1,1,22,11,0.0761647,"\int \sin (a+b x) \, dx","Integrate[Sin[a + b*x],x]","\frac{\sin (a) \sin (b x)}{b}-\frac{\cos (a) \cos (b x)}{b}","-\frac{\cos (a+b x)}{b}",1,"-((Cos[a]*Cos[b*x])/b) + (Sin[a]*Sin[b*x])/b","A",1
2,1,23,25,0.0305569,"\int \sin ^2(a+b x) \, dx","Integrate[Sin[a + b*x]^2,x]","-\frac{\sin (2 (a+b x))-2 (a+b x)}{4 b}","\frac{x}{2}-\frac{\sin (a+b x) \cos (a+b x)}{2 b}",1,"-1/4*(-2*(a + b*x) + Sin[2*(a + b*x)])/b","A",1
3,1,29,27,0.0101842,"\int \sin ^3(a+b x) \, dx","Integrate[Sin[a + b*x]^3,x]","\frac{\cos (3 (a+b x))}{12 b}-\frac{3 \cos (a+b x)}{4 b}","\frac{\cos ^3(a+b x)}{3 b}-\frac{\cos (a+b x)}{b}",1,"(-3*Cos[a + b*x])/(4*b) + Cos[3*(a + b*x)]/(12*b)","A",1
4,1,33,46,0.0384119,"\int \sin ^4(a+b x) \, dx","Integrate[Sin[a + b*x]^4,x]","\frac{12 (a+b x)-8 \sin (2 (a+b x))+\sin (4 (a+b x))}{32 b}","-\frac{\sin ^3(a+b x) \cos (a+b x)}{4 b}-\frac{3 \sin (a+b x) \cos (a+b x)}{8 b}+\frac{3 x}{8}",1,"(12*(a + b*x) - 8*Sin[2*(a + b*x)] + Sin[4*(a + b*x)])/(32*b)","A",1
5,1,44,42,0.0119383,"\int \sin ^5(a+b x) \, dx","Integrate[Sin[a + b*x]^5,x]","-\frac{5 \cos (a+b x)}{8 b}+\frac{5 \cos (3 (a+b x))}{48 b}-\frac{\cos (5 (a+b x))}{80 b}","-\frac{\cos ^5(a+b x)}{5 b}+\frac{2 \cos ^3(a+b x)}{3 b}-\frac{\cos (a+b x)}{b}",1,"(-5*Cos[a + b*x])/(8*b) + (5*Cos[3*(a + b*x)])/(48*b) - Cos[5*(a + b*x)]/(80*b)","A",1
6,1,45,67,0.0400816,"\int \sin ^6(a+b x) \, dx","Integrate[Sin[a + b*x]^6,x]","\frac{-45 \sin (2 (a+b x))+9 \sin (4 (a+b x))-\sin (6 (a+b x))+60 a+60 b x}{192 b}","-\frac{\sin ^5(a+b x) \cos (a+b x)}{6 b}-\frac{5 \sin ^3(a+b x) \cos (a+b x)}{24 b}-\frac{5 \sin (a+b x) \cos (a+b x)}{16 b}+\frac{5 x}{16}",1,"(60*a + 60*b*x - 45*Sin[2*(a + b*x)] + 9*Sin[4*(a + b*x)] - Sin[6*(a + b*x)])/(192*b)","A",1
7,1,59,54,0.0082383,"\int \sin ^7(a+b x) \, dx","Integrate[Sin[a + b*x]^7,x]","-\frac{35 \cos (a+b x)}{64 b}+\frac{7 \cos (3 (a+b x))}{64 b}-\frac{7 \cos (5 (a+b x))}{320 b}+\frac{\cos (7 (a+b x))}{448 b}","\frac{\cos ^7(a+b x)}{7 b}-\frac{3 \cos ^5(a+b x)}{5 b}+\frac{\cos ^3(a+b x)}{b}-\frac{\cos (a+b x)}{b}",1,"(-35*Cos[a + b*x])/(64*b) + (7*Cos[3*(a + b*x)])/(64*b) - (7*Cos[5*(a + b*x)])/(320*b) + Cos[7*(a + b*x)]/(448*b)","A",1
8,1,55,88,0.0514382,"\int \sin ^8(a+b x) \, dx","Integrate[Sin[a + b*x]^8,x]","\frac{-672 \sin (2 (a+b x))+168 \sin (4 (a+b x))-32 \sin (6 (a+b x))+3 \sin (8 (a+b x))+840 a+840 b x}{3072 b}","-\frac{\sin ^7(a+b x) \cos (a+b x)}{8 b}-\frac{7 \sin ^5(a+b x) \cos (a+b x)}{48 b}-\frac{35 \sin ^3(a+b x) \cos (a+b x)}{192 b}-\frac{35 \sin (a+b x) \cos (a+b x)}{128 b}+\frac{35 x}{128}",1,"(840*a + 840*b*x - 672*Sin[2*(a + b*x)] + 168*Sin[4*(a + b*x)] - 32*Sin[6*(a + b*x)] + 3*Sin[8*(a + b*x)])/(3072*b)","A",1
9,1,45,60,0.202097,"\int \sin ^{\frac{7}{2}}(b x) \, dx","Integrate[Sin[b*x]^(7/2),x]","\frac{\sqrt{\sin (b x)} (3 \cos (3 b x)-23 \cos (b x))-20 F\left(\left.\frac{1}{4} (\pi -2 b x)\right|2\right)}{42 b}","-\frac{10 F\left(\left.\frac{\pi }{4}-\frac{b x}{2}\right|2\right)}{21 b}-\frac{2 \sin ^{\frac{5}{2}}(b x) \cos (b x)}{7 b}-\frac{10 \sqrt{\sin (b x)} \cos (b x)}{21 b}",1,"(-20*EllipticF[(Pi - 2*b*x)/4, 2] + (-23*Cos[b*x] + 3*Cos[3*b*x])*Sqrt[Sin[b*x]])/(42*b)","A",1
10,1,35,41,0.0358988,"\int \sin ^{\frac{5}{2}}(b x) \, dx","Integrate[Sin[b*x]^(5/2),x]","-\frac{2 \left(3 E\left(\left.\frac{1}{4} (\pi -2 b x)\right|2\right)+\sin ^{\frac{3}{2}}(b x) \cos (b x)\right)}{5 b}","-\frac{6 E\left(\left.\frac{\pi }{4}-\frac{b x}{2}\right|2\right)}{5 b}-\frac{2 \sin ^{\frac{3}{2}}(b x) \cos (b x)}{5 b}",1,"(-2*(3*EllipticE[(Pi - 2*b*x)/4, 2] + Cos[b*x]*Sin[b*x]^(3/2)))/(5*b)","A",1
11,1,33,41,0.0334375,"\int \sin ^{\frac{3}{2}}(b x) \, dx","Integrate[Sin[b*x]^(3/2),x]","-\frac{2 \left(F\left(\left.\frac{1}{4} (\pi -2 b x)\right|2\right)+\sqrt{\sin (b x)} \cos (b x)\right)}{3 b}","-\frac{2 F\left(\left.\frac{\pi }{4}-\frac{b x}{2}\right|2\right)}{3 b}-\frac{2 \sqrt{\sin (b x)} \cos (b x)}{3 b}",1,"(-2*(EllipticF[(Pi - 2*b*x)/4, 2] + Cos[b*x]*Sqrt[Sin[b*x]]))/(3*b)","A",1
12,1,21,19,0.022098,"\int \sqrt{\sin (b x)} \, dx","Integrate[Sqrt[Sin[b*x]],x]","-\frac{2 E\left(\left.\frac{1}{2} \left(\frac{\pi }{2}-b x\right)\right|2\right)}{b}","-\frac{2 E\left(\left.\frac{\pi }{4}-\frac{b x}{2}\right|2\right)}{b}",1,"(-2*EllipticE[(Pi/2 - b*x)/2, 2])/b","A",1
13,1,21,19,0.0236085,"\int \frac{1}{\sqrt{\sin (b x)}} \, dx","Integrate[1/Sqrt[Sin[b*x]],x]","-\frac{2 F\left(\left.\frac{1}{2} \left(\frac{\pi }{2}-b x\right)\right|2\right)}{b}","-\frac{2 F\left(\left.\frac{\pi }{4}-\frac{b x}{2}\right|2\right)}{b}",1,"(-2*EllipticF[(Pi/2 - b*x)/2, 2])/b","A",1
14,1,32,37,0.0483961,"\int \frac{1}{\sin ^{\frac{3}{2}}(b x)} \, dx","Integrate[Sin[b*x]^(-3/2),x]","\frac{2 \left(E\left(\left.\frac{1}{4} (\pi -2 b x)\right|2\right)-\frac{\cos (b x)}{\sqrt{\sin (b x)}}\right)}{b}","\frac{2 E\left(\left.\frac{\pi }{4}-\frac{b x}{2}\right|2\right)}{b}-\frac{2 \cos (b x)}{b \sqrt{\sin (b x)}}",1,"(2*(EllipticE[(Pi - 2*b*x)/4, 2] - Cos[b*x]/Sqrt[Sin[b*x]]))/b","A",1
15,1,33,41,0.0470859,"\int \frac{1}{\sin ^{\frac{5}{2}}(b x)} \, dx","Integrate[Sin[b*x]^(-5/2),x]","-\frac{2 \left(F\left(\left.\frac{1}{4} (\pi -2 b x)\right|2\right)+\frac{\cos (b x)}{\sin ^{\frac{3}{2}}(b x)}\right)}{3 b}","-\frac{2 F\left(\left.\frac{\pi }{4}-\frac{b x}{2}\right|2\right)}{3 b}-\frac{2 \cos (b x)}{3 b \sin ^{\frac{3}{2}}(b x)}",1,"(-2*(EllipticF[(Pi - 2*b*x)/4, 2] + Cos[b*x]/Sin[b*x]^(3/2)))/(3*b)","A",1
16,1,51,60,0.0466386,"\int \frac{1}{\sin ^{\frac{7}{2}}(b x)} \, dx","Integrate[Sin[b*x]^(-7/2),x]","\frac{-7 \cos (b x)+3 \cos (3 b x)+12 \sin ^{\frac{5}{2}}(b x) E\left(\left.\frac{1}{4} (\pi -2 b x)\right|2\right)}{10 b \sin ^{\frac{5}{2}}(b x)}","\frac{6 E\left(\left.\frac{\pi }{4}-\frac{b x}{2}\right|2\right)}{5 b}-\frac{2 \cos (b x)}{5 b \sin ^{\frac{5}{2}}(b x)}-\frac{6 \cos (b x)}{5 b \sqrt{\sin (b x)}}",1,"(-7*Cos[b*x] + 3*Cos[3*b*x] + 12*EllipticE[(Pi - 2*b*x)/4, 2]*Sin[b*x]^(5/2))/(10*b*Sin[b*x]^(5/2))","A",1
17,1,55,70,0.113344,"\int \sin ^{\frac{7}{2}}(a+b x) \, dx","Integrate[Sin[a + b*x]^(7/2),x]","\frac{\sqrt{\sin (a+b x)} (3 \cos (3 (a+b x))-23 \cos (a+b x))-20 F\left(\left.\frac{1}{4} (-2 a-2 b x+\pi )\right|2\right)}{42 b}","\frac{10 F\left(\left.\frac{1}{2} \left(a+b x-\frac{\pi }{2}\right)\right|2\right)}{21 b}-\frac{2 \sin ^{\frac{5}{2}}(a+b x) \cos (a+b x)}{7 b}-\frac{10 \sqrt{\sin (a+b x)} \cos (a+b x)}{21 b}",1,"(-20*EllipticF[(-2*a + Pi - 2*b*x)/4, 2] + (-23*Cos[a + b*x] + 3*Cos[3*(a + b*x)])*Sqrt[Sin[a + b*x]])/(42*b)","A",1
18,1,44,47,0.0813796,"\int \sin ^{\frac{5}{2}}(a+b x) \, dx","Integrate[Sin[a + b*x]^(5/2),x]","-\frac{\sqrt{\sin (a+b x)} \sin (2 (a+b x))+6 E\left(\left.\frac{1}{4} (-2 a-2 b x+\pi )\right|2\right)}{5 b}","\frac{6 E\left(\left.\frac{1}{2} \left(a+b x-\frac{\pi }{2}\right)\right|2\right)}{5 b}-\frac{2 \sin ^{\frac{3}{2}}(a+b x) \cos (a+b x)}{5 b}",1,"-1/5*(6*EllipticE[(-2*a + Pi - 2*b*x)/4, 2] + Sqrt[Sin[a + b*x]]*Sin[2*(a + b*x)])/b","A",1
19,1,40,47,0.0366841,"\int \sin ^{\frac{3}{2}}(a+b x) \, dx","Integrate[Sin[a + b*x]^(3/2),x]","-\frac{2 \left(F\left(\left.\frac{1}{4} (-2 a-2 b x+\pi )\right|2\right)+\sqrt{\sin (a+b x)} \cos (a+b x)\right)}{3 b}","\frac{2 F\left(\left.\frac{1}{2} \left(a+b x-\frac{\pi }{2}\right)\right|2\right)}{3 b}-\frac{2 \sqrt{\sin (a+b x)} \cos (a+b x)}{3 b}",1,"(-2*(EllipticF[(-2*a + Pi - 2*b*x)/4, 2] + Cos[a + b*x]*Sqrt[Sin[a + b*x]]))/(3*b)","A",1
20,1,24,21,0.0151512,"\int \sqrt{\sin (a+b x)} \, dx","Integrate[Sqrt[Sin[a + b*x]],x]","-\frac{2 E\left(\left.\frac{1}{2} \left(-a-b x+\frac{\pi }{2}\right)\right|2\right)}{b}","\frac{2 E\left(\left.\frac{1}{2} \left(a+b x-\frac{\pi }{2}\right)\right|2\right)}{b}",1,"(-2*EllipticE[(-a + Pi/2 - b*x)/2, 2])/b","A",1
21,1,24,21,0.0184818,"\int \frac{1}{\sqrt{\sin (a+b x)}} \, dx","Integrate[1/Sqrt[Sin[a + b*x]],x]","-\frac{2 F\left(\left.\frac{1}{2} \left(-a-b x+\frac{\pi }{2}\right)\right|2\right)}{b}","\frac{2 F\left(\left.\frac{1}{2} \left(a+b x-\frac{\pi }{2}\right)\right|2\right)}{b}",1,"(-2*EllipticF[(-a + Pi/2 - b*x)/2, 2])/b","A",1
22,1,39,43,0.0753725,"\int \frac{1}{\sin ^{\frac{3}{2}}(a+b x)} \, dx","Integrate[Sin[a + b*x]^(-3/2),x]","\frac{2 \left(E\left(\left.\frac{1}{4} (-2 a-2 b x+\pi )\right|2\right)-\frac{\cos (a+b x)}{\sqrt{\sin (a+b x)}}\right)}{b}","-\frac{2 E\left(\left.\frac{1}{2} \left(a+b x-\frac{\pi }{2}\right)\right|2\right)}{b}-\frac{2 \cos (a+b x)}{b \sqrt{\sin (a+b x)}}",1,"(2*(EllipticE[(-2*a + Pi - 2*b*x)/4, 2] - Cos[a + b*x]/Sqrt[Sin[a + b*x]]))/b","A",1
23,1,43,47,0.1019766,"\int \frac{1}{\sin ^{\frac{5}{2}}(a+b x)} \, dx","Integrate[Sin[a + b*x]^(-5/2),x]","\frac{2 \left(F\left(\left.\frac{1}{4} (2 a+2 b x-\pi )\right|2\right)-\frac{\cos (a+b x)}{\sin ^{\frac{3}{2}}(a+b x)}\right)}{3 b}","\frac{2 F\left(\left.\frac{1}{2} \left(a+b x-\frac{\pi }{2}\right)\right|2\right)}{3 b}-\frac{2 \cos (a+b x)}{3 b \sin ^{\frac{3}{2}}(a+b x)}",1,"(2*(EllipticF[(2*a - Pi + 2*b*x)/4, 2] - Cos[a + b*x]/Sin[a + b*x]^(3/2)))/(3*b)","A",1
24,1,55,70,0.2696486,"\int \frac{1}{\sin ^{\frac{7}{2}}(a+b x)} \, dx","Integrate[Sin[a + b*x]^(-7/2),x]","\frac{2 \left(3 E\left(\left.\frac{1}{4} (-2 a-2 b x+\pi )\right|2\right)-\frac{\left(3 \sin ^2(a+b x)+1\right) \cos (a+b x)}{\sin ^{\frac{5}{2}}(a+b x)}\right)}{5 b}","-\frac{6 E\left(\left.\frac{1}{2} \left(a+b x-\frac{\pi }{2}\right)\right|2\right)}{5 b}-\frac{2 \cos (a+b x)}{5 b \sin ^{\frac{5}{2}}(a+b x)}-\frac{6 \cos (a+b x)}{5 b \sqrt{\sin (a+b x)}}",1,"(2*(3*EllipticE[(-2*a + Pi - 2*b*x)/4, 2] - (Cos[a + b*x]*(1 + 3*Sin[a + b*x]^2))/Sin[a + b*x]^(5/2)))/(5*b)","A",1
25,1,80,103,0.1505264,"\int (c \sin (a+b x))^{7/2} \, dx","Integrate[(c*Sin[a + b*x])^(7/2),x]","\frac{c^3 \sqrt{c \sin (a+b x)} \left(\sqrt{\sin (a+b x)} (3 \cos (3 (a+b x))-23 \cos (a+b x))-20 F\left(\left.\frac{1}{4} (-2 a-2 b x+\pi )\right|2\right)\right)}{42 b \sqrt{\sin (a+b x)}}","\frac{10 c^4 \sqrt{\sin (a+b x)} F\left(\left.\frac{1}{2} \left(a+b x-\frac{\pi }{2}\right)\right|2\right)}{21 b \sqrt{c \sin (a+b x)}}-\frac{10 c^3 \cos (a+b x) \sqrt{c \sin (a+b x)}}{21 b}-\frac{2 c \cos (a+b x) (c \sin (a+b x))^{5/2}}{7 b}",1,"(c^3*(-20*EllipticF[(-2*a + Pi - 2*b*x)/4, 2] + (-23*Cos[a + b*x] + 3*Cos[3*(a + b*x)])*Sqrt[Sin[a + b*x]])*Sqrt[c*Sin[a + b*x]])/(42*b*Sqrt[Sin[a + b*x]])","A",1
26,1,66,75,0.0984221,"\int (c \sin (a+b x))^{5/2} \, dx","Integrate[(c*Sin[a + b*x])^(5/2),x]","-\frac{(c \sin (a+b x))^{5/2} \left(\sqrt{\sin (a+b x)} \sin (2 (a+b x))+6 E\left(\left.\frac{1}{4} (-2 a-2 b x+\pi )\right|2\right)\right)}{5 b \sin ^{\frac{5}{2}}(a+b x)}","\frac{6 c^2 E\left(\left.\frac{1}{2} \left(a+b x-\frac{\pi }{2}\right)\right|2\right) \sqrt{c \sin (a+b x)}}{5 b \sqrt{\sin (a+b x)}}-\frac{2 c \cos (a+b x) (c \sin (a+b x))^{3/2}}{5 b}",1,"-1/5*((c*Sin[a + b*x])^(5/2)*(6*EllipticE[(-2*a + Pi - 2*b*x)/4, 2] + Sqrt[Sin[a + b*x]]*Sin[2*(a + b*x)]))/(b*Sin[a + b*x]^(5/2))","A",1
27,1,62,75,0.0499305,"\int (c \sin (a+b x))^{3/2} \, dx","Integrate[(c*Sin[a + b*x])^(3/2),x]","-\frac{2 (c \sin (a+b x))^{3/2} \left(F\left(\left.\frac{1}{4} (-2 a-2 b x+\pi )\right|2\right)+\sqrt{\sin (a+b x)} \cos (a+b x)\right)}{3 b \sin ^{\frac{3}{2}}(a+b x)}","\frac{2 c^2 \sqrt{\sin (a+b x)} F\left(\left.\frac{1}{2} \left(a+b x-\frac{\pi }{2}\right)\right|2\right)}{3 b \sqrt{c \sin (a+b x)}}-\frac{2 c \cos (a+b x) \sqrt{c \sin (a+b x)}}{3 b}",1,"(-2*(EllipticF[(-2*a + Pi - 2*b*x)/4, 2] + Cos[a + b*x]*Sqrt[Sin[a + b*x]])*(c*Sin[a + b*x])^(3/2))/(3*b*Sin[a + b*x]^(3/2))","A",1
28,1,42,43,0.019948,"\int \sqrt{c \sin (a+b x)} \, dx","Integrate[Sqrt[c*Sin[a + b*x]],x]","-\frac{2 E\left(\left.\frac{1}{4} (-2 a-2 b x+\pi )\right|2\right) \sqrt{c \sin (a+b x)}}{b \sqrt{\sin (a+b x)}}","\frac{2 E\left(\left.\frac{1}{2} \left(a+b x-\frac{\pi }{2}\right)\right|2\right) \sqrt{c \sin (a+b x)}}{b \sqrt{\sin (a+b x)}}",1,"(-2*EllipticE[(-2*a + Pi - 2*b*x)/4, 2]*Sqrt[c*Sin[a + b*x]])/(b*Sqrt[Sin[a + b*x]])","A",1
29,1,42,43,0.0261939,"\int \frac{1}{\sqrt{c \sin (a+b x)}} \, dx","Integrate[1/Sqrt[c*Sin[a + b*x]],x]","-\frac{2 \sqrt{\sin (a+b x)} F\left(\left.\frac{1}{4} (-2 a-2 b x+\pi )\right|2\right)}{b \sqrt{c \sin (a+b x)}}","\frac{2 \sqrt{\sin (a+b x)} F\left(\left.\frac{1}{2} \left(a+b x-\frac{\pi }{2}\right)\right|2\right)}{b \sqrt{c \sin (a+b x)}}",1,"(-2*EllipticF[(-2*a + Pi - 2*b*x)/4, 2]*Sqrt[Sin[a + b*x]])/(b*Sqrt[c*Sin[a + b*x]])","A",1
30,1,54,73,0.0392081,"\int \frac{1}{(c \sin (a+b x))^{3/2}} \, dx","Integrate[(c*Sin[a + b*x])^(-3/2),x]","-\frac{2 \left(\cos (a+b x)-\sqrt{\sin (a+b x)} E\left(\left.\frac{1}{4} (-2 a-2 b x+\pi )\right|2\right)\right)}{b c \sqrt{c \sin (a+b x)}}","-\frac{2 E\left(\left.\frac{1}{2} \left(a+b x-\frac{\pi }{2}\right)\right|2\right) \sqrt{c \sin (a+b x)}}{b c^2 \sqrt{\sin (a+b x)}}-\frac{2 \cos (a+b x)}{b c \sqrt{c \sin (a+b x)}}",1,"(-2*(Cos[a + b*x] - EllipticE[(-2*a + Pi - 2*b*x)/4, 2]*Sqrt[Sin[a + b*x]]))/(b*c*Sqrt[c*Sin[a + b*x]])","A",1
31,1,55,77,0.0636867,"\int \frac{1}{(c \sin (a+b x))^{5/2}} \, dx","Integrate[(c*Sin[a + b*x])^(-5/2),x]","-\frac{2 \left(\cos (a+b x)+\sin ^{\frac{3}{2}}(a+b x) F\left(\left.\frac{1}{4} (-2 a-2 b x+\pi )\right|2\right)\right)}{3 b c (c \sin (a+b x))^{3/2}}","\frac{2 \sqrt{\sin (a+b x)} F\left(\left.\frac{1}{2} \left(a+b x-\frac{\pi }{2}\right)\right|2\right)}{3 b c^2 \sqrt{c \sin (a+b x)}}-\frac{2 \cos (a+b x)}{3 b c (c \sin (a+b x))^{3/2}}",1,"(-2*(Cos[a + b*x] + EllipticF[(-2*a + Pi - 2*b*x)/4, 2]*Sin[a + b*x]^(3/2)))/(3*b*c*(c*Sin[a + b*x])^(3/2))","A",1
32,1,68,105,0.1478267,"\int \frac{1}{(c \sin (a+b x))^{7/2}} \, dx","Integrate[(c*Sin[a + b*x])^(-7/2),x]","-\frac{2 \left(\frac{3}{2} \sin (2 (a+b x))+\cot (a+b x)-3 \sin ^{\frac{3}{2}}(a+b x) E\left(\left.\frac{1}{4} (-2 a-2 b x+\pi )\right|2\right)\right)}{5 b c^2 (c \sin (a+b x))^{3/2}}","-\frac{6 E\left(\left.\frac{1}{2} \left(a+b x-\frac{\pi }{2}\right)\right|2\right) \sqrt{c \sin (a+b x)}}{5 b c^4 \sqrt{\sin (a+b x)}}-\frac{6 \cos (a+b x)}{5 b c^3 \sqrt{c \sin (a+b x)}}-\frac{2 \cos (a+b x)}{5 b c (c \sin (a+b x))^{5/2}}",1,"(-2*(Cot[a + b*x] - 3*EllipticE[(-2*a + Pi - 2*b*x)/4, 2]*Sin[a + b*x]^(3/2) + (3*Sin[2*(a + b*x)])/2))/(5*b*c^2*(c*Sin[a + b*x])^(3/2))","A",1
33,1,55,58,0.0579254,"\int (c \sin (a+b x))^{4/3} \, dx","Integrate[(c*Sin[a + b*x])^(4/3),x]","\frac{3 \sqrt{\cos ^2(a+b x)} \tan (a+b x) (c \sin (a+b x))^{4/3} \, _2F_1\left(\frac{1}{2},\frac{7}{6};\frac{13}{6};\sin ^2(a+b x)\right)}{7 b}","\frac{3 \cos (a+b x) (c \sin (a+b x))^{7/3} \, _2F_1\left(\frac{1}{2},\frac{7}{6};\frac{13}{6};\sin ^2(a+b x)\right)}{7 b c \sqrt{\cos ^2(a+b x)}}",1,"(3*Sqrt[Cos[a + b*x]^2]*Hypergeometric2F1[1/2, 7/6, 13/6, Sin[a + b*x]^2]*(c*Sin[a + b*x])^(4/3)*Tan[a + b*x])/(7*b)","A",1
34,1,55,58,0.034771,"\int (c \sin (a+b x))^{2/3} \, dx","Integrate[(c*Sin[a + b*x])^(2/3),x]","\frac{3 \sqrt{\cos ^2(a+b x)} \tan (a+b x) (c \sin (a+b x))^{2/3} \, _2F_1\left(\frac{1}{2},\frac{5}{6};\frac{11}{6};\sin ^2(a+b x)\right)}{5 b}","\frac{3 \cos (a+b x) (c \sin (a+b x))^{5/3} \, _2F_1\left(\frac{1}{2},\frac{5}{6};\frac{11}{6};\sin ^2(a+b x)\right)}{5 b c \sqrt{\cos ^2(a+b x)}}",1,"(3*Sqrt[Cos[a + b*x]^2]*Hypergeometric2F1[1/2, 5/6, 11/6, Sin[a + b*x]^2]*(c*Sin[a + b*x])^(2/3)*Tan[a + b*x])/(5*b)","A",1
35,1,55,517,0.0333768,"\int \sqrt[3]{c \sin (a+b x)} \, dx","Integrate[(c*Sin[a + b*x])^(1/3),x]","\frac{3 \sqrt{\cos ^2(a+b x)} \tan (a+b x) \sqrt[3]{c \sin (a+b x)} \, _2F_1\left(\frac{1}{2},\frac{2}{3};\frac{5}{3};\sin ^2(a+b x)\right)}{4 b}","\frac{3 \left(1-i \sqrt{3}\right) \sqrt{3-i \sqrt{3}} \sqrt[3]{c} \sec (a+b x) \sqrt{1-\frac{(c \sin (a+b x))^{2/3}}{c^{2/3}}} \sqrt{\frac{2 (c \sin (a+b x))^{2/3}}{\left(3-i \sqrt{3}\right) c^{2/3}}+\frac{\sqrt{3}+i}{\sqrt{3}+3 i}} \sqrt{\frac{2 (c \sin (a+b x))^{2/3}}{\left(3+i \sqrt{3}\right) c^{2/3}}+\frac{-\sqrt{3}+i}{-\sqrt{3}+3 i}} F\left(\sin ^{-1}\left(\frac{\sqrt{2} \sqrt{1-\frac{(c \sin (a+b x))^{2/3}}{c^{2/3}}}}{\sqrt{3-i \sqrt{3}}}\right)|\frac{3 i+\sqrt{3}}{3 i-\sqrt{3}}\right)}{2 \sqrt{2} b}-\frac{3 \sqrt{\frac{3}{2} \left(3-i \sqrt{3}\right)} \sqrt[3]{c} \sec (a+b x) \sqrt{1-\frac{(c \sin (a+b x))^{2/3}}{c^{2/3}}} \sqrt{\frac{2 (c \sin (a+b x))^{2/3}}{\left(3-i \sqrt{3}\right) c^{2/3}}+\frac{\sqrt{3}+i}{\sqrt{3}+3 i}} \sqrt{\frac{2 (c \sin (a+b x))^{2/3}}{\left(3+i \sqrt{3}\right) c^{2/3}}+\frac{-\sqrt{3}+i}{-\sqrt{3}+3 i}} E\left(\sin ^{-1}\left(\frac{\sqrt{2} \sqrt{1-\frac{(c \sin (a+b x))^{2/3}}{c^{2/3}}}}{\sqrt{3+i \sqrt{3}}}\right)|\frac{3 i-\sqrt{3}}{3 i+\sqrt{3}}\right)}{b}",1,"(3*Sqrt[Cos[a + b*x]^2]*Hypergeometric2F1[1/2, 2/3, 5/3, Sin[a + b*x]^2]*(c*Sin[a + b*x])^(1/3)*Tan[a + b*x])/(4*b)","C",1
36,1,55,252,0.0374841,"\int \frac{1}{\sqrt[3]{c \sin (a+b x)}} \, dx","Integrate[(c*Sin[a + b*x])^(-1/3),x]","\frac{3 \sqrt{\cos ^2(a+b x)} \tan (a+b x) \, _2F_1\left(\frac{1}{3},\frac{1}{2};\frac{4}{3};\sin ^2(a+b x)\right)}{2 b \sqrt[3]{c \sin (a+b x)}}","-\frac{3 \sqrt{3-i \sqrt{3}} \sec (a+b x) \sqrt{1-\frac{(c \sin (a+b x))^{2/3}}{c^{2/3}}} \sqrt{\frac{2 (c \sin (a+b x))^{2/3}}{\left(3-i \sqrt{3}\right) c^{2/3}}+\frac{\sqrt{3}+i}{\sqrt{3}+3 i}} \sqrt{\frac{2 (c \sin (a+b x))^{2/3}}{\left(3+i \sqrt{3}\right) c^{2/3}}+\frac{-\sqrt{3}+i}{-\sqrt{3}+3 i}} F\left(\sin ^{-1}\left(\frac{\sqrt{2} \sqrt{1-\frac{(c \sin (a+b x))^{2/3}}{c^{2/3}}}}{\sqrt{3-i \sqrt{3}}}\right)|\frac{3 i+\sqrt{3}}{3 i-\sqrt{3}}\right)}{\sqrt{2} b \sqrt[3]{c}}",1,"(3*Sqrt[Cos[a + b*x]^2]*Hypergeometric2F1[1/3, 1/2, 4/3, Sin[a + b*x]^2]*Tan[a + b*x])/(2*b*(c*Sin[a + b*x])^(1/3))","C",1
37,1,53,271,0.0369564,"\int \frac{1}{(c \sin (a+b x))^{2/3}} \, dx","Integrate[(c*Sin[a + b*x])^(-2/3),x]","\frac{3 \sqrt{\cos ^2(a+b x)} \tan (a+b x) \, _2F_1\left(\frac{1}{6},\frac{1}{2};\frac{7}{6};\sin ^2(a+b x)\right)}{b (c \sin (a+b x))^{2/3}}","\frac{3^{3/4} \sec (a+b x) \sqrt[3]{c \sin (a+b x)} \left(c^{2/3}-(c \sin (a+b x))^{2/3}\right) \sqrt{\frac{c^{4/3} \left(\frac{(c \sin (a+b x))^{4/3}}{c^{4/3}}+\frac{(c \sin (a+b x))^{2/3}}{c^{2/3}}+1\right)}{\left(c^{2/3}-\left(1+\sqrt{3}\right) (c \sin (a+b x))^{2/3}\right)^2}} F\left(\cos ^{-1}\left(\frac{c^{2/3}-\left(1-\sqrt{3}\right) (c \sin (a+b x))^{2/3}}{c^{2/3}-\left(1+\sqrt{3}\right) (c \sin (a+b x))^{2/3}}\right)|\frac{1}{4} \left(2+\sqrt{3}\right)\right)}{2 b c^{5/3} \sqrt{-\frac{(c \sin (a+b x))^{2/3} \left(c^{2/3}-(c \sin (a+b x))^{2/3}\right)}{\left(c^{2/3}-\left(1+\sqrt{3}\right) (c \sin (a+b x))^{2/3}\right)^2}}}",1,"(3*Sqrt[Cos[a + b*x]^2]*Hypergeometric2F1[1/6, 1/2, 7/6, Sin[a + b*x]^2]*Tan[a + b*x])/(b*(c*Sin[a + b*x])^(2/3))","C",1
38,1,53,56,0.0383072,"\int \frac{1}{(c \sin (a+b x))^{4/3}} \, dx","Integrate[(c*Sin[a + b*x])^(-4/3),x]","-\frac{3 \sqrt{\cos ^2(a+b x)} \tan (a+b x) \, _2F_1\left(-\frac{1}{6},\frac{1}{2};\frac{5}{6};\sin ^2(a+b x)\right)}{b (c \sin (a+b x))^{4/3}}","-\frac{3 \cos (a+b x) \, _2F_1\left(-\frac{1}{6},\frac{1}{2};\frac{5}{6};\sin ^2(a+b x)\right)}{b c \sqrt{\cos ^2(a+b x)} \sqrt[3]{c \sin (a+b x)}}",1,"(-3*Sqrt[Cos[a + b*x]^2]*Hypergeometric2F1[-1/6, 1/2, 5/6, Sin[a + b*x]^2]*Tan[a + b*x])/(b*(c*Sin[a + b*x])^(4/3))","A",1
39,1,63,63,0.0400301,"\int \sin ^n(a+b x) \, dx","Integrate[Sin[a + b*x]^n,x]","\frac{\sqrt{\cos ^2(a+b x)} \sec (a+b x) \sin ^{n+1}(a+b x) \, _2F_1\left(\frac{1}{2},\frac{n+1}{2};\frac{n+3}{2};\sin ^2(a+b x)\right)}{b (n+1)}","\frac{\cos (a+b x) \sin ^{n+1}(a+b x) \, _2F_1\left(\frac{1}{2},\frac{n+1}{2};\frac{n+3}{2};\sin ^2(a+b x)\right)}{b (n+1) \sqrt{\cos ^2(a+b x)}}",1,"(Sqrt[Cos[a + b*x]^2]*Hypergeometric2F1[1/2, (1 + n)/2, (3 + n)/2, Sin[a + b*x]^2]*Sec[a + b*x]*Sin[a + b*x]^(1 + n))/(b*(1 + n))","A",1
40,1,63,68,0.0378445,"\int (c \sin (a+b x))^n \, dx","Integrate[(c*Sin[a + b*x])^n,x]","\frac{\sqrt{\cos ^2(a+b x)} \tan (a+b x) (c \sin (a+b x))^n \, _2F_1\left(\frac{1}{2},\frac{n+1}{2};\frac{n+3}{2};\sin ^2(a+b x)\right)}{b (n+1)}","\frac{\cos (a+b x) (c \sin (a+b x))^{n+1} \, _2F_1\left(\frac{1}{2},\frac{n+1}{2};\frac{n+3}{2};\sin ^2(a+b x)\right)}{b c (n+1) \sqrt{\cos ^2(a+b x)}}",1,"(Sqrt[Cos[a + b*x]^2]*Hypergeometric2F1[1/2, (1 + n)/2, (3 + n)/2, Sin[a + b*x]^2]*(c*Sin[a + b*x])^n*Tan[a + b*x])/(b*(1 + n))","A",1
41,1,76,81,0.0738611,"\int (a \sin (e+f x))^m (b \sin (e+f x))^n \, dx","Integrate[(a*Sin[e + f*x])^m*(b*Sin[e + f*x])^n,x]","\frac{\sqrt{\cos ^2(e+f x)} \tan (e+f x) (a \sin (e+f x))^m (b \sin (e+f x))^n \, _2F_1\left(\frac{1}{2},\frac{1}{2} (m+n+1);\frac{1}{2} (m+n+3);\sin ^2(e+f x)\right)}{f (m+n+1)}","\frac{\cos (e+f x) (a \sin (e+f x))^{m+1} (b \sin (e+f x))^n \, _2F_1\left(\frac{1}{2},\frac{1}{2} (m+n+1);\frac{1}{2} (m+n+3);\sin ^2(e+f x)\right)}{a f (m+n+1) \sqrt{\cos ^2(e+f x)}}",1,"(Sqrt[Cos[e + f*x]^2]*Hypergeometric2F1[1/2, (1 + m + n)/2, (3 + m + n)/2, Sin[e + f*x]^2]*(a*Sin[e + f*x])^m*(b*Sin[e + f*x])^n*Tan[e + f*x])/(f*(1 + m + n))","A",1
42,1,15,15,0.004033,"\int \cos ^3(a+b x) \sin (a+b x) \, dx","Integrate[Cos[a + b*x]^3*Sin[a + b*x],x]","-\frac{\cos ^4(a+b x)}{4 b}","-\frac{\cos ^4(a+b x)}{4 b}",1,"-1/4*Cos[a + b*x]^4/b","A",1
43,1,15,15,0.0036032,"\int \cos ^2(a+b x) \sin (a+b x) \, dx","Integrate[Cos[a + b*x]^2*Sin[a + b*x],x]","-\frac{\cos ^3(a+b x)}{3 b}","-\frac{\cos ^3(a+b x)}{3 b}",1,"-1/3*Cos[a + b*x]^3/b","A",1
44,1,37,15,0.0108397,"\int \cos (a+b x) \sin (a+b x) \, dx","Integrate[Cos[a + b*x]*Sin[a + b*x],x]","\frac{1}{2} \left(\frac{\sin (2 a) \sin (2 b x)}{2 b}-\frac{\cos (2 a) \cos (2 b x)}{2 b}\right)","\frac{\sin ^2(a+b x)}{2 b}",1,"(-1/2*(Cos[2*a]*Cos[2*b*x])/b + (Sin[2*a]*Sin[2*b*x])/(2*b))/2","B",1
45,1,12,12,0.0068432,"\int \tan (a+b x) \, dx","Integrate[Tan[a + b*x],x]","-\frac{\log (\cos (a+b x))}{b}","-\frac{\log (\cos (a+b x))}{b}",1,"-(Log[Cos[a + b*x]]/b)","A",1
46,1,10,10,0.0062305,"\int \sec (a+b x) \tan (a+b x) \, dx","Integrate[Sec[a + b*x]*Tan[a + b*x],x]","\frac{\sec (a+b x)}{b}","\frac{\sec (a+b x)}{b}",1,"Sec[a + b*x]/b","A",1
47,1,15,15,0.0091946,"\int \sec ^2(a+b x) \tan (a+b x) \, dx","Integrate[Sec[a + b*x]^2*Tan[a + b*x],x]","\frac{\sec ^2(a+b x)}{2 b}","\frac{\sec ^2(a+b x)}{2 b}",1,"Sec[a + b*x]^2/(2*b)","A",1
48,1,15,15,0.0070393,"\int \sec ^3(a+b x) \tan (a+b x) \, dx","Integrate[Sec[a + b*x]^3*Tan[a + b*x],x]","\frac{\sec ^3(a+b x)}{3 b}","\frac{\sec ^3(a+b x)}{3 b}",1,"Sec[a + b*x]^3/(3*b)","A",1
49,1,47,61,0.1445078,"\int \cos ^7(a+b x) \sin ^2(a+b x) \, dx","Integrate[Cos[a + b*x]^7*Sin[a + b*x]^2,x]","\frac{\sin ^3(a+b x) (1389 \cos (2 (a+b x))+330 \cos (4 (a+b x))+35 \cos (6 (a+b x))+1606)}{10080 b}","-\frac{\sin ^9(a+b x)}{9 b}+\frac{3 \sin ^7(a+b x)}{7 b}-\frac{3 \sin ^5(a+b x)}{5 b}+\frac{\sin ^3(a+b x)}{3 b}",1,"((1606 + 1389*Cos[2*(a + b*x)] + 330*Cos[4*(a + b*x)] + 35*Cos[6*(a + b*x)])*Sin[a + b*x]^3)/(10080*b)","A",1
50,1,37,46,0.0847978,"\int \cos ^5(a+b x) \sin ^2(a+b x) \, dx","Integrate[Cos[a + b*x]^5*Sin[a + b*x]^2,x]","\frac{\sin ^3(a+b x) (108 \cos (2 (a+b x))+15 \cos (4 (a+b x))+157)}{840 b}","\frac{\sin ^7(a+b x)}{7 b}-\frac{2 \sin ^5(a+b x)}{5 b}+\frac{\sin ^3(a+b x)}{3 b}",1,"((157 + 108*Cos[2*(a + b*x)] + 15*Cos[4*(a + b*x)])*Sin[a + b*x]^3)/(840*b)","A",1
51,1,27,31,0.055734,"\int \cos ^3(a+b x) \sin ^2(a+b x) \, dx","Integrate[Cos[a + b*x]^3*Sin[a + b*x]^2,x]","\frac{\sin ^3(a+b x) (3 \cos (2 (a+b x))+7)}{30 b}","\frac{\sin ^3(a+b x)}{3 b}-\frac{\sin ^5(a+b x)}{5 b}",1,"((7 + 3*Cos[2*(a + b*x)])*Sin[a + b*x]^3)/(30*b)","A",1
52,1,15,15,0.0034077,"\int \cos (a+b x) \sin ^2(a+b x) \, dx","Integrate[Cos[a + b*x]*Sin[a + b*x]^2,x]","\frac{\sin ^3(a+b x)}{3 b}","\frac{\sin ^3(a+b x)}{3 b}",1,"Sin[a + b*x]^3/(3*b)","A",1
53,1,23,14,0.00638,"\int \tan ^2(a+b x) \, dx","Integrate[Tan[a + b*x]^2,x]","\frac{\tan (a+b x)}{b}-\frac{\tan ^{-1}(\tan (a+b x))}{b}","\frac{\tan (a+b x)}{b}-x",1,"-(ArcTan[Tan[a + b*x]]/b) + Tan[a + b*x]/b","A",1
54,1,15,15,0.0061834,"\int \sec ^2(a+b x) \tan ^2(a+b x) \, dx","Integrate[Sec[a + b*x]^2*Tan[a + b*x]^2,x]","\frac{\tan ^3(a+b x)}{3 b}","\frac{\tan ^3(a+b x)}{3 b}",1,"Tan[a + b*x]^3/(3*b)","A",1
55,1,56,31,0.0366863,"\int \sec ^4(a+b x) \tan ^2(a+b x) \, dx","Integrate[Sec[a + b*x]^4*Tan[a + b*x]^2,x]","-\frac{2 \tan (a+b x)}{15 b}+\frac{\tan (a+b x) \sec ^4(a+b x)}{5 b}-\frac{\tan (a+b x) \sec ^2(a+b x)}{15 b}","\frac{\tan ^5(a+b x)}{5 b}+\frac{\tan ^3(a+b x)}{3 b}",1,"(-2*Tan[a + b*x])/(15*b) - (Sec[a + b*x]^2*Tan[a + b*x])/(15*b) + (Sec[a + b*x]^4*Tan[a + b*x])/(5*b)","A",1
56,1,77,46,0.0361518,"\int \sec ^6(a+b x) \tan ^2(a+b x) \, dx","Integrate[Sec[a + b*x]^6*Tan[a + b*x]^2,x]","-\frac{8 \tan (a+b x)}{105 b}+\frac{\tan (a+b x) \sec ^6(a+b x)}{7 b}-\frac{\tan (a+b x) \sec ^4(a+b x)}{35 b}-\frac{4 \tan (a+b x) \sec ^2(a+b x)}{105 b}","\frac{\tan ^7(a+b x)}{7 b}+\frac{2 \tan ^5(a+b x)}{5 b}+\frac{\tan ^3(a+b x)}{3 b}",1,"(-8*Tan[a + b*x])/(105*b) - (4*Sec[a + b*x]^2*Tan[a + b*x])/(105*b) - (Sec[a + b*x]^4*Tan[a + b*x])/(35*b) + (Sec[a + b*x]^6*Tan[a + b*x])/(7*b)","A",1
57,1,98,61,0.0322885,"\int \sec ^8(a+b x) \tan ^2(a+b x) \, dx","Integrate[Sec[a + b*x]^8*Tan[a + b*x]^2,x]","-\frac{16 \tan (a+b x)}{315 b}+\frac{\tan (a+b x) \sec ^8(a+b x)}{9 b}-\frac{\tan (a+b x) \sec ^6(a+b x)}{63 b}-\frac{2 \tan (a+b x) \sec ^4(a+b x)}{105 b}-\frac{8 \tan (a+b x) \sec ^2(a+b x)}{315 b}","\frac{\tan ^9(a+b x)}{9 b}+\frac{3 \tan ^7(a+b x)}{7 b}+\frac{3 \tan ^5(a+b x)}{5 b}+\frac{\tan ^3(a+b x)}{3 b}",1,"(-16*Tan[a + b*x])/(315*b) - (8*Sec[a + b*x]^2*Tan[a + b*x])/(315*b) - (2*Sec[a + b*x]^4*Tan[a + b*x])/(105*b) - (Sec[a + b*x]^6*Tan[a + b*x])/(63*b) + (Sec[a + b*x]^8*Tan[a + b*x])/(9*b)","A",1
58,1,52,88,0.125018,"\int \cos ^6(a+b x) \sin ^2(a+b x) \, dx","Integrate[Cos[a + b*x]^6*Sin[a + b*x]^2,x]","\frac{48 \sin (2 (a+b x))-24 \sin (4 (a+b x))-16 \sin (6 (a+b x))-3 \sin (8 (a+b x))+120 b x}{3072 b}","-\frac{\sin (a+b x) \cos ^7(a+b x)}{8 b}+\frac{\sin (a+b x) \cos ^5(a+b x)}{48 b}+\frac{5 \sin (a+b x) \cos ^3(a+b x)}{192 b}+\frac{5 \sin (a+b x) \cos (a+b x)}{128 b}+\frac{5 x}{128}",1,"(120*b*x + 48*Sin[2*(a + b*x)] - 24*Sin[4*(a + b*x)] - 16*Sin[6*(a + b*x)] - 3*Sin[8*(a + b*x)])/(3072*b)","A",1
59,1,40,67,0.0758073,"\int \cos ^4(a+b x) \sin ^2(a+b x) \, dx","Integrate[Cos[a + b*x]^4*Sin[a + b*x]^2,x]","-\frac{-3 \sin (2 (a+b x))+3 \sin (4 (a+b x))+\sin (6 (a+b x))-12 b x}{192 b}","-\frac{\sin (a+b x) \cos ^5(a+b x)}{6 b}+\frac{\sin (a+b x) \cos ^3(a+b x)}{24 b}+\frac{\sin (a+b x) \cos (a+b x)}{16 b}+\frac{x}{16}",1,"-1/192*(-12*b*x - 3*Sin[2*(a + b*x)] + 3*Sin[4*(a + b*x)] + Sin[6*(a + b*x)])/b","A",1
60,1,23,46,0.0302085,"\int \cos ^2(a+b x) \sin ^2(a+b x) \, dx","Integrate[Cos[a + b*x]^2*Sin[a + b*x]^2,x]","-\frac{\sin (4 (a+b x))-4 (a+b x)}{32 b}","-\frac{\sin (a+b x) \cos ^3(a+b x)}{4 b}+\frac{\sin (a+b x) \cos (a+b x)}{8 b}+\frac{x}{8}",1,"-1/32*(-4*(a + b*x) + Sin[4*(a + b*x)])/b","A",1
61,1,23,25,0.0089394,"\int \sin ^2(a+b x) \, dx","Integrate[Sin[a + b*x]^2,x]","-\frac{\sin (2 (a+b x))-2 (a+b x)}{4 b}","\frac{x}{2}-\frac{\sin (a+b x) \cos (a+b x)}{2 b}",1,"-1/4*(-2*(a + b*x) + Sin[2*(a + b*x)])/b","A",1
62,1,23,23,0.009182,"\int \sin (a+b x) \tan (a+b x) \, dx","Integrate[Sin[a + b*x]*Tan[a + b*x],x]","\frac{\tanh ^{-1}(\sin (a+b x))}{b}-\frac{\sin (a+b x)}{b}","\frac{\tanh ^{-1}(\sin (a+b x))}{b}-\frac{\sin (a+b x)}{b}",1,"ArcTanh[Sin[a + b*x]]/b - Sin[a + b*x]/b","A",1
63,1,34,34,0.0136419,"\int \sec (a+b x) \tan ^2(a+b x) \, dx","Integrate[Sec[a + b*x]*Tan[a + b*x]^2,x]","\frac{\tan (a+b x) \sec (a+b x)}{2 b}-\frac{\tanh ^{-1}(\sin (a+b x))}{2 b}","\frac{\tan (a+b x) \sec (a+b x)}{2 b}-\frac{\tanh ^{-1}(\sin (a+b x))}{2 b}",1,"-1/2*ArcTanh[Sin[a + b*x]]/b + (Sec[a + b*x]*Tan[a + b*x])/(2*b)","A",1
64,1,55,55,0.04145,"\int \sec ^3(a+b x) \tan ^2(a+b x) \, dx","Integrate[Sec[a + b*x]^3*Tan[a + b*x]^2,x]","-\frac{\tanh ^{-1}(\sin (a+b x))}{8 b}+\frac{\tan (a+b x) \sec ^3(a+b x)}{4 b}-\frac{\tan (a+b x) \sec (a+b x)}{8 b}","-\frac{\tanh ^{-1}(\sin (a+b x))}{8 b}+\frac{\tan (a+b x) \sec ^3(a+b x)}{4 b}-\frac{\tan (a+b x) \sec (a+b x)}{8 b}",1,"-1/8*ArcTanh[Sin[a + b*x]]/b - (Sec[a + b*x]*Tan[a + b*x])/(8*b) + (Sec[a + b*x]^3*Tan[a + b*x])/(4*b)","A",1
65,1,76,76,0.0593261,"\int \sec ^5(a+b x) \tan ^2(a+b x) \, dx","Integrate[Sec[a + b*x]^5*Tan[a + b*x]^2,x]","-\frac{\tanh ^{-1}(\sin (a+b x))}{16 b}+\frac{\tan (a+b x) \sec ^5(a+b x)}{6 b}-\frac{\tan (a+b x) \sec ^3(a+b x)}{24 b}-\frac{\tan (a+b x) \sec (a+b x)}{16 b}","-\frac{\tanh ^{-1}(\sin (a+b x))}{16 b}+\frac{\tan (a+b x) \sec ^5(a+b x)}{6 b}-\frac{\tan (a+b x) \sec ^3(a+b x)}{24 b}-\frac{\tan (a+b x) \sec (a+b x)}{16 b}",1,"-1/16*ArcTanh[Sin[a + b*x]]/b - (Sec[a + b*x]*Tan[a + b*x])/(16*b) - (Sec[a + b*x]^3*Tan[a + b*x])/(24*b) + (Sec[a + b*x]^5*Tan[a + b*x])/(6*b)","A",1
66,1,48,31,0.1164203,"\int \cos ^5(a+b x) \sin ^3(a+b x) \, dx","Integrate[Cos[a + b*x]^5*Sin[a + b*x]^3,x]","\frac{-72 \cos (2 (a+b x))-12 \cos (4 (a+b x))+8 \cos (6 (a+b x))+3 \cos (8 (a+b x))}{3072 b}","\frac{\cos ^8(a+b x)}{8 b}-\frac{\cos ^6(a+b x)}{6 b}",1,"(-72*Cos[2*(a + b*x)] - 12*Cos[4*(a + b*x)] + 8*Cos[6*(a + b*x)] + 3*Cos[8*(a + b*x)])/(3072*b)","A",1
67,1,27,31,0.085848,"\int \cos ^4(a+b x) \sin ^3(a+b x) \, dx","Integrate[Cos[a + b*x]^4*Sin[a + b*x]^3,x]","\frac{\cos ^5(a+b x) (5 \cos (2 (a+b x))-9)}{70 b}","\frac{\cos ^7(a+b x)}{7 b}-\frac{\cos ^5(a+b x)}{5 b}",1,"(Cos[a + b*x]^5*(-9 + 5*Cos[2*(a + b*x)]))/(70*b)","A",1
68,1,35,31,0.0143119,"\int \cos ^3(a+b x) \sin ^3(a+b x) \, dx","Integrate[Cos[a + b*x]^3*Sin[a + b*x]^3,x]","\frac{1}{8} \left(\frac{\cos (6 (a+b x))}{24 b}-\frac{3 \cos (2 (a+b x))}{8 b}\right)","\frac{\sin ^4(a+b x)}{4 b}-\frac{\sin ^6(a+b x)}{6 b}",1,"((-3*Cos[2*(a + b*x)])/(8*b) + Cos[6*(a + b*x)]/(24*b))/8","A",1
69,1,27,31,0.0539126,"\int \cos ^2(a+b x) \sin ^3(a+b x) \, dx","Integrate[Cos[a + b*x]^2*Sin[a + b*x]^3,x]","\frac{\cos ^3(a+b x) (3 \cos (2 (a+b x))-7)}{30 b}","\frac{\cos ^5(a+b x)}{5 b}-\frac{\cos ^3(a+b x)}{3 b}",1,"(Cos[a + b*x]^3*(-7 + 3*Cos[2*(a + b*x)]))/(30*b)","A",1
70,1,15,15,0.002561,"\int \cos (a+b x) \sin ^3(a+b x) \, dx","Integrate[Cos[a + b*x]*Sin[a + b*x]^3,x]","\frac{\sin ^4(a+b x)}{4 b}","\frac{\sin ^4(a+b x)}{4 b}",1,"Sin[a + b*x]^4/(4*b)","A",1
71,1,25,28,0.0161377,"\int \sin ^2(a+b x) \tan (a+b x) \, dx","Integrate[Sin[a + b*x]^2*Tan[a + b*x],x]","-\frac{\log (\cos (a+b x))-\frac{1}{2} \cos ^2(a+b x)}{b}","\frac{\cos ^2(a+b x)}{2 b}-\frac{\log (\cos (a+b x))}{b}",1,"-((-1/2*Cos[a + b*x]^2 + Log[Cos[a + b*x]])/b)","A",1
72,1,21,21,0.0197814,"\int \sin (a+b x) \tan ^2(a+b x) \, dx","Integrate[Sin[a + b*x]*Tan[a + b*x]^2,x]","\frac{\cos (a+b x)}{b}+\frac{\sec (a+b x)}{b}","\frac{\cos (a+b x)}{b}+\frac{\sec (a+b x)}{b}",1,"Cos[a + b*x]/b + Sec[a + b*x]/b","A",1
73,1,25,27,0.0229353,"\int \tan ^3(a+b x) \, dx","Integrate[Tan[a + b*x]^3,x]","\frac{\tan ^2(a+b x)+2 \log (\cos (a+b x))}{2 b}","\frac{\tan ^2(a+b x)}{2 b}+\frac{\log (\cos (a+b x))}{b}",1,"(2*Log[Cos[a + b*x]] + Tan[a + b*x]^2)/(2*b)","A",1
74,1,27,27,0.022832,"\int \sec (a+b x) \tan ^3(a+b x) \, dx","Integrate[Sec[a + b*x]*Tan[a + b*x]^3,x]","\frac{\sec ^3(a+b x)}{3 b}-\frac{\sec (a+b x)}{b}","\frac{\sec ^3(a+b x)}{3 b}-\frac{\sec (a+b x)}{b}",1,"-(Sec[a + b*x]/b) + Sec[a + b*x]^3/(3*b)","A",1
75,1,15,15,0.0051288,"\int \sec ^2(a+b x) \tan ^3(a+b x) \, dx","Integrate[Sec[a + b*x]^2*Tan[a + b*x]^3,x]","\frac{\tan ^4(a+b x)}{4 b}","\frac{\tan ^4(a+b x)}{4 b}",1,"Tan[a + b*x]^4/(4*b)","A",1
76,1,31,31,0.0439498,"\int \sec ^3(a+b x) \tan ^3(a+b x) \, dx","Integrate[Sec[a + b*x]^3*Tan[a + b*x]^3,x]","\frac{\sec ^5(a+b x)}{5 b}-\frac{\sec ^3(a+b x)}{3 b}","\frac{\sec ^5(a+b x)}{5 b}-\frac{\sec ^3(a+b x)}{3 b}",1,"-1/3*Sec[a + b*x]^3/b + Sec[a + b*x]^5/(5*b)","A",1
77,1,28,31,0.0324334,"\int \sec ^4(a+b x) \tan ^3(a+b x) \, dx","Integrate[Sec[a + b*x]^4*Tan[a + b*x]^3,x]","-\frac{3 \sec ^4(a+b x)-2 \sec ^6(a+b x)}{12 b}","\frac{\sec ^6(a+b x)}{6 b}-\frac{\sec ^4(a+b x)}{4 b}",1,"-1/12*(3*Sec[a + b*x]^4 - 2*Sec[a + b*x]^6)/b","A",1
78,1,31,31,0.0258282,"\int \sec ^5(a+b x) \tan ^3(a+b x) \, dx","Integrate[Sec[a + b*x]^5*Tan[a + b*x]^3,x]","\frac{\sec ^7(a+b x)}{7 b}-\frac{\sec ^5(a+b x)}{5 b}","\frac{\sec ^7(a+b x)}{7 b}-\frac{\sec ^5(a+b x)}{5 b}",1,"-1/5*Sec[a + b*x]^5/b + Sec[a + b*x]^7/(7*b)","A",1
79,1,28,31,0.0334495,"\int \sec ^6(a+b x) \tan ^3(a+b x) \, dx","Integrate[Sec[a + b*x]^6*Tan[a + b*x]^3,x]","-\frac{4 \sec ^6(a+b x)-3 \sec ^8(a+b x)}{24 b}","\frac{\sec ^8(a+b x)}{8 b}-\frac{\sec ^6(a+b x)}{6 b}",1,"-1/24*(4*Sec[a + b*x]^6 - 3*Sec[a + b*x]^8)/b","A",1
80,1,47,61,0.1847938,"\int \cos ^7(a+b x) \sin ^4(a+b x) \, dx","Integrate[Cos[a + b*x]^7*Sin[a + b*x]^4,x]","\frac{\sin ^5(a+b x) (3335 \cos (2 (a+b x))+910 \cos (4 (a+b x))+105 \cos (6 (a+b x))+3042)}{36960 b}","-\frac{\sin ^{11}(a+b x)}{11 b}+\frac{\sin ^9(a+b x)}{3 b}-\frac{3 \sin ^7(a+b x)}{7 b}+\frac{\sin ^5(a+b x)}{5 b}",1,"((3042 + 3335*Cos[2*(a + b*x)] + 910*Cos[4*(a + b*x)] + 105*Cos[6*(a + b*x)])*Sin[a + b*x]^5)/(36960*b)","A",1
81,1,37,46,0.1081017,"\int \cos ^5(a+b x) \sin ^4(a+b x) \, dx","Integrate[Cos[a + b*x]^5*Sin[a + b*x]^4,x]","\frac{\sin ^5(a+b x) (220 \cos (2 (a+b x))+35 \cos (4 (a+b x))+249)}{2520 b}","\frac{\sin ^9(a+b x)}{9 b}-\frac{2 \sin ^7(a+b x)}{7 b}+\frac{\sin ^5(a+b x)}{5 b}",1,"((249 + 220*Cos[2*(a + b*x)] + 35*Cos[4*(a + b*x)])*Sin[a + b*x]^5)/(2520*b)","A",1
82,1,27,31,0.0688098,"\int \cos ^3(a+b x) \sin ^4(a+b x) \, dx","Integrate[Cos[a + b*x]^3*Sin[a + b*x]^4,x]","\frac{\sin ^5(a+b x) (5 \cos (2 (a+b x))+9)}{70 b}","\frac{\sin ^5(a+b x)}{5 b}-\frac{\sin ^7(a+b x)}{7 b}",1,"((9 + 5*Cos[2*(a + b*x)])*Sin[a + b*x]^5)/(70*b)","A",1
83,1,15,15,0.0031939,"\int \cos (a+b x) \sin ^4(a+b x) \, dx","Integrate[Cos[a + b*x]*Sin[a + b*x]^4,x]","\frac{\sin ^5(a+b x)}{5 b}","\frac{\sin ^5(a+b x)}{5 b}",1,"Sin[a + b*x]^5/(5*b)","A",1
84,1,31,40,0.1006847,"\int \sin ^2(a+b x) \tan ^2(a+b x) \, dx","Integrate[Sin[a + b*x]^2*Tan[a + b*x]^2,x]","\frac{-6 (a+b x)+\sin (2 (a+b x))+4 \tan (a+b x)}{4 b}","\frac{3 \tan (a+b x)}{2 b}-\frac{\sin ^2(a+b x) \tan (a+b x)}{2 b}-\frac{3 x}{2}",1,"(-6*(a + b*x) + Sin[2*(a + b*x)] + 4*Tan[a + b*x])/(4*b)","A",1
85,1,38,28,0.0076102,"\int \tan ^4(a+b x) \, dx","Integrate[Tan[a + b*x]^4,x]","\frac{\tan ^{-1}(\tan (a+b x))}{b}+\frac{\tan ^3(a+b x)}{3 b}-\frac{\tan (a+b x)}{b}","\frac{\tan ^3(a+b x)}{3 b}-\frac{\tan (a+b x)}{b}+x",1,"ArcTan[Tan[a + b*x]]/b - Tan[a + b*x]/b + Tan[a + b*x]^3/(3*b)","A",1
86,1,15,15,0.006585,"\int \sec ^2(a+b x) \tan ^4(a+b x) \, dx","Integrate[Sec[a + b*x]^2*Tan[a + b*x]^4,x]","\frac{\tan ^5(a+b x)}{5 b}","\frac{\tan ^5(a+b x)}{5 b}",1,"Tan[a + b*x]^5/(5*b)","A",1
87,1,77,31,0.0299419,"\int \sec ^4(a+b x) \tan ^4(a+b x) \, dx","Integrate[Sec[a + b*x]^4*Tan[a + b*x]^4,x]","\frac{2 \tan (a+b x)}{35 b}+\frac{\tan (a+b x) \sec ^6(a+b x)}{7 b}-\frac{8 \tan (a+b x) \sec ^4(a+b x)}{35 b}+\frac{\tan (a+b x) \sec ^2(a+b x)}{35 b}","\frac{\tan ^7(a+b x)}{7 b}+\frac{\tan ^5(a+b x)}{5 b}",1,"(2*Tan[a + b*x])/(35*b) + (Sec[a + b*x]^2*Tan[a + b*x])/(35*b) - (8*Sec[a + b*x]^4*Tan[a + b*x])/(35*b) + (Sec[a + b*x]^6*Tan[a + b*x])/(7*b)","B",1
88,1,98,46,0.0343753,"\int \sec ^6(a+b x) \tan ^4(a+b x) \, dx","Integrate[Sec[a + b*x]^6*Tan[a + b*x]^4,x]","\frac{8 \tan (a+b x)}{315 b}+\frac{\tan (a+b x) \sec ^8(a+b x)}{9 b}-\frac{10 \tan (a+b x) \sec ^6(a+b x)}{63 b}+\frac{\tan (a+b x) \sec ^4(a+b x)}{105 b}+\frac{4 \tan (a+b x) \sec ^2(a+b x)}{315 b}","\frac{\tan ^9(a+b x)}{9 b}+\frac{2 \tan ^7(a+b x)}{7 b}+\frac{\tan ^5(a+b x)}{5 b}",1,"(8*Tan[a + b*x])/(315*b) + (4*Sec[a + b*x]^2*Tan[a + b*x])/(315*b) + (Sec[a + b*x]^4*Tan[a + b*x])/(105*b) - (10*Sec[a + b*x]^6*Tan[a + b*x])/(63*b) + (Sec[a + b*x]^8*Tan[a + b*x])/(9*b)","B",1
89,1,62,111,0.1802115,"\int \cos ^6(a+b x) \sin ^4(a+b x) \, dx","Integrate[Cos[a + b*x]^6*Sin[a + b*x]^4,x]","\frac{20 \sin (2 (a+b x))-40 \sin (4 (a+b x))-10 \sin (6 (a+b x))+5 \sin (8 (a+b x))+2 \sin (10 (a+b x))+120 b x}{10240 b}","-\frac{\sin ^3(a+b x) \cos ^7(a+b x)}{10 b}-\frac{3 \sin (a+b x) \cos ^7(a+b x)}{80 b}+\frac{\sin (a+b x) \cos ^5(a+b x)}{160 b}+\frac{\sin (a+b x) \cos ^3(a+b x)}{128 b}+\frac{3 \sin (a+b x) \cos (a+b x)}{256 b}+\frac{3 x}{256}",1,"(120*b*x + 20*Sin[2*(a + b*x)] - 40*Sin[4*(a + b*x)] - 10*Sin[6*(a + b*x)] + 5*Sin[8*(a + b*x)] + 2*Sin[10*(a + b*x)])/(10240*b)","A",1
90,1,33,90,0.0439745,"\int \cos ^4(a+b x) \sin ^4(a+b x) \, dx","Integrate[Cos[a + b*x]^4*Sin[a + b*x]^4,x]","\frac{24 (a+b x)-8 \sin (4 (a+b x))+\sin (8 (a+b x))}{1024 b}","-\frac{\sin ^3(a+b x) \cos ^5(a+b x)}{8 b}-\frac{\sin (a+b x) \cos ^5(a+b x)}{16 b}+\frac{\sin (a+b x) \cos ^3(a+b x)}{64 b}+\frac{3 \sin (a+b x) \cos (a+b x)}{128 b}+\frac{3 x}{128}",1,"(24*(a + b*x) - 8*Sin[4*(a + b*x)] + Sin[8*(a + b*x)])/(1024*b)","A",1
91,1,40,69,0.0543057,"\int \cos ^2(a+b x) \sin ^4(a+b x) \, dx","Integrate[Cos[a + b*x]^2*Sin[a + b*x]^4,x]","\frac{-3 \sin (2 (a+b x))-3 \sin (4 (a+b x))+\sin (6 (a+b x))+12 b x}{192 b}","-\frac{\sin ^3(a+b x) \cos ^3(a+b x)}{6 b}-\frac{\sin (a+b x) \cos ^3(a+b x)}{8 b}+\frac{\sin (a+b x) \cos (a+b x)}{16 b}+\frac{x}{16}",1,"(12*b*x - 3*Sin[2*(a + b*x)] - 3*Sin[4*(a + b*x)] + Sin[6*(a + b*x)])/(192*b)","A",1
92,1,33,46,0.0088682,"\int \sin ^4(a+b x) \, dx","Integrate[Sin[a + b*x]^4,x]","\frac{12 (a+b x)-8 \sin (2 (a+b x))+\sin (4 (a+b x))}{32 b}","-\frac{\sin ^3(a+b x) \cos (a+b x)}{4 b}-\frac{3 \sin (a+b x) \cos (a+b x)}{8 b}+\frac{3 x}{8}",1,"(12*(a + b*x) - 8*Sin[2*(a + b*x)] + Sin[4*(a + b*x)])/(32*b)","A",1
93,1,38,38,0.0125074,"\int \sin ^3(a+b x) \tan (a+b x) \, dx","Integrate[Sin[a + b*x]^3*Tan[a + b*x],x]","-\frac{\sin ^3(a+b x)}{3 b}-\frac{\sin (a+b x)}{b}+\frac{\tanh ^{-1}(\sin (a+b x))}{b}","-\frac{\sin ^3(a+b x)}{3 b}-\frac{\sin (a+b x)}{b}+\frac{\tanh ^{-1}(\sin (a+b x))}{b}",1,"ArcTanh[Sin[a + b*x]]/b - Sin[a + b*x]/b - Sin[a + b*x]^3/(3*b)","A",1
94,1,40,49,0.0903239,"\int \sin (a+b x) \tan ^3(a+b x) \, dx","Integrate[Sin[a + b*x]*Tan[a + b*x]^3,x]","\frac{(\cos (2 (a+b x))+2) \tan (a+b x) \sec (a+b x)-3 \tanh ^{-1}(\sin (a+b x))}{2 b}","\frac{3 \sin (a+b x)}{2 b}+\frac{\sin (a+b x) \tan ^2(a+b x)}{2 b}-\frac{3 \tanh ^{-1}(\sin (a+b x))}{2 b}",1,"(-3*ArcTanh[Sin[a + b*x]] + (2 + Cos[2*(a + b*x)])*Sec[a + b*x]*Tan[a + b*x])/(2*b)","A",1
95,1,45,55,0.1148488,"\int \sec (a+b x) \tan ^4(a+b x) \, dx","Integrate[Sec[a + b*x]*Tan[a + b*x]^4,x]","\frac{6 \tanh ^{-1}(\sin (a+b x))-(5 \cos (2 (a+b x))+1) \tan (a+b x) \sec ^3(a+b x)}{16 b}","\frac{3 \tanh ^{-1}(\sin (a+b x))}{8 b}+\frac{\tan ^3(a+b x) \sec (a+b x)}{4 b}-\frac{3 \tan (a+b x) \sec (a+b x)}{8 b}",1,"(6*ArcTanh[Sin[a + b*x]] - (1 + 5*Cos[2*(a + b*x)])*Sec[a + b*x]^3*Tan[a + b*x])/(16*b)","A",1
96,1,99,78,0.0264766,"\int \sec ^3(a+b x) \tan ^4(a+b x) \, dx","Integrate[Sec[a + b*x]^3*Tan[a + b*x]^4,x]","\frac{\tanh ^{-1}(\sin (a+b x))}{16 b}-\frac{\tan (a+b x) \sec ^5(a+b x)}{6 b}+\frac{\tan ^3(a+b x) \sec ^3(a+b x)}{3 b}+\frac{\tan (a+b x) \sec ^3(a+b x)}{24 b}+\frac{\tan (a+b x) \sec (a+b x)}{16 b}","\frac{\tanh ^{-1}(\sin (a+b x))}{16 b}+\frac{\tan ^3(a+b x) \sec ^3(a+b x)}{6 b}-\frac{\tan (a+b x) \sec ^3(a+b x)}{8 b}+\frac{\tan (a+b x) \sec (a+b x)}{16 b}",1,"ArcTanh[Sin[a + b*x]]/(16*b) + (Sec[a + b*x]*Tan[a + b*x])/(16*b) + (Sec[a + b*x]^3*Tan[a + b*x])/(24*b) - (Sec[a + b*x]^5*Tan[a + b*x])/(6*b) + (Sec[a + b*x]^3*Tan[a + b*x]^3)/(3*b)","A",1
97,1,64,99,0.2899114,"\int \sec ^5(a+b x) \tan ^4(a+b x) \, dx","Integrate[Sec[a + b*x]^5*Tan[a + b*x]^4,x]","\frac{96 \tanh ^{-1}(\sin (a+b x))+(-307 \cos (2 (a+b x))+26 \cos (4 (a+b x))+3 \cos (6 (a+b x))+182) \tan (a+b x) \sec ^7(a+b x)}{4096 b}","\frac{3 \tanh ^{-1}(\sin (a+b x))}{128 b}+\frac{\tan ^3(a+b x) \sec ^5(a+b x)}{8 b}-\frac{\tan (a+b x) \sec ^5(a+b x)}{16 b}+\frac{\tan (a+b x) \sec ^3(a+b x)}{64 b}+\frac{3 \tan (a+b x) \sec (a+b x)}{128 b}",1,"(96*ArcTanh[Sin[a + b*x]] + (182 - 307*Cos[2*(a + b*x)] + 26*Cos[4*(a + b*x)] + 3*Cos[6*(a + b*x)])*Sec[a + b*x]^7*Tan[a + b*x])/(4096*b)","A",1
98,1,68,46,0.3318635,"\int \cos ^7(a+b x) \sin ^5(a+b x) \, dx","Integrate[Cos[a + b*x]^7*Sin[a + b*x]^5,x]","-\frac{600 \cos (2 (a+b x))+75 \cos (4 (a+b x))-100 \cos (6 (a+b x))-30 \cos (8 (a+b x))+12 \cos (10 (a+b x))+5 \cos (12 (a+b x))}{122880 b}","-\frac{\cos ^{12}(a+b x)}{12 b}+\frac{\cos ^{10}(a+b x)}{5 b}-\frac{\cos ^8(a+b x)}{8 b}",1,"-1/122880*(600*Cos[2*(a + b*x)] + 75*Cos[4*(a + b*x)] - 100*Cos[6*(a + b*x)] - 30*Cos[8*(a + b*x)] + 12*Cos[10*(a + b*x)] + 5*Cos[12*(a + b*x)])/b","A",1
99,1,37,46,0.2651395,"\int \cos ^6(a+b x) \sin ^5(a+b x) \, dx","Integrate[Cos[a + b*x]^6*Sin[a + b*x]^5,x]","\frac{\cos ^7(a+b x) (364 \cos (2 (a+b x))-63 \cos (4 (a+b x))-365)}{5544 b}","-\frac{\cos ^{11}(a+b x)}{11 b}+\frac{2 \cos ^9(a+b x)}{9 b}-\frac{\cos ^7(a+b x)}{7 b}",1,"(Cos[a + b*x]^7*(-365 + 364*Cos[2*(a + b*x)] - 63*Cos[4*(a + b*x)]))/(5544*b)","A",1
100,1,50,46,0.0274728,"\int \cos ^5(a+b x) \sin ^5(a+b x) \, dx","Integrate[Cos[a + b*x]^5*Sin[a + b*x]^5,x]","\frac{1}{32} \left(-\frac{5 \cos (2 (a+b x))}{16 b}+\frac{5 \cos (6 (a+b x))}{96 b}-\frac{\cos (10 (a+b x))}{160 b}\right)","\frac{\sin ^{10}(a+b x)}{10 b}-\frac{\sin ^8(a+b x)}{4 b}+\frac{\sin ^6(a+b x)}{6 b}",1,"((-5*Cos[2*(a + b*x)])/(16*b) + (5*Cos[6*(a + b*x)])/(96*b) - Cos[10*(a + b*x)]/(160*b))/32","A",1
101,1,37,46,0.1156399,"\int \cos ^4(a+b x) \sin ^5(a+b x) \, dx","Integrate[Cos[a + b*x]^4*Sin[a + b*x]^5,x]","\frac{\cos ^5(a+b x) (220 \cos (2 (a+b x))-35 \cos (4 (a+b x))-249)}{2520 b}","-\frac{\cos ^9(a+b x)}{9 b}+\frac{2 \cos ^7(a+b x)}{7 b}-\frac{\cos ^5(a+b x)}{5 b}",1,"(Cos[a + b*x]^5*(-249 + 220*Cos[2*(a + b*x)] - 35*Cos[4*(a + b*x)]))/(2520*b)","A",1
102,1,48,31,0.0906509,"\int \cos ^3(a+b x) \sin ^5(a+b x) \, dx","Integrate[Cos[a + b*x]^3*Sin[a + b*x]^5,x]","\frac{-72 \cos (2 (a+b x))+12 \cos (4 (a+b x))+8 \cos (6 (a+b x))-3 \cos (8 (a+b x))}{3072 b}","\frac{\sin ^6(a+b x)}{6 b}-\frac{\sin ^8(a+b x)}{8 b}",1,"(-72*Cos[2*(a + b*x)] + 12*Cos[4*(a + b*x)] + 8*Cos[6*(a + b*x)] - 3*Cos[8*(a + b*x)])/(3072*b)","A",1
103,1,37,46,0.0786276,"\int \cos ^2(a+b x) \sin ^5(a+b x) \, dx","Integrate[Cos[a + b*x]^2*Sin[a + b*x]^5,x]","\frac{\cos ^3(a+b x) (108 \cos (2 (a+b x))-15 \cos (4 (a+b x))-157)}{840 b}","-\frac{\cos ^7(a+b x)}{7 b}+\frac{2 \cos ^5(a+b x)}{5 b}-\frac{\cos ^3(a+b x)}{3 b}",1,"(Cos[a + b*x]^3*(-157 + 108*Cos[2*(a + b*x)] - 15*Cos[4*(a + b*x)]))/(840*b)","A",1
104,1,15,15,0.0037264,"\int \cos (a+b x) \sin ^5(a+b x) \, dx","Integrate[Cos[a + b*x]*Sin[a + b*x]^5,x]","\frac{\sin ^6(a+b x)}{6 b}","\frac{\sin ^6(a+b x)}{6 b}",1,"Sin[a + b*x]^6/(6*b)","A",1
105,1,35,40,0.0292024,"\int \sin ^4(a+b x) \tan (a+b x) \, dx","Integrate[Sin[a + b*x]^4*Tan[a + b*x],x]","-\frac{\frac{1}{4} \cos ^4(a+b x)-\cos ^2(a+b x)+\log (\cos (a+b x))}{b}","-\frac{\cos ^4(a+b x)}{4 b}+\frac{\cos ^2(a+b x)}{b}-\frac{\log (\cos (a+b x))}{b}",1,"-((-Cos[a + b*x]^2 + Cos[a + b*x]^4/4 + Log[Cos[a + b*x]])/b)","A",1
106,1,39,37,0.0254783,"\int \sin ^3(a+b x) \tan ^2(a+b x) \, dx","Integrate[Sin[a + b*x]^3*Tan[a + b*x]^2,x]","\frac{7 \cos (a+b x)}{4 b}-\frac{\cos (3 (a+b x))}{12 b}+\frac{\sec (a+b x)}{b}","-\frac{\cos ^3(a+b x)}{3 b}+\frac{2 \cos (a+b x)}{b}+\frac{\sec (a+b x)}{b}",1,"(7*Cos[a + b*x])/(4*b) - Cos[3*(a + b*x)]/(12*b) + Sec[a + b*x]/b","A",1
107,1,33,43,0.0345243,"\int \sin ^2(a+b x) \tan ^3(a+b x) \, dx","Integrate[Sin[a + b*x]^2*Tan[a + b*x]^3,x]","\frac{\sin ^2(a+b x)+\sec ^2(a+b x)+4 \log (\cos (a+b x))}{2 b}","-\frac{\cos ^2(a+b x)}{2 b}+\frac{\sec ^2(a+b x)}{2 b}+\frac{2 \log (\cos (a+b x))}{b}",1,"(4*Log[Cos[a + b*x]] + Sec[a + b*x]^2 + Sin[a + b*x]^2)/(2*b)","A",1
108,1,38,38,0.0223533,"\int \sin (a+b x) \tan ^4(a+b x) \, dx","Integrate[Sin[a + b*x]*Tan[a + b*x]^4,x]","-\frac{\cos (a+b x)}{b}+\frac{\sec ^3(a+b x)}{3 b}-\frac{2 \sec (a+b x)}{b}","-\frac{\cos (a+b x)}{b}+\frac{\sec ^3(a+b x)}{3 b}-\frac{2 \sec (a+b x)}{b}",1,"-(Cos[a + b*x]/b) - (2*Sec[a + b*x])/b + Sec[a + b*x]^3/(3*b)","A",1
109,1,37,43,0.0413826,"\int \tan ^5(a+b x) \, dx","Integrate[Tan[a + b*x]^5,x]","-\frac{-\tan ^4(a+b x)+2 \tan ^2(a+b x)+4 \log (\cos (a+b x))}{4 b}","\frac{\tan ^4(a+b x)}{4 b}-\frac{\tan ^2(a+b x)}{2 b}-\frac{\log (\cos (a+b x))}{b}",1,"-1/4*(4*Log[Cos[a + b*x]] + 2*Tan[a + b*x]^2 - Tan[a + b*x]^4)/b","A",1
110,1,41,41,0.0257185,"\int \sec (a+b x) \tan ^5(a+b x) \, dx","Integrate[Sec[a + b*x]*Tan[a + b*x]^5,x]","\frac{\sec ^5(a+b x)}{5 b}-\frac{2 \sec ^3(a+b x)}{3 b}+\frac{\sec (a+b x)}{b}","\frac{\sec ^5(a+b x)}{5 b}-\frac{2 \sec ^3(a+b x)}{3 b}+\frac{\sec (a+b x)}{b}",1,"Sec[a + b*x]/b - (2*Sec[a + b*x]^3)/(3*b) + Sec[a + b*x]^5/(5*b)","A",1
111,1,15,15,0.0077683,"\int \sec ^2(a+b x) \tan ^5(a+b x) \, dx","Integrate[Sec[a + b*x]^2*Tan[a + b*x]^5,x]","\frac{\tan ^6(a+b x)}{6 b}","\frac{\tan ^6(a+b x)}{6 b}",1,"Tan[a + b*x]^6/(6*b)","A",1
112,1,46,46,0.0289187,"\int \sec ^3(a+b x) \tan ^5(a+b x) \, dx","Integrate[Sec[a + b*x]^3*Tan[a + b*x]^5,x]","\frac{\sec ^7(a+b x)}{7 b}-\frac{2 \sec ^5(a+b x)}{5 b}+\frac{\sec ^3(a+b x)}{3 b}","\frac{\sec ^7(a+b x)}{7 b}-\frac{2 \sec ^5(a+b x)}{5 b}+\frac{\sec ^3(a+b x)}{3 b}",1,"Sec[a + b*x]^3/(3*b) - (2*Sec[a + b*x]^5)/(5*b) + Sec[a + b*x]^7/(7*b)","A",1
113,1,38,31,0.0508821,"\int \sec ^4(a+b x) \tan ^5(a+b x) \, dx","Integrate[Sec[a + b*x]^4*Tan[a + b*x]^5,x]","\frac{3 \sec ^8(a+b x)-8 \sec ^6(a+b x)+6 \sec ^4(a+b x)}{24 b}","\frac{\tan ^8(a+b x)}{8 b}+\frac{\tan ^6(a+b x)}{6 b}",1,"(6*Sec[a + b*x]^4 - 8*Sec[a + b*x]^6 + 3*Sec[a + b*x]^8)/(24*b)","A",1
114,1,46,46,0.0305354,"\int \sec ^5(a+b x) \tan ^5(a+b x) \, dx","Integrate[Sec[a + b*x]^5*Tan[a + b*x]^5,x]","\frac{\sec ^9(a+b x)}{9 b}-\frac{2 \sec ^7(a+b x)}{7 b}+\frac{\sec ^5(a+b x)}{5 b}","\frac{\sec ^9(a+b x)}{9 b}-\frac{2 \sec ^7(a+b x)}{7 b}+\frac{\sec ^5(a+b x)}{5 b}",1,"Sec[a + b*x]^5/(5*b) - (2*Sec[a + b*x]^7)/(7*b) + Sec[a + b*x]^9/(9*b)","A",1
115,1,38,46,0.0518719,"\int \sec ^6(a+b x) \tan ^5(a+b x) \, dx","Integrate[Sec[a + b*x]^6*Tan[a + b*x]^5,x]","\frac{6 \sec ^{10}(a+b x)-15 \sec ^8(a+b x)+10 \sec ^6(a+b x)}{60 b}","\frac{\sec ^{10}(a+b x)}{10 b}-\frac{\sec ^8(a+b x)}{4 b}+\frac{\sec ^6(a+b x)}{6 b}",1,"(10*Sec[a + b*x]^6 - 15*Sec[a + b*x]^8 + 6*Sec[a + b*x]^10)/(60*b)","A",1
116,1,46,46,0.0300102,"\int \sec ^7(a+b x) \tan ^5(a+b x) \, dx","Integrate[Sec[a + b*x]^7*Tan[a + b*x]^5,x]","\frac{\sec ^{11}(a+b x)}{11 b}-\frac{2 \sec ^9(a+b x)}{9 b}+\frac{\sec ^7(a+b x)}{7 b}","\frac{\sec ^{11}(a+b x)}{11 b}-\frac{2 \sec ^9(a+b x)}{9 b}+\frac{\sec ^7(a+b x)}{7 b}",1,"Sec[a + b*x]^7/(7*b) - (2*Sec[a + b*x]^9)/(9*b) + Sec[a + b*x]^11/(11*b)","A",1
117,1,38,46,0.1261732,"\int \sec ^8(a+b x) \tan ^5(a+b x) \, dx","Integrate[Sec[a + b*x]^8*Tan[a + b*x]^5,x]","\frac{10 \sec ^{12}(a+b x)-24 \sec ^{10}(a+b x)+15 \sec ^8(a+b x)}{120 b}","\frac{\sec ^{12}(a+b x)}{12 b}-\frac{\sec ^{10}(a+b x)}{5 b}+\frac{\sec ^8(a+b x)}{8 b}",1,"(15*Sec[a + b*x]^8 - 24*Sec[a + b*x]^10 + 10*Sec[a + b*x]^12)/(120*b)","A",1
118,1,52,66,0.1679406,"\int \sin ^3(a+b x) \tan ^3(a+b x) \, dx","Integrate[Sin[a + b*x]^3*Tan[a + b*x]^3,x]","\frac{(24 \cos (2 (a+b x))-\cos (4 (a+b x))+37) \tan (a+b x) \sec (a+b x)-60 \tanh ^{-1}(\sin (a+b x))}{24 b}","\frac{5 \sin ^3(a+b x)}{6 b}+\frac{5 \sin (a+b x)}{2 b}+\frac{\sin ^3(a+b x) \tan ^2(a+b x)}{2 b}-\frac{5 \tanh ^{-1}(\sin (a+b x))}{2 b}",1,"(-60*ArcTanh[Sin[a + b*x]] + (37 + 24*Cos[2*(a + b*x)] - Cos[4*(a + b*x)])*Sec[a + b*x]*Tan[a + b*x])/(24*b)","A",1
119,1,50,50,0.0285166,"\int \sin (a+b x) \tan ^6(a+b x) \, dx","Integrate[Sin[a + b*x]*Tan[a + b*x]^6,x]","\frac{\cos (a+b x)}{b}+\frac{\sec ^5(a+b x)}{5 b}-\frac{\sec ^3(a+b x)}{b}+\frac{3 \sec (a+b x)}{b}","\frac{\cos (a+b x)}{b}+\frac{\sec ^5(a+b x)}{5 b}-\frac{\sec ^3(a+b x)}{b}+\frac{3 \sec (a+b x)}{b}",1,"Cos[a + b*x]/b + (3*Sec[a + b*x])/b - Sec[a + b*x]^3/b + Sec[a + b*x]^5/(5*b)","A",1
120,1,75,53,0.0270607,"\int \cos ^5(a+b x) \cot (a+b x) \, dx","Integrate[Cos[a + b*x]^5*Cot[a + b*x],x]","\frac{11 \cos (a+b x)}{8 b}+\frac{7 \cos (3 (a+b x))}{48 b}+\frac{\cos (5 (a+b x))}{80 b}+\frac{\log \left(\sin \left(\frac{1}{2} (a+b x)\right)\right)}{b}-\frac{\log \left(\cos \left(\frac{1}{2} (a+b x)\right)\right)}{b}","\frac{\cos ^5(a+b x)}{5 b}+\frac{\cos ^3(a+b x)}{3 b}+\frac{\cos (a+b x)}{b}-\frac{\tanh ^{-1}(\cos (a+b x))}{b}",1,"(11*Cos[a + b*x])/(8*b) + (7*Cos[3*(a + b*x)])/(48*b) + Cos[5*(a + b*x)]/(80*b) - Log[Cos[(a + b*x)/2]]/b + Log[Sin[(a + b*x)/2]]/b","A",1
121,1,40,40,0.013377,"\int \cos ^4(a+b x) \cot (a+b x) \, dx","Integrate[Cos[a + b*x]^4*Cot[a + b*x],x]","\frac{\sin ^4(a+b x)}{4 b}-\frac{\sin ^2(a+b x)}{b}+\frac{\log (\sin (a+b x))}{b}","\frac{\sin ^4(a+b x)}{4 b}-\frac{\sin ^2(a+b x)}{b}+\frac{\log (\sin (a+b x))}{b}",1,"Log[Sin[a + b*x]]/b - Sin[a + b*x]^2/b + Sin[a + b*x]^4/(4*b)","A",1
122,1,60,38,0.0236526,"\int \cos ^3(a+b x) \cot (a+b x) \, dx","Integrate[Cos[a + b*x]^3*Cot[a + b*x],x]","\frac{5 \cos (a+b x)}{4 b}+\frac{\cos (3 (a+b x))}{12 b}+\frac{\log \left(\sin \left(\frac{1}{2} (a+b x)\right)\right)}{b}-\frac{\log \left(\cos \left(\frac{1}{2} (a+b x)\right)\right)}{b}","\frac{\cos ^3(a+b x)}{3 b}+\frac{\cos (a+b x)}{b}-\frac{\tanh ^{-1}(\cos (a+b x))}{b}",1,"(5*Cos[a + b*x])/(4*b) + Cos[3*(a + b*x)]/(12*b) - Log[Cos[(a + b*x)/2]]/b + Log[Sin[(a + b*x)/2]]/b","A",1
123,1,27,27,0.0121562,"\int \cos ^2(a+b x) \cot (a+b x) \, dx","Integrate[Cos[a + b*x]^2*Cot[a + b*x],x]","\frac{\log (\sin (a+b x))}{b}-\frac{\sin ^2(a+b x)}{2 b}","\frac{\log (\sin (a+b x))}{b}-\frac{\sin ^2(a+b x)}{2 b}",1,"Log[Sin[a + b*x]]/b - Sin[a + b*x]^2/(2*b)","A",1
124,1,42,23,0.0148866,"\int \cos (a+b x) \cot (a+b x) \, dx","Integrate[Cos[a + b*x]*Cot[a + b*x],x]","\frac{\cos (a+b x)}{b}+\frac{\log \left(\sin \left(\frac{1}{2} (a+b x)\right)\right)}{b}-\frac{\log \left(\cos \left(\frac{1}{2} (a+b x)\right)\right)}{b}","\frac{\cos (a+b x)}{b}-\frac{\tanh ^{-1}(\cos (a+b x))}{b}",1,"Cos[a + b*x]/b - Log[Cos[(a + b*x)/2]]/b + Log[Sin[(a + b*x)/2]]/b","A",1
125,1,19,11,0.0077982,"\int \cot (a+b x) \, dx","Integrate[Cot[a + b*x],x]","\frac{\log (\tan (a+b x))+\log (\cos (a+b x))}{b}","\frac{\log (\sin (a+b x))}{b}",1,"(Log[Cos[a + b*x]] + Log[Tan[a + b*x]])/b","A",1
126,1,31,11,0.0207724,"\int \csc (a+b x) \sec (a+b x) \, dx","Integrate[Csc[a + b*x]*Sec[a + b*x],x]","2 \left(\frac{\log (\sin (a+b x))}{2 b}-\frac{\log (\cos (a+b x))}{2 b}\right)","\frac{\log (\tan (a+b x))}{b}",1,"2*(-1/2*Log[Cos[a + b*x]]/b + Log[Sin[a + b*x]]/(2*b))","B",1
127,1,42,23,0.0248311,"\int \csc (a+b x) \sec ^2(a+b x) \, dx","Integrate[Csc[a + b*x]*Sec[a + b*x]^2,x]","\frac{\sec (a+b x)}{b}+\frac{\log \left(\sin \left(\frac{1}{2} (a+b x)\right)\right)}{b}-\frac{\log \left(\cos \left(\frac{1}{2} (a+b x)\right)\right)}{b}","\frac{\sec (a+b x)}{b}-\frac{\tanh ^{-1}(\cos (a+b x))}{b}",1,"-(Log[Cos[(a + b*x)/2]]/b) + Log[Sin[(a + b*x)/2]]/b + Sec[a + b*x]/b","A",1
128,1,36,27,0.0290958,"\int \csc (a+b x) \sec ^3(a+b x) \, dx","Integrate[Csc[a + b*x]*Sec[a + b*x]^3,x]","-\frac{-\sec ^2(a+b x)-2 \log (\sin (a+b x))+2 \log (\cos (a+b x))}{2 b}","\frac{\tan ^2(a+b x)}{2 b}+\frac{\log (\tan (a+b x))}{b}",1,"-1/2*(2*Log[Cos[a + b*x]] - 2*Log[Sin[a + b*x]] - Sec[a + b*x]^2)/b","A",1
129,1,57,38,0.0214181,"\int \csc (a+b x) \sec ^4(a+b x) \, dx","Integrate[Csc[a + b*x]*Sec[a + b*x]^4,x]","\frac{\sec ^3(a+b x)}{3 b}+\frac{\sec (a+b x)}{b}+\frac{\log \left(\sin \left(\frac{1}{2} (a+b x)\right)\right)}{b}-\frac{\log \left(\cos \left(\frac{1}{2} (a+b x)\right)\right)}{b}","\frac{\sec ^3(a+b x)}{3 b}+\frac{\sec (a+b x)}{b}-\frac{\tanh ^{-1}(\cos (a+b x))}{b}",1,"-(Log[Cos[(a + b*x)/2]]/b) + Log[Sin[(a + b*x)/2]]/b + Sec[a + b*x]/b + Sec[a + b*x]^3/(3*b)","A",1
130,1,46,39,0.0837169,"\int \csc (a+b x) \sec ^5(a+b x) \, dx","Integrate[Csc[a + b*x]*Sec[a + b*x]^5,x]","-\frac{-\sec ^4(a+b x)-2 \sec ^2(a+b x)-4 \log (\sin (a+b x))+4 \log (\cos (a+b x))}{4 b}","\frac{\tan ^4(a+b x)}{4 b}+\frac{\tan ^2(a+b x)}{b}+\frac{\log (\tan (a+b x))}{b}",1,"-1/4*(4*Log[Cos[a + b*x]] - 4*Log[Sin[a + b*x]] - 2*Sec[a + b*x]^2 - Sec[a + b*x]^4)/b","A",1
131,1,72,53,0.0214134,"\int \csc (a+b x) \sec ^6(a+b x) \, dx","Integrate[Csc[a + b*x]*Sec[a + b*x]^6,x]","\frac{\sec ^5(a+b x)}{5 b}+\frac{\sec ^3(a+b x)}{3 b}+\frac{\sec (a+b x)}{b}+\frac{\log \left(\sin \left(\frac{1}{2} (a+b x)\right)\right)}{b}-\frac{\log \left(\cos \left(\frac{1}{2} (a+b x)\right)\right)}{b}","\frac{\sec ^5(a+b x)}{5 b}+\frac{\sec ^3(a+b x)}{3 b}+\frac{\sec (a+b x)}{b}-\frac{\tanh ^{-1}(\cos (a+b x))}{b}",1,"-(Log[Cos[(a + b*x)/2]]/b) + Log[Sin[(a + b*x)/2]]/b + Sec[a + b*x]/b + Sec[a + b*x]^3/(3*b) + Sec[a + b*x]^5/(5*b)","A",1
132,1,56,57,0.1366488,"\int \csc (a+b x) \sec ^7(a+b x) \, dx","Integrate[Csc[a + b*x]*Sec[a + b*x]^7,x]","-\frac{-2 \sec ^6(a+b x)-3 \sec ^4(a+b x)-6 \sec ^2(a+b x)-12 \log (\sin (a+b x))+12 \log (\cos (a+b x))}{12 b}","\frac{\tan ^6(a+b x)}{6 b}+\frac{3 \tan ^4(a+b x)}{4 b}+\frac{3 \tan ^2(a+b x)}{2 b}+\frac{\log (\tan (a+b x))}{b}",1,"-1/12*(12*Log[Cos[a + b*x]] - 12*Log[Sin[a + b*x]] - 6*Sec[a + b*x]^2 - 3*Sec[a + b*x]^4 - 2*Sec[a + b*x]^6)/b","A",1
133,1,50,50,0.0226654,"\int \cos ^5(a+b x) \cot ^2(a+b x) \, dx","Integrate[Cos[a + b*x]^5*Cot[a + b*x]^2,x]","-\frac{\sin ^5(a+b x)}{5 b}+\frac{\sin ^3(a+b x)}{b}-\frac{3 \sin (a+b x)}{b}-\frac{\csc (a+b x)}{b}","-\frac{\sin ^5(a+b x)}{5 b}+\frac{\sin ^3(a+b x)}{b}-\frac{3 \sin (a+b x)}{b}-\frac{\csc (a+b x)}{b}",1,"-(Csc[a + b*x]/b) - (3*Sin[a + b*x])/b + Sin[a + b*x]^3/b - Sin[a + b*x]^5/(5*b)","A",1
134,1,41,61,0.1289691,"\int \cos ^4(a+b x) \cot ^2(a+b x) \, dx","Integrate[Cos[a + b*x]^4*Cot[a + b*x]^2,x]","-\frac{16 \sin (2 (a+b x))+\sin (4 (a+b x))+32 \cot (a+b x)+60 a+60 b x}{32 b}","-\frac{15 \cot (a+b x)}{8 b}+\frac{\cos ^4(a+b x) \cot (a+b x)}{4 b}+\frac{5 \cos ^2(a+b x) \cot (a+b x)}{8 b}-\frac{15 x}{8}",1,"-1/32*(60*a + 60*b*x + 32*Cot[a + b*x] + 16*Sin[2*(a + b*x)] + Sin[4*(a + b*x)])/b","A",1
135,1,38,38,0.0158182,"\int \cos ^3(a+b x) \cot ^2(a+b x) \, dx","Integrate[Cos[a + b*x]^3*Cot[a + b*x]^2,x]","\frac{\sin ^3(a+b x)}{3 b}-\frac{2 \sin (a+b x)}{b}-\frac{\csc (a+b x)}{b}","\frac{\sin ^3(a+b x)}{3 b}-\frac{2 \sin (a+b x)}{b}-\frac{\csc (a+b x)}{b}",1,"-(Csc[a + b*x]/b) - (2*Sin[a + b*x])/b + Sin[a + b*x]^3/(3*b)","A",1
136,1,31,40,0.123507,"\int \cos ^2(a+b x) \cot ^2(a+b x) \, dx","Integrate[Cos[a + b*x]^2*Cot[a + b*x]^2,x]","-\frac{6 (a+b x)+\sin (2 (a+b x))+4 \cot (a+b x)}{4 b}","-\frac{3 \cot (a+b x)}{2 b}+\frac{\cos ^2(a+b x) \cot (a+b x)}{2 b}-\frac{3 x}{2}",1,"-1/4*(6*(a + b*x) + 4*Cot[a + b*x] + Sin[2*(a + b*x)])/b","A",1
137,1,23,23,0.0114398,"\int \cos (a+b x) \cot ^2(a+b x) \, dx","Integrate[Cos[a + b*x]*Cot[a + b*x]^2,x]","-\frac{\sin (a+b x)}{b}-\frac{\csc (a+b x)}{b}","-\frac{\sin (a+b x)}{b}-\frac{\csc (a+b x)}{b}",1,"-(Csc[a + b*x]/b) - Sin[a + b*x]/b","A",1
138,1,29,15,0.0132863,"\int \cot ^2(a+b x) \, dx","Integrate[Cot[a + b*x]^2,x]","-\frac{\cot (a+b x) \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};-\tan ^2(a+b x)\right)}{b}","-\frac{\cot (a+b x)}{b}-x",1,"-((Cot[a + b*x]*Hypergeometric2F1[-1/2, 1, 1/2, -Tan[a + b*x]^2])/b)","C",1
139,1,11,11,0.0073596,"\int \cot (a+b x) \csc (a+b x) \, dx","Integrate[Cot[a + b*x]*Csc[a + b*x],x]","-\frac{\csc (a+b x)}{b}","-\frac{\csc (a+b x)}{b}",1,"-(Csc[a + b*x]/b)","A",1
140,1,27,23,0.0139745,"\int \csc ^2(a+b x) \sec (a+b x) \, dx","Integrate[Csc[a + b*x]^2*Sec[a + b*x],x]","-\frac{\csc (a+b x) \, _2F_1\left(-\frac{1}{2},1;\frac{1}{2};\sin ^2(a+b x)\right)}{b}","\frac{\tanh ^{-1}(\sin (a+b x))}{b}-\frac{\csc (a+b x)}{b}",1,"-((Csc[a + b*x]*Hypergeometric2F1[-1/2, 1, 1/2, Sin[a + b*x]^2])/b)","C",1
141,1,13,22,0.0094095,"\int \csc ^2(a+b x) \sec ^2(a+b x) \, dx","Integrate[Csc[a + b*x]^2*Sec[a + b*x]^2,x]","-\frac{2 \cot (2 (a+b x))}{b}","\frac{\tan (a+b x)}{b}-\frac{\cot (a+b x)}{b}",1,"(-2*Cot[2*(a + b*x)])/b","A",1
142,1,27,49,0.0124796,"\int \csc ^2(a+b x) \sec ^3(a+b x) \, dx","Integrate[Csc[a + b*x]^2*Sec[a + b*x]^3,x]","-\frac{\csc (a+b x) \, _2F_1\left(-\frac{1}{2},2;\frac{1}{2};\sin ^2(a+b x)\right)}{b}","-\frac{3 \csc (a+b x)}{2 b}+\frac{3 \tanh ^{-1}(\sin (a+b x))}{2 b}+\frac{\csc (a+b x) \sec ^2(a+b x)}{2 b}",1,"-((Csc[a + b*x]*Hypergeometric2F1[-1/2, 2, 1/2, Sin[a + b*x]^2])/b)","C",1
143,1,46,38,0.0308487,"\int \csc ^2(a+b x) \sec ^4(a+b x) \, dx","Integrate[Csc[a + b*x]^2*Sec[a + b*x]^4,x]","\frac{5 \tan (a+b x)}{3 b}-\frac{\cot (a+b x)}{b}+\frac{\tan (a+b x) \sec ^2(a+b x)}{3 b}","\frac{\tan ^3(a+b x)}{3 b}+\frac{2 \tan (a+b x)}{b}-\frac{\cot (a+b x)}{b}",1,"-(Cot[a + b*x]/b) + (5*Tan[a + b*x])/(3*b) + (Sec[a + b*x]^2*Tan[a + b*x])/(3*b)","A",1
144,1,27,70,0.0130378,"\int \csc ^2(a+b x) \sec ^5(a+b x) \, dx","Integrate[Csc[a + b*x]^2*Sec[a + b*x]^5,x]","-\frac{\csc (a+b x) \, _2F_1\left(-\frac{1}{2},3;\frac{1}{2};\sin ^2(a+b x)\right)}{b}","-\frac{15 \csc (a+b x)}{8 b}+\frac{15 \tanh ^{-1}(\sin (a+b x))}{8 b}+\frac{\csc (a+b x) \sec ^4(a+b x)}{4 b}+\frac{5 \csc (a+b x) \sec ^2(a+b x)}{8 b}",1,"-((Csc[a + b*x]*Hypergeometric2F1[-1/2, 3, 1/2, Sin[a + b*x]^2])/b)","C",1
145,1,45,58,0.0953337,"\int \cos ^4(a+b x) \cot ^3(a+b x) \, dx","Integrate[Cos[a + b*x]^4*Cot[a + b*x]^3,x]","-\frac{\sin ^4(a+b x)-6 \sin ^2(a+b x)+2 \csc ^2(a+b x)+12 \log (\sin (a+b x))}{4 b}","-\frac{\sin ^4(a+b x)}{4 b}+\frac{3 \sin ^2(a+b x)}{2 b}-\frac{\csc ^2(a+b x)}{2 b}-\frac{3 \log (\sin (a+b x))}{b}",1,"-1/4*(2*Csc[a + b*x]^2 + 12*Log[Sin[a + b*x]] - 6*Sin[a + b*x]^2 + Sin[a + b*x]^4)/b","A",1
146,1,103,66,0.0349418,"\int \cos ^3(a+b x) \cot ^3(a+b x) \, dx","Integrate[Cos[a + b*x]^3*Cot[a + b*x]^3,x]","-\frac{9 \cos (a+b x)}{4 b}-\frac{\cos (3 (a+b x))}{12 b}-\frac{\csc ^2\left(\frac{1}{2} (a+b x)\right)}{8 b}+\frac{\sec ^2\left(\frac{1}{2} (a+b x)\right)}{8 b}-\frac{5 \log \left(\sin \left(\frac{1}{2} (a+b x)\right)\right)}{2 b}+\frac{5 \log \left(\cos \left(\frac{1}{2} (a+b x)\right)\right)}{2 b}","-\frac{5 \cos ^3(a+b x)}{6 b}-\frac{5 \cos (a+b x)}{2 b}-\frac{\cos ^3(a+b x) \cot ^2(a+b x)}{2 b}+\frac{5 \tanh ^{-1}(\cos (a+b x))}{2 b}",1,"(-9*Cos[a + b*x])/(4*b) - Cos[3*(a + b*x)]/(12*b) - Csc[(a + b*x)/2]^2/(8*b) + (5*Log[Cos[(a + b*x)/2]])/(2*b) - (5*Log[Sin[(a + b*x)/2]])/(2*b) + Sec[(a + b*x)/2]^2/(8*b)","A",1
147,1,35,43,0.0576749,"\int \cos ^2(a+b x) \cot ^3(a+b x) \, dx","Integrate[Cos[a + b*x]^2*Cot[a + b*x]^3,x]","-\frac{-\sin ^2(a+b x)+\csc ^2(a+b x)+4 \log (\sin (a+b x))}{2 b}","\frac{\sin ^2(a+b x)}{2 b}-\frac{\csc ^2(a+b x)}{2 b}-\frac{2 \log (\sin (a+b x))}{b}",1,"-1/2*(Csc[a + b*x]^2 + 4*Log[Sin[a + b*x]] - Sin[a + b*x]^2)/b","A",1
148,1,86,49,0.025089,"\int \cos (a+b x) \cot ^3(a+b x) \, dx","Integrate[Cos[a + b*x]*Cot[a + b*x]^3,x]","-\frac{\cos (a+b x)}{b}-\frac{\csc ^2\left(\frac{1}{2} (a+b x)\right)}{8 b}+\frac{\sec ^2\left(\frac{1}{2} (a+b x)\right)}{8 b}-\frac{3 \log \left(\sin \left(\frac{1}{2} (a+b x)\right)\right)}{2 b}+\frac{3 \log \left(\cos \left(\frac{1}{2} (a+b x)\right)\right)}{2 b}","-\frac{3 \cos (a+b x)}{2 b}-\frac{\cos (a+b x) \cot ^2(a+b x)}{2 b}+\frac{3 \tanh ^{-1}(\cos (a+b x))}{2 b}",1,"-(Cos[a + b*x]/b) - Csc[(a + b*x)/2]^2/(8*b) + (3*Log[Cos[(a + b*x)/2]])/(2*b) - (3*Log[Sin[(a + b*x)/2]])/(2*b) + Sec[(a + b*x)/2]^2/(8*b)","A",1
149,1,34,28,0.0755896,"\int \cot ^3(a+b x) \, dx","Integrate[Cot[a + b*x]^3,x]","-\frac{\cot ^2(a+b x)+2 \log (\tan (a+b x))+2 \log (\cos (a+b x))}{2 b}","-\frac{\cot ^2(a+b x)}{2 b}-\frac{\log (\sin (a+b x))}{b}",1,"-1/2*(Cot[a + b*x]^2 + 2*Log[Cos[a + b*x]] + 2*Log[Tan[a + b*x]])/b","A",1
150,1,75,34,0.0249763,"\int \cot ^2(a+b x) \csc (a+b x) \, dx","Integrate[Cot[a + b*x]^2*Csc[a + b*x],x]","-\frac{\csc ^2\left(\frac{1}{2} (a+b x)\right)}{8 b}+\frac{\sec ^2\left(\frac{1}{2} (a+b x)\right)}{8 b}-\frac{\log \left(\sin \left(\frac{1}{2} (a+b x)\right)\right)}{2 b}+\frac{\log \left(\cos \left(\frac{1}{2} (a+b x)\right)\right)}{2 b}","\frac{\tanh ^{-1}(\cos (a+b x))}{2 b}-\frac{\cot (a+b x) \csc (a+b x)}{2 b}",1,"-1/8*Csc[(a + b*x)/2]^2/b + Log[Cos[(a + b*x)/2]]/(2*b) - Log[Sin[(a + b*x)/2]]/(2*b) + Sec[(a + b*x)/2]^2/(8*b)","B",1
151,1,15,15,0.0111162,"\int \cot (a+b x) \csc ^2(a+b x) \, dx","Integrate[Cot[a + b*x]*Csc[a + b*x]^2,x]","-\frac{\csc ^2(a+b x)}{2 b}","-\frac{\csc ^2(a+b x)}{2 b}",1,"-1/2*Csc[a + b*x]^2/b","A",1
152,1,34,27,0.0355561,"\int \csc ^3(a+b x) \sec (a+b x) \, dx","Integrate[Csc[a + b*x]^3*Sec[a + b*x],x]","-\frac{\csc ^2(a+b x)-2 \log (\sin (a+b x))+2 \log (\cos (a+b x))}{2 b}","\frac{\log (\tan (a+b x))}{b}-\frac{\cot ^2(a+b x)}{2 b}",1,"-1/2*(Csc[a + b*x]^2 + 2*Log[Cos[a + b*x]] - 2*Log[Sin[a + b*x]])/b","A",1
153,1,143,49,0.2252124,"\int \csc ^3(a+b x) \sec ^2(a+b x) \, dx","Integrate[Csc[a + b*x]^3*Sec[a + b*x]^2,x]","\frac{\csc ^4(a+b x) \left(-6 \cos (2 (a+b x))+2 \cos (3 (a+b x))+3 \cos (3 (a+b x)) \log \left(\cos \left(\frac{1}{2} (a+b x)\right)\right)-3 \cos (3 (a+b x)) \log \left(\sin \left(\frac{1}{2} (a+b x)\right)\right)+\cos (a+b x) \left(3 \log \left(\sin \left(\frac{1}{2} (a+b x)\right)\right)-3 \log \left(\cos \left(\frac{1}{2} (a+b x)\right)\right)-2\right)+2\right)}{2 b \left(\csc ^2\left(\frac{1}{2} (a+b x)\right)-\sec ^2\left(\frac{1}{2} (a+b x)\right)\right)}","\frac{3 \sec (a+b x)}{2 b}-\frac{3 \tanh ^{-1}(\cos (a+b x))}{2 b}-\frac{\csc ^2(a+b x) \sec (a+b x)}{2 b}",1,"(Csc[a + b*x]^4*(2 - 6*Cos[2*(a + b*x)] + 2*Cos[3*(a + b*x)] + 3*Cos[3*(a + b*x)]*Log[Cos[(a + b*x)/2]] - 3*Cos[3*(a + b*x)]*Log[Sin[(a + b*x)/2]] + Cos[a + b*x]*(-2 - 3*Log[Cos[(a + b*x)/2]] + 3*Log[Sin[(a + b*x)/2]])))/(2*b*(Csc[(a + b*x)/2]^2 - Sec[(a + b*x)/2]^2))","B",1
154,1,61,43,0.0120216,"\int \csc ^3(a+b x) \sec ^3(a+b x) \, dx","Integrate[Csc[a + b*x]^3*Sec[a + b*x]^3,x]","8 \left(-\frac{\csc ^2(a+b x)}{16 b}+\frac{\sec ^2(a+b x)}{16 b}+\frac{\log (\sin (a+b x))}{4 b}-\frac{\log (\cos (a+b x))}{4 b}\right)","\frac{\tan ^2(a+b x)}{2 b}-\frac{\cot ^2(a+b x)}{2 b}+\frac{2 \log (\tan (a+b x))}{b}",1,"8*(-1/16*Csc[a + b*x]^2/b - Log[Cos[a + b*x]]/(4*b) + Log[Sin[a + b*x]]/(4*b) + Sec[a + b*x]^2/(16*b))","A",1
155,1,205,66,0.3896428,"\int \csc ^3(a+b x) \sec ^4(a+b x) \, dx","Integrate[Csc[a + b*x]^3*Sec[a + b*x]^4,x]","\frac{2 \csc ^8(a+b x) \left(-40 \cos (2 (a+b x))+13 \cos (3 (a+b x))-30 \cos (4 (a+b x))+13 \cos (5 (a+b x))+15 \cos (3 (a+b x)) \log \left(\cos \left(\frac{1}{2} (a+b x)\right)\right)+15 \cos (5 (a+b x)) \log \left(\cos \left(\frac{1}{2} (a+b x)\right)\right)-15 \cos (3 (a+b x)) \log \left(\sin \left(\frac{1}{2} (a+b x)\right)\right)-15 \cos (5 (a+b x)) \log \left(\sin \left(\frac{1}{2} (a+b x)\right)\right)+\cos (a+b x) \left(30 \log \left(\sin \left(\frac{1}{2} (a+b x)\right)\right)-30 \log \left(\cos \left(\frac{1}{2} (a+b x)\right)\right)-26\right)+22\right)}{3 b \left(\csc ^2\left(\frac{1}{2} (a+b x)\right)-\sec ^2\left(\frac{1}{2} (a+b x)\right)\right)^3}","\frac{5 \sec ^3(a+b x)}{6 b}+\frac{5 \sec (a+b x)}{2 b}-\frac{5 \tanh ^{-1}(\cos (a+b x))}{2 b}-\frac{\csc ^2(a+b x) \sec ^3(a+b x)}{2 b}",1,"(2*Csc[a + b*x]^8*(22 - 40*Cos[2*(a + b*x)] + 13*Cos[3*(a + b*x)] - 30*Cos[4*(a + b*x)] + 13*Cos[5*(a + b*x)] + 15*Cos[3*(a + b*x)]*Log[Cos[(a + b*x)/2]] + 15*Cos[5*(a + b*x)]*Log[Cos[(a + b*x)/2]] - 15*Cos[3*(a + b*x)]*Log[Sin[(a + b*x)/2]] - 15*Cos[5*(a + b*x)]*Log[Sin[(a + b*x)/2]] + Cos[a + b*x]*(-26 - 30*Log[Cos[(a + b*x)/2]] + 30*Log[Sin[(a + b*x)/2]])))/(3*b*(Csc[(a + b*x)/2]^2 - Sec[(a + b*x)/2]^2)^3)","B",1
156,1,56,58,0.205956,"\int \csc ^3(a+b x) \sec ^5(a+b x) \, dx","Integrate[Csc[a + b*x]^3*Sec[a + b*x]^5,x]","-\frac{2 \csc ^2(a+b x)-\sec ^4(a+b x)-4 \sec ^2(a+b x)-12 \log (\sin (a+b x))+12 \log (\cos (a+b x))}{4 b}","\frac{\tan ^4(a+b x)}{4 b}+\frac{3 \tan ^2(a+b x)}{2 b}-\frac{\cot ^2(a+b x)}{2 b}+\frac{3 \log (\tan (a+b x))}{b}",1,"-1/4*(2*Csc[a + b*x]^2 + 12*Log[Cos[a + b*x]] - 12*Log[Sin[a + b*x]] - 4*Sec[a + b*x]^2 - Sec[a + b*x]^4)/b","A",1
157,1,68,68,0.0295586,"\int \cos ^5(a+b x) \cot ^4(a+b x) \, dx","Integrate[Cos[a + b*x]^5*Cot[a + b*x]^4,x]","\frac{\sin ^5(a+b x)}{5 b}-\frac{4 \sin ^3(a+b x)}{3 b}+\frac{6 \sin (a+b x)}{b}-\frac{\csc ^3(a+b x)}{3 b}+\frac{4 \csc (a+b x)}{b}","\frac{\sin ^5(a+b x)}{5 b}-\frac{4 \sin ^3(a+b x)}{3 b}+\frac{6 \sin (a+b x)}{b}-\frac{\csc ^3(a+b x)}{3 b}+\frac{4 \csc (a+b x)}{b}",1,"(4*Csc[a + b*x])/b - Csc[a + b*x]^3/(3*b) + (6*Sin[a + b*x])/b - (4*Sin[a + b*x]^3)/(3*b) + Sin[a + b*x]^5/(5*b)","A",1
158,1,53,80,0.2643595,"\int \cos ^4(a+b x) \cot ^4(a+b x) \, dx","Integrate[Cos[a + b*x]^4*Cot[a + b*x]^4,x]","\frac{420 (a+b x)+72 \sin (2 (a+b x))+3 \sin (4 (a+b x))-32 \cot (a+b x) \left(\csc ^2(a+b x)-10\right)}{96 b}","-\frac{35 \cot ^3(a+b x)}{24 b}+\frac{35 \cot (a+b x)}{8 b}+\frac{\cos ^4(a+b x) \cot ^3(a+b x)}{4 b}+\frac{7 \cos ^2(a+b x) \cot ^3(a+b x)}{8 b}+\frac{35 x}{8}",1,"(420*(a + b*x) - 32*Cot[a + b*x]*(-10 + Csc[a + b*x]^2) + 72*Sin[2*(a + b*x)] + 3*Sin[4*(a + b*x)])/(96*b)","A",1
159,1,53,53,0.0234745,"\int \cos ^3(a+b x) \cot ^4(a+b x) \, dx","Integrate[Cos[a + b*x]^3*Cot[a + b*x]^4,x]","-\frac{\sin ^3(a+b x)}{3 b}+\frac{3 \sin (a+b x)}{b}-\frac{\csc ^3(a+b x)}{3 b}+\frac{3 \csc (a+b x)}{b}","-\frac{\sin ^3(a+b x)}{3 b}+\frac{3 \sin (a+b x)}{b}-\frac{\csc ^3(a+b x)}{3 b}+\frac{3 \csc (a+b x)}{b}",1,"(3*Csc[a + b*x])/b - Csc[a + b*x]^3/(3*b) + (3*Sin[a + b*x])/b - Sin[a + b*x]^3/(3*b)","A",1
160,1,43,57,0.1668496,"\int \cos ^2(a+b x) \cot ^4(a+b x) \, dx","Integrate[Cos[a + b*x]^2*Cot[a + b*x]^4,x]","\frac{30 (a+b x)+3 \sin (2 (a+b x))-4 \cot (a+b x) \left(\csc ^2(a+b x)-7\right)}{12 b}","-\frac{5 \cot ^3(a+b x)}{6 b}+\frac{5 \cot (a+b x)}{2 b}+\frac{\cos ^2(a+b x) \cot ^3(a+b x)}{2 b}+\frac{5 x}{2}",1,"(30*(a + b*x) - 4*Cot[a + b*x]*(-7 + Csc[a + b*x]^2) + 3*Sin[2*(a + b*x)])/(12*b)","A",1
161,1,37,37,0.0170677,"\int \cos (a+b x) \cot ^4(a+b x) \, dx","Integrate[Cos[a + b*x]*Cot[a + b*x]^4,x]","\frac{\sin (a+b x)}{b}-\frac{\csc ^3(a+b x)}{3 b}+\frac{2 \csc (a+b x)}{b}","\frac{\sin (a+b x)}{b}-\frac{\csc ^3(a+b x)}{3 b}+\frac{2 \csc (a+b x)}{b}",1,"(2*Csc[a + b*x])/b - Csc[a + b*x]^3/(3*b) + Sin[a + b*x]/b","A",1
162,1,33,27,0.0089844,"\int \cot ^4(a+b x) \, dx","Integrate[Cot[a + b*x]^4,x]","-\frac{\cot ^3(a+b x) \, _2F_1\left(-\frac{3}{2},1;-\frac{1}{2};-\tan ^2(a+b x)\right)}{3 b}","-\frac{\cot ^3(a+b x)}{3 b}+\frac{\cot (a+b x)}{b}+x",1,"-1/3*(Cot[a + b*x]^3*Hypergeometric2F1[-3/2, 1, -1/2, -Tan[a + b*x]^2])/b","C",1
163,1,26,26,0.01243,"\int \cot ^3(a+b x) \csc (a+b x) \, dx","Integrate[Cot[a + b*x]^3*Csc[a + b*x],x]","\frac{\csc (a+b x)}{b}-\frac{\csc ^3(a+b x)}{3 b}","\frac{\csc (a+b x)}{b}-\frac{\csc ^3(a+b x)}{3 b}",1,"Csc[a + b*x]/b - Csc[a + b*x]^3/(3*b)","A",1
164,1,15,15,0.0058073,"\int \cot ^2(a+b x) \csc ^2(a+b x) \, dx","Integrate[Cot[a + b*x]^2*Csc[a + b*x]^2,x]","-\frac{\cot ^3(a+b x)}{3 b}","-\frac{\cot ^3(a+b x)}{3 b}",1,"-1/3*Cot[a + b*x]^3/b","A",1
165,1,15,15,0.0089128,"\int \cot (a+b x) \csc ^3(a+b x) \, dx","Integrate[Cot[a + b*x]*Csc[a + b*x]^3,x]","-\frac{\csc ^3(a+b x)}{3 b}","-\frac{\csc ^3(a+b x)}{3 b}",1,"-1/3*Csc[a + b*x]^3/b","A",1
166,1,31,38,0.0129159,"\int \csc ^4(a+b x) \sec (a+b x) \, dx","Integrate[Csc[a + b*x]^4*Sec[a + b*x],x]","-\frac{\csc ^3(a+b x) \, _2F_1\left(-\frac{3}{2},1;-\frac{1}{2};\sin ^2(a+b x)\right)}{3 b}","-\frac{\csc ^3(a+b x)}{3 b}-\frac{\csc (a+b x)}{b}+\frac{\tanh ^{-1}(\sin (a+b x))}{b}",1,"-1/3*(Csc[a + b*x]^3*Hypergeometric2F1[-3/2, 1, -1/2, Sin[a + b*x]^2])/b","C",1
167,1,45,37,0.0307294,"\int \csc ^4(a+b x) \sec ^2(a+b x) \, dx","Integrate[Csc[a + b*x]^4*Sec[a + b*x]^2,x]","\frac{\tan (a+b x)}{b}-\frac{5 \cot (a+b x)}{3 b}-\frac{\cot (a+b x) \csc ^2(a+b x)}{3 b}","\frac{\tan (a+b x)}{b}-\frac{\cot ^3(a+b x)}{3 b}-\frac{2 \cot (a+b x)}{b}",1,"(-5*Cot[a + b*x])/(3*b) - (Cot[a + b*x]*Csc[a + b*x]^2)/(3*b) + Tan[a + b*x]/b","A",1
168,1,31,66,0.0140731,"\int \csc ^4(a+b x) \sec ^3(a+b x) \, dx","Integrate[Csc[a + b*x]^4*Sec[a + b*x]^3,x]","-\frac{\csc ^3(a+b x) \, _2F_1\left(-\frac{3}{2},2;-\frac{1}{2};\sin ^2(a+b x)\right)}{3 b}","-\frac{5 \csc ^3(a+b x)}{6 b}-\frac{5 \csc (a+b x)}{2 b}+\frac{5 \tanh ^{-1}(\sin (a+b x))}{2 b}+\frac{\csc ^3(a+b x) \sec ^2(a+b x)}{2 b}",1,"-1/3*(Csc[a + b*x]^3*Hypergeometric2F1[-3/2, 2, -1/2, Sin[a + b*x]^2])/b","C",1
169,1,43,53,0.010234,"\int \csc ^4(a+b x) \sec ^4(a+b x) \, dx","Integrate[Csc[a + b*x]^4*Sec[a + b*x]^4,x]","16 \left(-\frac{\cot (2 (a+b x))}{3 b}-\frac{\cot (2 (a+b x)) \csc ^2(2 (a+b x))}{6 b}\right)","\frac{\tan ^3(a+b x)}{3 b}+\frac{3 \tan (a+b x)}{b}-\frac{\cot ^3(a+b x)}{3 b}-\frac{3 \cot (a+b x)}{b}",1,"16*(-1/3*Cot[2*(a + b*x)]/b - (Cot[2*(a + b*x)]*Csc[2*(a + b*x)]^2)/(6*b))","A",1
170,1,31,89,0.013533,"\int \csc ^4(a+b x) \sec ^5(a+b x) \, dx","Integrate[Csc[a + b*x]^4*Sec[a + b*x]^5,x]","-\frac{\csc ^3(a+b x) \, _2F_1\left(-\frac{3}{2},3;-\frac{1}{2};\sin ^2(a+b x)\right)}{3 b}","-\frac{35 \csc ^3(a+b x)}{24 b}-\frac{35 \csc (a+b x)}{8 b}+\frac{35 \tanh ^{-1}(\sin (a+b x))}{8 b}+\frac{\csc ^3(a+b x) \sec ^4(a+b x)}{4 b}+\frac{7 \csc ^3(a+b x) \sec ^2(a+b x)}{8 b}",1,"-1/3*(Csc[a + b*x]^3*Hypergeometric2F1[-3/2, 3, -1/2, Sin[a + b*x]^2])/b","C",1
171,1,55,69,0.1057729,"\int \cos ^4(a+b x) \cot ^5(a+b x) \, dx","Integrate[Cos[a + b*x]^4*Cot[a + b*x]^5,x]","\frac{\sin ^4(a+b x)-8 \sin ^2(a+b x)-\csc ^4(a+b x)+8 \csc ^2(a+b x)+24 \log (\sin (a+b x))}{4 b}","\frac{\sin ^4(a+b x)}{4 b}-\frac{2 \sin ^2(a+b x)}{b}-\frac{\csc ^4(a+b x)}{4 b}+\frac{2 \csc ^2(a+b x)}{b}+\frac{6 \log (\sin (a+b x))}{b}",1,"(8*Csc[a + b*x]^2 - Csc[a + b*x]^4 + 24*Log[Sin[a + b*x]] - 8*Sin[a + b*x]^2 + Sin[a + b*x]^4)/(4*b)","A",1
172,1,141,89,0.0408856,"\int \cos ^3(a+b x) \cot ^5(a+b x) \, dx","Integrate[Cos[a + b*x]^3*Cot[a + b*x]^5,x]","\frac{13 \cos (a+b x)}{4 b}+\frac{\cos (3 (a+b x))}{12 b}-\frac{\csc ^4\left(\frac{1}{2} (a+b x)\right)}{64 b}+\frac{13 \csc ^2\left(\frac{1}{2} (a+b x)\right)}{32 b}+\frac{\sec ^4\left(\frac{1}{2} (a+b x)\right)}{64 b}-\frac{13 \sec ^2\left(\frac{1}{2} (a+b x)\right)}{32 b}+\frac{35 \log \left(\sin \left(\frac{1}{2} (a+b x)\right)\right)}{8 b}-\frac{35 \log \left(\cos \left(\frac{1}{2} (a+b x)\right)\right)}{8 b}","\frac{35 \cos ^3(a+b x)}{24 b}+\frac{35 \cos (a+b x)}{8 b}-\frac{\cos ^3(a+b x) \cot ^4(a+b x)}{4 b}+\frac{7 \cos ^3(a+b x) \cot ^2(a+b x)}{8 b}-\frac{35 \tanh ^{-1}(\cos (a+b x))}{8 b}",1,"(13*Cos[a + b*x])/(4*b) + Cos[3*(a + b*x)]/(12*b) + (13*Csc[(a + b*x)/2]^2)/(32*b) - Csc[(a + b*x)/2]^4/(64*b) - (35*Log[Cos[(a + b*x)/2]])/(8*b) + (35*Log[Sin[(a + b*x)/2]])/(8*b) - (13*Sec[(a + b*x)/2]^2)/(32*b) + Sec[(a + b*x)/2]^4/(64*b)","A",1
173,1,47,58,0.1504506,"\int \cos ^2(a+b x) \cot ^5(a+b x) \, dx","Integrate[Cos[a + b*x]^2*Cot[a + b*x]^5,x]","\frac{-2 \sin ^2(a+b x)-\csc ^4(a+b x)+6 \csc ^2(a+b x)+12 \log (\sin (a+b x))}{4 b}","-\frac{\sin ^2(a+b x)}{2 b}-\frac{\csc ^4(a+b x)}{4 b}+\frac{3 \csc ^2(a+b x)}{2 b}+\frac{3 \log (\sin (a+b x))}{b}",1,"(6*Csc[a + b*x]^2 - Csc[a + b*x]^4 + 12*Log[Sin[a + b*x]] - 2*Sin[a + b*x]^2)/(4*b)","A",1
174,1,123,70,0.028825,"\int \cos (a+b x) \cot ^5(a+b x) \, dx","Integrate[Cos[a + b*x]*Cot[a + b*x]^5,x]","\frac{\cos (a+b x)}{b}-\frac{\csc ^4\left(\frac{1}{2} (a+b x)\right)}{64 b}+\frac{9 \csc ^2\left(\frac{1}{2} (a+b x)\right)}{32 b}+\frac{\sec ^4\left(\frac{1}{2} (a+b x)\right)}{64 b}-\frac{9 \sec ^2\left(\frac{1}{2} (a+b x)\right)}{32 b}+\frac{15 \log \left(\sin \left(\frac{1}{2} (a+b x)\right)\right)}{8 b}-\frac{15 \log \left(\cos \left(\frac{1}{2} (a+b x)\right)\right)}{8 b}","\frac{15 \cos (a+b x)}{8 b}-\frac{\cos (a+b x) \cot ^4(a+b x)}{4 b}+\frac{5 \cos (a+b x) \cot ^2(a+b x)}{8 b}-\frac{15 \tanh ^{-1}(\cos (a+b x))}{8 b}",1,"Cos[a + b*x]/b + (9*Csc[(a + b*x)/2]^2)/(32*b) - Csc[(a + b*x)/2]^4/(64*b) - (15*Log[Cos[(a + b*x)/2]])/(8*b) + (15*Log[Sin[(a + b*x)/2]])/(8*b) - (9*Sec[(a + b*x)/2]^2)/(32*b) + Sec[(a + b*x)/2]^4/(64*b)","A",1
175,1,46,42,0.1000712,"\int \cot ^5(a+b x) \, dx","Integrate[Cot[a + b*x]^5,x]","\frac{-\cot ^4(a+b x)+2 \cot ^2(a+b x)+4 \log (\tan (a+b x))+4 \log (\cos (a+b x))}{4 b}","-\frac{\cot ^4(a+b x)}{4 b}+\frac{\cot ^2(a+b x)}{2 b}+\frac{\log (\sin (a+b x))}{b}",1,"(2*Cot[a + b*x]^2 - Cot[a + b*x]^4 + 4*Log[Cos[a + b*x]] + 4*Log[Tan[a + b*x]])/(4*b)","A",1
176,1,113,55,0.0285375,"\int \cot ^4(a+b x) \csc (a+b x) \, dx","Integrate[Cot[a + b*x]^4*Csc[a + b*x],x]","-\frac{\csc ^4\left(\frac{1}{2} (a+b x)\right)}{64 b}+\frac{5 \csc ^2\left(\frac{1}{2} (a+b x)\right)}{32 b}+\frac{\sec ^4\left(\frac{1}{2} (a+b x)\right)}{64 b}-\frac{5 \sec ^2\left(\frac{1}{2} (a+b x)\right)}{32 b}+\frac{3 \log \left(\sin \left(\frac{1}{2} (a+b x)\right)\right)}{8 b}-\frac{3 \log \left(\cos \left(\frac{1}{2} (a+b x)\right)\right)}{8 b}","-\frac{3 \tanh ^{-1}(\cos (a+b x))}{8 b}-\frac{\cot ^3(a+b x) \csc (a+b x)}{4 b}+\frac{3 \cot (a+b x) \csc (a+b x)}{8 b}",1,"(5*Csc[(a + b*x)/2]^2)/(32*b) - Csc[(a + b*x)/2]^4/(64*b) - (3*Log[Cos[(a + b*x)/2]])/(8*b) + (3*Log[Sin[(a + b*x)/2]])/(8*b) - (5*Sec[(a + b*x)/2]^2)/(32*b) + Sec[(a + b*x)/2]^4/(64*b)","B",1
177,1,15,15,0.0049722,"\int \cot ^3(a+b x) \csc ^2(a+b x) \, dx","Integrate[Cot[a + b*x]^3*Csc[a + b*x]^2,x]","-\frac{\cot ^4(a+b x)}{4 b}","-\frac{\cot ^4(a+b x)}{4 b}",1,"-1/4*Cot[a + b*x]^4/b","A",1
178,1,113,55,0.032987,"\int \cot ^2(a+b x) \csc ^3(a+b x) \, dx","Integrate[Cot[a + b*x]^2*Csc[a + b*x]^3,x]","-\frac{\csc ^4\left(\frac{1}{2} (a+b x)\right)}{64 b}+\frac{\csc ^2\left(\frac{1}{2} (a+b x)\right)}{32 b}+\frac{\sec ^4\left(\frac{1}{2} (a+b x)\right)}{64 b}-\frac{\sec ^2\left(\frac{1}{2} (a+b x)\right)}{32 b}-\frac{\log \left(\sin \left(\frac{1}{2} (a+b x)\right)\right)}{8 b}+\frac{\log \left(\cos \left(\frac{1}{2} (a+b x)\right)\right)}{8 b}","\frac{\tanh ^{-1}(\cos (a+b x))}{8 b}-\frac{\cot (a+b x) \csc ^3(a+b x)}{4 b}+\frac{\cot (a+b x) \csc (a+b x)}{8 b}",1,"Csc[(a + b*x)/2]^2/(32*b) - Csc[(a + b*x)/2]^4/(64*b) + Log[Cos[(a + b*x)/2]]/(8*b) - Log[Sin[(a + b*x)/2]]/(8*b) - Sec[(a + b*x)/2]^2/(32*b) + Sec[(a + b*x)/2]^4/(64*b)","B",1
179,1,15,15,0.0085492,"\int \cot (a+b x) \csc ^4(a+b x) \, dx","Integrate[Cot[a + b*x]*Csc[a + b*x]^4,x]","-\frac{\csc ^4(a+b x)}{4 b}","-\frac{\csc ^4(a+b x)}{4 b}",1,"-1/4*Csc[a + b*x]^4/b","A",1
180,1,44,40,0.1085014,"\int \csc ^5(a+b x) \sec (a+b x) \, dx","Integrate[Csc[a + b*x]^5*Sec[a + b*x],x]","-\frac{\csc ^4(a+b x)+2 \csc ^2(a+b x)-4 \log (\sin (a+b x))+4 \log (\cos (a+b x))}{4 b}","-\frac{\cot ^4(a+b x)}{4 b}-\frac{\cot ^2(a+b x)}{b}+\frac{\log (\tan (a+b x))}{b}",1,"-1/4*(2*Csc[a + b*x]^2 + Csc[a + b*x]^4 + 4*Log[Cos[a + b*x]] - 4*Log[Sin[a + b*x]])/b","A",1
181,1,129,70,3.9875078,"\int \csc ^5(a+b x) \sec ^2(a+b x) \, dx","Integrate[Csc[a + b*x]^5*Sec[a + b*x]^2,x]","-\frac{\csc ^4\left(\frac{1}{2} (a+b x)\right)+14 \csc ^2\left(\frac{1}{2} (a+b x)\right)+\frac{\sec ^2\left(\frac{1}{2} (a+b x)\right) \left(-14 \tan ^2\left(\frac{1}{2} (a+b x)\right)+\cos (a+b x) \left(\sec ^4\left(\frac{1}{2} (a+b x)\right)-8 \left(-15 \log \left(\sin \left(\frac{1}{2} (a+b x)\right)\right)+15 \log \left(\cos \left(\frac{1}{2} (a+b x)\right)\right)+8\right)\right)+78\right)}{\tan ^2\left(\frac{1}{2} (a+b x)\right)-1}}{64 b}","\frac{15 \sec (a+b x)}{8 b}-\frac{15 \tanh ^{-1}(\cos (a+b x))}{8 b}-\frac{\csc ^4(a+b x) \sec (a+b x)}{4 b}-\frac{5 \csc ^2(a+b x) \sec (a+b x)}{8 b}",1,"-1/64*(14*Csc[(a + b*x)/2]^2 + Csc[(a + b*x)/2]^4 + (Sec[(a + b*x)/2]^2*(78 + Cos[a + b*x]*(-8*(8 + 15*Log[Cos[(a + b*x)/2]] - 15*Log[Sin[(a + b*x)/2]]) + Sec[(a + b*x)/2]^4) - 14*Tan[(a + b*x)/2]^2))/(-1 + Tan[(a + b*x)/2]^2))/b","A",1
182,1,54,58,0.301915,"\int \csc ^5(a+b x) \sec ^3(a+b x) \, dx","Integrate[Csc[a + b*x]^5*Sec[a + b*x]^3,x]","-\frac{\csc ^4(a+b x)+4 \csc ^2(a+b x)-2 \sec ^2(a+b x)-12 \log (\sin (a+b x))+12 \log (\cos (a+b x))}{4 b}","\frac{\tan ^2(a+b x)}{2 b}-\frac{\cot ^4(a+b x)}{4 b}-\frac{3 \cot ^2(a+b x)}{2 b}+\frac{3 \log (\tan (a+b x))}{b}",1,"-1/4*(4*Csc[a + b*x]^2 + Csc[a + b*x]^4 + 12*Log[Cos[a + b*x]] - 12*Log[Sin[a + b*x]] - 2*Sec[a + b*x]^2)/b","A",1
183,1,268,89,0.4365166,"\int \csc ^5(a+b x) \sec ^4(a+b x) \, dx","Integrate[Csc[a + b*x]^5*Sec[a + b*x]^4,x]","-\frac{\csc ^{10}(a+b x) \left(658 \cos (2 (a+b x))-228 \cos (3 (a+b x))+140 \cos (4 (a+b x))-76 \cos (5 (a+b x))-210 \cos (6 (a+b x))+76 \cos (7 (a+b x))-315 \cos (3 (a+b x)) \log \left(\cos \left(\frac{1}{2} (a+b x)\right)\right)-105 \cos (5 (a+b x)) \log \left(\cos \left(\frac{1}{2} (a+b x)\right)\right)+105 \cos (7 (a+b x)) \log \left(\cos \left(\frac{1}{2} (a+b x)\right)\right)+3 \cos (a+b x) \left(-105 \log \left(\sin \left(\frac{1}{2} (a+b x)\right)\right)+105 \log \left(\cos \left(\frac{1}{2} (a+b x)\right)\right)+76\right)+315 \cos (3 (a+b x)) \log \left(\sin \left(\frac{1}{2} (a+b x)\right)\right)+105 \cos (5 (a+b x)) \log \left(\sin \left(\frac{1}{2} (a+b x)\right)\right)-105 \cos (7 (a+b x)) \log \left(\sin \left(\frac{1}{2} (a+b x)\right)\right)-204\right)}{24 b \left(\csc ^2\left(\frac{1}{2} (a+b x)\right)-\sec ^2\left(\frac{1}{2} (a+b x)\right)\right)^3}","\frac{35 \sec ^3(a+b x)}{24 b}+\frac{35 \sec (a+b x)}{8 b}-\frac{35 \tanh ^{-1}(\cos (a+b x))}{8 b}-\frac{\csc ^4(a+b x) \sec ^3(a+b x)}{4 b}-\frac{7 \csc ^2(a+b x) \sec ^3(a+b x)}{8 b}",1,"-1/24*(Csc[a + b*x]^10*(-204 + 658*Cos[2*(a + b*x)] - 228*Cos[3*(a + b*x)] + 140*Cos[4*(a + b*x)] - 76*Cos[5*(a + b*x)] - 210*Cos[6*(a + b*x)] + 76*Cos[7*(a + b*x)] - 315*Cos[3*(a + b*x)]*Log[Cos[(a + b*x)/2]] - 105*Cos[5*(a + b*x)]*Log[Cos[(a + b*x)/2]] + 105*Cos[7*(a + b*x)]*Log[Cos[(a + b*x)/2]] + 3*Cos[a + b*x]*(76 + 105*Log[Cos[(a + b*x)/2]] - 105*Log[Sin[(a + b*x)/2]]) + 315*Cos[3*(a + b*x)]*Log[Sin[(a + b*x)/2]] + 105*Cos[5*(a + b*x)]*Log[Sin[(a + b*x)/2]] - 105*Cos[7*(a + b*x)]*Log[Sin[(a + b*x)/2]]))/(b*(Csc[(a + b*x)/2]^2 - Sec[(a + b*x)/2]^2)^3)","B",1
184,1,91,69,0.0140923,"\int \csc ^5(a+b x) \sec ^5(a+b x) \, dx","Integrate[Csc[a + b*x]^5*Sec[a + b*x]^5,x]","32 \left(-\frac{\csc ^4(a+b x)}{128 b}-\frac{3 \csc ^2(a+b x)}{64 b}+\frac{\sec ^4(a+b x)}{128 b}+\frac{3 \sec ^2(a+b x)}{64 b}+\frac{3 \log (\sin (a+b x))}{16 b}-\frac{3 \log (\cos (a+b x))}{16 b}\right)","\frac{\tan ^4(a+b x)}{4 b}+\frac{2 \tan ^2(a+b x)}{b}-\frac{\cot ^4(a+b x)}{4 b}-\frac{2 \cot ^2(a+b x)}{b}+\frac{6 \log (\tan (a+b x))}{b}",1,"32*((-3*Csc[a + b*x]^2)/(64*b) - Csc[a + b*x]^4/(128*b) - (3*Log[Cos[a + b*x]])/(16*b) + (3*Log[Sin[a + b*x]])/(16*b) + (3*Sec[a + b*x]^2)/(64*b) + Sec[a + b*x]^4/(128*b))","A",1
185,1,27,17,0.0207352,"\int \cot ^2(x) \csc ^4(x) \, dx","Integrate[Cot[x]^2*Csc[x]^4,x]","\frac{2 \cot (x)}{15}-\frac{1}{5} \cot (x) \csc ^4(x)+\frac{1}{15} \cot (x) \csc ^2(x)","-\frac{1}{5} \cot ^5(x)-\frac{\cot ^3(x)}{3}",1,"(2*Cot[x])/15 + (Cot[x]*Csc[x]^2)/15 - (Cot[x]*Csc[x]^4)/5","A",1
186,1,17,17,0.0079748,"\int \cot ^3(x) \csc ^4(x) \, dx","Integrate[Cot[x]^3*Csc[x]^4,x]","\frac{\csc ^4(x)}{4}-\frac{\csc ^6(x)}{6}","\frac{\csc ^4(x)}{4}-\frac{\csc ^6(x)}{6}",1,"Csc[x]^4/4 - Csc[x]^6/6","A",1
187,1,22,22,0.0351949,"\int (d \cos (a+b x))^{3/2} \sin (a+b x) \, dx","Integrate[(d*Cos[a + b*x])^(3/2)*Sin[a + b*x],x]","-\frac{2 (d \cos (a+b x))^{5/2}}{5 b d}","-\frac{2 (d \cos (a+b x))^{5/2}}{5 b d}",1,"(-2*(d*Cos[a + b*x])^(5/2))/(5*b*d)","A",1
188,1,22,22,0.0176033,"\int \sqrt{d \cos (a+b x)} \sin (a+b x) \, dx","Integrate[Sqrt[d*Cos[a + b*x]]*Sin[a + b*x],x]","-\frac{2 (d \cos (a+b x))^{3/2}}{3 b d}","-\frac{2 (d \cos (a+b x))^{3/2}}{3 b d}",1,"(-2*(d*Cos[a + b*x])^(3/2))/(3*b*d)","A",1
189,1,20,20,0.0128388,"\int \frac{\sin (a+b x)}{\sqrt{d \cos (a+b x)}} \, dx","Integrate[Sin[a + b*x]/Sqrt[d*Cos[a + b*x]],x]","-\frac{2 \sqrt{d \cos (a+b x)}}{b d}","-\frac{2 \sqrt{d \cos (a+b x)}}{b d}",1,"(-2*Sqrt[d*Cos[a + b*x]])/(b*d)","A",1
190,1,20,20,0.0221342,"\int \frac{\sin (a+b x)}{(d \cos (a+b x))^{3/2}} \, dx","Integrate[Sin[a + b*x]/(d*Cos[a + b*x])^(3/2),x]","\frac{2}{b d \sqrt{d \cos (a+b x)}}","\frac{2}{b d \sqrt{d \cos (a+b x)}}",1,"2/(b*d*Sqrt[d*Cos[a + b*x]])","A",1
191,1,22,22,0.0251283,"\int \frac{\sin (a+b x)}{(d \cos (a+b x))^{5/2}} \, dx","Integrate[Sin[a + b*x]/(d*Cos[a + b*x])^(5/2),x]","\frac{2}{3 b d (d \cos (a+b x))^{3/2}}","\frac{2}{3 b d (d \cos (a+b x))^{3/2}}",1,"2/(3*b*d*(d*Cos[a + b*x])^(3/2))","A",1
192,1,22,22,0.0394991,"\int \frac{\sin (a+b x)}{(d \cos (a+b x))^{7/2}} \, dx","Integrate[Sin[a + b*x]/(d*Cos[a + b*x])^(7/2),x]","\frac{2}{5 b d (d \cos (a+b x))^{5/2}}","\frac{2}{5 b d (d \cos (a+b x))^{5/2}}",1,"2/(5*b*d*(d*Cos[a + b*x])^(5/2))","A",1
193,1,22,22,0.058778,"\int \frac{\sin (a+b x)}{(d \cos (a+b x))^{9/2}} \, dx","Integrate[Sin[a + b*x]/(d*Cos[a + b*x])^(9/2),x]","\frac{2}{7 b d (d \cos (a+b x))^{7/2}}","\frac{2}{7 b d (d \cos (a+b x))^{7/2}}",1,"2/(7*b*d*(d*Cos[a + b*x])^(7/2))","A",1
194,1,60,126,0.1304414,"\int (d \cos (a+b x))^{9/2} \sin ^2(a+b x) \, dx","Integrate[(d*Cos[a + b*x])^(9/2)*Sin[a + b*x]^2,x]","\frac{d^2 \sqrt[4]{\cos ^2(a+b x)} \tan ^3(a+b x) (d \cos (a+b x))^{5/2} \, _2F_1\left(-\frac{7}{4},\frac{3}{2};\frac{5}{2};\sin ^2(a+b x)\right)}{3 b}","\frac{28 d^4 E\left(\left.\frac{1}{2} (a+b x)\right|2\right) \sqrt{d \cos (a+b x)}}{195 b \sqrt{\cos (a+b x)}}+\frac{28 d^3 \sin (a+b x) (d \cos (a+b x))^{3/2}}{585 b}-\frac{2 \sin (a+b x) (d \cos (a+b x))^{11/2}}{13 b d}+\frac{4 d \sin (a+b x) (d \cos (a+b x))^{7/2}}{117 b}",1,"(d^2*(d*Cos[a + b*x])^(5/2)*(Cos[a + b*x]^2)^(1/4)*Hypergeometric2F1[-7/4, 3/2, 5/2, Sin[a + b*x]^2]*Tan[a + b*x]^3)/(3*b)","C",1
195,1,60,126,0.1282388,"\int (d \cos (a+b x))^{7/2} \sin ^2(a+b x) \, dx","Integrate[(d*Cos[a + b*x])^(7/2)*Sin[a + b*x]^2,x]","\frac{d^2 \cos ^2(a+b x)^{3/4} \tan ^3(a+b x) (d \cos (a+b x))^{3/2} \, _2F_1\left(-\frac{5}{4},\frac{3}{2};\frac{5}{2};\sin ^2(a+b x)\right)}{3 b}","\frac{20 d^4 \sqrt{\cos (a+b x)} F\left(\left.\frac{1}{2} (a+b x)\right|2\right)}{231 b \sqrt{d \cos (a+b x)}}+\frac{20 d^3 \sin (a+b x) \sqrt{d \cos (a+b x)}}{231 b}-\frac{2 \sin (a+b x) (d \cos (a+b x))^{9/2}}{11 b d}+\frac{4 d \sin (a+b x) (d \cos (a+b x))^{5/2}}{77 b}",1,"(d^2*(d*Cos[a + b*x])^(3/2)*(Cos[a + b*x]^2)^(3/4)*Hypergeometric2F1[-5/4, 3/2, 5/2, Sin[a + b*x]^2]*Tan[a + b*x]^3)/(3*b)","C",1
196,1,57,98,0.0485192,"\int (d \cos (a+b x))^{5/2} \sin ^2(a+b x) \, dx","Integrate[(d*Cos[a + b*x])^(5/2)*Sin[a + b*x]^2,x]","\frac{\sqrt[4]{\cos ^2(a+b x)} \tan ^3(a+b x) (d \cos (a+b x))^{5/2} \, _2F_1\left(-\frac{3}{4},\frac{3}{2};\frac{5}{2};\sin ^2(a+b x)\right)}{3 b}","\frac{4 d^2 E\left(\left.\frac{1}{2} (a+b x)\right|2\right) \sqrt{d \cos (a+b x)}}{15 b \sqrt{\cos (a+b x)}}-\frac{2 \sin (a+b x) (d \cos (a+b x))^{7/2}}{9 b d}+\frac{4 d \sin (a+b x) (d \cos (a+b x))^{3/2}}{45 b}",1,"((d*Cos[a + b*x])^(5/2)*(Cos[a + b*x]^2)^(1/4)*Hypergeometric2F1[-3/4, 3/2, 5/2, Sin[a + b*x]^2]*Tan[a + b*x]^3)/(3*b)","C",1
197,1,57,98,0.0640599,"\int (d \cos (a+b x))^{3/2} \sin ^2(a+b x) \, dx","Integrate[(d*Cos[a + b*x])^(3/2)*Sin[a + b*x]^2,x]","\frac{\cos ^2(a+b x)^{3/4} \tan ^3(a+b x) (d \cos (a+b x))^{3/2} \, _2F_1\left(-\frac{1}{4},\frac{3}{2};\frac{5}{2};\sin ^2(a+b x)\right)}{3 b}","\frac{4 d^2 \sqrt{\cos (a+b x)} F\left(\left.\frac{1}{2} (a+b x)\right|2\right)}{21 b \sqrt{d \cos (a+b x)}}-\frac{2 \sin (a+b x) (d \cos (a+b x))^{5/2}}{7 b d}+\frac{4 d \sin (a+b x) \sqrt{d \cos (a+b x)}}{21 b}",1,"((d*Cos[a + b*x])^(3/2)*(Cos[a + b*x]^2)^(3/4)*Hypergeometric2F1[-1/4, 3/2, 5/2, Sin[a + b*x]^2]*Tan[a + b*x]^3)/(3*b)","C",1
198,1,58,69,0.0856912,"\int \sqrt{d \cos (a+b x)} \sin ^2(a+b x) \, dx","Integrate[Sqrt[d*Cos[a + b*x]]*Sin[a + b*x]^2,x]","\frac{d \sin ^3(a+b x) \sqrt[4]{\cos ^2(a+b x)} \, _2F_1\left(\frac{1}{4},\frac{3}{2};\frac{5}{2};\sin ^2(a+b x)\right)}{3 b \sqrt{d \cos (a+b x)}}","\frac{4 E\left(\left.\frac{1}{2} (a+b x)\right|2\right) \sqrt{d \cos (a+b x)}}{5 b \sqrt{\cos (a+b x)}}-\frac{2 \sin (a+b x) (d \cos (a+b x))^{3/2}}{5 b d}",1,"(d*(Cos[a + b*x]^2)^(1/4)*Hypergeometric2F1[1/4, 3/2, 5/2, Sin[a + b*x]^2]*Sin[a + b*x]^3)/(3*b*Sqrt[d*Cos[a + b*x]])","C",1
199,1,58,69,0.1000599,"\int \frac{\sin ^2(a+b x)}{\sqrt{d \cos (a+b x)}} \, dx","Integrate[Sin[a + b*x]^2/Sqrt[d*Cos[a + b*x]],x]","\frac{d \sin ^3(a+b x) \cos ^2(a+b x)^{3/4} \, _2F_1\left(\frac{3}{4},\frac{3}{2};\frac{5}{2};\sin ^2(a+b x)\right)}{3 b (d \cos (a+b x))^{3/2}}","\frac{4 \sqrt{\cos (a+b x)} F\left(\left.\frac{1}{2} (a+b x)\right|2\right)}{3 b \sqrt{d \cos (a+b x)}}-\frac{2 \sin (a+b x) \sqrt{d \cos (a+b x)}}{3 b d}",1,"(d*(Cos[a + b*x]^2)^(3/4)*Hypergeometric2F1[3/4, 3/2, 5/2, Sin[a + b*x]^2]*Sin[a + b*x]^3)/(3*b*(d*Cos[a + b*x])^(3/2))","C",1
200,1,60,68,0.0840807,"\int \frac{\sin ^2(a+b x)}{(d \cos (a+b x))^{3/2}} \, dx","Integrate[Sin[a + b*x]^2/(d*Cos[a + b*x])^(3/2),x]","\frac{\sin ^3(a+b x) \sqrt[4]{\cos ^2(a+b x)} \, _2F_1\left(\frac{5}{4},\frac{3}{2};\frac{5}{2};\sin ^2(a+b x)\right)}{3 b d \sqrt{d \cos (a+b x)}}","\frac{2 \sin (a+b x)}{b d \sqrt{d \cos (a+b x)}}-\frac{4 E\left(\left.\frac{1}{2} (a+b x)\right|2\right) \sqrt{d \cos (a+b x)}}{b d^2 \sqrt{\cos (a+b x)}}",1,"((Cos[a + b*x]^2)^(1/4)*Hypergeometric2F1[5/4, 3/2, 5/2, Sin[a + b*x]^2]*Sin[a + b*x]^3)/(3*b*d*Sqrt[d*Cos[a + b*x]])","C",1
201,1,60,72,0.0788609,"\int \frac{\sin ^2(a+b x)}{(d \cos (a+b x))^{5/2}} \, dx","Integrate[Sin[a + b*x]^2/(d*Cos[a + b*x])^(5/2),x]","\frac{\sin ^3(a+b x) \cos ^2(a+b x)^{3/4} \, _2F_1\left(\frac{3}{2},\frac{7}{4};\frac{5}{2};\sin ^2(a+b x)\right)}{3 b d (d \cos (a+b x))^{3/2}}","\frac{2 \sin (a+b x)}{3 b d (d \cos (a+b x))^{3/2}}-\frac{4 \sqrt{\cos (a+b x)} F\left(\left.\frac{1}{2} (a+b x)\right|2\right)}{3 b d^2 \sqrt{d \cos (a+b x)}}",1,"((Cos[a + b*x]^2)^(3/4)*Hypergeometric2F1[3/2, 7/4, 5/2, Sin[a + b*x]^2]*Sin[a + b*x]^3)/(3*b*d*(d*Cos[a + b*x])^(3/2))","C",1
202,1,59,100,0.0636159,"\int \frac{\sin ^2(a+b x)}{(d \cos (a+b x))^{7/2}} \, dx","Integrate[Sin[a + b*x]^2/(d*Cos[a + b*x])^(7/2),x]","\frac{\sin ^3(2 (a+b x)) \sqrt[4]{\cos ^2(a+b x)} \, _2F_1\left(\frac{3}{2},\frac{9}{4};\frac{5}{2};\sin ^2(a+b x)\right)}{24 b (d \cos (a+b x))^{7/2}}","\frac{4 E\left(\left.\frac{1}{2} (a+b x)\right|2\right) \sqrt{d \cos (a+b x)}}{5 b d^4 \sqrt{\cos (a+b x)}}-\frac{4 \sin (a+b x)}{5 b d^3 \sqrt{d \cos (a+b x)}}+\frac{2 \sin (a+b x)}{5 b d (d \cos (a+b x))^{5/2}}",1,"((Cos[a + b*x]^2)^(1/4)*Hypergeometric2F1[3/2, 9/4, 5/2, Sin[a + b*x]^2]*Sin[2*(a + b*x)]^3)/(24*b*(d*Cos[a + b*x])^(7/2))","C",1
203,1,59,100,0.0621725,"\int \frac{\sin ^2(a+b x)}{(d \cos (a+b x))^{9/2}} \, dx","Integrate[Sin[a + b*x]^2/(d*Cos[a + b*x])^(9/2),x]","\frac{\sin ^3(2 (a+b x)) \cos ^2(a+b x)^{3/4} \, _2F_1\left(\frac{3}{2},\frac{11}{4};\frac{5}{2};\sin ^2(a+b x)\right)}{24 b (d \cos (a+b x))^{9/2}}","-\frac{4 \sqrt{\cos (a+b x)} F\left(\left.\frac{1}{2} (a+b x)\right|2\right)}{21 b d^4 \sqrt{d \cos (a+b x)}}-\frac{4 \sin (a+b x)}{21 b d^3 (d \cos (a+b x))^{3/2}}+\frac{2 \sin (a+b x)}{7 b d (d \cos (a+b x))^{7/2}}",1,"((Cos[a + b*x]^2)^(3/4)*Hypergeometric2F1[3/2, 11/4, 5/2, Sin[a + b*x]^2]*Sin[2*(a + b*x)]^3)/(24*b*(d*Cos[a + b*x])^(9/2))","C",1
204,1,57,45,0.3054645,"\int \sqrt{d \cos (a+b x)} \sin ^3(a+b x) \, dx","Integrate[Sqrt[d*Cos[a + b*x]]*Sin[a + b*x]^3,x]","-\frac{d \left(3 \sin ^2(2 (a+b x))+16 \cos ^2(a+b x)-16 \sqrt[4]{\cos ^2(a+b x)}\right)}{42 b \sqrt{d \cos (a+b x)}}","\frac{2 (d \cos (a+b x))^{7/2}}{7 b d^3}-\frac{2 (d \cos (a+b x))^{3/2}}{3 b d}",1,"-1/42*(d*(16*Cos[a + b*x]^2 - 16*(Cos[a + b*x]^2)^(1/4) + 3*Sin[2*(a + b*x)]^2))/(b*Sqrt[d*Cos[a + b*x]])","A",1
205,1,57,43,0.1751448,"\int \frac{\sin ^3(a+b x)}{\sqrt{d \cos (a+b x)}} \, dx","Integrate[Sin[a + b*x]^3/Sqrt[d*Cos[a + b*x]],x]","\frac{\cos (a+b x) (\cos (2 (a+b x))-9)+8 \cos ^2(a+b x)^{3/4} \sec (a+b x)}{5 b \sqrt{d \cos (a+b x)}}","\frac{2 (d \cos (a+b x))^{5/2}}{5 b d^3}-\frac{2 \sqrt{d \cos (a+b x)}}{b d}",1,"(Cos[a + b*x]*(-9 + Cos[2*(a + b*x)]) + 8*(Cos[a + b*x]^2)^(3/4)*Sec[a + b*x])/(5*b*Sqrt[d*Cos[a + b*x]])","A",1
206,1,46,43,0.0736177,"\int \frac{\sin ^3(a+b x)}{(d \cos (a+b x))^{3/2}} \, dx","Integrate[Sin[a + b*x]^3/(d*Cos[a + b*x])^(3/2),x]","-\frac{2 \left(\sin ^2(a+b x)+4 \sqrt[4]{\cos ^2(a+b x)}-4\right)}{3 b d \sqrt{d \cos (a+b x)}}","\frac{2 (d \cos (a+b x))^{3/2}}{3 b d^3}+\frac{2}{b d \sqrt{d \cos (a+b x)}}",1,"(-2*(-4 + 4*(Cos[a + b*x]^2)^(1/4) + Sin[a + b*x]^2))/(3*b*d*Sqrt[d*Cos[a + b*x]])","A",1
207,1,48,43,0.0947669,"\int \frac{\sin ^3(a+b x)}{(d \cos (a+b x))^{5/2}} \, dx","Integrate[Sin[a + b*x]^3/(d*Cos[a + b*x])^(5/2),x]","-\frac{2 \left(3 \sin ^2(a+b x)+4 \cos ^2(a+b x)^{3/4}-4\right)}{3 b d (d \cos (a+b x))^{3/2}}","\frac{2 \sqrt{d \cos (a+b x)}}{b d^3}+\frac{2}{3 b d (d \cos (a+b x))^{3/2}}",1,"(-2*(-4 + 4*(Cos[a + b*x]^2)^(3/4) + 3*Sin[a + b*x]^2))/(3*b*d*(d*Cos[a + b*x])^(3/2))","A",1
208,1,70,43,0.2482731,"\int \frac{\sin ^3(a+b x)}{(d \cos (a+b x))^{7/2}} \, dx","Integrate[Sin[a + b*x]^3/(d*Cos[a + b*x])^(7/2),x]","\frac{2 \tan ^2(a+b x) \left(-4 \sqrt[4]{\cos ^2(a+b x)}+4 \left(\sqrt[4]{\cos ^2(a+b x)}-1\right) \csc ^2(a+b x)+5\right)}{5 b d^3 \sqrt{d \cos (a+b x)}}","\frac{2}{5 b d (d \cos (a+b x))^{5/2}}-\frac{2}{b d^3 \sqrt{d \cos (a+b x)}}",1,"(2*(5 - 4*(Cos[a + b*x]^2)^(1/4) + 4*(-1 + (Cos[a + b*x]^2)^(1/4))*Csc[a + b*x]^2)*Tan[a + b*x]^2)/(5*b*d^3*Sqrt[d*Cos[a + b*x]])","A",1
209,1,70,45,0.2709079,"\int \frac{\sin ^3(a+b x)}{(d \cos (a+b x))^{9/2}} \, dx","Integrate[Sin[a + b*x]^3/(d*Cos[a + b*x])^(9/2),x]","\frac{2 \tan ^2(a+b x) \left(-4 \cos ^2(a+b x)^{3/4}+4 \left(\cos ^2(a+b x)^{3/4}-1\right) \csc ^2(a+b x)+7\right)}{21 b d^3 (d \cos (a+b x))^{3/2}}","\frac{2}{7 b d (d \cos (a+b x))^{7/2}}-\frac{2}{3 b d^3 (d \cos (a+b x))^{3/2}}",1,"(2*(7 - 4*(Cos[a + b*x]^2)^(3/4) + 4*(-1 + (Cos[a + b*x]^2)^(3/4))*Csc[a + b*x]^2)*Tan[a + b*x]^2)/(21*b*d^3*(d*Cos[a + b*x])^(3/2))","A",1
210,1,94,45,0.527865,"\int \frac{\sin ^3(a+b x)}{(d \cos (a+b x))^{11/2}} \, dx","Integrate[Sin[a + b*x]^3/(d*Cos[a + b*x])^(11/2),x]","\frac{2 \tan ^4(a+b x) \left(4 \sqrt[4]{\cos ^2(a+b x)}+4 \left(\sqrt[4]{\cos ^2(a+b x)}-1\right) \csc ^4(a+b x)+\left(9-8 \sqrt[4]{\cos ^2(a+b x)}\right) \csc ^2(a+b x)\right)}{45 b d^5 \sqrt{d \cos (a+b x)}}","\frac{2}{9 b d (d \cos (a+b x))^{9/2}}-\frac{2}{5 b d^3 (d \cos (a+b x))^{5/2}}",1,"(2*(4*(Cos[a + b*x]^2)^(1/4) + (9 - 8*(Cos[a + b*x]^2)^(1/4))*Csc[a + b*x]^2 + 4*(-1 + (Cos[a + b*x]^2)^(1/4))*Csc[a + b*x]^4)*Tan[a + b*x]^4)/(45*b*d^5*Sqrt[d*Cos[a + b*x]])","B",1
211,1,57,156,0.1258765,"\int (d \cos (a+b x))^{9/2} \sin ^4(a+b x) \, dx","Integrate[(d*Cos[a + b*x])^(9/2)*Sin[a + b*x]^4,x]","\frac{\sqrt[4]{\cos ^2(a+b x)} \tan ^5(a+b x) (d \cos (a+b x))^{9/2} \, _2F_1\left(-\frac{7}{4},\frac{5}{2};\frac{7}{2};\sin ^2(a+b x)\right)}{5 b}","\frac{56 d^4 E\left(\left.\frac{1}{2} (a+b x)\right|2\right) \sqrt{d \cos (a+b x)}}{1105 b \sqrt{\cos (a+b x)}}+\frac{56 d^3 \sin (a+b x) (d \cos (a+b x))^{3/2}}{3315 b}-\frac{2 \sin ^3(a+b x) (d \cos (a+b x))^{11/2}}{17 b d}-\frac{12 \sin (a+b x) (d \cos (a+b x))^{11/2}}{221 b d}+\frac{8 d \sin (a+b x) (d \cos (a+b x))^{7/2}}{663 b}",1,"((d*Cos[a + b*x])^(9/2)*(Cos[a + b*x]^2)^(1/4)*Hypergeometric2F1[-7/4, 5/2, 7/2, Sin[a + b*x]^2]*Tan[a + b*x]^5)/(5*b)","C",1
212,1,57,156,0.090143,"\int (d \cos (a+b x))^{7/2} \sin ^4(a+b x) \, dx","Integrate[(d*Cos[a + b*x])^(7/2)*Sin[a + b*x]^4,x]","\frac{\cos ^2(a+b x)^{3/4} \tan ^5(a+b x) (d \cos (a+b x))^{7/2} \, _2F_1\left(-\frac{5}{4},\frac{5}{2};\frac{7}{2};\sin ^2(a+b x)\right)}{5 b}","\frac{8 d^4 \sqrt{\cos (a+b x)} F\left(\left.\frac{1}{2} (a+b x)\right|2\right)}{231 b \sqrt{d \cos (a+b x)}}+\frac{8 d^3 \sin (a+b x) \sqrt{d \cos (a+b x)}}{231 b}-\frac{2 \sin ^3(a+b x) (d \cos (a+b x))^{9/2}}{15 b d}-\frac{4 \sin (a+b x) (d \cos (a+b x))^{9/2}}{55 b d}+\frac{8 d \sin (a+b x) (d \cos (a+b x))^{5/2}}{385 b}",1,"((d*Cos[a + b*x])^(7/2)*(Cos[a + b*x]^2)^(3/4)*Hypergeometric2F1[-5/4, 5/2, 7/2, Sin[a + b*x]^2]*Tan[a + b*x]^5)/(5*b)","C",1
213,1,65,128,0.0750904,"\int (d \cos (a+b x))^{5/2} \sin ^4(a+b x) \, dx","Integrate[(d*Cos[a + b*x])^(5/2)*Sin[a + b*x]^4,x]","\frac{\sin ^2(a+b x) \sqrt[4]{\cos ^2(a+b x)} \tan ^3(a+b x) (d \cos (a+b x))^{5/2} \, _2F_1\left(-\frac{3}{4},\frac{5}{2};\frac{7}{2};\sin ^2(a+b x)\right)}{5 b}","\frac{8 d^2 E\left(\left.\frac{1}{2} (a+b x)\right|2\right) \sqrt{d \cos (a+b x)}}{65 b \sqrt{\cos (a+b x)}}-\frac{2 \sin ^3(a+b x) (d \cos (a+b x))^{7/2}}{13 b d}-\frac{4 \sin (a+b x) (d \cos (a+b x))^{7/2}}{39 b d}+\frac{8 d \sin (a+b x) (d \cos (a+b x))^{3/2}}{195 b}",1,"((d*Cos[a + b*x])^(5/2)*(Cos[a + b*x]^2)^(1/4)*Hypergeometric2F1[-3/4, 5/2, 7/2, Sin[a + b*x]^2]*Sin[a + b*x]^2*Tan[a + b*x]^3)/(5*b)","C",1
214,1,65,128,0.1084041,"\int (d \cos (a+b x))^{3/2} \sin ^4(a+b x) \, dx","Integrate[(d*Cos[a + b*x])^(3/2)*Sin[a + b*x]^4,x]","\frac{\sin ^2(a+b x) \cos ^2(a+b x)^{3/4} \tan ^3(a+b x) (d \cos (a+b x))^{3/2} \, _2F_1\left(-\frac{1}{4},\frac{5}{2};\frac{7}{2};\sin ^2(a+b x)\right)}{5 b}","\frac{8 d^2 \sqrt{\cos (a+b x)} F\left(\left.\frac{1}{2} (a+b x)\right|2\right)}{77 b \sqrt{d \cos (a+b x)}}-\frac{2 \sin ^3(a+b x) (d \cos (a+b x))^{5/2}}{11 b d}-\frac{12 \sin (a+b x) (d \cos (a+b x))^{5/2}}{77 b d}+\frac{8 d \sin (a+b x) \sqrt{d \cos (a+b x)}}{77 b}",1,"((d*Cos[a + b*x])^(3/2)*(Cos[a + b*x]^2)^(3/4)*Hypergeometric2F1[-1/4, 5/2, 7/2, Sin[a + b*x]^2]*Sin[a + b*x]^2*Tan[a + b*x]^3)/(5*b)","C",1
215,1,58,99,0.0614523,"\int \sqrt{d \cos (a+b x)} \sin ^4(a+b x) \, dx","Integrate[Sqrt[d*Cos[a + b*x]]*Sin[a + b*x]^4,x]","\frac{d \sin ^5(a+b x) \sqrt[4]{\cos ^2(a+b x)} \, _2F_1\left(\frac{1}{4},\frac{5}{2};\frac{7}{2};\sin ^2(a+b x)\right)}{5 b \sqrt{d \cos (a+b x)}}","-\frac{2 \sin ^3(a+b x) (d \cos (a+b x))^{3/2}}{9 b d}-\frac{4 \sin (a+b x) (d \cos (a+b x))^{3/2}}{15 b d}+\frac{8 E\left(\left.\frac{1}{2} (a+b x)\right|2\right) \sqrt{d \cos (a+b x)}}{15 b \sqrt{\cos (a+b x)}}",1,"(d*(Cos[a + b*x]^2)^(1/4)*Hypergeometric2F1[1/4, 5/2, 7/2, Sin[a + b*x]^2]*Sin[a + b*x]^5)/(5*b*Sqrt[d*Cos[a + b*x]])","C",1
216,1,58,99,0.074942,"\int \frac{\sin ^4(a+b x)}{\sqrt{d \cos (a+b x)}} \, dx","Integrate[Sin[a + b*x]^4/Sqrt[d*Cos[a + b*x]],x]","\frac{d \sin ^5(a+b x) \cos ^2(a+b x)^{3/4} \, _2F_1\left(\frac{3}{4},\frac{5}{2};\frac{7}{2};\sin ^2(a+b x)\right)}{5 b (d \cos (a+b x))^{3/2}}","-\frac{2 \sin ^3(a+b x) \sqrt{d \cos (a+b x)}}{7 b d}-\frac{4 \sin (a+b x) \sqrt{d \cos (a+b x)}}{7 b d}+\frac{8 \sqrt{\cos (a+b x)} F\left(\left.\frac{1}{2} (a+b x)\right|2\right)}{7 b \sqrt{d \cos (a+b x)}}",1,"(d*(Cos[a + b*x]^2)^(3/4)*Hypergeometric2F1[3/4, 5/2, 7/2, Sin[a + b*x]^2]*Sin[a + b*x]^5)/(5*b*(d*Cos[a + b*x])^(3/2))","C",1
217,1,60,100,0.0656152,"\int \frac{\sin ^4(a+b x)}{(d \cos (a+b x))^{3/2}} \, dx","Integrate[Sin[a + b*x]^4/(d*Cos[a + b*x])^(3/2),x]","\frac{\sin ^5(a+b x) \sqrt[4]{\cos ^2(a+b x)} \, _2F_1\left(\frac{5}{4},\frac{5}{2};\frac{7}{2};\sin ^2(a+b x)\right)}{5 b d \sqrt{d \cos (a+b x)}}","\frac{12 \sin (a+b x) (d \cos (a+b x))^{3/2}}{5 b d^3}-\frac{24 E\left(\left.\frac{1}{2} (a+b x)\right|2\right) \sqrt{d \cos (a+b x)}}{5 b d^2 \sqrt{\cos (a+b x)}}+\frac{2 \sin ^3(a+b x)}{b d \sqrt{d \cos (a+b x)}}",1,"((Cos[a + b*x]^2)^(1/4)*Hypergeometric2F1[5/4, 5/2, 7/2, Sin[a + b*x]^2]*Sin[a + b*x]^5)/(5*b*d*Sqrt[d*Cos[a + b*x]])","C",1
218,1,60,102,0.0656149,"\int \frac{\sin ^4(a+b x)}{(d \cos (a+b x))^{5/2}} \, dx","Integrate[Sin[a + b*x]^4/(d*Cos[a + b*x])^(5/2),x]","\frac{\sin ^5(a+b x) \cos ^2(a+b x)^{3/4} \, _2F_1\left(\frac{7}{4},\frac{5}{2};\frac{7}{2};\sin ^2(a+b x)\right)}{5 b d (d \cos (a+b x))^{3/2}}","\frac{4 \sin (a+b x) \sqrt{d \cos (a+b x)}}{3 b d^3}-\frac{8 \sqrt{\cos (a+b x)} F\left(\left.\frac{1}{2} (a+b x)\right|2\right)}{3 b d^2 \sqrt{d \cos (a+b x)}}+\frac{2 \sin ^3(a+b x)}{3 b d (d \cos (a+b x))^{3/2}}",1,"((Cos[a + b*x]^2)^(3/4)*Hypergeometric2F1[7/4, 5/2, 7/2, Sin[a + b*x]^2]*Sin[a + b*x]^5)/(5*b*d*(d*Cos[a + b*x])^(3/2))","C",1
219,1,65,102,0.0577382,"\int \frac{\sin ^4(a+b x)}{(d \cos (a+b x))^{7/2}} \, dx","Integrate[Sin[a + b*x]^4/(d*Cos[a + b*x])^(7/2),x]","\frac{\sin ^5(a+b x) \cos ^3(a+b x) \sqrt[4]{\cos ^2(a+b x)} \, _2F_1\left(\frac{9}{4},\frac{5}{2};\frac{7}{2};\sin ^2(a+b x)\right)}{5 b (d \cos (a+b x))^{7/2}}","\frac{24 E\left(\left.\frac{1}{2} (a+b x)\right|2\right) \sqrt{d \cos (a+b x)}}{5 b d^4 \sqrt{\cos (a+b x)}}-\frac{12 \sin (a+b x)}{5 b d^3 \sqrt{d \cos (a+b x)}}+\frac{2 \sin ^3(a+b x)}{5 b d (d \cos (a+b x))^{5/2}}",1,"(Cos[a + b*x]^3*(Cos[a + b*x]^2)^(1/4)*Hypergeometric2F1[9/4, 5/2, 7/2, Sin[a + b*x]^2]*Sin[a + b*x]^5)/(5*b*(d*Cos[a + b*x])^(7/2))","C",1
220,1,65,102,0.0639744,"\int \frac{\sin ^4(a+b x)}{(d \cos (a+b x))^{9/2}} \, dx","Integrate[Sin[a + b*x]^4/(d*Cos[a + b*x])^(9/2),x]","\frac{\sin ^5(a+b x) \cos ^3(a+b x) \cos ^2(a+b x)^{3/4} \, _2F_1\left(\frac{5}{2},\frac{11}{4};\frac{7}{2};\sin ^2(a+b x)\right)}{5 b (d \cos (a+b x))^{9/2}}","\frac{8 \sqrt{\cos (a+b x)} F\left(\left.\frac{1}{2} (a+b x)\right|2\right)}{7 b d^4 \sqrt{d \cos (a+b x)}}-\frac{4 \sin (a+b x)}{7 b d^3 (d \cos (a+b x))^{3/2}}+\frac{2 \sin ^3(a+b x)}{7 b d (d \cos (a+b x))^{7/2}}",1,"(Cos[a + b*x]^3*(Cos[a + b*x]^2)^(3/4)*Hypergeometric2F1[5/2, 11/4, 7/2, Sin[a + b*x]^2]*Sin[a + b*x]^5)/(5*b*(d*Cos[a + b*x])^(9/2))","C",1
221,1,111,52,0.2612215,"\int \cos ^{\frac{3}{2}}(a+b x) \sin ^5(a+b x) \, dx","Integrate[Cos[a + b*x]^(3/2)*Sin[a + b*x]^5,x]","\frac{2 \sqrt{\cos (a+b x)} \left(-32 \sqrt[4]{\cos ^2(a+b x)}+45 \sin ^6(a+b x) \sqrt[4]{\cos ^2(a+b x)}-5 \sin ^4(a+b x) \sqrt[4]{\cos ^2(a+b x)}-8 \sin ^2(a+b x) \sqrt[4]{\cos ^2(a+b x)}+32\right)}{585 b \sqrt[4]{\cos ^2(a+b x)}}","-\frac{2 \cos ^{\frac{13}{2}}(a+b x)}{13 b}+\frac{4 \cos ^{\frac{9}{2}}(a+b x)}{9 b}-\frac{2 \cos ^{\frac{5}{2}}(a+b x)}{5 b}",1,"(2*Sqrt[Cos[a + b*x]]*(32 - 32*(Cos[a + b*x]^2)^(1/4) - 8*(Cos[a + b*x]^2)^(1/4)*Sin[a + b*x]^2 - 5*(Cos[a + b*x]^2)^(1/4)*Sin[a + b*x]^4 + 45*(Cos[a + b*x]^2)^(1/4)*Sin[a + b*x]^6))/(585*b*(Cos[a + b*x]^2)^(1/4))","B",1
222,1,83,100,0.1871253,"\int (d \cos (a+b x))^{9/2} \csc (a+b x) \, dx","Integrate[(d*Cos[a + b*x])^(9/2)*Csc[a + b*x],x]","\frac{d^4 \sqrt{d \cos (a+b x)} \left(2 \left(3 \cos ^2(a+b x)+7\right) \cos ^{\frac{3}{2}}(a+b x)+21 \tan ^{-1}\left(\sqrt{\cos (a+b x)}\right)-21 \tanh ^{-1}\left(\sqrt{\cos (a+b x)}\right)\right)}{21 b \sqrt{\cos (a+b x)}}","\frac{d^{9/2} \tan ^{-1}\left(\frac{\sqrt{d \cos (a+b x)}}{\sqrt{d}}\right)}{b}-\frac{d^{9/2} \tanh ^{-1}\left(\frac{\sqrt{d \cos (a+b x)}}{\sqrt{d}}\right)}{b}+\frac{2 d^3 (d \cos (a+b x))^{3/2}}{3 b}+\frac{2 d (d \cos (a+b x))^{7/2}}{7 b}",1,"(d^4*Sqrt[d*Cos[a + b*x]]*(21*ArcTan[Sqrt[Cos[a + b*x]]] - 21*ArcTanh[Sqrt[Cos[a + b*x]]] + 2*Cos[a + b*x]^(3/2)*(7 + 3*Cos[a + b*x]^2)))/(21*b*Sqrt[Cos[a + b*x]])","A",1
223,1,80,99,0.1789465,"\int (d \cos (a+b x))^{7/2} \csc (a+b x) \, dx","Integrate[(d*Cos[a + b*x])^(7/2)*Csc[a + b*x],x]","\frac{d^3 \sqrt{d \cos (a+b x)} \left(\sqrt{\cos (a+b x)} (\cos (2 (a+b x))+11)-5 \tan ^{-1}\left(\sqrt{\cos (a+b x)}\right)-5 \tanh ^{-1}\left(\sqrt{\cos (a+b x)}\right)\right)}{5 b \sqrt{\cos (a+b x)}}","-\frac{d^{7/2} \tan ^{-1}\left(\frac{\sqrt{d \cos (a+b x)}}{\sqrt{d}}\right)}{b}-\frac{d^{7/2} \tanh ^{-1}\left(\frac{\sqrt{d \cos (a+b x)}}{\sqrt{d}}\right)}{b}+\frac{2 d^3 \sqrt{d \cos (a+b x)}}{b}+\frac{2 d (d \cos (a+b x))^{5/2}}{5 b}",1,"(d^3*Sqrt[d*Cos[a + b*x]]*(-5*ArcTan[Sqrt[Cos[a + b*x]]] - 5*ArcTanh[Sqrt[Cos[a + b*x]]] + Sqrt[Cos[a + b*x]]*(11 + Cos[2*(a + b*x)])))/(5*b*Sqrt[Cos[a + b*x]])","A",1
224,1,68,78,0.1043916,"\int (d \cos (a+b x))^{5/2} \csc (a+b x) \, dx","Integrate[(d*Cos[a + b*x])^(5/2)*Csc[a + b*x],x]","\frac{(d \cos (a+b x))^{5/2} \left(2 \cos ^{\frac{3}{2}}(a+b x)+3 \tan ^{-1}\left(\sqrt{\cos (a+b x)}\right)-3 \tanh ^{-1}\left(\sqrt{\cos (a+b x)}\right)\right)}{3 b \cos ^{\frac{5}{2}}(a+b x)}","\frac{d^{5/2} \tan ^{-1}\left(\frac{\sqrt{d \cos (a+b x)}}{\sqrt{d}}\right)}{b}-\frac{d^{5/2} \tanh ^{-1}\left(\frac{\sqrt{d \cos (a+b x)}}{\sqrt{d}}\right)}{b}+\frac{2 d (d \cos (a+b x))^{3/2}}{3 b}",1,"((d*Cos[a + b*x])^(5/2)*(3*ArcTan[Sqrt[Cos[a + b*x]]] - 3*ArcTanh[Sqrt[Cos[a + b*x]]] + 2*Cos[a + b*x]^(3/2)))/(3*b*Cos[a + b*x]^(5/2))","A",1
225,1,65,77,0.0606999,"\int (d \cos (a+b x))^{3/2} \csc (a+b x) \, dx","Integrate[(d*Cos[a + b*x])^(3/2)*Csc[a + b*x],x]","\frac{(d \cos (a+b x))^{3/2} \left(2 \sqrt{\cos (a+b x)}-\tan ^{-1}\left(\sqrt{\cos (a+b x)}\right)-\tanh ^{-1}\left(\sqrt{\cos (a+b x)}\right)\right)}{b \cos ^{\frac{3}{2}}(a+b x)}","-\frac{d^{3/2} \tan ^{-1}\left(\frac{\sqrt{d \cos (a+b x)}}{\sqrt{d}}\right)}{b}-\frac{d^{3/2} \tanh ^{-1}\left(\frac{\sqrt{d \cos (a+b x)}}{\sqrt{d}}\right)}{b}+\frac{2 d \sqrt{d \cos (a+b x)}}{b}",1,"((-ArcTan[Sqrt[Cos[a + b*x]]] - ArcTanh[Sqrt[Cos[a + b*x]]] + 2*Sqrt[Cos[a + b*x]])*(d*Cos[a + b*x])^(3/2))/(b*Cos[a + b*x]^(3/2))","A",1
226,1,51,58,0.0356462,"\int \sqrt{d \cos (a+b x)} \csc (a+b x) \, dx","Integrate[Sqrt[d*Cos[a + b*x]]*Csc[a + b*x],x]","\frac{\sqrt{d \cos (a+b x)} \left(\tan ^{-1}\left(\sqrt{\cos (a+b x)}\right)-\tanh ^{-1}\left(\sqrt{\cos (a+b x)}\right)\right)}{b \sqrt{\cos (a+b x)}}","\frac{\sqrt{d} \tan ^{-1}\left(\frac{\sqrt{d \cos (a+b x)}}{\sqrt{d}}\right)}{b}-\frac{\sqrt{d} \tanh ^{-1}\left(\frac{\sqrt{d \cos (a+b x)}}{\sqrt{d}}\right)}{b}",1,"((ArcTan[Sqrt[Cos[a + b*x]]] - ArcTanh[Sqrt[Cos[a + b*x]]])*Sqrt[d*Cos[a + b*x]])/(b*Sqrt[Cos[a + b*x]])","A",1
227,1,50,59,0.0367458,"\int \frac{\csc (a+b x)}{\sqrt{d \cos (a+b x)}} \, dx","Integrate[Csc[a + b*x]/Sqrt[d*Cos[a + b*x]],x]","-\frac{\sqrt{\cos (a+b x)} \left(\tan ^{-1}\left(\sqrt{\cos (a+b x)}\right)+\tanh ^{-1}\left(\sqrt{\cos (a+b x)}\right)\right)}{b \sqrt{d \cos (a+b x)}}","-\frac{\tan ^{-1}\left(\frac{\sqrt{d \cos (a+b x)}}{\sqrt{d}}\right)}{b \sqrt{d}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{d \cos (a+b x)}}{\sqrt{d}}\right)}{b \sqrt{d}}",1,"-(((ArcTan[Sqrt[Cos[a + b*x]]] + ArcTanh[Sqrt[Cos[a + b*x]]])*Sqrt[Cos[a + b*x]])/(b*Sqrt[d*Cos[a + b*x]]))","A",1
228,1,36,78,0.0519451,"\int \frac{\csc (a+b x)}{(d \cos (a+b x))^{3/2}} \, dx","Integrate[Csc[a + b*x]/(d*Cos[a + b*x])^(3/2),x]","\frac{2 \, _2F_1\left(-\frac{1}{4},1;\frac{3}{4};\cos ^2(a+b x)\right)}{b d \sqrt{d \cos (a+b x)}}","\frac{\tan ^{-1}\left(\frac{\sqrt{d \cos (a+b x)}}{\sqrt{d}}\right)}{b d^{3/2}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{d \cos (a+b x)}}{\sqrt{d}}\right)}{b d^{3/2}}+\frac{2}{b d \sqrt{d \cos (a+b x)}}",1,"(2*Hypergeometric2F1[-1/4, 1, 3/4, Cos[a + b*x]^2])/(b*d*Sqrt[d*Cos[a + b*x]])","C",1
229,1,38,81,0.0482346,"\int \frac{\csc (a+b x)}{(d \cos (a+b x))^{5/2}} \, dx","Integrate[Csc[a + b*x]/(d*Cos[a + b*x])^(5/2),x]","\frac{2 \, _2F_1\left(-\frac{3}{4},1;\frac{1}{4};\cos ^2(a+b x)\right)}{3 b d (d \cos (a+b x))^{3/2}}","-\frac{\tan ^{-1}\left(\frac{\sqrt{d \cos (a+b x)}}{\sqrt{d}}\right)}{b d^{5/2}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{d \cos (a+b x)}}{\sqrt{d}}\right)}{b d^{5/2}}+\frac{2}{3 b d (d \cos (a+b x))^{3/2}}",1,"(2*Hypergeometric2F1[-3/4, 1, 1/4, Cos[a + b*x]^2])/(3*b*d*(d*Cos[a + b*x])^(3/2))","C",1
230,1,38,100,0.0565548,"\int \frac{\csc (a+b x)}{(d \cos (a+b x))^{7/2}} \, dx","Integrate[Csc[a + b*x]/(d*Cos[a + b*x])^(7/2),x]","\frac{2 \, _2F_1\left(-\frac{5}{4},1;-\frac{1}{4};\cos ^2(a+b x)\right)}{5 b d (d \cos (a+b x))^{5/2}}","\frac{\tan ^{-1}\left(\frac{\sqrt{d \cos (a+b x)}}{\sqrt{d}}\right)}{b d^{7/2}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{d \cos (a+b x)}}{\sqrt{d}}\right)}{b d^{7/2}}+\frac{2}{b d^3 \sqrt{d \cos (a+b x)}}+\frac{2}{5 b d (d \cos (a+b x))^{5/2}}",1,"(2*Hypergeometric2F1[-5/4, 1, -1/4, Cos[a + b*x]^2])/(5*b*d*(d*Cos[a + b*x])^(5/2))","C",1
231,1,38,103,0.0659119,"\int \frac{\csc (a+b x)}{(d \cos (a+b x))^{9/2}} \, dx","Integrate[Csc[a + b*x]/(d*Cos[a + b*x])^(9/2),x]","\frac{2 \, _2F_1\left(-\frac{7}{4},1;-\frac{3}{4};\cos ^2(a+b x)\right)}{7 b d (d \cos (a+b x))^{7/2}}","-\frac{\tan ^{-1}\left(\frac{\sqrt{d \cos (a+b x)}}{\sqrt{d}}\right)}{b d^{9/2}}-\frac{\tanh ^{-1}\left(\frac{\sqrt{d \cos (a+b x)}}{\sqrt{d}}\right)}{b d^{9/2}}+\frac{2}{3 b d^3 (d \cos (a+b x))^{3/2}}+\frac{2}{7 b d (d \cos (a+b x))^{7/2}}",1,"(2*Hypergeometric2F1[-7/4, 1, -3/4, Cos[a + b*x]^2])/(7*b*d*(d*Cos[a + b*x])^(7/2))","C",1
232,1,89,124,0.355879,"\int (d \cos (a+b x))^{11/2} \csc ^2(a+b x) \, dx","Integrate[(d*Cos[a + b*x])^(11/2)*Csc[a + b*x]^2,x]","\frac{d^5 \csc (a+b x) \sqrt{d \cos (a+b x)} \left(\sqrt{\cos (a+b x)} (16 \cos (2 (a+b x))+\cos (4 (a+b x))-45)-60 \sin (a+b x) F\left(\left.\frac{1}{2} (a+b x)\right|2\right)\right)}{28 b \sqrt{\cos (a+b x)}}","-\frac{15 d^6 \sqrt{\cos (a+b x)} F\left(\left.\frac{1}{2} (a+b x)\right|2\right)}{7 b \sqrt{d \cos (a+b x)}}-\frac{15 d^5 \sin (a+b x) \sqrt{d \cos (a+b x)}}{7 b}-\frac{9 d^3 \sin (a+b x) (d \cos (a+b x))^{5/2}}{7 b}-\frac{d \csc (a+b x) (d \cos (a+b x))^{9/2}}{b}",1,"(d^5*Sqrt[d*Cos[a + b*x]]*Csc[a + b*x]*(Sqrt[Cos[a + b*x]]*(-45 + 16*Cos[2*(a + b*x)] + Cos[4*(a + b*x)]) - 60*EllipticF[(a + b*x)/2, 2]*Sin[a + b*x]))/(28*b*Sqrt[Cos[a + b*x]])","A",1
233,1,74,96,0.2359251,"\int (d \cos (a+b x))^{9/2} \csc ^2(a+b x) \, dx","Integrate[(d*Cos[a + b*x])^(9/2)*Csc[a + b*x]^2,x]","-\frac{d^4 \sqrt{d \cos (a+b x)} \left(21 E\left(\left.\frac{1}{2} (a+b x)\right|2\right)+\sqrt{\cos (a+b x)} (\sin (2 (a+b x))+5 \cot (a+b x))\right)}{5 b \sqrt{\cos (a+b x)}}","-\frac{21 d^4 E\left(\left.\frac{1}{2} (a+b x)\right|2\right) \sqrt{d \cos (a+b x)}}{5 b \sqrt{\cos (a+b x)}}-\frac{7 d^3 \sin (a+b x) (d \cos (a+b x))^{3/2}}{5 b}-\frac{d \csc (a+b x) (d \cos (a+b x))^{7/2}}{b}",1,"-1/5*(d^4*Sqrt[d*Cos[a + b*x]]*(21*EllipticE[(a + b*x)/2, 2] + Sqrt[Cos[a + b*x]]*(5*Cot[a + b*x] + Sin[2*(a + b*x)])))/(b*Sqrt[Cos[a + b*x]])","A",1
234,1,73,96,0.2195475,"\int (d \cos (a+b x))^{7/2} \csc ^2(a+b x) \, dx","Integrate[(d*Cos[a + b*x])^(7/2)*Csc[a + b*x]^2,x]","\frac{d^3 \sqrt{d \cos (a+b x)} \left(\sqrt{\cos (a+b x)} (\cos (2 (a+b x))-4) \csc (a+b x)-5 F\left(\left.\frac{1}{2} (a+b x)\right|2\right)\right)}{3 b \sqrt{\cos (a+b x)}}","-\frac{5 d^4 \sqrt{\cos (a+b x)} F\left(\left.\frac{1}{2} (a+b x)\right|2\right)}{3 b \sqrt{d \cos (a+b x)}}-\frac{5 d^3 \sin (a+b x) \sqrt{d \cos (a+b x)}}{3 b}-\frac{d \csc (a+b x) (d \cos (a+b x))^{5/2}}{b}",1,"(d^3*Sqrt[d*Cos[a + b*x]]*(Sqrt[Cos[a + b*x]]*(-4 + Cos[2*(a + b*x)])*Csc[a + b*x] - 5*EllipticF[(a + b*x)/2, 2]))/(3*b*Sqrt[Cos[a + b*x]])","A",1
235,1,58,66,0.1345099,"\int (d \cos (a+b x))^{5/2} \csc ^2(a+b x) \, dx","Integrate[(d*Cos[a + b*x])^(5/2)*Csc[a + b*x]^2,x]","-\frac{(d \cos (a+b x))^{5/2} \left(3 E\left(\left.\frac{1}{2} (a+b x)\right|2\right)+\cos ^{\frac{3}{2}}(a+b x) \csc (a+b x)\right)}{b \cos ^{\frac{5}{2}}(a+b x)}","-\frac{3 d^2 E\left(\left.\frac{1}{2} (a+b x)\right|2\right) \sqrt{d \cos (a+b x)}}{b \sqrt{\cos (a+b x)}}-\frac{d \csc (a+b x) (d \cos (a+b x))^{3/2}}{b}",1,"-(((d*Cos[a + b*x])^(5/2)*(Cos[a + b*x]^(3/2)*Csc[a + b*x] + 3*EllipticE[(a + b*x)/2, 2]))/(b*Cos[a + b*x]^(5/2)))","A",1
236,1,56,66,0.0939974,"\int (d \cos (a+b x))^{3/2} \csc ^2(a+b x) \, dx","Integrate[(d*Cos[a + b*x])^(3/2)*Csc[a + b*x]^2,x]","-\frac{(d \cos (a+b x))^{3/2} \left(F\left(\left.\frac{1}{2} (a+b x)\right|2\right)+\sqrt{\cos (a+b x)} \csc (a+b x)\right)}{b \cos ^{\frac{3}{2}}(a+b x)}","-\frac{d^2 \sqrt{\cos (a+b x)} F\left(\left.\frac{1}{2} (a+b x)\right|2\right)}{b \sqrt{d \cos (a+b x)}}-\frac{d \csc (a+b x) \sqrt{d \cos (a+b x)}}{b}",1,"-(((d*Cos[a + b*x])^(3/2)*(Sqrt[Cos[a + b*x]]*Csc[a + b*x] + EllipticF[(a + b*x)/2, 2]))/(b*Cos[a + b*x]^(3/2)))","A",1
237,1,56,65,0.0779913,"\int \sqrt{d \cos (a+b x)} \csc ^2(a+b x) \, dx","Integrate[Sqrt[d*Cos[a + b*x]]*Csc[a + b*x]^2,x]","-\frac{\sqrt{d \cos (a+b x)} \left(E\left(\left.\frac{1}{2} (a+b x)\right|2\right)+\cos ^{\frac{3}{2}}(a+b x) \csc (a+b x)\right)}{b \sqrt{\cos (a+b x)}}","-\frac{\csc (a+b x) (d \cos (a+b x))^{3/2}}{b d}-\frac{E\left(\left.\frac{1}{2} (a+b x)\right|2\right) \sqrt{d \cos (a+b x)}}{b \sqrt{\cos (a+b x)}}",1,"-((Sqrt[d*Cos[a + b*x]]*(Cos[a + b*x]^(3/2)*Csc[a + b*x] + EllipticE[(a + b*x)/2, 2]))/(b*Sqrt[Cos[a + b*x]]))","A",1
238,1,47,64,0.0781163,"\int \frac{\csc ^2(a+b x)}{\sqrt{d \cos (a+b x)}} \, dx","Integrate[Csc[a + b*x]^2/Sqrt[d*Cos[a + b*x]],x]","\frac{\sqrt{\cos (a+b x)} F\left(\left.\frac{1}{2} (a+b x)\right|2\right)-\cot (a+b x)}{b \sqrt{d \cos (a+b x)}}","\frac{\sqrt{\cos (a+b x)} F\left(\left.\frac{1}{2} (a+b x)\right|2\right)}{b \sqrt{d \cos (a+b x)}}-\frac{\csc (a+b x) \sqrt{d \cos (a+b x)}}{b d}",1,"(-Cot[a + b*x] + Sqrt[Cos[a + b*x]]*EllipticF[(a + b*x)/2, 2])/(b*Sqrt[d*Cos[a + b*x]])","A",1
239,1,65,94,0.1683761,"\int \frac{\csc ^2(a+b x)}{(d \cos (a+b x))^{3/2}} \, dx","Integrate[Csc[a + b*x]^2/(d*Cos[a + b*x])^(3/2),x]","\frac{2 \sin (a+b x)-\cos (a+b x) \cot (a+b x)-3 \sqrt{\cos (a+b x)} E\left(\left.\frac{1}{2} (a+b x)\right|2\right)}{b d \sqrt{d \cos (a+b x)}}","-\frac{3 E\left(\left.\frac{1}{2} (a+b x)\right|2\right) \sqrt{d \cos (a+b x)}}{b d^2 \sqrt{\cos (a+b x)}}+\frac{3 \sin (a+b x)}{b d \sqrt{d \cos (a+b x)}}-\frac{\csc (a+b x)}{b d \sqrt{d \cos (a+b x)}}",1,"(-(Cos[a + b*x]*Cot[a + b*x]) - 3*Sqrt[Cos[a + b*x]]*EllipticE[(a + b*x)/2, 2] + 2*Sin[a + b*x])/(b*d*Sqrt[d*Cos[a + b*x]])","A",1
240,1,62,98,0.1369779,"\int \frac{\csc ^2(a+b x)}{(d \cos (a+b x))^{5/2}} \, dx","Integrate[Csc[a + b*x]^2/(d*Cos[a + b*x])^(5/2),x]","\frac{2 \tan (a+b x)-3 \cot (a+b x)+5 \sqrt{\cos (a+b x)} F\left(\left.\frac{1}{2} (a+b x)\right|2\right)}{3 b d^2 \sqrt{d \cos (a+b x)}}","\frac{5 \sqrt{\cos (a+b x)} F\left(\left.\frac{1}{2} (a+b x)\right|2\right)}{3 b d^2 \sqrt{d \cos (a+b x)}}+\frac{5 \sin (a+b x)}{3 b d (d \cos (a+b x))^{3/2}}-\frac{\csc (a+b x)}{b d (d \cos (a+b x))^{3/2}}",1,"(-3*Cot[a + b*x] + 5*Sqrt[Cos[a + b*x]]*EllipticF[(a + b*x)/2, 2] + 2*Tan[a + b*x])/(3*b*d^2*Sqrt[d*Cos[a + b*x]])","A",1
241,1,82,126,0.1598886,"\int \frac{\csc ^2(a+b x)}{(d \cos (a+b x))^{7/2}} \, dx","Integrate[Csc[a + b*x]^2/(d*Cos[a + b*x])^(7/2),x]","\frac{16 \sin (a+b x)-5 \cos (a+b x) \cot (a+b x)-21 \sqrt{\cos (a+b x)} E\left(\left.\frac{1}{2} (a+b x)\right|2\right)+2 \tan (a+b x) \sec (a+b x)}{5 b d^3 \sqrt{d \cos (a+b x)}}","-\frac{21 E\left(\left.\frac{1}{2} (a+b x)\right|2\right) \sqrt{d \cos (a+b x)}}{5 b d^4 \sqrt{\cos (a+b x)}}+\frac{21 \sin (a+b x)}{5 b d^3 \sqrt{d \cos (a+b x)}}+\frac{7 \sin (a+b x)}{5 b d (d \cos (a+b x))^{5/2}}-\frac{\csc (a+b x)}{b d (d \cos (a+b x))^{5/2}}",1,"(-5*Cos[a + b*x]*Cot[a + b*x] - 21*Sqrt[Cos[a + b*x]]*EllipticE[(a + b*x)/2, 2] + 16*Sin[a + b*x] + 2*Sec[a + b*x]*Tan[a + b*x])/(5*b*d^3*Sqrt[d*Cos[a + b*x]])","A",1
242,1,137,135,2.1730915,"\int (d \cos (a+b x))^{11/2} \csc ^3(a+b x) \, dx","Integrate[(d*Cos[a + b*x])^(11/2)*Csc[a + b*x]^3,x]","\frac{d (d \cos (a+b x))^{9/2} \left(-\frac{21}{2} \left(8 \sqrt{\cos (a+b x)}+\log \left(1-\sqrt{\cos (a+b x)}\right)-\log \left(\sqrt{\cos (a+b x)}+1\right)\right)+45 \tan ^{-1}\left(\sqrt{\cos (a+b x)}\right)-2 \sqrt{\cos (a+b x)} \left(2 \cos (2 (a+b x))+5 \csc ^2(a+b x)\right)+24 \tanh ^{-1}\left(\sqrt{\cos (a+b x)}\right)\right)}{20 b \cos ^{\frac{9}{2}}(a+b x)}","\frac{9 d^{11/2} \tan ^{-1}\left(\frac{\sqrt{d \cos (a+b x)}}{\sqrt{d}}\right)}{4 b}+\frac{9 d^{11/2} \tanh ^{-1}\left(\frac{\sqrt{d \cos (a+b x)}}{\sqrt{d}}\right)}{4 b}-\frac{9 d^5 \sqrt{d \cos (a+b x)}}{2 b}-\frac{9 d^3 (d \cos (a+b x))^{5/2}}{10 b}-\frac{d \csc ^2(a+b x) (d \cos (a+b x))^{9/2}}{2 b}",1,"(d*(d*Cos[a + b*x])^(9/2)*(45*ArcTan[Sqrt[Cos[a + b*x]]] + 24*ArcTanh[Sqrt[Cos[a + b*x]]] - 2*Sqrt[Cos[a + b*x]]*(2*Cos[2*(a + b*x)] + 5*Csc[a + b*x]^2) - (21*(8*Sqrt[Cos[a + b*x]] + Log[1 - Sqrt[Cos[a + b*x]]] - Log[1 + Sqrt[Cos[a + b*x]]]))/2))/(20*b*Cos[a + b*x]^(9/2))","A",1
243,1,78,113,0.6354692,"\int (d \cos (a+b x))^{9/2} \csc ^3(a+b x) \, dx","Integrate[(d*Cos[a + b*x])^(9/2)*Csc[a + b*x]^3,x]","\frac{d^5 \left(21 \sqrt[4]{-\cot ^2(a+b x)} \, _2F_1\left(\frac{1}{4},\frac{1}{4};\frac{5}{4};\csc ^2(a+b x)\right)+(2 \cos (2 (a+b x))-5) \cot ^2(a+b x)\right)}{6 b \sqrt{d \cos (a+b x)}}","-\frac{7 d^{9/2} \tan ^{-1}\left(\frac{\sqrt{d \cos (a+b x)}}{\sqrt{d}}\right)}{4 b}+\frac{7 d^{9/2} \tanh ^{-1}\left(\frac{\sqrt{d \cos (a+b x)}}{\sqrt{d}}\right)}{4 b}-\frac{7 d^3 (d \cos (a+b x))^{3/2}}{6 b}-\frac{d \csc ^2(a+b x) (d \cos (a+b x))^{7/2}}{2 b}",1,"(d^5*((-5 + 2*Cos[2*(a + b*x)])*Cot[a + b*x]^2 + 21*(-Cot[a + b*x]^2)^(1/4)*Hypergeometric2F1[1/4, 1/4, 5/4, Csc[a + b*x]^2]))/(6*b*Sqrt[d*Cos[a + b*x]])","C",1
244,1,118,113,1.1594607,"\int (d \cos (a+b x))^{7/2} \csc ^3(a+b x) \, dx","Integrate[(d*Cos[a + b*x])^(7/2)*Csc[a + b*x]^3,x]","\frac{(d \cos (a+b x))^{7/2} \left(-8 \sqrt{\cos (a+b x)}-\log \left(1-\sqrt{\cos (a+b x)}\right)+\log \left(\sqrt{\cos (a+b x)}+1\right)+5 \tan ^{-1}\left(\sqrt{\cos (a+b x)}\right)-2 \sqrt{\cos (a+b x)} \csc ^2(a+b x)+3 \tanh ^{-1}\left(\sqrt{\cos (a+b x)}\right)\right)}{4 b \cos ^{\frac{7}{2}}(a+b x)}","\frac{5 d^{7/2} \tan ^{-1}\left(\frac{\sqrt{d \cos (a+b x)}}{\sqrt{d}}\right)}{4 b}+\frac{5 d^{7/2} \tanh ^{-1}\left(\frac{\sqrt{d \cos (a+b x)}}{\sqrt{d}}\right)}{4 b}-\frac{5 d^3 \sqrt{d \cos (a+b x)}}{2 b}-\frac{d \csc ^2(a+b x) (d \cos (a+b x))^{5/2}}{2 b}",1,"((d*Cos[a + b*x])^(7/2)*(5*ArcTan[Sqrt[Cos[a + b*x]]] + 3*ArcTanh[Sqrt[Cos[a + b*x]]] - 8*Sqrt[Cos[a + b*x]] - 2*Sqrt[Cos[a + b*x]]*Csc[a + b*x]^2 - Log[1 - Sqrt[Cos[a + b*x]]] + Log[1 + Sqrt[Cos[a + b*x]]]))/(4*b*Cos[a + b*x]^(7/2))","A",1
245,1,65,91,0.2864668,"\int (d \cos (a+b x))^{5/2} \csc ^3(a+b x) \, dx","Integrate[(d*Cos[a + b*x])^(5/2)*Csc[a + b*x]^3,x]","-\frac{d^3 \left(\cot ^2(a+b x)-3 \sqrt[4]{-\cot ^2(a+b x)} \, _2F_1\left(\frac{1}{4},\frac{1}{4};\frac{5}{4};\csc ^2(a+b x)\right)\right)}{2 b \sqrt{d \cos (a+b x)}}","-\frac{3 d^{5/2} \tan ^{-1}\left(\frac{\sqrt{d \cos (a+b x)}}{\sqrt{d}}\right)}{4 b}+\frac{3 d^{5/2} \tanh ^{-1}\left(\frac{\sqrt{d \cos (a+b x)}}{\sqrt{d}}\right)}{4 b}-\frac{d \csc ^2(a+b x) (d \cos (a+b x))^{3/2}}{2 b}",1,"-1/2*(d^3*(Cot[a + b*x]^2 - 3*(-Cot[a + b*x]^2)^(1/4)*Hypergeometric2F1[1/4, 1/4, 5/4, Csc[a + b*x]^2]))/(b*Sqrt[d*Cos[a + b*x]])","C",1
246,1,76,91,0.1784787,"\int (d \cos (a+b x))^{3/2} \csc ^3(a+b x) \, dx","Integrate[(d*Cos[a + b*x])^(3/2)*Csc[a + b*x]^3,x]","\frac{\left(-\cot ^2(a+b x)\right)^{3/4} \sec ^3(a+b x) (d \cos (a+b x))^{3/2} \left(\, _2F_1\left(\frac{3}{4},\frac{3}{4};\frac{7}{4};\csc ^2(a+b x)\right)+3 \sqrt[4]{-\cot ^2(a+b x)}\right)}{6 b}","\frac{d^{3/2} \tan ^{-1}\left(\frac{\sqrt{d \cos (a+b x)}}{\sqrt{d}}\right)}{4 b}+\frac{d^{3/2} \tanh ^{-1}\left(\frac{\sqrt{d \cos (a+b x)}}{\sqrt{d}}\right)}{4 b}-\frac{d \csc ^2(a+b x) \sqrt{d \cos (a+b x)}}{2 b}",1,"((d*Cos[a + b*x])^(3/2)*(-Cot[a + b*x]^2)^(3/4)*(3*(-Cot[a + b*x]^2)^(1/4) + Hypergeometric2F1[3/4, 3/4, 7/4, Csc[a + b*x]^2])*Sec[a + b*x]^3)/(6*b)","C",1
247,1,62,93,0.2515018,"\int \sqrt{d \cos (a+b x)} \csc ^3(a+b x) \, dx","Integrate[Sqrt[d*Cos[a + b*x]]*Csc[a + b*x]^3,x]","-\frac{d \left(\sqrt[4]{-\cot ^2(a+b x)} \, _2F_1\left(\frac{1}{4},\frac{1}{4};\frac{5}{4};\csc ^2(a+b x)\right)+\cot ^2(a+b x)\right)}{2 b \sqrt{d \cos (a+b x)}}","\frac{\sqrt{d} \tan ^{-1}\left(\frac{\sqrt{d \cos (a+b x)}}{\sqrt{d}}\right)}{4 b}-\frac{\csc ^2(a+b x) (d \cos (a+b x))^{3/2}}{2 b d}-\frac{\sqrt{d} \tanh ^{-1}\left(\frac{\sqrt{d \cos (a+b x)}}{\sqrt{d}}\right)}{4 b}",1,"-1/2*(d*(Cot[a + b*x]^2 + (-Cot[a + b*x]^2)^(1/4)*Hypergeometric2F1[1/4, 1/4, 5/4, Csc[a + b*x]^2]))/(b*Sqrt[d*Cos[a + b*x]])","C",1
248,1,69,93,0.2162265,"\int \frac{\csc ^3(a+b x)}{\sqrt{d \cos (a+b x)}} \, dx","Integrate[Csc[a + b*x]^3/Sqrt[d*Cos[a + b*x]],x]","\frac{d \left(-\cot ^2(a+b x)\right)^{3/4} \left(\sqrt[4]{-\cot ^2(a+b x)}-\, _2F_1\left(\frac{3}{4},\frac{3}{4};\frac{7}{4};\csc ^2(a+b x)\right)\right)}{2 b (d \cos (a+b x))^{3/2}}","-\frac{3 \tan ^{-1}\left(\frac{\sqrt{d \cos (a+b x)}}{\sqrt{d}}\right)}{4 b \sqrt{d}}-\frac{\csc ^2(a+b x) \sqrt{d \cos (a+b x)}}{2 b d}-\frac{3 \tanh ^{-1}\left(\frac{\sqrt{d \cos (a+b x)}}{\sqrt{d}}\right)}{4 b \sqrt{d}}",1,"(d*(-Cot[a + b*x]^2)^(3/4)*((-Cot[a + b*x]^2)^(1/4) - Hypergeometric2F1[3/4, 3/4, 7/4, Csc[a + b*x]^2]))/(2*b*(d*Cos[a + b*x])^(3/2))","C",1
249,1,91,115,0.2447729,"\int \frac{\csc ^3(a+b x)}{(d \cos (a+b x))^{3/2}} \, dx","Integrate[Csc[a + b*x]^3/(d*Cos[a + b*x])^(3/2),x]","\frac{5 \cot ^2(a+b x) \, _2F_1\left(\frac{1}{4},\frac{1}{4};\frac{5}{4};\csc ^2(a+b x)\right)-\left(-\cot ^2(a+b x)\right)^{3/4} \left(\cot ^2(a+b x)-4\right)}{2 b d \left(-\cot ^2(a+b x)\right)^{3/4} \sqrt{d \cos (a+b x)}}","\frac{5 \tan ^{-1}\left(\frac{\sqrt{d \cos (a+b x)}}{\sqrt{d}}\right)}{4 b d^{3/2}}-\frac{5 \tanh ^{-1}\left(\frac{\sqrt{d \cos (a+b x)}}{\sqrt{d}}\right)}{4 b d^{3/2}}+\frac{5}{2 b d \sqrt{d \cos (a+b x)}}-\frac{\csc ^2(a+b x)}{2 b d \sqrt{d \cos (a+b x)}}",1,"(-((-Cot[a + b*x]^2)^(3/4)*(-4 + Cot[a + b*x]^2)) + 5*Cot[a + b*x]^2*Hypergeometric2F1[1/4, 1/4, 5/4, Csc[a + b*x]^2])/(2*b*d*Sqrt[d*Cos[a + b*x]]*(-Cot[a + b*x]^2)^(3/4))","C",1
250,1,92,115,0.3627158,"\int \frac{\csc ^3(a+b x)}{(d \cos (a+b x))^{5/2}} \, dx","Integrate[Csc[a + b*x]^3/(d*Cos[a + b*x])^(5/2),x]","\frac{7 \cot ^2(a+b x) \, _2F_1\left(\frac{3}{4},\frac{3}{4};\frac{7}{4};\csc ^2(a+b x)\right)+\sqrt[4]{-\cot ^2(a+b x)} \left(4-3 \cot ^2(a+b x)\right)}{6 b d \sqrt[4]{-\cot ^2(a+b x)} (d \cos (a+b x))^{3/2}}","-\frac{7 \tan ^{-1}\left(\frac{\sqrt{d \cos (a+b x)}}{\sqrt{d}}\right)}{4 b d^{5/2}}-\frac{7 \tanh ^{-1}\left(\frac{\sqrt{d \cos (a+b x)}}{\sqrt{d}}\right)}{4 b d^{5/2}}+\frac{7}{6 b d (d \cos (a+b x))^{3/2}}-\frac{\csc ^2(a+b x)}{2 b d (d \cos (a+b x))^{3/2}}",1,"((-Cot[a + b*x]^2)^(1/4)*(4 - 3*Cot[a + b*x]^2) + 7*Cot[a + b*x]^2*Hypergeometric2F1[3/4, 3/4, 7/4, Csc[a + b*x]^2])/(6*b*d*(d*Cos[a + b*x])^(3/2)*(-Cot[a + b*x]^2)^(1/4))","C",1
251,1,102,137,0.4722861,"\int \frac{\csc ^3(a+b x)}{(d \cos (a+b x))^{7/2}} \, dx","Integrate[Csc[a + b*x]^3/(d*Cos[a + b*x])^(7/2),x]","\frac{45 \cot ^2(a+b x) \, _2F_1\left(\frac{1}{4},\frac{1}{4};\frac{5}{4};\csc ^2(a+b x)\right)+\left(-\cot ^2(a+b x)\right)^{3/4} \left(-5 \cot ^2(a+b x)+4 \sec ^2(a+b x)+40\right)}{10 b d^3 \left(-\cot ^2(a+b x)\right)^{3/4} \sqrt{d \cos (a+b x)}}","\frac{9 \tan ^{-1}\left(\frac{\sqrt{d \cos (a+b x)}}{\sqrt{d}}\right)}{4 b d^{7/2}}-\frac{9 \tanh ^{-1}\left(\frac{\sqrt{d \cos (a+b x)}}{\sqrt{d}}\right)}{4 b d^{7/2}}+\frac{9}{2 b d^3 \sqrt{d \cos (a+b x)}}+\frac{9}{10 b d (d \cos (a+b x))^{5/2}}-\frac{\csc ^2(a+b x)}{2 b d (d \cos (a+b x))^{5/2}}",1,"(45*Cot[a + b*x]^2*Hypergeometric2F1[1/4, 1/4, 5/4, Csc[a + b*x]^2] + (-Cot[a + b*x]^2)^(3/4)*(40 - 5*Cot[a + b*x]^2 + 4*Sec[a + b*x]^2))/(10*b*d^3*Sqrt[d*Cos[a + b*x]]*(-Cot[a + b*x]^2)^(3/4))","C",1
252,1,22,22,0.0195736,"\int \sqrt[5]{d \cos (a+b x)} \sin (a+b x) \, dx","Integrate[(d*Cos[a + b*x])^(1/5)*Sin[a + b*x],x]","-\frac{5 (d \cos (a+b x))^{6/5}}{6 b d}","-\frac{5 (d \cos (a+b x))^{6/5}}{6 b d}",1,"(-5*(d*Cos[a + b*x])^(6/5))/(6*b*d)","A",1
253,1,18,21,0.0136292,"\int \cos ^3(x) \sqrt{\sin (x)} \, dx","Integrate[Cos[x]^3*Sqrt[Sin[x]],x]","\frac{1}{21} \sin ^{\frac{3}{2}}(x) (3 \cos (2 x)+11)","\frac{2}{3} \sin ^{\frac{3}{2}}(x)-\frac{2}{7} \sin ^{\frac{7}{2}}(x)",1,"((11 + 3*Cos[2*x])*Sin[x]^(3/2))/21","A",1
254,1,18,21,0.0325756,"\int \cos ^3(x) \sin ^{\frac{3}{2}}(x) \, dx","Integrate[Cos[x]^3*Sin[x]^(3/2),x]","\frac{1}{45} \sin ^{\frac{5}{2}}(x) (5 \cos (2 x)+13)","\frac{2}{5} \sin ^{\frac{5}{2}}(x)-\frac{2}{9} \sin ^{\frac{9}{2}}(x)",1,"((13 + 5*Cos[2*x])*Sin[x]^(5/2))/45","A",1
255,1,18,21,0.0116564,"\int \cos ^3(x) \sin ^{\frac{5}{2}}(x) \, dx","Integrate[Cos[x]^3*Sin[x]^(5/2),x]","\frac{1}{77} \sin ^{\frac{7}{2}}(x) (7 \cos (2 x)+15)","\frac{2}{7} \sin ^{\frac{7}{2}}(x)-\frac{2}{11} \sin ^{\frac{11}{2}}(x)",1,"((15 + 7*Cos[2*x])*Sin[x]^(7/2))/77","A",1
256,1,16,19,0.0087279,"\int \frac{\cos ^3(x)}{\sqrt{\sin (x)}} \, dx","Integrate[Cos[x]^3/Sqrt[Sin[x]],x]","\frac{1}{5} \sqrt{\sin (x)} (\cos (2 x)+9)","2 \sqrt{\sin (x)}-\frac{2}{5} \sin ^{\frac{5}{2}}(x)",1,"((9 + Cos[2*x])*Sqrt[Sin[x]])/5","A",1
257,1,70,132,0.0893765,"\int (d \cos (a+b x))^{9/2} \sqrt{c \sin (a+b x)} \, dx","Integrate[(d*Cos[a + b*x])^(9/2)*Sqrt[c*Sin[a + b*x]],x]","\frac{2 d^4 \sqrt[4]{\cos ^2(a+b x)} \tan (a+b x) \sqrt{c \sin (a+b x)} \sqrt{d \cos (a+b x)} \, _2F_1\left(-\frac{7}{4},\frac{3}{4};\frac{7}{4};\sin ^2(a+b x)\right)}{3 b}","\frac{7 d^4 E\left(\left.a+b x-\frac{\pi }{4}\right|2\right) \sqrt{c \sin (a+b x)} \sqrt{d \cos (a+b x)}}{20 b \sqrt{\sin (2 a+2 b x)}}+\frac{7 d^3 (c \sin (a+b x))^{3/2} (d \cos (a+b x))^{3/2}}{30 b c}+\frac{d (c \sin (a+b x))^{3/2} (d \cos (a+b x))^{7/2}}{5 b c}",1,"(2*d^4*Sqrt[d*Cos[a + b*x]]*(Cos[a + b*x]^2)^(1/4)*Hypergeometric2F1[-7/4, 3/4, 7/4, Sin[a + b*x]^2]*Sqrt[c*Sin[a + b*x]]*Tan[a + b*x])/(3*b)","C",1
258,1,70,95,0.0840263,"\int (d \cos (a+b x))^{5/2} \sqrt{c \sin (a+b x)} \, dx","Integrate[(d*Cos[a + b*x])^(5/2)*Sqrt[c*Sin[a + b*x]],x]","\frac{2 d^2 \sqrt[4]{\cos ^2(a+b x)} \tan (a+b x) \sqrt{c \sin (a+b x)} \sqrt{d \cos (a+b x)} \, _2F_1\left(-\frac{3}{4},\frac{3}{4};\frac{7}{4};\sin ^2(a+b x)\right)}{3 b}","\frac{d^2 E\left(\left.a+b x-\frac{\pi }{4}\right|2\right) \sqrt{c \sin (a+b x)} \sqrt{d \cos (a+b x)}}{2 b \sqrt{\sin (2 a+2 b x)}}+\frac{d (c \sin (a+b x))^{3/2} (d \cos (a+b x))^{3/2}}{3 b c}",1,"(2*d^2*Sqrt[d*Cos[a + b*x]]*(Cos[a + b*x]^2)^(1/4)*Hypergeometric2F1[-3/4, 3/4, 7/4, Sin[a + b*x]^2]*Sqrt[c*Sin[a + b*x]]*Tan[a + b*x])/(3*b)","C",1
259,1,67,53,0.0605628,"\int \sqrt{d \cos (a+b x)} \sqrt{c \sin (a+b x)} \, dx","Integrate[Sqrt[d*Cos[a + b*x]]*Sqrt[c*Sin[a + b*x]],x]","\frac{2 \sqrt[4]{\cos ^2(a+b x)} \tan (a+b x) \sqrt{c \sin (a+b x)} \sqrt{d \cos (a+b x)} \, _2F_1\left(\frac{1}{4},\frac{3}{4};\frac{7}{4};\sin ^2(a+b x)\right)}{3 b}","\frac{E\left(\left.a+b x-\frac{\pi }{4}\right|2\right) \sqrt{c \sin (a+b x)} \sqrt{d \cos (a+b x)}}{b \sqrt{\sin (2 a+2 b x)}}",1,"(2*Sqrt[d*Cos[a + b*x]]*(Cos[a + b*x]^2)^(1/4)*Hypergeometric2F1[1/4, 3/4, 7/4, Sin[a + b*x]^2]*Sqrt[c*Sin[a + b*x]]*Tan[a + b*x])/(3*b)","C",1
260,1,70,93,0.1016305,"\int \frac{\sqrt{c \sin (a+b x)}}{(d \cos (a+b x))^{3/2}} \, dx","Integrate[Sqrt[c*Sin[a + b*x]]/(d*Cos[a + b*x])^(3/2),x]","\frac{2 \sqrt[4]{\cos ^2(a+b x)} \tan (a+b x) \sqrt{c \sin (a+b x)} \sqrt{d \cos (a+b x)} \, _2F_1\left(\frac{3}{4},\frac{5}{4};\frac{7}{4};\sin ^2(a+b x)\right)}{3 b d^2}","\frac{2 (c \sin (a+b x))^{3/2}}{b c d \sqrt{d \cos (a+b x)}}-\frac{2 E\left(\left.a+b x-\frac{\pi }{4}\right|2\right) \sqrt{c \sin (a+b x)} \sqrt{d \cos (a+b x)}}{b d^2 \sqrt{\sin (2 a+2 b x)}}",1,"(2*Sqrt[d*Cos[a + b*x]]*(Cos[a + b*x]^2)^(1/4)*Hypergeometric2F1[3/4, 5/4, 7/4, Sin[a + b*x]^2]*Sqrt[c*Sin[a + b*x]]*Tan[a + b*x])/(3*b*d^2)","C",1
261,1,70,134,0.1312351,"\int \frac{\sqrt{c \sin (a+b x)}}{(d \cos (a+b x))^{7/2}} \, dx","Integrate[Sqrt[c*Sin[a + b*x]]/(d*Cos[a + b*x])^(7/2),x]","\frac{2 \sqrt[4]{\cos ^2(a+b x)} \tan (a+b x) \sqrt{c \sin (a+b x)} \sqrt{d \cos (a+b x)} \, _2F_1\left(\frac{3}{4},\frac{9}{4};\frac{7}{4};\sin ^2(a+b x)\right)}{3 b d^4}","-\frac{4 E\left(\left.a+b x-\frac{\pi }{4}\right|2\right) \sqrt{c \sin (a+b x)} \sqrt{d \cos (a+b x)}}{5 b d^4 \sqrt{\sin (2 a+2 b x)}}+\frac{4 (c \sin (a+b x))^{3/2}}{5 b c d^3 \sqrt{d \cos (a+b x)}}+\frac{2 (c \sin (a+b x))^{3/2}}{5 b c d (d \cos (a+b x))^{5/2}}",1,"(2*Sqrt[d*Cos[a + b*x]]*(Cos[a + b*x]^2)^(1/4)*Hypergeometric2F1[3/4, 9/4, 7/4, Sin[a + b*x]^2]*Sqrt[c*Sin[a + b*x]]*Tan[a + b*x])/(3*b*d^4)","C",1
262,1,70,320,0.1111978,"\int (d \cos (a+b x))^{3/2} \sqrt{c \sin (a+b x)} \, dx","Integrate[(d*Cos[a + b*x])^(3/2)*Sqrt[c*Sin[a + b*x]],x]","\frac{2 d^2 \cos ^2(a+b x)^{3/4} \tan (a+b x) \sqrt{c \sin (a+b x)} \, _2F_1\left(-\frac{1}{4},\frac{3}{4};\frac{7}{4};\sin ^2(a+b x)\right)}{3 b \sqrt{d \cos (a+b x)}}","-\frac{\sqrt{c} d^{3/2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{d} \sqrt{c \sin (a+b x)}}{\sqrt{c} \sqrt{d \cos (a+b x)}}\right)}{4 \sqrt{2} b}+\frac{\sqrt{c} d^{3/2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{d} \sqrt{c \sin (a+b x)}}{\sqrt{c} \sqrt{d \cos (a+b x)}}+1\right)}{4 \sqrt{2} b}+\frac{\sqrt{c} d^{3/2} \log \left(-\frac{\sqrt{2} \sqrt{d} \sqrt{c \sin (a+b x)}}{\sqrt{d \cos (a+b x)}}+\sqrt{c} \tan (a+b x)+\sqrt{c}\right)}{8 \sqrt{2} b}-\frac{\sqrt{c} d^{3/2} \log \left(\frac{\sqrt{2} \sqrt{d} \sqrt{c \sin (a+b x)}}{\sqrt{d \cos (a+b x)}}+\sqrt{c} \tan (a+b x)+\sqrt{c}\right)}{8 \sqrt{2} b}+\frac{d (c \sin (a+b x))^{3/2} \sqrt{d \cos (a+b x)}}{2 b c}",1,"(2*d^2*(Cos[a + b*x]^2)^(3/4)*Hypergeometric2F1[-1/4, 3/4, 7/4, Sin[a + b*x]^2]*Sqrt[c*Sin[a + b*x]]*Tan[a + b*x])/(3*b*Sqrt[d*Cos[a + b*x]])","C",1
263,1,67,280,0.0581782,"\int \frac{\sqrt{c \sin (a+b x)}}{\sqrt{d \cos (a+b x)}} \, dx","Integrate[Sqrt[c*Sin[a + b*x]]/Sqrt[d*Cos[a + b*x]],x]","\frac{2 \cos ^2(a+b x)^{3/4} \tan (a+b x) \sqrt{c \sin (a+b x)} \, _2F_1\left(\frac{3}{4},\frac{3}{4};\frac{7}{4};\sin ^2(a+b x)\right)}{3 b \sqrt{d \cos (a+b x)}}","-\frac{\sqrt{c} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{d} \sqrt{c \sin (a+b x)}}{\sqrt{c} \sqrt{d \cos (a+b x)}}\right)}{\sqrt{2} b \sqrt{d}}+\frac{\sqrt{c} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{d} \sqrt{c \sin (a+b x)}}{\sqrt{c} \sqrt{d \cos (a+b x)}}+1\right)}{\sqrt{2} b \sqrt{d}}+\frac{\sqrt{c} \log \left(-\frac{\sqrt{2} \sqrt{d} \sqrt{c \sin (a+b x)}}{\sqrt{d \cos (a+b x)}}+\sqrt{c} \tan (a+b x)+\sqrt{c}\right)}{2 \sqrt{2} b \sqrt{d}}-\frac{\sqrt{c} \log \left(\frac{\sqrt{2} \sqrt{d} \sqrt{c \sin (a+b x)}}{\sqrt{d \cos (a+b x)}}+\sqrt{c} \tan (a+b x)+\sqrt{c}\right)}{2 \sqrt{2} b \sqrt{d}}",1,"(2*(Cos[a + b*x]^2)^(3/4)*Hypergeometric2F1[3/4, 3/4, 7/4, Sin[a + b*x]^2]*Sqrt[c*Sin[a + b*x]]*Tan[a + b*x])/(3*b*Sqrt[d*Cos[a + b*x]])","C",1
264,1,37,37,0.0766699,"\int \frac{\sqrt{c \sin (a+b x)}}{(d \cos (a+b x))^{5/2}} \, dx","Integrate[Sqrt[c*Sin[a + b*x]]/(d*Cos[a + b*x])^(5/2),x]","\frac{2 (c \sin (a+b x))^{3/2}}{3 b c d (d \cos (a+b x))^{3/2}}","\frac{2 (c \sin (a+b x))^{3/2}}{3 b c d (d \cos (a+b x))^{3/2}}",1,"(2*(c*Sin[a + b*x])^(3/2))/(3*b*c*d*(d*Cos[a + b*x])^(3/2))","A",1
265,1,57,75,0.2176866,"\int \frac{\sqrt{c \sin (a+b x)}}{(d \cos (a+b x))^{9/2}} \, dx","Integrate[Sqrt[c*Sin[a + b*x]]/(d*Cos[a + b*x])^(9/2),x]","\frac{2 (2 \cos (2 (a+b x))+5) \sec ^4(a+b x) (c \sin (a+b x))^{3/2} \sqrt{d \cos (a+b x)}}{21 b c d^5}","\frac{8 (c \sin (a+b x))^{3/2}}{21 b c d^3 (d \cos (a+b x))^{3/2}}+\frac{2 (c \sin (a+b x))^{3/2}}{7 b c d (d \cos (a+b x))^{7/2}}",1,"(2*Sqrt[d*Cos[a + b*x]]*(5 + 2*Cos[2*(a + b*x)])*Sec[a + b*x]^4*(c*Sin[a + b*x])^(3/2))/(21*b*c*d^5)","A",1
266,1,67,112,0.2281145,"\int \frac{\sqrt{c \sin (a+b x)}}{(d \cos (a+b x))^{13/2}} \, dx","Integrate[Sqrt[c*Sin[a + b*x]]/(d*Cos[a + b*x])^(13/2),x]","\frac{2 (28 \cos (2 (a+b x))+4 \cos (4 (a+b x))+45) \sec ^6(a+b x) (c \sin (a+b x))^{3/2} \sqrt{d \cos (a+b x)}}{231 b c d^7}","\frac{64 (c \sin (a+b x))^{3/2}}{231 b c d^5 (d \cos (a+b x))^{3/2}}+\frac{16 (c \sin (a+b x))^{3/2}}{77 b c d^3 (d \cos (a+b x))^{7/2}}+\frac{2 (c \sin (a+b x))^{3/2}}{11 b c d (d \cos (a+b x))^{11/2}}",1,"(2*Sqrt[d*Cos[a + b*x]]*(45 + 28*Cos[2*(a + b*x)] + 4*Cos[4*(a + b*x)])*Sec[a + b*x]^6*(c*Sin[a + b*x])^(3/2))/(231*b*c*d^7)","A",1
267,1,71,131,0.1167806,"\int (d \cos (a+b x))^{3/2} (c \sin (a+b x))^{3/2} \, dx","Integrate[(d*Cos[a + b*x])^(3/2)*(c*Sin[a + b*x])^(3/2),x]","\frac{2 c d \cos ^2(a+b x)^{3/4} \tan ^2(a+b x) \sqrt{c \sin (a+b x)} \sqrt{d \cos (a+b x)} \, _2F_1\left(-\frac{1}{4},\frac{5}{4};\frac{9}{4};\sin ^2(a+b x)\right)}{5 b}","\frac{c^2 d^2 \sqrt{\sin (2 a+2 b x)} F\left(\left.a+b x-\frac{\pi }{4}\right|2\right)}{12 b \sqrt{c \sin (a+b x)} \sqrt{d \cos (a+b x)}}-\frac{c \sqrt{c \sin (a+b x)} (d \cos (a+b x))^{5/2}}{3 b d}+\frac{c d \sqrt{c \sin (a+b x)} \sqrt{d \cos (a+b x)}}{6 b}",1,"(2*c*d*Sqrt[d*Cos[a + b*x]]*(Cos[a + b*x]^2)^(3/4)*Hypergeometric2F1[-1/4, 5/4, 9/4, Sin[a + b*x]^2]*Sqrt[c*Sin[a + b*x]]*Tan[a + b*x]^2)/(5*b)","C",1
268,1,67,93,0.0804187,"\int \frac{(c \sin (a+b x))^{3/2}}{\sqrt{d \cos (a+b x)}} \, dx","Integrate[(c*Sin[a + b*x])^(3/2)/Sqrt[d*Cos[a + b*x]],x]","\frac{2 \cos ^2(a+b x)^{3/4} \tan (a+b x) (c \sin (a+b x))^{3/2} \, _2F_1\left(\frac{3}{4},\frac{5}{4};\frac{9}{4};\sin ^2(a+b x)\right)}{5 b \sqrt{d \cos (a+b x)}}","\frac{c^2 \sqrt{\sin (2 a+2 b x)} F\left(\left.a+b x-\frac{\pi }{4}\right|2\right)}{2 b \sqrt{c \sin (a+b x)} \sqrt{d \cos (a+b x)}}-\frac{c \sqrt{c \sin (a+b x)} \sqrt{d \cos (a+b x)}}{b d}",1,"(2*(Cos[a + b*x]^2)^(3/4)*Hypergeometric2F1[3/4, 5/4, 9/4, Sin[a + b*x]^2]*(c*Sin[a + b*x])^(3/2)*Tan[a + b*x])/(5*b*Sqrt[d*Cos[a + b*x]])","C",1
269,1,67,98,0.1597052,"\int \frac{(c \sin (a+b x))^{3/2}}{(d \cos (a+b x))^{5/2}} \, dx","Integrate[(c*Sin[a + b*x])^(3/2)/(d*Cos[a + b*x])^(5/2),x]","\frac{2 \cos ^2(a+b x)^{3/4} (c \sin (a+b x))^{5/2} \, _2F_1\left(\frac{5}{4},\frac{7}{4};\frac{9}{4};\sin ^2(a+b x)\right)}{5 b c d (d \cos (a+b x))^{3/2}}","\frac{2 c \sqrt{c \sin (a+b x)}}{3 b d (d \cos (a+b x))^{3/2}}-\frac{c^2 \sqrt{\sin (2 a+2 b x)} F\left(\left.a+b x-\frac{\pi }{4}\right|2\right)}{3 b d^2 \sqrt{c \sin (a+b x)} \sqrt{d \cos (a+b x)}}",1,"(2*(Cos[a + b*x]^2)^(3/4)*Hypergeometric2F1[5/4, 7/4, 9/4, Sin[a + b*x]^2]*(c*Sin[a + b*x])^(5/2))/(5*b*c*d*(d*Cos[a + b*x])^(3/2))","C",1
270,1,70,133,0.1434618,"\int \frac{(c \sin (a+b x))^{3/2}}{(d \cos (a+b x))^{9/2}} \, dx","Integrate[(c*Sin[a + b*x])^(3/2)/(d*Cos[a + b*x])^(9/2),x]","\frac{2 \cos ^2(a+b x)^{7/4} \cot (a+b x) (c \sin (a+b x))^{7/2} \, _2F_1\left(\frac{5}{4},\frac{11}{4};\frac{9}{4};\sin ^2(a+b x)\right)}{5 b c^2 (d \cos (a+b x))^{9/2}}","-\frac{2 c^2 \sqrt{\sin (2 a+2 b x)} F\left(\left.a+b x-\frac{\pi }{4}\right|2\right)}{21 b d^4 \sqrt{c \sin (a+b x)} \sqrt{d \cos (a+b x)}}-\frac{2 c \sqrt{c \sin (a+b x)}}{21 b d^3 (d \cos (a+b x))^{3/2}}+\frac{2 c \sqrt{c \sin (a+b x)}}{7 b d (d \cos (a+b x))^{7/2}}",1,"(2*(Cos[a + b*x]^2)^(7/4)*Cot[a + b*x]*Hypergeometric2F1[5/4, 11/4, 9/4, Sin[a + b*x]^2]*(c*Sin[a + b*x])^(7/2))/(5*b*c^2*(d*Cos[a + b*x])^(9/2))","C",1
271,1,67,320,0.0645759,"\int \sqrt{d \cos (a+b x)} (c \sin (a+b x))^{3/2} \, dx","Integrate[Sqrt[d*Cos[a + b*x]]*(c*Sin[a + b*x])^(3/2),x]","\frac{2 \sqrt[4]{\cos ^2(a+b x)} \tan (a+b x) (c \sin (a+b x))^{3/2} \sqrt{d \cos (a+b x)} \, _2F_1\left(\frac{1}{4},\frac{5}{4};\frac{9}{4};\sin ^2(a+b x)\right)}{5 b}","\frac{c^{3/2} \sqrt{d} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{c} \sqrt{d \cos (a+b x)}}{\sqrt{d} \sqrt{c \sin (a+b x)}}\right)}{4 \sqrt{2} b}-\frac{c^{3/2} \sqrt{d} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{d \cos (a+b x)}}{\sqrt{d} \sqrt{c \sin (a+b x)}}+1\right)}{4 \sqrt{2} b}-\frac{c^{3/2} \sqrt{d} \log \left(-\frac{\sqrt{2} \sqrt{c} \sqrt{d \cos (a+b x)}}{\sqrt{c \sin (a+b x)}}+\sqrt{d} \cot (a+b x)+\sqrt{d}\right)}{8 \sqrt{2} b}+\frac{c^{3/2} \sqrt{d} \log \left(\frac{\sqrt{2} \sqrt{c} \sqrt{d \cos (a+b x)}}{\sqrt{c \sin (a+b x)}}+\sqrt{d} \cot (a+b x)+\sqrt{d}\right)}{8 \sqrt{2} b}-\frac{c \sqrt{c \sin (a+b x)} (d \cos (a+b x))^{3/2}}{2 b d}",1,"(2*Sqrt[d*Cos[a + b*x]]*(Cos[a + b*x]^2)^(1/4)*Hypergeometric2F1[1/4, 5/4, 9/4, Sin[a + b*x]^2]*(c*Sin[a + b*x])^(3/2)*Tan[a + b*x])/(5*b)","C",1
272,1,67,313,0.1529853,"\int \frac{(c \sin (a+b x))^{3/2}}{(d \cos (a+b x))^{3/2}} \, dx","Integrate[(c*Sin[a + b*x])^(3/2)/(d*Cos[a + b*x])^(3/2),x]","\frac{2 \sqrt[4]{\cos ^2(a+b x)} (c \sin (a+b x))^{5/2} \, _2F_1\left(\frac{5}{4},\frac{5}{4};\frac{9}{4};\sin ^2(a+b x)\right)}{5 b c d \sqrt{d \cos (a+b x)}}","-\frac{c^{3/2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{c} \sqrt{d \cos (a+b x)}}{\sqrt{d} \sqrt{c \sin (a+b x)}}\right)}{\sqrt{2} b d^{3/2}}+\frac{c^{3/2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{d \cos (a+b x)}}{\sqrt{d} \sqrt{c \sin (a+b x)}}+1\right)}{\sqrt{2} b d^{3/2}}+\frac{c^{3/2} \log \left(-\frac{\sqrt{2} \sqrt{c} \sqrt{d \cos (a+b x)}}{\sqrt{c \sin (a+b x)}}+\sqrt{d} \cot (a+b x)+\sqrt{d}\right)}{2 \sqrt{2} b d^{3/2}}-\frac{c^{3/2} \log \left(\frac{\sqrt{2} \sqrt{c} \sqrt{d \cos (a+b x)}}{\sqrt{c \sin (a+b x)}}+\sqrt{d} \cot (a+b x)+\sqrt{d}\right)}{2 \sqrt{2} b d^{3/2}}+\frac{2 c \sqrt{c \sin (a+b x)}}{b d \sqrt{d \cos (a+b x)}}",1,"(2*(Cos[a + b*x]^2)^(1/4)*Hypergeometric2F1[5/4, 5/4, 9/4, Sin[a + b*x]^2]*(c*Sin[a + b*x])^(5/2))/(5*b*c*d*Sqrt[d*Cos[a + b*x]])","C",1
273,1,40,37,0.1062178,"\int \frac{(c \sin (a+b x))^{3/2}}{(d \cos (a+b x))^{7/2}} \, dx","Integrate[(c*Sin[a + b*x])^(3/2)/(d*Cos[a + b*x])^(7/2),x]","\frac{2 \cot (a+b x) (c \sin (a+b x))^{7/2}}{5 b c^2 (d \cos (a+b x))^{7/2}}","\frac{2 (c \sin (a+b x))^{5/2}}{5 b c d (d \cos (a+b x))^{5/2}}",1,"(2*Cot[a + b*x]*(c*Sin[a + b*x])^(7/2))/(5*b*c^2*(d*Cos[a + b*x])^(7/2))","A",1
274,1,57,106,0.2881273,"\int \frac{(c \sin (a+b x))^{3/2}}{(d \cos (a+b x))^{11/2}} \, dx","Integrate[(c*Sin[a + b*x])^(3/2)/(d*Cos[a + b*x])^(11/2),x]","\frac{2 (2 \cos (2 (a+b x))+7) \sec ^5(a+b x) (c \sin (a+b x))^{5/2} \sqrt{d \cos (a+b x)}}{45 b c d^6}","-\frac{8 c \sqrt{c \sin (a+b x)}}{45 b d^5 \sqrt{d \cos (a+b x)}}-\frac{2 c \sqrt{c \sin (a+b x)}}{45 b d^3 (d \cos (a+b x))^{5/2}}+\frac{2 c \sqrt{c \sin (a+b x)}}{9 b d (d \cos (a+b x))^{9/2}}",1,"(2*Sqrt[d*Cos[a + b*x]]*(7 + 2*Cos[2*(a + b*x)])*Sec[a + b*x]^5*(c*Sin[a + b*x])^(5/2))/(45*b*c*d^6)","A",1
275,1,67,141,0.3085051,"\int \frac{(c \sin (a+b x))^{3/2}}{(d \cos (a+b x))^{15/2}} \, dx","Integrate[(c*Sin[a + b*x])^(3/2)/(d*Cos[a + b*x])^(15/2),x]","\frac{2 (36 \cos (2 (a+b x))+4 \cos (4 (a+b x))+77) \sec ^7(a+b x) (c \sin (a+b x))^{5/2} \sqrt{d \cos (a+b x)}}{585 b c d^8}","-\frac{64 c \sqrt{c \sin (a+b x)}}{585 b d^7 \sqrt{d \cos (a+b x)}}-\frac{16 c \sqrt{c \sin (a+b x)}}{585 b d^5 (d \cos (a+b x))^{5/2}}-\frac{2 c \sqrt{c \sin (a+b x)}}{117 b d^3 (d \cos (a+b x))^{9/2}}+\frac{2 c \sqrt{c \sin (a+b x)}}{13 b d (d \cos (a+b x))^{13/2}}",1,"(2*Sqrt[d*Cos[a + b*x]]*(77 + 36*Cos[2*(a + b*x)] + 4*Cos[4*(a + b*x)])*Sec[a + b*x]^7*(c*Sin[a + b*x])^(5/2))/(585*b*c*d^8)","A",1
276,1,72,166,0.1696807,"\int (d \cos (a+b x))^{9/2} (c \sin (a+b x))^{5/2} \, dx","Integrate[(d*Cos[a + b*x])^(9/2)*(c*Sin[a + b*x])^(5/2),x]","\frac{2 \sqrt[4]{\cos ^2(a+b x)} \sec ^5(a+b x) (c \sin (a+b x))^{7/2} (d \cos (a+b x))^{9/2} \, _2F_1\left(-\frac{7}{4},\frac{7}{4};\frac{11}{4};\sin ^2(a+b x)\right)}{7 b c}","\frac{3 c^2 d^4 E\left(\left.a+b x-\frac{\pi }{4}\right|2\right) \sqrt{c \sin (a+b x)} \sqrt{d \cos (a+b x)}}{40 b \sqrt{\sin (2 a+2 b x)}}+\frac{c d^3 (c \sin (a+b x))^{3/2} (d \cos (a+b x))^{3/2}}{20 b}-\frac{c (c \sin (a+b x))^{3/2} (d \cos (a+b x))^{11/2}}{7 b d}+\frac{3 c d (c \sin (a+b x))^{3/2} (d \cos (a+b x))^{7/2}}{70 b}",1,"(2*(d*Cos[a + b*x])^(9/2)*(Cos[a + b*x]^2)^(1/4)*Hypergeometric2F1[-7/4, 7/4, 11/4, Sin[a + b*x]^2]*Sec[a + b*x]^5*(c*Sin[a + b*x])^(7/2))/(7*b*c)","C",1
277,1,70,131,0.1780677,"\int (d \cos (a+b x))^{5/2} (c \sin (a+b x))^{5/2} \, dx","Integrate[(d*Cos[a + b*x])^(5/2)*(c*Sin[a + b*x])^(5/2),x]","\frac{2 d^2 \sqrt[4]{\cos ^2(a+b x)} \tan (a+b x) (c \sin (a+b x))^{5/2} \sqrt{d \cos (a+b x)} \, _2F_1\left(-\frac{3}{4},\frac{7}{4};\frac{11}{4};\sin ^2(a+b x)\right)}{7 b}","\frac{3 c^2 d^2 E\left(\left.a+b x-\frac{\pi }{4}\right|2\right) \sqrt{c \sin (a+b x)} \sqrt{d \cos (a+b x)}}{20 b \sqrt{\sin (2 a+2 b x)}}-\frac{c (c \sin (a+b x))^{3/2} (d \cos (a+b x))^{7/2}}{5 b d}+\frac{c d (c \sin (a+b x))^{3/2} (d \cos (a+b x))^{3/2}}{10 b}",1,"(2*d^2*Sqrt[d*Cos[a + b*x]]*(Cos[a + b*x]^2)^(1/4)*Hypergeometric2F1[-3/4, 7/4, 11/4, Sin[a + b*x]^2]*(c*Sin[a + b*x])^(5/2)*Tan[a + b*x])/(7*b)","C",1
278,1,67,95,0.0897685,"\int \sqrt{d \cos (a+b x)} (c \sin (a+b x))^{5/2} \, dx","Integrate[Sqrt[d*Cos[a + b*x]]*(c*Sin[a + b*x])^(5/2),x]","\frac{2 \sqrt[4]{\cos ^2(a+b x)} \tan (a+b x) (c \sin (a+b x))^{5/2} \sqrt{d \cos (a+b x)} \, _2F_1\left(\frac{1}{4},\frac{7}{4};\frac{11}{4};\sin ^2(a+b x)\right)}{7 b}","\frac{c^2 E\left(\left.a+b x-\frac{\pi }{4}\right|2\right) \sqrt{c \sin (a+b x)} \sqrt{d \cos (a+b x)}}{2 b \sqrt{\sin (2 a+2 b x)}}-\frac{c (c \sin (a+b x))^{3/2} (d \cos (a+b x))^{3/2}}{3 b d}",1,"(2*Sqrt[d*Cos[a + b*x]]*(Cos[a + b*x]^2)^(1/4)*Hypergeometric2F1[1/4, 7/4, 11/4, Sin[a + b*x]^2]*(c*Sin[a + b*x])^(5/2)*Tan[a + b*x])/(7*b)","C",1
279,1,67,94,0.120389,"\int \frac{(c \sin (a+b x))^{5/2}}{(d \cos (a+b x))^{3/2}} \, dx","Integrate[(c*Sin[a + b*x])^(5/2)/(d*Cos[a + b*x])^(3/2),x]","\frac{2 \sqrt[4]{\cos ^2(a+b x)} (c \sin (a+b x))^{7/2} \, _2F_1\left(\frac{5}{4},\frac{7}{4};\frac{11}{4};\sin ^2(a+b x)\right)}{7 b c d \sqrt{d \cos (a+b x)}}","\frac{2 c (c \sin (a+b x))^{3/2}}{b d \sqrt{d \cos (a+b x)}}-\frac{3 c^2 E\left(\left.a+b x-\frac{\pi }{4}\right|2\right) \sqrt{c \sin (a+b x)} \sqrt{d \cos (a+b x)}}{b d^2 \sqrt{\sin (2 a+2 b x)}}",1,"(2*(Cos[a + b*x]^2)^(1/4)*Hypergeometric2F1[5/4, 7/4, 11/4, Sin[a + b*x]^2]*(c*Sin[a + b*x])^(7/2))/(7*b*c*d*Sqrt[d*Cos[a + b*x]])","C",1
280,1,70,133,0.1709128,"\int \frac{(c \sin (a+b x))^{5/2}}{(d \cos (a+b x))^{7/2}} \, dx","Integrate[(c*Sin[a + b*x])^(5/2)/(d*Cos[a + b*x])^(7/2),x]","\frac{2 \cos ^2(a+b x)^{5/4} \cot (a+b x) (c \sin (a+b x))^{9/2} \, _2F_1\left(\frac{7}{4},\frac{9}{4};\frac{11}{4};\sin ^2(a+b x)\right)}{7 b c^2 (d \cos (a+b x))^{7/2}}","\frac{6 c^2 E\left(\left.a+b x-\frac{\pi }{4}\right|2\right) \sqrt{c \sin (a+b x)} \sqrt{d \cos (a+b x)}}{5 b d^4 \sqrt{\sin (2 a+2 b x)}}-\frac{6 c (c \sin (a+b x))^{3/2}}{5 b d^3 \sqrt{d \cos (a+b x)}}+\frac{2 c (c \sin (a+b x))^{3/2}}{5 b d (d \cos (a+b x))^{5/2}}",1,"(2*(Cos[a + b*x]^2)^(5/4)*Cot[a + b*x]*Hypergeometric2F1[7/4, 9/4, 11/4, Sin[a + b*x]^2]*(c*Sin[a + b*x])^(9/2))/(7*b*c^2*(d*Cos[a + b*x])^(7/2))","C",1
281,1,72,168,0.1715032,"\int \frac{(c \sin (a+b x))^{5/2}}{(d \cos (a+b x))^{11/2}} \, dx","Integrate[(c*Sin[a + b*x])^(5/2)/(d*Cos[a + b*x])^(11/2),x]","\frac{2 \cos ^5(a+b x) \sqrt[4]{\cos ^2(a+b x)} (c \sin (a+b x))^{7/2} \, _2F_1\left(\frac{7}{4},\frac{13}{4};\frac{11}{4};\sin ^2(a+b x)\right)}{7 b c (d \cos (a+b x))^{11/2}}","\frac{4 c^2 E\left(\left.a+b x-\frac{\pi }{4}\right|2\right) \sqrt{c \sin (a+b x)} \sqrt{d \cos (a+b x)}}{15 b d^6 \sqrt{\sin (2 a+2 b x)}}-\frac{4 c (c \sin (a+b x))^{3/2}}{15 b d^5 \sqrt{d \cos (a+b x)}}-\frac{2 c (c \sin (a+b x))^{3/2}}{15 b d^3 (d \cos (a+b x))^{5/2}}+\frac{2 c (c \sin (a+b x))^{3/2}}{9 b d (d \cos (a+b x))^{9/2}}",1,"(2*Cos[a + b*x]^5*(Cos[a + b*x]^2)^(1/4)*Hypergeometric2F1[7/4, 13/4, 11/4, Sin[a + b*x]^2]*(c*Sin[a + b*x])^(7/2))/(7*b*c*(d*Cos[a + b*x])^(11/2))","C",1
282,1,67,320,0.1167814,"\int \frac{(c \sin (a+b x))^{5/2}}{\sqrt{d \cos (a+b x)}} \, dx","Integrate[(c*Sin[a + b*x])^(5/2)/Sqrt[d*Cos[a + b*x]],x]","\frac{2 \cos ^2(a+b x)^{3/4} \tan (a+b x) (c \sin (a+b x))^{5/2} \, _2F_1\left(\frac{3}{4},\frac{7}{4};\frac{11}{4};\sin ^2(a+b x)\right)}{7 b \sqrt{d \cos (a+b x)}}","-\frac{3 c^{5/2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{d} \sqrt{c \sin (a+b x)}}{\sqrt{c} \sqrt{d \cos (a+b x)}}\right)}{4 \sqrt{2} b \sqrt{d}}+\frac{3 c^{5/2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{d} \sqrt{c \sin (a+b x)}}{\sqrt{c} \sqrt{d \cos (a+b x)}}+1\right)}{4 \sqrt{2} b \sqrt{d}}+\frac{3 c^{5/2} \log \left(-\frac{\sqrt{2} \sqrt{d} \sqrt{c \sin (a+b x)}}{\sqrt{d \cos (a+b x)}}+\sqrt{c} \tan (a+b x)+\sqrt{c}\right)}{8 \sqrt{2} b \sqrt{d}}-\frac{3 c^{5/2} \log \left(\frac{\sqrt{2} \sqrt{d} \sqrt{c \sin (a+b x)}}{\sqrt{d \cos (a+b x)}}+\sqrt{c} \tan (a+b x)+\sqrt{c}\right)}{8 \sqrt{2} b \sqrt{d}}-\frac{c (c \sin (a+b x))^{3/2} \sqrt{d \cos (a+b x)}}{2 b d}",1,"(2*(Cos[a + b*x]^2)^(3/4)*Hypergeometric2F1[3/4, 7/4, 11/4, Sin[a + b*x]^2]*(c*Sin[a + b*x])^(5/2)*Tan[a + b*x])/(7*b*Sqrt[d*Cos[a + b*x]])","C",1
283,1,67,315,0.123588,"\int \frac{(c \sin (a+b x))^{5/2}}{(d \cos (a+b x))^{5/2}} \, dx","Integrate[(c*Sin[a + b*x])^(5/2)/(d*Cos[a + b*x])^(5/2),x]","\frac{2 \cos ^2(a+b x)^{3/4} (c \sin (a+b x))^{7/2} \, _2F_1\left(\frac{7}{4},\frac{7}{4};\frac{11}{4};\sin ^2(a+b x)\right)}{7 b c d (d \cos (a+b x))^{3/2}}","\frac{c^{5/2} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{d} \sqrt{c \sin (a+b x)}}{\sqrt{c} \sqrt{d \cos (a+b x)}}\right)}{\sqrt{2} b d^{5/2}}-\frac{c^{5/2} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{d} \sqrt{c \sin (a+b x)}}{\sqrt{c} \sqrt{d \cos (a+b x)}}+1\right)}{\sqrt{2} b d^{5/2}}-\frac{c^{5/2} \log \left(-\frac{\sqrt{2} \sqrt{d} \sqrt{c \sin (a+b x)}}{\sqrt{d \cos (a+b x)}}+\sqrt{c} \tan (a+b x)+\sqrt{c}\right)}{2 \sqrt{2} b d^{5/2}}+\frac{c^{5/2} \log \left(\frac{\sqrt{2} \sqrt{d} \sqrt{c \sin (a+b x)}}{\sqrt{d \cos (a+b x)}}+\sqrt{c} \tan (a+b x)+\sqrt{c}\right)}{2 \sqrt{2} b d^{5/2}}+\frac{2 c (c \sin (a+b x))^{3/2}}{3 b d (d \cos (a+b x))^{3/2}}",1,"(2*(Cos[a + b*x]^2)^(3/4)*Hypergeometric2F1[7/4, 7/4, 11/4, Sin[a + b*x]^2]*(c*Sin[a + b*x])^(7/2))/(7*b*c*d*(d*Cos[a + b*x])^(3/2))","C",1
284,1,40,37,0.142979,"\int \frac{(c \sin (a+b x))^{5/2}}{(d \cos (a+b x))^{9/2}} \, dx","Integrate[(c*Sin[a + b*x])^(5/2)/(d*Cos[a + b*x])^(9/2),x]","\frac{2 \cot (a+b x) (c \sin (a+b x))^{9/2}}{7 b c^2 (d \cos (a+b x))^{9/2}}","\frac{2 (c \sin (a+b x))^{7/2}}{7 b c d (d \cos (a+b x))^{7/2}}",1,"(2*Cot[a + b*x]*(c*Sin[a + b*x])^(9/2))/(7*b*c^2*(d*Cos[a + b*x])^(9/2))","A",1
285,1,57,106,0.2863695,"\int \frac{(c \sin (a+b x))^{5/2}}{(d \cos (a+b x))^{13/2}} \, dx","Integrate[(c*Sin[a + b*x])^(5/2)/(d*Cos[a + b*x])^(13/2),x]","\frac{2 c^4 (2 \cos (2 (a+b x))+9) \tan ^5(a+b x)}{77 b d^6 (c \sin (a+b x))^{3/2} \sqrt{d \cos (a+b x)}}","-\frac{8 c (c \sin (a+b x))^{3/2}}{77 b d^5 (d \cos (a+b x))^{3/2}}-\frac{6 c (c \sin (a+b x))^{3/2}}{77 b d^3 (d \cos (a+b x))^{7/2}}+\frac{2 c (c \sin (a+b x))^{3/2}}{11 b d (d \cos (a+b x))^{11/2}}",1,"(2*c^4*(9 + 2*Cos[2*(a + b*x)])*Tan[a + b*x]^5)/(77*b*d^6*Sqrt[d*Cos[a + b*x]]*(c*Sin[a + b*x])^(3/2))","A",1
286,1,67,141,0.4673427,"\int \frac{(c \sin (a+b x))^{5/2}}{(d \cos (a+b x))^{17/2}} \, dx","Integrate[(c*Sin[a + b*x])^(5/2)/(d*Cos[a + b*x])^(17/2),x]","\frac{2 (44 \cos (2 (a+b x))+4 \cos (4 (a+b x))+117) \sec ^8(a+b x) (c \sin (a+b x))^{7/2} \sqrt{d \cos (a+b x)}}{1155 b c d^9}","-\frac{64 c (c \sin (a+b x))^{3/2}}{1155 b d^7 (d \cos (a+b x))^{3/2}}-\frac{16 c (c \sin (a+b x))^{3/2}}{385 b d^5 (d \cos (a+b x))^{7/2}}-\frac{2 c (c \sin (a+b x))^{3/2}}{55 b d^3 (d \cos (a+b x))^{11/2}}+\frac{2 c (c \sin (a+b x))^{3/2}}{15 b d (d \cos (a+b x))^{15/2}}",1,"(2*Sqrt[d*Cos[a + b*x]]*(117 + 44*Cos[2*(a + b*x)] + 4*Cos[4*(a + b*x)])*Sec[a + b*x]^8*(c*Sin[a + b*x])^(7/2))/(1155*b*c*d^9)","A",1
287,1,57,226,0.0547833,"\int \frac{\sin ^{\frac{7}{2}}(a+b x)}{\cos ^{\frac{7}{2}}(a+b x)} \, dx","Integrate[Sin[a + b*x]^(7/2)/Cos[a + b*x]^(7/2),x]","\frac{2 \sin ^{\frac{9}{2}}(a+b x) \sqrt[4]{\cos ^2(a+b x)} \, _2F_1\left(\frac{9}{4},\frac{9}{4};\frac{13}{4};\sin ^2(a+b x)\right)}{9 b \sqrt{\cos (a+b x)}}","\frac{2 \sin ^{\frac{5}{2}}(a+b x)}{5 b \cos ^{\frac{5}{2}}(a+b x)}-\frac{2 \sqrt{\sin (a+b x)}}{b \sqrt{\cos (a+b x)}}+\frac{\tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{\cos (a+b x)}}{\sqrt{\sin (a+b x)}}\right)}{\sqrt{2} b}-\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt{\cos (a+b x)}}{\sqrt{\sin (a+b x)}}+1\right)}{\sqrt{2} b}-\frac{\log \left(\cot (a+b x)-\frac{\sqrt{2} \sqrt{\cos (a+b x)}}{\sqrt{\sin (a+b x)}}+1\right)}{2 \sqrt{2} b}+\frac{\log \left(\cot (a+b x)+\frac{\sqrt{2} \sqrt{\cos (a+b x)}}{\sqrt{\sin (a+b x)}}+1\right)}{2 \sqrt{2} b}",1,"(2*(Cos[a + b*x]^2)^(1/4)*Hypergeometric2F1[9/4, 9/4, 13/4, Sin[a + b*x]^2]*Sin[a + b*x]^(9/2))/(9*b*Sqrt[Cos[a + b*x]])","C",1
288,1,16,16,0.0186096,"\int \frac{\sin ^{\frac{3}{2}}(x)}{\cos ^{\frac{7}{2}}(x)} \, dx","Integrate[Sin[x]^(3/2)/Cos[x]^(7/2),x]","\frac{2 \sin ^{\frac{5}{2}}(x)}{5 \cos ^{\frac{5}{2}}(x)}","\frac{2 \sin ^{\frac{5}{2}}(x)}{5 \cos ^{\frac{5}{2}}(x)}",1,"(2*Sin[x]^(5/2))/(5*Cos[x]^(5/2))","A",1
289,1,38,122,0.0130865,"\int \frac{\sqrt{\sin (x)}}{\sqrt{\cos (x)}} \, dx","Integrate[Sqrt[Sin[x]]/Sqrt[Cos[x]],x]","\frac{2 \sin ^{\frac{3}{2}}(x) \cos ^2(x)^{3/4} \, _2F_1\left(\frac{3}{4},\frac{3}{4};\frac{7}{4};\sin ^2(x)\right)}{3 \cos ^{\frac{3}{2}}(x)}","-\frac{\tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{\sin (x)}}{\sqrt{\cos (x)}}\right)}{\sqrt{2}}+\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt{\sin (x)}}{\sqrt{\cos (x)}}+1\right)}{\sqrt{2}}+\frac{\log \left(\tan (x)-\frac{\sqrt{2} \sqrt{\sin (x)}}{\sqrt{\cos (x)}}+1\right)}{2 \sqrt{2}}-\frac{\log \left(\tan (x)+\frac{\sqrt{2} \sqrt{\sin (x)}}{\sqrt{\cos (x)}}+1\right)}{2 \sqrt{2}}",1,"(2*(Cos[x]^2)^(3/4)*Hypergeometric2F1[3/4, 3/4, 7/4, Sin[x]^2]*Sin[x]^(3/2))/(3*Cos[x]^(3/2))","C",1
290,1,38,143,0.0114186,"\int \frac{\sin ^{\frac{5}{2}}(x)}{\sqrt{\cos (x)}} \, dx","Integrate[Sin[x]^(5/2)/Sqrt[Cos[x]],x]","\frac{2 \sin ^{\frac{7}{2}}(x) \cos ^2(x)^{3/4} \, _2F_1\left(\frac{3}{4},\frac{7}{4};\frac{11}{4};\sin ^2(x)\right)}{7 \cos ^{\frac{3}{2}}(x)}","-\frac{1}{2} \sin ^{\frac{3}{2}}(x) \sqrt{\cos (x)}-\frac{3 \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{\sin (x)}}{\sqrt{\cos (x)}}\right)}{4 \sqrt{2}}+\frac{3 \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{\sin (x)}}{\sqrt{\cos (x)}}+1\right)}{4 \sqrt{2}}+\frac{3 \log \left(\tan (x)-\frac{\sqrt{2} \sqrt{\sin (x)}}{\sqrt{\cos (x)}}+1\right)}{8 \sqrt{2}}-\frac{3 \log \left(\tan (x)+\frac{\sqrt{2} \sqrt{\sin (x)}}{\sqrt{\cos (x)}}+1\right)}{8 \sqrt{2}}",1,"(2*(Cos[x]^2)^(3/4)*Hypergeometric2F1[3/4, 7/4, 11/4, Sin[x]^2]*Sin[x]^(7/2))/(7*Cos[x]^(3/2))","C",1
291,1,70,132,0.1014405,"\int \frac{(d \cos (a+b x))^{7/2}}{\sqrt{c \sin (a+b x)}} \, dx","Integrate[(d*Cos[a + b*x])^(7/2)/Sqrt[c*Sin[a + b*x]],x]","\frac{2 \cos ^2(a+b x)^{3/4} \sec ^5(a+b x) \sqrt{c \sin (a+b x)} (d \cos (a+b x))^{7/2} \, _2F_1\left(-\frac{5}{4},\frac{1}{4};\frac{5}{4};\sin ^2(a+b x)\right)}{b c}","\frac{5 d^4 \sqrt{\sin (2 a+2 b x)} F\left(\left.a+b x-\frac{\pi }{4}\right|2\right)}{12 b \sqrt{c \sin (a+b x)} \sqrt{d \cos (a+b x)}}+\frac{5 d^3 \sqrt{c \sin (a+b x)} \sqrt{d \cos (a+b x)}}{6 b c}+\frac{d \sqrt{c \sin (a+b x)} (d \cos (a+b x))^{5/2}}{3 b c}",1,"(2*(d*Cos[a + b*x])^(7/2)*(Cos[a + b*x]^2)^(3/4)*Hypergeometric2F1[-5/4, 1/4, 5/4, Sin[a + b*x]^2]*Sec[a + b*x]^5*Sqrt[c*Sin[a + b*x]])/(b*c)","C",1
292,1,68,92,0.1065522,"\int \frac{(d \cos (a+b x))^{3/2}}{\sqrt{c \sin (a+b x)}} \, dx","Integrate[(d*Cos[a + b*x])^(3/2)/Sqrt[c*Sin[a + b*x]],x]","\frac{2 d^2 \cos ^2(a+b x)^{3/4} \tan (a+b x) \, _2F_1\left(-\frac{1}{4},\frac{1}{4};\frac{5}{4};\sin ^2(a+b x)\right)}{b \sqrt{c \sin (a+b x)} \sqrt{d \cos (a+b x)}}","\frac{d^2 \sqrt{\sin (2 a+2 b x)} F\left(\left.a+b x-\frac{\pi }{4}\right|2\right)}{2 b \sqrt{c \sin (a+b x)} \sqrt{d \cos (a+b x)}}+\frac{d \sqrt{c \sin (a+b x)} \sqrt{d \cos (a+b x)}}{b c}",1,"(2*d^2*(Cos[a + b*x]^2)^(3/4)*Hypergeometric2F1[-1/4, 1/4, 5/4, Sin[a + b*x]^2]*Tan[a + b*x])/(b*Sqrt[d*Cos[a + b*x]]*Sqrt[c*Sin[a + b*x]])","C",1
293,1,65,53,0.0576444,"\int \frac{1}{\sqrt{d \cos (a+b x)} \sqrt{c \sin (a+b x)}} \, dx","Integrate[1/(Sqrt[d*Cos[a + b*x]]*Sqrt[c*Sin[a + b*x]]),x]","\frac{2 \cos ^2(a+b x)^{3/4} \tan (a+b x) \, _2F_1\left(\frac{1}{4},\frac{3}{4};\frac{5}{4};\sin ^2(a+b x)\right)}{b \sqrt{c \sin (a+b x)} \sqrt{d \cos (a+b x)}}","\frac{\sqrt{\sin (2 a+2 b x)} F\left(\left.a+b x-\frac{\pi }{4}\right|2\right)}{b \sqrt{c \sin (a+b x)} \sqrt{d \cos (a+b x)}}",1,"(2*(Cos[a + b*x]^2)^(3/4)*Hypergeometric2F1[1/4, 3/4, 5/4, Sin[a + b*x]^2]*Tan[a + b*x])/(b*Sqrt[d*Cos[a + b*x]]*Sqrt[c*Sin[a + b*x]])","C",1
294,1,65,97,0.0990248,"\int \frac{1}{(d \cos (a+b x))^{5/2} \sqrt{c \sin (a+b x)}} \, dx","Integrate[1/((d*Cos[a + b*x])^(5/2)*Sqrt[c*Sin[a + b*x]]),x]","\frac{2 \cos ^2(a+b x)^{3/4} \sqrt{c \sin (a+b x)} \, _2F_1\left(\frac{1}{4},\frac{7}{4};\frac{5}{4};\sin ^2(a+b x)\right)}{b c d (d \cos (a+b x))^{3/2}}","\frac{2 \sqrt{\sin (2 a+2 b x)} F\left(\left.a+b x-\frac{\pi }{4}\right|2\right)}{3 b d^2 \sqrt{c \sin (a+b x)} \sqrt{d \cos (a+b x)}}+\frac{2 \sqrt{c \sin (a+b x)}}{3 b c d (d \cos (a+b x))^{3/2}}",1,"(2*(Cos[a + b*x]^2)^(3/4)*Hypergeometric2F1[1/4, 7/4, 5/4, Sin[a + b*x]^2]*Sqrt[c*Sin[a + b*x]])/(b*c*d*(d*Cos[a + b*x])^(3/2))","C",1
295,1,70,134,0.1175426,"\int \frac{1}{(d \cos (a+b x))^{9/2} \sqrt{c \sin (a+b x)}} \, dx","Integrate[1/((d*Cos[a + b*x])^(9/2)*Sqrt[c*Sin[a + b*x]]),x]","\frac{2 \cos ^3(a+b x) \cos ^2(a+b x)^{3/4} \sqrt{c \sin (a+b x)} \, _2F_1\left(\frac{1}{4},\frac{11}{4};\frac{5}{4};\sin ^2(a+b x)\right)}{b c (d \cos (a+b x))^{9/2}}","\frac{4 \sqrt{\sin (2 a+2 b x)} F\left(\left.a+b x-\frac{\pi }{4}\right|2\right)}{7 b d^4 \sqrt{c \sin (a+b x)} \sqrt{d \cos (a+b x)}}+\frac{4 \sqrt{c \sin (a+b x)}}{7 b c d^3 (d \cos (a+b x))^{3/2}}+\frac{2 \sqrt{c \sin (a+b x)}}{7 b c d (d \cos (a+b x))^{7/2}}",1,"(2*Cos[a + b*x]^3*(Cos[a + b*x]^2)^(3/4)*Hypergeometric2F1[1/4, 11/4, 5/4, Sin[a + b*x]^2]*Sqrt[c*Sin[a + b*x]])/(b*c*(d*Cos[a + b*x])^(9/2))","C",1
296,1,65,280,0.057623,"\int \frac{\sqrt{d \cos (a+b x)}}{\sqrt{c \sin (a+b x)}} \, dx","Integrate[Sqrt[d*Cos[a + b*x]]/Sqrt[c*Sin[a + b*x]],x]","\frac{2 \sqrt[4]{\cos ^2(a+b x)} \tan (a+b x) \sqrt{d \cos (a+b x)} \, _2F_1\left(\frac{1}{4},\frac{1}{4};\frac{5}{4};\sin ^2(a+b x)\right)}{b \sqrt{c \sin (a+b x)}}","\frac{\sqrt{d} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{c} \sqrt{d \cos (a+b x)}}{\sqrt{d} \sqrt{c \sin (a+b x)}}\right)}{\sqrt{2} b \sqrt{c}}-\frac{\sqrt{d} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{c} \sqrt{d \cos (a+b x)}}{\sqrt{d} \sqrt{c \sin (a+b x)}}+1\right)}{\sqrt{2} b \sqrt{c}}-\frac{\sqrt{d} \log \left(-\frac{\sqrt{2} \sqrt{c} \sqrt{d \cos (a+b x)}}{\sqrt{c \sin (a+b x)}}+\sqrt{d} \cot (a+b x)+\sqrt{d}\right)}{2 \sqrt{2} b \sqrt{c}}+\frac{\sqrt{d} \log \left(\frac{\sqrt{2} \sqrt{c} \sqrt{d \cos (a+b x)}}{\sqrt{c \sin (a+b x)}}+\sqrt{d} \cot (a+b x)+\sqrt{d}\right)}{2 \sqrt{2} b \sqrt{c}}",1,"(2*Sqrt[d*Cos[a + b*x]]*(Cos[a + b*x]^2)^(1/4)*Hypergeometric2F1[1/4, 1/4, 5/4, Sin[a + b*x]^2]*Tan[a + b*x])/(b*Sqrt[c*Sin[a + b*x]])","C",1
297,1,36,35,0.057551,"\int \frac{1}{(d \cos (a+b x))^{3/2} \sqrt{c \sin (a+b x)}} \, dx","Integrate[1/((d*Cos[a + b*x])^(3/2)*Sqrt[c*Sin[a + b*x]]),x]","\frac{\sin (2 (a+b x))}{b \sqrt{c \sin (a+b x)} (d \cos (a+b x))^{3/2}}","\frac{2 \sqrt{c \sin (a+b x)}}{b c d \sqrt{d \cos (a+b x)}}",1,"Sin[2*(a + b*x)]/(b*(d*Cos[a + b*x])^(3/2)*Sqrt[c*Sin[a + b*x]])","A",1
298,1,52,75,0.1493945,"\int \frac{1}{(d \cos (a+b x))^{7/2} \sqrt{c \sin (a+b x)}} \, dx","Integrate[1/((d*Cos[a + b*x])^(7/2)*Sqrt[c*Sin[a + b*x]]),x]","\frac{2 (2 \cos (2 (a+b x))+3) \tan (a+b x)}{5 b d^2 \sqrt{c \sin (a+b x)} (d \cos (a+b x))^{3/2}}","\frac{8 \sqrt{c \sin (a+b x)}}{5 b c d^3 \sqrt{d \cos (a+b x)}}+\frac{2 \sqrt{c \sin (a+b x)}}{5 b c d (d \cos (a+b x))^{5/2}}",1,"(2*(3 + 2*Cos[2*(a + b*x)])*Tan[a + b*x])/(5*b*d^2*(d*Cos[a + b*x])^(3/2)*Sqrt[c*Sin[a + b*x]])","A",1
299,1,67,112,0.2093516,"\int \frac{1}{(d \cos (a+b x))^{11/2} \sqrt{c \sin (a+b x)}} \, dx","Integrate[1/((d*Cos[a + b*x])^(11/2)*Sqrt[c*Sin[a + b*x]]),x]","\frac{2 (20 \cos (2 (a+b x))+4 \cos (4 (a+b x))+21) \sec ^5(a+b x) \sqrt{c \sin (a+b x)} \sqrt{d \cos (a+b x)}}{45 b c d^6}","\frac{64 \sqrt{c \sin (a+b x)}}{45 b c d^5 \sqrt{d \cos (a+b x)}}+\frac{16 \sqrt{c \sin (a+b x)}}{45 b c d^3 (d \cos (a+b x))^{5/2}}+\frac{2 \sqrt{c \sin (a+b x)}}{9 b c d (d \cos (a+b x))^{9/2}}",1,"(2*Sqrt[d*Cos[a + b*x]]*(21 + 20*Cos[2*(a + b*x)] + 4*Cos[4*(a + b*x)])*Sec[a + b*x]^5*Sqrt[c*Sin[a + b*x]])/(45*b*c*d^6)","A",1
300,1,55,174,0.0238652,"\int \frac{\sqrt{\cos (a+b x)}}{\sqrt{\sin (a+b x)}} \, dx","Integrate[Sqrt[Cos[a + b*x]]/Sqrt[Sin[a + b*x]],x]","\frac{2 \sqrt{\sin (a+b x)} \sqrt[4]{\cos ^2(a+b x)} \, _2F_1\left(\frac{1}{4},\frac{1}{4};\frac{5}{4};\sin ^2(a+b x)\right)}{b \sqrt{\cos (a+b x)}}","\frac{\tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{\cos (a+b x)}}{\sqrt{\sin (a+b x)}}\right)}{\sqrt{2} b}-\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt{\cos (a+b x)}}{\sqrt{\sin (a+b x)}}+1\right)}{\sqrt{2} b}-\frac{\log \left(\cot (a+b x)-\frac{\sqrt{2} \sqrt{\cos (a+b x)}}{\sqrt{\sin (a+b x)}}+1\right)}{2 \sqrt{2} b}+\frac{\log \left(\cot (a+b x)+\frac{\sqrt{2} \sqrt{\cos (a+b x)}}{\sqrt{\sin (a+b x)}}+1\right)}{2 \sqrt{2} b}",1,"(2*(Cos[a + b*x]^2)^(1/4)*Hypergeometric2F1[1/4, 1/4, 5/4, Sin[a + b*x]^2]*Sqrt[Sin[a + b*x]])/(b*Sqrt[Cos[a + b*x]])","C",1
301,1,55,199,0.0321708,"\int \frac{\cos ^{\frac{3}{2}}(a+b x)}{\sin ^{\frac{3}{2}}(a+b x)} \, dx","Integrate[Cos[a + b*x]^(3/2)/Sin[a + b*x]^(3/2),x]","-\frac{2 \cos ^2(a+b x)^{3/4} \, _2F_1\left(-\frac{1}{4},-\frac{1}{4};\frac{3}{4};\sin ^2(a+b x)\right)}{b \sqrt{\sin (a+b x)} \cos ^{\frac{3}{2}}(a+b x)}","-\frac{2 \sqrt{\cos (a+b x)}}{b \sqrt{\sin (a+b x)}}+\frac{\tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{\sin (a+b x)}}{\sqrt{\cos (a+b x)}}\right)}{\sqrt{2} b}-\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt{\sin (a+b x)}}{\sqrt{\cos (a+b x)}}+1\right)}{\sqrt{2} b}-\frac{\log \left(\tan (a+b x)-\frac{\sqrt{2} \sqrt{\sin (a+b x)}}{\sqrt{\cos (a+b x)}}+1\right)}{2 \sqrt{2} b}+\frac{\log \left(\tan (a+b x)+\frac{\sqrt{2} \sqrt{\sin (a+b x)}}{\sqrt{\cos (a+b x)}}+1\right)}{2 \sqrt{2} b}",1,"(-2*(Cos[a + b*x]^2)^(3/4)*Hypergeometric2F1[-1/4, -1/4, 3/4, Sin[a + b*x]^2])/(b*Cos[a + b*x]^(3/2)*Sqrt[Sin[a + b*x]])","C",1
302,1,57,201,0.030341,"\int \frac{\cos ^{\frac{5}{2}}(a+b x)}{\sin ^{\frac{5}{2}}(a+b x)} \, dx","Integrate[Cos[a + b*x]^(5/2)/Sin[a + b*x]^(5/2),x]","-\frac{2 \sqrt[4]{\cos ^2(a+b x)} \, _2F_1\left(-\frac{3}{4},-\frac{3}{4};\frac{1}{4};\sin ^2(a+b x)\right)}{3 b \sin ^{\frac{3}{2}}(a+b x) \sqrt{\cos (a+b x)}}","-\frac{2 \cos ^{\frac{3}{2}}(a+b x)}{3 b \sin ^{\frac{3}{2}}(a+b x)}-\frac{\tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{\cos (a+b x)}}{\sqrt{\sin (a+b x)}}\right)}{\sqrt{2} b}+\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt{\cos (a+b x)}}{\sqrt{\sin (a+b x)}}+1\right)}{\sqrt{2} b}+\frac{\log \left(\cot (a+b x)-\frac{\sqrt{2} \sqrt{\cos (a+b x)}}{\sqrt{\sin (a+b x)}}+1\right)}{2 \sqrt{2} b}-\frac{\log \left(\cot (a+b x)+\frac{\sqrt{2} \sqrt{\cos (a+b x)}}{\sqrt{\sin (a+b x)}}+1\right)}{2 \sqrt{2} b}",1,"(-2*(Cos[a + b*x]^2)^(1/4)*Hypergeometric2F1[-3/4, -3/4, 1/4, Sin[a + b*x]^2])/(3*b*Sqrt[Cos[a + b*x]]*Sin[a + b*x]^(3/2))","C",1
303,1,57,226,0.0416187,"\int \frac{\cos ^{\frac{7}{2}}(a+b x)}{\sin ^{\frac{7}{2}}(a+b x)} \, dx","Integrate[Cos[a + b*x]^(7/2)/Sin[a + b*x]^(7/2),x]","-\frac{2 \cos ^2(a+b x)^{3/4} \, _2F_1\left(-\frac{5}{4},-\frac{5}{4};-\frac{1}{4};\sin ^2(a+b x)\right)}{5 b \sin ^{\frac{5}{2}}(a+b x) \cos ^{\frac{3}{2}}(a+b x)}","-\frac{2 \cos ^{\frac{5}{2}}(a+b x)}{5 b \sin ^{\frac{5}{2}}(a+b x)}+\frac{2 \sqrt{\cos (a+b x)}}{b \sqrt{\sin (a+b x)}}-\frac{\tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{\sin (a+b x)}}{\sqrt{\cos (a+b x)}}\right)}{\sqrt{2} b}+\frac{\tan ^{-1}\left(\frac{\sqrt{2} \sqrt{\sin (a+b x)}}{\sqrt{\cos (a+b x)}}+1\right)}{\sqrt{2} b}+\frac{\log \left(\tan (a+b x)-\frac{\sqrt{2} \sqrt{\sin (a+b x)}}{\sqrt{\cos (a+b x)}}+1\right)}{2 \sqrt{2} b}-\frac{\log \left(\tan (a+b x)+\frac{\sqrt{2} \sqrt{\sin (a+b x)}}{\sqrt{\cos (a+b x)}}+1\right)}{2 \sqrt{2} b}",1,"(-2*(Cos[a + b*x]^2)^(3/4)*Hypergeometric2F1[-5/4, -5/4, -1/4, Sin[a + b*x]^2])/(5*b*Cos[a + b*x]^(3/2)*Sin[a + b*x]^(5/2))","C",1
304,1,55,58,0.0551971,"\int \cos ^4(e+f x) \sqrt[3]{b \sin (e+f x)} \, dx","Integrate[Cos[e + f*x]^4*(b*Sin[e + f*x])^(1/3),x]","\frac{3 \sqrt{\cos ^2(e+f x)} \tan (e+f x) \sqrt[3]{b \sin (e+f x)} \, _2F_1\left(-\frac{3}{2},\frac{2}{3};\frac{5}{3};\sin ^2(e+f x)\right)}{4 f}","\frac{3 \cos (e+f x) (b \sin (e+f x))^{4/3} \, _2F_1\left(-\frac{3}{2},\frac{2}{3};\frac{5}{3};\sin ^2(e+f x)\right)}{4 b f \sqrt{\cos ^2(e+f x)}}",1,"(3*Sqrt[Cos[e + f*x]^2]*Hypergeometric2F1[-3/2, 2/3, 5/3, Sin[e + f*x]^2]*(b*Sin[e + f*x])^(1/3)*Tan[e + f*x])/(4*f)","A",1
305,1,55,58,0.043892,"\int \cos ^2(e+f x) \sqrt[3]{b \sin (e+f x)} \, dx","Integrate[Cos[e + f*x]^2*(b*Sin[e + f*x])^(1/3),x]","\frac{3 \sqrt{\cos ^2(e+f x)} \tan (e+f x) \sqrt[3]{b \sin (e+f x)} \, _2F_1\left(-\frac{1}{2},\frac{2}{3};\frac{5}{3};\sin ^2(e+f x)\right)}{4 f}","\frac{3 \cos (e+f x) (b \sin (e+f x))^{4/3} \, _2F_1\left(-\frac{1}{2},\frac{2}{3};\frac{5}{3};\sin ^2(e+f x)\right)}{4 b f \sqrt{\cos ^2(e+f x)}}",1,"(3*Sqrt[Cos[e + f*x]^2]*Hypergeometric2F1[-1/2, 2/3, 5/3, Sin[e + f*x]^2]*(b*Sin[e + f*x])^(1/3)*Tan[e + f*x])/(4*f)","A",1
306,1,55,58,0.034782,"\int \sqrt[3]{b \sin (e+f x)} \, dx","Integrate[(b*Sin[e + f*x])^(1/3),x]","\frac{3 \sqrt{\cos ^2(e+f x)} \tan (e+f x) \sqrt[3]{b \sin (e+f x)} \, _2F_1\left(\frac{1}{2},\frac{2}{3};\frac{5}{3};\sin ^2(e+f x)\right)}{4 f}","\frac{3 \cos (e+f x) (b \sin (e+f x))^{4/3} \, _2F_1\left(\frac{1}{2},\frac{2}{3};\frac{5}{3};\sin ^2(e+f x)\right)}{4 b f \sqrt{\cos ^2(e+f x)}}",1,"(3*Sqrt[Cos[e + f*x]^2]*Hypergeometric2F1[1/2, 2/3, 5/3, Sin[e + f*x]^2]*(b*Sin[e + f*x])^(1/3)*Tan[e + f*x])/(4*f)","A",1
307,1,55,58,0.0410858,"\int \sec ^2(e+f x) \sqrt[3]{b \sin (e+f x)} \, dx","Integrate[Sec[e + f*x]^2*(b*Sin[e + f*x])^(1/3),x]","\frac{3 \sqrt{\cos ^2(e+f x)} \tan (e+f x) \sqrt[3]{b \sin (e+f x)} \, _2F_1\left(\frac{2}{3},\frac{3}{2};\frac{5}{3};\sin ^2(e+f x)\right)}{4 f}","\frac{3 \sqrt{\cos ^2(e+f x)} \sec (e+f x) (b \sin (e+f x))^{4/3} \, _2F_1\left(\frac{2}{3},\frac{3}{2};\frac{5}{3};\sin ^2(e+f x)\right)}{4 b f}",1,"(3*Sqrt[Cos[e + f*x]^2]*Hypergeometric2F1[2/3, 3/2, 5/3, Sin[e + f*x]^2]*(b*Sin[e + f*x])^(1/3)*Tan[e + f*x])/(4*f)","A",1
308,1,55,58,0.0382727,"\int \sec ^4(e+f x) \sqrt[3]{b \sin (e+f x)} \, dx","Integrate[Sec[e + f*x]^4*(b*Sin[e + f*x])^(1/3),x]","\frac{3 \sqrt{\cos ^2(e+f x)} \tan (e+f x) \sqrt[3]{b \sin (e+f x)} \, _2F_1\left(\frac{2}{3},\frac{5}{2};\frac{5}{3};\sin ^2(e+f x)\right)}{4 f}","\frac{3 \sqrt{\cos ^2(e+f x)} \sec (e+f x) (b \sin (e+f x))^{4/3} \, _2F_1\left(\frac{2}{3},\frac{5}{2};\frac{5}{3};\sin ^2(e+f x)\right)}{4 b f}",1,"(3*Sqrt[Cos[e + f*x]^2]*Hypergeometric2F1[2/3, 5/2, 5/3, Sin[e + f*x]^2]*(b*Sin[e + f*x])^(1/3)*Tan[e + f*x])/(4*f)","A",1
309,1,55,58,0.0542605,"\int \cos ^4(e+f x) (b \sin (e+f x))^{5/3} \, dx","Integrate[Cos[e + f*x]^4*(b*Sin[e + f*x])^(5/3),x]","\frac{3 \sqrt{\cos ^2(e+f x)} \tan (e+f x) (b \sin (e+f x))^{5/3} \, _2F_1\left(-\frac{3}{2},\frac{4}{3};\frac{7}{3};\sin ^2(e+f x)\right)}{8 f}","\frac{3 \cos (e+f x) (b \sin (e+f x))^{8/3} \, _2F_1\left(-\frac{3}{2},\frac{4}{3};\frac{7}{3};\sin ^2(e+f x)\right)}{8 b f \sqrt{\cos ^2(e+f x)}}",1,"(3*Sqrt[Cos[e + f*x]^2]*Hypergeometric2F1[-3/2, 4/3, 7/3, Sin[e + f*x]^2]*(b*Sin[e + f*x])^(5/3)*Tan[e + f*x])/(8*f)","A",1
310,1,55,58,0.0519377,"\int \cos ^2(e+f x) (b \sin (e+f x))^{5/3} \, dx","Integrate[Cos[e + f*x]^2*(b*Sin[e + f*x])^(5/3),x]","\frac{3 \sqrt{\cos ^2(e+f x)} \tan (e+f x) (b \sin (e+f x))^{5/3} \, _2F_1\left(-\frac{1}{2},\frac{4}{3};\frac{7}{3};\sin ^2(e+f x)\right)}{8 f}","\frac{3 \cos (e+f x) (b \sin (e+f x))^{8/3} \, _2F_1\left(-\frac{1}{2},\frac{4}{3};\frac{7}{3};\sin ^2(e+f x)\right)}{8 b f \sqrt{\cos ^2(e+f x)}}",1,"(3*Sqrt[Cos[e + f*x]^2]*Hypergeometric2F1[-1/2, 4/3, 7/3, Sin[e + f*x]^2]*(b*Sin[e + f*x])^(5/3)*Tan[e + f*x])/(8*f)","A",1
311,1,55,58,0.0422629,"\int (b \sin (e+f x))^{5/3} \, dx","Integrate[(b*Sin[e + f*x])^(5/3),x]","\frac{3 \sqrt{\cos ^2(e+f x)} \tan (e+f x) (b \sin (e+f x))^{5/3} \, _2F_1\left(\frac{1}{2},\frac{4}{3};\frac{7}{3};\sin ^2(e+f x)\right)}{8 f}","\frac{3 \cos (e+f x) (b \sin (e+f x))^{8/3} \, _2F_1\left(\frac{1}{2},\frac{4}{3};\frac{7}{3};\sin ^2(e+f x)\right)}{8 b f \sqrt{\cos ^2(e+f x)}}",1,"(3*Sqrt[Cos[e + f*x]^2]*Hypergeometric2F1[1/2, 4/3, 7/3, Sin[e + f*x]^2]*(b*Sin[e + f*x])^(5/3)*Tan[e + f*x])/(8*f)","A",1
312,1,55,58,0.0521246,"\int \sec ^2(e+f x) (b \sin (e+f x))^{5/3} \, dx","Integrate[Sec[e + f*x]^2*(b*Sin[e + f*x])^(5/3),x]","\frac{3 \sqrt{\cos ^2(e+f x)} \tan (e+f x) (b \sin (e+f x))^{5/3} \, _2F_1\left(\frac{4}{3},\frac{3}{2};\frac{7}{3};\sin ^2(e+f x)\right)}{8 f}","\frac{3 \sqrt{\cos ^2(e+f x)} \sec (e+f x) (b \sin (e+f x))^{8/3} \, _2F_1\left(\frac{4}{3},\frac{3}{2};\frac{7}{3};\sin ^2(e+f x)\right)}{8 b f}",1,"(3*Sqrt[Cos[e + f*x]^2]*Hypergeometric2F1[4/3, 3/2, 7/3, Sin[e + f*x]^2]*(b*Sin[e + f*x])^(5/3)*Tan[e + f*x])/(8*f)","A",1
313,1,55,58,0.0483096,"\int \sec ^4(e+f x) (b \sin (e+f x))^{5/3} \, dx","Integrate[Sec[e + f*x]^4*(b*Sin[e + f*x])^(5/3),x]","\frac{3 \sqrt{\cos ^2(e+f x)} \tan (e+f x) (b \sin (e+f x))^{5/3} \, _2F_1\left(\frac{4}{3},\frac{5}{2};\frac{7}{3};\sin ^2(e+f x)\right)}{8 f}","\frac{3 \sqrt{\cos ^2(e+f x)} \sec (e+f x) (b \sin (e+f x))^{8/3} \, _2F_1\left(\frac{4}{3},\frac{5}{2};\frac{7}{3};\sin ^2(e+f x)\right)}{8 b f}",1,"(3*Sqrt[Cos[e + f*x]^2]*Hypergeometric2F1[4/3, 5/2, 7/3, Sin[e + f*x]^2]*(b*Sin[e + f*x])^(5/3)*Tan[e + f*x])/(8*f)","A",1
314,1,55,58,0.0533283,"\int \frac{\cos ^4(e+f x)}{\sqrt[3]{b \sin (e+f x)}} \, dx","Integrate[Cos[e + f*x]^4/(b*Sin[e + f*x])^(1/3),x]","\frac{3 \sqrt{\cos ^2(e+f x)} \tan (e+f x) \, _2F_1\left(-\frac{3}{2},\frac{1}{3};\frac{4}{3};\sin ^2(e+f x)\right)}{2 f \sqrt[3]{b \sin (e+f x)}}","\frac{3 \cos (e+f x) (b \sin (e+f x))^{2/3} \, _2F_1\left(-\frac{3}{2},\frac{1}{3};\frac{4}{3};\sin ^2(e+f x)\right)}{2 b f \sqrt{\cos ^2(e+f x)}}",1,"(3*Sqrt[Cos[e + f*x]^2]*Hypergeometric2F1[-3/2, 1/3, 4/3, Sin[e + f*x]^2]*Tan[e + f*x])/(2*f*(b*Sin[e + f*x])^(1/3))","A",1
315,1,55,58,0.044709,"\int \frac{\cos ^2(e+f x)}{\sqrt[3]{b \sin (e+f x)}} \, dx","Integrate[Cos[e + f*x]^2/(b*Sin[e + f*x])^(1/3),x]","\frac{3 \sqrt{\cos ^2(e+f x)} \tan (e+f x) \, _2F_1\left(-\frac{1}{2},\frac{1}{3};\frac{4}{3};\sin ^2(e+f x)\right)}{2 f \sqrt[3]{b \sin (e+f x)}}","\frac{3 \cos (e+f x) (b \sin (e+f x))^{2/3} \, _2F_1\left(-\frac{1}{2},\frac{1}{3};\frac{4}{3};\sin ^2(e+f x)\right)}{2 b f \sqrt{\cos ^2(e+f x)}}",1,"(3*Sqrt[Cos[e + f*x]^2]*Hypergeometric2F1[-1/2, 1/3, 4/3, Sin[e + f*x]^2]*Tan[e + f*x])/(2*f*(b*Sin[e + f*x])^(1/3))","A",1
316,1,55,58,0.0351435,"\int \frac{1}{\sqrt[3]{b \sin (e+f x)}} \, dx","Integrate[(b*Sin[e + f*x])^(-1/3),x]","\frac{3 \sqrt{\cos ^2(e+f x)} \tan (e+f x) \, _2F_1\left(\frac{1}{3},\frac{1}{2};\frac{4}{3};\sin ^2(e+f x)\right)}{2 f \sqrt[3]{b \sin (e+f x)}}","\frac{3 \cos (e+f x) (b \sin (e+f x))^{2/3} \, _2F_1\left(\frac{1}{3},\frac{1}{2};\frac{4}{3};\sin ^2(e+f x)\right)}{2 b f \sqrt{\cos ^2(e+f x)}}",1,"(3*Sqrt[Cos[e + f*x]^2]*Hypergeometric2F1[1/3, 1/2, 4/3, Sin[e + f*x]^2]*Tan[e + f*x])/(2*f*(b*Sin[e + f*x])^(1/3))","A",1
317,1,55,58,0.0413571,"\int \frac{\sec ^2(e+f x)}{\sqrt[3]{b \sin (e+f x)}} \, dx","Integrate[Sec[e + f*x]^2/(b*Sin[e + f*x])^(1/3),x]","\frac{3 \sqrt{\cos ^2(e+f x)} \tan (e+f x) \, _2F_1\left(\frac{1}{3},\frac{3}{2};\frac{4}{3};\sin ^2(e+f x)\right)}{2 f \sqrt[3]{b \sin (e+f x)}}","\frac{3 \sqrt{\cos ^2(e+f x)} \sec (e+f x) (b \sin (e+f x))^{2/3} \, _2F_1\left(\frac{1}{3},\frac{3}{2};\frac{4}{3};\sin ^2(e+f x)\right)}{2 b f}",1,"(3*Sqrt[Cos[e + f*x]^2]*Hypergeometric2F1[1/3, 3/2, 4/3, Sin[e + f*x]^2]*Tan[e + f*x])/(2*f*(b*Sin[e + f*x])^(1/3))","A",1
318,1,55,58,0.0396839,"\int \frac{\sec ^4(e+f x)}{\sqrt[3]{b \sin (e+f x)}} \, dx","Integrate[Sec[e + f*x]^4/(b*Sin[e + f*x])^(1/3),x]","\frac{3 \sqrt{\cos ^2(e+f x)} \tan (e+f x) \, _2F_1\left(\frac{1}{3},\frac{5}{2};\frac{4}{3};\sin ^2(e+f x)\right)}{2 f \sqrt[3]{b \sin (e+f x)}}","\frac{3 \sqrt{\cos ^2(e+f x)} \sec (e+f x) (b \sin (e+f x))^{2/3} \, _2F_1\left(\frac{1}{3},\frac{5}{2};\frac{4}{3};\sin ^2(e+f x)\right)}{2 b f}",1,"(3*Sqrt[Cos[e + f*x]^2]*Hypergeometric2F1[1/3, 5/2, 4/3, Sin[e + f*x]^2]*Tan[e + f*x])/(2*f*(b*Sin[e + f*x])^(1/3))","A",1
319,1,55,58,0.0469179,"\int \frac{\cos ^4(e+f x)}{(b \sin (e+f x))^{5/3}} \, dx","Integrate[Cos[e + f*x]^4/(b*Sin[e + f*x])^(5/3),x]","-\frac{3 \sqrt{\cos ^2(e+f x)} \tan (e+f x) \, _2F_1\left(-\frac{3}{2},-\frac{1}{3};\frac{2}{3};\sin ^2(e+f x)\right)}{2 f (b \sin (e+f x))^{5/3}}","-\frac{3 \cos (e+f x) \, _2F_1\left(-\frac{3}{2},-\frac{1}{3};\frac{2}{3};\sin ^2(e+f x)\right)}{2 b f \sqrt{\cos ^2(e+f x)} (b \sin (e+f x))^{2/3}}",1,"(-3*Sqrt[Cos[e + f*x]^2]*Hypergeometric2F1[-3/2, -1/3, 2/3, Sin[e + f*x]^2]*Tan[e + f*x])/(2*f*(b*Sin[e + f*x])^(5/3))","A",1
320,1,55,58,0.0437547,"\int \frac{\cos ^2(e+f x)}{(b \sin (e+f x))^{5/3}} \, dx","Integrate[Cos[e + f*x]^2/(b*Sin[e + f*x])^(5/3),x]","-\frac{3 \sqrt{\cos ^2(e+f x)} \tan (e+f x) \, _2F_1\left(-\frac{1}{2},-\frac{1}{3};\frac{2}{3};\sin ^2(e+f x)\right)}{2 f (b \sin (e+f x))^{5/3}}","-\frac{3 \cos (e+f x) \, _2F_1\left(-\frac{1}{2},-\frac{1}{3};\frac{2}{3};\sin ^2(e+f x)\right)}{2 b f \sqrt{\cos ^2(e+f x)} (b \sin (e+f x))^{2/3}}",1,"(-3*Sqrt[Cos[e + f*x]^2]*Hypergeometric2F1[-1/2, -1/3, 2/3, Sin[e + f*x]^2]*Tan[e + f*x])/(2*f*(b*Sin[e + f*x])^(5/3))","A",1
321,1,55,58,0.0325334,"\int \frac{1}{(b \sin (e+f x))^{5/3}} \, dx","Integrate[(b*Sin[e + f*x])^(-5/3),x]","-\frac{3 \sqrt{\cos ^2(e+f x)} \tan (e+f x) \, _2F_1\left(-\frac{1}{3},\frac{1}{2};\frac{2}{3};\sin ^2(e+f x)\right)}{2 f (b \sin (e+f x))^{5/3}}","-\frac{3 \cos (e+f x) \, _2F_1\left(-\frac{1}{3},\frac{1}{2};\frac{2}{3};\sin ^2(e+f x)\right)}{2 b f \sqrt{\cos ^2(e+f x)} (b \sin (e+f x))^{2/3}}",1,"(-3*Sqrt[Cos[e + f*x]^2]*Hypergeometric2F1[-1/3, 1/2, 2/3, Sin[e + f*x]^2]*Tan[e + f*x])/(2*f*(b*Sin[e + f*x])^(5/3))","A",1
322,1,55,58,0.0435849,"\int \frac{\sec ^2(e+f x)}{(b \sin (e+f x))^{5/3}} \, dx","Integrate[Sec[e + f*x]^2/(b*Sin[e + f*x])^(5/3),x]","-\frac{3 \sqrt{\cos ^2(e+f x)} \tan (e+f x) \, _2F_1\left(-\frac{1}{3},\frac{3}{2};\frac{2}{3};\sin ^2(e+f x)\right)}{2 f (b \sin (e+f x))^{5/3}}","-\frac{3 \sqrt{\cos ^2(e+f x)} \sec (e+f x) \, _2F_1\left(-\frac{1}{3},\frac{3}{2};\frac{2}{3};\sin ^2(e+f x)\right)}{2 b f (b \sin (e+f x))^{2/3}}",1,"(-3*Sqrt[Cos[e + f*x]^2]*Hypergeometric2F1[-1/3, 3/2, 2/3, Sin[e + f*x]^2]*Tan[e + f*x])/(2*f*(b*Sin[e + f*x])^(5/3))","A",1
323,1,55,58,0.0392916,"\int \frac{\sec ^4(e+f x)}{(b \sin (e+f x))^{5/3}} \, dx","Integrate[Sec[e + f*x]^4/(b*Sin[e + f*x])^(5/3),x]","-\frac{3 \sqrt{\cos ^2(e+f x)} \tan (e+f x) \, _2F_1\left(-\frac{1}{3},\frac{5}{2};\frac{2}{3};\sin ^2(e+f x)\right)}{2 f (b \sin (e+f x))^{5/3}}","-\frac{3 \sqrt{\cos ^2(e+f x)} \sec (e+f x) \, _2F_1\left(-\frac{1}{3},\frac{5}{2};\frac{2}{3};\sin ^2(e+f x)\right)}{2 b f (b \sin (e+f x))^{2/3}}",1,"(-3*Sqrt[Cos[e + f*x]^2]*Hypergeometric2F1[-1/3, 5/2, 2/3, Sin[e + f*x]^2]*Tan[e + f*x])/(2*f*(b*Sin[e + f*x])^(5/3))","A",1
324,1,57,128,0.0441283,"\int \frac{\sqrt[3]{\sin (a+b x)}}{\sqrt[3]{\cos (a+b x)}} \, dx","Integrate[Sin[a + b*x]^(1/3)/Cos[a + b*x]^(1/3),x]","\frac{3 \sin ^{\frac{4}{3}}(a+b x) \cos ^2(a+b x)^{2/3} \, _2F_1\left(\frac{2}{3},\frac{2}{3};\frac{5}{3};\sin ^2(a+b x)\right)}{4 b \cos ^{\frac{4}{3}}(a+b x)}","-\frac{\log \left(\frac{\sin ^{\frac{2}{3}}(a+b x)}{\cos ^{\frac{2}{3}}(a+b x)}+1\right)}{2 b}+\frac{\log \left(\frac{\sin ^{\frac{4}{3}}(a+b x)}{\cos ^{\frac{4}{3}}(a+b x)}-\frac{\sin ^{\frac{2}{3}}(a+b x)}{\cos ^{\frac{2}{3}}(a+b x)}+1\right)}{4 b}-\frac{\sqrt{3} \tan ^{-1}\left(\frac{1-\frac{2 \sin ^{\frac{2}{3}}(a+b x)}{\cos ^{\frac{2}{3}}(a+b x)}}{\sqrt{3}}\right)}{2 b}",1,"(3*(Cos[a + b*x]^2)^(2/3)*Hypergeometric2F1[2/3, 2/3, 5/3, Sin[a + b*x]^2]*Sin[a + b*x]^(4/3))/(4*b*Cos[a + b*x]^(4/3))","C",1
325,1,57,224,0.0428437,"\int \frac{\sin ^{\frac{2}{3}}(a+b x)}{\cos ^{\frac{2}{3}}(a+b x)} \, dx","Integrate[Sin[a + b*x]^(2/3)/Cos[a + b*x]^(2/3),x]","\frac{3 \sin ^{\frac{5}{3}}(a+b x) \cos ^2(a+b x)^{5/6} \, _2F_1\left(\frac{5}{6},\frac{5}{6};\frac{11}{6};\sin ^2(a+b x)\right)}{5 b \cos ^{\frac{5}{3}}(a+b x)}","\frac{\sqrt{3} \log \left(\frac{\sin ^{\frac{2}{3}}(a+b x)}{\cos ^{\frac{2}{3}}(a+b x)}-\frac{\sqrt{3} \sqrt[3]{\sin (a+b x)}}{\sqrt[3]{\cos (a+b x)}}+1\right)}{4 b}-\frac{\sqrt{3} \log \left(\frac{\sin ^{\frac{2}{3}}(a+b x)}{\cos ^{\frac{2}{3}}(a+b x)}+\frac{\sqrt{3} \sqrt[3]{\sin (a+b x)}}{\sqrt[3]{\cos (a+b x)}}+1\right)}{4 b}-\frac{\tan ^{-1}\left(\sqrt{3}-\frac{2 \sqrt[3]{\sin (a+b x)}}{\sqrt[3]{\cos (a+b x)}}\right)}{2 b}+\frac{\tan ^{-1}\left(\frac{2 \sqrt[3]{\sin (a+b x)}}{\sqrt[3]{\cos (a+b x)}}+\sqrt{3}\right)}{2 b}+\frac{\tan ^{-1}\left(\frac{\sqrt[3]{\sin (a+b x)}}{\sqrt[3]{\cos (a+b x)}}\right)}{b}",1,"(3*(Cos[a + b*x]^2)^(5/6)*Hypergeometric2F1[5/6, 5/6, 11/6, Sin[a + b*x]^2]*Sin[a + b*x]^(5/3))/(5*b*Cos[a + b*x]^(5/3))","C",1
326,1,57,249,0.0542662,"\int \frac{\sin ^{\frac{4}{3}}(a+b x)}{\cos ^{\frac{4}{3}}(a+b x)} \, dx","Integrate[Sin[a + b*x]^(4/3)/Cos[a + b*x]^(4/3),x]","\frac{3 \sin ^{\frac{7}{3}}(a+b x) \sqrt[6]{\cos ^2(a+b x)} \, _2F_1\left(\frac{7}{6},\frac{7}{6};\frac{13}{6};\sin ^2(a+b x)\right)}{7 b \sqrt[3]{\cos (a+b x)}}","\frac{3 \sqrt[3]{\sin (a+b x)}}{b \sqrt[3]{\cos (a+b x)}}+\frac{\sqrt{3} \log \left(\frac{\cos ^{\frac{2}{3}}(a+b x)}{\sin ^{\frac{2}{3}}(a+b x)}-\frac{\sqrt{3} \sqrt[3]{\cos (a+b x)}}{\sqrt[3]{\sin (a+b x)}}+1\right)}{4 b}-\frac{\sqrt{3} \log \left(\frac{\cos ^{\frac{2}{3}}(a+b x)}{\sin ^{\frac{2}{3}}(a+b x)}+\frac{\sqrt{3} \sqrt[3]{\cos (a+b x)}}{\sqrt[3]{\sin (a+b x)}}+1\right)}{4 b}-\frac{\tan ^{-1}\left(\sqrt{3}-\frac{2 \sqrt[3]{\cos (a+b x)}}{\sqrt[3]{\sin (a+b x)}}\right)}{2 b}+\frac{\tan ^{-1}\left(\frac{2 \sqrt[3]{\cos (a+b x)}}{\sqrt[3]{\sin (a+b x)}}+\sqrt{3}\right)}{2 b}+\frac{\tan ^{-1}\left(\frac{\sqrt[3]{\cos (a+b x)}}{\sqrt[3]{\sin (a+b x)}}\right)}{b}",1,"(3*(Cos[a + b*x]^2)^(1/6)*Hypergeometric2F1[7/6, 7/6, 13/6, Sin[a + b*x]^2]*Sin[a + b*x]^(7/3))/(7*b*Cos[a + b*x]^(1/3))","C",1
327,1,57,155,0.0575022,"\int \frac{\sin ^{\frac{5}{3}}(a+b x)}{\cos ^{\frac{5}{3}}(a+b x)} \, dx","Integrate[Sin[a + b*x]^(5/3)/Cos[a + b*x]^(5/3),x]","\frac{3 \sin ^{\frac{8}{3}}(a+b x) \sqrt[3]{\cos ^2(a+b x)} \, _2F_1\left(\frac{4}{3},\frac{4}{3};\frac{7}{3};\sin ^2(a+b x)\right)}{8 b \cos ^{\frac{2}{3}}(a+b x)}","\frac{3 \sin ^{\frac{2}{3}}(a+b x)}{2 b \cos ^{\frac{2}{3}}(a+b x)}+\frac{\log \left(\frac{\cos ^{\frac{4}{3}}(a+b x)}{\sin ^{\frac{4}{3}}(a+b x)}-\frac{\cos ^{\frac{2}{3}}(a+b x)}{\sin ^{\frac{2}{3}}(a+b x)}+1\right)}{4 b}-\frac{\log \left(\frac{\cos ^{\frac{2}{3}}(a+b x)}{\sin ^{\frac{2}{3}}(a+b x)}+1\right)}{2 b}-\frac{\sqrt{3} \tan ^{-1}\left(\frac{1-\frac{2 \cos ^{\frac{2}{3}}(a+b x)}{\sin ^{\frac{2}{3}}(a+b x)}}{\sqrt{3}}\right)}{2 b}",1,"(3*(Cos[a + b*x]^2)^(1/3)*Hypergeometric2F1[4/3, 4/3, 7/3, Sin[a + b*x]^2]*Sin[a + b*x]^(8/3))/(8*b*Cos[a + b*x]^(2/3))","C",1
328,1,57,155,0.0525734,"\int \frac{\sin ^{\frac{7}{3}}(a+b x)}{\cos ^{\frac{7}{3}}(a+b x)} \, dx","Integrate[Sin[a + b*x]^(7/3)/Cos[a + b*x]^(7/3),x]","\frac{3 \sin ^{\frac{10}{3}}(a+b x) \cos ^2(a+b x)^{2/3} \, _2F_1\left(\frac{5}{3},\frac{5}{3};\frac{8}{3};\sin ^2(a+b x)\right)}{10 b \cos ^{\frac{4}{3}}(a+b x)}","\frac{3 \sin ^{\frac{4}{3}}(a+b x)}{4 b \cos ^{\frac{4}{3}}(a+b x)}+\frac{\log \left(\frac{\sin ^{\frac{2}{3}}(a+b x)}{\cos ^{\frac{2}{3}}(a+b x)}+1\right)}{2 b}-\frac{\log \left(\frac{\sin ^{\frac{4}{3}}(a+b x)}{\cos ^{\frac{4}{3}}(a+b x)}-\frac{\sin ^{\frac{2}{3}}(a+b x)}{\cos ^{\frac{2}{3}}(a+b x)}+1\right)}{4 b}+\frac{\sqrt{3} \tan ^{-1}\left(\frac{1-\frac{2 \sin ^{\frac{2}{3}}(a+b x)}{\cos ^{\frac{2}{3}}(a+b x)}}{\sqrt{3}}\right)}{2 b}",1,"(3*(Cos[a + b*x]^2)^(2/3)*Hypergeometric2F1[5/3, 5/3, 8/3, Sin[a + b*x]^2]*Sin[a + b*x]^(10/3))/(10*b*Cos[a + b*x]^(4/3))","C",1
329,1,57,128,0.0268032,"\int \frac{\sqrt[3]{\cos (a+b x)}}{\sqrt[3]{\sin (a+b x)}} \, dx","Integrate[Cos[a + b*x]^(1/3)/Sin[a + b*x]^(1/3),x]","\frac{3 \sin ^{\frac{2}{3}}(a+b x) \sqrt[3]{\cos ^2(a+b x)} \, _2F_1\left(\frac{1}{3},\frac{1}{3};\frac{4}{3};\sin ^2(a+b x)\right)}{2 b \cos ^{\frac{2}{3}}(a+b x)}","-\frac{\log \left(\frac{\cos ^{\frac{4}{3}}(a+b x)}{\sin ^{\frac{4}{3}}(a+b x)}-\frac{\cos ^{\frac{2}{3}}(a+b x)}{\sin ^{\frac{2}{3}}(a+b x)}+1\right)}{4 b}+\frac{\log \left(\frac{\cos ^{\frac{2}{3}}(a+b x)}{\sin ^{\frac{2}{3}}(a+b x)}+1\right)}{2 b}+\frac{\sqrt{3} \tan ^{-1}\left(\frac{1-\frac{2 \cos ^{\frac{2}{3}}(a+b x)}{\sin ^{\frac{2}{3}}(a+b x)}}{\sqrt{3}}\right)}{2 b}",1,"(3*(Cos[a + b*x]^2)^(1/3)*Hypergeometric2F1[1/3, 1/3, 4/3, Sin[a + b*x]^2]*Sin[a + b*x]^(2/3))/(2*b*Cos[a + b*x]^(2/3))","C",1
330,1,55,225,0.0277593,"\int \frac{\cos ^{\frac{2}{3}}(a+b x)}{\sin ^{\frac{2}{3}}(a+b x)} \, dx","Integrate[Cos[a + b*x]^(2/3)/Sin[a + b*x]^(2/3),x]","\frac{3 \sqrt[3]{\sin (a+b x)} \sqrt[6]{\cos ^2(a+b x)} \, _2F_1\left(\frac{1}{6},\frac{1}{6};\frac{7}{6};\sin ^2(a+b x)\right)}{b \sqrt[3]{\cos (a+b x)}}","-\frac{\sqrt{3} \log \left(\frac{\cos ^{\frac{2}{3}}(a+b x)}{\sin ^{\frac{2}{3}}(a+b x)}-\frac{\sqrt{3} \sqrt[3]{\cos (a+b x)}}{\sqrt[3]{\sin (a+b x)}}+1\right)}{4 b}+\frac{\sqrt{3} \log \left(\frac{\cos ^{\frac{2}{3}}(a+b x)}{\sin ^{\frac{2}{3}}(a+b x)}+\frac{\sqrt{3} \sqrt[3]{\cos (a+b x)}}{\sqrt[3]{\sin (a+b x)}}+1\right)}{4 b}+\frac{\tan ^{-1}\left(\sqrt{3}-\frac{2 \sqrt[3]{\cos (a+b x)}}{\sqrt[3]{\sin (a+b x)}}\right)}{2 b}-\frac{\tan ^{-1}\left(\frac{2 \sqrt[3]{\cos (a+b x)}}{\sqrt[3]{\sin (a+b x)}}+\sqrt{3}\right)}{2 b}-\frac{\tan ^{-1}\left(\frac{\sqrt[3]{\cos (a+b x)}}{\sqrt[3]{\sin (a+b x)}}\right)}{b}",1,"(3*(Cos[a + b*x]^2)^(1/6)*Hypergeometric2F1[1/6, 1/6, 7/6, Sin[a + b*x]^2]*Sin[a + b*x]^(1/3))/(b*Cos[a + b*x]^(1/3))","C",1
331,1,55,250,0.0299168,"\int \frac{\cos ^{\frac{4}{3}}(a+b x)}{\sin ^{\frac{4}{3}}(a+b x)} \, dx","Integrate[Cos[a + b*x]^(4/3)/Sin[a + b*x]^(4/3),x]","-\frac{3 \cos ^2(a+b x)^{5/6} \, _2F_1\left(-\frac{1}{6},-\frac{1}{6};\frac{5}{6};\sin ^2(a+b x)\right)}{b \sqrt[3]{\sin (a+b x)} \cos ^{\frac{5}{3}}(a+b x)}","-\frac{3 \sqrt[3]{\cos (a+b x)}}{b \sqrt[3]{\sin (a+b x)}}-\frac{\sqrt{3} \log \left(\frac{\sin ^{\frac{2}{3}}(a+b x)}{\cos ^{\frac{2}{3}}(a+b x)}-\frac{\sqrt{3} \sqrt[3]{\sin (a+b x)}}{\sqrt[3]{\cos (a+b x)}}+1\right)}{4 b}+\frac{\sqrt{3} \log \left(\frac{\sin ^{\frac{2}{3}}(a+b x)}{\cos ^{\frac{2}{3}}(a+b x)}+\frac{\sqrt{3} \sqrt[3]{\sin (a+b x)}}{\sqrt[3]{\cos (a+b x)}}+1\right)}{4 b}+\frac{\tan ^{-1}\left(\sqrt{3}-\frac{2 \sqrt[3]{\sin (a+b x)}}{\sqrt[3]{\cos (a+b x)}}\right)}{2 b}-\frac{\tan ^{-1}\left(\frac{2 \sqrt[3]{\sin (a+b x)}}{\sqrt[3]{\cos (a+b x)}}+\sqrt{3}\right)}{2 b}-\frac{\tan ^{-1}\left(\frac{\sqrt[3]{\sin (a+b x)}}{\sqrt[3]{\cos (a+b x)}}\right)}{b}",1,"(-3*(Cos[a + b*x]^2)^(5/6)*Hypergeometric2F1[-1/6, -1/6, 5/6, Sin[a + b*x]^2])/(b*Cos[a + b*x]^(5/3)*Sin[a + b*x]^(1/3))","C",1
332,1,57,155,0.0328145,"\int \frac{\cos ^{\frac{5}{3}}(a+b x)}{\sin ^{\frac{5}{3}}(a+b x)} \, dx","Integrate[Cos[a + b*x]^(5/3)/Sin[a + b*x]^(5/3),x]","-\frac{3 \cos ^2(a+b x)^{2/3} \, _2F_1\left(-\frac{1}{3},-\frac{1}{3};\frac{2}{3};\sin ^2(a+b x)\right)}{2 b \sin ^{\frac{2}{3}}(a+b x) \cos ^{\frac{4}{3}}(a+b x)}","-\frac{3 \cos ^{\frac{2}{3}}(a+b x)}{2 b \sin ^{\frac{2}{3}}(a+b x)}+\frac{\log \left(\frac{\sin ^{\frac{2}{3}}(a+b x)}{\cos ^{\frac{2}{3}}(a+b x)}+1\right)}{2 b}-\frac{\log \left(\frac{\sin ^{\frac{4}{3}}(a+b x)}{\cos ^{\frac{4}{3}}(a+b x)}-\frac{\sin ^{\frac{2}{3}}(a+b x)}{\cos ^{\frac{2}{3}}(a+b x)}+1\right)}{4 b}+\frac{\sqrt{3} \tan ^{-1}\left(\frac{1-\frac{2 \sin ^{\frac{2}{3}}(a+b x)}{\cos ^{\frac{2}{3}}(a+b x)}}{\sqrt{3}}\right)}{2 b}",1,"(-3*(Cos[a + b*x]^2)^(2/3)*Hypergeometric2F1[-1/3, -1/3, 2/3, Sin[a + b*x]^2])/(2*b*Cos[a + b*x]^(4/3)*Sin[a + b*x]^(2/3))","C",1
333,1,57,155,0.0338068,"\int \frac{\cos ^{\frac{7}{3}}(a+b x)}{\sin ^{\frac{7}{3}}(a+b x)} \, dx","Integrate[Cos[a + b*x]^(7/3)/Sin[a + b*x]^(7/3),x]","-\frac{3 \sqrt[3]{\cos ^2(a+b x)} \, _2F_1\left(-\frac{2}{3},-\frac{2}{3};\frac{1}{3};\sin ^2(a+b x)\right)}{4 b \sin ^{\frac{4}{3}}(a+b x) \cos ^{\frac{2}{3}}(a+b x)}","-\frac{3 \cos ^{\frac{4}{3}}(a+b x)}{4 b \sin ^{\frac{4}{3}}(a+b x)}+\frac{\log \left(\frac{\cos ^{\frac{4}{3}}(a+b x)}{\sin ^{\frac{4}{3}}(a+b x)}-\frac{\cos ^{\frac{2}{3}}(a+b x)}{\sin ^{\frac{2}{3}}(a+b x)}+1\right)}{4 b}-\frac{\log \left(\frac{\cos ^{\frac{2}{3}}(a+b x)}{\sin ^{\frac{2}{3}}(a+b x)}+1\right)}{2 b}-\frac{\sqrt{3} \tan ^{-1}\left(\frac{1-\frac{2 \cos ^{\frac{2}{3}}(a+b x)}{\sin ^{\frac{2}{3}}(a+b x)}}{\sqrt{3}}\right)}{2 b}",1,"(-3*(Cos[a + b*x]^2)^(1/3)*Hypergeometric2F1[-2/3, -2/3, 1/3, Sin[a + b*x]^2])/(4*b*Cos[a + b*x]^(2/3)*Sin[a + b*x]^(4/3))","C",1
334,1,16,16,0.0107815,"\int \frac{\cos ^{\frac{2}{3}}(x)}{\sin ^{\frac{8}{3}}(x)} \, dx","Integrate[Cos[x]^(2/3)/Sin[x]^(8/3),x]","-\frac{3 \cos ^{\frac{5}{3}}(x)}{5 \sin ^{\frac{5}{3}}(x)}","-\frac{3 \cos ^{\frac{5}{3}}(x)}{5 \sin ^{\frac{5}{3}}(x)}",1,"(-3*Cos[x]^(5/3))/(5*Sin[x]^(5/3))","A",1
335,1,16,16,0.0155162,"\int \frac{\sin ^{\frac{2}{3}}(x)}{\cos ^{\frac{8}{3}}(x)} \, dx","Integrate[Sin[x]^(2/3)/Cos[x]^(8/3),x]","\frac{3 \sin ^{\frac{5}{3}}(x)}{5 \cos ^{\frac{5}{3}}(x)}","\frac{3 \sin ^{\frac{5}{3}}(x)}{5 \cos ^{\frac{5}{3}}(x)}",1,"(3*Sin[x]^(5/3))/(5*Cos[x]^(5/3))","A",1
336,1,79,80,0.1054058,"\int \cos ^n(e+f x) \sin ^m(e+f x) \, dx","Integrate[Cos[e + f*x]^n*Sin[e + f*x]^m,x]","\frac{\sin ^{m+1}(e+f x) \cos ^{n-1}(e+f x) \cos ^2(e+f x)^{\frac{1-n}{2}} \, _2F_1\left(\frac{m+1}{2},\frac{1-n}{2};\frac{m+3}{2};\sin ^2(e+f x)\right)}{f (m+1)}","-\frac{\sin ^{m-1}(e+f x) \sin ^2(e+f x)^{\frac{1-m}{2}} \cos ^{n+1}(e+f x) \, _2F_1\left(\frac{1-m}{2},\frac{n+1}{2};\frac{n+3}{2};\cos ^2(e+f x)\right)}{f (n+1)}",1,"(Cos[e + f*x]^(-1 + n)*(Cos[e + f*x]^2)^((1 - n)/2)*Hypergeometric2F1[(1 + m)/2, (1 - n)/2, (3 + m)/2, Sin[e + f*x]^2]*Sin[e + f*x]^(1 + m))/(f*(1 + m))","A",1
337,1,82,85,0.116114,"\int (d \cos (e+f x))^n \sin ^m(e+f x) \, dx","Integrate[(d*Cos[e + f*x])^n*Sin[e + f*x]^m,x]","\frac{d \sin ^{m+1}(e+f x) \cos ^2(e+f x)^{\frac{1-n}{2}} (d \cos (e+f x))^{n-1} \, _2F_1\left(\frac{m+1}{2},\frac{1-n}{2};\frac{m+3}{2};\sin ^2(e+f x)\right)}{f (m+1)}","-\frac{\sin ^{m-1}(e+f x) \sin ^2(e+f x)^{\frac{1-m}{2}} (d \cos (e+f x))^{n+1} \, _2F_1\left(\frac{1-m}{2},\frac{n+1}{2};\frac{n+3}{2};\cos ^2(e+f x)\right)}{d f (n+1)}",1,"(d*(d*Cos[e + f*x])^(-1 + n)*(Cos[e + f*x]^2)^((1 - n)/2)*Hypergeometric2F1[(1 + m)/2, (1 - n)/2, (3 + m)/2, Sin[e + f*x]^2]*Sin[e + f*x]^(1 + m))/(f*(1 + m))","A",1
338,1,85,83,0.0741643,"\int \cos ^n(e+f x) (b \sin (e+f x))^m \, dx","Integrate[Cos[e + f*x]^n*(b*Sin[e + f*x])^m,x]","\frac{\sin (e+f x) \cos ^{n-1}(e+f x) \cos ^2(e+f x)^{\frac{1-n}{2}} (b \sin (e+f x))^m \, _2F_1\left(\frac{m+1}{2},\frac{1-n}{2};\frac{m+3}{2};\sin ^2(e+f x)\right)}{f (m+1)}","-\frac{b \sin ^2(e+f x)^{\frac{1-m}{2}} \cos ^{n+1}(e+f x) (b \sin (e+f x))^{m-1} \, _2F_1\left(\frac{1-m}{2},\frac{n+1}{2};\frac{n+3}{2};\cos ^2(e+f x)\right)}{f (n+1)}",1,"(Cos[e + f*x]^(-1 + n)*(Cos[e + f*x]^2)^((1 - n)/2)*Hypergeometric2F1[(1 + m)/2, (1 - n)/2, (3 + m)/2, Sin[e + f*x]^2]*Sin[e + f*x]*(b*Sin[e + f*x])^m)/(f*(1 + m))","A",1
339,1,85,88,0.0945822,"\int (d \cos (e+f x))^n (b \sin (e+f x))^m \, dx","Integrate[(d*Cos[e + f*x])^n*(b*Sin[e + f*x])^m,x]","\frac{\tan (e+f x) \cos ^2(e+f x)^{\frac{1-n}{2}} (b \sin (e+f x))^m (d \cos (e+f x))^n \, _2F_1\left(\frac{m+1}{2},\frac{1-n}{2};\frac{m+3}{2};\sin ^2(e+f x)\right)}{f (m+1)}","-\frac{b \sin ^2(e+f x)^{\frac{1-m}{2}} (b \sin (e+f x))^{m-1} (d \cos (e+f x))^{n+1} \, _2F_1\left(\frac{1-m}{2},\frac{n+1}{2};\frac{n+3}{2};\cos ^2(e+f x)\right)}{d f (n+1)}",1,"((d*Cos[e + f*x])^n*(Cos[e + f*x]^2)^((1 - n)/2)*Hypergeometric2F1[(1 + m)/2, (1 - n)/2, (3 + m)/2, Sin[e + f*x]^2]*(b*Sin[e + f*x])^m*Tan[e + f*x])/(f*(1 + m))","A",1
340,1,55,74,0.3123957,"\int \cos ^5(a+b x) (c \sin (a+b x))^m \, dx","Integrate[Cos[a + b*x]^5*(c*Sin[a + b*x])^m,x]","\frac{\sin (a+b x) \left(\frac{\sin ^4(a+b x)}{m+5}-\frac{2 \sin ^2(a+b x)}{m+3}+\frac{1}{m+1}\right) (c \sin (a+b x))^m}{b}","\frac{(c \sin (a+b x))^{m+5}}{b c^5 (m+5)}-\frac{2 (c \sin (a+b x))^{m+3}}{b c^3 (m+3)}+\frac{(c \sin (a+b x))^{m+1}}{b c (m+1)}",1,"(Sin[a + b*x]*(c*Sin[a + b*x])^m*((1 + m)^(-1) - (2*Sin[a + b*x]^2)/(3 + m) + Sin[a + b*x]^4/(5 + m)))/b","A",1
341,1,48,50,0.0914358,"\int \cos ^3(a+b x) (c \sin (a+b x))^m \, dx","Integrate[Cos[a + b*x]^3*(c*Sin[a + b*x])^m,x]","\frac{\sin (a+b x) ((m+1) \cos (2 (a+b x))+m+5) (c \sin (a+b x))^m}{2 b (m+1) (m+3)}","\frac{(c \sin (a+b x))^{m+1}}{b c (m+1)}-\frac{(c \sin (a+b x))^{m+3}}{b c^3 (m+3)}",1,"((5 + m + (1 + m)*Cos[2*(a + b*x)])*Sin[a + b*x]*(c*Sin[a + b*x])^m)/(2*b*(1 + m)*(3 + m))","A",1
342,1,25,24,0.0100832,"\int \cos (a+b x) (c \sin (a+b x))^m \, dx","Integrate[Cos[a + b*x]*(c*Sin[a + b*x])^m,x]","\frac{\sin (a+b x) (c \sin (a+b x))^m}{b (m+1)}","\frac{(c \sin (a+b x))^{m+1}}{b c (m+1)}",1,"(Sin[a + b*x]*(c*Sin[a + b*x])^m)/(b*(1 + m))","A",1
343,1,51,48,0.0201247,"\int \sec (a+b x) (c \sin (a+b x))^m \, dx","Integrate[Sec[a + b*x]*(c*Sin[a + b*x])^m,x]","\frac{\sin (a+b x) (c \sin (a+b x))^m \, _2F_1\left(1,\frac{m+1}{2};\frac{m+1}{2}+1;\sin ^2(a+b x)\right)}{b (m+1)}","\frac{(c \sin (a+b x))^{m+1} \, _2F_1\left(1,\frac{m+1}{2};\frac{m+3}{2};\sin ^2(a+b x)\right)}{b c (m+1)}",1,"(Hypergeometric2F1[1, (1 + m)/2, 1 + (1 + m)/2, Sin[a + b*x]^2]*Sin[a + b*x]*(c*Sin[a + b*x])^m)/(b*(1 + m))","A",1
344,1,51,48,0.023289,"\int \sec ^3(a+b x) (c \sin (a+b x))^m \, dx","Integrate[Sec[a + b*x]^3*(c*Sin[a + b*x])^m,x]","\frac{\sin (a+b x) (c \sin (a+b x))^m \, _2F_1\left(2,\frac{m+1}{2};\frac{m+1}{2}+1;\sin ^2(a+b x)\right)}{b (m+1)}","\frac{(c \sin (a+b x))^{m+1} \, _2F_1\left(2,\frac{m+1}{2};\frac{m+3}{2};\sin ^2(a+b x)\right)}{b c (m+1)}",1,"(Hypergeometric2F1[2, (1 + m)/2, 1 + (1 + m)/2, Sin[a + b*x]^2]*Sin[a + b*x]*(c*Sin[a + b*x])^m)/(b*(1 + m))","A",1
345,1,63,68,0.046082,"\int \cos ^4(a+b x) (c \sin (a+b x))^m \, dx","Integrate[Cos[a + b*x]^4*(c*Sin[a + b*x])^m,x]","\frac{\sqrt{\cos ^2(a+b x)} \tan (a+b x) (c \sin (a+b x))^m \, _2F_1\left(-\frac{3}{2},\frac{m+1}{2};\frac{m+3}{2};\sin ^2(a+b x)\right)}{b (m+1)}","\frac{\cos (a+b x) (c \sin (a+b x))^{m+1} \, _2F_1\left(-\frac{3}{2},\frac{m+1}{2};\frac{m+3}{2};\sin ^2(a+b x)\right)}{b c (m+1) \sqrt{\cos ^2(a+b x)}}",1,"(Sqrt[Cos[a + b*x]^2]*Hypergeometric2F1[-3/2, (1 + m)/2, (3 + m)/2, Sin[a + b*x]^2]*(c*Sin[a + b*x])^m*Tan[a + b*x])/(b*(1 + m))","A",1
346,1,63,68,0.047317,"\int \cos ^2(a+b x) (c \sin (a+b x))^m \, dx","Integrate[Cos[a + b*x]^2*(c*Sin[a + b*x])^m,x]","\frac{\sqrt{\cos ^2(a+b x)} \tan (a+b x) (c \sin (a+b x))^m \, _2F_1\left(-\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};\sin ^2(a+b x)\right)}{b (m+1)}","\frac{\cos (a+b x) (c \sin (a+b x))^{m+1} \, _2F_1\left(-\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};\sin ^2(a+b x)\right)}{b c (m+1) \sqrt{\cos ^2(a+b x)}}",1,"(Sqrt[Cos[a + b*x]^2]*Hypergeometric2F1[-1/2, (1 + m)/2, (3 + m)/2, Sin[a + b*x]^2]*(c*Sin[a + b*x])^m*Tan[a + b*x])/(b*(1 + m))","A",1
347,1,63,68,0.0389737,"\int (c \sin (a+b x))^m \, dx","Integrate[(c*Sin[a + b*x])^m,x]","\frac{\sqrt{\cos ^2(a+b x)} \tan (a+b x) (c \sin (a+b x))^m \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};\sin ^2(a+b x)\right)}{b (m+1)}","\frac{\cos (a+b x) (c \sin (a+b x))^{m+1} \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};\sin ^2(a+b x)\right)}{b c (m+1) \sqrt{\cos ^2(a+b x)}}",1,"(Sqrt[Cos[a + b*x]^2]*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, Sin[a + b*x]^2]*(c*Sin[a + b*x])^m*Tan[a + b*x])/(b*(1 + m))","A",1
348,1,63,68,0.0444551,"\int \sec ^2(a+b x) (c \sin (a+b x))^m \, dx","Integrate[Sec[a + b*x]^2*(c*Sin[a + b*x])^m,x]","\frac{\sqrt{\cos ^2(a+b x)} \tan (a+b x) (c \sin (a+b x))^m \, _2F_1\left(\frac{3}{2},\frac{m+1}{2};\frac{m+3}{2};\sin ^2(a+b x)\right)}{b (m+1)}","\frac{\sqrt{\cos ^2(a+b x)} \sec (a+b x) (c \sin (a+b x))^{m+1} \, _2F_1\left(\frac{3}{2},\frac{m+1}{2};\frac{m+3}{2};\sin ^2(a+b x)\right)}{b c (m+1)}",1,"(Sqrt[Cos[a + b*x]^2]*Hypergeometric2F1[3/2, (1 + m)/2, (3 + m)/2, Sin[a + b*x]^2]*(c*Sin[a + b*x])^m*Tan[a + b*x])/(b*(1 + m))","A",1
349,1,63,68,0.0443678,"\int \sec ^4(a+b x) (c \sin (a+b x))^m \, dx","Integrate[Sec[a + b*x]^4*(c*Sin[a + b*x])^m,x]","\frac{\sqrt{\cos ^2(a+b x)} \tan (a+b x) (c \sin (a+b x))^m \, _2F_1\left(\frac{5}{2},\frac{m+1}{2};\frac{m+3}{2};\sin ^2(a+b x)\right)}{b (m+1)}","\frac{\sqrt{\cos ^2(a+b x)} \sec (a+b x) (c \sin (a+b x))^{m+1} \, _2F_1\left(\frac{5}{2},\frac{m+1}{2};\frac{m+3}{2};\sin ^2(a+b x)\right)}{b c (m+1)}",1,"(Sqrt[Cos[a + b*x]^2]*Hypergeometric2F1[5/2, (1 + m)/2, (3 + m)/2, Sin[a + b*x]^2]*(c*Sin[a + b*x])^m*Tan[a + b*x])/(b*(1 + m))","A",1
350,1,78,75,0.1291302,"\int (d \cos (a+b x))^{3/2} (c \sin (a+b x))^m \, dx","Integrate[(d*Cos[a + b*x])^(3/2)*(c*Sin[a + b*x])^m,x]","\frac{d^2 \cos ^2(a+b x)^{3/4} \tan (a+b x) (c \sin (a+b x))^m \, _2F_1\left(-\frac{1}{4},\frac{m+1}{2};\frac{m+3}{2};\sin ^2(a+b x)\right)}{b (m+1) \sqrt{d \cos (a+b x)}}","\frac{d \sqrt{d \cos (a+b x)} (c \sin (a+b x))^{m+1} \, _2F_1\left(-\frac{1}{4},\frac{m+1}{2};\frac{m+3}{2};\sin ^2(a+b x)\right)}{b c (m+1) \sqrt[4]{\cos ^2(a+b x)}}",1,"(d^2*(Cos[a + b*x]^2)^(3/4)*Hypergeometric2F1[-1/4, (1 + m)/2, (3 + m)/2, Sin[a + b*x]^2]*(c*Sin[a + b*x])^m*Tan[a + b*x])/(b*(1 + m)*Sqrt[d*Cos[a + b*x]])","A",1
351,1,75,75,0.0648919,"\int \sqrt{d \cos (a+b x)} (c \sin (a+b x))^m \, dx","Integrate[Sqrt[d*Cos[a + b*x]]*(c*Sin[a + b*x])^m,x]","\frac{\sqrt[4]{\cos ^2(a+b x)} \tan (a+b x) \sqrt{d \cos (a+b x)} (c \sin (a+b x))^m \, _2F_1\left(\frac{1}{4},\frac{m+1}{2};\frac{m+3}{2};\sin ^2(a+b x)\right)}{b (m+1)}","\frac{d \sqrt[4]{\cos ^2(a+b x)} (c \sin (a+b x))^{m+1} \, _2F_1\left(\frac{1}{4},\frac{m+1}{2};\frac{m+3}{2};\sin ^2(a+b x)\right)}{b c (m+1) \sqrt{d \cos (a+b x)}}",1,"(Sqrt[d*Cos[a + b*x]]*(Cos[a + b*x]^2)^(1/4)*Hypergeometric2F1[1/4, (1 + m)/2, (3 + m)/2, Sin[a + b*x]^2]*(c*Sin[a + b*x])^m*Tan[a + b*x])/(b*(1 + m))","A",1
352,1,75,75,0.0700525,"\int \frac{(c \sin (a+b x))^m}{\sqrt{d \cos (a+b x)}} \, dx","Integrate[(c*Sin[a + b*x])^m/Sqrt[d*Cos[a + b*x]],x]","\frac{\cos ^2(a+b x)^{3/4} \tan (a+b x) (c \sin (a+b x))^m \, _2F_1\left(\frac{3}{4},\frac{m+1}{2};\frac{m+3}{2};\sin ^2(a+b x)\right)}{b (m+1) \sqrt{d \cos (a+b x)}}","\frac{d \cos ^2(a+b x)^{3/4} (c \sin (a+b x))^{m+1} \, _2F_1\left(\frac{3}{4},\frac{m+1}{2};\frac{m+3}{2};\sin ^2(a+b x)\right)}{b c (m+1) (d \cos (a+b x))^{3/2}}",1,"((Cos[a + b*x]^2)^(3/4)*Hypergeometric2F1[3/4, (1 + m)/2, (3 + m)/2, Sin[a + b*x]^2]*(c*Sin[a + b*x])^m*Tan[a + b*x])/(b*(1 + m)*Sqrt[d*Cos[a + b*x]])","A",1
353,1,78,77,0.0905593,"\int \frac{(c \sin (a+b x))^m}{(d \cos (a+b x))^{3/2}} \, dx","Integrate[(c*Sin[a + b*x])^m/(d*Cos[a + b*x])^(3/2),x]","\frac{\sqrt[4]{\cos ^2(a+b x)} \tan (a+b x) \sqrt{d \cos (a+b x)} (c \sin (a+b x))^m \, _2F_1\left(\frac{5}{4},\frac{m+1}{2};\frac{m+3}{2};\sin ^2(a+b x)\right)}{b d^2 (m+1)}","\frac{\sqrt[4]{\cos ^2(a+b x)} (c \sin (a+b x))^{m+1} \, _2F_1\left(\frac{5}{4},\frac{m+1}{2};\frac{m+3}{2};\sin ^2(a+b x)\right)}{b c d (m+1) \sqrt{d \cos (a+b x)}}",1,"(Sqrt[d*Cos[a + b*x]]*(Cos[a + b*x]^2)^(1/4)*Hypergeometric2F1[5/4, (1 + m)/2, (3 + m)/2, Sin[a + b*x]^2]*(c*Sin[a + b*x])^m*Tan[a + b*x])/(b*d^2*(1 + m))","A",1
354,1,78,77,0.0811069,"\int \frac{(c \sin (a+b x))^m}{(d \cos (a+b x))^{5/2}} \, dx","Integrate[(c*Sin[a + b*x])^m/(d*Cos[a + b*x])^(5/2),x]","\frac{\cos ^2(a+b x)^{3/4} \tan (a+b x) (c \sin (a+b x))^m \, _2F_1\left(\frac{7}{4},\frac{m+1}{2};\frac{m+3}{2};\sin ^2(a+b x)\right)}{b d^2 (m+1) \sqrt{d \cos (a+b x)}}","\frac{\cos ^2(a+b x)^{3/4} (c \sin (a+b x))^{m+1} \, _2F_1\left(\frac{7}{4},\frac{m+1}{2};\frac{m+3}{2};\sin ^2(a+b x)\right)}{b c d (m+1) (d \cos (a+b x))^{3/2}}",1,"((Cos[a + b*x]^2)^(3/4)*Hypergeometric2F1[7/4, (1 + m)/2, (3 + m)/2, Sin[a + b*x]^2]*(c*Sin[a + b*x])^m*Tan[a + b*x])/(b*d^2*(1 + m)*Sqrt[d*Cos[a + b*x]])","A",1
355,1,83,76,0.3067454,"\int (d \cos (a+b x))^n \sin ^5(a+b x) \, dx","Integrate[(d*Cos[a + b*x])^n*Sin[a + b*x]^5,x]","-\frac{\cos (a+b x) \left(-4 \left(n^2+8 n+7\right) \cos (2 (a+b x))+\left(n^2+4 n+3\right) \cos (4 (a+b x))+3 n^2+28 n+89\right) (d \cos (a+b x))^n}{8 b (n+1) (n+3) (n+5)}","-\frac{(d \cos (a+b x))^{n+5}}{b d^5 (n+5)}+\frac{2 (d \cos (a+b x))^{n+3}}{b d^3 (n+3)}-\frac{(d \cos (a+b x))^{n+1}}{b d (n+1)}",1,"-1/8*(Cos[a + b*x]*(d*Cos[a + b*x])^n*(89 + 28*n + 3*n^2 - 4*(7 + 8*n + n^2)*Cos[2*(a + b*x)] + (3 + 4*n + n^2)*Cos[4*(a + b*x)]))/(b*(1 + n)*(3 + n)*(5 + n))","A",1
356,1,50,50,0.1233429,"\int (d \cos (a+b x))^n \sin ^3(a+b x) \, dx","Integrate[(d*Cos[a + b*x])^n*Sin[a + b*x]^3,x]","\frac{\cos (a+b x) ((n+1) \cos (2 (a+b x))-n-5) (d \cos (a+b x))^n}{2 b (n+1) (n+3)}","\frac{(d \cos (a+b x))^{n+3}}{b d^3 (n+3)}-\frac{(d \cos (a+b x))^{n+1}}{b d (n+1)}",1,"(Cos[a + b*x]*(d*Cos[a + b*x])^n*(-5 - n + (1 + n)*Cos[2*(a + b*x)]))/(2*b*(1 + n)*(3 + n))","A",1
357,1,26,25,0.010478,"\int (d \cos (a+b x))^n \sin (a+b x) \, dx","Integrate[(d*Cos[a + b*x])^n*Sin[a + b*x],x]","-\frac{\cos (a+b x) (d \cos (a+b x))^n}{b (n+1)}","-\frac{(d \cos (a+b x))^{n+1}}{b d (n+1)}",1,"-((Cos[a + b*x]*(d*Cos[a + b*x])^n)/(b*(1 + n)))","A",1
358,1,52,49,0.0324399,"\int (d \cos (a+b x))^n \csc (a+b x) \, dx","Integrate[(d*Cos[a + b*x])^n*Csc[a + b*x],x]","-\frac{\cos (a+b x) (d \cos (a+b x))^n \, _2F_1\left(1,\frac{n+1}{2};\frac{n+1}{2}+1;\cos ^2(a+b x)\right)}{b (n+1)}","-\frac{(d \cos (a+b x))^{n+1} \, _2F_1\left(1,\frac{n+1}{2};\frac{n+3}{2};\cos ^2(a+b x)\right)}{b d (n+1)}",1,"-((Cos[a + b*x]*(d*Cos[a + b*x])^n*Hypergeometric2F1[1, (1 + n)/2, 1 + (1 + n)/2, Cos[a + b*x]^2])/(b*(1 + n)))","A",1
359,1,154,49,2.7001657,"\int (d \cos (a+b x))^n \csc ^3(a+b x) \, dx","Integrate[(d*Cos[a + b*x])^n*Csc[a + b*x]^3,x]","-\frac{2^{-n-3} \cos (a+b x) (d \cos (a+b x))^n \left(2^{n+1} \, _2F_1(1,n+1;n+2;\cos (a+b x))+2^{n+1} \, _2F_1(2,n+1;n+2;\cos (a+b x))+\sec ^2\left(\frac{1}{2} (a+b x)\right)^{n+1} \left(\, _2F_1\left(n,n+1;n+2;\frac{1}{2} \cos (a+b x) \sec ^2\left(\frac{1}{2} (a+b x)\right)\right)+\, _2F_1\left(n+1,n+1;n+2;\frac{1}{2} \cos (a+b x) \sec ^2\left(\frac{1}{2} (a+b x)\right)\right)\right)\right)}{b (n+1)}","-\frac{(d \cos (a+b x))^{n+1} \, _2F_1\left(2,\frac{n+1}{2};\frac{n+3}{2};\cos ^2(a+b x)\right)}{b d (n+1)}",1,"-((2^(-3 - n)*Cos[a + b*x]*(d*Cos[a + b*x])^n*(2^(1 + n)*Hypergeometric2F1[1, 1 + n, 2 + n, Cos[a + b*x]] + 2^(1 + n)*Hypergeometric2F1[2, 1 + n, 2 + n, Cos[a + b*x]] + (Hypergeometric2F1[n, 1 + n, 2 + n, (Cos[a + b*x]*Sec[(a + b*x)/2]^2)/2] + Hypergeometric2F1[1 + n, 1 + n, 2 + n, (Cos[a + b*x]*Sec[(a + b*x)/2]^2)/2])*(Sec[(a + b*x)/2]^2)^(1 + n)))/(b*(1 + n)))","B",1
360,1,244,49,4.0200134,"\int (d \cos (a+b x))^n \csc ^5(a+b x) \, dx","Integrate[(d*Cos[a + b*x])^n*Csc[a + b*x]^5,x]","-\frac{2^{-n-5} \cos (a+b x) (d \cos (a+b x))^n \left(3\ 2^{n+1} \, _2F_1(1,n+1;n+2;\cos (a+b x))+3\ 2^{n+1} \, _2F_1(2,n+1;n+2;\cos (a+b x))+2^{n+2} \, _2F_1(3,n+1;n+2;\cos (a+b x))+2 \sec ^2\left(\frac{1}{2} (a+b x)\right)^{n+1} \, _2F_1\left(n-1,n+1;n+2;\frac{1}{2} \cos (a+b x) \sec ^2\left(\frac{1}{2} (a+b x)\right)\right)+3 \sec ^2\left(\frac{1}{2} (a+b x)\right)^{n+1} \, _2F_1\left(n,n+1;n+2;\frac{1}{2} \cos (a+b x) \sec ^2\left(\frac{1}{2} (a+b x)\right)\right)+3 \sec ^2\left(\frac{1}{2} (a+b x)\right)^{n+1} \, _2F_1\left(n+1,n+1;n+2;\frac{1}{2} \cos (a+b x) \sec ^2\left(\frac{1}{2} (a+b x)\right)\right)\right)}{b (n+1)}","-\frac{(d \cos (a+b x))^{n+1} \, _2F_1\left(3,\frac{n+1}{2};\frac{n+3}{2};\cos ^2(a+b x)\right)}{b d (n+1)}",1,"-((2^(-5 - n)*Cos[a + b*x]*(d*Cos[a + b*x])^n*(3*2^(1 + n)*Hypergeometric2F1[1, 1 + n, 2 + n, Cos[a + b*x]] + 3*2^(1 + n)*Hypergeometric2F1[2, 1 + n, 2 + n, Cos[a + b*x]] + 2^(2 + n)*Hypergeometric2F1[3, 1 + n, 2 + n, Cos[a + b*x]] + 2*Hypergeometric2F1[-1 + n, 1 + n, 2 + n, (Cos[a + b*x]*Sec[(a + b*x)/2]^2)/2]*(Sec[(a + b*x)/2]^2)^(1 + n) + 3*Hypergeometric2F1[n, 1 + n, 2 + n, (Cos[a + b*x]*Sec[(a + b*x)/2]^2)/2]*(Sec[(a + b*x)/2]^2)^(1 + n) + 3*Hypergeometric2F1[1 + n, 1 + n, 2 + n, (Cos[a + b*x]*Sec[(a + b*x)/2]^2)/2]*(Sec[(a + b*x)/2]^2)^(1 + n)))/(b*(1 + n)))","B",1
361,1,68,69,0.1130832,"\int (d \cos (a+b x))^n \sin ^4(a+b x) \, dx","Integrate[(d*Cos[a + b*x])^n*Sin[a + b*x]^4,x]","-\frac{\sin (2 (a+b x)) (d \cos (a+b x))^n \, _2F_1\left(-\frac{3}{2},\frac{n+1}{2};\frac{n+3}{2};\cos ^2(a+b x)\right)}{2 b (n+1) \sqrt{\sin ^2(a+b x)}}","-\frac{\sin (a+b x) (d \cos (a+b x))^{n+1} \, _2F_1\left(-\frac{3}{2},\frac{n+1}{2};\frac{n+3}{2};\cos ^2(a+b x)\right)}{b d (n+1) \sqrt{\sin ^2(a+b x)}}",1,"-1/2*((d*Cos[a + b*x])^n*Hypergeometric2F1[-3/2, (1 + n)/2, (3 + n)/2, Cos[a + b*x]^2]*Sin[2*(a + b*x)])/(b*(1 + n)*Sqrt[Sin[a + b*x]^2])","A",1
362,1,68,69,0.0835867,"\int (d \cos (a+b x))^n \sin ^2(a+b x) \, dx","Integrate[(d*Cos[a + b*x])^n*Sin[a + b*x]^2,x]","-\frac{\sin (2 (a+b x)) (d \cos (a+b x))^n \, _2F_1\left(-\frac{1}{2},\frac{n+1}{2};\frac{n+3}{2};\cos ^2(a+b x)\right)}{2 b (n+1) \sqrt{\sin ^2(a+b x)}}","-\frac{\sin (a+b x) (d \cos (a+b x))^{n+1} \, _2F_1\left(-\frac{1}{2},\frac{n+1}{2};\frac{n+3}{2};\cos ^2(a+b x)\right)}{b d (n+1) \sqrt{\sin ^2(a+b x)}}",1,"-1/2*((d*Cos[a + b*x])^n*Hypergeometric2F1[-1/2, (1 + n)/2, (3 + n)/2, Cos[a + b*x]^2]*Sin[2*(a + b*x)])/(b*(1 + n)*Sqrt[Sin[a + b*x]^2])","A",1
363,1,64,69,0.0438656,"\int (d \cos (a+b x))^n \, dx","Integrate[(d*Cos[a + b*x])^n,x]","-\frac{\sqrt{\sin ^2(a+b x)} \cot (a+b x) (d \cos (a+b x))^n \, _2F_1\left(\frac{1}{2},\frac{n+1}{2};\frac{n+3}{2};\cos ^2(a+b x)\right)}{b (n+1)}","-\frac{\sin (a+b x) (d \cos (a+b x))^{n+1} \, _2F_1\left(\frac{1}{2},\frac{n+1}{2};\frac{n+3}{2};\cos ^2(a+b x)\right)}{b d (n+1) \sqrt{\sin ^2(a+b x)}}",1,"-(((d*Cos[a + b*x])^n*Cot[a + b*x]*Hypergeometric2F1[1/2, (1 + n)/2, (3 + n)/2, Cos[a + b*x]^2]*Sqrt[Sin[a + b*x]^2])/(b*(1 + n)))","A",1
364,1,80,69,0.1935486,"\int (d \cos (a+b x))^n \csc ^2(a+b x) \, dx","Integrate[(d*Cos[a + b*x])^n*Csc[a + b*x]^2,x]","\frac{d \csc (a+b x) \left(-\cot ^2(a+b x)\right)^{\frac{1-n}{2}} (d \cos (a+b x))^{n-1} \, _2F_1\left(\frac{1-n}{2},1-\frac{n}{2};2-\frac{n}{2};\csc ^2(a+b x)\right)}{b (n-2)}","-\frac{\sqrt{\sin ^2(a+b x)} \csc (a+b x) (d \cos (a+b x))^{n+1} \, _2F_1\left(\frac{3}{2},\frac{n+1}{2};\frac{n+3}{2};\cos ^2(a+b x)\right)}{b d (n+1)}",1,"(d*(d*Cos[a + b*x])^(-1 + n)*(-Cot[a + b*x]^2)^((1 - n)/2)*Csc[a + b*x]*Hypergeometric2F1[(1 - n)/2, 1 - n/2, 2 - n/2, Csc[a + b*x]^2])/(b*(-2 + n))","A",1
365,1,82,69,0.2114106,"\int (d \cos (a+b x))^n \csc ^4(a+b x) \, dx","Integrate[(d*Cos[a + b*x])^n*Csc[a + b*x]^4,x]","\frac{d \csc ^3(a+b x) \left(-\cot ^2(a+b x)\right)^{\frac{1-n}{2}} (d \cos (a+b x))^{n-1} \, _2F_1\left(\frac{1-n}{2},2-\frac{n}{2};3-\frac{n}{2};\csc ^2(a+b x)\right)}{b (n-4)}","-\frac{\sqrt{\sin ^2(a+b x)} \csc (a+b x) (d \cos (a+b x))^{n+1} \, _2F_1\left(\frac{5}{2},\frac{n+1}{2};\frac{n+3}{2};\cos ^2(a+b x)\right)}{b d (n+1)}",1,"(d*(d*Cos[a + b*x])^(-1 + n)*(-Cot[a + b*x]^2)^((1 - n)/2)*Csc[a + b*x]^3*Hypergeometric2F1[(1 - n)/2, 2 - n/2, 3 - n/2, Csc[a + b*x]^2])/(b*(-4 + n))","A",1
366,1,158,76,0.4157921,"\int (d \cos (a+b x))^n (c \sin (a+b x))^{5/2} \, dx","Integrate[(d*Cos[a + b*x])^n*(c*Sin[a + b*x])^(5/2),x]","\frac{\cot (a+b x) (c \sin (a+b x))^{5/2} (d \cos (a+b x))^n \left((n+1) \cos ^2(a+b x) \, _2F_1\left(\frac{1}{4},\frac{n+3}{2};\frac{n+5}{2};\cos ^2(a+b x)\right)-(n+3) \, _2F_1\left(-\frac{3}{4},\frac{n+1}{2};\frac{n+3}{2};\cos ^2(a+b x)\right)-(n+3) \, _2F_1\left(\frac{1}{4},\frac{n+1}{2};\frac{n+3}{2};\cos ^2(a+b x)\right)\right)}{2 b (n+1) (n+3) \sin ^2(a+b x)^{3/4}}","-\frac{c (c \sin (a+b x))^{3/2} (d \cos (a+b x))^{n+1} \, _2F_1\left(-\frac{3}{4},\frac{n+1}{2};\frac{n+3}{2};\cos ^2(a+b x)\right)}{b d (n+1) \sin ^2(a+b x)^{3/4}}",1,"((d*Cos[a + b*x])^n*Cot[a + b*x]*(-((3 + n)*Hypergeometric2F1[-3/4, (1 + n)/2, (3 + n)/2, Cos[a + b*x]^2]) - (3 + n)*Hypergeometric2F1[1/4, (1 + n)/2, (3 + n)/2, Cos[a + b*x]^2] + (1 + n)*Cos[a + b*x]^2*Hypergeometric2F1[1/4, (3 + n)/2, (5 + n)/2, Cos[a + b*x]^2])*(c*Sin[a + b*x])^(5/2))/(2*b*(1 + n)*(3 + n)*(Sin[a + b*x]^2)^(3/4))","B",1
367,1,76,76,0.1594855,"\int (d \cos (a+b x))^n (c \sin (a+b x))^{3/2} \, dx","Integrate[(d*Cos[a + b*x])^n*(c*Sin[a + b*x])^(3/2),x]","-\frac{\cot (a+b x) (c \sin (a+b x))^{3/2} (d \cos (a+b x))^n \, _2F_1\left(-\frac{1}{4},\frac{n+1}{2};\frac{n+3}{2};\cos ^2(a+b x)\right)}{b (n+1) \sqrt[4]{\sin ^2(a+b x)}}","-\frac{c \sqrt{c \sin (a+b x)} (d \cos (a+b x))^{n+1} \, _2F_1\left(-\frac{1}{4},\frac{n+1}{2};\frac{n+3}{2};\cos ^2(a+b x)\right)}{b d (n+1) \sqrt[4]{\sin ^2(a+b x)}}",1,"-(((d*Cos[a + b*x])^n*Cot[a + b*x]*Hypergeometric2F1[-1/4, (1 + n)/2, (3 + n)/2, Cos[a + b*x]^2]*(c*Sin[a + b*x])^(3/2))/(b*(1 + n)*(Sin[a + b*x]^2)^(1/4)))","A",1
368,1,82,76,0.1037845,"\int (d \cos (a+b x))^n \sqrt{c \sin (a+b x)} \, dx","Integrate[(d*Cos[a + b*x])^n*Sqrt[c*Sin[a + b*x]],x]","-\frac{\sin (a+b x) \cos (a+b x) \sqrt{c \sin (a+b x)} (d \cos (a+b x))^n \, _2F_1\left(\frac{1}{4},\frac{n+1}{2};\frac{n+3}{2};\cos ^2(a+b x)\right)}{b (n+1) \sin ^2(a+b x)^{3/4}}","-\frac{c \sqrt[4]{\sin ^2(a+b x)} (d \cos (a+b x))^{n+1} \, _2F_1\left(\frac{1}{4},\frac{n+1}{2};\frac{n+3}{2};\cos ^2(a+b x)\right)}{b d (n+1) \sqrt{c \sin (a+b x)}}",1,"-((Cos[a + b*x]*(d*Cos[a + b*x])^n*Hypergeometric2F1[1/4, (1 + n)/2, (3 + n)/2, Cos[a + b*x]^2]*Sin[a + b*x]*Sqrt[c*Sin[a + b*x]])/(b*(1 + n)*(Sin[a + b*x]^2)^(3/4)))","A",1
369,1,82,76,0.1128178,"\int \frac{(d \cos (a+b x))^n}{\sqrt{c \sin (a+b x)}} \, dx","Integrate[(d*Cos[a + b*x])^n/Sqrt[c*Sin[a + b*x]],x]","-\frac{\sin (a+b x) \cos (a+b x) (d \cos (a+b x))^n \, _2F_1\left(\frac{3}{4},\frac{n+1}{2};\frac{n+3}{2};\cos ^2(a+b x)\right)}{b (n+1) \sqrt[4]{\sin ^2(a+b x)} \sqrt{c \sin (a+b x)}}","-\frac{c \sin ^2(a+b x)^{3/4} (d \cos (a+b x))^{n+1} \, _2F_1\left(\frac{3}{4},\frac{n+1}{2};\frac{n+3}{2};\cos ^2(a+b x)\right)}{b d (n+1) (c \sin (a+b x))^{3/2}}",1,"-((Cos[a + b*x]*(d*Cos[a + b*x])^n*Hypergeometric2F1[3/4, (1 + n)/2, (3 + n)/2, Cos[a + b*x]^2]*Sin[a + b*x])/(b*(1 + n)*Sqrt[c*Sin[a + b*x]]*(Sin[a + b*x]^2)^(1/4)))","A",1
370,1,79,78,0.1332951,"\int \frac{(d \cos (a+b x))^n}{(c \sin (a+b x))^{3/2}} \, dx","Integrate[(d*Cos[a + b*x])^n/(c*Sin[a + b*x])^(3/2),x]","-\frac{\sqrt[4]{\sin ^2(a+b x)} \cot (a+b x) \sqrt{c \sin (a+b x)} (d \cos (a+b x))^n \, _2F_1\left(\frac{5}{4},\frac{n+1}{2};\frac{n+3}{2};\cos ^2(a+b x)\right)}{b c^2 (n+1)}","-\frac{\sqrt[4]{\sin ^2(a+b x)} (d \cos (a+b x))^{n+1} \, _2F_1\left(\frac{5}{4},\frac{n+1}{2};\frac{n+3}{2};\cos ^2(a+b x)\right)}{b c d (n+1) \sqrt{c \sin (a+b x)}}",1,"-(((d*Cos[a + b*x])^n*Cot[a + b*x]*Hypergeometric2F1[5/4, (1 + n)/2, (3 + n)/2, Cos[a + b*x]^2]*Sqrt[c*Sin[a + b*x]]*(Sin[a + b*x]^2)^(1/4))/(b*c^2*(1 + n)))","A",1
371,1,58,85,0.3396286,"\int \sqrt{b \sec (e+f x)} \sin ^7(e+f x) \, dx","Integrate[Sqrt[b*Sec[e + f*x]]*Sin[e + f*x]^7,x]","\frac{(-8939 \cos (e+f x)+887 \cos (3 (e+f x))-155 \cos (5 (e+f x))+15 \cos (7 (e+f x))) \sqrt{b \sec (e+f x)}}{6240 f}","\frac{2 b^7}{13 f (b \sec (e+f x))^{13/2}}-\frac{2 b^5}{3 f (b \sec (e+f x))^{9/2}}+\frac{6 b^3}{5 f (b \sec (e+f x))^{5/2}}-\frac{2 b}{f \sqrt{b \sec (e+f x)}}",1,"((-8939*Cos[e + f*x] + 887*Cos[3*(e + f*x)] - 155*Cos[5*(e + f*x)] + 15*Cos[7*(e + f*x)])*Sqrt[b*Sec[e + f*x]])/(6240*f)","A",1
372,1,48,63,0.2068877,"\int \sqrt{b \sec (e+f x)} \sin ^5(e+f x) \, dx","Integrate[Sqrt[b*Sec[e + f*x]]*Sin[e + f*x]^5,x]","-\frac{(554 \cos (e+f x)-47 \cos (3 (e+f x))+5 \cos (5 (e+f x))) \sqrt{b \sec (e+f x)}}{360 f}","-\frac{2 b^5}{9 f (b \sec (e+f x))^{9/2}}+\frac{4 b^3}{5 f (b \sec (e+f x))^{5/2}}-\frac{2 b}{f \sqrt{b \sec (e+f x)}}",1,"-1/360*((554*Cos[e + f*x] - 47*Cos[3*(e + f*x)] + 5*Cos[5*(e + f*x)])*Sqrt[b*Sec[e + f*x]])/f","A",1
373,1,36,41,0.1583773,"\int \sqrt{b \sec (e+f x)} \sin ^3(e+f x) \, dx","Integrate[Sqrt[b*Sec[e + f*x]]*Sin[e + f*x]^3,x]","\frac{(\cos (3 (e+f x))-17 \cos (e+f x)) \sqrt{b \sec (e+f x)}}{10 f}","\frac{2 b^3}{5 f (b \sec (e+f x))^{5/2}}-\frac{2 b}{f \sqrt{b \sec (e+f x)}}",1,"((-17*Cos[e + f*x] + Cos[3*(e + f*x)])*Sqrt[b*Sec[e + f*x]])/(10*f)","A",1
374,1,18,18,0.0345427,"\int \sqrt{b \sec (e+f x)} \sin (e+f x) \, dx","Integrate[Sqrt[b*Sec[e + f*x]]*Sin[e + f*x],x]","-\frac{2 b}{f \sqrt{b \sec (e+f x)}}","-\frac{2 b}{f \sqrt{b \sec (e+f x)}}",1,"(-2*b)/(f*Sqrt[b*Sec[e + f*x]])","A",1
375,1,73,58,0.3513408,"\int \csc (e+f x) \sqrt{b \sec (e+f x)} \, dx","Integrate[Csc[e + f*x]*Sqrt[b*Sec[e + f*x]],x]","\frac{\sqrt{b \sec (e+f x)} \left(\log \left(1-\sqrt{\sec (e+f x)}\right)-\log \left(\sqrt{\sec (e+f x)}+1\right)+2 \tan ^{-1}\left(\sqrt{\sec (e+f x)}\right)\right)}{2 f \sqrt{\sec (e+f x)}}","\frac{\sqrt{b} \tan ^{-1}\left(\frac{\sqrt{b \sec (e+f x)}}{\sqrt{b}}\right)}{f}-\frac{\sqrt{b} \tanh ^{-1}\left(\frac{\sqrt{b \sec (e+f x)}}{\sqrt{b}}\right)}{f}",1,"((2*ArcTan[Sqrt[Sec[e + f*x]]] + Log[1 - Sqrt[Sec[e + f*x]]] - Log[1 + Sqrt[Sec[e + f*x]]])*Sqrt[b*Sec[e + f*x]])/(2*f*Sqrt[Sec[e + f*x]])","A",1
376,1,95,93,0.6234969,"\int \csc ^3(e+f x) \sqrt{b \sec (e+f x)} \, dx","Integrate[Csc[e + f*x]^3*Sqrt[b*Sec[e + f*x]],x]","-\frac{\sqrt{b \sec (e+f x)} \left(-3 \log \left(1-\sqrt{\sec (e+f x)}\right)+3 \log \left(\sqrt{\sec (e+f x)}+1\right)+\frac{4 \csc ^2(e+f x)}{\sqrt{\sec (e+f x)}}-6 \tan ^{-1}\left(\sqrt{\sec (e+f x)}\right)\right)}{8 f \sqrt{\sec (e+f x)}}","-\frac{\cot ^2(e+f x) (b \sec (e+f x))^{3/2}}{2 b f}+\frac{3 \sqrt{b} \tan ^{-1}\left(\frac{\sqrt{b \sec (e+f x)}}{\sqrt{b}}\right)}{4 f}-\frac{3 \sqrt{b} \tanh ^{-1}\left(\frac{\sqrt{b \sec (e+f x)}}{\sqrt{b}}\right)}{4 f}",1,"-1/8*((-6*ArcTan[Sqrt[Sec[e + f*x]]] - 3*Log[1 - Sqrt[Sec[e + f*x]]] + 3*Log[1 + Sqrt[Sec[e + f*x]]] + (4*Csc[e + f*x]^2)/Sqrt[Sec[e + f*x]])*Sqrt[b*Sec[e + f*x]])/(f*Sqrt[Sec[e + f*x]])","A",1
377,1,107,123,1.0098001,"\int \csc ^5(e+f x) \sqrt{b \sec (e+f x)} \, dx","Integrate[Csc[e + f*x]^5*Sqrt[b*Sec[e + f*x]],x]","\frac{b \left(-16 \csc ^4(e+f x)-28 \csc ^2(e+f x)+21 \sqrt{\sec (e+f x)} \left(\log \left(1-\sqrt{\sec (e+f x)}\right)-\log \left(\sqrt{\sec (e+f x)}+1\right)\right)+42 \sqrt{\sec (e+f x)} \tan ^{-1}\left(\sqrt{\sec (e+f x)}\right)\right)}{64 f \sqrt{b \sec (e+f x)}}","-\frac{\cot ^4(e+f x) (b \sec (e+f x))^{7/2}}{4 b^3 f}-\frac{7 \cot ^2(e+f x) (b \sec (e+f x))^{3/2}}{16 b f}+\frac{21 \sqrt{b} \tan ^{-1}\left(\frac{\sqrt{b \sec (e+f x)}}{\sqrt{b}}\right)}{32 f}-\frac{21 \sqrt{b} \tanh ^{-1}\left(\frac{\sqrt{b \sec (e+f x)}}{\sqrt{b}}\right)}{32 f}",1,"(b*(-28*Csc[e + f*x]^2 - 16*Csc[e + f*x]^4 + 42*ArcTan[Sqrt[Sec[e + f*x]]]*Sqrt[Sec[e + f*x]] + 21*(Log[1 - Sqrt[Sec[e + f*x]]] - Log[1 + Sqrt[Sec[e + f*x]]])*Sqrt[Sec[e + f*x]]))/(64*f*Sqrt[b*Sec[e + f*x]])","A",1
378,1,73,123,0.1532075,"\int \sqrt{b \sec (e+f x)} \sin ^6(e+f x) \, dx","Integrate[Sqrt[b*Sec[e + f*x]]*Sin[e + f*x]^6,x]","\frac{\sqrt{b \sec (e+f x)} \left(-435 \sin (2 (e+f x))+68 \sin (4 (e+f x))-7 \sin (6 (e+f x))+1280 \sqrt{\cos (e+f x)} F\left(\left.\frac{1}{2} (e+f x)\right|2\right)\right)}{1232 f}","-\frac{2 b \sin ^5(e+f x)}{11 f \sqrt{b \sec (e+f x)}}-\frac{20 b \sin ^3(e+f x)}{77 f \sqrt{b \sec (e+f x)}}-\frac{40 b \sin (e+f x)}{77 f \sqrt{b \sec (e+f x)}}+\frac{80 \sqrt{\cos (e+f x)} F\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{b \sec (e+f x)}}{77 f}",1,"(Sqrt[b*Sec[e + f*x]]*(1280*Sqrt[Cos[e + f*x]]*EllipticF[(e + f*x)/2, 2] - 435*Sin[2*(e + f*x)] + 68*Sin[4*(e + f*x)] - 7*Sin[6*(e + f*x)]))/(1232*f)","A",1
379,1,61,95,0.1102896,"\int \sqrt{b \sec (e+f x)} \sin ^4(e+f x) \, dx","Integrate[Sqrt[b*Sec[e + f*x]]*Sin[e + f*x]^4,x]","\frac{\sqrt{b \sec (e+f x)} \left(-10 \sin (2 (e+f x))+\sin (4 (e+f x))+32 \sqrt{\cos (e+f x)} F\left(\left.\frac{1}{2} (e+f x)\right|2\right)\right)}{28 f}","-\frac{2 b \sin ^3(e+f x)}{7 f \sqrt{b \sec (e+f x)}}-\frac{4 b \sin (e+f x)}{7 f \sqrt{b \sec (e+f x)}}+\frac{8 \sqrt{\cos (e+f x)} F\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{b \sec (e+f x)}}{7 f}",1,"(Sqrt[b*Sec[e + f*x]]*(32*Sqrt[Cos[e + f*x]]*EllipticF[(e + f*x)/2, 2] - 10*Sin[2*(e + f*x)] + Sin[4*(e + f*x)]))/(28*f)","A",1
380,1,51,67,0.1245731,"\int \sqrt{b \sec (e+f x)} \sin ^2(e+f x) \, dx","Integrate[Sqrt[b*Sec[e + f*x]]*Sin[e + f*x]^2,x]","-\frac{\sqrt{b \sec (e+f x)} \left(\sin (2 (e+f x))-4 \sqrt{\cos (e+f x)} F\left(\left.\frac{1}{2} (e+f x)\right|2\right)\right)}{3 f}","\frac{4 \sqrt{\cos (e+f x)} F\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{b \sec (e+f x)}}{3 f}-\frac{2 b \sin (e+f x)}{3 f \sqrt{b \sec (e+f x)}}",1,"-1/3*(Sqrt[b*Sec[e + f*x]]*(-4*Sqrt[Cos[e + f*x]]*EllipticF[(e + f*x)/2, 2] + Sin[2*(e + f*x)]))/f","A",1
381,1,38,38,0.0382387,"\int \sqrt{b \sec (e+f x)} \, dx","Integrate[Sqrt[b*Sec[e + f*x]],x]","\frac{2 \sqrt{\cos (e+f x)} F\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{b \sec (e+f x)}}{f}","\frac{2 \sqrt{\cos (e+f x)} F\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{b \sec (e+f x)}}{f}",1,"(2*Sqrt[Cos[e + f*x]]*EllipticF[(e + f*x)/2, 2]*Sqrt[b*Sec[e + f*x]])/f","A",1
382,1,47,62,0.1157094,"\int \csc ^2(e+f x) \sqrt{b \sec (e+f x)} \, dx","Integrate[Csc[e + f*x]^2*Sqrt[b*Sec[e + f*x]],x]","\frac{\sqrt{b \sec (e+f x)} \left(\sqrt{\cos (e+f x)} F\left(\left.\frac{1}{2} (e+f x)\right|2\right)-\cot (e+f x)\right)}{f}","\frac{\sqrt{\cos (e+f x)} F\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{b \sec (e+f x)}}{f}-\frac{b \csc (e+f x)}{f \sqrt{b \sec (e+f x)}}",1,"((-Cot[e + f*x] + Sqrt[Cos[e + f*x]]*EllipticF[(e + f*x)/2, 2])*Sqrt[b*Sec[e + f*x]])/f","A",1
383,1,63,95,0.2416752,"\int \csc ^4(e+f x) \sqrt{b \sec (e+f x)} \, dx","Integrate[Csc[e + f*x]^4*Sqrt[b*Sec[e + f*x]],x]","\frac{\sqrt{b \sec (e+f x)} \left(5 \sqrt{\cos (e+f x)} F\left(\left.\frac{1}{2} (e+f x)\right|2\right)-\cot (e+f x) \left(2 \csc ^2(e+f x)+5\right)\right)}{6 f}","-\frac{b \csc ^3(e+f x)}{3 f \sqrt{b \sec (e+f x)}}-\frac{5 b \csc (e+f x)}{6 f \sqrt{b \sec (e+f x)}}+\frac{5 \sqrt{\cos (e+f x)} F\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{b \sec (e+f x)}}{6 f}",1,"((-(Cot[e + f*x]*(5 + 2*Csc[e + f*x]^2)) + 5*Sqrt[Cos[e + f*x]]*EllipticF[(e + f*x)/2, 2])*Sqrt[b*Sec[e + f*x]])/(6*f)","A",1
384,1,73,123,0.4818985,"\int \csc ^6(e+f x) \sqrt{b \sec (e+f x)} \, dx","Integrate[Csc[e + f*x]^6*Sqrt[b*Sec[e + f*x]],x]","\frac{\sqrt{b \sec (e+f x)} \left(15 \sqrt{\cos (e+f x)} F\left(\left.\frac{1}{2} (e+f x)\right|2\right)-\cot (e+f x) \left(4 \csc ^4(e+f x)+6 \csc ^2(e+f x)+15\right)\right)}{20 f}","-\frac{b \csc ^5(e+f x)}{5 f \sqrt{b \sec (e+f x)}}-\frac{3 b \csc ^3(e+f x)}{10 f \sqrt{b \sec (e+f x)}}-\frac{3 b \csc (e+f x)}{4 f \sqrt{b \sec (e+f x)}}+\frac{3 \sqrt{\cos (e+f x)} F\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{b \sec (e+f x)}}{4 f}",1,"((-(Cot[e + f*x]*(15 + 6*Csc[e + f*x]^2 + 4*Csc[e + f*x]^4)) + 15*Sqrt[Cos[e + f*x]]*EllipticF[(e + f*x)/2, 2])*Sqrt[b*Sec[e + f*x]])/(20*f)","A",1
385,1,52,83,0.1700267,"\int (b \sec (e+f x))^{3/2} \sin ^7(e+f x) \, dx","Integrate[(b*Sec[e + f*x])^(3/2)*Sin[e + f*x]^7,x]","\frac{b (809 \cos (2 (e+f x))-90 \cos (4 (e+f x))+7 \cos (6 (e+f x))+3370) \sqrt{b \sec (e+f x)}}{1232 f}","\frac{2 b^7}{11 f (b \sec (e+f x))^{11/2}}-\frac{6 b^5}{7 f (b \sec (e+f x))^{7/2}}+\frac{2 b^3}{f (b \sec (e+f x))^{3/2}}+\frac{2 b \sqrt{b \sec (e+f x)}}{f}",1,"(b*(3370 + 809*Cos[2*(e + f*x)] - 90*Cos[4*(e + f*x)] + 7*Cos[6*(e + f*x)])*Sqrt[b*Sec[e + f*x]])/(1232*f)","A",1
386,1,42,63,0.1095643,"\int (b \sec (e+f x))^{3/2} \sin ^5(e+f x) \, dx","Integrate[(b*Sec[e + f*x])^(3/2)*Sin[e + f*x]^5,x]","\frac{b (44 \cos (2 (e+f x))-3 \cos (4 (e+f x))+215) \sqrt{b \sec (e+f x)}}{84 f}","-\frac{2 b^5}{7 f (b \sec (e+f x))^{7/2}}+\frac{4 b^3}{3 f (b \sec (e+f x))^{3/2}}+\frac{2 b \sqrt{b \sec (e+f x)}}{f}",1,"(b*(215 + 44*Cos[2*(e + f*x)] - 3*Cos[4*(e + f*x)])*Sqrt[b*Sec[e + f*x]])/(84*f)","A",1
387,1,30,41,0.0744941,"\int (b \sec (e+f x))^{3/2} \sin ^3(e+f x) \, dx","Integrate[(b*Sec[e + f*x])^(3/2)*Sin[e + f*x]^3,x]","\frac{b (\cos (2 (e+f x))+7) \sqrt{b \sec (e+f x)}}{3 f}","\frac{2 b^3}{3 f (b \sec (e+f x))^{3/2}}+\frac{2 b \sqrt{b \sec (e+f x)}}{f}",1,"(b*(7 + Cos[2*(e + f*x)])*Sqrt[b*Sec[e + f*x]])/(3*f)","A",1
388,1,18,18,0.0326522,"\int (b \sec (e+f x))^{3/2} \sin (e+f x) \, dx","Integrate[(b*Sec[e + f*x])^(3/2)*Sin[e + f*x],x]","\frac{2 b \sqrt{b \sec (e+f x)}}{f}","\frac{2 b \sqrt{b \sec (e+f x)}}{f}",1,"(2*b*Sqrt[b*Sec[e + f*x]])/f","A",1
389,1,85,77,0.9592675,"\int \csc (e+f x) (b \sec (e+f x))^{3/2} \, dx","Integrate[Csc[e + f*x]*(b*Sec[e + f*x])^(3/2),x]","\frac{(b \sec (e+f x))^{3/2} \left(4 \sqrt{\sec (e+f x)}+\log \left(1-\sqrt{\sec (e+f x)}\right)-\log \left(\sqrt{\sec (e+f x)}+1\right)-2 \tan ^{-1}\left(\sqrt{\sec (e+f x)}\right)\right)}{2 f \sec ^{\frac{3}{2}}(e+f x)}","-\frac{b^{3/2} \tan ^{-1}\left(\frac{\sqrt{b \sec (e+f x)}}{\sqrt{b}}\right)}{f}-\frac{b^{3/2} \tanh ^{-1}\left(\frac{\sqrt{b \sec (e+f x)}}{\sqrt{b}}\right)}{f}+\frac{2 b \sqrt{b \sec (e+f x)}}{f}",1,"((-2*ArcTan[Sqrt[Sec[e + f*x]]] + Log[1 - Sqrt[Sec[e + f*x]]] - Log[1 + Sqrt[Sec[e + f*x]]] + 4*Sqrt[Sec[e + f*x]])*(b*Sec[e + f*x])^(3/2))/(2*f*Sec[e + f*x]^(3/2))","A",1
390,1,97,113,2.3272626,"\int \csc ^3(e+f x) (b \sec (e+f x))^{3/2} \, dx","Integrate[Csc[e + f*x]^3*(b*Sec[e + f*x])^(3/2),x]","-\frac{(b \sec (e+f x))^{3/2} \left(-5 \log \left(1-\sqrt{\sec (e+f x)}\right)+5 \log \left(\sqrt{\sec (e+f x)}+1\right)+4 \left(\csc ^2(e+f x)-5\right) \sqrt{\sec (e+f x)}+10 \tan ^{-1}\left(\sqrt{\sec (e+f x)}\right)\right)}{8 f \sec ^{\frac{3}{2}}(e+f x)}","-\frac{5 b^{3/2} \tan ^{-1}\left(\frac{\sqrt{b \sec (e+f x)}}{\sqrt{b}}\right)}{4 f}-\frac{5 b^{3/2} \tanh ^{-1}\left(\frac{\sqrt{b \sec (e+f x)}}{\sqrt{b}}\right)}{4 f}+\frac{5 b \sqrt{b \sec (e+f x)}}{2 f}-\frac{\cot ^2(e+f x) (b \sec (e+f x))^{5/2}}{2 b f}",1,"-1/8*((10*ArcTan[Sqrt[Sec[e + f*x]]] - 5*Log[1 - Sqrt[Sec[e + f*x]]] + 5*Log[1 + Sqrt[Sec[e + f*x]]] + 4*(-5 + Csc[e + f*x]^2)*Sqrt[Sec[e + f*x]])*(b*Sec[e + f*x])^(3/2))/(f*Sec[e + f*x]^(3/2))","A",1
391,1,70,128,0.1462611,"\int (b \sec (e+f x))^{3/2} \sin ^6(e+f x) \, dx","Integrate[(b*Sec[e + f*x])^(3/2)*Sin[e + f*x]^6,x]","-\frac{b \sqrt{b \sec (e+f x)} \left(-158 \sin (e+f x)-13 \sin (3 (e+f x))+\sin (5 (e+f x))+384 \sqrt{\cos (e+f x)} E\left(\left.\frac{1}{2} (e+f x)\right|2\right)\right)}{72 f}","\frac{20 b^3 \sin ^3(e+f x)}{9 f (b \sec (e+f x))^{3/2}}+\frac{8 b^3 \sin (e+f x)}{3 f (b \sec (e+f x))^{3/2}}-\frac{16 b^2 E\left(\left.\frac{1}{2} (e+f x)\right|2\right)}{3 f \sqrt{\cos (e+f x)} \sqrt{b \sec (e+f x)}}+\frac{2 b \sin ^5(e+f x) \sqrt{b \sec (e+f x)}}{f}",1,"-1/72*(b*Sqrt[b*Sec[e + f*x]]*(384*Sqrt[Cos[e + f*x]]*EllipticE[(e + f*x)/2, 2] - 158*Sin[e + f*x] - 13*Sin[3*(e + f*x)] + Sin[5*(e + f*x)]))/f","A",1
392,1,60,98,0.1292109,"\int (b \sec (e+f x))^{3/2} \sin ^4(e+f x) \, dx","Integrate[(b*Sec[e + f*x])^(3/2)*Sin[e + f*x]^4,x]","\frac{b \sqrt{b \sec (e+f x)} \left(21 \sin (e+f x)+\sin (3 (e+f x))-48 \sqrt{\cos (e+f x)} E\left(\left.\frac{1}{2} (e+f x)\right|2\right)\right)}{10 f}","\frac{12 b^3 \sin (e+f x)}{5 f (b \sec (e+f x))^{3/2}}-\frac{24 b^2 E\left(\left.\frac{1}{2} (e+f x)\right|2\right)}{5 f \sqrt{\cos (e+f x)} \sqrt{b \sec (e+f x)}}+\frac{2 b \sin ^3(e+f x) \sqrt{b \sec (e+f x)}}{f}",1,"(b*Sqrt[b*Sec[e + f*x]]*(-48*Sqrt[Cos[e + f*x]]*EllipticE[(e + f*x)/2, 2] + 21*Sin[e + f*x] + Sin[3*(e + f*x)]))/(10*f)","A",1
393,1,48,66,0.0831482,"\int (b \sec (e+f x))^{3/2} \sin ^2(e+f x) \, dx","Integrate[(b*Sec[e + f*x])^(3/2)*Sin[e + f*x]^2,x]","\frac{2 b \sqrt{b \sec (e+f x)} \left(\sin (e+f x)-2 \sqrt{\cos (e+f x)} E\left(\left.\frac{1}{2} (e+f x)\right|2\right)\right)}{f}","\frac{2 b \sin (e+f x) \sqrt{b \sec (e+f x)}}{f}-\frac{4 b^2 E\left(\left.\frac{1}{2} (e+f x)\right|2\right)}{f \sqrt{\cos (e+f x)} \sqrt{b \sec (e+f x)}}",1,"(2*b*Sqrt[b*Sec[e + f*x]]*(-2*Sqrt[Cos[e + f*x]]*EllipticE[(e + f*x)/2, 2] + Sin[e + f*x]))/f","A",1
394,1,48,66,0.057148,"\int (b \sec (e+f x))^{3/2} \, dx","Integrate[(b*Sec[e + f*x])^(3/2),x]","\frac{2 b \sqrt{b \sec (e+f x)} \left(\sin (e+f x)-\sqrt{\cos (e+f x)} E\left(\left.\frac{1}{2} (e+f x)\right|2\right)\right)}{f}","\frac{2 b \sin (e+f x) \sqrt{b \sec (e+f x)}}{f}-\frac{2 b^2 E\left(\left.\frac{1}{2} (e+f x)\right|2\right)}{f \sqrt{\cos (e+f x)} \sqrt{b \sec (e+f x)}}",1,"(2*b*Sqrt[b*Sec[e + f*x]]*(-(Sqrt[Cos[e + f*x]]*EllipticE[(e + f*x)/2, 2]) + Sin[e + f*x]))/f","A",1
395,1,57,90,0.1519199,"\int \csc ^2(e+f x) (b \sec (e+f x))^{3/2} \, dx","Integrate[Csc[e + f*x]^2*(b*Sec[e + f*x])^(3/2),x]","\frac{b \sqrt{b \sec (e+f x)} \left(3 \sin (e+f x)-\csc (e+f x)-3 \sqrt{\cos (e+f x)} E\left(\left.\frac{1}{2} (e+f x)\right|2\right)\right)}{f}","-\frac{3 b^2 E\left(\left.\frac{1}{2} (e+f x)\right|2\right)}{f \sqrt{\cos (e+f x)} \sqrt{b \sec (e+f x)}}-\frac{b \csc (e+f x) \sqrt{b \sec (e+f x)}}{f}+\frac{3 b \sin (e+f x) \sqrt{b \sec (e+f x)}}{f}",1,"(b*Sqrt[b*Sec[e + f*x]]*(-Csc[e + f*x] - 3*Sqrt[Cos[e + f*x]]*EllipticE[(e + f*x)/2, 2] + 3*Sin[e + f*x]))/f","A",1
396,1,77,124,0.2073668,"\int \csc ^4(e+f x) (b \sec (e+f x))^{3/2} \, dx","Integrate[Csc[e + f*x]^4*(b*Sec[e + f*x])^(3/2),x]","-\frac{b \sin (e+f x) \sqrt{b \sec (e+f x)} \left(2 \csc ^4(e+f x)+7 \csc ^2(e+f x)+21 \sqrt{\cos (e+f x)} \csc (e+f x) E\left(\left.\frac{1}{2} (e+f x)\right|2\right)-21\right)}{6 f}","-\frac{7 b^2 E\left(\left.\frac{1}{2} (e+f x)\right|2\right)}{2 f \sqrt{\cos (e+f x)} \sqrt{b \sec (e+f x)}}-\frac{b \csc ^3(e+f x) \sqrt{b \sec (e+f x)}}{3 f}-\frac{7 b \csc (e+f x) \sqrt{b \sec (e+f x)}}{6 f}+\frac{7 b \sin (e+f x) \sqrt{b \sec (e+f x)}}{2 f}",1,"-1/6*(b*(-21 + 7*Csc[e + f*x]^2 + 2*Csc[e + f*x]^4 + 21*Sqrt[Cos[e + f*x]]*Csc[e + f*x]*EllipticE[(e + f*x)/2, 2])*Sqrt[b*Sec[e + f*x]]*Sin[e + f*x])/f","A",1
397,1,52,85,0.4210278,"\int (b \sec (e+f x))^{5/2} \sin ^7(e+f x) \, dx","Integrate[(b*Sec[e + f*x])^(5/2)*Sin[e + f*x]^7,x]","\frac{b (1803 \cos (2 (e+f x))-78 \cos (4 (e+f x))+5 \cos (6 (e+f x))+2366) (b \sec (e+f x))^{3/2}}{720 f}","\frac{2 b^7}{9 f (b \sec (e+f x))^{9/2}}-\frac{6 b^5}{5 f (b \sec (e+f x))^{5/2}}+\frac{6 b^3}{f \sqrt{b \sec (e+f x)}}+\frac{2 b (b \sec (e+f x))^{3/2}}{3 f}",1,"(b*(2366 + 1803*Cos[2*(e + f*x)] - 78*Cos[4*(e + f*x)] + 5*Cos[6*(e + f*x)])*(b*Sec[e + f*x])^(3/2))/(720*f)","A",1
398,1,42,63,0.3317038,"\int (b \sec (e+f x))^{5/2} \sin ^5(e+f x) \, dx","Integrate[(b*Sec[e + f*x])^(5/2)*Sin[e + f*x]^5,x]","\frac{b (108 \cos (2 (e+f x))-3 \cos (4 (e+f x))+151) (b \sec (e+f x))^{3/2}}{60 f}","-\frac{2 b^5}{5 f (b \sec (e+f x))^{5/2}}+\frac{4 b^3}{f \sqrt{b \sec (e+f x)}}+\frac{2 b (b \sec (e+f x))^{3/2}}{3 f}",1,"(b*(151 + 108*Cos[2*(e + f*x)] - 3*Cos[4*(e + f*x)])*(b*Sec[e + f*x])^(3/2))/(60*f)","A",1
399,1,32,41,0.1966417,"\int (b \sec (e+f x))^{5/2} \sin ^3(e+f x) \, dx","Integrate[(b*Sec[e + f*x])^(5/2)*Sin[e + f*x]^3,x]","\frac{b (3 \cos (2 (e+f x))+5) (b \sec (e+f x))^{3/2}}{3 f}","\frac{2 b^3}{f \sqrt{b \sec (e+f x)}}+\frac{2 b (b \sec (e+f x))^{3/2}}{3 f}",1,"(b*(5 + 3*Cos[2*(e + f*x)])*(b*Sec[e + f*x])^(3/2))/(3*f)","A",1
400,1,20,20,0.035696,"\int (b \sec (e+f x))^{5/2} \sin (e+f x) \, dx","Integrate[(b*Sec[e + f*x])^(5/2)*Sin[e + f*x],x]","\frac{2 b (b \sec (e+f x))^{3/2}}{3 f}","\frac{2 b (b \sec (e+f x))^{3/2}}{3 f}",1,"(2*b*(b*Sec[e + f*x])^(3/2))/(3*f)","A",1
401,1,87,78,0.180059,"\int \csc (e+f x) (b \sec (e+f x))^{5/2} \, dx","Integrate[Csc[e + f*x]*(b*Sec[e + f*x])^(5/2),x]","\frac{(b \sec (e+f x))^{5/2} \left(4 \sec ^{\frac{3}{2}}(e+f x)+3 \log \left(1-\sqrt{\sec (e+f x)}\right)-3 \log \left(\sqrt{\sec (e+f x)}+1\right)+6 \tan ^{-1}\left(\sqrt{\sec (e+f x)}\right)\right)}{6 f \sec ^{\frac{5}{2}}(e+f x)}","\frac{b^{5/2} \tan ^{-1}\left(\frac{\sqrt{b \sec (e+f x)}}{\sqrt{b}}\right)}{f}-\frac{b^{5/2} \tanh ^{-1}\left(\frac{\sqrt{b \sec (e+f x)}}{\sqrt{b}}\right)}{f}+\frac{2 b (b \sec (e+f x))^{3/2}}{3 f}",1,"((b*Sec[e + f*x])^(5/2)*(6*ArcTan[Sqrt[Sec[e + f*x]]] + 3*Log[1 - Sqrt[Sec[e + f*x]]] - 3*Log[1 + Sqrt[Sec[e + f*x]]] + 4*Sec[e + f*x]^(3/2)))/(6*f*Sec[e + f*x]^(5/2))","A",1
402,1,109,113,1.8962909,"\int \csc ^3(e+f x) (b \sec (e+f x))^{5/2} \, dx","Integrate[Csc[e + f*x]^3*(b*Sec[e + f*x])^(5/2),x]","\frac{b^3 \left(-12 \csc ^2(e+f x)+16 \sec ^2(e+f x)+21 \sqrt{\sec (e+f x)} \left(\log \left(1-\sqrt{\sec (e+f x)}\right)-\log \left(\sqrt{\sec (e+f x)}+1\right)\right)+42 \sqrt{\sec (e+f x)} \tan ^{-1}\left(\sqrt{\sec (e+f x)}\right)\right)}{24 f \sqrt{b \sec (e+f x)}}","\frac{7 b^{5/2} \tan ^{-1}\left(\frac{\sqrt{b \sec (e+f x)}}{\sqrt{b}}\right)}{4 f}-\frac{7 b^{5/2} \tanh ^{-1}\left(\frac{\sqrt{b \sec (e+f x)}}{\sqrt{b}}\right)}{4 f}+\frac{7 b (b \sec (e+f x))^{3/2}}{6 f}-\frac{\cot ^2(e+f x) (b \sec (e+f x))^{7/2}}{2 b f}",1,"(b^3*(-12*Csc[e + f*x]^2 + 42*ArcTan[Sqrt[Sec[e + f*x]]]*Sqrt[Sec[e + f*x]] + 21*(Log[1 - Sqrt[Sec[e + f*x]]] - Log[1 + Sqrt[Sec[e + f*x]]])*Sqrt[Sec[e + f*x]] + 16*Sec[e + f*x]^2))/(24*f*Sqrt[b*Sec[e + f*x]])","A",1
403,1,119,143,1.3183398,"\int \csc ^5(e+f x) (b \sec (e+f x))^{5/2} \, dx","Integrate[Csc[e + f*x]^5*(b*Sec[e + f*x])^(5/2),x]","\frac{b^3 \left(-48 \csc ^4(e+f x)-180 \csc ^2(e+f x)+128 \sec ^2(e+f x)+231 \sqrt{\sec (e+f x)} \left(\log \left(1-\sqrt{\sec (e+f x)}\right)-\log \left(\sqrt{\sec (e+f x)}+1\right)\right)+462 \sqrt{\sec (e+f x)} \tan ^{-1}\left(\sqrt{\sec (e+f x)}\right)\right)}{192 f \sqrt{b \sec (e+f x)}}","\frac{77 b^{5/2} \tan ^{-1}\left(\frac{\sqrt{b \sec (e+f x)}}{\sqrt{b}}\right)}{32 f}-\frac{77 b^{5/2} \tanh ^{-1}\left(\frac{\sqrt{b \sec (e+f x)}}{\sqrt{b}}\right)}{32 f}-\frac{\cot ^4(e+f x) (b \sec (e+f x))^{11/2}}{4 b^3 f}+\frac{77 b (b \sec (e+f x))^{3/2}}{48 f}-\frac{11 \cot ^2(e+f x) (b \sec (e+f x))^{7/2}}{16 b f}",1,"(b^3*(-180*Csc[e + f*x]^2 - 48*Csc[e + f*x]^4 + 462*ArcTan[Sqrt[Sec[e + f*x]]]*Sqrt[Sec[e + f*x]] + 231*(Log[1 - Sqrt[Sec[e + f*x]]] - Log[1 + Sqrt[Sec[e + f*x]]])*Sqrt[Sec[e + f*x]] + 128*Sec[e + f*x]^2))/(192*f*Sqrt[b*Sec[e + f*x]])","A",1
404,1,74,130,0.1949433,"\int (b \sec (e+f x))^{5/2} \sin ^6(e+f x) \, dx","Integrate[(b*Sec[e + f*x])^(5/2)*Sin[e + f*x]^6,x]","-\frac{b^2 \sqrt{b \sec (e+f x)} \left(-58 \sin (2 (e+f x))+3 \sin (4 (e+f x))-56 \tan (e+f x)+320 \sqrt{\cos (e+f x)} F\left(\left.\frac{1}{2} (e+f x)\right|2\right)\right)}{84 f}","\frac{20 b^3 \sin ^3(e+f x)}{21 f \sqrt{b \sec (e+f x)}}+\frac{40 b^3 \sin (e+f x)}{21 f \sqrt{b \sec (e+f x)}}-\frac{80 b^2 \sqrt{\cos (e+f x)} F\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{b \sec (e+f x)}}{21 f}+\frac{2 b \sin ^5(e+f x) (b \sec (e+f x))^{3/2}}{3 f}",1,"-1/84*(b^2*Sqrt[b*Sec[e + f*x]]*(320*Sqrt[Cos[e + f*x]]*EllipticF[(e + f*x)/2, 2] - 58*Sin[2*(e + f*x)] + 3*Sin[4*(e + f*x)] - 56*Tan[e + f*x]))/f","A",1
405,1,64,100,0.12799,"\int (b \sec (e+f x))^{5/2} \sin ^4(e+f x) \, dx","Integrate[(b*Sec[e + f*x])^(5/2)*Sin[e + f*x]^4,x]","-\frac{b^2 \sqrt{b \sec (e+f x)} \left(-\sin (2 (e+f x))-2 \tan (e+f x)+8 \sqrt{\cos (e+f x)} F\left(\left.\frac{1}{2} (e+f x)\right|2\right)\right)}{3 f}","\frac{4 b^3 \sin (e+f x)}{3 f \sqrt{b \sec (e+f x)}}-\frac{8 b^2 \sqrt{\cos (e+f x)} F\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{b \sec (e+f x)}}{3 f}+\frac{2 b \sin ^3(e+f x) (b \sec (e+f x))^{3/2}}{3 f}",1,"-1/3*(b^2*Sqrt[b*Sec[e + f*x]]*(8*Sqrt[Cos[e + f*x]]*EllipticF[(e + f*x)/2, 2] - Sin[2*(e + f*x)] - 2*Tan[e + f*x]))/f","A",1
406,1,52,70,0.1186445,"\int (b \sec (e+f x))^{5/2} \sin ^2(e+f x) \, dx","Integrate[(b*Sec[e + f*x])^(5/2)*Sin[e + f*x]^2,x]","\frac{2 b^2 \sqrt{b \sec (e+f x)} \left(\tan (e+f x)-2 \sqrt{\cos (e+f x)} F\left(\left.\frac{1}{2} (e+f x)\right|2\right)\right)}{3 f}","\frac{2 b \sin (e+f x) (b \sec (e+f x))^{3/2}}{3 f}-\frac{4 b^2 \sqrt{\cos (e+f x)} F\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{b \sec (e+f x)}}{3 f}",1,"(2*b^2*Sqrt[b*Sec[e + f*x]]*(-2*Sqrt[Cos[e + f*x]]*EllipticF[(e + f*x)/2, 2] + Tan[e + f*x]))/(3*f)","A",1
407,1,51,70,0.0772159,"\int (b \sec (e+f x))^{5/2} \, dx","Integrate[(b*Sec[e + f*x])^(5/2),x]","\frac{2 b^2 \sqrt{b \sec (e+f x)} \left(\tan (e+f x)+\sqrt{\cos (e+f x)} F\left(\left.\frac{1}{2} (e+f x)\right|2\right)\right)}{3 f}","\frac{2 b^2 \sqrt{\cos (e+f x)} F\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{b \sec (e+f x)}}{3 f}+\frac{2 b \sin (e+f x) (b \sec (e+f x))^{3/2}}{3 f}",1,"(2*b^2*Sqrt[b*Sec[e + f*x]]*(Sqrt[Cos[e + f*x]]*EllipticF[(e + f*x)/2, 2] + Tan[e + f*x]))/(3*f)","A",1
408,1,67,98,0.1675473,"\int \csc ^2(e+f x) (b \sec (e+f x))^{5/2} \, dx","Integrate[Csc[e + f*x]^2*(b*Sec[e + f*x])^(5/2),x]","\frac{b \sin (e+f x) (b \sec (e+f x))^{3/2} \left(-3 \cot ^2(e+f x)+5 \cos ^{\frac{3}{2}}(e+f x) \csc (e+f x) F\left(\left.\frac{1}{2} (e+f x)\right|2\right)+2\right)}{3 f}","-\frac{5 b^3 \csc (e+f x)}{3 f \sqrt{b \sec (e+f x)}}+\frac{5 b^2 \sqrt{\cos (e+f x)} F\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{b \sec (e+f x)}}{3 f}+\frac{2 b \csc (e+f x) (b \sec (e+f x))^{3/2}}{3 f}",1,"(b*(2 - 3*Cot[e + f*x]^2 + 5*Cos[e + f*x]^(3/2)*Csc[e + f*x]*EllipticF[(e + f*x)/2, 2])*(b*Sec[e + f*x])^(3/2)*Sin[e + f*x])/(3*f)","A",1
409,1,79,123,0.3943913,"\int \csc ^4(e+f x) (b \sec (e+f x))^{5/2} \, dx","Integrate[Csc[e + f*x]^4*(b*Sec[e + f*x])^(5/2),x]","\frac{b \sin (e+f x) (b \sec (e+f x))^{3/2} \left(-\left(\cot ^2(e+f x) \left(2 \csc ^2(e+f x)+11\right)\right)+15 \cos ^{\frac{3}{2}}(e+f x) \csc (e+f x) F\left(\left.\frac{1}{2} (e+f x)\right|2\right)+4\right)}{6 f}","-\frac{5 b^3 \csc (e+f x)}{2 f \sqrt{b \sec (e+f x)}}+\frac{5 b^2 \sqrt{\cos (e+f x)} F\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{b \sec (e+f x)}}{2 f}-\frac{b \csc ^3(e+f x) (b \sec (e+f x))^{3/2}}{3 f}+\frac{b \csc (e+f x) (b \sec (e+f x))^{3/2}}{f}",1,"(b*(4 - Cot[e + f*x]^2*(11 + 2*Csc[e + f*x]^2) + 15*Cos[e + f*x]^(3/2)*Csc[e + f*x]*EllipticF[(e + f*x)/2, 2])*(b*Sec[e + f*x])^(3/2)*Sin[e + f*x])/(6*f)","A",1
410,1,52,87,0.2120464,"\int \frac{\sin ^7(e+f x)}{\sqrt{b \sec (e+f x)}} \, dx","Integrate[Sin[e + f*x]^7/Sqrt[b*Sec[e + f*x]],x]","\frac{b (4035 \cos (2 (e+f x))-798 \cos (4 (e+f x))+77 \cos (6 (e+f x))-7410)}{18480 f (b \sec (e+f x))^{3/2}}","\frac{2 b^7}{15 f (b \sec (e+f x))^{15/2}}-\frac{6 b^5}{11 f (b \sec (e+f x))^{11/2}}+\frac{6 b^3}{7 f (b \sec (e+f x))^{7/2}}-\frac{2 b}{3 f (b \sec (e+f x))^{3/2}}",1,"(b*(-7410 + 4035*Cos[2*(e + f*x)] - 798*Cos[4*(e + f*x)] + 77*Cos[6*(e + f*x)]))/(18480*f*(b*Sec[e + f*x])^(3/2))","A",1
411,1,42,65,0.1665048,"\int \frac{\sin ^5(e+f x)}{\sqrt{b \sec (e+f x)}} \, dx","Integrate[Sin[e + f*x]^5/Sqrt[b*Sec[e + f*x]],x]","\frac{b (180 \cos (2 (e+f x))-21 \cos (4 (e+f x))-415)}{924 f (b \sec (e+f x))^{3/2}}","-\frac{2 b^5}{11 f (b \sec (e+f x))^{11/2}}+\frac{4 b^3}{7 f (b \sec (e+f x))^{7/2}}-\frac{2 b}{3 f (b \sec (e+f x))^{3/2}}",1,"(b*(-415 + 180*Cos[2*(e + f*x)] - 21*Cos[4*(e + f*x)]))/(924*f*(b*Sec[e + f*x])^(3/2))","A",1
412,1,32,43,0.1029784,"\int \frac{\sin ^3(e+f x)}{\sqrt{b \sec (e+f x)}} \, dx","Integrate[Sin[e + f*x]^3/Sqrt[b*Sec[e + f*x]],x]","\frac{b (3 \cos (2 (e+f x))-11)}{21 f (b \sec (e+f x))^{3/2}}","\frac{2 b^3}{7 f (b \sec (e+f x))^{7/2}}-\frac{2 b}{3 f (b \sec (e+f x))^{3/2}}",1,"(b*(-11 + 3*Cos[2*(e + f*x)]))/(21*f*(b*Sec[e + f*x])^(3/2))","A",1
413,1,20,20,0.0411592,"\int \frac{\sin (e+f x)}{\sqrt{b \sec (e+f x)}} \, dx","Integrate[Sin[e + f*x]/Sqrt[b*Sec[e + f*x]],x]","-\frac{2 b}{3 f (b \sec (e+f x))^{3/2}}","-\frac{2 b}{3 f (b \sec (e+f x))^{3/2}}",1,"(-2*b)/(3*f*(b*Sec[e + f*x])^(3/2))","A",1
414,1,73,59,0.0976567,"\int \frac{\csc (e+f x)}{\sqrt{b \sec (e+f x)}} \, dx","Integrate[Csc[e + f*x]/Sqrt[b*Sec[e + f*x]],x]","-\frac{\sqrt{\sec (e+f x)} \left(-\log \left(1-\sqrt{\sec (e+f x)}\right)+\log \left(\sqrt{\sec (e+f x)}+1\right)+2 \tan ^{-1}\left(\sqrt{\sec (e+f x)}\right)\right)}{2 f \sqrt{b \sec (e+f x)}}","-\frac{\tan ^{-1}\left(\frac{\sqrt{b \sec (e+f x)}}{\sqrt{b}}\right)}{\sqrt{b} f}-\frac{\tanh ^{-1}\left(\frac{\sqrt{b \sec (e+f x)}}{\sqrt{b}}\right)}{\sqrt{b} f}",1,"-1/2*((2*ArcTan[Sqrt[Sec[e + f*x]]] - Log[1 - Sqrt[Sec[e + f*x]]] + Log[1 + Sqrt[Sec[e + f*x]]])*Sqrt[Sec[e + f*x]])/(f*Sqrt[b*Sec[e + f*x]])","A",1
415,1,93,93,1.0972946,"\int \frac{\csc ^3(e+f x)}{\sqrt{b \sec (e+f x)}} \, dx","Integrate[Csc[e + f*x]^3/Sqrt[b*Sec[e + f*x]],x]","\frac{\sqrt{\sec (e+f x)} \left(\log \left(1-\sqrt{\sec (e+f x)}\right)-\log \left(\sqrt{\sec (e+f x)}+1\right)-\frac{4 \csc ^2(e+f x)}{\sec ^{\frac{3}{2}}(e+f x)}-2 \tan ^{-1}\left(\sqrt{\sec (e+f x)}\right)\right)}{8 f \sqrt{b \sec (e+f x)}}","-\frac{\cot ^2(e+f x) \sqrt{b \sec (e+f x)}}{2 b f}-\frac{\tan ^{-1}\left(\frac{\sqrt{b \sec (e+f x)}}{\sqrt{b}}\right)}{4 \sqrt{b} f}-\frac{\tanh ^{-1}\left(\frac{\sqrt{b \sec (e+f x)}}{\sqrt{b}}\right)}{4 \sqrt{b} f}",1,"((-2*ArcTan[Sqrt[Sec[e + f*x]]] + Log[1 - Sqrt[Sec[e + f*x]]] - Log[1 + Sqrt[Sec[e + f*x]]] - (4*Csc[e + f*x]^2)/Sec[e + f*x]^(3/2))*Sqrt[Sec[e + f*x]])/(8*f*Sqrt[b*Sec[e + f*x]])","A",1
416,1,107,123,1.9199024,"\int \frac{\csc ^5(e+f x)}{\sqrt{b \sec (e+f x)}} \, dx","Integrate[Csc[e + f*x]^5/Sqrt[b*Sec[e + f*x]],x]","-\frac{\sqrt{\sec (e+f x)} \left(-5 \log \left(1-\sqrt{\sec (e+f x)}\right)+5 \log \left(\sqrt{\sec (e+f x)}+1\right)+4 \left(4 \csc ^4(e+f x)+\csc ^2(e+f x)-5\right) \sqrt{\sec (e+f x)}+10 \tan ^{-1}\left(\sqrt{\sec (e+f x)}\right)\right)}{64 f \sqrt{b \sec (e+f x)}}","-\frac{\cot ^4(e+f x) (b \sec (e+f x))^{5/2}}{4 b^3 f}-\frac{5 \cot ^2(e+f x) \sqrt{b \sec (e+f x)}}{16 b f}-\frac{5 \tan ^{-1}\left(\frac{\sqrt{b \sec (e+f x)}}{\sqrt{b}}\right)}{32 \sqrt{b} f}-\frac{5 \tanh ^{-1}\left(\frac{\sqrt{b \sec (e+f x)}}{\sqrt{b}}\right)}{32 \sqrt{b} f}",1,"-1/64*((10*ArcTan[Sqrt[Sec[e + f*x]]] - 5*Log[1 - Sqrt[Sec[e + f*x]]] + 5*Log[1 + Sqrt[Sec[e + f*x]]] + 4*(-5 + Csc[e + f*x]^2 + 4*Csc[e + f*x]^4)*Sqrt[Sec[e + f*x]])*Sqrt[Sec[e + f*x]])/(f*Sqrt[b*Sec[e + f*x]])","A",1
417,1,73,123,0.4609598,"\int \frac{\sin ^6(e+f x)}{\sqrt{b \sec (e+f x)}} \, dx","Integrate[Sin[e + f*x]^6/Sqrt[b*Sec[e + f*x]],x]","\frac{-317 \sin (2 (e+f x))+76 \sin (4 (e+f x))-9 \sin (6 (e+f x))+\frac{768 E\left(\left.\frac{1}{2} (e+f x)\right|2\right)}{\sqrt{\cos (e+f x)}}}{1872 f \sqrt{b \sec (e+f x)}}","-\frac{2 b \sin ^5(e+f x)}{13 f (b \sec (e+f x))^{3/2}}-\frac{20 b \sin ^3(e+f x)}{117 f (b \sec (e+f x))^{3/2}}-\frac{8 b \sin (e+f x)}{39 f (b \sec (e+f x))^{3/2}}+\frac{16 E\left(\left.\frac{1}{2} (e+f x)\right|2\right)}{39 f \sqrt{\cos (e+f x)} \sqrt{b \sec (e+f x)}}",1,"((768*EllipticE[(e + f*x)/2, 2])/Sqrt[Cos[e + f*x]] - 317*Sin[2*(e + f*x)] + 76*Sin[4*(e + f*x)] - 9*Sin[6*(e + f*x)])/(1872*f*Sqrt[b*Sec[e + f*x]])","A",1
418,1,63,95,0.333997,"\int \frac{\sin ^4(e+f x)}{\sqrt{b \sec (e+f x)}} \, dx","Integrate[Sin[e + f*x]^4/Sqrt[b*Sec[e + f*x]],x]","\frac{-68 \sin (2 (e+f x))+10 \sin (4 (e+f x))+\frac{192 E\left(\left.\frac{1}{2} (e+f x)\right|2\right)}{\sqrt{\cos (e+f x)}}}{360 f \sqrt{b \sec (e+f x)}}","-\frac{2 b \sin ^3(e+f x)}{9 f (b \sec (e+f x))^{3/2}}-\frac{4 b \sin (e+f x)}{15 f (b \sec (e+f x))^{3/2}}+\frac{8 E\left(\left.\frac{1}{2} (e+f x)\right|2\right)}{15 f \sqrt{\cos (e+f x)} \sqrt{b \sec (e+f x)}}",1,"((192*EllipticE[(e + f*x)/2, 2])/Sqrt[Cos[e + f*x]] - 68*Sin[2*(e + f*x)] + 10*Sin[4*(e + f*x)])/(360*f*Sqrt[b*Sec[e + f*x]])","A",1
419,1,60,67,0.1361182,"\int \frac{\sin ^2(e+f x)}{\sqrt{b \sec (e+f x)}} \, dx","Integrate[Sin[e + f*x]^2/Sqrt[b*Sec[e + f*x]],x]","-\frac{\sqrt{b \sec (e+f x)} \left(\sin (e+f x)+\sin (3 (e+f x))-8 \sqrt{\cos (e+f x)} E\left(\left.\frac{1}{2} (e+f x)\right|2\right)\right)}{10 b f}","\frac{4 E\left(\left.\frac{1}{2} (e+f x)\right|2\right)}{5 f \sqrt{\cos (e+f x)} \sqrt{b \sec (e+f x)}}-\frac{2 b \sin (e+f x)}{5 f (b \sec (e+f x))^{3/2}}",1,"-1/10*(Sqrt[b*Sec[e + f*x]]*(-8*Sqrt[Cos[e + f*x]]*EllipticE[(e + f*x)/2, 2] + Sin[e + f*x] + Sin[3*(e + f*x)]))/(b*f)","A",1
420,1,38,38,0.0364066,"\int \frac{1}{\sqrt{b \sec (e+f x)}} \, dx","Integrate[1/Sqrt[b*Sec[e + f*x]],x]","\frac{2 E\left(\left.\frac{1}{2} (e+f x)\right|2\right)}{f \sqrt{\cos (e+f x)} \sqrt{b \sec (e+f x)}}","\frac{2 E\left(\left.\frac{1}{2} (e+f x)\right|2\right)}{f \sqrt{\cos (e+f x)} \sqrt{b \sec (e+f x)}}",1,"(2*EllipticE[(e + f*x)/2, 2])/(f*Sqrt[Cos[e + f*x]]*Sqrt[b*Sec[e + f*x]])","A",1
421,1,48,63,0.1533679,"\int \frac{\csc ^2(e+f x)}{\sqrt{b \sec (e+f x)}} \, dx","Integrate[Csc[e + f*x]^2/Sqrt[b*Sec[e + f*x]],x]","\frac{-\cot (e+f x)-\frac{E\left(\left.\frac{1}{2} (e+f x)\right|2\right)}{\sqrt{\cos (e+f x)}}}{f \sqrt{b \sec (e+f x)}}","-\frac{b \csc (e+f x)}{f (b \sec (e+f x))^{3/2}}-\frac{E\left(\left.\frac{1}{2} (e+f x)\right|2\right)}{f \sqrt{\cos (e+f x)} \sqrt{b \sec (e+f x)}}",1,"(-Cot[e + f*x] - EllipticE[(e + f*x)/2, 2]/Sqrt[Cos[e + f*x]])/(f*Sqrt[b*Sec[e + f*x]])","A",1
422,1,74,95,0.1858619,"\int \frac{\csc ^4(e+f x)}{\sqrt{b \sec (e+f x)}} \, dx","Integrate[Csc[e + f*x]^4/Sqrt[b*Sec[e + f*x]],x]","-\frac{\tan (e+f x) \left(2 \csc ^4(e+f x)+\csc ^2(e+f x)+3 \sqrt{\cos (e+f x)} \csc (e+f x) E\left(\left.\frac{1}{2} (e+f x)\right|2\right)-3\right)}{6 f \sqrt{b \sec (e+f x)}}","-\frac{b \csc ^3(e+f x)}{3 f (b \sec (e+f x))^{3/2}}-\frac{b \csc (e+f x)}{2 f (b \sec (e+f x))^{3/2}}-\frac{E\left(\left.\frac{1}{2} (e+f x)\right|2\right)}{2 f \sqrt{\cos (e+f x)} \sqrt{b \sec (e+f x)}}",1,"-1/6*((-3 + Csc[e + f*x]^2 + 2*Csc[e + f*x]^4 + 3*Sqrt[Cos[e + f*x]]*Csc[e + f*x]*EllipticE[(e + f*x)/2, 2])*Tan[e + f*x])/(f*Sqrt[b*Sec[e + f*x]])","A",1
423,1,86,123,0.1343575,"\int \frac{\csc ^6(e+f x)}{\sqrt{b \sec (e+f x)}} \, dx","Integrate[Csc[e + f*x]^6/Sqrt[b*Sec[e + f*x]],x]","-\frac{\tan (e+f x) \left(12 \csc ^6(e+f x)+2 \csc ^4(e+f x)+7 \csc ^2(e+f x)+21 \sqrt{\cos (e+f x)} \csc (e+f x) E\left(\left.\frac{1}{2} (e+f x)\right|2\right)-21\right)}{60 f \sqrt{b \sec (e+f x)}}","-\frac{b \csc ^5(e+f x)}{5 f (b \sec (e+f x))^{3/2}}-\frac{7 b \csc ^3(e+f x)}{30 f (b \sec (e+f x))^{3/2}}-\frac{7 b \csc (e+f x)}{20 f (b \sec (e+f x))^{3/2}}-\frac{7 E\left(\left.\frac{1}{2} (e+f x)\right|2\right)}{20 f \sqrt{\cos (e+f x)} \sqrt{b \sec (e+f x)}}",1,"-1/60*((-21 + 7*Csc[e + f*x]^2 + 2*Csc[e + f*x]^4 + 12*Csc[e + f*x]^6 + 21*Sqrt[Cos[e + f*x]]*Csc[e + f*x]*EllipticE[(e + f*x)/2, 2])*Tan[e + f*x])/(f*Sqrt[b*Sec[e + f*x]])","A",1
424,1,52,87,0.4274498,"\int \frac{\sin ^7(e+f x)}{(b \sec (e+f x))^{3/2}} \, dx","Integrate[Sin[e + f*x]^7/(b*Sec[e + f*x])^(3/2),x]","\frac{b (8365 \cos (2 (e+f x))-1890 \cos (4 (e+f x))+195 \cos (6 (e+f x))-10766)}{53040 f (b \sec (e+f x))^{5/2}}","\frac{2 b^7}{17 f (b \sec (e+f x))^{17/2}}-\frac{6 b^5}{13 f (b \sec (e+f x))^{13/2}}+\frac{2 b^3}{3 f (b \sec (e+f x))^{9/2}}-\frac{2 b}{5 f (b \sec (e+f x))^{5/2}}",1,"(b*(-10766 + 8365*Cos[2*(e + f*x)] - 1890*Cos[4*(e + f*x)] + 195*Cos[6*(e + f*x)]))/(53040*f*(b*Sec[e + f*x])^(5/2))","A",1
425,1,42,65,0.2515876,"\int \frac{\sin ^5(e+f x)}{(b \sec (e+f x))^{3/2}} \, dx","Integrate[Sin[e + f*x]^5/(b*Sec[e + f*x])^(3/2),x]","\frac{b (340 \cos (2 (e+f x))-45 \cos (4 (e+f x))-551)}{2340 f (b \sec (e+f x))^{5/2}}","-\frac{2 b^5}{13 f (b \sec (e+f x))^{13/2}}+\frac{4 b^3}{9 f (b \sec (e+f x))^{9/2}}-\frac{2 b}{5 f (b \sec (e+f x))^{5/2}}",1,"(b*(-551 + 340*Cos[2*(e + f*x)] - 45*Cos[4*(e + f*x)]))/(2340*f*(b*Sec[e + f*x])^(5/2))","A",1
426,1,32,43,0.1741407,"\int \frac{\sin ^3(e+f x)}{(b \sec (e+f x))^{3/2}} \, dx","Integrate[Sin[e + f*x]^3/(b*Sec[e + f*x])^(3/2),x]","\frac{b (5 \cos (2 (e+f x))-13)}{45 f (b \sec (e+f x))^{5/2}}","\frac{2 b^3}{9 f (b \sec (e+f x))^{9/2}}-\frac{2 b}{5 f (b \sec (e+f x))^{5/2}}",1,"(b*(-13 + 5*Cos[2*(e + f*x)]))/(45*f*(b*Sec[e + f*x])^(5/2))","A",1
427,1,20,20,0.0530425,"\int \frac{\sin (e+f x)}{(b \sec (e+f x))^{3/2}} \, dx","Integrate[Sin[e + f*x]/(b*Sec[e + f*x])^(3/2),x]","-\frac{2 b}{5 f (b \sec (e+f x))^{5/2}}","-\frac{2 b}{5 f (b \sec (e+f x))^{5/2}}",1,"(-2*b)/(5*f*(b*Sec[e + f*x])^(5/2))","A",1
428,1,89,78,1.9992614,"\int \frac{\csc (e+f x)}{(b \sec (e+f x))^{3/2}} \, dx","Integrate[Csc[e + f*x]/(b*Sec[e + f*x])^(3/2),x]","\frac{\sqrt{\sec (e+f x)} \left(\log \left(1-\sqrt{\sec (e+f x)}\right)-\log \left(\sqrt{\sec (e+f x)}+1\right)\right)+2 \sqrt{\sec (e+f x)} \tan ^{-1}\left(\sqrt{\sec (e+f x)}\right)+4}{2 b f \sqrt{b \sec (e+f x)}}","\frac{\tan ^{-1}\left(\frac{\sqrt{b \sec (e+f x)}}{\sqrt{b}}\right)}{b^{3/2} f}-\frac{\tanh ^{-1}\left(\frac{\sqrt{b \sec (e+f x)}}{\sqrt{b}}\right)}{b^{3/2} f}+\frac{2}{b f \sqrt{b \sec (e+f x)}}",1,"(4 + 2*ArcTan[Sqrt[Sec[e + f*x]]]*Sqrt[Sec[e + f*x]] + (Log[1 - Sqrt[Sec[e + f*x]]] - Log[1 + Sqrt[Sec[e + f*x]]])*Sqrt[Sec[e + f*x]])/(2*b*f*Sqrt[b*Sec[e + f*x]])","A",1
429,1,98,93,0.4589208,"\int \frac{\csc ^3(e+f x)}{(b \sec (e+f x))^{3/2}} \, dx","Integrate[Csc[e + f*x]^3/(b*Sec[e + f*x])^(3/2),x]","\frac{-4 \csc ^2(e+f x)+\sqrt{\sec (e+f x)} \left(\log \left(\sqrt{\sec (e+f x)}+1\right)-\log \left(1-\sqrt{\sec (e+f x)}\right)\right)-2 \sqrt{\sec (e+f x)} \tan ^{-1}\left(\sqrt{\sec (e+f x)}\right)}{8 b f \sqrt{b \sec (e+f x)}}","-\frac{\tan ^{-1}\left(\frac{\sqrt{b \sec (e+f x)}}{\sqrt{b}}\right)}{4 b^{3/2} f}+\frac{\tanh ^{-1}\left(\frac{\sqrt{b \sec (e+f x)}}{\sqrt{b}}\right)}{4 b^{3/2} f}-\frac{\cot ^2(e+f x) (b \sec (e+f x))^{3/2}}{2 b^3 f}",1,"(-4*Csc[e + f*x]^2 - 2*ArcTan[Sqrt[Sec[e + f*x]]]*Sqrt[Sec[e + f*x]] + (-Log[1 - Sqrt[Sec[e + f*x]]] + Log[1 + Sqrt[Sec[e + f*x]]])*Sqrt[Sec[e + f*x]])/(8*b*f*Sqrt[b*Sec[e + f*x]])","A",1
430,1,109,123,0.6266732,"\int \frac{\csc ^5(e+f x)}{(b \sec (e+f x))^{3/2}} \, dx","Integrate[Csc[e + f*x]^5/(b*Sec[e + f*x])^(3/2),x]","\frac{-16 \csc ^4(e+f x)+4 \csc ^2(e+f x)+3 \sqrt{\sec (e+f x)} \left(\log \left(\sqrt{\sec (e+f x)}+1\right)-\log \left(1-\sqrt{\sec (e+f x)}\right)\right)-6 \sqrt{\sec (e+f x)} \tan ^{-1}\left(\sqrt{\sec (e+f x)}\right)}{64 b f \sqrt{b \sec (e+f x)}}","-\frac{3 \tan ^{-1}\left(\frac{\sqrt{b \sec (e+f x)}}{\sqrt{b}}\right)}{32 b^{3/2} f}+\frac{3 \tanh ^{-1}\left(\frac{\sqrt{b \sec (e+f x)}}{\sqrt{b}}\right)}{32 b^{3/2} f}-\frac{\cot ^4(e+f x) (b \sec (e+f x))^{3/2}}{4 b^3 f}-\frac{3 \cot ^2(e+f x) (b \sec (e+f x))^{3/2}}{16 b^3 f}",1,"(4*Csc[e + f*x]^2 - 16*Csc[e + f*x]^4 - 6*ArcTan[Sqrt[Sec[e + f*x]]]*Sqrt[Sec[e + f*x]] + 3*(-Log[1 - Sqrt[Sec[e + f*x]]] + Log[1 + Sqrt[Sec[e + f*x]]])*Sqrt[Sec[e + f*x]])/(64*b*f*Sqrt[b*Sec[e + f*x]])","A",1
431,1,81,126,0.1634948,"\int \frac{\sin ^4(e+f x)}{(b \sec (e+f x))^{3/2}} \, dx","Integrate[Sin[e + f*x]^4/(b*Sec[e + f*x])^(3/2),x]","\frac{\sec ^2(e+f x) \left(-5 \sin (2 (e+f x))-24 \sin (4 (e+f x))+7 \sin (6 (e+f x))+128 \sqrt{\cos (e+f x)} F\left(\left.\frac{1}{2} (e+f x)\right|2\right)\right)}{1232 f (b \sec (e+f x))^{3/2}}","\frac{8 \sqrt{\cos (e+f x)} F\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{b \sec (e+f x)}}{77 b^2 f}-\frac{2 b \sin ^3(e+f x)}{11 f (b \sec (e+f x))^{5/2}}+\frac{8 \sin (e+f x)}{77 b f \sqrt{b \sec (e+f x)}}-\frac{12 b \sin (e+f x)}{77 f (b \sec (e+f x))^{5/2}}",1,"(Sec[e + f*x]^2*(128*Sqrt[Cos[e + f*x]]*EllipticF[(e + f*x)/2, 2] - 5*Sin[2*(e + f*x)] - 24*Sin[4*(e + f*x)] + 7*Sin[6*(e + f*x)]))/(1232*f*(b*Sec[e + f*x])^(3/2))","A",1
432,1,71,98,0.1322272,"\int \frac{\sin ^2(e+f x)}{(b \sec (e+f x))^{3/2}} \, dx","Integrate[Sin[e + f*x]^2/(b*Sec[e + f*x])^(3/2),x]","\frac{\sec ^2(e+f x) \left(2 \sin (2 (e+f x))-3 \sin (4 (e+f x))+16 \sqrt{\cos (e+f x)} F\left(\left.\frac{1}{2} (e+f x)\right|2\right)\right)}{84 f (b \sec (e+f x))^{3/2}}","\frac{4 \sqrt{\cos (e+f x)} F\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{b \sec (e+f x)}}{21 b^2 f}+\frac{4 \sin (e+f x)}{21 b f \sqrt{b \sec (e+f x)}}-\frac{2 b \sin (e+f x)}{7 f (b \sec (e+f x))^{5/2}}",1,"(Sec[e + f*x]^2*(16*Sqrt[Cos[e + f*x]]*EllipticF[(e + f*x)/2, 2] + 2*Sin[2*(e + f*x)] - 3*Sin[4*(e + f*x)]))/(84*f*(b*Sec[e + f*x])^(3/2))","A",1
433,1,59,72,0.0619877,"\int \frac{1}{(b \sec (e+f x))^{3/2}} \, dx","Integrate[(b*Sec[e + f*x])^(-3/2),x]","\frac{\sec ^2(e+f x) \left(\sin (2 (e+f x))+2 \sqrt{\cos (e+f x)} F\left(\left.\frac{1}{2} (e+f x)\right|2\right)\right)}{3 f (b \sec (e+f x))^{3/2}}","\frac{2 \sqrt{\cos (e+f x)} F\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{b \sec (e+f x)}}{3 b^2 f}+\frac{2 \sin (e+f x)}{3 b f \sqrt{b \sec (e+f x)}}",1,"(Sec[e + f*x]^2*(2*Sqrt[Cos[e + f*x]]*EllipticF[(e + f*x)/2, 2] + Sin[2*(e + f*x)]))/(3*f*(b*Sec[e + f*x])^(3/2))","A",1
434,1,58,68,0.098216,"\int \frac{\csc ^2(e+f x)}{(b \sec (e+f x))^{3/2}} \, dx","Integrate[Csc[e + f*x]^2/(b*Sec[e + f*x])^(3/2),x]","\frac{-F\left(\left.\frac{1}{2} (e+f x)\right|2\right)-\sqrt{\cos (e+f x)} \csc (e+f x)}{f \cos ^{\frac{3}{2}}(e+f x) (b \sec (e+f x))^{3/2}}","-\frac{\sqrt{\cos (e+f x)} F\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{b \sec (e+f x)}}{b^2 f}-\frac{\csc (e+f x)}{b f \sqrt{b \sec (e+f x)}}",1,"(-(Sqrt[Cos[e + f*x]]*Csc[e + f*x]) - EllipticF[(e + f*x)/2, 2])/(f*Cos[e + f*x]^(3/2)*(b*Sec[e + f*x])^(3/2))","A",1
435,1,62,102,0.2196171,"\int \frac{\csc ^4(e+f x)}{(b \sec (e+f x))^{3/2}} \, dx","Integrate[Csc[e + f*x]^4/(b*Sec[e + f*x])^(3/2),x]","\frac{-2 \csc ^3(e+f x)+\csc (e+f x)-\frac{F\left(\left.\frac{1}{2} (e+f x)\right|2\right)}{\sqrt{\cos (e+f x)}}}{6 b f \sqrt{b \sec (e+f x)}}","-\frac{\sqrt{\cos (e+f x)} F\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{b \sec (e+f x)}}{6 b^2 f}-\frac{\csc ^3(e+f x)}{3 b f \sqrt{b \sec (e+f x)}}+\frac{\csc (e+f x)}{6 b f \sqrt{b \sec (e+f x)}}",1,"(Csc[e + f*x] - 2*Csc[e + f*x]^3 - EllipticF[(e + f*x)/2, 2]/Sqrt[Cos[e + f*x]])/(6*b*f*Sqrt[b*Sec[e + f*x]])","A",1
436,1,74,132,0.3069648,"\int \frac{\csc ^6(e+f x)}{(b \sec (e+f x))^{3/2}} \, dx","Integrate[Csc[e + f*x]^6/(b*Sec[e + f*x])^(3/2),x]","\frac{-12 \csc ^5(e+f x)+2 \csc ^3(e+f x)+5 \csc (e+f x)-\frac{5 F\left(\left.\frac{1}{2} (e+f x)\right|2\right)}{\sqrt{\cos (e+f x)}}}{60 b f \sqrt{b \sec (e+f x)}}","-\frac{\sqrt{\cos (e+f x)} F\left(\left.\frac{1}{2} (e+f x)\right|2\right) \sqrt{b \sec (e+f x)}}{12 b^2 f}-\frac{\csc ^5(e+f x)}{5 b f \sqrt{b \sec (e+f x)}}+\frac{\csc ^3(e+f x)}{30 b f \sqrt{b \sec (e+f x)}}+\frac{\csc (e+f x)}{12 b f \sqrt{b \sec (e+f x)}}",1,"(5*Csc[e + f*x] + 2*Csc[e + f*x]^3 - 12*Csc[e + f*x]^5 - (5*EllipticF[(e + f*x)/2, 2])/Sqrt[Cos[e + f*x]])/(60*b*f*Sqrt[b*Sec[e + f*x]])","A",1
437,1,62,87,0.3462584,"\int \frac{\sin ^7(e+f x)}{(b \sec (e+f x))^{5/2}} \, dx","Integrate[Sin[e + f*x]^7/(b*Sec[e + f*x])^(5/2),x]","\frac{\cos ^4(e+f x) (14287 \cos (2 (e+f x))-3542 \cos (4 (e+f x))+385 \cos (6 (e+f x))-15226) \sqrt{b \sec (e+f x)}}{117040 b^3 f}","\frac{2 b^7}{19 f (b \sec (e+f x))^{19/2}}-\frac{2 b^5}{5 f (b \sec (e+f x))^{15/2}}+\frac{6 b^3}{11 f (b \sec (e+f x))^{11/2}}-\frac{2 b}{7 f (b \sec (e+f x))^{7/2}}",1,"(Cos[e + f*x]^4*(-15226 + 14287*Cos[2*(e + f*x)] - 3542*Cos[4*(e + f*x)] + 385*Cos[6*(e + f*x)])*Sqrt[b*Sec[e + f*x]])/(117040*b^3*f)","A",1
438,1,52,65,0.2149545,"\int \frac{\sin ^5(e+f x)}{(b \sec (e+f x))^{5/2}} \, dx","Integrate[Sin[e + f*x]^5/(b*Sec[e + f*x])^(5/2),x]","\frac{\cos ^4(e+f x) (532 \cos (2 (e+f x))-77 \cos (4 (e+f x))-711) \sqrt{b \sec (e+f x)}}{4620 b^3 f}","-\frac{2 b^5}{15 f (b \sec (e+f x))^{15/2}}+\frac{4 b^3}{11 f (b \sec (e+f x))^{11/2}}-\frac{2 b}{7 f (b \sec (e+f x))^{7/2}}",1,"(Cos[e + f*x]^4*(-711 + 532*Cos[2*(e + f*x)] - 77*Cos[4*(e + f*x)])*Sqrt[b*Sec[e + f*x]])/(4620*b^3*f)","A",1
439,1,42,43,0.1584422,"\int \frac{\sin ^3(e+f x)}{(b \sec (e+f x))^{5/2}} \, dx","Integrate[Sin[e + f*x]^3/(b*Sec[e + f*x])^(5/2),x]","\frac{\cos ^4(e+f x) (7 \cos (2 (e+f x))-15) \sqrt{b \sec (e+f x)}}{77 b^3 f}","\frac{2 b^3}{11 f (b \sec (e+f x))^{11/2}}-\frac{2 b}{7 f (b \sec (e+f x))^{7/2}}",1,"(Cos[e + f*x]^4*(-15 + 7*Cos[2*(e + f*x)])*Sqrt[b*Sec[e + f*x]])/(77*b^3*f)","A",1
440,1,20,20,0.0775727,"\int \frac{\sin (e+f x)}{(b \sec (e+f x))^{5/2}} \, dx","Integrate[Sin[e + f*x]/(b*Sec[e + f*x])^(5/2),x]","-\frac{2 b}{7 f (b \sec (e+f x))^{7/2}}","-\frac{2 b}{7 f (b \sec (e+f x))^{7/2}}",1,"(-2*b)/(7*f*(b*Sec[e + f*x])^(7/2))","A",1
441,1,90,81,0.203881,"\int \frac{\csc (e+f x)}{(b \sec (e+f x))^{5/2}} \, dx","Integrate[Csc[e + f*x]/(b*Sec[e + f*x])^(5/2),x]","\frac{\sqrt{\sec (e+f x)} \left(\frac{4}{\sec ^{\frac{3}{2}}(e+f x)}+3 \log \left(1-\sqrt{\sec (e+f x)}\right)-3 \log \left(\sqrt{\sec (e+f x)}+1\right)-6 \tan ^{-1}\left(\sqrt{\sec (e+f x)}\right)\right)}{6 b^2 f \sqrt{b \sec (e+f x)}}","-\frac{\tan ^{-1}\left(\frac{\sqrt{b \sec (e+f x)}}{\sqrt{b}}\right)}{b^{5/2} f}-\frac{\tanh ^{-1}\left(\frac{\sqrt{b \sec (e+f x)}}{\sqrt{b}}\right)}{b^{5/2} f}+\frac{2}{3 b f (b \sec (e+f x))^{3/2}}",1,"((-6*ArcTan[Sqrt[Sec[e + f*x]]] + 3*Log[1 - Sqrt[Sec[e + f*x]]] - 3*Log[1 + Sqrt[Sec[e + f*x]]] + 4/Sec[e + f*x]^(3/2))*Sqrt[Sec[e + f*x]])/(6*b^2*f*Sqrt[b*Sec[e + f*x]])","A",1
442,1,98,93,2.4339998,"\int \frac{\csc ^3(e+f x)}{(b \sec (e+f x))^{5/2}} \, dx","Integrate[Csc[e + f*x]^3/(b*Sec[e + f*x])^(5/2),x]","\frac{\sqrt{\sec (e+f x)} \left(-3 \log \left(1-\sqrt{\sec (e+f x)}\right)+3 \log \left(\sqrt{\sec (e+f x)}+1\right)-\frac{4 \csc ^2(e+f x)}{\sec ^{\frac{3}{2}}(e+f x)}+6 \tan ^{-1}\left(\sqrt{\sec (e+f x)}\right)\right)}{8 b^2 f \sqrt{b \sec (e+f x)}}","\frac{3 \tan ^{-1}\left(\frac{\sqrt{b \sec (e+f x)}}{\sqrt{b}}\right)}{4 b^{5/2} f}+\frac{3 \tanh ^{-1}\left(\frac{\sqrt{b \sec (e+f x)}}{\sqrt{b}}\right)}{4 b^{5/2} f}-\frac{\cot ^2(e+f x) \sqrt{b \sec (e+f x)}}{2 b^3 f}",1,"((6*ArcTan[Sqrt[Sec[e + f*x]]] - 3*Log[1 - Sqrt[Sec[e + f*x]]] + 3*Log[1 + Sqrt[Sec[e + f*x]]] - (4*Csc[e + f*x]^2)/Sec[e + f*x]^(3/2))*Sqrt[Sec[e + f*x]])/(8*b^2*f*Sqrt[b*Sec[e + f*x]])","A",1
443,1,110,123,2.4011438,"\int \frac{\csc ^5(e+f x)}{(b \sec (e+f x))^{5/2}} \, dx","Integrate[Csc[e + f*x]^5/(b*Sec[e + f*x])^(5/2),x]","\frac{\sqrt{\sec (e+f x)} \left(-3 \log \left(1-\sqrt{\sec (e+f x)}\right)+3 \log \left(\sqrt{\sec (e+f x)}+1\right)+6 \tan ^{-1}\left(\sqrt{\sec (e+f x)}\right)-\frac{2 (3 \cos (2 (e+f x))+5) \csc ^4(e+f x)}{\sec ^{\frac{3}{2}}(e+f x)}\right)}{64 b^2 f \sqrt{b \sec (e+f x)}}","\frac{3 \tan ^{-1}\left(\frac{\sqrt{b \sec (e+f x)}}{\sqrt{b}}\right)}{32 b^{5/2} f}+\frac{3 \tanh ^{-1}\left(\frac{\sqrt{b \sec (e+f x)}}{\sqrt{b}}\right)}{32 b^{5/2} f}-\frac{\cot ^4(e+f x) \sqrt{b \sec (e+f x)}}{4 b^3 f}-\frac{\cot ^2(e+f x) \sqrt{b \sec (e+f x)}}{16 b^3 f}",1,"((6*ArcTan[Sqrt[Sec[e + f*x]]] - 3*Log[1 - Sqrt[Sec[e + f*x]]] + 3*Log[1 + Sqrt[Sec[e + f*x]]] - (2*(5 + 3*Cos[2*(e + f*x)])*Csc[e + f*x]^4)/Sec[e + f*x]^(3/2))*Sqrt[Sec[e + f*x]])/(64*b^2*f*Sqrt[b*Sec[e + f*x]])","A",1
444,1,83,126,0.4991027,"\int \frac{\sin ^4(e+f x)}{(b \sec (e+f x))^{5/2}} \, dx","Integrate[Sin[e + f*x]^4/(b*Sec[e + f*x])^(5/2),x]","\frac{192 E\left(\left.\frac{1}{2} (e+f x)\right|2\right)+(-6 \sin (e+f x)-55 \sin (3 (e+f x))+15 \sin (5 (e+f x))) \cos ^{\frac{3}{2}}(e+f x)}{1560 f \cos ^{\frac{5}{2}}(e+f x) (b \sec (e+f x))^{5/2}}","\frac{8 E\left(\left.\frac{1}{2} (e+f x)\right|2\right)}{65 b^2 f \sqrt{\cos (e+f x)} \sqrt{b \sec (e+f x)}}-\frac{2 b \sin ^3(e+f x)}{13 f (b \sec (e+f x))^{7/2}}+\frac{8 \sin (e+f x)}{195 b f (b \sec (e+f x))^{3/2}}-\frac{4 b \sin (e+f x)}{39 f (b \sec (e+f x))^{7/2}}",1,"(192*EllipticE[(e + f*x)/2, 2] + Cos[e + f*x]^(3/2)*(-6*Sin[e + f*x] - 55*Sin[3*(e + f*x)] + 15*Sin[5*(e + f*x)]))/(1560*f*Cos[e + f*x]^(5/2)*(b*Sec[e + f*x])^(5/2))","A",1
445,1,66,98,0.3954821,"\int \frac{\sin ^2(e+f x)}{(b \sec (e+f x))^{5/2}} \, dx","Integrate[Sin[e + f*x]^2/(b*Sec[e + f*x])^(5/2),x]","\frac{-4 \sin (2 (e+f x))-10 \sin (4 (e+f x))+\frac{96 E\left(\left.\frac{1}{2} (e+f x)\right|2\right)}{\sqrt{\cos (e+f x)}}}{360 b^2 f \sqrt{b \sec (e+f x)}}","\frac{4 E\left(\left.\frac{1}{2} (e+f x)\right|2\right)}{15 b^2 f \sqrt{\cos (e+f x)} \sqrt{b \sec (e+f x)}}+\frac{4 \sin (e+f x)}{45 b f (b \sec (e+f x))^{3/2}}-\frac{2 b \sin (e+f x)}{9 f (b \sec (e+f x))^{7/2}}",1,"((96*EllipticE[(e + f*x)/2, 2])/Sqrt[Cos[e + f*x]] - 4*Sin[2*(e + f*x)] - 10*Sin[4*(e + f*x)])/(360*b^2*f*Sqrt[b*Sec[e + f*x]])","A",1
446,1,60,72,0.0875909,"\int \frac{1}{(b \sec (e+f x))^{5/2}} \, dx","Integrate[(b*Sec[e + f*x])^(-5/2),x]","\frac{\sqrt{b \sec (e+f x)} \left(\sin (e+f x)+\sin (3 (e+f x))+12 \sqrt{\cos (e+f x)} E\left(\left.\frac{1}{2} (e+f x)\right|2\right)\right)}{10 b^3 f}","\frac{6 E\left(\left.\frac{1}{2} (e+f x)\right|2\right)}{5 b^2 f \sqrt{\cos (e+f x)} \sqrt{b \sec (e+f x)}}+\frac{2 \sin (e+f x)}{5 b f (b \sec (e+f x))^{3/2}}",1,"(Sqrt[b*Sec[e + f*x]]*(12*Sqrt[Cos[e + f*x]]*EllipticE[(e + f*x)/2, 2] + Sin[e + f*x] + Sin[3*(e + f*x)]))/(10*b^3*f)","A",1
447,1,51,68,0.1557604,"\int \frac{\csc ^2(e+f x)}{(b \sec (e+f x))^{5/2}} \, dx","Integrate[Csc[e + f*x]^2/(b*Sec[e + f*x])^(5/2),x]","\frac{-\cot (e+f x)-\frac{3 E\left(\left.\frac{1}{2} (e+f x)\right|2\right)}{\sqrt{\cos (e+f x)}}}{b^2 f \sqrt{b \sec (e+f x)}}","-\frac{3 E\left(\left.\frac{1}{2} (e+f x)\right|2\right)}{b^2 f \sqrt{\cos (e+f x)} \sqrt{b \sec (e+f x)}}-\frac{\csc (e+f x)}{b f (b \sec (e+f x))^{3/2}}",1,"(-Cot[e + f*x] - (3*EllipticE[(e + f*x)/2, 2])/Sqrt[Cos[e + f*x]])/(b^2*f*Sqrt[b*Sec[e + f*x]])","A",1
448,1,79,102,0.2371955,"\int \frac{\csc ^4(e+f x)}{(b \sec (e+f x))^{5/2}} \, dx","Integrate[Csc[e + f*x]^4/(b*Sec[e + f*x])^(5/2),x]","\frac{\sin (e+f x) \sqrt{b \sec (e+f x)} \left(-2 \csc ^4(e+f x)+5 \csc ^2(e+f x)+3 \sqrt{\cos (e+f x)} \csc (e+f x) E\left(\left.\frac{1}{2} (e+f x)\right|2\right)-3\right)}{6 b^3 f}","\frac{E\left(\left.\frac{1}{2} (e+f x)\right|2\right)}{2 b^2 f \sqrt{\cos (e+f x)} \sqrt{b \sec (e+f x)}}-\frac{\csc ^3(e+f x)}{3 b f (b \sec (e+f x))^{3/2}}+\frac{\csc (e+f x)}{2 b f (b \sec (e+f x))^{3/2}}",1,"((-3 + 5*Csc[e + f*x]^2 - 2*Csc[e + f*x]^4 + 3*Sqrt[Cos[e + f*x]]*Csc[e + f*x]*EllipticE[(e + f*x)/2, 2])*Sqrt[b*Sec[e + f*x]]*Sin[e + f*x])/(6*b^3*f)","A",1
449,1,87,132,0.1798692,"\int \frac{\csc ^6(e+f x)}{(b \sec (e+f x))^{5/2}} \, dx","Integrate[Csc[e + f*x]^6/(b*Sec[e + f*x])^(5/2),x]","\frac{\sin (e+f x) \sqrt{b \sec (e+f x)} \left(-4 \csc ^6(e+f x)+6 \csc ^4(e+f x)+\csc ^2(e+f x)+3 \sqrt{\cos (e+f x)} \csc (e+f x) E\left(\left.\frac{1}{2} (e+f x)\right|2\right)-3\right)}{20 b^3 f}","\frac{3 E\left(\left.\frac{1}{2} (e+f x)\right|2\right)}{20 b^2 f \sqrt{\cos (e+f x)} \sqrt{b \sec (e+f x)}}-\frac{\csc ^5(e+f x)}{5 b f (b \sec (e+f x))^{3/2}}+\frac{\csc ^3(e+f x)}{10 b f (b \sec (e+f x))^{3/2}}+\frac{3 \csc (e+f x)}{20 b f (b \sec (e+f x))^{3/2}}",1,"((-3 + Csc[e + f*x]^2 + 6*Csc[e + f*x]^4 - 4*Csc[e + f*x]^6 + 3*Sqrt[Cos[e + f*x]]*Csc[e + f*x]*EllipticE[(e + f*x)/2, 2])*Sqrt[b*Sec[e + f*x]]*Sin[e + f*x])/(20*b^3*f)","A",1
450,1,80,449,0.3697296,"\int \sqrt{b \sec (e+f x)} (a \sin (e+f x))^{9/2} \, dx","Integrate[Sqrt[b*Sec[e + f*x]]*(a*Sin[e + f*x])^(9/2),x]","\frac{a^4 \tan (e+f x) \sqrt{a \sin (e+f x)} \sqrt{b \sec (e+f x)} \left(14 \, _2F_1\left(\frac{3}{4},1;\frac{7}{4};-\tan ^2(e+f x)\right)-7 \cos (2 (e+f x))+\cos (4 (e+f x))-8\right)}{32 f}","-\frac{21 a^{9/2} \sqrt{b \cos (e+f x)} \sqrt{b \sec (e+f x)} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{b} \sqrt{a \sin (e+f x)}}{\sqrt{a} \sqrt{b \cos (e+f x)}}\right)}{32 \sqrt{2} \sqrt{b} f}+\frac{21 a^{9/2} \sqrt{b \cos (e+f x)} \sqrt{b \sec (e+f x)} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{b} \sqrt{a \sin (e+f x)}}{\sqrt{a} \sqrt{b \cos (e+f x)}}+1\right)}{32 \sqrt{2} \sqrt{b} f}+\frac{21 a^{9/2} \sqrt{b \cos (e+f x)} \sqrt{b \sec (e+f x)} \log \left(-\frac{\sqrt{2} \sqrt{b} \sqrt{a \sin (e+f x)}}{\sqrt{b \cos (e+f x)}}+\sqrt{a} \tan (e+f x)+\sqrt{a}\right)}{64 \sqrt{2} \sqrt{b} f}-\frac{21 a^{9/2} \sqrt{b \cos (e+f x)} \sqrt{b \sec (e+f x)} \log \left(\frac{\sqrt{2} \sqrt{b} \sqrt{a \sin (e+f x)}}{\sqrt{b \cos (e+f x)}}+\sqrt{a} \tan (e+f x)+\sqrt{a}\right)}{64 \sqrt{2} \sqrt{b} f}-\frac{7 a^3 b (a \sin (e+f x))^{3/2}}{16 f \sqrt{b \sec (e+f x)}}-\frac{a b (a \sin (e+f x))^{7/2}}{4 f \sqrt{b \sec (e+f x)}}",1,"(a^4*(-8 - 7*Cos[2*(e + f*x)] + Cos[4*(e + f*x)] + 14*Hypergeometric2F1[3/4, 1, 7/4, -Tan[e + f*x]^2])*Sqrt[b*Sec[e + f*x]]*Sqrt[a*Sin[e + f*x]]*Tan[e + f*x])/(32*f)","C",1
451,1,65,414,0.2558259,"\int \sqrt{b \sec (e+f x)} (a \sin (e+f x))^{5/2} \, dx","Integrate[Sqrt[b*Sec[e + f*x]]*(a*Sin[e + f*x])^(5/2),x]","-\frac{a (a \sin (e+f x))^{3/2} (b \sec (e+f x))^{3/2} \left(-2 \, _2F_1\left(\frac{3}{4},1;\frac{7}{4};-\tan ^2(e+f x)\right)+\cos (2 (e+f x))+1\right)}{4 b f}","-\frac{3 a^{5/2} \sqrt{b \cos (e+f x)} \sqrt{b \sec (e+f x)} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{b} \sqrt{a \sin (e+f x)}}{\sqrt{a} \sqrt{b \cos (e+f x)}}\right)}{4 \sqrt{2} \sqrt{b} f}+\frac{3 a^{5/2} \sqrt{b \cos (e+f x)} \sqrt{b \sec (e+f x)} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{b} \sqrt{a \sin (e+f x)}}{\sqrt{a} \sqrt{b \cos (e+f x)}}+1\right)}{4 \sqrt{2} \sqrt{b} f}+\frac{3 a^{5/2} \sqrt{b \cos (e+f x)} \sqrt{b \sec (e+f x)} \log \left(-\frac{\sqrt{2} \sqrt{b} \sqrt{a \sin (e+f x)}}{\sqrt{b \cos (e+f x)}}+\sqrt{a} \tan (e+f x)+\sqrt{a}\right)}{8 \sqrt{2} \sqrt{b} f}-\frac{3 a^{5/2} \sqrt{b \cos (e+f x)} \sqrt{b \sec (e+f x)} \log \left(\frac{\sqrt{2} \sqrt{b} \sqrt{a \sin (e+f x)}}{\sqrt{b \cos (e+f x)}}+\sqrt{a} \tan (e+f x)+\sqrt{a}\right)}{8 \sqrt{2} \sqrt{b} f}-\frac{a b (a \sin (e+f x))^{3/2}}{2 f \sqrt{b \sec (e+f x)}}",1,"-1/4*(a*(1 + Cos[2*(e + f*x)] - 2*Hypergeometric2F1[3/4, 1, 7/4, -Tan[e + f*x]^2])*(b*Sec[e + f*x])^(3/2)*(a*Sin[e + f*x])^(3/2))/(b*f)","C",1
452,1,55,376,0.1314041,"\int \sqrt{b \sec (e+f x)} \sqrt{a \sin (e+f x)} \, dx","Integrate[Sqrt[b*Sec[e + f*x]]*Sqrt[a*Sin[e + f*x]],x]","\frac{2 \tan (e+f x) \sqrt{a \sin (e+f x)} \sqrt{b \sec (e+f x)} \, _2F_1\left(\frac{3}{4},1;\frac{7}{4};-\tan ^2(e+f x)\right)}{3 f}","-\frac{\sqrt{a} \sqrt{b \cos (e+f x)} \sqrt{b \sec (e+f x)} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{b} \sqrt{a \sin (e+f x)}}{\sqrt{a} \sqrt{b \cos (e+f x)}}\right)}{\sqrt{2} \sqrt{b} f}+\frac{\sqrt{a} \sqrt{b \cos (e+f x)} \sqrt{b \sec (e+f x)} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{b} \sqrt{a \sin (e+f x)}}{\sqrt{a} \sqrt{b \cos (e+f x)}}+1\right)}{\sqrt{2} \sqrt{b} f}+\frac{\sqrt{a} \sqrt{b \cos (e+f x)} \sqrt{b \sec (e+f x)} \log \left(-\frac{\sqrt{2} \sqrt{b} \sqrt{a \sin (e+f x)}}{\sqrt{b \cos (e+f x)}}+\sqrt{a} \tan (e+f x)+\sqrt{a}\right)}{2 \sqrt{2} \sqrt{b} f}-\frac{\sqrt{a} \sqrt{b \cos (e+f x)} \sqrt{b \sec (e+f x)} \log \left(\frac{\sqrt{2} \sqrt{b} \sqrt{a \sin (e+f x)}}{\sqrt{b \cos (e+f x)}}+\sqrt{a} \tan (e+f x)+\sqrt{a}\right)}{2 \sqrt{2} \sqrt{b} f}",1,"(2*Hypergeometric2F1[3/4, 1, 7/4, -Tan[e + f*x]^2]*Sqrt[b*Sec[e + f*x]]*Sqrt[a*Sin[e + f*x]]*Tan[e + f*x])/(3*f)","C",1
453,1,37,33,0.0755012,"\int \frac{\sqrt{b \sec (e+f x)}}{(a \sin (e+f x))^{3/2}} \, dx","Integrate[Sqrt[b*Sec[e + f*x]]/(a*Sin[e + f*x])^(3/2),x]","-\frac{\sin (2 (e+f x)) \sqrt{b \sec (e+f x)}}{f (a \sin (e+f x))^{3/2}}","-\frac{2 b}{a f \sqrt{a \sin (e+f x)} \sqrt{b \sec (e+f x)}}",1,"-((Sqrt[b*Sec[e + f*x]]*Sin[2*(e + f*x)])/(f*(a*Sin[e + f*x])^(3/2)))","A",1
454,1,52,71,0.2043272,"\int \frac{\sqrt{b \sec (e+f x)}}{(a \sin (e+f x))^{7/2}} \, dx","Integrate[Sqrt[b*Sec[e + f*x]]/(a*Sin[e + f*x])^(7/2),x]","\frac{2 (2 \cos (2 (e+f x))-3) \cot (e+f x) \sqrt{b \sec (e+f x)}}{5 a^2 f (a \sin (e+f x))^{3/2}}","-\frac{8 b}{5 a^3 f \sqrt{a \sin (e+f x)} \sqrt{b \sec (e+f x)}}-\frac{2 b}{5 a f (a \sin (e+f x))^{5/2} \sqrt{b \sec (e+f x)}}",1,"(2*(-3 + 2*Cos[2*(e + f*x)])*Cot[e + f*x]*Sqrt[b*Sec[e + f*x]])/(5*a^2*f*(a*Sin[e + f*x])^(3/2))","A",1
455,1,65,106,0.2125377,"\int \frac{\sqrt{b \sec (e+f x)}}{(a \sin (e+f x))^{11/2}} \, dx","Integrate[Sqrt[b*Sec[e + f*x]]/(a*Sin[e + f*x])^(11/2),x]","\frac{2 b (20 \cos (2 (e+f x))-4 \cos (4 (e+f x))-21) \csc ^5(e+f x) \sqrt{a \sin (e+f x)}}{45 a^6 f \sqrt{b \sec (e+f x)}}","-\frac{64 b}{45 a^5 f \sqrt{a \sin (e+f x)} \sqrt{b \sec (e+f x)}}-\frac{16 b}{45 a^3 f (a \sin (e+f x))^{5/2} \sqrt{b \sec (e+f x)}}-\frac{2 b}{9 a f (a \sin (e+f x))^{9/2} \sqrt{b \sec (e+f x)}}",1,"(2*b*(-21 + 20*Cos[2*(e + f*x)] - 4*Cos[4*(e + f*x)])*Csc[e + f*x]^5*Sqrt[a*Sin[e + f*x]])/(45*a^6*f*Sqrt[b*Sec[e + f*x]])","A",1
456,1,90,128,0.8710412,"\int \sqrt{b \sec (e+f x)} (a \sin (e+f x))^{7/2} \, dx","Integrate[Sqrt[b*Sec[e + f*x]]*(a*Sin[e + f*x])^(7/2),x]","\frac{a^3 b \sqrt{a \sin (e+f x)} \left(5 \left(-\tan ^2(e+f x)\right)^{3/4} \csc ^2(e+f x) \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{3}{2};\sec ^2(e+f x)\right)+2 (\cos (2 (e+f x))-6)\right)}{12 f \sqrt{b \sec (e+f x)}}","\frac{5 a^4 \sqrt{\sin (2 e+2 f x)} F\left(\left.e+f x-\frac{\pi }{4}\right|2\right) \sqrt{b \sec (e+f x)}}{12 f \sqrt{a \sin (e+f x)}}-\frac{5 a^3 b \sqrt{a \sin (e+f x)}}{6 f \sqrt{b \sec (e+f x)}}-\frac{a b (a \sin (e+f x))^{5/2}}{3 f \sqrt{b \sec (e+f x)}}",1,"(a^3*b*Sqrt[a*Sin[e + f*x]]*(2*(-6 + Cos[2*(e + f*x)]) + 5*Csc[e + f*x]^2*Hypergeometric2F1[1/2, 3/4, 3/2, Sec[e + f*x]^2]*(-Tan[e + f*x]^2)^(3/4)))/(12*f*Sqrt[b*Sec[e + f*x]])","C",1
457,1,66,91,1.2839276,"\int \sqrt{b \sec (e+f x)} (a \sin (e+f x))^{3/2} \, dx","Integrate[Sqrt[b*Sec[e + f*x]]*(a*Sin[e + f*x])^(3/2),x]","\frac{(a \sin (e+f x))^{5/2} (b \sec (e+f x))^{3/2} \, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{1}{2};\sec ^2(e+f x)\right)}{a b f \left(-\tan ^2(e+f x)\right)^{5/4}}","\frac{a^2 \sqrt{\sin (2 e+2 f x)} F\left(\left.e+f x-\frac{\pi }{4}\right|2\right) \sqrt{b \sec (e+f x)}}{2 f \sqrt{a \sin (e+f x)}}-\frac{a b \sqrt{a \sin (e+f x)}}{f \sqrt{b \sec (e+f x)}}",1,"(Hypergeometric2F1[-1/2, -1/4, 1/2, Sec[e + f*x]^2]*(b*Sec[e + f*x])^(3/2)*(a*Sin[e + f*x])^(5/2))/(a*b*f*(-Tan[e + f*x]^2)^(5/4))","C",1
458,1,66,53,0.3750713,"\int \frac{\sqrt{b \sec (e+f x)}}{\sqrt{a \sin (e+f x)}} \, dx","Integrate[Sqrt[b*Sec[e + f*x]]/Sqrt[a*Sin[e + f*x]],x]","\frac{\left(-\tan ^2(e+f x)\right)^{3/4} \cot (e+f x) \sqrt{b \sec (e+f x)} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{3}{2};\sec ^2(e+f x)\right)}{f \sqrt{a \sin (e+f x)}}","\frac{\sqrt{\sin (2 e+2 f x)} F\left(\left.e+f x-\frac{\pi }{4}\right|2\right) \sqrt{b \sec (e+f x)}}{f \sqrt{a \sin (e+f x)}}",1,"(Cot[e + f*x]*Hypergeometric2F1[1/2, 3/4, 3/2, Sec[e + f*x]^2]*Sqrt[b*Sec[e + f*x]]*(-Tan[e + f*x]^2)^(3/4))/(f*Sqrt[a*Sin[e + f*x]])","C",1
459,1,75,95,0.4345774,"\int \frac{\sqrt{b \sec (e+f x)}}{(a \sin (e+f x))^{5/2}} \, dx","Integrate[Sqrt[b*Sec[e + f*x]]/(a*Sin[e + f*x])^(5/2),x]","\frac{2 \cot (e+f x) \sqrt{b \sec (e+f x)} \left(\left(-\tan ^2(e+f x)\right)^{3/4} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{3}{2};\sec ^2(e+f x)\right)-1\right)}{3 a^2 f \sqrt{a \sin (e+f x)}}","\frac{2 \sqrt{\sin (2 e+2 f x)} F\left(\left.e+f x-\frac{\pi }{4}\right|2\right) \sqrt{b \sec (e+f x)}}{3 a^2 f \sqrt{a \sin (e+f x)}}-\frac{2 b}{3 a f (a \sin (e+f x))^{3/2} \sqrt{b \sec (e+f x)}}",1,"(2*Cot[e + f*x]*Sqrt[b*Sec[e + f*x]]*(-1 + Hypergeometric2F1[1/2, 3/4, 3/2, Sec[e + f*x]^2]*(-Tan[e + f*x]^2)^(3/4)))/(3*a^2*f*Sqrt[a*Sin[e + f*x]])","C",1
460,1,111,130,1.0201447,"\int \frac{\sqrt{b \sec (e+f x)}}{(a \sin (e+f x))^{9/2}} \, dx","Integrate[Sqrt[b*Sec[e + f*x]]/(a*Sin[e + f*x])^(9/2),x]","-\frac{2 \cos (2 (e+f x)) (b \sec (e+f x))^{3/2} \left(2 \left(-\tan ^2(e+f x)\right)^{3/4} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{3}{2};\sec ^2(e+f x)\right)+(\cos (2 (e+f x))-2) \csc ^2(e+f x)\right)}{7 a^3 b f \left(\sec ^2(e+f x)-2\right) (a \sin (e+f x))^{3/2}}","\frac{4 \sqrt{\sin (2 e+2 f x)} F\left(\left.e+f x-\frac{\pi }{4}\right|2\right) \sqrt{b \sec (e+f x)}}{7 a^4 f \sqrt{a \sin (e+f x)}}-\frac{4 b}{7 a^3 f (a \sin (e+f x))^{3/2} \sqrt{b \sec (e+f x)}}-\frac{2 b}{7 a f (a \sin (e+f x))^{7/2} \sqrt{b \sec (e+f x)}}",1,"(-2*Cos[2*(e + f*x)]*(b*Sec[e + f*x])^(3/2)*((-2 + Cos[2*(e + f*x)])*Csc[e + f*x]^2 + 2*Hypergeometric2F1[1/2, 3/4, 3/2, Sec[e + f*x]^2]*(-Tan[e + f*x]^2)^(3/4)))/(7*a^3*b*f*(-2 + Sec[e + f*x]^2)*(a*Sin[e + f*x])^(3/2))","C",1
461,1,86,115,0.5327563,"\int \frac{\sin ^{\frac{9}{2}}(e+f x)}{\sqrt{b \sec (e+f x)}} \, dx","Integrate[Sin[e + f*x]^(9/2)/Sqrt[b*Sec[e + f*x]],x]","-\frac{b \left(42 \sqrt[4]{-\tan ^2(e+f x)} \, _2F_1\left(-\frac{1}{2},\frac{1}{4};\frac{1}{2};\sec ^2(e+f x)\right)-26 \cos (2 (e+f x))+3 \cos (4 (e+f x))+23\right)}{120 f \sqrt{\sin (e+f x)} (b \sec (e+f x))^{3/2}}","-\frac{b \sin ^{\frac{7}{2}}(e+f x)}{5 f (b \sec (e+f x))^{3/2}}-\frac{7 b \sin ^{\frac{3}{2}}(e+f x)}{30 f (b \sec (e+f x))^{3/2}}+\frac{7 \sqrt{\sin (e+f x)} E\left(\left.e+f x-\frac{\pi }{4}\right|2\right)}{20 f \sqrt{\sin (2 e+2 f x)} \sqrt{b \sec (e+f x)}}",1,"-1/120*(b*(23 - 26*Cos[2*(e + f*x)] + 3*Cos[4*(e + f*x)] + 42*Hypergeometric2F1[-1/2, 1/4, 1/2, Sec[e + f*x]^2]*(-Tan[e + f*x]^2)^(1/4)))/(f*(b*Sec[e + f*x])^(3/2)*Sqrt[Sin[e + f*x]])","C",1
462,1,74,85,0.2995159,"\int \frac{\sin ^{\frac{5}{2}}(e+f x)}{\sqrt{b \sec (e+f x)}} \, dx","Integrate[Sin[e + f*x]^(5/2)/Sqrt[b*Sec[e + f*x]],x]","\frac{b \left(-3 \sqrt[4]{-\tan ^2(e+f x)} \, _2F_1\left(-\frac{1}{2},\frac{1}{4};\frac{1}{2};\sec ^2(e+f x)\right)+\cos (2 (e+f x))-1\right)}{6 f \sqrt{\sin (e+f x)} (b \sec (e+f x))^{3/2}}","\frac{\sqrt{\sin (e+f x)} E\left(\left.e+f x-\frac{\pi }{4}\right|2\right)}{2 f \sqrt{\sin (2 e+2 f x)} \sqrt{b \sec (e+f x)}}-\frac{b \sin ^{\frac{3}{2}}(e+f x)}{3 f (b \sec (e+f x))^{3/2}}",1,"(b*(-1 + Cos[2*(e + f*x)] - 3*Hypergeometric2F1[-1/2, 1/4, 1/2, Sec[e + f*x]^2]*(-Tan[e + f*x]^2)^(1/4)))/(6*f*(b*Sec[e + f*x])^(3/2)*Sqrt[Sin[e + f*x]])","C",1
463,1,60,51,1.1021132,"\int \frac{\sqrt{\sin (e+f x)}}{\sqrt{b \sec (e+f x)}} \, dx","Integrate[Sqrt[Sin[e + f*x]]/Sqrt[b*Sec[e + f*x]],x]","-\frac{b \sqrt[4]{-\tan ^2(e+f x)} \, _2F_1\left(-\frac{1}{2},\frac{1}{4};\frac{1}{2};\sec ^2(e+f x)\right)}{f \sqrt{\sin (e+f x)} (b \sec (e+f x))^{3/2}}","\frac{\sqrt{\sin (e+f x)} E\left(\left.e+f x-\frac{\pi }{4}\right|2\right)}{f \sqrt{\sin (2 e+2 f x)} \sqrt{b \sec (e+f x)}}",1,"-((b*Hypergeometric2F1[-1/2, 1/4, 1/2, Sec[e + f*x]^2]*(-Tan[e + f*x]^2)^(1/4))/(f*(b*Sec[e + f*x])^(3/2)*Sqrt[Sin[e + f*x]]))","C",1
464,1,63,81,0.2508685,"\int \frac{1}{\sqrt{b \sec (e+f x)} \sin ^{\frac{3}{2}}(e+f x)} \, dx","Integrate[1/(Sqrt[b*Sec[e + f*x]]*Sin[e + f*x]^(3/2)),x]","\frac{2 b \left(\sqrt[4]{-\tan ^2(e+f x)} \, _2F_1\left(-\frac{1}{2},\frac{1}{4};\frac{1}{2};\sec ^2(e+f x)\right)-1\right)}{f \sqrt{\sin (e+f x)} (b \sec (e+f x))^{3/2}}","-\frac{2 b}{f \sqrt{\sin (e+f x)} (b \sec (e+f x))^{3/2}}-\frac{2 \sqrt{\sin (e+f x)} E\left(\left.e+f x-\frac{\pi }{4}\right|2\right)}{f \sqrt{\sin (2 e+2 f x)} \sqrt{b \sec (e+f x)}}",1,"(2*b*(-1 + Hypergeometric2F1[-1/2, 1/4, 1/2, Sec[e + f*x]^2]*(-Tan[e + f*x]^2)^(1/4)))/(f*(b*Sec[e + f*x])^(3/2)*Sqrt[Sin[e + f*x]])","C",1
465,1,82,115,0.4709376,"\int \frac{1}{\sqrt{b \sec (e+f x)} \sin ^{\frac{7}{2}}(e+f x)} \, dx","Integrate[1/(Sqrt[b*Sec[e + f*x]]*Sin[e + f*x]^(7/2)),x]","\frac{2 b \left(2 \sin ^2(e+f x) \sqrt[4]{-\tan ^2(e+f x)} \, _2F_1\left(-\frac{1}{2},\frac{1}{4};\frac{1}{2};\sec ^2(e+f x)\right)+\cos (2 (e+f x))-2\right)}{5 f \sin ^{\frac{5}{2}}(e+f x) (b \sec (e+f x))^{3/2}}","-\frac{2 b}{5 f \sin ^{\frac{5}{2}}(e+f x) (b \sec (e+f x))^{3/2}}-\frac{4 b}{5 f \sqrt{\sin (e+f x)} (b \sec (e+f x))^{3/2}}-\frac{4 \sqrt{\sin (e+f x)} E\left(\left.e+f x-\frac{\pi }{4}\right|2\right)}{5 f \sqrt{\sin (2 e+2 f x)} \sqrt{b \sec (e+f x)}}",1,"(2*b*(-2 + Cos[2*(e + f*x)] + 2*Hypergeometric2F1[-1/2, 1/4, 1/2, Sec[e + f*x]^2]*Sin[e + f*x]^2*(-Tan[e + f*x]^2)^(1/4)))/(5*f*(b*Sec[e + f*x])^(3/2)*Sin[e + f*x]^(5/2))","C",1
466,1,218,363,1.5594261,"\int \frac{\sin ^{\frac{3}{2}}(e+f x)}{\sqrt{b \sec (e+f x)}} \, dx","Integrate[Sin[e + f*x]^(3/2)/Sqrt[b*Sec[e + f*x]],x]","-\frac{\sqrt{\sin (e+f x)} \sqrt{b \sec (e+f x)} \left(4 \sqrt[4]{\tan ^2(e+f x)}+2 \sqrt{2} \tan ^{-1}\left(1-\sqrt{2} \sqrt[4]{\tan ^2(e+f x)}\right)-2 \sqrt{2} \tan ^{-1}\left(\sqrt{2} \sqrt[4]{\tan ^2(e+f x)}+1\right)+\sqrt{2} \log \left(\sqrt{\tan ^2(e+f x)}-\sqrt{2} \sqrt[4]{\tan ^2(e+f x)}+1\right)-\sqrt{2} \log \left(\sqrt{\tan ^2(e+f x)}+\sqrt{2} \sqrt[4]{\tan ^2(e+f x)}+1\right)+4 \cos (2 (e+f x)) \sqrt[4]{\tan ^2(e+f x)}\right)}{16 b f \sqrt[4]{\tan ^2(e+f x)}}","-\frac{b \sqrt{\sin (e+f x)}}{2 f (b \sec (e+f x))^{3/2}}+\frac{\sqrt{b} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{b \cos (e+f x)}}{\sqrt{b} \sqrt{\sin (e+f x)}}\right)}{4 \sqrt{2} f \sqrt{b \cos (e+f x)} \sqrt{b \sec (e+f x)}}-\frac{\sqrt{b} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{b \cos (e+f x)}}{\sqrt{b} \sqrt{\sin (e+f x)}}+1\right)}{4 \sqrt{2} f \sqrt{b \cos (e+f x)} \sqrt{b \sec (e+f x)}}-\frac{\sqrt{b} \log \left(\sqrt{b} \cot (e+f x)-\frac{\sqrt{2} \sqrt{b \cos (e+f x)}}{\sqrt{\sin (e+f x)}}+\sqrt{b}\right)}{8 \sqrt{2} f \sqrt{b \cos (e+f x)} \sqrt{b \sec (e+f x)}}+\frac{\sqrt{b} \log \left(\sqrt{b} \cot (e+f x)+\frac{\sqrt{2} \sqrt{b \cos (e+f x)}}{\sqrt{\sin (e+f x)}}+\sqrt{b}\right)}{8 \sqrt{2} f \sqrt{b \cos (e+f x)} \sqrt{b \sec (e+f x)}}",1,"-1/16*(Sqrt[b*Sec[e + f*x]]*Sqrt[Sin[e + f*x]]*(2*Sqrt[2]*ArcTan[1 - Sqrt[2]*(Tan[e + f*x]^2)^(1/4)] - 2*Sqrt[2]*ArcTan[1 + Sqrt[2]*(Tan[e + f*x]^2)^(1/4)] + Sqrt[2]*Log[1 - Sqrt[2]*(Tan[e + f*x]^2)^(1/4) + Sqrt[Tan[e + f*x]^2]] - Sqrt[2]*Log[1 + Sqrt[2]*(Tan[e + f*x]^2)^(1/4) + Sqrt[Tan[e + f*x]^2]] + 4*(Tan[e + f*x]^2)^(1/4) + 4*Cos[2*(e + f*x)]*(Tan[e + f*x]^2)^(1/4)))/(b*f*(Tan[e + f*x]^2)^(1/4))","A",1
467,1,166,328,0.8612252,"\int \frac{1}{\sqrt{b \sec (e+f x)} \sqrt{\sin (e+f x)}} \, dx","Integrate[1/(Sqrt[b*Sec[e + f*x]]*Sqrt[Sin[e + f*x]]),x]","\frac{\sqrt{\sin (e+f x)} \sqrt{b \sec (e+f x)} \left(-2 \tan ^{-1}\left(1-\sqrt{2} \sqrt[4]{\tan ^2(e+f x)}\right)+2 \tan ^{-1}\left(\sqrt{2} \sqrt[4]{\tan ^2(e+f x)}+1\right)-\log \left(\sqrt{\tan ^2(e+f x)}-\sqrt{2} \sqrt[4]{\tan ^2(e+f x)}+1\right)+\log \left(\sqrt{\tan ^2(e+f x)}+\sqrt{2} \sqrt[4]{\tan ^2(e+f x)}+1\right)\right)}{2 \sqrt{2} b f \sqrt[4]{\tan ^2(e+f x)}}","\frac{\sqrt{b} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{b \cos (e+f x)}}{\sqrt{b} \sqrt{\sin (e+f x)}}\right)}{\sqrt{2} f \sqrt{b \cos (e+f x)} \sqrt{b \sec (e+f x)}}-\frac{\sqrt{b} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{b \cos (e+f x)}}{\sqrt{b} \sqrt{\sin (e+f x)}}+1\right)}{\sqrt{2} f \sqrt{b \cos (e+f x)} \sqrt{b \sec (e+f x)}}-\frac{\sqrt{b} \log \left(\sqrt{b} \cot (e+f x)-\frac{\sqrt{2} \sqrt{b \cos (e+f x)}}{\sqrt{\sin (e+f x)}}+\sqrt{b}\right)}{2 \sqrt{2} f \sqrt{b \cos (e+f x)} \sqrt{b \sec (e+f x)}}+\frac{\sqrt{b} \log \left(\sqrt{b} \cot (e+f x)+\frac{\sqrt{2} \sqrt{b \cos (e+f x)}}{\sqrt{\sin (e+f x)}}+\sqrt{b}\right)}{2 \sqrt{2} f \sqrt{b \cos (e+f x)} \sqrt{b \sec (e+f x)}}",1,"((-2*ArcTan[1 - Sqrt[2]*(Tan[e + f*x]^2)^(1/4)] + 2*ArcTan[1 + Sqrt[2]*(Tan[e + f*x]^2)^(1/4)] - Log[1 - Sqrt[2]*(Tan[e + f*x]^2)^(1/4) + Sqrt[Tan[e + f*x]^2]] + Log[1 + Sqrt[2]*(Tan[e + f*x]^2)^(1/4) + Sqrt[Tan[e + f*x]^2]])*Sqrt[b*Sec[e + f*x]]*Sqrt[Sin[e + f*x]])/(2*Sqrt[2]*b*f*(Tan[e + f*x]^2)^(1/4))","A",1
468,1,30,30,0.0987968,"\int \frac{1}{\sqrt{b \sec (e+f x)} \sin ^{\frac{5}{2}}(e+f x)} \, dx","Integrate[1/(Sqrt[b*Sec[e + f*x]]*Sin[e + f*x]^(5/2)),x]","-\frac{2 b}{3 f \sin ^{\frac{3}{2}}(e+f x) (b \sec (e+f x))^{3/2}}","-\frac{2 b}{3 f \sin ^{\frac{3}{2}}(e+f x) (b \sec (e+f x))^{3/2}}",1,"(-2*b)/(3*f*(b*Sec[e + f*x])^(3/2)*Sin[e + f*x]^(3/2))","A",1
469,1,42,61,0.1356566,"\int \frac{1}{\sqrt{b \sec (e+f x)} \sin ^{\frac{9}{2}}(e+f x)} \, dx","Integrate[1/(Sqrt[b*Sec[e + f*x]]*Sin[e + f*x]^(9/2)),x]","\frac{2 b (2 \cos (2 (e+f x))-5)}{21 f \sin ^{\frac{7}{2}}(e+f x) (b \sec (e+f x))^{3/2}}","-\frac{8 b}{21 f \sin ^{\frac{3}{2}}(e+f x) (b \sec (e+f x))^{3/2}}-\frac{2 b}{7 f \sin ^{\frac{7}{2}}(e+f x) (b \sec (e+f x))^{3/2}}",1,"(2*b*(-5 + 2*Cos[2*(e + f*x)]))/(21*f*(b*Sec[e + f*x])^(3/2)*Sin[e + f*x]^(7/2))","A",1
470,1,52,91,0.2110942,"\int \frac{1}{\sqrt{b \sec (e+f x)} \sin ^{\frac{13}{2}}(e+f x)} \, dx","Integrate[1/(Sqrt[b*Sec[e + f*x]]*Sin[e + f*x]^(13/2)),x]","\frac{2 b (28 \cos (2 (e+f x))-4 \cos (4 (e+f x))-45)}{231 f \sin ^{\frac{11}{2}}(e+f x) (b \sec (e+f x))^{3/2}}","-\frac{64 b}{231 f \sin ^{\frac{3}{2}}(e+f x) (b \sec (e+f x))^{3/2}}-\frac{16 b}{77 f \sin ^{\frac{7}{2}}(e+f x) (b \sec (e+f x))^{3/2}}-\frac{2 b}{11 f \sin ^{\frac{11}{2}}(e+f x) (b \sec (e+f x))^{3/2}}",1,"(2*b*(-45 + 28*Cos[2*(e + f*x)] - 4*Cos[4*(e + f*x)]))/(231*f*(b*Sec[e + f*x])^(3/2)*Sin[e + f*x]^(11/2))","A",1
471,1,62,121,0.2621904,"\int \frac{1}{\sqrt{b \sec (e+f x)} \sin ^{\frac{17}{2}}(e+f x)} \, dx","Integrate[1/(Sqrt[b*Sec[e + f*x]]*Sin[e + f*x]^(17/2)),x]","\frac{2 b (150 \cos (2 (e+f x))-36 \cos (4 (e+f x))+4 \cos (6 (e+f x))-195)}{1155 f \sin ^{\frac{15}{2}}(e+f x) (b \sec (e+f x))^{3/2}}","-\frac{256 b}{1155 f \sin ^{\frac{3}{2}}(e+f x) (b \sec (e+f x))^{3/2}}-\frac{64 b}{385 f \sin ^{\frac{7}{2}}(e+f x) (b \sec (e+f x))^{3/2}}-\frac{8 b}{55 f \sin ^{\frac{11}{2}}(e+f x) (b \sec (e+f x))^{3/2}}-\frac{2 b}{15 f \sin ^{\frac{15}{2}}(e+f x) (b \sec (e+f x))^{3/2}}",1,"(2*b*(-195 + 150*Cos[2*(e + f*x)] - 36*Cos[4*(e + f*x)] + 4*Cos[6*(e + f*x)]))/(1155*f*(b*Sec[e + f*x])^(3/2)*Sin[e + f*x]^(15/2))","A",1
472,1,97,490,0.4330956,"\int \frac{(a \sin (e+f x))^{9/2}}{(b \sec (e+f x))^{3/2}} \, dx","Integrate[(a*Sin[e + f*x])^(9/2)/(b*Sec[e + f*x])^(3/2),x]","-\frac{a^5 \tan ^2(e+f x) \left(-14 \, _2F_1\left(\frac{3}{4},1;\frac{7}{4};-\tan ^2(e+f x)\right)+9 \cos (2 (e+f x))+3 \cos (4 (e+f x))-2 \cos (6 (e+f x))+4\right)}{384 b f \sqrt{a \sin (e+f x)} \sqrt{b \sec (e+f x)}}","-\frac{7 a^{9/2} \sqrt{b \cos (e+f x)} \sqrt{b \sec (e+f x)} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{b} \sqrt{a \sin (e+f x)}}{\sqrt{a} \sqrt{b \cos (e+f x)}}\right)}{128 \sqrt{2} b^{5/2} f}+\frac{7 a^{9/2} \sqrt{b \cos (e+f x)} \sqrt{b \sec (e+f x)} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{b} \sqrt{a \sin (e+f x)}}{\sqrt{a} \sqrt{b \cos (e+f x)}}+1\right)}{128 \sqrt{2} b^{5/2} f}+\frac{7 a^{9/2} \sqrt{b \cos (e+f x)} \sqrt{b \sec (e+f x)} \log \left(-\frac{\sqrt{2} \sqrt{b} \sqrt{a \sin (e+f x)}}{\sqrt{b \cos (e+f x)}}+\sqrt{a} \tan (e+f x)+\sqrt{a}\right)}{256 \sqrt{2} b^{5/2} f}-\frac{7 a^{9/2} \sqrt{b \cos (e+f x)} \sqrt{b \sec (e+f x)} \log \left(\frac{\sqrt{2} \sqrt{b} \sqrt{a \sin (e+f x)}}{\sqrt{b \cos (e+f x)}}+\sqrt{a} \tan (e+f x)+\sqrt{a}\right)}{256 \sqrt{2} b^{5/2} f}-\frac{7 a^3 (a \sin (e+f x))^{3/2}}{192 b f \sqrt{b \sec (e+f x)}}+\frac{(a \sin (e+f x))^{11/2}}{6 a b f \sqrt{b \sec (e+f x)}}-\frac{a (a \sin (e+f x))^{7/2}}{48 b f \sqrt{b \sec (e+f x)}}",1,"-1/384*(a^5*(4 + 9*Cos[2*(e + f*x)] + 3*Cos[4*(e + f*x)] - 2*Cos[6*(e + f*x)] - 14*Hypergeometric2F1[3/4, 1, 7/4, -Tan[e + f*x]^2])*Tan[e + f*x]^2)/(b*f*Sqrt[b*Sec[e + f*x]]*Sqrt[a*Sin[e + f*x]])","C",1
473,1,82,453,0.3241745,"\int \frac{(a \sin (e+f x))^{5/2}}{(b \sec (e+f x))^{3/2}} \, dx","Integrate[(a*Sin[e + f*x])^(5/2)/(b*Sec[e + f*x])^(3/2),x]","-\frac{a^3 \tan ^2(e+f x) \left(-2 \, _2F_1\left(\frac{3}{4},1;\frac{7}{4};-\tan ^2(e+f x)\right)+\cos (2 (e+f x))+\cos (4 (e+f x))\right)}{32 b f \sqrt{a \sin (e+f x)} \sqrt{b \sec (e+f x)}}","-\frac{3 a^{5/2} \sqrt{b \cos (e+f x)} \sqrt{b \sec (e+f x)} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{b} \sqrt{a \sin (e+f x)}}{\sqrt{a} \sqrt{b \cos (e+f x)}}\right)}{32 \sqrt{2} b^{5/2} f}+\frac{3 a^{5/2} \sqrt{b \cos (e+f x)} \sqrt{b \sec (e+f x)} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{b} \sqrt{a \sin (e+f x)}}{\sqrt{a} \sqrt{b \cos (e+f x)}}+1\right)}{32 \sqrt{2} b^{5/2} f}+\frac{3 a^{5/2} \sqrt{b \cos (e+f x)} \sqrt{b \sec (e+f x)} \log \left(-\frac{\sqrt{2} \sqrt{b} \sqrt{a \sin (e+f x)}}{\sqrt{b \cos (e+f x)}}+\sqrt{a} \tan (e+f x)+\sqrt{a}\right)}{64 \sqrt{2} b^{5/2} f}-\frac{3 a^{5/2} \sqrt{b \cos (e+f x)} \sqrt{b \sec (e+f x)} \log \left(\frac{\sqrt{2} \sqrt{b} \sqrt{a \sin (e+f x)}}{\sqrt{b \cos (e+f x)}}+\sqrt{a} \tan (e+f x)+\sqrt{a}\right)}{64 \sqrt{2} b^{5/2} f}+\frac{(a \sin (e+f x))^{7/2}}{4 a b f \sqrt{b \sec (e+f x)}}-\frac{a (a \sin (e+f x))^{3/2}}{16 b f \sqrt{b \sec (e+f x)}}",1,"-1/32*(a^3*(Cos[2*(e + f*x)] + Cos[4*(e + f*x)] - 2*Hypergeometric2F1[3/4, 1, 7/4, -Tan[e + f*x]^2])*Tan[e + f*x]^2)/(b*f*Sqrt[b*Sec[e + f*x]]*Sqrt[a*Sin[e + f*x]])","C",1
474,1,76,418,0.2281823,"\int \frac{\sqrt{a \sin (e+f x)}}{(b \sec (e+f x))^{3/2}} \, dx","Integrate[Sqrt[a*Sin[e + f*x]]/(b*Sec[e + f*x])^(3/2),x]","\frac{\sec ^2(e+f x) \sqrt{a \sin (e+f x)} \left(2 \tan (e+f x) \, _2F_1\left(\frac{3}{4},1;\frac{7}{4};-\tan ^2(e+f x)\right)+3 \sin (2 (e+f x))\right)}{12 f (b \sec (e+f x))^{3/2}}","-\frac{\sqrt{a} \sqrt{b \cos (e+f x)} \sqrt{b \sec (e+f x)} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{b} \sqrt{a \sin (e+f x)}}{\sqrt{a} \sqrt{b \cos (e+f x)}}\right)}{4 \sqrt{2} b^{5/2} f}+\frac{\sqrt{a} \sqrt{b \cos (e+f x)} \sqrt{b \sec (e+f x)} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{b} \sqrt{a \sin (e+f x)}}{\sqrt{a} \sqrt{b \cos (e+f x)}}+1\right)}{4 \sqrt{2} b^{5/2} f}+\frac{\sqrt{a} \sqrt{b \cos (e+f x)} \sqrt{b \sec (e+f x)} \log \left(-\frac{\sqrt{2} \sqrt{b} \sqrt{a \sin (e+f x)}}{\sqrt{b \cos (e+f x)}}+\sqrt{a} \tan (e+f x)+\sqrt{a}\right)}{8 \sqrt{2} b^{5/2} f}-\frac{\sqrt{a} \sqrt{b \cos (e+f x)} \sqrt{b \sec (e+f x)} \log \left(\frac{\sqrt{2} \sqrt{b} \sqrt{a \sin (e+f x)}}{\sqrt{b \cos (e+f x)}}+\sqrt{a} \tan (e+f x)+\sqrt{a}\right)}{8 \sqrt{2} b^{5/2} f}+\frac{(a \sin (e+f x))^{3/2}}{2 a b f \sqrt{b \sec (e+f x)}}",1,"(Sec[e + f*x]^2*Sqrt[a*Sin[e + f*x]]*(3*Sin[2*(e + f*x)] + 2*Hypergeometric2F1[3/4, 1, 7/4, -Tan[e + f*x]^2]*Tan[e + f*x]))/(12*f*(b*Sec[e + f*x])^(3/2))","C",1
475,1,66,411,0.1914352,"\int \frac{1}{(b \sec (e+f x))^{3/2} (a \sin (e+f x))^{3/2}} \, dx","Integrate[1/((b*Sec[e + f*x])^(3/2)*(a*Sin[e + f*x])^(3/2)),x]","-\frac{2 \left(\tan ^2(e+f x) \, _2F_1\left(\frac{3}{4},1;\frac{7}{4};-\tan ^2(e+f x)\right)+3\right)}{3 a b f \sqrt{a \sin (e+f x)} \sqrt{b \sec (e+f x)}}","\frac{\sqrt{b \cos (e+f x)} \sqrt{b \sec (e+f x)} \tan ^{-1}\left(1-\frac{\sqrt{2} \sqrt{b} \sqrt{a \sin (e+f x)}}{\sqrt{a} \sqrt{b \cos (e+f x)}}\right)}{\sqrt{2} a^{3/2} b^{5/2} f}-\frac{\sqrt{b \cos (e+f x)} \sqrt{b \sec (e+f x)} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{b} \sqrt{a \sin (e+f x)}}{\sqrt{a} \sqrt{b \cos (e+f x)}}+1\right)}{\sqrt{2} a^{3/2} b^{5/2} f}-\frac{\sqrt{b \cos (e+f x)} \sqrt{b \sec (e+f x)} \log \left(-\frac{\sqrt{2} \sqrt{b} \sqrt{a \sin (e+f x)}}{\sqrt{b \cos (e+f x)}}+\sqrt{a} \tan (e+f x)+\sqrt{a}\right)}{2 \sqrt{2} a^{3/2} b^{5/2} f}+\frac{\sqrt{b \cos (e+f x)} \sqrt{b \sec (e+f x)} \log \left(\frac{\sqrt{2} \sqrt{b} \sqrt{a \sin (e+f x)}}{\sqrt{b \cos (e+f x)}}+\sqrt{a} \tan (e+f x)+\sqrt{a}\right)}{2 \sqrt{2} a^{3/2} b^{5/2} f}-\frac{2}{a b f \sqrt{a \sin (e+f x)} \sqrt{b \sec (e+f x)}}",1,"(-2*(3 + Hypergeometric2F1[3/4, 1, 7/4, -Tan[e + f*x]^2]*Tan[e + f*x]^2))/(3*a*b*f*Sqrt[b*Sec[e + f*x]]*Sqrt[a*Sin[e + f*x]])","C",1
476,1,45,35,0.1077865,"\int \frac{1}{(b \sec (e+f x))^{3/2} (a \sin (e+f x))^{7/2}} \, dx","Integrate[1/((b*Sec[e + f*x])^(3/2)*(a*Sin[e + f*x])^(7/2)),x]","-\frac{2 \cot ^3(e+f x) \sqrt{a \sin (e+f x)} \sqrt{b \sec (e+f x)}}{5 a^4 b^2 f}","-\frac{2 b}{5 a f (a \sin (e+f x))^{5/2} (b \sec (e+f x))^{5/2}}",1,"(-2*Cot[e + f*x]^3*Sqrt[b*Sec[e + f*x]]*Sqrt[a*Sin[e + f*x]])/(5*a^4*b^2*f)","A",1
477,1,103,172,0.8468684,"\int \frac{(a \sin (e+f x))^{7/2}}{(b \sec (e+f x))^{3/2}} \, dx","Integrate[(a*Sin[e + f*x])^(7/2)/(b*Sec[e + f*x])^(3/2),x]","-\frac{a^5 \left(-20 \left(-\tan ^2(e+f x)\right)^{3/4} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{3}{2};\sec ^2(e+f x)\right)+17 \cos (2 (e+f x))-16 \cos (4 (e+f x))+3 \cos (6 (e+f x))-4\right)}{480 b f (a \sin (e+f x))^{3/2} \sqrt{b \sec (e+f x)}}","\frac{a^4 \sqrt{\sin (2 e+2 f x)} F\left(\left.e+f x-\frac{\pi }{4}\right|2\right) \sqrt{b \sec (e+f x)}}{24 b^2 f \sqrt{a \sin (e+f x)}}-\frac{a^3 \sqrt{a \sin (e+f x)}}{12 b f \sqrt{b \sec (e+f x)}}+\frac{(a \sin (e+f x))^{9/2}}{5 a b f \sqrt{b \sec (e+f x)}}-\frac{a (a \sin (e+f x))^{5/2}}{30 b f \sqrt{b \sec (e+f x)}}",1,"-1/480*(a^5*(-4 + 17*Cos[2*(e + f*x)] - 16*Cos[4*(e + f*x)] + 3*Cos[6*(e + f*x)] - 20*Hypergeometric2F1[1/2, 3/4, 3/2, Sec[e + f*x]^2]*(-Tan[e + f*x]^2)^(3/4)))/(b*f*Sqrt[b*Sec[e + f*x]]*(a*Sin[e + f*x])^(3/2))","C",1
478,1,87,135,0.4051217,"\int \frac{(a \sin (e+f x))^{3/2}}{(b \sec (e+f x))^{3/2}} \, dx","Integrate[(a*Sin[e + f*x])^(3/2)/(b*Sec[e + f*x])^(3/2),x]","\frac{a \sqrt{a \sin (e+f x)} \left(\left(-\tan ^2(e+f x)\right)^{3/4} \csc ^2(e+f x) \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{3}{2};\sec ^2(e+f x)\right)-2 \cos (2 (e+f x))\right)}{12 b f \sqrt{b \sec (e+f x)}}","\frac{a^2 \sqrt{\sin (2 e+2 f x)} F\left(\left.e+f x-\frac{\pi }{4}\right|2\right) \sqrt{b \sec (e+f x)}}{12 b^2 f \sqrt{a \sin (e+f x)}}+\frac{(a \sin (e+f x))^{5/2}}{3 a b f \sqrt{b \sec (e+f x)}}-\frac{a \sqrt{a \sin (e+f x)}}{6 b f \sqrt{b \sec (e+f x)}}",1,"(a*Sqrt[a*Sin[e + f*x]]*(-2*Cos[2*(e + f*x)] + Csc[e + f*x]^2*Hypergeometric2F1[1/2, 3/4, 3/2, Sec[e + f*x]^2]*(-Tan[e + f*x]^2)^(3/4)))/(12*b*f*Sqrt[b*Sec[e + f*x]])","C",1
479,1,84,94,0.5269395,"\int \frac{1}{(b \sec (e+f x))^{3/2} \sqrt{a \sin (e+f x)}} \, dx","Integrate[1/((b*Sec[e + f*x])^(3/2)*Sqrt[a*Sin[e + f*x]]),x]","-\frac{\cot (e+f x) \sqrt{b \sec (e+f x)} \left(-\left(-\tan ^2(e+f x)\right)^{3/4} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{3}{2};\sec ^2(e+f x)\right)+\cos (2 (e+f x))-1\right)}{2 b^2 f \sqrt{a \sin (e+f x)}}","\frac{\sqrt{\sin (2 e+2 f x)} F\left(\left.e+f x-\frac{\pi }{4}\right|2\right) \sqrt{b \sec (e+f x)}}{2 b^2 f \sqrt{a \sin (e+f x)}}+\frac{\sqrt{a \sin (e+f x)}}{a b f \sqrt{b \sec (e+f x)}}",1,"-1/2*(Cot[e + f*x]*Sqrt[b*Sec[e + f*x]]*(-1 + Cos[2*(e + f*x)] - Hypergeometric2F1[1/2, 3/4, 3/2, Sec[e + f*x]^2]*(-Tan[e + f*x]^2)^(3/4)))/(b^2*f*Sqrt[a*Sin[e + f*x]])","C",1
480,1,78,100,0.4757047,"\int \frac{1}{(b \sec (e+f x))^{3/2} (a \sin (e+f x))^{5/2}} \, dx","Integrate[1/((b*Sec[e + f*x])^(3/2)*(a*Sin[e + f*x])^(5/2)),x]","-\frac{\cot (e+f x) \sqrt{b \sec (e+f x)} \left(\left(-\tan ^2(e+f x)\right)^{3/4} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{3}{2};\sec ^2(e+f x)\right)+2\right)}{3 a^2 b^2 f \sqrt{a \sin (e+f x)}}","-\frac{\sqrt{\sin (2 e+2 f x)} F\left(\left.e+f x-\frac{\pi }{4}\right|2\right) \sqrt{b \sec (e+f x)}}{3 a^2 b^2 f \sqrt{a \sin (e+f x)}}-\frac{2}{3 a b f (a \sin (e+f x))^{3/2} \sqrt{b \sec (e+f x)}}",1,"-1/3*(Cot[e + f*x]*Sqrt[b*Sec[e + f*x]]*(2 + Hypergeometric2F1[1/2, 3/4, 3/2, Sec[e + f*x]^2]*(-Tan[e + f*x]^2)^(3/4)))/(a^2*b^2*f*Sqrt[a*Sin[e + f*x]])","C",1
481,1,119,137,0.8394151,"\int \frac{1}{(b \sec (e+f x))^{3/2} (a \sin (e+f x))^{9/2}} \, dx","Integrate[1/((b*Sec[e + f*x])^(3/2)*(a*Sin[e + f*x])^(9/2)),x]","\frac{\cos (2 (e+f x)) \csc ^4(e+f x) \sqrt{a \sin (e+f x)} \left((\cos (2 (e+f x))+5) \sec ^2(e+f x)-2 \left(-\tan ^2(e+f x)\right)^{7/4} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{3}{2};\sec ^2(e+f x)\right)\right)}{21 a^5 b f \left(\sec ^2(e+f x)-2\right) \sqrt{b \sec (e+f x)}}","-\frac{2 \sqrt{\sin (2 e+2 f x)} F\left(\left.e+f x-\frac{\pi }{4}\right|2\right) \sqrt{b \sec (e+f x)}}{21 a^4 b^2 f \sqrt{a \sin (e+f x)}}+\frac{2}{21 a^3 b f (a \sin (e+f x))^{3/2} \sqrt{b \sec (e+f x)}}-\frac{2}{7 a b f (a \sin (e+f x))^{7/2} \sqrt{b \sec (e+f x)}}",1,"(Cos[2*(e + f*x)]*Csc[e + f*x]^4*Sqrt[a*Sin[e + f*x]]*((5 + Cos[2*(e + f*x)])*Sec[e + f*x]^2 - 2*Hypergeometric2F1[1/2, 3/4, 3/2, Sec[e + f*x]^2]*(-Tan[e + f*x]^2)^(7/4)))/(21*a^5*b*f*Sqrt[b*Sec[e + f*x]]*(-2 + Sec[e + f*x]^2))","C",1
482,1,131,174,1.214783,"\int \frac{1}{(b \sec (e+f x))^{3/2} (a \sin (e+f x))^{13/2}} \, dx","Integrate[1/((b*Sec[e + f*x])^(3/2)*(a*Sin[e + f*x])^(13/2)),x]","\frac{2 \cot (2 (e+f x)) \csc (2 (e+f x)) \sqrt{a \sin (e+f x)} \left(8 \left(-\tan ^2(e+f x)\right)^{3/4} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{3}{2};\sec ^2(e+f x)\right)+(6 \cos (2 (e+f x))-\cos (4 (e+f x))+23) \csc ^4(e+f x)\right)}{77 a^7 b f \left(\sec ^2(e+f x)-2\right) \sqrt{b \sec (e+f x)}}","-\frac{4 \sqrt{\sin (2 e+2 f x)} F\left(\left.e+f x-\frac{\pi }{4}\right|2\right) \sqrt{b \sec (e+f x)}}{77 a^6 b^2 f \sqrt{a \sin (e+f x)}}+\frac{4}{77 a^5 b f (a \sin (e+f x))^{3/2} \sqrt{b \sec (e+f x)}}+\frac{2}{77 a^3 b f (a \sin (e+f x))^{7/2} \sqrt{b \sec (e+f x)}}-\frac{2}{11 a b f (a \sin (e+f x))^{11/2} \sqrt{b \sec (e+f x)}}",1,"(2*Cot[2*(e + f*x)]*Csc[2*(e + f*x)]*Sqrt[a*Sin[e + f*x]]*((23 + 6*Cos[2*(e + f*x)] - Cos[4*(e + f*x)])*Csc[e + f*x]^4 + 8*Hypergeometric2F1[1/2, 3/4, 3/2, Sec[e + f*x]^2]*(-Tan[e + f*x]^2)^(3/4)))/(77*a^7*b*f*Sqrt[b*Sec[e + f*x]]*(-2 + Sec[e + f*x]^2))","C",1
483,1,96,75,8.677672,"\int (d \sec (a+b x))^{5/2} (c \sin (a+b x))^m \, dx","Integrate[(d*Sec[a + b*x])^(5/2)*(c*Sin[a + b*x])^m,x]","-\frac{2 \cot (a+b x) (d \sec (a+b x))^{5/2} \left(-\tan ^2(a+b x)\right)^{\frac{1-m}{2}} (c \sin (a+b x))^m \, _2F_1\left(\frac{1}{4} (5-2 m),\frac{1-m}{2};\frac{1}{4} (9-2 m);\sec ^2(a+b x)\right)}{b (2 m-5)}","\frac{d \cos ^2(a+b x)^{3/4} (d \sec (a+b x))^{3/2} (c \sin (a+b x))^{m+1} \, _2F_1\left(\frac{7}{4},\frac{m+1}{2};\frac{m+3}{2};\sin ^2(a+b x)\right)}{b c (m+1)}",1,"(-2*Cot[a + b*x]*Hypergeometric2F1[(5 - 2*m)/4, (1 - m)/2, (9 - 2*m)/4, Sec[a + b*x]^2]*(d*Sec[a + b*x])^(5/2)*(c*Sin[a + b*x])^m*(-Tan[a + b*x]^2)^((1 - m)/2))/(b*(-5 + 2*m))","A",1
484,1,96,75,1.355765,"\int (d \sec (a+b x))^{3/2} (c \sin (a+b x))^m \, dx","Integrate[(d*Sec[a + b*x])^(3/2)*(c*Sin[a + b*x])^m,x]","-\frac{2 \cot (a+b x) (d \sec (a+b x))^{3/2} \left(-\tan ^2(a+b x)\right)^{\frac{1-m}{2}} (c \sin (a+b x))^m \, _2F_1\left(\frac{1}{4} (3-2 m),\frac{1-m}{2};\frac{1}{4} (7-2 m);\sec ^2(a+b x)\right)}{b (2 m-3)}","\frac{d \sqrt[4]{\cos ^2(a+b x)} \sqrt{d \sec (a+b x)} (c \sin (a+b x))^{m+1} \, _2F_1\left(\frac{5}{4},\frac{m+1}{2};\frac{m+3}{2};\sin ^2(a+b x)\right)}{b c (m+1)}",1,"(-2*Cot[a + b*x]*Hypergeometric2F1[(3 - 2*m)/4, (1 - m)/2, (7 - 2*m)/4, Sec[a + b*x]^2]*(d*Sec[a + b*x])^(3/2)*(c*Sin[a + b*x])^m*(-Tan[a + b*x]^2)^((1 - m)/2))/(b*(-3 + 2*m))","A",1
485,1,106,77,1.367964,"\int \sqrt{d \sec (a+b x)} (c \sin (a+b x))^m \, dx","Integrate[Sqrt[d*Sec[a + b*x]]*(c*Sin[a + b*x])^m,x]","-\frac{\sin (2 (a+b x)) \csc ^2(a+b x) \sqrt{d \sec (a+b x)} \left(-\tan ^2(a+b x)\right)^{\frac{1-m}{2}} (c \sin (a+b x))^m \, _2F_1\left(\frac{1}{4} (1-2 m),\frac{1-m}{2};\frac{1}{4} (5-2 m);\sec ^2(a+b x)\right)}{b (2 m-1)}","\frac{\cos ^2(a+b x)^{3/4} (d \sec (a+b x))^{3/2} (c \sin (a+b x))^{m+1} \, _2F_1\left(\frac{3}{4},\frac{m+1}{2};\frac{m+3}{2};\sin ^2(a+b x)\right)}{b c d (m+1)}",1,"-((Csc[a + b*x]^2*Hypergeometric2F1[(1 - 2*m)/4, (1 - m)/2, (5 - 2*m)/4, Sec[a + b*x]^2]*Sqrt[d*Sec[a + b*x]]*(c*Sin[a + b*x])^m*Sin[2*(a + b*x)]*(-Tan[a + b*x]^2)^((1 - m)/2))/(b*(-1 + 2*m)))","A",1
486,1,289,77,1.7533189,"\int \frac{(c \sin (a+b x))^m}{\sqrt{d \sec (a+b x)}} \, dx","Integrate[(c*Sin[a + b*x])^m/Sqrt[d*Sec[a + b*x]],x]","\frac{8 c (m+3) \sin ^2\left(\frac{1}{2} (a+b x)\right) \cos ^4\left(\frac{1}{2} (a+b x)\right) F_1\left(\frac{m+1}{2};-\frac{1}{2},m+\frac{3}{2};\frac{m+3}{2};\tan ^2\left(\frac{1}{2} (a+b x)\right),-\tan ^2\left(\frac{1}{2} (a+b x)\right)\right) (c \sin (a+b x))^{m-1}}{b (m+1) \sqrt{d \sec (a+b x)} \left((\cos (a+b x)-1) \left((2 m+3) F_1\left(\frac{m+3}{2};-\frac{1}{2},m+\frac{5}{2};\frac{m+5}{2};\tan ^2\left(\frac{1}{2} (a+b x)\right),-\tan ^2\left(\frac{1}{2} (a+b x)\right)\right)+F_1\left(\frac{m+3}{2};\frac{1}{2},m+\frac{3}{2};\frac{m+5}{2};\tan ^2\left(\frac{1}{2} (a+b x)\right),-\tan ^2\left(\frac{1}{2} (a+b x)\right)\right)\right)+(m+3) (\cos (a+b x)+1) F_1\left(\frac{m+1}{2};-\frac{1}{2},m+\frac{3}{2};\frac{m+3}{2};\tan ^2\left(\frac{1}{2} (a+b x)\right),-\tan ^2\left(\frac{1}{2} (a+b x)\right)\right)\right)}","\frac{\sqrt[4]{\cos ^2(a+b x)} \sqrt{d \sec (a+b x)} (c \sin (a+b x))^{m+1} \, _2F_1\left(\frac{1}{4},\frac{m+1}{2};\frac{m+3}{2};\sin ^2(a+b x)\right)}{b c d (m+1)}",1,"(8*c*(3 + m)*AppellF1[(1 + m)/2, -1/2, 3/2 + m, (3 + m)/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2]*Cos[(a + b*x)/2]^4*Sin[(a + b*x)/2]^2*(c*Sin[a + b*x])^(-1 + m))/(b*(1 + m)*(((3 + 2*m)*AppellF1[(3 + m)/2, -1/2, 5/2 + m, (5 + m)/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2] + AppellF1[(3 + m)/2, 1/2, 3/2 + m, (5 + m)/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2])*(-1 + Cos[a + b*x]) + (3 + m)*AppellF1[(1 + m)/2, -1/2, 3/2 + m, (3 + m)/2, Tan[(a + b*x)/2]^2, -Tan[(a + b*x)/2]^2]*(1 + Cos[a + b*x]))*Sqrt[d*Sec[a + b*x]])","C",0
487,1,116,77,4.0348584,"\int \frac{(c \sin (a+b x))^m}{(d \sec (a+b x))^{3/2}} \, dx","Integrate[(c*Sin[a + b*x])^m/(d*Sec[a + b*x])^(3/2),x]","\frac{2 c \cos (2 (a+b x)) \left(-\tan ^2(a+b x)\right)^{\frac{1-m}{2}} (c \sin (a+b x))^{m-1} \, _2F_1\left(\frac{1}{4} (-2 m-3),\frac{1-m}{2};\frac{1}{4} (1-2 m);\sec ^2(a+b x)\right)}{b d (2 m+3) \left(\sec ^2(a+b x)-2\right) \sqrt{d \sec (a+b x)}}","\frac{(c \sin (a+b x))^{m+1} \, _2F_1\left(-\frac{1}{4},\frac{m+1}{2};\frac{m+3}{2};\sin ^2(a+b x)\right)}{b c d (m+1) \sqrt[4]{\cos ^2(a+b x)} \sqrt{d \sec (a+b x)}}",1,"(2*c*Cos[2*(a + b*x)]*Hypergeometric2F1[(-3 - 2*m)/4, (1 - m)/2, (1 - 2*m)/4, Sec[a + b*x]^2]*(c*Sin[a + b*x])^(-1 + m)*(-Tan[a + b*x]^2)^((1 - m)/2))/(b*d*(3 + 2*m)*Sqrt[d*Sec[a + b*x]]*(-2 + Sec[a + b*x]^2))","A",1
488,1,285,86,1.4664936,"\int \sec ^n(e+f x) \sin ^m(e+f x) \, dx","Integrate[Sec[e + f*x]^n*Sin[e + f*x]^m,x]","\frac{4 (m+3) \sin \left(\frac{1}{2} (e+f x)\right) \cos ^3\left(\frac{1}{2} (e+f x)\right) \sin ^m(e+f x) \sec ^n(e+f x) F_1\left(\frac{m+1}{2};n,m-n+1;\frac{m+3}{2};\tan ^2\left(\frac{1}{2} (e+f x)\right),-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)}{f (m+1) \left((m+3) (\cos (e+f x)+1) F_1\left(\frac{m+1}{2};n,m-n+1;\frac{m+3}{2};\tan ^2\left(\frac{1}{2} (e+f x)\right),-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)-4 \sin ^2\left(\frac{1}{2} (e+f x)\right) \left((m-n+1) F_1\left(\frac{m+3}{2};n,m-n+2;\frac{m+5}{2};\tan ^2\left(\frac{1}{2} (e+f x)\right),-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)-n F_1\left(\frac{m+3}{2};n+1,m-n+1;\frac{m+5}{2};\tan ^2\left(\frac{1}{2} (e+f x)\right),-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)\right)\right)}","-\frac{\sin ^{m-1}(e+f x) \sin ^2(e+f x)^{\frac{1-m}{2}} \sec ^{n-1}(e+f x) \, _2F_1\left(\frac{1-m}{2},\frac{1-n}{2};\frac{3-n}{2};\cos ^2(e+f x)\right)}{f (1-n)}",1,"(4*(3 + m)*AppellF1[(1 + m)/2, n, 1 + m - n, (3 + m)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Cos[(e + f*x)/2]^3*Sec[e + f*x]^n*Sin[(e + f*x)/2]*Sin[e + f*x]^m)/(f*(1 + m)*((3 + m)*AppellF1[(1 + m)/2, n, 1 + m - n, (3 + m)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*(1 + Cos[e + f*x]) - 4*((1 + m - n)*AppellF1[(3 + m)/2, n, 2 + m - n, (5 + m)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] - n*AppellF1[(3 + m)/2, 1 + n, 1 + m - n, (5 + m)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Sin[(e + f*x)/2]^2))","C",0
489,1,287,89,0.1492672,"\int \sec ^n(e+f x) (a \sin (e+f x))^m \, dx","Integrate[Sec[e + f*x]^n*(a*Sin[e + f*x])^m,x]","\frac{4 (m+3) \sin \left(\frac{1}{2} (e+f x)\right) \cos ^3\left(\frac{1}{2} (e+f x)\right) \sec ^n(e+f x) (a \sin (e+f x))^m F_1\left(\frac{m+1}{2};n,m-n+1;\frac{m+3}{2};\tan ^2\left(\frac{1}{2} (e+f x)\right),-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)}{f (m+1) \left((m+3) (\cos (e+f x)+1) F_1\left(\frac{m+1}{2};n,m-n+1;\frac{m+3}{2};\tan ^2\left(\frac{1}{2} (e+f x)\right),-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)-4 \sin ^2\left(\frac{1}{2} (e+f x)\right) \left((m-n+1) F_1\left(\frac{m+3}{2};n,m-n+2;\frac{m+5}{2};\tan ^2\left(\frac{1}{2} (e+f x)\right),-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)-n F_1\left(\frac{m+3}{2};n+1,m-n+1;\frac{m+5}{2};\tan ^2\left(\frac{1}{2} (e+f x)\right),-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)\right)\right)}","-\frac{a \sin ^2(e+f x)^{\frac{1-m}{2}} \sec ^{n-1}(e+f x) (a \sin (e+f x))^{m-1} \, _2F_1\left(\frac{1-m}{2},\frac{1-n}{2};\frac{3-n}{2};\cos ^2(e+f x)\right)}{f (1-n)}",1,"(4*(3 + m)*AppellF1[(1 + m)/2, n, 1 + m - n, (3 + m)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Cos[(e + f*x)/2]^3*Sec[e + f*x]^n*Sin[(e + f*x)/2]*(a*Sin[e + f*x])^m)/(f*(1 + m)*((3 + m)*AppellF1[(1 + m)/2, n, 1 + m - n, (3 + m)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*(1 + Cos[e + f*x]) - 4*((1 + m - n)*AppellF1[(3 + m)/2, n, 2 + m - n, (5 + m)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] - n*AppellF1[(3 + m)/2, 1 + n, 1 + m - n, (5 + m)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Sin[(e + f*x)/2]^2))","C",0
490,1,287,89,0.1369759,"\int (b \sec (e+f x))^n \sin ^m(e+f x) \, dx","Integrate[(b*Sec[e + f*x])^n*Sin[e + f*x]^m,x]","\frac{4 (m+3) \sin \left(\frac{1}{2} (e+f x)\right) \cos ^3\left(\frac{1}{2} (e+f x)\right) \sin ^m(e+f x) (b \sec (e+f x))^n F_1\left(\frac{m+1}{2};n,m-n+1;\frac{m+3}{2};\tan ^2\left(\frac{1}{2} (e+f x)\right),-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)}{f (m+1) \left((m+3) (\cos (e+f x)+1) F_1\left(\frac{m+1}{2};n,m-n+1;\frac{m+3}{2};\tan ^2\left(\frac{1}{2} (e+f x)\right),-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)-4 \sin ^2\left(\frac{1}{2} (e+f x)\right) \left((m-n+1) F_1\left(\frac{m+3}{2};n,m-n+2;\frac{m+5}{2};\tan ^2\left(\frac{1}{2} (e+f x)\right),-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)-n F_1\left(\frac{m+3}{2};n+1,m-n+1;\frac{m+5}{2};\tan ^2\left(\frac{1}{2} (e+f x)\right),-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)\right)\right)}","-\frac{b \sin ^{m-1}(e+f x) \sin ^2(e+f x)^{\frac{1-m}{2}} (b \sec (e+f x))^{n-1} \, _2F_1\left(\frac{1-m}{2},\frac{1-n}{2};\frac{3-n}{2};\cos ^2(e+f x)\right)}{f (1-n)}",1,"(4*(3 + m)*AppellF1[(1 + m)/2, n, 1 + m - n, (3 + m)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Cos[(e + f*x)/2]^3*(b*Sec[e + f*x])^n*Sin[(e + f*x)/2]*Sin[e + f*x]^m)/(f*(1 + m)*((3 + m)*AppellF1[(1 + m)/2, n, 1 + m - n, (3 + m)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*(1 + Cos[e + f*x]) - 4*((1 + m - n)*AppellF1[(3 + m)/2, n, 2 + m - n, (5 + m)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] - n*AppellF1[(3 + m)/2, 1 + n, 1 + m - n, (5 + m)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Sin[(e + f*x)/2]^2))","C",0
491,1,289,92,0.1324077,"\int (b \sec (e+f x))^n (a \sin (e+f x))^m \, dx","Integrate[(b*Sec[e + f*x])^n*(a*Sin[e + f*x])^m,x]","\frac{4 (m+3) \sin \left(\frac{1}{2} (e+f x)\right) \cos ^3\left(\frac{1}{2} (e+f x)\right) (a \sin (e+f x))^m (b \sec (e+f x))^n F_1\left(\frac{m+1}{2};n,m-n+1;\frac{m+3}{2};\tan ^2\left(\frac{1}{2} (e+f x)\right),-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)}{f (m+1) \left((m+3) (\cos (e+f x)+1) F_1\left(\frac{m+1}{2};n,m-n+1;\frac{m+3}{2};\tan ^2\left(\frac{1}{2} (e+f x)\right),-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)-4 \sin ^2\left(\frac{1}{2} (e+f x)\right) \left((m-n+1) F_1\left(\frac{m+3}{2};n,m-n+2;\frac{m+5}{2};\tan ^2\left(\frac{1}{2} (e+f x)\right),-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)-n F_1\left(\frac{m+3}{2};n+1,m-n+1;\frac{m+5}{2};\tan ^2\left(\frac{1}{2} (e+f x)\right),-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)\right)\right)}","-\frac{a b \sin ^2(e+f x)^{\frac{1-m}{2}} (a \sin (e+f x))^{m-1} (b \sec (e+f x))^{n-1} \, _2F_1\left(\frac{1-m}{2},\frac{1-n}{2};\frac{3-n}{2};\cos ^2(e+f x)\right)}{f (1-n)}",1,"(4*(3 + m)*AppellF1[(1 + m)/2, n, 1 + m - n, (3 + m)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Cos[(e + f*x)/2]^3*(b*Sec[e + f*x])^n*Sin[(e + f*x)/2]*(a*Sin[e + f*x])^m)/(f*(1 + m)*((3 + m)*AppellF1[(1 + m)/2, n, 1 + m - n, (3 + m)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*(1 + Cos[e + f*x]) - 4*((1 + m - n)*AppellF1[(3 + m)/2, n, 2 + m - n, (5 + m)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] - n*AppellF1[(3 + m)/2, 1 + n, 1 + m - n, (5 + m)/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Sin[(e + f*x)/2]^2))","C",0
492,1,80,80,0.3653457,"\int (b \sec (e+f x))^n \sin ^5(e+f x) \, dx","Integrate[(b*Sec[e + f*x])^n*Sin[e + f*x]^5,x]","\frac{b \left(-4 \left(n^2-8 n+7\right) \cos (2 (e+f x))+\left(n^2-4 n+3\right) \cos (4 (e+f x))+3 n^2-28 n+89\right) (b \sec (e+f x))^{n-1}}{8 f (n-5) (n-3) (n-1)}","-\frac{b^5 (b \sec (e+f x))^{n-5}}{f (5-n)}+\frac{2 b^3 (b \sec (e+f x))^{n-3}}{f (3-n)}-\frac{b (b \sec (e+f x))^{n-1}}{f (1-n)}",1,"(b*(89 - 28*n + 3*n^2 - 4*(7 - 8*n + n^2)*Cos[2*(e + f*x)] + (3 - 4*n + n^2)*Cos[4*(e + f*x)])*(b*Sec[e + f*x])^(-1 + n))/(8*f*(-5 + n)*(-3 + n)*(-1 + n))","A",1
493,1,47,52,0.1527062,"\int (b \sec (e+f x))^n \sin ^3(e+f x) \, dx","Integrate[(b*Sec[e + f*x])^n*Sin[e + f*x]^3,x]","-\frac{b ((n-1) \cos (2 (e+f x))-n+5) (b \sec (e+f x))^{n-1}}{2 f (n-3) (n-1)}","\frac{b^3 (b \sec (e+f x))^{n-3}}{f (3-n)}-\frac{b (b \sec (e+f x))^{n-1}}{f (1-n)}",1,"-1/2*(b*(5 - n + (-1 + n)*Cos[2*(e + f*x)])*(b*Sec[e + f*x])^(-1 + n))/(f*(-3 + n)*(-1 + n))","A",1
494,1,22,25,0.0206968,"\int (b \sec (e+f x))^n \sin (e+f x) \, dx","Integrate[(b*Sec[e + f*x])^n*Sin[e + f*x],x]","\frac{b (b \sec (e+f x))^{n-1}}{f (n-1)}","-\frac{b (b \sec (e+f x))^{n-1}}{f (1-n)}",1,"(b*(b*Sec[e + f*x])^(-1 + n))/(f*(-1 + n))","A",1
495,1,92,49,0.3311225,"\int \csc (e+f x) (b \sec (e+f x))^n \, dx","Integrate[Csc[e + f*x]*(b*Sec[e + f*x])^n,x]","\frac{(b \sec (e+f x))^n \left(\, _2F_1(1,-n;1-n;\cos (e+f x))-2^n \sec ^2\left(\frac{1}{2} (e+f x)\right)^{-n} \, _2F_1\left(-n,-n;1-n;\frac{1}{2} \cos (e+f x) \sec ^2\left(\frac{1}{2} (e+f x)\right)\right)\right)}{2 f n}","-\frac{(b \sec (e+f x))^{n+1} \, _2F_1\left(1,\frac{n+1}{2};\frac{n+3}{2};\sec ^2(e+f x)\right)}{b f (n+1)}",1,"((Hypergeometric2F1[1, -n, 1 - n, Cos[e + f*x]] - (2^n*Hypergeometric2F1[-n, -n, 1 - n, (Cos[e + f*x]*Sec[(e + f*x)/2]^2)/2])/(Sec[(e + f*x)/2]^2)^n)*(b*Sec[e + f*x])^n)/(2*f*n)","A",1
496,1,201,48,4.516276,"\int \csc ^3(e+f x) (b \sec (e+f x))^n \, dx","Integrate[Csc[e + f*x]^3*(b*Sec[e + f*x])^n,x]","\frac{b (b \sec (e+f x))^{n-1} \left(2 \, _2F_1(1,1-n;2-n;\cos (e+f x))+2 \, _2F_1(2,1-n;2-n;\cos (e+f x))+2^n \sec ^{1-n}(e+f x) \left(\cos ^2\left(\frac{1}{2} (e+f x)\right) \sec (e+f x)\right)^{n-1} \, _2F_1\left(1-n,1-n;2-n;\frac{1}{2} \cos (e+f x) \sec ^2\left(\frac{1}{2} (e+f x)\right)\right)+2^n \sec ^2\left(\frac{1}{2} (e+f x)\right)^{1-n} \, _2F_1\left(1-n,-n;2-n;\frac{1}{2} \cos (e+f x) \sec ^2\left(\frac{1}{2} (e+f x)\right)\right)\right)}{8 f (n-1)}","\frac{(b \sec (e+f x))^{n+3} \, _2F_1\left(2,\frac{n+3}{2};\frac{n+5}{2};\sec ^2(e+f x)\right)}{b^3 f (n+3)}",1,"(b*(b*Sec[e + f*x])^(-1 + n)*(2*Hypergeometric2F1[1, 1 - n, 2 - n, Cos[e + f*x]] + 2*Hypergeometric2F1[2, 1 - n, 2 - n, Cos[e + f*x]] + 2^n*Hypergeometric2F1[1 - n, -n, 2 - n, (Cos[e + f*x]*Sec[(e + f*x)/2]^2)/2]*(Sec[(e + f*x)/2]^2)^(1 - n) + 2^n*Hypergeometric2F1[1 - n, 1 - n, 2 - n, (Cos[e + f*x]*Sec[(e + f*x)/2]^2)/2]*Sec[e + f*x]^(1 - n)*(Cos[(e + f*x)/2]^2*Sec[e + f*x])^(-1 + n)))/(8*f*(-1 + n))","B",0
497,1,8327,73,25.4592718,"\int (b \sec (e+f x))^n \sin ^6(e+f x) \, dx","Integrate[(b*Sec[e + f*x])^n*Sin[e + f*x]^6,x]","\text{Result too large to show}","-\frac{b \sin (e+f x) (b \sec (e+f x))^{n-1} \, _2F_1\left(-\frac{5}{2},\frac{1-n}{2};\frac{3-n}{2};\cos ^2(e+f x)\right)}{f (1-n) \sqrt{\sin ^2(e+f x)}}",1,"Result too large to show","C",0
498,1,6192,73,24.1828461,"\int (b \sec (e+f x))^n \sin ^4(e+f x) \, dx","Integrate[(b*Sec[e + f*x])^n*Sin[e + f*x]^4,x]","\text{Result too large to show}","-\frac{b \sin (e+f x) (b \sec (e+f x))^{n-1} \, _2F_1\left(-\frac{3}{2},\frac{1-n}{2};\frac{3-n}{2};\cos ^2(e+f x)\right)}{f (1-n) \sqrt{\sin ^2(e+f x)}}",1,"Result too large to show","C",0
499,1,4143,73,18.6320568,"\int (b \sec (e+f x))^n \sin ^2(e+f x) \, dx","Integrate[(b*Sec[e + f*x])^n*Sin[e + f*x]^2,x]","\text{Result too large to show}","-\frac{b \sin (e+f x) (b \sec (e+f x))^{n-1} \, _2F_1\left(-\frac{1}{2},\frac{1-n}{2};\frac{3-n}{2};\cos ^2(e+f x)\right)}{f (1-n) \sqrt{\sin ^2(e+f x)}}",1,"(24*(Sec[(e + f*x)/2]^2)^(-3 + n)*(b*Sec[e + f*x])^n*(Cos[(e + f*x)/2]^2*Sec[e + f*x])^n*Sin[e + f*x]^2*Tan[(e + f*x)/2]*((AppellF1[1/2, n, 2 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2)/(3*AppellF1[1/2, n, 2 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + 2*((-2 + n)*AppellF1[3/2, n, 3 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + n*AppellF1[3/2, 1 + n, 2 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2) - AppellF1[1/2, n, 3 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]/(3*AppellF1[1/2, n, 3 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + 2*((-3 + n)*AppellF1[3/2, n, 4 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + n*AppellF1[3/2, 1 + n, 3 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2)))/(f*(12*(Sec[(e + f*x)/2]^2)^(-2 + n)*(Cos[(e + f*x)/2]^2*Sec[e + f*x])^n*((AppellF1[1/2, n, 2 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2)/(3*AppellF1[1/2, n, 2 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + 2*((-2 + n)*AppellF1[3/2, n, 3 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + n*AppellF1[3/2, 1 + n, 2 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2) - AppellF1[1/2, n, 3 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]/(3*AppellF1[1/2, n, 3 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + 2*((-3 + n)*AppellF1[3/2, n, 4 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + n*AppellF1[3/2, 1 + n, 3 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2)) + 24*(-3 + n)*(Sec[(e + f*x)/2]^2)^(-3 + n)*(Cos[(e + f*x)/2]^2*Sec[e + f*x])^n*Tan[(e + f*x)/2]^2*((AppellF1[1/2, n, 2 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2)/(3*AppellF1[1/2, n, 2 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + 2*((-2 + n)*AppellF1[3/2, n, 3 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + n*AppellF1[3/2, 1 + n, 2 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2) - AppellF1[1/2, n, 3 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]/(3*AppellF1[1/2, n, 3 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + 2*((-3 + n)*AppellF1[3/2, n, 4 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + n*AppellF1[3/2, 1 + n, 3 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2)) + 24*(Sec[(e + f*x)/2]^2)^(-3 + n)*(Cos[(e + f*x)/2]^2*Sec[e + f*x])^n*Tan[(e + f*x)/2]*((AppellF1[1/2, n, 2 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/(3*AppellF1[1/2, n, 2 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + 2*((-2 + n)*AppellF1[3/2, n, 3 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + n*AppellF1[3/2, 1 + n, 2 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2) + (Sec[(e + f*x)/2]^2*(-1/3*((2 - n)*AppellF1[3/2, n, 3 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2]) + (n*AppellF1[3/2, 1 + n, 2 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/3))/(3*AppellF1[1/2, n, 2 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + 2*((-2 + n)*AppellF1[3/2, n, 3 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + n*AppellF1[3/2, 1 + n, 2 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2) - (-1/3*((3 - n)*AppellF1[3/2, n, 4 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2]) + (n*AppellF1[3/2, 1 + n, 3 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/3)/(3*AppellF1[1/2, n, 3 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + 2*((-3 + n)*AppellF1[3/2, n, 4 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + n*AppellF1[3/2, 1 + n, 3 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2) - (AppellF1[1/2, n, 2 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*(2*((-2 + n)*AppellF1[3/2, n, 3 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + n*AppellF1[3/2, 1 + n, 2 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2] + 3*(-1/3*((2 - n)*AppellF1[3/2, n, 3 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2]) + (n*AppellF1[3/2, 1 + n, 2 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/3) + 2*Tan[(e + f*x)/2]^2*((-2 + n)*((-3*(3 - n)*AppellF1[5/2, n, 4 - n, 7/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/5 + (3*n*AppellF1[5/2, 1 + n, 3 - n, 7/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/5) + n*((-3*(2 - n)*AppellF1[5/2, 1 + n, 3 - n, 7/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/5 + (3*(1 + n)*AppellF1[5/2, 2 + n, 2 - n, 7/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/5))))/(3*AppellF1[1/2, n, 2 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + 2*((-2 + n)*AppellF1[3/2, n, 3 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + n*AppellF1[3/2, 1 + n, 2 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2)^2 + (AppellF1[1/2, n, 3 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*(2*((-3 + n)*AppellF1[3/2, n, 4 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + n*AppellF1[3/2, 1 + n, 3 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2] + 3*(-1/3*((3 - n)*AppellF1[3/2, n, 4 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2]) + (n*AppellF1[3/2, 1 + n, 3 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/3) + 2*Tan[(e + f*x)/2]^2*((-3 + n)*((-3*(4 - n)*AppellF1[5/2, n, 5 - n, 7/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/5 + (3*n*AppellF1[5/2, 1 + n, 4 - n, 7/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/5) + n*((-3*(3 - n)*AppellF1[5/2, 1 + n, 4 - n, 7/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/5 + (3*(1 + n)*AppellF1[5/2, 2 + n, 3 - n, 7/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/5))))/(3*AppellF1[1/2, n, 3 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + 2*((-3 + n)*AppellF1[3/2, n, 4 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + n*AppellF1[3/2, 1 + n, 3 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2)^2) + 24*n*(Sec[(e + f*x)/2]^2)^(-3 + n)*(Cos[(e + f*x)/2]^2*Sec[e + f*x])^(-1 + n)*Tan[(e + f*x)/2]*((AppellF1[1/2, n, 2 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2)/(3*AppellF1[1/2, n, 2 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + 2*((-2 + n)*AppellF1[3/2, n, 3 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + n*AppellF1[3/2, 1 + n, 2 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2) - AppellF1[1/2, n, 3 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]/(3*AppellF1[1/2, n, 3 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + 2*((-3 + n)*AppellF1[3/2, n, 4 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + n*AppellF1[3/2, 1 + n, 3 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2))*(-(Cos[(e + f*x)/2]*Sec[e + f*x]*Sin[(e + f*x)/2]) + Cos[(e + f*x)/2]^2*Sec[e + f*x]*Tan[e + f*x])))","C",0
500,1,61,73,0.0585907,"\int (b \sec (e+f x))^n \, dx","Integrate[(b*Sec[e + f*x])^n,x]","\frac{\sqrt{-\tan ^2(e+f x)} \cot (e+f x) (b \sec (e+f x))^n \, _2F_1\left(\frac{1}{2},\frac{n}{2};\frac{n+2}{2};\sec ^2(e+f x)\right)}{f n}","-\frac{b \sin (e+f x) (b \sec (e+f x))^{n-1} \, _2F_1\left(\frac{1}{2},\frac{1-n}{2};\frac{3-n}{2};\cos ^2(e+f x)\right)}{f (1-n) \sqrt{\sin ^2(e+f x)}}",1,"(Cot[e + f*x]*Hypergeometric2F1[1/2, n/2, (2 + n)/2, Sec[e + f*x]^2]*(b*Sec[e + f*x])^n*Sqrt[-Tan[e + f*x]^2])/(f*n)","A",1
501,1,2638,73,15.2245756,"\int \csc ^2(e+f x) (b \sec (e+f x))^n \, dx","Integrate[Csc[e + f*x]^2*(b*Sec[e + f*x])^n,x]","\text{Result too large to show}","-\frac{b \sqrt{\sin ^2(e+f x)} \csc (e+f x) (b \sec (e+f x))^{n-1} \, _2F_1\left(\frac{3}{2},\frac{1-n}{2};\frac{3-n}{2};\cos ^2(e+f x)\right)}{f (1-n)}",1,"(Cot[(e + f*x)/2]*Csc[e + f*x]^2*(b*Sec[e + f*x])^n*(Cos[(e + f*x)/2]^2*Sec[e + f*x])^n*(-(AppellF1[-1/2, n, -n, 1/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*(Cos[e + f*x]*Sec[(e + f*x)/2]^2)^n) + (3*AppellF1[1/2, n, -n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*(Sec[(e + f*x)/2]^2)^n*Tan[(e + f*x)/2]^2)/(3*AppellF1[1/2, n, -n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + 2*n*(AppellF1[3/2, n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + AppellF1[3/2, 1 + n, -n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2)))/(2*f*(-1/4*(Csc[(e + f*x)/2]^2*(Cos[(e + f*x)/2]^2*Sec[e + f*x])^n*(-(AppellF1[-1/2, n, -n, 1/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*(Cos[e + f*x]*Sec[(e + f*x)/2]^2)^n) + (3*AppellF1[1/2, n, -n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*(Sec[(e + f*x)/2]^2)^n*Tan[(e + f*x)/2]^2)/(3*AppellF1[1/2, n, -n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + 2*n*(AppellF1[3/2, n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + AppellF1[3/2, 1 + n, -n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2))) + (Cot[(e + f*x)/2]*(Cos[(e + f*x)/2]^2*Sec[e + f*x])^n*(-((Cos[e + f*x]*Sec[(e + f*x)/2]^2)^n*(-(n*AppellF1[1/2, n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2]) - n*AppellF1[1/2, 1 + n, -n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])) - n*AppellF1[-1/2, n, -n, 1/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*(Cos[e + f*x]*Sec[(e + f*x)/2]^2)^(-1 + n)*(-(Sec[(e + f*x)/2]^2*Sin[e + f*x]) + Cos[e + f*x]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2]) + (3*AppellF1[1/2, n, -n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*(Sec[(e + f*x)/2]^2)^(1 + n)*Tan[(e + f*x)/2])/(3*AppellF1[1/2, n, -n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + 2*n*(AppellF1[3/2, n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + AppellF1[3/2, 1 + n, -n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2) + (3*n*AppellF1[1/2, n, -n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*(Sec[(e + f*x)/2]^2)^n*Tan[(e + f*x)/2]^3)/(3*AppellF1[1/2, n, -n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + 2*n*(AppellF1[3/2, n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + AppellF1[3/2, 1 + n, -n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2) + (3*(Sec[(e + f*x)/2]^2)^n*Tan[(e + f*x)/2]^2*((n*AppellF1[3/2, n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/3 + (n*AppellF1[3/2, 1 + n, -n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/3))/(3*AppellF1[1/2, n, -n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + 2*n*(AppellF1[3/2, n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + AppellF1[3/2, 1 + n, -n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2) - (3*AppellF1[1/2, n, -n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*(Sec[(e + f*x)/2]^2)^n*Tan[(e + f*x)/2]^2*(2*n*(AppellF1[3/2, n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + AppellF1[3/2, 1 + n, -n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2] + 3*((n*AppellF1[3/2, n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/3 + (n*AppellF1[3/2, 1 + n, -n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/3) + 2*n*Tan[(e + f*x)/2]^2*((-3*(1 - n)*AppellF1[5/2, n, 2 - n, 7/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/5 + (6*n*AppellF1[5/2, 1 + n, 1 - n, 7/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/5 + (3*(1 + n)*AppellF1[5/2, 2 + n, -n, 7/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/5)))/(3*AppellF1[1/2, n, -n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + 2*n*(AppellF1[3/2, n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + AppellF1[3/2, 1 + n, -n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2)^2))/2 + (n*Cot[(e + f*x)/2]*(Cos[(e + f*x)/2]^2*Sec[e + f*x])^(-1 + n)*(-(AppellF1[-1/2, n, -n, 1/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*(Cos[e + f*x]*Sec[(e + f*x)/2]^2)^n) + (3*AppellF1[1/2, n, -n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*(Sec[(e + f*x)/2]^2)^n*Tan[(e + f*x)/2]^2)/(3*AppellF1[1/2, n, -n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + 2*n*(AppellF1[3/2, n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + AppellF1[3/2, 1 + n, -n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2))*(-(Cos[(e + f*x)/2]*Sec[e + f*x]*Sin[(e + f*x)/2]) + Cos[(e + f*x)/2]^2*Sec[e + f*x]*Tan[e + f*x]))/2))","C",0
502,1,3833,73,17.5508454,"\int \csc ^4(e+f x) (b \sec (e+f x))^n \, dx","Integrate[Csc[e + f*x]^4*(b*Sec[e + f*x])^n,x]","\text{Result too large to show}","-\frac{b \sqrt{\sin ^2(e+f x)} \csc (e+f x) (b \sec (e+f x))^{n-1} \, _2F_1\left(\frac{5}{2},\frac{1-n}{2};\frac{3-n}{2};\cos ^2(e+f x)\right)}{f (1-n)}",1,"(Cot[(e + f*x)/2]^3*Csc[e + f*x]^4*(b*Sec[e + f*x])^n*(Cos[(e + f*x)/2]^2*Sec[e + f*x])^n*(-(AppellF1[-3/2, n, -n, -1/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*(Cos[e + f*x]*Sec[(e + f*x)/2]^2)^n) - 9*AppellF1[-1/2, n, -n, 1/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*(Cos[e + f*x]*Sec[(e + f*x)/2]^2)^n*Tan[(e + f*x)/2]^2 + AppellF1[3/2, n, -n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*(Cos[e + f*x]*Sec[(e + f*x)/2]^2)^n*Tan[(e + f*x)/2]^6 + (27*AppellF1[1/2, n, -n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*(Sec[(e + f*x)/2]^2)^n*Tan[(e + f*x)/2]^4)/(3*AppellF1[1/2, n, -n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + 2*n*(AppellF1[3/2, n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + AppellF1[3/2, 1 + n, -n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2)))/(24*f*(-1/16*(Cot[(e + f*x)/2]^2*Csc[(e + f*x)/2]^2*(Cos[(e + f*x)/2]^2*Sec[e + f*x])^n*(-(AppellF1[-3/2, n, -n, -1/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*(Cos[e + f*x]*Sec[(e + f*x)/2]^2)^n) - 9*AppellF1[-1/2, n, -n, 1/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*(Cos[e + f*x]*Sec[(e + f*x)/2]^2)^n*Tan[(e + f*x)/2]^2 + AppellF1[3/2, n, -n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*(Cos[e + f*x]*Sec[(e + f*x)/2]^2)^n*Tan[(e + f*x)/2]^6 + (27*AppellF1[1/2, n, -n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*(Sec[(e + f*x)/2]^2)^n*Tan[(e + f*x)/2]^4)/(3*AppellF1[1/2, n, -n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + 2*n*(AppellF1[3/2, n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + AppellF1[3/2, 1 + n, -n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2))) + (Cot[(e + f*x)/2]^3*(Cos[(e + f*x)/2]^2*Sec[e + f*x])^n*(-9*AppellF1[-1/2, n, -n, 1/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*(Cos[e + f*x]*Sec[(e + f*x)/2]^2)^n*Tan[(e + f*x)/2] + 3*AppellF1[3/2, n, -n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*(Cos[e + f*x]*Sec[(e + f*x)/2]^2)^n*Tan[(e + f*x)/2]^5 - (Cos[e + f*x]*Sec[(e + f*x)/2]^2)^n*(3*n*AppellF1[-1/2, n, 1 - n, 1/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2] + 3*n*AppellF1[-1/2, 1 + n, -n, 1/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2]) - 9*(Cos[e + f*x]*Sec[(e + f*x)/2]^2)^n*Tan[(e + f*x)/2]^2*(-(n*AppellF1[1/2, n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2]) - n*AppellF1[1/2, 1 + n, -n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2]) + (Cos[e + f*x]*Sec[(e + f*x)/2]^2)^n*Tan[(e + f*x)/2]^6*((3*n*AppellF1[5/2, n, 1 - n, 7/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/5 + (3*n*AppellF1[5/2, 1 + n, -n, 7/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/5) - n*AppellF1[-3/2, n, -n, -1/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*(Cos[e + f*x]*Sec[(e + f*x)/2]^2)^(-1 + n)*(-(Sec[(e + f*x)/2]^2*Sin[e + f*x]) + Cos[e + f*x]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2]) - 9*n*AppellF1[-1/2, n, -n, 1/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*(Cos[e + f*x]*Sec[(e + f*x)/2]^2)^(-1 + n)*Tan[(e + f*x)/2]^2*(-(Sec[(e + f*x)/2]^2*Sin[e + f*x]) + Cos[e + f*x]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2]) + n*AppellF1[3/2, n, -n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*(Cos[e + f*x]*Sec[(e + f*x)/2]^2)^(-1 + n)*Tan[(e + f*x)/2]^6*(-(Sec[(e + f*x)/2]^2*Sin[e + f*x]) + Cos[e + f*x]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2]) + (54*AppellF1[1/2, n, -n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*(Sec[(e + f*x)/2]^2)^(1 + n)*Tan[(e + f*x)/2]^3)/(3*AppellF1[1/2, n, -n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + 2*n*(AppellF1[3/2, n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + AppellF1[3/2, 1 + n, -n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2) + (27*n*AppellF1[1/2, n, -n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*(Sec[(e + f*x)/2]^2)^n*Tan[(e + f*x)/2]^5)/(3*AppellF1[1/2, n, -n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + 2*n*(AppellF1[3/2, n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + AppellF1[3/2, 1 + n, -n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2) + (27*(Sec[(e + f*x)/2]^2)^n*Tan[(e + f*x)/2]^4*((n*AppellF1[3/2, n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/3 + (n*AppellF1[3/2, 1 + n, -n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/3))/(3*AppellF1[1/2, n, -n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + 2*n*(AppellF1[3/2, n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + AppellF1[3/2, 1 + n, -n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2) - (27*AppellF1[1/2, n, -n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*(Sec[(e + f*x)/2]^2)^n*Tan[(e + f*x)/2]^4*(2*n*(AppellF1[3/2, n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + AppellF1[3/2, 1 + n, -n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2] + 3*((n*AppellF1[3/2, n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/3 + (n*AppellF1[3/2, 1 + n, -n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/3) + 2*n*Tan[(e + f*x)/2]^2*((-3*(1 - n)*AppellF1[5/2, n, 2 - n, 7/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/5 + (6*n*AppellF1[5/2, 1 + n, 1 - n, 7/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/5 + (3*(1 + n)*AppellF1[5/2, 2 + n, -n, 7/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/5)))/(3*AppellF1[1/2, n, -n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + 2*n*(AppellF1[3/2, n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + AppellF1[3/2, 1 + n, -n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2)^2))/24 + (n*Cot[(e + f*x)/2]^3*(Cos[(e + f*x)/2]^2*Sec[e + f*x])^(-1 + n)*(-(AppellF1[-3/2, n, -n, -1/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*(Cos[e + f*x]*Sec[(e + f*x)/2]^2)^n) - 9*AppellF1[-1/2, n, -n, 1/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*(Cos[e + f*x]*Sec[(e + f*x)/2]^2)^n*Tan[(e + f*x)/2]^2 + AppellF1[3/2, n, -n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*(Cos[e + f*x]*Sec[(e + f*x)/2]^2)^n*Tan[(e + f*x)/2]^6 + (27*AppellF1[1/2, n, -n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*(Sec[(e + f*x)/2]^2)^n*Tan[(e + f*x)/2]^4)/(3*AppellF1[1/2, n, -n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + 2*n*(AppellF1[3/2, n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + AppellF1[3/2, 1 + n, -n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2))*(-(Cos[(e + f*x)/2]*Sec[e + f*x]*Sin[(e + f*x)/2]) + Cos[(e + f*x)/2]^2*Sec[e + f*x]*Tan[e + f*x]))/24))","C",0
503,1,104,76,0.8145081,"\int (b \sec (a+b x))^n (c \sin (a+b x))^{3/2} \, dx","Integrate[(b*Sec[a + b*x])^n*(c*Sin[a + b*x])^(3/2),x]","\frac{2 (c \sin (a+b x))^{5/2} \cos ^2(a+b x)^{\frac{n-1}{2}} (b \sec (a+b x))^{n-1} \left(5 \sin ^2(a+b x) \, _2F_1\left(\frac{9}{4},\frac{n+1}{2};\frac{13}{4};\sin ^2(a+b x)\right)+9 \, _2F_1\left(\frac{5}{4},\frac{n-1}{2};\frac{9}{4};\sin ^2(a+b x)\right)\right)}{45 c}","-\frac{c \sqrt{c \sin (a+b x)} (b \sec (a+b x))^{n-1} \, _2F_1\left(-\frac{1}{4},\frac{1-n}{2};\frac{3-n}{2};\cos ^2(a+b x)\right)}{(1-n) \sqrt[4]{\sin ^2(a+b x)}}",1,"(2*(Cos[a + b*x]^2)^((-1 + n)/2)*(b*Sec[a + b*x])^(-1 + n)*(c*Sin[a + b*x])^(5/2)*(9*Hypergeometric2F1[5/4, (-1 + n)/2, 9/4, Sin[a + b*x]^2] + 5*Hypergeometric2F1[9/4, (1 + n)/2, 13/4, Sin[a + b*x]^2]*Sin[a + b*x]^2))/(45*c)","A",1
504,1,75,76,0.1231232,"\int (b \sec (a+b x))^n \sqrt{c \sin (a+b x)} \, dx","Integrate[(b*Sec[a + b*x])^n*Sqrt[c*Sin[a + b*x]],x]","\frac{\sin (2 (a+b x)) \sqrt{c \sin (a+b x)} \cos ^2(a+b x)^{\frac{n-1}{2}} (b \sec (a+b x))^n \, _2F_1\left(\frac{3}{4},\frac{n+1}{2};\frac{7}{4};\sin ^2(a+b x)\right)}{3 b}","-\frac{c \sqrt[4]{\sin ^2(a+b x)} (b \sec (a+b x))^{n-1} \, _2F_1\left(\frac{1}{4},\frac{1-n}{2};\frac{3-n}{2};\cos ^2(a+b x)\right)}{(1-n) \sqrt{c \sin (a+b x)}}",1,"((Cos[a + b*x]^2)^((-1 + n)/2)*Hypergeometric2F1[3/4, (1 + n)/2, 7/4, Sin[a + b*x]^2]*(b*Sec[a + b*x])^n*Sqrt[c*Sin[a + b*x]]*Sin[2*(a + b*x)])/(3*b)","A",1
505,1,72,76,0.1393496,"\int \frac{(b \sec (a+b x))^n}{\sqrt{c \sin (a+b x)}} \, dx","Integrate[(b*Sec[a + b*x])^n/Sqrt[c*Sin[a + b*x]],x]","\frac{\sin (2 (a+b x)) \cos ^2(a+b x)^{\frac{n-1}{2}} (b \sec (a+b x))^n \, _2F_1\left(\frac{1}{4},\frac{n+1}{2};\frac{5}{4};\sin ^2(a+b x)\right)}{b \sqrt{c \sin (a+b x)}}","-\frac{c \sin ^2(a+b x)^{3/4} (b \sec (a+b x))^{n-1} \, _2F_1\left(\frac{3}{4},\frac{1-n}{2};\frac{3-n}{2};\cos ^2(a+b x)\right)}{(1-n) (c \sin (a+b x))^{3/2}}",1,"((Cos[a + b*x]^2)^((-1 + n)/2)*Hypergeometric2F1[1/4, (1 + n)/2, 5/4, Sin[a + b*x]^2]*(b*Sec[a + b*x])^n*Sin[2*(a + b*x)])/(b*Sqrt[c*Sin[a + b*x]])","A",1
506,1,73,78,0.1482215,"\int \frac{(b \sec (a+b x))^n}{(c \sin (a+b x))^{3/2}} \, dx","Integrate[(b*Sec[a + b*x])^n/(c*Sin[a + b*x])^(3/2),x]","-\frac{\sin (2 (a+b x)) \cos ^2(a+b x)^{\frac{n-1}{2}} (b \sec (a+b x))^n \, _2F_1\left(-\frac{1}{4},\frac{n+1}{2};\frac{3}{4};\sin ^2(a+b x)\right)}{b (c \sin (a+b x))^{3/2}}","-\frac{\sqrt[4]{\sin ^2(a+b x)} (b \sec (a+b x))^{n-1} \, _2F_1\left(\frac{5}{4},\frac{1-n}{2};\frac{3-n}{2};\cos ^2(a+b x)\right)}{c (1-n) \sqrt{c \sin (a+b x)}}",1,"-(((Cos[a + b*x]^2)^((-1 + n)/2)*Hypergeometric2F1[-1/4, (1 + n)/2, 3/4, Sin[a + b*x]^2]*(b*Sec[a + b*x])^n*Sin[2*(a + b*x)])/(b*(c*Sin[a + b*x])^(3/2)))","A",1
507,1,67,100,0.1820937,"\int \sqrt{d \csc (e+f x)} \sin ^4(e+f x) \, dx","Integrate[Sqrt[d*Csc[e + f*x]]*Sin[e + f*x]^4,x]","-\frac{\sqrt{d \csc (e+f x)} \left(26 \sin (2 (e+f x))-3 \sin (4 (e+f x))+40 \sqrt{\sin (e+f x)} F\left(\left.\frac{1}{4} (-2 e-2 f x+\pi )\right|2\right)\right)}{84 f}","-\frac{2 d^3 \cos (e+f x)}{7 f (d \csc (e+f x))^{5/2}}-\frac{10 d \cos (e+f x)}{21 f \sqrt{d \csc (e+f x)}}+\frac{10 \sqrt{\sin (e+f x)} F\left(\left.\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)\right|2\right) \sqrt{d \csc (e+f x)}}{21 f}",1,"-1/84*(Sqrt[d*Csc[e + f*x]]*(40*EllipticF[(-2*e + Pi - 2*f*x)/4, 2]*Sqrt[Sin[e + f*x]] + 26*Sin[2*(e + f*x)] - 3*Sin[4*(e + f*x)]))/f","A",1
508,1,62,75,0.1236177,"\int \sqrt{d \csc (e+f x)} \sin ^3(e+f x) \, dx","Integrate[Sqrt[d*Csc[e + f*x]]*Sin[e + f*x]^3,x]","-\frac{2 \sqrt{d \csc (e+f x)} \left(\sin ^2(e+f x) \cos (e+f x)+3 \sqrt{\sin (e+f x)} E\left(\left.\frac{1}{4} (-2 e-2 f x+\pi )\right|2\right)\right)}{5 f}","\frac{6 d E\left(\left.\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)\right|2\right)}{5 f \sqrt{\sin (e+f x)} \sqrt{d \csc (e+f x)}}-\frac{2 d^2 \cos (e+f x)}{5 f (d \csc (e+f x))^{3/2}}",1,"(-2*Sqrt[d*Csc[e + f*x]]*(3*EllipticE[(-2*e + Pi - 2*f*x)/4, 2]*Sqrt[Sin[e + f*x]] + Cos[e + f*x]*Sin[e + f*x]^2))/(5*f)","A",1
509,1,55,72,0.0830303,"\int \sqrt{d \csc (e+f x)} \sin ^2(e+f x) \, dx","Integrate[Sqrt[d*Csc[e + f*x]]*Sin[e + f*x]^2,x]","-\frac{\sqrt{d \csc (e+f x)} \left(\sin (2 (e+f x))+2 \sqrt{\sin (e+f x)} F\left(\left.\frac{1}{4} (-2 e-2 f x+\pi )\right|2\right)\right)}{3 f}","\frac{2 \sqrt{\sin (e+f x)} F\left(\left.\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)\right|2\right) \sqrt{d \csc (e+f x)}}{3 f}-\frac{2 d \cos (e+f x)}{3 f \sqrt{d \csc (e+f x)}}",1,"-1/3*(Sqrt[d*Csc[e + f*x]]*(2*EllipticF[(-2*e + Pi - 2*f*x)/4, 2]*Sqrt[Sin[e + f*x]] + Sin[2*(e + f*x)]))/f","A",1
510,1,43,44,0.0455922,"\int \sqrt{d \csc (e+f x)} \sin (e+f x) \, dx","Integrate[Sqrt[d*Csc[e + f*x]]*Sin[e + f*x],x]","-\frac{2 d E\left(\left.\frac{1}{4} (-2 e-2 f x+\pi )\right|2\right)}{f \sqrt{\sin (e+f x)} \sqrt{d \csc (e+f x)}}","\frac{2 d E\left(\left.\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)\right|2\right)}{f \sqrt{\sin (e+f x)} \sqrt{d \csc (e+f x)}}",1,"(-2*d*EllipticE[(-2*e + Pi - 2*f*x)/4, 2])/(f*Sqrt[d*Csc[e + f*x]]*Sqrt[Sin[e + f*x]])","A",1
511,1,42,43,0.0367162,"\int \sqrt{d \csc (e+f x)} \, dx","Integrate[Sqrt[d*Csc[e + f*x]],x]","-\frac{2 \sqrt{\sin (e+f x)} F\left(\left.\frac{1}{4} (-2 e-2 f x+\pi )\right|2\right) \sqrt{d \csc (e+f x)}}{f}","\frac{2 \sqrt{\sin (e+f x)} F\left(\left.\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)\right|2\right) \sqrt{d \csc (e+f x)}}{f}",1,"(-2*Sqrt[d*Csc[e + f*x]]*EllipticF[(-2*e + Pi - 2*f*x)/4, 2]*Sqrt[Sin[e + f*x]])/f","A",1
512,1,57,68,0.1344642,"\int \csc (e+f x) \sqrt{d \csc (e+f x)} \, dx","Integrate[Csc[e + f*x]*Sqrt[d*Csc[e + f*x]],x]","\frac{(d \csc (e+f x))^{3/2} \left(2 \sin ^{\frac{3}{2}}(e+f x) E\left(\left.\frac{1}{4} (-2 e-2 f x+\pi )\right|2\right)-\sin (2 (e+f x))\right)}{d f}","-\frac{2 \cos (e+f x) \sqrt{d \csc (e+f x)}}{f}-\frac{2 d E\left(\left.\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)\right|2\right)}{f \sqrt{\sin (e+f x)} \sqrt{d \csc (e+f x)}}",1,"((d*Csc[e + f*x])^(3/2)*(2*EllipticE[(-2*e + Pi - 2*f*x)/4, 2]*Sin[e + f*x]^(3/2) - Sin[2*(e + f*x)]))/(d*f)","A",1
513,1,55,74,0.0973068,"\int \csc ^2(e+f x) \sqrt{d \csc (e+f x)} \, dx","Integrate[Csc[e + f*x]^2*Sqrt[d*Csc[e + f*x]],x]","-\frac{2 (d \csc (e+f x))^{3/2} \left(\cos (e+f x)+\sin ^{\frac{3}{2}}(e+f x) F\left(\left.\frac{1}{4} (-2 e-2 f x+\pi )\right|2\right)\right)}{3 d f}","\frac{2 \sqrt{\sin (e+f x)} F\left(\left.\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)\right|2\right) \sqrt{d \csc (e+f x)}}{3 f}-\frac{2 \cos (e+f x) (d \csc (e+f x))^{3/2}}{3 d f}",1,"(-2*(d*Csc[e + f*x])^(3/2)*(Cos[e + f*x] + EllipticF[(-2*e + Pi - 2*f*x)/4, 2]*Sin[e + f*x]^(3/2)))/(3*d*f)","A",1
514,1,68,100,0.1667226,"\int \csc ^3(e+f x) \sqrt{d \csc (e+f x)} \, dx","Integrate[Csc[e + f*x]^3*Sqrt[d*Csc[e + f*x]],x]","-\frac{2 \sqrt{d \csc (e+f x)} \left(3 \cos (e+f x)+\cot (e+f x) \csc (e+f x)-3 \sqrt{\sin (e+f x)} E\left(\left.\frac{1}{4} (-2 e-2 f x+\pi )\right|2\right)\right)}{5 f}","-\frac{2 \cos (e+f x) (d \csc (e+f x))^{5/2}}{5 d^2 f}-\frac{6 \cos (e+f x) \sqrt{d \csc (e+f x)}}{5 f}-\frac{6 d E\left(\left.\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)\right|2\right)}{5 f \sqrt{\sin (e+f x)} \sqrt{d \csc (e+f x)}}",1,"(-2*Sqrt[d*Csc[e + f*x]]*(3*Cos[e + f*x] + Cot[e + f*x]*Csc[e + f*x] - 3*EllipticE[(-2*e + Pi - 2*f*x)/4, 2]*Sqrt[Sin[e + f*x]]))/(5*f)","A",1
515,1,68,103,0.1368382,"\int (d \csc (e+f x))^{3/2} \sin ^5(e+f x) \, dx","Integrate[(d*Csc[e + f*x])^(3/2)*Sin[e + f*x]^5,x]","-\frac{d \sqrt{d \csc (e+f x)} \left(26 \sin (2 (e+f x))-3 \sin (4 (e+f x))+40 \sqrt{\sin (e+f x)} F\left(\left.\frac{1}{4} (-2 e-2 f x+\pi )\right|2\right)\right)}{84 f}","-\frac{2 d^4 \cos (e+f x)}{7 f (d \csc (e+f x))^{5/2}}-\frac{10 d^2 \cos (e+f x)}{21 f \sqrt{d \csc (e+f x)}}+\frac{10 d \sqrt{\sin (e+f x)} F\left(\left.\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)\right|2\right) \sqrt{d \csc (e+f x)}}{21 f}",1,"-1/84*(d*Sqrt[d*Csc[e + f*x]]*(40*EllipticF[(-2*e + Pi - 2*f*x)/4, 2]*Sqrt[Sin[e + f*x]] + 26*Sin[2*(e + f*x)] - 3*Sin[4*(e + f*x)]))/f","A",1
516,1,62,77,0.1948766,"\int (d \csc (e+f x))^{3/2} \sin ^4(e+f x) \, dx","Integrate[(d*Csc[e + f*x])^(3/2)*Sin[e + f*x]^4,x]","-\frac{2 (d \csc (e+f x))^{3/2} \left(\sin ^3(e+f x) \cos (e+f x)+3 \sin ^{\frac{3}{2}}(e+f x) E\left(\left.\frac{1}{4} (-2 e-2 f x+\pi )\right|2\right)\right)}{5 f}","\frac{6 d^2 E\left(\left.\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)\right|2\right)}{5 f \sqrt{\sin (e+f x)} \sqrt{d \csc (e+f x)}}-\frac{2 d^3 \cos (e+f x)}{5 f (d \csc (e+f x))^{3/2}}",1,"(-2*(d*Csc[e + f*x])^(3/2)*(3*EllipticE[(-2*e + Pi - 2*f*x)/4, 2]*Sin[e + f*x]^(3/2) + Cos[e + f*x]*Sin[e + f*x]^3))/(5*f)","A",1
517,1,56,75,0.0616359,"\int (d \csc (e+f x))^{3/2} \sin ^3(e+f x) \, dx","Integrate[(d*Csc[e + f*x])^(3/2)*Sin[e + f*x]^3,x]","-\frac{d \sqrt{d \csc (e+f x)} \left(\sin (2 (e+f x))+2 \sqrt{\sin (e+f x)} F\left(\left.\frac{1}{4} (-2 e-2 f x+\pi )\right|2\right)\right)}{3 f}","\frac{2 d \sqrt{\sin (e+f x)} F\left(\left.\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)\right|2\right) \sqrt{d \csc (e+f x)}}{3 f}-\frac{2 d^2 \cos (e+f x)}{3 f \sqrt{d \csc (e+f x)}}",1,"-1/3*(d*Sqrt[d*Csc[e + f*x]]*(2*EllipticF[(-2*e + Pi - 2*f*x)/4, 2]*Sqrt[Sin[e + f*x]] + Sin[2*(e + f*x)]))/f","A",1
518,1,45,46,0.0309228,"\int (d \csc (e+f x))^{3/2} \sin ^2(e+f x) \, dx","Integrate[(d*Csc[e + f*x])^(3/2)*Sin[e + f*x]^2,x]","-\frac{2 d^2 E\left(\left.\frac{1}{4} (-2 e-2 f x+\pi )\right|2\right)}{f \sqrt{\sin (e+f x)} \sqrt{d \csc (e+f x)}}","\frac{2 d^2 E\left(\left.\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)\right|2\right)}{f \sqrt{\sin (e+f x)} \sqrt{d \csc (e+f x)}}",1,"(-2*d^2*EllipticE[(-2*e + Pi - 2*f*x)/4, 2])/(f*Sqrt[d*Csc[e + f*x]]*Sqrt[Sin[e + f*x]])","A",1
519,1,43,44,0.0179101,"\int (d \csc (e+f x))^{3/2} \sin (e+f x) \, dx","Integrate[(d*Csc[e + f*x])^(3/2)*Sin[e + f*x],x]","-\frac{2 d \sqrt{\sin (e+f x)} F\left(\left.\frac{1}{4} (-2 e-2 f x+\pi )\right|2\right) \sqrt{d \csc (e+f x)}}{f}","\frac{2 d \sqrt{\sin (e+f x)} F\left(\left.\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)\right|2\right) \sqrt{d \csc (e+f x)}}{f}",1,"(-2*d*Sqrt[d*Csc[e + f*x]]*EllipticF[(-2*e + Pi - 2*f*x)/4, 2]*Sqrt[Sin[e + f*x]])/f","A",1
520,1,54,71,0.0186471,"\int (d \csc (e+f x))^{3/2} \, dx","Integrate[(d*Csc[e + f*x])^(3/2),x]","\frac{(d \csc (e+f x))^{3/2} \left(2 \sin ^{\frac{3}{2}}(e+f x) E\left(\left.\frac{1}{4} (-2 e-2 f x+\pi )\right|2\right)-\sin (2 (e+f x))\right)}{f}","-\frac{2 d^2 E\left(\left.\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)\right|2\right)}{f \sqrt{\sin (e+f x)} \sqrt{d \csc (e+f x)}}-\frac{2 d \cos (e+f x) \sqrt{d \csc (e+f x)}}{f}",1,"((d*Csc[e + f*x])^(3/2)*(2*EllipticE[(-2*e + Pi - 2*f*x)/4, 2]*Sin[e + f*x]^(3/2) - Sin[2*(e + f*x)]))/f","A",1
521,1,58,72,0.185205,"\int \csc (e+f x) (d \csc (e+f x))^{3/2} \, dx","Integrate[Csc[e + f*x]*(d*Csc[e + f*x])^(3/2),x]","-\frac{(d \csc (e+f x))^{5/2} \left(\sin (2 (e+f x))+2 \sin ^{\frac{5}{2}}(e+f x) F\left(\left.\frac{1}{4} (-2 e-2 f x+\pi )\right|2\right)\right)}{3 d f}","\frac{2 d \sqrt{\sin (e+f x)} F\left(\left.\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)\right|2\right) \sqrt{d \csc (e+f x)}}{3 f}-\frac{2 \cos (e+f x) (d \csc (e+f x))^{3/2}}{3 f}",1,"-1/3*((d*Csc[e + f*x])^(5/2)*(2*EllipticF[(-2*e + Pi - 2*f*x)/4, 2]*Sin[e + f*x]^(5/2) + Sin[2*(e + f*x)]))/(d*f)","A",1
522,1,68,103,0.2642829,"\int \csc ^2(e+f x) (d \csc (e+f x))^{3/2} \, dx","Integrate[Csc[e + f*x]^2*(d*Csc[e + f*x])^(3/2),x]","\frac{(d \csc (e+f x))^{5/2} \left(-7 \cos (e+f x)+3 \cos (3 (e+f x))+12 \sin ^{\frac{5}{2}}(e+f x) E\left(\left.\frac{1}{4} (-2 e-2 f x+\pi )\right|2\right)\right)}{10 d f}","-\frac{6 d^2 E\left(\left.\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)\right|2\right)}{5 f \sqrt{\sin (e+f x)} \sqrt{d \csc (e+f x)}}-\frac{2 \cos (e+f x) (d \csc (e+f x))^{5/2}}{5 d f}-\frac{6 d \cos (e+f x) \sqrt{d \csc (e+f x)}}{5 f}",1,"((d*Csc[e + f*x])^(5/2)*(-7*Cos[e + f*x] + 3*Cos[3*(e + f*x)] + 12*EllipticE[(-2*e + Pi - 2*f*x)/4, 2]*Sin[e + f*x]^(5/2)))/(10*d*f)","A",1
523,1,70,102,0.1080156,"\int \frac{\sin ^3(e+f x)}{\sqrt{d \csc (e+f x)}} \, dx","Integrate[Sin[e + f*x]^3/Sqrt[d*Csc[e + f*x]],x]","-\frac{\sqrt{d \csc (e+f x)} \left(26 \sin (2 (e+f x))-3 \sin (4 (e+f x))+40 \sqrt{\sin (e+f x)} F\left(\left.\frac{1}{4} (-2 e-2 f x+\pi )\right|2\right)\right)}{84 d f}","-\frac{2 d^2 \cos (e+f x)}{7 f (d \csc (e+f x))^{5/2}}-\frac{10 \cos (e+f x)}{21 f \sqrt{d \csc (e+f x)}}+\frac{10 \sqrt{\sin (e+f x)} F\left(\left.\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)\right|2\right) \sqrt{d \csc (e+f x)}}{21 d f}",1,"-1/84*(Sqrt[d*Csc[e + f*x]]*(40*EllipticF[(-2*e + Pi - 2*f*x)/4, 2]*Sqrt[Sin[e + f*x]] + 26*Sin[2*(e + f*x)] - 3*Sin[4*(e + f*x)]))/(d*f)","A",1
524,1,57,72,0.1761364,"\int \frac{\sin ^2(e+f x)}{\sqrt{d \csc (e+f x)}} \, dx","Integrate[Sin[e + f*x]^2/Sqrt[d*Csc[e + f*x]],x]","\frac{-2 \sin (2 (e+f x))-\frac{12 E\left(\left.\frac{1}{4} (-2 e-2 f x+\pi )\right|2\right)}{\sqrt{\sin (e+f x)}}}{10 f \sqrt{d \csc (e+f x)}}","\frac{6 E\left(\left.\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)\right|2\right)}{5 f \sqrt{\sin (e+f x)} \sqrt{d \csc (e+f x)}}-\frac{2 d \cos (e+f x)}{5 f (d \csc (e+f x))^{3/2}}",1,"((-12*EllipticE[(-2*e + Pi - 2*f*x)/4, 2])/Sqrt[Sin[e + f*x]] - 2*Sin[2*(e + f*x)])/(10*f*Sqrt[d*Csc[e + f*x]])","A",1
525,1,64,74,0.0798248,"\int \frac{\sin (e+f x)}{\sqrt{d \csc (e+f x)}} \, dx","Integrate[Sin[e + f*x]/Sqrt[d*Csc[e + f*x]],x]","-\frac{d \csc ^2(e+f x) \left(\sin (2 (e+f x))+2 \sqrt{\sin (e+f x)} F\left(\left.\frac{1}{4} (-2 e-2 f x+\pi )\right|2\right)\right)}{3 f (d \csc (e+f x))^{3/2}}","\frac{2 \sqrt{\sin (e+f x)} F\left(\left.\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)\right|2\right) \sqrt{d \csc (e+f x)}}{3 d f}-\frac{2 \cos (e+f x)}{3 f \sqrt{d \csc (e+f x)}}",1,"-1/3*(d*Csc[e + f*x]^2*(2*EllipticF[(-2*e + Pi - 2*f*x)/4, 2]*Sqrt[Sin[e + f*x]] + Sin[2*(e + f*x)]))/(f*(d*Csc[e + f*x])^(3/2))","A",1
526,1,42,43,0.0162055,"\int \frac{1}{\sqrt{d \csc (e+f x)}} \, dx","Integrate[1/Sqrt[d*Csc[e + f*x]],x]","-\frac{2 E\left(\left.\frac{1}{4} (-2 e-2 f x+\pi )\right|2\right)}{f \sqrt{\sin (e+f x)} \sqrt{d \csc (e+f x)}}","\frac{2 E\left(\left.\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)\right|2\right)}{f \sqrt{\sin (e+f x)} \sqrt{d \csc (e+f x)}}",1,"(-2*EllipticE[(-2*e + Pi - 2*f*x)/4, 2])/(f*Sqrt[d*Csc[e + f*x]]*Sqrt[Sin[e + f*x]])","A",1
527,1,45,46,0.0167429,"\int \frac{\csc (e+f x)}{\sqrt{d \csc (e+f x)}} \, dx","Integrate[Csc[e + f*x]/Sqrt[d*Csc[e + f*x]],x]","-\frac{2 \sqrt{\sin (e+f x)} F\left(\left.\frac{1}{4} (-2 e-2 f x+\pi )\right|2\right) \sqrt{d \csc (e+f x)}}{d f}","\frac{2 \sqrt{\sin (e+f x)} F\left(\left.\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)\right|2\right) \sqrt{d \csc (e+f x)}}{d f}",1,"(-2*Sqrt[d*Csc[e + f*x]]*EllipticF[(-2*e + Pi - 2*f*x)/4, 2]*Sqrt[Sin[e + f*x]])/(d*f)","A",1
528,1,52,70,0.0912308,"\int \frac{\csc ^2(e+f x)}{\sqrt{d \csc (e+f x)}} \, dx","Integrate[Csc[e + f*x]^2/Sqrt[d*Csc[e + f*x]],x]","\frac{\frac{2 E\left(\left.\frac{1}{4} (-2 e-2 f x+\pi )\right|2\right)}{\sqrt{\sin (e+f x)}}-2 \cot (e+f x)}{f \sqrt{d \csc (e+f x)}}","-\frac{2 \cos (e+f x) \sqrt{d \csc (e+f x)}}{d f}-\frac{2 E\left(\left.\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)\right|2\right)}{f \sqrt{\sin (e+f x)} \sqrt{d \csc (e+f x)}}",1,"(-2*Cot[e + f*x] + (2*EllipticE[(-2*e + Pi - 2*f*x)/4, 2])/Sqrt[Sin[e + f*x]])/(f*Sqrt[d*Csc[e + f*x]])","A",1
529,1,60,77,0.0639249,"\int \frac{\csc ^3(e+f x)}{\sqrt{d \csc (e+f x)}} \, dx","Integrate[Csc[e + f*x]^3/Sqrt[d*Csc[e + f*x]],x]","-\frac{2 \csc ^2(e+f x) \left(\cos (e+f x)+\sin ^{\frac{3}{2}}(e+f x) F\left(\left.\frac{1}{4} (-2 e-2 f x+\pi )\right|2\right)\right)}{3 f \sqrt{d \csc (e+f x)}}","\frac{2 \sqrt{\sin (e+f x)} F\left(\left.\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)\right|2\right) \sqrt{d \csc (e+f x)}}{3 d f}-\frac{2 \cos (e+f x) (d \csc (e+f x))^{3/2}}{3 d^2 f}",1,"(-2*Csc[e + f*x]^2*(Cos[e + f*x] + EllipticF[(-2*e + Pi - 2*f*x)/4, 2]*Sin[e + f*x]^(3/2)))/(3*f*Sqrt[d*Csc[e + f*x]])","A",1
530,1,70,103,0.1154535,"\int \frac{\sin ^2(e+f x)}{(d \csc (e+f x))^{3/2}} \, dx","Integrate[Sin[e + f*x]^2/(d*Csc[e + f*x])^(3/2),x]","-\frac{\sqrt{d \csc (e+f x)} \left(26 \sin (2 (e+f x))-3 \sin (4 (e+f x))+40 \sqrt{\sin (e+f x)} F\left(\left.\frac{1}{4} (-2 e-2 f x+\pi )\right|2\right)\right)}{84 d^2 f}","\frac{10 \sqrt{\sin (e+f x)} F\left(\left.\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)\right|2\right) \sqrt{d \csc (e+f x)}}{21 d^2 f}-\frac{10 \cos (e+f x)}{21 d f \sqrt{d \csc (e+f x)}}-\frac{2 d \cos (e+f x)}{7 f (d \csc (e+f x))^{5/2}}",1,"-1/84*(Sqrt[d*Csc[e + f*x]]*(40*EllipticF[(-2*e + Pi - 2*f*x)/4, 2]*Sqrt[Sin[e + f*x]] + 26*Sin[2*(e + f*x)] - 3*Sin[4*(e + f*x)]))/(d^2*f)","A",1
531,1,60,74,0.0333743,"\int \frac{\sin (e+f x)}{(d \csc (e+f x))^{3/2}} \, dx","Integrate[Sin[e + f*x]/(d*Csc[e + f*x])^(3/2),x]","\frac{-2 \sin (2 (e+f x))-\frac{12 E\left(\left.\frac{1}{4} (-2 e-2 f x+\pi )\right|2\right)}{\sqrt{\sin (e+f x)}}}{10 d f \sqrt{d \csc (e+f x)}}","\frac{6 E\left(\left.\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)\right|2\right)}{5 d f \sqrt{\sin (e+f x)} \sqrt{d \csc (e+f x)}}-\frac{2 \cos (e+f x)}{5 f (d \csc (e+f x))^{3/2}}",1,"((-12*EllipticE[(-2*e + Pi - 2*f*x)/4, 2])/Sqrt[Sin[e + f*x]] - 2*Sin[2*(e + f*x)])/(10*d*f*Sqrt[d*Csc[e + f*x]])","A",1
532,1,63,77,0.0184972,"\int \frac{1}{(d \csc (e+f x))^{3/2}} \, dx","Integrate[(d*Csc[e + f*x])^(-3/2),x]","-\frac{\csc ^2(e+f x) \left(\sin (2 (e+f x))+2 \sqrt{\sin (e+f x)} F\left(\left.\frac{1}{4} (-2 e-2 f x+\pi )\right|2\right)\right)}{3 f (d \csc (e+f x))^{3/2}}","\frac{2 \sqrt{\sin (e+f x)} F\left(\left.\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)\right|2\right) \sqrt{d \csc (e+f x)}}{3 d^2 f}-\frac{2 \cos (e+f x)}{3 d f \sqrt{d \csc (e+f x)}}",1,"-1/3*(Csc[e + f*x]^2*(2*EllipticF[(-2*e + Pi - 2*f*x)/4, 2]*Sqrt[Sin[e + f*x]] + Sin[2*(e + f*x)]))/(f*(d*Csc[e + f*x])^(3/2))","A",1
533,1,45,46,0.0295722,"\int \frac{\csc (e+f x)}{(d \csc (e+f x))^{3/2}} \, dx","Integrate[Csc[e + f*x]/(d*Csc[e + f*x])^(3/2),x]","-\frac{2 E\left(\left.\frac{1}{4} (-2 e-2 f x+\pi )\right|2\right)}{d f \sqrt{\sin (e+f x)} \sqrt{d \csc (e+f x)}}","\frac{2 E\left(\left.\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)\right|2\right)}{d f \sqrt{\sin (e+f x)} \sqrt{d \csc (e+f x)}}",1,"(-2*EllipticE[(-2*e + Pi - 2*f*x)/4, 2])/(d*f*Sqrt[d*Csc[e + f*x]]*Sqrt[Sin[e + f*x]])","A",1
534,1,45,46,0.0195292,"\int \frac{\csc ^2(e+f x)}{(d \csc (e+f x))^{3/2}} \, dx","Integrate[Csc[e + f*x]^2/(d*Csc[e + f*x])^(3/2),x]","-\frac{2 \sqrt{\sin (e+f x)} F\left(\left.\frac{1}{4} (-2 e-2 f x+\pi )\right|2\right) \sqrt{d \csc (e+f x)}}{d^2 f}","\frac{2 \sqrt{\sin (e+f x)} F\left(\left.\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)\right|2\right) \sqrt{d \csc (e+f x)}}{d^2 f}",1,"(-2*Sqrt[d*Csc[e + f*x]]*EllipticF[(-2*e + Pi - 2*f*x)/4, 2]*Sqrt[Sin[e + f*x]])/(d^2*f)","A",1
535,1,55,73,0.047265,"\int \frac{\csc ^3(e+f x)}{(d \csc (e+f x))^{3/2}} \, dx","Integrate[Csc[e + f*x]^3/(d*Csc[e + f*x])^(3/2),x]","\frac{\frac{2 E\left(\left.\frac{1}{4} (-2 e-2 f x+\pi )\right|2\right)}{\sqrt{\sin (e+f x)}}-2 \cot (e+f x)}{d f \sqrt{d \csc (e+f x)}}","-\frac{2 \cos (e+f x) \sqrt{d \csc (e+f x)}}{d^2 f}-\frac{2 E\left(\left.\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)\right|2\right)}{d f \sqrt{\sin (e+f x)} \sqrt{d \csc (e+f x)}}",1,"(-2*Cot[e + f*x] + (2*EllipticE[(-2*e + Pi - 2*f*x)/4, 2])/Sqrt[Sin[e + f*x]])/(d*f*Sqrt[d*Csc[e + f*x]])","A",1
536,1,60,77,0.0679949,"\int \frac{\csc ^4(e+f x)}{(d \csc (e+f x))^{3/2}} \, dx","Integrate[Csc[e + f*x]^4/(d*Csc[e + f*x])^(3/2),x]","-\frac{2 \csc ^3(e+f x) \left(\cos (e+f x)+\sin ^{\frac{3}{2}}(e+f x) F\left(\left.\frac{1}{4} (-2 e-2 f x+\pi )\right|2\right)\right)}{3 f (d \csc (e+f x))^{3/2}}","\frac{2 \sqrt{\sin (e+f x)} F\left(\left.\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)\right|2\right) \sqrt{d \csc (e+f x)}}{3 d^2 f}-\frac{2 \cos (e+f x) (d \csc (e+f x))^{3/2}}{3 d^3 f}",1,"(-2*Csc[e + f*x]^3*(Cos[e + f*x] + EllipticF[(-2*e + Pi - 2*f*x)/4, 2]*Sin[e + f*x]^(3/2)))/(3*f*(d*Csc[e + f*x])^(3/2))","A",1
537,1,73,105,0.1326545,"\int \frac{\csc ^5(e+f x)}{(d \csc (e+f x))^{3/2}} \, dx","Integrate[Csc[e + f*x]^5/(d*Csc[e + f*x])^(3/2),x]","\frac{\csc ^4(e+f x) \left(-7 \cos (e+f x)+3 \cos (3 (e+f x))+12 \sin ^{\frac{5}{2}}(e+f x) E\left(\left.\frac{1}{4} (-2 e-2 f x+\pi )\right|2\right)\right)}{10 f (d \csc (e+f x))^{3/2}}","-\frac{2 \cos (e+f x) (d \csc (e+f x))^{5/2}}{5 d^4 f}-\frac{6 \cos (e+f x) \sqrt{d \csc (e+f x)}}{5 d^2 f}-\frac{6 E\left(\left.\frac{1}{2} \left(e+f x-\frac{\pi }{2}\right)\right|2\right)}{5 d f \sqrt{\sin (e+f x)} \sqrt{d \csc (e+f x)}}",1,"(Csc[e + f*x]^4*(-7*Cos[e + f*x] + 3*Cos[3*(e + f*x)] + 12*EllipticE[(-2*e + Pi - 2*f*x)/4, 2]*Sin[e + f*x]^(5/2)))/(10*f*(d*Csc[e + f*x])^(3/2))","A",1
538,1,102,87,10.4710004,"\int (b \csc (e+f x))^n (a \sin (e+f x))^m \, dx","Integrate[(b*Csc[e + f*x])^n*(a*Sin[e + f*x])^m,x]","\frac{2 \tan \left(\frac{1}{2} (e+f x)\right) (a \sin (e+f x))^m (b \csc (e+f x))^n \sec ^2\left(\frac{1}{2} (e+f x)\right)^{m-n} \, _2F_1\left(\frac{1}{2} (m-n+1),m-n+1;\frac{1}{2} (m-n+3);-\tan ^2\left(\frac{1}{2} (e+f x)\right)\right)}{f (m-n+1)}","\frac{\cos (e+f x) (a \sin (e+f x))^{m+1} (b \csc (e+f x))^n \, _2F_1\left(\frac{1}{2},\frac{1}{2} (m-n+1);\frac{1}{2} (m-n+3);\sin ^2(e+f x)\right)}{a f (m-n+1) \sqrt{\cos ^2(e+f x)}}",1,"(2*(b*Csc[e + f*x])^n*Hypergeometric2F1[(1 + m - n)/2, 1 + m - n, (3 + m - n)/2, -Tan[(e + f*x)/2]^2]*(Sec[(e + f*x)/2]^2)^(m - n)*(a*Sin[e + f*x])^m*Tan[(e + f*x)/2])/(f*(1 + m - n))","A",1