1,1,148,112,3.5707615,"\int \left(a+b \csc ^2(c+d x)\right)^4 \, dx","Integrate[(a + b*Csc[c + d*x]^2)^4,x]","-\frac{16 \sin ^8(c+d x) \left(a+b \csc ^2(c+d x)\right)^4 \left(b \cot (c+d x) \left(420 a^3+2 b \left(105 a^2+56 a b+12 b^2\right) \csc ^2(c+d x)+420 a^2 b+6 b^2 (14 a+3 b) \csc ^4(c+d x)+224 a b^2+15 b^3 \csc ^6(c+d x)+48 b^3\right)-105 a^4 (c+d x)\right)}{105 d (a (-\cos (2 (c+d x)))+a+2 b)^4}","a^4 x-\frac{b^2 \left(6 a^2+8 a b+3 b^2\right) \cot ^3(c+d x)}{3 d}-\frac{b (2 a+b) \left(2 a^2+2 a b+b^2\right) \cot (c+d x)}{d}-\frac{b^3 (4 a+3 b) \cot ^5(c+d x)}{5 d}-\frac{b^4 \cot ^7(c+d x)}{7 d}",1,"(-16*(a + b*Csc[c + d*x]^2)^4*(-105*a^4*(c + d*x) + b*Cot[c + d*x]*(420*a^3 + 420*a^2*b + 224*a*b^2 + 48*b^3 + 2*b*(105*a^2 + 56*a*b + 12*b^2)*Csc[c + d*x]^2 + 6*b^2*(14*a + 3*b)*Csc[c + d*x]^4 + 15*b^3*Csc[c + d*x]^6))*Sin[c + d*x]^8)/(105*d*(a + 2*b - a*Cos[2*(c + d*x)])^4)","A",1
2,1,112,74,3.6731728,"\int \left(a+b \csc ^2(c+d x)\right)^3 \, dx","Integrate[(a + b*Csc[c + d*x]^2)^3,x]","\frac{8 \sin ^6(c+d x) \left(a+b \csc ^2(c+d x)\right)^3 \left(b \cot (c+d x) \left(45 a^2+b (15 a+4 b) \csc ^2(c+d x)+30 a b+3 b^2 \csc ^4(c+d x)+8 b^2\right)-15 a^3 (c+d x)\right)}{15 d (a \cos (2 (c+d x))-a-2 b)^3}","a^3 x-\frac{b \left(3 a^2+3 a b+b^2\right) \cot (c+d x)}{d}-\frac{b^2 (3 a+2 b) \cot ^3(c+d x)}{3 d}-\frac{b^3 \cot ^5(c+d x)}{5 d}",1,"(8*(a + b*Csc[c + d*x]^2)^3*(-15*a^3*(c + d*x) + b*Cot[c + d*x]*(45*a^2 + 30*a*b + 8*b^2 + b*(15*a + 4*b)*Csc[c + d*x]^2 + 3*b^2*Csc[c + d*x]^4))*Sin[c + d*x]^6)/(15*d*(-a - 2*b + a*Cos[2*(c + d*x)])^3)","A",1
3,1,83,41,0.6379727,"\int \left(a+b \csc ^2(c+d x)\right)^2 \, dx","Integrate[(a + b*Csc[c + d*x]^2)^2,x]","-\frac{4 \sin ^4(c+d x) \left(a+b \csc ^2(c+d x)\right)^2 \left(b \cot (c+d x) \left(6 a+b \csc ^2(c+d x)+2 b\right)-3 a^2 (c+d x)\right)}{3 d (a (-\cos (2 (c+d x)))+a+2 b)^2}","a^2 x-\frac{b (2 a+b) \cot (c+d x)}{d}-\frac{b^2 \cot ^3(c+d x)}{3 d}",1,"(-4*(a + b*Csc[c + d*x]^2)^2*(-3*a^2*(c + d*x) + b*Cot[c + d*x]*(6*a + 2*b + b*Csc[c + d*x]^2))*Sin[c + d*x]^4)/(3*d*(a + 2*b - a*Cos[2*(c + d*x)])^2)","B",1
4,1,16,16,0.0086614,"\int \left(a+b \csc ^2(c+d x)\right) \, dx","Integrate[a + b*Csc[c + d*x]^2,x]","a x-\frac{b \cot (c+d x)}{d}","a x-\frac{b \cot (c+d x)}{d}",1,"a*x - (b*Cot[c + d*x])/d","A",1
5,1,46,46,0.2079009,"\int \frac{1}{a+b \csc ^2(c+d x)} \, dx","Integrate[(a + b*Csc[c + d*x]^2)^(-1),x]","\frac{-\frac{\sqrt{b} \tan ^{-1}\left(\frac{\sqrt{a+b} \tan (c+d x)}{\sqrt{b}}\right)}{\sqrt{a+b}}+c+d x}{a d}","\frac{x}{a}-\frac{\sqrt{b} \tan ^{-1}\left(\frac{\sqrt{a+b} \tan (c+d x)}{\sqrt{b}}\right)}{a d \sqrt{a+b}}",1,"(c + d*x - (Sqrt[b]*ArcTan[(Sqrt[a + b]*Tan[c + d*x])/Sqrt[b]])/Sqrt[a + b])/(a*d)","A",1
6,1,166,92,0.8329554,"\int \frac{1}{\left(a+b \csc ^2(c+d x)\right)^2} \, dx","Integrate[(a + b*Csc[c + d*x]^2)^(-2),x]","-\frac{\csc ^4(c+d x) (a \cos (2 (c+d x))-a-2 b) \left(\sqrt{a+b} \left(2 \left(a^2+3 a b+2 b^2\right) (c+d x)+a b \sin (2 (c+d x))-2 a (a+b) (c+d x) \cos (2 (c+d x))\right)-\sqrt{b} (3 a+2 b) (a (-\cos (2 (c+d x)))+a+2 b) \tan ^{-1}\left(\frac{\sqrt{a+b} \tan (c+d x)}{\sqrt{b}}\right)\right)}{8 a^2 d (a+b)^{3/2} \left(a+b \csc ^2(c+d x)\right)^2}","\frac{\sqrt{b} (3 a+2 b) \tan ^{-1}\left(\frac{\sqrt{b} \cot (c+d x)}{\sqrt{a+b}}\right)}{2 a^2 d (a+b)^{3/2}}+\frac{x}{a^2}+\frac{b \cot (c+d x)}{2 a d (a+b) \left(a+b \cot ^2(c+d x)+b\right)}",1,"-1/8*((-a - 2*b + a*Cos[2*(c + d*x)])*Csc[c + d*x]^4*(-(Sqrt[b]*(3*a + 2*b)*ArcTan[(Sqrt[a + b]*Tan[c + d*x])/Sqrt[b]]*(a + 2*b - a*Cos[2*(c + d*x)])) + Sqrt[a + b]*(2*(a^2 + 3*a*b + 2*b^2)*(c + d*x) - 2*a*(a + b)*(c + d*x)*Cos[2*(c + d*x)] + a*b*Sin[2*(c + d*x)])))/(a^2*(a + b)^(3/2)*d*(a + b*Csc[c + d*x]^2)^2)","A",1
7,1,201,144,1.5938101,"\int \frac{1}{\left(a+b \csc ^2(c+d x)\right)^3} \, dx","Integrate[(a + b*Csc[c + d*x]^2)^(-3),x]","\frac{\csc ^6(c+d x) (a \cos (2 (c+d x))-a-2 b) \left(\frac{\sqrt{b} \left(15 a^2+20 a b+8 b^2\right) (a (-\cos (2 (c+d x)))+a+2 b)^2 \tan ^{-1}\left(\frac{\sqrt{a+b} \tan (c+d x)}{\sqrt{b}}\right)}{(a+b)^{5/2}}+\frac{4 a b^2 \sin (2 (c+d x))}{a+b}-8 (c+d x) (a (-\cos (2 (c+d x)))+a+2 b)^2+\frac{3 a b (3 a+2 b) \sin (2 (c+d x)) (a \cos (2 (c+d x))-a-2 b)}{(a+b)^2}\right)}{64 a^3 d \left(a+b \csc ^2(c+d x)\right)^3}","\frac{x}{a^3}+\frac{b (7 a+4 b) \cot (c+d x)}{8 a^2 d (a+b)^2 \left(a+b \cot ^2(c+d x)+b\right)}+\frac{\sqrt{b} \left(15 a^2+20 a b+8 b^2\right) \tan ^{-1}\left(\frac{\sqrt{b} \cot (c+d x)}{\sqrt{a+b}}\right)}{8 a^3 d (a+b)^{5/2}}+\frac{b \cot (c+d x)}{4 a d (a+b) \left(a+b \cot ^2(c+d x)+b\right)^2}",1,"((-a - 2*b + a*Cos[2*(c + d*x)])*Csc[c + d*x]^6*(-8*(c + d*x)*(a + 2*b - a*Cos[2*(c + d*x)])^2 + (Sqrt[b]*(15*a^2 + 20*a*b + 8*b^2)*ArcTan[(Sqrt[a + b]*Tan[c + d*x])/Sqrt[b]]*(a + 2*b - a*Cos[2*(c + d*x)])^2)/(a + b)^(5/2) + (4*a*b^2*Sin[2*(c + d*x)])/(a + b) + (3*a*b*(3*a + 2*b)*(-a - 2*b + a*Cos[2*(c + d*x)])*Sin[2*(c + d*x)])/(a + b)^2))/(64*a^3*d*(a + b*Csc[c + d*x]^2)^3)","A",1
8,1,263,204,3.3571191,"\int \frac{1}{\left(a+b \csc ^2(c+d x)\right)^4} \, dx","Integrate[(a + b*Csc[c + d*x]^2)^(-4),x]","\frac{\csc ^8(c+d x) (a \cos (2 (c+d x))-a-2 b) \left(-\frac{a b \left(87 a^2+116 a b+44 b^2\right) \sin (2 (c+d x)) (a (-\cos (2 (c+d x)))+a+2 b)^2}{(a+b)^3}+\frac{3 \sqrt{b} \left(35 a^3+70 a^2 b+56 a b^2+16 b^3\right) (a (-\cos (2 (c+d x)))+a+2 b)^3 \tan ^{-1}\left(\frac{\sqrt{a+b} \tan (c+d x)}{\sqrt{b}}\right)}{(a+b)^{7/2}}-\frac{32 a b^3 \sin (2 (c+d x))}{a+b}-\frac{4 a b^2 (19 a+14 b) \sin (2 (c+d x)) (a \cos (2 (c+d x))-a-2 b)}{(a+b)^2}+48 (c+d x) (a \cos (2 (c+d x))-a-2 b)^3\right)}{768 a^4 d \left(a+b \csc ^2(c+d x)\right)^4}","\frac{x}{a^4}+\frac{b (11 a+6 b) \cot (c+d x)}{24 a^2 d (a+b)^2 \left(a+b \cot ^2(c+d x)+b\right)^2}+\frac{b \left(19 a^2+22 a b+8 b^2\right) \cot (c+d x)}{16 a^3 d (a+b)^3 \left(a+b \cot ^2(c+d x)+b\right)}+\frac{\sqrt{b} \left(35 a^3+70 a^2 b+56 a b^2+16 b^3\right) \tan ^{-1}\left(\frac{\sqrt{b} \cot (c+d x)}{\sqrt{a+b}}\right)}{16 a^4 d (a+b)^{7/2}}+\frac{b \cot (c+d x)}{6 a d (a+b) \left(a+b \cot ^2(c+d x)+b\right)^3}",1,"((-a - 2*b + a*Cos[2*(c + d*x)])*Csc[c + d*x]^8*((3*Sqrt[b]*(35*a^3 + 70*a^2*b + 56*a*b^2 + 16*b^3)*ArcTan[(Sqrt[a + b]*Tan[c + d*x])/Sqrt[b]]*(a + 2*b - a*Cos[2*(c + d*x)])^3)/(a + b)^(7/2) + 48*(c + d*x)*(-a - 2*b + a*Cos[2*(c + d*x)])^3 - (32*a*b^3*Sin[2*(c + d*x)])/(a + b) - (a*b*(87*a^2 + 116*a*b + 44*b^2)*(a + 2*b - a*Cos[2*(c + d*x)])^2*Sin[2*(c + d*x)])/(a + b)^3 - (4*a*b^2*(19*a + 14*b)*(-a - 2*b + a*Cos[2*(c + d*x)])*Sin[2*(c + d*x)])/(a + b)^2))/(768*a^4*d*(a + b*Csc[c + d*x]^2)^4)","A",1
9,1,246,167,3.5851579,"\int \left(a+b \csc ^2(c+d x)\right)^{5/2} \, dx","Integrate[(a + b*Csc[c + d*x]^2)^(5/2),x]","\frac{\sin ^5(c+d x) \left(a+b \csc ^2(c+d x)\right)^{5/2} \left(\frac{1}{2} \sqrt{-b} \left(16 \sqrt{2} a^{5/2} \log \left(\sqrt{a \cos (2 (c+d x))-a-2 b}+\sqrt{2} \sqrt{a} \cos (c+d x)\right)+b \cot (c+d x) \csc ^3(c+d x) \sqrt{a \cos (2 (c+d x))-a-2 b} (3 (3 a+b) \cos (2 (c+d x))-9 a-7 b)\right)+\sqrt{2} b \left(15 a^2+10 a b+3 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{-b} \cos (c+d x)}{\sqrt{a \cos (2 (c+d x))-a-2 b}}\right)\right)}{2 \sqrt{-b} d (a \cos (2 (c+d x))-a-2 b)^{5/2}}","-\frac{a^{5/2} \tan ^{-1}\left(\frac{\sqrt{a} \cot (c+d x)}{\sqrt{a+b \cot ^2(c+d x)+b}}\right)}{d}-\frac{\sqrt{b} \left(15 a^2+10 a b+3 b^2\right) \tanh ^{-1}\left(\frac{\sqrt{b} \cot (c+d x)}{\sqrt{a+b \cot ^2(c+d x)+b}}\right)}{8 d}-\frac{b \cot (c+d x) \left(a+b \cot ^2(c+d x)+b\right)^{3/2}}{4 d}-\frac{b (7 a+3 b) \cot (c+d x) \sqrt{a+b \cot ^2(c+d x)+b}}{8 d}",1,"((a + b*Csc[c + d*x]^2)^(5/2)*(Sqrt[2]*b*(15*a^2 + 10*a*b + 3*b^2)*ArcTanh[(Sqrt[2]*Sqrt[-b]*Cos[c + d*x])/Sqrt[-a - 2*b + a*Cos[2*(c + d*x)]]] + (Sqrt[-b]*(b*Sqrt[-a - 2*b + a*Cos[2*(c + d*x)]]*(-9*a - 7*b + 3*(3*a + b)*Cos[2*(c + d*x)])*Cot[c + d*x]*Csc[c + d*x]^3 + 16*Sqrt[2]*a^(5/2)*Log[Sqrt[2]*Sqrt[a]*Cos[c + d*x] + Sqrt[-a - 2*b + a*Cos[2*(c + d*x)]]]))/2)*Sin[c + d*x]^5)/(2*Sqrt[-b]*d*(-a - 2*b + a*Cos[2*(c + d*x)])^(5/2))","A",1
10,1,208,119,0.910441,"\int \left(a+b \csc ^2(c+d x)\right)^{3/2} \, dx","Integrate[(a + b*Csc[c + d*x]^2)^(3/2),x]","\frac{b \sin ^3(c+d x) \left(a+b \csc ^2(c+d x)\right)^{3/2} \left(\sqrt{-b} \left(2 \sqrt{2} a^{3/2} \log \left(\sqrt{a \cos (2 (c+d x))-a-2 b}+\sqrt{2} \sqrt{a} \cos (c+d x)\right)-b \cot (c+d x) \csc (c+d x) \sqrt{a \cos (2 (c+d x))-a-2 b}\right)+\sqrt{2} b (3 a+b) \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{-b} \cos (c+d x)}{\sqrt{a \cos (2 (c+d x))-a-2 b}}\right)\right)}{(-b)^{3/2} d (a \cos (2 (c+d x))-a-2 b)^{3/2}}","-\frac{a^{3/2} \tan ^{-1}\left(\frac{\sqrt{a} \cot (c+d x)}{\sqrt{a+b \cot ^2(c+d x)+b}}\right)}{d}-\frac{b \cot (c+d x) \sqrt{a+b \cot ^2(c+d x)+b}}{2 d}-\frac{\sqrt{b} (3 a+b) \tanh ^{-1}\left(\frac{\sqrt{b} \cot (c+d x)}{\sqrt{a+b \cot ^2(c+d x)+b}}\right)}{2 d}",1,"(b*(a + b*Csc[c + d*x]^2)^(3/2)*(Sqrt[2]*b*(3*a + b)*ArcTanh[(Sqrt[2]*Sqrt[-b]*Cos[c + d*x])/Sqrt[-a - 2*b + a*Cos[2*(c + d*x)]]] + Sqrt[-b]*(-(b*Sqrt[-a - 2*b + a*Cos[2*(c + d*x)]]*Cot[c + d*x]*Csc[c + d*x]) + 2*Sqrt[2]*a^(3/2)*Log[Sqrt[2]*Sqrt[a]*Cos[c + d*x] + Sqrt[-a - 2*b + a*Cos[2*(c + d*x)]]]))*Sin[c + d*x]^3)/((-b)^(3/2)*d*(-a - 2*b + a*Cos[2*(c + d*x)])^(3/2))","A",1
11,1,149,81,0.2275008,"\int \sqrt{a+b \csc ^2(c+d x)} \, dx","Integrate[Sqrt[a + b*Csc[c + d*x]^2],x]","\frac{\sqrt{2} \sin (c+d x) \sqrt{a+b \csc ^2(c+d x)} \left(\sqrt{a} \log \left(\sqrt{a \cos (2 (c+d x))-a-2 b}+\sqrt{2} \sqrt{a} \cos (c+d x)\right)-\sqrt{-b} \tanh ^{-1}\left(\frac{\sqrt{2} \sqrt{-b} \cos (c+d x)}{\sqrt{a \cos (2 (c+d x))-a-2 b}}\right)\right)}{d \sqrt{a \cos (2 (c+d x))-a-2 b}}","-\frac{\sqrt{a} \tan ^{-1}\left(\frac{\sqrt{a} \cot (c+d x)}{\sqrt{a+b \cot ^2(c+d x)+b}}\right)}{d}-\frac{\sqrt{b} \tanh ^{-1}\left(\frac{\sqrt{b} \cot (c+d x)}{\sqrt{a+b \cot ^2(c+d x)+b}}\right)}{d}",1,"(Sqrt[2]*Sqrt[a + b*Csc[c + d*x]^2]*(-(Sqrt[-b]*ArcTanh[(Sqrt[2]*Sqrt[-b]*Cos[c + d*x])/Sqrt[-a - 2*b + a*Cos[2*(c + d*x)]]]) + Sqrt[a]*Log[Sqrt[2]*Sqrt[a]*Cos[c + d*x] + Sqrt[-a - 2*b + a*Cos[2*(c + d*x)]]])*Sin[c + d*x])/(d*Sqrt[-a - 2*b + a*Cos[2*(c + d*x)]])","A",1
12,1,98,39,0.1496631,"\int \frac{1}{\sqrt{a+b \csc ^2(c+d x)}} \, dx","Integrate[1/Sqrt[a + b*Csc[c + d*x]^2],x]","-\frac{\csc (c+d x) \sqrt{a \cos (2 (c+d x))-a-2 b} \log \left(\sqrt{a \cos (2 (c+d x))-a-2 b}+\sqrt{2} \sqrt{a} \cos (c+d x)\right)}{\sqrt{2} \sqrt{a} d \sqrt{a+b \csc ^2(c+d x)}}","-\frac{\tan ^{-1}\left(\frac{\sqrt{a} \cot (c+d x)}{\sqrt{a+b \csc ^2(c+d x)}}\right)}{\sqrt{a} d}",1,"-((Sqrt[-a - 2*b + a*Cos[2*(c + d*x)]]*Csc[c + d*x]*Log[Sqrt[2]*Sqrt[a]*Cos[c + d*x] + Sqrt[-a - 2*b + a*Cos[2*(c + d*x)]]])/(Sqrt[2]*Sqrt[a]*d*Sqrt[a + b*Csc[c + d*x]^2]))","B",1
13,1,146,77,0.3525112,"\int \frac{1}{\left(a+b \csc ^2(c+d x)\right)^{3/2}} \, dx","Integrate[(a + b*Csc[c + d*x]^2)^(-3/2),x]","\frac{\csc ^2(c+d x) \left(\sqrt{2} \csc (c+d x) (a \cos (2 (c+d x))-a-2 b)^{3/2} \log \left(\sqrt{a \cos (2 (c+d x))-a-2 b}+\sqrt{2} \sqrt{a} \cos (c+d x)\right)-\frac{2 \sqrt{a} b \cot (c+d x) (a \cos (2 (c+d x))-a-2 b)}{a+b}\right)}{4 a^{3/2} d \left(a+b \csc ^2(c+d x)\right)^{3/2}}","\frac{b \cot (c+d x)}{a d (a+b) \sqrt{a+b \cot ^2(c+d x)+b}}-\frac{\tan ^{-1}\left(\frac{\sqrt{a} \cot (c+d x)}{\sqrt{a+b \cot ^2(c+d x)+b}}\right)}{a^{3/2} d}",1,"(Csc[c + d*x]^2*((-2*Sqrt[a]*b*(-a - 2*b + a*Cos[2*(c + d*x)])*Cot[c + d*x])/(a + b) + Sqrt[2]*(-a - 2*b + a*Cos[2*(c + d*x)])^(3/2)*Csc[c + d*x]*Log[Sqrt[2]*Sqrt[a]*Cos[c + d*x] + Sqrt[-a - 2*b + a*Cos[2*(c + d*x)]]]))/(4*a^(3/2)*d*(a + b*Csc[c + d*x]^2)^(3/2))","A",1
14,1,173,126,1.7002679,"\int \frac{1}{\left(a+b \csc ^2(c+d x)\right)^{5/2}} \, dx","Integrate[(a + b*Csc[c + d*x]^2)^(-5/2),x]","\frac{\csc ^5(c+d x) \left(\frac{4 b \cos (c+d x) (a (-\cos (2 (c+d x)))+a+2 b) \left(3 a^2-a (3 a+2 b) \cos (2 (c+d x))+7 a b+3 b^2\right)}{3 a^2 (a+b)^2}-\frac{\sqrt{2} (a \cos (2 (c+d x))-a-2 b)^{5/2} \log \left(\sqrt{a \cos (2 (c+d x))-a-2 b}+\sqrt{2} \sqrt{a} \cos (c+d x)\right)}{a^{5/2}}\right)}{8 d \left(a+b \csc ^2(c+d x)\right)^{5/2}}","-\frac{\tan ^{-1}\left(\frac{\sqrt{a} \cot (c+d x)}{\sqrt{a+b \cot ^2(c+d x)+b}}\right)}{a^{5/2} d}+\frac{b (5 a+3 b) \cot (c+d x)}{3 a^2 d (a+b)^2 \sqrt{a+b \cot ^2(c+d x)+b}}+\frac{b \cot (c+d x)}{3 a d (a+b) \left(a+b \cot ^2(c+d x)+b\right)^{3/2}}",1,"(Csc[c + d*x]^5*((4*b*Cos[c + d*x]*(a + 2*b - a*Cos[2*(c + d*x)])*(3*a^2 + 7*a*b + 3*b^2 - a*(3*a + 2*b)*Cos[2*(c + d*x)]))/(3*a^2*(a + b)^2) - (Sqrt[2]*(-a - 2*b + a*Cos[2*(c + d*x)])^(5/2)*Log[Sqrt[2]*Sqrt[a]*Cos[c + d*x] + Sqrt[-a - 2*b + a*Cos[2*(c + d*x)]]])/a^(5/2)))/(8*d*(a + b*Csc[c + d*x]^2)^(5/2))","A",1
15,1,231,180,1.4438916,"\int \frac{1}{\left(a+b \csc ^2(c+d x)\right)^{7/2}} \, dx","Integrate[(a + b*Csc[c + d*x]^2)^(-7/2),x]","\frac{\csc ^7(c+d x) \left(\frac{\sqrt{2} (a \cos (2 (c+d x))-a-2 b)^{7/2} \log \left(\sqrt{a \cos (2 (c+d x))-a-2 b}+\sqrt{2} \sqrt{a} \cos (c+d x)\right)}{a^{7/2}}+\frac{b \cos (c+d x) (a (-\cos (2 (c+d x)))+a+2 b) \left(135 a^4+480 a^3 b+a^2 \left(45 a^2+60 a b+23 b^2\right) \cos (4 (c+d x))+709 a^2 b^2-4 a \left(45 a^3+135 a^2 b+117 a b^2+35 b^3\right) \cos (2 (c+d x))+460 a b^3+120 b^4\right)}{15 a^3 (a+b)^3}\right)}{16 d \left(a+b \csc ^2(c+d x)\right)^{7/2}}","-\frac{\tan ^{-1}\left(\frac{\sqrt{a} \cot (c+d x)}{\sqrt{a+b \cot ^2(c+d x)+b}}\right)}{a^{7/2} d}+\frac{b (9 a+5 b) \cot (c+d x)}{15 a^2 d (a+b)^2 \left(a+b \cot ^2(c+d x)+b\right)^{3/2}}+\frac{b \left(33 a^2+40 a b+15 b^2\right) \cot (c+d x)}{15 a^3 d (a+b)^3 \sqrt{a+b \cot ^2(c+d x)+b}}+\frac{b \cot (c+d x)}{5 a d (a+b) \left(a+b \cot ^2(c+d x)+b\right)^{5/2}}",1,"(Csc[c + d*x]^7*((b*Cos[c + d*x]*(a + 2*b - a*Cos[2*(c + d*x)])*(135*a^4 + 480*a^3*b + 709*a^2*b^2 + 460*a*b^3 + 120*b^4 - 4*a*(45*a^3 + 135*a^2*b + 117*a*b^2 + 35*b^3)*Cos[2*(c + d*x)] + a^2*(45*a^2 + 60*a*b + 23*b^2)*Cos[4*(c + d*x)]))/(15*a^3*(a + b)^3) + (Sqrt[2]*(-a - 2*b + a*Cos[2*(c + d*x)])^(7/2)*Log[Sqrt[2]*Sqrt[a]*Cos[c + d*x] + Sqrt[-a - 2*b + a*Cos[2*(c + d*x)]]])/a^(7/2)))/(16*d*(a + b*Csc[c + d*x]^2)^(7/2))","A",1
16,1,94,44,0.175763,"\int \left(1+\csc ^2(x)\right)^{3/2} \, dx","Integrate[(1 + Csc[x]^2)^(3/2),x]","\frac{\sin ^3(x) \left(\csc ^2(x)+1\right)^{3/2} \left(-2 \sqrt{2} \log \left(\sqrt{2} \cos (x)+\sqrt{\cos (2 x)-3}\right)-4 \sqrt{2} \tan ^{-1}\left(\frac{\sqrt{2} \cos (x)}{\sqrt{\cos (2 x)-3}}\right)+\sqrt{\cos (2 x)-3} \cot (x) \csc (x)\right)}{(\cos (2 x)-3)^{3/2}}","-\frac{1}{2} \cot (x) \sqrt{\cot ^2(x)+2}-\tan ^{-1}\left(\frac{\cot (x)}{\sqrt{\cot ^2(x)+2}}\right)-2 \sinh ^{-1}\left(\frac{\cot (x)}{\sqrt{2}}\right)",1,"((1 + Csc[x]^2)^(3/2)*(-4*Sqrt[2]*ArcTan[(Sqrt[2]*Cos[x])/Sqrt[-3 + Cos[2*x]]] + Sqrt[-3 + Cos[2*x]]*Cot[x]*Csc[x] - 2*Sqrt[2]*Log[Sqrt[2]*Cos[x] + Sqrt[-3 + Cos[2*x]]])*Sin[x]^3)/(-3 + Cos[2*x])^(3/2)","B",1
17,1,68,28,0.0549823,"\int \sqrt{1+\csc ^2(x)} \, dx","Integrate[Sqrt[1 + Csc[x]^2],x]","\frac{\sqrt{2} \sin (x) \sqrt{\csc ^2(x)+1} \left(\log \left(\sqrt{2} \cos (x)+\sqrt{\cos (2 x)-3}\right)+\tan ^{-1}\left(\frac{\sqrt{2} \cos (x)}{\sqrt{\cos (2 x)-3}}\right)\right)}{\sqrt{\cos (2 x)-3}}","-\tan ^{-1}\left(\frac{\cot (x)}{\sqrt{\cot ^2(x)+2}}\right)-\sinh ^{-1}\left(\frac{\cot (x)}{\sqrt{2}}\right)",1,"(Sqrt[2]*Sqrt[1 + Csc[x]^2]*(ArcTan[(Sqrt[2]*Cos[x])/Sqrt[-3 + Cos[2*x]]] + Log[Sqrt[2]*Cos[x] + Sqrt[-3 + Cos[2*x]]])*Sin[x])/Sqrt[-3 + Cos[2*x]]","B",1
18,1,49,16,0.0508006,"\int \frac{1}{\sqrt{1+\csc ^2(x)}} \, dx","Integrate[1/Sqrt[1 + Csc[x]^2],x]","-\frac{\sqrt{\cos (2 x)-3} \csc (x) \log \left(\sqrt{2} \cos (x)+\sqrt{\cos (2 x)-3}\right)}{\sqrt{2} \sqrt{\csc ^2(x)+1}}","-\tan ^{-1}\left(\frac{\cot (x)}{\sqrt{\cot ^2(x)+2}}\right)",1,"-((Sqrt[-3 + Cos[2*x]]*Csc[x]*Log[Sqrt[2]*Cos[x] + Sqrt[-3 + Cos[2*x]]])/(Sqrt[2]*Sqrt[1 + Csc[x]^2]))","B",1
19,1,26,33,0.0166916,"\int \left(1-\csc ^2(x)\right)^{3/2} \, dx","Integrate[(1 - Csc[x]^2)^(3/2),x]","\frac{1}{2} \tan (x) \sqrt{-\cot ^2(x)} \left(\csc ^2(x)+2 \log (\sin (x))\right)","\frac{1}{2} \sqrt{-\cot ^2(x)} \cot (x)+\tan (x) \sqrt{-\cot ^2(x)} \log (\sin (x))",1,"(Sqrt[-Cot[x]^2]*(Csc[x]^2 + 2*Log[Sin[x]])*Tan[x])/2","A",1
20,1,16,16,0.0063858,"\int \sqrt{1-\csc ^2(x)} \, dx","Integrate[Sqrt[1 - Csc[x]^2],x]","\tan (x) \sqrt{-\cot ^2(x)} \log (\sin (x))","\tan (x) \sqrt{-\cot ^2(x)} \log (\sin (x))",1,"Sqrt[-Cot[x]^2]*Log[Sin[x]]*Tan[x]","A",1
21,1,17,17,0.0090405,"\int \frac{1}{\sqrt{1-\csc ^2(x)}} \, dx","Integrate[1/Sqrt[1 - Csc[x]^2],x]","-\frac{\cot (x) \log (\cos (x))}{\sqrt{-\cot ^2(x)}}","-\frac{\cot (x) \log (\cos (x))}{\sqrt{-\cot ^2(x)}}",1,"-((Cot[x]*Log[Cos[x]])/Sqrt[-Cot[x]^2])","A",1
22,1,24,30,0.0145921,"\int \left(-1+\csc ^2(x)\right)^{3/2} \, dx","Integrate[(-1 + Csc[x]^2)^(3/2),x]","-\frac{1}{2} \tan (x) \sqrt{\cot ^2(x)} \left(\csc ^2(x)+2 \log (\sin (x))\right)","-\frac{1}{2} \tan (x) \cot ^2(x)^{3/2}-\tan (x) \sqrt{\cot ^2(x)} \log (\sin (x))",1,"-1/2*(Sqrt[Cot[x]^2]*(Csc[x]^2 + 2*Log[Sin[x]])*Tan[x])","A",1
23,1,14,14,0.0060032,"\int \sqrt{-1+\csc ^2(x)} \, dx","Integrate[Sqrt[-1 + Csc[x]^2],x]","\tan (x) \sqrt{\cot ^2(x)} \log (\sin (x))","\tan (x) \sqrt{\cot ^2(x)} \log (\sin (x))",1,"Sqrt[Cot[x]^2]*Log[Sin[x]]*Tan[x]","A",1
24,1,15,15,0.0090466,"\int \frac{1}{\sqrt{-1+\csc ^2(x)}} \, dx","Integrate[1/Sqrt[-1 + Csc[x]^2],x]","-\frac{\cot (x) \log (\cos (x))}{\sqrt{\cot ^2(x)}}","-\frac{\cot (x) \log (\cos (x))}{\sqrt{\cot ^2(x)}}",1,"-((Cot[x]*Log[Cos[x]])/Sqrt[Cot[x]^2])","A",1
25,1,96,55,0.1166234,"\int \left(-1-\csc ^2(x)\right)^{3/2} \, dx","Integrate[(-1 - Csc[x]^2)^(3/2),x]","\frac{\sin ^3(x) \left(-\csc ^2(x)-1\right)^{3/2} \left(-2 \sqrt{2} \log \left(\sqrt{2} \cos (x)+\sqrt{\cos (2 x)-3}\right)-4 \sqrt{2} \tan ^{-1}\left(\frac{\sqrt{2} \cos (x)}{\sqrt{\cos (2 x)-3}}\right)+\sqrt{\cos (2 x)-3} \cot (x) \csc (x)\right)}{(\cos (2 x)-3)^{3/2}}","\frac{1}{2} \cot (x) \sqrt{-\cot ^2(x)-2}-2 \tan ^{-1}\left(\frac{\cot (x)}{\sqrt{-\cot ^2(x)-2}}\right)-\tanh ^{-1}\left(\frac{\cot (x)}{\sqrt{-\cot ^2(x)-2}}\right)",1,"((-1 - Csc[x]^2)^(3/2)*(-4*Sqrt[2]*ArcTan[(Sqrt[2]*Cos[x])/Sqrt[-3 + Cos[2*x]]] + Sqrt[-3 + Cos[2*x]]*Cot[x]*Csc[x] - 2*Sqrt[2]*Log[Sqrt[2]*Cos[x] + Sqrt[-3 + Cos[2*x]]])*Sin[x]^3)/(-3 + Cos[2*x])^(3/2)","A",1
26,1,70,33,0.0388948,"\int \sqrt{-1-\csc ^2(x)} \, dx","Integrate[Sqrt[-1 - Csc[x]^2],x]","\frac{\sqrt{2} \sin (x) \sqrt{-\csc ^2(x)-1} \left(\log \left(\sqrt{2} \cos (x)+\sqrt{\cos (2 x)-3}\right)+\tan ^{-1}\left(\frac{\sqrt{2} \cos (x)}{\sqrt{\cos (2 x)-3}}\right)\right)}{\sqrt{\cos (2 x)-3}}","\tan ^{-1}\left(\frac{\cot (x)}{\sqrt{-\cot ^2(x)-2}}\right)+\tanh ^{-1}\left(\frac{\cot (x)}{\sqrt{-\cot ^2(x)-2}}\right)",1,"(Sqrt[2]*Sqrt[-1 - Csc[x]^2]*(ArcTan[(Sqrt[2]*Cos[x])/Sqrt[-3 + Cos[2*x]]] + Log[Sqrt[2]*Cos[x] + Sqrt[-3 + Cos[2*x]]])*Sin[x])/Sqrt[-3 + Cos[2*x]]","B",1
27,1,51,18,0.031463,"\int \frac{1}{\sqrt{-1-\csc ^2(x)}} \, dx","Integrate[1/Sqrt[-1 - Csc[x]^2],x]","-\frac{\sqrt{\cos (2 x)-3} \csc (x) \log \left(\sqrt{2} \cos (x)+\sqrt{\cos (2 x)-3}\right)}{\sqrt{2} \sqrt{-\csc ^2(x)-1}}","-\tanh ^{-1}\left(\frac{\cot (x)}{\sqrt{-\cot ^2(x)-2}}\right)",1,"-((Sqrt[-3 + Cos[2*x]]*Csc[x]*Log[Sqrt[2]*Cos[x] + Sqrt[-3 + Cos[2*x]]])/(Sqrt[2]*Sqrt[-1 - Csc[x]^2]))","B",1