1,1,38,12,0.0168105,"\int \csc (a+b x) \, dx","Integrate[Csc[a + b*x],x]","\frac{\log \left(\sin \left(\frac{a}{2}+\frac{b x}{2}\right)\right)}{b}-\frac{\log \left(\cos \left(\frac{a}{2}+\frac{b x}{2}\right)\right)}{b}","-\frac{\tanh ^{-1}(\cos (a+b x))}{b}",1,"-(Log[Cos[a/2 + (b*x)/2]]/b) + Log[Sin[a/2 + (b*x)/2]]/b","B",1
2,1,11,11,0.0087223,"\int \csc ^2(a+b x) \, dx","Integrate[Csc[a + b*x]^2,x]","-\frac{\cot (a+b x)}{b}","-\frac{\cot (a+b x)}{b}",1,"-(Cot[a + b*x]/b)","A",1
3,1,75,34,0.0125639,"\int \csc ^3(a+b x) \, dx","Integrate[Csc[a + b*x]^3,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
4,1,35,27,0.011409,"\int \csc ^4(a+b x) \, dx","Integrate[Csc[a + b*x]^4,x]","-\frac{2 \cot (a+b x)}{3 b}-\frac{\cot (a+b x) \csc ^2(a+b x)}{3 b}","-\frac{\cot ^3(a+b x)}{3 b}-\frac{\cot (a+b x)}{b}",1,"(-2*Cot[a + b*x])/(3*b) - (Cot[a + b*x]*Csc[a + b*x]^2)/(3*b)","A",1
5,1,113,55,0.0161671,"\int \csc ^5(a+b x) \, dx","Integrate[Csc[a + b*x]^5,x]","-\frac{\csc ^4\left(\frac{1}{2} (a+b x)\right)}{64 b}-\frac{3 \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{3 \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 (a+b x) \csc ^3(a+b x)}{4 b}-\frac{3 \cot (a+b x) \csc (a+b x)}{8 b}",1,"(-3*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) + (3*Sec[(a + b*x)/2]^2)/(32*b) + Sec[(a + b*x)/2]^4/(64*b)","B",1
6,1,56,42,0.0151495,"\int \csc ^6(a+b x) \, dx","Integrate[Csc[a + b*x]^6,x]","-\frac{8 \cot (a+b x)}{15 b}-\frac{\cot (a+b x) \csc ^4(a+b x)}{5 b}-\frac{4 \cot (a+b x) \csc ^2(a+b x)}{15 b}","-\frac{\cot ^5(a+b x)}{5 b}-\frac{2 \cot ^3(a+b x)}{3 b}-\frac{\cot (a+b x)}{b}",1,"(-8*Cot[a + b*x])/(15*b) - (4*Cot[a + b*x]*Csc[a + b*x]^2)/(15*b) - (Cot[a + b*x]*Csc[a + b*x]^4)/(5*b)","A",1
7,1,151,76,0.0140373,"\int \csc ^7(a+b x) \, dx","Integrate[Csc[a + b*x]^7,x]","-\frac{\csc ^6\left(\frac{1}{2} (a+b x)\right)}{384 b}-\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)}{64 b}+\frac{\sec ^6\left(\frac{1}{2} (a+b x)\right)}{384 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)}{64 b}+\frac{5 \log \left(\sin \left(\frac{1}{2} (a+b x)\right)\right)}{16 b}-\frac{5 \log \left(\cos \left(\frac{1}{2} (a+b x)\right)\right)}{16 b}","-\frac{5 \tanh ^{-1}(\cos (a+b x))}{16 b}-\frac{\cot (a+b x) \csc ^5(a+b x)}{6 b}-\frac{5 \cot (a+b x) \csc ^3(a+b x)}{24 b}-\frac{5 \cot (a+b x) \csc (a+b x)}{16 b}",1,"(-5*Csc[(a + b*x)/2]^2)/(64*b) - Csc[(a + b*x)/2]^4/(64*b) - Csc[(a + b*x)/2]^6/(384*b) - (5*Log[Cos[(a + b*x)/2]])/(16*b) + (5*Log[Sin[(a + b*x)/2]])/(16*b) + (5*Sec[(a + b*x)/2]^2)/(64*b) + Sec[(a + b*x)/2]^4/(64*b) + Sec[(a + b*x)/2]^6/(384*b)","A",1
8,1,77,55,0.009898,"\int \csc ^8(a+b x) \, dx","Integrate[Csc[a + b*x]^8,x]","-\frac{16 \cot (a+b x)}{35 b}-\frac{\cot (a+b x) \csc ^6(a+b x)}{7 b}-\frac{6 \cot (a+b x) \csc ^4(a+b x)}{35 b}-\frac{8 \cot (a+b x) \csc ^2(a+b x)}{35 b}","-\frac{\cot ^7(a+b x)}{7 b}-\frac{3 \cot ^5(a+b x)}{5 b}-\frac{\cot ^3(a+b x)}{b}-\frac{\cot (a+b x)}{b}",1,"(-16*Cot[a + b*x])/(35*b) - (8*Cot[a + b*x]*Csc[a + b*x]^2)/(35*b) - (6*Cot[a + b*x]*Csc[a + b*x]^4)/(35*b) - (Cot[a + b*x]*Csc[a + b*x]^6)/(7*b)","A",1
9,1,63,90,0.1821626,"\int \csc ^{\frac{7}{2}}(a+b x) \, dx","Integrate[Csc[a + b*x]^(7/2),x]","\frac{\csc ^{\frac{5}{2}}(a+b x) \left(-7 \cos (a+b x)+3 \cos (3 (a+b x))+12 \sin ^{\frac{5}{2}}(a+b x) E\left(\left.\frac{1}{4} (-2 a-2 b x+\pi )\right|2\right)\right)}{10 b}","-\frac{2 \cos (a+b x) \csc ^{\frac{5}{2}}(a+b x)}{5 b}-\frac{6 \cos (a+b x) \sqrt{\csc (a+b x)}}{5 b}-\frac{6 \sqrt{\sin (a+b x)} \sqrt{\csc (a+b x)} E\left(\left.\frac{1}{2} \left(a+b x-\frac{\pi }{2}\right)\right|2\right)}{5 b}",1,"(Csc[a + b*x]^(5/2)*(-7*Cos[a + b*x] + 3*Cos[3*(a + b*x)] + 12*EllipticE[(-2*a + Pi - 2*b*x)/4, 2]*Sin[a + b*x]^(5/2)))/(10*b)","A",1
10,1,50,67,0.0749757,"\int \csc ^{\frac{5}{2}}(a+b x) \, dx","Integrate[Csc[a + b*x]^(5/2),x]","-\frac{2 \csc ^{\frac{3}{2}}(a+b x) \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}","\frac{2 \sqrt{\sin (a+b x)} \sqrt{\csc (a+b x)} 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) \csc ^{\frac{3}{2}}(a+b x)}{3 b}",1,"(-2*Csc[a + b*x]^(3/2)*(Cos[a + b*x] + EllipticF[(-2*a + Pi - 2*b*x)/4, 2]*Sin[a + b*x]^(3/2)))/(3*b)","A",1
11,1,49,63,0.0509465,"\int \csc ^{\frac{3}{2}}(a+b x) \, dx","Integrate[Csc[a + b*x]^(3/2),x]","-\frac{2 \sqrt{\csc (a+b x)} \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}","-\frac{2 \cos (a+b x) \sqrt{\csc (a+b x)}}{b}-\frac{2 \sqrt{\sin (a+b x)} \sqrt{\csc (a+b x)} E\left(\left.\frac{1}{2} \left(a+b x-\frac{\pi }{2}\right)\right|2\right)}{b}",1,"(-2*Sqrt[Csc[a + b*x]]*(Cos[a + b*x] - EllipticE[(-2*a + Pi - 2*b*x)/4, 2]*Sqrt[Sin[a + b*x]]))/b","A",1
12,1,40,41,0.03177,"\int \sqrt{\csc (a+b x)} \, dx","Integrate[Sqrt[Csc[a + b*x]],x]","-\frac{2 \sqrt{\sin (a+b x)} \sqrt{\csc (a+b x)} F\left(\left.\frac{1}{4} (-2 a-2 b x+\pi )\right|2\right)}{b}","\frac{2 \sqrt{\sin (a+b x)} \sqrt{\csc (a+b x)} F\left(\left.\frac{1}{2} \left(a+b x-\frac{\pi }{2}\right)\right|2\right)}{b}",1,"(-2*Sqrt[Csc[a + b*x]]*EllipticF[(-2*a + Pi - 2*b*x)/4, 2]*Sqrt[Sin[a + b*x]])/b","A",1
13,1,40,41,0.0354505,"\int \frac{1}{\sqrt{\csc (a+b x)}} \, dx","Integrate[1/Sqrt[Csc[a + b*x]],x]","-\frac{2 \sqrt{\sin (a+b x)} \sqrt{\csc (a+b x)} E\left(\left.\frac{1}{4} (-2 a-2 b x+\pi )\right|2\right)}{b}","\frac{2 \sqrt{\sin (a+b x)} \sqrt{\csc (a+b x)} E\left(\left.\frac{1}{2} \left(a+b x-\frac{\pi }{2}\right)\right|2\right)}{b}",1,"(-2*Sqrt[Csc[a + b*x]]*EllipticE[(-2*a + Pi - 2*b*x)/4, 2]*Sqrt[Sin[a + b*x]])/b","A",1
14,1,53,67,0.0530605,"\int \frac{1}{\csc ^{\frac{3}{2}}(a+b x)} \, dx","Integrate[Csc[a + b*x]^(-3/2),x]","-\frac{\sqrt{\csc (a+b x)} \left(\sin (2 (a+b x))+2 \sqrt{\sin (a+b x)} F\left(\left.\frac{1}{4} (-2 a-2 b x+\pi )\right|2\right)\right)}{3 b}","\frac{2 \sqrt{\sin (a+b x)} \sqrt{\csc (a+b x)} 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 \sqrt{\csc (a+b x)}}",1,"-1/3*(Sqrt[Csc[a + b*x]]*(2*EllipticF[(-2*a + Pi - 2*b*x)/4, 2]*Sqrt[Sin[a + b*x]] + Sin[2*(a + b*x)]))/b","A",1
15,1,60,67,0.0970012,"\int \frac{1}{\csc ^{\frac{5}{2}}(a+b x)} \, dx","Integrate[Csc[a + b*x]^(-5/2),x]","-\frac{2 \sqrt{\csc (a+b x)} \left(\sin ^2(a+b x) \cos (a+b x)+3 \sqrt{\sin (a+b x)} E\left(\left.\frac{1}{4} (-2 a-2 b x+\pi )\right|2\right)\right)}{5 b}","\frac{6 \sqrt{\sin (a+b x)} \sqrt{\csc (a+b x)} 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 \csc ^{\frac{3}{2}}(a+b x)}",1,"(-2*Sqrt[Csc[a + b*x]]*(3*EllipticE[(-2*a + Pi - 2*b*x)/4, 2]*Sqrt[Sin[a + b*x]] + Cos[a + b*x]*Sin[a + b*x]^2))/(5*b)","A",1
16,1,65,90,0.1501064,"\int \frac{1}{\csc ^{\frac{7}{2}}(a+b x)} \, dx","Integrate[Csc[a + b*x]^(-7/2),x]","-\frac{\sqrt{\csc (a+b x)} \left(26 \sin (2 (a+b x))-3 \sin (4 (a+b x))+40 \sqrt{\sin (a+b x)} F\left(\left.\frac{1}{4} (-2 a-2 b x+\pi )\right|2\right)\right)}{84 b}","-\frac{2 \cos (a+b x)}{7 b \csc ^{\frac{5}{2}}(a+b x)}-\frac{10 \cos (a+b x)}{21 b \sqrt{\csc (a+b x)}}+\frac{10 \sqrt{\sin (a+b x)} \sqrt{\csc (a+b x)} F\left(\left.\frac{1}{2} \left(a+b x-\frac{\pi }{2}\right)\right|2\right)}{21 b}",1,"-1/84*(Sqrt[Csc[a + b*x]]*(40*EllipticF[(-2*a + Pi - 2*b*x)/4, 2]*Sqrt[Sin[a + b*x]] + 26*Sin[2*(a + b*x)] - 3*Sin[4*(a + b*x)]))/b","A",1
17,1,67,103,0.1808424,"\int (c \csc (a+b x))^{7/2} \, dx","Integrate[(c*Csc[a + b*x])^(7/2),x]","\frac{(c \csc (a+b x))^{7/2} \left(-10 \sin (2 (a+b x))+3 \sin (4 (a+b x))+24 \sin ^{\frac{7}{2}}(a+b x) E\left(\left.\frac{1}{4} (-2 a-2 b x+\pi )\right|2\right)\right)}{20 b}","-\frac{6 c^4 E\left(\left.\frac{1}{2} \left(a+b x-\frac{\pi }{2}\right)\right|2\right)}{5 b \sqrt{\sin (a+b x)} \sqrt{c \csc (a+b x)}}-\frac{6 c^3 \cos (a+b x) \sqrt{c \csc (a+b x)}}{5 b}-\frac{2 c \cos (a+b x) (c \csc (a+b x))^{5/2}}{5 b}",1,"((c*Csc[a + b*x])^(7/2)*(24*EllipticE[(-2*a + Pi - 2*b*x)/4, 2]*Sin[a + b*x]^(7/2) - 10*Sin[2*(a + b*x)] + 3*Sin[4*(a + b*x)]))/(20*b)","A",1
18,1,55,75,0.1646437,"\int (c \csc (a+b x))^{5/2} \, dx","Integrate[(c*Csc[a + b*x])^(5/2),x]","-\frac{(c \csc (a+b x))^{5/2} \left(\sin (2 (a+b x))+2 \sin ^{\frac{5}{2}}(a+b x) F\left(\left.\frac{1}{4} (-2 a-2 b x+\pi )\right|2\right)\right)}{3 b}","\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) \sqrt{c \csc (a+b x)}}{3 b}-\frac{2 c \cos (a+b x) (c \csc (a+b x))^{3/2}}{3 b}",1,"-1/3*((c*Csc[a + b*x])^(5/2)*(2*EllipticF[(-2*a + Pi - 2*b*x)/4, 2]*Sin[a + b*x]^(5/2) + Sin[2*(a + b*x)]))/b","A",1
19,1,54,71,0.1069319,"\int (c \csc (a+b x))^{3/2} \, dx","Integrate[(c*Csc[a + b*x])^(3/2),x]","\frac{(c \csc (a+b x))^{3/2} \left(2 \sin ^{\frac{3}{2}}(a+b x) E\left(\left.\frac{1}{4} (-2 a-2 b x+\pi )\right|2\right)-\sin (2 (a+b x))\right)}{b}","-\frac{2 c^2 E\left(\left.\frac{1}{2} \left(a+b x-\frac{\pi }{2}\right)\right|2\right)}{b \sqrt{\sin (a+b x)} \sqrt{c \csc (a+b x)}}-\frac{2 c \cos (a+b x) \sqrt{c \csc (a+b x)}}{b}",1,"((c*Csc[a + b*x])^(3/2)*(2*EllipticE[(-2*a + Pi - 2*b*x)/4, 2]*Sin[a + b*x]^(3/2) - Sin[2*(a + b*x)]))/b","A",1
20,1,42,43,0.0253496,"\int \sqrt{c \csc (a+b x)} \, dx","Integrate[Sqrt[c*Csc[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) \sqrt{c \csc (a+b x)}}{b}","\frac{2 \sqrt{\sin (a+b x)} F\left(\left.\frac{1}{2} \left(a+b x-\frac{\pi }{2}\right)\right|2\right) \sqrt{c \csc (a+b x)}}{b}",1,"(-2*Sqrt[c*Csc[a + b*x]]*EllipticF[(-2*a + Pi - 2*b*x)/4, 2]*Sqrt[Sin[a + b*x]])/b","A",1
21,1,42,43,0.0282637,"\int \frac{1}{\sqrt{c \csc (a+b x)}} \, dx","Integrate[1/Sqrt[c*Csc[a + b*x]],x]","-\frac{2 E\left(\left.\frac{1}{4} (-2 a-2 b x+\pi )\right|2\right)}{b \sqrt{\sin (a+b x)} \sqrt{c \csc (a+b x)}}","\frac{2 E\left(\left.\frac{1}{2} \left(a+b x-\frac{\pi }{2}\right)\right|2\right)}{b \sqrt{\sin (a+b x)} \sqrt{c \csc (a+b x)}}",1,"(-2*EllipticE[(-2*a + Pi - 2*b*x)/4, 2])/(b*Sqrt[c*Csc[a + b*x]]*Sqrt[Sin[a + b*x]])","A",1
22,1,63,77,0.0712779,"\int \frac{1}{(c \csc (a+b x))^{3/2}} \, dx","Integrate[(c*Csc[a + b*x])^(-3/2),x]","-\frac{\csc ^2(a+b x) \left(\sin (2 (a+b x))+2 \sqrt{\sin (a+b x)} F\left(\left.\frac{1}{4} (-2 a-2 b x+\pi )\right|2\right)\right)}{3 b (c \csc (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) \sqrt{c \csc (a+b x)}}{3 b c^2}-\frac{2 \cos (a+b x)}{3 b c \sqrt{c \csc (a+b x)}}",1,"-1/3*(Csc[a + b*x]^2*(2*EllipticF[(-2*a + Pi - 2*b*x)/4, 2]*Sqrt[Sin[a + b*x]] + Sin[2*(a + b*x)]))/(b*(c*Csc[a + b*x])^(3/2))","A",1
23,1,60,77,0.1358408,"\int \frac{1}{(c \csc (a+b x))^{5/2}} \, dx","Integrate[(c*Csc[a + b*x])^(-5/2),x]","\frac{-2 \sin (2 (a+b x))-\frac{12 E\left(\left.\frac{1}{4} (-2 a-2 b x+\pi )\right|2\right)}{\sqrt{\sin (a+b x)}}}{10 b c^2 \sqrt{c \csc (a+b x)}}","\frac{6 E\left(\left.\frac{1}{2} \left(a+b x-\frac{\pi }{2}\right)\right|2\right)}{5 b c^2 \sqrt{\sin (a+b x)} \sqrt{c \csc (a+b x)}}-\frac{2 \cos (a+b x)}{5 b c (c \csc (a+b x))^{3/2}}",1,"((-12*EllipticE[(-2*a + Pi - 2*b*x)/4, 2])/Sqrt[Sin[a + b*x]] - 2*Sin[2*(a + b*x)])/(10*b*c^2*Sqrt[c*Csc[a + b*x]])","A",1
24,1,70,105,0.1198502,"\int \frac{1}{(c \csc (a+b x))^{7/2}} \, dx","Integrate[(c*Csc[a + b*x])^(-7/2),x]","-\frac{\sqrt{c \csc (a+b x)} \left(26 \sin (2 (a+b x))-3 \sin (4 (a+b x))+40 \sqrt{\sin (a+b x)} F\left(\left.\frac{1}{4} (-2 a-2 b x+\pi )\right|2\right)\right)}{84 b c^4}","\frac{10 \sqrt{\sin (a+b x)} F\left(\left.\frac{1}{2} \left(a+b x-\frac{\pi }{2}\right)\right|2\right) \sqrt{c \csc (a+b x)}}{21 b c^4}-\frac{10 \cos (a+b x)}{21 b c^3 \sqrt{c \csc (a+b x)}}-\frac{2 \cos (a+b x)}{7 b c (c \csc (a+b x))^{5/2}}",1,"-1/84*(Sqrt[c*Csc[a + b*x]]*(40*EllipticF[(-2*a + Pi - 2*b*x)/4, 2]*Sqrt[Sin[a + b*x]] + 26*Sin[2*(a + b*x)] - 3*Sin[4*(a + b*x)]))/(b*c^4)","A",1
25,1,54,51,0.0700368,"\int \csc ^{\frac{4}{3}}(a+b x) \, dx","Integrate[Csc[a + b*x]^(4/3),x]","\frac{\cos (a+b x) \sqrt[3]{\csc (a+b x)} \left(2 \sqrt[6]{\sin ^2(a+b x)} \, _2F_1\left(\frac{1}{6},\frac{1}{2};\frac{3}{2};\cos ^2(a+b x)\right)-3\right)}{b}","-\frac{3 \cos (a+b x) \sqrt[3]{\csc (a+b x)} \, _2F_1\left(-\frac{1}{6},\frac{1}{2};\frac{5}{6};\sin ^2(a+b x)\right)}{b \sqrt{\cos ^2(a+b x)}}",1,"(Cos[a + b*x]*Csc[a + b*x]^(1/3)*(-3 + 2*Hypergeometric2F1[1/6, 1/2, 3/2, Cos[a + b*x]^2]*(Sin[a + b*x]^2)^(1/6)))/b","A",1
26,1,51,51,0.0505381,"\int \csc ^{\frac{2}{3}}(a+b x) \, dx","Integrate[Csc[a + b*x]^(2/3),x]","-\frac{\cos (a+b x) \, _2F_1\left(\frac{1}{2},\frac{5}{6};\frac{3}{2};\cos ^2(a+b x)\right)}{b \sqrt[6]{\sin ^2(a+b x)} \sqrt[3]{\csc (a+b x)}}","\frac{3 \cos (a+b x) \, _2F_1\left(\frac{1}{6},\frac{1}{2};\frac{7}{6};\sin ^2(a+b x)\right)}{b \sqrt{\cos ^2(a+b x)} \sqrt[3]{\csc (a+b x)}}",1,"-((Cos[a + b*x]*Hypergeometric2F1[1/2, 5/6, 3/2, Cos[a + b*x]^2])/(b*Csc[a + b*x]^(1/3)*(Sin[a + b*x]^2)^(1/6)))","A",1
27,1,51,53,0.0515718,"\int \sqrt[3]{\csc (a+b x)} \, dx","Integrate[Csc[a + b*x]^(1/3),x]","-\frac{\cos (a+b x) \, _2F_1\left(\frac{1}{2},\frac{2}{3};\frac{3}{2};\cos ^2(a+b x)\right)}{b \sqrt[3]{\sin ^2(a+b x)} \csc ^{\frac{2}{3}}(a+b x)}","\frac{3 \cos (a+b x) \, _2F_1\left(\frac{1}{3},\frac{1}{2};\frac{4}{3};\sin ^2(a+b x)\right)}{2 b \sqrt{\cos ^2(a+b x)} \csc ^{\frac{2}{3}}(a+b x)}",1,"-((Cos[a + b*x]*Hypergeometric2F1[1/2, 2/3, 3/2, Cos[a + b*x]^2])/(b*Csc[a + b*x]^(2/3)*(Sin[a + b*x]^2)^(1/3)))","A",1
28,1,51,53,0.0454279,"\int \frac{1}{\sqrt[3]{\csc (a+b x)}} \, dx","Integrate[Csc[a + b*x]^(-1/3),x]","-\frac{\cos (a+b x) \, _2F_1\left(\frac{1}{3},\frac{1}{2};\frac{3}{2};\cos ^2(a+b x)\right)}{b \sin ^2(a+b x)^{2/3} \csc ^{\frac{4}{3}}(a+b x)}","\frac{3 \cos (a+b x) \, _2F_1\left(\frac{1}{2},\frac{2}{3};\frac{5}{3};\sin ^2(a+b x)\right)}{4 b \sqrt{\cos ^2(a+b x)} \csc ^{\frac{4}{3}}(a+b x)}",1,"-((Cos[a + b*x]*Hypergeometric2F1[1/3, 1/2, 3/2, Cos[a + b*x]^2])/(b*Csc[a + b*x]^(4/3)*(Sin[a + b*x]^2)^(2/3)))","A",1
29,1,51,53,0.0456648,"\int \frac{1}{\csc ^{\frac{2}{3}}(a+b x)} \, dx","Integrate[Csc[a + b*x]^(-2/3),x]","-\frac{\cos (a+b x) \, _2F_1\left(\frac{1}{6},\frac{1}{2};\frac{3}{2};\cos ^2(a+b x)\right)}{b \sin ^2(a+b x)^{5/6} \csc ^{\frac{5}{3}}(a+b x)}","\frac{3 \cos (a+b x) \, _2F_1\left(\frac{1}{2},\frac{5}{6};\frac{11}{6};\sin ^2(a+b x)\right)}{5 b \sqrt{\cos ^2(a+b x)} \csc ^{\frac{5}{3}}(a+b x)}",1,"-((Cos[a + b*x]*Hypergeometric2F1[1/6, 1/2, 3/2, Cos[a + b*x]^2])/(b*Csc[a + b*x]^(5/3)*(Sin[a + b*x]^2)^(5/6)))","A",1
30,1,68,53,0.1125074,"\int \frac{1}{\csc ^{\frac{4}{3}}(a+b x)} \, dx","Integrate[Csc[a + b*x]^(-4/3),x]","-\frac{\cos (a+b x) \left(\, _2F_1\left(\frac{1}{2},\frac{5}{6};\frac{3}{2};\cos ^2(a+b x)\right)+3 \sqrt[6]{\sin ^2(a+b x)}\right)}{4 b \sqrt[6]{\sin ^2(a+b x)} \sqrt[3]{\csc (a+b x)}}","\frac{3 \cos (a+b x) \, _2F_1\left(\frac{1}{2},\frac{7}{6};\frac{13}{6};\sin ^2(a+b x)\right)}{7 b \sqrt{\cos ^2(a+b x)} \csc ^{\frac{7}{3}}(a+b x)}",1,"-1/4*(Cos[a + b*x]*(Hypergeometric2F1[1/2, 5/6, 3/2, Cos[a + b*x]^2] + 3*(Sin[a + b*x]^2)^(1/6)))/(b*Csc[a + b*x]^(1/3)*(Sin[a + b*x]^2)^(1/6))","A",1
31,1,57,54,0.0451847,"\int (c \csc (a+b x))^{4/3} \, dx","Integrate[(c*Csc[a + b*x])^(4/3),x]","\frac{c \cos (a+b x) \sqrt[3]{c \csc (a+b x)} \left(2 \sqrt[6]{\sin ^2(a+b x)} \, _2F_1\left(\frac{1}{6},\frac{1}{2};\frac{3}{2};\cos ^2(a+b x)\right)-3\right)}{b}","-\frac{3 c \cos (a+b x) \sqrt[3]{c \csc (a+b x)} \, _2F_1\left(-\frac{1}{6},\frac{1}{2};\frac{5}{6};\sin ^2(a+b x)\right)}{b \sqrt{\cos ^2(a+b x)}}",1,"(c*Cos[a + b*x]*(c*Csc[a + b*x])^(1/3)*(-3 + 2*Hypergeometric2F1[1/6, 1/2, 3/2, Cos[a + b*x]^2]*(Sin[a + b*x]^2)^(1/6)))/b","A",1
32,1,59,54,0.0670301,"\int (c \csc (a+b x))^{2/3} \, dx","Integrate[(c*Csc[a + b*x])^(2/3),x]","-\frac{\sin (a+b x) \cos (a+b x) (c \csc (a+b x))^{2/3} \, _2F_1\left(\frac{1}{2},\frac{5}{6};\frac{3}{2};\cos ^2(a+b x)\right)}{b \sqrt[6]{\sin ^2(a+b x)}}","\frac{3 c \cos (a+b x) \, _2F_1\left(\frac{1}{6},\frac{1}{2};\frac{7}{6};\sin ^2(a+b x)\right)}{b \sqrt{\cos ^2(a+b x)} \sqrt[3]{c \csc (a+b x)}}",1,"-((Cos[a + b*x]*(c*Csc[a + b*x])^(2/3)*Hypergeometric2F1[1/2, 5/6, 3/2, Cos[a + b*x]^2]*Sin[a + b*x])/(b*(Sin[a + b*x]^2)^(1/6)))","A",1
33,1,59,56,0.0664879,"\int \sqrt[3]{c \csc (a+b x)} \, dx","Integrate[(c*Csc[a + b*x])^(1/3),x]","-\frac{\sin (a+b x) \cos (a+b x) \sqrt[3]{c \csc (a+b x)} \, _2F_1\left(\frac{1}{2},\frac{2}{3};\frac{3}{2};\cos ^2(a+b x)\right)}{b \sqrt[3]{\sin ^2(a+b x)}}","\frac{3 c \cos (a+b x) \, _2F_1\left(\frac{1}{3},\frac{1}{2};\frac{4}{3};\sin ^2(a+b x)\right)}{2 b \sqrt{\cos ^2(a+b x)} (c \csc (a+b x))^{2/3}}",1,"-((Cos[a + b*x]*(c*Csc[a + b*x])^(1/3)*Hypergeometric2F1[1/2, 2/3, 3/2, Cos[a + b*x]^2]*Sin[a + b*x])/(b*(Sin[a + b*x]^2)^(1/3)))","A",1
34,1,59,56,0.0647348,"\int \frac{1}{\sqrt[3]{c \csc (a+b x)}} \, dx","Integrate[(c*Csc[a + b*x])^(-1/3),x]","-\frac{\sin (a+b x) \cos (a+b x) \, _2F_1\left(\frac{1}{3},\frac{1}{2};\frac{3}{2};\cos ^2(a+b x)\right)}{b \sin ^2(a+b x)^{2/3} \sqrt[3]{c \csc (a+b x)}}","\frac{3 c \cos (a+b x) \, _2F_1\left(\frac{1}{2},\frac{2}{3};\frac{5}{3};\sin ^2(a+b x)\right)}{4 b \sqrt{\cos ^2(a+b x)} (c \csc (a+b x))^{4/3}}",1,"-((Cos[a + b*x]*Hypergeometric2F1[1/3, 1/2, 3/2, Cos[a + b*x]^2]*Sin[a + b*x])/(b*(c*Csc[a + b*x])^(1/3)*(Sin[a + b*x]^2)^(2/3)))","A",1
35,1,59,56,0.063585,"\int \frac{1}{(c \csc (a+b x))^{2/3}} \, dx","Integrate[(c*Csc[a + b*x])^(-2/3),x]","-\frac{\sin (a+b x) \cos (a+b x) \, _2F_1\left(\frac{1}{6},\frac{1}{2};\frac{3}{2};\cos ^2(a+b x)\right)}{b \sin ^2(a+b x)^{5/6} (c \csc (a+b x))^{2/3}}","\frac{3 c \cos (a+b x) \, _2F_1\left(\frac{1}{2},\frac{5}{6};\frac{11}{6};\sin ^2(a+b x)\right)}{5 b \sqrt{\cos ^2(a+b x)} (c \csc (a+b x))^{5/3}}",1,"-((Cos[a + b*x]*Hypergeometric2F1[1/6, 1/2, 3/2, Cos[a + b*x]^2]*Sin[a + b*x])/(b*(c*Csc[a + b*x])^(2/3)*(Sin[a + b*x]^2)^(5/6)))","A",1
36,1,76,56,0.150498,"\int \frac{1}{(c \csc (a+b x))^{4/3}} \, dx","Integrate[(c*Csc[a + b*x])^(-4/3),x]","-\frac{\csc ^2(a+b x) \left(2 \sin ^2(a+b x)^{5/6} \cot (a+b x) \, _2F_1\left(\frac{1}{2},\frac{5}{6};\frac{3}{2};\cos ^2(a+b x)\right)+3 \sin (2 (a+b x))\right)}{8 b (c \csc (a+b x))^{4/3}}","\frac{3 c \cos (a+b x) \, _2F_1\left(\frac{1}{2},\frac{7}{6};\frac{13}{6};\sin ^2(a+b x)\right)}{7 b \sqrt{\cos ^2(a+b x)} (c \csc (a+b x))^{7/3}}",1,"-1/8*(Csc[a + b*x]^2*(2*Cot[a + b*x]*Hypergeometric2F1[1/2, 5/6, 3/2, Cos[a + b*x]^2]*(Sin[a + b*x]^2)^(5/6) + 3*Sin[2*(a + b*x)]))/(b*(c*Csc[a + b*x])^(4/3))","A",1
37,1,59,69,0.090153,"\int \csc ^n(a+b x) \, dx","Integrate[Csc[a + b*x]^n,x]","-\frac{\cos (a+b x) \sin ^2(a+b x)^{\frac{n-1}{2}} \csc ^{n-1}(a+b x) \, _2F_1\left(\frac{1}{2},\frac{n+1}{2};\frac{3}{2};\cos ^2(a+b x)\right)}{b}","\frac{\cos (a+b x) \csc ^{n-1}(a+b x) \, _2F_1\left(\frac{1}{2},\frac{1-n}{2};\frac{3-n}{2};\sin ^2(a+b x)\right)}{b (1-n) \sqrt{\cos ^2(a+b x)}}",1,"-((Cos[a + b*x]*Csc[a + b*x]^(-1 + n)*Hypergeometric2F1[1/2, (1 + n)/2, 3/2, Cos[a + b*x]^2]*(Sin[a + b*x]^2)^((-1 + n)/2))/b)","A",1
38,1,65,72,0.072618,"\int (c \csc (a+b x))^n \, dx","Integrate[(c*Csc[a + b*x])^n,x]","-\frac{\sin (a+b x) \cos (a+b x) \sin ^2(a+b x)^{\frac{n-1}{2}} (c \csc (a+b x))^n \, _2F_1\left(\frac{1}{2},\frac{n+1}{2};\frac{3}{2};\cos ^2(a+b x)\right)}{b}","\frac{c \cos (a+b x) (c \csc (a+b x))^{n-1} \, _2F_1\left(\frac{1}{2},\frac{1-n}{2};\frac{3-n}{2};\sin ^2(a+b x)\right)}{b (1-n) \sqrt{\cos ^2(a+b x)}}",1,"-((Cos[a + b*x]*(c*Csc[a + b*x])^n*Hypergeometric2F1[1/2, (1 + n)/2, 3/2, Cos[a + b*x]^2]*Sin[a + b*x]*(Sin[a + b*x]^2)^((-1 + n)/2))/b)","A",1
39,1,92,50,0.2548219,"\int \csc ^2(x)^{7/2} \, dx","Integrate[(Csc[x]^2)^(7/2),x]","\frac{1}{384} \sin (x) \sqrt{\csc ^2(x)} \left(-\csc ^6\left(\frac{x}{2}\right)-6 \csc ^4\left(\frac{x}{2}\right)-30 \csc ^2\left(\frac{x}{2}\right)+\sec ^6\left(\frac{x}{2}\right)+6 \sec ^4\left(\frac{x}{2}\right)+30 \sec ^2\left(\frac{x}{2}\right)-120 \left(\log \left(\cos \left(\frac{x}{2}\right)\right)-\log \left(\sin \left(\frac{x}{2}\right)\right)\right)\right)","-\frac{1}{6} \cot (x) \csc ^2(x)^{5/2}-\frac{5}{24} \cot (x) \csc ^2(x)^{3/2}-\frac{5}{16} \cot (x) \sqrt{\csc ^2(x)}-\frac{5}{16} \sinh ^{-1}(\cot (x))",1,"(Sqrt[Csc[x]^2]*(-30*Csc[x/2]^2 - 6*Csc[x/2]^4 - Csc[x/2]^6 - 120*(Log[Cos[x/2]] - Log[Sin[x/2]]) + 30*Sec[x/2]^2 + 6*Sec[x/2]^4 + Sec[x/2]^6)*Sin[x])/384","A",1
40,1,72,36,0.2552218,"\int \csc ^2(x)^{5/2} \, dx","Integrate[(Csc[x]^2)^(5/2),x]","\frac{1}{64} \sin (x) \sqrt{\csc ^2(x)} \left(-\csc ^4\left(\frac{x}{2}\right)-6 \csc ^2\left(\frac{x}{2}\right)+\sec ^4\left(\frac{x}{2}\right)+6 \sec ^2\left(\frac{x}{2}\right)+24 \left(\log \left(\sin \left(\frac{x}{2}\right)\right)-\log \left(\cos \left(\frac{x}{2}\right)\right)\right)\right)","-\frac{1}{4} \cot (x) \csc ^2(x)^{3/2}-\frac{3}{8} \cot (x) \sqrt{\csc ^2(x)}-\frac{3}{8} \sinh ^{-1}(\cot (x))",1,"(Sqrt[Csc[x]^2]*(-6*Csc[x/2]^2 - Csc[x/2]^4 + 24*(-Log[Cos[x/2]] + Log[Sin[x/2]]) + 6*Sec[x/2]^2 + Sec[x/2]^4)*Sin[x])/64","A",1
41,1,51,22,0.0925999,"\int \csc ^2(x)^{3/2} \, dx","Integrate[(Csc[x]^2)^(3/2),x]","\frac{1}{8} \sin (x) \sqrt{\csc ^2(x)} \left(-\csc ^2\left(\frac{x}{2}\right)+\sec ^2\left(\frac{x}{2}\right)+4 \log \left(\sin \left(\frac{x}{2}\right)\right)-4 \log \left(\cos \left(\frac{x}{2}\right)\right)\right)","-\frac{1}{2} \cot (x) \sqrt{\csc ^2(x)}-\frac{1}{2} \sinh ^{-1}(\cot (x))",1,"(Sqrt[Csc[x]^2]*(-Csc[x/2]^2 - 4*Log[Cos[x/2]] + 4*Log[Sin[x/2]] + Sec[x/2]^2)*Sin[x])/8","B",1
42,1,28,5,0.0131602,"\int \sqrt{\csc ^2(x)} \, dx","Integrate[Sqrt[Csc[x]^2],x]","\sin (x) \sqrt{\csc ^2(x)} \left(\log \left(\sin \left(\frac{x}{2}\right)\right)-\log \left(\cos \left(\frac{x}{2}\right)\right)\right)","-\sinh ^{-1}(\cot (x))",1,"Sqrt[Csc[x]^2]*(-Log[Cos[x/2]] + Log[Sin[x/2]])*Sin[x]","B",1
43,1,12,12,0.0073244,"\int \frac{1}{\sqrt{\csc ^2(x)}} \, dx","Integrate[1/Sqrt[Csc[x]^2],x]","-\frac{\cot (x)}{\sqrt{\csc ^2(x)}}","-\frac{\cot (x)}{\sqrt{\csc ^2(x)}}",1,"-(Cot[x]/Sqrt[Csc[x]^2])","A",1
44,1,23,29,0.0163203,"\int \frac{1}{\csc ^2(x)^{3/2}} \, dx","Integrate[(Csc[x]^2)^(-3/2),x]","\frac{(\cos (3 x)-9 \cos (x)) \csc (x)}{12 \sqrt{\csc ^2(x)}}","-\frac{2 \cot (x)}{3 \sqrt{\csc ^2(x)}}-\frac{\cot (x)}{3 \csc ^2(x)^{3/2}}",1,"((-9*Cos[x] + Cos[3*x])*Csc[x])/(12*Sqrt[Csc[x]^2])","A",1
45,1,31,43,0.0261534,"\int \frac{1}{\csc ^2(x)^{5/2}} \, dx","Integrate[(Csc[x]^2)^(-5/2),x]","-\frac{(150 \cos (x)-25 \cos (3 x)+3 \cos (5 x)) \csc (x)}{240 \sqrt{\csc ^2(x)}}","-\frac{8 \cot (x)}{15 \sqrt{\csc ^2(x)}}-\frac{4 \cot (x)}{15 \csc ^2(x)^{3/2}}-\frac{\cot (x)}{5 \csc ^2(x)^{5/2}}",1,"-1/240*((150*Cos[x] - 25*Cos[3*x] + 3*Cos[5*x])*Csc[x])/Sqrt[Csc[x]^2]","A",1
46,1,37,57,0.0403762,"\int \frac{1}{\csc ^2(x)^{7/2}} \, dx","Integrate[(Csc[x]^2)^(-7/2),x]","\frac{(-1225 \cos (x)+245 \cos (3 x)-49 \cos (5 x)+5 \cos (7 x)) \csc (x)}{2240 \sqrt{\csc ^2(x)}}","-\frac{16 \cot (x)}{35 \sqrt{\csc ^2(x)}}-\frac{8 \cot (x)}{35 \csc ^2(x)^{3/2}}-\frac{6 \cot (x)}{35 \csc ^2(x)^{5/2}}-\frac{\cot (x)}{7 \csc ^2(x)^{7/2}}",1,"((-1225*Cos[x] + 245*Cos[3*x] - 49*Cos[5*x] + 5*Cos[7*x])*Csc[x])/(2240*Sqrt[Csc[x]^2])","A",1
47,1,61,84,0.3220333,"\int \left(a \csc ^2(x)\right)^{7/2} \, dx","Integrate[(a*Csc[x]^2)^(7/2),x]","\frac{a^3 \csc ^5(x) \sqrt{a \csc ^2(x)} \left(-396 \cos (x)+170 \cos (3 x)-30 \cos (5 x)+480 \sin ^6(x) \left(\log \left(\sin \left(\frac{x}{2}\right)\right)-\log \left(\cos \left(\frac{x}{2}\right)\right)\right)\right)}{1536}","-\frac{5}{16} a^{7/2} \tanh ^{-1}\left(\frac{\sqrt{a} \cot (x)}{\sqrt{a \csc ^2(x)}}\right)-\frac{5}{16} a^3 \cot (x) \sqrt{a \csc ^2(x)}-\frac{5}{24} a^2 \cot (x) \left(a \csc ^2(x)\right)^{3/2}-\frac{1}{6} a \cot (x) \left(a \csc ^2(x)\right)^{5/2}",1,"(a^3*Csc[x]^5*Sqrt[a*Csc[x]^2]*(-396*Cos[x] + 170*Cos[3*x] - 30*Cos[5*x] + 480*(-Log[Cos[x/2]] + Log[Sin[x/2]])*Sin[x]^6))/1536","A",1
48,1,51,65,0.2136336,"\int \left(a \csc ^2(x)\right)^{5/2} \, dx","Integrate[(a*Csc[x]^2)^(5/2),x]","\frac{1}{64} \sin (x) \left(a \csc ^2(x)\right)^{5/2} \left(6 \left(\cos (3 x)+4 \sin ^4(x) \left(\log \left(\sin \left(\frac{x}{2}\right)\right)-\log \left(\cos \left(\frac{x}{2}\right)\right)\right)\right)-22 \cos (x)\right)","-\frac{3}{8} a^{5/2} \tanh ^{-1}\left(\frac{\sqrt{a} \cot (x)}{\sqrt{a \csc ^2(x)}}\right)-\frac{3}{8} a^2 \cot (x) \sqrt{a \csc ^2(x)}-\frac{1}{4} a \cot (x) \left(a \csc ^2(x)\right)^{3/2}",1,"((a*Csc[x]^2)^(5/2)*Sin[x]*(-22*Cos[x] + 6*(Cos[3*x] + 4*(-Log[Cos[x/2]] + Log[Sin[x/2]])*Sin[x]^4)))/64","A",1
49,1,39,46,0.0726873,"\int \left(a \csc ^2(x)\right)^{3/2} \, dx","Integrate[(a*Csc[x]^2)^(3/2),x]","-\frac{1}{2} a \sin (x) \sqrt{a \csc ^2(x)} \left(-\log \left(\sin \left(\frac{x}{2}\right)\right)+\log \left(\cos \left(\frac{x}{2}\right)\right)+\cot (x) \csc (x)\right)","-\frac{1}{2} a^{3/2} \tanh ^{-1}\left(\frac{\sqrt{a} \cot (x)}{\sqrt{a \csc ^2(x)}}\right)-\frac{1}{2} a \cot (x) \sqrt{a \csc ^2(x)}",1,"-1/2*(a*Sqrt[a*Csc[x]^2]*(Cot[x]*Csc[x] + Log[Cos[x/2]] - Log[Sin[x/2]])*Sin[x])","A",1
50,1,30,26,0.0155487,"\int \sqrt{a \csc ^2(x)} \, dx","Integrate[Sqrt[a*Csc[x]^2],x]","\sin (x) \sqrt{a \csc ^2(x)} \left(\log \left(\sin \left(\frac{x}{2}\right)\right)-\log \left(\cos \left(\frac{x}{2}\right)\right)\right)","-\sqrt{a} \tanh ^{-1}\left(\frac{\sqrt{a} \cot (x)}{\sqrt{a \csc ^2(x)}}\right)",1,"Sqrt[a*Csc[x]^2]*(-Log[Cos[x/2]] + Log[Sin[x/2]])*Sin[x]","A",1
51,1,14,14,0.0061821,"\int \frac{1}{\sqrt{a \csc ^2(x)}} \, dx","Integrate[1/Sqrt[a*Csc[x]^2],x]","-\frac{\cot (x)}{\sqrt{a \csc ^2(x)}}","-\frac{\cot (x)}{\sqrt{a \csc ^2(x)}}",1,"-(Cot[x]/Sqrt[a*Csc[x]^2])","A",1
52,1,27,36,0.0176017,"\int \frac{1}{\left(a \csc ^2(x)\right)^{3/2}} \, dx","Integrate[(a*Csc[x]^2)^(-3/2),x]","\frac{(\cos (3 x)-9 \cos (x)) \csc ^3(x)}{12 \left(a \csc ^2(x)\right)^{3/2}}","-\frac{2 \cot (x)}{3 a \sqrt{a \csc ^2(x)}}-\frac{\cot (x)}{3 \left(a \csc ^2(x)\right)^{3/2}}",1,"((-9*Cos[x] + Cos[3*x])*Csc[x]^3)/(12*(a*Csc[x]^2)^(3/2))","A",1
53,1,36,55,0.0322379,"\int \frac{1}{\left(a \csc ^2(x)\right)^{5/2}} \, dx","Integrate[(a*Csc[x]^2)^(-5/2),x]","-\frac{\sin (x) (150 \cos (x)-25 \cos (3 x)+3 \cos (5 x)) \sqrt{a \csc ^2(x)}}{240 a^3}","-\frac{8 \cot (x)}{15 a^2 \sqrt{a \csc ^2(x)}}-\frac{4 \cot (x)}{15 a \left(a \csc ^2(x)\right)^{3/2}}-\frac{\cot (x)}{5 \left(a \csc ^2(x)\right)^{5/2}}",1,"-1/240*((150*Cos[x] - 25*Cos[3*x] + 3*Cos[5*x])*Sqrt[a*Csc[x]^2]*Sin[x])/a^3","A",1
54,1,42,74,0.0448234,"\int \frac{1}{\left(a \csc ^2(x)\right)^{7/2}} \, dx","Integrate[(a*Csc[x]^2)^(-7/2),x]","\frac{\sin (x) (-1225 \cos (x)+245 \cos (3 x)-49 \cos (5 x)+5 \cos (7 x)) \sqrt{a \csc ^2(x)}}{2240 a^4}","-\frac{16 \cot (x)}{35 a^3 \sqrt{a \csc ^2(x)}}-\frac{8 \cot (x)}{35 a^2 \left(a \csc ^2(x)\right)^{3/2}}-\frac{6 \cot (x)}{35 a \left(a \csc ^2(x)\right)^{5/2}}-\frac{\cot (x)}{7 \left(a \csc ^2(x)\right)^{7/2}}",1,"((-1225*Cos[x] + 245*Cos[3*x] - 49*Cos[5*x] + 5*Cos[7*x])*Sqrt[a*Csc[x]^2]*Sin[x])/(2240*a^4)","A",1
55,1,58,123,0.1838001,"\int \left(a \csc ^3(x)\right)^{5/2} \, dx","Integrate[(a*Csc[x]^3)^(5/2),x]","\frac{\left(a \csc ^3(x)\right)^{5/2} \left(-9414 \sin (2 x)+5346 \sin (4 x)-1694 \sin (6 x)+231 \sin (8 x)+29568 \sin ^{\frac{15}{2}}(x) E\left(\left.\frac{1}{4} (\pi -2 x)\right|2\right)\right)}{37440}","-\frac{154}{585} a^2 \cot (x) \sqrt{a \csc ^3(x)}-\frac{2}{13} a^2 \cot (x) \csc ^4(x) \sqrt{a \csc ^3(x)}-\frac{22}{117} a^2 \cot (x) \csc ^2(x) \sqrt{a \csc ^3(x)}-\frac{154}{195} a^2 \sin (x) \cos (x) \sqrt{a \csc ^3(x)}+\frac{154}{195} a^2 \sin ^{\frac{3}{2}}(x) E\left(\left.\frac{\pi }{4}-\frac{x}{2}\right|2\right) \sqrt{a \csc ^3(x)}",1,"((a*Csc[x]^3)^(5/2)*(29568*EllipticE[(Pi - 2*x)/4, 2]*Sin[x]^(15/2) - 9414*Sin[2*x] + 5346*Sin[4*x] - 1694*Sin[6*x] + 231*Sin[8*x]))/37440","A",1
56,1,46,71,0.1133928,"\int \left(a \csc ^3(x)\right)^{3/2} \, dx","Integrate[(a*Csc[x]^3)^(3/2),x]","-\frac{1}{84} \left(a \csc ^3(x)\right)^{3/2} \left(22 \sin (2 x)-5 \sin (4 x)+40 \sin ^{\frac{9}{2}}(x) F\left(\left.\frac{1}{4} (\pi -2 x)\right|2\right)\right)","-\frac{10}{21} a \cos (x) \sqrt{a \csc ^3(x)}-\frac{2}{7} a \cot (x) \csc (x) \sqrt{a \csc ^3(x)}-\frac{10}{21} a \sin ^{\frac{3}{2}}(x) F\left(\left.\frac{\pi }{4}-\frac{x}{2}\right|2\right) \sqrt{a \csc ^3(x)}",1,"-1/84*((a*Csc[x]^3)^(3/2)*(40*EllipticF[(Pi - 2*x)/4, 2]*Sin[x]^(9/2) + 22*Sin[2*x] - 5*Sin[4*x]))","A",1
57,1,46,48,0.0395008,"\int \sqrt{a \csc ^3(x)} \, dx","Integrate[Sqrt[a*Csc[x]^3],x]","2 \sin ^{\frac{3}{2}}(x) E\left(\left.\frac{1}{4} (\pi -2 x)\right|2\right) \sqrt{a \csc ^3(x)}-2 \sin (x) \cos (x) \sqrt{a \csc ^3(x)}","2 \sin ^{\frac{3}{2}}(x) E\left(\left.\frac{\pi }{4}-\frac{x}{2}\right|2\right) \sqrt{a \csc ^3(x)}-2 \sin (x) \cos (x) \sqrt{a \csc ^3(x)}",1,"-2*Cos[x]*Sqrt[a*Csc[x]^3]*Sin[x] + 2*Sqrt[a*Csc[x]^3]*EllipticE[(Pi - 2*x)/4, 2]*Sin[x]^(3/2)","A",1
58,1,38,50,0.0489017,"\int \frac{1}{\sqrt{a \csc ^3(x)}} \, dx","Integrate[1/Sqrt[a*Csc[x]^3],x]","\frac{-2 \cot (x)-\frac{2 F\left(\left.\frac{1}{4} (\pi -2 x)\right|2\right)}{\sin ^{\frac{3}{2}}(x)}}{3 \sqrt{a \csc ^3(x)}}","-\frac{2 \cot (x)}{3 \sqrt{a \csc ^3(x)}}-\frac{2 F\left(\left.\frac{\pi }{4}-\frac{x}{2}\right|2\right)}{3 \sin ^{\frac{3}{2}}(x) \sqrt{a \csc ^3(x)}}",1,"(-2*Cot[x] - (2*EllipticF[(Pi - 2*x)/4, 2])/Sin[x]^(3/2))/(3*Sqrt[a*Csc[x]^3])","A",1
59,1,52,79,0.0944628,"\int \frac{1}{\left(a \csc ^3(x)\right)^{3/2}} \, dx","Integrate[(a*Csc[x]^3)^(-3/2),x]","\frac{\sin ^{\frac{3}{2}}(x) (5 \cos (3 x)-33 \cos (x))-84 E\left(\left.\frac{1}{4} (\pi -2 x)\right|2\right)}{90 \sin ^{\frac{9}{2}}(x) \left(a \csc ^3(x)\right)^{3/2}}","-\frac{14 \cos (x)}{45 a \sqrt{a \csc ^3(x)}}-\frac{2 \sin ^2(x) \cos (x)}{9 a \sqrt{a \csc ^3(x)}}-\frac{14 E\left(\left.\frac{\pi }{4}-\frac{x}{2}\right|2\right)}{15 a \sin ^{\frac{3}{2}}(x) \sqrt{a \csc ^3(x)}}",1,"(-84*EllipticE[(Pi - 2*x)/4, 2] + (-33*Cos[x] + 5*Cos[3*x])*Sin[x]^(3/2))/(90*(a*Csc[x]^3)^(3/2)*Sin[x]^(9/2))","A",1
60,1,63,123,0.1444543,"\int \frac{1}{\left(a \csc ^3(x)\right)^{5/2}} \, dx","Integrate[(a*Csc[x]^3)^(-5/2),x]","-\frac{\sin (x) \sqrt{a \csc ^3(x)} \left(19122 \sin (2 x)-4406 \sin (4 x)+826 \sin (6 x)-77 \sin (8 x)+24960 \sqrt{\sin (x)} F\left(\left.\frac{1}{4} (\pi -2 x)\right|2\right)\right)}{73920 a^3}","-\frac{26 \cot (x)}{77 a^2 \sqrt{a \csc ^3(x)}}-\frac{2 \sin ^5(x) \cos (x)}{15 a^2 \sqrt{a \csc ^3(x)}}-\frac{26 \sin ^3(x) \cos (x)}{165 a^2 \sqrt{a \csc ^3(x)}}-\frac{78 \sin (x) \cos (x)}{385 a^2 \sqrt{a \csc ^3(x)}}-\frac{26 F\left(\left.\frac{\pi }{4}-\frac{x}{2}\right|2\right)}{77 a^2 \sin ^{\frac{3}{2}}(x) \sqrt{a \csc ^3(x)}}",1,"-1/73920*(Sqrt[a*Csc[x]^3]*Sin[x]*(24960*EllipticF[(Pi - 2*x)/4, 2]*Sqrt[Sin[x]] + 19122*Sin[2*x] - 4406*Sin[4*x] + 826*Sin[6*x] - 77*Sin[8*x]))/a^3","A",1
61,1,59,164,0.068975,"\int \left(a \csc ^4(x)\right)^{7/2} \, dx","Integrate[(a*Csc[x]^4)^(7/2),x]","-\frac{a^3 \sin (x) \cos (x) \left(231 \csc ^{12}(x)+252 \csc ^{10}(x)+280 \csc ^8(x)+320 \csc ^6(x)+384 \csc ^4(x)+512 \csc ^2(x)+1024\right) \sqrt{a \csc ^4(x)}}{3003}","-\frac{1}{13} a^3 \cos ^2(x) \cot ^{11}(x) \sqrt{a \csc ^4(x)}-\frac{6}{11} a^3 \cos ^2(x) \cot ^9(x) \sqrt{a \csc ^4(x)}-\frac{5}{3} a^3 \cos ^2(x) \cot ^7(x) \sqrt{a \csc ^4(x)}-\frac{20}{7} a^3 \cos ^2(x) \cot ^5(x) \sqrt{a \csc ^4(x)}-3 a^3 \cos ^2(x) \cot ^3(x) \sqrt{a \csc ^4(x)}-2 a^3 \cos ^2(x) \cot (x) \sqrt{a \csc ^4(x)}-a^3 \sin (x) \cos (x) \sqrt{a \csc ^4(x)}",1,"-1/3003*(a^3*Cos[x]*Sqrt[a*Csc[x]^4]*(1024 + 512*Csc[x]^2 + 384*Csc[x]^4 + 320*Csc[x]^6 + 280*Csc[x]^8 + 252*Csc[x]^10 + 231*Csc[x]^12)*Sin[x])","A",1
62,1,47,118,0.0413215,"\int \left(a \csc ^4(x)\right)^{5/2} \, dx","Integrate[(a*Csc[x]^4)^(5/2),x]","-\frac{1}{315} a^2 \sin (x) \cos (x) \left(35 \csc ^8(x)+40 \csc ^6(x)+48 \csc ^4(x)+64 \csc ^2(x)+128\right) \sqrt{a \csc ^4(x)}","-\frac{1}{9} a^2 \cos ^2(x) \cot ^7(x) \sqrt{a \csc ^4(x)}-\frac{4}{7} a^2 \cos ^2(x) \cot ^5(x) \sqrt{a \csc ^4(x)}-\frac{6}{5} a^2 \cos ^2(x) \cot ^3(x) \sqrt{a \csc ^4(x)}-\frac{4}{3} a^2 \cos ^2(x) \cot (x) \sqrt{a \csc ^4(x)}-a^2 \sin (x) \cos (x) \sqrt{a \csc ^4(x)}",1,"-1/315*(a^2*Cos[x]*Sqrt[a*Csc[x]^4]*(128 + 64*Csc[x]^2 + 48*Csc[x]^4 + 40*Csc[x]^6 + 35*Csc[x]^8)*Sin[x])","A",1
63,1,33,62,0.0186367,"\int \left(a \csc ^4(x)\right)^{3/2} \, dx","Integrate[(a*Csc[x]^4)^(3/2),x]","-\frac{1}{15} a \sin (x) \cos (x) \left(3 \csc ^4(x)+4 \csc ^2(x)+8\right) \sqrt{a \csc ^4(x)}","-\frac{1}{5} a \cos ^2(x) \cot ^3(x) \sqrt{a \csc ^4(x)}-\frac{2}{3} a \cos ^2(x) \cot (x) \sqrt{a \csc ^4(x)}-a \sin (x) \cos (x) \sqrt{a \csc ^4(x)}",1,"-1/15*(a*Cos[x]*Sqrt[a*Csc[x]^4]*(8 + 4*Csc[x]^2 + 3*Csc[x]^4)*Sin[x])","A",1
64,1,16,16,0.0055203,"\int \sqrt{a \csc ^4(x)} \, dx","Integrate[Sqrt[a*Csc[x]^4],x]","\sin (x) (-\cos (x)) \sqrt{a \csc ^4(x)}","\sin (x) (-\cos (x)) \sqrt{a \csc ^4(x)}",1,"-(Cos[x]*Sqrt[a*Csc[x]^4]*Sin[x])","A",1
65,1,25,36,0.0260119,"\int \frac{1}{\sqrt{a \csc ^4(x)}} \, dx","Integrate[1/Sqrt[a*Csc[x]^4],x]","\frac{x \csc ^2(x)-\cot (x)}{2 \sqrt{a \csc ^4(x)}}","\frac{x \csc ^2(x)}{2 \sqrt{a \csc ^4(x)}}-\frac{\cot (x)}{2 \sqrt{a \csc ^4(x)}}",1,"(-Cot[x] + x*Csc[x]^2)/(2*Sqrt[a*Csc[x]^4])","A",1
66,1,38,86,0.0440163,"\int \frac{1}{\left(a \csc ^4(x)\right)^{3/2}} \, dx","Integrate[(a*Csc[x]^4)^(-3/2),x]","-\frac{(-60 x+45 \sin (2 x)-9 \sin (4 x)+\sin (6 x)) \csc ^6(x)}{192 \left(a \csc ^4(x)\right)^{3/2}}","\frac{5 x \csc ^2(x)}{16 a \sqrt{a \csc ^4(x)}}-\frac{5 \cot (x)}{16 a \sqrt{a \csc ^4(x)}}-\frac{\sin ^3(x) \cos (x)}{6 a \sqrt{a \csc ^4(x)}}-\frac{5 \sin (x) \cos (x)}{24 a \sqrt{a \csc ^4(x)}}",1,"-1/192*(Csc[x]^6*(-60*x + 45*Sin[2*x] - 9*Sin[4*x] + Sin[6*x]))/(a*Csc[x]^4)^(3/2)","A",1
67,1,55,132,0.0993015,"\int \frac{1}{\left(a \csc ^4(x)\right)^{5/2}} \, dx","Integrate[(a*Csc[x]^4)^(-5/2),x]","\frac{\sin ^2(x) (2520 x-2100 \sin (2 x)+600 \sin (4 x)-150 \sin (6 x)+25 \sin (8 x)-2 \sin (10 x)) \sqrt{a \csc ^4(x)}}{10240 a^3}","\frac{63 x \csc ^2(x)}{256 a^2 \sqrt{a \csc ^4(x)}}-\frac{63 \cot (x)}{256 a^2 \sqrt{a \csc ^4(x)}}-\frac{\sin ^7(x) \cos (x)}{10 a^2 \sqrt{a \csc ^4(x)}}-\frac{9 \sin ^5(x) \cos (x)}{80 a^2 \sqrt{a \csc ^4(x)}}-\frac{21 \sin ^3(x) \cos (x)}{160 a^2 \sqrt{a \csc ^4(x)}}-\frac{21 \sin (x) \cos (x)}{128 a^2 \sqrt{a \csc ^4(x)}}",1,"(Sqrt[a*Csc[x]^4]*Sin[x]^2*(2520*x - 2100*Sin[2*x] + 600*Sin[4*x] - 150*Sin[6*x] + 25*Sin[8*x] - 2*Sin[10*x]))/(10240*a^3)","A",1
68,1,71,80,0.1218458,"\int \left((b \csc (c+d x))^p\right)^n \, dx","Integrate[((b*Csc[c + d*x])^p)^n,x]","-\frac{\sin (c+d x) \cos (c+d x) \sin ^2(c+d x)^{\frac{1}{2} (n p-1)} \left((b \csc (c+d x))^p\right)^n \, _2F_1\left(\frac{1}{2},\frac{1}{2} (n p+1);\frac{3}{2};\cos ^2(c+d x)\right)}{d}","\frac{\sin (c+d x) \cos (c+d x) \left((b \csc (c+d x))^p\right)^n \, _2F_1\left(\frac{1}{2},\frac{1}{2} (1-n p);\frac{1}{2} (3-n p);\sin ^2(c+d x)\right)}{d (1-n p) \sqrt{\cos ^2(c+d x)}}",1,"-((Cos[c + d*x]*((b*Csc[c + d*x])^p)^n*Hypergeometric2F1[1/2, (1 + n*p)/2, 3/2, Cos[c + d*x]^2]*Sin[c + d*x]*(Sin[c + d*x]^2)^((-1 + n*p)/2))/d)","A",1
69,1,73,82,0.088286,"\int \left(a (b \csc (c+d x))^p\right)^n \, dx","Integrate[(a*(b*Csc[c + d*x])^p)^n,x]","-\frac{\sin (c+d x) \cos (c+d x) \sin ^2(c+d x)^{\frac{1}{2} (n p-1)} \, _2F_1\left(\frac{1}{2},\frac{1}{2} (n p+1);\frac{3}{2};\cos ^2(c+d x)\right) \left(a (b \csc (c+d x))^p\right)^n}{d}","\frac{\sin (c+d x) \cos (c+d x) \, _2F_1\left(\frac{1}{2},\frac{1}{2} (1-n p);\frac{1}{2} (3-n p);\sin ^2(c+d x)\right) \left(a (b \csc (c+d x))^p\right)^n}{d (1-n p) \sqrt{\cos ^2(c+d x)}}",1,"-((Cos[c + d*x]*(a*(b*Csc[c + d*x])^p)^n*Hypergeometric2F1[1/2, (1 + n*p)/2, 3/2, Cos[c + d*x]^2]*Sin[c + d*x]*(Sin[c + d*x]^2)^((-1 + n*p)/2))/d)","A",1
70,1,77,91,0.1418636,"\int (a \csc (e+f x))^m (b \csc (e+f x))^n \, dx","Integrate[(a*Csc[e + f*x])^m*(b*Csc[e + f*x])^n,x]","-\frac{\sin (e+f x) \cos (e+f x) (a \csc (e+f x))^m (b \csc (e+f x))^n \sin ^2(e+f x)^{\frac{1}{2} (m+n-1)} \, _2F_1\left(\frac{1}{2},\frac{1}{2} (m+n+1);\frac{3}{2};\cos ^2(e+f x)\right)}{f}","\frac{a \cos (e+f x) (a \csc (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)}{f (-m-n+1) \sqrt{\cos ^2(e+f x)}}",1,"-((Cos[e + f*x]*(a*Csc[e + f*x])^m*(b*Csc[e + f*x])^n*Hypergeometric2F1[1/2, (1 + m + n)/2, 3/2, Cos[e + f*x]^2]*Sin[e + f*x]*(Sin[e + f*x]^2)^((-1 + m + n)/2))/f)","A",1