)}{f n (n+1)}","-\frac{a \sin (e+f x) (d \cos (e+f x))^n \, _2F_1\left(\frac{1}{2},\frac{n}{2};\frac{n+2}{2};\cos ^2(e+f x)\right)}{f n \sqrt{\sin ^2(e+f x)}}-\frac{a \sin (e+f x) (d \cos (e+f x))^{n+1} \, _2F_1\left(\frac{1}{2},\frac{n+1}{2};\frac{n+3}{2};\cos ^2(e+f x)\right)}{d f (n+1) \sqrt{\sin ^2(e+f x)}}",1,"-((a*(d*Cos[e + f*x])^n*((1 + n)*Csc[e + f*x]*Hypergeometric2F1[1/2, n/2, (2 + n)/2, Cos[e + f*x]^2] + n*Cot[e + f*x]*Hypergeometric2F1[1/2, (1 + n)/2, (3 + n)/2, Cos[e + f*x]^2])*Sqrt[Sin[e + f*x]^2])/(f*n*(1 + n)))","A",1
444,0,0,178,0.9762216,"\int \frac{(d \cos (e+f x))^n}{a+a \sec (e+f x)} \, dx","Integrate[(d*Cos[e + f*x])^n/(a + a*Sec[e + f*x]),x]","\int \frac{(d \cos (e+f x))^n}{a+a \sec (e+f x)} \, dx","-\frac{\sin (e+f x) \cos (e+f x) (d \cos (e+f x))^n \, _2F_1\left(\frac{1}{2},\frac{n+1}{2};\frac{n+3}{2};\cos ^2(e+f x)\right)}{a f \sqrt{\sin ^2(e+f x)}}+\frac{(n+1) \sin (e+f x) \cos ^2(e+f x) (d \cos (e+f x))^n \, _2F_1\left(\frac{1}{2},\frac{n+2}{2};\frac{n+4}{2};\cos ^2(e+f x)\right)}{a f (n+2) \sqrt{\sin ^2(e+f x)}}+\frac{\sin (e+f x) (d \cos (e+f x))^n}{f (a \sec (e+f x)+a)}",1,"Integrate[(d*Cos[e + f*x])^n/(a + a*Sec[e + f*x]), x]","F",-1
445,0,0,215,9.0495283,"\int \frac{(d \cos (e+f x))^n}{(a+a \sec (e+f x))^2} \, dx","Integrate[(d*Cos[e + f*x])^n/(a + a*Sec[e + f*x])^2,x]","\int \frac{(d \cos (e+f x))^n}{(a+a \sec (e+f x))^2} \, dx","\frac{2 (n+2) \sin (e+f x) (d \cos (e+f x))^n \, _2F_1\left(\frac{1}{2},\frac{n}{2};\frac{n+2}{2};\cos ^2(e+f x)\right)}{3 a^2 f \sqrt{\sin ^2(e+f x)}}-\frac{(2 n+3) \sin (e+f x) \cos (e+f x) (d \cos (e+f x))^n \, _2F_1\left(\frac{1}{2},\frac{n+1}{2};\frac{n+3}{2};\cos ^2(e+f x)\right)}{3 a^2 f \sqrt{\sin ^2(e+f x)}}-\frac{2 (n+2) \tan (e+f x) (d \cos (e+f x))^n}{3 a^2 f (\sec (e+f x)+1)}-\frac{\tan (e+f x) (d \cos (e+f x))^n}{3 f (a \sec (e+f x)+a)^2}",1,"Integrate[(d*Cos[e + f*x])^n/(a + a*Sec[e + f*x])^2, x]","F",-1
446,1,76,85,0.2484247,"\int \sec ^4(c+d x) (a+b \sec (c+d x)) \, dx","Integrate[Sec[c + d*x]^4*(a + b*Sec[c + d*x]),x]","\frac{a \left(\frac{1}{3} \tan ^3(c+d x)+\tan (c+d x)\right)}{d}+\frac{b \tan (c+d x) \sec ^3(c+d x)}{4 d}+\frac{3 b \left(\tanh ^{-1}(\sin (c+d x))+\tan (c+d x) \sec (c+d x)\right)}{8 d}","\frac{a \tan ^3(c+d x)}{3 d}+\frac{a \tan (c+d x)}{d}+\frac{3 b \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{b \tan (c+d x) \sec ^3(c+d x)}{4 d}+\frac{3 b \tan (c+d x) \sec (c+d x)}{8 d}",1,"(b*Sec[c + d*x]^3*Tan[c + d*x])/(4*d) + (3*b*(ArcTanh[Sin[c + d*x]] + Sec[c + d*x]*Tan[c + d*x]))/(8*d) + (a*(Tan[c + d*x] + Tan[c + d*x]^3/3))/d","A",1
447,1,60,63,0.1605539,"\int \sec ^3(c+d x) (a+b \sec (c+d x)) \, dx","Integrate[Sec[c + d*x]^3*(a + b*Sec[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 \left(\frac{1}{3} \tan ^3(c+d x)+\tan (c+d x)\right)}{d}","\frac{a \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{a \tan (c+d x) \sec (c+d x)}{2 d}+\frac{b \tan ^3(c+d x)}{3 d}+\frac{b \tan (c+d x)}{d}",1,"(a*ArcTanh[Sin[c + d*x]])/(2*d) + (a*Sec[c + d*x]*Tan[c + d*x])/(2*d) + (b*(Tan[c + d*x] + Tan[c + d*x]^3/3))/d","A",1
448,1,47,47,0.019407,"\int \sec ^2(c+d x) (a+b \sec (c+d x)) \, dx","Integrate[Sec[c + d*x]^2*(a + b*Sec[c + d*x]),x]","\frac{a \tan (c+d x)}{d}+\frac{b \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{b \tan (c+d x) \sec (c+d x)}{2 d}","\frac{a \tan (c+d x)}{d}+\frac{b \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{b \tan (c+d x) \sec (c+d x)}{2 d}",1,"(b*ArcTanh[Sin[c + d*x]])/(2*d) + (a*Tan[c + d*x])/d + (b*Sec[c + d*x]*Tan[c + d*x])/(2*d)","A",1
449,1,24,24,0.011588,"\int \sec (c+d x) (a+b \sec (c+d x)) \, dx","Integrate[Sec[c + d*x]*(a + b*Sec[c + d*x]),x]","\frac{a \tanh ^{-1}(\sin (c+d x))}{d}+\frac{b \tan (c+d x)}{d}","\frac{a \tanh ^{-1}(\sin (c+d x))}{d}+\frac{b \tan (c+d x)}{d}",1,"(a*ArcTanh[Sin[c + d*x]])/d + (b*Tan[c + d*x])/d","A",1
450,1,16,16,0.0017362,"\int (a+b \sec (c+d x)) \, dx","Integrate[a + b*Sec[c + d*x],x]","a x+\frac{b \tanh ^{-1}(\sin (c+d x))}{d}","a x+\frac{b \tanh ^{-1}(\sin (c+d x))}{d}",1,"a*x + (b*ArcTanh[Sin[c + d*x]])/d","A",1
451,1,26,15,0.0084494,"\int \cos (c+d x) (a+b \sec (c+d x)) \, dx","Integrate[Cos[c + d*x]*(a + b*Sec[c + d*x]),x]","\frac{a \sin (c) \cos (d x)}{d}+\frac{a \cos (c) \sin (d x)}{d}+b x","\frac{a \sin (c+d x)}{d}+b x",1,"b*x + (a*Cos[d*x]*Sin[c])/d + (a*Cos[c]*Sin[d*x])/d","A",1
452,1,35,38,0.064199,"\int \cos ^2(c+d x) (a+b \sec (c+d x)) \, dx","Integrate[Cos[c + d*x]^2*(a + b*Sec[c + d*x]),x]","\frac{a (2 (c+d x)+\sin (2 (c+d x)))+4 b \sin (c+d x)}{4 d}","\frac{a \sin (c+d x) \cos (c+d x)}{2 d}+\frac{a x}{2}+\frac{b \sin (c+d x)}{d}",1,"(4*b*Sin[c + d*x] + a*(2*(c + d*x) + Sin[2*(c + d*x)]))/(4*d)","A",1
453,1,57,54,0.0677375,"\int \cos ^3(c+d x) (a+b \sec (c+d x)) \, dx","Integrate[Cos[c + d*x]^3*(a + b*Sec[c + d*x]),x]","-\frac{a \sin ^3(c+d x)}{3 d}+\frac{a \sin (c+d x)}{d}+\frac{b (c+d x)}{2 d}+\frac{b \sin (2 (c+d x))}{4 d}","-\frac{a \sin ^3(c+d x)}{3 d}+\frac{a \sin (c+d x)}{d}+\frac{b \sin (c+d x) \cos (c+d x)}{2 d}+\frac{b x}{2}",1,"(b*(c + d*x))/(2*d) + (a*Sin[c + d*x])/d - (a*Sin[c + d*x]^3)/(3*d) + (b*Sin[2*(c + d*x)])/(4*d)","A",1
454,1,73,76,0.1284905,"\int \cos ^4(c+d x) (a+b \sec (c+d x)) \, dx","Integrate[Cos[c + d*x]^4*(a + b*Sec[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 \sin ^3(c+d x)}{3 d}+\frac{b \sin (c+d x)}{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 \sin ^3(c+d x)}{3 d}+\frac{b \sin (c+d x)}{d}",1,"(3*a*(c + d*x))/(8*d) + (b*Sin[c + d*x])/d - (b*Sin[c + d*x]^3)/(3*d) + (a*Sin[2*(c + d*x)])/(4*d) + (a*Sin[4*(c + d*x)])/(32*d)","A",1
455,1,89,92,0.1168208,"\int \cos ^5(c+d x) (a+b \sec (c+d x)) \, dx","Integrate[Cos[c + d*x]^5*(a + b*Sec[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{3 b (c+d x)}{8 d}+\frac{b \sin (2 (c+d x))}{4 d}+\frac{b \sin (4 (c+d x))}{32 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 \sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac{3 b \sin (c+d x) \cos (c+d x)}{8 d}+\frac{3 b x}{8}",1,"(3*b*(c + d*x))/(8*d) + (a*Sin[c + d*x])/d - (2*a*Sin[c + d*x]^3)/(3*d) + (a*Sin[c + d*x]^5)/(5*d) + (b*Sin[2*(c + d*x)])/(4*d) + (b*Sin[4*(c + d*x)])/(32*d)","A",1
456,1,118,135,0.5830972,"\int \sec ^4(c+d x) (a+b \sec (c+d x))^2 \, dx","Integrate[Sec[c + d*x]^4*(a + b*Sec[c + d*x])^2,x]","\frac{a^2 \left(\frac{1}{3} \tan ^3(c+d x)+\tan (c+d x)\right)}{d}+\frac{a b \tan (c+d x) \sec ^3(c+d x)}{2 d}+\frac{3 a b \left(\tanh ^{-1}(\sin (c+d x))+\tan (c+d x) \sec (c+d x)\right)}{4 d}+\frac{b^2 \left(\frac{1}{5} \tan ^5(c+d x)+\frac{2}{3} \tan ^3(c+d x)+\tan (c+d x)\right)}{d}","\frac{\left(5 a^2+4 b^2\right) \tan ^3(c+d x)}{15 d}+\frac{\left(5 a^2+4 b^2\right) \tan (c+d x)}{5 d}+\frac{3 a b \tanh ^{-1}(\sin (c+d x))}{4 d}+\frac{a b \tan (c+d x) \sec ^3(c+d x)}{2 d}+\frac{3 a b \tan (c+d x) \sec (c+d x)}{4 d}+\frac{b^2 \tan (c+d x) \sec ^4(c+d x)}{5 d}",1,"(a*b*Sec[c + d*x]^3*Tan[c + d*x])/(2*d) + (3*a*b*(ArcTanh[Sin[c + d*x]] + Sec[c + d*x]*Tan[c + d*x]))/(4*d) + (a^2*(Tan[c + d*x] + Tan[c + d*x]^3/3))/d + (b^2*(Tan[c + d*x] + (2*Tan[c + d*x]^3)/3 + Tan[c + d*x]^5/5))/d","A",1
457,1,82,110,0.2835871,"\int \sec ^3(c+d x) (a+b \sec (c+d x))^2 \, dx","Integrate[Sec[c + d*x]^3*(a + b*Sec[c + d*x])^2,x]","\frac{3 \left(4 a^2+3 b^2\right) \tanh ^{-1}(\sin (c+d x))+\tan (c+d x) \left(3 \left(4 a^2+3 b^2\right) \sec (c+d x)+16 a b \left(\tan ^2(c+d x)+3\right)+6 b^2 \sec ^3(c+d x)\right)}{24 d}","\frac{\left(4 a^2+3 b^2\right) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{\left(4 a^2+3 b^2\right) \tan (c+d x) \sec (c+d x)}{8 d}+\frac{2 a b \tan ^3(c+d x)}{3 d}+\frac{2 a b \tan (c+d x)}{d}+\frac{b^2 \tan (c+d x) \sec ^3(c+d x)}{4 d}",1,"(3*(4*a^2 + 3*b^2)*ArcTanh[Sin[c + d*x]] + Tan[c + d*x]*(3*(4*a^2 + 3*b^2)*Sec[c + d*x] + 6*b^2*Sec[c + d*x]^3 + 16*a*b*(3 + Tan[c + d*x]^2)))/(24*d)","A",1
458,1,71,80,0.2239558,"\int \sec ^2(c+d x) (a+b \sec (c+d x))^2 \, dx","Integrate[Sec[c + d*x]^2*(a + b*Sec[c + d*x])^2,x]","\frac{a^2 \tan (c+d x)}{d}+\frac{a b \tanh ^{-1}(\sin (c+d x))}{d}+\frac{a b \tan (c+d x) \sec (c+d x)}{d}+\frac{b^2 \left(\frac{1}{3} \tan ^3(c+d x)+\tan (c+d x)\right)}{d}","\frac{\left(3 a^2+2 b^2\right) \tan (c+d x)}{3 d}+\frac{a b \tanh ^{-1}(\sin (c+d x))}{d}+\frac{a b \tan (c+d x) \sec (c+d x)}{d}+\frac{b^2 \tan (c+d x) \sec ^2(c+d x)}{3 d}",1,"(a*b*ArcTanh[Sin[c + d*x]])/d + (a^2*Tan[c + d*x])/d + (a*b*Sec[c + d*x]*Tan[c + d*x])/d + (b^2*(Tan[c + d*x] + Tan[c + d*x]^3/3))/d","A",1
459,1,45,59,0.1079317,"\int \sec (c+d x) (a+b \sec (c+d x))^2 \, dx","Integrate[Sec[c + d*x]*(a + b*Sec[c + d*x])^2,x]","\frac{\left(2 a^2+b^2\right) \tanh ^{-1}(\sin (c+d x))+b \tan (c+d x) (4 a+b \sec (c+d x))}{2 d}","\frac{\left(2 a^2+b^2\right) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{2 a b \tan (c+d x)}{d}+\frac{b^2 \tan (c+d x) \sec (c+d x)}{2 d}",1,"((2*a^2 + b^2)*ArcTanh[Sin[c + d*x]] + b*(4*a + b*Sec[c + d*x])*Tan[c + d*x])/(2*d)","A",1
460,1,32,33,0.075973,"\int (a+b \sec (c+d x))^2 \, dx","Integrate[(a + b*Sec[c + d*x])^2,x]","\frac{a^2 d x+2 a b \tanh ^{-1}(\sin (c+d x))+b^2 \tan (c+d x)}{d}","a^2 x+\frac{2 a b \tanh ^{-1}(\sin (c+d x))}{d}+\frac{b^2 \tan (c+d x)}{d}",1,"(a^2*d*x + 2*a*b*ArcTanh[Sin[c + d*x]] + b^2*Tan[c + d*x])/d","A",1
461,1,46,33,0.0167585,"\int \cos (c+d x) (a+b \sec (c+d x))^2 \, dx","Integrate[Cos[c + d*x]*(a + b*Sec[c + d*x])^2,x]","\frac{a^2 \sin (c) \cos (d x)}{d}+\frac{a^2 \cos (c) \sin (d x)}{d}+2 a b x+\frac{b^2 \tanh ^{-1}(\sin (c+d x))}{d}","\frac{a^2 \sin (c+d x)}{d}+2 a b x+\frac{b^2 \tanh ^{-1}(\sin (c+d x))}{d}",1,"2*a*b*x + (b^2*ArcTanh[Sin[c + d*x]])/d + (a^2*Cos[d*x]*Sin[c])/d + (a^2*Cos[c]*Sin[d*x])/d","A",1
462,1,46,50,0.0755936,"\int \cos ^2(c+d x) (a+b \sec (c+d x))^2 \, dx","Integrate[Cos[c + d*x]^2*(a + b*Sec[c + d*x])^2,x]","\frac{2 \left(a^2+2 b^2\right) (c+d x)+a^2 \sin (2 (c+d x))+8 a b \sin (c+d x)}{4 d}","\frac{1}{2} x \left(a^2+2 b^2\right)+\frac{a^2 \sin (c+d x) \cos (c+d x)}{2 d}+\frac{2 a b \sin (c+d x)}{d}",1,"(2*(a^2 + 2*b^2)*(c + d*x) + 8*a*b*Sin[c + d*x] + a^2*Sin[2*(c + d*x)])/(4*d)","A",1
463,1,59,58,0.1538684,"\int \cos ^3(c+d x) (a+b \sec (c+d x))^2 \, dx","Integrate[Cos[c + d*x]^3*(a + b*Sec[c + d*x])^2,x]","\frac{3 \left(3 a^2+4 b^2\right) \sin (c+d x)+a (a \sin (3 (c+d x))+12 b (c+d x)+6 b \sin (2 (c+d x)))}{12 d}","\frac{\left(a^2+b^2\right) \sin (c+d x)}{d}-\frac{a^2 \sin ^3(c+d x)}{3 d}+\frac{a b \sin (c+d x) \cos (c+d x)}{d}+a b x",1,"(3*(3*a^2 + 4*b^2)*Sin[c + d*x] + a*(12*b*(c + d*x) + 6*b*Sin[2*(c + d*x)] + a*Sin[3*(c + d*x)]))/(12*d)","A",1
464,1,86,101,0.172908,"\int \cos ^4(c+d x) (a+b \sec (c+d x))^2 \, dx","Integrate[Cos[c + d*x]^4*(a + b*Sec[c + d*x])^2,x]","\frac{24 \left(a^2+b^2\right) \sin (2 (c+d x))+3 a^2 \sin (4 (c+d x))+36 a^2 c+36 a^2 d x-64 a b \sin ^3(c+d x)+192 a b \sin (c+d x)+48 b^2 c+48 b^2 d x}{96 d}","\frac{\left(3 a^2+4 b^2\right) \sin (c+d x) \cos (c+d x)}{8 d}+\frac{1}{8} x \left(3 a^2+4 b^2\right)+\frac{a^2 \sin (c+d x) \cos ^3(c+d x)}{4 d}-\frac{2 a b \sin ^3(c+d x)}{3 d}+\frac{2 a b \sin (c+d x)}{d}",1,"(36*a^2*c + 48*b^2*c + 36*a^2*d*x + 48*b^2*d*x + 192*a*b*Sin[c + d*x] - 64*a*b*Sin[c + d*x]^3 + 24*(a^2 + b^2)*Sin[2*(c + d*x)] + 3*a^2*Sin[4*(c + d*x)])/(96*d)","A",1
465,1,85,111,0.1624617,"\int \cos ^5(c+d x) (a+b \sec (c+d x))^2 \, dx","Integrate[Cos[c + d*x]^5*(a + b*Sec[c + d*x])^2,x]","\frac{-80 \left(2 a^2+b^2\right) \sin ^3(c+d x)+240 \left(a^2+b^2\right) \sin (c+d x)+48 a^2 \sin ^5(c+d x)+15 a b (12 (c+d x)+8 \sin (2 (c+d x))+\sin (4 (c+d x)))}{240 d}","-\frac{\left(2 a^2+b^2\right) \sin ^3(c+d x)}{3 d}+\frac{\left(a^2+b^2\right) \sin (c+d x)}{d}+\frac{a^2 \sin ^5(c+d x)}{5 d}+\frac{a b \sin (c+d x) \cos ^3(c+d x)}{2 d}+\frac{3 a b \sin (c+d x) \cos (c+d x)}{4 d}+\frac{3 a b x}{4}",1,"(240*(a^2 + b^2)*Sin[c + d*x] - 80*(2*a^2 + b^2)*Sin[c + d*x]^3 + 48*a^2*Sin[c + d*x]^5 + 15*a*b*(12*(c + d*x) + 8*Sin[2*(c + d*x)] + Sin[4*(c + d*x)]))/(240*d)","A",1
466,1,120,189,0.8986918,"\int \sec ^3(c+d x) (a+b \sec (c+d x))^3 \, dx","Integrate[Sec[c + d*x]^3*(a + b*Sec[c + d*x])^3,x]","\frac{15 a \left(4 a^2+9 b^2\right) \tanh ^{-1}(\sin (c+d x))+\tan (c+d x) \left(8 b \left(5 \left(3 a^2+2 b^2\right) \tan ^2(c+d x)+15 \left(3 a^2+b^2\right)+3 b^2 \tan ^4(c+d x)\right)+15 a \left(4 a^2+9 b^2\right) \sec (c+d x)+90 a b^2 \sec ^3(c+d x)\right)}{120 d}","\frac{a \left(4 a^2+9 b^2\right) \tanh ^{-1}(\sin (c+d x))}{8 d}-\frac{\left(3 a^2-16 b^2\right) \tan (c+d x) (a+b \sec (c+d x))^2}{60 b d}-\frac{a \left(6 a^2-71 b^2\right) \tan (c+d x) \sec (c+d x)}{120 d}-\frac{\left(3 a^4-52 a^2 b^2-16 b^4\right) \tan (c+d x)}{30 b d}+\frac{\tan (c+d x) (a+b \sec (c+d x))^4}{5 b d}-\frac{a \tan (c+d x) (a+b \sec (c+d x))^3}{20 b d}",1,"(15*a*(4*a^2 + 9*b^2)*ArcTanh[Sin[c + d*x]] + Tan[c + d*x]*(15*a*(4*a^2 + 9*b^2)*Sec[c + d*x] + 90*a*b^2*Sec[c + d*x]^3 + 8*b*(15*(3*a^2 + b^2) + 5*(3*a^2 + 2*b^2)*Tan[c + d*x]^2 + 3*b^2*Tan[c + d*x]^4)))/(120*d)","A",1
467,1,90,130,0.4497986,"\int \sec ^2(c+d x) (a+b \sec (c+d x))^3 \, dx","Integrate[Sec[c + d*x]^2*(a + b*Sec[c + d*x])^3,x]","\frac{3 b \left(4 a^2+b^2\right) \tanh ^{-1}(\sin (c+d x))+\tan (c+d x) \left(8 a \left(a^2+b^2 \tan ^2(c+d x)+3 b^2\right)+3 b \left(4 a^2+b^2\right) \sec (c+d x)+2 b^3 \sec ^3(c+d x)\right)}{8 d}","\frac{a \left(a^2+4 b^2\right) \tan (c+d x)}{2 d}+\frac{3 b \left(4 a^2+b^2\right) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{b \left(2 a^2+3 b^2\right) \tan (c+d x) \sec (c+d x)}{8 d}+\frac{\tan (c+d x) (a+b \sec (c+d x))^3}{4 d}+\frac{a \tan (c+d x) (a+b \sec (c+d x))^2}{4 d}",1,"(3*b*(4*a^2 + b^2)*ArcTanh[Sin[c + d*x]] + Tan[c + d*x]*(3*b*(4*a^2 + b^2)*Sec[c + d*x] + 2*b^3*Sec[c + d*x]^3 + 8*a*(a^2 + 3*b^2 + b^2*Tan[c + d*x]^2)))/(8*d)","A",1
468,1,70,99,0.2510258,"\int \sec (c+d x) (a+b \sec (c+d x))^3 \, dx","Integrate[Sec[c + d*x]*(a + b*Sec[c + d*x])^3,x]","\frac{\left(6 a^3+9 a b^2\right) \tanh ^{-1}(\sin (c+d x))+b \tan (c+d x) \left(18 a^2+9 a b \sec (c+d x)+2 b^2 \tan ^2(c+d x)+6 b^2\right)}{6 d}","\frac{2 b \left(4 a^2+b^2\right) \tan (c+d x)}{3 d}+\frac{a \left(2 a^2+3 b^2\right) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{5 a b^2 \tan (c+d x) \sec (c+d x)}{6 d}+\frac{b \tan (c+d x) (a+b \sec (c+d x))^2}{3 d}",1,"((6*a^3 + 9*a*b^2)*ArcTanh[Sin[c + d*x]] + b*Tan[c + d*x]*(18*a^2 + 6*b^2 + 9*a*b*Sec[c + d*x] + 2*b^2*Tan[c + d*x]^2))/(6*d)","A",1
469,1,55,73,0.1634536,"\int (a+b \sec (c+d x))^3 \, dx","Integrate[(a + b*Sec[c + d*x])^3,x]","\frac{2 a^3 d x+b \left(6 a^2+b^2\right) \tanh ^{-1}(\sin (c+d x))+b^2 \tan (c+d x) (6 a+b \sec (c+d x))}{2 d}","a^3 x+\frac{b \left(6 a^2+b^2\right) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{5 a b^2 \tan (c+d x)}{2 d}+\frac{b^2 \tan (c+d x) (a+b \sec (c+d x))}{2 d}",1,"(2*a^3*d*x + b*(6*a^2 + b^2)*ArcTanh[Sin[c + d*x]] + b^2*(6*a + b*Sec[c + d*x])*Tan[c + d*x])/(2*d)","A",1
470,1,88,67,0.3423864,"\int \cos (c+d x) (a+b \sec (c+d x))^3 \, dx","Integrate[Cos[c + d*x]*(a + b*Sec[c + d*x])^3,x]","\frac{a^3 \sin (c+d x)+3 a b \left(a c+a d x-b \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+b \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)+b^3 \tan (c+d x)}{d}","\frac{a \left(a^2-b^2\right) \sin (c+d x)}{d}+3 a^2 b x+\frac{3 a b^2 \tanh ^{-1}(\sin (c+d x))}{d}+\frac{b^2 \sin (c+d x) (a+b \sec (c+d x))}{d}",1,"(3*a*b*(a*c + a*d*x - b*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + b*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) + a^3*Sin[c + d*x] + b^3*Tan[c + d*x])/d","A",1
471,1,105,79,0.1434786,"\int \cos ^2(c+d x) (a+b \sec (c+d x))^3 \, dx","Integrate[Cos[c + d*x]^2*(a + b*Sec[c + d*x])^3,x]","\frac{a^3 \sin (2 (c+d x))+2 a \left(a^2+6 b^2\right) (c+d x)+12 a^2 b \sin (c+d x)-4 b^3 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+4 b^3 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}{4 d}","\frac{1}{2} a x \left(a^2+6 b^2\right)+\frac{5 a^2 b \sin (c+d x)}{2 d}+\frac{a^2 \sin (c+d x) \cos (c+d x) (a+b \sec (c+d x))}{2 d}+\frac{b^3 \tanh ^{-1}(\sin (c+d x))}{d}",1,"(2*a*(a^2 + 6*b^2)*(c + d*x) - 4*b^3*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 4*b^3*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + 12*a^2*b*Sin[c + d*x] + a^3*Sin[2*(c + d*x)])/(4*d)","A",1
472,1,80,100,0.1207146,"\int \cos ^3(c+d x) (a+b \sec (c+d x))^3 \, dx","Integrate[Cos[c + d*x]^3*(a + b*Sec[c + d*x])^3,x]","\frac{a^3 \sin (3 (c+d x))+9 a \left(a^2+4 b^2\right) \sin (c+d x)+9 a^2 b \sin (2 (c+d x))+18 a^2 b c+18 a^2 b d x+12 b^3 c+12 b^3 d x}{12 d}","\frac{a \left(2 a^2+9 b^2\right) \sin (c+d x)}{3 d}+\frac{1}{2} b x \left(3 a^2+2 b^2\right)+\frac{7 a^2 b \sin (c+d x) \cos (c+d x)}{6 d}+\frac{a^2 \sin (c+d x) \cos ^2(c+d x) (a+b \sec (c+d x))}{3 d}",1,"(18*a^2*b*c + 12*b^3*c + 18*a^2*b*d*x + 12*b^3*d*x + 9*a*(a^2 + 4*b^2)*Sin[c + d*x] + 9*a^2*b*Sin[2*(c + d*x)] + a^3*Sin[3*(c + d*x)])/(12*d)","A",1
473,1,100,123,0.2823141,"\int \cos ^4(c+d x) (a+b \sec (c+d x))^3 \, dx","Integrate[Cos[c + d*x]^4*(a + b*Sec[c + d*x])^3,x]","\frac{8 b \left(9 a^2+4 b^2\right) \sin (c+d x)+a \left(8 \left(a^2+3 b^2\right) \sin (2 (c+d x))+a^2 \sin (4 (c+d x))+12 a^2 c+12 a^2 d x+8 a b \sin (3 (c+d x))+48 b^2 c+48 b^2 d x\right)}{32 d}","\frac{b \left(11 a^2+4 b^2\right) \sin (c+d x)}{4 d}+\frac{3 a \left(a^2+4 b^2\right) \sin (c+d x) \cos (c+d x)}{8 d}+\frac{3}{8} a x \left(a^2+4 b^2\right)-\frac{3 a^2 b \sin ^3(c+d x)}{4 d}+\frac{a^2 \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))}{4 d}",1,"(8*b*(9*a^2 + 4*b^2)*Sin[c + d*x] + a*(12*a^2*c + 48*b^2*c + 12*a^2*d*x + 48*b^2*d*x + 8*(a^2 + 3*b^2)*Sin[2*(c + d*x)] + 8*a*b*Sin[3*(c + d*x)] + a^2*Sin[4*(c + d*x)]))/(32*d)","A",1
474,1,130,160,0.3004964,"\int \cos ^5(c+d x) (a+b \sec (c+d x))^3 \, dx","Integrate[Cos[c + d*x]^5*(a + b*Sec[c + d*x])^3,x]","\frac{50 a^3 \sin (3 (c+d x))+6 a^3 \sin (5 (c+d x))+120 \left(3 a^2 b+b^3\right) \sin (2 (c+d x))+60 a \left(5 a^2+18 b^2\right) \sin (c+d x)+45 a^2 b \sin (4 (c+d x))+540 a^2 b c+540 a^2 b d x+120 a b^2 \sin (3 (c+d x))+240 b^3 c+240 b^3 d x}{480 d}","-\frac{a \left(4 a^2+15 b^2\right) \sin ^3(c+d x)}{15 d}+\frac{a \left(4 a^2+15 b^2\right) \sin (c+d x)}{5 d}+\frac{b \left(9 a^2+4 b^2\right) \sin (c+d x) \cos (c+d x)}{8 d}+\frac{1}{8} b x \left(9 a^2+4 b^2\right)+\frac{11 a^2 b \sin (c+d x) \cos ^3(c+d x)}{20 d}+\frac{a^2 \sin (c+d x) \cos ^4(c+d x) (a+b \sec (c+d x))}{5 d}",1,"(540*a^2*b*c + 240*b^3*c + 540*a^2*b*d*x + 240*b^3*d*x + 60*a*(5*a^2 + 18*b^2)*Sin[c + d*x] + 120*(3*a^2*b + b^3)*Sin[2*(c + d*x)] + 50*a^3*Sin[3*(c + d*x)] + 120*a*b^2*Sin[3*(c + d*x)] + 45*a^2*b*Sin[4*(c + d*x)] + 6*a^3*Sin[5*(c + d*x)])/(480*d)","A",1
475,1,159,185,0.3396018,"\int \cos ^6(c+d x) (a+b \sec (c+d x))^3 \, dx","Integrate[Cos[c + d*x]^6*(a + b*Sec[c + d*x])^3,x]","\frac{45 \left(5 a^3+16 a b^2\right) \sin (2 (c+d x))+45 a^3 \sin (4 (c+d x))+5 a^3 \sin (6 (c+d x))+300 a^3 c+300 a^3 d x+360 b \left(5 a^2+2 b^2\right) \sin (c+d x)+300 a^2 b \sin (3 (c+d x))+36 a^2 b \sin (5 (c+d x))+90 a b^2 \sin (4 (c+d x))+1080 a b^2 c+1080 a b^2 d x+80 b^3 \sin (3 (c+d x))}{960 d}","-\frac{b \left(5 a^2+b^2\right) \sin ^3(c+d x)}{3 d}+\frac{b \left(17 a^2+6 b^2\right) \sin (c+d x)}{6 d}+\frac{a \left(5 a^2+18 b^2\right) \sin (c+d x) \cos ^3(c+d x)}{24 d}+\frac{a \left(5 a^2+18 b^2\right) \sin (c+d x) \cos (c+d x)}{16 d}+\frac{1}{16} a x \left(5 a^2+18 b^2\right)+\frac{13 a^2 b \sin ^5(c+d x)}{30 d}+\frac{a^2 \sin (c+d x) \cos ^5(c+d x) (a+b \sec (c+d x))}{6 d}",1,"(300*a^3*c + 1080*a*b^2*c + 300*a^3*d*x + 1080*a*b^2*d*x + 360*b*(5*a^2 + 2*b^2)*Sin[c + d*x] + 45*(5*a^3 + 16*a*b^2)*Sin[2*(c + d*x)] + 300*a^2*b*Sin[3*(c + d*x)] + 80*b^3*Sin[3*(c + d*x)] + 45*a^3*Sin[4*(c + d*x)] + 90*a*b^2*Sin[4*(c + d*x)] + 36*a^2*b*Sin[5*(c + d*x)] + 5*a^3*Sin[6*(c + d*x)])/(960*d)","A",1
476,1,154,244,0.9629338,"\int \sec ^3(c+d x) (a+b \sec (c+d x))^4 \, dx","Integrate[Sec[c + d*x]^3*(a + b*Sec[c + d*x])^4,x]","\frac{15 \left(8 a^4+36 a^2 b^2+5 b^4\right) \tanh ^{-1}(\sin (c+d x))+\tan (c+d x) \left(64 a b \left(5 \left(a^2+2 b^2\right) \tan ^2(c+d x)+15 \left(a^2+b^2\right)+3 b^2 \tan ^4(c+d x)\right)+10 b^2 \left(36 a^2+5 b^2\right) \sec ^3(c+d x)+15 \left(8 a^4+36 a^2 b^2+5 b^4\right) \sec (c+d x)+40 b^4 \sec ^5(c+d x)\right)}{240 d}","-\frac{\left(4 a^2-25 b^2\right) \tan (c+d x) (a+b \sec (c+d x))^3}{120 b d}-\frac{a \left(4 a^2-53 b^2\right) \tan (c+d x) (a+b \sec (c+d x))^2}{120 b d}-\frac{a \left(4 a^4-121 a^2 b^2-128 b^4\right) \tan (c+d x)}{60 b d}+\frac{\left(8 a^4+36 a^2 b^2+5 b^4\right) \tanh ^{-1}(\sin (c+d x))}{16 d}-\frac{\left(8 a^4-178 a^2 b^2-75 b^4\right) \tan (c+d x) \sec (c+d x)}{240 d}+\frac{\tan (c+d x) (a+b \sec (c+d x))^5}{6 b d}-\frac{a \tan (c+d x) (a+b \sec (c+d x))^4}{30 b d}",1,"(15*(8*a^4 + 36*a^2*b^2 + 5*b^4)*ArcTanh[Sin[c + d*x]] + Tan[c + d*x]*(15*(8*a^4 + 36*a^2*b^2 + 5*b^4)*Sec[c + d*x] + 10*b^2*(36*a^2 + 5*b^2)*Sec[c + d*x]^3 + 40*b^4*Sec[c + d*x]^5 + 64*a*b*(15*(a^2 + b^2) + 5*(a^2 + 2*b^2)*Tan[c + d*x]^2 + 3*b^2*Tan[c + d*x]^4)))/(240*d)","A",1
477,1,125,179,0.7890523,"\int \sec ^2(c+d x) (a+b \sec (c+d x))^4 \, dx","Integrate[Sec[c + d*x]^2*(a + b*Sec[c + d*x])^4,x]","\frac{15 a b \left(4 a^2+3 b^2\right) \tanh ^{-1}(\sin (c+d x))+\tan (c+d x) \left(20 b^2 \left(3 a^2+b^2\right) \tan ^2(c+d x)+15 a b \left(4 a^2+3 b^2\right) \sec (c+d x)+30 \left(a^4+6 a^2 b^2+b^4\right)+30 a b^3 \sec ^3(c+d x)+6 b^4 \tan ^4(c+d x)\right)}{30 d}","\frac{a b \left(4 a^2+3 b^2\right) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{\left(3 a^2+4 b^2\right) \tan (c+d x) (a+b \sec (c+d x))^2}{15 d}+\frac{a b \left(6 a^2+29 b^2\right) \tan (c+d x) \sec (c+d x)}{30 d}+\frac{2 \left(3 a^4+28 a^2 b^2+4 b^4\right) \tan (c+d x)}{15 d}+\frac{\tan (c+d x) (a+b \sec (c+d x))^4}{5 d}+\frac{a \tan (c+d x) (a+b \sec (c+d x))^3}{5 d}",1,"(15*a*b*(4*a^2 + 3*b^2)*ArcTanh[Sin[c + d*x]] + Tan[c + d*x]*(30*(a^4 + 6*a^2*b^2 + b^4) + 15*a*b*(4*a^2 + 3*b^2)*Sec[c + d*x] + 30*a*b^3*Sec[c + d*x]^3 + 20*b^2*(3*a^2 + b^2)*Tan[c + d*x]^2 + 6*b^4*Tan[c + d*x]^4))/(30*d)","A",1
478,1,101,146,0.5433468,"\int \sec (c+d x) (a+b \sec (c+d x))^4 \, dx","Integrate[Sec[c + d*x]*(a + b*Sec[c + d*x])^4,x]","\frac{b \tan (c+d x) \left(32 a \left(3 \left(a^2+b^2\right)+b^2 \tan ^2(c+d x)\right)+9 b \left(8 a^2+b^2\right) \sec (c+d x)+6 b^3 \sec ^3(c+d x)\right)+3 \left(8 a^4+24 a^2 b^2+3 b^4\right) \tanh ^{-1}(\sin (c+d x))}{24 d}","\frac{a b \left(19 a^2+16 b^2\right) \tan (c+d x)}{6 d}+\frac{b^2 \left(26 a^2+9 b^2\right) \tan (c+d x) \sec (c+d x)}{24 d}+\frac{\left(8 a^4+24 a^2 b^2+3 b^4\right) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{b \tan (c+d x) (a+b \sec (c+d x))^3}{4 d}+\frac{7 a b \tan (c+d x) (a+b \sec (c+d x))^2}{12 d}",1,"(3*(8*a^4 + 24*a^2*b^2 + 3*b^4)*ArcTanh[Sin[c + d*x]] + b*Tan[c + d*x]*(9*b*(8*a^2 + b^2)*Sec[c + d*x] + 6*b^3*Sec[c + d*x]^3 + 32*a*(3*(a^2 + b^2) + b^2*Tan[c + d*x]^2)))/(24*d)","A",1
479,1,77,107,0.3033217,"\int (a+b \sec (c+d x))^4 \, dx","Integrate[(a + b*Sec[c + d*x])^4,x]","\frac{3 a^4 d x+6 a b \left(2 a^2+b^2\right) \tanh ^{-1}(\sin (c+d x))+3 b^2 \tan (c+d x) \left(6 a^2+2 a b \sec (c+d x)+b^2\right)+b^4 \tan ^3(c+d x)}{3 d}","a^4 x+\frac{b^2 \left(17 a^2+2 b^2\right) \tan (c+d x)}{3 d}+\frac{2 a b \left(2 a^2+b^2\right) \tanh ^{-1}(\sin (c+d x))}{d}+\frac{4 a b^3 \tan (c+d x) \sec (c+d x)}{3 d}+\frac{b^2 \tan (c+d x) (a+b \sec (c+d x))^2}{3 d}",1,"(3*a^4*d*x + 6*a*b*(2*a^2 + b^2)*ArcTanh[Sin[c + d*x]] + 3*b^2*(6*a^2 + b^2 + 2*a*b*Sec[c + d*x])*Tan[c + d*x] + b^4*Tan[c + d*x]^3)/(3*d)","A",1
480,1,280,104,0.5409157,"\int \cos (c+d x) (a+b \sec (c+d x))^4 \, dx","Integrate[Cos[c + d*x]*(a + b*Sec[c + d*x])^4,x]","\frac{\sec ^2(c+d x) \left(\left(a^4+2 b^4\right) \sin (c+d x)+a^4 \sin (3 (c+d x))+8 a^3 b c+8 a^3 b d x-12 a^2 b^2 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+12 a^2 b^2 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+b \cos (2 (c+d x)) \left(8 a^3 (c+d x)-b \left(12 a^2+b^2\right) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+b \left(12 a^2+b^2\right) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)+8 a b^3 \sin (2 (c+d x))-b^4 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+b^4 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)}{4 d}","4 a^3 b x+\frac{a^2 \left(2 a^2-b^2\right) \sin (c+d x)}{2 d}+\frac{b^2 \left(12 a^2+b^2\right) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{3 a b^3 \tan (c+d x)}{d}+\frac{b^2 \sin (c+d x) (a+b \sec (c+d x))^2}{2 d}",1,"(Sec[c + d*x]^2*(8*a^3*b*c + 8*a^3*b*d*x - 12*a^2*b^2*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - b^4*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 12*a^2*b^2*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + b^4*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + b*Cos[2*(c + d*x)]*(8*a^3*(c + d*x) - b*(12*a^2 + b^2)*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + b*(12*a^2 + b^2)*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) + (a^4 + 2*b^4)*Sin[c + d*x] + 8*a*b^3*Sin[2*(c + d*x)] + a^4*Sin[3*(c + d*x)]))/(4*d)","B",1
481,1,119,108,0.6840797,"\int \cos ^2(c+d x) (a+b \sec (c+d x))^4 \, dx","Integrate[Cos[c + d*x]^2*(a + b*Sec[c + d*x])^4,x]","\frac{a^4 \sin (2 (c+d x))+16 a^3 b \sin (c+d x)+2 a \left(a \left(a^2+12 b^2\right) (c+d x)-8 b^3 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+8 b^3 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)+4 b^4 \tan (c+d x)}{4 d}","\frac{3 a^3 b \sin (c+d x)}{d}-\frac{b^2 \left(a^2-2 b^2\right) \tan (c+d x)}{2 d}+\frac{1}{2} a^2 x \left(a^2+12 b^2\right)+\frac{a^2 \sin (c+d x) \cos (c+d x) (a+b \sec (c+d x))^2}{2 d}+\frac{4 a b^3 \tanh ^{-1}(\sin (c+d x))}{d}",1,"(2*a*(a*(a^2 + 12*b^2)*(c + d*x) - 8*b^3*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 8*b^3*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) + 16*a^3*b*Sin[c + d*x] + a^4*Sin[2*(c + d*x)] + 4*b^4*Tan[c + d*x])/(4*d)","A",1
482,1,128,115,0.1645773,"\int \cos ^3(c+d x) (a+b \sec (c+d x))^4 \, dx","Integrate[Cos[c + d*x]^3*(a + b*Sec[c + d*x])^4,x]","\frac{a^4 \sin (3 (c+d x))+12 a^3 b \sin (2 (c+d x))+24 a b \left(a^2+2 b^2\right) (c+d x)+9 a^2 \left(a^2+8 b^2\right) \sin (c+d x)-12 b^4 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+12 b^4 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}{12 d}","\frac{4 a^3 b \sin (c+d x) \cos (c+d x)}{3 d}+\frac{a^2 \left(2 a^2+17 b^2\right) \sin (c+d x)}{3 d}+2 a b x \left(a^2+2 b^2\right)+\frac{a^2 \sin (c+d x) \cos ^2(c+d x) (a+b \sec (c+d x))^2}{3 d}+\frac{b^4 \tanh ^{-1}(\sin (c+d x))}{d}",1,"(24*a*b*(a^2 + 2*b^2)*(c + d*x) - 12*b^4*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 12*b^4*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + 9*a^2*(a^2 + 8*b^2)*Sin[c + d*x] + 12*a^3*b*Sin[2*(c + d*x)] + a^4*Sin[3*(c + d*x)])/(12*d)","A",1
483,1,104,145,0.2361183,"\int \cos ^4(c+d x) (a+b \sec (c+d x))^4 \, dx","Integrate[Cos[c + d*x]^4*(a + b*Sec[c + d*x])^4,x]","\frac{3 a^4 \sin (4 (c+d x))+32 a^3 b \sin (3 (c+d x))+24 a^2 \left(a^2+6 b^2\right) \sin (2 (c+d x))+96 a b \left(3 a^2+4 b^2\right) \sin (c+d x)+12 \left(3 a^4+24 a^2 b^2+8 b^4\right) (c+d x)}{96 d}","\frac{5 a^3 b \sin (c+d x) \cos ^2(c+d x)}{6 d}+\frac{4 a b \left(2 a^2+3 b^2\right) \sin (c+d x)}{3 d}+\frac{a^2 \left(3 a^2+22 b^2\right) \sin (c+d x) \cos (c+d x)}{8 d}+\frac{a^2 \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^2}{4 d}+\frac{1}{8} x \left(3 a^4+24 a^2 b^2+8 b^4\right)",1,"(12*(3*a^4 + 24*a^2*b^2 + 8*b^4)*(c + d*x) + 96*a*b*(3*a^2 + 4*b^2)*Sin[c + d*x] + 24*a^2*(a^2 + 6*b^2)*Sin[2*(c + d*x)] + 32*a^3*b*Sin[3*(c + d*x)] + 3*a^4*Sin[4*(c + d*x)])/(96*d)","A",1
484,1,133,173,0.5184028,"\int \cos ^5(c+d x) (a+b \sec (c+d x))^4 \, dx","Integrate[Cos[c + d*x]^5*(a + b*Sec[c + d*x])^4,x]","\frac{30 \left(5 a^4+36 a^2 b^2+8 b^4\right) \sin (c+d x)+a \left(5 \left(5 a^3+24 a b^2\right) \sin (3 (c+d x))+3 a^3 \sin (5 (c+d x))+240 b \left(a^2+b^2\right) \sin (2 (c+d x))+30 a^2 b \sin (4 (c+d x))+360 a^2 b c+360 a^2 b d x+480 b^3 c+480 b^3 d x\right)}{240 d}","\frac{3 a^3 b \sin (c+d x) \cos ^3(c+d x)}{5 d}-\frac{a^2 \left(4 a^2+27 b^2\right) \sin ^3(c+d x)}{15 d}+\frac{a b \left(3 a^2+4 b^2\right) \sin (c+d x) \cos (c+d x)}{2 d}+\frac{1}{2} a b x \left(3 a^2+4 b^2\right)+\frac{a^2 \sin (c+d x) \cos ^4(c+d x) (a+b \sec (c+d x))^2}{5 d}+\frac{\left(4 a^4+29 a^2 b^2+5 b^4\right) \sin (c+d x)}{5 d}",1,"(30*(5*a^4 + 36*a^2*b^2 + 8*b^4)*Sin[c + d*x] + a*(360*a^2*b*c + 480*b^3*c + 360*a^2*b*d*x + 480*b^3*d*x + 240*b*(a^2 + b^2)*Sin[2*(c + d*x)] + 5*(5*a^3 + 24*a*b^2)*Sin[3*(c + d*x)] + 30*a^2*b*Sin[4*(c + d*x)] + 3*a^3*Sin[5*(c + d*x)]))/(240*d)","A",1
485,1,156,213,0.4582224,"\int \cos ^6(c+d x) (a+b \sec (c+d x))^4 \, dx","Integrate[Cos[c + d*x]^6*(a + b*Sec[c + d*x])^4,x]","\frac{5 a^4 \sin (6 (c+d x))+48 a^3 b \sin (5 (c+d x))+45 a^2 \left(a^2+4 b^2\right) \sin (4 (c+d x))+480 a b \left(5 a^2+6 b^2\right) \sin (c+d x)+80 a b \left(5 a^2+4 b^2\right) \sin (3 (c+d x))+60 \left(5 a^4+36 a^2 b^2+8 b^4\right) (c+d x)+15 \left(15 a^4+96 a^2 b^2+16 b^4\right) \sin (2 (c+d x))}{960 d}","\frac{7 a^3 b \sin (c+d x) \cos ^4(c+d x)}{15 d}-\frac{4 a b \left(4 a^2+5 b^2\right) \sin ^3(c+d x)}{15 d}+\frac{4 a b \left(4 a^2+5 b^2\right) \sin (c+d x)}{5 d}+\frac{a^2 \left(5 a^2+32 b^2\right) \sin (c+d x) \cos ^3(c+d x)}{24 d}+\frac{a^2 \sin (c+d x) \cos ^5(c+d x) (a+b \sec (c+d x))^2}{6 d}+\frac{\left(5 a^4+36 a^2 b^2+8 b^4\right) \sin (c+d x) \cos (c+d x)}{16 d}+\frac{1}{16} x \left(5 a^4+36 a^2 b^2+8 b^4\right)",1,"(60*(5*a^4 + 36*a^2*b^2 + 8*b^4)*(c + d*x) + 480*a*b*(5*a^2 + 6*b^2)*Sin[c + d*x] + 15*(15*a^4 + 96*a^2*b^2 + 16*b^4)*Sin[2*(c + d*x)] + 80*a*b*(5*a^2 + 4*b^2)*Sin[3*(c + d*x)] + 45*a^2*(a^2 + 4*b^2)*Sin[4*(c + d*x)] + 48*a^3*b*Sin[5*(c + d*x)] + 5*a^4*Sin[6*(c + d*x)])/(960*d)","A",1
486,1,114,158,0.620683,"\int (a+b \sec (c+d x))^5 \, dx","Integrate[(a + b*Sec[c + d*x])^5,x]","\frac{24 a^5 d x+3 b^2 \tan (c+d x) \left(b \left(40 a^2+3 b^2\right) \sec (c+d x)+40 a \left(2 a^2+b^2\right)+2 b^3 \sec ^3(c+d x)\right)+3 b \left(40 a^4+40 a^2 b^2+3 b^4\right) \tanh ^{-1}(\sin (c+d x))+40 a b^4 \tan ^3(c+d x)}{24 d}","a^5 x+\frac{a b^2 \left(53 a^2+20 b^2\right) \tan (c+d x)}{6 d}+\frac{b^3 \left(58 a^2+9 b^2\right) \tan (c+d x) \sec (c+d x)}{24 d}+\frac{b \left(40 a^4+40 a^2 b^2+3 b^4\right) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{11 a b^2 \tan (c+d x) (a+b \sec (c+d x))^2}{12 d}+\frac{b^2 \tan (c+d x) (a+b \sec (c+d x))^3}{4 d}",1,"(24*a^5*d*x + 3*b*(40*a^4 + 40*a^2*b^2 + 3*b^4)*ArcTanh[Sin[c + d*x]] + 3*b^2*(40*a*(2*a^2 + b^2) + b*(40*a^2 + 3*b^2)*Sec[c + d*x] + 2*b^3*Sec[c + d*x]^3)*Tan[c + d*x] + 40*a*b^4*Tan[c + d*x]^3)/(24*d)","A",1
487,1,258,157,2.7013339,"\int \frac{\sec ^5(c+d x)}{a+b \sec (c+d x)} \, dx","Integrate[Sec[c + d*x]^5/(a + b*Sec[c + d*x]),x]","\frac{\frac{1}{2} \sec ^3(c+d x) \left(4 b \sin (c+d x) \left(\left(3 a^2+2 b^2\right) \cos (2 (c+d x))+3 a^2-3 a b \cos (c+d x)+4 b^2\right)+9 a \left(2 a^2+b^2\right) \cos (c+d x) \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)+3 a \left(2 a^2+b^2\right) \cos (3 (c+d x)) \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)\right)-\frac{24 a^4 \tanh ^{-1}\left(\frac{(b-a) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a^2-b^2}}\right)}{\sqrt{a^2-b^2}}}{12 b^4 d}","\frac{2 a^4 \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{b^4 d \sqrt{a-b} \sqrt{a+b}}-\frac{a \left(2 a^2+b^2\right) \tanh ^{-1}(\sin (c+d x))}{2 b^4 d}+\frac{\left(3 a^2+2 b^2\right) \tan (c+d x)}{3 b^3 d}-\frac{a \tan (c+d x) \sec (c+d x)}{2 b^2 d}+\frac{\tan (c+d x) \sec ^2(c+d x)}{3 b d}",1,"((-24*a^4*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/Sqrt[a^2 - b^2] + (Sec[c + d*x]^3*(9*a*(2*a^2 + b^2)*Cos[c + d*x]*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) + 3*a*(2*a^2 + b^2)*Cos[3*(c + d*x)]*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) + 4*b*(3*a^2 + 4*b^2 - 3*a*b*Cos[c + d*x] + (3*a^2 + 2*b^2)*Cos[2*(c + d*x)])*Sin[c + d*x]))/2)/(12*b^4*d)","A",1
488,1,238,119,1.1400162,"\int \frac{\sec ^4(c+d x)}{a+b \sec (c+d x)} \, dx","Integrate[Sec[c + d*x]^4/(a + b*Sec[c + d*x]),x]","\frac{-4 a^2 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+4 a^2 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+\frac{8 a^3 \tanh ^{-1}\left(\frac{(b-a) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a^2-b^2}}\right)}{\sqrt{a^2-b^2}}-4 a b \tan (c+d x)+\frac{b^2}{\left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^2}-\frac{b^2}{\left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2}-2 b^2 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+2 b^2 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}{4 b^3 d}","-\frac{2 a^3 \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{b^3 d \sqrt{a-b} \sqrt{a+b}}+\frac{\left(2 a^2+b^2\right) \tanh ^{-1}(\sin (c+d x))}{2 b^3 d}-\frac{a \tan (c+d x)}{b^2 d}+\frac{\tan (c+d x) \sec (c+d x)}{2 b d}",1,"((8*a^3*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/Sqrt[a^2 - b^2] - 4*a^2*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - 2*b^2*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 4*a^2*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + 2*b^2*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + b^2/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2 - b^2/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2 - 4*a*b*Tan[c + d*x])/(4*b^3*d)","A",1
489,1,115,85,0.3944539,"\int \frac{\sec ^3(c+d x)}{a+b \sec (c+d x)} \, dx","Integrate[Sec[c + d*x]^3/(a + b*Sec[c + d*x]),x]","\frac{-\frac{2 a^2 \tanh ^{-1}\left(\frac{(b-a) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a^2-b^2}}\right)}{\sqrt{a^2-b^2}}+a \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)+b \tan (c+d x)}{b^2 d}","\frac{2 a^2 \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{b^2 d \sqrt{a-b} \sqrt{a+b}}-\frac{a \tanh ^{-1}(\sin (c+d x))}{b^2 d}+\frac{\tan (c+d x)}{b d}",1,"((-2*a^2*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/Sqrt[a^2 - b^2] + a*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) + b*Tan[c + d*x])/(b^2*d)","A",1
490,1,102,68,0.0852257,"\int \frac{\sec ^2(c+d x)}{a+b \sec (c+d x)} \, dx","Integrate[Sec[c + d*x]^2/(a + b*Sec[c + d*x]),x]","\frac{\frac{2 a \tanh ^{-1}\left(\frac{(b-a) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a^2-b^2}}\right)}{\sqrt{a^2-b^2}}-\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}{b d}","\frac{\tanh ^{-1}(\sin (c+d x))}{b d}-\frac{2 a \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{b d \sqrt{a-b} \sqrt{a+b}}",1,"((2*a*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/Sqrt[a^2 - b^2] - Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]])/(b*d)","A",1
491,1,48,49,0.0391288,"\int \frac{\sec (c+d x)}{a+b \sec (c+d x)} \, dx","Integrate[Sec[c + d*x]/(a + b*Sec[c + d*x]),x]","-\frac{2 \tanh ^{-1}\left(\frac{(b-a) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a^2-b^2}}\right)}{d \sqrt{a^2-b^2}}","\frac{2 \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{d \sqrt{a-b} \sqrt{a+b}}",1,"(-2*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(Sqrt[a^2 - b^2]*d)","A",1
492,1,60,59,0.0882655,"\int \frac{1}{a+b \sec (c+d x)} \, dx","Integrate[(a + b*Sec[c + d*x])^(-1),x]","\frac{\frac{2 b \tanh ^{-1}\left(\frac{(b-a) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a^2-b^2}}\right)}{d \sqrt{a^2-b^2}}+\frac{c}{d}+x}{a}","\frac{x}{a}-\frac{2 b \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a d \sqrt{a-b} \sqrt{a+b}}",1,"(c/d + x + (2*b*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(Sqrt[a^2 - b^2]*d))/a","A",1
493,1,72,76,0.1537076,"\int \frac{\cos (c+d x)}{a+b \sec (c+d x)} \, dx","Integrate[Cos[c + d*x]/(a + b*Sec[c + d*x]),x]","\frac{-\frac{2 b^2 \tanh ^{-1}\left(\frac{(b-a) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a^2-b^2}}\right)}{\sqrt{a^2-b^2}}+a \sin (c+d x)-b (c+d x)}{a^2 d}","\frac{2 b^2 \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^2 d \sqrt{a-b} \sqrt{a+b}}-\frac{b x}{a^2}+\frac{\sin (c+d x)}{a d}",1,"(-(b*(c + d*x)) - (2*b^2*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/Sqrt[a^2 - b^2] + a*Sin[c + d*x])/(a^2*d)","A",1
494,1,97,110,0.2417371,"\int \frac{\cos ^2(c+d x)}{a+b \sec (c+d x)} \, dx","Integrate[Cos[c + d*x]^2/(a + b*Sec[c + d*x]),x]","\frac{2 \left(a^2+2 b^2\right) (c+d x)+\frac{8 b^3 \tanh ^{-1}\left(\frac{(b-a) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a^2-b^2}}\right)}{\sqrt{a^2-b^2}}+a^2 \sin (2 (c+d x))-4 a b \sin (c+d x)}{4 a^3 d}","-\frac{2 b^3 \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^3 d \sqrt{a-b} \sqrt{a+b}}-\frac{b \sin (c+d x)}{a^2 d}+\frac{x \left(a^2+2 b^2\right)}{2 a^3}+\frac{\sin (c+d x) \cos (c+d x)}{2 a d}",1,"(2*(a^2 + 2*b^2)*(c + d*x) + (8*b^3*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/Sqrt[a^2 - b^2] - 4*a*b*Sin[c + d*x] + a^2*Sin[2*(c + d*x)])/(4*a^3*d)","A",1
495,1,122,148,0.332657,"\int \frac{\cos ^3(c+d x)}{a+b \sec (c+d x)} \, dx","Integrate[Cos[c + d*x]^3/(a + b*Sec[c + d*x]),x]","\frac{a^3 \sin (3 (c+d x))-6 b \left(a^2+2 b^2\right) (c+d x)+3 a \left(3 a^2+4 b^2\right) \sin (c+d x)-\frac{24 b^4 \tanh ^{-1}\left(\frac{(b-a) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a^2-b^2}}\right)}{\sqrt{a^2-b^2}}-3 a^2 b \sin (2 (c+d x))}{12 a^4 d}","\frac{2 b^4 \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^4 d \sqrt{a-b} \sqrt{a+b}}-\frac{b \sin (c+d x) \cos (c+d x)}{2 a^2 d}-\frac{b x \left(a^2+2 b^2\right)}{2 a^4}+\frac{\left(2 a^2+3 b^2\right) \sin (c+d x)}{3 a^3 d}+\frac{\sin (c+d x) \cos ^2(c+d x)}{3 a d}",1,"(-6*b*(a^2 + 2*b^2)*(c + d*x) - (24*b^4*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/Sqrt[a^2 - b^2] + 3*a*(3*a^2 + 4*b^2)*Sin[c + d*x] - 3*a^2*b*Sin[2*(c + d*x)] + a^3*Sin[3*(c + d*x)])/(12*a^4*d)","A",1
496,1,153,193,0.5920523,"\int \frac{\cos ^4(c+d x)}{a+b \sec (c+d x)} \, dx","Integrate[Cos[c + d*x]^4/(a + b*Sec[c + d*x]),x]","\frac{3 a^4 \sin (4 (c+d x))-8 a^3 b \sin (3 (c+d x))-24 a b \left(3 a^2+4 b^2\right) \sin (c+d x)+24 a^2 \left(a^2+b^2\right) \sin (2 (c+d x))+\frac{192 b^5 \tanh ^{-1}\left(\frac{(b-a) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a^2-b^2}}\right)}{\sqrt{a^2-b^2}}+12 \left(3 a^4+4 a^2 b^2+8 b^4\right) (c+d x)}{96 a^5 d}","-\frac{2 b^5 \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^5 d \sqrt{a-b} \sqrt{a+b}}-\frac{b \sin (c+d x) \cos ^2(c+d x)}{3 a^2 d}-\frac{b \left(2 a^2+3 b^2\right) \sin (c+d x)}{3 a^4 d}+\frac{\left(3 a^2+4 b^2\right) \sin (c+d x) \cos (c+d x)}{8 a^3 d}+\frac{x \left(3 a^4+4 a^2 b^2+8 b^4\right)}{8 a^5}+\frac{\sin (c+d x) \cos ^3(c+d x)}{4 a d}",1,"(12*(3*a^4 + 4*a^2*b^2 + 8*b^4)*(c + d*x) + (192*b^5*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/Sqrt[a^2 - b^2] - 24*a*b*(3*a^2 + 4*b^2)*Sin[c + d*x] + 24*a^2*(a^2 + b^2)*Sin[2*(c + d*x)] - 8*a^3*b*Sin[3*(c + d*x)] + 3*a^4*Sin[4*(c + d*x)])/(96*a^5*d)","A",1
497,1,357,222,6.1318419,"\int \frac{\sec ^5(c+d x)}{(a+b \sec (c+d x))^2} \, dx","Integrate[Sec[c + d*x]^5/(a + b*Sec[c + d*x])^2,x]","\frac{a^4 \sin (c+d x)}{b^3 d (b-a) (a+b) (a \cos (c+d x)+b)}+\frac{\left(-6 a^2-b^2\right) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}{2 b^4 d}+\frac{\left(6 a^2+b^2\right) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}{2 b^4 d}+\frac{2 a^3 \left(4 b^2-3 a^2\right) \tanh ^{-1}\left(\frac{(b-a) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a^2-b^2}}\right)}{b^4 d \sqrt{a^2-b^2} \left(b^2-a^2\right)}-\frac{2 a \sin \left(\frac{1}{2} (c+d x)\right)}{b^3 d \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}-\frac{2 a \sin \left(\frac{1}{2} (c+d x)\right)}{b^3 d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}+\frac{1}{4 b^2 d \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^2}-\frac{1}{4 b^2 d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2}","-\frac{a^2 \tan (c+d x) \sec ^2(c+d x)}{b d \left(a^2-b^2\right) (a+b \sec (c+d x))}+\frac{\left(3 a^2-b^2\right) \tan (c+d x) \sec (c+d x)}{2 b^2 d \left(a^2-b^2\right)}+\frac{\left(6 a^2+b^2\right) \tanh ^{-1}(\sin (c+d x))}{2 b^4 d}-\frac{a \left(3 a^2-2 b^2\right) \tan (c+d x)}{b^3 d \left(a^2-b^2\right)}-\frac{2 a^3 \left(3 a^2-4 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{b^4 d (a-b)^{3/2} (a+b)^{3/2}}",1,"(2*a^3*(-3*a^2 + 4*b^2)*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(b^4*Sqrt[a^2 - b^2]*(-a^2 + b^2)*d) + ((-6*a^2 - b^2)*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]])/(2*b^4*d) + ((6*a^2 + b^2)*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]])/(2*b^4*d) + 1/(4*b^2*d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2) - (2*a*Sin[(c + d*x)/2])/(b^3*d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])) - 1/(4*b^2*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2) - (2*a*Sin[(c + d*x)/2])/(b^3*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])) + (a^4*Sin[c + d*x])/(b^3*(-a + b)*(a + b)*d*(b + a*Cos[c + d*x]))","A",0
498,1,162,164,1.5537366,"\int \frac{\sec ^4(c+d x)}{(a+b \sec (c+d x))^2} \, dx","Integrate[Sec[c + d*x]^4/(a + b*Sec[c + d*x])^2,x]","\frac{\frac{a^3 b \sin (c+d x)}{(a-b) (a+b) (a \cos (c+d x)+b)}-\frac{2 a^2 \left(2 a^2-3 b^2\right) \tanh ^{-1}\left(\frac{(b-a) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a^2-b^2}}\right)}{\left(a^2-b^2\right)^{3/2}}+2 a \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-2 a \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+b \tan (c+d x)}{b^3 d}","\frac{\left(2 a^2-b^2\right) \tan (c+d x)}{b^2 d \left(a^2-b^2\right)}-\frac{a^2 \tan (c+d x) \sec (c+d x)}{b d \left(a^2-b^2\right) (a+b \sec (c+d x))}+\frac{2 a^2 \left(2 a^2-3 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{b^3 d (a-b)^{3/2} (a+b)^{3/2}}-\frac{2 a \tanh ^{-1}(\sin (c+d x))}{b^3 d}",1,"((-2*a^2*(2*a^2 - 3*b^2)*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(3/2) + 2*a*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - 2*a*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + (a^3*b*Sin[c + d*x])/((a - b)*(a + b)*(b + a*Cos[c + d*x])) + b*Tan[c + d*x])/(b^3*d)","A",1
499,1,146,117,0.3870725,"\int \frac{\sec ^3(c+d x)}{(a+b \sec (c+d x))^2} \, dx","Integrate[Sec[c + d*x]^3/(a + b*Sec[c + d*x])^2,x]","\frac{\frac{2 a \left(a^2-2 b^2\right) \tanh ^{-1}\left(\frac{(b-a) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a^2-b^2}}\right)}{\left(a^2-b^2\right)^{3/2}}+\frac{a^2 b \sin (c+d x)}{(b-a) (a+b) (a \cos (c+d x)+b)}-\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}{b^2 d}","-\frac{2 a \left(a^2-2 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{b^2 d (a-b)^{3/2} (a+b)^{3/2}}-\frac{a^2 \tan (c+d x)}{b d \left(a^2-b^2\right) (a+b \sec (c+d x))}+\frac{\tanh ^{-1}(\sin (c+d x))}{b^2 d}",1,"((2*a*(a^2 - 2*b^2)*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(3/2) - Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + (a^2*b*Sin[c + d*x])/((-a + b)*(a + b)*(b + a*Cos[c + d*x])))/(b^2*d)","A",1
500,1,83,85,0.2001419,"\int \frac{\sec ^2(c+d x)}{(a+b \sec (c+d x))^2} \, dx","Integrate[Sec[c + d*x]^2/(a + b*Sec[c + d*x])^2,x]","\frac{\frac{2 b \tanh ^{-1}\left(\frac{(b-a) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a^2-b^2}}\right)}{\left(a^2-b^2\right)^{3/2}}+\frac{a \sin (c+d x)}{(a-b) (a+b) (a \cos (c+d x)+b)}}{d}","\frac{a \tan (c+d x)}{d \left(a^2-b^2\right) (a+b \sec (c+d x))}-\frac{2 b \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{d (a-b)^{3/2} (a+b)^{3/2}}",1,"((2*b*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(3/2) + (a*Sin[c + d*x])/((a - b)*(a + b)*(b + a*Cos[c + d*x])))/d","A",1
501,1,83,86,0.2293233,"\int \frac{\sec (c+d x)}{(a+b \sec (c+d x))^2} \, dx","Integrate[Sec[c + d*x]/(a + b*Sec[c + d*x])^2,x]","\frac{\frac{b \sin (c+d x)}{(b-a) (a+b) (a \cos (c+d x)+b)}-\frac{2 a \tanh ^{-1}\left(\frac{(b-a) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a^2-b^2}}\right)}{\left(a^2-b^2\right)^{3/2}}}{d}","\frac{2 a \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{d (a-b)^{3/2} (a+b)^{3/2}}-\frac{b \tan (c+d x)}{d \left(a^2-b^2\right) (a+b \sec (c+d x))}",1,"((-2*a*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(3/2) + (b*Sin[c + d*x])/((-a + b)*(a + b)*(b + a*Cos[c + d*x])))/d","A",1
502,1,138,109,0.4758126,"\int \frac{1}{(a+b \sec (c+d x))^2} \, dx","Integrate[(a + b*Sec[c + d*x])^(-2),x]","\frac{\frac{b \left(\left(a^2-b^2\right) (c+d x)+a b \sin (c+d x)\right)+a \left(a^2-b^2\right) (c+d x) \cos (c+d x)}{a \cos (c+d x)+b}-\frac{2 b \left(b^2-2 a^2\right) \tanh ^{-1}\left(\frac{(b-a) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a^2-b^2}}\right)}{\sqrt{a^2-b^2}}}{a^2 d (a-b) (a+b)}","-\frac{2 b \left(2 a^2-b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^2 d (a-b)^{3/2} (a+b)^{3/2}}+\frac{b^2 \tan (c+d x)}{a d \left(a^2-b^2\right) (a+b \sec (c+d x))}+\frac{x}{a^2}",1,"((-2*b*(-2*a^2 + b^2)*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/Sqrt[a^2 - b^2] + (a*(a^2 - b^2)*(c + d*x)*Cos[c + d*x] + b*((a^2 - b^2)*(c + d*x) + a*b*Sin[c + d*x]))/(b + a*Cos[c + d*x]))/(a^2*(a - b)*(a + b)*d)","A",1
503,1,172,146,0.7563173,"\int \frac{\cos (c+d x)}{(a+b \sec (c+d x))^2} \, dx","Integrate[Cos[c + d*x]/(a + b*Sec[c + d*x])^2,x]","\frac{\frac{2 a b \left(a^2-2 b^2\right) \sin (c+d x)+\left(a^2-b^2\right) \left(a^2 \sin (2 (c+d x))-4 b^2 (c+d x)\right)-4 a b \left(a^2-b^2\right) (c+d x) \cos (c+d x)}{a \cos (c+d x)+b}+\frac{4 b^2 \left(2 b^2-3 a^2\right) \tanh ^{-1}\left(\frac{(b-a) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a^2-b^2}}\right)}{\sqrt{a^2-b^2}}}{2 a^3 d (a-b) (a+b)}","-\frac{2 b x}{a^3}+\frac{\left(a^2-2 b^2\right) \sin (c+d x)}{a^2 d \left(a^2-b^2\right)}+\frac{b^2 \sin (c+d x)}{a d \left(a^2-b^2\right) (a+b \sec (c+d x))}+\frac{2 b^2 \left(3 a^2-2 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^3 d (a-b)^{3/2} (a+b)^{3/2}}",1,"((4*b^2*(-3*a^2 + 2*b^2)*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/Sqrt[a^2 - b^2] + (-4*a*b*(a^2 - b^2)*(c + d*x)*Cos[c + d*x] + 2*a*b*(a^2 - 2*b^2)*Sin[c + d*x] + (a^2 - b^2)*(-4*b^2*(c + d*x) + a^2*Sin[2*(c + d*x)]))/(b + a*Cos[c + d*x]))/(2*a^3*(a - b)*(a + b)*d)","A",1
504,1,144,208,0.7689084,"\int \frac{\cos ^2(c+d x)}{(a+b \sec (c+d x))^2} \, dx","Integrate[Cos[c + d*x]^2/(a + b*Sec[c + d*x])^2,x]","\frac{2 \left(a^2+6 b^2\right) (c+d x)-\frac{8 b^3 \left(3 b^2-4 a^2\right) \tanh ^{-1}\left(\frac{(b-a) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a^2-b^2}}\right)}{\left(a^2-b^2\right)^{3/2}}+a^2 \sin (2 (c+d x))+\frac{4 a b^4 \sin (c+d x)}{(a-b) (a+b) (a \cos (c+d x)+b)}-8 a b \sin (c+d x)}{4 a^4 d}","\frac{\left(a^2-3 b^2\right) \sin (c+d x) \cos (c+d x)}{2 a^2 d \left(a^2-b^2\right)}+\frac{b^2 \sin (c+d x) \cos (c+d x)}{a d \left(a^2-b^2\right) (a+b \sec (c+d x))}+\frac{x \left(a^2+6 b^2\right)}{2 a^4}-\frac{2 b^3 \left(4 a^2-3 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^4 d (a-b)^{3/2} (a+b)^{3/2}}-\frac{b \left(2 a^2-3 b^2\right) \sin (c+d x)}{a^3 d \left(a^2-b^2\right)}",1,"(2*(a^2 + 6*b^2)*(c + d*x) - (8*b^3*(-4*a^2 + 3*b^2)*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(3/2) - 8*a*b*Sin[c + d*x] + (4*a*b^4*Sin[c + d*x])/((a - b)*(a + b)*(b + a*Cos[c + d*x])) + a^2*Sin[2*(c + d*x)])/(4*a^4*d)","A",1
505,1,176,261,1.0695144,"\int \frac{\cos ^3(c+d x)}{(a+b \sec (c+d x))^2} \, dx","Integrate[Cos[c + d*x]^3/(a + b*Sec[c + d*x])^2,x]","\frac{a^3 \sin (3 (c+d x))+9 a \left(a^2+4 b^2\right) \sin (c+d x)+\frac{24 b^4 \left(4 b^2-5 a^2\right) \tanh ^{-1}\left(\frac{(b-a) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a^2-b^2}}\right)}{\left(a^2-b^2\right)^{3/2}}-6 a^2 b \sin (2 (c+d x))+\frac{12 a b^5 \sin (c+d x)}{(b-a) (a+b) (a \cos (c+d x)+b)}-12 b (2 b-i a) (2 b+i a) (c+d x)}{12 a^5 d}","\frac{\left(a^2-4 b^2\right) \sin (c+d x) \cos ^2(c+d x)}{3 a^2 d \left(a^2-b^2\right)}+\frac{b^2 \sin (c+d x) \cos ^2(c+d x)}{a d \left(a^2-b^2\right) (a+b \sec (c+d x))}-\frac{b x \left(a^2+4 b^2\right)}{a^5}+\frac{2 b^4 \left(5 a^2-4 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^5 d (a-b)^{3/2} (a+b)^{3/2}}+\frac{\left(2 a^4+7 a^2 b^2-12 b^4\right) \sin (c+d x)}{3 a^4 d \left(a^2-b^2\right)}-\frac{b \left(a^2-2 b^2\right) \sin (c+d x) \cos (c+d x)}{a^3 d \left(a^2-b^2\right)}",1,"(-12*b*((-I)*a + 2*b)*(I*a + 2*b)*(c + d*x) + (24*b^4*(-5*a^2 + 4*b^2)*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(3/2) + 9*a*(a^2 + 4*b^2)*Sin[c + d*x] + (12*a*b^5*Sin[c + d*x])/((-a + b)*(a + b)*(b + a*Cos[c + d*x])) - 6*a^2*b*Sin[2*(c + d*x)] + a^3*Sin[3*(c + d*x)])/(12*a^5*d)","C",1
506,1,205,230,4.8509591,"\int \frac{\sec ^5(c+d x)}{(a+b \sec (c+d x))^3} \, dx","Integrate[Sec[c + d*x]^5/(a + b*Sec[c + d*x])^3,x]","\frac{-\frac{6 a^2 \left(2 a^4-5 a^2 b^2+4 b^4\right) \tanh ^{-1}\left(\frac{(b-a) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a^2-b^2}}\right)}{\left(a^2-b^2\right)^{5/2}}+\frac{a^3 b \sin (c+d x) \left(a \left(4 a^2-7 b^2\right) \cos (c+d x)+5 a^2 b-8 b^3\right)}{(a-b)^2 (a+b)^2 (a \cos (c+d x)+b)^2}+6 a \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-6 a \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+2 b \tan (c+d x)}{2 b^4 d}","-\frac{a^2 \tan (c+d x) \sec ^2(c+d x)}{2 b d \left(a^2-b^2\right) (a+b \sec (c+d x))^2}+\frac{\left(3 a^2-2 b^2\right) \tan (c+d x)}{2 b^3 d \left(a^2-b^2\right)}+\frac{3 a^2 \left(2 a^4-5 a^2 b^2+4 b^4\right) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{b^4 d (a-b)^{5/2} (a+b)^{5/2}}+\frac{3 a^3 \left(a^2-2 b^2\right) \tan (c+d x)}{2 b^3 d \left(a^2-b^2\right)^2 (a+b \sec (c+d x))}-\frac{3 a \tanh ^{-1}(\sin (c+d x))}{b^4 d}",1,"((-6*a^2*(2*a^4 - 5*a^2*b^2 + 4*b^4)*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(5/2) + 6*a*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - 6*a*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + (a^3*b*(5*a^2*b - 8*b^3 + a*(4*a^2 - 7*b^2)*Cos[c + d*x])*Sin[c + d*x])/((a - b)^2*(a + b)^2*(b + a*Cos[c + d*x])^2) + 2*b*Tan[c + d*x])/(2*b^4*d)","A",1
507,1,194,188,1.361981,"\int \frac{\sec ^4(c+d x)}{(a+b \sec (c+d x))^3} \, dx","Integrate[Sec[c + d*x]^4/(a + b*Sec[c + d*x])^3,x]","\frac{-\frac{a^2 b \sin (c+d x) \left(a \left(2 a^2-5 b^2\right) \cos (c+d x)+3 b \left(a^2-2 b^2\right)\right)}{(a-b)^2 (a+b)^2 (a \cos (c+d x)+b)^2}+\frac{2 a \left(2 a^4-5 a^2 b^2+6 b^4\right) \tanh ^{-1}\left(\frac{(b-a) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a^2-b^2}}\right)}{\left(a^2-b^2\right)^{5/2}}-2 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+2 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}{2 b^3 d}","-\frac{a^2 \left(2 a^2-5 b^2\right) \tan (c+d x)}{2 b^2 d \left(a^2-b^2\right)^2 (a+b \sec (c+d x))}-\frac{a^2 \tan (c+d x) \sec (c+d x)}{2 b d \left(a^2-b^2\right) (a+b \sec (c+d x))^2}-\frac{a \left(2 a^4-5 a^2 b^2+6 b^4\right) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{b^3 d (a-b)^{5/2} (a+b)^{5/2}}+\frac{\tanh ^{-1}(\sin (c+d x))}{b^3 d}",1,"((2*a*(2*a^4 - 5*a^2*b^2 + 6*b^4)*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(5/2) - 2*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 2*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] - (a^2*b*(3*b*(a^2 - 2*b^2) + a*(2*a^2 - 5*b^2)*Cos[c + d*x])*Sin[c + d*x])/((a - b)^2*(a + b)^2*(b + a*Cos[c + d*x])^2))/(2*b^3*d)","A",1
508,1,113,149,0.4361796,"\int \frac{\sec ^3(c+d x)}{(a+b \sec (c+d x))^3} \, dx","Integrate[Sec[c + d*x]^3/(a + b*Sec[c + d*x])^3,x]","\frac{\frac{a \sin (c+d x) \left(a^2-3 a b \cos (c+d x)-4 b^2\right)}{(a-b)^2 (a+b)^2 (a \cos (c+d x)+b)^2}-\frac{2 \left(a^2+2 b^2\right) \tanh ^{-1}\left(\frac{(b-a) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a^2-b^2}}\right)}{\left(a^2-b^2\right)^{5/2}}}{2 d}","\frac{\left(a^2+2 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{d (a-b)^{5/2} (a+b)^{5/2}}-\frac{a^2 \tan (c+d x)}{2 b d \left(a^2-b^2\right) (a+b \sec (c+d x))^2}+\frac{a \left(a^2-4 b^2\right) \tan (c+d x)}{2 b d \left(a^2-b^2\right)^2 (a+b \sec (c+d x))}",1,"((-2*(a^2 + 2*b^2)*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(5/2) + (a*(a^2 - 4*b^2 - 3*a*b*Cos[c + d*x])*Sin[c + d*x])/((a - b)^2*(a + b)^2*(b + a*Cos[c + d*x])^2))/(2*d)","A",1
509,1,115,134,0.3970433,"\int \frac{\sec ^2(c+d x)}{(a+b \sec (c+d x))^3} \, dx","Integrate[Sec[c + d*x]^2/(a + b*Sec[c + d*x])^3,x]","\frac{\frac{\sin (c+d x) \left(a \left(2 a^2+b^2\right) \cos (c+d x)+b \left(a^2+2 b^2\right)\right)}{(a \cos (c+d x)+b)^2}+\frac{6 a b \tanh ^{-1}\left(\frac{(b-a) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a^2-b^2}}\right)}{\sqrt{a^2-b^2}}}{2 d (a-b)^2 (a+b)^2}","\frac{\left(a^2+2 b^2\right) \tan (c+d x)}{2 d \left(a^2-b^2\right)^2 (a+b \sec (c+d x))}+\frac{a \tan (c+d x)}{2 d \left(a^2-b^2\right) (a+b \sec (c+d x))^2}-\frac{3 a b \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{d (a-b)^{5/2} (a+b)^{5/2}}",1,"((6*a*b*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/Sqrt[a^2 - b^2] + ((b*(a^2 + 2*b^2) + a*(2*a^2 + b^2)*Cos[c + d*x])*Sin[c + d*x])/(b + a*Cos[c + d*x])^2)/(2*(a - b)^2*(a + b)^2*d)","A",1
510,1,115,133,0.4695901,"\int \frac{\sec (c+d x)}{(a+b \sec (c+d x))^3} \, dx","Integrate[Sec[c + d*x]/(a + b*Sec[c + d*x])^3,x]","\frac{\frac{b \sin (c+d x) \left(\left(b^2-4 a^2\right) \cos (c+d x)-3 a b\right)}{(a \cos (c+d x)+b)^2}-\frac{2 \left(2 a^2+b^2\right) \tanh ^{-1}\left(\frac{(b-a) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a^2-b^2}}\right)}{\sqrt{a^2-b^2}}}{2 d (a-b)^2 (a+b)^2}","\frac{\left(2 a^2+b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{d (a-b)^{5/2} (a+b)^{5/2}}-\frac{3 a b \tan (c+d x)}{2 d \left(a^2-b^2\right)^2 (a+b \sec (c+d x))}-\frac{b \tan (c+d x)}{2 d \left(a^2-b^2\right) (a+b \sec (c+d x))^2}",1,"((-2*(2*a^2 + b^2)*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/Sqrt[a^2 - b^2] + (b*(-3*a*b + (-4*a^2 + b^2)*Cos[c + d*x])*Sin[c + d*x])/(b + a*Cos[c + d*x])^2)/(2*(a - b)^2*(a + b)^2*d)","A",1
511,1,205,173,0.7647524,"\int \frac{1}{(a+b \sec (c+d x))^3} \, dx","Integrate[(a + b*Sec[c + d*x])^(-3),x]","\frac{\sec ^3(c+d x) (a \cos (c+d x)+b) \left(\frac{3 a b^2 \left(2 a^2-b^2\right) \sin (c+d x) (a \cos (c+d x)+b)}{(a-b)^2 (a+b)^2}+\frac{2 b \left(6 a^4-5 a^2 b^2+2 b^4\right) (a \cos (c+d x)+b)^2 \tanh ^{-1}\left(\frac{(b-a) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a^2-b^2}}\right)}{\left(a^2-b^2\right)^{5/2}}+\frac{a b^3 \sin (c+d x)}{(b-a) (a+b)}+2 (c+d x) (a \cos (c+d x)+b)^2\right)}{2 a^3 d (a+b \sec (c+d x))^3}","\frac{x}{a^3}+\frac{b^2 \left(5 a^2-2 b^2\right) \tan (c+d x)}{2 a^2 d \left(a^2-b^2\right)^2 (a+b \sec (c+d x))}+\frac{b^2 \tan (c+d x)}{2 a d \left(a^2-b^2\right) (a+b \sec (c+d x))^2}-\frac{b \left(6 a^4-5 a^2 b^2+2 b^4\right) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^3 d (a-b)^{5/2} (a+b)^{5/2}}",1,"((b + a*Cos[c + d*x])*Sec[c + d*x]^3*(2*(c + d*x)*(b + a*Cos[c + d*x])^2 + (2*b*(6*a^4 - 5*a^2*b^2 + 2*b^4)*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]]*(b + a*Cos[c + d*x])^2)/(a^2 - b^2)^(5/2) + (a*b^3*Sin[c + d*x])/((-a + b)*(a + b)) + (3*a*b^2*(2*a^2 - b^2)*(b + a*Cos[c + d*x])*Sin[c + d*x])/((a - b)^2*(a + b)^2)))/(2*a^3*d*(a + b*Sec[c + d*x])^3)","A",1
512,1,229,223,0.8798791,"\int \frac{\cos (c+d x)}{(a+b \sec (c+d x))^3} \, dx","Integrate[Cos[c + d*x]/(a + b*Sec[c + d*x])^3,x]","\frac{\sec ^3(c+d x) (a \cos (c+d x)+b) \left(-\frac{a b^3 \left(8 a^2-5 b^2\right) \sin (c+d x) (a \cos (c+d x)+b)}{(a-b)^2 (a+b)^2}-\frac{6 b^2 \left(4 a^4-5 a^2 b^2+2 b^4\right) (a \cos (c+d x)+b)^2 \tanh ^{-1}\left(\frac{(b-a) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a^2-b^2}}\right)}{\left(a^2-b^2\right)^{5/2}}+\frac{a b^4 \sin (c+d x)}{(a-b) (a+b)}-6 b (c+d x) (a \cos (c+d x)+b)^2+2 a \sin (c+d x) (a \cos (c+d x)+b)^2\right)}{2 a^4 d (a+b \sec (c+d x))^3}","-\frac{3 b x}{a^4}+\frac{3 b^2 \left(2 a^2-b^2\right) \sin (c+d x)}{2 a^2 d \left(a^2-b^2\right)^2 (a+b \sec (c+d x))}+\frac{b^2 \sin (c+d x)}{2 a d \left(a^2-b^2\right) (a+b \sec (c+d x))^2}+\frac{3 b^2 \left(4 a^4-5 a^2 b^2+2 b^4\right) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^4 d (a-b)^{5/2} (a+b)^{5/2}}+\frac{\left(2 a^4-11 a^2 b^2+6 b^4\right) \sin (c+d x)}{2 a^3 d \left(a^2-b^2\right)^2}",1,"((b + a*Cos[c + d*x])*Sec[c + d*x]^3*(-6*b*(c + d*x)*(b + a*Cos[c + d*x])^2 - (6*b^2*(4*a^4 - 5*a^2*b^2 + 2*b^4)*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]]*(b + a*Cos[c + d*x])^2)/(a^2 - b^2)^(5/2) + (a*b^4*Sin[c + d*x])/((a - b)*(a + b)) - (a*b^3*(8*a^2 - 5*b^2)*(b + a*Cos[c + d*x])*Sin[c + d*x])/((a - b)^2*(a + b)^2) + 2*a*(b + a*Cos[c + d*x])^2*Sin[c + d*x]))/(2*a^4*d*(a + b*Sec[c + d*x])^3)","A",1
513,1,199,296,2.0423489,"\int \frac{\cos ^2(c+d x)}{(a+b \sec (c+d x))^3} \, dx","Integrate[Cos[c + d*x]^2/(a + b*Sec[c + d*x])^3,x]","\frac{2 \left(a^2+12 b^2\right) (c+d x)+\frac{2 a b^4 \left(10 a^2-7 b^2\right) \sin (c+d x)}{(a-b)^2 (a+b)^2 (a \cos (c+d x)+b)}+a^2 \sin (2 (c+d x))+\frac{4 b^3 \left(20 a^4-29 a^2 b^2+12 b^4\right) \tanh ^{-1}\left(\frac{(b-a) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a^2-b^2}}\right)}{\left(a^2-b^2\right)^{5/2}}+\frac{2 a b^5 \sin (c+d x)}{(b-a) (a+b) (a \cos (c+d x)+b)^2}-12 a b \sin (c+d x)}{4 a^5 d}","\frac{b^2 \left(7 a^2-4 b^2\right) \sin (c+d x) \cos (c+d x)}{2 a^2 d \left(a^2-b^2\right)^2 (a+b \sec (c+d x))}+\frac{b^2 \sin (c+d x) \cos (c+d x)}{2 a d \left(a^2-b^2\right) (a+b \sec (c+d x))^2}+\frac{x \left(a^2+12 b^2\right)}{2 a^5}-\frac{3 b \left(2 a^4-7 a^2 b^2+4 b^4\right) \sin (c+d x)}{2 a^4 d \left(a^2-b^2\right)^2}-\frac{b^3 \left(20 a^4-29 a^2 b^2+12 b^4\right) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^5 d (a-b)^{5/2} (a+b)^{5/2}}+\frac{\left(a^4-10 a^2 b^2+6 b^4\right) \sin (c+d x) \cos (c+d x)}{2 a^3 d \left(a^2-b^2\right)^2}",1,"(2*(a^2 + 12*b^2)*(c + d*x) + (4*b^3*(20*a^4 - 29*a^2*b^2 + 12*b^4)*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(5/2) - 12*a*b*Sin[c + d*x] + (2*a*b^5*Sin[c + d*x])/((-a + b)*(a + b)*(b + a*Cos[c + d*x])^2) + (2*a*b^4*(10*a^2 - 7*b^2)*Sin[c + d*x])/((a - b)^2*(a + b)^2*(b + a*Cos[c + d*x])) + a^2*Sin[2*(c + d*x)])/(4*a^5*d)","A",1
514,1,416,316,6.2349993,"\int \frac{\sec ^6(c+d x)}{(a+b \sec (c+d x))^4} \, dx","Integrate[Sec[c + d*x]^6/(a + b*Sec[c + d*x])^4,x]","-\frac{a^3 \sin (c+d x)}{3 b^2 d (b-a) (a+b) (a \cos (c+d x)+b)^3}+\frac{6 a^5 \sin (c+d x)-11 a^3 b^2 \sin (c+d x)}{6 b^3 d (b-a)^2 (a+b)^2 (a \cos (c+d x)+b)^2}+\frac{-18 a^7 \sin (c+d x)+50 a^5 b^2 \sin (c+d x)-47 a^3 b^4 \sin (c+d x)}{6 b^4 d (b-a)^3 (a+b)^3 (a \cos (c+d x)+b)}-\frac{a^2 \left(-8 a^6+28 a^4 b^2-35 a^2 b^4+20 b^6\right) \tanh ^{-1}\left(\frac{(b-a) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a^2-b^2}}\right)}{b^5 d \sqrt{a^2-b^2} \left(b^2-a^2\right)^3}+\frac{4 a \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}{b^5 d}-\frac{4 a \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}{b^5 d}+\frac{\sin \left(\frac{1}{2} (c+d x)\right)}{b^4 d \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}+\frac{\sin \left(\frac{1}{2} (c+d x)\right)}{b^4 d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}","-\frac{a^2 \tan (c+d x) \sec ^3(c+d x)}{3 b d \left(a^2-b^2\right) (a+b \sec (c+d x))^3}-\frac{a^2 \left(4 a^2-9 b^2\right) \tan (c+d x) \sec ^2(c+d x)}{6 b^2 d \left(a^2-b^2\right)^2 (a+b \sec (c+d x))^2}+\frac{\left(12 a^4-23 a^2 b^2+6 b^4\right) \tan (c+d x)}{6 b^4 d \left(a^2-b^2\right)^2}+\frac{a^2 \left(8 a^6-28 a^4 b^2+35 a^2 b^4-20 b^6\right) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{b^5 d (a-b)^{7/2} (a+b)^{7/2}}+\frac{a^3 \left(4 a^4-11 a^2 b^2+12 b^4\right) \tan (c+d x)}{2 b^4 d \left(a^2-b^2\right)^3 (a+b \sec (c+d x))}-\frac{4 a \tanh ^{-1}(\sin (c+d x))}{b^5 d}",1,"-((a^2*(-8*a^6 + 28*a^4*b^2 - 35*a^2*b^4 + 20*b^6)*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(b^5*Sqrt[a^2 - b^2]*(-a^2 + b^2)^3*d)) + (4*a*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]])/(b^5*d) - (4*a*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]])/(b^5*d) + Sin[(c + d*x)/2]/(b^4*d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])) + Sin[(c + d*x)/2]/(b^4*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])) - (a^3*Sin[c + d*x])/(3*b^2*(-a + b)*(a + b)*d*(b + a*Cos[c + d*x])^3) + (6*a^5*Sin[c + d*x] - 11*a^3*b^2*Sin[c + d*x])/(6*b^3*(-a + b)^2*(a + b)^2*d*(b + a*Cos[c + d*x])^2) + (-18*a^7*Sin[c + d*x] + 50*a^5*b^2*Sin[c + d*x] - 47*a^3*b^4*Sin[c + d*x])/(6*b^4*(-a + b)^3*(a + b)^3*d*(b + a*Cos[c + d*x]))","A",1
515,1,250,259,4.1435606,"\int \frac{\sec ^5(c+d x)}{(a+b \sec (c+d x))^4} \, dx","Integrate[Sec[c + d*x]^5/(a + b*Sec[c + d*x])^4,x]","\frac{-\frac{a^2 b \sin (c+d x) \left(11 a^4 b^2-32 a^2 b^4+a^2 \left(6 a^4-17 a^2 b^2+26 b^4\right) \cos ^2(c+d x)+15 a b \left(a^4-3 a^2 b^2+4 b^4\right) \cos (c+d x)+36 b^6\right)}{(a-b)^3 (a+b)^3 (a \cos (c+d x)+b)^3}+\frac{6 a \left(2 a^6-7 a^4 b^2+8 a^2 b^4-8 b^6\right) \tanh ^{-1}\left(\frac{(b-a) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a^2-b^2}}\right)}{\left(a^2-b^2\right)^{7/2}}-6 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+6 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}{6 b^4 d}","-\frac{a^2 \tan (c+d x) \sec ^2(c+d x)}{3 b d \left(a^2-b^2\right) (a+b \sec (c+d x))^3}-\frac{a^2 \left(9 a^4-28 a^2 b^2+34 b^4\right) \tan (c+d x)}{6 b^3 d \left(a^2-b^2\right)^3 (a+b \sec (c+d x))}+\frac{a^3 \left(3 a^2-8 b^2\right) \tan (c+d x)}{6 b^3 d \left(a^2-b^2\right)^2 (a+b \sec (c+d x))^2}-\frac{a \left(2 a^6-7 a^4 b^2+8 a^2 b^4-8 b^6\right) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{b^4 d (a-b)^{7/2} (a+b)^{7/2}}+\frac{\tanh ^{-1}(\sin (c+d x))}{b^4 d}",1,"((6*a*(2*a^6 - 7*a^4*b^2 + 8*a^2*b^4 - 8*b^6)*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(7/2) - 6*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 6*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] - (a^2*b*(11*a^4*b^2 - 32*a^2*b^4 + 36*b^6 + 15*a*b*(a^4 - 3*a^2*b^2 + 4*b^4)*Cos[c + d*x] + a^2*(6*a^4 - 17*a^2*b^2 + 26*b^4)*Cos[c + d*x]^2)*Sin[c + d*x])/((a - b)^3*(a + b)^3*(b + a*Cos[c + d*x])^3))/(6*b^4*d)","A",1
516,1,158,222,1.0274131,"\int \frac{\sec ^4(c+d x)}{(a+b \sec (c+d x))^4} \, dx","Integrate[Sec[c + d*x]^4/(a + b*Sec[c + d*x])^4,x]","\frac{\frac{6 b \left(3 a^2+2 b^2\right) \tanh ^{-1}\left(\frac{(b-a) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a^2-b^2}}\right)}{\left(a^2-b^2\right)^{7/2}}+\frac{a \sin (c+d x) \left(2 a^4+a^2 \left(4 a^2+11 b^2\right) \cos ^2(c+d x)+3 a b \left(a^2+9 b^2\right) \cos (c+d x)-5 a^2 b^2+18 b^4\right)}{(a-b)^3 (a+b)^3 (a \cos (c+d x)+b)^3}}{6 d}","-\frac{b \left(3 a^2+2 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{d (a-b)^{7/2} (a+b)^{7/2}}-\frac{a^2 \left(2 a^2-7 b^2\right) \tan (c+d x)}{6 b^2 d \left(a^2-b^2\right)^2 (a+b \sec (c+d x))^2}-\frac{a^2 \tan (c+d x) \sec (c+d x)}{3 b d \left(a^2-b^2\right) (a+b \sec (c+d x))^3}+\frac{a \left(2 a^4-5 a^2 b^2+18 b^4\right) \tan (c+d x)}{6 b^2 d \left(a^2-b^2\right)^3 (a+b \sec (c+d x))}",1,"((6*b*(3*a^2 + 2*b^2)*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(7/2) + (a*(2*a^4 - 5*a^2*b^2 + 18*b^4 + 3*a*b*(a^2 + 9*b^2)*Cos[c + d*x] + a^2*(4*a^2 + 11*b^2)*Cos[c + d*x]^2)*Sin[c + d*x])/((a - b)^3*(a + b)^3*(b + a*Cos[c + d*x])^3))/(6*d)","A",1
517,1,165,206,1.0975824,"\int \frac{\sec ^3(c+d x)}{(a+b \sec (c+d x))^4} \, dx","Integrate[Sec[c + d*x]^3/(a + b*Sec[c + d*x])^4,x]","\frac{\frac{\sin (c+d x) \left(-a^2 b \left(13 a^2+2 b^2\right) \cos ^2(c+d x)+3 a \left(a^4-9 a^2 b^2-2 b^4\right) \cos (c+d x)+b \left(a^4-10 a^2 b^2-6 b^4\right)\right)}{(a-b)^3 (a+b)^3 (a \cos (c+d x)+b)^3}-\frac{6 a \left(a^2+4 b^2\right) \tanh ^{-1}\left(\frac{(b-a) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a^2-b^2}}\right)}{\left(a^2-b^2\right)^{7/2}}}{6 d}","\frac{a \left(a^2+4 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{d (a-b)^{7/2} (a+b)^{7/2}}-\frac{a^2 \tan (c+d x)}{3 b d \left(a^2-b^2\right) (a+b \sec (c+d x))^3}+\frac{a \left(a^2-6 b^2\right) \tan (c+d x)}{6 b d \left(a^2-b^2\right)^2 (a+b \sec (c+d x))^2}+\frac{\left(a^4-10 a^2 b^2-6 b^4\right) \tan (c+d x)}{6 b d \left(a^2-b^2\right)^3 (a+b \sec (c+d x))}",1,"((-6*a*(a^2 + 4*b^2)*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(7/2) + ((b*(a^4 - 10*a^2*b^2 - 6*b^4) + 3*a*(a^4 - 9*a^2*b^2 - 2*b^4)*Cos[c + d*x] - a^2*b*(13*a^2 + 2*b^2)*Cos[c + d*x]^2)*Sin[c + d*x])/((a - b)^3*(a + b)^3*(b + a*Cos[c + d*x])^3))/(6*d)","A",1
518,1,164,192,1.2834714,"\int \frac{\sec ^2(c+d x)}{(a+b \sec (c+d x))^4} \, dx","Integrate[Sec[c + d*x]^2/(a + b*Sec[c + d*x])^4,x]","\frac{\frac{6 b \left(4 a^2+b^2\right) \tanh ^{-1}\left(\frac{(b-a) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a^2-b^2}}\right)}{\left(a^2-b^2\right)^{7/2}}+\frac{\sin (c+d x) \left(2 a^3 b^2+a \left(6 a^4+10 a^2 b^2-b^4\right) \cos ^2(c+d x)-3 b \left(-2 a^4-9 a^2 b^2+b^4\right) \cos (c+d x)+13 a b^4\right)}{(a-b)^3 (a+b)^3 (a \cos (c+d x)+b)^3}}{6 d}","-\frac{b \left(4 a^2+b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{d (a-b)^{7/2} (a+b)^{7/2}}+\frac{a \left(2 a^2+13 b^2\right) \tan (c+d x)}{6 d \left(a^2-b^2\right)^3 (a+b \sec (c+d x))}+\frac{\left(2 a^2+3 b^2\right) \tan (c+d x)}{6 d \left(a^2-b^2\right)^2 (a+b \sec (c+d x))^2}+\frac{a \tan (c+d x)}{3 d \left(a^2-b^2\right) (a+b \sec (c+d x))^3}",1,"((6*b*(4*a^2 + b^2)*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(7/2) + ((2*a^3*b^2 + 13*a*b^4 - 3*b*(-2*a^4 - 9*a^2*b^2 + b^4)*Cos[c + d*x] + a*(6*a^4 + 10*a^2*b^2 - b^4)*Cos[c + d*x]^2)*Sin[c + d*x])/((a - b)^3*(a + b)^3*(b + a*Cos[c + d*x])^3))/(6*d)","A",1
519,1,163,184,1.1662402,"\int \frac{\sec (c+d x)}{(a+b \sec (c+d x))^4} \, dx","Integrate[Sec[c + d*x]/(a + b*Sec[c + d*x])^4,x]","-\frac{\frac{12 a \left(2 a^2+3 b^2\right) \tanh ^{-1}\left(\frac{(b-a) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a^2-b^2}}\right)}{\sqrt{a^2-b^2}}+\frac{b \sin (c+d x) \left(18 a^4+6 a b \left(9 a^2+b^2\right) \cos (c+d x)+17 a^2 b^2+\left(18 a^4-5 a^2 b^2+2 b^4\right) \cos (2 (c+d x))+10 b^4\right)}{(a \cos (c+d x)+b)^3}}{12 d (a-b)^3 (a+b)^3}","\frac{a \left(2 a^2+3 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{d (a-b)^{7/2} (a+b)^{7/2}}-\frac{b \left(11 a^2+4 b^2\right) \tan (c+d x)}{6 d \left(a^2-b^2\right)^3 (a+b \sec (c+d x))}-\frac{5 a b \tan (c+d x)}{6 d \left(a^2-b^2\right)^2 (a+b \sec (c+d x))^2}-\frac{b \tan (c+d x)}{3 d \left(a^2-b^2\right) (a+b \sec (c+d x))^3}",1,"-1/12*((12*a*(2*a^2 + 3*b^2)*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/Sqrt[a^2 - b^2] + (b*(18*a^4 + 17*a^2*b^2 + 10*b^4 + 6*a*b*(9*a^2 + b^2)*Cos[c + d*x] + (18*a^4 - 5*a^2*b^2 + 2*b^4)*Cos[2*(c + d*x)])*Sin[c + d*x])/(b + a*Cos[c + d*x])^3)/((a - b)^3*(a + b)^3*d)","A",1
520,1,268,242,1.4950632,"\int \frac{1}{(a+b \sec (c+d x))^4} \, dx","Integrate[(a + b*Sec[c + d*x])^(-4),x]","\frac{\sec ^4(c+d x) (a \cos (c+d x)+b) \left(-\frac{a b^3 \left(12 a^2-7 b^2\right) \sin (c+d x) (a \cos (c+d x)+b)}{(a-b)^2 (a+b)^2}+\frac{a b^2 \left(36 a^4-32 a^2 b^2+11 b^4\right) \sin (c+d x) (a \cos (c+d x)+b)^2}{(a-b)^3 (a+b)^3}-\frac{6 b \left(-8 a^6+8 a^4 b^2-7 a^2 b^4+2 b^6\right) (a \cos (c+d x)+b)^3 \tanh ^{-1}\left(\frac{(b-a) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a^2-b^2}}\right)}{\left(a^2-b^2\right)^{7/2}}+\frac{2 a b^4 \sin (c+d x)}{(a-b) (a+b)}+6 (c+d x) (a \cos (c+d x)+b)^3\right)}{6 a^4 d (a+b \sec (c+d x))^4}","\frac{x}{a^4}+\frac{b^2 \left(8 a^2-3 b^2\right) \tan (c+d x)}{6 a^2 d \left(a^2-b^2\right)^2 (a+b \sec (c+d x))^2}+\frac{b^2 \tan (c+d x)}{3 a d \left(a^2-b^2\right) (a+b \sec (c+d x))^3}-\frac{b \left(8 a^6-8 a^4 b^2+7 a^2 b^4-2 b^6\right) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^4 d (a-b)^{7/2} (a+b)^{7/2}}+\frac{b^2 \left(26 a^4-17 a^2 b^2+6 b^4\right) \tan (c+d x)}{6 a^3 d \left(a^2-b^2\right)^3 (a+b \sec (c+d x))}",1,"((b + a*Cos[c + d*x])*Sec[c + d*x]^4*(6*(c + d*x)*(b + a*Cos[c + d*x])^3 - (6*b*(-8*a^6 + 8*a^4*b^2 - 7*a^2*b^4 + 2*b^6)*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]]*(b + a*Cos[c + d*x])^3)/(a^2 - b^2)^(7/2) + (2*a*b^4*Sin[c + d*x])/((a - b)*(a + b)) - (a*b^3*(12*a^2 - 7*b^2)*(b + a*Cos[c + d*x])*Sin[c + d*x])/((a - b)^2*(a + b)^2) + (a*b^2*(36*a^4 - 32*a^2*b^2 + 11*b^4)*(b + a*Cos[c + d*x])^2*Sin[c + d*x])/((a - b)^3*(a + b)^3)))/(6*a^4*d*(a + b*Sec[c + d*x])^4)","A",1
521,1,293,299,1.7004571,"\int \frac{\cos (c+d x)}{(a+b \sec (c+d x))^4} \, dx","Integrate[Cos[c + d*x]/(a + b*Sec[c + d*x])^4,x]","\frac{\sec ^4(c+d x) (a \cos (c+d x)+b) \left(\frac{5 a b^4 \left(3 a^2-2 b^2\right) \sin (c+d x) (a \cos (c+d x)+b)}{(a-b)^2 (a+b)^2}-\frac{a b^3 \left(60 a^4-71 a^2 b^2+26 b^4\right) \sin (c+d x) (a \cos (c+d x)+b)^2}{(a-b)^3 (a+b)^3}+\frac{6 b^2 \left(-20 a^6+35 a^4 b^2-28 a^2 b^4+8 b^6\right) (a \cos (c+d x)+b)^3 \tanh ^{-1}\left(\frac{(b-a) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a^2-b^2}}\right)}{\left(a^2-b^2\right)^{7/2}}+\frac{2 a b^5 \sin (c+d x)}{(b-a) (a+b)}-24 b (c+d x) (a \cos (c+d x)+b)^3+6 a \sin (c+d x) (a \cos (c+d x)+b)^3\right)}{6 a^5 d (a+b \sec (c+d x))^4}","-\frac{4 b x}{a^5}+\frac{b^2 \left(9 a^2-4 b^2\right) \sin (c+d x)}{6 a^2 d \left(a^2-b^2\right)^2 (a+b \sec (c+d x))^2}+\frac{b^2 \sin (c+d x)}{3 a d \left(a^2-b^2\right) (a+b \sec (c+d x))^3}+\frac{\left(6 a^6-65 a^4 b^2+68 a^2 b^4-24 b^6\right) \sin (c+d x)}{6 a^4 d \left(a^2-b^2\right)^3}+\frac{b^2 \left(12 a^4-11 a^2 b^2+4 b^4\right) \sin (c+d x)}{2 a^3 d \left(a^2-b^2\right)^3 (a+b \sec (c+d x))}+\frac{b^2 \left(20 a^6-35 a^4 b^2+28 a^2 b^4-8 b^6\right) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^5 d (a-b)^{7/2} (a+b)^{7/2}}",1,"((b + a*Cos[c + d*x])*Sec[c + d*x]^4*(-24*b*(c + d*x)*(b + a*Cos[c + d*x])^3 + (6*b^2*(-20*a^6 + 35*a^4*b^2 - 28*a^2*b^4 + 8*b^6)*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]]*(b + a*Cos[c + d*x])^3)/(a^2 - b^2)^(7/2) + (2*a*b^5*Sin[c + d*x])/((-a + b)*(a + b)) + (5*a*b^4*(3*a^2 - 2*b^2)*(b + a*Cos[c + d*x])*Sin[c + d*x])/((a - b)^2*(a + b)^2) - (a*b^3*(60*a^4 - 71*a^2*b^2 + 26*b^4)*(b + a*Cos[c + d*x])^2*Sin[c + d*x])/((a - b)^3*(a + b)^3) + 6*a*(b + a*Cos[c + d*x])^3*Sin[c + d*x]))/(6*a^5*d*(a + b*Sec[c + d*x])^4)","A",1
522,1,326,387,6.3838407,"\int \frac{\cos ^2(c+d x)}{(a+b \sec (c+d x))^4} \, dx","Integrate[Cos[c + d*x]^2/(a + b*Sec[c + d*x])^4,x]","-\frac{b^6 \sin (c+d x)}{3 a^5 d (b-a) (a+b) (a \cos (c+d x)+b)^3}-\frac{4 b \sin (c+d x)}{a^5 d}+\frac{\sin (2 (c+d x))}{4 a^4 d}+\frac{\left(a^2+20 b^2\right) (c+d x)}{2 a^6 d}+\frac{13 b^7 \sin (c+d x)-18 a^2 b^5 \sin (c+d x)}{6 a^5 d (b-a)^2 (a+b)^2 (a \cos (c+d x)+b)^2}+\frac{b^3 \left(-40 a^6+84 a^4 b^2-69 a^2 b^4+20 b^6\right) \tanh ^{-1}\left(\frac{(b-a) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a^2-b^2}}\right)}{a^6 d \sqrt{a^2-b^2} \left(b^2-a^2\right)^3}+\frac{-90 a^4 b^4 \sin (c+d x)+122 a^2 b^6 \sin (c+d x)-47 b^8 \sin (c+d x)}{6 a^5 d (b-a)^3 (a+b)^3 (a \cos (c+d x)+b)}","\frac{5 b^2 \left(2 a^2-b^2\right) \sin (c+d x) \cos (c+d x)}{6 a^2 d \left(a^2-b^2\right)^2 (a+b \sec (c+d x))^2}+\frac{b^2 \sin (c+d x) \cos (c+d x)}{3 a d \left(a^2-b^2\right) (a+b \sec (c+d x))^3}+\frac{x \left(a^2+20 b^2\right)}{2 a^6}+\frac{\left(a^6-23 a^4 b^2+27 a^2 b^4-10 b^6\right) \sin (c+d x) \cos (c+d x)}{2 a^4 d \left(a^2-b^2\right)^3}-\frac{b^3 \left(40 a^6-84 a^4 b^2+69 a^2 b^4-20 b^6\right) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^6 d (a-b)^{7/2} (a+b)^{7/2}}+\frac{b^2 \left(48 a^4-53 a^2 b^2+20 b^4\right) \sin (c+d x) \cos (c+d x)}{6 a^3 d \left(a^2-b^2\right)^3 (a+b \sec (c+d x))}-\frac{b \left(24 a^6-146 a^4 b^2+167 a^2 b^4-60 b^6\right) \sin (c+d x)}{6 a^5 d \left(a^2-b^2\right)^3}",1,"((a^2 + 20*b^2)*(c + d*x))/(2*a^6*d) + (b^3*(-40*a^6 + 84*a^4*b^2 - 69*a^2*b^4 + 20*b^6)*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^6*Sqrt[a^2 - b^2]*(-a^2 + b^2)^3*d) - (4*b*Sin[c + d*x])/(a^5*d) - (b^6*Sin[c + d*x])/(3*a^5*(-a + b)*(a + b)*d*(b + a*Cos[c + d*x])^3) + (-18*a^2*b^5*Sin[c + d*x] + 13*b^7*Sin[c + d*x])/(6*a^5*(-a + b)^2*(a + b)^2*d*(b + a*Cos[c + d*x])^2) + (-90*a^4*b^4*Sin[c + d*x] + 122*a^2*b^6*Sin[c + d*x] - 47*b^8*Sin[c + d*x])/(6*a^5*(-a + b)^3*(a + b)^3*d*(b + a*Cos[c + d*x])) + Sin[2*(c + d*x)]/(4*a^4*d)","A",1
523,1,30,31,0.053214,"\int \frac{1}{3+5 \sec (c+d x)} \, dx","Integrate[(3 + 5*Sec[c + d*x])^(-1),x]","\frac{2 (c+d x)+5 \tan ^{-1}\left(2 \cot \left(\frac{1}{2} (c+d x)\right)\right)}{6 d}","\frac{5 \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)+3}\right)}{6 d}-\frac{x}{12}",1,"(2*(c + d*x) + 5*ArcTan[2*Cot[(c + d*x)/2]])/(6*d)","A",1
524,1,73,56,0.1722496,"\int \frac{1}{(3+5 \sec (c+d x))^2} \, dx","Integrate[(3 + 5*Sec[c + d*x])^(-2),x]","\frac{160 (c+d x)-150 \sin (c+d x)+96 (c+d x) \cos (c+d x)+35 (3 \cos (c+d x)+5) \tan ^{-1}\left(2 \cot \left(\frac{1}{2} (c+d x)\right)\right)}{288 d (3 \cos (c+d x)+5)}","-\frac{25 \tan (c+d x)}{48 d (5 \sec (c+d x)+3)}+\frac{35 \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)+3}\right)}{288 d}+\frac{29 x}{576}",1,"(160*(c + d*x) + 96*(c + d*x)*Cos[c + d*x] + 35*ArcTan[2*Cot[(c + d*x)/2]]*(5 + 3*Cos[c + d*x]) - 150*Sin[c + d*x])/(288*d*(5 + 3*Cos[c + d*x]))","A",1
525,1,108,81,0.345702,"\int \frac{1}{(3+5 \sec (c+d x))^3} \, dx","Integrate[(3 + 5*Sec[c + d*x])^(-3),x]","\frac{-3750 \sin (c+d x)-4725 \sin (2 (c+d x))+30720 (c+d x) \cos (c+d x)+4608 c \cos (2 (c+d x))+4608 d x \cos (2 (c+d x))+3055 (3 \cos (c+d x)+5)^2 \tan ^{-1}\left(2 \cot \left(\frac{1}{2} (c+d x)\right)\right)+30208 c+30208 d x}{27648 d (3 \cos (c+d x)+5)^2}","-\frac{125 \tan (c+d x)}{4608 d (5 \sec (c+d x)+3)}-\frac{25 \tan (c+d x)}{96 d (5 \sec (c+d x)+3)^2}+\frac{3055 \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)+3}\right)}{27648 d}-\frac{1007 x}{55296}",1,"(30208*c + 30208*d*x + 30720*(c + d*x)*Cos[c + d*x] + 3055*ArcTan[2*Cot[(c + d*x)/2]]*(5 + 3*Cos[c + d*x])^2 + 4608*c*Cos[2*(c + d*x)] + 4608*d*x*Cos[2*(c + d*x)] - 3750*Sin[c + d*x] - 4725*Sin[2*(c + d*x)])/(27648*d*(5 + 3*Cos[c + d*x])^2)","A",1
526,1,141,106,0.5340413,"\int \frac{1}{(3+5 \sec (c+d x))^4} \, dx","Integrate[(3 + 5*Sec[c + d*x])^(-4),x]","\frac{-5660475 \sin (c+d x)-3082500 \sin (2 (c+d x))-582975 \sin (3 (c+d x))+8036352 (c+d x) \cos (c+d x)+2211840 c \cos (2 (c+d x))+2211840 d x \cos (2 (c+d x))+221184 c \cos (3 (c+d x))+221184 d x \cos (3 (c+d x))+22430 (3 \cos (c+d x)+5)^3 \tan ^{-1}\left(2 \cot \left(\frac{1}{2} (c+d x)\right)\right)+6307840 c+6307840 d x}{2654208 d (3 \cos (c+d x)+5)^3}","-\frac{16925 \tan (c+d x)}{221184 d (5 \sec (c+d x)+3)}-\frac{25 \tan (c+d x)}{4608 d (5 \sec (c+d x)+3)^2}-\frac{25 \tan (c+d x)}{144 d (5 \sec (c+d x)+3)^3}+\frac{11215 \tan ^{-1}\left(\frac{\sin (c+d x)}{\cos (c+d x)+3}\right)}{1327104 d}+\frac{21553 x}{2654208}",1,"(6307840*c + 6307840*d*x + 8036352*(c + d*x)*Cos[c + d*x] + 22430*ArcTan[2*Cot[(c + d*x)/2]]*(5 + 3*Cos[c + d*x])^3 + 2211840*c*Cos[2*(c + d*x)] + 2211840*d*x*Cos[2*(c + d*x)] + 221184*c*Cos[3*(c + d*x)] + 221184*d*x*Cos[3*(c + d*x)] - 5660475*Sin[c + d*x] - 3082500*Sin[2*(c + d*x)] - 582975*Sin[3*(c + d*x)])/(2654208*d*(5 + 3*Cos[c + d*x])^3)","A",1
527,1,69,70,0.0676079,"\int \frac{1}{5+3 \sec (c+d x)} \, dx","Integrate[(5 + 3*Sec[c + d*x])^(-1),x]","\frac{4 (c+d x)+3 \log \left(2 \cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-3 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+2 \cos \left(\frac{1}{2} (c+d x)\right)\right)}{20 d}","\frac{3 \log \left(2 \cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}{20 d}-\frac{3 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+2 \cos \left(\frac{1}{2} (c+d x)\right)\right)}{20 d}+\frac{x}{5}",1,"(4*(c + d*x) + 3*Log[2*Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - 3*Log[2*Cos[(c + d*x)/2] + Sin[(c + d*x)/2]])/(20*d)","A",1
528,1,162,95,0.1616316,"\int \frac{1}{(5+3 \sec (c+d x))^2} \, dx","Integrate[(5 + 3*Sec[c + d*x])^(-2),x]","\frac{5 \cos (c+d x) \left(64 (c+d x)+123 \log \left(2 \cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-123 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+2 \cos \left(\frac{1}{2} (c+d x)\right)\right)\right)+3 \left(60 \sin (c+d x)+123 \log \left(2 \cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-123 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+2 \cos \left(\frac{1}{2} (c+d x)\right)\right)+64 c+64 d x\right)}{1600 d (5 \cos (c+d x)+3)}","\frac{9 \tan (c+d x)}{80 d (3 \sec (c+d x)+5)}+\frac{123 \log \left(2 \cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}{1600 d}-\frac{123 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+2 \cos \left(\frac{1}{2} (c+d x)\right)\right)}{1600 d}+\frac{x}{25}",1,"(5*Cos[c + d*x]*(64*(c + d*x) + 123*Log[2*Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - 123*Log[2*Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) + 3*(64*c + 64*d*x + 123*Log[2*Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - 123*Log[2*Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + 60*Sin[c + d*x]))/(1600*d*(3 + 5*Cos[c + d*x]))","A",1
529,1,241,120,0.3152463,"\int \frac{1}{(5+3 \sec (c+d x))^3} \, dx","Integrate[(5 + 3*Sec[c + d*x])^(-3),x]","\frac{115560 \sin (c+d x)+110700 \sin (2 (c+d x))+359523 \log \left(2 \cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+60 \cos (c+d x) \left(2048 (c+d x)+8361 \log \left(2 \cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-8361 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+2 \cos \left(\frac{1}{2} (c+d x)\right)\right)\right)+25 \cos (2 (c+d x)) \left(2048 (c+d x)+8361 \log \left(2 \cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-8361 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+2 \cos \left(\frac{1}{2} (c+d x)\right)\right)\right)-359523 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+2 \cos \left(\frac{1}{2} (c+d x)\right)\right)+88064 c+88064 d x}{512000 d (5 \cos (c+d x)+3)^2}","\frac{963 \tan (c+d x)}{12800 d (3 \sec (c+d x)+5)}+\frac{9 \tan (c+d x)}{160 d (3 \sec (c+d x)+5)^2}+\frac{8361 \log \left(2 \cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}{256000 d}-\frac{8361 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+2 \cos \left(\frac{1}{2} (c+d x)\right)\right)}{256000 d}+\frac{x}{125}",1,"(88064*c + 88064*d*x + 359523*Log[2*Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 60*Cos[c + d*x]*(2048*(c + d*x) + 8361*Log[2*Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - 8361*Log[2*Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) + 25*Cos[2*(c + d*x)]*(2048*(c + d*x) + 8361*Log[2*Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - 8361*Log[2*Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) - 359523*Log[2*Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + 115560*Sin[c + d*x] + 110700*Sin[2*(c + d*x)])/(512000*d*(3 + 5*Cos[c + d*x])^2)","B",1
530,1,344,145,0.5159134,"\int \frac{1}{(5+3 \sec (c+d x))^4} \, dx","Integrate[(5 + 3*Sec[c + d*x])^(-4),x]","\frac{52174260 \sin (c+d x)+51462000 \sin (2 (c+d x))+24286500 \sin (3 (c+d x))+4096000 c \cos (3 (c+d x))+4096000 d x \cos (3 (c+d x))+34768875 \cos (3 (c+d x)) \log \left(2 \cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+155208258 \log \left(2 \cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+915 \cos (c+d x) \left(32768 (c+d x)+278151 \log \left(2 \cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-278151 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+2 \cos \left(\frac{1}{2} (c+d x)\right)\right)\right)+450 \cos (2 (c+d x)) \left(32768 (c+d x)+278151 \log \left(2 \cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-278151 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+2 \cos \left(\frac{1}{2} (c+d x)\right)\right)\right)-34768875 \cos (3 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+2 \cos \left(\frac{1}{2} (c+d x)\right)\right)-155208258 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+2 \cos \left(\frac{1}{2} (c+d x)\right)\right)+18284544 c+18284544 d x}{81920000 d (5 \cos (c+d x)+3)^3}","\frac{38733 \tan (c+d x)}{1024000 d (3 \sec (c+d x)+5)}+\frac{519 \tan (c+d x)}{12800 d (3 \sec (c+d x)+5)^2}+\frac{3 \tan (c+d x)}{80 d (3 \sec (c+d x)+5)^3}+\frac{278151 \log \left(2 \cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}{20480000 d}-\frac{278151 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+2 \cos \left(\frac{1}{2} (c+d x)\right)\right)}{20480000 d}+\frac{x}{625}",1,"(18284544*c + 18284544*d*x + 4096000*c*Cos[3*(c + d*x)] + 4096000*d*x*Cos[3*(c + d*x)] + 155208258*Log[2*Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 34768875*Cos[3*(c + d*x)]*Log[2*Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 915*Cos[c + d*x]*(32768*(c + d*x) + 278151*Log[2*Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - 278151*Log[2*Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) + 450*Cos[2*(c + d*x)]*(32768*(c + d*x) + 278151*Log[2*Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - 278151*Log[2*Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) - 155208258*Log[2*Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] - 34768875*Cos[3*(c + d*x)]*Log[2*Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + 52174260*Sin[c + d*x] + 51462000*Sin[2*(c + d*x)] + 24286500*Sin[3*(c + d*x)])/(81920000*d*(3 + 5*Cos[c + d*x])^3)","B",1
531,1,401,292,13.5006588,"\int \sec ^3(c+d x) \sqrt{a+b \sec (c+d x)} \, dx","Integrate[Sec[c + d*x]^3*Sqrt[a + b*Sec[c + d*x]],x]","\frac{\sqrt{a+b \sec (c+d x)} \left(\frac{2 \left(9 b^2-2 a^2\right) \sin (c+d x)}{15 b^2}+\frac{2 a \tan (c+d x)}{15 b}+\frac{2}{5} \tan (c+d x) \sec (c+d x)\right)}{d}+\frac{2 \sqrt{\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)} \sqrt{a+b \sec (c+d x)} \left(\left(2 a^2-9 b^2\right) \cos (c+d x) \tan \left(\frac{1}{2} (c+d x)\right) \sec ^2\left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b)+2 b \left(-2 a^2+7 a b+9 b^2\right) \sqrt{\frac{\cos (c+d x)}{\cos (c+d x)+1}} \sqrt{\frac{a \cos (c+d x)+b}{(a+b) (\cos (c+d x)+1)}} F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right)+2 \left(2 a^3+2 a^2 b-9 a b^2-9 b^3\right) \sqrt{\frac{\cos (c+d x)}{\cos (c+d x)+1}} \sqrt{\frac{a \cos (c+d x)+b}{(a+b) (\cos (c+d x)+1)}} E\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right)\right)}{15 b^2 d \sqrt{\sec ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\sec (c+d x)} (a \cos (c+d x)+b)}","\frac{2 (a-b) \sqrt{a+b} \left(2 a^2-9 b^2\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{15 b^3 d}+\frac{2 (a-b) \sqrt{a+b} (2 a+9 b) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{15 b^2 d}+\frac{2 \tan (c+d x) (a+b \sec (c+d x))^{3/2}}{5 b d}-\frac{4 a \tan (c+d x) \sqrt{a+b \sec (c+d x)}}{15 b d}",1,"(2*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*(2*(2*a^3 + 2*a^2*b - 9*a*b^2 - 9*b^3)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + 2*b*(-2*a^2 + 7*a*b + 9*b^2)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + (2*a^2 - 9*b^2)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(15*b^2*d*(b + a*Cos[c + d*x])*Sqrt[Sec[(c + d*x)/2]^2]*Sqrt[Sec[c + d*x]]) + (Sqrt[a + b*Sec[c + d*x]]*((2*(-2*a^2 + 9*b^2)*Sin[c + d*x])/(15*b^2) + (2*a*Tan[c + d*x])/(15*b) + (2*Sec[c + d*x]*Tan[c + d*x])/5))/d","A",0
532,1,293,241,10.7925746,"\int \sec ^2(c+d x) \sqrt{a+b \sec (c+d x)} \, dx","Integrate[Sec[c + d*x]^2*Sqrt[a + b*Sec[c + d*x]],x]","\frac{\sqrt{a+b \sec (c+d x)} \left(\frac{2 a \sin (c+d x)}{3 b}+\frac{2}{3} \tan (c+d x)\right)}{d}-\frac{2 \cos ^2\left(\frac{1}{2} (c+d x)\right) \sqrt{a+b \sec (c+d x)} \left(a \cos (c+d x) \tan \left(\frac{1}{2} (c+d x)\right) \sec ^2\left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b)-2 b (a+b) \sqrt{\frac{\cos (c+d x)}{\cos (c+d x)+1}} \sqrt{\frac{a \cos (c+d x)+b}{(a+b) (\cos (c+d x)+1)}} F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right)+2 a (a+b) \sqrt{\frac{\cos (c+d x)}{\cos (c+d x)+1}} \sqrt{\frac{a \cos (c+d x)+b}{(a+b) (\cos (c+d x)+1)}} E\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right)\right)}{3 b d (a \cos (c+d x)+b)}","-\frac{2 a (a-b) \sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{3 b^2 d}+\frac{2 \tan (c+d x) \sqrt{a+b \sec (c+d x)}}{3 d}-\frac{2 (a-b) \sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{3 b d}",1,"(-2*Cos[(c + d*x)/2]^2*Sqrt[a + b*Sec[c + d*x]]*(2*a*(a + b)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] - 2*b*(a + b)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + a*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(3*b*d*(b + a*Cos[c + d*x])) + (Sqrt[a + b*Sec[c + d*x]]*((2*a*Sin[c + d*x])/(3*b) + (2*Tan[c + d*x])/3))/d","A",1
533,1,232,209,10.344,"\int \sec (c+d x) \sqrt{a+b \sec (c+d x)} \, dx","Integrate[Sec[c + d*x]*Sqrt[a + b*Sec[c + d*x]],x]","\frac{2 \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{d}-\frac{2 \sqrt{a+b \sec (c+d x)} \left(\tan \left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b)+\frac{(a+b) \sqrt{\frac{a \cos (c+d x)+b}{(a+b) (\cos (c+d x)+1)}} \left(E\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right)-F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right)\right)}{\sqrt{\frac{\cos (c+d x)}{\cos (c+d x)+1}}}\right)}{d \sqrt{\sec ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\sec (c+d x)} \sqrt{\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)} (a \cos (c+d x)+b)}","\frac{2 (a-b) \sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{b d}-\frac{2 (a-b) \sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{b d}",1,"(2*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/d - (2*Sqrt[a + b*Sec[c + d*x]]*(((a + b)*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*(EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] - EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]))/Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])] + (b + a*Cos[c + d*x])*Tan[(c + d*x)/2]))/(d*(b + a*Cos[c + d*x])*Sqrt[Sec[(c + d*x)/2]^2]*Sqrt[Sec[c + d*x]]*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]])","A",1
534,1,151,125,1.6123924,"\int \sqrt{a+b \sec (c+d x)} \, dx","Integrate[Sqrt[a + b*Sec[c + d*x]],x]","\frac{4 \cos ^2\left(\frac{1}{2} (c+d x)\right) \sqrt{\frac{\cos (c+d x)}{\cos (c+d x)+1}} \sqrt{\frac{a \cos (c+d x)+b}{(a+b) (\cos (c+d x)+1)}} \sqrt{a+b \sec (c+d x)} \left((b-a) F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right)+2 a \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right)\right)}{d (a \cos (c+d x)+b)}","-\frac{2 \cot (c+d x) \sqrt{-\frac{b (1-\sec (c+d x))}{a+b \sec (c+d x)}} \sqrt{\frac{b (\sec (c+d x)+1)}{a+b \sec (c+d x)}} (a+b \sec (c+d x)) \Pi \left(\frac{a}{a+b};\sin ^{-1}\left(\frac{\sqrt{a+b}}{\sqrt{a+b \sec (c+d x)}}\right)|\frac{a-b}{a+b}\right)}{d \sqrt{a+b}}",1,"(4*Cos[(c + d*x)/2]^2*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*((-a + b)*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + 2*a*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)])*Sqrt[a + b*Sec[c + d*x]])/(d*(b + a*Cos[c + d*x]))","A",1
535,1,2713,330,18.250893,"\int \cos (c+d x) \sqrt{a+b \sec (c+d x)} \, dx","Integrate[Cos[c + d*x]*Sqrt[a + b*Sec[c + d*x]],x]","\text{Result too large to show}","\frac{\sin (c+d x) \sqrt{a+b \sec (c+d x)}}{d}+\frac{\sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{d}+\frac{(a-b) \sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{b d}-\frac{b \sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{a d}",1,"(Cos[c + d*x]*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*(I*(a - b)*EllipticE[I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] + (2*I)*b*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] - Sqrt[2]*Sqrt[(-a + b)/(a + b)]*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*(b + a*Cos[c + d*x])*Tan[(c + d*x)/2])*(-1 + Tan[(c + d*x)/2]^2))/(Sqrt[(-a + b)/(a + b)]*d*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Cos[c + d*x]*Sec[(c + d*x)/2]^4]*((Sec[(c + d*x)/2]^2*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Tan[(c + d*x)/2]*(I*(a - b)*EllipticE[I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] + (2*I)*b*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] - Sqrt[2]*Sqrt[(-a + b)/(a + b)]*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*(b + a*Cos[c + d*x])*Tan[(c + d*x)/2]))/(Sqrt[(-a + b)/(a + b)]*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Cos[c + d*x]*Sec[(c + d*x)/2]^4]) + (a*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sin[c + d*x]*(I*(a - b)*EllipticE[I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] + (2*I)*b*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] - Sqrt[2]*Sqrt[(-a + b)/(a + b)]*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*(b + a*Cos[c + d*x])*Tan[(c + d*x)/2])*(-1 + Tan[(c + d*x)/2]^2))/(2*Sqrt[(-a + b)/(a + b)]*(b + a*Cos[c + d*x])^(3/2)*Sqrt[Cos[c + d*x]*Sec[(c + d*x)/2]^4]) - (Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(I*(a - b)*EllipticE[I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] + (2*I)*b*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] - Sqrt[2]*Sqrt[(-a + b)/(a + b)]*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*(b + a*Cos[c + d*x])*Tan[(c + d*x)/2])*(-(Sec[(c + d*x)/2]^4*Sin[c + d*x]) + 2*Cos[c + d*x]*Sec[(c + d*x)/2]^4*Tan[(c + d*x)/2])*(-1 + Tan[(c + d*x)/2]^2))/(2*Sqrt[(-a + b)/(a + b)]*Sqrt[b + a*Cos[c + d*x]]*(Cos[c + d*x]*Sec[(c + d*x)/2]^4)^(3/2)) + (Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(-1 + Tan[(c + d*x)/2]^2)*(-((Sqrt[(-a + b)/(a + b)]*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/Sqrt[2]) + Sqrt[2]*a*Sqrt[(-a + b)/(a + b)]*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sin[c + d*x]*Tan[(c + d*x)/2] - (Sqrt[(-a + b)/(a + b)]*(b + a*Cos[c + d*x])*((Cos[c + d*x]*Sin[c + d*x])/(1 + Cos[c + d*x])^2 - Sin[c + d*x]/(1 + Cos[c + d*x]))*Tan[(c + d*x)/2])/(Sqrt[2]*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]) + ((I/2)*(a - b)*EllipticE[I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*(-((a*Sec[(c + d*x)/2]^2*Sin[c + d*x])/(a + b)) + ((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/(a + b)))/Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] + (I*b*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*(-((a*Sec[(c + d*x)/2]^2*Sin[c + d*x])/(a + b)) + ((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/(a + b)))/Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] - (b*Sqrt[(-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)])/((1 - ((-a + b)*Tan[(c + d*x)/2]^2)/(a - b))*Sqrt[1 + ((-a + b)*Tan[(c + d*x)/2]^2)/(a - b)]*Sqrt[1 + ((-a + b)*Tan[(c + d*x)/2]^2)/(a + b)]) - ((a - b)*Sqrt[(-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)]*Sqrt[1 + ((-a + b)*Tan[(c + d*x)/2]^2)/(a - b)])/(2*Sqrt[1 + ((-a + b)*Tan[(c + d*x)/2]^2)/(a + b)])))/(Sqrt[(-a + b)/(a + b)]*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Cos[c + d*x]*Sec[(c + d*x)/2]^4]) + ((I*(a - b)*EllipticE[I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] + (2*I)*b*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] - Sqrt[2]*Sqrt[(-a + b)/(a + b)]*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*(b + a*Cos[c + d*x])*Tan[(c + d*x)/2])*(-1 + Tan[(c + d*x)/2]^2)*(-(Cos[(c + d*x)/2]*Sec[c + d*x]*Sin[(c + d*x)/2]) + Cos[(c + d*x)/2]^2*Sec[c + d*x]*Tan[c + d*x]))/(2*Sqrt[(-a + b)/(a + b)]*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Cos[c + d*x]*Sec[(c + d*x)/2]^4]*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]])))","C",0
536,1,1173,396,18.7719748,"\int \cos ^2(c+d x) \sqrt{a+b \sec (c+d x)} \, dx","Integrate[Cos[c + d*x]^2*Sqrt[a + b*Sec[c + d*x]],x]","\frac{\sqrt{a+b \sec (c+d x)} \sin (2 (c+d x))}{4 d}+\frac{\sqrt{a+b \sec (c+d x)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{\tan ^2\left(\frac{1}{2} (c+d x)\right)+1}} \left(-b^2 \sqrt{\frac{b-a}{a+b}} \tan ^5\left(\frac{1}{2} (c+d x)\right)+a b \sqrt{\frac{b-a}{a+b}} \tan ^5\left(\frac{1}{2} (c+d x)\right)-2 a b \sqrt{\frac{b-a}{a+b}} \tan ^3\left(\frac{1}{2} (c+d x)\right)-8 i a^2 \Pi \left(-\frac{a+b}{a-b};i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)+2 i b^2 \Pi \left(-\frac{a+b}{a-b};i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)+b^2 \sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)+a b \sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)-i (a-b) b E\left(i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}+2 i \left(2 a^2-b a-b^2\right) F\left(i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}-8 i a^2 \Pi \left(-\frac{a+b}{a-b};i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}+2 i b^2 \Pi \left(-\frac{a+b}{a-b};i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}\right)}{4 a \sqrt{\frac{b-a}{a+b}} d \sqrt{b+a \cos (c+d x)} \sqrt{\sec (c+d x)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)-1\right) \sqrt{\frac{\tan ^2\left(\frac{1}{2} (c+d x)\right)+1}{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)}} \left(a \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)-1\right)-b \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right)\right)}","-\frac{\sqrt{a+b} \left(4 a^2-b^2\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{4 a^2 d}+\frac{b \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{4 a d}+\frac{\sin (c+d x) \cos (c+d x) \sqrt{a+b \sec (c+d x)}}{2 d}+\frac{\sqrt{a+b} (2 a+b) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{4 a d}+\frac{(a-b) \sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{4 a d}",1,"(Sqrt[a + b*Sec[c + d*x]]*Sin[2*(c + d*x)])/(4*d) + (Sqrt[a + b*Sec[c + d*x]]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(1 + Tan[(c + d*x)/2]^2)]*(a*b*Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2] + b^2*Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2] - 2*a*b*Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]^3 + a*b*Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]^5 - b^2*Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]^5 - (8*I)*a^2*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + (2*I)*b^2*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - (8*I)*a^2*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + (2*I)*b^2*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - I*(a - b)*b*EllipticE[I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + (2*I)*(2*a^2 - a*b - b^2)*EllipticF[I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)]))/(4*a*Sqrt[(-a + b)/(a + b)]*d*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*(-1 + Tan[(c + d*x)/2]^2)*Sqrt[(1 + Tan[(c + d*x)/2]^2)/(1 - Tan[(c + d*x)/2]^2)]*(a*(-1 + Tan[(c + d*x)/2]^2) - b*(1 + Tan[(c + d*x)/2]^2)))","C",0
537,1,550,405,17.8265201,"\int \sec ^4(c+d x) (a+b \sec (c+d x))^{3/2} \, dx","Integrate[Sec[c + d*x]^4*(a + b*Sec[c + d*x])^(3/2),x]","\frac{\cos (c+d x) (a+b \sec (c+d x))^{3/2} \left(\frac{8 \sec (c+d x) \left(22 a b^2 \sin (c+d x)-a^3 \sin (c+d x)\right)}{315 b^2}+\frac{2 \sec ^2(c+d x) \left(3 a^2 \sin (c+d x)+49 b^2 \sin (c+d x)\right)}{315 b}+\frac{2 \left(8 a^4+33 a^2 b^2+147 b^4\right) \sin (c+d x)}{315 b^3}+\frac{20}{63} a \tan (c+d x) \sec ^2(c+d x)+\frac{2}{9} b \tan (c+d x) \sec ^3(c+d x)\right)}{d (a \cos (c+d x)+b)}-\frac{2 \sqrt{\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)} (a+b \sec (c+d x))^{3/2} \left(\left(8 a^4+33 a^2 b^2+147 b^4\right) \cos (c+d x) \tan \left(\frac{1}{2} (c+d x)\right) \sec ^2\left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b)-2 b \left(8 a^4+2 a^3 b+33 a^2 b^2+186 a b^3+147 b^4\right) \sqrt{\frac{\cos (c+d x)}{\cos (c+d x)+1}} \sqrt{\frac{a \cos (c+d x)+b}{(a+b) (\cos (c+d x)+1)}} F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right)+2 \left(8 a^5+8 a^4 b+33 a^3 b^2+33 a^2 b^3+147 a b^4+147 b^5\right) \sqrt{\frac{\cos (c+d x)}{\cos (c+d x)+1}} \sqrt{\frac{a \cos (c+d x)+b}{(a+b) (\cos (c+d x)+1)}} E\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right)\right)}{315 b^3 d \sqrt{\sec ^2\left(\frac{1}{2} (c+d x)\right)} \sec ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+b)^2}","\frac{2 \left(8 a^2+49 b^2\right) \tan (c+d x) (a+b \sec (c+d x))^{3/2}}{315 b^2 d}+\frac{2 a \left(8 a^2+39 b^2\right) \tan (c+d x) \sqrt{a+b \sec (c+d x)}}{315 b^2 d}-\frac{2 (a-b) \sqrt{a+b} \left(8 a^4+33 a^2 b^2+147 b^4\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{315 b^4 d}-\frac{2 (a-b) \sqrt{a+b} \left(8 a^3+6 a^2 b+39 a b^2-147 b^3\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{315 b^3 d}-\frac{8 a \tan (c+d x) (a+b \sec (c+d x))^{5/2}}{63 b^2 d}+\frac{2 \tan (c+d x) \sec (c+d x) (a+b \sec (c+d x))^{5/2}}{9 b d}",1,"(-2*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(a + b*Sec[c + d*x])^(3/2)*(2*(8*a^5 + 8*a^4*b + 33*a^3*b^2 + 33*a^2*b^3 + 147*a*b^4 + 147*b^5)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] - 2*b*(8*a^4 + 2*a^3*b + 33*a^2*b^2 + 186*a*b^3 + 147*b^4)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + (8*a^4 + 33*a^2*b^2 + 147*b^4)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(315*b^3*d*(b + a*Cos[c + d*x])^2*Sqrt[Sec[(c + d*x)/2]^2]*Sec[c + d*x]^(3/2)) + (Cos[c + d*x]*(a + b*Sec[c + d*x])^(3/2)*((2*(8*a^4 + 33*a^2*b^2 + 147*b^4)*Sin[c + d*x])/(315*b^3) + (2*Sec[c + d*x]^2*(3*a^2*Sin[c + d*x] + 49*b^2*Sin[c + d*x]))/(315*b) + (8*Sec[c + d*x]*(-(a^3*Sin[c + d*x]) + 22*a*b^2*Sin[c + d*x]))/(315*b^2) + (20*a*Sec[c + d*x]^2*Tan[c + d*x])/63 + (2*b*Sec[c + d*x]^3*Tan[c + d*x])/9))/(d*(b + a*Cos[c + d*x]))","A",0
538,1,471,342,14.0465544,"\int \sec ^3(c+d x) (a+b \sec (c+d x))^{3/2} \, dx","Integrate[Sec[c + d*x]^3*(a + b*Sec[c + d*x])^(3/2),x]","\frac{\cos (c+d x) (a+b \sec (c+d x))^{3/2} \left(-\frac{4 a \left(3 a^2-41 b^2\right) \sin (c+d x)}{105 b^2}+\frac{2 \sec (c+d x) \left(3 a^2 \sin (c+d x)+25 b^2 \sin (c+d x)\right)}{105 b}+\frac{16}{35} a \tan (c+d x) \sec (c+d x)+\frac{2}{7} b \tan (c+d x) \sec ^2(c+d x)\right)}{d (a \cos (c+d x)+b)}+\frac{4 \sqrt{\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)} (a+b \sec (c+d x))^{3/2} \left(a \left(3 a^2-41 b^2\right) \cos (c+d x) \tan \left(\frac{1}{2} (c+d x)\right) \sec ^2\left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b)+b \left(-6 a^3+51 a^2 b+82 a b^2+25 b^3\right) \sqrt{\frac{\cos (c+d x)}{\cos (c+d x)+1}} \sqrt{\frac{a \cos (c+d x)+b}{(a+b) (\cos (c+d x)+1)}} F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right)+2 a \left(3 a^3+3 a^2 b-41 a b^2-41 b^3\right) \sqrt{\frac{\cos (c+d x)}{\cos (c+d x)+1}} \sqrt{\frac{a \cos (c+d x)+b}{(a+b) (\cos (c+d x)+1)}} E\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right)\right)}{105 b^2 d \sqrt{\sec ^2\left(\frac{1}{2} (c+d x)\right)} \sec ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+b)^2}","-\frac{2 \left(6 a^2-25 b^2\right) \tan (c+d x) \sqrt{a+b \sec (c+d x)}}{105 b d}+\frac{2 (a-b) \sqrt{a+b} \left(6 a^2+57 a b-25 b^2\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{105 b^2 d}+\frac{4 a (a-b) \sqrt{a+b} \left(3 a^2-41 b^2\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{105 b^3 d}+\frac{2 \tan (c+d x) (a+b \sec (c+d x))^{5/2}}{7 b d}-\frac{4 a \tan (c+d x) (a+b \sec (c+d x))^{3/2}}{35 b d}",1,"(4*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(a + b*Sec[c + d*x])^(3/2)*(2*a*(3*a^3 + 3*a^2*b - 41*a*b^2 - 41*b^3)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + b*(-6*a^3 + 51*a^2*b + 82*a*b^2 + 25*b^3)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + a*(3*a^2 - 41*b^2)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(105*b^2*d*(b + a*Cos[c + d*x])^2*Sqrt[Sec[(c + d*x)/2]^2]*Sec[c + d*x]^(3/2)) + (Cos[c + d*x]*(a + b*Sec[c + d*x])^(3/2)*((-4*a*(3*a^2 - 41*b^2)*Sin[c + d*x])/(105*b^2) + (2*Sec[c + d*x]*(3*a^2*Sin[c + d*x] + 25*b^2*Sin[c + d*x]))/(105*b) + (16*a*Sec[c + d*x]*Tan[c + d*x])/35 + (2*b*Sec[c + d*x]^2*Tan[c + d*x])/7))/(d*(b + a*Cos[c + d*x]))","A",0
539,1,408,282,13.2604373,"\int \sec ^2(c+d x) (a+b \sec (c+d x))^{3/2} \, dx","Integrate[Sec[c + d*x]^2*(a + b*Sec[c + d*x])^(3/2),x]","\frac{\cos (c+d x) (a+b \sec (c+d x))^{3/2} \left(\frac{2 \left(a^2+3 b^2\right) \sin (c+d x)}{5 b}+\frac{4}{5} a \tan (c+d x)+\frac{2}{5} b \tan (c+d x) \sec (c+d x)\right)}{d (a \cos (c+d x)+b)}-\frac{2 \sqrt{\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)} (a+b \sec (c+d x))^{3/2} \left(\left(a^2+3 b^2\right) \cos (c+d x) \tan \left(\frac{1}{2} (c+d x)\right) \sec ^2\left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b)-2 b \left(a^2+4 a b+3 b^2\right) \sqrt{\frac{\cos (c+d x)}{\cos (c+d x)+1}} \sqrt{\frac{a \cos (c+d x)+b}{(a+b) (\cos (c+d x)+1)}} F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right)+2 \left(a^3+a^2 b+3 a b^2+3 b^3\right) \sqrt{\frac{\cos (c+d x)}{\cos (c+d x)+1}} \sqrt{\frac{a \cos (c+d x)+b}{(a+b) (\cos (c+d x)+1)}} E\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right)\right)}{5 b d \sqrt{\sec ^2\left(\frac{1}{2} (c+d x)\right)} \sec ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+b)^2}","-\frac{2 (a-b) \sqrt{a+b} \left(a^2+3 b^2\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{5 b^2 d}+\frac{2 \tan (c+d x) (a+b \sec (c+d x))^{3/2}}{5 d}+\frac{2 a \tan (c+d x) \sqrt{a+b \sec (c+d x)}}{5 d}-\frac{2 (a-3 b) (a-b) \sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{5 b d}",1,"(-2*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(a + b*Sec[c + d*x])^(3/2)*(2*(a^3 + a^2*b + 3*a*b^2 + 3*b^3)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] - 2*b*(a^2 + 4*a*b + 3*b^2)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + (a^2 + 3*b^2)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(5*b*d*(b + a*Cos[c + d*x])^2*Sqrt[Sec[(c + d*x)/2]^2]*Sec[c + d*x]^(3/2)) + (Cos[c + d*x]*(a + b*Sec[c + d*x])^(3/2)*((2*(a^2 + 3*b^2)*Sin[c + d*x])/(5*b) + (4*a*Tan[c + d*x])/5 + (2*b*Sec[c + d*x]*Tan[c + d*x])/5))/(d*(b + a*Cos[c + d*x]))","A",0
540,1,304,249,10.5104075,"\int \sec (c+d x) (a+b \sec (c+d x))^{3/2} \, dx","Integrate[Sec[c + d*x]*(a + b*Sec[c + d*x])^(3/2),x]","-\frac{2 \sqrt{a+b \sec (c+d x)} \left(-2 \left(3 a^2+4 a b+b^2\right) \cos ^2\left(\frac{1}{2} (c+d x)\right) \sqrt{\frac{\cos (c+d x)}{\cos (c+d x)+1}} \sqrt{\frac{a \cos (c+d x)+b}{(a+b) (\cos (c+d x)+1)}} F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right)-2 a^2 \sin (2 (c+d x))+4 a^2 \cos ^2(c+d x) \tan \left(\frac{1}{2} (c+d x)\right)-5 a b \sin (c+d x)+4 a b \cos (c+d x) \tan \left(\frac{1}{2} (c+d x)\right)+8 a (a+b) \cos ^2\left(\frac{1}{2} (c+d x)\right) \sqrt{\frac{\cos (c+d x)}{\cos (c+d x)+1}} \sqrt{\frac{a \cos (c+d x)+b}{(a+b) (\cos (c+d x)+1)}} E\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right)-b^2 \tan (c+d x)\right)}{3 d (a \cos (c+d x)+b)}","\frac{2 b \tan (c+d x) \sqrt{a+b \sec (c+d x)}}{3 d}+\frac{2 (a-b) (3 a-b) \sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{3 b d}-\frac{8 a (a-b) \sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{3 b d}",1,"(-2*Sqrt[a + b*Sec[c + d*x]]*(8*a*(a + b)*Cos[(c + d*x)/2]^2*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] - 2*(3*a^2 + 4*a*b + b^2)*Cos[(c + d*x)/2]^2*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] - 5*a*b*Sin[c + d*x] - 2*a^2*Sin[2*(c + d*x)] + 4*a*b*Cos[c + d*x]*Tan[(c + d*x)/2] + 4*a^2*Cos[c + d*x]^2*Tan[(c + d*x)/2] - b^2*Tan[c + d*x]))/(3*d*(b + a*Cos[c + d*x]))","A",1
541,1,882,309,18.1953932,"\int (a+b \sec (c+d x))^{3/2} \, dx","Integrate[(a + b*Sec[c + d*x])^(3/2),x]","\frac{2 b \cos (c+d x) \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{d (b+a \cos (c+d x))}+\frac{2 \left(-b^2 \sqrt{\frac{b-a}{a+b}} \tan ^5\left(\frac{1}{2} (c+d x)\right)+a b \sqrt{\frac{b-a}{a+b}} \tan ^5\left(\frac{1}{2} (c+d x)\right)-2 a b \sqrt{\frac{b-a}{a+b}} \tan ^3\left(\frac{1}{2} (c+d x)\right)+2 i a^2 \Pi \left(-\frac{a+b}{a-b};i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)+b^2 \sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)+a b \sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)-i (a-b) b E\left(i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}-i (a-b)^2 F\left(i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}+2 i a^2 \Pi \left(-\frac{a+b}{a-b};i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}\right) (a+b \sec (c+d x))^{3/2}}{\sqrt{\frac{b-a}{a+b}} d (b+a \cos (c+d x))^{3/2} \sec ^{\frac{3}{2}}(c+d x) \sqrt{\frac{1}{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)}} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)-1\right) \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right)^{3/2} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{\tan ^2\left(\frac{1}{2} (c+d x)\right)+1}}}","\frac{2 (2 a-b) \sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{d}-\frac{2 (a-b) \sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{d}-\frac{2 a \sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{d}",1,"(2*b*Cos[c + d*x]*(a + b*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(d*(b + a*Cos[c + d*x])) + (2*(a + b*Sec[c + d*x])^(3/2)*(a*b*Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2] + b^2*Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2] - 2*a*b*Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]^3 + a*b*Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]^5 - b^2*Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]^5 + (2*I)*a^2*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + (2*I)*a^2*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - I*(a - b)*b*EllipticE[I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - I*(a - b)^2*EllipticF[I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)]))/(Sqrt[(-a + b)/(a + b)]*d*(b + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(3/2)*Sqrt[(1 - Tan[(c + d*x)/2]^2)^(-1)]*(-1 + Tan[(c + d*x)/2]^2)*(1 + Tan[(c + d*x)/2]^2)^(3/2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(1 + Tan[(c + d*x)/2]^2)])","C",1
542,1,439,334,11.9980903,"\int \cos (c+d x) (a+b \sec (c+d x))^{3/2} \, dx","Integrate[Cos[c + d*x]*(a + b*Sec[c + d*x])^(3/2),x]","\frac{\cos ^2\left(\frac{1}{2} (c+d x)\right) \cos (c+d x) (a+b \sec (c+d x))^{3/2} \left(a \sqrt{\frac{b-a}{a+b}} \cos (c+d x) \tan \left(\frac{1}{2} (c+d x)\right) \sec ^2\left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b)+4 i b (a-b) \sqrt{\frac{\cos (c+d x)}{\cos (c+d x)+1}} \sqrt{\frac{a \cos (c+d x)+b}{(a+b) (\cos (c+d x)+1)}} F\left(i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a+b}{a-b}\right)-2 i a (a-b) \sqrt{\frac{\cos (c+d x)}{\cos (c+d x)+1}} \sqrt{\frac{a \cos (c+d x)+b}{(a+b) (\cos (c+d x)+1)}} E\left(i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a+b}{a-b}\right)-12 i a b \sqrt{\frac{\cos (c+d x)}{\cos (c+d x)+1}} \sqrt{\frac{a \cos (c+d x)+b}{(a+b) (\cos (c+d x)+1)}} \Pi \left(-\frac{a+b}{a-b};i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a+b}{a-b}\right)\right)}{d \sqrt{\frac{b-a}{a+b}} (a \cos (c+d x)+b)^2}","\frac{a \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{d}+\frac{\sqrt{a+b} (a+2 b) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{d}+\frac{a (a-b) \sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{b d}-\frac{3 b \sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{d}",1,"(Cos[(c + d*x)/2]^2*Cos[c + d*x]*(a + b*Sec[c + d*x])^(3/2)*((-2*I)*a*(a - b)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)] + (4*I)*(a - b)*b*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)] - (12*I)*a*b*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)] + a*Sqrt[(-a + b)/(a + b)]*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(Sqrt[(-a + b)/(a + b)]*d*(b + a*Cos[c + d*x])^2)","C",1
543,1,1159,390,18.5870649,"\int \cos ^2(c+d x) (a+b \sec (c+d x))^{3/2} \, dx","Integrate[Cos[c + d*x]^2*(a + b*Sec[c + d*x])^(3/2),x]","\frac{a \cos (c+d x) (a+b \sec (c+d x))^{3/2} \sin (2 (c+d x))}{4 d (b+a \cos (c+d x))}-\frac{(a+b \sec (c+d x))^{3/2} \left(-5 b^2 \sqrt{\frac{b-a}{a+b}} \tan ^5\left(\frac{1}{2} (c+d x)\right)+5 a b \sqrt{\frac{b-a}{a+b}} \tan ^5\left(\frac{1}{2} (c+d x)\right)-10 a b \sqrt{\frac{b-a}{a+b}} \tan ^3\left(\frac{1}{2} (c+d x)\right)-8 i a^2 \Pi \left(-\frac{a+b}{a-b};i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)-6 i b^2 \Pi \left(-\frac{a+b}{a-b};i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)+5 b^2 \sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)+5 a b \sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)-5 i (a-b) b E\left(i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}+2 i \left(2 a^2-b a-b^2\right) F\left(i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}-8 i a^2 \Pi \left(-\frac{a+b}{a-b};i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}-6 i b^2 \Pi \left(-\frac{a+b}{a-b};i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}\right)}{4 \sqrt{\frac{b-a}{a+b}} d (b+a \cos (c+d x))^{3/2} \sec ^{\frac{3}{2}}(c+d x) \sqrt{\frac{1}{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)}} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)-1\right) \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right)^{3/2} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{\tan ^2\left(\frac{1}{2} (c+d x)\right)+1}}}","-\frac{\sqrt{a+b} \left(4 a^2+3 b^2\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{4 a d}+\frac{5 b \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{4 d}+\frac{a \sin (c+d x) \cos (c+d x) \sqrt{a+b \sec (c+d x)}}{2 d}+\frac{\sqrt{a+b} (2 a+5 b) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{4 d}+\frac{5 (a-b) \sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{4 d}",1,"(a*Cos[c + d*x]*(a + b*Sec[c + d*x])^(3/2)*Sin[2*(c + d*x)])/(4*d*(b + a*Cos[c + d*x])) - ((a + b*Sec[c + d*x])^(3/2)*(5*a*b*Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2] + 5*b^2*Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2] - 10*a*b*Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]^3 + 5*a*b*Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]^5 - 5*b^2*Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]^5 - (8*I)*a^2*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - (6*I)*b^2*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - (8*I)*a^2*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - (6*I)*b^2*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - (5*I)*(a - b)*b*EllipticE[I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + (2*I)*(2*a^2 - a*b - b^2)*EllipticF[I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)]))/(4*Sqrt[(-a + b)/(a + b)]*d*(b + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(3/2)*Sqrt[(1 - Tan[(c + d*x)/2]^2)^(-1)]*(-1 + Tan[(c + d*x)/2]^2)*(1 + Tan[(c + d*x)/2]^2)^(3/2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(1 + Tan[(c + d*x)/2]^2)])","C",0
544,1,615,463,16.9624734,"\int \sec ^4(c+d x) (a+b \sec (c+d x))^{5/2} \, dx","Integrate[Sec[c + d*x]^4*(a + b*Sec[c + d*x])^(5/2),x]","\frac{\cos ^2(c+d x) (a+b \sec (c+d x))^{5/2} \left(\frac{2 \sec ^2(c+d x) \left(3 a^3 \sin (c+d x)+229 a b^2 \sin (c+d x)\right)}{693 b}+\frac{2}{693} \sec ^3(c+d x) \left(113 a^2 \sin (c+d x)+81 b^2 \sin (c+d x)\right)+\frac{2 \sec (c+d x) \left(-4 a^4 \sin (c+d x)+205 a^2 b^2 \sin (c+d x)+135 b^4 \sin (c+d x)\right)}{693 b^2}+\frac{2 a \left(8 a^4+51 a^2 b^2+741 b^4\right) \sin (c+d x)}{693 b^3}+\frac{46}{99} a b \tan (c+d x) \sec ^3(c+d x)+\frac{2}{11} b^2 \tan (c+d x) \sec ^4(c+d x)\right)}{d (a \cos (c+d x)+b)^2}-\frac{2 \sqrt{\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)} (a+b \sec (c+d x))^{5/2} \left(a \left(8 a^4+51 a^2 b^2+741 b^4\right) \cos (c+d x) \tan \left(\frac{1}{2} (c+d x)\right) \sec ^2\left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b)-2 b \left(8 a^5+2 a^4 b+51 a^3 b^2+663 a^2 b^3+741 a b^4+135 b^5\right) \sqrt{\frac{\cos (c+d x)}{\cos (c+d x)+1}} \sqrt{\frac{a \cos (c+d x)+b}{(a+b) (\cos (c+d x)+1)}} F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right)+2 a \left(8 a^5+8 a^4 b+51 a^3 b^2+51 a^2 b^3+741 a b^4+741 b^5\right) \sqrt{\frac{\cos (c+d x)}{\cos (c+d x)+1}} \sqrt{\frac{a \cos (c+d x)+b}{(a+b) (\cos (c+d x)+1)}} E\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right)\right)}{693 b^3 d \sqrt{\sec ^2\left(\frac{1}{2} (c+d x)\right)} \sec ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+b)^3}","\frac{2 \left(8 a^2+81 b^2\right) \tan (c+d x) (a+b \sec (c+d x))^{5/2}}{693 b^2 d}+\frac{2 a \left(8 a^2+67 b^2\right) \tan (c+d x) (a+b \sec (c+d x))^{3/2}}{693 b^2 d}+\frac{2 \left(8 a^4+57 a^2 b^2+135 b^4\right) \tan (c+d x) \sqrt{a+b \sec (c+d x)}}{693 b^2 d}-\frac{2 a (a-b) \sqrt{a+b} \left(8 a^4+51 a^2 b^2+741 b^4\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{693 b^4 d}-\frac{2 (a-b) \sqrt{a+b} \left(8 a^4+6 a^3 b+57 a^2 b^2-606 a b^3+135 b^4\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{693 b^3 d}-\frac{8 a \tan (c+d x) (a+b \sec (c+d x))^{7/2}}{99 b^2 d}+\frac{2 \tan (c+d x) \sec (c+d x) (a+b \sec (c+d x))^{7/2}}{11 b d}",1,"(-2*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(a + b*Sec[c + d*x])^(5/2)*(2*a*(8*a^5 + 8*a^4*b + 51*a^3*b^2 + 51*a^2*b^3 + 741*a*b^4 + 741*b^5)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] - 2*b*(8*a^5 + 2*a^4*b + 51*a^3*b^2 + 663*a^2*b^3 + 741*a*b^4 + 135*b^5)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + a*(8*a^4 + 51*a^2*b^2 + 741*b^4)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(693*b^3*d*(b + a*Cos[c + d*x])^3*Sqrt[Sec[(c + d*x)/2]^2]*Sec[c + d*x]^(5/2)) + (Cos[c + d*x]^2*(a + b*Sec[c + d*x])^(5/2)*((2*a*(8*a^4 + 51*a^2*b^2 + 741*b^4)*Sin[c + d*x])/(693*b^3) + (2*Sec[c + d*x]^3*(113*a^2*Sin[c + d*x] + 81*b^2*Sin[c + d*x]))/693 + (2*Sec[c + d*x]^2*(3*a^3*Sin[c + d*x] + 229*a*b^2*Sin[c + d*x]))/(693*b) + (2*Sec[c + d*x]*(-4*a^4*Sin[c + d*x] + 205*a^2*b^2*Sin[c + d*x] + 135*b^4*Sin[c + d*x]))/(693*b^2) + (46*a*b*Sec[c + d*x]^3*Tan[c + d*x])/99 + (2*b^2*Sec[c + d*x]^4*Tan[c + d*x])/11))/(d*(b + a*Cos[c + d*x])^2)","A",0
545,1,552,399,16.5678668,"\int \sec ^3(c+d x) (a+b \sec (c+d x))^{5/2} \, dx","Integrate[Sec[c + d*x]^3*(a + b*Sec[c + d*x])^(5/2),x]","\frac{\cos ^2(c+d x) (a+b \sec (c+d x))^{5/2} \left(\frac{2 \sec (c+d x) \left(5 a^3 \sin (c+d x)+163 a b^2 \sin (c+d x)\right)}{315 b}+\frac{2}{315} \sec ^2(c+d x) \left(75 a^2 \sin (c+d x)+49 b^2 \sin (c+d x)\right)+\frac{2 \left(-10 a^4+279 a^2 b^2+147 b^4\right) \sin (c+d x)}{315 b^2}+\frac{38}{63} a b \tan (c+d x) \sec ^2(c+d x)+\frac{2}{9} b^2 \tan (c+d x) \sec ^3(c+d x)\right)}{d (a \cos (c+d x)+b)^2}+\frac{2 \sqrt{\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)} (a+b \sec (c+d x))^{5/2} \left(\left(10 a^4-279 a^2 b^2-147 b^4\right) \cos (c+d x) \tan \left(\frac{1}{2} (c+d x)\right) \sec ^2\left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b)+2 b \left(-10 a^4+155 a^3 b+279 a^2 b^2+261 a b^3+147 b^4\right) \sqrt{\frac{\cos (c+d x)}{\cos (c+d x)+1}} \sqrt{\frac{a \cos (c+d x)+b}{(a+b) (\cos (c+d x)+1)}} F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right)+2 \left(10 a^5+10 a^4 b-279 a^3 b^2-279 a^2 b^3-147 a b^4-147 b^5\right) \sqrt{\frac{\cos (c+d x)}{\cos (c+d x)+1}} \sqrt{\frac{a \cos (c+d x)+b}{(a+b) (\cos (c+d x)+1)}} E\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right)\right)}{315 b^2 d \sqrt{\sec ^2\left(\frac{1}{2} (c+d x)\right)} \sec ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+b)^3}","-\frac{2 \left(10 a^2-49 b^2\right) \tan (c+d x) (a+b \sec (c+d x))^{3/2}}{315 b d}-\frac{4 a \left(5 a^2-57 b^2\right) \tan (c+d x) \sqrt{a+b \sec (c+d x)}}{315 b d}+\frac{2 (a-b) \sqrt{a+b} \left(10 a^4-279 a^2 b^2-147 b^4\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{315 b^3 d}+\frac{2 (a-b) \sqrt{a+b} \left(10 a^3+165 a^2 b-114 a b^2+147 b^3\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{315 b^2 d}+\frac{2 \tan (c+d x) (a+b \sec (c+d x))^{7/2}}{9 b d}-\frac{4 a \tan (c+d x) (a+b \sec (c+d x))^{5/2}}{63 b d}",1,"(2*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(a + b*Sec[c + d*x])^(5/2)*(2*(10*a^5 + 10*a^4*b - 279*a^3*b^2 - 279*a^2*b^3 - 147*a*b^4 - 147*b^5)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + 2*b*(-10*a^4 + 155*a^3*b + 279*a^2*b^2 + 261*a*b^3 + 147*b^4)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + (10*a^4 - 279*a^2*b^2 - 147*b^4)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(315*b^2*d*(b + a*Cos[c + d*x])^3*Sqrt[Sec[(c + d*x)/2]^2]*Sec[c + d*x]^(5/2)) + (Cos[c + d*x]^2*(a + b*Sec[c + d*x])^(5/2)*((2*(-10*a^4 + 279*a^2*b^2 + 147*b^4)*Sin[c + d*x])/(315*b^2) + (2*Sec[c + d*x]^2*(75*a^2*Sin[c + d*x] + 49*b^2*Sin[c + d*x]))/315 + (2*Sec[c + d*x]*(5*a^3*Sin[c + d*x] + 163*a*b^2*Sin[c + d*x]))/(315*b) + (38*a*b*Sec[c + d*x]^2*Tan[c + d*x])/63 + (2*b^2*Sec[c + d*x]^3*Tan[c + d*x])/9))/(d*(b + a*Cos[c + d*x])^2)","A",0
546,1,474,333,13.9583539,"\int \sec ^2(c+d x) (a+b \sec (c+d x))^{5/2} \, dx","Integrate[Sec[c + d*x]^2*(a + b*Sec[c + d*x])^(5/2),x]","\frac{\cos ^2(c+d x) (a+b \sec (c+d x))^{5/2} \left(\frac{2 a \left(3 a^2+29 b^2\right) \sin (c+d x)}{21 b}+\frac{2}{21} \sec (c+d x) \left(9 a^2 \sin (c+d x)+5 b^2 \sin (c+d x)\right)+\frac{6}{7} a b \tan (c+d x) \sec (c+d x)+\frac{2}{7} b^2 \tan (c+d x) \sec ^2(c+d x)\right)}{d (a \cos (c+d x)+b)^2}-\frac{2 \sqrt{\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)} (a+b \sec (c+d x))^{5/2} \left(a \left(3 a^2+29 b^2\right) \cos (c+d x) \tan \left(\frac{1}{2} (c+d x)\right) \sec ^2\left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b)-2 b \left(3 a^3+27 a^2 b+29 a b^2+5 b^3\right) \sqrt{\frac{\cos (c+d x)}{\cos (c+d x)+1}} \sqrt{\frac{a \cos (c+d x)+b}{(a+b) (\cos (c+d x)+1)}} F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right)+2 a \left(3 a^3+3 a^2 b+29 a b^2+29 b^3\right) \sqrt{\frac{\cos (c+d x)}{\cos (c+d x)+1}} \sqrt{\frac{a \cos (c+d x)+b}{(a+b) (\cos (c+d x)+1)}} E\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right)\right)}{21 b d \sqrt{\sec ^2\left(\frac{1}{2} (c+d x)\right)} \sec ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+b)^3}","\frac{2 \left(3 a^2+5 b^2\right) \tan (c+d x) \sqrt{a+b \sec (c+d x)}}{21 d}-\frac{2 (a-b) \sqrt{a+b} \left(3 a^2-24 a b+5 b^2\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{21 b d}-\frac{2 a (a-b) \sqrt{a+b} \left(3 a^2+29 b^2\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{21 b^2 d}+\frac{2 \tan (c+d x) (a+b \sec (c+d x))^{5/2}}{7 d}+\frac{2 a \tan (c+d x) (a+b \sec (c+d x))^{3/2}}{7 d}",1,"(-2*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(a + b*Sec[c + d*x])^(5/2)*(2*a*(3*a^3 + 3*a^2*b + 29*a*b^2 + 29*b^3)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] - 2*b*(3*a^3 + 27*a^2*b + 29*a*b^2 + 5*b^3)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + a*(3*a^2 + 29*b^2)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(21*b*d*(b + a*Cos[c + d*x])^3*Sqrt[Sec[(c + d*x)/2]^2]*Sec[c + d*x]^(5/2)) + (Cos[c + d*x]^2*(a + b*Sec[c + d*x])^(5/2)*((2*a*(3*a^2 + 29*b^2)*Sin[c + d*x])/(21*b) + (2*Sec[c + d*x]*(9*a^2*Sin[c + d*x] + 5*b^2*Sin[c + d*x]))/21 + (6*a*b*Sec[c + d*x]*Tan[c + d*x])/7 + (2*b^2*Sec[c + d*x]^2*Tan[c + d*x])/7))/(d*(b + a*Cos[c + d*x])^2)","A",0
547,1,440,296,16.2524823,"\int \sec (c+d x) (a+b \sec (c+d x))^{5/2} \, dx","Integrate[Sec[c + d*x]*(a + b*Sec[c + d*x])^(5/2),x]","\frac{\cos ^2(c+d x) (a+b \sec (c+d x))^{5/2} \left(\frac{2}{15} \left(23 a^2+9 b^2\right) \sin (c+d x)+\frac{22}{15} a b \tan (c+d x)+\frac{2}{5} b^2 \tan (c+d x) \sec (c+d x)\right)}{d (a \cos (c+d x)+b)^2}-\frac{2 (a+b \sec (c+d x))^{5/2} \left(-\left(\left(23 a^2+9 b^2\right) \cos (c+d x) \tan \left(\frac{1}{2} (c+d x)\right) \sec ^2\left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b)\right)+2 \left(15 a^3+23 a^2 b+17 a b^2+9 b^3\right) \sqrt{\frac{\cos (c+d x)}{\cos (c+d x)+1}} \sqrt{\frac{a \cos (c+d x)+b}{(a+b) (\cos (c+d x)+1)}} F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right)-2 \left(23 a^3+23 a^2 b+9 a b^2+9 b^3\right) \sqrt{\frac{\cos (c+d x)}{\cos (c+d x)+1}} \sqrt{\frac{a \cos (c+d x)+b}{(a+b) (\cos (c+d x)+1)}} E\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right)\right)}{15 d \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)-1\right) \sqrt{\sec ^2\left(\frac{1}{2} (c+d x)\right)} \sec ^{\frac{5}{2}}(c+d x) \sqrt{\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)} (a \cos (c+d x)+b)^3}","\frac{2 (a-b) \sqrt{a+b} \left(15 a^2-8 a b+9 b^2\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{15 b d}-\frac{2 (a-b) \sqrt{a+b} \left(23 a^2+9 b^2\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{15 b d}+\frac{2 b \tan (c+d x) (a+b \sec (c+d x))^{3/2}}{5 d}+\frac{16 a b \tan (c+d x) \sqrt{a+b \sec (c+d x)}}{15 d}",1,"(-2*(a + b*Sec[c + d*x])^(5/2)*(-2*(23*a^3 + 23*a^2*b + 9*a*b^2 + 9*b^3)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + 2*(15*a^3 + 23*a^2*b + 17*a*b^2 + 9*b^3)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] - (23*a^2 + 9*b^2)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(15*d*(b + a*Cos[c + d*x])^3*Sqrt[Sec[(c + d*x)/2]^2]*Sec[c + d*x]^(5/2)*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(-1 + Tan[(c + d*x)/2]^2)) + (Cos[c + d*x]^2*(a + b*Sec[c + d*x])^(5/2)*((2*(23*a^2 + 9*b^2)*Sin[c + d*x])/15 + (22*a*b*Tan[c + d*x])/15 + (2*b^2*Sec[c + d*x]*Tan[c + d*x])/5))/(d*(b + a*Cos[c + d*x])^2)","A",0
548,1,713,352,17.9165531,"\int (a+b \sec (c+d x))^{5/2} \, dx","Integrate[(a + b*Sec[c + d*x])^(5/2),x]","\frac{\cos ^2(c+d x) (a+b \sec (c+d x))^{5/2} \left(\frac{14}{3} a b \sin (c+d x)+\frac{2}{3} b^2 \tan (c+d x)\right)}{d (a \cos (c+d x)+b)^2}+\frac{2 (a+b \sec (c+d x))^{5/2} \left(6 i a^3 \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+b}{a+b}} \Pi \left(-\frac{a+b}{a-b};i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a+b}{a-b}\right)-i \left(3 a^3-9 a^2 b+7 a b^2-b^3\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+b}{a+b}} F\left(i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a+b}{a-b}\right)+7 a b \sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right) \left(a \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)-1\right)^2-b \tan ^4\left(\frac{1}{2} (c+d x)\right)+b\right)-7 i a b (a-b) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+b}{a+b}} E\left(i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a+b}{a-b}\right)\right)}{3 d \sqrt{\frac{b-a}{a+b}} \sqrt{\frac{1}{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)}} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)-1\right) \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right)^{3/2} \sec ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+b)^{5/2} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+b}{\tan ^2\left(\frac{1}{2} (c+d x)\right)+1}}}","\frac{2 \sqrt{a+b} \left(9 a^2-7 a b+b^2\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{3 d}-\frac{2 a^2 \sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{d}+\frac{2 b^2 \tan (c+d x) \sqrt{a+b \sec (c+d x)}}{3 d}-\frac{14 a (a-b) \sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{3 d}",1,"(2*(a + b*Sec[c + d*x])^(5/2)*((-7*I)*a*(a - b)*b*EllipticE[I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - I*(3*a^3 - 9*a^2*b + 7*a*b^2 - b^3)*EllipticF[I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + (6*I)*a^3*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + 7*a*b*Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]*(b - b*Tan[(c + d*x)/2]^4 + a*(-1 + Tan[(c + d*x)/2]^2)^2)))/(3*Sqrt[(-a + b)/(a + b)]*d*(b + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(5/2)*Sqrt[(1 - Tan[(c + d*x)/2]^2)^(-1)]*(-1 + Tan[(c + d*x)/2]^2)*(1 + Tan[(c + d*x)/2]^2)^(3/2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(1 + Tan[(c + d*x)/2]^2)]) + (Cos[c + d*x]^2*(a + b*Sec[c + d*x])^(5/2)*((14*a*b*Sin[c + d*x])/3 + (2*b^2*Tan[c + d*x])/3))/(d*(b + a*Cos[c + d*x])^2)","C",1
549,1,780,353,17.0880147,"\int \cos (c+d x) (a+b \sec (c+d x))^{5/2} \, dx","Integrate[Cos[c + d*x]*(a + b*Sec[c + d*x])^(5/2),x]","\frac{\sqrt{\frac{1}{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)}} (a+b \sec (c+d x))^{5/2} \left(a^3 \tan ^5\left(\frac{1}{2} (c+d x)\right)-2 a^3 \tan ^3\left(\frac{1}{2} (c+d x)\right)+a^3 \tan \left(\frac{1}{2} (c+d x)\right)+2 b \left(-3 a^2+3 a b+b^2\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+b}{a+b}} F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right)-a^2 b \tan ^5\left(\frac{1}{2} (c+d x)\right)+a^2 b \tan \left(\frac{1}{2} (c+d x)\right)+10 a^2 b \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \tan ^2\left(\frac{1}{2} (c+d x)\right) \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+b}{a+b}} \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right)+10 a^2 b \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+b}{a+b}} \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right)+\left(a^3+a^2 b-2 a b^2-2 b^3\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+b}{a+b}} E\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right)-2 a b^2 \tan ^5\left(\frac{1}{2} (c+d x)\right)+4 a b^2 \tan ^3\left(\frac{1}{2} (c+d x)\right)-2 a b^2 \tan \left(\frac{1}{2} (c+d x)\right)+2 b^3 \tan ^5\left(\frac{1}{2} (c+d x)\right)-2 b^3 \tan \left(\frac{1}{2} (c+d x)\right)\right)}{d \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right)^{3/2} \sec ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+b)^{5/2} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+b}{\tan ^2\left(\frac{1}{2} (c+d x)\right)+1}}}+\frac{2 b^2 \sin (c+d x) \cos ^2(c+d x) (a+b \sec (c+d x))^{5/2}}{d (a \cos (c+d x)+b)^2}","\frac{\sqrt{a+b} \left(a^2+6 a b-2 b^2\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{d}+\frac{(a-b) \sqrt{a+b} \left(a^2-2 b^2\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{b d}+\frac{a^2 \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{d}-\frac{5 a b \sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{d}",1,"(2*b^2*Cos[c + d*x]^2*(a + b*Sec[c + d*x])^(5/2)*Sin[c + d*x])/(d*(b + a*Cos[c + d*x])^2) + ((a + b*Sec[c + d*x])^(5/2)*Sqrt[(1 - Tan[(c + d*x)/2]^2)^(-1)]*(a^3*Tan[(c + d*x)/2] + a^2*b*Tan[(c + d*x)/2] - 2*a*b^2*Tan[(c + d*x)/2] - 2*b^3*Tan[(c + d*x)/2] - 2*a^3*Tan[(c + d*x)/2]^3 + 4*a*b^2*Tan[(c + d*x)/2]^3 + a^3*Tan[(c + d*x)/2]^5 - a^2*b*Tan[(c + d*x)/2]^5 - 2*a*b^2*Tan[(c + d*x)/2]^5 + 2*b^3*Tan[(c + d*x)/2]^5 + 10*a^2*b*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + 10*a^2*b*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + (a^3 + a^2*b - 2*a*b^2 - 2*b^3)*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + 2*b*(-3*a^2 + 3*a*b + b^2)*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)]))/(d*(b + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(5/2)*(1 + Tan[(c + d*x)/2]^2)^(3/2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(1 + Tan[(c + d*x)/2]^2)])","B",0
550,1,4588,399,23.1702593,"\int \cos ^2(c+d x) (a+b \sec (c+d x))^{5/2} \, dx","Integrate[Cos[c + d*x]^2*(a + b*Sec[c + d*x])^(5/2),x]","\text{Result too large to show}","\frac{\sqrt{a+b} \left(2 a^2+9 a b+8 b^2\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{4 d}-\frac{\sqrt{a+b} \left(4 a^2+15 b^2\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{4 d}+\frac{a^2 \sin (c+d x) \cos (c+d x) \sqrt{a+b \sec (c+d x)}}{2 d}+\frac{9 a b \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{4 d}+\frac{9 a (a-b) \sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{4 d}",1,"(a^2*Cos[c + d*x]^2*(a + b*Sec[c + d*x])^(5/2)*Sin[2*(c + d*x)])/(4*d*(b + a*Cos[c + d*x])^2) + ((a^3/(2*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (3*a*b^2)/(Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (11*a^2*b*Sqrt[Sec[c + d*x]])/(8*Sqrt[b + a*Cos[c + d*x]]) + (b^3*Sqrt[Sec[c + d*x]])/Sqrt[b + a*Cos[c + d*x]] + (9*a^2*b*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(8*Sqrt[b + a*Cos[c + d*x]]))*(a + b*Sec[c + d*x])^(5/2)*((18*I)*a*(a - b)*b*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)] - (4*I)*(2*a^3 - a^2*b + 3*a*b^2 - 4*b^3)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)] + (4*I)*a*(4*a^2 + 15*b^2)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)] - 9*a*b*Sqrt[(-a + b)/(a + b)]*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(4*Sqrt[(-a + b)/(a + b)]*d*(b + a*Cos[c + d*x])^3*Sqrt[Sec[(c + d*x)/2]^2]*Sec[c + d*x]^(5/2)*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(-1 + Tan[(c + d*x)/2]^2)*(-1/4*(Sqrt[Sec[(c + d*x)/2]^2]*Tan[(c + d*x)/2]*((18*I)*a*(a - b)*b*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)] - (4*I)*(2*a^3 - a^2*b + 3*a*b^2 - 4*b^3)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)] + (4*I)*a*(4*a^2 + 15*b^2)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)] - 9*a*b*Sqrt[(-a + b)/(a + b)]*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(Sqrt[(-a + b)/(a + b)]*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(-1 + Tan[(c + d*x)/2]^2)^2) + (a*Sin[c + d*x]*((18*I)*a*(a - b)*b*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)] - (4*I)*(2*a^3 - a^2*b + 3*a*b^2 - 4*b^3)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)] + (4*I)*a*(4*a^2 + 15*b^2)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)] - 9*a*b*Sqrt[(-a + b)/(a + b)]*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(8*Sqrt[(-a + b)/(a + b)]*(b + a*Cos[c + d*x])^(3/2)*Sqrt[Sec[(c + d*x)/2]^2]*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(-1 + Tan[(c + d*x)/2]^2)) - (Tan[(c + d*x)/2]*((18*I)*a*(a - b)*b*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)] - (4*I)*(2*a^3 - a^2*b + 3*a*b^2 - 4*b^3)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)] + (4*I)*a*(4*a^2 + 15*b^2)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)] - 9*a*b*Sqrt[(-a + b)/(a + b)]*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(8*Sqrt[(-a + b)/(a + b)]*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(-1 + Tan[(c + d*x)/2]^2)) + ((-9*a*b*Sqrt[(-a + b)/(a + b)]*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^4)/2 + ((9*I)*a*(a - b)*b*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*((Cos[c + d*x]*Sin[c + d*x])/(1 + Cos[c + d*x])^2 - Sin[c + d*x]/(1 + Cos[c + d*x])))/Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])] - ((2*I)*(2*a^3 - a^2*b + 3*a*b^2 - 4*b^3)*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*((Cos[c + d*x]*Sin[c + d*x])/(1 + Cos[c + d*x])^2 - Sin[c + d*x]/(1 + Cos[c + d*x])))/Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])] + ((2*I)*a*(4*a^2 + 15*b^2)*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*((Cos[c + d*x]*Sin[c + d*x])/(1 + Cos[c + d*x])^2 - Sin[c + d*x]/(1 + Cos[c + d*x])))/Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])] + ((9*I)*a*(a - b)*b*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*EllipticE[I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*(-((a*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x]))) + ((b + a*Cos[c + d*x])*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x])^2)))/Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))] - ((2*I)*(2*a^3 - a^2*b + 3*a*b^2 - 4*b^3)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*EllipticF[I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*(-((a*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x]))) + ((b + a*Cos[c + d*x])*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x])^2)))/Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))] + ((2*I)*a*(4*a^2 + 15*b^2)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*(-((a*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x]))) + ((b + a*Cos[c + d*x])*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x])^2)))/Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))] + 9*a^2*b*Sqrt[(-a + b)/(a + b)]*Cos[c + d*x]*Sec[(c + d*x)/2]^2*Sin[c + d*x]*Tan[(c + d*x)/2] + 9*a*b*Sqrt[(-a + b)/(a + b)]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Sin[c + d*x]*Tan[(c + d*x)/2] - 9*a*b*Sqrt[(-a + b)/(a + b)]*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]^2 + (2*Sqrt[(-a + b)/(a + b)]*(2*a^3 - a^2*b + 3*a*b^2 - 4*b^3)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*Sec[(c + d*x)/2]^2)/(Sqrt[1 + ((-a + b)*Tan[(c + d*x)/2]^2)/(a - b)]*Sqrt[1 + ((-a + b)*Tan[(c + d*x)/2]^2)/(a + b)]) - (2*a*Sqrt[(-a + b)/(a + b)]*(4*a^2 + 15*b^2)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*Sec[(c + d*x)/2]^2)/((1 - ((-a + b)*Tan[(c + d*x)/2]^2)/(a - b))*Sqrt[1 + ((-a + b)*Tan[(c + d*x)/2]^2)/(a - b)]*Sqrt[1 + ((-a + b)*Tan[(c + d*x)/2]^2)/(a + b)]) - (9*a*(a - b)*b*Sqrt[(-a + b)/(a + b)]*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*Sec[(c + d*x)/2]^2*Sqrt[1 + ((-a + b)*Tan[(c + d*x)/2]^2)/(a - b)])/Sqrt[1 + ((-a + b)*Tan[(c + d*x)/2]^2)/(a + b)])/(4*Sqrt[(-a + b)/(a + b)]*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(-1 + Tan[(c + d*x)/2]^2)) - (((18*I)*a*(a - b)*b*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)] - (4*I)*(2*a^3 - a^2*b + 3*a*b^2 - 4*b^3)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)] + (4*I)*a*(4*a^2 + 15*b^2)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)] - 9*a*b*Sqrt[(-a + b)/(a + b)]*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])*(-(Cos[(c + d*x)/2]*Sec[c + d*x]*Sin[(c + d*x)/2]) + Cos[(c + d*x)/2]^2*Sec[c + d*x]*Tan[c + d*x]))/(8*Sqrt[(-a + b)/(a + b)]*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^(3/2)*(-1 + Tan[(c + d*x)/2]^2))))","C",0
551,1,1018,460,17.4914523,"\int \cos ^3(c+d x) (a+b \sec (c+d x))^{5/2} \, dx","Integrate[Cos[c + d*x]^3*(a + b*Sec[c + d*x])^(5/2),x]","\frac{\cos ^2(c+d x) \left(\frac{1}{12} \sin (c+d x) a^2+\frac{1}{12} \sin (3 (c+d x)) a^2+\frac{13}{24} b \sin (2 (c+d x)) a\right) (a+b \sec (c+d x))^{5/2}}{d (b+a \cos (c+d x))^2}+\frac{\sqrt{\frac{1}{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)}} \left(16 a^3 \tan ^5\left(\frac{1}{2} (c+d x)\right)-33 b^3 \tan ^5\left(\frac{1}{2} (c+d x)\right)+33 a b^2 \tan ^5\left(\frac{1}{2} (c+d x)\right)-16 a^2 b \tan ^5\left(\frac{1}{2} (c+d x)\right)-32 a^3 \tan ^3\left(\frac{1}{2} (c+d x)\right)-66 a b^2 \tan ^3\left(\frac{1}{2} (c+d x)\right)+30 b^3 \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)+120 a^2 b \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)+16 a^3 \tan \left(\frac{1}{2} (c+d x)\right)+33 b^3 \tan \left(\frac{1}{2} (c+d x)\right)+33 a b^2 \tan \left(\frac{1}{2} (c+d x)\right)+16 a^2 b \tan \left(\frac{1}{2} (c+d x)\right)+\left(16 a^3+16 b a^2+33 b^2 a+33 b^3\right) E\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}-2 b \left(38 a^2-13 b a+24 b^2\right) F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}+30 b^3 \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}+120 a^2 b \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}\right) (a+b \sec (c+d x))^{5/2}}{24 d (b+a \cos (c+d x))^{5/2} \sec ^{\frac{5}{2}}(c+d x) \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right)^{3/2} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{\tan ^2\left(\frac{1}{2} (c+d x)\right)+1}}}","\frac{\left(16 a^2+33 b^2\right) \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{24 d}+\frac{\sqrt{a+b} \left(16 a^2+26 a b+33 b^2\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{24 d}+\frac{(a-b) \sqrt{a+b} \left(16 a^2+33 b^2\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{24 b d}-\frac{5 b \sqrt{a+b} \left(4 a^2+b^2\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{8 a d}+\frac{a^2 \sin (c+d x) \cos ^2(c+d x) \sqrt{a+b \sec (c+d x)}}{3 d}+\frac{13 a b \sin (c+d x) \cos (c+d x) \sqrt{a+b \sec (c+d x)}}{12 d}",1,"(Cos[c + d*x]^2*(a + b*Sec[c + d*x])^(5/2)*((a^2*Sin[c + d*x])/12 + (13*a*b*Sin[2*(c + d*x)])/24 + (a^2*Sin[3*(c + d*x)])/12))/(d*(b + a*Cos[c + d*x])^2) + ((a + b*Sec[c + d*x])^(5/2)*Sqrt[(1 - Tan[(c + d*x)/2]^2)^(-1)]*(16*a^3*Tan[(c + d*x)/2] + 16*a^2*b*Tan[(c + d*x)/2] + 33*a*b^2*Tan[(c + d*x)/2] + 33*b^3*Tan[(c + d*x)/2] - 32*a^3*Tan[(c + d*x)/2]^3 - 66*a*b^2*Tan[(c + d*x)/2]^3 + 16*a^3*Tan[(c + d*x)/2]^5 - 16*a^2*b*Tan[(c + d*x)/2]^5 + 33*a*b^2*Tan[(c + d*x)/2]^5 - 33*b^3*Tan[(c + d*x)/2]^5 + 120*a^2*b*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + 30*b^3*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + 120*a^2*b*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + 30*b^3*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + (16*a^3 + 16*a^2*b + 33*a*b^2 + 33*b^3)*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - 2*b*(38*a^2 - 13*a*b + 24*b^2)*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)]))/(24*d*(b + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(5/2)*(1 + Tan[(c + d*x)/2]^2)^(3/2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(1 + Tan[(c + d*x)/2]^2)])","B",0
552,1,1688,530,16.8625883,"\int \cos ^4(c+d x) (a+b \sec (c+d x))^{5/2} \, dx","Integrate[Cos[c + d*x]^4*(a + b*Sec[c + d*x])^(5/2),x]","\frac{\cos ^2(c+d x) \left(\frac{1}{32} \sin (4 (c+d x)) a^2+\frac{17}{96} b \sin (c+d x) a+\frac{17}{96} b \sin (3 (c+d x)) a+\frac{1}{192} \left(48 a^2+59 b^2\right) \sin (2 (c+d x))\right) (a+b \sec (c+d x))^{5/2}}{d (b+a \cos (c+d x))^2}+\frac{\left(15 b^4 \sqrt{\frac{b-a}{a+b}} \tan ^5\left(\frac{1}{2} (c+d x)\right)-15 a b^3 \sqrt{\frac{b-a}{a+b}} \tan ^5\left(\frac{1}{2} (c+d x)\right)+284 a^2 b^2 \sqrt{\frac{b-a}{a+b}} \tan ^5\left(\frac{1}{2} (c+d x)\right)-284 a^3 b \sqrt{\frac{b-a}{a+b}} \tan ^5\left(\frac{1}{2} (c+d x)\right)+30 a b^3 \sqrt{\frac{b-a}{a+b}} \tan ^3\left(\frac{1}{2} (c+d x)\right)+568 a^3 b \sqrt{\frac{b-a}{a+b}} \tan ^3\left(\frac{1}{2} (c+d x)\right)+288 i a^4 \Pi \left(-\frac{a+b}{a-b};i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)-30 i b^4 \Pi \left(-\frac{a+b}{a-b};i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)+720 i a^2 b^2 \Pi \left(-\frac{a+b}{a-b};i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)-15 b^4 \sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)-15 a b^3 \sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)-284 a^2 b^2 \sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)-284 a^3 b \sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)+i b \left(284 a^3-284 b a^2+15 b^2 a-15 b^3\right) E\left(i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}-2 i \left(72 a^4-36 b a^3+38 b^2 a^2-59 b^3 a-15 b^4\right) F\left(i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}+288 i a^4 \Pi \left(-\frac{a+b}{a-b};i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}-30 i b^4 \Pi \left(-\frac{a+b}{a-b};i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}+720 i a^2 b^2 \Pi \left(-\frac{a+b}{a-b};i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}\right) (a+b \sec (c+d x))^{5/2}}{192 a \sqrt{\frac{b-a}{a+b}} d (b+a \cos (c+d x))^{5/2} \sec ^{\frac{5}{2}}(c+d x) \sqrt{\frac{1}{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)}} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)-1\right) \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right)^{3/2} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{\tan ^2\left(\frac{1}{2} (c+d x)\right)+1}}}","\frac{b \left(284 a^2+15 b^2\right) \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{192 a d}+\frac{\left(36 a^2+59 b^2\right) \sin (c+d x) \cos (c+d x) \sqrt{a+b \sec (c+d x)}}{96 d}+\frac{(a-b) \sqrt{a+b} \left(284 a^2+15 b^2\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{192 a d}+\frac{a^2 \sin (c+d x) \cos ^3(c+d x) \sqrt{a+b \sec (c+d x)}}{4 d}-\frac{\sqrt{a+b} \left(48 a^4+120 a^2 b^2-5 b^4\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{64 a^2 d}+\frac{\sqrt{a+b} \left(72 a^3+284 a^2 b+118 a b^2+15 b^3\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{192 a d}+\frac{17 a b \sin (c+d x) \cos ^2(c+d x) \sqrt{a+b \sec (c+d x)}}{24 d}",1,"(Cos[c + d*x]^2*(a + b*Sec[c + d*x])^(5/2)*((17*a*b*Sin[c + d*x])/96 + ((48*a^2 + 59*b^2)*Sin[2*(c + d*x)])/192 + (17*a*b*Sin[3*(c + d*x)])/96 + (a^2*Sin[4*(c + d*x)])/32))/(d*(b + a*Cos[c + d*x])^2) + ((a + b*Sec[c + d*x])^(5/2)*(-284*a^3*b*Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2] - 284*a^2*b^2*Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2] - 15*a*b^3*Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2] - 15*b^4*Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2] + 568*a^3*b*Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]^3 + 30*a*b^3*Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]^3 - 284*a^3*b*Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]^5 + 284*a^2*b^2*Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]^5 - 15*a*b^3*Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]^5 + 15*b^4*Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]^5 + (288*I)*a^4*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + (720*I)*a^2*b^2*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - (30*I)*b^4*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + (288*I)*a^4*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + (720*I)*a^2*b^2*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - (30*I)*b^4*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + I*b*(284*a^3 - 284*a^2*b + 15*a*b^2 - 15*b^3)*EllipticE[I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - (2*I)*(72*a^4 - 36*a^3*b + 38*a^2*b^2 - 59*a*b^3 - 15*b^4)*EllipticF[I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)]))/(192*a*Sqrt[(-a + b)/(a + b)]*d*(b + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(5/2)*Sqrt[(1 - Tan[(c + d*x)/2]^2)^(-1)]*(-1 + Tan[(c + d*x)/2]^2)*(1 + Tan[(c + d*x)/2]^2)^(3/2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(1 + Tan[(c + d*x)/2]^2)])","C",0
553,1,1150,403,15.6982648,"\int (a+b \sec (c+d x))^{7/2} \, dx","Integrate[(a + b*Sec[c + d*x])^(7/2),x]","\frac{2 \left(-9 b^4 \sqrt{\frac{b-a}{a+b}} \tan ^5\left(\frac{1}{2} (c+d x)\right)+9 a b^3 \sqrt{\frac{b-a}{a+b}} \tan ^5\left(\frac{1}{2} (c+d x)\right)-58 a^2 b^2 \sqrt{\frac{b-a}{a+b}} \tan ^5\left(\frac{1}{2} (c+d x)\right)+58 a^3 b \sqrt{\frac{b-a}{a+b}} \tan ^5\left(\frac{1}{2} (c+d x)\right)-18 a b^3 \sqrt{\frac{b-a}{a+b}} \tan ^3\left(\frac{1}{2} (c+d x)\right)-116 a^3 b \sqrt{\frac{b-a}{a+b}} \tan ^3\left(\frac{1}{2} (c+d x)\right)+30 i a^4 \Pi \left(-\frac{a+b}{a-b};i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)+9 b^4 \sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)+9 a b^3 \sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)+58 a^2 b^2 \sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)+58 a^3 b \sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)+i b \left(-58 a^3+58 b a^2-9 b^2 a+9 b^3\right) E\left(i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}-i \left(15 a^4-60 b a^3+58 b^2 a^2-22 b^3 a+9 b^4\right) F\left(i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}+30 i a^4 \Pi \left(-\frac{a+b}{a-b};i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}\right) (a+b \sec (c+d x))^{7/2}}{15 \sqrt{\frac{b-a}{a+b}} d (b+a \cos (c+d x))^{7/2} \sec ^{\frac{7}{2}}(c+d x) \sqrt{\frac{1}{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)}} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)-1\right) \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right)^{3/2} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{\tan ^2\left(\frac{1}{2} (c+d x)\right)+1}}}+\frac{\cos ^3(c+d x) \left(\frac{2}{5} \sec (c+d x) \tan (c+d x) b^3+\frac{32}{15} a \tan (c+d x) b^2+\frac{2}{15} \left(58 a^2+9 b^2\right) \sin (c+d x) b\right) (a+b \sec (c+d x))^{7/2}}{d (b+a \cos (c+d x))^3}","-\frac{2 a^3 \sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{d}-\frac{2 (a-b) \sqrt{a+b} \left(58 a^2+9 b^2\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{15 d}+\frac{2 \sqrt{a+b} \left(60 a^3-58 a^2 b+22 a b^2-9 b^3\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{15 d}+\frac{26 a b^2 \tan (c+d x) \sqrt{a+b \sec (c+d x)}}{15 d}+\frac{2 b^2 \tan (c+d x) (a+b \sec (c+d x))^{3/2}}{5 d}",1,"(2*(a + b*Sec[c + d*x])^(7/2)*(58*a^3*b*Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2] + 58*a^2*b^2*Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2] + 9*a*b^3*Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2] + 9*b^4*Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2] - 116*a^3*b*Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]^3 - 18*a*b^3*Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]^3 + 58*a^3*b*Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]^5 - 58*a^2*b^2*Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]^5 + 9*a*b^3*Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]^5 - 9*b^4*Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]^5 + (30*I)*a^4*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + (30*I)*a^4*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + I*b*(-58*a^3 + 58*a^2*b - 9*a*b^2 + 9*b^3)*EllipticE[I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - I*(15*a^4 - 60*a^3*b + 58*a^2*b^2 - 22*a*b^3 + 9*b^4)*EllipticF[I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)]))/(15*Sqrt[(-a + b)/(a + b)]*d*(b + a*Cos[c + d*x])^(7/2)*Sec[c + d*x]^(7/2)*Sqrt[(1 - Tan[(c + d*x)/2]^2)^(-1)]*(-1 + Tan[(c + d*x)/2]^2)*(1 + Tan[(c + d*x)/2]^2)^(3/2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(1 + Tan[(c + d*x)/2]^2)]) + (Cos[c + d*x]^3*(a + b*Sec[c + d*x])^(7/2)*((2*b*(58*a^2 + 9*b^2)*Sin[c + d*x])/15 + (32*a*b^2*Tan[c + d*x])/15 + (2*b^3*Sec[c + d*x]*Tan[c + d*x])/5))/(d*(b + a*Cos[c + d*x])^3)","C",0
554,1,463,359,14.6680914,"\int \frac{\sec ^5(c+d x)}{\sqrt{a+b \sec (c+d x)}} \, dx","Integrate[Sec[c + d*x]^5/Sqrt[a + b*Sec[c + d*x]],x]","\frac{\sec (c+d x) (a \cos (c+d x)+b) \left(-\frac{8 a \left(12 a^2+11 b^2\right) \sin (c+d x)}{105 b^4}+\frac{2 \sec (c+d x) \left(24 a^2 \sin (c+d x)+25 b^2 \sin (c+d x)\right)}{105 b^3}-\frac{12 a \tan (c+d x) \sec (c+d x)}{35 b^2}+\frac{2 \tan (c+d x) \sec ^2(c+d x)}{7 b}\right)}{d \sqrt{a+b \sec (c+d x)}}+\frac{4 \sqrt{\sec (c+d x)} \sqrt{\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)} \left(2 a \left(12 a^2+11 b^2\right) \cos (c+d x) \tan \left(\frac{1}{2} (c+d x)\right) \sec ^2\left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b)+b \left(-48 a^3-12 a^2 b-44 a b^2+25 b^3\right) \sqrt{\frac{\cos (c+d x)}{\cos (c+d x)+1}} \sqrt{\frac{a \cos (c+d x)+b}{(a+b) (\cos (c+d x)+1)}} F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right)+4 a \left(12 a^3+12 a^2 b+11 a b^2+11 b^3\right) \sqrt{\frac{\cos (c+d x)}{\cos (c+d x)+1}} \sqrt{\frac{a \cos (c+d x)+b}{(a+b) (\cos (c+d x)+1)}} E\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right)\right)}{105 b^4 d \sqrt{\sec ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{a+b \sec (c+d x)}}","\frac{8 a (a-b) \sqrt{a+b} \left(12 a^2+11 b^2\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{105 b^5 d}+\frac{2 \left(24 a^2+25 b^2\right) \tan (c+d x) \sqrt{a+b \sec (c+d x)}}{105 b^3 d}+\frac{2 \sqrt{a+b} \left(48 a^3-12 a^2 b+44 a b^2+25 b^3\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{105 b^4 d}-\frac{12 a \tan (c+d x) \sec (c+d x) \sqrt{a+b \sec (c+d x)}}{35 b^2 d}+\frac{2 \tan (c+d x) \sec ^2(c+d x) \sqrt{a+b \sec (c+d x)}}{7 b d}",1,"(4*Sqrt[Sec[c + d*x]]*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(4*a*(12*a^3 + 12*a^2*b + 11*a*b^2 + 11*b^3)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + b*(-48*a^3 - 12*a^2*b - 44*a*b^2 + 25*b^3)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + 2*a*(12*a^2 + 11*b^2)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(105*b^4*d*Sqrt[Sec[(c + d*x)/2]^2]*Sqrt[a + b*Sec[c + d*x]]) + ((b + a*Cos[c + d*x])*Sec[c + d*x]*((-8*a*(12*a^2 + 11*b^2)*Sin[c + d*x])/(105*b^4) + (2*Sec[c + d*x]*(24*a^2*Sin[c + d*x] + 25*b^2*Sin[c + d*x]))/(105*b^3) - (12*a*Sec[c + d*x]*Tan[c + d*x])/(35*b^2) + (2*Sec[c + d*x]^2*Tan[c + d*x])/(7*b)))/(d*Sqrt[a + b*Sec[c + d*x]])","A",0
555,1,365,301,14.5864806,"\int \frac{\sec ^4(c+d x)}{\sqrt{a+b \sec (c+d x)}} \, dx","Integrate[Sec[c + d*x]^4/Sqrt[a + b*Sec[c + d*x]],x]","\frac{2 \sqrt{\sec (c+d x)} \left(\sqrt{\sec (c+d x)} (a \cos (c+d x)+b) \left(\left(8 a^2+9 b^2\right) \sin (c+d x)+b \tan (c+d x) (3 b \sec (c+d x)-4 a)\right)-\frac{\sqrt{\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)} \left(\left(8 a^2+9 b^2\right) \cos (c+d x) \tan \left(\frac{1}{2} (c+d x)\right) \sec ^2\left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b)-2 b \left(8 a^2+2 a b+9 b^2\right) \sqrt{\frac{1}{\sec (c+d x)+1}} \sqrt{\frac{a+b \sec (c+d x)}{(a+b) (\sec (c+d x)+1)}} F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right)+2 \left(8 a^3+8 a^2 b+9 a b^2+9 b^3\right) \sqrt{\frac{1}{\sec (c+d x)+1}} \sqrt{\frac{a+b \sec (c+d x)}{(a+b) (\sec (c+d x)+1)}} E\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right)\right)}{\sqrt{\sec ^2\left(\frac{1}{2} (c+d x)\right)}}\right)}{15 b^3 d \sqrt{a+b \sec (c+d x)}}","-\frac{2 (a-b) \sqrt{a+b} \left(8 a^2+9 b^2\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{15 b^4 d}-\frac{2 \sqrt{a+b} \left(8 a^2-2 a b+9 b^2\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{15 b^3 d}-\frac{8 a \tan (c+d x) \sqrt{a+b \sec (c+d x)}}{15 b^2 d}+\frac{2 \tan (c+d x) \sec (c+d x) \sqrt{a+b \sec (c+d x)}}{5 b d}",1,"(2*Sqrt[Sec[c + d*x]]*(-((Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(2*(8*a^3 + 8*a^2*b + 9*a*b^2 + 9*b^3)*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[(1 + Sec[c + d*x])^(-1)]*Sqrt[(a + b*Sec[c + d*x])/((a + b)*(1 + Sec[c + d*x]))] - 2*b*(8*a^2 + 2*a*b + 9*b^2)*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[(1 + Sec[c + d*x])^(-1)]*Sqrt[(a + b*Sec[c + d*x])/((a + b)*(1 + Sec[c + d*x]))] + (8*a^2 + 9*b^2)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/Sqrt[Sec[(c + d*x)/2]^2]) + (b + a*Cos[c + d*x])*Sqrt[Sec[c + d*x]]*((8*a^2 + 9*b^2)*Sin[c + d*x] + b*(-4*a + 3*b*Sec[c + d*x])*Tan[c + d*x])))/(15*b^3*d*Sqrt[a + b*Sec[c + d*x]])","A",0
556,1,341,244,13.6464128,"\int \frac{\sec ^3(c+d x)}{\sqrt{a+b \sec (c+d x)}} \, dx","Integrate[Sec[c + d*x]^3/Sqrt[a + b*Sec[c + d*x]],x]","\frac{\sec (c+d x) (a \cos (c+d x)+b) \left(\frac{2 \tan (c+d x)}{3 b}-\frac{4 a \sin (c+d x)}{3 b^2}\right)}{d \sqrt{a+b \sec (c+d x)}}+\frac{4 \sqrt{\sec (c+d x)} \sqrt{\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)} \left(a \cos (c+d x) \tan \left(\frac{1}{2} (c+d x)\right) \sec ^2\left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b)-b (2 a-b) \sqrt{\frac{\cos (c+d x)}{\cos (c+d x)+1}} \sqrt{\frac{a \cos (c+d x)+b}{(a+b) (\cos (c+d x)+1)}} F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right)+2 a (a+b) \sqrt{\frac{\cos (c+d x)}{\cos (c+d x)+1}} \sqrt{\frac{a \cos (c+d x)+b}{(a+b) (\cos (c+d x)+1)}} E\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right)\right)}{3 b^2 d \sqrt{\sec ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{a+b \sec (c+d x)}}","\frac{4 a (a-b) \sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{3 b^3 d}+\frac{2 \sqrt{a+b} (2 a+b) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{3 b^2 d}+\frac{2 \tan (c+d x) \sqrt{a+b \sec (c+d x)}}{3 b d}",1,"(4*Sqrt[Sec[c + d*x]]*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(2*a*(a + b)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] - (2*a - b)*b*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + a*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(3*b^2*d*Sqrt[Sec[(c + d*x)/2]^2]*Sqrt[a + b*Sec[c + d*x]]) + ((b + a*Cos[c + d*x])*Sec[c + d*x]*((-4*a*Sin[c + d*x])/(3*b^2) + (2*Tan[c + d*x])/(3*b)))/(d*Sqrt[a + b*Sec[c + d*x]])","A",0
557,1,2189,204,19.3041257,"\int \frac{\sec ^2(c+d x)}{\sqrt{a+b \sec (c+d x)}} \, dx","Integrate[Sec[c + d*x]^2/Sqrt[a + b*Sec[c + d*x]],x]","\text{Result too large to show}","-\frac{2 (a-b) \sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{b^2 d}-\frac{2 \sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{b d}",1,"(2*(b + a*Cos[c + d*x])*Tan[c + d*x])/(b*d*Sqrt[a + b*Sec[c + d*x]]) + ((-(1/(Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]])) - (a*Sqrt[Sec[c + d*x]])/(b*Sqrt[b + a*Cos[c + d*x]]) - (a*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(b*Sqrt[b + a*Cos[c + d*x]]))*Sqrt[Sec[c + d*x]]*(-2*(a + b)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sec[c + d*x] + 2*b*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sec[c + d*x]*Sqrt[(1 + Sec[c + d*x])^(-1)]*Sqrt[(a + b*Sec[c + d*x])/((a + b)*(1 + Sec[c + d*x]))] - (b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(b*d*((1 + Cos[c + d*x])^(-1))^(3/2)*Sqrt[1 + Sec[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*((a*Sin[c + d*x]*(-2*(a + b)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sec[c + d*x] + 2*b*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sec[c + d*x]*Sqrt[(1 + Sec[c + d*x])^(-1)]*Sqrt[(a + b*Sec[c + d*x])/((a + b)*(1 + Sec[c + d*x]))] - (b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(2*b*((1 + Cos[c + d*x])^(-1))^(3/2)*(b + a*Cos[c + d*x])^(3/2)*Sqrt[1 + Sec[c + d*x]]) - (3*Sin[c + d*x]*(-2*(a + b)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sec[c + d*x] + 2*b*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sec[c + d*x]*Sqrt[(1 + Sec[c + d*x])^(-1)]*Sqrt[(a + b*Sec[c + d*x])/((a + b)*(1 + Sec[c + d*x]))] - (b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(2*b*Sqrt[(1 + Cos[c + d*x])^(-1)]*Sqrt[b + a*Cos[c + d*x]]*Sqrt[1 + Sec[c + d*x]]) - (Sec[c + d*x]*(-2*(a + b)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sec[c + d*x] + 2*b*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sec[c + d*x]*Sqrt[(1 + Sec[c + d*x])^(-1)]*Sqrt[(a + b*Sec[c + d*x])/((a + b)*(1 + Sec[c + d*x]))] - (b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])*Tan[c + d*x])/(2*b*((1 + Cos[c + d*x])^(-1))^(3/2)*Sqrt[b + a*Cos[c + d*x]]*(1 + Sec[c + d*x])^(3/2)) + (-1/2*((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^4) - ((a + b)*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sec[c + d*x]*((Cos[c + d*x]*Sin[c + d*x])/(1 + Cos[c + d*x])^2 - Sin[c + d*x]/(1 + Cos[c + d*x])))/Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])] - ((a + b)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sec[c + d*x]*(-((a*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x]))) + ((b + a*Cos[c + d*x])*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x])^2)))/Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))] + a*Sec[(c + d*x)/2]^2*Sin[c + d*x]*Tan[(c + d*x)/2] - (b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]^2 + (b*Sec[(c + d*x)/2]^2*Sec[c + d*x]*Sqrt[(1 + Sec[c + d*x])^(-1)]*Sqrt[(a + b*Sec[c + d*x])/((a + b)*(1 + Sec[c + d*x]))])/(Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[1 - ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)]) - ((a + b)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*Sec[(c + d*x)/2]^2*Sec[c + d*x]*Sqrt[1 - ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)])/Sqrt[1 - Tan[(c + d*x)/2]^2] - 2*(a + b)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sec[c + d*x]*Tan[c + d*x] + 2*b*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sec[c + d*x]*Sqrt[(1 + Sec[c + d*x])^(-1)]*Sqrt[(a + b*Sec[c + d*x])/((a + b)*(1 + Sec[c + d*x]))]*Tan[c + d*x] - b*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sec[c + d*x]^2*((1 + Sec[c + d*x])^(-1))^(3/2)*Sqrt[(a + b*Sec[c + d*x])/((a + b)*(1 + Sec[c + d*x]))]*Tan[c + d*x] + (b*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sec[c + d*x]*Sqrt[(1 + Sec[c + d*x])^(-1)]*((b*Sec[c + d*x]*Tan[c + d*x])/((a + b)*(1 + Sec[c + d*x])) - (Sec[c + d*x]*(a + b*Sec[c + d*x])*Tan[c + d*x])/((a + b)*(1 + Sec[c + d*x])^2)))/Sqrt[(a + b*Sec[c + d*x])/((a + b)*(1 + Sec[c + d*x]))])/(b*((1 + Cos[c + d*x])^(-1))^(3/2)*Sqrt[b + a*Cos[c + d*x]]*Sqrt[1 + Sec[c + d*x]])))","B",0
558,1,93,99,1.6628503,"\int \frac{\sec (c+d x)}{\sqrt{a+b \sec (c+d x)}} \, dx","Integrate[Sec[c + d*x]/Sqrt[a + b*Sec[c + d*x]],x]","\frac{2 \sqrt{\frac{a \cos (c+d x)+b}{(a+b) (\cos (c+d x)+1)}} F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right)}{d \sqrt{\frac{\cos (c+d x)}{\cos (c+d x)+1}} \sqrt{a+b \sec (c+d x)}}","\frac{2 \sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{b d}",1,"(2*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)])/(d*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[a + b*Sec[c + d*x]])","A",1
559,1,138,106,1.5702505,"\int \frac{1}{\sqrt{a+b \sec (c+d x)}} \, dx","Integrate[1/Sqrt[a + b*Sec[c + d*x]],x]","-\frac{4 \cos ^2\left(\frac{1}{2} (c+d x)\right) \sqrt{\frac{\cos (c+d x)}{\cos (c+d x)+1}} \sec (c+d x) \sqrt{\frac{a \cos (c+d x)+b}{(a+b) (\cos (c+d x)+1)}} \left(F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right)-2 \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right)\right)}{d \sqrt{a+b \sec (c+d x)}}","-\frac{2 \sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{a d}",1,"(-4*Cos[(c + d*x)/2]^2*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*(EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] - 2*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)])*Sec[c + d*x])/(d*Sqrt[a + b*Sec[c + d*x]])","A",1
560,1,5060,338,24.4216719,"\int \frac{\cos (c+d x)}{\sqrt{a+b \sec (c+d x)}} \, dx","Integrate[Cos[c + d*x]/Sqrt[a + b*Sec[c + d*x]],x]","\text{Result too large to show}","\frac{b \sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{a^2 d}+\frac{\sin (c+d x) \sqrt{a+b \sec (c+d x)}}{a d}+\frac{\sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{a d}+\frac{(a-b) \sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{a b d}",1,"Result too large to show","C",0
561,1,1195,401,19.3788956,"\int \frac{\cos ^2(c+d x)}{\sqrt{a+b \sec (c+d x)}} \, dx","Integrate[Cos[c + d*x]^2/Sqrt[a + b*Sec[c + d*x]],x]","\frac{(b+a \cos (c+d x)) \sec (c+d x) \sin (2 (c+d x))}{4 a d \sqrt{a+b \sec (c+d x)}}-\frac{\sqrt{b+a \cos (c+d x)} \sqrt{\sec (c+d x)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{\tan ^2\left(\frac{1}{2} (c+d x)\right)+1}} \left(-3 b^2 \sqrt{\frac{b-a}{a+b}} \tan ^5\left(\frac{1}{2} (c+d x)\right)+3 a b \sqrt{\frac{b-a}{a+b}} \tan ^5\left(\frac{1}{2} (c+d x)\right)-6 a b \sqrt{\frac{b-a}{a+b}} \tan ^3\left(\frac{1}{2} (c+d x)\right)+8 i a^2 \Pi \left(-\frac{a+b}{a-b};i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)+6 i b^2 \Pi \left(-\frac{a+b}{a-b};i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)+3 b^2 \sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)+3 a b \sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)-3 i (a-b) b E\left(i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}-2 i \left(2 a^2-b a+3 b^2\right) F\left(i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}+8 i a^2 \Pi \left(-\frac{a+b}{a-b};i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}+6 i b^2 \Pi \left(-\frac{a+b}{a-b};i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}\right)}{4 a^2 \sqrt{\frac{b-a}{a+b}} d \sqrt{a+b \sec (c+d x)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)-1\right) \sqrt{\frac{\tan ^2\left(\frac{1}{2} (c+d x)\right)+1}{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)}} \left(a \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)-1\right)-b \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right)\right)}","-\frac{3 b \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{4 a^2 d}+\frac{(2 a-3 b) \sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{4 a^2 d}-\frac{3 (a-b) \sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{4 a^2 d}-\frac{\sqrt{a+b} \left(4 a^2+3 b^2\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{4 a^3 d}+\frac{\sin (c+d x) \cos (c+d x) \sqrt{a+b \sec (c+d x)}}{2 a d}",1,"((b + a*Cos[c + d*x])*Sec[c + d*x]*Sin[2*(c + d*x)])/(4*a*d*Sqrt[a + b*Sec[c + d*x]]) - (Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(1 + Tan[(c + d*x)/2]^2)]*(3*a*b*Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2] + 3*b^2*Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2] - 6*a*b*Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]^3 + 3*a*b*Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]^5 - 3*b^2*Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]^5 + (8*I)*a^2*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + (6*I)*b^2*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + (8*I)*a^2*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + (6*I)*b^2*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - (3*I)*(a - b)*b*EllipticE[I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - (2*I)*(2*a^2 - a*b + 3*b^2)*EllipticF[I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)]))/(4*a^2*Sqrt[(-a + b)/(a + b)]*d*Sqrt[a + b*Sec[c + d*x]]*(-1 + Tan[(c + d*x)/2]^2)*Sqrt[(1 + Tan[(c + d*x)/2]^2)/(1 - Tan[(c + d*x)/2]^2)]*(a*(-1 + Tan[(c + d*x)/2]^2) - b*(1 + Tan[(c + d*x)/2]^2)))","C",0
562,1,498,399,17.0780354,"\int \frac{\sec ^5(c+d x)}{(a+b \sec (c+d x))^{3/2}} \, dx","Integrate[Sec[c + d*x]^5/(a + b*Sec[c + d*x])^(3/2),x]","\frac{\sec ^2(c+d x) (a \cos (c+d x)+b)^2 \left(\frac{2 \left(-16 a^4+8 a^2 b^2+3 b^4\right) \sin (c+d x)}{5 b^4 \left(b^2-a^2\right)}+\frac{2 a^4 \sin (c+d x)}{b^3 \left(b^2-a^2\right) (a \cos (c+d x)+b)}-\frac{6 a \tan (c+d x)}{5 b^3}+\frac{2 \tan (c+d x) \sec (c+d x)}{5 b^2}\right)}{d (a+b \sec (c+d x))^{3/2}}+\frac{2 \left(4 a^2+b^2\right) \sec ^{\frac{3}{2}}(c+d x) \sqrt{\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)} (a \cos (c+d x)+b) \left(\left(4 a^2-3 b^2\right) \cos (c+d x) \tan \left(\frac{1}{2} (c+d x)\right) \sec ^2\left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b)+2 b \left(-4 a^2-a b+3 b^2\right) \sqrt{\frac{\cos (c+d x)}{\cos (c+d x)+1}} \sqrt{\frac{a \cos (c+d x)+b}{(a+b) (\cos (c+d x)+1)}} F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right)+2 \left(4 a^3+4 a^2 b-3 a b^2-3 b^3\right) \sqrt{\frac{\cos (c+d x)}{\cos (c+d x)+1}} \sqrt{\frac{a \cos (c+d x)+b}{(a+b) (\cos (c+d x)+1)}} E\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right)\right)}{5 b^4 d \left(b^2-a^2\right) \sqrt{\sec ^2\left(\frac{1}{2} (c+d x)\right)} (a+b \sec (c+d x))^{3/2}}","-\frac{2 a^2 \tan (c+d x) \sec ^2(c+d x)}{b d \left(a^2-b^2\right) \sqrt{a+b \sec (c+d x)}}+\frac{2 \left(6 a^2-b^2\right) \tan (c+d x) \sec (c+d x) \sqrt{a+b \sec (c+d x)}}{5 b^2 d \left(a^2-b^2\right)}-\frac{2 (4 a+3 b) \left(4 a^2+b^2\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{5 b^4 d \sqrt{a+b}}-\frac{2 a \left(8 a^2-3 b^2\right) \tan (c+d x) \sqrt{a+b \sec (c+d x)}}{5 b^3 d \left(a^2-b^2\right)}-\frac{2 \left(16 a^4-8 a^2 b^2-3 b^4\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{5 b^5 d \sqrt{a+b}}",1,"(2*(4*a^2 + b^2)*(b + a*Cos[c + d*x])*Sec[c + d*x]^(3/2)*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(2*(4*a^3 + 4*a^2*b - 3*a*b^2 - 3*b^3)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + 2*b*(-4*a^2 - a*b + 3*b^2)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + (4*a^2 - 3*b^2)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(5*b^4*(-a^2 + b^2)*d*Sqrt[Sec[(c + d*x)/2]^2]*(a + b*Sec[c + d*x])^(3/2)) + ((b + a*Cos[c + d*x])^2*Sec[c + d*x]^2*((2*(-16*a^4 + 8*a^2*b^2 + 3*b^4)*Sin[c + d*x])/(5*b^4*(-a^2 + b^2)) + (2*a^4*Sin[c + d*x])/(b^3*(-a^2 + b^2)*(b + a*Cos[c + d*x])) - (6*a*Tan[c + d*x])/(5*b^3) + (2*Sec[c + d*x]*Tan[c + d*x])/(5*b^2)))/(d*(a + b*Sec[c + d*x])^(3/2))","A",0
563,1,470,325,15.8959491,"\int \frac{\sec ^4(c+d x)}{(a+b \sec (c+d x))^{3/2}} \, dx","Integrate[Sec[c + d*x]^4/(a + b*Sec[c + d*x])^(3/2),x]","\frac{\sec ^2(c+d x) (a \cos (c+d x)+b)^2 \left(-\frac{2 a \left(5 b^2-8 a^2\right) \sin (c+d x)}{3 b^3 \left(b^2-a^2\right)}-\frac{2 a^3 \sin (c+d x)}{b^2 \left(b^2-a^2\right) (a \cos (c+d x)+b)}+\frac{2 \tan (c+d x)}{3 b^2}\right)}{d (a+b \sec (c+d x))^{3/2}}-\frac{2 \sec ^{\frac{3}{2}}(c+d x) \sqrt{\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)} (a \cos (c+d x)+b) \left(a \left(8 a^2-5 b^2\right) \cos (c+d x) \tan \left(\frac{1}{2} (c+d x)\right) \sec ^2\left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b)-2 b \left(8 a^3+2 a^2 b-5 a b^2+b^3\right) \sqrt{\frac{\cos (c+d x)}{\cos (c+d x)+1}} \sqrt{\frac{a \cos (c+d x)+b}{(a+b) (\cos (c+d x)+1)}} F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right)+2 a \left(8 a^3+8 a^2 b-5 a b^2-5 b^3\right) \sqrt{\frac{\cos (c+d x)}{\cos (c+d x)+1}} \sqrt{\frac{a \cos (c+d x)+b}{(a+b) (\cos (c+d x)+1)}} E\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right)\right)}{3 b^3 d \left(b^2-a^2\right) \sqrt{\sec ^2\left(\frac{1}{2} (c+d x)\right)} (a+b \sec (c+d x))^{3/2}}","-\frac{2 a^2 \tan (c+d x) \sec (c+d x)}{b d \left(a^2-b^2\right) \sqrt{a+b \sec (c+d x)}}+\frac{2 \left(4 a^2-b^2\right) \tan (c+d x) \sqrt{a+b \sec (c+d x)}}{3 b^2 d \left(a^2-b^2\right)}+\frac{2 a \left(8 a^2-5 b^2\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{3 b^4 d \sqrt{a+b}}+\frac{2 (2 a+b) (4 a+b) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{3 b^3 d \sqrt{a+b}}",1,"(-2*(b + a*Cos[c + d*x])*Sec[c + d*x]^(3/2)*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(2*a*(8*a^3 + 8*a^2*b - 5*a*b^2 - 5*b^3)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] - 2*b*(8*a^3 + 2*a^2*b - 5*a*b^2 + b^3)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + a*(8*a^2 - 5*b^2)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(3*b^3*(-a^2 + b^2)*d*Sqrt[Sec[(c + d*x)/2]^2]*(a + b*Sec[c + d*x])^(3/2)) + ((b + a*Cos[c + d*x])^2*Sec[c + d*x]^2*((-2*a*(-8*a^2 + 5*b^2)*Sin[c + d*x])/(3*b^3*(-a^2 + b^2)) - (2*a^3*Sin[c + d*x])/(b^2*(-a^2 + b^2)*(b + a*Cos[c + d*x])) + (2*Tan[c + d*x])/(3*b^2)))/(d*(a + b*Sec[c + d*x])^(3/2))","A",0
564,1,440,257,14.325814,"\int \frac{\sec ^3(c+d x)}{(a+b \sec (c+d x))^{3/2}} \, dx","Integrate[Sec[c + d*x]^3/(a + b*Sec[c + d*x])^(3/2),x]","\frac{\sec ^2(c+d x) (a \cos (c+d x)+b)^2 \left(\frac{2 \left(b^2-2 a^2\right) \sin (c+d x)}{b^2 \left(b^2-a^2\right)}+\frac{2 a^2 \sin (c+d x)}{b \left(b^2-a^2\right) (a \cos (c+d x)+b)}\right)}{d (a+b \sec (c+d x))^{3/2}}+\frac{2 \sec ^{\frac{3}{2}}(c+d x) \sqrt{\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)} (a \cos (c+d x)+b) \left(\left(2 a^2-b^2\right) \cos (c+d x) \tan \left(\frac{1}{2} (c+d x)\right) \sec ^2\left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b)+2 b \left(-2 a^2-a b+b^2\right) \sqrt{\frac{\cos (c+d x)}{\cos (c+d x)+1}} \sqrt{\frac{a \cos (c+d x)+b}{(a+b) (\cos (c+d x)+1)}} F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right)+2 \left(2 a^3+2 a^2 b-a b^2-b^3\right) \sqrt{\frac{\cos (c+d x)}{\cos (c+d x)+1}} \sqrt{\frac{a \cos (c+d x)+b}{(a+b) (\cos (c+d x)+1)}} E\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right)\right)}{b^2 d \left(b^2-a^2\right) \sqrt{\sec ^2\left(\frac{1}{2} (c+d x)\right)} (a+b \sec (c+d x))^{3/2}}","-\frac{2 a^2 \tan (c+d x)}{b d \left(a^2-b^2\right) \sqrt{a+b \sec (c+d x)}}-\frac{2 \left(2 a^2-b^2\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{b^3 d \sqrt{a+b}}-\frac{2 (2 a+b) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{b^2 d \sqrt{a+b}}",1,"((b + a*Cos[c + d*x])^2*Sec[c + d*x]^2*((2*(-2*a^2 + b^2)*Sin[c + d*x])/(b^2*(-a^2 + b^2)) + (2*a^2*Sin[c + d*x])/(b*(-a^2 + b^2)*(b + a*Cos[c + d*x]))))/(d*(a + b*Sec[c + d*x])^(3/2)) + (2*(b + a*Cos[c + d*x])*Sec[c + d*x]^(3/2)*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(2*(2*a^3 + 2*a^2*b - a*b^2 - b^3)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + 2*b*(-2*a^2 - a*b + b^2)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + (2*a^2 - b^2)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(b^2*(-a^2 + b^2)*d*Sqrt[Sec[(c + d*x)/2]^2]*(a + b*Sec[c + d*x])^(3/2))","A",0
565,1,249,237,9.4256196,"\int \frac{\sec ^2(c+d x)}{(a+b \sec (c+d x))^{3/2}} \, dx","Integrate[Sec[c + d*x]^2/(a + b*Sec[c + d*x])^(3/2),x]","\frac{\sec \left(\frac{1}{2} (c+d x)\right) \sec (c+d x) \left(a (a-b) \left(\sin \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{3}{2} (c+d x)\right)\right)-4 b (a+b) \cos ^3\left(\frac{1}{2} (c+d x)\right) \sqrt{\frac{1}{\sec (c+d x)+1}} \sqrt{\frac{a \cos (c+d x)+b}{(a+b) (\cos (c+d x)+1)}} F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right)+4 a (a+b) \cos ^3\left(\frac{1}{2} (c+d x)\right) \sqrt{\frac{1}{\sec (c+d x)+1}} \sqrt{\frac{a \cos (c+d x)+b}{(a+b) (\cos (c+d x)+1)}} E\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right)\right)}{b d \left(a^2-b^2\right) \sqrt{a+b \sec (c+d x)}}","\frac{2 a \tan (c+d x)}{d \left(a^2-b^2\right) \sqrt{a+b \sec (c+d x)}}+\frac{2 a \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{b^2 d \sqrt{a+b}}+\frac{2 \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{b d \sqrt{a+b}}",1,"(Sec[(c + d*x)/2]*Sec[c + d*x]*(4*a*(a + b)*Cos[(c + d*x)/2]^3*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[(1 + Sec[c + d*x])^(-1)] - 4*b*(a + b)*Cos[(c + d*x)/2]^3*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[(1 + Sec[c + d*x])^(-1)] + a*(a - b)*(Sin[(c + d*x)/2] - Sin[(3*(c + d*x))/2])))/(b*(a^2 - b^2)*d*Sqrt[a + b*Sec[c + d*x]])","A",1
566,1,244,236,10.3116879,"\int \frac{\sec (c+d x)}{(a+b \sec (c+d x))^{3/2}} \, dx","Integrate[Sec[c + d*x]/(a + b*Sec[c + d*x])^(3/2),x]","-\frac{\sec \left(\frac{1}{2} (c+d x)\right) \sec (c+d x) \left((a-b) \left(\sin \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{3}{2} (c+d x)\right)\right)-4 (a+b) \cos ^3\left(\frac{1}{2} (c+d x)\right) \sqrt{\frac{1}{\sec (c+d x)+1}} \sqrt{\frac{a \cos (c+d x)+b}{(a+b) (\cos (c+d x)+1)}} F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right)+4 (a+b) \cos ^3\left(\frac{1}{2} (c+d x)\right) \sqrt{\frac{1}{\sec (c+d x)+1}} \sqrt{\frac{a \cos (c+d x)+b}{(a+b) (\cos (c+d x)+1)}} E\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right)\right)}{d \left(a^2-b^2\right) \sqrt{a+b \sec (c+d x)}}","-\frac{2 b \tan (c+d x)}{d \left(a^2-b^2\right) \sqrt{a+b \sec (c+d x)}}+\frac{2 \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{b d \sqrt{a+b}}-\frac{2 \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{b d \sqrt{a+b}}",1,"-((Sec[(c + d*x)/2]*Sec[c + d*x]*(4*(a + b)*Cos[(c + d*x)/2]^3*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[(1 + Sec[c + d*x])^(-1)] - 4*(a + b)*Cos[(c + d*x)/2]^3*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[(1 + Sec[c + d*x])^(-1)] + (a - b)*(Sin[(c + d*x)/2] - Sin[(3*(c + d*x))/2])))/((a^2 - b^2)*d*Sqrt[a + b*Sec[c + d*x]]))","A",1
567,1,1249,347,18.7311228,"\int \frac{1}{(a+b \sec (c+d x))^{3/2}} \, dx","Integrate[(a + b*Sec[c + d*x])^(-3/2),x]","\frac{(b+a \cos (c+d x))^2 \left(\frac{2 \sin (c+d x) b^2}{a \left(a^2-b^2\right) (b+a \cos (c+d x))}+\frac{2 \sin (c+d x) b}{a \left(b^2-a^2\right)}\right) \sec ^2(c+d x)}{d (a+b \sec (c+d x))^{3/2}}+\frac{2 (b+a \cos (c+d x))^{3/2} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{\tan ^2\left(\frac{1}{2} (c+d x)\right)+1}} \left(-b^2 \sqrt{\frac{b-a}{a+b}} \tan ^5\left(\frac{1}{2} (c+d x)\right)+a b \sqrt{\frac{b-a}{a+b}} \tan ^5\left(\frac{1}{2} (c+d x)\right)-2 a b \sqrt{\frac{b-a}{a+b}} \tan ^3\left(\frac{1}{2} (c+d x)\right)-2 i a^2 \Pi \left(-\frac{a+b}{a-b};i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)+2 i b^2 \Pi \left(-\frac{a+b}{a-b};i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)+b^2 \sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)+a b \sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)-i (a-b) b E\left(i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}+i \left(a^2+b a-2 b^2\right) F\left(i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}-2 i a^2 \Pi \left(-\frac{a+b}{a-b};i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}+2 i b^2 \Pi \left(-\frac{a+b}{a-b};i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}\right) \sec ^{\frac{3}{2}}(c+d x)}{a \sqrt{\frac{b-a}{a+b}} \left(a^2-b^2\right) d (a+b \sec (c+d x))^{3/2} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)-1\right) \sqrt{\frac{\tan ^2\left(\frac{1}{2} (c+d x)\right)+1}{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)}} \left(a \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)-1\right)-b \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right)\right)}","\frac{2 b^2 \tan (c+d x)}{a d \left(a^2-b^2\right) \sqrt{a+b \sec (c+d x)}}-\frac{2 \sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{a^2 d}-\frac{2 \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{a d \sqrt{a+b}}+\frac{2 \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{a d \sqrt{a+b}}",1,"((b + a*Cos[c + d*x])^2*Sec[c + d*x]^2*((2*b*Sin[c + d*x])/(a*(-a^2 + b^2)) + (2*b^2*Sin[c + d*x])/(a*(a^2 - b^2)*(b + a*Cos[c + d*x]))))/(d*(a + b*Sec[c + d*x])^(3/2)) + (2*(b + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(3/2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(1 + Tan[(c + d*x)/2]^2)]*(a*b*Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2] + b^2*Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2] - 2*a*b*Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]^3 + a*b*Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]^5 - b^2*Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]^5 - (2*I)*a^2*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + (2*I)*b^2*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - (2*I)*a^2*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + (2*I)*b^2*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - I*(a - b)*b*EllipticE[I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + I*(a^2 + a*b - 2*b^2)*EllipticF[I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)]))/(a*Sqrt[(-a + b)/(a + b)]*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^(3/2)*(-1 + Tan[(c + d*x)/2]^2)*Sqrt[(1 + Tan[(c + d*x)/2]^2)/(1 - Tan[(c + d*x)/2]^2)]*(a*(-1 + Tan[(c + d*x)/2]^2) - b*(1 + Tan[(c + d*x)/2]^2)))","C",0
568,1,1069,396,15.6197654,"\int \frac{\cos (c+d x)}{(a+b \sec (c+d x))^{3/2}} \, dx","Integrate[Cos[c + d*x]/(a + b*Sec[c + d*x])^(3/2),x]","\frac{(b+a \cos (c+d x))^2 \sec ^2(c+d x) \left(-\frac{2 \sin (c+d x) b^3}{a^2 \left(a^2-b^2\right) (b+a \cos (c+d x))}-\frac{2 \sin (c+d x) b^2}{a^2 \left(b^2-a^2\right)}\right)}{d (a+b \sec (c+d x))^{3/2}}-\frac{(b+a \cos (c+d x))^{3/2} \sec ^{\frac{3}{2}}(c+d x) \sqrt{\frac{1}{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)}} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{\tan ^2\left(\frac{1}{2} (c+d x)\right)+1}} \left(a^3 \tan ^5\left(\frac{1}{2} (c+d x)\right)+3 b^3 \tan ^5\left(\frac{1}{2} (c+d x)\right)-3 a b^2 \tan ^5\left(\frac{1}{2} (c+d x)\right)-a^2 b \tan ^5\left(\frac{1}{2} (c+d x)\right)-2 a^3 \tan ^3\left(\frac{1}{2} (c+d x)\right)+6 a b^2 \tan ^3\left(\frac{1}{2} (c+d x)\right)+6 b^3 \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)-6 a^2 b \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)+a^3 \tan \left(\frac{1}{2} (c+d x)\right)-3 b^3 \tan \left(\frac{1}{2} (c+d x)\right)-3 a b^2 \tan \left(\frac{1}{2} (c+d x)\right)+a^2 b \tan \left(\frac{1}{2} (c+d x)\right)+\left(a^3+b a^2-3 b^2 a-3 b^3\right) E\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}+2 a b (a+b) F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}+6 b^3 \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}-6 a^2 b \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}\right)}{a^2 \left(a^2-b^2\right) d (a+b \sec (c+d x))^{3/2} \sqrt{\tan ^2\left(\frac{1}{2} (c+d x)\right)+1} \left(a \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)-1\right)-b \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right)\right)}","\frac{3 b \sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{a^3 d}+\frac{b \left(a^2-3 b^2\right) \tan (c+d x)}{a^2 d \left(a^2-b^2\right) \sqrt{a+b \sec (c+d x)}}+\frac{\left(a^2-3 b^2\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{a^2 b d \sqrt{a+b}}+\frac{(a+3 b) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{a^2 d \sqrt{a+b}}+\frac{\sin (c+d x)}{a d \sqrt{a+b \sec (c+d x)}}",1,"((b + a*Cos[c + d*x])^2*Sec[c + d*x]^2*((-2*b^2*Sin[c + d*x])/(a^2*(-a^2 + b^2)) - (2*b^3*Sin[c + d*x])/(a^2*(a^2 - b^2)*(b + a*Cos[c + d*x]))))/(d*(a + b*Sec[c + d*x])^(3/2)) - ((b + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(3/2)*Sqrt[(1 - Tan[(c + d*x)/2]^2)^(-1)]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(1 + Tan[(c + d*x)/2]^2)]*(a^3*Tan[(c + d*x)/2] + a^2*b*Tan[(c + d*x)/2] - 3*a*b^2*Tan[(c + d*x)/2] - 3*b^3*Tan[(c + d*x)/2] - 2*a^3*Tan[(c + d*x)/2]^3 + 6*a*b^2*Tan[(c + d*x)/2]^3 + a^3*Tan[(c + d*x)/2]^5 - a^2*b*Tan[(c + d*x)/2]^5 - 3*a*b^2*Tan[(c + d*x)/2]^5 + 3*b^3*Tan[(c + d*x)/2]^5 - 6*a^2*b*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + 6*b^3*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - 6*a^2*b*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + 6*b^3*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + (a^3 + a^2*b - 3*a*b^2 - 3*b^3)*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + 2*a*b*(a + b)*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)]))/(a^2*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^(3/2)*Sqrt[1 + Tan[(c + d*x)/2]^2]*(a*(-1 + Tan[(c + d*x)/2]^2) - b*(1 + Tan[(c + d*x)/2]^2)))","B",0
569,1,1745,470,14.65893,"\int \frac{\cos ^2(c+d x)}{(a+b \sec (c+d x))^{3/2}} \, dx","Integrate[Cos[c + d*x]^2/(a + b*Sec[c + d*x])^(3/2),x]","\frac{(b+a \cos (c+d x))^2 \left(\frac{2 \sin (c+d x) b^4}{a^3 \left(a^2-b^2\right) (b+a \cos (c+d x))}+\frac{2 \sin (c+d x) b^3}{a^3 \left(b^2-a^2\right)}+\frac{\sin (2 (c+d x))}{4 a^2}\right) \sec ^2(c+d x)}{d (a+b \sec (c+d x))^{3/2}}+\frac{(b+a \cos (c+d x))^{3/2} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{\tan ^2\left(\frac{1}{2} (c+d x)\right)+1}} \left(-15 b^4 \sqrt{\frac{b-a}{a+b}} \tan ^5\left(\frac{1}{2} (c+d x)\right)+15 a b^3 \sqrt{\frac{b-a}{a+b}} \tan ^5\left(\frac{1}{2} (c+d x)\right)+7 a^2 b^2 \sqrt{\frac{b-a}{a+b}} \tan ^5\left(\frac{1}{2} (c+d x)\right)-7 a^3 b \sqrt{\frac{b-a}{a+b}} \tan ^5\left(\frac{1}{2} (c+d x)\right)-30 a b^3 \sqrt{\frac{b-a}{a+b}} \tan ^3\left(\frac{1}{2} (c+d x)\right)+14 a^3 b \sqrt{\frac{b-a}{a+b}} \tan ^3\left(\frac{1}{2} (c+d x)\right)-8 i a^4 \Pi \left(-\frac{a+b}{a-b};i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)+30 i b^4 \Pi \left(-\frac{a+b}{a-b};i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)-22 i a^2 b^2 \Pi \left(-\frac{a+b}{a-b};i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)+15 b^4 \sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)+15 a b^3 \sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)-7 a^2 b^2 \sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)-7 a^3 b \sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)+i b \left(7 a^3-7 b a^2-15 b^2 a+15 b^3\right) E\left(i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}+2 i \left(2 a^4-b a^3+9 b^2 a^2+5 b^3 a-15 b^4\right) F\left(i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}-8 i a^4 \Pi \left(-\frac{a+b}{a-b};i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}+30 i b^4 \Pi \left(-\frac{a+b}{a-b};i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}-22 i a^2 b^2 \Pi \left(-\frac{a+b}{a-b};i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}\right) \sec ^{\frac{3}{2}}(c+d x)}{4 a^3 \sqrt{\frac{b-a}{a+b}} \left(a^2-b^2\right) d (a+b \sec (c+d x))^{3/2} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)-1\right) \sqrt{\frac{\tan ^2\left(\frac{1}{2} (c+d x)\right)+1}{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)}} \left(a \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)-1\right)-b \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right)\right)}","-\frac{5 b \sin (c+d x)}{4 a^2 d \sqrt{a+b \sec (c+d x)}}-\frac{\sqrt{a+b} \left(4 a^2+15 b^2\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{4 a^4 d}-\frac{b^2 \left(7 a^2-15 b^2\right) \tan (c+d x)}{4 a^3 d \left(a^2-b^2\right) \sqrt{a+b \sec (c+d x)}}+\frac{\left(2 a^2-5 a b-15 b^2\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{4 a^3 d \sqrt{a+b}}-\frac{\left(7 a^2-15 b^2\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{4 a^3 d \sqrt{a+b}}+\frac{\sin (c+d x) \cos (c+d x)}{2 a d \sqrt{a+b \sec (c+d x)}}",1,"((b + a*Cos[c + d*x])^2*Sec[c + d*x]^2*((2*b^3*Sin[c + d*x])/(a^3*(-a^2 + b^2)) + (2*b^4*Sin[c + d*x])/(a^3*(a^2 - b^2)*(b + a*Cos[c + d*x])) + Sin[2*(c + d*x)]/(4*a^2)))/(d*(a + b*Sec[c + d*x])^(3/2)) + ((b + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(3/2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(1 + Tan[(c + d*x)/2]^2)]*(-7*a^3*b*Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2] - 7*a^2*b^2*Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2] + 15*a*b^3*Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2] + 15*b^4*Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2] + 14*a^3*b*Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]^3 - 30*a*b^3*Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]^3 - 7*a^3*b*Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]^5 + 7*a^2*b^2*Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]^5 + 15*a*b^3*Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]^5 - 15*b^4*Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]^5 - (8*I)*a^4*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - (22*I)*a^2*b^2*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + (30*I)*b^4*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - (8*I)*a^4*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - (22*I)*a^2*b^2*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + (30*I)*b^4*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + I*b*(7*a^3 - 7*a^2*b - 15*a*b^2 + 15*b^3)*EllipticE[I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + (2*I)*(2*a^4 - a^3*b + 9*a^2*b^2 + 5*a*b^3 - 15*b^4)*EllipticF[I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)]))/(4*a^3*Sqrt[(-a + b)/(a + b)]*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^(3/2)*(-1 + Tan[(c + d*x)/2]^2)*Sqrt[(1 + Tan[(c + d*x)/2]^2)/(1 - Tan[(c + d*x)/2]^2)]*(a*(-1 + Tan[(c + d*x)/2]^2) - b*(1 + Tan[(c + d*x)/2]^2)))","C",0
570,1,578,427,19.0882739,"\int \frac{\sec ^5(c+d x)}{(a+b \sec (c+d x))^{5/2}} \, dx","Integrate[Sec[c + d*x]^5/(a + b*Sec[c + d*x])^(5/2),x]","\frac{\sec ^3(c+d x) (a \cos (c+d x)+b)^3 \left(-\frac{8 a \left(4 a^4-7 a^2 b^2+2 b^4\right) \sin (c+d x)}{3 b^4 \left(b^2-a^2\right)^2}-\frac{2 a^3 \sin (c+d x)}{3 b^2 \left(b^2-a^2\right) (a \cos (c+d x)+b)^2}-\frac{2 \left(11 a^3 b^2 \sin (c+d x)-7 a^5 \sin (c+d x)\right)}{3 b^3 \left(b^2-a^2\right)^2 (a \cos (c+d x)+b)}+\frac{2 \tan (c+d x)}{3 b^3}\right)}{d (a+b \sec (c+d x))^{5/2}}+\frac{4 \sec ^{\frac{5}{2}}(c+d x) \sqrt{\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)} (a \cos (c+d x)+b)^2 \left(2 a \left(4 a^4-7 a^2 b^2+2 b^4\right) \cos (c+d x) \tan \left(\frac{1}{2} (c+d x)\right) \sec ^2\left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b)+b \left(-16 a^5-4 a^4 b+28 a^3 b^2+7 a^2 b^3-8 a b^4+b^5\right) \sqrt{\frac{\cos (c+d x)}{\cos (c+d x)+1}} \sqrt{\frac{a \cos (c+d x)+b}{(a+b) (\cos (c+d x)+1)}} F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right)+4 a \left(4 a^5+4 a^4 b-7 a^3 b^2-7 a^2 b^3+2 a b^4+2 b^5\right) \sqrt{\frac{\cos (c+d x)}{\cos (c+d x)+1}} \sqrt{\frac{a \cos (c+d x)+b}{(a+b) (\cos (c+d x)+1)}} E\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right)\right)}{3 b^4 d \left(a^2-b^2\right)^2 \sqrt{\sec ^2\left(\frac{1}{2} (c+d x)\right)} (a+b \sec (c+d x))^{5/2}}","-\frac{2 a^2 \tan (c+d x) \sec ^2(c+d x)}{3 b d \left(a^2-b^2\right) (a+b \sec (c+d x))^{3/2}}+\frac{2 \left(2 a^2-b^2\right) \tan (c+d x) \sqrt{a+b \sec (c+d x)}}{3 b^3 d \left(a^2-b^2\right)}+\frac{8 a \left(4 a^4-7 a^2 b^2+2 b^4\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{3 b^5 d (a-b) (a+b)^{3/2}}+\frac{4 a^3 \left(3 a^2-5 b^2\right) \tan (c+d x)}{3 b^3 d \left(a^2-b^2\right)^2 \sqrt{a+b \sec (c+d x)}}+\frac{2 \left(16 a^4+12 a^3 b-16 a^2 b^2-9 a b^3-b^4\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{3 b^4 d (a-b) (a+b)^{3/2}}",1,"(4*(b + a*Cos[c + d*x])^2*Sec[c + d*x]^(5/2)*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(4*a*(4*a^5 + 4*a^4*b - 7*a^3*b^2 - 7*a^2*b^3 + 2*a*b^4 + 2*b^5)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + b*(-16*a^5 - 4*a^4*b + 28*a^3*b^2 + 7*a^2*b^3 - 8*a*b^4 + b^5)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + 2*a*(4*a^4 - 7*a^2*b^2 + 2*b^4)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(3*b^4*(a^2 - b^2)^2*d*Sqrt[Sec[(c + d*x)/2]^2]*(a + b*Sec[c + d*x])^(5/2)) + ((b + a*Cos[c + d*x])^3*Sec[c + d*x]^3*((-8*a*(4*a^4 - 7*a^2*b^2 + 2*b^4)*Sin[c + d*x])/(3*b^4*(-a^2 + b^2)^2) - (2*a^3*Sin[c + d*x])/(3*b^2*(-a^2 + b^2)*(b + a*Cos[c + d*x])^2) - (2*(-7*a^5*Sin[c + d*x] + 11*a^3*b^2*Sin[c + d*x]))/(3*b^3*(-a^2 + b^2)^2*(b + a*Cos[c + d*x])) + (2*Tan[c + d*x])/(3*b^3)))/(d*(a + b*Sec[c + d*x])^(5/2))","A",0
571,1,3345,362,23.5265008,"\int \frac{\sec ^4(c+d x)}{(a+b \sec (c+d x))^{5/2}} \, dx","Integrate[Sec[c + d*x]^4/(a + b*Sec[c + d*x])^(5/2),x]","\text{Result too large to show}","-\frac{8 a^2 \left(a^2-2 b^2\right) \tan (c+d x)}{3 b^2 d \left(a^2-b^2\right)^2 \sqrt{a+b \sec (c+d x)}}-\frac{2 a^2 \tan (c+d x) \sec (c+d x)}{3 b d \left(a^2-b^2\right) (a+b \sec (c+d x))^{3/2}}-\frac{2 \left(8 a^4-15 a^2 b^2+3 b^4\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{3 b^4 d (a-b) (a+b)^{3/2}}-\frac{2 \left(8 a^3+6 a^2 b-9 a b^2-3 b^3\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{3 b^3 d (a-b) (a+b)^{3/2}}",1,"((b + a*Cos[c + d*x])^3*Sec[c + d*x]^3*((2*(8*a^4 - 15*a^2*b^2 + 3*b^4)*Sin[c + d*x])/(3*b^3*(-a^2 + b^2)^2) + (2*a^2*Sin[c + d*x])/(3*b*(-a^2 + b^2)*(b + a*Cos[c + d*x])^2) + (8*(-(a^4*Sin[c + d*x]) + 2*a^2*b^2*Sin[c + d*x]))/(3*b^2*(-a^2 + b^2)^2*(b + a*Cos[c + d*x]))))/(d*(a + b*Sec[c + d*x])^(5/2)) - (2*(b + a*Cos[c + d*x])^2*((5*a^2)/((-a^2 + b^2)^2*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (8*a^4)/(3*b^2*(-a^2 + b^2)^2*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - b^2/((-a^2 + b^2)^2*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (8*a^5*Sqrt[Sec[c + d*x]])/(3*b^3*(-a^2 + b^2)^2*Sqrt[b + a*Cos[c + d*x]]) + (17*a^3*Sqrt[Sec[c + d*x]])/(3*b*(-a^2 + b^2)^2*Sqrt[b + a*Cos[c + d*x]]) - (3*a*b*Sqrt[Sec[c + d*x]])/((-a^2 + b^2)^2*Sqrt[b + a*Cos[c + d*x]]) - (8*a^5*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(3*b^3*(-a^2 + b^2)^2*Sqrt[b + a*Cos[c + d*x]]) + (5*a^3*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(b*(-a^2 + b^2)^2*Sqrt[b + a*Cos[c + d*x]]) - (a*b*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/((-a^2 + b^2)^2*Sqrt[b + a*Cos[c + d*x]]))*Sec[c + d*x]^(5/2)*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(2*(8*a^5 + 8*a^4*b - 15*a^3*b^2 - 15*a^2*b^3 + 3*a*b^4 + 3*b^5)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] - 2*b*(8*a^4 + 2*a^3*b - 15*a^2*b^2 - 6*a*b^3 + 3*b^4)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + (8*a^4 - 15*a^2*b^2 + 3*b^4)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(3*b^3*(a^2 - b^2)^2*d*Sqrt[Sec[(c + d*x)/2]^2]*(a + b*Sec[c + d*x])^(5/2)*((Cos[(c + d*x)/2]*Sqrt[Sec[(c + d*x)/2]^2]*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sin[(c + d*x)/2]*(2*(8*a^5 + 8*a^4*b - 15*a^3*b^2 - 15*a^2*b^3 + 3*a*b^4 + 3*b^5)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] - 2*b*(8*a^4 + 2*a^3*b - 15*a^2*b^2 - 6*a*b^3 + 3*b^4)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + (8*a^4 - 15*a^2*b^2 + 3*b^4)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(3*b^3*(a^2 - b^2)^2*Sqrt[b + a*Cos[c + d*x]]) - (a*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sin[c + d*x]*(2*(8*a^5 + 8*a^4*b - 15*a^3*b^2 - 15*a^2*b^3 + 3*a*b^4 + 3*b^5)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] - 2*b*(8*a^4 + 2*a^3*b - 15*a^2*b^2 - 6*a*b^3 + 3*b^4)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + (8*a^4 - 15*a^2*b^2 + 3*b^4)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(3*b^3*(a^2 - b^2)^2*(b + a*Cos[c + d*x])^(3/2)*Sqrt[Sec[(c + d*x)/2]^2]) - (2*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(((8*a^4 - 15*a^2*b^2 + 3*b^4)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^4)/2 + ((8*a^5 + 8*a^4*b - 15*a^3*b^2 - 15*a^2*b^3 + 3*a*b^4 + 3*b^5)*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*((Cos[c + d*x]*Sin[c + d*x])/(1 + Cos[c + d*x])^2 - Sin[c + d*x]/(1 + Cos[c + d*x])))/Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])] - (b*(8*a^4 + 2*a^3*b - 15*a^2*b^2 - 6*a*b^3 + 3*b^4)*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*((Cos[c + d*x]*Sin[c + d*x])/(1 + Cos[c + d*x])^2 - Sin[c + d*x]/(1 + Cos[c + d*x])))/Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])] + ((8*a^5 + 8*a^4*b - 15*a^3*b^2 - 15*a^2*b^3 + 3*a*b^4 + 3*b^5)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*(-((a*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x]))) + ((b + a*Cos[c + d*x])*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x])^2)))/Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))] - (b*(8*a^4 + 2*a^3*b - 15*a^2*b^2 - 6*a*b^3 + 3*b^4)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*(-((a*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x]))) + ((b + a*Cos[c + d*x])*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x])^2)))/Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))] - a*(8*a^4 - 15*a^2*b^2 + 3*b^4)*Cos[c + d*x]*Sec[(c + d*x)/2]^2*Sin[c + d*x]*Tan[(c + d*x)/2] - (8*a^4 - 15*a^2*b^2 + 3*b^4)*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Sin[c + d*x]*Tan[(c + d*x)/2] + (8*a^4 - 15*a^2*b^2 + 3*b^4)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]^2 - (b*(8*a^4 + 2*a^3*b - 15*a^2*b^2 - 6*a*b^3 + 3*b^4)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*Sec[(c + d*x)/2]^2)/(Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[1 - ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)]) + ((8*a^5 + 8*a^4*b - 15*a^3*b^2 - 15*a^2*b^3 + 3*a*b^4 + 3*b^5)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*Sec[(c + d*x)/2]^2*Sqrt[1 - ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)])/Sqrt[1 - Tan[(c + d*x)/2]^2]))/(3*b^3*(a^2 - b^2)^2*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]) - ((2*(8*a^5 + 8*a^4*b - 15*a^3*b^2 - 15*a^2*b^3 + 3*a*b^4 + 3*b^5)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] - 2*b*(8*a^4 + 2*a^3*b - 15*a^2*b^2 - 6*a*b^3 + 3*b^4)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + (8*a^4 - 15*a^2*b^2 + 3*b^4)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])*(-(Cos[(c + d*x)/2]*Sec[c + d*x]*Sin[(c + d*x)/2]) + Cos[(c + d*x)/2]^2*Sec[c + d*x]*Tan[c + d*x]))/(3*b^3*(a^2 - b^2)^2*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]])))","B",0
572,1,503,337,16.0861933,"\int \frac{\sec ^3(c+d x)}{(a+b \sec (c+d x))^{5/2}} \, dx","Integrate[Sec[c + d*x]^3/(a + b*Sec[c + d*x])^(5/2),x]","\frac{\sec ^3(c+d x) (a \cos (c+d x)+b)^3 \left(\frac{4 a \left(3 b^2-a^2\right) \sin (c+d x)}{3 b^2 \left(b^2-a^2\right)^2}-\frac{2 a \sin (c+d x)}{3 \left(b^2-a^2\right) (a \cos (c+d x)+b)^2}-\frac{2 \left(5 a b^2 \sin (c+d x)-a^3 \sin (c+d x)\right)}{3 b \left(b^2-a^2\right)^2 (a \cos (c+d x)+b)}\right)}{d (a+b \sec (c+d x))^{5/2}}+\frac{4 \sec ^{\frac{5}{2}}(c+d x) \sqrt{\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)} (a \cos (c+d x)+b)^2 \left(a \left(a^2-3 b^2\right) \cos (c+d x) \tan \left(\frac{1}{2} (c+d x)\right) \sec ^2\left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b)+b \left(-2 a^3+a^2 b+6 a b^2+3 b^3\right) \sqrt{\frac{\cos (c+d x)}{\cos (c+d x)+1}} \sqrt{\frac{a \cos (c+d x)+b}{(a+b) (\cos (c+d x)+1)}} F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right)+2 a \left(a^3+a^2 b-3 a b^2-3 b^3\right) \sqrt{\frac{\cos (c+d x)}{\cos (c+d x)+1}} \sqrt{\frac{a \cos (c+d x)+b}{(a+b) (\cos (c+d x)+1)}} E\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right)\right)}{3 d \left(b^3-a^2 b\right)^2 \sqrt{\sec ^2\left(\frac{1}{2} (c+d x)\right)} (a+b \sec (c+d x))^{5/2}}","-\frac{2 a^2 \tan (c+d x)}{3 b d \left(a^2-b^2\right) (a+b \sec (c+d x))^{3/2}}+\frac{4 a \left(a^2-3 b^2\right) \tan (c+d x)}{3 b d \left(a^2-b^2\right)^2 \sqrt{a+b \sec (c+d x)}}+\frac{2 \left(2 a^2+3 a b-3 b^2\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{3 b^2 d (a-b) (a+b)^{3/2}}+\frac{4 a \left(a^2-3 b^2\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{3 b^3 d (a-b) (a+b)^{3/2}}",1,"((b + a*Cos[c + d*x])^3*Sec[c + d*x]^3*((4*a*(-a^2 + 3*b^2)*Sin[c + d*x])/(3*b^2*(-a^2 + b^2)^2) - (2*a*Sin[c + d*x])/(3*(-a^2 + b^2)*(b + a*Cos[c + d*x])^2) - (2*(-(a^3*Sin[c + d*x]) + 5*a*b^2*Sin[c + d*x]))/(3*b*(-a^2 + b^2)^2*(b + a*Cos[c + d*x]))))/(d*(a + b*Sec[c + d*x])^(5/2)) + (4*(b + a*Cos[c + d*x])^2*Sec[c + d*x]^(5/2)*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(2*a*(a^3 + a^2*b - 3*a*b^2 - 3*b^3)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + b*(-2*a^3 + a^2*b + 6*a*b^2 + 3*b^3)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + a*(a^2 - 3*b^2)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(3*(-(a^2*b) + b^3)^2*d*Sqrt[Sec[(c + d*x)/2]^2]*(a + b*Sec[c + d*x])^(5/2))","A",0
573,1,486,317,13.9416278,"\int \frac{\sec ^2(c+d x)}{(a+b \sec (c+d x))^{5/2}} \, dx","Integrate[Sec[c + d*x]^2/(a + b*Sec[c + d*x])^(5/2),x]","\frac{\sec ^3(c+d x) (a \cos (c+d x)+b)^3 \left(-\frac{2 \left(a^2+3 b^2\right) \sin (c+d x)}{3 b \left(b^2-a^2\right)^2}+\frac{2 b \sin (c+d x)}{3 \left(b^2-a^2\right) (a \cos (c+d x)+b)^2}+\frac{4 \left(a^2 \sin (c+d x)+b^2 \sin (c+d x)\right)}{3 \left(b^2-a^2\right)^2 (a \cos (c+d x)+b)}\right)}{d (a+b \sec (c+d x))^{5/2}}+\frac{2 \sec ^{\frac{5}{2}}(c+d x) \sqrt{\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)} (a \cos (c+d x)+b)^2 \left(\left(a^2+3 b^2\right) \cos (c+d x) \tan \left(\frac{1}{2} (c+d x)\right) \sec ^2\left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b)-2 b \left(a^2+4 a b+3 b^2\right) \sqrt{\frac{\cos (c+d x)}{\cos (c+d x)+1}} \sqrt{\frac{a \cos (c+d x)+b}{(a+b) (\cos (c+d x)+1)}} F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right)+2 \left(a^3+a^2 b+3 a b^2+3 b^3\right) \sqrt{\frac{\cos (c+d x)}{\cos (c+d x)+1}} \sqrt{\frac{a \cos (c+d x)+b}{(a+b) (\cos (c+d x)+1)}} E\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right)\right)}{3 b d \left(a^2-b^2\right)^2 \sqrt{\sec ^2\left(\frac{1}{2} (c+d x)\right)} (a+b \sec (c+d x))^{5/2}}","\frac{2 \left(a^2+3 b^2\right) \tan (c+d x)}{3 d \left(a^2-b^2\right)^2 \sqrt{a+b \sec (c+d x)}}+\frac{2 a \tan (c+d x)}{3 d \left(a^2-b^2\right) (a+b \sec (c+d x))^{3/2}}+\frac{2 \left(a^2+3 b^2\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{3 b^2 d (a-b) (a+b)^{3/2}}+\frac{2 (a-3 b) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{3 b d (a-b) (a+b)^{3/2}}",1,"((b + a*Cos[c + d*x])^3*Sec[c + d*x]^3*((-2*(a^2 + 3*b^2)*Sin[c + d*x])/(3*b*(-a^2 + b^2)^2) + (2*b*Sin[c + d*x])/(3*(-a^2 + b^2)*(b + a*Cos[c + d*x])^2) + (4*(a^2*Sin[c + d*x] + b^2*Sin[c + d*x]))/(3*(-a^2 + b^2)^2*(b + a*Cos[c + d*x]))))/(d*(a + b*Sec[c + d*x])^(5/2)) + (2*(b + a*Cos[c + d*x])^2*Sec[c + d*x]^(5/2)*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(2*(a^3 + a^2*b + 3*a*b^2 + 3*b^3)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] - 2*b*(a^2 + 4*a*b + 3*b^2)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + (a^2 + 3*b^2)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(3*b*(a^2 - b^2)^2*d*Sqrt[Sec[(c + d*x)/2]^2]*(a + b*Sec[c + d*x])^(5/2))","A",0
574,1,360,304,8.8346559,"\int \frac{\sec (c+d x)}{(a+b \sec (c+d x))^{5/2}} \, dx","Integrate[Sec[c + d*x]/(a + b*Sec[c + d*x])^(5/2),x]","-\frac{2 \sec ^3(c+d x) (a \cos (c+d x)+b) \left(b^2 \left(b^2-a^2\right) \sin (c+d x)-b \left(b^2-5 a^2\right) \sin (c+d x) (a \cos (c+d x)+b)+2 a \cos ^2\left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b) \left(-\left(3 a^2+4 a b+b^2\right) \sqrt{\frac{\cos (c+d x)}{\cos (c+d x)+1}} \sqrt{\frac{a \cos (c+d x)+b}{(a+b) (\cos (c+d x)+1)}} F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right)+2 a \cos (c+d x) \tan \left(\frac{1}{2} (c+d x)\right) \sec ^2\left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b)+4 a (a+b) \sqrt{\frac{\cos (c+d x)}{\cos (c+d x)+1}} \sqrt{\frac{a \cos (c+d x)+b}{(a+b) (\cos (c+d x)+1)}} E\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right)\right)-4 a^2 \sin (c+d x) (a \cos (c+d x)+b)^2\right)}{3 a d \left(a^2-b^2\right)^2 (a+b \sec (c+d x))^{5/2}}","-\frac{8 a b \tan (c+d x)}{3 d \left(a^2-b^2\right)^2 \sqrt{a+b \sec (c+d x)}}-\frac{2 b \tan (c+d x)}{3 d \left(a^2-b^2\right) (a+b \sec (c+d x))^{3/2}}+\frac{2 (3 a-b) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{3 b d (a-b) (a+b)^{3/2}}-\frac{8 a \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{3 b d (a-b) (a+b)^{3/2}}",1,"(-2*(b + a*Cos[c + d*x])*Sec[c + d*x]^3*(b^2*(-a^2 + b^2)*Sin[c + d*x] - b*(-5*a^2 + b^2)*(b + a*Cos[c + d*x])*Sin[c + d*x] - 4*a^2*(b + a*Cos[c + d*x])^2*Sin[c + d*x] + 2*a*Cos[(c + d*x)/2]^2*(b + a*Cos[c + d*x])*(4*a*(a + b)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] - (3*a^2 + 4*a*b + b^2)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + 2*a*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])))/(3*a*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x])^(5/2))","A",1
575,1,1798,448,15.5931306,"\int \frac{1}{(a+b \sec (c+d x))^{5/2}} \, dx","Integrate[(a + b*Sec[c + d*x])^(-5/2),x]","\frac{(b+a \cos (c+d x))^3 \left(-\frac{2 \sin (c+d x) b^3}{3 a^2 \left(a^2-b^2\right) (b+a \cos (c+d x))^2}+\frac{2 \left(3 b^2-7 a^2\right) \sin (c+d x) b}{3 a^2 \left(b^2-a^2\right)^2}-\frac{8 \left(b^4 \sin (c+d x)-2 a^2 b^2 \sin (c+d x)\right)}{3 a^2 \left(a^2-b^2\right)^2 (b+a \cos (c+d x))}\right) \sec ^3(c+d x)}{d (a+b \sec (c+d x))^{5/2}}+\frac{2 (b+a \cos (c+d x))^{5/2} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{\tan ^2\left(\frac{1}{2} (c+d x)\right)+1}} \left(3 b^4 \sqrt{\frac{b-a}{a+b}} \tan ^5\left(\frac{1}{2} (c+d x)\right)-3 a b^3 \sqrt{\frac{b-a}{a+b}} \tan ^5\left(\frac{1}{2} (c+d x)\right)-7 a^2 b^2 \sqrt{\frac{b-a}{a+b}} \tan ^5\left(\frac{1}{2} (c+d x)\right)+7 a^3 b \sqrt{\frac{b-a}{a+b}} \tan ^5\left(\frac{1}{2} (c+d x)\right)+6 a b^3 \sqrt{\frac{b-a}{a+b}} \tan ^3\left(\frac{1}{2} (c+d x)\right)-14 a^3 b \sqrt{\frac{b-a}{a+b}} \tan ^3\left(\frac{1}{2} (c+d x)\right)-6 i a^4 \Pi \left(-\frac{a+b}{a-b};i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)-6 i b^4 \Pi \left(-\frac{a+b}{a-b};i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)+12 i a^2 b^2 \Pi \left(-\frac{a+b}{a-b};i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)-3 b^4 \sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)-3 a b^3 \sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)+7 a^2 b^2 \sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)+7 a^3 b \sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)-i b \left(7 a^3-7 b a^2-3 b^2 a+3 b^3\right) E\left(i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}+i \left(3 a^4+6 b a^3-13 b^2 a^2-2 b^3 a+6 b^4\right) F\left(i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}-6 i a^4 \Pi \left(-\frac{a+b}{a-b};i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}-6 i b^4 \Pi \left(-\frac{a+b}{a-b};i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}+12 i a^2 b^2 \Pi \left(-\frac{a+b}{a-b};i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}\right) \sec ^{\frac{5}{2}}(c+d x)}{3 a^2 \sqrt{\frac{b-a}{a+b}} \left(a^2-b^2\right)^2 d (a+b \sec (c+d x))^{5/2} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)-1\right) \sqrt{\frac{\tan ^2\left(\frac{1}{2} (c+d x)\right)+1}{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)}} \left(a \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)-1\right)-b \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right)\right)}","-\frac{2 \sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{a^3 d}+\frac{2 b^2 \left(7 a^2-3 b^2\right) \tan (c+d x)}{3 a^2 d \left(a^2-b^2\right)^2 \sqrt{a+b \sec (c+d x)}}+\frac{2 b^2 \tan (c+d x)}{3 a d \left(a^2-b^2\right) (a+b \sec (c+d x))^{3/2}}-\frac{2 \left(6 a^2-a b-3 b^2\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{3 a^2 d (a-b) (a+b)^{3/2}}+\frac{2 \left(7 a^2-3 b^2\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{3 a^2 d (a-b) (a+b)^{3/2}}",1,"((b + a*Cos[c + d*x])^3*Sec[c + d*x]^3*((2*b*(-7*a^2 + 3*b^2)*Sin[c + d*x])/(3*a^2*(-a^2 + b^2)^2) - (2*b^3*Sin[c + d*x])/(3*a^2*(a^2 - b^2)*(b + a*Cos[c + d*x])^2) - (8*(-2*a^2*b^2*Sin[c + d*x] + b^4*Sin[c + d*x]))/(3*a^2*(a^2 - b^2)^2*(b + a*Cos[c + d*x]))))/(d*(a + b*Sec[c + d*x])^(5/2)) + (2*(b + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(5/2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(1 + Tan[(c + d*x)/2]^2)]*(7*a^3*b*Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2] + 7*a^2*b^2*Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2] - 3*a*b^3*Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2] - 3*b^4*Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2] - 14*a^3*b*Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]^3 + 6*a*b^3*Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]^3 + 7*a^3*b*Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]^5 - 7*a^2*b^2*Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]^5 - 3*a*b^3*Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]^5 + 3*b^4*Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]^5 - (6*I)*a^4*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + (12*I)*a^2*b^2*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - (6*I)*b^4*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - (6*I)*a^4*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + (12*I)*a^2*b^2*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - (6*I)*b^4*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - I*b*(7*a^3 - 7*a^2*b - 3*a*b^2 + 3*b^3)*EllipticE[I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + I*(3*a^4 + 6*a^3*b - 13*a^2*b^2 - 2*a*b^3 + 6*b^4)*EllipticF[I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)]))/(3*a^2*Sqrt[(-a + b)/(a + b)]*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x])^(5/2)*(-1 + Tan[(c + d*x)/2]^2)*Sqrt[(1 + Tan[(c + d*x)/2]^2)/(1 - Tan[(c + d*x)/2]^2)]*(a*(-1 + Tan[(c + d*x)/2]^2) - b*(1 + Tan[(c + d*x)/2]^2)))","C",0
576,1,1481,510,17.6663086,"\int \frac{\cos (c+d x)}{(a+b \sec (c+d x))^{5/2}} \, dx","Integrate[Cos[c + d*x]/(a + b*Sec[c + d*x])^(5/2),x]","\frac{(b+a \cos (c+d x))^3 \sec ^3(c+d x) \left(\frac{2 \sin (c+d x) b^4}{3 a^3 \left(a^2-b^2\right) (b+a \cos (c+d x))^2}-\frac{4 \left(3 b^2-5 a^2\right) \sin (c+d x) b^2}{3 a^3 \left(b^2-a^2\right)^2}+\frac{2 \left(7 b^5 \sin (c+d x)-11 a^2 b^3 \sin (c+d x)\right)}{3 a^3 \left(a^2-b^2\right)^2 (b+a \cos (c+d x))}\right)}{d (a+b \sec (c+d x))^{5/2}}-\frac{(b+a \cos (c+d x))^{5/2} \sec ^{\frac{5}{2}}(c+d x) \sqrt{\frac{1}{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)}} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{\tan ^2\left(\frac{1}{2} (c+d x)\right)+1}} \left(3 \tan ^5\left(\frac{1}{2} (c+d x)\right) a^5-6 \tan ^3\left(\frac{1}{2} (c+d x)\right) a^5+3 \tan \left(\frac{1}{2} (c+d x)\right) a^5-3 b \tan ^5\left(\frac{1}{2} (c+d x)\right) a^4+3 b \tan \left(\frac{1}{2} (c+d x)\right) a^4-30 b \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \tan ^2\left(\frac{1}{2} (c+d x)\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} a^4-30 b \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} a^4-26 b^2 \tan ^5\left(\frac{1}{2} (c+d x)\right) a^3+52 b^2 \tan ^3\left(\frac{1}{2} (c+d x)\right) a^3-26 b^2 \tan \left(\frac{1}{2} (c+d x)\right) a^3+26 b^3 \tan ^5\left(\frac{1}{2} (c+d x)\right) a^2-26 b^3 \tan \left(\frac{1}{2} (c+d x)\right) a^2+60 b^3 \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \tan ^2\left(\frac{1}{2} (c+d x)\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} a^2+60 b^3 \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} a^2+15 b^4 \tan ^5\left(\frac{1}{2} (c+d x)\right) a-30 b^4 \tan ^3\left(\frac{1}{2} (c+d x)\right) a+15 b^4 \tan \left(\frac{1}{2} (c+d x)\right) a+2 b \left(6 a^3+9 b a^2-2 b^2 a-5 b^3\right) F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} a-15 b^5 \tan ^5\left(\frac{1}{2} (c+d x)\right)+15 b^5 \tan \left(\frac{1}{2} (c+d x)\right)+\left(3 a^5+3 b a^4-26 b^2 a^3-26 b^3 a^2+15 b^4 a+15 b^5\right) E\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}-30 b^5 \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \tan ^2\left(\frac{1}{2} (c+d x)\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}-30 b^5 \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}\right)}{3 a \left(a^3-a b^2\right)^2 d (a+b \sec (c+d x))^{5/2} \sqrt{\tan ^2\left(\frac{1}{2} (c+d x)\right)+1} \left(a \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)-1\right)-b \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right)\right)}","\frac{5 b \sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{a^4 d}+\frac{b \left(3 a^2-5 b^2\right) \tan (c+d x)}{3 a^2 d \left(a^2-b^2\right) (a+b \sec (c+d x))^{3/2}}+\frac{\left(3 a^3+21 a^2 b-5 a b^2-15 b^3\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{3 a^3 d (a-b) (a+b)^{3/2}}+\frac{b \left(3 a^4-26 a^2 b^2+15 b^4\right) \tan (c+d x)}{3 a^3 d \left(a^2-b^2\right)^2 \sqrt{a+b \sec (c+d x)}}+\frac{\left(3 a^4-26 a^2 b^2+15 b^4\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{3 a^3 b d (a-b) (a+b)^{3/2}}+\frac{\sin (c+d x)}{a d (a+b \sec (c+d x))^{3/2}}",1,"((b + a*Cos[c + d*x])^3*Sec[c + d*x]^3*((-4*b^2*(-5*a^2 + 3*b^2)*Sin[c + d*x])/(3*a^3*(-a^2 + b^2)^2) + (2*b^4*Sin[c + d*x])/(3*a^3*(a^2 - b^2)*(b + a*Cos[c + d*x])^2) + (2*(-11*a^2*b^3*Sin[c + d*x] + 7*b^5*Sin[c + d*x]))/(3*a^3*(a^2 - b^2)^2*(b + a*Cos[c + d*x]))))/(d*(a + b*Sec[c + d*x])^(5/2)) - ((b + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(5/2)*Sqrt[(1 - Tan[(c + d*x)/2]^2)^(-1)]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(1 + Tan[(c + d*x)/2]^2)]*(3*a^5*Tan[(c + d*x)/2] + 3*a^4*b*Tan[(c + d*x)/2] - 26*a^3*b^2*Tan[(c + d*x)/2] - 26*a^2*b^3*Tan[(c + d*x)/2] + 15*a*b^4*Tan[(c + d*x)/2] + 15*b^5*Tan[(c + d*x)/2] - 6*a^5*Tan[(c + d*x)/2]^3 + 52*a^3*b^2*Tan[(c + d*x)/2]^3 - 30*a*b^4*Tan[(c + d*x)/2]^3 + 3*a^5*Tan[(c + d*x)/2]^5 - 3*a^4*b*Tan[(c + d*x)/2]^5 - 26*a^3*b^2*Tan[(c + d*x)/2]^5 + 26*a^2*b^3*Tan[(c + d*x)/2]^5 + 15*a*b^4*Tan[(c + d*x)/2]^5 - 15*b^5*Tan[(c + d*x)/2]^5 - 30*a^4*b*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + 60*a^2*b^3*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - 30*b^5*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - 30*a^4*b*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + 60*a^2*b^3*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - 30*b^5*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + (3*a^5 + 3*a^4*b - 26*a^3*b^2 - 26*a^2*b^3 + 15*a*b^4 + 15*b^5)*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + 2*a*b*(6*a^3 + 9*a^2*b - 2*a*b^2 - 5*b^3)*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)]))/(3*a*(a^3 - a*b^2)^2*d*(a + b*Sec[c + d*x])^(5/2)*Sqrt[1 + Tan[(c + d*x)/2]^2]*(a*(-1 + Tan[(c + d*x)/2]^2) - b*(1 + Tan[(c + d*x)/2]^2)))","B",0
577,1,2285,562,16.3794048,"\int \frac{\cos ^2(c+d x)}{(a+b \sec (c+d x))^{5/2}} \, dx","Integrate[Cos[c + d*x]^2/(a + b*Sec[c + d*x])^(5/2),x]","\text{Result too large to show}","-\frac{7 b \sin (c+d x)}{4 a^2 d (a+b \sec (c+d x))^{3/2}}-\frac{\sqrt{a+b} \left(4 a^2+35 b^2\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{4 a^5 d}-\frac{b^2 \left(33 a^4-170 a^2 b^2+105 b^4\right) \tan (c+d x)}{12 a^4 d \left(a^2-b^2\right)^2 \sqrt{a+b \sec (c+d x)}}-\frac{\left(33 a^4-170 a^2 b^2+105 b^4\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{12 a^4 d (a-b) (a+b)^{3/2}}-\frac{b^2 \left(27 a^2-35 b^2\right) \tan (c+d x)}{12 a^3 d \left(a^2-b^2\right) (a+b \sec (c+d x))^{3/2}}+\frac{(a+3 b) \left(6 a^3-45 a^2 b+35 b^3\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{12 a^4 d (a-b) (a+b)^{3/2}}+\frac{\sin (c+d x) \cos (c+d x)}{2 a d (a+b \sec (c+d x))^{3/2}}",1,"((b + a*Cos[c + d*x])^3*Sec[c + d*x]^3*((2*b^3*(-13*a^2 + 9*b^2)*Sin[c + d*x])/(3*a^4*(-a^2 + b^2)^2) - (2*b^5*Sin[c + d*x])/(3*a^4*(a^2 - b^2)*(b + a*Cos[c + d*x])^2) - (4*(-7*a^2*b^4*Sin[c + d*x] + 5*b^6*Sin[c + d*x]))/(3*a^4*(a^2 - b^2)^2*(b + a*Cos[c + d*x])) + Sin[2*(c + d*x)]/(4*a^3)))/(d*(a + b*Sec[c + d*x])^(5/2)) - ((b + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(5/2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(1 + Tan[(c + d*x)/2]^2)]*(33*a^5*b*Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2] + 33*a^4*b^2*Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2] - 170*a^3*b^3*Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2] - 170*a^2*b^4*Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2] + 105*a*b^5*Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2] + 105*b^6*Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2] - 66*a^5*b*Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]^3 + 340*a^3*b^3*Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]^3 - 210*a*b^5*Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]^3 + 33*a^5*b*Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]^5 - 33*a^4*b^2*Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]^5 - 170*a^3*b^3*Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]^5 + 170*a^2*b^4*Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]^5 + 105*a*b^5*Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]^5 - 105*b^6*Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]^5 + (24*I)*a^6*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + (162*I)*a^4*b^2*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - (396*I)*a^2*b^4*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + (210*I)*b^6*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + (24*I)*a^6*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + (162*I)*a^4*b^2*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - (396*I)*a^2*b^4*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + (210*I)*b^6*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + I*b*(-33*a^5 + 33*a^4*b + 170*a^3*b^2 - 170*a^2*b^3 - 105*a*b^4 + 105*b^5)*EllipticE[I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - (2*I)*(6*a^6 - 3*a^5*b + 57*a^4*b^2 + 54*a^3*b^3 - 184*a^2*b^4 - 35*a*b^5 + 105*b^6)*EllipticF[I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)]))/(12*a^4*Sqrt[(-a + b)/(a + b)]*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x])^(5/2)*(-1 + Tan[(c + d*x)/2]^2)*Sqrt[(1 + Tan[(c + d*x)/2]^2)/(1 - Tan[(c + d*x)/2]^2)]*(a*(-1 + Tan[(c + d*x)/2]^2) - b*(1 + Tan[(c + d*x)/2]^2)))","C",0
578,1,2346,535,16.5311244,"\int \frac{1}{(a+b \sec (c+d x))^{7/2}} \, dx","Integrate[(a + b*Sec[c + d*x])^(-7/2),x]","\text{Result too large to show}","-\frac{2 \sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{a^4 d}+\frac{2 b^2 \left(13 a^2-5 b^2\right) \tan (c+d x)}{15 a^2 d \left(a^2-b^2\right)^2 (a+b \sec (c+d x))^{3/2}}+\frac{2 b^2 \tan (c+d x)}{5 a d \left(a^2-b^2\right) (a+b \sec (c+d x))^{5/2}}+\frac{2 b^2 \left(58 a^4-41 a^2 b^2+15 b^4\right) \tan (c+d x)}{15 a^3 d \left(a^2-b^2\right)^3 \sqrt{a+b \sec (c+d x)}}+\frac{2 \left(58 a^4-41 a^2 b^2+15 b^4\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{15 a^3 d (a-b)^2 (a+b)^{5/2}}-\frac{2 \left(45 a^4-13 a^3 b-36 a^2 b^2+5 a b^3+15 b^4\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{15 a^3 d (a-b)^2 (a+b)^{5/2}}",1,"((b + a*Cos[c + d*x])^4*Sec[c + d*x]^4*((2*b*(58*a^4 - 41*a^2*b^2 + 15*b^4)*Sin[c + d*x])/(15*a^3*(-a^2 + b^2)^3) + (2*b^4*Sin[c + d*x])/(5*a^3*(a^2 - b^2)*(b + a*Cos[c + d*x])^3) + (2*(-19*a^2*b^3*Sin[c + d*x] + 11*b^5*Sin[c + d*x]))/(15*a^3*(a^2 - b^2)^2*(b + a*Cos[c + d*x])^2) + (2*(74*a^4*b^2*Sin[c + d*x] - 65*a^2*b^4*Sin[c + d*x] + 23*b^6*Sin[c + d*x]))/(15*a^3*(a^2 - b^2)^3*(b + a*Cos[c + d*x]))))/(d*(a + b*Sec[c + d*x])^(7/2)) + (2*(b + a*Cos[c + d*x])^(7/2)*Sec[c + d*x]^(7/2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(1 + Tan[(c + d*x)/2]^2)]*(58*a^5*b*Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2] + 58*a^4*b^2*Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2] - 41*a^3*b^3*Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2] - 41*a^2*b^4*Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2] + 15*a*b^5*Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2] + 15*b^6*Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2] - 116*a^5*b*Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]^3 + 82*a^3*b^3*Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]^3 - 30*a*b^5*Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]^3 + 58*a^5*b*Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]^5 - 58*a^4*b^2*Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]^5 - 41*a^3*b^3*Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]^5 + 41*a^2*b^4*Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]^5 + 15*a*b^5*Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]^5 - 15*b^6*Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]^5 - (30*I)*a^6*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + (90*I)*a^4*b^2*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - (90*I)*a^2*b^4*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + (30*I)*b^6*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - (30*I)*a^6*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + (90*I)*a^4*b^2*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - (90*I)*a^2*b^4*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + (30*I)*b^6*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + I*b*(-58*a^5 + 58*a^4*b + 41*a^3*b^2 - 41*a^2*b^3 - 15*a*b^4 + 15*b^5)*EllipticE[I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + I*(15*a^6 + 45*a^5*b - 103*a^4*b^2 - 23*a^3*b^3 + 86*a^2*b^4 + 10*a*b^5 - 30*b^6)*EllipticF[I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)]))/(15*a^3*Sqrt[(-a + b)/(a + b)]*(a^2 - b^2)^3*d*(a + b*Sec[c + d*x])^(7/2)*(-1 + Tan[(c + d*x)/2]^2)*Sqrt[(1 + Tan[(c + d*x)/2]^2)/(1 - Tan[(c + d*x)/2]^2)]*(a*(-1 + Tan[(c + d*x)/2]^2) - b*(1 + Tan[(c + d*x)/2]^2)))","C",0
579,1,97,151,0.3649305,"\int \sec ^{\frac{5}{2}}(c+d x) (a+b \sec (c+d x)) \, dx","Integrate[Sec[c + d*x]^(5/2)*(a + b*Sec[c + d*x]),x]","\frac{\sec ^{\frac{5}{2}}(c+d x) \left(10 a \sin (2 (c+d x))+20 a \cos ^{\frac{5}{2}}(c+d x) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+21 b \sin (c+d x)+9 b \sin (3 (c+d x))-36 b \cos ^{\frac{5}{2}}(c+d x) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{30 d}","\frac{2 a \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 d}+\frac{2 a \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{2 b \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{5 d}+\frac{6 b \sin (c+d x) \sqrt{\sec (c+d x)}}{5 d}-\frac{6 b \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}",1,"(Sec[c + d*x]^(5/2)*(-36*b*Cos[c + d*x]^(5/2)*EllipticE[(c + d*x)/2, 2] + 20*a*Cos[c + d*x]^(5/2)*EllipticF[(c + d*x)/2, 2] + 21*b*Sin[c + d*x] + 10*a*Sin[2*(c + d*x)] + 9*b*Sin[3*(c + d*x)]))/(30*d)","A",1
580,1,85,123,0.2688939,"\int \sec ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x)) \, dx","Integrate[Sec[c + d*x]^(3/2)*(a + b*Sec[c + d*x]),x]","\frac{\sec ^{\frac{3}{2}}(c+d x) \left(2 \sin (c+d x) (3 a \cos (c+d x)+b)-6 a \cos ^{\frac{3}{2}}(c+d x) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+2 b \cos ^{\frac{3}{2}}(c+d x) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{3 d}","\frac{2 a \sin (c+d x) \sqrt{\sec (c+d x)}}{d}-\frac{2 a \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 b \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 d}+\frac{2 b \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}",1,"(Sec[c + d*x]^(3/2)*(-6*a*Cos[c + d*x]^(3/2)*EllipticE[(c + d*x)/2, 2] + 2*b*Cos[c + d*x]^(3/2)*EllipticF[(c + d*x)/2, 2] + 2*(b + 3*a*Cos[c + d*x])*Sin[c + d*x]))/(3*d)","A",1
581,1,71,97,0.128134,"\int \sqrt{\sec (c+d x)} (a+b \sec (c+d x)) \, dx","Integrate[Sqrt[Sec[c + d*x]]*(a + b*Sec[c + d*x]),x]","\frac{2 \sqrt{\sec (c+d x)} \left(a \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+b \sin (c+d x)-b \sqrt{\cos (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{d}","\frac{2 a \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 b \sin (c+d x) \sqrt{\sec (c+d x)}}{d}-\frac{2 b \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}",1,"(2*Sqrt[Sec[c + d*x]]*(-(b*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]) + a*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + b*Sin[c + d*x]))/d","A",1
582,1,52,75,0.0862039,"\int \frac{a+b \sec (c+d x)}{\sqrt{\sec (c+d x)}} \, dx","Integrate[(a + b*Sec[c + d*x])/Sqrt[Sec[c + d*x]],x]","\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \left(a E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+b F\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{d}","\frac{2 a \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 b \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}",1,"(2*Sqrt[Cos[c + d*x]]*(a*EllipticE[(c + d*x)/2, 2] + b*EllipticF[(c + d*x)/2, 2])*Sqrt[Sec[c + d*x]])/d","A",1
583,1,76,101,0.1583734,"\int \frac{a+b \sec (c+d x)}{\sec ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[(a + b*Sec[c + d*x])/Sec[c + d*x]^(3/2),x]","\frac{\sqrt{\sec (c+d x)} \left(a \left(\sin (2 (c+d x))+2 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)+6 b \sqrt{\cos (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{3 d}","\frac{2 a \sin (c+d x)}{3 d \sqrt{\sec (c+d x)}}+\frac{2 a \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{2 b \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}",1,"(Sqrt[Sec[c + d*x]]*(6*b*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2] + a*(2*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + Sin[2*(c + d*x)])))/(3*d)","A",1
584,1,88,127,0.399272,"\int \frac{a+b \sec (c+d x)}{\sec ^{\frac{5}{2}}(c+d x)} \, dx","Integrate[(a + b*Sec[c + d*x])/Sec[c + d*x]^(5/2),x]","\frac{\sqrt{\sec (c+d x)} \left(\sin (2 (c+d x)) (3 a \cos (c+d x)+5 b)+18 a \sqrt{\cos (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+10 b \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{15 d}","\frac{2 a \sin (c+d x)}{5 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{6 a \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 b \sin (c+d x)}{3 d \sqrt{\sec (c+d x)}}+\frac{2 b \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}",1,"(Sqrt[Sec[c + d*x]]*(18*a*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2] + 10*b*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + (5*b + 3*a*Cos[c + d*x])*Sin[2*(c + d*x)]))/(15*d)","A",1
585,1,99,151,0.6206538,"\int \frac{a+b \sec (c+d x)}{\sec ^{\frac{7}{2}}(c+d x)} \, dx","Integrate[(a + b*Sec[c + d*x])/Sec[c + d*x]^(7/2),x]","\frac{\sqrt{\sec (c+d x)} \left(\sin (2 (c+d x)) (15 a \cos (2 (c+d x))+65 a+42 b \cos (c+d x))+100 a \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+252 b \sqrt{\cos (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{210 d}","\frac{2 a \sin (c+d x)}{7 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{10 a \sin (c+d x)}{21 d \sqrt{\sec (c+d x)}}+\frac{10 a \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{2 b \sin (c+d x)}{5 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{6 b \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}",1,"(Sqrt[Sec[c + d*x]]*(252*b*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2] + 100*a*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + (65*a + 42*b*Cos[c + d*x] + 15*a*Cos[2*(c + d*x)])*Sin[2*(c + d*x)]))/(210*d)","A",1
586,1,139,200,0.9874893,"\int \sec ^{\frac{5}{2}}(c+d x) (a+b \sec (c+d x))^2 \, dx","Integrate[Sec[c + d*x]^(5/2)*(a + b*Sec[c + d*x])^2,x]","\frac{\sec ^{\frac{7}{2}}(c+d x) \left(20 \left(7 a^2+5 b^2\right) \cos ^{\frac{7}{2}}(c+d x) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+2 \sin (c+d x) \left(5 \left(7 a^2+5 b^2\right) \cos (2 (c+d x))+35 a^2+273 a b \cos (c+d x)+63 a b \cos (3 (c+d x))+55 b^2\right)-504 a b \cos ^{\frac{7}{2}}(c+d x) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{210 d}","\frac{2 \left(7 a^2+5 b^2\right) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{21 d}+\frac{2 \left(7 a^2+5 b^2\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{4 a b \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{5 d}+\frac{12 a b \sin (c+d x) \sqrt{\sec (c+d x)}}{5 d}-\frac{12 a b \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 b^2 \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x)}{7 d}",1,"(Sec[c + d*x]^(7/2)*(-504*a*b*Cos[c + d*x]^(7/2)*EllipticE[(c + d*x)/2, 2] + 20*(7*a^2 + 5*b^2)*Cos[c + d*x]^(7/2)*EllipticF[(c + d*x)/2, 2] + 2*(35*a^2 + 55*b^2 + 273*a*b*Cos[c + d*x] + 5*(7*a^2 + 5*b^2)*Cos[2*(c + d*x)] + 63*a*b*Cos[3*(c + d*x)])*Sin[c + d*x]))/(210*d)","A",1
587,1,126,175,1.3999076,"\int \sec ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^2 \, dx","Integrate[Sec[c + d*x]^(3/2)*(a + b*Sec[c + d*x])^2,x]","\frac{\sec ^{\frac{5}{2}}(c+d x) \left(-12 \left(5 a^2+3 b^2\right) \cos ^{\frac{5}{2}}(c+d x) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+2 \sin (c+d x) \left(3 \left(5 a^2+3 b^2\right) \cos (2 (c+d x))+15 \left(a^2+b^2\right)+20 a b \cos (c+d x)\right)+40 a b \cos ^{\frac{5}{2}}(c+d x) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{30 d}","\frac{2 \left(5 a^2+3 b^2\right) \sin (c+d x) \sqrt{\sec (c+d x)}}{5 d}-\frac{2 \left(5 a^2+3 b^2\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{4 a b \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 d}+\frac{4 a b \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{2 b^2 \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{5 d}",1,"(Sec[c + d*x]^(5/2)*(-12*(5*a^2 + 3*b^2)*Cos[c + d*x]^(5/2)*EllipticE[(c + d*x)/2, 2] + 40*a*b*Cos[c + d*x]^(5/2)*EllipticF[(c + d*x)/2, 2] + 2*(15*(a^2 + b^2) + 20*a*b*Cos[c + d*x] + 3*(5*a^2 + 3*b^2)*Cos[2*(c + d*x)])*Sin[c + d*x]))/(30*d)","A",1
588,1,93,135,0.3615879,"\int \sqrt{\sec (c+d x)} (a+b \sec (c+d x))^2 \, dx","Integrate[Sqrt[Sec[c + d*x]]*(a + b*Sec[c + d*x])^2,x]","\frac{2 \sec ^{\frac{3}{2}}(c+d x) \left(\left(3 a^2+b^2\right) \cos ^{\frac{3}{2}}(c+d x) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-6 a b \cos ^{\frac{3}{2}}(c+d x) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+b \sin (c+d x) (6 a \cos (c+d x)+b)\right)}{3 d}","\frac{2 \left(3 a^2+b^2\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{4 a b \sin (c+d x) \sqrt{\sec (c+d x)}}{d}-\frac{4 a b \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 b^2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 d}",1,"(2*Sec[c + d*x]^(3/2)*(-6*a*b*Cos[c + d*x]^(3/2)*EllipticE[(c + d*x)/2, 2] + (3*a^2 + b^2)*Cos[c + d*x]^(3/2)*EllipticF[(c + d*x)/2, 2] + b*(b + 6*a*Cos[c + d*x])*Sin[c + d*x]))/(3*d)","A",1
589,1,82,108,0.2044359,"\int \frac{(a+b \sec (c+d x))^2}{\sqrt{\sec (c+d x)}} \, dx","Integrate[(a + b*Sec[c + d*x])^2/Sqrt[Sec[c + d*x]],x]","\frac{2 \sqrt{\sec (c+d x)} \left(\left(a^2-b^2\right) \sqrt{\cos (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+b \left(2 a \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+b \sin (c+d x)\right)\right)}{d}","\frac{2 \left(a^2-b^2\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{4 a b \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 b^2 \sin (c+d x) \sqrt{\sec (c+d x)}}{d}",1,"(2*Sqrt[Sec[c + d*x]]*((a^2 - b^2)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2] + b*(2*a*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + b*Sin[c + d*x])))/d","A",1
590,1,87,112,0.206221,"\int \frac{(a+b \sec (c+d x))^2}{\sec ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[(a + b*Sec[c + d*x])^2/Sec[c + d*x]^(3/2),x]","\frac{\sqrt{\sec (c+d x)} \left(2 \left(a^2+3 b^2\right) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+a^2 \sin (2 (c+d x))+12 a b \sqrt{\cos (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{3 d}","\frac{2 \left(a^2+3 b^2\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{2 a^2 \sin (c+d x)}{3 d \sqrt{\sec (c+d x)}}+\frac{4 a b \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}",1,"(Sqrt[Sec[c + d*x]]*(12*a*b*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2] + 2*(a^2 + 3*b^2)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + a^2*Sin[2*(c + d*x)]))/(3*d)","A",1
591,1,100,141,0.4815783,"\int \frac{(a+b \sec (c+d x))^2}{\sec ^{\frac{5}{2}}(c+d x)} \, dx","Integrate[(a + b*Sec[c + d*x])^2/Sec[c + d*x]^(5/2),x]","\frac{\sqrt{\sec (c+d x)} \left(6 \left(3 a^2+5 b^2\right) \sqrt{\cos (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+a \sin (2 (c+d x)) (3 a \cos (c+d x)+10 b)+20 a b \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{15 d}","\frac{2 \left(3 a^2+5 b^2\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 a^2 \sin (c+d x)}{5 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{4 a b \sin (c+d x)}{3 d \sqrt{\sec (c+d x)}}+\frac{4 a b \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}",1,"(Sqrt[Sec[c + d*x]]*(6*(3*a^2 + 5*b^2)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2] + 20*a*b*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + a*(10*b + 3*a*Cos[c + d*x])*Sin[2*(c + d*x)]))/(15*d)","A",1
592,1,120,175,0.8146563,"\int \frac{(a+b \sec (c+d x))^2}{\sec ^{\frac{7}{2}}(c+d x)} \, dx","Integrate[(a + b*Sec[c + d*x])^2/Sec[c + d*x]^(7/2),x]","\frac{\sqrt{\sec (c+d x)} \left(\sin (2 (c+d x)) \left(15 a^2 \cos (2 (c+d x))+65 a^2+84 a b \cos (c+d x)+70 b^2\right)+20 \left(5 a^2+7 b^2\right) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+504 a b \sqrt{\cos (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{210 d}","\frac{2 \left(5 a^2+7 b^2\right) \sin (c+d x)}{21 d \sqrt{\sec (c+d x)}}+\frac{2 \left(5 a^2+7 b^2\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{2 a^2 \sin (c+d x)}{7 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{4 a b \sin (c+d x)}{5 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{12 a b \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}",1,"(Sqrt[Sec[c + d*x]]*(504*a*b*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2] + 20*(5*a^2 + 7*b^2)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + (65*a^2 + 70*b^2 + 84*a*b*Cos[c + d*x] + 15*a^2*Cos[2*(c + d*x)])*Sin[2*(c + d*x)]))/(210*d)","A",1
593,1,177,234,3.741393,"\int \sec ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^3 \, dx","Integrate[Sec[c + d*x]^(3/2)*(a + b*Sec[c + d*x])^3,x]","\frac{\sec ^{\frac{7}{2}}(c+d x) \left(40 b \left(21 a^2+5 b^2\right) \cos ^{\frac{7}{2}}(c+d x) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-168 a \left(5 a^2+9 b^2\right) \cos ^{\frac{7}{2}}(c+d x) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+2 \sin (c+d x) \left(105 a^3 \cos (3 (c+d x))+10 \left(21 a^2 b+5 b^3\right) \cos (2 (c+d x))+63 a \left(5 a^2+13 b^2\right) \cos (c+d x)+210 a^2 b+189 a b^2 \cos (3 (c+d x))+110 b^3\right)\right)}{420 d}","\frac{2 b \left(21 a^2+5 b^2\right) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{21 d}+\frac{2 a \left(5 a^2+9 b^2\right) \sin (c+d x) \sqrt{\sec (c+d x)}}{5 d}+\frac{2 b \left(21 a^2+5 b^2\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}-\frac{2 a \left(5 a^2+9 b^2\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 b^2 \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) (a+b \sec (c+d x))}{7 d}+\frac{32 a b^2 \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{35 d}",1,"(Sec[c + d*x]^(7/2)*(-168*a*(5*a^2 + 9*b^2)*Cos[c + d*x]^(7/2)*EllipticE[(c + d*x)/2, 2] + 40*b*(21*a^2 + 5*b^2)*Cos[c + d*x]^(7/2)*EllipticF[(c + d*x)/2, 2] + 2*(210*a^2*b + 110*b^3 + 63*a*(5*a^2 + 13*b^2)*Cos[c + d*x] + 10*(21*a^2*b + 5*b^3)*Cos[2*(c + d*x)] + 105*a^3*Cos[3*(c + d*x)] + 189*a*b^2*Cos[3*(c + d*x)])*Sin[c + d*x]))/(420*d)","A",1
594,1,134,189,1.6541134,"\int \sqrt{\sec (c+d x)} (a+b \sec (c+d x))^3 \, dx","Integrate[Sqrt[Sec[c + d*x]]*(a + b*Sec[c + d*x])^3,x]","\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \left(5 a \left(a^2+b^2\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-3 b \left(5 a^2+b^2\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+\frac{b \sin (c+d x) \left(3 \left(5 a^2+b^2\right) \cos (2 (c+d x))+5 \left(3 a^2+b^2\right)+10 a b \cos (c+d x)\right)}{2 \cos ^{\frac{5}{2}}(c+d x)}\right)}{5 d}","\frac{6 b \left(5 a^2+b^2\right) \sin (c+d x) \sqrt{\sec (c+d x)}}{5 d}+\frac{2 a \left(a^2+b^2\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}-\frac{6 b \left(5 a^2+b^2\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{8 a b^2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{5 d}+\frac{2 b^2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))}{5 d}",1,"(2*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*(-3*b*(5*a^2 + b^2)*EllipticE[(c + d*x)/2, 2] + 5*a*(a^2 + b^2)*EllipticF[(c + d*x)/2, 2] + (b*(5*(3*a^2 + b^2) + 10*a*b*Cos[c + d*x] + 3*(5*a^2 + b^2)*Cos[2*(c + d*x)])*Sin[c + d*x])/(2*Cos[c + d*x]^(5/2))))/(5*d)","A",1
595,1,106,158,0.5400055,"\int \frac{(a+b \sec (c+d x))^3}{\sqrt{\sec (c+d x)}} \, dx","Integrate[(a + b*Sec[c + d*x])^3/Sqrt[Sec[c + d*x]],x]","\frac{\sec ^{\frac{3}{2}}(c+d x) \left(6 a \left(a^2-3 b^2\right) \cos ^{\frac{3}{2}}(c+d x) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+b \left(2 \left(9 a^2+b^2\right) \cos ^{\frac{3}{2}}(c+d x) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+2 b \sin (c+d x) (9 a \cos (c+d x)+b)\right)\right)}{3 d}","\frac{2 b \left(9 a^2+b^2\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{2 a \left(a^2-3 b^2\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 b^2 \sin (c+d x) \sqrt{\sec (c+d x)} (a+b \sec (c+d x))}{3 d}+\frac{16 a b^2 \sin (c+d x) \sqrt{\sec (c+d x)}}{3 d}",1,"(Sec[c + d*x]^(3/2)*(6*a*(a^2 - 3*b^2)*Cos[c + d*x]^(3/2)*EllipticE[(c + d*x)/2, 2] + b*(2*(9*a^2 + b^2)*Cos[c + d*x]^(3/2)*EllipticF[(c + d*x)/2, 2] + 2*b*(b + 9*a*Cos[c + d*x])*Sin[c + d*x])))/(3*d)","A",1
596,1,108,166,0.6259966,"\int \frac{(a+b \sec (c+d x))^3}{\sec ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[(a + b*Sec[c + d*x])^3/Sec[c + d*x]^(3/2),x]","\frac{\sqrt{\sec (c+d x)} \left(2 \sin (c+d x) \left(a^3 \cos (c+d x)+3 b^3\right)+2 a \left(a^2+9 b^2\right) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-6 b \left(b^2-3 a^2\right) \sqrt{\cos (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{3 d}","-\frac{2 b \left(a^2-3 b^2\right) \sin (c+d x) \sqrt{\sec (c+d x)}}{3 d}+\frac{2 a \left(a^2+9 b^2\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{2 b \left(3 a^2-b^2\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 a^2 \sin (c+d x) (a+b \sec (c+d x))}{3 d \sqrt{\sec (c+d x)}}",1,"(Sqrt[Sec[c + d*x]]*(-6*b*(-3*a^2 + b^2)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2] + 2*a*(a^2 + 9*b^2)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + 2*(3*b^3 + a^3*Cos[c + d*x])*Sin[c + d*x]))/(3*d)","A",1
597,1,106,156,0.4993029,"\int \frac{(a+b \sec (c+d x))^3}{\sec ^{\frac{5}{2}}(c+d x)} \, dx","Integrate[(a + b*Sec[c + d*x])^3/Sec[c + d*x]^(5/2),x]","\frac{\sqrt{\sec (c+d x)} \left(10 b \left(a^2+b^2\right) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+6 a \left(a^2+5 b^2\right) \sqrt{\cos (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+a^2 \sin (2 (c+d x)) (a \cos (c+d x)+5 b)\right)}{5 d}","\frac{2 b \left(a^2+b^2\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{6 a \left(a^2+5 b^2\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 a^2 \sin (c+d x) (a+b \sec (c+d x))}{5 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{8 a^2 b \sin (c+d x)}{5 d \sqrt{\sec (c+d x)}}",1,"(Sqrt[Sec[c + d*x]]*(6*a*(a^2 + 5*b^2)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2] + 10*b*(a^2 + b^2)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + a^2*(5*b + a*Cos[c + d*x])*Sin[2*(c + d*x)]))/(5*d)","A",1
598,1,132,199,1.07356,"\int \frac{(a+b \sec (c+d x))^3}{\sec ^{\frac{7}{2}}(c+d x)} \, dx","Integrate[(a + b*Sec[c + d*x])^3/Sec[c + d*x]^(7/2),x]","\frac{\sqrt{\sec (c+d x)} \left(a \sin (2 (c+d x)) \left(15 a^2 \cos (2 (c+d x))+65 a^2+126 a b \cos (c+d x)+210 b^2\right)+20 a \left(5 a^2+21 b^2\right) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+84 b \left(9 a^2+5 b^2\right) \sqrt{\cos (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{210 d}","\frac{2 a \left(5 a^2+21 b^2\right) \sin (c+d x)}{21 d \sqrt{\sec (c+d x)}}+\frac{2 a \left(5 a^2+21 b^2\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{2 b \left(9 a^2+5 b^2\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 a^2 \sin (c+d x) (a+b \sec (c+d x))}{7 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{32 a^2 b \sin (c+d x)}{35 d \sec ^{\frac{3}{2}}(c+d x)}",1,"(Sqrt[Sec[c + d*x]]*(84*b*(9*a^2 + 5*b^2)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2] + 20*a*(5*a^2 + 21*b^2)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + a*(65*a^2 + 210*b^2 + 126*a*b*Cos[c + d*x] + 15*a^2*Cos[2*(c + d*x)])*Sin[2*(c + d*x)]))/(210*d)","A",1
599,1,159,234,1.4415277,"\int \frac{(a+b \sec (c+d x))^3}{\sec ^{\frac{9}{2}}(c+d x)} \, dx","Integrate[(a + b*Sec[c + d*x])^3/Sec[c + d*x]^(9/2),x]","\frac{\sqrt{\sec (c+d x)} \left(120 b \left(15 a^2+7 b^2\right) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+168 a \left(7 a^2+27 b^2\right) \sqrt{\cos (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+\sin (2 (c+d x)) \left(7 a \left(43 a^2+108 b^2\right) \cos (c+d x)+5 \left(7 a^3 \cos (3 (c+d x))+54 a^2 b \cos (2 (c+d x))+234 a^2 b+84 b^3\right)\right)\right)}{1260 d}","\frac{2 a \left(7 a^2+27 b^2\right) \sin (c+d x)}{45 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 b \left(15 a^2+7 b^2\right) \sin (c+d x)}{21 d \sqrt{\sec (c+d x)}}+\frac{2 b \left(15 a^2+7 b^2\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{2 a \left(7 a^2+27 b^2\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{2 a^2 \sin (c+d x) (a+b \sec (c+d x))}{9 d \sec ^{\frac{7}{2}}(c+d x)}+\frac{40 a^2 b \sin (c+d x)}{63 d \sec ^{\frac{5}{2}}(c+d x)}",1,"(Sqrt[Sec[c + d*x]]*(168*a*(7*a^2 + 27*b^2)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2] + 120*b*(15*a^2 + 7*b^2)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + (7*a*(43*a^2 + 108*b^2)*Cos[c + d*x] + 5*(234*a^2*b + 84*b^3 + 54*a^2*b*Cos[2*(c + d*x)] + 7*a^3*Cos[3*(c + d*x)]))*Sin[2*(c + d*x)]))/(1260*d)","A",1
600,1,256,287,2.4300648,"\int \sec ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^4 \, dx","Integrate[Sec[c + d*x]^(3/2)*(a + b*Sec[c + d*x])^4,x]","-\frac{2 (a+b \sec (c+d x))^4 \left(-315 a^4 \sin (c+d x)-420 a^3 b \tan (c+d x)-1134 a^2 b^2 \sin (c+d x)-60 a b \left(7 a^2+5 b^2\right) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-378 a^2 b^2 \tan (c+d x) \sec (c+d x)+21 \left(15 a^4+54 a^2 b^2+7 b^4\right) \sqrt{\cos (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)-300 a b^3 \tan (c+d x)-180 a b^3 \tan (c+d x) \sec ^2(c+d x)-147 b^4 \sin (c+d x)-35 b^4 \tan (c+d x) \sec ^3(c+d x)-49 b^4 \tan (c+d x) \sec (c+d x)\right)}{315 d \sec ^{\frac{7}{2}}(c+d x) (a \cos (c+d x)+b)^4}","\frac{14 b^2 \left(7 a^2+b^2\right) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{45 d}+\frac{8 a b \left(7 a^2+5 b^2\right) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{21 d}+\frac{8 a b \left(7 a^2+5 b^2\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{2 \left(15 a^4+54 a^2 b^2+7 b^4\right) \sin (c+d x) \sqrt{\sec (c+d x)}}{15 d}-\frac{2 \left(15 a^4+54 a^2 b^2+7 b^4\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{44 a b^3 \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x)}{63 d}+\frac{2 b^2 \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) (a+b \sec (c+d x))^2}{9 d}",1,"(-2*(a + b*Sec[c + d*x])^4*(21*(15*a^4 + 54*a^2*b^2 + 7*b^4)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2] - 60*a*b*(7*a^2 + 5*b^2)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] - 315*a^4*Sin[c + d*x] - 1134*a^2*b^2*Sin[c + d*x] - 147*b^4*Sin[c + d*x] - 420*a^3*b*Tan[c + d*x] - 300*a*b^3*Tan[c + d*x] - 378*a^2*b^2*Sec[c + d*x]*Tan[c + d*x] - 49*b^4*Sec[c + d*x]*Tan[c + d*x] - 180*a*b^3*Sec[c + d*x]^2*Tan[c + d*x] - 35*b^4*Sec[c + d*x]^3*Tan[c + d*x]))/(315*d*(b + a*Cos[c + d*x])^4*Sec[c + d*x]^(7/2))","A",1
601,1,168,247,1.7151592,"\int \sqrt{\sec (c+d x)} (a+b \sec (c+d x))^4 \, dx","Integrate[Sqrt[Sec[c + d*x]]*(a + b*Sec[c + d*x])^4,x]","\frac{2 \sec ^{\frac{7}{2}}(c+d x) \left(-84 a b \left(5 a^2+3 b^2\right) \cos ^{\frac{7}{2}}(c+d x) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+b \sin (c+d x) \left(84 a \left(5 a^2+3 b^2\right) \cos ^3(c+d x)+5 b \left(42 a^2+5 b^2\right) \cos ^2(c+d x)+15 b^3\right)+5 \left(21 a^4+42 a^2 b^2+5 b^4\right) \cos ^{\frac{7}{2}}(c+d x) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+42 a b^3 \sin (2 (c+d x))\right)}{105 d}","\frac{2 b^2 \left(39 a^2+5 b^2\right) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{21 d}+\frac{8 a b \left(5 a^2+3 b^2\right) \sin (c+d x) \sqrt{\sec (c+d x)}}{5 d}-\frac{8 a b \left(5 a^2+3 b^2\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 \left(21 a^4+42 a^2 b^2+5 b^4\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{36 a b^3 \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{35 d}+\frac{2 b^2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^2}{7 d}",1,"(2*Sec[c + d*x]^(7/2)*(-84*a*b*(5*a^2 + 3*b^2)*Cos[c + d*x]^(7/2)*EllipticE[(c + d*x)/2, 2] + 5*(21*a^4 + 42*a^2*b^2 + 5*b^4)*Cos[c + d*x]^(7/2)*EllipticF[(c + d*x)/2, 2] + b*(15*b^3 + 5*b*(42*a^2 + 5*b^2)*Cos[c + d*x]^2 + 84*a*(5*a^2 + 3*b^2)*Cos[c + d*x]^3)*Sin[c + d*x] + 42*a*b^3*Sin[2*(c + d*x)]))/(105*d)","A",1
602,1,146,209,2.384162,"\int \frac{(a+b \sec (c+d x))^4}{\sqrt{\sec (c+d x)}} \, dx","Integrate[(a + b*Sec[c + d*x])^4/Sqrt[Sec[c + d*x]],x]","\frac{\sec ^{\frac{5}{2}}(c+d x) \left(b \left(80 a \left(3 a^2+b^2\right) \cos ^{\frac{5}{2}}(c+d x) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+2 b \sin (c+d x) \left(9 \left(10 a^2+b^2\right) \cos (2 (c+d x))+15 \left(6 a^2+b^2\right)+40 a b \cos (c+d x)\right)\right)+12 \left(5 a^4-30 a^2 b^2-3 b^4\right) \cos ^{\frac{5}{2}}(c+d x) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{30 d}","\frac{2 b^2 \left(29 a^2+3 b^2\right) \sin (c+d x) \sqrt{\sec (c+d x)}}{5 d}+\frac{8 a b \left(3 a^2+b^2\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{2 \left(5 a^4-30 a^2 b^2-3 b^4\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{28 a b^3 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{15 d}+\frac{2 b^2 \sin (c+d x) \sqrt{\sec (c+d x)} (a+b \sec (c+d x))^2}{5 d}",1,"(Sec[c + d*x]^(5/2)*(12*(5*a^4 - 30*a^2*b^2 - 3*b^4)*Cos[c + d*x]^(5/2)*EllipticE[(c + d*x)/2, 2] + b*(80*a*(3*a^2 + b^2)*Cos[c + d*x]^(5/2)*EllipticF[(c + d*x)/2, 2] + 2*b*(15*(6*a^2 + b^2) + 40*a*b*Cos[c + d*x] + 9*(10*a^2 + b^2)*Cos[2*(c + d*x)])*Sin[c + d*x])))/(30*d)","A",1
603,1,130,208,1.2359567,"\int \frac{(a+b \sec (c+d x))^4}{\sec ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[(a + b*Sec[c + d*x])^4/Sec[c + d*x]^(3/2),x]","\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \left(\frac{\sin (c+d x) \left(a^4 \cos (2 (c+d x))+a^4+24 a b^3 \cos (c+d x)+2 b^4\right)}{\cos ^{\frac{3}{2}}(c+d x)}+24 a b \left(a^2-b^2\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+2 \left(a^4+18 a^2 b^2+b^4\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{3 d}","-\frac{2 b^2 \left(a^2-b^2\right) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 d}-\frac{4 a b \left(a^2-6 b^2\right) \sin (c+d x) \sqrt{\sec (c+d x)}}{3 d}+\frac{8 a b \left(a^2-b^2\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 a^2 \sin (c+d x) (a+b \sec (c+d x))^2}{3 d \sqrt{\sec (c+d x)}}+\frac{2 \left(a^4+18 a^2 b^2+b^4\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}",1,"(Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*(24*a*b*(a^2 - b^2)*EllipticE[(c + d*x)/2, 2] + 2*(a^4 + 18*a^2*b^2 + b^4)*EllipticF[(c + d*x)/2, 2] + ((a^4 + 2*b^4 + 24*a*b^3*Cos[c + d*x] + a^4*Cos[2*(c + d*x)])*Sin[c + d*x])/Cos[c + d*x]^(3/2)))/(3*d)","A",1
604,1,138,207,0.6661355,"\int \frac{(a+b \sec (c+d x))^4}{\sec ^{\frac{5}{2}}(c+d x)} \, dx","Integrate[(a + b*Sec[c + d*x])^4/Sec[c + d*x]^(5/2),x]","\frac{\sqrt{\sec (c+d x)} \left(80 a b \left(a^2+3 b^2\right) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+2 \sin (c+d x) \left(3 a^4 \cos (2 (c+d x))+3 a^4+40 a^3 b \cos (c+d x)+30 b^4\right)+12 \left(3 a^4+30 a^2 b^2-5 b^4\right) \sqrt{\cos (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{30 d}","\frac{28 a^3 b \sin (c+d x)}{15 d \sqrt{\sec (c+d x)}}-\frac{2 b^2 \left(a^2-5 b^2\right) \sin (c+d x) \sqrt{\sec (c+d x)}}{5 d}+\frac{8 a b \left(a^2+3 b^2\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{2 a^2 \sin (c+d x) (a+b \sec (c+d x))^2}{5 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 \left(3 a^4+30 a^2 b^2-5 b^4\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}",1,"(Sqrt[Sec[c + d*x]]*(12*(3*a^4 + 30*a^2*b^2 - 5*b^4)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2] + 80*a*b*(a^2 + 3*b^2)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + 2*(3*a^4 + 30*b^4 + 40*a^3*b*Cos[c + d*x] + 3*a^4*Cos[2*(c + d*x)])*Sin[c + d*x]))/(30*d)","A",1
605,1,142,211,0.9351857,"\int \frac{(a+b \sec (c+d x))^4}{\sec ^{\frac{7}{2}}(c+d x)} \, dx","Integrate[(a + b*Sec[c + d*x])^4/Sec[c + d*x]^(7/2),x]","\frac{\sqrt{\sec (c+d x)} \left(a^2 \sin (2 (c+d x)) \left(15 a^2 \cos (2 (c+d x))+65 a^2+168 a b \cos (c+d x)+420 b^2\right)+336 a b \left(3 a^2+5 b^2\right) \sqrt{\cos (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+20 \left(5 a^4+42 a^2 b^2+21 b^4\right) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{210 d}","\frac{36 a^3 b \sin (c+d x)}{35 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 a^2 \left(5 a^2+39 b^2\right) \sin (c+d x)}{21 d \sqrt{\sec (c+d x)}}+\frac{8 a b \left(3 a^2+5 b^2\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 a^2 \sin (c+d x) (a+b \sec (c+d x))^2}{7 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 \left(5 a^4+42 a^2 b^2+21 b^4\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}",1,"(Sqrt[Sec[c + d*x]]*(336*a*b*(3*a^2 + 5*b^2)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2] + 20*(5*a^4 + 42*a^2*b^2 + 21*b^4)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + a^2*(65*a^2 + 420*b^2 + 168*a*b*Cos[c + d*x] + 15*a^2*Cos[2*(c + d*x)])*Sin[2*(c + d*x)]))/(210*d)","A",1
606,1,168,245,1.6935028,"\int \frac{(a+b \sec (c+d x))^4}{\sec ^{\frac{9}{2}}(c+d x)} \, dx","Integrate[(a + b*Sec[c + d*x])^4/Sec[c + d*x]^(9/2),x]","\frac{\sqrt{\sec (c+d x)} \left(480 a b \left(5 a^2+7 b^2\right) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+168 \left(7 a^4+54 a^2 b^2+15 b^4\right) \sqrt{\cos (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+a \sin (2 (c+d x)) \left(7 a \left(43 a^2+216 b^2\right) \cos (c+d x)+5 \left(7 a^3 \cos (3 (c+d x))+72 a^2 b \cos (2 (c+d x))+312 a^2 b+336 b^3\right)\right)\right)}{1260 d}","\frac{44 a^3 b \sin (c+d x)}{63 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{14 a^2 \left(a^2+7 b^2\right) \sin (c+d x)}{45 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{8 a b \left(5 a^2+7 b^2\right) \sin (c+d x)}{21 d \sqrt{\sec (c+d x)}}+\frac{8 a b \left(5 a^2+7 b^2\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{2 a^2 \sin (c+d x) (a+b \sec (c+d x))^2}{9 d \sec ^{\frac{7}{2}}(c+d x)}+\frac{2 \left(7 a^4+54 a^2 b^2+15 b^4\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}",1,"(Sqrt[Sec[c + d*x]]*(168*(7*a^4 + 54*a^2*b^2 + 15*b^4)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2] + 480*a*b*(5*a^2 + 7*b^2)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + a*(7*a*(43*a^2 + 216*b^2)*Cos[c + d*x] + 5*(312*a^2*b + 336*b^3 + 72*a^2*b*Cos[2*(c + d*x)] + 7*a^3*Cos[3*(c + d*x)]))*Sin[2*(c + d*x)]))/(1260*d)","A",1
607,1,199,289,2.1149047,"\int \frac{(a+b \sec (c+d x))^4}{\sec ^{\frac{11}{2}}(c+d x)} \, dx","Integrate[(a + b*Sec[c + d*x])^4/Sec[c + d*x]^(11/2),x]","\frac{\sqrt{\sec (c+d x)} \left(14784 a b \left(7 a^2+9 b^2\right) \sqrt{\cos (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+240 \left(45 a^4+330 a^2 b^2+77 b^4\right) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+\sin (2 (c+d x)) \left(616 a b \left(43 a^2+36 b^2\right) \cos (c+d x)+5 \left(63 a^4 \cos (4 (c+d x))+1593 a^4+616 a^3 b \cos (3 (c+d x))+10296 a^2 b^2+72 \left(8 a^4+33 a^2 b^2\right) \cos (2 (c+d x))+1848 b^4\right)\right)\right)}{27720 d}","\frac{52 a^3 b \sin (c+d x)}{99 d \sec ^{\frac{7}{2}}(c+d x)}+\frac{2 a^2 \left(9 a^2+59 b^2\right) \sin (c+d x)}{77 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{8 a b \left(7 a^2+9 b^2\right) \sin (c+d x)}{45 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{8 a b \left(7 a^2+9 b^2\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{2 a^2 \sin (c+d x) (a+b \sec (c+d x))^2}{11 d \sec ^{\frac{9}{2}}(c+d x)}+\frac{2 \left(45 a^4+330 a^2 b^2+77 b^4\right) \sin (c+d x)}{231 d \sqrt{\sec (c+d x)}}+\frac{2 \left(45 a^4+330 a^2 b^2+77 b^4\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{231 d}",1,"(Sqrt[Sec[c + d*x]]*(14784*a*b*(7*a^2 + 9*b^2)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2] + 240*(45*a^4 + 330*a^2*b^2 + 77*b^4)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + (616*a*b*(43*a^2 + 36*b^2)*Cos[c + d*x] + 5*(1593*a^4 + 10296*a^2*b^2 + 1848*b^4 + 72*(8*a^4 + 33*a^2*b^2)*Cos[2*(c + d*x)] + 616*a^3*b*Cos[3*(c + d*x)] + 63*a^4*Cos[4*(c + d*x)]))*Sin[2*(c + d*x)]))/(27720*d)","A",1
608,1,165,188,3.2500202,"\int \frac{\sec ^{\frac{7}{2}}(c+d x)}{a+b \sec (c+d x)} \, dx","Integrate[Sec[c + d*x]^(7/2)/(a + b*Sec[c + d*x]),x]","-\frac{\cot (c+d x) \left(-2 \left(3 a^2+3 a b+b^2\right) \sqrt{-\tan ^2(c+d x)} F\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)+6 a^2 \sqrt{-\tan ^2(c+d x)} \Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)+6 a b \sqrt{-\tan ^2(c+d x)} E\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)-b^2 \sec ^{\frac{5}{2}}(c+d x)+b^2 \cos (2 (c+d x)) \sec ^{\frac{5}{2}}(c+d x)\right)}{3 b^3 d}","\frac{2 a^2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{b^2 d (a+b)}-\frac{2 a \sin (c+d x) \sqrt{\sec (c+d x)}}{b^2 d}+\frac{2 a \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b^2 d}+\frac{2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 b d}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 b d}",1,"-1/3*(Cot[c + d*x]*(-(b^2*Sec[c + d*x]^(5/2)) + b^2*Cos[2*(c + d*x)]*Sec[c + d*x]^(5/2) + 6*a*b*EllipticE[ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[-Tan[c + d*x]^2] - 2*(3*a^2 + 3*a*b + b^2)*EllipticF[ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[-Tan[c + d*x]^2] + 6*a^2*EllipticPi[-(b/a), ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[-Tan[c + d*x]^2]))/(b^3*d)","A",1
609,1,83,117,5.2268769,"\int \frac{\sec ^{\frac{5}{2}}(c+d x)}{a+b \sec (c+d x)} \, dx","Integrate[Sec[c + d*x]^(5/2)/(a + b*Sec[c + d*x]),x]","\frac{2 \sqrt{-\tan ^2(c+d x)} \cot (c+d x) \left(-(a+b) F\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)+a \Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)+b E\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)\right)}{b^2 d}","-\frac{2 a \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{b d (a+b)}+\frac{2 \sin (c+d x) \sqrt{\sec (c+d x)}}{b d}-\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b d}",1,"(2*Cot[c + d*x]*(b*EllipticE[ArcSin[Sqrt[Sec[c + d*x]]], -1] - (a + b)*EllipticF[ArcSin[Sqrt[Sec[c + d*x]]], -1] + a*EllipticPi[-(b/a), ArcSin[Sqrt[Sec[c + d*x]]], -1])*Sqrt[-Tan[c + d*x]^2])/(b^2*d)","A",1
610,1,63,49,0.3727405,"\int \frac{\sec ^{\frac{3}{2}}(c+d x)}{a+b \sec (c+d x)} \, dx","Integrate[Sec[c + d*x]^(3/2)/(a + b*Sec[c + d*x]),x]","\frac{2 \sqrt{-\tan ^2(c+d x)} \cot (c+d x) \left(F\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)-\Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)\right)}{b d}","\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{d (a+b)}",1,"(2*Cot[c + d*x]*(EllipticF[ArcSin[Sqrt[Sec[c + d*x]]], -1] - EllipticPi[-(b/a), ArcSin[Sqrt[Sec[c + d*x]]], -1])*Sqrt[-Tan[c + d*x]^2])/(b*d)","A",1
611,1,47,93,0.2775366,"\int \frac{\sqrt{\sec (c+d x)}}{a+b \sec (c+d x)} \, dx","Integrate[Sqrt[Sec[c + d*x]]/(a + b*Sec[c + d*x]),x]","\frac{2 \sqrt{-\tan ^2(c+d x)} \cot (c+d x) \Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)}{a d}","\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}-\frac{2 b \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a d (a+b)}",1,"(2*Cot[c + d*x]*EllipticPi[-(b/a), ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[-Tan[c + d*x]^2])/(a*d)","A",1
612,1,176,135,7.1976158,"\int \frac{1}{\sqrt{\sec (c+d x)} (a+b \sec (c+d x))} \, dx","Integrate[1/(Sqrt[Sec[c + d*x]]*(a + b*Sec[c + d*x])),x]","\frac{\cot (c+d x) \left(-2 b \sqrt{-\tan ^2(c+d x)} \Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)+a \sec ^{\frac{7}{2}}(c+d x)-a \sec ^{\frac{3}{2}}(c+d x)+a \cos (2 (c+d x)) \sec ^{\frac{7}{2}}(c+d x)-a \cos (2 (c+d x)) \sec ^{\frac{3}{2}}(c+d x)+2 a \sqrt{-\tan ^2(c+d x)} F\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)-2 a \sqrt{-\tan ^2(c+d x)} E\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)\right)}{a^2 d}","\frac{2 b^2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d (a+b)}-\frac{2 b \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}",1,"(Cot[c + d*x]*(-(a*Sec[c + d*x]^(3/2)) - a*Cos[2*(c + d*x)]*Sec[c + d*x]^(3/2) + a*Sec[c + d*x]^(7/2) + a*Cos[2*(c + d*x)]*Sec[c + d*x]^(7/2) - 2*a*EllipticE[ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[-Tan[c + d*x]^2] + 2*a*EllipticF[ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[-Tan[c + d*x]^2] - 2*b*EllipticPi[-(b/a), ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[-Tan[c + d*x]^2]))/(a^2*d)","A",1
613,1,194,172,7.1332717,"\int \frac{1}{\sec ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))} \, dx","Integrate[1/(Sec[c + d*x]^(3/2)*(a + b*Sec[c + d*x])),x]","-\frac{\cot (c+d x) \left(-a^2 \sqrt{\sec (c+d x)}+a^2 \cos (3 (c+d x)) \sec ^{\frac{3}{2}}(c+d x)-12 b^2 \sqrt{-\tan ^2(c+d x)} \Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)+6 a b \sec ^{\frac{3}{2}}(c+d x)-6 a b \cos (2 (c+d x)) \sec ^{\frac{3}{2}}(c+d x)-4 a (a-3 b) \sqrt{-\tan ^2(c+d x)} F\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)-12 a b \sqrt{-\tan ^2(c+d x)} E\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)\right)}{6 a^3 d}","-\frac{2 b^3 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a^3 d (a+b)}-\frac{2 b \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d}+\frac{2 \left(a^2+3 b^2\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^3 d}+\frac{2 \sin (c+d x)}{3 a d \sqrt{\sec (c+d x)}}",1,"-1/6*(Cot[c + d*x]*(-(a^2*Sqrt[Sec[c + d*x]]) + 6*a*b*Sec[c + d*x]^(3/2) - 6*a*b*Cos[2*(c + d*x)]*Sec[c + d*x]^(3/2) + a^2*Cos[3*(c + d*x)]*Sec[c + d*x]^(3/2) - 12*a*b*EllipticE[ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[-Tan[c + d*x]^2] - 4*a*(a - 3*b)*EllipticF[ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[-Tan[c + d*x]^2] - 12*b^2*EllipticPi[-(b/a), ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[-Tan[c + d*x]^2]))/(a^3*d)","A",1
614,1,653,342,6.8666492,"\int \frac{\sec ^{\frac{9}{2}}(c+d x)}{(a+b \sec (c+d x))^2} \, dx","Integrate[Sec[c + d*x]^(9/2)/(a + b*Sec[c + d*x])^2,x]","\frac{\sqrt{\sec (c+d x)} \left(-\frac{a \left(4 b^2-5 a^2\right) \sin (c+d x)}{b^3 \left(b^2-a^2\right)}-\frac{a^3 \sin (c+d x)}{b^2 \left(b^2-a^2\right) (a \cos (c+d x)+b)}+\frac{2 \tan (c+d x)}{3 b^2}\right)}{d}+\frac{\frac{2 \left(40 a^3 b-28 a b^3\right) \sin (c+d x) \cos ^2(c+d x) \sqrt{1-\sec ^2(c+d x)} (a+b \sec (c+d x)) \Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)}{a \left(1-\cos ^2(c+d x)\right) (a \cos (c+d x)+b)}+\frac{\left(15 a^4-12 a^2 b^2\right) \sin (c+d x) \cos (2 (c+d x)) (a+b \sec (c+d x)) \left(2 a^2 \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} \Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)-4 b^2 \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} \Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)+4 a b \sec ^2(c+d x)-2 a (a-2 b) \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} F\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)-4 a b \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} E\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)-4 a b\right)}{a^2 b \left(1-\cos ^2(c+d x)\right) \sqrt{\sec (c+d x)} \left(2-\sec ^2(c+d x)\right) (a \cos (c+d x)+b)}+\frac{2 \left(45 a^4-44 a^2 b^2-4 b^4\right) \sin (c+d x) \cos ^2(c+d x) \sqrt{1-\sec ^2(c+d x)} (a+b \sec (c+d x)) \left(F\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)-\Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)\right)}{b \left(1-\cos ^2(c+d x)\right) (a \cos (c+d x)+b)}}{12 b^3 d (a-b) (a+b)}","-\frac{a^2 \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{b d \left(a^2-b^2\right) (a+b \sec (c+d x))}+\frac{\left(5 a^2-2 b^2\right) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 b^2 d \left(a^2-b^2\right)}+\frac{\left(5 a^2-2 b^2\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 b^2 d \left(a^2-b^2\right)}-\frac{a \left(5 a^2-4 b^2\right) \sin (c+d x) \sqrt{\sec (c+d x)}}{b^3 d \left(a^2-b^2\right)}+\frac{a \left(5 a^2-4 b^2\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b^3 d \left(a^2-b^2\right)}+\frac{a^2 \left(5 a^2-7 b^2\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{b^3 d (a-b) (a+b)^2}",1,"((2*(45*a^4 - 44*a^2*b^2 - 4*b^4)*Cos[c + d*x]^2*(EllipticF[ArcSin[Sqrt[Sec[c + d*x]]], -1] - EllipticPi[-(b/a), ArcSin[Sqrt[Sec[c + d*x]]], -1])*(a + b*Sec[c + d*x])*Sqrt[1 - Sec[c + d*x]^2]*Sin[c + d*x])/(b*(b + a*Cos[c + d*x])*(1 - Cos[c + d*x]^2)) + (2*(40*a^3*b - 28*a*b^3)*Cos[c + d*x]^2*EllipticPi[-(b/a), ArcSin[Sqrt[Sec[c + d*x]]], -1]*(a + b*Sec[c + d*x])*Sqrt[1 - Sec[c + d*x]^2]*Sin[c + d*x])/(a*(b + a*Cos[c + d*x])*(1 - Cos[c + d*x]^2)) + ((15*a^4 - 12*a^2*b^2)*Cos[2*(c + d*x)]*(a + b*Sec[c + d*x])*(-4*a*b + 4*a*b*Sec[c + d*x]^2 - 4*a*b*EllipticE[ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] - 2*a*(a - 2*b)*EllipticF[ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] + 2*a^2*EllipticPi[-(b/a), ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] - 4*b^2*EllipticPi[-(b/a), ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2])*Sin[c + d*x])/(a^2*b*(b + a*Cos[c + d*x])*(1 - Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]]*(2 - Sec[c + d*x]^2)))/(12*(a - b)*b^3*(a + b)*d) + (Sqrt[Sec[c + d*x]]*(-((a*(-5*a^2 + 4*b^2)*Sin[c + d*x])/(b^3*(-a^2 + b^2))) - (a^3*Sin[c + d*x])/(b^2*(-a^2 + b^2)*(b + a*Cos[c + d*x])) + (2*Tan[c + d*x])/(3*b^2)))/d","A",0
615,1,351,279,6.3614889,"\int \frac{\sec ^{\frac{7}{2}}(c+d x)}{(a+b \sec (c+d x))^2} \, dx","Integrate[Sec[c + d*x]^(7/2)/(a + b*Sec[c + d*x])^2,x]","\frac{\frac{2 b \sin (c+d x) \left(-3 a^3+2 b \left(b^2-a^2\right) \sec (c+d x)+2 a b^2\right)}{\left(b^2-a^2\right) \sqrt{\sec (c+d x)} (a \cos (c+d x)+b)}+\frac{\cot (c+d x) \left(6 a^3 \sqrt{-\tan ^2(c+d x)} \Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)+2 b \left(3 a^2-2 b^2\right) \sqrt{-\tan ^2(c+d x)} E\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)-3 a^2 b \sec ^{\frac{3}{2}}(c+d x)+3 a^2 b \cos (2 (c+d x)) \sec ^{\frac{3}{2}}(c+d x)-2 \left(3 a^3+3 a^2 b-4 a b^2-2 b^3\right) \sqrt{-\tan ^2(c+d x)} F\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)-10 a b^2 \sqrt{-\tan ^2(c+d x)} \Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)+2 b^3 \sec ^{\frac{3}{2}}(c+d x)-2 b^3 \cos (2 (c+d x)) \sec ^{\frac{3}{2}}(c+d x)\right)}{(a-b) (a+b)}}{2 b^3 d}","-\frac{a^2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{b d \left(a^2-b^2\right) (a+b \sec (c+d x))}+\frac{\left(3 a^2-2 b^2\right) \sin (c+d x) \sqrt{\sec (c+d x)}}{b^2 d \left(a^2-b^2\right)}-\frac{a \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b d \left(a^2-b^2\right)}-\frac{\left(3 a^2-2 b^2\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b^2 d \left(a^2-b^2\right)}-\frac{a \left(3 a^2-5 b^2\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{b^2 d (a-b) (a+b)^2}",1,"((2*b*(-3*a^3 + 2*a*b^2 + 2*b*(-a^2 + b^2)*Sec[c + d*x])*Sin[c + d*x])/((-a^2 + b^2)*(b + a*Cos[c + d*x])*Sqrt[Sec[c + d*x]]) + (Cot[c + d*x]*(-3*a^2*b*Sec[c + d*x]^(3/2) + 2*b^3*Sec[c + d*x]^(3/2) + 3*a^2*b*Cos[2*(c + d*x)]*Sec[c + d*x]^(3/2) - 2*b^3*Cos[2*(c + d*x)]*Sec[c + d*x]^(3/2) + 2*b*(3*a^2 - 2*b^2)*EllipticE[ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[-Tan[c + d*x]^2] - 2*(3*a^3 + 3*a^2*b - 4*a*b^2 - 2*b^3)*EllipticF[ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[-Tan[c + d*x]^2] + 6*a^3*EllipticPi[-(b/a), ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[-Tan[c + d*x]^2] - 10*a*b^2*EllipticPi[-(b/a), ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[-Tan[c + d*x]^2]))/((a - b)*(a + b)))/(2*b^3*d)","A",1
616,1,582,214,6.7229591,"\int \frac{\sec ^{\frac{5}{2}}(c+d x)}{(a+b \sec (c+d x))^2} \, dx","Integrate[Sec[c + d*x]^(5/2)/(a + b*Sec[c + d*x])^2,x]","\frac{\sqrt{\sec (c+d x)} \left(\frac{a \sin (c+d x)}{b \left(b^2-a^2\right)}+\frac{a \sin (c+d x)}{\left(a^2-b^2\right) (a \cos (c+d x)+b)}\right)}{d}+\frac{\frac{2 \left(3 a^2-4 b^2\right) \sin (c+d x) \cos ^2(c+d x) \sqrt{1-\sec ^2(c+d x)} (a+b \sec (c+d x)) \left(F\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)-\Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)\right)}{b \left(1-\cos ^2(c+d x)\right) (a \cos (c+d x)+b)}+\frac{\sin (c+d x) \cos (2 (c+d x)) (a+b \sec (c+d x)) \left(2 a^2 \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} \Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)-4 b^2 \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} \Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)+4 a b \sec ^2(c+d x)-2 a (a-2 b) \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} F\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)-4 a b \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} E\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)-4 a b\right)}{b \left(1-\cos ^2(c+d x)\right) \sqrt{\sec (c+d x)} \left(2-\sec ^2(c+d x)\right) (a \cos (c+d x)+b)}+\frac{8 b \sin (c+d x) \cos ^2(c+d x) \sqrt{1-\sec ^2(c+d x)} (a+b \sec (c+d x)) \Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)}{\left(1-\cos ^2(c+d x)\right) (a \cos (c+d x)+b)}}{4 b d (a-b) (a+b)}","-\frac{a^2 \sin (c+d x) \sqrt{\sec (c+d x)}}{b d \left(a^2-b^2\right) (a+b \sec (c+d x))}+\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d \left(a^2-b^2\right)}+\frac{a \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b d \left(a^2-b^2\right)}+\frac{\left(a^2-3 b^2\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{b d (a-b) (a+b)^2}",1,"(Sqrt[Sec[c + d*x]]*((a*Sin[c + d*x])/(b*(-a^2 + b^2)) + (a*Sin[c + d*x])/((a^2 - b^2)*(b + a*Cos[c + d*x]))))/d + ((2*(3*a^2 - 4*b^2)*Cos[c + d*x]^2*(EllipticF[ArcSin[Sqrt[Sec[c + d*x]]], -1] - EllipticPi[-(b/a), ArcSin[Sqrt[Sec[c + d*x]]], -1])*(a + b*Sec[c + d*x])*Sqrt[1 - Sec[c + d*x]^2]*Sin[c + d*x])/(b*(b + a*Cos[c + d*x])*(1 - Cos[c + d*x]^2)) + (8*b*Cos[c + d*x]^2*EllipticPi[-(b/a), ArcSin[Sqrt[Sec[c + d*x]]], -1]*(a + b*Sec[c + d*x])*Sqrt[1 - Sec[c + d*x]^2]*Sin[c + d*x])/((b + a*Cos[c + d*x])*(1 - Cos[c + d*x]^2)) + (Cos[2*(c + d*x)]*(a + b*Sec[c + d*x])*(-4*a*b + 4*a*b*Sec[c + d*x]^2 - 4*a*b*EllipticE[ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] - 2*a*(a - 2*b)*EllipticF[ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] + 2*a^2*EllipticPi[-(b/a), ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] - 4*b^2*EllipticPi[-(b/a), ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2])*Sin[c + d*x])/(b*(b + a*Cos[c + d*x])*(1 - Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]]*(2 - Sec[c + d*x]^2)))/(4*(a - b)*b*(a + b)*d)","B",0
617,1,628,208,6.7250972,"\int \frac{\sec ^{\frac{3}{2}}(c+d x)}{(a+b \sec (c+d x))^2} \, dx","Integrate[Sec[c + d*x]^(3/2)/(a + b*Sec[c + d*x])^2,x]","\frac{\sec ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+b)^2 \left(\frac{b \sin (c+d x)}{\left(b^2-a^2\right) (a \cos (c+d x)+b)}-\frac{\sin (c+d x)}{b^2-a^2}\right)}{d (a+b \sec (c+d x))^2}+\frac{\sec ^2(c+d x) (a \cos (c+d x)+b)^2 \left(\frac{\sin (c+d x) \cos (2 (c+d x)) (a+b \sec (c+d x)) \left(2 a^2 \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} \Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)-4 b^2 \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} \Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)+4 a b \sec ^2(c+d x)-2 a (a-2 b) \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} F\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)-4 a b \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} E\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)-4 a b\right)}{a b \left(1-\cos ^2(c+d x)\right) \sqrt{\sec (c+d x)} \left(2-\sec ^2(c+d x)\right) (a \cos (c+d x)+b)}+\frac{8 b \sin (c+d x) \cos ^2(c+d x) \sqrt{1-\sec ^2(c+d x)} (a+b \sec (c+d x)) \Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)}{a \left(1-\cos ^2(c+d x)\right) (a \cos (c+d x)+b)}-\frac{2 a \sin (c+d x) \cos ^2(c+d x) \sqrt{1-\sec ^2(c+d x)} (a+b \sec (c+d x)) \left(F\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)-\Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)\right)}{b \left(1-\cos ^2(c+d x)\right) (a \cos (c+d x)+b)}\right)}{4 d (b-a) (a+b) (a+b \sec (c+d x))^2}","\frac{a \sin (c+d x) \sqrt{\sec (c+d x)}}{d \left(a^2-b^2\right) (a+b \sec (c+d x))}-\frac{b \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d \left(a^2-b^2\right)}-\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d \left(a^2-b^2\right)}+\frac{\left(a^2+b^2\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a d (a-b) (a+b)^2}",1,"((b + a*Cos[c + d*x])^2*Sec[c + d*x]^(5/2)*(-(Sin[c + d*x]/(-a^2 + b^2)) + (b*Sin[c + d*x])/((-a^2 + b^2)*(b + a*Cos[c + d*x]))))/(d*(a + b*Sec[c + d*x])^2) + ((b + a*Cos[c + d*x])^2*Sec[c + d*x]^2*((-2*a*Cos[c + d*x]^2*(EllipticF[ArcSin[Sqrt[Sec[c + d*x]]], -1] - EllipticPi[-(b/a), ArcSin[Sqrt[Sec[c + d*x]]], -1])*(a + b*Sec[c + d*x])*Sqrt[1 - Sec[c + d*x]^2]*Sin[c + d*x])/(b*(b + a*Cos[c + d*x])*(1 - Cos[c + d*x]^2)) + (8*b*Cos[c + d*x]^2*EllipticPi[-(b/a), ArcSin[Sqrt[Sec[c + d*x]]], -1]*(a + b*Sec[c + d*x])*Sqrt[1 - Sec[c + d*x]^2]*Sin[c + d*x])/(a*(b + a*Cos[c + d*x])*(1 - Cos[c + d*x]^2)) + (Cos[2*(c + d*x)]*(a + b*Sec[c + d*x])*(-4*a*b + 4*a*b*Sec[c + d*x]^2 - 4*a*b*EllipticE[ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] - 2*a*(a - 2*b)*EllipticF[ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] + 2*a^2*EllipticPi[-(b/a), ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] - 4*b^2*EllipticPi[-(b/a), ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2])*Sin[c + d*x])/(a*b*(b + a*Cos[c + d*x])*(1 - Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]]*(2 - Sec[c + d*x]^2))))/(4*(-a + b)*(a + b)*d*(a + b*Sec[c + d*x])^2)","B",0
618,1,251,227,5.1368177,"\int \frac{\sqrt{\sec (c+d x)}}{(a+b \sec (c+d x))^2} \, dx","Integrate[Sqrt[Sec[c + d*x]]/(a + b*Sec[c + d*x])^2,x]","\frac{\cos (2 (c+d x)) \csc (c+d x) \sqrt{\sec (c+d x)} \left(-\left(3 a^2-b^2\right) \sqrt{-\tan ^2(c+d x)} \sqrt{\sec (c+d x)} (a \cos (c+d x)+b) \Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)+a (a-b) \sqrt{-\tan ^2(c+d x)} \sqrt{\sec (c+d x)} (a \cos (c+d x)+b) F\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)+a b \left(\sqrt{-\tan ^2(c+d x)} \sqrt{\sec (c+d x)} (a \cos (c+d x)+b) E\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)-b \tan ^2(c+d x)\right)\right)}{a^2 d (a-b) (a+b) \left(\sec ^2(c+d x)-2\right) (a \cos (c+d x)+b)}","-\frac{b \sin (c+d x) \sqrt{\sec (c+d x)}}{d \left(a^2-b^2\right) (a+b \sec (c+d x))}+\frac{\left(2 a^2-b^2\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d \left(a^2-b^2\right)}+\frac{b \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d \left(a^2-b^2\right)}-\frac{b \left(3 a^2-b^2\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d (a-b) (a+b)^2}",1,"(Cos[2*(c + d*x)]*Csc[c + d*x]*Sqrt[Sec[c + d*x]]*(a*(a - b)*(b + a*Cos[c + d*x])*EllipticF[ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[-Tan[c + d*x]^2] - (3*a^2 - b^2)*(b + a*Cos[c + d*x])*EllipticPi[-(b/a), ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[-Tan[c + d*x]^2] + a*b*(-(b*Tan[c + d*x]^2) + (b + a*Cos[c + d*x])*EllipticE[ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[-Tan[c + d*x]^2])))/(a^2*(a - b)*(a + b)*d*(b + a*Cos[c + d*x])*(-2 + Sec[c + d*x]^2))","A",0
619,1,603,244,6.768915,"\int \frac{1}{\sqrt{\sec (c+d x)} (a+b \sec (c+d x))^2} \, dx","Integrate[1/(Sqrt[Sec[c + d*x]]*(a + b*Sec[c + d*x])^2),x]","\frac{\frac{2 \left(b^2-2 a^2\right) \sin (c+d x) \cos ^2(c+d x) \sqrt{1-\sec ^2(c+d x)} (a+b \sec (c+d x)) \left(F\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)-\Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)\right)}{b \left(1-\cos ^2(c+d x)\right) (a \cos (c+d x)+b)}+\frac{\left(3 b^2-2 a^2\right) \sin (c+d x) \cos (2 (c+d x)) (a+b \sec (c+d x)) \left(2 a^2 \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} \Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)-4 b^2 \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} \Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)+4 a b \sec ^2(c+d x)-2 a (a-2 b) \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} F\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)-4 a b \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} E\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)-4 a b\right)}{a^2 b \left(1-\cos ^2(c+d x)\right) \sqrt{\sec (c+d x)} \left(2-\sec ^2(c+d x)\right) (a \cos (c+d x)+b)}+\frac{8 b \sin (c+d x) \cos ^2(c+d x) \sqrt{1-\sec ^2(c+d x)} (a+b \sec (c+d x)) \Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)}{\left(1-\cos ^2(c+d x)\right) (a \cos (c+d x)+b)}}{4 a d (b-a) (a+b)}+\frac{\sqrt{\sec (c+d x)} \left(-\frac{b^2 \sin (c+d x)}{a^2 \left(b^2-a^2\right)}-\frac{b^3 \sin (c+d x)}{a^2 \left(a^2-b^2\right) (a \cos (c+d x)+b)}\right)}{d}","\frac{b^2 \sin (c+d x) \sqrt{\sec (c+d x)}}{a d \left(a^2-b^2\right) (a+b \sec (c+d x))}+\frac{\left(2 a^2-3 b^2\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d \left(a^2-b^2\right)}-\frac{b \left(4 a^2-3 b^2\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^3 d \left(a^2-b^2\right)}+\frac{b^2 \left(5 a^2-3 b^2\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a^3 d (a-b) (a+b)^2}",1,"(Sqrt[Sec[c + d*x]]*(-((b^2*Sin[c + d*x])/(a^2*(-a^2 + b^2))) - (b^3*Sin[c + d*x])/(a^2*(a^2 - b^2)*(b + a*Cos[c + d*x]))))/d + ((2*(-2*a^2 + b^2)*Cos[c + d*x]^2*(EllipticF[ArcSin[Sqrt[Sec[c + d*x]]], -1] - EllipticPi[-(b/a), ArcSin[Sqrt[Sec[c + d*x]]], -1])*(a + b*Sec[c + d*x])*Sqrt[1 - Sec[c + d*x]^2]*Sin[c + d*x])/(b*(b + a*Cos[c + d*x])*(1 - Cos[c + d*x]^2)) + (8*b*Cos[c + d*x]^2*EllipticPi[-(b/a), ArcSin[Sqrt[Sec[c + d*x]]], -1]*(a + b*Sec[c + d*x])*Sqrt[1 - Sec[c + d*x]^2]*Sin[c + d*x])/((b + a*Cos[c + d*x])*(1 - Cos[c + d*x]^2)) + ((-2*a^2 + 3*b^2)*Cos[2*(c + d*x)]*(a + b*Sec[c + d*x])*(-4*a*b + 4*a*b*Sec[c + d*x]^2 - 4*a*b*EllipticE[ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] - 2*a*(a - 2*b)*EllipticF[ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] + 2*a^2*EllipticPi[-(b/a), ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] - 4*b^2*EllipticPi[-(b/a), ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2])*Sin[c + d*x])/(a^2*b*(b + a*Cos[c + d*x])*(1 - Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]]*(2 - Sec[c + d*x]^2)))/(4*a*(-a + b)*(a + b)*d)","B",0
620,1,634,304,6.8510114,"\int \frac{1}{\sec ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^2} \, dx","Integrate[1/(Sec[c + d*x]^(3/2)*(a + b*Sec[c + d*x])^2),x]","\frac{\frac{2 \left(4 a^3+8 a b^2\right) \sin (c+d x) \cos ^2(c+d x) \sqrt{1-\sec ^2(c+d x)} (a+b \sec (c+d x)) \Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)}{a \left(1-\cos ^2(c+d x)\right) (a \cos (c+d x)+b)}+\frac{2 \left(5 b^3-8 a^2 b\right) \sin (c+d x) \cos ^2(c+d x) \sqrt{1-\sec ^2(c+d x)} (a+b \sec (c+d x)) \left(F\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)-\Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)\right)}{b \left(1-\cos ^2(c+d x)\right) (a \cos (c+d x)+b)}+\frac{\left(15 b^3-12 a^2 b\right) \sin (c+d x) \cos (2 (c+d x)) (a+b \sec (c+d x)) \left(2 a^2 \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} \Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)-4 b^2 \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} \Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)+4 a b \sec ^2(c+d x)-2 a (a-2 b) \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} F\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)-4 a b \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} E\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)-4 a b\right)}{a^2 b \left(1-\cos ^2(c+d x)\right) \sqrt{\sec (c+d x)} \left(2-\sec ^2(c+d x)\right) (a \cos (c+d x)+b)}}{12 a^2 d (a-b) (a+b)}+\frac{\sqrt{\sec (c+d x)} \left(\frac{\sin (2 (c+d x))}{3 a^2}+\frac{b^4 \sin (c+d x)}{a^3 \left(a^2-b^2\right) (a \cos (c+d x)+b)}+\frac{b^3 \sin (c+d x)}{a^3 \left(b^2-a^2\right)}\right)}{d}","\frac{b^2 \sin (c+d x)}{a d \left(a^2-b^2\right) \sqrt{\sec (c+d x)} (a+b \sec (c+d x))}+\frac{\left(2 a^2-5 b^2\right) \sin (c+d x)}{3 a^2 d \left(a^2-b^2\right) \sqrt{\sec (c+d x)}}+\frac{\left(2 a^4+16 a^2 b^2-15 b^4\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^4 d \left(a^2-b^2\right)}-\frac{b^3 \left(7 a^2-5 b^2\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a^4 d (a-b) (a+b)^2}-\frac{b \left(4 a^2-5 b^2\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^3 d \left(a^2-b^2\right)}",1,"((2*(-8*a^2*b + 5*b^3)*Cos[c + d*x]^2*(EllipticF[ArcSin[Sqrt[Sec[c + d*x]]], -1] - EllipticPi[-(b/a), ArcSin[Sqrt[Sec[c + d*x]]], -1])*(a + b*Sec[c + d*x])*Sqrt[1 - Sec[c + d*x]^2]*Sin[c + d*x])/(b*(b + a*Cos[c + d*x])*(1 - Cos[c + d*x]^2)) + (2*(4*a^3 + 8*a*b^2)*Cos[c + d*x]^2*EllipticPi[-(b/a), ArcSin[Sqrt[Sec[c + d*x]]], -1]*(a + b*Sec[c + d*x])*Sqrt[1 - Sec[c + d*x]^2]*Sin[c + d*x])/(a*(b + a*Cos[c + d*x])*(1 - Cos[c + d*x]^2)) + ((-12*a^2*b + 15*b^3)*Cos[2*(c + d*x)]*(a + b*Sec[c + d*x])*(-4*a*b + 4*a*b*Sec[c + d*x]^2 - 4*a*b*EllipticE[ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] - 2*a*(a - 2*b)*EllipticF[ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] + 2*a^2*EllipticPi[-(b/a), ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] - 4*b^2*EllipticPi[-(b/a), ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2])*Sin[c + d*x])/(a^2*b*(b + a*Cos[c + d*x])*(1 - Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]]*(2 - Sec[c + d*x]^2)))/(12*a^2*(a - b)*(a + b)*d) + (Sqrt[Sec[c + d*x]]*((b^3*Sin[c + d*x])/(a^3*(-a^2 + b^2)) + (b^4*Sin[c + d*x])/(a^3*(a^2 - b^2)*(b + a*Cos[c + d*x])) + Sin[2*(c + d*x)]/(3*a^2)))/d","B",0
621,1,721,388,6.8847445,"\int \frac{\sec ^{\frac{9}{2}}(c+d x)}{(a+b \sec (c+d x))^3} \, dx","Integrate[Sec[c + d*x]^(9/2)/(a + b*Sec[c + d*x])^3,x]","\frac{\sqrt{\sec (c+d x)} \left(\frac{a^2 \sin (c+d x)}{2 b \left(b^2-a^2\right) (a \cos (c+d x)+b)^2}+\frac{11 a^2 b^2 \sin (c+d x)-5 a^4 \sin (c+d x)}{4 b^2 \left(b^2-a^2\right)^2 (a \cos (c+d x)+b)}+\frac{\left(15 a^4-29 a^2 b^2+8 b^4\right) \sin (c+d x)}{4 b^3 \left(b^2-a^2\right)^2}\right)}{d}-\frac{\frac{2 \left(45 a^5-95 a^3 b^2+56 a b^4\right) \sin (c+d x) \cos ^2(c+d x) \sqrt{1-\sec ^2(c+d x)} (a+b \sec (c+d x)) \left(F\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)-\Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)\right)}{b \left(1-\cos ^2(c+d x)\right) (a \cos (c+d x)+b)}+\frac{2 \left(40 a^4 b-80 a^2 b^3+16 b^5\right) \sin (c+d x) \cos ^2(c+d x) \sqrt{1-\sec ^2(c+d x)} (a+b \sec (c+d x)) \Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)}{a \left(1-\cos ^2(c+d x)\right) (a \cos (c+d x)+b)}+\frac{\left(15 a^5-29 a^3 b^2+8 a b^4\right) \sin (c+d x) \cos (2 (c+d x)) (a+b \sec (c+d x)) \left(2 a^2 \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} \Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)-4 b^2 \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} \Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)+4 a b \sec ^2(c+d x)-2 a (a-2 b) \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} F\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)-4 a b \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} E\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)-4 a b\right)}{a^2 b \left(1-\cos ^2(c+d x)\right) \sqrt{\sec (c+d x)} \left(2-\sec ^2(c+d x)\right) (a \cos (c+d x)+b)}}{16 b^3 d (a-b)^2 (a+b)^2}","-\frac{a^2 \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{2 b d \left(a^2-b^2\right) (a+b \sec (c+d x))^2}-\frac{a^2 \left(5 a^2-11 b^2\right) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{4 b^2 d \left(a^2-b^2\right)^2 (a+b \sec (c+d x))}-\frac{a \left(5 a^2-11 b^2\right) \sqrt{\cos (c+d x)} \sqrt{\sec (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)^2}+\frac{\left(15 a^4-29 a^2 b^2+8 b^4\right) \sin (c+d x) \sqrt{\sec (c+d x)}}{4 b^3 d \left(a^2-b^2\right)^2}-\frac{\left(15 a^4-29 a^2 b^2+8 b^4\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{4 b^3 d \left(a^2-b^2\right)^2}-\frac{a \left(15 a^4-38 a^2 b^2+35 b^4\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{4 b^3 d (a-b)^2 (a+b)^3}",1,"-1/16*((2*(45*a^5 - 95*a^3*b^2 + 56*a*b^4)*Cos[c + d*x]^2*(EllipticF[ArcSin[Sqrt[Sec[c + d*x]]], -1] - EllipticPi[-(b/a), ArcSin[Sqrt[Sec[c + d*x]]], -1])*(a + b*Sec[c + d*x])*Sqrt[1 - Sec[c + d*x]^2]*Sin[c + d*x])/(b*(b + a*Cos[c + d*x])*(1 - Cos[c + d*x]^2)) + (2*(40*a^4*b - 80*a^2*b^3 + 16*b^5)*Cos[c + d*x]^2*EllipticPi[-(b/a), ArcSin[Sqrt[Sec[c + d*x]]], -1]*(a + b*Sec[c + d*x])*Sqrt[1 - Sec[c + d*x]^2]*Sin[c + d*x])/(a*(b + a*Cos[c + d*x])*(1 - Cos[c + d*x]^2)) + ((15*a^5 - 29*a^3*b^2 + 8*a*b^4)*Cos[2*(c + d*x)]*(a + b*Sec[c + d*x])*(-4*a*b + 4*a*b*Sec[c + d*x]^2 - 4*a*b*EllipticE[ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] - 2*a*(a - 2*b)*EllipticF[ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] + 2*a^2*EllipticPi[-(b/a), ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] - 4*b^2*EllipticPi[-(b/a), ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2])*Sin[c + d*x])/(a^2*b*(b + a*Cos[c + d*x])*(1 - Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]]*(2 - Sec[c + d*x]^2)))/((a - b)^2*b^3*(a + b)^2*d) + (Sqrt[Sec[c + d*x]]*(((15*a^4 - 29*a^2*b^2 + 8*b^4)*Sin[c + d*x])/(4*b^3*(-a^2 + b^2)^2) + (a^2*Sin[c + d*x])/(2*b*(-a^2 + b^2)*(b + a*Cos[c + d*x])^2) + (-5*a^4*Sin[c + d*x] + 11*a^2*b^2*Sin[c + d*x])/(4*b^2*(-a^2 + b^2)^2*(b + a*Cos[c + d*x]))))/d","A",0
622,1,692,315,6.8189791,"\int \frac{\sec ^{\frac{7}{2}}(c+d x)}{(a+b \sec (c+d x))^3} \, dx","Integrate[Sec[c + d*x]^(7/2)/(a + b*Sec[c + d*x])^3,x]","\frac{\sqrt{\sec (c+d x)} \left(\frac{3 a \left(3 b^2-a^2\right) \sin (c+d x)}{4 b^2 \left(b^2-a^2\right)^2}-\frac{a \sin (c+d x)}{2 \left(b^2-a^2\right) (a \cos (c+d x)+b)^2}+\frac{a^3 \sin (c+d x)-7 a b^2 \sin (c+d x)}{4 b \left(b^2-a^2\right)^2 (a \cos (c+d x)+b)}\right)}{d}+\frac{\frac{2 \left(8 a^3 b-32 a b^3\right) \sin (c+d x) \cos ^2(c+d x) \sqrt{1-\sec ^2(c+d x)} (a+b \sec (c+d x)) \Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)}{a \left(1-\cos ^2(c+d x)\right) (a \cos (c+d x)+b)}+\frac{\left(3 a^4-9 a^2 b^2\right) \sin (c+d x) \cos (2 (c+d x)) (a+b \sec (c+d x)) \left(2 a^2 \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} \Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)-4 b^2 \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} \Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)+4 a b \sec ^2(c+d x)-2 a (a-2 b) \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} F\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)-4 a b \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} E\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)-4 a b\right)}{a^2 b \left(1-\cos ^2(c+d x)\right) \sqrt{\sec (c+d x)} \left(2-\sec ^2(c+d x)\right) (a \cos (c+d x)+b)}+\frac{2 \left(9 a^4-19 a^2 b^2+16 b^4\right) \sin (c+d x) \cos ^2(c+d x) \sqrt{1-\sec ^2(c+d x)} (a+b \sec (c+d x)) \left(F\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)-\Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)\right)}{b \left(1-\cos ^2(c+d x)\right) (a \cos (c+d x)+b)}}{16 b^2 d (a-b)^2 (a+b)^2}","-\frac{a^2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{2 b d \left(a^2-b^2\right) (a+b \sec (c+d x))^2}-\frac{3 a^2 \left(a^2-3 b^2\right) \sin (c+d x) \sqrt{\sec (c+d x)}}{4 b^2 d \left(a^2-b^2\right)^2 (a+b \sec (c+d x))}+\frac{\left(a^2-7 b^2\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{4 b d \left(a^2-b^2\right)^2}+\frac{3 a \left(a^2-3 b^2\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{4 b^2 d \left(a^2-b^2\right)^2}+\frac{3 \left(a^4-2 a^2 b^2+5 b^4\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{4 b^2 d (a-b)^2 (a+b)^3}",1,"((2*(9*a^4 - 19*a^2*b^2 + 16*b^4)*Cos[c + d*x]^2*(EllipticF[ArcSin[Sqrt[Sec[c + d*x]]], -1] - EllipticPi[-(b/a), ArcSin[Sqrt[Sec[c + d*x]]], -1])*(a + b*Sec[c + d*x])*Sqrt[1 - Sec[c + d*x]^2]*Sin[c + d*x])/(b*(b + a*Cos[c + d*x])*(1 - Cos[c + d*x]^2)) + (2*(8*a^3*b - 32*a*b^3)*Cos[c + d*x]^2*EllipticPi[-(b/a), ArcSin[Sqrt[Sec[c + d*x]]], -1]*(a + b*Sec[c + d*x])*Sqrt[1 - Sec[c + d*x]^2]*Sin[c + d*x])/(a*(b + a*Cos[c + d*x])*(1 - Cos[c + d*x]^2)) + ((3*a^4 - 9*a^2*b^2)*Cos[2*(c + d*x)]*(a + b*Sec[c + d*x])*(-4*a*b + 4*a*b*Sec[c + d*x]^2 - 4*a*b*EllipticE[ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] - 2*a*(a - 2*b)*EllipticF[ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] + 2*a^2*EllipticPi[-(b/a), ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] - 4*b^2*EllipticPi[-(b/a), ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2])*Sin[c + d*x])/(a^2*b*(b + a*Cos[c + d*x])*(1 - Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]]*(2 - Sec[c + d*x]^2)))/(16*(a - b)^2*b^2*(a + b)^2*d) + (Sqrt[Sec[c + d*x]]*((3*a*(-a^2 + 3*b^2)*Sin[c + d*x])/(4*b^2*(-a^2 + b^2)^2) - (a*Sin[c + d*x])/(2*(-a^2 + b^2)*(b + a*Cos[c + d*x])^2) + (a^3*Sin[c + d*x] - 7*a*b^2*Sin[c + d*x])/(4*b*(-a^2 + b^2)^2*(b + a*Cos[c + d*x]))))/d","B",0
623,1,728,313,6.8090739,"\int \frac{\sec ^{\frac{5}{2}}(c+d x)}{(a+b \sec (c+d x))^3} \, dx","Integrate[Sec[c + d*x]^(5/2)/(a + b*Sec[c + d*x])^3,x]","\frac{\sec ^{\frac{7}{2}}(c+d x) (a \cos (c+d x)+b)^3 \left(-\frac{\left(a^2+5 b^2\right) \sin (c+d x)}{4 b \left(b^2-a^2\right)^2}+\frac{b \sin (c+d x)}{2 \left(b^2-a^2\right) (a \cos (c+d x)+b)^2}+\frac{3 \left(a^2 \sin (c+d x)+b^2 \sin (c+d x)\right)}{4 \left(b^2-a^2\right)^2 (a \cos (c+d x)+b)}\right)}{d (a+b \sec (c+d x))^3}+\frac{\sec ^3(c+d x) (a \cos (c+d x)+b)^3 \left(\frac{2 \left(3 a^3-9 a b^2\right) \sin (c+d x) \cos ^2(c+d x) \sqrt{1-\sec ^2(c+d x)} (a+b \sec (c+d x)) \left(F\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)-\Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)\right)}{b \left(1-\cos ^2(c+d x)\right) (a \cos (c+d x)+b)}+\frac{2 \left(8 a^2 b+16 b^3\right) \sin (c+d x) \cos ^2(c+d x) \sqrt{1-\sec ^2(c+d x)} (a+b \sec (c+d x)) \Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)}{a \left(1-\cos ^2(c+d x)\right) (a \cos (c+d x)+b)}+\frac{\left(a^3+5 a b^2\right) \sin (c+d x) \cos (2 (c+d x)) (a+b \sec (c+d x)) \left(2 a^2 \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} \Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)-4 b^2 \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} \Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)+4 a b \sec ^2(c+d x)-2 a (a-2 b) \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} F\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)-4 a b \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} E\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)-4 a b\right)}{a^2 b \left(1-\cos ^2(c+d x)\right) \sqrt{\sec (c+d x)} \left(2-\sec ^2(c+d x)\right) (a \cos (c+d x)+b)}\right)}{16 b d (a-b)^2 (a+b)^2 (a+b \sec (c+d x))^3}","-\frac{a^2 \sin (c+d x) \sqrt{\sec (c+d x)}}{2 b d \left(a^2-b^2\right) (a+b \sec (c+d x))^2}+\frac{a \left(a^2-7 b^2\right) \sin (c+d x) \sqrt{\sec (c+d x)}}{4 b d \left(a^2-b^2\right)^2 (a+b \sec (c+d x))}+\frac{3 \left(a^2+b^2\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a d \left(a^2-b^2\right)^2}+\frac{\left(a^2+5 b^2\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{4 b d \left(a^2-b^2\right)^2}+\frac{\left(a^4-10 a^2 b^2-3 b^4\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a b d (a-b)^2 (a+b)^3}",1,"((b + a*Cos[c + d*x])^3*Sec[c + d*x]^3*((2*(3*a^3 - 9*a*b^2)*Cos[c + d*x]^2*(EllipticF[ArcSin[Sqrt[Sec[c + d*x]]], -1] - EllipticPi[-(b/a), ArcSin[Sqrt[Sec[c + d*x]]], -1])*(a + b*Sec[c + d*x])*Sqrt[1 - Sec[c + d*x]^2]*Sin[c + d*x])/(b*(b + a*Cos[c + d*x])*(1 - Cos[c + d*x]^2)) + (2*(8*a^2*b + 16*b^3)*Cos[c + d*x]^2*EllipticPi[-(b/a), ArcSin[Sqrt[Sec[c + d*x]]], -1]*(a + b*Sec[c + d*x])*Sqrt[1 - Sec[c + d*x]^2]*Sin[c + d*x])/(a*(b + a*Cos[c + d*x])*(1 - Cos[c + d*x]^2)) + ((a^3 + 5*a*b^2)*Cos[2*(c + d*x)]*(a + b*Sec[c + d*x])*(-4*a*b + 4*a*b*Sec[c + d*x]^2 - 4*a*b*EllipticE[ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] - 2*a*(a - 2*b)*EllipticF[ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] + 2*a^2*EllipticPi[-(b/a), ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] - 4*b^2*EllipticPi[-(b/a), ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2])*Sin[c + d*x])/(a^2*b*(b + a*Cos[c + d*x])*(1 - Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]]*(2 - Sec[c + d*x]^2))))/(16*(a - b)^2*b*(a + b)^2*d*(a + b*Sec[c + d*x])^3) + ((b + a*Cos[c + d*x])^3*Sec[c + d*x]^(7/2)*(-1/4*((a^2 + 5*b^2)*Sin[c + d*x])/(b*(-a^2 + b^2)^2) + (b*Sin[c + d*x])/(2*(-a^2 + b^2)*(b + a*Cos[c + d*x])^2) + (3*(a^2*Sin[c + d*x] + b^2*Sin[c + d*x]))/(4*(-a^2 + b^2)^2*(b + a*Cos[c + d*x]))))/(d*(a + b*Sec[c + d*x])^3)","B",0
624,1,719,306,6.7729828,"\int \frac{\sec ^{\frac{3}{2}}(c+d x)}{(a+b \sec (c+d x))^3} \, dx","Integrate[Sec[c + d*x]^(3/2)/(a + b*Sec[c + d*x])^3,x]","\frac{\sec ^{\frac{7}{2}}(c+d x) (a \cos (c+d x)+b)^3 \left(\frac{\left(5 a^2+b^2\right) \sin (c+d x)}{4 a \left(a^2-b^2\right)^2}+\frac{b^2 \sin (c+d x)}{2 a \left(a^2-b^2\right) (a \cos (c+d x)+b)^2}+\frac{b^3 \sin (c+d x)-7 a^2 b \sin (c+d x)}{4 a \left(a^2-b^2\right)^2 (a \cos (c+d x)+b)}\right)}{d (a+b \sec (c+d x))^3}-\frac{\sec ^3(c+d x) (a \cos (c+d x)+b)^3 \left(\frac{2 \left(-a^2-5 b^2\right) \sin (c+d x) \cos ^2(c+d x) \sqrt{1-\sec ^2(c+d x)} (a+b \sec (c+d x)) \left(F\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)-\Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)\right)}{b \left(1-\cos ^2(c+d x)\right) (a \cos (c+d x)+b)}+\frac{\left(5 a^2+b^2\right) \sin (c+d x) \cos (2 (c+d x)) (a+b \sec (c+d x)) \left(2 a^2 \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} \Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)-4 b^2 \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} \Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)+4 a b \sec ^2(c+d x)-2 a (a-2 b) \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} F\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)-4 a b \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} E\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)-4 a b\right)}{a^2 b \left(1-\cos ^2(c+d x)\right) \sqrt{\sec (c+d x)} \left(2-\sec ^2(c+d x)\right) (a \cos (c+d x)+b)}+\frac{48 b \sin (c+d x) \cos ^2(c+d x) \sqrt{1-\sec ^2(c+d x)} (a+b \sec (c+d x)) \Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)}{\left(1-\cos ^2(c+d x)\right) (a \cos (c+d x)+b)}\right)}{16 d (a-b)^2 (a+b)^2 (a+b \sec (c+d x))^3}","\frac{3 \left(a^2+b^2\right) \sin (c+d x) \sqrt{\sec (c+d x)}}{4 d \left(a^2-b^2\right)^2 (a+b \sec (c+d x))}+\frac{a \sin (c+d x) \sqrt{\sec (c+d x)}}{2 d \left(a^2-b^2\right) (a+b \sec (c+d x))^2}-\frac{b \left(7 a^2-b^2\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a^2 d \left(a^2-b^2\right)^2}-\frac{\left(5 a^2+b^2\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a d \left(a^2-b^2\right)^2}+\frac{\left(3 a^4+10 a^2 b^2-b^4\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a^2 d (a-b)^2 (a+b)^3}",1,"-1/16*((b + a*Cos[c + d*x])^3*Sec[c + d*x]^3*((2*(-a^2 - 5*b^2)*Cos[c + d*x]^2*(EllipticF[ArcSin[Sqrt[Sec[c + d*x]]], -1] - EllipticPi[-(b/a), ArcSin[Sqrt[Sec[c + d*x]]], -1])*(a + b*Sec[c + d*x])*Sqrt[1 - Sec[c + d*x]^2]*Sin[c + d*x])/(b*(b + a*Cos[c + d*x])*(1 - Cos[c + d*x]^2)) + (48*b*Cos[c + d*x]^2*EllipticPi[-(b/a), ArcSin[Sqrt[Sec[c + d*x]]], -1]*(a + b*Sec[c + d*x])*Sqrt[1 - Sec[c + d*x]^2]*Sin[c + d*x])/((b + a*Cos[c + d*x])*(1 - Cos[c + d*x]^2)) + ((5*a^2 + b^2)*Cos[2*(c + d*x)]*(a + b*Sec[c + d*x])*(-4*a*b + 4*a*b*Sec[c + d*x]^2 - 4*a*b*EllipticE[ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] - 2*a*(a - 2*b)*EllipticF[ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] + 2*a^2*EllipticPi[-(b/a), ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] - 4*b^2*EllipticPi[-(b/a), ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2])*Sin[c + d*x])/(a^2*b*(b + a*Cos[c + d*x])*(1 - Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]]*(2 - Sec[c + d*x]^2))))/((a - b)^2*(a + b)^2*d*(a + b*Sec[c + d*x])^3) + ((b + a*Cos[c + d*x])^3*Sec[c + d*x]^(7/2)*(((5*a^2 + b^2)*Sin[c + d*x])/(4*a*(a^2 - b^2)^2) + (b^2*Sin[c + d*x])/(2*a*(a^2 - b^2)*(b + a*Cos[c + d*x])^2) + (-7*a^2*b*Sin[c + d*x] + b^3*Sin[c + d*x])/(4*a*(a^2 - b^2)^2*(b + a*Cos[c + d*x]))))/(d*(a + b*Sec[c + d*x])^3)","B",0
625,1,744,323,6.8553331,"\int \frac{\sqrt{\sec (c+d x)}}{(a+b \sec (c+d x))^3} \, dx","Integrate[Sqrt[Sec[c + d*x]]/(a + b*Sec[c + d*x])^3,x]","\frac{\sec ^{\frac{7}{2}}(c+d x) (a \cos (c+d x)+b)^3 \left(\frac{3 b \left(b^2-3 a^2\right) \sin (c+d x)}{4 a^2 \left(b^2-a^2\right)^2}+\frac{11 a^2 b^2 \sin (c+d x)-5 b^4 \sin (c+d x)}{4 a^2 \left(a^2-b^2\right)^2 (a \cos (c+d x)+b)}-\frac{b^3 \sin (c+d x)}{2 a^2 \left(a^2-b^2\right) (a \cos (c+d x)+b)^2}\right)}{d (a+b \sec (c+d x))^3}+\frac{\sec ^3(c+d x) (a \cos (c+d x)+b)^3 \left(\frac{2 \left(16 a^3+8 a b^2\right) \sin (c+d x) \cos ^2(c+d x) \sqrt{1-\sec ^2(c+d x)} (a+b \sec (c+d x)) \Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)}{a \left(1-\cos ^2(c+d x)\right) (a \cos (c+d x)+b)}+\frac{2 \left(-5 a^2 b-b^3\right) \sin (c+d x) \cos ^2(c+d x) \sqrt{1-\sec ^2(c+d x)} (a+b \sec (c+d x)) \left(F\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)-\Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)\right)}{b \left(1-\cos ^2(c+d x)\right) (a \cos (c+d x)+b)}+\frac{\left(9 a^2 b-3 b^3\right) \sin (c+d x) \cos (2 (c+d x)) (a+b \sec (c+d x)) \left(2 a^2 \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} \Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)-4 b^2 \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} \Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)+4 a b \sec ^2(c+d x)-2 a (a-2 b) \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} F\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)-4 a b \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} E\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)-4 a b\right)}{a^2 b \left(1-\cos ^2(c+d x)\right) \sqrt{\sec (c+d x)} \left(2-\sec ^2(c+d x)\right) (a \cos (c+d x)+b)}\right)}{16 a d (a-b)^2 (a+b)^2 (a+b \sec (c+d x))^3}","-\frac{b \left(7 a^2-b^2\right) \sin (c+d x) \sqrt{\sec (c+d x)}}{4 a d \left(a^2-b^2\right)^2 (a+b \sec (c+d x))}-\frac{b \sin (c+d x) \sqrt{\sec (c+d x)}}{2 d \left(a^2-b^2\right) (a+b \sec (c+d x))^2}+\frac{3 b \left(3 a^2-b^2\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a^2 d \left(a^2-b^2\right)^2}+\frac{\left(8 a^4-5 a^2 b^2+3 b^4\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a^3 d \left(a^2-b^2\right)^2}-\frac{3 b \left(5 a^4-2 a^2 b^2+b^4\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a^3 d (a-b)^2 (a+b)^3}",1,"((b + a*Cos[c + d*x])^3*Sec[c + d*x]^3*((2*(-5*a^2*b - b^3)*Cos[c + d*x]^2*(EllipticF[ArcSin[Sqrt[Sec[c + d*x]]], -1] - EllipticPi[-(b/a), ArcSin[Sqrt[Sec[c + d*x]]], -1])*(a + b*Sec[c + d*x])*Sqrt[1 - Sec[c + d*x]^2]*Sin[c + d*x])/(b*(b + a*Cos[c + d*x])*(1 - Cos[c + d*x]^2)) + (2*(16*a^3 + 8*a*b^2)*Cos[c + d*x]^2*EllipticPi[-(b/a), ArcSin[Sqrt[Sec[c + d*x]]], -1]*(a + b*Sec[c + d*x])*Sqrt[1 - Sec[c + d*x]^2]*Sin[c + d*x])/(a*(b + a*Cos[c + d*x])*(1 - Cos[c + d*x]^2)) + ((9*a^2*b - 3*b^3)*Cos[2*(c + d*x)]*(a + b*Sec[c + d*x])*(-4*a*b + 4*a*b*Sec[c + d*x]^2 - 4*a*b*EllipticE[ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] - 2*a*(a - 2*b)*EllipticF[ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] + 2*a^2*EllipticPi[-(b/a), ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] - 4*b^2*EllipticPi[-(b/a), ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2])*Sin[c + d*x])/(a^2*b*(b + a*Cos[c + d*x])*(1 - Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]]*(2 - Sec[c + d*x]^2))))/(16*a*(a - b)^2*(a + b)^2*d*(a + b*Sec[c + d*x])^3) + ((b + a*Cos[c + d*x])^3*Sec[c + d*x]^(7/2)*((3*b*(-3*a^2 + b^2)*Sin[c + d*x])/(4*a^2*(-a^2 + b^2)^2) - (b^3*Sin[c + d*x])/(2*a^2*(a^2 - b^2)*(b + a*Cos[c + d*x])^2) + (11*a^2*b^2*Sin[c + d*x] - 5*b^4*Sin[c + d*x])/(4*a^2*(a^2 - b^2)^2*(b + a*Cos[c + d*x]))))/(d*(a + b*Sec[c + d*x])^3)","B",0
626,1,707,342,6.9145172,"\int \frac{1}{\sqrt{\sec (c+d x)} (a+b \sec (c+d x))^3} \, dx","Integrate[1/(Sqrt[Sec[c + d*x]]*(a + b*Sec[c + d*x])^3),x]","\frac{\sqrt{\sec (c+d x)} \left(-\frac{b^2 \left(7 b^2-13 a^2\right) \sin (c+d x)}{4 a^3 \left(b^2-a^2\right)^2}+\frac{b^4 \sin (c+d x)}{2 a^3 \left(a^2-b^2\right) (a \cos (c+d x)+b)^2}+\frac{3 \left(3 b^5 \sin (c+d x)-5 a^2 b^3 \sin (c+d x)\right)}{4 a^3 \left(a^2-b^2\right)^2 (a \cos (c+d x)+b)}\right)}{d}+\frac{\frac{2 \left(8 a b^3-32 a^3 b\right) \sin (c+d x) \cos ^2(c+d x) \sqrt{1-\sec ^2(c+d x)} (a+b \sec (c+d x)) \Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)}{a \left(1-\cos ^2(c+d x)\right) (a \cos (c+d x)+b)}+\frac{2 \left(8 a^4-7 a^2 b^2+5 b^4\right) \sin (c+d x) \cos ^2(c+d x) \sqrt{1-\sec ^2(c+d x)} (a+b \sec (c+d x)) \left(F\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)-\Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)\right)}{b \left(1-\cos ^2(c+d x)\right) (a \cos (c+d x)+b)}+\frac{\left(8 a^4-29 a^2 b^2+15 b^4\right) \sin (c+d x) \cos (2 (c+d x)) (a+b \sec (c+d x)) \left(2 a^2 \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} \Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)-4 b^2 \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} \Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)+4 a b \sec ^2(c+d x)-2 a (a-2 b) \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} F\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)-4 a b \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} E\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)-4 a b\right)}{a^2 b \left(1-\cos ^2(c+d x)\right) \sqrt{\sec (c+d x)} \left(2-\sec ^2(c+d x)\right) (a \cos (c+d x)+b)}}{16 a^2 d (a-b)^2 (a+b)^2}","\frac{b^2 \left(11 a^2-5 b^2\right) \sin (c+d x) \sqrt{\sec (c+d x)}}{4 a^2 d \left(a^2-b^2\right)^2 (a+b \sec (c+d x))}+\frac{b^2 \sin (c+d x) \sqrt{\sec (c+d x)}}{2 a d \left(a^2-b^2\right) (a+b \sec (c+d x))^2}-\frac{3 b \left(8 a^4-11 a^2 b^2+5 b^4\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a^4 d \left(a^2-b^2\right)^2}+\frac{b^2 \left(35 a^4-38 a^2 b^2+15 b^4\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a^4 d (a-b)^2 (a+b)^3}+\frac{\left(8 a^4-29 a^2 b^2+15 b^4\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a^3 d \left(a^2-b^2\right)^2}",1,"((2*(8*a^4 - 7*a^2*b^2 + 5*b^4)*Cos[c + d*x]^2*(EllipticF[ArcSin[Sqrt[Sec[c + d*x]]], -1] - EllipticPi[-(b/a), ArcSin[Sqrt[Sec[c + d*x]]], -1])*(a + b*Sec[c + d*x])*Sqrt[1 - Sec[c + d*x]^2]*Sin[c + d*x])/(b*(b + a*Cos[c + d*x])*(1 - Cos[c + d*x]^2)) + (2*(-32*a^3*b + 8*a*b^3)*Cos[c + d*x]^2*EllipticPi[-(b/a), ArcSin[Sqrt[Sec[c + d*x]]], -1]*(a + b*Sec[c + d*x])*Sqrt[1 - Sec[c + d*x]^2]*Sin[c + d*x])/(a*(b + a*Cos[c + d*x])*(1 - Cos[c + d*x]^2)) + ((8*a^4 - 29*a^2*b^2 + 15*b^4)*Cos[2*(c + d*x)]*(a + b*Sec[c + d*x])*(-4*a*b + 4*a*b*Sec[c + d*x]^2 - 4*a*b*EllipticE[ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] - 2*a*(a - 2*b)*EllipticF[ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] + 2*a^2*EllipticPi[-(b/a), ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] - 4*b^2*EllipticPi[-(b/a), ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2])*Sin[c + d*x])/(a^2*b*(b + a*Cos[c + d*x])*(1 - Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]]*(2 - Sec[c + d*x]^2)))/(16*a^2*(a - b)^2*(a + b)^2*d) + (Sqrt[Sec[c + d*x]]*(-1/4*(b^2*(-13*a^2 + 7*b^2)*Sin[c + d*x])/(a^3*(-a^2 + b^2)^2) + (b^4*Sin[c + d*x])/(2*a^3*(a^2 - b^2)*(b + a*Cos[c + d*x])^2) + (3*(-5*a^2*b^3*Sin[c + d*x] + 3*b^5*Sin[c + d*x]))/(4*a^3*(a^2 - b^2)^2*(b + a*Cos[c + d*x]))))/d","B",0
627,1,731,406,6.9704187,"\int \frac{1}{\sec ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^3} \, dx","Integrate[1/(Sec[c + d*x]^(3/2)*(a + b*Sec[c + d*x])^3),x]","\frac{\sqrt{\sec (c+d x)} \left(\frac{\sin (2 (c+d x))}{3 a^3}-\frac{b^5 \sin (c+d x)}{2 a^4 \left(a^2-b^2\right) (a \cos (c+d x)+b)^2}+\frac{b^3 \left(11 b^2-17 a^2\right) \sin (c+d x)}{4 a^4 \left(b^2-a^2\right)^2}+\frac{19 a^2 b^4 \sin (c+d x)-13 b^6 \sin (c+d x)}{4 a^4 \left(a^2-b^2\right)^2 (a \cos (c+d x)+b)}\right)}{d}+\frac{\frac{2 \left(16 a^5+112 a^3 b^2-56 a b^4\right) \sin (c+d x) \cos ^2(c+d x) \sqrt{1-\sec ^2(c+d x)} (a+b \sec (c+d x)) \Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)}{a \left(1-\cos ^2(c+d x)\right) (a \cos (c+d x)+b)}+\frac{2 \left(-56 a^4 b+73 a^2 b^3-35 b^5\right) \sin (c+d x) \cos ^2(c+d x) \sqrt{1-\sec ^2(c+d x)} (a+b \sec (c+d x)) \left(F\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)-\Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)\right)}{b \left(1-\cos ^2(c+d x)\right) (a \cos (c+d x)+b)}+\frac{\left(-72 a^4 b+195 a^2 b^3-105 b^5\right) \sin (c+d x) \cos (2 (c+d x)) (a+b \sec (c+d x)) \left(2 a^2 \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} \Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)-4 b^2 \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} \Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)+4 a b \sec ^2(c+d x)-2 a (a-2 b) \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} F\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)-4 a b \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} E\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)-4 a b\right)}{a^2 b \left(1-\cos ^2(c+d x)\right) \sqrt{\sec (c+d x)} \left(2-\sec ^2(c+d x)\right) (a \cos (c+d x)+b)}}{48 a^3 d (a-b)^2 (a+b)^2}","\frac{b^2 \left(13 a^2-7 b^2\right) \sin (c+d x)}{4 a^2 d \left(a^2-b^2\right)^2 \sqrt{\sec (c+d x)} (a+b \sec (c+d x))}+\frac{b^2 \sin (c+d x)}{2 a d \left(a^2-b^2\right) \sqrt{\sec (c+d x)} (a+b \sec (c+d x))^2}-\frac{b \left(24 a^4-65 a^2 b^2+35 b^4\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a^4 d \left(a^2-b^2\right)^2}-\frac{b^3 \left(63 a^4-86 a^2 b^2+35 b^4\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a^5 d (a-b)^2 (a+b)^3}+\frac{\left(8 a^4-61 a^2 b^2+35 b^4\right) \sin (c+d x)}{12 a^3 d \left(a^2-b^2\right)^2 \sqrt{\sec (c+d x)}}+\frac{\left(8 a^6+128 a^4 b^2-223 a^2 b^4+105 b^6\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{12 a^5 d \left(a^2-b^2\right)^2}",1,"((2*(-56*a^4*b + 73*a^2*b^3 - 35*b^5)*Cos[c + d*x]^2*(EllipticF[ArcSin[Sqrt[Sec[c + d*x]]], -1] - EllipticPi[-(b/a), ArcSin[Sqrt[Sec[c + d*x]]], -1])*(a + b*Sec[c + d*x])*Sqrt[1 - Sec[c + d*x]^2]*Sin[c + d*x])/(b*(b + a*Cos[c + d*x])*(1 - Cos[c + d*x]^2)) + (2*(16*a^5 + 112*a^3*b^2 - 56*a*b^4)*Cos[c + d*x]^2*EllipticPi[-(b/a), ArcSin[Sqrt[Sec[c + d*x]]], -1]*(a + b*Sec[c + d*x])*Sqrt[1 - Sec[c + d*x]^2]*Sin[c + d*x])/(a*(b + a*Cos[c + d*x])*(1 - Cos[c + d*x]^2)) + ((-72*a^4*b + 195*a^2*b^3 - 105*b^5)*Cos[2*(c + d*x)]*(a + b*Sec[c + d*x])*(-4*a*b + 4*a*b*Sec[c + d*x]^2 - 4*a*b*EllipticE[ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] - 2*a*(a - 2*b)*EllipticF[ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] + 2*a^2*EllipticPi[-(b/a), ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] - 4*b^2*EllipticPi[-(b/a), ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2])*Sin[c + d*x])/(a^2*b*(b + a*Cos[c + d*x])*(1 - Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]]*(2 - Sec[c + d*x]^2)))/(48*a^3*(a - b)^2*(a + b)^2*d) + (Sqrt[Sec[c + d*x]]*((b^3*(-17*a^2 + 11*b^2)*Sin[c + d*x])/(4*a^4*(-a^2 + b^2)^2) - (b^5*Sin[c + d*x])/(2*a^4*(a^2 - b^2)*(b + a*Cos[c + d*x])^2) + (19*a^2*b^4*Sin[c + d*x] - 13*b^6*Sin[c + d*x])/(4*a^4*(a^2 - b^2)^2*(b + a*Cos[c + d*x])) + Sin[2*(c + d*x)]/(3*a^3)))/d","A",0
628,1,321,237,5.6421301,"\int \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+b \sec (c+d x)} \, dx","Integrate[Sec[c + d*x]^(3/2)*Sqrt[a + b*Sec[c + d*x]],x]","\frac{\sqrt{a+b \sec (c+d x)} \left(\frac{2 a \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{(a+b) \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}-\frac{2 i \csc (c+d x) \sqrt{-\frac{a (\cos (c+d x)-1)}{a+b}} \sqrt{\frac{a (\cos (c+d x)+1)}{a-b}} \left(a \left(2 b F\left(i \sinh ^{-1}\left(\sqrt{\frac{1}{a-b}} \sqrt{b+a \cos (c+d x)}\right)|\frac{b-a}{a+b}\right)+a \Pi \left(1-\frac{a}{b};i \sinh ^{-1}\left(\sqrt{\frac{1}{a-b}} \sqrt{b+a \cos (c+d x)}\right)|\frac{b-a}{a+b}\right)\right)-2 b (a+b) E\left(i \sinh ^{-1}\left(\sqrt{\frac{1}{a-b}} \sqrt{b+a \cos (c+d x)}\right)|\frac{b-a}{a+b}\right)\right)}{a b \sqrt{\frac{1}{a-b}} \sqrt{a \cos (c+d x)+b}}+4 \tan (c+d x)\right)}{4 d \sqrt{\sec (c+d x)}}","\frac{\sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a+b \sec (c+d x)}}{d}+\frac{b \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{d \sqrt{a+b \sec (c+d x)}}-\frac{\sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{d \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{a \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{d \sqrt{a+b \sec (c+d x)}}",1,"(Sqrt[a + b*Sec[c + d*x]]*((2*a*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)])/((a + b)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) - ((2*I)*Sqrt[-((a*(-1 + Cos[c + d*x]))/(a + b))]*Sqrt[(a*(1 + Cos[c + d*x]))/(a - b)]*Csc[c + d*x]*(-2*b*(a + b)*EllipticE[I*ArcSinh[Sqrt[(a - b)^(-1)]*Sqrt[b + a*Cos[c + d*x]]], (-a + b)/(a + b)] + a*(2*b*EllipticF[I*ArcSinh[Sqrt[(a - b)^(-1)]*Sqrt[b + a*Cos[c + d*x]]], (-a + b)/(a + b)] + a*EllipticPi[1 - a/b, I*ArcSinh[Sqrt[(a - b)^(-1)]*Sqrt[b + a*Cos[c + d*x]]], (-a + b)/(a + b)])))/(a*Sqrt[(a - b)^(-1)]*b*Sqrt[b + a*Cos[c + d*x]]) + 4*Tan[c + d*x]))/(4*d*Sqrt[Sec[c + d*x]])","C",1
629,1,96,138,2.3086603,"\int \sqrt{\sec (c+d x)} \sqrt{a+b \sec (c+d x)} \, dx","Integrate[Sqrt[Sec[c + d*x]]*Sqrt[a + b*Sec[c + d*x]],x]","\frac{2 \sqrt{a+b \sec (c+d x)} \left(a F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)+b \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)\right)}{d (a+b) \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}","\frac{2 a \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{d \sqrt{a+b \sec (c+d x)}}+\frac{2 b \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{d \sqrt{a+b \sec (c+d x)}}",1,"(2*(a*EllipticF[(c + d*x)/2, (2*a)/(a + b)] + b*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)])*Sqrt[a + b*Sec[c + d*x]])/((a + b)*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]])","A",1
630,1,67,67,0.1176031,"\int \frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{\sec (c+d x)}} \, dx","Integrate[Sqrt[a + b*Sec[c + d*x]]/Sqrt[Sec[c + d*x]],x]","\frac{2 \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{d \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}","\frac{2 \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{d \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}",1,"(2*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]])","A",1
631,1,156,192,0.6299493,"\int \frac{\sqrt{a+b \sec (c+d x)}}{\sec ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[Sqrt[a + b*Sec[c + d*x]]/Sec[c + d*x]^(3/2),x]","\frac{2 \sqrt{a+b \sec (c+d x)} \left(\left(a^2-b^2\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)+a \sin (c+d x) (a \cos (c+d x)+b)+b (a+b) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)\right)}{3 a d \sqrt{\sec (c+d x)} (a \cos (c+d x)+b)}","\frac{2 \left(a^2-b^2\right) \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{3 a d \sqrt{a+b \sec (c+d x)}}+\frac{2 \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{3 d \sqrt{\sec (c+d x)}}+\frac{2 b \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{3 a d \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}",1,"(2*Sqrt[a + b*Sec[c + d*x]]*(b*(a + b)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticE[(c + d*x)/2, (2*a)/(a + b)] + (a^2 - b^2)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)] + a*(b + a*Cos[c + d*x])*Sin[c + d*x]))/(3*a*d*(b + a*Cos[c + d*x])*Sqrt[Sec[c + d*x]])","A",1
632,1,203,244,0.9465533,"\int \frac{\sqrt{a+b \sec (c+d x)}}{\sec ^{\frac{5}{2}}(c+d x)} \, dx","Integrate[Sqrt[a + b*Sec[c + d*x]]/Sec[c + d*x]^(5/2),x]","\frac{\sqrt{a+b \sec (c+d x)} \left(2 a \sin (c+d x) \left(3 a^2 \cos (2 (c+d x))+3 a^2+8 a b \cos (c+d x)+2 b^2\right)+8 b \left(b^2-a^2\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)+4 \left(9 a^3+9 a^2 b-2 a b^2-2 b^3\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)\right)}{30 a^2 d \sqrt{\sec (c+d x)} (a \cos (c+d x)+b)}","-\frac{4 b \left(a^2-b^2\right) \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{15 a^2 d \sqrt{a+b \sec (c+d x)}}+\frac{2 \left(9 a^2-2 b^2\right) \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{15 a^2 d \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{2 \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{5 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 b \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{15 a d \sqrt{\sec (c+d x)}}",1,"(Sqrt[a + b*Sec[c + d*x]]*(4*(9*a^3 + 9*a^2*b - 2*a*b^2 - 2*b^3)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticE[(c + d*x)/2, (2*a)/(a + b)] + 8*b*(-a^2 + b^2)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)] + 2*a*(3*a^2 + 2*b^2 + 8*a*b*Cos[c + d*x] + 3*a^2*Cos[2*(c + d*x)])*Sin[c + d*x]))/(30*a^2*d*(b + a*Cos[c + d*x])*Sqrt[Sec[c + d*x]])","A",1
633,1,237,305,1.3406746,"\int \frac{\sqrt{a+b \sec (c+d x)}}{\sec ^{\frac{7}{2}}(c+d x)} \, dx","Integrate[Sqrt[a + b*Sec[c + d*x]]/Sec[c + d*x]^(7/2),x]","\frac{\sqrt{a+b \sec (c+d x)} \left(8 \left(25 a^4-17 a^2 b^2-8 b^4\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)+2 a \sin (c+d x) \left(15 a^3 \cos (3 (c+d x))+a \left(145 a^2-4 b^2\right) \cos (c+d x)+36 a^2 b \cos (2 (c+d x))+136 a^2 b-16 b^3\right)+8 b \left(19 a^3+19 a^2 b+8 a b^2+8 b^3\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)\right)}{420 a^3 d \sqrt{\sec (c+d x)} (a \cos (c+d x)+b)}","\frac{2 \left(25 a^2-4 b^2\right) \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{105 a^2 d \sqrt{\sec (c+d x)}}+\frac{2 b \left(19 a^2+8 b^2\right) \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{105 a^3 d \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{2 \left(25 a^4-17 a^2 b^2-8 b^4\right) \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{105 a^3 d \sqrt{a+b \sec (c+d x)}}+\frac{2 b \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{35 a d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{7 d \sec ^{\frac{5}{2}}(c+d x)}",1,"(Sqrt[a + b*Sec[c + d*x]]*(8*b*(19*a^3 + 19*a^2*b + 8*a*b^2 + 8*b^3)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticE[(c + d*x)/2, (2*a)/(a + b)] + 8*(25*a^4 - 17*a^2*b^2 - 8*b^4)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)] + 2*a*(136*a^2*b - 16*b^3 + a*(145*a^2 - 4*b^2)*Cos[c + d*x] + 36*a^2*b*Cos[2*(c + d*x)] + 15*a^3*Cos[3*(c + d*x)])*Sin[c + d*x]))/(420*a^3*d*(b + a*Cos[c + d*x])*Sqrt[Sec[c + d*x]])","A",1
634,1,549,299,6.6016725,"\int \sec ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2} \, dx","Integrate[Sec[c + d*x]^(3/2)*(a + b*Sec[c + d*x])^(3/2),x]","\frac{(a+b \sec (c+d x))^{3/2} \left(\frac{5}{4} a \tan (c+d x)+\frac{1}{2} b \tan (c+d x) \sec (c+d x)\right)}{d \sec ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+b)}+\frac{(a+b \sec (c+d x))^{3/2} \left(-\frac{2 \left(-a^2-8 b^2\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{\sqrt{a \cos (c+d x)+b}}-\frac{10 i a^2 \sin (c+d x) \cos (2 (c+d x)) \sqrt{\frac{a-a \cos (c+d x)}{a+b}} \sqrt{\frac{a \cos (c+d x)+a}{a-b}} \left(a \left(2 b F\left(i \sinh ^{-1}\left(\sqrt{\frac{1}{a-b}} \sqrt{b+a \cos (c+d x)}\right)|\frac{b-a}{a+b}\right)+a \Pi \left(1-\frac{a}{b};i \sinh ^{-1}\left(\sqrt{\frac{1}{a-b}} \sqrt{b+a \cos (c+d x)}\right)|\frac{b-a}{a+b}\right)\right)-2 b (a+b) E\left(i \sinh ^{-1}\left(\sqrt{\frac{1}{a-b}} \sqrt{b+a \cos (c+d x)}\right)|\frac{b-a}{a+b}\right)\right)}{b \sqrt{\frac{1}{a-b}} \sqrt{1-\cos ^2(c+d x)} \sqrt{\frac{a^2-a^2 \cos ^2(c+d x)}{a^2}} \left(-a^2+2 (a \cos (c+d x)+b)^2-4 b (a \cos (c+d x)+b)+2 b^2\right)}+\frac{8 a b \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{\sqrt{a \cos (c+d x)+b}}\right)}{16 d \sec ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+b)^{3/2}}","\frac{\left(3 a^2+4 b^2\right) \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{4 d \sqrt{a+b \sec (c+d x)}}+\frac{b \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+b \sec (c+d x)}}{2 d}+\frac{5 a \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a+b \sec (c+d x)}}{4 d}+\frac{7 a b \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{4 d \sqrt{a+b \sec (c+d x)}}-\frac{5 a \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{4 d \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}",1,"((a + b*Sec[c + d*x])^(3/2)*((8*a*b*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/Sqrt[b + a*Cos[c + d*x]] - (2*(-a^2 - 8*b^2)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)])/Sqrt[b + a*Cos[c + d*x]] - ((10*I)*a^2*Sqrt[(a - a*Cos[c + d*x])/(a + b)]*Sqrt[(a + a*Cos[c + d*x])/(a - b)]*Cos[2*(c + d*x)]*(-2*b*(a + b)*EllipticE[I*ArcSinh[Sqrt[(a - b)^(-1)]*Sqrt[b + a*Cos[c + d*x]]], (-a + b)/(a + b)] + a*(2*b*EllipticF[I*ArcSinh[Sqrt[(a - b)^(-1)]*Sqrt[b + a*Cos[c + d*x]]], (-a + b)/(a + b)] + a*EllipticPi[1 - a/b, I*ArcSinh[Sqrt[(a - b)^(-1)]*Sqrt[b + a*Cos[c + d*x]]], (-a + b)/(a + b)]))*Sin[c + d*x])/(Sqrt[(a - b)^(-1)]*b*Sqrt[1 - Cos[c + d*x]^2]*Sqrt[(a^2 - a^2*Cos[c + d*x]^2)/a^2]*(-a^2 + 2*b^2 - 4*b*(b + a*Cos[c + d*x]) + 2*(b + a*Cos[c + d*x])^2))))/(16*d*(b + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(3/2)) + ((a + b*Sec[c + d*x])^(3/2)*((5*a*Tan[c + d*x])/4 + (b*Sec[c + d*x]*Tan[c + d*x])/2))/(d*(b + a*Cos[c + d*x])*Sec[c + d*x]^(3/2))","C",1
635,1,394,249,8.383779,"\int \sqrt{\sec (c+d x)} (a+b \sec (c+d x))^{3/2} \, dx","Integrate[Sqrt[Sec[c + d*x]]*(a + b*Sec[c + d*x])^(3/2),x]","\frac{(a+b \sec (c+d x))^{3/2} \left(\frac{8 a^2 \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{(a \cos (c+d x)+b)^2}+\frac{4 b \tan (c+d x)}{a \cos (c+d x)+b}+\frac{10 a b \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{(a \cos (c+d x)+b)^2}-\frac{2 i \csc (c+d x) \sqrt{-\frac{a (\cos (c+d x)-1)}{a+b}} \sqrt{\frac{a (\cos (c+d x)+1)}{a-b}} \left(a \left(2 b F\left(i \sinh ^{-1}\left(\sqrt{\frac{1}{a-b}} \sqrt{b+a \cos (c+d x)}\right)|\frac{b-a}{a+b}\right)+a \Pi \left(1-\frac{a}{b};i \sinh ^{-1}\left(\sqrt{\frac{1}{a-b}} \sqrt{b+a \cos (c+d x)}\right)|\frac{b-a}{a+b}\right)\right)-2 b (a+b) E\left(i \sinh ^{-1}\left(\sqrt{\frac{1}{a-b}} \sqrt{b+a \cos (c+d x)}\right)|\frac{b-a}{a+b}\right)\right)}{a \sqrt{\frac{1}{a-b}} (a \cos (c+d x)+b)^{3/2}}\right)}{4 d \sec ^{\frac{3}{2}}(c+d x)}","\frac{\left(2 a^2+b^2\right) \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{d \sqrt{a+b \sec (c+d x)}}+\frac{b \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a+b \sec (c+d x)}}{d}-\frac{b \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{d \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{3 a b \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{d \sqrt{a+b \sec (c+d x)}}",1,"((a + b*Sec[c + d*x])^(3/2)*((8*a^2*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/(b + a*Cos[c + d*x])^2 + (10*a*b*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)])/(b + a*Cos[c + d*x])^2 - ((2*I)*Sqrt[-((a*(-1 + Cos[c + d*x]))/(a + b))]*Sqrt[(a*(1 + Cos[c + d*x]))/(a - b)]*Csc[c + d*x]*(-2*b*(a + b)*EllipticE[I*ArcSinh[Sqrt[(a - b)^(-1)]*Sqrt[b + a*Cos[c + d*x]]], (-a + b)/(a + b)] + a*(2*b*EllipticF[I*ArcSinh[Sqrt[(a - b)^(-1)]*Sqrt[b + a*Cos[c + d*x]]], (-a + b)/(a + b)] + a*EllipticPi[1 - a/b, I*ArcSinh[Sqrt[(a - b)^(-1)]*Sqrt[b + a*Cos[c + d*x]]], (-a + b)/(a + b)])))/(a*Sqrt[(a - b)^(-1)]*(b + a*Cos[c + d*x])^(3/2)) + (4*b*Tan[c + d*x])/(b + a*Cos[c + d*x])))/(4*d*Sec[c + d*x]^(3/2))","C",0
636,1,129,209,2.5535259,"\int \frac{(a+b \sec (c+d x))^{3/2}}{\sqrt{\sec (c+d x)}} \, dx","Integrate[(a + b*Sec[c + d*x])^(3/2)/Sqrt[Sec[c + d*x]],x]","\frac{2 \sqrt{\frac{a \cos (c+d x)+b}{a+b}} (a+b \sec (c+d x))^{3/2} \left(a (a+b) E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)+b \left(a F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)+b \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)\right)\right)}{d \sec ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+b)^2}","\frac{2 b^2 \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{d \sqrt{a+b \sec (c+d x)}}+\frac{2 a b \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{d \sqrt{a+b \sec (c+d x)}}+\frac{2 a \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{d \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}",1,"(2*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*(a*(a + b)*EllipticE[(c + d*x)/2, (2*a)/(a + b)] + b*(a*EllipticF[(c + d*x)/2, (2*a)/(a + b)] + b*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)]))*(a + b*Sec[c + d*x])^(3/2))/(d*(b + a*Cos[c + d*x])^2*Sec[c + d*x]^(3/2))","A",1
637,1,156,187,0.7492229,"\int \frac{(a+b \sec (c+d x))^{3/2}}{\sec ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[(a + b*Sec[c + d*x])^(3/2)/Sec[c + d*x]^(3/2),x]","\frac{(a+b \sec (c+d x))^{3/2} \left(2 \left(a^2-b^2\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)+2 a \sin (c+d x) (a \cos (c+d x)+b)+8 b (a+b) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)\right)}{3 d \sec ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+b)^2}","\frac{2 \left(a^2-b^2\right) \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{3 d \sqrt{a+b \sec (c+d x)}}+\frac{2 a \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{3 d \sqrt{\sec (c+d x)}}+\frac{8 b \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{3 d \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}",1,"((a + b*Sec[c + d*x])^(3/2)*(8*b*(a + b)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticE[(c + d*x)/2, (2*a)/(a + b)] + 2*(a^2 - b^2)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)] + 2*a*(b + a*Cos[c + d*x])*Sin[c + d*x]))/(3*d*(b + a*Cos[c + d*x])^2*Sec[c + d*x]^(3/2))","A",1
638,1,197,240,1.2332626,"\int \frac{(a+b \sec (c+d x))^{3/2}}{\sec ^{\frac{5}{2}}(c+d x)} \, dx","Integrate[(a + b*Sec[c + d*x])^(3/2)/Sec[c + d*x]^(5/2),x]","\frac{(a+b \sec (c+d x))^{3/2} \left(2 a \sin (c+d x) \left(a^2 \cos (2 (c+d x))+a^2+6 a b \cos (c+d x)+4 b^2\right)+4 b \left(a^2-b^2\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)+4 \left(3 a^3+3 a^2 b+a b^2+b^3\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)\right)}{10 a d \sec ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+b)^2}","\frac{2 b \left(a^2-b^2\right) \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{5 a d \sqrt{a+b \sec (c+d x)}}+\frac{2 \left(3 a^2+b^2\right) \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{5 a d \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{2 a \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{5 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{4 b \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{5 d \sqrt{\sec (c+d x)}}",1,"((a + b*Sec[c + d*x])^(3/2)*(4*(3*a^3 + 3*a^2*b + a*b^2 + b^3)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticE[(c + d*x)/2, (2*a)/(a + b)] + 4*b*(a^2 - b^2)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)] + 2*a*(a^2 + 4*b^2 + 6*a*b*Cos[c + d*x] + a^2*Cos[2*(c + d*x)])*Sin[c + d*x]))/(10*a*d*(b + a*Cos[c + d*x])^2*Sec[c + d*x]^(3/2))","A",1
639,1,237,303,1.9713047,"\int \frac{(a+b \sec (c+d x))^{3/2}}{\sec ^{\frac{7}{2}}(c+d x)} \, dx","Integrate[(a + b*Sec[c + d*x])^(3/2)/Sec[c + d*x]^(7/2),x]","\frac{(a+b \sec (c+d x))^{3/2} \left(8 \left(25 a^4-31 a^2 b^2+6 b^4\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)+2 a \sin (c+d x) \left(15 a^3 \cos (3 (c+d x))+a \left(145 a^2+108 b^2\right) \cos (c+d x)+78 a^2 b \cos (2 (c+d x))+178 a^2 b+12 b^3\right)+16 b \left(41 a^3+41 a^2 b-3 a b^2-3 b^3\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)\right)}{420 a^2 d \sec ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+b)^2}","\frac{2 \left(25 a^2+3 b^2\right) \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{105 a d \sqrt{\sec (c+d x)}}+\frac{4 b \left(41 a^2-3 b^2\right) \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{105 a^2 d \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{2 \left(25 a^4-31 a^2 b^2+6 b^4\right) \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{105 a^2 d \sqrt{a+b \sec (c+d x)}}+\frac{16 b \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{35 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 a \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{7 d \sec ^{\frac{5}{2}}(c+d x)}",1,"((a + b*Sec[c + d*x])^(3/2)*(16*b*(41*a^3 + 41*a^2*b - 3*a*b^2 - 3*b^3)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticE[(c + d*x)/2, (2*a)/(a + b)] + 8*(25*a^4 - 31*a^2*b^2 + 6*b^4)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)] + 2*a*(178*a^2*b + 12*b^3 + a*(145*a^2 + 108*b^2)*Cos[c + d*x] + 78*a^2*b*Cos[2*(c + d*x)] + 15*a^3*Cos[3*(c + d*x)])*Sin[c + d*x]))/(420*a^2*d*(b + a*Cos[c + d*x])^2*Sec[c + d*x]^(3/2))","A",1
640,1,602,369,6.686498,"\int \sec ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^{5/2} \, dx","Integrate[Sec[c + d*x]^(3/2)*(a + b*Sec[c + d*x])^(5/2),x]","\frac{(a+b \sec (c+d x))^{5/2} \left(\frac{1}{24} \sec (c+d x) \left(33 a^2 \sin (c+d x)+16 b^2 \sin (c+d x)\right)+\frac{13}{12} a b \tan (c+d x) \sec (c+d x)+\frac{1}{3} b^2 \tan (c+d x) \sec ^2(c+d x)\right)}{d \sec ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+b)^2}-\frac{a (a+b \sec (c+d x))^{5/2} \left(\frac{2 \left(3 a^2-104 b^2\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{\sqrt{a \cos (c+d x)+b}}+\frac{2 i \left(33 a^2+16 b^2\right) \sin (c+d x) \cos (2 (c+d x)) \sqrt{\frac{a-a \cos (c+d x)}{a+b}} \sqrt{\frac{a \cos (c+d x)+a}{a-b}} \left(a \left(2 b F\left(i \sinh ^{-1}\left(\sqrt{\frac{1}{a-b}} \sqrt{b+a \cos (c+d x)}\right)|\frac{b-a}{a+b}\right)+a \Pi \left(1-\frac{a}{b};i \sinh ^{-1}\left(\sqrt{\frac{1}{a-b}} \sqrt{b+a \cos (c+d x)}\right)|\frac{b-a}{a+b}\right)\right)-2 b (a+b) E\left(i \sinh ^{-1}\left(\sqrt{\frac{1}{a-b}} \sqrt{b+a \cos (c+d x)}\right)|\frac{b-a}{a+b}\right)\right)}{b \sqrt{\frac{1}{a-b}} \sqrt{1-\cos ^2(c+d x)} \sqrt{\frac{a^2-a^2 \cos ^2(c+d x)}{a^2}} \left(-a^2+2 (a \cos (c+d x)+b)^2-4 b (a \cos (c+d x)+b)+2 b^2\right)}-\frac{104 a b \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{\sqrt{a \cos (c+d x)+b}}\right)}{96 d \sec ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+b)^{5/2}}","\frac{\left(33 a^2+16 b^2\right) \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a+b \sec (c+d x)}}{24 d}+\frac{b \left(59 a^2+16 b^2\right) \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{24 d \sqrt{a+b \sec (c+d x)}}-\frac{\left(33 a^2+16 b^2\right) \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{24 d \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{5 a \left(a^2+4 b^2\right) \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{8 d \sqrt{a+b \sec (c+d x)}}+\frac{b^2 \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) \sqrt{a+b \sec (c+d x)}}{3 d}+\frac{13 a b \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+b \sec (c+d x)}}{12 d}",1,"-1/96*(a*(a + b*Sec[c + d*x])^(5/2)*((-104*a*b*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/Sqrt[b + a*Cos[c + d*x]] + (2*(3*a^2 - 104*b^2)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)])/Sqrt[b + a*Cos[c + d*x]] + ((2*I)*(33*a^2 + 16*b^2)*Sqrt[(a - a*Cos[c + d*x])/(a + b)]*Sqrt[(a + a*Cos[c + d*x])/(a - b)]*Cos[2*(c + d*x)]*(-2*b*(a + b)*EllipticE[I*ArcSinh[Sqrt[(a - b)^(-1)]*Sqrt[b + a*Cos[c + d*x]]], (-a + b)/(a + b)] + a*(2*b*EllipticF[I*ArcSinh[Sqrt[(a - b)^(-1)]*Sqrt[b + a*Cos[c + d*x]]], (-a + b)/(a + b)] + a*EllipticPi[1 - a/b, I*ArcSinh[Sqrt[(a - b)^(-1)]*Sqrt[b + a*Cos[c + d*x]]], (-a + b)/(a + b)]))*Sin[c + d*x])/(Sqrt[(a - b)^(-1)]*b*Sqrt[1 - Cos[c + d*x]^2]*Sqrt[(a^2 - a^2*Cos[c + d*x]^2)/a^2]*(-a^2 + 2*b^2 - 4*b*(b + a*Cos[c + d*x]) + 2*(b + a*Cos[c + d*x])^2))))/(d*(b + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(5/2)) + ((a + b*Sec[c + d*x])^(5/2)*((Sec[c + d*x]*(33*a^2*Sin[c + d*x] + 16*b^2*Sin[c + d*x]))/24 + (13*a*b*Sec[c + d*x]*Tan[c + d*x])/12 + (b^2*Sec[c + d*x]^2*Tan[c + d*x])/3))/(d*(b + a*Cos[c + d*x])^2*Sec[c + d*x]^(5/2))","C",1
641,1,560,314,6.5967211,"\int \sqrt{\sec (c+d x)} (a+b \sec (c+d x))^{5/2} \, dx","Integrate[Sqrt[Sec[c + d*x]]*(a + b*Sec[c + d*x])^(5/2),x]","\frac{(a+b \sec (c+d x))^{5/2} \left(\frac{9}{4} a b \tan (c+d x)+\frac{1}{2} b^2 \tan (c+d x) \sec (c+d x)\right)}{d \sec ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+b)^2}+\frac{(a+b \sec (c+d x))^{5/2} \left(\frac{2 \left(16 a^3+4 a b^2\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{\sqrt{a \cos (c+d x)+b}}+\frac{2 \left(21 a^2 b+8 b^3\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{\sqrt{a \cos (c+d x)+b}}-\frac{18 i a^2 \sin (c+d x) \cos (2 (c+d x)) \sqrt{\frac{a-a \cos (c+d x)}{a+b}} \sqrt{\frac{a \cos (c+d x)+a}{a-b}} \left(a \left(2 b F\left(i \sinh ^{-1}\left(\sqrt{\frac{1}{a-b}} \sqrt{b+a \cos (c+d x)}\right)|\frac{b-a}{a+b}\right)+a \Pi \left(1-\frac{a}{b};i \sinh ^{-1}\left(\sqrt{\frac{1}{a-b}} \sqrt{b+a \cos (c+d x)}\right)|\frac{b-a}{a+b}\right)\right)-2 b (a+b) E\left(i \sinh ^{-1}\left(\sqrt{\frac{1}{a-b}} \sqrt{b+a \cos (c+d x)}\right)|\frac{b-a}{a+b}\right)\right)}{\sqrt{\frac{1}{a-b}} \sqrt{1-\cos ^2(c+d x)} \sqrt{\frac{a^2-a^2 \cos ^2(c+d x)}{a^2}} \left(-a^2+2 (a \cos (c+d x)+b)^2-4 b (a \cos (c+d x)+b)+2 b^2\right)}\right)}{16 d \sec ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+b)^{5/2}}","\frac{a \left(8 a^2+11 b^2\right) \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{4 d \sqrt{a+b \sec (c+d x)}}+\frac{b \left(15 a^2+4 b^2\right) \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{4 d \sqrt{a+b \sec (c+d x)}}+\frac{b^2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+b \sec (c+d x)}}{2 d}+\frac{9 a b \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a+b \sec (c+d x)}}{4 d}-\frac{9 a b \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{4 d \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}",1,"((a + b*Sec[c + d*x])^(5/2)*((2*(16*a^3 + 4*a*b^2)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/Sqrt[b + a*Cos[c + d*x]] + (2*(21*a^2*b + 8*b^3)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)])/Sqrt[b + a*Cos[c + d*x]] - ((18*I)*a^2*Sqrt[(a - a*Cos[c + d*x])/(a + b)]*Sqrt[(a + a*Cos[c + d*x])/(a - b)]*Cos[2*(c + d*x)]*(-2*b*(a + b)*EllipticE[I*ArcSinh[Sqrt[(a - b)^(-1)]*Sqrt[b + a*Cos[c + d*x]]], (-a + b)/(a + b)] + a*(2*b*EllipticF[I*ArcSinh[Sqrt[(a - b)^(-1)]*Sqrt[b + a*Cos[c + d*x]]], (-a + b)/(a + b)] + a*EllipticPi[1 - a/b, I*ArcSinh[Sqrt[(a - b)^(-1)]*Sqrt[b + a*Cos[c + d*x]]], (-a + b)/(a + b)]))*Sin[c + d*x])/(Sqrt[(a - b)^(-1)]*Sqrt[1 - Cos[c + d*x]^2]*Sqrt[(a^2 - a^2*Cos[c + d*x]^2)/a^2]*(-a^2 + 2*b^2 - 4*b*(b + a*Cos[c + d*x]) + 2*(b + a*Cos[c + d*x])^2))))/(16*d*(b + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(5/2)) + ((a + b*Sec[c + d*x])^(5/2)*((9*a*b*Tan[c + d*x])/4 + (b^2*Sec[c + d*x]*Tan[c + d*x])/2))/(d*(b + a*Cos[c + d*x])^2*Sec[c + d*x]^(5/2))","C",1
642,1,538,263,6.6229451,"\int \frac{(a+b \sec (c+d x))^{5/2}}{\sqrt{\sec (c+d x)}} \, dx","Integrate[(a + b*Sec[c + d*x])^(5/2)/Sqrt[Sec[c + d*x]],x]","\frac{b^2 \sin (c+d x) (a+b \sec (c+d x))^{5/2}}{d \sec ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+b)^2}+\frac{a (a+b \sec (c+d x))^{5/2} \left(\frac{2 \left(2 a^2+9 b^2\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{\sqrt{a \cos (c+d x)+b}}+\frac{2 i \left(2 a^2-b^2\right) \sin (c+d x) \cos (2 (c+d x)) \sqrt{\frac{a-a \cos (c+d x)}{a+b}} \sqrt{\frac{a \cos (c+d x)+a}{a-b}} \left(a \left(2 b F\left(i \sinh ^{-1}\left(\sqrt{\frac{1}{a-b}} \sqrt{b+a \cos (c+d x)}\right)|\frac{b-a}{a+b}\right)+a \Pi \left(1-\frac{a}{b};i \sinh ^{-1}\left(\sqrt{\frac{1}{a-b}} \sqrt{b+a \cos (c+d x)}\right)|\frac{b-a}{a+b}\right)\right)-2 b (a+b) E\left(i \sinh ^{-1}\left(\sqrt{\frac{1}{a-b}} \sqrt{b+a \cos (c+d x)}\right)|\frac{b-a}{a+b}\right)\right)}{b \sqrt{\frac{1}{a-b}} \sqrt{1-\cos ^2(c+d x)} \sqrt{\frac{a^2-a^2 \cos ^2(c+d x)}{a^2}} \left(-a^2+2 (a \cos (c+d x)+b)^2-4 b (a \cos (c+d x)+b)+2 b^2\right)}+\frac{24 a b \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{\sqrt{a \cos (c+d x)+b}}\right)}{4 d \sec ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+b)^{5/2}}","\frac{b \left(4 a^2+b^2\right) \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{d \sqrt{a+b \sec (c+d x)}}+\frac{\left(2 a^2-b^2\right) \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{d \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{b^2 \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a+b \sec (c+d x)}}{d}+\frac{5 a b^2 \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{d \sqrt{a+b \sec (c+d x)}}",1,"(b^2*(a + b*Sec[c + d*x])^(5/2)*Sin[c + d*x])/(d*(b + a*Cos[c + d*x])^2*Sec[c + d*x]^(3/2)) + (a*(a + b*Sec[c + d*x])^(5/2)*((24*a*b*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/Sqrt[b + a*Cos[c + d*x]] + (2*(2*a^2 + 9*b^2)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)])/Sqrt[b + a*Cos[c + d*x]] + ((2*I)*(2*a^2 - b^2)*Sqrt[(a - a*Cos[c + d*x])/(a + b)]*Sqrt[(a + a*Cos[c + d*x])/(a - b)]*Cos[2*(c + d*x)]*(-2*b*(a + b)*EllipticE[I*ArcSinh[Sqrt[(a - b)^(-1)]*Sqrt[b + a*Cos[c + d*x]]], (-a + b)/(a + b)] + a*(2*b*EllipticF[I*ArcSinh[Sqrt[(a - b)^(-1)]*Sqrt[b + a*Cos[c + d*x]]], (-a + b)/(a + b)] + a*EllipticPi[1 - a/b, I*ArcSinh[Sqrt[(a - b)^(-1)]*Sqrt[b + a*Cos[c + d*x]]], (-a + b)/(a + b)]))*Sin[c + d*x])/(Sqrt[(a - b)^(-1)]*b*Sqrt[1 - Cos[c + d*x]^2]*Sqrt[(a^2 - a^2*Cos[c + d*x]^2)/a^2]*(-a^2 + 2*b^2 - 4*b*(b + a*Cos[c + d*x]) + 2*(b + a*Cos[c + d*x])^2))))/(4*d*(b + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(5/2))","C",0
643,1,409,262,6.5069383,"\int \frac{(a+b \sec (c+d x))^{5/2}}{\sec ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[(a + b*Sec[c + d*x])^(5/2)/Sec[c + d*x]^(3/2),x]","\frac{(a+b \sec (c+d x))^{5/2} \left(\frac{2 a \left(a^2+9 b^2\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{(a \cos (c+d x)+b)^3}+\frac{b \left(7 a^2+6 b^2\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{(a \cos (c+d x)+b)^3}+\frac{2 a^2 \sin (c+d x)}{(a \cos (c+d x)+b)^2}+\frac{7 i \csc (c+d x) \sqrt{-\frac{a (\cos (c+d x)-1)}{a+b}} \sqrt{\frac{a (\cos (c+d x)+1)}{a-b}} \left(a \left(2 b F\left(i \sinh ^{-1}\left(\sqrt{\frac{1}{a-b}} \sqrt{b+a \cos (c+d x)}\right)|\frac{b-a}{a+b}\right)+a \Pi \left(1-\frac{a}{b};i \sinh ^{-1}\left(\sqrt{\frac{1}{a-b}} \sqrt{b+a \cos (c+d x)}\right)|\frac{b-a}{a+b}\right)\right)-2 b (a+b) E\left(i \sinh ^{-1}\left(\sqrt{\frac{1}{a-b}} \sqrt{b+a \cos (c+d x)}\right)|\frac{b-a}{a+b}\right)\right)}{\sqrt{\frac{1}{a-b}} (a \cos (c+d x)+b)^{5/2}}\right)}{3 d \sec ^{\frac{5}{2}}(c+d x)}","\frac{2 a \left(a^2+2 b^2\right) \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{3 d \sqrt{a+b \sec (c+d x)}}+\frac{2 a^2 \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{3 d \sqrt{\sec (c+d x)}}+\frac{2 b^3 \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{d \sqrt{a+b \sec (c+d x)}}+\frac{14 a b \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{3 d \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}",1,"((a + b*Sec[c + d*x])^(5/2)*((2*a*(a^2 + 9*b^2)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/(b + a*Cos[c + d*x])^3 + (b*(7*a^2 + 6*b^2)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)])/(b + a*Cos[c + d*x])^3 + ((7*I)*Sqrt[-((a*(-1 + Cos[c + d*x]))/(a + b))]*Sqrt[(a*(1 + Cos[c + d*x]))/(a - b)]*Csc[c + d*x]*(-2*b*(a + b)*EllipticE[I*ArcSinh[Sqrt[(a - b)^(-1)]*Sqrt[b + a*Cos[c + d*x]]], (-a + b)/(a + b)] + a*(2*b*EllipticF[I*ArcSinh[Sqrt[(a - b)^(-1)]*Sqrt[b + a*Cos[c + d*x]]], (-a + b)/(a + b)] + a*EllipticPi[1 - a/b, I*ArcSinh[Sqrt[(a - b)^(-1)]*Sqrt[b + a*Cos[c + d*x]]], (-a + b)/(a + b)])))/(Sqrt[(a - b)^(-1)]*(b + a*Cos[c + d*x])^(5/2)) + (2*a^2*Sin[c + d*x])/(b + a*Cos[c + d*x])^2))/(3*d*Sec[c + d*x]^(5/2))","C",0
644,1,200,239,1.7981843,"\int \frac{(a+b \sec (c+d x))^{5/2}}{\sec ^{\frac{5}{2}}(c+d x)} \, dx","Integrate[(a + b*Sec[c + d*x])^(5/2)/Sec[c + d*x]^(5/2),x]","\frac{(a+b \sec (c+d x))^{5/2} \left(2 a \sin (c+d x) \left(3 a^2 \cos (2 (c+d x))+3 a^2+28 a b \cos (c+d x)+22 b^2\right)+32 b \left(a^2-b^2\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)+4 \left(9 a^3+9 a^2 b+23 a b^2+23 b^3\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)\right)}{30 d \sec ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+b)^3}","\frac{16 b \left(a^2-b^2\right) \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{15 d \sqrt{a+b \sec (c+d x)}}+\frac{2 \left(9 a^2+23 b^2\right) \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{15 d \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{2 a^2 \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{5 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{22 a b \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{15 d \sqrt{\sec (c+d x)}}",1,"((a + b*Sec[c + d*x])^(5/2)*(4*(9*a^3 + 9*a^2*b + 23*a*b^2 + 23*b^3)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticE[(c + d*x)/2, (2*a)/(a + b)] + 32*b*(a^2 - b^2)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)] + 2*a*(3*a^2 + 22*b^2 + 28*a*b*Cos[c + d*x] + 3*a^2*Cos[2*(c + d*x)])*Sin[c + d*x]))/(30*d*(b + a*Cos[c + d*x])^3*Sec[c + d*x]^(5/2))","A",1
645,1,237,303,2.4803849,"\int \frac{(a+b \sec (c+d x))^{5/2}}{\sec ^{\frac{7}{2}}(c+d x)} \, dx","Integrate[(a + b*Sec[c + d*x])^(5/2)/Sec[c + d*x]^(7/2),x]","\frac{(a+b \sec (c+d x))^{5/2} \left(8 \left(5 a^4-2 a^2 b^2-3 b^4\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)+2 a \sin (c+d x) \left(3 a^3 \cos (3 (c+d x))+a \left(29 a^2+72 b^2\right) \cos (c+d x)+24 a^2 b \cos (2 (c+d x))+44 a^2 b+36 b^3\right)+8 b \left(29 a^3+29 a^2 b+3 a b^2+3 b^3\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)\right)}{84 a d \sec ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+b)^3}","\frac{2 \left(5 a^2+9 b^2\right) \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{21 d \sqrt{\sec (c+d x)}}+\frac{2 b \left(29 a^2+3 b^2\right) \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{21 a d \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{2 a^2 \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{7 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 \left(5 a^4-2 a^2 b^2-3 b^4\right) \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{21 a d \sqrt{a+b \sec (c+d x)}}+\frac{6 a b \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{7 d \sec ^{\frac{3}{2}}(c+d x)}",1,"((a + b*Sec[c + d*x])^(5/2)*(8*b*(29*a^3 + 29*a^2*b + 3*a*b^2 + 3*b^3)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticE[(c + d*x)/2, (2*a)/(a + b)] + 8*(5*a^4 - 2*a^2*b^2 - 3*b^4)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)] + 2*a*(44*a^2*b + 36*b^3 + a*(29*a^2 + 72*b^2)*Cos[c + d*x] + 24*a^2*b*Cos[2*(c + d*x)] + 3*a^3*Cos[3*(c + d*x)])*Sin[c + d*x]))/(84*a*d*(b + a*Cos[c + d*x])^3*Sec[c + d*x]^(5/2))","A",1
646,1,286,363,2.8896646,"\int \frac{(a+b \sec (c+d x))^{5/2}}{\sec ^{\frac{9}{2}}(c+d x)} \, dx","Integrate[(a + b*Sec[c + d*x])^(5/2)/Sec[c + d*x]^(9/2),x]","\frac{(a+b \sec (c+d x))^{5/2} \left(32 b \left(57 a^4-62 a^2 b^2+5 b^4\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)+2 a \sin (c+d x) \left(35 a^4 \cos (4 (c+d x))+301 a^4+260 a^3 b \cos (3 (c+d x))+4 a b \left(619 a^2+160 b^2\right) \cos (c+d x)+1984 a^2 b^2+8 \left(42 a^4+85 a^2 b^2\right) \cos (2 (c+d x))+40 b^4\right)+16 \left(147 a^5+147 a^4 b+279 a^3 b^2+279 a^2 b^3-10 a b^4-10 b^5\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)\right)}{2520 a^2 d \sec ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+b)^3}","\frac{2 \left(49 a^2+75 b^2\right) \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{315 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 b \left(163 a^2+5 b^2\right) \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{315 a d \sqrt{\sec (c+d x)}}+\frac{2 a^2 \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{9 d \sec ^{\frac{7}{2}}(c+d x)}+\frac{4 b \left(57 a^4-62 a^2 b^2+5 b^4\right) \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{315 a^2 d \sqrt{a+b \sec (c+d x)}}+\frac{2 \left(147 a^4+279 a^2 b^2-10 b^4\right) \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{315 a^2 d \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{38 a b \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{63 d \sec ^{\frac{5}{2}}(c+d x)}",1,"((a + b*Sec[c + d*x])^(5/2)*(16*(147*a^5 + 147*a^4*b + 279*a^3*b^2 + 279*a^2*b^3 - 10*a*b^4 - 10*b^5)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticE[(c + d*x)/2, (2*a)/(a + b)] + 32*b*(57*a^4 - 62*a^2*b^2 + 5*b^4)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)] + 2*a*(301*a^4 + 1984*a^2*b^2 + 40*b^4 + 4*a*b*(619*a^2 + 160*b^2)*Cos[c + d*x] + 8*(42*a^4 + 85*a^2*b^2)*Cos[2*(c + d*x)] + 260*a^3*b*Cos[3*(c + d*x)] + 35*a^4*Cos[4*(c + d*x)])*Sin[c + d*x]))/(2520*a^2*d*(b + a*Cos[c + d*x])^3*Sec[c + d*x]^(5/2))","A",1
647,1,397,312,6.440598,"\int \frac{\sec ^{\frac{7}{2}}(c+d x)}{\sqrt{a+b \sec (c+d x)}} \, dx","Integrate[Sec[c + d*x]^(7/2)/Sqrt[a + b*Sec[c + d*x]],x]","\frac{\sqrt{\sec (c+d x)} \left(b \left(9 a^2+8 b^2\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)+4 a b^2 \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)+2 b \tan (c+d x) \sec (c+d x) (2 b-3 a \cos (c+d x)) (a \cos (c+d x)+b)+\frac{3 i \csc (c+d x) \sqrt{-\frac{a (\cos (c+d x)-1)}{a+b}} \sqrt{\frac{a (\cos (c+d x)+1)}{a-b}} \sqrt{a \cos (c+d x)+b} \left(a \left(2 b F\left(i \sinh ^{-1}\left(\sqrt{\frac{1}{a-b}} \sqrt{b+a \cos (c+d x)}\right)|\frac{b-a}{a+b}\right)+a \Pi \left(1-\frac{a}{b};i \sinh ^{-1}\left(\sqrt{\frac{1}{a-b}} \sqrt{b+a \cos (c+d x)}\right)|\frac{b-a}{a+b}\right)\right)-2 b (a+b) E\left(i \sinh ^{-1}\left(\sqrt{\frac{1}{a-b}} \sqrt{b+a \cos (c+d x)}\right)|\frac{b-a}{a+b}\right)\right)}{\sqrt{\frac{1}{a-b}}}\right)}{8 b^3 d \sqrt{a+b \sec (c+d x)}}","\frac{\left(3 a^2+4 b^2\right) \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{4 b^2 d \sqrt{a+b \sec (c+d x)}}-\frac{3 a \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a+b \sec (c+d x)}}{4 b^2 d}+\frac{3 a \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{4 b^2 d \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{\sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+b \sec (c+d x)}}{2 b d}-\frac{a \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{4 b d \sqrt{a+b \sec (c+d x)}}",1,"(Sqrt[Sec[c + d*x]]*(4*a*b^2*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)] + b*(9*a^2 + 8*b^2)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)] + ((3*I)*Sqrt[-((a*(-1 + Cos[c + d*x]))/(a + b))]*Sqrt[(a*(1 + Cos[c + d*x]))/(a - b)]*Sqrt[b + a*Cos[c + d*x]]*Csc[c + d*x]*(-2*b*(a + b)*EllipticE[I*ArcSinh[Sqrt[(a - b)^(-1)]*Sqrt[b + a*Cos[c + d*x]]], (-a + b)/(a + b)] + a*(2*b*EllipticF[I*ArcSinh[Sqrt[(a - b)^(-1)]*Sqrt[b + a*Cos[c + d*x]]], (-a + b)/(a + b)] + a*EllipticPi[1 - a/b, I*ArcSinh[Sqrt[(a - b)^(-1)]*Sqrt[b + a*Cos[c + d*x]]], (-a + b)/(a + b)])))/Sqrt[(a - b)^(-1)] + 2*b*(2*b - 3*a*Cos[c + d*x])*(b + a*Cos[c + d*x])*Sec[c + d*x]*Tan[c + d*x]))/(8*b^3*d*Sqrt[a + b*Sec[c + d*x]])","C",1
648,1,329,246,9.4250856,"\int \frac{\sec ^{\frac{5}{2}}(c+d x)}{\sqrt{a+b \sec (c+d x)}} \, dx","Integrate[Sec[c + d*x]^(5/2)/Sqrt[a + b*Sec[c + d*x]],x]","\frac{\sqrt{\sec (c+d x)} \left(4 \tan (c+d x) (a \cos (c+d x)+b)-6 a \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)-\frac{2 i \csc (c+d x) \sqrt{-\frac{a (\cos (c+d x)-1)}{a+b}} \sqrt{\frac{a (\cos (c+d x)+1)}{a-b}} \sqrt{a \cos (c+d x)+b} \left(a \left(2 b F\left(i \sinh ^{-1}\left(\sqrt{\frac{1}{a-b}} \sqrt{b+a \cos (c+d x)}\right)|\frac{b-a}{a+b}\right)+a \Pi \left(1-\frac{a}{b};i \sinh ^{-1}\left(\sqrt{\frac{1}{a-b}} \sqrt{b+a \cos (c+d x)}\right)|\frac{b-a}{a+b}\right)\right)-2 b (a+b) E\left(i \sinh ^{-1}\left(\sqrt{\frac{1}{a-b}} \sqrt{b+a \cos (c+d x)}\right)|\frac{b-a}{a+b}\right)\right)}{a b \sqrt{\frac{1}{a-b}}}\right)}{4 b d \sqrt{a+b \sec (c+d x)}}","\frac{\sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a+b \sec (c+d x)}}{b d}+\frac{\sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{d \sqrt{a+b \sec (c+d x)}}-\frac{\sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{b d \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}-\frac{a \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{b d \sqrt{a+b \sec (c+d x)}}",1,"(Sqrt[Sec[c + d*x]]*(-6*a*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)] - ((2*I)*Sqrt[-((a*(-1 + Cos[c + d*x]))/(a + b))]*Sqrt[(a*(1 + Cos[c + d*x]))/(a - b)]*Sqrt[b + a*Cos[c + d*x]]*Csc[c + d*x]*(-2*b*(a + b)*EllipticE[I*ArcSinh[Sqrt[(a - b)^(-1)]*Sqrt[b + a*Cos[c + d*x]]], (-a + b)/(a + b)] + a*(2*b*EllipticF[I*ArcSinh[Sqrt[(a - b)^(-1)]*Sqrt[b + a*Cos[c + d*x]]], (-a + b)/(a + b)] + a*EllipticPi[1 - a/b, I*ArcSinh[Sqrt[(a - b)^(-1)]*Sqrt[b + a*Cos[c + d*x]]], (-a + b)/(a + b)])))/(a*Sqrt[(a - b)^(-1)]*b) + 4*(b + a*Cos[c + d*x])*Tan[c + d*x]))/(4*b*d*Sqrt[a + b*Sec[c + d*x]])","C",1
649,1,68,68,0.1301322,"\int \frac{\sec ^{\frac{3}{2}}(c+d x)}{\sqrt{a+b \sec (c+d x)}} \, dx","Integrate[Sec[c + d*x]^(3/2)/Sqrt[a + b*Sec[c + d*x]],x]","\frac{2 \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{d \sqrt{a+b \sec (c+d x)}}","\frac{2 \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{d \sqrt{a+b \sec (c+d x)}}",1,"(2*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(d*Sqrt[a + b*Sec[c + d*x]])","A",1
650,1,67,67,0.0738389,"\int \frac{\sqrt{\sec (c+d x)}}{\sqrt{a+b \sec (c+d x)}} \, dx","Integrate[Sqrt[Sec[c + d*x]]/Sqrt[a + b*Sec[c + d*x]],x]","\frac{2 \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{d \sqrt{a+b \sec (c+d x)}}","\frac{2 \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{d \sqrt{a+b \sec (c+d x)}}",1,"(2*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(d*Sqrt[a + b*Sec[c + d*x]])","A",1
651,1,96,142,2.9566322,"\int \frac{1}{\sqrt{\sec (c+d x)} \sqrt{a+b \sec (c+d x)}} \, dx","Integrate[1/(Sqrt[Sec[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]),x]","\frac{2 \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \left((a+b) E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)-b F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)\right)}{a d \sqrt{a+b \sec (c+d x)}}","\frac{2 \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{a d \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}-\frac{2 b \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{a d \sqrt{a+b \sec (c+d x)}}",1,"(2*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*((a + b)*EllipticE[(c + d*x)/2, (2*a)/(a + b)] - b*EllipticF[(c + d*x)/2, (2*a)/(a + b)])*Sqrt[Sec[c + d*x]])/(a*d*Sqrt[a + b*Sec[c + d*x]])","A",1
652,1,147,195,0.7025907,"\int \frac{1}{\sec ^{\frac{3}{2}}(c+d x) \sqrt{a+b \sec (c+d x)}} \, dx","Integrate[1/(Sec[c + d*x]^(3/2)*Sqrt[a + b*Sec[c + d*x]]),x]","\frac{\sqrt{\sec (c+d x)} \left(2 \left(a^2+2 b^2\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)+2 a \sin (c+d x) (a \cos (c+d x)+b)-4 b (a+b) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)\right)}{3 a^2 d \sqrt{a+b \sec (c+d x)}}","\frac{2 \left(a^2+2 b^2\right) \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{3 a^2 d \sqrt{a+b \sec (c+d x)}}-\frac{4 b \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{3 a^2 d \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{2 \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{3 a d \sqrt{\sec (c+d x)}}",1,"(Sqrt[Sec[c + d*x]]*(-4*b*(a + b)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticE[(c + d*x)/2, (2*a)/(a + b)] + 2*(a^2 + 2*b^2)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)] + 2*a*(b + a*Cos[c + d*x])*Sin[c + d*x]))/(3*a^2*d*Sqrt[a + b*Sec[c + d*x]])","A",1
653,1,193,249,0.8450524,"\int \frac{1}{\sec ^{\frac{5}{2}}(c+d x) \sqrt{a+b \sec (c+d x)}} \, dx","Integrate[1/(Sec[c + d*x]^(5/2)*Sqrt[a + b*Sec[c + d*x]]),x]","\frac{\sqrt{\sec (c+d x)} \left(2 a \sin (c+d x) \left(3 a^2 \cos (2 (c+d x))+3 a^2-2 a b \cos (c+d x)-8 b^2\right)-4 b \left(7 a^2+8 b^2\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)+4 \left(9 a^3+9 a^2 b+8 a b^2+8 b^3\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)\right)}{30 a^3 d \sqrt{a+b \sec (c+d x)}}","-\frac{8 b \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{15 a^2 d \sqrt{\sec (c+d x)}}-\frac{2 b \left(7 a^2+8 b^2\right) \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{15 a^3 d \sqrt{a+b \sec (c+d x)}}+\frac{2 \left(9 a^2+8 b^2\right) \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{15 a^3 d \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{2 \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{5 a d \sec ^{\frac{3}{2}}(c+d x)}",1,"(Sqrt[Sec[c + d*x]]*(4*(9*a^3 + 9*a^2*b + 8*a*b^2 + 8*b^3)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticE[(c + d*x)/2, (2*a)/(a + b)] - 4*b*(7*a^2 + 8*b^2)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)] + 2*a*(3*a^2 - 8*b^2 - 2*a*b*Cos[c + d*x] + 3*a^2*Cos[2*(c + d*x)])*Sin[c + d*x]))/(30*a^3*d*Sqrt[a + b*Sec[c + d*x]])","A",1
654,1,478,345,4.850732,"\int \frac{\sec ^{\frac{7}{2}}(c+d x)}{(a+b \sec (c+d x))^{3/2}} \, dx","Integrate[Sec[c + d*x]^(7/2)/(a + b*Sec[c + d*x])^(3/2),x]","\frac{\sec ^{\frac{3}{2}}(c+d x) \left(\frac{4 \tan (c+d x) (a \cos (c+d x)+b) \left(\left(a b^2-3 a^3\right) \cos (c+d x)-a^2 b+b^3\right)}{b^4-a^2 b^2}-\frac{a (a \cos (c+d x)+b)^{3/2} \left(\frac{2 \left(9 a^2-7 b^2\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{\sqrt{a \cos (c+d x)+b}}+\frac{2 i \left(3 a^2-b^2\right) \csc (c+d x) \sqrt{-\frac{a (\cos (c+d x)-1)}{a+b}} \sqrt{\frac{a (\cos (c+d x)+1)}{a-b}} \left(a \left(2 b F\left(i \sinh ^{-1}\left(\sqrt{\frac{1}{a-b}} \sqrt{b+a \cos (c+d x)}\right)|\frac{b-a}{a+b}\right)+a \Pi \left(1-\frac{a}{b};i \sinh ^{-1}\left(\sqrt{\frac{1}{a-b}} \sqrt{b+a \cos (c+d x)}\right)|\frac{b-a}{a+b}\right)\right)-2 b (a+b) E\left(i \sinh ^{-1}\left(\sqrt{\frac{1}{a-b}} \sqrt{b+a \cos (c+d x)}\right)|\frac{b-a}{a+b}\right)\right)}{a^2 b \sqrt{\frac{1}{a-b}}}+\frac{8 a b \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{\sqrt{a \cos (c+d x)+b}}\right)}{b^2 (a-b) (a+b)}\right)}{4 d (a+b \sec (c+d x))^{3/2}}","-\frac{2 a^2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{b d \left(a^2-b^2\right) \sqrt{a+b \sec (c+d x)}}+\frac{\left(3 a^2-b^2\right) \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a+b \sec (c+d x)}}{b^2 d \left(a^2-b^2\right)}-\frac{\left(3 a^2-b^2\right) \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{b^2 d \left(a^2-b^2\right) \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}-\frac{3 a \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{b^2 d \sqrt{a+b \sec (c+d x)}}+\frac{\sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{b d \sqrt{a+b \sec (c+d x)}}",1,"(Sec[c + d*x]^(3/2)*(-((a*(b + a*Cos[c + d*x])^(3/2)*((8*a*b*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/Sqrt[b + a*Cos[c + d*x]] + (2*(9*a^2 - 7*b^2)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)])/Sqrt[b + a*Cos[c + d*x]] + ((2*I)*(3*a^2 - b^2)*Sqrt[-((a*(-1 + Cos[c + d*x]))/(a + b))]*Sqrt[(a*(1 + Cos[c + d*x]))/(a - b)]*Csc[c + d*x]*(-2*b*(a + b)*EllipticE[I*ArcSinh[Sqrt[(a - b)^(-1)]*Sqrt[b + a*Cos[c + d*x]]], (-a + b)/(a + b)] + a*(2*b*EllipticF[I*ArcSinh[Sqrt[(a - b)^(-1)]*Sqrt[b + a*Cos[c + d*x]]], (-a + b)/(a + b)] + a*EllipticPi[1 - a/b, I*ArcSinh[Sqrt[(a - b)^(-1)]*Sqrt[b + a*Cos[c + d*x]]], (-a + b)/(a + b)])))/(a^2*Sqrt[(a - b)^(-1)]*b)))/((a - b)*b^2*(a + b))) + (4*(b + a*Cos[c + d*x])*(-(a^2*b) + b^3 + (-3*a^3 + a*b^2)*Cos[c + d*x])*Tan[c + d*x])/(-(a^2*b^2) + b^4)))/(4*d*(a + b*Sec[c + d*x])^(3/2))","C",1
655,1,557,206,6.5235812,"\int \frac{\sec ^{\frac{5}{2}}(c+d x)}{(a+b \sec (c+d x))^{3/2}} \, dx","Integrate[Sec[c + d*x]^(5/2)/(a + b*Sec[c + d*x])^(3/2),x]","\frac{2 a^2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+b)}{b d \left(b^2-a^2\right) (a+b \sec (c+d x))^{3/2}}+\frac{\sec ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+b)^{3/2} \left(\frac{2 \left(3 a^2-2 b^2\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{\sqrt{a \cos (c+d x)+b}}+\frac{2 i a^2 \sin (c+d x) \cos (2 (c+d x)) \sqrt{\frac{a-a \cos (c+d x)}{a+b}} \sqrt{\frac{a \cos (c+d x)+a}{a-b}} \left(a \left(2 b F\left(i \sinh ^{-1}\left(\sqrt{\frac{1}{a-b}} \sqrt{b+a \cos (c+d x)}\right)|\frac{b-a}{a+b}\right)+a \Pi \left(1-\frac{a}{b};i \sinh ^{-1}\left(\sqrt{\frac{1}{a-b}} \sqrt{b+a \cos (c+d x)}\right)|\frac{b-a}{a+b}\right)\right)-2 b (a+b) E\left(i \sinh ^{-1}\left(\sqrt{\frac{1}{a-b}} \sqrt{b+a \cos (c+d x)}\right)|\frac{b-a}{a+b}\right)\right)}{b \sqrt{\frac{1}{a-b}} \sqrt{1-\cos ^2(c+d x)} \sqrt{\frac{a^2-a^2 \cos ^2(c+d x)}{a^2}} \left(-a^2+2 (a \cos (c+d x)+b)^2-4 b (a \cos (c+d x)+b)+2 b^2\right)}+\frac{4 a b \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{\sqrt{a \cos (c+d x)+b}}\right)}{2 b d (a-b) (a+b) (a+b \sec (c+d x))^{3/2}}","-\frac{2 a^2 \sin (c+d x) \sqrt{\sec (c+d x)}}{b d \left(a^2-b^2\right) \sqrt{a+b \sec (c+d x)}}+\frac{2 a \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{b d \left(a^2-b^2\right) \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{2 \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{b d \sqrt{a+b \sec (c+d x)}}",1,"(2*a^2*(b + a*Cos[c + d*x])*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(b*(-a^2 + b^2)*d*(a + b*Sec[c + d*x])^(3/2)) + ((b + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(3/2)*((4*a*b*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/Sqrt[b + a*Cos[c + d*x]] + (2*(3*a^2 - 2*b^2)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)])/Sqrt[b + a*Cos[c + d*x]] + ((2*I)*a^2*Sqrt[(a - a*Cos[c + d*x])/(a + b)]*Sqrt[(a + a*Cos[c + d*x])/(a - b)]*Cos[2*(c + d*x)]*(-2*b*(a + b)*EllipticE[I*ArcSinh[Sqrt[(a - b)^(-1)]*Sqrt[b + a*Cos[c + d*x]]], (-a + b)/(a + b)] + a*(2*b*EllipticF[I*ArcSinh[Sqrt[(a - b)^(-1)]*Sqrt[b + a*Cos[c + d*x]]], (-a + b)/(a + b)] + a*EllipticPi[1 - a/b, I*ArcSinh[Sqrt[(a - b)^(-1)]*Sqrt[b + a*Cos[c + d*x]]], (-a + b)/(a + b)]))*Sin[c + d*x])/(Sqrt[(a - b)^(-1)]*b*Sqrt[1 - Cos[c + d*x]^2]*Sqrt[(a^2 - a^2*Cos[c + d*x]^2)/a^2]*(-a^2 + 2*b^2 - 4*b*(b + a*Cos[c + d*x]) + 2*(b + a*Cos[c + d*x])^2))))/(2*(a - b)*b*(a + b)*d*(a + b*Sec[c + d*x])^(3/2))","C",1
656,1,103,126,0.3224481,"\int \frac{\sec ^{\frac{3}{2}}(c+d x)}{(a+b \sec (c+d x))^{3/2}} \, dx","Integrate[Sec[c + d*x]^(3/2)/(a + b*Sec[c + d*x])^(3/2),x]","-\frac{2 \sec ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+b) \left((a+b) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)-a \sin (c+d x)\right)}{d (a-b) (a+b) (a+b \sec (c+d x))^{3/2}}","\frac{2 a \sin (c+d x) \sqrt{\sec (c+d x)}}{d \left(a^2-b^2\right) \sqrt{a+b \sec (c+d x)}}-\frac{2 \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{d \left(a^2-b^2\right) \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}",1,"(-2*(b + a*Cos[c + d*x])*Sec[c + d*x]^(3/2)*((a + b)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticE[(c + d*x)/2, (2*a)/(a + b)] - a*Sin[c + d*x]))/((a - b)*(a + b)*d*(a + b*Sec[c + d*x])^(3/2))","A",1
657,1,156,200,0.623832,"\int \frac{\sqrt{\sec (c+d x)}}{(a+b \sec (c+d x))^{3/2}} \, dx","Integrate[Sqrt[Sec[c + d*x]]/(a + b*Sec[c + d*x])^(3/2),x]","\frac{2 \sec ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+b) \left(\left(a^2-b^2\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)-a b \sin (c+d x)+b (a+b) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)\right)}{a d (a-b) (a+b) (a+b \sec (c+d x))^{3/2}}","-\frac{2 b \sin (c+d x) \sqrt{\sec (c+d x)}}{d \left(a^2-b^2\right) \sqrt{a+b \sec (c+d x)}}+\frac{2 b \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{a d \left(a^2-b^2\right) \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{2 \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{a d \sqrt{a+b \sec (c+d x)}}",1,"(2*(b + a*Cos[c + d*x])*Sec[c + d*x]^(3/2)*(b*(a + b)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticE[(c + d*x)/2, (2*a)/(a + b)] + (a^2 - b^2)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)] - a*b*Sin[c + d*x]))/(a*(a - b)*(a + b)*d*(a + b*Sec[c + d*x])^(3/2))","A",1
658,1,165,214,0.7162103,"\int \frac{1}{\sqrt{\sec (c+d x)} (a+b \sec (c+d x))^{3/2}} \, dx","Integrate[1/(Sqrt[Sec[c + d*x]]*(a + b*Sec[c + d*x])^(3/2)),x]","\frac{2 \sqrt{\sec (c+d x)} \left(b \left(a b \sin (c+d x)-2 \left(a^2-b^2\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)\right)+\left(a^3+a^2 b-2 a b^2-2 b^3\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)\right)}{a^2 d (a-b) (a+b) \sqrt{a+b \sec (c+d x)}}","\frac{2 b^2 \sin (c+d x) \sqrt{\sec (c+d x)}}{a d \left(a^2-b^2\right) \sqrt{a+b \sec (c+d x)}}+\frac{2 \left(a^2-2 b^2\right) \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{a^2 d \left(a^2-b^2\right) \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}-\frac{4 b \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{a^2 d \sqrt{a+b \sec (c+d x)}}",1,"(2*Sqrt[Sec[c + d*x]]*((a^3 + a^2*b - 2*a*b^2 - 2*b^3)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticE[(c + d*x)/2, (2*a)/(a + b)] + b*(-2*(a^2 - b^2)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)] + a*b*Sin[c + d*x])))/(a^2*(a - b)*(a + b)*d*Sqrt[a + b*Sec[c + d*x]])","A",1
659,1,203,289,0.9578678,"\int \frac{1}{\sec ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}} \, dx","Integrate[1/(Sec[c + d*x]^(3/2)*(a + b*Sec[c + d*x])^(3/2)),x]","\frac{2 \sqrt{\sec (c+d x)} \left(a \sin (c+d x) \left(a \left(a^2-b^2\right) \cos (c+d x)+b \left(a^2-4 b^2\right)\right)+\left(a^4+7 a^2 b^2-8 b^4\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)+b \left(-5 a^3-5 a^2 b+8 a b^2+8 b^3\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)\right)}{3 a^3 d (a-b) (a+b) \sqrt{a+b \sec (c+d x)}}","\frac{2 b^2 \sin (c+d x)}{a d \left(a^2-b^2\right) \sqrt{\sec (c+d x)} \sqrt{a+b \sec (c+d x)}}+\frac{2 \left(a^2-4 b^2\right) \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{3 a^2 d \left(a^2-b^2\right) \sqrt{\sec (c+d x)}}+\frac{2 \left(a^2+8 b^2\right) \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{3 a^3 d \sqrt{a+b \sec (c+d x)}}-\frac{2 b \left(5 a^2-8 b^2\right) \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{3 a^3 d \left(a^2-b^2\right) \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}",1,"(2*Sqrt[Sec[c + d*x]]*(b*(-5*a^3 - 5*a^2*b + 8*a*b^2 + 8*b^3)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticE[(c + d*x)/2, (2*a)/(a + b)] + (a^4 + 7*a^2*b^2 - 8*b^4)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)] + a*(b*(a^2 - 4*b^2) + a*(a^2 - b^2)*Cos[c + d*x])*Sin[c + d*x]))/(3*a^3*(a - b)*(a + b)*d*Sqrt[a + b*Sec[c + d*x]])","A",1
660,1,250,360,1.4636269,"\int \frac{1}{\sec ^{\frac{5}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}} \, dx","Integrate[1/(Sec[c + d*x]^(5/2)*(a + b*Sec[c + d*x])^(3/2)),x]","\frac{\sqrt{\sec (c+d x)} \left(2 a \sin (c+d x) \left(a^4-4 a b \left(a^2-b^2\right) \cos (c+d x)-7 a^2 b^2+\left(a^4-a^2 b^2\right) \cos (2 (c+d x))+16 b^4\right)-16 b \left(a^4+3 a^2 b^2-4 b^4\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)+4 \left(3 a^5+3 a^4 b+8 a^3 b^2+8 a^2 b^3-16 a b^4-16 b^5\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)\right)}{10 a^4 d (a-b) (a+b) \sqrt{a+b \sec (c+d x)}}","\frac{2 b^2 \sin (c+d x)}{a d \left(a^2-b^2\right) \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+b \sec (c+d x)}}+\frac{2 \left(a^2-6 b^2\right) \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{5 a^2 d \left(a^2-b^2\right) \sec ^{\frac{3}{2}}(c+d x)}-\frac{8 b \left(a^2+4 b^2\right) \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{5 a^4 d \sqrt{a+b \sec (c+d x)}}+\frac{2 \left(3 a^4+8 a^2 b^2-16 b^4\right) \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{5 a^4 d \left(a^2-b^2\right) \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}-\frac{2 b \left(3 a^2-8 b^2\right) \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{5 a^3 d \left(a^2-b^2\right) \sqrt{\sec (c+d x)}}",1,"(Sqrt[Sec[c + d*x]]*(4*(3*a^5 + 3*a^4*b + 8*a^3*b^2 + 8*a^2*b^3 - 16*a*b^4 - 16*b^5)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticE[(c + d*x)/2, (2*a)/(a + b)] - 16*b*(a^4 + 3*a^2*b^2 - 4*b^4)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)] + 2*a*(a^4 - 7*a^2*b^2 + 16*b^4 - 4*a*b*(a^2 - b^2)*Cos[c + d*x] + (a^4 - a^2*b^2)*Cos[2*(c + d*x)])*Sin[c + d*x]))/(10*a^4*(a - b)*(a + b)*d*Sqrt[a + b*Sec[c + d*x]])","A",1
661,1,677,458,6.8172036,"\int \frac{\sec ^{\frac{9}{2}}(c+d x)}{(a+b \sec (c+d x))^{5/2}} \, dx","Integrate[Sec[c + d*x]^(9/2)/(a + b*Sec[c + d*x])^(5/2),x]","\frac{\sec ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+b)^3 \left(-\frac{2 a^3 \sin (c+d x)}{3 b^2 \left(b^2-a^2\right) (a \cos (c+d x)+b)^2}-\frac{4 \left(5 a^3 b^2 \sin (c+d x)-3 a^5 \sin (c+d x)\right)}{3 b^3 \left(b^2-a^2\right)^2 (a \cos (c+d x)+b)}+\frac{\tan (c+d x)}{b^3}\right)}{d (a+b \sec (c+d x))^{5/2}}-\frac{a \sec ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+b)^{5/2} \left(\frac{2 \left(20 a^3 b-36 a b^3\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{\sqrt{a \cos (c+d x)+b}}+\frac{2 \left(45 a^4-86 a^2 b^2+33 b^4\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{\sqrt{a \cos (c+d x)+b}}+\frac{2 i \left(15 a^4-26 a^2 b^2+3 b^4\right) \sin (c+d x) \cos (2 (c+d x)) \sqrt{\frac{a-a \cos (c+d x)}{a+b}} \sqrt{\frac{a \cos (c+d x)+a}{a-b}} \left(a \left(2 b F\left(i \sinh ^{-1}\left(\sqrt{\frac{1}{a-b}} \sqrt{b+a \cos (c+d x)}\right)|\frac{b-a}{a+b}\right)+a \Pi \left(1-\frac{a}{b};i \sinh ^{-1}\left(\sqrt{\frac{1}{a-b}} \sqrt{b+a \cos (c+d x)}\right)|\frac{b-a}{a+b}\right)\right)-2 b (a+b) E\left(i \sinh ^{-1}\left(\sqrt{\frac{1}{a-b}} \sqrt{b+a \cos (c+d x)}\right)|\frac{b-a}{a+b}\right)\right)}{b \sqrt{\frac{1}{a-b}} \sqrt{1-\cos ^2(c+d x)} \sqrt{\frac{a^2-a^2 \cos ^2(c+d x)}{a^2}} \left(-a^2+2 (a \cos (c+d x)+b)^2-4 b (a \cos (c+d x)+b)+2 b^2\right)}\right)}{12 b^3 d (a-b)^2 (a+b)^2 (a+b \sec (c+d x))^{5/2}}","-\frac{2 a^2 \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{3 b d \left(a^2-b^2\right) (a+b \sec (c+d x))^{3/2}}-\frac{2 a^2 \left(5 a^2-9 b^2\right) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 b^2 d \left(a^2-b^2\right)^2 \sqrt{a+b \sec (c+d x)}}+\frac{\left(5 a^2-3 b^2\right) \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{3 b^2 d \left(a^2-b^2\right) \sqrt{a+b \sec (c+d x)}}+\frac{\left(15 a^4-26 a^2 b^2+3 b^4\right) \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a+b \sec (c+d x)}}{3 b^3 d \left(a^2-b^2\right)^2}-\frac{\left(15 a^4-26 a^2 b^2+3 b^4\right) \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{3 b^3 d \left(a^2-b^2\right)^2 \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}-\frac{5 a \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{b^3 d \sqrt{a+b \sec (c+d x)}}",1,"-1/12*(a*(b + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(5/2)*((2*(20*a^3*b - 36*a*b^3)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/Sqrt[b + a*Cos[c + d*x]] + (2*(45*a^4 - 86*a^2*b^2 + 33*b^4)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)])/Sqrt[b + a*Cos[c + d*x]] + ((2*I)*(15*a^4 - 26*a^2*b^2 + 3*b^4)*Sqrt[(a - a*Cos[c + d*x])/(a + b)]*Sqrt[(a + a*Cos[c + d*x])/(a - b)]*Cos[2*(c + d*x)]*(-2*b*(a + b)*EllipticE[I*ArcSinh[Sqrt[(a - b)^(-1)]*Sqrt[b + a*Cos[c + d*x]]], (-a + b)/(a + b)] + a*(2*b*EllipticF[I*ArcSinh[Sqrt[(a - b)^(-1)]*Sqrt[b + a*Cos[c + d*x]]], (-a + b)/(a + b)] + a*EllipticPi[1 - a/b, I*ArcSinh[Sqrt[(a - b)^(-1)]*Sqrt[b + a*Cos[c + d*x]]], (-a + b)/(a + b)]))*Sin[c + d*x])/(Sqrt[(a - b)^(-1)]*b*Sqrt[1 - Cos[c + d*x]^2]*Sqrt[(a^2 - a^2*Cos[c + d*x]^2)/a^2]*(-a^2 + 2*b^2 - 4*b*(b + a*Cos[c + d*x]) + 2*(b + a*Cos[c + d*x])^2))))/((a - b)^2*b^3*(a + b)^2*d*(a + b*Sec[c + d*x])^(5/2)) + ((b + a*Cos[c + d*x])^3*Sec[c + d*x]^(5/2)*((-2*a^3*Sin[c + d*x])/(3*b^2*(-a^2 + b^2)*(b + a*Cos[c + d*x])^2) - (4*(-3*a^5*Sin[c + d*x] + 5*a^3*b^2*Sin[c + d*x]))/(3*b^3*(-a^2 + b^2)^2*(b + a*Cos[c + d*x])) + Tan[c + d*x]/b^3))/(d*(a + b*Sec[c + d*x])^(5/2))","C",1
662,1,487,370,5.9313013,"\int \frac{\sec ^{\frac{7}{2}}(c+d x)}{(a+b \sec (c+d x))^{5/2}} \, dx","Integrate[Sec[c + d*x]^(7/2)/(a + b*Sec[c + d*x])^(5/2),x]","\frac{\sec ^{\frac{5}{2}}(c+d x) \left(\frac{2 a^2 b \left(7 b^2-3 a^2\right) \sin (c+d x) (a \cos (c+d x)+b)^2}{\left(a^2-b^2\right)^2}+\frac{2 a^2 b^2 \sin (c+d x) (a \cos (c+d x)+b)}{b^2-a^2}+\frac{4 a b^2 \left(a^2-3 b^2\right) \left(\frac{a \cos (c+d x)+b}{a+b}\right)^{5/2} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{(a-b)^2}+\frac{i \left(\frac{1}{a-b}\right)^{3/2} \left(3 a^2-7 b^2\right) \csc (c+d x) \sqrt{-\frac{a (\cos (c+d x)-1)}{a+b}} \sqrt{\frac{a (\cos (c+d x)+1)}{a-b}} (a \cos (c+d x)+b)^{5/2} \left(a \left(2 b F\left(i \sinh ^{-1}\left(\sqrt{\frac{1}{a-b}} \sqrt{b+a \cos (c+d x)}\right)|\frac{b-a}{a+b}\right)+a \Pi \left(1-\frac{a}{b};i \sinh ^{-1}\left(\sqrt{\frac{1}{a-b}} \sqrt{b+a \cos (c+d x)}\right)|\frac{b-a}{a+b}\right)\right)-2 b (a+b) E\left(i \sinh ^{-1}\left(\sqrt{\frac{1}{a-b}} \sqrt{b+a \cos (c+d x)}\right)|\frac{b-a}{a+b}\right)\right)}{(a+b)^2}+\frac{b \left(9 a^4-19 a^2 b^2+6 b^4\right) \left(\frac{a \cos (c+d x)+b}{a+b}\right)^{5/2} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{(a-b)^2}\right)}{3 b^3 d (a+b \sec (c+d x))^{5/2}}","-\frac{2 a^2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 b d \left(a^2-b^2\right) (a+b \sec (c+d x))^{3/2}}-\frac{2 a^2 \left(3 a^2-7 b^2\right) \sin (c+d x) \sqrt{\sec (c+d x)}}{3 b^2 d \left(a^2-b^2\right)^2 \sqrt{a+b \sec (c+d x)}}-\frac{2 a \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{3 b d \left(a^2-b^2\right) \sqrt{a+b \sec (c+d x)}}+\frac{2 a \left(3 a^2-7 b^2\right) \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{3 b^2 d \left(a^2-b^2\right)^2 \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{2 \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{b^2 d \sqrt{a+b \sec (c+d x)}}",1,"(Sec[c + d*x]^(5/2)*((4*a*b^2*(a^2 - 3*b^2)*((b + a*Cos[c + d*x])/(a + b))^(5/2)*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/(a - b)^2 + (b*(9*a^4 - 19*a^2*b^2 + 6*b^4)*((b + a*Cos[c + d*x])/(a + b))^(5/2)*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)])/(a - b)^2 + (I*((a - b)^(-1))^(3/2)*(3*a^2 - 7*b^2)*Sqrt[-((a*(-1 + Cos[c + d*x]))/(a + b))]*Sqrt[(a*(1 + Cos[c + d*x]))/(a - b)]*(b + a*Cos[c + d*x])^(5/2)*Csc[c + d*x]*(-2*b*(a + b)*EllipticE[I*ArcSinh[Sqrt[(a - b)^(-1)]*Sqrt[b + a*Cos[c + d*x]]], (-a + b)/(a + b)] + a*(2*b*EllipticF[I*ArcSinh[Sqrt[(a - b)^(-1)]*Sqrt[b + a*Cos[c + d*x]]], (-a + b)/(a + b)] + a*EllipticPi[1 - a/b, I*ArcSinh[Sqrt[(a - b)^(-1)]*Sqrt[b + a*Cos[c + d*x]]], (-a + b)/(a + b)])))/(a + b)^2 + (2*a^2*b^2*(b + a*Cos[c + d*x])*Sin[c + d*x])/(-a^2 + b^2) + (2*a^2*b*(-3*a^2 + 7*b^2)*(b + a*Cos[c + d*x])^2*Sin[c + d*x])/(a^2 - b^2)^2))/(3*b^3*d*(a + b*Sec[c + d*x])^(5/2))","C",0
663,1,169,277,1.1544233,"\int \frac{\sec ^{\frac{5}{2}}(c+d x)}{(a+b \sec (c+d x))^{5/2}} \, dx","Integrate[Sec[c + d*x]^(5/2)/(a + b*Sec[c + d*x])^(5/2),x]","-\frac{2 \sec ^{\frac{3}{2}}(c+d x) \left(a \sin (c+d x) \left(-a^2+4 a b \cos (c+d x)+5 b^2\right)-(a-b) (a+b)^2 \left(\frac{a \cos (c+d x)+b}{a+b}\right)^{3/2} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)-4 b (a+b)^2 \left(\frac{a \cos (c+d x)+b}{a+b}\right)^{3/2} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)\right)}{3 d (a-b)^2 (a+b)^2 (a+b \sec (c+d x))^{3/2}}","-\frac{2 a^2 \sin (c+d x) \sqrt{\sec (c+d x)}}{3 b d \left(a^2-b^2\right) (a+b \sec (c+d x))^{3/2}}+\frac{2 a \left(a^2-5 b^2\right) \sin (c+d x) \sqrt{\sec (c+d x)}}{3 b d \left(a^2-b^2\right)^2 \sqrt{a+b \sec (c+d x)}}+\frac{2 \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{3 d \left(a^2-b^2\right) \sqrt{a+b \sec (c+d x)}}+\frac{8 b \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{3 d \left(a^2-b^2\right)^2 \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}",1,"(-2*Sec[c + d*x]^(3/2)*(-4*b*(a + b)^2*((b + a*Cos[c + d*x])/(a + b))^(3/2)*EllipticE[(c + d*x)/2, (2*a)/(a + b)] - (a - b)*(a + b)^2*((b + a*Cos[c + d*x])/(a + b))^(3/2)*EllipticF[(c + d*x)/2, (2*a)/(a + b)] + a*(-a^2 + 5*b^2 + 4*a*b*Cos[c + d*x])*Sin[c + d*x]))/(3*(a - b)^2*(a + b)^2*d*(a + b*Sec[c + d*x])^(3/2))","A",1
664,1,178,281,1.1060567,"\int \frac{\sec ^{\frac{3}{2}}(c+d x)}{(a+b \sec (c+d x))^{5/2}} \, dx","Integrate[Sec[c + d*x]^(3/2)/(a + b*Sec[c + d*x])^(5/2),x]","\frac{\sec ^{\frac{5}{2}}(c+d x) \left(\frac{2 \sin (c+d x) (a \cos (c+d x)+b) \left(a \left(3 a^2+b^2\right) \cos (c+d x)+2 b \left(a^2+b^2\right)\right)}{\left(a^2-b^2\right)^2}-\frac{2 (a+b) \left(\frac{a \cos (c+d x)+b}{a+b}\right)^{5/2} \left(\left(3 a^2+b^2\right) E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)+b (a-b) F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)\right)}{a (a-b)^2}\right)}{3 d (a+b \sec (c+d x))^{5/2}}","\frac{4 \left(a^2+b^2\right) \sin (c+d x) \sqrt{\sec (c+d x)}}{3 d \left(a^2-b^2\right)^2 \sqrt{a+b \sec (c+d x)}}+\frac{2 a \sin (c+d x) \sqrt{\sec (c+d x)}}{3 d \left(a^2-b^2\right) (a+b \sec (c+d x))^{3/2}}-\frac{2 b \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{3 a d \left(a^2-b^2\right) \sqrt{a+b \sec (c+d x)}}-\frac{2 \left(3 a^2+b^2\right) \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{3 a d \left(a^2-b^2\right)^2 \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}",1,"(Sec[c + d*x]^(5/2)*((-2*(a + b)*((b + a*Cos[c + d*x])/(a + b))^(5/2)*((3*a^2 + b^2)*EllipticE[(c + d*x)/2, (2*a)/(a + b)] + (a - b)*b*EllipticF[(c + d*x)/2, (2*a)/(a + b)]))/(a*(a - b)^2) + (2*(b + a*Cos[c + d*x])*(2*b*(a^2 + b^2) + a*(3*a^2 + b^2)*Cos[c + d*x])*Sin[c + d*x])/(a^2 - b^2)^2))/(3*d*(a + b*Sec[c + d*x])^(5/2))","A",1
665,1,196,302,1.3072146,"\int \frac{\sqrt{\sec (c+d x)}}{(a+b \sec (c+d x))^{5/2}} \, dx","Integrate[Sqrt[Sec[c + d*x]]/(a + b*Sec[c + d*x])^(5/2),x]","\frac{2 \sec ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+b) \left(\frac{a b \sin (c+d x) \left(\left(2 a b^2-6 a^3\right) \cos (c+d x)-5 a^2 b+b^3\right)}{\left(a^2-b^2\right)^2}+\frac{\left(\frac{a \cos (c+d x)+b}{a+b}\right)^{3/2} \left(\left(6 a^2 b-2 b^3\right) E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)+\left(3 a^3-3 a^2 b-2 a b^2+2 b^3\right) F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)\right)}{(a-b)^2}\right)}{3 a^2 d (a+b \sec (c+d x))^{5/2}}","-\frac{2 b \left(5 a^2-b^2\right) \sin (c+d x) \sqrt{\sec (c+d x)}}{3 a d \left(a^2-b^2\right)^2 \sqrt{a+b \sec (c+d x)}}-\frac{2 b \sin (c+d x) \sqrt{\sec (c+d x)}}{3 d \left(a^2-b^2\right) (a+b \sec (c+d x))^{3/2}}+\frac{2 \left(3 a^2-2 b^2\right) \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{3 a^2 d \left(a^2-b^2\right) \sqrt{a+b \sec (c+d x)}}+\frac{4 b \left(3 a^2-b^2\right) \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{3 a^2 d \left(a^2-b^2\right)^2 \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}",1,"(2*(b + a*Cos[c + d*x])*Sec[c + d*x]^(5/2)*((((b + a*Cos[c + d*x])/(a + b))^(3/2)*((6*a^2*b - 2*b^3)*EllipticE[(c + d*x)/2, (2*a)/(a + b)] + (3*a^3 - 3*a^2*b - 2*a*b^2 + 2*b^3)*EllipticF[(c + d*x)/2, (2*a)/(a + b)]))/(a - b)^2 + (a*b*(-5*a^2*b + b^3 + (-6*a^3 + 2*a*b^2)*Cos[c + d*x])*Sin[c + d*x])/(a^2 - b^2)^2))/(3*a^2*d*(a + b*Sec[c + d*x])^(5/2))","A",1
666,1,208,317,1.3819156,"\int \frac{1}{\sqrt{\sec (c+d x)} (a+b \sec (c+d x))^{5/2}} \, dx","Integrate[1/(Sqrt[Sec[c + d*x]]*(a + b*Sec[c + d*x])^(5/2)),x]","\frac{2 \sec ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+b) \left(\frac{a b^2 \sin (c+d x) \left(a \left(9 a^2-5 b^2\right) \cos (c+d x)+8 a^2 b-4 b^3\right)}{\left(a^2-b^2\right)^2}+\frac{\left(\frac{a \cos (c+d x)+b}{a+b}\right)^{3/2} \left(\left(3 a^4-15 a^2 b^2+8 b^4\right) E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)+b \left(-9 a^3+9 a^2 b+8 a b^2-8 b^3\right) F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)\right)}{(a-b)^2}\right)}{3 a^3 d (a+b \sec (c+d x))^{5/2}}","\frac{8 b^2 \left(2 a^2-b^2\right) \sin (c+d x) \sqrt{\sec (c+d x)}}{3 a^2 d \left(a^2-b^2\right)^2 \sqrt{a+b \sec (c+d x)}}+\frac{2 b^2 \sin (c+d x) \sqrt{\sec (c+d x)}}{3 a d \left(a^2-b^2\right) (a+b \sec (c+d x))^{3/2}}-\frac{2 b \left(9 a^2-8 b^2\right) \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{3 a^3 d \left(a^2-b^2\right) \sqrt{a+b \sec (c+d x)}}+\frac{2 \left(3 a^4-15 a^2 b^2+8 b^4\right) \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{3 a^3 d \left(a^2-b^2\right)^2 \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}",1,"(2*(b + a*Cos[c + d*x])*Sec[c + d*x]^(5/2)*((((b + a*Cos[c + d*x])/(a + b))^(3/2)*((3*a^4 - 15*a^2*b^2 + 8*b^4)*EllipticE[(c + d*x)/2, (2*a)/(a + b)] + b*(-9*a^3 + 9*a^2*b + 8*a*b^2 - 8*b^3)*EllipticF[(c + d*x)/2, (2*a)/(a + b)]))/(a - b)^2 + (a*b^2*(8*a^2*b - 4*b^3 + a*(9*a^2 - 5*b^2)*Cos[c + d*x])*Sin[c + d*x])/(a^2 - b^2)^2))/(3*a^3*d*(a + b*Sec[c + d*x])^(5/2))","A",1
667,1,257,391,1.7944822,"\int \frac{1}{\sec ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^{5/2}} \, dx","Integrate[1/(Sec[c + d*x]^(3/2)*(a + b*Sec[c + d*x])^(5/2)),x]","\frac{2 \sec ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+b) \left(\frac{a \sin (c+d x) \left(a^6+\left(a^3-a b^2\right)^2 \cos (2 (c+d x))-25 a^2 b^4+4 a b \left(a^4-8 a^2 b^2+5 b^4\right) \cos (c+d x)+16 b^6\right)}{2 \left(a^2-b^2\right)^2}+\frac{\left(\frac{a \cos (c+d x)+b}{a+b}\right)^{3/2} \left(\left(a^5-a^4 b+16 a^3 b^2-16 a^2 b^3-16 a b^4+16 b^5\right) F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)-4 \left(2 a^4 b-7 a^2 b^3+4 b^5\right) E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)\right)}{(a-b)^2}\right)}{3 a^4 d (a+b \sec (c+d x))^{5/2}}","\frac{4 b^2 \left(5 a^2-3 b^2\right) \sin (c+d x)}{3 a^2 d \left(a^2-b^2\right)^2 \sqrt{\sec (c+d x)} \sqrt{a+b \sec (c+d x)}}+\frac{2 b^2 \sin (c+d x)}{3 a d \left(a^2-b^2\right) \sqrt{\sec (c+d x)} (a+b \sec (c+d x))^{3/2}}+\frac{2 \left(a^4+16 a^2 b^2-16 b^4\right) \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{3 a^4 d \left(a^2-b^2\right) \sqrt{a+b \sec (c+d x)}}-\frac{8 b \left(2 a^4-7 a^2 b^2+4 b^4\right) \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{3 a^4 d \left(a^2-b^2\right)^2 \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{2 \left(a^4-13 a^2 b^2+8 b^4\right) \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{3 a^3 d \left(a^2-b^2\right)^2 \sqrt{\sec (c+d x)}}",1,"(2*(b + a*Cos[c + d*x])*Sec[c + d*x]^(5/2)*((((b + a*Cos[c + d*x])/(a + b))^(3/2)*(-4*(2*a^4*b - 7*a^2*b^3 + 4*b^5)*EllipticE[(c + d*x)/2, (2*a)/(a + b)] + (a^5 - a^4*b + 16*a^3*b^2 - 16*a^2*b^3 - 16*a*b^4 + 16*b^5)*EllipticF[(c + d*x)/2, (2*a)/(a + b)]))/(a - b)^2 + (a*(a^6 - 25*a^2*b^4 + 16*b^6 + 4*a*b*(a^4 - 8*a^2*b^2 + 5*b^4)*Cos[c + d*x] + (a^3 - a*b^2)^2*Cos[2*(c + d*x)])*Sin[c + d*x])/(2*(a^2 - b^2)^2)))/(3*a^4*d*(a + b*Sec[c + d*x])^(5/2))","A",1
668,1,292,474,2.2048668,"\int \frac{1}{\sec ^{\frac{5}{2}}(c+d x) (a+b \sec (c+d x))^{5/2}} \, dx","Integrate[1/(Sec[c + d*x]^(5/2)*(a + b*Sec[c + d*x])^(5/2)),x]","\frac{\sec ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+b) \left(a \left(\frac{10 b^5 \sin (c+d x)}{b^2-a^2}-\frac{10 b^4 \left(11 b^2-15 a^2\right) \sin (c+d x) (a \cos (c+d x)+b)}{\left(a^2-b^2\right)^2}-28 b \sin (c+d x) (a \cos (c+d x)+b)^2+3 a \sin (2 (c+d x)) (a \cos (c+d x)+b)^2\right)+\frac{2 \left(\frac{a \cos (c+d x)+b}{a+b}\right)^{3/2} \left(\left(9 a^6+55 a^4 b^2-212 a^2 b^4+128 b^6\right) E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)+b \left(-17 a^5+17 a^4 b-116 a^3 b^2+116 a^2 b^3+128 a b^4-128 b^5\right) F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)\right)}{(a-b)^2}\right)}{15 a^5 d (a+b \sec (c+d x))^{5/2}}","\frac{8 b^2 \left(3 a^2-2 b^2\right) \sin (c+d x)}{3 a^2 d \left(a^2-b^2\right)^2 \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+b \sec (c+d x)}}+\frac{2 b^2 \sin (c+d x)}{3 a d \left(a^2-b^2\right) \sec ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}-\frac{4 b \left(7 a^4-49 a^2 b^2+32 b^4\right) \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{15 a^4 d \left(a^2-b^2\right)^2 \sqrt{\sec (c+d x)}}-\frac{2 b \left(17 a^4+116 a^2 b^2-128 b^4\right) \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{15 a^5 d \left(a^2-b^2\right) \sqrt{a+b \sec (c+d x)}}+\frac{2 \left(3 a^4-71 a^2 b^2+48 b^4\right) \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{15 a^3 d \left(a^2-b^2\right)^2 \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 \left(9 a^6+55 a^4 b^2-212 a^2 b^4+128 b^6\right) \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{15 a^5 d \left(a^2-b^2\right)^2 \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}",1,"((b + a*Cos[c + d*x])*Sec[c + d*x]^(5/2)*((2*((b + a*Cos[c + d*x])/(a + b))^(3/2)*((9*a^6 + 55*a^4*b^2 - 212*a^2*b^4 + 128*b^6)*EllipticE[(c + d*x)/2, (2*a)/(a + b)] + b*(-17*a^5 + 17*a^4*b - 116*a^3*b^2 + 116*a^2*b^3 + 128*a*b^4 - 128*b^5)*EllipticF[(c + d*x)/2, (2*a)/(a + b)]))/(a - b)^2 + a*((10*b^5*Sin[c + d*x])/(-a^2 + b^2) - (10*b^4*(-15*a^2 + 11*b^2)*(b + a*Cos[c + d*x])*Sin[c + d*x])/(a^2 - b^2)^2 - 28*b*(b + a*Cos[c + d*x])^2*Sin[c + d*x] + 3*a*(b + a*Cos[c + d*x])^2*Sin[2*(c + d*x)])))/(15*a^5*d*(a + b*Sec[c + d*x])^(5/2))","A",1
669,1,78,122,0.1532331,"\int \frac{1}{\sqrt{\sec (c+d x)} \sqrt{2+3 \sec (c+d x)}} \, dx","Integrate[1/(Sqrt[Sec[c + d*x]]*Sqrt[2 + 3*Sec[c + d*x]]),x]","\frac{\sqrt{2 \cos (c+d x)+3} \sqrt{\sec (c+d x)} \left(5 E\left(\frac{1}{2} (c+d x)|\frac{4}{5}\right)-3 F\left(\frac{1}{2} (c+d x)|\frac{4}{5}\right)\right)}{\sqrt{5} d \sqrt{3 \sec (c+d x)+2}}","\frac{\sqrt{5} \sqrt{3 \sec (c+d x)+2} E\left(\frac{1}{2} (c+d x)|\frac{4}{5}\right)}{d \sqrt{2 \cos (c+d x)+3} \sqrt{\sec (c+d x)}}-\frac{3 \sqrt{2 \cos (c+d x)+3} \sqrt{\sec (c+d x)} F\left(\frac{1}{2} (c+d x)|\frac{4}{5}\right)}{\sqrt{5} d \sqrt{3 \sec (c+d x)+2}}",1,"(Sqrt[3 + 2*Cos[c + d*x]]*(5*EllipticE[(c + d*x)/2, 4/5] - 3*EllipticF[(c + d*x)/2, 4/5])*Sqrt[Sec[c + d*x]])/(Sqrt[5]*d*Sqrt[2 + 3*Sec[c + d*x]])","A",1
670,1,68,109,0.1266549,"\int \frac{1}{\sqrt{\sec (c+d x)} \sqrt{-2+3 \sec (c+d x)}} \, dx","Integrate[1/(Sqrt[Sec[c + d*x]]*Sqrt[-2 + 3*Sec[c + d*x]]),x]","-\frac{\sqrt{3-2 \cos (c+d x)} \sqrt{\sec (c+d x)} \left(E\left(\left.\frac{1}{2} (c+d x)\right|-4\right)-3 F\left(\left.\frac{1}{2} (c+d x)\right|-4\right)\right)}{d \sqrt{3 \sec (c+d x)-2}}","\frac{3 \sqrt{3-2 \cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|-4\right)}{d \sqrt{3 \sec (c+d x)-2}}-\frac{\sqrt{3 \sec (c+d x)-2} E\left(\left.\frac{1}{2} (c+d x)\right|-4\right)}{d \sqrt{3-2 \cos (c+d x)} \sqrt{\sec (c+d x)}}",1,"-((Sqrt[3 - 2*Cos[c + d*x]]*(EllipticE[(c + d*x)/2, -4] - 3*EllipticF[(c + d*x)/2, -4])*Sqrt[Sec[c + d*x]])/(d*Sqrt[-2 + 3*Sec[c + d*x]]))","A",1
671,1,68,108,0.0846867,"\int \frac{1}{\sqrt{2-3 \sec (c+d x)} \sqrt{\sec (c+d x)}} \, dx","Integrate[1/(Sqrt[2 - 3*Sec[c + d*x]]*Sqrt[Sec[c + d*x]]),x]","-\frac{\sqrt{3-2 \cos (c+d x)} \sqrt{\sec (c+d x)} \left(E\left(\left.\frac{1}{2} (c+d x)\right|-4\right)-3 F\left(\left.\frac{1}{2} (c+d x)\right|-4\right)\right)}{d \sqrt{2-3 \sec (c+d x)}}","\frac{3 \sqrt{3-2 \cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|-4\right)}{d \sqrt{2-3 \sec (c+d x)}}+\frac{\sqrt{2-3 \sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|-4\right)}{d \sqrt{3-2 \cos (c+d x)} \sqrt{\sec (c+d x)}}",1,"-((Sqrt[3 - 2*Cos[c + d*x]]*(EllipticE[(c + d*x)/2, -4] - 3*EllipticF[(c + d*x)/2, -4])*Sqrt[Sec[c + d*x]])/(d*Sqrt[2 - 3*Sec[c + d*x]]))","A",1
672,1,78,123,0.102423,"\int \frac{1}{\sqrt{-2-3 \sec (c+d x)} \sqrt{\sec (c+d x)}} \, dx","Integrate[1/(Sqrt[-2 - 3*Sec[c + d*x]]*Sqrt[Sec[c + d*x]]),x]","\frac{\sqrt{2 \cos (c+d x)+3} \sqrt{\sec (c+d x)} \left(5 E\left(\frac{1}{2} (c+d x)|\frac{4}{5}\right)-3 F\left(\frac{1}{2} (c+d x)|\frac{4}{5}\right)\right)}{\sqrt{5} d \sqrt{-3 \sec (c+d x)-2}}","-\frac{3 \sqrt{2 \cos (c+d x)+3} \sqrt{\sec (c+d x)} F\left(\frac{1}{2} (c+d x)|\frac{4}{5}\right)}{\sqrt{5} d \sqrt{-3 \sec (c+d x)-2}}-\frac{\sqrt{5} \sqrt{-3 \sec (c+d x)-2} E\left(\frac{1}{2} (c+d x)|\frac{4}{5}\right)}{d \sqrt{2 \cos (c+d x)+3} \sqrt{\sec (c+d x)}}",1,"(Sqrt[3 + 2*Cos[c + d*x]]*(5*EllipticE[(c + d*x)/2, 4/5] - 3*EllipticF[(c + d*x)/2, 4/5])*Sqrt[Sec[c + d*x]])/(Sqrt[5]*d*Sqrt[-2 - 3*Sec[c + d*x]])","A",1
673,1,81,127,0.1601866,"\int \frac{1}{\sqrt{\sec (c+d x)} \sqrt{3+2 \sec (c+d x)}} \, dx","Integrate[1/(Sqrt[Sec[c + d*x]]*Sqrt[3 + 2*Sec[c + d*x]]),x]","\frac{2 \sqrt{3 \cos (c+d x)+2} \sqrt{\sec (c+d x)} \left(5 E\left(\frac{1}{2} (c+d x)|\frac{6}{5}\right)-2 F\left(\frac{1}{2} (c+d x)|\frac{6}{5}\right)\right)}{3 \sqrt{5} d \sqrt{2 \sec (c+d x)+3}}","\frac{2 \sqrt{5} \sqrt{2 \sec (c+d x)+3} E\left(\frac{1}{2} (c+d x)|\frac{6}{5}\right)}{3 d \sqrt{3 \cos (c+d x)+2} \sqrt{\sec (c+d x)}}-\frac{4 \sqrt{3 \cos (c+d x)+2} \sqrt{\sec (c+d x)} F\left(\frac{1}{2} (c+d x)|\frac{6}{5}\right)}{3 \sqrt{5} d \sqrt{2 \sec (c+d x)+3}}",1,"(2*Sqrt[2 + 3*Cos[c + d*x]]*(5*EllipticE[(c + d*x)/2, 6/5] - 2*EllipticF[(c + d*x)/2, 6/5])*Sqrt[Sec[c + d*x]])/(3*Sqrt[5]*d*Sqrt[3 + 2*Sec[c + d*x]])","A",1
674,1,72,113,0.1288271,"\int \frac{1}{\sqrt{3-2 \sec (c+d x)} \sqrt{\sec (c+d x)}} \, dx","Integrate[1/(Sqrt[3 - 2*Sec[c + d*x]]*Sqrt[Sec[c + d*x]]),x]","\frac{\sqrt{3 \cos (c+d x)-2} \sqrt{\sec (c+d x)} \left(4 F\left(\left.\frac{1}{2} (c+d x)\right|6\right)+2 E\left(\left.\frac{1}{2} (c+d x)\right|6\right)\right)}{3 d \sqrt{3-2 \sec (c+d x)}}","\frac{4 \sqrt{3 \cos (c+d x)-2} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|6\right)}{3 d \sqrt{3-2 \sec (c+d x)}}+\frac{2 \sqrt{3-2 \sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|6\right)}{3 d \sqrt{3 \cos (c+d x)-2} \sqrt{\sec (c+d x)}}",1,"(Sqrt[-2 + 3*Cos[c + d*x]]*(2*EllipticE[(c + d*x)/2, 6] + 4*EllipticF[(c + d*x)/2, 6])*Sqrt[Sec[c + d*x]])/(3*d*Sqrt[3 - 2*Sec[c + d*x]])","A",1
675,1,72,129,0.0862496,"\int \frac{1}{\sqrt{\sec (c+d x)} \sqrt{-3+2 \sec (c+d x)}} \, dx","Integrate[1/(Sqrt[Sec[c + d*x]]*Sqrt[-3 + 2*Sec[c + d*x]]),x]","\frac{\sqrt{3 \cos (c+d x)-2} \sqrt{\sec (c+d x)} \left(4 F\left(\left.\frac{1}{2} (c+d x)\right|6\right)+2 E\left(\left.\frac{1}{2} (c+d x)\right|6\right)\right)}{3 d \sqrt{2 \sec (c+d x)-3}}","\frac{4 \sqrt{2-3 \cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\frac{1}{2} (c+d x+\pi )|\frac{6}{5}\right)}{3 \sqrt{5} d \sqrt{2 \sec (c+d x)-3}}-\frac{2 \sqrt{5} \sqrt{2 \sec (c+d x)-3} E\left(\frac{1}{2} (c+d x+\pi )|\frac{6}{5}\right)}{3 d \sqrt{2-3 \cos (c+d x)} \sqrt{\sec (c+d x)}}",1,"(Sqrt[-2 + 3*Cos[c + d*x]]*(2*EllipticE[(c + d*x)/2, 6] + 4*EllipticF[(c + d*x)/2, 6])*Sqrt[Sec[c + d*x]])/(3*d*Sqrt[-3 + 2*Sec[c + d*x]])","A",1
676,1,81,115,0.1183666,"\int \frac{1}{\sqrt{-3-2 \sec (c+d x)} \sqrt{\sec (c+d x)}} \, dx","Integrate[1/(Sqrt[-3 - 2*Sec[c + d*x]]*Sqrt[Sec[c + d*x]]),x]","\frac{2 \sqrt{3 \cos (c+d x)+2} \sqrt{\sec (c+d x)} \left(5 E\left(\frac{1}{2} (c+d x)|\frac{6}{5}\right)-2 F\left(\frac{1}{2} (c+d x)|\frac{6}{5}\right)\right)}{3 \sqrt{5} d \sqrt{-2 \sec (c+d x)-3}}","-\frac{4 \sqrt{-3 \cos (c+d x)-2} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x+\pi )\right|6\right)}{3 d \sqrt{-2 \sec (c+d x)-3}}-\frac{2 \sqrt{-2 \sec (c+d x)-3} E\left(\left.\frac{1}{2} (c+d x+\pi )\right|6\right)}{3 d \sqrt{-3 \cos (c+d x)-2} \sqrt{\sec (c+d x)}}",1,"(2*Sqrt[2 + 3*Cos[c + d*x]]*(5*EllipticE[(c + d*x)/2, 6/5] - 2*EllipticF[(c + d*x)/2, 6/5])*Sqrt[Sec[c + d*x]])/(3*Sqrt[5]*d*Sqrt[-3 - 2*Sec[c + d*x]])","A",1
677,1,61,61,0.0595779,"\int \frac{\sqrt{\sec (c+d x)}}{\sqrt{2+3 \sec (c+d x)}} \, dx","Integrate[Sqrt[Sec[c + d*x]]/Sqrt[2 + 3*Sec[c + d*x]],x]","\frac{2 \sqrt{2 \cos (c+d x)+3} \sqrt{\sec (c+d x)} F\left(\frac{1}{2} (c+d x)|\frac{4}{5}\right)}{\sqrt{5} d \sqrt{3 \sec (c+d x)+2}}","\frac{2 \sqrt{2 \cos (c+d x)+3} \sqrt{\sec (c+d x)} F\left(\frac{1}{2} (c+d x)|\frac{4}{5}\right)}{\sqrt{5} d \sqrt{3 \sec (c+d x)+2}}",1,"(2*Sqrt[3 + 2*Cos[c + d*x]]*EllipticF[(c + d*x)/2, 4/5]*Sqrt[Sec[c + d*x]])/(Sqrt[5]*d*Sqrt[2 + 3*Sec[c + d*x]])","A",1
678,1,54,54,0.0678894,"\int \frac{\sqrt{\sec (c+d x)}}{\sqrt{-2+3 \sec (c+d x)}} \, dx","Integrate[Sqrt[Sec[c + d*x]]/Sqrt[-2 + 3*Sec[c + d*x]],x]","\frac{2 \sqrt{3-2 \cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|-4\right)}{d \sqrt{3 \sec (c+d x)-2}}","\frac{2 \sqrt{3-2 \cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|-4\right)}{d \sqrt{3 \sec (c+d x)-2}}",1,"(2*Sqrt[3 - 2*Cos[c + d*x]]*EllipticF[(c + d*x)/2, -4]*Sqrt[Sec[c + d*x]])/(d*Sqrt[-2 + 3*Sec[c + d*x]])","A",1
679,1,54,54,0.0400256,"\int \frac{\sqrt{\sec (c+d x)}}{\sqrt{2-3 \sec (c+d x)}} \, dx","Integrate[Sqrt[Sec[c + d*x]]/Sqrt[2 - 3*Sec[c + d*x]],x]","\frac{2 \sqrt{3-2 \cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|-4\right)}{d \sqrt{2-3 \sec (c+d x)}}","\frac{2 \sqrt{3-2 \cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|-4\right)}{d \sqrt{2-3 \sec (c+d x)}}",1,"(2*Sqrt[3 - 2*Cos[c + d*x]]*EllipticF[(c + d*x)/2, -4]*Sqrt[Sec[c + d*x]])/(d*Sqrt[2 - 3*Sec[c + d*x]])","A",1
680,1,61,61,0.0461197,"\int \frac{\sqrt{\sec (c+d x)}}{\sqrt{-2-3 \sec (c+d x)}} \, dx","Integrate[Sqrt[Sec[c + d*x]]/Sqrt[-2 - 3*Sec[c + d*x]],x]","\frac{2 \sqrt{2 \cos (c+d x)+3} \sqrt{\sec (c+d x)} F\left(\frac{1}{2} (c+d x)|\frac{4}{5}\right)}{\sqrt{5} d \sqrt{-3 \sec (c+d x)-2}}","\frac{2 \sqrt{2 \cos (c+d x)+3} \sqrt{\sec (c+d x)} F\left(\frac{1}{2} (c+d x)|\frac{4}{5}\right)}{\sqrt{5} d \sqrt{-3 \sec (c+d x)-2}}",1,"(2*Sqrt[3 + 2*Cos[c + d*x]]*EllipticF[(c + d*x)/2, 4/5]*Sqrt[Sec[c + d*x]])/(Sqrt[5]*d*Sqrt[-2 - 3*Sec[c + d*x]])","A",1
681,1,61,61,0.0621327,"\int \frac{\sqrt{\sec (c+d x)}}{\sqrt{3+2 \sec (c+d x)}} \, dx","Integrate[Sqrt[Sec[c + d*x]]/Sqrt[3 + 2*Sec[c + d*x]],x]","\frac{2 \sqrt{3 \cos (c+d x)+2} \sqrt{\sec (c+d x)} F\left(\frac{1}{2} (c+d x)|\frac{6}{5}\right)}{\sqrt{5} d \sqrt{2 \sec (c+d x)+3}}","\frac{2 \sqrt{3 \cos (c+d x)+2} \sqrt{\sec (c+d x)} F\left(\frac{1}{2} (c+d x)|\frac{6}{5}\right)}{\sqrt{5} d \sqrt{2 \sec (c+d x)+3}}",1,"(2*Sqrt[2 + 3*Cos[c + d*x]]*EllipticF[(c + d*x)/2, 6/5]*Sqrt[Sec[c + d*x]])/(Sqrt[5]*d*Sqrt[3 + 2*Sec[c + d*x]])","A",1
682,1,54,54,0.061264,"\int \frac{\sqrt{\sec (c+d x)}}{\sqrt{3-2 \sec (c+d x)}} \, dx","Integrate[Sqrt[Sec[c + d*x]]/Sqrt[3 - 2*Sec[c + d*x]],x]","\frac{2 \sqrt{3 \cos (c+d x)-2} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|6\right)}{d \sqrt{3-2 \sec (c+d x)}}","\frac{2 \sqrt{3 \cos (c+d x)-2} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|6\right)}{d \sqrt{3-2 \sec (c+d x)}}",1,"(2*Sqrt[-2 + 3*Cos[c + d*x]]*EllipticF[(c + d*x)/2, 6]*Sqrt[Sec[c + d*x]])/(d*Sqrt[3 - 2*Sec[c + d*x]])","A",1
683,1,54,62,0.0462436,"\int \frac{\sqrt{\sec (c+d x)}}{\sqrt{-3+2 \sec (c+d x)}} \, dx","Integrate[Sqrt[Sec[c + d*x]]/Sqrt[-3 + 2*Sec[c + d*x]],x]","\frac{2 \sqrt{3 \cos (c+d x)-2} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|6\right)}{d \sqrt{2 \sec (c+d x)-3}}","\frac{2 \sqrt{2-3 \cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\frac{1}{2} (c+d x+\pi )|\frac{6}{5}\right)}{\sqrt{5} d \sqrt{2 \sec (c+d x)-3}}",1,"(2*Sqrt[-2 + 3*Cos[c + d*x]]*EllipticF[(c + d*x)/2, 6]*Sqrt[Sec[c + d*x]])/(d*Sqrt[-3 + 2*Sec[c + d*x]])","A",1
684,1,61,55,0.047055,"\int \frac{\sqrt{\sec (c+d x)}}{\sqrt{-3-2 \sec (c+d x)}} \, dx","Integrate[Sqrt[Sec[c + d*x]]/Sqrt[-3 - 2*Sec[c + d*x]],x]","\frac{2 \sqrt{3 \cos (c+d x)+2} \sqrt{\sec (c+d x)} F\left(\frac{1}{2} (c+d x)|\frac{6}{5}\right)}{\sqrt{5} d \sqrt{-2 \sec (c+d x)-3}}","\frac{2 \sqrt{-3 \cos (c+d x)-2} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x+\pi )\right|6\right)}{d \sqrt{-2 \sec (c+d x)-3}}",1,"(2*Sqrt[2 + 3*Cos[c + d*x]]*EllipticF[(c + d*x)/2, 6/5]*Sqrt[Sec[c + d*x]])/(Sqrt[5]*d*Sqrt[-3 - 2*Sec[c + d*x]])","A",1
685,1,7160,105,26.8575454,"\int \sec (c+d x) \sqrt[3]{a+b \sec (c+d x)} \, dx","Integrate[Sec[c + d*x]*(a + b*Sec[c + d*x])^(1/3),x]","\text{Result too large to show}","\frac{\sqrt{2} \tan (c+d x) \sqrt[3]{a+b \sec (c+d x)} F_1\left(\frac{1}{2};\frac{1}{2},-\frac{1}{3};\frac{3}{2};\frac{1}{2} (1-\sec (c+d x)),\frac{b (1-\sec (c+d x))}{a+b}\right)}{d \sqrt{\sec (c+d x)+1} \sqrt[3]{\frac{a+b \sec (c+d x)}{a+b}}}",1,"Result too large to show","B",0
686,0,0,17,2.089394,"\int \sqrt[3]{a+b \sec (c+d x)} \, dx","Integrate[(a + b*Sec[c + d*x])^(1/3),x]","\int \sqrt[3]{a+b \sec (c+d x)} \, dx","\text{Int}\left(\sqrt[3]{a+b \sec (c+d x)},x\right)",0,"Integrate[(a + b*Sec[c + d*x])^(1/3), x]","A",-1
687,1,21877,362,26.9519012,"\int \sec ^4(c+d x) (a+b \sec (c+d x))^{2/3} \, dx","Integrate[Sec[c + d*x]^4*(a + b*Sec[c + d*x])^(2/3),x]","\text{Result too large to show}","\frac{a \left(18 a^2+49 b^2\right) \tan (c+d x) (a+b \sec (c+d x))^{2/3} F_1\left(\frac{1}{2};\frac{1}{2},-\frac{2}{3};\frac{3}{2};\frac{1}{2} (1-\sec (c+d x)),\frac{b (1-\sec (c+d x))}{a+b}\right)}{110 \sqrt{2} b^3 d \sqrt{\sec (c+d x)+1} \left(\frac{a+b \sec (c+d x)}{a+b}\right)^{2/3}}+\frac{3 \left(9 a^2+32 b^2\right) \tan (c+d x) (a+b \sec (c+d x))^{2/3}}{220 b^2 d}-\frac{\left(9 a^4+23 a^2 b^2-32 b^4\right) \tan (c+d x) \sqrt[3]{\frac{a+b \sec (c+d x)}{a+b}} F_1\left(\frac{1}{2};\frac{1}{2},\frac{1}{3};\frac{3}{2};\frac{1}{2} (1-\sec (c+d x)),\frac{b (1-\sec (c+d x))}{a+b}\right)}{55 \sqrt{2} b^3 d \sqrt{\sec (c+d x)+1} \sqrt[3]{a+b \sec (c+d x)}}-\frac{9 a \tan (c+d x) (a+b \sec (c+d x))^{5/3}}{44 b^2 d}+\frac{3 \tan (c+d x) \sec (c+d x) (a+b \sec (c+d x))^{5/3}}{11 b d}",1,"Result too large to show","B",0
688,1,18991,305,26.774882,"\int \sec ^3(c+d x) (a+b \sec (c+d x))^{2/3} \, dx","Integrate[Sec[c + d*x]^3*(a + b*Sec[c + d*x])^(2/3),x]","\text{Result too large to show}","-\frac{\left(6 a^2-25 b^2\right) \tan (c+d x) (a+b \sec (c+d x))^{2/3} F_1\left(\frac{1}{2};\frac{1}{2},-\frac{2}{3};\frac{3}{2};\frac{1}{2} (1-\sec (c+d x)),\frac{b (1-\sec (c+d x))}{a+b}\right)}{20 \sqrt{2} b^2 d \sqrt{\sec (c+d x)+1} \left(\frac{a+b \sec (c+d x)}{a+b}\right)^{2/3}}+\frac{3 a \left(a^2-b^2\right) \tan (c+d x) \sqrt[3]{\frac{a+b \sec (c+d x)}{a+b}} F_1\left(\frac{1}{2};\frac{1}{2},\frac{1}{3};\frac{3}{2};\frac{1}{2} (1-\sec (c+d x)),\frac{b (1-\sec (c+d x))}{a+b}\right)}{10 \sqrt{2} b^2 d \sqrt{\sec (c+d x)+1} \sqrt[3]{a+b \sec (c+d x)}}+\frac{3 \tan (c+d x) (a+b \sec (c+d x))^{5/3}}{8 b d}-\frac{9 a \tan (c+d x) (a+b \sec (c+d x))^{2/3}}{40 b d}",1,"Result too large to show","B",0
689,1,2505,260,18.8969774,"\int \sec ^2(c+d x) (a+b \sec (c+d x))^{2/3} \, dx","Integrate[Sec[c + d*x]^2*(a + b*Sec[c + d*x])^(2/3),x]","\text{Result too large to show}","-\frac{2 \sqrt{2} \left(a^2-b^2\right) \tan (c+d x) \sqrt[3]{\frac{a+b \sec (c+d x)}{a+b}} F_1\left(\frac{1}{2};\frac{1}{2},\frac{1}{3};\frac{3}{2};\frac{1}{2} (1-\sec (c+d x)),\frac{b (1-\sec (c+d x))}{a+b}\right)}{5 b d \sqrt{\sec (c+d x)+1} \sqrt[3]{a+b \sec (c+d x)}}+\frac{2 \sqrt{2} a \tan (c+d x) (a+b \sec (c+d x))^{2/3} F_1\left(\frac{1}{2};\frac{1}{2},-\frac{2}{3};\frac{3}{2};\frac{1}{2} (1-\sec (c+d x)),\frac{b (1-\sec (c+d x))}{a+b}\right)}{5 b d \sqrt{\sec (c+d x)+1} \left(\frac{a+b \sec (c+d x)}{a+b}\right)^{2/3}}+\frac{3 \tan (c+d x) (a+b \sec (c+d x))^{2/3}}{5 d}",1,"((a + b*Sec[c + d*x])^(2/3)*((3*a*Sin[c + d*x])/(5*b) + (3*Tan[c + d*x])/5))/d - ((-2*b + 3*a*Cos[c + d*x])*(a + b*Sec[c + d*x])^(2/3)*(3*a*(b + a*Cos[c + d*x])^(2/3)*Sqrt[1 - Cos[c + d*x]^2]*Sec[c + d*x]^(2/3) - (3*(b + a*Cos[c + d*x])^(2/3)*Sqrt[(1 - Sqrt[b^(-2)]*b*Sec[c + d*x])/(1 + a*Sqrt[b^(-2)])]*Sqrt[(1 + Sqrt[b^(-2)]*b*Sec[c + d*x])/(1 - a*Sqrt[b^(-2)])]*(-5*(a^2 - b^2)*AppellF1[2/3, 1/2, 1/2, 5/3, -((a + b*Sec[c + d*x])/(-a + 1/Sqrt[b^(-2)])), (a + b*Sec[c + d*x])/(a + 1/Sqrt[b^(-2)])] + 2*a*AppellF1[5/3, 1/2, 1/2, 8/3, -((a + b*Sec[c + d*x])/(-a + 1/Sqrt[b^(-2)])), (a + b*Sec[c + d*x])/(a + 1/Sqrt[b^(-2)])]*(a + b*Sec[c + d*x])))/(5*b*Sqrt[1 - Cos[c + d*x]^2]*Sec[c + d*x]^(1/3))))/(5*b*d*((3*a*(b + a*Cos[c + d*x])^(2/3)*Sin[c + d*x])/(Sqrt[1 - Cos[c + d*x]^2]*Sec[c + d*x]^(1/3)) - (2*a^2*Sqrt[1 - Cos[c + d*x]^2]*Sec[c + d*x]^(2/3)*Sin[c + d*x])/(b + a*Cos[c + d*x])^(1/3) + 2*a*(b + a*Cos[c + d*x])^(2/3)*Sqrt[1 - Cos[c + d*x]^2]*Sec[c + d*x]^(5/3)*Sin[c + d*x] - (3*Sqrt[b^(-2)]*(b + a*Cos[c + d*x])^(2/3)*Sec[c + d*x]^(5/3)*Sqrt[(1 - Sqrt[b^(-2)]*b*Sec[c + d*x])/(1 + a*Sqrt[b^(-2)])]*(-5*(a^2 - b^2)*AppellF1[2/3, 1/2, 1/2, 5/3, -((a + b*Sec[c + d*x])/(-a + 1/Sqrt[b^(-2)])), (a + b*Sec[c + d*x])/(a + 1/Sqrt[b^(-2)])] + 2*a*AppellF1[5/3, 1/2, 1/2, 8/3, -((a + b*Sec[c + d*x])/(-a + 1/Sqrt[b^(-2)])), (a + b*Sec[c + d*x])/(a + 1/Sqrt[b^(-2)])]*(a + b*Sec[c + d*x]))*Sin[c + d*x])/(10*(1 - a*Sqrt[b^(-2)])*Sqrt[1 - Cos[c + d*x]^2]*Sqrt[(1 + Sqrt[b^(-2)]*b*Sec[c + d*x])/(1 - a*Sqrt[b^(-2)])]) + (3*Sqrt[b^(-2)]*(b + a*Cos[c + d*x])^(2/3)*Sec[c + d*x]^(5/3)*Sqrt[(1 + Sqrt[b^(-2)]*b*Sec[c + d*x])/(1 - a*Sqrt[b^(-2)])]*(-5*(a^2 - b^2)*AppellF1[2/3, 1/2, 1/2, 5/3, -((a + b*Sec[c + d*x])/(-a + 1/Sqrt[b^(-2)])), (a + b*Sec[c + d*x])/(a + 1/Sqrt[b^(-2)])] + 2*a*AppellF1[5/3, 1/2, 1/2, 8/3, -((a + b*Sec[c + d*x])/(-a + 1/Sqrt[b^(-2)])), (a + b*Sec[c + d*x])/(a + 1/Sqrt[b^(-2)])]*(a + b*Sec[c + d*x]))*Sin[c + d*x])/(10*(1 + a*Sqrt[b^(-2)])*Sqrt[1 - Cos[c + d*x]^2]*Sqrt[(1 - Sqrt[b^(-2)]*b*Sec[c + d*x])/(1 + a*Sqrt[b^(-2)])]) + (3*(b + a*Cos[c + d*x])^(2/3)*Sqrt[(1 - Sqrt[b^(-2)]*b*Sec[c + d*x])/(1 + a*Sqrt[b^(-2)])]*Sqrt[(1 + Sqrt[b^(-2)]*b*Sec[c + d*x])/(1 - a*Sqrt[b^(-2)])]*(-5*(a^2 - b^2)*AppellF1[2/3, 1/2, 1/2, 5/3, -((a + b*Sec[c + d*x])/(-a + 1/Sqrt[b^(-2)])), (a + b*Sec[c + d*x])/(a + 1/Sqrt[b^(-2)])] + 2*a*AppellF1[5/3, 1/2, 1/2, 8/3, -((a + b*Sec[c + d*x])/(-a + 1/Sqrt[b^(-2)])), (a + b*Sec[c + d*x])/(a + 1/Sqrt[b^(-2)])]*(a + b*Sec[c + d*x]))*Sin[c + d*x])/(5*b*(1 - Cos[c + d*x]^2)^(3/2)*Sec[c + d*x]^(4/3)) + (2*a*Sqrt[(1 - Sqrt[b^(-2)]*b*Sec[c + d*x])/(1 + a*Sqrt[b^(-2)])]*Sqrt[(1 + Sqrt[b^(-2)]*b*Sec[c + d*x])/(1 - a*Sqrt[b^(-2)])]*(-5*(a^2 - b^2)*AppellF1[2/3, 1/2, 1/2, 5/3, -((a + b*Sec[c + d*x])/(-a + 1/Sqrt[b^(-2)])), (a + b*Sec[c + d*x])/(a + 1/Sqrt[b^(-2)])] + 2*a*AppellF1[5/3, 1/2, 1/2, 8/3, -((a + b*Sec[c + d*x])/(-a + 1/Sqrt[b^(-2)])), (a + b*Sec[c + d*x])/(a + 1/Sqrt[b^(-2)])]*(a + b*Sec[c + d*x]))*Sin[c + d*x])/(5*b*(b + a*Cos[c + d*x])^(1/3)*Sqrt[1 - Cos[c + d*x]^2]*Sec[c + d*x]^(1/3)) + ((b + a*Cos[c + d*x])^(2/3)*Sec[c + d*x]^(2/3)*Sqrt[(1 - Sqrt[b^(-2)]*b*Sec[c + d*x])/(1 + a*Sqrt[b^(-2)])]*Sqrt[(1 + Sqrt[b^(-2)]*b*Sec[c + d*x])/(1 - a*Sqrt[b^(-2)])]*(-5*(a^2 - b^2)*AppellF1[2/3, 1/2, 1/2, 5/3, -((a + b*Sec[c + d*x])/(-a + 1/Sqrt[b^(-2)])), (a + b*Sec[c + d*x])/(a + 1/Sqrt[b^(-2)])] + 2*a*AppellF1[5/3, 1/2, 1/2, 8/3, -((a + b*Sec[c + d*x])/(-a + 1/Sqrt[b^(-2)])), (a + b*Sec[c + d*x])/(a + 1/Sqrt[b^(-2)])]*(a + b*Sec[c + d*x]))*Sin[c + d*x])/(5*b*Sqrt[1 - Cos[c + d*x]^2]) - (3*(b + a*Cos[c + d*x])^(2/3)*Sqrt[(1 - Sqrt[b^(-2)]*b*Sec[c + d*x])/(1 + a*Sqrt[b^(-2)])]*Sqrt[(1 + Sqrt[b^(-2)]*b*Sec[c + d*x])/(1 - a*Sqrt[b^(-2)])]*(2*a*b*AppellF1[5/3, 1/2, 1/2, 8/3, -((a + b*Sec[c + d*x])/(-a + 1/Sqrt[b^(-2)])), (a + b*Sec[c + d*x])/(a + 1/Sqrt[b^(-2)])]*Sec[c + d*x]*Tan[c + d*x] - 5*(a^2 - b^2)*((b*AppellF1[5/3, 1/2, 3/2, 8/3, -((a + b*Sec[c + d*x])/(-a + 1/Sqrt[b^(-2)])), (a + b*Sec[c + d*x])/(a + 1/Sqrt[b^(-2)])]*Sec[c + d*x]*Tan[c + d*x])/(5*(a + 1/Sqrt[b^(-2)])) - (b*AppellF1[5/3, 3/2, 1/2, 8/3, -((a + b*Sec[c + d*x])/(-a + 1/Sqrt[b^(-2)])), (a + b*Sec[c + d*x])/(a + 1/Sqrt[b^(-2)])]*Sec[c + d*x]*Tan[c + d*x])/(5*(-a + 1/Sqrt[b^(-2)]))) + 2*a*(a + b*Sec[c + d*x])*((5*b*AppellF1[8/3, 1/2, 3/2, 11/3, -((a + b*Sec[c + d*x])/(-a + 1/Sqrt[b^(-2)])), (a + b*Sec[c + d*x])/(a + 1/Sqrt[b^(-2)])]*Sec[c + d*x]*Tan[c + d*x])/(16*(a + 1/Sqrt[b^(-2)])) - (5*b*AppellF1[8/3, 3/2, 1/2, 11/3, -((a + b*Sec[c + d*x])/(-a + 1/Sqrt[b^(-2)])), (a + b*Sec[c + d*x])/(a + 1/Sqrt[b^(-2)])]*Sec[c + d*x]*Tan[c + d*x])/(16*(-a + 1/Sqrt[b^(-2)])))))/(5*b*Sqrt[1 - Cos[c + d*x]^2]*Sec[c + d*x]^(1/3))))","B",0
690,1,7142,105,26.3995912,"\int \sec (c+d x) (a+b \sec (c+d x))^{2/3} \, dx","Integrate[Sec[c + d*x]*(a + b*Sec[c + d*x])^(2/3),x]","\text{Result too large to show}","\frac{\sqrt{2} \tan (c+d x) (a+b \sec (c+d x))^{2/3} F_1\left(\frac{1}{2};\frac{1}{2},-\frac{2}{3};\frac{3}{2};\frac{1}{2} (1-\sec (c+d x)),\frac{b (1-\sec (c+d x))}{a+b}\right)}{d \sqrt{\sec (c+d x)+1} \left(\frac{a+b \sec (c+d x)}{a+b}\right)^{2/3}}",1,"Result too large to show","B",0
691,0,0,17,2.0584027,"\int (a+b \sec (c+d x))^{2/3} \, dx","Integrate[(a + b*Sec[c + d*x])^(2/3),x]","\int (a+b \sec (c+d x))^{2/3} \, dx","\text{Int}\left((a+b \sec (c+d x))^{2/3},x\right)",0,"Integrate[(a + b*Sec[c + d*x])^(2/3), x]","A",-1
692,1,8660,108,27.2972932,"\int \sec (c+d x) (a+b \sec (c+d x))^{4/3} \, dx","Integrate[Sec[c + d*x]*(a + b*Sec[c + d*x])^(4/3),x]","\text{Result too large to show}","\frac{\sqrt{2} (a+b) \tan (c+d x) \sqrt[3]{a+b \sec (c+d x)} F_1\left(\frac{1}{2};\frac{1}{2},-\frac{4}{3};\frac{3}{2};\frac{1}{2} (1-\sec (c+d x)),\frac{b (1-\sec (c+d x))}{a+b}\right)}{d \sqrt{\sec (c+d x)+1} \sqrt[3]{\frac{a+b \sec (c+d x)}{a+b}}}",1,"Result too large to show","B",0
693,0,0,17,20.4455344,"\int (a+b \sec (c+d x))^{4/3} \, dx","Integrate[(a + b*Sec[c + d*x])^(4/3),x]","\int (a+b \sec (c+d x))^{4/3} \, dx","\text{Int}\left((a+b \sec (c+d x))^{4/3},x\right)",0,"Integrate[(a + b*Sec[c + d*x])^(4/3), x]","A",-1
694,1,28057,412,27.4096314,"\int \sec ^4(c+d x) (a+b \sec (c+d x))^{5/3} \, dx","Integrate[Sec[c + d*x]^4*(a + b*Sec[c + d*x])^(5/3),x]","\text{Result too large to show}","\frac{3 \left(18 a^2+121 b^2\right) \tan (c+d x) (a+b \sec (c+d x))^{5/3}}{1232 b^2 d}+\frac{3 a \left(18 a^2+97 b^2\right) \tan (c+d x) (a+b \sec (c+d x))^{2/3}}{1232 b^2 d}+\frac{\left(36 a^4+164 a^2 b^2+605 b^4\right) \tan (c+d x) (a+b \sec (c+d x))^{2/3} F_1\left(\frac{1}{2};\frac{1}{2},-\frac{2}{3};\frac{3}{2};\frac{1}{2} (1-\sec (c+d x)),\frac{b (1-\sec (c+d x))}{a+b}\right)}{616 \sqrt{2} b^3 d \sqrt{\sec (c+d x)+1} \left(\frac{a+b \sec (c+d x)}{a+b}\right)^{2/3}}-\frac{a \left(18 a^4+79 a^2 b^2-97 b^4\right) \tan (c+d x) \sqrt[3]{\frac{a+b \sec (c+d x)}{a+b}} F_1\left(\frac{1}{2};\frac{1}{2},\frac{1}{3};\frac{3}{2};\frac{1}{2} (1-\sec (c+d x)),\frac{b (1-\sec (c+d x))}{a+b}\right)}{308 \sqrt{2} b^3 d \sqrt{\sec (c+d x)+1} \sqrt[3]{a+b \sec (c+d x)}}-\frac{9 a \tan (c+d x) (a+b \sec (c+d x))^{8/3}}{77 b^2 d}+\frac{3 \tan (c+d x) \sec (c+d x) (a+b \sec (c+d x))^{8/3}}{14 b d}",1,"Result too large to show","B",0
695,1,21890,356,27.3401588,"\int \sec ^3(c+d x) (a+b \sec (c+d x))^{5/3} \, dx","Integrate[Sec[c + d*x]^3*(a + b*Sec[c + d*x])^(5/3),x]","\text{Result too large to show}","-\frac{a \left(30 a^2-373 b^2\right) \tan (c+d x) (a+b \sec (c+d x))^{2/3} F_1\left(\frac{1}{2};\frac{1}{2},-\frac{2}{3};\frac{3}{2};\frac{1}{2} (1-\sec (c+d x)),\frac{b (1-\sec (c+d x))}{a+b}\right)}{220 \sqrt{2} b^2 d \sqrt{\sec (c+d x)+1} \left(\frac{a+b \sec (c+d x)}{a+b}\right)^{2/3}}-\frac{3 \left(15 a^2-64 b^2\right) \tan (c+d x) (a+b \sec (c+d x))^{2/3}}{440 b d}+\frac{\left(15 a^4-79 a^2 b^2+64 b^4\right) \tan (c+d x) \sqrt[3]{\frac{a+b \sec (c+d x)}{a+b}} F_1\left(\frac{1}{2};\frac{1}{2},\frac{1}{3};\frac{3}{2};\frac{1}{2} (1-\sec (c+d x)),\frac{b (1-\sec (c+d x))}{a+b}\right)}{110 \sqrt{2} b^2 d \sqrt{\sec (c+d x)+1} \sqrt[3]{a+b \sec (c+d x)}}+\frac{3 \tan (c+d x) (a+b \sec (c+d x))^{8/3}}{11 b d}-\frac{9 a \tan (c+d x) (a+b \sec (c+d x))^{5/3}}{88 b d}",1,"Result too large to show","B",0
696,1,19016,299,27.2760936,"\int \sec ^2(c+d x) (a+b \sec (c+d x))^{5/3} \, dx","Integrate[Sec[c + d*x]^2*(a + b*Sec[c + d*x])^(5/3),x]","\text{Result too large to show}","\frac{\left(2 a^2+5 b^2\right) \tan (c+d x) (a+b \sec (c+d x))^{2/3} F_1\left(\frac{1}{2};\frac{1}{2},-\frac{2}{3};\frac{3}{2};\frac{1}{2} (1-\sec (c+d x)),\frac{b (1-\sec (c+d x))}{a+b}\right)}{4 \sqrt{2} b d \sqrt{\sec (c+d x)+1} \left(\frac{a+b \sec (c+d x)}{a+b}\right)^{2/3}}-\frac{a \left(a^2-b^2\right) \tan (c+d x) \sqrt[3]{\frac{a+b \sec (c+d x)}{a+b}} F_1\left(\frac{1}{2};\frac{1}{2},\frac{1}{3};\frac{3}{2};\frac{1}{2} (1-\sec (c+d x)),\frac{b (1-\sec (c+d x))}{a+b}\right)}{2 \sqrt{2} b d \sqrt{\sec (c+d x)+1} \sqrt[3]{a+b \sec (c+d x)}}+\frac{3 \tan (c+d x) (a+b \sec (c+d x))^{5/3}}{8 d}+\frac{3 a \tan (c+d x) (a+b \sec (c+d x))^{2/3}}{8 d}",1,"Result too large to show","B",0
697,1,8668,108,27.7743461,"\int \sec (c+d x) (a+b \sec (c+d x))^{5/3} \, dx","Integrate[Sec[c + d*x]*(a + b*Sec[c + d*x])^(5/3),x]","\text{Result too large to show}","\frac{\sqrt{2} (a+b) \tan (c+d x) (a+b \sec (c+d x))^{2/3} F_1\left(\frac{1}{2};\frac{1}{2},-\frac{5}{3};\frac{3}{2};\frac{1}{2} (1-\sec (c+d x)),\frac{b (1-\sec (c+d x))}{a+b}\right)}{d \sqrt{\sec (c+d x)+1} \left(\frac{a+b \sec (c+d x)}{a+b}\right)^{2/3}}",1,"Result too large to show","B",0
698,0,0,17,26.4892384,"\int (a+b \sec (c+d x))^{5/3} \, dx","Integrate[(a + b*Sec[c + d*x])^(5/3),x]","\int (a+b \sec (c+d x))^{5/3} \, dx","\text{Int}\left((a+b \sec (c+d x))^{5/3},x\right)",0,"Integrate[(a + b*Sec[c + d*x])^(5/3), x]","A",-1
699,1,19015,313,26.6022792,"\int \frac{\sec ^4(c+d x)}{\sqrt[3]{a+b \sec (c+d x)}} \, dx","Integrate[Sec[c + d*x]^4/(a + b*Sec[c + d*x])^(1/3),x]","\text{Result too large to show}","-\frac{a \left(9 a^2+11 b^2\right) \tan (c+d x) \sqrt[3]{\frac{a+b \sec (c+d x)}{a+b}} F_1\left(\frac{1}{2};\frac{1}{2},\frac{1}{3};\frac{3}{2};\frac{1}{2} (1-\sec (c+d x)),\frac{b (1-\sec (c+d x))}{a+b}\right)}{10 \sqrt{2} b^3 d \sqrt{\sec (c+d x)+1} \sqrt[3]{a+b \sec (c+d x)}}+\frac{\left(18 a^2+25 b^2\right) \tan (c+d x) (a+b \sec (c+d x))^{2/3} F_1\left(\frac{1}{2};\frac{1}{2},-\frac{2}{3};\frac{3}{2};\frac{1}{2} (1-\sec (c+d x)),\frac{b (1-\sec (c+d x))}{a+b}\right)}{20 \sqrt{2} b^3 d \sqrt{\sec (c+d x)+1} \left(\frac{a+b \sec (c+d x)}{a+b}\right)^{2/3}}-\frac{9 a \tan (c+d x) (a+b \sec (c+d x))^{2/3}}{20 b^2 d}+\frac{3 \tan (c+d x) \sec (c+d x) (a+b \sec (c+d x))^{2/3}}{8 b d}",1,"Result too large to show","B",0
700,1,7195,265,26.7829712,"\int \frac{\sec ^3(c+d x)}{\sqrt[3]{a+b \sec (c+d x)}} \, dx","Integrate[Sec[c + d*x]^3/(a + b*Sec[c + d*x])^(1/3),x]","\text{Result too large to show}","\frac{\sqrt{2} \left(3 a^2+2 b^2\right) \tan (c+d x) \sqrt[3]{\frac{a+b \sec (c+d x)}{a+b}} F_1\left(\frac{1}{2};\frac{1}{2},\frac{1}{3};\frac{3}{2};\frac{1}{2} (1-\sec (c+d x)),\frac{b (1-\sec (c+d x))}{a+b}\right)}{5 b^2 d \sqrt{\sec (c+d x)+1} \sqrt[3]{a+b \sec (c+d x)}}-\frac{3 \sqrt{2} a \tan (c+d x) (a+b \sec (c+d x))^{2/3} F_1\left(\frac{1}{2};\frac{1}{2},-\frac{2}{3};\frac{3}{2};\frac{1}{2} (1-\sec (c+d x)),\frac{b (1-\sec (c+d x))}{a+b}\right)}{5 b^2 d \sqrt{\sec (c+d x)+1} \left(\frac{a+b \sec (c+d x)}{a+b}\right)^{2/3}}+\frac{3 \tan (c+d x) (a+b \sec (c+d x))^{2/3}}{5 b d}",1,"Result too large to show","B",0
701,1,2759,219,19.1889593,"\int \frac{\sec ^2(c+d x)}{\sqrt[3]{a+b \sec (c+d x)}} \, dx","Integrate[Sec[c + d*x]^2/(a + b*Sec[c + d*x])^(1/3),x]","\text{Result too large to show}","\frac{\sqrt{2} \tan (c+d x) (a+b \sec (c+d x))^{2/3} F_1\left(\frac{1}{2};\frac{1}{2},-\frac{2}{3};\frac{3}{2};\frac{1}{2} (1-\sec (c+d x)),\frac{b (1-\sec (c+d x))}{a+b}\right)}{b d \sqrt{\sec (c+d x)+1} \left(\frac{a+b \sec (c+d x)}{a+b}\right)^{2/3}}-\frac{\sqrt{2} a \tan (c+d x) \sqrt[3]{\frac{a+b \sec (c+d x)}{a+b}} F_1\left(\frac{1}{2};\frac{1}{2},\frac{1}{3};\frac{3}{2};\frac{1}{2} (1-\sec (c+d x)),\frac{b (1-\sec (c+d x))}{a+b}\right)}{b d \sqrt{\sec (c+d x)+1} \sqrt[3]{a+b \sec (c+d x)}}",1,"(3*(b + a*Cos[c + d*x])*Tan[c + d*x])/(2*b*d*(a + b*Sec[c + d*x])^(1/3)) - ((b + 3*a*Cos[c + d*x])*(3*(b + a*Cos[c + d*x])^(2/3)*Sqrt[1 - Cos[c + d*x]^2]*Sec[c + d*x]^(2/3) - (3*(a + b*Sec[c + d*x])*Sqrt[(1 - Sqrt[b^(-2)]*b*Sec[c + d*x])/(1 + a*Sqrt[b^(-2)])]*Sqrt[(1 + Sqrt[b^(-2)]*b*Sec[c + d*x])/(1 - a*Sqrt[b^(-2)])]*(-5*a*AppellF1[2/3, 1/2, 1/2, 5/3, -((a + b*Sec[c + d*x])/(-a + 1/Sqrt[b^(-2)])), (a + b*Sec[c + d*x])/(a + 1/Sqrt[b^(-2)])] + 2*AppellF1[5/3, 1/2, 1/2, 8/3, -((a + b*Sec[c + d*x])/(-a + 1/Sqrt[b^(-2)])), (a + b*Sec[c + d*x])/(a + 1/Sqrt[b^(-2)])]*(a + b*Sec[c + d*x])))/(5*b*(b + a*Cos[c + d*x])^(1/3)*Sqrt[1 - Cos[c + d*x]^2]*Sec[c + d*x]^(4/3))))/(2*b*d*(a + b*Sec[c + d*x])^(1/3)*((3*(b + a*Cos[c + d*x])^(2/3)*Sin[c + d*x])/(Sqrt[1 - Cos[c + d*x]^2]*Sec[c + d*x]^(1/3)) - (2*a*Sqrt[1 - Cos[c + d*x]^2]*Sec[c + d*x]^(2/3)*Sin[c + d*x])/(b + a*Cos[c + d*x])^(1/3) + 2*(b + a*Cos[c + d*x])^(2/3)*Sqrt[1 - Cos[c + d*x]^2]*Sec[c + d*x]^(5/3)*Sin[c + d*x] - (3*Sqrt[b^(-2)]*Sec[c + d*x]^(2/3)*(a + b*Sec[c + d*x])*Sqrt[(1 - Sqrt[b^(-2)]*b*Sec[c + d*x])/(1 + a*Sqrt[b^(-2)])]*(-5*a*AppellF1[2/3, 1/2, 1/2, 5/3, -((a + b*Sec[c + d*x])/(-a + 1/Sqrt[b^(-2)])), (a + b*Sec[c + d*x])/(a + 1/Sqrt[b^(-2)])] + 2*AppellF1[5/3, 1/2, 1/2, 8/3, -((a + b*Sec[c + d*x])/(-a + 1/Sqrt[b^(-2)])), (a + b*Sec[c + d*x])/(a + 1/Sqrt[b^(-2)])]*(a + b*Sec[c + d*x]))*Sin[c + d*x])/(10*(1 - a*Sqrt[b^(-2)])*(b + a*Cos[c + d*x])^(1/3)*Sqrt[1 - Cos[c + d*x]^2]*Sqrt[(1 + Sqrt[b^(-2)]*b*Sec[c + d*x])/(1 - a*Sqrt[b^(-2)])]) + (3*Sqrt[b^(-2)]*Sec[c + d*x]^(2/3)*(a + b*Sec[c + d*x])*Sqrt[(1 + Sqrt[b^(-2)]*b*Sec[c + d*x])/(1 - a*Sqrt[b^(-2)])]*(-5*a*AppellF1[2/3, 1/2, 1/2, 5/3, -((a + b*Sec[c + d*x])/(-a + 1/Sqrt[b^(-2)])), (a + b*Sec[c + d*x])/(a + 1/Sqrt[b^(-2)])] + 2*AppellF1[5/3, 1/2, 1/2, 8/3, -((a + b*Sec[c + d*x])/(-a + 1/Sqrt[b^(-2)])), (a + b*Sec[c + d*x])/(a + 1/Sqrt[b^(-2)])]*(a + b*Sec[c + d*x]))*Sin[c + d*x])/(10*(1 + a*Sqrt[b^(-2)])*(b + a*Cos[c + d*x])^(1/3)*Sqrt[1 - Cos[c + d*x]^2]*Sqrt[(1 - Sqrt[b^(-2)]*b*Sec[c + d*x])/(1 + a*Sqrt[b^(-2)])]) - (3*Sec[c + d*x]^(2/3)*Sqrt[(1 - Sqrt[b^(-2)]*b*Sec[c + d*x])/(1 + a*Sqrt[b^(-2)])]*Sqrt[(1 + Sqrt[b^(-2)]*b*Sec[c + d*x])/(1 - a*Sqrt[b^(-2)])]*(-5*a*AppellF1[2/3, 1/2, 1/2, 5/3, -((a + b*Sec[c + d*x])/(-a + 1/Sqrt[b^(-2)])), (a + b*Sec[c + d*x])/(a + 1/Sqrt[b^(-2)])] + 2*AppellF1[5/3, 1/2, 1/2, 8/3, -((a + b*Sec[c + d*x])/(-a + 1/Sqrt[b^(-2)])), (a + b*Sec[c + d*x])/(a + 1/Sqrt[b^(-2)])]*(a + b*Sec[c + d*x]))*Sin[c + d*x])/(5*(b + a*Cos[c + d*x])^(1/3)*Sqrt[1 - Cos[c + d*x]^2]) + (3*(a + b*Sec[c + d*x])*Sqrt[(1 - Sqrt[b^(-2)]*b*Sec[c + d*x])/(1 + a*Sqrt[b^(-2)])]*Sqrt[(1 + Sqrt[b^(-2)]*b*Sec[c + d*x])/(1 - a*Sqrt[b^(-2)])]*(-5*a*AppellF1[2/3, 1/2, 1/2, 5/3, -((a + b*Sec[c + d*x])/(-a + 1/Sqrt[b^(-2)])), (a + b*Sec[c + d*x])/(a + 1/Sqrt[b^(-2)])] + 2*AppellF1[5/3, 1/2, 1/2, 8/3, -((a + b*Sec[c + d*x])/(-a + 1/Sqrt[b^(-2)])), (a + b*Sec[c + d*x])/(a + 1/Sqrt[b^(-2)])]*(a + b*Sec[c + d*x]))*Sin[c + d*x])/(5*b*(b + a*Cos[c + d*x])^(1/3)*(1 - Cos[c + d*x]^2)^(3/2)*Sec[c + d*x]^(7/3)) - (a*(a + b*Sec[c + d*x])*Sqrt[(1 - Sqrt[b^(-2)]*b*Sec[c + d*x])/(1 + a*Sqrt[b^(-2)])]*Sqrt[(1 + Sqrt[b^(-2)]*b*Sec[c + d*x])/(1 - a*Sqrt[b^(-2)])]*(-5*a*AppellF1[2/3, 1/2, 1/2, 5/3, -((a + b*Sec[c + d*x])/(-a + 1/Sqrt[b^(-2)])), (a + b*Sec[c + d*x])/(a + 1/Sqrt[b^(-2)])] + 2*AppellF1[5/3, 1/2, 1/2, 8/3, -((a + b*Sec[c + d*x])/(-a + 1/Sqrt[b^(-2)])), (a + b*Sec[c + d*x])/(a + 1/Sqrt[b^(-2)])]*(a + b*Sec[c + d*x]))*Sin[c + d*x])/(5*b*(b + a*Cos[c + d*x])^(4/3)*Sqrt[1 - Cos[c + d*x]^2]*Sec[c + d*x]^(4/3)) + (4*(a + b*Sec[c + d*x])*Sqrt[(1 - Sqrt[b^(-2)]*b*Sec[c + d*x])/(1 + a*Sqrt[b^(-2)])]*Sqrt[(1 + Sqrt[b^(-2)]*b*Sec[c + d*x])/(1 - a*Sqrt[b^(-2)])]*(-5*a*AppellF1[2/3, 1/2, 1/2, 5/3, -((a + b*Sec[c + d*x])/(-a + 1/Sqrt[b^(-2)])), (a + b*Sec[c + d*x])/(a + 1/Sqrt[b^(-2)])] + 2*AppellF1[5/3, 1/2, 1/2, 8/3, -((a + b*Sec[c + d*x])/(-a + 1/Sqrt[b^(-2)])), (a + b*Sec[c + d*x])/(a + 1/Sqrt[b^(-2)])]*(a + b*Sec[c + d*x]))*Sin[c + d*x])/(5*b*(b + a*Cos[c + d*x])^(1/3)*Sqrt[1 - Cos[c + d*x]^2]*Sec[c + d*x]^(1/3)) - (3*(a + b*Sec[c + d*x])*Sqrt[(1 - Sqrt[b^(-2)]*b*Sec[c + d*x])/(1 + a*Sqrt[b^(-2)])]*Sqrt[(1 + Sqrt[b^(-2)]*b*Sec[c + d*x])/(1 - a*Sqrt[b^(-2)])]*(2*b*AppellF1[5/3, 1/2, 1/2, 8/3, -((a + b*Sec[c + d*x])/(-a + 1/Sqrt[b^(-2)])), (a + b*Sec[c + d*x])/(a + 1/Sqrt[b^(-2)])]*Sec[c + d*x]*Tan[c + d*x] - 5*a*((b*AppellF1[5/3, 1/2, 3/2, 8/3, -((a + b*Sec[c + d*x])/(-a + 1/Sqrt[b^(-2)])), (a + b*Sec[c + d*x])/(a + 1/Sqrt[b^(-2)])]*Sec[c + d*x]*Tan[c + d*x])/(5*(a + 1/Sqrt[b^(-2)])) - (b*AppellF1[5/3, 3/2, 1/2, 8/3, -((a + b*Sec[c + d*x])/(-a + 1/Sqrt[b^(-2)])), (a + b*Sec[c + d*x])/(a + 1/Sqrt[b^(-2)])]*Sec[c + d*x]*Tan[c + d*x])/(5*(-a + 1/Sqrt[b^(-2)]))) + 2*(a + b*Sec[c + d*x])*((5*b*AppellF1[8/3, 1/2, 3/2, 11/3, -((a + b*Sec[c + d*x])/(-a + 1/Sqrt[b^(-2)])), (a + b*Sec[c + d*x])/(a + 1/Sqrt[b^(-2)])]*Sec[c + d*x]*Tan[c + d*x])/(16*(a + 1/Sqrt[b^(-2)])) - (5*b*AppellF1[8/3, 3/2, 1/2, 11/3, -((a + b*Sec[c + d*x])/(-a + 1/Sqrt[b^(-2)])), (a + b*Sec[c + d*x])/(a + 1/Sqrt[b^(-2)])]*Sec[c + d*x]*Tan[c + d*x])/(16*(-a + 1/Sqrt[b^(-2)])))))/(5*b*(b + a*Cos[c + d*x])^(1/3)*Sqrt[1 - Cos[c + d*x]^2]*Sec[c + d*x]^(4/3))))","B",0
702,1,310,105,2.0668745,"\int \frac{\sec (c+d x)}{\sqrt[3]{a+b \sec (c+d x)}} \, dx","Integrate[Sec[c + d*x]/(a + b*Sec[c + d*x])^(1/3),x]","\frac{15 (a-b)^2 (a+b) \cos (c+d x) \cot ^3(c+d x) (\sec (c+d x)+1) (b-b \sec (c+d x)) (a+b \sec (c+d x))^{2/3} F_1\left(\frac{2}{3};\frac{1}{2},\frac{1}{2};\frac{5}{3};\frac{a+b \sec (c+d x)}{a-b},\frac{a+b \sec (c+d x)}{a+b}\right)}{b^2 d (b-a) \left(3 (a-b) (a \cos (c+d x)+b) F_1\left(\frac{5}{3};\frac{1}{2},\frac{3}{2};\frac{8}{3};\frac{a+b \sec (c+d x)}{a-b},\frac{a+b \sec (c+d x)}{a+b}\right)+(a+b) \left(10 (a-b) \cos (c+d x) F_1\left(\frac{2}{3};\frac{1}{2},\frac{1}{2};\frac{5}{3};\frac{a+b \sec (c+d x)}{a-b},\frac{a+b \sec (c+d x)}{a+b}\right)+3 (a \cos (c+d x)+b) F_1\left(\frac{5}{3};\frac{3}{2},\frac{1}{2};\frac{8}{3};\frac{a+b \sec (c+d x)}{a-b},\frac{a+b \sec (c+d x)}{a+b}\right)\right)\right)}","\frac{\sqrt{2} \tan (c+d x) \sqrt[3]{\frac{a+b \sec (c+d x)}{a+b}} F_1\left(\frac{1}{2};\frac{1}{2},\frac{1}{3};\frac{3}{2};\frac{1}{2} (1-\sec (c+d x)),\frac{b (1-\sec (c+d x))}{a+b}\right)}{d \sqrt{\sec (c+d x)+1} \sqrt[3]{a+b \sec (c+d x)}}",1,"(15*(a - b)^2*(a + b)*AppellF1[2/3, 1/2, 1/2, 5/3, (a + b*Sec[c + d*x])/(a - b), (a + b*Sec[c + d*x])/(a + b)]*Cos[c + d*x]*Cot[c + d*x]^3*(1 + Sec[c + d*x])*(b - b*Sec[c + d*x])*(a + b*Sec[c + d*x])^(2/3))/(b^2*(-a + b)*d*(3*(a - b)*AppellF1[5/3, 1/2, 3/2, 8/3, (a + b*Sec[c + d*x])/(a - b), (a + b*Sec[c + d*x])/(a + b)]*(b + a*Cos[c + d*x]) + (a + b)*(10*(a - b)*AppellF1[2/3, 1/2, 1/2, 5/3, (a + b*Sec[c + d*x])/(a - b), (a + b*Sec[c + d*x])/(a + b)]*Cos[c + d*x] + 3*AppellF1[5/3, 3/2, 1/2, 8/3, (a + b*Sec[c + d*x])/(a - b), (a + b*Sec[c + d*x])/(a + b)]*(b + a*Cos[c + d*x]))))","B",0
703,0,0,17,1.0247589,"\int \frac{1}{\sqrt[3]{a+b \sec (c+d x)}} \, dx","Integrate[(a + b*Sec[c + d*x])^(-1/3),x]","\int \frac{1}{\sqrt[3]{a+b \sec (c+d x)}} \, dx","\text{Int}\left(\frac{1}{\sqrt[3]{a+b \sec (c+d x)}},x\right)",0,"Integrate[(a + b*Sec[c + d*x])^(-1/3), x]","A",-1
704,1,310,105,1.9685537,"\int \frac{\sec (c+d x)}{(a+b \sec (c+d x))^{2/3}} \, dx","Integrate[Sec[c + d*x]/(a + b*Sec[c + d*x])^(2/3),x]","\frac{24 (a-b)^2 (a+b) \cos (c+d x) \cot ^3(c+d x) (\sec (c+d x)+1) (b-b \sec (c+d x)) \sqrt[3]{a+b \sec (c+d x)} F_1\left(\frac{1}{3};\frac{1}{2},\frac{1}{2};\frac{4}{3};\frac{a+b \sec (c+d x)}{a-b},\frac{a+b \sec (c+d x)}{a+b}\right)}{b^2 d (b-a) \left(3 (a-b) (a \cos (c+d x)+b) F_1\left(\frac{4}{3};\frac{1}{2},\frac{3}{2};\frac{7}{3};\frac{a+b \sec (c+d x)}{a-b},\frac{a+b \sec (c+d x)}{a+b}\right)+(a+b) \left(8 (a-b) \cos (c+d x) F_1\left(\frac{1}{3};\frac{1}{2},\frac{1}{2};\frac{4}{3};\frac{a+b \sec (c+d x)}{a-b},\frac{a+b \sec (c+d x)}{a+b}\right)+3 (a \cos (c+d x)+b) F_1\left(\frac{4}{3};\frac{3}{2},\frac{1}{2};\frac{7}{3};\frac{a+b \sec (c+d x)}{a-b},\frac{a+b \sec (c+d x)}{a+b}\right)\right)\right)}","\frac{\sqrt{2} \tan (c+d x) \left(\frac{a+b \sec (c+d x)}{a+b}\right)^{2/3} F_1\left(\frac{1}{2};\frac{1}{2},\frac{2}{3};\frac{3}{2};\frac{1}{2} (1-\sec (c+d x)),\frac{b (1-\sec (c+d x))}{a+b}\right)}{d \sqrt{\sec (c+d x)+1} (a+b \sec (c+d x))^{2/3}}",1,"(24*(a - b)^2*(a + b)*AppellF1[1/3, 1/2, 1/2, 4/3, (a + b*Sec[c + d*x])/(a - b), (a + b*Sec[c + d*x])/(a + b)]*Cos[c + d*x]*Cot[c + d*x]^3*(1 + Sec[c + d*x])*(b - b*Sec[c + d*x])*(a + b*Sec[c + d*x])^(1/3))/(b^2*(-a + b)*d*(3*(a - b)*AppellF1[4/3, 1/2, 3/2, 7/3, (a + b*Sec[c + d*x])/(a - b), (a + b*Sec[c + d*x])/(a + b)]*(b + a*Cos[c + d*x]) + (a + b)*(8*(a - b)*AppellF1[1/3, 1/2, 1/2, 4/3, (a + b*Sec[c + d*x])/(a - b), (a + b*Sec[c + d*x])/(a + b)]*Cos[c + d*x] + 3*AppellF1[4/3, 3/2, 1/2, 7/3, (a + b*Sec[c + d*x])/(a - b), (a + b*Sec[c + d*x])/(a + b)]*(b + a*Cos[c + d*x]))))","B",0
705,0,0,17,1.0636319,"\int \frac{1}{(a+b \sec (c+d x))^{2/3}} \, dx","Integrate[(a + b*Sec[c + d*x])^(-2/3),x]","\int \frac{1}{(a+b \sec (c+d x))^{2/3}} \, dx","\text{Int}\left(\frac{1}{(a+b \sec (c+d x))^{2/3}},x\right)",0,"Integrate[(a + b*Sec[c + d*x])^(-2/3), x]","A",-1
706,1,10343,110,27.2925711,"\int \frac{\sec (c+d x)}{(a+b \sec (c+d x))^{4/3}} \, dx","Integrate[Sec[c + d*x]/(a + b*Sec[c + d*x])^(4/3),x]","\text{Result too large to show}","\frac{\sqrt{2} \tan (c+d x) \sqrt[3]{\frac{a+b \sec (c+d x)}{a+b}} F_1\left(\frac{1}{2};\frac{1}{2},\frac{4}{3};\frac{3}{2};\frac{1}{2} (1-\sec (c+d x)),\frac{b (1-\sec (c+d x))}{a+b}\right)}{d (a+b) \sqrt{\sec (c+d x)+1} \sqrt[3]{a+b \sec (c+d x)}}",1,"Result too large to show","B",0
707,0,0,17,27.3575849,"\int \frac{1}{(a+b \sec (c+d x))^{4/3}} \, dx","Integrate[(a + b*Sec[c + d*x])^(-4/3),x]","\int \frac{1}{(a+b \sec (c+d x))^{4/3}} \, dx","\text{Int}\left(\frac{1}{(a+b \sec (c+d x))^{4/3}},x\right)",0,"Integrate[(a + b*Sec[c + d*x])^(-4/3), x]","A",-1
708,1,21987,378,26.8305465,"\int \frac{\sec ^4(c+d x)}{(a+b \sec (c+d x))^{5/3}} \, dx","Integrate[Sec[c + d*x]^4/(a + b*Sec[c + d*x])^(5/3),x]","\text{Result too large to show}","-\frac{a \left(9 a^2-7 b^2\right) \tan (c+d x) \sqrt[3]{a+b \sec (c+d x)} F_1\left(\frac{1}{2};\frac{1}{2},-\frac{1}{3};\frac{3}{2};\frac{1}{2} (1-\sec (c+d x)),\frac{b (1-\sec (c+d x))}{a+b}\right)}{2 \sqrt{2} b^3 d \left(a^2-b^2\right) \sqrt{\sec (c+d x)+1} \sqrt[3]{\frac{a+b \sec (c+d x)}{a+b}}}-\frac{3 a^2 \tan (c+d x) \sec (c+d x)}{2 b d \left(a^2-b^2\right) (a+b \sec (c+d x))^{2/3}}+\frac{3 \left(3 a^2-b^2\right) \tan (c+d x) \sqrt[3]{a+b \sec (c+d x)}}{4 b^2 d \left(a^2-b^2\right)}+\frac{\left(9 a^4-10 a^2 b^2-b^4\right) \tan (c+d x) \left(\frac{a+b \sec (c+d x)}{a+b}\right)^{2/3} F_1\left(\frac{1}{2};\frac{1}{2},\frac{2}{3};\frac{3}{2};\frac{1}{2} (1-\sec (c+d x)),\frac{b (1-\sec (c+d x))}{a+b}\right)}{2 \sqrt{2} b^3 d \left(a^2-b^2\right) \sqrt{\sec (c+d x)+1} (a+b \sec (c+d x))^{2/3}}",1,"Result too large to show","B",0
709,1,19126,307,26.6967014,"\int \frac{\sec ^3(c+d x)}{(a+b \sec (c+d x))^{5/3}} \, dx","Integrate[Sec[c + d*x]^3/(a + b*Sec[c + d*x])^(5/3),x]","\text{Result too large to show}","-\frac{a \left(3 a^2-4 b^2\right) \tan (c+d x) \left(\frac{a+b \sec (c+d x)}{a+b}\right)^{2/3} F_1\left(\frac{1}{2};\frac{1}{2},\frac{2}{3};\frac{3}{2};\frac{1}{2} (1-\sec (c+d x)),\frac{b (1-\sec (c+d x))}{a+b}\right)}{\sqrt{2} b^2 d \left(a^2-b^2\right) \sqrt{\sec (c+d x)+1} (a+b \sec (c+d x))^{2/3}}+\frac{\left(3 a^2-2 b^2\right) \tan (c+d x) \sqrt[3]{a+b \sec (c+d x)} F_1\left(\frac{1}{2};\frac{1}{2},-\frac{1}{3};\frac{3}{2};\frac{1}{2} (1-\sec (c+d x)),\frac{b (1-\sec (c+d x))}{a+b}\right)}{\sqrt{2} b^2 d \left(a^2-b^2\right) \sqrt{\sec (c+d x)+1} \sqrt[3]{\frac{a+b \sec (c+d x)}{a+b}}}-\frac{3 a^2 \tan (c+d x)}{2 b d \left(a^2-b^2\right) (a+b \sec (c+d x))^{2/3}}",1,"Result too large to show","B",0
710,1,7325,289,27.0038865,"\int \frac{\sec ^2(c+d x)}{(a+b \sec (c+d x))^{5/3}} \, dx","Integrate[Sec[c + d*x]^2/(a + b*Sec[c + d*x])^(5/3),x]","\text{Result too large to show}","\frac{\left(a^2-2 b^2\right) \tan (c+d x) \left(\frac{a+b \sec (c+d x)}{a+b}\right)^{2/3} F_1\left(\frac{1}{2};\frac{1}{2},\frac{2}{3};\frac{3}{2};\frac{1}{2} (1-\sec (c+d x)),\frac{b (1-\sec (c+d x))}{a+b}\right)}{\sqrt{2} b d \left(a^2-b^2\right) \sqrt{\sec (c+d x)+1} (a+b \sec (c+d x))^{2/3}}-\frac{a \tan (c+d x) \sqrt[3]{a+b \sec (c+d x)} F_1\left(\frac{1}{2};\frac{1}{2},-\frac{1}{3};\frac{3}{2};\frac{1}{2} (1-\sec (c+d x)),\frac{b (1-\sec (c+d x))}{a+b}\right)}{\sqrt{2} b d \left(a^2-b^2\right) \sqrt{\sec (c+d x)+1} \sqrt[3]{\frac{a+b \sec (c+d x)}{a+b}}}+\frac{3 a \tan (c+d x)}{2 d \left(a^2-b^2\right) (a+b \sec (c+d x))^{2/3}}",1,"Result too large to show","B",0
711,1,10363,110,27.2760231,"\int \frac{\sec (c+d x)}{(a+b \sec (c+d x))^{5/3}} \, dx","Integrate[Sec[c + d*x]/(a + b*Sec[c + d*x])^(5/3),x]","\text{Result too large to show}","\frac{\sqrt{2} \tan (c+d x) \left(\frac{a+b \sec (c+d x)}{a+b}\right)^{2/3} F_1\left(\frac{1}{2};\frac{1}{2},\frac{5}{3};\frac{3}{2};\frac{1}{2} (1-\sec (c+d x)),\frac{b (1-\sec (c+d x))}{a+b}\right)}{d (a+b) \sqrt{\sec (c+d x)+1} (a+b \sec (c+d x))^{2/3}}",1,"Result too large to show","B",0
712,0,0,17,23.06178,"\int \frac{1}{(a+b \sec (c+d x))^{5/3}} \, dx","Integrate[(a + b*Sec[c + d*x])^(-5/3),x]","\int \frac{1}{(a+b \sec (c+d x))^{5/3}} \, dx","\text{Int}\left(\frac{1}{(a+b \sec (c+d x))^{5/3}},x\right)",0,"Integrate[(a + b*Sec[c + d*x])^(-5/3), x]","A",-1
713,1,4543,174,21.738819,"\int \frac{\sec ^{\frac{2}{3}}(c+d x)}{a+b \sec (c+d x)} \, dx","Integrate[Sec[c + d*x]^(2/3)/(a + b*Sec[c + d*x]),x]","\text{Result too large to show}","\frac{a \sin (c+d x) F_1\left(\frac{1}{2};-\frac{1}{6},1;\frac{3}{2};\sin ^2(c+d x),\frac{a^2 \sin ^2(c+d x)}{a^2-b^2}\right)}{d \left(a^2-b^2\right) \sqrt[6]{\cos ^2(c+d x)} \sqrt[3]{\sec (c+d x)}}-\frac{b \sin (c+d x) \sqrt[3]{\cos ^2(c+d x)} \sec ^{\frac{2}{3}}(c+d x) F_1\left(\frac{1}{2};\frac{1}{3},1;\frac{3}{2};\sin ^2(c+d x),\frac{a^2 \sin ^2(c+d x)}{a^2-b^2}\right)}{d \left(a^2-b^2\right)}",1,"(9*(a^2 - b^2)*Sec[c + d*x]^(5/3)*Sin[c + d*x]*((b*AppellF1[1/2, 1/6, 1, 3/2, -Tan[c + d*x]^2, (b^2*Tan[c + d*x]^2)/(a^2 - b^2)]*Sqrt[Sec[c + d*x]^2])/(9*(a^2 - b^2)*AppellF1[1/2, 1/6, 1, 3/2, -Tan[c + d*x]^2, (b^2*Tan[c + d*x]^2)/(a^2 - b^2)] + (6*b^2*AppellF1[3/2, 1/6, 2, 5/2, -Tan[c + d*x]^2, (b^2*Tan[c + d*x]^2)/(a^2 - b^2)] + (-a^2 + b^2)*AppellF1[3/2, 7/6, 1, 5/2, -Tan[c + d*x]^2, (b^2*Tan[c + d*x]^2)/(a^2 - b^2)])*Tan[c + d*x]^2) + (a*AppellF1[1/2, 2/3, 1, 3/2, -Tan[c + d*x]^2, (b^2*Tan[c + d*x]^2)/(a^2 - b^2)])/(-9*(a^2 - b^2)*AppellF1[1/2, 2/3, 1, 3/2, -Tan[c + d*x]^2, (b^2*Tan[c + d*x]^2)/(a^2 - b^2)] - 2*(3*b^2*AppellF1[3/2, 2/3, 2, 5/2, -Tan[c + d*x]^2, (b^2*Tan[c + d*x]^2)/(a^2 - b^2)] + 2*(-a^2 + b^2)*AppellF1[3/2, 5/3, 1, 5/2, -Tan[c + d*x]^2, (b^2*Tan[c + d*x]^2)/(a^2 - b^2)])*Tan[c + d*x]^2)))/(d*(Sec[c + d*x]^2)^(2/3)*(a + b*Sec[c + d*x])*(-a^2 + b^2*Sec[c + d*x]^2)*((9*(a^2 - b^2)*(Sec[c + d*x]^2)^(1/3)*((b*AppellF1[1/2, 1/6, 1, 3/2, -Tan[c + d*x]^2, (b^2*Tan[c + d*x]^2)/(a^2 - b^2)]*Sqrt[Sec[c + d*x]^2])/(9*(a^2 - b^2)*AppellF1[1/2, 1/6, 1, 3/2, -Tan[c + d*x]^2, (b^2*Tan[c + d*x]^2)/(a^2 - b^2)] + (6*b^2*AppellF1[3/2, 1/6, 2, 5/2, -Tan[c + d*x]^2, (b^2*Tan[c + d*x]^2)/(a^2 - b^2)] + (-a^2 + b^2)*AppellF1[3/2, 7/6, 1, 5/2, -Tan[c + d*x]^2, (b^2*Tan[c + d*x]^2)/(a^2 - b^2)])*Tan[c + d*x]^2) + (a*AppellF1[1/2, 2/3, 1, 3/2, -Tan[c + d*x]^2, (b^2*Tan[c + d*x]^2)/(a^2 - b^2)])/(-9*(a^2 - b^2)*AppellF1[1/2, 2/3, 1, 3/2, -Tan[c + d*x]^2, (b^2*Tan[c + d*x]^2)/(a^2 - b^2)] - 2*(3*b^2*AppellF1[3/2, 2/3, 2, 5/2, -Tan[c + d*x]^2, (b^2*Tan[c + d*x]^2)/(a^2 - b^2)] + 2*(-a^2 + b^2)*AppellF1[3/2, 5/3, 1, 5/2, -Tan[c + d*x]^2, (b^2*Tan[c + d*x]^2)/(a^2 - b^2)])*Tan[c + d*x]^2)))/(-a^2 + b^2*Sec[c + d*x]^2) - (18*b^2*(a^2 - b^2)*(Sec[c + d*x]^2)^(1/3)*Tan[c + d*x]^2*((b*AppellF1[1/2, 1/6, 1, 3/2, -Tan[c + d*x]^2, (b^2*Tan[c + d*x]^2)/(a^2 - b^2)]*Sqrt[Sec[c + d*x]^2])/(9*(a^2 - b^2)*AppellF1[1/2, 1/6, 1, 3/2, -Tan[c + d*x]^2, (b^2*Tan[c + d*x]^2)/(a^2 - b^2)] + (6*b^2*AppellF1[3/2, 1/6, 2, 5/2, -Tan[c + d*x]^2, (b^2*Tan[c + d*x]^2)/(a^2 - b^2)] + (-a^2 + b^2)*AppellF1[3/2, 7/6, 1, 5/2, -Tan[c + d*x]^2, (b^2*Tan[c + d*x]^2)/(a^2 - b^2)])*Tan[c + d*x]^2) + (a*AppellF1[1/2, 2/3, 1, 3/2, -Tan[c + d*x]^2, (b^2*Tan[c + d*x]^2)/(a^2 - b^2)])/(-9*(a^2 - b^2)*AppellF1[1/2, 2/3, 1, 3/2, -Tan[c + d*x]^2, (b^2*Tan[c + d*x]^2)/(a^2 - b^2)] - 2*(3*b^2*AppellF1[3/2, 2/3, 2, 5/2, -Tan[c + d*x]^2, (b^2*Tan[c + d*x]^2)/(a^2 - b^2)] + 2*(-a^2 + b^2)*AppellF1[3/2, 5/3, 1, 5/2, -Tan[c + d*x]^2, (b^2*Tan[c + d*x]^2)/(a^2 - b^2)])*Tan[c + d*x]^2)))/(-a^2 + b^2*Sec[c + d*x]^2)^2 - (12*(a^2 - b^2)*Tan[c + d*x]^2*((b*AppellF1[1/2, 1/6, 1, 3/2, -Tan[c + d*x]^2, (b^2*Tan[c + d*x]^2)/(a^2 - b^2)]*Sqrt[Sec[c + d*x]^2])/(9*(a^2 - b^2)*AppellF1[1/2, 1/6, 1, 3/2, -Tan[c + d*x]^2, (b^2*Tan[c + d*x]^2)/(a^2 - b^2)] + (6*b^2*AppellF1[3/2, 1/6, 2, 5/2, -Tan[c + d*x]^2, (b^2*Tan[c + d*x]^2)/(a^2 - b^2)] + (-a^2 + b^2)*AppellF1[3/2, 7/6, 1, 5/2, -Tan[c + d*x]^2, (b^2*Tan[c + d*x]^2)/(a^2 - b^2)])*Tan[c + d*x]^2) + (a*AppellF1[1/2, 2/3, 1, 3/2, -Tan[c + d*x]^2, (b^2*Tan[c + d*x]^2)/(a^2 - b^2)])/(-9*(a^2 - b^2)*AppellF1[1/2, 2/3, 1, 3/2, -Tan[c + d*x]^2, (b^2*Tan[c + d*x]^2)/(a^2 - b^2)] - 2*(3*b^2*AppellF1[3/2, 2/3, 2, 5/2, -Tan[c + d*x]^2, (b^2*Tan[c + d*x]^2)/(a^2 - b^2)] + 2*(-a^2 + b^2)*AppellF1[3/2, 5/3, 1, 5/2, -Tan[c + d*x]^2, (b^2*Tan[c + d*x]^2)/(a^2 - b^2)])*Tan[c + d*x]^2)))/((Sec[c + d*x]^2)^(2/3)*(-a^2 + b^2*Sec[c + d*x]^2)) + (9*(a^2 - b^2)*Tan[c + d*x]*((b*AppellF1[1/2, 1/6, 1, 3/2, -Tan[c + d*x]^2, (b^2*Tan[c + d*x]^2)/(a^2 - b^2)]*Sqrt[Sec[c + d*x]^2]*Tan[c + d*x])/(9*(a^2 - b^2)*AppellF1[1/2, 1/6, 1, 3/2, -Tan[c + d*x]^2, (b^2*Tan[c + d*x]^2)/(a^2 - b^2)] + (6*b^2*AppellF1[3/2, 1/6, 2, 5/2, -Tan[c + d*x]^2, (b^2*Tan[c + d*x]^2)/(a^2 - b^2)] + (-a^2 + b^2)*AppellF1[3/2, 7/6, 1, 5/2, -Tan[c + d*x]^2, (b^2*Tan[c + d*x]^2)/(a^2 - b^2)])*Tan[c + d*x]^2) + (b*Sqrt[Sec[c + d*x]^2]*((2*b^2*AppellF1[3/2, 1/6, 2, 5/2, -Tan[c + d*x]^2, (b^2*Tan[c + d*x]^2)/(a^2 - b^2)]*Sec[c + d*x]^2*Tan[c + d*x])/(3*(a^2 - b^2)) - (AppellF1[3/2, 7/6, 1, 5/2, -Tan[c + d*x]^2, (b^2*Tan[c + d*x]^2)/(a^2 - b^2)]*Sec[c + d*x]^2*Tan[c + d*x])/9))/(9*(a^2 - b^2)*AppellF1[1/2, 1/6, 1, 3/2, -Tan[c + d*x]^2, (b^2*Tan[c + d*x]^2)/(a^2 - b^2)] + (6*b^2*AppellF1[3/2, 1/6, 2, 5/2, -Tan[c + d*x]^2, (b^2*Tan[c + d*x]^2)/(a^2 - b^2)] + (-a^2 + b^2)*AppellF1[3/2, 7/6, 1, 5/2, -Tan[c + d*x]^2, (b^2*Tan[c + d*x]^2)/(a^2 - b^2)])*Tan[c + d*x]^2) + (a*((2*b^2*AppellF1[3/2, 2/3, 2, 5/2, -Tan[c + d*x]^2, (b^2*Tan[c + d*x]^2)/(a^2 - b^2)]*Sec[c + d*x]^2*Tan[c + d*x])/(3*(a^2 - b^2)) - (4*AppellF1[3/2, 5/3, 1, 5/2, -Tan[c + d*x]^2, (b^2*Tan[c + d*x]^2)/(a^2 - b^2)]*Sec[c + d*x]^2*Tan[c + d*x])/9))/(-9*(a^2 - b^2)*AppellF1[1/2, 2/3, 1, 3/2, -Tan[c + d*x]^2, (b^2*Tan[c + d*x]^2)/(a^2 - b^2)] - 2*(3*b^2*AppellF1[3/2, 2/3, 2, 5/2, -Tan[c + d*x]^2, (b^2*Tan[c + d*x]^2)/(a^2 - b^2)] + 2*(-a^2 + b^2)*AppellF1[3/2, 5/3, 1, 5/2, -Tan[c + d*x]^2, (b^2*Tan[c + d*x]^2)/(a^2 - b^2)])*Tan[c + d*x]^2) - (b*AppellF1[1/2, 1/6, 1, 3/2, -Tan[c + d*x]^2, (b^2*Tan[c + d*x]^2)/(a^2 - b^2)]*Sqrt[Sec[c + d*x]^2]*(2*(6*b^2*AppellF1[3/2, 1/6, 2, 5/2, -Tan[c + d*x]^2, (b^2*Tan[c + d*x]^2)/(a^2 - b^2)] + (-a^2 + b^2)*AppellF1[3/2, 7/6, 1, 5/2, -Tan[c + d*x]^2, (b^2*Tan[c + d*x]^2)/(a^2 - b^2)])*Sec[c + d*x]^2*Tan[c + d*x] + 9*(a^2 - b^2)*((2*b^2*AppellF1[3/2, 1/6, 2, 5/2, -Tan[c + d*x]^2, (b^2*Tan[c + d*x]^2)/(a^2 - b^2)]*Sec[c + d*x]^2*Tan[c + d*x])/(3*(a^2 - b^2)) - (AppellF1[3/2, 7/6, 1, 5/2, -Tan[c + d*x]^2, (b^2*Tan[c + d*x]^2)/(a^2 - b^2)]*Sec[c + d*x]^2*Tan[c + d*x])/9) + Tan[c + d*x]^2*(6*b^2*((12*b^2*AppellF1[5/2, 1/6, 3, 7/2, -Tan[c + d*x]^2, (b^2*Tan[c + d*x]^2)/(a^2 - b^2)]*Sec[c + d*x]^2*Tan[c + d*x])/(5*(a^2 - b^2)) - (AppellF1[5/2, 7/6, 2, 7/2, -Tan[c + d*x]^2, (b^2*Tan[c + d*x]^2)/(a^2 - b^2)]*Sec[c + d*x]^2*Tan[c + d*x])/5) + (-a^2 + b^2)*((6*b^2*AppellF1[5/2, 7/6, 2, 7/2, -Tan[c + d*x]^2, (b^2*Tan[c + d*x]^2)/(a^2 - b^2)]*Sec[c + d*x]^2*Tan[c + d*x])/(5*(a^2 - b^2)) - (7*AppellF1[5/2, 13/6, 1, 7/2, -Tan[c + d*x]^2, (b^2*Tan[c + d*x]^2)/(a^2 - b^2)]*Sec[c + d*x]^2*Tan[c + d*x])/5))))/(9*(a^2 - b^2)*AppellF1[1/2, 1/6, 1, 3/2, -Tan[c + d*x]^2, (b^2*Tan[c + d*x]^2)/(a^2 - b^2)] + (6*b^2*AppellF1[3/2, 1/6, 2, 5/2, -Tan[c + d*x]^2, (b^2*Tan[c + d*x]^2)/(a^2 - b^2)] + (-a^2 + b^2)*AppellF1[3/2, 7/6, 1, 5/2, -Tan[c + d*x]^2, (b^2*Tan[c + d*x]^2)/(a^2 - b^2)])*Tan[c + d*x]^2)^2 - (a*AppellF1[1/2, 2/3, 1, 3/2, -Tan[c + d*x]^2, (b^2*Tan[c + d*x]^2)/(a^2 - b^2)]*(-4*(3*b^2*AppellF1[3/2, 2/3, 2, 5/2, -Tan[c + d*x]^2, (b^2*Tan[c + d*x]^2)/(a^2 - b^2)] + 2*(-a^2 + b^2)*AppellF1[3/2, 5/3, 1, 5/2, -Tan[c + d*x]^2, (b^2*Tan[c + d*x]^2)/(a^2 - b^2)])*Sec[c + d*x]^2*Tan[c + d*x] - 9*(a^2 - b^2)*((2*b^2*AppellF1[3/2, 2/3, 2, 5/2, -Tan[c + d*x]^2, (b^2*Tan[c + d*x]^2)/(a^2 - b^2)]*Sec[c + d*x]^2*Tan[c + d*x])/(3*(a^2 - b^2)) - (4*AppellF1[3/2, 5/3, 1, 5/2, -Tan[c + d*x]^2, (b^2*Tan[c + d*x]^2)/(a^2 - b^2)]*Sec[c + d*x]^2*Tan[c + d*x])/9) - 2*Tan[c + d*x]^2*(3*b^2*((12*b^2*AppellF1[5/2, 2/3, 3, 7/2, -Tan[c + d*x]^2, (b^2*Tan[c + d*x]^2)/(a^2 - b^2)]*Sec[c + d*x]^2*Tan[c + d*x])/(5*(a^2 - b^2)) - (4*AppellF1[5/2, 5/3, 2, 7/2, -Tan[c + d*x]^2, (b^2*Tan[c + d*x]^2)/(a^2 - b^2)]*Sec[c + d*x]^2*Tan[c + d*x])/5) + 2*(-a^2 + b^2)*((6*b^2*AppellF1[5/2, 5/3, 2, 7/2, -Tan[c + d*x]^2, (b^2*Tan[c + d*x]^2)/(a^2 - b^2)]*Sec[c + d*x]^2*Tan[c + d*x])/(5*(a^2 - b^2)) - 2*AppellF1[5/2, 8/3, 1, 7/2, -Tan[c + d*x]^2, (b^2*Tan[c + d*x]^2)/(a^2 - b^2)]*Sec[c + d*x]^2*Tan[c + d*x]))))/(-9*(a^2 - b^2)*AppellF1[1/2, 2/3, 1, 3/2, -Tan[c + d*x]^2, (b^2*Tan[c + d*x]^2)/(a^2 - b^2)] - 2*(3*b^2*AppellF1[3/2, 2/3, 2, 5/2, -Tan[c + d*x]^2, (b^2*Tan[c + d*x]^2)/(a^2 - b^2)] + 2*(-a^2 + b^2)*AppellF1[3/2, 5/3, 1, 5/2, -Tan[c + d*x]^2, (b^2*Tan[c + d*x]^2)/(a^2 - b^2)])*Tan[c + d*x]^2)^2))/((Sec[c + d*x]^2)^(2/3)*(-a^2 + b^2*Sec[c + d*x]^2))))","B",0
714,1,4544,174,21.5444986,"\int \frac{\sqrt[3]{\sec (c+d x)}}{a+b \sec (c+d x)} \, dx","Integrate[Sec[c + d*x]^(1/3)/(a + b*Sec[c + d*x]),x]","\text{Result too large to show}","\frac{a \sin (c+d x) F_1\left(\frac{1}{2};-\frac{1}{3},1;\frac{3}{2};\sin ^2(c+d x),\frac{a^2 \sin ^2(c+d x)}{a^2-b^2}\right)}{d \left(a^2-b^2\right) \sqrt[3]{\cos ^2(c+d x)} \sec ^{\frac{2}{3}}(c+d x)}-\frac{b \sin (c+d x) \sqrt[6]{\cos ^2(c+d x)} \sqrt[3]{\sec (c+d x)} F_1\left(\frac{1}{2};\frac{1}{6},1;\frac{3}{2};\sin ^2(c+d x),\frac{a^2 \sin ^2(c+d x)}{a^2-b^2}\right)}{d \left(a^2-b^2\right)}",1,"(9*(a^2 - b^2)*Sec[c + d*x]^(4/3)*Sin[c + d*x]*((b*AppellF1[1/2, 1/3, 1, 3/2, -Tan[c + d*x]^2, (b^2*Tan[c + d*x]^2)/(a^2 - b^2)]*Sqrt[Sec[c + d*x]^2])/(9*(a^2 - b^2)*AppellF1[1/2, 1/3, 1, 3/2, -Tan[c + d*x]^2, (b^2*Tan[c + d*x]^2)/(a^2 - b^2)] + 2*(3*b^2*AppellF1[3/2, 1/3, 2, 5/2, -Tan[c + d*x]^2, (b^2*Tan[c + d*x]^2)/(a^2 - b^2)] + (-a^2 + b^2)*AppellF1[3/2, 4/3, 1, 5/2, -Tan[c + d*x]^2, (b^2*Tan[c + d*x]^2)/(a^2 - b^2)])*Tan[c + d*x]^2) + (a*AppellF1[1/2, 5/6, 1, 3/2, -Tan[c + d*x]^2, (b^2*Tan[c + d*x]^2)/(a^2 - b^2)])/(-9*(a^2 - b^2)*AppellF1[1/2, 5/6, 1, 3/2, -Tan[c + d*x]^2, (b^2*Tan[c + d*x]^2)/(a^2 - b^2)] + (-6*b^2*AppellF1[3/2, 5/6, 2, 5/2, -Tan[c + d*x]^2, (b^2*Tan[c + d*x]^2)/(a^2 - b^2)] + 5*(a^2 - b^2)*AppellF1[3/2, 11/6, 1, 5/2, -Tan[c + d*x]^2, (b^2*Tan[c + d*x]^2)/(a^2 - b^2)])*Tan[c + d*x]^2)))/(d*(Sec[c + d*x]^2)^(5/6)*(a + b*Sec[c + d*x])*(-a^2 + b^2*Sec[c + d*x]^2)*((9*(a^2 - b^2)*(Sec[c + d*x]^2)^(1/6)*((b*AppellF1[1/2, 1/3, 1, 3/2, -Tan[c + d*x]^2, (b^2*Tan[c + d*x]^2)/(a^2 - b^2)]*Sqrt[Sec[c + d*x]^2])/(9*(a^2 - b^2)*AppellF1[1/2, 1/3, 1, 3/2, -Tan[c + d*x]^2, (b^2*Tan[c + d*x]^2)/(a^2 - b^2)] + 2*(3*b^2*AppellF1[3/2, 1/3, 2, 5/2, -Tan[c + d*x]^2, (b^2*Tan[c + d*x]^2)/(a^2 - b^2)] + (-a^2 + b^2)*AppellF1[3/2, 4/3, 1, 5/2, -Tan[c + d*x]^2, (b^2*Tan[c + d*x]^2)/(a^2 - b^2)])*Tan[c + d*x]^2) + (a*AppellF1[1/2, 5/6, 1, 3/2, -Tan[c + d*x]^2, (b^2*Tan[c + d*x]^2)/(a^2 - b^2)])/(-9*(a^2 - b^2)*AppellF1[1/2, 5/6, 1, 3/2, -Tan[c + d*x]^2, (b^2*Tan[c + d*x]^2)/(a^2 - b^2)] + (-6*b^2*AppellF1[3/2, 5/6, 2, 5/2, -Tan[c + d*x]^2, (b^2*Tan[c + d*x]^2)/(a^2 - b^2)] + 5*(a^2 - b^2)*AppellF1[3/2, 11/6, 1, 5/2, -Tan[c + d*x]^2, (b^2*Tan[c + d*x]^2)/(a^2 - b^2)])*Tan[c + d*x]^2)))/(-a^2 + b^2*Sec[c + d*x]^2) - (18*b^2*(a^2 - b^2)*(Sec[c + d*x]^2)^(1/6)*Tan[c + d*x]^2*((b*AppellF1[1/2, 1/3, 1, 3/2, -Tan[c + d*x]^2, (b^2*Tan[c + d*x]^2)/(a^2 - b^2)]*Sqrt[Sec[c + d*x]^2])/(9*(a^2 - b^2)*AppellF1[1/2, 1/3, 1, 3/2, -Tan[c + d*x]^2, (b^2*Tan[c + d*x]^2)/(a^2 - b^2)] + 2*(3*b^2*AppellF1[3/2, 1/3, 2, 5/2, -Tan[c + d*x]^2, (b^2*Tan[c + d*x]^2)/(a^2 - b^2)] + (-a^2 + b^2)*AppellF1[3/2, 4/3, 1, 5/2, -Tan[c + d*x]^2, (b^2*Tan[c + d*x]^2)/(a^2 - b^2)])*Tan[c + d*x]^2) + (a*AppellF1[1/2, 5/6, 1, 3/2, -Tan[c + d*x]^2, (b^2*Tan[c + d*x]^2)/(a^2 - b^2)])/(-9*(a^2 - b^2)*AppellF1[1/2, 5/6, 1, 3/2, -Tan[c + d*x]^2, (b^2*Tan[c + d*x]^2)/(a^2 - b^2)] + (-6*b^2*AppellF1[3/2, 5/6, 2, 5/2, -Tan[c + d*x]^2, (b^2*Tan[c + d*x]^2)/(a^2 - b^2)] + 5*(a^2 - b^2)*AppellF1[3/2, 11/6, 1, 5/2, -Tan[c + d*x]^2, (b^2*Tan[c + d*x]^2)/(a^2 - b^2)])*Tan[c + d*x]^2)))/(-a^2 + b^2*Sec[c + d*x]^2)^2 - (15*(a^2 - b^2)*Tan[c + d*x]^2*((b*AppellF1[1/2, 1/3, 1, 3/2, -Tan[c + d*x]^2, (b^2*Tan[c + d*x]^2)/(a^2 - b^2)]*Sqrt[Sec[c + d*x]^2])/(9*(a^2 - b^2)*AppellF1[1/2, 1/3, 1, 3/2, -Tan[c + d*x]^2, (b^2*Tan[c + d*x]^2)/(a^2 - b^2)] + 2*(3*b^2*AppellF1[3/2, 1/3, 2, 5/2, -Tan[c + d*x]^2, (b^2*Tan[c + d*x]^2)/(a^2 - b^2)] + (-a^2 + b^2)*AppellF1[3/2, 4/3, 1, 5/2, -Tan[c + d*x]^2, (b^2*Tan[c + d*x]^2)/(a^2 - b^2)])*Tan[c + d*x]^2) + (a*AppellF1[1/2, 5/6, 1, 3/2, -Tan[c + d*x]^2, (b^2*Tan[c + d*x]^2)/(a^2 - b^2)])/(-9*(a^2 - b^2)*AppellF1[1/2, 5/6, 1, 3/2, -Tan[c + d*x]^2, (b^2*Tan[c + d*x]^2)/(a^2 - b^2)] + (-6*b^2*AppellF1[3/2, 5/6, 2, 5/2, -Tan[c + d*x]^2, (b^2*Tan[c + d*x]^2)/(a^2 - b^2)] + 5*(a^2 - b^2)*AppellF1[3/2, 11/6, 1, 5/2, -Tan[c + d*x]^2, (b^2*Tan[c + d*x]^2)/(a^2 - b^2)])*Tan[c + d*x]^2)))/((Sec[c + d*x]^2)^(5/6)*(-a^2 + b^2*Sec[c + d*x]^2)) + (9*(a^2 - b^2)*Tan[c + d*x]*((b*AppellF1[1/2, 1/3, 1, 3/2, -Tan[c + d*x]^2, (b^2*Tan[c + d*x]^2)/(a^2 - b^2)]*Sqrt[Sec[c + d*x]^2]*Tan[c + d*x])/(9*(a^2 - b^2)*AppellF1[1/2, 1/3, 1, 3/2, -Tan[c + d*x]^2, (b^2*Tan[c + d*x]^2)/(a^2 - b^2)] + 2*(3*b^2*AppellF1[3/2, 1/3, 2, 5/2, -Tan[c + d*x]^2, (b^2*Tan[c + d*x]^2)/(a^2 - b^2)] + (-a^2 + b^2)*AppellF1[3/2, 4/3, 1, 5/2, -Tan[c + d*x]^2, (b^2*Tan[c + d*x]^2)/(a^2 - b^2)])*Tan[c + d*x]^2) + (b*Sqrt[Sec[c + d*x]^2]*((2*b^2*AppellF1[3/2, 1/3, 2, 5/2, -Tan[c + d*x]^2, (b^2*Tan[c + d*x]^2)/(a^2 - b^2)]*Sec[c + d*x]^2*Tan[c + d*x])/(3*(a^2 - b^2)) - (2*AppellF1[3/2, 4/3, 1, 5/2, -Tan[c + d*x]^2, (b^2*Tan[c + d*x]^2)/(a^2 - b^2)]*Sec[c + d*x]^2*Tan[c + d*x])/9))/(9*(a^2 - b^2)*AppellF1[1/2, 1/3, 1, 3/2, -Tan[c + d*x]^2, (b^2*Tan[c + d*x]^2)/(a^2 - b^2)] + 2*(3*b^2*AppellF1[3/2, 1/3, 2, 5/2, -Tan[c + d*x]^2, (b^2*Tan[c + d*x]^2)/(a^2 - b^2)] + (-a^2 + b^2)*AppellF1[3/2, 4/3, 1, 5/2, -Tan[c + d*x]^2, (b^2*Tan[c + d*x]^2)/(a^2 - b^2)])*Tan[c + d*x]^2) + (a*((2*b^2*AppellF1[3/2, 5/6, 2, 5/2, -Tan[c + d*x]^2, (b^2*Tan[c + d*x]^2)/(a^2 - b^2)]*Sec[c + d*x]^2*Tan[c + d*x])/(3*(a^2 - b^2)) - (5*AppellF1[3/2, 11/6, 1, 5/2, -Tan[c + d*x]^2, (b^2*Tan[c + d*x]^2)/(a^2 - b^2)]*Sec[c + d*x]^2*Tan[c + d*x])/9))/(-9*(a^2 - b^2)*AppellF1[1/2, 5/6, 1, 3/2, -Tan[c + d*x]^2, (b^2*Tan[c + d*x]^2)/(a^2 - b^2)] + (-6*b^2*AppellF1[3/2, 5/6, 2, 5/2, -Tan[c + d*x]^2, (b^2*Tan[c + d*x]^2)/(a^2 - b^2)] + 5*(a^2 - b^2)*AppellF1[3/2, 11/6, 1, 5/2, -Tan[c + d*x]^2, (b^2*Tan[c + d*x]^2)/(a^2 - b^2)])*Tan[c + d*x]^2) - (b*AppellF1[1/2, 1/3, 1, 3/2, -Tan[c + d*x]^2, (b^2*Tan[c + d*x]^2)/(a^2 - b^2)]*Sqrt[Sec[c + d*x]^2]*(4*(3*b^2*AppellF1[3/2, 1/3, 2, 5/2, -Tan[c + d*x]^2, (b^2*Tan[c + d*x]^2)/(a^2 - b^2)] + (-a^2 + b^2)*AppellF1[3/2, 4/3, 1, 5/2, -Tan[c + d*x]^2, (b^2*Tan[c + d*x]^2)/(a^2 - b^2)])*Sec[c + d*x]^2*Tan[c + d*x] + 9*(a^2 - b^2)*((2*b^2*AppellF1[3/2, 1/3, 2, 5/2, -Tan[c + d*x]^2, (b^2*Tan[c + d*x]^2)/(a^2 - b^2)]*Sec[c + d*x]^2*Tan[c + d*x])/(3*(a^2 - b^2)) - (2*AppellF1[3/2, 4/3, 1, 5/2, -Tan[c + d*x]^2, (b^2*Tan[c + d*x]^2)/(a^2 - b^2)]*Sec[c + d*x]^2*Tan[c + d*x])/9) + 2*Tan[c + d*x]^2*(3*b^2*((12*b^2*AppellF1[5/2, 1/3, 3, 7/2, -Tan[c + d*x]^2, (b^2*Tan[c + d*x]^2)/(a^2 - b^2)]*Sec[c + d*x]^2*Tan[c + d*x])/(5*(a^2 - b^2)) - (2*AppellF1[5/2, 4/3, 2, 7/2, -Tan[c + d*x]^2, (b^2*Tan[c + d*x]^2)/(a^2 - b^2)]*Sec[c + d*x]^2*Tan[c + d*x])/5) + (-a^2 + b^2)*((6*b^2*AppellF1[5/2, 4/3, 2, 7/2, -Tan[c + d*x]^2, (b^2*Tan[c + d*x]^2)/(a^2 - b^2)]*Sec[c + d*x]^2*Tan[c + d*x])/(5*(a^2 - b^2)) - (8*AppellF1[5/2, 7/3, 1, 7/2, -Tan[c + d*x]^2, (b^2*Tan[c + d*x]^2)/(a^2 - b^2)]*Sec[c + d*x]^2*Tan[c + d*x])/5))))/(9*(a^2 - b^2)*AppellF1[1/2, 1/3, 1, 3/2, -Tan[c + d*x]^2, (b^2*Tan[c + d*x]^2)/(a^2 - b^2)] + 2*(3*b^2*AppellF1[3/2, 1/3, 2, 5/2, -Tan[c + d*x]^2, (b^2*Tan[c + d*x]^2)/(a^2 - b^2)] + (-a^2 + b^2)*AppellF1[3/2, 4/3, 1, 5/2, -Tan[c + d*x]^2, (b^2*Tan[c + d*x]^2)/(a^2 - b^2)])*Tan[c + d*x]^2)^2 - (a*AppellF1[1/2, 5/6, 1, 3/2, -Tan[c + d*x]^2, (b^2*Tan[c + d*x]^2)/(a^2 - b^2)]*(2*(-6*b^2*AppellF1[3/2, 5/6, 2, 5/2, -Tan[c + d*x]^2, (b^2*Tan[c + d*x]^2)/(a^2 - b^2)] + 5*(a^2 - b^2)*AppellF1[3/2, 11/6, 1, 5/2, -Tan[c + d*x]^2, (b^2*Tan[c + d*x]^2)/(a^2 - b^2)])*Sec[c + d*x]^2*Tan[c + d*x] - 9*(a^2 - b^2)*((2*b^2*AppellF1[3/2, 5/6, 2, 5/2, -Tan[c + d*x]^2, (b^2*Tan[c + d*x]^2)/(a^2 - b^2)]*Sec[c + d*x]^2*Tan[c + d*x])/(3*(a^2 - b^2)) - (5*AppellF1[3/2, 11/6, 1, 5/2, -Tan[c + d*x]^2, (b^2*Tan[c + d*x]^2)/(a^2 - b^2)]*Sec[c + d*x]^2*Tan[c + d*x])/9) + Tan[c + d*x]^2*(-6*b^2*((12*b^2*AppellF1[5/2, 5/6, 3, 7/2, -Tan[c + d*x]^2, (b^2*Tan[c + d*x]^2)/(a^2 - b^2)]*Sec[c + d*x]^2*Tan[c + d*x])/(5*(a^2 - b^2)) - AppellF1[5/2, 11/6, 2, 7/2, -Tan[c + d*x]^2, (b^2*Tan[c + d*x]^2)/(a^2 - b^2)]*Sec[c + d*x]^2*Tan[c + d*x]) + 5*(a^2 - b^2)*((6*b^2*AppellF1[5/2, 11/6, 2, 7/2, -Tan[c + d*x]^2, (b^2*Tan[c + d*x]^2)/(a^2 - b^2)]*Sec[c + d*x]^2*Tan[c + d*x])/(5*(a^2 - b^2)) - (11*AppellF1[5/2, 17/6, 1, 7/2, -Tan[c + d*x]^2, (b^2*Tan[c + d*x]^2)/(a^2 - b^2)]*Sec[c + d*x]^2*Tan[c + d*x])/5))))/(-9*(a^2 - b^2)*AppellF1[1/2, 5/6, 1, 3/2, -Tan[c + d*x]^2, (b^2*Tan[c + d*x]^2)/(a^2 - b^2)] + (-6*b^2*AppellF1[3/2, 5/6, 2, 5/2, -Tan[c + d*x]^2, (b^2*Tan[c + d*x]^2)/(a^2 - b^2)] + 5*(a^2 - b^2)*AppellF1[3/2, 11/6, 1, 5/2, -Tan[c + d*x]^2, (b^2*Tan[c + d*x]^2)/(a^2 - b^2)])*Tan[c + d*x]^2)^2))/((Sec[c + d*x]^2)^(5/6)*(-a^2 + b^2*Sec[c + d*x]^2))))","B",0
715,1,7542,174,29.3259002,"\int \frac{1}{\sqrt[3]{\sec (c+d x)} (a+b \sec (c+d x))} \, dx","Integrate[1/(Sec[c + d*x]^(1/3)*(a + b*Sec[c + d*x])),x]","\text{Result too large to show}","\frac{a \sin (c+d x) \sqrt[3]{\cos ^2(c+d x)} \sec ^{\frac{2}{3}}(c+d x) F_1\left(\frac{1}{2};-\frac{2}{3},1;\frac{3}{2};\sin ^2(c+d x),\frac{a^2 \sin ^2(c+d x)}{a^2-b^2}\right)}{d \left(a^2-b^2\right)}-\frac{b \sin (c+d x) F_1\left(\frac{1}{2};-\frac{1}{6},1;\frac{3}{2};\sin ^2(c+d x),\frac{a^2 \sin ^2(c+d x)}{a^2-b^2}\right)}{d \left(a^2-b^2\right) \sqrt[6]{\cos ^2(c+d x)} \sqrt[3]{\sec (c+d x)}}",1,"Result too large to show","B",0
716,1,7588,174,29.2564143,"\int \frac{1}{\sec ^{\frac{2}{3}}(c+d x) (a+b \sec (c+d x))} \, dx","Integrate[1/(Sec[c + d*x]^(2/3)*(a + b*Sec[c + d*x])),x]","\text{Result too large to show}","\frac{a \sin (c+d x) \sqrt[6]{\cos ^2(c+d x)} \sqrt[3]{\sec (c+d x)} F_1\left(\frac{1}{2};-\frac{5}{6},1;\frac{3}{2};\sin ^2(c+d x),\frac{a^2 \sin ^2(c+d x)}{a^2-b^2}\right)}{d \left(a^2-b^2\right)}-\frac{b \sin (c+d x) F_1\left(\frac{1}{2};-\frac{1}{3},1;\frac{3}{2};\sin ^2(c+d x),\frac{a^2 \sin ^2(c+d x)}{a^2-b^2}\right)}{d \left(a^2-b^2\right) \sqrt[3]{\cos ^2(c+d x)} \sec ^{\frac{2}{3}}(c+d x)}",1,"Result too large to show","B",0
717,0,0,28,37.7256552,"\int \sec ^{\frac{7}{3}}(c+d x) \sqrt{a+b \sec (c+d x)} \, dx","Integrate[Sec[c + d*x]^(7/3)*Sqrt[a + b*Sec[c + d*x]],x]","\int \sec ^{\frac{7}{3}}(c+d x) \sqrt{a+b \sec (c+d x)} \, dx","\text{Int}\left(\sec ^{\frac{7}{3}}(c+d x) \sqrt{a+b \sec (c+d x)},x\right)",0,"Integrate[Sec[c + d*x]^(7/3)*Sqrt[a + b*Sec[c + d*x]], x]","A",-1
718,0,0,28,49.5900558,"\int \sec ^{\frac{5}{3}}(c+d x) \sqrt{a+b \sec (c+d x)} \, dx","Integrate[Sec[c + d*x]^(5/3)*Sqrt[a + b*Sec[c + d*x]],x]","\int \sec ^{\frac{5}{3}}(c+d x) \sqrt{a+b \sec (c+d x)} \, dx","\text{Int}\left(\sec ^{\frac{5}{3}}(c+d x) \sqrt{a+b \sec (c+d x)},x\right)",0,"Integrate[Sec[c + d*x]^(5/3)*Sqrt[a + b*Sec[c + d*x]], x]","A",-1
719,0,0,28,32.1165176,"\int \sec ^{\frac{4}{3}}(c+d x) \sqrt{a+b \sec (c+d x)} \, dx","Integrate[Sec[c + d*x]^(4/3)*Sqrt[a + b*Sec[c + d*x]],x]","\int \sec ^{\frac{4}{3}}(c+d x) \sqrt{a+b \sec (c+d x)} \, dx","\text{Int}\left(\sec ^{\frac{4}{3}}(c+d x) \sqrt{a+b \sec (c+d x)},x\right)",0,"Integrate[Sec[c + d*x]^(4/3)*Sqrt[a + b*Sec[c + d*x]], x]","A",-1
720,0,0,28,39.8486172,"\int \sec ^{\frac{2}{3}}(c+d x) \sqrt{a+b \sec (c+d x)} \, dx","Integrate[Sec[c + d*x]^(2/3)*Sqrt[a + b*Sec[c + d*x]],x]","\int \sec ^{\frac{2}{3}}(c+d x) \sqrt{a+b \sec (c+d x)} \, dx","\text{Int}\left(\sec ^{\frac{2}{3}}(c+d x) \sqrt{a+b \sec (c+d x)},x\right)",0,"Integrate[Sec[c + d*x]^(2/3)*Sqrt[a + b*Sec[c + d*x]], x]","A",-1
721,0,0,28,2.6745676,"\int \sqrt[3]{\sec (c+d x)} \sqrt{a+b \sec (c+d x)} \, dx","Integrate[Sec[c + d*x]^(1/3)*Sqrt[a + b*Sec[c + d*x]],x]","\int \sqrt[3]{\sec (c+d x)} \sqrt{a+b \sec (c+d x)} \, dx","\text{Int}\left(\sqrt[3]{\sec (c+d x)} \sqrt{a+b \sec (c+d x)},x\right)",0,"Integrate[Sec[c + d*x]^(1/3)*Sqrt[a + b*Sec[c + d*x]], x]","A",-1
722,0,0,28,9.3874312,"\int \frac{\sqrt{a+b \sec (c+d x)}}{\sqrt[3]{\sec (c+d x)}} \, dx","Integrate[Sqrt[a + b*Sec[c + d*x]]/Sec[c + d*x]^(1/3),x]","\int \frac{\sqrt{a+b \sec (c+d x)}}{\sqrt[3]{\sec (c+d x)}} \, dx","\text{Int}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt[3]{\sec (c+d x)}},x\right)",0,"Integrate[Sqrt[a + b*Sec[c + d*x]]/Sec[c + d*x]^(1/3), x]","A",-1
723,0,0,28,18.3345733,"\int \frac{\sqrt{a+b \sec (c+d x)}}{\sec ^{\frac{2}{3}}(c+d x)} \, dx","Integrate[Sqrt[a + b*Sec[c + d*x]]/Sec[c + d*x]^(2/3),x]","\int \frac{\sqrt{a+b \sec (c+d x)}}{\sec ^{\frac{2}{3}}(c+d x)} \, dx","\text{Int}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sec ^{\frac{2}{3}}(c+d x)},x\right)",0,"Integrate[Sqrt[a + b*Sec[c + d*x]]/Sec[c + d*x]^(2/3), x]","A",-1
724,0,0,28,24.4334862,"\int \frac{\sqrt{a+b \sec (c+d x)}}{\sec ^{\frac{4}{3}}(c+d x)} \, dx","Integrate[Sqrt[a + b*Sec[c + d*x]]/Sec[c + d*x]^(4/3),x]","\int \frac{\sqrt{a+b \sec (c+d x)}}{\sec ^{\frac{4}{3}}(c+d x)} \, dx","\text{Int}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sec ^{\frac{4}{3}}(c+d x)},x\right)",0,"Integrate[Sqrt[a + b*Sec[c + d*x]]/Sec[c + d*x]^(4/3), x]","A",-1
725,0,0,28,35.5103896,"\int \frac{\sqrt{a+b \sec (c+d x)}}{\sec ^{\frac{5}{3}}(c+d x)} \, dx","Integrate[Sqrt[a + b*Sec[c + d*x]]/Sec[c + d*x]^(5/3),x]","\int \frac{\sqrt{a+b \sec (c+d x)}}{\sec ^{\frac{5}{3}}(c+d x)} \, dx","\text{Int}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sec ^{\frac{5}{3}}(c+d x)},x\right)",0,"Integrate[Sqrt[a + b*Sec[c + d*x]]/Sec[c + d*x]^(5/3), x]","A",-1
726,0,0,28,41.6109382,"\int \frac{\sqrt{a+b \sec (c+d x)}}{\sec ^{\frac{7}{3}}(c+d x)} \, dx","Integrate[Sqrt[a + b*Sec[c + d*x]]/Sec[c + d*x]^(7/3),x]","\int \frac{\sqrt{a+b \sec (c+d x)}}{\sec ^{\frac{7}{3}}(c+d x)} \, dx","\text{Int}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sec ^{\frac{7}{3}}(c+d x)},x\right)",0,"Integrate[Sqrt[a + b*Sec[c + d*x]]/Sec[c + d*x]^(7/3), x]","A",-1
727,0,0,28,39.3287285,"\int \sec ^{\frac{7}{3}}(c+d x) (a+b \sec (c+d x))^{3/2} \, dx","Integrate[Sec[c + d*x]^(7/3)*(a + b*Sec[c + d*x])^(3/2),x]","\int \sec ^{\frac{7}{3}}(c+d x) (a+b \sec (c+d x))^{3/2} \, dx","\text{Int}\left(\sec ^{\frac{7}{3}}(c+d x) (a+b \sec (c+d x))^{3/2},x\right)",0,"Integrate[Sec[c + d*x]^(7/3)*(a + b*Sec[c + d*x])^(3/2), x]","A",-1
728,0,0,28,41.2687203,"\int \sec ^{\frac{5}{3}}(c+d x) (a+b \sec (c+d x))^{3/2} \, dx","Integrate[Sec[c + d*x]^(5/3)*(a + b*Sec[c + d*x])^(3/2),x]","\int \sec ^{\frac{5}{3}}(c+d x) (a+b \sec (c+d x))^{3/2} \, dx","\text{Int}\left(\sec ^{\frac{5}{3}}(c+d x) (a+b \sec (c+d x))^{3/2},x\right)",0,"Integrate[Sec[c + d*x]^(5/3)*(a + b*Sec[c + d*x])^(3/2), x]","A",-1
729,0,0,28,36.0595714,"\int \sec ^{\frac{4}{3}}(c+d x) (a+b \sec (c+d x))^{3/2} \, dx","Integrate[Sec[c + d*x]^(4/3)*(a + b*Sec[c + d*x])^(3/2),x]","\int \sec ^{\frac{4}{3}}(c+d x) (a+b \sec (c+d x))^{3/2} \, dx","\text{Int}\left(\sec ^{\frac{4}{3}}(c+d x) (a+b \sec (c+d x))^{3/2},x\right)",0,"Integrate[Sec[c + d*x]^(4/3)*(a + b*Sec[c + d*x])^(3/2), x]","A",-1
730,0,0,28,42.1309782,"\int \sec ^{\frac{2}{3}}(c+d x) (a+b \sec (c+d x))^{3/2} \, dx","Integrate[Sec[c + d*x]^(2/3)*(a + b*Sec[c + d*x])^(3/2),x]","\int \sec ^{\frac{2}{3}}(c+d x) (a+b \sec (c+d x))^{3/2} \, dx","\text{Int}\left(\sec ^{\frac{2}{3}}(c+d x) (a+b \sec (c+d x))^{3/2},x\right)",0,"Integrate[Sec[c + d*x]^(2/3)*(a + b*Sec[c + d*x])^(3/2), x]","A",-1
731,0,0,28,31.8386795,"\int \sqrt[3]{\sec (c+d x)} (a+b \sec (c+d x))^{3/2} \, dx","Integrate[Sec[c + d*x]^(1/3)*(a + b*Sec[c + d*x])^(3/2),x]","\int \sqrt[3]{\sec (c+d x)} (a+b \sec (c+d x))^{3/2} \, dx","\text{Int}\left(\sqrt[3]{\sec (c+d x)} (a+b \sec (c+d x))^{3/2},x\right)",0,"Integrate[Sec[c + d*x]^(1/3)*(a + b*Sec[c + d*x])^(3/2), x]","A",-1
732,0,0,28,42.6492283,"\int \frac{(a+b \sec (c+d x))^{3/2}}{\sqrt[3]{\sec (c+d x)}} \, dx","Integrate[(a + b*Sec[c + d*x])^(3/2)/Sec[c + d*x]^(1/3),x]","\int \frac{(a+b \sec (c+d x))^{3/2}}{\sqrt[3]{\sec (c+d x)}} \, dx","\text{Int}\left(\frac{(a+b \sec (c+d x))^{3/2}}{\sqrt[3]{\sec (c+d x)}},x\right)",0,"Integrate[(a + b*Sec[c + d*x])^(3/2)/Sec[c + d*x]^(1/3), x]","A",-1
733,0,0,28,23.9032964,"\int \frac{(a+b \sec (c+d x))^{3/2}}{\sec ^{\frac{2}{3}}(c+d x)} \, dx","Integrate[(a + b*Sec[c + d*x])^(3/2)/Sec[c + d*x]^(2/3),x]","\int \frac{(a+b \sec (c+d x))^{3/2}}{\sec ^{\frac{2}{3}}(c+d x)} \, dx","\text{Int}\left(\frac{(a+b \sec (c+d x))^{3/2}}{\sec ^{\frac{2}{3}}(c+d x)},x\right)",0,"Integrate[(a + b*Sec[c + d*x])^(3/2)/Sec[c + d*x]^(2/3), x]","A",-1
734,0,0,28,28.9787371,"\int \frac{(a+b \sec (c+d x))^{3/2}}{\sec ^{\frac{4}{3}}(c+d x)} \, dx","Integrate[(a + b*Sec[c + d*x])^(3/2)/Sec[c + d*x]^(4/3),x]","\int \frac{(a+b \sec (c+d x))^{3/2}}{\sec ^{\frac{4}{3}}(c+d x)} \, dx","\text{Int}\left(\frac{(a+b \sec (c+d x))^{3/2}}{\sec ^{\frac{4}{3}}(c+d x)},x\right)",0,"Integrate[(a + b*Sec[c + d*x])^(3/2)/Sec[c + d*x]^(4/3), x]","A",-1
735,0,0,28,31.4651482,"\int \frac{(a+b \sec (c+d x))^{3/2}}{\sec ^{\frac{5}{3}}(c+d x)} \, dx","Integrate[(a + b*Sec[c + d*x])^(3/2)/Sec[c + d*x]^(5/3),x]","\int \frac{(a+b \sec (c+d x))^{3/2}}{\sec ^{\frac{5}{3}}(c+d x)} \, dx","\text{Int}\left(\frac{(a+b \sec (c+d x))^{3/2}}{\sec ^{\frac{5}{3}}(c+d x)},x\right)",0,"Integrate[(a + b*Sec[c + d*x])^(3/2)/Sec[c + d*x]^(5/3), x]","A",-1
736,0,0,28,39.2186514,"\int \frac{(a+b \sec (c+d x))^{3/2}}{\sec ^{\frac{7}{3}}(c+d x)} \, dx","Integrate[(a + b*Sec[c + d*x])^(3/2)/Sec[c + d*x]^(7/3),x]","\int \frac{(a+b \sec (c+d x))^{3/2}}{\sec ^{\frac{7}{3}}(c+d x)} \, dx","\text{Int}\left(\frac{(a+b \sec (c+d x))^{3/2}}{\sec ^{\frac{7}{3}}(c+d x)},x\right)",0,"Integrate[(a + b*Sec[c + d*x])^(3/2)/Sec[c + d*x]^(7/3), x]","A",-1
737,0,0,28,43.7686396,"\int \sec ^{\frac{7}{3}}(c+d x) (a+b \sec (c+d x))^{5/2} \, dx","Integrate[Sec[c + d*x]^(7/3)*(a + b*Sec[c + d*x])^(5/2),x]","\int \sec ^{\frac{7}{3}}(c+d x) (a+b \sec (c+d x))^{5/2} \, dx","\text{Int}\left(\sec ^{\frac{7}{3}}(c+d x) (a+b \sec (c+d x))^{5/2},x\right)",0,"Integrate[Sec[c + d*x]^(7/3)*(a + b*Sec[c + d*x])^(5/2), x]","A",-1
738,0,0,28,48.4424373,"\int \sec ^{\frac{5}{3}}(c+d x) (a+b \sec (c+d x))^{5/2} \, dx","Integrate[Sec[c + d*x]^(5/3)*(a + b*Sec[c + d*x])^(5/2),x]","\int \sec ^{\frac{5}{3}}(c+d x) (a+b \sec (c+d x))^{5/2} \, dx","\text{Int}\left(\sec ^{\frac{5}{3}}(c+d x) (a+b \sec (c+d x))^{5/2},x\right)",0,"Integrate[Sec[c + d*x]^(5/3)*(a + b*Sec[c + d*x])^(5/2), x]","A",-1
739,0,0,28,42.2330386,"\int \sec ^{\frac{4}{3}}(c+d x) (a+b \sec (c+d x))^{5/2} \, dx","Integrate[Sec[c + d*x]^(4/3)*(a + b*Sec[c + d*x])^(5/2),x]","\int \sec ^{\frac{4}{3}}(c+d x) (a+b \sec (c+d x))^{5/2} \, dx","\text{Int}\left(\sec ^{\frac{4}{3}}(c+d x) (a+b \sec (c+d x))^{5/2},x\right)",0,"Integrate[Sec[c + d*x]^(4/3)*(a + b*Sec[c + d*x])^(5/2), x]","A",-1
740,0,0,28,49.0996068,"\int \sec ^{\frac{2}{3}}(c+d x) (a+b \sec (c+d x))^{5/2} \, dx","Integrate[Sec[c + d*x]^(2/3)*(a + b*Sec[c + d*x])^(5/2),x]","\int \sec ^{\frac{2}{3}}(c+d x) (a+b \sec (c+d x))^{5/2} \, dx","\text{Int}\left(\sec ^{\frac{2}{3}}(c+d x) (a+b \sec (c+d x))^{5/2},x\right)",0,"Integrate[Sec[c + d*x]^(2/3)*(a + b*Sec[c + d*x])^(5/2), x]","A",-1
741,0,0,28,36.3963534,"\int \sqrt[3]{\sec (c+d x)} (a+b \sec (c+d x))^{5/2} \, dx","Integrate[Sec[c + d*x]^(1/3)*(a + b*Sec[c + d*x])^(5/2),x]","\int \sqrt[3]{\sec (c+d x)} (a+b \sec (c+d x))^{5/2} \, dx","\text{Int}\left(\sqrt[3]{\sec (c+d x)} (a+b \sec (c+d x))^{5/2},x\right)",0,"Integrate[Sec[c + d*x]^(1/3)*(a + b*Sec[c + d*x])^(5/2), x]","A",-1
742,0,0,28,50.1440917,"\int \frac{(a+b \sec (c+d x))^{5/2}}{\sqrt[3]{\sec (c+d x)}} \, dx","Integrate[(a + b*Sec[c + d*x])^(5/2)/Sec[c + d*x]^(1/3),x]","\int \frac{(a+b \sec (c+d x))^{5/2}}{\sqrt[3]{\sec (c+d x)}} \, dx","\text{Int}\left(\frac{(a+b \sec (c+d x))^{5/2}}{\sqrt[3]{\sec (c+d x)}},x\right)",0,"Integrate[(a + b*Sec[c + d*x])^(5/2)/Sec[c + d*x]^(1/3), x]","A",-1
743,0,0,28,37.7071292,"\int \frac{(a+b \sec (c+d x))^{5/2}}{\sec ^{\frac{2}{3}}(c+d x)} \, dx","Integrate[(a + b*Sec[c + d*x])^(5/2)/Sec[c + d*x]^(2/3),x]","\int \frac{(a+b \sec (c+d x))^{5/2}}{\sec ^{\frac{2}{3}}(c+d x)} \, dx","\text{Int}\left(\frac{(a+b \sec (c+d x))^{5/2}}{\sec ^{\frac{2}{3}}(c+d x)},x\right)",0,"Integrate[(a + b*Sec[c + d*x])^(5/2)/Sec[c + d*x]^(2/3), x]","A",-1
744,0,0,28,43.8394284,"\int \frac{(a+b \sec (c+d x))^{5/2}}{\sec ^{\frac{4}{3}}(c+d x)} \, dx","Integrate[(a + b*Sec[c + d*x])^(5/2)/Sec[c + d*x]^(4/3),x]","\int \frac{(a+b \sec (c+d x))^{5/2}}{\sec ^{\frac{4}{3}}(c+d x)} \, dx","\text{Int}\left(\frac{(a+b \sec (c+d x))^{5/2}}{\sec ^{\frac{4}{3}}(c+d x)},x\right)",0,"Integrate[(a + b*Sec[c + d*x])^(5/2)/Sec[c + d*x]^(4/3), x]","A",-1
745,0,0,28,36.2048032,"\int \frac{(a+b \sec (c+d x))^{5/2}}{\sec ^{\frac{5}{3}}(c+d x)} \, dx","Integrate[(a + b*Sec[c + d*x])^(5/2)/Sec[c + d*x]^(5/3),x]","\int \frac{(a+b \sec (c+d x))^{5/2}}{\sec ^{\frac{5}{3}}(c+d x)} \, dx","\text{Int}\left(\frac{(a+b \sec (c+d x))^{5/2}}{\sec ^{\frac{5}{3}}(c+d x)},x\right)",0,"Integrate[(a + b*Sec[c + d*x])^(5/2)/Sec[c + d*x]^(5/3), x]","A",-1
746,0,0,28,42.9381417,"\int \frac{(a+b \sec (c+d x))^{5/2}}{\sec ^{\frac{7}{3}}(c+d x)} \, dx","Integrate[(a + b*Sec[c + d*x])^(5/2)/Sec[c + d*x]^(7/3),x]","\int \frac{(a+b \sec (c+d x))^{5/2}}{\sec ^{\frac{7}{3}}(c+d x)} \, dx","\text{Int}\left(\frac{(a+b \sec (c+d x))^{5/2}}{\sec ^{\frac{7}{3}}(c+d x)},x\right)",0,"Integrate[(a + b*Sec[c + d*x])^(5/2)/Sec[c + d*x]^(7/3), x]","A",-1
747,0,0,28,30.2843083,"\int \frac{\sec ^{\frac{7}{3}}(c+d x)}{\sqrt{a+b \sec (c+d x)}} \, dx","Integrate[Sec[c + d*x]^(7/3)/Sqrt[a + b*Sec[c + d*x]],x]","\int \frac{\sec ^{\frac{7}{3}}(c+d x)}{\sqrt{a+b \sec (c+d x)}} \, dx","\text{Int}\left(\frac{\sec ^{\frac{7}{3}}(c+d x)}{\sqrt{a+b \sec (c+d x)}},x\right)",0,"Integrate[Sec[c + d*x]^(7/3)/Sqrt[a + b*Sec[c + d*x]], x]","A",-1
748,0,0,28,36.1039902,"\int \frac{\sec ^{\frac{5}{3}}(c+d x)}{\sqrt{a+b \sec (c+d x)}} \, dx","Integrate[Sec[c + d*x]^(5/3)/Sqrt[a + b*Sec[c + d*x]],x]","\int \frac{\sec ^{\frac{5}{3}}(c+d x)}{\sqrt{a+b \sec (c+d x)}} \, dx","\text{Int}\left(\frac{\sec ^{\frac{5}{3}}(c+d x)}{\sqrt{a+b \sec (c+d x)}},x\right)",0,"Integrate[Sec[c + d*x]^(5/3)/Sqrt[a + b*Sec[c + d*x]], x]","A",-1
749,0,0,28,1.9481934,"\int \frac{\sec ^{\frac{4}{3}}(c+d x)}{\sqrt{a+b \sec (c+d x)}} \, dx","Integrate[Sec[c + d*x]^(4/3)/Sqrt[a + b*Sec[c + d*x]],x]","\int \frac{\sec ^{\frac{4}{3}}(c+d x)}{\sqrt{a+b \sec (c+d x)}} \, dx","\text{Int}\left(\frac{\sec ^{\frac{4}{3}}(c+d x)}{\sqrt{a+b \sec (c+d x)}},x\right)",0,"Integrate[Sec[c + d*x]^(4/3)/Sqrt[a + b*Sec[c + d*x]], x]","A",-1
750,0,0,28,2.2328684,"\int \frac{\sec ^{\frac{2}{3}}(c+d x)}{\sqrt{a+b \sec (c+d x)}} \, dx","Integrate[Sec[c + d*x]^(2/3)/Sqrt[a + b*Sec[c + d*x]],x]","\int \frac{\sec ^{\frac{2}{3}}(c+d x)}{\sqrt{a+b \sec (c+d x)}} \, dx","\text{Int}\left(\frac{\sec ^{\frac{2}{3}}(c+d x)}{\sqrt{a+b \sec (c+d x)}},x\right)",0,"Integrate[Sec[c + d*x]^(2/3)/Sqrt[a + b*Sec[c + d*x]], x]","A",-1
751,0,0,28,1.4709419,"\int \frac{\sqrt[3]{\sec (c+d x)}}{\sqrt{a+b \sec (c+d x)}} \, dx","Integrate[Sec[c + d*x]^(1/3)/Sqrt[a + b*Sec[c + d*x]],x]","\int \frac{\sqrt[3]{\sec (c+d x)}}{\sqrt{a+b \sec (c+d x)}} \, dx","\text{Int}\left(\frac{\sqrt[3]{\sec (c+d x)}}{\sqrt{a+b \sec (c+d x)}},x\right)",0,"Integrate[Sec[c + d*x]^(1/3)/Sqrt[a + b*Sec[c + d*x]], x]","A",-1
752,0,0,28,3.2014487,"\int \frac{1}{\sqrt[3]{\sec (c+d x)} \sqrt{a+b \sec (c+d x)}} \, dx","Integrate[1/(Sec[c + d*x]^(1/3)*Sqrt[a + b*Sec[c + d*x]]),x]","\int \frac{1}{\sqrt[3]{\sec (c+d x)} \sqrt{a+b \sec (c+d x)}} \, dx","\text{Int}\left(\frac{1}{\sqrt[3]{\sec (c+d x)} \sqrt{a+b \sec (c+d x)}},x\right)",0,"Integrate[1/(Sec[c + d*x]^(1/3)*Sqrt[a + b*Sec[c + d*x]]), x]","A",-1
753,0,0,28,26.0251427,"\int \frac{1}{\sec ^{\frac{2}{3}}(c+d x) \sqrt{a+b \sec (c+d x)}} \, dx","Integrate[1/(Sec[c + d*x]^(2/3)*Sqrt[a + b*Sec[c + d*x]]),x]","\int \frac{1}{\sec ^{\frac{2}{3}}(c+d x) \sqrt{a+b \sec (c+d x)}} \, dx","\text{Int}\left(\frac{1}{\sec ^{\frac{2}{3}}(c+d x) \sqrt{a+b \sec (c+d x)}},x\right)",0,"Integrate[1/(Sec[c + d*x]^(2/3)*Sqrt[a + b*Sec[c + d*x]]), x]","A",-1
754,0,0,28,31.2423171,"\int \frac{1}{\sec ^{\frac{4}{3}}(c+d x) \sqrt{a+b \sec (c+d x)}} \, dx","Integrate[1/(Sec[c + d*x]^(4/3)*Sqrt[a + b*Sec[c + d*x]]),x]","\int \frac{1}{\sec ^{\frac{4}{3}}(c+d x) \sqrt{a+b \sec (c+d x)}} \, dx","\text{Int}\left(\frac{1}{\sec ^{\frac{4}{3}}(c+d x) \sqrt{a+b \sec (c+d x)}},x\right)",0,"Integrate[1/(Sec[c + d*x]^(4/3)*Sqrt[a + b*Sec[c + d*x]]), x]","A",-1
755,0,0,28,34.4798004,"\int \frac{1}{\sec ^{\frac{5}{3}}(c+d x) \sqrt{a+b \sec (c+d x)}} \, dx","Integrate[1/(Sec[c + d*x]^(5/3)*Sqrt[a + b*Sec[c + d*x]]),x]","\int \frac{1}{\sec ^{\frac{5}{3}}(c+d x) \sqrt{a+b \sec (c+d x)}} \, dx","\text{Int}\left(\frac{1}{\sec ^{\frac{5}{3}}(c+d x) \sqrt{a+b \sec (c+d x)}},x\right)",0,"Integrate[1/(Sec[c + d*x]^(5/3)*Sqrt[a + b*Sec[c + d*x]]), x]","A",-1
756,0,0,28,41.7948465,"\int \frac{1}{\sec ^{\frac{7}{3}}(c+d x) \sqrt{a+b \sec (c+d x)}} \, dx","Integrate[1/(Sec[c + d*x]^(7/3)*Sqrt[a + b*Sec[c + d*x]]),x]","\int \frac{1}{\sec ^{\frac{7}{3}}(c+d x) \sqrt{a+b \sec (c+d x)}} \, dx","\text{Int}\left(\frac{1}{\sec ^{\frac{7}{3}}(c+d x) \sqrt{a+b \sec (c+d x)}},x\right)",0,"Integrate[1/(Sec[c + d*x]^(7/3)*Sqrt[a + b*Sec[c + d*x]]), x]","A",-1
757,0,0,28,34.9657865,"\int \frac{\sec ^{\frac{7}{3}}(c+d x)}{(a+b \sec (c+d x))^{3/2}} \, dx","Integrate[Sec[c + d*x]^(7/3)/(a + b*Sec[c + d*x])^(3/2),x]","\int \frac{\sec ^{\frac{7}{3}}(c+d x)}{(a+b \sec (c+d x))^{3/2}} \, dx","\text{Int}\left(\frac{\sec ^{\frac{7}{3}}(c+d x)}{(a+b \sec (c+d x))^{3/2}},x\right)",0,"Integrate[Sec[c + d*x]^(7/3)/(a + b*Sec[c + d*x])^(3/2), x]","A",-1
758,0,0,28,42.0358527,"\int \frac{\sec ^{\frac{5}{3}}(c+d x)}{(a+b \sec (c+d x))^{3/2}} \, dx","Integrate[Sec[c + d*x]^(5/3)/(a + b*Sec[c + d*x])^(3/2),x]","\int \frac{\sec ^{\frac{5}{3}}(c+d x)}{(a+b \sec (c+d x))^{3/2}} \, dx","\text{Int}\left(\frac{\sec ^{\frac{5}{3}}(c+d x)}{(a+b \sec (c+d x))^{3/2}},x\right)",0,"Integrate[Sec[c + d*x]^(5/3)/(a + b*Sec[c + d*x])^(3/2), x]","A",-1
759,0,0,28,36.2166119,"\int \frac{\sec ^{\frac{4}{3}}(c+d x)}{(a+b \sec (c+d x))^{3/2}} \, dx","Integrate[Sec[c + d*x]^(4/3)/(a + b*Sec[c + d*x])^(3/2),x]","\int \frac{\sec ^{\frac{4}{3}}(c+d x)}{(a+b \sec (c+d x))^{3/2}} \, dx","\text{Int}\left(\frac{\sec ^{\frac{4}{3}}(c+d x)}{(a+b \sec (c+d x))^{3/2}},x\right)",0,"Integrate[Sec[c + d*x]^(4/3)/(a + b*Sec[c + d*x])^(3/2), x]","A",-1
760,0,0,28,45.1103962,"\int \frac{\sec ^{\frac{2}{3}}(c+d x)}{(a+b \sec (c+d x))^{3/2}} \, dx","Integrate[Sec[c + d*x]^(2/3)/(a + b*Sec[c + d*x])^(3/2),x]","\int \frac{\sec ^{\frac{2}{3}}(c+d x)}{(a+b \sec (c+d x))^{3/2}} \, dx","\text{Int}\left(\frac{\sec ^{\frac{2}{3}}(c+d x)}{(a+b \sec (c+d x))^{3/2}},x\right)",0,"Integrate[Sec[c + d*x]^(2/3)/(a + b*Sec[c + d*x])^(3/2), x]","A",-1
761,0,0,28,37.4994398,"\int \frac{\sqrt[3]{\sec (c+d x)}}{(a+b \sec (c+d x))^{3/2}} \, dx","Integrate[Sec[c + d*x]^(1/3)/(a + b*Sec[c + d*x])^(3/2),x]","\int \frac{\sqrt[3]{\sec (c+d x)}}{(a+b \sec (c+d x))^{3/2}} \, dx","\text{Int}\left(\frac{\sqrt[3]{\sec (c+d x)}}{(a+b \sec (c+d x))^{3/2}},x\right)",0,"Integrate[Sec[c + d*x]^(1/3)/(a + b*Sec[c + d*x])^(3/2), x]","A",-1
762,0,0,28,44.292703,"\int \frac{1}{\sqrt[3]{\sec (c+d x)} (a+b \sec (c+d x))^{3/2}} \, dx","Integrate[1/(Sec[c + d*x]^(1/3)*(a + b*Sec[c + d*x])^(3/2)),x]","\int \frac{1}{\sqrt[3]{\sec (c+d x)} (a+b \sec (c+d x))^{3/2}} \, dx","\text{Int}\left(\frac{1}{\sqrt[3]{\sec (c+d x)} (a+b \sec (c+d x))^{3/2}},x\right)",0,"Integrate[1/(Sec[c + d*x]^(1/3)*(a + b*Sec[c + d*x])^(3/2)), x]","A",-1
763,0,0,28,42.3112097,"\int \frac{1}{\sec ^{\frac{2}{3}}(c+d x) (a+b \sec (c+d x))^{3/2}} \, dx","Integrate[1/(Sec[c + d*x]^(2/3)*(a + b*Sec[c + d*x])^(3/2)),x]","\int \frac{1}{\sec ^{\frac{2}{3}}(c+d x) (a+b \sec (c+d x))^{3/2}} \, dx","\text{Int}\left(\frac{1}{\sec ^{\frac{2}{3}}(c+d x) (a+b \sec (c+d x))^{3/2}},x\right)",0,"Integrate[1/(Sec[c + d*x]^(2/3)*(a + b*Sec[c + d*x])^(3/2)), x]","A",-1
764,0,0,28,49.1005778,"\int \frac{1}{\sec ^{\frac{4}{3}}(c+d x) (a+b \sec (c+d x))^{3/2}} \, dx","Integrate[1/(Sec[c + d*x]^(4/3)*(a + b*Sec[c + d*x])^(3/2)),x]","\int \frac{1}{\sec ^{\frac{4}{3}}(c+d x) (a+b \sec (c+d x))^{3/2}} \, dx","\text{Int}\left(\frac{1}{\sec ^{\frac{4}{3}}(c+d x) (a+b \sec (c+d x))^{3/2}},x\right)",0,"Integrate[1/(Sec[c + d*x]^(4/3)*(a + b*Sec[c + d*x])^(3/2)), x]","A",-1
765,0,0,28,18.8334149,"\int \frac{1}{\sec ^{\frac{5}{3}}(c+d x) (a+b \sec (c+d x))^{3/2}} \, dx","Integrate[1/(Sec[c + d*x]^(5/3)*(a + b*Sec[c + d*x])^(3/2)),x]","\int \frac{1}{\sec ^{\frac{5}{3}}(c+d x) (a+b \sec (c+d x))^{3/2}} \, dx","\text{Int}\left(\frac{1}{\sec ^{\frac{5}{3}}(c+d x) (a+b \sec (c+d x))^{3/2}},x\right)",0,"Integrate[1/(Sec[c + d*x]^(5/3)*(a + b*Sec[c + d*x])^(3/2)), x]","A",-1
766,0,0,28,51.4930002,"\int \frac{1}{\sec ^{\frac{7}{3}}(c+d x) (a+b \sec (c+d x))^{3/2}} \, dx","Integrate[1/(Sec[c + d*x]^(7/3)*(a + b*Sec[c + d*x])^(3/2)),x]","\int \frac{1}{\sec ^{\frac{7}{3}}(c+d x) (a+b \sec (c+d x))^{3/2}} \, dx","\text{Int}\left(\frac{1}{\sec ^{\frac{7}{3}}(c+d x) (a+b \sec (c+d x))^{3/2}},x\right)",0,"Integrate[1/(Sec[c + d*x]^(7/3)*(a + b*Sec[c + d*x])^(3/2)), x]","A",-1
767,0,0,28,41.3053152,"\int \frac{\sec ^{\frac{7}{3}}(c+d x)}{(a+b \sec (c+d x))^{5/2}} \, dx","Integrate[Sec[c + d*x]^(7/3)/(a + b*Sec[c + d*x])^(5/2),x]","\int \frac{\sec ^{\frac{7}{3}}(c+d x)}{(a+b \sec (c+d x))^{5/2}} \, dx","\text{Int}\left(\frac{\sec ^{\frac{7}{3}}(c+d x)}{(a+b \sec (c+d x))^{5/2}},x\right)",0,"Integrate[Sec[c + d*x]^(7/3)/(a + b*Sec[c + d*x])^(5/2), x]","A",-1
768,0,0,28,49.2578025,"\int \frac{\sec ^{\frac{5}{3}}(c+d x)}{(a+b \sec (c+d x))^{5/2}} \, dx","Integrate[Sec[c + d*x]^(5/3)/(a + b*Sec[c + d*x])^(5/2),x]","\int \frac{\sec ^{\frac{5}{3}}(c+d x)}{(a+b \sec (c+d x))^{5/2}} \, dx","\text{Int}\left(\frac{\sec ^{\frac{5}{3}}(c+d x)}{(a+b \sec (c+d x))^{5/2}},x\right)",0,"Integrate[Sec[c + d*x]^(5/3)/(a + b*Sec[c + d*x])^(5/2), x]","A",-1
769,0,0,28,40.6947997,"\int \frac{\sec ^{\frac{4}{3}}(c+d x)}{(a+b \sec (c+d x))^{5/2}} \, dx","Integrate[Sec[c + d*x]^(4/3)/(a + b*Sec[c + d*x])^(5/2),x]","\int \frac{\sec ^{\frac{4}{3}}(c+d x)}{(a+b \sec (c+d x))^{5/2}} \, dx","\text{Int}\left(\frac{\sec ^{\frac{4}{3}}(c+d x)}{(a+b \sec (c+d x))^{5/2}},x\right)",0,"Integrate[Sec[c + d*x]^(4/3)/(a + b*Sec[c + d*x])^(5/2), x]","A",-1
770,0,0,28,46.5021207,"\int \frac{\sec ^{\frac{2}{3}}(c+d x)}{(a+b \sec (c+d x))^{5/2}} \, dx","Integrate[Sec[c + d*x]^(2/3)/(a + b*Sec[c + d*x])^(5/2),x]","\int \frac{\sec ^{\frac{2}{3}}(c+d x)}{(a+b \sec (c+d x))^{5/2}} \, dx","\text{Int}\left(\frac{\sec ^{\frac{2}{3}}(c+d x)}{(a+b \sec (c+d x))^{5/2}},x\right)",0,"Integrate[Sec[c + d*x]^(2/3)/(a + b*Sec[c + d*x])^(5/2), x]","A",-1
771,0,0,28,44.4545994,"\int \frac{\sqrt[3]{\sec (c+d x)}}{(a+b \sec (c+d x))^{5/2}} \, dx","Integrate[Sec[c + d*x]^(1/3)/(a + b*Sec[c + d*x])^(5/2),x]","\int \frac{\sqrt[3]{\sec (c+d x)}}{(a+b \sec (c+d x))^{5/2}} \, dx","\text{Int}\left(\frac{\sqrt[3]{\sec (c+d x)}}{(a+b \sec (c+d x))^{5/2}},x\right)",0,"Integrate[Sec[c + d*x]^(1/3)/(a + b*Sec[c + d*x])^(5/2), x]","A",-1
772,0,0,28,51.2965167,"\int \frac{1}{\sqrt[3]{\sec (c+d x)} (a+b \sec (c+d x))^{5/2}} \, dx","Integrate[1/(Sec[c + d*x]^(1/3)*(a + b*Sec[c + d*x])^(5/2)),x]","\int \frac{1}{\sqrt[3]{\sec (c+d x)} (a+b \sec (c+d x))^{5/2}} \, dx","\text{Int}\left(\frac{1}{\sqrt[3]{\sec (c+d x)} (a+b \sec (c+d x))^{5/2}},x\right)",0,"Integrate[1/(Sec[c + d*x]^(1/3)*(a + b*Sec[c + d*x])^(5/2)), x]","A",-1
773,0,0,28,49.969653,"\int \frac{1}{\sec ^{\frac{2}{3}}(c+d x) (a+b \sec (c+d x))^{5/2}} \, dx","Integrate[1/(Sec[c + d*x]^(2/3)*(a + b*Sec[c + d*x])^(5/2)),x]","\int \frac{1}{\sec ^{\frac{2}{3}}(c+d x) (a+b \sec (c+d x))^{5/2}} \, dx","\text{Int}\left(\frac{1}{\sec ^{\frac{2}{3}}(c+d x) (a+b \sec (c+d x))^{5/2}},x\right)",0,"Integrate[1/(Sec[c + d*x]^(2/3)*(a + b*Sec[c + d*x])^(5/2)), x]","A",-1
774,0,0,28,58.9129735,"\int \frac{1}{\sec ^{\frac{4}{3}}(c+d x) (a+b \sec (c+d x))^{5/2}} \, dx","Integrate[1/(Sec[c + d*x]^(4/3)*(a + b*Sec[c + d*x])^(5/2)),x]","\int \frac{1}{\sec ^{\frac{4}{3}}(c+d x) (a+b \sec (c+d x))^{5/2}} \, dx","\text{Int}\left(\frac{1}{\sec ^{\frac{4}{3}}(c+d x) (a+b \sec (c+d x))^{5/2}},x\right)",0,"Integrate[1/(Sec[c + d*x]^(4/3)*(a + b*Sec[c + d*x])^(5/2)), x]","A",-1
775,0,0,28,23.2759495,"\int \frac{1}{\sec ^{\frac{5}{3}}(c+d x) (a+b \sec (c+d x))^{5/2}} \, dx","Integrate[1/(Sec[c + d*x]^(5/3)*(a + b*Sec[c + d*x])^(5/2)),x]","\int \frac{1}{\sec ^{\frac{5}{3}}(c+d x) (a+b \sec (c+d x))^{5/2}} \, dx","\text{Int}\left(\frac{1}{\sec ^{\frac{5}{3}}(c+d x) (a+b \sec (c+d x))^{5/2}},x\right)",0,"Integrate[1/(Sec[c + d*x]^(5/3)*(a + b*Sec[c + d*x])^(5/2)), x]","A",-1
776,0,0,28,65.9611147,"\int \frac{1}{\sec ^{\frac{7}{3}}(c+d x) (a+b \sec (c+d x))^{5/2}} \, dx","Integrate[1/(Sec[c + d*x]^(7/3)*(a + b*Sec[c + d*x])^(5/2)),x]","\int \frac{1}{\sec ^{\frac{7}{3}}(c+d x) (a+b \sec (c+d x))^{5/2}} \, dx","\text{Int}\left(\frac{1}{\sec ^{\frac{7}{3}}(c+d x) (a+b \sec (c+d x))^{5/2}},x\right)",0,"Integrate[1/(Sec[c + d*x]^(7/3)*(a + b*Sec[c + d*x])^(5/2)), x]","A",-1
777,1,231,251,0.9669502,"\int (d \sec (e+f x))^n (a+b \sec (e+f x))^3 \, dx","Integrate[(d*Sec[e + f*x])^n*(a + b*Sec[e + f*x])^3,x]","-\frac{\left(-\tan ^2(e+f x)\right)^{3/2} \csc ^3(e+f x) (d \sec (e+f x))^n \left(a^3 \left(n^3+6 n^2+11 n+6\right) \cos ^3(e+f x) \, _2F_1\left(\frac{1}{2},\frac{n}{2};\frac{n+2}{2};\sec ^2(e+f x)\right)+b n \left(3 a^2 \left(n^2+5 n+6\right) \cos ^2(e+f x) \, _2F_1\left(\frac{1}{2},\frac{n+1}{2};\frac{n+3}{2};\sec ^2(e+f x)\right)+b (n+1) \left(3 a (n+3) \cos (e+f x) \, _2F_1\left(\frac{1}{2},\frac{n+2}{2};\frac{n+4}{2};\sec ^2(e+f x)\right)+b (n+2) \, _2F_1\left(\frac{1}{2},\frac{n+3}{2};\frac{n+5}{2};\sec ^2(e+f x)\right)\right)\right)\right)}{f n (n+1) (n+2) (n+3)}","-\frac{a d \left(a^2 (n+1)+3 b^2 n\right) \sin (e+f x) (d \sec (e+f x))^{n-1} \, _2F_1\left(\frac{1}{2},\frac{1-n}{2};\frac{3-n}{2};\cos ^2(e+f x)\right)}{f \left(1-n^2\right) \sqrt{\sin ^2(e+f x)}}+\frac{b \left(3 a^2 (n+2)+b^2 (n+1)\right) \sin (e+f x) (d \sec (e+f x))^n \, _2F_1\left(\frac{1}{2},-\frac{n}{2};\frac{2-n}{2};\cos ^2(e+f x)\right)}{f n (n+2) \sqrt{\sin ^2(e+f x)}}+\frac{a b^2 (2 n+5) \tan (e+f x) (d \sec (e+f x))^n}{f (n+1) (n+2)}+\frac{b^2 \tan (e+f x) (a+b \sec (e+f x)) (d \sec (e+f x))^n}{f (n+2)}",1,"-((Csc[e + f*x]^3*(a^3*(6 + 11*n + 6*n^2 + n^3)*Cos[e + f*x]^3*Hypergeometric2F1[1/2, n/2, (2 + n)/2, Sec[e + f*x]^2] + b*n*(3*a^2*(6 + 5*n + n^2)*Cos[e + f*x]^2*Hypergeometric2F1[1/2, (1 + n)/2, (3 + n)/2, Sec[e + f*x]^2] + b*(1 + n)*(3*a*(3 + n)*Cos[e + f*x]*Hypergeometric2F1[1/2, (2 + n)/2, (4 + n)/2, Sec[e + f*x]^2] + b*(2 + n)*Hypergeometric2F1[1/2, (3 + n)/2, (5 + n)/2, Sec[e + f*x]^2])))*(d*Sec[e + f*x])^n*(-Tan[e + f*x]^2)^(3/2))/(f*n*(1 + n)*(2 + n)*(3 + n)))","A",1
778,1,171,181,0.3610194,"\int (d \sec (e+f x))^n (a+b \sec (e+f x))^2 \, dx","Integrate[(d*Sec[e + f*x])^n*(a + b*Sec[e + f*x])^2,x]","\frac{\sqrt{-\tan ^2(e+f x)} \csc (e+f x) \sec (e+f x) (d \sec (e+f x))^n \left(a^2 \left(n^2+3 n+2\right) \cos ^2(e+f x) \, _2F_1\left(\frac{1}{2},\frac{n}{2};\frac{n+2}{2};\sec ^2(e+f x)\right)+b n \left(2 a (n+2) \cos (e+f x) \, _2F_1\left(\frac{1}{2},\frac{n+1}{2};\frac{n+3}{2};\sec ^2(e+f x)\right)+b (n+1) \, _2F_1\left(\frac{1}{2},\frac{n+2}{2};\frac{n+4}{2};\sec ^2(e+f x)\right)\right)\right)}{f n (n+1) (n+2)}","-\frac{d \left(a^2 (n+1)+b^2 n\right) \sin (e+f x) (d \sec (e+f x))^{n-1} \, _2F_1\left(\frac{1}{2},\frac{1-n}{2};\frac{3-n}{2};\cos ^2(e+f x)\right)}{f \left(1-n^2\right) \sqrt{\sin ^2(e+f x)}}+\frac{2 a b \sin (e+f x) (d \sec (e+f x))^n \, _2F_1\left(\frac{1}{2},-\frac{n}{2};\frac{2-n}{2};\cos ^2(e+f x)\right)}{f n \sqrt{\sin ^2(e+f x)}}+\frac{b^2 \tan (e+f x) (d \sec (e+f x))^n}{f (n+1)}",1,"(Csc[e + f*x]*(a^2*(2 + 3*n + n^2)*Cos[e + f*x]^2*Hypergeometric2F1[1/2, n/2, (2 + n)/2, Sec[e + f*x]^2] + b*n*(2*a*(2 + n)*Cos[e + f*x]*Hypergeometric2F1[1/2, (1 + n)/2, (3 + n)/2, Sec[e + f*x]^2] + b*(1 + n)*Hypergeometric2F1[1/2, (2 + n)/2, (4 + n)/2, Sec[e + f*x]^2]))*Sec[e + f*x]*(d*Sec[e + f*x])^n*Sqrt[-Tan[e + f*x]^2])/(f*n*(1 + n)*(2 + n))","A",1
779,1,107,137,0.1725547,"\int (d \sec (e+f x))^n (a+b \sec (e+f x)) \, dx","Integrate[(d*Sec[e + f*x])^n*(a + b*Sec[e + f*x]),x]","\frac{\sqrt{-\tan ^2(e+f x)} \csc (e+f x) (d \sec (e+f x))^n \left(a (n+1) \cos (e+f x) \, _2F_1\left(\frac{1}{2},\frac{n}{2};\frac{n+2}{2};\sec ^2(e+f x)\right)+b n \, _2F_1\left(\frac{1}{2},\frac{n+1}{2};\frac{n+3}{2};\sec ^2(e+f x)\right)\right)}{f n (n+1)}","\frac{b \sin (e+f x) (d \sec (e+f x))^n \, _2F_1\left(\frac{1}{2},-\frac{n}{2};\frac{2-n}{2};\cos ^2(e+f x)\right)}{f n \sqrt{\sin ^2(e+f x)}}-\frac{a d \sin (e+f x) (d \sec (e+f x))^{n-1} \, _2F_1\left(\frac{1}{2},\frac{1-n}{2};\frac{3-n}{2};\cos ^2(e+f x)\right)}{f (1-n) \sqrt{\sin ^2(e+f x)}}",1,"(Csc[e + f*x]*(a*(1 + n)*Cos[e + f*x]*Hypergeometric2F1[1/2, n/2, (2 + n)/2, Sec[e + f*x]^2] + b*n*Hypergeometric2F1[1/2, (1 + n)/2, (3 + n)/2, Sec[e + f*x]^2])*(d*Sec[e + f*x])^n*Sqrt[-Tan[e + f*x]^2])/(f*n*(1 + n))","A",1
780,1,5280,192,25.8451895,"\int \frac{(d \sec (e+f x))^n}{a+b \sec (e+f x)} \, dx","Integrate[(d*Sec[e + f*x])^n/(a + b*Sec[e + f*x]),x]","\text{Result too large to show}","\frac{a \sin (e+f x) \cos (e+f x) \cos ^2(e+f x)^{\frac{n-1}{2}} (d \sec (e+f x))^n F_1\left(\frac{1}{2};\frac{n-1}{2},1;\frac{3}{2};\sin ^2(e+f x),\frac{a^2 \sin ^2(e+f x)}{a^2-b^2}\right)}{f \left(a^2-b^2\right)}-\frac{b \sin (e+f x) \cos ^2(e+f x)^{n/2} (d \sec (e+f x))^n F_1\left(\frac{1}{2};\frac{n}{2},1;\frac{3}{2};\sin ^2(e+f x),\frac{a^2 \sin ^2(e+f x)}{a^2-b^2}\right)}{f \left(a^2-b^2\right)}",1,"Result too large to show","B",0
781,1,13940,299,46.5759871,"\int \frac{(d \sec (e+f x))^n}{(a+b \sec (e+f x))^2} \, dx","Integrate[(d*Sec[e + f*x])^n/(a + b*Sec[e + f*x])^2,x]","\text{Result too large to show}","\frac{a^2 \sin (e+f x) \cos (e+f x) \cos ^2(e+f x)^{\frac{n-1}{2}} (d \sec (e+f x))^n F_1\left(\frac{1}{2};\frac{n-3}{2},2;\frac{3}{2};\sin ^2(e+f x),\frac{a^2 \sin ^2(e+f x)}{a^2-b^2}\right)}{f \left(a^2-b^2\right)^2}+\frac{b^2 \sin (e+f x) \cos (e+f x) \cos ^2(e+f x)^{\frac{n-1}{2}} (d \sec (e+f x))^n F_1\left(\frac{1}{2};\frac{n-1}{2},2;\frac{3}{2};\sin ^2(e+f x),\frac{a^2 \sin ^2(e+f x)}{a^2-b^2}\right)}{f \left(a^2-b^2\right)^2}-\frac{2 a b \sin (e+f x) \cos ^2(e+f x)^{n/2} (d \sec (e+f x))^n F_1\left(\frac{1}{2};\frac{n-2}{2},2;\frac{3}{2};\sin ^2(e+f x),\frac{a^2 \sin ^2(e+f x)}{a^2-b^2}\right)}{f \left(a^2-b^2\right)^2}",1,"Result too large to show","B",0
782,0,0,28,16.8204686,"\int (d \sec (e+f x))^n (a+b \sec (e+f x))^{3/2} \, dx","Integrate[(d*Sec[e + f*x])^n*(a + b*Sec[e + f*x])^(3/2),x]","\int (d \sec (e+f x))^n (a+b \sec (e+f x))^{3/2} \, dx","\text{Int}\left((a+b \sec (e+f x))^{3/2} (d \sec (e+f x))^n,x\right)",0,"Integrate[(d*Sec[e + f*x])^n*(a + b*Sec[e + f*x])^(3/2), x]","A",-1
783,0,0,28,0.5359918,"\int (d \sec (e+f x))^n \sqrt{a+b \sec (e+f x)} \, dx","Integrate[(d*Sec[e + f*x])^n*Sqrt[a + b*Sec[e + f*x]],x]","\int (d \sec (e+f x))^n \sqrt{a+b \sec (e+f x)} \, dx","\text{Int}\left(\sqrt{a+b \sec (e+f x)} (d \sec (e+f x))^n,x\right)",0,"Integrate[(d*Sec[e + f*x])^n*Sqrt[a + b*Sec[e + f*x]], x]","A",-1
784,0,0,28,3.0078922,"\int \frac{(d \sec (e+f x))^n}{\sqrt{a+b \sec (e+f x)}} \, dx","Integrate[(d*Sec[e + f*x])^n/Sqrt[a + b*Sec[e + f*x]],x]","\int \frac{(d \sec (e+f x))^n}{\sqrt{a+b \sec (e+f x)}} \, dx","\text{Int}\left(\frac{(d \sec (e+f x))^n}{\sqrt{a+b \sec (e+f x)}},x\right)",0,"Integrate[(d*Sec[e + f*x])^n/Sqrt[a + b*Sec[e + f*x]], x]","A",-1
785,0,0,28,2.7412188,"\int \frac{(d \sec (e+f x))^n}{(a+b \sec (e+f x))^{3/2}} \, dx","Integrate[(d*Sec[e + f*x])^n/(a + b*Sec[e + f*x])^(3/2),x]","\int \frac{(d \sec (e+f x))^n}{(a+b \sec (e+f x))^{3/2}} \, dx","\text{Int}\left(\frac{(d \sec (e+f x))^n}{(a+b \sec (e+f x))^{3/2}},x\right)",0,"Integrate[(d*Sec[e + f*x])^n/(a + b*Sec[e + f*x])^(3/2), x]","A",-1
786,0,0,24,2.695591,"\int \sec ^n(e+f x) (a+b \sec (e+f x))^m \, dx","Integrate[Sec[e + f*x]^n*(a + b*Sec[e + f*x])^m,x]","\int \sec ^n(e+f x) (a+b \sec (e+f x))^m \, dx","\text{Int}\left(\sec ^n(e+f x) (a+b \sec (e+f x))^m,x\right)",0,"Integrate[Sec[e + f*x]^n*(a + b*Sec[e + f*x])^m, x]","A",-1
787,0,0,26,0.6434253,"\int (d \sec (e+f x))^n (a+b \sec (e+f x))^m \, dx","Integrate[(d*Sec[e + f*x])^n*(a + b*Sec[e + f*x])^m,x]","\int (d \sec (e+f x))^n (a+b \sec (e+f x))^m \, dx","\text{Int}\left((d \sec (e+f x))^n (a+b \sec (e+f x))^m,x\right)",0,"Integrate[(d*Sec[e + f*x])^n*(a + b*Sec[e + f*x])^m, x]","A",-1
788,1,8899,273,26.7747895,"\int \sec ^3(e+f x) (a+b \sec (e+f x))^m \, dx","Integrate[Sec[e + f*x]^3*(a + b*Sec[e + f*x])^m,x]","\text{Result too large to show}","\frac{\sqrt{2} \left(a^2+b^2 (m+1)\right) \tan (e+f x) (a+b \sec (e+f x))^m \left(\frac{a+b \sec (e+f x)}{a+b}\right)^{-m} F_1\left(\frac{1}{2};\frac{1}{2},-m;\frac{3}{2};\frac{1}{2} (1-\sec (e+f x)),\frac{b (1-\sec (e+f x))}{a+b}\right)}{b^2 f (m+2) \sqrt{\sec (e+f x)+1}}-\frac{\sqrt{2} a (a+b) \tan (e+f x) (a+b \sec (e+f x))^m \left(\frac{a+b \sec (e+f x)}{a+b}\right)^{-m} F_1\left(\frac{1}{2};\frac{1}{2},-m-1;\frac{3}{2};\frac{1}{2} (1-\sec (e+f x)),\frac{b (1-\sec (e+f x))}{a+b}\right)}{b^2 f (m+2) \sqrt{\sec (e+f x)+1}}+\frac{\tan (e+f x) (a+b \sec (e+f x))^{m+1}}{b f (m+2)}",1,"Result too large to show","B",0
789,1,5564,220,23.1638242,"\int \sec ^2(e+f x) (a+b \sec (e+f x))^m \, dx","Integrate[Sec[e + f*x]^2*(a + b*Sec[e + f*x])^m,x]","\text{Result too large to show}","\frac{\sqrt{2} (a+b) \tan (e+f x) (a+b \sec (e+f x))^m \left(\frac{a+b \sec (e+f x)}{a+b}\right)^{-m} F_1\left(\frac{1}{2};\frac{1}{2},-m-1;\frac{3}{2};\frac{1}{2} (1-\sec (e+f x)),\frac{b (1-\sec (e+f x))}{a+b}\right)}{b f \sqrt{\sec (e+f x)+1}}-\frac{\sqrt{2} a \tan (e+f x) (a+b \sec (e+f x))^m \left(\frac{a+b \sec (e+f x)}{a+b}\right)^{-m} F_1\left(\frac{1}{2};\frac{1}{2},-m;\frac{3}{2};\frac{1}{2} (1-\sec (e+f x)),\frac{b (1-\sec (e+f x))}{a+b}\right)}{b f \sqrt{\sec (e+f x)+1}}",1,"Result too large to show","B",0
790,1,2828,103,14.8966442,"\int \sec (e+f x) (a+b \sec (e+f x))^m \, dx","Integrate[Sec[e + f*x]*(a + b*Sec[e + f*x])^m,x]","\text{Result too large to show}","\frac{\sqrt{2} \tan (e+f x) (a+b \sec (e+f x))^m \left(\frac{a+b \sec (e+f x)}{a+b}\right)^{-m} F_1\left(\frac{1}{2};\frac{1}{2},-m;\frac{3}{2};\frac{1}{2} (1-\sec (e+f x)),\frac{b (1-\sec (e+f x))}{a+b}\right)}{f \sqrt{\sec (e+f x)+1}}",1,"(-6*(a + b)*AppellF1[1/2, 1 + m, -m, 3/2, Tan[(e + f*x)/2]^2, ((a - b)*Tan[(e + f*x)/2]^2)/(a + b)]*(b + a*Cos[e + f*x])^m*Sec[e + f*x]^(1 + m)*(a + b*Sec[e + f*x])^m*Tan[(e + f*x)/2])/(f*(-1 + Tan[(e + f*x)/2]^2)*(3*(a + b)*AppellF1[1/2, 1 + m, -m, 3/2, Tan[(e + f*x)/2]^2, ((a - b)*Tan[(e + f*x)/2]^2)/(a + b)] + 2*(-((a - b)*m*AppellF1[3/2, 1 + m, 1 - m, 5/2, Tan[(e + f*x)/2]^2, ((a - b)*Tan[(e + f*x)/2]^2)/(a + b)]) + (a + b)*(1 + m)*AppellF1[3/2, 2 + m, -m, 5/2, Tan[(e + f*x)/2]^2, ((a - b)*Tan[(e + f*x)/2]^2)/(a + b)])*Tan[(e + f*x)/2]^2)*((6*(a + b)*AppellF1[1/2, 1 + m, -m, 3/2, Tan[(e + f*x)/2]^2, ((a - b)*Tan[(e + f*x)/2]^2)/(a + b)]*(b + a*Cos[e + f*x])^m*Sec[(e + f*x)/2]^2*Sec[e + f*x]^m*Tan[(e + f*x)/2]^2)/((-1 + Tan[(e + f*x)/2]^2)^2*(3*(a + b)*AppellF1[1/2, 1 + m, -m, 3/2, Tan[(e + f*x)/2]^2, ((a - b)*Tan[(e + f*x)/2]^2)/(a + b)] + 2*(-((a - b)*m*AppellF1[3/2, 1 + m, 1 - m, 5/2, Tan[(e + f*x)/2]^2, ((a - b)*Tan[(e + f*x)/2]^2)/(a + b)]) + (a + b)*(1 + m)*AppellF1[3/2, 2 + m, -m, 5/2, Tan[(e + f*x)/2]^2, ((a - b)*Tan[(e + f*x)/2]^2)/(a + b)])*Tan[(e + f*x)/2]^2)) - (3*(a + b)*AppellF1[1/2, 1 + m, -m, 3/2, Tan[(e + f*x)/2]^2, ((a - b)*Tan[(e + f*x)/2]^2)/(a + b)]*(b + a*Cos[e + f*x])^m*Sec[(e + f*x)/2]^2*Sec[e + f*x]^m)/((-1 + Tan[(e + f*x)/2]^2)*(3*(a + b)*AppellF1[1/2, 1 + m, -m, 3/2, Tan[(e + f*x)/2]^2, ((a - b)*Tan[(e + f*x)/2]^2)/(a + b)] + 2*(-((a - b)*m*AppellF1[3/2, 1 + m, 1 - m, 5/2, Tan[(e + f*x)/2]^2, ((a - b)*Tan[(e + f*x)/2]^2)/(a + b)]) + (a + b)*(1 + m)*AppellF1[3/2, 2 + m, -m, 5/2, Tan[(e + f*x)/2]^2, ((a - b)*Tan[(e + f*x)/2]^2)/(a + b)])*Tan[(e + f*x)/2]^2)) + (6*a*(a + b)*m*AppellF1[1/2, 1 + m, -m, 3/2, Tan[(e + f*x)/2]^2, ((a - b)*Tan[(e + f*x)/2]^2)/(a + b)]*(b + a*Cos[e + f*x])^(-1 + m)*Sec[e + f*x]^m*Sin[e + f*x]*Tan[(e + f*x)/2])/((-1 + Tan[(e + f*x)/2]^2)*(3*(a + b)*AppellF1[1/2, 1 + m, -m, 3/2, Tan[(e + f*x)/2]^2, ((a - b)*Tan[(e + f*x)/2]^2)/(a + b)] + 2*(-((a - b)*m*AppellF1[3/2, 1 + m, 1 - m, 5/2, Tan[(e + f*x)/2]^2, ((a - b)*Tan[(e + f*x)/2]^2)/(a + b)]) + (a + b)*(1 + m)*AppellF1[3/2, 2 + m, -m, 5/2, Tan[(e + f*x)/2]^2, ((a - b)*Tan[(e + f*x)/2]^2)/(a + b)])*Tan[(e + f*x)/2]^2)) - (6*(a + b)*m*AppellF1[1/2, 1 + m, -m, 3/2, Tan[(e + f*x)/2]^2, ((a - b)*Tan[(e + f*x)/2]^2)/(a + b)]*(b + a*Cos[e + f*x])^m*Sec[e + f*x]^(1 + m)*Sin[e + f*x]*Tan[(e + f*x)/2])/((-1 + Tan[(e + f*x)/2]^2)*(3*(a + b)*AppellF1[1/2, 1 + m, -m, 3/2, Tan[(e + f*x)/2]^2, ((a - b)*Tan[(e + f*x)/2]^2)/(a + b)] + 2*(-((a - b)*m*AppellF1[3/2, 1 + m, 1 - m, 5/2, Tan[(e + f*x)/2]^2, ((a - b)*Tan[(e + f*x)/2]^2)/(a + b)]) + (a + b)*(1 + m)*AppellF1[3/2, 2 + m, -m, 5/2, Tan[(e + f*x)/2]^2, ((a - b)*Tan[(e + f*x)/2]^2)/(a + b)])*Tan[(e + f*x)/2]^2)) - (6*(a + b)*(b + a*Cos[e + f*x])^m*Sec[e + f*x]^m*Tan[(e + f*x)/2]*(-1/3*((a - b)*m*AppellF1[3/2, 1 + m, 1 - m, 5/2, Tan[(e + f*x)/2]^2, ((a - b)*Tan[(e + f*x)/2]^2)/(a + b)]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/(a + b) + ((1 + m)*AppellF1[3/2, 2 + m, -m, 5/2, Tan[(e + f*x)/2]^2, ((a - b)*Tan[(e + f*x)/2]^2)/(a + b)]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/3))/((-1 + Tan[(e + f*x)/2]^2)*(3*(a + b)*AppellF1[1/2, 1 + m, -m, 3/2, Tan[(e + f*x)/2]^2, ((a - b)*Tan[(e + f*x)/2]^2)/(a + b)] + 2*(-((a - b)*m*AppellF1[3/2, 1 + m, 1 - m, 5/2, Tan[(e + f*x)/2]^2, ((a - b)*Tan[(e + f*x)/2]^2)/(a + b)]) + (a + b)*(1 + m)*AppellF1[3/2, 2 + m, -m, 5/2, Tan[(e + f*x)/2]^2, ((a - b)*Tan[(e + f*x)/2]^2)/(a + b)])*Tan[(e + f*x)/2]^2)) + (6*(a + b)*AppellF1[1/2, 1 + m, -m, 3/2, Tan[(e + f*x)/2]^2, ((a - b)*Tan[(e + f*x)/2]^2)/(a + b)]*(b + a*Cos[e + f*x])^m*Sec[e + f*x]^m*Tan[(e + f*x)/2]*(2*(-((a - b)*m*AppellF1[3/2, 1 + m, 1 - m, 5/2, Tan[(e + f*x)/2]^2, ((a - b)*Tan[(e + f*x)/2]^2)/(a + b)]) + (a + b)*(1 + m)*AppellF1[3/2, 2 + m, -m, 5/2, Tan[(e + f*x)/2]^2, ((a - b)*Tan[(e + f*x)/2]^2)/(a + b)])*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2] + 3*(a + b)*(-1/3*((a - b)*m*AppellF1[3/2, 1 + m, 1 - m, 5/2, Tan[(e + f*x)/2]^2, ((a - b)*Tan[(e + f*x)/2]^2)/(a + b)]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/(a + b) + ((1 + m)*AppellF1[3/2, 2 + m, -m, 5/2, Tan[(e + f*x)/2]^2, ((a - b)*Tan[(e + f*x)/2]^2)/(a + b)]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/3) + 2*Tan[(e + f*x)/2]^2*(-((a - b)*m*((3*(a - b)*(1 - m)*AppellF1[5/2, 1 + m, 2 - m, 7/2, Tan[(e + f*x)/2]^2, ((a - b)*Tan[(e + f*x)/2]^2)/(a + b)]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/(5*(a + b)) + (3*(1 + m)*AppellF1[5/2, 2 + m, 1 - m, 7/2, Tan[(e + f*x)/2]^2, ((a - b)*Tan[(e + f*x)/2]^2)/(a + b)]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/5)) + (a + b)*(1 + m)*((-3*(a - b)*m*AppellF1[5/2, 2 + m, 1 - m, 7/2, Tan[(e + f*x)/2]^2, ((a - b)*Tan[(e + f*x)/2]^2)/(a + b)]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/(5*(a + b)) + (3*(2 + m)*AppellF1[5/2, 3 + m, -m, 7/2, Tan[(e + f*x)/2]^2, ((a - b)*Tan[(e + f*x)/2]^2)/(a + b)]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/5))))/((-1 + Tan[(e + f*x)/2]^2)*(3*(a + b)*AppellF1[1/2, 1 + m, -m, 3/2, Tan[(e + f*x)/2]^2, ((a - b)*Tan[(e + f*x)/2]^2)/(a + b)] + 2*(-((a - b)*m*AppellF1[3/2, 1 + m, 1 - m, 5/2, Tan[(e + f*x)/2]^2, ((a - b)*Tan[(e + f*x)/2]^2)/(a + b)]) + (a + b)*(1 + m)*AppellF1[3/2, 2 + m, -m, 5/2, Tan[(e + f*x)/2]^2, ((a - b)*Tan[(e + f*x)/2]^2)/(a + b)])*Tan[(e + f*x)/2]^2)^2)))","B",0
791,0,0,15,2.0839113,"\int (a+b \sec (e+f x))^m \, dx","Integrate[(a + b*Sec[e + f*x])^m,x]","\int (a+b \sec (e+f x))^m \, dx","\text{Int}\left((a+b \sec (e+f x))^m,x\right)",0,"Integrate[(a + b*Sec[e + f*x])^m, x]","A",-1
792,0,0,22,6.7238318,"\int \cos (e+f x) (a+b \sec (e+f x))^m \, dx","Integrate[Cos[e + f*x]*(a + b*Sec[e + f*x])^m,x]","\int \cos (e+f x) (a+b \sec (e+f x))^m \, dx","\text{Int}\left(\cos (e+f x) (a+b \sec (e+f x))^m,x\right)",0,"Integrate[Cos[e + f*x]*(a + b*Sec[e + f*x])^m, x]","A",-1
793,0,0,24,5.8951381,"\int \cos ^2(e+f x) (a+b \sec (e+f x))^m \, dx","Integrate[Cos[e + f*x]^2*(a + b*Sec[e + f*x])^m,x]","\int \cos ^2(e+f x) (a+b \sec (e+f x))^m \, dx","\text{Int}\left(\cos ^2(e+f x) (a+b \sec (e+f x))^m,x\right)",0,"Integrate[Cos[e + f*x]^2*(a + b*Sec[e + f*x])^m, x]","A",-1
794,1,90,135,0.3913541,"\int \cos ^{\frac{9}{2}}(c+d x) (a+b \sec (c+d x)) \, dx","Integrate[Cos[c + d*x]^(9/2)*(a + b*Sec[c + d*x]),x]","\frac{\sqrt{\cos (c+d x)} (266 a \sin (2 (c+d x))+35 a \sin (4 (c+d x))+690 b \sin (c+d x)+90 b \sin (3 (c+d x)))+1176 a E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+600 b F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{1260 d}","\frac{14 a E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{2 a \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x)}{9 d}+\frac{14 a \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{45 d}+\frac{10 b F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{2 b \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{7 d}+\frac{10 b \sin (c+d x) \sqrt{\cos (c+d x)}}{21 d}",1,"(1176*a*EllipticE[(c + d*x)/2, 2] + 600*b*EllipticF[(c + d*x)/2, 2] + Sqrt[Cos[c + d*x]]*(690*b*Sin[c + d*x] + 266*a*Sin[2*(c + d*x)] + 90*b*Sin[3*(c + d*x)] + 35*a*Sin[4*(c + d*x)]))/(1260*d)","A",1
795,1,77,111,0.5640497,"\int \cos ^{\frac{7}{2}}(c+d x) (a+b \sec (c+d x)) \, dx","Integrate[Cos[c + d*x]^(7/2)*(a + b*Sec[c + d*x]),x]","\frac{\sin (c+d x) \sqrt{\cos (c+d x)} (15 a \cos (2 (c+d x))+65 a+42 b \cos (c+d x))+50 a F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+126 b E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{105 d}","\frac{10 a F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{2 a \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{7 d}+\frac{10 a \sin (c+d x) \sqrt{\cos (c+d x)}}{21 d}+\frac{6 b E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 b \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 d}",1,"(126*b*EllipticE[(c + d*x)/2, 2] + 50*a*EllipticF[(c + d*x)/2, 2] + Sqrt[Cos[c + d*x]]*(65*a + 42*b*Cos[c + d*x] + 15*a*Cos[2*(c + d*x)])*Sin[c + d*x])/(105*d)","A",1
796,1,66,87,0.2744577,"\int \cos ^{\frac{5}{2}}(c+d x) (a+b \sec (c+d x)) \, dx","Integrate[Cos[c + d*x]^(5/2)*(a + b*Sec[c + d*x]),x]","\frac{2 \left(\sin (c+d x) \sqrt{\cos (c+d x)} (3 a \cos (c+d x)+5 b)+9 a E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+5 b F\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{15 d}","\frac{6 a E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 a \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 d}+\frac{2 b F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{2 b \sin (c+d x) \sqrt{\cos (c+d x)}}{3 d}",1,"(2*(9*a*EllipticE[(c + d*x)/2, 2] + 5*b*EllipticF[(c + d*x)/2, 2] + Sqrt[Cos[c + d*x]]*(5*b + 3*a*Cos[c + d*x])*Sin[c + d*x]))/(15*d)","A",1
797,1,53,61,0.1219146,"\int \cos ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x)) \, dx","Integrate[Cos[c + d*x]^(3/2)*(a + b*Sec[c + d*x]),x]","\frac{2 \left(a \left(F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+\sin (c+d x) \sqrt{\cos (c+d x)}\right)+3 b E\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{3 d}","\frac{2 a F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{2 a \sin (c+d x) \sqrt{\cos (c+d x)}}{3 d}+\frac{2 b E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}",1,"(2*(3*b*EllipticE[(c + d*x)/2, 2] + a*(EllipticF[(c + d*x)/2, 2] + Sqrt[Cos[c + d*x]]*Sin[c + d*x])))/(3*d)","A",1
798,1,32,35,0.0872698,"\int \sqrt{\cos (c+d x)} (a+b \sec (c+d x)) \, dx","Integrate[Sqrt[Cos[c + d*x]]*(a + b*Sec[c + d*x]),x]","\frac{2 \left(a E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+b F\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{d}","\frac{2 a E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 b F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}",1,"(2*(a*EllipticE[(c + d*x)/2, 2] + b*EllipticF[(c + d*x)/2, 2]))/d","A",1
799,1,51,57,0.1672481,"\int \frac{a+b \sec (c+d x)}{\sqrt{\cos (c+d x)}} \, dx","Integrate[(a + b*Sec[c + d*x])/Sqrt[Cos[c + d*x]],x]","\frac{2 \left(a F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-b E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+\frac{b \sin (c+d x)}{\sqrt{\cos (c+d x)}}\right)}{d}","\frac{2 a F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}-\frac{2 b E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 b \sin (c+d x)}{d \sqrt{\cos (c+d x)}}",1,"(2*(-(b*EllipticE[(c + d*x)/2, 2]) + a*EllipticF[(c + d*x)/2, 2] + (b*Sin[c + d*x])/Sqrt[Cos[c + d*x]]))/d","A",1
800,1,65,83,0.4465318,"\int \frac{a+b \sec (c+d x)}{\cos ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[(a + b*Sec[c + d*x])/Cos[c + d*x]^(3/2),x]","\frac{\frac{2 \sin (c+d x) (3 a \cos (c+d x)+b)}{\cos ^{\frac{3}{2}}(c+d x)}-6 a E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+2 b F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}","-\frac{2 a E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 a \sin (c+d x)}{d \sqrt{\cos (c+d x)}}+\frac{2 b F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{2 b \sin (c+d x)}{3 d \cos ^{\frac{3}{2}}(c+d x)}",1,"(-6*a*EllipticE[(c + d*x)/2, 2] + 2*b*EllipticF[(c + d*x)/2, 2] + (2*(b + 3*a*Cos[c + d*x])*Sin[c + d*x])/Cos[c + d*x]^(3/2))/(3*d)","A",1
801,1,95,111,0.3488158,"\int \frac{a+b \sec (c+d x)}{\cos ^{\frac{5}{2}}(c+d x)} \, dx","Integrate[(a + b*Sec[c + d*x])/Cos[c + d*x]^(5/2),x]","\frac{10 a \sin (c+d x)+10 a \cos ^{\frac{3}{2}}(c+d x) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+9 b \sin (2 (c+d x))+6 b \tan (c+d x)-18 b \cos ^{\frac{3}{2}}(c+d x) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d \cos ^{\frac{3}{2}}(c+d x)}","\frac{2 a F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{2 a \sin (c+d x)}{3 d \cos ^{\frac{3}{2}}(c+d x)}-\frac{6 b E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 b \sin (c+d x)}{5 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{6 b \sin (c+d x)}{5 d \sqrt{\cos (c+d x)}}",1,"(-18*b*Cos[c + d*x]^(3/2)*EllipticE[(c + d*x)/2, 2] + 10*a*Cos[c + d*x]^(3/2)*EllipticF[(c + d*x)/2, 2] + 10*a*Sin[c + d*x] + 9*b*Sin[2*(c + d*x)] + 6*b*Tan[c + d*x])/(15*d*Cos[c + d*x]^(3/2))","A",1
802,1,113,160,0.869528,"\int \cos ^{\frac{9}{2}}(c+d x) (a+b \sec (c+d x))^2 \, dx","Integrate[Cos[c + d*x]^(9/2)*(a + b*Sec[c + d*x])^2,x]","\frac{84 \left(7 a^2+9 b^2\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+\sin (c+d x) \sqrt{\cos (c+d x)} \left(7 \left(43 a^2+36 b^2\right) \cos (c+d x)+5 a (7 a \cos (3 (c+d x))+36 b \cos (2 (c+d x))+156 b)\right)+600 a b F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{630 d}","\frac{2 \left(7 a^2+9 b^2\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{2 \left(7 a^2+9 b^2\right) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{45 d}+\frac{2 a^2 \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x)}{9 d}+\frac{20 a b F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{4 a b \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{7 d}+\frac{20 a b \sin (c+d x) \sqrt{\cos (c+d x)}}{21 d}",1,"(84*(7*a^2 + 9*b^2)*EllipticE[(c + d*x)/2, 2] + 600*a*b*EllipticF[(c + d*x)/2, 2] + Sqrt[Cos[c + d*x]]*(7*(43*a^2 + 36*b^2)*Cos[c + d*x] + 5*a*(156*b + 36*b*Cos[2*(c + d*x)] + 7*a*Cos[3*(c + d*x)]))*Sin[c + d*x])/(630*d)","A",1
803,1,98,135,0.6525497,"\int \cos ^{\frac{7}{2}}(c+d x) (a+b \sec (c+d x))^2 \, dx","Integrate[Cos[c + d*x]^(7/2)*(a + b*Sec[c + d*x])^2,x]","\frac{10 \left(5 a^2+7 b^2\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+\sin (c+d x) \sqrt{\cos (c+d x)} \left(15 a^2 \cos (2 (c+d x))+65 a^2+84 a b \cos (c+d x)+70 b^2\right)+252 a b E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{105 d}","\frac{2 \left(5 a^2+7 b^2\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{2 \left(5 a^2+7 b^2\right) \sin (c+d x) \sqrt{\cos (c+d x)}}{21 d}+\frac{2 a^2 \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{7 d}+\frac{12 a b E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{4 a b \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 d}",1,"(252*a*b*EllipticE[(c + d*x)/2, 2] + 10*(5*a^2 + 7*b^2)*EllipticF[(c + d*x)/2, 2] + Sqrt[Cos[c + d*x]]*(65*a^2 + 70*b^2 + 84*a*b*Cos[c + d*x] + 15*a^2*Cos[2*(c + d*x)])*Sin[c + d*x])/(105*d)","A",1
804,1,79,101,0.3247506,"\int \cos ^{\frac{5}{2}}(c+d x) (a+b \sec (c+d x))^2 \, dx","Integrate[Cos[c + d*x]^(5/2)*(a + b*Sec[c + d*x])^2,x]","\frac{6 \left(3 a^2+5 b^2\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+20 a b F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+2 a \sin (c+d x) \sqrt{\cos (c+d x)} (3 a \cos (c+d x)+10 b)}{15 d}","\frac{2 \left(3 a^2+5 b^2\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 a^2 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 d}+\frac{4 a b F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{4 a b \sin (c+d x) \sqrt{\cos (c+d x)}}{3 d}",1,"(6*(3*a^2 + 5*b^2)*EllipticE[(c + d*x)/2, 2] + 20*a*b*EllipticF[(c + d*x)/2, 2] + 2*a*Sqrt[Cos[c + d*x]]*(10*b + 3*a*Cos[c + d*x])*Sin[c + d*x])/(15*d)","A",1
805,1,64,72,0.1730286,"\int \cos ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^2 \, dx","Integrate[Cos[c + d*x]^(3/2)*(a + b*Sec[c + d*x])^2,x]","\frac{2 \left(\left(a^2+3 b^2\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+a^2 \sin (c+d x) \sqrt{\cos (c+d x)}+6 a b E\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{3 d}","\frac{2 \left(a^2+3 b^2\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{2 a^2 \sin (c+d x) \sqrt{\cos (c+d x)}}{3 d}+\frac{4 a b E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}",1,"(2*(6*a*b*EllipticE[(c + d*x)/2, 2] + (a^2 + 3*b^2)*EllipticF[(c + d*x)/2, 2] + a^2*Sqrt[Cos[c + d*x]]*Sin[c + d*x]))/(3*d)","A",1
806,1,62,68,0.355073,"\int \sqrt{\cos (c+d x)} (a+b \sec (c+d x))^2 \, dx","Integrate[Sqrt[Cos[c + d*x]]*(a + b*Sec[c + d*x])^2,x]","\frac{2 \left(\left(a^2-b^2\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+b \left(2 a F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+\frac{b \sin (c+d x)}{\sqrt{\cos (c+d x)}}\right)\right)}{d}","\frac{2 \left(a^2-b^2\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{4 a b F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 b^2 \sin (c+d x)}{d \sqrt{\cos (c+d x)}}",1,"(2*((a^2 - b^2)*EllipticE[(c + d*x)/2, 2] + b*(2*a*EllipticF[(c + d*x)/2, 2] + (b*Sin[c + d*x])/Sqrt[Cos[c + d*x]])))/d","A",1
807,1,73,95,0.6730264,"\int \frac{(a+b \sec (c+d x))^2}{\sqrt{\cos (c+d x)}} \, dx","Integrate[(a + b*Sec[c + d*x])^2/Sqrt[Cos[c + d*x]],x]","\frac{2 \left(\left(3 a^2+b^2\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-6 a b E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+\frac{b \sin (c+d x) (6 a \cos (c+d x)+b)}{\cos ^{\frac{3}{2}}(c+d x)}\right)}{3 d}","\frac{2 \left(3 a^2+b^2\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{4 a b E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{4 a b \sin (c+d x)}{d \sqrt{\cos (c+d x)}}+\frac{2 b^2 \sin (c+d x)}{3 d \cos ^{\frac{3}{2}}(c+d x)}",1,"(2*(-6*a*b*EllipticE[(c + d*x)/2, 2] + (3*a^2 + b^2)*EllipticF[(c + d*x)/2, 2] + (b*(b + 6*a*Cos[c + d*x])*Sin[c + d*x])/Cos[c + d*x]^(3/2)))/(3*d)","A",1
808,1,124,135,0.4435186,"\int \frac{(a+b \sec (c+d x))^2}{\cos ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[(a + b*Sec[c + d*x])^2/Cos[c + d*x]^(3/2),x]","\frac{-6 \left(5 a^2+3 b^2\right) \cos ^{\frac{3}{2}}(c+d x) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+15 a^2 \sin (2 (c+d x))+20 a b \sin (c+d x)+20 a b \cos ^{\frac{3}{2}}(c+d x) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+9 b^2 \sin (2 (c+d x))+6 b^2 \tan (c+d x)}{15 d \cos ^{\frac{3}{2}}(c+d x)}","-\frac{2 \left(5 a^2+3 b^2\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 \left(5 a^2+3 b^2\right) \sin (c+d x)}{5 d \sqrt{\cos (c+d x)}}+\frac{4 a b F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{4 a b \sin (c+d x)}{3 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 b^2 \sin (c+d x)}{5 d \cos ^{\frac{5}{2}}(c+d x)}",1,"(-6*(5*a^2 + 3*b^2)*Cos[c + d*x]^(3/2)*EllipticE[(c + d*x)/2, 2] + 20*a*b*Cos[c + d*x]^(3/2)*EllipticF[(c + d*x)/2, 2] + 20*a*b*Sin[c + d*x] + 15*a^2*Sin[2*(c + d*x)] + 9*b^2*Sin[2*(c + d*x)] + 6*b^2*Tan[c + d*x])/(15*d*Cos[c + d*x]^(3/2))","A",1
809,1,142,160,0.6205614,"\int \frac{(a+b \sec (c+d x))^2}{\cos ^{\frac{5}{2}}(c+d x)} \, dx","Integrate[(a + b*Sec[c + d*x])^2/Cos[c + d*x]^(5/2),x]","\frac{10 \left(7 a^2+5 b^2\right) \cos ^{\frac{5}{2}}(c+d x) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+35 a^2 \sin (2 (c+d x))+84 a b \sin (c+d x)-252 a b \cos ^{\frac{5}{2}}(c+d x) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+252 a b \sin (c+d x) \cos ^2(c+d x)+25 b^2 \sin (2 (c+d x))+30 b^2 \tan (c+d x)}{105 d \cos ^{\frac{5}{2}}(c+d x)}","\frac{2 \left(7 a^2+5 b^2\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{2 \left(7 a^2+5 b^2\right) \sin (c+d x)}{21 d \cos ^{\frac{3}{2}}(c+d x)}-\frac{12 a b E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{4 a b \sin (c+d x)}{5 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{12 a b \sin (c+d x)}{5 d \sqrt{\cos (c+d x)}}+\frac{2 b^2 \sin (c+d x)}{7 d \cos ^{\frac{7}{2}}(c+d x)}",1,"(-252*a*b*Cos[c + d*x]^(5/2)*EllipticE[(c + d*x)/2, 2] + 10*(7*a^2 + 5*b^2)*Cos[c + d*x]^(5/2)*EllipticF[(c + d*x)/2, 2] + 84*a*b*Sin[c + d*x] + 252*a*b*Cos[c + d*x]^2*Sin[c + d*x] + 35*a^2*Sin[2*(c + d*x)] + 25*b^2*Sin[2*(c + d*x)] + 30*b^2*Tan[c + d*x])/(105*d*Cos[c + d*x]^(5/2))","A",1
810,1,137,194,1.0886067,"\int \cos ^{\frac{9}{2}}(c+d x) (a+b \sec (c+d x))^3 \, dx","Integrate[Cos[c + d*x]^(9/2)*(a + b*Sec[c + d*x])^3,x]","\frac{84 \left(7 a^3+27 a b^2\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+60 \left(15 a^2 b+7 b^3\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+\sin (c+d x) \sqrt{\cos (c+d x)} \left(7 a \left(43 a^2+108 b^2\right) \cos (c+d x)+5 \left(7 a^3 \cos (3 (c+d x))+54 a^2 b \cos (2 (c+d x))+234 a^2 b+84 b^3\right)\right)}{630 d}","\frac{2 b \left(15 a^2+7 b^2\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{2 a \left(7 a^2+27 b^2\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{2 a \left(7 a^2+27 b^2\right) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{45 d}+\frac{2 b \left(15 a^2+7 b^2\right) \sin (c+d x) \sqrt{\cos (c+d x)}}{21 d}+\frac{40 a^2 b \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{63 d}+\frac{2 a^2 \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x) (a+b \sec (c+d x))}{9 d}",1,"(84*(7*a^3 + 27*a*b^2)*EllipticE[(c + d*x)/2, 2] + 60*(15*a^2*b + 7*b^3)*EllipticF[(c + d*x)/2, 2] + Sqrt[Cos[c + d*x]]*(7*a*(43*a^2 + 108*b^2)*Cos[c + d*x] + 5*(234*a^2*b + 84*b^3 + 54*a^2*b*Cos[2*(c + d*x)] + 7*a^3*Cos[3*(c + d*x)]))*Sin[c + d*x])/(630*d)","A",1
811,1,110,159,0.850586,"\int \cos ^{\frac{7}{2}}(c+d x) (a+b \sec (c+d x))^3 \, dx","Integrate[Cos[c + d*x]^(7/2)*(a + b*Sec[c + d*x])^3,x]","\frac{10 \left(5 a^3+21 a b^2\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+42 \left(9 a^2 b+5 b^3\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+a \sin (c+d x) \sqrt{\cos (c+d x)} \left(15 a^2 \cos (2 (c+d x))+65 a^2+126 a b \cos (c+d x)+210 b^2\right)}{105 d}","\frac{2 a \left(5 a^2+21 b^2\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{2 b \left(9 a^2+5 b^2\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 a \left(5 a^2+21 b^2\right) \sin (c+d x) \sqrt{\cos (c+d x)}}{21 d}+\frac{32 a^2 b \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{35 d}+\frac{2 a^2 \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) (a+b \sec (c+d x))}{7 d}",1,"(42*(9*a^2*b + 5*b^3)*EllipticE[(c + d*x)/2, 2] + 10*(5*a^3 + 21*a*b^2)*EllipticF[(c + d*x)/2, 2] + a*Sqrt[Cos[c + d*x]]*(65*a^2 + 210*b^2 + 126*a*b*Cos[c + d*x] + 15*a^2*Cos[2*(c + d*x)])*Sin[c + d*x])/(105*d)","A",1
812,1,84,116,0.4231699,"\int \cos ^{\frac{5}{2}}(c+d x) (a+b \sec (c+d x))^3 \, dx","Integrate[Cos[c + d*x]^(5/2)*(a + b*Sec[c + d*x])^3,x]","\frac{2 \left(3 \left(a^3+5 a b^2\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+5 b \left(a^2+b^2\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+a^2 \sin (c+d x) \sqrt{\cos (c+d x)} (a \cos (c+d x)+5 b)\right)}{5 d}","\frac{2 b \left(a^2+b^2\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{6 a \left(a^2+5 b^2\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{8 a^2 b \sin (c+d x) \sqrt{\cos (c+d x)}}{5 d}+\frac{2 a^2 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))}{5 d}",1,"(2*(3*(a^3 + 5*a*b^2)*EllipticE[(c + d*x)/2, 2] + 5*b*(a^2 + b^2)*EllipticF[(c + d*x)/2, 2] + a^2*Sqrt[Cos[c + d*x]]*(5*b + a*Cos[c + d*x])*Sin[c + d*x]))/(5*d)","A",1
813,1,87,126,0.5994157,"\int \cos ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^3 \, dx","Integrate[Cos[c + d*x]^(3/2)*(a + b*Sec[c + d*x])^3,x]","\frac{2 \left(\frac{\sin (c+d x) \left(a^3 \cos (c+d x)+3 b^3\right)}{\sqrt{\cos (c+d x)}}+\left(a^3+9 a b^2\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+\left(9 a^2 b-3 b^3\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{3 d}","\frac{2 a \left(a^2+9 b^2\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{2 b \left(3 a^2-b^2\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}-\frac{2 b \left(a^2-3 b^2\right) \sin (c+d x)}{3 d \sqrt{\cos (c+d x)}}+\frac{2 a^2 \sin (c+d x) \sqrt{\cos (c+d x)} (a+b \sec (c+d x))}{3 d}",1,"(2*((9*a^2*b - 3*b^3)*EllipticE[(c + d*x)/2, 2] + (a^3 + 9*a*b^2)*EllipticF[(c + d*x)/2, 2] + ((3*b^3 + a^3*Cos[c + d*x])*Sin[c + d*x])/Sqrt[Cos[c + d*x]]))/(3*d)","A",1
814,1,84,118,1.2859932,"\int \sqrt{\cos (c+d x)} (a+b \sec (c+d x))^3 \, dx","Integrate[Sqrt[Cos[c + d*x]]*(a + b*Sec[c + d*x])^3,x]","\frac{2 \left(3 \left(a^3-3 a b^2\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+b \left(\left(9 a^2+b^2\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+\frac{b \sin (c+d x) (9 a \cos (c+d x)+b)}{\cos ^{\frac{3}{2}}(c+d x)}\right)\right)}{3 d}","\frac{2 b \left(9 a^2+b^2\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{2 a \left(a^2-3 b^2\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{16 a b^2 \sin (c+d x)}{3 d \sqrt{\cos (c+d x)}}+\frac{2 b^2 \sin (c+d x) (a+b \sec (c+d x))}{3 d \sqrt{\cos (c+d x)}}",1,"(2*(3*(a^3 - 3*a*b^2)*EllipticE[(c + d*x)/2, 2] + b*((9*a^2 + b^2)*EllipticF[(c + d*x)/2, 2] + (b*(b + 9*a*Cos[c + d*x])*Sin[c + d*x])/Cos[c + d*x]^(3/2))))/(3*d)","A",1
815,1,125,149,1.0925301,"\int \frac{(a+b \sec (c+d x))^3}{\sqrt{\cos (c+d x)}} \, dx","Integrate[(a + b*Sec[c + d*x])^3/Sqrt[Cos[c + d*x]],x]","\frac{3 \left(5 a^2 b+b^3\right) \sin (2 (c+d x))+10 a \left(a^2+b^2\right) \cos ^{\frac{3}{2}}(c+d x) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-6 b \left(5 a^2+b^2\right) \cos ^{\frac{3}{2}}(c+d x) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+10 a b^2 \sin (c+d x)+2 b^3 \tan (c+d x)}{5 d \cos ^{\frac{3}{2}}(c+d x)}","\frac{2 a \left(a^2+b^2\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}-\frac{6 b \left(5 a^2+b^2\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{6 b \left(5 a^2+b^2\right) \sin (c+d x)}{5 d \sqrt{\cos (c+d x)}}+\frac{8 a b^2 \sin (c+d x)}{5 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 b^2 \sin (c+d x) (a+b \sec (c+d x))}{5 d \cos ^{\frac{3}{2}}(c+d x)}",1,"(-6*b*(5*a^2 + b^2)*Cos[c + d*x]^(3/2)*EllipticE[(c + d*x)/2, 2] + 10*a*(a^2 + b^2)*Cos[c + d*x]^(3/2)*EllipticF[(c + d*x)/2, 2] + 10*a*b^2*Sin[c + d*x] + 3*(5*a^2*b + b^3)*Sin[2*(c + d*x)] + 2*b^3*Tan[c + d*x])/(5*d*Cos[c + d*x]^(3/2))","A",1
816,1,177,194,0.9365015,"\int \frac{(a+b \sec (c+d x))^3}{\cos ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[(a + b*Sec[c + d*x])^3/Cos[c + d*x]^(3/2),x]","\frac{210 a^3 \sin (c+d x) \cos ^2(c+d x)+10 b \left(21 a^2+5 b^2\right) \cos ^{\frac{5}{2}}(c+d x) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-42 a \left(5 a^2+9 b^2\right) \cos ^{\frac{5}{2}}(c+d x) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+105 a^2 b \sin (2 (c+d x))+126 a b^2 \sin (c+d x)+378 a b^2 \sin (c+d x) \cos ^2(c+d x)+25 b^3 \sin (2 (c+d x))+30 b^3 \tan (c+d x)}{105 d \cos ^{\frac{5}{2}}(c+d x)}","\frac{2 b \left(21 a^2+5 b^2\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}-\frac{2 a \left(5 a^2+9 b^2\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 b \left(21 a^2+5 b^2\right) \sin (c+d x)}{21 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 a \left(5 a^2+9 b^2\right) \sin (c+d x)}{5 d \sqrt{\cos (c+d x)}}+\frac{32 a b^2 \sin (c+d x)}{35 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 b^2 \sin (c+d x) (a+b \sec (c+d x))}{7 d \cos ^{\frac{5}{2}}(c+d x)}",1,"(-42*a*(5*a^2 + 9*b^2)*Cos[c + d*x]^(5/2)*EllipticE[(c + d*x)/2, 2] + 10*b*(21*a^2 + 5*b^2)*Cos[c + d*x]^(5/2)*EllipticF[(c + d*x)/2, 2] + 126*a*b^2*Sin[c + d*x] + 210*a^3*Cos[c + d*x]^2*Sin[c + d*x] + 378*a*b^2*Cos[c + d*x]^2*Sin[c + d*x] + 105*a^2*b*Sin[2*(c + d*x)] + 25*b^3*Sin[2*(c + d*x)] + 30*b^3*Tan[c + d*x])/(105*d*Cos[c + d*x]^(5/2))","A",1
817,1,226,152,1.7962829,"\int \frac{\cos ^{\frac{5}{2}}(c+d x)}{a+b \sec (c+d x)} \, dx","Integrate[Cos[c + d*x]^(5/2)/(a + b*Sec[c + d*x]),x]","\frac{\frac{2 \left(9 a^2+5 b^2\right) \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a+b}+\frac{6 \left(3 a^2+5 b^2\right) \sin (c+d x) \left(\left(a^2-2 b^2\right) \Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)+2 b (a+b) F\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)-2 a b E\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)\right)}{a^2 b \sqrt{\sin ^2(c+d x)}}+4 \sin (c+d x) \sqrt{\cos (c+d x)} (3 a \cos (c+d x)-5 b)+8 b \left(2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-\frac{2 b \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a+b}\right)}{30 a^2 d}","\frac{2 b^4 \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a^4 d (a+b)}-\frac{2 b \sin (c+d x) \sqrt{\cos (c+d x)}}{3 a^2 d}-\frac{2 b \left(a^2+3 b^2\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^4 d}+\frac{2 \left(3 a^2+5 b^2\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 a^3 d}+\frac{2 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 a d}",1,"((2*(9*a^2 + 5*b^2)*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2])/(a + b) + 8*b*(2*EllipticF[(c + d*x)/2, 2] - (2*b*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2])/(a + b)) + 4*Sqrt[Cos[c + d*x]]*(-5*b + 3*a*Cos[c + d*x])*Sin[c + d*x] + (6*(3*a^2 + 5*b^2)*(-2*a*b*EllipticE[ArcSin[Sqrt[Cos[c + d*x]]], -1] + 2*b*(a + b)*EllipticF[ArcSin[Sqrt[Cos[c + d*x]]], -1] + (a^2 - 2*b^2)*EllipticPi[-(a/b), ArcSin[Sqrt[Cos[c + d*x]]], -1])*Sin[c + d*x])/(a^2*b*Sqrt[Sin[c + d*x]^2]))/(30*a^2*d)","A",1
818,1,158,112,1.9293376,"\int \frac{\cos ^{\frac{3}{2}}(c+d x)}{a+b \sec (c+d x)} \, dx","Integrate[Cos[c + d*x]^(3/2)/(a + b*Sec[c + d*x]),x]","\frac{-\frac{6 \sin (c+d x) \left(\left(a^2-2 b^2\right) \Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)+2 b (a+b) F\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)-2 a b E\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)\right)}{a^2 \sqrt{\sin ^2(c+d x)}}-\frac{6 b \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a+b}+4 F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+4 \sin (c+d x) \sqrt{\cos (c+d x)}}{6 a d}","-\frac{2 b^3 \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a^3 d (a+b)}-\frac{2 b E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d}+\frac{2 \left(a^2+3 b^2\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^3 d}+\frac{2 \sin (c+d x) \sqrt{\cos (c+d x)}}{3 a d}",1,"(4*EllipticF[(c + d*x)/2, 2] - (6*b*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2])/(a + b) + 4*Sqrt[Cos[c + d*x]]*Sin[c + d*x] - (6*(-2*a*b*EllipticE[ArcSin[Sqrt[Cos[c + d*x]]], -1] + 2*b*(a + b)*EllipticF[ArcSin[Sqrt[Cos[c + d*x]]], -1] + (a^2 - 2*b^2)*EllipticPi[-(a/b), ArcSin[Sqrt[Cos[c + d*x]]], -1])*Sin[c + d*x])/(a^2*Sqrt[Sin[c + d*x]^2]))/(6*a*d)","A",1
819,1,81,75,0.3366985,"\int \frac{\sqrt{\cos (c+d x)}}{a+b \sec (c+d x)} \, dx","Integrate[Sqrt[Cos[c + d*x]]/(a + b*Sec[c + d*x]),x]","-\frac{2 \sin (c+d x) \left(-(a+b) F\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)+b \Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)+a E\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)\right)}{a^2 d \sqrt{\sin ^2(c+d x)}}","\frac{2 b^2 \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d (a+b)}-\frac{2 b F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d}+\frac{2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}",1,"(-2*(a*EllipticE[ArcSin[Sqrt[Cos[c + d*x]]], -1] - (a + b)*EllipticF[ArcSin[Sqrt[Cos[c + d*x]]], -1] + b*EllipticPi[-(a/b), ArcSin[Sqrt[Cos[c + d*x]]], -1])*Sin[c + d*x])/(a^2*d*Sqrt[Sin[c + d*x]^2])","A",1
820,1,48,53,0.0799486,"\int \frac{1}{\sqrt{\cos (c+d x)} (a+b \sec (c+d x))} \, dx","Integrate[1/(Sqrt[Cos[c + d*x]]*(a + b*Sec[c + d*x])),x]","\frac{2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-\frac{2 b \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a+b}}{a d}","\frac{2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}-\frac{2 b \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a d (a+b)}",1,"(2*EllipticF[(c + d*x)/2, 2] - (2*b*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2])/(a + b))/(a*d)","A",1
821,1,29,29,0.0898891,"\int \frac{1}{\cos ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))} \, dx","Integrate[1/(Cos[c + d*x]^(3/2)*(a + b*Sec[c + d*x])),x]","\frac{2 \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{d (a+b)}","\frac{2 \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{d (a+b)}",1,"(2*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2])/((a + b)*d)","A",1
822,1,195,77,3.3408063,"\int \frac{1}{\cos ^{\frac{5}{2}}(c+d x) (a+b \sec (c+d x))} \, dx","Integrate[1/(Cos[c + d*x]^(5/2)*(a + b*Sec[c + d*x])),x]","-\frac{\frac{2 \sin (c+d x) \left(\left(a^2-2 b^2\right) \Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)+2 b (a+b) F\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)-2 a b E\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)\right)}{a b \sqrt{\sin ^2(c+d x)}}+\frac{6 a \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a+b}+\frac{2 b \left(2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-\frac{2 b \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a+b}\right)}{a}-\frac{4 \sin (c+d x)}{\sqrt{\cos (c+d x)}}}{2 b d}","-\frac{2 a \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{b d (a+b)}-\frac{2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b d}+\frac{2 \sin (c+d x)}{b d \sqrt{\cos (c+d x)}}",1,"-1/2*((6*a*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2])/(a + b) + (2*b*(2*EllipticF[(c + d*x)/2, 2] - (2*b*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2])/(a + b)))/a - (4*Sin[c + d*x])/Sqrt[Cos[c + d*x]] + (2*(-2*a*b*EllipticE[ArcSin[Sqrt[Cos[c + d*x]]], -1] + 2*b*(a + b)*EllipticF[ArcSin[Sqrt[Cos[c + d*x]]], -1] + (a^2 - 2*b^2)*EllipticPi[-(a/b), ArcSin[Sqrt[Cos[c + d*x]]], -1])*Sin[c + d*x])/(a*b*Sqrt[Sin[c + d*x]^2]))/(b*d)","B",1
823,1,210,128,4.515652,"\int \frac{1}{\cos ^{\frac{7}{2}}(c+d x) (a+b \sec (c+d x))} \, dx","Integrate[1/(Cos[c + d*x]^(7/2)*(a + b*Sec[c + d*x])),x]","\frac{\frac{2 \left(9 a^2+2 b^2\right) \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a+b}+\frac{6 \sin (c+d x) \left(\left(a^2-2 b^2\right) \Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)+2 b (a+b) F\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)-2 a b E\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)\right)}{b \sqrt{\sin ^2(c+d x)}}+\frac{4 \sin (c+d x) (b-3 a \cos (c+d x))}{\cos ^{\frac{3}{2}}(c+d x)}+8 b \left(2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-\frac{2 b \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a+b}\right)}{6 b^2 d}","\frac{2 a^2 \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{b^2 d (a+b)}+\frac{2 a E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b^2 d}-\frac{2 a \sin (c+d x)}{b^2 d \sqrt{\cos (c+d x)}}+\frac{2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 b d}+\frac{2 \sin (c+d x)}{3 b d \cos ^{\frac{3}{2}}(c+d x)}",1,"((2*(9*a^2 + 2*b^2)*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2])/(a + b) + 8*b*(2*EllipticF[(c + d*x)/2, 2] - (2*b*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2])/(a + b)) + (4*(b - 3*a*Cos[c + d*x])*Sin[c + d*x])/Cos[c + d*x]^(3/2) + (6*(-2*a*b*EllipticE[ArcSin[Sqrt[Cos[c + d*x]]], -1] + 2*b*(a + b)*EllipticF[ArcSin[Sqrt[Cos[c + d*x]]], -1] + (a^2 - 2*b^2)*EllipticPi[-(a/b), ArcSin[Sqrt[Cos[c + d*x]]], -1])*Sin[c + d*x])/(b*Sqrt[Sin[c + d*x]^2]))/(6*b^2*d)","A",1
824,1,266,244,2.1284125,"\int \frac{\cos ^{\frac{3}{2}}(c+d x)}{(a+b \sec (c+d x))^2} \, dx","Integrate[Cos[c + d*x]^(3/2)/(a + b*Sec[c + d*x])^2,x]","\frac{4 \sin (c+d x) \sqrt{\cos (c+d x)} \left(\frac{3 b^3}{\left(b^2-a^2\right) (a \cos (c+d x)+b)}+2\right)-\frac{\frac{2 \left(5 b^3-8 a^2 b\right) \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a+b}+\frac{8 \left(a^2+2 b^2\right) \left((a+b) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-b \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{a+b}-\frac{6 \left(4 a^2-5 b^2\right) \sin (c+d x) \left(\left(a^2-2 b^2\right) \Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)+2 b (a+b) F\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)-2 a b E\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)\right)}{a^2 \sqrt{\sin ^2(c+d x)}}}{(b-a) (a+b)}}{12 a^2 d}","\frac{\left(2 a^2-5 b^2\right) \sin (c+d x) \sqrt{\cos (c+d x)}}{3 a^2 d \left(a^2-b^2\right)}+\frac{b^2 \sin (c+d x) \sqrt{\cos (c+d x)}}{a d \left(a^2-b^2\right) (a+b \sec (c+d x))}+\frac{\left(2 a^4+16 a^2 b^2-15 b^4\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^4 d \left(a^2-b^2\right)}-\frac{b^3 \left(7 a^2-5 b^2\right) \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a^4 d (a-b) (a+b)^2}-\frac{b \left(4 a^2-5 b^2\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^3 d \left(a^2-b^2\right)}",1,"(4*Sqrt[Cos[c + d*x]]*(2 + (3*b^3)/((-a^2 + b^2)*(b + a*Cos[c + d*x])))*Sin[c + d*x] - ((2*(-8*a^2*b + 5*b^3)*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2])/(a + b) + (8*(a^2 + 2*b^2)*((a + b)*EllipticF[(c + d*x)/2, 2] - b*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2]))/(a + b) - (6*(4*a^2 - 5*b^2)*(-2*a*b*EllipticE[ArcSin[Sqrt[Cos[c + d*x]]], -1] + 2*b*(a + b)*EllipticF[ArcSin[Sqrt[Cos[c + d*x]]], -1] + (a^2 - 2*b^2)*EllipticPi[-(a/b), ArcSin[Sqrt[Cos[c + d*x]]], -1])*Sin[c + d*x])/(a^2*Sqrt[Sin[c + d*x]^2]))/((-a + b)*(a + b)))/(12*a^2*d)","A",1
825,1,252,184,2.0845599,"\int \frac{\sqrt{\cos (c+d x)}}{(a+b \sec (c+d x))^2} \, dx","Integrate[Sqrt[Cos[c + d*x]]/(a + b*Sec[c + d*x])^2,x]","\frac{\frac{4 b^2 \sin (c+d x) \sqrt{\cos (c+d x)}}{\left(a^2-b^2\right) (a \cos (c+d x)+b)}+\frac{\frac{2 \left(2 a^2-b^2\right) \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a+b}+\frac{2 \left(2 a^2-3 b^2\right) \sin (c+d x) \left(\left(a^2-2 b^2\right) \Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)+2 b (a+b) F\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)-2 a b E\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)\right)}{a^2 b \sqrt{\sin ^2(c+d x)}}+8 b \left(\frac{b \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a+b}-F\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{(a-b) (a+b)}}{4 a d}","\frac{\left(2 a^2-3 b^2\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d \left(a^2-b^2\right)}+\frac{b^2 \sin (c+d x)}{a d \left(a^2-b^2\right) \sqrt{\cos (c+d x)} (a+b \sec (c+d x))}-\frac{b \left(4 a^2-3 b^2\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^3 d \left(a^2-b^2\right)}+\frac{b^2 \left(5 a^2-3 b^2\right) \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a^3 d (a-b) (a+b)^2}",1,"((4*b^2*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/((a^2 - b^2)*(b + a*Cos[c + d*x])) + ((2*(2*a^2 - b^2)*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2])/(a + b) + 8*b*(-EllipticF[(c + d*x)/2, 2] + (b*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2])/(a + b)) + (2*(2*a^2 - 3*b^2)*(-2*a*b*EllipticE[ArcSin[Sqrt[Cos[c + d*x]]], -1] + 2*b*(a + b)*EllipticF[ArcSin[Sqrt[Cos[c + d*x]]], -1] + (a^2 - 2*b^2)*EllipticPi[-(a/b), ArcSin[Sqrt[Cos[c + d*x]]], -1])*Sin[c + d*x])/(a^2*b*Sqrt[Sin[c + d*x]^2]))/((a - b)*(a + b)))/(4*a*d)","A",1
826,1,194,167,4.2135363,"\int \frac{1}{\sqrt{\cos (c+d x)} (a+b \sec (c+d x))^2} \, dx","Integrate[1/(Sqrt[Cos[c + d*x]]*(a + b*Sec[c + d*x])^2),x]","\frac{\frac{4 b \sin (c+d x) \sqrt{\cos (c+d x)}}{\left(b^2-a^2\right) (a \cos (c+d x)+b)}-\frac{\frac{2 \sin (c+d x) \left(\left(a^2-2 b^2\right) \Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)+2 b (a+b) F\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)-2 a b E\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)\right)}{a^2 \sqrt{\sin ^2(c+d x)}}-\frac{10 b \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a+b}+8 F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{(b-a) (a+b)}}{4 d}","\frac{\left(2 a^2-b^2\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d \left(a^2-b^2\right)}+\frac{b E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d \left(a^2-b^2\right)}-\frac{b \left(3 a^2-b^2\right) \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d (a-b) (a+b)^2}-\frac{b \sin (c+d x)}{d \left(a^2-b^2\right) \sqrt{\cos (c+d x)} (a+b \sec (c+d x))}",1,"((4*b*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/((-a^2 + b^2)*(b + a*Cos[c + d*x])) - (8*EllipticF[(c + d*x)/2, 2] - (10*b*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2])/(a + b) + (2*(-2*a*b*EllipticE[ArcSin[Sqrt[Cos[c + d*x]]], -1] + 2*b*(a + b)*EllipticF[ArcSin[Sqrt[Cos[c + d*x]]], -1] + (a^2 - 2*b^2)*EllipticPi[-(a/b), ArcSin[Sqrt[Cos[c + d*x]]], -1])*Sin[c + d*x])/(a^2*Sqrt[Sin[c + d*x]^2]))/((-a + b)*(a + b)))/(4*d)","A",1
827,1,229,148,3.2667861,"\int \frac{1}{\cos ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^2} \, dx","Integrate[1/(Cos[c + d*x]^(3/2)*(a + b*Sec[c + d*x])^2),x]","\frac{\frac{4 a \sin (c+d x) \sqrt{\cos (c+d x)}}{\left(a^2-b^2\right) (a \cos (c+d x)+b)}-\frac{2 \left(\frac{\sin (c+d x) \left(\left(a^2-2 b^2\right) \Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)+2 b (a+b) F\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)-2 a b E\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)\right)}{b \sqrt{\sin ^2(c+d x)}}-\frac{a^2 \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a+b}+2 b \left(2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-\frac{2 b \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a+b}\right)\right)}{a (a-b) (a+b)}}{4 d}","-\frac{b F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d \left(a^2-b^2\right)}-\frac{E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d \left(a^2-b^2\right)}+\frac{\left(a^2+b^2\right) \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a d (a-b) (a+b)^2}+\frac{a \sin (c+d x)}{d \left(a^2-b^2\right) \sqrt{\cos (c+d x)} (a+b \sec (c+d x))}",1,"((4*a*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/((a^2 - b^2)*(b + a*Cos[c + d*x])) - (2*(-((a^2*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2])/(a + b)) + 2*b*(2*EllipticF[(c + d*x)/2, 2] - (2*b*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2])/(a + b)) + ((-2*a*b*EllipticE[ArcSin[Sqrt[Cos[c + d*x]]], -1] + 2*b*(a + b)*EllipticF[ArcSin[Sqrt[Cos[c + d*x]]], -1] + (a^2 - 2*b^2)*EllipticPi[-(a/b), ArcSin[Sqrt[Cos[c + d*x]]], -1])*Sin[c + d*x])/(b*Sqrt[Sin[c + d*x]^2])))/(a*(a - b)*(a + b)))/(4*d)","A",1
828,1,239,154,3.8346606,"\int \frac{1}{\cos ^{\frac{5}{2}}(c+d x) (a+b \sec (c+d x))^2} \, dx","Integrate[1/(Cos[c + d*x]^(5/2)*(a + b*Sec[c + d*x])^2),x]","\frac{\frac{4 a^2 \sin (c+d x) \sqrt{\cos (c+d x)}}{\left(b^2-a^2\right) (a \cos (c+d x)+b)}+\frac{\frac{2 \left(3 a^2-4 b^2\right) \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a+b}+\frac{2 \sin (c+d x) \left(\left(a^2-2 b^2\right) \Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)+2 b (a+b) F\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)-2 a b E\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)\right)}{b \sqrt{\sin ^2(c+d x)}}+4 b \left(2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-\frac{2 b \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a+b}\right)}{(a-b) (a+b)}}{4 b d}","\frac{F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d \left(a^2-b^2\right)}+\frac{a E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b d \left(a^2-b^2\right)}+\frac{\left(a^2-3 b^2\right) \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{b d (a-b) (a+b)^2}-\frac{a^2 \sin (c+d x)}{b d \left(a^2-b^2\right) \sqrt{\cos (c+d x)} (a+b \sec (c+d x))}",1,"((4*a^2*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/((-a^2 + b^2)*(b + a*Cos[c + d*x])) + ((2*(3*a^2 - 4*b^2)*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2])/(a + b) + 4*b*(2*EllipticF[(c + d*x)/2, 2] - (2*b*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2])/(a + b)) + (2*(-2*a*b*EllipticE[ArcSin[Sqrt[Cos[c + d*x]]], -1] + 2*b*(a + b)*EllipticF[ArcSin[Sqrt[Cos[c + d*x]]], -1] + (a^2 - 2*b^2)*EllipticPi[-(a/b), ArcSin[Sqrt[Cos[c + d*x]]], -1])*Sin[c + d*x])/(b*Sqrt[Sin[c + d*x]^2]))/((a - b)*(a + b)))/(4*b*d)","A",1
829,1,278,219,3.5685734,"\int \frac{1}{\cos ^{\frac{7}{2}}(c+d x) (a+b \sec (c+d x))^2} \, dx","Integrate[1/(Cos[c + d*x]^(7/2)*(a + b*Sec[c + d*x])^2),x]","\frac{4 \sqrt{\cos (c+d x)} \left(\frac{a^3 \sin (c+d x)}{\left(a^2-b^2\right) (a \cos (c+d x)+b)}+2 \tan (c+d x)\right)-\frac{\frac{2 \left(9 a^3-10 a b^2\right) \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a+b}+\frac{\left(8 a^2 b-4 b^3\right) \left(2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-\frac{2 b \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a+b}\right)}{a}+\frac{2 \left(3 a^2-2 b^2\right) \sin (c+d x) \left(\left(a^2-2 b^2\right) \Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)+2 b (a+b) F\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)-2 a b E\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)\right)}{a b \sqrt{\sin ^2(c+d x)}}}{(a-b) (a+b)}}{4 b^2 d}","-\frac{a F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b d \left(a^2-b^2\right)}-\frac{\left(3 a^2-2 b^2\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b^2 d \left(a^2-b^2\right)}-\frac{a \left(3 a^2-5 b^2\right) \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{b^2 d (a-b) (a+b)^2}+\frac{\left(3 a^2-2 b^2\right) \sin (c+d x)}{b^2 d \left(a^2-b^2\right) \sqrt{\cos (c+d x)}}-\frac{a^2 \sin (c+d x)}{b d \left(a^2-b^2\right) \cos ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))}",1,"(-(((2*(9*a^3 - 10*a*b^2)*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2])/(a + b) + ((8*a^2*b - 4*b^3)*(2*EllipticF[(c + d*x)/2, 2] - (2*b*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2])/(a + b)))/a + (2*(3*a^2 - 2*b^2)*(-2*a*b*EllipticE[ArcSin[Sqrt[Cos[c + d*x]]], -1] + 2*b*(a + b)*EllipticF[ArcSin[Sqrt[Cos[c + d*x]]], -1] + (a^2 - 2*b^2)*EllipticPi[-(a/b), ArcSin[Sqrt[Cos[c + d*x]]], -1])*Sin[c + d*x])/(a*b*Sqrt[Sin[c + d*x]^2]))/((a - b)*(a + b))) + 4*Sqrt[Cos[c + d*x]]*((a^3*Sin[c + d*x])/((a^2 - b^2)*(b + a*Cos[c + d*x])) + 2*Tan[c + d*x]))/(4*b^2*d)","A",1
830,1,353,346,4.3472613,"\int \frac{\cos ^{\frac{3}{2}}(c+d x)}{(a+b \sec (c+d x))^3} \, dx","Integrate[Cos[c + d*x]^(3/2)/(a + b*Sec[c + d*x])^3,x]","\frac{\frac{-\frac{2 \left(56 a^4 b-73 a^2 b^3+35 b^5\right) \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a+b}+\frac{16 \left(2 a^4+14 a^2 b^2-7 b^4\right) \left((a+b) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-b \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{a+b}-\frac{6 \left(24 a^4-65 a^2 b^2+35 b^4\right) \sin (c+d x) \left(\left(a^2-2 b^2\right) \Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)+2 b (a+b) F\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)-2 a b E\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)\right)}{a^2 \sqrt{\sin ^2(c+d x)}}}{(a-b)^2 (a+b)^2}+\frac{4 \sin (c+d x) \sqrt{\cos (c+d x)} \left(4 a^6+4 \left(a^3-a b^2\right)^2 \cos (2 (c+d x))-57 a^2 b^4+a b \left(16 a^4-83 a^2 b^2+49 b^4\right) \cos (c+d x)+35 b^6\right)}{\left(a^2-b^2\right)^2 (a \cos (c+d x)+b)^2}}{48 a^3 d}","\frac{b^2 \left(13 a^2-7 b^2\right) \sin (c+d x) \sqrt{\cos (c+d x)}}{4 a^2 d \left(a^2-b^2\right)^2 (a+b \sec (c+d x))}+\frac{b^2 \sin (c+d x) \sqrt{\cos (c+d x)}}{2 a d \left(a^2-b^2\right) (a+b \sec (c+d x))^2}-\frac{b \left(24 a^4-65 a^2 b^2+35 b^4\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a^4 d \left(a^2-b^2\right)^2}-\frac{b^3 \left(63 a^4-86 a^2 b^2+35 b^4\right) \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a^5 d (a-b)^2 (a+b)^3}+\frac{\left(8 a^4-61 a^2 b^2+35 b^4\right) \sin (c+d x) \sqrt{\cos (c+d x)}}{12 a^3 d \left(a^2-b^2\right)^2}+\frac{\left(8 a^6+128 a^4 b^2-223 a^2 b^4+105 b^6\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{12 a^5 d \left(a^2-b^2\right)^2}",1,"((4*Sqrt[Cos[c + d*x]]*(4*a^6 - 57*a^2*b^4 + 35*b^6 + a*b*(16*a^4 - 83*a^2*b^2 + 49*b^4)*Cos[c + d*x] + 4*(a^3 - a*b^2)^2*Cos[2*(c + d*x)])*Sin[c + d*x])/((a^2 - b^2)^2*(b + a*Cos[c + d*x])^2) + ((-2*(56*a^4*b - 73*a^2*b^3 + 35*b^5)*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2])/(a + b) + (16*(2*a^4 + 14*a^2*b^2 - 7*b^4)*((a + b)*EllipticF[(c + d*x)/2, 2] - b*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2]))/(a + b) - (6*(24*a^4 - 65*a^2*b^2 + 35*b^4)*(-2*a*b*EllipticE[ArcSin[Sqrt[Cos[c + d*x]]], -1] + 2*b*(a + b)*EllipticF[ArcSin[Sqrt[Cos[c + d*x]]], -1] + (a^2 - 2*b^2)*EllipticPi[-(a/b), ArcSin[Sqrt[Cos[c + d*x]]], -1])*Sin[c + d*x])/(a^2*Sqrt[Sin[c + d*x]^2]))/((a - b)^2*(a + b)^2))/(48*a^3*d)","A",1
831,1,311,282,3.2659941,"\int \frac{\sqrt{\cos (c+d x)}}{(a+b \sec (c+d x))^3} \, dx","Integrate[Sqrt[Cos[c + d*x]]/(a + b*Sec[c + d*x])^3,x]","\frac{\frac{2 b^2 \sin (c+d x) \sqrt{\cos (c+d x)} \left(a \left(13 a^2-7 b^2\right) \cos (c+d x)+11 a^2 b-5 b^3\right)}{\left(a^2-b^2\right)^2 (a \cos (c+d x)+b)^2}+\frac{-\frac{8 \left(4 a^2 b-b^3\right) \left((a+b) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-b \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{a+b}+\frac{\left(8 a^4-7 a^2 b^2+5 b^4\right) \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a+b}+\frac{\left(8 a^4-29 a^2 b^2+15 b^4\right) \sin (c+d x) \left(\left(a^2-2 b^2\right) \Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)+2 b (a+b) F\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)-2 a b E\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)\right)}{a^2 b \sqrt{\sin ^2(c+d x)}}}{(a-b)^2 (a+b)^2}}{8 a^2 d}","\frac{b^2 \left(11 a^2-5 b^2\right) \sin (c+d x)}{4 a^2 d \left(a^2-b^2\right)^2 \sqrt{\cos (c+d x)} (a+b \sec (c+d x))}+\frac{b^2 \sin (c+d x)}{2 a d \left(a^2-b^2\right) \sqrt{\cos (c+d x)} (a+b \sec (c+d x))^2}-\frac{3 b \left(8 a^4-11 a^2 b^2+5 b^4\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a^4 d \left(a^2-b^2\right)^2}+\frac{b^2 \left(35 a^4-38 a^2 b^2+15 b^4\right) \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a^4 d (a-b)^2 (a+b)^3}+\frac{\left(8 a^4-29 a^2 b^2+15 b^4\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a^3 d \left(a^2-b^2\right)^2}",1,"((2*b^2*Sqrt[Cos[c + d*x]]*(11*a^2*b - 5*b^3 + a*(13*a^2 - 7*b^2)*Cos[c + d*x])*Sin[c + d*x])/((a^2 - b^2)^2*(b + a*Cos[c + d*x])^2) + (((8*a^4 - 7*a^2*b^2 + 5*b^4)*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2])/(a + b) - (8*(4*a^2*b - b^3)*((a + b)*EllipticF[(c + d*x)/2, 2] - b*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2]))/(a + b) + ((8*a^4 - 29*a^2*b^2 + 15*b^4)*(-2*a*b*EllipticE[ArcSin[Sqrt[Cos[c + d*x]]], -1] + 2*b*(a + b)*EllipticF[ArcSin[Sqrt[Cos[c + d*x]]], -1] + (a^2 - 2*b^2)*EllipticPi[-(a/b), ArcSin[Sqrt[Cos[c + d*x]]], -1])*Sin[c + d*x])/(a^2*b*Sqrt[Sin[c + d*x]^2]))/((a - b)^2*(a + b)^2))/(8*a^2*d)","A",1
832,1,286,263,3.1478486,"\int \frac{1}{\sqrt{\cos (c+d x)} (a+b \sec (c+d x))^3} \, dx","Integrate[1/(Sqrt[Cos[c + d*x]]*(a + b*Sec[c + d*x])^3),x]","\frac{\frac{-\frac{2 \left(5 a^2 b+b^3\right) \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a+b}+\frac{16 \left(2 a^2+b^2\right) \left((a+b) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-b \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{a+b}+\frac{6 \left(3 a^2-b^2\right) \sin (c+d x) \left(\left(a^2-2 b^2\right) \Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)+2 b (a+b) F\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)-2 a b E\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)\right)}{a^2 \sqrt{\sin ^2(c+d x)}}}{(a-b)^2 (a+b)^2}+\frac{4 b \sin (c+d x) \sqrt{\cos (c+d x)} \left(\left(3 a b^2-9 a^3\right) \cos (c+d x)-7 a^2 b+b^3\right)}{\left(a^2-b^2\right)^2 (a \cos (c+d x)+b)^2}}{16 a d}","\frac{3 b \left(3 a^2-b^2\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a^2 d \left(a^2-b^2\right)^2}-\frac{b \left(7 a^2-b^2\right) \sin (c+d x)}{4 a d \left(a^2-b^2\right)^2 \sqrt{\cos (c+d x)} (a+b \sec (c+d x))}-\frac{b \sin (c+d x)}{2 d \left(a^2-b^2\right) \sqrt{\cos (c+d x)} (a+b \sec (c+d x))^2}+\frac{\left(8 a^4-5 a^2 b^2+3 b^4\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a^3 d \left(a^2-b^2\right)^2}-\frac{3 b \left(5 a^4-2 a^2 b^2+b^4\right) \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a^3 d (a-b)^2 (a+b)^3}",1,"((4*b*Sqrt[Cos[c + d*x]]*(-7*a^2*b + b^3 + (-9*a^3 + 3*a*b^2)*Cos[c + d*x])*Sin[c + d*x])/((a^2 - b^2)^2*(b + a*Cos[c + d*x])^2) + ((-2*(5*a^2*b + b^3)*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2])/(a + b) + (16*(2*a^2 + b^2)*((a + b)*EllipticF[(c + d*x)/2, 2] - b*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2]))/(a + b) + (6*(3*a^2 - b^2)*(-2*a*b*EllipticE[ArcSin[Sqrt[Cos[c + d*x]]], -1] + 2*b*(a + b)*EllipticF[ArcSin[Sqrt[Cos[c + d*x]]], -1] + (a^2 - 2*b^2)*EllipticPi[-(a/b), ArcSin[Sqrt[Cos[c + d*x]]], -1])*Sin[c + d*x])/(a^2*Sqrt[Sin[c + d*x]^2]))/((a - b)^2*(a + b)^2))/(16*a*d)","A",1
833,1,272,246,2.2334623,"\int \frac{1}{\cos ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^3} \, dx","Integrate[1/(Cos[c + d*x]^(3/2)*(a + b*Sec[c + d*x])^3),x]","\frac{\frac{4 \sin (c+d x) \sqrt{\cos (c+d x)} \left(a \left(5 a^2+b^2\right) \cos (c+d x)+3 b \left(a^2+b^2\right)\right)}{\left(a^2-b^2\right)^2 (a \cos (c+d x)+b)^2}-\frac{-\frac{2 \left(a^2+5 b^2\right) \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a+b}+\frac{2 \left(5 a^2+b^2\right) \sin (c+d x) \left(\left(a^2-2 b^2\right) \Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)+2 b (a+b) F\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)-2 a b E\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)\right)}{a^2 b \sqrt{\sin ^2(c+d x)}}+24 b \left(2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-\frac{2 b \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a+b}\right)}{(a-b)^2 (a+b)^2}}{16 d}","-\frac{b \left(7 a^2-b^2\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a^2 d \left(a^2-b^2\right)^2}-\frac{\left(5 a^2+b^2\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a d \left(a^2-b^2\right)^2}+\frac{3 \left(a^2+b^2\right) \sin (c+d x)}{4 d \left(a^2-b^2\right)^2 \sqrt{\cos (c+d x)} (a+b \sec (c+d x))}+\frac{a \sin (c+d x)}{2 d \left(a^2-b^2\right) \sqrt{\cos (c+d x)} (a+b \sec (c+d x))^2}+\frac{\left(3 a^4+10 a^2 b^2-b^4\right) \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a^2 d (a-b)^2 (a+b)^3}",1,"((4*Sqrt[Cos[c + d*x]]*(3*b*(a^2 + b^2) + a*(5*a^2 + b^2)*Cos[c + d*x])*Sin[c + d*x])/((a^2 - b^2)^2*(b + a*Cos[c + d*x])^2) - ((-2*(a^2 + 5*b^2)*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2])/(a + b) + 24*b*(2*EllipticF[(c + d*x)/2, 2] - (2*b*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2])/(a + b)) + (2*(5*a^2 + b^2)*(-2*a*b*EllipticE[ArcSin[Sqrt[Cos[c + d*x]]], -1] + 2*b*(a + b)*EllipticF[ArcSin[Sqrt[Cos[c + d*x]]], -1] + (a^2 - 2*b^2)*EllipticPi[-(a/b), ArcSin[Sqrt[Cos[c + d*x]]], -1])*Sin[c + d*x])/(a^2*b*Sqrt[Sin[c + d*x]^2]))/((a - b)^2*(a + b)^2))/(16*d)","A",1
834,1,289,253,3.1872429,"\int \frac{1}{\cos ^{\frac{5}{2}}(c+d x) (a+b \sec (c+d x))^3} \, dx","Integrate[1/(Cos[c + d*x]^(5/2)*(a + b*Sec[c + d*x])^3),x]","\frac{\frac{\frac{6 \left(a^3-3 a b^2\right) \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a+b}+\frac{8 b \left(a^2+2 b^2\right) \left(2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-\frac{2 b \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a+b}\right)}{a}+\frac{2 \left(a^2+5 b^2\right) \sin (c+d x) \left(\left(a^2-2 b^2\right) \Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)+2 b (a+b) F\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)-2 a b E\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)\right)}{a b \sqrt{\sin ^2(c+d x)}}}{(a-b)^2 (a+b)^2}-\frac{4 a \sin (c+d x) \sqrt{\cos (c+d x)} \left(a \left(a^2+5 b^2\right) \cos (c+d x)-a^2 b+7 b^3\right)}{\left(a^2-b^2\right)^2 (a \cos (c+d x)+b)^2}}{16 b d}","\frac{3 \left(a^2+b^2\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a d \left(a^2-b^2\right)^2}+\frac{\left(a^2+5 b^2\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{4 b d \left(a^2-b^2\right)^2}-\frac{a^2 \sin (c+d x)}{2 b d \left(a^2-b^2\right) \sqrt{\cos (c+d x)} (a+b \sec (c+d x))^2}+\frac{a \left(a^2-7 b^2\right) \sin (c+d x)}{4 b d \left(a^2-b^2\right)^2 \sqrt{\cos (c+d x)} (a+b \sec (c+d x))}+\frac{\left(a^4-10 a^2 b^2-3 b^4\right) \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a b d (a-b)^2 (a+b)^3}",1,"((-4*a*Sqrt[Cos[c + d*x]]*(-(a^2*b) + 7*b^3 + a*(a^2 + 5*b^2)*Cos[c + d*x])*Sin[c + d*x])/((a^2 - b^2)^2*(b + a*Cos[c + d*x])^2) + ((6*(a^3 - 3*a*b^2)*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2])/(a + b) + (8*b*(a^2 + 2*b^2)*(2*EllipticF[(c + d*x)/2, 2] - (2*b*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2])/(a + b)))/a + (2*(a^2 + 5*b^2)*(-2*a*b*EllipticE[ArcSin[Sqrt[Cos[c + d*x]]], -1] + 2*b*(a + b)*EllipticF[ArcSin[Sqrt[Cos[c + d*x]]], -1] + (a^2 - 2*b^2)*EllipticPi[-(a/b), ArcSin[Sqrt[Cos[c + d*x]]], -1])*Sin[c + d*x])/(a*b*Sqrt[Sin[c + d*x]^2]))/((a - b)^2*(a + b)^2))/(16*b*d)","A",1
835,1,297,255,3.161572,"\int \frac{1}{\cos ^{\frac{7}{2}}(c+d x) (a+b \sec (c+d x))^3} \, dx","Integrate[1/(Cos[c + d*x]^(7/2)*(a + b*Sec[c + d*x])^3),x]","\frac{\frac{\frac{16 b \left(a^2-4 b^2\right) \left((a+b) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-b \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{a+b}+\frac{6 \left(a^2-3 b^2\right) \sin (c+d x) \left(\left(a^2-2 b^2\right) \Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)+2 b (a+b) F\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)-2 a b E\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)\right)}{b \sqrt{\sin ^2(c+d x)}}+\frac{2 \left(9 a^4-19 a^2 b^2+16 b^4\right) \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a+b}}{(a-b)^2 (a+b)^2}-\frac{4 a^2 \sin (c+d x) \sqrt{\cos (c+d x)} \left(3 a \left(a^2-3 b^2\right) \cos (c+d x)+5 a^2 b-11 b^3\right)}{\left(a^2-b^2\right)^2 (a \cos (c+d x)+b)^2}}{16 b^2 d}","\frac{\left(a^2-7 b^2\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{4 b d \left(a^2-b^2\right)^2}+\frac{3 a \left(a^2-3 b^2\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{4 b^2 d \left(a^2-b^2\right)^2}-\frac{a^2 \sin (c+d x)}{2 b d \left(a^2-b^2\right) \cos ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^2}-\frac{3 a^2 \left(a^2-3 b^2\right) \sin (c+d x)}{4 b^2 d \left(a^2-b^2\right)^2 \sqrt{\cos (c+d x)} (a+b \sec (c+d x))}+\frac{3 \left(a^4-2 a^2 b^2+5 b^4\right) \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{4 b^2 d (a-b)^2 (a+b)^3}",1,"((-4*a^2*Sqrt[Cos[c + d*x]]*(5*a^2*b - 11*b^3 + 3*a*(a^2 - 3*b^2)*Cos[c + d*x])*Sin[c + d*x])/((a^2 - b^2)^2*(b + a*Cos[c + d*x])^2) + ((2*(9*a^4 - 19*a^2*b^2 + 16*b^4)*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2])/(a + b) + (16*b*(a^2 - 4*b^2)*((a + b)*EllipticF[(c + d*x)/2, 2] - b*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2]))/(a + b) + (6*(a^2 - 3*b^2)*(-2*a*b*EllipticE[ArcSin[Sqrt[Cos[c + d*x]]], -1] + 2*b*(a + b)*EllipticF[ArcSin[Sqrt[Cos[c + d*x]]], -1] + (a^2 - 2*b^2)*EllipticPi[-(a/b), ArcSin[Sqrt[Cos[c + d*x]]], -1])*Sin[c + d*x])/(b*Sqrt[Sin[c + d*x]^2]))/((a - b)^2*(a + b)^2))/(16*b^2*d)","A",1
836,1,334,328,3.7432315,"\int \frac{1}{\cos ^{\frac{9}{2}}(c+d x) (a+b \sec (c+d x))^3} \, dx","Integrate[1/(Cos[c + d*x]^(9/2)*(a + b*Sec[c + d*x])^3),x]","\frac{4 \sqrt{\cos (c+d x)} \left(\frac{a^3 \sin (c+d x) \left(a \left(7 a^2-13 b^2\right) \cos (c+d x)+9 a^2 b-15 b^3\right)}{\left(a^2-b^2\right)^2 (a \cos (c+d x)+b)^2}+8 \tan (c+d x)\right)-\frac{\frac{2 \left(45 a^5-95 a^3 b^2+56 a b^4\right) \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a+b}+\frac{8 b \left(5 a^4-10 a^2 b^2+2 b^4\right) \left(2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-\frac{2 b \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a+b}\right)}{a}+\frac{2 \left(15 a^4-29 a^2 b^2+8 b^4\right) \sin (c+d x) \left(\left(a^2-2 b^2\right) \Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)+2 b (a+b) F\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)-2 a b E\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)\right)}{a b \sqrt{\sin ^2(c+d x)}}}{(a-b)^2 (a+b)^2}}{16 b^3 d}","-\frac{a \left(5 a^2-11 b^2\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{4 b^2 d \left(a^2-b^2\right)^2}-\frac{a^2 \left(5 a^2-11 b^2\right) \sin (c+d x)}{4 b^2 d \left(a^2-b^2\right)^2 \cos ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))}-\frac{a^2 \sin (c+d x)}{2 b d \left(a^2-b^2\right) \cos ^{\frac{5}{2}}(c+d x) (a+b \sec (c+d x))^2}-\frac{\left(15 a^4-29 a^2 b^2+8 b^4\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{4 b^3 d \left(a^2-b^2\right)^2}-\frac{a \left(15 a^4-38 a^2 b^2+35 b^4\right) \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{4 b^3 d (a-b)^2 (a+b)^3}+\frac{\left(15 a^4-29 a^2 b^2+8 b^4\right) \sin (c+d x)}{4 b^3 d \left(a^2-b^2\right)^2 \sqrt{\cos (c+d x)}}",1,"(-(((2*(45*a^5 - 95*a^3*b^2 + 56*a*b^4)*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2])/(a + b) + (8*b*(5*a^4 - 10*a^2*b^2 + 2*b^4)*(2*EllipticF[(c + d*x)/2, 2] - (2*b*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2])/(a + b)))/a + (2*(15*a^4 - 29*a^2*b^2 + 8*b^4)*(-2*a*b*EllipticE[ArcSin[Sqrt[Cos[c + d*x]]], -1] + 2*b*(a + b)*EllipticF[ArcSin[Sqrt[Cos[c + d*x]]], -1] + (a^2 - 2*b^2)*EllipticPi[-(a/b), ArcSin[Sqrt[Cos[c + d*x]]], -1])*Sin[c + d*x])/(a*b*Sqrt[Sin[c + d*x]^2]))/((a - b)^2*(a + b)^2)) + 4*Sqrt[Cos[c + d*x]]*((a^3*(9*a^2*b - 15*b^3 + a*(7*a^2 - 13*b^2)*Cos[c + d*x])*Sin[c + d*x])/((a^2 - b^2)^2*(b + a*Cos[c + d*x])^2) + 8*Tan[c + d*x]))/(16*b^3*d)","A",1
837,1,340,244,9.5795685,"\int \cos ^{\frac{5}{2}}(c+d x) \sqrt{a+b \sec (c+d x)} \, dx","Integrate[Cos[c + d*x]^(5/2)*Sqrt[a + b*Sec[c + d*x]],x]","\frac{2 \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)} \left(a \sin (c+d x) (3 a \cos (c+d x)+b)-\frac{\left(\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)\right)^{3/2} \left(-\left(9 a^2-2 b^2\right) \tan \left(\frac{1}{2} (c+d x)\right) \sec ^2\left(\frac{1}{2} (c+d x)\right)^{3/2} (a \cos (c+d x)+b)+i a \left(9 a^2+7 a b-2 b^2\right) \sec ^2\left(\frac{1}{2} (c+d x)\right) \sqrt{\frac{\sec ^2\left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b)}{a+b}} F\left(i \sinh ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)-i \left(9 a^3+9 a^2 b-2 a b^2-2 b^3\right) \sec ^2\left(\frac{1}{2} (c+d x)\right) \sqrt{\frac{\sec ^2\left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b)}{a+b}} E\left(i \sinh ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)\right)}{\sec ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+b)}\right)}{15 a^2 d}","-\frac{4 b \left(a^2-b^2\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{15 a^2 d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}+\frac{2 \left(9 a^2-2 b^2\right) \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{15 a^2 d \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{2 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \sqrt{a+b \sec (c+d x)}}{5 d}+\frac{2 b \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}{15 a d}",1,"(2*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*(a*(b + 3*a*Cos[c + d*x])*Sin[c + d*x] - ((Cos[(c + d*x)/2]^2*Sec[c + d*x])^(3/2)*((-I)*(9*a^3 + 9*a^2*b - 2*a*b^2 - 2*b^3)*EllipticE[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] + I*a*(9*a^2 + 7*a*b - 2*b^2)*EllipticF[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] - (9*a^2 - 2*b^2)*(b + a*Cos[c + d*x])*(Sec[(c + d*x)/2]^2)^(3/2)*Tan[(c + d*x)/2]))/((b + a*Cos[c + d*x])*Sec[c + d*x]^(3/2))))/(15*a^2*d)","C",0
838,1,273,192,8.8967797,"\int \cos ^{\frac{3}{2}}(c+d x) \sqrt{a+b \sec (c+d x)} \, dx","Integrate[Cos[c + d*x]^(3/2)*Sqrt[a + b*Sec[c + d*x]],x]","\frac{2 \cos ^{\frac{3}{2}}(c+d x) \sqrt{a+b \sec (c+d x)} \left(a^2 \sin (c+d x)+a b \tan \left(\frac{1}{2} (c+d x)\right)+a b \tan (c+d x)-i a (a+b) \sqrt{\sec (c+d x)} \sqrt{\sec (c+d x)+1} \sqrt{\frac{a \cos (c+d x)+b}{(a+b) (\cos (c+d x)+1)}} F\left(i \sinh ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)+i b (a+b) \sqrt{\sec (c+d x)} \sqrt{\sec (c+d x)+1} \sqrt{\frac{a \cos (c+d x)+b}{(a+b) (\cos (c+d x)+1)}} E\left(i \sinh ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)+b^2 \tan \left(\frac{1}{2} (c+d x)\right) \sec (c+d x)\right)}{3 a d (a \cos (c+d x)+b)}","\frac{2 \left(a^2-b^2\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{3 a d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}+\frac{2 \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}{3 d}+\frac{2 b \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{3 a d \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}",1,"(2*Cos[c + d*x]^(3/2)*Sqrt[a + b*Sec[c + d*x]]*(I*b*(a + b)*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sqrt[Sec[c + d*x]]*Sqrt[1 + Sec[c + d*x]] - I*a*(a + b)*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sqrt[Sec[c + d*x]]*Sqrt[1 + Sec[c + d*x]] + a^2*Sin[c + d*x] + a*b*Tan[(c + d*x)/2] + b^2*Sec[c + d*x]*Tan[(c + d*x)/2] + a*b*Tan[c + d*x]))/(3*a*d*(b + a*Cos[c + d*x]))","C",0
839,1,198,67,3.8761635,"\int \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)} \, dx","Integrate[Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]],x]","\frac{\sqrt{\cos (c+d x)} \sec ^2\left(\frac{1}{2} (c+d x)\right) \sqrt{a+b \sec (c+d x)} \left(\sin (c+d x) \sqrt{\frac{1}{\cos (c+d x)+1}} \sqrt{\frac{a \cos (c+d x)+b}{(a+b) (\cos (c+d x)+1)}}-i F\left(i \sinh ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)+i E\left(i \sinh ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)\right)}{d \sqrt{\frac{1}{\cos (c+d x)+1}} \sqrt{\frac{a \cos (c+d x)+b}{(a+b) (\cos (c+d x)+1)}}}","\frac{2 \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{d \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}",1,"(Sqrt[Cos[c + d*x]]*Sec[(c + d*x)/2]^2*Sqrt[a + b*Sec[c + d*x]]*(I*EllipticE[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)] - I*EllipticF[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)] + Sqrt[(1 + Cos[c + d*x])^(-1)]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*Sin[c + d*x]))/(d*Sqrt[(1 + Cos[c + d*x])^(-1)]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))])","C",1
840,1,14885,138,28.8414101,"\int \frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{\cos (c+d x)}} \, dx","Integrate[Sqrt[a + b*Sec[c + d*x]]/Sqrt[Cos[c + d*x]],x]","\text{Result too large to show}","\frac{2 a \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}+\frac{2 b \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}",1,"Result too large to show","C",0
841,1,33277,237,31.5371014,"\int \frac{\sqrt{a+b \sec (c+d x)}}{\cos ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[Sqrt[a + b*Sec[c + d*x]]/Cos[c + d*x]^(3/2),x]","\text{Result too large to show}","\frac{\sin (c+d x) \sqrt{a+b \sec (c+d x)}}{d \sqrt{\cos (c+d x)}}+\frac{b \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}-\frac{\sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{d \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{a \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}",1,"Result too large to show","C",0
842,1,383,303,11.5865468,"\int \cos ^{\frac{7}{2}}(c+d x) (a+b \sec (c+d x))^{3/2} \, dx","Integrate[Cos[c + d*x]^(7/2)*(a + b*Sec[c + d*x])^(3/2),x]","\frac{\cos ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2} \left(a \sin (c+d x) (a \cos (c+d x)+b) \left(15 a^2 \cos (2 (c+d x))+65 a^2+48 a b \cos (c+d x)+6 b^2\right)-\frac{2 \left(\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)\right)^{3/2} \left(2 b \left(3 b^2-41 a^2\right) \tan \left(\frac{1}{2} (c+d x)\right) \sec ^2\left(\frac{1}{2} (c+d x)\right)^{3/2} (a \cos (c+d x)+b)+i a \left(25 a^3+82 a^2 b+51 a b^2-6 b^3\right) \sec ^2\left(\frac{1}{2} (c+d x)\right) \sqrt{\frac{\sec ^2\left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b)}{a+b}} F\left(i \sinh ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)+2 i b \left(-41 a^3-41 a^2 b+3 a b^2+3 b^3\right) \sec ^2\left(\frac{1}{2} (c+d x)\right) \sqrt{\frac{\sec ^2\left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b)}{a+b}} E\left(i \sinh ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)\right)}{\sec ^{\frac{3}{2}}(c+d x)}\right)}{105 a^2 d (a \cos (c+d x)+b)^2}","\frac{2 \left(25 a^2+3 b^2\right) \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}{105 a d}+\frac{4 b \left(41 a^2-3 b^2\right) \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{105 a^2 d \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{2 \left(25 a^4-31 a^2 b^2+6 b^4\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{105 a^2 d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}+\frac{2 a \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) \sqrt{a+b \sec (c+d x)}}{7 d}+\frac{16 b \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \sqrt{a+b \sec (c+d x)}}{35 d}",1,"(Cos[c + d*x]^(3/2)*(a + b*Sec[c + d*x])^(3/2)*(a*(b + a*Cos[c + d*x])*(65*a^2 + 6*b^2 + 48*a*b*Cos[c + d*x] + 15*a^2*Cos[2*(c + d*x)])*Sin[c + d*x] - (2*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^(3/2)*((2*I)*b*(-41*a^3 - 41*a^2*b + 3*a*b^2 + 3*b^3)*EllipticE[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] + I*a*(25*a^3 + 82*a^2*b + 51*a*b^2 - 6*b^3)*EllipticF[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] + 2*b*(-41*a^2 + 3*b^2)*(b + a*Cos[c + d*x])*(Sec[(c + d*x)/2]^2)^(3/2)*Tan[(c + d*x)/2]))/Sec[c + d*x]^(3/2)))/(105*a^2*d*(b + a*Cos[c + d*x])^2)","C",0
843,1,344,240,9.2611431,"\int \cos ^{\frac{5}{2}}(c+d x) (a+b \sec (c+d x))^{3/2} \, dx","Integrate[Cos[c + d*x]^(5/2)*(a + b*Sec[c + d*x])^(3/2),x]","\frac{\cos ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2} \left(2 \sin (c+d x) (a \cos (c+d x)+b) (a \cos (c+d x)+2 b)-\frac{2 \left(\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)\right)^{3/2} \left(-\left(3 a^2+b^2\right) \tan \left(\frac{1}{2} (c+d x)\right) \sec ^2\left(\frac{1}{2} (c+d x)\right)^{3/2} (a \cos (c+d x)+b)+i a \left(3 a^2+4 a b+b^2\right) \sec ^2\left(\frac{1}{2} (c+d x)\right) \sqrt{\frac{\sec ^2\left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b)}{a+b}} F\left(i \sinh ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)-i \left(3 a^3+3 a^2 b+a b^2+b^3\right) \sec ^2\left(\frac{1}{2} (c+d x)\right) \sqrt{\frac{\sec ^2\left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b)}{a+b}} E\left(i \sinh ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)\right)}{a \sec ^{\frac{3}{2}}(c+d x)}\right)}{5 d (a \cos (c+d x)+b)^2}","\frac{2 b \left(a^2-b^2\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{5 a d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}+\frac{2 \left(3 a^2+b^2\right) \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{5 a d \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{2 a \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \sqrt{a+b \sec (c+d x)}}{5 d}+\frac{4 b \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}{5 d}",1,"(Cos[c + d*x]^(3/2)*(a + b*Sec[c + d*x])^(3/2)*(2*(b + a*Cos[c + d*x])*(2*b + a*Cos[c + d*x])*Sin[c + d*x] - (2*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^(3/2)*((-I)*(3*a^3 + 3*a^2*b + a*b^2 + b^3)*EllipticE[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] + I*a*(3*a^2 + 4*a*b + b^2)*EllipticF[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] - (3*a^2 + b^2)*(b + a*Cos[c + d*x])*(Sec[(c + d*x)/2]^2)^(3/2)*Tan[(c + d*x)/2]))/(a*Sec[c + d*x]^(3/2))))/(5*d*(b + a*Cos[c + d*x])^2)","C",0
844,1,284,187,7.4406604,"\int \cos ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2} \, dx","Integrate[Cos[c + d*x]^(3/2)*(a + b*Sec[c + d*x])^(3/2),x]","\frac{2 \sqrt{\cos (c+d x)} (a+b \sec (c+d x))^{3/2} \left(\frac{1}{2} a \sin (2 (c+d x)) (a \cos (c+d x)+b)+\frac{\sqrt{\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)} \left(-i \left(a^2+4 a b+3 b^2\right) \sqrt{\frac{\sec ^2\left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b)}{a+b}} F\left(i \sinh ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)+4 b \tan \left(\frac{1}{2} (c+d x)\right) \sqrt{\sec ^2\left(\frac{1}{2} (c+d x)\right)} (a \cos (c+d x)+b)+4 i b (a+b) \sqrt{\frac{\sec ^2\left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b)}{a+b}} E\left(i \sinh ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)\right)}{\sec ^{\frac{3}{2}}(c+d x)}\right)}{3 d (a \cos (c+d x)+b)^2}","\frac{2 \left(a^2-b^2\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{3 d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}+\frac{2 a \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}{3 d}+\frac{8 b \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{3 d \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}",1,"(2*Sqrt[Cos[c + d*x]]*(a + b*Sec[c + d*x])^(3/2)*((a*(b + a*Cos[c + d*x])*Sin[2*(c + d*x)])/2 + (Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*((4*I)*b*(a + b)*EllipticE[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] - I*(a^2 + 4*a*b + 3*b^2)*EllipticF[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] + 4*b*(b + a*Cos[c + d*x])*Sqrt[Sec[(c + d*x)/2]^2]*Tan[(c + d*x)/2]))/Sec[c + d*x]^(3/2)))/(3*d*(b + a*Cos[c + d*x])^2)","C",0
845,1,25369,209,30.2142428,"\int \sqrt{\cos (c+d x)} (a+b \sec (c+d x))^{3/2} \, dx","Integrate[Sqrt[Cos[c + d*x]]*(a + b*Sec[c + d*x])^(3/2),x]","\text{Result too large to show}","\frac{2 b^2 \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}+\frac{2 a b \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}+\frac{2 a \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{d \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}",1,"Result too large to show","C",0
846,1,34674,249,31.9251381,"\int \frac{(a+b \sec (c+d x))^{3/2}}{\sqrt{\cos (c+d x)}} \, dx","Integrate[(a + b*Sec[c + d*x])^(3/2)/Sqrt[Cos[c + d*x]],x]","\text{Result too large to show}","\frac{\left(2 a^2+b^2\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}+\frac{b \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{d \sqrt{\cos (c+d x)}}-\frac{b \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{d \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{3 a b \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}",1,"Result too large to show","C",0
847,1,51315,299,32.8682524,"\int \frac{(a+b \sec (c+d x))^{3/2}}{\cos ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[(a + b*Sec[c + d*x])^(3/2)/Cos[c + d*x]^(3/2),x]","\text{Result too large to show}","\frac{\left(3 a^2+4 b^2\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{4 d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}+\frac{b \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{2 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{5 a \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{4 d \sqrt{\cos (c+d x)}}+\frac{7 a b \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{4 d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}-\frac{5 a \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{4 d \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}",1,"Result too large to show","C",0
848,1,477,363,14.5797314,"\int \cos ^{\frac{9}{2}}(c+d x) (a+b \sec (c+d x))^{5/2} \, dx","Integrate[Cos[c + d*x]^(9/2)*(a + b*Sec[c + d*x])^(5/2),x]","\frac{\cos ^{\frac{5}{2}}(c+d x) \left(\frac{1}{630} \left(133 a^2+150 b^2\right) \sin (2 (c+d x))+\frac{b \left(747 a^2+20 b^2\right) \sin (c+d x)}{630 a}+\frac{1}{36} a^2 \sin (4 (c+d x))+\frac{19}{126} a b \sin (3 (c+d x))\right) (a+b \sec (c+d x))^{5/2}}{d (a \cos (c+d x)+b)^2}-\frac{2 \cos ^{\frac{3}{2}}(c+d x) \left(\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)\right)^{3/2} (a+b \sec (c+d x))^{5/2} \left(-\left(147 a^4+279 a^2 b^2-10 b^4\right) \tan \left(\frac{1}{2} (c+d x)\right) \sec ^2\left(\frac{1}{2} (c+d x)\right)^{3/2} (a \cos (c+d x)+b)+i a \left(147 a^4+261 a^3 b+279 a^2 b^2+155 a b^3-10 b^4\right) \sec ^2\left(\frac{1}{2} (c+d x)\right) \sqrt{\frac{\sec ^2\left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b)}{a+b}} F\left(i \sinh ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)-i \left(147 a^5+147 a^4 b+279 a^3 b^2+279 a^2 b^3-10 a b^4-10 b^5\right) \sec ^2\left(\frac{1}{2} (c+d x)\right) \sqrt{\frac{\sec ^2\left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b)}{a+b}} E\left(i \sinh ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)\right)}{315 a^2 d \sec ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+b)^3}","\frac{2 \left(49 a^2+75 b^2\right) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \sqrt{a+b \sec (c+d x)}}{315 d}+\frac{2 b \left(163 a^2+5 b^2\right) \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}{315 a d}+\frac{2 a^2 \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x) \sqrt{a+b \sec (c+d x)}}{9 d}+\frac{4 b \left(57 a^4-62 a^2 b^2+5 b^4\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{315 a^2 d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}+\frac{2 \left(147 a^4+279 a^2 b^2-10 b^4\right) \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{315 a^2 d \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{38 a b \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) \sqrt{a+b \sec (c+d x)}}{63 d}",1,"(Cos[c + d*x]^(5/2)*(a + b*Sec[c + d*x])^(5/2)*((b*(747*a^2 + 20*b^2)*Sin[c + d*x])/(630*a) + ((133*a^2 + 150*b^2)*Sin[2*(c + d*x)])/630 + (19*a*b*Sin[3*(c + d*x)])/126 + (a^2*Sin[4*(c + d*x)])/36))/(d*(b + a*Cos[c + d*x])^2) - (2*Cos[c + d*x]^(3/2)*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^(3/2)*(a + b*Sec[c + d*x])^(5/2)*((-I)*(147*a^5 + 147*a^4*b + 279*a^3*b^2 + 279*a^2*b^3 - 10*a*b^4 - 10*b^5)*EllipticE[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] + I*a*(147*a^4 + 261*a^3*b + 279*a^2*b^2 + 155*a*b^3 - 10*b^4)*EllipticF[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] - (147*a^4 + 279*a^2*b^2 - 10*b^4)*(b + a*Cos[c + d*x])*(Sec[(c + d*x)/2]^2)^(3/2)*Tan[(c + d*x)/2]))/(315*a^2*d*(b + a*Cos[c + d*x])^3*Sec[c + d*x]^(5/2))","C",0
849,1,419,303,12.1369308,"\int \cos ^{\frac{7}{2}}(c+d x) (a+b \sec (c+d x))^{5/2} \, dx","Integrate[Cos[c + d*x]^(7/2)*(a + b*Sec[c + d*x])^(5/2),x]","\frac{\cos ^{\frac{5}{2}}(c+d x) \left(\frac{1}{42} \left(23 a^2+36 b^2\right) \sin (c+d x)+\frac{1}{14} a^2 \sin (3 (c+d x))+\frac{3}{7} a b \sin (2 (c+d x))\right) (a+b \sec (c+d x))^{5/2}}{d (a \cos (c+d x)+b)^2}+\frac{2 \cos ^{\frac{3}{2}}(c+d x) \left(\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)\right)^{3/2} (a+b \sec (c+d x))^{5/2} \left(b \left(29 a^2+3 b^2\right) \tan \left(\frac{1}{2} (c+d x)\right) \sec ^2\left(\frac{1}{2} (c+d x)\right)^{3/2} (a \cos (c+d x)+b)-i a \left(5 a^3+29 a^2 b+27 a b^2+3 b^3\right) \sec ^2\left(\frac{1}{2} (c+d x)\right) \sqrt{\frac{\sec ^2\left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b)}{a+b}} F\left(i \sinh ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)+i b \left(29 a^3+29 a^2 b+3 a b^2+3 b^3\right) \sec ^2\left(\frac{1}{2} (c+d x)\right) \sqrt{\frac{\sec ^2\left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b)}{a+b}} E\left(i \sinh ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)\right)}{21 a d \sec ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+b)^3}","\frac{2 \left(5 a^2+9 b^2\right) \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}{21 d}+\frac{2 b \left(29 a^2+3 b^2\right) \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{21 a d \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{2 a^2 \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) \sqrt{a+b \sec (c+d x)}}{7 d}+\frac{2 \left(5 a^4-2 a^2 b^2-3 b^4\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{21 a d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}+\frac{6 a b \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \sqrt{a+b \sec (c+d x)}}{7 d}",1,"(Cos[c + d*x]^(5/2)*(a + b*Sec[c + d*x])^(5/2)*(((23*a^2 + 36*b^2)*Sin[c + d*x])/42 + (3*a*b*Sin[2*(c + d*x)])/7 + (a^2*Sin[3*(c + d*x)])/14))/(d*(b + a*Cos[c + d*x])^2) + (2*Cos[c + d*x]^(3/2)*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^(3/2)*(a + b*Sec[c + d*x])^(5/2)*(I*b*(29*a^3 + 29*a^2*b + 3*a*b^2 + 3*b^3)*EllipticE[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] - I*a*(5*a^3 + 29*a^2*b + 27*a*b^2 + 3*b^3)*EllipticF[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] + b*(29*a^2 + 3*b^2)*(b + a*Cos[c + d*x])*(Sec[(c + d*x)/2]^2)^(3/2)*Tan[(c + d*x)/2]))/(21*a*d*(b + a*Cos[c + d*x])^3*Sec[c + d*x]^(5/2))","C",0
850,1,391,239,12.4920632,"\int \cos ^{\frac{5}{2}}(c+d x) (a+b \sec (c+d x))^{5/2} \, dx","Integrate[Cos[c + d*x]^(5/2)*(a + b*Sec[c + d*x])^(5/2),x]","\frac{\cos ^{\frac{5}{2}}(c+d x) \left(\frac{1}{5} a^2 \sin (2 (c+d x))+\frac{22}{15} a b \sin (c+d x)\right) (a+b \sec (c+d x))^{5/2}}{d (a \cos (c+d x)+b)^2}-\frac{2 \cos ^{\frac{3}{2}}(c+d x) \left(\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)\right)^{3/2} (a+b \sec (c+d x))^{5/2} \left(-\left(9 a^2+23 b^2\right) \tan \left(\frac{1}{2} (c+d x)\right) \sec ^2\left(\frac{1}{2} (c+d x)\right)^{3/2} (a \cos (c+d x)+b)+i \left(9 a^3+17 a^2 b+23 a b^2+15 b^3\right) \sec ^2\left(\frac{1}{2} (c+d x)\right) \sqrt{\frac{\sec ^2\left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b)}{a+b}} F\left(i \sinh ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)-i \left(9 a^3+9 a^2 b+23 a b^2+23 b^3\right) \sec ^2\left(\frac{1}{2} (c+d x)\right) \sqrt{\frac{\sec ^2\left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b)}{a+b}} E\left(i \sinh ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)\right)}{15 d \sec ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+b)^3}","\frac{16 b \left(a^2-b^2\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{15 d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}+\frac{2 \left(9 a^2+23 b^2\right) \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{15 d \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{2 a^2 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \sqrt{a+b \sec (c+d x)}}{5 d}+\frac{22 a b \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}{15 d}",1,"(Cos[c + d*x]^(5/2)*(a + b*Sec[c + d*x])^(5/2)*((22*a*b*Sin[c + d*x])/15 + (a^2*Sin[2*(c + d*x)])/5))/(d*(b + a*Cos[c + d*x])^2) - (2*Cos[c + d*x]^(3/2)*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^(3/2)*(a + b*Sec[c + d*x])^(5/2)*((-I)*(9*a^3 + 9*a^2*b + 23*a*b^2 + 23*b^3)*EllipticE[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] + I*(9*a^3 + 17*a^2*b + 23*a*b^2 + 15*b^3)*EllipticF[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] - (9*a^2 + 23*b^2)*(b + a*Cos[c + d*x])*(Sec[(c + d*x)/2]^2)^(3/2)*Tan[(c + d*x)/2]))/(15*d*(b + a*Cos[c + d*x])^3*Sec[c + d*x]^(5/2))","C",0
851,1,36372,262,32.8850751,"\int \cos ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^{5/2} \, dx","Integrate[Cos[c + d*x]^(3/2)*(a + b*Sec[c + d*x])^(5/2),x]","\text{Result too large to show}","\frac{2 a \left(a^2+2 b^2\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{3 d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}+\frac{2 a^2 \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}{3 d}+\frac{2 b^3 \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}+\frac{14 a b \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{3 d \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}",1,"Result too large to show","C",0
852,1,44191,263,32.1704249,"\int \sqrt{\cos (c+d x)} (a+b \sec (c+d x))^{5/2} \, dx","Integrate[Sqrt[Cos[c + d*x]]*(a + b*Sec[c + d*x])^(5/2),x]","\text{Result too large to show}","\frac{b \left(4 a^2+b^2\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}+\frac{\left(2 a^2-b^2\right) \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{d \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{b^2 \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{d \sqrt{\cos (c+d x)}}+\frac{5 a b^2 \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}",1,"Result too large to show","C",0
853,1,52888,314,32.3314858,"\int \frac{(a+b \sec (c+d x))^{5/2}}{\sqrt{\cos (c+d x)}} \, dx","Integrate[(a + b*Sec[c + d*x])^(5/2)/Sqrt[Cos[c + d*x]],x]","\text{Result too large to show}","\frac{a \left(8 a^2+11 b^2\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{4 d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}+\frac{b \left(15 a^2+4 b^2\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{4 d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}+\frac{b^2 \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{2 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{9 a b \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{4 d \sqrt{\cos (c+d x)}}-\frac{9 a b \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{4 d \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}",1,"Result too large to show","C",0
854,1,61979,369,32.6410104,"\int \frac{(a+b \sec (c+d x))^{5/2}}{\cos ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[(a + b*Sec[c + d*x])^(5/2)/Cos[c + d*x]^(3/2),x]","\text{Result too large to show}","\frac{\left(33 a^2+16 b^2\right) \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{24 d \sqrt{\cos (c+d x)}}+\frac{b \left(59 a^2+16 b^2\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{24 d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}-\frac{\left(33 a^2+16 b^2\right) \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{24 d \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{5 a \left(a^2+4 b^2\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{8 d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}+\frac{b^2 \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{3 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{13 a b \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{12 d \cos ^{\frac{3}{2}}(c+d x)}",1,"Result too large to show","C",0
855,1,340,249,9.6767588,"\int \frac{\cos ^{\frac{5}{2}}(c+d x)}{\sqrt{a+b \sec (c+d x)}} \, dx","Integrate[Cos[c + d*x]^(5/2)/Sqrt[a + b*Sec[c + d*x]],x]","\frac{2 a \sin (c+d x) (a \cos (c+d x)+b) (3 a \cos (c+d x)-4 b)+\frac{2 \left(\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)\right)^{3/2} \left(\left(9 a^2+8 b^2\right) \tan \left(\frac{1}{2} (c+d x)\right) \sec ^2\left(\frac{1}{2} (c+d x)\right)^{3/2} (a \cos (c+d x)+b)-i a \left(9 a^2+2 a b+8 b^2\right) \sec ^2\left(\frac{1}{2} (c+d x)\right) \sqrt{\frac{\sec ^2\left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b)}{a+b}} F\left(i \sinh ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)+i \left(9 a^3+9 a^2 b+8 a b^2+8 b^3\right) \sec ^2\left(\frac{1}{2} (c+d x)\right) \sqrt{\frac{\sec ^2\left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b)}{a+b}} E\left(i \sinh ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)\right)}{\sec ^{\frac{3}{2}}(c+d x)}}{15 a^3 d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}","-\frac{8 b \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}{15 a^2 d}-\frac{2 b \left(7 a^2+8 b^2\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{15 a^3 d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}+\frac{2 \left(9 a^2+8 b^2\right) \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{15 a^3 d \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{2 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \sqrt{a+b \sec (c+d x)}}{5 a d}",1,"(2*a*(b + a*Cos[c + d*x])*(-4*b + 3*a*Cos[c + d*x])*Sin[c + d*x] + (2*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^(3/2)*(I*(9*a^3 + 9*a^2*b + 8*a*b^2 + 8*b^3)*EllipticE[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] - I*a*(9*a^2 + 2*a*b + 8*b^2)*EllipticF[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] + (9*a^2 + 8*b^2)*(b + a*Cos[c + d*x])*(Sec[(c + d*x)/2]^2)^(3/2)*Tan[(c + d*x)/2]))/Sec[c + d*x]^(3/2))/(15*a^3*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]])","C",0
856,1,265,195,7.6433407,"\int \frac{\cos ^{\frac{3}{2}}(c+d x)}{\sqrt{a+b \sec (c+d x)}} \, dx","Integrate[Cos[c + d*x]^(3/2)/Sqrt[a + b*Sec[c + d*x]],x]","\frac{2 \sqrt{\cos (c+d x)} \left(a^2 \sin (c+d x)-2 a b \tan \left(\frac{1}{2} (c+d x)\right)+a b \tan (c+d x)-i a (a-2 b) \sqrt{\sec (c+d x)} \sqrt{\sec (c+d x)+1} \sqrt{\frac{a \cos (c+d x)+b}{(a+b) (\cos (c+d x)+1)}} F\left(i \sinh ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)-2 i b (a+b) \sqrt{\sec (c+d x)} \sqrt{\sec (c+d x)+1} \sqrt{\frac{a \cos (c+d x)+b}{(a+b) (\cos (c+d x)+1)}} E\left(i \sinh ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)-2 b^2 \tan \left(\frac{1}{2} (c+d x)\right) \sec (c+d x)\right)}{3 a^2 d \sqrt{a+b \sec (c+d x)}}","\frac{2 \left(a^2+2 b^2\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{3 a^2 d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}-\frac{4 b \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{3 a^2 d \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{2 \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}{3 a d}",1,"(2*Sqrt[Cos[c + d*x]]*((-2*I)*b*(a + b)*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sqrt[Sec[c + d*x]]*Sqrt[1 + Sec[c + d*x]] - I*a*(a - 2*b)*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sqrt[Sec[c + d*x]]*Sqrt[1 + Sec[c + d*x]] + a^2*Sin[c + d*x] - 2*a*b*Tan[(c + d*x)/2] - 2*b^2*Sec[c + d*x]*Tan[(c + d*x)/2] + a*b*Tan[c + d*x]))/(3*a^2*d*Sqrt[a + b*Sec[c + d*x]])","C",0
857,1,216,142,4.0261159,"\int \frac{\sqrt{\cos (c+d x)}}{\sqrt{a+b \sec (c+d x)}} \, dx","Integrate[Sqrt[Cos[c + d*x]]/Sqrt[a + b*Sec[c + d*x]],x]","\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sqrt{\sec (c+d x)+1} \left(\sqrt{\frac{1}{\cos (c+d x)+1}} \tan \left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b)-i a \sqrt{\frac{a \cos (c+d x)+b}{(a+b) (\cos (c+d x)+1)}} F\left(i \sinh ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)+i (a+b) \sqrt{\frac{a \cos (c+d x)+b}{(a+b) (\cos (c+d x)+1)}} E\left(i \sinh ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)\right)}{a d \sqrt{a+b \sec (c+d x)}}","\frac{2 \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{a d \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}-\frac{2 b \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{a d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}",1,"(2*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*Sqrt[1 + Sec[c + d*x]]*(I*(a + b)*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)] - I*a*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)] + Sqrt[(1 + Cos[c + d*x])^(-1)]*(b + a*Cos[c + d*x])*Tan[(c + d*x)/2]))/(a*d*Sqrt[a + b*Sec[c + d*x]])","C",1
858,1,102,67,0.893028,"\int \frac{1}{\sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}} \, dx","Integrate[1/(Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]),x]","-\frac{2 i \sqrt{\frac{a \cos (c+d x)+b}{(a+b) (\cos (c+d x)+1)}} F\left(i \sinh ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)}{d \sqrt{\cos (c+d x)} \sqrt{\frac{1}{\cos (c+d x)+1}} \sqrt{a+b \sec (c+d x)}}","\frac{2 \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}",1,"((-2*I)*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)])/(d*Sqrt[Cos[c + d*x]]*Sqrt[(1 + Cos[c + d*x])^(-1)]*Sqrt[a + b*Sec[c + d*x]])","C",1
859,1,14986,68,29.0522956,"\int \frac{1}{\cos ^{\frac{3}{2}}(c+d x) \sqrt{a+b \sec (c+d x)}} \, dx","Integrate[1/(Cos[c + d*x]^(3/2)*Sqrt[a + b*Sec[c + d*x]]),x]","\text{Result too large to show}","\frac{2 \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}",1,"Result too large to show","C",0
860,1,21698,246,30.9923611,"\int \frac{1}{\cos ^{\frac{5}{2}}(c+d x) \sqrt{a+b \sec (c+d x)}} \, dx","Integrate[1/(Cos[c + d*x]^(5/2)*Sqrt[a + b*Sec[c + d*x]]),x]","\text{Result too large to show}","\frac{\sin (c+d x) \sqrt{a+b \sec (c+d x)}}{b d \sqrt{\cos (c+d x)}}+\frac{\sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}-\frac{\sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{b d \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}-\frac{a \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{b d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}",1,"Result too large to show","C",0
861,1,51323,312,32.3065086,"\int \frac{1}{\cos ^{\frac{7}{2}}(c+d x) \sqrt{a+b \sec (c+d x)}} \, dx","Integrate[1/(Cos[c + d*x]^(7/2)*Sqrt[a + b*Sec[c + d*x]]),x]","\text{Result too large to show}","\frac{\left(3 a^2+4 b^2\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{4 b^2 d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}-\frac{3 a \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{4 b^2 d \sqrt{\cos (c+d x)}}+\frac{3 a \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{4 b^2 d \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{\sin (c+d x) \sqrt{a+b \sec (c+d x)}}{2 b d \cos ^{\frac{3}{2}}(c+d x)}-\frac{a \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{4 b d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}",1,"Result too large to show","C",0
862,1,419,360,12.7513547,"\int \frac{\cos ^{\frac{5}{2}}(c+d x)}{(a+b \sec (c+d x))^{3/2}} \, dx","Integrate[Cos[c + d*x]^(5/2)/(a + b*Sec[c + d*x])^(3/2),x]","\frac{(a \cos (c+d x)+b) \left(a \sec ^{\frac{3}{2}}(c+d x) \left(6 b \left(b^2-a^2\right) \sin (c+d x) (a \cos (c+d x)+b)+a \left(a^2-b^2\right) \sin (2 (c+d x)) (a \cos (c+d x)+b)+10 b^4 \sin (c+d x)\right)+2 \left(a^2+4 b^2\right) \left(\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)\right)^{3/2} \left(\left(3 a^2-4 b^2\right) \tan \left(\frac{1}{2} (c+d x)\right) \sec ^2\left(\frac{1}{2} (c+d x)\right)^{3/2} (a \cos (c+d x)+b)-i a \left(3 a^2-a b-4 b^2\right) \sec ^2\left(\frac{1}{2} (c+d x)\right) \sqrt{\frac{\sec ^2\left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b)}{a+b}} F\left(i \sinh ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)+i \left(3 a^3+3 a^2 b-4 a b^2-4 b^3\right) \sec ^2\left(\frac{1}{2} (c+d x)\right) \sqrt{\frac{\sec ^2\left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b)}{a+b}} E\left(i \sinh ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)\right)\right)}{5 a^4 d \left(a^2-b^2\right) \cos ^{\frac{3}{2}}(c+d x) \sec ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}","\frac{2 b^2 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{a d \left(a^2-b^2\right) \sqrt{a+b \sec (c+d x)}}+\frac{2 \left(a^2-6 b^2\right) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \sqrt{a+b \sec (c+d x)}}{5 a^2 d \left(a^2-b^2\right)}-\frac{8 b \left(a^2+4 b^2\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{5 a^4 d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}+\frac{2 \left(3 a^4+8 a^2 b^2-16 b^4\right) \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{5 a^4 d \left(a^2-b^2\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}-\frac{2 b \left(3 a^2-8 b^2\right) \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}{5 a^3 d \left(a^2-b^2\right)}",1,"((b + a*Cos[c + d*x])*(a*Sec[c + d*x]^(3/2)*(10*b^4*Sin[c + d*x] + 6*b*(-a^2 + b^2)*(b + a*Cos[c + d*x])*Sin[c + d*x] + a*(a^2 - b^2)*(b + a*Cos[c + d*x])*Sin[2*(c + d*x)]) + 2*(a^2 + 4*b^2)*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^(3/2)*(I*(3*a^3 + 3*a^2*b - 4*a*b^2 - 4*b^3)*EllipticE[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] - I*a*(3*a^2 - a*b - 4*b^2)*EllipticF[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] + (3*a^2 - 4*b^2)*(b + a*Cos[c + d*x])*(Sec[(c + d*x)/2]^2)^(3/2)*Tan[(c + d*x)/2])))/(5*a^4*(a^2 - b^2)*d*Cos[c + d*x]^(3/2)*Sec[c + d*x]^(3/2)*(a + b*Sec[c + d*x])^(3/2))","C",0
863,1,382,289,9.3161067,"\int \frac{\cos ^{\frac{3}{2}}(c+d x)}{(a+b \sec (c+d x))^{3/2}} \, dx","Integrate[Cos[c + d*x]^(3/2)/(a + b*Sec[c + d*x])^(3/2),x]","-\frac{2 (a \cos (c+d x)+b) \left(a \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \left(3 b^3-\left(a^2-b^2\right) (a \cos (c+d x)+b)\right)-\left(\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)\right)^{3/2} \left(b \left(8 b^2-5 a^2\right) \tan \left(\frac{1}{2} (c+d x)\right) \sec ^2\left(\frac{1}{2} (c+d x)\right)^{3/2} (a \cos (c+d x)+b)-i a \left(a^3-5 a^2 b+2 a b^2+8 b^3\right) \sec ^2\left(\frac{1}{2} (c+d x)\right) \sqrt{\frac{\sec ^2\left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b)}{a+b}} F\left(i \sinh ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)+i b \left(-5 a^3-5 a^2 b+8 a b^2+8 b^3\right) \sec ^2\left(\frac{1}{2} (c+d x)\right) \sqrt{\frac{\sec ^2\left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b)}{a+b}} E\left(i \sinh ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)\right)\right)}{3 a^3 d \left(a^2-b^2\right) \cos ^{\frac{3}{2}}(c+d x) \sec ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}","\frac{2 b^2 \sin (c+d x) \sqrt{\cos (c+d x)}}{a d \left(a^2-b^2\right) \sqrt{a+b \sec (c+d x)}}+\frac{2 \left(a^2-4 b^2\right) \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}{3 a^2 d \left(a^2-b^2\right)}+\frac{2 \left(a^2+8 b^2\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{3 a^3 d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}-\frac{2 b \left(5 a^2-8 b^2\right) \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{3 a^3 d \left(a^2-b^2\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}",1,"(-2*(b + a*Cos[c + d*x])*(a*(3*b^3 - (a^2 - b^2)*(b + a*Cos[c + d*x]))*Sec[c + d*x]^(3/2)*Sin[c + d*x] - (Cos[(c + d*x)/2]^2*Sec[c + d*x])^(3/2)*(I*b*(-5*a^3 - 5*a^2*b + 8*a*b^2 + 8*b^3)*EllipticE[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] - I*a*(a^3 - 5*a^2*b + 2*a*b^2 + 8*b^3)*EllipticF[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] + b*(-5*a^2 + 8*b^2)*(b + a*Cos[c + d*x])*(Sec[(c + d*x)/2]^2)^(3/2)*Tan[(c + d*x)/2])))/(3*a^3*(a^2 - b^2)*d*Cos[c + d*x]^(3/2)*Sec[c + d*x]^(3/2)*(a + b*Sec[c + d*x])^(3/2))","C",0
864,1,330,214,9.6593393,"\int \frac{\sqrt{\cos (c+d x)}}{(a+b \sec (c+d x))^{3/2}} \, dx","Integrate[Sqrt[Cos[c + d*x]]/(a + b*Sec[c + d*x])^(3/2),x]","\frac{2 (a \cos (c+d x)+b) \left(a b^2 \sin (c+d x)+\frac{\left(\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)\right)^{3/2} \left(\left(a^2-2 b^2\right) \tan \left(\frac{1}{2} (c+d x)\right) \sec ^2\left(\frac{1}{2} (c+d x)\right)^{3/2} (a \cos (c+d x)+b)-i a \left(a^2-a b-2 b^2\right) \sec ^2\left(\frac{1}{2} (c+d x)\right) \sqrt{\frac{\sec ^2\left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b)}{a+b}} F\left(i \sinh ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)+i \left(a^3+a^2 b-2 a b^2-2 b^3\right) \sec ^2\left(\frac{1}{2} (c+d x)\right) \sqrt{\frac{\sec ^2\left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b)}{a+b}} E\left(i \sinh ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)\right)}{\sec ^{\frac{3}{2}}(c+d x)}\right)}{a^2 d \left(a^2-b^2\right) \cos ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}","\frac{2 b^2 \sin (c+d x)}{a d \left(a^2-b^2\right) \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}+\frac{2 \left(a^2-2 b^2\right) \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{a^2 d \left(a^2-b^2\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}-\frac{4 b \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{a^2 d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}",1,"(2*(b + a*Cos[c + d*x])*(a*b^2*Sin[c + d*x] + ((Cos[(c + d*x)/2]^2*Sec[c + d*x])^(3/2)*(I*(a^3 + a^2*b - 2*a*b^2 - 2*b^3)*EllipticE[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] - I*a*(a^2 - a*b - 2*b^2)*EllipticF[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] + (a^2 - 2*b^2)*(b + a*Cos[c + d*x])*(Sec[(c + d*x)/2]^2)^(3/2)*Tan[(c + d*x)/2]))/Sec[c + d*x]^(3/2)))/(a^2*(a^2 - b^2)*d*Cos[c + d*x]^(3/2)*(a + b*Sec[c + d*x])^(3/2))","C",0
865,1,245,200,7.8005957,"\int \frac{1}{\sqrt{\cos (c+d x)} (a+b \sec (c+d x))^{3/2}} \, dx","Integrate[1/(Sqrt[Cos[c + d*x]]*(a + b*Sec[c + d*x])^(3/2)),x]","\frac{2 \sqrt{\cos (c+d x)} \sec ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+b) \left(b (b-a) \tan \left(\frac{1}{2} (c+d x)\right) \sqrt{\sec (c+d x)}-i a (a+b) \sqrt{\sec (c+d x)+1} \sqrt{\frac{a \cos (c+d x)+b}{(a+b) (\cos (c+d x)+1)}} F\left(i \sinh ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)+i b (a+b) \sqrt{\sec (c+d x)+1} \sqrt{\frac{a \cos (c+d x)+b}{(a+b) (\cos (c+d x)+1)}} E\left(i \sinh ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)\right)}{a d \left(a^2-b^2\right) (a+b \sec (c+d x))^{3/2}}","-\frac{2 b \sin (c+d x)}{d \left(a^2-b^2\right) \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}+\frac{2 b \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{a d \left(a^2-b^2\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{2 \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{a d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}",1,"(2*Sqrt[Cos[c + d*x]]*(b + a*Cos[c + d*x])*Sec[c + d*x]^(3/2)*(I*b*(a + b)*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sqrt[1 + Sec[c + d*x]] - I*a*(a + b)*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sqrt[1 + Sec[c + d*x]] + b*(-a + b)*Sqrt[Sec[c + d*x]]*Tan[(c + d*x)/2]))/(a*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^(3/2))","C",1
866,1,260,126,7.8453577,"\int \frac{1}{\cos ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}} \, dx","Integrate[1/(Cos[c + d*x]^(3/2)*(a + b*Sec[c + d*x])^(3/2)),x]","\frac{\sqrt{\cos (c+d x)} \sec ^2\left(\frac{1}{2} (c+d x)\right) \sec ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+b) \left((a-b) \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{\frac{a+b \sec (c+d x)}{(a+b) (\sec (c+d x)+1)}}+i \sqrt{\sec (c+d x)+1} (a \cos (c+d x)+b) F\left(i \sinh ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)-i \sqrt{\sec (c+d x)+1} (a \cos (c+d x)+b) E\left(i \sinh ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)\right)}{d \left(a^2-b^2\right) \sqrt{\frac{a \cos (c+d x)+b}{(a+b) (\cos (c+d x)+1)}} (a+b \sec (c+d x))^{3/2}}","\frac{2 a \sin (c+d x)}{d \left(a^2-b^2\right) \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}-\frac{2 \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{d \left(a^2-b^2\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}",1,"(Sqrt[Cos[c + d*x]]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Sec[c + d*x]^(3/2)*((-I)*(b + a*Cos[c + d*x])*EllipticE[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sqrt[1 + Sec[c + d*x]] + I*(b + a*Cos[c + d*x])*EllipticF[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sqrt[1 + Sec[c + d*x]] + (a - b)*Sqrt[Sec[c + d*x]]*Sqrt[(a + b*Sec[c + d*x])/((a + b)*(1 + Sec[c + d*x]))]*Sin[c + d*x]))/((a^2 - b^2)*d*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*(a + b*Sec[c + d*x])^(3/2))","C",0
867,1,47811,206,32.2707166,"\int \frac{1}{\cos ^{\frac{5}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}} \, dx","Integrate[1/(Cos[c + d*x]^(5/2)*(a + b*Sec[c + d*x])^(3/2)),x]","\text{Result too large to show}","-\frac{2 a^2 \sin (c+d x)}{b d \left(a^2-b^2\right) \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}+\frac{2 a \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{b d \left(a^2-b^2\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{2 \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{b d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}",1,"Result too large to show","C",0
868,1,51610,345,32.5648867,"\int \frac{1}{\cos ^{\frac{7}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}} \, dx","Integrate[1/(Cos[c + d*x]^(7/2)*(a + b*Sec[c + d*x])^(3/2)),x]","\text{Result too large to show}","-\frac{2 a^2 \sin (c+d x)}{b d \left(a^2-b^2\right) \cos ^{\frac{3}{2}}(c+d x) \sqrt{a+b \sec (c+d x)}}+\frac{\left(3 a^2-b^2\right) \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{b^2 d \left(a^2-b^2\right) \sqrt{\cos (c+d x)}}-\frac{\left(3 a^2-b^2\right) \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{b^2 d \left(a^2-b^2\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}-\frac{3 a \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{b^2 d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}+\frac{\sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{b d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}",1,"Result too large to show","C",0
869,1,527,391,14.7112847,"\int \frac{\cos ^{\frac{3}{2}}(c+d x)}{(a+b \sec (c+d x))^{5/2}} \, dx","Integrate[Cos[c + d*x]^(3/2)/(a + b*Sec[c + d*x])^(5/2),x]","\frac{(a \cos (c+d x)+b)^3 \left(\frac{2 \sin (c+d x)}{3 a^3}+\frac{2 b^4 \sin (c+d x)}{3 a^3 \left(a^2-b^2\right) (a \cos (c+d x)+b)^2}+\frac{8 \left(2 b^5 \sin (c+d x)-3 a^2 b^3 \sin (c+d x)\right)}{3 a^3 \left(a^2-b^2\right)^2 (a \cos (c+d x)+b)}\right)}{d \cos ^{\frac{5}{2}}(c+d x) (a+b \sec (c+d x))^{5/2}}+\frac{2 \cos ^{\frac{3}{2}}(c+d x) \sec ^{\frac{5}{2}}(c+d x) \left(\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)\right)^{3/2} (a \cos (c+d x)+b)^2 \left(-4 b \left(2 a^4-7 a^2 b^2+4 b^4\right) \tan \left(\frac{1}{2} (c+d x)\right) \sec ^2\left(\frac{1}{2} (c+d x)\right)^{3/2} (a \cos (c+d x)+b)-i a \left(a^5-8 a^4 b+7 a^3 b^2+28 a^2 b^3-4 a b^4-16 b^5\right) \sec ^2\left(\frac{1}{2} (c+d x)\right) \sqrt{\frac{\sec ^2\left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b)}{a+b}} F\left(i \sinh ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)-4 i b \left(2 a^5+2 a^4 b-7 a^3 b^2-7 a^2 b^3+4 a b^4+4 b^5\right) \sec ^2\left(\frac{1}{2} (c+d x)\right) \sqrt{\frac{\sec ^2\left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b)}{a+b}} E\left(i \sinh ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)\right)}{3 a^4 d \left(a^2-b^2\right)^2 (a+b \sec (c+d x))^{5/2}}","\frac{4 b^2 \left(5 a^2-3 b^2\right) \sin (c+d x) \sqrt{\cos (c+d x)}}{3 a^2 d \left(a^2-b^2\right)^2 \sqrt{a+b \sec (c+d x)}}+\frac{2 b^2 \sin (c+d x) \sqrt{\cos (c+d x)}}{3 a d \left(a^2-b^2\right) (a+b \sec (c+d x))^{3/2}}+\frac{2 \left(a^4+16 a^2 b^2-16 b^4\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{3 a^4 d \left(a^2-b^2\right) \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}-\frac{8 b \left(2 a^4-7 a^2 b^2+4 b^4\right) \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{3 a^4 d \left(a^2-b^2\right)^2 \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{2 \left(a^4-13 a^2 b^2+8 b^4\right) \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}{3 a^3 d \left(a^2-b^2\right)^2}",1,"((b + a*Cos[c + d*x])^3*((2*Sin[c + d*x])/(3*a^3) + (2*b^4*Sin[c + d*x])/(3*a^3*(a^2 - b^2)*(b + a*Cos[c + d*x])^2) + (8*(-3*a^2*b^3*Sin[c + d*x] + 2*b^5*Sin[c + d*x]))/(3*a^3*(a^2 - b^2)^2*(b + a*Cos[c + d*x]))))/(d*Cos[c + d*x]^(5/2)*(a + b*Sec[c + d*x])^(5/2)) + (2*Cos[c + d*x]^(3/2)*(b + a*Cos[c + d*x])^2*Sec[c + d*x]^(5/2)*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^(3/2)*((-4*I)*b*(2*a^5 + 2*a^4*b - 7*a^3*b^2 - 7*a^2*b^3 + 4*a*b^4 + 4*b^5)*EllipticE[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] - I*a*(a^5 - 8*a^4*b + 7*a^3*b^2 + 28*a^2*b^3 - 4*a*b^4 - 16*b^5)*EllipticF[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] - 4*b*(2*a^4 - 7*a^2*b^2 + 4*b^4)*(b + a*Cos[c + d*x])*(Sec[(c + d*x)/2]^2)^(3/2)*Tan[(c + d*x)/2]))/(3*a^4*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x])^(5/2))","C",0
870,1,507,317,14.5906352,"\int \frac{\sqrt{\cos (c+d x)}}{(a+b \sec (c+d x))^{5/2}} \, dx","Integrate[Sqrt[Cos[c + d*x]]/(a + b*Sec[c + d*x])^(5/2),x]","\frac{(a \cos (c+d x)+b)^3 \left(-\frac{2 \left(5 b^4 \sin (c+d x)-9 a^2 b^2 \sin (c+d x)\right)}{3 a^2 \left(a^2-b^2\right)^2 (a \cos (c+d x)+b)}-\frac{2 b^3 \sin (c+d x)}{3 a^2 \left(a^2-b^2\right) (a \cos (c+d x)+b)^2}\right)}{d \cos ^{\frac{5}{2}}(c+d x) (a+b \sec (c+d x))^{5/2}}-\frac{2 \cos ^{\frac{3}{2}}(c+d x) \sec ^{\frac{5}{2}}(c+d x) \left(\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)\right)^{3/2} (a \cos (c+d x)+b)^2 \left(-\left(3 a^4-15 a^2 b^2+8 b^4\right) \tan \left(\frac{1}{2} (c+d x)\right) \sec ^2\left(\frac{1}{2} (c+d x)\right)^{3/2} (a \cos (c+d x)+b)+i a \left(3 a^4-6 a^3 b-15 a^2 b^2+2 a b^3+8 b^4\right) \sec ^2\left(\frac{1}{2} (c+d x)\right) \sqrt{\frac{\sec ^2\left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b)}{a+b}} F\left(i \sinh ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)-i \left(3 a^5+3 a^4 b-15 a^3 b^2-15 a^2 b^3+8 a b^4+8 b^5\right) \sec ^2\left(\frac{1}{2} (c+d x)\right) \sqrt{\frac{\sec ^2\left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b)}{a+b}} E\left(i \sinh ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)\right)}{3 a d \left(a^3-a b^2\right)^2 (a+b \sec (c+d x))^{5/2}}","\frac{8 b^2 \left(2 a^2-b^2\right) \sin (c+d x)}{3 a^2 d \left(a^2-b^2\right)^2 \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}+\frac{2 b^2 \sin (c+d x)}{3 a d \left(a^2-b^2\right) \sqrt{\cos (c+d x)} (a+b \sec (c+d x))^{3/2}}-\frac{2 b \left(9 a^2-8 b^2\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{3 a^3 d \left(a^2-b^2\right) \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}+\frac{2 \left(3 a^4-15 a^2 b^2+8 b^4\right) \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{3 a^3 d \left(a^2-b^2\right)^2 \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}",1,"((b + a*Cos[c + d*x])^3*((-2*b^3*Sin[c + d*x])/(3*a^2*(a^2 - b^2)*(b + a*Cos[c + d*x])^2) - (2*(-9*a^2*b^2*Sin[c + d*x] + 5*b^4*Sin[c + d*x]))/(3*a^2*(a^2 - b^2)^2*(b + a*Cos[c + d*x]))))/(d*Cos[c + d*x]^(5/2)*(a + b*Sec[c + d*x])^(5/2)) - (2*Cos[c + d*x]^(3/2)*(b + a*Cos[c + d*x])^2*Sec[c + d*x]^(5/2)*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^(3/2)*((-I)*(3*a^5 + 3*a^4*b - 15*a^3*b^2 - 15*a^2*b^3 + 8*a*b^4 + 8*b^5)*EllipticE[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] + I*a*(3*a^4 - 6*a^3*b - 15*a^2*b^2 + 2*a*b^3 + 8*b^4)*EllipticF[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] - (3*a^4 - 15*a^2*b^2 + 8*b^4)*(b + a*Cos[c + d*x])*(Sec[(c + d*x)/2]^2)^(3/2)*Tan[(c + d*x)/2]))/(3*a*(a^3 - a*b^2)^2*d*(a + b*Sec[c + d*x])^(5/2))","C",0
871,1,398,302,9.2427393,"\int \frac{1}{\sqrt{\cos (c+d x)} (a+b \sec (c+d x))^{5/2}} \, dx","Integrate[1/(Sqrt[Cos[c + d*x]]*(a + b*Sec[c + d*x])^(5/2)),x]","\frac{(a \cos (c+d x)+b)^2 \left(\frac{2 b \sin (c+d x) \left(\left(2 a b^2-6 a^3\right) \cos (c+d x)-5 a^2 b+b^3\right)}{a \left(a^2-b^2\right)^2 (a \cos (c+d x)+b)}-\frac{2 \left(\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)\right)^{3/2} \left(2 b \left(b^2-3 a^2\right) \tan \left(\frac{1}{2} (c+d x)\right) \sec ^2\left(\frac{1}{2} (c+d x)\right)^{3/2} (a \cos (c+d x)+b)+i a \left(3 a^3+6 a^2 b+a b^2-2 b^3\right) \sec ^2\left(\frac{1}{2} (c+d x)\right) \sqrt{\frac{\sec ^2\left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b)}{a+b}} F\left(i \sinh ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)+2 i b \left(-3 a^3-3 a^2 b+a b^2+b^3\right) \sec ^2\left(\frac{1}{2} (c+d x)\right) \sqrt{\frac{\sec ^2\left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b)}{a+b}} E\left(i \sinh ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)\right)}{\left(a^3-a b^2\right)^2 \sec ^{\frac{3}{2}}(c+d x)}\right)}{3 d \cos ^{\frac{5}{2}}(c+d x) (a+b \sec (c+d x))^{5/2}}","-\frac{2 b \left(5 a^2-b^2\right) \sin (c+d x)}{3 a d \left(a^2-b^2\right)^2 \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}-\frac{2 b \sin (c+d x)}{3 d \left(a^2-b^2\right) \sqrt{\cos (c+d x)} (a+b \sec (c+d x))^{3/2}}+\frac{2 \left(3 a^2-2 b^2\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{3 a^2 d \left(a^2-b^2\right) \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}+\frac{4 b \left(3 a^2-b^2\right) \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{3 a^2 d \left(a^2-b^2\right)^2 \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}",1,"((b + a*Cos[c + d*x])^2*((2*b*(-5*a^2*b + b^3 + (-6*a^3 + 2*a*b^2)*Cos[c + d*x])*Sin[c + d*x])/(a*(a^2 - b^2)^2*(b + a*Cos[c + d*x])) - (2*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^(3/2)*((2*I)*b*(-3*a^3 - 3*a^2*b + a*b^2 + b^3)*EllipticE[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] + I*a*(3*a^3 + 6*a^2*b + a*b^2 - 2*b^3)*EllipticF[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] + 2*b*(-3*a^2 + b^2)*(b + a*Cos[c + d*x])*(Sec[(c + d*x)/2]^2)^(3/2)*Tan[(c + d*x)/2]))/((a^3 - a*b^2)^2*Sec[c + d*x]^(3/2))))/(3*d*Cos[c + d*x]^(5/2)*(a + b*Sec[c + d*x])^(5/2))","C",0
872,1,447,281,11.7040957,"\int \frac{1}{\cos ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^{5/2}} \, dx","Integrate[1/(Cos[c + d*x]^(3/2)*(a + b*Sec[c + d*x])^(5/2)),x]","\frac{(a \cos (c+d x)+b)^3 \left(\frac{2 b \sin (c+d x)}{3 \left(b^2-a^2\right) (a \cos (c+d x)+b)^2}+\frac{2 \left(3 a^2 \sin (c+d x)+b^2 \sin (c+d x)\right)}{3 \left(b^2-a^2\right)^2 (a \cos (c+d x)+b)}\right)}{d \cos ^{\frac{5}{2}}(c+d x) (a+b \sec (c+d x))^{5/2}}+\frac{2 \cos ^{\frac{3}{2}}(c+d x) \sec ^{\frac{5}{2}}(c+d x) \left(\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)\right)^{3/2} (a \cos (c+d x)+b)^2 \left(-\left(3 a^2+b^2\right) \tan \left(\frac{1}{2} (c+d x)\right) \sec ^2\left(\frac{1}{2} (c+d x)\right)^{3/2} (a \cos (c+d x)+b)+i a \left(3 a^2+4 a b+b^2\right) \sec ^2\left(\frac{1}{2} (c+d x)\right) \sqrt{\frac{\sec ^2\left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b)}{a+b}} F\left(i \sinh ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)-i \left(3 a^3+3 a^2 b+a b^2+b^3\right) \sec ^2\left(\frac{1}{2} (c+d x)\right) \sqrt{\frac{\sec ^2\left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b)}{a+b}} E\left(i \sinh ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)\right)}{3 a d \left(a^2-b^2\right)^2 (a+b \sec (c+d x))^{5/2}}","\frac{4 \left(a^2+b^2\right) \sin (c+d x)}{3 d \left(a^2-b^2\right)^2 \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}+\frac{2 a \sin (c+d x)}{3 d \left(a^2-b^2\right) \sqrt{\cos (c+d x)} (a+b \sec (c+d x))^{3/2}}-\frac{2 b \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{3 a d \left(a^2-b^2\right) \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}-\frac{2 \left(3 a^2+b^2\right) \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{3 a d \left(a^2-b^2\right)^2 \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}",1,"((b + a*Cos[c + d*x])^3*((2*b*Sin[c + d*x])/(3*(-a^2 + b^2)*(b + a*Cos[c + d*x])^2) + (2*(3*a^2*Sin[c + d*x] + b^2*Sin[c + d*x]))/(3*(-a^2 + b^2)^2*(b + a*Cos[c + d*x]))))/(d*Cos[c + d*x]^(5/2)*(a + b*Sec[c + d*x])^(5/2)) + (2*Cos[c + d*x]^(3/2)*(b + a*Cos[c + d*x])^2*Sec[c + d*x]^(5/2)*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^(3/2)*((-I)*(3*a^3 + 3*a^2*b + a*b^2 + b^3)*EllipticE[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] + I*a*(3*a^2 + 4*a*b + b^2)*EllipticF[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] - (3*a^2 + b^2)*(b + a*Cos[c + d*x])*(Sec[(c + d*x)/2]^2)^(3/2)*Tan[(c + d*x)/2]))/(3*a*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x])^(5/2))","C",0
873,1,311,277,9.5178958,"\int \frac{1}{\cos ^{\frac{5}{2}}(c+d x) (a+b \sec (c+d x))^{5/2}} \, dx","Integrate[1/(Cos[c + d*x]^(5/2)*(a + b*Sec[c + d*x])^(5/2)),x]","\frac{2 (a \cos (c+d x)+b)^2 \left(\frac{a \sin (c+d x) \left(a^2-4 a b \cos (c+d x)-5 b^2\right)}{a \cos (c+d x)+b}+\frac{\sqrt{\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)} \left(-i \left(a^2+4 a b+3 b^2\right) \sqrt{\frac{\sec ^2\left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b)}{a+b}} F\left(i \sinh ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)+4 b \tan \left(\frac{1}{2} (c+d x)\right) \sqrt{\sec ^2\left(\frac{1}{2} (c+d x)\right)} (a \cos (c+d x)+b)+4 i b (a+b) \sqrt{\frac{\sec ^2\left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b)}{a+b}} E\left(i \sinh ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)\right)}{\sqrt{\sec (c+d x)}}\right)}{3 d \left(a^2-b^2\right)^2 \cos ^{\frac{5}{2}}(c+d x) (a+b \sec (c+d x))^{5/2}}","-\frac{2 a^2 \sin (c+d x)}{3 b d \left(a^2-b^2\right) \sqrt{\cos (c+d x)} (a+b \sec (c+d x))^{3/2}}+\frac{2 a \left(a^2-5 b^2\right) \sin (c+d x)}{3 b d \left(a^2-b^2\right)^2 \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}+\frac{2 \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{3 d \left(a^2-b^2\right) \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}+\frac{8 b \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{3 d \left(a^2-b^2\right)^2 \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}",1,"(2*(b + a*Cos[c + d*x])^2*((a*(a^2 - 5*b^2 - 4*a*b*Cos[c + d*x])*Sin[c + d*x])/(b + a*Cos[c + d*x]) + (Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*((4*I)*b*(a + b)*EllipticE[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] - I*(a^2 + 4*a*b + 3*b^2)*EllipticF[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] + 4*b*(b + a*Cos[c + d*x])*Sqrt[Sec[(c + d*x)/2]^2]*Tan[(c + d*x)/2]))/Sqrt[Sec[c + d*x]]))/(3*(a^2 - b^2)^2*d*Cos[c + d*x]^(5/2)*(a + b*Sec[c + d*x])^(5/2))","C",0
874,1,92128,370,33.4675797,"\int \frac{1}{\cos ^{\frac{7}{2}}(c+d x) (a+b \sec (c+d x))^{5/2}} \, dx","Integrate[1/(Cos[c + d*x]^(7/2)*(a + b*Sec[c + d*x])^(5/2)),x]","\text{Result too large to show}","-\frac{2 a^2 \sin (c+d x)}{3 b d \left(a^2-b^2\right) \cos ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}-\frac{2 a^2 \left(3 a^2-7 b^2\right) \sin (c+d x)}{3 b^2 d \left(a^2-b^2\right)^2 \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}-\frac{2 a \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{3 b d \left(a^2-b^2\right) \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}+\frac{2 a \left(3 a^2-7 b^2\right) \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{3 b^2 d \left(a^2-b^2\right)^2 \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{2 \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{b^2 d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}",1,"Result too large to show","C",0
875,1,222,266,0.7453033,"\int (d \cos (e+f x))^n (a+b \sec (e+f x))^3 \, dx","Integrate[(d*Cos[e + f*x])^n*(a + b*Sec[e + f*x])^3,x]","-\frac{\sqrt{\sin ^2(e+f x)} \csc (e+f x) \sec ^2(e+f x) (d \cos (e+f x))^n \left(\frac{1}{2} a (n-2) \cos (e+f x) \left(2 a (n-1) \cos (e+f x) \left(a n \cos (e+f x) \, _2F_1\left(\frac{1}{2},\frac{n+1}{2};\frac{n+3}{2};\cos ^2(e+f x)\right)+3 b (n+1) \, _2F_1\left(\frac{1}{2},\frac{n}{2};\frac{n+2}{2};\cos ^2(e+f x)\right)\right)+6 b^2 n (n+1) \, _2F_1\left(\frac{1}{2},\frac{n-1}{2};\frac{n+1}{2};\cos ^2(e+f x)\right)\right)+b^3 n \left(n^2-1\right) \, _2F_1\left(\frac{1}{2},\frac{n-2}{2};\frac{n}{2};\cos ^2(e+f x)\right)\right)}{f (n-2) (n-1) n (n+1)}","-\frac{b \left(3 a^2 (2-n)+b^2 (1-n)\right) \sin (e+f x) (d \cos (e+f x))^n \, _2F_1\left(\frac{1}{2},\frac{n}{2};\frac{n+2}{2};\cos ^2(e+f x)\right)}{f (2-n) n \sqrt{\sin ^2(e+f x)}}-\frac{a \left(a^2 (1-n)-3 b^2 n\right) \sin (e+f x) \cos (e+f x) (d \cos (e+f x))^n \, _2F_1\left(\frac{1}{2},\frac{n+1}{2};\frac{n+3}{2};\cos ^2(e+f x)\right)}{f (1-n) (n+1) \sqrt{\sin ^2(e+f x)}}+\frac{a b^2 (5-2 n) \tan (e+f x) (d \cos (e+f x))^n}{f (1-n) (2-n)}+\frac{b^2 \tan (e+f x) (a+b \sec (e+f x)) (d \cos (e+f x))^n}{f (2-n)}",1,"-(((d*Cos[e + f*x])^n*Csc[e + f*x]*(b^3*n*(-1 + n^2)*Hypergeometric2F1[1/2, (-2 + n)/2, n/2, Cos[e + f*x]^2] + (a*(-2 + n)*Cos[e + f*x]*(6*b^2*n*(1 + n)*Hypergeometric2F1[1/2, (-1 + n)/2, (1 + n)/2, Cos[e + f*x]^2] + 2*a*(-1 + n)*Cos[e + f*x]*(3*b*(1 + n)*Hypergeometric2F1[1/2, n/2, (2 + n)/2, Cos[e + f*x]^2] + a*n*Cos[e + f*x]*Hypergeometric2F1[1/2, (1 + n)/2, (3 + n)/2, Cos[e + f*x]^2])))/2)*Sec[e + f*x]^2*Sqrt[Sin[e + f*x]^2])/(f*(-2 + n)*(-1 + n)*n*(1 + n)))","A",1
876,1,161,186,0.4303644,"\int (d \cos (e+f x))^n (a+b \sec (e+f x))^2 \, dx","Integrate[(d*Cos[e + f*x])^n*(a + b*Sec[e + f*x])^2,x]","-\frac{d \sqrt{\sin ^2(e+f x)} \csc (e+f x) (d \cos (e+f x))^{n-1} \left(a (n-1) \cos (e+f x) \left(a n \cos (e+f x) \, _2F_1\left(\frac{1}{2},\frac{n+1}{2};\frac{n+3}{2};\cos ^2(e+f x)\right)+2 b (n+1) \, _2F_1\left(\frac{1}{2},\frac{n}{2};\frac{n+2}{2};\cos ^2(e+f x)\right)\right)+b^2 n (n+1) \, _2F_1\left(\frac{1}{2},\frac{n-1}{2};\frac{n+1}{2};\cos ^2(e+f x)\right)\right)}{f (n-1) n (n+1)}","-\frac{\left(a^2 (1-n)-b^2 n\right) \sin (e+f x) \cos (e+f x) (d \cos (e+f x))^n \, _2F_1\left(\frac{1}{2},\frac{n+1}{2};\frac{n+3}{2};\cos ^2(e+f x)\right)}{f (1-n) (n+1) \sqrt{\sin ^2(e+f x)}}-\frac{2 a b \sin (e+f x) (d \cos (e+f x))^n \, _2F_1\left(\frac{1}{2},\frac{n}{2};\frac{n+2}{2};\cos ^2(e+f x)\right)}{f n \sqrt{\sin ^2(e+f x)}}+\frac{b^2 \tan (e+f x) (d \cos (e+f x))^n}{f (1-n)}",1,"-((d*(d*Cos[e + f*x])^(-1 + n)*Csc[e + f*x]*(b^2*n*(1 + n)*Hypergeometric2F1[1/2, (-1 + n)/2, (1 + n)/2, Cos[e + f*x]^2] + a*(-1 + n)*Cos[e + f*x]*(2*b*(1 + n)*Hypergeometric2F1[1/2, n/2, (2 + n)/2, Cos[e + f*x]^2] + a*n*Cos[e + f*x]*Hypergeometric2F1[1/2, (1 + n)/2, (3 + n)/2, Cos[e + f*x]^2]))*Sqrt[Sin[e + f*x]^2])/(f*(-1 + n)*n*(1 + n)))","A",1
877,1,106,132,0.1292173,"\int (d \cos (e+f x))^n (a+b \sec (e+f x)) \, dx","Integrate[(d*Cos[e + f*x])^n*(a + b*Sec[e + f*x]),x]","-\frac{\sqrt{\sin ^2(e+f x)} \csc (e+f x) (d \cos (e+f x))^n \left(a n \cos (e+f x) \, _2F_1\left(\frac{1}{2},\frac{n+1}{2};\frac{n+3}{2};\cos ^2(e+f x)\right)+b (n+1) \, _2F_1\left(\frac{1}{2},\frac{n}{2};\frac{n+2}{2};\cos ^2(e+f x)\right)\right)}{f n (n+1)}","-\frac{a \sin (e+f x) (d \cos (e+f x))^{n+1} \, _2F_1\left(\frac{1}{2},\frac{n+1}{2};\frac{n+3}{2};\cos ^2(e+f x)\right)}{d f (n+1) \sqrt{\sin ^2(e+f x)}}-\frac{b \sin (e+f x) (d \cos (e+f x))^n \, _2F_1\left(\frac{1}{2},\frac{n}{2};\frac{n+2}{2};\cos ^2(e+f x)\right)}{f n \sqrt{\sin ^2(e+f x)}}",1,"-(((d*Cos[e + f*x])^n*Csc[e + f*x]*(b*(1 + n)*Hypergeometric2F1[1/2, n/2, (2 + n)/2, Cos[e + f*x]^2] + a*n*Cos[e + f*x]*Hypergeometric2F1[1/2, (1 + n)/2, (3 + n)/2, Cos[e + f*x]^2])*Sqrt[Sin[e + f*x]^2])/(f*n*(1 + n)))","A",1
878,1,5216,196,25.7281901,"\int \frac{(d \cos (e+f x))^n}{a+b \sec (e+f x)} \, dx","Integrate[(d*Cos[e + f*x])^n/(a + b*Sec[e + f*x]),x]","\text{Result too large to show}","\frac{a \sin (e+f x) \cos (e+f x) \cos ^2(e+f x)^{\frac{1}{2} (-n-1)} (d \cos (e+f x))^n F_1\left(\frac{1}{2};\frac{1}{2} (-n-1),1;\frac{3}{2};\sin ^2(e+f x),\frac{a^2 \sin ^2(e+f x)}{a^2-b^2}\right)}{f \left(a^2-b^2\right)}-\frac{b \sin (e+f x) \cos ^2(e+f x)^{-n/2} (d \cos (e+f x))^n F_1\left(\frac{1}{2};-\frac{n}{2},1;\frac{3}{2};\sin ^2(e+f x),\frac{a^2 \sin ^2(e+f x)}{a^2-b^2}\right)}{f \left(a^2-b^2\right)}",1,"Result too large to show","B",0
879,1,14144,309,45.5758455,"\int \frac{(d \cos (e+f x))^n}{(a+b \sec (e+f x))^2} \, dx","Integrate[(d*Cos[e + f*x])^n/(a + b*Sec[e + f*x])^2,x]","\text{Result too large to show}","\frac{a^2 \sin (e+f x) \cos (e+f x) \cos ^2(e+f x)^{\frac{1}{2} (-n-1)} (d \cos (e+f x))^n F_1\left(\frac{1}{2};\frac{1}{2} (-n-3),2;\frac{3}{2};\sin ^2(e+f x),\frac{a^2 \sin ^2(e+f x)}{a^2-b^2}\right)}{f \left(a^2-b^2\right)^2}+\frac{b^2 \sin (e+f x) \cos (e+f x) \cos ^2(e+f x)^{\frac{1}{2} (-n-1)} (d \cos (e+f x))^n F_1\left(\frac{1}{2};\frac{1}{2} (-n-1),2;\frac{3}{2};\sin ^2(e+f x),\frac{a^2 \sin ^2(e+f x)}{a^2-b^2}\right)}{f \left(a^2-b^2\right)^2}-\frac{2 a b \sin (e+f x) \cos ^2(e+f x)^{-n/2} (d \cos (e+f x))^n F_1\left(\frac{1}{2};\frac{1}{2} (-n-2),2;\frac{3}{2};\sin ^2(e+f x),\frac{a^2 \sin ^2(e+f x)}{a^2-b^2}\right)}{f \left(a^2-b^2\right)^2}",1,"Result too large to show","B",0