1,1,106,152,0.218794,"\int (a+a \sec (c+d x)) \sin ^9(c+d x) \, dx","Integrate[(a + a*Sec[c + d*x])*Sin[c + d*x]^9,x]","-\frac{a \left(10080 \cos ^8(c+d x)-53760 \cos ^6(c+d x)+120960 \cos ^4(c+d x)-161280 \cos ^2(c+d x)+39690 \cos (c+d x)-8820 \cos (3 (c+d x))+2268 \cos (5 (c+d x))-405 \cos (7 (c+d x))+35 \cos (9 (c+d x))+80640 \log (\cos (c+d x))\right)}{80640 d}","-\frac{a \cos ^9(c+d x)}{9 d}-\frac{a \cos ^8(c+d x)}{8 d}+\frac{4 a \cos ^7(c+d x)}{7 d}+\frac{2 a \cos ^6(c+d x)}{3 d}-\frac{6 a \cos ^5(c+d x)}{5 d}-\frac{3 a \cos ^4(c+d x)}{2 d}+\frac{4 a \cos ^3(c+d x)}{3 d}+\frac{2 a \cos ^2(c+d x)}{d}-\frac{a \cos (c+d x)}{d}-\frac{a \log (\cos (c+d x))}{d}",1,"-1/80640*(a*(39690*Cos[c + d*x] - 161280*Cos[c + d*x]^2 + 120960*Cos[c + d*x]^4 - 53760*Cos[c + d*x]^6 + 10080*Cos[c + d*x]^8 - 8820*Cos[3*(c + d*x)] + 2268*Cos[5*(c + d*x)] - 405*Cos[7*(c + d*x)] + 35*Cos[9*(c + d*x)] + 80640*Log[Cos[c + d*x]]))/d","A",1
2,1,86,119,0.1386208,"\int (a+a \sec (c+d x)) \sin ^7(c+d x) \, dx","Integrate[(a + a*Sec[c + d*x])*Sin[c + d*x]^7,x]","\frac{a \left(1120 \cos ^6(c+d x)-5040 \cos ^4(c+d x)+10080 \cos ^2(c+d x)-3675 \cos (c+d x)+735 \cos (3 (c+d x))-147 \cos (5 (c+d x))+15 \cos (7 (c+d x))-6720 \log (\cos (c+d x))\right)}{6720 d}","\frac{a \cos ^7(c+d x)}{7 d}+\frac{a \cos ^6(c+d x)}{6 d}-\frac{3 a \cos ^5(c+d x)}{5 d}-\frac{3 a \cos ^4(c+d x)}{4 d}+\frac{a \cos ^3(c+d x)}{d}+\frac{3 a \cos ^2(c+d x)}{2 d}-\frac{a \cos (c+d x)}{d}-\frac{a \log (\cos (c+d x))}{d}",1,"(a*(-3675*Cos[c + d*x] + 10080*Cos[c + d*x]^2 - 5040*Cos[c + d*x]^4 + 1120*Cos[c + d*x]^6 + 735*Cos[3*(c + d*x)] - 147*Cos[5*(c + d*x)] + 15*Cos[7*(c + d*x)] - 6720*Log[Cos[c + d*x]]))/(6720*d)","A",1
3,1,83,87,0.0868985,"\int (a+a \sec (c+d x)) \sin ^5(c+d x) \, dx","Integrate[(a + a*Sec[c + d*x])*Sin[c + d*x]^5,x]","-\frac{5 a \cos (c+d x)}{8 d}+\frac{5 a \cos (3 (c+d x))}{48 d}-\frac{a \cos (5 (c+d x))}{80 d}-\frac{a \left(\frac{1}{4} \cos ^4(c+d x)-\cos ^2(c+d x)+\log (\cos (c+d x))\right)}{d}","-\frac{a \cos ^5(c+d x)}{5 d}-\frac{a \cos ^4(c+d x)}{4 d}+\frac{2 a \cos ^3(c+d x)}{3 d}+\frac{a \cos ^2(c+d x)}{d}-\frac{a \cos (c+d x)}{d}-\frac{a \log (\cos (c+d x))}{d}",1,"(-5*a*Cos[c + d*x])/(8*d) + (5*a*Cos[3*(c + d*x)])/(48*d) - (a*Cos[5*(c + d*x)])/(80*d) - (a*(-Cos[c + d*x]^2 + Cos[c + d*x]^4/4 + Log[Cos[c + d*x]]))/d","A",1
4,1,57,58,0.0575594,"\int (a+a \sec (c+d x)) \sin ^3(c+d x) \, dx","Integrate[(a + a*Sec[c + d*x])*Sin[c + d*x]^3,x]","-\frac{3 a \cos (c+d x)}{4 d}+\frac{a \cos (3 (c+d x))}{12 d}-\frac{a \left(\log (\cos (c+d x))-\frac{1}{2} \cos ^2(c+d x)\right)}{d}","\frac{a \cos ^3(c+d x)}{3 d}+\frac{a \cos ^2(c+d x)}{2 d}-\frac{a \cos (c+d x)}{d}-\frac{a \log (\cos (c+d x))}{d}",1,"(-3*a*Cos[c + d*x])/(4*d) + (a*Cos[3*(c + d*x)])/(12*d) - (a*(-1/2*Cos[c + d*x]^2 + Log[Cos[c + d*x]]))/d","A",1
5,1,37,26,0.0196344,"\int (a+a \sec (c+d x)) \sin (c+d x) \, dx","Integrate[(a + a*Sec[c + d*x])*Sin[c + d*x],x]","\frac{a \sin (c) \sin (d x)}{d}-\frac{a \cos (c) \cos (d x)}{d}-\frac{a \log (\cos (c+d x))}{d}","-\frac{a \cos (c+d x)}{d}-\frac{a \log (\cos (c+d x))}{d}",1,"-((a*Cos[c]*Cos[d*x])/d) - (a*Log[Cos[c + d*x]])/d + (a*Sin[c]*Sin[d*x])/d","A",1
6,1,63,30,0.0353931,"\int \csc (c+d x) (a+a \sec (c+d x)) \, dx","Integrate[Csc[c + d*x]*(a + a*Sec[c + d*x]),x]","\frac{a \log \left(\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)}{d}-\frac{a \log \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)\right)}{d}-\frac{a (\log (\cos (c+d x))-\log (\sin (c+d x)))}{d}","\frac{a \log (1-\cos (c+d x))}{d}-\frac{a \log (\cos (c+d x))}{d}",1,"-((a*Log[Cos[c/2 + (d*x)/2]])/d) + (a*Log[Sin[c/2 + (d*x)/2]])/d - (a*(Log[Cos[c + d*x]] - Log[Sin[c + d*x]]))/d","B",1
7,1,114,73,0.8434241,"\int \csc ^3(c+d x) (a+a \sec (c+d x)) \, dx","Integrate[Csc[c + d*x]^3*(a + a*Sec[c + d*x]),x]","-\frac{a \csc ^2\left(\frac{1}{2} (c+d x)\right)}{8 d}+\frac{a \sec ^2\left(\frac{1}{2} (c+d x)\right)}{8 d}+\frac{a \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)}{2 d}-\frac{a \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)}{2 d}-\frac{a \left(\csc ^2(c+d x)-2 \log (\sin (c+d x))+2 \log (\cos (c+d x))\right)}{2 d}","-\frac{a^2}{2 d (a-a \cos (c+d x))}+\frac{3 a \log (1-\cos (c+d x))}{4 d}-\frac{a \log (\cos (c+d x))}{d}+\frac{a \log (\cos (c+d x)+1)}{4 d}",1,"-1/8*(a*Csc[(c + d*x)/2]^2)/d - (a*Log[Cos[(c + d*x)/2]])/(2*d) + (a*Log[Sin[(c + d*x)/2]])/(2*d) - (a*(Csc[c + d*x]^2 + 2*Log[Cos[c + d*x]] - 2*Log[Sin[c + d*x]]))/(2*d) + (a*Sec[(c + d*x)/2]^2)/(8*d)","A",1
8,1,164,118,0.3581103,"\int \csc ^5(c+d x) (a+a \sec (c+d x)) \, dx","Integrate[Csc[c + d*x]^5*(a + a*Sec[c + d*x]),x]","-\frac{a \csc ^4\left(\frac{1}{2} (c+d x)\right)}{64 d}-\frac{3 a \csc ^2\left(\frac{1}{2} (c+d x)\right)}{32 d}+\frac{a \sec ^4\left(\frac{1}{2} (c+d x)\right)}{64 d}+\frac{3 a \sec ^2\left(\frac{1}{2} (c+d x)\right)}{32 d}+\frac{3 a \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)}{8 d}-\frac{3 a \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)}{8 d}-\frac{a \left(\csc ^4(c+d x)+2 \csc ^2(c+d x)-4 \log (\sin (c+d x))+4 \log (\cos (c+d x))\right)}{4 d}","-\frac{a^3}{8 d (a-a \cos (c+d x))^2}-\frac{a^2}{2 d (a-a \cos (c+d x))}-\frac{a^2}{8 d (a \cos (c+d x)+a)}+\frac{11 a \log (1-\cos (c+d x))}{16 d}-\frac{a \log (\cos (c+d x))}{d}+\frac{5 a \log (\cos (c+d x)+1)}{16 d}",1,"(-3*a*Csc[(c + d*x)/2]^2)/(32*d) - (a*Csc[(c + d*x)/2]^4)/(64*d) - (3*a*Log[Cos[(c + d*x)/2]])/(8*d) + (3*a*Log[Sin[(c + d*x)/2]])/(8*d) - (a*(2*Csc[c + d*x]^2 + Csc[c + d*x]^4 + 4*Log[Cos[c + d*x]] - 4*Log[Sin[c + d*x]]))/(4*d) + (3*a*Sec[(c + d*x)/2]^2)/(32*d) + (a*Sec[(c + d*x)/2]^4)/(64*d)","A",1
9,1,165,163,0.4601329,"\int \csc ^7(c+d x) (a+a \sec (c+d x)) \, dx","Integrate[Csc[c + d*x]^7*(a + a*Sec[c + d*x]),x]","-\frac{a \left(\csc ^6\left(\frac{1}{2} (c+d x)\right)+64 \csc ^6(c+d x)+6 \csc ^4\left(\frac{1}{2} (c+d x)\right)+96 \csc ^4(c+d x)+30 \csc ^2\left(\frac{1}{2} (c+d x)\right)+192 \csc ^2(c+d x)-\sec ^6\left(\frac{1}{2} (c+d x)\right)-6 \sec ^4\left(\frac{1}{2} (c+d x)\right)-30 \sec ^2\left(\frac{1}{2} (c+d x)\right)-120 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-384 \log (\sin (c+d x))+120 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)+384 \log (\cos (c+d x))\right)}{384 d}","-\frac{a^4}{24 d (a-a \cos (c+d x))^3}-\frac{5 a^3}{32 d (a-a \cos (c+d x))^2}-\frac{a^3}{32 d (a \cos (c+d x)+a)^2}-\frac{a^2}{2 d (a-a \cos (c+d x))}-\frac{3 a^2}{16 d (a \cos (c+d x)+a)}+\frac{21 a \log (1-\cos (c+d x))}{32 d}-\frac{a \log (\cos (c+d x))}{d}+\frac{11 a \log (\cos (c+d x)+1)}{32 d}",1,"-1/384*(a*(30*Csc[(c + d*x)/2]^2 + 6*Csc[(c + d*x)/2]^4 + Csc[(c + d*x)/2]^6 + 192*Csc[c + d*x]^2 + 96*Csc[c + d*x]^4 + 64*Csc[c + d*x]^6 + 120*Log[Cos[(c + d*x)/2]] + 384*Log[Cos[c + d*x]] - 120*Log[Sin[(c + d*x)/2]] - 384*Log[Sin[c + d*x]] - 30*Sec[(c + d*x)/2]^2 - 6*Sec[(c + d*x)/2]^4 - Sec[(c + d*x)/2]^6))/d","A",1
10,1,106,165,0.3476311,"\int (a+a \sec (c+d x)) \sin ^8(c+d x) \, dx","Integrate[(a + a*Sec[c + d*x])*Sin[c + d*x]^8,x]","\frac{a \left(-15360 \sin ^7(c+d x)-21504 \sin ^5(c+d x)-35840 \sin ^3(c+d x)-107520 \sin (c+d x)+35 (-672 \sin (2 (c+d x))+168 \sin (4 (c+d x))-32 \sin (6 (c+d x))+3 \sin (8 (c+d x))+840 c+840 d x)+107520 \tanh ^{-1}(\sin (c+d x))\right)}{107520 d}","-\frac{a \sin ^7(c+d x)}{7 d}-\frac{a \sin ^5(c+d x)}{5 d}-\frac{a \sin ^3(c+d x)}{3 d}-\frac{a \sin (c+d x)}{d}+\frac{a \tanh ^{-1}(\sin (c+d x))}{d}-\frac{a \sin ^7(c+d x) \cos (c+d x)}{8 d}-\frac{7 a \sin ^5(c+d x) \cos (c+d x)}{48 d}-\frac{35 a \sin ^3(c+d x) \cos (c+d x)}{192 d}-\frac{35 a \sin (c+d x) \cos (c+d x)}{128 d}+\frac{35 a x}{128}",1,"(a*(107520*ArcTanh[Sin[c + d*x]] - 107520*Sin[c + d*x] - 35840*Sin[c + d*x]^3 - 21504*Sin[c + d*x]^5 - 15360*Sin[c + d*x]^7 + 35*(840*c + 840*d*x - 672*Sin[2*(c + d*x)] + 168*Sin[4*(c + d*x)] - 32*Sin[6*(c + d*x)] + 3*Sin[8*(c + d*x)])))/(107520*d)","A",1
11,1,86,127,0.1948505,"\int (a+a \sec (c+d x)) \sin ^6(c+d x) \, dx","Integrate[(a + a*Sec[c + d*x])*Sin[c + d*x]^6,x]","\frac{a \left(-192 \sin ^5(c+d x)-320 \sin ^3(c+d x)-960 \sin (c+d x)+5 (-45 \sin (2 (c+d x))+9 \sin (4 (c+d x))-\sin (6 (c+d x))+60 c+60 d x)+960 \tanh ^{-1}(\sin (c+d x))\right)}{960 d}","-\frac{a \sin ^5(c+d x)}{5 d}-\frac{a \sin ^3(c+d x)}{3 d}-\frac{a \sin (c+d x)}{d}+\frac{a \tanh ^{-1}(\sin (c+d x))}{d}-\frac{a \sin ^5(c+d x) \cos (c+d x)}{6 d}-\frac{5 a \sin ^3(c+d x) \cos (c+d x)}{24 d}-\frac{5 a \sin (c+d x) \cos (c+d x)}{16 d}+\frac{5 a x}{16}",1,"(a*(960*ArcTanh[Sin[c + d*x]] - 960*Sin[c + d*x] - 320*Sin[c + d*x]^3 - 192*Sin[c + d*x]^5 + 5*(60*c + 60*d*x - 45*Sin[2*(c + d*x)] + 9*Sin[4*(c + d*x)] - Sin[6*(c + d*x)])))/(960*d)","A",1
12,1,86,89,0.1190353,"\int (a+a \sec (c+d x)) \sin ^4(c+d x) \, dx","Integrate[(a + a*Sec[c + d*x])*Sin[c + d*x]^4,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 \tanh ^{-1}(\sin (c+d x))}{d}","-\frac{a \sin ^3(c+d x)}{3 d}-\frac{a \sin (c+d x)}{d}+\frac{a \tanh ^{-1}(\sin (c+d x))}{d}-\frac{a \sin ^3(c+d x) \cos (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*ArcTanh[Sin[c + d*x]])/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
13,1,54,51,0.056835,"\int (a+a \sec (c+d x)) \sin ^2(c+d x) \, dx","Integrate[(a + a*Sec[c + d*x])*Sin[c + d*x]^2,x]","\frac{a (c+d x)}{2 d}-\frac{a \sin (c+d x)}{d}-\frac{a \sin (2 (c+d x))}{4 d}+\frac{a \tanh ^{-1}(\sin (c+d x))}{d}","-\frac{a \sin (c+d x)}{d}+\frac{a \tanh ^{-1}(\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*ArcTanh[Sin[c + d*x]])/d - (a*Sin[c + d*x])/d - (a*Sin[2*(c + d*x)])/(4*d)","A",1
14,1,41,37,0.0296922,"\int \csc ^2(c+d x) (a+a \sec (c+d x)) \, dx","Integrate[Csc[c + d*x]^2*(a + a*Sec[c + d*x]),x]","-\frac{a \csc (c+d x) \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\sin ^2(c+d x)\right)}{d}-\frac{a \cot (c+d x)}{d}","-\frac{a \cot (c+d x)}{d}-\frac{a \csc (c+d x)}{d}+\frac{a \tanh ^{-1}(\sin (c+d x))}{d}",1,"-((a*Cot[c + d*x])/d) - (a*Csc[c + d*x]*Hypergeometric2F1[-1/2, 1, 1/2, Sin[c + d*x]^2])/d","C",1
15,1,69,69,0.0286697,"\int \csc ^4(c+d x) (a+a \sec (c+d x)) \, dx","Integrate[Csc[c + d*x]^4*(a + a*Sec[c + d*x]),x]","-\frac{a \csc ^3(c+d x) \, _2F_1\left(-\frac{3}{2},1;-\frac{1}{2};\sin ^2(c+d x)\right)}{3 d}-\frac{2 a \cot (c+d x)}{3 d}-\frac{a \cot (c+d x) \csc ^2(c+d x)}{3 d}","-\frac{a \cot ^3(c+d x)}{3 d}-\frac{a \cot (c+d x)}{d}-\frac{a \csc ^3(c+d x)}{3 d}-\frac{a \csc (c+d x)}{d}+\frac{a \tanh ^{-1}(\sin (c+d x))}{d}",1,"(-2*a*Cot[c + d*x])/(3*d) - (a*Cot[c + d*x]*Csc[c + d*x]^2)/(3*d) - (a*Csc[c + d*x]^3*Hypergeometric2F1[-3/2, 1, -1/2, Sin[c + d*x]^2])/(3*d)","C",1
16,1,91,101,0.033413,"\int \csc ^6(c+d x) (a+a \sec (c+d x)) \, dx","Integrate[Csc[c + d*x]^6*(a + a*Sec[c + d*x]),x]","-\frac{a \csc ^5(c+d x) \, _2F_1\left(-\frac{5}{2},1;-\frac{3}{2};\sin ^2(c+d x)\right)}{5 d}-\frac{8 a \cot (c+d x)}{15 d}-\frac{a \cot (c+d x) \csc ^4(c+d x)}{5 d}-\frac{4 a \cot (c+d x) \csc ^2(c+d x)}{15 d}","-\frac{a \cot ^5(c+d x)}{5 d}-\frac{2 a \cot ^3(c+d x)}{3 d}-\frac{a \cot (c+d x)}{d}-\frac{a \csc ^5(c+d x)}{5 d}-\frac{a \csc ^3(c+d x)}{3 d}-\frac{a \csc (c+d x)}{d}+\frac{a \tanh ^{-1}(\sin (c+d x))}{d}",1,"(-8*a*Cot[c + d*x])/(15*d) - (4*a*Cot[c + d*x]*Csc[c + d*x]^2)/(15*d) - (a*Cot[c + d*x]*Csc[c + d*x]^4)/(5*d) - (a*Csc[c + d*x]^5*Hypergeometric2F1[-5/2, 1, -3/2, Sin[c + d*x]^2])/(5*d)","C",1
17,1,113,131,0.0506882,"\int \csc ^8(c+d x) (a+a \sec (c+d x)) \, dx","Integrate[Csc[c + d*x]^8*(a + a*Sec[c + d*x]),x]","-\frac{a \csc ^7(c+d x) \, _2F_1\left(-\frac{7}{2},1;-\frac{5}{2};\sin ^2(c+d x)\right)}{7 d}-\frac{16 a \cot (c+d x)}{35 d}-\frac{a \cot (c+d x) \csc ^6(c+d x)}{7 d}-\frac{6 a \cot (c+d x) \csc ^4(c+d x)}{35 d}-\frac{8 a \cot (c+d x) \csc ^2(c+d x)}{35 d}","-\frac{a \cot ^7(c+d x)}{7 d}-\frac{3 a \cot ^5(c+d x)}{5 d}-\frac{a \cot ^3(c+d x)}{d}-\frac{a \cot (c+d x)}{d}-\frac{a \csc ^7(c+d x)}{7 d}-\frac{a \csc ^5(c+d x)}{5 d}-\frac{a \csc ^3(c+d x)}{3 d}-\frac{a \csc (c+d x)}{d}+\frac{a \tanh ^{-1}(\sin (c+d x))}{d}",1,"(-16*a*Cot[c + d*x])/(35*d) - (8*a*Cot[c + d*x]*Csc[c + d*x]^2)/(35*d) - (6*a*Cot[c + d*x]*Csc[c + d*x]^4)/(35*d) - (a*Cot[c + d*x]*Csc[c + d*x]^6)/(7*d) - (a*Csc[c + d*x]^7*Hypergeometric2F1[-7/2, 1, -5/2, Sin[c + d*x]^2])/(7*d)","C",1
18,1,135,165,0.0567862,"\int \csc ^{10}(c+d x) (a+a \sec (c+d x)) \, dx","Integrate[Csc[c + d*x]^10*(a + a*Sec[c + d*x]),x]","-\frac{a \csc ^9(c+d x) \, _2F_1\left(-\frac{9}{2},1;-\frac{7}{2};\sin ^2(c+d x)\right)}{9 d}-\frac{128 a \cot (c+d x)}{315 d}-\frac{a \cot (c+d x) \csc ^8(c+d x)}{9 d}-\frac{8 a \cot (c+d x) \csc ^6(c+d x)}{63 d}-\frac{16 a \cot (c+d x) \csc ^4(c+d x)}{105 d}-\frac{64 a \cot (c+d x) \csc ^2(c+d x)}{315 d}","-\frac{a \cot ^9(c+d x)}{9 d}-\frac{4 a \cot ^7(c+d x)}{7 d}-\frac{6 a \cot ^5(c+d x)}{5 d}-\frac{4 a \cot ^3(c+d x)}{3 d}-\frac{a \cot (c+d x)}{d}-\frac{a \csc ^9(c+d x)}{9 d}-\frac{a \csc ^7(c+d x)}{7 d}-\frac{a \csc ^5(c+d x)}{5 d}-\frac{a \csc ^3(c+d x)}{3 d}-\frac{a \csc (c+d x)}{d}+\frac{a \tanh ^{-1}(\sin (c+d x))}{d}",1,"(-128*a*Cot[c + d*x])/(315*d) - (64*a*Cot[c + d*x]*Csc[c + d*x]^2)/(315*d) - (16*a*Cot[c + d*x]*Csc[c + d*x]^4)/(105*d) - (8*a*Cot[c + d*x]*Csc[c + d*x]^6)/(63*d) - (a*Cot[c + d*x]*Csc[c + d*x]^8)/(9*d) - (a*Csc[c + d*x]^9*Hypergeometric2F1[-9/2, 1, -7/2, Sin[c + d*x]^2])/(9*d)","C",1
19,1,127,183,0.9365831,"\int (a+a \sec (c+d x))^2 \sin ^9(c+d x) \, dx","Integrate[(a + a*Sec[c + d*x])^2*Sin[c + d*x]^9,x]","-\frac{a^2 \sec (c+d x) (-361620 \cos (2 (c+d x))-134820 \cos (3 (c+d x))+29232 \cos (4 (c+d x))+24780 \cos (5 (c+d x))-1458 \cos (6 (c+d x))-3885 \cos (7 (c+d x))-380 \cos (8 (c+d x))+315 \cos (9 (c+d x))+70 \cos (10 (c+d x))+210 \cos (c+d x) (3072 \log (\cos (c+d x))+205)-714420)}{322560 d}","-\frac{a^2 \cos ^9(c+d x)}{9 d}-\frac{a^2 \cos ^8(c+d x)}{4 d}+\frac{3 a^2 \cos ^7(c+d x)}{7 d}+\frac{4 a^2 \cos ^6(c+d x)}{3 d}-\frac{2 a^2 \cos ^5(c+d x)}{5 d}-\frac{3 a^2 \cos ^4(c+d x)}{d}-\frac{2 a^2 \cos ^3(c+d x)}{3 d}+\frac{4 a^2 \cos ^2(c+d x)}{d}+\frac{3 a^2 \cos (c+d x)}{d}+\frac{a^2 \sec (c+d x)}{d}-\frac{2 a^2 \log (\cos (c+d x))}{d}",1,"-1/322560*(a^2*(-714420 - 361620*Cos[2*(c + d*x)] - 134820*Cos[3*(c + d*x)] + 29232*Cos[4*(c + d*x)] + 24780*Cos[5*(c + d*x)] - 1458*Cos[6*(c + d*x)] - 3885*Cos[7*(c + d*x)] - 380*Cos[8*(c + d*x)] + 315*Cos[9*(c + d*x)] + 70*Cos[10*(c + d*x)] + 210*Cos[c + d*x]*(205 + 3072*Log[Cos[c + d*x]]))*Sec[c + d*x])/d","A",1
20,1,107,131,0.5607367,"\int (a+a \sec (c+d x))^2 \sin ^7(c+d x) \, dx","Integrate[(a + a*Sec[c + d*x])^2*Sin[c + d*x]^7,x]","\frac{a^2 \sec (c+d x) (11760 \cos (2 (c+d x))+5250 \cos (3 (c+d x))-588 \cos (4 (c+d x))-770 \cos (5 (c+d x))-48 \cos (6 (c+d x))+70 \cos (7 (c+d x))+15 \cos (8 (c+d x))-70 \cos (c+d x) (384 \log (\cos (c+d x))+5)+25725)}{13440 d}","\frac{a^2 \cos ^7(c+d x)}{7 d}+\frac{a^2 \cos ^6(c+d x)}{3 d}-\frac{2 a^2 \cos ^5(c+d x)}{5 d}-\frac{3 a^2 \cos ^4(c+d x)}{2 d}+\frac{3 a^2 \cos ^2(c+d x)}{d}+\frac{2 a^2 \cos (c+d x)}{d}+\frac{a^2 \sec (c+d x)}{d}-\frac{2 a^2 \log (\cos (c+d x))}{d}",1,"(a^2*(25725 + 11760*Cos[2*(c + d*x)] + 5250*Cos[3*(c + d*x)] - 588*Cos[4*(c + d*x)] - 770*Cos[5*(c + d*x)] - 48*Cos[6*(c + d*x)] + 70*Cos[7*(c + d*x)] + 15*Cos[8*(c + d*x)] - 70*Cos[c + d*x]*(5 + 384*Log[Cos[c + d*x]]))*Sec[c + d*x])/(13440*d)","A",1
21,1,87,112,0.3093823,"\int (a+a \sec (c+d x))^2 \sin ^5(c+d x) \, dx","Integrate[(a + a*Sec[c + d*x])^2*Sin[c + d*x]^5,x]","-\frac{a^2 \sec (c+d x) (-275 \cos (2 (c+d x))-165 \cos (3 (c+d x))-2 \cos (4 (c+d x))+15 \cos (5 (c+d x))+3 \cos (6 (c+d x))+30 \cos (c+d x) (32 \log (\cos (c+d x))-3)-750)}{480 d}","-\frac{a^2 \cos ^5(c+d x)}{5 d}-\frac{a^2 \cos ^4(c+d x)}{2 d}+\frac{a^2 \cos ^3(c+d x)}{3 d}+\frac{2 a^2 \cos ^2(c+d x)}{d}+\frac{a^2 \cos (c+d x)}{d}+\frac{a^2 \sec (c+d x)}{d}-\frac{2 a^2 \log (\cos (c+d x))}{d}",1,"-1/480*(a^2*(-750 - 275*Cos[2*(c + d*x)] - 165*Cos[3*(c + d*x)] - 2*Cos[4*(c + d*x)] + 15*Cos[5*(c + d*x)] + 3*Cos[6*(c + d*x)] + 30*Cos[c + d*x]*(-3 + 32*Log[Cos[c + d*x]]))*Sec[c + d*x])/d","A",1
22,1,65,62,0.2130145,"\int (a+a \sec (c+d x))^2 \sin ^3(c+d x) \, dx","Integrate[(a + a*Sec[c + d*x])^2*Sin[c + d*x]^3,x]","\frac{a^2 \sec (c+d x) (4 \cos (2 (c+d x))+6 \cos (3 (c+d x))+\cos (4 (c+d x))-6 \cos (c+d x) (8 \log (\cos (c+d x))+1)+27)}{24 d}","\frac{a^2 \cos ^3(c+d x)}{3 d}+\frac{a^2 \cos ^2(c+d x)}{d}+\frac{a^2 \sec (c+d x)}{d}-\frac{2 a^2 \log (\cos (c+d x))}{d}",1,"(a^2*(27 + 4*Cos[2*(c + d*x)] + 6*Cos[3*(c + d*x)] + Cos[4*(c + d*x)] - 6*Cos[c + d*x]*(1 + 8*Log[Cos[c + d*x]]))*Sec[c + d*x])/(24*d)","A",1
23,1,31,43,0.1192903,"\int (a+a \sec (c+d x))^2 \sin (c+d x) \, dx","Integrate[(a + a*Sec[c + d*x])^2*Sin[c + d*x],x]","\frac{a^2 (\sin (c+d x) \tan (c+d x)-2 \log (\cos (c+d x))+1)}{d}","-\frac{a^2 \cos (c+d x)}{d}+\frac{a^2 \sec (c+d x)}{d}-\frac{2 a^2 \log (\cos (c+d x))}{d}",1,"(a^2*(1 - 2*Log[Cos[c + d*x]] + Sin[c + d*x]*Tan[c + d*x]))/d","A",1
24,1,36,48,0.0852439,"\int \csc (c+d x) (a+a \sec (c+d x))^2 \, dx","Integrate[Csc[c + d*x]*(a + a*Sec[c + d*x])^2,x]","\frac{a^2 \left(\sec (c+d x)+4 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-2 \log (\cos (c+d x))\right)}{d}","\frac{a^2 \sec (c+d x)}{d}+\frac{2 a^2 \log (1-\cos (c+d x))}{d}-\frac{2 a^2 \log (\cos (c+d x))}{d}",1,"(a^2*(-2*Log[Cos[c + d*x]] + 4*Log[Sin[(c + d*x)/2]] + Sec[c + d*x]))/d","A",1
25,1,75,69,0.5577874,"\int \csc ^3(c+d x) (a+a \sec (c+d x))^2 \, dx","Integrate[Csc[c + d*x]^3*(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(\csc ^2\left(\frac{1}{2} (c+d x)\right)-2 \sec (c+d x)-8 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+4 \log (\cos (c+d x))\right)}{8 d}","-\frac{a^3}{d (a-a \cos (c+d x))}+\frac{a^2 \sec (c+d x)}{d}+\frac{2 a^2 \log (1-\cos (c+d x))}{d}-\frac{2 a^2 \log (\cos (c+d x))}{d}",1,"-1/8*(a^2*(1 + Cos[c + d*x])^2*Sec[(c + d*x)/2]^4*(Csc[(c + d*x)/2]^2 + 4*Log[Cos[c + d*x]] - 8*Log[Sin[(c + d*x)/2]] - 2*Sec[c + d*x]))/d","A",1
26,1,103,115,1.572247,"\int \csc ^5(c+d x) (a+a \sec (c+d x))^2 \, dx","Integrate[Csc[c + d*x]^5*(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(\csc ^4\left(\frac{1}{2} (c+d x)\right)+10 \csc ^2\left(\frac{1}{2} (c+d x)\right)+4 \left(-4 \sec (c+d x)-17 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+\log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)+8 \log (\cos (c+d x))\right)\right)}{64 d}","-\frac{a^4}{4 d (a-a \cos (c+d x))^2}-\frac{5 a^3}{4 d (a-a \cos (c+d x))}+\frac{a^2 \sec (c+d x)}{d}+\frac{17 a^2 \log (1-\cos (c+d x))}{8 d}-\frac{2 a^2 \log (\cos (c+d x))}{d}-\frac{a^2 \log (\cos (c+d x)+1)}{8 d}",1,"-1/64*(a^2*(1 + Cos[c + d*x])^2*Sec[(c + d*x)/2]^4*(10*Csc[(c + d*x)/2]^2 + Csc[(c + d*x)/2]^4 + 4*(Log[Cos[(c + d*x)/2]] + 8*Log[Cos[c + d*x]] - 17*Log[Sin[(c + d*x)/2]] - 4*Sec[c + d*x])))/d","A",1
27,1,136,160,1.3379034,"\int \csc ^7(c+d x) (a+a \sec (c+d x))^2 \, dx","Integrate[Csc[c + d*x]^7*(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(36 \csc ^4\left(\frac{1}{2} (c+d x)\right)+120 \csc ^2\left(\frac{1}{2} (c+d x)\right)+\csc ^6\left(\frac{1}{2} (c+d x)\right) \left(16-3 \sec ^2\left(\frac{1}{2} (c+d x)\right) (2 \sec (c+d x)+3)\right)+48 \left(-9 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+\log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)+4 \log (\cos (c+d x))\right)\right)}{384 d}","-\frac{a^5}{12 d (a-a \cos (c+d x))^3}-\frac{3 a^4}{8 d (a-a \cos (c+d x))^2}-\frac{23 a^3}{16 d (a-a \cos (c+d x))}+\frac{a^3}{16 d (a \cos (c+d x)+a)}+\frac{a^2 \sec (c+d x)}{d}+\frac{9 a^2 \log (1-\cos (c+d x))}{4 d}-\frac{2 a^2 \log (\cos (c+d x))}{d}-\frac{a^2 \log (\cos (c+d x)+1)}{4 d}",1,"-1/384*(a^2*(1 + Cos[c + d*x])^2*Sec[(c + d*x)/2]^4*(120*Csc[(c + d*x)/2]^2 + 36*Csc[(c + d*x)/2]^4 + 48*(Log[Cos[(c + d*x)/2]] + 4*Log[Cos[c + d*x]] - 9*Log[Sin[(c + d*x)/2]]) + Csc[(c + d*x)/2]^6*(16 - 3*Sec[(c + d*x)/2]^2*(3 + 2*Sec[c + d*x]))))/d","A",1
28,1,164,205,3.4550226,"\int \csc ^9(c+d x) (a+a \sec (c+d x))^2 \, dx","Integrate[Csc[c + d*x]^9*(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(3 \csc ^8\left(\frac{1}{2} (c+d x)\right)+28 \csc ^6\left(\frac{1}{2} (c+d x)\right)+180 \csc ^4\left(\frac{1}{2} (c+d x)\right)+1224 \csc ^2\left(\frac{1}{2} (c+d x)\right)-6 \left(\sec ^4\left(\frac{1}{2} (c+d x)\right)+18 \sec ^2\left(\frac{1}{2} (c+d x)\right)+4 \left(64 \sec (c+d x)+303 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-47 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)-128 \log (\cos (c+d x))\right)\right)\right)}{6144 d}","-\frac{a^6}{32 d (a-a \cos (c+d x))^4}-\frac{7 a^5}{48 d (a-a \cos (c+d x))^3}-\frac{15 a^4}{32 d (a-a \cos (c+d x))^2}+\frac{a^4}{64 d (a \cos (c+d x)+a)^2}-\frac{51 a^3}{32 d (a-a \cos (c+d x))}+\frac{9 a^3}{64 d (a \cos (c+d x)+a)}+\frac{a^2 \sec (c+d x)}{d}+\frac{303 a^2 \log (1-\cos (c+d x))}{128 d}-\frac{2 a^2 \log (\cos (c+d x))}{d}-\frac{47 a^2 \log (\cos (c+d x)+1)}{128 d}",1,"-1/6144*(a^2*(1 + Cos[c + d*x])^2*Sec[(c + d*x)/2]^4*(1224*Csc[(c + d*x)/2]^2 + 180*Csc[(c + d*x)/2]^4 + 28*Csc[(c + d*x)/2]^6 + 3*Csc[(c + d*x)/2]^8 - 6*(18*Sec[(c + d*x)/2]^2 + Sec[(c + d*x)/2]^4 + 4*(-47*Log[Cos[(c + d*x)/2]] - 128*Log[Cos[c + d*x]] + 303*Log[Sin[(c + d*x)/2]] + 64*Sec[c + d*x]))))/d","A",1
29,1,144,199,0.9625597,"\int (a+a \sec (c+d x))^2 \sin ^8(c+d x) \, dx","Integrate[(a + a*Sec[c + d*x])^2*Sin[c + d*x]^8,x]","-\frac{a^2 (\cos (c+d x)+1)^2 \sec ^4\left(\frac{1}{2} (c+d x)\right) \left(30720 \sin ^7(c+d x)+43008 \sin ^5(c+d x)+71680 \sin ^3(c+d x)+215040 \sin (c+d x)-55440 \sin (2 (c+d x))+2520 \sin (4 (c+d x))+560 \sin (6 (c+d x))-105 \sin (8 (c+d x))+37800 \tan ^{-1}(\tan (c+d x))-107520 \tan (c+d x)-215040 \tanh ^{-1}(\sin (c+d x))+168000 c+168000 d x\right)}{430080 d}","-\frac{2 a^2 \sin ^7(c+d x)}{7 d}-\frac{2 a^2 \sin ^5(c+d x)}{5 d}-\frac{2 a^2 \sin ^3(c+d x)}{3 d}-\frac{2 a^2 \sin (c+d x)}{d}+\frac{a^2 \tan (c+d x)}{d}+\frac{2 a^2 \tanh ^{-1}(\sin (c+d x))}{d}+\frac{a^2 \sin (c+d x) \cos ^7(c+d x)}{8 d}-\frac{17 a^2 \sin (c+d x) \cos ^5(c+d x)}{48 d}+\frac{11 a^2 \sin (c+d x) \cos ^3(c+d x)}{192 d}+\frac{139 a^2 \sin (c+d x) \cos (c+d x)}{128 d}-\frac{245 a^2 x}{128}",1,"-1/430080*(a^2*(1 + Cos[c + d*x])^2*Sec[(c + d*x)/2]^4*(168000*c + 168000*d*x + 37800*ArcTan[Tan[c + d*x]] - 215040*ArcTanh[Sin[c + d*x]] + 215040*Sin[c + d*x] + 71680*Sin[c + d*x]^3 + 43008*Sin[c + d*x]^5 + 30720*Sin[c + d*x]^7 - 55440*Sin[2*(c + d*x)] + 2520*Sin[4*(c + d*x)] + 560*Sin[6*(c + d*x)] - 105*Sin[8*(c + d*x)] - 107520*Tan[c + d*x]))/d","A",1
30,1,124,157,0.5608624,"\int (a+a \sec (c+d x))^2 \sin ^6(c+d x) \, dx","Integrate[(a + a*Sec[c + d*x])^2*Sin[c + d*x]^6,x]","-\frac{a^2 (\cos (c+d x)+1)^2 \sec ^4\left(\frac{1}{2} (c+d x)\right) \left(384 \sin ^5(c+d x)+640 \sin ^3(c+d x)+1920 \sin (c+d x)-255 \sin (2 (c+d x))-15 \sin (4 (c+d x))+5 \sin (6 (c+d x))+420 \tan ^{-1}(\tan (c+d x))-960 \tan (c+d x)-1920 \tanh ^{-1}(\sin (c+d x))+1080 c+1080 d x\right)}{3840 d}","-\frac{2 a^2 \sin ^5(c+d x)}{5 d}-\frac{2 a^2 \sin ^3(c+d x)}{3 d}-\frac{2 a^2 \sin (c+d x)}{d}+\frac{a^2 \tan (c+d x)}{d}+\frac{2 a^2 \tanh ^{-1}(\sin (c+d x))}{d}-\frac{a^2 \sin (c+d x) \cos ^5(c+d x)}{6 d}+\frac{7 a^2 \sin (c+d x) \cos ^3(c+d x)}{24 d}+\frac{7 a^2 \sin (c+d x) \cos (c+d x)}{16 d}-\frac{25 a^2 x}{16}",1,"-1/3840*(a^2*(1 + Cos[c + d*x])^2*Sec[(c + d*x)/2]^4*(1080*c + 1080*d*x + 420*ArcTan[Tan[c + d*x]] - 1920*ArcTanh[Sin[c + d*x]] + 1920*Sin[c + d*x] + 640*Sin[c + d*x]^3 + 384*Sin[c + d*x]^5 - 255*Sin[2*(c + d*x)] - 15*Sin[4*(c + d*x)] + 5*Sin[6*(c + d*x)] - 960*Tan[c + d*x]))/d","A",1
31,1,94,115,0.2426659,"\int (a+a \sec (c+d x))^2 \sin ^4(c+d x) \, dx","Integrate[(a + a*Sec[c + d*x])^2*Sin[c + d*x]^4,x]","-\frac{a^2 (\cos (c+d x)+1)^2 \sec ^4\left(\frac{1}{2} (c+d x)\right) \left(64 \sin ^3(c+d x)+192 \sin (c+d x)-3 \sin (4 (c+d x))+60 \tan ^{-1}(\tan (c+d x))-96 \tan (c+d x)-192 \tanh ^{-1}(\sin (c+d x))+48 c+48 d x\right)}{384 d}","-\frac{2 a^2 \sin ^3(c+d x)}{3 d}-\frac{2 a^2 \sin (c+d x)}{d}+\frac{a^2 \tan (c+d x)}{d}+\frac{2 a^2 \tanh ^{-1}(\sin (c+d x))}{d}+\frac{a^2 \sin (c+d x) \cos ^3(c+d x)}{4 d}-\frac{a^2 \sin (c+d x) \cos (c+d x)}{8 d}-\frac{9 a^2 x}{8}",1,"-1/384*(a^2*(1 + Cos[c + d*x])^2*Sec[(c + d*x)/2]^4*(48*c + 48*d*x + 60*ArcTan[Tan[c + d*x]] - 192*ArcTanh[Sin[c + d*x]] + 192*Sin[c + d*x] + 64*Sin[c + d*x]^3 - 3*Sin[4*(c + d*x)] - 96*Tan[c + d*x]))/d","A",1
32,1,243,73,1.1568537,"\int (a+a \sec (c+d x))^2 \sin ^2(c+d x) \, dx","Integrate[(a + a*Sec[c + d*x])^2*Sin[c + d*x]^2,x]","\frac{1}{16} a^2 (\cos (c+d x)+1)^2 \sec ^4\left(\frac{1}{2} (c+d x)\right) \left(-\frac{8 \sin (c) \cos (d x)}{d}-\frac{\sin (2 c) \cos (2 d x)}{d}-\frac{8 \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{8 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}{d}+\frac{8 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}{d}-2 x\right)","-\frac{2 a^2 \sin (c+d x)}{d}+\frac{a^2 \tan (c+d x)}{d}+\frac{2 a^2 \tanh ^{-1}(\sin (c+d x))}{d}-\frac{a^2 \sin (c+d x) \cos (c+d x)}{2 d}-\frac{a^2 x}{2}",1,"(a^2*(1 + Cos[c + d*x])^2*Sec[(c + d*x)/2]^4*(-2*x - (8*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]])/d + (8*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]])/d - (8*Cos[d*x]*Sin[c])/d - (Cos[2*d*x]*Sin[2*c])/d - (8*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]))))/16","B",1
33,1,401,57,6.158444,"\int \csc ^2(c+d x) (a+a \sec (c+d x))^2 \, dx","Integrate[Csc[c + d*x]^2*(a + a*Sec[c + d*x])^2,x]","\frac{\sin \left(\frac{d x}{2}\right) \cos ^2(c+d x) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right) (a \sec (c+d x)+a)^2}{4 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{\sin \left(\frac{d x}{2}\right) \cos ^2(c+d x) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right) (a \sec (c+d x)+a)^2}{4 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{\cos ^2(c+d x) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right) (a \sec (c+d x)+a)^2 \log \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)-\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)}{2 d}+\frac{\cos ^2(c+d x) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right) (a \sec (c+d x)+a)^2 \log \left(\sin \left(\frac{c}{2}+\frac{d x}{2}\right)+\cos \left(\frac{c}{2}+\frac{d x}{2}\right)\right)}{2 d}+\frac{\csc \left(\frac{c}{2}\right) \sin \left(\frac{d x}{2}\right) \cos ^2(c+d x) \csc \left(\frac{c}{2}+\frac{d x}{2}\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right) (a \sec (c+d x)+a)^2}{2 d}","\frac{a^2 \tan (c+d x)}{d}-\frac{2 a^2 \cot (c+d x)}{d}-\frac{2 a^2 \csc (c+d x)}{d}+\frac{2 a^2 \tanh ^{-1}(\sin (c+d x))}{d}",1,"-1/2*(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)/d + (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)/(2*d) + (Cos[c + d*x]^2*Csc[c/2]*Csc[c/2 + (d*x)/2]*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2*Sin[(d*x)/2])/(2*d) + (Cos[c + d*x]^2*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2*Sin[(d*x)/2])/(4*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*Sin[(d*x)/2])/(4*d*(Cos[c/2] + Sin[c/2])*(Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2]))","B",1
34,1,228,87,1.8525266,"\int \csc ^4(c+d x) (a+a \sec (c+d x))^2 \, dx","Integrate[Csc[c + d*x]^4*(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(-\cot \left(\frac{c}{2}\right) \csc ^2\left(\frac{1}{2} (c+d x)\right)+6 \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)-\left(\csc \left(\frac{c}{2}\right) \sin \left(\frac{d x}{2}\right) (7 \cos (c+d x)-8) \csc ^3\left(\frac{1}{2} (c+d x)\right)\right)\right)}{24 d}","-\frac{a^4 \tan (c+d x)}{3 d (a-a \cos (c+d x))^2}+\frac{10 a^2 \tan (c+d x)}{3 d}+\frac{2 a^2 \tanh ^{-1}(\sin (c+d x))}{d}-\frac{2 a^2 \tan (c+d x)}{d (1-\cos (c+d x))}",1,"(a^2*(1 + Cos[c + d*x])^2*Sec[(c + d*x)/2]^4*(-(Cot[c/2]*Csc[(c + d*x)/2]^2) - (-8 + 7*Cos[c + d*x])*Csc[c/2]*Csc[(c + d*x)/2]^3*Sin[(d*x)/2] + 6*(-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])))))/(24*d)","B",1
35,1,317,129,0.9953167,"\int \csc ^6(c+d x) (a+a \sec (c+d x))^2 \, dx","Integrate[Csc[c + d*x]^6*(a + a*Sec[c + d*x])^2,x]","\frac{a^2 \cos (c+d x) \sec ^4\left(\frac{1}{2} (c+d x)\right) (\sec (c+d x)+1)^2 \left(\csc (2 c) (216 \sin (c-d x)-416 \sin (c+d x)+624 \sin (2 (c+d x))-416 \sin (3 (c+d x))+104 \sin (4 (c+d x))-596 \sin (2 c+d x)-680 \sin (3 c+d x)+894 \sin (c+2 d x)+224 \sin (2 (c+2 d x))+894 \sin (3 c+2 d x)+480 \sin (4 c+2 d x)-776 \sin (c+3 d x)-596 \sin (2 c+3 d x)-596 \sin (4 c+3 d x)-120 \sin (5 c+3 d x)+149 \sin (3 c+4 d x)+149 \sin (5 c+4 d x)+320 \sin (2 c)-596 \sin (d x)+864 \sin (2 d x)) \csc (c+d x) \csc ^4\left(\frac{1}{2} (c+d x)\right)-3840 \cos (c+d x) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+3840 \cos (c+d x) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)}{7680 d}","\frac{a^2 \tan (c+d x)}{d}-\frac{2 a^2 \cot ^5(c+d x)}{5 d}-\frac{5 a^2 \cot ^3(c+d x)}{3 d}-\frac{4 a^2 \cot (c+d x)}{d}-\frac{2 a^2 \csc ^5(c+d x)}{5 d}-\frac{2 a^2 \csc ^3(c+d x)}{3 d}-\frac{2 a^2 \csc (c+d x)}{d}+\frac{2 a^2 \tanh ^{-1}(\sin (c+d x))}{d}",1,"(a^2*Cos[c + d*x]*Sec[(c + d*x)/2]^4*(1 + Sec[c + d*x])^2*(-3840*Cos[c + d*x]*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 3840*Cos[c + d*x]*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + Csc[2*c]*Csc[(c + d*x)/2]^4*Csc[c + d*x]*(320*Sin[2*c] - 596*Sin[d*x] + 864*Sin[2*d*x] + 216*Sin[c - d*x] - 416*Sin[c + d*x] + 624*Sin[2*(c + d*x)] - 416*Sin[3*(c + d*x)] + 104*Sin[4*(c + d*x)] - 596*Sin[2*c + d*x] - 680*Sin[3*c + d*x] + 894*Sin[c + 2*d*x] + 224*Sin[2*(c + 2*d*x)] + 894*Sin[3*c + 2*d*x] + 480*Sin[4*c + 2*d*x] - 776*Sin[c + 3*d*x] - 596*Sin[2*c + 3*d*x] - 596*Sin[4*c + 3*d*x] - 120*Sin[5*c + 3*d*x] + 149*Sin[3*c + 4*d*x] + 149*Sin[5*c + 4*d*x])))/(7680*d)","B",1
36,1,428,163,1.301999,"\int \csc ^8(c+d x) (a+a \sec (c+d x))^2 \, dx","Integrate[Csc[c + d*x]^8*(a + a*Sec[c + d*x])^2,x]","\frac{a^2 \cos (c+d x) \sec ^4\left(\frac{1}{2} (c+d x)\right) (\sec (c+d x)+1)^2 \left(-32 \csc (2 c) (-7264 \sin (c-d x)+14208 \sin (c+d x)-19536 \sin (2 (c+d x))+7104 \sin (3 (c+d x))+7104 \sin (4 (c+d x))-7104 \sin (5 (c+d x))+1776 \sin (6 (c+d x))+17288 \sin (2 c+d x)+20384 \sin (3 c+d x)-23771 \sin (c+2 d x)+7104 \sin (2 (c+2 d x))-23771 \sin (3 c+2 d x)-8960 \sin (4 c+2 d x)+19984 \sin (c+3 d x)+8644 \sin (2 c+3 d x)+8644 \sin (4 c+3 d x)-6160 \sin (5 c+3 d x)+8644 \sin (3 c+4 d x)+8644 \sin (5 c+4 d x)+6720 \sin (6 c+4 d x)-12144 \sin (3 c+5 d x)-8644 \sin (4 c+5 d x)-8644 \sin (6 c+5 d x)-1680 \sin (7 c+5 d x)+3456 \sin (4 c+6 d x)+2161 \sin (5 c+6 d x)+2161 \sin (7 c+6 d x)-9856 \sin (2 c)+17288 \sin (d x)-29056 \sin (2 d x)) \csc ^3(c+d x) \csc ^4\left(\frac{1}{2} (c+d x)\right)-6881280 \cos (c+d x) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+6881280 \cos (c+d x) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)}{13762560 d}","\frac{a^2 \tan (c+d x)}{d}-\frac{2 a^2 \cot ^7(c+d x)}{7 d}-\frac{7 a^2 \cot ^5(c+d x)}{5 d}-\frac{3 a^2 \cot ^3(c+d x)}{d}-\frac{5 a^2 \cot (c+d x)}{d}-\frac{2 a^2 \csc ^7(c+d x)}{7 d}-\frac{2 a^2 \csc ^5(c+d x)}{5 d}-\frac{2 a^2 \csc ^3(c+d x)}{3 d}-\frac{2 a^2 \csc (c+d x)}{d}+\frac{2 a^2 \tanh ^{-1}(\sin (c+d x))}{d}",1,"(a^2*Cos[c + d*x]*Sec[(c + d*x)/2]^4*(1 + Sec[c + d*x])^2*(-6881280*Cos[c + d*x]*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 6881280*Cos[c + d*x]*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] - 32*Csc[2*c]*Csc[(c + d*x)/2]^4*Csc[c + d*x]^3*(-9856*Sin[2*c] + 17288*Sin[d*x] - 29056*Sin[2*d*x] - 7264*Sin[c - d*x] + 14208*Sin[c + d*x] - 19536*Sin[2*(c + d*x)] + 7104*Sin[3*(c + d*x)] + 7104*Sin[4*(c + d*x)] - 7104*Sin[5*(c + d*x)] + 1776*Sin[6*(c + d*x)] + 17288*Sin[2*c + d*x] + 20384*Sin[3*c + d*x] - 23771*Sin[c + 2*d*x] + 7104*Sin[2*(c + 2*d*x)] - 23771*Sin[3*c + 2*d*x] - 8960*Sin[4*c + 2*d*x] + 19984*Sin[c + 3*d*x] + 8644*Sin[2*c + 3*d*x] + 8644*Sin[4*c + 3*d*x] - 6160*Sin[5*c + 3*d*x] + 8644*Sin[3*c + 4*d*x] + 8644*Sin[5*c + 4*d*x] + 6720*Sin[6*c + 4*d*x] - 12144*Sin[3*c + 5*d*x] - 8644*Sin[4*c + 5*d*x] - 8644*Sin[6*c + 5*d*x] - 1680*Sin[7*c + 5*d*x] + 3456*Sin[4*c + 6*d*x] + 2161*Sin[5*c + 6*d*x] + 2161*Sin[7*c + 6*d*x])))/(13762560*d)","B",1
37,1,1050,201,6.9458862,"\int \csc ^{10}(c+d x) (a+a \sec (c+d x))^2 \, dx","Integrate[Csc[c + d*x]^10*(a + a*Sec[c + d*x])^2,x]","\frac{\cos ^2(c+d x) \csc \left(\frac{c}{2}\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right) (\sec (c+d x) a+a)^2 \sin \left(\frac{d x}{2}\right) \csc ^9\left(\frac{c}{2}+\frac{d x}{2}\right)}{4608 d}-\frac{\cos ^2(c+d x) \cot \left(\frac{c}{2}\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right) (\sec (c+d x) a+a)^2 \csc ^8\left(\frac{c}{2}+\frac{d x}{2}\right)}{4608 d}+\frac{71 \cos ^2(c+d x) \csc \left(\frac{c}{2}\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right) (\sec (c+d x) a+a)^2 \sin \left(\frac{d x}{2}\right) \csc ^7\left(\frac{c}{2}+\frac{d x}{2}\right)}{32256 d}-\frac{71 \cos ^2(c+d x) \cot \left(\frac{c}{2}\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right) (\sec (c+d x) a+a)^2 \csc ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{32256 d}+\frac{193 \cos ^2(c+d x) \csc \left(\frac{c}{2}\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right) (\sec (c+d x) a+a)^2 \sin \left(\frac{d x}{2}\right) \csc ^5\left(\frac{c}{2}+\frac{d x}{2}\right)}{13440 d}-\frac{193 \cos ^2(c+d x) \cot \left(\frac{c}{2}\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right) (\sec (c+d x) a+a)^2 \csc ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{13440 d}+\frac{6899 \cos ^2(c+d x) \csc \left(\frac{c}{2}\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right) (\sec (c+d x) a+a)^2 \sin \left(\frac{d x}{2}\right) \csc ^3\left(\frac{c}{2}+\frac{d x}{2}\right)}{80640 d}-\frac{6899 \cos ^2(c+d x) \cot \left(\frac{c}{2}\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right) (\sec (c+d x) a+a)^2 \csc ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{80640 d}+\frac{123041 \cos ^2(c+d x) \csc \left(\frac{c}{2}\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right) (\sec (c+d x) a+a)^2 \sin \left(\frac{d x}{2}\right) \csc \left(\frac{c}{2}+\frac{d x}{2}\right)}{161280 d}-\frac{\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 ^4\left(\frac{c}{2}+\frac{d x}{2}\right) (\sec (c+d x) a+a)^2}{2 d}+\frac{\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 ^4\left(\frac{c}{2}+\frac{d x}{2}\right) (\sec (c+d x) a+a)^2}{2 d}+\frac{\cos ^2(c+d x) \sec \left(\frac{c}{2}\right) \sec ^9\left(\frac{c}{2}+\frac{d x}{2}\right) (\sec (c+d x) a+a)^2 \sin \left(\frac{d x}{2}\right)}{2560 d}+\frac{49 \cos ^2(c+d x) \sec \left(\frac{c}{2}\right) \sec ^7\left(\frac{c}{2}+\frac{d x}{2}\right) (\sec (c+d x) a+a)^2 \sin \left(\frac{d x}{2}\right)}{7680 d}+\frac{803 \cos ^2(c+d x) \sec \left(\frac{c}{2}\right) \sec ^5\left(\frac{c}{2}+\frac{d x}{2}\right) (\sec (c+d x) a+a)^2 \sin \left(\frac{d x}{2}\right)}{7680 d}+\frac{\cos (c+d x) \sec (c) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right) (\sec (c+d x) a+a)^2 \sin (d x)}{4 d}+\frac{\cos ^2(c+d x) \sec ^8\left(\frac{c}{2}+\frac{d x}{2}\right) (\sec (c+d x) a+a)^2 \tan \left(\frac{c}{2}\right)}{2560 d}+\frac{49 \cos ^2(c+d x) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right) (\sec (c+d x) a+a)^2 \tan \left(\frac{c}{2}\right)}{7680 d}","\frac{a^2 \tan (c+d x)}{d}-\frac{2 a^2 \cot ^9(c+d x)}{9 d}-\frac{9 a^2 \cot ^7(c+d x)}{7 d}-\frac{16 a^2 \cot ^5(c+d x)}{5 d}-\frac{14 a^2 \cot ^3(c+d x)}{3 d}-\frac{6 a^2 \cot (c+d x)}{d}-\frac{2 a^2 \csc ^9(c+d x)}{9 d}-\frac{2 a^2 \csc ^7(c+d x)}{7 d}-\frac{2 a^2 \csc ^5(c+d x)}{5 d}-\frac{2 a^2 \csc ^3(c+d x)}{3 d}-\frac{2 a^2 \csc (c+d x)}{d}+\frac{2 a^2 \tanh ^{-1}(\sin (c+d x))}{d}",1,"(-6899*Cos[c + d*x]^2*Cot[c/2]*Csc[c/2 + (d*x)/2]^2*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2)/(80640*d) - (193*Cos[c + d*x]^2*Cot[c/2]*Csc[c/2 + (d*x)/2]^4*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2)/(13440*d) - (71*Cos[c + d*x]^2*Cot[c/2]*Csc[c/2 + (d*x)/2]^6*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2)/(32256*d) - (Cos[c + d*x]^2*Cot[c/2]*Csc[c/2 + (d*x)/2]^8*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2)/(4608*d) - (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)/(2*d) + (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)/(2*d) + (123041*Cos[c + d*x]^2*Csc[c/2]*Csc[c/2 + (d*x)/2]*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2*Sin[(d*x)/2])/(161280*d) + (6899*Cos[c + d*x]^2*Csc[c/2]*Csc[c/2 + (d*x)/2]^3*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2*Sin[(d*x)/2])/(80640*d) + (193*Cos[c + d*x]^2*Csc[c/2]*Csc[c/2 + (d*x)/2]^5*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2*Sin[(d*x)/2])/(13440*d) + (71*Cos[c + d*x]^2*Csc[c/2]*Csc[c/2 + (d*x)/2]^7*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2*Sin[(d*x)/2])/(32256*d) + (Cos[c + d*x]^2*Csc[c/2]*Csc[c/2 + (d*x)/2]^9*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2*Sin[(d*x)/2])/(4608*d) + (803*Cos[c + d*x]^2*Sec[c/2]*Sec[c/2 + (d*x)/2]^5*(a + a*Sec[c + d*x])^2*Sin[(d*x)/2])/(7680*d) + (49*Cos[c + d*x]^2*Sec[c/2]*Sec[c/2 + (d*x)/2]^7*(a + a*Sec[c + d*x])^2*Sin[(d*x)/2])/(7680*d) + (Cos[c + d*x]^2*Sec[c/2]*Sec[c/2 + (d*x)/2]^9*(a + a*Sec[c + d*x])^2*Sin[(d*x)/2])/(2560*d) + (Cos[c + d*x]*Sec[c]*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2*Sin[d*x])/(4*d) + (49*Cos[c + d*x]^2*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^2*Tan[c/2])/(7680*d) + (Cos[c + d*x]^2*Sec[c/2 + (d*x)/2]^8*(a + a*Sec[c + d*x])^2*Tan[c/2])/(2560*d)","B",0
38,1,148,203,1.7896215,"\int (a+a \sec (c+d x))^3 \sin ^9(c+d x) \, dx","Integrate[(a + a*Sec[c + d*x])^3*Sin[c + d*x]^9,x]","\frac{a^3 \sec ^2(c+d x) (11624760 \cos (c+d x)+2188872 \cos (3 (c+d x))+41160 \cos (4 (c+d x))-204156 \cos (5 (c+d x))-35805 \cos (6 (c+d x))+22972 \cos (7 (c+d x))+9030 \cos (8 (c+d x))-820 \cos (9 (c+d x))-945 \cos (10 (c+d x))-140 \cos (11 (c+d x))+645120 \log (\cos (c+d x))+210 \cos (2 (c+d x)) (3072 \log (\cos (c+d x))-413)+471450)}{1290240 d}","-\frac{a^3 \cos ^9(c+d x)}{9 d}-\frac{3 a^3 \cos ^8(c+d x)}{8 d}+\frac{a^3 \cos ^7(c+d x)}{7 d}+\frac{11 a^3 \cos ^6(c+d x)}{6 d}+\frac{6 a^3 \cos ^5(c+d x)}{5 d}-\frac{7 a^3 \cos ^4(c+d x)}{2 d}-\frac{14 a^3 \cos ^3(c+d x)}{3 d}+\frac{3 a^3 \cos ^2(c+d x)}{d}+\frac{11 a^3 \cos (c+d x)}{d}+\frac{a^3 \sec ^2(c+d x)}{2 d}+\frac{3 a^3 \sec (c+d x)}{d}+\frac{a^3 \log (\cos (c+d x))}{d}",1,"(a^3*(471450 + 11624760*Cos[c + d*x] + 2188872*Cos[3*(c + d*x)] + 41160*Cos[4*(c + d*x)] - 204156*Cos[5*(c + d*x)] - 35805*Cos[6*(c + d*x)] + 22972*Cos[7*(c + d*x)] + 9030*Cos[8*(c + d*x)] - 820*Cos[9*(c + d*x)] - 945*Cos[10*(c + d*x)] - 140*Cos[11*(c + d*x)] + 645120*Log[Cos[c + d*x]] + 210*Cos[2*(c + d*x)]*(-413 + 3072*Log[Cos[c + d*x]]))*Sec[c + d*x]^2)/(1290240*d)","A",1
39,1,106,131,1.0087209,"\int (a+a \sec (c+d x))^3 \sin ^7(c+d x) \, dx","Integrate[(a + a*Sec[c + d*x])^3*Sin[c + d*x]^7,x]","\frac{a^3 (14014 \cos (c+d x)-210 \cos (2 (c+d x))+2548 \cos (3 (c+d x))+196 \cos (4 (c+d x))-188 \cos (5 (c+d x))-56 \cos (6 (c+d x))+9 \cos (7 (c+d x))+7 \cos (8 (c+d x))+\cos (9 (c+d x))+427) \sec ^2(c+d x)}{1792 d}","\frac{a^3 \cos ^7(c+d x)}{7 d}+\frac{a^3 \cos ^6(c+d x)}{2 d}-\frac{2 a^3 \cos ^4(c+d x)}{d}-\frac{2 a^3 \cos ^3(c+d x)}{d}+\frac{3 a^3 \cos ^2(c+d x)}{d}+\frac{8 a^3 \cos (c+d x)}{d}+\frac{a^3 \sec ^2(c+d x)}{2 d}+\frac{3 a^3 \sec (c+d x)}{d}",1,"(a^3*(427 + 14014*Cos[c + d*x] - 210*Cos[2*(c + d*x)] + 2548*Cos[3*(c + d*x)] + 196*Cos[4*(c + d*x)] - 188*Cos[5*(c + d*x)] - 56*Cos[6*(c + d*x)] + 9*Cos[7*(c + d*x)] + 7*Cos[8*(c + d*x)] + Cos[9*(c + d*x)])*Sec[c + d*x]^2)/(1792*d)","A",1
40,1,108,134,0.6551198,"\int (a+a \sec (c+d x))^3 \sin ^5(c+d x) \, dx","Integrate[(a + a*Sec[c + d*x])^3*Sin[c + d*x]^5,x]","-\frac{a^3 \sec ^2(c+d x) (-12350 \cos (c+d x)-2074 \cos (3 (c+d x))-330 \cos (4 (c+d x))+82 \cos (5 (c+d x))+45 \cos (6 (c+d x))+6 \cos (7 (c+d x))+960 \log (\cos (c+d x))+15 \cos (2 (c+d x)) (64 \log (\cos (c+d x))+31)-120)}{1920 d}","-\frac{a^3 \cos ^5(c+d x)}{5 d}-\frac{3 a^3 \cos ^4(c+d x)}{4 d}-\frac{a^3 \cos ^3(c+d x)}{3 d}+\frac{5 a^3 \cos ^2(c+d x)}{2 d}+\frac{5 a^3 \cos (c+d x)}{d}+\frac{a^3 \sec ^2(c+d x)}{2 d}+\frac{3 a^3 \sec (c+d x)}{d}-\frac{a^3 \log (\cos (c+d x))}{d}",1,"-1/1920*(a^3*(-120 - 12350*Cos[c + d*x] - 2074*Cos[3*(c + d*x)] - 330*Cos[4*(c + d*x)] + 82*Cos[5*(c + d*x)] + 45*Cos[6*(c + d*x)] + 6*Cos[7*(c + d*x)] + 960*Log[Cos[c + d*x]] + 15*Cos[2*(c + d*x)]*(31 + 64*Log[Cos[c + d*x]]))*Sec[c + d*x]^2)/d","A",1
41,1,86,98,0.203061,"\int (a+a \sec (c+d x))^3 \sin ^3(c+d x) \, dx","Integrate[(a + a*Sec[c + d*x])^3*Sin[c + d*x]^3,x]","\frac{a^3 \sec ^2(c+d x) (226 \cos (c+d x)+29 \cos (3 (c+d x))+9 \cos (4 (c+d x))+\cos (5 (c+d x))-48 \log (\cos (c+d x))-8 \cos (2 (c+d x)) (6 \log (\cos (c+d x))+7)-41)}{48 d}","\frac{a^3 \cos ^3(c+d x)}{3 d}+\frac{3 a^3 \cos ^2(c+d x)}{2 d}+\frac{2 a^3 \cos (c+d x)}{d}+\frac{a^3 \sec ^2(c+d x)}{2 d}+\frac{3 a^3 \sec (c+d x)}{d}-\frac{2 a^3 \log (\cos (c+d x))}{d}",1,"(a^3*(-41 + 226*Cos[c + d*x] + 29*Cos[3*(c + d*x)] + 9*Cos[4*(c + d*x)] + Cos[5*(c + d*x)] - 48*Log[Cos[c + d*x]] - 8*Cos[2*(c + d*x)]*(7 + 6*Log[Cos[c + d*x]]))*Sec[c + d*x]^2)/(48*d)","A",1
42,1,65,62,0.2504941,"\int (a+a \sec (c+d x))^3 \sin (c+d x) \, dx","Integrate[(a + a*Sec[c + d*x])^3*Sin[c + d*x],x]","-\frac{a^3 \sec ^2(c+d x) (-9 \cos (c+d x)+\cos (3 (c+d x))+6 \log (\cos (c+d x))+\cos (2 (c+d x)) (6 \log (\cos (c+d x))-2)-4)}{4 d}","-\frac{a^3 \cos (c+d x)}{d}+\frac{a^3 \sec ^2(c+d x)}{2 d}+\frac{3 a^3 \sec (c+d x)}{d}-\frac{3 a^3 \log (\cos (c+d x))}{d}",1,"-1/4*(a^3*(-4 - 9*Cos[c + d*x] + Cos[3*(c + d*x)] + 6*Log[Cos[c + d*x]] + Cos[2*(c + d*x)]*(-2 + 6*Log[Cos[c + d*x]]))*Sec[c + d*x]^2)/d","A",1
43,1,81,67,0.1456994,"\int \csc (c+d x) (a+a \sec (c+d x))^3 \, dx","Integrate[Csc[c + d*x]*(a + a*Sec[c + d*x])^3,x]","\frac{a^3 \sec ^2(c+d x) \left(6 \cos (c+d x)+8 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-4 \log (\cos (c+d x))-4 \cos (2 (c+d x)) \left(\log (\cos (c+d x))-2 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)\right)+1\right)}{2 d}","\frac{a^3 \sec ^2(c+d x)}{2 d}+\frac{3 a^3 \sec (c+d x)}{d}+\frac{4 a^3 \log (1-\cos (c+d x))}{d}-\frac{4 a^3 \log (\cos (c+d x))}{d}",1,"(a^3*(1 + 6*Cos[c + d*x] - 4*Log[Cos[c + d*x]] - 4*Cos[2*(c + d*x)]*(Log[Cos[c + d*x]] - 2*Log[Sin[(c + d*x)/2]]) + 8*Log[Sin[(c + d*x)/2]])*Sec[c + d*x]^2)/(2*d)","A",1
44,1,88,88,0.9041945,"\int \csc ^3(c+d x) (a+a \sec (c+d x))^3 \, dx","Integrate[Csc[c + d*x]^3*(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(2 \csc ^2\left(\frac{1}{2} (c+d x)\right)-\sec ^2(c+d x)-6 \sec (c+d x)+10 \left(\log (\cos (c+d x))-2 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)\right)\right)}{16 d}","-\frac{2 a^4}{d (a-a \cos (c+d x))}+\frac{a^3 \sec ^2(c+d x)}{2 d}+\frac{3 a^3 \sec (c+d x)}{d}+\frac{5 a^3 \log (1-\cos (c+d x))}{d}-\frac{5 a^3 \log (\cos (c+d x))}{d}",1,"-1/16*(a^3*(1 + Cos[c + d*x])^3*Sec[(c + d*x)/2]^6*(2*Csc[(c + d*x)/2]^2 + 10*(Log[Cos[c + d*x]] - 2*Log[Sin[(c + d*x)/2]]) - 6*Sec[c + d*x] - Sec[c + d*x]^2))/d","A",1
45,1,100,111,0.9472644,"\int \csc ^5(c+d x) (a+a \sec (c+d x))^3 \, dx","Integrate[Csc[c + d*x]^5*(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(\csc ^4\left(\frac{1}{2} (c+d x)\right)+12 \csc ^2\left(\frac{1}{2} (c+d x)\right)-4 \sec ^2(c+d x)-24 \sec (c+d x)+48 \left(\log (\cos (c+d x))-2 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)\right)\right)}{64 d}","-\frac{a^5}{2 d (a-a \cos (c+d x))^2}-\frac{3 a^4}{d (a-a \cos (c+d x))}+\frac{a^3 \sec ^2(c+d x)}{2 d}+\frac{3 a^3 \sec (c+d x)}{d}+\frac{6 a^3 \log (1-\cos (c+d x))}{d}-\frac{6 a^3 \log (\cos (c+d x))}{d}",1,"-1/64*(a^3*(1 + Cos[c + d*x])^3*Sec[(c + d*x)/2]^6*(12*Csc[(c + d*x)/2]^2 + Csc[(c + d*x)/2]^4 + 48*(Log[Cos[c + d*x]] - 2*Log[Sin[(c + d*x)/2]]) - 24*Sec[c + d*x] - 4*Sec[c + d*x]^2))/d","A",1
46,1,129,157,1.0559426,"\int \csc ^7(c+d x) (a+a \sec (c+d x))^3 \, dx","Integrate[Csc[c + d*x]^7*(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(2 \csc ^6\left(\frac{1}{2} (c+d x)\right)+21 \csc ^4\left(\frac{1}{2} (c+d x)\right)+186 \csc ^2\left(\frac{1}{2} (c+d x)\right)-12 \left(4 \sec ^2(c+d x)+24 \sec (c+d x)+111 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+\log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)-56 \log (\cos (c+d x))\right)\right)}{768 d}","-\frac{a^6}{6 d (a-a \cos (c+d x))^3}-\frac{7 a^5}{8 d (a-a \cos (c+d x))^2}-\frac{31 a^4}{8 d (a-a \cos (c+d x))}+\frac{a^3 \sec ^2(c+d x)}{2 d}+\frac{3 a^3 \sec (c+d x)}{d}+\frac{111 a^3 \log (1-\cos (c+d x))}{16 d}-\frac{7 a^3 \log (\cos (c+d x))}{d}+\frac{a^3 \log (\cos (c+d x)+1)}{16 d}",1,"-1/768*(a^3*(1 + Cos[c + d*x])^3*Sec[(c + d*x)/2]^6*(186*Csc[(c + d*x)/2]^2 + 21*Csc[(c + d*x)/2]^4 + 2*Csc[(c + d*x)/2]^6 - 12*(Log[Cos[(c + d*x)/2]] - 56*Log[Cos[c + d*x]] + 111*Log[Sin[(c + d*x)/2]] + 24*Sec[c + d*x] + 4*Sec[c + d*x]^2)))/d","A",1
47,1,159,202,1.2336544,"\int \csc ^9(c+d x) (a+a \sec (c+d x))^3 \, dx","Integrate[Csc[c + d*x]^9*(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(3 \csc ^8\left(\frac{1}{2} (c+d x)\right)+32 \csc ^6\left(\frac{1}{2} (c+d x)\right)+234 \csc ^4\left(\frac{1}{2} (c+d x)\right)+1800 \csc ^2\left(\frac{1}{2} (c+d x)\right)-12 \left(-\sec ^2\left(\frac{1}{2} (c+d x)\right)+32 \sec ^2(c+d x)+192 \sec (c+d x)+1002 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+22 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)-512 \log (\cos (c+d x))\right)\right)}{6144 d}","-\frac{a^7}{16 d (a-a \cos (c+d x))^4}-\frac{a^6}{3 d (a-a \cos (c+d x))^3}-\frac{39 a^5}{32 d (a-a \cos (c+d x))^2}-\frac{75 a^4}{16 d (a-a \cos (c+d x))}-\frac{a^4}{32 d (a \cos (c+d x)+a)}+\frac{a^3 \sec ^2(c+d x)}{2 d}+\frac{3 a^3 \sec (c+d x)}{d}+\frac{501 a^3 \log (1-\cos (c+d x))}{64 d}-\frac{8 a^3 \log (\cos (c+d x))}{d}+\frac{11 a^3 \log (\cos (c+d x)+1)}{64 d}",1,"-1/6144*(a^3*(1 + Cos[c + d*x])^3*Sec[(c + d*x)/2]^6*(1800*Csc[(c + d*x)/2]^2 + 234*Csc[(c + d*x)/2]^4 + 32*Csc[(c + d*x)/2]^6 + 3*Csc[(c + d*x)/2]^8 - 12*(22*Log[Cos[(c + d*x)/2]] - 512*Log[Cos[c + d*x]] + 1002*Log[Sin[(c + d*x)/2]] - Sec[(c + d*x)/2]^2 + 192*Sec[c + d*x] + 32*Sec[c + d*x]^2)))/d","A",1
48,1,156,210,2.1039571,"\int (a+a \sec (c+d x))^3 \sin ^8(c+d x) \, dx","Integrate[(a + a*Sec[c + d*x])^3*Sin[c + d*x]^8,x]","\frac{a^3 \sec ^2(c+d x) \left(173600 \sin (c+d x)+1052520 \sin (2 (c+d x))-11648 \sin (3 (c+d x))+175280 \sin (4 (c+d x))+22784 \sin (5 (c+d x))-18095 \sin (6 (c+d x))-6288 \sin (7 (c+d x))+770 \sin (8 (c+d x))+720 \sin (9 (c+d x))+105 \sin (10 (c+d x))-1352400 (c+d x) \cos (2 (c+d x))-215040 \cos ^2(c+d x) \tanh ^{-1}(\sin (c+d x))-1352400 c-1352400 d x\right)}{430080 d}","-\frac{3 a^3 \sin ^7(c+d x)}{7 d}-\frac{2 a^3 \sin ^5(c+d x)}{5 d}-\frac{a^3 \sin ^3(c+d x)}{3 d}+\frac{3 a^3 \tan (c+d x)}{d}-\frac{a^3 \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{a^3 \sin (c+d x) \cos ^7(c+d x)}{8 d}-\frac{a^3 \sin (c+d x) \cos ^5(c+d x)}{48 d}-\frac{293 a^3 \sin (c+d x) \cos ^3(c+d x)}{192 d}+\frac{603 a^3 \sin (c+d x) \cos (c+d x)}{128 d}+\frac{a^3 \tan (c+d x) \sec (c+d x)}{2 d}-\frac{805 a^3 x}{128}",1,"(a^3*Sec[c + d*x]^2*(-1352400*c - 1352400*d*x - 215040*ArcTanh[Sin[c + d*x]]*Cos[c + d*x]^2 - 1352400*(c + d*x)*Cos[2*(c + d*x)] + 173600*Sin[c + d*x] + 1052520*Sin[2*(c + d*x)] - 11648*Sin[3*(c + d*x)] + 175280*Sin[4*(c + d*x)] + 22784*Sin[5*(c + d*x)] - 18095*Sin[6*(c + d*x)] - 6288*Sin[7*(c + d*x)] + 770*Sin[8*(c + d*x)] + 720*Sin[9*(c + d*x)] + 105*Sin[10*(c + d*x)]))/(430080*d)","A",1
49,1,136,182,0.9527744,"\int (a+a \sec (c+d x))^3 \sin ^6(c+d x) \, dx","Integrate[(a + a*Sec[c + d*x])^3*Sin[c + d*x]^6,x]","-\frac{a^3 \sec ^2(c+d x) \left(-460 \sin (c+d x)-8145 \sin (2 (c+d x))+1156 \sin (3 (c+d x))-1120 \sin (4 (c+d x))-268 \sin (5 (c+d x))+55 \sin (6 (c+d x))+36 \sin (7 (c+d x))+5 \sin (8 (c+d x))+10200 (c+d x) \cos (2 (c+d x))-1920 \cos ^2(c+d x) \tanh ^{-1}(\sin (c+d x))+10200 c+10200 d x\right)}{3840 d}","-\frac{3 a^3 \sin ^5(c+d x)}{5 d}-\frac{2 a^3 \sin ^3(c+d x)}{3 d}-\frac{a^3 \sin (c+d x)}{d}+\frac{3 a^3 \tan (c+d x)}{d}+\frac{a^3 \tanh ^{-1}(\sin (c+d x))}{2 d}-\frac{a^3 \sin (c+d x) \cos ^5(c+d x)}{6 d}-\frac{5 a^3 \sin (c+d x) \cos ^3(c+d x)}{24 d}+\frac{43 a^3 \sin (c+d x) \cos (c+d x)}{16 d}+\frac{a^3 \tan (c+d x) \sec (c+d x)}{2 d}-\frac{85 a^3 x}{16}",1,"-1/3840*(a^3*Sec[c + d*x]^2*(10200*c + 10200*d*x - 1920*ArcTanh[Sin[c + d*x]]*Cos[c + d*x]^2 + 10200*(c + d*x)*Cos[2*(c + d*x)] - 460*Sin[c + d*x] - 8145*Sin[2*(c + d*x)] + 1156*Sin[3*(c + d*x)] - 1120*Sin[4*(c + d*x)] - 268*Sin[5*(c + d*x)] + 55*Sin[6*(c + d*x)] + 36*Sin[7*(c + d*x)] + 5*Sin[8*(c + d*x)]))/d","A",1
50,1,114,138,0.4124326,"\int (a+a \sec (c+d x))^3 \sin ^4(c+d x) \, dx","Integrate[(a + a*Sec[c + d*x])^3*Sin[c + d*x]^4,x]","\frac{a^3 \sec ^2(c+d x) \left(-16 \sin (c+d x)+225 \sin (2 (c+d x))-72 \sin (3 (c+d x))+18 \sin (4 (c+d x))+8 \sin (5 (c+d x))+\sin (6 (c+d x))-264 (c+d x) \cos (2 (c+d x))+192 \cos ^2(c+d x) \tanh ^{-1}(\sin (c+d x))-264 c-264 d x\right)}{128 d}","-\frac{a^3 \sin ^3(c+d x)}{d}-\frac{2 a^3 \sin (c+d x)}{d}+\frac{3 a^3 \tan (c+d x)}{d}+\frac{3 a^3 \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{a^3 \sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac{7 a^3 \sin (c+d x) \cos (c+d x)}{8 d}+\frac{a^3 \tan (c+d x) \sec (c+d x)}{2 d}-\frac{33 a^3 x}{8}",1,"(a^3*Sec[c + d*x]^2*(-264*c - 264*d*x + 192*ArcTanh[Sin[c + d*x]]*Cos[c + d*x]^2 - 264*(c + d*x)*Cos[2*(c + d*x)] - 16*Sin[c + d*x] + 225*Sin[2*(c + d*x)] - 72*Sin[3*(c + d*x)] + 18*Sin[4*(c + d*x)] + 8*Sin[5*(c + d*x)] + Sin[6*(c + d*x)]))/(128*d)","A",1
51,1,300,98,2.4171334,"\int (a+a \sec (c+d x))^3 \sin ^2(c+d x) \, dx","Integrate[(a + a*Sec[c + d*x])^3*Sin[c + d*x]^2,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 (c) \cos (d x)}{d}-\frac{\sin (2 c) \cos (2 d x)}{d}-\frac{12 \cos (c) \sin (d x)}{d}-\frac{\cos (2 c) \sin (2 d x)}{d}+\frac{12 \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{12 \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{10 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}{d}+\frac{10 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}{d}-10 x\right)","-\frac{3 a^3 \sin (c+d x)}{d}+\frac{3 a^3 \tan (c+d x)}{d}+\frac{5 a^3 \tanh ^{-1}(\sin (c+d x))}{2 d}-\frac{a^3 \sin (c+d x) \cos (c+d x)}{2 d}+\frac{a^3 \tan (c+d x) \sec (c+d x)}{2 d}-\frac{5 a^3 x}{2}",1,"(a^3*(1 + Cos[c + d*x])^3*Sec[(c + d*x)/2]^6*(-10*x - (10*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]])/d + (10*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]])/d - (12*Cos[d*x]*Sin[c])/d - (Cos[2*d*x]*Sin[2*c])/d - (12*Cos[c]*Sin[d*x])/d - (Cos[2*c]*Sin[2*d*x])/d + 1/(d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2) + (12*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) + (12*Sin[(d*x)/2])/(d*(Cos[c/2] + Sin[c/2])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))))/32","B",1
52,1,244,80,1.1604267,"\int \csc ^2(c+d x) (a+a \sec (c+d x))^3 \, dx","Integrate[Csc[c + d*x]^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(\frac{12 \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}+16 \csc \left(\frac{c}{2}\right) \sin \left(\frac{d x}{2}\right) \csc \left(\frac{1}{2} (c+d x)\right)-18 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+18 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)}{32 d}","\frac{3 a^3 \tan (c+d x)}{d}+\frac{9 a^3 \tanh ^{-1}(\sin (c+d x))}{2 d}-\frac{4 a^3 \sin (c+d x)}{d (1-\cos (c+d x))}+\frac{a^3 \tan (c+d x) \sec (c+d x)}{2 d}",1,"(a^3*(1 + Cos[c + d*x])^3*Sec[(c + d*x)/2]^6*(-18*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 18*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + 16*Csc[c/2]*Csc[(c + d*x)/2]*Sin[(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) + (12*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]))))/(32*d)","B",1
53,1,678,110,6.2451885,"\int \csc ^4(c+d x) (a+a \sec (c+d x))^3 \, dx","Integrate[Csc[c + d*x]^4*(a + a*Sec[c + d*x])^3,x]","\frac{3 \sin \left(\frac{d x}{2}\right) \cos ^3(c+d x) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right) (a \sec (c+d x)+a)^3}{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 \sin \left(\frac{d x}{2}\right) \cos ^3(c+d x) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right) (a \sec (c+d x)+a)^3}{8 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{\cos ^3(c+d x) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right) (a \sec (c+d x)+a)^3}{32 d \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 ^6\left(\frac{c}{2}+\frac{d x}{2}\right) (a \sec (c+d x)+a)^3}{32 d \left(\sin \left(\frac{c}{2}+\frac{d x}{2}\right)+\cos \left(\frac{c}{2}+\frac{d x}{2}\right)\right)^2}-\frac{11 \cos ^3(c+d x) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right) (a \sec (c+d x)+a)^3 \log \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)-\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)}{16 d}+\frac{11 \cos ^3(c+d x) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right) (a \sec (c+d x)+a)^3 \log \left(\sin \left(\frac{c}{2}+\frac{d x}{2}\right)+\cos \left(\frac{c}{2}+\frac{d x}{2}\right)\right)}{16 d}-\frac{\cot \left(\frac{c}{2}\right) \cos ^3(c+d x) \csc ^2\left(\frac{c}{2}+\frac{d x}{2}\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right) (a \sec (c+d x)+a)^3}{24 d}+\frac{\csc \left(\frac{c}{2}\right) \sin \left(\frac{d x}{2}\right) \cos ^3(c+d x) \csc ^3\left(\frac{c}{2}+\frac{d x}{2}\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right) (a \sec (c+d x)+a)^3}{24 d}+\frac{17 \csc \left(\frac{c}{2}\right) \sin \left(\frac{d x}{2}\right) \cos ^3(c+d x) \csc \left(\frac{c}{2}+\frac{d x}{2}\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right) (a \sec (c+d x)+a)^3}{24 d}","\frac{3 a^3 \tan (c+d x)}{d}+\frac{11 a^3 \tanh ^{-1}(\sin (c+d x))}{2 d}-\frac{17 a^3 \sin (c+d x)}{3 d (1-\cos (c+d x))}-\frac{2 a^3 \sin (c+d x)}{3 d (1-\cos (c+d x))^2}+\frac{a^3 \tan (c+d x) \sec (c+d x)}{2 d}",1,"-1/24*(Cos[c + d*x]^3*Cot[c/2]*Csc[c/2 + (d*x)/2]^2*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3)/d - (11*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)/(16*d) + (11*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)/(16*d) + (17*Cos[c + d*x]^3*Csc[c/2]*Csc[c/2 + (d*x)/2]*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3*Sin[(d*x)/2])/(24*d) + (Cos[c + d*x]^3*Csc[c/2]*Csc[c/2 + (d*x)/2]^3*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3*Sin[(d*x)/2])/(24*d) + (Cos[c + d*x]^3*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3)/(32*d*(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)/(32*d*(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",1
54,1,353,165,1.2240505,"\int \csc ^6(c+d x) (a+a \sec (c+d x))^3 \, dx","Integrate[Csc[c + d*x]^6*(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) \sec ^2(c+d x) \left(24960 \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)+\csc \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) \csc ^5\left(\frac{1}{2} (c+d x)\right)\right)}{30720 d}","-\frac{a^6 \tan (c+d x) \sec (c+d x)}{5 d (a-a \cos (c+d x))^3}-\frac{11 a^5 \tan (c+d x) \sec (c+d x)}{15 d (a-a \cos (c+d x))^2}+\frac{152 a^3 \tan (c+d x)}{15 d}+\frac{13 a^3 \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{13 a^3 \tan (c+d x) \sec (c+d x)}{2 d}-\frac{76 a^6 \tan (c+d x) \sec (c+d x)}{15 d \left(a^3-a^3 \cos (c+d x)\right)}",1,"-1/30720*(a^3*(1 + Cos[c + d*x])^3*Sec[(c + d*x)/2]^6*Sec[c + d*x]^2*(24960*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]]) + Csc[c/2]*Csc[(c + d*x)/2]^5*Sec[c]*(-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])))/d","B",1
55,1,430,192,1.2496823,"\int \csc ^8(c+d x) (a+a \sec (c+d x))^3 \, dx","Integrate[Csc[c + d*x]^8*(a + a*Sec[c + d*x])^3,x]","\frac{a^3 \cos (c+d x) \sec ^6\left(\frac{1}{2} (c+d x)\right) (\sec (c+d x)+1)^3 \left(-8 \csc (2 c) (2776 \sin (c-d x)-6080 \sin (c+d x)+8816 \sin (2 (c+d x))-7904 \sin (3 (c+d x))+4864 \sin (4 (c+d x))-1824 \sin (5 (c+d x))+304 \sin (6 (c+d x))-9580 \sin (2 c+d x)-10024 \sin (3 c+d x)+13891 \sin (c+2 d x)+7720 \sin (2 (c+2 d x))+13891 \sin (3 c+2 d x)+10080 \sin (4 c+2 d x)-10060 \sin (c+3 d x)-12454 \sin (2 c+3 d x)-12454 \sin (4 c+3 d x)-6580 \sin (5 c+3 d x)+7664 \sin (3 c+4 d x)+7664 \sin (5 c+4 d x)+2520 \sin (6 c+4 d x)-3420 \sin (3 c+5 d x)-2874 \sin (4 c+5 d x)-2874 \sin (6 c+5 d x)-420 \sin (7 c+5 d x)+640 \sin (4 c+6 d x)+479 \sin (5 c+6 d x)+479 \sin (7 c+6 d x)+5264 \sin (2 c)-9580 \sin (d x)+8480 \sin (2 d x)) \csc (c+d x) \csc ^6\left(\frac{1}{2} (c+d x)\right)-860160 \cos ^2(c+d x) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+860160 \cos ^2(c+d x) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)}{917504 d}","\frac{3 a^3 \tan (c+d x)}{d}-\frac{4 a^3 \cot ^7(c+d x)}{7 d}-\frac{3 a^3 \cot ^5(c+d x)}{d}-\frac{7 a^3 \cot ^3(c+d x)}{d}-\frac{13 a^3 \cot (c+d x)}{d}-\frac{15 a^3 \csc ^7(c+d x)}{14 d}-\frac{3 a^3 \csc ^5(c+d x)}{2 d}-\frac{5 a^3 \csc ^3(c+d x)}{2 d}-\frac{15 a^3 \csc (c+d x)}{2 d}+\frac{15 a^3 \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{a^3 \csc ^7(c+d x) \sec ^2(c+d x)}{2 d}",1,"(a^3*Cos[c + d*x]*Sec[(c + d*x)/2]^6*(1 + Sec[c + d*x])^3*(-860160*Cos[c + d*x]^2*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 860160*Cos[c + d*x]^2*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] - 8*Csc[2*c]*Csc[(c + d*x)/2]^6*Csc[c + d*x]*(5264*Sin[2*c] - 9580*Sin[d*x] + 8480*Sin[2*d*x] + 2776*Sin[c - d*x] - 6080*Sin[c + d*x] + 8816*Sin[2*(c + d*x)] - 7904*Sin[3*(c + d*x)] + 4864*Sin[4*(c + d*x)] - 1824*Sin[5*(c + d*x)] + 304*Sin[6*(c + d*x)] - 9580*Sin[2*c + d*x] - 10024*Sin[3*c + d*x] + 13891*Sin[c + 2*d*x] + 7720*Sin[2*(c + 2*d*x)] + 13891*Sin[3*c + 2*d*x] + 10080*Sin[4*c + 2*d*x] - 10060*Sin[c + 3*d*x] - 12454*Sin[2*c + 3*d*x] - 12454*Sin[4*c + 3*d*x] - 6580*Sin[5*c + 3*d*x] + 7664*Sin[3*c + 4*d*x] + 7664*Sin[5*c + 4*d*x] + 2520*Sin[6*c + 4*d*x] - 3420*Sin[3*c + 5*d*x] - 2874*Sin[4*c + 5*d*x] - 2874*Sin[6*c + 5*d*x] - 420*Sin[7*c + 5*d*x] + 640*Sin[4*c + 6*d*x] + 479*Sin[5*c + 6*d*x] + 479*Sin[7*c + 6*d*x])))/(917504*d)","B",1
56,1,1000,232,6.7138149,"\int \csc ^{10}(c+d x) (a+a \sec (c+d x))^3 \, dx","Integrate[Csc[c + d*x]^10*(a + a*Sec[c + d*x])^3,x]","\frac{\cos ^3(c+d x) \csc \left(\frac{c}{2}\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right) (\sec (c+d x) a+a)^3 \sin \left(\frac{d x}{2}\right) \csc ^9\left(\frac{c}{2}+\frac{d x}{2}\right)}{4608 d}-\frac{\cos ^3(c+d x) \cot \left(\frac{c}{2}\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right) (\sec (c+d x) a+a)^3 \csc ^8\left(\frac{c}{2}+\frac{d x}{2}\right)}{4608 d}+\frac{5 \cos ^3(c+d x) \csc \left(\frac{c}{2}\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right) (\sec (c+d x) a+a)^3 \sin \left(\frac{d x}{2}\right) \csc ^7\left(\frac{c}{2}+\frac{d x}{2}\right)}{2016 d}-\frac{5 \cos ^3(c+d x) \cot \left(\frac{c}{2}\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right) (\sec (c+d x) a+a)^3 \csc ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{2016 d}+\frac{979 \cos ^3(c+d x) \csc \left(\frac{c}{2}\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right) (\sec (c+d x) a+a)^3 \sin \left(\frac{d x}{2}\right) \csc ^5\left(\frac{c}{2}+\frac{d x}{2}\right)}{53760 d}-\frac{979 \cos ^3(c+d x) \cot \left(\frac{c}{2}\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right) (\sec (c+d x) a+a)^3 \csc ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{53760 d}+\frac{9833 \cos ^3(c+d x) \csc \left(\frac{c}{2}\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right) (\sec (c+d x) a+a)^3 \sin \left(\frac{d x}{2}\right) \csc ^3\left(\frac{c}{2}+\frac{d x}{2}\right)}{80640 d}-\frac{9833 \cos ^3(c+d x) \cot \left(\frac{c}{2}\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right) (\sec (c+d x) a+a)^3 \csc ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{80640 d}+\frac{197147 \cos ^3(c+d x) \csc \left(\frac{c}{2}\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right) (\sec (c+d x) a+a)^3 \sin \left(\frac{d x}{2}\right) \csc \left(\frac{c}{2}+\frac{d x}{2}\right)}{161280 d}-\frac{17 \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 ^6\left(\frac{c}{2}+\frac{d x}{2}\right) (\sec (c+d x) a+a)^3}{16 d}+\frac{17 \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 ^6\left(\frac{c}{2}+\frac{d x}{2}\right) (\sec (c+d x) a+a)^3}{16 d}-\frac{\cos ^3(c+d x) \sec \left(\frac{c}{2}\right) \sec ^9\left(\frac{c}{2}+\frac{d x}{2}\right) (\sec (c+d x) a+a)^3 \sin \left(\frac{d x}{2}\right)}{1536 d}-\frac{35 \cos ^3(c+d x) \sec \left(\frac{c}{2}\right) \sec ^7\left(\frac{c}{2}+\frac{d x}{2}\right) (\sec (c+d x) a+a)^3 \sin \left(\frac{d x}{2}\right)}{1536 d}+\frac{\cos (c+d x) \sec (c) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right) (\sec (c+d x) a+a)^3 \sin (d x)}{16 d}+\frac{\cos ^2(c+d x) \sec (c) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right) (\sec (c+d x) a+a)^3 (\sin (c)+6 \sin (d x))}{16 d}-\frac{\cos ^3(c+d x) \sec ^8\left(\frac{c}{2}+\frac{d x}{2}\right) (\sec (c+d x) a+a)^3 \tan \left(\frac{c}{2}\right)}{1536 d}","\frac{3 a^3 \tan (c+d x)}{d}-\frac{4 a^3 \cot ^9(c+d x)}{9 d}-\frac{19 a^3 \cot ^7(c+d x)}{7 d}-\frac{36 a^3 \cot ^5(c+d x)}{5 d}-\frac{34 a^3 \cot ^3(c+d x)}{3 d}-\frac{16 a^3 \cot (c+d x)}{d}-\frac{17 a^3 \csc ^9(c+d x)}{18 d}-\frac{17 a^3 \csc ^7(c+d x)}{14 d}-\frac{17 a^3 \csc ^5(c+d x)}{10 d}-\frac{17 a^3 \csc ^3(c+d x)}{6 d}-\frac{17 a^3 \csc (c+d x)}{2 d}+\frac{17 a^3 \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{a^3 \csc ^9(c+d x) \sec ^2(c+d x)}{2 d}",1,"(-9833*Cos[c + d*x]^3*Cot[c/2]*Csc[c/2 + (d*x)/2]^2*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3)/(80640*d) - (979*Cos[c + d*x]^3*Cot[c/2]*Csc[c/2 + (d*x)/2]^4*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3)/(53760*d) - (5*Cos[c + d*x]^3*Cot[c/2]*Csc[c/2 + (d*x)/2]^6*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3)/(2016*d) - (Cos[c + d*x]^3*Cot[c/2]*Csc[c/2 + (d*x)/2]^8*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3)/(4608*d) - (17*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)/(16*d) + (17*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)/(16*d) + (197147*Cos[c + d*x]^3*Csc[c/2]*Csc[c/2 + (d*x)/2]*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3*Sin[(d*x)/2])/(161280*d) + (9833*Cos[c + d*x]^3*Csc[c/2]*Csc[c/2 + (d*x)/2]^3*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3*Sin[(d*x)/2])/(80640*d) + (979*Cos[c + d*x]^3*Csc[c/2]*Csc[c/2 + (d*x)/2]^5*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3*Sin[(d*x)/2])/(53760*d) + (5*Cos[c + d*x]^3*Csc[c/2]*Csc[c/2 + (d*x)/2]^7*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3*Sin[(d*x)/2])/(2016*d) + (Cos[c + d*x]^3*Csc[c/2]*Csc[c/2 + (d*x)/2]^9*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3*Sin[(d*x)/2])/(4608*d) - (35*Cos[c + d*x]^3*Sec[c/2]*Sec[c/2 + (d*x)/2]^7*(a + a*Sec[c + d*x])^3*Sin[(d*x)/2])/(1536*d) - (Cos[c + d*x]^3*Sec[c/2]*Sec[c/2 + (d*x)/2]^9*(a + a*Sec[c + d*x])^3*Sin[(d*x)/2])/(1536*d) + (Cos[c + d*x]*Sec[c]*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3*Sin[d*x])/(16*d) + (Cos[c + d*x]^2*Sec[c]*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3*(Sin[c] + 6*Sin[d*x]))/(16*d) - (Cos[c + d*x]^3*Sec[c/2 + (d*x)/2]^8*(a + a*Sec[c + d*x])^3*Tan[c/2])/(1536*d)","B",0
57,1,62,91,4.8691914,"\int \frac{\sin ^9(c+d x)}{a+a \sec (c+d x)} \, dx","Integrate[Sin[c + d*x]^9/(a + a*Sec[c + d*x]),x]","\frac{\sin ^{10}\left(\frac{1}{2} (c+d x)\right) (6995 \cos (c+d x)+3650 \cos (2 (c+d x))+1085 \cos (3 (c+d x))+140 \cos (4 (c+d x))+4258)}{315 a d}","\frac{\sin ^8(c+d x)}{8 a d}-\frac{\cos ^9(c+d x)}{9 a d}+\frac{3 \cos ^7(c+d x)}{7 a d}-\frac{3 \cos ^5(c+d x)}{5 a d}+\frac{\cos ^3(c+d x)}{3 a d}",1,"((4258 + 6995*Cos[c + d*x] + 3650*Cos[2*(c + d*x)] + 1085*Cos[3*(c + d*x)] + 140*Cos[4*(c + d*x)])*Sin[(c + d*x)/2]^10)/(315*a*d)","A",1
58,1,52,73,1.7202324,"\int \frac{\sin ^7(c+d x)}{a+a \sec (c+d x)} \, dx","Integrate[Sin[c + d*x]^7/(a + a*Sec[c + d*x]),x]","\frac{4 \sin ^8\left(\frac{1}{2} (c+d x)\right) (197 \cos (c+d x)+85 \cos (2 (c+d x))+15 \cos (3 (c+d x))+123)}{105 a d}","\frac{\sin ^6(c+d x)}{6 a d}+\frac{\cos ^7(c+d x)}{7 a d}-\frac{2 \cos ^5(c+d x)}{5 a d}+\frac{\cos ^3(c+d x)}{3 a d}",1,"(4*(123 + 197*Cos[c + d*x] + 85*Cos[2*(c + d*x)] + 15*Cos[3*(c + d*x)])*Sin[(c + d*x)/2]^8)/(105*a*d)","A",1
59,1,42,55,0.3975035,"\int \frac{\sin ^5(c+d x)}{a+a \sec (c+d x)} \, dx","Integrate[Sin[c + d*x]^5/(a + a*Sec[c + d*x]),x]","\frac{2 \sin ^6\left(\frac{1}{2} (c+d x)\right) (21 \cos (c+d x)+6 \cos (2 (c+d x))+13)}{15 a d}","\frac{\sin ^4(c+d x)}{4 a d}-\frac{\cos ^5(c+d x)}{5 a d}+\frac{\cos ^3(c+d x)}{3 a d}",1,"(2*(13 + 21*Cos[c + d*x] + 6*Cos[2*(c + d*x)])*Sin[(c + d*x)/2]^6)/(15*a*d)","A",1
60,1,32,37,0.127842,"\int \frac{\sin ^3(c+d x)}{a+a \sec (c+d x)} \, dx","Integrate[Sin[c + d*x]^3/(a + a*Sec[c + d*x]),x]","\frac{2 \sin ^4\left(\frac{1}{2} (c+d x)\right) (2 \cos (c+d x)+1)}{3 a d}","\frac{\sin ^2(c+d x)}{2 a d}+\frac{\cos ^3(c+d x)}{3 a d}",1,"(2*(1 + 2*Cos[c + d*x])*Sin[(c + d*x)/2]^4)/(3*a*d)","A",1
61,1,28,31,0.0868303,"\int \frac{\sin (c+d x)}{a+a \sec (c+d x)} \, dx","Integrate[Sin[c + d*x]/(a + a*Sec[c + d*x]),x]","-\frac{\cos (c+d x)-2 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)}{a d}","\frac{\log (\cos (c+d x)+1)}{a d}-\frac{\cos (c+d x)}{a d}",1,"-((Cos[c + d*x] - 2*Log[Cos[(c + d*x)/2]])/(a*d))","A",1
62,1,67,58,0.1060627,"\int \frac{\csc (c+d x)}{a+a \sec (c+d x)} \, dx","Integrate[Csc[c + d*x]/(a + a*Sec[c + d*x]),x]","-\frac{\sec (c+d x) \left(2 \cos ^2\left(\frac{1}{2} (c+d x)\right) \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)\right)+1\right)}{2 a d (\sec (c+d x)+1)}","-\frac{\csc ^2(c+d x)}{2 a d}-\frac{\tanh ^{-1}(\cos (c+d x))}{2 a d}+\frac{\cot (c+d x) \csc (c+d x)}{2 a d}",1,"-1/2*((1 + 2*Cos[(c + d*x)/2]^2*(Log[Cos[(c + d*x)/2]] - Log[Sin[(c + d*x)/2]]))*Sec[c + d*x])/(a*d*(1 + Sec[c + d*x]))","A",1
63,1,91,82,0.3940442,"\int \frac{\csc ^3(c+d x)}{a+a \sec (c+d x)} \, dx","Integrate[Csc[c + d*x]^3/(a + a*Sec[c + d*x]),x]","-\frac{\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x) \left(2 \csc ^2\left(\frac{1}{2} (c+d x)\right)+\sec ^4\left(\frac{1}{2} (c+d x)\right)-4 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+4 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)}{16 a d (\sec (c+d x)+1)}","-\frac{\csc ^4(c+d x)}{4 a d}-\frac{\tanh ^{-1}(\cos (c+d x))}{8 a d}+\frac{\cot (c+d x) \csc ^3(c+d x)}{4 a d}-\frac{\cot (c+d x) \csc (c+d x)}{8 a d}",1,"-1/16*(Cos[(c + d*x)/2]^2*(2*Csc[(c + d*x)/2]^2 + 4*Log[Cos[(c + d*x)/2]] - 4*Log[Sin[(c + d*x)/2]] + Sec[(c + d*x)/2]^4)*Sec[c + d*x])/(a*d*(1 + Sec[c + d*x]))","A",1
64,1,122,106,0.4998727,"\int \frac{\csc ^5(c+d x)}{a+a \sec (c+d x)} \, dx","Integrate[Csc[c + d*x]^5/(a + a*Sec[c + d*x]),x]","-\frac{\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x) \left(3 \csc ^4\left(\frac{1}{2} (c+d x)\right)+12 \csc ^2\left(\frac{1}{2} (c+d x)\right)+2 \sec ^6\left(\frac{1}{2} (c+d x)\right)+3 \sec ^4\left(\frac{1}{2} (c+d x)\right)+24 \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)\right)\right)}{192 a d (\sec (c+d x)+1)}","-\frac{\csc ^6(c+d x)}{6 a d}-\frac{\tanh ^{-1}(\cos (c+d x))}{16 a d}+\frac{\cot (c+d x) \csc ^5(c+d x)}{6 a d}-\frac{\cot (c+d x) \csc ^3(c+d x)}{24 a d}-\frac{\cot (c+d x) \csc (c+d x)}{16 a d}",1,"-1/192*(Cos[(c + d*x)/2]^2*(12*Csc[(c + d*x)/2]^2 + 3*Csc[(c + d*x)/2]^4 + 24*(Log[Cos[(c + d*x)/2]] - Log[Sin[(c + d*x)/2]]) + 3*Sec[(c + d*x)/2]^4 + 2*Sec[(c + d*x)/2]^6)*Sec[c + d*x])/(a*d*(1 + Sec[c + d*x]))","A",1
65,1,132,125,1.3071122,"\int \frac{\sin ^8(c+d x)}{a+a \sec (c+d x)} \, dx","Integrate[Sin[c + d*x]^8/(a + a*Sec[c + d*x]),x]","\frac{\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x) \left(1680 \sin (c+d x)+336 \sin (2 (c+d x))-1008 \sin (3 (c+d x))+168 \sin (4 (c+d x))+336 \sin (5 (c+d x))-112 \sin (6 (c+d x))-48 \sin (7 (c+d x))+21 \sin (8 (c+d x))+1176 c-1176 \tan \left(\frac{c}{2}\right)-840 d x\right)}{10752 a d (\sec (c+d x)+1)}","\frac{\sin ^7(c+d x)}{7 a d}+\frac{\sin ^5(c+d x) \cos ^3(c+d x)}{8 a d}+\frac{5 \sin ^3(c+d x) \cos ^3(c+d x)}{48 a d}+\frac{5 \sin (c+d x) \cos ^3(c+d x)}{64 a d}-\frac{5 \sin (c+d x) \cos (c+d x)}{128 a d}-\frac{5 x}{128 a}",1,"(Cos[(c + d*x)/2]^2*Sec[c + d*x]*(1176*c - 840*d*x + 1680*Sin[c + d*x] + 336*Sin[2*(c + d*x)] - 1008*Sin[3*(c + d*x)] + 168*Sin[4*(c + d*x)] + 336*Sin[5*(c + d*x)] - 112*Sin[6*(c + d*x)] - 48*Sin[7*(c + d*x)] + 21*Sin[8*(c + d*x)] - 1176*Tan[c/2]))/(10752*a*d*(1 + Sec[c + d*x]))","A",1
66,1,112,99,0.7416826,"\int \frac{\sin ^6(c+d x)}{a+a \sec (c+d x)} \, dx","Integrate[Sin[c + d*x]^6/(a + a*Sec[c + d*x]),x]","\frac{\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x) \left(120 \sin (c+d x)+15 \sin (2 (c+d x))-60 \sin (3 (c+d x))+15 \sin (4 (c+d x))+12 \sin (5 (c+d x))-5 \sin (6 (c+d x))+75 c-75 \tan \left(\frac{c}{2}\right)-60 d x\right)}{480 a d (\sec (c+d x)+1)}","\frac{\sin ^5(c+d x)}{5 a d}+\frac{\sin ^3(c+d x) \cos ^3(c+d x)}{6 a d}+\frac{\sin (c+d x) \cos ^3(c+d x)}{8 a d}-\frac{\sin (c+d x) \cos (c+d x)}{16 a d}-\frac{x}{16 a}",1,"(Cos[(c + d*x)/2]^2*Sec[c + d*x]*(75*c - 60*d*x + 120*Sin[c + d*x] + 15*Sin[2*(c + d*x)] - 60*Sin[3*(c + d*x)] + 15*Sin[4*(c + d*x)] + 12*Sin[5*(c + d*x)] - 5*Sin[6*(c + d*x)] - 75*Tan[c/2]))/(480*a*d*(1 + Sec[c + d*x]))","A",1
67,1,83,73,0.6492753,"\int \frac{\sin ^4(c+d x)}{a+a \sec (c+d x)} \, dx","Integrate[Sin[c + d*x]^4/(a + a*Sec[c + d*x]),x]","\frac{\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x) \left(24 \sin (c+d x)-8 \sin (3 (c+d x))+3 \left(\sin (4 (c+d x))+4 c-4 \tan \left(\frac{c}{2}\right)-4 d x\right)\right)}{48 a d (\sec (c+d x)+1)}","\frac{\sin ^3(c+d x)}{3 a d}+\frac{\sin (c+d x) \cos ^3(c+d x)}{4 a d}-\frac{\sin (c+d x) \cos (c+d x)}{8 a d}-\frac{x}{8 a}",1,"(Cos[(c + d*x)/2]^2*Sec[c + d*x]*(24*Sin[c + d*x] - 8*Sin[3*(c + d*x)] + 3*(4*c - 4*d*x + Sin[4*(c + d*x)] - 4*Tan[c/2])))/(48*a*d*(1 + Sec[c + d*x]))","A",1
68,1,68,44,0.282179,"\int \frac{\sin ^2(c+d x)}{a+a \sec (c+d x)} \, dx","Integrate[Sin[c + d*x]^2/(a + a*Sec[c + d*x]),x]","-\frac{\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x) \left(-4 \sin (c+d x)+\sin (2 (c+d x))-c+\tan \left(\frac{c}{2}\right)+2 d x\right)}{2 a d (\sec (c+d x)+1)}","\frac{\sin (c+d x)}{a d}-\frac{\sin (c+d x) \cos (c+d x)}{2 a d}-\frac{x}{2 a}",1,"-1/2*(Cos[(c + d*x)/2]^2*Sec[c + d*x]*(-c + 2*d*x - 4*Sin[c + d*x] + Sin[2*(c + d*x)] + Tan[c/2]))/(a*d*(1 + Sec[c + d*x]))","A",1
69,1,66,37,0.2246918,"\int \frac{\csc ^2(c+d x)}{a+a \sec (c+d x)} \, dx","Integrate[Csc[c + d*x]^2/(a + a*Sec[c + d*x]),x]","\frac{\csc (c) (2 \sin (c+d x)+\sin (2 (c+d x))+2 \sin (c+2 d x)-6 \sin (c)+4 \sin (d x)) \csc (2 (c+d x))}{6 a d (\sec (c+d x)+1)}","\frac{\cot ^3(c+d x)}{3 a d}-\frac{\csc ^3(c+d x)}{3 a d}",1,"(Csc[c]*Csc[2*(c + d*x)]*(-6*Sin[c] + 4*Sin[d*x] + 2*Sin[c + d*x] + Sin[2*(c + d*x)] + 2*Sin[c + 2*d*x]))/(6*a*d*(1 + Sec[c + d*x]))","A",1
70,1,116,55,0.5355802,"\int \frac{\csc ^4(c+d x)}{a+a \sec (c+d x)} \, dx","Integrate[Csc[c + d*x]^4/(a + a*Sec[c + d*x]),x]","-\frac{\csc (c) (-54 \sin (c+d x)-18 \sin (2 (c+d x))+18 \sin (3 (c+d x))+9 \sin (4 (c+d x))-32 \sin (c+2 d x)+32 \sin (2 c+3 d x)+16 \sin (3 c+4 d x)+240 \sin (c)-96 \sin (d x)) \csc ^3(c+d x) \sec (c+d x)}{960 a d (\sec (c+d x)+1)}","\frac{\cot ^5(c+d x)}{5 a d}+\frac{\cot ^3(c+d x)}{3 a d}-\frac{\csc ^5(c+d x)}{5 a d}",1,"-1/960*(Csc[c]*Csc[c + d*x]^3*Sec[c + d*x]*(240*Sin[c] - 96*Sin[d*x] - 54*Sin[c + d*x] - 18*Sin[2*(c + d*x)] + 18*Sin[3*(c + d*x)] + 9*Sin[4*(c + d*x)] - 32*Sin[c + 2*d*x] + 32*Sin[2*c + 3*d*x] + 16*Sin[3*c + 4*d*x]))/(a*d*(1 + Sec[c + d*x]))","B",1
71,1,158,73,0.6551395,"\int \frac{\csc ^6(c+d x)}{a+a \sec (c+d x)} \, dx","Integrate[Csc[c + d*x]^6/(a + a*Sec[c + d*x]),x]","\frac{\csc (c) (1500 \sin (c+d x)+375 \sin (2 (c+d x))-750 \sin (3 (c+d x))-300 \sin (4 (c+d x))+150 \sin (5 (c+d x))+75 \sin (6 (c+d x))+640 \sin (c+2 d x)-1280 \sin (2 c+3 d x)-512 \sin (3 c+4 d x)+256 \sin (4 c+5 d x)+128 \sin (5 c+6 d x)-8960 \sin (c)+2560 \sin (d x)) \csc ^5(c+d x) \sec (c+d x)}{53760 a d (\sec (c+d x)+1)}","\frac{\cot ^7(c+d x)}{7 a d}+\frac{2 \cot ^5(c+d x)}{5 a d}+\frac{\cot ^3(c+d x)}{3 a d}-\frac{\csc ^7(c+d x)}{7 a d}",1,"(Csc[c]*Csc[c + d*x]^5*Sec[c + d*x]*(-8960*Sin[c] + 2560*Sin[d*x] + 1500*Sin[c + d*x] + 375*Sin[2*(c + d*x)] - 750*Sin[3*(c + d*x)] - 300*Sin[4*(c + d*x)] + 150*Sin[5*(c + d*x)] + 75*Sin[6*(c + d*x)] + 640*Sin[c + 2*d*x] - 1280*Sin[2*c + 3*d*x] - 512*Sin[3*c + 4*d*x] + 256*Sin[4*c + 5*d*x] + 128*Sin[5*c + 6*d*x]))/(53760*a*d*(1 + Sec[c + d*x]))","B",1
72,1,200,91,1.0613477,"\int \frac{\csc ^8(c+d x)}{a+a \sec (c+d x)} \, dx","Integrate[Csc[c + d*x]^8/(a + a*Sec[c + d*x]),x]","-\frac{\csc (c) (-85750 \sin (c+d x)-17150 \sin (2 (c+d x))+51450 \sin (3 (c+d x))+17150 \sin (4 (c+d x))-17150 \sin (5 (c+d x))-7350 \sin (6 (c+d x))+2450 \sin (7 (c+d x))+1225 \sin (8 (c+d x))-28672 \sin (c+2 d x)+86016 \sin (2 c+3 d x)+28672 \sin (3 c+4 d x)-28672 \sin (4 c+5 d x)-12288 \sin (5 c+6 d x)+4096 \sin (6 c+7 d x)+2048 \sin (7 c+8 d x)+645120 \sin (c)-143360 \sin (d x)) \csc ^7(c+d x) \sec (c+d x)}{5160960 a d (\sec (c+d x)+1)}","\frac{\cot ^9(c+d x)}{9 a d}+\frac{3 \cot ^7(c+d x)}{7 a d}+\frac{3 \cot ^5(c+d x)}{5 a d}+\frac{\cot ^3(c+d x)}{3 a d}-\frac{\csc ^9(c+d x)}{9 a d}",1,"-1/5160960*(Csc[c]*Csc[c + d*x]^7*Sec[c + d*x]*(645120*Sin[c] - 143360*Sin[d*x] - 85750*Sin[c + d*x] - 17150*Sin[2*(c + d*x)] + 51450*Sin[3*(c + d*x)] + 17150*Sin[4*(c + d*x)] - 17150*Sin[5*(c + d*x)] - 7350*Sin[6*(c + d*x)] + 2450*Sin[7*(c + d*x)] + 1225*Sin[8*(c + d*x)] - 28672*Sin[c + 2*d*x] + 86016*Sin[2*c + 3*d*x] + 28672*Sin[3*c + 4*d*x] - 28672*Sin[4*c + 5*d*x] - 12288*Sin[5*c + 6*d*x] + 4096*Sin[6*c + 7*d*x] + 2048*Sin[7*c + 8*d*x]))/(a*d*(1 + Sec[c + d*x]))","B",1
73,1,242,109,1.5755183,"\int \frac{\csc ^{10}(c+d x)}{a+a \sec (c+d x)} \, dx","Integrate[Csc[c + d*x]^10/(a + a*Sec[c + d*x]),x]","\frac{\csc (c) (5000940 \sin (c+d x)+833490 \sin (2 (c+d x))-3333960 \sin (3 (c+d x))-952560 \sin (4 (c+d x))+1428840 \sin (5 (c+d x))+535815 \sin (6 (c+d x))-357210 \sin (7 (c+d x))-158760 \sin (8 (c+d x))+39690 \sin (9 (c+d x))+19845 \sin (10 (c+d x))+1376256 \sin (c+2 d x)-5505024 \sin (2 c+3 d x)-1572864 \sin (3 c+4 d x)+2359296 \sin (4 c+5 d x)+884736 \sin (5 c+6 d x)-589824 \sin (6 c+7 d x)-262144 \sin (7 c+8 d x)+65536 \sin (8 c+9 d x)+32768 \sin (9 c+10 d x)-45416448 \sin (c)+8257536 \sin (d x)) \csc ^9(c+d x) \sec (c+d x)}{454164480 a d (\sec (c+d x)+1)}","\frac{\cot ^{11}(c+d x)}{11 a d}+\frac{4 \cot ^9(c+d x)}{9 a d}+\frac{6 \cot ^7(c+d x)}{7 a d}+\frac{4 \cot ^5(c+d x)}{5 a d}+\frac{\cot ^3(c+d x)}{3 a d}-\frac{\csc ^{11}(c+d x)}{11 a d}",1,"(Csc[c]*Csc[c + d*x]^9*Sec[c + d*x]*(-45416448*Sin[c] + 8257536*Sin[d*x] + 5000940*Sin[c + d*x] + 833490*Sin[2*(c + d*x)] - 3333960*Sin[3*(c + d*x)] - 952560*Sin[4*(c + d*x)] + 1428840*Sin[5*(c + d*x)] + 535815*Sin[6*(c + d*x)] - 357210*Sin[7*(c + d*x)] - 158760*Sin[8*(c + d*x)] + 39690*Sin[9*(c + d*x)] + 19845*Sin[10*(c + d*x)] + 1376256*Sin[c + 2*d*x] - 5505024*Sin[2*c + 3*d*x] - 1572864*Sin[3*c + 4*d*x] + 2359296*Sin[4*c + 5*d*x] + 884736*Sin[5*c + 6*d*x] - 589824*Sin[6*c + 7*d*x] - 262144*Sin[7*c + 8*d*x] + 65536*Sin[8*c + 9*d*x] + 32768*Sin[9*c + 10*d*x]))/(454164480*a*d*(1 + Sec[c + d*x]))","B",1
74,1,72,137,5.6307866,"\int \frac{\sin ^{11}(c+d x)}{(a+a \sec (c+d x))^2} \, dx","Integrate[Sin[c + d*x]^11/(a + a*Sec[c + d*x])^2,x]","\frac{4 \sin ^{12}\left(\frac{1}{2} (c+d x)\right) (4038 \cos (c+d x)+2586 \cos (2 (c+d x))+1189 \cos (3 (c+d x))+342 \cos (4 (c+d x))+45 \cos (5 (c+d x))+2360)}{495 a^2 d}","-\frac{(a-a \cos (c+d x))^{11}}{11 a^{13} d}+\frac{4 (a-a \cos (c+d x))^{10}}{5 a^{12} d}-\frac{25 (a-a \cos (c+d x))^9}{9 a^{11} d}+\frac{19 (a-a \cos (c+d x))^8}{4 a^{10} d}-\frac{4 (a-a \cos (c+d x))^7}{a^9 d}+\frac{4 (a-a \cos (c+d x))^6}{3 a^8 d}",1,"(4*(2360 + 4038*Cos[c + d*x] + 2586*Cos[2*(c + d*x)] + 1189*Cos[3*(c + d*x)] + 342*Cos[4*(c + d*x)] + 45*Cos[5*(c + d*x)])*Sin[(c + d*x)/2]^12)/(495*a^2*d)","A",1
75,1,62,114,3.9532864,"\int \frac{\sin ^9(c+d x)}{(a+a \sec (c+d x))^2} \, dx","Integrate[Sin[c + d*x]^9/(a + a*Sec[c + d*x])^2,x]","\frac{2 \sin ^{10}\left(\frac{1}{2} (c+d x)\right) (1615 \cos (c+d x)+970 \cos (2 (c+d x))+385 \cos (3 (c+d x))+70 \cos (4 (c+d x))+992)}{315 a^2 d}","\frac{(a-a \cos (c+d x))^9}{9 a^{11} d}-\frac{3 (a-a \cos (c+d x))^8}{4 a^{10} d}+\frac{13 (a-a \cos (c+d x))^7}{7 a^9 d}-\frac{2 (a-a \cos (c+d x))^6}{a^8 d}+\frac{4 (a-a \cos (c+d x))^5}{5 a^7 d}",1,"(2*(992 + 1615*Cos[c + d*x] + 970*Cos[2*(c + d*x)] + 385*Cos[3*(c + d*x)] + 70*Cos[4*(c + d*x)])*Sin[(c + d*x)/2]^10)/(315*a^2*d)","A",1
76,1,53,73,1.9671352,"\int \frac{\sin ^7(c+d x)}{(a+a \sec (c+d x))^2} \, dx","Integrate[Sin[c + d*x]^7/(a + a*Sec[c + d*x])^2,x]","\frac{4 \sin ^8\left(\frac{1}{2} (c+d x)\right) (17 \cos (c+d x)+10 \cos (2 (c+d x))+3 (\cos (3 (c+d x))+4))}{21 a^2 d}","\frac{\cos ^7(c+d x)}{7 a^2 d}-\frac{\cos ^6(c+d x)}{3 a^2 d}+\frac{\cos ^4(c+d x)}{2 a^2 d}-\frac{\cos ^3(c+d x)}{3 a^2 d}",1,"(4*(17*Cos[c + d*x] + 10*Cos[2*(c + d*x)] + 3*(4 + Cos[3*(c + d*x)]))*Sin[(c + d*x)/2]^8)/(21*a^2*d)","A",1
77,1,42,55,0.5978851,"\int \frac{\sin ^5(c+d x)}{(a+a \sec (c+d x))^2} \, dx","Integrate[Sin[c + d*x]^5/(a + a*Sec[c + d*x])^2,x]","\frac{4 \sin ^6\left(\frac{1}{2} (c+d x)\right) (3 \cos (c+d x)+3 \cos (2 (c+d x))+4)}{15 a^2 d}","-\frac{\cos ^5(c+d x)}{5 a^2 d}+\frac{\cos ^4(c+d x)}{2 a^2 d}-\frac{\cos ^3(c+d x)}{3 a^2 d}",1,"(4*(4 + 3*Cos[c + d*x] + 3*Cos[2*(c + d*x)])*Sin[(c + d*x)/2]^6)/(15*a^2*d)","A",1
78,1,51,66,0.2431254,"\int \frac{\sin ^3(c+d x)}{(a+a \sec (c+d x))^2} \, dx","Integrate[Sin[c + d*x]^3/(a + a*Sec[c + d*x])^2,x]","\frac{27 \cos (c+d x)-6 \cos (2 (c+d x))+\cos (3 (c+d x))-48 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)-22}{12 a^2 d}","\frac{\cos ^3(c+d x)}{3 a^2 d}-\frac{\cos ^2(c+d x)}{a^2 d}+\frac{2 \cos (c+d x)}{a^2 d}-\frac{2 \log (\cos (c+d x)+1)}{a^2 d}",1,"(-22 + 27*Cos[c + d*x] - 6*Cos[2*(c + d*x)] + Cos[3*(c + d*x)] - 48*Log[Cos[(c + d*x)/2]])/(12*a^2*d)","A",1
79,1,64,52,0.2442141,"\int \frac{\sin (c+d x)}{(a+a \sec (c+d x))^2} \, dx","Integrate[Sin[c + d*x]/(a + a*Sec[c + d*x])^2,x]","-\frac{\sec ^2\left(\frac{1}{2} (c+d x)\right) \left(\cos (2 (c+d x))-8 \cos (c+d x) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)-8 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)-3\right)}{4 a^2 d}","-\frac{\cos (c+d x)}{a^2 d}+\frac{1}{d \left(a^2 \cos (c+d x)+a^2\right)}+\frac{2 \log (\cos (c+d x)+1)}{a^2 d}",1,"-1/4*((-3 + Cos[2*(c + d*x)] - 8*Log[Cos[(c + d*x)/2]] - 8*Cos[c + d*x]*Log[Cos[(c + d*x)/2]])*Sec[(c + d*x)/2]^2)/(a^2*d)","A",1
80,1,83,60,0.2058632,"\int \frac{\csc (c+d x)}{(a+a \sec (c+d x))^2} \, dx","Integrate[Csc[c + d*x]/(a + a*Sec[c + d*x])^2,x]","-\frac{\sec ^2(c+d x) \left(6 \cos ^2\left(\frac{1}{2} (c+d x)\right)+4 \cos ^4\left(\frac{1}{2} (c+d x)\right) \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)\right)-1\right)}{4 a^2 d (\sec (c+d x)+1)^2}","-\frac{3}{4 d \left(a^2 \cos (c+d x)+a^2\right)}-\frac{\tanh ^{-1}(\cos (c+d x))}{4 a^2 d}+\frac{1}{4 d (a \cos (c+d x)+a)^2}",1,"-1/4*((-1 + 6*Cos[(c + d*x)/2]^2 + 4*Cos[(c + d*x)/2]^4*(Log[Cos[(c + d*x)/2]] - Log[Sin[(c + d*x)/2]]))*Sec[c + d*x]^2)/(a^2*d*(1 + Sec[c + d*x])^2)","A",1
81,1,38,42,0.1032677,"\int \frac{\csc ^3(c+d x)}{(a+a \sec (c+d x))^2} \, dx","Integrate[Csc[c + d*x]^3/(a + a*Sec[c + d*x])^2,x]","-\frac{(2 \cos (c+d x)+1) \csc ^2(c+d x)}{6 a^2 d (\cos (c+d x)+1)^2}","-\frac{2 a \cos (c+d x)+a}{6 d (1-\cos (c+d x)) (a \cos (c+d x)+a)^3}",1,"-1/6*((1 + 2*Cos[c + d*x])*Csc[c + d*x]^2)/(a^2*d*(1 + Cos[c + d*x])^2)","A",1
82,1,152,146,0.7905896,"\int \frac{\csc ^5(c+d x)}{(a+a \sec (c+d x))^2} \, dx","Integrate[Csc[c + d*x]^5/(a + a*Sec[c + d*x])^2,x]","-\frac{\cos ^4\left(\frac{1}{2} (c+d x)\right) \sec ^2(c+d x) \left(6 \csc ^4\left(\frac{1}{2} (c+d x)\right)+12 \csc ^2\left(\frac{1}{2} (c+d x)\right)-3 \sec ^8\left(\frac{1}{2} (c+d x)\right)+4 \sec ^6\left(\frac{1}{2} (c+d x)\right)+12 \sec ^4\left(\frac{1}{2} (c+d x)\right)+24 \sec ^2\left(\frac{1}{2} (c+d x)\right)+24 \left(\log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)\right)}{384 a^2 d (\sec (c+d x)+1)^2}","\frac{a^2}{32 d (a \cos (c+d x)+a)^4}-\frac{1}{64 d \left(a^2-a^2 \cos (c+d x)\right)}-\frac{1}{32 d \left(a^2 \cos (c+d x)+a^2\right)}+\frac{\tanh ^{-1}(\cos (c+d x))}{64 a^2 d}-\frac{a}{48 d (a \cos (c+d x)+a)^3}-\frac{1}{64 d (a-a \cos (c+d x))^2}-\frac{1}{32 d (a \cos (c+d x)+a)^2}",1,"-1/384*(Cos[(c + d*x)/2]^4*(12*Csc[(c + d*x)/2]^2 + 6*Csc[(c + d*x)/2]^4 + 24*(-Log[Cos[(c + d*x)/2]] + Log[Sin[(c + d*x)/2]]) + 24*Sec[(c + d*x)/2]^2 + 12*Sec[(c + d*x)/2]^4 + 4*Sec[(c + d*x)/2]^6 - 3*Sec[(c + d*x)/2]^8)*Sec[c + d*x]^2)/(a^2*d*(1 + Sec[c + d*x])^2)","A",1
83,1,131,167,3.1374549,"\int \frac{\sin ^8(c+d x)}{(a+a \sec (c+d x))^2} \, dx","Integrate[Sin[c + d*x]^8/(a + a*Sec[c + d*x])^2,x]","\frac{\cos ^4\left(\frac{1}{2} (c+d x)\right) \sec ^2(c+d x) \left(-10080 \sin (c+d x)-1680 \sin (2 (c+d x))+3360 \sin (3 (c+d x))-2520 \sin (4 (c+d x))+672 \sin (5 (c+d x))+560 \sin (6 (c+d x))-480 \sin (7 (c+d x))+105 \sin (8 (c+d x))+980 \tan \left(\frac{c}{2}\right)+9240 d x\right)}{26880 a^2 d (\sec (c+d x)+1)^2}","\frac{2 \sin ^7(c+d x)}{7 a^2 d}-\frac{2 \sin ^5(c+d x)}{5 a^2 d}-\frac{\sin ^3(c+d x) \cos ^5(c+d x)}{8 a^2 d}-\frac{\sin (c+d x) \cos ^5(c+d x)}{16 a^2 d}-\frac{\sin ^3(c+d x) \cos ^3(c+d x)}{6 a^2 d}-\frac{7 \sin (c+d x) \cos ^3(c+d x)}{64 a^2 d}+\frac{11 \sin (c+d x) \cos (c+d x)}{128 a^2 d}+\frac{11 x}{128 a^2}",1,"(Cos[(c + d*x)/2]^4*Sec[c + d*x]^2*(9240*d*x - 10080*Sin[c + d*x] - 1680*Sin[2*(c + d*x)] + 3360*Sin[3*(c + d*x)] - 2520*Sin[4*(c + d*x)] + 672*Sin[5*(c + d*x)] + 560*Sin[6*(c + d*x)] - 480*Sin[7*(c + d*x)] + 105*Sin[8*(c + d*x)] + 980*Tan[c/2]))/(26880*a^2*d*(1 + Sec[c + d*x])^2)","A",1
84,1,111,104,0.9478086,"\int \frac{\sin ^6(c+d x)}{(a+a \sec (c+d x))^2} \, dx","Integrate[Sin[c + d*x]^6/(a + a*Sec[c + d*x])^2,x]","\frac{\cos ^4\left(\frac{1}{2} (c+d x)\right) \sec ^2(c+d x) \left(-480 \sin (c+d x)+30 \sin (2 (c+d x))+80 \sin (3 (c+d x))-90 \sin (4 (c+d x))+48 \sin (5 (c+d x))-10 \sin (6 (c+d x))+25 \tan \left(\frac{c}{2}\right)+360 d x\right)}{480 a^2 d (\sec (c+d x)+1)^2}","-\frac{\sin ^3(c+d x) (a-a \cos (c+d x))^3}{6 a^5 d}-\frac{\sin ^5(c+d x)}{10 a^2 d}-\frac{\sin ^3(c+d x) \cos (c+d x)}{8 a^2 d}-\frac{3 \sin (c+d x) \cos (c+d x)}{16 a^2 d}+\frac{3 x}{16 a^2}",1,"(Cos[(c + d*x)/2]^4*Sec[c + d*x]^2*(360*d*x - 480*Sin[c + d*x] + 30*Sin[2*(c + d*x)] + 80*Sin[3*(c + d*x)] - 90*Sin[4*(c + d*x)] + 48*Sin[5*(c + d*x)] - 10*Sin[6*(c + d*x)] + 25*Tan[c/2]))/(480*a^2*d*(1 + Sec[c + d*x])^2)","A",1
85,1,91,87,0.5887319,"\int \frac{\sin ^4(c+d x)}{(a+a \sec (c+d x))^2} \, dx","Integrate[Sin[c + d*x]^4/(a + a*Sec[c + d*x])^2,x]","\frac{\cos ^4\left(\frac{1}{2} (c+d x)\right) \sec ^2(c+d x) \left(-144 \sin (c+d x)+48 \sin (2 (c+d x))-16 \sin (3 (c+d x))+3 \sin (4 (c+d x))+2 \tan \left(\frac{c}{2}\right)+84 d x\right)}{24 a^2 d (\sec (c+d x)+1)^2}","\frac{2 \sin ^3(c+d x)}{3 a^2 d}-\frac{2 \sin (c+d x)}{a^2 d}+\frac{\sin (c+d x) \cos ^3(c+d x)}{4 a^2 d}+\frac{7 \sin (c+d x) \cos (c+d x)}{8 a^2 d}+\frac{7 x}{8 a^2}",1,"(Cos[(c + d*x)/2]^4*Sec[c + d*x]^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)] + 2*Tan[c/2]))/(24*a^2*d*(1 + Sec[c + d*x])^2)","A",1
86,1,121,69,0.3593626,"\int \frac{\sin ^2(c+d x)}{(a+a \sec (c+d x))^2} \, dx","Integrate[Sin[c + d*x]^2/(a + a*Sec[c + d*x])^2,x]","-\frac{\sec \left(\frac{c}{2}\right) \sec \left(\frac{1}{2} (c+d x)\right) \left(-25 \sin \left(c+\frac{d x}{2}\right)-21 \sin \left(c+\frac{3 d x}{2}\right)-21 \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)+60 d x \cos \left(c+\frac{d x}{2}\right)-119 \sin \left(\frac{d x}{2}\right)+60 d x \cos \left(\frac{d x}{2}\right)\right)}{48 a^2 d}","\frac{2 \sin (c+d x)}{a^2 d}-\frac{\sin (c+d x) \cos (c+d x)}{2 a^2 d}+\frac{2 \sin (c+d x)}{a^2 d (\cos (c+d x)+1)}-\frac{5 x}{2 a^2}",1,"-1/48*(Sec[c/2]*Sec[(c + d*x)/2]*(60*d*x*Cos[(d*x)/2] + 60*d*x*Cos[c + (d*x)/2] - 119*Sin[(d*x)/2] - 25*Sin[c + (d*x)/2] - 21*Sin[c + (3*d*x)/2] - 21*Sin[2*c + (3*d*x)/2] + 3*Sin[2*c + (5*d*x)/2] + 3*Sin[3*c + (5*d*x)/2]))/(a^2*d)","A",1
87,1,105,73,0.4757523,"\int \frac{\csc ^2(c+d x)}{(a+a \sec (c+d x))^2} \, dx","Integrate[Csc[c + d*x]^2/(a + a*Sec[c + d*x])^2,x]","\frac{\csc (c) (55 \sin (c+d x)+44 \sin (2 (c+d x))+11 \sin (3 (c+d x))-60 \sin (2 c+d x)+16 \sin (c+2 d x)+4 \sin (2 c+3 d x)-80 \sin (c)+80 \sin (d x)) \csc (c+d x) \sec ^2(c+d x)}{240 a^2 d (\sec (c+d x)+1)^2}","-\frac{2 \cot ^5(c+d x)}{5 a^2 d}-\frac{\cot ^3(c+d x)}{3 a^2 d}+\frac{2 \csc ^5(c+d x)}{5 a^2 d}-\frac{2 \csc ^3(c+d x)}{3 a^2 d}",1,"(Csc[c]*Csc[c + d*x]*Sec[c + d*x]^2*(-80*Sin[c] + 80*Sin[d*x] + 55*Sin[c + d*x] + 44*Sin[2*(c + d*x)] + 11*Sin[3*(c + d*x)] - 60*Sin[2*c + d*x] + 16*Sin[c + 2*d*x] + 4*Sin[2*c + 3*d*x]))/(240*a^2*d*(1 + Sec[c + d*x])^2)","A",1
88,1,149,91,0.6998337,"\int \frac{\csc ^4(c+d x)}{(a+a \sec (c+d x))^2} \, dx","Integrate[Csc[c + d*x]^4/(a + a*Sec[c + d*x])^2,x]","-\frac{\csc (c) (-714 \sin (c+d x)-408 \sin (2 (c+d x))+153 \sin (3 (c+d x))+204 \sin (4 (c+d x))+51 \sin (5 (c+d x))+1680 \sin (2 c+d x)+128 \sin (c+2 d x)-48 \sin (2 c+3 d x)-64 \sin (3 c+4 d x)-16 \sin (4 c+5 d x)+1344 \sin (c)-1456 \sin (d x)) \csc ^3(c+d x) \sec ^2(c+d x)}{13440 a^2 d (\sec (c+d x)+1)^2}","-\frac{2 \cot ^7(c+d x)}{7 a^2 d}-\frac{3 \cot ^5(c+d x)}{5 a^2 d}-\frac{\cot ^3(c+d x)}{3 a^2 d}+\frac{2 \csc ^7(c+d x)}{7 a^2 d}-\frac{2 \csc ^5(c+d x)}{5 a^2 d}",1,"-1/13440*(Csc[c]*Csc[c + d*x]^3*Sec[c + d*x]^2*(1344*Sin[c] - 1456*Sin[d*x] - 714*Sin[c + d*x] - 408*Sin[2*(c + d*x)] + 153*Sin[3*(c + d*x)] + 204*Sin[4*(c + d*x)] + 51*Sin[5*(c + d*x)] + 1680*Sin[2*c + d*x] + 128*Sin[c + 2*d*x] - 48*Sin[2*c + 3*d*x] - 64*Sin[3*c + 4*d*x] - 16*Sin[4*c + 5*d*x]))/(a^2*d*(1 + Sec[c + d*x])^2)","A",1
89,1,191,109,1.0244515,"\int \frac{\csc ^6(c+d x)}{(a+a \sec (c+d x))^2} \, dx","Integrate[Csc[c + d*x]^6/(a + a*Sec[c + d*x])^2,x]","\frac{\csc (c) (25875 \sin (c+d x)+11500 \sin (2 (c+d x))-10925 \sin (3 (c+d x))-9200 \sin (4 (c+d x))+575 \sin (5 (c+d x))+2300 \sin (6 (c+d x))+575 \sin (7 (c+d x))-107520 \sin (2 c+d x)-10240 \sin (c+2 d x)+9728 \sin (2 c+3 d x)+8192 \sin (3 c+4 d x)-512 \sin (4 c+5 d x)-2048 \sin (5 c+6 d x)-512 \sin (6 c+7 d x)-61440 \sin (c)+84480 \sin (d x)) \csc ^5(c+d x) \sec ^2(c+d x)}{1290240 a^2 d (\sec (c+d x)+1)^2}","-\frac{2 \cot ^9(c+d x)}{9 a^2 d}-\frac{5 \cot ^7(c+d x)}{7 a^2 d}-\frac{4 \cot ^5(c+d x)}{5 a^2 d}-\frac{\cot ^3(c+d x)}{3 a^2 d}+\frac{2 \csc ^9(c+d x)}{9 a^2 d}-\frac{2 \csc ^7(c+d x)}{7 a^2 d}",1,"(Csc[c]*Csc[c + d*x]^5*Sec[c + d*x]^2*(-61440*Sin[c] + 84480*Sin[d*x] + 25875*Sin[c + d*x] + 11500*Sin[2*(c + d*x)] - 10925*Sin[3*(c + d*x)] - 9200*Sin[4*(c + d*x)] + 575*Sin[5*(c + d*x)] + 2300*Sin[6*(c + d*x)] + 575*Sin[7*(c + d*x)] - 107520*Sin[2*c + d*x] - 10240*Sin[c + 2*d*x] + 9728*Sin[2*c + 3*d*x] + 8192*Sin[3*c + 4*d*x] - 512*Sin[4*c + 5*d*x] - 2048*Sin[5*c + 6*d*x] - 512*Sin[6*c + 7*d*x]))/(1290240*a^2*d*(1 + Sec[c + d*x])^2)","A",1
90,1,233,125,1.4998868,"\int \frac{\csc ^8(c+d x)}{(a+a \sec (c+d x))^2} \, dx","Integrate[Csc[c + d*x]^8/(a + a*Sec[c + d*x])^2,x]","-\frac{\csc (c) (-218834 \sin (c+d x)-79576 \sin (2 (c+d x))+119364 \sin (3 (c+d x))+79576 \sin (4 (c+d x))-28420 \sin (5 (c+d x))-34104 \sin (6 (c+d x))-1421 \sin (7 (c+d x))+5684 \sin (8 (c+d x))+1421 \sin (9 (c+d x))+1419264 \sin (2 c+d x)+114688 \sin (c+2 d x)-172032 \sin (2 c+3 d x)-114688 \sin (3 c+4 d x)+40960 \sin (4 c+5 d x)+49152 \sin (5 c+6 d x)+2048 \sin (6 c+7 d x)-8192 \sin (7 c+8 d x)-2048 \sin (8 c+9 d x)+630784 \sin (c)-1103872 \sin (d x)) \csc ^7(c+d x) \sec ^2(c+d x)}{22708224 a^2 d (\sec (c+d x)+1)^2}","-\frac{2 \cot ^{11}(c+d x)}{11 a^2 d}-\frac{7 \cot ^9(c+d x)}{9 a^2 d}-\frac{9 \cot ^7(c+d x)}{7 a^2 d}-\frac{\cot ^5(c+d x)}{a^2 d}-\frac{\cot ^3(c+d x)}{3 a^2 d}+\frac{2 \csc ^{11}(c+d x)}{11 a^2 d}-\frac{2 \csc ^9(c+d x)}{9 a^2 d}",1,"-1/22708224*(Csc[c]*Csc[c + d*x]^7*Sec[c + d*x]^2*(630784*Sin[c] - 1103872*Sin[d*x] - 218834*Sin[c + d*x] - 79576*Sin[2*(c + d*x)] + 119364*Sin[3*(c + d*x)] + 79576*Sin[4*(c + d*x)] - 28420*Sin[5*(c + d*x)] - 34104*Sin[6*(c + d*x)] - 1421*Sin[7*(c + d*x)] + 5684*Sin[8*(c + d*x)] + 1421*Sin[9*(c + d*x)] + 1419264*Sin[2*c + d*x] + 114688*Sin[c + 2*d*x] - 172032*Sin[2*c + 3*d*x] - 114688*Sin[3*c + 4*d*x] + 40960*Sin[4*c + 5*d*x] + 49152*Sin[5*c + 6*d*x] + 2048*Sin[6*c + 7*d*x] - 8192*Sin[7*c + 8*d*x] - 2048*Sin[8*c + 9*d*x]))/(a^2*d*(1 + Sec[c + d*x])^2)","A",1
91,1,120,139,4.6434933,"\int \frac{\sin ^{11}(c+d x)}{(a+a \sec (c+d x))^3} \, dx","Integrate[Sin[c + d*x]^11/(a + a*Sec[c + d*x])^3,x]","\frac{2273040 \cos (c+d x)-1496880 \cos (2 (c+d x))+535920 \cos (3 (c+d x))+110880 \cos (4 (c+d x))-293832 \cos (5 (c+d x))+212520 \cos (6 (c+d x))-67320 \cos (7 (c+d x))-27720 \cos (8 (c+d x))+40040 \cos (9 (c+d x))-16632 \cos (10 (c+d x))+2520 \cos (11 (c+d x))-1615571}{28385280 a^3 d}","-\frac{(a-a \cos (c+d x))^{11}}{11 a^{14} d}+\frac{7 (a-a \cos (c+d x))^{10}}{10 a^{13} d}-\frac{19 (a-a \cos (c+d x))^9}{9 a^{12} d}+\frac{25 (a-a \cos (c+d x))^8}{8 a^{11} d}-\frac{16 (a-a \cos (c+d x))^7}{7 a^{10} d}+\frac{2 (a-a \cos (c+d x))^6}{3 a^9 d}",1,"(-1615571 + 2273040*Cos[c + d*x] - 1496880*Cos[2*(c + d*x)] + 535920*Cos[3*(c + d*x)] + 110880*Cos[4*(c + d*x)] - 293832*Cos[5*(c + d*x)] + 212520*Cos[6*(c + d*x)] - 67320*Cos[7*(c + d*x)] - 27720*Cos[8*(c + d*x)] + 40040*Cos[9*(c + d*x)] - 16632*Cos[10*(c + d*x)] + 2520*Cos[11*(c + d*x)])/(28385280*a^3*d)","A",1
92,1,100,109,3.0950199,"\int \frac{\sin ^9(c+d x)}{(a+a \sec (c+d x))^3} \, dx","Integrate[Sin[c + d*x]^9/(a + a*Sec[c + d*x])^3,x]","-\frac{-52920 \cos (c+d x)+37800 \cos (2 (c+d x))-18480 \cos (3 (c+d x))+3780 \cos (4 (c+d x))+3024 \cos (5 (c+d x))-4200 \cos (6 (c+d x))+2700 \cos (7 (c+d x))-945 \cos (8 (c+d x))+140 \cos (9 (c+d x))+34771}{322560 a^3 d}","-\frac{\cos ^9(c+d x)}{9 a^3 d}+\frac{3 \cos ^8(c+d x)}{8 a^3 d}-\frac{2 \cos ^7(c+d x)}{7 a^3 d}-\frac{\cos ^6(c+d x)}{3 a^3 d}+\frac{3 \cos ^5(c+d x)}{5 a^3 d}-\frac{\cos ^4(c+d x)}{4 a^3 d}",1,"-1/322560*(34771 - 52920*Cos[c + d*x] + 37800*Cos[2*(c + d*x)] - 18480*Cos[3*(c + d*x)] + 3780*Cos[4*(c + d*x)] + 3024*Cos[5*(c + d*x)] - 4200*Cos[6*(c + d*x)] + 2700*Cos[7*(c + d*x)] - 945*Cos[8*(c + d*x)] + 140*Cos[9*(c + d*x)])/(a^3*d)","A",1
93,1,80,73,1.7558927,"\int \frac{\sin ^7(c+d x)}{(a+a \sec (c+d x))^3} \, dx","Integrate[Sin[c + d*x]^7/(a + a*Sec[c + d*x])^3,x]","\frac{4060 \cos (c+d x)-3220 \cos (2 (c+d x))+2100 \cos (3 (c+d x))-1120 \cos (4 (c+d x))+476 \cos (5 (c+d x))-140 \cos (6 (c+d x))+20 \cos (7 (c+d x))-2421}{8960 a^3 d}","\frac{\cos ^7(c+d x)}{7 a^3 d}-\frac{\cos ^6(c+d x)}{2 a^3 d}+\frac{3 \cos ^5(c+d x)}{5 a^3 d}-\frac{\cos ^4(c+d x)}{4 a^3 d}",1,"(-2421 + 4060*Cos[c + d*x] - 3220*Cos[2*(c + d*x)] + 2100*Cos[3*(c + d*x)] - 1120*Cos[4*(c + d*x)] + 476*Cos[5*(c + d*x)] - 140*Cos[6*(c + d*x)] + 20*Cos[7*(c + d*x)])/(8960*a^3*d)","A",1
94,1,73,102,0.9566161,"\int \frac{\sin ^5(c+d x)}{(a+a \sec (c+d x))^3} \, dx","Integrate[Sin[c + d*x]^5/(a + a*Sec[c + d*x])^3,x]","\frac{-4920 \cos (c+d x)+1320 \cos (2 (c+d x))-380 \cos (3 (c+d x))+90 \cos (4 (c+d x))-12 \cos (5 (c+d x))+7680 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)+3857}{960 a^3 d}","-\frac{\cos ^5(c+d x)}{5 a^3 d}+\frac{3 \cos ^4(c+d x)}{4 a^3 d}-\frac{4 \cos ^3(c+d x)}{3 a^3 d}+\frac{2 \cos ^2(c+d x)}{a^3 d}-\frac{4 \cos (c+d x)}{a^3 d}+\frac{4 \log (\cos (c+d x)+1)}{a^3 d}",1,"(3857 - 4920*Cos[c + d*x] + 1320*Cos[2*(c + d*x)] - 380*Cos[3*(c + d*x)] + 90*Cos[4*(c + d*x)] - 12*Cos[5*(c + d*x)] + 7680*Log[Cos[(c + d*x)/2]])/(960*a^3*d)","A",1
95,1,99,89,0.4391157,"\int \frac{\sin ^3(c+d x)}{(a+a \sec (c+d x))^3} \, dx","Integrate[Sin[c + d*x]^3/(a + a*Sec[c + d*x])^3,x]","-\frac{\cos ^4\left(\frac{1}{2} (c+d x)\right) \left(-184 \cos (2 (c+d x))+28 \cos (3 (c+d x))-4 \cos (4 (c+d x))+1344 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)+\cos (c+d x) \left(1344 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)-19\right)+389\right)}{24 a^3 d (\cos (c+d x)+1)^3}","\frac{\cos ^3(c+d x)}{3 a^3 d}-\frac{3 \cos ^2(c+d x)}{2 a^3 d}+\frac{5 \cos (c+d x)}{a^3 d}-\frac{2}{d \left(a^3 \cos (c+d x)+a^3\right)}-\frac{7 \log (\cos (c+d x)+1)}{a^3 d}",1,"-1/24*(Cos[(c + d*x)/2]^4*(389 - 184*Cos[2*(c + d*x)] + 28*Cos[3*(c + d*x)] - 4*Cos[4*(c + d*x)] + 1344*Log[Cos[(c + d*x)/2]] + Cos[c + d*x]*(-19 + 1344*Log[Cos[(c + d*x)/2]])))/(a^3*d*(1 + Cos[c + d*x])^3)","A",1
96,1,103,75,0.3626214,"\int \frac{\sin (c+d x)}{(a+a \sec (c+d x))^3} \, dx","Integrate[Sin[c + d*x]/(a + a*Sec[c + d*x])^3,x]","\frac{\cos ^2\left(\frac{1}{2} (c+d x)\right) \left(-2 \cos (3 (c+d x))+72 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)+\cos (2 (c+d x)) \left(24 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)-5\right)+\cos (c+d x) \left(96 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)+22\right)+21\right)}{4 a^3 d (\cos (c+d x)+1)^3}","-\frac{\cos (c+d x)}{a^3 d}+\frac{3}{d \left(a^3 \cos (c+d x)+a^3\right)}+\frac{3 \log (\cos (c+d x)+1)}{a^3 d}-\frac{1}{2 a d (a \cos (c+d x)+a)^2}",1,"(Cos[(c + d*x)/2]^2*(21 - 2*Cos[3*(c + d*x)] + 72*Log[Cos[(c + d*x)/2]] + Cos[2*(c + d*x)]*(-5 + 24*Log[Cos[(c + d*x)/2]]) + Cos[c + d*x]*(22 + 96*Log[Cos[(c + d*x)/2]])))/(4*a^3*d*(1 + Cos[c + d*x])^3)","A",1
97,1,97,82,0.3459924,"\int \frac{\csc (c+d x)}{(a+a \sec (c+d x))^3} \, dx","Integrate[Csc[c + d*x]/(a + a*Sec[c + d*x])^3,x]","-\frac{\sec ^3(c+d x) \left(42 \cos ^4\left(\frac{1}{2} (c+d x)\right)-15 \cos ^2\left(\frac{1}{2} (c+d x)\right)+12 \cos ^6\left(\frac{1}{2} (c+d x)\right) \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)\right)+2\right)}{12 a^3 d (\sec (c+d x)+1)^3}","-\frac{7}{8 d \left(a^3 \cos (c+d x)+a^3\right)}-\frac{\tanh ^{-1}(\cos (c+d x))}{8 a^3 d}+\frac{5}{8 a d (a \cos (c+d x)+a)^2}-\frac{1}{6 d (a \cos (c+d x)+a)^3}",1,"-1/12*((2 - 15*Cos[(c + d*x)/2]^2 + 42*Cos[(c + d*x)/2]^4 + 12*Cos[(c + d*x)/2]^6*(Log[Cos[(c + d*x)/2]] - Log[Sin[(c + d*x)/2]]))*Sec[c + d*x]^3)/(a^3*d*(1 + Sec[c + d*x])^3)","A",1
98,1,138,126,0.6507729,"\int \frac{\csc ^3(c+d x)}{(a+a \sec (c+d x))^3} \, dx","Integrate[Csc[c + d*x]^3/(a + a*Sec[c + d*x])^3,x]","-\frac{\cos ^6\left(\frac{1}{2} (c+d x)\right) \sec ^3(c+d x) \left(12 \csc ^2\left(\frac{1}{2} (c+d x)\right)+3 \sec ^8\left(\frac{1}{2} (c+d x)\right)-16 \sec ^6\left(\frac{1}{2} (c+d x)\right)+18 \sec ^4\left(\frac{1}{2} (c+d x)\right)+24 \sec ^2\left(\frac{1}{2} (c+d x)\right)+24 \left(\log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)\right)}{96 a^3 d (\sec (c+d x)+1)^3}","-\frac{1}{32 d \left(a^3-a^3 \cos (c+d x)\right)}-\frac{1}{16 d \left(a^3 \cos (c+d x)+a^3\right)}+\frac{\tanh ^{-1}(\cos (c+d x))}{32 a^3 d}-\frac{a}{16 d (a \cos (c+d x)+a)^4}+\frac{1}{6 d (a \cos (c+d x)+a)^3}-\frac{3}{32 a d (a \cos (c+d x)+a)^2}",1,"-1/96*(Cos[(c + d*x)/2]^6*(12*Csc[(c + d*x)/2]^2 + 24*(-Log[Cos[(c + d*x)/2]] + Log[Sin[(c + d*x)/2]]) + 24*Sec[(c + d*x)/2]^2 + 18*Sec[(c + d*x)/2]^4 - 16*Sec[(c + d*x)/2]^6 + 3*Sec[(c + d*x)/2]^8)*Sec[c + d*x]^3)/(a^3*d*(1 + Sec[c + d*x])^3)","A",1
99,1,137,128,5.2896746,"\int \frac{\csc ^5(c+d x)}{(a+a \sec (c+d x))^3} \, dx","Integrate[Csc[c + d*x]^5/(a + a*Sec[c + d*x])^3,x]","-\frac{\sec ^4\left(\frac{1}{2} (c+d x)\right) \sec ^3(c+d x) \left(60 \cos ^8\left(\frac{1}{2} (c+d x)\right)-15 \cos ^2\left(\frac{1}{2} (c+d x)\right)+10 \cos ^6\left(\frac{1}{2} (c+d x)\right) \left(\cot ^4\left(\frac{1}{2} (c+d x)\right)+2\right)-120 \cos ^{10}\left(\frac{1}{2} (c+d x)\right) \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)\right)+4\right)}{640 a^3 d (\sec (c+d x)+1)^3}","-\frac{3}{128 d \left(a^3 \cos (c+d x)+a^3\right)}+\frac{3 \tanh ^{-1}(\cos (c+d x))}{128 a^3 d}-\frac{a^2}{40 d (a \cos (c+d x)+a)^5}+\frac{3 a}{64 d (a \cos (c+d x)+a)^4}-\frac{1}{128 a d (a-a \cos (c+d x))^2}-\frac{1}{64 a d (a \cos (c+d x)+a)^2}",1,"-1/640*((4 - 15*Cos[(c + d*x)/2]^2 + 60*Cos[(c + d*x)/2]^8 + 10*Cos[(c + d*x)/2]^6*(2 + Cot[(c + d*x)/2]^4) - 120*Cos[(c + d*x)/2]^10*(Log[Cos[(c + d*x)/2]] - Log[Sin[(c + d*x)/2]]))*Sec[(c + d*x)/2]^4*Sec[c + d*x]^3)/(a^3*d*(1 + Sec[c + d*x])^3)","A",1
100,1,131,157,4.9976704,"\int \frac{\sin ^8(c+d x)}{(a+a \sec (c+d x))^3} \, dx","Integrate[Sin[c + d*x]^8/(a + a*Sec[c + d*x])^3,x]","\frac{\cos ^6\left(\frac{1}{2} (c+d x)\right) \sec ^3(c+d x) \left(38640 \sin (c+d x)-6720 \sin (2 (c+d x))-3920 \sin (3 (c+d x))+5880 \sin (4 (c+d x))-4368 \sin (5 (c+d x))+2240 \sin (6 (c+d x))-720 \sin (7 (c+d x))+105 \sin (8 (c+d x))+294 \tan \left(\frac{c}{2}\right)-24360 d x\right)}{13440 a^3 d (\sec (c+d x)+1)^3}","\frac{3 \sin ^7(c+d x)}{7 a^3 d}-\frac{7 \sin ^5(c+d x)}{5 a^3 d}+\frac{4 \sin ^3(c+d x)}{3 a^3 d}+\frac{\sin (c+d x) \cos ^7(c+d x)}{8 a^3 d}+\frac{23 \sin (c+d x) \cos ^5(c+d x)}{48 a^3 d}-\frac{29 \sin (c+d x) \cos ^3(c+d x)}{192 a^3 d}-\frac{29 \sin (c+d x) \cos (c+d x)}{128 a^3 d}-\frac{29 x}{128 a^3}",1,"(Cos[(c + d*x)/2]^6*Sec[c + d*x]^3*(-24360*d*x + 38640*Sin[c + d*x] - 6720*Sin[2*(c + d*x)] - 3920*Sin[3*(c + d*x)] + 5880*Sin[4*(c + d*x)] - 4368*Sin[5*(c + d*x)] + 2240*Sin[6*(c + d*x)] - 720*Sin[7*(c + d*x)] + 105*Sin[8*(c + d*x)] + 294*Tan[c/2]))/(13440*a^3*d*(1 + Sec[c + d*x])^3)","A",1
101,1,111,129,1.9640623,"\int \frac{\sin ^6(c+d x)}{(a+a \sec (c+d x))^3} \, dx","Integrate[Sin[c + d*x]^6/(a + a*Sec[c + d*x])^3,x]","\frac{\cos ^6\left(\frac{1}{2} (c+d x)\right) \sec ^3(c+d x) \left(5040 \sin (c+d x)-1890 \sin (2 (c+d x))+760 \sin (3 (c+d x))-270 \sin (4 (c+d x))+72 \sin (5 (c+d x))-10 \sin (6 (c+d x))+9 \tan \left(\frac{c}{2}\right)-2760 d x\right)}{240 a^3 d (\sec (c+d x)+1)^3}","\frac{3 \sin ^5(c+d x)}{5 a^3 d}-\frac{7 \sin ^3(c+d x)}{3 a^3 d}+\frac{4 \sin (c+d x)}{a^3 d}-\frac{\sin (c+d x) \cos ^5(c+d x)}{6 a^3 d}-\frac{23 \sin (c+d x) \cos ^3(c+d x)}{24 a^3 d}-\frac{23 \sin (c+d x) \cos (c+d x)}{16 a^3 d}-\frac{23 x}{16 a^3}",1,"(Cos[(c + d*x)/2]^6*Sec[c + d*x]^3*(-2760*d*x + 5040*Sin[c + d*x] - 1890*Sin[2*(c + d*x)] + 760*Sin[3*(c + d*x)] - 270*Sin[4*(c + d*x)] + 72*Sin[5*(c + d*x)] - 10*Sin[6*(c + d*x)] + 9*Tan[c/2]))/(240*a^3*d*(1 + Sec[c + d*x])^3)","A",1
102,1,173,108,0.6839104,"\int \frac{\sin ^4(c+d x)}{(a+a \sec (c+d x))^3} \, dx","Integrate[Sin[c + d*x]^4/(a + a*Sec[c + d*x])^3,x]","\frac{\sec \left(\frac{c}{2}\right) \sec \left(\frac{1}{2} (c+d x)\right) \left(-997 \sin \left(c+\frac{d x}{2}\right)-800 \sin \left(c+\frac{3 d x}{2}\right)-800 \sin \left(2 c+\frac{3 d x}{2}\right)+160 \sin \left(2 c+\frac{5 d x}{2}\right)+160 \sin \left(3 c+\frac{5 d x}{2}\right)-35 \sin \left(3 c+\frac{7 d x}{2}\right)-35 \sin \left(4 c+\frac{7 d x}{2}\right)+5 \sin \left(4 c+\frac{9 d x}{2}\right)+5 \sin \left(5 c+\frac{9 d x}{2}\right)+2040 d x \cos \left(c+\frac{d x}{2}\right)-3563 \sin \left(\frac{d x}{2}\right)+2040 d x \cos \left(\frac{d x}{2}\right)\right)}{640 a^3 d}","\frac{\sin ^3(c+d x)}{a^3 d}-\frac{7 \sin (c+d x)}{a^3 d}+\frac{\sin (c+d x) \cos ^3(c+d x)}{4 a^3 d}+\frac{19 \sin (c+d x) \cos (c+d x)}{8 a^3 d}-\frac{4 \sin (c+d x)}{a^3 d (\cos (c+d x)+1)}+\frac{51 x}{8 a^3}",1,"(Sec[c/2]*Sec[(c + d*x)/2]*(2040*d*x*Cos[(d*x)/2] + 2040*d*x*Cos[c + (d*x)/2] - 3563*Sin[(d*x)/2] - 997*Sin[c + (d*x)/2] - 800*Sin[c + (3*d*x)/2] - 800*Sin[2*c + (3*d*x)/2] + 160*Sin[2*c + (5*d*x)/2] + 160*Sin[3*c + (5*d*x)/2] - 35*Sin[3*c + (7*d*x)/2] - 35*Sin[4*c + (7*d*x)/2] + 5*Sin[4*c + (9*d*x)/2] + 5*Sin[5*c + (9*d*x)/2]))/(640*a^3*d)","A",1
103,1,177,97,0.4669034,"\int \frac{\sin ^2(c+d x)}{(a+a \sec (c+d x))^3} \, dx","Integrate[Sin[c + d*x]^2/(a + a*Sec[c + d*x])^3,x]","-\frac{\sec \left(\frac{c}{2}\right) \sec ^3\left(\frac{1}{2} (c+d x)\right) \left(1326 \sin \left(c+\frac{d x}{2}\right)-2012 \sin \left(c+\frac{3 d x}{2}\right)-498 \sin \left(2 c+\frac{3 d x}{2}\right)-135 \sin \left(2 c+\frac{5 d x}{2}\right)-135 \sin \left(3 c+\frac{5 d x}{2}\right)+15 \sin \left(3 c+\frac{7 d x}{2}\right)+15 \sin \left(4 c+\frac{7 d x}{2}\right)+1980 d x \cos \left(c+\frac{d x}{2}\right)+660 d x \cos \left(c+\frac{3 d x}{2}\right)+660 d x \cos \left(2 c+\frac{3 d x}{2}\right)-3216 \sin \left(\frac{d x}{2}\right)+1980 d x \cos \left(\frac{d x}{2}\right)\right)}{960 a^3 d}","\frac{3 \sin (c+d x)}{a^3 d}-\frac{\sin (c+d x) \cos (c+d x)}{2 a^3 d}+\frac{19 \sin (c+d x)}{3 a^3 d (\cos (c+d x)+1)}-\frac{2 \sin (c+d x)}{3 a^3 d (\cos (c+d x)+1)^2}-\frac{11 x}{2 a^3}",1,"-1/960*(Sec[c/2]*Sec[(c + d*x)/2]^3*(1980*d*x*Cos[(d*x)/2] + 1980*d*x*Cos[c + (d*x)/2] + 660*d*x*Cos[c + (3*d*x)/2] + 660*d*x*Cos[2*c + (3*d*x)/2] - 3216*Sin[(d*x)/2] + 1326*Sin[c + (d*x)/2] - 2012*Sin[c + (3*d*x)/2] - 498*Sin[2*c + (3*d*x)/2] - 135*Sin[2*c + (5*d*x)/2] - 135*Sin[3*c + (5*d*x)/2] + 15*Sin[3*c + (7*d*x)/2] + 15*Sin[4*c + (7*d*x)/2]))/(a^3*d)","A",1
104,1,137,89,0.6444066,"\int \frac{\csc ^2(c+d x)}{(a+a \sec (c+d x))^3} \, dx","Integrate[Csc[c + d*x]^2/(a + a*Sec[c + d*x])^3,x]","\frac{\csc (c) (602 \sin (c+d x)+602 \sin (2 (c+d x))+258 \sin (3 (c+d x))+43 \sin (4 (c+d x))-560 \sin (2 c+d x)+168 \sin (c+2 d x)-280 \sin (3 c+2 d x)-48 \sin (2 c+3 d x)-8 \sin (3 c+4 d x)-840 \sin (c)+448 \sin (d x)) \csc (c+d x) \sec ^3(c+d x)}{2240 a^3 d (\sec (c+d x)+1)^3}","\frac{4 \cot ^7(c+d x)}{7 a^3 d}+\frac{3 \cot ^5(c+d x)}{5 a^3 d}-\frac{4 \csc ^7(c+d x)}{7 a^3 d}+\frac{7 \csc ^5(c+d x)}{5 a^3 d}-\frac{\csc ^3(c+d x)}{a^3 d}",1,"(Csc[c]*Csc[c + d*x]*Sec[c + d*x]^3*(-840*Sin[c] + 448*Sin[d*x] + 602*Sin[c + d*x] + 602*Sin[2*(c + d*x)] + 258*Sin[3*(c + d*x)] + 43*Sin[4*(c + d*x)] - 560*Sin[2*c + d*x] + 168*Sin[c + 2*d*x] - 280*Sin[3*c + 2*d*x] - 48*Sin[2*c + 3*d*x] - 8*Sin[3*c + 4*d*x]))/(2240*a^3*d*(1 + Sec[c + d*x])^3)","A",1
105,1,175,103,0.87132,"\int \frac{\csc ^4(c+d x)}{(a+a \sec (c+d x))^3} \, dx","Integrate[Csc[c + d*x]^4/(a + a*Sec[c + d*x])^3,x]","-\frac{\csc (c) (-1764 \sin (c+d x)-1323 \sin (2 (c+d x))+98 \sin (3 (c+d x))+588 \sin (4 (c+d x))+294 \sin (5 (c+d x))+49 \sin (6 (c+d x))+3456 \sin (2 c+d x)-1152 \sin (c+2 d x)+2880 \sin (3 c+2 d x)-128 \sin (2 c+3 d x)-768 \sin (3 c+4 d x)-384 \sin (4 c+5 d x)-64 \sin (5 c+6 d x)+5376 \sin (c)-1152 \sin (d x)) \csc ^3(2 (c+d x))}{5760 a^3 d (\sec (c+d x)+1)^3}","\frac{4 \cot ^9(c+d x)}{9 a^3 d}+\frac{\cot ^7(c+d x)}{a^3 d}+\frac{3 \cot ^5(c+d x)}{5 a^3 d}-\frac{4 \csc ^9(c+d x)}{9 a^3 d}+\frac{\csc ^7(c+d x)}{a^3 d}-\frac{3 \csc ^5(c+d x)}{5 a^3 d}",1,"-1/5760*(Csc[c]*Csc[2*(c + d*x)]^3*(5376*Sin[c] - 1152*Sin[d*x] - 1764*Sin[c + d*x] - 1323*Sin[2*(c + d*x)] + 98*Sin[3*(c + d*x)] + 588*Sin[4*(c + d*x)] + 294*Sin[5*(c + d*x)] + 49*Sin[6*(c + d*x)] + 3456*Sin[2*c + d*x] - 1152*Sin[c + 2*d*x] + 2880*Sin[3*c + 2*d*x] - 128*Sin[2*c + 3*d*x] - 768*Sin[3*c + 4*d*x] - 384*Sin[4*c + 5*d*x] - 64*Sin[5*c + 6*d*x]))/(a^3*d*(1 + Sec[c + d*x])^3)","A",1
106,1,223,127,1.3480645,"\int \frac{\csc ^6(c+d x)}{(a+a \sec (c+d x))^3} \, dx","Integrate[Csc[c + d*x]^6/(a + a*Sec[c + d*x])^3,x]","\frac{\csc (c) (524150 \sin (c+d x)+314490 \sin (2 (c+d x))-162010 \sin (3 (c+d x))-238250 \sin (4 (c+d x))-47650 \sin (5 (c+d x))+47650 \sin (6 (c+d x))+28590 \sin (7 (c+d x))+4765 \sin (8 (c+d x))-2027520 \sin (2 c+d x)+1486848 \sin (c+2 d x)-2365440 \sin (3 c+2 d x)+452608 \sin (2 c+3 d x)+665600 \sin (3 c+4 d x)+133120 \sin (4 c+5 d x)-133120 \sin (5 c+6 d x)-79872 \sin (6 c+7 d x)-13312 \sin (7 c+8 d x)-3886080 \sin (c)+563200 \sin (d x)) \csc ^5(c+d x) \sec ^3(c+d x)}{56770560 a^3 d (\sec (c+d x)+1)^3}","\frac{4 \cot ^{11}(c+d x)}{11 a^3 d}+\frac{11 \cot ^9(c+d x)}{9 a^3 d}+\frac{10 \cot ^7(c+d x)}{7 a^3 d}+\frac{3 \cot ^5(c+d x)}{5 a^3 d}-\frac{4 \csc ^{11}(c+d x)}{11 a^3 d}+\frac{7 \csc ^9(c+d x)}{9 a^3 d}-\frac{3 \csc ^7(c+d x)}{7 a^3 d}",1,"(Csc[c]*Csc[c + d*x]^5*Sec[c + d*x]^3*(-3886080*Sin[c] + 563200*Sin[d*x] + 524150*Sin[c + d*x] + 314490*Sin[2*(c + d*x)] - 162010*Sin[3*(c + d*x)] - 238250*Sin[4*(c + d*x)] - 47650*Sin[5*(c + d*x)] + 47650*Sin[6*(c + d*x)] + 28590*Sin[7*(c + d*x)] + 4765*Sin[8*(c + d*x)] - 2027520*Sin[2*c + d*x] + 1486848*Sin[c + 2*d*x] - 2365440*Sin[3*c + 2*d*x] + 452608*Sin[2*c + 3*d*x] + 665600*Sin[3*c + 4*d*x] + 133120*Sin[4*c + 5*d*x] - 133120*Sin[5*c + 6*d*x] - 79872*Sin[6*c + 7*d*x] - 13312*Sin[7*c + 8*d*x]))/(56770560*a^3*d*(1 + Sec[c + d*x])^3)","A",1
107,1,265,145,1.9318385,"\int \frac{\csc ^8(c+d x)}{(a+a \sec (c+d x))^3} \, dx","Integrate[Csc[c + d*x]^8/(a + a*Sec[c + d*x])^3,x]","-\frac{\csc (c) (-2764580 \sin (c+d x)-1382290 \sin (2 (c+d x))+1275960 \sin (3 (c+d x))+1336720 \sin (4 (c+d x))-60760 \sin (5 (c+d x))-524055 \sin (6 (c+d x))-167090 \sin (7 (c+d x))+60760 \sin (8 (c+d x))+45570 \sin (9 (c+d x))+7595 \sin (10 (c+d x))+20500480 \sin (2 c+d x)-23668736 \sin (c+2 d x)+30750720 \sin (3 c+2 d x)-6537216 \sin (2 c+3 d x)-6848512 \sin (3 c+4 d x)+311296 \sin (4 c+5 d x)+2684928 \sin (5 c+6 d x)+856064 \sin (6 c+7 d x)-311296 \sin (7 c+8 d x)-233472 \sin (8 c+9 d x)-38912 \sin (9 c+10 d x)+49201152 \sin (c)-6336512 \sin (d x)) \csc ^7(c+d x) \sec ^3(c+d x)}{984023040 a^3 d (\sec (c+d x)+1)^3}","\frac{4 \cot ^{13}(c+d x)}{13 a^3 d}+\frac{15 \cot ^{11}(c+d x)}{11 a^3 d}+\frac{7 \cot ^9(c+d x)}{3 a^3 d}+\frac{13 \cot ^7(c+d x)}{7 a^3 d}+\frac{3 \cot ^5(c+d x)}{5 a^3 d}-\frac{4 \csc ^{13}(c+d x)}{13 a^3 d}+\frac{7 \csc ^{11}(c+d x)}{11 a^3 d}-\frac{\csc ^9(c+d x)}{3 a^3 d}",1,"-1/984023040*(Csc[c]*Csc[c + d*x]^7*Sec[c + d*x]^3*(49201152*Sin[c] - 6336512*Sin[d*x] - 2764580*Sin[c + d*x] - 1382290*Sin[2*(c + d*x)] + 1275960*Sin[3*(c + d*x)] + 1336720*Sin[4*(c + d*x)] - 60760*Sin[5*(c + d*x)] - 524055*Sin[6*(c + d*x)] - 167090*Sin[7*(c + d*x)] + 60760*Sin[8*(c + d*x)] + 45570*Sin[9*(c + d*x)] + 7595*Sin[10*(c + d*x)] + 20500480*Sin[2*c + d*x] - 23668736*Sin[c + 2*d*x] + 30750720*Sin[3*c + 2*d*x] - 6537216*Sin[2*c + 3*d*x] - 6848512*Sin[3*c + 4*d*x] + 311296*Sin[4*c + 5*d*x] + 2684928*Sin[5*c + 6*d*x] + 856064*Sin[6*c + 7*d*x] - 311296*Sin[7*c + 8*d*x] - 233472*Sin[8*c + 9*d*x] - 38912*Sin[9*c + 10*d*x]))/(a^3*d*(1 + Sec[c + d*x])^3)","A",1
108,1,106,157,0.3070606,"\int (a+a \sec (c+d x)) (e \sin (c+d x))^{5/2} \, dx","Integrate[(a + a*Sec[c + d*x])*(e*Sin[c + d*x])^(5/2),x]","-\frac{a (e \sin (c+d x))^{5/2} \left(10 \sin ^{\frac{3}{2}}(c+d x)+3 \sin (2 (c+d x)) \sqrt{\sin (c+d x)}+18 E\left(\left.\frac{1}{4} (-2 c-2 d x+\pi )\right|2\right)+15 \tan ^{-1}\left(\sqrt{\sin (c+d x)}\right)-15 \tanh ^{-1}\left(\sqrt{\sin (c+d x)}\right)\right)}{15 d \sin ^{\frac{5}{2}}(c+d x)}","-\frac{a e^{5/2} \tan ^{-1}\left(\frac{\sqrt{e \sin (c+d x)}}{\sqrt{e}}\right)}{d}+\frac{a e^{5/2} \tanh ^{-1}\left(\frac{\sqrt{e \sin (c+d x)}}{\sqrt{e}}\right)}{d}+\frac{6 a e^2 E\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{e \sin (c+d x)}}{5 d \sqrt{\sin (c+d x)}}-\frac{2 a e (e \sin (c+d x))^{3/2}}{3 d}-\frac{2 a e \cos (c+d x) (e \sin (c+d x))^{3/2}}{5 d}",1,"-1/15*(a*(e*Sin[c + d*x])^(5/2)*(15*ArcTan[Sqrt[Sin[c + d*x]]] - 15*ArcTanh[Sqrt[Sin[c + d*x]]] + 18*EllipticE[(-2*c + Pi - 2*d*x)/4, 2] + 10*Sin[c + d*x]^(3/2) + 3*Sqrt[Sin[c + d*x]]*Sin[2*(c + d*x)]))/(d*Sin[c + d*x]^(5/2))","A",1
109,1,170,154,0.5948938,"\int (a+a \sec (c+d x)) (e \sin (c+d x))^{3/2} \, dx","Integrate[(a + a*Sec[c + d*x])*(e*Sin[c + d*x])^(3/2),x]","\frac{a (e \sin (c+d x))^{3/2} \left(-24 \sqrt{\sin (c+d x)}-8 F\left(\left.\frac{1}{4} (-2 c-2 d x+\pi )\right|2\right)-3 \log \left(1-\sqrt{\sin (c+d x)}\right)+3 \log \left(\sqrt{\sin (c+d x)}+1\right)+12 \tan ^{-1}\left(\sqrt{\sin (c+d x)}\right)+6 \tanh ^{-1}\left(\sqrt{\sin (c+d x)}\right)+16 \sin ^{\frac{5}{2}}(c+d x) \cos (c+d x) \sec (2 (c+d x))-8 \sqrt{\sin (c+d x)} \cos (c+d x) \sec (2 (c+d x))\right)}{12 d \sin ^{\frac{3}{2}}(c+d x)}","\frac{a e^{3/2} \tan ^{-1}\left(\frac{\sqrt{e \sin (c+d x)}}{\sqrt{e}}\right)}{d}+\frac{a e^{3/2} \tanh ^{-1}\left(\frac{\sqrt{e \sin (c+d x)}}{\sqrt{e}}\right)}{d}+\frac{2 a e^2 \sqrt{\sin (c+d x)} F\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{3 d \sqrt{e \sin (c+d x)}}-\frac{2 a e \sqrt{e \sin (c+d x)}}{d}-\frac{2 a e \cos (c+d x) \sqrt{e \sin (c+d x)}}{3 d}",1,"(a*(e*Sin[c + d*x])^(3/2)*(12*ArcTan[Sqrt[Sin[c + d*x]]] + 6*ArcTanh[Sqrt[Sin[c + d*x]]] - 8*EllipticF[(-2*c + Pi - 2*d*x)/4, 2] - 3*Log[1 - Sqrt[Sin[c + d*x]]] + 3*Log[1 + Sqrt[Sin[c + d*x]]] - 24*Sqrt[Sin[c + d*x]] - 8*Cos[c + d*x]*Sec[2*(c + d*x)]*Sqrt[Sin[c + d*x]] + 16*Cos[c + d*x]*Sec[2*(c + d*x)]*Sin[c + d*x]^(5/2)))/(12*d*Sin[c + d*x]^(3/2))","A",1
110,1,69,104,0.1150071,"\int (a+a \sec (c+d x)) \sqrt{e \sin (c+d x)} \, dx","Integrate[(a + a*Sec[c + d*x])*Sqrt[e*Sin[c + d*x]],x]","\frac{a \sqrt{e \sin (c+d x)} \left(-2 E\left(\left.\frac{1}{4} (-2 c-2 d x+\pi )\right|2\right)-\tan ^{-1}\left(\sqrt{\sin (c+d x)}\right)+\tanh ^{-1}\left(\sqrt{\sin (c+d x)}\right)\right)}{d \sqrt{\sin (c+d x)}}","-\frac{a \sqrt{e} \tan ^{-1}\left(\frac{\sqrt{e \sin (c+d x)}}{\sqrt{e}}\right)}{d}+\frac{a \sqrt{e} \tanh ^{-1}\left(\frac{\sqrt{e \sin (c+d x)}}{\sqrt{e}}\right)}{d}+\frac{2 a E\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{e \sin (c+d x)}}{d \sqrt{\sin (c+d x)}}",1,"(a*(-ArcTan[Sqrt[Sin[c + d*x]]] + ArcTanh[Sqrt[Sin[c + d*x]]] - 2*EllipticE[(-2*c + Pi - 2*d*x)/4, 2])*Sqrt[e*Sin[c + d*x]])/(d*Sqrt[Sin[c + d*x]])","A",1
111,1,193,103,3.1763523,"\int \frac{a+a \sec (c+d x)}{\sqrt{e \sin (c+d x)}} \, dx","Integrate[(a + a*Sec[c + d*x])/Sqrt[e*Sin[c + d*x]],x]","\frac{4 a \cos \left(\frac{1}{2} (c+d x)\right) \left(4 F\left(\left.\sin ^{-1}\left(\frac{1}{\sqrt{\tan \left(\frac{1}{4} (c+d x)\right)}}\right)\right|-1\right)+\sqrt{2} \left(\Pi \left(-1-\sqrt{2};\left.\sin ^{-1}\left(\frac{1}{\sqrt{\tan \left(\frac{1}{4} (c+d x)\right)}}\right)\right|-1\right)-\Pi \left(1-\sqrt{2};\left.\sin ^{-1}\left(\frac{1}{\sqrt{\tan \left(\frac{1}{4} (c+d x)\right)}}\right)\right|-1\right)-\Pi \left(-1+\sqrt{2};\left.\sin ^{-1}\left(\frac{1}{\sqrt{\tan \left(\frac{1}{4} (c+d x)\right)}}\right)\right|-1\right)+\Pi \left(1+\sqrt{2};\left.\sin ^{-1}\left(\frac{1}{\sqrt{\tan \left(\frac{1}{4} (c+d x)\right)}}\right)\right|-1\right)\right)\right)}{d \sqrt{\tan \left(\frac{1}{4} (c+d x)\right)} \sqrt{1-\cot ^2\left(\frac{1}{4} (c+d x)\right)} \sqrt{e \sin (c+d x)}}","\frac{a \tan ^{-1}\left(\frac{\sqrt{e \sin (c+d x)}}{\sqrt{e}}\right)}{d \sqrt{e}}+\frac{a \tanh ^{-1}\left(\frac{\sqrt{e \sin (c+d x)}}{\sqrt{e}}\right)}{d \sqrt{e}}+\frac{2 a \sqrt{\sin (c+d x)} F\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{d \sqrt{e \sin (c+d x)}}",1,"(4*a*Cos[(c + d*x)/2]*(4*EllipticF[ArcSin[1/Sqrt[Tan[(c + d*x)/4]]], -1] + Sqrt[2]*(EllipticPi[-1 - Sqrt[2], ArcSin[1/Sqrt[Tan[(c + d*x)/4]]], -1] - EllipticPi[1 - Sqrt[2], ArcSin[1/Sqrt[Tan[(c + d*x)/4]]], -1] - EllipticPi[-1 + Sqrt[2], ArcSin[1/Sqrt[Tan[(c + d*x)/4]]], -1] + EllipticPi[1 + Sqrt[2], ArcSin[1/Sqrt[Tan[(c + d*x)/4]]], -1])))/(d*Sqrt[1 - Cot[(c + d*x)/4]^2]*Sqrt[e*Sin[c + d*x]]*Sqrt[Tan[(c + d*x)/4]])","A",1
112,1,143,155,0.3973067,"\int \frac{a+a \sec (c+d x)}{(e \sin (c+d x))^{3/2}} \, dx","Integrate[(a + a*Sec[c + d*x])/(e*Sin[c + d*x])^(3/2),x]","-\frac{a \sin ^{\frac{3}{2}}(c+d x) (\cos (c+d x)+1) \sec \left(\frac{1}{2} (c+d x)\right) \left(2 \sqrt{\sin (c+d x)} \csc \left(\frac{1}{2} (c+d x)\right)-2 \sec \left(\frac{1}{2} (c+d x)\right) E\left(\left.\frac{1}{4} (-2 c-2 d x+\pi )\right|2\right)+\sec \left(\frac{1}{2} (c+d x)\right) \tan ^{-1}\left(\sqrt{\sin (c+d x)}\right)-\sec \left(\frac{1}{2} (c+d x)\right) \tanh ^{-1}\left(\sqrt{\sin (c+d x)}\right)\right)}{2 d (e \sin (c+d x))^{3/2}}","-\frac{a \tan ^{-1}\left(\frac{\sqrt{e \sin (c+d x)}}{\sqrt{e}}\right)}{d e^{3/2}}+\frac{a \tanh ^{-1}\left(\frac{\sqrt{e \sin (c+d x)}}{\sqrt{e}}\right)}{d e^{3/2}}-\frac{2 a E\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{e \sin (c+d x)}}{d e^2 \sqrt{\sin (c+d x)}}-\frac{2 a}{d e \sqrt{e \sin (c+d x)}}-\frac{2 a \cos (c+d x)}{d e \sqrt{e \sin (c+d x)}}",1,"-1/2*(a*(1 + Cos[c + d*x])*Sec[(c + d*x)/2]*(ArcTan[Sqrt[Sin[c + d*x]]]*Sec[(c + d*x)/2] - ArcTanh[Sqrt[Sin[c + d*x]]]*Sec[(c + d*x)/2] - 2*EllipticE[(-2*c + Pi - 2*d*x)/4, 2]*Sec[(c + d*x)/2] + 2*Csc[(c + d*x)/2]*Sqrt[Sin[c + d*x]])*Sin[c + d*x]^(3/2))/(d*(e*Sin[c + d*x])^(3/2))","A",1
113,1,120,160,0.3685516,"\int \frac{a+a \sec (c+d x)}{(e \sin (c+d x))^{5/2}} \, dx","Integrate[(a + a*Sec[c + d*x])/(e*Sin[c + d*x])^(5/2),x]","-\frac{a \sqrt{\sin (c+d x)} (\cos (c+d x)+1) \sec ^2\left(\frac{1}{2} (c+d x)\right) \left(2 F\left(\left.\frac{1}{4} (-2 c-2 d x+\pi )\right|2\right)-3 \tan ^{-1}\left(\sqrt{\sin (c+d x)}\right)-3 \tanh ^{-1}\left(\sqrt{\sin (c+d x)}\right)+\sqrt{\sin (c+d x)} \csc ^2\left(\frac{1}{2} (c+d x)\right)\right)}{6 d e^2 \sqrt{e \sin (c+d x)}}","\frac{a \tan ^{-1}\left(\frac{\sqrt{e \sin (c+d x)}}{\sqrt{e}}\right)}{d e^{5/2}}+\frac{a \tanh ^{-1}\left(\frac{\sqrt{e \sin (c+d x)}}{\sqrt{e}}\right)}{d e^{5/2}}+\frac{2 a \sqrt{\sin (c+d x)} F\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{3 d e^2 \sqrt{e \sin (c+d x)}}-\frac{2 a}{3 d e (e \sin (c+d x))^{3/2}}-\frac{2 a \cos (c+d x)}{3 d e (e \sin (c+d x))^{3/2}}",1,"-1/6*(a*(1 + Cos[c + d*x])*Sec[(c + d*x)/2]^2*(-3*ArcTan[Sqrt[Sin[c + d*x]]] - 3*ArcTanh[Sqrt[Sin[c + d*x]]] + 2*EllipticF[(-2*c + Pi - 2*d*x)/4, 2] + Csc[(c + d*x)/2]^2*Sqrt[Sin[c + d*x]])*Sqrt[Sin[c + d*x]])/(d*e^2*Sqrt[e*Sin[c + d*x]])","A",1
114,1,205,194,17.7893977,"\int (a+a \sec (c+d x))^2 (e \sin (c+d x))^{5/2} \, dx","Integrate[(a + a*Sec[c + d*x])^2*(e*Sin[c + d*x])^(5/2),x]","\frac{2 a^2 \cos ^4\left(\frac{1}{2} (c+d x)\right) \sec (c+d x) (e \sin (c+d x))^{5/2} \sec ^4\left(\frac{1}{2} \sin ^{-1}(\sin (c+d x))\right) \left(9 \sin ^{\frac{3}{2}}(c+d x) \sqrt{\cos ^2(c+d x)} \, _2F_1\left(\frac{3}{4},\frac{3}{2};\frac{7}{4};\sin ^2(c+d x)\right)+3 \sin ^{\frac{7}{2}}(c+d x)-9 \sin ^{\frac{3}{2}}(c+d x)-10 \sin ^{\frac{3}{2}}(c+d x) \sqrt{\cos ^2(c+d x)}-15 \sqrt{\cos ^2(c+d x)} \tan ^{-1}\left(\sqrt{\sin (c+d x)}\right)+15 \sqrt{\cos ^2(c+d x)} \tanh ^{-1}\left(\sqrt{\sin (c+d x)}\right)\right)}{15 d \sin ^{\frac{5}{2}}(c+d x)}","-\frac{2 a^2 e^{5/2} \tan ^{-1}\left(\frac{\sqrt{e \sin (c+d x)}}{\sqrt{e}}\right)}{d}+\frac{2 a^2 e^{5/2} \tanh ^{-1}\left(\frac{\sqrt{e \sin (c+d x)}}{\sqrt{e}}\right)}{d}-\frac{9 a^2 e^2 E\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{e \sin (c+d x)}}{5 d \sqrt{\sin (c+d x)}}-\frac{4 a^2 e (e \sin (c+d x))^{3/2}}{3 d}-\frac{2 a^2 e \cos (c+d x) (e \sin (c+d x))^{3/2}}{5 d}+\frac{a^2 e \sec (c+d x) (e \sin (c+d x))^{3/2}}{d}",1,"(2*a^2*Cos[(c + d*x)/2]^4*Sec[c + d*x]*Sec[ArcSin[Sin[c + d*x]]/2]^4*(e*Sin[c + d*x])^(5/2)*(-15*ArcTan[Sqrt[Sin[c + d*x]]]*Sqrt[Cos[c + d*x]^2] + 15*ArcTanh[Sqrt[Sin[c + d*x]]]*Sqrt[Cos[c + d*x]^2] - 9*Sin[c + d*x]^(3/2) - 10*Sqrt[Cos[c + d*x]^2]*Sin[c + d*x]^(3/2) + 9*Sqrt[Cos[c + d*x]^2]*Hypergeometric2F1[3/4, 3/2, 7/4, Sin[c + d*x]^2]*Sin[c + d*x]^(3/2) + 3*Sin[c + d*x]^(7/2)))/(15*d*Sin[c + d*x]^(5/2))","C",0
115,1,204,192,15.3856783,"\int (a+a \sec (c+d x))^2 (e \sin (c+d x))^{3/2} \, dx","Integrate[(a + a*Sec[c + d*x])^2*(e*Sin[c + d*x])^(3/2),x]","\frac{16 a^2 e \sin ^4\left(\frac{1}{2} \sin ^{-1}(\sin (c+d x))\right) \cos ^4\left(\frac{1}{2} (c+d x)\right) \sec (c+d x) \sqrt{e \sin (c+d x)} \left(-\sqrt{\sin (c+d x)} \sqrt{\cos ^2(c+d x)} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2(c+d x)\right)+2 \sin ^{\frac{5}{2}}(c+d x)+\sqrt{\sin (c+d x)}-12 \sqrt{\sin (c+d x)} \sqrt{\cos ^2(c+d x)}+6 \sqrt{\cos ^2(c+d x)} \tan ^{-1}\left(\sqrt{\sin (c+d x)}\right)+6 \sqrt{\cos ^2(c+d x)} \tanh ^{-1}\left(\sqrt{\sin (c+d x)}\right)\right)}{3 d \sin ^{\frac{9}{2}}(c+d x)}","\frac{2 a^2 e^{3/2} \tan ^{-1}\left(\frac{\sqrt{e \sin (c+d x)}}{\sqrt{e}}\right)}{d}+\frac{2 a^2 e^{3/2} \tanh ^{-1}\left(\frac{\sqrt{e \sin (c+d x)}}{\sqrt{e}}\right)}{d}-\frac{a^2 e^2 \sqrt{\sin (c+d x)} F\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{3 d \sqrt{e \sin (c+d x)}}-\frac{4 a^2 e \sqrt{e \sin (c+d x)}}{d}-\frac{2 a^2 e \cos (c+d x) \sqrt{e \sin (c+d x)}}{3 d}+\frac{a^2 e \sec (c+d x) \sqrt{e \sin (c+d x)}}{d}",1,"(16*a^2*e*Cos[(c + d*x)/2]^4*Sec[c + d*x]*Sqrt[e*Sin[c + d*x]]*(6*ArcTan[Sqrt[Sin[c + d*x]]]*Sqrt[Cos[c + d*x]^2] + 6*ArcTanh[Sqrt[Sin[c + d*x]]]*Sqrt[Cos[c + d*x]^2] + Sqrt[Sin[c + d*x]] - 12*Sqrt[Cos[c + d*x]^2]*Sqrt[Sin[c + d*x]] - Sqrt[Cos[c + d*x]^2]*Hypergeometric2F1[1/4, 1/2, 5/4, Sin[c + d*x]^2]*Sqrt[Sin[c + d*x]] + 2*Sin[c + d*x]^(5/2))*Sin[ArcSin[Sin[c + d*x]]/2]^4)/(3*d*Sin[c + d*x]^(9/2))","C",0
116,1,168,138,2.177224,"\int (a+a \sec (c+d x))^2 \sqrt{e \sin (c+d x)} \, dx","Integrate[(a + a*Sec[c + d*x])^2*Sqrt[e*Sin[c + d*x]],x]","-\frac{2 a^2 \cos ^4\left(\frac{1}{2} (c+d x)\right) \sec (c+d x) \sqrt{e \sin (c+d x)} \sec ^4\left(\frac{1}{2} \sin ^{-1}(\sin (c+d x))\right) \left(\sin ^{\frac{3}{2}}(c+d x) \sqrt{\cos ^2(c+d x)} \, _2F_1\left(\frac{3}{4},\frac{3}{2};\frac{7}{4};\sin ^2(c+d x)\right)-3 \sin ^{\frac{3}{2}}(c+d x)+3 \sqrt{\cos ^2(c+d x)} \tan ^{-1}\left(\sqrt{\sin (c+d x)}\right)-3 \sqrt{\cos ^2(c+d x)} \tanh ^{-1}\left(\sqrt{\sin (c+d x)}\right)\right)}{3 d \sqrt{\sin (c+d x)}}","-\frac{2 a^2 \sqrt{e} \tan ^{-1}\left(\frac{\sqrt{e \sin (c+d x)}}{\sqrt{e}}\right)}{d}+\frac{2 a^2 \sqrt{e} \tanh ^{-1}\left(\frac{\sqrt{e \sin (c+d x)}}{\sqrt{e}}\right)}{d}+\frac{a^2 \sec (c+d x) (e \sin (c+d x))^{3/2}}{d e}+\frac{a^2 E\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{e \sin (c+d x)}}{d \sqrt{\sin (c+d x)}}",1,"(-2*a^2*Cos[(c + d*x)/2]^4*Sec[c + d*x]*Sec[ArcSin[Sin[c + d*x]]/2]^4*Sqrt[e*Sin[c + d*x]]*(3*ArcTan[Sqrt[Sin[c + d*x]]]*Sqrt[Cos[c + d*x]^2] - 3*ArcTanh[Sqrt[Sin[c + d*x]]]*Sqrt[Cos[c + d*x]^2] - 3*Sin[c + d*x]^(3/2) + Sqrt[Cos[c + d*x]^2]*Hypergeometric2F1[3/4, 3/2, 7/4, Sin[c + d*x]^2]*Sin[c + d*x]^(3/2)))/(3*d*Sqrt[Sin[c + d*x]])","C",0
117,1,164,139,71.8892977,"\int \frac{(a+a \sec (c+d x))^2}{\sqrt{e \sin (c+d x)}} \, dx","Integrate[(a + a*Sec[c + d*x])^2/Sqrt[e*Sin[c + d*x]],x]","\frac{a^2 \sqrt{\sin (c+d x)} \cos ^4\left(\frac{1}{2} (c+d x)\right) \sec (c+d x) \sec ^4\left(\frac{1}{2} \sin ^{-1}(\sin (c+d x))\right) \left(3 \sqrt{\sin (c+d x)} \sqrt{\cos ^2(c+d x)} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2(c+d x)\right)+\sqrt{\sin (c+d x)}+2 \sqrt{\cos ^2(c+d x)} \tan ^{-1}\left(\sqrt{\sin (c+d x)}\right)+2 \sqrt{\cos ^2(c+d x)} \tanh ^{-1}\left(\sqrt{\sin (c+d x)}\right)\right)}{d \sqrt{e \sin (c+d x)}}","\frac{2 a^2 \tan ^{-1}\left(\frac{\sqrt{e \sin (c+d x)}}{\sqrt{e}}\right)}{d \sqrt{e}}+\frac{2 a^2 \tanh ^{-1}\left(\frac{\sqrt{e \sin (c+d x)}}{\sqrt{e}}\right)}{d \sqrt{e}}+\frac{a^2 \sec (c+d x) \sqrt{e \sin (c+d x)}}{d e}+\frac{3 a^2 \sqrt{\sin (c+d x)} F\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{d \sqrt{e \sin (c+d x)}}",1,"(a^2*Cos[(c + d*x)/2]^4*Sec[c + d*x]*Sec[ArcSin[Sin[c + d*x]]/2]^4*(2*ArcTan[Sqrt[Sin[c + d*x]]]*Sqrt[Cos[c + d*x]^2] + 2*ArcTanh[Sqrt[Sin[c + d*x]]]*Sqrt[Cos[c + d*x]^2] + Sqrt[Sin[c + d*x]] + 3*Sqrt[Cos[c + d*x]^2]*Hypergeometric2F1[1/4, 1/2, 5/4, Sin[c + d*x]^2]*Sqrt[Sin[c + d*x]])*Sqrt[Sin[c + d*x]])/(d*Sqrt[e*Sin[c + d*x]])","C",0
118,1,135,224,11.0557832,"\int \frac{(a+a \sec (c+d x))^2}{(e \sin (c+d x))^{3/2}} \, dx","Integrate[(a + a*Sec[c + d*x])^2/(e*Sin[c + d*x])^(3/2),x]","-\frac{2 a^2 \cos ^4\left(\frac{1}{2} (c+d x)\right) \cot (c+d x) \sqrt{e \sin (c+d x)} \left(\sin ^2(c+d x) \, _2F_1\left(\frac{3}{4},\frac{3}{2};\frac{7}{4};\sin ^2(c+d x)\right)+6 \, _2F_1\left(-\frac{1}{4},1;\frac{3}{4};\sin ^2(c+d x)\right)+6 \, _2F_1\left(-\frac{1}{4},\frac{3}{2};\frac{3}{4};\sin ^2(c+d x)\right)\right) \sec ^4\left(\frac{1}{2} \sin ^{-1}(\sin (c+d x))\right)}{3 d e^2 \sqrt{\cos ^2(c+d x)}}","-\frac{2 a^2 \tan ^{-1}\left(\frac{\sqrt{e \sin (c+d x)}}{\sqrt{e}}\right)}{d e^{3/2}}+\frac{2 a^2 \tanh ^{-1}\left(\frac{\sqrt{e \sin (c+d x)}}{\sqrt{e}}\right)}{d e^{3/2}}+\frac{3 a^2 \sec (c+d x) (e \sin (c+d x))^{3/2}}{d e^3}-\frac{5 a^2 E\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{e \sin (c+d x)}}{d e^2 \sqrt{\sin (c+d x)}}-\frac{4 a^2}{d e \sqrt{e \sin (c+d x)}}-\frac{2 a^2 \cos (c+d x)}{d e \sqrt{e \sin (c+d x)}}-\frac{2 a^2 \sec (c+d x)}{d e \sqrt{e \sin (c+d x)}}",1,"(-2*a^2*Cos[(c + d*x)/2]^4*Cot[c + d*x]*Sec[ArcSin[Sin[c + d*x]]/2]^4*Sqrt[e*Sin[c + d*x]]*(6*Hypergeometric2F1[-1/4, 1, 3/4, Sin[c + d*x]^2] + 6*Hypergeometric2F1[-1/4, 3/2, 3/4, Sin[c + d*x]^2] + Hypergeometric2F1[3/4, 3/2, 7/4, Sin[c + d*x]^2]*Sin[c + d*x]^2))/(3*d*e^2*Sqrt[Cos[c + d*x]^2])","C",0
119,1,169,234,48.8764125,"\int \frac{(a+a \sec (c+d x))^2}{(e \sin (c+d x))^{5/2}} \, dx","Integrate[(a + a*Sec[c + d*x])^2/(e*Sin[c + d*x])^(5/2),x]","-\frac{a^2 \cos ^4\left(\frac{1}{2} (c+d x)\right) \sec (c+d x) \sqrt{e \sin (c+d x)} \sec ^4\left(\frac{1}{2} \sin ^{-1}(\sin (c+d x))\right) \left(3 \sqrt{\cos ^2(c+d x)} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2(c+d x)\right)+4 \sqrt{\cos ^2(c+d x)} \csc ^2(c+d x) \, _2F_1\left(-\frac{3}{4},1;\frac{1}{4};\sin ^2(c+d x)\right)+4 \sqrt{\cos ^2(c+d x)} \csc ^2(c+d x) \, _2F_1\left(-\frac{3}{4},\frac{3}{2};\frac{1}{4};\sin ^2(c+d x)\right)+3\right)}{3 d e^3}","\frac{2 a^2 \tan ^{-1}\left(\frac{\sqrt{e \sin (c+d x)}}{\sqrt{e}}\right)}{d e^{5/2}}+\frac{2 a^2 \tanh ^{-1}\left(\frac{\sqrt{e \sin (c+d x)}}{\sqrt{e}}\right)}{d e^{5/2}}+\frac{5 a^2 \sec (c+d x) \sqrt{e \sin (c+d x)}}{3 d e^3}+\frac{7 a^2 \sqrt{\sin (c+d x)} F\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{3 d e^2 \sqrt{e \sin (c+d x)}}-\frac{4 a^2}{3 d e (e \sin (c+d x))^{3/2}}-\frac{2 a^2 \cos (c+d x)}{3 d e (e \sin (c+d x))^{3/2}}-\frac{2 a^2 \sec (c+d x)}{3 d e (e \sin (c+d x))^{3/2}}",1,"-1/3*(a^2*Cos[(c + d*x)/2]^4*(3 + 4*Sqrt[Cos[c + d*x]^2]*Csc[c + d*x]^2*Hypergeometric2F1[-3/4, 1, 1/4, Sin[c + d*x]^2] + 4*Sqrt[Cos[c + d*x]^2]*Csc[c + d*x]^2*Hypergeometric2F1[-3/4, 3/2, 1/4, Sin[c + d*x]^2] + 3*Sqrt[Cos[c + d*x]^2]*Hypergeometric2F1[1/4, 1/2, 5/4, Sin[c + d*x]^2])*Sec[c + d*x]*Sec[ArcSin[Sin[c + d*x]]/2]^4*Sqrt[e*Sin[c + d*x]])/(d*e^3)","C",0
120,1,122,139,0.6910983,"\int \frac{(e \sin (c+d x))^{7/2}}{a+a \sec (c+d x)} \, dx","Integrate[(e*Sin[c + d*x])^(7/2)/(a + a*Sec[c + d*x]),x]","\frac{e^3 \cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x) \sqrt{e \sin (c+d x)} \left(40 F\left(\left.\frac{1}{4} (-2 c-2 d x+\pi )\right|2\right)+\sqrt{\sin (c+d x)} (25 \cos (c+d x)-42 \cos (2 (c+d x))+15 \cos (3 (c+d x))+42)\right)}{105 a d \sqrt{\sin (c+d x)} (\sec (c+d x)+1)}","-\frac{4 e^4 \sqrt{\sin (c+d x)} F\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{21 a d \sqrt{e \sin (c+d x)}}+\frac{2 e^3 \cos ^3(c+d x) \sqrt{e \sin (c+d x)}}{7 a d}-\frac{2 e^3 \cos (c+d x) \sqrt{e \sin (c+d x)}}{21 a d}+\frac{2 e (e \sin (c+d x))^{5/2}}{5 a d}",1,"(e^3*Cos[(c + d*x)/2]^2*Sec[c + d*x]*(40*EllipticF[(-2*c + Pi - 2*d*x)/4, 2] + (42 + 25*Cos[c + d*x] - 42*Cos[2*(c + d*x)] + 15*Cos[3*(c + d*x)])*Sqrt[Sin[c + d*x]])*Sqrt[e*Sin[c + d*x]])/(105*a*d*(1 + Sec[c + d*x])*Sqrt[Sin[c + d*x]])","A",1
121,1,232,104,4.8971547,"\int \frac{(e \sin (c+d x))^{5/2}}{a+a \sec (c+d x)} \, dx","Integrate[(e*Sin[c + d*x])^(5/2)/(a + a*Sec[c + d*x]),x]","\frac{2 \cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x) (e \sin (c+d x))^{5/2} \left(\sqrt{\sin (c+d x)} (10 \sin (c) \cos (d x)-3 \sin (2 c) \cos (2 d x)+10 \cos (c) \sin (d x)-3 \cos (2 c) \sin (2 d x)-12 \tan (c))+\frac{2 \sec (c) e^{-i d x} \sqrt{2-2 e^{2 i (c+d x)}} \left(3 \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};e^{2 i (c+d x)}\right)+e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};e^{2 i (c+d x)}\right)\right)}{\sqrt{-i e^{-i (c+d x)} \left(-1+e^{2 i (c+d x)}\right)}}\right)}{15 a d \sin ^{\frac{5}{2}}(c+d x) (\sec (c+d x)+1)}","-\frac{4 e^2 E\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{e \sin (c+d x)}}{5 a d \sqrt{\sin (c+d x)}}+\frac{2 e (e \sin (c+d x))^{3/2}}{3 a d}-\frac{2 e \cos (c+d x) (e \sin (c+d x))^{3/2}}{5 a d}",1,"(2*Cos[(c + d*x)/2]^2*Sec[c + d*x]*(e*Sin[c + d*x])^(5/2)*((2*Sqrt[2 - 2*E^((2*I)*(c + d*x))]*(3*Hypergeometric2F1[-1/4, 1/2, 3/4, E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, E^((2*I)*(c + d*x))])*Sec[c])/(E^(I*d*x)*Sqrt[((-I)*(-1 + E^((2*I)*(c + d*x))))/E^(I*(c + d*x))]) + Sqrt[Sin[c + d*x]]*(10*Cos[d*x]*Sin[c] - 3*Cos[2*d*x]*Sin[2*c] + 10*Cos[c]*Sin[d*x] - 3*Cos[2*c]*Sin[2*d*x] - 12*Tan[c])))/(15*a*d*(1 + Sec[c + d*x])*Sin[c + d*x]^(5/2))","C",1
122,1,69,102,20.6506074,"\int \frac{(e \sin (c+d x))^{3/2}}{a+a \sec (c+d x)} \, dx","Integrate[(e*Sin[c + d*x])^(3/2)/(a + a*Sec[c + d*x]),x]","-\frac{2 (e \sin (c+d x))^{3/2} \left(\sqrt{\sin (c+d x)} (\cos (c+d x)-3)-2 F\left(\left.\frac{1}{4} (-2 c-2 d x+\pi )\right|2\right)\right)}{3 a d \sin ^{\frac{3}{2}}(c+d x)}","-\frac{4 e^2 \sqrt{\sin (c+d x)} F\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{3 a d \sqrt{e \sin (c+d x)}}+\frac{2 e \sqrt{e \sin (c+d x)}}{a d}-\frac{2 e \cos (c+d x) \sqrt{e \sin (c+d x)}}{3 a d}",1,"(-2*(-2*EllipticF[(-2*c + Pi - 2*d*x)/4, 2] + (-3 + Cos[c + d*x])*Sqrt[Sin[c + d*x]])*(e*Sin[c + d*x])^(3/2))/(3*a*d*Sin[c + d*x]^(3/2))","A",1
123,1,249,95,0.6122325,"\int \frac{\sqrt{e \sin (c+d x)}}{a+a \sec (c+d x)} \, dx","Integrate[Sqrt[e*Sin[c + d*x]]/(a + a*Sec[c + d*x]),x]","\frac{2 \left(12 e^{2 i c} \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)+4 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)+6 e^{i (c+d x)}-9 e^{2 i (c+d x)}+3 e^{2 i (2 c+d x)}+6 e^{i (3 c+d x)}-9 e^{2 i c}+3\right) \sqrt{e \sin (c+d x)}}{3 a \left(1+i e^{i c}\right) \left(e^{i c}+i\right) d \left(-1+e^{i (c+d x)}\right) \left(1+e^{i (c+d x)}\right)}","-\frac{2 e}{a d \sqrt{e \sin (c+d x)}}+\frac{2 e \cos (c+d x)}{a d \sqrt{e \sin (c+d x)}}+\frac{4 E\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{e \sin (c+d x)}}{a d \sqrt{\sin (c+d x)}}",1,"(2*(3 - 9*E^((2*I)*c) + 6*E^(I*(c + d*x)) - 9*E^((2*I)*(c + d*x)) + 3*E^((2*I)*(2*c + d*x)) + 6*E^(I*(3*c + d*x)) + 12*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))] + 4*E^((2*I)*(c + d*x))*Sqrt[1 - E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/2, 3/4, 7/4, E^((2*I)*(c + d*x))])*Sqrt[e*Sin[c + d*x]])/(3*a*d*(1 + I*E^(I*c))*(I + E^(I*c))*(-1 + E^(I*(c + d*x)))*(1 + E^(I*(c + d*x))))","C",1
124,1,77,101,0.5798826,"\int \frac{1}{(a+a \sec (c+d x)) \sqrt{e \sin (c+d x)}} \, dx","Integrate[1/((a + a*Sec[c + d*x])*Sqrt[e*Sin[c + d*x]]),x]","\frac{2 \cot \left(\frac{1}{2} (c+d x)\right) \left(\cos (c+d x)-2 \sin ^{\frac{3}{2}}(c+d x) F\left(\left.\frac{1}{4} (-2 c-2 d x+\pi )\right|2\right)-1\right)}{3 a d (\cos (c+d x)+1) \sqrt{e \sin (c+d x)}}","-\frac{2 e}{3 a d (e \sin (c+d x))^{3/2}}+\frac{2 e \cos (c+d x)}{3 a d (e \sin (c+d x))^{3/2}}+\frac{4 \sqrt{\sin (c+d x)} F\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{3 a d \sqrt{e \sin (c+d x)}}",1,"(2*Cot[(c + d*x)/2]*(-1 + Cos[c + d*x] - 2*EllipticF[(-2*c + Pi - 2*d*x)/4, 2]*Sin[c + d*x]^(3/2)))/(3*a*d*(1 + Cos[c + d*x])*Sqrt[e*Sin[c + d*x]])","A",1
125,1,124,135,1.095533,"\int \frac{1}{(a+a \sec (c+d x)) (e \sin (c+d x))^{3/2}} \, dx","Integrate[1/((a + a*Sec[c + d*x])*(e*Sin[c + d*x])^(3/2)),x]","\frac{\sec ^2\left(\frac{1}{2} (c+d x)\right) (\cos (c+d x)+i \sin (c+d x)) \left(2 \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 (c+d x)+1)+3 i \sin (c+d x)-9 \cos (c+d x)-6\right)}{15 a d e \sqrt{e \sin (c+d x)}}","-\frac{4 E\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{e \sin (c+d x)}}{5 a d e^2 \sqrt{\sin (c+d x)}}-\frac{2 e}{5 a d (e \sin (c+d x))^{5/2}}+\frac{2 e \cos (c+d x)}{5 a d (e \sin (c+d x))^{5/2}}-\frac{4 \cos (c+d x)}{5 a d e \sqrt{e \sin (c+d x)}}",1,"(Sec[(c + d*x)/2]^2*(Cos[c + d*x] + I*Sin[c + d*x])*(-6 - 9*Cos[c + d*x] + 2*Sqrt[1 - E^((2*I)*(c + d*x))]*(1 + Cos[c + d*x])*Hypergeometric2F1[1/2, 3/4, 7/4, E^((2*I)*(c + d*x))] + (3*I)*Sin[c + d*x]))/(15*a*d*e*Sqrt[e*Sin[c + d*x]])","C",1
126,1,91,135,1.2745623,"\int \frac{1}{(a+a \sec (c+d x)) (e \sin (c+d x))^{5/2}} \, dx","Integrate[1/((a + a*Sec[c + d*x])*(e*Sin[c + d*x])^(5/2)),x]","-\frac{2 \left(2 \cos (c+d x)+\cos (2 (c+d x))+\sin ^{\frac{7}{2}}(c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right) F\left(\left.\frac{1}{4} (-2 c-2 d x+\pi )\right|2\right)+4\right)}{21 a d e (\cos (c+d x)+1) (e \sin (c+d x))^{3/2}}","\frac{4 \sqrt{\sin (c+d x)} F\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{21 a d e^2 \sqrt{e \sin (c+d x)}}-\frac{2 e}{7 a d (e \sin (c+d x))^{7/2}}+\frac{2 e \cos (c+d x)}{7 a d (e \sin (c+d x))^{7/2}}-\frac{4 \cos (c+d x)}{21 a d e (e \sin (c+d x))^{3/2}}",1,"(-2*(4 + 2*Cos[c + d*x] + Cos[2*(c + d*x)] + Csc[(c + d*x)/2]^2*EllipticF[(-2*c + Pi - 2*d*x)/4, 2]*Sin[c + d*x]^(7/2)))/(21*a*d*e*(1 + Cos[c + d*x])*(e*Sin[c + d*x])^(3/2))","A",1
127,1,94,162,1.5505288,"\int \frac{(e \sin (c+d x))^{7/2}}{(a+a \sec (c+d x))^2} \, dx","Integrate[(e*Sin[c + d*x])^(7/2)/(a + a*Sec[c + d*x])^2,x]","-\frac{e^3 \sqrt{e \sin (c+d x)} \left(520 F\left(\left.\frac{1}{4} (-2 c-2 d x+\pi )\right|2\right)+\sqrt{\sin (c+d x)} (-305 \cos (c+d x)+84 \cos (2 (c+d x))-15 \cos (3 (c+d x))+756)\right)}{210 a^2 d \sqrt{\sin (c+d x)}}","\frac{52 e^4 \sqrt{\sin (c+d x)} F\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{21 a^2 d \sqrt{e \sin (c+d x)}}-\frac{4 e^3 \sqrt{e \sin (c+d x)}}{a^2 d}+\frac{2 e^3 \cos ^3(c+d x) \sqrt{e \sin (c+d x)}}{7 a^2 d}+\frac{26 e^3 \cos (c+d x) \sqrt{e \sin (c+d x)}}{21 a^2 d}+\frac{4 e (e \sin (c+d x))^{5/2}}{5 a^2 d}",1,"-1/210*(e^3*(520*EllipticF[(-2*c + Pi - 2*d*x)/4, 2] + (756 - 305*Cos[c + d*x] + 84*Cos[2*(c + d*x)] - 15*Cos[3*(c + d*x)])*Sqrt[Sin[c + d*x]])*Sqrt[e*Sin[c + d*x]])/(a^2*d*Sqrt[Sin[c + d*x]])","A",1
128,1,249,187,2.987585,"\int \frac{(e \sin (c+d x))^{5/2}}{(a+a \sec (c+d x))^2} \, dx","Integrate[(e*Sin[c + d*x])^(5/2)/(a + a*Sec[c + d*x])^2,x]","\frac{4 \cos ^4\left(\frac{1}{2} (c+d x)\right) \sec ^2(c+d x) (e \sin (c+d x))^{5/2} \left(\csc ^2(c+d x) \left(20 \sin (c) \cos (d x)-3 \sin (2 c) \cos (2 d x)+20 \cos (c) \sin (d x)-3 \cos (2 c) \sin (2 d x)+\sec \left(\frac{c}{2}\right) \left(60 \sin \left(\frac{d x}{2}\right) \sec \left(\frac{1}{2} (c+d x)\right)-36 \sin \left(\frac{3 c}{2}\right) \sec (c)\right)-96 \tan \left(\frac{c}{2}\right) \sec (c)\right)+\frac{352 i e^{2 i (2 c+d x)} \left(3 \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};e^{2 i (c+d x)}\right)+e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};e^{2 i (c+d x)}\right)\right)}{\left(1+e^{2 i c}\right) \left(1-e^{2 i (c+d x)}\right)^{5/2}}\right)}{15 a^2 d (\sec (c+d x)+1)^2}","\frac{4 e^3}{a^2 d \sqrt{e \sin (c+d x)}}-\frac{2 e^3 \cos ^3(c+d x)}{a^2 d \sqrt{e \sin (c+d x)}}-\frac{2 e^3 \cos (c+d x)}{a^2 d \sqrt{e \sin (c+d x)}}-\frac{44 e^2 E\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{e \sin (c+d x)}}{5 a^2 d \sqrt{\sin (c+d x)}}+\frac{4 e (e \sin (c+d x))^{3/2}}{3 a^2 d}-\frac{12 e \cos (c+d x) (e \sin (c+d x))^{3/2}}{5 a^2 d}",1,"(4*Cos[(c + d*x)/2]^4*Sec[c + d*x]^2*(e*Sin[c + d*x])^(5/2)*(((352*I)*E^((2*I)*(2*c + d*x))*(3*Hypergeometric2F1[-1/4, 1/2, 3/4, E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, E^((2*I)*(c + d*x))]))/((1 + E^((2*I)*c))*(1 - E^((2*I)*(c + d*x)))^(5/2)) + Csc[c + d*x]^2*(20*Cos[d*x]*Sin[c] - 3*Cos[2*d*x]*Sin[2*c] + Sec[c/2]*(-36*Sec[c]*Sin[(3*c)/2] + 60*Sec[(c + d*x)/2]*Sin[(d*x)/2]) + 20*Cos[c]*Sin[d*x] - 3*Cos[2*c]*Sin[2*d*x] - 96*Sec[c]*Tan[c/2])))/(15*a^2*d*(1 + Sec[c + d*x])^2)","C",1
129,1,119,189,1.9132547,"\int \frac{(e \sin (c+d x))^{3/2}}{(a+a \sec (c+d x))^2} \, dx","Integrate[(e*Sin[c + d*x])^(3/2)/(a + a*Sec[c + d*x])^2,x]","\frac{2 \cos ^4\left(\frac{1}{2} (c+d x)\right) \sec ^2(c+d x) (e \sin (c+d x))^{3/2} \left(\frac{24 F\left(\left.\frac{1}{4} (-2 c-2 d x+\pi )\right|2\right)}{\sin ^{\frac{3}{2}}(c+d x)}+(10 \cos (c+d x)-\cos (2 (c+d x))+15) \csc (c+d x) \sec ^2\left(\frac{1}{2} (c+d x)\right)\right)}{3 a^2 d (\sec (c+d x)+1)^2}","\frac{4 e^3}{3 a^2 d (e \sin (c+d x))^{3/2}}-\frac{2 e^3 \cos ^3(c+d x)}{3 a^2 d (e \sin (c+d x))^{3/2}}-\frac{2 e^3 \cos (c+d x)}{3 a^2 d (e \sin (c+d x))^{3/2}}-\frac{4 e^2 \sqrt{\sin (c+d x)} F\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{a^2 d \sqrt{e \sin (c+d x)}}+\frac{4 e \sqrt{e \sin (c+d x)}}{a^2 d}-\frac{4 e \cos (c+d x) \sqrt{e \sin (c+d x)}}{3 a^2 d}",1,"(2*Cos[(c + d*x)/2]^4*Sec[c + d*x]^2*((15 + 10*Cos[c + d*x] - Cos[2*(c + d*x)])*Csc[c + d*x]*Sec[(c + d*x)/2]^2 + (24*EllipticF[(-2*c + Pi - 2*d*x)/4, 2])/Sin[c + d*x]^(3/2))*(e*Sin[c + d*x])^(3/2))/(3*a^2*d*(1 + Sec[c + d*x])^2)","A",1
130,1,222,188,1.3629036,"\int \frac{\sqrt{e \sin (c+d x)}}{(a+a \sec (c+d x))^2} \, dx","Integrate[Sqrt[e*Sin[c + d*x]]/(a + a*Sec[c + d*x])^2,x]","\frac{4 \cos ^4\left(\frac{1}{2} (c+d x)\right) \sec ^2(c+d x) \sqrt{e \sin (c+d x)} \left(\frac{3}{4} \sec (c) \left(49 \sin \left(\frac{1}{2} (c-d x)\right)+35 \sin \left(\frac{1}{2} (3 c+d x)\right)-23 \sin \left(\frac{1}{2} (c+3 d x)\right)+5 \sin \left(\frac{1}{2} (5 c+3 d x)\right)\right) \sec ^3\left(\frac{1}{2} (c+d x)\right)+\frac{56 i e^{2 i c} \left(3 \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};e^{2 i (c+d x)}\right)+e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};e^{2 i (c+d x)}\right)\right)}{\left(1+e^{2 i c}\right) \sqrt{1-e^{2 i (c+d x)}}}\right)}{15 a^2 d (\sec (c+d x)+1)^2}","\frac{4 e^3}{5 a^2 d (e \sin (c+d x))^{5/2}}-\frac{2 e^3 \cos ^3(c+d x)}{5 a^2 d (e \sin (c+d x))^{5/2}}-\frac{2 e^3 \cos (c+d x)}{5 a^2 d (e \sin (c+d x))^{5/2}}-\frac{4 e}{a^2 d \sqrt{e \sin (c+d x)}}+\frac{16 e \cos (c+d x)}{5 a^2 d \sqrt{e \sin (c+d x)}}+\frac{28 E\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{e \sin (c+d x)}}{5 a^2 d \sqrt{\sin (c+d x)}}",1,"(4*Cos[(c + d*x)/2]^4*Sec[c + d*x]^2*Sqrt[e*Sin[c + d*x]]*(((56*I)*E^((2*I)*c)*(3*Hypergeometric2F1[-1/4, 1/2, 3/4, E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, E^((2*I)*(c + d*x))]))/((1 + E^((2*I)*c))*Sqrt[1 - E^((2*I)*(c + d*x))]) + (3*Sec[c]*Sec[(c + d*x)/2]^3*(49*Sin[(c - d*x)/2] + 35*Sin[(3*c + d*x)/2] - 23*Sin[(c + 3*d*x)/2] + 5*Sin[(5*c + 3*d*x)/2]))/4))/(15*a^2*d*(1 + Sec[c + d*x])^2)","C",1
131,1,82,190,1.4041712,"\int \frac{1}{(a+a \sec (c+d x))^2 \sqrt{e \sin (c+d x)}} \, dx","Integrate[1/((a + a*Sec[c + d*x])^2*Sqrt[e*Sin[c + d*x]]),x]","-\frac{\csc ^3(c+d x) \left(16 \sin ^4\left(\frac{1}{2} (c+d x)\right) (11 \cos (c+d x)+8)+40 \sin ^{\frac{7}{2}}(c+d x) F\left(\left.\frac{1}{4} (-2 c-2 d x+\pi )\right|2\right)\right)}{42 a^2 d \sqrt{e \sin (c+d x)}}","\frac{4 e^3}{7 a^2 d (e \sin (c+d x))^{7/2}}-\frac{2 e^3 \cos ^3(c+d x)}{7 a^2 d (e \sin (c+d x))^{7/2}}-\frac{2 e^3 \cos (c+d x)}{7 a^2 d (e \sin (c+d x))^{7/2}}-\frac{4 e}{3 a^2 d (e \sin (c+d x))^{3/2}}+\frac{16 e \cos (c+d x)}{21 a^2 d (e \sin (c+d x))^{3/2}}+\frac{20 \sqrt{\sin (c+d x)} F\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{21 a^2 d \sqrt{e \sin (c+d x)}}",1,"-1/42*(Csc[c + d*x]^3*(16*(8 + 11*Cos[c + d*x])*Sin[(c + d*x)/2]^4 + 40*EllipticF[(-2*c + Pi - 2*d*x)/4, 2]*Sin[c + d*x]^(7/2)))/(a^2*d*Sqrt[e*Sin[c + d*x]])","A",1
132,1,163,224,1.4433625,"\int \frac{1}{(a+a \sec (c+d x))^2 (e \sin (c+d x))^{3/2}} \, dx","Integrate[1/((a + a*Sec[c + d*x])^2*(e*Sin[c + d*x])^(3/2)),x]","\frac{\sec ^4\left(\frac{1}{2} (c+d x)\right) (\cos (c+d x)+i \sin (c+d x)) \left(e^{-2 i (c+d x)} \sqrt{1-e^{2 i (c+d x)}} \left(1+e^{i (c+d x)}\right)^4 \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};e^{2 i (c+d x)}\right)+16 i \sin (c+d x)+13 i \sin (2 (c+d x))-40 \cos (c+d x)-19 \cos (2 (c+d x))-31\right)}{180 a^2 d e \sqrt{e \sin (c+d x)}}","\frac{4 e^3}{9 a^2 d (e \sin (c+d x))^{9/2}}-\frac{2 e^3 \cos ^3(c+d x)}{9 a^2 d (e \sin (c+d x))^{9/2}}-\frac{2 e^3 \cos (c+d x)}{9 a^2 d (e \sin (c+d x))^{9/2}}-\frac{4 E\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{e \sin (c+d x)}}{15 a^2 d e^2 \sqrt{\sin (c+d x)}}-\frac{4 e}{5 a^2 d (e \sin (c+d x))^{5/2}}+\frac{16 e \cos (c+d x)}{45 a^2 d (e \sin (c+d x))^{5/2}}-\frac{4 \cos (c+d x)}{15 a^2 d e \sqrt{e \sin (c+d x)}}",1,"(Sec[(c + d*x)/2]^4*(Cos[c + d*x] + I*Sin[c + d*x])*(-31 - 40*Cos[c + d*x] - 19*Cos[2*(c + d*x)] + ((1 + E^(I*(c + d*x)))^4*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)) + (16*I)*Sin[c + d*x] + (13*I)*Sin[2*(c + d*x)]))/(180*a^2*d*e*Sqrt[e*Sin[c + d*x]])","C",1
133,1,113,224,0.9936292,"\int \frac{1}{(a+a \sec (c+d x))^2 (e \sin (c+d x))^{5/2}} \, dx","Integrate[1/((a + a*Sec[c + d*x])^2*(e*Sin[c + d*x])^(5/2)),x]","-\frac{\csc \left(\frac{1}{2} (c+d x)\right) \sec ^5\left(\frac{1}{2} (c+d x)\right) \left(97 \cos (c+d x)+4 \cos (2 (c+d x))+\cos (3 (c+d x))+\sin ^{\frac{11}{2}}(c+d x) \csc ^4\left(\frac{1}{2} (c+d x)\right) F\left(\left.\frac{1}{4} (-2 c-2 d x+\pi )\right|2\right)+52\right)}{1848 a^2 d e^2 \sqrt{e \sin (c+d x)}}","\frac{4 e^3}{11 a^2 d (e \sin (c+d x))^{11/2}}-\frac{2 e^3 \cos ^3(c+d x)}{11 a^2 d (e \sin (c+d x))^{11/2}}-\frac{2 e^3 \cos (c+d x)}{11 a^2 d (e \sin (c+d x))^{11/2}}+\frac{4 \sqrt{\sin (c+d x)} F\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{231 a^2 d e^2 \sqrt{e \sin (c+d x)}}-\frac{4 e}{7 a^2 d (e \sin (c+d x))^{7/2}}+\frac{16 e \cos (c+d x)}{77 a^2 d (e \sin (c+d x))^{7/2}}-\frac{4 \cos (c+d x)}{231 a^2 d e (e \sin (c+d x))^{3/2}}",1,"-1/1848*(Csc[(c + d*x)/2]*Sec[(c + d*x)/2]^5*(52 + 97*Cos[c + d*x] + 4*Cos[2*(c + d*x)] + Cos[3*(c + d*x)] + Csc[(c + d*x)/2]^4*EllipticF[(-2*c + Pi - 2*d*x)/4, 2]*Sin[c + d*x]^(11/2)))/(a^2*d*e^2*Sqrt[e*Sin[c + d*x]])","A",1
134,1,287,247,7.348428,"\int (a+a \sec (c+d x))^3 (e \sin (c+d x))^m \, dx","Integrate[(a + a*Sec[c + d*x])^3*(e*Sin[c + d*x])^m,x]","\frac{a^3 (\cos (c+d x)+1)^3 \tan (c+d x) \sec ^6\left(\frac{1}{2} (c+d x)\right) (e \sin (c+d x))^m \left(3 \sqrt{\cos ^2(c+d x)} \, _2F_1\left(\frac{3}{2},\frac{m+1}{2};\frac{m+3}{2};\sin ^2(c+d x)\right)+3 \cos (c+d x) \, _2F_1\left(1,\frac{m+1}{2};\frac{m+3}{2};\sin ^2(c+d x)\right)+\cos (c+d x) \, _2F_1\left(2,\frac{m+1}{2};\frac{m+3}{2};\sin ^2(c+d x)\right)\right)}{8 d (m+1)}+\frac{a^3 2^{-m-3} e^{i (c+d x)} \left(-i e^{-i (c+d x)} \left(-1+e^{2 i (c+d x)}\right)\right)^{m+1} (\cos (c+d x)+1)^3 \sec ^6\left(\frac{1}{2} (c+d x)\right) \, _2F_1\left(1,\frac{m+2}{2};1-\frac{m}{2};e^{2 i (c+d x)}\right) \sin ^{-m}(c+d x) (e \sin (c+d x))^m}{d m}","\frac{3 a^3 (e \sin (c+d x))^{m+1} \, _2F_1\left(1,\frac{m+1}{2};\frac{m+3}{2};\sin ^2(c+d x)\right)}{d e (m+1)}+\frac{a^3 (e \sin (c+d x))^{m+1} \, _2F_1\left(2,\frac{m+1}{2};\frac{m+3}{2};\sin ^2(c+d x)\right)}{d e (m+1)}+\frac{a^3 \cos (c+d x) (e \sin (c+d x))^{m+1} \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};\sin ^2(c+d x)\right)}{d e (m+1) \sqrt{\cos ^2(c+d x)}}+\frac{3 a^3 \sqrt{\cos ^2(c+d x)} \sec (c+d x) (e \sin (c+d x))^{m+1} \, _2F_1\left(\frac{3}{2},\frac{m+1}{2};\frac{m+3}{2};\sin ^2(c+d x)\right)}{d e (m+1)}",1,"(2^(-3 - m)*a^3*E^(I*(c + d*x))*(((-I)*(-1 + E^((2*I)*(c + d*x))))/E^(I*(c + d*x)))^(1 + m)*(1 + Cos[c + d*x])^3*Hypergeometric2F1[1, (2 + m)/2, 1 - m/2, E^((2*I)*(c + d*x))]*Sec[(c + d*x)/2]^6*(e*Sin[c + d*x])^m)/(d*m*Sin[c + d*x]^m) + (a^3*(1 + Cos[c + d*x])^3*(3*Cos[c + d*x]*Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, Sin[c + d*x]^2] + 3*Sqrt[Cos[c + d*x]^2]*Hypergeometric2F1[3/2, (1 + m)/2, (3 + m)/2, Sin[c + d*x]^2] + Cos[c + d*x]*Hypergeometric2F1[2, (1 + m)/2, (3 + m)/2, Sin[c + d*x]^2])*Sec[(c + d*x)/2]^6*(e*Sin[c + d*x])^m*Tan[c + d*x])/(8*d*(1 + m))","C",0
135,1,230,195,4.6109169,"\int (a+a \sec (c+d x))^2 (e \sin (c+d x))^m \, dx","Integrate[(a + a*Sec[c + d*x])^2*(e*Sin[c + d*x])^m,x]","\frac{a^2 \tan (c+d x) (e \sin (c+d x))^m \left(\sqrt{\cos ^2(c+d x)} \, _2F_1\left(\frac{3}{2},\frac{m+1}{2};\frac{m+3}{2};\sin ^2(c+d x)\right)+2 \cos (c+d x) \, _2F_1\left(1,\frac{m+1}{2};\frac{m+3}{2};\sin ^2(c+d x)\right)\right)}{d (m+1)}+\frac{a^2 2^{-m-2} e^{i (c+d x)} \left(-i e^{-i (c+d x)} \left(-1+e^{2 i (c+d x)}\right)\right)^{m+1} (\cos (c+d x)+1)^2 \sec ^4\left(\frac{1}{2} (c+d x)\right) \, _2F_1\left(1,\frac{m+2}{2};1-\frac{m}{2};e^{2 i (c+d x)}\right) \sin ^{-m}(c+d x) (e \sin (c+d x))^m}{d m}","\frac{2 a^2 (e \sin (c+d x))^{m+1} \, _2F_1\left(1,\frac{m+1}{2};\frac{m+3}{2};\sin ^2(c+d x)\right)}{d e (m+1)}+\frac{a^2 \cos (c+d x) (e \sin (c+d x))^{m+1} \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};\sin ^2(c+d x)\right)}{d e (m+1) \sqrt{\cos ^2(c+d x)}}+\frac{a^2 \sqrt{\cos ^2(c+d x)} \sec (c+d x) (e \sin (c+d x))^{m+1} \, _2F_1\left(\frac{3}{2},\frac{m+1}{2};\frac{m+3}{2};\sin ^2(c+d x)\right)}{d e (m+1)}",1,"(2^(-2 - m)*a^2*E^(I*(c + d*x))*(((-I)*(-1 + E^((2*I)*(c + d*x))))/E^(I*(c + d*x)))^(1 + m)*(1 + Cos[c + d*x])^2*Hypergeometric2F1[1, (2 + m)/2, 1 - m/2, E^((2*I)*(c + d*x))]*Sec[(c + d*x)/2]^4*(e*Sin[c + d*x])^m)/(d*m*Sin[c + d*x]^m) + (a^2*(2*Cos[c + d*x]*Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, Sin[c + d*x]^2] + Sqrt[Cos[c + d*x]^2]*Hypergeometric2F1[3/2, (1 + m)/2, (3 + m)/2, Sin[c + d*x]^2])*(e*Sin[c + d*x])^m*Tan[c + d*x])/(d*(1 + m))","C",0
136,1,97,119,0.1473712,"\int (a+a \sec (c+d x)) (e \sin (c+d x))^m \, dx","Integrate[(a + a*Sec[c + d*x])*(e*Sin[c + d*x])^m,x]","\frac{a (e \sin (c+d x))^m \left(\sin (c+d x) \, _2F_1\left(1,\frac{m+1}{2};\frac{m+3}{2};\sin ^2(c+d x)\right)+\sqrt{\cos ^2(c+d x)} \tan (c+d x) \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};\sin ^2(c+d x)\right)\right)}{d (m+1)}","\frac{a (e \sin (c+d x))^{m+1} \, _2F_1\left(1,\frac{m+1}{2};\frac{m+3}{2};\sin ^2(c+d x)\right)}{d e (m+1)}+\frac{a \cos (c+d x) (e \sin (c+d x))^{m+1} \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};\sin ^2(c+d x)\right)}{d e (m+1) \sqrt{\cos ^2(c+d x)}}",1,"(a*(e*Sin[c + d*x])^m*(Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, Sin[c + d*x]^2]*Sin[c + d*x] + Sqrt[Cos[c + d*x]^2]*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, Sin[c + d*x]^2]*Tan[c + d*x]))/(d*(1 + m))","A",1
137,0,0,100,31.6427608,"\int \frac{(e \sin (c+d x))^m}{a+a \sec (c+d x)} \, dx","Integrate[(e*Sin[c + d*x])^m/(a + a*Sec[c + d*x]),x]","\int \frac{(e \sin (c+d x))^m}{a+a \sec (c+d x)} \, dx","\frac{e \cos (c+d x) (e \sin (c+d x))^{m-1} \, _2F_1\left(-\frac{1}{2},\frac{m-1}{2};\frac{m+1}{2};\sin ^2(c+d x)\right)}{a d (1-m) \sqrt{\cos ^2(c+d x)}}-\frac{e (e \sin (c+d x))^{m-1}}{a d (1-m)}",1,"Integrate[(e*Sin[c + d*x])^m/(a + a*Sec[c + d*x]), x]","F",-1
138,1,2833,207,22.698924,"\int \frac{(e \sin (c+d x))^m}{(a+a \sec (c+d x))^2} \, dx","Integrate[(e*Sin[c + d*x])^m/(a + a*Sec[c + d*x])^2,x]","\text{Result too large to show}","-\frac{e^3 \cos (c+d x) (e \sin (c+d x))^{m-3} \, _2F_1\left(-\frac{3}{2},\frac{m-3}{2};\frac{m-1}{2};\sin ^2(c+d x)\right)}{a^2 d (3-m) \sqrt{\cos ^2(c+d x)}}-\frac{e^3 \cos (c+d x) (e \sin (c+d x))^{m-3} \, _2F_1\left(-\frac{1}{2},\frac{m-3}{2};\frac{m-1}{2};\sin ^2(c+d x)\right)}{a^2 d (3-m) \sqrt{\cos ^2(c+d x)}}+\frac{2 e^3 (e \sin (c+d x))^{m-3}}{a^2 d (3-m)}-\frac{2 e (e \sin (c+d x))^{m-1}}{a^2 d (1-m)}",1,"(2^(2 - m)*E^(I*(c + d*x))*(((-I)*(-1 + E^((2*I)*(c + d*x))))/E^(I*(c + d*x)))^(1 + m)*Cos[c/2 + (d*x)/2]^4*Hypergeometric2F1[1, (2 + m)/2, 1 - m/2, E^((2*I)*(c + d*x))]*Sec[c + d*x]^2*(e*Sin[c + d*x])^m)/(d*m*(a + a*Sec[c + d*x])^2*Sin[c + d*x]^m) + ((AppellF1[(1 + m)/2, 1 - m, 2*m, (3 + m)/2, Tan[(c + d*x)/4]^2, -Tan[(c + d*x)/4]^2] - 6*AppellF1[(1 + m)/2, 2 - m, 2*m, (3 + m)/2, Tan[(c + d*x)/4]^2, -Tan[(c + d*x)/4]^2] + 12*AppellF1[(1 + m)/2, 3 - m, 2*m, (3 + m)/2, Tan[(c + d*x)/4]^2, -Tan[(c + d*x)/4]^2] - 8*AppellF1[(1 + m)/2, 4 - m, 2*m, (3 + m)/2, Tan[(c + d*x)/4]^2, -Tan[(c + d*x)/4]^2])*(Sec[(c + d*x)/4]^2)^(2*m)*Sec[(c + d*x)/2]^4*Sin[c + d*x]^m*(e*Sin[c + d*x])^m*Tan[(c + d*x)/4])/(4*a^2*d*(1 + m)*(1 - Tan[(c + d*x)/4]^2)^m*((m*(AppellF1[(1 + m)/2, 1 - m, 2*m, (3 + m)/2, Tan[(c + d*x)/4]^2, -Tan[(c + d*x)/4]^2] - 6*AppellF1[(1 + m)/2, 2 - m, 2*m, (3 + m)/2, Tan[(c + d*x)/4]^2, -Tan[(c + d*x)/4]^2] + 12*AppellF1[(1 + m)/2, 3 - m, 2*m, (3 + m)/2, Tan[(c + d*x)/4]^2, -Tan[(c + d*x)/4]^2] - 8*AppellF1[(1 + m)/2, 4 - m, 2*m, (3 + m)/2, Tan[(c + d*x)/4]^2, -Tan[(c + d*x)/4]^2])*(Sec[(c + d*x)/4]^2)^(1 + 2*m)*Sin[c + d*x]^m*Tan[(c + d*x)/4]^2*(1 - Tan[(c + d*x)/4]^2)^(-1 - m))/(2*(1 + m)) + ((AppellF1[(1 + m)/2, 1 - m, 2*m, (3 + m)/2, Tan[(c + d*x)/4]^2, -Tan[(c + d*x)/4]^2] - 6*AppellF1[(1 + m)/2, 2 - m, 2*m, (3 + m)/2, Tan[(c + d*x)/4]^2, -Tan[(c + d*x)/4]^2] + 12*AppellF1[(1 + m)/2, 3 - m, 2*m, (3 + m)/2, Tan[(c + d*x)/4]^2, -Tan[(c + d*x)/4]^2] - 8*AppellF1[(1 + m)/2, 4 - m, 2*m, (3 + m)/2, Tan[(c + d*x)/4]^2, -Tan[(c + d*x)/4]^2])*(Sec[(c + d*x)/4]^2)^(1 + 2*m)*Sin[c + d*x]^m)/(4*(1 + m)*(1 - Tan[(c + d*x)/4]^2)^m) + (m*(AppellF1[(1 + m)/2, 1 - m, 2*m, (3 + m)/2, Tan[(c + d*x)/4]^2, -Tan[(c + d*x)/4]^2] - 6*AppellF1[(1 + m)/2, 2 - m, 2*m, (3 + m)/2, Tan[(c + d*x)/4]^2, -Tan[(c + d*x)/4]^2] + 12*AppellF1[(1 + m)/2, 3 - m, 2*m, (3 + m)/2, Tan[(c + d*x)/4]^2, -Tan[(c + d*x)/4]^2] - 8*AppellF1[(1 + m)/2, 4 - m, 2*m, (3 + m)/2, Tan[(c + d*x)/4]^2, -Tan[(c + d*x)/4]^2])*Cos[c + d*x]*(Sec[(c + d*x)/4]^2)^(2*m)*Sin[c + d*x]^(-1 + m)*Tan[(c + d*x)/4])/((1 + m)*(1 - Tan[(c + d*x)/4]^2)^m) + (m*(AppellF1[(1 + m)/2, 1 - m, 2*m, (3 + m)/2, Tan[(c + d*x)/4]^2, -Tan[(c + d*x)/4]^2] - 6*AppellF1[(1 + m)/2, 2 - m, 2*m, (3 + m)/2, Tan[(c + d*x)/4]^2, -Tan[(c + d*x)/4]^2] + 12*AppellF1[(1 + m)/2, 3 - m, 2*m, (3 + m)/2, Tan[(c + d*x)/4]^2, -Tan[(c + d*x)/4]^2] - 8*AppellF1[(1 + m)/2, 4 - m, 2*m, (3 + m)/2, Tan[(c + d*x)/4]^2, -Tan[(c + d*x)/4]^2])*(Sec[(c + d*x)/4]^2)^(2*m)*Sin[c + d*x]^m*Tan[(c + d*x)/4]^2)/((1 + m)*(1 - Tan[(c + d*x)/4]^2)^m) + ((Sec[(c + d*x)/4]^2)^(2*m)*Sin[c + d*x]^m*Tan[(c + d*x)/4]*(-((m*(1 + m)*AppellF1[1 + (1 + m)/2, 1 - m, 1 + 2*m, 1 + (3 + m)/2, Tan[(c + d*x)/4]^2, -Tan[(c + d*x)/4]^2]*Sec[(c + d*x)/4]^2*Tan[(c + d*x)/4])/(3 + m)) + ((1 - m)*(1 + m)*AppellF1[1 + (1 + m)/2, 2 - m, 2*m, 1 + (3 + m)/2, Tan[(c + d*x)/4]^2, -Tan[(c + d*x)/4]^2]*Sec[(c + d*x)/4]^2*Tan[(c + d*x)/4])/(2*(3 + m)) - 6*(-((m*(1 + m)*AppellF1[1 + (1 + m)/2, 2 - m, 1 + 2*m, 1 + (3 + m)/2, Tan[(c + d*x)/4]^2, -Tan[(c + d*x)/4]^2]*Sec[(c + d*x)/4]^2*Tan[(c + d*x)/4])/(3 + m)) + ((2 - m)*(1 + m)*AppellF1[1 + (1 + m)/2, 3 - m, 2*m, 1 + (3 + m)/2, Tan[(c + d*x)/4]^2, -Tan[(c + d*x)/4]^2]*Sec[(c + d*x)/4]^2*Tan[(c + d*x)/4])/(2*(3 + m))) + 12*(-((m*(1 + m)*AppellF1[1 + (1 + m)/2, 3 - m, 1 + 2*m, 1 + (3 + m)/2, Tan[(c + d*x)/4]^2, -Tan[(c + d*x)/4]^2]*Sec[(c + d*x)/4]^2*Tan[(c + d*x)/4])/(3 + m)) + ((3 - m)*(1 + m)*AppellF1[1 + (1 + m)/2, 4 - m, 2*m, 1 + (3 + m)/2, Tan[(c + d*x)/4]^2, -Tan[(c + d*x)/4]^2]*Sec[(c + d*x)/4]^2*Tan[(c + d*x)/4])/(2*(3 + m))) - 8*(-((m*(1 + m)*AppellF1[1 + (1 + m)/2, 4 - m, 1 + 2*m, 1 + (3 + m)/2, Tan[(c + d*x)/4]^2, -Tan[(c + d*x)/4]^2]*Sec[(c + d*x)/4]^2*Tan[(c + d*x)/4])/(3 + m)) + ((4 - m)*(1 + m)*AppellF1[1 + (1 + m)/2, 5 - m, 2*m, 1 + (3 + m)/2, Tan[(c + d*x)/4]^2, -Tan[(c + d*x)/4]^2]*Sec[(c + d*x)/4]^2*Tan[(c + d*x)/4])/(2*(3 + m)))))/((1 + m)*(1 - Tan[(c + d*x)/4]^2)^m))) - (2*(3 + m)*(AppellF1[(1 + m)/2, 1 - m, 2*m, (3 + m)/2, Tan[(c + d*x)/4]^2, -Tan[(c + d*x)/4]^2] - 2*AppellF1[(1 + m)/2, 2 - m, 2*m, (3 + m)/2, Tan[(c + d*x)/4]^2, -Tan[(c + d*x)/4]^2])*Sec[(c + d*x)/2]*(e*Sin[c + d*x])^m*Tan[(c + d*x)/2])/(a^2*d*(1 + m)*((3 + m)*AppellF1[(1 + m)/2, 1 - m, 2*m, (3 + m)/2, Tan[(c + d*x)/4]^2, -Tan[(c + d*x)/4]^2] - 2*((3 + m)*AppellF1[(1 + m)/2, 2 - m, 2*m, (3 + m)/2, Tan[(c + d*x)/4]^2, -Tan[(c + d*x)/4]^2] + (2*m*AppellF1[(3 + m)/2, 1 - m, 1 + 2*m, (5 + m)/2, Tan[(c + d*x)/4]^2, -Tan[(c + d*x)/4]^2] + (-1 + m)*AppellF1[(3 + m)/2, 2 - m, 2*m, (5 + m)/2, Tan[(c + d*x)/4]^2, -Tan[(c + d*x)/4]^2] - 4*m*AppellF1[(3 + m)/2, 2 - m, 1 + 2*m, (5 + m)/2, Tan[(c + d*x)/4]^2, -Tan[(c + d*x)/4]^2] + 4*AppellF1[(3 + m)/2, 3 - m, 2*m, (5 + m)/2, Tan[(c + d*x)/4]^2, -Tan[(c + d*x)/4]^2] - 2*m*AppellF1[(3 + m)/2, 3 - m, 2*m, (5 + m)/2, Tan[(c + d*x)/4]^2, -Tan[(c + d*x)/4]^2])*Tan[(c + d*x)/4]^2)))","C",0
139,0,0,236,10.7417596,"\int \frac{(e \sin (c+d x))^m}{(a+a \sec (c+d x))^3} \, dx","Integrate[(e*Sin[c + d*x])^m/(a + a*Sec[c + d*x])^3,x]","\int \frac{(e \sin (c+d x))^m}{(a+a \sec (c+d x))^3} \, dx","\frac{e^5 \cos (c+d x) (e \sin (c+d x))^{m-5} \, _2F_1\left(-\frac{5}{2},\frac{m-5}{2};\frac{m-3}{2};\sin ^2(c+d x)\right)}{a^3 d (5-m) \sqrt{\cos ^2(c+d x)}}+\frac{3 e^5 \cos (c+d x) (e \sin (c+d x))^{m-5} \, _2F_1\left(-\frac{3}{2},\frac{m-5}{2};\frac{m-3}{2};\sin ^2(c+d x)\right)}{a^3 d (5-m) \sqrt{\cos ^2(c+d x)}}-\frac{4 e^5 (e \sin (c+d x))^{m-5}}{a^3 d (5-m)}+\frac{7 e^3 (e \sin (c+d x))^{m-3}}{a^3 d (3-m)}-\frac{3 e (e \sin (c+d x))^{m-1}}{a^3 d (1-m)}",1,"Integrate[(e*Sin[c + d*x])^m/(a + a*Sec[c + d*x])^3, x]","F",-1
140,1,1243,106,9.7459869,"\int (a+a \sec (c+d x))^{3/2} (e \sin (c+d x))^m \, dx","Integrate[(a + a*Sec[c + d*x])^(3/2)*(e*Sin[c + d*x])^m,x]","\frac{4 (m+3) \left(F_1\left(\frac{m+1}{2};-\frac{1}{2},m+1;\frac{m+3}{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{m+1}{2};\frac{1}{2},m;\frac{m+3}{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{m+1}{2};\frac{3}{2},m;\frac{m+3}{2};\tan ^2\left(\frac{1}{2} (c+d x)\right),-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right)\right) \cos ^3\left(\frac{1}{2} (c+d x)\right) (a (\sec (c+d x)+1))^{3/2} \sin \left(\frac{1}{2} (c+d x)\right) (e \sin (c+d x))^m}{d (m+1) \left(2 m F_1\left(\frac{m+1}{2};\frac{3}{2},m;\frac{m+3}{2};\tan ^2\left(\frac{1}{2} (c+d x)\right),-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right)+2 m \cos (c+d x) F_1\left(\frac{m+1}{2};\frac{3}{2},m;\frac{m+3}{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) F_1\left(\frac{m+1}{2};\frac{3}{2},m;\frac{m+3}{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{m+1}{2};\frac{3}{2},m;\frac{m+3}{2};\tan ^2\left(\frac{1}{2} (c+d x)\right),-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right)-2 m F_1\left(\frac{m+3}{2};-\frac{1}{2},m+2;\frac{m+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{m+3}{2};-\frac{1}{2},m+2;\frac{m+5}{2};\tan ^2\left(\frac{1}{2} (c+d x)\right),-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right)-2 m F_1\left(\frac{m+3}{2};\frac{1}{2},m+1;\frac{m+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{m+3}{2};\frac{1}{2},m+1;\frac{m+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{m+3}{2};\frac{3}{2},m;\frac{m+5}{2};\tan ^2\left(\frac{1}{2} (c+d x)\right),-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right)-4 m F_1\left(\frac{m+3}{2};\frac{3}{2},m+1;\frac{m+5}{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{m+3}{2};\frac{5}{2},m;\frac{m+5}{2};\tan ^2\left(\frac{1}{2} (c+d x)\right),-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right)+2 m F_1\left(\frac{m+3}{2};-\frac{1}{2},m+2;\frac{m+5}{2};\tan ^2\left(\frac{1}{2} (c+d x)\right),-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right) \cos (c+d x)+2 F_1\left(\frac{m+3}{2};-\frac{1}{2},m+2;\frac{m+5}{2};\tan ^2\left(\frac{1}{2} (c+d x)\right),-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right) \cos (c+d x)+2 m F_1\left(\frac{m+3}{2};\frac{1}{2},m+1;\frac{m+5}{2};\tan ^2\left(\frac{1}{2} (c+d x)\right),-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right) \cos (c+d x)+F_1\left(\frac{m+3}{2};\frac{1}{2},m+1;\frac{m+5}{2};\tan ^2\left(\frac{1}{2} (c+d x)\right),-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right) \cos (c+d x)-F_1\left(\frac{m+3}{2};\frac{3}{2},m;\frac{m+5}{2};\tan ^2\left(\frac{1}{2} (c+d x)\right),-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right) \cos (c+d x)+4 m F_1\left(\frac{m+3}{2};\frac{3}{2},m+1;\frac{m+5}{2};\tan ^2\left(\frac{1}{2} (c+d x)\right),-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right) \cos (c+d x)-6 F_1\left(\frac{m+3}{2};\frac{5}{2},m;\frac{m+5}{2};\tan ^2\left(\frac{1}{2} (c+d x)\right),-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right) \cos (c+d x)+(m+3) F_1\left(\frac{m+1}{2};-\frac{1}{2},m+1;\frac{m+3}{2};\tan ^2\left(\frac{1}{2} (c+d x)\right),-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right) (\cos (c+d x)+1)+(m+3) F_1\left(\frac{m+1}{2};\frac{1}{2},m;\frac{m+3}{2};\tan ^2\left(\frac{1}{2} (c+d x)\right),-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right) (\cos (c+d x)+1)\right)}","\frac{2 a e \sqrt{a \sec (c+d x)+a} (1-\cos (c+d x))^{\frac{1-m}{2}} (\cos (c+d x)+1)^{-m/2} F_1\left(-\frac{1}{2};\frac{1-m}{2},\frac{1}{2} (-m-2);\frac{1}{2};\cos (c+d x),-\cos (c+d x)\right) (e \sin (c+d x))^{m-1}}{d}",1,"(4*(3 + m)*(AppellF1[(1 + m)/2, -1/2, 1 + m, (3 + m)/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + AppellF1[(1 + m)/2, 1/2, m, (3 + m)/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + 2*AppellF1[(1 + m)/2, 3/2, m, (3 + m)/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Cos[(c + d*x)/2]^3*(a*(1 + Sec[c + d*x]))^(3/2)*Sin[(c + d*x)/2]*(e*Sin[c + d*x])^m)/(d*(1 + m)*(6*AppellF1[(1 + m)/2, 3/2, m, (3 + m)/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + 2*m*AppellF1[(1 + m)/2, 3/2, m, (3 + m)/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] - 2*AppellF1[(3 + m)/2, -1/2, 2 + m, (5 + m)/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] - 2*m*AppellF1[(3 + m)/2, -1/2, 2 + m, (5 + m)/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] - AppellF1[(3 + m)/2, 1/2, 1 + m, (5 + m)/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] - 2*m*AppellF1[(3 + m)/2, 1/2, 1 + m, (5 + m)/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + AppellF1[(3 + m)/2, 3/2, m, (5 + m)/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] - 4*m*AppellF1[(3 + m)/2, 3/2, 1 + m, (5 + m)/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + 6*AppellF1[(3 + m)/2, 5/2, m, (5 + m)/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + 6*AppellF1[(1 + m)/2, 3/2, m, (3 + m)/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Cos[c + d*x] + 2*m*AppellF1[(1 + m)/2, 3/2, m, (3 + m)/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Cos[c + d*x] + 2*AppellF1[(3 + m)/2, -1/2, 2 + m, (5 + m)/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Cos[c + d*x] + 2*m*AppellF1[(3 + m)/2, -1/2, 2 + m, (5 + m)/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Cos[c + d*x] + AppellF1[(3 + m)/2, 1/2, 1 + m, (5 + m)/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Cos[c + d*x] + 2*m*AppellF1[(3 + m)/2, 1/2, 1 + m, (5 + m)/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Cos[c + d*x] - AppellF1[(3 + m)/2, 3/2, m, (5 + m)/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Cos[c + d*x] + 4*m*AppellF1[(3 + m)/2, 3/2, 1 + m, (5 + m)/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Cos[c + d*x] - 6*AppellF1[(3 + m)/2, 5/2, m, (5 + m)/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Cos[c + d*x] + (3 + m)*AppellF1[(1 + m)/2, -1/2, 1 + m, (3 + m)/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*(1 + Cos[c + d*x]) + (3 + m)*AppellF1[(1 + m)/2, 1/2, m, (3 + m)/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*(1 + Cos[c + d*x])))","B",0
141,1,433,107,2.899798,"\int \sqrt{a+a \sec (c+d x)} (e \sin (c+d x))^m \, dx","Integrate[Sqrt[a + a*Sec[c + d*x]]*(e*Sin[c + d*x])^m,x]","\frac{4 (m+3) \sin \left(\frac{1}{2} (c+d x)\right) \cos ^3\left(\frac{1}{2} (c+d x)\right) \sqrt{a (\sec (c+d x)+1)} \left(F_1\left(\frac{m+1}{2};-\frac{1}{2},m+1;\frac{m+3}{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{m+1}{2};\frac{1}{2},m;\frac{m+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 \sin (c+d x))^m}{d (m+1) \left((\cos (c+d x)-1) \left(2 (m+1) F_1\left(\frac{m+3}{2};-\frac{1}{2},m+2;\frac{m+5}{2};\tan ^2\left(\frac{1}{2} (c+d x)\right),-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right)+(2 m+1) F_1\left(\frac{m+3}{2};\frac{1}{2},m+1;\frac{m+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{m+3}{2};\frac{3}{2},m;\frac{m+5}{2};\tan ^2\left(\frac{1}{2} (c+d x)\right),-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right)\right)+(m+3) (\cos (c+d x)+1) F_1\left(\frac{m+1}{2};-\frac{1}{2},m+1;\frac{m+3}{2};\tan ^2\left(\frac{1}{2} (c+d x)\right),-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right)+(m+3) (\cos (c+d x)+1) F_1\left(\frac{m+1}{2};\frac{1}{2},m;\frac{m+3}{2};\tan ^2\left(\frac{1}{2} (c+d x)\right),-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right)\right)}","-\frac{2 e \cos (c+d x) \sqrt{a \sec (c+d x)+a} (1-\cos (c+d x))^{\frac{1-m}{2}} (\cos (c+d x)+1)^{-m/2} F_1\left(\frac{1}{2};\frac{1-m}{2},-\frac{m}{2};\frac{3}{2};\cos (c+d x),-\cos (c+d x)\right) (e \sin (c+d x))^{m-1}}{d}",1,"(4*(3 + m)*(AppellF1[(1 + m)/2, -1/2, 1 + m, (3 + m)/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + AppellF1[(1 + m)/2, 1/2, m, (3 + m)/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Cos[(c + d*x)/2]^3*Sqrt[a*(1 + Sec[c + d*x])]*Sin[(c + d*x)/2]*(e*Sin[c + d*x])^m)/(d*(1 + m)*((2*(1 + m)*AppellF1[(3 + m)/2, -1/2, 2 + m, (5 + m)/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + (1 + 2*m)*AppellF1[(3 + m)/2, 1/2, 1 + m, (5 + m)/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] - AppellF1[(3 + m)/2, 3/2, m, (5 + m)/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*(-1 + Cos[c + d*x]) + (3 + m)*AppellF1[(1 + m)/2, -1/2, 1 + m, (3 + m)/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*(1 + Cos[c + d*x]) + (3 + m)*AppellF1[(1 + m)/2, 1/2, m, (3 + m)/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*(1 + Cos[c + d*x])))","B",0
142,1,277,115,2.0662525,"\int \frac{(e \sin (c+d x))^m}{\sqrt{a+a \sec (c+d x)}} \, dx","Integrate[(e*Sin[c + d*x])^m/Sqrt[a + a*Sec[c + d*x]],x]","\frac{4 (m+3) \sin \left(\frac{1}{2} (c+d x)\right) \cos ^3\left(\frac{1}{2} (c+d x)\right) F_1\left(\frac{m+1}{2};-\frac{1}{2},m+1;\frac{m+3}{2};\tan ^2\left(\frac{1}{2} (c+d x)\right),-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right) (e \sin (c+d x))^m}{d (m+1) \sqrt{a (\sec (c+d x)+1)} \left((\cos (c+d x)-1) \left(2 (m+1) F_1\left(\frac{m+3}{2};-\frac{1}{2},m+2;\frac{m+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{m+3}{2};\frac{1}{2},m+1;\frac{m+5}{2};\tan ^2\left(\frac{1}{2} (c+d x)\right),-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right)\right)+(m+3) (\cos (c+d x)+1) F_1\left(\frac{m+1}{2};-\frac{1}{2},m+1;\frac{m+3}{2};\tan ^2\left(\frac{1}{2} (c+d x)\right),-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right)\right)}","-\frac{2 e \cos (c+d x) (1-\cos (c+d x))^{\frac{1-m}{2}} (\cos (c+d x)+1)^{1-\frac{m}{2}} F_1\left(\frac{3}{2};\frac{1-m}{2},\frac{2-m}{2};\frac{5}{2};\cos (c+d x),-\cos (c+d x)\right) (e \sin (c+d x))^{m-1}}{3 d \sqrt{a \sec (c+d x)+a}}",1,"(4*(3 + m)*AppellF1[(1 + m)/2, -1/2, 1 + m, (3 + m)/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Cos[(c + d*x)/2]^3*Sin[(c + d*x)/2]*(e*Sin[c + d*x])^m)/(d*(1 + m)*((2*(1 + m)*AppellF1[(3 + m)/2, -1/2, 2 + m, (5 + m)/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + AppellF1[(3 + m)/2, 1/2, 1 + m, (5 + m)/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*(-1 + Cos[c + d*x]) + (3 + m)*AppellF1[(1 + m)/2, -1/2, 1 + m, (3 + m)/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*(1 + Cos[c + d*x]))*Sqrt[a*(1 + Sec[c + d*x])])","B",0
143,1,484,120,3.0091164,"\int \frac{(e \sin (c+d x))^m}{(a+a \sec (c+d x))^{3/2}} \, dx","Integrate[(e*Sin[c + d*x])^m/(a + a*Sec[c + d*x])^(3/2),x]","\frac{4 (m+3) \sin \left(\frac{1}{2} (c+d x)\right) \cos ^3\left(\frac{1}{2} (c+d x)\right) \left(F_1\left(\frac{m+1}{2};-\frac{1}{2},m;\frac{m+3}{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{m+1}{2};-\frac{1}{2},m+1;\frac{m+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 \sin (c+d x))^m}{d (m+1) (a (\sec (c+d x)+1))^{3/2} \left(-4 (m+3) \cos ^2\left(\frac{1}{2} (c+d x)\right) F_1\left(\frac{m+1}{2};-\frac{1}{2},m+1;\frac{m+3}{2};\tan ^2\left(\frac{1}{2} (c+d x)\right),-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right)+(\cos (c+d x)-1) \left(2 m F_1\left(\frac{m+3}{2};-\frac{1}{2},m+1;\frac{m+5}{2};\tan ^2\left(\frac{1}{2} (c+d x)\right),-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right)-4 (m+1) F_1\left(\frac{m+3}{2};-\frac{1}{2},m+2;\frac{m+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{m+3}{2};\frac{1}{2},m;\frac{m+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{m+3}{2};\frac{1}{2},m+1;\frac{m+5}{2};\tan ^2\left(\frac{1}{2} (c+d x)\right),-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right)\right)+(m+3) (\cos (c+d x)+1) F_1\left(\frac{m+1}{2};-\frac{1}{2},m;\frac{m+3}{2};\tan ^2\left(\frac{1}{2} (c+d x)\right),-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right)\right)}","-\frac{2 e \cos ^2(c+d x) (1-\cos (c+d x))^{\frac{1-m}{2}} (\cos (c+d x)+1)^{1-\frac{m}{2}} F_1\left(\frac{5}{2};\frac{1-m}{2},\frac{4-m}{2};\frac{7}{2};\cos (c+d x),-\cos (c+d x)\right) (e \sin (c+d x))^{m-1}}{5 a d \sqrt{a \sec (c+d x)+a}}",1,"(4*(3 + m)*(AppellF1[(1 + m)/2, -1/2, m, (3 + m)/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] - 2*AppellF1[(1 + m)/2, -1/2, 1 + m, (3 + m)/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Cos[(c + d*x)/2]^3*Sin[(c + d*x)/2]*(e*Sin[c + d*x])^m)/(d*(1 + m)*(-4*(3 + m)*AppellF1[(1 + m)/2, -1/2, 1 + m, (3 + m)/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Cos[(c + d*x)/2]^2 + (2*m*AppellF1[(3 + m)/2, -1/2, 1 + m, (5 + m)/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] - 4*(1 + m)*AppellF1[(3 + m)/2, -1/2, 2 + m, (5 + m)/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + AppellF1[(3 + m)/2, 1/2, m, (5 + m)/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] - 2*AppellF1[(3 + m)/2, 1/2, 1 + m, (5 + m)/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*(-1 + Cos[c + d*x]) + (3 + m)*AppellF1[(1 + m)/2, -1/2, m, (3 + m)/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*(1 + Cos[c + d*x]))*(a*(1 + Sec[c + d*x]))^(3/2))","B",0
144,1,276,130,1.924084,"\int (a+a \sec (c+d x))^n (e \sin (c+d x))^m \, dx","Integrate[(a + a*Sec[c + d*x])^n*(e*Sin[c + d*x])^m,x]","\frac{4 (m+3) \sin \left(\frac{1}{2} (c+d x)\right) \cos ^3\left(\frac{1}{2} (c+d x)\right) (a (\sec (c+d x)+1))^n (e \sin (c+d x))^m F_1\left(\frac{m+1}{2};n,m+1;\frac{m+3}{2};\tan ^2\left(\frac{1}{2} (c+d x)\right),-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right)}{d (m+1) \left((m+3) (\cos (c+d x)+1) F_1\left(\frac{m+1}{2};n,m+1;\frac{m+3}{2};\tan ^2\left(\frac{1}{2} (c+d x)\right),-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right)-4 \sin ^2\left(\frac{1}{2} (c+d x)\right) \left((m+1) F_1\left(\frac{m+3}{2};n,m+2;\frac{m+5}{2};\tan ^2\left(\frac{1}{2} (c+d x)\right),-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right)-n F_1\left(\frac{m+3}{2};n+1,m+1;\frac{m+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{e \cos (c+d x) (1-\cos (c+d x))^{\frac{1-m}{2}} (a \sec (c+d x)+a)^n (e \sin (c+d x))^{m-1} (\cos (c+d x)+1)^{\frac{1}{2} (-m-2 n+1)} F_1\left(1-n;\frac{1-m}{2},\frac{1}{2} (-m-2 n+1);2-n;\cos (c+d x),-\cos (c+d x)\right)}{d (1-n)}",1,"(4*(3 + m)*AppellF1[(1 + m)/2, n, 1 + m, (3 + m)/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Cos[(c + d*x)/2]^3*(a*(1 + Sec[c + d*x]))^n*Sin[(c + d*x)/2]*(e*Sin[c + d*x])^m)/(d*(1 + m)*((3 + m)*AppellF1[(1 + m)/2, n, 1 + m, (3 + m)/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*(1 + Cos[c + d*x]) - 4*((1 + m)*AppellF1[(3 + m)/2, n, 2 + m, (5 + m)/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] - n*AppellF1[(3 + m)/2, 1 + n, 1 + m, (5 + m)/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Sin[(c + d*x)/2]^2))","B",0
145,1,113,180,1.5131691,"\int (a+a \sec (c+d x))^n \sin ^7(c+d x) \, dx","Integrate[(a + a*Sec[c + d*x])^n*Sin[c + d*x]^7,x]","\frac{(\sec (c+d x)+1)^4 (a (\sec (c+d x)+1))^n \left((n+4) \cos ^5(c+d x) \left(\left(n^2-25 n+24\right) \cos (c+d x)+6 (n-1) \cos ^2(c+d x)+42\right)-\left(n^3-27 n^2+200 n-384\right) \, _2F_1(6,n+4;n+5;\sec (c+d x)+1)\right)}{42 d (n-1) (n+4)}","-\frac{(3-n) (8-n) (16-n) (a \sec (c+d x)+a)^{n+4} \, _2F_1(6,n+4;n+5;\sec (c+d x)+1)}{42 a^4 d (1-n) (n+4)}+\frac{\cos ^7(c+d x) \left(6 (8-n)-\left(n^2-25 n+108\right) \sec (c+d x)\right) (a \sec (c+d x)+a)^{n+4}}{42 a^4 d (1-n)}-\frac{\cos ^7(c+d x) (1-\sec (c+d x))^2 (a \sec (c+d x)+a)^{n+4}}{a^4 d (1-n)}",1,"(((4 + n)*Cos[c + d*x]^5*(42 + (24 - 25*n + n^2)*Cos[c + d*x] + 6*(-1 + n)*Cos[c + d*x]^2) - (-384 + 200*n - 27*n^2 + n^3)*Hypergeometric2F1[6, 4 + n, 5 + n, 1 + Sec[c + d*x]])*(1 + Sec[c + d*x])^4*(a*(1 + Sec[c + d*x]))^n)/(42*d*(-1 + n)*(4 + n))","A",1
146,1,84,123,0.5199535,"\int (a+a \sec (c+d x))^n \sin ^5(c+d x) \, dx","Integrate[(a + a*Sec[c + d*x])^n*Sin[c + d*x]^5,x]","-\frac{(\sec (c+d x)+1)^3 (a (\sec (c+d x)+1))^n \left((n+3) \cos ^4(c+d x) (4 \cos (c+d x)+n-12)-\left(n^2-13 n+32\right) \, _2F_1(4,n+3;n+4;\sec (c+d x)+1)\right)}{20 d (n+3)}","\frac{\left(n^2-13 n+32\right) (a \sec (c+d x)+a)^{n+3} \, _2F_1(4,n+3;n+4;\sec (c+d x)+1)}{20 a^3 d (n+3)}-\frac{\cos ^5(c+d x) (a \sec (c+d x)+a)^{n+3}}{5 a^3 d}+\frac{(12-n) \cos ^4(c+d x) (a \sec (c+d x)+a)^{n+3}}{20 a^3 d}",1,"-1/20*(((3 + n)*Cos[c + d*x]^4*(-12 + n + 4*Cos[c + d*x]) - (32 - 13*n + n^2)*Hypergeometric2F1[4, 3 + n, 4 + n, 1 + Sec[c + d*x]])*(1 + Sec[c + d*x])^3*(a*(1 + Sec[c + d*x]))^n)/(d*(3 + n))","A",1
147,1,67,83,0.1553727,"\int (a+a \sec (c+d x))^n \sin ^3(c+d x) \, dx","Integrate[(a + a*Sec[c + d*x])^n*Sin[c + d*x]^3,x]","\frac{(\sec (c+d x)+1)^2 (a (\sec (c+d x)+1))^n \left((n-4) \, _2F_1(3,n+2;n+3;\sec (c+d x)+1)+(n+2) \cos ^3(c+d x)\right)}{3 d (n+2)}","\frac{\cos ^3(c+d x) (a \sec (c+d x)+a)^{n+2}}{3 a^2 d}-\frac{(4-n) (a \sec (c+d x)+a)^{n+2} \, _2F_1(3,n+2;n+3;\sec (c+d x)+1)}{3 a^2 d (n+2)}",1,"(((2 + n)*Cos[c + d*x]^3 + (-4 + n)*Hypergeometric2F1[3, 2 + n, 3 + n, 1 + Sec[c + d*x]])*(1 + Sec[c + d*x])^2*(a*(1 + Sec[c + d*x]))^n)/(3*d*(2 + n))","A",1
148,1,42,42,0.0415463,"\int (a+a \sec (c+d x))^n \sin (c+d x) \, dx","Integrate[(a + a*Sec[c + d*x])^n*Sin[c + d*x],x]","\frac{(a (\sec (c+d x)+1))^{n+1} \, _2F_1(2,n+1;n+2;\sec (c+d x)+1)}{a d (n+1)}","\frac{(a \sec (c+d x)+a)^{n+1} \, _2F_1(2,n+1;n+2;\sec (c+d x)+1)}{a d (n+1)}",1,"(Hypergeometric2F1[2, 1 + n, 2 + n, 1 + Sec[c + d*x]]*(a*(1 + Sec[c + d*x]))^(1 + n))/(a*d*(1 + n))","A",1
149,1,92,40,0.6844518,"\int \csc (c+d x) (a+a \sec (c+d x))^n \, dx","Integrate[Csc[c + d*x]*(a + a*Sec[c + d*x])^n,x]","\frac{2^{n-1} (\sec (c+d x)+1)^{-n} (a (\sec (c+d x)+1))^n \left(\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)\right)^{n-1} \, _2F_1\left(1,1-n;2-n;\cos (c+d x) \sec ^2\left(\frac{1}{2} (c+d x)\right)\right)}{d (n-1)}","-\frac{(a \sec (c+d x)+a)^n \, _2F_1\left(1,n;n+1;\frac{1}{2} (\sec (c+d x)+1)\right)}{2 d n}",1,"(2^(-1 + n)*Hypergeometric2F1[1, 1 - n, 2 - n, Cos[c + d*x]*Sec[(c + d*x)/2]^2]*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^(-1 + n)*(a*(1 + Sec[c + d*x]))^n)/(d*(-1 + n)*(1 + Sec[c + d*x])^n)","B",0
150,1,179,112,2.1732995,"\int \csc ^3(c+d x) (a+a \sec (c+d x))^n \, dx","Integrate[Csc[c + d*x]^3*(a + a*Sec[c + d*x])^n,x]","\frac{\cos (c+d x) \sec ^2\left(\frac{1}{2} (c+d x)\right) (\sec (c+d x)+1)^{-n} (a (\sec (c+d x)+1))^n \left(2^{n+1} \left(\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)\right)^n \, _2F_1\left(1,1-n;2-n;\cos (c+d x) \sec ^2\left(\frac{1}{2} (c+d x)\right)\right)+2^n \left(\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)\right)^n \, _2F_1\left(2,1-n;2-n;\cos (c+d x) \sec ^2\left(\frac{1}{2} (c+d x)\right)\right)+(\sec (c+d x)+1)^n\right)}{8 d (n-1)}","-\frac{(n+2) (a \sec (c+d x)+a)^n \, _2F_1\left(1,n;n+1;\frac{1}{2} (\sec (c+d x)+1)\right)}{8 d n}-\frac{a (2-n) (a \sec (c+d x)+a)^{n-1}}{4 d (1-n)}+\frac{a (a \sec (c+d x)+a)^{n-1}}{2 d (1-\sec (c+d x))}",1,"(Cos[c + d*x]*Sec[(c + d*x)/2]^2*(a*(1 + Sec[c + d*x]))^n*(2^(1 + n)*Hypergeometric2F1[1, 1 - n, 2 - n, Cos[c + d*x]*Sec[(c + d*x)/2]^2]*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^n + 2^n*Hypergeometric2F1[2, 1 - n, 2 - n, Cos[c + d*x]*Sec[(c + d*x)/2]^2]*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^n + (1 + Sec[c + d*x])^n))/(8*d*(-1 + n)*(1 + Sec[c + d*x])^n)","A",0
151,1,492,240,6.5290815,"\int \csc ^5(c+d x) (a+a \sec (c+d x))^n \, dx","Integrate[Csc[c + d*x]^5*(a + a*Sec[c + d*x])^n,x]","-\frac{\cos (c+d x) (\sec (c+d x)+1)^{-n} (a (\sec (c+d x)+1))^n \left(-2^n \left(n^2+7 n-18\right) \sec (c+d x) \left(\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)\right)^{n-1} \, _2F_1\left(2,1-n;2-n;\cos (c+d x) \sec ^2\left(\frac{1}{2} (c+d x)\right)\right)-3\ 2^{n+2} (n-2) \sec (c+d x) \left(\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)\right)^{n-1} \, _2F_1\left(1,1-n;2-n;\cos (c+d x) \sec ^2\left(\frac{1}{2} (c+d x)\right)\right)+2^n n^2 \cot ^4\left(\frac{1}{2} (c+d x)\right) \sec (c+d x) \left(\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)\right)^{n-1}+2 n \sec ^4\left(\frac{1}{2} (c+d x)\right) \sec (c+d x) (\sec (c+d x)+1)^n-2 \sec ^4\left(\frac{1}{2} (c+d x)\right) \sec (c+d x) (\sec (c+d x)+1)^n+2 n \sec ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x) (\sec (c+d x)+1)^n-12 \sec ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x) (\sec (c+d x)+1)^n-12 n \sec (c+d x) (\sec (c+d x)+1)^n+32 \sec (c+d x) (\sec (c+d x)+1)^n+2^{n+1} \cot ^4\left(\frac{1}{2} (c+d x)\right) \sec (c+d x) \left(\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)\right)^{n-1}-3\ 2^n n \cot ^4\left(\frac{1}{2} (c+d x)\right) \sec (c+d x) \left(\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)\right)^{n-1}\right)}{64 d (n-2) (n-1)}","\frac{a^2 \left(n^2+9 n+12\right) (a \sec (c+d x)+a)^{n-2} \, _2F_1\left(1,n-2;n-1;\frac{1}{2} (\sec (c+d x)+1)\right)}{16 d (2-n)}-\frac{a^2 \left(-2 (1-n) (n+6) \sec (c+d x)-n^3-7 n^2+4 n+12\right) (a \sec (c+d x)+a)^{n-2}}{8 d \left(n^2-3 n+2\right) (1-\sec (c+d x))}-\frac{a^2 \sec ^3(c+d x) (a \sec (c+d x)+a)^{n-2}}{d (1-n) (1-\sec (c+d x))^2}+\frac{a^2 (n+3) \sec ^2(c+d x) (a \sec (c+d x)+a)^{n-2}}{4 d (1-n) (1-\sec (c+d x))^2}",1,"-1/64*(Cos[c + d*x]*(a*(1 + Sec[c + d*x]))^n*(2^(1 + n)*Cot[(c + d*x)/2]^4*Sec[c + d*x]*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^(-1 + n) - 3*2^n*n*Cot[(c + d*x)/2]^4*Sec[c + d*x]*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^(-1 + n) + 2^n*n^2*Cot[(c + d*x)/2]^4*Sec[c + d*x]*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^(-1 + n) - 3*2^(2 + n)*(-2 + n)*Hypergeometric2F1[1, 1 - n, 2 - n, Cos[c + d*x]*Sec[(c + d*x)/2]^2]*Sec[c + d*x]*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^(-1 + n) - 2^n*(-18 + 7*n + n^2)*Hypergeometric2F1[2, 1 - n, 2 - n, Cos[c + d*x]*Sec[(c + d*x)/2]^2]*Sec[c + d*x]*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^(-1 + n) + 32*Sec[c + d*x]*(1 + Sec[c + d*x])^n - 12*n*Sec[c + d*x]*(1 + Sec[c + d*x])^n - 12*Sec[(c + d*x)/2]^2*Sec[c + d*x]*(1 + Sec[c + d*x])^n + 2*n*Sec[(c + d*x)/2]^2*Sec[c + d*x]*(1 + Sec[c + d*x])^n - 2*Sec[(c + d*x)/2]^4*Sec[c + d*x]*(1 + Sec[c + d*x])^n + 2*n*Sec[(c + d*x)/2]^4*Sec[c + d*x]*(1 + Sec[c + d*x])^n))/(d*(-2 + n)*(-1 + n)*(1 + Sec[c + d*x])^n)","B",0
152,1,7069,230,22.865115,"\int (a+a \sec (c+d x))^n \sin ^4(c+d x) \, dx","Integrate[(a + a*Sec[c + d*x])^n*Sin[c + d*x]^4,x]","\text{Result too large to show}","\frac{2^{n+\frac{1}{2}} \sin (c+d x) \cos ^n(c+d x) (\cos (c+d x)+1)^{-n-\frac{1}{2}} (a \sec (c+d x)+a)^n F_1\left(\frac{1}{2};n-4,\frac{1}{2}-n;\frac{3}{2};1-\cos (c+d x),\frac{1}{2} (1-\cos (c+d x))\right)}{d}-\frac{\cot (c+d x) (n-n \cos (c+d x)) (\cos (c+d x)+1)^{\frac{1}{2}-n} (a \sec (c+d x)+a)^n F_1\left(1-n;-\frac{1}{2},\frac{1}{2}-n;2-n;\cos (c+d x),-\cos (c+d x)\right)}{d (1-n) \sqrt{1-\cos (c+d x)}}-\frac{\sin (c+d x) \cos (c+d x) (a \sec (c+d x)+a)^n}{d}",1,"Result too large to show","C",0
153,1,4297,95,17.1432491,"\int (a+a \sec (c+d x))^n \sin ^2(c+d x) \, dx","Integrate[(a + a*Sec[c + d*x])^n*Sin[c + d*x]^2,x]","\text{Result too large to show}","-\frac{\sqrt{1-\cos (c+d x)} \cot (c+d x) (\cos (c+d x)+1)^{\frac{1}{2}-n} (a \sec (c+d x)+a)^n F_1\left(1-n;-\frac{1}{2},-n-\frac{1}{2};2-n;\cos (c+d x),-\cos (c+d x)\right)}{d (1-n)}",1,"(2^(3 + n)*Cos[(c + d*x)/2]^5*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^n*(a*(1 + Sec[c + d*x]))^n*Sin[(c + d*x)/2]*(Cos[2*(c + d*x)]*(-1/4*(1 + Sec[c + d*x])^n - ((1 + Sec[c + d*x])^n*Sin[c + d*x]^2)/2 - ((1 + Sec[c + d*x])^n*Sin[c + d*x]^4)/4) + (I/4)*(1 + Sec[c + d*x])^n*Sin[2*(c + d*x)] + (I/2)*(1 + Sec[c + d*x])^n*Sin[c + d*x]^2*Sin[2*(c + d*x)] + (I/4)*(1 + Sec[c + d*x])^n*Sin[c + d*x]^4*Sin[2*(c + d*x)] + Cos[c + d*x]^4*(-1/4*(Cos[2*(c + d*x)]*(1 + Sec[c + d*x])^n) + (I/4)*(1 + Sec[c + d*x])^n*Sin[2*(c + d*x)]) + Cos[c + d*x]^3*((-I)*Cos[2*(c + d*x)]*(1 + Sec[c + d*x])^n*Sin[c + d*x] - (1 + Sec[c + d*x])^n*Sin[c + d*x]*Sin[2*(c + d*x)]) + Cos[c + d*x]^2*(Cos[2*(c + d*x)]*((1 + Sec[c + d*x])^n/2 + (3*(1 + Sec[c + d*x])^n*Sin[c + d*x]^2)/2) - (I/2)*(1 + Sec[c + d*x])^n*Sin[2*(c + d*x)] - ((3*I)/2)*(1 + Sec[c + d*x])^n*Sin[c + d*x]^2*Sin[2*(c + d*x)]) + Cos[c + d*x]*(Cos[2*(c + d*x)]*(I*(1 + Sec[c + d*x])^n*Sin[c + d*x] + I*(1 + Sec[c + d*x])^n*Sin[c + d*x]^3) + (1 + Sec[c + d*x])^n*Sin[c + d*x]*Sin[2*(c + d*x)] + (1 + Sec[c + d*x])^n*Sin[c + d*x]^3*Sin[2*(c + d*x)]))*((3*AppellF1[1/2, n, 2, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2)/(3*AppellF1[1/2, n, 2, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + 2*(-2*AppellF1[3/2, n, 3, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + n*AppellF1[3/2, 1 + n, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Tan[(c + d*x)/2]^2) - AppellF1[1/2, n, 3, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]/(AppellF1[1/2, n, 3, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + (2*(-3*AppellF1[3/2, n, 4, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + n*AppellF1[3/2, 1 + n, 3, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Tan[(c + d*x)/2]^2)/3)))/(d*(1 + Sec[c + d*x])^n*(2^(2 + n)*Cos[(c + d*x)/2]^6*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^n*((3*AppellF1[1/2, n, 2, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2)/(3*AppellF1[1/2, n, 2, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + 2*(-2*AppellF1[3/2, n, 3, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + n*AppellF1[3/2, 1 + n, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Tan[(c + d*x)/2]^2) - AppellF1[1/2, n, 3, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]/(AppellF1[1/2, n, 3, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + (2*(-3*AppellF1[3/2, n, 4, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + n*AppellF1[3/2, 1 + n, 3, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Tan[(c + d*x)/2]^2)/3)) - 5*2^(2 + n)*Cos[(c + d*x)/2]^4*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^n*Sin[(c + d*x)/2]^2*((3*AppellF1[1/2, n, 2, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2)/(3*AppellF1[1/2, n, 2, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + 2*(-2*AppellF1[3/2, n, 3, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + n*AppellF1[3/2, 1 + n, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Tan[(c + d*x)/2]^2) - AppellF1[1/2, n, 3, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]/(AppellF1[1/2, n, 3, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + (2*(-3*AppellF1[3/2, n, 4, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + n*AppellF1[3/2, 1 + n, 3, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Tan[(c + d*x)/2]^2)/3)) + 2^(3 + n)*Cos[(c + d*x)/2]^5*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^n*Sin[(c + d*x)/2]*((3*AppellF1[1/2, n, 2, 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])/(3*AppellF1[1/2, n, 2, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + 2*(-2*AppellF1[3/2, n, 3, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + n*AppellF1[3/2, 1 + n, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Tan[(c + d*x)/2]^2) + (3*Sec[(c + d*x)/2]^2*((-2*AppellF1[3/2, n, 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])/3 + (n*AppellF1[3/2, 1 + n, 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])/3))/(3*AppellF1[1/2, n, 2, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + 2*(-2*AppellF1[3/2, n, 3, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + n*AppellF1[3/2, 1 + n, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Tan[(c + d*x)/2]^2) - (-(AppellF1[3/2, n, 4, 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]) + (n*AppellF1[3/2, 1 + n, 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])/3)/(AppellF1[1/2, n, 3, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + (2*(-3*AppellF1[3/2, n, 4, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + n*AppellF1[3/2, 1 + n, 3, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Tan[(c + d*x)/2]^2)/3) - (3*AppellF1[1/2, n, 2, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*(2*(-2*AppellF1[3/2, n, 3, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + n*AppellF1[3/2, 1 + n, 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] + 3*((-2*AppellF1[3/2, n, 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])/3 + (n*AppellF1[3/2, 1 + n, 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])/3) + 2*Tan[(c + d*x)/2]^2*(-2*((-9*AppellF1[5/2, n, 4, 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*n*AppellF1[5/2, 1 + n, 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) + n*((-6*AppellF1[5/2, 1 + n, 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*(1 + n)*AppellF1[5/2, 2 + n, 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))))/(3*AppellF1[1/2, n, 2, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + 2*(-2*AppellF1[3/2, n, 3, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + n*AppellF1[3/2, 1 + n, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Tan[(c + d*x)/2]^2)^2 + (AppellF1[1/2, n, 3, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*(-(AppellF1[3/2, n, 4, 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]) + (n*AppellF1[3/2, 1 + n, 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])/3 + (2*(-3*AppellF1[3/2, n, 4, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + n*AppellF1[3/2, 1 + n, 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])/3 + (2*Tan[(c + d*x)/2]^2*(-3*((-12*AppellF1[5/2, n, 5, 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*n*AppellF1[5/2, 1 + n, 4, 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) + n*((-9*AppellF1[5/2, 1 + n, 4, 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*(1 + n)*AppellF1[5/2, 2 + n, 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))/(AppellF1[1/2, n, 3, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + (2*(-3*AppellF1[3/2, n, 4, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + n*AppellF1[3/2, 1 + n, 3, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Tan[(c + d*x)/2]^2)/3)^2) + 2^(3 + n)*n*Cos[(c + d*x)/2]^5*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^(-1 + n)*Sin[(c + d*x)/2]*((3*AppellF1[1/2, n, 2, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2)/(3*AppellF1[1/2, n, 2, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + 2*(-2*AppellF1[3/2, n, 3, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + n*AppellF1[3/2, 1 + n, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Tan[(c + d*x)/2]^2) - AppellF1[1/2, n, 3, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]/(AppellF1[1/2, n, 3, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + (2*(-3*AppellF1[3/2, n, 4, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + n*AppellF1[3/2, 1 + n, 3, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Tan[(c + d*x)/2]^2)/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])))","C",0
154,1,142,98,1.1641428,"\int \csc ^2(c+d x) (a+a \sec (c+d x))^n \, dx","Integrate[Csc[c + d*x]^2*(a + a*Sec[c + d*x])^n,x]","-\frac{2^{n-1} \tan \left(\frac{1}{2} (c+d x)\right) (\sec (c+d x)+1)^{-n} (a (\sec (c+d x)+1))^n \left(\cos (c+d x) \sec ^2\left(\frac{1}{2} (c+d x)\right)\right)^n \left(\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)\right)^n \left(\cot ^2\left(\frac{1}{2} (c+d x)\right) \, _2F_1\left(-\frac{1}{2},n;\frac{1}{2};\tan ^2\left(\frac{1}{2} (c+d x)\right)\right)-\, _2F_1\left(\frac{1}{2},n;\frac{3}{2};\tan ^2\left(\frac{1}{2} (c+d x)\right)\right)\right)}{d}","\frac{2^{n-\frac{1}{2}} n \tan (c+d x) (\sec (c+d x)+1)^{-n-\frac{1}{2}} (a \sec (c+d x)+a)^n \, _2F_1\left(\frac{1}{2},\frac{3}{2}-n;\frac{3}{2};\frac{1}{2} (1-\sec (c+d x))\right)}{d}-\frac{\cot (c+d x) (a \sec (c+d x)+a)^n}{d}",1,"-((2^(-1 + n)*(Cot[(c + d*x)/2]^2*Hypergeometric2F1[-1/2, n, 1/2, Tan[(c + d*x)/2]^2] - Hypergeometric2F1[1/2, n, 3/2, Tan[(c + d*x)/2]^2])*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^n*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^n*(a*(1 + Sec[c + d*x]))^n*Tan[(c + d*x)/2])/(d*(1 + Sec[c + d*x])^n))","A",0
155,1,350,349,7.0390518,"\int \csc ^4(c+d x) (a+a \sec (c+d x))^n \, dx","Integrate[Csc[c + d*x]^4*(a + a*Sec[c + d*x])^n,x]","\frac{\tan \left(\frac{1}{2} (c+d x)\right) (a (\sec (c+d x)+1))^n \left(\frac{24 (\sec (c+d x)+1)^{-n} \, _2F_1\left(\frac{1}{2},n;\frac{3}{2};\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)^n \left(-2 (\sec (c+d x)+1)^n-3\ 2^n \left(\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)\right)^n+n \left((\sec (c+d x)+1)^n+2^{n+1} \left(\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)\right)^n\right)\right)-\cos (c+d x) \csc ^4\left(\frac{1}{2} (c+d x)\right) \sec ^2\left(\frac{1}{2} (c+d x)\right) (4 n \cos (c+d x)+(n-3) (\cos (2 (c+d x))+3))}{4 (2 n-3)}-2 \cot ^2\left(\frac{1}{2} (c+d x)\right) (\sec (c+d x)+1)^{-n} \, _2F_1\left(-\frac{1}{2},n;\frac{1}{2};\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)^n \left(n (\sec (c+d x)+1)^n+2 (\sec (c+d x)+1)^n+3\ 2^n \left(\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)\right)^n\right)\right)}{24 d}","-\frac{a^4 \sin (c+d x) \cos (c+d x) (a \sec (c+d x)+a)^n}{d (3-2 n) (a-a \cos (c+d x))^2 (a \cos (c+d x)+a)^2}-\frac{a^3 (4-n) \sin (c+d x) \cos (c+d x) (a \sec (c+d x)+a)^n}{d \left(4 n^2-8 n+3\right) (a-a \cos (c+d x))^2 (a \cos (c+d x)+a)}+\frac{n \left(-n^2-3 n+7\right) \sin (c+d x) \cos (c+d x) \left(\frac{\cos (c+d x)+1}{1-\cos (c+d x)}\right)^{-n-\frac{1}{2}} (a \sec (c+d x)+a)^n \, _2F_1\left(-n-\frac{1}{2},1-n;2-n;-\frac{2 \cos (c+d x)}{1-\cos (c+d x)}\right)}{d (1-2 n) (3-2 n) (1-n) (2 n+1) (1-\cos (c+d x))^2}+\frac{\left(n^2-n+2\right) \sin (c+d x) \cos (c+d x) (a \sec (c+d x)+a)^n}{d (3-2 n) \left(1-4 n^2\right) (1-\cos (c+d x))^2}",1,"((a*(1 + Sec[c + d*x]))^n*((-2*Cot[(c + d*x)/2]^2*Hypergeometric2F1[-1/2, n, 1/2, Tan[(c + d*x)/2]^2]*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^n*(3*2^n*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^n + 2*(1 + Sec[c + d*x])^n + n*(1 + Sec[c + d*x])^n))/(1 + Sec[c + d*x])^n + (-(Cos[c + d*x]*(4*n*Cos[c + d*x] + (-3 + n)*(3 + Cos[2*(c + d*x)]))*Csc[(c + d*x)/2]^4*Sec[(c + d*x)/2]^2) + (24*Hypergeometric2F1[1/2, n, 3/2, Tan[(c + d*x)/2]^2]*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^n*(-3*2^n*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^n - 2*(1 + Sec[c + d*x])^n + n*(2^(1 + n)*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^n + (1 + Sec[c + d*x])^n)))/(1 + Sec[c + d*x])^n)/(4*(-3 + 2*n)))*Tan[(c + d*x)/2])/(24*d)","A",0
156,1,382,105,3.3565968,"\int (a+a \sec (c+d x))^n \sin ^{\frac{3}{2}}(c+d x) \, dx","Integrate[(a + a*Sec[c + d*x])^n*Sin[c + d*x]^(3/2),x]","\frac{10 \sin ^{\frac{5}{2}}(c+d x) (\cos (c+d x)+1) (a (\sec (c+d x)+1))^n \left(F_1\left(\frac{1}{4};n,\frac{3}{2};\frac{5}{4};\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{1}{4};n,\frac{5}{2};\frac{5}{4};\tan ^2\left(\frac{1}{2} (c+d x)\right),-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right)\right)}{d \left(2 (\cos (c+d x)-1) \left(3 F_1\left(\frac{5}{4};n,\frac{5}{2};\frac{9}{4};\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}{4};n,\frac{7}{2};\frac{9}{4};\tan ^2\left(\frac{1}{2} (c+d x)\right),-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right)-2 n F_1\left(\frac{5}{4};n+1,\frac{3}{2};\frac{9}{4};\tan ^2\left(\frac{1}{2} (c+d x)\right),-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right)+2 n F_1\left(\frac{5}{4};n+1,\frac{5}{2};\frac{9}{4};\tan ^2\left(\frac{1}{2} (c+d x)\right),-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right)\right)+5 (\cos (c+d x)+1) F_1\left(\frac{1}{4};n,\frac{3}{2};\frac{5}{4};\tan ^2\left(\frac{1}{2} (c+d x)\right),-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right)-5 (\cos (c+d x)+1) F_1\left(\frac{1}{4};n,\frac{5}{2};\frac{5}{4};\tan ^2\left(\frac{1}{2} (c+d x)\right),-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right)\right)}","-\frac{\sqrt{\sin (c+d x)} \cos (c+d x) (\cos (c+d x)+1)^{-n-\frac{1}{4}} (a \sec (c+d x)+a)^n F_1\left(1-n;-\frac{1}{4},-n-\frac{1}{4};2-n;\cos (c+d x),-\cos (c+d x)\right)}{d (1-n) \sqrt[4]{1-\cos (c+d x)}}",1,"(10*(AppellF1[1/4, n, 3/2, 5/4, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] - AppellF1[1/4, n, 5/2, 5/4, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*(1 + Cos[c + d*x])*(a*(1 + Sec[c + d*x]))^n*Sin[c + d*x]^(5/2))/(d*(2*(3*AppellF1[5/4, n, 5/2, 9/4, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] - 5*AppellF1[5/4, n, 7/2, 9/4, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] - 2*n*AppellF1[5/4, 1 + n, 3/2, 9/4, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + 2*n*AppellF1[5/4, 1 + n, 5/2, 9/4, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*(-1 + Cos[c + d*x]) + 5*AppellF1[1/4, n, 3/2, 5/4, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*(1 + Cos[c + d*x]) - 5*AppellF1[1/4, n, 5/2, 5/4, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*(1 + Cos[c + d*x])))","B",0
157,1,214,105,1.4444315,"\int (a+a \sec (c+d x))^n \sqrt{\sin (c+d x)} \, dx","Integrate[(a + a*Sec[c + d*x])^n*Sqrt[Sin[c + d*x]],x]","\frac{14 \sin ^{\frac{3}{2}}(c+d x) (\cos (c+d x)+1) (a (\sec (c+d x)+1))^n F_1\left(\frac{3}{4};n,\frac{3}{2};\frac{7}{4};\tan ^2\left(\frac{1}{2} (c+d x)\right),-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right)}{d \left(6 (\cos (c+d x)-1) \left(3 F_1\left(\frac{7}{4};n,\frac{5}{2};\frac{11}{4};\tan ^2\left(\frac{1}{2} (c+d x)\right),-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right)-2 n F_1\left(\frac{7}{4};n+1,\frac{3}{2};\frac{11}{4};\tan ^2\left(\frac{1}{2} (c+d x)\right),-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right)\right)+21 (\cos (c+d x)+1) F_1\left(\frac{3}{4};n,\frac{3}{2};\frac{7}{4};\tan ^2\left(\frac{1}{2} (c+d x)\right),-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right)\right)}","-\frac{\sqrt[4]{1-\cos (c+d x)} \cos (c+d x) (\cos (c+d x)+1)^{\frac{1}{4}-n} (a \sec (c+d x)+a)^n F_1\left(1-n;\frac{1}{4},\frac{1}{4}-n;2-n;\cos (c+d x),-\cos (c+d x)\right)}{d (1-n) \sqrt{\sin (c+d x)}}",1,"(14*AppellF1[3/4, n, 3/2, 7/4, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*(1 + Cos[c + d*x])*(a*(1 + Sec[c + d*x]))^n*Sin[c + d*x]^(3/2))/(d*(6*(3*AppellF1[7/4, n, 5/2, 11/4, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] - 2*n*AppellF1[7/4, 1 + n, 3/2, 11/4, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*(-1 + Cos[c + d*x]) + 21*AppellF1[3/4, n, 3/2, 7/4, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*(1 + Cos[c + d*x])))","B",0
158,1,212,105,1.0416545,"\int \frac{(a+a \sec (c+d x))^n}{\sqrt{\sin (c+d x)}} \, dx","Integrate[(a + a*Sec[c + d*x])^n/Sqrt[Sin[c + d*x]],x]","\frac{10 \sqrt{\sin (c+d x)} (\cos (c+d x)+1) (a (\sec (c+d x)+1))^n F_1\left(\frac{1}{4};n,\frac{1}{2};\frac{5}{4};\tan ^2\left(\frac{1}{2} (c+d x)\right),-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right)}{d \left(2 (\cos (c+d x)-1) \left(F_1\left(\frac{5}{4};n,\frac{3}{2};\frac{9}{4};\tan ^2\left(\frac{1}{2} (c+d x)\right),-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right)-2 n F_1\left(\frac{5}{4};n+1,\frac{1}{2};\frac{9}{4};\tan ^2\left(\frac{1}{2} (c+d x)\right),-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right)\right)+5 (\cos (c+d x)+1) F_1\left(\frac{1}{4};n,\frac{1}{2};\frac{5}{4};\tan ^2\left(\frac{1}{2} (c+d x)\right),-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right)\right)}","-\frac{(1-\cos (c+d x))^{3/4} \cos (c+d x) (\cos (c+d x)+1)^{\frac{3}{4}-n} (a \sec (c+d x)+a)^n F_1\left(1-n;\frac{3}{4},\frac{3}{4}-n;2-n;\cos (c+d x),-\cos (c+d x)\right)}{d (1-n) \sin ^{\frac{3}{2}}(c+d x)}",1,"(10*AppellF1[1/4, n, 1/2, 5/4, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*(1 + Cos[c + d*x])*(a*(1 + Sec[c + d*x]))^n*Sqrt[Sin[c + d*x]])/(d*(2*(AppellF1[5/4, n, 3/2, 9/4, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] - 2*n*AppellF1[5/4, 1 + n, 1/2, 9/4, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*(-1 + Cos[c + d*x]) + 5*AppellF1[1/4, n, 1/2, 5/4, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*(1 + Cos[c + d*x])))","B",0
159,1,212,105,1.2363128,"\int \frac{(a+a \sec (c+d x))^n}{\sin ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[(a + a*Sec[c + d*x])^n/Sin[c + d*x]^(3/2),x]","-\frac{6 (\cos (c+d x)+1) (a (\sec (c+d x)+1))^n F_1\left(-\frac{1}{4};n,-\frac{1}{2};\frac{3}{4};\tan ^2\left(\frac{1}{2} (c+d x)\right),-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right)}{d \sqrt{\sin (c+d x)} \left(3 (\cos (c+d x)+1) F_1\left(-\frac{1}{4};n,-\frac{1}{2};\frac{3}{4};\tan ^2\left(\frac{1}{2} (c+d x)\right),-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right)-2 (\cos (c+d x)-1) \left(F_1\left(\frac{3}{4};n,\frac{1}{2};\frac{7}{4};\tan ^2\left(\frac{1}{2} (c+d x)\right),-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right)+2 n F_1\left(\frac{3}{4};n+1,-\frac{1}{2};\frac{7}{4};\tan ^2\left(\frac{1}{2} (c+d x)\right),-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right)\right)\right)}","-\frac{(1-\cos (c+d x))^{5/4} \cos (c+d x) (\cos (c+d x)+1)^{\frac{5}{4}-n} (a \sec (c+d x)+a)^n F_1\left(1-n;\frac{5}{4},\frac{5}{4}-n;2-n;\cos (c+d x),-\cos (c+d x)\right)}{d (1-n) \sin ^{\frac{5}{2}}(c+d x)}",1,"(-6*AppellF1[-1/4, n, -1/2, 3/4, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*(1 + Cos[c + d*x])*(a*(1 + Sec[c + d*x]))^n)/(d*(-2*(AppellF1[3/4, n, 1/2, 7/4, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + 2*n*AppellF1[3/4, 1 + n, -1/2, 7/4, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*(-1 + Cos[c + d*x]) + 3*AppellF1[-1/4, n, -1/2, 3/4, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*(1 + Cos[c + d*x]))*Sqrt[Sin[c + d*x]])","B",0
160,1,115,119,0.1480171,"\int (a+b \sec (c+d x)) \sin ^7(c+d x) \, dx","Integrate[(a + b*Sec[c + d*x])*Sin[c + d*x]^7,x]","-\frac{35 a \cos (c+d x)}{64 d}+\frac{7 a \cos (3 (c+d x))}{64 d}-\frac{7 a \cos (5 (c+d x))}{320 d}+\frac{a \cos (7 (c+d x))}{448 d}-\frac{b \left(-\frac{1}{3} \cos ^6(c+d x)+\frac{3}{2} \cos ^4(c+d x)-3 \cos ^2(c+d x)+2 \log (\cos (c+d x))\right)}{2 d}","\frac{a \cos ^7(c+d x)}{7 d}-\frac{3 a \cos ^5(c+d x)}{5 d}+\frac{a \cos ^3(c+d x)}{d}-\frac{a \cos (c+d x)}{d}+\frac{b \cos ^6(c+d x)}{6 d}-\frac{3 b \cos ^4(c+d x)}{4 d}+\frac{3 b \cos ^2(c+d x)}{2 d}-\frac{b \log (\cos (c+d x))}{d}",1,"(-35*a*Cos[c + d*x])/(64*d) + (7*a*Cos[3*(c + d*x)])/(64*d) - (7*a*Cos[5*(c + d*x)])/(320*d) + (a*Cos[7*(c + d*x)])/(448*d) - (b*(-3*Cos[c + d*x]^2 + (3*Cos[c + d*x]^4)/2 - Cos[c + d*x]^6/3 + 2*Log[Cos[c + d*x]]))/(2*d)","A",1
161,1,83,87,0.0844599,"\int (a+b \sec (c+d x)) \sin ^5(c+d x) \, dx","Integrate[(a + b*Sec[c + d*x])*Sin[c + d*x]^5,x]","-\frac{5 a \cos (c+d x)}{8 d}+\frac{5 a \cos (3 (c+d x))}{48 d}-\frac{a \cos (5 (c+d x))}{80 d}-\frac{b \left(\frac{1}{4} \cos ^4(c+d x)-\cos ^2(c+d x)+\log (\cos (c+d x))\right)}{d}","-\frac{a \cos ^5(c+d x)}{5 d}+\frac{2 a \cos ^3(c+d x)}{3 d}-\frac{a \cos (c+d x)}{d}-\frac{b \cos ^4(c+d x)}{4 d}+\frac{b \cos ^2(c+d x)}{d}-\frac{b \log (\cos (c+d x))}{d}",1,"(-5*a*Cos[c + d*x])/(8*d) + (5*a*Cos[3*(c + d*x)])/(48*d) - (a*Cos[5*(c + d*x)])/(80*d) - (b*(-Cos[c + d*x]^2 + Cos[c + d*x]^4/4 + Log[Cos[c + d*x]]))/d","A",1
162,1,57,58,0.0472816,"\int (a+b \sec (c+d x)) \sin ^3(c+d x) \, dx","Integrate[(a + b*Sec[c + d*x])*Sin[c + d*x]^3,x]","-\frac{3 a \cos (c+d x)}{4 d}+\frac{a \cos (3 (c+d x))}{12 d}-\frac{b \left(\log (\cos (c+d x))-\frac{1}{2} \cos ^2(c+d x)\right)}{d}","\frac{a \cos ^3(c+d x)}{3 d}-\frac{a \cos (c+d x)}{d}+\frac{b \cos ^2(c+d x)}{2 d}-\frac{b \log (\cos (c+d x))}{d}",1,"(-3*a*Cos[c + d*x])/(4*d) + (a*Cos[3*(c + d*x)])/(12*d) - (b*(-1/2*Cos[c + d*x]^2 + Log[Cos[c + d*x]]))/d","A",1
163,1,37,26,0.026437,"\int (a+b \sec (c+d x)) \sin (c+d x) \, dx","Integrate[(a + b*Sec[c + d*x])*Sin[c + d*x],x]","\frac{a \sin (c) \sin (d x)}{d}-\frac{a \cos (c) \cos (d x)}{d}-\frac{b \log (\cos (c+d x))}{d}","-\frac{a \cos (c+d x)}{d}-\frac{b \log (\cos (c+d x))}{d}",1,"-((a*Cos[c]*Cos[d*x])/d) - (b*Log[Cos[c + d*x]])/d + (a*Sin[c]*Sin[d*x])/d","A",1
164,1,63,26,0.0359447,"\int \csc (c+d x) (a+b \sec (c+d x)) \, dx","Integrate[Csc[c + d*x]*(a + b*Sec[c + d*x]),x]","\frac{a \log \left(\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)}{d}-\frac{a \log \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)\right)}{d}-\frac{b (\log (\cos (c+d x))-\log (\sin (c+d x)))}{d}","\frac{b \log (\tan (c+d x))}{d}-\frac{a \tanh ^{-1}(\cos (c+d x))}{d}",1,"-((a*Log[Cos[c/2 + (d*x)/2]])/d) + (a*Log[Sin[c/2 + (d*x)/2]])/d - (b*(Log[Cos[c + d*x]] - Log[Sin[c + d*x]]))/d","B",1
165,1,114,64,0.5110251,"\int \csc ^3(c+d x) (a+b \sec (c+d x)) \, dx","Integrate[Csc[c + d*x]^3*(a + b*Sec[c + d*x]),x]","-\frac{a \csc ^2\left(\frac{1}{2} (c+d x)\right)}{8 d}+\frac{a \sec ^2\left(\frac{1}{2} (c+d x)\right)}{8 d}+\frac{a \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)}{2 d}-\frac{a \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)}{2 d}-\frac{b \left(\csc ^2(c+d x)-2 \log (\sin (c+d x))+2 \log (\cos (c+d x))\right)}{2 d}","-\frac{a \tanh ^{-1}(\cos (c+d x))}{2 d}-\frac{a \cot (c+d x) \csc (c+d x)}{2 d}-\frac{b \cot ^2(c+d x)}{2 d}+\frac{b \log (\tan (c+d x))}{d}",1,"-1/8*(a*Csc[(c + d*x)/2]^2)/d - (a*Log[Cos[(c + d*x)/2]])/(2*d) + (a*Log[Sin[(c + d*x)/2]])/(2*d) - (b*(Csc[c + d*x]^2 + 2*Log[Cos[c + d*x]] - 2*Log[Sin[c + d*x]]))/(2*d) + (a*Sec[(c + d*x)/2]^2)/(8*d)","A",1
166,1,164,100,0.6511666,"\int \csc ^5(c+d x) (a+b \sec (c+d x)) \, dx","Integrate[Csc[c + d*x]^5*(a + b*Sec[c + d*x]),x]","-\frac{a \csc ^4\left(\frac{1}{2} (c+d x)\right)}{64 d}-\frac{3 a \csc ^2\left(\frac{1}{2} (c+d x)\right)}{32 d}+\frac{a \sec ^4\left(\frac{1}{2} (c+d x)\right)}{64 d}+\frac{3 a \sec ^2\left(\frac{1}{2} (c+d x)\right)}{32 d}+\frac{3 a \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)}{8 d}-\frac{3 a \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)}{8 d}-\frac{b \left(\csc ^4(c+d x)+2 \csc ^2(c+d x)-4 \log (\sin (c+d x))+4 \log (\cos (c+d x))\right)}{4 d}","-\frac{3 a \tanh ^{-1}(\cos (c+d x))}{8 d}-\frac{a \cot (c+d x) \csc ^3(c+d x)}{4 d}-\frac{3 a \cot (c+d x) \csc (c+d x)}{8 d}-\frac{b \cot ^4(c+d x)}{4 d}-\frac{b \cot ^2(c+d x)}{d}+\frac{b \log (\tan (c+d x))}{d}",1,"(-3*a*Csc[(c + d*x)/2]^2)/(32*d) - (a*Csc[(c + d*x)/2]^4)/(64*d) - (3*a*Log[Cos[(c + d*x)/2]])/(8*d) + (3*a*Log[Sin[(c + d*x)/2]])/(8*d) - (b*(2*Csc[c + d*x]^2 + Csc[c + d*x]^4 + 4*Log[Cos[c + d*x]] - 4*Log[Sin[c + d*x]]))/(4*d) + (3*a*Sec[(c + d*x)/2]^2)/(32*d) + (a*Sec[(c + d*x)/2]^4)/(64*d)","A",1
167,1,216,140,0.6213671,"\int \csc ^7(c+d x) (a+b \sec (c+d x)) \, dx","Integrate[Csc[c + d*x]^7*(a + b*Sec[c + d*x]),x]","-\frac{a \csc ^6\left(\frac{1}{2} (c+d x)\right)}{384 d}-\frac{a \csc ^4\left(\frac{1}{2} (c+d x)\right)}{64 d}-\frac{5 a \csc ^2\left(\frac{1}{2} (c+d x)\right)}{64 d}+\frac{a \sec ^6\left(\frac{1}{2} (c+d x)\right)}{384 d}+\frac{a \sec ^4\left(\frac{1}{2} (c+d x)\right)}{64 d}+\frac{5 a \sec ^2\left(\frac{1}{2} (c+d x)\right)}{64 d}+\frac{5 a \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)}{16 d}-\frac{5 a \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)}{16 d}-\frac{b \left(2 \csc ^6(c+d x)+3 \csc ^4(c+d x)+6 \csc ^2(c+d x)-12 \log (\sin (c+d x))+12 \log (\cos (c+d x))\right)}{12 d}","-\frac{5 a \tanh ^{-1}(\cos (c+d x))}{16 d}-\frac{a \cot (c+d x) \csc ^5(c+d x)}{6 d}-\frac{5 a \cot (c+d x) \csc ^3(c+d x)}{24 d}-\frac{5 a \cot (c+d x) \csc (c+d x)}{16 d}-\frac{b \cot ^6(c+d x)}{6 d}-\frac{3 b \cot ^4(c+d x)}{4 d}-\frac{3 b \cot ^2(c+d x)}{2 d}+\frac{b \log (\tan (c+d x))}{d}",1,"(-5*a*Csc[(c + d*x)/2]^2)/(64*d) - (a*Csc[(c + d*x)/2]^4)/(64*d) - (a*Csc[(c + d*x)/2]^6)/(384*d) - (5*a*Log[Cos[(c + d*x)/2]])/(16*d) + (5*a*Log[Sin[(c + d*x)/2]])/(16*d) - (b*(6*Csc[c + d*x]^2 + 3*Csc[c + d*x]^4 + 2*Csc[c + d*x]^6 + 12*Log[Cos[c + d*x]] - 12*Log[Sin[c + d*x]]))/(12*d) + (5*a*Sec[(c + d*x)/2]^2)/(64*d) + (a*Sec[(c + d*x)/2]^4)/(64*d) + (a*Sec[(c + d*x)/2]^6)/(384*d)","A",1
168,1,118,127,0.2116298,"\int (a+b \sec (c+d x)) \sin ^6(c+d x) \, dx","Integrate[(a + b*Sec[c + d*x])*Sin[c + d*x]^6,x]","\frac{5 a (c+d x)}{16 d}-\frac{15 a \sin (2 (c+d x))}{64 d}+\frac{3 a \sin (4 (c+d x))}{64 d}-\frac{a \sin (6 (c+d x))}{192 d}-\frac{b \sin ^5(c+d x)}{5 d}-\frac{b \sin ^3(c+d x)}{3 d}-\frac{b \sin (c+d x)}{d}+\frac{b \tanh ^{-1}(\sin (c+d x))}{d}","-\frac{a \sin ^5(c+d x) \cos (c+d x)}{6 d}-\frac{5 a \sin ^3(c+d x) \cos (c+d x)}{24 d}-\frac{5 a \sin (c+d x) \cos (c+d x)}{16 d}+\frac{5 a x}{16}-\frac{b \sin ^5(c+d x)}{5 d}-\frac{b \sin ^3(c+d x)}{3 d}-\frac{b \sin (c+d x)}{d}+\frac{b \tanh ^{-1}(\sin (c+d x))}{d}",1,"(5*a*(c + d*x))/(16*d) + (b*ArcTanh[Sin[c + d*x]])/d - (b*Sin[c + d*x])/d - (b*Sin[c + d*x]^3)/(3*d) - (b*Sin[c + d*x]^5)/(5*d) - (15*a*Sin[2*(c + d*x)])/(64*d) + (3*a*Sin[4*(c + d*x)])/(64*d) - (a*Sin[6*(c + d*x)])/(192*d)","A",1
169,1,86,89,0.1549838,"\int (a+b \sec (c+d x)) \sin ^4(c+d x) \, dx","Integrate[(a + b*Sec[c + d*x])*Sin[c + d*x]^4,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{b \tanh ^{-1}(\sin (c+d x))}{d}","-\frac{a \sin ^3(c+d x) \cos (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}+\frac{b \tanh ^{-1}(\sin (c+d x))}{d}",1,"(3*a*(c + d*x))/(8*d) + (b*ArcTanh[Sin[c + d*x]])/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
170,1,54,51,0.0633505,"\int (a+b \sec (c+d x)) \sin ^2(c+d x) \, dx","Integrate[(a + b*Sec[c + d*x])*Sin[c + d*x]^2,x]","\frac{a (c+d x)}{2 d}-\frac{a \sin (2 (c+d x))}{4 d}-\frac{b \sin (c+d x)}{d}+\frac{b \tanh ^{-1}(\sin (c+d x))}{d}","-\frac{a \sin (c+d x) \cos (c+d x)}{2 d}+\frac{a x}{2}-\frac{b \sin (c+d x)}{d}+\frac{b \tanh ^{-1}(\sin (c+d x))}{d}",1,"(a*(c + d*x))/(2*d) + (b*ArcTanh[Sin[c + d*x]])/d - (b*Sin[c + d*x])/d - (a*Sin[2*(c + d*x)])/(4*d)","A",1
171,1,41,37,0.0271534,"\int \csc ^2(c+d x) (a+b \sec (c+d x)) \, dx","Integrate[Csc[c + d*x]^2*(a + b*Sec[c + d*x]),x]","-\frac{a \cot (c+d x)}{d}-\frac{b \csc (c+d x) \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\sin ^2(c+d x)\right)}{d}","-\frac{a \cot (c+d x)}{d}-\frac{b \csc (c+d x)}{d}+\frac{b \tanh ^{-1}(\sin (c+d x))}{d}",1,"-((a*Cot[c + d*x])/d) - (b*Csc[c + d*x]*Hypergeometric2F1[-1/2, 1, 1/2, Sin[c + d*x]^2])/d","C",1
172,1,69,69,0.0264001,"\int \csc ^4(c+d x) (a+b \sec (c+d x)) \, dx","Integrate[Csc[c + d*x]^4*(a + b*Sec[c + d*x]),x]","-\frac{2 a \cot (c+d x)}{3 d}-\frac{a \cot (c+d x) \csc ^2(c+d x)}{3 d}-\frac{b \csc ^3(c+d x) \, _2F_1\left(-\frac{3}{2},1;-\frac{1}{2};\sin ^2(c+d x)\right)}{3 d}","-\frac{a \cot ^3(c+d x)}{3 d}-\frac{a \cot (c+d x)}{d}-\frac{b \csc ^3(c+d x)}{3 d}-\frac{b \csc (c+d x)}{d}+\frac{b \tanh ^{-1}(\sin (c+d x))}{d}",1,"(-2*a*Cot[c + d*x])/(3*d) - (a*Cot[c + d*x]*Csc[c + d*x]^2)/(3*d) - (b*Csc[c + d*x]^3*Hypergeometric2F1[-3/2, 1, -1/2, Sin[c + d*x]^2])/(3*d)","C",1
173,1,91,101,0.0273412,"\int \csc ^6(c+d x) (a+b \sec (c+d x)) \, dx","Integrate[Csc[c + d*x]^6*(a + b*Sec[c + d*x]),x]","-\frac{8 a \cot (c+d x)}{15 d}-\frac{a \cot (c+d x) \csc ^4(c+d x)}{5 d}-\frac{4 a \cot (c+d x) \csc ^2(c+d x)}{15 d}-\frac{b \csc ^5(c+d x) \, _2F_1\left(-\frac{5}{2},1;-\frac{3}{2};\sin ^2(c+d x)\right)}{5 d}","-\frac{a \cot ^5(c+d x)}{5 d}-\frac{2 a \cot ^3(c+d x)}{3 d}-\frac{a \cot (c+d x)}{d}-\frac{b \csc ^5(c+d x)}{5 d}-\frac{b \csc ^3(c+d x)}{3 d}-\frac{b \csc (c+d x)}{d}+\frac{b \tanh ^{-1}(\sin (c+d x))}{d}",1,"(-8*a*Cot[c + d*x])/(15*d) - (4*a*Cot[c + d*x]*Csc[c + d*x]^2)/(15*d) - (a*Cot[c + d*x]*Csc[c + d*x]^4)/(5*d) - (b*Csc[c + d*x]^5*Hypergeometric2F1[-5/2, 1, -3/2, Sin[c + d*x]^2])/(5*d)","C",1
174,1,112,124,0.3611018,"\int (a+b \sec (c+d x))^2 \sin ^5(c+d x) \, dx","Integrate[(a + b*Sec[c + d*x])^2*Sin[c + d*x]^5,x]","-\frac{30 \left(5 a^2-14 b^2\right) \cos (c+d x)-25 a^2 \cos (3 (c+d x))+3 a^2 \cos (5 (c+d x))-180 a b \cos (2 (c+d x))+15 a b \cos (4 (c+d x))+480 a b \log (\cos (c+d x))+20 b^2 \cos (3 (c+d x))-240 b^2 \sec (c+d x)}{240 d}","\frac{\left(2 a^2-b^2\right) \cos ^3(c+d x)}{3 d}-\frac{\left(a^2-2 b^2\right) \cos (c+d x)}{d}-\frac{a^2 \cos ^5(c+d x)}{5 d}-\frac{a b \cos ^4(c+d x)}{2 d}+\frac{2 a b \cos ^2(c+d x)}{d}-\frac{2 a b \log (\cos (c+d x))}{d}+\frac{b^2 \sec (c+d x)}{d}",1,"-1/240*(30*(5*a^2 - 14*b^2)*Cos[c + d*x] - 180*a*b*Cos[2*(c + d*x)] - 25*a^2*Cos[3*(c + d*x)] + 20*b^2*Cos[3*(c + d*x)] + 15*a*b*Cos[4*(c + d*x)] + 3*a^2*Cos[5*(c + d*x)] + 480*a*b*Log[Cos[c + d*x]] - 240*b^2*Sec[c + d*x])/d","A",1
175,1,72,80,0.1819081,"\int (a+b \sec (c+d x))^2 \sin ^3(c+d x) \, dx","Integrate[(a + b*Sec[c + d*x])^2*Sin[c + d*x]^3,x]","\frac{\left(12 b^2-9 a^2\right) \cos (c+d x)+a^2 \cos (3 (c+d x))+6 a b \cos (2 (c+d x))-24 a b \log (\cos (c+d x))+12 b^2 \sec (c+d x)}{12 d}","-\frac{\left(a^2-b^2\right) \cos (c+d x)}{d}+\frac{a^2 \cos ^3(c+d x)}{3 d}+\frac{a b \cos ^2(c+d x)}{d}-\frac{2 a b \log (\cos (c+d x))}{d}+\frac{b^2 \sec (c+d x)}{d}",1,"((-9*a^2 + 12*b^2)*Cos[c + d*x] + 6*a*b*Cos[2*(c + d*x)] + a^2*Cos[3*(c + d*x)] - 24*a*b*Log[Cos[c + d*x]] + 12*b^2*Sec[c + d*x])/(12*d)","A",1
176,1,37,42,0.0580499,"\int (a+b \sec (c+d x))^2 \sin (c+d x) \, dx","Integrate[(a + b*Sec[c + d*x])^2*Sin[c + d*x],x]","\frac{b (b \sec (c+d x)-2 a \log (\cos (c+d x)))-a^2 \cos (c+d x)}{d}","-\frac{a^2 \cos (c+d x)}{d}-\frac{2 a b \log (\cos (c+d x))}{d}+\frac{b^2 \sec (c+d x)}{d}",1,"(-(a^2*Cos[c + d*x]) + b*(-2*a*Log[Cos[c + d*x]] + b*Sec[c + d*x]))/d","A",1
177,1,91,74,0.1606275,"\int \csc (c+d x) (a+b \sec (c+d x))^2 \, dx","Integrate[Csc[c + d*x]*(a + b*Sec[c + d*x])^2,x]","\frac{a^2 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+2 a b \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-2 a b \log (\cos (c+d x))-(a-b)^2 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)+b^2 \sec (c+d x)+b^2 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)}{d}","-\frac{(a-b)^2 \log (\cos (c+d x)+1)}{2 d}+\frac{(a+b)^2 \log (1-\cos (c+d x))}{2 d}-\frac{2 a b \log (\cos (c+d x))}{d}+\frac{b^2 \sec (c+d x)}{d}",1,"(-((a - b)^2*Log[Cos[(c + d*x)/2]]) - 2*a*b*Log[Cos[c + d*x]] + a^2*Log[Sin[(c + d*x)/2]] + 2*a*b*Log[Sin[(c + d*x)/2]] + b^2*Log[Sin[(c + d*x)/2]] + b^2*Sec[c + d*x])/d","A",1
178,1,329,114,0.6333878,"\int \csc ^3(c+d x) (a+b \sec (c+d x))^2 \, dx","Integrate[Csc[c + d*x]^3*(a + b*Sec[c + d*x])^2,x]","-\frac{\csc ^4(c+d x) \left(2 \left(a^2+3 b^2\right) \cos (2 (c+d x))+\cos (c+d x) \left(\left(a^2-4 a b+3 b^2\right) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)+a^2 \left(-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)\right)-4 a b \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+4 a b \log (\cos (c+d x))+8 a b-3 b^2 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)\right)+a^2 (-\cos (3 (c+d x))) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)+a^2 \cos (3 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+2 a^2+4 a b \cos (3 (c+d x)) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)-4 a b \cos (3 (c+d x)) \log (\cos (c+d x))+4 a b \cos (3 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-3 b^2 \cos (3 (c+d x)) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)+3 b^2 \cos (3 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-2 b^2\right)}{2 d \left(\csc ^2\left(\frac{1}{2} (c+d x)\right)-\sec ^2\left(\frac{1}{2} (c+d x)\right)\right)}","-\frac{\csc ^2(c+d x) \left(\left(a^2+b^2\right) \cos (c+d x)+2 a b\right)}{2 d}-\frac{2 a b \log (\cos (c+d x))}{d}+\frac{(a+b) (a+3 b) \log (1-\cos (c+d x))}{4 d}-\frac{(a-3 b) (a-b) \log (\cos (c+d x)+1)}{4 d}+\frac{b^2 \sec (c+d x)}{d}",1,"-1/2*(Csc[c + d*x]^4*(2*a^2 - 2*b^2 + 2*(a^2 + 3*b^2)*Cos[2*(c + d*x)] - a^2*Cos[3*(c + d*x)]*Log[Cos[(c + d*x)/2]] + 4*a*b*Cos[3*(c + d*x)]*Log[Cos[(c + d*x)/2]] - 3*b^2*Cos[3*(c + d*x)]*Log[Cos[(c + d*x)/2]] - 4*a*b*Cos[3*(c + d*x)]*Log[Cos[c + d*x]] + a^2*Cos[3*(c + d*x)]*Log[Sin[(c + d*x)/2]] + 4*a*b*Cos[3*(c + d*x)]*Log[Sin[(c + d*x)/2]] + 3*b^2*Cos[3*(c + d*x)]*Log[Sin[(c + d*x)/2]] + Cos[c + d*x]*(8*a*b + (a^2 - 4*a*b + 3*b^2)*Log[Cos[(c + d*x)/2]] + 4*a*b*Log[Cos[c + d*x]] - a^2*Log[Sin[(c + d*x)/2]] - 4*a*b*Log[Sin[(c + d*x)/2]] - 3*b^2*Log[Sin[(c + d*x)/2]])))/(d*(Csc[(c + d*x)/2]^2 - Sec[(c + d*x)/2]^2))","B",1
179,1,193,175,1.6811526,"\int (a+b \sec (c+d x))^2 \sin ^6(c+d x) \, dx","Integrate[(a + b*Sec[c + d*x])^2*Sin[c + d*x]^6,x]","\frac{\tan (c+d x) \left(-5 \left(29 a^2-84 b^2\right) \cos (2 (c+d x))+35 a^2 \cos (4 (c+d x))-5 a^2 \cos (6 (c+d x))-185 a^2+232 a b \cos (3 (c+d x))-24 a b \cos (5 (c+d x))-30 b^2 \cos (4 (c+d x))+1410 b^2\right)+60 \left(5 \left(a^2-6 b^2\right) (c+d x)-32 a b \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+32 a b \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)-2128 a b \sin (c+d x)}{960 d}","\frac{\left(13 a^2-6 b^2\right) \sin (c+d x) \cos ^3(c+d x)}{24 d}-\frac{\left(11 a^2-18 b^2\right) \sin (c+d x) \cos (c+d x)}{16 d}+\frac{5}{16} x \left(a^2-6 b^2\right)-\frac{a^2 \sin (c+d x) \cos ^5(c+d x)}{6 d}-\frac{2 a b \sin ^5(c+d x)}{5 d}-\frac{2 a b \sin ^3(c+d x)}{3 d}-\frac{2 a b \sin (c+d x)}{d}+\frac{2 a b \tanh ^{-1}(\sin (c+d x))}{d}+\frac{b^2 \tan (c+d x)}{d}",1,"(60*(5*(a^2 - 6*b^2)*(c + d*x) - 32*a*b*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 32*a*b*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) - 2128*a*b*Sin[c + d*x] + (-185*a^2 + 1410*b^2 - 5*(29*a^2 - 84*b^2)*Cos[2*(c + d*x)] + 232*a*b*Cos[3*(c + d*x)] + 35*a^2*Cos[4*(c + d*x)] - 30*b^2*Cos[4*(c + d*x)] - 24*a*b*Cos[5*(c + d*x)] - 5*a^2*Cos[6*(c + d*x)])*Tan[c + d*x])/(960*d)","A",1
180,1,157,178,1.0462937,"\int (a+b \sec (c+d x))^2 \sin ^4(c+d x) \, dx","Integrate[(a + b*Sec[c + d*x])^2*Sin[c + d*x]^4,x]","\frac{\tan (c+d x) \left(-6 \left(3 a^2-4 b^2\right) \cos (2 (c+d x))+3 \left(a^2 \cos (4 (c+d x))-7 a^2+40 b^2\right)+16 a b \cos (3 (c+d x))\right)+12 \left(3 \left(a^2-4 b^2\right) (c+d x)-16 a b \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+16 a b \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)-208 a b \sin (c+d x)}{96 d}","-\frac{b \left(28 a^2+b^2\right) \sin (c+d x)}{6 a d}-\frac{\left(12 a^2+b^2\right) \sin (c+d x) (a \cos (c+d x)+b)^2}{12 a b d}-\frac{\left(39 a^2+2 b^2\right) \sin (c+d x) \cos (c+d x)}{24 d}+\frac{3}{8} x \left(a^2-4 b^2\right)+\frac{2 a b \tanh ^{-1}(\sin (c+d x))}{d}+\frac{\sin (c+d x) (a \cos (c+d x)+b)^3}{4 a d}+\frac{\tan (c+d x) (a \cos (c+d x)+b)^3}{b d}",1,"(12*(3*(a^2 - 4*b^2)*(c + d*x) - 16*a*b*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 16*a*b*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) - 208*a*b*Sin[c + d*x] + (-6*(3*a^2 - 4*b^2)*Cos[2*(c + d*x)] + 16*a*b*Cos[3*(c + d*x)] + 3*(-7*a^2 + 40*b^2 + a^2*Cos[4*(c + d*x)]))*Tan[c + d*x])/(96*d)","A",1
181,1,121,77,0.6036086,"\int (a+b \sec (c+d x))^2 \sin ^2(c+d x) \, dx","Integrate[(a + b*Sec[c + d*x])^2*Sin[c + d*x]^2,x]","-\frac{a^2 \sin (2 (c+d x))-2 a^2 c-2 a^2 d x+8 a b \sin (c+d x)+8 a b \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-8 a b \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)-4 b^2 \tan (c+d x)+4 b^2 c+4 b^2 d x}{4 d}","-\frac{a^2 \sin (c+d x) \cos (c+d x)}{2 d}+\frac{a^2 x}{2}-\frac{2 a b \sin (c+d x)}{d}+\frac{2 a b \tanh ^{-1}(\sin (c+d x))}{d}+\frac{b^2 \tan (c+d x)}{d}-b^2 x",1,"-1/4*(-2*a^2*c + 4*b^2*c - 2*a^2*d*x + 4*b^2*d*x + 8*a*b*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - 8*a*b*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + 8*a*b*Sin[c + d*x] + a^2*Sin[2*(c + d*x)] - 4*b^2*Tan[c + d*x])/d","A",1
182,1,138,59,0.4882144,"\int \csc ^2(c+d x) (a+b \sec (c+d x))^2 \, dx","Integrate[Csc[c + d*x]^2*(a + b*Sec[c + d*x])^2,x]","-\frac{\csc ^3\left(\frac{1}{2} (c+d x)\right) \sec \left(\frac{1}{2} (c+d x)\right) \left(\left(a^2+2 b^2\right) \cos (2 (c+d x))+4 a b \cos (c+d x)+a \left(a+2 b \sin (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)\right)\right)}{4 d \left(\cot ^2\left(\frac{1}{2} (c+d x)\right)-1\right)}","-\frac{\left(a^2+b^2\right) \cot (c+d x)}{d}-\frac{2 a b \csc (c+d x)}{d}+\frac{2 a b \tanh ^{-1}(\sin (c+d x))}{d}+\frac{b^2 \tan (c+d x)}{d}",1,"-1/4*(Csc[(c + d*x)/2]^3*Sec[(c + d*x)/2]*(4*a*b*Cos[c + d*x] + (a^2 + 2*b^2)*Cos[2*(c + d*x)] + a*(a + 2*b*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]])*Sin[2*(c + d*x)])))/(d*(-1 + Cot[(c + d*x)/2]^2))","B",1
183,1,259,100,0.6615213,"\int \csc ^4(c+d x) (a+b \sec (c+d x))^2 \, dx","Integrate[Csc[c + d*x]^4*(a + b*Sec[c + d*x])^2,x]","\frac{\csc ^5\left(\frac{1}{2} (c+d x)\right) \sec ^3\left(\frac{1}{2} (c+d x)\right) \left(-2 \left(a^2+4 b^2\right) \cos (2 (c+d x))+a^2 \cos (4 (c+d x))-3 a^2-14 a b \cos (c+d x)+6 a b \cos (3 (c+d x))-6 a b \sin (2 (c+d x)) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+6 a b \sin (2 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+3 a b \sin (4 (c+d x)) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-3 a b \sin (4 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+4 b^2 \cos (4 (c+d x))\right)}{96 d \left(\cot ^2\left(\frac{1}{2} (c+d x)\right)-1\right)}","-\frac{\left(a^2+b^2\right) \cot ^3(c+d x)}{3 d}-\frac{\left(a^2+2 b^2\right) \cot (c+d x)}{d}-\frac{2 a b \csc ^3(c+d x)}{3 d}-\frac{2 a b \csc (c+d x)}{d}+\frac{2 a b \tanh ^{-1}(\sin (c+d x))}{d}+\frac{b^2 \tan (c+d x)}{d}",1,"(Csc[(c + d*x)/2]^5*Sec[(c + d*x)/2]^3*(-3*a^2 - 14*a*b*Cos[c + d*x] - 2*(a^2 + 4*b^2)*Cos[2*(c + d*x)] + 6*a*b*Cos[3*(c + d*x)] + a^2*Cos[4*(c + d*x)] + 4*b^2*Cos[4*(c + d*x)] - 6*a*b*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]]*Sin[2*(c + d*x)] + 6*a*b*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]*Sin[2*(c + d*x)] + 3*a*b*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]]*Sin[4*(c + d*x)] - 3*a*b*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]*Sin[4*(c + d*x)]))/(96*d*(-1 + Cot[(c + d*x)/2]^2))","B",1
184,1,368,143,0.7390944,"\int \csc ^6(c+d x) (a+b \sec (c+d x))^2 \, dx","Integrate[Csc[c + d*x]^6*(a + b*Sec[c + d*x])^2,x]","-\frac{\csc ^7\left(\frac{1}{2} (c+d x)\right) \sec ^5\left(\frac{1}{2} (c+d x)\right) \left(20 \left(a^2+6 b^2\right) \cos (2 (c+d x))-16 a^2 \cos (4 (c+d x))+4 a^2 \cos (6 (c+d x))+40 a^2+196 a b \cos (c+d x)-130 a b \cos (3 (c+d x))+30 a b \cos (5 (c+d x))+75 a b \sin (2 (c+d x)) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-75 a b \sin (2 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)-60 a b \sin (4 (c+d x)) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+60 a b \sin (4 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+15 a b \sin (6 (c+d x)) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-15 a b \sin (6 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)-96 b^2 \cos (4 (c+d x))+24 b^2 \cos (6 (c+d x))\right)}{7680 d \left(\cot ^2\left(\frac{1}{2} (c+d x)\right)-1\right)}","-\frac{\left(a^2+b^2\right) \cot ^5(c+d x)}{5 d}-\frac{\left(2 a^2+3 b^2\right) \cot ^3(c+d x)}{3 d}-\frac{\left(a^2+3 b^2\right) \cot (c+d x)}{d}-\frac{2 a b \csc ^5(c+d x)}{5 d}-\frac{2 a b \csc ^3(c+d x)}{3 d}-\frac{2 a b \csc (c+d x)}{d}+\frac{2 a b \tanh ^{-1}(\sin (c+d x))}{d}+\frac{b^2 \tan (c+d x)}{d}",1,"-1/7680*(Csc[(c + d*x)/2]^7*Sec[(c + d*x)/2]^5*(40*a^2 + 196*a*b*Cos[c + d*x] + 20*(a^2 + 6*b^2)*Cos[2*(c + d*x)] - 130*a*b*Cos[3*(c + d*x)] - 16*a^2*Cos[4*(c + d*x)] - 96*b^2*Cos[4*(c + d*x)] + 30*a*b*Cos[5*(c + d*x)] + 4*a^2*Cos[6*(c + d*x)] + 24*b^2*Cos[6*(c + d*x)] + 75*a*b*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]]*Sin[2*(c + d*x)] - 75*a*b*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]*Sin[2*(c + d*x)] - 60*a*b*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]]*Sin[4*(c + d*x)] + 60*a*b*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]*Sin[4*(c + d*x)] + 15*a*b*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]]*Sin[6*(c + d*x)] - 15*a*b*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]*Sin[6*(c + d*x)]))/(d*(-1 + Cot[(c + d*x)/2]^2))","B",1
185,1,154,170,0.690685,"\int (a+b \sec (c+d x))^3 \sin ^5(c+d x) \, dx","Integrate[(a + b*Sec[c + d*x])^3*Sin[c + d*x]^5,x]","\frac{50 a^3 \cos (3 (c+d x))-6 a^3 \cos (5 (c+d x))+60 \left(9 a^2 b-2 b^3\right) \cos (2 (c+d x))-60 a \left(5 a^2-42 b^2\right) \cos (c+d x)-45 a^2 b \cos (4 (c+d x))-1440 a^2 b \log (\cos (c+d x))-120 a b^2 \cos (3 (c+d x))+1440 a b^2 \sec (c+d x)+240 b^3 \sec ^2(c+d x)+960 b^3 \log (\cos (c+d x))}{480 d}","-\frac{a^3 \cos ^5(c+d x)}{5 d}+\frac{a \left(2 a^2-3 b^2\right) \cos ^3(c+d x)}{3 d}+\frac{b \left(6 a^2-b^2\right) \cos ^2(c+d x)}{2 d}-\frac{a \left(a^2-6 b^2\right) \cos (c+d x)}{d}-\frac{b \left(3 a^2-2 b^2\right) \log (\cos (c+d x))}{d}-\frac{3 a^2 b \cos ^4(c+d x)}{4 d}+\frac{3 a b^2 \sec (c+d x)}{d}+\frac{b^3 \sec ^2(c+d x)}{2 d}",1,"(-60*a*(5*a^2 - 42*b^2)*Cos[c + d*x] + 60*(9*a^2*b - 2*b^3)*Cos[2*(c + d*x)] + 50*a^3*Cos[3*(c + d*x)] - 120*a*b^2*Cos[3*(c + d*x)] - 45*a^2*b*Cos[4*(c + d*x)] - 6*a^3*Cos[5*(c + d*x)] - 1440*a^2*b*Log[Cos[c + d*x]] + 960*b^3*Log[Cos[c + d*x]] + 1440*a*b^2*Sec[c + d*x] + 240*b^3*Sec[c + d*x]^2)/(480*d)","A",1
186,1,102,116,0.3503667,"\int (a+b \sec (c+d x))^3 \sin ^3(c+d x) \, dx","Integrate[(a + b*Sec[c + d*x])^3*Sin[c + d*x]^3,x]","\frac{a^3 \cos (3 (c+d x))-9 a \left(a^2-4 b^2\right) \cos (c+d x)+9 a^2 b \cos (2 (c+d x))-36 a^2 b \log (\cos (c+d x))+36 a b^2 \sec (c+d x)+6 b^3 \sec ^2(c+d x)+12 b^3 \log (\cos (c+d x))}{12 d}","\frac{a^3 \cos ^3(c+d x)}{3 d}-\frac{a \left(a^2-3 b^2\right) \cos (c+d x)}{d}-\frac{b \left(3 a^2-b^2\right) \log (\cos (c+d x))}{d}+\frac{3 a^2 b \cos ^2(c+d x)}{2 d}+\frac{3 a b^2 \sec (c+d x)}{d}+\frac{b^3 \sec ^2(c+d x)}{2 d}",1,"(-9*a*(a^2 - 4*b^2)*Cos[c + d*x] + 9*a^2*b*Cos[2*(c + d*x)] + a^3*Cos[3*(c + d*x)] - 36*a^2*b*Log[Cos[c + d*x]] + 12*b^3*Log[Cos[c + d*x]] + 36*a*b^2*Sec[c + d*x] + 6*b^3*Sec[c + d*x]^2)/(12*d)","A",1
187,1,56,64,0.1186531,"\int (a+b \sec (c+d x))^3 \sin (c+d x) \, dx","Integrate[(a + b*Sec[c + d*x])^3*Sin[c + d*x],x]","\frac{b \left(-6 a^2 \log (\cos (c+d x))+6 a b \sec (c+d x)+b^2 \sec ^2(c+d x)\right)-2 a^3 \cos (c+d x)}{2 d}","-\frac{a^3 \cos (c+d x)}{d}-\frac{3 a^2 b \log (\cos (c+d x))}{d}+\frac{3 a b^2 \sec (c+d x)}{d}+\frac{b^3 \sec ^2(c+d x)}{2 d}",1,"(-2*a^3*Cos[c + d*x] + b*(-6*a^2*Log[Cos[c + d*x]] + 6*a*b*Sec[c + d*x] + b^2*Sec[c + d*x]^2))/(2*d)","A",1
188,1,89,102,0.3003453,"\int \csc (c+d x) (a+b \sec (c+d x))^3 \, dx","Integrate[Csc[c + d*x]*(a + b*Sec[c + d*x])^3,x]","\frac{-2 b \left(3 a^2+b^2\right) \log (\cos (c+d x))+6 a b^2 \sec (c+d x)+2 (a+b)^3 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-2 (a-b)^3 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)+b^3 \sec ^2(c+d x)}{2 d}","-\frac{b \left(3 a^2+b^2\right) \log (\cos (c+d x))}{d}+\frac{3 a b^2 \sec (c+d x)}{d}-\frac{(a-b)^3 \log (\cos (c+d x)+1)}{2 d}+\frac{(a+b)^3 \log (1-\cos (c+d x))}{2 d}+\frac{b^3 \sec ^2(c+d x)}{2 d}",1,"(-2*(a - b)^3*Log[Cos[(c + d*x)/2]] - 2*b*(3*a^2 + b^2)*Log[Cos[c + d*x]] + 2*(a + b)^3*Log[Sin[(c + d*x)/2]] + 6*a*b^2*Sec[c + d*x] + b^3*Sec[c + d*x]^2)/(2*d)","A",1
189,1,669,162,6.1989367,"\int \csc ^3(c+d x) (a+b \sec (c+d x))^3 \, dx","Integrate[Csc[c + d*x]^3*(a + b*Sec[c + d*x])^3,x]","\frac{\left(-3 a^2 b-2 b^3\right) \cos ^3(c+d x) \log (\cos (c+d x)) (a+b \sec (c+d x))^3}{d (a \cos (c+d x)+b)^3}+\frac{\left(a^3-3 a^2 b+3 a b^2-b^3\right) \cos ^3(c+d x) \sec ^2\left(\frac{1}{2} (c+d x)\right) (a+b \sec (c+d x))^3}{8 d (a \cos (c+d x)+b)^3}+\frac{\left(-a^3+6 a^2 b-9 a b^2+4 b^3\right) \cos ^3(c+d x) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right) (a+b \sec (c+d x))^3}{2 d (a \cos (c+d x)+b)^3}+\frac{\left(-a^3-3 a^2 b-3 a b^2-b^3\right) \cos ^3(c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right) (a+b \sec (c+d x))^3}{8 d (a \cos (c+d x)+b)^3}+\frac{\left(a^3+6 a^2 b+9 a b^2+4 b^3\right) \cos ^3(c+d x) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right) (a+b \sec (c+d x))^3}{2 d (a \cos (c+d x)+b)^3}+\frac{b^3 \cos ^3(c+d x) (a+b \sec (c+d x))^3}{4 d \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^2 (a \cos (c+d x)+b)^3}+\frac{b^3 \cos ^3(c+d x) (a+b \sec (c+d x))^3}{4 d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2 (a \cos (c+d x)+b)^3}+\frac{3 a b^2 \cos ^3(c+d x) (a+b \sec (c+d x))^3}{d (a \cos (c+d x)+b)^3}+\frac{3 a b^2 \sin \left(\frac{1}{2} (c+d x)\right) \cos ^3(c+d x) (a+b \sec (c+d x))^3}{d \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right) (a \cos (c+d x)+b)^3}-\frac{3 a b^2 \sin \left(\frac{1}{2} (c+d x)\right) \cos ^3(c+d x) (a+b \sec (c+d x))^3}{d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right) (a \cos (c+d x)+b)^3}","-\frac{b \left(3 a^2+2 b^2\right) \log (\cos (c+d x))}{d}-\frac{a^2 \csc ^2(c+d x) \left(a \left(\frac{3 b^2}{a^2}+1\right) \cos (c+d x)+b \left(\frac{b^2}{a^2}+3\right)\right)}{2 d}+\frac{3 a b^2 \sec (c+d x)}{d}+\frac{(a+b)^2 (a+4 b) \log (1-\cos (c+d x))}{4 d}-\frac{(a-4 b) (a-b)^2 \log (\cos (c+d x)+1)}{4 d}+\frac{b^3 \sec ^2(c+d x)}{2 d}",1,"(3*a*b^2*Cos[c + d*x]^3*(a + b*Sec[c + d*x])^3)/(d*(b + a*Cos[c + d*x])^3) + ((-a^3 - 3*a^2*b - 3*a*b^2 - b^3)*Cos[c + d*x]^3*Csc[(c + d*x)/2]^2*(a + b*Sec[c + d*x])^3)/(8*d*(b + a*Cos[c + d*x])^3) + ((-a^3 + 6*a^2*b - 9*a*b^2 + 4*b^3)*Cos[c + d*x]^3*Log[Cos[(c + d*x)/2]]*(a + b*Sec[c + d*x])^3)/(2*d*(b + a*Cos[c + d*x])^3) + ((-3*a^2*b - 2*b^3)*Cos[c + d*x]^3*Log[Cos[c + d*x]]*(a + b*Sec[c + d*x])^3)/(d*(b + a*Cos[c + d*x])^3) + ((a^3 + 6*a^2*b + 9*a*b^2 + 4*b^3)*Cos[c + d*x]^3*Log[Sin[(c + d*x)/2]]*(a + b*Sec[c + d*x])^3)/(2*d*(b + a*Cos[c + d*x])^3) + ((a^3 - 3*a^2*b + 3*a*b^2 - b^3)*Cos[c + d*x]^3*Sec[(c + d*x)/2]^2*(a + b*Sec[c + d*x])^3)/(8*d*(b + a*Cos[c + d*x])^3) + (b^3*Cos[c + d*x]^3*(a + b*Sec[c + d*x])^3)/(4*d*(b + a*Cos[c + d*x])^3*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2) + (3*a*b^2*Cos[c + d*x]^3*(a + b*Sec[c + d*x])^3*Sin[(c + d*x)/2])/(d*(b + a*Cos[c + d*x])^3*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])) + (b^3*Cos[c + d*x]^3*(a + b*Sec[c + d*x])^3)/(4*d*(b + a*Cos[c + d*x])^3*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2) - (3*a*b^2*Cos[c + d*x]^3*(a + b*Sec[c + d*x])^3*Sin[(c + d*x)/2])/(d*(b + a*Cos[c + d*x])^3*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))","B",1
190,1,818,299,6.2583659,"\int (a+b \sec (c+d x))^3 \sin ^6(c+d x) \, dx","Integrate[(a + b*Sec[c + d*x])^3*Sin[c + d*x]^6,x]","\frac{\left(5 b^3-6 a^2 b\right) \cos ^3(c+d x) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right) (a+b \sec (c+d x))^3}{2 d (b+a \cos (c+d x))^3}+\frac{\left(6 a^2 b-5 b^3\right) \cos ^3(c+d x) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)+\sin \left(\frac{1}{2} (c+d x)\right)\right) (a+b \sec (c+d x))^3}{2 d (b+a \cos (c+d x))^3}+\frac{3 a b^2 \cos ^3(c+d x) \sin \left(\frac{1}{2} (c+d x)\right) (a+b \sec (c+d x))^3}{d (b+a \cos (c+d x))^3 \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}+\frac{3 b \left(6 b^2-11 a^2\right) \cos ^3(c+d x) \sin (c+d x) (a+b \sec (c+d x))^3}{8 d (b+a \cos (c+d x))^3}-\frac{3 a \left(5 a^2-32 b^2\right) \cos ^3(c+d x) \sin (2 (c+d x)) (a+b \sec (c+d x))^3}{64 d (b+a \cos (c+d x))^3}-\frac{b \left(4 b^2-21 a^2\right) \cos ^3(c+d x) \sin (3 (c+d x)) (a+b \sec (c+d x))^3}{48 d (b+a \cos (c+d x))^3}+\frac{3 a \left(a^2-2 b^2\right) \cos ^3(c+d x) \sin (4 (c+d x)) (a+b \sec (c+d x))^3}{64 d (b+a \cos (c+d x))^3}-\frac{3 a^2 b \cos ^3(c+d x) \sin (5 (c+d x)) (a+b \sec (c+d x))^3}{80 d (b+a \cos (c+d x))^3}-\frac{a^3 \cos ^3(c+d x) \sin (6 (c+d x)) (a+b \sec (c+d x))^3}{192 d (b+a \cos (c+d x))^3}+\frac{3 a b^2 \cos ^3(c+d x) \sin \left(\frac{1}{2} (c+d x)\right) (a+b \sec (c+d x))^3}{d (b+a \cos (c+d x))^3 \left(\cos \left(\frac{1}{2} (c+d x)\right)+\sin \left(\frac{1}{2} (c+d x)\right)\right)}+\frac{b^3 \cos ^3(c+d x) (a+b \sec (c+d x))^3}{4 d (b+a \cos (c+d x))^3 \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^2}-\frac{b^3 \cos ^3(c+d x) (a+b \sec (c+d x))^3}{4 d (b+a \cos (c+d x))^3 \left(\cos \left(\frac{1}{2} (c+d x)\right)+\sin \left(\frac{1}{2} (c+d x)\right)\right)^2}+\frac{5 a \left(a^2-18 b^2\right) (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^3}{16 d (b+a \cos (c+d x))^3}","-\frac{a^3 \sin ^5(c+d x) \cos (c+d x)}{6 d}-\frac{5 a^3 \sin ^3(c+d x) \cos (c+d x)}{24 d}-\frac{5 a^3 \sin (c+d x) \cos (c+d x)}{16 d}+\frac{5 a^3 x}{16}-\frac{3 a^2 b \sin ^5(c+d x)}{5 d}-\frac{a^2 b \sin ^3(c+d x)}{d}-\frac{3 a^2 b \sin (c+d x)}{d}+\frac{3 a^2 b \tanh ^{-1}(\sin (c+d x))}{d}+\frac{45 a b^2 \tan (c+d x)}{8 d}-\frac{3 a b^2 \sin ^4(c+d x) \tan (c+d x)}{4 d}-\frac{15 a b^2 \sin ^2(c+d x) \tan (c+d x)}{8 d}-\frac{45}{8} a b^2 x+\frac{5 b^3 \sin ^3(c+d x)}{6 d}+\frac{5 b^3 \sin (c+d x)}{2 d}+\frac{b^3 \sin ^3(c+d x) \tan ^2(c+d x)}{2 d}-\frac{5 b^3 \tanh ^{-1}(\sin (c+d x))}{2 d}",1,"(5*a*(a^2 - 18*b^2)*(c + d*x)*Cos[c + d*x]^3*(a + b*Sec[c + d*x])^3)/(16*d*(b + a*Cos[c + d*x])^3) + ((-6*a^2*b + 5*b^3)*Cos[c + d*x]^3*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]]*(a + b*Sec[c + d*x])^3)/(2*d*(b + a*Cos[c + d*x])^3) + ((6*a^2*b - 5*b^3)*Cos[c + d*x]^3*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]*(a + b*Sec[c + d*x])^3)/(2*d*(b + a*Cos[c + d*x])^3) + (b^3*Cos[c + d*x]^3*(a + b*Sec[c + d*x])^3)/(4*d*(b + a*Cos[c + d*x])^3*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2) + (3*a*b^2*Cos[c + d*x]^3*(a + b*Sec[c + d*x])^3*Sin[(c + d*x)/2])/(d*(b + a*Cos[c + d*x])^3*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])) - (b^3*Cos[c + d*x]^3*(a + b*Sec[c + d*x])^3)/(4*d*(b + a*Cos[c + d*x])^3*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2) + (3*a*b^2*Cos[c + d*x]^3*(a + b*Sec[c + d*x])^3*Sin[(c + d*x)/2])/(d*(b + a*Cos[c + d*x])^3*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])) + (3*b*(-11*a^2 + 6*b^2)*Cos[c + d*x]^3*(a + b*Sec[c + d*x])^3*Sin[c + d*x])/(8*d*(b + a*Cos[c + d*x])^3) - (3*a*(5*a^2 - 32*b^2)*Cos[c + d*x]^3*(a + b*Sec[c + d*x])^3*Sin[2*(c + d*x)])/(64*d*(b + a*Cos[c + d*x])^3) - (b*(-21*a^2 + 4*b^2)*Cos[c + d*x]^3*(a + b*Sec[c + d*x])^3*Sin[3*(c + d*x)])/(48*d*(b + a*Cos[c + d*x])^3) + (3*a*(a^2 - 2*b^2)*Cos[c + d*x]^3*(a + b*Sec[c + d*x])^3*Sin[4*(c + d*x)])/(64*d*(b + a*Cos[c + d*x])^3) - (3*a^2*b*Cos[c + d*x]^3*(a + b*Sec[c + d*x])^3*Sin[5*(c + d*x)])/(80*d*(b + a*Cos[c + d*x])^3) - (a^3*Cos[c + d*x]^3*(a + b*Sec[c + d*x])^3*Sin[6*(c + d*x)])/(192*d*(b + a*Cos[c + d*x])^3)","B",1
191,1,696,236,6.192225,"\int (a+b \sec (c+d x))^3 \sin ^4(c+d x) \, dx","Integrate[(a + b*Sec[c + d*x])^3*Sin[c + d*x]^4,x]","\frac{a^3 \sin (4 (c+d x)) \cos ^3(c+d x) (a+b \sec (c+d x))^3}{32 d (a \cos (c+d x)+b)^3}+\frac{3 \left(b^3-2 a^2 b\right) \cos ^3(c+d x) (a+b \sec (c+d x))^3 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}{2 d (a \cos (c+d x)+b)^3}-\frac{3 \left(b^3-2 a^2 b\right) \cos ^3(c+d x) (a+b \sec (c+d x))^3 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}{2 d (a \cos (c+d x)+b)^3}+\frac{3 a \left(a^2-12 b^2\right) (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^3}{8 d (a \cos (c+d x)+b)^3}+\frac{b \left(4 b^2-15 a^2\right) \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^3}{4 d (a \cos (c+d x)+b)^3}-\frac{a \left(a^2-3 b^2\right) \sin (2 (c+d x)) \cos ^3(c+d x) (a+b \sec (c+d x))^3}{4 d (a \cos (c+d x)+b)^3}+\frac{a^2 b \sin (3 (c+d x)) \cos ^3(c+d x) (a+b \sec (c+d x))^3}{4 d (a \cos (c+d x)+b)^3}+\frac{b^3 \cos ^3(c+d x) (a+b \sec (c+d x))^3}{4 d \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^2 (a \cos (c+d x)+b)^3}-\frac{b^3 \cos ^3(c+d x) (a+b \sec (c+d x))^3}{4 d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2 (a \cos (c+d x)+b)^3}+\frac{3 a b^2 \sin \left(\frac{1}{2} (c+d x)\right) \cos ^3(c+d x) (a+b \sec (c+d x))^3}{d \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right) (a \cos (c+d x)+b)^3}+\frac{3 a b^2 \sin \left(\frac{1}{2} (c+d x)\right) \cos ^3(c+d x) (a+b \sec (c+d x))^3}{d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right) (a \cos (c+d x)+b)^3}","-\frac{b \left(17 a^2-b^2\right) \sin (c+d x)}{2 d}+\frac{3 b \left(2 a^2-b^2\right) \tanh ^{-1}(\sin (c+d x))}{2 d}-\frac{\left(4 a^2-b^2\right) \sin (c+d x) (a \cos (c+d x)+b)^3}{4 b^2 d}-\frac{\left(6 a^2-b^2\right) \sin (c+d x) (a \cos (c+d x)+b)^2}{4 b d}-\frac{a \left(21 a^2-2 b^2\right) \sin (c+d x) \cos (c+d x)}{8 d}+\frac{3}{8} a x \left(a^2-12 b^2\right)+\frac{a \tan (c+d x) (a \cos (c+d x)+b)^4}{b^2 d}+\frac{\tan (c+d x) \sec (c+d x) (a \cos (c+d x)+b)^4}{2 b d}",1,"(3*a*(a^2 - 12*b^2)*(c + d*x)*Cos[c + d*x]^3*(a + b*Sec[c + d*x])^3)/(8*d*(b + a*Cos[c + d*x])^3) + (3*(-2*a^2*b + b^3)*Cos[c + d*x]^3*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]]*(a + b*Sec[c + d*x])^3)/(2*d*(b + a*Cos[c + d*x])^3) - (3*(-2*a^2*b + b^3)*Cos[c + d*x]^3*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]*(a + b*Sec[c + d*x])^3)/(2*d*(b + a*Cos[c + d*x])^3) + (b^3*Cos[c + d*x]^3*(a + b*Sec[c + d*x])^3)/(4*d*(b + a*Cos[c + d*x])^3*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2) + (3*a*b^2*Cos[c + d*x]^3*(a + b*Sec[c + d*x])^3*Sin[(c + d*x)/2])/(d*(b + a*Cos[c + d*x])^3*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])) - (b^3*Cos[c + d*x]^3*(a + b*Sec[c + d*x])^3)/(4*d*(b + a*Cos[c + d*x])^3*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2) + (3*a*b^2*Cos[c + d*x]^3*(a + b*Sec[c + d*x])^3*Sin[(c + d*x)/2])/(d*(b + a*Cos[c + d*x])^3*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])) + (b*(-15*a^2 + 4*b^2)*Cos[c + d*x]^3*(a + b*Sec[c + d*x])^3*Sin[c + d*x])/(4*d*(b + a*Cos[c + d*x])^3) - (a*(a^2 - 3*b^2)*Cos[c + d*x]^3*(a + b*Sec[c + d*x])^3*Sin[2*(c + d*x)])/(4*d*(b + a*Cos[c + d*x])^3) + (a^2*b*Cos[c + d*x]^3*(a + b*Sec[c + d*x])^3*Sin[3*(c + d*x)])/(4*d*(b + a*Cos[c + d*x])^3) + (a^3*Cos[c + d*x]^3*(a + b*Sec[c + d*x])^3*Sin[4*(c + d*x)])/(32*d*(b + a*Cos[c + d*x])^3)","B",1
192,1,327,138,0.9670809,"\int (a+b \sec (c+d x))^3 \sin ^2(c+d x) \, dx","Integrate[(a + b*Sec[c + d*x])^3*Sin[c + d*x]^2,x]","\frac{\sec ^2(c+d x) \left(-\frac{1}{2} a^3 \sin (2 (c+d x))-\frac{1}{4} a^3 \sin (4 (c+d x))+a^3 c+a^3 d x+\left(2 b^3-3 a^2 b\right) \sin (c+d x)+\cos (2 (c+d x)) \left(\left(b^3-6 a^2 b\right) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+a \left(a^2-6 b^2\right) (c+d x)-b \left(b^2-6 a^2\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^2 b \sin (3 (c+d x))-6 a^2 b \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+6 a^2 b \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+6 a b^2 \sin (2 (c+d x))-6 a b^2 c-6 a b^2 d x+b^3 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-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 d}","-\frac{5 a^3 \sin (c+d x) \cos (c+d x)}{2 d}+\frac{b \left(6 a^2-b^2\right) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{1}{2} a x \left(a^2-6 b^2\right)-\frac{15 a^2 b \sin (c+d x)}{2 d}+\frac{3 a \tan (c+d x) (a \cos (c+d x)+b)^2}{2 d}+\frac{\tan (c+d x) \sec (c+d x) (a \cos (c+d x)+b)^3}{2 d}",1,"(Sec[c + d*x]^2*(a^3*c - 6*a*b^2*c + a^3*d*x - 6*a*b^2*d*x - 6*a^2*b*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + b^3*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 6*a^2*b*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] - b^3*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + Cos[2*(c + d*x)]*(a*(a^2 - 6*b^2)*(c + d*x) + (-6*a^2*b + b^3)*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - b*(-6*a^2 + b^2)*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) + (-3*a^2*b + 2*b^3)*Sin[c + d*x] - (a^3*Sin[2*(c + d*x)])/2 + 6*a*b^2*Sin[2*(c + d*x)] - 3*a^2*b*Sin[3*(c + d*x)] - (a^3*Sin[4*(c + d*x)])/4))/(4*d)","B",1
193,1,406,133,0.659232,"\int \csc ^2(c+d x) (a+b \sec (c+d x))^3 \, dx","Integrate[Csc[c + d*x]^2*(a + b*Sec[c + d*x])^3,x]","-\frac{\csc ^5\left(\frac{1}{2} (c+d x)\right) \sec \left(\frac{1}{2} (c+d x)\right) \left(2 a^3 \cos (3 (c+d x))+6 \left(2 a^2 b+b^3\right) \cos (2 (c+d x))+6 a \left(a^2+2 b^2\right) \cos (c+d x)+6 a^2 b \sin (c+d x) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-6 a^2 b \sin (c+d x) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+6 a^2 b \sin (3 (c+d x)) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-6 a^2 b \sin (3 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+12 a^2 b+12 a b^2 \cos (3 (c+d x))+3 b^3 \sin (c+d x) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-3 b^3 \sin (c+d x) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+3 b^3 \sin (3 (c+d x)) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-3 b^3 \sin (3 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+2 b^3\right)}{16 d \left(\cot ^2\left(\frac{1}{2} (c+d x)\right)-1\right)^2}","-\frac{a^3 \cot (c+d x)}{d}-\frac{3 a^2 b \csc (c+d x)}{d}+\frac{3 a^2 b \tanh ^{-1}(\sin (c+d x))}{d}+\frac{3 a b^2 \tan (c+d x)}{d}-\frac{3 a b^2 \cot (c+d x)}{d}-\frac{3 b^3 \csc (c+d x)}{2 d}+\frac{3 b^3 \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{b^3 \csc (c+d x) \sec ^2(c+d x)}{2 d}",1,"-1/16*(Csc[(c + d*x)/2]^5*Sec[(c + d*x)/2]*(12*a^2*b + 2*b^3 + 6*a*(a^2 + 2*b^2)*Cos[c + d*x] + 6*(2*a^2*b + b^3)*Cos[2*(c + d*x)] + 2*a^3*Cos[3*(c + d*x)] + 12*a*b^2*Cos[3*(c + d*x)] + 6*a^2*b*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]]*Sin[c + d*x] + 3*b^3*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]]*Sin[c + d*x] - 6*a^2*b*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]*Sin[c + d*x] - 3*b^3*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]*Sin[c + d*x] + 6*a^2*b*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]]*Sin[3*(c + d*x)] + 3*b^3*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]]*Sin[3*(c + d*x)] - 6*a^2*b*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]*Sin[3*(c + d*x)] - 3*b^3*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]*Sin[3*(c + d*x)]))/(d*(-1 + Cot[(c + d*x)/2]^2)^2)","B",1
194,1,610,205,0.9717788,"\int \csc ^4(c+d x) (a+b \sec (c+d x))^3 \, dx","Integrate[Csc[c + d*x]^4*(a + b*Sec[c + d*x])^3,x]","-\frac{\csc ^7\left(\frac{1}{2} (c+d x)\right) \sec ^3\left(\frac{1}{2} (c+d x)\right) \left(4 a^3 \cos (3 (c+d x))-4 a^3 \cos (5 (c+d x))+8 \left(6 a^2 b+5 b^3\right) \cos (2 (c+d x))+32 a \left(a^2+3 b^2\right) \cos (c+d x)-36 a^2 b \cos (4 (c+d x))+36 a^2 b \sin (c+d x) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-36 a^2 b \sin (c+d x) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+18 a^2 b \sin (3 (c+d x)) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-18 a^2 b \sin (3 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)-18 a^2 b \sin (5 (c+d x)) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+18 a^2 b \sin (5 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+84 a^2 b+48 a b^2 \cos (3 (c+d x))-48 a b^2 \cos (5 (c+d x))-30 b^3 \cos (4 (c+d x))+30 b^3 \sin (c+d x) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-30 b^3 \sin (c+d x) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+15 b^3 \sin (3 (c+d x)) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-15 b^3 \sin (3 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)-15 b^3 \sin (5 (c+d x)) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+15 b^3 \sin (5 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+22 b^3\right)}{768 d \left(\cot ^2\left(\frac{1}{2} (c+d x)\right)-1\right)^2}","-\frac{a^3 \cot ^3(c+d x)}{3 d}-\frac{a^3 \cot (c+d x)}{d}-\frac{a^2 b \csc ^3(c+d x)}{d}-\frac{3 a^2 b \csc (c+d x)}{d}+\frac{3 a^2 b \tanh ^{-1}(\sin (c+d x))}{d}+\frac{3 a b^2 \tan (c+d x)}{d}-\frac{a b^2 \cot ^3(c+d x)}{d}-\frac{6 a b^2 \cot (c+d x)}{d}-\frac{5 b^3 \csc ^3(c+d x)}{6 d}-\frac{5 b^3 \csc (c+d x)}{2 d}+\frac{5 b^3 \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{b^3 \csc ^3(c+d x) \sec ^2(c+d x)}{2 d}",1,"-1/768*(Csc[(c + d*x)/2]^7*Sec[(c + d*x)/2]^3*(84*a^2*b + 22*b^3 + 32*a*(a^2 + 3*b^2)*Cos[c + d*x] + 8*(6*a^2*b + 5*b^3)*Cos[2*(c + d*x)] + 4*a^3*Cos[3*(c + d*x)] + 48*a*b^2*Cos[3*(c + d*x)] - 36*a^2*b*Cos[4*(c + d*x)] - 30*b^3*Cos[4*(c + d*x)] - 4*a^3*Cos[5*(c + d*x)] - 48*a*b^2*Cos[5*(c + d*x)] + 36*a^2*b*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]]*Sin[c + d*x] + 30*b^3*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]]*Sin[c + d*x] - 36*a^2*b*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]*Sin[c + d*x] - 30*b^3*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]*Sin[c + d*x] + 18*a^2*b*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]]*Sin[3*(c + d*x)] + 15*b^3*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]]*Sin[3*(c + d*x)] - 18*a^2*b*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]*Sin[3*(c + d*x)] - 15*b^3*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]*Sin[3*(c + d*x)] - 18*a^2*b*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]]*Sin[5*(c + d*x)] - 15*b^3*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]]*Sin[5*(c + d*x)] + 18*a^2*b*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]*Sin[5*(c + d*x)] + 15*b^3*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]*Sin[5*(c + d*x)]))/(d*(-1 + Cot[(c + d*x)/2]^2)^2)","B",1
195,1,812,279,1.4997465,"\int \csc ^6(c+d x) (a+b \sec (c+d x))^3 \, dx","Integrate[Csc[c + d*x]^6*(a + b*Sec[c + d*x])^3,x]","-\frac{\csc ^9\left(\frac{1}{2} (c+d x)\right) \sec ^5\left(\frac{1}{2} (c+d x)\right) \left(16 \cos (3 (c+d x)) a^3-48 \cos (5 (c+d x)) a^3+16 \cos (7 (c+d x)) a^3+1176 b a^2-600 b \cos (4 (c+d x)) a^2+180 b \cos (6 (c+d x)) a^2+450 b \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right) \sin (c+d x) a^2-450 b \log \left(\cos \left(\frac{1}{2} (c+d x)\right)+\sin \left(\frac{1}{2} (c+d x)\right)\right) \sin (c+d x) a^2+90 b \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right) \sin (3 (c+d x)) a^2-90 b \log \left(\cos \left(\frac{1}{2} (c+d x)\right)+\sin \left(\frac{1}{2} (c+d x)\right)\right) \sin (3 (c+d x)) a^2-270 b \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right) \sin (5 (c+d x)) a^2+270 b \log \left(\cos \left(\frac{1}{2} (c+d x)\right)+\sin \left(\frac{1}{2} (c+d x)\right)\right) \sin (5 (c+d x)) a^2+90 b \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right) \sin (7 (c+d x)) a^2-90 b \log \left(\cos \left(\frac{1}{2} (c+d x)\right)+\sin \left(\frac{1}{2} (c+d x)\right)\right) \sin (7 (c+d x)) a^2+80 \left(5 a^2+18 b^2\right) \cos (c+d x) a+288 b^2 \cos (3 (c+d x)) a-864 b^2 \cos (5 (c+d x)) a+288 b^2 \cos (7 (c+d x)) a+412 b^3+66 \left(7 b^3+6 a^2 b\right) \cos (2 (c+d x))-700 b^3 \cos (4 (c+d x))+210 b^3 \cos (6 (c+d x))+525 b^3 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right) \sin (c+d x)-525 b^3 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)+\sin \left(\frac{1}{2} (c+d x)\right)\right) \sin (c+d x)+105 b^3 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right) \sin (3 (c+d x))-105 b^3 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)+\sin \left(\frac{1}{2} (c+d x)\right)\right) \sin (3 (c+d x))-315 b^3 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right) \sin (5 (c+d x))+315 b^3 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)+\sin \left(\frac{1}{2} (c+d x)\right)\right) \sin (5 (c+d x))+105 b^3 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right) \sin (7 (c+d x))-105 b^3 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)+\sin \left(\frac{1}{2} (c+d x)\right)\right) \sin (7 (c+d x))\right)}{61440 d \left(\cot ^2\left(\frac{1}{2} (c+d x)\right)-1\right)^2}","-\frac{a^3 \cot ^5(c+d x)}{5 d}-\frac{2 a^3 \cot ^3(c+d x)}{3 d}-\frac{a^3 \cot (c+d x)}{d}-\frac{3 a^2 b \csc ^5(c+d x)}{5 d}-\frac{a^2 b \csc ^3(c+d x)}{d}-\frac{3 a^2 b \csc (c+d x)}{d}+\frac{3 a^2 b \tanh ^{-1}(\sin (c+d x))}{d}+\frac{3 a b^2 \tan (c+d x)}{d}-\frac{3 a b^2 \cot ^5(c+d x)}{5 d}-\frac{3 a b^2 \cot ^3(c+d x)}{d}-\frac{9 a b^2 \cot (c+d x)}{d}-\frac{7 b^3 \csc ^5(c+d x)}{10 d}-\frac{7 b^3 \csc ^3(c+d x)}{6 d}-\frac{7 b^3 \csc (c+d x)}{2 d}+\frac{7 b^3 \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{b^3 \csc ^5(c+d x) \sec ^2(c+d x)}{2 d}",1,"-1/61440*(Csc[(c + d*x)/2]^9*Sec[(c + d*x)/2]^5*(1176*a^2*b + 412*b^3 + 80*a*(5*a^2 + 18*b^2)*Cos[c + d*x] + 66*(6*a^2*b + 7*b^3)*Cos[2*(c + d*x)] + 16*a^3*Cos[3*(c + d*x)] + 288*a*b^2*Cos[3*(c + d*x)] - 600*a^2*b*Cos[4*(c + d*x)] - 700*b^3*Cos[4*(c + d*x)] - 48*a^3*Cos[5*(c + d*x)] - 864*a*b^2*Cos[5*(c + d*x)] + 180*a^2*b*Cos[6*(c + d*x)] + 210*b^3*Cos[6*(c + d*x)] + 16*a^3*Cos[7*(c + d*x)] + 288*a*b^2*Cos[7*(c + d*x)] + 450*a^2*b*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]]*Sin[c + d*x] + 525*b^3*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]]*Sin[c + d*x] - 450*a^2*b*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]*Sin[c + d*x] - 525*b^3*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]*Sin[c + d*x] + 90*a^2*b*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]]*Sin[3*(c + d*x)] + 105*b^3*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]]*Sin[3*(c + d*x)] - 90*a^2*b*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]*Sin[3*(c + d*x)] - 105*b^3*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]*Sin[3*(c + d*x)] - 270*a^2*b*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]]*Sin[5*(c + d*x)] - 315*b^3*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]]*Sin[5*(c + d*x)] + 270*a^2*b*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]*Sin[5*(c + d*x)] + 315*b^3*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]*Sin[5*(c + d*x)] + 90*a^2*b*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]]*Sin[7*(c + d*x)] + 105*b^3*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]]*Sin[7*(c + d*x)] - 90*a^2*b*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]*Sin[7*(c + d*x)] - 105*b^3*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]*Sin[7*(c + d*x)]))/(d*(-1 + Cot[(c + d*x)/2]^2)^2)","B",1
196,1,282,223,1.5620708,"\int \frac{\sin ^7(c+d x)}{a+b \sec (c+d x)} \, dx","Integrate[Sin[c + d*x]^7/(a + b*Sec[c + d*x]),x]","\frac{735 a^7 \cos (3 (c+d x))-147 a^7 \cos (5 (c+d x))+15 a^7 \cos (7 (c+d x))+420 a^6 b \cos (4 (c+d x))-35 a^6 b \cos (6 (c+d x))+6720 a^6 b \log (a \cos (c+d x)+b)-1260 a^5 b^2 \cos (3 (c+d x))+84 a^5 b^2 \cos (5 (c+d x))-210 a^4 b^3 \cos (4 (c+d x))-20160 a^4 b^3 \log (a \cos (c+d x)+b)+560 a^3 b^4 \cos (3 (c+d x))+20160 a^2 b^5 \log (a \cos (c+d x)+b)-105 \left(29 a^6 b-40 a^4 b^3+16 a^2 b^5\right) \cos (2 (c+d x))-105 a \left(35 a^6-152 a^4 b^2+176 a^2 b^4-64 b^6\right) \cos (c+d x)-6720 b^7 \log (a \cos (c+d x)+b)}{6720 a^8 d}","-\frac{b \cos ^6(c+d x)}{6 a^2 d}+\frac{b \left(a^2-b^2\right)^3 \log (a \cos (c+d x)+b)}{a^8 d}-\frac{\left(a^2-b^2\right)^3 \cos (c+d x)}{a^7 d}+\frac{b \left(3 a^2-b^2\right) \cos ^4(c+d x)}{4 a^4 d}-\frac{\left(3 a^2-b^2\right) \cos ^5(c+d x)}{5 a^3 d}-\frac{b \left(3 a^4-3 a^2 b^2+b^4\right) \cos ^2(c+d x)}{2 a^6 d}+\frac{\left(3 a^4-3 a^2 b^2+b^4\right) \cos ^3(c+d x)}{3 a^5 d}+\frac{\cos ^7(c+d x)}{7 a d}",1,"(-105*a*(35*a^6 - 152*a^4*b^2 + 176*a^2*b^4 - 64*b^6)*Cos[c + d*x] - 105*(29*a^6*b - 40*a^4*b^3 + 16*a^2*b^5)*Cos[2*(c + d*x)] + 735*a^7*Cos[3*(c + d*x)] - 1260*a^5*b^2*Cos[3*(c + d*x)] + 560*a^3*b^4*Cos[3*(c + d*x)] + 420*a^6*b*Cos[4*(c + d*x)] - 210*a^4*b^3*Cos[4*(c + d*x)] - 147*a^7*Cos[5*(c + d*x)] + 84*a^5*b^2*Cos[5*(c + d*x)] - 35*a^6*b*Cos[6*(c + d*x)] + 15*a^7*Cos[7*(c + d*x)] + 6720*a^6*b*Log[b + a*Cos[c + d*x]] - 20160*a^4*b^3*Log[b + a*Cos[c + d*x]] + 20160*a^2*b^5*Log[b + a*Cos[c + d*x]] - 6720*b^7*Log[b + a*Cos[c + d*x]])/(6720*a^8*d)","A",1
197,1,172,152,0.3767712,"\int \frac{\sin ^5(c+d x)}{a+b \sec (c+d x)} \, dx","Integrate[Sin[c + d*x]^5/(a + b*Sec[c + d*x]),x]","\frac{50 a^5 \cos (3 (c+d x))-6 a^5 \cos (5 (c+d x))+15 a^4 b \cos (4 (c+d x))+480 a^4 b \log (a \cos (c+d x)+b)-40 a^3 b^2 \cos (3 (c+d x))-960 a^2 b^3 \log (a \cos (c+d x)+b)-60 \left(3 a^4 b-2 a^2 b^3\right) \cos (2 (c+d x))-60 a \left(5 a^4-14 a^2 b^2+8 b^4\right) \cos (c+d x)+480 b^5 \log (a \cos (c+d x)+b)}{480 a^6 d}","\frac{b \cos ^4(c+d x)}{4 a^2 d}+\frac{b \left(a^2-b^2\right)^2 \log (a \cos (c+d x)+b)}{a^6 d}-\frac{\left(a^2-b^2\right)^2 \cos (c+d x)}{a^5 d}-\frac{b \left(2 a^2-b^2\right) \cos ^2(c+d x)}{2 a^4 d}+\frac{\left(2 a^2-b^2\right) \cos ^3(c+d x)}{3 a^3 d}-\frac{\cos ^5(c+d x)}{5 a d}",1,"(-60*a*(5*a^4 - 14*a^2*b^2 + 8*b^4)*Cos[c + d*x] - 60*(3*a^4*b - 2*a^2*b^3)*Cos[2*(c + d*x)] + 50*a^5*Cos[3*(c + d*x)] - 40*a^3*b^2*Cos[3*(c + d*x)] + 15*a^4*b*Cos[4*(c + d*x)] - 6*a^5*Cos[5*(c + d*x)] + 480*a^4*b*Log[b + a*Cos[c + d*x]] - 960*a^2*b^3*Log[b + a*Cos[c + d*x]] + 480*b^5*Log[b + a*Cos[c + d*x]])/(480*a^6*d)","A",1
198,1,89,89,0.2097217,"\int \frac{\sin ^3(c+d x)}{a+b \sec (c+d x)} \, dx","Integrate[Sin[c + d*x]^3/(a + b*Sec[c + d*x]),x]","\frac{\left(12 a b^2-9 a^3\right) \cos (c+d x)+a^3 \cos (3 (c+d x))-3 a^2 b \cos (2 (c+d x))+12 a^2 b \log (a \cos (c+d x)+b)-12 b^3 \log (a \cos (c+d x)+b)}{12 a^4 d}","-\frac{b \cos ^2(c+d x)}{2 a^2 d}+\frac{b \left(a^2-b^2\right) \log (a \cos (c+d x)+b)}{a^4 d}-\frac{\left(a^2-b^2\right) \cos (c+d x)}{a^3 d}+\frac{\cos ^3(c+d x)}{3 a d}",1,"((-9*a^3 + 12*a*b^2)*Cos[c + d*x] - 3*a^2*b*Cos[2*(c + d*x)] + a^3*Cos[3*(c + d*x)] + 12*a^2*b*Log[b + a*Cos[c + d*x]] - 12*b^3*Log[b + a*Cos[c + d*x]])/(12*a^4*d)","A",1
199,1,30,34,0.0202646,"\int \frac{\sin (c+d x)}{a+b \sec (c+d x)} \, dx","Integrate[Sin[c + d*x]/(a + b*Sec[c + d*x]),x]","\frac{b \log (a \cos (c+d x)+b)-a \cos (c+d x)}{a^2 d}","\frac{b \log (a \cos (c+d x)+b)}{a^2 d}-\frac{\cos (c+d x)}{a d}",1,"(-(a*Cos[c + d*x]) + b*Log[b + a*Cos[c + d*x]])/(a^2*d)","A",1
200,1,63,74,0.101244,"\int \frac{\csc (c+d x)}{a+b \sec (c+d x)} \, dx","Integrate[Csc[c + d*x]/(a + b*Sec[c + d*x]),x]","\frac{(a-b) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-\left((a+b) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)+b \log (a \cos (c+d x)+b)}{d (a-b) (a+b)}","\frac{b \log (a \cos (c+d x)+b)}{d \left(a^2-b^2\right)}+\frac{\log (1-\cos (c+d x))}{2 d (a+b)}-\frac{\log (\cos (c+d x)+1)}{2 d (a-b)}",1,"(-((a + b)*Log[Cos[(c + d*x)/2]]) + b*Log[b + a*Cos[c + d*x]] + (a - b)*Log[Sin[(c + d*x)/2]])/((a - b)*(a + b)*d)","A",1
201,1,123,116,0.6265087,"\int \frac{\csc ^3(c+d x)}{a+b \sec (c+d x)} \, dx","Integrate[Csc[c + d*x]^3/(a + b*Sec[c + d*x]),x]","\frac{-(a-b)^2 (a+b) \csc ^2\left(\frac{1}{2} (c+d x)\right)+(a-b) (a+b)^2 \sec ^2\left(\frac{1}{2} (c+d x)\right)-4 a \left((a-b)^2 \left(-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)\right)+(a+b)^2 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)-2 a b \log (a \cos (c+d x)+b)\right)}{8 d (a-b)^2 (a+b)^2}","\frac{a^2 b \log (a \cos (c+d x)+b)}{d \left(a^2-b^2\right)^2}+\frac{\csc ^2(c+d x) (b-a \cos (c+d x))}{2 d \left(a^2-b^2\right)}+\frac{a \log (1-\cos (c+d x))}{4 d (a+b)^2}-\frac{a \log (\cos (c+d x)+1)}{4 d (a-b)^2}",1,"(-((a - b)^2*(a + b)*Csc[(c + d*x)/2]^2) - 4*a*((a + b)^2*Log[Cos[(c + d*x)/2]] - 2*a*b*Log[b + a*Cos[c + d*x]] - (a - b)^2*Log[Sin[(c + d*x)/2]]) + (a - b)*(a + b)^2*Sec[(c + d*x)/2]^2)/(8*(a - b)^2*(a + b)^2*d)","A",1
202,1,207,179,5.2181424,"\int \frac{\csc ^5(c+d x)}{a+b \sec (c+d x)} \, dx","Integrate[Csc[c + d*x]^5/(a + b*Sec[c + d*x]),x]","\frac{8 a \left(8 a^3 b \log (a \cos (c+d x)+b)+(a-b)^3 (3 a+b) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-(3 a-b) (a+b)^3 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)-2 (a-b)^3 \left(3 a^2+4 a b+b^2\right) \csc ^2\left(\frac{1}{2} (c+d x)\right)+2 (a+b)^3 \left(3 a^2-4 a b+b^2\right) \sec ^2\left(\frac{1}{2} (c+d x)\right)-(a-b)^3 (a+b)^2 \csc ^4\left(\frac{1}{2} (c+d x)\right)+(a-b)^2 (a+b)^3 \sec ^4\left(\frac{1}{2} (c+d x)\right)}{64 d (a-b)^3 (a+b)^3}","\frac{\csc ^4(c+d x) (b-a \cos (c+d x))}{4 d \left(a^2-b^2\right)}+\frac{\csc ^2(c+d x) \left(4 a^2 b-a \left(3 a^2+b^2\right) \cos (c+d x)\right)}{8 d \left(a^2-b^2\right)^2}+\frac{a^4 b \log (a \cos (c+d x)+b)}{d \left(a^2-b^2\right)^3}+\frac{a (3 a+b) \log (1-\cos (c+d x))}{16 d (a+b)^3}-\frac{a (3 a-b) \log (\cos (c+d x)+1)}{16 d (a-b)^3}",1,"(-2*(a - b)^3*(3*a^2 + 4*a*b + b^2)*Csc[(c + d*x)/2]^2 - (a - b)^3*(a + b)^2*Csc[(c + d*x)/2]^4 + 8*a*(-((3*a - b)*(a + b)^3*Log[Cos[(c + d*x)/2]]) + 8*a^3*b*Log[b + a*Cos[c + d*x]] + (a - b)^3*(3*a + b)*Log[Sin[(c + d*x)/2]]) + 2*(a + b)^3*(3*a^2 - 4*a*b + b^2)*Sec[(c + d*x)/2]^2 + (a - b)^2*(a + b)^3*Sec[(c + d*x)/2]^4)/(64*(a - b)^3*(a + b)^3*d)","A",1
203,1,268,230,2.3995455,"\int \frac{\sin ^6(c+d x)}{a+b \sec (c+d x)} \, dx","Integrate[Sin[c + d*x]^6/(a + b*Sec[c + d*x]),x]","\frac{45 a^6 \sin (4 (c+d x))-5 a^6 \sin (6 (c+d x))+300 a^6 c+300 a^6 d x-140 a^5 b \sin (3 (c+d x))+12 a^5 b \sin (5 (c+d x))-30 a^4 b^2 \sin (4 (c+d x))-1800 a^4 b^2 c-1800 a^4 b^2 d x+80 a^3 b^3 \sin (3 (c+d x))+2400 a^2 b^4 c+2400 a^2 b^4 d x+1920 b \left(a^2-b^2\right)^{5/2} \tanh ^{-1}\left(\frac{(b-a) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a^2-b^2}}\right)+120 a b \left(11 a^4-18 a^2 b^2+8 b^4\right) \sin (c+d x)-15 \left(15 a^6-32 a^4 b^2+16 a^2 b^4\right) \sin (2 (c+d x))-960 b^6 c-960 b^6 d x}{960 a^7 d}","-\frac{2 b (a-b)^{5/2} (a+b)^{5/2} \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^7 d}+\frac{\sin ^5(c+d x) (6 b-5 a \cos (c+d x))}{30 a^2 d}+\frac{\sin ^3(c+d x) \left(8 b \left(a^2-b^2\right)-a \left(5 a^2-6 b^2\right) \cos (c+d x)\right)}{24 a^4 d}+\frac{\sin (c+d x) \left(16 b \left(a^2-b^2\right)^2-a \left(5 a^4-14 a^2 b^2+8 b^4\right) \cos (c+d x)\right)}{16 a^6 d}+\frac{x \left(5 a^6-30 a^4 b^2+40 a^2 b^4-16 b^6\right)}{16 a^7}",1,"(300*a^6*c - 1800*a^4*b^2*c + 2400*a^2*b^4*c - 960*b^6*c + 300*a^6*d*x - 1800*a^4*b^2*d*x + 2400*a^2*b^4*d*x - 960*b^6*d*x + 1920*b*(a^2 - b^2)^(5/2)*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]] + 120*a*b*(11*a^4 - 18*a^2*b^2 + 8*b^4)*Sin[c + d*x] - 15*(15*a^6 - 32*a^4*b^2 + 16*a^2*b^4)*Sin[2*(c + d*x)] - 140*a^5*b*Sin[3*(c + d*x)] + 80*a^3*b^3*Sin[3*(c + d*x)] + 45*a^6*Sin[4*(c + d*x)] - 30*a^4*b^2*Sin[4*(c + d*x)] + 12*a^5*b*Sin[5*(c + d*x)] - 5*a^6*Sin[6*(c + d*x)])/(960*a^7*d)","A",1
204,1,172,161,0.8223405,"\int \frac{\sin ^4(c+d x)}{a+b \sec (c+d x)} \, dx","Integrate[Sin[c + d*x]^4/(a + b*Sec[c + d*x]),x]","\frac{3 a^4 \sin (4 (c+d x))+36 a^4 c+36 a^4 d x-8 a^3 b \sin (3 (c+d x))+24 a b \left(5 a^2-4 b^2\right) \sin (c+d x)+192 b \left(a^2-b^2\right)^{3/2} \tanh ^{-1}\left(\frac{(b-a) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a^2-b^2}}\right)-144 a^2 b^2 c-144 a^2 b^2 d x-24 \left(a^4-a^2 b^2\right) \sin (2 (c+d x))+96 b^4 c+96 b^4 d x}{96 a^5 d}","-\frac{2 b (a-b)^{3/2} (a+b)^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^5 d}+\frac{\sin ^3(c+d x) (4 b-3 a \cos (c+d x))}{12 a^2 d}+\frac{\sin (c+d x) \left(8 b \left(a^2-b^2\right)-a \left(3 a^2-4 b^2\right) \cos (c+d x)\right)}{8 a^4 d}+\frac{x \left(3 a^4-12 a^2 b^2+8 b^4\right)}{8 a^5}",1,"(36*a^4*c - 144*a^2*b^2*c + 96*b^4*c + 36*a^4*d*x - 144*a^2*b^2*d*x + 96*b^4*d*x + 192*b*(a^2 - b^2)^(3/2)*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]] + 24*a*b*(5*a^2 - 4*b^2)*Sin[c + d*x] - 24*(a^4 - 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
205,1,96,100,0.3178357,"\int \frac{\sin ^2(c+d x)}{a+b \sec (c+d x)} \, dx","Integrate[Sin[c + d*x]^2/(a + b*Sec[c + d*x]),x]","\frac{2 \left(a^2-2 b^2\right) (c+d x)+8 b \sqrt{a^2-b^2} \tanh ^{-1}\left(\frac{(b-a) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a^2-b^2}}\right)+a^2 (-\sin (2 (c+d x)))+4 a b \sin (c+d x)}{4 a^3 d}","-\frac{2 b \sqrt{a-b} \sqrt{a+b} \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^3 d}+\frac{\sin (c+d x) (2 b-a \cos (c+d x))}{2 a^2 d}+\frac{x \left(a^2-2 b^2\right)}{2 a^3}",1,"(2*(a^2 - 2*b^2)*(c + d*x) + 8*b*Sqrt[a^2 - b^2]*ArcTanh[((-a + b)*Tan[(c + d*x)/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
206,1,118,84,0.2001509,"\int \frac{\csc ^2(c+d x)}{a+b \sec (c+d x)} \, dx","Integrate[Csc[c + d*x]^2/(a + b*Sec[c + d*x]),x]","\frac{\csc \left(\frac{1}{2} (c+d x)\right) \sec \left(\frac{1}{2} (c+d x)\right) \left(\sqrt{a^2-b^2} (b-a \cos (c+d x))+2 a b \sin (c+d x) \tanh ^{-1}\left(\frac{(b-a) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a^2-b^2}}\right)\right)}{2 d (a-b) (a+b) \sqrt{a^2-b^2}}","\frac{\csc (c+d x) (b-a \cos (c+d x))}{d \left(a^2-b^2\right)}-\frac{2 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)^{3/2} (a+b)^{3/2}}",1,"(Csc[(c + d*x)/2]*Sec[(c + d*x)/2]*(Sqrt[a^2 - b^2]*(b - a*Cos[c + d*x]) + 2*a*b*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]]*Sin[c + d*x]))/(2*(a - b)*(a + b)*Sqrt[a^2 - b^2]*d)","A",1
207,1,162,140,0.9229858,"\int \frac{\csc ^4(c+d x)}{a+b \sec (c+d x)} \, dx","Integrate[Csc[c + d*x]^4/(a + b*Sec[c + d*x]),x]","\frac{24 a^3 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} \csc ^3(c+d x) \left(\left(3 a b^2-6 a^3\right) \cos (c+d x)+2 a^3 \cos (3 (c+d x))-6 a^2 b \cos (2 (c+d x))+10 a^2 b+a b^2 \cos (3 (c+d x))-4 b^3\right)}{12 d (a-b)^2 (a+b)^2 \sqrt{a^2-b^2}}","-\frac{2 a^3 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}}+\frac{\csc ^3(c+d x) (b-a \cos (c+d x))}{3 d \left(a^2-b^2\right)}+\frac{\csc (c+d x) \left(3 a^2 b-a \left(2 a^2+b^2\right) \cos (c+d x)\right)}{3 d \left(a^2-b^2\right)^2}",1,"(24*a^3*b*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]] + Sqrt[a^2 - b^2]*(10*a^2*b - 4*b^3 + (-6*a^3 + 3*a*b^2)*Cos[c + d*x] - 6*a^2*b*Cos[2*(c + d*x)] + 2*a^3*Cos[3*(c + d*x)] + a*b^2*Cos[3*(c + d*x)])*Csc[c + d*x]^3)/(12*(a - b)^2*(a + b)^2*Sqrt[a^2 - b^2]*d)","A",1
208,1,277,201,1.2922787,"\int \frac{\csc ^6(c+d x)}{a+b \sec (c+d x)} \, dx","Integrate[Csc[c + d*x]^6/(a + b*Sec[c + d*x]),x]","\frac{\sec (c+d x) (a \cos (c+d x)+b) \left(\frac{2 \left(64 a^2-43 a b+9 b^2\right) \tan \left(\frac{1}{2} (c+d x)\right)}{(a-b)^3}-\frac{2 \left(64 a^2+43 a b+9 b^2\right) \cot \left(\frac{1}{2} (c+d x)\right)}{(a+b)^3}+\frac{960 a^5 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)^{7/2}}-\frac{3 \sin (c+d x) \csc ^6\left(\frac{1}{2} (c+d x)\right)}{2 (a+b)}+\frac{96 \sin ^6\left(\frac{1}{2} (c+d x)\right) \csc ^5(c+d x)}{a-b}-\frac{(19 a+9 b) \sin (c+d x) \csc ^4\left(\frac{1}{2} (c+d x)\right)}{2 (a+b)^2}+\frac{8 (19 a-9 b) \sin ^4\left(\frac{1}{2} (c+d x)\right) \csc ^3(c+d x)}{(a-b)^2}\right)}{480 d (a+b \sec (c+d x))}","-\frac{2 a^5 b \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{\csc ^5(c+d x) (b-a \cos (c+d x))}{5 d \left(a^2-b^2\right)}+\frac{\csc ^3(c+d x) \left(5 a^2 b-a \left(4 a^2+b^2\right) \cos (c+d x)\right)}{15 d \left(a^2-b^2\right)^2}+\frac{\csc (c+d x) \left(15 a^4 b-a \left(8 a^4+9 a^2 b^2-2 b^4\right) \cos (c+d x)\right)}{15 d \left(a^2-b^2\right)^3}",1,"((b + a*Cos[c + d*x])*Sec[c + d*x]*((960*a^5*b*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(7/2) - (2*(64*a^2 + 43*a*b + 9*b^2)*Cot[(c + d*x)/2])/(a + b)^3 + (8*(19*a - 9*b)*Csc[c + d*x]^3*Sin[(c + d*x)/2]^4)/(a - b)^2 + (96*Csc[c + d*x]^5*Sin[(c + d*x)/2]^6)/(a - b) - ((19*a + 9*b)*Csc[(c + d*x)/2]^4*Sin[c + d*x])/(2*(a + b)^2) - (3*Csc[(c + d*x)/2]^6*Sin[c + d*x])/(2*(a + b)) + (2*(64*a^2 - 43*a*b + 9*b^2)*Tan[(c + d*x)/2])/(a - b)^3))/(480*d*(a + b*Sec[c + d*x]))","A",1
209,1,417,267,3.6120109,"\int \frac{\sin ^7(c+d x)}{(a+b \sec (c+d x))^2} \, dx","Integrate[Sin[c + d*x]^7/(a + b*Sec[c + d*x])^2,x]","\frac{588 a^8 \cos (4 (c+d x))-132 a^8 \cos (6 (c+d x))+15 a^8 \cos (8 (c+d x))-3675 a^8-3780 a^7 b \cos (3 (c+d x))+476 a^7 b \cos (5 (c+d x))-40 a^7 b \cos (7 (c+d x))-1848 a^6 b^2 \cos (4 (c+d x))+112 a^6 b^2 \cos (6 (c+d x))+26880 a^6 b^2 \log (a \cos (c+d x)+b)+61320 a^6 b^2+8400 a^5 b^3 \cos (3 (c+d x))-336 a^5 b^3 \cos (5 (c+d x))+1120 a^4 b^4 \cos (4 (c+d x))-161280 a^4 b^4 \log (a \cos (c+d x)+b)-132720 a^4 b^4-4480 a^3 b^5 \cos (3 (c+d x))+241920 a^2 b^6 \log (a \cos (c+d x)+b)+87360 a^2 b^6+1680 a b \cos (c+d x) \left(-8 a^6+67 a^4 b^2-116 a^2 b^4+16 \left(a^2-4 b^2\right) \left(a^2-b^2\right)^2 \log (a \cos (c+d x)+b)+56 b^6\right)-140 \left(21 a^8-228 a^6 b^2+400 a^4 b^4-192 a^2 b^6\right) \cos (2 (c+d x))-107520 b^8 \log (a \cos (c+d x)+b)-13440 b^8}{13440 a^9 d (a \cos (c+d x)+b)}","-\frac{b \cos ^6(c+d x)}{3 a^3 d}+\frac{\cos ^7(c+d x)}{7 a^2 d}+\frac{b^2 \left(a^2-b^2\right)^3}{a^9 d (a \cos (c+d x)+b)}+\frac{2 b \left(a^2-4 b^2\right) \left(a^2-b^2\right)^2 \log (a \cos (c+d x)+b)}{a^9 d}-\frac{\left(a^2-7 b^2\right) \left(a^2-b^2\right)^2 \cos (c+d x)}{a^8 d}-\frac{3 b \left(a^2-b^2\right)^2 \cos ^2(c+d x)}{a^7 d}+\frac{b \left(3 a^2-2 b^2\right) \cos ^4(c+d x)}{2 a^5 d}-\frac{3 \left(a^2-b^2\right) \cos ^5(c+d x)}{5 a^4 d}+\frac{\left(3 a^4-9 a^2 b^2+5 b^4\right) \cos ^3(c+d x)}{3 a^6 d}",1,"(-3675*a^8 + 61320*a^6*b^2 - 132720*a^4*b^4 + 87360*a^2*b^6 - 13440*b^8 - 140*(21*a^8 - 228*a^6*b^2 + 400*a^4*b^4 - 192*a^2*b^6)*Cos[2*(c + d*x)] - 3780*a^7*b*Cos[3*(c + d*x)] + 8400*a^5*b^3*Cos[3*(c + d*x)] - 4480*a^3*b^5*Cos[3*(c + d*x)] + 588*a^8*Cos[4*(c + d*x)] - 1848*a^6*b^2*Cos[4*(c + d*x)] + 1120*a^4*b^4*Cos[4*(c + d*x)] + 476*a^7*b*Cos[5*(c + d*x)] - 336*a^5*b^3*Cos[5*(c + d*x)] - 132*a^8*Cos[6*(c + d*x)] + 112*a^6*b^2*Cos[6*(c + d*x)] - 40*a^7*b*Cos[7*(c + d*x)] + 15*a^8*Cos[8*(c + d*x)] + 26880*a^6*b^2*Log[b + a*Cos[c + d*x]] - 161280*a^4*b^4*Log[b + a*Cos[c + d*x]] + 241920*a^2*b^6*Log[b + a*Cos[c + d*x]] - 107520*b^8*Log[b + a*Cos[c + d*x]] + 1680*a*b*Cos[c + d*x]*(-8*a^6 + 67*a^4*b^2 - 116*a^2*b^4 + 56*b^6 + 16*(a^2 - 4*b^2)*(a^2 - b^2)^2*Log[b + a*Cos[c + d*x]]))/(13440*a^9*d*(b + a*Cos[c + d*x]))","A",1
210,1,280,194,1.6373365,"\int \frac{\sin ^5(c+d x)}{(a+b \sec (c+d x))^2} \, dx","Integrate[Sin[c + d*x]^5/(a + b*Sec[c + d*x])^2,x]","\frac{22 a^6 \cos (4 (c+d x))-3 a^6 \cos (6 (c+d x))-150 a^6-115 a^5 b \cos (3 (c+d x))+9 a^5 b \cos (5 (c+d x))-30 a^4 b^2 \cos (4 (c+d x))+960 a^4 b^2 \log (a \cos (c+d x)+b)+1740 a^4 b^2+120 a^3 b^3 \cos (3 (c+d x))-3840 a^2 b^4 \log (a \cos (c+d x)+b)-2160 a^2 b^4+120 a b \cos (c+d x) \left(-4 a^4+23 a^2 b^2+8 \left(a^4-4 a^2 b^2+3 b^4\right) \log (a \cos (c+d x)+b)-20 b^4\right)-5 \left(25 a^6-168 a^4 b^2+144 a^2 b^4\right) \cos (2 (c+d x))+2880 b^6 \log (a \cos (c+d x)+b)+480 b^6}{480 a^7 d (a \cos (c+d x)+b)}","\frac{b \cos ^4(c+d x)}{2 a^3 d}-\frac{\cos ^5(c+d x)}{5 a^2 d}+\frac{b^2 \left(a^2-b^2\right)^2}{a^7 d (a \cos (c+d x)+b)}-\frac{2 b \left(a^2-b^2\right) \cos ^2(c+d x)}{a^5 d}+\frac{\left(2 a^2-3 b^2\right) \cos ^3(c+d x)}{3 a^4 d}+\frac{2 b \left(a^4-4 a^2 b^2+3 b^4\right) \log (a \cos (c+d x)+b)}{a^7 d}-\frac{\left(a^4-6 a^2 b^2+5 b^4\right) \cos (c+d x)}{a^6 d}",1,"(-150*a^6 + 1740*a^4*b^2 - 2160*a^2*b^4 + 480*b^6 - 5*(25*a^6 - 168*a^4*b^2 + 144*a^2*b^4)*Cos[2*(c + d*x)] - 115*a^5*b*Cos[3*(c + d*x)] + 120*a^3*b^3*Cos[3*(c + d*x)] + 22*a^6*Cos[4*(c + d*x)] - 30*a^4*b^2*Cos[4*(c + d*x)] + 9*a^5*b*Cos[5*(c + d*x)] - 3*a^6*Cos[6*(c + d*x)] + 960*a^4*b^2*Log[b + a*Cos[c + d*x]] - 3840*a^2*b^4*Log[b + a*Cos[c + d*x]] + 2880*b^6*Log[b + a*Cos[c + d*x]] + 120*a*b*Cos[c + d*x]*(-4*a^4 + 23*a^2*b^2 - 20*b^4 + 8*(a^4 - 4*a^2*b^2 + 3*b^4)*Log[b + a*Cos[c + d*x]]))/(480*a^7*d*(b + a*Cos[c + d*x]))","A",1
211,1,167,119,0.5562571,"\int \frac{\sin ^3(c+d x)}{(a+b \sec (c+d x))^2} \, dx","Integrate[Sin[c + d*x]^3/(a + b*Sec[c + d*x])^2,x]","\frac{a^4 \cos (4 (c+d x))-9 a^4-4 a^3 b \cos (3 (c+d x))+48 a^2 b^2 \log (a \cos (c+d x)+b)+24 a b \cos (c+d x) \left(2 \left(a^2-2 b^2\right) \log (a \cos (c+d x)+b)-a^2+3 b^2\right)+60 a^2 b^2-8 \left(a^4-3 a^2 b^2\right) \cos (2 (c+d x))-96 b^4 \log (a \cos (c+d x)+b)-24 b^4}{24 a^5 d (a \cos (c+d x)+b)}","-\frac{b \cos ^2(c+d x)}{a^3 d}+\frac{\cos ^3(c+d x)}{3 a^2 d}+\frac{b^2 \left(a^2-b^2\right)}{a^5 d (a \cos (c+d x)+b)}+\frac{2 b \left(a^2-2 b^2\right) \log (a \cos (c+d x)+b)}{a^5 d}-\frac{\left(a^2-3 b^2\right) \cos (c+d x)}{a^4 d}",1,"(-9*a^4 + 60*a^2*b^2 - 24*b^4 - 8*(a^4 - 3*a^2*b^2)*Cos[2*(c + d*x)] - 4*a^3*b*Cos[3*(c + d*x)] + a^4*Cos[4*(c + d*x)] + 48*a^2*b^2*Log[b + a*Cos[c + d*x]] - 96*b^4*Log[b + a*Cos[c + d*x]] + 24*a*b*Cos[c + d*x]*(-a^2 + 3*b^2 + 2*(a^2 - 2*b^2)*Log[b + a*Cos[c + d*x]]))/(24*a^5*d*(b + a*Cos[c + d*x]))","A",1
212,1,76,57,0.1773052,"\int \frac{\sin (c+d x)}{(a+b \sec (c+d x))^2} \, dx","Integrate[Sin[c + d*x]/(a + b*Sec[c + d*x])^2,x]","\frac{-a^2 \cos ^2(c+d x)+b^2 (2 \log (a \cos (c+d x)+b)+1)+a b \cos (c+d x) (2 \log (a \cos (c+d x)+b)-1)}{a^3 d (a \cos (c+d x)+b)}","\frac{b^2}{a^3 d (a \cos (c+d x)+b)}+\frac{2 b \log (a \cos (c+d x)+b)}{a^3 d}-\frac{\cos (c+d x)}{a^2 d}",1,"(-(a^2*Cos[c + d*x]^2) + a*b*Cos[c + d*x]*(-1 + 2*Log[b + a*Cos[c + d*x]]) + b^2*(1 + 2*Log[b + a*Cos[c + d*x]]))/(a^3*d*(b + a*Cos[c + d*x]))","A",1
213,1,165,109,0.3511612,"\int \frac{\csc (c+d x)}{(a+b \sec (c+d x))^2} \, dx","Integrate[Csc[c + d*x]/(a + b*Sec[c + d*x])^2,x]","\frac{b \left(2 a^2 b \log (a \cos (c+d x)+b)+(a-b) \left(a (a-b) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+b (a+b)\right)-a (a+b)^2 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)-a^2 \cos (c+d x) \left((a-b)^2 \left(-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right)\right)+(a+b)^2 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right)-2 a b \log (a \cos (c+d x)+b)\right)}{a d (a-b)^2 (a+b)^2 (a \cos (c+d x)+b)}","\frac{b^2}{a d \left(a^2-b^2\right) (a \cos (c+d x)+b)}+\frac{2 a b \log (a \cos (c+d x)+b)}{d \left(a^2-b^2\right)^2}+\frac{\log (1-\cos (c+d x))}{2 d (a+b)^2}-\frac{\log (\cos (c+d x)+1)}{2 d (a-b)^2}",1,"(-(a^2*Cos[c + d*x]*((a + b)^2*Log[Cos[(c + d*x)/2]] - 2*a*b*Log[b + a*Cos[c + d*x]] - (a - b)^2*Log[Sin[(c + d*x)/2]])) + b*(-(a*(a + b)^2*Log[Cos[(c + d*x)/2]]) + 2*a^2*b*Log[b + a*Cos[c + d*x]] + (a - b)*(b*(a + b) + a*(a - b)*Log[Sin[(c + d*x)/2]])))/(a*(a - b)^2*(a + b)^2*d*(b + a*Cos[c + d*x]))","A",1
214,1,224,168,1.3073787,"\int \frac{\csc ^3(c+d x)}{(a+b \sec (c+d x))^2} \, dx","Integrate[Csc[c + d*x]^3/(a + b*Sec[c + d*x])^2,x]","\frac{\sec ^2(c+d x) (a \cos (c+d x)+b) \left(\frac{16 a b \left(a^2+b^2\right) (a \cos (c+d x)+b) \log (a \cos (c+d x)+b)}{\left(a^2-b^2\right)^3}+\frac{8 a b^2}{(a-b)^2 (a+b)^2}+\frac{4 (a+b) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right) (a \cos (c+d x)+b)}{(b-a)^3}-\frac{\csc ^2\left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b)}{(a+b)^2}+\frac{\sec ^2\left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b)}{(a-b)^2}+\frac{4 (a-b) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right) (a \cos (c+d x)+b)}{(a+b)^3}\right)}{8 d (a+b \sec (c+d x))^2}","\frac{a b^2}{d \left(a^2-b^2\right)^2 (a \cos (c+d x)+b)}+\frac{2 a b \left(a^2+b^2\right) \log (a \cos (c+d x)+b)}{d \left(a^2-b^2\right)^3}+\frac{\csc ^2(c+d x) \left(2 a b-\left(a^2+b^2\right) \cos (c+d x)\right)}{2 d \left(a^2-b^2\right)^2}+\frac{(a-b) \log (1-\cos (c+d x))}{4 d (a+b)^3}-\frac{(a+b) \log (\cos (c+d x)+1)}{4 d (a-b)^3}",1,"((b + a*Cos[c + d*x])*((8*a*b^2)/((a - b)^2*(a + b)^2) - ((b + a*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/(a + b)^2 + (4*(a + b)*(b + a*Cos[c + d*x])*Log[Cos[(c + d*x)/2]])/(-a + b)^3 + (16*a*b*(a^2 + b^2)*(b + a*Cos[c + d*x])*Log[b + a*Cos[c + d*x]])/(a^2 - b^2)^3 + (4*(a - b)*(b + a*Cos[c + d*x])*Log[Sin[(c + d*x)/2]])/(a + b)^3 + ((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a - b)^2)*Sec[c + d*x]^2)/(8*d*(a + b*Sec[c + d*x])^2)","A",1
215,1,320,259,1.4676402,"\int \frac{\csc ^5(c+d x)}{(a+b \sec (c+d x))^2} \, dx","Integrate[Csc[c + d*x]^5/(a + b*Sec[c + d*x])^2,x]","\frac{\sec ^2(c+d x) (a \cos (c+d x)+b) \left(\frac{64 a^3 b^2}{(a-b)^3 (a+b)^3}+\frac{8 \left(-3 a^2-4 a b+b^2\right) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right) (a \cos (c+d x)+b)}{(a-b)^4}+\frac{8 \left(3 a^2-4 a b-b^2\right) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right) (a \cos (c+d x)+b)}{(a+b)^4}+\frac{128 a^3 b \left(a^2+2 b^2\right) (a \cos (c+d x)+b) \log (a \cos (c+d x)+b)}{\left(a^2-b^2\right)^4}-\frac{\csc ^4\left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b)}{(a+b)^2}+\frac{2 (b-3 a) \csc ^2\left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b)}{(a+b)^3}+\frac{\sec ^4\left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b)}{(a-b)^2}+\frac{2 (3 a+b) \sec ^2\left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b)}{(a-b)^3}\right)}{64 d (a+b \sec (c+d x))^2}","\frac{\left(3 a^2-4 a b-b^2\right) \log (1-\cos (c+d x))}{16 d (a+b)^4}-\frac{\left(3 a^2+4 a b-b^2\right) \log (\cos (c+d x)+1)}{16 d (a-b)^4}+\frac{\csc ^4(c+d x) \left(2 a b-\left(a^2+b^2\right) \cos (c+d x)\right)}{4 d \left(a^2-b^2\right)^2}+\frac{\csc ^2(c+d x) \left(8 a b \left(a^2+b^2\right)-\left(3 a^4+12 a^2 b^2+b^4\right) \cos (c+d x)\right)}{8 d \left(a^2-b^2\right)^3}+\frac{a^3 b^2}{d \left(a^2-b^2\right)^3 (a \cos (c+d x)+b)}+\frac{2 a^3 b \left(a^2+2 b^2\right) \log (a \cos (c+d x)+b)}{d \left(a^2-b^2\right)^4}",1,"((b + a*Cos[c + d*x])*((64*a^3*b^2)/((a - b)^3*(a + b)^3) + (2*(-3*a + b)*(b + a*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/(a + b)^3 - ((b + a*Cos[c + d*x])*Csc[(c + d*x)/2]^4)/(a + b)^2 + (8*(-3*a^2 - 4*a*b + b^2)*(b + a*Cos[c + d*x])*Log[Cos[(c + d*x)/2]])/(a - b)^4 + (128*a^3*b*(a^2 + 2*b^2)*(b + a*Cos[c + d*x])*Log[b + a*Cos[c + d*x]])/(a^2 - b^2)^4 + (8*(3*a^2 - 4*a*b - b^2)*(b + a*Cos[c + d*x])*Log[Sin[(c + d*x)/2]])/(a + b)^4 + (2*(3*a + b)*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a - b)^3 + ((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^4)/(a - b)^2)*Sec[c + d*x]^2)/(64*d*(a + b*Sec[c + d*x])^2)","A",1
216,1,402,473,7.1870767,"\int \frac{\sin ^6(c+d x)}{(a+b \sec (c+d x))^2} \, dx","Integrate[Sin[c + d*x]^6/(a + b*Sec[c + d*x])^2,x]","\frac{3840 b \left(2 a^2-7 b^2\right) \left(a^2-b^2\right)^{3/2} \tanh ^{-1}\left(\frac{(b-a) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a^2-b^2}}\right)+\frac{-180 a^7 \sin (3 (c+d x))+40 a^7 \sin (5 (c+d x))-5 a^7 \sin (7 (c+d x))+1910 a^6 b \sin (2 (c+d x))-166 a^6 b \sin (4 (c+d x))+14 a^6 b \sin (6 (c+d x))+600 a^6 b c+600 a^6 b d x+790 a^5 b^2 \sin (3 (c+d x))-42 a^5 b^2 \sin (5 (c+d x))-5440 a^4 b^3 \sin (2 (c+d x))+140 a^4 b^3 \sin (4 (c+d x))-10800 a^4 b^3 c-10800 a^4 b^3 d x-560 a^3 b^4 \sin (3 (c+d x))+3360 a^2 b^5 \sin (2 (c+d x))+24000 a^2 b^5 c+24000 a^2 b^5 d x-15 a \left(15 a^6-576 a^4 b^2+1488 a^2 b^4-896 b^6\right) \sin (c+d x)+120 a \left(5 a^6-90 a^4 b^2+200 a^2 b^4-112 b^6\right) (c+d x) \cos (c+d x)-13440 b^7 c-13440 b^7 d x}{a \cos (c+d x)+b}}{1920 a^8 d}","\frac{7 b \sin (c+d x) \cos ^5(c+d x)}{30 a^2 d (a \cos (c+d x)+b)}-\frac{2 b (a-b)^{3/2} (a+b)^{3/2} \left(2 a^2-7 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^8 d}-\frac{\left(16 a^4-61 a^2 b^2+42 b^4\right) \sin (c+d x) \cos ^3(c+d x)}{24 a^4 b^2 d}+\frac{b \left(61 a^4-170 a^2 b^2+105 b^4\right) \sin (c+d x)}{15 a^7 d}-\frac{\left(27 a^4-86 a^2 b^2+56 b^4\right) \sin (c+d x) \cos (c+d x)}{16 a^6 d}+\frac{\left(15 a^4-52 a^2 b^2+35 b^4\right) \sin (c+d x) \cos ^2(c+d x)}{15 a^5 b d}+\frac{\left(5 a^4-20 a^2 b^2+14 b^4\right) \sin (c+d x) \cos ^4(c+d x)}{10 a^3 b^2 d (a \cos (c+d x)+b)}+\frac{x \left(5 a^6-90 a^4 b^2+200 a^2 b^4-112 b^6\right)}{16 a^8}+\frac{a \sin (c+d x) \cos ^4(c+d x)}{6 b^2 d (a \cos (c+d x)+b)}-\frac{\sin (c+d x) \cos ^6(c+d x)}{6 a d (a \cos (c+d x)+b)}-\frac{\sin (c+d x) \cos ^3(c+d x)}{3 b d (a \cos (c+d x)+b)}",1,"(3840*b*(2*a^2 - 7*b^2)*(a^2 - b^2)^(3/2)*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]] + (600*a^6*b*c - 10800*a^4*b^3*c + 24000*a^2*b^5*c - 13440*b^7*c + 600*a^6*b*d*x - 10800*a^4*b^3*d*x + 24000*a^2*b^5*d*x - 13440*b^7*d*x + 120*a*(5*a^6 - 90*a^4*b^2 + 200*a^2*b^4 - 112*b^6)*(c + d*x)*Cos[c + d*x] - 15*a*(15*a^6 - 576*a^4*b^2 + 1488*a^2*b^4 - 896*b^6)*Sin[c + d*x] + 1910*a^6*b*Sin[2*(c + d*x)] - 5440*a^4*b^3*Sin[2*(c + d*x)] + 3360*a^2*b^5*Sin[2*(c + d*x)] - 180*a^7*Sin[3*(c + d*x)] + 790*a^5*b^2*Sin[3*(c + d*x)] - 560*a^3*b^4*Sin[3*(c + d*x)] - 166*a^6*b*Sin[4*(c + d*x)] + 140*a^4*b^3*Sin[4*(c + d*x)] + 40*a^7*Sin[5*(c + d*x)] - 42*a^5*b^2*Sin[5*(c + d*x)] + 14*a^6*b*Sin[6*(c + d*x)] - 5*a^7*Sin[7*(c + d*x)])/(b + a*Cos[c + d*x]))/(1920*a^8*d)","A",1
217,1,282,261,3.3610074,"\int \frac{\sin ^4(c+d x)}{(a+b \sec (c+d x))^2} \, dx","Integrate[Sin[c + d*x]^4/(a + b*Sec[c + d*x])^2,x]","\frac{\frac{384 b \left(2 a^4-7 a^2 b^2+5 b^4\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{-21 a^5 \sin (3 (c+d x))+3 a^5 \sin (5 (c+d x))+176 a^4 b \sin (2 (c+d x))-10 a^4 b \sin (4 (c+d x))+72 a^4 b c+72 a^4 b d x+40 a^3 b^2 \sin (3 (c+d x))-240 a^2 b^3 \sin (2 (c+d x))-864 a^2 b^3 c-864 a^2 b^3 d x-24 a \left(a^4-31 a^2 b^2+40 b^4\right) \sin (c+d x)+24 a \left(3 a^4-36 a^2 b^2+40 b^4\right) (c+d x) \cos (c+d x)+960 b^5 c+960 b^5 d x}{a \cos (c+d x)+b}}{192 a^6 d}","-\frac{\left(a^2-b^2\right) \sin (c+d x) \cos ^3(c+d x)}{a^2 b d (a \cos (c+d x)+b)}+\frac{\sin (c+d x) \cos ^3(c+d x)}{4 a^2 d}-\frac{2 b \sqrt{a-b} \sqrt{a+b} \left(2 a^2-5 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^6 d}+\frac{b \left(11 a^2-15 b^2\right) \sin (c+d x)}{3 a^5 d}-\frac{\left(13 a^2-20 b^2\right) \sin (c+d x) \cos (c+d x)}{8 a^4 d}+\frac{\left(3 a^2-5 b^2\right) \sin (c+d x) \cos ^2(c+d x)}{3 a^3 b d}+\frac{x \left(3 a^4-36 a^2 b^2+40 b^4\right)}{8 a^6}",1,"((384*b*(2*a^4 - 7*a^2*b^2 + 5*b^4)*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/Sqrt[a^2 - b^2] + (72*a^4*b*c - 864*a^2*b^3*c + 960*b^5*c + 72*a^4*b*d*x - 864*a^2*b^3*d*x + 960*b^5*d*x + 24*a*(3*a^4 - 36*a^2*b^2 + 40*b^4)*(c + d*x)*Cos[c + d*x] - 24*a*(a^4 - 31*a^2*b^2 + 40*b^4)*Sin[c + d*x] + 176*a^4*b*Sin[2*(c + d*x)] - 240*a^2*b^3*Sin[2*(c + d*x)] - 21*a^5*Sin[3*(c + d*x)] + 40*a^3*b^2*Sin[3*(c + d*x)] - 10*a^4*b*Sin[4*(c + d*x)] + 3*a^5*Sin[5*(c + d*x)])/(b + a*Cos[c + d*x]))/(192*a^6*d)","A",1
218,1,178,152,1.3325322,"\int \frac{\sin ^2(c+d x)}{(a+b \sec (c+d x))^2} \, dx","Integrate[Sin[c + d*x]^2/(a + b*Sec[c + d*x])^2,x]","\frac{\frac{16 b \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{a^3 (-\sin (3 (c+d x)))-a \left(a^2-24 b^2\right) \sin (c+d x)+4 a \left(a^2-6 b^2\right) (c+d x) \cos (c+d x)+6 a^2 b \sin (2 (c+d x))+4 a^2 b c+4 a^2 b d x-24 b^3 c-24 b^3 d x}{a \cos (c+d x)+b}}{8 a^4 d}","\frac{3 b \sin (c+d x)}{a^3 d}-\frac{3 \sin (c+d x) \cos (c+d x)}{2 a^2 d}-\frac{2 b \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)}{a^4 d \sqrt{a-b} \sqrt{a+b}}+\frac{x \left(a^2-6 b^2\right)}{2 a^4}+\frac{\sin (c+d x) \cos ^2(c+d x)}{a d (a \cos (c+d x)+b)}",1,"((16*b*(2*a^2 - 3*b^2)*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/Sqrt[a^2 - b^2] + (4*a^2*b*c - 24*b^3*c + 4*a^2*b*d*x - 24*b^3*d*x + 4*a*(a^2 - 6*b^2)*(c + d*x)*Cos[c + d*x] - a*(a^2 - 24*b^2)*Sin[c + d*x] + 6*a^2*b*Sin[2*(c + d*x)] - a^3*Sin[3*(c + d*x)])/(b + a*Cos[c + d*x]))/(8*a^4*d)","A",1
219,1,128,203,0.9068355,"\int \frac{\csc ^2(c+d x)}{(a+b \sec (c+d x))^2} \, dx","Integrate[Csc[c + d*x]^2/(a + b*Sec[c + d*x])^2,x]","\frac{\frac{4 b \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)}{\left(a^2-b^2\right)^{5/2}}+\frac{\frac{2 a b^2 \sin (c+d x)}{(a+b)^2 (a \cos (c+d x)+b)}+\tan \left(\frac{1}{2} (c+d x)\right)}{(a-b)^2}-\frac{\cot \left(\frac{1}{2} (c+d x)\right)}{(a+b)^2}}{2 d}","\frac{a b^2 \sin (c+d x)}{d \left(a^2-b^2\right)^2 (a \cos (c+d x)+b)}-\frac{4 a^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)^{5/2} (a+b)^{5/2}}-\frac{2 b^3 \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{\sin (c+d x)}{2 d (a+b)^2 (1-\cos (c+d x))}+\frac{\sin (c+d x)}{2 d (a-b)^2 (\cos (c+d x)+1)}",1,"((4*b*(2*a^2 + b^2)*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(5/2) - Cot[(c + d*x)/2]/(a + b)^2 + ((2*a*b^2*Sin[c + d*x])/((a + b)^2*(b + a*Cos[c + d*x])) + Tan[(c + d*x)/2])/(a - b)^2)/(2*d)","A",1
220,1,281,343,1.130699,"\int \frac{\csc ^4(c+d x)}{(a+b \sec (c+d x))^2} \, dx","Integrate[Csc[c + d*x]^4/(a + b*Sec[c + d*x])^2,x]","\frac{\sec ^2(c+d x) (a \cos (c+d x)+b) \left(\frac{24 a^3 b^2 \sin (c+d x)}{(a-b)^3 (a+b)^3}+\frac{48 a^2 b \left(2 a^2+3 b^2\right) (a \cos (c+d x)+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)^{7/2}}+\frac{4 (2 a+b) \tan \left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b)}{(a-b)^3}-\frac{4 (2 a-b) \cot \left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b)}{(a+b)^3}-\frac{\cot \left(\frac{1}{2} (c+d x)\right) \csc ^2\left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b)}{(a+b)^2}+\frac{\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)}{(a-b)^2}\right)}{24 d (a+b \sec (c+d x))^2}","-\frac{2 a^2 b^3 \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{4 a^2 b \left(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^3 b^2 \sin (c+d x)}{d \left(a^2-b^2\right)^3 (a \cos (c+d x)+b)}-\frac{\sin (c+d x)}{12 d (a+b)^2 (1-\cos (c+d x))}-\frac{(a-b) \sin (c+d x)}{4 d (a+b)^3 (1-\cos (c+d x))}+\frac{(a+b) \sin (c+d x)}{4 d (a-b)^3 (\cos (c+d x)+1)}+\frac{\sin (c+d x)}{12 d (a-b)^2 (\cos (c+d x)+1)}-\frac{\sin (c+d x)}{12 d (a+b)^2 (1-\cos (c+d x))^2}+\frac{\sin (c+d x)}{12 d (a-b)^2 (\cos (c+d x)+1)^2}",1,"((b + a*Cos[c + d*x])*Sec[c + d*x]^2*((48*a^2*b*(2*a^2 + 3*b^2)*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]]*(b + a*Cos[c + d*x]))/(a^2 - b^2)^(7/2) - (4*(2*a - b)*(b + a*Cos[c + d*x])*Cot[(c + d*x)/2])/(a + b)^3 - ((b + a*Cos[c + d*x])*Cot[(c + d*x)/2]*Csc[(c + d*x)/2]^2)/(a + b)^2 + (24*a^3*b^2*Sin[c + d*x])/((a - b)^3*(a + b)^3) + (4*(2*a + b)*(b + a*Cos[c + d*x])*Tan[(c + d*x)/2])/(a - b)^3 + ((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/(a - b)^2))/(24*d*(a + b*Sec[c + d*x])^2)","A",1
221,1,550,329,5.0884201,"\int \frac{\sin ^7(c+d x)}{(a+b \sec (c+d x))^3} \, dx","Integrate[Sin[c + d*x]^7/(a + b*Sec[c + d*x])^3,x]","\frac{-784 a^9 \cos (3 (c+d x))+152 a^9 \cos (5 (c+d x))-39 a^9 \cos (7 (c+d x))+5 a^9 \cos (9 (c+d x))-1456 a^8 b \cos (4 (c+d x))+174 a^8 b \cos (6 (c+d x))-15 a^8 b \cos (8 (c+d x))+13440 a^8 b \log (a \cos (c+d x)+b)-7945 a^8 b+17528 a^7 b^2 \cos (3 (c+d x))-840 a^7 b^2 \cos (5 (c+d x))+48 a^7 b^2 \cos (7 (c+d x))+4872 a^6 b^3 \cos (4 (c+d x))-168 a^6 b^3 \cos (6 (c+d x))-107520 a^6 b^3 \log (a \cos (c+d x)+b)+164080 a^6 b^3-43680 a^5 b^4 \cos (3 (c+d x))+672 a^5 b^4 \cos (5 (c+d x))-3360 a^4 b^5 \cos (4 (c+d x))+13440 a^4 b^5 \log (a \cos (c+d x)+b)-502320 a^4 b^5+26880 a^3 b^6 \cos (3 (c+d x))+403200 a^2 b^7 \log (a \cos (c+d x)+b)+425600 a^2 b^7+70 a^2 b \cos (2 (c+d x)) \left(-137 a^6+1896 a^4 b^2-4656 a^2 b^4+192 \left(a^6-10 a^4 b^2+21 a^2 b^4-12 b^6\right) \log (a \cos (c+d x)+b)+2912 b^6\right)-70 a \cos (c+d x) \left(49 a^8-1472 a^6 b^2+3216 a^4 b^4+576 a^2 b^6-768 b^2 \left(a^6-10 a^4 b^2+21 a^2 b^4-12 b^6\right) \log (a \cos (c+d x)+b)-2432 b^8\right)-322560 b^9 \log (a \cos (c+d x)+b)-76160 b^9}{8960 a^{10} d (a \cos (c+d x)+b)^2}","-\frac{b \cos ^6(c+d x)}{2 a^4 d}+\frac{\cos ^7(c+d x)}{7 a^3 d}+\frac{3 b^2 \left(a^2-3 b^2\right) \left(a^2-b^2\right)^2}{a^{10} d (a \cos (c+d x)+b)}-\frac{b^3 \left(a^2-b^2\right)^3}{2 a^{10} d (a \cos (c+d x)+b)^2}+\frac{b \left(9 a^2-10 b^2\right) \cos ^4(c+d x)}{4 a^6 d}-\frac{3 \left(a^2-2 b^2\right) \cos ^5(c+d x)}{5 a^5 d}+\frac{3 b \left(a^2-b^2\right) \left(a^4-9 a^2 b^2+12 b^4\right) \log (a \cos (c+d x)+b)}{a^{10} d}-\frac{3 b \left(3 a^4-10 a^2 b^2+7 b^4\right) \cos ^2(c+d x)}{2 a^8 d}+\frac{\left(a^4-6 a^2 b^2+5 b^4\right) \cos ^3(c+d x)}{a^7 d}-\frac{\left(a^6-18 a^4 b^2+45 a^2 b^4-28 b^6\right) \cos (c+d x)}{a^9 d}",1,"(-7945*a^8*b + 164080*a^6*b^3 - 502320*a^4*b^5 + 425600*a^2*b^7 - 76160*b^9 - 784*a^9*Cos[3*(c + d*x)] + 17528*a^7*b^2*Cos[3*(c + d*x)] - 43680*a^5*b^4*Cos[3*(c + d*x)] + 26880*a^3*b^6*Cos[3*(c + d*x)] - 1456*a^8*b*Cos[4*(c + d*x)] + 4872*a^6*b^3*Cos[4*(c + d*x)] - 3360*a^4*b^5*Cos[4*(c + d*x)] + 152*a^9*Cos[5*(c + d*x)] - 840*a^7*b^2*Cos[5*(c + d*x)] + 672*a^5*b^4*Cos[5*(c + d*x)] + 174*a^8*b*Cos[6*(c + d*x)] - 168*a^6*b^3*Cos[6*(c + d*x)] - 39*a^9*Cos[7*(c + d*x)] + 48*a^7*b^2*Cos[7*(c + d*x)] - 15*a^8*b*Cos[8*(c + d*x)] + 5*a^9*Cos[9*(c + d*x)] + 13440*a^8*b*Log[b + a*Cos[c + d*x]] - 107520*a^6*b^3*Log[b + a*Cos[c + d*x]] + 13440*a^4*b^5*Log[b + a*Cos[c + d*x]] + 403200*a^2*b^7*Log[b + a*Cos[c + d*x]] - 322560*b^9*Log[b + a*Cos[c + d*x]] + 70*a^2*b*Cos[2*(c + d*x)]*(-137*a^6 + 1896*a^4*b^2 - 4656*a^2*b^4 + 2912*b^6 + 192*(a^6 - 10*a^4*b^2 + 21*a^2*b^4 - 12*b^6)*Log[b + a*Cos[c + d*x]]) - 70*a*Cos[c + d*x]*(49*a^8 - 1472*a^6*b^2 + 3216*a^4*b^4 + 576*a^2*b^6 - 2432*b^8 - 768*b^2*(a^6 - 10*a^4*b^2 + 21*a^2*b^4 - 12*b^6)*Log[b + a*Cos[c + d*x]]))/(8960*a^10*d*(b + a*Cos[c + d*x])^2)","A",1
222,1,388,239,3.0007617,"\int \frac{\sin ^5(c+d x)}{(a+b \sec (c+d x))^3} \, dx","Integrate[Sin[c + d*x]^5/(a + b*Sec[c + d*x])^3,x]","\frac{-206 a^7 \cos (3 (c+d x))+38 a^7 \cos (5 (c+d x))-6 a^7 \cos (7 (c+d x))-274 a^6 b \cos (4 (c+d x))+21 a^6 b \cos (6 (c+d x))+2880 a^6 b \log (a \cos (c+d x)+b)-1740 a^6 b+2780 a^5 b^2 \cos (3 (c+d x))-84 a^5 b^2 \cos (5 (c+d x))+420 a^4 b^3 \cos (4 (c+d x))-13440 a^4 b^3 \log (a \cos (c+d x)+b)+26160 a^4 b^3-3360 a^3 b^4 \cos (3 (c+d x))-18240 a^2 b^5 \log (a \cos (c+d x)+b)-46080 a^2 b^5+5 a^2 b \cos (2 (c+d x)) \left(-407 a^4+3888 a^2 b^2+192 \left(3 a^4-20 a^2 b^2+21 b^4\right) \log (a \cos (c+d x)+b)-4800 b^4\right)-10 a \cos (c+d x) \left(85 a^6-1728 a^4 b^2+1584 a^2 b^4-384 b^2 \left(3 a^4-20 a^2 b^2+21 b^4\right) \log (a \cos (c+d x)+b)+1536 b^6\right)+40320 b^7 \log (a \cos (c+d x)+b)+12480 b^7}{1920 a^8 d (a \cos (c+d x)+b)^2}","\frac{3 b \cos ^4(c+d x)}{4 a^4 d}-\frac{\cos ^5(c+d x)}{5 a^3 d}-\frac{b^3 \left(a^2-b^2\right)^2}{2 a^8 d (a \cos (c+d x)+b)^2}-\frac{b \left(3 a^2-5 b^2\right) \cos ^2(c+d x)}{a^6 d}+\frac{2 \left(a^2-3 b^2\right) \cos ^3(c+d x)}{3 a^5 d}+\frac{b^2 \left(3 a^4-10 a^2 b^2+7 b^4\right)}{a^8 d (a \cos (c+d x)+b)}+\frac{b \left(3 a^4-20 a^2 b^2+21 b^4\right) \log (a \cos (c+d x)+b)}{a^8 d}-\frac{\left(a^4-12 a^2 b^2+15 b^4\right) \cos (c+d x)}{a^7 d}",1,"(-1740*a^6*b + 26160*a^4*b^3 - 46080*a^2*b^5 + 12480*b^7 - 206*a^7*Cos[3*(c + d*x)] + 2780*a^5*b^2*Cos[3*(c + d*x)] - 3360*a^3*b^4*Cos[3*(c + d*x)] - 274*a^6*b*Cos[4*(c + d*x)] + 420*a^4*b^3*Cos[4*(c + d*x)] + 38*a^7*Cos[5*(c + d*x)] - 84*a^5*b^2*Cos[5*(c + d*x)] + 21*a^6*b*Cos[6*(c + d*x)] - 6*a^7*Cos[7*(c + d*x)] + 2880*a^6*b*Log[b + a*Cos[c + d*x]] - 13440*a^4*b^3*Log[b + a*Cos[c + d*x]] - 18240*a^2*b^5*Log[b + a*Cos[c + d*x]] + 40320*b^7*Log[b + a*Cos[c + d*x]] + 5*a^2*b*Cos[2*(c + d*x)]*(-407*a^4 + 3888*a^2*b^2 - 4800*b^4 + 192*(3*a^4 - 20*a^2*b^2 + 21*b^4)*Log[b + a*Cos[c + d*x]]) - 10*a*Cos[c + d*x]*(85*a^6 - 1728*a^4*b^2 + 1584*a^2*b^4 + 1536*b^6 - 384*b^2*(3*a^4 - 20*a^2*b^2 + 21*b^4)*Log[b + a*Cos[c + d*x]]))/(1920*a^8*d*(b + a*Cos[c + d*x])^2)","A",1
223,1,208,158,0.783054,"\int \frac{\sin ^3(c+d x)}{(a+b \sec (c+d x))^3} \, dx","Integrate[Sin[c + d*x]^3/(a + b*Sec[c + d*x])^3,x]","\frac{\sec ^3(c+d x) (a \cos (c+d x)+b) \left(9 a^4 (2 a \cos (c+d x)+b)-(a \cos (c+d x)+b)^2 \left(-8 a^3 \cos (3 (c+d x))+96 \left(10 b^3-3 a^2 b\right) \log (a \cos (c+d x)+b)+72 a \left(a^2-8 b^2\right) \cos (c+d x)+72 a^2 b \cos (2 (c+d x))+\frac{-9 a^4 b+48 a^2 b^3-48 b^5}{(a \cos (c+d x)+b)^2}+\frac{6 \left(3 a^4-48 a^2 b^2+80 b^4\right)}{a \cos (c+d x)+b}\right)\right)}{96 a^6 d (a+b \sec (c+d x))^3}","-\frac{3 b \cos ^2(c+d x)}{2 a^4 d}+\frac{\cos ^3(c+d x)}{3 a^3 d}+\frac{b^2 \left(3 a^2-5 b^2\right)}{a^6 d (a \cos (c+d x)+b)}+\frac{b \left(3 a^2-10 b^2\right) \log (a \cos (c+d x)+b)}{a^6 d}-\frac{b^3 \left(a^2-b^2\right)}{2 a^6 d (a \cos (c+d x)+b)^2}-\frac{\left(a^2-6 b^2\right) \cos (c+d x)}{a^5 d}",1,"((b + a*Cos[c + d*x])*(9*a^4*(b + 2*a*Cos[c + d*x]) - (b + a*Cos[c + d*x])^2*(72*a*(a^2 - 8*b^2)*Cos[c + d*x] + (-9*a^4*b + 48*a^2*b^3 - 48*b^5)/(b + a*Cos[c + d*x])^2 + (6*(3*a^4 - 48*a^2*b^2 + 80*b^4))/(b + a*Cos[c + d*x]) + 72*a^2*b*Cos[2*(c + d*x)] - 8*a^3*Cos[3*(c + d*x)] + 96*(-3*a^2*b + 10*b^3)*Log[b + a*Cos[c + d*x]]))*Sec[c + d*x]^3)/(96*a^6*d*(a + b*Sec[c + d*x])^3)","A",1
224,1,111,83,0.4586379,"\int \frac{\sin (c+d x)}{(a+b \sec (c+d x))^3} \, dx","Integrate[Sin[c + d*x]/(a + b*Sec[c + d*x])^3,x]","\frac{-2 a^3 \cos ^3(c+d x)+2 a^2 b \cos ^2(c+d x) (3 \log (a \cos (c+d x)+b)-2)+b^3 (6 \log (a \cos (c+d x)+b)+5)+4 a b^2 \cos (c+d x) (3 \log (a \cos (c+d x)+b)+1)}{2 a^4 d (a \cos (c+d x)+b)^2}","-\frac{b^3}{2 a^4 d (a \cos (c+d x)+b)^2}+\frac{3 b^2}{a^4 d (a \cos (c+d x)+b)}+\frac{3 b \log (a \cos (c+d x)+b)}{a^4 d}-\frac{\cos (c+d x)}{a^3 d}",1,"(-2*a^3*Cos[c + d*x]^3 + 2*a^2*b*Cos[c + d*x]^2*(-2 + 3*Log[b + a*Cos[c + d*x]]) + 4*a*b^2*Cos[c + d*x]*(1 + 3*Log[b + a*Cos[c + d*x]]) + b^3*(5 + 6*Log[b + a*Cos[c + d*x]]))/(2*a^4*d*(b + a*Cos[c + d*x])^2)","A",1
225,1,203,163,0.6056801,"\int \frac{\csc (c+d x)}{(a+b \sec (c+d x))^3} \, dx","Integrate[Csc[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{2 b^2 \left(b^2-3 a^2\right) (a \cos (c+d x)+b)}{a^2 (a-b)^2 (a+b)^2}+\frac{2 b \left(3 a^2+b^2\right) (a \cos (c+d x)+b)^2 \log (a \cos (c+d x)+b)}{\left(a^2-b^2\right)^3}+\frac{b^3}{a^2 \left(b^2-a^2\right)}+\frac{2 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)\right) (a \cos (c+d x)+b)^2}{(b-a)^3}+\frac{2 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)\right) (a \cos (c+d x)+b)^2}{(a+b)^3}\right)}{2 d (a+b \sec (c+d x))^3}","\frac{b^2 \left(3 a^2-b^2\right)}{a^2 d \left(a^2-b^2\right)^2 (a \cos (c+d x)+b)}+\frac{b \left(3 a^2+b^2\right) \log (a \cos (c+d x)+b)}{d \left(a^2-b^2\right)^3}-\frac{b^3}{2 a^2 d \left(a^2-b^2\right) (a \cos (c+d x)+b)^2}+\frac{\log (1-\cos (c+d x))}{2 d (a+b)^3}-\frac{\log (\cos (c+d x)+1)}{2 d (a-b)^3}",1,"((b + a*Cos[c + d*x])*(b^3/(a^2*(-a^2 + b^2)) - (2*b^2*(-3*a^2 + b^2)*(b + a*Cos[c + d*x]))/(a^2*(a - b)^2*(a + b)^2) + (2*(b + a*Cos[c + d*x])^2*Log[Cos[(c + d*x)/2]])/(-a + b)^3 + (2*b*(3*a^2 + b^2)*(b + a*Cos[c + d*x])^2*Log[b + a*Cos[c + d*x]])/(a^2 - b^2)^3 + (2*(b + a*Cos[c + d*x])^2*Log[Sin[(c + d*x)/2]])/(a + b)^3)*Sec[c + d*x]^3)/(2*d*(a + b*Sec[c + d*x])^3)","A",1
226,1,332,229,6.3370687,"\int \frac{\csc ^3(c+d x)}{(a+b \sec (c+d x))^3} \, dx","Integrate[Csc[c + d*x]^3/(a + b*Sec[c + d*x])^3,x]","-\frac{b^2 \left(3 a^2+b^2\right)}{d (b-a)^3 (a+b)^3 (a \cos (c+d x)+b)}-\frac{2 i \left(3 a^4 b+8 a^2 b^3+b^5\right) (c+d x)}{d (a-b)^4 (a+b)^4}+\frac{\left(3 a^4 b+8 a^2 b^3+b^5\right) \log (a \cos (c+d x)+b)}{d \left(b^2-a^2\right)^4}-\frac{b^3}{2 d (b-a)^2 (a+b)^2 (a \cos (c+d x)+b)^2}-\frac{i (-a-2 b) \tan ^{-1}(\tan (c+d x))}{2 d (b-a)^4}-\frac{i (a-2 b) \tan ^{-1}(\tan (c+d x))}{2 d (a+b)^4}-\frac{\csc ^2\left(\frac{1}{2} (c+d x)\right)}{8 d (a+b)^3}-\frac{\sec ^2\left(\frac{1}{2} (c+d x)\right)}{8 d (b-a)^3}+\frac{(a-2 b) \log \left(\sin ^2\left(\frac{1}{2} (c+d x)\right)\right)}{4 d (a+b)^4}+\frac{(-a-2 b) \log \left(\cos ^2\left(\frac{1}{2} (c+d x)\right)\right)}{4 d (b-a)^4}","\frac{b^2 \left(3 a^2+b^2\right)}{d \left(a^2-b^2\right)^3 (a \cos (c+d x)+b)}+\frac{\csc ^2(c+d x) \left(b \left(3 a^2+b^2\right)-a \left(a^2+3 b^2\right) \cos (c+d x)\right)}{2 d \left(a^2-b^2\right)^3}-\frac{b^3}{2 d \left(a^2-b^2\right)^2 (a \cos (c+d x)+b)^2}+\frac{b \left(3 a^4+8 a^2 b^2+b^4\right) \log (a \cos (c+d x)+b)}{d \left(a^2-b^2\right)^4}+\frac{(a-2 b) \log (1-\cos (c+d x))}{4 d (a+b)^4}-\frac{(a+2 b) \log (\cos (c+d x)+1)}{4 d (a-b)^4}",1,"((-2*I)*(3*a^4*b + 8*a^2*b^3 + b^5)*(c + d*x))/((a - b)^4*(a + b)^4*d) - ((I/2)*(-a - 2*b)*ArcTan[Tan[c + d*x]])/((-a + b)^4*d) - ((I/2)*(a - 2*b)*ArcTan[Tan[c + d*x]])/((a + b)^4*d) - b^3/(2*(-a + b)^2*(a + b)^2*d*(b + a*Cos[c + d*x])^2) - (b^2*(3*a^2 + b^2))/((-a + b)^3*(a + b)^3*d*(b + a*Cos[c + d*x])) - Csc[(c + d*x)/2]^2/(8*(a + b)^3*d) + ((-a - 2*b)*Log[Cos[(c + d*x)/2]^2])/(4*(-a + b)^4*d) + ((3*a^4*b + 8*a^2*b^3 + b^5)*Log[b + a*Cos[c + d*x]])/((-a^2 + b^2)^4*d) + ((a - 2*b)*Log[Sin[(c + d*x)/2]^2])/(4*(a + b)^4*d) - Sec[(c + d*x)/2]^2/(8*(-a + b)^3*d)","C",1
227,1,496,313,4.8885601,"\int \frac{\csc ^5(c+d x)}{(a+b \sec (c+d x))^3} \, dx","Integrate[Csc[c + d*x]^5/(a + b*Sec[c + d*x])^3,x]","\frac{\sec ^3(c+d x) (a \cos (c+d x)+b) \left(\frac{32 a^2 b^3}{(b-a)^3 (a+b)^3}+\frac{192 a^2 b^2 (a-i b) (a+i b) (a \cos (c+d x)+b)}{(a-b)^4 (a+b)^4}-\frac{384 i a^2 b \left(a^4+5 a^2 b^2+2 b^4\right) (c+d x) (a \cos (c+d x)+b)^2}{(a-b)^5 (a+b)^5}+\frac{192 a^2 b \left(a^4+5 a^2 b^2+2 b^4\right) (a \cos (c+d x)+b)^2 \log (a \cos (c+d x)+b)}{\left(a^2-b^2\right)^5}-\frac{12 a (a+3 b) \log \left(\cos ^2\left(\frac{1}{2} (c+d x)\right)\right) (a \cos (c+d x)+b)^2}{(a-b)^5}+\frac{24 i a (a+3 b) \tan ^{-1}(\tan (c+d x)) (a \cos (c+d x)+b)^2}{(a-b)^5}-\frac{24 i a (a-3 b) \tan ^{-1}(\tan (c+d x)) (a \cos (c+d x)+b)^2}{(a+b)^5}-\frac{\csc ^4\left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b)^2}{(a+b)^3}+\frac{6 (b-a) \csc ^2\left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b)^2}{(a+b)^4}+\frac{\sec ^4\left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b)^2}{(a-b)^3}+\frac{6 (a+b) \sec ^2\left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b)^2}{(a-b)^4}+\frac{12 a (a-3 b) \log \left(\sin ^2\left(\frac{1}{2} (c+d x)\right)\right) (a \cos (c+d x)+b)^2}{(a+b)^5}\right)}{64 d (a+b \sec (c+d x))^3}","\frac{3 a^2 b^2 \left(a^2+b^2\right)}{d \left(a^2-b^2\right)^4 (a \cos (c+d x)+b)}+\frac{\csc ^4(c+d x) \left(b \left(3 a^2+b^2\right)-a \left(a^2+3 b^2\right) \cos (c+d x)\right)}{4 d \left(a^2-b^2\right)^3}-\frac{a^2 b^3}{2 d \left(a^2-b^2\right)^3 (a \cos (c+d x)+b)^2}+\frac{3 a^2 b \left(a^4+5 a^2 b^2+2 b^4\right) \log (a \cos (c+d x)+b)}{d \left(a^2-b^2\right)^5}+\frac{\csc ^2(c+d x) \left(4 b \left(3 a^4+8 a^2 b^2+b^4\right)-3 a \left(a^4+10 a^2 b^2+5 b^4\right) \cos (c+d x)\right)}{8 d \left(a^2-b^2\right)^4}+\frac{3 a (a-3 b) \log (1-\cos (c+d x))}{16 d (a+b)^5}-\frac{3 a (a+3 b) \log (\cos (c+d x)+1)}{16 d (a-b)^5}",1,"((b + a*Cos[c + d*x])*((32*a^2*b^3)/((-a + b)^3*(a + b)^3) + (192*a^2*(a - I*b)*(a + I*b)*b^2*(b + a*Cos[c + d*x]))/((a - b)^4*(a + b)^4) - ((384*I)*a^2*b*(a^4 + 5*a^2*b^2 + 2*b^4)*(c + d*x)*(b + a*Cos[c + d*x])^2)/((a - b)^5*(a + b)^5) - ((24*I)*a*(a - 3*b)*ArcTan[Tan[c + d*x]]*(b + a*Cos[c + d*x])^2)/(a + b)^5 + ((24*I)*a*(a + 3*b)*ArcTan[Tan[c + d*x]]*(b + a*Cos[c + d*x])^2)/(a - b)^5 + (6*(-a + b)*(b + a*Cos[c + d*x])^2*Csc[(c + d*x)/2]^2)/(a + b)^4 - ((b + a*Cos[c + d*x])^2*Csc[(c + d*x)/2]^4)/(a + b)^3 - (12*a*(a + 3*b)*(b + a*Cos[c + d*x])^2*Log[Cos[(c + d*x)/2]^2])/(a - b)^5 + (192*a^2*b*(a^4 + 5*a^2*b^2 + 2*b^4)*(b + a*Cos[c + d*x])^2*Log[b + a*Cos[c + d*x]])/(a^2 - b^2)^5 + (12*a*(a - 3*b)*(b + a*Cos[c + d*x])^2*Log[Sin[(c + d*x)/2]^2])/(a + b)^5 + (6*(a + b)*(b + a*Cos[c + d*x])^2*Sec[(c + d*x)/2]^2)/(a - b)^4 + ((b + a*Cos[c + d*x])^2*Sec[(c + d*x)/2]^4)/(a - b)^3)*Sec[c + d*x]^3)/(64*d*(a + b*Sec[c + d*x])^3)","C",1
228,1,599,539,12.6952078,"\int \frac{\sin ^6(c+d x)}{(a+b \sec (c+d x))^3} \, dx","Integrate[Sin[c + d*x]^6/(a + b*Sec[c + d*x])^3,x]","\frac{2 \left(a^2-b^2\right)^{5/2} \left(-405 a^8 \sin (2 (c+d x))-140 a^8 \sin (4 (c+d x))+35 a^8 \sin (6 (c+d x))-5 a^8 \sin (8 (c+d x))+600 a^8 c+600 a^8 d x+2640 a^7 b \sin (c+d x)+2436 a^7 b \sin (3 (c+d x))-188 a^7 b \sin (5 (c+d x))+16 a^7 b \sin (7 (c+d x))+24600 a^6 b^2 \sin (2 (c+d x))+1164 a^6 b^2 \sin (4 (c+d x))-56 a^6 b^2 \sin (6 (c+d x))-20400 a^6 b^2 c-20400 a^6 b^2 d x+16160 a^5 b^3 \sin (c+d x)-10880 a^5 b^3 \sin (3 (c+d x))+224 a^5 b^3 \sin (5 (c+d x))-99040 a^4 b^4 \sin (2 (c+d x))-1120 a^4 b^4 \sin (4 (c+d x))+28800 a^4 b^4 c+28800 a^4 b^4 d x-117120 a^3 b^5 \sin (c+d x)+8960 a^3 b^5 \sin (3 (c+d x))+80640 a^2 b^6 \sin (2 (c+d x))+90240 a^2 b^6 c+90240 a^2 b^6 d x+120 a^2 \left(5 a^6-180 a^4 b^2+600 a^2 b^4-448 b^6\right) (c+d x) \cos (2 (c+d x))+480 a b \left(5 a^6-180 a^4 b^2+600 a^2 b^4-448 b^6\right) (c+d x) \cos (c+d x)+107520 a b^7 \sin (c+d x)-107520 b^8 c-107520 b^8 d x\right)-7680 b \left(b^2-a^2\right)^3 \left(6 a^4-47 a^2 b^2+56 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)}{7680 a^9 d (a-b)^2 (a+b)^2 \sqrt{a^2-b^2} (a \cos (c+d x)+b)^2}","\frac{4 b \sin (c+d x) \cos ^6(c+d x)}{15 a^2 d (a \cos (c+d x)+b)^2}+\frac{\left(15 a^4-110 a^2 b^2+112 b^4\right) \sin (c+d x) \cos ^4(c+d x)}{20 a^4 b^2 d (a \cos (c+d x)+b)}-\frac{b \sqrt{a-b} \sqrt{a+b} \left(6 a^4-47 a^2 b^2+56 b^4\right) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^9 d}+\frac{b \left(213 a^4-985 a^2 b^2+840 b^4\right) \sin (c+d x)}{30 a^8 d}-\frac{\left(43 a^4-244 a^2 b^2+224 b^4\right) \sin (c+d x) \cos (c+d x)}{16 a^7 d}+\frac{\left(45 a^4-291 a^2 b^2+280 b^4\right) \sin (c+d x) \cos ^2(c+d x)}{30 a^6 b d}-\frac{\left(24 a^4-169 a^2 b^2+168 b^4\right) \sin (c+d x) \cos ^3(c+d x)}{24 a^5 b^2 d}+\frac{\left(9 a^4-60 a^2 b^2+56 b^4\right) \sin (c+d x) \cos ^5(c+d x)}{60 a^3 b^2 d (a \cos (c+d x)+b)^2}+\frac{x \left(5 a^6-180 a^4 b^2+600 a^2 b^4-448 b^6\right)}{16 a^9}+\frac{a \sin (c+d x) \cos ^5(c+d x)}{10 b^2 d (a \cos (c+d x)+b)^2}-\frac{\sin (c+d x) \cos ^7(c+d x)}{6 a d (a \cos (c+d x)+b)^2}-\frac{\sin (c+d x) \cos ^4(c+d x)}{4 b d (a \cos (c+d x)+b)^2}",1,"(-7680*b*(-a^2 + b^2)^3*(6*a^4 - 47*a^2*b^2 + 56*b^4)*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]]*(b + a*Cos[c + d*x])^2 + 2*(a^2 - b^2)^(5/2)*(600*a^8*c - 20400*a^6*b^2*c + 28800*a^4*b^4*c + 90240*a^2*b^6*c - 107520*b^8*c + 600*a^8*d*x - 20400*a^6*b^2*d*x + 28800*a^4*b^4*d*x + 90240*a^2*b^6*d*x - 107520*b^8*d*x + 480*a*b*(5*a^6 - 180*a^4*b^2 + 600*a^2*b^4 - 448*b^6)*(c + d*x)*Cos[c + d*x] + 120*a^2*(5*a^6 - 180*a^4*b^2 + 600*a^2*b^4 - 448*b^6)*(c + d*x)*Cos[2*(c + d*x)] + 2640*a^7*b*Sin[c + d*x] + 16160*a^5*b^3*Sin[c + d*x] - 117120*a^3*b^5*Sin[c + d*x] + 107520*a*b^7*Sin[c + d*x] - 405*a^8*Sin[2*(c + d*x)] + 24600*a^6*b^2*Sin[2*(c + d*x)] - 99040*a^4*b^4*Sin[2*(c + d*x)] + 80640*a^2*b^6*Sin[2*(c + d*x)] + 2436*a^7*b*Sin[3*(c + d*x)] - 10880*a^5*b^3*Sin[3*(c + d*x)] + 8960*a^3*b^5*Sin[3*(c + d*x)] - 140*a^8*Sin[4*(c + d*x)] + 1164*a^6*b^2*Sin[4*(c + d*x)] - 1120*a^4*b^4*Sin[4*(c + d*x)] - 188*a^7*b*Sin[5*(c + d*x)] + 224*a^5*b^3*Sin[5*(c + d*x)] + 35*a^8*Sin[6*(c + d*x)] - 56*a^6*b^2*Sin[6*(c + d*x)] + 16*a^7*b*Sin[7*(c + d*x)] - 5*a^8*Sin[8*(c + d*x)]))/(7680*a^9*(a - b)^2*(a + b)^2*Sqrt[a^2 - b^2]*d*(b + a*Cos[c + d*x])^2)","A",1
229,1,1178,333,9.0462093,"\int \frac{\sin ^4(c+d x)}{(a+b \sec (c+d x))^3} \, dx","Integrate[Sin[c + d*x]^4/(a + b*Sec[c + d*x])^3,x]","\frac{-\frac{6 \left(8 (c+d x)+\frac{2 b \left(15 a^4-20 b^2 a^2+8 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{3 a \left(2 a^4-7 b^2 a^2+4 b^4\right) \sin (c+d x)}{(a-b)^2 (a+b)^2 (b+a \cos (c+d x))}+\frac{a b \left(3 a^2-4 b^2\right) \sin (c+d x)}{(a-b) (a+b) (b+a \cos (c+d x))^2}\right)}{a^3}+\frac{6 \left(\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}}+\frac{\left(b \left(a^2+2 b^2\right)+a \left(2 a^2+b^2\right) \cos (c+d x)\right) \sin (c+d x)}{(b+a \cos (c+d x))^2}\right)}{(a-b)^2 (a+b)^2}-\frac{2 \left(8 \sin (2 (c+d x)) a^2-96 b \sin (c+d x) a+\frac{\left(10 a^6-115 b^2 a^4+220 b^4 a^2-112 b^6\right) \sin (c+d x) a}{(a-b)^2 (a+b)^2 (b+a \cos (c+d x))}+\frac{b \left(-5 a^4+20 b^2 a^2-16 b^4\right) \sin (c+d x) a}{(a-b) (a+b) (b+a \cos (c+d x))^2}-24 \left(a^2-8 b^2\right) (c+d x)+\frac{6 b \left(-35 a^6+140 b^2 a^4-168 b^4 a^2+64 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)^{5/2}}\right)}{a^5}+\frac{\frac{12 b \left(105 a^8-840 b^2 a^6+2016 b^4 a^4-1920 b^6 a^2+640 b^8\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{48 c a^{10}+48 d x a^{10}-36 \sin (2 (c+d x)) a^{10}-8 \sin (4 (c+d x)) a^{10}+2 \sin (6 (c+d x)) a^{10}+114 b \sin (c+d x) a^9+120 b \sin (3 (c+d x)) a^9-8 b \sin (5 (c+d x)) a^9-960 b^2 c a^8-960 b^2 d x a^8+1221 b^2 \sin (2 (c+d x)) a^8+56 b^2 \sin (4 (c+d x)) a^8-4 b^2 \sin (6 (c+d x)) a^8+788 b^3 \sin (c+d x) a^7-560 b^3 \sin (3 (c+d x)) a^7+16 b^3 \sin (5 (c+d x)) a^7+1776 b^4 c a^6+1776 b^4 d x a^6-5182 b^4 \sin (2 (c+d x)) a^6-88 b^4 \sin (4 (c+d x)) a^6+2 b^4 \sin (6 (c+d x)) a^6-5696 b^5 \sin (c+d x) a^5+760 b^5 \sin (3 (c+d x)) a^5-8 b^5 \sin (5 (c+d x)) a^5+2976 b^6 c a^4+2976 b^6 d x a^4+6880 b^6 \sin (2 (c+d x)) a^4+40 b^6 \sin (4 (c+d x)) a^4+8640 b^7 \sin (c+d x) a^3-320 b^7 \sin (3 (c+d x)) a^3-7680 b^8 c a^2-7680 b^8 d x a^2-2880 b^8 \sin (2 (c+d x)) a^2+192 b \left(a^2-b^2\right)^2 \left(a^4-20 b^2 a^2+40 b^4\right) (c+d x) \cos (c+d x) a-3840 b^9 \sin (c+d x) a+3840 b^{10} c+3840 b^{10} d x+48 \left(a^3-a b^2\right)^2 \left(a^4-20 b^2 a^2+40 b^4\right) (c+d x) \cos (2 (c+d x))}{\left(a^2-b^2\right)^2 (b+a \cos (c+d x))^2}}{a^7}}{256 d}","\frac{\left(2 a^2-7 b^2\right) \sin (c+d x) \cos ^4(c+d x)}{2 a^2 b^2 d (a \cos (c+d x)+b)}-\frac{\left(a^2-b^2\right) \sin (c+d x) \cos ^4(c+d x)}{2 a^2 b d (a \cos (c+d x)+b)^2}+\frac{b \left(13 a^2-30 b^2\right) \sin (c+d x)}{2 a^6 d}-\frac{3 \left(7 a^2-20 b^2\right) \sin (c+d x) \cos (c+d x)}{8 a^5 d}+\frac{\left(3 a^2-10 b^2\right) \sin (c+d x) \cos ^2(c+d x)}{2 a^4 b d}-\frac{\left(4 a^2-15 b^2\right) \sin (c+d x) \cos ^3(c+d x)}{4 a^3 b^2 d}-\frac{3 b \left(2 a^4-11 a^2 b^2+10 b^4\right) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^7 d \sqrt{a-b} \sqrt{a+b}}+\frac{3 x \left(a^4-24 a^2 b^2+40 b^4\right)}{8 a^7}",1,"((-6*(8*(c + d*x) + (2*b*(15*a^4 - 20*a^2*b^2 + 8*b^4)*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(5/2) + (a*b*(3*a^2 - 4*b^2)*Sin[c + d*x])/((a - b)*(a + b)*(b + a*Cos[c + d*x])^2) - (3*a*(2*a^4 - 7*a^2*b^2 + 4*b^4)*Sin[c + d*x])/((a - b)^2*(a + b)^2*(b + a*Cos[c + d*x]))))/a^3 + (6*((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))/((a - b)^2*(a + b)^2) - (2*(-24*(a^2 - 8*b^2)*(c + d*x) + (6*b*(-35*a^6 + 140*a^4*b^2 - 168*a^2*b^4 + 64*b^6)*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(5/2) - 96*a*b*Sin[c + d*x] + (a*b*(-5*a^4 + 20*a^2*b^2 - 16*b^4)*Sin[c + d*x])/((a - b)*(a + b)*(b + a*Cos[c + d*x])^2) + (a*(10*a^6 - 115*a^4*b^2 + 220*a^2*b^4 - 112*b^6)*Sin[c + d*x])/((a - b)^2*(a + b)^2*(b + a*Cos[c + d*x])) + 8*a^2*Sin[2*(c + d*x)]))/a^5 + ((12*b*(105*a^8 - 840*a^6*b^2 + 2016*a^4*b^4 - 1920*a^2*b^6 + 640*b^8)*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(5/2) + (48*a^10*c - 960*a^8*b^2*c + 1776*a^6*b^4*c + 2976*a^4*b^6*c - 7680*a^2*b^8*c + 3840*b^10*c + 48*a^10*d*x - 960*a^8*b^2*d*x + 1776*a^6*b^4*d*x + 2976*a^4*b^6*d*x - 7680*a^2*b^8*d*x + 3840*b^10*d*x + 192*a*b*(a^2 - b^2)^2*(a^4 - 20*a^2*b^2 + 40*b^4)*(c + d*x)*Cos[c + d*x] + 48*(a^3 - a*b^2)^2*(a^4 - 20*a^2*b^2 + 40*b^4)*(c + d*x)*Cos[2*(c + d*x)] + 114*a^9*b*Sin[c + d*x] + 788*a^7*b^3*Sin[c + d*x] - 5696*a^5*b^5*Sin[c + d*x] + 8640*a^3*b^7*Sin[c + d*x] - 3840*a*b^9*Sin[c + d*x] - 36*a^10*Sin[2*(c + d*x)] + 1221*a^8*b^2*Sin[2*(c + d*x)] - 5182*a^6*b^4*Sin[2*(c + d*x)] + 6880*a^4*b^6*Sin[2*(c + d*x)] - 2880*a^2*b^8*Sin[2*(c + d*x)] + 120*a^9*b*Sin[3*(c + d*x)] - 560*a^7*b^3*Sin[3*(c + d*x)] + 760*a^5*b^5*Sin[3*(c + d*x)] - 320*a^3*b^7*Sin[3*(c + d*x)] - 8*a^10*Sin[4*(c + d*x)] + 56*a^8*b^2*Sin[4*(c + d*x)] - 88*a^6*b^4*Sin[4*(c + d*x)] + 40*a^4*b^6*Sin[4*(c + d*x)] - 8*a^9*b*Sin[5*(c + d*x)] + 16*a^7*b^3*Sin[5*(c + d*x)] - 8*a^5*b^5*Sin[5*(c + d*x)] + 2*a^10*Sin[6*(c + d*x)] - 4*a^8*b^2*Sin[6*(c + d*x)] + 2*a^6*b^4*Sin[6*(c + d*x)])/((a^2 - b^2)^2*(b + a*Cos[c + d*x])^2))/a^7)/(256*d)","B",1
230,1,282,267,4.196014,"\int \frac{\sin ^2(c+d x)}{(a+b \sec (c+d x))^3} \, dx","Integrate[Sin[c + d*x]^2/(a + b*Sec[c + d*x])^3,x]","\frac{\frac{4 b \left(6 a^4-19 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)}{\sqrt{a^2-b^2}}+\frac{2 a^2 \left(a^2-b^2\right) \cos ^2(c+d x) \left(\left(a^2-12 b^2\right) (c+d x)+4 a b \sin (c+d x)\right)-2 a^4 \left(a^2-b^2\right) \sin (c+d x) \cos ^3(c+d x)+4 a b \left(a^4-13 a^2 b^2+12 b^4\right) (c+d x) \cos (c+d x)+b^2 \left(\left(22 a^3 b-24 a b^3\right) \sin (c+d x)+\left(17 a^4-18 a^2 b^2\right) \sin (2 (c+d x))+2 \left(a^4-13 a^2 b^2+12 b^4\right) (c+d x)\right)}{(a \cos (c+d x)+b)^2}}{4 a^5 d (a-b) (a+b)}","\frac{\left(3 a^2-4 b^2\right) \sin (c+d x) \cos ^2(c+d x)}{2 a^2 d \left(a^2-b^2\right) (a \cos (c+d x)+b)}+\frac{x \left(a^2-12 b^2\right)}{2 a^5}+\frac{b \left(11 a^2-12 b^2\right) \sin (c+d x)}{2 a^4 d \left(a^2-b^2\right)}-\frac{\left(5 a^2-6 b^2\right) \sin (c+d x) \cos (c+d x)}{2 a^3 d \left(a^2-b^2\right)}-\frac{b \left(6 a^4-19 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)^{3/2} (a+b)^{3/2}}+\frac{\sin (c+d x) \cos ^3(c+d x)}{2 a d (a \cos (c+d x)+b)^2}",1,"((4*b*(6*a^4 - 19*a^2*b^2 + 12*b^4)*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/Sqrt[a^2 - b^2] + (4*a*b*(a^4 - 13*a^2*b^2 + 12*b^4)*(c + d*x)*Cos[c + d*x] - 2*a^4*(a^2 - b^2)*Cos[c + d*x]^3*Sin[c + d*x] + 2*a^2*(a^2 - b^2)*Cos[c + d*x]^2*((a^2 - 12*b^2)*(c + d*x) + 4*a*b*Sin[c + d*x]) + b^2*(2*(a^4 - 13*a^2*b^2 + 12*b^4)*(c + d*x) + (22*a^3*b - 24*a*b^3)*Sin[c + d*x] + (17*a^4 - 18*a^2*b^2)*Sin[2*(c + d*x)]))/(b + a*Cos[c + d*x])^2)/(4*a^5*(a - b)*(a + b)*d)","A",1
231,1,231,376,1.1074665,"\int \frac{\csc ^2(c+d x)}{(a+b \sec (c+d x))^3} \, dx","Integrate[Csc[c + d*x]^2/(a + b*Sec[c + d*x])^3,x]","\frac{\sec ^3(c+d x) (a \cos (c+d x)+b) \left(\frac{b^2 \left(6 a^2+b^2\right) \sin (c+d x) (a \cos (c+d x)+b)}{(a-b)^3 (a+b)^3}+\frac{6 a b \left(2 a^2+3 b^2\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)^{7/2}}-\frac{b^3 \sin (c+d x)}{(a-b)^2 (a+b)^2}+\frac{\tan \left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b)^2}{(a-b)^3}-\frac{\cot \left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b)^2}{(a+b)^3}\right)}{2 d (a+b \sec (c+d x))^3}","\frac{b^2 \left(3 a^2-b^2\right) \sin (c+d x)}{d \left(a^2-b^2\right)^3 (a \cos (c+d x)+b)}-\frac{2 a b \left(3 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{3 b^4 \sin (c+d x)}{2 d \left(a^2-b^2\right)^3 (a \cos (c+d x)+b)}-\frac{b^3 \sin (c+d x)}{2 d \left(a^2-b^2\right)^2 (a \cos (c+d x)+b)^2}-\frac{2 b^3 \left(3 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 d (a-b)^{7/2} (a+b)^{7/2}}-\frac{b^3 \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)}{a d (a-b)^{7/2} (a+b)^{7/2}}-\frac{\sin (c+d x)}{2 d (a+b)^3 (1-\cos (c+d x))}+\frac{\sin (c+d x)}{2 d (a-b)^3 (\cos (c+d x)+1)}",1,"((b + a*Cos[c + d*x])*Sec[c + d*x]^3*((6*a*b*(2*a^2 + 3*b^2)*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]]*(b + a*Cos[c + d*x])^2)/(a^2 - b^2)^(7/2) - ((b + a*Cos[c + d*x])^2*Cot[(c + d*x)/2])/(a + b)^3 - (b^3*Sin[c + d*x])/((a - b)^2*(a + b)^2) + (b^2*(6*a^2 + b^2)*(b + a*Cos[c + d*x])*Sin[c + d*x])/((a - b)^3*(a + b)^3) + ((b + a*Cos[c + d*x])^2*Tan[(c + d*x)/2])/(a - b)^3))/(2*d*(a + b*Sec[c + d*x])^3)","A",1
232,1,388,515,1.0816213,"\int \frac{\csc ^4(c+d x)}{(a+b \sec (c+d x))^3} \, dx","Integrate[Csc[c + d*x]^4/(a + b*Sec[c + d*x])^3,x]","\frac{\sec ^3(c+d x) (a \cos (c+d x)+b) \left(\frac{96 a b \left(6 a^4+23 a^2 b^2+6 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)}{\sqrt{a^2-b^2}}+\csc ^3(c+d x) \left(-4 a^7 \cos (3 (c+d x))+4 a^7 \cos (5 (c+d x))-20 a^6 b \cos (4 (c+d x))+36 a^6 b-154 a^5 b^2 \cos (3 (c+d x))+62 a^5 b^2 \cos (5 (c+d x))+110 a^4 b^3 \cos (4 (c+d x))+154 a^4 b^3-205 a^3 b^4 \cos (3 (c+d x))+39 a^3 b^4 \cos (5 (c+d x))+120 a^2 b^5 \cos (4 (c+d x))+424 a^2 b^5+8 \left(2 a^6 b-45 a^4 b^3-56 a^2 b^5-6 b^7\right) \cos (2 (c+d x))-2 a \left(16 a^6-94 a^4 b^2-35 a^2 b^4+8 b^6\right) \cos (c+d x)+48 a b^6 \cos (3 (c+d x))+16 b^7\right)\right)}{96 d \left(a^2-b^2\right)^4 (a+b \sec (c+d x))^3}","\frac{a^2 b^2 \left(3 a^2+b^2\right) \sin (c+d x)}{d \left(a^2-b^2\right)^4 (a \cos (c+d x)+b)}+\frac{3 a^2 b^4 \sin (c+d x)}{2 d \left(a^2-b^2\right)^4 (a \cos (c+d x)+b)}-\frac{a^2 b^3 \sin (c+d x)}{2 d \left(a^2-b^2\right)^3 (a \cos (c+d x)+b)^2}-\frac{2 a b^3 \left(3 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)^{9/2} (a+b)^{9/2}}-\frac{a b^3 \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)^{9/2} (a+b)^{9/2}}-\frac{2 a b \left(3 a^4+8 a^2 b^2+b^4\right) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{d (a-b)^{9/2} (a+b)^{9/2}}-\frac{\sin (c+d x)}{12 d (a+b)^3 (1-\cos (c+d x))}-\frac{(a-2 b) \sin (c+d x)}{4 d (a+b)^4 (1-\cos (c+d x))}+\frac{(a+2 b) \sin (c+d x)}{4 d (a-b)^4 (\cos (c+d x)+1)}+\frac{\sin (c+d x)}{12 d (a-b)^3 (\cos (c+d x)+1)}-\frac{\sin (c+d x)}{12 d (a+b)^3 (1-\cos (c+d x))^2}+\frac{\sin (c+d x)}{12 d (a-b)^3 (\cos (c+d x)+1)^2}",1,"((b + a*Cos[c + d*x])*((96*a*b*(6*a^4 + 23*a^2*b^2 + 6*b^4)*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]]*(b + a*Cos[c + d*x])^2)/Sqrt[a^2 - b^2] + (36*a^6*b + 154*a^4*b^3 + 424*a^2*b^5 + 16*b^7 - 2*a*(16*a^6 - 94*a^4*b^2 - 35*a^2*b^4 + 8*b^6)*Cos[c + d*x] + 8*(2*a^6*b - 45*a^4*b^3 - 56*a^2*b^5 - 6*b^7)*Cos[2*(c + d*x)] - 4*a^7*Cos[3*(c + d*x)] - 154*a^5*b^2*Cos[3*(c + d*x)] - 205*a^3*b^4*Cos[3*(c + d*x)] + 48*a*b^6*Cos[3*(c + d*x)] - 20*a^6*b*Cos[4*(c + d*x)] + 110*a^4*b^3*Cos[4*(c + d*x)] + 120*a^2*b^5*Cos[4*(c + d*x)] + 4*a^7*Cos[5*(c + d*x)] + 62*a^5*b^2*Cos[5*(c + d*x)] + 39*a^3*b^4*Cos[5*(c + d*x)])*Csc[c + d*x]^3)*Sec[c + d*x]^3)/(96*(a^2 - b^2)^4*d*(a + b*Sec[c + d*x])^3)","A",1
233,1,2049,516,17.5743351,"\int \frac{(e \sin (c+d x))^{7/2}}{a+b \sec (c+d x)} \, dx","Integrate[(e*Sin[c + d*x])^(7/2)/(a + b*Sec[c + d*x]),x]","\text{Result too large to show}","\frac{2 e (e \sin (c+d x))^{5/2} (7 b-5 a \cos (c+d x))}{35 a^2 d}-\frac{b e^{7/2} \left(a^2-b^2\right)^{5/4} \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt{e} \sqrt[4]{a^2-b^2}}\right)}{a^{9/2} d}-\frac{b e^{7/2} \left(a^2-b^2\right)^{5/4} \tanh ^{-1}\left(\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt{e} \sqrt[4]{a^2-b^2}}\right)}{a^{9/2} d}+\frac{b^2 e^4 \left(a^2-b^2\right)^2 \sqrt{\sin (c+d x)} \Pi \left(\frac{2 a}{a-\sqrt{a^2-b^2}};\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{a^5 d \left(-a \sqrt{a^2-b^2}+a^2-b^2\right) \sqrt{e \sin (c+d x)}}+\frac{b^2 e^4 \left(a^2-b^2\right)^2 \sqrt{\sin (c+d x)} \Pi \left(\frac{2 a}{a+\sqrt{a^2-b^2}};\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{a^5 d \left(a \sqrt{a^2-b^2}+a^2-b^2\right) \sqrt{e \sin (c+d x)}}+\frac{2 e^3 \sqrt{e \sin (c+d x)} \left(21 b \left(a^2-b^2\right)-a \left(5 a^2-7 b^2\right) \cos (c+d x)\right)}{21 a^4 d}+\frac{2 e^4 \left(5 a^4-28 a^2 b^2+21 b^4\right) \sqrt{\sin (c+d x)} F\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{21 a^5 d \sqrt{e \sin (c+d x)}}",1,"((b + a*Cos[c + d*x])*(-1/42*((23*a^2 - 28*b^2)*Cos[c + d*x])/a^3 - (b*Cos[2*(c + d*x)])/(5*a^2) + Cos[3*(c + d*x)]/(14*a))*Csc[c + d*x]^3*Sec[c + d*x]*(e*Sin[c + d*x])^(7/2))/(d*(a + b*Sec[c + d*x])) - ((b + a*Cos[c + d*x])*Sec[c + d*x]*(e*Sin[c + d*x])^(7/2)*((2*(-100*a^3 + 98*a*b^2)*Cos[c + d*x]^2*(b + a*Sqrt[1 - Sin[c + d*x]^2])*((b*(-2*ArcTan[1 - (Sqrt[2]*Sqrt[a]*Sqrt[Sin[c + d*x]])/(-a^2 + b^2)^(1/4)] + 2*ArcTan[1 + (Sqrt[2]*Sqrt[a]*Sqrt[Sin[c + d*x]])/(-a^2 + b^2)^(1/4)] - Log[Sqrt[-a^2 + b^2] - Sqrt[2]*Sqrt[a]*(-a^2 + b^2)^(1/4)*Sqrt[Sin[c + d*x]] + a*Sin[c + d*x]] + Log[Sqrt[-a^2 + b^2] + Sqrt[2]*Sqrt[a]*(-a^2 + b^2)^(1/4)*Sqrt[Sin[c + d*x]] + a*Sin[c + d*x]]))/(4*Sqrt[2]*Sqrt[a]*(-a^2 + b^2)^(3/4)) - (5*a*(a^2 - b^2)*AppellF1[1/4, -1/2, 1, 5/4, Sin[c + d*x]^2, (a^2*Sin[c + d*x]^2)/(a^2 - b^2)]*Sqrt[Sin[c + d*x]]*Sqrt[1 - Sin[c + d*x]^2])/((5*(a^2 - b^2)*AppellF1[1/4, -1/2, 1, 5/4, Sin[c + d*x]^2, (a^2*Sin[c + d*x]^2)/(a^2 - b^2)] + 2*(2*a^2*AppellF1[5/4, -1/2, 2, 9/4, Sin[c + d*x]^2, (a^2*Sin[c + d*x]^2)/(a^2 - b^2)] + (-a^2 + b^2)*AppellF1[5/4, 1/2, 1, 9/4, Sin[c + d*x]^2, (a^2*Sin[c + d*x]^2)/(a^2 - b^2)])*Sin[c + d*x]^2)*(b^2 + a^2*(-1 + Sin[c + d*x]^2)))))/((b + a*Cos[c + d*x])*(1 - Sin[c + d*x]^2)) + (2*(89*a^2*b - 70*b^3)*Cos[c + d*x]*(b + a*Sqrt[1 - Sin[c + d*x]^2])*(((-1/8 + I/8)*Sqrt[a]*(2*ArcTan[1 - ((1 + I)*Sqrt[a]*Sqrt[Sin[c + d*x]])/(a^2 - b^2)^(1/4)] - 2*ArcTan[1 + ((1 + I)*Sqrt[a]*Sqrt[Sin[c + d*x]])/(a^2 - b^2)^(1/4)] + Log[Sqrt[a^2 - b^2] - (1 + I)*Sqrt[a]*(a^2 - b^2)^(1/4)*Sqrt[Sin[c + d*x]] + I*a*Sin[c + d*x]] - Log[Sqrt[a^2 - b^2] + (1 + I)*Sqrt[a]*(a^2 - b^2)^(1/4)*Sqrt[Sin[c + d*x]] + I*a*Sin[c + d*x]]))/(a^2 - b^2)^(3/4) + (5*b*(a^2 - b^2)*AppellF1[1/4, 1/2, 1, 5/4, Sin[c + d*x]^2, (a^2*Sin[c + d*x]^2)/(a^2 - b^2)]*Sqrt[Sin[c + d*x]])/(Sqrt[1 - Sin[c + d*x]^2]*(5*(a^2 - b^2)*AppellF1[1/4, 1/2, 1, 5/4, Sin[c + d*x]^2, (a^2*Sin[c + d*x]^2)/(a^2 - b^2)] + 2*(2*a^2*AppellF1[5/4, 1/2, 2, 9/4, Sin[c + d*x]^2, (a^2*Sin[c + d*x]^2)/(a^2 - b^2)] + (a^2 - b^2)*AppellF1[5/4, 3/2, 1, 9/4, Sin[c + d*x]^2, (a^2*Sin[c + d*x]^2)/(a^2 - b^2)])*Sin[c + d*x]^2)*(b^2 + a^2*(-1 + Sin[c + d*x]^2)))))/((b + a*Cos[c + d*x])*Sqrt[1 - Sin[c + d*x]^2]) + ((-231*a^2*b + 210*b^3)*Cos[c + d*x]*Cos[2*(c + d*x)]*(b + a*Sqrt[1 - Sin[c + d*x]^2])*(((1/2 - I/2)*(a^2 - 2*b^2)*ArcTan[1 - ((1 + I)*Sqrt[a]*Sqrt[Sin[c + d*x]])/(a^2 - b^2)^(1/4)])/(a^(3/2)*(a^2 - b^2)^(3/4)) - ((1/2 - I/2)*(a^2 - 2*b^2)*ArcTan[1 + ((1 + I)*Sqrt[a]*Sqrt[Sin[c + d*x]])/(a^2 - b^2)^(1/4)])/(a^(3/2)*(a^2 - b^2)^(3/4)) + ((1/4 - I/4)*(a^2 - 2*b^2)*Log[Sqrt[a^2 - b^2] - (1 + I)*Sqrt[a]*(a^2 - b^2)^(1/4)*Sqrt[Sin[c + d*x]] + I*a*Sin[c + d*x]])/(a^(3/2)*(a^2 - b^2)^(3/4)) - ((1/4 - I/4)*(a^2 - 2*b^2)*Log[Sqrt[a^2 - b^2] + (1 + I)*Sqrt[a]*(a^2 - b^2)^(1/4)*Sqrt[Sin[c + d*x]] + I*a*Sin[c + d*x]])/(a^(3/2)*(a^2 - b^2)^(3/4)) + (4*Sqrt[Sin[c + d*x]])/a + (4*b*AppellF1[5/4, 1/2, 1, 9/4, Sin[c + d*x]^2, (a^2*Sin[c + d*x]^2)/(a^2 - b^2)]*Sin[c + d*x]^(5/2))/(5*(a^2 - b^2)) + (10*b*(a^2 - b^2)*AppellF1[1/4, 1/2, 1, 5/4, Sin[c + d*x]^2, (a^2*Sin[c + d*x]^2)/(a^2 - b^2)]*Sqrt[Sin[c + d*x]])/(Sqrt[1 - Sin[c + d*x]^2]*(5*(a^2 - b^2)*AppellF1[1/4, 1/2, 1, 5/4, Sin[c + d*x]^2, (a^2*Sin[c + d*x]^2)/(a^2 - b^2)] + 2*(2*a^2*AppellF1[5/4, 1/2, 2, 9/4, Sin[c + d*x]^2, (a^2*Sin[c + d*x]^2)/(a^2 - b^2)] + (a^2 - b^2)*AppellF1[5/4, 3/2, 1, 9/4, Sin[c + d*x]^2, (a^2*Sin[c + d*x]^2)/(a^2 - b^2)])*Sin[c + d*x]^2)*(b^2 + a^2*(-1 + Sin[c + d*x]^2)))))/((b + a*Cos[c + d*x])*(1 - 2*Sin[c + d*x]^2)*Sqrt[1 - Sin[c + d*x]^2])))/(420*a^3*d*(a + b*Sec[c + d*x])*Sin[c + d*x]^(7/2))","C",0
234,1,853,430,15.0399007,"\int \frac{(e \sin (c+d x))^{5/2}}{a+b \sec (c+d x)} \, dx","Integrate[(e*Sin[c + d*x])^(5/2)/(a + b*Sec[c + d*x]),x]","\frac{(b+a \cos (c+d x)) \csc ^2(c+d x) \sec (c+d x) (e \sin (c+d x))^{5/2} \left(\frac{2 b \sin (c+d x)}{3 a^2}-\frac{\sin (2 (c+d x))}{5 a}\right)}{d (a+b \sec (c+d x))}-\frac{(b+a \cos (c+d x)) \sec (c+d x) (e \sin (c+d x))^{5/2} \left(\frac{\left(5 b^2-3 a^2\right) \left(8 F_1\left(\frac{3}{4};-\frac{1}{2},1;\frac{7}{4};\sin ^2(c+d x),\frac{a^2 \sin ^2(c+d x)}{a^2-b^2}\right) \sin ^{\frac{3}{2}}(c+d x) a^{5/2}+3 \sqrt{2} b \left(b^2-a^2\right)^{3/4} \left(2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{a} \sqrt{\sin (c+d x)}}{\sqrt[4]{b^2-a^2}}\right)-2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{a} \sqrt{\sin (c+d x)}}{\sqrt[4]{b^2-a^2}}+1\right)-\log \left(a \sin (c+d x)-\sqrt{2} \sqrt{a} \sqrt[4]{b^2-a^2} \sqrt{\sin (c+d x)}+\sqrt{b^2-a^2}\right)+\log \left(a \sin (c+d x)+\sqrt{2} \sqrt{a} \sqrt[4]{b^2-a^2} \sqrt{\sin (c+d x)}+\sqrt{b^2-a^2}\right)\right)\right) \left(\sqrt{1-\sin ^2(c+d x)} a+b\right) \cos ^2(c+d x)}{12 a^{3/2} \left(a^2-b^2\right) (b+a \cos (c+d x)) \left(1-\sin ^2(c+d x)\right)}+\frac{4 a b \left(\frac{b F_1\left(\frac{3}{4};\frac{1}{2},1;\frac{7}{4};\sin ^2(c+d x),\frac{a^2 \sin ^2(c+d x)}{a^2-b^2}\right) \sin ^{\frac{3}{2}}(c+d x)}{3 \left(b^2-a^2\right)}+\frac{\left(\frac{1}{8}+\frac{i}{8}\right) \left(2 \tan ^{-1}\left(1-\frac{(1+i) \sqrt{a} \sqrt{\sin (c+d x)}}{\sqrt[4]{a^2-b^2}}\right)-2 \tan ^{-1}\left(\frac{(1+i) \sqrt{a} \sqrt{\sin (c+d x)}}{\sqrt[4]{a^2-b^2}}+1\right)-\log \left(i a \sin (c+d x)-(1+i) \sqrt{a} \sqrt[4]{a^2-b^2} \sqrt{\sin (c+d x)}+\sqrt{a^2-b^2}\right)+\log \left(i a \sin (c+d x)+(1+i) \sqrt{a} \sqrt[4]{a^2-b^2} \sqrt{\sin (c+d x)}+\sqrt{a^2-b^2}\right)\right)}{\sqrt{a} \sqrt[4]{a^2-b^2}}\right) \left(\sqrt{1-\sin ^2(c+d x)} a+b\right) \cos (c+d x)}{(b+a \cos (c+d x)) \sqrt{1-\sin ^2(c+d x)}}\right)}{5 a^2 d (a+b \sec (c+d x)) \sin ^{\frac{5}{2}}(c+d x)}","\frac{2 e (e \sin (c+d x))^{3/2} (5 b-3 a \cos (c+d x))}{15 a^2 d}+\frac{b e^{5/2} \left(a^2-b^2\right)^{3/4} \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt{e} \sqrt[4]{a^2-b^2}}\right)}{a^{7/2} d}-\frac{b e^{5/2} \left(a^2-b^2\right)^{3/4} \tanh ^{-1}\left(\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt{e} \sqrt[4]{a^2-b^2}}\right)}{a^{7/2} d}-\frac{b^2 e^3 \left(a^2-b^2\right) \sqrt{\sin (c+d x)} \Pi \left(\frac{2 a}{a-\sqrt{a^2-b^2}};\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{a^4 d \left(a-\sqrt{a^2-b^2}\right) \sqrt{e \sin (c+d x)}}-\frac{b^2 e^3 \left(a^2-b^2\right) \sqrt{\sin (c+d x)} \Pi \left(\frac{2 a}{a+\sqrt{a^2-b^2}};\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{a^4 d \left(\sqrt{a^2-b^2}+a\right) \sqrt{e \sin (c+d x)}}+\frac{2 e^2 \left(3 a^2-5 b^2\right) E\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{e \sin (c+d x)}}{5 a^3 d \sqrt{\sin (c+d x)}}",1,"-1/5*((b + a*Cos[c + d*x])*Sec[c + d*x]*(e*Sin[c + d*x])^(5/2)*(((-3*a^2 + 5*b^2)*Cos[c + d*x]^2*(3*Sqrt[2]*b*(-a^2 + b^2)^(3/4)*(2*ArcTan[1 - (Sqrt[2]*Sqrt[a]*Sqrt[Sin[c + d*x]])/(-a^2 + b^2)^(1/4)] - 2*ArcTan[1 + (Sqrt[2]*Sqrt[a]*Sqrt[Sin[c + d*x]])/(-a^2 + b^2)^(1/4)] - Log[Sqrt[-a^2 + b^2] - Sqrt[2]*Sqrt[a]*(-a^2 + b^2)^(1/4)*Sqrt[Sin[c + d*x]] + a*Sin[c + d*x]] + Log[Sqrt[-a^2 + b^2] + Sqrt[2]*Sqrt[a]*(-a^2 + b^2)^(1/4)*Sqrt[Sin[c + d*x]] + a*Sin[c + d*x]]) + 8*a^(5/2)*AppellF1[3/4, -1/2, 1, 7/4, Sin[c + d*x]^2, (a^2*Sin[c + d*x]^2)/(a^2 - b^2)]*Sin[c + d*x]^(3/2))*(b + a*Sqrt[1 - Sin[c + d*x]^2]))/(12*a^(3/2)*(a^2 - b^2)*(b + a*Cos[c + d*x])*(1 - Sin[c + d*x]^2)) + (4*a*b*Cos[c + d*x]*(((1/8 + I/8)*(2*ArcTan[1 - ((1 + I)*Sqrt[a]*Sqrt[Sin[c + d*x]])/(a^2 - b^2)^(1/4)] - 2*ArcTan[1 + ((1 + I)*Sqrt[a]*Sqrt[Sin[c + d*x]])/(a^2 - b^2)^(1/4)] - Log[Sqrt[a^2 - b^2] - (1 + I)*Sqrt[a]*(a^2 - b^2)^(1/4)*Sqrt[Sin[c + d*x]] + I*a*Sin[c + d*x]] + Log[Sqrt[a^2 - b^2] + (1 + I)*Sqrt[a]*(a^2 - b^2)^(1/4)*Sqrt[Sin[c + d*x]] + I*a*Sin[c + d*x]]))/(Sqrt[a]*(a^2 - b^2)^(1/4)) + (b*AppellF1[3/4, 1/2, 1, 7/4, Sin[c + d*x]^2, (a^2*Sin[c + d*x]^2)/(a^2 - b^2)]*Sin[c + d*x]^(3/2))/(3*(-a^2 + b^2)))*(b + a*Sqrt[1 - Sin[c + d*x]^2]))/((b + a*Cos[c + d*x])*Sqrt[1 - Sin[c + d*x]^2])))/(a^2*d*(a + b*Sec[c + d*x])*Sin[c + d*x]^(5/2)) + ((b + a*Cos[c + d*x])*Csc[c + d*x]^2*Sec[c + d*x]*(e*Sin[c + d*x])^(5/2)*((2*b*Sin[c + d*x])/(3*a^2) - Sin[2*(c + d*x)]/(5*a)))/(d*(a + b*Sec[c + d*x]))","C",0
235,1,1959,444,17.345177,"\int \frac{(e \sin (c+d x))^{3/2}}{a+b \sec (c+d x)} \, dx","Integrate[(e*Sin[c + d*x])^(3/2)/(a + b*Sec[c + d*x]),x]","\frac{(b+a \cos (c+d x)) \sec (c+d x) (e \sin (c+d x))^{3/2} \left(\frac{4 a \left(\sqrt{1-\sin ^2(c+d x)} a+b\right) \left(\frac{b \left(-2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{a} \sqrt{\sin (c+d x)}}{\sqrt[4]{b^2-a^2}}\right)+2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{a} \sqrt{\sin (c+d x)}}{\sqrt[4]{b^2-a^2}}+1\right)-\log \left(a \sin (c+d x)-\sqrt{2} \sqrt{a} \sqrt[4]{b^2-a^2} \sqrt{\sin (c+d x)}+\sqrt{b^2-a^2}\right)+\log \left(a \sin (c+d x)+\sqrt{2} \sqrt{a} \sqrt[4]{b^2-a^2} \sqrt{\sin (c+d x)}+\sqrt{b^2-a^2}\right)\right)}{4 \sqrt{2} \sqrt{a} \left(b^2-a^2\right)^{3/4}}-\frac{5 a \left(a^2-b^2\right) F_1\left(\frac{1}{4};-\frac{1}{2},1;\frac{5}{4};\sin ^2(c+d x),\frac{a^2 \sin ^2(c+d x)}{a^2-b^2}\right) \sqrt{\sin (c+d x)} \sqrt{1-\sin ^2(c+d x)}}{\left(2 \left(2 F_1\left(\frac{5}{4};-\frac{1}{2},2;\frac{9}{4};\sin ^2(c+d x),\frac{a^2 \sin ^2(c+d x)}{a^2-b^2}\right) a^2+\left(b^2-a^2\right) F_1\left(\frac{5}{4};\frac{1}{2},1;\frac{9}{4};\sin ^2(c+d x),\frac{a^2 \sin ^2(c+d x)}{a^2-b^2}\right)\right) \sin ^2(c+d x)+5 \left(a^2-b^2\right) F_1\left(\frac{1}{4};-\frac{1}{2},1;\frac{5}{4};\sin ^2(c+d x),\frac{a^2 \sin ^2(c+d x)}{a^2-b^2}\right)\right) \left(\left(\sin ^2(c+d x)-1\right) a^2+b^2\right)}\right) \cos ^2(c+d x)}{(b+a \cos (c+d x)) \left(1-\sin ^2(c+d x)\right)}-\frac{2 b \left(\sqrt{1-\sin ^2(c+d x)} a+b\right) \left(\frac{5 b \left(a^2-b^2\right) F_1\left(\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\sin ^2(c+d x),\frac{a^2 \sin ^2(c+d x)}{a^2-b^2}\right) \sqrt{\sin (c+d x)}}{\sqrt{1-\sin ^2(c+d x)} \left(2 \left(2 F_1\left(\frac{5}{4};\frac{1}{2},2;\frac{9}{4};\sin ^2(c+d x),\frac{a^2 \sin ^2(c+d x)}{a^2-b^2}\right) a^2+\left(a^2-b^2\right) F_1\left(\frac{5}{4};\frac{3}{2},1;\frac{9}{4};\sin ^2(c+d x),\frac{a^2 \sin ^2(c+d x)}{a^2-b^2}\right)\right) \sin ^2(c+d x)+5 \left(a^2-b^2\right) F_1\left(\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\sin ^2(c+d x),\frac{a^2 \sin ^2(c+d x)}{a^2-b^2}\right)\right) \left(\left(\sin ^2(c+d x)-1\right) a^2+b^2\right)}-\frac{\left(\frac{1}{8}-\frac{i}{8}\right) \sqrt{a} \left(2 \tan ^{-1}\left(1-\frac{(1+i) \sqrt{a} \sqrt{\sin (c+d x)}}{\sqrt[4]{a^2-b^2}}\right)-2 \tan ^{-1}\left(\frac{(1+i) \sqrt{a} \sqrt{\sin (c+d x)}}{\sqrt[4]{a^2-b^2}}+1\right)+\log \left(i a \sin (c+d x)-(1+i) \sqrt{a} \sqrt[4]{a^2-b^2} \sqrt{\sin (c+d x)}+\sqrt{a^2-b^2}\right)-\log \left(i a \sin (c+d x)+(1+i) \sqrt{a} \sqrt[4]{a^2-b^2} \sqrt{\sin (c+d x)}+\sqrt{a^2-b^2}\right)\right)}{\left(a^2-b^2\right)^{3/4}}\right) \cos (c+d x)}{(b+a \cos (c+d x)) \sqrt{1-\sin ^2(c+d x)}}+\frac{3 b \cos (2 (c+d x)) \left(\sqrt{1-\sin ^2(c+d x)} a+b\right) \left(\frac{4 b F_1\left(\frac{5}{4};\frac{1}{2},1;\frac{9}{4};\sin ^2(c+d x),\frac{a^2 \sin ^2(c+d x)}{a^2-b^2}\right) \sin ^{\frac{5}{2}}(c+d x)}{5 \left(a^2-b^2\right)}+\frac{4 \sqrt{\sin (c+d x)}}{a}+\frac{10 b \left(a^2-b^2\right) F_1\left(\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\sin ^2(c+d x),\frac{a^2 \sin ^2(c+d x)}{a^2-b^2}\right) \sqrt{\sin (c+d x)}}{\sqrt{1-\sin ^2(c+d x)} \left(2 \left(2 F_1\left(\frac{5}{4};\frac{1}{2},2;\frac{9}{4};\sin ^2(c+d x),\frac{a^2 \sin ^2(c+d x)}{a^2-b^2}\right) a^2+\left(a^2-b^2\right) F_1\left(\frac{5}{4};\frac{3}{2},1;\frac{9}{4};\sin ^2(c+d x),\frac{a^2 \sin ^2(c+d x)}{a^2-b^2}\right)\right) \sin ^2(c+d x)+5 \left(a^2-b^2\right) F_1\left(\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\sin ^2(c+d x),\frac{a^2 \sin ^2(c+d x)}{a^2-b^2}\right)\right) \left(\left(\sin ^2(c+d x)-1\right) a^2+b^2\right)}+\frac{\left(\frac{1}{2}-\frac{i}{2}\right) \left(a^2-2 b^2\right) \tan ^{-1}\left(1-\frac{(1+i) \sqrt{a} \sqrt{\sin (c+d x)}}{\sqrt[4]{a^2-b^2}}\right)}{a^{3/2} \left(a^2-b^2\right)^{3/4}}-\frac{\left(\frac{1}{2}-\frac{i}{2}\right) \left(a^2-2 b^2\right) \tan ^{-1}\left(\frac{(1+i) \sqrt{a} \sqrt{\sin (c+d x)}}{\sqrt[4]{a^2-b^2}}+1\right)}{a^{3/2} \left(a^2-b^2\right)^{3/4}}+\frac{\left(\frac{1}{4}-\frac{i}{4}\right) \left(a^2-2 b^2\right) \log \left(i a \sin (c+d x)-(1+i) \sqrt{a} \sqrt[4]{a^2-b^2} \sqrt{\sin (c+d x)}+\sqrt{a^2-b^2}\right)}{a^{3/2} \left(a^2-b^2\right)^{3/4}}-\frac{\left(\frac{1}{4}-\frac{i}{4}\right) \left(a^2-2 b^2\right) \log \left(i a \sin (c+d x)+(1+i) \sqrt{a} \sqrt[4]{a^2-b^2} \sqrt{\sin (c+d x)}+\sqrt{a^2-b^2}\right)}{a^{3/2} \left(a^2-b^2\right)^{3/4}}\right) \cos (c+d x)}{(b+a \cos (c+d x)) \left(1-2 \sin ^2(c+d x)\right) \sqrt{1-\sin ^2(c+d x)}}\right)}{6 a d (a+b \sec (c+d x)) \sin ^{\frac{3}{2}}(c+d x)}-\frac{2 (b+a \cos (c+d x)) \csc (c+d x) (e \sin (c+d x))^{3/2}}{3 a d (a+b \sec (c+d x))}","\frac{2 e \sqrt{e \sin (c+d x)} (3 b-a \cos (c+d x))}{3 a^2 d}-\frac{b e^{3/2} \sqrt[4]{a^2-b^2} \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt{e} \sqrt[4]{a^2-b^2}}\right)}{a^{5/2} d}-\frac{b e^{3/2} \sqrt[4]{a^2-b^2} \tanh ^{-1}\left(\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt{e} \sqrt[4]{a^2-b^2}}\right)}{a^{5/2} d}+\frac{2 e^2 \left(a^2-3 b^2\right) \sqrt{\sin (c+d x)} F\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{3 a^3 d \sqrt{e \sin (c+d x)}}+\frac{b^2 e^2 \left(a^2-b^2\right) \sqrt{\sin (c+d x)} \Pi \left(\frac{2 a}{a-\sqrt{a^2-b^2}};\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{a^3 d \left(-a \sqrt{a^2-b^2}+a^2-b^2\right) \sqrt{e \sin (c+d x)}}+\frac{b^2 e^2 \left(a^2-b^2\right) \sqrt{\sin (c+d x)} \Pi \left(\frac{2 a}{a+\sqrt{a^2-b^2}};\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{a^3 d \left(a \sqrt{a^2-b^2}+a^2-b^2\right) \sqrt{e \sin (c+d x)}}",1,"(-2*(b + a*Cos[c + d*x])*Csc[c + d*x]*(e*Sin[c + d*x])^(3/2))/(3*a*d*(a + b*Sec[c + d*x])) + ((b + a*Cos[c + d*x])*Sec[c + d*x]*(e*Sin[c + d*x])^(3/2)*((4*a*Cos[c + d*x]^2*(b + a*Sqrt[1 - Sin[c + d*x]^2])*((b*(-2*ArcTan[1 - (Sqrt[2]*Sqrt[a]*Sqrt[Sin[c + d*x]])/(-a^2 + b^2)^(1/4)] + 2*ArcTan[1 + (Sqrt[2]*Sqrt[a]*Sqrt[Sin[c + d*x]])/(-a^2 + b^2)^(1/4)] - Log[Sqrt[-a^2 + b^2] - Sqrt[2]*Sqrt[a]*(-a^2 + b^2)^(1/4)*Sqrt[Sin[c + d*x]] + a*Sin[c + d*x]] + Log[Sqrt[-a^2 + b^2] + Sqrt[2]*Sqrt[a]*(-a^2 + b^2)^(1/4)*Sqrt[Sin[c + d*x]] + a*Sin[c + d*x]]))/(4*Sqrt[2]*Sqrt[a]*(-a^2 + b^2)^(3/4)) - (5*a*(a^2 - b^2)*AppellF1[1/4, -1/2, 1, 5/4, Sin[c + d*x]^2, (a^2*Sin[c + d*x]^2)/(a^2 - b^2)]*Sqrt[Sin[c + d*x]]*Sqrt[1 - Sin[c + d*x]^2])/((5*(a^2 - b^2)*AppellF1[1/4, -1/2, 1, 5/4, Sin[c + d*x]^2, (a^2*Sin[c + d*x]^2)/(a^2 - b^2)] + 2*(2*a^2*AppellF1[5/4, -1/2, 2, 9/4, Sin[c + d*x]^2, (a^2*Sin[c + d*x]^2)/(a^2 - b^2)] + (-a^2 + b^2)*AppellF1[5/4, 1/2, 1, 9/4, Sin[c + d*x]^2, (a^2*Sin[c + d*x]^2)/(a^2 - b^2)])*Sin[c + d*x]^2)*(b^2 + a^2*(-1 + Sin[c + d*x]^2)))))/((b + a*Cos[c + d*x])*(1 - Sin[c + d*x]^2)) - (2*b*Cos[c + d*x]*(b + a*Sqrt[1 - Sin[c + d*x]^2])*(((-1/8 + I/8)*Sqrt[a]*(2*ArcTan[1 - ((1 + I)*Sqrt[a]*Sqrt[Sin[c + d*x]])/(a^2 - b^2)^(1/4)] - 2*ArcTan[1 + ((1 + I)*Sqrt[a]*Sqrt[Sin[c + d*x]])/(a^2 - b^2)^(1/4)] + Log[Sqrt[a^2 - b^2] - (1 + I)*Sqrt[a]*(a^2 - b^2)^(1/4)*Sqrt[Sin[c + d*x]] + I*a*Sin[c + d*x]] - Log[Sqrt[a^2 - b^2] + (1 + I)*Sqrt[a]*(a^2 - b^2)^(1/4)*Sqrt[Sin[c + d*x]] + I*a*Sin[c + d*x]]))/(a^2 - b^2)^(3/4) + (5*b*(a^2 - b^2)*AppellF1[1/4, 1/2, 1, 5/4, Sin[c + d*x]^2, (a^2*Sin[c + d*x]^2)/(a^2 - b^2)]*Sqrt[Sin[c + d*x]])/(Sqrt[1 - Sin[c + d*x]^2]*(5*(a^2 - b^2)*AppellF1[1/4, 1/2, 1, 5/4, Sin[c + d*x]^2, (a^2*Sin[c + d*x]^2)/(a^2 - b^2)] + 2*(2*a^2*AppellF1[5/4, 1/2, 2, 9/4, Sin[c + d*x]^2, (a^2*Sin[c + d*x]^2)/(a^2 - b^2)] + (a^2 - b^2)*AppellF1[5/4, 3/2, 1, 9/4, Sin[c + d*x]^2, (a^2*Sin[c + d*x]^2)/(a^2 - b^2)])*Sin[c + d*x]^2)*(b^2 + a^2*(-1 + Sin[c + d*x]^2)))))/((b + a*Cos[c + d*x])*Sqrt[1 - Sin[c + d*x]^2]) + (3*b*Cos[c + d*x]*Cos[2*(c + d*x)]*(b + a*Sqrt[1 - Sin[c + d*x]^2])*(((1/2 - I/2)*(a^2 - 2*b^2)*ArcTan[1 - ((1 + I)*Sqrt[a]*Sqrt[Sin[c + d*x]])/(a^2 - b^2)^(1/4)])/(a^(3/2)*(a^2 - b^2)^(3/4)) - ((1/2 - I/2)*(a^2 - 2*b^2)*ArcTan[1 + ((1 + I)*Sqrt[a]*Sqrt[Sin[c + d*x]])/(a^2 - b^2)^(1/4)])/(a^(3/2)*(a^2 - b^2)^(3/4)) + ((1/4 - I/4)*(a^2 - 2*b^2)*Log[Sqrt[a^2 - b^2] - (1 + I)*Sqrt[a]*(a^2 - b^2)^(1/4)*Sqrt[Sin[c + d*x]] + I*a*Sin[c + d*x]])/(a^(3/2)*(a^2 - b^2)^(3/4)) - ((1/4 - I/4)*(a^2 - 2*b^2)*Log[Sqrt[a^2 - b^2] + (1 + I)*Sqrt[a]*(a^2 - b^2)^(1/4)*Sqrt[Sin[c + d*x]] + I*a*Sin[c + d*x]])/(a^(3/2)*(a^2 - b^2)^(3/4)) + (4*Sqrt[Sin[c + d*x]])/a + (4*b*AppellF1[5/4, 1/2, 1, 9/4, Sin[c + d*x]^2, (a^2*Sin[c + d*x]^2)/(a^2 - b^2)]*Sin[c + d*x]^(5/2))/(5*(a^2 - b^2)) + (10*b*(a^2 - b^2)*AppellF1[1/4, 1/2, 1, 5/4, Sin[c + d*x]^2, (a^2*Sin[c + d*x]^2)/(a^2 - b^2)]*Sqrt[Sin[c + d*x]])/(Sqrt[1 - Sin[c + d*x]^2]*(5*(a^2 - b^2)*AppellF1[1/4, 1/2, 1, 5/4, Sin[c + d*x]^2, (a^2*Sin[c + d*x]^2)/(a^2 - b^2)] + 2*(2*a^2*AppellF1[5/4, 1/2, 2, 9/4, Sin[c + d*x]^2, (a^2*Sin[c + d*x]^2)/(a^2 - b^2)] + (a^2 - b^2)*AppellF1[5/4, 3/2, 1, 9/4, Sin[c + d*x]^2, (a^2*Sin[c + d*x]^2)/(a^2 - b^2)])*Sin[c + d*x]^2)*(b^2 + a^2*(-1 + Sin[c + d*x]^2)))))/((b + a*Cos[c + d*x])*(1 - 2*Sin[c + d*x]^2)*Sqrt[1 - Sin[c + d*x]^2])))/(6*a*d*(a + b*Sec[c + d*x])*Sin[c + d*x]^(3/2))","C",0
236,1,351,356,20.7488873,"\int \frac{\sqrt{e \sin (c+d x)}}{a+b \sec (c+d x)} \, dx","Integrate[Sqrt[e*Sin[c + d*x]]/(a + b*Sec[c + d*x]),x]","\frac{\sqrt{e \sin (c+d x)} \left(a \sqrt{\cos ^2(c+d x)}+b\right) \left(3 \sqrt{2} b \left(b^2-a^2\right)^{3/4} \left(-\log \left(-\sqrt{2} \sqrt{a} \sqrt[4]{b^2-a^2} \sqrt{\sin (c+d x)}+\sqrt{b^2-a^2}+a \sin (c+d x)\right)+\log \left(\sqrt{2} \sqrt{a} \sqrt[4]{b^2-a^2} \sqrt{\sin (c+d x)}+\sqrt{b^2-a^2}+a \sin (c+d x)\right)+2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{a} \sqrt{\sin (c+d x)}}{\sqrt[4]{b^2-a^2}}\right)-2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{a} \sqrt{\sin (c+d x)}}{\sqrt[4]{b^2-a^2}}+1\right)\right)+8 a^{5/2} \sin ^{\frac{3}{2}}(c+d x) F_1\left(\frac{3}{4};-\frac{1}{2},1;\frac{7}{4};\sin ^2(c+d x),\frac{a^2 \sin ^2(c+d x)}{a^2-b^2}\right)\right)}{12 a^{3/2} d \left(a^2-b^2\right) \sqrt{\sin (c+d x)} (a \cos (c+d x)+b)}","-\frac{b^2 e \sqrt{\sin (c+d x)} \Pi \left(\frac{2 a}{a-\sqrt{a^2-b^2}};\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{a^2 d \left(a-\sqrt{a^2-b^2}\right) \sqrt{e \sin (c+d x)}}-\frac{b^2 e \sqrt{\sin (c+d x)} \Pi \left(\frac{2 a}{a+\sqrt{a^2-b^2}};\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{a^2 d \left(\sqrt{a^2-b^2}+a\right) \sqrt{e \sin (c+d x)}}+\frac{b \sqrt{e} \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt{e} \sqrt[4]{a^2-b^2}}\right)}{a^{3/2} d \sqrt[4]{a^2-b^2}}-\frac{b \sqrt{e} \tanh ^{-1}\left(\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt{e} \sqrt[4]{a^2-b^2}}\right)}{a^{3/2} d \sqrt[4]{a^2-b^2}}+\frac{2 E\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{e \sin (c+d x)}}{a d \sqrt{\sin (c+d x)}}",1,"((b + a*Sqrt[Cos[c + d*x]^2])*Sqrt[e*Sin[c + d*x]]*(3*Sqrt[2]*b*(-a^2 + b^2)^(3/4)*(2*ArcTan[1 - (Sqrt[2]*Sqrt[a]*Sqrt[Sin[c + d*x]])/(-a^2 + b^2)^(1/4)] - 2*ArcTan[1 + (Sqrt[2]*Sqrt[a]*Sqrt[Sin[c + d*x]])/(-a^2 + b^2)^(1/4)] - Log[Sqrt[-a^2 + b^2] - Sqrt[2]*Sqrt[a]*(-a^2 + b^2)^(1/4)*Sqrt[Sin[c + d*x]] + a*Sin[c + d*x]] + Log[Sqrt[-a^2 + b^2] + Sqrt[2]*Sqrt[a]*(-a^2 + b^2)^(1/4)*Sqrt[Sin[c + d*x]] + a*Sin[c + d*x]]) + 8*a^(5/2)*AppellF1[3/4, -1/2, 1, 7/4, Sin[c + d*x]^2, (a^2*Sin[c + d*x]^2)/(a^2 - b^2)]*Sin[c + d*x]^(3/2)))/(12*a^(3/2)*(a^2 - b^2)*d*(b + a*Cos[c + d*x])*Sqrt[Sin[c + d*x]])","C",0
237,1,546,370,6.3083691,"\int \frac{1}{(a+b \sec (c+d x)) \sqrt{e \sin (c+d x)}} \, dx","Integrate[1/((a + b*Sec[c + d*x])*Sqrt[e*Sin[c + d*x]]),x]","\frac{2 \sqrt{\sin (c+d x)} \left(a \sqrt{\cos ^2(c+d x)}+b\right) \left(\frac{b \left(-\log \left(-\sqrt{2} \sqrt{a} \sqrt[4]{b^2-a^2} \sqrt{\sin (c+d x)}+\sqrt{b^2-a^2}+a \sin (c+d x)\right)+\log \left(\sqrt{2} \sqrt{a} \sqrt[4]{b^2-a^2} \sqrt{\sin (c+d x)}+\sqrt{b^2-a^2}+a \sin (c+d x)\right)-2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{a} \sqrt{\sin (c+d x)}}{\sqrt[4]{b^2-a^2}}\right)+2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{a} \sqrt{\sin (c+d x)}}{\sqrt[4]{b^2-a^2}}+1\right)\right)}{4 \sqrt{2} \sqrt{a} \left(b^2-a^2\right)^{3/4}}-\frac{5 a \left(a^2-b^2\right) \sqrt{\sin (c+d x)} \sqrt{\cos ^2(c+d x)} F_1\left(\frac{1}{4};-\frac{1}{2},1;\frac{5}{4};\sin ^2(c+d x),\frac{a^2 \sin ^2(c+d x)}{a^2-b^2}\right)}{\left(a^2 \sin ^2(c+d x)-a^2+b^2\right) \left(2 \sin ^2(c+d x) \left(2 a^2 F_1\left(\frac{5}{4};-\frac{1}{2},2;\frac{9}{4};\sin ^2(c+d x),\frac{a^2 \sin ^2(c+d x)}{a^2-b^2}\right)+\left(b^2-a^2\right) F_1\left(\frac{5}{4};\frac{1}{2},1;\frac{9}{4};\sin ^2(c+d x),\frac{a^2 \sin ^2(c+d x)}{a^2-b^2}\right)\right)+5 \left(a^2-b^2\right) F_1\left(\frac{1}{4};-\frac{1}{2},1;\frac{5}{4};\sin ^2(c+d x),\frac{a^2 \sin ^2(c+d x)}{a^2-b^2}\right)\right)}\right)}{d \sqrt{e \sin (c+d x)} (a \cos (c+d x)+b)}","-\frac{b \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt{e} \sqrt[4]{a^2-b^2}}\right)}{\sqrt{a} d \sqrt{e} \left(a^2-b^2\right)^{3/4}}-\frac{b \tanh ^{-1}\left(\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt{e} \sqrt[4]{a^2-b^2}}\right)}{\sqrt{a} d \sqrt{e} \left(a^2-b^2\right)^{3/4}}+\frac{b^2 \sqrt{\sin (c+d x)} \Pi \left(\frac{2 a}{a-\sqrt{a^2-b^2}};\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{a d \left(-a \sqrt{a^2-b^2}+a^2-b^2\right) \sqrt{e \sin (c+d x)}}+\frac{b^2 \sqrt{\sin (c+d x)} \Pi \left(\frac{2 a}{a+\sqrt{a^2-b^2}};\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{a d \left(a \sqrt{a^2-b^2}+a^2-b^2\right) \sqrt{e \sin (c+d x)}}+\frac{2 \sqrt{\sin (c+d x)} F\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{a d \sqrt{e \sin (c+d x)}}",1,"(2*(b + a*Sqrt[Cos[c + d*x]^2])*Sqrt[Sin[c + d*x]]*((b*(-2*ArcTan[1 - (Sqrt[2]*Sqrt[a]*Sqrt[Sin[c + d*x]])/(-a^2 + b^2)^(1/4)] + 2*ArcTan[1 + (Sqrt[2]*Sqrt[a]*Sqrt[Sin[c + d*x]])/(-a^2 + b^2)^(1/4)] - Log[Sqrt[-a^2 + b^2] - Sqrt[2]*Sqrt[a]*(-a^2 + b^2)^(1/4)*Sqrt[Sin[c + d*x]] + a*Sin[c + d*x]] + Log[Sqrt[-a^2 + b^2] + Sqrt[2]*Sqrt[a]*(-a^2 + b^2)^(1/4)*Sqrt[Sin[c + d*x]] + a*Sin[c + d*x]]))/(4*Sqrt[2]*Sqrt[a]*(-a^2 + b^2)^(3/4)) - (5*a*(a^2 - b^2)*AppellF1[1/4, -1/2, 1, 5/4, Sin[c + d*x]^2, (a^2*Sin[c + d*x]^2)/(a^2 - b^2)]*Sqrt[Cos[c + d*x]^2]*Sqrt[Sin[c + d*x]])/((-a^2 + b^2 + a^2*Sin[c + d*x]^2)*(5*(a^2 - b^2)*AppellF1[1/4, -1/2, 1, 5/4, Sin[c + d*x]^2, (a^2*Sin[c + d*x]^2)/(a^2 - b^2)] + 2*(2*a^2*AppellF1[5/4, -1/2, 2, 9/4, Sin[c + d*x]^2, (a^2*Sin[c + d*x]^2)/(a^2 - b^2)] + (-a^2 + b^2)*AppellF1[5/4, 1/2, 1, 9/4, Sin[c + d*x]^2, (a^2*Sin[c + d*x]^2)/(a^2 - b^2)])*Sin[c + d*x]^2))))/(d*(b + a*Cos[c + d*x])*Sqrt[e*Sin[c + d*x]])","C",0
238,1,834,430,14.47532,"\int \frac{1}{(a+b \sec (c+d x)) (e \sin (c+d x))^{3/2}} \, dx","Integrate[1/((a + b*Sec[c + d*x])*(e*Sin[c + d*x])^(3/2)),x]","-\frac{a (b+a \cos (c+d x)) \sec (c+d x) \left(\frac{\left(8 F_1\left(\frac{3}{4};-\frac{1}{2},1;\frac{7}{4};\sin ^2(c+d x),\frac{a^2 \sin ^2(c+d x)}{a^2-b^2}\right) \sin ^{\frac{3}{2}}(c+d x) a^{5/2}+3 \sqrt{2} b \left(b^2-a^2\right)^{3/4} \left(2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{a} \sqrt{\sin (c+d x)}}{\sqrt[4]{b^2-a^2}}\right)-2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{a} \sqrt{\sin (c+d x)}}{\sqrt[4]{b^2-a^2}}+1\right)-\log \left(a \sin (c+d x)-\sqrt{2} \sqrt{a} \sqrt[4]{b^2-a^2} \sqrt{\sin (c+d x)}+\sqrt{b^2-a^2}\right)+\log \left(a \sin (c+d x)+\sqrt{2} \sqrt{a} \sqrt[4]{b^2-a^2} \sqrt{\sin (c+d x)}+\sqrt{b^2-a^2}\right)\right)\right) \left(\sqrt{1-\sin ^2(c+d x)} a+b\right) \cos ^2(c+d x)}{12 \sqrt{a} \left(a^2-b^2\right) (b+a \cos (c+d x)) \left(1-\sin ^2(c+d x)\right)}+\frac{4 b \left(\frac{b F_1\left(\frac{3}{4};\frac{1}{2},1;\frac{7}{4};\sin ^2(c+d x),\frac{a^2 \sin ^2(c+d x)}{a^2-b^2}\right) \sin ^{\frac{3}{2}}(c+d x)}{3 \left(b^2-a^2\right)}+\frac{\left(\frac{1}{8}+\frac{i}{8}\right) \left(2 \tan ^{-1}\left(1-\frac{(1+i) \sqrt{a} \sqrt{\sin (c+d x)}}{\sqrt[4]{a^2-b^2}}\right)-2 \tan ^{-1}\left(\frac{(1+i) \sqrt{a} \sqrt{\sin (c+d x)}}{\sqrt[4]{a^2-b^2}}+1\right)-\log \left(i a \sin (c+d x)-(1+i) \sqrt{a} \sqrt[4]{a^2-b^2} \sqrt{\sin (c+d x)}+\sqrt{a^2-b^2}\right)+\log \left(i a \sin (c+d x)+(1+i) \sqrt{a} \sqrt[4]{a^2-b^2} \sqrt{\sin (c+d x)}+\sqrt{a^2-b^2}\right)\right)}{\sqrt{a} \sqrt[4]{a^2-b^2}}\right) \left(\sqrt{1-\sin ^2(c+d x)} a+b\right) \cos (c+d x)}{(b+a \cos (c+d x)) \sqrt{1-\sin ^2(c+d x)}}\right) \sin ^{\frac{3}{2}}(c+d x)}{(a-b) (a+b) d (a+b \sec (c+d x)) (e \sin (c+d x))^{3/2}}-\frac{2 (b-a \cos (c+d x)) (b+a \cos (c+d x)) \tan (c+d x)}{\left(b^2-a^2\right) d (a+b \sec (c+d x)) (e \sin (c+d x))^{3/2}}","\frac{\sqrt{a} b \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt{e} \sqrt[4]{a^2-b^2}}\right)}{d e^{3/2} \left(a^2-b^2\right)^{5/4}}-\frac{\sqrt{a} b \tanh ^{-1}\left(\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt{e} \sqrt[4]{a^2-b^2}}\right)}{d e^{3/2} \left(a^2-b^2\right)^{5/4}}-\frac{2 a E\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{e \sin (c+d x)}}{d e^2 \left(a^2-b^2\right) \sqrt{\sin (c+d x)}}+\frac{2 (b-a \cos (c+d x))}{d e \left(a^2-b^2\right) \sqrt{e \sin (c+d x)}}-\frac{b^2 \sqrt{\sin (c+d x)} \Pi \left(\frac{2 a}{a-\sqrt{a^2-b^2}};\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{d e \left(a^2-b^2\right) \left(a-\sqrt{a^2-b^2}\right) \sqrt{e \sin (c+d x)}}-\frac{b^2 \sqrt{\sin (c+d x)} \Pi \left(\frac{2 a}{a+\sqrt{a^2-b^2}};\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{d e \left(a^2-b^2\right) \left(\sqrt{a^2-b^2}+a\right) \sqrt{e \sin (c+d x)}}",1,"-((a*(b + a*Cos[c + d*x])*Sec[c + d*x]*Sin[c + d*x]^(3/2)*((Cos[c + d*x]^2*(3*Sqrt[2]*b*(-a^2 + b^2)^(3/4)*(2*ArcTan[1 - (Sqrt[2]*Sqrt[a]*Sqrt[Sin[c + d*x]])/(-a^2 + b^2)^(1/4)] - 2*ArcTan[1 + (Sqrt[2]*Sqrt[a]*Sqrt[Sin[c + d*x]])/(-a^2 + b^2)^(1/4)] - Log[Sqrt[-a^2 + b^2] - Sqrt[2]*Sqrt[a]*(-a^2 + b^2)^(1/4)*Sqrt[Sin[c + d*x]] + a*Sin[c + d*x]] + Log[Sqrt[-a^2 + b^2] + Sqrt[2]*Sqrt[a]*(-a^2 + b^2)^(1/4)*Sqrt[Sin[c + d*x]] + a*Sin[c + d*x]]) + 8*a^(5/2)*AppellF1[3/4, -1/2, 1, 7/4, Sin[c + d*x]^2, (a^2*Sin[c + d*x]^2)/(a^2 - b^2)]*Sin[c + d*x]^(3/2))*(b + a*Sqrt[1 - Sin[c + d*x]^2]))/(12*Sqrt[a]*(a^2 - b^2)*(b + a*Cos[c + d*x])*(1 - Sin[c + d*x]^2)) + (4*b*Cos[c + d*x]*(((1/8 + I/8)*(2*ArcTan[1 - ((1 + I)*Sqrt[a]*Sqrt[Sin[c + d*x]])/(a^2 - b^2)^(1/4)] - 2*ArcTan[1 + ((1 + I)*Sqrt[a]*Sqrt[Sin[c + d*x]])/(a^2 - b^2)^(1/4)] - Log[Sqrt[a^2 - b^2] - (1 + I)*Sqrt[a]*(a^2 - b^2)^(1/4)*Sqrt[Sin[c + d*x]] + I*a*Sin[c + d*x]] + Log[Sqrt[a^2 - b^2] + (1 + I)*Sqrt[a]*(a^2 - b^2)^(1/4)*Sqrt[Sin[c + d*x]] + I*a*Sin[c + d*x]]))/(Sqrt[a]*(a^2 - b^2)^(1/4)) + (b*AppellF1[3/4, 1/2, 1, 7/4, Sin[c + d*x]^2, (a^2*Sin[c + d*x]^2)/(a^2 - b^2)]*Sin[c + d*x]^(3/2))/(3*(-a^2 + b^2)))*(b + a*Sqrt[1 - Sin[c + d*x]^2]))/((b + a*Cos[c + d*x])*Sqrt[1 - Sin[c + d*x]^2])))/((a - b)*(a + b)*d*(a + b*Sec[c + d*x])*(e*Sin[c + d*x])^(3/2))) - (2*(b - a*Cos[c + d*x])*(b + a*Cos[c + d*x])*Tan[c + d*x])/((-a^2 + b^2)*d*(a + b*Sec[c + d*x])*(e*Sin[c + d*x])^(3/2))","C",0
239,1,1233,452,12.7078246,"\int \frac{1}{(a+b \sec (c+d x)) (e \sin (c+d x))^{5/2}} \, dx","Integrate[1/((a + b*Sec[c + d*x])*(e*Sin[c + d*x])^(5/2)),x]","-\frac{a (b+a \cos (c+d x)) \sec (c+d x) \left(\frac{4 b \cos (c+d x) \left(\sqrt{1-\sin ^2(c+d x)} a+b\right) \left(\frac{5 b \left(a^2-b^2\right) F_1\left(\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\sin ^2(c+d x),\frac{a^2 \sin ^2(c+d x)}{a^2-b^2}\right) \sqrt{\sin (c+d x)}}{\sqrt{1-\sin ^2(c+d x)} \left(2 \left(2 F_1\left(\frac{5}{4};\frac{1}{2},2;\frac{9}{4};\sin ^2(c+d x),\frac{a^2 \sin ^2(c+d x)}{a^2-b^2}\right) a^2+\left(a^2-b^2\right) F_1\left(\frac{5}{4};\frac{3}{2},1;\frac{9}{4};\sin ^2(c+d x),\frac{a^2 \sin ^2(c+d x)}{a^2-b^2}\right)\right) \sin ^2(c+d x)+5 \left(a^2-b^2\right) F_1\left(\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\sin ^2(c+d x),\frac{a^2 \sin ^2(c+d x)}{a^2-b^2}\right)\right) \left(\left(\sin ^2(c+d x)-1\right) a^2+b^2\right)}-\frac{\left(\frac{1}{8}-\frac{i}{8}\right) \sqrt{a} \left(2 \tan ^{-1}\left(1-\frac{(1+i) \sqrt{a} \sqrt{\sin (c+d x)}}{\sqrt[4]{a^2-b^2}}\right)-2 \tan ^{-1}\left(\frac{(1+i) \sqrt{a} \sqrt{\sin (c+d x)}}{\sqrt[4]{a^2-b^2}}+1\right)+\log \left(i a \sin (c+d x)-(1+i) \sqrt{a} \sqrt[4]{a^2-b^2} \sqrt{\sin (c+d x)}+\sqrt{a^2-b^2}\right)-\log \left(i a \sin (c+d x)+(1+i) \sqrt{a} \sqrt[4]{a^2-b^2} \sqrt{\sin (c+d x)}+\sqrt{a^2-b^2}\right)\right)}{\left(a^2-b^2\right)^{3/4}}\right)}{(b+a \cos (c+d x)) \sqrt{1-\sin ^2(c+d x)}}-\frac{2 a \cos ^2(c+d x) \left(\sqrt{1-\sin ^2(c+d x)} a+b\right) \left(\frac{b \left(-2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{a} \sqrt{\sin (c+d x)}}{\sqrt[4]{b^2-a^2}}\right)+2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{a} \sqrt{\sin (c+d x)}}{\sqrt[4]{b^2-a^2}}+1\right)-\log \left(a \sin (c+d x)-\sqrt{2} \sqrt{a} \sqrt[4]{b^2-a^2} \sqrt{\sin (c+d x)}+\sqrt{b^2-a^2}\right)+\log \left(a \sin (c+d x)+\sqrt{2} \sqrt{a} \sqrt[4]{b^2-a^2} \sqrt{\sin (c+d x)}+\sqrt{b^2-a^2}\right)\right)}{4 \sqrt{2} \sqrt{a} \left(b^2-a^2\right)^{3/4}}-\frac{5 a \left(a^2-b^2\right) F_1\left(\frac{1}{4};-\frac{1}{2},1;\frac{5}{4};\sin ^2(c+d x),\frac{a^2 \sin ^2(c+d x)}{a^2-b^2}\right) \sqrt{\sin (c+d x)} \sqrt{1-\sin ^2(c+d x)}}{\left(2 \left(2 F_1\left(\frac{5}{4};-\frac{1}{2},2;\frac{9}{4};\sin ^2(c+d x),\frac{a^2 \sin ^2(c+d x)}{a^2-b^2}\right) a^2+\left(b^2-a^2\right) F_1\left(\frac{5}{4};\frac{1}{2},1;\frac{9}{4};\sin ^2(c+d x),\frac{a^2 \sin ^2(c+d x)}{a^2-b^2}\right)\right) \sin ^2(c+d x)+5 \left(a^2-b^2\right) F_1\left(\frac{1}{4};-\frac{1}{2},1;\frac{5}{4};\sin ^2(c+d x),\frac{a^2 \sin ^2(c+d x)}{a^2-b^2}\right)\right) \left(\left(\sin ^2(c+d x)-1\right) a^2+b^2\right)}\right)}{(b+a \cos (c+d x)) \left(1-\sin ^2(c+d x)\right)}\right) \sin ^{\frac{5}{2}}(c+d x)}{3 (a-b) (a+b) d (a+b \sec (c+d x)) (e \sin (c+d x))^{5/2}}-\frac{2 (b-a \cos (c+d x)) (b+a \cos (c+d x)) \tan (c+d x)}{3 \left(b^2-a^2\right) d (a+b \sec (c+d x)) (e \sin (c+d x))^{5/2}}","\frac{2 a \sqrt{\sin (c+d x)} F\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{3 d e^2 \left(a^2-b^2\right) \sqrt{e \sin (c+d x)}}+\frac{a b^2 \sqrt{\sin (c+d x)} \Pi \left(\frac{2 a}{a-\sqrt{a^2-b^2}};\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{d e^2 \left(a^2-b^2\right) \left(-a \sqrt{a^2-b^2}+a^2-b^2\right) \sqrt{e \sin (c+d x)}}+\frac{a b^2 \sqrt{\sin (c+d x)} \Pi \left(\frac{2 a}{a+\sqrt{a^2-b^2}};\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{d e^2 \left(a^2-b^2\right) \left(a \sqrt{a^2-b^2}+a^2-b^2\right) \sqrt{e \sin (c+d x)}}+\frac{2 (b-a \cos (c+d x))}{3 d e \left(a^2-b^2\right) (e \sin (c+d x))^{3/2}}-\frac{a^{3/2} b \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt{e} \sqrt[4]{a^2-b^2}}\right)}{d e^{5/2} \left(a^2-b^2\right)^{7/4}}-\frac{a^{3/2} b \tanh ^{-1}\left(\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt{e} \sqrt[4]{a^2-b^2}}\right)}{d e^{5/2} \left(a^2-b^2\right)^{7/4}}",1,"-1/3*(a*(b + a*Cos[c + d*x])*Sec[c + d*x]*Sin[c + d*x]^(5/2)*((-2*a*Cos[c + d*x]^2*(b + a*Sqrt[1 - Sin[c + d*x]^2])*((b*(-2*ArcTan[1 - (Sqrt[2]*Sqrt[a]*Sqrt[Sin[c + d*x]])/(-a^2 + b^2)^(1/4)] + 2*ArcTan[1 + (Sqrt[2]*Sqrt[a]*Sqrt[Sin[c + d*x]])/(-a^2 + b^2)^(1/4)] - Log[Sqrt[-a^2 + b^2] - Sqrt[2]*Sqrt[a]*(-a^2 + b^2)^(1/4)*Sqrt[Sin[c + d*x]] + a*Sin[c + d*x]] + Log[Sqrt[-a^2 + b^2] + Sqrt[2]*Sqrt[a]*(-a^2 + b^2)^(1/4)*Sqrt[Sin[c + d*x]] + a*Sin[c + d*x]]))/(4*Sqrt[2]*Sqrt[a]*(-a^2 + b^2)^(3/4)) - (5*a*(a^2 - b^2)*AppellF1[1/4, -1/2, 1, 5/4, Sin[c + d*x]^2, (a^2*Sin[c + d*x]^2)/(a^2 - b^2)]*Sqrt[Sin[c + d*x]]*Sqrt[1 - Sin[c + d*x]^2])/((5*(a^2 - b^2)*AppellF1[1/4, -1/2, 1, 5/4, Sin[c + d*x]^2, (a^2*Sin[c + d*x]^2)/(a^2 - b^2)] + 2*(2*a^2*AppellF1[5/4, -1/2, 2, 9/4, Sin[c + d*x]^2, (a^2*Sin[c + d*x]^2)/(a^2 - b^2)] + (-a^2 + b^2)*AppellF1[5/4, 1/2, 1, 9/4, Sin[c + d*x]^2, (a^2*Sin[c + d*x]^2)/(a^2 - b^2)])*Sin[c + d*x]^2)*(b^2 + a^2*(-1 + Sin[c + d*x]^2)))))/((b + a*Cos[c + d*x])*(1 - Sin[c + d*x]^2)) + (4*b*Cos[c + d*x]*(b + a*Sqrt[1 - Sin[c + d*x]^2])*(((-1/8 + I/8)*Sqrt[a]*(2*ArcTan[1 - ((1 + I)*Sqrt[a]*Sqrt[Sin[c + d*x]])/(a^2 - b^2)^(1/4)] - 2*ArcTan[1 + ((1 + I)*Sqrt[a]*Sqrt[Sin[c + d*x]])/(a^2 - b^2)^(1/4)] + Log[Sqrt[a^2 - b^2] - (1 + I)*Sqrt[a]*(a^2 - b^2)^(1/4)*Sqrt[Sin[c + d*x]] + I*a*Sin[c + d*x]] - Log[Sqrt[a^2 - b^2] + (1 + I)*Sqrt[a]*(a^2 - b^2)^(1/4)*Sqrt[Sin[c + d*x]] + I*a*Sin[c + d*x]]))/(a^2 - b^2)^(3/4) + (5*b*(a^2 - b^2)*AppellF1[1/4, 1/2, 1, 5/4, Sin[c + d*x]^2, (a^2*Sin[c + d*x]^2)/(a^2 - b^2)]*Sqrt[Sin[c + d*x]])/(Sqrt[1 - Sin[c + d*x]^2]*(5*(a^2 - b^2)*AppellF1[1/4, 1/2, 1, 5/4, Sin[c + d*x]^2, (a^2*Sin[c + d*x]^2)/(a^2 - b^2)] + 2*(2*a^2*AppellF1[5/4, 1/2, 2, 9/4, Sin[c + d*x]^2, (a^2*Sin[c + d*x]^2)/(a^2 - b^2)] + (a^2 - b^2)*AppellF1[5/4, 3/2, 1, 9/4, Sin[c + d*x]^2, (a^2*Sin[c + d*x]^2)/(a^2 - b^2)])*Sin[c + d*x]^2)*(b^2 + a^2*(-1 + Sin[c + d*x]^2)))))/((b + a*Cos[c + d*x])*Sqrt[1 - Sin[c + d*x]^2])))/((a - b)*(a + b)*d*(a + b*Sec[c + d*x])*(e*Sin[c + d*x])^(5/2)) - (2*(b - a*Cos[c + d*x])*(b + a*Cos[c + d*x])*Tan[c + d*x])/(3*(-a^2 + b^2)*d*(a + b*Sec[c + d*x])*(e*Sin[c + d*x])^(5/2))","C",0
240,1,930,511,6.8898048,"\int \frac{1}{(a+b \sec (c+d x)) (e \sin (c+d x))^{7/2}} \, dx","Integrate[1/((a + b*Sec[c + d*x])*(e*Sin[c + d*x])^(7/2)),x]","\frac{(b+a \cos (c+d x)) \left(-\frac{2 (b-a \cos (c+d x)) \csc ^3(c+d x)}{5 \left(b^2-a^2\right)}-\frac{2 \left(3 \cos (c+d x) a^3-5 b a^2+2 b^2 \cos (c+d x) a\right) \csc (c+d x)}{5 \left(b^2-a^2\right)^2}\right) \sin ^3(c+d x) \tan (c+d x)}{d (a+b \sec (c+d x)) (e \sin (c+d x))^{7/2}}-\frac{a (b+a \cos (c+d x)) \sec (c+d x) \sin ^{\frac{7}{2}}(c+d x) \left(\frac{\left(3 a^3+2 b^2 a\right) \left(8 F_1\left(\frac{3}{4};-\frac{1}{2},1;\frac{7}{4};\sin ^2(c+d x),\frac{a^2 \sin ^2(c+d x)}{a^2-b^2}\right) \sin ^{\frac{3}{2}}(c+d x) a^{5/2}+3 \sqrt{2} b \left(b^2-a^2\right)^{3/4} \left(2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{a} \sqrt{\sin (c+d x)}}{\sqrt[4]{b^2-a^2}}\right)-2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{a} \sqrt{\sin (c+d x)}}{\sqrt[4]{b^2-a^2}}+1\right)-\log \left(a \sin (c+d x)-\sqrt{2} \sqrt{a} \sqrt[4]{b^2-a^2} \sqrt{\sin (c+d x)}+\sqrt{b^2-a^2}\right)+\log \left(a \sin (c+d x)+\sqrt{2} \sqrt{a} \sqrt[4]{b^2-a^2} \sqrt{\sin (c+d x)}+\sqrt{b^2-a^2}\right)\right)\right) \left(\sqrt{1-\sin ^2(c+d x)} a+b\right) \cos ^2(c+d x)}{12 a^{3/2} \left(a^2-b^2\right) (b+a \cos (c+d x)) \left(1-\sin ^2(c+d x)\right)}+\frac{2 \left(2 b^3+8 a^2 b\right) \left(\frac{b F_1\left(\frac{3}{4};\frac{1}{2},1;\frac{7}{4};\sin ^2(c+d x),\frac{a^2 \sin ^2(c+d x)}{a^2-b^2}\right) \sin ^{\frac{3}{2}}(c+d x)}{3 \left(b^2-a^2\right)}+\frac{\left(\frac{1}{8}+\frac{i}{8}\right) \left(2 \tan ^{-1}\left(1-\frac{(1+i) \sqrt{a} \sqrt{\sin (c+d x)}}{\sqrt[4]{a^2-b^2}}\right)-2 \tan ^{-1}\left(\frac{(1+i) \sqrt{a} \sqrt{\sin (c+d x)}}{\sqrt[4]{a^2-b^2}}+1\right)-\log \left(i a \sin (c+d x)-(1+i) \sqrt{a} \sqrt[4]{a^2-b^2} \sqrt{\sin (c+d x)}+\sqrt{a^2-b^2}\right)+\log \left(i a \sin (c+d x)+(1+i) \sqrt{a} \sqrt[4]{a^2-b^2} \sqrt{\sin (c+d x)}+\sqrt{a^2-b^2}\right)\right)}{\sqrt{a} \sqrt[4]{a^2-b^2}}\right) \left(\sqrt{1-\sin ^2(c+d x)} a+b\right) \cos (c+d x)}{(b+a \cos (c+d x)) \sqrt{1-\sin ^2(c+d x)}}\right)}{5 (a-b)^2 (a+b)^2 d (a+b \sec (c+d x)) (e \sin (c+d x))^{7/2}}","-\frac{2 a \left(3 a^2+2 b^2\right) E\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{e \sin (c+d x)}}{5 d e^4 \left(a^2-b^2\right)^2 \sqrt{\sin (c+d x)}}+\frac{2 \left(5 a^2 b-a \left(3 a^2+2 b^2\right) \cos (c+d x)\right)}{5 d e^3 \left(a^2-b^2\right)^2 \sqrt{e \sin (c+d x)}}-\frac{a^2 b^2 \sqrt{\sin (c+d x)} \Pi \left(\frac{2 a}{a-\sqrt{a^2-b^2}};\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{d e^3 \left(a^2-b^2\right)^2 \left(a-\sqrt{a^2-b^2}\right) \sqrt{e \sin (c+d x)}}-\frac{a^2 b^2 \sqrt{\sin (c+d x)} \Pi \left(\frac{2 a}{a+\sqrt{a^2-b^2}};\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{d e^3 \left(a^2-b^2\right)^2 \left(\sqrt{a^2-b^2}+a\right) \sqrt{e \sin (c+d x)}}+\frac{2 (b-a \cos (c+d x))}{5 d e \left(a^2-b^2\right) (e \sin (c+d x))^{5/2}}+\frac{a^{5/2} b \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt{e} \sqrt[4]{a^2-b^2}}\right)}{d e^{7/2} \left(a^2-b^2\right)^{9/4}}-\frac{a^{5/2} b \tanh ^{-1}\left(\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt{e} \sqrt[4]{a^2-b^2}}\right)}{d e^{7/2} \left(a^2-b^2\right)^{9/4}}",1,"-1/5*(a*(b + a*Cos[c + d*x])*Sec[c + d*x]*Sin[c + d*x]^(7/2)*(((3*a^3 + 2*a*b^2)*Cos[c + d*x]^2*(3*Sqrt[2]*b*(-a^2 + b^2)^(3/4)*(2*ArcTan[1 - (Sqrt[2]*Sqrt[a]*Sqrt[Sin[c + d*x]])/(-a^2 + b^2)^(1/4)] - 2*ArcTan[1 + (Sqrt[2]*Sqrt[a]*Sqrt[Sin[c + d*x]])/(-a^2 + b^2)^(1/4)] - Log[Sqrt[-a^2 + b^2] - Sqrt[2]*Sqrt[a]*(-a^2 + b^2)^(1/4)*Sqrt[Sin[c + d*x]] + a*Sin[c + d*x]] + Log[Sqrt[-a^2 + b^2] + Sqrt[2]*Sqrt[a]*(-a^2 + b^2)^(1/4)*Sqrt[Sin[c + d*x]] + a*Sin[c + d*x]]) + 8*a^(5/2)*AppellF1[3/4, -1/2, 1, 7/4, Sin[c + d*x]^2, (a^2*Sin[c + d*x]^2)/(a^2 - b^2)]*Sin[c + d*x]^(3/2))*(b + a*Sqrt[1 - Sin[c + d*x]^2]))/(12*a^(3/2)*(a^2 - b^2)*(b + a*Cos[c + d*x])*(1 - Sin[c + d*x]^2)) + (2*(8*a^2*b + 2*b^3)*Cos[c + d*x]*(((1/8 + I/8)*(2*ArcTan[1 - ((1 + I)*Sqrt[a]*Sqrt[Sin[c + d*x]])/(a^2 - b^2)^(1/4)] - 2*ArcTan[1 + ((1 + I)*Sqrt[a]*Sqrt[Sin[c + d*x]])/(a^2 - b^2)^(1/4)] - Log[Sqrt[a^2 - b^2] - (1 + I)*Sqrt[a]*(a^2 - b^2)^(1/4)*Sqrt[Sin[c + d*x]] + I*a*Sin[c + d*x]] + Log[Sqrt[a^2 - b^2] + (1 + I)*Sqrt[a]*(a^2 - b^2)^(1/4)*Sqrt[Sin[c + d*x]] + I*a*Sin[c + d*x]]))/(Sqrt[a]*(a^2 - b^2)^(1/4)) + (b*AppellF1[3/4, 1/2, 1, 7/4, Sin[c + d*x]^2, (a^2*Sin[c + d*x]^2)/(a^2 - b^2)]*Sin[c + d*x]^(3/2))/(3*(-a^2 + b^2)))*(b + a*Sqrt[1 - Sin[c + d*x]^2]))/((b + a*Cos[c + d*x])*Sqrt[1 - Sin[c + d*x]^2])))/((a - b)^2*(a + b)^2*d*(a + b*Sec[c + d*x])*(e*Sin[c + d*x])^(7/2)) + ((b + a*Cos[c + d*x])*((-2*(-5*a^2*b + 3*a^3*Cos[c + d*x] + 2*a*b^2*Cos[c + d*x])*Csc[c + d*x])/(5*(-a^2 + b^2)^2) - (2*(b - a*Cos[c + d*x])*Csc[c + d*x]^3)/(5*(-a^2 + b^2)))*Sin[c + d*x]^3*Tan[c + d*x])/(d*(a + b*Sec[c + d*x])*(e*Sin[c + d*x])^(7/2))","C",0
241,1,974,1070,15.5933029,"\int \frac{(e \sin (c+d x))^{9/2}}{(a+b \sec (c+d x))^2} \, dx","Integrate[(e*Sin[c + d*x])^(9/2)/(a + b*Sec[c + d*x])^2,x]","\frac{(b+a \cos (c+d x))^2 \sec ^2(c+d x) \left(\frac{\left(14 a^4-159 b^2 a^2+165 b^4\right) \left(8 F_1\left(\frac{3}{4};-\frac{1}{2},1;\frac{7}{4};\sin ^2(c+d x),\frac{a^2 \sin ^2(c+d x)}{a^2-b^2}\right) \sin ^{\frac{3}{2}}(c+d x) a^{5/2}+3 \sqrt{2} b \left(b^2-a^2\right)^{3/4} \left(2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{a} \sqrt{\sin (c+d x)}}{\sqrt[4]{b^2-a^2}}\right)-2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{a} \sqrt{\sin (c+d x)}}{\sqrt[4]{b^2-a^2}}+1\right)-\log \left(a \sin (c+d x)-\sqrt{2} \sqrt{a} \sqrt[4]{b^2-a^2} \sqrt{\sin (c+d x)}+\sqrt{b^2-a^2}\right)+\log \left(a \sin (c+d x)+\sqrt{2} \sqrt{a} \sqrt[4]{b^2-a^2} \sqrt{\sin (c+d x)}+\sqrt{b^2-a^2}\right)\right)\right) \left(\sqrt{1-\sin ^2(c+d x)} a+b\right) \cos ^2(c+d x)}{12 a^{3/2} \left(a^2-b^2\right) (b+a \cos (c+d x)) \left(1-\sin ^2(c+d x)\right)}+\frac{2 \left(66 a b^3-46 a^3 b\right) \left(\frac{b F_1\left(\frac{3}{4};\frac{1}{2},1;\frac{7}{4};\sin ^2(c+d x),\frac{a^2 \sin ^2(c+d x)}{a^2-b^2}\right) \sin ^{\frac{3}{2}}(c+d x)}{3 \left(b^2-a^2\right)}+\frac{\left(\frac{1}{8}+\frac{i}{8}\right) \left(2 \tan ^{-1}\left(1-\frac{(1+i) \sqrt{a} \sqrt{\sin (c+d x)}}{\sqrt[4]{a^2-b^2}}\right)-2 \tan ^{-1}\left(\frac{(1+i) \sqrt{a} \sqrt{\sin (c+d x)}}{\sqrt[4]{a^2-b^2}}+1\right)-\log \left(i a \sin (c+d x)-(1+i) \sqrt{a} \sqrt[4]{a^2-b^2} \sqrt{\sin (c+d x)}+\sqrt{a^2-b^2}\right)+\log \left(i a \sin (c+d x)+(1+i) \sqrt{a} \sqrt[4]{a^2-b^2} \sqrt{\sin (c+d x)}+\sqrt{a^2-b^2}\right)\right)}{\sqrt{a} \sqrt[4]{a^2-b^2}}\right) \left(\sqrt{1-\sin ^2(c+d x)} a+b\right) \cos (c+d x)}{(b+a \cos (c+d x)) \sqrt{1-\sin ^2(c+d x)}}\right) (e \sin (c+d x))^{9/2}}{30 a^5 d (a+b \sec (c+d x))^2 \sin ^{\frac{9}{2}}(c+d x)}+\frac{(b+a \cos (c+d x))^2 \csc ^4(c+d x) \sec ^2(c+d x) \left(-\frac{b \left(56 b^2-37 a^2\right) \sin (c+d x)}{21 a^5}+\frac{a^2 b^2 \sin (c+d x)-b^4 \sin (c+d x)}{a^5 (b+a \cos (c+d x))}-\frac{\left(19 a^2-54 b^2\right) \sin (2 (c+d x))}{90 a^4}-\frac{b \sin (3 (c+d x))}{7 a^3}+\frac{\sin (4 (c+d x))}{36 a^2}\right) (e \sin (c+d x))^{9/2}}{d (a+b \sec (c+d x))^2}","-\frac{2 b^2 \left(a^2-b^2\right)^2 \Pi \left(\frac{2 a}{a-\sqrt{a^2-b^2}};\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{\sin (c+d x)} e^5}{a^7 \left(a-\sqrt{a^2-b^2}\right) d \sqrt{e \sin (c+d x)}}+\frac{7 b^4 \left(a^2-b^2\right) \Pi \left(\frac{2 a}{a-\sqrt{a^2-b^2}};\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{\sin (c+d x)} e^5}{2 a^7 \left(a-\sqrt{a^2-b^2}\right) d \sqrt{e \sin (c+d x)}}-\frac{2 b^2 \left(a^2-b^2\right)^2 \Pi \left(\frac{2 a}{a+\sqrt{a^2-b^2}};\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{\sin (c+d x)} e^5}{a^7 \left(a+\sqrt{a^2-b^2}\right) d \sqrt{e \sin (c+d x)}}+\frac{7 b^4 \left(a^2-b^2\right) \Pi \left(\frac{2 a}{a+\sqrt{a^2-b^2}};\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{\sin (c+d x)} e^5}{2 a^7 \left(a+\sqrt{a^2-b^2}\right) d \sqrt{e \sin (c+d x)}}+\frac{2 b \left(a^2-b^2\right)^{7/4} \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt[4]{a^2-b^2} \sqrt{e}}\right) e^{9/2}}{a^{13/2} d}-\frac{7 b^3 \left(a^2-b^2\right)^{3/4} \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt[4]{a^2-b^2} \sqrt{e}}\right) e^{9/2}}{2 a^{13/2} d}-\frac{2 b \left(a^2-b^2\right)^{7/4} \tanh ^{-1}\left(\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt[4]{a^2-b^2} \sqrt{e}}\right) e^{9/2}}{a^{13/2} d}+\frac{7 b^3 \left(a^2-b^2\right)^{3/4} \tanh ^{-1}\left(\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt[4]{a^2-b^2} \sqrt{e}}\right) e^{9/2}}{2 a^{13/2} d}-\frac{7 b^2 \left(3 a^2-5 b^2\right) E\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{e \sin (c+d x)} e^4}{5 a^6 d \sqrt{\sin (c+d x)}}-\frac{4 b^2 \left(8 a^2-5 b^2\right) E\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{e \sin (c+d x)} e^4}{5 a^6 d \sqrt{\sin (c+d x)}}+\frac{14 E\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{e \sin (c+d x)} e^4}{15 a^2 d \sqrt{\sin (c+d x)}}-\frac{14 \cos (c+d x) (e \sin (c+d x))^{3/2} e^3}{45 a^2 d}-\frac{7 b^2 (5 b-3 a \cos (c+d x)) (e \sin (c+d x))^{3/2} e^3}{15 a^5 d}+\frac{4 b \left(5 \left(a^2-b^2\right)+3 a b \cos (c+d x)\right) (e \sin (c+d x))^{3/2} e^3}{15 a^5 d}-\frac{2 \cos (c+d x) (e \sin (c+d x))^{7/2} e}{9 a^2 d}+\frac{4 b (e \sin (c+d x))^{7/2} e}{7 a^3 d}+\frac{b^2 (e \sin (c+d x))^{7/2} e}{a^3 d (b+a \cos (c+d x))}",1,"((b + a*Cos[c + d*x])^2*Sec[c + d*x]^2*(e*Sin[c + d*x])^(9/2)*(((14*a^4 - 159*a^2*b^2 + 165*b^4)*Cos[c + d*x]^2*(3*Sqrt[2]*b*(-a^2 + b^2)^(3/4)*(2*ArcTan[1 - (Sqrt[2]*Sqrt[a]*Sqrt[Sin[c + d*x]])/(-a^2 + b^2)^(1/4)] - 2*ArcTan[1 + (Sqrt[2]*Sqrt[a]*Sqrt[Sin[c + d*x]])/(-a^2 + b^2)^(1/4)] - Log[Sqrt[-a^2 + b^2] - Sqrt[2]*Sqrt[a]*(-a^2 + b^2)^(1/4)*Sqrt[Sin[c + d*x]] + a*Sin[c + d*x]] + Log[Sqrt[-a^2 + b^2] + Sqrt[2]*Sqrt[a]*(-a^2 + b^2)^(1/4)*Sqrt[Sin[c + d*x]] + a*Sin[c + d*x]]) + 8*a^(5/2)*AppellF1[3/4, -1/2, 1, 7/4, Sin[c + d*x]^2, (a^2*Sin[c + d*x]^2)/(a^2 - b^2)]*Sin[c + d*x]^(3/2))*(b + a*Sqrt[1 - Sin[c + d*x]^2]))/(12*a^(3/2)*(a^2 - b^2)*(b + a*Cos[c + d*x])*(1 - Sin[c + d*x]^2)) + (2*(-46*a^3*b + 66*a*b^3)*Cos[c + d*x]*(((1/8 + I/8)*(2*ArcTan[1 - ((1 + I)*Sqrt[a]*Sqrt[Sin[c + d*x]])/(a^2 - b^2)^(1/4)] - 2*ArcTan[1 + ((1 + I)*Sqrt[a]*Sqrt[Sin[c + d*x]])/(a^2 - b^2)^(1/4)] - Log[Sqrt[a^2 - b^2] - (1 + I)*Sqrt[a]*(a^2 - b^2)^(1/4)*Sqrt[Sin[c + d*x]] + I*a*Sin[c + d*x]] + Log[Sqrt[a^2 - b^2] + (1 + I)*Sqrt[a]*(a^2 - b^2)^(1/4)*Sqrt[Sin[c + d*x]] + I*a*Sin[c + d*x]]))/(Sqrt[a]*(a^2 - b^2)^(1/4)) + (b*AppellF1[3/4, 1/2, 1, 7/4, Sin[c + d*x]^2, (a^2*Sin[c + d*x]^2)/(a^2 - b^2)]*Sin[c + d*x]^(3/2))/(3*(-a^2 + b^2)))*(b + a*Sqrt[1 - Sin[c + d*x]^2]))/((b + a*Cos[c + d*x])*Sqrt[1 - Sin[c + d*x]^2])))/(30*a^5*d*(a + b*Sec[c + d*x])^2*Sin[c + d*x]^(9/2)) + ((b + a*Cos[c + d*x])^2*Csc[c + d*x]^4*Sec[c + d*x]^2*(e*Sin[c + d*x])^(9/2)*(-1/21*(b*(-37*a^2 + 56*b^2)*Sin[c + d*x])/a^5 + (a^2*b^2*Sin[c + d*x] - b^4*Sin[c + d*x])/(a^5*(b + a*Cos[c + d*x])) - ((19*a^2 - 54*b^2)*Sin[2*(c + d*x)])/(90*a^4) - (b*Sin[3*(c + d*x)])/(7*a^3) + Sin[4*(c + d*x)]/(36*a^2)))/(d*(a + b*Sec[c + d*x])^2)","C",0
242,1,2095,1101,17.6160513,"\int \frac{(e \sin (c+d x))^{7/2}}{(a+b \sec (c+d x))^2} \, dx","Integrate[(e*Sin[c + d*x])^(7/2)/(a + b*Sec[c + d*x])^2,x]","\text{Result too large to show}","-\frac{5 b^2 \left(a^2-3 b^2\right) F\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{\sin (c+d x)} e^4}{3 a^6 d \sqrt{e \sin (c+d x)}}-\frac{4 b^2 \left(4 a^2-3 b^2\right) F\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{\sin (c+d x)} e^4}{3 a^6 d \sqrt{e \sin (c+d x)}}+\frac{10 F\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{\sin (c+d x)} e^4}{21 a^2 d \sqrt{e \sin (c+d x)}}+\frac{2 b^2 \left(a^2-b^2\right)^2 \Pi \left(\frac{2 a}{a-\sqrt{a^2-b^2}};\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{\sin (c+d x)} e^4}{a^6 \left(a^2-\sqrt{a^2-b^2} a-b^2\right) d \sqrt{e \sin (c+d x)}}-\frac{5 b^4 \left(a^2-b^2\right) \Pi \left(\frac{2 a}{a-\sqrt{a^2-b^2}};\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{\sin (c+d x)} e^4}{2 a^6 \left(a^2-\sqrt{a^2-b^2} a-b^2\right) d \sqrt{e \sin (c+d x)}}+\frac{2 b^2 \left(a^2-b^2\right)^2 \Pi \left(\frac{2 a}{a+\sqrt{a^2-b^2}};\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{\sin (c+d x)} e^4}{a^6 \left(a^2+\sqrt{a^2-b^2} a-b^2\right) d \sqrt{e \sin (c+d x)}}-\frac{5 b^4 \left(a^2-b^2\right) \Pi \left(\frac{2 a}{a+\sqrt{a^2-b^2}};\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{\sin (c+d x)} e^4}{2 a^6 \left(a^2+\sqrt{a^2-b^2} a-b^2\right) d \sqrt{e \sin (c+d x)}}-\frac{2 b \left(a^2-b^2\right)^{5/4} \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt[4]{a^2-b^2} \sqrt{e}}\right) e^{7/2}}{a^{11/2} d}+\frac{5 b^3 \sqrt[4]{a^2-b^2} \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt[4]{a^2-b^2} \sqrt{e}}\right) e^{7/2}}{2 a^{11/2} d}-\frac{2 b \left(a^2-b^2\right)^{5/4} \tanh ^{-1}\left(\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt[4]{a^2-b^2} \sqrt{e}}\right) e^{7/2}}{a^{11/2} d}+\frac{5 b^3 \sqrt[4]{a^2-b^2} \tanh ^{-1}\left(\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt[4]{a^2-b^2} \sqrt{e}}\right) e^{7/2}}{2 a^{11/2} d}-\frac{10 \cos (c+d x) \sqrt{e \sin (c+d x)} e^3}{21 a^2 d}-\frac{5 b^2 (3 b-a \cos (c+d x)) \sqrt{e \sin (c+d x)} e^3}{3 a^5 d}+\frac{4 b \left(3 \left(a^2-b^2\right)+a b \cos (c+d x)\right) \sqrt{e \sin (c+d x)} e^3}{3 a^5 d}-\frac{2 \cos (c+d x) (e \sin (c+d x))^{5/2} e}{7 a^2 d}+\frac{4 b (e \sin (c+d x))^{5/2} e}{5 a^3 d}+\frac{b^2 (e \sin (c+d x))^{5/2} e}{a^3 d (b+a \cos (c+d x))}",1,"((b + a*Cos[c + d*x])^2*(-1/42*((23*a^2 - 84*b^2)*Cos[c + d*x])/a^4 - (b^2*(-a^2 + b^2))/(a^5*(b + a*Cos[c + d*x])) - (2*b*Cos[2*(c + d*x)])/(5*a^3) + Cos[3*(c + d*x)]/(14*a^2))*Csc[c + d*x]^3*Sec[c + d*x]^2*(e*Sin[c + d*x])^(7/2))/(d*(a + b*Sec[c + d*x])^2) + ((b + a*Cos[c + d*x])^2*Sec[c + d*x]^2*(e*Sin[c + d*x])^(7/2)*((2*(50*a^4 - 273*a^2*b^2 + 105*b^4)*Cos[c + d*x]^2*(b + a*Sqrt[1 - Sin[c + d*x]^2])*((b*(-2*ArcTan[1 - (Sqrt[2]*Sqrt[a]*Sqrt[Sin[c + d*x]])/(-a^2 + b^2)^(1/4)] + 2*ArcTan[1 + (Sqrt[2]*Sqrt[a]*Sqrt[Sin[c + d*x]])/(-a^2 + b^2)^(1/4)] - Log[Sqrt[-a^2 + b^2] - Sqrt[2]*Sqrt[a]*(-a^2 + b^2)^(1/4)*Sqrt[Sin[c + d*x]] + a*Sin[c + d*x]] + Log[Sqrt[-a^2 + b^2] + Sqrt[2]*Sqrt[a]*(-a^2 + b^2)^(1/4)*Sqrt[Sin[c + d*x]] + a*Sin[c + d*x]]))/(4*Sqrt[2]*Sqrt[a]*(-a^2 + b^2)^(3/4)) - (5*a*(a^2 - b^2)*AppellF1[1/4, -1/2, 1, 5/4, Sin[c + d*x]^2, (a^2*Sin[c + d*x]^2)/(a^2 - b^2)]*Sqrt[Sin[c + d*x]]*Sqrt[1 - Sin[c + d*x]^2])/((5*(a^2 - b^2)*AppellF1[1/4, -1/2, 1, 5/4, Sin[c + d*x]^2, (a^2*Sin[c + d*x]^2)/(a^2 - b^2)] + 2*(2*a^2*AppellF1[5/4, -1/2, 2, 9/4, Sin[c + d*x]^2, (a^2*Sin[c + d*x]^2)/(a^2 - b^2)] + (-a^2 + b^2)*AppellF1[5/4, 1/2, 1, 9/4, Sin[c + d*x]^2, (a^2*Sin[c + d*x]^2)/(a^2 - b^2)])*Sin[c + d*x]^2)*(b^2 + a^2*(-1 + Sin[c + d*x]^2)))))/((b + a*Cos[c + d*x])*(1 - Sin[c + d*x]^2)) + (2*(-139*a^3*b + 210*a*b^3)*Cos[c + d*x]*(b + a*Sqrt[1 - Sin[c + d*x]^2])*(((-1/8 + I/8)*Sqrt[a]*(2*ArcTan[1 - ((1 + I)*Sqrt[a]*Sqrt[Sin[c + d*x]])/(a^2 - b^2)^(1/4)] - 2*ArcTan[1 + ((1 + I)*Sqrt[a]*Sqrt[Sin[c + d*x]])/(a^2 - b^2)^(1/4)] + Log[Sqrt[a^2 - b^2] - (1 + I)*Sqrt[a]*(a^2 - b^2)^(1/4)*Sqrt[Sin[c + d*x]] + I*a*Sin[c + d*x]] - Log[Sqrt[a^2 - b^2] + (1 + I)*Sqrt[a]*(a^2 - b^2)^(1/4)*Sqrt[Sin[c + d*x]] + I*a*Sin[c + d*x]]))/(a^2 - b^2)^(3/4) + (5*b*(a^2 - b^2)*AppellF1[1/4, 1/2, 1, 5/4, Sin[c + d*x]^2, (a^2*Sin[c + d*x]^2)/(a^2 - b^2)]*Sqrt[Sin[c + d*x]])/(Sqrt[1 - Sin[c + d*x]^2]*(5*(a^2 - b^2)*AppellF1[1/4, 1/2, 1, 5/4, Sin[c + d*x]^2, (a^2*Sin[c + d*x]^2)/(a^2 - b^2)] + 2*(2*a^2*AppellF1[5/4, 1/2, 2, 9/4, Sin[c + d*x]^2, (a^2*Sin[c + d*x]^2)/(a^2 - b^2)] + (a^2 - b^2)*AppellF1[5/4, 3/2, 1, 9/4, Sin[c + d*x]^2, (a^2*Sin[c + d*x]^2)/(a^2 - b^2)])*Sin[c + d*x]^2)*(b^2 + a^2*(-1 + Sin[c + d*x]^2)))))/((b + a*Cos[c + d*x])*Sqrt[1 - Sin[c + d*x]^2]) + ((231*a^3*b - 420*a*b^3)*Cos[c + d*x]*Cos[2*(c + d*x)]*(b + a*Sqrt[1 - Sin[c + d*x]^2])*(((1/2 - I/2)*(a^2 - 2*b^2)*ArcTan[1 - ((1 + I)*Sqrt[a]*Sqrt[Sin[c + d*x]])/(a^2 - b^2)^(1/4)])/(a^(3/2)*(a^2 - b^2)^(3/4)) - ((1/2 - I/2)*(a^2 - 2*b^2)*ArcTan[1 + ((1 + I)*Sqrt[a]*Sqrt[Sin[c + d*x]])/(a^2 - b^2)^(1/4)])/(a^(3/2)*(a^2 - b^2)^(3/4)) + ((1/4 - I/4)*(a^2 - 2*b^2)*Log[Sqrt[a^2 - b^2] - (1 + I)*Sqrt[a]*(a^2 - b^2)^(1/4)*Sqrt[Sin[c + d*x]] + I*a*Sin[c + d*x]])/(a^(3/2)*(a^2 - b^2)^(3/4)) - ((1/4 - I/4)*(a^2 - 2*b^2)*Log[Sqrt[a^2 - b^2] + (1 + I)*Sqrt[a]*(a^2 - b^2)^(1/4)*Sqrt[Sin[c + d*x]] + I*a*Sin[c + d*x]])/(a^(3/2)*(a^2 - b^2)^(3/4)) + (4*Sqrt[Sin[c + d*x]])/a + (4*b*AppellF1[5/4, 1/2, 1, 9/4, Sin[c + d*x]^2, (a^2*Sin[c + d*x]^2)/(a^2 - b^2)]*Sin[c + d*x]^(5/2))/(5*(a^2 - b^2)) + (10*b*(a^2 - b^2)*AppellF1[1/4, 1/2, 1, 5/4, Sin[c + d*x]^2, (a^2*Sin[c + d*x]^2)/(a^2 - b^2)]*Sqrt[Sin[c + d*x]])/(Sqrt[1 - Sin[c + d*x]^2]*(5*(a^2 - b^2)*AppellF1[1/4, 1/2, 1, 5/4, Sin[c + d*x]^2, (a^2*Sin[c + d*x]^2)/(a^2 - b^2)] + 2*(2*a^2*AppellF1[5/4, 1/2, 2, 9/4, Sin[c + d*x]^2, (a^2*Sin[c + d*x]^2)/(a^2 - b^2)] + (a^2 - b^2)*AppellF1[5/4, 3/2, 1, 9/4, Sin[c + d*x]^2, (a^2*Sin[c + d*x]^2)/(a^2 - b^2)])*Sin[c + d*x]^2)*(b^2 + a^2*(-1 + Sin[c + d*x]^2)))))/((b + a*Cos[c + d*x])*(1 - 2*Sin[c + d*x]^2)*Sqrt[1 - Sin[c + d*x]^2])))/(210*a^5*d*(a + b*Sec[c + d*x])^2*Sin[c + d*x]^(7/2))","C",0
243,1,886,850,15.1498953,"\int \frac{(e \sin (c+d x))^{5/2}}{(a+b \sec (c+d x))^2} \, dx","Integrate[(e*Sin[c + d*x])^(5/2)/(a + b*Sec[c + d*x])^2,x]","\frac{(b+a \cos (c+d x))^2 \csc ^2(c+d x) \sec ^2(c+d x) (e \sin (c+d x))^{5/2} \left(\frac{\sin (c+d x) b^2}{a^3 (b+a \cos (c+d x))}+\frac{4 \sin (c+d x) b}{3 a^3}-\frac{\sin (2 (c+d x))}{5 a^2}\right)}{d (a+b \sec (c+d x))^2}-\frac{(b+a \cos (c+d x))^2 \sec ^2(c+d x) (e \sin (c+d x))^{5/2} \left(\frac{\left(35 b^2-6 a^2\right) \left(8 F_1\left(\frac{3}{4};-\frac{1}{2},1;\frac{7}{4};\sin ^2(c+d x),\frac{a^2 \sin ^2(c+d x)}{a^2-b^2}\right) \sin ^{\frac{3}{2}}(c+d x) a^{5/2}+3 \sqrt{2} b \left(b^2-a^2\right)^{3/4} \left(2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{a} \sqrt{\sin (c+d x)}}{\sqrt[4]{b^2-a^2}}\right)-2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{a} \sqrt{\sin (c+d x)}}{\sqrt[4]{b^2-a^2}}+1\right)-\log \left(a \sin (c+d x)-\sqrt{2} \sqrt{a} \sqrt[4]{b^2-a^2} \sqrt{\sin (c+d x)}+\sqrt{b^2-a^2}\right)+\log \left(a \sin (c+d x)+\sqrt{2} \sqrt{a} \sqrt[4]{b^2-a^2} \sqrt{\sin (c+d x)}+\sqrt{b^2-a^2}\right)\right)\right) \left(\sqrt{1-\sin ^2(c+d x)} a+b\right) \cos ^2(c+d x)}{12 a^{3/2} \left(a^2-b^2\right) (b+a \cos (c+d x)) \left(1-\sin ^2(c+d x)\right)}+\frac{28 a b \left(\frac{b F_1\left(\frac{3}{4};\frac{1}{2},1;\frac{7}{4};\sin ^2(c+d x),\frac{a^2 \sin ^2(c+d x)}{a^2-b^2}\right) \sin ^{\frac{3}{2}}(c+d x)}{3 \left(b^2-a^2\right)}+\frac{\left(\frac{1}{8}+\frac{i}{8}\right) \left(2 \tan ^{-1}\left(1-\frac{(1+i) \sqrt{a} \sqrt{\sin (c+d x)}}{\sqrt[4]{a^2-b^2}}\right)-2 \tan ^{-1}\left(\frac{(1+i) \sqrt{a} \sqrt{\sin (c+d x)}}{\sqrt[4]{a^2-b^2}}+1\right)-\log \left(i a \sin (c+d x)-(1+i) \sqrt{a} \sqrt[4]{a^2-b^2} \sqrt{\sin (c+d x)}+\sqrt{a^2-b^2}\right)+\log \left(i a \sin (c+d x)+(1+i) \sqrt{a} \sqrt[4]{a^2-b^2} \sqrt{\sin (c+d x)}+\sqrt{a^2-b^2}\right)\right)}{\sqrt{a} \sqrt[4]{a^2-b^2}}\right) \left(\sqrt{1-\sin ^2(c+d x)} a+b\right) \cos (c+d x)}{(b+a \cos (c+d x)) \sqrt{1-\sin ^2(c+d x)}}\right)}{10 a^3 d (a+b \sec (c+d x))^2 \sin ^{\frac{5}{2}}(c+d x)}","\frac{3 e^3 \Pi \left(\frac{2 a}{a-\sqrt{a^2-b^2}};\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{\sin (c+d x)} b^4}{2 a^5 \left(a-\sqrt{a^2-b^2}\right) d \sqrt{e \sin (c+d x)}}+\frac{3 e^3 \Pi \left(\frac{2 a}{a+\sqrt{a^2-b^2}};\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{\sin (c+d x)} b^4}{2 a^5 \left(a+\sqrt{a^2-b^2}\right) d \sqrt{e \sin (c+d x)}}-\frac{3 e^{5/2} \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt[4]{a^2-b^2} \sqrt{e}}\right) b^3}{2 a^{9/2} \sqrt[4]{a^2-b^2} d}+\frac{3 e^{5/2} \tanh ^{-1}\left(\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt[4]{a^2-b^2} \sqrt{e}}\right) b^3}{2 a^{9/2} \sqrt[4]{a^2-b^2} d}+\frac{e (e \sin (c+d x))^{3/2} b^2}{a^3 d (b+a \cos (c+d x))}-\frac{7 e^2 E\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{e \sin (c+d x)} b^2}{a^4 d \sqrt{\sin (c+d x)}}-\frac{2 \left(a^2-b^2\right) e^3 \Pi \left(\frac{2 a}{a-\sqrt{a^2-b^2}};\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{\sin (c+d x)} b^2}{a^5 \left(a-\sqrt{a^2-b^2}\right) d \sqrt{e \sin (c+d x)}}-\frac{2 \left(a^2-b^2\right) e^3 \Pi \left(\frac{2 a}{a+\sqrt{a^2-b^2}};\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{\sin (c+d x)} b^2}{a^5 \left(a+\sqrt{a^2-b^2}\right) d \sqrt{e \sin (c+d x)}}+\frac{4 e (e \sin (c+d x))^{3/2} b}{3 a^3 d}+\frac{2 \left(a^2-b^2\right)^{3/4} e^{5/2} \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt[4]{a^2-b^2} \sqrt{e}}\right) b}{a^{9/2} d}-\frac{2 \left(a^2-b^2\right)^{3/4} e^{5/2} \tanh ^{-1}\left(\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt[4]{a^2-b^2} \sqrt{e}}\right) b}{a^{9/2} d}-\frac{2 e \cos (c+d x) (e \sin (c+d x))^{3/2}}{5 a^2 d}+\frac{6 e^2 E\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{e \sin (c+d x)}}{5 a^2 d \sqrt{\sin (c+d x)}}",1,"-1/10*((b + a*Cos[c + d*x])^2*Sec[c + d*x]^2*(e*Sin[c + d*x])^(5/2)*(((-6*a^2 + 35*b^2)*Cos[c + d*x]^2*(3*Sqrt[2]*b*(-a^2 + b^2)^(3/4)*(2*ArcTan[1 - (Sqrt[2]*Sqrt[a]*Sqrt[Sin[c + d*x]])/(-a^2 + b^2)^(1/4)] - 2*ArcTan[1 + (Sqrt[2]*Sqrt[a]*Sqrt[Sin[c + d*x]])/(-a^2 + b^2)^(1/4)] - Log[Sqrt[-a^2 + b^2] - Sqrt[2]*Sqrt[a]*(-a^2 + b^2)^(1/4)*Sqrt[Sin[c + d*x]] + a*Sin[c + d*x]] + Log[Sqrt[-a^2 + b^2] + Sqrt[2]*Sqrt[a]*(-a^2 + b^2)^(1/4)*Sqrt[Sin[c + d*x]] + a*Sin[c + d*x]]) + 8*a^(5/2)*AppellF1[3/4, -1/2, 1, 7/4, Sin[c + d*x]^2, (a^2*Sin[c + d*x]^2)/(a^2 - b^2)]*Sin[c + d*x]^(3/2))*(b + a*Sqrt[1 - Sin[c + d*x]^2]))/(12*a^(3/2)*(a^2 - b^2)*(b + a*Cos[c + d*x])*(1 - Sin[c + d*x]^2)) + (28*a*b*Cos[c + d*x]*(((1/8 + I/8)*(2*ArcTan[1 - ((1 + I)*Sqrt[a]*Sqrt[Sin[c + d*x]])/(a^2 - b^2)^(1/4)] - 2*ArcTan[1 + ((1 + I)*Sqrt[a]*Sqrt[Sin[c + d*x]])/(a^2 - b^2)^(1/4)] - Log[Sqrt[a^2 - b^2] - (1 + I)*Sqrt[a]*(a^2 - b^2)^(1/4)*Sqrt[Sin[c + d*x]] + I*a*Sin[c + d*x]] + Log[Sqrt[a^2 - b^2] + (1 + I)*Sqrt[a]*(a^2 - b^2)^(1/4)*Sqrt[Sin[c + d*x]] + I*a*Sin[c + d*x]]))/(Sqrt[a]*(a^2 - b^2)^(1/4)) + (b*AppellF1[3/4, 1/2, 1, 7/4, Sin[c + d*x]^2, (a^2*Sin[c + d*x]^2)/(a^2 - b^2)]*Sin[c + d*x]^(3/2))/(3*(-a^2 + b^2)))*(b + a*Sqrt[1 - Sin[c + d*x]^2]))/((b + a*Cos[c + d*x])*Sqrt[1 - Sin[c + d*x]^2])))/(a^3*d*(a + b*Sec[c + d*x])^2*Sin[c + d*x]^(5/2)) + ((b + a*Cos[c + d*x])^2*Csc[c + d*x]^2*Sec[c + d*x]^2*(e*Sin[c + d*x])^(5/2)*((4*b*Sin[c + d*x])/(3*a^3) + (b^2*Sin[c + d*x])/(a^3*(b + a*Cos[c + d*x])) - Sin[2*(c + d*x)]/(5*a^2)))/(d*(a + b*Sec[c + d*x])^2)","C",0
244,1,2012,882,16.9658149,"\int \frac{(e \sin (c+d x))^{3/2}}{(a+b \sec (c+d x))^2} \, dx","Integrate[(e*Sin[c + d*x])^(3/2)/(a + b*Sec[c + d*x])^2,x]","\text{Result too large to show}","-\frac{e^2 \Pi \left(\frac{2 a}{a-\sqrt{a^2-b^2}};\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{\sin (c+d x)} b^4}{2 a^4 \left(a^2-\sqrt{a^2-b^2} a-b^2\right) d \sqrt{e \sin (c+d x)}}-\frac{e^2 \Pi \left(\frac{2 a}{a+\sqrt{a^2-b^2}};\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{\sin (c+d x)} b^4}{2 a^4 \left(a^2+\sqrt{a^2-b^2} a-b^2\right) d \sqrt{e \sin (c+d x)}}+\frac{e^{3/2} \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt[4]{a^2-b^2} \sqrt{e}}\right) b^3}{2 a^{7/2} \left(a^2-b^2\right)^{3/4} d}+\frac{e^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt[4]{a^2-b^2} \sqrt{e}}\right) b^3}{2 a^{7/2} \left(a^2-b^2\right)^{3/4} d}+\frac{e \sqrt{e \sin (c+d x)} b^2}{a^3 d (b+a \cos (c+d x))}-\frac{5 e^2 F\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{\sin (c+d x)} b^2}{a^4 d \sqrt{e \sin (c+d x)}}+\frac{2 \left(a^2-b^2\right) e^2 \Pi \left(\frac{2 a}{a-\sqrt{a^2-b^2}};\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{\sin (c+d x)} b^2}{a^4 \left(a^2-\sqrt{a^2-b^2} a-b^2\right) d \sqrt{e \sin (c+d x)}}+\frac{2 \left(a^2-b^2\right) e^2 \Pi \left(\frac{2 a}{a+\sqrt{a^2-b^2}};\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{\sin (c+d x)} b^2}{a^4 \left(a^2+\sqrt{a^2-b^2} a-b^2\right) d \sqrt{e \sin (c+d x)}}-\frac{2 \sqrt[4]{a^2-b^2} e^{3/2} \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt[4]{a^2-b^2} \sqrt{e}}\right) b}{a^{7/2} d}-\frac{2 \sqrt[4]{a^2-b^2} e^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt[4]{a^2-b^2} \sqrt{e}}\right) b}{a^{7/2} d}+\frac{4 e \sqrt{e \sin (c+d x)} b}{a^3 d}-\frac{2 e \cos (c+d x) \sqrt{e \sin (c+d x)}}{3 a^2 d}+\frac{2 e^2 F\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{\sin (c+d x)}}{3 a^2 d \sqrt{e \sin (c+d x)}}",1,"((b + a*Cos[c + d*x])^2*((-2*Cos[c + d*x])/(3*a^2) + b^2/(a^3*(b + a*Cos[c + d*x])))*Csc[c + d*x]*Sec[c + d*x]^2*(e*Sin[c + d*x])^(3/2))/(d*(a + b*Sec[c + d*x])^2) - ((b + a*Cos[c + d*x])^2*Sec[c + d*x]^2*(e*Sin[c + d*x])^(3/2)*((2*(-2*a^2 + 3*b^2)*Cos[c + d*x]^2*(b + a*Sqrt[1 - Sin[c + d*x]^2])*((b*(-2*ArcTan[1 - (Sqrt[2]*Sqrt[a]*Sqrt[Sin[c + d*x]])/(-a^2 + b^2)^(1/4)] + 2*ArcTan[1 + (Sqrt[2]*Sqrt[a]*Sqrt[Sin[c + d*x]])/(-a^2 + b^2)^(1/4)] - Log[Sqrt[-a^2 + b^2] - Sqrt[2]*Sqrt[a]*(-a^2 + b^2)^(1/4)*Sqrt[Sin[c + d*x]] + a*Sin[c + d*x]] + Log[Sqrt[-a^2 + b^2] + Sqrt[2]*Sqrt[a]*(-a^2 + b^2)^(1/4)*Sqrt[Sin[c + d*x]] + a*Sin[c + d*x]]))/(4*Sqrt[2]*Sqrt[a]*(-a^2 + b^2)^(3/4)) - (5*a*(a^2 - b^2)*AppellF1[1/4, -1/2, 1, 5/4, Sin[c + d*x]^2, (a^2*Sin[c + d*x]^2)/(a^2 - b^2)]*Sqrt[Sin[c + d*x]]*Sqrt[1 - Sin[c + d*x]^2])/((5*(a^2 - b^2)*AppellF1[1/4, -1/2, 1, 5/4, Sin[c + d*x]^2, (a^2*Sin[c + d*x]^2)/(a^2 - b^2)] + 2*(2*a^2*AppellF1[5/4, -1/2, 2, 9/4, Sin[c + d*x]^2, (a^2*Sin[c + d*x]^2)/(a^2 - b^2)] + (-a^2 + b^2)*AppellF1[5/4, 1/2, 1, 9/4, Sin[c + d*x]^2, (a^2*Sin[c + d*x]^2)/(a^2 - b^2)])*Sin[c + d*x]^2)*(b^2 + a^2*(-1 + Sin[c + d*x]^2)))))/((b + a*Cos[c + d*x])*(1 - Sin[c + d*x]^2)) + (8*a*b*Cos[c + d*x]*(b + a*Sqrt[1 - Sin[c + d*x]^2])*(((-1/8 + I/8)*Sqrt[a]*(2*ArcTan[1 - ((1 + I)*Sqrt[a]*Sqrt[Sin[c + d*x]])/(a^2 - b^2)^(1/4)] - 2*ArcTan[1 + ((1 + I)*Sqrt[a]*Sqrt[Sin[c + d*x]])/(a^2 - b^2)^(1/4)] + Log[Sqrt[a^2 - b^2] - (1 + I)*Sqrt[a]*(a^2 - b^2)^(1/4)*Sqrt[Sin[c + d*x]] + I*a*Sin[c + d*x]] - Log[Sqrt[a^2 - b^2] + (1 + I)*Sqrt[a]*(a^2 - b^2)^(1/4)*Sqrt[Sin[c + d*x]] + I*a*Sin[c + d*x]]))/(a^2 - b^2)^(3/4) + (5*b*(a^2 - b^2)*AppellF1[1/4, 1/2, 1, 5/4, Sin[c + d*x]^2, (a^2*Sin[c + d*x]^2)/(a^2 - b^2)]*Sqrt[Sin[c + d*x]])/(Sqrt[1 - Sin[c + d*x]^2]*(5*(a^2 - b^2)*AppellF1[1/4, 1/2, 1, 5/4, Sin[c + d*x]^2, (a^2*Sin[c + d*x]^2)/(a^2 - b^2)] + 2*(2*a^2*AppellF1[5/4, 1/2, 2, 9/4, Sin[c + d*x]^2, (a^2*Sin[c + d*x]^2)/(a^2 - b^2)] + (a^2 - b^2)*AppellF1[5/4, 3/2, 1, 9/4, Sin[c + d*x]^2, (a^2*Sin[c + d*x]^2)/(a^2 - b^2)])*Sin[c + d*x]^2)*(b^2 + a^2*(-1 + Sin[c + d*x]^2)))))/((b + a*Cos[c + d*x])*Sqrt[1 - Sin[c + d*x]^2]) - (6*a*b*Cos[c + d*x]*Cos[2*(c + d*x)]*(b + a*Sqrt[1 - Sin[c + d*x]^2])*(((1/2 - I/2)*(a^2 - 2*b^2)*ArcTan[1 - ((1 + I)*Sqrt[a]*Sqrt[Sin[c + d*x]])/(a^2 - b^2)^(1/4)])/(a^(3/2)*(a^2 - b^2)^(3/4)) - ((1/2 - I/2)*(a^2 - 2*b^2)*ArcTan[1 + ((1 + I)*Sqrt[a]*Sqrt[Sin[c + d*x]])/(a^2 - b^2)^(1/4)])/(a^(3/2)*(a^2 - b^2)^(3/4)) + ((1/4 - I/4)*(a^2 - 2*b^2)*Log[Sqrt[a^2 - b^2] - (1 + I)*Sqrt[a]*(a^2 - b^2)^(1/4)*Sqrt[Sin[c + d*x]] + I*a*Sin[c + d*x]])/(a^(3/2)*(a^2 - b^2)^(3/4)) - ((1/4 - I/4)*(a^2 - 2*b^2)*Log[Sqrt[a^2 - b^2] + (1 + I)*Sqrt[a]*(a^2 - b^2)^(1/4)*Sqrt[Sin[c + d*x]] + I*a*Sin[c + d*x]])/(a^(3/2)*(a^2 - b^2)^(3/4)) + (4*Sqrt[Sin[c + d*x]])/a + (4*b*AppellF1[5/4, 1/2, 1, 9/4, Sin[c + d*x]^2, (a^2*Sin[c + d*x]^2)/(a^2 - b^2)]*Sin[c + d*x]^(5/2))/(5*(a^2 - b^2)) + (10*b*(a^2 - b^2)*AppellF1[1/4, 1/2, 1, 5/4, Sin[c + d*x]^2, (a^2*Sin[c + d*x]^2)/(a^2 - b^2)]*Sqrt[Sin[c + d*x]])/(Sqrt[1 - Sin[c + d*x]^2]*(5*(a^2 - b^2)*AppellF1[1/4, 1/2, 1, 5/4, Sin[c + d*x]^2, (a^2*Sin[c + d*x]^2)/(a^2 - b^2)] + 2*(2*a^2*AppellF1[5/4, 1/2, 2, 9/4, Sin[c + d*x]^2, (a^2*Sin[c + d*x]^2)/(a^2 - b^2)] + (a^2 - b^2)*AppellF1[5/4, 3/2, 1, 9/4, Sin[c + d*x]^2, (a^2*Sin[c + d*x]^2)/(a^2 - b^2)])*Sin[c + d*x]^2)*(b^2 + a^2*(-1 + Sin[c + d*x]^2)))))/((b + a*Cos[c + d*x])*(1 - 2*Sin[c + d*x]^2)*Sqrt[1 - Sin[c + d*x]^2])))/(6*a^3*d*(a + b*Sec[c + d*x])^2*Sin[c + d*x]^(3/2))","C",0
245,1,854,809,15.2691605,"\int \frac{\sqrt{e \sin (c+d x)}}{(a+b \sec (c+d x))^2} \, dx","Integrate[Sqrt[e*Sin[c + d*x]]/(a + b*Sec[c + d*x])^2,x]","\frac{(b+a \cos (c+d x)) \sec (c+d x) \sqrt{e \sin (c+d x)} \tan (c+d x) b^2}{a \left(a^2-b^2\right) d (a+b \sec (c+d x))^2}+\frac{(b+a \cos (c+d x))^2 \sec ^2(c+d x) \sqrt{e \sin (c+d x)} \left(\frac{\left(3 b^2-2 a^2\right) \left(8 F_1\left(\frac{3}{4};-\frac{1}{2},1;\frac{7}{4};\sin ^2(c+d x),\frac{a^2 \sin ^2(c+d x)}{a^2-b^2}\right) \sin ^{\frac{3}{2}}(c+d x) a^{5/2}+3 \sqrt{2} b \left(b^2-a^2\right)^{3/4} \left(2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{a} \sqrt{\sin (c+d x)}}{\sqrt[4]{b^2-a^2}}\right)-2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{a} \sqrt{\sin (c+d x)}}{\sqrt[4]{b^2-a^2}}+1\right)-\log \left(a \sin (c+d x)-\sqrt{2} \sqrt{a} \sqrt[4]{b^2-a^2} \sqrt{\sin (c+d x)}+\sqrt{b^2-a^2}\right)+\log \left(a \sin (c+d x)+\sqrt{2} \sqrt{a} \sqrt[4]{b^2-a^2} \sqrt{\sin (c+d x)}+\sqrt{b^2-a^2}\right)\right)\right) \left(\sqrt{1-\sin ^2(c+d x)} a+b\right) \cos ^2(c+d x)}{12 a^{3/2} \left(a^2-b^2\right) (b+a \cos (c+d x)) \left(1-\sin ^2(c+d x)\right)}+\frac{4 a b \left(\frac{b F_1\left(\frac{3}{4};\frac{1}{2},1;\frac{7}{4};\sin ^2(c+d x),\frac{a^2 \sin ^2(c+d x)}{a^2-b^2}\right) \sin ^{\frac{3}{2}}(c+d x)}{3 \left(b^2-a^2\right)}+\frac{\left(\frac{1}{8}+\frac{i}{8}\right) \left(2 \tan ^{-1}\left(1-\frac{(1+i) \sqrt{a} \sqrt{\sin (c+d x)}}{\sqrt[4]{a^2-b^2}}\right)-2 \tan ^{-1}\left(\frac{(1+i) \sqrt{a} \sqrt{\sin (c+d x)}}{\sqrt[4]{a^2-b^2}}+1\right)-\log \left(i a \sin (c+d x)-(1+i) \sqrt{a} \sqrt[4]{a^2-b^2} \sqrt{\sin (c+d x)}+\sqrt{a^2-b^2}\right)+\log \left(i a \sin (c+d x)+(1+i) \sqrt{a} \sqrt[4]{a^2-b^2} \sqrt{\sin (c+d x)}+\sqrt{a^2-b^2}\right)\right)}{\sqrt{a} \sqrt[4]{a^2-b^2}}\right) \left(\sqrt{1-\sin ^2(c+d x)} a+b\right) \cos (c+d x)}{(b+a \cos (c+d x)) \sqrt{1-\sin ^2(c+d x)}}\right)}{2 a (b-a) (a+b) d (a+b \sec (c+d x))^2 \sqrt{\sin (c+d x)}}","-\frac{e \Pi \left(\frac{2 a}{a-\sqrt{a^2-b^2}};\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{\sin (c+d x)} b^4}{2 a^3 \left(a^2-b^2\right) \left(a-\sqrt{a^2-b^2}\right) d \sqrt{e \sin (c+d x)}}-\frac{e \Pi \left(\frac{2 a}{a+\sqrt{a^2-b^2}};\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{\sin (c+d x)} b^4}{2 a^3 \left(a^2-b^2\right) \left(a+\sqrt{a^2-b^2}\right) d \sqrt{e \sin (c+d x)}}+\frac{\sqrt{e} \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt[4]{a^2-b^2} \sqrt{e}}\right) b^3}{2 a^{5/2} \left(a^2-b^2\right)^{5/4} d}-\frac{\sqrt{e} \tanh ^{-1}\left(\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt[4]{a^2-b^2} \sqrt{e}}\right) b^3}{2 a^{5/2} \left(a^2-b^2\right)^{5/4} d}+\frac{(e \sin (c+d x))^{3/2} b^2}{a \left(a^2-b^2\right) d e (b+a \cos (c+d x))}-\frac{E\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{e \sin (c+d x)} b^2}{a^2 \left(a^2-b^2\right) d \sqrt{\sin (c+d x)}}-\frac{2 e \Pi \left(\frac{2 a}{a-\sqrt{a^2-b^2}};\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{\sin (c+d x)} b^2}{a^3 \left(a-\sqrt{a^2-b^2}\right) d \sqrt{e \sin (c+d x)}}-\frac{2 e \Pi \left(\frac{2 a}{a+\sqrt{a^2-b^2}};\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{\sin (c+d x)} b^2}{a^3 \left(a+\sqrt{a^2-b^2}\right) d \sqrt{e \sin (c+d x)}}+\frac{2 \sqrt{e} \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt[4]{a^2-b^2} \sqrt{e}}\right) b}{a^{5/2} \sqrt[4]{a^2-b^2} d}-\frac{2 \sqrt{e} \tanh ^{-1}\left(\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt[4]{a^2-b^2} \sqrt{e}}\right) b}{a^{5/2} \sqrt[4]{a^2-b^2} d}+\frac{2 E\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{e \sin (c+d x)}}{a^2 d \sqrt{\sin (c+d x)}}",1,"((b + a*Cos[c + d*x])^2*Sec[c + d*x]^2*Sqrt[e*Sin[c + d*x]]*(((-2*a^2 + 3*b^2)*Cos[c + d*x]^2*(3*Sqrt[2]*b*(-a^2 + b^2)^(3/4)*(2*ArcTan[1 - (Sqrt[2]*Sqrt[a]*Sqrt[Sin[c + d*x]])/(-a^2 + b^2)^(1/4)] - 2*ArcTan[1 + (Sqrt[2]*Sqrt[a]*Sqrt[Sin[c + d*x]])/(-a^2 + b^2)^(1/4)] - Log[Sqrt[-a^2 + b^2] - Sqrt[2]*Sqrt[a]*(-a^2 + b^2)^(1/4)*Sqrt[Sin[c + d*x]] + a*Sin[c + d*x]] + Log[Sqrt[-a^2 + b^2] + Sqrt[2]*Sqrt[a]*(-a^2 + b^2)^(1/4)*Sqrt[Sin[c + d*x]] + a*Sin[c + d*x]]) + 8*a^(5/2)*AppellF1[3/4, -1/2, 1, 7/4, Sin[c + d*x]^2, (a^2*Sin[c + d*x]^2)/(a^2 - b^2)]*Sin[c + d*x]^(3/2))*(b + a*Sqrt[1 - Sin[c + d*x]^2]))/(12*a^(3/2)*(a^2 - b^2)*(b + a*Cos[c + d*x])*(1 - Sin[c + d*x]^2)) + (4*a*b*Cos[c + d*x]*(((1/8 + I/8)*(2*ArcTan[1 - ((1 + I)*Sqrt[a]*Sqrt[Sin[c + d*x]])/(a^2 - b^2)^(1/4)] - 2*ArcTan[1 + ((1 + I)*Sqrt[a]*Sqrt[Sin[c + d*x]])/(a^2 - b^2)^(1/4)] - Log[Sqrt[a^2 - b^2] - (1 + I)*Sqrt[a]*(a^2 - b^2)^(1/4)*Sqrt[Sin[c + d*x]] + I*a*Sin[c + d*x]] + Log[Sqrt[a^2 - b^2] + (1 + I)*Sqrt[a]*(a^2 - b^2)^(1/4)*Sqrt[Sin[c + d*x]] + I*a*Sin[c + d*x]]))/(Sqrt[a]*(a^2 - b^2)^(1/4)) + (b*AppellF1[3/4, 1/2, 1, 7/4, Sin[c + d*x]^2, (a^2*Sin[c + d*x]^2)/(a^2 - b^2)]*Sin[c + d*x]^(3/2))/(3*(-a^2 + b^2)))*(b + a*Sqrt[1 - Sin[c + d*x]^2]))/((b + a*Cos[c + d*x])*Sqrt[1 - Sin[c + d*x]^2])))/(2*a*(-a + b)*(a + b)*d*(a + b*Sec[c + d*x])^2*Sqrt[Sin[c + d*x]]) + (b^2*(b + a*Cos[c + d*x])*Sec[c + d*x]*Sqrt[e*Sin[c + d*x]]*Tan[c + d*x])/(a*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^2)","C",0
246,1,1246,838,13.3064249,"\int \frac{1}{(a+b \sec (c+d x))^2 \sqrt{e \sin (c+d x)}} \, dx","Integrate[1/((a + b*Sec[c + d*x])^2*Sqrt[e*Sin[c + d*x]]),x]","\frac{(b+a \cos (c+d x)) \sec (c+d x) \tan (c+d x) b^2}{a \left(a^2-b^2\right) d (a+b \sec (c+d x))^2 \sqrt{e \sin (c+d x)}}+\frac{(b+a \cos (c+d x))^2 \sec ^2(c+d x) \sqrt{\sin (c+d x)} \left(\frac{2 \left(b^2-2 a^2\right) \left(\sqrt{1-\sin ^2(c+d x)} a+b\right) \left(\frac{b \left(-2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{a} \sqrt{\sin (c+d x)}}{\sqrt[4]{b^2-a^2}}\right)+2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{a} \sqrt{\sin (c+d x)}}{\sqrt[4]{b^2-a^2}}+1\right)-\log \left(a \sin (c+d x)-\sqrt{2} \sqrt{a} \sqrt[4]{b^2-a^2} \sqrt{\sin (c+d x)}+\sqrt{b^2-a^2}\right)+\log \left(a \sin (c+d x)+\sqrt{2} \sqrt{a} \sqrt[4]{b^2-a^2} \sqrt{\sin (c+d x)}+\sqrt{b^2-a^2}\right)\right)}{4 \sqrt{2} \sqrt{a} \left(b^2-a^2\right)^{3/4}}-\frac{5 a \left(a^2-b^2\right) F_1\left(\frac{1}{4};-\frac{1}{2},1;\frac{5}{4};\sin ^2(c+d x),\frac{a^2 \sin ^2(c+d x)}{a^2-b^2}\right) \sqrt{\sin (c+d x)} \sqrt{1-\sin ^2(c+d x)}}{\left(2 \left(2 F_1\left(\frac{5}{4};-\frac{1}{2},2;\frac{9}{4};\sin ^2(c+d x),\frac{a^2 \sin ^2(c+d x)}{a^2-b^2}\right) a^2+\left(b^2-a^2\right) F_1\left(\frac{5}{4};\frac{1}{2},1;\frac{9}{4};\sin ^2(c+d x),\frac{a^2 \sin ^2(c+d x)}{a^2-b^2}\right)\right) \sin ^2(c+d x)+5 \left(a^2-b^2\right) F_1\left(\frac{1}{4};-\frac{1}{2},1;\frac{5}{4};\sin ^2(c+d x),\frac{a^2 \sin ^2(c+d x)}{a^2-b^2}\right)\right) \left(\left(\sin ^2(c+d x)-1\right) a^2+b^2\right)}\right) \cos ^2(c+d x)}{(b+a \cos (c+d x)) \left(1-\sin ^2(c+d x)\right)}+\frac{4 a b \left(\sqrt{1-\sin ^2(c+d x)} a+b\right) \left(\frac{5 b \left(a^2-b^2\right) F_1\left(\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\sin ^2(c+d x),\frac{a^2 \sin ^2(c+d x)}{a^2-b^2}\right) \sqrt{\sin (c+d x)}}{\sqrt{1-\sin ^2(c+d x)} \left(2 \left(2 F_1\left(\frac{5}{4};\frac{1}{2},2;\frac{9}{4};\sin ^2(c+d x),\frac{a^2 \sin ^2(c+d x)}{a^2-b^2}\right) a^2+\left(a^2-b^2\right) F_1\left(\frac{5}{4};\frac{3}{2},1;\frac{9}{4};\sin ^2(c+d x),\frac{a^2 \sin ^2(c+d x)}{a^2-b^2}\right)\right) \sin ^2(c+d x)+5 \left(a^2-b^2\right) F_1\left(\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\sin ^2(c+d x),\frac{a^2 \sin ^2(c+d x)}{a^2-b^2}\right)\right) \left(\left(\sin ^2(c+d x)-1\right) a^2+b^2\right)}-\frac{\left(\frac{1}{8}-\frac{i}{8}\right) \sqrt{a} \left(2 \tan ^{-1}\left(1-\frac{(1+i) \sqrt{a} \sqrt{\sin (c+d x)}}{\sqrt[4]{a^2-b^2}}\right)-2 \tan ^{-1}\left(\frac{(1+i) \sqrt{a} \sqrt{\sin (c+d x)}}{\sqrt[4]{a^2-b^2}}+1\right)+\log \left(i a \sin (c+d x)-(1+i) \sqrt{a} \sqrt[4]{a^2-b^2} \sqrt{\sin (c+d x)}+\sqrt{a^2-b^2}\right)-\log \left(i a \sin (c+d x)+(1+i) \sqrt{a} \sqrt[4]{a^2-b^2} \sqrt{\sin (c+d x)}+\sqrt{a^2-b^2}\right)\right)}{\left(a^2-b^2\right)^{3/4}}\right) \cos (c+d x)}{(b+a \cos (c+d x)) \sqrt{1-\sin ^2(c+d x)}}\right)}{2 a (b-a) (a+b) d (a+b \sec (c+d x))^2 \sqrt{e \sin (c+d x)}}","\frac{3 \Pi \left(\frac{2 a}{a-\sqrt{a^2-b^2}};\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{\sin (c+d x)} b^4}{2 a^2 \left(a^2-b^2\right) \left(a^2-\sqrt{a^2-b^2} a-b^2\right) d \sqrt{e \sin (c+d x)}}+\frac{3 \Pi \left(\frac{2 a}{a+\sqrt{a^2-b^2}};\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{\sin (c+d x)} b^4}{2 a^2 \left(a^2-b^2\right) \left(a^2+\sqrt{a^2-b^2} a-b^2\right) d \sqrt{e \sin (c+d x)}}-\frac{3 \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt[4]{a^2-b^2} \sqrt{e}}\right) b^3}{2 a^{3/2} \left(a^2-b^2\right)^{7/4} d \sqrt{e}}-\frac{3 \tanh ^{-1}\left(\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt[4]{a^2-b^2} \sqrt{e}}\right) b^3}{2 a^{3/2} \left(a^2-b^2\right)^{7/4} d \sqrt{e}}+\frac{\sqrt{e \sin (c+d x)} b^2}{a \left(a^2-b^2\right) d e (b+a \cos (c+d x))}+\frac{F\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{\sin (c+d x)} b^2}{a^2 \left(a^2-b^2\right) d \sqrt{e \sin (c+d x)}}+\frac{2 \Pi \left(\frac{2 a}{a-\sqrt{a^2-b^2}};\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{\sin (c+d x)} b^2}{a^2 \left(a^2-\sqrt{a^2-b^2} a-b^2\right) d \sqrt{e \sin (c+d x)}}+\frac{2 \Pi \left(\frac{2 a}{a+\sqrt{a^2-b^2}};\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{\sin (c+d x)} b^2}{a^2 \left(a^2+\sqrt{a^2-b^2} a-b^2\right) d \sqrt{e \sin (c+d x)}}-\frac{2 \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt[4]{a^2-b^2} \sqrt{e}}\right) b}{a^{3/2} \left(a^2-b^2\right)^{3/4} d \sqrt{e}}-\frac{2 \tanh ^{-1}\left(\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt[4]{a^2-b^2} \sqrt{e}}\right) b}{a^{3/2} \left(a^2-b^2\right)^{3/4} d \sqrt{e}}+\frac{2 F\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{\sin (c+d x)}}{a^2 d \sqrt{e \sin (c+d x)}}",1,"((b + a*Cos[c + d*x])^2*Sec[c + d*x]^2*Sqrt[Sin[c + d*x]]*((2*(-2*a^2 + b^2)*Cos[c + d*x]^2*(b + a*Sqrt[1 - Sin[c + d*x]^2])*((b*(-2*ArcTan[1 - (Sqrt[2]*Sqrt[a]*Sqrt[Sin[c + d*x]])/(-a^2 + b^2)^(1/4)] + 2*ArcTan[1 + (Sqrt[2]*Sqrt[a]*Sqrt[Sin[c + d*x]])/(-a^2 + b^2)^(1/4)] - Log[Sqrt[-a^2 + b^2] - Sqrt[2]*Sqrt[a]*(-a^2 + b^2)^(1/4)*Sqrt[Sin[c + d*x]] + a*Sin[c + d*x]] + Log[Sqrt[-a^2 + b^2] + Sqrt[2]*Sqrt[a]*(-a^2 + b^2)^(1/4)*Sqrt[Sin[c + d*x]] + a*Sin[c + d*x]]))/(4*Sqrt[2]*Sqrt[a]*(-a^2 + b^2)^(3/4)) - (5*a*(a^2 - b^2)*AppellF1[1/4, -1/2, 1, 5/4, Sin[c + d*x]^2, (a^2*Sin[c + d*x]^2)/(a^2 - b^2)]*Sqrt[Sin[c + d*x]]*Sqrt[1 - Sin[c + d*x]^2])/((5*(a^2 - b^2)*AppellF1[1/4, -1/2, 1, 5/4, Sin[c + d*x]^2, (a^2*Sin[c + d*x]^2)/(a^2 - b^2)] + 2*(2*a^2*AppellF1[5/4, -1/2, 2, 9/4, Sin[c + d*x]^2, (a^2*Sin[c + d*x]^2)/(a^2 - b^2)] + (-a^2 + b^2)*AppellF1[5/4, 1/2, 1, 9/4, Sin[c + d*x]^2, (a^2*Sin[c + d*x]^2)/(a^2 - b^2)])*Sin[c + d*x]^2)*(b^2 + a^2*(-1 + Sin[c + d*x]^2)))))/((b + a*Cos[c + d*x])*(1 - Sin[c + d*x]^2)) + (4*a*b*Cos[c + d*x]*(b + a*Sqrt[1 - Sin[c + d*x]^2])*(((-1/8 + I/8)*Sqrt[a]*(2*ArcTan[1 - ((1 + I)*Sqrt[a]*Sqrt[Sin[c + d*x]])/(a^2 - b^2)^(1/4)] - 2*ArcTan[1 + ((1 + I)*Sqrt[a]*Sqrt[Sin[c + d*x]])/(a^2 - b^2)^(1/4)] + Log[Sqrt[a^2 - b^2] - (1 + I)*Sqrt[a]*(a^2 - b^2)^(1/4)*Sqrt[Sin[c + d*x]] + I*a*Sin[c + d*x]] - Log[Sqrt[a^2 - b^2] + (1 + I)*Sqrt[a]*(a^2 - b^2)^(1/4)*Sqrt[Sin[c + d*x]] + I*a*Sin[c + d*x]]))/(a^2 - b^2)^(3/4) + (5*b*(a^2 - b^2)*AppellF1[1/4, 1/2, 1, 5/4, Sin[c + d*x]^2, (a^2*Sin[c + d*x]^2)/(a^2 - b^2)]*Sqrt[Sin[c + d*x]])/(Sqrt[1 - Sin[c + d*x]^2]*(5*(a^2 - b^2)*AppellF1[1/4, 1/2, 1, 5/4, Sin[c + d*x]^2, (a^2*Sin[c + d*x]^2)/(a^2 - b^2)] + 2*(2*a^2*AppellF1[5/4, 1/2, 2, 9/4, Sin[c + d*x]^2, (a^2*Sin[c + d*x]^2)/(a^2 - b^2)] + (a^2 - b^2)*AppellF1[5/4, 3/2, 1, 9/4, Sin[c + d*x]^2, (a^2*Sin[c + d*x]^2)/(a^2 - b^2)])*Sin[c + d*x]^2)*(b^2 + a^2*(-1 + Sin[c + d*x]^2)))))/((b + a*Cos[c + d*x])*Sqrt[1 - Sin[c + d*x]^2])))/(2*a*(-a + b)*(a + b)*d*(a + b*Sec[c + d*x])^2*Sqrt[e*Sin[c + d*x]]) + (b^2*(b + a*Cos[c + d*x])*Sec[c + d*x]*Tan[c + d*x])/(a*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^2*Sqrt[e*Sin[c + d*x]])","C",0
247,1,922,1054,6.9315581,"\int \frac{1}{(a+b \sec (c+d x))^2 (e \sin (c+d x))^{3/2}} \, dx","Integrate[1/((a + b*Sec[c + d*x])^2*(e*Sin[c + d*x])^(3/2)),x]","\frac{(b+a \cos (c+d x))^2 \left(\frac{a b^2 \sin (c+d x)}{\left(b^2-a^2\right)^2 (b+a \cos (c+d x))}-\frac{2 \left(\cos (c+d x) a^2-2 b a+b^2 \cos (c+d x)\right) \csc (c+d x)}{\left(b^2-a^2\right)^2}\right) \tan ^2(c+d x)}{d (a+b \sec (c+d x))^2 (e \sin (c+d x))^{3/2}}-\frac{(b+a \cos (c+d x))^2 \sec ^2(c+d x) \sin ^{\frac{3}{2}}(c+d x) \left(\frac{\left(2 a^3+3 b^2 a\right) \left(8 F_1\left(\frac{3}{4};-\frac{1}{2},1;\frac{7}{4};\sin ^2(c+d x),\frac{a^2 \sin ^2(c+d x)}{a^2-b^2}\right) \sin ^{\frac{3}{2}}(c+d x) a^{5/2}+3 \sqrt{2} b \left(b^2-a^2\right)^{3/4} \left(2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{a} \sqrt{\sin (c+d x)}}{\sqrt[4]{b^2-a^2}}\right)-2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{a} \sqrt{\sin (c+d x)}}{\sqrt[4]{b^2-a^2}}+1\right)-\log \left(a \sin (c+d x)-\sqrt{2} \sqrt{a} \sqrt[4]{b^2-a^2} \sqrt{\sin (c+d x)}+\sqrt{b^2-a^2}\right)+\log \left(a \sin (c+d x)+\sqrt{2} \sqrt{a} \sqrt[4]{b^2-a^2} \sqrt{\sin (c+d x)}+\sqrt{b^2-a^2}\right)\right)\right) \left(\sqrt{1-\sin ^2(c+d x)} a+b\right) \cos ^2(c+d x)}{12 a^{3/2} \left(a^2-b^2\right) (b+a \cos (c+d x)) \left(1-\sin ^2(c+d x)\right)}+\frac{2 \left(4 b^3+6 a^2 b\right) \left(\frac{b F_1\left(\frac{3}{4};\frac{1}{2},1;\frac{7}{4};\sin ^2(c+d x),\frac{a^2 \sin ^2(c+d x)}{a^2-b^2}\right) \sin ^{\frac{3}{2}}(c+d x)}{3 \left(b^2-a^2\right)}+\frac{\left(\frac{1}{8}+\frac{i}{8}\right) \left(2 \tan ^{-1}\left(1-\frac{(1+i) \sqrt{a} \sqrt{\sin (c+d x)}}{\sqrt[4]{a^2-b^2}}\right)-2 \tan ^{-1}\left(\frac{(1+i) \sqrt{a} \sqrt{\sin (c+d x)}}{\sqrt[4]{a^2-b^2}}+1\right)-\log \left(i a \sin (c+d x)-(1+i) \sqrt{a} \sqrt[4]{a^2-b^2} \sqrt{\sin (c+d x)}+\sqrt{a^2-b^2}\right)+\log \left(i a \sin (c+d x)+(1+i) \sqrt{a} \sqrt[4]{a^2-b^2} \sqrt{\sin (c+d x)}+\sqrt{a^2-b^2}\right)\right)}{\sqrt{a} \sqrt[4]{a^2-b^2}}\right) \left(\sqrt{1-\sin ^2(c+d x)} a+b\right) \cos (c+d x)}{(b+a \cos (c+d x)) \sqrt{1-\sin ^2(c+d x)}}\right)}{2 (a-b)^2 (a+b)^2 d (a+b \sec (c+d x))^2 (e \sin (c+d x))^{3/2}}","-\frac{5 \Pi \left(\frac{2 a}{a-\sqrt{a^2-b^2}};\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{\sin (c+d x)} b^4}{2 a \left(a^2-b^2\right)^2 \left(a-\sqrt{a^2-b^2}\right) d e \sqrt{e \sin (c+d x)}}-\frac{5 \Pi \left(\frac{2 a}{a+\sqrt{a^2-b^2}};\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{\sin (c+d x)} b^4}{2 a \left(a^2-b^2\right)^2 \left(a+\sqrt{a^2-b^2}\right) d e \sqrt{e \sin (c+d x)}}+\frac{5 \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt[4]{a^2-b^2} \sqrt{e}}\right) b^3}{2 \sqrt{a} \left(a^2-b^2\right)^{9/4} d e^{3/2}}-\frac{5 \tanh ^{-1}\left(\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt[4]{a^2-b^2} \sqrt{e}}\right) b^3}{2 \sqrt{a} \left(a^2-b^2\right)^{9/4} d e^{3/2}}-\frac{\left(3 a^2+2 b^2\right) E\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{e \sin (c+d x)} b^2}{a^2 \left(a^2-b^2\right)^2 d e^2 \sqrt{\sin (c+d x)}}-\frac{4 E\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{e \sin (c+d x)} b^2}{a^2 \left(a^2-b^2\right) d e^2 \sqrt{\sin (c+d x)}}+\frac{\left(5 a b-\left(3 a^2+2 b^2\right) \cos (c+d x)\right) b^2}{a^2 \left(a^2-b^2\right)^2 d e \sqrt{e \sin (c+d x)}}-\frac{2 \Pi \left(\frac{2 a}{a-\sqrt{a^2-b^2}};\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{\sin (c+d x)} b^2}{a \left(a^2-b^2\right) \left(a-\sqrt{a^2-b^2}\right) d e \sqrt{e \sin (c+d x)}}-\frac{2 \Pi \left(\frac{2 a}{a+\sqrt{a^2-b^2}};\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{\sin (c+d x)} b^2}{a \left(a^2-b^2\right) \left(a+\sqrt{a^2-b^2}\right) d e \sqrt{e \sin (c+d x)}}+\frac{b^2}{a \left(a^2-b^2\right) d e (b+a \cos (c+d x)) \sqrt{e \sin (c+d x)}}+\frac{2 \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt[4]{a^2-b^2} \sqrt{e}}\right) b}{\sqrt{a} \left(a^2-b^2\right)^{5/4} d e^{3/2}}-\frac{2 \tanh ^{-1}\left(\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt[4]{a^2-b^2} \sqrt{e}}\right) b}{\sqrt{a} \left(a^2-b^2\right)^{5/4} d e^{3/2}}+\frac{4 (a-b \cos (c+d x)) b}{a^2 \left(a^2-b^2\right) d e \sqrt{e \sin (c+d x)}}-\frac{2 E\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{e \sin (c+d x)}}{a^2 d e^2 \sqrt{\sin (c+d x)}}-\frac{2 \cos (c+d x)}{a^2 d e \sqrt{e \sin (c+d x)}}",1,"-1/2*((b + a*Cos[c + d*x])^2*Sec[c + d*x]^2*Sin[c + d*x]^(3/2)*(((2*a^3 + 3*a*b^2)*Cos[c + d*x]^2*(3*Sqrt[2]*b*(-a^2 + b^2)^(3/4)*(2*ArcTan[1 - (Sqrt[2]*Sqrt[a]*Sqrt[Sin[c + d*x]])/(-a^2 + b^2)^(1/4)] - 2*ArcTan[1 + (Sqrt[2]*Sqrt[a]*Sqrt[Sin[c + d*x]])/(-a^2 + b^2)^(1/4)] - Log[Sqrt[-a^2 + b^2] - Sqrt[2]*Sqrt[a]*(-a^2 + b^2)^(1/4)*Sqrt[Sin[c + d*x]] + a*Sin[c + d*x]] + Log[Sqrt[-a^2 + b^2] + Sqrt[2]*Sqrt[a]*(-a^2 + b^2)^(1/4)*Sqrt[Sin[c + d*x]] + a*Sin[c + d*x]]) + 8*a^(5/2)*AppellF1[3/4, -1/2, 1, 7/4, Sin[c + d*x]^2, (a^2*Sin[c + d*x]^2)/(a^2 - b^2)]*Sin[c + d*x]^(3/2))*(b + a*Sqrt[1 - Sin[c + d*x]^2]))/(12*a^(3/2)*(a^2 - b^2)*(b + a*Cos[c + d*x])*(1 - Sin[c + d*x]^2)) + (2*(6*a^2*b + 4*b^3)*Cos[c + d*x]*(((1/8 + I/8)*(2*ArcTan[1 - ((1 + I)*Sqrt[a]*Sqrt[Sin[c + d*x]])/(a^2 - b^2)^(1/4)] - 2*ArcTan[1 + ((1 + I)*Sqrt[a]*Sqrt[Sin[c + d*x]])/(a^2 - b^2)^(1/4)] - Log[Sqrt[a^2 - b^2] - (1 + I)*Sqrt[a]*(a^2 - b^2)^(1/4)*Sqrt[Sin[c + d*x]] + I*a*Sin[c + d*x]] + Log[Sqrt[a^2 - b^2] + (1 + I)*Sqrt[a]*(a^2 - b^2)^(1/4)*Sqrt[Sin[c + d*x]] + I*a*Sin[c + d*x]]))/(Sqrt[a]*(a^2 - b^2)^(1/4)) + (b*AppellF1[3/4, 1/2, 1, 7/4, Sin[c + d*x]^2, (a^2*Sin[c + d*x]^2)/(a^2 - b^2)]*Sin[c + d*x]^(3/2))/(3*(-a^2 + b^2)))*(b + a*Sqrt[1 - Sin[c + d*x]^2]))/((b + a*Cos[c + d*x])*Sqrt[1 - Sin[c + d*x]^2])))/((a - b)^2*(a + b)^2*d*(a + b*Sec[c + d*x])^2*(e*Sin[c + d*x])^(3/2)) + ((b + a*Cos[c + d*x])^2*((-2*(-2*a*b + a^2*Cos[c + d*x] + b^2*Cos[c + d*x])*Csc[c + d*x])/(-a^2 + b^2)^2 + (a*b^2*Sin[c + d*x])/((-a^2 + b^2)^2*(b + a*Cos[c + d*x])))*Tan[c + d*x]^2)/(d*(a + b*Sec[c + d*x])^2*(e*Sin[c + d*x])^(3/2))","C",0
248,1,1320,1089,15.7697417,"\int \frac{1}{(a+b \sec (c+d x))^2 (e \sin (c+d x))^{5/2}} \, dx","Integrate[1/((a + b*Sec[c + d*x])^2*(e*Sin[c + d*x])^(5/2)),x]","\frac{(b+a \cos (c+d x))^2 \left(\frac{a b^2}{\left(b^2-a^2\right)^2 (b+a \cos (c+d x))}-\frac{2 \left(\cos (c+d x) a^2-2 b a+b^2 \cos (c+d x)\right) \csc ^2(c+d x)}{3 \left(b^2-a^2\right)^2}\right) \sin (c+d x) \tan ^2(c+d x)}{d (a+b \sec (c+d x))^2 (e \sin (c+d x))^{5/2}}-\frac{(b+a \cos (c+d x))^2 \sec ^2(c+d x) \sin ^{\frac{5}{2}}(c+d x) \left(\frac{2 \left(-2 a^3-5 b^2 a\right) \left(\sqrt{1-\sin ^2(c+d x)} a+b\right) \left(\frac{b \left(-2 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{a} \sqrt{\sin (c+d x)}}{\sqrt[4]{b^2-a^2}}\right)+2 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{a} \sqrt{\sin (c+d x)}}{\sqrt[4]{b^2-a^2}}+1\right)-\log \left(a \sin (c+d x)-\sqrt{2} \sqrt{a} \sqrt[4]{b^2-a^2} \sqrt{\sin (c+d x)}+\sqrt{b^2-a^2}\right)+\log \left(a \sin (c+d x)+\sqrt{2} \sqrt{a} \sqrt[4]{b^2-a^2} \sqrt{\sin (c+d x)}+\sqrt{b^2-a^2}\right)\right)}{4 \sqrt{2} \sqrt{a} \left(b^2-a^2\right)^{3/4}}-\frac{5 a \left(a^2-b^2\right) F_1\left(\frac{1}{4};-\frac{1}{2},1;\frac{5}{4};\sin ^2(c+d x),\frac{a^2 \sin ^2(c+d x)}{a^2-b^2}\right) \sqrt{\sin (c+d x)} \sqrt{1-\sin ^2(c+d x)}}{\left(2 \left(2 F_1\left(\frac{5}{4};-\frac{1}{2},2;\frac{9}{4};\sin ^2(c+d x),\frac{a^2 \sin ^2(c+d x)}{a^2-b^2}\right) a^2+\left(b^2-a^2\right) F_1\left(\frac{5}{4};\frac{1}{2},1;\frac{9}{4};\sin ^2(c+d x),\frac{a^2 \sin ^2(c+d x)}{a^2-b^2}\right)\right) \sin ^2(c+d x)+5 \left(a^2-b^2\right) F_1\left(\frac{1}{4};-\frac{1}{2},1;\frac{5}{4};\sin ^2(c+d x),\frac{a^2 \sin ^2(c+d x)}{a^2-b^2}\right)\right) \left(\left(\sin ^2(c+d x)-1\right) a^2+b^2\right)}\right) \cos ^2(c+d x)}{(b+a \cos (c+d x)) \left(1-\sin ^2(c+d x)\right)}+\frac{2 \left(4 b^3+10 a^2 b\right) \left(\sqrt{1-\sin ^2(c+d x)} a+b\right) \left(\frac{5 b \left(a^2-b^2\right) F_1\left(\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\sin ^2(c+d x),\frac{a^2 \sin ^2(c+d x)}{a^2-b^2}\right) \sqrt{\sin (c+d x)}}{\sqrt{1-\sin ^2(c+d x)} \left(2 \left(2 F_1\left(\frac{5}{4};\frac{1}{2},2;\frac{9}{4};\sin ^2(c+d x),\frac{a^2 \sin ^2(c+d x)}{a^2-b^2}\right) a^2+\left(a^2-b^2\right) F_1\left(\frac{5}{4};\frac{3}{2},1;\frac{9}{4};\sin ^2(c+d x),\frac{a^2 \sin ^2(c+d x)}{a^2-b^2}\right)\right) \sin ^2(c+d x)+5 \left(a^2-b^2\right) F_1\left(\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\sin ^2(c+d x),\frac{a^2 \sin ^2(c+d x)}{a^2-b^2}\right)\right) \left(\left(\sin ^2(c+d x)-1\right) a^2+b^2\right)}-\frac{\left(\frac{1}{8}-\frac{i}{8}\right) \sqrt{a} \left(2 \tan ^{-1}\left(1-\frac{(1+i) \sqrt{a} \sqrt{\sin (c+d x)}}{\sqrt[4]{a^2-b^2}}\right)-2 \tan ^{-1}\left(\frac{(1+i) \sqrt{a} \sqrt{\sin (c+d x)}}{\sqrt[4]{a^2-b^2}}+1\right)+\log \left(i a \sin (c+d x)-(1+i) \sqrt{a} \sqrt[4]{a^2-b^2} \sqrt{\sin (c+d x)}+\sqrt{a^2-b^2}\right)-\log \left(i a \sin (c+d x)+(1+i) \sqrt{a} \sqrt[4]{a^2-b^2} \sqrt{\sin (c+d x)}+\sqrt{a^2-b^2}\right)\right)}{\left(a^2-b^2\right)^{3/4}}\right) \cos (c+d x)}{(b+a \cos (c+d x)) \sqrt{1-\sin ^2(c+d x)}}\right)}{6 (a-b)^2 (a+b)^2 d (a+b \sec (c+d x))^2 (e \sin (c+d x))^{5/2}}","\frac{7 \Pi \left(\frac{2 a}{a-\sqrt{a^2-b^2}};\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{\sin (c+d x)} b^4}{2 \left(a^2-b^2\right)^2 \left(a^2-\sqrt{a^2-b^2} a-b^2\right) d e^2 \sqrt{e \sin (c+d x)}}+\frac{7 \Pi \left(\frac{2 a}{a+\sqrt{a^2-b^2}};\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{\sin (c+d x)} b^4}{2 \left(a^2-b^2\right)^2 \left(a^2+\sqrt{a^2-b^2} a-b^2\right) d e^2 \sqrt{e \sin (c+d x)}}-\frac{7 \sqrt{a} \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt[4]{a^2-b^2} \sqrt{e}}\right) b^3}{2 \left(a^2-b^2\right)^{11/4} d e^{5/2}}-\frac{7 \sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt[4]{a^2-b^2} \sqrt{e}}\right) b^3}{2 \left(a^2-b^2\right)^{11/4} d e^{5/2}}+\frac{\left(5 a^2+2 b^2\right) F\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{\sin (c+d x)} b^2}{3 a^2 \left(a^2-b^2\right)^2 d e^2 \sqrt{e \sin (c+d x)}}+\frac{4 F\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{\sin (c+d x)} b^2}{3 a^2 \left(a^2-b^2\right) d e^2 \sqrt{e \sin (c+d x)}}+\frac{2 \Pi \left(\frac{2 a}{a-\sqrt{a^2-b^2}};\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{\sin (c+d x)} b^2}{\left(a^2-b^2\right) \left(a^2-\sqrt{a^2-b^2} a-b^2\right) d e^2 \sqrt{e \sin (c+d x)}}+\frac{2 \Pi \left(\frac{2 a}{a+\sqrt{a^2-b^2}};\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{\sin (c+d x)} b^2}{\left(a^2-b^2\right) \left(a^2+\sqrt{a^2-b^2} a-b^2\right) d e^2 \sqrt{e \sin (c+d x)}}+\frac{\left(7 a b-\left(5 a^2+2 b^2\right) \cos (c+d x)\right) b^2}{3 a^2 \left(a^2-b^2\right)^2 d e (e \sin (c+d x))^{3/2}}+\frac{b^2}{a \left(a^2-b^2\right) d e (b+a \cos (c+d x)) (e \sin (c+d x))^{3/2}}-\frac{2 \sqrt{a} \tan ^{-1}\left(\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt[4]{a^2-b^2} \sqrt{e}}\right) b}{\left(a^2-b^2\right)^{7/4} d e^{5/2}}-\frac{2 \sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt[4]{a^2-b^2} \sqrt{e}}\right) b}{\left(a^2-b^2\right)^{7/4} d e^{5/2}}+\frac{4 (a-b \cos (c+d x)) b}{3 a^2 \left(a^2-b^2\right) d e (e \sin (c+d x))^{3/2}}+\frac{2 F\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{\sin (c+d x)}}{3 a^2 d e^2 \sqrt{e \sin (c+d x)}}-\frac{2 \cos (c+d x)}{3 a^2 d e (e \sin (c+d x))^{3/2}}",1,"-1/6*((b + a*Cos[c + d*x])^2*Sec[c + d*x]^2*Sin[c + d*x]^(5/2)*((2*(-2*a^3 - 5*a*b^2)*Cos[c + d*x]^2*(b + a*Sqrt[1 - Sin[c + d*x]^2])*((b*(-2*ArcTan[1 - (Sqrt[2]*Sqrt[a]*Sqrt[Sin[c + d*x]])/(-a^2 + b^2)^(1/4)] + 2*ArcTan[1 + (Sqrt[2]*Sqrt[a]*Sqrt[Sin[c + d*x]])/(-a^2 + b^2)^(1/4)] - Log[Sqrt[-a^2 + b^2] - Sqrt[2]*Sqrt[a]*(-a^2 + b^2)^(1/4)*Sqrt[Sin[c + d*x]] + a*Sin[c + d*x]] + Log[Sqrt[-a^2 + b^2] + Sqrt[2]*Sqrt[a]*(-a^2 + b^2)^(1/4)*Sqrt[Sin[c + d*x]] + a*Sin[c + d*x]]))/(4*Sqrt[2]*Sqrt[a]*(-a^2 + b^2)^(3/4)) - (5*a*(a^2 - b^2)*AppellF1[1/4, -1/2, 1, 5/4, Sin[c + d*x]^2, (a^2*Sin[c + d*x]^2)/(a^2 - b^2)]*Sqrt[Sin[c + d*x]]*Sqrt[1 - Sin[c + d*x]^2])/((5*(a^2 - b^2)*AppellF1[1/4, -1/2, 1, 5/4, Sin[c + d*x]^2, (a^2*Sin[c + d*x]^2)/(a^2 - b^2)] + 2*(2*a^2*AppellF1[5/4, -1/2, 2, 9/4, Sin[c + d*x]^2, (a^2*Sin[c + d*x]^2)/(a^2 - b^2)] + (-a^2 + b^2)*AppellF1[5/4, 1/2, 1, 9/4, Sin[c + d*x]^2, (a^2*Sin[c + d*x]^2)/(a^2 - b^2)])*Sin[c + d*x]^2)*(b^2 + a^2*(-1 + Sin[c + d*x]^2)))))/((b + a*Cos[c + d*x])*(1 - Sin[c + d*x]^2)) + (2*(10*a^2*b + 4*b^3)*Cos[c + d*x]*(b + a*Sqrt[1 - Sin[c + d*x]^2])*(((-1/8 + I/8)*Sqrt[a]*(2*ArcTan[1 - ((1 + I)*Sqrt[a]*Sqrt[Sin[c + d*x]])/(a^2 - b^2)^(1/4)] - 2*ArcTan[1 + ((1 + I)*Sqrt[a]*Sqrt[Sin[c + d*x]])/(a^2 - b^2)^(1/4)] + Log[Sqrt[a^2 - b^2] - (1 + I)*Sqrt[a]*(a^2 - b^2)^(1/4)*Sqrt[Sin[c + d*x]] + I*a*Sin[c + d*x]] - Log[Sqrt[a^2 - b^2] + (1 + I)*Sqrt[a]*(a^2 - b^2)^(1/4)*Sqrt[Sin[c + d*x]] + I*a*Sin[c + d*x]]))/(a^2 - b^2)^(3/4) + (5*b*(a^2 - b^2)*AppellF1[1/4, 1/2, 1, 5/4, Sin[c + d*x]^2, (a^2*Sin[c + d*x]^2)/(a^2 - b^2)]*Sqrt[Sin[c + d*x]])/(Sqrt[1 - Sin[c + d*x]^2]*(5*(a^2 - b^2)*AppellF1[1/4, 1/2, 1, 5/4, Sin[c + d*x]^2, (a^2*Sin[c + d*x]^2)/(a^2 - b^2)] + 2*(2*a^2*AppellF1[5/4, 1/2, 2, 9/4, Sin[c + d*x]^2, (a^2*Sin[c + d*x]^2)/(a^2 - b^2)] + (a^2 - b^2)*AppellF1[5/4, 3/2, 1, 9/4, Sin[c + d*x]^2, (a^2*Sin[c + d*x]^2)/(a^2 - b^2)])*Sin[c + d*x]^2)*(b^2 + a^2*(-1 + Sin[c + d*x]^2)))))/((b + a*Cos[c + d*x])*Sqrt[1 - Sin[c + d*x]^2])))/((a - b)^2*(a + b)^2*d*(a + b*Sec[c + d*x])^2*(e*Sin[c + d*x])^(5/2)) + ((b + a*Cos[c + d*x])^2*((a*b^2)/((-a^2 + b^2)^2*(b + a*Cos[c + d*x])) - (2*(-2*a*b + a^2*Cos[c + d*x] + b^2*Cos[c + d*x])*Csc[c + d*x]^2)/(3*(-a^2 + b^2)^2))*Sin[c + d*x]*Tan[c + d*x]^2)/(d*(a + b*Sec[c + d*x])^2*(e*Sin[c + d*x])^(5/2))","C",0
249,1,151,125,0.2727984,"\int \sqrt{a+b \sec (e+f x)} \, dx","Integrate[Sqrt[a + b*Sec[e + f*x]],x]","\frac{4 \cos ^2\left(\frac{1}{2} (e+f x)\right) \sqrt{\frac{\cos (e+f x)}{\cos (e+f x)+1}} \sqrt{\frac{a \cos (e+f x)+b}{(a+b) (\cos (e+f x)+1)}} \sqrt{a+b \sec (e+f x)} \left((b-a) F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (e+f x)\right)\right)|\frac{a-b}{a+b}\right)+2 a \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (e+f x)\right)\right)|\frac{a-b}{a+b}\right)\right)}{f (a \cos (e+f x)+b)}","-\frac{2 \cot (e+f x) \sqrt{-\frac{b (1-\sec (e+f x))}{a+b \sec (e+f x)}} \sqrt{\frac{b (\sec (e+f x)+1)}{a+b \sec (e+f x)}} (a+b \sec (e+f x)) \Pi \left(\frac{a}{a+b};\sin ^{-1}\left(\frac{\sqrt{a+b}}{\sqrt{a+b \sec (e+f x)}}\right)|\frac{a-b}{a+b}\right)}{f \sqrt{a+b}}",1,"(4*Cos[(e + f*x)/2]^2*Sqrt[Cos[e + f*x]/(1 + Cos[e + f*x])]*Sqrt[(b + a*Cos[e + f*x])/((a + b)*(1 + Cos[e + f*x]))]*((-a + b)*EllipticF[ArcSin[Tan[(e + f*x)/2]], (a - b)/(a + b)] + 2*a*EllipticPi[-1, ArcSin[Tan[(e + f*x)/2]], (a - b)/(a + b)])*Sqrt[a + b*Sec[e + f*x]])/(f*(b + a*Cos[e + f*x]))","A",1
250,1,120,121,1.274397,"\int \csc ^2(e+f x) \sqrt{a+b \sec (e+f x)} \, dx","Integrate[Csc[e + f*x]^2*Sqrt[a + b*Sec[e + f*x]],x]","\frac{b \sqrt{\frac{a+b \sec (e+f x)}{(a+b) (\sec (e+f x)+1)}} F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (e+f x)\right)\right)|\frac{a-b}{a+b}\right)-\csc (e+f x) \sqrt{\frac{1}{\sec (e+f x)+1}} (a \cos (e+f x)+b)}{f \sqrt{\frac{1}{\sec (e+f x)+1}} \sqrt{a+b \sec (e+f x)}}","\frac{\sqrt{a+b} \cot (e+f x) \sqrt{\frac{b (1-\sec (e+f x))}{a+b}} \sqrt{-\frac{b (\sec (e+f x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (e+f x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{f}-\frac{\cot (e+f x) \sqrt{a+b \sec (e+f x)}}{f}",1,"(-((b + a*Cos[e + f*x])*Csc[e + f*x]*Sqrt[(1 + Sec[e + f*x])^(-1)]) + b*EllipticF[ArcSin[Tan[(e + f*x)/2]], (a - b)/(a + b)]*Sqrt[(a + b*Sec[e + f*x])/((a + b)*(1 + Sec[e + f*x]))])/(f*Sqrt[(1 + Sec[e + f*x])^(-1)]*Sqrt[a + b*Sec[e + f*x]])","A",1
251,1,882,309,6.1390246,"\int (a+b \sec (e+f x))^{3/2} \, dx","Integrate[(a + b*Sec[e + f*x])^(3/2),x]","\frac{2 b \cos (e+f x) \sin (e+f x) (a+b \sec (e+f x))^{3/2}}{f (b+a \cos (e+f x))}+\frac{2 \left(-b^2 \sqrt{\frac{b-a}{a+b}} \tan ^5\left(\frac{1}{2} (e+f x)\right)+a b \sqrt{\frac{b-a}{a+b}} \tan ^5\left(\frac{1}{2} (e+f x)\right)-2 a b \sqrt{\frac{b-a}{a+b}} \tan ^3\left(\frac{1}{2} (e+f 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} (e+f x)\right)\right)|\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (e+f x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (e+f x)\right)+b \tan ^2\left(\frac{1}{2} (e+f x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (e+f x)\right)+b^2 \sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (e+f x)\right)+a b \sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (e+f x)\right)-i (a-b) b E\left(i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (e+f x)\right)\right)|\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (e+f x)\right)} \left(\tan ^2\left(\frac{1}{2} (e+f x)\right)+1\right) \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (e+f x)\right)+b \tan ^2\left(\frac{1}{2} (e+f 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} (e+f x)\right)\right)|\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (e+f x)\right)} \left(\tan ^2\left(\frac{1}{2} (e+f x)\right)+1\right) \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (e+f x)\right)+b \tan ^2\left(\frac{1}{2} (e+f 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} (e+f x)\right)\right)|\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (e+f x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (e+f x)\right)+b \tan ^2\left(\frac{1}{2} (e+f x)\right)+a+b}{a+b}}\right) (a+b \sec (e+f x))^{3/2}}{\sqrt{\frac{b-a}{a+b}} f (b+a \cos (e+f x))^{3/2} \sec ^{\frac{3}{2}}(e+f x) \sqrt{\frac{1}{1-\tan ^2\left(\frac{1}{2} (e+f x)\right)}} \left(\tan ^2\left(\frac{1}{2} (e+f x)\right)-1\right) \left(\tan ^2\left(\frac{1}{2} (e+f x)\right)+1\right)^{3/2} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (e+f x)\right)+b \tan ^2\left(\frac{1}{2} (e+f x)\right)+a+b}{\tan ^2\left(\frac{1}{2} (e+f x)\right)+1}}}","\frac{2 (2 a-b) \sqrt{a+b} \cot (e+f x) \sqrt{\frac{b (1-\sec (e+f x))}{a+b}} \sqrt{-\frac{b (\sec (e+f x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (e+f x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{f}-\frac{2 (a-b) \sqrt{a+b} \cot (e+f x) \sqrt{\frac{b (1-\sec (e+f x))}{a+b}} \sqrt{-\frac{b (\sec (e+f x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (e+f x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{f}-\frac{2 a \sqrt{a+b} \cot (e+f x) \sqrt{\frac{b (1-\sec (e+f x))}{a+b}} \sqrt{-\frac{b (\sec (e+f x)+1)}{a-b}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (e+f x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{f}",1,"(2*b*Cos[e + f*x]*(a + b*Sec[e + f*x])^(3/2)*Sin[e + f*x])/(f*(b + a*Cos[e + f*x])) + (2*(a + b*Sec[e + f*x])^(3/2)*(a*b*Sqrt[(-a + b)/(a + b)]*Tan[(e + f*x)/2] + b^2*Sqrt[(-a + b)/(a + b)]*Tan[(e + f*x)/2] - 2*a*b*Sqrt[(-a + b)/(a + b)]*Tan[(e + f*x)/2]^3 + a*b*Sqrt[(-a + b)/(a + b)]*Tan[(e + f*x)/2]^5 - b^2*Sqrt[(-a + b)/(a + b)]*Tan[(e + f*x)/2]^5 + (2*I)*a^2*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(e + f*x)/2]], (a + b)/(a - b)]*Sqrt[1 - Tan[(e + f*x)/2]^2]*Sqrt[(a + b - a*Tan[(e + f*x)/2]^2 + b*Tan[(e + f*x)/2]^2)/(a + b)] + (2*I)*a^2*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(e + f*x)/2]], (a + b)/(a - b)]*Tan[(e + f*x)/2]^2*Sqrt[1 - Tan[(e + f*x)/2]^2]*Sqrt[(a + b - a*Tan[(e + f*x)/2]^2 + b*Tan[(e + f*x)/2]^2)/(a + b)] - I*(a - b)*b*EllipticE[I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(e + f*x)/2]], (a + b)/(a - b)]*Sqrt[1 - Tan[(e + f*x)/2]^2]*(1 + Tan[(e + f*x)/2]^2)*Sqrt[(a + b - a*Tan[(e + f*x)/2]^2 + b*Tan[(e + f*x)/2]^2)/(a + b)] - I*(a - b)^2*EllipticF[I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(e + f*x)/2]], (a + b)/(a - b)]*Sqrt[1 - Tan[(e + f*x)/2]^2]*(1 + Tan[(e + f*x)/2]^2)*Sqrt[(a + b - a*Tan[(e + f*x)/2]^2 + b*Tan[(e + f*x)/2]^2)/(a + b)]))/(Sqrt[(-a + b)/(a + b)]*f*(b + a*Cos[e + f*x])^(3/2)*Sec[e + f*x]^(3/2)*Sqrt[(1 - Tan[(e + f*x)/2]^2)^(-1)]*(-1 + Tan[(e + f*x)/2]^2)*(1 + Tan[(e + f*x)/2]^2)^(3/2)*Sqrt[(a + b - a*Tan[(e + f*x)/2]^2 + b*Tan[(e + f*x)/2]^2)/(1 + Tan[(e + f*x)/2]^2)])","C",1
252,1,276,228,11.2549277,"\int \csc ^2(e+f x) (a+b \sec (e+f x))^{3/2} \, dx","Integrate[Csc[e + f*x]^2*(a + b*Sec[e + f*x])^(3/2),x]","\frac{\cos (e+f x) (a+b \sec (e+f x))^{3/2} (\csc (e+f x) (-a \cos (e+f x)-b)+3 b \sin (e+f x))}{f (a \cos (e+f x)+b)}+\frac{3 b (a+b \sec (e+f x))^{3/2} \left(-\tan \left(\frac{1}{2} (e+f x)\right) (a \cos (e+f x)+b)-\frac{(a+b) \sqrt{\frac{a \cos (e+f x)+b}{(a+b) (\cos (e+f x)+1)}} \left(E\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (e+f x)\right)\right)|\frac{a-b}{a+b}\right)-F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (e+f x)\right)\right)|\frac{a-b}{a+b}\right)\right)}{\sqrt{\frac{\cos (e+f x)}{\cos (e+f x)+1}}}\right)}{f \sqrt{\sec ^2\left(\frac{1}{2} (e+f x)\right)} \sec ^{\frac{3}{2}}(e+f x) \sqrt{\cos ^2\left(\frac{1}{2} (e+f x)\right) \sec (e+f x)} (a \cos (e+f x)+b)^2}","-\frac{\cot (e+f x) (a+b \sec (e+f x))^{3/2}}{f}+\frac{3 (a-b) \sqrt{a+b} \cot (e+f x) \sqrt{\frac{b (1-\sec (e+f x))}{a+b}} \sqrt{-\frac{b (\sec (e+f x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (e+f x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{f}-\frac{3 (a-b) \sqrt{a+b} \cot (e+f x) \sqrt{\frac{b (1-\sec (e+f x))}{a+b}} \sqrt{-\frac{b (\sec (e+f x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (e+f x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{f}",1,"(Cos[e + f*x]*(a + b*Sec[e + f*x])^(3/2)*((-b - a*Cos[e + f*x])*Csc[e + f*x] + 3*b*Sin[e + f*x]))/(f*(b + a*Cos[e + f*x])) + (3*b*(a + b*Sec[e + f*x])^(3/2)*(-(((a + b)*Sqrt[(b + a*Cos[e + f*x])/((a + b)*(1 + Cos[e + f*x]))]*(EllipticE[ArcSin[Tan[(e + f*x)/2]], (a - b)/(a + b)] - EllipticF[ArcSin[Tan[(e + f*x)/2]], (a - b)/(a + b)]))/Sqrt[Cos[e + f*x]/(1 + Cos[e + f*x])]) - (b + a*Cos[e + f*x])*Tan[(e + f*x)/2]))/(f*(b + a*Cos[e + f*x])^2*Sqrt[Sec[(e + f*x)/2]^2]*Sec[e + f*x]^(3/2)*Sqrt[Cos[(e + f*x)/2]^2*Sec[e + f*x]])","A",1
253,1,138,106,0.2219891,"\int \frac{1}{\sqrt{a+b \sec (e+f x)}} \, dx","Integrate[1/Sqrt[a + b*Sec[e + f*x]],x]","-\frac{4 \cos ^2\left(\frac{1}{2} (e+f x)\right) \sqrt{\frac{\cos (e+f x)}{\cos (e+f x)+1}} \sec (e+f x) \sqrt{\frac{a \cos (e+f x)+b}{(a+b) (\cos (e+f x)+1)}} \left(F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (e+f x)\right)\right)|\frac{a-b}{a+b}\right)-2 \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (e+f x)\right)\right)|\frac{a-b}{a+b}\right)\right)}{f \sqrt{a+b \sec (e+f x)}}","-\frac{2 \sqrt{a+b} \cot (e+f x) \sqrt{\frac{b (1-\sec (e+f x))}{a+b}} \sqrt{-\frac{b (\sec (e+f x)+1)}{a-b}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (e+f x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{a f}",1,"(-4*Cos[(e + f*x)/2]^2*Sqrt[Cos[e + f*x]/(1 + Cos[e + f*x])]*Sqrt[(b + a*Cos[e + f*x])/((a + b)*(1 + Cos[e + f*x]))]*(EllipticF[ArcSin[Tan[(e + f*x)/2]], (a - b)/(a + b)] - 2*EllipticPi[-1, ArcSin[Tan[(e + f*x)/2]], (a - b)/(a + b)])*Sec[e + f*x])/(f*Sqrt[a + b*Sec[e + f*x]])","A",1
254,1,259,255,7.7473261,"\int \frac{\csc ^2(e+f x)}{\sqrt{a+b \sec (e+f x)}} \, dx","Integrate[Csc[e + f*x]^2/Sqrt[a + b*Sec[e + f*x]],x]","\frac{\sqrt{\sec (e+f x)} \left(\frac{\csc (e+f x) (a \cos (e+f x)+b) (b \cos (e+f x)-a)}{\left(a^2-b^2\right) \sqrt{\sec (e+f x)}}+\frac{b \left(-\tan \left(\frac{1}{2} (e+f x)\right) (a \cos (e+f x)+b)-\frac{(a+b) \sqrt{\frac{a \cos (e+f x)+b}{(a+b) (\cos (e+f x)+1)}} \left(E\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (e+f x)\right)\right)|\frac{a-b}{a+b}\right)-F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (e+f x)\right)\right)|\frac{a-b}{a+b}\right)\right)}{\sqrt{\frac{\cos (e+f x)}{\cos (e+f x)+1}}}\right)}{\left(b^2-a^2\right) \sqrt{\sec ^2\left(\frac{1}{2} (e+f x)\right)} \sqrt{\cos ^2\left(\frac{1}{2} (e+f x)\right) \sec (e+f x)}}\right)}{f \sqrt{a+b \sec (e+f x)}}","\frac{b^2 \tan (e+f x)}{f \left(a^2-b^2\right) \sqrt{a+b \sec (e+f x)}}-\frac{\cot (e+f x)}{f \sqrt{a+b \sec (e+f x)}}-\frac{\cot (e+f x) \sqrt{\frac{b (1-\sec (e+f x))}{a+b}} \sqrt{-\frac{b (\sec (e+f x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (e+f x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{f \sqrt{a+b}}+\frac{\cot (e+f x) \sqrt{\frac{b (1-\sec (e+f x))}{a+b}} \sqrt{-\frac{b (\sec (e+f x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (e+f x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{f \sqrt{a+b}}",1,"(Sqrt[Sec[e + f*x]]*(((b + a*Cos[e + f*x])*(-a + b*Cos[e + f*x])*Csc[e + f*x])/((a^2 - b^2)*Sqrt[Sec[e + f*x]]) + (b*(-(((a + b)*Sqrt[(b + a*Cos[e + f*x])/((a + b)*(1 + Cos[e + f*x]))]*(EllipticE[ArcSin[Tan[(e + f*x)/2]], (a - b)/(a + b)] - EllipticF[ArcSin[Tan[(e + f*x)/2]], (a - b)/(a + b)]))/Sqrt[Cos[e + f*x]/(1 + Cos[e + f*x])]) - (b + a*Cos[e + f*x])*Tan[(e + f*x)/2]))/((-a^2 + b^2)*Sqrt[Sec[(e + f*x)/2]^2]*Sqrt[Cos[(e + f*x)/2]^2*Sec[e + f*x]])))/(f*Sqrt[a + b*Sec[e + f*x]])","A",0
255,1,1249,347,6.1532658,"\int \frac{1}{(a+b \sec (e+f x))^{3/2}} \, dx","Integrate[(a + b*Sec[e + f*x])^(-3/2),x]","\frac{(b+a \cos (e+f x))^2 \left(\frac{2 \sin (e+f x) b^2}{a \left(a^2-b^2\right) (b+a \cos (e+f x))}+\frac{2 \sin (e+f x) b}{a \left(b^2-a^2\right)}\right) \sec ^2(e+f x)}{f (a+b \sec (e+f x))^{3/2}}+\frac{2 (b+a \cos (e+f x))^{3/2} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (e+f x)\right)+b \tan ^2\left(\frac{1}{2} (e+f x)\right)+a+b}{\tan ^2\left(\frac{1}{2} (e+f x)\right)+1}} \left(-b^2 \sqrt{\frac{b-a}{a+b}} \tan ^5\left(\frac{1}{2} (e+f x)\right)+a b \sqrt{\frac{b-a}{a+b}} \tan ^5\left(\frac{1}{2} (e+f x)\right)-2 a b \sqrt{\frac{b-a}{a+b}} \tan ^3\left(\frac{1}{2} (e+f 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} (e+f x)\right)\right)|\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (e+f x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (e+f x)\right)+b \tan ^2\left(\frac{1}{2} (e+f x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (e+f 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} (e+f x)\right)\right)|\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (e+f x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (e+f x)\right)+b \tan ^2\left(\frac{1}{2} (e+f x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (e+f x)\right)+b^2 \sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (e+f x)\right)+a b \sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (e+f x)\right)-i (a-b) b E\left(i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (e+f x)\right)\right)|\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (e+f x)\right)} \left(\tan ^2\left(\frac{1}{2} (e+f x)\right)+1\right) \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (e+f x)\right)+b \tan ^2\left(\frac{1}{2} (e+f 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} (e+f x)\right)\right)|\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (e+f x)\right)} \left(\tan ^2\left(\frac{1}{2} (e+f x)\right)+1\right) \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (e+f x)\right)+b \tan ^2\left(\frac{1}{2} (e+f 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} (e+f x)\right)\right)|\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (e+f x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (e+f x)\right)+b \tan ^2\left(\frac{1}{2} (e+f 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} (e+f x)\right)\right)|\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (e+f x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (e+f x)\right)+b \tan ^2\left(\frac{1}{2} (e+f x)\right)+a+b}{a+b}}\right) \sec ^{\frac{3}{2}}(e+f x)}{a \sqrt{\frac{b-a}{a+b}} \left(a^2-b^2\right) f (a+b \sec (e+f x))^{3/2} \left(\tan ^2\left(\frac{1}{2} (e+f x)\right)-1\right) \sqrt{\frac{\tan ^2\left(\frac{1}{2} (e+f x)\right)+1}{1-\tan ^2\left(\frac{1}{2} (e+f x)\right)}} \left(a \left(\tan ^2\left(\frac{1}{2} (e+f x)\right)-1\right)-b \left(\tan ^2\left(\frac{1}{2} (e+f x)\right)+1\right)\right)}","\frac{2 b^2 \tan (e+f x)}{a f \left(a^2-b^2\right) \sqrt{a+b \sec (e+f x)}}-\frac{2 \sqrt{a+b} \cot (e+f x) \sqrt{\frac{b (1-\sec (e+f x))}{a+b}} \sqrt{-\frac{b (\sec (e+f x)+1)}{a-b}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (e+f x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{a^2 f}-\frac{2 \cot (e+f x) \sqrt{\frac{b (1-\sec (e+f x))}{a+b}} \sqrt{-\frac{b (\sec (e+f x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (e+f x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{a f \sqrt{a+b}}+\frac{2 \cot (e+f x) \sqrt{\frac{b (1-\sec (e+f x))}{a+b}} \sqrt{-\frac{b (\sec (e+f x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (e+f x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{a f \sqrt{a+b}}",1,"((b + a*Cos[e + f*x])^2*Sec[e + f*x]^2*((2*b*Sin[e + f*x])/(a*(-a^2 + b^2)) + (2*b^2*Sin[e + f*x])/(a*(a^2 - b^2)*(b + a*Cos[e + f*x]))))/(f*(a + b*Sec[e + f*x])^(3/2)) + (2*(b + a*Cos[e + f*x])^(3/2)*Sec[e + f*x]^(3/2)*Sqrt[(a + b - a*Tan[(e + f*x)/2]^2 + b*Tan[(e + f*x)/2]^2)/(1 + Tan[(e + f*x)/2]^2)]*(a*b*Sqrt[(-a + b)/(a + b)]*Tan[(e + f*x)/2] + b^2*Sqrt[(-a + b)/(a + b)]*Tan[(e + f*x)/2] - 2*a*b*Sqrt[(-a + b)/(a + b)]*Tan[(e + f*x)/2]^3 + a*b*Sqrt[(-a + b)/(a + b)]*Tan[(e + f*x)/2]^5 - b^2*Sqrt[(-a + b)/(a + b)]*Tan[(e + f*x)/2]^5 - (2*I)*a^2*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(e + f*x)/2]], (a + b)/(a - b)]*Sqrt[1 - Tan[(e + f*x)/2]^2]*Sqrt[(a + b - a*Tan[(e + f*x)/2]^2 + b*Tan[(e + f*x)/2]^2)/(a + b)] + (2*I)*b^2*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(e + f*x)/2]], (a + b)/(a - b)]*Sqrt[1 - Tan[(e + f*x)/2]^2]*Sqrt[(a + b - a*Tan[(e + f*x)/2]^2 + b*Tan[(e + f*x)/2]^2)/(a + b)] - (2*I)*a^2*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(e + f*x)/2]], (a + b)/(a - b)]*Tan[(e + f*x)/2]^2*Sqrt[1 - Tan[(e + f*x)/2]^2]*Sqrt[(a + b - a*Tan[(e + f*x)/2]^2 + b*Tan[(e + f*x)/2]^2)/(a + b)] + (2*I)*b^2*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(e + f*x)/2]], (a + b)/(a - b)]*Tan[(e + f*x)/2]^2*Sqrt[1 - Tan[(e + f*x)/2]^2]*Sqrt[(a + b - a*Tan[(e + f*x)/2]^2 + b*Tan[(e + f*x)/2]^2)/(a + b)] - I*(a - b)*b*EllipticE[I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(e + f*x)/2]], (a + b)/(a - b)]*Sqrt[1 - Tan[(e + f*x)/2]^2]*(1 + Tan[(e + f*x)/2]^2)*Sqrt[(a + b - a*Tan[(e + f*x)/2]^2 + b*Tan[(e + f*x)/2]^2)/(a + b)] + I*(a^2 + a*b - 2*b^2)*EllipticF[I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(e + f*x)/2]], (a + b)/(a - b)]*Sqrt[1 - Tan[(e + f*x)/2]^2]*(1 + Tan[(e + f*x)/2]^2)*Sqrt[(a + b - a*Tan[(e + f*x)/2]^2 + b*Tan[(e + f*x)/2]^2)/(a + b)]))/(a*Sqrt[(-a + b)/(a + b)]*(a^2 - b^2)*f*(a + b*Sec[e + f*x])^(3/2)*(-1 + Tan[(e + f*x)/2]^2)*Sqrt[(1 + Tan[(e + f*x)/2]^2)/(1 - Tan[(e + f*x)/2]^2)]*(a*(-1 + Tan[(e + f*x)/2]^2) - b*(1 + Tan[(e + f*x)/2]^2)))","C",0
256,1,259,318,8.4889352,"\int \frac{\csc ^2(e+f x)}{(a+b \sec (e+f x))^{3/2}} \, dx","Integrate[Csc[e + f*x]^2/(a + b*Sec[e + f*x])^(3/2),x]","\frac{-2 b \left(3 a^2+4 a b+b^2\right) \cos ^2\left(\frac{1}{2} (e+f x)\right) \sec (e+f x) \sqrt{\frac{1}{\sec (e+f x)+1}} \sqrt{\frac{a \cos (e+f x)+b}{(a+b) (\cos (e+f x)+1)}} F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (e+f x)\right)\right)|\frac{a-b}{a+b}\right)-(a-b) \csc (e+f x) (a (a-3 b) \cos (e+f x)+b (3 a-b))+8 a b (a+b) \cos ^2\left(\frac{1}{2} (e+f x)\right) \sec (e+f x) \sqrt{\frac{1}{\sec (e+f x)+1}} \sqrt{\frac{a \cos (e+f x)+b}{(a+b) (\cos (e+f x)+1)}} E\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (e+f x)\right)\right)|\frac{a-b}{a+b}\right)}{f \left(a^2-b^2\right)^2 \sqrt{a+b \sec (e+f x)}}","\frac{4 a b^2 \tan (e+f x)}{f \left(a^2-b^2\right)^2 \sqrt{a+b \sec (e+f x)}}+\frac{b^2 \tan (e+f x)}{f \left(a^2-b^2\right) (a+b \sec (e+f x))^{3/2}}-\frac{\cot (e+f x)}{f (a+b \sec (e+f x))^{3/2}}-\frac{(3 a-b) \cot (e+f x) \sqrt{\frac{b (1-\sec (e+f x))}{a+b}} \sqrt{-\frac{b (\sec (e+f x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (e+f x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{f (a-b) (a+b)^{3/2}}+\frac{4 a \cot (e+f x) \sqrt{\frac{b (1-\sec (e+f x))}{a+b}} \sqrt{-\frac{b (\sec (e+f x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (e+f x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{f (a-b) (a+b)^{3/2}}",1,"(-((a - b)*((3*a - b)*b + a*(a - 3*b)*Cos[e + f*x])*Csc[e + f*x]) + 8*a*b*(a + b)*Cos[(e + f*x)/2]^2*Sqrt[(b + a*Cos[e + f*x])/((a + b)*(1 + Cos[e + f*x]))]*EllipticE[ArcSin[Tan[(e + f*x)/2]], (a - b)/(a + b)]*Sec[e + f*x]*Sqrt[(1 + Sec[e + f*x])^(-1)] - 2*b*(3*a^2 + 4*a*b + b^2)*Cos[(e + f*x)/2]^2*Sqrt[(b + a*Cos[e + f*x])/((a + b)*(1 + Cos[e + f*x]))]*EllipticF[ArcSin[Tan[(e + f*x)/2]], (a - b)/(a + b)]*Sec[e + f*x]*Sqrt[(1 + Sec[e + f*x])^(-1)])/((a^2 - b^2)^2*f*Sqrt[a + b*Sec[e + f*x]])","A",1
257,1,182,249,0.3357137,"\int (a+b \sec (c+d x))^3 (e \sin (c+d x))^m \, dx","Integrate[(a + b*Sec[c + d*x])^3*(e*Sin[c + d*x])^m,x]","\frac{\tan (c+d x) (e \sin (c+d x))^m \left(a^3 \sqrt{\cos ^2(c+d x)} \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};\sin ^2(c+d x)\right)+b \left(3 a^2 \cos (c+d x) \, _2F_1\left(1,\frac{m+1}{2};\frac{m+3}{2};\sin ^2(c+d x)\right)+b \left(3 a \sqrt{\cos ^2(c+d x)} \, _2F_1\left(\frac{3}{2},\frac{m+1}{2};\frac{m+3}{2};\sin ^2(c+d x)\right)+b \cos (c+d x) \, _2F_1\left(2,\frac{m+1}{2};\frac{m+3}{2};\sin ^2(c+d x)\right)\right)\right)\right)}{d (m+1)}","\frac{a^3 \cos (c+d x) (e \sin (c+d x))^{m+1} \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};\sin ^2(c+d x)\right)}{d e (m+1) \sqrt{\cos ^2(c+d x)}}+\frac{3 a^2 b (e \sin (c+d x))^{m+1} \, _2F_1\left(1,\frac{m+1}{2};\frac{m+3}{2};\sin ^2(c+d x)\right)}{d e (m+1)}+\frac{3 a b^2 \sqrt{\cos ^2(c+d x)} \sec (c+d x) (e \sin (c+d x))^{m+1} \, _2F_1\left(\frac{3}{2},\frac{m+1}{2};\frac{m+3}{2};\sin ^2(c+d x)\right)}{d e (m+1)}+\frac{b^3 (e \sin (c+d x))^{m+1} \, _2F_1\left(2,\frac{m+1}{2};\frac{m+3}{2};\sin ^2(c+d x)\right)}{d e (m+1)}",1,"((a^3*Sqrt[Cos[c + d*x]^2]*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, Sin[c + d*x]^2] + b*(3*a^2*Cos[c + d*x]*Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, Sin[c + d*x]^2] + b*(3*a*Sqrt[Cos[c + d*x]^2]*Hypergeometric2F1[3/2, (1 + m)/2, (3 + m)/2, Sin[c + d*x]^2] + b*Cos[c + d*x]*Hypergeometric2F1[2, (1 + m)/2, (3 + m)/2, Sin[c + d*x]^2])))*(e*Sin[c + d*x])^m*Tan[c + d*x])/(d*(1 + m))","A",1
258,1,134,190,0.2957473,"\int (a+b \sec (c+d x))^2 (e \sin (c+d x))^m \, dx","Integrate[(a + b*Sec[c + d*x])^2*(e*Sin[c + d*x])^m,x]","\frac{(e \sin (c+d x))^m \left(\sqrt{\cos ^2(c+d x)} \tan (c+d x) \left(a^2 \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};\sin ^2(c+d x)\right)+b^2 \, _2F_1\left(\frac{3}{2},\frac{m+1}{2};\frac{m+3}{2};\sin ^2(c+d x)\right)\right)+2 a b \sin (c+d x) \, _2F_1\left(1,\frac{m+1}{2};\frac{m+3}{2};\sin ^2(c+d x)\right)\right)}{d (m+1)}","\frac{a^2 \sin (c+d x) \cos (c+d x) (e \sin (c+d x))^m \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};\sin ^2(c+d x)\right)}{d (m+1) \sqrt{\cos ^2(c+d x)}}+\frac{2 a b (e \sin (c+d x))^{m+1} \, _2F_1\left(1,\frac{m+1}{2};\frac{m+3}{2};\sin ^2(c+d x)\right)}{d e (m+1)}+\frac{b^2 \sqrt{\cos ^2(c+d x)} \tan (c+d x) (e \sin (c+d x))^m \, _2F_1\left(\frac{3}{2},\frac{m+1}{2};\frac{m+3}{2};\sin ^2(c+d x)\right)}{d (m+1)}",1,"((e*Sin[c + d*x])^m*(2*a*b*Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, Sin[c + d*x]^2]*Sin[c + d*x] + Sqrt[Cos[c + d*x]^2]*(a^2*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, Sin[c + d*x]^2] + b^2*Hypergeometric2F1[3/2, (1 + m)/2, (3 + m)/2, Sin[c + d*x]^2])*Tan[c + d*x]))/(d*(1 + m))","A",1
259,1,98,119,0.1107804,"\int (a+b \sec (c+d x)) (e \sin (c+d x))^m \, dx","Integrate[(a + b*Sec[c + d*x])*(e*Sin[c + d*x])^m,x]","\frac{\tan (c+d x) (e \sin (c+d x))^m \left(a \sqrt{\cos ^2(c+d x)} \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};\sin ^2(c+d x)\right)+b \cos (c+d x) \, _2F_1\left(1,\frac{m+1}{2};\frac{m+3}{2};\sin ^2(c+d x)\right)\right)}{d (m+1)}","\frac{a \cos (c+d x) (e \sin (c+d x))^{m+1} \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};\sin ^2(c+d x)\right)}{d e (m+1) \sqrt{\cos ^2(c+d x)}}+\frac{b (e \sin (c+d x))^{m+1} \, _2F_1\left(1,\frac{m+1}{2};\frac{m+3}{2};\sin ^2(c+d x)\right)}{d e (m+1)}",1,"((a*Sqrt[Cos[c + d*x]^2]*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, Sin[c + d*x]^2] + b*Cos[c + d*x]*Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, Sin[c + d*x]^2])*(e*Sin[c + d*x])^m*Tan[c + d*x])/(d*(1 + m))","A",1
260,1,687,232,5.9669677,"\int \frac{(e \sin (c+d x))^m}{a+b \sec (c+d x)} \, dx","Integrate[(e*Sin[c + d*x])^m/(a + b*Sec[c + d*x]),x]","\frac{2 \tan \left(\frac{1}{2} (c+d x)\right) (e \sin (c+d x))^m \left((a+b) \, _2F_1\left(\frac{m+1}{2},m+1;\frac{m+3}{2};-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right)-b F_1\left(\frac{m+1}{2};m,1;\frac{m+3}{2};-\tan ^2\left(\frac{1}{2} (c+d x)\right),\frac{(a-b) \tan ^2\left(\frac{1}{2} (c+d x)\right)}{a+b}\right)\right)}{d (a+b \sec (c+d x)) \left(2 m \tan ^2\left(\frac{1}{2} (c+d x)\right) \left((a+b) \, _2F_1\left(\frac{m+1}{2},m+1;\frac{m+3}{2};-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right)-b F_1\left(\frac{m+1}{2};m,1;\frac{m+3}{2};-\tan ^2\left(\frac{1}{2} (c+d x)\right),\frac{(a-b) \tan ^2\left(\frac{1}{2} (c+d x)\right)}{a+b}\right)\right)+2 m \tan \left(\frac{1}{2} (c+d x)\right) \cot (c+d x) \left((a+b) \, _2F_1\left(\frac{m+1}{2},m+1;\frac{m+3}{2};-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right)-b F_1\left(\frac{m+1}{2};m,1;\frac{m+3}{2};-\tan ^2\left(\frac{1}{2} (c+d x)\right),\frac{(a-b) \tan ^2\left(\frac{1}{2} (c+d x)\right)}{a+b}\right)\right)+\sec ^2\left(\frac{1}{2} (c+d x)\right) \left((a+b) \, _2F_1\left(\frac{m+1}{2},m+1;\frac{m+3}{2};-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right)-b F_1\left(\frac{m+1}{2};m,1;\frac{m+3}{2};-\tan ^2\left(\frac{1}{2} (c+d x)\right),\frac{(a-b) \tan ^2\left(\frac{1}{2} (c+d x)\right)}{a+b}\right)\right)+\frac{(m+1) \sec ^2\left(\frac{1}{2} (c+d x)\right) \left(\frac{2 b \tan ^2\left(\frac{1}{2} (c+d x)\right) \left((b-a) F_1\left(\frac{m+3}{2};m,2;\frac{m+5}{2};-\tan ^2\left(\frac{1}{2} (c+d x)\right),\frac{(a-b) \tan ^2\left(\frac{1}{2} (c+d x)\right)}{a+b}\right)+m (a+b) F_1\left(\frac{m+3}{2};m+1,1;\frac{m+5}{2};-\tan ^2\left(\frac{1}{2} (c+d x)\right),\frac{(a-b) \tan ^2\left(\frac{1}{2} (c+d x)\right)}{a+b}\right)\right)}{m+3}-(a+b)^2 \left(\, _2F_1\left(\frac{m+1}{2},m+1;\frac{m+3}{2};-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right)-\sec ^2\left(\frac{1}{2} (c+d x)\right)^{-m-1}\right)\right)}{a+b}\right)}","\frac{\cos (c+d x) (e \sin (c+d x))^{m+1} \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};\sin ^2(c+d x)\right)}{a d e (m+1) \sqrt{\cos ^2(c+d x)}}-\frac{b e (e \sin (c+d x))^{m-1} \left(-\frac{a (1-\cos (c+d x))}{a \cos (c+d x)+b}\right)^{\frac{1-m}{2}} \left(\frac{a (\cos (c+d x)+1)}{a \cos (c+d x)+b}\right)^{\frac{1-m}{2}} F_1\left(1-m;\frac{1-m}{2},\frac{1-m}{2};2-m;-\frac{a-b}{b+a \cos (c+d x)},\frac{a+b}{b+a \cos (c+d x)}\right)}{a^2 d (1-m)}",1,"(2*(-(b*AppellF1[(1 + m)/2, m, 1, (3 + m)/2, -Tan[(c + d*x)/2]^2, ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)]) + (a + b)*Hypergeometric2F1[(1 + m)/2, 1 + m, (3 + m)/2, -Tan[(c + d*x)/2]^2])*(e*Sin[c + d*x])^m*Tan[(c + d*x)/2])/(d*(a + b*Sec[c + d*x])*((-(b*AppellF1[(1 + m)/2, m, 1, (3 + m)/2, -Tan[(c + d*x)/2]^2, ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)]) + (a + b)*Hypergeometric2F1[(1 + m)/2, 1 + m, (3 + m)/2, -Tan[(c + d*x)/2]^2])*Sec[(c + d*x)/2]^2 + 2*m*Cot[c + d*x]*(-(b*AppellF1[(1 + m)/2, m, 1, (3 + m)/2, -Tan[(c + d*x)/2]^2, ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)]) + (a + b)*Hypergeometric2F1[(1 + m)/2, 1 + m, (3 + m)/2, -Tan[(c + d*x)/2]^2])*Tan[(c + d*x)/2] + 2*m*(-(b*AppellF1[(1 + m)/2, m, 1, (3 + m)/2, -Tan[(c + d*x)/2]^2, ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)]) + (a + b)*Hypergeometric2F1[(1 + m)/2, 1 + m, (3 + m)/2, -Tan[(c + d*x)/2]^2])*Tan[(c + d*x)/2]^2 + ((1 + m)*Sec[(c + d*x)/2]^2*(-((a + b)^2*(Hypergeometric2F1[(1 + m)/2, 1 + m, (3 + m)/2, -Tan[(c + d*x)/2]^2] - (Sec[(c + d*x)/2]^2)^(-1 - m))) + (2*b*((-a + b)*AppellF1[(3 + m)/2, m, 2, (5 + m)/2, -Tan[(c + d*x)/2]^2, ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)] + (a + b)*m*AppellF1[(3 + m)/2, 1 + m, 1, (5 + m)/2, -Tan[(c + d*x)/2]^2, ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)])*Tan[(c + d*x)/2]^2)/(3 + m)))/(a + b)))","B",0
261,1,1433,405,11.7475235,"\int \frac{(e \sin (c+d x))^m}{(a+b \sec (c+d x))^2} \, dx","Integrate[(e*Sin[c + d*x])^m/(a + b*Sec[c + d*x])^2,x]","-\frac{(b+a \cos (c+d x))^2 \, _2F_1\left(\frac{1}{2},\frac{1-m}{2};\frac{3}{2};\cos ^2(c+d x)\right) (e \sin (c+d x))^m \tan (c+d x) \sin ^2(c+d x)^{\frac{1}{2} (-m-1)}}{a^2 d (a+b \sec (c+d x))^2}-\frac{4 b F_1\left(\frac{m+1}{2};m,1;\frac{m+3}{2};-\tan ^2\left(\frac{1}{2} (c+d x)\right),\frac{(a-b) \tan ^2\left(\frac{1}{2} (c+d x)\right)}{a+b}\right) (b+a \cos (c+d x)) \sec ^2(c+d x) (e \sin (c+d x))^m \tan \left(\frac{1}{2} (c+d x)\right)}{a^2 d (a+b \sec (c+d x))^2 \left(-\frac{2 (m+1) \left((b-a) F_1\left(\frac{m+3}{2};m,2;\frac{m+5}{2};-\tan ^2\left(\frac{1}{2} (c+d x)\right),\frac{(a-b) \tan ^2\left(\frac{1}{2} (c+d x)\right)}{a+b}\right)+(a+b) m F_1\left(\frac{m+3}{2};m+1,1;\frac{m+5}{2};-\tan ^2\left(\frac{1}{2} (c+d x)\right),\frac{(a-b) \tan ^2\left(\frac{1}{2} (c+d x)\right)}{a+b}\right)\right) \tan ^2\left(\frac{1}{2} (c+d x)\right) \sec ^2\left(\frac{1}{2} (c+d x)\right)}{(a+b) (m+3)}+F_1\left(\frac{m+1}{2};m,1;\frac{m+3}{2};-\tan ^2\left(\frac{1}{2} (c+d x)\right),\frac{(a-b) \tan ^2\left(\frac{1}{2} (c+d x)\right)}{a+b}\right) \sec ^2\left(\frac{1}{2} (c+d x)\right)+2 m F_1\left(\frac{m+1}{2};m,1;\frac{m+3}{2};-\tan ^2\left(\frac{1}{2} (c+d x)\right),\frac{(a-b) \tan ^2\left(\frac{1}{2} (c+d x)\right)}{a+b}\right) \tan ^2\left(\frac{1}{2} (c+d x)\right)+2 m F_1\left(\frac{m+1}{2};m,1;\frac{m+3}{2};-\tan ^2\left(\frac{1}{2} (c+d x)\right),\frac{(a-b) \tan ^2\left(\frac{1}{2} (c+d x)\right)}{a+b}\right) \cot (c+d x) \tan \left(\frac{1}{2} (c+d x)\right)\right)}+\frac{2 b^2 \left((a+b) F_1\left(\frac{m+1}{2};m,1;\frac{m+3}{2};-\tan ^2\left(\frac{1}{2} (c+d x)\right),\frac{(a-b) \tan ^2\left(\frac{1}{2} (c+d x)\right)}{a+b}\right)-2 a F_1\left(\frac{m+1}{2};m,2;\frac{m+3}{2};-\tan ^2\left(\frac{1}{2} (c+d x)\right),\frac{(a-b) \tan ^2\left(\frac{1}{2} (c+d x)\right)}{a+b}\right)\right) \sec ^2(c+d x) (e \sin (c+d x))^m \tan \left(\frac{1}{2} (c+d x)\right)}{a^2 d (a+b \sec (c+d x))^2 \left(-\frac{2 (m+1) \left(\left(b^2-a^2\right) F_1\left(\frac{m+3}{2};m,2;\frac{m+5}{2};-\tan ^2\left(\frac{1}{2} (c+d x)\right),\frac{(a-b) \tan ^2\left(\frac{1}{2} (c+d x)\right)}{a+b}\right)+4 a (a-b) F_1\left(\frac{m+3}{2};m,3;\frac{m+5}{2};-\tan ^2\left(\frac{1}{2} (c+d x)\right),\frac{(a-b) \tan ^2\left(\frac{1}{2} (c+d x)\right)}{a+b}\right)+(a+b) m \left((a+b) F_1\left(\frac{m+3}{2};m+1,1;\frac{m+5}{2};-\tan ^2\left(\frac{1}{2} (c+d x)\right),\frac{(a-b) \tan ^2\left(\frac{1}{2} (c+d x)\right)}{a+b}\right)-2 a F_1\left(\frac{m+3}{2};m+1,2;\frac{m+5}{2};-\tan ^2\left(\frac{1}{2} (c+d x)\right),\frac{(a-b) \tan ^2\left(\frac{1}{2} (c+d x)\right)}{a+b}\right)\right)\right) \tan ^2\left(\frac{1}{2} (c+d x)\right) \sec ^2\left(\frac{1}{2} (c+d x)\right)}{(a+b) (m+3)}+\left((a+b) F_1\left(\frac{m+1}{2};m,1;\frac{m+3}{2};-\tan ^2\left(\frac{1}{2} (c+d x)\right),\frac{(a-b) \tan ^2\left(\frac{1}{2} (c+d x)\right)}{a+b}\right)-2 a F_1\left(\frac{m+1}{2};m,2;\frac{m+3}{2};-\tan ^2\left(\frac{1}{2} (c+d x)\right),\frac{(a-b) \tan ^2\left(\frac{1}{2} (c+d x)\right)}{a+b}\right)\right) \sec ^2\left(\frac{1}{2} (c+d x)\right)+2 m \left((a+b) F_1\left(\frac{m+1}{2};m,1;\frac{m+3}{2};-\tan ^2\left(\frac{1}{2} (c+d x)\right),\frac{(a-b) \tan ^2\left(\frac{1}{2} (c+d x)\right)}{a+b}\right)-2 a F_1\left(\frac{m+1}{2};m,2;\frac{m+3}{2};-\tan ^2\left(\frac{1}{2} (c+d x)\right),\frac{(a-b) \tan ^2\left(\frac{1}{2} (c+d x)\right)}{a+b}\right)\right) \tan ^2\left(\frac{1}{2} (c+d x)\right)+2 m \left((a+b) F_1\left(\frac{m+1}{2};m,1;\frac{m+3}{2};-\tan ^2\left(\frac{1}{2} (c+d x)\right),\frac{(a-b) \tan ^2\left(\frac{1}{2} (c+d x)\right)}{a+b}\right)-2 a F_1\left(\frac{m+1}{2};m,2;\frac{m+3}{2};-\tan ^2\left(\frac{1}{2} (c+d x)\right),\frac{(a-b) \tan ^2\left(\frac{1}{2} (c+d x)\right)}{a+b}\right)\right) \cot (c+d x) \tan \left(\frac{1}{2} (c+d x)\right)\right)}","\frac{b^2 e (e \sin (c+d x))^{m-1} \left(-\frac{a (1-\cos (c+d x))}{a \cos (c+d x)+b}\right)^{\frac{1-m}{2}} \left(\frac{a (\cos (c+d x)+1)}{a \cos (c+d x)+b}\right)^{\frac{1-m}{2}} F_1\left(2-m;\frac{1-m}{2},\frac{1-m}{2};3-m;-\frac{a-b}{b+a \cos (c+d x)},\frac{a+b}{b+a \cos (c+d x)}\right)}{a^3 d (2-m) (a \cos (c+d x)+b)}-\frac{2 b e (e \sin (c+d x))^{m-1} \left(-\frac{a (1-\cos (c+d x))}{a \cos (c+d x)+b}\right)^{\frac{1-m}{2}} \left(\frac{a (\cos (c+d x)+1)}{a \cos (c+d x)+b}\right)^{\frac{1-m}{2}} F_1\left(1-m;\frac{1-m}{2},\frac{1-m}{2};2-m;-\frac{a-b}{b+a \cos (c+d x)},\frac{a+b}{b+a \cos (c+d x)}\right)}{a^3 d (1-m)}+\frac{\cos (c+d x) (e \sin (c+d x))^{m+1} \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};\sin ^2(c+d x)\right)}{a^2 d e (m+1) \sqrt{\cos ^2(c+d x)}}",1,"(-4*b*AppellF1[(1 + m)/2, m, 1, (3 + m)/2, -Tan[(c + d*x)/2]^2, ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)]*(b + a*Cos[c + d*x])*Sec[c + d*x]^2*(e*Sin[c + d*x])^m*Tan[(c + d*x)/2])/(a^2*d*(a + b*Sec[c + d*x])^2*(AppellF1[(1 + m)/2, m, 1, (3 + m)/2, -Tan[(c + d*x)/2]^2, ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)]*Sec[(c + d*x)/2]^2 + 2*m*AppellF1[(1 + m)/2, m, 1, (3 + m)/2, -Tan[(c + d*x)/2]^2, ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)]*Cot[c + d*x]*Tan[(c + d*x)/2] + 2*m*AppellF1[(1 + m)/2, m, 1, (3 + m)/2, -Tan[(c + d*x)/2]^2, ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)]*Tan[(c + d*x)/2]^2 - (2*(1 + m)*((-a + b)*AppellF1[(3 + m)/2, m, 2, (5 + m)/2, -Tan[(c + d*x)/2]^2, ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)] + (a + b)*m*AppellF1[(3 + m)/2, 1 + m, 1, (5 + m)/2, -Tan[(c + d*x)/2]^2, ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]^2)/((a + b)*(3 + m)))) + (2*b^2*((a + b)*AppellF1[(1 + m)/2, m, 1, (3 + m)/2, -Tan[(c + d*x)/2]^2, ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)] - 2*a*AppellF1[(1 + m)/2, m, 2, (3 + m)/2, -Tan[(c + d*x)/2]^2, ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)])*Sec[c + d*x]^2*(e*Sin[c + d*x])^m*Tan[(c + d*x)/2])/(a^2*d*(a + b*Sec[c + d*x])^2*(((a + b)*AppellF1[(1 + m)/2, m, 1, (3 + m)/2, -Tan[(c + d*x)/2]^2, ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)] - 2*a*AppellF1[(1 + m)/2, m, 2, (3 + m)/2, -Tan[(c + d*x)/2]^2, ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)])*Sec[(c + d*x)/2]^2 + 2*m*((a + b)*AppellF1[(1 + m)/2, m, 1, (3 + m)/2, -Tan[(c + d*x)/2]^2, ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)] - 2*a*AppellF1[(1 + m)/2, m, 2, (3 + m)/2, -Tan[(c + d*x)/2]^2, ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)])*Cot[c + d*x]*Tan[(c + d*x)/2] + 2*m*((a + b)*AppellF1[(1 + m)/2, m, 1, (3 + m)/2, -Tan[(c + d*x)/2]^2, ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)] - 2*a*AppellF1[(1 + m)/2, m, 2, (3 + m)/2, -Tan[(c + d*x)/2]^2, ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)])*Tan[(c + d*x)/2]^2 - (2*(1 + m)*((-a^2 + b^2)*AppellF1[(3 + m)/2, m, 2, (5 + m)/2, -Tan[(c + d*x)/2]^2, ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)] + 4*a*(a - b)*AppellF1[(3 + m)/2, m, 3, (5 + m)/2, -Tan[(c + d*x)/2]^2, ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)] + (a + b)*m*((a + b)*AppellF1[(3 + m)/2, 1 + m, 1, (5 + m)/2, -Tan[(c + d*x)/2]^2, ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)] - 2*a*AppellF1[(3 + m)/2, 1 + m, 2, (5 + m)/2, -Tan[(c + d*x)/2]^2, ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)]))*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]^2)/((a + b)*(3 + m)))) - ((b + a*Cos[c + d*x])^2*Hypergeometric2F1[1/2, (1 - m)/2, 3/2, Cos[c + d*x]^2]*(e*Sin[c + d*x])^m*(Sin[c + d*x]^2)^((-1 - m)/2)*Tan[c + d*x])/(a^2*d*(a + b*Sec[c + d*x])^2)","B",0
262,1,2700,580,17.9726256,"\int \frac{(e \sin (c+d x))^m}{(a+b \sec (c+d x))^3} \, dx","Integrate[(e*Sin[c + d*x])^m/(a + b*Sec[c + d*x])^3,x]","\text{Result too large to show}","-\frac{b^3 e (e \sin (c+d x))^{m-1} \left(-\frac{a (1-\cos (c+d x))}{a \cos (c+d x)+b}\right)^{\frac{1-m}{2}} \left(\frac{a (\cos (c+d x)+1)}{a \cos (c+d x)+b}\right)^{\frac{1-m}{2}} F_1\left(3-m;\frac{1-m}{2},\frac{1-m}{2};4-m;-\frac{a-b}{b+a \cos (c+d x)},\frac{a+b}{b+a \cos (c+d x)}\right)}{a^4 d (3-m) (a \cos (c+d x)+b)^2}+\frac{3 b^2 e (e \sin (c+d x))^{m-1} \left(-\frac{a (1-\cos (c+d x))}{a \cos (c+d x)+b}\right)^{\frac{1-m}{2}} \left(\frac{a (\cos (c+d x)+1)}{a \cos (c+d x)+b}\right)^{\frac{1-m}{2}} F_1\left(2-m;\frac{1-m}{2},\frac{1-m}{2};3-m;-\frac{a-b}{b+a \cos (c+d x)},\frac{a+b}{b+a \cos (c+d x)}\right)}{a^4 d (2-m) (a \cos (c+d x)+b)}-\frac{3 b e (e \sin (c+d x))^{m-1} \left(-\frac{a (1-\cos (c+d x))}{a \cos (c+d x)+b}\right)^{\frac{1-m}{2}} \left(\frac{a (\cos (c+d x)+1)}{a \cos (c+d x)+b}\right)^{\frac{1-m}{2}} F_1\left(1-m;\frac{1-m}{2},\frac{1-m}{2};2-m;-\frac{a-b}{b+a \cos (c+d x)},\frac{a+b}{b+a \cos (c+d x)}\right)}{a^4 d (1-m)}+\frac{\cos (c+d x) (e \sin (c+d x))^{m+1} \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};\sin ^2(c+d x)\right)}{a^3 d e (m+1) \sqrt{\cos ^2(c+d x)}}",1,"(-6*b*AppellF1[(1 + m)/2, m, 1, (3 + m)/2, -Tan[(c + d*x)/2]^2, ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)]*(b + a*Cos[c + d*x])^2*Sec[c + d*x]^3*(e*Sin[c + d*x])^m*Tan[(c + d*x)/2])/(a^3*d*(a + b*Sec[c + d*x])^3*(AppellF1[(1 + m)/2, m, 1, (3 + m)/2, -Tan[(c + d*x)/2]^2, ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)]*Sec[(c + d*x)/2]^2 + 2*m*AppellF1[(1 + m)/2, m, 1, (3 + m)/2, -Tan[(c + d*x)/2]^2, ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)]*Cot[c + d*x]*Tan[(c + d*x)/2] + 2*m*AppellF1[(1 + m)/2, m, 1, (3 + m)/2, -Tan[(c + d*x)/2]^2, ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)]*Tan[(c + d*x)/2]^2 - (2*(1 + m)*((-a + b)*AppellF1[(3 + m)/2, m, 2, (5 + m)/2, -Tan[(c + d*x)/2]^2, ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)] + (a + b)*m*AppellF1[(3 + m)/2, 1 + m, 1, (5 + m)/2, -Tan[(c + d*x)/2]^2, ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]^2)/((a + b)*(3 + m)))) + (6*b^2*((a + b)*AppellF1[(1 + m)/2, m, 1, (3 + m)/2, -Tan[(c + d*x)/2]^2, ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)] - 2*a*AppellF1[(1 + m)/2, m, 2, (3 + m)/2, -Tan[(c + d*x)/2]^2, ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)])*(b + a*Cos[c + d*x])*Sec[c + d*x]^3*(e*Sin[c + d*x])^m*Tan[(c + d*x)/2])/(a^3*d*(a + b*Sec[c + d*x])^3*(((a + b)*AppellF1[(1 + m)/2, m, 1, (3 + m)/2, -Tan[(c + d*x)/2]^2, ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)] - 2*a*AppellF1[(1 + m)/2, m, 2, (3 + m)/2, -Tan[(c + d*x)/2]^2, ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)])*Sec[(c + d*x)/2]^2 + 2*m*((a + b)*AppellF1[(1 + m)/2, m, 1, (3 + m)/2, -Tan[(c + d*x)/2]^2, ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)] - 2*a*AppellF1[(1 + m)/2, m, 2, (3 + m)/2, -Tan[(c + d*x)/2]^2, ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)])*Cot[c + d*x]*Tan[(c + d*x)/2] + 2*m*((a + b)*AppellF1[(1 + m)/2, m, 1, (3 + m)/2, -Tan[(c + d*x)/2]^2, ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)] - 2*a*AppellF1[(1 + m)/2, m, 2, (3 + m)/2, -Tan[(c + d*x)/2]^2, ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)])*Tan[(c + d*x)/2]^2 - (2*(1 + m)*((-a^2 + b^2)*AppellF1[(3 + m)/2, m, 2, (5 + m)/2, -Tan[(c + d*x)/2]^2, ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)] + 4*a*(a - b)*AppellF1[(3 + m)/2, m, 3, (5 + m)/2, -Tan[(c + d*x)/2]^2, ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)] + (a + b)*m*((a + b)*AppellF1[(3 + m)/2, 1 + m, 1, (5 + m)/2, -Tan[(c + d*x)/2]^2, ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)] - 2*a*AppellF1[(3 + m)/2, 1 + m, 2, (5 + m)/2, -Tan[(c + d*x)/2]^2, ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)]))*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]^2)/((a + b)*(3 + m)))) - (2*b^3*((a + b)^2*AppellF1[(1 + m)/2, m, 1, (3 + m)/2, -Tan[(c + d*x)/2]^2, ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)] - 4*a*(a + b)*AppellF1[(1 + m)/2, m, 2, (3 + m)/2, -Tan[(c + d*x)/2]^2, ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)] + 4*a^2*AppellF1[(1 + m)/2, m, 3, (3 + m)/2, -Tan[(c + d*x)/2]^2, ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)])*Sec[c + d*x]^3*(e*Sin[c + d*x])^m*Tan[(c + d*x)/2])/(a^3*d*(a + b*Sec[c + d*x])^3*(((a + b)^2*AppellF1[(1 + m)/2, m, 1, (3 + m)/2, -Tan[(c + d*x)/2]^2, ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)] - 4*a*(a + b)*AppellF1[(1 + m)/2, m, 2, (3 + m)/2, -Tan[(c + d*x)/2]^2, ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)] + 4*a^2*AppellF1[(1 + m)/2, m, 3, (3 + m)/2, -Tan[(c + d*x)/2]^2, ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)])*Sec[(c + d*x)/2]^2 + 2*m*((a + b)^2*AppellF1[(1 + m)/2, m, 1, (3 + m)/2, -Tan[(c + d*x)/2]^2, ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)] - 4*a*(a + b)*AppellF1[(1 + m)/2, m, 2, (3 + m)/2, -Tan[(c + d*x)/2]^2, ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)] + 4*a^2*AppellF1[(1 + m)/2, m, 3, (3 + m)/2, -Tan[(c + d*x)/2]^2, ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)])*Cot[c + d*x]*Tan[(c + d*x)/2] + 2*m*((a + b)^2*AppellF1[(1 + m)/2, m, 1, (3 + m)/2, -Tan[(c + d*x)/2]^2, ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)] - 4*a*(a + b)*AppellF1[(1 + m)/2, m, 2, (3 + m)/2, -Tan[(c + d*x)/2]^2, ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)] + 4*a^2*AppellF1[(1 + m)/2, m, 3, (3 + m)/2, -Tan[(c + d*x)/2]^2, ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)])*Tan[(c + d*x)/2]^2 + (2*(1 + m)*((a + b)^2*((a - b)*AppellF1[(3 + m)/2, m, 2, (5 + m)/2, -Tan[(c + d*x)/2]^2, ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)] - (a + b)*m*AppellF1[(3 + m)/2, 1 + m, 1, (5 + m)/2, -Tan[(c + d*x)/2]^2, ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)]) - 4*a*(a + b)*(2*(a - b)*AppellF1[(3 + m)/2, m, 3, (5 + m)/2, -Tan[(c + d*x)/2]^2, ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)] - (a + b)*m*AppellF1[(3 + m)/2, 1 + m, 2, (5 + m)/2, -Tan[(c + d*x)/2]^2, ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)]) + 4*a^2*(3*(a - b)*AppellF1[(3 + m)/2, m, 4, (5 + m)/2, -Tan[(c + d*x)/2]^2, ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)] - (a + b)*m*AppellF1[(3 + m)/2, 1 + m, 3, (5 + m)/2, -Tan[(c + d*x)/2]^2, ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)]))*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]^2)/((a + b)*(3 + m)))) - ((b + a*Cos[c + d*x])^3*Hypergeometric2F1[1/2, (1 - m)/2, 3/2, Cos[c + d*x]^2]*Sec[c + d*x]*(e*Sin[c + d*x])^m*(Sin[c + d*x]^2)^((-1 - m)/2)*Tan[c + d*x])/(a^3*d*(a + b*Sec[c + d*x])^3)","B",0
263,0,0,28,7.694556,"\int (a+b \sec (c+d x))^{3/2} (e \sin (c+d x))^m \, dx","Integrate[(a + b*Sec[c + d*x])^(3/2)*(e*Sin[c + d*x])^m,x]","\int (a+b \sec (c+d x))^{3/2} (e \sin (c+d x))^m \, dx","\text{Int}\left((a+b \sec (c+d x))^{3/2} (e \sin (c+d x))^m,x\right)",0,"Integrate[(a + b*Sec[c + d*x])^(3/2)*(e*Sin[c + d*x])^m, x]","A",-1
264,0,0,28,0.5208506,"\int \sqrt{a+b \sec (c+d x)} (e \sin (c+d x))^m \, dx","Integrate[Sqrt[a + b*Sec[c + d*x]]*(e*Sin[c + d*x])^m,x]","\int \sqrt{a+b \sec (c+d x)} (e \sin (c+d x))^m \, dx","\text{Int}\left(\sqrt{a+b \sec (c+d x)} (e \sin (c+d x))^m,x\right)",0,"Integrate[Sqrt[a + b*Sec[c + d*x]]*(e*Sin[c + d*x])^m, x]","A",-1
265,0,0,28,2.6330059,"\int \frac{(e \sin (c+d x))^m}{\sqrt{a+b \sec (c+d x)}} \, dx","Integrate[(e*Sin[c + d*x])^m/Sqrt[a + b*Sec[c + d*x]],x]","\int \frac{(e \sin (c+d x))^m}{\sqrt{a+b \sec (c+d x)}} \, dx","\text{Int}\left(\frac{(e \sin (c+d x))^m}{\sqrt{a+b \sec (c+d x)}},x\right)",0,"Integrate[(e*Sin[c + d*x])^m/Sqrt[a + b*Sec[c + d*x]], x]","A",-1
266,0,0,28,2.7981318,"\int \frac{(e \sin (c+d x))^m}{(a+b \sec (c+d x))^{3/2}} \, dx","Integrate[(e*Sin[c + d*x])^m/(a + b*Sec[c + d*x])^(3/2),x]","\int \frac{(e \sin (c+d x))^m}{(a+b \sec (c+d x))^{3/2}} \, dx","\text{Int}\left(\frac{(e \sin (c+d x))^m}{(a+b \sec (c+d x))^{3/2}},x\right)",0,"Integrate[(e*Sin[c + d*x])^m/(a + b*Sec[c + d*x])^(3/2), x]","A",-1
267,0,0,26,3.3781097,"\int (a+b \sec (c+d x))^n (e \sin (c+d x))^m \, dx","Integrate[(a + b*Sec[c + d*x])^n*(e*Sin[c + d*x])^m,x]","\int (a+b \sec (c+d x))^n (e \sin (c+d x))^m \, dx","\text{Int}\left((e \sin (c+d x))^m (a+b \sec (c+d x))^n,x\right)",0,"Integrate[(a + b*Sec[c + d*x])^n*(e*Sin[c + d*x])^m, x]","A",-1
268,1,562,150,8.9344647,"\int (a+b \sec (c+d x))^n \sin ^5(c+d x) \, dx","Integrate[(a + b*Sec[c + d*x])^n*Sin[c + d*x]^5,x]","-\frac{\cos ^6\left(\frac{1}{2} (c+d x)\right) \cos (c+d x) (a+b \sec (c+d x))^n \left(192 a^3 (n-1) (a \cos (c+d x)+b)^2-240 a^3 (n-1) \sec ^2\left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b)^2+a (1-n) \left(96 a^2+4 a b (6-4 n)-4 b^2 \left(n^2-7 n+12\right)\right) \sec ^4\left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b)^2+40 a^2 (n-1) (2 a-b (n-3)) \sec ^4\left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b)^2-24 a^2 (n-1) (2 a-b (n-4)) \sec ^2\left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b)^2-10 a \sec ^6\left(\frac{1}{2} (c+d x)\right) \left((n-1) \left(-14 a^2+2 a b (n-1)+b^2 \left(n^2-5 n+6\right)\right) (a \cos (c+d x)+b)^2+b \left(24 a^3+12 a^2 b (n-1)-4 a b^2 \left(n^2-3 n+2\right)-b^3 \left(n^3-6 n^2+11 n-6\right)\right) \, _2F_1\left(2,1-n;2-n;\frac{a \cos (c+d x)}{b+a \cos (c+d x)}\right)\right)+\sec ^6\left(\frac{1}{2} (c+d x)\right) \left(b \left(120 a^4+120 a^3 b (n-1)-10 a b^3 \left(n^3-6 n^2+11 n-6\right)-b^4 \left(n^4-10 n^3+35 n^2-50 n+24\right)\right) \, _2F_1\left(2,1-n;2-n;\frac{a \cos (c+d x)}{b+a \cos (c+d x)}\right)+(n-1) \left(-84 a^3+2 a^2 b (18-7 n)+4 a b^2 \left(2 n^2-9 n+9\right)+b^3 \left(n^3-9 n^2+26 n-24\right)\right) (a \cos (c+d x)+b)^2\right)\right)}{120 a^4 d (n-1) (a \cos (c+d x)+b)}","\frac{b^5 (a+b \sec (c+d x))^{n+1} \, _2F_1\left(6,n+1;n+2;\frac{b \sec (c+d x)}{a}+1\right)}{a^6 d (n+1)}-\frac{2 b^3 (a+b \sec (c+d x))^{n+1} \, _2F_1\left(4,n+1;n+2;\frac{b \sec (c+d x)}{a}+1\right)}{a^4 d (n+1)}+\frac{b (a+b \sec (c+d x))^{n+1} \, _2F_1\left(2,n+1;n+2;\frac{b \sec (c+d x)}{a}+1\right)}{a^2 d (n+1)}",1,"-1/120*(Cos[(c + d*x)/2]^6*Cos[c + d*x]*(192*a^3*(-1 + n)*(b + a*Cos[c + d*x])^2 - 240*a^3*(-1 + n)*(b + a*Cos[c + d*x])^2*Sec[(c + d*x)/2]^2 - 24*a^2*(2*a - b*(-4 + n))*(-1 + n)*(b + a*Cos[c + d*x])^2*Sec[(c + d*x)/2]^2 + 40*a^2*(2*a - b*(-3 + n))*(-1 + n)*(b + a*Cos[c + d*x])^2*Sec[(c + d*x)/2]^4 + a*(1 - n)*(96*a^2 + 4*a*b*(6 - 4*n) - 4*b^2*(12 - 7*n + n^2))*(b + a*Cos[c + d*x])^2*Sec[(c + d*x)/2]^4 - 10*a*((-1 + n)*(-14*a^2 + 2*a*b*(-1 + n) + b^2*(6 - 5*n + n^2))*(b + a*Cos[c + d*x])^2 + b*(24*a^3 + 12*a^2*b*(-1 + n) - 4*a*b^2*(2 - 3*n + n^2) - b^3*(-6 + 11*n - 6*n^2 + n^3))*Hypergeometric2F1[2, 1 - n, 2 - n, (a*Cos[c + d*x])/(b + a*Cos[c + d*x])])*Sec[(c + d*x)/2]^6 + ((-1 + n)*(-84*a^3 + 2*a^2*b*(18 - 7*n) + 4*a*b^2*(9 - 9*n + 2*n^2) + b^3*(-24 + 26*n - 9*n^2 + n^3))*(b + a*Cos[c + d*x])^2 + b*(120*a^4 + 120*a^3*b*(-1 + n) - 10*a*b^3*(-6 + 11*n - 6*n^2 + n^3) - b^4*(24 - 50*n + 35*n^2 - 10*n^3 + n^4))*Hypergeometric2F1[2, 1 - n, 2 - n, (a*Cos[c + d*x])/(b + a*Cos[c + d*x])])*Sec[(c + d*x)/2]^6)*(a + b*Sec[c + d*x])^n)/(a^4*d*(-1 + n)*(b + a*Cos[c + d*x]))","B",1
269,1,155,121,1.8616532,"\int (a+b \sec (c+d x))^n \sin ^3(c+d x) \, dx","Integrate[(a + b*Sec[c + d*x])^n*Sin[c + d*x]^3,x]","\frac{\cos (c+d x) (a+b \sec (c+d x))^n \left(-\frac{2 b \left(b^2 \left(n^2-3 n+2\right)-6 a^2\right) \, _2F_1\left(2,1-n;2-n;\frac{a \cos (c+d x)}{b+a \cos (c+d x)}\right)}{a (n-1)}-\frac{2 (2 a-b (n-2)) (a \cos (c+d x)+b)^2}{a}+8 \cos ^2\left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b)^2\right)}{12 a d (a \cos (c+d x)+b)}","\frac{\cos ^3(c+d x) (2 a-b (2-n) \sec (c+d x)) (a+b \sec (c+d x))^{n+1}}{6 a^2 d}+\frac{b \left(6 a^2-b^2 \left(n^2-3 n+2\right)\right) (a+b \sec (c+d x))^{n+1} \, _2F_1\left(2,n+1;n+2;\frac{b \sec (c+d x)}{a}+1\right)}{6 a^4 d (n+1)}",1,"(Cos[c + d*x]*((-2*(2*a - b*(-2 + n))*(b + a*Cos[c + d*x])^2)/a + 8*Cos[(c + d*x)/2]^2*(b + a*Cos[c + d*x])^2 - (2*b*(-6*a^2 + b^2*(2 - 3*n + n^2))*Hypergeometric2F1[2, 1 - n, 2 - n, (a*Cos[c + d*x])/(b + a*Cos[c + d*x])])/(a*(-1 + n)))*(a + b*Sec[c + d*x])^n)/(12*a*d*(b + a*Cos[c + d*x]))","A",1
270,1,72,48,0.5357464,"\int (a+b \sec (c+d x))^n \sin (c+d x) \, dx","Integrate[(a + b*Sec[c + d*x])^n*Sin[c + d*x],x]","\frac{b \cos (c+d x) (a+b \sec (c+d x))^n \, _2F_1\left(2,1-n;2-n;\frac{a \cos (c+d x)}{b+a \cos (c+d x)}\right)}{d (n-1) (a \cos (c+d x)+b)}","\frac{b (a+b \sec (c+d x))^{n+1} \, _2F_1\left(2,n+1;n+2;\frac{b \sec (c+d x)}{a}+1\right)}{a^2 d (n+1)}",1,"(b*Cos[c + d*x]*Hypergeometric2F1[2, 1 - n, 2 - n, (a*Cos[c + d*x])/(b + a*Cos[c + d*x])]*(a + b*Sec[c + d*x])^n)/(d*(-1 + n)*(b + a*Cos[c + d*x]))","A",1
271,1,132,115,0.9764097,"\int \csc (c+d x) (a+b \sec (c+d x))^n \, dx","Integrate[Csc[c + d*x]*(a + b*Sec[c + d*x])^n,x]","\frac{(a+b \sec (c+d x))^n \left(\, _2F_1\left(1,-n;1-n;\frac{(a+b) \cos (c+d x)}{b+a \cos (c+d x)}\right)-2^n \left(\frac{\sec ^2\left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b)}{b}\right)^{-n} \, _2F_1\left(-n,-n;1-n;\frac{(b-a) \cos (c+d x) \sec ^2\left(\frac{1}{2} (c+d x)\right)}{2 b}\right)\right)}{2 d n}","\frac{(a+b \sec (c+d x))^{n+1} \, _2F_1\left(1,n+1;n+2;\frac{a+b \sec (c+d x)}{a-b}\right)}{2 d (n+1) (a-b)}-\frac{(a+b \sec (c+d x))^{n+1} \, _2F_1\left(1,n+1;n+2;\frac{a+b \sec (c+d x)}{a+b}\right)}{2 d (n+1) (a+b)}",1,"((Hypergeometric2F1[1, -n, 1 - n, ((a + b)*Cos[c + d*x])/(b + a*Cos[c + d*x])] - (2^n*Hypergeometric2F1[-n, -n, 1 - n, ((-a + b)*Cos[c + d*x]*Sec[(c + d*x)/2]^2)/(2*b)])/(((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/b)^n)*(a + b*Sec[c + d*x])^n)/(2*d*n)","A",1
272,1,710,231,17.0540839,"\int \csc ^3(c+d x) (a+b \sec (c+d x))^n \, dx","Integrate[Csc[c + d*x]^3*(a + b*Sec[c + d*x])^n,x]","\frac{\left(\frac{1}{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)}\right)^n \left(1-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right)^{-2 n} \left(1-\tan ^4\left(\frac{1}{2} (c+d x)\right)\right)^n \left(\cos (c+d x) \sec ^2\left(\frac{1}{2} (c+d x)\right)\right)^n \left(\cos (c+d x) \sec ^4\left(\frac{1}{2} (c+d x)\right)\right)^{-n} \left(\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)\right)^{-n} (a \cos (c+d x)+b)^{-n} \left(\frac{a-a \tan ^2\left(\frac{1}{2} (c+d x)\right)}{\tan ^2\left(\frac{1}{2} (c+d x)\right)+1}+b\right)^n (a+b \sec (c+d x))^n \left(2 (a+b n+b) \left(1-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right)^n \, _2F_1\left(1,-n;1-n;\frac{(a+b) \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)-1\right)}{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^{-n} \cot ^2\left(\frac{1}{2} (c+d x)\right) \left(\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}{b}\right)^{-n} \left(n \left(1-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right)^n \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) \left(2^n (n+1) (a-b) \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)-1\right) \left(\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}{b}\right)^n-2 a \tan ^2\left(\frac{1}{2} (c+d x)\right) \left(-\frac{\left(\tan ^2\left(\frac{1}{2} (c+d x)\right)-1\right) \left(a^2 \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)-1\right)-2 a b \tan ^2\left(\frac{1}{2} (c+d x)\right)+b^2 \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right)\right)}{b^2}\right)^n \, _2F_1\left(n,n+1;n+2;\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}{2 b}\right)\right)+2^{n+1} (n+1) (a-b) (a+b n+b) \tan ^2\left(\frac{1}{2} (c+d x)\right) \left(2-2 \tan ^2\left(\frac{1}{2} (c+d x)\right)\right)^n \, _2F_1\left(-n,-n;1-n;\frac{(a-b) \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)-1\right)}{2 b}\right)\right)}{(n+1) (a-b)}\right)}{8 d n (a+b)}","\frac{(a+b \sec (c+d x))^{n+1} \, _2F_1\left(1,n+1;n+2;\frac{a+b \sec (c+d x)}{a-b}\right)}{4 d (n+1) (a-b)}-\frac{(a+b \sec (c+d x))^{n+1} \, _2F_1\left(1,n+1;n+2;\frac{a+b \sec (c+d x)}{a+b}\right)}{4 d (n+1) (a+b)}+\frac{b (a+b \sec (c+d x))^{n+1} \, _2F_1\left(2,n+1;n+2;\frac{a+b \sec (c+d x)}{a-b}\right)}{4 d (n+1) (a-b)^2}+\frac{b (a+b \sec (c+d x))^{n+1} \, _2F_1\left(2,n+1;n+2;\frac{a+b \sec (c+d x)}{a+b}\right)}{4 d (n+1) (a+b)^2}",1,"((Cos[c + d*x]*Sec[(c + d*x)/2]^2)^n*(a + b*Sec[c + d*x])^n*((1 - Tan[(c + d*x)/2]^2)^(-1))^n*(1 - Tan[(c + d*x)/2]^4)^n*(b + (a - a*Tan[(c + d*x)/2]^2)/(1 + Tan[(c + d*x)/2]^2))^n*(2*(a + b + b*n)*Hypergeometric2F1[1, -n, 1 - n, ((a + b)*(-1 + Tan[(c + d*x)/2]^2))/(a*(-1 + Tan[(c + d*x)/2]^2) - b*(1 + Tan[(c + d*x)/2]^2))]*(1 - Tan[(c + d*x)/2]^2)^n - (Cot[(c + d*x)/2]^2*(2^(1 + n)*(a - b)*(1 + n)*(a + b + b*n)*Hypergeometric2F1[-n, -n, 1 - n, ((a - b)*(-1 + Tan[(c + d*x)/2]^2))/(2*b)]*Tan[(c + d*x)/2]^2*(2 - 2*Tan[(c + d*x)/2]^2)^n + n*(1 - Tan[(c + d*x)/2]^2)^n*(a*(-1 + Tan[(c + d*x)/2]^2) - b*(1 + Tan[(c + d*x)/2]^2))*(2^n*(a - b)*(1 + n)*(-1 + Tan[(c + d*x)/2]^2)*((a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/b)^n - 2*a*Hypergeometric2F1[n, 1 + n, 2 + n, (a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(2*b)]*Tan[(c + d*x)/2]^2*(-(((-1 + Tan[(c + d*x)/2]^2)*(-2*a*b*Tan[(c + d*x)/2]^2 + a^2*(-1 + Tan[(c + d*x)/2]^2) + b^2*(1 + Tan[(c + d*x)/2]^2)))/b^2))^n)))/(2^n*(a - b)*(1 + n)*((a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/b)^n)))/(8*(a + b)*d*n*(b + a*Cos[c + d*x])^n*(Cos[c + d*x]*Sec[(c + d*x)/2]^4)^n*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^n*(1 - Tan[(c + d*x)/2]^2)^(2*n))","B",0
273,0,0,24,14.436454,"\int (a+b \sec (c+d x))^n \sin ^4(c+d x) \, dx","Integrate[(a + b*Sec[c + d*x])^n*Sin[c + d*x]^4,x]","\int (a+b \sec (c+d x))^n \sin ^4(c+d x) \, dx","\text{Int}\left(\sin ^4(c+d x) (a+b \sec (c+d x))^n,x\right)",0,"Integrate[(a + b*Sec[c + d*x])^n*Sin[c + d*x]^4, x]","A",-1
274,0,0,24,3.9034351,"\int (a+b \sec (c+d x))^n \sin ^2(c+d x) \, dx","Integrate[(a + b*Sec[c + d*x])^n*Sin[c + d*x]^2,x]","\int (a+b \sec (c+d x))^n \sin ^2(c+d x) \, dx","\text{Int}\left(\sin ^2(c+d x) (a+b \sec (c+d x))^n,x\right)",0,"Integrate[(a + b*Sec[c + d*x])^n*Sin[c + d*x]^2, x]","A",-1
275,1,3614,136,18.4590339,"\int \csc ^2(c+d x) (a+b \sec (c+d x))^n \, dx","Integrate[Csc[c + d*x]^2*(a + b*Sec[c + d*x])^n,x]","\text{Result too large to show}","\frac{\sqrt{2} b n \tan (c+d x) (a+b \sec (c+d x))^n \left(\frac{a+b \sec (c+d x)}{a+b}\right)^{-n} F_1\left(\frac{1}{2};\frac{1}{2},1-n;\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}}-\frac{\cot (c+d x) (a+b \sec (c+d x))^n}{d}",1,"((b + a*Cos[c + d*x])^n*Cot[(c + d*x)/2]*Csc[c + d*x]^2*Sec[c + d*x]^n*(a + b*Sec[c + d*x])^n*(-((AppellF1[-1/2, n, -n, 1/2, Tan[(c + d*x)/2]^2, ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)]*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^n)/(((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b))^n) + (3*(a + b)*AppellF1[1/2, n, -n, 3/2, Tan[(c + d*x)/2]^2, ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)]*Tan[(c + d*x)/2]^2)/(3*(a + b)*AppellF1[1/2, n, -n, 3/2, Tan[(c + d*x)/2]^2, ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)] + 2*n*((-a + b)*AppellF1[3/2, n, 1 - n, 5/2, Tan[(c + d*x)/2]^2, ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)] + (a + b)*AppellF1[3/2, 1 + n, -n, 5/2, Tan[(c + d*x)/2]^2, ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)])*Tan[(c + d*x)/2]^2)))/(2*d*(-1/4*((b + a*Cos[c + d*x])^n*Csc[(c + d*x)/2]^2*Sec[c + d*x]^n*(-((AppellF1[-1/2, n, -n, 1/2, Tan[(c + d*x)/2]^2, ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)]*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^n)/(((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b))^n) + (3*(a + b)*AppellF1[1/2, n, -n, 3/2, Tan[(c + d*x)/2]^2, ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)]*Tan[(c + d*x)/2]^2)/(3*(a + b)*AppellF1[1/2, n, -n, 3/2, Tan[(c + d*x)/2]^2, ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)] + 2*n*((-a + b)*AppellF1[3/2, n, 1 - n, 5/2, Tan[(c + d*x)/2]^2, ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)] + (a + b)*AppellF1[3/2, 1 + n, -n, 5/2, Tan[(c + d*x)/2]^2, ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)])*Tan[(c + d*x)/2]^2))) - (a*n*(b + a*Cos[c + d*x])^(-1 + n)*Cot[(c + d*x)/2]*Sec[c + d*x]^n*Sin[c + d*x]*(-((AppellF1[-1/2, n, -n, 1/2, Tan[(c + d*x)/2]^2, ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)]*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^n)/(((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b))^n) + (3*(a + b)*AppellF1[1/2, n, -n, 3/2, Tan[(c + d*x)/2]^2, ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)]*Tan[(c + d*x)/2]^2)/(3*(a + b)*AppellF1[1/2, n, -n, 3/2, Tan[(c + d*x)/2]^2, ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)] + 2*n*((-a + b)*AppellF1[3/2, n, 1 - n, 5/2, Tan[(c + d*x)/2]^2, ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)] + (a + b)*AppellF1[3/2, 1 + n, -n, 5/2, Tan[(c + d*x)/2]^2, ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)])*Tan[(c + d*x)/2]^2)))/2 + (n*(b + a*Cos[c + d*x])^n*Cot[(c + d*x)/2]*Sec[c + d*x]^(1 + n)*Sin[c + d*x]*(-((AppellF1[-1/2, n, -n, 1/2, Tan[(c + d*x)/2]^2, ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)]*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^n)/(((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b))^n) + (3*(a + b)*AppellF1[1/2, n, -n, 3/2, Tan[(c + d*x)/2]^2, ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)]*Tan[(c + d*x)/2]^2)/(3*(a + b)*AppellF1[1/2, n, -n, 3/2, Tan[(c + d*x)/2]^2, ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)] + 2*n*((-a + b)*AppellF1[3/2, n, 1 - n, 5/2, Tan[(c + d*x)/2]^2, ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)] + (a + b)*AppellF1[3/2, 1 + n, -n, 5/2, Tan[(c + d*x)/2]^2, ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)])*Tan[(c + d*x)/2]^2)))/2 + ((b + a*Cos[c + d*x])^n*Cot[(c + d*x)/2]*Sec[c + d*x]^n*(-(((Cos[c + d*x]*Sec[(c + d*x)/2]^2)^n*(((a - b)*n*AppellF1[1/2, n, 1 - n, 3/2, Tan[(c + d*x)/2]^2, ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/(a + b) - n*AppellF1[1/2, 1 + n, -n, 3/2, Tan[(c + d*x)/2]^2, ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b))^n) - (n*AppellF1[-1/2, n, -n, 1/2, Tan[(c + d*x)/2]^2, ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)]*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(-1 + n)*(-(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]))/(((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b))^n + n*AppellF1[-1/2, n, -n, 1/2, Tan[(c + d*x)/2]^2, ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)]*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^n*(((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b))^(-1 - n)*(-((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)) + (3*(a + b)*AppellF1[1/2, n, -n, 3/2, Tan[(c + d*x)/2]^2, ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/(3*(a + b)*AppellF1[1/2, n, -n, 3/2, Tan[(c + d*x)/2]^2, ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)] + 2*n*((-a + b)*AppellF1[3/2, n, 1 - n, 5/2, Tan[(c + d*x)/2]^2, ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)] + (a + b)*AppellF1[3/2, 1 + n, -n, 5/2, Tan[(c + d*x)/2]^2, ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)])*Tan[(c + d*x)/2]^2) + (3*(a + b)*Tan[(c + d*x)/2]^2*(-1/3*((a - b)*n*AppellF1[3/2, n, 1 - n, 5/2, Tan[(c + d*x)/2]^2, ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/(a + b) + (n*AppellF1[3/2, 1 + n, -n, 5/2, Tan[(c + d*x)/2]^2, ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/3))/(3*(a + b)*AppellF1[1/2, n, -n, 3/2, Tan[(c + d*x)/2]^2, ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)] + 2*n*((-a + b)*AppellF1[3/2, n, 1 - n, 5/2, Tan[(c + d*x)/2]^2, ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)] + (a + b)*AppellF1[3/2, 1 + n, -n, 5/2, Tan[(c + d*x)/2]^2, ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)])*Tan[(c + d*x)/2]^2) - (3*(a + b)*AppellF1[1/2, n, -n, 3/2, Tan[(c + d*x)/2]^2, ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)]*Tan[(c + d*x)/2]^2*(2*n*((-a + b)*AppellF1[3/2, n, 1 - n, 5/2, Tan[(c + d*x)/2]^2, ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)] + (a + b)*AppellF1[3/2, 1 + n, -n, 5/2, Tan[(c + d*x)/2]^2, ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2] + 3*(a + b)*(-1/3*((a - b)*n*AppellF1[3/2, n, 1 - n, 5/2, Tan[(c + d*x)/2]^2, ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/(a + b) + (n*AppellF1[3/2, 1 + n, -n, 5/2, Tan[(c + d*x)/2]^2, ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/3) + 2*n*Tan[(c + d*x)/2]^2*((-a + b)*((3*(a - b)*(1 - n)*AppellF1[5/2, n, 2 - n, 7/2, Tan[(c + d*x)/2]^2, ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/(5*(a + b)) + (3*n*AppellF1[5/2, 1 + n, 1 - n, 7/2, Tan[(c + d*x)/2]^2, ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/5) + (a + b)*((-3*(a - b)*n*AppellF1[5/2, 1 + n, 1 - n, 7/2, Tan[(c + d*x)/2]^2, ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/(5*(a + b)) + (3*(1 + n)*AppellF1[5/2, 2 + n, -n, 7/2, Tan[(c + d*x)/2]^2, ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/5))))/(3*(a + b)*AppellF1[1/2, n, -n, 3/2, Tan[(c + d*x)/2]^2, ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)] + 2*n*((-a + b)*AppellF1[3/2, n, 1 - n, 5/2, Tan[(c + d*x)/2]^2, ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)] + (a + b)*AppellF1[3/2, 1 + n, -n, 5/2, Tan[(c + d*x)/2]^2, ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)])*Tan[(c + d*x)/2]^2)^2))/2))","B",0
276,1,6403,424,23.7247814,"\int \csc ^4(c+d x) (a+b \sec (c+d x))^n \, dx","Integrate[Csc[c + d*x]^4*(a + b*Sec[c + d*x])^n,x]","\text{Result too large to show}","-\frac{\cot ^3(c+d x) (\sec (c+d x)+1)^{3/2} (a+b \sec (c+d x))^n \left(\frac{a+b \sec (c+d x)}{a+b}\right)^{-n} F_1\left(-\frac{3}{2};\frac{5}{2},-n;-\frac{1}{2};\frac{1}{2} (1-\sec (c+d x)),\frac{b (1-\sec (c+d x))}{a+b}\right)}{6 \sqrt{2} d}-\frac{3 \cot (c+d x) \sqrt{\sec (c+d x)+1} (a+b \sec (c+d x))^n \left(\frac{a+b \sec (c+d x)}{a+b}\right)^{-n} F_1\left(-\frac{1}{2};\frac{5}{2},-n;\frac{1}{2};\frac{1}{2} (1-\sec (c+d x)),\frac{b (1-\sec (c+d x))}{a+b}\right)}{2 \sqrt{2} d}+\frac{\tan (c+d x) (a+b \sec (c+d x))^n \left(\frac{a+b \sec (c+d x)}{a+b}\right)^{-n} F_1\left(\frac{1}{2};\frac{3}{2},-n;\frac{3}{2};\frac{1}{2} (1-\sec (c+d x)),\frac{b (1-\sec (c+d x))}{a+b}\right)}{\sqrt{2} d \sqrt{\sec (c+d x)+1}}+\frac{\tan (c+d x) (a+b \sec (c+d x))^n \left(\frac{a+b \sec (c+d x)}{a+b}\right)^{-n} F_1\left(\frac{1}{2};\frac{5}{2},-n;\frac{3}{2};\frac{1}{2} (1-\sec (c+d x)),\frac{b (1-\sec (c+d x))}{a+b}\right)}{2 \sqrt{2} d \sqrt{\sec (c+d x)+1}}",1,"Result too large to show","B",0
277,0,0,26,1.8749344,"\int (a+b \sec (c+d x))^n \sin ^{\frac{3}{2}}(c+d x) \, dx","Integrate[(a + b*Sec[c + d*x])^n*Sin[c + d*x]^(3/2),x]","\int (a+b \sec (c+d x))^n \sin ^{\frac{3}{2}}(c+d x) \, dx","\text{Int}\left(\sin ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^n,x\right)",0,"Integrate[(a + b*Sec[c + d*x])^n*Sin[c + d*x]^(3/2), x]","A",-1
278,0,0,26,4.9882486,"\int (a+b \sec (c+d x))^n \sqrt{\sin (c+d x)} \, dx","Integrate[(a + b*Sec[c + d*x])^n*Sqrt[Sin[c + d*x]],x]","\int (a+b \sec (c+d x))^n \sqrt{\sin (c+d x)} \, dx","\text{Int}\left(\sqrt{\sin (c+d x)} (a+b \sec (c+d x))^n,x\right)",0,"Integrate[(a + b*Sec[c + d*x])^n*Sqrt[Sin[c + d*x]], x]","A",-1
279,0,0,26,2.7489806,"\int \frac{(a+b \sec (c+d x))^n}{\sqrt{\sin (c+d x)}} \, dx","Integrate[(a + b*Sec[c + d*x])^n/Sqrt[Sin[c + d*x]],x]","\int \frac{(a+b \sec (c+d x))^n}{\sqrt{\sin (c+d x)}} \, dx","\text{Int}\left(\frac{(a+b \sec (c+d x))^n}{\sqrt{\sin (c+d x)}},x\right)",0,"Integrate[(a + b*Sec[c + d*x])^n/Sqrt[Sin[c + d*x]], x]","A",-1
280,0,0,26,2.9131988,"\int \frac{(a+b \sec (c+d x))^n}{\sin ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[(a + b*Sec[c + d*x])^n/Sin[c + d*x]^(3/2),x]","\int \frac{(a+b \sec (c+d x))^n}{\sin ^{\frac{3}{2}}(c+d x)} \, dx","\text{Int}\left(\frac{(a+b \sec (c+d x))^n}{\sin ^{\frac{3}{2}}(c+d x)},x\right)",0,"Integrate[(a + b*Sec[c + d*x])^n/Sin[c + d*x]^(3/2), x]","A",-1
281,1,135,190,1.515439,"\int (e \csc (c+d x))^{5/2} (a+a \sec (c+d x)) \, dx","Integrate[(e*Csc[c + d*x])^(5/2)*(a + a*Sec[c + d*x]),x]","-\frac{a (e \csc (c+d x))^{5/2} \left(4 \cot \left(\frac{1}{2} (c+d x)\right) \sqrt{\csc (c+d x)}+3 \log \left(1-\sqrt{\csc (c+d x)}\right)-3 \log \left(\sqrt{\csc (c+d x)}+1\right)+6 \tan ^{-1}\left(\sqrt{\csc (c+d x)}\right)+4 \sqrt{\sin (c+d x)} \sqrt{\csc (c+d x)} F\left(\left.\frac{1}{4} (-2 c-2 d x+\pi )\right|2\right)\right)}{6 d \csc ^{\frac{5}{2}}(c+d x)}","-\frac{2 a e^2 \csc (c+d x) \sqrt{e \csc (c+d x)}}{3 d}-\frac{2 a e^2 \cot (c+d x) \sqrt{e \csc (c+d x)}}{3 d}+\frac{a e^2 \sqrt{\sin (c+d x)} \sqrt{e \csc (c+d x)} \tan ^{-1}\left(\sqrt{\sin (c+d x)}\right)}{d}+\frac{a e^2 \sqrt{\sin (c+d x)} \sqrt{e \csc (c+d x)} \tanh ^{-1}\left(\sqrt{\sin (c+d x)}\right)}{d}+\frac{2 a e^2 \sqrt{\sin (c+d x)} F\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{e \csc (c+d x)}}{3 d}",1,"-1/6*(a*(e*Csc[c + d*x])^(5/2)*(6*ArcTan[Sqrt[Csc[c + d*x]]] + 4*Cot[(c + d*x)/2]*Sqrt[Csc[c + d*x]] + 3*Log[1 - Sqrt[Csc[c + d*x]]] - 3*Log[1 + Sqrt[Csc[c + d*x]]] + 4*Sqrt[Csc[c + d*x]]*EllipticF[(-2*c + Pi - 2*d*x)/4, 2]*Sqrt[Sin[c + d*x]]))/(d*Csc[c + d*x]^(5/2))","A",1
282,1,146,169,1.2796474,"\int (e \csc (c+d x))^{3/2} (a+a \sec (c+d x)) \, dx","Integrate[(e*Csc[c + d*x])^(3/2)*(a + a*Sec[c + d*x]),x]","\frac{a (e \csc (c+d x))^{3/2} \left(\frac{2 \sin (2 (c+d x)) \csc ^{\frac{3}{2}}(c+d x) \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};\csc ^2(c+d x)\right)}{\sqrt{-\cot ^2(c+d x)}}-4 (\cos (c+d x)+1) \sqrt{\csc (c+d x)}-\log \left(1-\sqrt{\csc (c+d x)}\right)+\log \left(\sqrt{\csc (c+d x)}+1\right)+2 \tan ^{-1}\left(\sqrt{\csc (c+d x)}\right)\right)}{2 d \csc ^{\frac{3}{2}}(c+d x)}","-\frac{2 a e \sqrt{e \csc (c+d x)}}{d}-\frac{2 a e \cos (c+d x) \sqrt{e \csc (c+d x)}}{d}-\frac{a e \sqrt{\sin (c+d x)} \sqrt{e \csc (c+d x)} \tan ^{-1}\left(\sqrt{\sin (c+d x)}\right)}{d}+\frac{a e \sqrt{\sin (c+d x)} \sqrt{e \csc (c+d x)} \tanh ^{-1}\left(\sqrt{\sin (c+d x)}\right)}{d}-\frac{2 a e \sqrt{\sin (c+d x)} E\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{e \csc (c+d x)}}{d}",1,"(a*(e*Csc[c + d*x])^(3/2)*(2*ArcTan[Sqrt[Csc[c + d*x]]] - 4*(1 + Cos[c + d*x])*Sqrt[Csc[c + d*x]] - Log[1 - Sqrt[Csc[c + d*x]]] + Log[1 + Sqrt[Csc[c + d*x]]] + (2*Csc[c + d*x]^(3/2)*Hypergeometric2F1[-1/4, 1/2, 3/4, Csc[c + d*x]^2]*Sin[2*(c + d*x)])/Sqrt[-Cot[c + d*x]^2]))/(2*d*Csc[c + d*x]^(3/2))","C",1
283,1,111,121,0.8539372,"\int \sqrt{e \csc (c+d x)} (a+a \sec (c+d x)) \, dx","Integrate[Sqrt[e*Csc[c + d*x]]*(a + a*Sec[c + d*x]),x]","-\frac{a \sqrt{e \csc (c+d x)} \left(\log \left(1-\sqrt{\csc (c+d x)}\right)-\log \left(\sqrt{\csc (c+d x)}+1\right)+2 \tan ^{-1}\left(\sqrt{\csc (c+d x)}\right)+4 \sqrt{\sin (c+d x)} \sqrt{\csc (c+d x)} F\left(\left.\frac{1}{4} (-2 c-2 d x+\pi )\right|2\right)\right)}{2 d \sqrt{\csc (c+d x)}}","\frac{a \sqrt{\sin (c+d x)} \sqrt{e \csc (c+d x)} \tan ^{-1}\left(\sqrt{\sin (c+d x)}\right)}{d}+\frac{a \sqrt{\sin (c+d x)} \sqrt{e \csc (c+d x)} \tanh ^{-1}\left(\sqrt{\sin (c+d x)}\right)}{d}+\frac{2 a \sqrt{\sin (c+d x)} F\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{e \csc (c+d x)}}{d}",1,"-1/2*(a*Sqrt[e*Csc[c + d*x]]*(2*ArcTan[Sqrt[Csc[c + d*x]]] + Log[1 - Sqrt[Csc[c + d*x]]] - Log[1 + Sqrt[Csc[c + d*x]]] + 4*Sqrt[Csc[c + d*x]]*EllipticF[(-2*c + Pi - 2*d*x)/4, 2]*Sqrt[Sin[c + d*x]]))/(d*Sqrt[Csc[c + d*x]])","A",1
284,1,130,122,0.826149,"\int \frac{a+a \sec (c+d x)}{\sqrt{e \csc (c+d x)}} \, dx","Integrate[(a + a*Sec[c + d*x])/Sqrt[e*Csc[c + d*x]],x]","\frac{a \left(\sqrt{-\cot ^2(c+d x)} \sqrt{\csc (c+d x)} \left(-\log \left(1-\sqrt{\csc (c+d x)}\right)+\log \left(\sqrt{\csc (c+d x)}+1\right)+2 \tan ^{-1}\left(\sqrt{\csc (c+d x)}\right)\right)-4 \cot (c+d x) \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};\csc ^2(c+d x)\right)\right)}{2 d \sqrt{-\cot ^2(c+d x)} \sqrt{e \csc (c+d x)}}","-\frac{a \tan ^{-1}\left(\sqrt{\sin (c+d x)}\right)}{d \sqrt{\sin (c+d x)} \sqrt{e \csc (c+d x)}}+\frac{a \tanh ^{-1}\left(\sqrt{\sin (c+d x)}\right)}{d \sqrt{\sin (c+d x)} \sqrt{e \csc (c+d x)}}+\frac{2 a E\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{d \sqrt{\sin (c+d x)} \sqrt{e \csc (c+d x)}}",1,"(a*(-4*Cot[c + d*x]*Hypergeometric2F1[-1/4, 1/2, 3/4, Csc[c + d*x]^2] + Sqrt[-Cot[c + d*x]^2]*Sqrt[Csc[c + d*x]]*(2*ArcTan[Sqrt[Csc[c + d*x]]] - Log[1 - Sqrt[Csc[c + d*x]]] + Log[1 + Sqrt[Csc[c + d*x]]])))/(2*d*Sqrt[-Cot[c + d*x]^2]*Sqrt[e*Csc[c + d*x]])","C",1
285,1,135,182,11.0316761,"\int \frac{a+a \sec (c+d x)}{(e \csc (c+d x))^{3/2}} \, dx","Integrate[(a + a*Sec[c + d*x])/(e*Csc[c + d*x])^(3/2),x]","-\frac{a \left(4 \cos (c+d x)+3 \sqrt{\csc (c+d x)} \log \left(1-\sqrt{\csc (c+d x)}\right)-3 \sqrt{\csc (c+d x)} \log \left(\sqrt{\csc (c+d x)}+1\right)+6 \sqrt{\csc (c+d x)} \tan ^{-1}\left(\sqrt{\csc (c+d x)}\right)+\frac{4 F\left(\left.\frac{1}{4} (-2 c-2 d x+\pi )\right|2\right)}{\sqrt{\sin (c+d x)}}+12\right)}{6 d e \sqrt{e \csc (c+d x)}}","-\frac{2 a}{d e \sqrt{e \csc (c+d x)}}-\frac{2 a \cos (c+d x)}{3 d e \sqrt{e \csc (c+d x)}}+\frac{a \tan ^{-1}\left(\sqrt{\sin (c+d x)}\right)}{d e \sqrt{\sin (c+d x)} \sqrt{e \csc (c+d x)}}+\frac{a \tanh ^{-1}\left(\sqrt{\sin (c+d x)}\right)}{d e \sqrt{\sin (c+d x)} \sqrt{e \csc (c+d x)}}+\frac{2 a F\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{3 d e \sqrt{\sin (c+d x)} \sqrt{e \csc (c+d x)}}",1,"-1/6*(a*(12 + 4*Cos[c + d*x] + 6*ArcTan[Sqrt[Csc[c + d*x]]]*Sqrt[Csc[c + d*x]] + 3*Sqrt[Csc[c + d*x]]*Log[1 - Sqrt[Csc[c + d*x]]] - 3*Sqrt[Csc[c + d*x]]*Log[1 + Sqrt[Csc[c + d*x]]] + (4*EllipticF[(-2*c + Pi - 2*d*x)/4, 2])/Sqrt[Sin[c + d*x]]))/(d*e*Sqrt[e*Csc[c + d*x]])","A",1
286,1,165,197,1.5074575,"\int \frac{a+a \sec (c+d x)}{(e \csc (c+d x))^{5/2}} \, dx","Integrate[(a + a*Sec[c + d*x])/(e*Csc[c + d*x])^(5/2),x]","\frac{a \left(-72 \cot (c+d x) \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};\csc ^2(c+d x)\right)-2 \sqrt{-\cot ^2(c+d x)} \left(20 \sin (c+d x)+6 \sin (2 (c+d x))+15 \sqrt{\csc (c+d x)} \left(\log \left(1-\sqrt{\csc (c+d x)}\right)-\log \left(\sqrt{\csc (c+d x)}+1\right)\right)-30 \sqrt{\csc (c+d x)} \tan ^{-1}\left(\sqrt{\csc (c+d x)}\right)\right)\right)}{60 d e^2 \sqrt{-\cot ^2(c+d x)} \sqrt{e \csc (c+d x)}}","-\frac{2 a \sin (c+d x)}{3 d e^2 \sqrt{e \csc (c+d x)}}-\frac{2 a \sin (c+d x) \cos (c+d x)}{5 d e^2 \sqrt{e \csc (c+d x)}}-\frac{a \tan ^{-1}\left(\sqrt{\sin (c+d x)}\right)}{d e^2 \sqrt{\sin (c+d x)} \sqrt{e \csc (c+d x)}}+\frac{a \tanh ^{-1}\left(\sqrt{\sin (c+d x)}\right)}{d e^2 \sqrt{\sin (c+d x)} \sqrt{e \csc (c+d x)}}+\frac{6 a E\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{5 d e^2 \sqrt{\sin (c+d x)} \sqrt{e \csc (c+d x)}}",1,"(a*(-72*Cot[c + d*x]*Hypergeometric2F1[-1/4, 1/2, 3/4, Csc[c + d*x]^2] - 2*Sqrt[-Cot[c + d*x]^2]*(-30*ArcTan[Sqrt[Csc[c + d*x]]]*Sqrt[Csc[c + d*x]] + 15*Sqrt[Csc[c + d*x]]*(Log[1 - Sqrt[Csc[c + d*x]]] - Log[1 + Sqrt[Csc[c + d*x]]]) + 20*Sin[c + d*x] + 6*Sin[2*(c + d*x)])))/(60*d*e^2*Sqrt[-Cot[c + d*x]^2]*Sqrt[e*Csc[c + d*x]])","C",1
287,1,195,270,3.8618813,"\int (e \csc (c+d x))^{5/2} (a+a \sec (c+d x))^2 \, dx","Integrate[(e*Csc[c + d*x])^(5/2)*(a + a*Sec[c + d*x])^2,x]","-\frac{a^2 e^2 \cos ^4\left(\frac{1}{2} (c+d x)\right) \tan (c+d x) \sqrt{e \csc (c+d x)} \sec ^4\left(\frac{1}{2} \csc ^{-1}(\csc (c+d x))\right) \left(7 \sqrt{-\cot ^2(c+d x)} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\csc ^2(c+d x)\right)+4 \csc ^2(c+d x)+4 \sqrt{\cos ^2(c+d x)} \csc ^2(c+d x)+6 \sqrt{\cos ^2(c+d x)} \sqrt{\csc (c+d x)} \tan ^{-1}\left(\sqrt{\csc (c+d x)}\right)-6 \sqrt{\cos ^2(c+d x)} \sqrt{\csc (c+d x)} \tanh ^{-1}\left(\sqrt{\csc (c+d x)}\right)-7\right)}{3 d}","-\frac{4 a^2 e^2 \csc (c+d x) \sqrt{e \csc (c+d x)}}{3 d}-\frac{2 a^2 e^2 \cot (c+d x) \sqrt{e \csc (c+d x)}}{3 d}+\frac{5 a^2 e^2 \tan (c+d x) \sqrt{e \csc (c+d x)}}{3 d}-\frac{2 a^2 e^2 \csc (c+d x) \sec (c+d x) \sqrt{e \csc (c+d x)}}{3 d}+\frac{2 a^2 e^2 \sqrt{\sin (c+d x)} \sqrt{e \csc (c+d x)} \tan ^{-1}\left(\sqrt{\sin (c+d x)}\right)}{d}+\frac{2 a^2 e^2 \sqrt{\sin (c+d x)} \sqrt{e \csc (c+d x)} \tanh ^{-1}\left(\sqrt{\sin (c+d x)}\right)}{d}+\frac{7 a^2 e^2 \sqrt{\sin (c+d x)} F\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{e \csc (c+d x)}}{3 d}",1,"-1/3*(a^2*e^2*Cos[(c + d*x)/2]^4*Sqrt[e*Csc[c + d*x]]*(-7 + 6*ArcTan[Sqrt[Csc[c + d*x]]]*Sqrt[Cos[c + d*x]^2]*Sqrt[Csc[c + d*x]] - 6*ArcTanh[Sqrt[Csc[c + d*x]]]*Sqrt[Cos[c + d*x]^2]*Sqrt[Csc[c + d*x]] + 4*Csc[c + d*x]^2 + 4*Sqrt[Cos[c + d*x]^2]*Csc[c + d*x]^2 + 7*Sqrt[-Cot[c + d*x]^2]*Hypergeometric2F1[1/4, 1/2, 5/4, Csc[c + d*x]^2])*Sec[ArcCsc[Csc[c + d*x]]/2]^4*Tan[c + d*x])/d","C",0
288,1,195,240,4.9227844,"\int (e \csc (c+d x))^{3/2} (a+a \sec (c+d x))^2 \, dx","Integrate[(e*Csc[c + d*x])^(3/2)*(a + a*Sec[c + d*x])^2,x]","\frac{2 a^2 \cos ^4\left(\frac{1}{2} (c+d x)\right) \sec (c+d x) (e \csc (c+d x))^{3/2} \sec ^4\left(\frac{1}{2} \csc ^{-1}(\csc (c+d x))\right) \left(5 \sqrt{-\cot ^2(c+d x)} \sqrt{\csc (c+d x)} \, _2F_1\left(\frac{3}{4},\frac{3}{2};\frac{7}{4};\csc ^2(c+d x)\right)-6 \sqrt{\csc (c+d x)}-6 \sqrt{\cos ^2(c+d x)} \sqrt{\csc (c+d x)}+3 \sqrt{\cos ^2(c+d x)} \tan ^{-1}\left(\sqrt{\csc (c+d x)}\right)+3 \sqrt{\cos ^2(c+d x)} \tanh ^{-1}\left(\sqrt{\csc (c+d x)}\right)\right)}{3 d \csc ^{\frac{3}{2}}(c+d x)}","-\frac{4 a^2 e \sqrt{e \csc (c+d x)}}{d}-\frac{2 a^2 e \cos (c+d x) \sqrt{e \csc (c+d x)}}{d}-\frac{2 a^2 e \sec (c+d x) \sqrt{e \csc (c+d x)}}{d}-\frac{2 a^2 e \sqrt{\sin (c+d x)} \sqrt{e \csc (c+d x)} \tan ^{-1}\left(\sqrt{\sin (c+d x)}\right)}{d}+\frac{3 a^2 e \sin (c+d x) \tan (c+d x) \sqrt{e \csc (c+d x)}}{d}+\frac{2 a^2 e \sqrt{\sin (c+d x)} \sqrt{e \csc (c+d x)} \tanh ^{-1}\left(\sqrt{\sin (c+d x)}\right)}{d}-\frac{5 a^2 e \sqrt{\sin (c+d x)} E\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{e \csc (c+d x)}}{d}",1,"(2*a^2*Cos[(c + d*x)/2]^4*(e*Csc[c + d*x])^(3/2)*(3*ArcTan[Sqrt[Csc[c + d*x]]]*Sqrt[Cos[c + d*x]^2] + 3*ArcTanh[Sqrt[Csc[c + d*x]]]*Sqrt[Cos[c + d*x]^2] - 6*Sqrt[Csc[c + d*x]] - 6*Sqrt[Cos[c + d*x]^2]*Sqrt[Csc[c + d*x]] + 5*Sqrt[-Cot[c + d*x]^2]*Sqrt[Csc[c + d*x]]*Hypergeometric2F1[3/4, 3/2, 7/4, Csc[c + d*x]^2])*Sec[c + d*x]*Sec[ArcCsc[Csc[c + d*x]]/2]^4)/(3*d*Csc[c + d*x]^(3/2))","C",0
289,1,168,154,2.4249478,"\int \sqrt{e \csc (c+d x)} (a+a \sec (c+d x))^2 \, dx","Integrate[Sqrt[e*Csc[c + d*x]]*(a + a*Sec[c + d*x])^2,x]","-\frac{2 a^2 \sin \left(\frac{1}{2} (c+d x)\right) \cos ^5\left(\frac{1}{2} (c+d x)\right) \sec (c+d x) \sqrt{e \csc (c+d x)} \sec ^4\left(\frac{1}{2} \csc ^{-1}(\csc (c+d x))\right) \left(3 \sqrt{-\cot ^2(c+d x)} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\csc ^2(c+d x)\right)+2 \sqrt{\cos ^2(c+d x)} \sqrt{\csc (c+d x)} \tan ^{-1}\left(\sqrt{\csc (c+d x)}\right)-2 \sqrt{\cos ^2(c+d x)} \sqrt{\csc (c+d x)} \tanh ^{-1}\left(\sqrt{\csc (c+d x)}\right)-1\right)}{d}","\frac{a^2 \tan (c+d x) \sqrt{e \csc (c+d x)}}{d}+\frac{2 a^2 \sqrt{\sin (c+d x)} \sqrt{e \csc (c+d x)} \tan ^{-1}\left(\sqrt{\sin (c+d x)}\right)}{d}+\frac{2 a^2 \sqrt{\sin (c+d x)} \sqrt{e \csc (c+d x)} \tanh ^{-1}\left(\sqrt{\sin (c+d x)}\right)}{d}+\frac{3 a^2 \sqrt{\sin (c+d x)} F\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{e \csc (c+d x)}}{d}",1,"(-2*a^2*Cos[(c + d*x)/2]^5*Sqrt[e*Csc[c + d*x]]*(-1 + 2*ArcTan[Sqrt[Csc[c + d*x]]]*Sqrt[Cos[c + d*x]^2]*Sqrt[Csc[c + d*x]] - 2*ArcTanh[Sqrt[Csc[c + d*x]]]*Sqrt[Cos[c + d*x]^2]*Sqrt[Csc[c + d*x]] + 3*Sqrt[-Cot[c + d*x]^2]*Hypergeometric2F1[1/4, 1/2, 5/4, Csc[c + d*x]^2])*Sec[c + d*x]*Sec[ArcCsc[Csc[c + d*x]]/2]^4*Sin[(c + d*x)/2])/d","C",0
290,1,287,153,8.5230951,"\int \frac{(a+a \sec (c+d x))^2}{\sqrt{e \csc (c+d x)}} \, dx","Integrate[(a + a*Sec[c + d*x])^2/Sqrt[e*Csc[c + d*x]],x]","-\frac{\left(\cos \left(2 \left(\frac{c}{2}+\frac{d x}{2}\right)\right)+1\right)^2 \cos (c+d x) \left(\csc ^2(c+d x)-1\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right) (a \sec (c+d x)+a)^2 \left(\frac{\sqrt{1-\sin ^2(c+d x)} \sqrt{\csc (c+d x)} \, _2F_1\left(-\frac{1}{4},\frac{3}{2};\frac{3}{4};\csc ^2(c+d x)\right)}{\sqrt{1-\csc ^2(c+d x)}}-\frac{2 \sqrt{\csc (c+d x)} \sqrt{1-\csc ^2(c+d x)} \, _2F_1\left(\frac{3}{4},\frac{3}{2};\frac{7}{4};\csc ^2(c+d x)\right)}{3 \sqrt{1-\sin ^2(c+d x)}}-\tan ^{-1}\left(\sqrt{\csc (c+d x)}\right)-\tanh ^{-1}\left(\sqrt{\csc (c+d x)}\right)\right)}{2 d \sqrt{1-\sin ^2(c+d x)} \csc ^{\frac{3}{2}}(c+d x) \sqrt{e \csc (c+d x)} \left(\cos \left(2 \left(\frac{1}{2} \left(\csc ^{-1}(\csc (c+d x))-c\right)+\frac{c}{2}\right)\right)+1\right)^2}","\frac{a^2 \tan (c+d x)}{d \sqrt{e \csc (c+d x)}}-\frac{2 a^2 \tan ^{-1}\left(\sqrt{\sin (c+d x)}\right)}{d \sqrt{\sin (c+d x)} \sqrt{e \csc (c+d x)}}+\frac{2 a^2 \tanh ^{-1}\left(\sqrt{\sin (c+d x)}\right)}{d \sqrt{\sin (c+d x)} \sqrt{e \csc (c+d x)}}+\frac{a^2 E\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{d \sqrt{\sin (c+d x)} \sqrt{e \csc (c+d x)}}",1,"-1/2*((1 + Cos[2*(c/2 + (d*x)/2)])^2*Cos[c + d*x]*(-1 + Csc[c + d*x]^2)*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2*(-ArcTan[Sqrt[Csc[c + d*x]]] - ArcTanh[Sqrt[Csc[c + d*x]]] - (2*Sqrt[Csc[c + d*x]]*Sqrt[1 - Csc[c + d*x]^2]*Hypergeometric2F1[3/4, 3/2, 7/4, Csc[c + d*x]^2])/(3*Sqrt[1 - Sin[c + d*x]^2]) + (Sqrt[Csc[c + d*x]]*Hypergeometric2F1[-1/4, 3/2, 3/4, Csc[c + d*x]^2]*Sqrt[1 - Sin[c + d*x]^2])/Sqrt[1 - Csc[c + d*x]^2]))/(d*(1 + Cos[2*(c/2 + (-c + ArcCsc[Csc[c + d*x]])/2)])^2*Csc[c + d*x]^(3/2)*Sqrt[e*Csc[c + d*x]]*Sqrt[1 - Sin[c + d*x]^2])","C",0
291,1,164,222,7.8158487,"\int \frac{(a+a \sec (c+d x))^2}{(e \csc (c+d x))^{3/2}} \, dx","Integrate[(a + a*Sec[c + d*x])^2/(e*Csc[c + d*x])^(3/2),x]","\frac{2 a^2 \cos ^4\left(\frac{1}{2} (c+d x)\right) \tan (c+d x) \sqrt{e \csc (c+d x)} \sec ^4\left(\frac{1}{2} \csc ^{-1}(\csc (c+d x))\right) \left(-6 \sqrt{\cos ^2(c+d x)} \, _2F_1\left(-\frac{1}{4},1;\frac{3}{4};\csc ^2(c+d x)\right)+3 \sqrt{-\cot ^2(c+d x)} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\csc ^2(c+d x)\right)+\sin ^2(c+d x) \sqrt{-\cot ^2(c+d x)} \, _2F_1\left(-\frac{3}{4},\frac{3}{2};\frac{1}{4};\csc ^2(c+d x)\right)+3\right)}{3 d e^2}","-\frac{4 a^2}{d e \sqrt{e \csc (c+d x)}}-\frac{2 a^2 \cos (c+d x)}{3 d e \sqrt{e \csc (c+d x)}}+\frac{a^2 \sec (c+d x)}{d e \sqrt{e \csc (c+d x)}}+\frac{2 a^2 \tan ^{-1}\left(\sqrt{\sin (c+d x)}\right)}{d e \sqrt{\sin (c+d x)} \sqrt{e \csc (c+d x)}}+\frac{2 a^2 \tanh ^{-1}\left(\sqrt{\sin (c+d x)}\right)}{d e \sqrt{\sin (c+d x)} \sqrt{e \csc (c+d x)}}-\frac{a^2 F\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{3 d e \sqrt{\sin (c+d x)} \sqrt{e \csc (c+d x)}}",1,"(2*a^2*Cos[(c + d*x)/2]^4*Sqrt[e*Csc[c + d*x]]*Sec[ArcCsc[Csc[c + d*x]]/2]^4*(3 - 6*Sqrt[Cos[c + d*x]^2]*Hypergeometric2F1[-1/4, 1, 3/4, Csc[c + d*x]^2] + 3*Sqrt[-Cot[c + d*x]^2]*Hypergeometric2F1[1/4, 1/2, 5/4, Csc[c + d*x]^2] + Sqrt[-Cot[c + d*x]^2]*Hypergeometric2F1[-3/4, 3/2, 1/4, Csc[c + d*x]^2]*Sin[c + d*x]^2)*Tan[c + d*x])/(3*d*e^2)","C",0
292,1,152,236,11.3120243,"\int \frac{(a+a \sec (c+d x))^2}{(e \csc (c+d x))^{5/2}} \, dx","Integrate[(a + a*Sec[c + d*x])^2/(e*Csc[c + d*x])^(5/2),x]","\frac{2 a^2 \cos ^4\left(\frac{1}{2} (c+d x)\right) \tan (c+d x) \sec ^4\left(\frac{1}{2} \csc ^{-1}(\csc (c+d x))\right) \left(3 \sqrt{-\cot ^2(c+d x)} \left(\sin ^2(c+d x) \, _2F_1\left(-\frac{5}{4},\frac{3}{2};-\frac{1}{4};\csc ^2(c+d x)\right)-10 \, _2F_1\left(-\frac{1}{4},\frac{3}{2};\frac{3}{4};\csc ^2(c+d x)\right)\right)-10 \sqrt{\cos ^2(c+d x)} \, _2F_1\left(-\frac{3}{4},1;\frac{1}{4};\csc ^2(c+d x)\right)\right)}{15 d e^2 \sqrt{e \csc (c+d x)}}","-\frac{4 a^2 \sin (c+d x)}{3 d e^2 \sqrt{e \csc (c+d x)}}+\frac{a^2 \tan (c+d x)}{d e^2 \sqrt{e \csc (c+d x)}}-\frac{2 a^2 \sin (c+d x) \cos (c+d x)}{5 d e^2 \sqrt{e \csc (c+d x)}}-\frac{2 a^2 \tan ^{-1}\left(\sqrt{\sin (c+d x)}\right)}{d e^2 \sqrt{\sin (c+d x)} \sqrt{e \csc (c+d x)}}+\frac{2 a^2 \tanh ^{-1}\left(\sqrt{\sin (c+d x)}\right)}{d e^2 \sqrt{\sin (c+d x)} \sqrt{e \csc (c+d x)}}-\frac{9 a^2 E\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{5 d e^2 \sqrt{\sin (c+d x)} \sqrt{e \csc (c+d x)}}",1,"(2*a^2*Cos[(c + d*x)/2]^4*Sec[ArcCsc[Csc[c + d*x]]/2]^4*(-10*Sqrt[Cos[c + d*x]^2]*Hypergeometric2F1[-3/4, 1, 1/4, Csc[c + d*x]^2] + 3*Sqrt[-Cot[c + d*x]^2]*(-10*Hypergeometric2F1[-1/4, 3/2, 3/4, Csc[c + d*x]^2] + Hypergeometric2F1[-5/4, 3/2, -1/4, Csc[c + d*x]^2]*Sin[c + d*x]^2))*Tan[c + d*x])/(15*d*e^2*Sqrt[e*Csc[c + d*x]])","C",0
293,1,131,155,1.035497,"\int \frac{(e \csc (c+d x))^{5/2}}{a+a \sec (c+d x)} \, dx","Integrate[(e*Csc[c + d*x])^(5/2)/(a + a*Sec[c + d*x]),x]","-\frac{\sin ^{\frac{5}{2}}(c+d x) \csc ^2\left(\frac{1}{2} (c+d x)\right) \sec ^4\left(\frac{1}{2} (c+d x)\right) (e \csc (c+d x))^{5/2} \left(2 \sqrt{\sin (c+d x)} (2 \cos (c+d x)+\cos (2 (c+d x))+4)+(\cos (c+d x)-2 \cos (2 (c+d x))-\cos (3 (c+d x))+2) F\left(\left.\frac{1}{4} (-2 c-2 d x+\pi )\right|2\right)\right)}{168 a d}","-\frac{2 e^2 \csc ^3(c+d x) \sqrt{e \csc (c+d x)}}{7 a d}+\frac{2 e^2 \cot (c+d x) \csc ^2(c+d x) \sqrt{e \csc (c+d x)}}{7 a d}-\frac{4 e^2 \cot (c+d x) \sqrt{e \csc (c+d x)}}{21 a d}+\frac{4 e^2 \sqrt{\sin (c+d x)} F\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{e \csc (c+d x)}}{21 a d}",1,"-1/168*(Csc[(c + d*x)/2]^2*(e*Csc[c + d*x])^(5/2)*Sec[(c + d*x)/2]^4*((2 + Cos[c + d*x] - 2*Cos[2*(c + d*x)] - Cos[3*(c + d*x)])*EllipticF[(-2*c + Pi - 2*d*x)/4, 2] + 2*(4 + 2*Cos[c + d*x] + Cos[2*(c + d*x)])*Sqrt[Sin[c + d*x]])*Sin[c + d*x]^(5/2))/(a*d)","A",1
294,1,230,145,1.3859345,"\int \frac{(e \csc (c+d x))^{3/2}}{a+a \sec (c+d x)} \, dx","Integrate[(e*Csc[c + d*x])^(3/2)/(a + a*Sec[c + d*x]),x]","\frac{\cos ^2\left(\frac{1}{2} (c+d x)\right) (e \csc (c+d x))^{3/2} \left(-\frac{6 \tan (c+d x) \left(\sec ^2\left(\frac{1}{2} (c+d x)\right)+4 \sec (c) \cos (d x)\right)}{d}+\frac{8 \sqrt{2} e^{i (c-d x)} \sqrt{\frac{i e^{i (c+d x)}}{-1+e^{2 i (c+d x)}}} \left(\left(1+e^{2 i c}\right) e^{2 i 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)-3 e^{2 i (c+d x)}+3\right) \sec (c+d x)}{\left(1+e^{2 i c}\right) d \csc ^{\frac{3}{2}}(c+d x)}\right)}{15 a (\sec (c+d x)+1)}","-\frac{2 e \csc ^2(c+d x) \sqrt{e \csc (c+d x)}}{5 a d}-\frac{4 e \cos (c+d x) \sqrt{e \csc (c+d x)}}{5 a d}+\frac{2 e \cot (c+d x) \csc (c+d x) \sqrt{e \csc (c+d x)}}{5 a d}-\frac{4 e \sqrt{\sin (c+d x)} E\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{e \csc (c+d x)}}{5 a d}",1,"(Cos[(c + d*x)/2]^2*(e*Csc[c + d*x])^(3/2)*((8*Sqrt[2]*E^(I*(c - d*x))*Sqrt[(I*E^(I*(c + d*x)))/(-1 + E^((2*I)*(c + d*x)))]*(3 - 3*E^((2*I)*(c + d*x)) + E^((2*I)*d*x)*(1 + E^((2*I)*c))*Sqrt[1 - E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/2, 3/4, 7/4, E^((2*I)*(c + d*x))])*Sec[c + d*x])/(d*(1 + E^((2*I)*c))*Csc[c + d*x]^(3/2)) - (6*(4*Cos[d*x]*Sec[c] + Sec[(c + d*x)/2]^2)*Tan[c + d*x])/d))/(15*a*(1 + Sec[c + d*x]))","C",1
295,1,60,105,0.3774338,"\int \frac{\sqrt{e \csc (c+d x)}}{a+a \sec (c+d x)} \, dx","Integrate[Sqrt[e*Csc[c + d*x]]/(a + a*Sec[c + d*x]),x]","\frac{2 (e \csc (c+d x))^{3/2} \left(\cos (c+d x)-2 \sin ^{\frac{3}{2}}(c+d x) F\left(\left.\frac{1}{4} (-2 c-2 d x+\pi )\right|2\right)-1\right)}{3 a d e}","-\frac{2 \csc (c+d x) \sqrt{e \csc (c+d x)}}{3 a d}+\frac{2 \cot (c+d x) \sqrt{e \csc (c+d x)}}{3 a d}+\frac{4 \sqrt{\sin (c+d x)} F\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{e \csc (c+d x)}}{3 a d}",1,"(2*(e*Csc[c + d*x])^(3/2)*(-1 + Cos[c + d*x] - 2*EllipticF[(-2*c + Pi - 2*d*x)/4, 2]*Sin[c + d*x]^(3/2)))/(3*a*d*e)","A",1
296,1,95,99,0.6169527,"\int \frac{1}{\sqrt{e \csc (c+d x)} (a+a \sec (c+d x))} \, dx","Integrate[1/(Sqrt[e*Csc[c + d*x]]*(a + a*Sec[c + d*x])),x]","\frac{6 (\cot (c+d x)-\csc (c+d x)+2 i)-4 \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) (\cot (c+d x)+i)}{3 a d \sqrt{e \csc (c+d x)}}","-\frac{2 \csc (c+d x)}{a d \sqrt{e \csc (c+d x)}}+\frac{2 \cot (c+d x)}{a d \sqrt{e \csc (c+d x)}}+\frac{4 E\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{a d \sqrt{\sin (c+d x)} \sqrt{e \csc (c+d x)}}",1,"(6*(2*I + Cot[c + d*x] - Csc[c + d*x]) - 4*Sqrt[1 - E^((2*I)*(c + d*x))]*(I + Cot[c + d*x])*Hypergeometric2F1[1/2, 3/4, 7/4, E^((2*I)*(c + d*x))])/(3*a*d*Sqrt[e*Csc[c + d*x]])","C",1
297,1,70,106,0.3953411,"\int \frac{1}{(e \csc (c+d x))^{3/2} (a+a \sec (c+d x))} \, dx","Integrate[1/((e*Csc[c + d*x])^(3/2)*(a + a*Sec[c + d*x])),x]","\frac{4 F\left(\left.\frac{1}{4} (-2 c-2 d x+\pi )\right|2\right)-2 \sqrt{\sin (c+d x)} (\cos (c+d x)-3)}{3 a d \sin ^{\frac{3}{2}}(c+d x) (e \csc (c+d x))^{3/2}}","\frac{2}{a d e \sqrt{e \csc (c+d x)}}-\frac{2 \cos (c+d x)}{3 a d e \sqrt{e \csc (c+d x)}}-\frac{4 F\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{3 a d e \sqrt{\sin (c+d x)} \sqrt{e \csc (c+d x)}}",1,"(4*EllipticF[(-2*c + Pi - 2*d*x)/4, 2] - 2*(-3 + Cos[c + d*x])*Sqrt[Sin[c + d*x]])/(3*a*d*(e*Csc[c + d*x])^(3/2)*Sin[c + d*x]^(3/2))","A",1
298,1,100,120,0.829905,"\int \frac{1}{(e \csc (c+d x))^{5/2} (a+a \sec (c+d x))} \, dx","Integrate[1/((e*Csc[c + d*x])^(5/2)*(a + a*Sec[c + d*x])),x]","\frac{8 \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) (\cot (c+d x)+i)+20 \sin (c+d x)-6 (\sin (2 (c+d x))+4 i)}{30 a d e^2 \sqrt{e \csc (c+d x)}}","\frac{2 \sin (c+d x)}{3 a d e^2 \sqrt{e \csc (c+d x)}}-\frac{2 \sin (c+d x) \cos (c+d x)}{5 a d e^2 \sqrt{e \csc (c+d x)}}-\frac{4 E\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{5 a d e^2 \sqrt{\sin (c+d x)} \sqrt{e \csc (c+d x)}}",1,"(8*Sqrt[1 - E^((2*I)*(c + d*x))]*(I + Cot[c + d*x])*Hypergeometric2F1[1/2, 3/4, 7/4, E^((2*I)*(c + d*x))] + 20*Sin[c + d*x] - 6*(4*I + Sin[2*(c + d*x)]))/(30*a*d*e^2*Sqrt[e*Csc[c + d*x]])","C",1
299,1,91,149,0.5777169,"\int \frac{1}{(e \csc (c+d x))^{7/2} (a+a \sec (c+d x))} \, dx","Integrate[1/((e*Csc[c + d*x])^(7/2)*(a + a*Sec[c + d*x])),x]","\frac{\sqrt{e \csc (c+d x)} \left(126 \sin (c+d x)+10 \sin (2 (c+d x))-42 \sin (3 (c+d x))+15 \sin (4 (c+d x))+80 \sqrt{\sin (c+d x)} F\left(\left.\frac{1}{4} (-2 c-2 d x+\pi )\right|2\right)\right)}{420 a d e^4}","\frac{2 \cos ^3(c+d x)}{7 a d e^3 \sqrt{e \csc (c+d x)}}-\frac{2 \cos (c+d x)}{21 a d e^3 \sqrt{e \csc (c+d x)}}+\frac{2 \sin ^2(c+d x)}{5 a d e^3 \sqrt{e \csc (c+d x)}}-\frac{4 F\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{21 a d e^3 \sqrt{\sin (c+d x)} \sqrt{e \csc (c+d x)}}",1,"(Sqrt[e*Csc[c + d*x]]*(80*EllipticF[(-2*c + Pi - 2*d*x)/4, 2]*Sqrt[Sin[c + d*x]] + 126*Sin[c + d*x] + 10*Sin[2*(c + d*x)] - 42*Sin[3*(c + d*x)] + 15*Sin[4*(c + d*x)]))/(420*a*d*e^4)","A",1
300,1,115,268,1.1993836,"\int \frac{(e \csc (c+d x))^{5/2}}{(a+a \sec (c+d x))^2} \, dx","Integrate[(e*Csc[c + d*x])^(5/2)/(a + a*Sec[c + d*x])^2,x]","-\frac{e^3 \csc ^2\left(\frac{1}{2} (c+d x)\right) \sec ^6\left(\frac{1}{2} (c+d x)\right) \left(97 \cos (c+d x)+4 \cos (2 (c+d x))+\cos (3 (c+d x))+\sin ^{\frac{11}{2}}(c+d x) \csc ^4\left(\frac{1}{2} (c+d x)\right) F\left(\left.\frac{1}{4} (-2 c-2 d x+\pi )\right|2\right)+52\right)}{3696 a^2 d \sqrt{e \csc (c+d x)}}","\frac{4 e^2 \csc ^5(c+d x) \sqrt{e \csc (c+d x)}}{11 a^2 d}-\frac{4 e^2 \csc ^3(c+d x) \sqrt{e \csc (c+d x)}}{7 a^2 d}-\frac{2 e^2 \cot ^3(c+d x) \csc ^2(c+d x) \sqrt{e \csc (c+d x)}}{11 a^2 d}-\frac{2 e^2 \cot (c+d x) \csc ^4(c+d x) \sqrt{e \csc (c+d x)}}{11 a^2 d}+\frac{16 e^2 \cot (c+d x) \csc ^2(c+d x) \sqrt{e \csc (c+d x)}}{77 a^2 d}-\frac{4 e^2 \cot (c+d x) \sqrt{e \csc (c+d x)}}{231 a^2 d}+\frac{4 e^2 \sqrt{\sin (c+d x)} F\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{e \csc (c+d x)}}{231 a^2 d}",1,"-1/3696*(e^3*Csc[(c + d*x)/2]^2*Sec[(c + d*x)/2]^6*(52 + 97*Cos[c + d*x] + 4*Cos[2*(c + d*x)] + Cos[3*(c + d*x)] + Csc[(c + d*x)/2]^4*EllipticF[(-2*c + Pi - 2*d*x)/4, 2]*Sin[c + d*x]^(11/2)))/(a^2*d*Sqrt[e*Csc[c + d*x]])","A",1
301,1,247,250,1.8323365,"\int \frac{(e \csc (c+d x))^{3/2}}{(a+a \sec (c+d x))^2} \, dx","Integrate[(e*Csc[c + d*x])^(3/2)/(a + a*Sec[c + d*x])^2,x]","\frac{\cos ^4\left(\frac{1}{2} (c+d x)\right) \sec (c+d x) (e \csc (c+d x))^{3/2} \left(-\frac{2 \tan (c+d x) \left((13 \cos (c+d x)+8) \sec ^4\left(\frac{1}{2} (c+d x)\right)+24 \sec (c) \cos (d x)\right)}{d}+\frac{16 \sqrt{2} e^{i (c-d x)} \sqrt{\frac{i e^{i (c+d x)}}{-1+e^{2 i (c+d x)}}} \left(\left(1+e^{2 i c}\right) e^{2 i 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)-3 e^{2 i (c+d x)}+3\right) \sec (c+d x)}{\left(1+e^{2 i c}\right) d \csc ^{\frac{3}{2}}(c+d x)}\right)}{45 a^2 (\sec (c+d x)+1)^2}","\frac{4 e \csc ^4(c+d x) \sqrt{e \csc (c+d x)}}{9 a^2 d}-\frac{4 e \csc ^2(c+d x) \sqrt{e \csc (c+d x)}}{5 a^2 d}-\frac{4 e \cos (c+d x) \sqrt{e \csc (c+d x)}}{15 a^2 d}-\frac{2 e \cot ^3(c+d x) \csc (c+d x) \sqrt{e \csc (c+d x)}}{9 a^2 d}-\frac{2 e \cot (c+d x) \csc ^3(c+d x) \sqrt{e \csc (c+d x)}}{9 a^2 d}+\frac{16 e \cot (c+d x) \csc (c+d x) \sqrt{e \csc (c+d x)}}{45 a^2 d}-\frac{4 e \sqrt{\sin (c+d x)} E\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{e \csc (c+d x)}}{15 a^2 d}",1,"(Cos[(c + d*x)/2]^4*(e*Csc[c + d*x])^(3/2)*Sec[c + d*x]*((16*Sqrt[2]*E^(I*(c - d*x))*Sqrt[(I*E^(I*(c + d*x)))/(-1 + E^((2*I)*(c + d*x)))]*(3 - 3*E^((2*I)*(c + d*x)) + E^((2*I)*d*x)*(1 + E^((2*I)*c))*Sqrt[1 - E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/2, 3/4, 7/4, E^((2*I)*(c + d*x))])*Sec[c + d*x])/(d*(1 + E^((2*I)*c))*Csc[c + d*x]^(3/2)) - (2*(24*Cos[d*x]*Sec[c] + (8 + 13*Cos[c + d*x])*Sec[(c + d*x)/2]^4)*Tan[c + d*x])/d))/(45*a^2*(1 + Sec[c + d*x])^2)","C",1
302,1,82,201,0.7471021,"\int \frac{\sqrt{e \csc (c+d x)}}{(a+a \sec (c+d x))^2} \, dx","Integrate[Sqrt[e*Csc[c + d*x]]/(a + a*Sec[c + d*x])^2,x]","-\frac{4 \csc ^3(c+d x) \sqrt{e \csc (c+d x)} \left(2 \sin ^4\left(\frac{1}{2} (c+d x)\right) (11 \cos (c+d x)+8)+5 \sin ^{\frac{7}{2}}(c+d x) F\left(\left.\frac{1}{4} (-2 c-2 d x+\pi )\right|2\right)\right)}{21 a^2 d}","\frac{4 \csc ^3(c+d x) \sqrt{e \csc (c+d x)}}{7 a^2 d}-\frac{4 \csc (c+d x) \sqrt{e \csc (c+d x)}}{3 a^2 d}-\frac{2 \cot ^3(c+d x) \sqrt{e \csc (c+d x)}}{7 a^2 d}-\frac{2 \cot (c+d x) \csc ^2(c+d x) \sqrt{e \csc (c+d x)}}{7 a^2 d}+\frac{16 \cot (c+d x) \sqrt{e \csc (c+d x)}}{21 a^2 d}+\frac{20 \sqrt{\sin (c+d x)} F\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right) \sqrt{e \csc (c+d x)}}{21 a^2 d}",1,"(-4*Csc[c + d*x]^3*Sqrt[e*Csc[c + d*x]]*(2*(8 + 11*Cos[c + d*x])*Sin[(c + d*x)/2]^4 + 5*EllipticF[(-2*c + Pi - 2*d*x)/4, 2]*Sin[c + d*x]^(7/2)))/(21*a^2*d)","A",1
303,1,252,199,1.5988313,"\int \frac{1}{\sqrt{e \csc (c+d x)} (a+a \sec (c+d x))^2} \, dx","Integrate[1/(Sqrt[e*Csc[c + d*x]]*(a + a*Sec[c + d*x])^2),x]","\frac{4 \cos ^4\left(\frac{1}{2} (c+d x)\right) \sqrt{\csc (c+d x)} \sec ^2(c+d x) \left(-3 \sqrt{\csc (c+d x)} \left((5 \cos (2 c)-23) \sec (c) \cos (d x)-2 \left(5 \sin (c) \sin (d x)+\sec ^2\left(\frac{1}{2} (c+d x)\right)-10\right)\right)-\frac{28 \sqrt{2} e^{i (c-d x)} \sqrt{\frac{i e^{i (c+d x)}}{-1+e^{2 i (c+d x)}}} \left(\left(1+e^{2 i c}\right) e^{2 i 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)-3 e^{2 i (c+d x)}+3\right)}{1+e^{2 i c}}\right)}{15 a^2 d (\sec (c+d x)+1)^2 \sqrt{e \csc (c+d x)}}","\frac{4 \csc ^3(c+d x)}{5 a^2 d \sqrt{e \csc (c+d x)}}-\frac{4 \csc (c+d x)}{a^2 d \sqrt{e \csc (c+d x)}}-\frac{2 \cot ^3(c+d x)}{5 a^2 d \sqrt{e \csc (c+d x)}}-\frac{2 \cot (c+d x) \csc ^2(c+d x)}{5 a^2 d \sqrt{e \csc (c+d x)}}+\frac{16 \cot (c+d x)}{5 a^2 d \sqrt{e \csc (c+d x)}}+\frac{28 E\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{5 a^2 d \sqrt{\sin (c+d x)} \sqrt{e \csc (c+d x)}}",1,"(4*Cos[(c + d*x)/2]^4*Sqrt[Csc[c + d*x]]*Sec[c + d*x]^2*((-28*Sqrt[2]*E^(I*(c - d*x))*Sqrt[(I*E^(I*(c + d*x)))/(-1 + E^((2*I)*(c + d*x)))]*(3 - 3*E^((2*I)*(c + d*x)) + E^((2*I)*d*x)*(1 + E^((2*I)*c))*Sqrt[1 - E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/2, 3/4, 7/4, E^((2*I)*(c + d*x))]))/(1 + E^((2*I)*c)) - 3*Sqrt[Csc[c + d*x]]*((-23 + 5*Cos[2*c])*Cos[d*x]*Sec[c] - 2*(-10 + Sec[(c + d*x)/2]^2 + 5*Sin[c]*Sin[d*x]))))/(15*a^2*d*Sqrt[e*Csc[c + d*x]]*(1 + Sec[c + d*x])^2)","C",1
304,1,101,213,0.5574147,"\int \frac{1}{(e \csc (c+d x))^{3/2} (a+a \sec (c+d x))^2} \, dx","Integrate[1/((e*Csc[c + d*x])^(3/2)*(a + a*Sec[c + d*x])^2),x]","\frac{\sec ^2\left(\frac{1}{2} (c+d x)\right) \left(\sqrt{\sin (c+d x)} (10 \cos (c+d x)-\cos (2 (c+d x))+15)+12 (\cos (c+d x)+1) F\left(\left.\frac{1}{4} (-2 c-2 d x+\pi )\right|2\right)\right)}{6 a^2 d \sin ^{\frac{3}{2}}(c+d x) (e \csc (c+d x))^{3/2}}","\frac{4 \csc ^2(c+d x)}{3 a^2 d e \sqrt{e \csc (c+d x)}}+\frac{4}{a^2 d e \sqrt{e \csc (c+d x)}}-\frac{4 \cos (c+d x)}{3 a^2 d e \sqrt{e \csc (c+d x)}}-\frac{2 \cot (c+d x) \csc (c+d x)}{3 a^2 d e \sqrt{e \csc (c+d x)}}-\frac{2 \cos (c+d x) \cot ^2(c+d x)}{3 a^2 d e \sqrt{e \csc (c+d x)}}-\frac{4 F\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{a^2 d e \sqrt{\sin (c+d x)} \sqrt{e \csc (c+d x)}}",1,"(Sec[(c + d*x)/2]^2*(12*(1 + Cos[c + d*x])*EllipticF[(-2*c + Pi - 2*d*x)/4, 2] + (15 + 10*Cos[c + d*x] - Cos[2*(c + d*x)])*Sqrt[Sin[c + d*x]]))/(6*a^2*d*(e*Csc[c + d*x])^(3/2)*Sin[c + d*x]^(3/2))","A",1
305,1,125,215,2.1448951,"\int \frac{1}{(e \csc (c+d x))^{5/2} (a+a \sec (c+d x))^2} \, dx","Integrate[1/((e*Csc[c + d*x])^(5/2)*(a + a*Sec[c + d*x])^2),x]","\frac{88 \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) (\cot (c+d x)+i)-123 \cot (c+d x)+\csc (c+d x) (-264 i \sin (c+d x)-20 \cos (2 (c+d x))+3 \cos (3 (c+d x))+140)}{30 a^2 d e^2 \sqrt{e \csc (c+d x)}}","\frac{4 \csc (c+d x)}{a^2 d e^2 \sqrt{e \csc (c+d x)}}-\frac{2 \cot (c+d x)}{a^2 d e^2 \sqrt{e \csc (c+d x)}}+\frac{4 \sin (c+d x)}{3 a^2 d e^2 \sqrt{e \csc (c+d x)}}-\frac{2 \cos ^2(c+d x) \cot (c+d x)}{a^2 d e^2 \sqrt{e \csc (c+d x)}}-\frac{12 \sin (c+d x) \cos (c+d x)}{5 a^2 d e^2 \sqrt{e \csc (c+d x)}}-\frac{44 E\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{5 a^2 d e^2 \sqrt{\sin (c+d x)} \sqrt{e \csc (c+d x)}}",1,"(-123*Cot[c + d*x] + 88*Sqrt[1 - E^((2*I)*(c + d*x))]*(I + Cot[c + d*x])*Hypergeometric2F1[1/2, 3/4, 7/4, E^((2*I)*(c + d*x))] + Csc[c + d*x]*(140 - 20*Cos[2*(c + d*x)] + 3*Cos[3*(c + d*x)] - (264*I)*Sin[c + d*x]))/(30*a^2*d*e^2*Sqrt[e*Csc[c + d*x]])","C",1
306,1,94,172,2.2016591,"\int \frac{1}{(e \csc (c+d x))^{7/2} (a+a \sec (c+d x))^2} \, dx","Integrate[1/((e*Csc[c + d*x])^(7/2)*(a + a*Sec[c + d*x])^2),x]","\frac{\sqrt{\sin (c+d x)} \sqrt{e \csc (c+d x)} \left(\sqrt{\sin (c+d x)} (305 \cos (c+d x)-84 \cos (2 (c+d x))+15 \cos (3 (c+d x))-756)-520 F\left(\left.\frac{1}{4} (-2 c-2 d x+\pi )\right|2\right)\right)}{210 a^2 d e^4}","-\frac{4}{a^2 d e^3 \sqrt{e \csc (c+d x)}}+\frac{2 \cos ^3(c+d x)}{7 a^2 d e^3 \sqrt{e \csc (c+d x)}}+\frac{26 \cos (c+d x)}{21 a^2 d e^3 \sqrt{e \csc (c+d x)}}+\frac{4 \sin ^2(c+d x)}{5 a^2 d e^3 \sqrt{e \csc (c+d x)}}+\frac{52 F\left(\left.\frac{1}{2} \left(c+d x-\frac{\pi }{2}\right)\right|2\right)}{21 a^2 d e^3 \sqrt{\sin (c+d x)} \sqrt{e \csc (c+d x)}}",1,"(Sqrt[e*Csc[c + d*x]]*(-520*EllipticF[(-2*c + Pi - 2*d*x)/4, 2] + (-756 + 305*Cos[c + d*x] - 84*Cos[2*(c + d*x)] + 15*Cos[3*(c + d*x)])*Sqrt[Sin[c + d*x]])*Sqrt[Sin[c + d*x]])/(210*a^2*d*e^4)","A",1