1,1,159,129,0.2575863,"\int x^5 \left(a+b \csc \left(c+d x^2\right)\right) \, dx","Integrate[x^5*(a + b*Csc[c + d*x^2]),x]","\frac{a x^6}{6}-\frac{b \left(d^2 x^4 \tanh ^{-1}\left(\cos \left(c+d x^2\right)+i \sin \left(c+d x^2\right)\right)-i d x^2 \text{Li}_2\left(-\cos \left(d x^2+c\right)-i \sin \left(d x^2+c\right)\right)+i d x^2 \text{Li}_2\left(\cos \left(d x^2+c\right)+i \sin \left(d x^2+c\right)\right)+\text{Li}_3\left(-\cos \left(d x^2+c\right)-i \sin \left(d x^2+c\right)\right)-\text{Li}_3\left(\cos \left(d x^2+c\right)+i \sin \left(d x^2+c\right)\right)\right)}{d^3}","\frac{a x^6}{6}-\frac{b \text{Li}_3\left(-e^{i \left(d x^2+c\right)}\right)}{d^3}+\frac{b \text{Li}_3\left(e^{i \left(d x^2+c\right)}\right)}{d^3}+\frac{i b x^2 \text{Li}_2\left(-e^{i \left(d x^2+c\right)}\right)}{d^2}-\frac{i b x^2 \text{Li}_2\left(e^{i \left(d x^2+c\right)}\right)}{d^2}-\frac{b x^4 \tanh ^{-1}\left(e^{i \left(c+d x^2\right)}\right)}{d}",1,"(a*x^6)/6 - (b*(d^2*x^4*ArcTanh[Cos[c + d*x^2] + I*Sin[c + d*x^2]] - I*d*x^2*PolyLog[2, -Cos[c + d*x^2] - I*Sin[c + d*x^2]] + I*d*x^2*PolyLog[2, Cos[c + d*x^2] + I*Sin[c + d*x^2]] + PolyLog[3, -Cos[c + d*x^2] - I*Sin[c + d*x^2]] - PolyLog[3, Cos[c + d*x^2] + I*Sin[c + d*x^2]]))/d^3","A",0
2,0,0,26,3.2277037,"\int x^4 \left(a+b \csc \left(c+d x^2\right)\right) \, dx","Integrate[x^4*(a + b*Csc[c + d*x^2]),x]","\int x^4 \left(a+b \csc \left(c+d x^2\right)\right) \, dx","b \text{Int}\left(x^4 \csc \left(c+d x^2\right),x\right)+\frac{a x^5}{5}",0,"Integrate[x^4*(a + b*Csc[c + d*x^2]), x]","A",-1
3,1,118,84,0.0909512,"\int x^3 \left(a+b \csc \left(c+d x^2\right)\right) \, dx","Integrate[x^3*(a + b*Csc[c + d*x^2]),x]","\frac{a x^4}{4}+\frac{b \left(i \left(\text{Li}_2\left(-e^{i \left(d x^2+c\right)}\right)-\text{Li}_2\left(e^{i \left(d x^2+c\right)}\right)\right)+\left(c+d x^2\right) \left(\log \left(1-e^{i \left(c+d x^2\right)}\right)-\log \left(1+e^{i \left(c+d x^2\right)}\right)\right)-c \log \left(\tan \left(\frac{1}{2} \left(c+d x^2\right)\right)\right)\right)}{2 d^2}","\frac{a x^4}{4}+\frac{i b \text{Li}_2\left(-e^{i \left(d x^2+c\right)}\right)}{2 d^2}-\frac{i b \text{Li}_2\left(e^{i \left(d x^2+c\right)}\right)}{2 d^2}-\frac{b x^2 \tanh ^{-1}\left(e^{i \left(c+d x^2\right)}\right)}{d}",1,"(a*x^4)/4 + (b*((c + d*x^2)*(Log[1 - E^(I*(c + d*x^2))] - Log[1 + E^(I*(c + d*x^2))]) - c*Log[Tan[(c + d*x^2)/2]] + I*(PolyLog[2, -E^(I*(c + d*x^2))] - PolyLog[2, E^(I*(c + d*x^2))])))/(2*d^2)","A",1
4,0,0,26,2.5525698,"\int x^2 \left(a+b \csc \left(c+d x^2\right)\right) \, dx","Integrate[x^2*(a + b*Csc[c + d*x^2]),x]","\int x^2 \left(a+b \csc \left(c+d x^2\right)\right) \, dx","b \text{Int}\left(x^2 \csc \left(c+d x^2\right),x\right)+\frac{a x^3}{3}",0,"Integrate[x^2*(a + b*Csc[c + d*x^2]), x]","A",-1
5,1,57,26,0.0258126,"\int x \left(a+b \csc \left(c+d x^2\right)\right) \, dx","Integrate[x*(a + b*Csc[c + d*x^2]),x]","\frac{a x^2}{2}+\frac{b \log \left(\sin \left(\frac{c}{2}+\frac{d x^2}{2}\right)\right)}{2 d}-\frac{b \log \left(\cos \left(\frac{c}{2}+\frac{d x^2}{2}\right)\right)}{2 d}","\frac{a x^2}{2}-\frac{b \tanh ^{-1}\left(\cos \left(c+d x^2\right)\right)}{2 d}",1,"(a*x^2)/2 - (b*Log[Cos[c/2 + (d*x^2)/2]])/(2*d) + (b*Log[Sin[c/2 + (d*x^2)/2]])/(2*d)","B",1
6,0,0,22,2.28388,"\int \frac{a+b \csc \left(c+d x^2\right)}{x} \, dx","Integrate[(a + b*Csc[c + d*x^2])/x,x]","\int \frac{a+b \csc \left(c+d x^2\right)}{x} \, dx","b \text{Int}\left(\frac{\csc \left(c+d x^2\right)}{x},x\right)+a \log (x)",0,"Integrate[(a + b*Csc[c + d*x^2])/x, x]","A",-1
7,0,0,24,2.3269134,"\int \frac{a+b \csc \left(c+d x^2\right)}{x^2} \, dx","Integrate[(a + b*Csc[c + d*x^2])/x^2,x]","\int \frac{a+b \csc \left(c+d x^2\right)}{x^2} \, dx","b \text{Int}\left(\frac{\csc \left(c+d x^2\right)}{x^2},x\right)-\frac{a}{x}",0,"Integrate[(a + b*Csc[c + d*x^2])/x^2, x]","A",-1
8,1,639,228,3.2199594,"\int x^5 \left(a+b \csc \left(c+d x^2\right)\right)^2 \, dx","Integrate[x^5*(a + b*Csc[c + d*x^2])^2,x]","\frac{2 a^2 e^{2 i c} d^3 x^6-2 a^2 d^3 x^6+12 a b e^{2 i c} d^2 x^4 \log \left(1-e^{-i \left(c+d x^2\right)}\right)-12 a b d^2 x^4 \log \left(1-e^{-i \left(c+d x^2\right)}\right)-12 a b e^{2 i c} d^2 x^4 \log \left(1+e^{-i \left(c+d x^2\right)}\right)+12 a b d^2 x^4 \log \left(1+e^{-i \left(c+d x^2\right)}\right)+12 i b \left(-1+e^{2 i c}\right) \left(b-2 a d x^2\right) \text{Li}_2\left(-e^{-i \left(d x^2+c\right)}\right)+12 i b \left(-1+e^{2 i c}\right) \left(2 a d x^2+b\right) \text{Li}_2\left(e^{-i \left(d x^2+c\right)}\right)-24 a b e^{2 i c} \text{Li}_3\left(-e^{-i \left(d x^2+c\right)}\right)+24 a b \text{Li}_3\left(-e^{-i \left(d x^2+c\right)}\right)+24 a b e^{2 i c} \text{Li}_3\left(e^{-i \left(d x^2+c\right)}\right)-24 a b \text{Li}_3\left(e^{-i \left(d x^2+c\right)}\right)+3 b^2 e^{2 i c} d^2 x^4 \csc \left(\frac{c}{2}\right) \sin \left(\frac{d x^2}{2}\right) \csc \left(\frac{1}{2} \left(c+d x^2\right)\right)-3 b^2 d^2 x^4 \csc \left(\frac{c}{2}\right) \sin \left(\frac{d x^2}{2}\right) \csc \left(\frac{1}{2} \left(c+d x^2\right)\right)+3 b^2 e^{2 i c} d^2 x^4 \sec \left(\frac{c}{2}\right) \sin \left(\frac{d x^2}{2}\right) \sec \left(\frac{1}{2} \left(c+d x^2\right)\right)-3 b^2 d^2 x^4 \sec \left(\frac{c}{2}\right) \sin \left(\frac{d x^2}{2}\right) \sec \left(\frac{1}{2} \left(c+d x^2\right)\right)+12 b^2 e^{2 i c} d x^2 \log \left(1-e^{-i \left(c+d x^2\right)}\right)-12 b^2 d x^2 \log \left(1-e^{-i \left(c+d x^2\right)}\right)+12 b^2 e^{2 i c} d x^2 \log \left(1+e^{-i \left(c+d x^2\right)}\right)-12 b^2 d x^2 \log \left(1+e^{-i \left(c+d x^2\right)}\right)-12 i b^2 d^2 x^4}{12 \left(-1+e^{2 i c}\right) d^3}","\frac{a^2 x^6}{6}-\frac{2 a b \text{Li}_3\left(-e^{i \left(d x^2+c\right)}\right)}{d^3}+\frac{2 a b \text{Li}_3\left(e^{i \left(d x^2+c\right)}\right)}{d^3}+\frac{2 i a b x^2 \text{Li}_2\left(-e^{i \left(d x^2+c\right)}\right)}{d^2}-\frac{2 i a b x^2 \text{Li}_2\left(e^{i \left(d x^2+c\right)}\right)}{d^2}-\frac{2 a b x^4 \tanh ^{-1}\left(e^{i \left(c+d x^2\right)}\right)}{d}-\frac{i b^2 \text{Li}_2\left(e^{2 i \left(d x^2+c\right)}\right)}{2 d^3}+\frac{b^2 x^2 \log \left(1-e^{2 i \left(c+d x^2\right)}\right)}{d^2}-\frac{b^2 x^4 \cot \left(c+d x^2\right)}{2 d}-\frac{i b^2 x^4}{2 d}",1,"((-12*I)*b^2*d^2*x^4 - 2*a^2*d^3*x^6 + 2*a^2*d^3*E^((2*I)*c)*x^6 - 12*b^2*d*x^2*Log[1 - E^((-I)*(c + d*x^2))] + 12*b^2*d*E^((2*I)*c)*x^2*Log[1 - E^((-I)*(c + d*x^2))] - 12*a*b*d^2*x^4*Log[1 - E^((-I)*(c + d*x^2))] + 12*a*b*d^2*E^((2*I)*c)*x^4*Log[1 - E^((-I)*(c + d*x^2))] - 12*b^2*d*x^2*Log[1 + E^((-I)*(c + d*x^2))] + 12*b^2*d*E^((2*I)*c)*x^2*Log[1 + E^((-I)*(c + d*x^2))] + 12*a*b*d^2*x^4*Log[1 + E^((-I)*(c + d*x^2))] - 12*a*b*d^2*E^((2*I)*c)*x^4*Log[1 + E^((-I)*(c + d*x^2))] + (12*I)*b*(-1 + E^((2*I)*c))*(b - 2*a*d*x^2)*PolyLog[2, -E^((-I)*(c + d*x^2))] + (12*I)*b*(-1 + E^((2*I)*c))*(b + 2*a*d*x^2)*PolyLog[2, E^((-I)*(c + d*x^2))] + 24*a*b*PolyLog[3, -E^((-I)*(c + d*x^2))] - 24*a*b*E^((2*I)*c)*PolyLog[3, -E^((-I)*(c + d*x^2))] - 24*a*b*PolyLog[3, E^((-I)*(c + d*x^2))] + 24*a*b*E^((2*I)*c)*PolyLog[3, E^((-I)*(c + d*x^2))] - 3*b^2*d^2*x^4*Csc[c/2]*Csc[(c + d*x^2)/2]*Sin[(d*x^2)/2] + 3*b^2*d^2*E^((2*I)*c)*x^4*Csc[c/2]*Csc[(c + d*x^2)/2]*Sin[(d*x^2)/2] - 3*b^2*d^2*x^4*Sec[c/2]*Sec[(c + d*x^2)/2]*Sin[(d*x^2)/2] + 3*b^2*d^2*E^((2*I)*c)*x^4*Sec[c/2]*Sec[(c + d*x^2)/2]*Sin[(d*x^2)/2])/(12*d^3*(-1 + E^((2*I)*c)))","B",1
9,0,0,21,14.0669132,"\int x^4 \left(a+b \csc \left(c+d x^2\right)\right)^2 \, dx","Integrate[x^4*(a + b*Csc[c + d*x^2])^2,x]","\int x^4 \left(a+b \csc \left(c+d x^2\right)\right)^2 \, dx","\text{Int}\left(x^4 \left(a+b \csc \left(c+d x^2\right)\right)^2,x\right)",0,"Integrate[x^4*(a + b*Csc[c + d*x^2])^2, x]","A",-1
10,1,268,125,4.7275605,"\int x^3 \left(a+b \csc \left(c+d x^2\right)\right)^2 \, dx","Integrate[x^3*(a + b*Csc[c + d*x^2])^2,x]","\frac{d x^2 \left(a^2 d x^2-2 b^2 \cot (c)\right)+4 a b \left(2 \tan ^{-1}(\tan (c)) \tanh ^{-1}\left(\cos (c)-\sin (c) \tan \left(\frac{d x^2}{2}\right)\right)+\frac{\sec (c) \left(i \text{Li}_2\left(-e^{i \left(d x^2+\tan ^{-1}(\tan (c))\right)}\right)-i \text{Li}_2\left(e^{i \left(d x^2+\tan ^{-1}(\tan (c))\right)}\right)+\left(\tan ^{-1}(\tan (c))+d x^2\right) \left(\log \left(1-e^{i \left(\tan ^{-1}(\tan (c))+d x^2\right)}\right)-\log \left(1+e^{i \left(\tan ^{-1}(\tan (c))+d x^2\right)}\right)\right)\right)}{\sqrt{\sec ^2(c)}}\right)+2 b^2 d x^2 \cot (c)+b^2 d x^2 \csc \left(\frac{c}{2}\right) \sin \left(\frac{d x^2}{2}\right) \csc \left(\frac{1}{2} \left(c+d x^2\right)\right)+b^2 d x^2 \sec \left(\frac{c}{2}\right) \sin \left(\frac{d x^2}{2}\right) \sec \left(\frac{1}{2} \left(c+d x^2\right)\right)-2 b^2 \left(d x^2 \cot (c)-\log \left(\sin \left(c+d x^2\right)\right)\right)}{4 d^2}","\frac{a^2 x^4}{4}+\frac{i a b \text{Li}_2\left(-e^{i \left(d x^2+c\right)}\right)}{d^2}-\frac{i a b \text{Li}_2\left(e^{i \left(d x^2+c\right)}\right)}{d^2}-\frac{2 a b x^2 \tanh ^{-1}\left(e^{i \left(c+d x^2\right)}\right)}{d}+\frac{b^2 \log \left(\sin \left(c+d x^2\right)\right)}{2 d^2}-\frac{b^2 x^2 \cot \left(c+d x^2\right)}{2 d}",1,"(2*b^2*d*x^2*Cot[c] + d*x^2*(a^2*d*x^2 - 2*b^2*Cot[c]) - 2*b^2*(d*x^2*Cot[c] - Log[Sin[c + d*x^2]]) + 4*a*b*(2*ArcTan[Tan[c]]*ArcTanh[Cos[c] - Sin[c]*Tan[(d*x^2)/2]] + (((d*x^2 + ArcTan[Tan[c]])*(Log[1 - E^(I*(d*x^2 + ArcTan[Tan[c]]))] - Log[1 + E^(I*(d*x^2 + ArcTan[Tan[c]]))]) + I*PolyLog[2, -E^(I*(d*x^2 + ArcTan[Tan[c]]))] - I*PolyLog[2, E^(I*(d*x^2 + ArcTan[Tan[c]]))])*Sec[c])/Sqrt[Sec[c]^2]) + b^2*d*x^2*Csc[c/2]*Csc[(c + d*x^2)/2]*Sin[(d*x^2)/2] + b^2*d*x^2*Sec[c/2]*Sec[(c + d*x^2)/2]*Sin[(d*x^2)/2])/(4*d^2)","B",0
11,0,0,21,14.0607436,"\int x^2 \left(a+b \csc \left(c+d x^2\right)\right)^2 \, dx","Integrate[x^2*(a + b*Csc[c + d*x^2])^2,x]","\int x^2 \left(a+b \csc \left(c+d x^2\right)\right)^2 \, dx","\text{Int}\left(x^2 \left(a+b \csc \left(c+d x^2\right)\right)^2,x\right)",0,"Integrate[x^2*(a + b*Csc[c + d*x^2])^2, x]","A",-1
12,1,86,45,0.4003407,"\int x \left(a+b \csc \left(c+d x^2\right)\right)^2 \, dx","Integrate[x*(a + b*Csc[c + d*x^2])^2,x]","\frac{2 a \left(a c+a d x^2+2 b \log \left(\sin \left(\frac{1}{2} \left(c+d x^2\right)\right)\right)-2 b \log \left(\cos \left(\frac{1}{2} \left(c+d x^2\right)\right)\right)\right)+b^2 \tan \left(\frac{1}{2} \left(c+d x^2\right)\right)+b^2 \left(-\cot \left(\frac{1}{2} \left(c+d x^2\right)\right)\right)}{4 d}","\frac{a^2 x^2}{2}-\frac{a b \tanh ^{-1}\left(\cos \left(c+d x^2\right)\right)}{d}-\frac{b^2 \cot \left(c+d x^2\right)}{2 d}",1,"(-(b^2*Cot[(c + d*x^2)/2]) + 2*a*(a*c + a*d*x^2 - 2*b*Log[Cos[(c + d*x^2)/2]] + 2*b*Log[Sin[(c + d*x^2)/2]]) + b^2*Tan[(c + d*x^2)/2])/(4*d)","A",1
13,0,0,21,31.5232503,"\int \frac{\left(a+b \csc \left(c+d x^2\right)\right)^2}{x} \, dx","Integrate[(a + b*Csc[c + d*x^2])^2/x,x]","\int \frac{\left(a+b \csc \left(c+d x^2\right)\right)^2}{x} \, dx","\text{Int}\left(\frac{\left(a+b \csc \left(c+d x^2\right)\right)^2}{x},x\right)",0,"Integrate[(a + b*Csc[c + d*x^2])^2/x, x]","A",-1
14,0,0,21,17.4419662,"\int \frac{\left(a+b \csc \left(c+d x^2\right)\right)^2}{x^2} \, dx","Integrate[(a + b*Csc[c + d*x^2])^2/x^2,x]","\int \frac{\left(a+b \csc \left(c+d x^2\right)\right)^2}{x^2} \, dx","\text{Int}\left(\frac{\left(a+b \csc \left(c+d x^2\right)\right)^2}{x^2},x\right)",0,"Integrate[(a + b*Csc[c + d*x^2])^2/x^2, x]","A",-1
15,1,167,90,0.0859006,"\int x \csc ^7\left(a+b x^2\right) \, dx","Integrate[x*Csc[a + b*x^2]^7,x]","-\frac{\csc ^6\left(\frac{1}{2} \left(a+b x^2\right)\right)}{768 b}-\frac{\csc ^4\left(\frac{1}{2} \left(a+b x^2\right)\right)}{128 b}-\frac{5 \csc ^2\left(\frac{1}{2} \left(a+b x^2\right)\right)}{128 b}+\frac{\sec ^6\left(\frac{1}{2} \left(a+b x^2\right)\right)}{768 b}+\frac{\sec ^4\left(\frac{1}{2} \left(a+b x^2\right)\right)}{128 b}+\frac{5 \sec ^2\left(\frac{1}{2} \left(a+b x^2\right)\right)}{128 b}+\frac{5 \log \left(\sin \left(\frac{1}{2} \left(a+b x^2\right)\right)\right)}{32 b}-\frac{5 \log \left(\cos \left(\frac{1}{2} \left(a+b x^2\right)\right)\right)}{32 b}","-\frac{5 \tanh ^{-1}\left(\cos \left(a+b x^2\right)\right)}{32 b}-\frac{\cot \left(a+b x^2\right) \csc ^5\left(a+b x^2\right)}{12 b}-\frac{5 \cot \left(a+b x^2\right) \csc ^3\left(a+b x^2\right)}{48 b}-\frac{5 \cot \left(a+b x^2\right) \csc \left(a+b x^2\right)}{32 b}",1,"(-5*Csc[(a + b*x^2)/2]^2)/(128*b) - Csc[(a + b*x^2)/2]^4/(128*b) - Csc[(a + b*x^2)/2]^6/(768*b) - (5*Log[Cos[(a + b*x^2)/2]])/(32*b) + (5*Log[Sin[(a + b*x^2)/2]])/(32*b) + (5*Sec[(a + b*x^2)/2]^2)/(128*b) + Sec[(a + b*x^2)/2]^4/(128*b) + Sec[(a + b*x^2)/2]^6/(768*b)","A",1
16,1,488,396,1.1902307,"\int \frac{x^5}{a+b \csc \left(c+d x^2\right)} \, dx","Integrate[x^5/(a + b*Csc[c + d*x^2]),x]","\frac{d^3 x^6 \sqrt{e^{2 i c} \left(a^2-b^2\right)}-3 b e^{i c} d^2 x^4 \log \left(1+\frac{a e^{i \left(2 c+d x^2\right)}}{i b e^{i c}-\sqrt{e^{2 i c} \left(a^2-b^2\right)}}\right)+3 b e^{i c} d^2 x^4 \log \left(1+\frac{a e^{i \left(2 c+d x^2\right)}}{\sqrt{e^{2 i c} \left(a^2-b^2\right)}+i b e^{i c}}\right)+6 i b e^{i c} d x^2 \text{Li}_2\left(\frac{i a e^{i \left(d x^2+2 c\right)}}{e^{i c} b+i \sqrt{\left(a^2-b^2\right) e^{2 i c}}}\right)-6 i b e^{i c} d x^2 \text{Li}_2\left(-\frac{a e^{i \left(d x^2+2 c\right)}}{i e^{i c} b+\sqrt{\left(a^2-b^2\right) e^{2 i c}}}\right)-6 b e^{i c} \text{Li}_3\left(\frac{i a e^{i \left(d x^2+2 c\right)}}{e^{i c} b+i \sqrt{\left(a^2-b^2\right) e^{2 i c}}}\right)+6 b e^{i c} \text{Li}_3\left(-\frac{a e^{i \left(d x^2+2 c\right)}}{i e^{i c} b+\sqrt{\left(a^2-b^2\right) e^{2 i c}}}\right)}{6 a d^3 \sqrt{e^{2 i c} \left(a^2-b^2\right)}}","\frac{i b \text{Li}_3\left(\frac{i a e^{i \left(d x^2+c\right)}}{b-\sqrt{b^2-a^2}}\right)}{a d^3 \sqrt{b^2-a^2}}-\frac{i b \text{Li}_3\left(\frac{i a e^{i \left(d x^2+c\right)}}{b+\sqrt{b^2-a^2}}\right)}{a d^3 \sqrt{b^2-a^2}}+\frac{b x^2 \text{Li}_2\left(\frac{i a e^{i \left(d x^2+c\right)}}{b-\sqrt{b^2-a^2}}\right)}{a d^2 \sqrt{b^2-a^2}}-\frac{b x^2 \text{Li}_2\left(\frac{i a e^{i \left(d x^2+c\right)}}{b+\sqrt{b^2-a^2}}\right)}{a d^2 \sqrt{b^2-a^2}}+\frac{i b x^4 \log \left(1-\frac{i a e^{i \left(c+d x^2\right)}}{b-\sqrt{b^2-a^2}}\right)}{2 a d \sqrt{b^2-a^2}}-\frac{i b x^4 \log \left(1-\frac{i a e^{i \left(c+d x^2\right)}}{\sqrt{b^2-a^2}+b}\right)}{2 a d \sqrt{b^2-a^2}}+\frac{x^6}{6 a}",1,"(d^3*Sqrt[(a^2 - b^2)*E^((2*I)*c)]*x^6 - 3*b*d^2*E^(I*c)*x^4*Log[1 + (a*E^(I*(2*c + d*x^2)))/(I*b*E^(I*c) - Sqrt[(a^2 - b^2)*E^((2*I)*c)])] + 3*b*d^2*E^(I*c)*x^4*Log[1 + (a*E^(I*(2*c + d*x^2)))/(I*b*E^(I*c) + Sqrt[(a^2 - b^2)*E^((2*I)*c)])] + (6*I)*b*d*E^(I*c)*x^2*PolyLog[2, (I*a*E^(I*(2*c + d*x^2)))/(b*E^(I*c) + I*Sqrt[(a^2 - b^2)*E^((2*I)*c)])] - (6*I)*b*d*E^(I*c)*x^2*PolyLog[2, -((a*E^(I*(2*c + d*x^2)))/(I*b*E^(I*c) + Sqrt[(a^2 - b^2)*E^((2*I)*c)]))] - 6*b*E^(I*c)*PolyLog[3, (I*a*E^(I*(2*c + d*x^2)))/(b*E^(I*c) + I*Sqrt[(a^2 - b^2)*E^((2*I)*c)])] + 6*b*E^(I*c)*PolyLog[3, -((a*E^(I*(2*c + d*x^2)))/(I*b*E^(I*c) + Sqrt[(a^2 - b^2)*E^((2*I)*c)]))])/(6*a*d^3*Sqrt[(a^2 - b^2)*E^((2*I)*c)])","A",1
17,0,0,21,1.4942459,"\int \frac{x^4}{a+b \csc \left(c+d x^2\right)} \, dx","Integrate[x^4/(a + b*Csc[c + d*x^2]),x]","\int \frac{x^4}{a+b \csc \left(c+d x^2\right)} \, dx","\text{Int}\left(\frac{x^4}{a+b \csc \left(c+d x^2\right)},x\right)",0,"Integrate[x^4/(a + b*Csc[c + d*x^2]), x]","A",-1
18,1,987,271,3.9098214,"\int \frac{x^3}{a+b \csc \left(c+d x^2\right)} \, dx","Integrate[x^3/(a + b*Csc[c + d*x^2]),x]","\frac{\csc \left(d x^2+c\right) \left(x^4-\frac{2 b \left(\frac{\pi  \tan ^{-1}\left(\frac{a+b \tan \left(\frac{1}{2} \left(d x^2+c\right)\right)}{\sqrt{b^2-a^2}}\right)}{\sqrt{b^2-a^2}}+\frac{2 \left(c-\cos ^{-1}\left(-\frac{b}{a}\right)\right) \tanh ^{-1}\left(\frac{(a-b) \cot \left(\frac{1}{4} \left(2 d x^2+2 c+\pi \right)\right)}{\sqrt{a^2-b^2}}\right)+\left(-2 d x^2-2 c+\pi \right) \tanh ^{-1}\left(\frac{(a+b) \tan \left(\frac{1}{4} \left(2 d x^2+2 c+\pi \right)\right)}{\sqrt{a^2-b^2}}\right)-\left(\cos ^{-1}\left(-\frac{b}{a}\right)-2 i \tanh ^{-1}\left(\frac{(a-b) \cot \left(\frac{1}{4} \left(2 d x^2+2 c+\pi \right)\right)}{\sqrt{a^2-b^2}}\right)\right) \log \left(\frac{(a+b) \left(a-b-i \sqrt{a^2-b^2}\right) \left(i \cot \left(\frac{1}{4} \left(2 d x^2+2 c+\pi \right)\right)+1\right)}{a \left(a+b+\sqrt{a^2-b^2} \cot \left(\frac{1}{4} \left(2 d x^2+2 c+\pi \right)\right)\right)}\right)+\left(\cos ^{-1}\left(-\frac{b}{a}\right)+2 i \left(\tanh ^{-1}\left(\frac{(a+b) \tan \left(\frac{1}{4} \left(2 d x^2+2 c+\pi \right)\right)}{\sqrt{a^2-b^2}}\right)-\tanh ^{-1}\left(\frac{(a-b) \cot \left(\frac{1}{4} \left(2 d x^2+2 c+\pi \right)\right)}{\sqrt{a^2-b^2}}\right)\right)\right) \log \left(\frac{\sqrt[4]{-1} \sqrt{a^2-b^2} e^{-\frac{1}{2} i \left(d x^2+c\right)}}{\sqrt{2} \sqrt{a} \sqrt{b+a \sin \left(d x^2+c\right)}}\right)+\left(\cos ^{-1}\left(-\frac{b}{a}\right)+2 i \tanh ^{-1}\left(\frac{(a-b) \cot \left(\frac{1}{4} \left(2 d x^2+2 c+\pi \right)\right)}{\sqrt{a^2-b^2}}\right)-2 i \tanh ^{-1}\left(\frac{(a+b) \tan \left(\frac{1}{4} \left(2 d x^2+2 c+\pi \right)\right)}{\sqrt{a^2-b^2}}\right)\right) \log \left(-\frac{(-1)^{3/4} \sqrt{a^2-b^2} e^{\frac{1}{2} i \left(d x^2+c\right)}}{\sqrt{2} \sqrt{a} \sqrt{b+a \sin \left(d x^2+c\right)}}\right)-\left(\cos ^{-1}\left(-\frac{b}{a}\right)+2 i \tanh ^{-1}\left(\frac{(a-b) \cot \left(\frac{1}{4} \left(2 d x^2+2 c+\pi \right)\right)}{\sqrt{a^2-b^2}}\right)\right) \log \left(\frac{i \left(i b+\sqrt{a^2-b^2}\right) \left(a+b+\sqrt{a^2-b^2} \tan \left(\frac{1}{4} \left(2 d x^2+2 c-\pi \right)\right)\right)}{a \left(a+b+\sqrt{a^2-b^2} \cot \left(\frac{1}{4} \left(2 d x^2+2 c+\pi \right)\right)\right)}+1\right)+i \left(\text{Li}_2\left(\frac{\left(b-i \sqrt{a^2-b^2}\right) \left(a+b+\sqrt{a^2-b^2} \tan \left(\frac{1}{4} \left(2 d x^2+2 c-\pi \right)\right)\right)}{a \left(a+b+\sqrt{a^2-b^2} \cot \left(\frac{1}{4} \left(2 d x^2+2 c+\pi \right)\right)\right)}\right)-\text{Li}_2\left(\frac{\left(b+i \sqrt{a^2-b^2}\right) \left(a+b+\sqrt{a^2-b^2} \tan \left(\frac{1}{4} \left(2 d x^2+2 c-\pi \right)\right)\right)}{a \left(a+b+\sqrt{a^2-b^2} \cot \left(\frac{1}{4} \left(2 d x^2+2 c+\pi \right)\right)\right)}\right)\right)}{\sqrt{a^2-b^2}}\right)}{d^2}\right) \left(b+a \sin \left(d x^2+c\right)\right)}{4 a \left(a+b \csc \left(d x^2+c\right)\right)}","\frac{b \text{Li}_2\left(\frac{i a e^{i \left(d x^2+c\right)}}{b-\sqrt{b^2-a^2}}\right)}{2 a d^2 \sqrt{b^2-a^2}}-\frac{b \text{Li}_2\left(\frac{i a e^{i \left(d x^2+c\right)}}{b+\sqrt{b^2-a^2}}\right)}{2 a d^2 \sqrt{b^2-a^2}}+\frac{i b x^2 \log \left(1-\frac{i a e^{i \left(c+d x^2\right)}}{b-\sqrt{b^2-a^2}}\right)}{2 a d \sqrt{b^2-a^2}}-\frac{i b x^2 \log \left(1-\frac{i a e^{i \left(c+d x^2\right)}}{\sqrt{b^2-a^2}+b}\right)}{2 a d \sqrt{b^2-a^2}}+\frac{x^4}{4 a}",1,"(Csc[c + d*x^2]*(x^4 - (2*b*((Pi*ArcTan[(a + b*Tan[(c + d*x^2)/2])/Sqrt[-a^2 + b^2]])/Sqrt[-a^2 + b^2] + (2*(c - ArcCos[-(b/a)])*ArcTanh[((a - b)*Cot[(2*c + Pi + 2*d*x^2)/4])/Sqrt[a^2 - b^2]] + (-2*c + Pi - 2*d*x^2)*ArcTanh[((a + b)*Tan[(2*c + Pi + 2*d*x^2)/4])/Sqrt[a^2 - b^2]] - (ArcCos[-(b/a)] - (2*I)*ArcTanh[((a - b)*Cot[(2*c + Pi + 2*d*x^2)/4])/Sqrt[a^2 - b^2]])*Log[((a + b)*(a - b - I*Sqrt[a^2 - b^2])*(1 + I*Cot[(2*c + Pi + 2*d*x^2)/4]))/(a*(a + b + Sqrt[a^2 - b^2]*Cot[(2*c + Pi + 2*d*x^2)/4]))] + (ArcCos[-(b/a)] + (2*I)*(-ArcTanh[((a - b)*Cot[(2*c + Pi + 2*d*x^2)/4])/Sqrt[a^2 - b^2]] + ArcTanh[((a + b)*Tan[(2*c + Pi + 2*d*x^2)/4])/Sqrt[a^2 - b^2]]))*Log[((-1)^(1/4)*Sqrt[a^2 - b^2])/(Sqrt[2]*Sqrt[a]*E^((I/2)*(c + d*x^2))*Sqrt[b + a*Sin[c + d*x^2]])] + (ArcCos[-(b/a)] + (2*I)*ArcTanh[((a - b)*Cot[(2*c + Pi + 2*d*x^2)/4])/Sqrt[a^2 - b^2]] - (2*I)*ArcTanh[((a + b)*Tan[(2*c + Pi + 2*d*x^2)/4])/Sqrt[a^2 - b^2]])*Log[-(((-1)^(3/4)*Sqrt[a^2 - b^2]*E^((I/2)*(c + d*x^2)))/(Sqrt[2]*Sqrt[a]*Sqrt[b + a*Sin[c + d*x^2]]))] - (ArcCos[-(b/a)] + (2*I)*ArcTanh[((a - b)*Cot[(2*c + Pi + 2*d*x^2)/4])/Sqrt[a^2 - b^2]])*Log[1 + (I*(I*b + Sqrt[a^2 - b^2])*(a + b + Sqrt[a^2 - b^2]*Tan[(2*c - Pi + 2*d*x^2)/4]))/(a*(a + b + Sqrt[a^2 - b^2]*Cot[(2*c + Pi + 2*d*x^2)/4]))] + I*(PolyLog[2, ((b - I*Sqrt[a^2 - b^2])*(a + b + Sqrt[a^2 - b^2]*Tan[(2*c - Pi + 2*d*x^2)/4]))/(a*(a + b + Sqrt[a^2 - b^2]*Cot[(2*c + Pi + 2*d*x^2)/4]))] - PolyLog[2, ((b + I*Sqrt[a^2 - b^2])*(a + b + Sqrt[a^2 - b^2]*Tan[(2*c - Pi + 2*d*x^2)/4]))/(a*(a + b + Sqrt[a^2 - b^2]*Cot[(2*c + Pi + 2*d*x^2)/4]))]))/Sqrt[a^2 - b^2]))/d^2)*(b + a*Sin[c + d*x^2]))/(4*a*(a + b*Csc[c + d*x^2]))","B",0
19,0,0,21,1.2444901,"\int \frac{x^2}{a+b \csc \left(c+d x^2\right)} \, dx","Integrate[x^2/(a + b*Csc[c + d*x^2]),x]","\int \frac{x^2}{a+b \csc \left(c+d x^2\right)} \, dx","\text{Int}\left(\frac{x^2}{a+b \csc \left(c+d x^2\right)},x\right)",0,"Integrate[x^2/(a + b*Csc[c + d*x^2]), x]","A",-1
20,1,66,63,0.1488547,"\int \frac{x}{a+b \csc \left(c+d x^2\right)} \, dx","Integrate[x/(a + b*Csc[c + d*x^2]),x]","\frac{-\frac{2 b \tan ^{-1}\left(\frac{a+b \tan \left(\frac{1}{2} \left(c+d x^2\right)\right)}{\sqrt{b^2-a^2}}\right)}{d \sqrt{b^2-a^2}}+\frac{c}{d}+x^2}{2 a}","\frac{b \tanh ^{-1}\left(\frac{a+b \tan \left(\frac{1}{2} \left(c+d x^2\right)\right)}{\sqrt{a^2-b^2}}\right)}{a d \sqrt{a^2-b^2}}+\frac{x^2}{2 a}",1,"(c/d + x^2 - (2*b*ArcTan[(a + b*Tan[(c + d*x^2)/2])/Sqrt[-a^2 + b^2]])/(Sqrt[-a^2 + b^2]*d))/(2*a)","A",1
21,0,0,21,1.0365434,"\int \frac{1}{x \left(a+b \csc \left(c+d x^2\right)\right)} \, dx","Integrate[1/(x*(a + b*Csc[c + d*x^2])),x]","\int \frac{1}{x \left(a+b \csc \left(c+d x^2\right)\right)} \, dx","\text{Int}\left(\frac{1}{x \left(a+b \csc \left(c+d x^2\right)\right)},x\right)",0,"Integrate[1/(x*(a + b*Csc[c + d*x^2])), x]","A",-1
22,0,0,24,0.2471028,"\int \frac{a+b \csc \left(c+d x^2\right)}{x^2} \, dx","Integrate[(a + b*Csc[c + d*x^2])/x^2,x]","\int \frac{a+b \csc \left(c+d x^2\right)}{x^2} \, dx","b \text{Int}\left(\frac{\csc \left(c+d x^2\right)}{x^2},x\right)-\frac{a}{x}",0,"Integrate[(a + b*Csc[c + d*x^2])/x^2, x]","A",-1
23,1,2033,1124,9.3895941,"\int \frac{x^5}{\left(a+b \csc \left(c+d x^2\right)\right)^2} \, dx","Integrate[x^5/(a + b*Csc[c + d*x^2])^2,x]","\text{Result too large to show}","\frac{x^6}{6 a^2}+\frac{i b \log \left(1-\frac{i a e^{i \left(d x^2+c\right)}}{b-\sqrt{b^2-a^2}}\right) x^4}{a^2 \sqrt{b^2-a^2} d}-\frac{i b^3 \log \left(1-\frac{i a e^{i \left(d x^2+c\right)}}{b-\sqrt{b^2-a^2}}\right) x^4}{2 a^2 \left(b^2-a^2\right)^{3/2} d}-\frac{i b \log \left(1-\frac{i a e^{i \left(d x^2+c\right)}}{b+\sqrt{b^2-a^2}}\right) x^4}{a^2 \sqrt{b^2-a^2} d}+\frac{i b^3 \log \left(1-\frac{i a e^{i \left(d x^2+c\right)}}{b+\sqrt{b^2-a^2}}\right) x^4}{2 a^2 \left(b^2-a^2\right)^{3/2} d}-\frac{i b^2 x^4}{2 a^2 \left(a^2-b^2\right) d}-\frac{b^2 \cos \left(d x^2+c\right) x^4}{2 a \left(a^2-b^2\right) d \left(b+a \sin \left(d x^2+c\right)\right)}+\frac{b^2 \log \left(\frac{e^{i \left(d x^2+c\right)} a}{i b-\sqrt{a^2-b^2}}+1\right) x^2}{a^2 \left(a^2-b^2\right) d^2}+\frac{b^2 \log \left(\frac{e^{i \left(d x^2+c\right)} a}{i b+\sqrt{a^2-b^2}}+1\right) x^2}{a^2 \left(a^2-b^2\right) d^2}+\frac{2 b \text{Li}_2\left(\frac{i a e^{i \left(d x^2+c\right)}}{b-\sqrt{b^2-a^2}}\right) x^2}{a^2 \sqrt{b^2-a^2} d^2}-\frac{b^3 \text{Li}_2\left(\frac{i a e^{i \left(d x^2+c\right)}}{b-\sqrt{b^2-a^2}}\right) x^2}{a^2 \left(b^2-a^2\right)^{3/2} d^2}-\frac{2 b \text{Li}_2\left(\frac{i a e^{i \left(d x^2+c\right)}}{b+\sqrt{b^2-a^2}}\right) x^2}{a^2 \sqrt{b^2-a^2} d^2}+\frac{b^3 \text{Li}_2\left(\frac{i a e^{i \left(d x^2+c\right)}}{b+\sqrt{b^2-a^2}}\right) x^2}{a^2 \left(b^2-a^2\right)^{3/2} d^2}-\frac{i b^2 \text{Li}_2\left(-\frac{a e^{i \left(d x^2+c\right)}}{i b-\sqrt{a^2-b^2}}\right)}{a^2 \left(a^2-b^2\right) d^3}-\frac{i b^2 \text{Li}_2\left(-\frac{a e^{i \left(d x^2+c\right)}}{i b+\sqrt{a^2-b^2}}\right)}{a^2 \left(a^2-b^2\right) d^3}+\frac{2 i b \text{Li}_3\left(\frac{i a e^{i \left(d x^2+c\right)}}{b-\sqrt{b^2-a^2}}\right)}{a^2 \sqrt{b^2-a^2} d^3}-\frac{i b^3 \text{Li}_3\left(\frac{i a e^{i \left(d x^2+c\right)}}{b-\sqrt{b^2-a^2}}\right)}{a^2 \left(b^2-a^2\right)^{3/2} d^3}-\frac{2 i b \text{Li}_3\left(\frac{i a e^{i \left(d x^2+c\right)}}{b+\sqrt{b^2-a^2}}\right)}{a^2 \sqrt{b^2-a^2} d^3}+\frac{i b^3 \text{Li}_3\left(\frac{i a e^{i \left(d x^2+c\right)}}{b+\sqrt{b^2-a^2}}\right)}{a^2 \left(b^2-a^2\right)^{3/2} d^3}",1,"(Csc[c/2]*Csc[c + d*x^2]^2*Sec[c/2]*(-(b^3*x^4*Cos[c]) - a*b^2*x^4*Sin[d*x^2])*(b + a*Sin[c + d*x^2]))/(4*a^2*(-a + b)*(a + b)*d*(a + b*Csc[c + d*x^2])^2) + (x^6*Csc[c + d*x^2]^2*(b + a*Sin[c + d*x^2])^2)/(6*a^2*(a + b*Csc[c + d*x^2])^2) + (b*E^((2*I)*c)*Csc[c + d*x^2]^2*((-2*I)*b*d^2*E^((2*I)*c)*Sqrt[(a^2 - b^2)*E^((2*I)*c)]*x^4 - 2*b*d*Sqrt[(a^2 - b^2)*E^((2*I)*c)]*x^2*Log[1 + (a*E^(I*(2*c + d*x^2)))/(I*b*E^(I*c) - Sqrt[(a^2 - b^2)*E^((2*I)*c)])] + 2*b*d*E^((2*I)*c)*Sqrt[(a^2 - b^2)*E^((2*I)*c)]*x^2*Log[1 + (a*E^(I*(2*c + d*x^2)))/(I*b*E^(I*c) - Sqrt[(a^2 - b^2)*E^((2*I)*c)])] + 2*a^2*d^2*E^(I*c)*x^4*Log[1 + (a*E^(I*(2*c + d*x^2)))/(I*b*E^(I*c) - Sqrt[(a^2 - b^2)*E^((2*I)*c)])] - b^2*d^2*E^(I*c)*x^4*Log[1 + (a*E^(I*(2*c + d*x^2)))/(I*b*E^(I*c) - Sqrt[(a^2 - b^2)*E^((2*I)*c)])] - 2*a^2*d^2*E^((3*I)*c)*x^4*Log[1 + (a*E^(I*(2*c + d*x^2)))/(I*b*E^(I*c) - Sqrt[(a^2 - b^2)*E^((2*I)*c)])] + b^2*d^2*E^((3*I)*c)*x^4*Log[1 + (a*E^(I*(2*c + d*x^2)))/(I*b*E^(I*c) - Sqrt[(a^2 - b^2)*E^((2*I)*c)])] - 2*b*d*Sqrt[(a^2 - b^2)*E^((2*I)*c)]*x^2*Log[1 + (a*E^(I*(2*c + d*x^2)))/(I*b*E^(I*c) + Sqrt[(a^2 - b^2)*E^((2*I)*c)])] + 2*b*d*E^((2*I)*c)*Sqrt[(a^2 - b^2)*E^((2*I)*c)]*x^2*Log[1 + (a*E^(I*(2*c + d*x^2)))/(I*b*E^(I*c) + Sqrt[(a^2 - b^2)*E^((2*I)*c)])] - 2*a^2*d^2*E^(I*c)*x^4*Log[1 + (a*E^(I*(2*c + d*x^2)))/(I*b*E^(I*c) + Sqrt[(a^2 - b^2)*E^((2*I)*c)])] + b^2*d^2*E^(I*c)*x^4*Log[1 + (a*E^(I*(2*c + d*x^2)))/(I*b*E^(I*c) + Sqrt[(a^2 - b^2)*E^((2*I)*c)])] + 2*a^2*d^2*E^((3*I)*c)*x^4*Log[1 + (a*E^(I*(2*c + d*x^2)))/(I*b*E^(I*c) + Sqrt[(a^2 - b^2)*E^((2*I)*c)])] - b^2*d^2*E^((3*I)*c)*x^4*Log[1 + (a*E^(I*(2*c + d*x^2)))/(I*b*E^(I*c) + Sqrt[(a^2 - b^2)*E^((2*I)*c)])] - (2*I)*(-1 + E^((2*I)*c))*(b*Sqrt[(a^2 - b^2)*E^((2*I)*c)] - 2*a^2*d*E^(I*c)*x^2 + b^2*d*E^(I*c)*x^2)*PolyLog[2, (I*a*E^(I*(2*c + d*x^2)))/(b*E^(I*c) + I*Sqrt[(a^2 - b^2)*E^((2*I)*c)])] + (2*I)*(-1 + E^((2*I)*c))*(-(b*Sqrt[(a^2 - b^2)*E^((2*I)*c)]) - 2*a^2*d*E^(I*c)*x^2 + b^2*d*E^(I*c)*x^2)*PolyLog[2, -((a*E^(I*(2*c + d*x^2)))/(I*b*E^(I*c) + Sqrt[(a^2 - b^2)*E^((2*I)*c)]))] + 4*a^2*E^(I*c)*PolyLog[3, (I*a*E^(I*(2*c + d*x^2)))/(b*E^(I*c) + I*Sqrt[(a^2 - b^2)*E^((2*I)*c)])] - 2*b^2*E^(I*c)*PolyLog[3, (I*a*E^(I*(2*c + d*x^2)))/(b*E^(I*c) + I*Sqrt[(a^2 - b^2)*E^((2*I)*c)])] - 4*a^2*E^((3*I)*c)*PolyLog[3, (I*a*E^(I*(2*c + d*x^2)))/(b*E^(I*c) + I*Sqrt[(a^2 - b^2)*E^((2*I)*c)])] + 2*b^2*E^((3*I)*c)*PolyLog[3, (I*a*E^(I*(2*c + d*x^2)))/(b*E^(I*c) + I*Sqrt[(a^2 - b^2)*E^((2*I)*c)])] - 4*a^2*E^(I*c)*PolyLog[3, -((a*E^(I*(2*c + d*x^2)))/(I*b*E^(I*c) + Sqrt[(a^2 - b^2)*E^((2*I)*c)]))] + 2*b^2*E^(I*c)*PolyLog[3, -((a*E^(I*(2*c + d*x^2)))/(I*b*E^(I*c) + Sqrt[(a^2 - b^2)*E^((2*I)*c)]))] + 4*a^2*E^((3*I)*c)*PolyLog[3, -((a*E^(I*(2*c + d*x^2)))/(I*b*E^(I*c) + Sqrt[(a^2 - b^2)*E^((2*I)*c)]))] - 2*b^2*E^((3*I)*c)*PolyLog[3, -((a*E^(I*(2*c + d*x^2)))/(I*b*E^(I*c) + Sqrt[(a^2 - b^2)*E^((2*I)*c)]))])*(b + a*Sin[c + d*x^2])^2)/(2*a^2*d^3*((a^2 - b^2)*E^((2*I)*c))^(3/2)*(-1 + E^((2*I)*c))*(a + b*Csc[c + d*x^2])^2)","A",1
24,0,0,21,15.4609177,"\int \frac{x^4}{\left(a+b \csc \left(c+d x^2\right)\right)^2} \, dx","Integrate[x^4/(a + b*Csc[c + d*x^2])^2,x]","\int \frac{x^4}{\left(a+b \csc \left(c+d x^2\right)\right)^2} \, dx","\text{Int}\left(\frac{x^4}{\left(a+b \csc \left(c+d x^2\right)\right)^2},x\right)",0,"Integrate[x^4/(a + b*Csc[c + d*x^2])^2, x]","A",-1
25,1,2446,616,15.2912234,"\int \frac{x^3}{\left(a+b \csc \left(c+d x^2\right)\right)^2} \, dx","Integrate[x^3/(a + b*Csc[c + d*x^2])^2,x]","\text{Result too large to show}","\frac{b \text{Li}_2\left(\frac{i a e^{i \left(d x^2+c\right)}}{b-\sqrt{b^2-a^2}}\right)}{a^2 d^2 \sqrt{b^2-a^2}}-\frac{b \text{Li}_2\left(\frac{i a e^{i \left(d x^2+c\right)}}{b+\sqrt{b^2-a^2}}\right)}{a^2 d^2 \sqrt{b^2-a^2}}+\frac{b^2 \log \left(a \sin \left(c+d x^2\right)+b\right)}{2 a^2 d^2 \left(a^2-b^2\right)}+\frac{i b x^2 \log \left(1-\frac{i a e^{i \left(c+d x^2\right)}}{b-\sqrt{b^2-a^2}}\right)}{a^2 d \sqrt{b^2-a^2}}-\frac{i b x^2 \log \left(1-\frac{i a e^{i \left(c+d x^2\right)}}{\sqrt{b^2-a^2}+b}\right)}{a^2 d \sqrt{b^2-a^2}}-\frac{b^2 x^2 \cos \left(c+d x^2\right)}{2 a d \left(a^2-b^2\right) \left(a \sin \left(c+d x^2\right)+b\right)}-\frac{b^3 \text{Li}_2\left(\frac{i a e^{i \left(d x^2+c\right)}}{b-\sqrt{b^2-a^2}}\right)}{2 a^2 d^2 \left(b^2-a^2\right)^{3/2}}+\frac{b^3 \text{Li}_2\left(\frac{i a e^{i \left(d x^2+c\right)}}{b+\sqrt{b^2-a^2}}\right)}{2 a^2 d^2 \left(b^2-a^2\right)^{3/2}}-\frac{i b^3 x^2 \log \left(1-\frac{i a e^{i \left(c+d x^2\right)}}{b-\sqrt{b^2-a^2}}\right)}{2 a^2 d \left(b^2-a^2\right)^{3/2}}+\frac{i b^3 x^2 \log \left(1-\frac{i a e^{i \left(c+d x^2\right)}}{\sqrt{b^2-a^2}+b}\right)}{2 a^2 d \left(b^2-a^2\right)^{3/2}}+\frac{x^4}{4 a^2}",1,"((-(b^2*c*Cos[c + d*x^2]) + b^2*(c + d*x^2)*Cos[c + d*x^2])*Csc[c + d*x^2]^2*(b + a*Sin[c + d*x^2]))/(2*a*(-a + b)*(a + b)*d^2*(a + b*Csc[c + d*x^2])^2) + ((-c + d*x^2)*(c + d*x^2)*Csc[c + d*x^2]^2*(b + a*Sin[c + d*x^2])^2)/(4*a^2*d^2*(a + b*Csc[c + d*x^2])^2) + (Csc[c + d*x^2]^2*(-2*a*b*ArcTanh[(a + b*Tan[(c + d*x^2)/2])/Sqrt[a^2 - b^2]] + 2*(a*b + 2*a^2*c - b^2*c)*ArcTanh[(a + b*Tan[(c + d*x^2)/2])/Sqrt[a^2 - b^2]] + b*Sqrt[a^2 - b^2]*Log[Sec[(c + d*x^2)/2]^2] - b*Sqrt[a^2 - b^2]*Log[Sec[(c + d*x^2)/2]^2*(b + a*Sin[c + d*x^2])] - I*(2*a^2 - b^2)*(Log[1 + I*Tan[(c + d*x^2)/2]]*Log[(a - Sqrt[a^2 - b^2] + b*Tan[(c + d*x^2)/2])/(a + I*b - Sqrt[a^2 - b^2])] + PolyLog[2, (b*(1 + I*Tan[(c + d*x^2)/2]))/((-I)*a + b + I*Sqrt[a^2 - b^2])]) + I*(2*a^2 - b^2)*(Log[1 + I*Tan[(c + d*x^2)/2]]*Log[(a + Sqrt[a^2 - b^2] + b*Tan[(c + d*x^2)/2])/(a + I*b + Sqrt[a^2 - b^2])] + PolyLog[2, (b*(1 + I*Tan[(c + d*x^2)/2]))/(b - I*(a + Sqrt[a^2 - b^2]))]) - I*(2*a^2 - b^2)*(Log[1 - I*Tan[(c + d*x^2)/2]]*Log[(a + Sqrt[a^2 - b^2] + b*Tan[(c + d*x^2)/2])/(a - I*b + Sqrt[a^2 - b^2])] + PolyLog[2, -((b*(I + Tan[(c + d*x^2)/2]))/(a - I*b + Sqrt[a^2 - b^2]))]) + I*(2*a^2 - b^2)*(Log[1 - I*Tan[(c + d*x^2)/2]]*Log[(a - Sqrt[a^2 - b^2] + b*Tan[(c + d*x^2)/2])/(a - I*b - Sqrt[a^2 - b^2])] + PolyLog[2, (b*(I + Tan[(c + d*x^2)/2]))/(-a + I*b + Sqrt[a^2 - b^2])]))*(b + a*Sin[c + d*x^2])^2*((2*b*c)/((a^2 - b^2)*d*(b + a*Sin[c + d*x^2])) - (b^3*c)/(a^2*(a^2 - b^2)*d*(b + a*Sin[c + d*x^2])) - (2*b*(c + d*x^2))/((a^2 - b^2)*d*(b + a*Sin[c + d*x^2])) + (b^3*(c + d*x^2))/(a^2*(a^2 - b^2)*d*(b + a*Sin[c + d*x^2])) + (b^2*Cos[c + d*x^2])/(a*(a^2 - b^2)*d*(b + a*Sin[c + d*x^2]))))/(2*d*(a + b*Csc[c + d*x^2])^2*(b*Sqrt[a^2 - b^2]*Tan[(c + d*x^2)/2] - (b*Sqrt[a^2 - b^2]*Cos[(c + d*x^2)/2]^2*(a*Cos[c + d*x^2]*Sec[(c + d*x^2)/2]^2 + Sec[(c + d*x^2)/2]^2*(b + a*Sin[c + d*x^2])*Tan[(c + d*x^2)/2]))/(b + a*Sin[c + d*x^2]) - (a*b^2*Sec[(c + d*x^2)/2]^2)/(Sqrt[a^2 - b^2]*(1 - (a + b*Tan[(c + d*x^2)/2])^2/(a^2 - b^2))) + (b*(a*b + 2*a^2*c - b^2*c)*Sec[(c + d*x^2)/2]^2)/(Sqrt[a^2 - b^2]*(1 - (a + b*Tan[(c + d*x^2)/2])^2/(a^2 - b^2))) + I*(2*a^2 - b^2)*(((-1/2*I)*Log[(a - Sqrt[a^2 - b^2] + b*Tan[(c + d*x^2)/2])/(a - I*b - Sqrt[a^2 - b^2])]*Sec[(c + d*x^2)/2]^2)/(1 - I*Tan[(c + d*x^2)/2]) - (Log[1 - (b*(I + Tan[(c + d*x^2)/2]))/(-a + I*b + Sqrt[a^2 - b^2])]*Sec[(c + d*x^2)/2]^2)/(2*(I + Tan[(c + d*x^2)/2])) + (b*Log[1 - I*Tan[(c + d*x^2)/2]]*Sec[(c + d*x^2)/2]^2)/(2*(a - Sqrt[a^2 - b^2] + b*Tan[(c + d*x^2)/2]))) - I*(2*a^2 - b^2)*(((-1/2*I)*Log[1 - (b*(1 + I*Tan[(c + d*x^2)/2]))/((-I)*a + b + I*Sqrt[a^2 - b^2])]*Sec[(c + d*x^2)/2]^2)/(1 + I*Tan[(c + d*x^2)/2]) + ((I/2)*Log[(a - Sqrt[a^2 - b^2] + b*Tan[(c + d*x^2)/2])/(a + I*b - Sqrt[a^2 - b^2])]*Sec[(c + d*x^2)/2]^2)/(1 + I*Tan[(c + d*x^2)/2]) + (b*Log[1 + I*Tan[(c + d*x^2)/2]]*Sec[(c + d*x^2)/2]^2)/(2*(a - Sqrt[a^2 - b^2] + b*Tan[(c + d*x^2)/2]))) - I*(2*a^2 - b^2)*(((-1/2*I)*Log[(a + Sqrt[a^2 - b^2] + b*Tan[(c + d*x^2)/2])/(a - I*b + Sqrt[a^2 - b^2])]*Sec[(c + d*x^2)/2]^2)/(1 - I*Tan[(c + d*x^2)/2]) - (Log[1 + (b*(I + Tan[(c + d*x^2)/2]))/(a - I*b + Sqrt[a^2 - b^2])]*Sec[(c + d*x^2)/2]^2)/(2*(I + Tan[(c + d*x^2)/2])) + (b*Log[1 - I*Tan[(c + d*x^2)/2]]*Sec[(c + d*x^2)/2]^2)/(2*(a + Sqrt[a^2 - b^2] + b*Tan[(c + d*x^2)/2]))) + I*(2*a^2 - b^2)*(((-1/2*I)*Log[1 - (b*(1 + I*Tan[(c + d*x^2)/2]))/(b - I*(a + Sqrt[a^2 - b^2]))]*Sec[(c + d*x^2)/2]^2)/(1 + I*Tan[(c + d*x^2)/2]) + ((I/2)*Log[(a + Sqrt[a^2 - b^2] + b*Tan[(c + d*x^2)/2])/(a + I*b + Sqrt[a^2 - b^2])]*Sec[(c + d*x^2)/2]^2)/(1 + I*Tan[(c + d*x^2)/2]) + (b*Log[1 + I*Tan[(c + d*x^2)/2]]*Sec[(c + d*x^2)/2]^2)/(2*(a + Sqrt[a^2 - b^2] + b*Tan[(c + d*x^2)/2])))))","B",0
26,0,0,21,11.8736503,"\int \frac{x^2}{\left(a+b \csc \left(c+d x^2\right)\right)^2} \, dx","Integrate[x^2/(a + b*Csc[c + d*x^2])^2,x]","\int \frac{x^2}{\left(a+b \csc \left(c+d x^2\right)\right)^2} \, dx","\text{Int}\left(\frac{x^2}{\left(a+b \csc \left(c+d x^2\right)\right)^2},x\right)",0,"Integrate[x^2/(a + b*Csc[c + d*x^2])^2, x]","A",-1
27,1,158,120,0.6787661,"\int \frac{x}{\left(a+b \csc \left(c+d x^2\right)\right)^2} \, dx","Integrate[x/(a + b*Csc[c + d*x^2])^2,x]","\frac{\csc \left(c+d x^2\right) \left(a \sin \left(c+d x^2\right)+b\right) \left(-\frac{2 b \left(b^2-2 a^2\right) \tan ^{-1}\left(\frac{a+b \tan \left(\frac{1}{2} \left(c+d x^2\right)\right)}{\sqrt{b^2-a^2}}\right) \left(a+b \csc \left(c+d x^2\right)\right)}{\left(b^2-a^2\right)^{3/2}}+\frac{a b^2 \cot \left(c+d x^2\right)}{(b-a) (a+b)}+\left(c+d x^2\right) \left(a+b \csc \left(c+d x^2\right)\right)\right)}{2 a^2 d \left(a+b \csc \left(c+d x^2\right)\right)^2}","\frac{b \left(2 a^2-b^2\right) \tanh ^{-1}\left(\frac{a+b \tan \left(\frac{1}{2} \left(c+d x^2\right)\right)}{\sqrt{a^2-b^2}}\right)}{a^2 d \left(a^2-b^2\right)^{3/2}}-\frac{b^2 \cot \left(c+d x^2\right)}{2 a d \left(a^2-b^2\right) \left(a+b \csc \left(c+d x^2\right)\right)}+\frac{x^2}{2 a^2}",1,"(Csc[c + d*x^2]*((a*b^2*Cot[c + d*x^2])/((-a + b)*(a + b)) + (c + d*x^2)*(a + b*Csc[c + d*x^2]) - (2*b*(-2*a^2 + b^2)*ArcTan[(a + b*Tan[(c + d*x^2)/2])/Sqrt[-a^2 + b^2]]*(a + b*Csc[c + d*x^2]))/(-a^2 + b^2)^(3/2))*(b + a*Sin[c + d*x^2]))/(2*a^2*d*(a + b*Csc[c + d*x^2])^2)","A",1
28,0,0,21,22.8406314,"\int \frac{1}{x \left(a+b \csc \left(c+d x^2\right)\right)^2} \, dx","Integrate[1/(x*(a + b*Csc[c + d*x^2])^2),x]","\int \frac{1}{x \left(a+b \csc \left(c+d x^2\right)\right)^2} \, dx","\text{Int}\left(\frac{1}{x \left(a+b \csc \left(c+d x^2\right)\right)^2},x\right)",0,"Integrate[1/(x*(a + b*Csc[c + d*x^2])^2), x]","A",-1
29,0,0,21,17.9568573,"\int \frac{1}{x^2 \left(a+b \csc \left(c+d x^2\right)\right)^2} \, dx","Integrate[1/(x^2*(a + b*Csc[c + d*x^2])^2),x]","\int \frac{1}{x^2 \left(a+b \csc \left(c+d x^2\right)\right)^2} \, dx","\text{Int}\left(\frac{1}{x^2 \left(a+b \csc \left(c+d x^2\right)\right)^2},x\right)",0,"Integrate[1/(x^2*(a + b*Csc[c + d*x^2])^2), x]","A",-1
30,0,0,21,18.2863192,"\int \frac{1}{x^3 \left(a+b \csc \left(c+d x^2\right)\right)^2} \, dx","Integrate[1/(x^3*(a + b*Csc[c + d*x^2])^2),x]","\int \frac{1}{x^3 \left(a+b \csc \left(c+d x^2\right)\right)^2} \, dx","\text{Int}\left(\frac{1}{x^3 \left(a+b \csc \left(c+d x^2\right)\right)^2},x\right)",0,"Integrate[1/(x^3*(a + b*Csc[c + d*x^2])^2), x]","A",-1
31,1,445,432,0.5735179,"\int x^3 \left(a+b \csc \left(c+d \sqrt{x}\right)\right) \, dx","Integrate[x^3*(a + b*Csc[c + d*Sqrt[x]]),x]","\frac{a x^4}{4}+\frac{2 b \left(d^7 x^{7/2} \log \left(1-e^{i \left(c+d \sqrt{x}\right)}\right)-d^7 x^{7/2} \log \left(1+e^{i \left(c+d \sqrt{x}\right)}\right)+7 i d^6 x^3 \text{Li}_2\left(-e^{i \left(c+d \sqrt{x}\right)}\right)-7 i d^6 x^3 \text{Li}_2\left(e^{i \left(c+d \sqrt{x}\right)}\right)-42 d^5 x^{5/2} \text{Li}_3\left(-e^{i \left(c+d \sqrt{x}\right)}\right)+42 d^5 x^{5/2} \text{Li}_3\left(e^{i \left(c+d \sqrt{x}\right)}\right)-210 i d^4 x^2 \text{Li}_4\left(-e^{i \left(c+d \sqrt{x}\right)}\right)+210 i d^4 x^2 \text{Li}_4\left(e^{i \left(c+d \sqrt{x}\right)}\right)+840 d^3 x^{3/2} \text{Li}_5\left(-e^{i \left(c+d \sqrt{x}\right)}\right)-840 d^3 x^{3/2} \text{Li}_5\left(e^{i \left(c+d \sqrt{x}\right)}\right)+2520 i d^2 x \text{Li}_6\left(-e^{i \left(c+d \sqrt{x}\right)}\right)-2520 i d^2 x \text{Li}_6\left(e^{i \left(c+d \sqrt{x}\right)}\right)-5040 d \sqrt{x} \text{Li}_7\left(-e^{i \left(c+d \sqrt{x}\right)}\right)+5040 d \sqrt{x} \text{Li}_7\left(e^{i \left(c+d \sqrt{x}\right)}\right)-5040 i \text{Li}_8\left(-e^{i \left(c+d \sqrt{x}\right)}\right)+5040 i \text{Li}_8\left(e^{i \left(c+d \sqrt{x}\right)}\right)\right)}{d^8}","\frac{a x^4}{4}-\frac{10080 i b \text{Li}_8\left(-e^{i \left(c+d \sqrt{x}\right)}\right)}{d^8}+\frac{10080 i b \text{Li}_8\left(e^{i \left(c+d \sqrt{x}\right)}\right)}{d^8}-\frac{10080 b \sqrt{x} \text{Li}_7\left(-e^{i \left(c+d \sqrt{x}\right)}\right)}{d^7}+\frac{10080 b \sqrt{x} \text{Li}_7\left(e^{i \left(c+d \sqrt{x}\right)}\right)}{d^7}+\frac{5040 i b x \text{Li}_6\left(-e^{i \left(c+d \sqrt{x}\right)}\right)}{d^6}-\frac{5040 i b x \text{Li}_6\left(e^{i \left(c+d \sqrt{x}\right)}\right)}{d^6}+\frac{1680 b x^{3/2} \text{Li}_5\left(-e^{i \left(c+d \sqrt{x}\right)}\right)}{d^5}-\frac{1680 b x^{3/2} \text{Li}_5\left(e^{i \left(c+d \sqrt{x}\right)}\right)}{d^5}-\frac{420 i b x^2 \text{Li}_4\left(-e^{i \left(c+d \sqrt{x}\right)}\right)}{d^4}+\frac{420 i b x^2 \text{Li}_4\left(e^{i \left(c+d \sqrt{x}\right)}\right)}{d^4}-\frac{84 b x^{5/2} \text{Li}_3\left(-e^{i \left(c+d \sqrt{x}\right)}\right)}{d^3}+\frac{84 b x^{5/2} \text{Li}_3\left(e^{i \left(c+d \sqrt{x}\right)}\right)}{d^3}+\frac{14 i b x^3 \text{Li}_2\left(-e^{i \left(c+d \sqrt{x}\right)}\right)}{d^2}-\frac{14 i b x^3 \text{Li}_2\left(e^{i \left(c+d \sqrt{x}\right)}\right)}{d^2}-\frac{4 b x^{7/2} \tanh ^{-1}\left(e^{i \left(c+d \sqrt{x}\right)}\right)}{d}",1,"(a*x^4)/4 + (2*b*(d^7*x^(7/2)*Log[1 - E^(I*(c + d*Sqrt[x]))] - d^7*x^(7/2)*Log[1 + E^(I*(c + d*Sqrt[x]))] + (7*I)*d^6*x^3*PolyLog[2, -E^(I*(c + d*Sqrt[x]))] - (7*I)*d^6*x^3*PolyLog[2, E^(I*(c + d*Sqrt[x]))] - 42*d^5*x^(5/2)*PolyLog[3, -E^(I*(c + d*Sqrt[x]))] + 42*d^5*x^(5/2)*PolyLog[3, E^(I*(c + d*Sqrt[x]))] - (210*I)*d^4*x^2*PolyLog[4, -E^(I*(c + d*Sqrt[x]))] + (210*I)*d^4*x^2*PolyLog[4, E^(I*(c + d*Sqrt[x]))] + 840*d^3*x^(3/2)*PolyLog[5, -E^(I*(c + d*Sqrt[x]))] - 840*d^3*x^(3/2)*PolyLog[5, E^(I*(c + d*Sqrt[x]))] + (2520*I)*d^2*x*PolyLog[6, -E^(I*(c + d*Sqrt[x]))] - (2520*I)*d^2*x*PolyLog[6, E^(I*(c + d*Sqrt[x]))] - 5040*d*Sqrt[x]*PolyLog[7, -E^(I*(c + d*Sqrt[x]))] + 5040*d*Sqrt[x]*PolyLog[7, E^(I*(c + d*Sqrt[x]))] - (5040*I)*PolyLog[8, -E^(I*(c + d*Sqrt[x]))] + (5040*I)*PolyLog[8, E^(I*(c + d*Sqrt[x]))]))/d^8","A",1
32,1,333,316,0.3837632,"\int x^2 \left(a+b \csc \left(c+d \sqrt{x}\right)\right) \, dx","Integrate[x^2*(a + b*Csc[c + d*Sqrt[x]]),x]","\frac{a x^3}{3}+\frac{2 b \left(d^5 x^{5/2} \log \left(1-e^{i \left(c+d \sqrt{x}\right)}\right)-d^5 x^{5/2} \log \left(1+e^{i \left(c+d \sqrt{x}\right)}\right)+5 i d^4 x^2 \text{Li}_2\left(-e^{i \left(c+d \sqrt{x}\right)}\right)-5 i d^4 x^2 \text{Li}_2\left(e^{i \left(c+d \sqrt{x}\right)}\right)-20 d^3 x^{3/2} \text{Li}_3\left(-e^{i \left(c+d \sqrt{x}\right)}\right)+20 d^3 x^{3/2} \text{Li}_3\left(e^{i \left(c+d \sqrt{x}\right)}\right)-60 i d^2 x \text{Li}_4\left(-e^{i \left(c+d \sqrt{x}\right)}\right)+60 i d^2 x \text{Li}_4\left(e^{i \left(c+d \sqrt{x}\right)}\right)+120 d \sqrt{x} \text{Li}_5\left(-e^{i \left(c+d \sqrt{x}\right)}\right)-120 d \sqrt{x} \text{Li}_5\left(e^{i \left(c+d \sqrt{x}\right)}\right)+120 i \text{Li}_6\left(-e^{i \left(c+d \sqrt{x}\right)}\right)-120 i \text{Li}_6\left(e^{i \left(c+d \sqrt{x}\right)}\right)\right)}{d^6}","\frac{a x^3}{3}+\frac{240 i b \text{Li}_6\left(-e^{i \left(c+d \sqrt{x}\right)}\right)}{d^6}-\frac{240 i b \text{Li}_6\left(e^{i \left(c+d \sqrt{x}\right)}\right)}{d^6}+\frac{240 b \sqrt{x} \text{Li}_5\left(-e^{i \left(c+d \sqrt{x}\right)}\right)}{d^5}-\frac{240 b \sqrt{x} \text{Li}_5\left(e^{i \left(c+d \sqrt{x}\right)}\right)}{d^5}-\frac{120 i b x \text{Li}_4\left(-e^{i \left(c+d \sqrt{x}\right)}\right)}{d^4}+\frac{120 i b x \text{Li}_4\left(e^{i \left(c+d \sqrt{x}\right)}\right)}{d^4}-\frac{40 b x^{3/2} \text{Li}_3\left(-e^{i \left(c+d \sqrt{x}\right)}\right)}{d^3}+\frac{40 b x^{3/2} \text{Li}_3\left(e^{i \left(c+d \sqrt{x}\right)}\right)}{d^3}+\frac{10 i b x^2 \text{Li}_2\left(-e^{i \left(c+d \sqrt{x}\right)}\right)}{d^2}-\frac{10 i b x^2 \text{Li}_2\left(e^{i \left(c+d \sqrt{x}\right)}\right)}{d^2}-\frac{4 b x^{5/2} \tanh ^{-1}\left(e^{i \left(c+d \sqrt{x}\right)}\right)}{d}",1,"(a*x^3)/3 + (2*b*(d^5*x^(5/2)*Log[1 - E^(I*(c + d*Sqrt[x]))] - d^5*x^(5/2)*Log[1 + E^(I*(c + d*Sqrt[x]))] + (5*I)*d^4*x^2*PolyLog[2, -E^(I*(c + d*Sqrt[x]))] - (5*I)*d^4*x^2*PolyLog[2, E^(I*(c + d*Sqrt[x]))] - 20*d^3*x^(3/2)*PolyLog[3, -E^(I*(c + d*Sqrt[x]))] + 20*d^3*x^(3/2)*PolyLog[3, E^(I*(c + d*Sqrt[x]))] - (60*I)*d^2*x*PolyLog[4, -E^(I*(c + d*Sqrt[x]))] + (60*I)*d^2*x*PolyLog[4, E^(I*(c + d*Sqrt[x]))] + 120*d*Sqrt[x]*PolyLog[5, -E^(I*(c + d*Sqrt[x]))] - 120*d*Sqrt[x]*PolyLog[5, E^(I*(c + d*Sqrt[x]))] + (120*I)*PolyLog[6, -E^(I*(c + d*Sqrt[x]))] - (120*I)*PolyLog[6, E^(I*(c + d*Sqrt[x]))]))/d^6","A",1
33,1,260,200,0.4301134,"\int x \left(a+b \csc \left(c+d \sqrt{x}\right)\right) \, dx","Integrate[x*(a + b*Csc[c + d*Sqrt[x]]),x]","\frac{a x^2}{2}-\frac{2 b \left(2 d^3 x^{3/2} \tanh ^{-1}\left(\cos \left(c+d \sqrt{x}\right)+i \sin \left(c+d \sqrt{x}\right)\right)-3 i d^2 x \text{Li}_2\left(-\cos \left(c+d \sqrt{x}\right)-i \sin \left(c+d \sqrt{x}\right)\right)+3 i d^2 x \text{Li}_2\left(\cos \left(c+d \sqrt{x}\right)+i \sin \left(c+d \sqrt{x}\right)\right)+6 d \sqrt{x} \text{Li}_3\left(-\cos \left(c+d \sqrt{x}\right)-i \sin \left(c+d \sqrt{x}\right)\right)-6 d \sqrt{x} \text{Li}_3\left(\cos \left(c+d \sqrt{x}\right)+i \sin \left(c+d \sqrt{x}\right)\right)+6 i \text{Li}_4\left(-\cos \left(c+d \sqrt{x}\right)-i \sin \left(c+d \sqrt{x}\right)\right)-6 i \text{Li}_4\left(\cos \left(c+d \sqrt{x}\right)+i \sin \left(c+d \sqrt{x}\right)\right)\right)}{d^4}","\frac{a x^2}{2}-\frac{12 i b \text{Li}_4\left(-e^{i \left(c+d \sqrt{x}\right)}\right)}{d^4}+\frac{12 i b \text{Li}_4\left(e^{i \left(c+d \sqrt{x}\right)}\right)}{d^4}-\frac{12 b \sqrt{x} \text{Li}_3\left(-e^{i \left(c+d \sqrt{x}\right)}\right)}{d^3}+\frac{12 b \sqrt{x} \text{Li}_3\left(e^{i \left(c+d \sqrt{x}\right)}\right)}{d^3}+\frac{6 i b x \text{Li}_2\left(-e^{i \left(c+d \sqrt{x}\right)}\right)}{d^2}-\frac{6 i b x \text{Li}_2\left(e^{i \left(c+d \sqrt{x}\right)}\right)}{d^2}-\frac{4 b x^{3/2} \tanh ^{-1}\left(e^{i \left(c+d \sqrt{x}\right)}\right)}{d}",1,"(a*x^2)/2 - (2*b*(2*d^3*x^(3/2)*ArcTanh[Cos[c + d*Sqrt[x]] + I*Sin[c + d*Sqrt[x]]] - (3*I)*d^2*x*PolyLog[2, -Cos[c + d*Sqrt[x]] - I*Sin[c + d*Sqrt[x]]] + (3*I)*d^2*x*PolyLog[2, Cos[c + d*Sqrt[x]] + I*Sin[c + d*Sqrt[x]]] + 6*d*Sqrt[x]*PolyLog[3, -Cos[c + d*Sqrt[x]] - I*Sin[c + d*Sqrt[x]]] - 6*d*Sqrt[x]*PolyLog[3, Cos[c + d*Sqrt[x]] + I*Sin[c + d*Sqrt[x]]] + (6*I)*PolyLog[4, -Cos[c + d*Sqrt[x]] - I*Sin[c + d*Sqrt[x]]] - (6*I)*PolyLog[4, Cos[c + d*Sqrt[x]] + I*Sin[c + d*Sqrt[x]]]))/d^4","A",0
34,0,0,24,5.8405483,"\int \frac{a+b \csc \left(c+d \sqrt{x}\right)}{x} \, dx","Integrate[(a + b*Csc[c + d*Sqrt[x]])/x,x]","\int \frac{a+b \csc \left(c+d \sqrt{x}\right)}{x} \, dx","b \text{Int}\left(\frac{\csc \left(c+d \sqrt{x}\right)}{x},x\right)+a \log (x)",0,"Integrate[(a + b*Csc[c + d*Sqrt[x]])/x, x]","A",-1
35,0,0,26,6.7946643,"\int \frac{a+b \csc \left(c+d \sqrt{x}\right)}{x^2} \, dx","Integrate[(a + b*Csc[c + d*Sqrt[x]])/x^2,x]","\int \frac{a+b \csc \left(c+d \sqrt{x}\right)}{x^2} \, dx","b \text{Int}\left(\frac{\csc \left(c+d \sqrt{x}\right)}{x^2},x\right)-\frac{a}{x}",0,"Integrate[(a + b*Csc[c + d*Sqrt[x]])/x^2, x]","A",-1
36,1,1242,695,22.4076893,"\int x^3 \left(a+b \csc \left(c+d \sqrt{x}\right)\right)^2 \, dx","Integrate[x^3*(a + b*Csc[c + d*Sqrt[x]])^2,x]","\frac{a^2 \left(a+b \csc \left(c+d \sqrt{x}\right)\right)^2 \sin ^2\left(c+d \sqrt{x}\right) x^4}{4 \left(b+a \sin \left(c+d \sqrt{x}\right)\right)^2}+\frac{b^2 \csc \left(\frac{c}{2}\right) \csc \left(\frac{c}{2}+\frac{d \sqrt{x}}{2}\right) \left(a+b \csc \left(c+d \sqrt{x}\right)\right)^2 \sin ^2\left(c+d \sqrt{x}\right) \sin \left(\frac{d \sqrt{x}}{2}\right) x^{7/2}}{d \left(b+a \sin \left(c+d \sqrt{x}\right)\right)^2}+\frac{b^2 \left(a+b \csc \left(c+d \sqrt{x}\right)\right)^2 \sec \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}+\frac{d \sqrt{x}}{2}\right) \sin ^2\left(c+d \sqrt{x}\right) \sin \left(\frac{d \sqrt{x}}{2}\right) x^{7/2}}{d \left(b+a \sin \left(c+d \sqrt{x}\right)\right)^2}-\frac{7 b^2 e^{i c} \csc (c) \left(a+b \csc \left(c+d \sqrt{x}\right)\right)^2 \left(\frac{2}{7} e^{-2 i c} x^{7/2}+\frac{i \left(1-e^{-2 i c}\right) \log \left(1-e^{-i \left(c+d \sqrt{x}\right)}\right) x^3}{d}+\frac{i \left(1-e^{-2 i c}\right) \log \left(1+e^{-i \left(c+d \sqrt{x}\right)}\right) x^3}{d}-\frac{6 e^{-2 i c} \left(-1+e^{2 i c}\right) \left(d^5 x^{5/2} \text{Li}_2\left(-e^{-i \left(c+d \sqrt{x}\right)}\right)-5 i \left(x^2 \text{Li}_3\left(-e^{-i \left(c+d \sqrt{x}\right)}\right) d^4+4 \left(-i x^{3/2} \text{Li}_4\left(-e^{-i \left(c+d \sqrt{x}\right)}\right) d^3-3 x \text{Li}_5\left(-e^{-i \left(c+d \sqrt{x}\right)}\right) d^2+6 i \sqrt{x} \text{Li}_6\left(-e^{-i \left(c+d \sqrt{x}\right)}\right) d+6 \text{Li}_7\left(-e^{-i \left(c+d \sqrt{x}\right)}\right)\right)\right)\right)}{d^7}-\frac{6 e^{-2 i c} \left(-1+e^{2 i c}\right) \left(d^5 x^{5/2} \text{Li}_2\left(e^{-i \left(c+d \sqrt{x}\right)}\right)-5 i \left(x^2 \text{Li}_3\left(e^{-i \left(c+d \sqrt{x}\right)}\right) d^4+4 \left(-i x^{3/2} \text{Li}_4\left(e^{-i \left(c+d \sqrt{x}\right)}\right) d^3-3 x \text{Li}_5\left(e^{-i \left(c+d \sqrt{x}\right)}\right) d^2+6 i \sqrt{x} \text{Li}_6\left(e^{-i \left(c+d \sqrt{x}\right)}\right) d+6 \text{Li}_7\left(e^{-i \left(c+d \sqrt{x}\right)}\right)\right)\right)\right)}{d^7}\right) \sin ^2\left(c+d \sqrt{x}\right)}{d \left(b+a \sin \left(c+d \sqrt{x}\right)\right)^2}+\frac{4 a b \left(a+b \csc \left(c+d \sqrt{x}\right)\right)^2 \left(x^{7/2} \log \left(1-e^{i \left(c+d \sqrt{x}\right)}\right) d^7-x^{7/2} \log \left(1+e^{i \left(c+d \sqrt{x}\right)}\right) d^7+7 i x^3 \text{Li}_2\left(-e^{i \left(c+d \sqrt{x}\right)}\right) d^6-7 i x^3 \text{Li}_2\left(e^{i \left(c+d \sqrt{x}\right)}\right) d^6-42 x^{5/2} \text{Li}_3\left(-e^{i \left(c+d \sqrt{x}\right)}\right) d^5+42 x^{5/2} \text{Li}_3\left(e^{i \left(c+d \sqrt{x}\right)}\right) d^5-210 i x^2 \text{Li}_4\left(-e^{i \left(c+d \sqrt{x}\right)}\right) d^4+210 i x^2 \text{Li}_4\left(e^{i \left(c+d \sqrt{x}\right)}\right) d^4+840 x^{3/2} \text{Li}_5\left(-e^{i \left(c+d \sqrt{x}\right)}\right) d^3-840 x^{3/2} \text{Li}_5\left(e^{i \left(c+d \sqrt{x}\right)}\right) d^3+2520 i x \text{Li}_6\left(-e^{i \left(c+d \sqrt{x}\right)}\right) d^2-2520 i x \text{Li}_6\left(e^{i \left(c+d \sqrt{x}\right)}\right) d^2-5040 \sqrt{x} \text{Li}_7\left(-e^{i \left(c+d \sqrt{x}\right)}\right) d+5040 \sqrt{x} \text{Li}_7\left(e^{i \left(c+d \sqrt{x}\right)}\right) d-5040 i \text{Li}_8\left(-e^{i \left(c+d \sqrt{x}\right)}\right)+5040 i \text{Li}_8\left(e^{i \left(c+d \sqrt{x}\right)}\right)\right) \sin ^2\left(c+d \sqrt{x}\right)}{d^8 \left(b+a \sin \left(c+d \sqrt{x}\right)\right)^2}","\frac{a^2 x^4}{4}-\frac{20160 i a b \text{Li}_8\left(-e^{i \left(c+d \sqrt{x}\right)}\right)}{d^8}+\frac{20160 i a b \text{Li}_8\left(e^{i \left(c+d \sqrt{x}\right)}\right)}{d^8}-\frac{20160 a b \sqrt{x} \text{Li}_7\left(-e^{i \left(c+d \sqrt{x}\right)}\right)}{d^7}+\frac{20160 a b \sqrt{x} \text{Li}_7\left(e^{i \left(c+d \sqrt{x}\right)}\right)}{d^7}+\frac{10080 i a b x \text{Li}_6\left(-e^{i \left(c+d \sqrt{x}\right)}\right)}{d^6}-\frac{10080 i a b x \text{Li}_6\left(e^{i \left(c+d \sqrt{x}\right)}\right)}{d^6}+\frac{3360 a b x^{3/2} \text{Li}_5\left(-e^{i \left(c+d \sqrt{x}\right)}\right)}{d^5}-\frac{3360 a b x^{3/2} \text{Li}_5\left(e^{i \left(c+d \sqrt{x}\right)}\right)}{d^5}-\frac{840 i a b x^2 \text{Li}_4\left(-e^{i \left(c+d \sqrt{x}\right)}\right)}{d^4}+\frac{840 i a b x^2 \text{Li}_4\left(e^{i \left(c+d \sqrt{x}\right)}\right)}{d^4}-\frac{168 a b x^{5/2} \text{Li}_3\left(-e^{i \left(c+d \sqrt{x}\right)}\right)}{d^3}+\frac{168 a b x^{5/2} \text{Li}_3\left(e^{i \left(c+d \sqrt{x}\right)}\right)}{d^3}+\frac{28 i a b x^3 \text{Li}_2\left(-e^{i \left(c+d \sqrt{x}\right)}\right)}{d^2}-\frac{28 i a b x^3 \text{Li}_2\left(e^{i \left(c+d \sqrt{x}\right)}\right)}{d^2}-\frac{8 a b x^{7/2} \tanh ^{-1}\left(e^{i \left(c+d \sqrt{x}\right)}\right)}{d}+\frac{315 b^2 \text{Li}_7\left(e^{2 i \left(c+d \sqrt{x}\right)}\right)}{2 d^8}-\frac{315 i b^2 \sqrt{x} \text{Li}_6\left(e^{2 i \left(c+d \sqrt{x}\right)}\right)}{d^7}-\frac{315 b^2 x \text{Li}_5\left(e^{2 i \left(c+d \sqrt{x}\right)}\right)}{d^6}+\frac{210 i b^2 x^{3/2} \text{Li}_4\left(e^{2 i \left(c+d \sqrt{x}\right)}\right)}{d^5}+\frac{105 b^2 x^2 \text{Li}_3\left(e^{2 i \left(c+d \sqrt{x}\right)}\right)}{d^4}-\frac{42 i b^2 x^{5/2} \text{Li}_2\left(e^{2 i \left(c+d \sqrt{x}\right)}\right)}{d^3}+\frac{14 b^2 x^3 \log \left(1-e^{2 i \left(c+d \sqrt{x}\right)}\right)}{d^2}-\frac{2 b^2 x^{7/2} \cot \left(c+d \sqrt{x}\right)}{d}-\frac{2 i b^2 x^{7/2}}{d}",1,"(a^2*x^4*(a + b*Csc[c + d*Sqrt[x]])^2*Sin[c + d*Sqrt[x]]^2)/(4*(b + a*Sin[c + d*Sqrt[x]])^2) - (7*b^2*E^(I*c)*Csc[c]*(a + b*Csc[c + d*Sqrt[x]])^2*((2*x^(7/2))/(7*E^((2*I)*c)) + (I*(1 - E^((-2*I)*c))*x^3*Log[1 - E^((-I)*(c + d*Sqrt[x]))])/d + (I*(1 - E^((-2*I)*c))*x^3*Log[1 + E^((-I)*(c + d*Sqrt[x]))])/d - (6*(-1 + E^((2*I)*c))*(d^5*x^(5/2)*PolyLog[2, -E^((-I)*(c + d*Sqrt[x]))] - (5*I)*(d^4*x^2*PolyLog[3, -E^((-I)*(c + d*Sqrt[x]))] + 4*((-I)*d^3*x^(3/2)*PolyLog[4, -E^((-I)*(c + d*Sqrt[x]))] - 3*d^2*x*PolyLog[5, -E^((-I)*(c + d*Sqrt[x]))] + (6*I)*d*Sqrt[x]*PolyLog[6, -E^((-I)*(c + d*Sqrt[x]))] + 6*PolyLog[7, -E^((-I)*(c + d*Sqrt[x]))]))))/(d^7*E^((2*I)*c)) - (6*(-1 + E^((2*I)*c))*(d^5*x^(5/2)*PolyLog[2, E^((-I)*(c + d*Sqrt[x]))] - (5*I)*(d^4*x^2*PolyLog[3, E^((-I)*(c + d*Sqrt[x]))] + 4*((-I)*d^3*x^(3/2)*PolyLog[4, E^((-I)*(c + d*Sqrt[x]))] - 3*d^2*x*PolyLog[5, E^((-I)*(c + d*Sqrt[x]))] + (6*I)*d*Sqrt[x]*PolyLog[6, E^((-I)*(c + d*Sqrt[x]))] + 6*PolyLog[7, E^((-I)*(c + d*Sqrt[x]))]))))/(d^7*E^((2*I)*c)))*Sin[c + d*Sqrt[x]]^2)/(d*(b + a*Sin[c + d*Sqrt[x]])^2) + (4*a*b*(a + b*Csc[c + d*Sqrt[x]])^2*(d^7*x^(7/2)*Log[1 - E^(I*(c + d*Sqrt[x]))] - d^7*x^(7/2)*Log[1 + E^(I*(c + d*Sqrt[x]))] + (7*I)*d^6*x^3*PolyLog[2, -E^(I*(c + d*Sqrt[x]))] - (7*I)*d^6*x^3*PolyLog[2, E^(I*(c + d*Sqrt[x]))] - 42*d^5*x^(5/2)*PolyLog[3, -E^(I*(c + d*Sqrt[x]))] + 42*d^5*x^(5/2)*PolyLog[3, E^(I*(c + d*Sqrt[x]))] - (210*I)*d^4*x^2*PolyLog[4, -E^(I*(c + d*Sqrt[x]))] + (210*I)*d^4*x^2*PolyLog[4, E^(I*(c + d*Sqrt[x]))] + 840*d^3*x^(3/2)*PolyLog[5, -E^(I*(c + d*Sqrt[x]))] - 840*d^3*x^(3/2)*PolyLog[5, E^(I*(c + d*Sqrt[x]))] + (2520*I)*d^2*x*PolyLog[6, -E^(I*(c + d*Sqrt[x]))] - (2520*I)*d^2*x*PolyLog[6, E^(I*(c + d*Sqrt[x]))] - 5040*d*Sqrt[x]*PolyLog[7, -E^(I*(c + d*Sqrt[x]))] + 5040*d*Sqrt[x]*PolyLog[7, E^(I*(c + d*Sqrt[x]))] - (5040*I)*PolyLog[8, -E^(I*(c + d*Sqrt[x]))] + (5040*I)*PolyLog[8, E^(I*(c + d*Sqrt[x]))])*Sin[c + d*Sqrt[x]]^2)/(d^8*(b + a*Sin[c + d*Sqrt[x]])^2) + (b^2*x^(7/2)*Csc[c/2]*Csc[c/2 + (d*Sqrt[x])/2]*(a + b*Csc[c + d*Sqrt[x]])^2*Sin[c + d*Sqrt[x]]^2*Sin[(d*Sqrt[x])/2])/(d*(b + a*Sin[c + d*Sqrt[x]])^2) + (b^2*x^(7/2)*(a + b*Csc[c + d*Sqrt[x]])^2*Sec[c/2]*Sec[c/2 + (d*Sqrt[x])/2]*Sin[c + d*Sqrt[x]]^2*Sin[(d*Sqrt[x])/2])/(d*(b + a*Sin[c + d*Sqrt[x]])^2)","A",1
37,1,779,513,14.1022944,"\int x^2 \left(a+b \csc \left(c+d \sqrt{x}\right)\right)^2 \, dx","Integrate[x^2*(a + b*Csc[c + d*Sqrt[x]])^2,x]","\frac{a^2 x^3 \sin ^2\left(c+d \sqrt{x}\right) \left(a+b \csc \left(c+d \sqrt{x}\right)\right)^2}{3 \left(a \sin \left(c+d \sqrt{x}\right)+b\right)^2}+\frac{b^2 x^{5/2} \csc \left(\frac{c}{2}\right) \sin \left(\frac{d \sqrt{x}}{2}\right) \sin ^2\left(c+d \sqrt{x}\right) \csc \left(\frac{c}{2}+\frac{d \sqrt{x}}{2}\right) \left(a+b \csc \left(c+d \sqrt{x}\right)\right)^2}{d \left(a \sin \left(c+d \sqrt{x}\right)+b\right)^2}+\frac{b^2 x^{5/2} \sec \left(\frac{c}{2}\right) \sin \left(\frac{d \sqrt{x}}{2}\right) \sin ^2\left(c+d \sqrt{x}\right) \sec \left(\frac{c}{2}+\frac{d \sqrt{x}}{2}\right) \left(a+b \csc \left(c+d \sqrt{x}\right)\right)^2}{d \left(a \sin \left(c+d \sqrt{x}\right)+b\right)^2}-\frac{i b \sin ^2\left(c+d \sqrt{x}\right) \left(a+b \csc \left(c+d \sqrt{x}\right)\right)^2 \left(i \left(4 a d^5 x^{5/2} \log \left(1-e^{i \left(c+d \sqrt{x}\right)}\right)-4 a d^5 x^{5/2} \log \left(1+e^{i \left(c+d \sqrt{x}\right)}\right)-80 a d^3 x^{3/2} \text{Li}_3\left(-e^{i \left(c+d \sqrt{x}\right)}\right)+80 a d^3 x^{3/2} \text{Li}_3\left(e^{i \left(c+d \sqrt{x}\right)}\right)-240 i a d^2 x \text{Li}_4\left(-e^{i \left(c+d \sqrt{x}\right)}\right)+240 i a d^2 x \text{Li}_4\left(e^{i \left(c+d \sqrt{x}\right)}\right)+480 a d \sqrt{x} \text{Li}_5\left(-e^{i \left(c+d \sqrt{x}\right)}\right)-480 a d \sqrt{x} \text{Li}_5\left(e^{i \left(c+d \sqrt{x}\right)}\right)+480 i a \text{Li}_6\left(-e^{i \left(c+d \sqrt{x}\right)}\right)-480 i a \text{Li}_6\left(e^{i \left(c+d \sqrt{x}\right)}\right)+10 b d^4 x^2 \log \left(1-e^{2 i \left(c+d \sqrt{x}\right)}\right)-20 i b d^3 x^{3/2} \text{Li}_2\left(e^{2 i \left(c+d \sqrt{x}\right)}\right)+30 b d^2 x \text{Li}_3\left(e^{2 i \left(c+d \sqrt{x}\right)}\right)+30 i b d \sqrt{x} \text{Li}_4\left(e^{2 i \left(c+d \sqrt{x}\right)}\right)-15 b \text{Li}_5\left(e^{2 i \left(c+d \sqrt{x}\right)}\right)\right)-20 a d^4 x^2 \text{Li}_2\left(-e^{i \left(c+d \sqrt{x}\right)}\right)+20 a d^4 x^2 \text{Li}_2\left(e^{i \left(c+d \sqrt{x}\right)}\right)+\frac{4 b e^{2 i c} d^5 x^{5/2}}{-1+e^{2 i c}}\right)}{d^6 \left(a \sin \left(c+d \sqrt{x}\right)+b\right)^2}","\frac{a^2 x^3}{3}+\frac{480 i a b \text{Li}_6\left(-e^{i \left(c+d \sqrt{x}\right)}\right)}{d^6}-\frac{480 i a b \text{Li}_6\left(e^{i \left(c+d \sqrt{x}\right)}\right)}{d^6}+\frac{480 a b \sqrt{x} \text{Li}_5\left(-e^{i \left(c+d \sqrt{x}\right)}\right)}{d^5}-\frac{480 a b \sqrt{x} \text{Li}_5\left(e^{i \left(c+d \sqrt{x}\right)}\right)}{d^5}-\frac{240 i a b x \text{Li}_4\left(-e^{i \left(c+d \sqrt{x}\right)}\right)}{d^4}+\frac{240 i a b x \text{Li}_4\left(e^{i \left(c+d \sqrt{x}\right)}\right)}{d^4}-\frac{80 a b x^{3/2} \text{Li}_3\left(-e^{i \left(c+d \sqrt{x}\right)}\right)}{d^3}+\frac{80 a b x^{3/2} \text{Li}_3\left(e^{i \left(c+d \sqrt{x}\right)}\right)}{d^3}+\frac{20 i a b x^2 \text{Li}_2\left(-e^{i \left(c+d \sqrt{x}\right)}\right)}{d^2}-\frac{20 i a b x^2 \text{Li}_2\left(e^{i \left(c+d \sqrt{x}\right)}\right)}{d^2}-\frac{8 a b x^{5/2} \tanh ^{-1}\left(e^{i \left(c+d \sqrt{x}\right)}\right)}{d}-\frac{15 b^2 \text{Li}_5\left(e^{2 i \left(c+d \sqrt{x}\right)}\right)}{d^6}+\frac{30 i b^2 \sqrt{x} \text{Li}_4\left(e^{2 i \left(c+d \sqrt{x}\right)}\right)}{d^5}+\frac{30 b^2 x \text{Li}_3\left(e^{2 i \left(c+d \sqrt{x}\right)}\right)}{d^4}-\frac{20 i b^2 x^{3/2} \text{Li}_2\left(e^{2 i \left(c+d \sqrt{x}\right)}\right)}{d^3}+\frac{10 b^2 x^2 \log \left(1-e^{2 i \left(c+d \sqrt{x}\right)}\right)}{d^2}-\frac{2 b^2 x^{5/2} \cot \left(c+d \sqrt{x}\right)}{d}-\frac{2 i b^2 x^{5/2}}{d}",1,"(a^2*x^3*(a + b*Csc[c + d*Sqrt[x]])^2*Sin[c + d*Sqrt[x]]^2)/(3*(b + a*Sin[c + d*Sqrt[x]])^2) - (I*b*(a + b*Csc[c + d*Sqrt[x]])^2*((4*b*d^5*E^((2*I)*c)*x^(5/2))/(-1 + E^((2*I)*c)) - 20*a*d^4*x^2*PolyLog[2, -E^(I*(c + d*Sqrt[x]))] + 20*a*d^4*x^2*PolyLog[2, E^(I*(c + d*Sqrt[x]))] + I*(4*a*d^5*x^(5/2)*Log[1 - E^(I*(c + d*Sqrt[x]))] - 4*a*d^5*x^(5/2)*Log[1 + E^(I*(c + d*Sqrt[x]))] + 10*b*d^4*x^2*Log[1 - E^((2*I)*(c + d*Sqrt[x]))] - (20*I)*b*d^3*x^(3/2)*PolyLog[2, E^((2*I)*(c + d*Sqrt[x]))] - 80*a*d^3*x^(3/2)*PolyLog[3, -E^(I*(c + d*Sqrt[x]))] + 80*a*d^3*x^(3/2)*PolyLog[3, E^(I*(c + d*Sqrt[x]))] + 30*b*d^2*x*PolyLog[3, E^((2*I)*(c + d*Sqrt[x]))] - (240*I)*a*d^2*x*PolyLog[4, -E^(I*(c + d*Sqrt[x]))] + (240*I)*a*d^2*x*PolyLog[4, E^(I*(c + d*Sqrt[x]))] + (30*I)*b*d*Sqrt[x]*PolyLog[4, E^((2*I)*(c + d*Sqrt[x]))] + 480*a*d*Sqrt[x]*PolyLog[5, -E^(I*(c + d*Sqrt[x]))] - 480*a*d*Sqrt[x]*PolyLog[5, E^(I*(c + d*Sqrt[x]))] - 15*b*PolyLog[5, E^((2*I)*(c + d*Sqrt[x]))] + (480*I)*a*PolyLog[6, -E^(I*(c + d*Sqrt[x]))] - (480*I)*a*PolyLog[6, E^(I*(c + d*Sqrt[x]))]))*Sin[c + d*Sqrt[x]]^2)/(d^6*(b + a*Sin[c + d*Sqrt[x]])^2) + (b^2*x^(5/2)*Csc[c/2]*Csc[c/2 + (d*Sqrt[x])/2]*(a + b*Csc[c + d*Sqrt[x]])^2*Sin[c + d*Sqrt[x]]^2*Sin[(d*Sqrt[x])/2])/(d*(b + a*Sin[c + d*Sqrt[x]])^2) + (b^2*x^(5/2)*(a + b*Csc[c + d*Sqrt[x]])^2*Sec[c/2]*Sec[c/2 + (d*Sqrt[x])/2]*Sin[c + d*Sqrt[x]]^2*Sin[(d*Sqrt[x])/2])/(d*(b + a*Sin[c + d*Sqrt[x]])^2)","A",1
38,1,449,333,7.6591999,"\int x \left(a+b \csc \left(c+d \sqrt{x}\right)\right)^2 \, dx","Integrate[x*(a + b*Csc[c + d*Sqrt[x]])^2,x]","\frac{a^2 x^2}{2}-\frac{2 i b \left(\left(6 b d \sqrt{x}-6 a d^2 x\right) \text{Li}_2\left(-e^{i \left(c+d \sqrt{x}\right)}\right)+i \left(-6 i \left(a d^2 x+b d \sqrt{x}\right) \text{Li}_2\left(e^{i \left(c+d \sqrt{x}\right)}\right)+6 \left(b-2 a d \sqrt{x}\right) \text{Li}_3\left(-e^{i \left(c+d \sqrt{x}\right)}\right)+2 a d^3 x^{3/2} \log \left(1-e^{i \left(c+d \sqrt{x}\right)}\right)-2 a d^3 x^{3/2} \log \left(1+e^{i \left(c+d \sqrt{x}\right)}\right)+12 a d \sqrt{x} \text{Li}_3\left(e^{i \left(c+d \sqrt{x}\right)}\right)-12 i a \text{Li}_4\left(-e^{i \left(c+d \sqrt{x}\right)}\right)+12 i a \text{Li}_4\left(e^{i \left(c+d \sqrt{x}\right)}\right)+3 b d^2 x \log \left(1-e^{i \left(c+d \sqrt{x}\right)}\right)+3 b d^2 x \log \left(1+e^{i \left(c+d \sqrt{x}\right)}\right)+6 b \text{Li}_3\left(e^{i \left(c+d \sqrt{x}\right)}\right)\right)+\frac{2 b e^{2 i c} d^3 x^{3/2}}{-1+e^{2 i c}}\right)}{d^4}+\frac{b^2 x^{3/2} \csc \left(\frac{c}{2}\right) \sin \left(\frac{d \sqrt{x}}{2}\right) \csc \left(\frac{1}{2} \left(c+d \sqrt{x}\right)\right)}{d}+\frac{b^2 x^{3/2} \sec \left(\frac{c}{2}\right) \sin \left(\frac{d \sqrt{x}}{2}\right) \sec \left(\frac{1}{2} \left(c+d \sqrt{x}\right)\right)}{d}","\frac{a^2 x^2}{2}-\frac{24 i a b \text{Li}_4\left(-e^{i \left(c+d \sqrt{x}\right)}\right)}{d^4}+\frac{24 i a b \text{Li}_4\left(e^{i \left(c+d \sqrt{x}\right)}\right)}{d^4}-\frac{24 a b \sqrt{x} \text{Li}_3\left(-e^{i \left(c+d \sqrt{x}\right)}\right)}{d^3}+\frac{24 a b \sqrt{x} \text{Li}_3\left(e^{i \left(c+d \sqrt{x}\right)}\right)}{d^3}+\frac{12 i a b x \text{Li}_2\left(-e^{i \left(c+d \sqrt{x}\right)}\right)}{d^2}-\frac{12 i a b x \text{Li}_2\left(e^{i \left(c+d \sqrt{x}\right)}\right)}{d^2}-\frac{8 a b x^{3/2} \tanh ^{-1}\left(e^{i \left(c+d \sqrt{x}\right)}\right)}{d}+\frac{3 b^2 \text{Li}_3\left(e^{2 i \left(c+d \sqrt{x}\right)}\right)}{d^4}-\frac{6 i b^2 \sqrt{x} \text{Li}_2\left(e^{2 i \left(c+d \sqrt{x}\right)}\right)}{d^3}+\frac{6 b^2 x \log \left(1-e^{2 i \left(c+d \sqrt{x}\right)}\right)}{d^2}-\frac{2 b^2 x^{3/2} \cot \left(c+d \sqrt{x}\right)}{d}-\frac{2 i b^2 x^{3/2}}{d}",1,"(a^2*x^2)/2 - ((2*I)*b*((2*b*d^3*E^((2*I)*c)*x^(3/2))/(-1 + E^((2*I)*c)) + (6*b*d*Sqrt[x] - 6*a*d^2*x)*PolyLog[2, -E^(I*(c + d*Sqrt[x]))] + I*(3*b*d^2*x*Log[1 - E^(I*(c + d*Sqrt[x]))] + 2*a*d^3*x^(3/2)*Log[1 - E^(I*(c + d*Sqrt[x]))] + 3*b*d^2*x*Log[1 + E^(I*(c + d*Sqrt[x]))] - 2*a*d^3*x^(3/2)*Log[1 + E^(I*(c + d*Sqrt[x]))] - (6*I)*(b*d*Sqrt[x] + a*d^2*x)*PolyLog[2, E^(I*(c + d*Sqrt[x]))] + 6*(b - 2*a*d*Sqrt[x])*PolyLog[3, -E^(I*(c + d*Sqrt[x]))] + 6*b*PolyLog[3, E^(I*(c + d*Sqrt[x]))] + 12*a*d*Sqrt[x]*PolyLog[3, E^(I*(c + d*Sqrt[x]))] - (12*I)*a*PolyLog[4, -E^(I*(c + d*Sqrt[x]))] + (12*I)*a*PolyLog[4, E^(I*(c + d*Sqrt[x]))])))/d^4 + (b^2*x^(3/2)*Csc[c/2]*Csc[(c + d*Sqrt[x])/2]*Sin[(d*Sqrt[x])/2])/d + (b^2*x^(3/2)*Sec[c/2]*Sec[(c + d*Sqrt[x])/2]*Sin[(d*Sqrt[x])/2])/d","A",1
39,0,0,23,39.9076087,"\int \frac{\left(a+b \csc \left(c+d \sqrt{x}\right)\right)^2}{x} \, dx","Integrate[(a + b*Csc[c + d*Sqrt[x]])^2/x,x]","\int \frac{\left(a+b \csc \left(c+d \sqrt{x}\right)\right)^2}{x} \, dx","\text{Int}\left(\frac{\left(a+b \csc \left(c+d \sqrt{x}\right)\right)^2}{x},x\right)",0,"Integrate[(a + b*Csc[c + d*Sqrt[x]])^2/x, x]","A",-1
40,0,0,23,21.0494431,"\int \frac{\left(a+b \csc \left(c+d \sqrt{x}\right)\right)^2}{x^2} \, dx","Integrate[(a + b*Csc[c + d*Sqrt[x]])^2/x^2,x]","\int \frac{\left(a+b \csc \left(c+d \sqrt{x}\right)\right)^2}{x^2} \, dx","\text{Int}\left(\frac{\left(a+b \csc \left(c+d \sqrt{x}\right)\right)^2}{x^2},x\right)",0,"Integrate[(a + b*Csc[c + d*Sqrt[x]])^2/x^2, x]","A",-1
41,1,1176,1075,2.3065805,"\int \frac{x^3}{a+b \csc \left(c+d \sqrt{x}\right)} \, dx","Integrate[x^3/(a + b*Csc[c + d*Sqrt[x]]),x]","\frac{\csc \left(c+d \sqrt{x}\right) \left(x^4-\frac{8 b e^{i c} \left(x^{7/2} \log \left(\frac{e^{i \left(2 c+d \sqrt{x}\right)} a}{i b e^{i c}-\sqrt{\left(a^2-b^2\right) e^{2 i c}}}+1\right) d^7-x^{7/2} \log \left(\frac{e^{i \left(2 c+d \sqrt{x}\right)} a}{i e^{i c} b+\sqrt{\left(a^2-b^2\right) e^{2 i c}}}+1\right) d^7-7 i x^3 \text{Li}_2\left(\frac{i a e^{i \left(2 c+d \sqrt{x}\right)}}{e^{i c} b+i \sqrt{\left(a^2-b^2\right) e^{2 i c}}}\right) d^6+7 i x^3 \text{Li}_2\left(-\frac{a e^{i \left(2 c+d \sqrt{x}\right)}}{i e^{i c} b+\sqrt{\left(a^2-b^2\right) e^{2 i c}}}\right) d^6+42 x^{5/2} \text{Li}_3\left(\frac{i a e^{i \left(2 c+d \sqrt{x}\right)}}{e^{i c} b+i \sqrt{\left(a^2-b^2\right) e^{2 i c}}}\right) d^5-42 x^{5/2} \text{Li}_3\left(-\frac{a e^{i \left(2 c+d \sqrt{x}\right)}}{i e^{i c} b+\sqrt{\left(a^2-b^2\right) e^{2 i c}}}\right) d^5+210 i x^2 \text{Li}_4\left(\frac{i a e^{i \left(2 c+d \sqrt{x}\right)}}{e^{i c} b+i \sqrt{\left(a^2-b^2\right) e^{2 i c}}}\right) d^4-210 i x^2 \text{Li}_4\left(-\frac{a e^{i \left(2 c+d \sqrt{x}\right)}}{i e^{i c} b+\sqrt{\left(a^2-b^2\right) e^{2 i c}}}\right) d^4-840 x^{3/2} \text{Li}_5\left(\frac{i a e^{i \left(2 c+d \sqrt{x}\right)}}{e^{i c} b+i \sqrt{\left(a^2-b^2\right) e^{2 i c}}}\right) d^3+840 x^{3/2} \text{Li}_5\left(-\frac{a e^{i \left(2 c+d \sqrt{x}\right)}}{i e^{i c} b+\sqrt{\left(a^2-b^2\right) e^{2 i c}}}\right) d^3-2520 i x \text{Li}_6\left(\frac{i a e^{i \left(2 c+d \sqrt{x}\right)}}{e^{i c} b+i \sqrt{\left(a^2-b^2\right) e^{2 i c}}}\right) d^2+2520 i x \text{Li}_6\left(-\frac{a e^{i \left(2 c+d \sqrt{x}\right)}}{i e^{i c} b+\sqrt{\left(a^2-b^2\right) e^{2 i c}}}\right) d^2+5040 \sqrt{x} \text{Li}_7\left(\frac{i a e^{i \left(2 c+d \sqrt{x}\right)}}{e^{i c} b+i \sqrt{\left(a^2-b^2\right) e^{2 i c}}}\right) d-5040 \sqrt{x} \text{Li}_7\left(-\frac{a e^{i \left(2 c+d \sqrt{x}\right)}}{i e^{i c} b+\sqrt{\left(a^2-b^2\right) e^{2 i c}}}\right) d+5040 i \text{Li}_8\left(\frac{i a e^{i \left(2 c+d \sqrt{x}\right)}}{e^{i c} b+i \sqrt{\left(a^2-b^2\right) e^{2 i c}}}\right)-5040 i \text{Li}_8\left(-\frac{a e^{i \left(2 c+d \sqrt{x}\right)}}{i e^{i c} b+\sqrt{\left(a^2-b^2\right) e^{2 i c}}}\right)\right)}{d^8 \sqrt{\left(a^2-b^2\right) e^{2 i c}}}\right) \left(b+a \sin \left(c+d \sqrt{x}\right)\right)}{4 a \left(a+b \csc \left(c+d \sqrt{x}\right)\right)}","\frac{x^4}{4 a}+\frac{2 i b \log \left(1-\frac{i a e^{i \left(c+d \sqrt{x}\right)}}{b-\sqrt{b^2-a^2}}\right) x^{7/2}}{a \sqrt{b^2-a^2} d}-\frac{2 i b \log \left(1-\frac{i a e^{i \left(c+d \sqrt{x}\right)}}{b+\sqrt{b^2-a^2}}\right) x^{7/2}}{a \sqrt{b^2-a^2} d}+\frac{14 b \text{Li}_2\left(\frac{i a e^{i \left(c+d \sqrt{x}\right)}}{b-\sqrt{b^2-a^2}}\right) x^3}{a \sqrt{b^2-a^2} d^2}-\frac{14 b \text{Li}_2\left(\frac{i a e^{i \left(c+d \sqrt{x}\right)}}{b+\sqrt{b^2-a^2}}\right) x^3}{a \sqrt{b^2-a^2} d^2}+\frac{84 i b \text{Li}_3\left(\frac{i a e^{i \left(c+d \sqrt{x}\right)}}{b-\sqrt{b^2-a^2}}\right) x^{5/2}}{a \sqrt{b^2-a^2} d^3}-\frac{84 i b \text{Li}_3\left(\frac{i a e^{i \left(c+d \sqrt{x}\right)}}{b+\sqrt{b^2-a^2}}\right) x^{5/2}}{a \sqrt{b^2-a^2} d^3}-\frac{420 b \text{Li}_4\left(\frac{i a e^{i \left(c+d \sqrt{x}\right)}}{b-\sqrt{b^2-a^2}}\right) x^2}{a \sqrt{b^2-a^2} d^4}+\frac{420 b \text{Li}_4\left(\frac{i a e^{i \left(c+d \sqrt{x}\right)}}{b+\sqrt{b^2-a^2}}\right) x^2}{a \sqrt{b^2-a^2} d^4}-\frac{1680 i b \text{Li}_5\left(\frac{i a e^{i \left(c+d \sqrt{x}\right)}}{b-\sqrt{b^2-a^2}}\right) x^{3/2}}{a \sqrt{b^2-a^2} d^5}+\frac{1680 i b \text{Li}_5\left(\frac{i a e^{i \left(c+d \sqrt{x}\right)}}{b+\sqrt{b^2-a^2}}\right) x^{3/2}}{a \sqrt{b^2-a^2} d^5}+\frac{5040 b \text{Li}_6\left(\frac{i a e^{i \left(c+d \sqrt{x}\right)}}{b-\sqrt{b^2-a^2}}\right) x}{a \sqrt{b^2-a^2} d^6}-\frac{5040 b \text{Li}_6\left(\frac{i a e^{i \left(c+d \sqrt{x}\right)}}{b+\sqrt{b^2-a^2}}\right) x}{a \sqrt{b^2-a^2} d^6}+\frac{10080 i b \text{Li}_7\left(\frac{i a e^{i \left(c+d \sqrt{x}\right)}}{b-\sqrt{b^2-a^2}}\right) \sqrt{x}}{a \sqrt{b^2-a^2} d^7}-\frac{10080 i b \text{Li}_7\left(\frac{i a e^{i \left(c+d \sqrt{x}\right)}}{b+\sqrt{b^2-a^2}}\right) \sqrt{x}}{a \sqrt{b^2-a^2} d^7}-\frac{10080 b \text{Li}_8\left(\frac{i a e^{i \left(c+d \sqrt{x}\right)}}{b-\sqrt{b^2-a^2}}\right)}{a \sqrt{b^2-a^2} d^8}+\frac{10080 b \text{Li}_8\left(\frac{i a e^{i \left(c+d \sqrt{x}\right)}}{b+\sqrt{b^2-a^2}}\right)}{a \sqrt{b^2-a^2} d^8}",1,"(Csc[c + d*Sqrt[x]]*(x^4 - (8*b*E^(I*c)*(d^7*x^(7/2)*Log[1 + (a*E^(I*(2*c + d*Sqrt[x])))/(I*b*E^(I*c) - Sqrt[(a^2 - b^2)*E^((2*I)*c)])] - d^7*x^(7/2)*Log[1 + (a*E^(I*(2*c + d*Sqrt[x])))/(I*b*E^(I*c) + Sqrt[(a^2 - b^2)*E^((2*I)*c)])] - (7*I)*d^6*x^3*PolyLog[2, (I*a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) + I*Sqrt[(a^2 - b^2)*E^((2*I)*c)])] + (7*I)*d^6*x^3*PolyLog[2, -((a*E^(I*(2*c + d*Sqrt[x])))/(I*b*E^(I*c) + Sqrt[(a^2 - b^2)*E^((2*I)*c)]))] + 42*d^5*x^(5/2)*PolyLog[3, (I*a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) + I*Sqrt[(a^2 - b^2)*E^((2*I)*c)])] - 42*d^5*x^(5/2)*PolyLog[3, -((a*E^(I*(2*c + d*Sqrt[x])))/(I*b*E^(I*c) + Sqrt[(a^2 - b^2)*E^((2*I)*c)]))] + (210*I)*d^4*x^2*PolyLog[4, (I*a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) + I*Sqrt[(a^2 - b^2)*E^((2*I)*c)])] - (210*I)*d^4*x^2*PolyLog[4, -((a*E^(I*(2*c + d*Sqrt[x])))/(I*b*E^(I*c) + Sqrt[(a^2 - b^2)*E^((2*I)*c)]))] - 840*d^3*x^(3/2)*PolyLog[5, (I*a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) + I*Sqrt[(a^2 - b^2)*E^((2*I)*c)])] + 840*d^3*x^(3/2)*PolyLog[5, -((a*E^(I*(2*c + d*Sqrt[x])))/(I*b*E^(I*c) + Sqrt[(a^2 - b^2)*E^((2*I)*c)]))] - (2520*I)*d^2*x*PolyLog[6, (I*a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) + I*Sqrt[(a^2 - b^2)*E^((2*I)*c)])] + (2520*I)*d^2*x*PolyLog[6, -((a*E^(I*(2*c + d*Sqrt[x])))/(I*b*E^(I*c) + Sqrt[(a^2 - b^2)*E^((2*I)*c)]))] + 5040*d*Sqrt[x]*PolyLog[7, (I*a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) + I*Sqrt[(a^2 - b^2)*E^((2*I)*c)])] - 5040*d*Sqrt[x]*PolyLog[7, -((a*E^(I*(2*c + d*Sqrt[x])))/(I*b*E^(I*c) + Sqrt[(a^2 - b^2)*E^((2*I)*c)]))] + (5040*I)*PolyLog[8, (I*a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) + I*Sqrt[(a^2 - b^2)*E^((2*I)*c)])] - (5040*I)*PolyLog[8, -((a*E^(I*(2*c + d*Sqrt[x])))/(I*b*E^(I*c) + Sqrt[(a^2 - b^2)*E^((2*I)*c)]))]))/(d^8*Sqrt[(a^2 - b^2)*E^((2*I)*c)]))*(b + a*Sin[c + d*Sqrt[x]]))/(4*a*(a + b*Csc[c + d*Sqrt[x]]))","A",1
42,1,898,807,1.8019812,"\int \frac{x^2}{a+b \csc \left(c+d \sqrt{x}\right)} \, dx","Integrate[x^2/(a + b*Csc[c + d*Sqrt[x]]),x]","\frac{\csc \left(c+d \sqrt{x}\right) \left(x^3-\frac{6 b e^{i c} \left(x^{5/2} \log \left(\frac{e^{i \left(2 c+d \sqrt{x}\right)} a}{i b e^{i c}-\sqrt{\left(a^2-b^2\right) e^{2 i c}}}+1\right) d^5-x^{5/2} \log \left(\frac{e^{i \left(2 c+d \sqrt{x}\right)} a}{i e^{i c} b+\sqrt{\left(a^2-b^2\right) e^{2 i c}}}+1\right) d^5-5 i x^2 \text{Li}_2\left(\frac{i a e^{i \left(2 c+d \sqrt{x}\right)}}{e^{i c} b+i \sqrt{\left(a^2-b^2\right) e^{2 i c}}}\right) d^4+5 i x^2 \text{Li}_2\left(-\frac{a e^{i \left(2 c+d \sqrt{x}\right)}}{i e^{i c} b+\sqrt{\left(a^2-b^2\right) e^{2 i c}}}\right) d^4+20 x^{3/2} \text{Li}_3\left(\frac{i a e^{i \left(2 c+d \sqrt{x}\right)}}{e^{i c} b+i \sqrt{\left(a^2-b^2\right) e^{2 i c}}}\right) d^3-20 x^{3/2} \text{Li}_3\left(-\frac{a e^{i \left(2 c+d \sqrt{x}\right)}}{i e^{i c} b+\sqrt{\left(a^2-b^2\right) e^{2 i c}}}\right) d^3+60 i x \text{Li}_4\left(\frac{i a e^{i \left(2 c+d \sqrt{x}\right)}}{e^{i c} b+i \sqrt{\left(a^2-b^2\right) e^{2 i c}}}\right) d^2-60 i x \text{Li}_4\left(-\frac{a e^{i \left(2 c+d \sqrt{x}\right)}}{i e^{i c} b+\sqrt{\left(a^2-b^2\right) e^{2 i c}}}\right) d^2-120 \sqrt{x} \text{Li}_5\left(\frac{i a e^{i \left(2 c+d \sqrt{x}\right)}}{e^{i c} b+i \sqrt{\left(a^2-b^2\right) e^{2 i c}}}\right) d+120 \sqrt{x} \text{Li}_5\left(-\frac{a e^{i \left(2 c+d \sqrt{x}\right)}}{i e^{i c} b+\sqrt{\left(a^2-b^2\right) e^{2 i c}}}\right) d-120 i \text{Li}_6\left(\frac{i a e^{i \left(2 c+d \sqrt{x}\right)}}{e^{i c} b+i \sqrt{\left(a^2-b^2\right) e^{2 i c}}}\right)+120 i \text{Li}_6\left(-\frac{a e^{i \left(2 c+d \sqrt{x}\right)}}{i e^{i c} b+\sqrt{\left(a^2-b^2\right) e^{2 i c}}}\right)\right)}{d^6 \sqrt{\left(a^2-b^2\right) e^{2 i c}}}\right) \left(b+a \sin \left(c+d \sqrt{x}\right)\right)}{3 a \left(a+b \csc \left(c+d \sqrt{x}\right)\right)}","\frac{x^3}{3 a}+\frac{2 i b \log \left(1-\frac{i a e^{i \left(c+d \sqrt{x}\right)}}{b-\sqrt{b^2-a^2}}\right) x^{5/2}}{a \sqrt{b^2-a^2} d}-\frac{2 i b \log \left(1-\frac{i a e^{i \left(c+d \sqrt{x}\right)}}{b+\sqrt{b^2-a^2}}\right) x^{5/2}}{a \sqrt{b^2-a^2} d}+\frac{10 b \text{Li}_2\left(\frac{i a e^{i \left(c+d \sqrt{x}\right)}}{b-\sqrt{b^2-a^2}}\right) x^2}{a \sqrt{b^2-a^2} d^2}-\frac{10 b \text{Li}_2\left(\frac{i a e^{i \left(c+d \sqrt{x}\right)}}{b+\sqrt{b^2-a^2}}\right) x^2}{a \sqrt{b^2-a^2} d^2}+\frac{40 i b \text{Li}_3\left(\frac{i a e^{i \left(c+d \sqrt{x}\right)}}{b-\sqrt{b^2-a^2}}\right) x^{3/2}}{a \sqrt{b^2-a^2} d^3}-\frac{40 i b \text{Li}_3\left(\frac{i a e^{i \left(c+d \sqrt{x}\right)}}{b+\sqrt{b^2-a^2}}\right) x^{3/2}}{a \sqrt{b^2-a^2} d^3}-\frac{120 b \text{Li}_4\left(\frac{i a e^{i \left(c+d \sqrt{x}\right)}}{b-\sqrt{b^2-a^2}}\right) x}{a \sqrt{b^2-a^2} d^4}+\frac{120 b \text{Li}_4\left(\frac{i a e^{i \left(c+d \sqrt{x}\right)}}{b+\sqrt{b^2-a^2}}\right) x}{a \sqrt{b^2-a^2} d^4}-\frac{240 i b \text{Li}_5\left(\frac{i a e^{i \left(c+d \sqrt{x}\right)}}{b-\sqrt{b^2-a^2}}\right) \sqrt{x}}{a \sqrt{b^2-a^2} d^5}+\frac{240 i b \text{Li}_5\left(\frac{i a e^{i \left(c+d \sqrt{x}\right)}}{b+\sqrt{b^2-a^2}}\right) \sqrt{x}}{a \sqrt{b^2-a^2} d^5}+\frac{240 b \text{Li}_6\left(\frac{i a e^{i \left(c+d \sqrt{x}\right)}}{b-\sqrt{b^2-a^2}}\right)}{a \sqrt{b^2-a^2} d^6}-\frac{240 b \text{Li}_6\left(\frac{i a e^{i \left(c+d \sqrt{x}\right)}}{b+\sqrt{b^2-a^2}}\right)}{a \sqrt{b^2-a^2} d^6}",1,"(Csc[c + d*Sqrt[x]]*(x^3 - (6*b*E^(I*c)*(d^5*x^(5/2)*Log[1 + (a*E^(I*(2*c + d*Sqrt[x])))/(I*b*E^(I*c) - Sqrt[(a^2 - b^2)*E^((2*I)*c)])] - d^5*x^(5/2)*Log[1 + (a*E^(I*(2*c + d*Sqrt[x])))/(I*b*E^(I*c) + Sqrt[(a^2 - b^2)*E^((2*I)*c)])] - (5*I)*d^4*x^2*PolyLog[2, (I*a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) + I*Sqrt[(a^2 - b^2)*E^((2*I)*c)])] + (5*I)*d^4*x^2*PolyLog[2, -((a*E^(I*(2*c + d*Sqrt[x])))/(I*b*E^(I*c) + Sqrt[(a^2 - b^2)*E^((2*I)*c)]))] + 20*d^3*x^(3/2)*PolyLog[3, (I*a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) + I*Sqrt[(a^2 - b^2)*E^((2*I)*c)])] - 20*d^3*x^(3/2)*PolyLog[3, -((a*E^(I*(2*c + d*Sqrt[x])))/(I*b*E^(I*c) + Sqrt[(a^2 - b^2)*E^((2*I)*c)]))] + (60*I)*d^2*x*PolyLog[4, (I*a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) + I*Sqrt[(a^2 - b^2)*E^((2*I)*c)])] - (60*I)*d^2*x*PolyLog[4, -((a*E^(I*(2*c + d*Sqrt[x])))/(I*b*E^(I*c) + Sqrt[(a^2 - b^2)*E^((2*I)*c)]))] - 120*d*Sqrt[x]*PolyLog[5, (I*a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) + I*Sqrt[(a^2 - b^2)*E^((2*I)*c)])] + 120*d*Sqrt[x]*PolyLog[5, -((a*E^(I*(2*c + d*Sqrt[x])))/(I*b*E^(I*c) + Sqrt[(a^2 - b^2)*E^((2*I)*c)]))] - (120*I)*PolyLog[6, (I*a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) + I*Sqrt[(a^2 - b^2)*E^((2*I)*c)])] + (120*I)*PolyLog[6, -((a*E^(I*(2*c + d*Sqrt[x])))/(I*b*E^(I*c) + Sqrt[(a^2 - b^2)*E^((2*I)*c)]))]))/(d^6*Sqrt[(a^2 - b^2)*E^((2*I)*c)]))*(b + a*Sin[c + d*Sqrt[x]]))/(3*a*(a + b*Csc[c + d*Sqrt[x]]))","A",1
43,1,659,539,1.4932682,"\int \frac{x}{a+b \csc \left(c+d \sqrt{x}\right)} \, dx","Integrate[x/(a + b*Csc[c + d*Sqrt[x]]),x]","\frac{d^4 x^2 \sqrt{e^{2 i c} \left(a^2-b^2\right)}-4 b e^{i c} d^3 x^{3/2} \log \left(1+\frac{a e^{i \left(2 c+d \sqrt{x}\right)}}{i b e^{i c}-\sqrt{e^{2 i c} \left(a^2-b^2\right)}}\right)+4 b e^{i c} d^3 x^{3/2} \log \left(1+\frac{a e^{i \left(2 c+d \sqrt{x}\right)}}{\sqrt{e^{2 i c} \left(a^2-b^2\right)}+i b e^{i c}}\right)+12 i b e^{i c} d^2 x \text{Li}_2\left(\frac{i a e^{i \left(2 c+d \sqrt{x}\right)}}{e^{i c} b+i \sqrt{\left(a^2-b^2\right) e^{2 i c}}}\right)-12 i b e^{i c} d^2 x \text{Li}_2\left(-\frac{a e^{i \left(2 c+d \sqrt{x}\right)}}{i e^{i c} b+\sqrt{\left(a^2-b^2\right) e^{2 i c}}}\right)-24 b e^{i c} d \sqrt{x} \text{Li}_3\left(\frac{i a e^{i \left(2 c+d \sqrt{x}\right)}}{e^{i c} b+i \sqrt{\left(a^2-b^2\right) e^{2 i c}}}\right)+24 b e^{i c} d \sqrt{x} \text{Li}_3\left(-\frac{a e^{i \left(2 c+d \sqrt{x}\right)}}{i e^{i c} b+\sqrt{\left(a^2-b^2\right) e^{2 i c}}}\right)-24 i b e^{i c} \text{Li}_4\left(\frac{i a e^{i \left(2 c+d \sqrt{x}\right)}}{e^{i c} b+i \sqrt{\left(a^2-b^2\right) e^{2 i c}}}\right)+24 i b e^{i c} \text{Li}_4\left(-\frac{a e^{i \left(2 c+d \sqrt{x}\right)}}{i e^{i c} b+\sqrt{\left(a^2-b^2\right) e^{2 i c}}}\right)}{2 a d^4 \sqrt{e^{2 i c} \left(a^2-b^2\right)}}","-\frac{12 b \text{Li}_4\left(\frac{i a e^{i \left(c+d \sqrt{x}\right)}}{b-\sqrt{b^2-a^2}}\right)}{a d^4 \sqrt{b^2-a^2}}+\frac{12 b \text{Li}_4\left(\frac{i a e^{i \left(c+d \sqrt{x}\right)}}{b+\sqrt{b^2-a^2}}\right)}{a d^4 \sqrt{b^2-a^2}}+\frac{12 i b \sqrt{x} \text{Li}_3\left(\frac{i a e^{i \left(c+d \sqrt{x}\right)}}{b-\sqrt{b^2-a^2}}\right)}{a d^3 \sqrt{b^2-a^2}}-\frac{12 i b \sqrt{x} \text{Li}_3\left(\frac{i a e^{i \left(c+d \sqrt{x}\right)}}{b+\sqrt{b^2-a^2}}\right)}{a d^3 \sqrt{b^2-a^2}}+\frac{6 b x \text{Li}_2\left(\frac{i a e^{i \left(c+d \sqrt{x}\right)}}{b-\sqrt{b^2-a^2}}\right)}{a d^2 \sqrt{b^2-a^2}}-\frac{6 b x \text{Li}_2\left(\frac{i a e^{i \left(c+d \sqrt{x}\right)}}{b+\sqrt{b^2-a^2}}\right)}{a d^2 \sqrt{b^2-a^2}}+\frac{2 i b x^{3/2} \log \left(1-\frac{i a e^{i \left(c+d \sqrt{x}\right)}}{b-\sqrt{b^2-a^2}}\right)}{a d \sqrt{b^2-a^2}}-\frac{2 i b x^{3/2} \log \left(1-\frac{i a e^{i \left(c+d \sqrt{x}\right)}}{\sqrt{b^2-a^2}+b}\right)}{a d \sqrt{b^2-a^2}}+\frac{x^2}{2 a}",1,"(d^4*Sqrt[(a^2 - b^2)*E^((2*I)*c)]*x^2 - 4*b*d^3*E^(I*c)*x^(3/2)*Log[1 + (a*E^(I*(2*c + d*Sqrt[x])))/(I*b*E^(I*c) - Sqrt[(a^2 - b^2)*E^((2*I)*c)])] + 4*b*d^3*E^(I*c)*x^(3/2)*Log[1 + (a*E^(I*(2*c + d*Sqrt[x])))/(I*b*E^(I*c) + Sqrt[(a^2 - b^2)*E^((2*I)*c)])] + (12*I)*b*d^2*E^(I*c)*x*PolyLog[2, (I*a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) + I*Sqrt[(a^2 - b^2)*E^((2*I)*c)])] - (12*I)*b*d^2*E^(I*c)*x*PolyLog[2, -((a*E^(I*(2*c + d*Sqrt[x])))/(I*b*E^(I*c) + Sqrt[(a^2 - b^2)*E^((2*I)*c)]))] - 24*b*d*E^(I*c)*Sqrt[x]*PolyLog[3, (I*a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) + I*Sqrt[(a^2 - b^2)*E^((2*I)*c)])] + 24*b*d*E^(I*c)*Sqrt[x]*PolyLog[3, -((a*E^(I*(2*c + d*Sqrt[x])))/(I*b*E^(I*c) + Sqrt[(a^2 - b^2)*E^((2*I)*c)]))] - (24*I)*b*E^(I*c)*PolyLog[4, (I*a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) + I*Sqrt[(a^2 - b^2)*E^((2*I)*c)])] + (24*I)*b*E^(I*c)*PolyLog[4, -((a*E^(I*(2*c + d*Sqrt[x])))/(I*b*E^(I*c) + Sqrt[(a^2 - b^2)*E^((2*I)*c)]))])/(2*a*d^4*Sqrt[(a^2 - b^2)*E^((2*I)*c)])","A",1
44,0,0,23,2.2490276,"\int \frac{1}{x \left(a+b \csc \left(c+d \sqrt{x}\right)\right)} \, dx","Integrate[1/(x*(a + b*Csc[c + d*Sqrt[x]])),x]","\int \frac{1}{x \left(a+b \csc \left(c+d \sqrt{x}\right)\right)} \, dx","\text{Int}\left(\frac{1}{x \left(a+b \csc \left(c+d \sqrt{x}\right)\right)},x\right)",0,"Integrate[1/(x*(a + b*Csc[c + d*Sqrt[x]])), x]","A",-1
45,0,0,26,0.5805521,"\int \frac{a+b \csc \left(c+d \sqrt{x}\right)}{x^2} \, dx","Integrate[(a + b*Csc[c + d*Sqrt[x]])/x^2,x]","\int \frac{a+b \csc \left(c+d \sqrt{x}\right)}{x^2} \, dx","b \text{Int}\left(\frac{\csc \left(c+d \sqrt{x}\right)}{x^2},x\right)-\frac{a}{x}",0,"Integrate[(a + b*Csc[c + d*Sqrt[x]])/x^2, x]","A",-1
46,1,3831,3205,14.9418472,"\int \frac{x^3}{\left(a+b \csc \left(c+d \sqrt{x}\right)\right)^2} \, dx","Integrate[x^3/(a + b*Csc[c + d*Sqrt[x]])^2,x]","\text{Result too large to show}","\frac{x^4}{4 a^2}+\frac{4 i b \log \left(1-\frac{i a e^{i \left(c+d \sqrt{x}\right)}}{b-\sqrt{b^2-a^2}}\right) x^{7/2}}{a^2 \sqrt{b^2-a^2} d}-\frac{2 i b^3 \log \left(1-\frac{i a e^{i \left(c+d \sqrt{x}\right)}}{b-\sqrt{b^2-a^2}}\right) x^{7/2}}{a^2 \left(b^2-a^2\right)^{3/2} d}-\frac{4 i b \log \left(1-\frac{i a e^{i \left(c+d \sqrt{x}\right)}}{b+\sqrt{b^2-a^2}}\right) x^{7/2}}{a^2 \sqrt{b^2-a^2} d}+\frac{2 i b^3 \log \left(1-\frac{i a e^{i \left(c+d \sqrt{x}\right)}}{b+\sqrt{b^2-a^2}}\right) x^{7/2}}{a^2 \left(b^2-a^2\right)^{3/2} d}-\frac{2 i b^2 x^{7/2}}{a^2 \left(a^2-b^2\right) d}-\frac{2 b^2 \cos \left(c+d \sqrt{x}\right) x^{7/2}}{a \left(a^2-b^2\right) d \left(b+a \sin \left(c+d \sqrt{x}\right)\right)}+\frac{14 b^2 \log \left(\frac{e^{i \left(c+d \sqrt{x}\right)} a}{i b-\sqrt{a^2-b^2}}+1\right) x^3}{a^2 \left(a^2-b^2\right) d^2}+\frac{14 b^2 \log \left(\frac{e^{i \left(c+d \sqrt{x}\right)} a}{i b+\sqrt{a^2-b^2}}+1\right) x^3}{a^2 \left(a^2-b^2\right) d^2}+\frac{28 b \text{Li}_2\left(\frac{i a e^{i \left(c+d \sqrt{x}\right)}}{b-\sqrt{b^2-a^2}}\right) x^3}{a^2 \sqrt{b^2-a^2} d^2}-\frac{14 b^3 \text{Li}_2\left(\frac{i a e^{i \left(c+d \sqrt{x}\right)}}{b-\sqrt{b^2-a^2}}\right) x^3}{a^2 \left(b^2-a^2\right)^{3/2} d^2}-\frac{28 b \text{Li}_2\left(\frac{i a e^{i \left(c+d \sqrt{x}\right)}}{b+\sqrt{b^2-a^2}}\right) x^3}{a^2 \sqrt{b^2-a^2} d^2}+\frac{14 b^3 \text{Li}_2\left(\frac{i a e^{i \left(c+d \sqrt{x}\right)}}{b+\sqrt{b^2-a^2}}\right) x^3}{a^2 \left(b^2-a^2\right)^{3/2} d^2}-\frac{84 i b^2 \text{Li}_2\left(-\frac{a e^{i \left(c+d \sqrt{x}\right)}}{i b-\sqrt{a^2-b^2}}\right) x^{5/2}}{a^2 \left(a^2-b^2\right) d^3}-\frac{84 i b^2 \text{Li}_2\left(-\frac{a e^{i \left(c+d \sqrt{x}\right)}}{i b+\sqrt{a^2-b^2}}\right) x^{5/2}}{a^2 \left(a^2-b^2\right) d^3}+\frac{168 i b \text{Li}_3\left(\frac{i a e^{i \left(c+d \sqrt{x}\right)}}{b-\sqrt{b^2-a^2}}\right) x^{5/2}}{a^2 \sqrt{b^2-a^2} d^3}-\frac{84 i b^3 \text{Li}_3\left(\frac{i a e^{i \left(c+d \sqrt{x}\right)}}{b-\sqrt{b^2-a^2}}\right) x^{5/2}}{a^2 \left(b^2-a^2\right)^{3/2} d^3}-\frac{168 i b \text{Li}_3\left(\frac{i a e^{i \left(c+d \sqrt{x}\right)}}{b+\sqrt{b^2-a^2}}\right) x^{5/2}}{a^2 \sqrt{b^2-a^2} d^3}+\frac{84 i b^3 \text{Li}_3\left(\frac{i a e^{i \left(c+d \sqrt{x}\right)}}{b+\sqrt{b^2-a^2}}\right) x^{5/2}}{a^2 \left(b^2-a^2\right)^{3/2} d^3}+\frac{420 b^2 \text{Li}_3\left(-\frac{a e^{i \left(c+d \sqrt{x}\right)}}{i b-\sqrt{a^2-b^2}}\right) x^2}{a^2 \left(a^2-b^2\right) d^4}+\frac{420 b^2 \text{Li}_3\left(-\frac{a e^{i \left(c+d \sqrt{x}\right)}}{i b+\sqrt{a^2-b^2}}\right) x^2}{a^2 \left(a^2-b^2\right) d^4}-\frac{840 b \text{Li}_4\left(\frac{i a e^{i \left(c+d \sqrt{x}\right)}}{b-\sqrt{b^2-a^2}}\right) x^2}{a^2 \sqrt{b^2-a^2} d^4}+\frac{420 b^3 \text{Li}_4\left(\frac{i a e^{i \left(c+d \sqrt{x}\right)}}{b-\sqrt{b^2-a^2}}\right) x^2}{a^2 \left(b^2-a^2\right)^{3/2} d^4}+\frac{840 b \text{Li}_4\left(\frac{i a e^{i \left(c+d \sqrt{x}\right)}}{b+\sqrt{b^2-a^2}}\right) x^2}{a^2 \sqrt{b^2-a^2} d^4}-\frac{420 b^3 \text{Li}_4\left(\frac{i a e^{i \left(c+d \sqrt{x}\right)}}{b+\sqrt{b^2-a^2}}\right) x^2}{a^2 \left(b^2-a^2\right)^{3/2} d^4}+\frac{1680 i b^2 \text{Li}_4\left(-\frac{a e^{i \left(c+d \sqrt{x}\right)}}{i b-\sqrt{a^2-b^2}}\right) x^{3/2}}{a^2 \left(a^2-b^2\right) d^5}+\frac{1680 i b^2 \text{Li}_4\left(-\frac{a e^{i \left(c+d \sqrt{x}\right)}}{i b+\sqrt{a^2-b^2}}\right) x^{3/2}}{a^2 \left(a^2-b^2\right) d^5}-\frac{3360 i b \text{Li}_5\left(\frac{i a e^{i \left(c+d \sqrt{x}\right)}}{b-\sqrt{b^2-a^2}}\right) x^{3/2}}{a^2 \sqrt{b^2-a^2} d^5}+\frac{1680 i b^3 \text{Li}_5\left(\frac{i a e^{i \left(c+d \sqrt{x}\right)}}{b-\sqrt{b^2-a^2}}\right) x^{3/2}}{a^2 \left(b^2-a^2\right)^{3/2} d^5}+\frac{3360 i b \text{Li}_5\left(\frac{i a e^{i \left(c+d \sqrt{x}\right)}}{b+\sqrt{b^2-a^2}}\right) x^{3/2}}{a^2 \sqrt{b^2-a^2} d^5}-\frac{1680 i b^3 \text{Li}_5\left(\frac{i a e^{i \left(c+d \sqrt{x}\right)}}{b+\sqrt{b^2-a^2}}\right) x^{3/2}}{a^2 \left(b^2-a^2\right)^{3/2} d^5}-\frac{5040 b^2 \text{Li}_5\left(-\frac{a e^{i \left(c+d \sqrt{x}\right)}}{i b-\sqrt{a^2-b^2}}\right) x}{a^2 \left(a^2-b^2\right) d^6}-\frac{5040 b^2 \text{Li}_5\left(-\frac{a e^{i \left(c+d \sqrt{x}\right)}}{i b+\sqrt{a^2-b^2}}\right) x}{a^2 \left(a^2-b^2\right) d^6}+\frac{10080 b \text{Li}_6\left(\frac{i a e^{i \left(c+d \sqrt{x}\right)}}{b-\sqrt{b^2-a^2}}\right) x}{a^2 \sqrt{b^2-a^2} d^6}-\frac{5040 b^3 \text{Li}_6\left(\frac{i a e^{i \left(c+d \sqrt{x}\right)}}{b-\sqrt{b^2-a^2}}\right) x}{a^2 \left(b^2-a^2\right)^{3/2} d^6}-\frac{10080 b \text{Li}_6\left(\frac{i a e^{i \left(c+d \sqrt{x}\right)}}{b+\sqrt{b^2-a^2}}\right) x}{a^2 \sqrt{b^2-a^2} d^6}+\frac{5040 b^3 \text{Li}_6\left(\frac{i a e^{i \left(c+d \sqrt{x}\right)}}{b+\sqrt{b^2-a^2}}\right) x}{a^2 \left(b^2-a^2\right)^{3/2} d^6}-\frac{10080 i b^2 \text{Li}_6\left(-\frac{a e^{i \left(c+d \sqrt{x}\right)}}{i b-\sqrt{a^2-b^2}}\right) \sqrt{x}}{a^2 \left(a^2-b^2\right) d^7}-\frac{10080 i b^2 \text{Li}_6\left(-\frac{a e^{i \left(c+d \sqrt{x}\right)}}{i b+\sqrt{a^2-b^2}}\right) \sqrt{x}}{a^2 \left(a^2-b^2\right) d^7}+\frac{20160 i b \text{Li}_7\left(\frac{i a e^{i \left(c+d \sqrt{x}\right)}}{b-\sqrt{b^2-a^2}}\right) \sqrt{x}}{a^2 \sqrt{b^2-a^2} d^7}-\frac{10080 i b^3 \text{Li}_7\left(\frac{i a e^{i \left(c+d \sqrt{x}\right)}}{b-\sqrt{b^2-a^2}}\right) \sqrt{x}}{a^2 \left(b^2-a^2\right)^{3/2} d^7}-\frac{20160 i b \text{Li}_7\left(\frac{i a e^{i \left(c+d \sqrt{x}\right)}}{b+\sqrt{b^2-a^2}}\right) \sqrt{x}}{a^2 \sqrt{b^2-a^2} d^7}+\frac{10080 i b^3 \text{Li}_7\left(\frac{i a e^{i \left(c+d \sqrt{x}\right)}}{b+\sqrt{b^2-a^2}}\right) \sqrt{x}}{a^2 \left(b^2-a^2\right)^{3/2} d^7}+\frac{10080 b^2 \text{Li}_7\left(-\frac{a e^{i \left(c+d \sqrt{x}\right)}}{i b-\sqrt{a^2-b^2}}\right)}{a^2 \left(a^2-b^2\right) d^8}+\frac{10080 b^2 \text{Li}_7\left(-\frac{a e^{i \left(c+d \sqrt{x}\right)}}{i b+\sqrt{a^2-b^2}}\right)}{a^2 \left(a^2-b^2\right) d^8}-\frac{20160 b \text{Li}_8\left(\frac{i a e^{i \left(c+d \sqrt{x}\right)}}{b-\sqrt{b^2-a^2}}\right)}{a^2 \sqrt{b^2-a^2} d^8}+\frac{10080 b^3 \text{Li}_8\left(\frac{i a e^{i \left(c+d \sqrt{x}\right)}}{b-\sqrt{b^2-a^2}}\right)}{a^2 \left(b^2-a^2\right)^{3/2} d^8}+\frac{20160 b \text{Li}_8\left(\frac{i a e^{i \left(c+d \sqrt{x}\right)}}{b+\sqrt{b^2-a^2}}\right)}{a^2 \sqrt{b^2-a^2} d^8}-\frac{10080 b^3 \text{Li}_8\left(\frac{i a e^{i \left(c+d \sqrt{x}\right)}}{b+\sqrt{b^2-a^2}}\right)}{a^2 \left(b^2-a^2\right)^{3/2} d^8}",1,"(x^4*Csc[c + d*Sqrt[x]]^2*(b + a*Sin[c + d*Sqrt[x]])^2)/(4*a^2*(a + b*Csc[c + d*Sqrt[x]])^2) - ((2*I)*b*E^(I*c)*Csc[c + d*Sqrt[x]]^2*(2*b*E^(I*c)*x^(7/2) - ((-1 + E^((2*I)*c))*((-7*I)*b*d^6*Sqrt[(a^2 - b^2)*E^((2*I)*c)]*x^3*Log[1 + (a*E^(I*(2*c + d*Sqrt[x])))/(I*b*E^(I*c) - Sqrt[(a^2 - b^2)*E^((2*I)*c)])] + (2*I)*a^2*d^7*E^(I*c)*x^(7/2)*Log[1 + (a*E^(I*(2*c + d*Sqrt[x])))/(I*b*E^(I*c) - Sqrt[(a^2 - b^2)*E^((2*I)*c)])] - I*b^2*d^7*E^(I*c)*x^(7/2)*Log[1 + (a*E^(I*(2*c + d*Sqrt[x])))/(I*b*E^(I*c) - Sqrt[(a^2 - b^2)*E^((2*I)*c)])] - (7*I)*b*d^6*Sqrt[(a^2 - b^2)*E^((2*I)*c)]*x^3*Log[1 + (a*E^(I*(2*c + d*Sqrt[x])))/(I*b*E^(I*c) + Sqrt[(a^2 - b^2)*E^((2*I)*c)])] - (2*I)*a^2*d^7*E^(I*c)*x^(7/2)*Log[1 + (a*E^(I*(2*c + d*Sqrt[x])))/(I*b*E^(I*c) + Sqrt[(a^2 - b^2)*E^((2*I)*c)])] + I*b^2*d^7*E^(I*c)*x^(7/2)*Log[1 + (a*E^(I*(2*c + d*Sqrt[x])))/(I*b*E^(I*c) + Sqrt[(a^2 - b^2)*E^((2*I)*c)])] - 7*d^5*(6*b*Sqrt[(a^2 - b^2)*E^((2*I)*c)] - 2*a^2*d*E^(I*c)*Sqrt[x] + b^2*d*E^(I*c)*Sqrt[x])*x^(5/2)*PolyLog[2, (I*a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) + I*Sqrt[(a^2 - b^2)*E^((2*I)*c)])] + 7*d^5*(-6*b*Sqrt[(a^2 - b^2)*E^((2*I)*c)] - 2*a^2*d*E^(I*c)*Sqrt[x] + b^2*d*E^(I*c)*Sqrt[x])*x^(5/2)*PolyLog[2, -((a*E^(I*(2*c + d*Sqrt[x])))/(I*b*E^(I*c) + Sqrt[(a^2 - b^2)*E^((2*I)*c)]))] - (210*I)*b*d^4*Sqrt[(a^2 - b^2)*E^((2*I)*c)]*x^2*PolyLog[3, (I*a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) + I*Sqrt[(a^2 - b^2)*E^((2*I)*c)])] + (84*I)*a^2*d^5*E^(I*c)*x^(5/2)*PolyLog[3, (I*a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) + I*Sqrt[(a^2 - b^2)*E^((2*I)*c)])] - (42*I)*b^2*d^5*E^(I*c)*x^(5/2)*PolyLog[3, (I*a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) + I*Sqrt[(a^2 - b^2)*E^((2*I)*c)])] - (210*I)*b*d^4*Sqrt[(a^2 - b^2)*E^((2*I)*c)]*x^2*PolyLog[3, -((a*E^(I*(2*c + d*Sqrt[x])))/(I*b*E^(I*c) + Sqrt[(a^2 - b^2)*E^((2*I)*c)]))] - (84*I)*a^2*d^5*E^(I*c)*x^(5/2)*PolyLog[3, -((a*E^(I*(2*c + d*Sqrt[x])))/(I*b*E^(I*c) + Sqrt[(a^2 - b^2)*E^((2*I)*c)]))] + (42*I)*b^2*d^5*E^(I*c)*x^(5/2)*PolyLog[3, -((a*E^(I*(2*c + d*Sqrt[x])))/(I*b*E^(I*c) + Sqrt[(a^2 - b^2)*E^((2*I)*c)]))] + 840*b*d^3*Sqrt[(a^2 - b^2)*E^((2*I)*c)]*x^(3/2)*PolyLog[4, (I*a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) + I*Sqrt[(a^2 - b^2)*E^((2*I)*c)])] - 420*a^2*d^4*E^(I*c)*x^2*PolyLog[4, (I*a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) + I*Sqrt[(a^2 - b^2)*E^((2*I)*c)])] + 210*b^2*d^4*E^(I*c)*x^2*PolyLog[4, (I*a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) + I*Sqrt[(a^2 - b^2)*E^((2*I)*c)])] + 840*b*d^3*Sqrt[(a^2 - b^2)*E^((2*I)*c)]*x^(3/2)*PolyLog[4, -((a*E^(I*(2*c + d*Sqrt[x])))/(I*b*E^(I*c) + Sqrt[(a^2 - b^2)*E^((2*I)*c)]))] + 420*a^2*d^4*E^(I*c)*x^2*PolyLog[4, -((a*E^(I*(2*c + d*Sqrt[x])))/(I*b*E^(I*c) + Sqrt[(a^2 - b^2)*E^((2*I)*c)]))] - 210*b^2*d^4*E^(I*c)*x^2*PolyLog[4, -((a*E^(I*(2*c + d*Sqrt[x])))/(I*b*E^(I*c) + Sqrt[(a^2 - b^2)*E^((2*I)*c)]))] + (2520*I)*b*d^2*Sqrt[(a^2 - b^2)*E^((2*I)*c)]*x*PolyLog[5, (I*a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) + I*Sqrt[(a^2 - b^2)*E^((2*I)*c)])] - (1680*I)*a^2*d^3*E^(I*c)*x^(3/2)*PolyLog[5, (I*a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) + I*Sqrt[(a^2 - b^2)*E^((2*I)*c)])] + (840*I)*b^2*d^3*E^(I*c)*x^(3/2)*PolyLog[5, (I*a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) + I*Sqrt[(a^2 - b^2)*E^((2*I)*c)])] + (2520*I)*b*d^2*Sqrt[(a^2 - b^2)*E^((2*I)*c)]*x*PolyLog[5, -((a*E^(I*(2*c + d*Sqrt[x])))/(I*b*E^(I*c) + Sqrt[(a^2 - b^2)*E^((2*I)*c)]))] + (1680*I)*a^2*d^3*E^(I*c)*x^(3/2)*PolyLog[5, -((a*E^(I*(2*c + d*Sqrt[x])))/(I*b*E^(I*c) + Sqrt[(a^2 - b^2)*E^((2*I)*c)]))] - (840*I)*b^2*d^3*E^(I*c)*x^(3/2)*PolyLog[5, -((a*E^(I*(2*c + d*Sqrt[x])))/(I*b*E^(I*c) + Sqrt[(a^2 - b^2)*E^((2*I)*c)]))] - 5040*b*d*Sqrt[(a^2 - b^2)*E^((2*I)*c)]*Sqrt[x]*PolyLog[6, (I*a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) + I*Sqrt[(a^2 - b^2)*E^((2*I)*c)])] + 5040*a^2*d^2*E^(I*c)*x*PolyLog[6, (I*a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) + I*Sqrt[(a^2 - b^2)*E^((2*I)*c)])] - 2520*b^2*d^2*E^(I*c)*x*PolyLog[6, (I*a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) + I*Sqrt[(a^2 - b^2)*E^((2*I)*c)])] - 5040*b*d*Sqrt[(a^2 - b^2)*E^((2*I)*c)]*Sqrt[x]*PolyLog[6, -((a*E^(I*(2*c + d*Sqrt[x])))/(I*b*E^(I*c) + Sqrt[(a^2 - b^2)*E^((2*I)*c)]))] - 5040*a^2*d^2*E^(I*c)*x*PolyLog[6, -((a*E^(I*(2*c + d*Sqrt[x])))/(I*b*E^(I*c) + Sqrt[(a^2 - b^2)*E^((2*I)*c)]))] + 2520*b^2*d^2*E^(I*c)*x*PolyLog[6, -((a*E^(I*(2*c + d*Sqrt[x])))/(I*b*E^(I*c) + Sqrt[(a^2 - b^2)*E^((2*I)*c)]))] - (5040*I)*b*Sqrt[(a^2 - b^2)*E^((2*I)*c)]*PolyLog[7, (I*a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) + I*Sqrt[(a^2 - b^2)*E^((2*I)*c)])] + (10080*I)*a^2*d*E^(I*c)*Sqrt[x]*PolyLog[7, (I*a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) + I*Sqrt[(a^2 - b^2)*E^((2*I)*c)])] - (5040*I)*b^2*d*E^(I*c)*Sqrt[x]*PolyLog[7, (I*a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) + I*Sqrt[(a^2 - b^2)*E^((2*I)*c)])] - (5040*I)*b*Sqrt[(a^2 - b^2)*E^((2*I)*c)]*PolyLog[7, -((a*E^(I*(2*c + d*Sqrt[x])))/(I*b*E^(I*c) + Sqrt[(a^2 - b^2)*E^((2*I)*c)]))] - (10080*I)*a^2*d*E^(I*c)*Sqrt[x]*PolyLog[7, -((a*E^(I*(2*c + d*Sqrt[x])))/(I*b*E^(I*c) + Sqrt[(a^2 - b^2)*E^((2*I)*c)]))] + (5040*I)*b^2*d*E^(I*c)*Sqrt[x]*PolyLog[7, -((a*E^(I*(2*c + d*Sqrt[x])))/(I*b*E^(I*c) + Sqrt[(a^2 - b^2)*E^((2*I)*c)]))] - 10080*a^2*E^(I*c)*PolyLog[8, (I*a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) + I*Sqrt[(a^2 - b^2)*E^((2*I)*c)])] + 5040*b^2*E^(I*c)*PolyLog[8, (I*a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) + I*Sqrt[(a^2 - b^2)*E^((2*I)*c)])] + 10080*a^2*E^(I*c)*PolyLog[8, -((a*E^(I*(2*c + d*Sqrt[x])))/(I*b*E^(I*c) + Sqrt[(a^2 - b^2)*E^((2*I)*c)]))] - 5040*b^2*E^(I*c)*PolyLog[8, -((a*E^(I*(2*c + d*Sqrt[x])))/(I*b*E^(I*c) + Sqrt[(a^2 - b^2)*E^((2*I)*c)]))]))/(d^7*E^(I*c)*Sqrt[(a^2 - b^2)*E^((2*I)*c)]))*(b + a*Sin[c + d*Sqrt[x]])^2)/(a^2*(a^2 - b^2)*d*(-1 + E^((2*I)*c))*(a + b*Csc[c + d*Sqrt[x]])^2) + (Csc[c/2]*Csc[c + d*Sqrt[x]]^2*Sec[c/2]*(b + a*Sin[c + d*Sqrt[x]])*(-(b^3*x^(7/2)*Cos[c]) - a*b^2*x^(7/2)*Sin[d*Sqrt[x]]))/(a^2*(-a + b)*(a + b)*d*(a + b*Csc[c + d*Sqrt[x]])^2)","A",0
47,1,2829,2385,14.2945889,"\int \frac{x^2}{\left(a+b \csc \left(c+d \sqrt{x}\right)\right)^2} \, dx","Integrate[x^2/(a + b*Csc[c + d*Sqrt[x]])^2,x]","\text{Result too large to show}","-\frac{2 i x^{5/2} \log \left(1-\frac{i a e^{i \left(c+d \sqrt{x}\right)}}{b-\sqrt{b^2-a^2}}\right) b^3}{a^2 \left(b^2-a^2\right)^{3/2} d}+\frac{2 i x^{5/2} \log \left(1-\frac{i a e^{i \left(c+d \sqrt{x}\right)}}{b+\sqrt{b^2-a^2}}\right) b^3}{a^2 \left(b^2-a^2\right)^{3/2} d}-\frac{10 x^2 \text{Li}_2\left(\frac{i a e^{i \left(c+d \sqrt{x}\right)}}{b-\sqrt{b^2-a^2}}\right) b^3}{a^2 \left(b^2-a^2\right)^{3/2} d^2}+\frac{10 x^2 \text{Li}_2\left(\frac{i a e^{i \left(c+d \sqrt{x}\right)}}{b+\sqrt{b^2-a^2}}\right) b^3}{a^2 \left(b^2-a^2\right)^{3/2} d^2}-\frac{40 i x^{3/2} \text{Li}_3\left(\frac{i a e^{i \left(c+d \sqrt{x}\right)}}{b-\sqrt{b^2-a^2}}\right) b^3}{a^2 \left(b^2-a^2\right)^{3/2} d^3}+\frac{40 i x^{3/2} \text{Li}_3\left(\frac{i a e^{i \left(c+d \sqrt{x}\right)}}{b+\sqrt{b^2-a^2}}\right) b^3}{a^2 \left(b^2-a^2\right)^{3/2} d^3}+\frac{120 x \text{Li}_4\left(\frac{i a e^{i \left(c+d \sqrt{x}\right)}}{b-\sqrt{b^2-a^2}}\right) b^3}{a^2 \left(b^2-a^2\right)^{3/2} d^4}-\frac{120 x \text{Li}_4\left(\frac{i a e^{i \left(c+d \sqrt{x}\right)}}{b+\sqrt{b^2-a^2}}\right) b^3}{a^2 \left(b^2-a^2\right)^{3/2} d^4}+\frac{240 i \sqrt{x} \text{Li}_5\left(\frac{i a e^{i \left(c+d \sqrt{x}\right)}}{b-\sqrt{b^2-a^2}}\right) b^3}{a^2 \left(b^2-a^2\right)^{3/2} d^5}-\frac{240 i \sqrt{x} \text{Li}_5\left(\frac{i a e^{i \left(c+d \sqrt{x}\right)}}{b+\sqrt{b^2-a^2}}\right) b^3}{a^2 \left(b^2-a^2\right)^{3/2} d^5}-\frac{240 \text{Li}_6\left(\frac{i a e^{i \left(c+d \sqrt{x}\right)}}{b-\sqrt{b^2-a^2}}\right) b^3}{a^2 \left(b^2-a^2\right)^{3/2} d^6}+\frac{240 \text{Li}_6\left(\frac{i a e^{i \left(c+d \sqrt{x}\right)}}{b+\sqrt{b^2-a^2}}\right) b^3}{a^2 \left(b^2-a^2\right)^{3/2} d^6}-\frac{2 i x^{5/2} b^2}{a^2 \left(a^2-b^2\right) d}+\frac{10 x^2 \log \left(\frac{e^{i \left(c+d \sqrt{x}\right)} a}{i b-\sqrt{a^2-b^2}}+1\right) b^2}{a^2 \left(a^2-b^2\right) d^2}+\frac{10 x^2 \log \left(\frac{e^{i \left(c+d \sqrt{x}\right)} a}{i b+\sqrt{a^2-b^2}}+1\right) b^2}{a^2 \left(a^2-b^2\right) d^2}-\frac{40 i x^{3/2} \text{Li}_2\left(-\frac{a e^{i \left(c+d \sqrt{x}\right)}}{i b-\sqrt{a^2-b^2}}\right) b^2}{a^2 \left(a^2-b^2\right) d^3}-\frac{40 i x^{3/2} \text{Li}_2\left(-\frac{a e^{i \left(c+d \sqrt{x}\right)}}{i b+\sqrt{a^2-b^2}}\right) b^2}{a^2 \left(a^2-b^2\right) d^3}+\frac{120 x \text{Li}_3\left(-\frac{a e^{i \left(c+d \sqrt{x}\right)}}{i b-\sqrt{a^2-b^2}}\right) b^2}{a^2 \left(a^2-b^2\right) d^4}+\frac{120 x \text{Li}_3\left(-\frac{a e^{i \left(c+d \sqrt{x}\right)}}{i b+\sqrt{a^2-b^2}}\right) b^2}{a^2 \left(a^2-b^2\right) d^4}+\frac{240 i \sqrt{x} \text{Li}_4\left(-\frac{a e^{i \left(c+d \sqrt{x}\right)}}{i b-\sqrt{a^2-b^2}}\right) b^2}{a^2 \left(a^2-b^2\right) d^5}+\frac{240 i \sqrt{x} \text{Li}_4\left(-\frac{a e^{i \left(c+d \sqrt{x}\right)}}{i b+\sqrt{a^2-b^2}}\right) b^2}{a^2 \left(a^2-b^2\right) d^5}-\frac{240 \text{Li}_5\left(-\frac{a e^{i \left(c+d \sqrt{x}\right)}}{i b-\sqrt{a^2-b^2}}\right) b^2}{a^2 \left(a^2-b^2\right) d^6}-\frac{240 \text{Li}_5\left(-\frac{a e^{i \left(c+d \sqrt{x}\right)}}{i b+\sqrt{a^2-b^2}}\right) b^2}{a^2 \left(a^2-b^2\right) d^6}-\frac{2 x^{5/2} \cos \left(c+d \sqrt{x}\right) b^2}{a \left(a^2-b^2\right) d \left(b+a \sin \left(c+d \sqrt{x}\right)\right)}+\frac{4 i x^{5/2} \log \left(1-\frac{i a e^{i \left(c+d \sqrt{x}\right)}}{b-\sqrt{b^2-a^2}}\right) b}{a^2 \sqrt{b^2-a^2} d}-\frac{4 i x^{5/2} \log \left(1-\frac{i a e^{i \left(c+d \sqrt{x}\right)}}{b+\sqrt{b^2-a^2}}\right) b}{a^2 \sqrt{b^2-a^2} d}+\frac{20 x^2 \text{Li}_2\left(\frac{i a e^{i \left(c+d \sqrt{x}\right)}}{b-\sqrt{b^2-a^2}}\right) b}{a^2 \sqrt{b^2-a^2} d^2}-\frac{20 x^2 \text{Li}_2\left(\frac{i a e^{i \left(c+d \sqrt{x}\right)}}{b+\sqrt{b^2-a^2}}\right) b}{a^2 \sqrt{b^2-a^2} d^2}+\frac{80 i x^{3/2} \text{Li}_3\left(\frac{i a e^{i \left(c+d \sqrt{x}\right)}}{b-\sqrt{b^2-a^2}}\right) b}{a^2 \sqrt{b^2-a^2} d^3}-\frac{80 i x^{3/2} \text{Li}_3\left(\frac{i a e^{i \left(c+d \sqrt{x}\right)}}{b+\sqrt{b^2-a^2}}\right) b}{a^2 \sqrt{b^2-a^2} d^3}-\frac{240 x \text{Li}_4\left(\frac{i a e^{i \left(c+d \sqrt{x}\right)}}{b-\sqrt{b^2-a^2}}\right) b}{a^2 \sqrt{b^2-a^2} d^4}+\frac{240 x \text{Li}_4\left(\frac{i a e^{i \left(c+d \sqrt{x}\right)}}{b+\sqrt{b^2-a^2}}\right) b}{a^2 \sqrt{b^2-a^2} d^4}-\frac{480 i \sqrt{x} \text{Li}_5\left(\frac{i a e^{i \left(c+d \sqrt{x}\right)}}{b-\sqrt{b^2-a^2}}\right) b}{a^2 \sqrt{b^2-a^2} d^5}+\frac{480 i \sqrt{x} \text{Li}_5\left(\frac{i a e^{i \left(c+d \sqrt{x}\right)}}{b+\sqrt{b^2-a^2}}\right) b}{a^2 \sqrt{b^2-a^2} d^5}+\frac{480 \text{Li}_6\left(\frac{i a e^{i \left(c+d \sqrt{x}\right)}}{b-\sqrt{b^2-a^2}}\right) b}{a^2 \sqrt{b^2-a^2} d^6}-\frac{480 \text{Li}_6\left(\frac{i a e^{i \left(c+d \sqrt{x}\right)}}{b+\sqrt{b^2-a^2}}\right) b}{a^2 \sqrt{b^2-a^2} d^6}+\frac{x^3}{3 a^2}",1,"(x^3*Csc[c + d*Sqrt[x]]^2*(b + a*Sin[c + d*Sqrt[x]])^2)/(3*a^2*(a + b*Csc[c + d*Sqrt[x]])^2) - ((2*I)*b*E^(I*c)*Csc[c + d*Sqrt[x]]^2*(2*b*E^(I*c)*x^(5/2) - ((-1 + E^((2*I)*c))*((-5*I)*b*d^4*Sqrt[(a^2 - b^2)*E^((2*I)*c)]*x^2*Log[1 + (a*E^(I*(2*c + d*Sqrt[x])))/(I*b*E^(I*c) - Sqrt[(a^2 - b^2)*E^((2*I)*c)])] + (2*I)*a^2*d^5*E^(I*c)*x^(5/2)*Log[1 + (a*E^(I*(2*c + d*Sqrt[x])))/(I*b*E^(I*c) - Sqrt[(a^2 - b^2)*E^((2*I)*c)])] - I*b^2*d^5*E^(I*c)*x^(5/2)*Log[1 + (a*E^(I*(2*c + d*Sqrt[x])))/(I*b*E^(I*c) - Sqrt[(a^2 - b^2)*E^((2*I)*c)])] - (5*I)*b*d^4*Sqrt[(a^2 - b^2)*E^((2*I)*c)]*x^2*Log[1 + (a*E^(I*(2*c + d*Sqrt[x])))/(I*b*E^(I*c) + Sqrt[(a^2 - b^2)*E^((2*I)*c)])] - (2*I)*a^2*d^5*E^(I*c)*x^(5/2)*Log[1 + (a*E^(I*(2*c + d*Sqrt[x])))/(I*b*E^(I*c) + Sqrt[(a^2 - b^2)*E^((2*I)*c)])] + I*b^2*d^5*E^(I*c)*x^(5/2)*Log[1 + (a*E^(I*(2*c + d*Sqrt[x])))/(I*b*E^(I*c) + Sqrt[(a^2 - b^2)*E^((2*I)*c)])] - 5*d^3*(4*b*Sqrt[(a^2 - b^2)*E^((2*I)*c)] - 2*a^2*d*E^(I*c)*Sqrt[x] + b^2*d*E^(I*c)*Sqrt[x])*x^(3/2)*PolyLog[2, (I*a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) + I*Sqrt[(a^2 - b^2)*E^((2*I)*c)])] + 5*d^3*(-4*b*Sqrt[(a^2 - b^2)*E^((2*I)*c)] - 2*a^2*d*E^(I*c)*Sqrt[x] + b^2*d*E^(I*c)*Sqrt[x])*x^(3/2)*PolyLog[2, -((a*E^(I*(2*c + d*Sqrt[x])))/(I*b*E^(I*c) + Sqrt[(a^2 - b^2)*E^((2*I)*c)]))] - (60*I)*b*d^2*Sqrt[(a^2 - b^2)*E^((2*I)*c)]*x*PolyLog[3, (I*a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) + I*Sqrt[(a^2 - b^2)*E^((2*I)*c)])] + (40*I)*a^2*d^3*E^(I*c)*x^(3/2)*PolyLog[3, (I*a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) + I*Sqrt[(a^2 - b^2)*E^((2*I)*c)])] - (20*I)*b^2*d^3*E^(I*c)*x^(3/2)*PolyLog[3, (I*a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) + I*Sqrt[(a^2 - b^2)*E^((2*I)*c)])] - (60*I)*b*d^2*Sqrt[(a^2 - b^2)*E^((2*I)*c)]*x*PolyLog[3, -((a*E^(I*(2*c + d*Sqrt[x])))/(I*b*E^(I*c) + Sqrt[(a^2 - b^2)*E^((2*I)*c)]))] - (40*I)*a^2*d^3*E^(I*c)*x^(3/2)*PolyLog[3, -((a*E^(I*(2*c + d*Sqrt[x])))/(I*b*E^(I*c) + Sqrt[(a^2 - b^2)*E^((2*I)*c)]))] + (20*I)*b^2*d^3*E^(I*c)*x^(3/2)*PolyLog[3, -((a*E^(I*(2*c + d*Sqrt[x])))/(I*b*E^(I*c) + Sqrt[(a^2 - b^2)*E^((2*I)*c)]))] + 120*b*d*Sqrt[(a^2 - b^2)*E^((2*I)*c)]*Sqrt[x]*PolyLog[4, (I*a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) + I*Sqrt[(a^2 - b^2)*E^((2*I)*c)])] - 120*a^2*d^2*E^(I*c)*x*PolyLog[4, (I*a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) + I*Sqrt[(a^2 - b^2)*E^((2*I)*c)])] + 60*b^2*d^2*E^(I*c)*x*PolyLog[4, (I*a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) + I*Sqrt[(a^2 - b^2)*E^((2*I)*c)])] + 120*b*d*Sqrt[(a^2 - b^2)*E^((2*I)*c)]*Sqrt[x]*PolyLog[4, -((a*E^(I*(2*c + d*Sqrt[x])))/(I*b*E^(I*c) + Sqrt[(a^2 - b^2)*E^((2*I)*c)]))] + 120*a^2*d^2*E^(I*c)*x*PolyLog[4, -((a*E^(I*(2*c + d*Sqrt[x])))/(I*b*E^(I*c) + Sqrt[(a^2 - b^2)*E^((2*I)*c)]))] - 60*b^2*d^2*E^(I*c)*x*PolyLog[4, -((a*E^(I*(2*c + d*Sqrt[x])))/(I*b*E^(I*c) + Sqrt[(a^2 - b^2)*E^((2*I)*c)]))] + (120*I)*b*Sqrt[(a^2 - b^2)*E^((2*I)*c)]*PolyLog[5, (I*a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) + I*Sqrt[(a^2 - b^2)*E^((2*I)*c)])] - (240*I)*a^2*d*E^(I*c)*Sqrt[x]*PolyLog[5, (I*a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) + I*Sqrt[(a^2 - b^2)*E^((2*I)*c)])] + (120*I)*b^2*d*E^(I*c)*Sqrt[x]*PolyLog[5, (I*a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) + I*Sqrt[(a^2 - b^2)*E^((2*I)*c)])] + (120*I)*b*Sqrt[(a^2 - b^2)*E^((2*I)*c)]*PolyLog[5, -((a*E^(I*(2*c + d*Sqrt[x])))/(I*b*E^(I*c) + Sqrt[(a^2 - b^2)*E^((2*I)*c)]))] + (240*I)*a^2*d*E^(I*c)*Sqrt[x]*PolyLog[5, -((a*E^(I*(2*c + d*Sqrt[x])))/(I*b*E^(I*c) + Sqrt[(a^2 - b^2)*E^((2*I)*c)]))] - (120*I)*b^2*d*E^(I*c)*Sqrt[x]*PolyLog[5, -((a*E^(I*(2*c + d*Sqrt[x])))/(I*b*E^(I*c) + Sqrt[(a^2 - b^2)*E^((2*I)*c)]))] + 240*a^2*E^(I*c)*PolyLog[6, (I*a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) + I*Sqrt[(a^2 - b^2)*E^((2*I)*c)])] - 120*b^2*E^(I*c)*PolyLog[6, (I*a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) + I*Sqrt[(a^2 - b^2)*E^((2*I)*c)])] - 240*a^2*E^(I*c)*PolyLog[6, -((a*E^(I*(2*c + d*Sqrt[x])))/(I*b*E^(I*c) + Sqrt[(a^2 - b^2)*E^((2*I)*c)]))] + 120*b^2*E^(I*c)*PolyLog[6, -((a*E^(I*(2*c + d*Sqrt[x])))/(I*b*E^(I*c) + Sqrt[(a^2 - b^2)*E^((2*I)*c)]))]))/(d^5*E^(I*c)*Sqrt[(a^2 - b^2)*E^((2*I)*c)]))*(b + a*Sin[c + d*Sqrt[x]])^2)/(a^2*(a^2 - b^2)*d*(-1 + E^((2*I)*c))*(a + b*Csc[c + d*Sqrt[x]])^2) + (Csc[c/2]*Csc[c + d*Sqrt[x]]^2*Sec[c/2]*(b + a*Sin[c + d*Sqrt[x]])*(-(b^3*x^(5/2)*Cos[c]) - a*b^2*x^(5/2)*Sin[d*Sqrt[x]]))/(a^2*(-a + b)*(a + b)*d*(a + b*Csc[c + d*Sqrt[x]])^2)","A",0
48,1,1729,1565,14.9238048,"\int \frac{x}{\left(a+b \csc \left(c+d \sqrt{x}\right)\right)^2} \, dx","Integrate[x/(a + b*Csc[c + d*Sqrt[x]])^2,x]","\frac{\csc ^2\left(c+d \sqrt{x}\right) \left(b+a \sin \left(c+d \sqrt{x}\right)\right) \left(\frac{4 x^{3/2} \csc (c) \left(b \cos (c)+a \sin \left(d \sqrt{x}\right)\right) b^2}{(a-b) (a+b) d}-\frac{4 i \left(\frac{2 b e^{2 i c} x^{3/2} d^3}{-1+e^{2 i c}}+\frac{-2 i a^2 e^{i c} x^{3/2} \log \left(\frac{e^{i \left(2 c+d \sqrt{x}\right)} a}{i b e^{i c}-\sqrt{\left(a^2-b^2\right) e^{2 i c}}}+1\right) d^3+i b^2 e^{i c} x^{3/2} \log \left(\frac{e^{i \left(2 c+d \sqrt{x}\right)} a}{i b e^{i c}-\sqrt{\left(a^2-b^2\right) e^{2 i c}}}+1\right) d^3+2 i a^2 e^{i c} x^{3/2} \log \left(\frac{e^{i \left(2 c+d \sqrt{x}\right)} a}{i e^{i c} b+\sqrt{\left(a^2-b^2\right) e^{2 i c}}}+1\right) d^3-i b^2 e^{i c} x^{3/2} \log \left(\frac{e^{i \left(2 c+d \sqrt{x}\right)} a}{i e^{i c} b+\sqrt{\left(a^2-b^2\right) e^{2 i c}}}+1\right) d^3+3 i b \sqrt{\left(a^2-b^2\right) e^{2 i c}} x \log \left(\frac{e^{i \left(2 c+d \sqrt{x}\right)} a}{i b e^{i c}-\sqrt{\left(a^2-b^2\right) e^{2 i c}}}+1\right) d^2+3 i b \sqrt{\left(a^2-b^2\right) e^{2 i c}} x \log \left(\frac{e^{i \left(2 c+d \sqrt{x}\right)} a}{i e^{i c} b+\sqrt{\left(a^2-b^2\right) e^{2 i c}}}+1\right) d^2+3 \left(-2 d e^{i c} \sqrt{x} a^2+2 b \sqrt{\left(a^2-b^2\right) e^{2 i c}}+b^2 d e^{i c} \sqrt{x}\right) \sqrt{x} \text{Li}_2\left(\frac{i a e^{i \left(2 c+d \sqrt{x}\right)}}{e^{i c} b+i \sqrt{\left(a^2-b^2\right) e^{2 i c}}}\right) d+3 \left(2 d e^{i c} \sqrt{x} a^2+2 b \sqrt{\left(a^2-b^2\right) e^{2 i c}}-b^2 d e^{i c} \sqrt{x}\right) \sqrt{x} \text{Li}_2\left(-\frac{a e^{i \left(2 c+d \sqrt{x}\right)}}{i e^{i c} b+\sqrt{\left(a^2-b^2\right) e^{2 i c}}}\right) d-12 i a^2 e^{i c} \sqrt{x} \text{Li}_3\left(\frac{i a e^{i \left(2 c+d \sqrt{x}\right)}}{e^{i c} b+i \sqrt{\left(a^2-b^2\right) e^{2 i c}}}\right) d+6 i b^2 e^{i c} \sqrt{x} \text{Li}_3\left(\frac{i a e^{i \left(2 c+d \sqrt{x}\right)}}{e^{i c} b+i \sqrt{\left(a^2-b^2\right) e^{2 i c}}}\right) d+12 i a^2 e^{i c} \sqrt{x} \text{Li}_3\left(-\frac{a e^{i \left(2 c+d \sqrt{x}\right)}}{i e^{i c} b+\sqrt{\left(a^2-b^2\right) e^{2 i c}}}\right) d-6 i b^2 e^{i c} \sqrt{x} \text{Li}_3\left(-\frac{a e^{i \left(2 c+d \sqrt{x}\right)}}{i e^{i c} b+\sqrt{\left(a^2-b^2\right) e^{2 i c}}}\right) d+6 i b \sqrt{\left(a^2-b^2\right) e^{2 i c}} \text{Li}_3\left(\frac{i a e^{i \left(2 c+d \sqrt{x}\right)}}{e^{i c} b+i \sqrt{\left(a^2-b^2\right) e^{2 i c}}}\right)+6 i b \sqrt{\left(a^2-b^2\right) e^{2 i c}} \text{Li}_3\left(-\frac{a e^{i \left(2 c+d \sqrt{x}\right)}}{i e^{i c} b+\sqrt{\left(a^2-b^2\right) e^{2 i c}}}\right)+12 a^2 e^{i c} \text{Li}_4\left(\frac{i a e^{i \left(2 c+d \sqrt{x}\right)}}{e^{i c} b+i \sqrt{\left(a^2-b^2\right) e^{2 i c}}}\right)-6 b^2 e^{i c} \text{Li}_4\left(\frac{i a e^{i \left(2 c+d \sqrt{x}\right)}}{e^{i c} b+i \sqrt{\left(a^2-b^2\right) e^{2 i c}}}\right)-12 a^2 e^{i c} \text{Li}_4\left(-\frac{a e^{i \left(2 c+d \sqrt{x}\right)}}{i e^{i c} b+\sqrt{\left(a^2-b^2\right) e^{2 i c}}}\right)+6 b^2 e^{i c} \text{Li}_4\left(-\frac{a e^{i \left(2 c+d \sqrt{x}\right)}}{i e^{i c} b+\sqrt{\left(a^2-b^2\right) e^{2 i c}}}\right)}{\sqrt{\left(a^2-b^2\right) e^{2 i c}}}\right) \left(b+a \sin \left(c+d \sqrt{x}\right)\right) b}{\left(a^2-b^2\right) d^4}+x^2 \left(b+a \sin \left(c+d \sqrt{x}\right)\right)\right)}{2 a^2 \left(a+b \csc \left(c+d \sqrt{x}\right)\right)^2}","-\frac{2 i x^{3/2} \log \left(1-\frac{i a e^{i \left(c+d \sqrt{x}\right)}}{b-\sqrt{b^2-a^2}}\right) b^3}{a^2 \left(b^2-a^2\right)^{3/2} d}+\frac{2 i x^{3/2} \log \left(1-\frac{i a e^{i \left(c+d \sqrt{x}\right)}}{b+\sqrt{b^2-a^2}}\right) b^3}{a^2 \left(b^2-a^2\right)^{3/2} d}-\frac{6 x \text{Li}_2\left(\frac{i a e^{i \left(c+d \sqrt{x}\right)}}{b-\sqrt{b^2-a^2}}\right) b^3}{a^2 \left(b^2-a^2\right)^{3/2} d^2}+\frac{6 x \text{Li}_2\left(\frac{i a e^{i \left(c+d \sqrt{x}\right)}}{b+\sqrt{b^2-a^2}}\right) b^3}{a^2 \left(b^2-a^2\right)^{3/2} d^2}-\frac{12 i \sqrt{x} \text{Li}_3\left(\frac{i a e^{i \left(c+d \sqrt{x}\right)}}{b-\sqrt{b^2-a^2}}\right) b^3}{a^2 \left(b^2-a^2\right)^{3/2} d^3}+\frac{12 i \sqrt{x} \text{Li}_3\left(\frac{i a e^{i \left(c+d \sqrt{x}\right)}}{b+\sqrt{b^2-a^2}}\right) b^3}{a^2 \left(b^2-a^2\right)^{3/2} d^3}+\frac{12 \text{Li}_4\left(\frac{i a e^{i \left(c+d \sqrt{x}\right)}}{b-\sqrt{b^2-a^2}}\right) b^3}{a^2 \left(b^2-a^2\right)^{3/2} d^4}-\frac{12 \text{Li}_4\left(\frac{i a e^{i \left(c+d \sqrt{x}\right)}}{b+\sqrt{b^2-a^2}}\right) b^3}{a^2 \left(b^2-a^2\right)^{3/2} d^4}-\frac{2 i x^{3/2} b^2}{a^2 \left(a^2-b^2\right) d}+\frac{6 x \log \left(\frac{e^{i \left(c+d \sqrt{x}\right)} a}{i b-\sqrt{a^2-b^2}}+1\right) b^2}{a^2 \left(a^2-b^2\right) d^2}+\frac{6 x \log \left(\frac{e^{i \left(c+d \sqrt{x}\right)} a}{i b+\sqrt{a^2-b^2}}+1\right) b^2}{a^2 \left(a^2-b^2\right) d^2}-\frac{12 i \sqrt{x} \text{Li}_2\left(-\frac{a e^{i \left(c+d \sqrt{x}\right)}}{i b-\sqrt{a^2-b^2}}\right) b^2}{a^2 \left(a^2-b^2\right) d^3}-\frac{12 i \sqrt{x} \text{Li}_2\left(-\frac{a e^{i \left(c+d \sqrt{x}\right)}}{i b+\sqrt{a^2-b^2}}\right) b^2}{a^2 \left(a^2-b^2\right) d^3}+\frac{12 \text{Li}_3\left(-\frac{a e^{i \left(c+d \sqrt{x}\right)}}{i b-\sqrt{a^2-b^2}}\right) b^2}{a^2 \left(a^2-b^2\right) d^4}+\frac{12 \text{Li}_3\left(-\frac{a e^{i \left(c+d \sqrt{x}\right)}}{i b+\sqrt{a^2-b^2}}\right) b^2}{a^2 \left(a^2-b^2\right) d^4}-\frac{2 x^{3/2} \cos \left(c+d \sqrt{x}\right) b^2}{a \left(a^2-b^2\right) d \left(b+a \sin \left(c+d \sqrt{x}\right)\right)}+\frac{4 i x^{3/2} \log \left(1-\frac{i a e^{i \left(c+d \sqrt{x}\right)}}{b-\sqrt{b^2-a^2}}\right) b}{a^2 \sqrt{b^2-a^2} d}-\frac{4 i x^{3/2} \log \left(1-\frac{i a e^{i \left(c+d \sqrt{x}\right)}}{b+\sqrt{b^2-a^2}}\right) b}{a^2 \sqrt{b^2-a^2} d}+\frac{12 x \text{Li}_2\left(\frac{i a e^{i \left(c+d \sqrt{x}\right)}}{b-\sqrt{b^2-a^2}}\right) b}{a^2 \sqrt{b^2-a^2} d^2}-\frac{12 x \text{Li}_2\left(\frac{i a e^{i \left(c+d \sqrt{x}\right)}}{b+\sqrt{b^2-a^2}}\right) b}{a^2 \sqrt{b^2-a^2} d^2}+\frac{24 i \sqrt{x} \text{Li}_3\left(\frac{i a e^{i \left(c+d \sqrt{x}\right)}}{b-\sqrt{b^2-a^2}}\right) b}{a^2 \sqrt{b^2-a^2} d^3}-\frac{24 i \sqrt{x} \text{Li}_3\left(\frac{i a e^{i \left(c+d \sqrt{x}\right)}}{b+\sqrt{b^2-a^2}}\right) b}{a^2 \sqrt{b^2-a^2} d^3}-\frac{24 \text{Li}_4\left(\frac{i a e^{i \left(c+d \sqrt{x}\right)}}{b-\sqrt{b^2-a^2}}\right) b}{a^2 \sqrt{b^2-a^2} d^4}+\frac{24 \text{Li}_4\left(\frac{i a e^{i \left(c+d \sqrt{x}\right)}}{b+\sqrt{b^2-a^2}}\right) b}{a^2 \sqrt{b^2-a^2} d^4}+\frac{x^2}{2 a^2}",1,"(Csc[c + d*Sqrt[x]]^2*(b + a*Sin[c + d*Sqrt[x]])*(x^2*(b + a*Sin[c + d*Sqrt[x]]) - ((4*I)*b*((2*b*d^3*E^((2*I)*c)*x^(3/2))/(-1 + E^((2*I)*c)) + ((3*I)*b*d^2*Sqrt[(a^2 - b^2)*E^((2*I)*c)]*x*Log[1 + (a*E^(I*(2*c + d*Sqrt[x])))/(I*b*E^(I*c) - Sqrt[(a^2 - b^2)*E^((2*I)*c)])] - (2*I)*a^2*d^3*E^(I*c)*x^(3/2)*Log[1 + (a*E^(I*(2*c + d*Sqrt[x])))/(I*b*E^(I*c) - Sqrt[(a^2 - b^2)*E^((2*I)*c)])] + I*b^2*d^3*E^(I*c)*x^(3/2)*Log[1 + (a*E^(I*(2*c + d*Sqrt[x])))/(I*b*E^(I*c) - Sqrt[(a^2 - b^2)*E^((2*I)*c)])] + (3*I)*b*d^2*Sqrt[(a^2 - b^2)*E^((2*I)*c)]*x*Log[1 + (a*E^(I*(2*c + d*Sqrt[x])))/(I*b*E^(I*c) + Sqrt[(a^2 - b^2)*E^((2*I)*c)])] + (2*I)*a^2*d^3*E^(I*c)*x^(3/2)*Log[1 + (a*E^(I*(2*c + d*Sqrt[x])))/(I*b*E^(I*c) + Sqrt[(a^2 - b^2)*E^((2*I)*c)])] - I*b^2*d^3*E^(I*c)*x^(3/2)*Log[1 + (a*E^(I*(2*c + d*Sqrt[x])))/(I*b*E^(I*c) + Sqrt[(a^2 - b^2)*E^((2*I)*c)])] + 3*d*(2*b*Sqrt[(a^2 - b^2)*E^((2*I)*c)] - 2*a^2*d*E^(I*c)*Sqrt[x] + b^2*d*E^(I*c)*Sqrt[x])*Sqrt[x]*PolyLog[2, (I*a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) + I*Sqrt[(a^2 - b^2)*E^((2*I)*c)])] + 3*d*(2*b*Sqrt[(a^2 - b^2)*E^((2*I)*c)] + 2*a^2*d*E^(I*c)*Sqrt[x] - b^2*d*E^(I*c)*Sqrt[x])*Sqrt[x]*PolyLog[2, -((a*E^(I*(2*c + d*Sqrt[x])))/(I*b*E^(I*c) + Sqrt[(a^2 - b^2)*E^((2*I)*c)]))] + (6*I)*b*Sqrt[(a^2 - b^2)*E^((2*I)*c)]*PolyLog[3, (I*a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) + I*Sqrt[(a^2 - b^2)*E^((2*I)*c)])] - (12*I)*a^2*d*E^(I*c)*Sqrt[x]*PolyLog[3, (I*a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) + I*Sqrt[(a^2 - b^2)*E^((2*I)*c)])] + (6*I)*b^2*d*E^(I*c)*Sqrt[x]*PolyLog[3, (I*a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) + I*Sqrt[(a^2 - b^2)*E^((2*I)*c)])] + (6*I)*b*Sqrt[(a^2 - b^2)*E^((2*I)*c)]*PolyLog[3, -((a*E^(I*(2*c + d*Sqrt[x])))/(I*b*E^(I*c) + Sqrt[(a^2 - b^2)*E^((2*I)*c)]))] + (12*I)*a^2*d*E^(I*c)*Sqrt[x]*PolyLog[3, -((a*E^(I*(2*c + d*Sqrt[x])))/(I*b*E^(I*c) + Sqrt[(a^2 - b^2)*E^((2*I)*c)]))] - (6*I)*b^2*d*E^(I*c)*Sqrt[x]*PolyLog[3, -((a*E^(I*(2*c + d*Sqrt[x])))/(I*b*E^(I*c) + Sqrt[(a^2 - b^2)*E^((2*I)*c)]))] + 12*a^2*E^(I*c)*PolyLog[4, (I*a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) + I*Sqrt[(a^2 - b^2)*E^((2*I)*c)])] - 6*b^2*E^(I*c)*PolyLog[4, (I*a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) + I*Sqrt[(a^2 - b^2)*E^((2*I)*c)])] - 12*a^2*E^(I*c)*PolyLog[4, -((a*E^(I*(2*c + d*Sqrt[x])))/(I*b*E^(I*c) + Sqrt[(a^2 - b^2)*E^((2*I)*c)]))] + 6*b^2*E^(I*c)*PolyLog[4, -((a*E^(I*(2*c + d*Sqrt[x])))/(I*b*E^(I*c) + Sqrt[(a^2 - b^2)*E^((2*I)*c)]))])/Sqrt[(a^2 - b^2)*E^((2*I)*c)])*(b + a*Sin[c + d*Sqrt[x]]))/((a^2 - b^2)*d^4) + (4*b^2*x^(3/2)*Csc[c]*(b*Cos[c] + a*Sin[d*Sqrt[x]]))/((a - b)*(a + b)*d)))/(2*a^2*(a + b*Csc[c + d*Sqrt[x]])^2)","A",0
49,0,0,23,41.7068754,"\int \frac{1}{x \left(a+b \csc \left(c+d \sqrt{x}\right)\right)^2} \, dx","Integrate[1/(x*(a + b*Csc[c + d*Sqrt[x]])^2),x]","\int \frac{1}{x \left(a+b \csc \left(c+d \sqrt{x}\right)\right)^2} \, dx","\text{Int}\left(\frac{1}{x \left(a+b \csc \left(c+d \sqrt{x}\right)\right)^2},x\right)",0,"Integrate[1/(x*(a + b*Csc[c + d*Sqrt[x]])^2), x]","A",-1
50,0,0,23,29.4058169,"\int \frac{1}{x^2 \left(a+b \csc \left(c+d \sqrt{x}\right)\right)^2} \, dx","Integrate[1/(x^2*(a + b*Csc[c + d*Sqrt[x]])^2),x]","\int \frac{1}{x^2 \left(a+b \csc \left(c+d \sqrt{x}\right)\right)^2} \, dx","\text{Int}\left(\frac{1}{x^2 \left(a+b \csc \left(c+d \sqrt{x}\right)\right)^2},x\right)",0,"Integrate[1/(x^2*(a + b*Csc[c + d*Sqrt[x]])^2), x]","A",-1
51,1,286,258,0.4564903,"\int x^{3/2} \left(a+b \csc \left(c+d \sqrt{x}\right)\right) \, dx","Integrate[x^(3/2)*(a + b*Csc[c + d*Sqrt[x]]),x]","\frac{2 \left(a d^5 x^{5/2}+5 b d^4 x^2 \log \left(1-e^{i \left(c+d \sqrt{x}\right)}\right)-5 b d^4 x^2 \log \left(1+e^{i \left(c+d \sqrt{x}\right)}\right)+20 i b d^3 x^{3/2} \text{Li}_2\left(-e^{i \left(c+d \sqrt{x}\right)}\right)-20 i b d^3 x^{3/2} \text{Li}_2\left(e^{i \left(c+d \sqrt{x}\right)}\right)-60 b d^2 x \text{Li}_3\left(-e^{i \left(c+d \sqrt{x}\right)}\right)+60 b d^2 x \text{Li}_3\left(e^{i \left(c+d \sqrt{x}\right)}\right)-120 i b d \sqrt{x} \text{Li}_4\left(-e^{i \left(c+d \sqrt{x}\right)}\right)+120 i b d \sqrt{x} \text{Li}_4\left(e^{i \left(c+d \sqrt{x}\right)}\right)+120 b \text{Li}_5\left(-e^{i \left(c+d \sqrt{x}\right)}\right)-120 b \text{Li}_5\left(e^{i \left(c+d \sqrt{x}\right)}\right)\right)}{5 d^5}","\frac{2}{5} a x^{5/2}+\frac{48 b \text{Li}_5\left(-e^{i \left(c+d \sqrt{x}\right)}\right)}{d^5}-\frac{48 b \text{Li}_5\left(e^{i \left(c+d \sqrt{x}\right)}\right)}{d^5}-\frac{48 i b \sqrt{x} \text{Li}_4\left(-e^{i \left(c+d \sqrt{x}\right)}\right)}{d^4}+\frac{48 i b \sqrt{x} \text{Li}_4\left(e^{i \left(c+d \sqrt{x}\right)}\right)}{d^4}-\frac{24 b x \text{Li}_3\left(-e^{i \left(c+d \sqrt{x}\right)}\right)}{d^3}+\frac{24 b x \text{Li}_3\left(e^{i \left(c+d \sqrt{x}\right)}\right)}{d^3}+\frac{8 i b x^{3/2} \text{Li}_2\left(-e^{i \left(c+d \sqrt{x}\right)}\right)}{d^2}-\frac{8 i b x^{3/2} \text{Li}_2\left(e^{i \left(c+d \sqrt{x}\right)}\right)}{d^2}-\frac{4 b x^2 \tanh ^{-1}\left(e^{i \left(c+d \sqrt{x}\right)}\right)}{d}",1,"(2*(a*d^5*x^(5/2) + 5*b*d^4*x^2*Log[1 - E^(I*(c + d*Sqrt[x]))] - 5*b*d^4*x^2*Log[1 + E^(I*(c + d*Sqrt[x]))] + (20*I)*b*d^3*x^(3/2)*PolyLog[2, -E^(I*(c + d*Sqrt[x]))] - (20*I)*b*d^3*x^(3/2)*PolyLog[2, E^(I*(c + d*Sqrt[x]))] - 60*b*d^2*x*PolyLog[3, -E^(I*(c + d*Sqrt[x]))] + 60*b*d^2*x*PolyLog[3, E^(I*(c + d*Sqrt[x]))] - (120*I)*b*d*Sqrt[x]*PolyLog[4, -E^(I*(c + d*Sqrt[x]))] + (120*I)*b*d*Sqrt[x]*PolyLog[4, E^(I*(c + d*Sqrt[x]))] + 120*b*PolyLog[5, -E^(I*(c + d*Sqrt[x]))] - 120*b*PolyLog[5, E^(I*(c + d*Sqrt[x]))]))/(5*d^5)","A",1
52,1,191,144,3.6525183,"\int \sqrt{x} \left(a+b \csc \left(c+d \sqrt{x}\right)\right) \, dx","Integrate[Sqrt[x]*(a + b*Csc[c + d*Sqrt[x]]),x]","\frac{2 \left(a d^3 x^{3/2}-6 b d^2 x \tanh ^{-1}\left(\cos \left(c+d \sqrt{x}\right)+i \sin \left(c+d \sqrt{x}\right)\right)+6 i b d \sqrt{x} \text{Li}_2\left(-\cos \left(c+d \sqrt{x}\right)-i \sin \left(c+d \sqrt{x}\right)\right)-6 i b d \sqrt{x} \text{Li}_2\left(\cos \left(c+d \sqrt{x}\right)+i \sin \left(c+d \sqrt{x}\right)\right)-6 b \text{Li}_3\left(-\cos \left(c+d \sqrt{x}\right)-i \sin \left(c+d \sqrt{x}\right)\right)+6 b \text{Li}_3\left(\cos \left(c+d \sqrt{x}\right)+i \sin \left(c+d \sqrt{x}\right)\right)\right)}{3 d^3}","\frac{2}{3} a x^{3/2}-\frac{4 b \text{Li}_3\left(-e^{i \left(c+d \sqrt{x}\right)}\right)}{d^3}+\frac{4 b \text{Li}_3\left(e^{i \left(c+d \sqrt{x}\right)}\right)}{d^3}+\frac{4 i b \sqrt{x} \text{Li}_2\left(-e^{i \left(c+d \sqrt{x}\right)}\right)}{d^2}-\frac{4 i b \sqrt{x} \text{Li}_2\left(e^{i \left(c+d \sqrt{x}\right)}\right)}{d^2}-\frac{4 b x \tanh ^{-1}\left(e^{i \left(c+d \sqrt{x}\right)}\right)}{d}",1,"(2*(a*d^3*x^(3/2) - 6*b*d^2*x*ArcTanh[Cos[c + d*Sqrt[x]] + I*Sin[c + d*Sqrt[x]]] + (6*I)*b*d*Sqrt[x]*PolyLog[2, -Cos[c + d*Sqrt[x]] - I*Sin[c + d*Sqrt[x]]] - (6*I)*b*d*Sqrt[x]*PolyLog[2, Cos[c + d*Sqrt[x]] + I*Sin[c + d*Sqrt[x]]] - 6*b*PolyLog[3, -Cos[c + d*Sqrt[x]] - I*Sin[c + d*Sqrt[x]]] + 6*b*PolyLog[3, Cos[c + d*Sqrt[x]] + I*Sin[c + d*Sqrt[x]]]))/(3*d^3)","A",0
53,1,52,26,0.0940608,"\int \frac{a+b \csc \left(c+d \sqrt{x}\right)}{\sqrt{x}} \, dx","Integrate[(a + b*Csc[c + d*Sqrt[x]])/Sqrt[x],x]","\frac{2 \left(a \left(c+d \sqrt{x}\right)+b \log \left(\sin \left(\frac{1}{2} \left(c+d \sqrt{x}\right)\right)\right)-b \log \left(\cos \left(\frac{1}{2} \left(c+d \sqrt{x}\right)\right)\right)\right)}{d}","2 a \sqrt{x}-\frac{2 b \tanh ^{-1}\left(\cos \left(c+d \sqrt{x}\right)\right)}{d}",1,"(2*(a*(c + d*Sqrt[x]) - b*Log[Cos[(c + d*Sqrt[x])/2]] + b*Log[Sin[(c + d*Sqrt[x])/2]]))/d","A",1
54,0,0,30,6.3537524,"\int \frac{a+b \csc \left(c+d \sqrt{x}\right)}{x^{3/2}} \, dx","Integrate[(a + b*Csc[c + d*Sqrt[x]])/x^(3/2),x]","\int \frac{a+b \csc \left(c+d \sqrt{x}\right)}{x^{3/2}} \, dx","b \text{Int}\left(\frac{\csc \left(c+d \sqrt{x}\right)}{x^{3/2}},x\right)-\frac{2 a}{\sqrt{x}}",0,"Integrate[(a + b*Csc[c + d*Sqrt[x]])/x^(3/2), x]","A",-1
55,0,0,32,6.3686835,"\int \frac{a+b \csc \left(c+d \sqrt{x}\right)}{x^{5/2}} \, dx","Integrate[(a + b*Csc[c + d*Sqrt[x]])/x^(5/2),x]","\int \frac{a+b \csc \left(c+d \sqrt{x}\right)}{x^{5/2}} \, dx","b \text{Int}\left(\frac{\csc \left(c+d \sqrt{x}\right)}{x^{5/2}},x\right)-\frac{2 a}{3 x^{3/2}}",0,"Integrate[(a + b*Csc[c + d*Sqrt[x]])/x^(5/2), x]","A",-1
56,1,749,421,8.3812484,"\int x^{3/2} \left(a+b \csc \left(c+d \sqrt{x}\right)\right)^2 \, dx","Integrate[x^(3/2)*(a + b*Csc[c + d*Sqrt[x]])^2,x]","\frac{2 a^2 x^{5/2} \sin ^2\left(c+d \sqrt{x}\right) \left(a+b \csc \left(c+d \sqrt{x}\right)\right)^2}{5 \left(a \sin \left(c+d \sqrt{x}\right)+b\right)^2}+\frac{b^2 x^2 \csc \left(\frac{c}{2}\right) \sin \left(\frac{d \sqrt{x}}{2}\right) \sin ^2\left(c+d \sqrt{x}\right) \csc \left(\frac{c}{2}+\frac{d \sqrt{x}}{2}\right) \left(a+b \csc \left(c+d \sqrt{x}\right)\right)^2}{d \left(a \sin \left(c+d \sqrt{x}\right)+b\right)^2}+\frac{b^2 x^2 \sec \left(\frac{c}{2}\right) \sin \left(\frac{d \sqrt{x}}{2}\right) \sin ^2\left(c+d \sqrt{x}\right) \sec \left(\frac{c}{2}+\frac{d \sqrt{x}}{2}\right) \left(a+b \csc \left(c+d \sqrt{x}\right)\right)^2}{d \left(a \sin \left(c+d \sqrt{x}\right)+b\right)^2}+\frac{4 b \sin ^2\left(c+d \sqrt{x}\right) \left(a+b \csc \left(c+d \sqrt{x}\right)\right)^2 \left(-2 i d^2 x \left(2 a d \sqrt{x}-3 b\right) \text{Li}_2\left(-e^{-i \left(c+d \sqrt{x}\right)}\right)+2 i d^2 x \left(2 a d \sqrt{x}+3 b\right) \text{Li}_2\left(e^{-i \left(c+d \sqrt{x}\right)}\right)+a d^4 x^2 \log \left(1-e^{-i \left(c+d \sqrt{x}\right)}\right)-a d^4 x^2 \log \left(1+e^{-i \left(c+d \sqrt{x}\right)}\right)-12 a d^2 x \text{Li}_3\left(-e^{-i \left(c+d \sqrt{x}\right)}\right)+12 a d^2 x \text{Li}_3\left(e^{-i \left(c+d \sqrt{x}\right)}\right)+24 i a d \sqrt{x} \text{Li}_4\left(-e^{-i \left(c+d \sqrt{x}\right)}\right)-24 i a d \sqrt{x} \text{Li}_4\left(e^{-i \left(c+d \sqrt{x}\right)}\right)+24 a \text{Li}_5\left(-e^{-i \left(c+d \sqrt{x}\right)}\right)-24 a \text{Li}_5\left(e^{-i \left(c+d \sqrt{x}\right)}\right)-\frac{i b d^4 x^2}{-1+e^{2 i c}}+2 b d^3 x^{3/2} \log \left(1-e^{-i \left(c+d \sqrt{x}\right)}\right)+2 b d^3 x^{3/2} \log \left(1+e^{-i \left(c+d \sqrt{x}\right)}\right)+12 b d \sqrt{x} \text{Li}_3\left(-e^{-i \left(c+d \sqrt{x}\right)}\right)+12 b d \sqrt{x} \text{Li}_3\left(e^{-i \left(c+d \sqrt{x}\right)}\right)-12 i b \text{Li}_4\left(-e^{-i \left(c+d \sqrt{x}\right)}\right)-12 i b \text{Li}_4\left(e^{-i \left(c+d \sqrt{x}\right)}\right)\right)}{d^5 \left(a \sin \left(c+d \sqrt{x}\right)+b\right)^2}","\frac{2}{5} a^2 x^{5/2}+\frac{96 a b \text{Li}_5\left(-e^{i \left(c+d \sqrt{x}\right)}\right)}{d^5}-\frac{96 a b \text{Li}_5\left(e^{i \left(c+d \sqrt{x}\right)}\right)}{d^5}-\frac{96 i a b \sqrt{x} \text{Li}_4\left(-e^{i \left(c+d \sqrt{x}\right)}\right)}{d^4}+\frac{96 i a b \sqrt{x} \text{Li}_4\left(e^{i \left(c+d \sqrt{x}\right)}\right)}{d^4}-\frac{48 a b x \text{Li}_3\left(-e^{i \left(c+d \sqrt{x}\right)}\right)}{d^3}+\frac{48 a b x \text{Li}_3\left(e^{i \left(c+d \sqrt{x}\right)}\right)}{d^3}+\frac{16 i a b x^{3/2} \text{Li}_2\left(-e^{i \left(c+d \sqrt{x}\right)}\right)}{d^2}-\frac{16 i a b x^{3/2} \text{Li}_2\left(e^{i \left(c+d \sqrt{x}\right)}\right)}{d^2}-\frac{8 a b x^2 \tanh ^{-1}\left(e^{i \left(c+d \sqrt{x}\right)}\right)}{d}+\frac{6 i b^2 \text{Li}_4\left(e^{2 i \left(c+d \sqrt{x}\right)}\right)}{d^5}+\frac{12 b^2 \sqrt{x} \text{Li}_3\left(e^{2 i \left(c+d \sqrt{x}\right)}\right)}{d^4}-\frac{12 i b^2 x \text{Li}_2\left(e^{2 i \left(c+d \sqrt{x}\right)}\right)}{d^3}+\frac{8 b^2 x^{3/2} \log \left(1-e^{2 i \left(c+d \sqrt{x}\right)}\right)}{d^2}-\frac{2 b^2 x^2 \cot \left(c+d \sqrt{x}\right)}{d}-\frac{2 i b^2 x^2}{d}",1,"(2*a^2*x^(5/2)*(a + b*Csc[c + d*Sqrt[x]])^2*Sin[c + d*Sqrt[x]]^2)/(5*(b + a*Sin[c + d*Sqrt[x]])^2) + (4*b*(a + b*Csc[c + d*Sqrt[x]])^2*(((-I)*b*d^4*x^2)/(-1 + E^((2*I)*c)) + 2*b*d^3*x^(3/2)*Log[1 - E^((-I)*(c + d*Sqrt[x]))] + a*d^4*x^2*Log[1 - E^((-I)*(c + d*Sqrt[x]))] + 2*b*d^3*x^(3/2)*Log[1 + E^((-I)*(c + d*Sqrt[x]))] - a*d^4*x^2*Log[1 + E^((-I)*(c + d*Sqrt[x]))] - (2*I)*d^2*(-3*b + 2*a*d*Sqrt[x])*x*PolyLog[2, -E^((-I)*(c + d*Sqrt[x]))] + (2*I)*d^2*(3*b + 2*a*d*Sqrt[x])*x*PolyLog[2, E^((-I)*(c + d*Sqrt[x]))] + 12*b*d*Sqrt[x]*PolyLog[3, -E^((-I)*(c + d*Sqrt[x]))] - 12*a*d^2*x*PolyLog[3, -E^((-I)*(c + d*Sqrt[x]))] + 12*b*d*Sqrt[x]*PolyLog[3, E^((-I)*(c + d*Sqrt[x]))] + 12*a*d^2*x*PolyLog[3, E^((-I)*(c + d*Sqrt[x]))] - (12*I)*b*PolyLog[4, -E^((-I)*(c + d*Sqrt[x]))] + (24*I)*a*d*Sqrt[x]*PolyLog[4, -E^((-I)*(c + d*Sqrt[x]))] - (12*I)*b*PolyLog[4, E^((-I)*(c + d*Sqrt[x]))] - (24*I)*a*d*Sqrt[x]*PolyLog[4, E^((-I)*(c + d*Sqrt[x]))] + 24*a*PolyLog[5, -E^((-I)*(c + d*Sqrt[x]))] - 24*a*PolyLog[5, E^((-I)*(c + d*Sqrt[x]))])*Sin[c + d*Sqrt[x]]^2)/(d^5*(b + a*Sin[c + d*Sqrt[x]])^2) + (b^2*x^2*Csc[c/2]*Csc[c/2 + (d*Sqrt[x])/2]*(a + b*Csc[c + d*Sqrt[x]])^2*Sin[c + d*Sqrt[x]]^2*Sin[(d*Sqrt[x])/2])/(d*(b + a*Sin[c + d*Sqrt[x]])^2) + (b^2*x^2*(a + b*Csc[c + d*Sqrt[x]])^2*Sec[c/2]*Sec[c/2 + (d*Sqrt[x])/2]*Sin[c + d*Sqrt[x]]^2*Sin[(d*Sqrt[x])/2])/(d*(b + a*Sin[c + d*Sqrt[x]])^2)","A",1
57,1,681,241,3.8421138,"\int \sqrt{x} \left(a+b \csc \left(c+d \sqrt{x}\right)\right)^2 \, dx","Integrate[Sqrt[x]*(a + b*Csc[c + d*Sqrt[x]])^2,x]","\frac{2 a^2 e^{2 i c} d^3 x^{3/2}-2 a^2 d^3 x^{3/2}+12 a b e^{2 i c} d^2 x \log \left(1-e^{-i \left(c+d \sqrt{x}\right)}\right)-12 a b d^2 x \log \left(1-e^{-i \left(c+d \sqrt{x}\right)}\right)-12 a b e^{2 i c} d^2 x \log \left(1+e^{-i \left(c+d \sqrt{x}\right)}\right)+12 a b d^2 x \log \left(1+e^{-i \left(c+d \sqrt{x}\right)}\right)+12 i b \left(-1+e^{2 i c}\right) \left(b-2 a d \sqrt{x}\right) \text{Li}_2\left(-e^{-i \left(c+d \sqrt{x}\right)}\right)+12 i b \left(-1+e^{2 i c}\right) \left(2 a d \sqrt{x}+b\right) \text{Li}_2\left(e^{-i \left(c+d \sqrt{x}\right)}\right)-24 a b e^{2 i c} \text{Li}_3\left(-e^{-i \left(c+d \sqrt{x}\right)}\right)+24 a b \text{Li}_3\left(-e^{-i \left(c+d \sqrt{x}\right)}\right)+24 a b e^{2 i c} \text{Li}_3\left(e^{-i \left(c+d \sqrt{x}\right)}\right)-24 a b \text{Li}_3\left(e^{-i \left(c+d \sqrt{x}\right)}\right)+3 b^2 e^{2 i c} d^2 x \csc \left(\frac{c}{2}\right) \sin \left(\frac{d \sqrt{x}}{2}\right) \csc \left(\frac{1}{2} \left(c+d \sqrt{x}\right)\right)-3 b^2 d^2 x \csc \left(\frac{c}{2}\right) \sin \left(\frac{d \sqrt{x}}{2}\right) \csc \left(\frac{1}{2} \left(c+d \sqrt{x}\right)\right)+3 b^2 e^{2 i c} d^2 x \sec \left(\frac{c}{2}\right) \sin \left(\frac{d \sqrt{x}}{2}\right) \sec \left(\frac{1}{2} \left(c+d \sqrt{x}\right)\right)-3 b^2 d^2 x \sec \left(\frac{c}{2}\right) \sin \left(\frac{d \sqrt{x}}{2}\right) \sec \left(\frac{1}{2} \left(c+d \sqrt{x}\right)\right)+12 b^2 e^{2 i c} d \sqrt{x} \log \left(1-e^{-i \left(c+d \sqrt{x}\right)}\right)-12 b^2 d \sqrt{x} \log \left(1-e^{-i \left(c+d \sqrt{x}\right)}\right)+12 b^2 e^{2 i c} d \sqrt{x} \log \left(1+e^{-i \left(c+d \sqrt{x}\right)}\right)-12 b^2 d \sqrt{x} \log \left(1+e^{-i \left(c+d \sqrt{x}\right)}\right)-12 i b^2 d^2 x}{3 \left(-1+e^{2 i c}\right) d^3}","\frac{2}{3} a^2 x^{3/2}-\frac{8 a b \text{Li}_3\left(-e^{i \left(c+d \sqrt{x}\right)}\right)}{d^3}+\frac{8 a b \text{Li}_3\left(e^{i \left(c+d \sqrt{x}\right)}\right)}{d^3}+\frac{8 i a b \sqrt{x} \text{Li}_2\left(-e^{i \left(c+d \sqrt{x}\right)}\right)}{d^2}-\frac{8 i a b \sqrt{x} \text{Li}_2\left(e^{i \left(c+d \sqrt{x}\right)}\right)}{d^2}-\frac{8 a b x \tanh ^{-1}\left(e^{i \left(c+d \sqrt{x}\right)}\right)}{d}-\frac{2 i b^2 \text{Li}_2\left(e^{2 i \left(c+d \sqrt{x}\right)}\right)}{d^3}+\frac{4 b^2 \sqrt{x} \log \left(1-e^{2 i \left(c+d \sqrt{x}\right)}\right)}{d^2}-\frac{2 b^2 x \cot \left(c+d \sqrt{x}\right)}{d}-\frac{2 i b^2 x}{d}",1,"((-12*I)*b^2*d^2*x - 2*a^2*d^3*x^(3/2) + 2*a^2*d^3*E^((2*I)*c)*x^(3/2) - 12*b^2*d*Sqrt[x]*Log[1 - E^((-I)*(c + d*Sqrt[x]))] + 12*b^2*d*E^((2*I)*c)*Sqrt[x]*Log[1 - E^((-I)*(c + d*Sqrt[x]))] - 12*a*b*d^2*x*Log[1 - E^((-I)*(c + d*Sqrt[x]))] + 12*a*b*d^2*E^((2*I)*c)*x*Log[1 - E^((-I)*(c + d*Sqrt[x]))] - 12*b^2*d*Sqrt[x]*Log[1 + E^((-I)*(c + d*Sqrt[x]))] + 12*b^2*d*E^((2*I)*c)*Sqrt[x]*Log[1 + E^((-I)*(c + d*Sqrt[x]))] + 12*a*b*d^2*x*Log[1 + E^((-I)*(c + d*Sqrt[x]))] - 12*a*b*d^2*E^((2*I)*c)*x*Log[1 + E^((-I)*(c + d*Sqrt[x]))] + (12*I)*b*(-1 + E^((2*I)*c))*(b - 2*a*d*Sqrt[x])*PolyLog[2, -E^((-I)*(c + d*Sqrt[x]))] + (12*I)*b*(-1 + E^((2*I)*c))*(b + 2*a*d*Sqrt[x])*PolyLog[2, E^((-I)*(c + d*Sqrt[x]))] + 24*a*b*PolyLog[3, -E^((-I)*(c + d*Sqrt[x]))] - 24*a*b*E^((2*I)*c)*PolyLog[3, -E^((-I)*(c + d*Sqrt[x]))] - 24*a*b*PolyLog[3, E^((-I)*(c + d*Sqrt[x]))] + 24*a*b*E^((2*I)*c)*PolyLog[3, E^((-I)*(c + d*Sqrt[x]))] - 3*b^2*d^2*x*Csc[c/2]*Csc[(c + d*Sqrt[x])/2]*Sin[(d*Sqrt[x])/2] + 3*b^2*d^2*E^((2*I)*c)*x*Csc[c/2]*Csc[(c + d*Sqrt[x])/2]*Sin[(d*Sqrt[x])/2] - 3*b^2*d^2*x*Sec[c/2]*Sec[(c + d*Sqrt[x])/2]*Sin[(d*Sqrt[x])/2] + 3*b^2*d^2*E^((2*I)*c)*x*Sec[c/2]*Sec[(c + d*Sqrt[x])/2]*Sin[(d*Sqrt[x])/2])/(3*d^3*(-1 + E^((2*I)*c)))","B",1
58,1,93,47,0.433031,"\int \frac{\left(a+b \csc \left(c+d \sqrt{x}\right)\right)^2}{\sqrt{x}} \, dx","Integrate[(a + b*Csc[c + d*Sqrt[x]])^2/Sqrt[x],x]","\frac{2 a \left(a c+a d \sqrt{x}+2 b \log \left(\sin \left(\frac{1}{2} \left(c+d \sqrt{x}\right)\right)\right)-2 b \log \left(\cos \left(\frac{1}{2} \left(c+d \sqrt{x}\right)\right)\right)\right)+b^2 \tan \left(\frac{1}{2} \left(c+d \sqrt{x}\right)\right)+b^2 \left(-\cot \left(\frac{1}{2} \left(c+d \sqrt{x}\right)\right)\right)}{d}","2 a^2 \sqrt{x}-\frac{4 a b \tanh ^{-1}\left(\cos \left(c+d \sqrt{x}\right)\right)}{d}-\frac{2 b^2 \cot \left(c+d \sqrt{x}\right)}{d}",1,"(-(b^2*Cot[(c + d*Sqrt[x])/2]) + 2*a*(a*c + a*d*Sqrt[x] - 2*b*Log[Cos[(c + d*Sqrt[x])/2]] + 2*b*Log[Sin[(c + d*Sqrt[x])/2]]) + b^2*Tan[(c + d*Sqrt[x])/2])/d","A",1
59,0,0,25,20.7281237,"\int \frac{\left(a+b \csc \left(c+d \sqrt{x}\right)\right)^2}{x^{3/2}} \, dx","Integrate[(a + b*Csc[c + d*Sqrt[x]])^2/x^(3/2),x]","\int \frac{\left(a+b \csc \left(c+d \sqrt{x}\right)\right)^2}{x^{3/2}} \, dx","\text{Int}\left(\frac{\left(a+b \csc \left(c+d \sqrt{x}\right)\right)^2}{x^{3/2}},x\right)",0,"Integrate[(a + b*Csc[c + d*Sqrt[x]])^2/x^(3/2), x]","A",-1
60,0,0,25,22.2110983,"\int \frac{\left(a+b \csc \left(c+d \sqrt{x}\right)\right)^2}{x^{5/2}} \, dx","Integrate[(a + b*Csc[c + d*Sqrt[x]])^2/x^(5/2),x]","\int \frac{\left(a+b \csc \left(c+d \sqrt{x}\right)\right)^2}{x^{5/2}} \, dx","\text{Int}\left(\frac{\left(a+b \csc \left(c+d \sqrt{x}\right)\right)^2}{x^{5/2}},x\right)",0,"Integrate[(a + b*Csc[c + d*Sqrt[x]])^2/x^(5/2), x]","A",-1
61,1,57,24,0.0381181,"\int \frac{\csc ^3\left(\sqrt{x}\right)}{\sqrt{x}} \, dx","Integrate[Csc[Sqrt[x]]^3/Sqrt[x],x]","-\frac{1}{4} \csc ^2\left(\frac{\sqrt{x}}{2}\right)+\frac{1}{4} \sec ^2\left(\frac{\sqrt{x}}{2}\right)+\log \left(\sin \left(\frac{\sqrt{x}}{2}\right)\right)-\log \left(\cos \left(\frac{\sqrt{x}}{2}\right)\right)","-\tanh ^{-1}\left(\cos \left(\sqrt{x}\right)\right)-\cot \left(\sqrt{x}\right) \csc \left(\sqrt{x}\right)",1,"-1/4*Csc[Sqrt[x]/2]^2 - Log[Cos[Sqrt[x]/2]] + Log[Sin[Sqrt[x]/2]] + Sec[Sqrt[x]/2]^2/4","B",1
62,1,757,675,1.9096018,"\int \frac{x^{3/2}}{a+b \csc \left(c+d \sqrt{x}\right)} \, dx","Integrate[x^(3/2)/(a + b*Csc[c + d*Sqrt[x]]),x]","\frac{2 \csc \left(c+d \sqrt{x}\right) \left(a \sin \left(c+d \sqrt{x}\right)+b\right) \left(x^{5/2}-\frac{5 b e^{i c} \left(d^4 x^2 \log \left(1+\frac{a e^{i \left(2 c+d \sqrt{x}\right)}}{i b e^{i c}-\sqrt{e^{2 i c} \left(a^2-b^2\right)}}\right)-d^4 x^2 \log \left(1+\frac{a e^{i \left(2 c+d \sqrt{x}\right)}}{\sqrt{e^{2 i c} \left(a^2-b^2\right)}+i b e^{i c}}\right)-4 i d^3 x^{3/2} \text{Li}_2\left(\frac{i a e^{i \left(2 c+d \sqrt{x}\right)}}{e^{i c} b+i \sqrt{\left(a^2-b^2\right) e^{2 i c}}}\right)+4 i d^3 x^{3/2} \text{Li}_2\left(-\frac{a e^{i \left(2 c+d \sqrt{x}\right)}}{i e^{i c} b+\sqrt{\left(a^2-b^2\right) e^{2 i c}}}\right)+12 d^2 x \text{Li}_3\left(\frac{i a e^{i \left(2 c+d \sqrt{x}\right)}}{e^{i c} b+i \sqrt{\left(a^2-b^2\right) e^{2 i c}}}\right)-12 d^2 x \text{Li}_3\left(-\frac{a e^{i \left(2 c+d \sqrt{x}\right)}}{i e^{i c} b+\sqrt{\left(a^2-b^2\right) e^{2 i c}}}\right)+24 i d \sqrt{x} \text{Li}_4\left(\frac{i a e^{i \left(2 c+d \sqrt{x}\right)}}{e^{i c} b+i \sqrt{\left(a^2-b^2\right) e^{2 i c}}}\right)-24 i d \sqrt{x} \text{Li}_4\left(-\frac{a e^{i \left(2 c+d \sqrt{x}\right)}}{i e^{i c} b+\sqrt{\left(a^2-b^2\right) e^{2 i c}}}\right)-24 \text{Li}_5\left(\frac{i a e^{i \left(2 c+d \sqrt{x}\right)}}{e^{i c} b+i \sqrt{\left(a^2-b^2\right) e^{2 i c}}}\right)+24 \text{Li}_5\left(-\frac{a e^{i \left(2 c+d \sqrt{x}\right)}}{i e^{i c} b+\sqrt{\left(a^2-b^2\right) e^{2 i c}}}\right)\right)}{d^5 \sqrt{e^{2 i c} \left(a^2-b^2\right)}}\right)}{5 a \left(a+b \csc \left(c+d \sqrt{x}\right)\right)}","-\frac{48 i b \text{Li}_5\left(\frac{i a e^{i \left(c+d \sqrt{x}\right)}}{b-\sqrt{b^2-a^2}}\right)}{a d^5 \sqrt{b^2-a^2}}+\frac{48 i b \text{Li}_5\left(\frac{i a e^{i \left(c+d \sqrt{x}\right)}}{b+\sqrt{b^2-a^2}}\right)}{a d^5 \sqrt{b^2-a^2}}-\frac{48 b \sqrt{x} \text{Li}_4\left(\frac{i a e^{i \left(c+d \sqrt{x}\right)}}{b-\sqrt{b^2-a^2}}\right)}{a d^4 \sqrt{b^2-a^2}}+\frac{48 b \sqrt{x} \text{Li}_4\left(\frac{i a e^{i \left(c+d \sqrt{x}\right)}}{b+\sqrt{b^2-a^2}}\right)}{a d^4 \sqrt{b^2-a^2}}+\frac{24 i b x \text{Li}_3\left(\frac{i a e^{i \left(c+d \sqrt{x}\right)}}{b-\sqrt{b^2-a^2}}\right)}{a d^3 \sqrt{b^2-a^2}}-\frac{24 i b x \text{Li}_3\left(\frac{i a e^{i \left(c+d \sqrt{x}\right)}}{b+\sqrt{b^2-a^2}}\right)}{a d^3 \sqrt{b^2-a^2}}+\frac{8 b x^{3/2} \text{Li}_2\left(\frac{i a e^{i \left(c+d \sqrt{x}\right)}}{b-\sqrt{b^2-a^2}}\right)}{a d^2 \sqrt{b^2-a^2}}-\frac{8 b x^{3/2} \text{Li}_2\left(\frac{i a e^{i \left(c+d \sqrt{x}\right)}}{b+\sqrt{b^2-a^2}}\right)}{a d^2 \sqrt{b^2-a^2}}+\frac{2 i b x^2 \log \left(1-\frac{i a e^{i \left(c+d \sqrt{x}\right)}}{b-\sqrt{b^2-a^2}}\right)}{a d \sqrt{b^2-a^2}}-\frac{2 i b x^2 \log \left(1-\frac{i a e^{i \left(c+d \sqrt{x}\right)}}{\sqrt{b^2-a^2}+b}\right)}{a d \sqrt{b^2-a^2}}+\frac{2 x^{5/2}}{5 a}",1,"(2*Csc[c + d*Sqrt[x]]*(x^(5/2) - (5*b*E^(I*c)*(d^4*x^2*Log[1 + (a*E^(I*(2*c + d*Sqrt[x])))/(I*b*E^(I*c) - Sqrt[(a^2 - b^2)*E^((2*I)*c)])] - d^4*x^2*Log[1 + (a*E^(I*(2*c + d*Sqrt[x])))/(I*b*E^(I*c) + Sqrt[(a^2 - b^2)*E^((2*I)*c)])] - (4*I)*d^3*x^(3/2)*PolyLog[2, (I*a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) + I*Sqrt[(a^2 - b^2)*E^((2*I)*c)])] + (4*I)*d^3*x^(3/2)*PolyLog[2, -((a*E^(I*(2*c + d*Sqrt[x])))/(I*b*E^(I*c) + Sqrt[(a^2 - b^2)*E^((2*I)*c)]))] + 12*d^2*x*PolyLog[3, (I*a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) + I*Sqrt[(a^2 - b^2)*E^((2*I)*c)])] - 12*d^2*x*PolyLog[3, -((a*E^(I*(2*c + d*Sqrt[x])))/(I*b*E^(I*c) + Sqrt[(a^2 - b^2)*E^((2*I)*c)]))] + (24*I)*d*Sqrt[x]*PolyLog[4, (I*a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) + I*Sqrt[(a^2 - b^2)*E^((2*I)*c)])] - (24*I)*d*Sqrt[x]*PolyLog[4, -((a*E^(I*(2*c + d*Sqrt[x])))/(I*b*E^(I*c) + Sqrt[(a^2 - b^2)*E^((2*I)*c)]))] - 24*PolyLog[5, (I*a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) + I*Sqrt[(a^2 - b^2)*E^((2*I)*c)])] + 24*PolyLog[5, -((a*E^(I*(2*c + d*Sqrt[x])))/(I*b*E^(I*c) + Sqrt[(a^2 - b^2)*E^((2*I)*c)]))]))/(d^5*Sqrt[(a^2 - b^2)*E^((2*I)*c)]))*(b + a*Sin[c + d*Sqrt[x]]))/(5*a*(a + b*Csc[c + d*Sqrt[x]]))","A",1
63,1,512,407,10.7017434,"\int \frac{\sqrt{x}}{a+b \csc \left(c+d \sqrt{x}\right)} \, dx","Integrate[Sqrt[x]/(a + b*Csc[c + d*Sqrt[x]]),x]","\frac{2 \csc \left(c+d \sqrt{x}\right) \left(a \sin \left(c+d \sqrt{x}\right)+b\right) \left(x^{3/2}+\frac{3 b e^{3 i c} \left(a^2-b^2\right) \left(d^2 x \log \left(1+\frac{a e^{i \left(2 c+d \sqrt{x}\right)}}{i b e^{i c}-\sqrt{e^{2 i c} \left(a^2-b^2\right)}}\right)-d^2 x \log \left(1+\frac{a e^{i \left(2 c+d \sqrt{x}\right)}}{\sqrt{e^{2 i c} \left(a^2-b^2\right)}+i b e^{i c}}\right)-2 i d \sqrt{x} \text{Li}_2\left(\frac{i a e^{i \left(2 c+d \sqrt{x}\right)}}{e^{i c} b+i \sqrt{\left(a^2-b^2\right) e^{2 i c}}}\right)+2 i d \sqrt{x} \text{Li}_2\left(-\frac{a e^{i \left(2 c+d \sqrt{x}\right)}}{i e^{i c} b+\sqrt{\left(a^2-b^2\right) e^{2 i c}}}\right)+2 \text{Li}_3\left(\frac{i a e^{i \left(2 c+d \sqrt{x}\right)}}{e^{i c} b+i \sqrt{\left(a^2-b^2\right) e^{2 i c}}}\right)-2 \text{Li}_3\left(-\frac{a e^{i \left(2 c+d \sqrt{x}\right)}}{i e^{i c} b+\sqrt{\left(a^2-b^2\right) e^{2 i c}}}\right)\right)}{d^3 \sqrt{e^{2 i c} \left(b^2-a^2\right)} \sqrt{e^{4 i c} \left(-\left(a^2-b^2\right)^2\right)}}\right)}{3 a \left(a+b \csc \left(c+d \sqrt{x}\right)\right)}","\frac{4 i b \text{Li}_3\left(\frac{i a e^{i \left(c+d \sqrt{x}\right)}}{b-\sqrt{b^2-a^2}}\right)}{a d^3 \sqrt{b^2-a^2}}-\frac{4 i b \text{Li}_3\left(\frac{i a e^{i \left(c+d \sqrt{x}\right)}}{b+\sqrt{b^2-a^2}}\right)}{a d^3 \sqrt{b^2-a^2}}+\frac{4 b \sqrt{x} \text{Li}_2\left(\frac{i a e^{i \left(c+d \sqrt{x}\right)}}{b-\sqrt{b^2-a^2}}\right)}{a d^2 \sqrt{b^2-a^2}}-\frac{4 b \sqrt{x} \text{Li}_2\left(\frac{i a e^{i \left(c+d \sqrt{x}\right)}}{b+\sqrt{b^2-a^2}}\right)}{a d^2 \sqrt{b^2-a^2}}+\frac{2 i b x \log \left(1-\frac{i a e^{i \left(c+d \sqrt{x}\right)}}{b-\sqrt{b^2-a^2}}\right)}{a d \sqrt{b^2-a^2}}-\frac{2 i b x \log \left(1-\frac{i a e^{i \left(c+d \sqrt{x}\right)}}{\sqrt{b^2-a^2}+b}\right)}{a d \sqrt{b^2-a^2}}+\frac{2 x^{3/2}}{3 a}",1,"(2*Csc[c + d*Sqrt[x]]*(x^(3/2) + (3*b*(a^2 - b^2)*E^((3*I)*c)*(d^2*x*Log[1 + (a*E^(I*(2*c + d*Sqrt[x])))/(I*b*E^(I*c) - Sqrt[(a^2 - b^2)*E^((2*I)*c)])] - d^2*x*Log[1 + (a*E^(I*(2*c + d*Sqrt[x])))/(I*b*E^(I*c) + Sqrt[(a^2 - b^2)*E^((2*I)*c)])] - (2*I)*d*Sqrt[x]*PolyLog[2, (I*a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) + I*Sqrt[(a^2 - b^2)*E^((2*I)*c)])] + (2*I)*d*Sqrt[x]*PolyLog[2, -((a*E^(I*(2*c + d*Sqrt[x])))/(I*b*E^(I*c) + Sqrt[(a^2 - b^2)*E^((2*I)*c)]))] + 2*PolyLog[3, (I*a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) + I*Sqrt[(a^2 - b^2)*E^((2*I)*c)])] - 2*PolyLog[3, -((a*E^(I*(2*c + d*Sqrt[x])))/(I*b*E^(I*c) + Sqrt[(a^2 - b^2)*E^((2*I)*c)]))]))/(d^3*Sqrt[(-a^2 + b^2)*E^((2*I)*c)]*Sqrt[-((a^2 - b^2)^2*E^((4*I)*c))]))*(b + a*Sin[c + d*Sqrt[x]]))/(3*a*(a + b*Csc[c + d*Sqrt[x]]))","A",0
64,1,68,66,0.2124877,"\int \frac{1}{\sqrt{x} \left(a+b \csc \left(c+d \sqrt{x}\right)\right)} \, dx","Integrate[1/(Sqrt[x]*(a + b*Csc[c + d*Sqrt[x]])),x]","\frac{2 \left(-\frac{2 b \tan ^{-1}\left(\frac{a+b \tan \left(\frac{1}{2} \left(c+d \sqrt{x}\right)\right)}{\sqrt{b^2-a^2}}\right)}{d \sqrt{b^2-a^2}}+\frac{c}{d}+\sqrt{x}\right)}{a}","\frac{4 b \tanh ^{-1}\left(\frac{a+b \tan \left(\frac{1}{2} \left(c+d \sqrt{x}\right)\right)}{\sqrt{a^2-b^2}}\right)}{a d \sqrt{a^2-b^2}}+\frac{2 \sqrt{x}}{a}",1,"(2*(c/d + Sqrt[x] - (2*b*ArcTan[(a + b*Tan[(c + d*Sqrt[x])/2])/Sqrt[-a^2 + b^2]])/(Sqrt[-a^2 + b^2]*d)))/a","A",1
65,0,0,25,3.5767247,"\int \frac{1}{x^{3/2} \left(a+b \csc \left(c+d \sqrt{x}\right)\right)} \, dx","Integrate[1/(x^(3/2)*(a + b*Csc[c + d*Sqrt[x]])),x]","\int \frac{1}{x^{3/2} \left(a+b \csc \left(c+d \sqrt{x}\right)\right)} \, dx","\text{Int}\left(\frac{1}{x^{3/2} \left(a+b \csc \left(c+d \sqrt{x}\right)\right)},x\right)",0,"Integrate[1/(x^(3/2)*(a + b*Csc[c + d*Sqrt[x]])), x]","A",-1
66,0,0,25,3.480424,"\int \frac{1}{x^{5/2} \left(a+b \csc \left(c+d \sqrt{x}\right)\right)} \, dx","Integrate[1/(x^(5/2)*(a + b*Csc[c + d*Sqrt[x]])),x]","\int \frac{1}{x^{5/2} \left(a+b \csc \left(c+d \sqrt{x}\right)\right)} \, dx","\text{Int}\left(\frac{1}{x^{5/2} \left(a+b \csc \left(c+d \sqrt{x}\right)\right)},x\right)",0,"Integrate[1/(x^(5/2)*(a + b*Csc[c + d*Sqrt[x]])), x]","A",-1
67,1,2293,1977,13.3458924,"\int \frac{x^{3/2}}{\left(a+b \csc \left(c+d \sqrt{x}\right)\right)^2} \, dx","Integrate[x^(3/2)/(a + b*Csc[c + d*Sqrt[x]])^2,x]","\text{Result too large to show}","-\frac{2 i x^2 \log \left(1-\frac{i a e^{i \left(c+d \sqrt{x}\right)}}{b-\sqrt{b^2-a^2}}\right) b^3}{a^2 \left(b^2-a^2\right)^{3/2} d}+\frac{2 i x^2 \log \left(1-\frac{i a e^{i \left(c+d \sqrt{x}\right)}}{b+\sqrt{b^2-a^2}}\right) b^3}{a^2 \left(b^2-a^2\right)^{3/2} d}-\frac{8 x^{3/2} \text{Li}_2\left(\frac{i a e^{i \left(c+d \sqrt{x}\right)}}{b-\sqrt{b^2-a^2}}\right) b^3}{a^2 \left(b^2-a^2\right)^{3/2} d^2}+\frac{8 x^{3/2} \text{Li}_2\left(\frac{i a e^{i \left(c+d \sqrt{x}\right)}}{b+\sqrt{b^2-a^2}}\right) b^3}{a^2 \left(b^2-a^2\right)^{3/2} d^2}-\frac{24 i x \text{Li}_3\left(\frac{i a e^{i \left(c+d \sqrt{x}\right)}}{b-\sqrt{b^2-a^2}}\right) b^3}{a^2 \left(b^2-a^2\right)^{3/2} d^3}+\frac{24 i x \text{Li}_3\left(\frac{i a e^{i \left(c+d \sqrt{x}\right)}}{b+\sqrt{b^2-a^2}}\right) b^3}{a^2 \left(b^2-a^2\right)^{3/2} d^3}+\frac{48 \sqrt{x} \text{Li}_4\left(\frac{i a e^{i \left(c+d \sqrt{x}\right)}}{b-\sqrt{b^2-a^2}}\right) b^3}{a^2 \left(b^2-a^2\right)^{3/2} d^4}-\frac{48 \sqrt{x} \text{Li}_4\left(\frac{i a e^{i \left(c+d \sqrt{x}\right)}}{b+\sqrt{b^2-a^2}}\right) b^3}{a^2 \left(b^2-a^2\right)^{3/2} d^4}+\frac{48 i \text{Li}_5\left(\frac{i a e^{i \left(c+d \sqrt{x}\right)}}{b-\sqrt{b^2-a^2}}\right) b^3}{a^2 \left(b^2-a^2\right)^{3/2} d^5}-\frac{48 i \text{Li}_5\left(\frac{i a e^{i \left(c+d \sqrt{x}\right)}}{b+\sqrt{b^2-a^2}}\right) b^3}{a^2 \left(b^2-a^2\right)^{3/2} d^5}-\frac{2 i x^2 b^2}{a^2 \left(a^2-b^2\right) d}+\frac{8 x^{3/2} \log \left(\frac{e^{i \left(c+d \sqrt{x}\right)} a}{i b-\sqrt{a^2-b^2}}+1\right) b^2}{a^2 \left(a^2-b^2\right) d^2}+\frac{8 x^{3/2} \log \left(\frac{e^{i \left(c+d \sqrt{x}\right)} a}{i b+\sqrt{a^2-b^2}}+1\right) b^2}{a^2 \left(a^2-b^2\right) d^2}-\frac{24 i x \text{Li}_2\left(-\frac{a e^{i \left(c+d \sqrt{x}\right)}}{i b-\sqrt{a^2-b^2}}\right) b^2}{a^2 \left(a^2-b^2\right) d^3}-\frac{24 i x \text{Li}_2\left(-\frac{a e^{i \left(c+d \sqrt{x}\right)}}{i b+\sqrt{a^2-b^2}}\right) b^2}{a^2 \left(a^2-b^2\right) d^3}+\frac{48 \sqrt{x} \text{Li}_3\left(-\frac{a e^{i \left(c+d \sqrt{x}\right)}}{i b-\sqrt{a^2-b^2}}\right) b^2}{a^2 \left(a^2-b^2\right) d^4}+\frac{48 \sqrt{x} \text{Li}_3\left(-\frac{a e^{i \left(c+d \sqrt{x}\right)}}{i b+\sqrt{a^2-b^2}}\right) b^2}{a^2 \left(a^2-b^2\right) d^4}+\frac{48 i \text{Li}_4\left(-\frac{a e^{i \left(c+d \sqrt{x}\right)}}{i b-\sqrt{a^2-b^2}}\right) b^2}{a^2 \left(a^2-b^2\right) d^5}+\frac{48 i \text{Li}_4\left(-\frac{a e^{i \left(c+d \sqrt{x}\right)}}{i b+\sqrt{a^2-b^2}}\right) b^2}{a^2 \left(a^2-b^2\right) d^5}-\frac{2 x^2 \cos \left(c+d \sqrt{x}\right) b^2}{a \left(a^2-b^2\right) d \left(b+a \sin \left(c+d \sqrt{x}\right)\right)}+\frac{4 i x^2 \log \left(1-\frac{i a e^{i \left(c+d \sqrt{x}\right)}}{b-\sqrt{b^2-a^2}}\right) b}{a^2 \sqrt{b^2-a^2} d}-\frac{4 i x^2 \log \left(1-\frac{i a e^{i \left(c+d \sqrt{x}\right)}}{b+\sqrt{b^2-a^2}}\right) b}{a^2 \sqrt{b^2-a^2} d}+\frac{16 x^{3/2} \text{Li}_2\left(\frac{i a e^{i \left(c+d \sqrt{x}\right)}}{b-\sqrt{b^2-a^2}}\right) b}{a^2 \sqrt{b^2-a^2} d^2}-\frac{16 x^{3/2} \text{Li}_2\left(\frac{i a e^{i \left(c+d \sqrt{x}\right)}}{b+\sqrt{b^2-a^2}}\right) b}{a^2 \sqrt{b^2-a^2} d^2}+\frac{48 i x \text{Li}_3\left(\frac{i a e^{i \left(c+d \sqrt{x}\right)}}{b-\sqrt{b^2-a^2}}\right) b}{a^2 \sqrt{b^2-a^2} d^3}-\frac{48 i x \text{Li}_3\left(\frac{i a e^{i \left(c+d \sqrt{x}\right)}}{b+\sqrt{b^2-a^2}}\right) b}{a^2 \sqrt{b^2-a^2} d^3}-\frac{96 \sqrt{x} \text{Li}_4\left(\frac{i a e^{i \left(c+d \sqrt{x}\right)}}{b-\sqrt{b^2-a^2}}\right) b}{a^2 \sqrt{b^2-a^2} d^4}+\frac{96 \sqrt{x} \text{Li}_4\left(\frac{i a e^{i \left(c+d \sqrt{x}\right)}}{b+\sqrt{b^2-a^2}}\right) b}{a^2 \sqrt{b^2-a^2} d^4}-\frac{96 i \text{Li}_5\left(\frac{i a e^{i \left(c+d \sqrt{x}\right)}}{b-\sqrt{b^2-a^2}}\right) b}{a^2 \sqrt{b^2-a^2} d^5}+\frac{96 i \text{Li}_5\left(\frac{i a e^{i \left(c+d \sqrt{x}\right)}}{b+\sqrt{b^2-a^2}}\right) b}{a^2 \sqrt{b^2-a^2} d^5}+\frac{2 x^{5/2}}{5 a^2}",1,"(2*x^(5/2)*Csc[c + d*Sqrt[x]]^2*(b + a*Sin[c + d*Sqrt[x]])^2)/(5*a^2*(a + b*Csc[c + d*Sqrt[x]])^2) - ((2*I)*b*Csc[c + d*Sqrt[x]]^2*((2*b*d^4*E^((2*I)*c)*x^2)/(-1 + E^((2*I)*c)) + ((4*I)*b*d^3*Sqrt[(a^2 - b^2)*E^((2*I)*c)]*x^(3/2)*Log[1 + (a*E^(I*(2*c + d*Sqrt[x])))/(I*b*E^(I*c) - Sqrt[(a^2 - b^2)*E^((2*I)*c)])] - (2*I)*a^2*d^4*E^(I*c)*x^2*Log[1 + (a*E^(I*(2*c + d*Sqrt[x])))/(I*b*E^(I*c) - Sqrt[(a^2 - b^2)*E^((2*I)*c)])] + I*b^2*d^4*E^(I*c)*x^2*Log[1 + (a*E^(I*(2*c + d*Sqrt[x])))/(I*b*E^(I*c) - Sqrt[(a^2 - b^2)*E^((2*I)*c)])] + (4*I)*b*d^3*Sqrt[(a^2 - b^2)*E^((2*I)*c)]*x^(3/2)*Log[1 + (a*E^(I*(2*c + d*Sqrt[x])))/(I*b*E^(I*c) + Sqrt[(a^2 - b^2)*E^((2*I)*c)])] + (2*I)*a^2*d^4*E^(I*c)*x^2*Log[1 + (a*E^(I*(2*c + d*Sqrt[x])))/(I*b*E^(I*c) + Sqrt[(a^2 - b^2)*E^((2*I)*c)])] - I*b^2*d^4*E^(I*c)*x^2*Log[1 + (a*E^(I*(2*c + d*Sqrt[x])))/(I*b*E^(I*c) + Sqrt[(a^2 - b^2)*E^((2*I)*c)])] + 4*d^2*(3*b*Sqrt[(a^2 - b^2)*E^((2*I)*c)] - 2*a^2*d*E^(I*c)*Sqrt[x] + b^2*d*E^(I*c)*Sqrt[x])*x*PolyLog[2, (I*a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) + I*Sqrt[(a^2 - b^2)*E^((2*I)*c)])] + 4*d^2*(3*b*Sqrt[(a^2 - b^2)*E^((2*I)*c)] + 2*a^2*d*E^(I*c)*Sqrt[x] - b^2*d*E^(I*c)*Sqrt[x])*x*PolyLog[2, -((a*E^(I*(2*c + d*Sqrt[x])))/(I*b*E^(I*c) + Sqrt[(a^2 - b^2)*E^((2*I)*c)]))] + (24*I)*b*d*Sqrt[(a^2 - b^2)*E^((2*I)*c)]*Sqrt[x]*PolyLog[3, (I*a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) + I*Sqrt[(a^2 - b^2)*E^((2*I)*c)])] - (24*I)*a^2*d^2*E^(I*c)*x*PolyLog[3, (I*a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) + I*Sqrt[(a^2 - b^2)*E^((2*I)*c)])] + (12*I)*b^2*d^2*E^(I*c)*x*PolyLog[3, (I*a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) + I*Sqrt[(a^2 - b^2)*E^((2*I)*c)])] + (24*I)*b*d*Sqrt[(a^2 - b^2)*E^((2*I)*c)]*Sqrt[x]*PolyLog[3, -((a*E^(I*(2*c + d*Sqrt[x])))/(I*b*E^(I*c) + Sqrt[(a^2 - b^2)*E^((2*I)*c)]))] + (24*I)*a^2*d^2*E^(I*c)*x*PolyLog[3, -((a*E^(I*(2*c + d*Sqrt[x])))/(I*b*E^(I*c) + Sqrt[(a^2 - b^2)*E^((2*I)*c)]))] - (12*I)*b^2*d^2*E^(I*c)*x*PolyLog[3, -((a*E^(I*(2*c + d*Sqrt[x])))/(I*b*E^(I*c) + Sqrt[(a^2 - b^2)*E^((2*I)*c)]))] - 24*b*Sqrt[(a^2 - b^2)*E^((2*I)*c)]*PolyLog[4, (I*a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) + I*Sqrt[(a^2 - b^2)*E^((2*I)*c)])] + 48*a^2*d*E^(I*c)*Sqrt[x]*PolyLog[4, (I*a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) + I*Sqrt[(a^2 - b^2)*E^((2*I)*c)])] - 24*b^2*d*E^(I*c)*Sqrt[x]*PolyLog[4, (I*a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) + I*Sqrt[(a^2 - b^2)*E^((2*I)*c)])] - 24*b*Sqrt[(a^2 - b^2)*E^((2*I)*c)]*PolyLog[4, -((a*E^(I*(2*c + d*Sqrt[x])))/(I*b*E^(I*c) + Sqrt[(a^2 - b^2)*E^((2*I)*c)]))] - 48*a^2*d*E^(I*c)*Sqrt[x]*PolyLog[4, -((a*E^(I*(2*c + d*Sqrt[x])))/(I*b*E^(I*c) + Sqrt[(a^2 - b^2)*E^((2*I)*c)]))] + 24*b^2*d*E^(I*c)*Sqrt[x]*PolyLog[4, -((a*E^(I*(2*c + d*Sqrt[x])))/(I*b*E^(I*c) + Sqrt[(a^2 - b^2)*E^((2*I)*c)]))] + (48*I)*a^2*E^(I*c)*PolyLog[5, (I*a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) + I*Sqrt[(a^2 - b^2)*E^((2*I)*c)])] - (24*I)*b^2*E^(I*c)*PolyLog[5, (I*a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) + I*Sqrt[(a^2 - b^2)*E^((2*I)*c)])] - (48*I)*a^2*E^(I*c)*PolyLog[5, -((a*E^(I*(2*c + d*Sqrt[x])))/(I*b*E^(I*c) + Sqrt[(a^2 - b^2)*E^((2*I)*c)]))] + (24*I)*b^2*E^(I*c)*PolyLog[5, -((a*E^(I*(2*c + d*Sqrt[x])))/(I*b*E^(I*c) + Sqrt[(a^2 - b^2)*E^((2*I)*c)]))])/Sqrt[(a^2 - b^2)*E^((2*I)*c)])*(b + a*Sin[c + d*Sqrt[x]])^2)/(a^2*(a^2 - b^2)*d^5*(a + b*Csc[c + d*Sqrt[x]])^2) + (Csc[c/2]*Csc[c + d*Sqrt[x]]^2*Sec[c/2]*(b + a*Sin[c + d*Sqrt[x]])*(-(b^3*x^2*Cos[c]) - a*b^2*x^2*Sin[d*Sqrt[x]]))/(a^2*(-a + b)*(a + b)*d*(a + b*Csc[c + d*Sqrt[x]])^2)","A",0
68,1,846,1157,8.0962417,"\int \frac{\sqrt{x}}{\left(a+b \csc \left(c+d \sqrt{x}\right)\right)^2} \, dx","Integrate[Sqrt[x]/(a + b*Csc[c + d*Sqrt[x]])^2,x]","\frac{\csc ^2\left(c+d \sqrt{x}\right) \left(b+a \sin \left(c+d \sqrt{x}\right)\right) \left(\frac{6 x \csc (c) \left(b \cos (c)+a \sin \left(d \sqrt{x}\right)\right) b^2}{(a-b) (a+b) d}-\frac{6 i \left(\frac{2 b e^{2 i c} x d^2}{-1+e^{2 i c}}+\frac{2 \left(-2 d e^{i c} \sqrt{x} a^2+b \sqrt{\left(a^2-b^2\right) e^{2 i c}}+b^2 d e^{i c} \sqrt{x}\right) \text{Li}_2\left(\frac{i a e^{i \left(2 c+d \sqrt{x}\right)}}{e^{i c} b+i \sqrt{\left(a^2-b^2\right) e^{2 i c}}}\right)+2 \left(2 d e^{i c} \sqrt{x} a^2+b \sqrt{\left(a^2-b^2\right) e^{2 i c}}-b^2 d e^{i c} \sqrt{x}\right) \text{Li}_2\left(-\frac{a e^{i \left(2 c+d \sqrt{x}\right)}}{i e^{i c} b+\sqrt{\left(a^2-b^2\right) e^{2 i c}}}\right)+i \left(d \sqrt{x} \left(\left(-2 d e^{i c} \sqrt{x} a^2+2 b \sqrt{\left(a^2-b^2\right) e^{2 i c}}+b^2 d e^{i c} \sqrt{x}\right) \log \left(\frac{e^{i \left(2 c+d \sqrt{x}\right)} a}{i b e^{i c}-\sqrt{\left(a^2-b^2\right) e^{2 i c}}}+1\right)+\left(2 d e^{i c} \sqrt{x} a^2+2 b \sqrt{\left(a^2-b^2\right) e^{2 i c}}-b^2 d e^{i c} \sqrt{x}\right) \log \left(\frac{e^{i \left(2 c+d \sqrt{x}\right)} a}{i e^{i c} b+\sqrt{\left(a^2-b^2\right) e^{2 i c}}}+1\right)\right)-2 \left(2 a^2-b^2\right) e^{i c} \text{Li}_3\left(\frac{i a e^{i \left(2 c+d \sqrt{x}\right)}}{e^{i c} b+i \sqrt{\left(a^2-b^2\right) e^{2 i c}}}\right)+2 \left(2 a^2-b^2\right) e^{i c} \text{Li}_3\left(-\frac{a e^{i \left(2 c+d \sqrt{x}\right)}}{i e^{i c} b+\sqrt{\left(a^2-b^2\right) e^{2 i c}}}\right)\right)}{\sqrt{\left(a^2-b^2\right) e^{2 i c}}}\right) \left(b+a \sin \left(c+d \sqrt{x}\right)\right) b}{\left(a^2-b^2\right) d^3}+2 x^{3/2} \left(b+a \sin \left(c+d \sqrt{x}\right)\right)\right)}{3 a^2 \left(a+b \csc \left(c+d \sqrt{x}\right)\right)^2}","-\frac{2 i x \log \left(1-\frac{i a e^{i \left(c+d \sqrt{x}\right)}}{b-\sqrt{b^2-a^2}}\right) b^3}{a^2 \left(b^2-a^2\right)^{3/2} d}+\frac{2 i x \log \left(1-\frac{i a e^{i \left(c+d \sqrt{x}\right)}}{b+\sqrt{b^2-a^2}}\right) b^3}{a^2 \left(b^2-a^2\right)^{3/2} d}-\frac{4 \sqrt{x} \text{Li}_2\left(\frac{i a e^{i \left(c+d \sqrt{x}\right)}}{b-\sqrt{b^2-a^2}}\right) b^3}{a^2 \left(b^2-a^2\right)^{3/2} d^2}+\frac{4 \sqrt{x} \text{Li}_2\left(\frac{i a e^{i \left(c+d \sqrt{x}\right)}}{b+\sqrt{b^2-a^2}}\right) b^3}{a^2 \left(b^2-a^2\right)^{3/2} d^2}-\frac{4 i \text{Li}_3\left(\frac{i a e^{i \left(c+d \sqrt{x}\right)}}{b-\sqrt{b^2-a^2}}\right) b^3}{a^2 \left(b^2-a^2\right)^{3/2} d^3}+\frac{4 i \text{Li}_3\left(\frac{i a e^{i \left(c+d \sqrt{x}\right)}}{b+\sqrt{b^2-a^2}}\right) b^3}{a^2 \left(b^2-a^2\right)^{3/2} d^3}-\frac{2 i x b^2}{a^2 \left(a^2-b^2\right) d}+\frac{4 \sqrt{x} \log \left(\frac{e^{i \left(c+d \sqrt{x}\right)} a}{i b-\sqrt{a^2-b^2}}+1\right) b^2}{a^2 \left(a^2-b^2\right) d^2}+\frac{4 \sqrt{x} \log \left(\frac{e^{i \left(c+d \sqrt{x}\right)} a}{i b+\sqrt{a^2-b^2}}+1\right) b^2}{a^2 \left(a^2-b^2\right) d^2}-\frac{4 i \text{Li}_2\left(-\frac{a e^{i \left(c+d \sqrt{x}\right)}}{i b-\sqrt{a^2-b^2}}\right) b^2}{a^2 \left(a^2-b^2\right) d^3}-\frac{4 i \text{Li}_2\left(-\frac{a e^{i \left(c+d \sqrt{x}\right)}}{i b+\sqrt{a^2-b^2}}\right) b^2}{a^2 \left(a^2-b^2\right) d^3}-\frac{2 x \cos \left(c+d \sqrt{x}\right) b^2}{a \left(a^2-b^2\right) d \left(b+a \sin \left(c+d \sqrt{x}\right)\right)}+\frac{4 i x \log \left(1-\frac{i a e^{i \left(c+d \sqrt{x}\right)}}{b-\sqrt{b^2-a^2}}\right) b}{a^2 \sqrt{b^2-a^2} d}-\frac{4 i x \log \left(1-\frac{i a e^{i \left(c+d \sqrt{x}\right)}}{b+\sqrt{b^2-a^2}}\right) b}{a^2 \sqrt{b^2-a^2} d}+\frac{8 \sqrt{x} \text{Li}_2\left(\frac{i a e^{i \left(c+d \sqrt{x}\right)}}{b-\sqrt{b^2-a^2}}\right) b}{a^2 \sqrt{b^2-a^2} d^2}-\frac{8 \sqrt{x} \text{Li}_2\left(\frac{i a e^{i \left(c+d \sqrt{x}\right)}}{b+\sqrt{b^2-a^2}}\right) b}{a^2 \sqrt{b^2-a^2} d^2}+\frac{8 i \text{Li}_3\left(\frac{i a e^{i \left(c+d \sqrt{x}\right)}}{b-\sqrt{b^2-a^2}}\right) b}{a^2 \sqrt{b^2-a^2} d^3}-\frac{8 i \text{Li}_3\left(\frac{i a e^{i \left(c+d \sqrt{x}\right)}}{b+\sqrt{b^2-a^2}}\right) b}{a^2 \sqrt{b^2-a^2} d^3}+\frac{2 x^{3/2}}{3 a^2}",1,"(Csc[c + d*Sqrt[x]]^2*(b + a*Sin[c + d*Sqrt[x]])*(2*x^(3/2)*(b + a*Sin[c + d*Sqrt[x]]) - ((6*I)*b*((2*b*d^2*E^((2*I)*c)*x)/(-1 + E^((2*I)*c)) + (2*(b*Sqrt[(a^2 - b^2)*E^((2*I)*c)] - 2*a^2*d*E^(I*c)*Sqrt[x] + b^2*d*E^(I*c)*Sqrt[x])*PolyLog[2, (I*a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) + I*Sqrt[(a^2 - b^2)*E^((2*I)*c)])] + 2*(b*Sqrt[(a^2 - b^2)*E^((2*I)*c)] + 2*a^2*d*E^(I*c)*Sqrt[x] - b^2*d*E^(I*c)*Sqrt[x])*PolyLog[2, -((a*E^(I*(2*c + d*Sqrt[x])))/(I*b*E^(I*c) + Sqrt[(a^2 - b^2)*E^((2*I)*c)]))] + I*(d*Sqrt[x]*((2*b*Sqrt[(a^2 - b^2)*E^((2*I)*c)] - 2*a^2*d*E^(I*c)*Sqrt[x] + b^2*d*E^(I*c)*Sqrt[x])*Log[1 + (a*E^(I*(2*c + d*Sqrt[x])))/(I*b*E^(I*c) - Sqrt[(a^2 - b^2)*E^((2*I)*c)])] + (2*b*Sqrt[(a^2 - b^2)*E^((2*I)*c)] + 2*a^2*d*E^(I*c)*Sqrt[x] - b^2*d*E^(I*c)*Sqrt[x])*Log[1 + (a*E^(I*(2*c + d*Sqrt[x])))/(I*b*E^(I*c) + Sqrt[(a^2 - b^2)*E^((2*I)*c)])]) - 2*(2*a^2 - b^2)*E^(I*c)*PolyLog[3, (I*a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) + I*Sqrt[(a^2 - b^2)*E^((2*I)*c)])] + 2*(2*a^2 - b^2)*E^(I*c)*PolyLog[3, -((a*E^(I*(2*c + d*Sqrt[x])))/(I*b*E^(I*c) + Sqrt[(a^2 - b^2)*E^((2*I)*c)]))]))/Sqrt[(a^2 - b^2)*E^((2*I)*c)])*(b + a*Sin[c + d*Sqrt[x]]))/((a^2 - b^2)*d^3) + (6*b^2*x*Csc[c]*(b*Cos[c] + a*Sin[d*Sqrt[x]]))/((a - b)*(a + b)*d)))/(3*a^2*(a + b*Csc[c + d*Sqrt[x]])^2)","A",1
69,1,172,125,0.6918531,"\int \frac{1}{\sqrt{x} \left(a+b \csc \left(c+d \sqrt{x}\right)\right)^2} \, dx","Integrate[1/(Sqrt[x]*(a + b*Csc[c + d*Sqrt[x]])^2),x]","\frac{2 \csc \left(c+d \sqrt{x}\right) \left(a \sin \left(c+d \sqrt{x}\right)+b\right) \left(-\frac{2 b \left(b^2-2 a^2\right) \tan ^{-1}\left(\frac{a+b \tan \left(\frac{1}{2} \left(c+d \sqrt{x}\right)\right)}{\sqrt{b^2-a^2}}\right) \left(a+b \csc \left(c+d \sqrt{x}\right)\right)}{\left(b^2-a^2\right)^{3/2}}+\frac{a b^2 \cot \left(c+d \sqrt{x}\right)}{(b-a) (a+b)}+\left(c+d \sqrt{x}\right) \left(a+b \csc \left(c+d \sqrt{x}\right)\right)\right)}{a^2 d \left(a+b \csc \left(c+d \sqrt{x}\right)\right)^2}","\frac{4 b \left(2 a^2-b^2\right) \tanh ^{-1}\left(\frac{a+b \tan \left(\frac{1}{2} \left(c+d \sqrt{x}\right)\right)}{\sqrt{a^2-b^2}}\right)}{a^2 d \left(a^2-b^2\right)^{3/2}}-\frac{2 b^2 \cot \left(c+d \sqrt{x}\right)}{a d \left(a^2-b^2\right) \left(a+b \csc \left(c+d \sqrt{x}\right)\right)}+\frac{2 \sqrt{x}}{a^2}",1,"(2*Csc[c + d*Sqrt[x]]*((a*b^2*Cot[c + d*Sqrt[x]])/((-a + b)*(a + b)) + (c + d*Sqrt[x])*(a + b*Csc[c + d*Sqrt[x]]) - (2*b*(-2*a^2 + b^2)*ArcTan[(a + b*Tan[(c + d*Sqrt[x])/2])/Sqrt[-a^2 + b^2]]*(a + b*Csc[c + d*Sqrt[x]]))/(-a^2 + b^2)^(3/2))*(b + a*Sin[c + d*Sqrt[x]]))/(a^2*d*(a + b*Csc[c + d*Sqrt[x]])^2)","A",1
70,0,0,25,27.3550568,"\int \frac{1}{x^{3/2} \left(a+b \csc \left(c+d \sqrt{x}\right)\right)^2} \, dx","Integrate[1/(x^(3/2)*(a + b*Csc[c + d*Sqrt[x]])^2),x]","\int \frac{1}{x^{3/2} \left(a+b \csc \left(c+d \sqrt{x}\right)\right)^2} \, dx","\text{Int}\left(\frac{1}{x^{3/2} \left(a+b \csc \left(c+d \sqrt{x}\right)\right)^2},x\right)",0,"Integrate[1/(x^(3/2)*(a + b*Csc[c + d*Sqrt[x]])^2), x]","A",-1
71,0,0,25,28.4975098,"\int \frac{1}{x^{5/2} \left(a+b \csc \left(c+d \sqrt{x}\right)\right)^2} \, dx","Integrate[1/(x^(5/2)*(a + b*Csc[c + d*Sqrt[x]])^2),x]","\int \frac{1}{x^{5/2} \left(a+b \csc \left(c+d \sqrt{x}\right)\right)^2} \, dx","\text{Int}\left(\frac{1}{x^{5/2} \left(a+b \csc \left(c+d \sqrt{x}\right)\right)^2},x\right)",0,"Integrate[1/(x^(5/2)*(a + b*Csc[c + d*Sqrt[x]])^2), x]","A",-1
72,0,0,32,2.8432056,"\int (e x)^m \left(a+b \csc \left(c+d x^n\right)\right)^p \, dx","Integrate[(e*x)^m*(a + b*Csc[c + d*x^n])^p,x]","\int (e x)^m \left(a+b \csc \left(c+d x^n\right)\right)^p \, dx","x^{-m} (e x)^m \text{Int}\left(x^m \left(a+b \csc \left(c+d x^n\right)\right)^p,x\right)",0,"Integrate[(e*x)^m*(a + b*Csc[c + d*x^n])^p, x]","A",-1
73,1,61,45,0.1325835,"\int (e x)^{-1+n} \left(a+b \csc \left(c+d x^n\right)\right) \, dx","Integrate[(e*x)^(-1 + n)*(a + b*Csc[c + d*x^n]),x]","\frac{x^{-n} (e x)^n \left(a \left(c+d x^n\right)+b \log \left(\sin \left(\frac{1}{2} \left(c+d x^n\right)\right)\right)-b \log \left(\cos \left(\frac{1}{2} \left(c+d x^n\right)\right)\right)\right)}{d e n}","\frac{a (e x)^n}{e n}-\frac{b x^{-n} (e x)^n \tanh ^{-1}\left(\cos \left(c+d x^n\right)\right)}{d e n}",1,"((e*x)^n*(a*(c + d*x^n) - b*Log[Cos[(c + d*x^n)/2]] + b*Log[Sin[(c + d*x^n)/2]]))/(d*e*n*x^n)","A",1
74,1,185,141,0.2194189,"\int (e x)^{-1+2 n} \left(a+b \csc \left(c+d x^n\right)\right) \, dx","Integrate[(e*x)^(-1 + 2*n)*(a + b*Csc[c + d*x^n]),x]","\frac{x^{-2 n} (e x)^{2 n} \left(a d^2 x^{2 n}+2 i b \text{Li}_2\left(-e^{i \left(d x^n+c\right)}\right)-2 i b \text{Li}_2\left(e^{i \left(d x^n+c\right)}\right)+2 b d x^n \log \left(1-e^{i \left(c+d x^n\right)}\right)-2 b d x^n \log \left(1+e^{i \left(c+d x^n\right)}\right)+2 b c \log \left(1-e^{i \left(c+d x^n\right)}\right)-2 b c \log \left(1+e^{i \left(c+d x^n\right)}\right)-2 b c \log \left(\tan \left(\frac{1}{2} \left(c+d x^n\right)\right)\right)\right)}{2 d^2 e n}","\frac{a (e x)^{2 n}}{2 e n}+\frac{i b x^{-2 n} (e x)^{2 n} \text{Li}_2\left(-e^{i \left(d x^n+c\right)}\right)}{d^2 e n}-\frac{i b x^{-2 n} (e x)^{2 n} \text{Li}_2\left(e^{i \left(d x^n+c\right)}\right)}{d^2 e n}-\frac{2 b x^{-n} (e x)^{2 n} \tanh ^{-1}\left(e^{i \left(c+d x^n\right)}\right)}{d e n}",1,"((e*x)^(2*n)*(a*d^2*x^(2*n) + 2*b*c*Log[1 - E^(I*(c + d*x^n))] + 2*b*d*x^n*Log[1 - E^(I*(c + d*x^n))] - 2*b*c*Log[1 + E^(I*(c + d*x^n))] - 2*b*d*x^n*Log[1 + E^(I*(c + d*x^n))] - 2*b*c*Log[Tan[(c + d*x^n)/2]] + (2*I)*b*PolyLog[2, -E^(I*(c + d*x^n))] - (2*I)*b*PolyLog[2, E^(I*(c + d*x^n))]))/(2*d^2*e*n*x^(2*n))","A",1
75,0,0,221,3.9601443,"\int (e x)^{-1+3 n} \left(a+b \csc \left(c+d x^n\right)\right) \, dx","Integrate[(e*x)^(-1 + 3*n)*(a + b*Csc[c + d*x^n]),x]","\int (e x)^{-1+3 n} \left(a+b \csc \left(c+d x^n\right)\right) \, dx","\frac{a (e x)^{3 n}}{3 e n}-\frac{2 b x^{-3 n} (e x)^{3 n} \text{Li}_3\left(-e^{i \left(d x^n+c\right)}\right)}{d^3 e n}+\frac{2 b x^{-3 n} (e x)^{3 n} \text{Li}_3\left(e^{i \left(d x^n+c\right)}\right)}{d^3 e n}+\frac{2 i b x^{-2 n} (e x)^{3 n} \text{Li}_2\left(-e^{i \left(d x^n+c\right)}\right)}{d^2 e n}-\frac{2 i b x^{-2 n} (e x)^{3 n} \text{Li}_2\left(e^{i \left(d x^n+c\right)}\right)}{d^2 e n}-\frac{2 b x^{-n} (e x)^{3 n} \tanh ^{-1}\left(e^{i \left(c+d x^n\right)}\right)}{d e n}",1,"Integrate[(e*x)^(-1 + 3*n)*(a + b*Csc[c + d*x^n]), x]","F",-1
76,1,102,80,0.7025265,"\int (e x)^{-1+n} \left(a+b \csc \left(c+d x^n\right)\right)^2 \, dx","Integrate[(e*x)^(-1 + n)*(a + b*Csc[c + d*x^n])^2,x]","\frac{x^{-n} (e x)^n \left(2 a \left(a c+a d x^n+2 b \log \left(\sin \left(\frac{1}{2} \left(c+d x^n\right)\right)\right)-2 b \log \left(\cos \left(\frac{1}{2} \left(c+d x^n\right)\right)\right)\right)+b^2 \tan \left(\frac{1}{2} \left(c+d x^n\right)\right)+b^2 \left(-\cot \left(\frac{1}{2} \left(c+d x^n\right)\right)\right)\right)}{2 d e n}","\frac{a^2 (e x)^n}{e n}-\frac{2 a b x^{-n} (e x)^n \tanh ^{-1}\left(\cos \left(c+d x^n\right)\right)}{d e n}-\frac{b^2 x^{-n} (e x)^n \cot \left(c+d x^n\right)}{d e n}",1,"((e*x)^n*(-(b^2*Cot[(c + d*x^n)/2]) + 2*a*(a*c + a*d*x^n - 2*b*Log[Cos[(c + d*x^n)/2]] + 2*b*Log[Sin[(c + d*x^n)/2]]) + b^2*Tan[(c + d*x^n)/2]))/(2*d*e*n*x^n)","A",1
77,1,286,214,6.2821406,"\int (e x)^{-1+2 n} \left(a+b \csc \left(c+d x^n\right)\right)^2 \, dx","Integrate[(e*x)^(-1 + 2*n)*(a + b*Csc[c + d*x^n])^2,x]","\frac{x^{-2 n} (e x)^{2 n} \left(d x^n \left(a^2 d x^n-2 b^2 \cot (c)\right)+4 a b \left(2 \tan ^{-1}(\tan (c)) \tanh ^{-1}\left(\cos (c)-\sin (c) \tan \left(\frac{d x^n}{2}\right)\right)+\frac{\sec (c) \left(i \text{Li}_2\left(-e^{i \left(d x^n+\tan ^{-1}(\tan (c))\right)}\right)-i \text{Li}_2\left(e^{i \left(d x^n+\tan ^{-1}(\tan (c))\right)}\right)+\left(\tan ^{-1}(\tan (c))+d x^n\right) \left(\log \left(1-e^{i \left(\tan ^{-1}(\tan (c))+d x^n\right)}\right)-\log \left(1+e^{i \left(\tan ^{-1}(\tan (c))+d x^n\right)}\right)\right)\right)}{\sqrt{\sec ^2(c)}}\right)+2 b^2 d \cot (c) x^n+b^2 d \csc \left(\frac{c}{2}\right) x^n \sin \left(\frac{d x^n}{2}\right) \csc \left(\frac{1}{2} \left(c+d x^n\right)\right)+b^2 d \sec \left(\frac{c}{2}\right) x^n \sin \left(\frac{d x^n}{2}\right) \sec \left(\frac{1}{2} \left(c+d x^n\right)\right)-2 b^2 \left(d \cot (c) x^n-\log \left(\sin \left(c+d x^n\right)\right)\right)\right)}{2 d^2 e n}","\frac{a^2 (e x)^{2 n}}{2 e n}+\frac{2 i a b x^{-2 n} (e x)^{2 n} \text{Li}_2\left(-e^{i \left(d x^n+c\right)}\right)}{d^2 e n}-\frac{2 i a b x^{-2 n} (e x)^{2 n} \text{Li}_2\left(e^{i \left(d x^n+c\right)}\right)}{d^2 e n}-\frac{4 a b x^{-n} (e x)^{2 n} \tanh ^{-1}\left(e^{i \left(c+d x^n\right)}\right)}{d e n}+\frac{b^2 x^{-2 n} (e x)^{2 n} \log \left(\sin \left(c+d x^n\right)\right)}{d^2 e n}-\frac{b^2 x^{-n} (e x)^{2 n} \cot \left(c+d x^n\right)}{d e n}",1,"((e*x)^(2*n)*(2*b^2*d*x^n*Cot[c] + d*x^n*(a^2*d*x^n - 2*b^2*Cot[c]) - 2*b^2*(d*x^n*Cot[c] - Log[Sin[c + d*x^n]]) + 4*a*b*(2*ArcTan[Tan[c]]*ArcTanh[Cos[c] - Sin[c]*Tan[(d*x^n)/2]] + (((d*x^n + ArcTan[Tan[c]])*(Log[1 - E^(I*(d*x^n + ArcTan[Tan[c]]))] - Log[1 + E^(I*(d*x^n + ArcTan[Tan[c]]))]) + I*PolyLog[2, -E^(I*(d*x^n + ArcTan[Tan[c]]))] - I*PolyLog[2, E^(I*(d*x^n + ArcTan[Tan[c]]))])*Sec[c])/Sqrt[Sec[c]^2]) + b^2*d*x^n*Csc[c/2]*Csc[(c + d*x^n)/2]*Sin[(d*x^n)/2] + b^2*d*x^n*Sec[c/2]*Sec[(c + d*x^n)/2]*Sin[(d*x^n)/2]))/(2*d^2*e*n*x^(2*n))","A",0
78,0,0,377,12.7389839,"\int (e x)^{-1+3 n} \left(a+b \csc \left(c+d x^n\right)\right)^2 \, dx","Integrate[(e*x)^(-1 + 3*n)*(a + b*Csc[c + d*x^n])^2,x]","\int (e x)^{-1+3 n} \left(a+b \csc \left(c+d x^n\right)\right)^2 \, dx","\frac{a^2 (e x)^{3 n}}{3 e n}-\frac{4 a b x^{-3 n} (e x)^{3 n} \text{Li}_3\left(-e^{i \left(d x^n+c\right)}\right)}{d^3 e n}+\frac{4 a b x^{-3 n} (e x)^{3 n} \text{Li}_3\left(e^{i \left(d x^n+c\right)}\right)}{d^3 e n}+\frac{4 i a b x^{-2 n} (e x)^{3 n} \text{Li}_2\left(-e^{i \left(d x^n+c\right)}\right)}{d^2 e n}-\frac{4 i a b x^{-2 n} (e x)^{3 n} \text{Li}_2\left(e^{i \left(d x^n+c\right)}\right)}{d^2 e n}-\frac{4 a b x^{-n} (e x)^{3 n} \tanh ^{-1}\left(e^{i \left(c+d x^n\right)}\right)}{d e n}-\frac{i b^2 x^{-3 n} (e x)^{3 n} \text{Li}_2\left(e^{2 i \left(d x^n+c\right)}\right)}{d^3 e n}+\frac{2 b^2 x^{-2 n} (e x)^{3 n} \log \left(1-e^{2 i \left(c+d x^n\right)}\right)}{d^2 e n}-\frac{b^2 x^{-n} (e x)^{3 n} \cot \left(c+d x^n\right)}{d e n}-\frac{i b^2 x^{-n} (e x)^{3 n}}{d e n}",1,"Integrate[(e*x)^(-1 + 3*n)*(a + b*Csc[c + d*x^n])^2, x]","F",-1
79,1,79,85,0.2546987,"\int \frac{(e x)^{-1+n}}{a+b \csc \left(c+d x^n\right)} \, dx","Integrate[(e*x)^(-1 + n)/(a + b*Csc[c + d*x^n]),x]","\frac{(e x)^n \left(-\frac{2 b x^{-n} \tan ^{-1}\left(\frac{a+b \tan \left(\frac{1}{2} \left(c+d x^n\right)\right)}{\sqrt{b^2-a^2}}\right)}{\sqrt{b^2-a^2}}+c x^{-n}+d\right)}{a d e n}","\frac{2 b x^{-n} (e x)^n \tanh ^{-1}\left(\frac{a+b \tan \left(\frac{1}{2} \left(c+d x^n\right)\right)}{\sqrt{a^2-b^2}}\right)}{a d e n \sqrt{a^2-b^2}}+\frac{(e x)^n}{a e n}",1,"((e*x)^n*(d + c/x^n - (2*b*ArcTan[(a + b*Tan[(c + d*x^n)/2])/Sqrt[-a^2 + b^2]])/(Sqrt[-a^2 + b^2]*x^n)))/(a*d*e*n)","A",1
80,1,1003,338,5.0691717,"\int \frac{(e x)^{-1+2 n}}{a+b \csc \left(c+d x^n\right)} \, dx","Integrate[(e*x)^(-1 + 2*n)/(a + b*Csc[c + d*x^n]),x]","\frac{(e x)^{2 n} \csc \left(d x^n+c\right) \left(1-\frac{2 b x^{-2 n} \left(\frac{\pi  \tan ^{-1}\left(\frac{a+b \tan \left(\frac{1}{2} \left(d x^n+c\right)\right)}{\sqrt{b^2-a^2}}\right)}{\sqrt{b^2-a^2}}+\frac{2 \left(c-\cos ^{-1}\left(-\frac{b}{a}\right)\right) \tanh ^{-1}\left(\frac{(a-b) \cot \left(\frac{1}{4} \left(2 d x^n+2 c+\pi \right)\right)}{\sqrt{a^2-b^2}}\right)+\left(-2 d x^n-2 c+\pi \right) \tanh ^{-1}\left(\frac{(a+b) \tan \left(\frac{1}{4} \left(2 d x^n+2 c+\pi \right)\right)}{\sqrt{a^2-b^2}}\right)-\left(\cos ^{-1}\left(-\frac{b}{a}\right)-2 i \tanh ^{-1}\left(\frac{(a-b) \cot \left(\frac{1}{4} \left(2 d x^n+2 c+\pi \right)\right)}{\sqrt{a^2-b^2}}\right)\right) \log \left(\frac{(a+b) \left(a-b-i \sqrt{a^2-b^2}\right) \left(i \cot \left(\frac{1}{4} \left(2 d x^n+2 c+\pi \right)\right)+1\right)}{a \left(a+b+\sqrt{a^2-b^2} \cot \left(\frac{1}{4} \left(2 d x^n+2 c+\pi \right)\right)\right)}\right)+\left(\cos ^{-1}\left(-\frac{b}{a}\right)+2 i \left(\tanh ^{-1}\left(\frac{(a+b) \tan \left(\frac{1}{4} \left(2 d x^n+2 c+\pi \right)\right)}{\sqrt{a^2-b^2}}\right)-\tanh ^{-1}\left(\frac{(a-b) \cot \left(\frac{1}{4} \left(2 d x^n+2 c+\pi \right)\right)}{\sqrt{a^2-b^2}}\right)\right)\right) \log \left(\frac{\sqrt[4]{-1} \sqrt{a^2-b^2} e^{-\frac{1}{2} i \left(d x^n+c\right)}}{\sqrt{2} \sqrt{a} \sqrt{b+a \sin \left(d x^n+c\right)}}\right)+\left(\cos ^{-1}\left(-\frac{b}{a}\right)+2 i \tanh ^{-1}\left(\frac{(a-b) \cot \left(\frac{1}{4} \left(2 d x^n+2 c+\pi \right)\right)}{\sqrt{a^2-b^2}}\right)-2 i \tanh ^{-1}\left(\frac{(a+b) \tan \left(\frac{1}{4} \left(2 d x^n+2 c+\pi \right)\right)}{\sqrt{a^2-b^2}}\right)\right) \log \left(-\frac{(-1)^{3/4} \sqrt{a^2-b^2} e^{\frac{1}{2} i \left(d x^n+c\right)}}{\sqrt{2} \sqrt{a} \sqrt{b+a \sin \left(d x^n+c\right)}}\right)-\left(\cos ^{-1}\left(-\frac{b}{a}\right)+2 i \tanh ^{-1}\left(\frac{(a-b) \cot \left(\frac{1}{4} \left(2 d x^n+2 c+\pi \right)\right)}{\sqrt{a^2-b^2}}\right)\right) \log \left(\frac{i \left(i b+\sqrt{a^2-b^2}\right) \left(a+b+\sqrt{a^2-b^2} \tan \left(\frac{1}{4} \left(2 d x^n+2 c-\pi \right)\right)\right)}{a \left(a+b+\sqrt{a^2-b^2} \cot \left(\frac{1}{4} \left(2 d x^n+2 c+\pi \right)\right)\right)}+1\right)+i \left(\text{Li}_2\left(\frac{\left(b-i \sqrt{a^2-b^2}\right) \left(a+b+\sqrt{a^2-b^2} \tan \left(\frac{1}{4} \left(2 d x^n+2 c-\pi \right)\right)\right)}{a \left(a+b+\sqrt{a^2-b^2} \cot \left(\frac{1}{4} \left(2 d x^n+2 c+\pi \right)\right)\right)}\right)-\text{Li}_2\left(\frac{\left(b+i \sqrt{a^2-b^2}\right) \left(a+b+\sqrt{a^2-b^2} \tan \left(\frac{1}{4} \left(2 d x^n+2 c-\pi \right)\right)\right)}{a \left(a+b+\sqrt{a^2-b^2} \cot \left(\frac{1}{4} \left(2 d x^n+2 c+\pi \right)\right)\right)}\right)\right)}{\sqrt{a^2-b^2}}\right)}{d^2}\right) \left(b+a \sin \left(d x^n+c\right)\right)}{2 a e n \left(a+b \csc \left(d x^n+c\right)\right)}","\frac{b x^{-2 n} (e x)^{2 n} \text{Li}_2\left(\frac{i a e^{i \left(d x^n+c\right)}}{b-\sqrt{b^2-a^2}}\right)}{a d^2 e n \sqrt{b^2-a^2}}-\frac{b x^{-2 n} (e x)^{2 n} \text{Li}_2\left(\frac{i a e^{i \left(d x^n+c\right)}}{b+\sqrt{b^2-a^2}}\right)}{a d^2 e n \sqrt{b^2-a^2}}+\frac{i b x^{-n} (e x)^{2 n} \log \left(1-\frac{i a e^{i \left(c+d x^n\right)}}{b-\sqrt{b^2-a^2}}\right)}{a d e n \sqrt{b^2-a^2}}-\frac{i b x^{-n} (e x)^{2 n} \log \left(1-\frac{i a e^{i \left(c+d x^n\right)}}{\sqrt{b^2-a^2}+b}\right)}{a d e n \sqrt{b^2-a^2}}+\frac{(e x)^{2 n}}{2 a e n}",1,"((e*x)^(2*n)*Csc[c + d*x^n]*(1 - (2*b*((Pi*ArcTan[(a + b*Tan[(c + d*x^n)/2])/Sqrt[-a^2 + b^2]])/Sqrt[-a^2 + b^2] + (2*(c - ArcCos[-(b/a)])*ArcTanh[((a - b)*Cot[(2*c + Pi + 2*d*x^n)/4])/Sqrt[a^2 - b^2]] + (-2*c + Pi - 2*d*x^n)*ArcTanh[((a + b)*Tan[(2*c + Pi + 2*d*x^n)/4])/Sqrt[a^2 - b^2]] - (ArcCos[-(b/a)] - (2*I)*ArcTanh[((a - b)*Cot[(2*c + Pi + 2*d*x^n)/4])/Sqrt[a^2 - b^2]])*Log[((a + b)*(a - b - I*Sqrt[a^2 - b^2])*(1 + I*Cot[(2*c + Pi + 2*d*x^n)/4]))/(a*(a + b + Sqrt[a^2 - b^2]*Cot[(2*c + Pi + 2*d*x^n)/4]))] + (ArcCos[-(b/a)] + (2*I)*(-ArcTanh[((a - b)*Cot[(2*c + Pi + 2*d*x^n)/4])/Sqrt[a^2 - b^2]] + ArcTanh[((a + b)*Tan[(2*c + Pi + 2*d*x^n)/4])/Sqrt[a^2 - b^2]]))*Log[((-1)^(1/4)*Sqrt[a^2 - b^2])/(Sqrt[2]*Sqrt[a]*E^((I/2)*(c + d*x^n))*Sqrt[b + a*Sin[c + d*x^n]])] + (ArcCos[-(b/a)] + (2*I)*ArcTanh[((a - b)*Cot[(2*c + Pi + 2*d*x^n)/4])/Sqrt[a^2 - b^2]] - (2*I)*ArcTanh[((a + b)*Tan[(2*c + Pi + 2*d*x^n)/4])/Sqrt[a^2 - b^2]])*Log[-(((-1)^(3/4)*Sqrt[a^2 - b^2]*E^((I/2)*(c + d*x^n)))/(Sqrt[2]*Sqrt[a]*Sqrt[b + a*Sin[c + d*x^n]]))] - (ArcCos[-(b/a)] + (2*I)*ArcTanh[((a - b)*Cot[(2*c + Pi + 2*d*x^n)/4])/Sqrt[a^2 - b^2]])*Log[1 + (I*(I*b + Sqrt[a^2 - b^2])*(a + b + Sqrt[a^2 - b^2]*Tan[(2*c - Pi + 2*d*x^n)/4]))/(a*(a + b + Sqrt[a^2 - b^2]*Cot[(2*c + Pi + 2*d*x^n)/4]))] + I*(PolyLog[2, ((b - I*Sqrt[a^2 - b^2])*(a + b + Sqrt[a^2 - b^2]*Tan[(2*c - Pi + 2*d*x^n)/4]))/(a*(a + b + Sqrt[a^2 - b^2]*Cot[(2*c + Pi + 2*d*x^n)/4]))] - PolyLog[2, ((b + I*Sqrt[a^2 - b^2])*(a + b + Sqrt[a^2 - b^2]*Tan[(2*c - Pi + 2*d*x^n)/4]))/(a*(a + b + Sqrt[a^2 - b^2]*Cot[(2*c + Pi + 2*d*x^n)/4]))]))/Sqrt[a^2 - b^2]))/(d^2*x^(2*n)))*(b + a*Sin[c + d*x^n]))/(2*a*e*n*(a + b*Csc[c + d*x^n]))","B",0
81,0,0,499,1.8788502,"\int \frac{(e x)^{-1+3 n}}{a+b \csc \left(c+d x^n\right)} \, dx","Integrate[(e*x)^(-1 + 3*n)/(a + b*Csc[c + d*x^n]),x]","\int \frac{(e x)^{-1+3 n}}{a+b \csc \left(c+d x^n\right)} \, dx","\frac{2 i b x^{-3 n} (e x)^{3 n} \text{Li}_3\left(\frac{i a e^{i \left(d x^n+c\right)}}{b-\sqrt{b^2-a^2}}\right)}{a d^3 e n \sqrt{b^2-a^2}}-\frac{2 i b x^{-3 n} (e x)^{3 n} \text{Li}_3\left(\frac{i a e^{i \left(d x^n+c\right)}}{b+\sqrt{b^2-a^2}}\right)}{a d^3 e n \sqrt{b^2-a^2}}+\frac{2 b x^{-2 n} (e x)^{3 n} \text{Li}_2\left(\frac{i a e^{i \left(d x^n+c\right)}}{b-\sqrt{b^2-a^2}}\right)}{a d^2 e n \sqrt{b^2-a^2}}-\frac{2 b x^{-2 n} (e x)^{3 n} \text{Li}_2\left(\frac{i a e^{i \left(d x^n+c\right)}}{b+\sqrt{b^2-a^2}}\right)}{a d^2 e n \sqrt{b^2-a^2}}+\frac{i b x^{-n} (e x)^{3 n} \log \left(1-\frac{i a e^{i \left(c+d x^n\right)}}{b-\sqrt{b^2-a^2}}\right)}{a d e n \sqrt{b^2-a^2}}-\frac{i b x^{-n} (e x)^{3 n} \log \left(1-\frac{i a e^{i \left(c+d x^n\right)}}{\sqrt{b^2-a^2}+b}\right)}{a d e n \sqrt{b^2-a^2}}+\frac{(e x)^{3 n}}{3 a e n}",1,"Integrate[(e*x)^(-1 + 3*n)/(a + b*Csc[c + d*x^n]), x]","F",-1
82,1,176,156,0.9478681,"\int \frac{(e x)^{-1+n}}{\left(a+b \csc \left(c+d x^n\right)\right)^2} \, dx","Integrate[(e*x)^(-1 + n)/(a + b*Csc[c + d*x^n])^2,x]","\frac{x^{-n} (e x)^n \left(\sqrt{b^2-a^2} \left(\left(a^2-b^2\right) \left(c+d x^n\right) \left(a+b \csc \left(c+d x^n\right)\right)-a b^2 \cot \left(c+d x^n\right)\right)+2 b \left(b^2-2 a^2\right) \tan ^{-1}\left(\frac{a+b \tan \left(\frac{1}{2} \left(c+d x^n\right)\right)}{\sqrt{b^2-a^2}}\right) \left(a+b \csc \left(c+d x^n\right)\right)\right)}{a^2 d e n (a-b) (a+b) \sqrt{b^2-a^2} \left(a+b \csc \left(c+d x^n\right)\right)}","\frac{2 b \left(2 a^2-b^2\right) x^{-n} (e x)^n \tanh ^{-1}\left(\frac{a+b \tan \left(\frac{1}{2} \left(c+d x^n\right)\right)}{\sqrt{a^2-b^2}}\right)}{a^2 d e n \left(a^2-b^2\right)^{3/2}}-\frac{b^2 x^{-n} (e x)^n \cot \left(c+d x^n\right)}{a d e n \left(a^2-b^2\right) \left(a+b \csc \left(c+d x^n\right)\right)}+\frac{(e x)^n}{a^2 e n}",1,"((e*x)^n*(2*b*(-2*a^2 + b^2)*ArcTan[(a + b*Tan[(c + d*x^n)/2])/Sqrt[-a^2 + b^2]]*(a + b*Csc[c + d*x^n]) + Sqrt[-a^2 + b^2]*(-(a*b^2*Cot[c + d*x^n]) + (a^2 - b^2)*(c + d*x^n)*(a + b*Csc[c + d*x^n]))))/(a^2*(a - b)*(a + b)*Sqrt[-a^2 + b^2]*d*e*n*x^n*(a + b*Csc[c + d*x^n]))","A",1
83,1,2839,778,10.3207508,"\int \frac{(e x)^{-1+2 n}}{\left(a+b \csc \left(c+d x^n\right)\right)^2} \, dx","Integrate[(e*x)^(-1 + 2*n)/(a + b*Csc[c + d*x^n])^2,x]","\text{Result too large to show}","\frac{2 b x^{-2 n} (e x)^{2 n} \text{Li}_2\left(\frac{i a e^{i \left(d x^n+c\right)}}{b-\sqrt{b^2-a^2}}\right)}{a^2 d^2 e n \sqrt{b^2-a^2}}-\frac{2 b x^{-2 n} (e x)^{2 n} \text{Li}_2\left(\frac{i a e^{i \left(d x^n+c\right)}}{b+\sqrt{b^2-a^2}}\right)}{a^2 d^2 e n \sqrt{b^2-a^2}}+\frac{b^2 x^{-2 n} (e x)^{2 n} \log \left(a \sin \left(c+d x^n\right)+b\right)}{a^2 d^2 e n \left(a^2-b^2\right)}+\frac{2 i b x^{-n} (e x)^{2 n} \log \left(1-\frac{i a e^{i \left(c+d x^n\right)}}{b-\sqrt{b^2-a^2}}\right)}{a^2 d e n \sqrt{b^2-a^2}}-\frac{2 i b x^{-n} (e x)^{2 n} \log \left(1-\frac{i a e^{i \left(c+d x^n\right)}}{\sqrt{b^2-a^2}+b}\right)}{a^2 d e n \sqrt{b^2-a^2}}-\frac{b^2 x^{-n} (e x)^{2 n} \cos \left(c+d x^n\right)}{a d e n \left(a^2-b^2\right) \left(a \sin \left(c+d x^n\right)+b\right)}-\frac{b^3 x^{-2 n} (e x)^{2 n} \text{Li}_2\left(\frac{i a e^{i \left(d x^n+c\right)}}{b-\sqrt{b^2-a^2}}\right)}{a^2 d^2 e n \left(b^2-a^2\right)^{3/2}}+\frac{b^3 x^{-2 n} (e x)^{2 n} \text{Li}_2\left(\frac{i a e^{i \left(d x^n+c\right)}}{b+\sqrt{b^2-a^2}}\right)}{a^2 d^2 e n \left(b^2-a^2\right)^{3/2}}-\frac{i b^3 x^{-n} (e x)^{2 n} \log \left(1-\frac{i a e^{i \left(c+d x^n\right)}}{b-\sqrt{b^2-a^2}}\right)}{a^2 d e n \left(b^2-a^2\right)^{3/2}}+\frac{i b^3 x^{-n} (e x)^{2 n} \log \left(1-\frac{i a e^{i \left(c+d x^n\right)}}{\sqrt{b^2-a^2}+b}\right)}{a^2 d e n \left(b^2-a^2\right)^{3/2}}+\frac{(e x)^{2 n}}{2 a^2 e n}",1,"-1/2*(b^2*x^(1 - n)*(e*x)^(-1 + 2*n)*Csc[c/2]*Csc[c + d*x^n]^2*Sec[c/2]*(b*Cos[c] + a*Sin[d*x^n])*(b + a*Sin[c + d*x^n]))/(a^2*(-a + b)*(a + b)*d*n*(a + b*Csc[c + d*x^n])^2) - (b^2*x^(1 - n)*(e*x)^(-1 + 2*n)*Cot[c]*Csc[c + d*x^n]^2*(b + a*Sin[c + d*x^n])^2)/(a^2*(-a^2 + b^2)*d*n*(a + b*Csc[c + d*x^n])^2) + (2*b^3*x^(1 - 2*n)*(e*x)^(-1 + 2*n)*ArcTanh[(a*Cos[c + d*x^n] + I*(b + a*Sin[c + d*x^n]))/Sqrt[a^2 - b^2]]*Cot[c]*Csc[c + d*x^n]^2*(b + a*Sin[c + d*x^n])^2)/(a^2*(a^2 - b^2)^(3/2)*d^2*n*(a + b*Csc[c + d*x^n])^2) - (2*b*x^(1 - 2*n)*(e*x)^(-1 + 2*n)*Csc[c + d*x^n]^2*((Pi*ArcTan[(a + b*Tan[(c + d*x^n)/2])/Sqrt[-a^2 + b^2]])/Sqrt[-a^2 + b^2] + (2*(-c + Pi/2 - d*x^n)*ArcTanh[((a + b)*Cot[(-c + Pi/2 - d*x^n)/2])/Sqrt[a^2 - b^2]] - 2*(-c + ArcCos[-(b/a)])*ArcTanh[((a - b)*Tan[(-c + Pi/2 - d*x^n)/2])/Sqrt[a^2 - b^2]] + (ArcCos[-(b/a)] - (2*I)*(ArcTanh[((a + b)*Cot[(-c + Pi/2 - d*x^n)/2])/Sqrt[a^2 - b^2]] - ArcTanh[((a - b)*Tan[(-c + Pi/2 - d*x^n)/2])/Sqrt[a^2 - b^2]]))*Log[Sqrt[a^2 - b^2]/(Sqrt[2]*Sqrt[a]*E^((I/2)*(-c + Pi/2 - d*x^n))*Sqrt[b + a*Sin[c + d*x^n]])] + (ArcCos[-(b/a)] + (2*I)*(ArcTanh[((a + b)*Cot[(-c + Pi/2 - d*x^n)/2])/Sqrt[a^2 - b^2]] - ArcTanh[((a - b)*Tan[(-c + Pi/2 - d*x^n)/2])/Sqrt[a^2 - b^2]]))*Log[(Sqrt[a^2 - b^2]*E^((I/2)*(-c + Pi/2 - d*x^n)))/(Sqrt[2]*Sqrt[a]*Sqrt[b + a*Sin[c + d*x^n]])] - (ArcCos[-(b/a)] + (2*I)*ArcTanh[((a - b)*Tan[(-c + Pi/2 - d*x^n)/2])/Sqrt[a^2 - b^2]])*Log[1 - ((b - I*Sqrt[a^2 - b^2])*(a + b - Sqrt[a^2 - b^2]*Tan[(-c + Pi/2 - d*x^n)/2]))/(a*(a + b + Sqrt[a^2 - b^2]*Tan[(-c + Pi/2 - d*x^n)/2]))] + (-ArcCos[-(b/a)] + (2*I)*ArcTanh[((a - b)*Tan[(-c + Pi/2 - d*x^n)/2])/Sqrt[a^2 - b^2]])*Log[1 - ((b + I*Sqrt[a^2 - b^2])*(a + b - Sqrt[a^2 - b^2]*Tan[(-c + Pi/2 - d*x^n)/2]))/(a*(a + b + Sqrt[a^2 - b^2]*Tan[(-c + Pi/2 - d*x^n)/2]))] + I*(PolyLog[2, ((b - I*Sqrt[a^2 - b^2])*(a + b - Sqrt[a^2 - b^2]*Tan[(-c + Pi/2 - d*x^n)/2]))/(a*(a + b + Sqrt[a^2 - b^2]*Tan[(-c + Pi/2 - d*x^n)/2]))] - PolyLog[2, ((b + I*Sqrt[a^2 - b^2])*(a + b - Sqrt[a^2 - b^2]*Tan[(-c + Pi/2 - d*x^n)/2]))/(a*(a + b + Sqrt[a^2 - b^2]*Tan[(-c + Pi/2 - d*x^n)/2]))]))/Sqrt[a^2 - b^2])*(b + a*Sin[c + d*x^n])^2)/((a^2 - b^2)*d^2*n*(a + b*Csc[c + d*x^n])^2) + (b^3*x^(1 - 2*n)*(e*x)^(-1 + 2*n)*Csc[c + d*x^n]^2*((Pi*ArcTan[(a + b*Tan[(c + d*x^n)/2])/Sqrt[-a^2 + b^2]])/Sqrt[-a^2 + b^2] + (2*(-c + Pi/2 - d*x^n)*ArcTanh[((a + b)*Cot[(-c + Pi/2 - d*x^n)/2])/Sqrt[a^2 - b^2]] - 2*(-c + ArcCos[-(b/a)])*ArcTanh[((a - b)*Tan[(-c + Pi/2 - d*x^n)/2])/Sqrt[a^2 - b^2]] + (ArcCos[-(b/a)] - (2*I)*(ArcTanh[((a + b)*Cot[(-c + Pi/2 - d*x^n)/2])/Sqrt[a^2 - b^2]] - ArcTanh[((a - b)*Tan[(-c + Pi/2 - d*x^n)/2])/Sqrt[a^2 - b^2]]))*Log[Sqrt[a^2 - b^2]/(Sqrt[2]*Sqrt[a]*E^((I/2)*(-c + Pi/2 - d*x^n))*Sqrt[b + a*Sin[c + d*x^n]])] + (ArcCos[-(b/a)] + (2*I)*(ArcTanh[((a + b)*Cot[(-c + Pi/2 - d*x^n)/2])/Sqrt[a^2 - b^2]] - ArcTanh[((a - b)*Tan[(-c + Pi/2 - d*x^n)/2])/Sqrt[a^2 - b^2]]))*Log[(Sqrt[a^2 - b^2]*E^((I/2)*(-c + Pi/2 - d*x^n)))/(Sqrt[2]*Sqrt[a]*Sqrt[b + a*Sin[c + d*x^n]])] - (ArcCos[-(b/a)] + (2*I)*ArcTanh[((a - b)*Tan[(-c + Pi/2 - d*x^n)/2])/Sqrt[a^2 - b^2]])*Log[1 - ((b - I*Sqrt[a^2 - b^2])*(a + b - Sqrt[a^2 - b^2]*Tan[(-c + Pi/2 - d*x^n)/2]))/(a*(a + b + Sqrt[a^2 - b^2]*Tan[(-c + Pi/2 - d*x^n)/2]))] + (-ArcCos[-(b/a)] + (2*I)*ArcTanh[((a - b)*Tan[(-c + Pi/2 - d*x^n)/2])/Sqrt[a^2 - b^2]])*Log[1 - ((b + I*Sqrt[a^2 - b^2])*(a + b - Sqrt[a^2 - b^2]*Tan[(-c + Pi/2 - d*x^n)/2]))/(a*(a + b + Sqrt[a^2 - b^2]*Tan[(-c + Pi/2 - d*x^n)/2]))] + I*(PolyLog[2, ((b - I*Sqrt[a^2 - b^2])*(a + b - Sqrt[a^2 - b^2]*Tan[(-c + Pi/2 - d*x^n)/2]))/(a*(a + b + Sqrt[a^2 - b^2]*Tan[(-c + Pi/2 - d*x^n)/2]))] - PolyLog[2, ((b + I*Sqrt[a^2 - b^2])*(a + b - Sqrt[a^2 - b^2]*Tan[(-c + Pi/2 - d*x^n)/2]))/(a*(a + b + Sqrt[a^2 - b^2]*Tan[(-c + Pi/2 - d*x^n)/2]))]))/Sqrt[a^2 - b^2])*(b + a*Sin[c + d*x^n])^2)/(a^2*(a^2 - b^2)*d^2*n*(a + b*Csc[c + d*x^n])^2) + (x^(1 - n)*(e*x)^(-1 + 2*n)*Csc[c/2]*Csc[c + d*x^n]^2*Sec[c/2]*(-2*b^2*Cos[c] + a^2*d*x^n*Sin[c] - b^2*d*x^n*Sin[c])*(b + a*Sin[c + d*x^n])^2)/(4*a^2*(a - b)*(a + b)*d*n*(a + b*Csc[c + d*x^n])^2) + (b^2*x^(1 - 2*n)*(e*x)^(-1 + 2*n)*Csc[c]*Csc[c + d*x^n]^2*(-(a*d*x^n*Cos[c]) + a*Log[b + a*Cos[d*x^n]*Sin[c] + a*Cos[c]*Sin[d*x^n]]*Sin[c] + ((2*I)*a*b*ArcTan[(I*a*Cos[c] - I*(-b + a*Sin[c])*Tan[(d*x^n)/2])/Sqrt[-b^2 + a^2*Cos[c]^2 + a^2*Sin[c]^2]]*Cos[c])/Sqrt[-b^2 + a^2*Cos[c]^2 + a^2*Sin[c]^2])*(b + a*Sin[c + d*x^n])^2)/(a*(a^2 - b^2)*d^2*n*(a + b*Csc[c + d*x^n])^2*(a^2*Cos[c]^2 + a^2*Sin[c]^2))","B",0
84,0,0,1417,10.6659604,"\int \frac{(e x)^{-1+3 n}}{\left(a+b \csc \left(c+d x^n\right)\right)^2} \, dx","Integrate[(e*x)^(-1 + 3*n)/(a + b*Csc[c + d*x^n])^2,x]","\int \frac{(e x)^{-1+3 n}}{\left(a+b \csc \left(c+d x^n\right)\right)^2} \, dx","-\frac{2 i b^2 (e x)^{3 n} \text{Li}_2\left(-\frac{a e^{i \left(d x^n+c\right)}}{i b-\sqrt{a^2-b^2}}\right) x^{-3 n}}{a^2 \left(a^2-b^2\right) d^3 e n}-\frac{2 i b^2 (e x)^{3 n} \text{Li}_2\left(-\frac{a e^{i \left(d x^n+c\right)}}{i b+\sqrt{a^2-b^2}}\right) x^{-3 n}}{a^2 \left(a^2-b^2\right) d^3 e n}+\frac{4 i b (e x)^{3 n} \text{Li}_3\left(\frac{i a e^{i \left(d x^n+c\right)}}{b-\sqrt{b^2-a^2}}\right) x^{-3 n}}{a^2 \sqrt{b^2-a^2} d^3 e n}-\frac{2 i b^3 (e x)^{3 n} \text{Li}_3\left(\frac{i a e^{i \left(d x^n+c\right)}}{b-\sqrt{b^2-a^2}}\right) x^{-3 n}}{a^2 \left(b^2-a^2\right)^{3/2} d^3 e n}-\frac{4 i b (e x)^{3 n} \text{Li}_3\left(\frac{i a e^{i \left(d x^n+c\right)}}{b+\sqrt{b^2-a^2}}\right) x^{-3 n}}{a^2 \sqrt{b^2-a^2} d^3 e n}+\frac{2 i b^3 (e x)^{3 n} \text{Li}_3\left(\frac{i a e^{i \left(d x^n+c\right)}}{b+\sqrt{b^2-a^2}}\right) x^{-3 n}}{a^2 \left(b^2-a^2\right)^{3/2} d^3 e n}+\frac{2 b^2 (e x)^{3 n} \log \left(\frac{e^{i \left(d x^n+c\right)} a}{i b-\sqrt{a^2-b^2}}+1\right) x^{-2 n}}{a^2 \left(a^2-b^2\right) d^2 e n}+\frac{2 b^2 (e x)^{3 n} \log \left(\frac{e^{i \left(d x^n+c\right)} a}{i b+\sqrt{a^2-b^2}}+1\right) x^{-2 n}}{a^2 \left(a^2-b^2\right) d^2 e n}+\frac{4 b (e x)^{3 n} \text{Li}_2\left(\frac{i a e^{i \left(d x^n+c\right)}}{b-\sqrt{b^2-a^2}}\right) x^{-2 n}}{a^2 \sqrt{b^2-a^2} d^2 e n}-\frac{2 b^3 (e x)^{3 n} \text{Li}_2\left(\frac{i a e^{i \left(d x^n+c\right)}}{b-\sqrt{b^2-a^2}}\right) x^{-2 n}}{a^2 \left(b^2-a^2\right)^{3/2} d^2 e n}-\frac{4 b (e x)^{3 n} \text{Li}_2\left(\frac{i a e^{i \left(d x^n+c\right)}}{b+\sqrt{b^2-a^2}}\right) x^{-2 n}}{a^2 \sqrt{b^2-a^2} d^2 e n}+\frac{2 b^3 (e x)^{3 n} \text{Li}_2\left(\frac{i a e^{i \left(d x^n+c\right)}}{b+\sqrt{b^2-a^2}}\right) x^{-2 n}}{a^2 \left(b^2-a^2\right)^{3/2} d^2 e n}-\frac{i b^2 (e x)^{3 n} x^{-n}}{a^2 \left(a^2-b^2\right) d e n}+\frac{2 i b (e x)^{3 n} \log \left(1-\frac{i a e^{i \left(d x^n+c\right)}}{b-\sqrt{b^2-a^2}}\right) x^{-n}}{a^2 \sqrt{b^2-a^2} d e n}-\frac{i b^3 (e x)^{3 n} \log \left(1-\frac{i a e^{i \left(d x^n+c\right)}}{b-\sqrt{b^2-a^2}}\right) x^{-n}}{a^2 \left(b^2-a^2\right)^{3/2} d e n}-\frac{2 i b (e x)^{3 n} \log \left(1-\frac{i a e^{i \left(d x^n+c\right)}}{b+\sqrt{b^2-a^2}}\right) x^{-n}}{a^2 \sqrt{b^2-a^2} d e n}+\frac{i b^3 (e x)^{3 n} \log \left(1-\frac{i a e^{i \left(d x^n+c\right)}}{b+\sqrt{b^2-a^2}}\right) x^{-n}}{a^2 \left(b^2-a^2\right)^{3/2} d e n}-\frac{b^2 (e x)^{3 n} \cos \left(d x^n+c\right) x^{-n}}{a \left(a^2-b^2\right) d e n \left(b+a \sin \left(d x^n+c\right)\right)}+\frac{(e x)^{3 n}}{3 a^2 e n}",1,"Integrate[(e*x)^(-1 + 3*n)/(a + b*Csc[c + d*x^n])^2, x]","F",-1