1,1,107,114,0.1440247,"\int x^6 \left(a+b \csc ^{-1}(c x)\right) \, dx","Integrate[x^6*(a + b*ArcCsc[c*x]),x]","\frac{a x^7}{7}+\frac{5 b \log \left(x \left(\sqrt{\frac{c^2 x^2-1}{c^2 x^2}}+1\right)\right)}{112 c^7}+b \sqrt{\frac{c^2 x^2-1}{c^2 x^2}} \left(\frac{5 x^2}{112 c^5}+\frac{5 x^4}{168 c^3}+\frac{x^6}{42 c}\right)+\frac{1}{7} b x^7 \csc ^{-1}(c x)","\frac{1}{7} x^7 \left(a+b \csc ^{-1}(c x)\right)+\frac{b x^6 \sqrt{1-\frac{1}{c^2 x^2}}}{42 c}+\frac{5 b \tanh ^{-1}\left(\sqrt{1-\frac{1}{c^2 x^2}}\right)}{112 c^7}+\frac{5 b x^2 \sqrt{1-\frac{1}{c^2 x^2}}}{112 c^5}+\frac{5 b x^4 \sqrt{1-\frac{1}{c^2 x^2}}}{168 c^3}",1,"(a*x^7)/7 + b*Sqrt[(-1 + c^2*x^2)/(c^2*x^2)]*((5*x^2)/(112*c^5) + (5*x^4)/(168*c^3) + x^6/(42*c)) + (b*x^7*ArcCsc[c*x])/7 + (5*b*Log[x*(1 + Sqrt[(-1 + c^2*x^2)/(c^2*x^2)])])/(112*c^7)","A",1
2,1,72,89,0.1295093,"\int x^5 \left(a+b \csc ^{-1}(c x)\right) \, dx","Integrate[x^5*(a + b*ArcCsc[c*x]),x]","\frac{a x^6}{6}+b \sqrt{\frac{c^2 x^2-1}{c^2 x^2}} \left(\frac{4 x}{45 c^5}+\frac{2 x^3}{45 c^3}+\frac{x^5}{30 c}\right)+\frac{1}{6} b x^6 \csc ^{-1}(c x)","\frac{1}{6} x^6 \left(a+b \csc ^{-1}(c x)\right)+\frac{b x^5 \sqrt{1-\frac{1}{c^2 x^2}}}{30 c}+\frac{4 b x \sqrt{1-\frac{1}{c^2 x^2}}}{45 c^5}+\frac{2 b x^3 \sqrt{1-\frac{1}{c^2 x^2}}}{45 c^3}",1,"(a*x^6)/6 + b*Sqrt[(-1 + c^2*x^2)/(c^2*x^2)]*((4*x)/(45*c^5) + (2*x^3)/(45*c^3) + x^5/(30*c)) + (b*x^6*ArcCsc[c*x])/6","A",1
3,1,97,89,0.0761211,"\int x^4 \left(a+b \csc ^{-1}(c x)\right) \, dx","Integrate[x^4*(a + b*ArcCsc[c*x]),x]","\frac{a x^5}{5}+\frac{3 b \log \left(x \left(\sqrt{\frac{c^2 x^2-1}{c^2 x^2}}+1\right)\right)}{40 c^5}+b \sqrt{\frac{c^2 x^2-1}{c^2 x^2}} \left(\frac{3 x^2}{40 c^3}+\frac{x^4}{20 c}\right)+\frac{1}{5} b x^5 \csc ^{-1}(c x)","\frac{1}{5} x^5 \left(a+b \csc ^{-1}(c x)\right)+\frac{b x^4 \sqrt{1-\frac{1}{c^2 x^2}}}{20 c}+\frac{3 b \tanh ^{-1}\left(\sqrt{1-\frac{1}{c^2 x^2}}\right)}{40 c^5}+\frac{3 b x^2 \sqrt{1-\frac{1}{c^2 x^2}}}{40 c^3}",1,"(a*x^5)/5 + b*Sqrt[(-1 + c^2*x^2)/(c^2*x^2)]*((3*x^2)/(40*c^3) + x^4/(20*c)) + (b*x^5*ArcCsc[c*x])/5 + (3*b*Log[x*(1 + Sqrt[(-1 + c^2*x^2)/(c^2*x^2)])])/(40*c^5)","A",1
4,1,62,64,0.1041098,"\int x^3 \left(a+b \csc ^{-1}(c x)\right) \, dx","Integrate[x^3*(a + b*ArcCsc[c*x]),x]","\frac{a x^4}{4}+b \sqrt{\frac{c^2 x^2-1}{c^2 x^2}} \left(\frac{x}{6 c^3}+\frac{x^3}{12 c}\right)+\frac{1}{4} b x^4 \csc ^{-1}(c x)","\frac{1}{4} x^4 \left(a+b \csc ^{-1}(c x)\right)+\frac{b x^3 \sqrt{1-\frac{1}{c^2 x^2}}}{12 c}+\frac{b x \sqrt{1-\frac{1}{c^2 x^2}}}{6 c^3}",1,"(a*x^4)/4 + b*Sqrt[(-1 + c^2*x^2)/(c^2*x^2)]*(x/(6*c^3) + x^3/(12*c)) + (b*x^4*ArcCsc[c*x])/4","A",1
5,1,85,64,0.0554445,"\int x^2 \left(a+b \csc ^{-1}(c x)\right) \, dx","Integrate[x^2*(a + b*ArcCsc[c*x]),x]","\frac{a x^3}{3}+\frac{b x^2 \sqrt{\frac{c^2 x^2-1}{c^2 x^2}}}{6 c}+\frac{b \log \left(x \left(\sqrt{\frac{c^2 x^2-1}{c^2 x^2}}+1\right)\right)}{6 c^3}+\frac{1}{3} b x^3 \csc ^{-1}(c x)","\frac{1}{3} x^3 \left(a+b \csc ^{-1}(c x)\right)+\frac{b x^2 \sqrt{1-\frac{1}{c^2 x^2}}}{6 c}+\frac{b \tanh ^{-1}\left(\sqrt{1-\frac{1}{c^2 x^2}}\right)}{6 c^3}",1,"(a*x^3)/3 + (b*x^2*Sqrt[(-1 + c^2*x^2)/(c^2*x^2)])/(6*c) + (b*x^3*ArcCsc[c*x])/3 + (b*Log[x*(1 + Sqrt[(-1 + c^2*x^2)/(c^2*x^2)])])/(6*c^3)","A",1
6,1,50,39,0.0228766,"\int x \left(a+b \csc ^{-1}(c x)\right) \, dx","Integrate[x*(a + b*ArcCsc[c*x]),x]","\frac{a x^2}{2}+\frac{b x \sqrt{\frac{c^2 x^2-1}{c^2 x^2}}}{2 c}+\frac{1}{2} b x^2 \csc ^{-1}(c x)","\frac{1}{2} x^2 \left(a+b \csc ^{-1}(c x)\right)+\frac{b x \sqrt{1-\frac{1}{c^2 x^2}}}{2 c}",1,"(a*x^2)/2 + (b*x*Sqrt[(-1 + c^2*x^2)/(c^2*x^2)])/(2*c) + (b*x^2*ArcCsc[c*x])/2","A",1
7,1,58,31,0.0662682,"\int \left(a+b \csc ^{-1}(c x)\right) \, dx","Integrate[a + b*ArcCsc[c*x],x]","a x+\frac{b x \sqrt{1-\frac{1}{c^2 x^2}} \tanh ^{-1}\left(\frac{c x}{\sqrt{c^2 x^2-1}}\right)}{\sqrt{c^2 x^2-1}}+b x \csc ^{-1}(c x)","a x+\frac{b \tanh ^{-1}\left(\sqrt{1-\frac{1}{c^2 x^2}}\right)}{c}+b x \csc ^{-1}(c x)",1,"a*x + b*x*ArcCsc[c*x] + (b*Sqrt[1 - 1/(c^2*x^2)]*x*ArcTanh[(c*x)/Sqrt[-1 + c^2*x^2]])/Sqrt[-1 + c^2*x^2]","A",1
8,1,53,64,0.0325858,"\int \frac{a+b \csc ^{-1}(c x)}{x} \, dx","Integrate[(a + b*ArcCsc[c*x])/x,x]","a \log (x)+\frac{1}{2} i b \left(\csc ^{-1}(c x)^2+\text{Li}_2\left(e^{2 i \csc ^{-1}(c x)}\right)\right)-b \csc ^{-1}(c x) \log \left(1-e^{2 i \csc ^{-1}(c x)}\right)","\frac{i \left(a+b \csc ^{-1}(c x)\right)^2}{2 b}-\log \left(1-e^{2 i \csc ^{-1}(c x)}\right) \left(a+b \csc ^{-1}(c x)\right)+\frac{1}{2} i b \text{Li}_2\left(e^{2 i \csc ^{-1}(c x)}\right)",1,"-(b*ArcCsc[c*x]*Log[1 - E^((2*I)*ArcCsc[c*x])]) + a*Log[x] + (I/2)*b*(ArcCsc[c*x]^2 + PolyLog[2, E^((2*I)*ArcCsc[c*x])])","A",0
9,1,41,32,0.0294181,"\int \frac{a+b \csc ^{-1}(c x)}{x^2} \, dx","Integrate[(a + b*ArcCsc[c*x])/x^2,x]","-\frac{a}{x}-b c \sqrt{\frac{c^2 x^2-1}{c^2 x^2}}-\frac{b \csc ^{-1}(c x)}{x}","-\frac{a+b \csc ^{-1}(c x)}{x}-b c \sqrt{1-\frac{1}{c^2 x^2}}",1,"-(a/x) - b*c*Sqrt[(-1 + c^2*x^2)/(c^2*x^2)] - (b*ArcCsc[c*x])/x","A",1
10,1,66,51,0.0374877,"\int \frac{a+b \csc ^{-1}(c x)}{x^3} \, dx","Integrate[(a + b*ArcCsc[c*x])/x^3,x]","-\frac{a}{2 x^2}-\frac{b c \sqrt{\frac{c^2 x^2-1}{c^2 x^2}}}{4 x}+\frac{1}{4} b c^2 \sin ^{-1}\left(\frac{1}{c x}\right)-\frac{b \csc ^{-1}(c x)}{2 x^2}","-\frac{a+b \csc ^{-1}(c x)}{2 x^2}-\frac{b c \sqrt{1-\frac{1}{c^2 x^2}}}{4 x}+\frac{1}{4} b c^2 \csc ^{-1}(c x)",1,"-1/2*a/x^2 - (b*c*Sqrt[(-1 + c^2*x^2)/(c^2*x^2)])/(4*x) - (b*ArcCsc[c*x])/(2*x^2) + (b*c^2*ArcSin[1/(c*x)])/4","A",1
11,1,59,60,0.068131,"\int \frac{a+b \csc ^{-1}(c x)}{x^4} \, dx","Integrate[(a + b*ArcCsc[c*x])/x^4,x]","-\frac{a}{3 x^3}+b \left(-\frac{2 c^3}{9}-\frac{c}{9 x^2}\right) \sqrt{\frac{c^2 x^2-1}{c^2 x^2}}-\frac{b \csc ^{-1}(c x)}{3 x^3}","-\frac{a+b \csc ^{-1}(c x)}{3 x^3}+\frac{1}{9} b c^3 \left(1-\frac{1}{c^2 x^2}\right)^{3/2}-\frac{1}{3} b c^3 \sqrt{1-\frac{1}{c^2 x^2}}",1,"-1/3*a/x^3 + b*((-2*c^3)/9 - c/(9*x^2))*Sqrt[(-1 + c^2*x^2)/(c^2*x^2)] - (b*ArcCsc[c*x])/(3*x^3)","A",1
12,1,78,76,0.060035,"\int \frac{a+b \csc ^{-1}(c x)}{x^5} \, dx","Integrate[(a + b*ArcCsc[c*x])/x^5,x]","-\frac{a}{4 x^4}+\frac{3}{32} b c^4 \sin ^{-1}\left(\frac{1}{c x}\right)+b \left(-\frac{3 c^3}{32 x}-\frac{c}{16 x^3}\right) \sqrt{\frac{c^2 x^2-1}{c^2 x^2}}-\frac{b \csc ^{-1}(c x)}{4 x^4}","-\frac{a+b \csc ^{-1}(c x)}{4 x^4}+\frac{3}{32} b c^4 \csc ^{-1}(c x)-\frac{b c \sqrt{1-\frac{1}{c^2 x^2}}}{16 x^3}-\frac{3 b c^3 \sqrt{1-\frac{1}{c^2 x^2}}}{32 x}",1,"-1/4*a/x^4 + b*(-1/16*c/x^3 - (3*c^3)/(32*x))*Sqrt[(-1 + c^2*x^2)/(c^2*x^2)] - (b*ArcCsc[c*x])/(4*x^4) + (3*b*c^4*ArcSin[1/(c*x)])/32","A",1
13,1,69,82,0.0870483,"\int \frac{a+b \csc ^{-1}(c x)}{x^6} \, dx","Integrate[(a + b*ArcCsc[c*x])/x^6,x]","-\frac{a}{5 x^5}+b \left(-\frac{8 c^5}{75}-\frac{4 c^3}{75 x^2}-\frac{c}{25 x^4}\right) \sqrt{\frac{c^2 x^2-1}{c^2 x^2}}-\frac{b \csc ^{-1}(c x)}{5 x^5}","-\frac{a+b \csc ^{-1}(c x)}{5 x^5}-\frac{1}{25} b c^5 \left(1-\frac{1}{c^2 x^2}\right)^{5/2}+\frac{2}{15} b c^5 \left(1-\frac{1}{c^2 x^2}\right)^{3/2}-\frac{1}{5} b c^5 \sqrt{1-\frac{1}{c^2 x^2}}",1,"-1/5*a/x^5 + b*((-8*c^5)/75 - c/(25*x^4) - (4*c^3)/(75*x^2))*Sqrt[(-1 + c^2*x^2)/(c^2*x^2)] - (b*ArcCsc[c*x])/(5*x^5)","A",1
14,1,88,101,0.1049731,"\int \frac{a+b \csc ^{-1}(c x)}{x^7} \, dx","Integrate[(a + b*ArcCsc[c*x])/x^7,x]","-\frac{a}{6 x^6}+\frac{5}{96} b c^6 \sin ^{-1}\left(\frac{1}{c x}\right)+b \left(-\frac{5 c^5}{96 x}-\frac{5 c^3}{144 x^3}-\frac{c}{36 x^5}\right) \sqrt{\frac{c^2 x^2-1}{c^2 x^2}}-\frac{b \csc ^{-1}(c x)}{6 x^6}","-\frac{a+b \csc ^{-1}(c x)}{6 x^6}+\frac{5}{96} b c^6 \csc ^{-1}(c x)-\frac{b c \sqrt{1-\frac{1}{c^2 x^2}}}{36 x^5}-\frac{5 b c^5 \sqrt{1-\frac{1}{c^2 x^2}}}{96 x}-\frac{5 b c^3 \sqrt{1-\frac{1}{c^2 x^2}}}{144 x^3}",1,"-1/6*a/x^6 + b*(-1/36*c/x^5 - (5*c^3)/(144*x^3) - (5*c^5)/(96*x))*Sqrt[(-1 + c^2*x^2)/(c^2*x^2)] - (b*ArcCsc[c*x])/(6*x^6) + (5*b*c^6*ArcSin[1/(c*x)])/96","A",1
15,1,124,107,0.2245005,"\int x^3 \left(a+b \csc ^{-1}(c x)\right)^2 \, dx","Integrate[x^3*(a + b*ArcCsc[c*x])^2,x]","\frac{c x \left(3 a^2 c^3 x^3+2 a b \sqrt{1-\frac{1}{c^2 x^2}} \left(c^2 x^2+2\right)+b^2 c x\right)+2 b c x \csc ^{-1}(c x) \left(3 a c^3 x^3+b \sqrt{1-\frac{1}{c^2 x^2}} \left(c^2 x^2+2\right)\right)+3 b^2 c^4 x^4 \csc ^{-1}(c x)^2+4 b^2 \log (x)}{12 c^4}","\frac{b x^3 \sqrt{1-\frac{1}{c^2 x^2}} \left(a+b \csc ^{-1}(c x)\right)}{6 c}+\frac{b x \sqrt{1-\frac{1}{c^2 x^2}} \left(a+b \csc ^{-1}(c x)\right)}{3 c^3}+\frac{1}{4} x^4 \left(a+b \csc ^{-1}(c x)\right)^2+\frac{b^2 \log (x)}{3 c^4}+\frac{b^2 x^2}{12 c^2}",1,"(c*x*(b^2*c*x + 3*a^2*c^3*x^3 + 2*a*b*Sqrt[1 - 1/(c^2*x^2)]*(2 + c^2*x^2)) + 2*b*c*x*(3*a*c^3*x^3 + b*Sqrt[1 - 1/(c^2*x^2)]*(2 + c^2*x^2))*ArcCsc[c*x] + 3*b^2*c^4*x^4*ArcCsc[c*x]^2 + 4*b^2*Log[x])/(12*c^4)","A",1
16,1,210,139,1.3649552,"\int x^2 \left(a+b \csc ^{-1}(c x)\right)^2 \, dx","Integrate[x^2*(a + b*ArcCsc[c*x])^2,x]","\frac{1}{3} \left(a^2 x^3+\frac{a b \left(c^3 x^3+\sqrt{c^2 x^2-1} \tanh ^{-1}\left(\frac{c x}{\sqrt{c^2 x^2-1}}\right)-c x\right)}{c^4 x \sqrt{1-\frac{1}{c^2 x^2}}}+2 a b x^3 \csc ^{-1}(c x)-\frac{i b^2 \text{Li}_2\left(-e^{i \csc ^{-1}(c x)}\right)}{c^3}+\frac{b^2 \left(c^3 x^3 \csc ^{-1}(c x)^2+\csc ^{-1}(c x) \left(c^2 x^2 \sqrt{1-\frac{1}{c^2 x^2}}-\log \left(1-e^{i \csc ^{-1}(c x)}\right)+\log \left(1+e^{i \csc ^{-1}(c x)}\right)\right)+i \text{Li}_2\left(e^{i \csc ^{-1}(c x)}\right)+c x\right)}{c^3}\right)","\frac{2 b \tanh ^{-1}\left(e^{i \csc ^{-1}(c x)}\right) \left(a+b \csc ^{-1}(c x)\right)}{3 c^3}+\frac{b x^2 \sqrt{1-\frac{1}{c^2 x^2}} \left(a+b \csc ^{-1}(c x)\right)}{3 c}+\frac{1}{3} x^3 \left(a+b \csc ^{-1}(c x)\right)^2-\frac{i b^2 \text{Li}_2\left(-e^{i \csc ^{-1}(c x)}\right)}{3 c^3}+\frac{i b^2 \text{Li}_2\left(e^{i \csc ^{-1}(c x)}\right)}{3 c^3}+\frac{b^2 x}{3 c^2}",1,"(a^2*x^3 + 2*a*b*x^3*ArcCsc[c*x] + (a*b*(-(c*x) + c^3*x^3 + Sqrt[-1 + c^2*x^2]*ArcTanh[(c*x)/Sqrt[-1 + c^2*x^2]]))/(c^4*Sqrt[1 - 1/(c^2*x^2)]*x) - (I*b^2*PolyLog[2, -E^(I*ArcCsc[c*x])])/c^3 + (b^2*(c*x + c^3*x^3*ArcCsc[c*x]^2 + ArcCsc[c*x]*(c^2*Sqrt[1 - 1/(c^2*x^2)]*x^2 - Log[1 - E^(I*ArcCsc[c*x])] + Log[1 + E^(I*ArcCsc[c*x])]) + I*PolyLog[2, E^(I*ArcCsc[c*x])]))/c^3)/3","A",0
17,1,89,55,0.1508709,"\int x \left(a+b \csc ^{-1}(c x)\right)^2 \, dx","Integrate[x*(a + b*ArcCsc[c*x])^2,x]","\frac{a c x \left(a c x+2 b \sqrt{1-\frac{1}{c^2 x^2}}\right)+2 b c x \csc ^{-1}(c x) \left(a c x+b \sqrt{1-\frac{1}{c^2 x^2}}\right)+b^2 c^2 x^2 \csc ^{-1}(c x)^2+2 b^2 \log (c x)}{2 c^2}","\frac{b x \sqrt{1-\frac{1}{c^2 x^2}} \left(a+b \csc ^{-1}(c x)\right)}{c}+\frac{1}{2} x^2 \left(a+b \csc ^{-1}(c x)\right)^2+\frac{b^2 \log (x)}{c^2}",1,"(a*c*x*(2*b*Sqrt[1 - 1/(c^2*x^2)] + a*c*x) + 2*b*c*x*(b*Sqrt[1 - 1/(c^2*x^2)] + a*c*x)*ArcCsc[c*x] + b^2*c^2*x^2*ArcCsc[c*x]^2 + 2*b^2*Log[c*x])/(2*c^2)","A",1
18,1,147,84,0.2100051,"\int \left(a+b \csc ^{-1}(c x)\right)^2 \, dx","Integrate[(a + b*ArcCsc[c*x])^2,x]","\frac{a^2 c x+2 a b c x \csc ^{-1}(c x)+2 a b \log \left(\cos \left(\frac{1}{2} \csc ^{-1}(c x)\right)\right)-2 a b \log \left(\sin \left(\frac{1}{2} \csc ^{-1}(c x)\right)\right)-2 i b^2 \text{Li}_2\left(-e^{i \csc ^{-1}(c x)}\right)+2 i b^2 \text{Li}_2\left(e^{i \csc ^{-1}(c x)}\right)+b^2 c x \csc ^{-1}(c x)^2-2 b^2 \csc ^{-1}(c x) \log \left(1-e^{i \csc ^{-1}(c x)}\right)+2 b^2 \csc ^{-1}(c x) \log \left(1+e^{i \csc ^{-1}(c x)}\right)}{c}","x \left(a+b \csc ^{-1}(c x)\right)^2+\frac{4 b \tanh ^{-1}\left(e^{i \csc ^{-1}(c x)}\right) \left(a+b \csc ^{-1}(c x)\right)}{c}-\frac{2 i b^2 \text{Li}_2\left(-e^{i \csc ^{-1}(c x)}\right)}{c}+\frac{2 i b^2 \text{Li}_2\left(e^{i \csc ^{-1}(c x)}\right)}{c}",1,"(a^2*c*x + 2*a*b*c*x*ArcCsc[c*x] + b^2*c*x*ArcCsc[c*x]^2 - 2*b^2*ArcCsc[c*x]*Log[1 - E^(I*ArcCsc[c*x])] + 2*b^2*ArcCsc[c*x]*Log[1 + E^(I*ArcCsc[c*x])] + 2*a*b*Log[Cos[ArcCsc[c*x]/2]] - 2*a*b*Log[Sin[ArcCsc[c*x]/2]] - (2*I)*b^2*PolyLog[2, -E^(I*ArcCsc[c*x])] + (2*I)*b^2*PolyLog[2, E^(I*ArcCsc[c*x])])/c","A",0
19,1,137,91,0.1552081,"\int \frac{\left(a+b \csc ^{-1}(c x)\right)^2}{x} \, dx","Integrate[(a + b*ArcCsc[c*x])^2/x,x]","a^2 \log (c x)+i a b \left(\csc ^{-1}(c x)^2+\text{Li}_2\left(e^{2 i \csc ^{-1}(c x)}\right)\right)-2 a b \csc ^{-1}(c x) \log \left(1-e^{2 i \csc ^{-1}(c x)}\right)+\frac{1}{24} i b^2 \left(-24 \csc ^{-1}(c x) \text{Li}_2\left(e^{-2 i \csc ^{-1}(c x)}\right)+12 i \text{Li}_3\left(e^{-2 i \csc ^{-1}(c x)}\right)-8 \csc ^{-1}(c x)^3+24 i \csc ^{-1}(c x)^2 \log \left(1-e^{-2 i \csc ^{-1}(c x)}\right)+\pi ^3\right)","i b \text{Li}_2\left(e^{2 i \csc ^{-1}(c x)}\right) \left(a+b \csc ^{-1}(c x)\right)+\frac{i \left(a+b \csc ^{-1}(c x)\right)^3}{3 b}-\log \left(1-e^{2 i \csc ^{-1}(c x)}\right) \left(a+b \csc ^{-1}(c x)\right)^2-\frac{1}{2} b^2 \text{Li}_3\left(e^{2 i \csc ^{-1}(c x)}\right)",1,"-2*a*b*ArcCsc[c*x]*Log[1 - E^((2*I)*ArcCsc[c*x])] + a^2*Log[c*x] + I*a*b*(ArcCsc[c*x]^2 + PolyLog[2, E^((2*I)*ArcCsc[c*x])]) + (I/24)*b^2*(Pi^3 - 8*ArcCsc[c*x]^3 + (24*I)*ArcCsc[c*x]^2*Log[1 - E^((-2*I)*ArcCsc[c*x])] - 24*ArcCsc[c*x]*PolyLog[2, E^((-2*I)*ArcCsc[c*x])] + (12*I)*PolyLog[3, E^((-2*I)*ArcCsc[c*x])])","A",0
20,1,71,50,0.1485594,"\int \frac{\left(a+b \csc ^{-1}(c x)\right)^2}{x^2} \, dx","Integrate[(a + b*ArcCsc[c*x])^2/x^2,x]","-\frac{a^2+2 a b c x \sqrt{1-\frac{1}{c^2 x^2}}+2 b \csc ^{-1}(c x) \left(a+b c x \sqrt{1-\frac{1}{c^2 x^2}}\right)+b^2 \csc ^{-1}(c x)^2-2 b^2}{x}","-2 b c \sqrt{1-\frac{1}{c^2 x^2}} \left(a+b \csc ^{-1}(c x)\right)-\frac{\left(a+b \csc ^{-1}(c x)\right)^2}{x}+\frac{2 b^2}{x}",1,"-((a^2 - 2*b^2 + 2*a*b*c*Sqrt[1 - 1/(c^2*x^2)]*x + 2*b*(a + b*c*Sqrt[1 - 1/(c^2*x^2)]*x)*ArcCsc[c*x] + b^2*ArcCsc[c*x]^2)/x)","A",1
21,1,102,88,0.120374,"\int \frac{\left(a+b \csc ^{-1}(c x)\right)^2}{x^3} \, dx","Integrate[(a + b*ArcCsc[c*x])^2/x^3,x]","\frac{-2 a^2-2 a b c x \sqrt{1-\frac{1}{c^2 x^2}}+2 a b c^2 x^2 \sin ^{-1}\left(\frac{1}{c x}\right)-2 b \csc ^{-1}(c x) \left(2 a+b c x \sqrt{1-\frac{1}{c^2 x^2}}\right)+b^2 \left(c^2 x^2-2\right) \csc ^{-1}(c x)^2+b^2}{4 x^2}","-\frac{b c \sqrt{1-\frac{1}{c^2 x^2}} \left(a+b \csc ^{-1}(c x)\right)}{2 x}+\frac{1}{2} a b c^2 \csc ^{-1}(c x)-\frac{\left(a+b \csc ^{-1}(c x)\right)^2}{2 x^2}+\frac{1}{4} b^2 c^2 \csc ^{-1}(c x)^2+\frac{b^2}{4 x^2}",1,"(-2*a^2 + b^2 - 2*a*b*c*Sqrt[1 - 1/(c^2*x^2)]*x - 2*b*(2*a + b*c*Sqrt[1 - 1/(c^2*x^2)]*x)*ArcCsc[c*x] + b^2*(-2 + c^2*x^2)*ArcCsc[c*x]^2 + 2*a*b*c^2*x^2*ArcSin[1/(c*x)])/(4*x^2)","A",1
22,1,108,102,0.2353805,"\int \frac{\left(a+b \csc ^{-1}(c x)\right)^2}{x^4} \, dx","Integrate[(a + b*ArcCsc[c*x])^2/x^4,x]","-\frac{9 a^2+6 a b c x \sqrt{1-\frac{1}{c^2 x^2}} \left(2 c^2 x^2+1\right)+6 b \csc ^{-1}(c x) \left(3 a+b c x \sqrt{1-\frac{1}{c^2 x^2}} \left(2 c^2 x^2+1\right)\right)-2 b^2 \left(6 c^2 x^2+1\right)+9 b^2 \csc ^{-1}(c x)^2}{27 x^3}","-\frac{2 b c \sqrt{1-\frac{1}{c^2 x^2}} \left(a+b \csc ^{-1}(c x)\right)}{9 x^2}-\frac{4}{9} b c^3 \sqrt{1-\frac{1}{c^2 x^2}} \left(a+b \csc ^{-1}(c x)\right)-\frac{\left(a+b \csc ^{-1}(c x)\right)^2}{3 x^3}+\frac{4 b^2 c^2}{9 x}+\frac{2 b^2}{27 x^3}",1,"-1/27*(9*a^2 + 6*a*b*c*Sqrt[1 - 1/(c^2*x^2)]*x*(1 + 2*c^2*x^2) - 2*b^2*(1 + 6*c^2*x^2) + 6*b*(3*a + b*c*Sqrt[1 - 1/(c^2*x^2)]*x*(1 + 2*c^2*x^2))*ArcCsc[c*x] + 9*b^2*ArcCsc[c*x]^2)/x^3","A",1
23,1,148,134,0.182704,"\int \frac{\left(a+b \csc ^{-1}(c x)\right)^2}{x^5} \, dx","Integrate[(a + b*ArcCsc[c*x])^2/x^5,x]","\frac{-8 a^2+6 a b c^4 x^4 \sin ^{-1}\left(\frac{1}{c x}\right)-4 a b c x \sqrt{1-\frac{1}{c^2 x^2}}-2 b \csc ^{-1}(c x) \left(8 a+b c x \sqrt{1-\frac{1}{c^2 x^2}} \left(3 c^2 x^2+2\right)\right)-6 a b c^3 x^3 \sqrt{1-\frac{1}{c^2 x^2}}+b^2 \left(3 c^4 x^4-8\right) \csc ^{-1}(c x)^2+3 b^2 c^2 x^2+b^2}{32 x^4}","\frac{3}{16} a b c^4 \csc ^{-1}(c x)-\frac{b c \sqrt{1-\frac{1}{c^2 x^2}} \left(a+b \csc ^{-1}(c x)\right)}{8 x^3}-\frac{3 b c^3 \sqrt{1-\frac{1}{c^2 x^2}} \left(a+b \csc ^{-1}(c x)\right)}{16 x}-\frac{\left(a+b \csc ^{-1}(c x)\right)^2}{4 x^4}+\frac{3}{32} b^2 c^4 \csc ^{-1}(c x)^2+\frac{3 b^2 c^2}{32 x^2}+\frac{b^2}{32 x^4}",1,"(-8*a^2 + b^2 - 4*a*b*c*Sqrt[1 - 1/(c^2*x^2)]*x + 3*b^2*c^2*x^2 - 6*a*b*c^3*Sqrt[1 - 1/(c^2*x^2)]*x^3 - 2*b*(8*a + b*c*Sqrt[1 - 1/(c^2*x^2)]*x*(2 + 3*c^2*x^2))*ArcCsc[c*x] + b^2*(-8 + 3*c^4*x^4)*ArcCsc[c*x]^2 + 6*a*b*c^4*x^4*ArcSin[1/(c*x)])/(32*x^4)","A",1
24,1,285,207,0.9051602,"\int x^3 \left(a+b \csc ^{-1}(c x)\right)^3 \, dx","Integrate[x^3*(a + b*ArcCsc[c*x])^3,x]","\frac{a^3 c^4 x^4+b \csc ^{-1}(c x) \left(c x \left(3 a^2 c^3 x^3+2 a b \sqrt{1-\frac{1}{c^2 x^2}} \left(c^2 x^2+2\right)+b^2 c x\right)-4 b^2 \log \left(1-e^{2 i \csc ^{-1}(c x)}\right)\right)+2 a^2 b c x \sqrt{1-\frac{1}{c^2 x^2}}+a^2 b c^3 x^3 \sqrt{1-\frac{1}{c^2 x^2}}+a b^2 c^2 x^2+b^2 \csc ^{-1}(c x)^2 \left(3 a c^4 x^4+b \left(2 c x \sqrt{1-\frac{1}{c^2 x^2}}+c^3 x^3 \sqrt{1-\frac{1}{c^2 x^2}}+2 i\right)\right)-4 a b^2 \log \left(\frac{1}{c x}\right)+b^3 c^4 x^4 \csc ^{-1}(c x)^3+b^3 c x \sqrt{1-\frac{1}{c^2 x^2}}+2 i b^3 \text{Li}_2\left(e^{2 i \csc ^{-1}(c x)}\right)}{4 c^4}","-\frac{b^2 \log \left(1-e^{2 i \csc ^{-1}(c x)}\right) \left(a+b \csc ^{-1}(c x)\right)}{c^4}+\frac{b^2 x^2 \left(a+b \csc ^{-1}(c x)\right)}{4 c^2}+\frac{i b \left(a+b \csc ^{-1}(c x)\right)^2}{2 c^4}+\frac{b x^3 \sqrt{1-\frac{1}{c^2 x^2}} \left(a+b \csc ^{-1}(c x)\right)^2}{4 c}+\frac{b x \sqrt{1-\frac{1}{c^2 x^2}} \left(a+b \csc ^{-1}(c x)\right)^2}{2 c^3}+\frac{1}{4} x^4 \left(a+b \csc ^{-1}(c x)\right)^3+\frac{i b^3 \text{Li}_2\left(e^{2 i \csc ^{-1}(c x)}\right)}{2 c^4}+\frac{b^3 x \sqrt{1-\frac{1}{c^2 x^2}}}{4 c^3}",1,"(2*a^2*b*c*Sqrt[1 - 1/(c^2*x^2)]*x + b^3*c*Sqrt[1 - 1/(c^2*x^2)]*x + a*b^2*c^2*x^2 + a^2*b*c^3*Sqrt[1 - 1/(c^2*x^2)]*x^3 + a^3*c^4*x^4 + b^2*(3*a*c^4*x^4 + b*(2*I + 2*c*Sqrt[1 - 1/(c^2*x^2)]*x + c^3*Sqrt[1 - 1/(c^2*x^2)]*x^3))*ArcCsc[c*x]^2 + b^3*c^4*x^4*ArcCsc[c*x]^3 + b*ArcCsc[c*x]*(c*x*(b^2*c*x + 3*a^2*c^3*x^3 + 2*a*b*Sqrt[1 - 1/(c^2*x^2)]*(2 + c^2*x^2)) - 4*b^2*Log[1 - E^((2*I)*ArcCsc[c*x])]) - 4*a*b^2*Log[1/(c*x)] + (2*I)*b^3*PolyLog[2, E^((2*I)*ArcCsc[c*x])])/(4*c^4)","A",0
25,1,580,220,7.8182204,"\int x^2 \left(a+b \csc ^{-1}(c x)\right)^3 \, dx","Integrate[x^2*(a + b*ArcCsc[c*x])^3,x]","\frac{a^3 x^3}{3}+\frac{a^2 b x^2 \sqrt{\frac{c^2 x^2-1}{c^2 x^2}}}{2 c}+\frac{a^2 b \log \left(x \left(\sqrt{\frac{c^2 x^2-1}{c^2 x^2}}+1\right)\right)}{2 c^3}+a^2 b x^3 \csc ^{-1}(c x)+\frac{a b^2 \left(2 c^3 x^3 \left(\frac{4 i \text{Li}_2\left(e^{i \csc ^{-1}(c x)}\right)}{c^3 x^3}+4 \csc ^{-1}(c x)^2-2 \cos \left(2 \csc ^{-1}(c x)\right)-\frac{3 \csc ^{-1}(c x) \log \left(1-e^{i \csc ^{-1}(c x)}\right)}{c x}+\frac{3 \csc ^{-1}(c x) \log \left(1+e^{i \csc ^{-1}(c x)}\right)}{c x}+2 \csc ^{-1}(c x) \sin \left(2 \csc ^{-1}(c x)\right)+\csc ^{-1}(c x) \log \left(1-e^{i \csc ^{-1}(c x)}\right) \sin \left(3 \csc ^{-1}(c x)\right)-\csc ^{-1}(c x) \log \left(1+e^{i \csc ^{-1}(c x)}\right) \sin \left(3 \csc ^{-1}(c x)\right)+2\right)-8 i \text{Li}_2\left(-e^{i \csc ^{-1}(c x)}\right)\right)}{8 c^3}+\frac{b^3 \left(16 c^3 x^3 \csc ^{-1}(c x)^3 \sin ^4\left(\frac{1}{2} \csc ^{-1}(c x)\right)-48 i \csc ^{-1}(c x) \text{Li}_2\left(-e^{i \csc ^{-1}(c x)}\right)+48 i \csc ^{-1}(c x) \text{Li}_2\left(e^{i \csc ^{-1}(c x)}\right)+48 \text{Li}_3\left(-e^{i \csc ^{-1}(c x)}\right)-48 \text{Li}_3\left(e^{i \csc ^{-1}(c x)}\right)+\frac{\csc ^{-1}(c x)^3 \csc ^4\left(\frac{1}{2} \csc ^{-1}(c x)\right)}{c x}+6 \csc ^{-1}(c x)^2 \csc ^2\left(\frac{1}{2} \csc ^{-1}(c x)\right)+4 \csc ^{-1}(c x)^3 \cot \left(\frac{1}{2} \csc ^{-1}(c x)\right)+24 \csc ^{-1}(c x) \cot \left(\frac{1}{2} \csc ^{-1}(c x)\right)-24 \csc ^{-1}(c x)^2 \log \left(1-e^{i \csc ^{-1}(c x)}\right)+24 \csc ^{-1}(c x)^2 \log \left(1+e^{i \csc ^{-1}(c x)}\right)+4 \csc ^{-1}(c x)^3 \tan \left(\frac{1}{2} \csc ^{-1}(c x)\right)+24 \csc ^{-1}(c x) \tan \left(\frac{1}{2} \csc ^{-1}(c x)\right)-6 \csc ^{-1}(c x)^2 \sec ^2\left(\frac{1}{2} \csc ^{-1}(c x)\right)-48 \log \left(\tan \left(\frac{1}{2} \csc ^{-1}(c x)\right)\right)\right)}{48 c^3}","-\frac{i b^2 \text{Li}_2\left(-e^{i \csc ^{-1}(c x)}\right) \left(a+b \csc ^{-1}(c x)\right)}{c^3}+\frac{i b^2 \text{Li}_2\left(e^{i \csc ^{-1}(c x)}\right) \left(a+b \csc ^{-1}(c x)\right)}{c^3}+\frac{b^2 x \left(a+b \csc ^{-1}(c x)\right)}{c^2}+\frac{b \tanh ^{-1}\left(e^{i \csc ^{-1}(c x)}\right) \left(a+b \csc ^{-1}(c x)\right)^2}{c^3}+\frac{b x^2 \sqrt{1-\frac{1}{c^2 x^2}} \left(a+b \csc ^{-1}(c x)\right)^2}{2 c}+\frac{1}{3} x^3 \left(a+b \csc ^{-1}(c x)\right)^3+\frac{b^3 \text{Li}_3\left(-e^{i \csc ^{-1}(c x)}\right)}{c^3}-\frac{b^3 \text{Li}_3\left(e^{i \csc ^{-1}(c x)}\right)}{c^3}+\frac{b^3 \tanh ^{-1}\left(\sqrt{1-\frac{1}{c^2 x^2}}\right)}{c^3}",1,"(a^3*x^3)/3 + (a^2*b*x^2*Sqrt[(-1 + c^2*x^2)/(c^2*x^2)])/(2*c) + a^2*b*x^3*ArcCsc[c*x] + (a^2*b*Log[x*(1 + Sqrt[(-1 + c^2*x^2)/(c^2*x^2)])])/(2*c^3) + (a*b^2*((-8*I)*PolyLog[2, -E^(I*ArcCsc[c*x])] + 2*c^3*x^3*(2 + 4*ArcCsc[c*x]^2 - 2*Cos[2*ArcCsc[c*x]] - (3*ArcCsc[c*x]*Log[1 - E^(I*ArcCsc[c*x])])/(c*x) + (3*ArcCsc[c*x]*Log[1 + E^(I*ArcCsc[c*x])])/(c*x) + ((4*I)*PolyLog[2, E^(I*ArcCsc[c*x])])/(c^3*x^3) + 2*ArcCsc[c*x]*Sin[2*ArcCsc[c*x]] + ArcCsc[c*x]*Log[1 - E^(I*ArcCsc[c*x])]*Sin[3*ArcCsc[c*x]] - ArcCsc[c*x]*Log[1 + E^(I*ArcCsc[c*x])]*Sin[3*ArcCsc[c*x]])))/(8*c^3) + (b^3*(24*ArcCsc[c*x]*Cot[ArcCsc[c*x]/2] + 4*ArcCsc[c*x]^3*Cot[ArcCsc[c*x]/2] + 6*ArcCsc[c*x]^2*Csc[ArcCsc[c*x]/2]^2 + (ArcCsc[c*x]^3*Csc[ArcCsc[c*x]/2]^4)/(c*x) - 24*ArcCsc[c*x]^2*Log[1 - E^(I*ArcCsc[c*x])] + 24*ArcCsc[c*x]^2*Log[1 + E^(I*ArcCsc[c*x])] - 48*Log[Tan[ArcCsc[c*x]/2]] - (48*I)*ArcCsc[c*x]*PolyLog[2, -E^(I*ArcCsc[c*x])] + (48*I)*ArcCsc[c*x]*PolyLog[2, E^(I*ArcCsc[c*x])] + 48*PolyLog[3, -E^(I*ArcCsc[c*x])] - 48*PolyLog[3, E^(I*ArcCsc[c*x])] - 6*ArcCsc[c*x]^2*Sec[ArcCsc[c*x]/2]^2 + 16*c^3*x^3*ArcCsc[c*x]^3*Sin[ArcCsc[c*x]/2]^4 + 24*ArcCsc[c*x]*Tan[ArcCsc[c*x]/2] + 4*ArcCsc[c*x]^3*Tan[ArcCsc[c*x]/2]))/(48*c^3)","B",0
26,1,182,126,0.4862477,"\int x \left(a+b \csc ^{-1}(c x)\right)^3 \, dx","Integrate[x*(a + b*ArcCsc[c*x])^3,x]","\frac{a \left(a c x \left(a c x+3 b \sqrt{1-\frac{1}{c^2 x^2}}\right)-6 b^2 \log \left(\frac{1}{c x}\right)\right)+3 b^2 \csc ^{-1}(c x)^2 \left(a c^2 x^2+b \left(c x \sqrt{1-\frac{1}{c^2 x^2}}+i\right)\right)+3 b \csc ^{-1}(c x) \left(a c x \left(a c x+2 b \sqrt{1-\frac{1}{c^2 x^2}}\right)-2 b^2 \log \left(1-e^{2 i \csc ^{-1}(c x)}\right)\right)+b^3 c^2 x^2 \csc ^{-1}(c x)^3+3 i b^3 \text{Li}_2\left(e^{2 i \csc ^{-1}(c x)}\right)}{2 c^2}","-\frac{3 b^2 \log \left(1-e^{2 i \csc ^{-1}(c x)}\right) \left(a+b \csc ^{-1}(c x)\right)}{c^2}+\frac{3 b x \sqrt{1-\frac{1}{c^2 x^2}} \left(a+b \csc ^{-1}(c x)\right)^2}{2 c}+\frac{3 i b \left(a+b \csc ^{-1}(c x)\right)^2}{2 c^2}+\frac{1}{2} x^2 \left(a+b \csc ^{-1}(c x)\right)^3+\frac{3 i b^3 \text{Li}_2\left(e^{2 i \csc ^{-1}(c x)}\right)}{2 c^2}",1,"(3*b^2*(a*c^2*x^2 + b*(I + c*Sqrt[1 - 1/(c^2*x^2)]*x))*ArcCsc[c*x]^2 + b^3*c^2*x^2*ArcCsc[c*x]^3 + 3*b*ArcCsc[c*x]*(a*c*x*(2*b*Sqrt[1 - 1/(c^2*x^2)] + a*c*x) - 2*b^2*Log[1 - E^((2*I)*ArcCsc[c*x])]) + a*(a*c*x*(3*b*Sqrt[1 - 1/(c^2*x^2)] + a*c*x) - 6*b^2*Log[1/(c*x)]) + (3*I)*b^3*PolyLog[2, E^((2*I)*ArcCsc[c*x])])/(2*c^2)","A",0
27,1,265,144,0.2965952,"\int \left(a+b \csc ^{-1}(c x)\right)^3 \, dx","Integrate[(a + b*ArcCsc[c*x])^3,x]","\frac{a^3 c x+3 a^2 b \log \left(c x \left(\sqrt{1-\frac{1}{c^2 x^2}}+1\right)\right)+3 a^2 b c x \csc ^{-1}(c x)-6 i b^2 \text{Li}_2\left(-e^{i \csc ^{-1}(c x)}\right) \left(a+b \csc ^{-1}(c x)\right)+6 i b^2 \text{Li}_2\left(e^{i \csc ^{-1}(c x)}\right) \left(a+b \csc ^{-1}(c x)\right)+3 a b^2 c x \csc ^{-1}(c x)^2-6 a b^2 \csc ^{-1}(c x) \log \left(1-e^{i \csc ^{-1}(c x)}\right)+6 a b^2 \csc ^{-1}(c x) \log \left(1+e^{i \csc ^{-1}(c x)}\right)+6 b^3 \text{Li}_3\left(-e^{i \csc ^{-1}(c x)}\right)-6 b^3 \text{Li}_3\left(e^{i \csc ^{-1}(c x)}\right)+b^3 c x \csc ^{-1}(c x)^3-3 b^3 \csc ^{-1}(c x)^2 \log \left(1-e^{i \csc ^{-1}(c x)}\right)+3 b^3 \csc ^{-1}(c x)^2 \log \left(1+e^{i \csc ^{-1}(c x)}\right)}{c}","-\frac{6 i b^2 \text{Li}_2\left(-e^{i \csc ^{-1}(c x)}\right) \left(a+b \csc ^{-1}(c x)\right)}{c}+\frac{6 i b^2 \text{Li}_2\left(e^{i \csc ^{-1}(c x)}\right) \left(a+b \csc ^{-1}(c x)\right)}{c}+x \left(a+b \csc ^{-1}(c x)\right)^3+\frac{6 b \tanh ^{-1}\left(e^{i \csc ^{-1}(c x)}\right) \left(a+b \csc ^{-1}(c x)\right)^2}{c}+\frac{6 b^3 \text{Li}_3\left(-e^{i \csc ^{-1}(c x)}\right)}{c}-\frac{6 b^3 \text{Li}_3\left(e^{i \csc ^{-1}(c x)}\right)}{c}",1,"(a^3*c*x + 3*a^2*b*c*x*ArcCsc[c*x] + 3*a*b^2*c*x*ArcCsc[c*x]^2 + b^3*c*x*ArcCsc[c*x]^3 - 6*a*b^2*ArcCsc[c*x]*Log[1 - E^(I*ArcCsc[c*x])] - 3*b^3*ArcCsc[c*x]^2*Log[1 - E^(I*ArcCsc[c*x])] + 6*a*b^2*ArcCsc[c*x]*Log[1 + E^(I*ArcCsc[c*x])] + 3*b^3*ArcCsc[c*x]^2*Log[1 + E^(I*ArcCsc[c*x])] + 3*a^2*b*Log[c*(1 + Sqrt[1 - 1/(c^2*x^2)])*x] - (6*I)*b^2*(a + b*ArcCsc[c*x])*PolyLog[2, -E^(I*ArcCsc[c*x])] + (6*I)*b^2*(a + b*ArcCsc[c*x])*PolyLog[2, E^(I*ArcCsc[c*x])] + 6*b^3*PolyLog[3, -E^(I*ArcCsc[c*x])] - 6*b^3*PolyLog[3, E^(I*ArcCsc[c*x])])/c","A",0
28,1,242,124,0.2321437,"\int \frac{\left(a+b \csc ^{-1}(c x)\right)^3}{x} \, dx","Integrate[(a + b*ArcCsc[c*x])^3/x,x]","a^3 \log (c x)+\frac{3}{2} i a^2 b \left(\text{Li}_2\left(e^{2 i \csc ^{-1}(c x)}\right)+\csc ^{-1}(c x) \left(\csc ^{-1}(c x)+2 i \log \left(1-e^{2 i \csc ^{-1}(c x)}\right)\right)\right)+\frac{1}{8} i a b^2 \left(-24 \csc ^{-1}(c x) \text{Li}_2\left(e^{-2 i \csc ^{-1}(c x)}\right)+12 i \text{Li}_3\left(e^{-2 i \csc ^{-1}(c x)}\right)-8 \csc ^{-1}(c x)^3+24 i \csc ^{-1}(c x)^2 \log \left(1-e^{-2 i \csc ^{-1}(c x)}\right)+\pi ^3\right)+\frac{1}{64} i b^3 \left(-96 \csc ^{-1}(c x)^2 \text{Li}_2\left(e^{-2 i \csc ^{-1}(c x)}\right)+96 i \csc ^{-1}(c x) \text{Li}_3\left(e^{-2 i \csc ^{-1}(c x)}\right)+48 \text{Li}_4\left(e^{-2 i \csc ^{-1}(c x)}\right)-16 \csc ^{-1}(c x)^4+64 i \csc ^{-1}(c x)^3 \log \left(1-e^{-2 i \csc ^{-1}(c x)}\right)+\pi ^4\right)","-\frac{3}{2} b^2 \text{Li}_3\left(e^{2 i \csc ^{-1}(c x)}\right) \left(a+b \csc ^{-1}(c x)\right)+\frac{3}{2} i b \text{Li}_2\left(e^{2 i \csc ^{-1}(c x)}\right) \left(a+b \csc ^{-1}(c x)\right)^2+\frac{i \left(a+b \csc ^{-1}(c x)\right)^4}{4 b}-\log \left(1-e^{2 i \csc ^{-1}(c x)}\right) \left(a+b \csc ^{-1}(c x)\right)^3-\frac{3}{4} i b^3 \text{Li}_4\left(e^{2 i \csc ^{-1}(c x)}\right)",1,"a^3*Log[c*x] + ((3*I)/2)*a^2*b*(ArcCsc[c*x]*(ArcCsc[c*x] + (2*I)*Log[1 - E^((2*I)*ArcCsc[c*x])]) + PolyLog[2, E^((2*I)*ArcCsc[c*x])]) + (I/8)*a*b^2*(Pi^3 - 8*ArcCsc[c*x]^3 + (24*I)*ArcCsc[c*x]^2*Log[1 - E^((-2*I)*ArcCsc[c*x])] - 24*ArcCsc[c*x]*PolyLog[2, E^((-2*I)*ArcCsc[c*x])] + (12*I)*PolyLog[3, E^((-2*I)*ArcCsc[c*x])]) + (I/64)*b^3*(Pi^4 - 16*ArcCsc[c*x]^4 + (64*I)*ArcCsc[c*x]^3*Log[1 - E^((-2*I)*ArcCsc[c*x])] - 96*ArcCsc[c*x]^2*PolyLog[2, E^((-2*I)*ArcCsc[c*x])] + (96*I)*ArcCsc[c*x]*PolyLog[3, E^((-2*I)*ArcCsc[c*x])] + 48*PolyLog[4, E^((-2*I)*ArcCsc[c*x])])","A",0
29,1,135,80,0.2105098,"\int \frac{\left(a+b \csc ^{-1}(c x)\right)^3}{x^2} \, dx","Integrate[(a + b*ArcCsc[c*x])^3/x^2,x]","-\frac{a^3+3 b \csc ^{-1}(c x) \left(a^2+2 a b c x \sqrt{1-\frac{1}{c^2 x^2}}-2 b^2\right)+3 a^2 b c x \sqrt{1-\frac{1}{c^2 x^2}}+3 b^2 \csc ^{-1}(c x)^2 \left(a+b c x \sqrt{1-\frac{1}{c^2 x^2}}\right)-6 a b^2-6 b^3 c x \sqrt{1-\frac{1}{c^2 x^2}}+b^3 \csc ^{-1}(c x)^3}{x}","\frac{6 b^2 \left(a+b \csc ^{-1}(c x)\right)}{x}-3 b c \sqrt{1-\frac{1}{c^2 x^2}} \left(a+b \csc ^{-1}(c x)\right)^2-\frac{\left(a+b \csc ^{-1}(c x)\right)^3}{x}+6 b^3 c \sqrt{1-\frac{1}{c^2 x^2}}",1,"-((a^3 - 6*a*b^2 + 3*a^2*b*c*Sqrt[1 - 1/(c^2*x^2)]*x - 6*b^3*c*Sqrt[1 - 1/(c^2*x^2)]*x + 3*b*(a^2 - 2*b^2 + 2*a*b*c*Sqrt[1 - 1/(c^2*x^2)]*x)*ArcCsc[c*x] + 3*b^2*(a + b*c*Sqrt[1 - 1/(c^2*x^2)]*x)*ArcCsc[c*x]^2 + b^3*ArcCsc[c*x]^3)/x)","A",1
30,1,186,125,0.2197946,"\int \frac{\left(a+b \csc ^{-1}(c x)\right)^3}{x^3} \, dx","Integrate[(a + b*ArcCsc[c*x])^3/x^3,x]","\frac{-4 a^3-3 b c^2 x^2 \left(b^2-2 a^2\right) \sin ^{-1}\left(\frac{1}{c x}\right)+6 b \csc ^{-1}(c x) \left(-2 a^2-2 a b c x \sqrt{1-\frac{1}{c^2 x^2}}+b^2\right)-6 a^2 b c x \sqrt{1-\frac{1}{c^2 x^2}}-6 b^2 \csc ^{-1}(c x)^2 \left(a \left(2-c^2 x^2\right)+b c x \sqrt{1-\frac{1}{c^2 x^2}}\right)+6 a b^2+3 b^3 c x \sqrt{1-\frac{1}{c^2 x^2}}+2 b^3 \left(c^2 x^2-2\right) \csc ^{-1}(c x)^3}{8 x^2}","\frac{3 b^2 \left(a+b \csc ^{-1}(c x)\right)}{4 x^2}-\frac{3 b c \sqrt{1-\frac{1}{c^2 x^2}} \left(a+b \csc ^{-1}(c x)\right)^2}{4 x}+\frac{1}{4} c^2 \left(a+b \csc ^{-1}(c x)\right)^3-\frac{\left(a+b \csc ^{-1}(c x)\right)^3}{2 x^2}+\frac{3 b^3 c \sqrt{1-\frac{1}{c^2 x^2}}}{8 x}-\frac{3}{8} b^3 c^2 \csc ^{-1}(c x)",1,"(-4*a^3 + 6*a*b^2 - 6*a^2*b*c*Sqrt[1 - 1/(c^2*x^2)]*x + 3*b^3*c*Sqrt[1 - 1/(c^2*x^2)]*x + 6*b*(-2*a^2 + b^2 - 2*a*b*c*Sqrt[1 - 1/(c^2*x^2)]*x)*ArcCsc[c*x] - 6*b^2*(b*c*Sqrt[1 - 1/(c^2*x^2)]*x + a*(2 - c^2*x^2))*ArcCsc[c*x]^2 + 2*b^3*(-2 + c^2*x^2)*ArcCsc[c*x]^3 - 3*b*(-2*a^2 + b^2)*c^2*x^2*ArcSin[1/(c*x)])/(8*x^2)","A",1
31,1,204,170,0.325627,"\int \frac{\left(a+b \csc ^{-1}(c x)\right)^3}{x^4} \, dx","Integrate[(a + b*ArcCsc[c*x])^3/x^4,x]","\frac{-9 a^3+3 b \csc ^{-1}(c x) \left(-9 a^2-6 a b c x \sqrt{1-\frac{1}{c^2 x^2}} \left(2 c^2 x^2+1\right)+2 b^2 \left(6 c^2 x^2+1\right)\right)-9 a^2 b c x \sqrt{1-\frac{1}{c^2 x^2}} \left(2 c^2 x^2+1\right)+6 a b^2 \left(6 c^2 x^2+1\right)-9 b^2 \csc ^{-1}(c x)^2 \left(3 a+b c x \sqrt{1-\frac{1}{c^2 x^2}} \left(2 c^2 x^2+1\right)\right)+2 b^3 c x \sqrt{1-\frac{1}{c^2 x^2}} \left(20 c^2 x^2+1\right)-9 b^3 \csc ^{-1}(c x)^3}{27 x^3}","\frac{4 b^2 c^2 \left(a+b \csc ^{-1}(c x)\right)}{3 x}+\frac{2 b^2 \left(a+b \csc ^{-1}(c x)\right)}{9 x^3}-\frac{b c \sqrt{1-\frac{1}{c^2 x^2}} \left(a+b \csc ^{-1}(c x)\right)^2}{3 x^2}-\frac{2}{3} b c^3 \sqrt{1-\frac{1}{c^2 x^2}} \left(a+b \csc ^{-1}(c x)\right)^2-\frac{\left(a+b \csc ^{-1}(c x)\right)^3}{3 x^3}-\frac{2}{27} b^3 c^3 \left(1-\frac{1}{c^2 x^2}\right)^{3/2}+\frac{14}{9} b^3 c^3 \sqrt{1-\frac{1}{c^2 x^2}}",1,"(-9*a^3 - 9*a^2*b*c*Sqrt[1 - 1/(c^2*x^2)]*x*(1 + 2*c^2*x^2) + 6*a*b^2*(1 + 6*c^2*x^2) + 2*b^3*c*Sqrt[1 - 1/(c^2*x^2)]*x*(1 + 20*c^2*x^2) + 3*b*(-9*a^2 - 6*a*b*c*Sqrt[1 - 1/(c^2*x^2)]*x*(1 + 2*c^2*x^2) + 2*b^2*(1 + 6*c^2*x^2))*ArcCsc[c*x] - 9*b^2*(3*a + b*c*Sqrt[1 - 1/(c^2*x^2)]*x*(1 + 2*c^2*x^2))*ArcCsc[c*x]^2 - 9*b^3*ArcCsc[c*x]^3)/(27*x^3)","A",1
32,1,283,208,0.3596908,"\int \frac{\left(a+b \csc ^{-1}(c x)\right)^3}{x^5} \, dx","Integrate[(a + b*ArcCsc[c*x])^3/x^5,x]","\frac{-64 a^3+9 b c^4 x^4 \left(8 a^2-5 b^2\right) \sin ^{-1}\left(\frac{1}{c x}\right)+24 b \csc ^{-1}(c x) \left(-8 a^2-2 a b c x \sqrt{1-\frac{1}{c^2 x^2}} \left(3 c^2 x^2+2\right)+b^2 \left(3 c^2 x^2+1\right)\right)-48 a^2 b c x \sqrt{1-\frac{1}{c^2 x^2}}-72 a^2 b c^3 x^3 \sqrt{1-\frac{1}{c^2 x^2}}+72 a b^2 c^2 x^2-24 b^2 \csc ^{-1}(c x)^2 \left(a \left(8-3 c^4 x^4\right)+b c x \sqrt{1-\frac{1}{c^2 x^2}} \left(3 c^2 x^2+2\right)\right)+24 a b^2+8 b^3 \left(3 c^4 x^4-8\right) \csc ^{-1}(c x)^3+6 b^3 c x \sqrt{1-\frac{1}{c^2 x^2}}+45 b^3 c^3 x^3 \sqrt{1-\frac{1}{c^2 x^2}}}{256 x^4}","\frac{9 b^2 c^2 \left(a+b \csc ^{-1}(c x)\right)}{32 x^2}+\frac{3 b^2 \left(a+b \csc ^{-1}(c x)\right)}{32 x^4}+\frac{3}{32} c^4 \left(a+b \csc ^{-1}(c x)\right)^3-\frac{3 b c \sqrt{1-\frac{1}{c^2 x^2}} \left(a+b \csc ^{-1}(c x)\right)^2}{16 x^3}-\frac{9 b c^3 \sqrt{1-\frac{1}{c^2 x^2}} \left(a+b \csc ^{-1}(c x)\right)^2}{32 x}-\frac{\left(a+b \csc ^{-1}(c x)\right)^3}{4 x^4}-\frac{45}{256} b^3 c^4 \csc ^{-1}(c x)+\frac{3 b^3 c \sqrt{1-\frac{1}{c^2 x^2}}}{128 x^3}+\frac{45 b^3 c^3 \sqrt{1-\frac{1}{c^2 x^2}}}{256 x}",1,"(-64*a^3 + 24*a*b^2 - 48*a^2*b*c*Sqrt[1 - 1/(c^2*x^2)]*x + 6*b^3*c*Sqrt[1 - 1/(c^2*x^2)]*x + 72*a*b^2*c^2*x^2 - 72*a^2*b*c^3*Sqrt[1 - 1/(c^2*x^2)]*x^3 + 45*b^3*c^3*Sqrt[1 - 1/(c^2*x^2)]*x^3 + 24*b*(-8*a^2 + b^2*(1 + 3*c^2*x^2) - 2*a*b*c*Sqrt[1 - 1/(c^2*x^2)]*x*(2 + 3*c^2*x^2))*ArcCsc[c*x] - 24*b^2*(b*c*Sqrt[1 - 1/(c^2*x^2)]*x*(2 + 3*c^2*x^2) + a*(8 - 3*c^4*x^4))*ArcCsc[c*x]^2 + 8*b^3*(-8 + 3*c^4*x^4)*ArcCsc[c*x]^3 + 9*b*(8*a^2 - 5*b^2)*c^4*x^4*ArcSin[1/(c*x)])/(256*x^4)","A",1
33,0,0,15,2.9129301,"\int \frac{x}{a+b \csc ^{-1}(c x)} \, dx","Integrate[x/(a + b*ArcCsc[c*x]),x]","\int \frac{x}{a+b \csc ^{-1}(c x)} \, dx","\text{Int}\left(\frac{x}{a+b \csc ^{-1}(c x)},x\right)",0,"Integrate[x/(a + b*ArcCsc[c*x]), x]","A",-1
34,0,0,13,3.4886209,"\int \frac{1}{a+b \csc ^{-1}(c x)} \, dx","Integrate[(a + b*ArcCsc[c*x])^(-1),x]","\int \frac{1}{a+b \csc ^{-1}(c x)} \, dx","\text{Int}\left(\frac{1}{a+b \csc ^{-1}(c x)},x\right)",0,"Integrate[(a + b*ArcCsc[c*x])^(-1), x]","A",-1
35,0,0,17,0.2987031,"\int \frac{1}{x \left(a+b \csc ^{-1}(c x)\right)} \, dx","Integrate[1/(x*(a + b*ArcCsc[c*x])),x]","\int \frac{1}{x \left(a+b \csc ^{-1}(c x)\right)} \, dx","\text{Int}\left(\frac{1}{x \left(a+b \csc ^{-1}(c x)\right)},x\right)",0,"Integrate[1/(x*(a + b*ArcCsc[c*x])), x]","A",-1
36,1,43,47,0.0788464,"\int \frac{1}{x^2 \left(a+b \csc ^{-1}(c x)\right)} \, dx","Integrate[1/(x^2*(a + b*ArcCsc[c*x])),x]","-\frac{c \left(\cos \left(\frac{a}{b}\right) \text{Ci}\left(\frac{a}{b}+\csc ^{-1}(c x)\right)+\sin \left(\frac{a}{b}\right) \text{Si}\left(\frac{a}{b}+\csc ^{-1}(c x)\right)\right)}{b}","-\frac{c \cos \left(\frac{a}{b}\right) \text{Ci}\left(\frac{a}{b}+\csc ^{-1}(c x)\right)}{b}-\frac{c \sin \left(\frac{a}{b}\right) \text{Si}\left(\frac{a}{b}+\csc ^{-1}(c x)\right)}{b}",1,"-((c*(Cos[a/b]*CosIntegral[a/b + ArcCsc[c*x]] + Sin[a/b]*SinIntegral[a/b + ArcCsc[c*x]]))/b)","A",1
37,1,56,63,0.0791067,"\int \frac{1}{x^3 \left(a+b \csc ^{-1}(c x)\right)} \, dx","Integrate[1/(x^3*(a + b*ArcCsc[c*x])),x]","-\frac{c^2 \left(\cos \left(\frac{2 a}{b}\right) \text{Si}\left(\frac{2 a}{b}+2 \csc ^{-1}(c x)\right)-\sin \left(\frac{2 a}{b}\right) \text{Ci}\left(\frac{2 a}{b}+2 \csc ^{-1}(c x)\right)\right)}{2 b}","\frac{c^2 \sin \left(\frac{2 a}{b}\right) \text{Ci}\left(\frac{2 a}{b}+2 \csc ^{-1}(c x)\right)}{2 b}-\frac{c^2 \cos \left(\frac{2 a}{b}\right) \text{Si}\left(\frac{2 a}{b}+2 \csc ^{-1}(c x)\right)}{2 b}",1,"-1/2*(c^2*(-(CosIntegral[(2*a)/b + 2*ArcCsc[c*x]]*Sin[(2*a)/b]) + Cos[(2*a)/b]*SinIntegral[(2*a)/b + 2*ArcCsc[c*x]]))/b","A",1
38,1,91,117,0.1758175,"\int \frac{1}{x^4 \left(a+b \csc ^{-1}(c x)\right)} \, dx","Integrate[1/(x^4*(a + b*ArcCsc[c*x])),x]","-\frac{c^3 \left(\cos \left(\frac{a}{b}\right) \text{Ci}\left(\frac{a}{b}+\csc ^{-1}(c x)\right)-\cos \left(\frac{3 a}{b}\right) \text{Ci}\left(3 \left(\frac{a}{b}+\csc ^{-1}(c x)\right)\right)+\sin \left(\frac{a}{b}\right) \text{Si}\left(\frac{a}{b}+\csc ^{-1}(c x)\right)-\sin \left(\frac{3 a}{b}\right) \text{Si}\left(3 \left(\frac{a}{b}+\csc ^{-1}(c x)\right)\right)\right)}{4 b}","-\frac{c^3 \cos \left(\frac{a}{b}\right) \text{Ci}\left(\frac{a}{b}+\csc ^{-1}(c x)\right)}{4 b}+\frac{c^3 \cos \left(\frac{3 a}{b}\right) \text{Ci}\left(\frac{3 a}{b}+3 \csc ^{-1}(c x)\right)}{4 b}-\frac{c^3 \sin \left(\frac{a}{b}\right) \text{Si}\left(\frac{a}{b}+\csc ^{-1}(c x)\right)}{4 b}+\frac{c^3 \sin \left(\frac{3 a}{b}\right) \text{Si}\left(\frac{3 a}{b}+3 \csc ^{-1}(c x)\right)}{4 b}",1,"-1/4*(c^3*(Cos[a/b]*CosIntegral[a/b + ArcCsc[c*x]] - Cos[(3*a)/b]*CosIntegral[3*(a/b + ArcCsc[c*x])] + Sin[a/b]*SinIntegral[a/b + ArcCsc[c*x]] - Sin[(3*a)/b]*SinIntegral[3*(a/b + ArcCsc[c*x])]))/b","A",1
39,0,0,19,7.3522569,"\int (d x)^m \left(a+b \csc ^{-1}(c x)\right)^3 \, dx","Integrate[(d*x)^m*(a + b*ArcCsc[c*x])^3,x]","\int (d x)^m \left(a+b \csc ^{-1}(c x)\right)^3 \, dx","\text{Int}\left((d x)^m \left(a+b \csc ^{-1}(c x)\right)^3,x\right)",0,"Integrate[(d*x)^m*(a + b*ArcCsc[c*x])^3, x]","A",-1
40,0,0,19,4.8633637,"\int (d x)^m \left(a+b \csc ^{-1}(c x)\right)^2 \, dx","Integrate[(d*x)^m*(a + b*ArcCsc[c*x])^2,x]","\int (d x)^m \left(a+b \csc ^{-1}(c x)\right)^2 \, dx","\text{Int}\left((d x)^m \left(a+b \csc ^{-1}(c x)\right)^2,x\right)",0,"Integrate[(d*x)^m*(a + b*ArcCsc[c*x])^2, x]","A",-1
41,1,83,66,0.1866608,"\int (d x)^m \left(a+b \csc ^{-1}(c x)\right) \, dx","Integrate[(d*x)^m*(a + b*ArcCsc[c*x]),x]","\frac{(d x)^m \left((m+1) x \left(a+b \csc ^{-1}(c x)\right)+\frac{b \sqrt{1-c^2 x^2} \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};c^2 x^2\right)}{c \sqrt{1-\frac{1}{c^2 x^2}}}\right)}{(m+1)^2}","\frac{(d x)^{m+1} \left(a+b \csc ^{-1}(c x)\right)}{d (m+1)}+\frac{b (d x)^m \, _2F_1\left(\frac{1}{2},-\frac{m}{2};1-\frac{m}{2};\frac{1}{c^2 x^2}\right)}{c m (m+1)}",1,"((d*x)^m*((1 + m)*x*(a + b*ArcCsc[c*x]) + (b*Sqrt[1 - c^2*x^2]*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, c^2*x^2])/(c*Sqrt[1 - 1/(c^2*x^2)])))/(1 + m)^2","A",1
42,0,0,19,0.8777017,"\int \frac{(d x)^m}{a+b \csc ^{-1}(c x)} \, dx","Integrate[(d*x)^m/(a + b*ArcCsc[c*x]),x]","\int \frac{(d x)^m}{a+b \csc ^{-1}(c x)} \, dx","\text{Int}\left(\frac{(d x)^m}{a+b \csc ^{-1}(c x)},x\right)",0,"Integrate[(d*x)^m/(a + b*ArcCsc[c*x]), x]","A",-1
43,0,0,19,1.7750372,"\int \frac{(d x)^m}{\left(a+b \csc ^{-1}(c x)\right)^2} \, dx","Integrate[(d*x)^m/(a + b*ArcCsc[c*x])^2,x]","\int \frac{(d x)^m}{\left(a+b \csc ^{-1}(c x)\right)^2} \, dx","\text{Int}\left(\frac{(d x)^m}{\left(a+b \csc ^{-1}(c x)\right)^2},x\right)",0,"Integrate[(d*x)^m/(a + b*ArcCsc[c*x])^2, x]","A",-1
44,1,165,167,0.3139691,"\int (d+e x)^3 \left(a+b \csc ^{-1}(c x)\right) \, dx","Integrate[(d + e*x)^3*(a + b*ArcCsc[c*x]),x]","\frac{3 a c^3 x \left(4 d^3+6 d^2 e x+4 d e^2 x^2+e^3 x^3\right)+3 b c^3 x \csc ^{-1}(c x) \left(4 d^3+6 d^2 e x+4 d e^2 x^2+e^3 x^3\right)+b e x \sqrt{1-\frac{1}{c^2 x^2}} \left(c^2 \left(18 d^2+6 d e x+e^2 x^2\right)+2 e^2\right)+6 b d \left(2 c^2 d^2+e^2\right) \log \left(x \left(\sqrt{1-\frac{1}{c^2 x^2}}+1\right)\right)}{12 c^3}","\frac{(d+e x)^4 \left(a+b \csc ^{-1}(c x)\right)}{4 e}+\frac{b d e^2 x^2 \sqrt{1-\frac{1}{c^2 x^2}}}{2 c}+\frac{b e^3 x^3 \sqrt{1-\frac{1}{c^2 x^2}}}{12 c}+\frac{b e x \sqrt{1-\frac{1}{c^2 x^2}} \left(9 c^2 d^2+e^2\right)}{6 c^3}+\frac{b d \left(2 c^2 d^2+e^2\right) \tanh ^{-1}\left(\sqrt{1-\frac{1}{c^2 x^2}}\right)}{2 c^3}-\frac{b d^4 \csc ^{-1}(c x)}{4 e}",1,"(3*a*c^3*x*(4*d^3 + 6*d^2*e*x + 4*d*e^2*x^2 + e^3*x^3) + b*e*Sqrt[1 - 1/(c^2*x^2)]*x*(2*e^2 + c^2*(18*d^2 + 6*d*e*x + e^2*x^2)) + 3*b*c^3*x*(4*d^3 + 6*d^2*e*x + 4*d*e^2*x^2 + e^3*x^3)*ArcCsc[c*x] + 6*b*d*(2*c^2*d^2 + e^2)*Log[(1 + Sqrt[1 - 1/(c^2*x^2)])*x])/(12*c^3)","A",1
45,1,122,123,0.1918572,"\int (d+e x)^2 \left(a+b \csc ^{-1}(c x)\right) \, dx","Integrate[(d + e*x)^2*(a + b*ArcCsc[c*x]),x]","\frac{c^2 x \left(2 a c \left(3 d^2+3 d e x+e^2 x^2\right)+b e \sqrt{1-\frac{1}{c^2 x^2}} (6 d+e x)\right)+2 b c^3 x \csc ^{-1}(c x) \left(3 d^2+3 d e x+e^2 x^2\right)+b \left(6 c^2 d^2+e^2\right) \log \left(x \left(\sqrt{1-\frac{1}{c^2 x^2}}+1\right)\right)}{6 c^3}","\frac{(d+e x)^3 \left(a+b \csc ^{-1}(c x)\right)}{3 e}+\frac{b d e x \sqrt{1-\frac{1}{c^2 x^2}}}{c}+\frac{b e^2 x^2 \sqrt{1-\frac{1}{c^2 x^2}}}{6 c}+\frac{b \left(6 c^2 d^2+e^2\right) \tanh ^{-1}\left(\sqrt{1-\frac{1}{c^2 x^2}}\right)}{6 c^3}-\frac{b d^3 \csc ^{-1}(c x)}{3 e}",1,"(c^2*x*(b*e*Sqrt[1 - 1/(c^2*x^2)]*(6*d + e*x) + 2*a*c*(3*d^2 + 3*d*e*x + e^2*x^2)) + 2*b*c^3*x*(3*d^2 + 3*d*e*x + e^2*x^2)*ArcCsc[c*x] + b*(6*c^2*d^2 + e^2)*Log[(1 + Sqrt[1 - 1/(c^2*x^2)])*x])/(6*c^3)","A",1
46,1,113,83,0.2031105,"\int (d+e x) \left(a+b \csc ^{-1}(c x)\right) \, dx","Integrate[(d + e*x)*(a + b*ArcCsc[c*x]),x]","a d x+\frac{1}{2} a e x^2+\frac{b d x \sqrt{1-\frac{1}{c^2 x^2}} \tanh ^{-1}\left(\frac{c x}{\sqrt{c^2 x^2-1}}\right)}{\sqrt{c^2 x^2-1}}+\frac{b e x \sqrt{\frac{c^2 x^2-1}{c^2 x^2}}}{2 c}+b d x \csc ^{-1}(c x)+\frac{1}{2} b e x^2 \csc ^{-1}(c x)","\frac{(d+e x)^2 \left(a+b \csc ^{-1}(c x)\right)}{2 e}+\frac{b d \tanh ^{-1}\left(\sqrt{1-\frac{1}{c^2 x^2}}\right)}{c}+\frac{b e x \sqrt{1-\frac{1}{c^2 x^2}}}{2 c}-\frac{b d^2 \csc ^{-1}(c x)}{2 e}",1,"a*d*x + (a*e*x^2)/2 + (b*e*x*Sqrt[(-1 + c^2*x^2)/(c^2*x^2)])/(2*c) + b*d*x*ArcCsc[c*x] + (b*e*x^2*ArcCsc[c*x])/2 + (b*d*Sqrt[1 - 1/(c^2*x^2)]*x*ArcTanh[(c*x)/Sqrt[-1 + c^2*x^2]])/Sqrt[-1 + c^2*x^2]","A",1
47,1,58,31,0.048158,"\int \left(a+b \csc ^{-1}(c x)\right) \, dx","Integrate[a + b*ArcCsc[c*x],x]","a x+\frac{b x \sqrt{1-\frac{1}{c^2 x^2}} \tanh ^{-1}\left(\frac{c x}{\sqrt{c^2 x^2-1}}\right)}{\sqrt{c^2 x^2-1}}+b x \csc ^{-1}(c x)","a x+\frac{b \tanh ^{-1}\left(\sqrt{1-\frac{1}{c^2 x^2}}\right)}{c}+b x \csc ^{-1}(c x)",1,"a*x + b*x*ArcCsc[c*x] + (b*Sqrt[1 - 1/(c^2*x^2)]*x*ArcTanh[(c*x)/Sqrt[-1 + c^2*x^2]])/Sqrt[-1 + c^2*x^2]","A",1
48,1,411,257,0.6940854,"\int \frac{a+b \csc ^{-1}(c x)}{d+e x} \, dx","Integrate[(a + b*ArcCsc[c*x])/(d + e*x),x]","\frac{a \log (d+e x)}{e}+\frac{b \left(8 i \left(\text{Li}_2\left(\frac{i \left(\sqrt{e^2-c^2 d^2}-e\right) e^{-i \csc ^{-1}(c x)}}{c d}\right)+\text{Li}_2\left(-\frac{i \left(e+\sqrt{e^2-c^2 d^2}\right) e^{-i \csc ^{-1}(c x)}}{c d}\right)\right)-4 \log \left(1+\frac{i \left(e-\sqrt{e^2-c^2 d^2}\right) e^{-i \csc ^{-1}(c x)}}{c d}\right) \left(4 \sin ^{-1}\left(\frac{\sqrt{\frac{e}{c d}+1}}{\sqrt{2}}\right)-2 \csc ^{-1}(c x)+\pi \right)-4 \log \left(1+\frac{i \left(\sqrt{e^2-c^2 d^2}+e\right) e^{-i \csc ^{-1}(c x)}}{c d}\right) \left(-4 \sin ^{-1}\left(\frac{\sqrt{\frac{e}{c d}+1}}{\sqrt{2}}\right)-2 \csc ^{-1}(c x)+\pi \right)+32 i \sin ^{-1}\left(\frac{\sqrt{\frac{e}{c d}+1}}{\sqrt{2}}\right) \tan ^{-1}\left(\frac{(c d-e) \cot \left(\frac{1}{4} \left(2 \csc ^{-1}(c x)+\pi \right)\right)}{\sqrt{e^2-c^2 d^2}}\right)+4 \left(\pi -2 \csc ^{-1}(c x)\right) \log \left(\frac{d}{x}+e\right)+8 \csc ^{-1}(c x) \log \left(\frac{d}{x}+e\right)+4 i \left(\csc ^{-1}(c x)^2+\text{Li}_2\left(e^{2 i \csc ^{-1}(c x)}\right)\right)+i \left(\pi -2 \csc ^{-1}(c x)\right)^2-8 \csc ^{-1}(c x) \log \left(1-e^{2 i \csc ^{-1}(c x)}\right)\right)}{8 e}","\frac{\left(a+b \csc ^{-1}(c x)\right) \log \left(1-\frac{i \left(e-\sqrt{e^2-c^2 d^2}\right) e^{i \csc ^{-1}(c x)}}{c d}\right)}{e}+\frac{\left(a+b \csc ^{-1}(c x)\right) \log \left(1-\frac{i \left(\sqrt{e^2-c^2 d^2}+e\right) e^{i \csc ^{-1}(c x)}}{c d}\right)}{e}-\frac{\log \left(1-e^{2 i \csc ^{-1}(c x)}\right) \left(a+b \csc ^{-1}(c x)\right)}{e}-\frac{i b \text{Li}_2\left(\frac{i \left(e-\sqrt{e^2-c^2 d^2}\right) e^{i \csc ^{-1}(c x)}}{c d}\right)}{e}-\frac{i b \text{Li}_2\left(\frac{i \left(e+\sqrt{e^2-c^2 d^2}\right) e^{i \csc ^{-1}(c x)}}{c d}\right)}{e}+\frac{i b \text{Li}_2\left(e^{2 i \csc ^{-1}(c x)}\right)}{2 e}",1,"(a*Log[d + e*x])/e + (b*(I*(Pi - 2*ArcCsc[c*x])^2 + (32*I)*ArcSin[Sqrt[1 + e/(c*d)]/Sqrt[2]]*ArcTan[((c*d - e)*Cot[(Pi + 2*ArcCsc[c*x])/4])/Sqrt[-(c^2*d^2) + e^2]] - 4*(Pi - 2*ArcCsc[c*x] + 4*ArcSin[Sqrt[1 + e/(c*d)]/Sqrt[2]])*Log[1 + (I*(e - Sqrt[-(c^2*d^2) + e^2]))/(c*d*E^(I*ArcCsc[c*x]))] - 4*(Pi - 2*ArcCsc[c*x] - 4*ArcSin[Sqrt[1 + e/(c*d)]/Sqrt[2]])*Log[1 + (I*(e + Sqrt[-(c^2*d^2) + e^2]))/(c*d*E^(I*ArcCsc[c*x]))] - 8*ArcCsc[c*x]*Log[1 - E^((2*I)*ArcCsc[c*x])] + 4*(Pi - 2*ArcCsc[c*x])*Log[e + d/x] + 8*ArcCsc[c*x]*Log[e + d/x] + (8*I)*(PolyLog[2, (I*(-e + Sqrt[-(c^2*d^2) + e^2]))/(c*d*E^(I*ArcCsc[c*x]))] + PolyLog[2, ((-I)*(e + Sqrt[-(c^2*d^2) + e^2]))/(c*d*E^(I*ArcCsc[c*x]))]) + (4*I)*(ArcCsc[c*x]^2 + PolyLog[2, E^((2*I)*ArcCsc[c*x])])))/(8*e)","A",0
49,1,141,102,0.2229818,"\int \frac{a+b \csc ^{-1}(c x)}{(d+e x)^2} \, dx","Integrate[(a + b*ArcCsc[c*x])/(d + e*x)^2,x]","-\frac{a}{e (d+e x)}-\frac{b \log \left(c x \left(c d-\sqrt{1-\frac{1}{c^2 x^2}} \sqrt{c^2 d^2-e^2}\right)+e\right)}{d \sqrt{c^2 d^2-e^2}}+\frac{b \log (d+e x)}{d \sqrt{c^2 d^2-e^2}}+\frac{b \sin ^{-1}\left(\frac{1}{c x}\right)}{d e}-\frac{b \csc ^{-1}(c x)}{e (d+e x)}","-\frac{a+b \csc ^{-1}(c x)}{e (d+e x)}+\frac{b \tanh ^{-1}\left(\frac{c^2 d+\frac{e}{x}}{c \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{c^2 d^2-e^2}}\right)}{d \sqrt{c^2 d^2-e^2}}+\frac{b \csc ^{-1}(c x)}{d e}",1,"-(a/(e*(d + e*x))) - (b*ArcCsc[c*x])/(e*(d + e*x)) + (b*ArcSin[1/(c*x)])/(d*e) + (b*Log[d + e*x])/(d*Sqrt[c^2*d^2 - e^2]) - (b*Log[e + c*(c*d - Sqrt[c^2*d^2 - e^2]*Sqrt[1 - 1/(c^2*x^2)])*x])/(d*Sqrt[c^2*d^2 - e^2])","A",1
50,1,250,172,0.5058383,"\int \frac{a+b \csc ^{-1}(c x)}{(d+e x)^3} \, dx","Integrate[(a + b*ArcCsc[c*x])/(d + e*x)^3,x]","\frac{1}{2} \left(-\frac{a}{e (d+e x)^2}-\frac{b c e x \sqrt{1-\frac{1}{c^2 x^2}}}{d \left(c^2 d^2-e^2\right) (d+e x)}-\frac{b \left(2 c^2 d^2-e^2\right) \log \left(c x \left(c d-\sqrt{1-\frac{1}{c^2 x^2}} \sqrt{c^2 d^2-e^2}\right)+e\right)}{d^2 (c d-e) (c d+e) \sqrt{c^2 d^2-e^2}}+\frac{b \left(2 c^2 d^2-e^2\right) \log (d+e x)}{d^2 (c d-e) (c d+e) \sqrt{c^2 d^2-e^2}}+\frac{b \sin ^{-1}\left(\frac{1}{c x}\right)}{d^2 e}-\frac{b \csc ^{-1}(c x)}{e (d+e x)^2}\right)","-\frac{a+b \csc ^{-1}(c x)}{2 e (d+e x)^2}-\frac{b c e \sqrt{1-\frac{1}{c^2 x^2}}}{2 d \left(c^2 d^2-e^2\right) \left(\frac{d}{x}+e\right)}+\frac{b \left(2 c^2 d^2-e^2\right) \tanh ^{-1}\left(\frac{c^2 d+\frac{e}{x}}{c \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{c^2 d^2-e^2}}\right)}{2 d^2 \left(c^2 d^2-e^2\right)^{3/2}}+\frac{b \csc ^{-1}(c x)}{2 d^2 e}",1,"(-(a/(e*(d + e*x)^2)) - (b*c*e*Sqrt[1 - 1/(c^2*x^2)]*x)/(d*(c^2*d^2 - e^2)*(d + e*x)) - (b*ArcCsc[c*x])/(e*(d + e*x)^2) + (b*ArcSin[1/(c*x)])/(d^2*e) + (b*(2*c^2*d^2 - e^2)*Log[d + e*x])/(d^2*(c*d - e)*(c*d + e)*Sqrt[c^2*d^2 - e^2]) - (b*(2*c^2*d^2 - e^2)*Log[e + c*(c*d - Sqrt[c^2*d^2 - e^2]*Sqrt[1 - 1/(c^2*x^2)])*x])/(d^2*(c*d - e)*(c*d + e)*Sqrt[c^2*d^2 - e^2]))/2","A",1
51,1,870,496,13.9339887,"\int x^2 \sqrt{d+e x} \left(a+b \csc ^{-1}(c x)\right) \, dx","Integrate[x^2*Sqrt[d + e*x]*(a + b*ArcCsc[c*x]),x]","\frac{b \left(-\frac{c \left(\frac{d}{x}+e\right) x \left(-\frac{16 c^3 \csc ^{-1}(c x) d^3}{105 e^3}-\frac{2}{7} c^3 x^3 \csc ^{-1}(c x)-\frac{2 c^2 x^2 \left(2 \sqrt{1-\frac{1}{c^2 x^2}} e+c d \csc ^{-1}(c x)\right)}{35 e}-\frac{8 c x \left(c d e \sqrt{1-\frac{1}{c^2 x^2}}-c^2 d^2 \csc ^{-1}(c x)\right)}{105 e^2}-\frac{4 \left(9 e^2-5 c^2 d^2\right) \sqrt{1-\frac{1}{c^2 x^2}}}{105 e^2}\right)}{\sqrt{d+e x}}-\frac{2 \sqrt{\frac{d}{x}+e} \sqrt{c x} \left(\frac{2 \left(9 c^3 d^3 e-c d e^3\right) \sqrt{\frac{c d+c e x}{c d+e}} \sqrt{1-c^2 x^2} F\left(\sin ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{2}}\right)|\frac{2 e}{c d+e}\right)}{\sqrt{1-\frac{1}{c^2 x^2}} \sqrt{\frac{d}{x}+e} (c x)^{3/2}}+\frac{2 \left(8 c^4 d^4+5 c^2 e^2 d^2-9 e^4\right) \sqrt{\frac{c d+c e x}{c d+e}} \sqrt{1-c^2 x^2} \Pi \left(2;\sin ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{2}}\right)|\frac{2 e}{c d+e}\right)}{\sqrt{1-\frac{1}{c^2 x^2}} \sqrt{\frac{d}{x}+e} (c x)^{3/2}}+\frac{2 \left(9 c d e^3-5 c^3 d^3 e\right) \cos \left(2 \csc ^{-1}(c x)\right) \left(d x \sqrt{\frac{c d+c e x}{c d+e}} \sqrt{1-c^2 x^2} F\left(\sin ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{2}}\right)|\frac{2 e}{c d+e}\right) c^2-\frac{x (c x+1) \sqrt{\frac{e-c e x}{c d+e}} \sqrt{\frac{c d+c e x}{c d-e}} \left((c d+e) E\left(\sin ^{-1}\left(\sqrt{\frac{c d+c e x}{c d-e}}\right)|\frac{c d-e}{c d+e}\right)-e F\left(\sin ^{-1}\left(\sqrt{\frac{c d+c e x}{c d-e}}\right)|\frac{c d-e}{c d+e}\right)\right) c}{\sqrt{\frac{e (c x+1)}{e-c d}}}+e x \sqrt{\frac{c d+c e x}{c d+e}} \sqrt{1-c^2 x^2} \Pi \left(2;\sin ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{2}}\right)|\frac{2 e}{c d+e}\right) c+(c d+c e x) \left(c^2 x^2-1\right)\right)}{c d \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{\frac{d}{x}+e} \sqrt{c x} \left(c^2 x^2-2\right)}\right)}{105 e^3 \sqrt{d+e x}}\right)}{c^4}-\frac{a d^3 \sqrt{d+e x} B_{-\frac{e x}{d}}\left(3,\frac{3}{2}\right)}{e^3 \sqrt{\frac{e x}{d}+1}}","\frac{2 d^2 (d+e x)^{3/2} \left(a+b \csc ^{-1}(c x)\right)}{3 e^3}-\frac{4 d (d+e x)^{5/2} \left(a+b \csc ^{-1}(c x)\right)}{5 e^3}+\frac{2 (d+e x)^{7/2} \left(a+b \csc ^{-1}(c x)\right)}{7 e^3}-\frac{32 b d^4 \sqrt{1-c^2 x^2} \sqrt{\frac{c (d+e x)}{c d+e}} \Pi \left(2;\sin ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{2}}\right)|\frac{2 e}{c d+e}\right)}{105 c e^3 x \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{d+e x}}-\frac{4 b d \sqrt{1-c^2 x^2} \left(9 c^2 d^2-e^2\right) \sqrt{\frac{c (d+e x)}{c d+e}} F\left(\sin ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{2}}\right)|\frac{2 e}{c d+e}\right)}{105 c^4 e^2 x \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{d+e x}}+\frac{4 b \sqrt{1-c^2 x^2} \left(5 c^2 d^2-9 e^2\right) \sqrt{d+e x} E\left(\sin ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{2}}\right)|\frac{2 e}{c d+e}\right)}{105 c^4 e^2 x \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{\frac{c (d+e x)}{c d+e}}}+\frac{4 b d \left(1-c^2 x^2\right) \sqrt{d+e x}}{105 c^3 e x \sqrt{1-\frac{1}{c^2 x^2}}}-\frac{4 b \left(1-c^2 x^2\right) (d+e x)^{3/2}}{35 c^3 e x \sqrt{1-\frac{1}{c^2 x^2}}}",1,"-((a*d^3*Sqrt[d + e*x]*Beta[-((e*x)/d), 3, 3/2])/(e^3*Sqrt[1 + (e*x)/d])) + (b*(-((c*(e + d/x)*x*((-4*(-5*c^2*d^2 + 9*e^2)*Sqrt[1 - 1/(c^2*x^2)])/(105*e^2) - (16*c^3*d^3*ArcCsc[c*x])/(105*e^3) - (2*c^3*x^3*ArcCsc[c*x])/7 - (2*c^2*x^2*(2*e*Sqrt[1 - 1/(c^2*x^2)] + c*d*ArcCsc[c*x]))/(35*e) - (8*c*x*(c*d*e*Sqrt[1 - 1/(c^2*x^2)] - c^2*d^2*ArcCsc[c*x]))/(105*e^2)))/Sqrt[d + e*x]) - (2*Sqrt[e + d/x]*Sqrt[c*x]*((2*(9*c^3*d^3*e - c*d*e^3)*Sqrt[(c*d + c*e*x)/(c*d + e)]*Sqrt[1 - c^2*x^2]*EllipticF[ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)])/(Sqrt[1 - 1/(c^2*x^2)]*Sqrt[e + d/x]*(c*x)^(3/2)) + (2*(8*c^4*d^4 + 5*c^2*d^2*e^2 - 9*e^4)*Sqrt[(c*d + c*e*x)/(c*d + e)]*Sqrt[1 - c^2*x^2]*EllipticPi[2, ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)])/(Sqrt[1 - 1/(c^2*x^2)]*Sqrt[e + d/x]*(c*x)^(3/2)) + (2*(-5*c^3*d^3*e + 9*c*d*e^3)*Cos[2*ArcCsc[c*x]]*((c*d + c*e*x)*(-1 + c^2*x^2) + c^2*d*x*Sqrt[(c*d + c*e*x)/(c*d + e)]*Sqrt[1 - c^2*x^2]*EllipticF[ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)] - (c*x*(1 + c*x)*Sqrt[(e - c*e*x)/(c*d + e)]*Sqrt[(c*d + c*e*x)/(c*d - e)]*((c*d + e)*EllipticE[ArcSin[Sqrt[(c*d + c*e*x)/(c*d - e)]], (c*d - e)/(c*d + e)] - e*EllipticF[ArcSin[Sqrt[(c*d + c*e*x)/(c*d - e)]], (c*d - e)/(c*d + e)]))/Sqrt[(e*(1 + c*x))/(-(c*d) + e)] + c*e*x*Sqrt[(c*d + c*e*x)/(c*d + e)]*Sqrt[1 - c^2*x^2]*EllipticPi[2, ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)]))/(c*d*Sqrt[1 - 1/(c^2*x^2)]*Sqrt[e + d/x]*Sqrt[c*x]*(-2 + c^2*x^2))))/(105*e^3*Sqrt[d + e*x])))/c^4","C",0
52,1,368,404,1.6554217,"\int x \sqrt{d+e x} \left(a+b \csc ^{-1}(c x)\right) \, dx","Integrate[x*Sqrt[d + e*x]*(a + b*ArcCsc[c*x]),x]","\frac{1}{15} \left(\frac{2 a \sqrt{d+e x} \left(-2 d^2+d e x+3 e^2 x^2\right)}{e^2}+\frac{4 b x \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{d+e x}}{c}-\frac{4 i b \sqrt{\frac{e (c x+1)}{e-c d}} \sqrt{\frac{e-c e x}{c d+e}} \left(\left(-c^2 d^2-2 c d e+e^2\right) F\left(i \sinh ^{-1}\left(\sqrt{-\frac{c}{c d+e}} \sqrt{d+e x}\right)|\frac{c d+e}{c d-e}\right)+2 c^2 d^2 \Pi \left(\frac{e}{c d}+1;i \sinh ^{-1}\left(\sqrt{-\frac{c}{c d+e}} \sqrt{d+e x}\right)|\frac{c d+e}{c d-e}\right)-2 c d (c d-e) E\left(i \sinh ^{-1}\left(\sqrt{-\frac{c}{c d+e}} \sqrt{d+e x}\right)|\frac{c d+e}{c d-e}\right)\right)}{c^3 e^2 x \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{-\frac{c}{c d+e}}}+\frac{2 b \csc ^{-1}(c x) \sqrt{d+e x} \left(-2 d^2+d e x+3 e^2 x^2\right)}{e^2}\right)","-\frac{2 d (d+e x)^{3/2} \left(a+b \csc ^{-1}(c x)\right)}{3 e^2}+\frac{2 (d+e x)^{5/2} \left(a+b \csc ^{-1}(c x)\right)}{5 e^2}+\frac{8 b d^3 \sqrt{1-c^2 x^2} \sqrt{\frac{c (d+e x)}{c d+e}} \Pi \left(2;\sin ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{2}}\right)|\frac{2 e}{c d+e}\right)}{15 c e^2 x \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{d+e x}}-\frac{8 b d \sqrt{1-c^2 x^2} \sqrt{d+e x} E\left(\sin ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{2}}\right)|\frac{2 e}{c d+e}\right)}{15 c^2 e x \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{\frac{c (d+e x)}{c d+e}}}+\frac{4 b \sqrt{1-c^2 x^2} \left(3 c^2 d^2-e^2\right) \sqrt{\frac{c (d+e x)}{c d+e}} F\left(\sin ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{2}}\right)|\frac{2 e}{c d+e}\right)}{15 c^4 e x \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{d+e x}}-\frac{4 b \left(1-c^2 x^2\right) \sqrt{d+e x}}{15 c^3 x \sqrt{1-\frac{1}{c^2 x^2}}}",1,"((4*b*Sqrt[1 - 1/(c^2*x^2)]*x*Sqrt[d + e*x])/c + (2*a*Sqrt[d + e*x]*(-2*d^2 + d*e*x + 3*e^2*x^2))/e^2 + (2*b*Sqrt[d + e*x]*(-2*d^2 + d*e*x + 3*e^2*x^2)*ArcCsc[c*x])/e^2 - ((4*I)*b*Sqrt[(e*(1 + c*x))/(-(c*d) + e)]*Sqrt[(e - c*e*x)/(c*d + e)]*(-2*c*d*(c*d - e)*EllipticE[I*ArcSinh[Sqrt[-(c/(c*d + e))]*Sqrt[d + e*x]], (c*d + e)/(c*d - e)] + (-(c^2*d^2) - 2*c*d*e + e^2)*EllipticF[I*ArcSinh[Sqrt[-(c/(c*d + e))]*Sqrt[d + e*x]], (c*d + e)/(c*d - e)] + 2*c^2*d^2*EllipticPi[1 + e/(c*d), I*ArcSinh[Sqrt[-(c/(c*d + e))]*Sqrt[d + e*x]], (c*d + e)/(c*d - e)]))/(c^3*e^2*Sqrt[-(c/(c*d + e))]*Sqrt[1 - 1/(c^2*x^2)]*x))/15","C",1
53,1,657,315,13.1962483,"\int \sqrt{d+e x} \left(a+b \csc ^{-1}(c x)\right) \, dx","Integrate[Sqrt[d + e*x]*(a + b*ArcCsc[c*x]),x]","\frac{2 a (d+e x)^{3/2}}{3 e}-\frac{b (c d+c e x) \left(-\frac{2 \left(2 e \sqrt{1-\frac{1}{c^2 x^2}}+c d \csc ^{-1}(c x)+c e x \csc ^{-1}(c x)\right)}{e}+\frac{4 d \sqrt{-c^2 x^2 \left(1-\frac{1}{c^2 x^2}\right)} F\left(\sin ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{2}}\right)|\frac{2 e}{c d+e}\right)}{x \sqrt{1-\frac{1}{c^2 x^2}} (c d+e) \sqrt{\frac{c d+c e x}{c d+e}}}-\frac{4 \sqrt{-c^2 x^2 \left(1-\frac{1}{c^2 x^2}\right)} (e-c d) \Pi \left(2;\sin ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{2}}\right)|\frac{2 e}{c d+e}\right)}{c e x \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{\frac{c d+c e x}{c d+e}}}+\frac{\sin \left(4 \csc ^{-1}(c x)\right) \sec \left(2 \csc ^{-1}(c x)\right) \left(c^2 x^2 \left(1-\frac{1}{c^2 x^2}\right) (c d+c e x)+c^2 d x \sqrt{-c^2 x^2 \left(1-\frac{1}{c^2 x^2}\right)} \sqrt{\frac{c d+c e x}{c d+e}} F\left(\sin ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{2}}\right)|\frac{2 e}{c d+e}\right)+c e x \sqrt{-c^2 x^2 \left(1-\frac{1}{c^2 x^2}\right)} \sqrt{\frac{c d+c e x}{c d+e}} \Pi \left(2;\sin ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{2}}\right)|\frac{2 e}{c d+e}\right)-\frac{c x (c x+1) \sqrt{\frac{e-c e x}{c d+e}} \sqrt{\frac{c d+c e x}{c d-e}} \left((c d+e) E\left(\sin ^{-1}\left(\sqrt{\frac{c d+c e x}{c d-e}}\right)|\frac{c d-e}{c d+e}\right)-e F\left(\sin ^{-1}\left(\sqrt{\frac{c d+c e x}{c d-e}}\right)|\frac{c d-e}{c d+e}\right)\right)}{\sqrt{\frac{e (c x+1)}{e-c d}}}\right)}{c^2 x^2 \left(1-\frac{1}{c^2 x^2}\right) \left(\frac{d}{x}+e\right)}\right)}{3 c^2 \sqrt{d+e x}}","\frac{2 (d+e x)^{3/2} \left(a+b \csc ^{-1}(c x)\right)}{3 e}-\frac{4 b d^2 \sqrt{1-c^2 x^2} \sqrt{\frac{c (d+e x)}{c d+e}} \Pi \left(2;\sin ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{2}}\right)|\frac{2 e}{c d+e}\right)}{3 c e x \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{d+e x}}-\frac{4 b d \sqrt{1-c^2 x^2} \sqrt{\frac{c (d+e x)}{c d+e}} F\left(\sin ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{2}}\right)|\frac{2 e}{c d+e}\right)}{3 c^2 x \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{d+e x}}-\frac{4 b \sqrt{1-c^2 x^2} \sqrt{d+e x} E\left(\sin ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{2}}\right)|\frac{2 e}{c d+e}\right)}{3 c^2 x \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{\frac{c (d+e x)}{c d+e}}}",1,"(2*a*(d + e*x)^(3/2))/(3*e) - (b*(c*d + c*e*x)*((-2*(2*e*Sqrt[1 - 1/(c^2*x^2)] + c*d*ArcCsc[c*x] + c*e*x*ArcCsc[c*x]))/e + (4*d*Sqrt[-(c^2*(1 - 1/(c^2*x^2))*x^2)]*EllipticF[ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)])/((c*d + e)*Sqrt[1 - 1/(c^2*x^2)]*x*Sqrt[(c*d + c*e*x)/(c*d + e)]) - (4*(-(c*d) + e)*Sqrt[-(c^2*(1 - 1/(c^2*x^2))*x^2)]*EllipticPi[2, ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)])/(c*e*Sqrt[1 - 1/(c^2*x^2)]*x*Sqrt[(c*d + c*e*x)/(c*d + e)]) + ((c^2*(1 - 1/(c^2*x^2))*x^2*(c*d + c*e*x) + c^2*d*x*Sqrt[-(c^2*(1 - 1/(c^2*x^2))*x^2)]*Sqrt[(c*d + c*e*x)/(c*d + e)]*EllipticF[ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)] - (c*x*(1 + c*x)*Sqrt[(e - c*e*x)/(c*d + e)]*Sqrt[(c*d + c*e*x)/(c*d - e)]*((c*d + e)*EllipticE[ArcSin[Sqrt[(c*d + c*e*x)/(c*d - e)]], (c*d - e)/(c*d + e)] - e*EllipticF[ArcSin[Sqrt[(c*d + c*e*x)/(c*d - e)]], (c*d - e)/(c*d + e)]))/Sqrt[(e*(1 + c*x))/(-(c*d) + e)] + c*e*x*Sqrt[-(c^2*(1 - 1/(c^2*x^2))*x^2)]*Sqrt[(c*d + c*e*x)/(c*d + e)]*EllipticPi[2, ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)])*Sec[2*ArcCsc[c*x]]*Sin[4*ArcCsc[c*x]])/(c^2*(1 - 1/(c^2*x^2))*(e + d/x)*x^2)))/(3*c^2*Sqrt[d + e*x])","B",0
54,0,0,24,18.5183049,"\int \frac{\sqrt{d+e x} \left(a+b \csc ^{-1}(c x)\right)}{x} \, dx","Integrate[(Sqrt[d + e*x]*(a + b*ArcCsc[c*x]))/x,x]","\int \frac{\sqrt{d+e x} \left(a+b \csc ^{-1}(c x)\right)}{x} \, dx","\text{Int}\left(\frac{\sqrt{d+e x} \left(a+b \csc ^{-1}(c x)\right)}{x},x\right)",0,"Integrate[(Sqrt[d + e*x]*(a + b*ArcCsc[c*x]))/x, x]","A",-1
55,0,0,24,8.3071124,"\int \frac{\sqrt{d+e x} \left(a+b \csc ^{-1}(c x)\right)}{x^2} \, dx","Integrate[(Sqrt[d + e*x]*(a + b*ArcCsc[c*x]))/x^2,x]","\int \frac{\sqrt{d+e x} \left(a+b \csc ^{-1}(c x)\right)}{x^2} \, dx","\text{Int}\left(\frac{\sqrt{d+e x} \left(a+b \csc ^{-1}(c x)\right)}{x^2},x\right)",0,"Integrate[(Sqrt[d + e*x]*(a + b*ArcCsc[c*x]))/x^2, x]","A",-1
56,1,333,372,1.4977433,"\int (d+e x)^{3/2} \left(a+b \csc ^{-1}(c x)\right) \, dx","Integrate[(d + e*x)^(3/2)*(a + b*ArcCsc[c*x]),x]","\frac{1}{15} \left(\frac{6 a (d+e x)^{5/2}}{e}+\frac{4 b e x \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{d+e x}}{c}-\frac{4 i b \sqrt{\frac{e (c x+1)}{e-c d}} \sqrt{\frac{e-c e x}{c d+e}} \left(\left(9 c^2 d^2-7 c d e+e^2\right) F\left(i \sinh ^{-1}\left(\sqrt{-\frac{c}{c d+e}} \sqrt{d+e x}\right)|\frac{c d+e}{c d-e}\right)-3 c^2 d^2 \Pi \left(\frac{e}{c d}+1;i \sinh ^{-1}\left(\sqrt{-\frac{c}{c d+e}} \sqrt{d+e x}\right)|\frac{c d+e}{c d-e}\right)-7 c d (c d-e) E\left(i \sinh ^{-1}\left(\sqrt{-\frac{c}{c d+e}} \sqrt{d+e x}\right)|\frac{c d+e}{c d-e}\right)\right)}{c^3 e x \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{-\frac{c}{c d+e}}}+\frac{6 b \csc ^{-1}(c x) (d+e x)^{5/2}}{e}\right)","\frac{2 (d+e x)^{5/2} \left(a+b \csc ^{-1}(c x)\right)}{5 e}-\frac{4 b d^3 \sqrt{1-c^2 x^2} \sqrt{\frac{c (d+e x)}{c d+e}} \Pi \left(2;\sin ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{2}}\right)|\frac{2 e}{c d+e}\right)}{5 c e x \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{d+e x}}-\frac{28 b d \sqrt{1-c^2 x^2} \sqrt{d+e x} E\left(\sin ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{2}}\right)|\frac{2 e}{c d+e}\right)}{15 c^2 x \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{\frac{c (d+e x)}{c d+e}}}-\frac{4 b \sqrt{1-c^2 x^2} \left(2 c^2 d^2+e^2\right) \sqrt{\frac{c (d+e x)}{c d+e}} F\left(\sin ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{2}}\right)|\frac{2 e}{c d+e}\right)}{15 c^4 x \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{d+e x}}-\frac{4 b e \left(1-c^2 x^2\right) \sqrt{d+e x}}{15 c^3 x \sqrt{1-\frac{1}{c^2 x^2}}}",1,"((4*b*e*Sqrt[1 - 1/(c^2*x^2)]*x*Sqrt[d + e*x])/c + (6*a*(d + e*x)^(5/2))/e + (6*b*(d + e*x)^(5/2)*ArcCsc[c*x])/e - ((4*I)*b*Sqrt[(e*(1 + c*x))/(-(c*d) + e)]*Sqrt[(e - c*e*x)/(c*d + e)]*(-7*c*d*(c*d - e)*EllipticE[I*ArcSinh[Sqrt[-(c/(c*d + e))]*Sqrt[d + e*x]], (c*d + e)/(c*d - e)] + (9*c^2*d^2 - 7*c*d*e + e^2)*EllipticF[I*ArcSinh[Sqrt[-(c/(c*d + e))]*Sqrt[d + e*x]], (c*d + e)/(c*d - e)] - 3*c^2*d^2*EllipticPi[1 + e/(c*d), I*ArcSinh[Sqrt[-(c/(c*d + e))]*Sqrt[d + e*x]], (c*d + e)/(c*d - e)]))/(c^3*e*Sqrt[-(c/(c*d + e))]*Sqrt[1 - 1/(c^2*x^2)]*x))/15","C",1
57,1,873,714,13.8481163,"\int \frac{x^3 \left(a+b \csc ^{-1}(c x)\right)}{\sqrt{d+e x}} \, dx","Integrate[(x^3*(a + b*ArcCsc[c*x]))/Sqrt[d + e*x],x]","\frac{a \sqrt{\frac{e x}{d}+1} B_{-\frac{e x}{d}}\left(4,\frac{1}{2}\right) d^4}{e^4 \sqrt{d+e x}}+\frac{b \left(\frac{2 \sqrt{\frac{d}{x}+e} \sqrt{c x} \left(\frac{2 \left(40 c^3 e d^3+8 c e^3 d\right) \sqrt{\frac{c d+c e x}{c d+e}} \sqrt{1-c^2 x^2} F\left(\sin ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{2}}\right)|\frac{2 e}{c d+e}\right)}{\sqrt{1-\frac{1}{c^2 x^2}} \sqrt{\frac{d}{x}+e} (c x)^{3/2}}+\frac{2 \left(48 c^4 d^4+16 c^2 e^2 d^2+9 e^4\right) \sqrt{\frac{c d+c e x}{c d+e}} \sqrt{1-c^2 x^2} \Pi \left(2;\sin ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{2}}\right)|\frac{2 e}{c d+e}\right)}{\sqrt{1-\frac{1}{c^2 x^2}} \sqrt{\frac{d}{x}+e} (c x)^{3/2}}+\frac{2 \left(-16 c^3 e d^3-9 c e^3 d\right) \cos \left(2 \csc ^{-1}(c x)\right) \left(d x \sqrt{\frac{c d+c e x}{c d+e}} \sqrt{1-c^2 x^2} F\left(\sin ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{2}}\right)|\frac{2 e}{c d+e}\right) c^2-\frac{x (c x+1) \sqrt{\frac{e-c e x}{c d+e}} \sqrt{\frac{c d+c e x}{c d-e}} \left((c d+e) E\left(\sin ^{-1}\left(\sqrt{\frac{c d+c e x}{c d-e}}\right)|\frac{c d-e}{c d+e}\right)-e F\left(\sin ^{-1}\left(\sqrt{\frac{c d+c e x}{c d-e}}\right)|\frac{c d-e}{c d+e}\right)\right) c}{\sqrt{\frac{e (c x+1)}{e-c d}}}+e x \sqrt{\frac{c d+c e x}{c d+e}} \sqrt{1-c^2 x^2} \Pi \left(2;\sin ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{2}}\right)|\frac{2 e}{c d+e}\right) c+(c d+c e x) \left(c^2 x^2-1\right)\right)}{c d \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{\frac{d}{x}+e} \sqrt{c x} \left(c^2 x^2-2\right)}\right)}{105 e^4 \sqrt{d+e x}}-\frac{c \left(\frac{d}{x}+e\right) x \left(\frac{32 c^3 \csc ^{-1}(c x) d^3}{35 e^4}-\frac{2 c^3 x^3 \csc ^{-1}(c x)}{7 e}-\frac{4 c^2 x^2 \left(e \sqrt{1-\frac{1}{c^2 x^2}}-3 c d \csc ^{-1}(c x)\right)}{35 e^2}+\frac{4 c x \left(5 c d e \sqrt{1-\frac{1}{c^2 x^2}}-12 c^2 d^2 \csc ^{-1}(c x)\right)}{105 e^3}-\frac{4 \left(16 c^2 d^2+9 e^2\right) \sqrt{1-\frac{1}{c^2 x^2}}}{105 e^3}\right)}{\sqrt{d+e x}}\right)}{c^4}","-\frac{2 d^3 \sqrt{d+e x} \left(a+b \csc ^{-1}(c x)\right)}{e^4}+\frac{2 d^2 (d+e x)^{3/2} \left(a+b \csc ^{-1}(c x)\right)}{e^4}-\frac{6 d (d+e x)^{5/2} \left(a+b \csc ^{-1}(c x)\right)}{5 e^4}+\frac{2 (d+e x)^{7/2} \left(a+b \csc ^{-1}(c x)\right)}{7 e^4}+\frac{64 b d^4 \sqrt{1-c^2 x^2} \sqrt{\frac{c (d+e x)}{c d+e}} \Pi \left(2;\sin ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{2}}\right)|\frac{2 e}{c d+e}\right)}{35 c e^4 x \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{d+e x}}+\frac{64 b d^3 \sqrt{1-c^2 x^2} \sqrt{\frac{c (d+e x)}{c d+e}} F\left(\sin ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{2}}\right)|\frac{2 e}{c d+e}\right)}{35 c^2 e^3 x \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{d+e x}}-\frac{24 b d^2 \sqrt{1-c^2 x^2} \sqrt{d+e x} E\left(\sin ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{2}}\right)|\frac{2 e}{c d+e}\right)}{35 c^2 e^3 x \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{\frac{c (d+e x)}{c d+e}}}+\frac{4 b \sqrt{1-c^2 x^2} \left(2 c^2 d^2-9 e^2\right) \sqrt{d+e x} E\left(\sin ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{2}}\right)|\frac{2 e}{c d+e}\right)}{105 c^4 e^3 x \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{\frac{c (d+e x)}{c d+e}}}-\frac{32 b d \sqrt{1-c^2 x^2} (c d-e) (c d+e) \sqrt{\frac{c (d+e x)}{c d+e}} F\left(\sin ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{2}}\right)|\frac{2 e}{c d+e}\right)}{105 c^4 e^3 x \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{d+e x}}+\frac{4 b d \left(1-c^2 x^2\right) \sqrt{d+e x}}{21 c^3 e^2 x \sqrt{1-\frac{1}{c^2 x^2}}}-\frac{4 b \left(1-c^2 x^2\right) \sqrt{d+e x}}{35 c^3 e \sqrt{1-\frac{1}{c^2 x^2}}}",1,"(a*d^4*Sqrt[1 + (e*x)/d]*Beta[-((e*x)/d), 4, 1/2])/(e^4*Sqrt[d + e*x]) + (b*(-((c*(e + d/x)*x*((-4*(16*c^2*d^2 + 9*e^2)*Sqrt[1 - 1/(c^2*x^2)])/(105*e^3) + (32*c^3*d^3*ArcCsc[c*x])/(35*e^4) - (2*c^3*x^3*ArcCsc[c*x])/(7*e) - (4*c^2*x^2*(e*Sqrt[1 - 1/(c^2*x^2)] - 3*c*d*ArcCsc[c*x]))/(35*e^2) + (4*c*x*(5*c*d*e*Sqrt[1 - 1/(c^2*x^2)] - 12*c^2*d^2*ArcCsc[c*x]))/(105*e^3)))/Sqrt[d + e*x]) + (2*Sqrt[e + d/x]*Sqrt[c*x]*((2*(40*c^3*d^3*e + 8*c*d*e^3)*Sqrt[(c*d + c*e*x)/(c*d + e)]*Sqrt[1 - c^2*x^2]*EllipticF[ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)])/(Sqrt[1 - 1/(c^2*x^2)]*Sqrt[e + d/x]*(c*x)^(3/2)) + (2*(48*c^4*d^4 + 16*c^2*d^2*e^2 + 9*e^4)*Sqrt[(c*d + c*e*x)/(c*d + e)]*Sqrt[1 - c^2*x^2]*EllipticPi[2, ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)])/(Sqrt[1 - 1/(c^2*x^2)]*Sqrt[e + d/x]*(c*x)^(3/2)) + (2*(-16*c^3*d^3*e - 9*c*d*e^3)*Cos[2*ArcCsc[c*x]]*((c*d + c*e*x)*(-1 + c^2*x^2) + c^2*d*x*Sqrt[(c*d + c*e*x)/(c*d + e)]*Sqrt[1 - c^2*x^2]*EllipticF[ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)] - (c*x*(1 + c*x)*Sqrt[(e - c*e*x)/(c*d + e)]*Sqrt[(c*d + c*e*x)/(c*d - e)]*((c*d + e)*EllipticE[ArcSin[Sqrt[(c*d + c*e*x)/(c*d - e)]], (c*d - e)/(c*d + e)] - e*EllipticF[ArcSin[Sqrt[(c*d + c*e*x)/(c*d - e)]], (c*d - e)/(c*d + e)]))/Sqrt[(e*(1 + c*x))/(-(c*d) + e)] + c*e*x*Sqrt[(c*d + c*e*x)/(c*d + e)]*Sqrt[1 - c^2*x^2]*EllipticPi[2, ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)]))/(c*d*Sqrt[1 - 1/(c^2*x^2)]*Sqrt[e + d/x]*Sqrt[c*x]*(-2 + c^2*x^2))))/(105*e^4*Sqrt[d + e*x])))/c^4","C",0
58,1,784,530,13.6707196,"\int \frac{x^2 \left(a+b \csc ^{-1}(c x)\right)}{\sqrt{d+e x}} \, dx","Integrate[(x^2*(a + b*ArcCsc[c*x]))/Sqrt[d + e*x],x]","\frac{b \left(-\frac{c x \left(\frac{d}{x}+e\right) \left(-\frac{16 c^2 d^2 \csc ^{-1}(c x)}{15 e^3}+\frac{4 c d \sqrt{1-\frac{1}{c^2 x^2}}}{5 e^2}-\frac{4 c x \left(e \sqrt{1-\frac{1}{c^2 x^2}}-2 c d \csc ^{-1}(c x)\right)}{15 e^2}-\frac{2 c^2 x^2 \csc ^{-1}(c x)}{5 e}\right)}{\sqrt{d+e x}}-\frac{2 \sqrt{c x} \sqrt{\frac{d}{x}+e} \left(\frac{2 \sqrt{1-c^2 x^2} \left(7 c^2 d^2 e+e^3\right) \sqrt{\frac{c d+c e x}{c d+e}} F\left(\sin ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{2}}\right)|\frac{2 e}{c d+e}\right)}{\sqrt{1-\frac{1}{c^2 x^2}} (c x)^{3/2} \sqrt{\frac{d}{x}+e}}-\frac{6 c d e \cos \left(2 \csc ^{-1}(c x)\right) \left(\left(c^2 x^2-1\right) (c d+c e x)+c^2 d x \sqrt{1-c^2 x^2} \sqrt{\frac{c d+c e x}{c d+e}} F\left(\sin ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{2}}\right)|\frac{2 e}{c d+e}\right)+c e x \sqrt{1-c^2 x^2} \sqrt{\frac{c d+c e x}{c d+e}} \Pi \left(2;\sin ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{2}}\right)|\frac{2 e}{c d+e}\right)-\frac{c x (c x+1) \sqrt{\frac{e-c e x}{c d+e}} \sqrt{\frac{c d+c e x}{c d-e}} \left((c d+e) E\left(\sin ^{-1}\left(\sqrt{\frac{c d+c e x}{c d-e}}\right)|\frac{c d-e}{c d+e}\right)-e F\left(\sin ^{-1}\left(\sqrt{\frac{c d+c e x}{c d-e}}\right)|\frac{c d-e}{c d+e}\right)\right)}{\sqrt{\frac{e (c x+1)}{e-c d}}}\right)}{\sqrt{1-\frac{1}{c^2 x^2}} \sqrt{c x} \left(c^2 x^2-2\right) \sqrt{\frac{d}{x}+e}}+\frac{2 \sqrt{1-c^2 x^2} \left(8 c^3 d^3+3 c d e^2\right) \sqrt{\frac{c d+c e x}{c d+e}} \Pi \left(2;\sin ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{2}}\right)|\frac{2 e}{c d+e}\right)}{\sqrt{1-\frac{1}{c^2 x^2}} (c x)^{3/2} \sqrt{\frac{d}{x}+e}}\right)}{15 e^3 \sqrt{d+e x}}\right)}{c^3}-\frac{a d^3 \sqrt{\frac{e x}{d}+1} B_{-\frac{e x}{d}}\left(3,\frac{1}{2}\right)}{e^3 \sqrt{d+e x}}","\frac{2 d^2 \sqrt{d+e x} \left(a+b \csc ^{-1}(c x)\right)}{e^3}-\frac{4 d (d+e x)^{3/2} \left(a+b \csc ^{-1}(c x)\right)}{3 e^3}+\frac{2 (d+e x)^{5/2} \left(a+b \csc ^{-1}(c x)\right)}{5 e^3}-\frac{32 b d^3 \sqrt{1-c^2 x^2} \sqrt{\frac{c (d+e x)}{c d+e}} \Pi \left(2;\sin ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{2}}\right)|\frac{2 e}{c d+e}\right)}{15 c e^3 x \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{d+e x}}-\frac{32 b d^2 \sqrt{1-c^2 x^2} \sqrt{\frac{c (d+e x)}{c d+e}} F\left(\sin ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{2}}\right)|\frac{2 e}{c d+e}\right)}{15 c^2 e^2 x \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{d+e x}}+\frac{4 b d \sqrt{1-c^2 x^2} \sqrt{d+e x} E\left(\sin ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{2}}\right)|\frac{2 e}{c d+e}\right)}{5 c^2 e^2 x \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{\frac{c (d+e x)}{c d+e}}}+\frac{4 b \sqrt{1-c^2 x^2} (c d-e) (c d+e) \sqrt{\frac{c (d+e x)}{c d+e}} F\left(\sin ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{2}}\right)|\frac{2 e}{c d+e}\right)}{15 c^4 e^2 x \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{d+e x}}-\frac{4 b \left(1-c^2 x^2\right) \sqrt{d+e x}}{15 c^3 e x \sqrt{1-\frac{1}{c^2 x^2}}}",1,"-((a*d^3*Sqrt[1 + (e*x)/d]*Beta[-((e*x)/d), 3, 1/2])/(e^3*Sqrt[d + e*x])) + (b*(-((c*(e + d/x)*x*((4*c*d*Sqrt[1 - 1/(c^2*x^2)])/(5*e^2) - (16*c^2*d^2*ArcCsc[c*x])/(15*e^3) - (2*c^2*x^2*ArcCsc[c*x])/(5*e) - (4*c*x*(e*Sqrt[1 - 1/(c^2*x^2)] - 2*c*d*ArcCsc[c*x]))/(15*e^2)))/Sqrt[d + e*x]) - (2*Sqrt[e + d/x]*Sqrt[c*x]*((2*(7*c^2*d^2*e + e^3)*Sqrt[(c*d + c*e*x)/(c*d + e)]*Sqrt[1 - c^2*x^2]*EllipticF[ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)])/(Sqrt[1 - 1/(c^2*x^2)]*Sqrt[e + d/x]*(c*x)^(3/2)) + (2*(8*c^3*d^3 + 3*c*d*e^2)*Sqrt[(c*d + c*e*x)/(c*d + e)]*Sqrt[1 - c^2*x^2]*EllipticPi[2, ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)])/(Sqrt[1 - 1/(c^2*x^2)]*Sqrt[e + d/x]*(c*x)^(3/2)) - (6*c*d*e*Cos[2*ArcCsc[c*x]]*((c*d + c*e*x)*(-1 + c^2*x^2) + c^2*d*x*Sqrt[(c*d + c*e*x)/(c*d + e)]*Sqrt[1 - c^2*x^2]*EllipticF[ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)] - (c*x*(1 + c*x)*Sqrt[(e - c*e*x)/(c*d + e)]*Sqrt[(c*d + c*e*x)/(c*d - e)]*((c*d + e)*EllipticE[ArcSin[Sqrt[(c*d + c*e*x)/(c*d - e)]], (c*d - e)/(c*d + e)] - e*EllipticF[ArcSin[Sqrt[(c*d + c*e*x)/(c*d - e)]], (c*d - e)/(c*d + e)]))/Sqrt[(e*(1 + c*x))/(-(c*d) + e)] + c*e*x*Sqrt[(c*d + c*e*x)/(c*d + e)]*Sqrt[1 - c^2*x^2]*EllipticPi[2, ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)]))/(Sqrt[1 - 1/(c^2*x^2)]*Sqrt[e + d/x]*Sqrt[c*x]*(-2 + c^2*x^2))))/(15*e^3*Sqrt[d + e*x])))/c^3","C",1
59,1,289,344,1.5091953,"\int \frac{x \left(a+b \csc ^{-1}(c x)\right)}{\sqrt{d+e x}} \, dx","Integrate[(x*(a + b*ArcCsc[c*x]))/Sqrt[d + e*x],x]","\frac{2 \left(a \sqrt{d+e x} (e x-2 d)+\frac{2 i b \sqrt{\frac{e (c x+1)}{e-c d}} \sqrt{\frac{e-c e x}{c d+e}} \left((c d+e) F\left(i \sinh ^{-1}\left(\sqrt{-\frac{c}{c d+e}} \sqrt{d+e x}\right)|\frac{c d+e}{c d-e}\right)+(c d-e) E\left(i \sinh ^{-1}\left(\sqrt{-\frac{c}{c d+e}} \sqrt{d+e x}\right)|\frac{c d+e}{c d-e}\right)-2 c d \Pi \left(\frac{e}{c d}+1;i \sinh ^{-1}\left(\sqrt{-\frac{c}{c d+e}} \sqrt{d+e x}\right)|\frac{c d+e}{c d-e}\right)\right)}{c^2 x \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{-\frac{c}{c d+e}}}+b \csc ^{-1}(c x) \sqrt{d+e x} (e x-2 d)\right)}{3 e^2}","-\frac{2 d \sqrt{d+e x} \left(a+b \csc ^{-1}(c x)\right)}{e^2}+\frac{2 (d+e x)^{3/2} \left(a+b \csc ^{-1}(c x)\right)}{3 e^2}+\frac{8 b d^2 \sqrt{1-c^2 x^2} \sqrt{\frac{c (d+e x)}{c d+e}} \Pi \left(2;\sin ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{2}}\right)|\frac{2 e}{c d+e}\right)}{3 c e^2 x \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{d+e x}}+\frac{8 b d \sqrt{1-c^2 x^2} \sqrt{\frac{c (d+e x)}{c d+e}} F\left(\sin ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{2}}\right)|\frac{2 e}{c d+e}\right)}{3 c^2 e x \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{d+e x}}-\frac{4 b \sqrt{1-c^2 x^2} \sqrt{d+e x} E\left(\sin ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{2}}\right)|\frac{2 e}{c d+e}\right)}{3 c^2 e x \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{\frac{c (d+e x)}{c d+e}}}",1,"(2*(a*(-2*d + e*x)*Sqrt[d + e*x] + b*(-2*d + e*x)*Sqrt[d + e*x]*ArcCsc[c*x] + ((2*I)*b*Sqrt[(e*(1 + c*x))/(-(c*d) + e)]*Sqrt[(e - c*e*x)/(c*d + e)]*((c*d - e)*EllipticE[I*ArcSinh[Sqrt[-(c/(c*d + e))]*Sqrt[d + e*x]], (c*d + e)/(c*d - e)] + (c*d + e)*EllipticF[I*ArcSinh[Sqrt[-(c/(c*d + e))]*Sqrt[d + e*x]], (c*d + e)/(c*d - e)] - 2*c*d*EllipticPi[1 + e/(c*d), I*ArcSinh[Sqrt[-(c/(c*d + e))]*Sqrt[d + e*x]], (c*d + e)/(c*d - e)]))/(c^2*Sqrt[-(c/(c*d + e))]*Sqrt[1 - 1/(c^2*x^2)]*x)))/(3*e^2)","C",1
60,1,243,212,3.6904452,"\int \frac{a+b \csc ^{-1}(c x)}{\sqrt{d+e x}} \, dx","Integrate[(a + b*ArcCsc[c*x])/Sqrt[d + e*x],x]","\frac{2 \left(\frac{a (d+e x)}{e}+\frac{b \left(\frac{2 c d x \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{\frac{c (d+e x)}{c d+e}} \Pi \left(2;\sin ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{2}}\right)|\frac{2 e}{c d+e}\right)}{\sqrt{1-c^2 x^2}}+\csc ^{-1}(c x) (d+e x)\right)}{e}+\frac{2 b c x^2 \sqrt{c x+1} \sqrt{\frac{c (d+e x)}{c d+e}} \left(\cos \left(\frac{1}{2} \csc ^{-1}(c x)\right)+\sin \left(\frac{1}{2} \csc ^{-1}(c x)\right)\right) \left(\cos \left(\frac{1}{2} \csc ^{-1}(c x)\right)-\sin \left(\frac{1}{2} \csc ^{-1}(c x)\right)\right)^3 F\left(\sin ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{2}}\right)|\frac{2 e}{c d+e}\right)}{\sqrt{1-c x} \left(c^2 x^2-1\right)}\right)}{\sqrt{d+e x}}","\frac{2 \sqrt{d+e x} \left(a+b \csc ^{-1}(c x)\right)}{e}-\frac{4 b \sqrt{1-c^2 x^2} \sqrt{\frac{c (d+e x)}{c d+e}} F\left(\sin ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{2}}\right)|\frac{2 e}{c d+e}\right)}{c^2 x \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{d+e x}}-\frac{4 b d \sqrt{1-c^2 x^2} \sqrt{\frac{c (d+e x)}{c d+e}} \Pi \left(2;\sin ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{2}}\right)|\frac{2 e}{c d+e}\right)}{c e x \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{d+e x}}",1,"(2*((a*(d + e*x))/e + (b*((d + e*x)*ArcCsc[c*x] + (2*c*d*Sqrt[1 - 1/(c^2*x^2)]*x*Sqrt[(c*(d + e*x))/(c*d + e)]*EllipticPi[2, ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)])/Sqrt[1 - c^2*x^2]))/e + (2*b*c*x^2*Sqrt[1 + c*x]*Sqrt[(c*(d + e*x))/(c*d + e)]*EllipticF[ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)]*(Cos[ArcCsc[c*x]/2] - Sin[ArcCsc[c*x]/2])^3*(Cos[ArcCsc[c*x]/2] + Sin[ArcCsc[c*x]/2]))/(Sqrt[1 - c*x]*(-1 + c^2*x^2))))/Sqrt[d + e*x]","A",0
61,0,0,24,6.6396773,"\int \frac{a+b \csc ^{-1}(c x)}{x \sqrt{d+e x}} \, dx","Integrate[(a + b*ArcCsc[c*x])/(x*Sqrt[d + e*x]),x]","\int \frac{a+b \csc ^{-1}(c x)}{x \sqrt{d+e x}} \, dx","\text{Int}\left(\frac{a+b \csc ^{-1}(c x)}{x \sqrt{d+e x}},x\right)",0,"Integrate[(a + b*ArcCsc[c*x])/(x*Sqrt[d + e*x]), x]","A",-1
62,0,0,24,9.214182,"\int \frac{a+b \csc ^{-1}(c x)}{x^2 \sqrt{d+e x}} \, dx","Integrate[(a + b*ArcCsc[c*x])/(x^2*Sqrt[d + e*x]),x]","\int \frac{a+b \csc ^{-1}(c x)}{x^2 \sqrt{d+e x}} \, dx","\text{Int}\left(\frac{a+b \csc ^{-1}(c x)}{x^2 \sqrt{d+e x}},x\right)",0,"Integrate[(a + b*ArcCsc[c*x])/(x^2*Sqrt[d + e*x]), x]","A",-1
63,1,814,551,14.0157776,"\int \frac{x^3 \left(a+b \csc ^{-1}(c x)\right)}{(d+e x)^{3/2}} \, dx","Integrate[(x^3*(a + b*ArcCsc[c*x]))/(d + e*x)^(3/2),x]","\frac{a \left(\frac{e x}{d}+1\right)^{3/2} B_{-\frac{e x}{d}}\left(4,-\frac{1}{2}\right) d^4}{e^4 (d+e x)^{3/2}}+\frac{b \left(-\frac{c^2 \left(\frac{d}{x}+e\right)^2 \left(\frac{2 c^2 \csc ^{-1}(c x) d^2}{e^3 \left(\frac{d}{x}+e\right)}-\frac{32 c^2 \csc ^{-1}(c x) d^2}{5 e^4}+\frac{32 c \sqrt{1-\frac{1}{c^2 x^2}} d}{15 e^3}-\frac{2 c^2 x^2 \csc ^{-1}(c x)}{5 e^2}-\frac{2 c x \left(2 e \sqrt{1-\frac{1}{c^2 x^2}}-9 c d \csc ^{-1}(c x)\right)}{15 e^3}\right) x^2}{(d+e x)^{3/2}}-\frac{2 \left(\frac{d}{x}+e\right)^{3/2} (c x)^{3/2} \left(\frac{2 \left(e^3+32 c^2 d^2 e\right) \sqrt{\frac{c d+c e x}{c d+e}} \sqrt{1-c^2 x^2} F\left(\sin ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{2}}\right)|\frac{2 e}{c d+e}\right)}{\sqrt{1-\frac{1}{c^2 x^2}} \sqrt{\frac{d}{x}+e} (c x)^{3/2}}+\frac{2 \left(48 c^3 d^3+8 c e^2 d\right) \sqrt{\frac{c d+c e x}{c d+e}} \sqrt{1-c^2 x^2} \Pi \left(2;\sin ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{2}}\right)|\frac{2 e}{c d+e}\right)}{\sqrt{1-\frac{1}{c^2 x^2}} \sqrt{\frac{d}{x}+e} (c x)^{3/2}}-\frac{16 c d e \cos \left(2 \csc ^{-1}(c x)\right) \left(d x \sqrt{\frac{c d+c e x}{c d+e}} \sqrt{1-c^2 x^2} F\left(\sin ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{2}}\right)|\frac{2 e}{c d+e}\right) c^2-\frac{x (c x+1) \sqrt{\frac{e-c e x}{c d+e}} \sqrt{\frac{c d+c e x}{c d-e}} \left((c d+e) E\left(\sin ^{-1}\left(\sqrt{\frac{c d+c e x}{c d-e}}\right)|\frac{c d-e}{c d+e}\right)-e F\left(\sin ^{-1}\left(\sqrt{\frac{c d+c e x}{c d-e}}\right)|\frac{c d-e}{c d+e}\right)\right) c}{\sqrt{\frac{e (c x+1)}{e-c d}}}+e x \sqrt{\frac{c d+c e x}{c d+e}} \sqrt{1-c^2 x^2} \Pi \left(2;\sin ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{2}}\right)|\frac{2 e}{c d+e}\right) c+(c d+c e x) \left(c^2 x^2-1\right)\right)}{\sqrt{1-\frac{1}{c^2 x^2}} \sqrt{\frac{d}{x}+e} \sqrt{c x} \left(c^2 x^2-2\right)}\right)}{15 e^4 (d+e x)^{3/2}}\right)}{c^4}","\frac{2 d^3 \left(a+b \csc ^{-1}(c x)\right)}{e^4 \sqrt{d+e x}}+\frac{6 d^2 \sqrt{d+e x} \left(a+b \csc ^{-1}(c x)\right)}{e^4}-\frac{2 d (d+e x)^{3/2} \left(a+b \csc ^{-1}(c x)\right)}{e^4}+\frac{2 (d+e x)^{5/2} \left(a+b \csc ^{-1}(c x)\right)}{5 e^4}-\frac{64 b d^3 \sqrt{1-c^2 x^2} \sqrt{\frac{c (d+e x)}{c d+e}} \Pi \left(2;\sin ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{2}}\right)|\frac{2 e}{c d+e}\right)}{5 c e^4 x \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{d+e x}}-\frac{8 b d^2 \sqrt{1-c^2 x^2} \sqrt{\frac{c (d+e x)}{c d+e}} F\left(\sin ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{2}}\right)|\frac{2 e}{c d+e}\right)}{c^2 e^3 x \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{d+e x}}+\frac{32 b d \sqrt{1-c^2 x^2} \sqrt{d+e x} E\left(\sin ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{2}}\right)|\frac{2 e}{c d+e}\right)}{15 c^2 e^3 x \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{\frac{c (d+e x)}{c d+e}}}-\frac{4 b \sqrt{1-c^2 x^2} \left(2 c^2 d^2+e^2\right) \sqrt{\frac{c (d+e x)}{c d+e}} F\left(\sin ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{2}}\right)|\frac{2 e}{c d+e}\right)}{15 c^4 e^3 x \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{d+e x}}-\frac{4 b \left(1-c^2 x^2\right) \sqrt{d+e x}}{15 c^3 e^2 x \sqrt{1-\frac{1}{c^2 x^2}}}",1,"(a*d^4*(1 + (e*x)/d)^(3/2)*Beta[-((e*x)/d), 4, -1/2])/(e^4*(d + e*x)^(3/2)) + (b*(-((c^2*(e + d/x)^2*x^2*((32*c*d*Sqrt[1 - 1/(c^2*x^2)])/(15*e^3) - (32*c^2*d^2*ArcCsc[c*x])/(5*e^4) + (2*c^2*d^2*ArcCsc[c*x])/(e^3*(e + d/x)) - (2*c^2*x^2*ArcCsc[c*x])/(5*e^2) - (2*c*x*(2*e*Sqrt[1 - 1/(c^2*x^2)] - 9*c*d*ArcCsc[c*x]))/(15*e^3)))/(d + e*x)^(3/2)) - (2*(e + d/x)^(3/2)*(c*x)^(3/2)*((2*(32*c^2*d^2*e + e^3)*Sqrt[(c*d + c*e*x)/(c*d + e)]*Sqrt[1 - c^2*x^2]*EllipticF[ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)])/(Sqrt[1 - 1/(c^2*x^2)]*Sqrt[e + d/x]*(c*x)^(3/2)) + (2*(48*c^3*d^3 + 8*c*d*e^2)*Sqrt[(c*d + c*e*x)/(c*d + e)]*Sqrt[1 - c^2*x^2]*EllipticPi[2, ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)])/(Sqrt[1 - 1/(c^2*x^2)]*Sqrt[e + d/x]*(c*x)^(3/2)) - (16*c*d*e*Cos[2*ArcCsc[c*x]]*((c*d + c*e*x)*(-1 + c^2*x^2) + c^2*d*x*Sqrt[(c*d + c*e*x)/(c*d + e)]*Sqrt[1 - c^2*x^2]*EllipticF[ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)] - (c*x*(1 + c*x)*Sqrt[(e - c*e*x)/(c*d + e)]*Sqrt[(c*d + c*e*x)/(c*d - e)]*((c*d + e)*EllipticE[ArcSin[Sqrt[(c*d + c*e*x)/(c*d - e)]], (c*d - e)/(c*d + e)] - e*EllipticF[ArcSin[Sqrt[(c*d + c*e*x)/(c*d - e)]], (c*d - e)/(c*d + e)]))/Sqrt[(e*(1 + c*x))/(-(c*d) + e)] + c*e*x*Sqrt[(c*d + c*e*x)/(c*d + e)]*Sqrt[1 - c^2*x^2]*EllipticPi[2, ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)]))/(Sqrt[1 - 1/(c^2*x^2)]*Sqrt[e + d/x]*Sqrt[c*x]*(-2 + c^2*x^2))))/(15*e^4*(d + e*x)^(3/2))))/c^4","C",1
64,1,750,369,13.8388893,"\int \frac{x^2 \left(a+b \csc ^{-1}(c x)\right)}{(d+e x)^{3/2}} \, dx","Integrate[(x^2*(a + b*ArcCsc[c*x]))/(d + e*x)^(3/2),x]","\frac{b \left(\frac{2 (c x)^{3/2} \left(\frac{d}{x}+e\right)^{3/2} \left(\frac{2 \sqrt{1-c^2 x^2} \left(8 c^2 d^2+e^2\right) \sqrt{\frac{c d+c e x}{c d+e}} \Pi \left(2;\sin ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{2}}\right)|\frac{2 e}{c d+e}\right)}{\sqrt{1-\frac{1}{c^2 x^2}} (c x)^{3/2} \sqrt{\frac{d}{x}+e}}+\frac{10 c d e \sqrt{1-c^2 x^2} \sqrt{\frac{c d+c e x}{c d+e}} F\left(\sin ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{2}}\right)|\frac{2 e}{c d+e}\right)}{\sqrt{1-\frac{1}{c^2 x^2}} (c x)^{3/2} \sqrt{\frac{d}{x}+e}}-\frac{2 e \cos \left(2 \csc ^{-1}(c x)\right) \left(\left(c^2 x^2-1\right) (c d+c e x)+c^2 d x \sqrt{1-c^2 x^2} \sqrt{\frac{c d+c e x}{c d+e}} F\left(\sin ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{2}}\right)|\frac{2 e}{c d+e}\right)+c e x \sqrt{1-c^2 x^2} \sqrt{\frac{c d+c e x}{c d+e}} \Pi \left(2;\sin ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{2}}\right)|\frac{2 e}{c d+e}\right)-\frac{c x (c x+1) \sqrt{\frac{e-c e x}{c d+e}} \sqrt{\frac{c d+c e x}{c d-e}} \left((c d+e) E\left(\sin ^{-1}\left(\sqrt{\frac{c d+c e x}{c d-e}}\right)|\frac{c d-e}{c d+e}\right)-e F\left(\sin ^{-1}\left(\sqrt{\frac{c d+c e x}{c d-e}}\right)|\frac{c d-e}{c d+e}\right)\right)}{\sqrt{\frac{e (c x+1)}{e-c d}}}\right)}{\sqrt{1-\frac{1}{c^2 x^2}} \sqrt{c x} \left(c^2 x^2-2\right) \sqrt{\frac{d}{x}+e}}\right)}{3 e^3 (d+e x)^{3/2}}-\frac{c^2 x^2 \left(\frac{d}{x}+e\right)^2 \left(-\frac{4 \sqrt{1-\frac{1}{c^2 x^2}}}{3 e^2}+\frac{16 c d \csc ^{-1}(c x)}{3 e^3}-\frac{2 c d \csc ^{-1}(c x)}{e^2 \left(\frac{d}{x}+e\right)}-\frac{2 c x \csc ^{-1}(c x)}{3 e^2}\right)}{(d+e x)^{3/2}}\right)}{c^3}-\frac{a d^3 \left(\frac{e x}{d}+1\right)^{3/2} B_{-\frac{e x}{d}}\left(3,-\frac{1}{2}\right)}{e^3 (d+e x)^{3/2}}","-\frac{2 d^2 \left(a+b \csc ^{-1}(c x)\right)}{e^3 \sqrt{d+e x}}-\frac{4 d \sqrt{d+e x} \left(a+b \csc ^{-1}(c x)\right)}{e^3}+\frac{2 (d+e x)^{3/2} \left(a+b \csc ^{-1}(c x)\right)}{3 e^3}+\frac{32 b d^2 \sqrt{1-c^2 x^2} \sqrt{\frac{c (d+e x)}{c d+e}} \Pi \left(2;\sin ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{2}}\right)|\frac{2 e}{c d+e}\right)}{3 c e^3 x \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{d+e x}}+\frac{20 b d \sqrt{1-c^2 x^2} \sqrt{\frac{c (d+e x)}{c d+e}} F\left(\sin ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{2}}\right)|\frac{2 e}{c d+e}\right)}{3 c^2 e^2 x \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{d+e x}}-\frac{4 b \sqrt{1-c^2 x^2} \sqrt{d+e x} E\left(\sin ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{2}}\right)|\frac{2 e}{c d+e}\right)}{3 c^2 e^2 x \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{\frac{c (d+e x)}{c d+e}}}",1,"-((a*d^3*(1 + (e*x)/d)^(3/2)*Beta[-((e*x)/d), 3, -1/2])/(e^3*(d + e*x)^(3/2))) + (b*(-((c^2*(e + d/x)^2*x^2*((-4*Sqrt[1 - 1/(c^2*x^2)])/(3*e^2) + (16*c*d*ArcCsc[c*x])/(3*e^3) - (2*c*d*ArcCsc[c*x])/(e^2*(e + d/x)) - (2*c*x*ArcCsc[c*x])/(3*e^2)))/(d + e*x)^(3/2)) + (2*(e + d/x)^(3/2)*(c*x)^(3/2)*((10*c*d*e*Sqrt[(c*d + c*e*x)/(c*d + e)]*Sqrt[1 - c^2*x^2]*EllipticF[ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)])/(Sqrt[1 - 1/(c^2*x^2)]*Sqrt[e + d/x]*(c*x)^(3/2)) + (2*(8*c^2*d^2 + e^2)*Sqrt[(c*d + c*e*x)/(c*d + e)]*Sqrt[1 - c^2*x^2]*EllipticPi[2, ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)])/(Sqrt[1 - 1/(c^2*x^2)]*Sqrt[e + d/x]*(c*x)^(3/2)) - (2*e*Cos[2*ArcCsc[c*x]]*((c*d + c*e*x)*(-1 + c^2*x^2) + c^2*d*x*Sqrt[(c*d + c*e*x)/(c*d + e)]*Sqrt[1 - c^2*x^2]*EllipticF[ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)] - (c*x*(1 + c*x)*Sqrt[(e - c*e*x)/(c*d + e)]*Sqrt[(c*d + c*e*x)/(c*d - e)]*((c*d + e)*EllipticE[ArcSin[Sqrt[(c*d + c*e*x)/(c*d - e)]], (c*d - e)/(c*d + e)] - e*EllipticF[ArcSin[Sqrt[(c*d + c*e*x)/(c*d - e)]], (c*d - e)/(c*d + e)]))/Sqrt[(e*(1 + c*x))/(-(c*d) + e)] + c*e*x*Sqrt[(c*d + c*e*x)/(c*d + e)]*Sqrt[1 - c^2*x^2]*EllipticPi[2, ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)]))/(Sqrt[1 - 1/(c^2*x^2)]*Sqrt[e + d/x]*Sqrt[c*x]*(-2 + c^2*x^2))))/(3*e^3*(d + e*x)^(3/2))))/c^3","C",1
65,1,226,238,1.8825296,"\int \frac{x \left(a+b \csc ^{-1}(c x)\right)}{(d+e x)^{3/2}} \, dx","Integrate[(x*(a + b*ArcCsc[c*x]))/(d + e*x)^(3/2),x]","\frac{2 \left(\frac{a (2 d+e x)}{\sqrt{d+e x}}-\frac{2 i b \sqrt{\frac{e (c x+1)}{e-c d}} \sqrt{\frac{e-c e x}{c d+e}} \left(F\left(i \sinh ^{-1}\left(\sqrt{-\frac{c}{c d+e}} \sqrt{d+e x}\right)|\frac{c d+e}{c d-e}\right)-2 \Pi \left(\frac{e}{c d}+1;i \sinh ^{-1}\left(\sqrt{-\frac{c}{c d+e}} \sqrt{d+e x}\right)|\frac{c d+e}{c d-e}\right)\right)}{c x \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{-\frac{c}{c d+e}}}+\frac{b \csc ^{-1}(c x) (2 d+e x)}{\sqrt{d+e x}}\right)}{e^2}","\frac{2 \sqrt{d+e x} \left(a+b \csc ^{-1}(c x)\right)}{e^2}+\frac{2 d \left(a+b \csc ^{-1}(c x)\right)}{e^2 \sqrt{d+e x}}-\frac{8 b d \sqrt{1-c^2 x^2} \sqrt{\frac{c (d+e x)}{c d+e}} \Pi \left(2;\sin ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{2}}\right)|\frac{2 e}{c d+e}\right)}{c e^2 x \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{d+e x}}-\frac{4 b \sqrt{1-c^2 x^2} \sqrt{\frac{c (d+e x)}{c d+e}} F\left(\sin ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{2}}\right)|\frac{2 e}{c d+e}\right)}{c^2 e x \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{d+e x}}",1,"(2*((a*(2*d + e*x))/Sqrt[d + e*x] + (b*(2*d + e*x)*ArcCsc[c*x])/Sqrt[d + e*x] - ((2*I)*b*Sqrt[(e*(1 + c*x))/(-(c*d) + e)]*Sqrt[(e - c*e*x)/(c*d + e)]*(EllipticF[I*ArcSinh[Sqrt[-(c/(c*d + e))]*Sqrt[d + e*x]], (c*d + e)/(c*d - e)] - 2*EllipticPi[1 + e/(c*d), I*ArcSinh[Sqrt[-(c/(c*d + e))]*Sqrt[d + e*x]], (c*d + e)/(c*d - e)]))/(c*Sqrt[-(c/(c*d + e))]*Sqrt[1 - 1/(c^2*x^2)]*x)))/e^2","C",1
66,1,124,119,0.2160486,"\int \frac{a+b \csc ^{-1}(c x)}{(d+e x)^{3/2}} \, dx","Integrate[(a + b*ArcCsc[c*x])/(d + e*x)^(3/2),x]","\frac{4 b c x \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{1-c^2 x^2} \sqrt{\frac{c (d+e x)}{c d+e}} \Pi \left(2;\sin ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{2}}\right)|\frac{2 e}{c d+e}\right)-2 \left(c^2 x^2-1\right) \left(a+b \csc ^{-1}(c x)\right)}{e \left(c^2 x^2-1\right) \sqrt{d+e x}}","\frac{4 b \sqrt{1-c^2 x^2} \sqrt{\frac{c (d+e x)}{c d+e}} \Pi \left(2;\sin ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{2}}\right)|\frac{2 e}{c d+e}\right)}{c e x \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{d+e x}}-\frac{2 \left(a+b \csc ^{-1}(c x)\right)}{e \sqrt{d+e x}}",1,"(-2*(-1 + c^2*x^2)*(a + b*ArcCsc[c*x]) + 4*b*c*Sqrt[1 - 1/(c^2*x^2)]*x*Sqrt[(c*(d + e*x))/(c*d + e)]*Sqrt[1 - c^2*x^2]*EllipticPi[2, ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)])/(e*Sqrt[d + e*x]*(-1 + c^2*x^2))","A",1
67,0,0,24,11.7826922,"\int \frac{a+b \csc ^{-1}(c x)}{x (d+e x)^{3/2}} \, dx","Integrate[(a + b*ArcCsc[c*x])/(x*(d + e*x)^(3/2)),x]","\int \frac{a+b \csc ^{-1}(c x)}{x (d+e x)^{3/2}} \, dx","\text{Int}\left(\frac{a+b \csc ^{-1}(c x)}{x (d+e x)^{3/2}},x\right)",0,"Integrate[(a + b*ArcCsc[c*x])/(x*(d + e*x)^(3/2)), x]","A",-1
68,0,0,24,14.9087656,"\int \frac{a+b \csc ^{-1}(c x)}{x^2 (d+e x)^{3/2}} \, dx","Integrate[(a + b*ArcCsc[c*x])/(x^2*(d + e*x)^(3/2)),x]","\int \frac{a+b \csc ^{-1}(c x)}{x^2 (d+e x)^{3/2}} \, dx","\text{Int}\left(\frac{a+b \csc ^{-1}(c x)}{x^2 (d+e x)^{3/2}},x\right)",0,"Integrate[(a + b*ArcCsc[c*x])/(x^2*(d + e*x)^(3/2)), x]","A",-1
69,1,887,602,14.1012785,"\int \frac{x^3 \left(a+b \csc ^{-1}(c x)\right)}{(d+e x)^{5/2}} \, dx","Integrate[(x^3*(a + b*ArcCsc[c*x]))/(d + e*x)^(5/2),x]","\frac{a \left(\frac{e x}{d}+1\right)^{5/2} B_{-\frac{e x}{d}}\left(4,-\frac{3}{2}\right) d^4}{e^4 (d+e x)^{5/2}}+\frac{b \left(\frac{2 \left(\frac{d}{x}+e\right)^{5/2} (c x)^{5/2} \left(\frac{2 \cos \left(2 \csc ^{-1}(c x)\right) \left(d x \sqrt{\frac{c d+c e x}{c d+e}} \sqrt{1-c^2 x^2} F\left(\sin ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{2}}\right)|\frac{2 e}{c d+e}\right) c^2-\frac{x (c x+1) \sqrt{\frac{e-c e x}{c d+e}} \sqrt{\frac{c d+c e x}{c d-e}} \left((c d+e) E\left(\sin ^{-1}\left(\sqrt{\frac{c d+c e x}{c d-e}}\right)|\frac{c d-e}{c d+e}\right)-e F\left(\sin ^{-1}\left(\sqrt{\frac{c d+c e x}{c d-e}}\right)|\frac{c d-e}{c d+e}\right)\right) c}{\sqrt{\frac{e (c x+1)}{e-c d}}}+e x \sqrt{\frac{c d+c e x}{c d+e}} \sqrt{1-c^2 x^2} \Pi \left(2;\sin ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{2}}\right)|\frac{2 e}{c d+e}\right) c+(c d+c e x) \left(c^2 x^2-1\right)\right) e^3}{\sqrt{1-\frac{1}{c^2 x^2}} \sqrt{\frac{d}{x}+e} \sqrt{c x} \left(c^2 x^2-2\right)}+\frac{2 \left(8 c^3 d^3 e-8 c d e^3\right) \sqrt{\frac{c d+c e x}{c d+e}} \sqrt{1-c^2 x^2} F\left(\sin ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{2}}\right)|\frac{2 e}{c d+e}\right)}{\sqrt{1-\frac{1}{c^2 x^2}} \sqrt{\frac{d}{x}+e} (c x)^{3/2}}+\frac{2 \left(16 c^4 d^4-16 c^2 e^2 d^2-e^4\right) \sqrt{\frac{c d+c e x}{c d+e}} \sqrt{1-c^2 x^2} \Pi \left(2;\sin ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{2}}\right)|\frac{2 e}{c d+e}\right)}{\sqrt{1-\frac{1}{c^2 x^2}} \sqrt{\frac{d}{x}+e} (c x)^{3/2}}\right)}{3 (c d-e) e^4 (c d+e) (d+e x)^{5/2}}-\frac{c^3 \left(\frac{d}{x}+e\right)^3 x^3 \left(-\frac{2 c x \csc ^{-1}(c x)}{3 e^3}-\frac{2 c d \csc ^{-1}(c x)}{3 e^2 \left(\frac{d}{x}+e\right)^2}+\frac{32 c d \csc ^{-1}(c x)}{3 e^4}-\frac{2 \left(-7 c^3 \csc ^{-1}(c x) d^3-2 c^2 e \sqrt{1-\frac{1}{c^2 x^2}} d^2+7 c e^2 \csc ^{-1}(c x) d\right)}{3 e^3 \left(e^2-c^2 d^2\right) \left(\frac{d}{x}+e\right)}-\frac{4 \sqrt{1-\frac{1}{c^2 x^2}}}{3 e \left(e^2-c^2 d^2\right)}\right)}{(d+e x)^{5/2}}\right)}{c^4}","\frac{2 d^3 \left(a+b \csc ^{-1}(c x)\right)}{3 e^4 (d+e x)^{3/2}}-\frac{6 d^2 \left(a+b \csc ^{-1}(c x)\right)}{e^4 \sqrt{d+e x}}-\frac{6 d \sqrt{d+e x} \left(a+b \csc ^{-1}(c x)\right)}{e^4}+\frac{2 (d+e x)^{3/2} \left(a+b \csc ^{-1}(c x)\right)}{3 e^4}+\frac{64 b d^2 \sqrt{1-c^2 x^2} \sqrt{\frac{c (d+e x)}{c d+e}} \Pi \left(2;\sin ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{2}}\right)|\frac{2 e}{c d+e}\right)}{3 c e^4 x \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{d+e x}}-\frac{4 b d^2 \left(1-c^2 x^2\right)}{3 c e^2 x \sqrt{1-\frac{1}{c^2 x^2}} \left(c^2 d^2-e^2\right) \sqrt{d+e x}}+\frac{8 b d^2 \sqrt{1-c^2 x^2} \sqrt{d+e x} E\left(\sin ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{2}}\right)|\frac{2 e}{c d+e}\right)}{3 e^3 x \sqrt{1-\frac{1}{c^2 x^2}} \left(c^2 d^2-e^2\right) \sqrt{\frac{c (d+e x)}{c d+e}}}-\frac{4 b \sqrt{1-c^2 x^2} \left(2 c^2 d^2-e^2\right) \sqrt{d+e x} E\left(\sin ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{2}}\right)|\frac{2 e}{c d+e}\right)}{3 c^2 e^3 x \sqrt{1-\frac{1}{c^2 x^2}} \left(c^2 d^2-e^2\right) \sqrt{\frac{c (d+e x)}{c d+e}}}+\frac{32 b d \sqrt{1-c^2 x^2} \sqrt{\frac{c (d+e x)}{c d+e}} F\left(\sin ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{2}}\right)|\frac{2 e}{c d+e}\right)}{3 c^2 e^3 x \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{d+e x}}",1,"(a*d^4*(1 + (e*x)/d)^(5/2)*Beta[-((e*x)/d), 4, -3/2])/(e^4*(d + e*x)^(5/2)) + (b*(-((c^3*(e + d/x)^3*x^3*((-4*Sqrt[1 - 1/(c^2*x^2)])/(3*e*(-(c^2*d^2) + e^2)) + (32*c*d*ArcCsc[c*x])/(3*e^4) - (2*c*d*ArcCsc[c*x])/(3*e^2*(e + d/x)^2) - (2*c*x*ArcCsc[c*x])/(3*e^3) - (2*(-2*c^2*d^2*e*Sqrt[1 - 1/(c^2*x^2)] - 7*c^3*d^3*ArcCsc[c*x] + 7*c*d*e^2*ArcCsc[c*x]))/(3*e^3*(-(c^2*d^2) + e^2)*(e + d/x))))/(d + e*x)^(5/2)) + (2*(e + d/x)^(5/2)*(c*x)^(5/2)*((2*(8*c^3*d^3*e - 8*c*d*e^3)*Sqrt[(c*d + c*e*x)/(c*d + e)]*Sqrt[1 - c^2*x^2]*EllipticF[ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)])/(Sqrt[1 - 1/(c^2*x^2)]*Sqrt[e + d/x]*(c*x)^(3/2)) + (2*(16*c^4*d^4 - 16*c^2*d^2*e^2 - e^4)*Sqrt[(c*d + c*e*x)/(c*d + e)]*Sqrt[1 - c^2*x^2]*EllipticPi[2, ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)])/(Sqrt[1 - 1/(c^2*x^2)]*Sqrt[e + d/x]*(c*x)^(3/2)) + (2*e^3*Cos[2*ArcCsc[c*x]]*((c*d + c*e*x)*(-1 + c^2*x^2) + c^2*d*x*Sqrt[(c*d + c*e*x)/(c*d + e)]*Sqrt[1 - c^2*x^2]*EllipticF[ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)] - (c*x*(1 + c*x)*Sqrt[(e - c*e*x)/(c*d + e)]*Sqrt[(c*d + c*e*x)/(c*d - e)]*((c*d + e)*EllipticE[ArcSin[Sqrt[(c*d + c*e*x)/(c*d - e)]], (c*d - e)/(c*d + e)] - e*EllipticF[ArcSin[Sqrt[(c*d + c*e*x)/(c*d - e)]], (c*d - e)/(c*d + e)]))/Sqrt[(e*(1 + c*x))/(-(c*d) + e)] + c*e*x*Sqrt[(c*d + c*e*x)/(c*d + e)]*Sqrt[1 - c^2*x^2]*EllipticPi[2, ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)]))/(Sqrt[1 - 1/(c^2*x^2)]*Sqrt[e + d/x]*Sqrt[c*x]*(-2 + c^2*x^2))))/(3*(c*d - e)*e^4*(c*d + e)*(d + e*x)^(5/2))))/c^4","C",1
70,1,856,440,13.9521937,"\int \frac{x^2 \left(a+b \csc ^{-1}(c x)\right)}{(d+e x)^{5/2}} \, dx","Integrate[(x^2*(a + b*ArcCsc[c*x]))/(d + e*x)^(5/2),x]","\frac{b \left(-\frac{c^3 \left(\frac{d}{x}+e\right)^3 \left(\frac{4 c \sqrt{1-\frac{1}{c^2 x^2}} d}{3 e^2 \left(e^2-c^2 d^2\right)}+\frac{2 \csc ^{-1}(c x)}{3 e \left(\frac{d}{x}+e\right)^2}-\frac{16 \csc ^{-1}(c x)}{3 e^3}+\frac{4 \left(-2 c^2 \csc ^{-1}(c x) d^2-c e \sqrt{1-\frac{1}{c^2 x^2}} d+2 e^2 \csc ^{-1}(c x)\right)}{3 e^2 \left(e^2-c^2 d^2\right) \left(\frac{d}{x}+e\right)}\right) x^3}{(d+e x)^{5/2}}-\frac{2 \left(\frac{d}{x}+e\right)^{5/2} (c x)^{5/2} \left(\frac{2 \left(3 c^2 d^2 e-3 e^3\right) \sqrt{\frac{c d+c e x}{c d+e}} \sqrt{1-c^2 x^2} F\left(\sin ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{2}}\right)|\frac{2 e}{c d+e}\right)}{\sqrt{1-\frac{1}{c^2 x^2}} \sqrt{\frac{d}{x}+e} (c x)^{3/2}}+\frac{2 \left(8 c^3 d^3-9 c d e^2\right) \sqrt{\frac{c d+c e x}{c d+e}} \sqrt{1-c^2 x^2} \Pi \left(2;\sin ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{2}}\right)|\frac{2 e}{c d+e}\right)}{\sqrt{1-\frac{1}{c^2 x^2}} \sqrt{\frac{d}{x}+e} (c x)^{3/2}}+\frac{2 c d e \cos \left(2 \csc ^{-1}(c x)\right) \left(d x \sqrt{\frac{c d+c e x}{c d+e}} \sqrt{1-c^2 x^2} F\left(\sin ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{2}}\right)|\frac{2 e}{c d+e}\right) c^2-\frac{x (c x+1) \sqrt{\frac{e-c e x}{c d+e}} \sqrt{\frac{c d+c e x}{c d-e}} \left((c d+e) E\left(\sin ^{-1}\left(\sqrt{\frac{c d+c e x}{c d-e}}\right)|\frac{c d-e}{c d+e}\right)-e F\left(\sin ^{-1}\left(\sqrt{\frac{c d+c e x}{c d-e}}\right)|\frac{c d-e}{c d+e}\right)\right) c}{\sqrt{\frac{e (c x+1)}{e-c d}}}+e x \sqrt{\frac{c d+c e x}{c d+e}} \sqrt{1-c^2 x^2} \Pi \left(2;\sin ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{2}}\right)|\frac{2 e}{c d+e}\right) c+(c d+c e x) \left(c^2 x^2-1\right)\right)}{\sqrt{1-\frac{1}{c^2 x^2}} \sqrt{\frac{d}{x}+e} \sqrt{c x} \left(c^2 x^2-2\right)}\right)}{3 (c d-e) e^3 (c d+e) (d+e x)^{5/2}}\right)}{c^3}-\frac{a d^3 \left(\frac{e x}{d}+1\right)^{5/2} B_{-\frac{e x}{d}}\left(3,-\frac{3}{2}\right)}{e^3 (d+e x)^{5/2}}","-\frac{2 d^2 \left(a+b \csc ^{-1}(c x)\right)}{3 e^3 (d+e x)^{3/2}}+\frac{4 d \left(a+b \csc ^{-1}(c x)\right)}{e^3 \sqrt{d+e x}}+\frac{2 \sqrt{d+e x} \left(a+b \csc ^{-1}(c x)\right)}{e^3}+\frac{4 b d \left(1-c^2 x^2\right)}{3 c e x \sqrt{1-\frac{1}{c^2 x^2}} \left(c^2 d^2-e^2\right) \sqrt{d+e x}}-\frac{4 b d \sqrt{1-c^2 x^2} \sqrt{d+e x} E\left(\sin ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{2}}\right)|\frac{2 e}{c d+e}\right)}{3 e^2 x \sqrt{1-\frac{1}{c^2 x^2}} \left(c^2 d^2-e^2\right) \sqrt{\frac{c (d+e x)}{c d+e}}}-\frac{32 b d \sqrt{1-c^2 x^2} \sqrt{\frac{c (d+e x)}{c d+e}} \Pi \left(2;\sin ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{2}}\right)|\frac{2 e}{c d+e}\right)}{3 c e^3 x \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{d+e x}}-\frac{4 b \sqrt{1-c^2 x^2} \sqrt{\frac{c (d+e x)}{c d+e}} F\left(\sin ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{2}}\right)|\frac{2 e}{c d+e}\right)}{c^2 e^2 x \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{d+e x}}",1,"-((a*d^3*(1 + (e*x)/d)^(5/2)*Beta[-((e*x)/d), 3, -3/2])/(e^3*(d + e*x)^(5/2))) + (b*(-((c^3*(e + d/x)^3*x^3*((4*c*d*Sqrt[1 - 1/(c^2*x^2)])/(3*e^2*(-(c^2*d^2) + e^2)) - (16*ArcCsc[c*x])/(3*e^3) + (2*ArcCsc[c*x])/(3*e*(e + d/x)^2) + (4*(-(c*d*e*Sqrt[1 - 1/(c^2*x^2)]) - 2*c^2*d^2*ArcCsc[c*x] + 2*e^2*ArcCsc[c*x]))/(3*e^2*(-(c^2*d^2) + e^2)*(e + d/x))))/(d + e*x)^(5/2)) - (2*(e + d/x)^(5/2)*(c*x)^(5/2)*((2*(3*c^2*d^2*e - 3*e^3)*Sqrt[(c*d + c*e*x)/(c*d + e)]*Sqrt[1 - c^2*x^2]*EllipticF[ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)])/(Sqrt[1 - 1/(c^2*x^2)]*Sqrt[e + d/x]*(c*x)^(3/2)) + (2*(8*c^3*d^3 - 9*c*d*e^2)*Sqrt[(c*d + c*e*x)/(c*d + e)]*Sqrt[1 - c^2*x^2]*EllipticPi[2, ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)])/(Sqrt[1 - 1/(c^2*x^2)]*Sqrt[e + d/x]*(c*x)^(3/2)) + (2*c*d*e*Cos[2*ArcCsc[c*x]]*((c*d + c*e*x)*(-1 + c^2*x^2) + c^2*d*x*Sqrt[(c*d + c*e*x)/(c*d + e)]*Sqrt[1 - c^2*x^2]*EllipticF[ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)] - (c*x*(1 + c*x)*Sqrt[(e - c*e*x)/(c*d + e)]*Sqrt[(c*d + c*e*x)/(c*d - e)]*((c*d + e)*EllipticE[ArcSin[Sqrt[(c*d + c*e*x)/(c*d - e)]], (c*d - e)/(c*d + e)] - e*EllipticF[ArcSin[Sqrt[(c*d + c*e*x)/(c*d - e)]], (c*d - e)/(c*d + e)]))/Sqrt[(e*(1 + c*x))/(-(c*d) + e)] + c*e*x*Sqrt[(c*d + c*e*x)/(c*d + e)]*Sqrt[1 - c^2*x^2]*EllipticPi[2, ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)]))/(Sqrt[1 - 1/(c^2*x^2)]*Sqrt[e + d/x]*Sqrt[c*x]*(-2 + c^2*x^2))))/(3*(c*d - e)*e^3*(c*d + e)*(d + e*x)^(5/2))))/c^3","C",1
71,1,345,314,1.7503442,"\int \frac{x \left(a+b \csc ^{-1}(c x)\right)}{(d+e x)^{5/2}} \, dx","Integrate[(x*(a + b*ArcCsc[c*x]))/(d + e*x)^(5/2),x]","-\frac{2 a (2 d+3 e x)}{3 e^2 (d+e x)^{3/2}}+\frac{4 b c x \sqrt{1-\frac{1}{c^2 x^2}}}{3 \left(c^2 d^2-e^2\right) \sqrt{d+e x}}+\frac{4 i b \sqrt{-\frac{c}{c d+e}} \sqrt{\frac{e (c x+1)}{e-c d}} \sqrt{\frac{e-c e x}{c d+e}} \left(-c d F\left(i \sinh ^{-1}\left(\sqrt{-\frac{c}{c d+e}} \sqrt{d+e x}\right)|\frac{c d+e}{c d-e}\right)+c d E\left(i \sinh ^{-1}\left(\sqrt{-\frac{c}{c d+e}} \sqrt{d+e x}\right)|\frac{c d+e}{c d-e}\right)+2 (c d+e) \Pi \left(\frac{e}{c d}+1;i \sinh ^{-1}\left(\sqrt{-\frac{c}{c d+e}} \sqrt{d+e x}\right)|\frac{c d+e}{c d-e}\right)\right)}{3 c^2 d e^2 x \sqrt{1-\frac{1}{c^2 x^2}}}-\frac{2 b \csc ^{-1}(c x) (2 d+3 e x)}{3 e^2 (d+e x)^{3/2}}","-\frac{2 \left(a+b \csc ^{-1}(c x)\right)}{e^2 \sqrt{d+e x}}+\frac{2 d \left(a+b \csc ^{-1}(c x)\right)}{3 e^2 (d+e x)^{3/2}}-\frac{4 b \left(1-c^2 x^2\right)}{3 c x \sqrt{1-\frac{1}{c^2 x^2}} \left(c^2 d^2-e^2\right) \sqrt{d+e x}}+\frac{4 b \sqrt{1-c^2 x^2} \sqrt{d+e x} E\left(\sin ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{2}}\right)|\frac{2 e}{c d+e}\right)}{3 e x \sqrt{1-\frac{1}{c^2 x^2}} \left(c^2 d^2-e^2\right) \sqrt{\frac{c (d+e x)}{c d+e}}}+\frac{8 b \sqrt{1-c^2 x^2} \sqrt{\frac{c (d+e x)}{c d+e}} \Pi \left(2;\sin ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{2}}\right)|\frac{2 e}{c d+e}\right)}{3 c e^2 x \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{d+e x}}",1,"(4*b*c*Sqrt[1 - 1/(c^2*x^2)]*x)/(3*(c^2*d^2 - e^2)*Sqrt[d + e*x]) - (2*a*(2*d + 3*e*x))/(3*e^2*(d + e*x)^(3/2)) - (2*b*(2*d + 3*e*x)*ArcCsc[c*x])/(3*e^2*(d + e*x)^(3/2)) + (((4*I)/3)*b*Sqrt[-(c/(c*d + e))]*Sqrt[(e*(1 + c*x))/(-(c*d) + e)]*Sqrt[(e - c*e*x)/(c*d + e)]*(c*d*EllipticE[I*ArcSinh[Sqrt[-(c/(c*d + e))]*Sqrt[d + e*x]], (c*d + e)/(c*d - e)] - c*d*EllipticF[I*ArcSinh[Sqrt[-(c/(c*d + e))]*Sqrt[d + e*x]], (c*d + e)/(c*d - e)] + 2*(c*d + e)*EllipticPi[1 + e/(c*d), I*ArcSinh[Sqrt[-(c/(c*d + e))]*Sqrt[d + e*x]], (c*d + e)/(c*d - e)]))/(c^2*d*e^2*Sqrt[1 - 1/(c^2*x^2)]*x)","C",1
72,1,725,298,13.6364122,"\int \frac{a+b \csc ^{-1}(c x)}{(d+e x)^{5/2}} \, dx","Integrate[(a + b*ArcCsc[c*x])/(d + e*x)^(5/2),x]","\frac{b \left(\frac{2 (c x)^{5/2} \left(\frac{d}{x}+e\right)^{5/2} \left(\frac{2 c d \sqrt{1-c^2 x^2} \sqrt{\frac{c d+c e x}{c d+e}} \Pi \left(2;\sin ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{2}}\right)|\frac{2 e}{c d+e}\right)}{\sqrt{1-\frac{1}{c^2 x^2}} (c x)^{3/2} \sqrt{\frac{d}{x}+e}}-\frac{2 e \cos \left(2 \csc ^{-1}(c x)\right) \left(\left(c^2 x^2-1\right) (c d+c e x)+c^2 d x \sqrt{1-c^2 x^2} \sqrt{\frac{c d+c e x}{c d+e}} F\left(\sin ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{2}}\right)|\frac{2 e}{c d+e}\right)+c e x \sqrt{1-c^2 x^2} \sqrt{\frac{c d+c e x}{c d+e}} \Pi \left(2;\sin ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{2}}\right)|\frac{2 e}{c d+e}\right)-\frac{c x (c x+1) \sqrt{\frac{e-c e x}{c d+e}} \sqrt{\frac{c d+c e x}{c d-e}} \left((c d+e) E\left(\sin ^{-1}\left(\sqrt{\frac{c d+c e x}{c d-e}}\right)|\frac{c d-e}{c d+e}\right)-e F\left(\sin ^{-1}\left(\sqrt{\frac{c d+c e x}{c d-e}}\right)|\frac{c d-e}{c d+e}\right)\right)}{\sqrt{\frac{e (c x+1)}{e-c d}}}\right)}{c d \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{c x} \left(c^2 x^2-2\right) \sqrt{\frac{d}{x}+e}}\right)}{3 e (c d-e) (c d+e) (d+e x)^{5/2}}-\frac{c^3 x^3 \left(\frac{d}{x}+e\right)^3 \left(-\frac{4 \sqrt{1-\frac{1}{c^2 x^2}}}{3 c d \left(c^2 d^2-e^2\right)}-\frac{4 \left(c^2 d^2 \csc ^{-1}(c x)-c d e \sqrt{1-\frac{1}{c^2 x^2}}-e^2 \csc ^{-1}(c x)\right)}{3 c^2 d^2 \left(c^2 d^2-e^2\right) \left(\frac{d}{x}+e\right)}+\frac{2 \csc ^{-1}(c x)}{3 c^2 d^2 e}+\frac{2 e \csc ^{-1}(c x)}{3 c^2 d^2 \left(\frac{d}{x}+e\right)^2}\right)}{(d+e x)^{5/2}}\right)}{c}-\frac{2 a}{3 e (d+e x)^{3/2}}","-\frac{2 \left(a+b \csc ^{-1}(c x)\right)}{3 e (d+e x)^{3/2}}+\frac{4 b e \left(1-c^2 x^2\right)}{3 c d x \sqrt{1-\frac{1}{c^2 x^2}} \left(c^2 d^2-e^2\right) \sqrt{d+e x}}-\frac{4 b \sqrt{1-c^2 x^2} \sqrt{d+e x} E\left(\sin ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{2}}\right)|\frac{2 e}{c d+e}\right)}{3 d x \sqrt{1-\frac{1}{c^2 x^2}} \left(c^2 d^2-e^2\right) \sqrt{\frac{c (d+e x)}{c d+e}}}+\frac{4 b \sqrt{1-c^2 x^2} \sqrt{\frac{c (d+e x)}{c d+e}} \Pi \left(2;\sin ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{2}}\right)|\frac{2 e}{c d+e}\right)}{3 c d e x \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{d+e x}}",1,"(-2*a)/(3*e*(d + e*x)^(3/2)) + (b*(-((c^3*(e + d/x)^3*x^3*((-4*Sqrt[1 - 1/(c^2*x^2)])/(3*c*d*(c^2*d^2 - e^2)) + (2*ArcCsc[c*x])/(3*c^2*d^2*e) + (2*e*ArcCsc[c*x])/(3*c^2*d^2*(e + d/x)^2) - (4*(-(c*d*e*Sqrt[1 - 1/(c^2*x^2)]) + c^2*d^2*ArcCsc[c*x] - e^2*ArcCsc[c*x]))/(3*c^2*d^2*(c^2*d^2 - e^2)*(e + d/x))))/(d + e*x)^(5/2)) + (2*(e + d/x)^(5/2)*(c*x)^(5/2)*((2*c*d*Sqrt[(c*d + c*e*x)/(c*d + e)]*Sqrt[1 - c^2*x^2]*EllipticPi[2, ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)])/(Sqrt[1 - 1/(c^2*x^2)]*Sqrt[e + d/x]*(c*x)^(3/2)) - (2*e*Cos[2*ArcCsc[c*x]]*((c*d + c*e*x)*(-1 + c^2*x^2) + c^2*d*x*Sqrt[(c*d + c*e*x)/(c*d + e)]*Sqrt[1 - c^2*x^2]*EllipticF[ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)] - (c*x*(1 + c*x)*Sqrt[(e - c*e*x)/(c*d + e)]*Sqrt[(c*d + c*e*x)/(c*d - e)]*((c*d + e)*EllipticE[ArcSin[Sqrt[(c*d + c*e*x)/(c*d - e)]], (c*d - e)/(c*d + e)] - e*EllipticF[ArcSin[Sqrt[(c*d + c*e*x)/(c*d - e)]], (c*d - e)/(c*d + e)]))/Sqrt[(e*(1 + c*x))/(-(c*d) + e)] + c*e*x*Sqrt[(c*d + c*e*x)/(c*d + e)]*Sqrt[1 - c^2*x^2]*EllipticPi[2, ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)]))/(c*d*Sqrt[1 - 1/(c^2*x^2)]*Sqrt[e + d/x]*Sqrt[c*x]*(-2 + c^2*x^2))))/(3*(c*d - e)*e*(c*d + e)*(d + e*x)^(5/2))))/c","B",1
73,0,0,24,29.9574302,"\int \frac{a+b \csc ^{-1}(c x)}{x (d+e x)^{5/2}} \, dx","Integrate[(a + b*ArcCsc[c*x])/(x*(d + e*x)^(5/2)),x]","\int \frac{a+b \csc ^{-1}(c x)}{x (d+e x)^{5/2}} \, dx","\text{Int}\left(\frac{a+b \csc ^{-1}(c x)}{x (d+e x)^{5/2}},x\right)",0,"Integrate[(a + b*ArcCsc[c*x])/(x*(d + e*x)^(5/2)), x]","A",-1
74,0,0,24,26.6770024,"\int \frac{a+b \csc ^{-1}(c x)}{x^2 (d+e x)^{5/2}} \, dx","Integrate[(a + b*ArcCsc[c*x])/(x^2*(d + e*x)^(5/2)),x]","\int \frac{a+b \csc ^{-1}(c x)}{x^2 (d+e x)^{5/2}} \, dx","\text{Int}\left(\frac{a+b \csc ^{-1}(c x)}{x^2 (d+e x)^{5/2}},x\right)",0,"Integrate[(a + b*ArcCsc[c*x])/(x^2*(d + e*x)^(5/2)), x]","A",-1
75,1,1002,540,13.9854837,"\int \frac{a+b \csc ^{-1}(c x)}{(d+e x)^{7/2}} \, dx","Integrate[(a + b*ArcCsc[c*x])/(d + e*x)^(7/2),x]","\frac{b \left(\frac{2 \left(\frac{d}{x}+e\right)^{7/2} (c x)^{7/2} \left(\frac{2 \left(c^2 d^2 e-e^3\right) \sqrt{\frac{c d+c e x}{c d+e}} \sqrt{1-c^2 x^2} F\left(\sin ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{2}}\right)|\frac{2 e}{c d+e}\right)}{\sqrt{1-\frac{1}{c^2 x^2}} \sqrt{\frac{d}{x}+e} (c x)^{3/2}}+\frac{2 \left(3 c^3 d^3+c e^2 d\right) \sqrt{\frac{c d+c e x}{c d+e}} \sqrt{1-c^2 x^2} \Pi \left(2;\sin ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{2}}\right)|\frac{2 e}{c d+e}\right)}{\sqrt{1-\frac{1}{c^2 x^2}} \sqrt{\frac{d}{x}+e} (c x)^{3/2}}+\frac{2 \left(3 e^3-7 c^2 d^2 e\right) \cos \left(2 \csc ^{-1}(c x)\right) \left(d x \sqrt{\frac{c d+c e x}{c d+e}} \sqrt{1-c^2 x^2} F\left(\sin ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{2}}\right)|\frac{2 e}{c d+e}\right) c^2-\frac{x (c x+1) \sqrt{\frac{e-c e x}{c d+e}} \sqrt{\frac{c d+c e x}{c d-e}} \left((c d+e) E\left(\sin ^{-1}\left(\sqrt{\frac{c d+c e x}{c d-e}}\right)|\frac{c d-e}{c d+e}\right)-e F\left(\sin ^{-1}\left(\sqrt{\frac{c d+c e x}{c d-e}}\right)|\frac{c d-e}{c d+e}\right)\right) c}{\sqrt{\frac{e (c x+1)}{e-c d}}}+e x \sqrt{\frac{c d+c e x}{c d+e}} \sqrt{1-c^2 x^2} \Pi \left(2;\sin ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{2}}\right)|\frac{2 e}{c d+e}\right) c+(c d+c e x) \left(c^2 x^2-1\right)\right)}{c d \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{\frac{d}{x}+e} \sqrt{c x} \left(c^2 x^2-2\right)}\right)}{15 c d (c d-e)^2 e (c d+e)^2 (d+e x)^{7/2}}-\frac{c^4 \left(\frac{d}{x}+e\right)^4 x^4 \left(-\frac{2 \csc ^{-1}(c x) e^2}{5 c^3 d^3 \left(\frac{d}{x}+e\right)^3}-\frac{2 \left(9 \csc ^{-1}(c x) e^3+2 c d \sqrt{1-\frac{1}{c^2 x^2}} e^2-9 c^2 d^2 \csc ^{-1}(c x) e\right)}{15 c^3 d^3 \left(c^2 d^2-e^2\right) \left(\frac{d}{x}+e\right)^2}-\frac{2 \left(9 c^4 \csc ^{-1}(c x) d^4-16 c^3 e \sqrt{1-\frac{1}{c^2 x^2}} d^3-18 c^2 e^2 \csc ^{-1}(c x) d^2+8 c e^3 \sqrt{1-\frac{1}{c^2 x^2}} d+9 e^4 \csc ^{-1}(c x)\right)}{15 c^3 d^3 \left(c^2 d^2-e^2\right)^2 \left(\frac{d}{x}+e\right)}+\frac{4 \left(3 e^2-7 c^2 d^2\right) \sqrt{1-\frac{1}{c^2 x^2}}}{15 c^2 d^2 \left(e^2-c^2 d^2\right)^2}+\frac{2 \csc ^{-1}(c x)}{5 c^3 d^3 e}\right)}{(d+e x)^{7/2}}\right)}{c}-\frac{2 a}{5 e (d+e x)^{5/2}}","-\frac{2 \left(a+b \csc ^{-1}(c x)\right)}{5 e (d+e x)^{5/2}}+\frac{4 b e \left(1-c^2 x^2\right)}{5 c d^2 x \sqrt{1-\frac{1}{c^2 x^2}} \left(c^2 d^2-e^2\right) \sqrt{d+e x}}+\frac{16 b c e \left(1-c^2 x^2\right)}{15 x \sqrt{1-\frac{1}{c^2 x^2}} \left(c^2 d^2-e^2\right)^2 \sqrt{d+e x}}+\frac{4 b e \left(1-c^2 x^2\right)}{15 c d x \sqrt{1-\frac{1}{c^2 x^2}} \left(c^2 d^2-e^2\right) (d+e x)^{3/2}}+\frac{4 b \sqrt{1-c^2 x^2} \sqrt{\frac{c (d+e x)}{c d+e}} F\left(\sin ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{2}}\right)|\frac{2 e}{c d+e}\right)}{15 d x \sqrt{1-\frac{1}{c^2 x^2}} \left(c^2 d^2-e^2\right) \sqrt{d+e x}}+\frac{4 b \sqrt{1-c^2 x^2} \sqrt{\frac{c (d+e x)}{c d+e}} \Pi \left(2;\sin ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{2}}\right)|\frac{2 e}{c d+e}\right)}{5 c d^2 e x \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{d+e x}}-\frac{4 b \sqrt{1-c^2 x^2} \left(7 c^2 d^2-3 e^2\right) \sqrt{d+e x} E\left(\sin ^{-1}\left(\frac{\sqrt{1-c x}}{\sqrt{2}}\right)|\frac{2 e}{c d+e}\right)}{15 x \sqrt{1-\frac{1}{c^2 x^2}} \left(c^2 d^3-d e^2\right)^2 \sqrt{\frac{c (d+e x)}{c d+e}}}",1,"(-2*a)/(5*e*(d + e*x)^(5/2)) + (b*(-((c^4*(e + d/x)^4*x^4*((4*(-7*c^2*d^2 + 3*e^2)*Sqrt[1 - 1/(c^2*x^2)])/(15*c^2*d^2*(-(c^2*d^2) + e^2)^2) + (2*ArcCsc[c*x])/(5*c^3*d^3*e) - (2*e^2*ArcCsc[c*x])/(5*c^3*d^3*(e + d/x)^3) - (2*(2*c*d*e^2*Sqrt[1 - 1/(c^2*x^2)] - 9*c^2*d^2*e*ArcCsc[c*x] + 9*e^3*ArcCsc[c*x]))/(15*c^3*d^3*(c^2*d^2 - e^2)*(e + d/x)^2) - (2*(-16*c^3*d^3*e*Sqrt[1 - 1/(c^2*x^2)] + 8*c*d*e^3*Sqrt[1 - 1/(c^2*x^2)] + 9*c^4*d^4*ArcCsc[c*x] - 18*c^2*d^2*e^2*ArcCsc[c*x] + 9*e^4*ArcCsc[c*x]))/(15*c^3*d^3*(c^2*d^2 - e^2)^2*(e + d/x))))/(d + e*x)^(7/2)) + (2*(e + d/x)^(7/2)*(c*x)^(7/2)*((2*(c^2*d^2*e - e^3)*Sqrt[(c*d + c*e*x)/(c*d + e)]*Sqrt[1 - c^2*x^2]*EllipticF[ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)])/(Sqrt[1 - 1/(c^2*x^2)]*Sqrt[e + d/x]*(c*x)^(3/2)) + (2*(3*c^3*d^3 + c*d*e^2)*Sqrt[(c*d + c*e*x)/(c*d + e)]*Sqrt[1 - c^2*x^2]*EllipticPi[2, ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)])/(Sqrt[1 - 1/(c^2*x^2)]*Sqrt[e + d/x]*(c*x)^(3/2)) + (2*(-7*c^2*d^2*e + 3*e^3)*Cos[2*ArcCsc[c*x]]*((c*d + c*e*x)*(-1 + c^2*x^2) + c^2*d*x*Sqrt[(c*d + c*e*x)/(c*d + e)]*Sqrt[1 - c^2*x^2]*EllipticF[ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)] - (c*x*(1 + c*x)*Sqrt[(e - c*e*x)/(c*d + e)]*Sqrt[(c*d + c*e*x)/(c*d - e)]*((c*d + e)*EllipticE[ArcSin[Sqrt[(c*d + c*e*x)/(c*d - e)]], (c*d - e)/(c*d + e)] - e*EllipticF[ArcSin[Sqrt[(c*d + c*e*x)/(c*d - e)]], (c*d - e)/(c*d + e)]))/Sqrt[(e*(1 + c*x))/(-(c*d) + e)] + c*e*x*Sqrt[(c*d + c*e*x)/(c*d + e)]*Sqrt[1 - c^2*x^2]*EllipticPi[2, ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)]))/(c*d*Sqrt[1 - 1/(c^2*x^2)]*Sqrt[e + d/x]*Sqrt[c*x]*(-2 + c^2*x^2))))/(15*c*d*(c*d - e)^2*e*(c*d + e)^2*(d + e*x)^(7/2))))/c","A",0
76,1,140,206,0.2299281,"\int x^4 \left(d+e x^2\right) \left(a+b \csc ^{-1}(c x)\right) \, dx","Integrate[x^4*(d + e*x^2)*(a + b*ArcCsc[c*x]),x]","\frac{48 a c^7 x^5 \left(7 d+5 e x^2\right)+48 b c^7 x^5 \csc ^{-1}(c x) \left(7 d+5 e x^2\right)+3 b \left(42 c^2 d+25 e\right) \log \left(x \left(\sqrt{1-\frac{1}{c^2 x^2}}+1\right)\right)+b c^2 x^2 \sqrt{1-\frac{1}{c^2 x^2}} \left(c^4 \left(84 d x^2+40 e x^4\right)+2 c^2 \left(63 d+25 e x^2\right)+75 e\right)}{1680 c^7}","\frac{1}{5} d x^5 \left(a+b \csc ^{-1}(c x)\right)+\frac{1}{7} e x^7 \left(a+b \csc ^{-1}(c x)\right)+\frac{b e x^6 \sqrt{c^2 x^2-1}}{42 c \sqrt{c^2 x^2}}+\frac{b x \left(42 c^2 d+25 e\right) \tanh ^{-1}\left(\frac{c x}{\sqrt{c^2 x^2-1}}\right)}{560 c^6 \sqrt{c^2 x^2}}+\frac{b x^2 \sqrt{c^2 x^2-1} \left(42 c^2 d+25 e\right)}{560 c^5 \sqrt{c^2 x^2}}+\frac{b x^4 \sqrt{c^2 x^2-1} \left(42 c^2 d+25 e\right)}{840 c^3 \sqrt{c^2 x^2}}",1,"(48*a*c^7*x^5*(7*d + 5*e*x^2) + b*c^2*Sqrt[1 - 1/(c^2*x^2)]*x^2*(75*e + 2*c^2*(63*d + 25*e*x^2) + c^4*(84*d*x^2 + 40*e*x^4)) + 48*b*c^7*x^5*(7*d + 5*e*x^2)*ArcCsc[c*x] + 3*b*(42*c^2*d + 25*e)*Log[(1 + Sqrt[1 - 1/(c^2*x^2)])*x])/(1680*c^7)","A",1
77,1,121,161,0.1763967,"\int x^2 \left(d+e x^2\right) \left(a+b \csc ^{-1}(c x)\right) \, dx","Integrate[x^2*(d + e*x^2)*(a + b*ArcCsc[c*x]),x]","\frac{c^2 x^2 \left(8 a c^3 x \left(5 d+3 e x^2\right)+b \sqrt{1-\frac{1}{c^2 x^2}} \left(c^2 \left(20 d+6 e x^2\right)+9 e\right)\right)+8 b c^5 x^3 \csc ^{-1}(c x) \left(5 d+3 e x^2\right)+b \left(20 c^2 d+9 e\right) \log \left(x \left(\sqrt{1-\frac{1}{c^2 x^2}}+1\right)\right)}{120 c^5}","\frac{1}{3} d x^3 \left(a+b \csc ^{-1}(c x)\right)+\frac{1}{5} e x^5 \left(a+b \csc ^{-1}(c x)\right)+\frac{b e x^4 \sqrt{c^2 x^2-1}}{20 c \sqrt{c^2 x^2}}+\frac{b x \left(20 c^2 d+9 e\right) \tanh ^{-1}\left(\frac{c x}{\sqrt{c^2 x^2-1}}\right)}{120 c^4 \sqrt{c^2 x^2}}+\frac{b x^2 \sqrt{c^2 x^2-1} \left(20 c^2 d+9 e\right)}{120 c^3 \sqrt{c^2 x^2}}",1,"(c^2*x^2*(8*a*c^3*x*(5*d + 3*e*x^2) + b*Sqrt[1 - 1/(c^2*x^2)]*(9*e + c^2*(20*d + 6*e*x^2))) + 8*b*c^5*x^3*(5*d + 3*e*x^2)*ArcCsc[c*x] + b*(20*c^2*d + 9*e)*Log[(1 + Sqrt[1 - 1/(c^2*x^2)])*x])/(120*c^5)","A",1
78,1,149,109,0.2756579,"\int \left(d+e x^2\right) \left(a+b \csc ^{-1}(c x)\right) \, dx","Integrate[(d + e*x^2)*(a + b*ArcCsc[c*x]),x]","a d x+\frac{1}{3} a e x^3+\frac{b d x \sqrt{1-\frac{1}{c^2 x^2}} \tanh ^{-1}\left(\frac{c x}{\sqrt{c^2 x^2-1}}\right)}{\sqrt{c^2 x^2-1}}+\frac{b e x^2 \sqrt{\frac{c^2 x^2-1}{c^2 x^2}}}{6 c}+\frac{b e \log \left(x \left(\sqrt{\frac{c^2 x^2-1}{c^2 x^2}}+1\right)\right)}{6 c^3}+b d x \csc ^{-1}(c x)+\frac{1}{3} b e x^3 \csc ^{-1}(c x)","d x \left(a+b \csc ^{-1}(c x)\right)+\frac{1}{3} e x^3 \left(a+b \csc ^{-1}(c x)\right)+\frac{b x \left(6 c^2 d+e\right) \tanh ^{-1}\left(\frac{c x}{\sqrt{c^2 x^2-1}}\right)}{6 c^2 \sqrt{c^2 x^2}}+\frac{b e x^2 \sqrt{c^2 x^2-1}}{6 c \sqrt{c^2 x^2}}",1,"a*d*x + (a*e*x^3)/3 + (b*e*x^2*Sqrt[(-1 + c^2*x^2)/(c^2*x^2)])/(6*c) + b*d*x*ArcCsc[c*x] + (b*e*x^3*ArcCsc[c*x])/3 + (b*d*Sqrt[1 - 1/(c^2*x^2)]*x*ArcTanh[(c*x)/Sqrt[-1 + c^2*x^2]])/Sqrt[-1 + c^2*x^2] + (b*e*Log[x*(1 + Sqrt[(-1 + c^2*x^2)/(c^2*x^2)])])/(6*c^3)","A",1
79,1,104,87,0.1241741,"\int \frac{\left(d+e x^2\right) \left(a+b \csc ^{-1}(c x)\right)}{x^2} \, dx","Integrate[((d + e*x^2)*(a + b*ArcCsc[c*x]))/x^2,x]","-\frac{a d}{x}+a e x-b c d \sqrt{\frac{c^2 x^2-1}{c^2 x^2}}+\frac{b e x \sqrt{1-\frac{1}{c^2 x^2}} \tanh ^{-1}\left(\frac{c x}{\sqrt{c^2 x^2-1}}\right)}{\sqrt{c^2 x^2-1}}-\frac{b d \csc ^{-1}(c x)}{x}+b e x \csc ^{-1}(c x)","-\frac{d \left(a+b \csc ^{-1}(c x)\right)}{x}+e x \left(a+b \csc ^{-1}(c x)\right)-\frac{b c d \sqrt{c^2 x^2-1}}{\sqrt{c^2 x^2}}+\frac{b e x \tanh ^{-1}\left(\frac{c x}{\sqrt{c^2 x^2-1}}\right)}{\sqrt{c^2 x^2}}",1,"-((a*d)/x) + a*e*x - b*c*d*Sqrt[(-1 + c^2*x^2)/(c^2*x^2)] - (b*d*ArcCsc[c*x])/x + b*e*x*ArcCsc[c*x] + (b*e*Sqrt[1 - 1/(c^2*x^2)]*x*ArcTanh[(c*x)/Sqrt[-1 + c^2*x^2]])/Sqrt[-1 + c^2*x^2]","A",1
80,1,69,105,0.1145706,"\int \frac{\left(d+e x^2\right) \left(a+b \csc ^{-1}(c x)\right)}{x^4} \, dx","Integrate[((d + e*x^2)*(a + b*ArcCsc[c*x]))/x^4,x]","-\frac{3 a \left(d+3 e x^2\right)+b c x \sqrt{1-\frac{1}{c^2 x^2}} \left(2 c^2 d x^2+d+9 e x^2\right)+3 b \csc ^{-1}(c x) \left(d+3 e x^2\right)}{9 x^3}","-\frac{d \left(a+b \csc ^{-1}(c x)\right)}{3 x^3}-\frac{e \left(a+b \csc ^{-1}(c x)\right)}{x}-\frac{b c \sqrt{c^2 x^2-1} \left(2 c^2 d+9 e\right)}{9 \sqrt{c^2 x^2}}-\frac{b c d \sqrt{c^2 x^2-1}}{9 x^2 \sqrt{c^2 x^2}}",1,"-1/9*(3*a*(d + 3*e*x^2) + b*c*Sqrt[1 - 1/(c^2*x^2)]*x*(d + 2*c^2*d*x^2 + 9*e*x^2) + 3*b*(d + 3*e*x^2)*ArcCsc[c*x])/x^3","A",1
81,1,94,152,0.1696943,"\int \frac{\left(d+e x^2\right) \left(a+b \csc ^{-1}(c x)\right)}{x^6} \, dx","Integrate[((d + e*x^2)*(a + b*ArcCsc[c*x]))/x^6,x]","-\frac{15 a \left(3 d+5 e x^2\right)+b c x \sqrt{1-\frac{1}{c^2 x^2}} \left(25 e x^2 \left(2 c^2 x^2+1\right)+3 d \left(8 c^4 x^4+4 c^2 x^2+3\right)\right)+15 b \csc ^{-1}(c x) \left(3 d+5 e x^2\right)}{225 x^5}","-\frac{d \left(a+b \csc ^{-1}(c x)\right)}{5 x^5}-\frac{e \left(a+b \csc ^{-1}(c x)\right)}{3 x^3}-\frac{b c \sqrt{c^2 x^2-1} \left(12 c^2 d+25 e\right)}{225 x^2 \sqrt{c^2 x^2}}-\frac{b c d \sqrt{c^2 x^2-1}}{25 x^4 \sqrt{c^2 x^2}}-\frac{2 b c^3 \sqrt{c^2 x^2-1} \left(12 c^2 d+25 e\right)}{225 \sqrt{c^2 x^2}}",1,"-1/225*(15*a*(3*d + 5*e*x^2) + b*c*Sqrt[1 - 1/(c^2*x^2)]*x*(25*e*x^2*(1 + 2*c^2*x^2) + 3*d*(3 + 4*c^2*x^2 + 8*c^4*x^4)) + 15*b*(3*d + 5*e*x^2)*ArcCsc[c*x])/x^5","A",1
82,1,110,197,0.1727223,"\int \frac{\left(d+e x^2\right) \left(a+b \csc ^{-1}(c x)\right)}{x^8} \, dx","Integrate[((d + e*x^2)*(a + b*ArcCsc[c*x]))/x^8,x]","-\frac{105 a \left(5 d+7 e x^2\right)+b c x \sqrt{1-\frac{1}{c^2 x^2}} \left(49 e x^2 \left(8 c^4 x^4+4 c^2 x^2+3\right)+15 d \left(16 c^6 x^6+8 c^4 x^4+6 c^2 x^2+5\right)\right)+105 b \csc ^{-1}(c x) \left(5 d+7 e x^2\right)}{3675 x^7}","-\frac{d \left(a+b \csc ^{-1}(c x)\right)}{7 x^7}-\frac{e \left(a+b \csc ^{-1}(c x)\right)}{5 x^5}-\frac{b c \sqrt{c^2 x^2-1} \left(30 c^2 d+49 e\right)}{1225 x^4 \sqrt{c^2 x^2}}-\frac{b c d \sqrt{c^2 x^2-1}}{49 x^6 \sqrt{c^2 x^2}}-\frac{8 b c^5 \sqrt{c^2 x^2-1} \left(30 c^2 d+49 e\right)}{3675 \sqrt{c^2 x^2}}-\frac{4 b c^3 \sqrt{c^2 x^2-1} \left(30 c^2 d+49 e\right)}{3675 x^2 \sqrt{c^2 x^2}}",1,"-1/3675*(105*a*(5*d + 7*e*x^2) + b*c*Sqrt[1 - 1/(c^2*x^2)]*x*(49*e*x^2*(3 + 4*c^2*x^2 + 8*c^4*x^4) + 15*d*(5 + 6*c^2*x^2 + 8*c^4*x^4 + 16*c^6*x^6)) + 105*b*(5*d + 7*e*x^2)*ArcCsc[c*x])/x^7","A",1
83,1,115,196,0.2486862,"\int x^5 \left(d+e x^2\right) \left(a+b \csc ^{-1}(c x)\right) \, dx","Integrate[x^5*(d + e*x^2)*(a + b*ArcCsc[c*x]),x]","\frac{x \left(105 a x^5 \left(4 d+3 e x^2\right)+\frac{b \sqrt{1-\frac{1}{c^2 x^2}} \left(c^6 \left(84 d x^4+45 e x^6\right)+2 c^4 \left(56 d x^2+27 e x^4\right)+8 c^2 \left(28 d+9 e x^2\right)+144 e\right)}{c^7}+105 b x^5 \csc ^{-1}(c x) \left(4 d+3 e x^2\right)\right)}{2520}","\frac{1}{6} d x^6 \left(a+b \csc ^{-1}(c x)\right)+\frac{1}{8} e x^8 \left(a+b \csc ^{-1}(c x)\right)+\frac{b x \left(c^2 x^2-1\right)^{5/2} \left(4 c^2 d+9 e\right)}{120 c^7 \sqrt{c^2 x^2}}+\frac{b x \left(c^2 x^2-1\right)^{3/2} \left(8 c^2 d+9 e\right)}{72 c^7 \sqrt{c^2 x^2}}+\frac{b x \sqrt{c^2 x^2-1} \left(4 c^2 d+3 e\right)}{24 c^7 \sqrt{c^2 x^2}}+\frac{b e x \left(c^2 x^2-1\right)^{7/2}}{56 c^7 \sqrt{c^2 x^2}}",1,"(x*(105*a*x^5*(4*d + 3*e*x^2) + (b*Sqrt[1 - 1/(c^2*x^2)]*(144*e + 8*c^2*(28*d + 9*e*x^2) + 2*c^4*(56*d*x^2 + 27*e*x^4) + c^6*(84*d*x^4 + 45*e*x^6)))/c^7 + 105*b*x^5*(4*d + 3*e*x^2)*ArcCsc[c*x]))/2520","A",1
84,1,97,153,0.2185824,"\int x^3 \left(d+e x^2\right) \left(a+b \csc ^{-1}(c x)\right) \, dx","Integrate[x^3*(d + e*x^2)*(a + b*ArcCsc[c*x]),x]","\frac{1}{180} x \left(15 a x^3 \left(3 d+2 e x^2\right)+\frac{b \sqrt{1-\frac{1}{c^2 x^2}} \left(3 c^4 \left(5 d x^2+2 e x^4\right)+c^2 \left(30 d+8 e x^2\right)+16 e\right)}{c^5}+15 b x^3 \csc ^{-1}(c x) \left(3 d+2 e x^2\right)\right)","\frac{1}{4} d x^4 \left(a+b \csc ^{-1}(c x)\right)+\frac{1}{6} e x^6 \left(a+b \csc ^{-1}(c x)\right)+\frac{b x \left(c^2 x^2-1\right)^{3/2} \left(3 c^2 d+4 e\right)}{36 c^5 \sqrt{c^2 x^2}}+\frac{b x \sqrt{c^2 x^2-1} \left(3 c^2 d+2 e\right)}{12 c^5 \sqrt{c^2 x^2}}+\frac{b e x \left(c^2 x^2-1\right)^{5/2}}{30 c^5 \sqrt{c^2 x^2}}",1,"(x*(15*a*x^3*(3*d + 2*e*x^2) + (b*Sqrt[1 - 1/(c^2*x^2)]*(16*e + c^2*(30*d + 8*e*x^2) + 3*c^4*(5*d*x^2 + 2*e*x^4)))/c^5 + 15*b*x^3*(3*d + 2*e*x^2)*ArcCsc[c*x]))/180","A",1
85,1,78,138,0.103606,"\int x \left(d+e x^2\right) \left(a+b \csc ^{-1}(c x)\right) \, dx","Integrate[x*(d + e*x^2)*(a + b*ArcCsc[c*x]),x]","\frac{x \left(3 a c^3 x \left(2 d+e x^2\right)+3 b c^3 x \csc ^{-1}(c x) \left(2 d+e x^2\right)+b \sqrt{1-\frac{1}{c^2 x^2}} \left(c^2 \left(6 d+e x^2\right)+2 e\right)\right)}{12 c^3}","\frac{\left(d+e x^2\right)^2 \left(a+b \csc ^{-1}(c x)\right)}{4 e}+\frac{b c d^2 x \tan ^{-1}\left(\sqrt{c^2 x^2-1}\right)}{4 e \sqrt{c^2 x^2}}+\frac{b x \sqrt{c^2 x^2-1} \left(2 c^2 d+e\right)}{4 c^3 \sqrt{c^2 x^2}}+\frac{b e x \left(c^2 x^2-1\right)^{3/2}}{12 c^3 \sqrt{c^2 x^2}}",1,"(x*(3*a*c^3*x*(2*d + e*x^2) + b*Sqrt[1 - 1/(c^2*x^2)]*(2*e + c^2*(6*d + e*x^2)) + 3*b*c^3*x*(2*d + e*x^2)*ArcCsc[c*x]))/(12*c^3)","A",1
86,1,108,124,0.0881803,"\int \frac{\left(d+e x^2\right) \left(a+b \csc ^{-1}(c x)\right)}{x} \, dx","Integrate[((d + e*x^2)*(a + b*ArcCsc[c*x]))/x,x]","a d \log (x)+\frac{1}{2} a e x^2+\frac{b e x \sqrt{\frac{c^2 x^2-1}{c^2 x^2}}}{2 c}+\frac{1}{2} i b d \left(\csc ^{-1}(c x)^2+\text{Li}_2\left(e^{2 i \csc ^{-1}(c x)}\right)\right)-b d \csc ^{-1}(c x) \log \left(1-e^{2 i \csc ^{-1}(c x)}\right)+\frac{1}{2} b e x^2 \csc ^{-1}(c x)","-d \log \left(\frac{1}{x}\right) \left(a+b \csc ^{-1}(c x)\right)+\frac{1}{2} e x^2 \left(a+b \csc ^{-1}(c x)\right)+\frac{b e x \sqrt{1-\frac{1}{c^2 x^2}}}{2 c}+\frac{1}{2} i b d \text{Li}_2\left(e^{2 i \csc ^{-1}(c x)}\right)+\frac{1}{2} i b d \csc ^{-1}(c x)^2-b d \csc ^{-1}(c x) \log \left(1-e^{2 i \csc ^{-1}(c x)}\right)+b d \log \left(\frac{1}{x}\right) \csc ^{-1}(c x)",1,"(a*e*x^2)/2 + (b*e*x*Sqrt[(-1 + c^2*x^2)/(c^2*x^2)])/(2*c) + (b*e*x^2*ArcCsc[c*x])/2 - b*d*ArcCsc[c*x]*Log[1 - E^((2*I)*ArcCsc[c*x])] + a*d*Log[x] + (I/2)*b*d*(ArcCsc[c*x]^2 + PolyLog[2, E^((2*I)*ArcCsc[c*x])])","A",0
87,1,125,137,0.1010356,"\int \frac{\left(d+e x^2\right) \left(a+b \csc ^{-1}(c x)\right)}{x^3} \, dx","Integrate[((d + e*x^2)*(a + b*ArcCsc[c*x]))/x^3,x]","-\frac{a d}{2 x^2}+a e \log (x)-\frac{b c d \sqrt{\frac{c^2 x^2-1}{c^2 x^2}}}{4 x}+\frac{1}{4} b c^2 d \sin ^{-1}\left(\frac{1}{c x}\right)-\frac{b d \csc ^{-1}(c x)}{2 x^2}+\frac{1}{2} i b e \left(\csc ^{-1}(c x)^2+\text{Li}_2\left(e^{2 i \csc ^{-1}(c x)}\right)\right)-b e \csc ^{-1}(c x) \log \left(1-e^{2 i \csc ^{-1}(c x)}\right)","-\frac{d \left(a+b \csc ^{-1}(c x)\right)}{2 x^2}-e \log \left(\frac{1}{x}\right) \left(a+b \csc ^{-1}(c x)\right)-\frac{b c d \sqrt{1-\frac{1}{c^2 x^2}}}{4 x}+\frac{1}{4} b c^2 d \csc ^{-1}(c x)+\frac{1}{2} i b e \text{Li}_2\left(e^{2 i \csc ^{-1}(c x)}\right)+\frac{1}{2} i b e \csc ^{-1}(c x)^2-b e \csc ^{-1}(c x) \log \left(1-e^{2 i \csc ^{-1}(c x)}\right)+b e \log \left(\frac{1}{x}\right) \csc ^{-1}(c x)",1,"-1/2*(a*d)/x^2 - (b*c*d*Sqrt[(-1 + c^2*x^2)/(c^2*x^2)])/(4*x) - (b*d*ArcCsc[c*x])/(2*x^2) + (b*c^2*d*ArcSin[1/(c*x)])/4 - b*e*ArcCsc[c*x]*Log[1 - E^((2*I)*ArcCsc[c*x])] + a*e*Log[x] + (I/2)*b*e*(ArcCsc[c*x]^2 + PolyLog[2, E^((2*I)*ArcCsc[c*x])])","A",0
88,1,184,252,0.3725796,"\int x^2 \left(d+e x^2\right)^2 \left(a+b \csc ^{-1}(c x)\right) \, dx","Integrate[x^2*(d + e*x^2)^2*(a + b*ArcCsc[c*x]),x]","\frac{c^2 x^2 \left(16 a c^5 x \left(35 d^2+42 d e x^2+15 e^2 x^4\right)+b \sqrt{1-\frac{1}{c^2 x^2}} \left(8 c^4 \left(35 d^2+21 d e x^2+5 e^2 x^4\right)+2 c^2 e \left(126 d+25 e x^2\right)+75 e^2\right)\right)+16 b c^7 x^3 \csc ^{-1}(c x) \left(35 d^2+42 d e x^2+15 e^2 x^4\right)+b \left(280 c^4 d^2+252 c^2 d e+75 e^2\right) \log \left(x \left(\sqrt{1-\frac{1}{c^2 x^2}}+1\right)\right)}{1680 c^7}","\frac{1}{3} d^2 x^3 \left(a+b \csc ^{-1}(c x)\right)+\frac{2}{5} d e x^5 \left(a+b \csc ^{-1}(c x)\right)+\frac{1}{7} e^2 x^7 \left(a+b \csc ^{-1}(c x)\right)+\frac{b e^2 x^6 \sqrt{c^2 x^2-1}}{42 c \sqrt{c^2 x^2}}+\frac{b e x^4 \sqrt{c^2 x^2-1} \left(84 c^2 d+25 e\right)}{840 c^3 \sqrt{c^2 x^2}}+\frac{b x \left(280 c^4 d^2+252 c^2 d e+75 e^2\right) \tanh ^{-1}\left(\frac{c x}{\sqrt{c^2 x^2-1}}\right)}{1680 c^6 \sqrt{c^2 x^2}}+\frac{b x^2 \sqrt{c^2 x^2-1} \left(280 c^4 d^2+252 c^2 d e+75 e^2\right)}{1680 c^5 \sqrt{c^2 x^2}}",1,"(c^2*x^2*(16*a*c^5*x*(35*d^2 + 42*d*e*x^2 + 15*e^2*x^4) + b*Sqrt[1 - 1/(c^2*x^2)]*(75*e^2 + 2*c^2*e*(126*d + 25*e*x^2) + 8*c^4*(35*d^2 + 21*d*e*x^2 + 5*e^2*x^4))) + 16*b*c^7*x^3*(35*d^2 + 42*d*e*x^2 + 15*e^2*x^4)*ArcCsc[c*x] + b*(280*c^4*d^2 + 252*c^2*d*e + 75*e^2)*Log[(1 + Sqrt[1 - 1/(c^2*x^2)])*x])/(1680*c^7)","A",1
89,1,151,191,0.2566599,"\int \left(d+e x^2\right)^2 \left(a+b \csc ^{-1}(c x)\right) \, dx","Integrate[(d + e*x^2)^2*(a + b*ArcCsc[c*x]),x]","\frac{c^2 x \left(8 a c^3 \left(15 d^2+10 d e x^2+3 e^2 x^4\right)+b e x \sqrt{1-\frac{1}{c^2 x^2}} \left(c^2 \left(40 d+6 e x^2\right)+9 e\right)\right)+8 b c^5 x \csc ^{-1}(c x) \left(15 d^2+10 d e x^2+3 e^2 x^4\right)+b \left(120 c^4 d^2+40 c^2 d e+9 e^2\right) \log \left(x \left(\sqrt{1-\frac{1}{c^2 x^2}}+1\right)\right)}{120 c^5}","d^2 x \left(a+b \csc ^{-1}(c x)\right)+\frac{2}{3} d e x^3 \left(a+b \csc ^{-1}(c x)\right)+\frac{1}{5} e^2 x^5 \left(a+b \csc ^{-1}(c x)\right)+\frac{b e^2 x^4 \sqrt{c^2 x^2-1}}{20 c \sqrt{c^2 x^2}}+\frac{b x \left(120 c^4 d^2+40 c^2 d e+9 e^2\right) \tanh ^{-1}\left(\frac{c x}{\sqrt{c^2 x^2-1}}\right)}{120 c^4 \sqrt{c^2 x^2}}+\frac{b e x^2 \sqrt{c^2 x^2-1} \left(40 c^2 d+9 e\right)}{120 c^3 \sqrt{c^2 x^2}}",1,"(c^2*x*(8*a*c^3*(15*d^2 + 10*d*e*x^2 + 3*e^2*x^4) + b*e*Sqrt[1 - 1/(c^2*x^2)]*x*(9*e + c^2*(40*d + 6*e*x^2))) + 8*b*c^5*x*(15*d^2 + 10*d*e*x^2 + 3*e^2*x^4)*ArcCsc[c*x] + b*(120*c^4*d^2 + 40*c^2*d*e + 9*e^2)*Log[(1 + Sqrt[1 - 1/(c^2*x^2)])*x])/(120*c^5)","A",1
90,1,134,163,0.2104191,"\int \frac{\left(d+e x^2\right)^2 \left(a+b \csc ^{-1}(c x)\right)}{x^2} \, dx","Integrate[((d + e*x^2)^2*(a + b*ArcCsc[c*x]))/x^2,x]","\frac{c^2 \left(2 a c \left(-3 d^2+6 d e x^2+e^2 x^4\right)+b x \sqrt{1-\frac{1}{c^2 x^2}} \left(e^2 x^2-6 c^2 d^2\right)\right)+2 b c^3 \csc ^{-1}(c x) \left(-3 d^2+6 d e x^2+e^2 x^4\right)+b e x \left(12 c^2 d+e\right) \log \left(x \left(\sqrt{1-\frac{1}{c^2 x^2}}+1\right)\right)}{6 c^3 x}","-\frac{d^2 \left(a+b \csc ^{-1}(c x)\right)}{x}+2 d e x \left(a+b \csc ^{-1}(c x)\right)+\frac{1}{3} e^2 x^3 \left(a+b \csc ^{-1}(c x)\right)-\frac{b c d^2 \sqrt{c^2 x^2-1}}{\sqrt{c^2 x^2}}+\frac{b e x \left(12 c^2 d+e\right) \tanh ^{-1}\left(\frac{c x}{\sqrt{c^2 x^2-1}}\right)}{6 c^2 \sqrt{c^2 x^2}}+\frac{b e^2 x^2 \sqrt{c^2 x^2-1}}{6 c \sqrt{c^2 x^2}}",1,"(c^2*(b*Sqrt[1 - 1/(c^2*x^2)]*x*(-6*c^2*d^2 + e^2*x^2) + 2*a*c*(-3*d^2 + 6*d*e*x^2 + e^2*x^4)) + 2*b*c^3*(-3*d^2 + 6*d*e*x^2 + e^2*x^4)*ArcCsc[c*x] + b*e*(12*c^2*d + e)*x*Log[(1 + Sqrt[1 - 1/(c^2*x^2)])*x])/(6*c^3*x)","A",1
91,1,125,157,0.289763,"\int \frac{\left(d+e x^2\right)^2 \left(a+b \csc ^{-1}(c x)\right)}{x^4} \, dx","Integrate[((d + e*x^2)^2*(a + b*ArcCsc[c*x]))/x^4,x]","-\frac{3 a \left(d^2+6 d e x^2-3 e^2 x^4\right)+b c d x \sqrt{1-\frac{1}{c^2 x^2}} \left(2 c^2 d x^2+d+18 e x^2\right)}{9 x^3}+\frac{b e^2 \log \left(x \left(\sqrt{1-\frac{1}{c^2 x^2}}+1\right)\right)}{c}-\frac{b \csc ^{-1}(c x) \left(d^2+6 d e x^2-3 e^2 x^4\right)}{3 x^3}","-\frac{d^2 \left(a+b \csc ^{-1}(c x)\right)}{3 x^3}-\frac{2 d e \left(a+b \csc ^{-1}(c x)\right)}{x}+e^2 x \left(a+b \csc ^{-1}(c x)\right)-\frac{b c d^2 \sqrt{c^2 x^2-1}}{9 x^2 \sqrt{c^2 x^2}}-\frac{2 b c d \sqrt{c^2 x^2-1} \left(c^2 d+9 e\right)}{9 \sqrt{c^2 x^2}}+\frac{b e^2 x \tanh ^{-1}\left(\frac{c x}{\sqrt{c^2 x^2-1}}\right)}{\sqrt{c^2 x^2}}",1,"-1/9*(b*c*d*Sqrt[1 - 1/(c^2*x^2)]*x*(d + 2*c^2*d*x^2 + 18*e*x^2) + 3*a*(d^2 + 6*d*e*x^2 - 3*e^2*x^4))/x^3 - (b*(d^2 + 6*d*e*x^2 - 3*e^2*x^4)*ArcCsc[c*x])/(3*x^3) + (b*e^2*Log[(1 + Sqrt[1 - 1/(c^2*x^2)])*x])/c","A",1
92,1,127,183,0.2781055,"\int \frac{\left(d+e x^2\right)^2 \left(a+b \csc ^{-1}(c x)\right)}{x^6} \, dx","Integrate[((d + e*x^2)^2*(a + b*ArcCsc[c*x]))/x^6,x]","-\frac{15 a \left(3 d^2+10 d e x^2+15 e^2 x^4\right)+b c x \sqrt{1-\frac{1}{c^2 x^2}} \left(50 d e x^2 \left(2 c^2 x^2+1\right)+3 d^2 \left(8 c^4 x^4+4 c^2 x^2+3\right)+225 e^2 x^4\right)+15 b \csc ^{-1}(c x) \left(3 d^2+10 d e x^2+15 e^2 x^4\right)}{225 x^5}","-\frac{d^2 \left(a+b \csc ^{-1}(c x)\right)}{5 x^5}-\frac{2 d e \left(a+b \csc ^{-1}(c x)\right)}{3 x^3}-\frac{e^2 \left(a+b \csc ^{-1}(c x)\right)}{x}-\frac{b c d^2 \sqrt{c^2 x^2-1}}{25 x^4 \sqrt{c^2 x^2}}-\frac{2 b c d \sqrt{c^2 x^2-1} \left(6 c^2 d+25 e\right)}{225 x^2 \sqrt{c^2 x^2}}-\frac{b c \sqrt{c^2 x^2-1} \left(24 c^4 d^2+100 c^2 d e+225 e^2\right)}{225 \sqrt{c^2 x^2}}",1,"-1/225*(15*a*(3*d^2 + 10*d*e*x^2 + 15*e^2*x^4) + b*c*Sqrt[1 - 1/(c^2*x^2)]*x*(225*e^2*x^4 + 50*d*e*x^2*(1 + 2*c^2*x^2) + 3*d^2*(3 + 4*c^2*x^2 + 8*c^4*x^4)) + 15*b*(3*d^2 + 10*d*e*x^2 + 15*e^2*x^4)*ArcCsc[c*x])/x^5","A",1
93,1,153,241,0.2782403,"\int \frac{\left(d+e x^2\right)^2 \left(a+b \csc ^{-1}(c x)\right)}{x^8} \, dx","Integrate[((d + e*x^2)^2*(a + b*ArcCsc[c*x]))/x^8,x]","-\frac{105 a \left(15 d^2+42 d e x^2+35 e^2 x^4\right)+b c x \sqrt{1-\frac{1}{c^2 x^2}} \left(1225 e^2 x^4 \left(2 c^2 x^2+1\right)+294 d e x^2 \left(8 c^4 x^4+4 c^2 x^2+3\right)+45 d^2 \left(16 c^6 x^6+8 c^4 x^4+6 c^2 x^2+5\right)\right)+105 b \csc ^{-1}(c x) \left(15 d^2+42 d e x^2+35 e^2 x^4\right)}{11025 x^7}","-\frac{d^2 \left(a+b \csc ^{-1}(c x)\right)}{7 x^7}-\frac{2 d e \left(a+b \csc ^{-1}(c x)\right)}{5 x^5}-\frac{e^2 \left(a+b \csc ^{-1}(c x)\right)}{3 x^3}-\frac{b c d^2 \sqrt{c^2 x^2-1}}{49 x^6 \sqrt{c^2 x^2}}-\frac{2 b c d \sqrt{c^2 x^2-1} \left(15 c^2 d+49 e\right)}{1225 x^4 \sqrt{c^2 x^2}}-\frac{b c \sqrt{c^2 x^2-1} \left(360 c^4 d^2+1176 c^2 d e+1225 e^2\right)}{11025 x^2 \sqrt{c^2 x^2}}-\frac{2 b c^3 \sqrt{c^2 x^2-1} \left(360 c^4 d^2+1176 c^2 d e+1225 e^2\right)}{11025 \sqrt{c^2 x^2}}",1,"-1/11025*(105*a*(15*d^2 + 42*d*e*x^2 + 35*e^2*x^4) + b*c*Sqrt[1 - 1/(c^2*x^2)]*x*(1225*e^2*x^4*(1 + 2*c^2*x^2) + 294*d*e*x^2*(3 + 4*c^2*x^2 + 8*c^4*x^4) + 45*d^2*(5 + 6*c^2*x^2 + 8*c^4*x^4 + 16*c^6*x^6)) + 105*b*(15*d^2 + 42*d*e*x^2 + 35*e^2*x^4)*ArcCsc[c*x])/x^7","A",1
94,1,159,242,0.3383112,"\int x^3 \left(d+e x^2\right)^2 \left(a+b \csc ^{-1}(c x)\right) \, dx","Integrate[x^3*(d + e*x^2)^2*(a + b*ArcCsc[c*x]),x]","\frac{x \left(105 a x^3 \left(6 d^2+8 d e x^2+3 e^2 x^4\right)+\frac{b \sqrt{1-\frac{1}{c^2 x^2}} \left(3 c^6 \left(70 d^2 x^2+56 d e x^4+15 e^2 x^6\right)+c^4 \left(420 d^2+224 d e x^2+54 e^2 x^4\right)+8 c^2 e \left(56 d+9 e x^2\right)+144 e^2\right)}{c^7}+105 b x^3 \csc ^{-1}(c x) \left(6 d^2+8 d e x^2+3 e^2 x^4\right)\right)}{2520}","\frac{1}{4} d^2 x^4 \left(a+b \csc ^{-1}(c x)\right)+\frac{1}{3} d e x^6 \left(a+b \csc ^{-1}(c x)\right)+\frac{1}{8} e^2 x^8 \left(a+b \csc ^{-1}(c x)\right)+\frac{b e x \left(c^2 x^2-1\right)^{5/2} \left(8 c^2 d+9 e\right)}{120 c^7 \sqrt{c^2 x^2}}+\frac{b e^2 x \left(c^2 x^2-1\right)^{7/2}}{56 c^7 \sqrt{c^2 x^2}}+\frac{b x \left(c^2 x^2-1\right)^{3/2} \left(6 c^4 d^2+16 c^2 d e+9 e^2\right)}{72 c^7 \sqrt{c^2 x^2}}+\frac{b x \sqrt{c^2 x^2-1} \left(6 c^4 d^2+8 c^2 d e+3 e^2\right)}{24 c^7 \sqrt{c^2 x^2}}",1,"(x*(105*a*x^3*(6*d^2 + 8*d*e*x^2 + 3*e^2*x^4) + (b*Sqrt[1 - 1/(c^2*x^2)]*(144*e^2 + 8*c^2*e*(56*d + 9*e*x^2) + c^4*(420*d^2 + 224*d*e*x^2 + 54*e^2*x^4) + 3*c^6*(70*d^2*x^2 + 56*d*e*x^4 + 15*e^2*x^6)))/c^7 + 105*b*x^3*(6*d^2 + 8*d*e*x^2 + 3*e^2*x^4)*ArcCsc[c*x]))/2520","A",1
95,1,124,195,0.2941719,"\int x \left(d+e x^2\right)^2 \left(a+b \csc ^{-1}(c x)\right) \, dx","Integrate[x*(d + e*x^2)^2*(a + b*ArcCsc[c*x]),x]","\frac{1}{90} x \left(15 a x \left(3 d^2+3 d e x^2+e^2 x^4\right)+\frac{b \sqrt{1-\frac{1}{c^2 x^2}} \left(3 c^4 \left(15 d^2+5 d e x^2+e^2 x^4\right)+2 c^2 e \left(15 d+2 e x^2\right)+8 e^2\right)}{c^5}+15 b x \csc ^{-1}(c x) \left(3 d^2+3 d e x^2+e^2 x^4\right)\right)","\frac{\left(d+e x^2\right)^3 \left(a+b \csc ^{-1}(c x)\right)}{6 e}+\frac{b c d^3 x \tan ^{-1}\left(\sqrt{c^2 x^2-1}\right)}{6 e \sqrt{c^2 x^2}}+\frac{b e x \left(c^2 x^2-1\right)^{3/2} \left(3 c^2 d+2 e\right)}{18 c^5 \sqrt{c^2 x^2}}+\frac{b e^2 x \left(c^2 x^2-1\right)^{5/2}}{30 c^5 \sqrt{c^2 x^2}}+\frac{b x \sqrt{c^2 x^2-1} \left(3 c^4 d^2+3 c^2 d e+e^2\right)}{6 c^5 \sqrt{c^2 x^2}}",1,"(x*(15*a*x*(3*d^2 + 3*d*e*x^2 + e^2*x^4) + (b*Sqrt[1 - 1/(c^2*x^2)]*(8*e^2 + 2*c^2*e*(15*d + 2*e*x^2) + 3*c^4*(15*d^2 + 5*d*e*x^2 + e^2*x^4)))/c^5 + 15*b*x*(3*d^2 + 3*d*e*x^2 + e^2*x^4)*ArcCsc[c*x]))/90","A",1
96,1,157,186,0.3871087,"\int \frac{\left(d+e x^2\right)^2 \left(a+b \csc ^{-1}(c x)\right)}{x} \, dx","Integrate[((d + e*x^2)^2*(a + b*ArcCsc[c*x]))/x,x]","a d^2 \log (x)+a d e x^2+\frac{1}{4} a e^2 x^4+\frac{b d e x \left(\sqrt{1-\frac{1}{c^2 x^2}}+c x \csc ^{-1}(c x)\right)}{c}+\frac{b e^2 x \left(3 c^3 x^3 \csc ^{-1}(c x)+\sqrt{1-\frac{1}{c^2 x^2}} \left(c^2 x^2+2\right)\right)}{12 c^3}+\frac{1}{2} i b d^2 \left(\text{Li}_2\left(e^{2 i \csc ^{-1}(c x)}\right)+\csc ^{-1}(c x) \left(\csc ^{-1}(c x)+2 i \log \left(1-e^{2 i \csc ^{-1}(c x)}\right)\right)\right)","-d^2 \log \left(\frac{1}{x}\right) \left(a+b \csc ^{-1}(c x)\right)+d e x^2 \left(a+b \csc ^{-1}(c x)\right)+\frac{1}{4} e^2 x^4 \left(a+b \csc ^{-1}(c x)\right)+\frac{b e^2 x^3 \sqrt{1-\frac{1}{c^2 x^2}}}{12 c}+\frac{b e x \sqrt{1-\frac{1}{c^2 x^2}} \left(6 c^2 d+e\right)}{6 c^3}+\frac{1}{2} i b d^2 \text{Li}_2\left(e^{2 i \csc ^{-1}(c x)}\right)+\frac{1}{2} i b d^2 \csc ^{-1}(c x)^2-b d^2 \csc ^{-1}(c x) \log \left(1-e^{2 i \csc ^{-1}(c x)}\right)+b d^2 \log \left(\frac{1}{x}\right) \csc ^{-1}(c x)",1,"a*d*e*x^2 + (a*e^2*x^4)/4 + (b*d*e*x*(Sqrt[1 - 1/(c^2*x^2)] + c*x*ArcCsc[c*x]))/c + (b*e^2*x*(Sqrt[1 - 1/(c^2*x^2)]*(2 + c^2*x^2) + 3*c^3*x^3*ArcCsc[c*x]))/(12*c^3) + a*d^2*Log[x] + (I/2)*b*d^2*(ArcCsc[c*x]*(ArcCsc[c*x] + (2*I)*Log[1 - E^((2*I)*ArcCsc[c*x])]) + PolyLog[2, E^((2*I)*ArcCsc[c*x])])","A",0
97,1,187,189,0.6776437,"\int \frac{\left(d+e x^2\right)^2 \left(a+b \csc ^{-1}(c x)\right)}{x^3} \, dx","Integrate[((d + e*x^2)^2*(a + b*ArcCsc[c*x]))/x^3,x]","\frac{1}{4} \left(-\frac{2 a d^2}{x^2}+8 a d e \log (x)+2 a e^2 x^2-\frac{b c d^2 \sqrt{1-\frac{1}{c^2 x^2}} \left(\frac{c^2 x^2 \tanh ^{-1}\left(\sqrt{1-c^2 x^2}\right)}{\sqrt{1-c^2 x^2}}+1\right)}{x}+\frac{2 b e^2 x \left(\sqrt{1-\frac{1}{c^2 x^2}}+c x \csc ^{-1}(c x)\right)}{c}-\frac{2 b d^2 \csc ^{-1}(c x)}{x^2}+4 i b d e \left(\csc ^{-1}(c x)^2+\text{Li}_2\left(e^{2 i \csc ^{-1}(c x)}\right)\right)-8 b d e \csc ^{-1}(c x) \log \left(1-e^{2 i \csc ^{-1}(c x)}\right)\right)","-\frac{d^2 \left(a+b \csc ^{-1}(c x)\right)}{2 x^2}-2 d e \log \left(\frac{1}{x}\right) \left(a+b \csc ^{-1}(c x)\right)+\frac{1}{2} e^2 x^2 \left(a+b \csc ^{-1}(c x)\right)-\frac{b c d^2 \sqrt{1-\frac{1}{c^2 x^2}}}{4 x}+\frac{1}{4} b c^2 d^2 \csc ^{-1}(c x)+\frac{b e^2 x \sqrt{1-\frac{1}{c^2 x^2}}}{2 c}+i b d e \text{Li}_2\left(e^{2 i \csc ^{-1}(c x)}\right)+i b d e \csc ^{-1}(c x)^2-2 b d e \csc ^{-1}(c x) \log \left(1-e^{2 i \csc ^{-1}(c x)}\right)+2 b d e \log \left(\frac{1}{x}\right) \csc ^{-1}(c x)",1,"((-2*a*d^2)/x^2 + 2*a*e^2*x^2 - (2*b*d^2*ArcCsc[c*x])/x^2 + (2*b*e^2*x*(Sqrt[1 - 1/(c^2*x^2)] + c*x*ArcCsc[c*x]))/c - (b*c*d^2*Sqrt[1 - 1/(c^2*x^2)]*(1 + (c^2*x^2*ArcTanh[Sqrt[1 - c^2*x^2]])/Sqrt[1 - c^2*x^2]))/x - 8*b*d*e*ArcCsc[c*x]*Log[1 - E^((2*I)*ArcCsc[c*x])] + 8*a*d*e*Log[x] + (4*I)*b*d*e*(ArcCsc[c*x]^2 + PolyLog[2, E^((2*I)*ArcCsc[c*x])]))/4","A",0
98,1,1260,565,1.8185959,"\int \frac{x^2 \left(a+b \csc ^{-1}(c x)\right)}{d+e x^2} \, dx","Integrate[(x^2*(a + b*ArcCsc[c*x]))/(d + e*x^2),x]","\frac{i \left(-4 i b c \sqrt{e} \csc ^{-1}(c x) x-4 i a c \sqrt{e} x+4 i a c \sqrt{d} \tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right)+8 i b c \sqrt{d} \sin ^{-1}\left(\frac{\sqrt{1-\frac{i \sqrt{e}}{c \sqrt{d}}}}{\sqrt{2}}\right) \tan ^{-1}\left(\frac{\left(\sqrt{e}-i c \sqrt{d}\right) \cot \left(\frac{1}{4} \left(2 \csc ^{-1}(c x)+\pi \right)\right)}{\sqrt{d c^2+e}}\right)-8 i b c \sqrt{d} \sin ^{-1}\left(\frac{\sqrt{\frac{i \sqrt{e}}{c \sqrt{d}}+1}}{\sqrt{2}}\right) \tan ^{-1}\left(\frac{\left(i \sqrt{d} c+\sqrt{e}\right) \cot \left(\frac{1}{4} \left(2 \csc ^{-1}(c x)+\pi \right)\right)}{\sqrt{d c^2+e}}\right)-2 b c \sqrt{d} \csc ^{-1}(c x) \log \left(\frac{e^{-i \csc ^{-1}(c x)} \left(\sqrt{e}-\sqrt{d c^2+e}\right)}{c \sqrt{d}}+1\right)+4 b c \sqrt{d} \sin ^{-1}\left(\frac{\sqrt{1-\frac{i \sqrt{e}}{c \sqrt{d}}}}{\sqrt{2}}\right) \log \left(\frac{e^{-i \csc ^{-1}(c x)} \left(\sqrt{e}-\sqrt{d c^2+e}\right)}{c \sqrt{d}}+1\right)+b c \sqrt{d} \pi  \log \left(\frac{e^{-i \csc ^{-1}(c x)} \left(\sqrt{e}-\sqrt{d c^2+e}\right)}{c \sqrt{d}}+1\right)+2 b c \sqrt{d} \csc ^{-1}(c x) \log \left(\frac{e^{-i \csc ^{-1}(c x)} \left(\sqrt{d c^2+e}-\sqrt{e}\right)}{c \sqrt{d}}+1\right)-4 b c \sqrt{d} \sin ^{-1}\left(\frac{\sqrt{\frac{i \sqrt{e}}{c \sqrt{d}}+1}}{\sqrt{2}}\right) \log \left(\frac{e^{-i \csc ^{-1}(c x)} \left(\sqrt{d c^2+e}-\sqrt{e}\right)}{c \sqrt{d}}+1\right)-b c \sqrt{d} \pi  \log \left(\frac{e^{-i \csc ^{-1}(c x)} \left(\sqrt{d c^2+e}-\sqrt{e}\right)}{c \sqrt{d}}+1\right)+2 b c \sqrt{d} \csc ^{-1}(c x) \log \left(1-\frac{\left(\sqrt{e}+\sqrt{d c^2+e}\right) e^{-i \csc ^{-1}(c x)}}{c \sqrt{d}}\right)+4 b c \sqrt{d} \sin ^{-1}\left(\frac{\sqrt{\frac{i \sqrt{e}}{c \sqrt{d}}+1}}{\sqrt{2}}\right) \log \left(1-\frac{\left(\sqrt{e}+\sqrt{d c^2+e}\right) e^{-i \csc ^{-1}(c x)}}{c \sqrt{d}}\right)-b c \sqrt{d} \pi  \log \left(1-\frac{\left(\sqrt{e}+\sqrt{d c^2+e}\right) e^{-i \csc ^{-1}(c x)}}{c \sqrt{d}}\right)-2 b c \sqrt{d} \csc ^{-1}(c x) \log \left(\frac{e^{-i \csc ^{-1}(c x)} \left(\sqrt{e}+\sqrt{d c^2+e}\right)}{c \sqrt{d}}+1\right)-4 b c \sqrt{d} \sin ^{-1}\left(\frac{\sqrt{1-\frac{i \sqrt{e}}{c \sqrt{d}}}}{\sqrt{2}}\right) \log \left(\frac{e^{-i \csc ^{-1}(c x)} \left(\sqrt{e}+\sqrt{d c^2+e}\right)}{c \sqrt{d}}+1\right)+b c \sqrt{d} \pi  \log \left(\frac{e^{-i \csc ^{-1}(c x)} \left(\sqrt{e}+\sqrt{d c^2+e}\right)}{c \sqrt{d}}+1\right)+b c \sqrt{d} \pi  \log \left(\sqrt{e}-\frac{i \sqrt{d}}{x}\right)-b c \sqrt{d} \pi  \log \left(\frac{i \sqrt{d}}{x}+\sqrt{e}\right)-4 i b \sqrt{e} \log \left(\cos \left(\frac{1}{2} \csc ^{-1}(c x)\right)\right)+4 i b \sqrt{e} \log \left(\sin \left(\frac{1}{2} \csc ^{-1}(c x)\right)\right)+2 i b c \sqrt{d} \text{Li}_2\left(\frac{\left(\sqrt{e}-\sqrt{d c^2+e}\right) e^{-i \csc ^{-1}(c x)}}{c \sqrt{d}}\right)-2 i b c \sqrt{d} \text{Li}_2\left(\frac{\left(\sqrt{d c^2+e}-\sqrt{e}\right) e^{-i \csc ^{-1}(c x)}}{c \sqrt{d}}\right)-2 i b c \sqrt{d} \text{Li}_2\left(-\frac{\left(\sqrt{e}+\sqrt{d c^2+e}\right) e^{-i \csc ^{-1}(c x)}}{c \sqrt{d}}\right)+2 i b c \sqrt{d} \text{Li}_2\left(\frac{\left(\sqrt{e}+\sqrt{d c^2+e}\right) e^{-i \csc ^{-1}(c x)}}{c \sqrt{d}}\right)\right)}{4 c e^{3/2}}","-\frac{\sqrt{-d} \left(a+b \csc ^{-1}(c x)\right) \log \left(1-\frac{i c \sqrt{-d} e^{i \csc ^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right)}{2 e^{3/2}}+\frac{\sqrt{-d} \left(a+b \csc ^{-1}(c x)\right) \log \left(1+\frac{i c \sqrt{-d} e^{i \csc ^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right)}{2 e^{3/2}}-\frac{\sqrt{-d} \left(a+b \csc ^{-1}(c x)\right) \log \left(1-\frac{i c \sqrt{-d} e^{i \csc ^{-1}(c x)}}{\sqrt{c^2 d+e}+\sqrt{e}}\right)}{2 e^{3/2}}+\frac{\sqrt{-d} \left(a+b \csc ^{-1}(c x)\right) \log \left(1+\frac{i c \sqrt{-d} e^{i \csc ^{-1}(c x)}}{\sqrt{c^2 d+e}+\sqrt{e}}\right)}{2 e^{3/2}}+\frac{x \left(a+b \csc ^{-1}(c x)\right)}{e}-\frac{i b \sqrt{-d} \text{Li}_2\left(-\frac{i c \sqrt{-d} e^{i \csc ^{-1}(c x)}}{\sqrt{e}-\sqrt{d c^2+e}}\right)}{2 e^{3/2}}+\frac{i b \sqrt{-d} \text{Li}_2\left(\frac{i c \sqrt{-d} e^{i \csc ^{-1}(c x)}}{\sqrt{e}-\sqrt{d c^2+e}}\right)}{2 e^{3/2}}-\frac{i b \sqrt{-d} \text{Li}_2\left(-\frac{i c \sqrt{-d} e^{i \csc ^{-1}(c x)}}{\sqrt{e}+\sqrt{d c^2+e}}\right)}{2 e^{3/2}}+\frac{i b \sqrt{-d} \text{Li}_2\left(\frac{i c \sqrt{-d} e^{i \csc ^{-1}(c x)}}{\sqrt{e}+\sqrt{d c^2+e}}\right)}{2 e^{3/2}}+\frac{b \tanh ^{-1}\left(\sqrt{1-\frac{1}{c^2 x^2}}\right)}{c e}",1,"((I/4)*((-4*I)*a*c*Sqrt[e]*x - (4*I)*b*c*Sqrt[e]*x*ArcCsc[c*x] + (4*I)*a*c*Sqrt[d]*ArcTan[(Sqrt[e]*x)/Sqrt[d]] + (8*I)*b*c*Sqrt[d]*ArcSin[Sqrt[1 - (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*ArcTan[(((-I)*c*Sqrt[d] + Sqrt[e])*Cot[(Pi + 2*ArcCsc[c*x])/4])/Sqrt[c^2*d + e]] - (8*I)*b*c*Sqrt[d]*ArcSin[Sqrt[1 + (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*ArcTan[((I*c*Sqrt[d] + Sqrt[e])*Cot[(Pi + 2*ArcCsc[c*x])/4])/Sqrt[c^2*d + e]] + b*c*Sqrt[d]*Pi*Log[1 + (Sqrt[e] - Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] - 2*b*c*Sqrt[d]*ArcCsc[c*x]*Log[1 + (Sqrt[e] - Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] + 4*b*c*Sqrt[d]*ArcSin[Sqrt[1 - (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*Log[1 + (Sqrt[e] - Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] - b*c*Sqrt[d]*Pi*Log[1 + (-Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] + 2*b*c*Sqrt[d]*ArcCsc[c*x]*Log[1 + (-Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] - 4*b*c*Sqrt[d]*ArcSin[Sqrt[1 + (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*Log[1 + (-Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] - b*c*Sqrt[d]*Pi*Log[1 - (Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] + 2*b*c*Sqrt[d]*ArcCsc[c*x]*Log[1 - (Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] + 4*b*c*Sqrt[d]*ArcSin[Sqrt[1 + (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*Log[1 - (Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] + b*c*Sqrt[d]*Pi*Log[1 + (Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] - 2*b*c*Sqrt[d]*ArcCsc[c*x]*Log[1 + (Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] - 4*b*c*Sqrt[d]*ArcSin[Sqrt[1 - (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*Log[1 + (Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] + b*c*Sqrt[d]*Pi*Log[Sqrt[e] - (I*Sqrt[d])/x] - b*c*Sqrt[d]*Pi*Log[Sqrt[e] + (I*Sqrt[d])/x] - (4*I)*b*Sqrt[e]*Log[Cos[ArcCsc[c*x]/2]] + (4*I)*b*Sqrt[e]*Log[Sin[ArcCsc[c*x]/2]] + (2*I)*b*c*Sqrt[d]*PolyLog[2, (Sqrt[e] - Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] - (2*I)*b*c*Sqrt[d]*PolyLog[2, (-Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] - (2*I)*b*c*Sqrt[d]*PolyLog[2, -((Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x])))] + (2*I)*b*c*Sqrt[d]*PolyLog[2, (Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))]))/(c*e^(3/2))","B",0
99,1,1123,507,0.4267699,"\int \frac{x \left(a+b \csc ^{-1}(c x)\right)}{d+e x^2} \, dx","Integrate[(x*(a + b*ArcCsc[c*x]))/(d + e*x^2),x]","\frac{8 i b \csc ^{-1}(c x)^2+4 b \log \left(\frac{e^{-i \csc ^{-1}(c x)} \left(\sqrt{e}-\sqrt{d c^2+e}\right)}{c \sqrt{d}}+1\right) \csc ^{-1}(c x)+4 b \log \left(\frac{e^{-i \csc ^{-1}(c x)} \left(\sqrt{d c^2+e}-\sqrt{e}\right)}{c \sqrt{d}}+1\right) \csc ^{-1}(c x)+4 b \log \left(1-\frac{\left(\sqrt{e}+\sqrt{d c^2+e}\right) e^{-i \csc ^{-1}(c x)}}{c \sqrt{d}}\right) \csc ^{-1}(c x)+4 b \log \left(\frac{e^{-i \csc ^{-1}(c x)} \left(\sqrt{e}+\sqrt{d c^2+e}\right)}{c \sqrt{d}}+1\right) \csc ^{-1}(c x)-8 b \log \left(1-e^{2 i \csc ^{-1}(c x)}\right) \csc ^{-1}(c x)-4 i b \pi  \csc ^{-1}(c x)-16 i b \sin ^{-1}\left(\frac{\sqrt{1-\frac{i \sqrt{e}}{c \sqrt{d}}}}{\sqrt{2}}\right) \tan ^{-1}\left(\frac{\left(\sqrt{e}-i c \sqrt{d}\right) \cot \left(\frac{1}{4} \left(2 \csc ^{-1}(c x)+\pi \right)\right)}{\sqrt{d c^2+e}}\right)-16 i b \sin ^{-1}\left(\frac{\sqrt{\frac{i \sqrt{e}}{c \sqrt{d}}+1}}{\sqrt{2}}\right) \tan ^{-1}\left(\frac{\left(i \sqrt{d} c+\sqrt{e}\right) \cot \left(\frac{1}{4} \left(2 \csc ^{-1}(c x)+\pi \right)\right)}{\sqrt{d c^2+e}}\right)-8 b \sin ^{-1}\left(\frac{\sqrt{1-\frac{i \sqrt{e}}{c \sqrt{d}}}}{\sqrt{2}}\right) \log \left(\frac{e^{-i \csc ^{-1}(c x)} \left(\sqrt{e}-\sqrt{d c^2+e}\right)}{c \sqrt{d}}+1\right)-2 b \pi  \log \left(\frac{e^{-i \csc ^{-1}(c x)} \left(\sqrt{e}-\sqrt{d c^2+e}\right)}{c \sqrt{d}}+1\right)-8 b \sin ^{-1}\left(\frac{\sqrt{\frac{i \sqrt{e}}{c \sqrt{d}}+1}}{\sqrt{2}}\right) \log \left(\frac{e^{-i \csc ^{-1}(c x)} \left(\sqrt{d c^2+e}-\sqrt{e}\right)}{c \sqrt{d}}+1\right)-2 b \pi  \log \left(\frac{e^{-i \csc ^{-1}(c x)} \left(\sqrt{d c^2+e}-\sqrt{e}\right)}{c \sqrt{d}}+1\right)+8 b \sin ^{-1}\left(\frac{\sqrt{\frac{i \sqrt{e}}{c \sqrt{d}}+1}}{\sqrt{2}}\right) \log \left(1-\frac{\left(\sqrt{e}+\sqrt{d c^2+e}\right) e^{-i \csc ^{-1}(c x)}}{c \sqrt{d}}\right)-2 b \pi  \log \left(1-\frac{\left(\sqrt{e}+\sqrt{d c^2+e}\right) e^{-i \csc ^{-1}(c x)}}{c \sqrt{d}}\right)+8 b \sin ^{-1}\left(\frac{\sqrt{1-\frac{i \sqrt{e}}{c \sqrt{d}}}}{\sqrt{2}}\right) \log \left(\frac{e^{-i \csc ^{-1}(c x)} \left(\sqrt{e}+\sqrt{d c^2+e}\right)}{c \sqrt{d}}+1\right)-2 b \pi  \log \left(\frac{e^{-i \csc ^{-1}(c x)} \left(\sqrt{e}+\sqrt{d c^2+e}\right)}{c \sqrt{d}}+1\right)+2 b \pi  \log \left(\sqrt{e}-\frac{i \sqrt{d}}{x}\right)+2 b \pi  \log \left(\frac{i \sqrt{d}}{x}+\sqrt{e}\right)+4 a \log \left(e x^2+d\right)+4 i b \text{Li}_2\left(\frac{\left(\sqrt{e}-\sqrt{d c^2+e}\right) e^{-i \csc ^{-1}(c x)}}{c \sqrt{d}}\right)+4 i b \text{Li}_2\left(\frac{\left(\sqrt{d c^2+e}-\sqrt{e}\right) e^{-i \csc ^{-1}(c x)}}{c \sqrt{d}}\right)+4 i b \text{Li}_2\left(-\frac{\left(\sqrt{e}+\sqrt{d c^2+e}\right) e^{-i \csc ^{-1}(c x)}}{c \sqrt{d}}\right)+4 i b \text{Li}_2\left(\frac{\left(\sqrt{e}+\sqrt{d c^2+e}\right) e^{-i \csc ^{-1}(c x)}}{c \sqrt{d}}\right)+4 i b \text{Li}_2\left(e^{2 i \csc ^{-1}(c x)}\right)+i b \pi ^2}{8 e}","\frac{\left(a+b \csc ^{-1}(c x)\right) \log \left(1-\frac{i c \sqrt{-d} e^{i \csc ^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right)}{2 e}+\frac{\left(a+b \csc ^{-1}(c x)\right) \log \left(1+\frac{i c \sqrt{-d} e^{i \csc ^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right)}{2 e}+\frac{\left(a+b \csc ^{-1}(c x)\right) \log \left(1-\frac{i c \sqrt{-d} e^{i \csc ^{-1}(c x)}}{\sqrt{c^2 d+e}+\sqrt{e}}\right)}{2 e}+\frac{\left(a+b \csc ^{-1}(c x)\right) \log \left(1+\frac{i c \sqrt{-d} e^{i \csc ^{-1}(c x)}}{\sqrt{c^2 d+e}+\sqrt{e}}\right)}{2 e}-\frac{\log \left(1-e^{2 i \csc ^{-1}(c x)}\right) \left(a+b \csc ^{-1}(c x)\right)}{e}-\frac{i b \text{Li}_2\left(-\frac{i c \sqrt{-d} e^{i \csc ^{-1}(c x)}}{\sqrt{e}-\sqrt{d c^2+e}}\right)}{2 e}-\frac{i b \text{Li}_2\left(\frac{i c \sqrt{-d} e^{i \csc ^{-1}(c x)}}{\sqrt{e}-\sqrt{d c^2+e}}\right)}{2 e}-\frac{i b \text{Li}_2\left(-\frac{i c \sqrt{-d} e^{i \csc ^{-1}(c x)}}{\sqrt{e}+\sqrt{d c^2+e}}\right)}{2 e}-\frac{i b \text{Li}_2\left(\frac{i c \sqrt{-d} e^{i \csc ^{-1}(c x)}}{\sqrt{e}+\sqrt{d c^2+e}}\right)}{2 e}+\frac{i b \text{Li}_2\left(e^{2 i \csc ^{-1}(c x)}\right)}{2 e}",1,"(I*b*Pi^2 - (4*I)*b*Pi*ArcCsc[c*x] + (8*I)*b*ArcCsc[c*x]^2 - (16*I)*b*ArcSin[Sqrt[1 - (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*ArcTan[(((-I)*c*Sqrt[d] + Sqrt[e])*Cot[(Pi + 2*ArcCsc[c*x])/4])/Sqrt[c^2*d + e]] - (16*I)*b*ArcSin[Sqrt[1 + (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*ArcTan[((I*c*Sqrt[d] + Sqrt[e])*Cot[(Pi + 2*ArcCsc[c*x])/4])/Sqrt[c^2*d + e]] - 2*b*Pi*Log[1 + (Sqrt[e] - Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] + 4*b*ArcCsc[c*x]*Log[1 + (Sqrt[e] - Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] - 8*b*ArcSin[Sqrt[1 - (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*Log[1 + (Sqrt[e] - Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] - 2*b*Pi*Log[1 + (-Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] + 4*b*ArcCsc[c*x]*Log[1 + (-Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] - 8*b*ArcSin[Sqrt[1 + (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*Log[1 + (-Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] - 2*b*Pi*Log[1 - (Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] + 4*b*ArcCsc[c*x]*Log[1 - (Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] + 8*b*ArcSin[Sqrt[1 + (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*Log[1 - (Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] - 2*b*Pi*Log[1 + (Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] + 4*b*ArcCsc[c*x]*Log[1 + (Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] + 8*b*ArcSin[Sqrt[1 - (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*Log[1 + (Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] - 8*b*ArcCsc[c*x]*Log[1 - E^((2*I)*ArcCsc[c*x])] + 2*b*Pi*Log[Sqrt[e] - (I*Sqrt[d])/x] + 2*b*Pi*Log[Sqrt[e] + (I*Sqrt[d])/x] + 4*a*Log[d + e*x^2] + (4*I)*b*PolyLog[2, (Sqrt[e] - Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] + (4*I)*b*PolyLog[2, (-Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] + (4*I)*b*PolyLog[2, -((Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x])))] + (4*I)*b*PolyLog[2, (Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] + (4*I)*b*PolyLog[2, E^((2*I)*ArcCsc[c*x])])/(8*e)","B",0
100,1,1068,529,0.4381742,"\int \frac{a+b \csc ^{-1}(c x)}{d+e x^2} \, dx","Integrate[(a + b*ArcCsc[c*x])/(d + e*x^2),x]","-\frac{i \left(4 i a \tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right)+8 i b \sin ^{-1}\left(\frac{\sqrt{1-\frac{i \sqrt{e}}{c \sqrt{d}}}}{\sqrt{2}}\right) \tan ^{-1}\left(\frac{\left(\sqrt{e}-i c \sqrt{d}\right) \cot \left(\frac{1}{4} \left(2 \csc ^{-1}(c x)+\pi \right)\right)}{\sqrt{d c^2+e}}\right)-8 i b \sin ^{-1}\left(\frac{\sqrt{\frac{i \sqrt{e}}{c \sqrt{d}}+1}}{\sqrt{2}}\right) \tan ^{-1}\left(\frac{\left(i \sqrt{d} c+\sqrt{e}\right) \cot \left(\frac{1}{4} \left(2 \csc ^{-1}(c x)+\pi \right)\right)}{\sqrt{d c^2+e}}\right)-2 b \csc ^{-1}(c x) \log \left(\frac{e^{-i \csc ^{-1}(c x)} \left(\sqrt{e}-\sqrt{d c^2+e}\right)}{c \sqrt{d}}+1\right)+4 b \sin ^{-1}\left(\frac{\sqrt{1-\frac{i \sqrt{e}}{c \sqrt{d}}}}{\sqrt{2}}\right) \log \left(\frac{e^{-i \csc ^{-1}(c x)} \left(\sqrt{e}-\sqrt{d c^2+e}\right)}{c \sqrt{d}}+1\right)+b \pi  \log \left(\frac{e^{-i \csc ^{-1}(c x)} \left(\sqrt{e}-\sqrt{d c^2+e}\right)}{c \sqrt{d}}+1\right)+2 b \csc ^{-1}(c x) \log \left(\frac{e^{-i \csc ^{-1}(c x)} \left(\sqrt{d c^2+e}-\sqrt{e}\right)}{c \sqrt{d}}+1\right)-4 b \sin ^{-1}\left(\frac{\sqrt{\frac{i \sqrt{e}}{c \sqrt{d}}+1}}{\sqrt{2}}\right) \log \left(\frac{e^{-i \csc ^{-1}(c x)} \left(\sqrt{d c^2+e}-\sqrt{e}\right)}{c \sqrt{d}}+1\right)-b \pi  \log \left(\frac{e^{-i \csc ^{-1}(c x)} \left(\sqrt{d c^2+e}-\sqrt{e}\right)}{c \sqrt{d}}+1\right)+2 b \csc ^{-1}(c x) \log \left(1-\frac{\left(\sqrt{e}+\sqrt{d c^2+e}\right) e^{-i \csc ^{-1}(c x)}}{c \sqrt{d}}\right)+4 b \sin ^{-1}\left(\frac{\sqrt{\frac{i \sqrt{e}}{c \sqrt{d}}+1}}{\sqrt{2}}\right) \log \left(1-\frac{\left(\sqrt{e}+\sqrt{d c^2+e}\right) e^{-i \csc ^{-1}(c x)}}{c \sqrt{d}}\right)-b \pi  \log \left(1-\frac{\left(\sqrt{e}+\sqrt{d c^2+e}\right) e^{-i \csc ^{-1}(c x)}}{c \sqrt{d}}\right)-2 b \csc ^{-1}(c x) \log \left(\frac{e^{-i \csc ^{-1}(c x)} \left(\sqrt{e}+\sqrt{d c^2+e}\right)}{c \sqrt{d}}+1\right)-4 b \sin ^{-1}\left(\frac{\sqrt{1-\frac{i \sqrt{e}}{c \sqrt{d}}}}{\sqrt{2}}\right) \log \left(\frac{e^{-i \csc ^{-1}(c x)} \left(\sqrt{e}+\sqrt{d c^2+e}\right)}{c \sqrt{d}}+1\right)+b \pi  \log \left(\frac{e^{-i \csc ^{-1}(c x)} \left(\sqrt{e}+\sqrt{d c^2+e}\right)}{c \sqrt{d}}+1\right)+b \pi  \log \left(\sqrt{e}-\frac{i \sqrt{d}}{x}\right)-b \pi  \log \left(\frac{i \sqrt{d}}{x}+\sqrt{e}\right)+2 i b \text{Li}_2\left(\frac{\left(\sqrt{e}-\sqrt{d c^2+e}\right) e^{-i \csc ^{-1}(c x)}}{c \sqrt{d}}\right)-2 i b \text{Li}_2\left(\frac{\left(\sqrt{d c^2+e}-\sqrt{e}\right) e^{-i \csc ^{-1}(c x)}}{c \sqrt{d}}\right)-2 i b \text{Li}_2\left(-\frac{\left(\sqrt{e}+\sqrt{d c^2+e}\right) e^{-i \csc ^{-1}(c x)}}{c \sqrt{d}}\right)+2 i b \text{Li}_2\left(\frac{\left(\sqrt{e}+\sqrt{d c^2+e}\right) e^{-i \csc ^{-1}(c x)}}{c \sqrt{d}}\right)\right)}{4 \sqrt{d} \sqrt{e}}","-\frac{\left(a+b \csc ^{-1}(c x)\right) \log \left(1-\frac{i c \sqrt{-d} e^{i \csc ^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right)}{2 \sqrt{-d} \sqrt{e}}+\frac{\left(a+b \csc ^{-1}(c x)\right) \log \left(1+\frac{i c \sqrt{-d} e^{i \csc ^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right)}{2 \sqrt{-d} \sqrt{e}}-\frac{\left(a+b \csc ^{-1}(c x)\right) \log \left(1-\frac{i c \sqrt{-d} e^{i \csc ^{-1}(c x)}}{\sqrt{c^2 d+e}+\sqrt{e}}\right)}{2 \sqrt{-d} \sqrt{e}}+\frac{\left(a+b \csc ^{-1}(c x)\right) \log \left(1+\frac{i c \sqrt{-d} e^{i \csc ^{-1}(c x)}}{\sqrt{c^2 d+e}+\sqrt{e}}\right)}{2 \sqrt{-d} \sqrt{e}}-\frac{i b \text{Li}_2\left(-\frac{i c \sqrt{-d} e^{i \csc ^{-1}(c x)}}{\sqrt{e}-\sqrt{d c^2+e}}\right)}{2 \sqrt{-d} \sqrt{e}}+\frac{i b \text{Li}_2\left(\frac{i c \sqrt{-d} e^{i \csc ^{-1}(c x)}}{\sqrt{e}-\sqrt{d c^2+e}}\right)}{2 \sqrt{-d} \sqrt{e}}-\frac{i b \text{Li}_2\left(-\frac{i c \sqrt{-d} e^{i \csc ^{-1}(c x)}}{\sqrt{e}+\sqrt{d c^2+e}}\right)}{2 \sqrt{-d} \sqrt{e}}+\frac{i b \text{Li}_2\left(\frac{i c \sqrt{-d} e^{i \csc ^{-1}(c x)}}{\sqrt{e}+\sqrt{d c^2+e}}\right)}{2 \sqrt{-d} \sqrt{e}}",1,"((-1/4*I)*((4*I)*a*ArcTan[(Sqrt[e]*x)/Sqrt[d]] + (8*I)*b*ArcSin[Sqrt[1 - (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*ArcTan[(((-I)*c*Sqrt[d] + Sqrt[e])*Cot[(Pi + 2*ArcCsc[c*x])/4])/Sqrt[c^2*d + e]] - (8*I)*b*ArcSin[Sqrt[1 + (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*ArcTan[((I*c*Sqrt[d] + Sqrt[e])*Cot[(Pi + 2*ArcCsc[c*x])/4])/Sqrt[c^2*d + e]] + b*Pi*Log[1 + (Sqrt[e] - Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] - 2*b*ArcCsc[c*x]*Log[1 + (Sqrt[e] - Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] + 4*b*ArcSin[Sqrt[1 - (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*Log[1 + (Sqrt[e] - Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] - b*Pi*Log[1 + (-Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] + 2*b*ArcCsc[c*x]*Log[1 + (-Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] - 4*b*ArcSin[Sqrt[1 + (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*Log[1 + (-Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] - b*Pi*Log[1 - (Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] + 2*b*ArcCsc[c*x]*Log[1 - (Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] + 4*b*ArcSin[Sqrt[1 + (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*Log[1 - (Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] + b*Pi*Log[1 + (Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] - 2*b*ArcCsc[c*x]*Log[1 + (Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] - 4*b*ArcSin[Sqrt[1 - (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*Log[1 + (Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] + b*Pi*Log[Sqrt[e] - (I*Sqrt[d])/x] - b*Pi*Log[Sqrt[e] + (I*Sqrt[d])/x] + (2*I)*b*PolyLog[2, (Sqrt[e] - Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] - (2*I)*b*PolyLog[2, (-Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] - (2*I)*b*PolyLog[2, -((Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x])))] + (2*I)*b*PolyLog[2, (Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))]))/(Sqrt[d]*Sqrt[e])","B",0
101,1,407,479,0.9396524,"\int \frac{a+b \csc ^{-1}(c x)}{x \left(d+e x^2\right)} \, dx","Integrate[(a + b*ArcCsc[c*x])/(x*(d + e*x^2)),x]","\frac{-2 a \log \left(d+e x^2\right)+4 a \log (x)+i b \left(\text{Li}_2\left(\frac{\left(d c^2+2 e-2 \sqrt{e \left(d c^2+e\right)}\right) e^{2 i \csc ^{-1}(c x)}}{c^2 d}\right)+\text{Li}_2\left(\frac{\left(d c^2+2 \left(e+\sqrt{e \left(d c^2+e\right)}\right)\right) e^{2 i \csc ^{-1}(c x)}}{c^2 d}\right)+4 \sin ^{-1}\left(\sqrt{-\frac{e}{c^2 d}}\right) \tan ^{-1}\left(\frac{\sqrt{e \left(c^2 d+e\right)}}{c e x \sqrt{1-\frac{1}{c^2 x^2}}}\right)+2 i \csc ^{-1}(c x) \log \left(1-\frac{\left(-2 \sqrt{e \left(c^2 d+e\right)}+c^2 d+2 e\right) e^{2 i \csc ^{-1}(c x)}}{c^2 d}\right)+2 i \csc ^{-1}(c x) \log \left(1-\frac{\left(2 \left(\sqrt{e \left(c^2 d+e\right)}+e\right)+c^2 d\right) e^{2 i \csc ^{-1}(c x)}}{c^2 d}\right)-2 i \sin ^{-1}\left(\sqrt{-\frac{e}{c^2 d}}\right) \log \left(1-\frac{\left(-2 \sqrt{e \left(c^2 d+e\right)}+c^2 d+2 e\right) e^{2 i \csc ^{-1}(c x)}}{c^2 d}\right)+2 i \sin ^{-1}\left(\sqrt{-\frac{e}{c^2 d}}\right) \log \left(1-\frac{\left(2 \left(\sqrt{e \left(c^2 d+e\right)}+e\right)+c^2 d\right) e^{2 i \csc ^{-1}(c x)}}{c^2 d}\right)+2 \csc ^{-1}(c x)^2\right)}{4 d}","-\frac{\left(a+b \csc ^{-1}(c x)\right) \log \left(1-\frac{i c \sqrt{-d} e^{i \csc ^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right)}{2 d}-\frac{\left(a+b \csc ^{-1}(c x)\right) \log \left(1+\frac{i c \sqrt{-d} e^{i \csc ^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right)}{2 d}-\frac{\left(a+b \csc ^{-1}(c x)\right) \log \left(1-\frac{i c \sqrt{-d} e^{i \csc ^{-1}(c x)}}{\sqrt{c^2 d+e}+\sqrt{e}}\right)}{2 d}-\frac{\left(a+b \csc ^{-1}(c x)\right) \log \left(1+\frac{i c \sqrt{-d} e^{i \csc ^{-1}(c x)}}{\sqrt{c^2 d+e}+\sqrt{e}}\right)}{2 d}+\frac{i \left(a+b \csc ^{-1}(c x)\right)^2}{2 b d}+\frac{i b \text{Li}_2\left(-\frac{i c \sqrt{-d} e^{i \csc ^{-1}(c x)}}{\sqrt{e}-\sqrt{d c^2+e}}\right)}{2 d}+\frac{i b \text{Li}_2\left(\frac{i c \sqrt{-d} e^{i \csc ^{-1}(c x)}}{\sqrt{e}-\sqrt{d c^2+e}}\right)}{2 d}+\frac{i b \text{Li}_2\left(-\frac{i c \sqrt{-d} e^{i \csc ^{-1}(c x)}}{\sqrt{e}+\sqrt{d c^2+e}}\right)}{2 d}+\frac{i b \text{Li}_2\left(\frac{i c \sqrt{-d} e^{i \csc ^{-1}(c x)}}{\sqrt{e}+\sqrt{d c^2+e}}\right)}{2 d}",1,"(4*a*Log[x] - 2*a*Log[d + e*x^2] + I*b*(2*ArcCsc[c*x]^2 + 4*ArcSin[Sqrt[-(e/(c^2*d))]]*ArcTan[Sqrt[e*(c^2*d + e)]/(c*e*Sqrt[1 - 1/(c^2*x^2)]*x)] + (2*I)*ArcCsc[c*x]*Log[1 - ((c^2*d + 2*e - 2*Sqrt[e*(c^2*d + e)])*E^((2*I)*ArcCsc[c*x]))/(c^2*d)] - (2*I)*ArcSin[Sqrt[-(e/(c^2*d))]]*Log[1 - ((c^2*d + 2*e - 2*Sqrt[e*(c^2*d + e)])*E^((2*I)*ArcCsc[c*x]))/(c^2*d)] + (2*I)*ArcCsc[c*x]*Log[1 - ((c^2*d + 2*(e + Sqrt[e*(c^2*d + e)]))*E^((2*I)*ArcCsc[c*x]))/(c^2*d)] + (2*I)*ArcSin[Sqrt[-(e/(c^2*d))]]*Log[1 - ((c^2*d + 2*(e + Sqrt[e*(c^2*d + e)]))*E^((2*I)*ArcCsc[c*x]))/(c^2*d)] + PolyLog[2, ((c^2*d + 2*e - 2*Sqrt[e*(c^2*d + e)])*E^((2*I)*ArcCsc[c*x]))/(c^2*d)] + PolyLog[2, ((c^2*d + 2*(e + Sqrt[e*(c^2*d + e)]))*E^((2*I)*ArcCsc[c*x]))/(c^2*d)]))/(4*d)","A",0
102,1,1241,572,1.7500977,"\int \frac{a+b \csc ^{-1}(c x)}{x^2 \left(d+e x^2\right)} \, dx","Integrate[(a + b*ArcCsc[c*x])/(x^2*(d + e*x^2)),x]","-\frac{\sqrt{e} \tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right) a}{d^{3/2}}-\frac{a}{d x}+b \left(-\frac{c \sqrt{1-\frac{1}{c^2 x^2}} x+\csc ^{-1}(c x)}{d x}+\frac{\sqrt{e} \left(8 \csc ^{-1}(c x)^2-8 i \log \left(\frac{e^{-i \csc ^{-1}(c x)} \left(\sqrt{e}-\sqrt{d c^2+e}\right)}{c \sqrt{d}}+1\right) \csc ^{-1}(c x)-8 i \log \left(\frac{e^{-i \csc ^{-1}(c x)} \left(\sqrt{e}+\sqrt{d c^2+e}\right)}{c \sqrt{d}}+1\right) \csc ^{-1}(c x)+8 i \log \left(1-e^{2 i \csc ^{-1}(c x)}\right) \csc ^{-1}(c x)-4 \pi  \csc ^{-1}(c x)-32 \sin ^{-1}\left(\frac{\sqrt{1-\frac{i \sqrt{e}}{c \sqrt{d}}}}{\sqrt{2}}\right) \tan ^{-1}\left(\frac{\left(\sqrt{e}-i c \sqrt{d}\right) \cot \left(\frac{1}{4} \left(2 \csc ^{-1}(c x)+\pi \right)\right)}{\sqrt{d c^2+e}}\right)+16 i \sin ^{-1}\left(\frac{\sqrt{1-\frac{i \sqrt{e}}{c \sqrt{d}}}}{\sqrt{2}}\right) \log \left(\frac{e^{-i \csc ^{-1}(c x)} \left(\sqrt{e}-\sqrt{d c^2+e}\right)}{c \sqrt{d}}+1\right)+4 i \pi  \log \left(\frac{e^{-i \csc ^{-1}(c x)} \left(\sqrt{e}-\sqrt{d c^2+e}\right)}{c \sqrt{d}}+1\right)-16 i \sin ^{-1}\left(\frac{\sqrt{1-\frac{i \sqrt{e}}{c \sqrt{d}}}}{\sqrt{2}}\right) \log \left(\frac{e^{-i \csc ^{-1}(c x)} \left(\sqrt{e}+\sqrt{d c^2+e}\right)}{c \sqrt{d}}+1\right)+4 i \pi  \log \left(\frac{e^{-i \csc ^{-1}(c x)} \left(\sqrt{e}+\sqrt{d c^2+e}\right)}{c \sqrt{d}}+1\right)-4 i \pi  \log \left(\frac{i \sqrt{d}}{x}+\sqrt{e}\right)+8 \text{Li}_2\left(\frac{\left(\sqrt{d c^2+e}-\sqrt{e}\right) e^{-i \csc ^{-1}(c x)}}{c \sqrt{d}}\right)+8 \text{Li}_2\left(-\frac{\left(\sqrt{e}+\sqrt{d c^2+e}\right) e^{-i \csc ^{-1}(c x)}}{c \sqrt{d}}\right)+4 \text{Li}_2\left(e^{2 i \csc ^{-1}(c x)}\right)+\pi ^2\right)}{16 d^{3/2}}-\frac{\sqrt{e} \left(8 \csc ^{-1}(c x)^2-8 i \log \left(\frac{e^{-i \csc ^{-1}(c x)} \left(\sqrt{d c^2+e}-\sqrt{e}\right)}{c \sqrt{d}}+1\right) \csc ^{-1}(c x)-8 i \log \left(1-\frac{\left(\sqrt{e}+\sqrt{d c^2+e}\right) e^{-i \csc ^{-1}(c x)}}{c \sqrt{d}}\right) \csc ^{-1}(c x)+8 i \log \left(1-e^{2 i \csc ^{-1}(c x)}\right) \csc ^{-1}(c x)-4 \pi  \csc ^{-1}(c x)-32 \sin ^{-1}\left(\frac{\sqrt{\frac{i \sqrt{e}}{c \sqrt{d}}+1}}{\sqrt{2}}\right) \tan ^{-1}\left(\frac{\left(i \sqrt{d} c+\sqrt{e}\right) \cot \left(\frac{1}{4} \left(2 \csc ^{-1}(c x)+\pi \right)\right)}{\sqrt{d c^2+e}}\right)+16 i \sin ^{-1}\left(\frac{\sqrt{\frac{i \sqrt{e}}{c \sqrt{d}}+1}}{\sqrt{2}}\right) \log \left(\frac{e^{-i \csc ^{-1}(c x)} \left(\sqrt{d c^2+e}-\sqrt{e}\right)}{c \sqrt{d}}+1\right)+4 i \pi  \log \left(\frac{e^{-i \csc ^{-1}(c x)} \left(\sqrt{d c^2+e}-\sqrt{e}\right)}{c \sqrt{d}}+1\right)-16 i \sin ^{-1}\left(\frac{\sqrt{\frac{i \sqrt{e}}{c \sqrt{d}}+1}}{\sqrt{2}}\right) \log \left(1-\frac{\left(\sqrt{e}+\sqrt{d c^2+e}\right) e^{-i \csc ^{-1}(c x)}}{c \sqrt{d}}\right)+4 i \pi  \log \left(1-\frac{\left(\sqrt{e}+\sqrt{d c^2+e}\right) e^{-i \csc ^{-1}(c x)}}{c \sqrt{d}}\right)-4 i \pi  \log \left(\sqrt{e}-\frac{i \sqrt{d}}{x}\right)+8 \text{Li}_2\left(\frac{\left(\sqrt{e}-\sqrt{d c^2+e}\right) e^{-i \csc ^{-1}(c x)}}{c \sqrt{d}}\right)+8 \text{Li}_2\left(\frac{\left(\sqrt{e}+\sqrt{d c^2+e}\right) e^{-i \csc ^{-1}(c x)}}{c \sqrt{d}}\right)+4 \text{Li}_2\left(e^{2 i \csc ^{-1}(c x)}\right)+\pi ^2\right)}{16 d^{3/2}}\right)","-\frac{\sqrt{e} \left(a+b \csc ^{-1}(c x)\right) \log \left(1-\frac{i c \sqrt{-d} e^{i \csc ^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right)}{2 (-d)^{3/2}}+\frac{\sqrt{e} \left(a+b \csc ^{-1}(c x)\right) \log \left(1+\frac{i c \sqrt{-d} e^{i \csc ^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right)}{2 (-d)^{3/2}}-\frac{\sqrt{e} \left(a+b \csc ^{-1}(c x)\right) \log \left(1-\frac{i c \sqrt{-d} e^{i \csc ^{-1}(c x)}}{\sqrt{c^2 d+e}+\sqrt{e}}\right)}{2 (-d)^{3/2}}+\frac{\sqrt{e} \left(a+b \csc ^{-1}(c x)\right) \log \left(1+\frac{i c \sqrt{-d} e^{i \csc ^{-1}(c x)}}{\sqrt{c^2 d+e}+\sqrt{e}}\right)}{2 (-d)^{3/2}}-\frac{a}{d x}-\frac{i b \sqrt{e} \text{Li}_2\left(-\frac{i c \sqrt{-d} e^{i \csc ^{-1}(c x)}}{\sqrt{e}-\sqrt{d c^2+e}}\right)}{2 (-d)^{3/2}}+\frac{i b \sqrt{e} \text{Li}_2\left(\frac{i c \sqrt{-d} e^{i \csc ^{-1}(c x)}}{\sqrt{e}-\sqrt{d c^2+e}}\right)}{2 (-d)^{3/2}}-\frac{i b \sqrt{e} \text{Li}_2\left(-\frac{i c \sqrt{-d} e^{i \csc ^{-1}(c x)}}{\sqrt{e}+\sqrt{d c^2+e}}\right)}{2 (-d)^{3/2}}+\frac{i b \sqrt{e} \text{Li}_2\left(\frac{i c \sqrt{-d} e^{i \csc ^{-1}(c x)}}{\sqrt{e}+\sqrt{d c^2+e}}\right)}{2 (-d)^{3/2}}-\frac{b c \sqrt{1-\frac{1}{c^2 x^2}}}{d}-\frac{b \csc ^{-1}(c x)}{d x}",1,"-(a/(d*x)) - (a*Sqrt[e]*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/d^(3/2) + b*(-((c*Sqrt[1 - 1/(c^2*x^2)]*x + ArcCsc[c*x])/(d*x)) + (Sqrt[e]*(Pi^2 - 4*Pi*ArcCsc[c*x] + 8*ArcCsc[c*x]^2 - 32*ArcSin[Sqrt[1 - (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*ArcTan[(((-I)*c*Sqrt[d] + Sqrt[e])*Cot[(Pi + 2*ArcCsc[c*x])/4])/Sqrt[c^2*d + e]] + (4*I)*Pi*Log[1 + (Sqrt[e] - Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] - (8*I)*ArcCsc[c*x]*Log[1 + (Sqrt[e] - Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] + (16*I)*ArcSin[Sqrt[1 - (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*Log[1 + (Sqrt[e] - Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] + (4*I)*Pi*Log[1 + (Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] - (8*I)*ArcCsc[c*x]*Log[1 + (Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] - (16*I)*ArcSin[Sqrt[1 - (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*Log[1 + (Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] + (8*I)*ArcCsc[c*x]*Log[1 - E^((2*I)*ArcCsc[c*x])] - (4*I)*Pi*Log[Sqrt[e] + (I*Sqrt[d])/x] + 8*PolyLog[2, (-Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] + 8*PolyLog[2, -((Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x])))] + 4*PolyLog[2, E^((2*I)*ArcCsc[c*x])]))/(16*d^(3/2)) - (Sqrt[e]*(Pi^2 - 4*Pi*ArcCsc[c*x] + 8*ArcCsc[c*x]^2 - 32*ArcSin[Sqrt[1 + (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*ArcTan[((I*c*Sqrt[d] + Sqrt[e])*Cot[(Pi + 2*ArcCsc[c*x])/4])/Sqrt[c^2*d + e]] + (4*I)*Pi*Log[1 + (-Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] - (8*I)*ArcCsc[c*x]*Log[1 + (-Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] + (16*I)*ArcSin[Sqrt[1 + (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*Log[1 + (-Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] + (4*I)*Pi*Log[1 - (Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] - (8*I)*ArcCsc[c*x]*Log[1 - (Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] - (16*I)*ArcSin[Sqrt[1 + (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*Log[1 - (Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] + (8*I)*ArcCsc[c*x]*Log[1 - E^((2*I)*ArcCsc[c*x])] - (4*I)*Pi*Log[Sqrt[e] - (I*Sqrt[d])/x] + 8*PolyLog[2, (Sqrt[e] - Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] + 8*PolyLog[2, (Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] + 4*PolyLog[2, E^((2*I)*ArcCsc[c*x])]))/(16*d^(3/2)))","B",0
103,1,1480,628,6.2364013,"\int \frac{x^5 \left(a+b \csc ^{-1}(c x)\right)}{\left(d+e x^2\right)^2} \, dx","Integrate[(x^5*(a + b*ArcCsc[c*x]))/(d + e*x^2)^2,x]","-\frac{\frac{2 a d^2}{e x^2+d}+4 a \log \left(e x^2+d\right) d-2 a e x^2+b \left(-2 e \csc ^{-1}(c x) x^2-\frac{2 e \sqrt{1-\frac{1}{c^2 x^2}} x}{c}+8 i d \csc ^{-1}(c x)^2+\frac{d^{3/2} \csc ^{-1}(c x)}{\sqrt{d}-i \sqrt{e} x}+\frac{d^{3/2} \csc ^{-1}(c x)}{i \sqrt{e} x+\sqrt{d}}-4 i d \pi  \csc ^{-1}(c x)-2 d \sin ^{-1}\left(\frac{1}{c x}\right)-16 i d \sin ^{-1}\left(\frac{\sqrt{1-\frac{i \sqrt{e}}{c \sqrt{d}}}}{\sqrt{2}}\right) \tan ^{-1}\left(\frac{\left(\sqrt{e}-i c \sqrt{d}\right) \cot \left(\frac{1}{4} \left(2 \csc ^{-1}(c x)+\pi \right)\right)}{\sqrt{d c^2+e}}\right)-16 i d \sin ^{-1}\left(\frac{\sqrt{\frac{i \sqrt{e}}{c \sqrt{d}}+1}}{\sqrt{2}}\right) \tan ^{-1}\left(\frac{\left(i \sqrt{d} c+\sqrt{e}\right) \cot \left(\frac{1}{4} \left(2 \csc ^{-1}(c x)+\pi \right)\right)}{\sqrt{d c^2+e}}\right)+4 d \csc ^{-1}(c x) \log \left(\frac{e^{-i \csc ^{-1}(c x)} \left(\sqrt{e}-\sqrt{d c^2+e}\right)}{c \sqrt{d}}+1\right)-8 d \sin ^{-1}\left(\frac{\sqrt{1-\frac{i \sqrt{e}}{c \sqrt{d}}}}{\sqrt{2}}\right) \log \left(\frac{e^{-i \csc ^{-1}(c x)} \left(\sqrt{e}-\sqrt{d c^2+e}\right)}{c \sqrt{d}}+1\right)-2 d \pi  \log \left(\frac{e^{-i \csc ^{-1}(c x)} \left(\sqrt{e}-\sqrt{d c^2+e}\right)}{c \sqrt{d}}+1\right)+4 d \csc ^{-1}(c x) \log \left(\frac{e^{-i \csc ^{-1}(c x)} \left(\sqrt{d c^2+e}-\sqrt{e}\right)}{c \sqrt{d}}+1\right)-8 d \sin ^{-1}\left(\frac{\sqrt{\frac{i \sqrt{e}}{c \sqrt{d}}+1}}{\sqrt{2}}\right) \log \left(\frac{e^{-i \csc ^{-1}(c x)} \left(\sqrt{d c^2+e}-\sqrt{e}\right)}{c \sqrt{d}}+1\right)-2 d \pi  \log \left(\frac{e^{-i \csc ^{-1}(c x)} \left(\sqrt{d c^2+e}-\sqrt{e}\right)}{c \sqrt{d}}+1\right)+4 d \csc ^{-1}(c x) \log \left(1-\frac{\left(\sqrt{e}+\sqrt{d c^2+e}\right) e^{-i \csc ^{-1}(c x)}}{c \sqrt{d}}\right)+8 d \sin ^{-1}\left(\frac{\sqrt{\frac{i \sqrt{e}}{c \sqrt{d}}+1}}{\sqrt{2}}\right) \log \left(1-\frac{\left(\sqrt{e}+\sqrt{d c^2+e}\right) e^{-i \csc ^{-1}(c x)}}{c \sqrt{d}}\right)-2 d \pi  \log \left(1-\frac{\left(\sqrt{e}+\sqrt{d c^2+e}\right) e^{-i \csc ^{-1}(c x)}}{c \sqrt{d}}\right)+4 d \csc ^{-1}(c x) \log \left(\frac{e^{-i \csc ^{-1}(c x)} \left(\sqrt{e}+\sqrt{d c^2+e}\right)}{c \sqrt{d}}+1\right)+8 d \sin ^{-1}\left(\frac{\sqrt{1-\frac{i \sqrt{e}}{c \sqrt{d}}}}{\sqrt{2}}\right) \log \left(\frac{e^{-i \csc ^{-1}(c x)} \left(\sqrt{e}+\sqrt{d c^2+e}\right)}{c \sqrt{d}}+1\right)-2 d \pi  \log \left(\frac{e^{-i \csc ^{-1}(c x)} \left(\sqrt{e}+\sqrt{d c^2+e}\right)}{c \sqrt{d}}+1\right)-8 d \csc ^{-1}(c x) \log \left(1-e^{2 i \csc ^{-1}(c x)}\right)+2 d \pi  \log \left(\sqrt{e}-\frac{i \sqrt{d}}{x}\right)+2 d \pi  \log \left(\frac{i \sqrt{d}}{x}+\sqrt{e}\right)+\frac{d \sqrt{e} \log \left(\frac{2 \sqrt{d} \sqrt{e} \left(c \left(-i \sqrt{d} c-\sqrt{-d c^2-e} \sqrt{1-\frac{1}{c^2 x^2}}\right) x+\sqrt{e}\right)}{\sqrt{-d c^2-e} \left(i \sqrt{e} x+\sqrt{d}\right)}\right)}{\sqrt{-d c^2-e}}+\frac{d \sqrt{e} \log \left(\frac{2 \sqrt{d} \sqrt{e} \left(c \left(\sqrt{-d c^2-e} \sqrt{1-\frac{1}{c^2 x^2}}-i c \sqrt{d}\right) x-\sqrt{e}\right)}{\sqrt{-d c^2-e} \left(\sqrt{d}-i \sqrt{e} x\right)}\right)}{\sqrt{-d c^2-e}}+4 i d \text{Li}_2\left(\frac{\left(\sqrt{e}-\sqrt{d c^2+e}\right) e^{-i \csc ^{-1}(c x)}}{c \sqrt{d}}\right)+4 i d \text{Li}_2\left(\frac{\left(\sqrt{d c^2+e}-\sqrt{e}\right) e^{-i \csc ^{-1}(c x)}}{c \sqrt{d}}\right)+4 i d \text{Li}_2\left(-\frac{\left(\sqrt{e}+\sqrt{d c^2+e}\right) e^{-i \csc ^{-1}(c x)}}{c \sqrt{d}}\right)+4 i d \text{Li}_2\left(\frac{\left(\sqrt{e}+\sqrt{d c^2+e}\right) e^{-i \csc ^{-1}(c x)}}{c \sqrt{d}}\right)+4 i d \text{Li}_2\left(e^{2 i \csc ^{-1}(c x)}\right)+i d \pi ^2\right)}{4 e^3}","-\frac{d \left(a+b \csc ^{-1}(c x)\right) \log \left(1-\frac{i c \sqrt{-d} e^{i \csc ^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right)}{e^3}-\frac{d \left(a+b \csc ^{-1}(c x)\right) \log \left(1+\frac{i c \sqrt{-d} e^{i \csc ^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right)}{e^3}-\frac{d \left(a+b \csc ^{-1}(c x)\right) \log \left(1-\frac{i c \sqrt{-d} e^{i \csc ^{-1}(c x)}}{\sqrt{c^2 d+e}+\sqrt{e}}\right)}{e^3}-\frac{d \left(a+b \csc ^{-1}(c x)\right) \log \left(1+\frac{i c \sqrt{-d} e^{i \csc ^{-1}(c x)}}{\sqrt{c^2 d+e}+\sqrt{e}}\right)}{e^3}+\frac{2 d \log \left(1-e^{2 i \csc ^{-1}(c x)}\right) \left(a+b \csc ^{-1}(c x)\right)}{e^3}+\frac{d \left(a+b \csc ^{-1}(c x)\right)}{2 e^2 \left(\frac{d}{x^2}+e\right)}+\frac{x^2 \left(a+b \csc ^{-1}(c x)\right)}{2 e^2}-\frac{b d \tan ^{-1}\left(\frac{\sqrt{c^2 d+e}}{c \sqrt{e} x \sqrt{1-\frac{1}{c^2 x^2}}}\right)}{2 e^{5/2} \sqrt{c^2 d+e}}+\frac{i b d \text{Li}_2\left(-\frac{i c \sqrt{-d} e^{i \csc ^{-1}(c x)}}{\sqrt{e}-\sqrt{d c^2+e}}\right)}{e^3}+\frac{i b d \text{Li}_2\left(\frac{i c \sqrt{-d} e^{i \csc ^{-1}(c x)}}{\sqrt{e}-\sqrt{d c^2+e}}\right)}{e^3}+\frac{i b d \text{Li}_2\left(-\frac{i c \sqrt{-d} e^{i \csc ^{-1}(c x)}}{\sqrt{e}+\sqrt{d c^2+e}}\right)}{e^3}+\frac{i b d \text{Li}_2\left(\frac{i c \sqrt{-d} e^{i \csc ^{-1}(c x)}}{\sqrt{e}+\sqrt{d c^2+e}}\right)}{e^3}+\frac{b x \sqrt{1-\frac{1}{c^2 x^2}}}{2 c e^2}-\frac{i b d \text{Li}_2\left(e^{2 i \csc ^{-1}(c x)}\right)}{e^3}",1,"-1/4*(-2*a*e*x^2 + (2*a*d^2)/(d + e*x^2) + 4*a*d*Log[d + e*x^2] + b*(I*d*Pi^2 - (2*e*Sqrt[1 - 1/(c^2*x^2)]*x)/c - (4*I)*d*Pi*ArcCsc[c*x] - 2*e*x^2*ArcCsc[c*x] + (d^(3/2)*ArcCsc[c*x])/(Sqrt[d] - I*Sqrt[e]*x) + (d^(3/2)*ArcCsc[c*x])/(Sqrt[d] + I*Sqrt[e]*x) + (8*I)*d*ArcCsc[c*x]^2 - 2*d*ArcSin[1/(c*x)] - (16*I)*d*ArcSin[Sqrt[1 - (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*ArcTan[(((-I)*c*Sqrt[d] + Sqrt[e])*Cot[(Pi + 2*ArcCsc[c*x])/4])/Sqrt[c^2*d + e]] - (16*I)*d*ArcSin[Sqrt[1 + (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*ArcTan[((I*c*Sqrt[d] + Sqrt[e])*Cot[(Pi + 2*ArcCsc[c*x])/4])/Sqrt[c^2*d + e]] - 2*d*Pi*Log[1 + (Sqrt[e] - Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] + 4*d*ArcCsc[c*x]*Log[1 + (Sqrt[e] - Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] - 8*d*ArcSin[Sqrt[1 - (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*Log[1 + (Sqrt[e] - Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] - 2*d*Pi*Log[1 + (-Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] + 4*d*ArcCsc[c*x]*Log[1 + (-Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] - 8*d*ArcSin[Sqrt[1 + (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*Log[1 + (-Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] - 2*d*Pi*Log[1 - (Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] + 4*d*ArcCsc[c*x]*Log[1 - (Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] + 8*d*ArcSin[Sqrt[1 + (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*Log[1 - (Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] - 2*d*Pi*Log[1 + (Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] + 4*d*ArcCsc[c*x]*Log[1 + (Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] + 8*d*ArcSin[Sqrt[1 - (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*Log[1 + (Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] - 8*d*ArcCsc[c*x]*Log[1 - E^((2*I)*ArcCsc[c*x])] + 2*d*Pi*Log[Sqrt[e] - (I*Sqrt[d])/x] + 2*d*Pi*Log[Sqrt[e] + (I*Sqrt[d])/x] + (d*Sqrt[e]*Log[(2*Sqrt[d]*Sqrt[e]*(Sqrt[e] + c*((-I)*c*Sqrt[d] - Sqrt[-(c^2*d) - e]*Sqrt[1 - 1/(c^2*x^2)])*x))/(Sqrt[-(c^2*d) - e]*(Sqrt[d] + I*Sqrt[e]*x))])/Sqrt[-(c^2*d) - e] + (d*Sqrt[e]*Log[(2*Sqrt[d]*Sqrt[e]*(-Sqrt[e] + c*((-I)*c*Sqrt[d] + Sqrt[-(c^2*d) - e]*Sqrt[1 - 1/(c^2*x^2)])*x))/(Sqrt[-(c^2*d) - e]*(Sqrt[d] - I*Sqrt[e]*x))])/Sqrt[-(c^2*d) - e] + (4*I)*d*PolyLog[2, (Sqrt[e] - Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] + (4*I)*d*PolyLog[2, (-Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] + (4*I)*d*PolyLog[2, -((Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x])))] + (4*I)*d*PolyLog[2, (Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] + (4*I)*d*PolyLog[2, E^((2*I)*ArcCsc[c*x])]))/e^3","B",0
104,1,1442,593,2.2552189,"\int \frac{x^3 \left(a+b \csc ^{-1}(c x)\right)}{\left(d+e x^2\right)^2} \, dx","Integrate[(x^3*(a + b*ArcCsc[c*x]))/(d + e*x^2)^2,x]","\frac{8 i b \csc ^{-1}(c x)^2+4 b \log \left(\frac{e^{-i \csc ^{-1}(c x)} \left(\sqrt{e}-\sqrt{d c^2+e}\right)}{c \sqrt{d}}+1\right) \csc ^{-1}(c x)+4 b \log \left(\frac{e^{-i \csc ^{-1}(c x)} \left(\sqrt{d c^2+e}-\sqrt{e}\right)}{c \sqrt{d}}+1\right) \csc ^{-1}(c x)+4 b \log \left(1-\frac{\left(\sqrt{e}+\sqrt{d c^2+e}\right) e^{-i \csc ^{-1}(c x)}}{c \sqrt{d}}\right) \csc ^{-1}(c x)+4 b \log \left(\frac{e^{-i \csc ^{-1}(c x)} \left(\sqrt{e}+\sqrt{d c^2+e}\right)}{c \sqrt{d}}+1\right) \csc ^{-1}(c x)-8 b \log \left(1-e^{2 i \csc ^{-1}(c x)}\right) \csc ^{-1}(c x)+\frac{2 b \sqrt{d} \csc ^{-1}(c x)}{\sqrt{d}-i \sqrt{e} x}+\frac{2 b \sqrt{d} \csc ^{-1}(c x)}{i \sqrt{e} x+\sqrt{d}}-4 i b \pi  \csc ^{-1}(c x)-4 b \sin ^{-1}\left(\frac{1}{c x}\right)-16 i b \sin ^{-1}\left(\frac{\sqrt{1-\frac{i \sqrt{e}}{c \sqrt{d}}}}{\sqrt{2}}\right) \tan ^{-1}\left(\frac{\left(\sqrt{e}-i c \sqrt{d}\right) \cot \left(\frac{1}{4} \left(2 \csc ^{-1}(c x)+\pi \right)\right)}{\sqrt{d c^2+e}}\right)-16 i b \sin ^{-1}\left(\frac{\sqrt{\frac{i \sqrt{e}}{c \sqrt{d}}+1}}{\sqrt{2}}\right) \tan ^{-1}\left(\frac{\left(i \sqrt{d} c+\sqrt{e}\right) \cot \left(\frac{1}{4} \left(2 \csc ^{-1}(c x)+\pi \right)\right)}{\sqrt{d c^2+e}}\right)-8 b \sin ^{-1}\left(\frac{\sqrt{1-\frac{i \sqrt{e}}{c \sqrt{d}}}}{\sqrt{2}}\right) \log \left(\frac{e^{-i \csc ^{-1}(c x)} \left(\sqrt{e}-\sqrt{d c^2+e}\right)}{c \sqrt{d}}+1\right)-2 b \pi  \log \left(\frac{e^{-i \csc ^{-1}(c x)} \left(\sqrt{e}-\sqrt{d c^2+e}\right)}{c \sqrt{d}}+1\right)-8 b \sin ^{-1}\left(\frac{\sqrt{\frac{i \sqrt{e}}{c \sqrt{d}}+1}}{\sqrt{2}}\right) \log \left(\frac{e^{-i \csc ^{-1}(c x)} \left(\sqrt{d c^2+e}-\sqrt{e}\right)}{c \sqrt{d}}+1\right)-2 b \pi  \log \left(\frac{e^{-i \csc ^{-1}(c x)} \left(\sqrt{d c^2+e}-\sqrt{e}\right)}{c \sqrt{d}}+1\right)+8 b \sin ^{-1}\left(\frac{\sqrt{\frac{i \sqrt{e}}{c \sqrt{d}}+1}}{\sqrt{2}}\right) \log \left(1-\frac{\left(\sqrt{e}+\sqrt{d c^2+e}\right) e^{-i \csc ^{-1}(c x)}}{c \sqrt{d}}\right)-2 b \pi  \log \left(1-\frac{\left(\sqrt{e}+\sqrt{d c^2+e}\right) e^{-i \csc ^{-1}(c x)}}{c \sqrt{d}}\right)+8 b \sin ^{-1}\left(\frac{\sqrt{1-\frac{i \sqrt{e}}{c \sqrt{d}}}}{\sqrt{2}}\right) \log \left(\frac{e^{-i \csc ^{-1}(c x)} \left(\sqrt{e}+\sqrt{d c^2+e}\right)}{c \sqrt{d}}+1\right)-2 b \pi  \log \left(\frac{e^{-i \csc ^{-1}(c x)} \left(\sqrt{e}+\sqrt{d c^2+e}\right)}{c \sqrt{d}}+1\right)+2 b \pi  \log \left(\sqrt{e}-\frac{i \sqrt{d}}{x}\right)+2 b \pi  \log \left(\frac{i \sqrt{d}}{x}+\sqrt{e}\right)+\frac{2 b \sqrt{e} \log \left(\frac{2 \sqrt{d} \sqrt{e} \left(c \left(-i \sqrt{d} c-\sqrt{-d c^2-e} \sqrt{1-\frac{1}{c^2 x^2}}\right) x+\sqrt{e}\right)}{\sqrt{-d c^2-e} \left(i \sqrt{e} x+\sqrt{d}\right)}\right)}{\sqrt{-d c^2-e}}+\frac{2 b \sqrt{e} \log \left(\frac{2 \sqrt{d} \sqrt{e} \left(c \left(\sqrt{-d c^2-e} \sqrt{1-\frac{1}{c^2 x^2}}-i c \sqrt{d}\right) x-\sqrt{e}\right)}{\sqrt{-d c^2-e} \left(\sqrt{d}-i \sqrt{e} x\right)}\right)}{\sqrt{-d c^2-e}}+4 a \log \left(e x^2+d\right)+4 i b \text{Li}_2\left(\frac{\left(\sqrt{e}-\sqrt{d c^2+e}\right) e^{-i \csc ^{-1}(c x)}}{c \sqrt{d}}\right)+4 i b \text{Li}_2\left(\frac{\left(\sqrt{d c^2+e}-\sqrt{e}\right) e^{-i \csc ^{-1}(c x)}}{c \sqrt{d}}\right)+4 i b \text{Li}_2\left(-\frac{\left(\sqrt{e}+\sqrt{d c^2+e}\right) e^{-i \csc ^{-1}(c x)}}{c \sqrt{d}}\right)+4 i b \text{Li}_2\left(\frac{\left(\sqrt{e}+\sqrt{d c^2+e}\right) e^{-i \csc ^{-1}(c x)}}{c \sqrt{d}}\right)+4 i b \text{Li}_2\left(e^{2 i \csc ^{-1}(c x)}\right)+\frac{4 a d}{e x^2+d}+i b \pi ^2}{8 e^2}","\frac{\left(a+b \csc ^{-1}(c x)\right) \log \left(1-\frac{i c \sqrt{-d} e^{i \csc ^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right)}{2 e^2}+\frac{\left(a+b \csc ^{-1}(c x)\right) \log \left(1+\frac{i c \sqrt{-d} e^{i \csc ^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right)}{2 e^2}+\frac{\left(a+b \csc ^{-1}(c x)\right) \log \left(1-\frac{i c \sqrt{-d} e^{i \csc ^{-1}(c x)}}{\sqrt{c^2 d+e}+\sqrt{e}}\right)}{2 e^2}+\frac{\left(a+b \csc ^{-1}(c x)\right) \log \left(1+\frac{i c \sqrt{-d} e^{i \csc ^{-1}(c x)}}{\sqrt{c^2 d+e}+\sqrt{e}}\right)}{2 e^2}+\frac{-a-b \csc ^{-1}(c x)}{2 e \left(\frac{d}{x^2}+e\right)}-\frac{\log \left(1-e^{2 i \csc ^{-1}(c x)}\right) \left(a+b \csc ^{-1}(c x)\right)}{e^2}+\frac{b \tan ^{-1}\left(\frac{\sqrt{c^2 d+e}}{c \sqrt{e} x \sqrt{1-\frac{1}{c^2 x^2}}}\right)}{2 e^{3/2} \sqrt{c^2 d+e}}-\frac{i b \text{Li}_2\left(-\frac{i c \sqrt{-d} e^{i \csc ^{-1}(c x)}}{\sqrt{e}-\sqrt{d c^2+e}}\right)}{2 e^2}-\frac{i b \text{Li}_2\left(\frac{i c \sqrt{-d} e^{i \csc ^{-1}(c x)}}{\sqrt{e}-\sqrt{d c^2+e}}\right)}{2 e^2}-\frac{i b \text{Li}_2\left(-\frac{i c \sqrt{-d} e^{i \csc ^{-1}(c x)}}{\sqrt{e}+\sqrt{d c^2+e}}\right)}{2 e^2}-\frac{i b \text{Li}_2\left(\frac{i c \sqrt{-d} e^{i \csc ^{-1}(c x)}}{\sqrt{e}+\sqrt{d c^2+e}}\right)}{2 e^2}+\frac{i b \text{Li}_2\left(e^{2 i \csc ^{-1}(c x)}\right)}{2 e^2}",1,"(I*b*Pi^2 + (4*a*d)/(d + e*x^2) - (4*I)*b*Pi*ArcCsc[c*x] + (2*b*Sqrt[d]*ArcCsc[c*x])/(Sqrt[d] - I*Sqrt[e]*x) + (2*b*Sqrt[d]*ArcCsc[c*x])/(Sqrt[d] + I*Sqrt[e]*x) + (8*I)*b*ArcCsc[c*x]^2 - 4*b*ArcSin[1/(c*x)] - (16*I)*b*ArcSin[Sqrt[1 - (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*ArcTan[(((-I)*c*Sqrt[d] + Sqrt[e])*Cot[(Pi + 2*ArcCsc[c*x])/4])/Sqrt[c^2*d + e]] - (16*I)*b*ArcSin[Sqrt[1 + (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*ArcTan[((I*c*Sqrt[d] + Sqrt[e])*Cot[(Pi + 2*ArcCsc[c*x])/4])/Sqrt[c^2*d + e]] - 2*b*Pi*Log[1 + (Sqrt[e] - Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] + 4*b*ArcCsc[c*x]*Log[1 + (Sqrt[e] - Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] - 8*b*ArcSin[Sqrt[1 - (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*Log[1 + (Sqrt[e] - Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] - 2*b*Pi*Log[1 + (-Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] + 4*b*ArcCsc[c*x]*Log[1 + (-Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] - 8*b*ArcSin[Sqrt[1 + (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*Log[1 + (-Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] - 2*b*Pi*Log[1 - (Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] + 4*b*ArcCsc[c*x]*Log[1 - (Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] + 8*b*ArcSin[Sqrt[1 + (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*Log[1 - (Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] - 2*b*Pi*Log[1 + (Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] + 4*b*ArcCsc[c*x]*Log[1 + (Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] + 8*b*ArcSin[Sqrt[1 - (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*Log[1 + (Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] - 8*b*ArcCsc[c*x]*Log[1 - E^((2*I)*ArcCsc[c*x])] + 2*b*Pi*Log[Sqrt[e] - (I*Sqrt[d])/x] + 2*b*Pi*Log[Sqrt[e] + (I*Sqrt[d])/x] + (2*b*Sqrt[e]*Log[(2*Sqrt[d]*Sqrt[e]*(Sqrt[e] + c*((-I)*c*Sqrt[d] - Sqrt[-(c^2*d) - e]*Sqrt[1 - 1/(c^2*x^2)])*x))/(Sqrt[-(c^2*d) - e]*(Sqrt[d] + I*Sqrt[e]*x))])/Sqrt[-(c^2*d) - e] + (2*b*Sqrt[e]*Log[(2*Sqrt[d]*Sqrt[e]*(-Sqrt[e] + c*((-I)*c*Sqrt[d] + Sqrt[-(c^2*d) - e]*Sqrt[1 - 1/(c^2*x^2)])*x))/(Sqrt[-(c^2*d) - e]*(Sqrt[d] - I*Sqrt[e]*x))])/Sqrt[-(c^2*d) - e] + 4*a*Log[d + e*x^2] + (4*I)*b*PolyLog[2, (Sqrt[e] - Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] + (4*I)*b*PolyLog[2, (-Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] + (4*I)*b*PolyLog[2, -((Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x])))] + (4*I)*b*PolyLog[2, (Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] + (4*I)*b*PolyLog[2, E^((2*I)*ArcCsc[c*x])])/(8*e^2)","B",0
105,1,286,134,0.7834847,"\int \frac{x \left(a+b \csc ^{-1}(c x)\right)}{\left(d+e x^2\right)^2} \, dx","Integrate[(x*(a + b*ArcCsc[c*x]))/(d + e*x^2)^2,x]","-\frac{\frac{2 a}{d+e x^2}+\frac{b \sqrt{e} \log \left(\frac{4 i d e-4 c d \sqrt{e} x \left(c \sqrt{d}+i \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{c^2 (-d)-e}\right)}{b \sqrt{c^2 (-d)-e} \left(\sqrt{d}-i \sqrt{e} x\right)}\right)}{d \sqrt{c^2 (-d)-e}}+\frac{b \sqrt{e} \log \left(\frac{4 i \left(-d e+c d \sqrt{e} x \left(\sqrt{1-\frac{1}{c^2 x^2}} \sqrt{c^2 (-d)-e}+i c \sqrt{d}\right)\right)}{b \sqrt{c^2 (-d)-e} \left(\sqrt{d}+i \sqrt{e} x\right)}\right)}{d \sqrt{c^2 (-d)-e}}+\frac{2 b \csc ^{-1}(c x)}{d+e x^2}-\frac{2 b \sin ^{-1}\left(\frac{1}{c x}\right)}{d}}{4 e}","\frac{-a-b \csc ^{-1}(c x)}{2 e \left(d+e x^2\right)}-\frac{b c x \tan ^{-1}\left(\sqrt{c^2 x^2-1}\right)}{2 d e \sqrt{c^2 x^2}}+\frac{b c x \tan ^{-1}\left(\frac{\sqrt{e} \sqrt{c^2 x^2-1}}{\sqrt{c^2 d+e}}\right)}{2 d \sqrt{e} \sqrt{c^2 x^2} \sqrt{c^2 d+e}}",1,"-1/4*((2*a)/(d + e*x^2) + (2*b*ArcCsc[c*x])/(d + e*x^2) - (2*b*ArcSin[1/(c*x)])/d + (b*Sqrt[e]*Log[((4*I)*d*e - 4*c*d*Sqrt[e]*(c*Sqrt[d] + I*Sqrt[-(c^2*d) - e]*Sqrt[1 - 1/(c^2*x^2)])*x)/(b*Sqrt[-(c^2*d) - e]*(Sqrt[d] - I*Sqrt[e]*x))])/(d*Sqrt[-(c^2*d) - e]) + (b*Sqrt[e]*Log[((4*I)*(-(d*e) + c*d*Sqrt[e]*(I*c*Sqrt[d] + Sqrt[-(c^2*d) - e]*Sqrt[1 - 1/(c^2*x^2)])*x))/(b*Sqrt[-(c^2*d) - e]*(Sqrt[d] + I*Sqrt[e]*x))])/(d*Sqrt[-(c^2*d) - e]))/e","C",1
106,0,0,566,43.451731,"\int \frac{a+b \csc ^{-1}(c x)}{x \left(d+e x^2\right)^2} \, dx","Integrate[(a + b*ArcCsc[c*x])/(x*(d + e*x^2)^2),x]","\int \frac{a+b \csc ^{-1}(c x)}{x \left(d+e x^2\right)^2} \, dx","-\frac{\left(a+b \csc ^{-1}(c x)\right) \log \left(1-\frac{i c \sqrt{-d} e^{i \csc ^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right)}{2 d^2}-\frac{\left(a+b \csc ^{-1}(c x)\right) \log \left(1+\frac{i c \sqrt{-d} e^{i \csc ^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right)}{2 d^2}-\frac{\left(a+b \csc ^{-1}(c x)\right) \log \left(1-\frac{i c \sqrt{-d} e^{i \csc ^{-1}(c x)}}{\sqrt{c^2 d+e}+\sqrt{e}}\right)}{2 d^2}-\frac{\left(a+b \csc ^{-1}(c x)\right) \log \left(1+\frac{i c \sqrt{-d} e^{i \csc ^{-1}(c x)}}{\sqrt{c^2 d+e}+\sqrt{e}}\right)}{2 d^2}-\frac{e \left(a+b \csc ^{-1}(c x)\right)}{2 d^2 \left(\frac{d}{x^2}+e\right)}+\frac{i \left(a+b \csc ^{-1}(c x)\right)^2}{2 b d^2}+\frac{i b \text{Li}_2\left(-\frac{i c \sqrt{-d} e^{i \csc ^{-1}(c x)}}{\sqrt{e}-\sqrt{d c^2+e}}\right)}{2 d^2}+\frac{i b \text{Li}_2\left(\frac{i c \sqrt{-d} e^{i \csc ^{-1}(c x)}}{\sqrt{e}-\sqrt{d c^2+e}}\right)}{2 d^2}+\frac{i b \text{Li}_2\left(-\frac{i c \sqrt{-d} e^{i \csc ^{-1}(c x)}}{\sqrt{e}+\sqrt{d c^2+e}}\right)}{2 d^2}+\frac{i b \text{Li}_2\left(\frac{i c \sqrt{-d} e^{i \csc ^{-1}(c x)}}{\sqrt{e}+\sqrt{d c^2+e}}\right)}{2 d^2}+\frac{b \sqrt{e} \tan ^{-1}\left(\frac{\sqrt{c^2 d+e}}{c \sqrt{e} x \sqrt{1-\frac{1}{c^2 x^2}}}\right)}{2 d^2 \sqrt{c^2 d+e}}",1,"Integrate[(a + b*ArcCsc[c*x])/(x*(d + e*x^2)^2), x]","F",-1
107,1,1634,803,6.1652024,"\int \frac{x^4 \left(a+b \csc ^{-1}(c x)\right)}{\left(d+e x^2\right)^2} \, dx","Integrate[(x^4*(a + b*ArcCsc[c*x]))/(d + e*x^2)^2,x]","\frac{a d x}{2 e^2 \left(e x^2+d\right)}+\frac{a x}{e^2}-\frac{3 a \sqrt{d} \tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right)}{2 e^{5/2}}+b \left(-\frac{d \left(\frac{i \left(\frac{\sin ^{-1}\left(\frac{1}{c x}\right)}{\sqrt{e}}-\frac{\log \left(\frac{2 \sqrt{d} \sqrt{e} \left(c \left(-i \sqrt{d} c-\sqrt{-d c^2-e} \sqrt{1-\frac{1}{c^2 x^2}}\right) x+\sqrt{e}\right)}{\sqrt{-d c^2-e} \left(i \sqrt{e} x+\sqrt{d}\right)}\right)}{\sqrt{-d c^2-e}}\right)}{\sqrt{d}}-\frac{\csc ^{-1}(c x)}{e x-i \sqrt{d} \sqrt{e}}\right)}{4 e^2}-\frac{d \left(-\frac{\csc ^{-1}(c x)}{e x+i \sqrt{d} \sqrt{e}}-\frac{i \left(\frac{\sin ^{-1}\left(\frac{1}{c x}\right)}{\sqrt{e}}-\frac{\log \left(\frac{2 \sqrt{d} \sqrt{e} \left(c \left(\sqrt{-d c^2-e} \sqrt{1-\frac{1}{c^2 x^2}}-i c \sqrt{d}\right) x-\sqrt{e}\right)}{\sqrt{-d c^2-e} \left(\sqrt{d}-i \sqrt{e} x\right)}\right)}{\sqrt{-d c^2-e}}\right)}{\sqrt{d}}\right)}{4 e^2}+\frac{3 \sqrt{d} \left(8 \csc ^{-1}(c x)^2-8 i \log \left(\frac{e^{-i \csc ^{-1}(c x)} \left(\sqrt{e}-\sqrt{d c^2+e}\right)}{c \sqrt{d}}+1\right) \csc ^{-1}(c x)-8 i \log \left(\frac{e^{-i \csc ^{-1}(c x)} \left(\sqrt{e}+\sqrt{d c^2+e}\right)}{c \sqrt{d}}+1\right) \csc ^{-1}(c x)+8 i \log \left(1-e^{2 i \csc ^{-1}(c x)}\right) \csc ^{-1}(c x)-4 \pi  \csc ^{-1}(c x)-32 \sin ^{-1}\left(\frac{\sqrt{1-\frac{i \sqrt{e}}{c \sqrt{d}}}}{\sqrt{2}}\right) \tan ^{-1}\left(\frac{\left(\sqrt{e}-i c \sqrt{d}\right) \cot \left(\frac{1}{4} \left(2 \csc ^{-1}(c x)+\pi \right)\right)}{\sqrt{d c^2+e}}\right)+16 i \sin ^{-1}\left(\frac{\sqrt{1-\frac{i \sqrt{e}}{c \sqrt{d}}}}{\sqrt{2}}\right) \log \left(\frac{e^{-i \csc ^{-1}(c x)} \left(\sqrt{e}-\sqrt{d c^2+e}\right)}{c \sqrt{d}}+1\right)+4 i \pi  \log \left(\frac{e^{-i \csc ^{-1}(c x)} \left(\sqrt{e}-\sqrt{d c^2+e}\right)}{c \sqrt{d}}+1\right)-16 i \sin ^{-1}\left(\frac{\sqrt{1-\frac{i \sqrt{e}}{c \sqrt{d}}}}{\sqrt{2}}\right) \log \left(\frac{e^{-i \csc ^{-1}(c x)} \left(\sqrt{e}+\sqrt{d c^2+e}\right)}{c \sqrt{d}}+1\right)+4 i \pi  \log \left(\frac{e^{-i \csc ^{-1}(c x)} \left(\sqrt{e}+\sqrt{d c^2+e}\right)}{c \sqrt{d}}+1\right)-4 i \pi  \log \left(\frac{i \sqrt{d}}{x}+\sqrt{e}\right)+8 \text{Li}_2\left(\frac{\left(\sqrt{d c^2+e}-\sqrt{e}\right) e^{-i \csc ^{-1}(c x)}}{c \sqrt{d}}\right)+8 \text{Li}_2\left(-\frac{\left(\sqrt{e}+\sqrt{d c^2+e}\right) e^{-i \csc ^{-1}(c x)}}{c \sqrt{d}}\right)+4 \text{Li}_2\left(e^{2 i \csc ^{-1}(c x)}\right)+\pi ^2\right)}{32 e^{5/2}}-\frac{3 \sqrt{d} \left(8 \csc ^{-1}(c x)^2-8 i \log \left(\frac{e^{-i \csc ^{-1}(c x)} \left(\sqrt{d c^2+e}-\sqrt{e}\right)}{c \sqrt{d}}+1\right) \csc ^{-1}(c x)-8 i \log \left(1-\frac{\left(\sqrt{e}+\sqrt{d c^2+e}\right) e^{-i \csc ^{-1}(c x)}}{c \sqrt{d}}\right) \csc ^{-1}(c x)+8 i \log \left(1-e^{2 i \csc ^{-1}(c x)}\right) \csc ^{-1}(c x)-4 \pi  \csc ^{-1}(c x)-32 \sin ^{-1}\left(\frac{\sqrt{\frac{i \sqrt{e}}{c \sqrt{d}}+1}}{\sqrt{2}}\right) \tan ^{-1}\left(\frac{\left(i \sqrt{d} c+\sqrt{e}\right) \cot \left(\frac{1}{4} \left(2 \csc ^{-1}(c x)+\pi \right)\right)}{\sqrt{d c^2+e}}\right)+16 i \sin ^{-1}\left(\frac{\sqrt{\frac{i \sqrt{e}}{c \sqrt{d}}+1}}{\sqrt{2}}\right) \log \left(\frac{e^{-i \csc ^{-1}(c x)} \left(\sqrt{d c^2+e}-\sqrt{e}\right)}{c \sqrt{d}}+1\right)+4 i \pi  \log \left(\frac{e^{-i \csc ^{-1}(c x)} \left(\sqrt{d c^2+e}-\sqrt{e}\right)}{c \sqrt{d}}+1\right)-16 i \sin ^{-1}\left(\frac{\sqrt{\frac{i \sqrt{e}}{c \sqrt{d}}+1}}{\sqrt{2}}\right) \log \left(1-\frac{\left(\sqrt{e}+\sqrt{d c^2+e}\right) e^{-i \csc ^{-1}(c x)}}{c \sqrt{d}}\right)+4 i \pi  \log \left(1-\frac{\left(\sqrt{e}+\sqrt{d c^2+e}\right) e^{-i \csc ^{-1}(c x)}}{c \sqrt{d}}\right)-4 i \pi  \log \left(\sqrt{e}-\frac{i \sqrt{d}}{x}\right)+8 \text{Li}_2\left(\frac{\left(\sqrt{e}-\sqrt{d c^2+e}\right) e^{-i \csc ^{-1}(c x)}}{c \sqrt{d}}\right)+8 \text{Li}_2\left(\frac{\left(\sqrt{e}+\sqrt{d c^2+e}\right) e^{-i \csc ^{-1}(c x)}}{c \sqrt{d}}\right)+4 \text{Li}_2\left(e^{2 i \csc ^{-1}(c x)}\right)+\pi ^2\right)}{32 e^{5/2}}+\frac{\frac{1}{2} \csc ^{-1}(c x) \cot \left(\frac{1}{2} \csc ^{-1}(c x)\right)+\log \left(\cos \left(\frac{1}{2} \csc ^{-1}(c x)\right)\right)-\log \left(\sin \left(\frac{1}{2} \csc ^{-1}(c x)\right)\right)+\frac{1}{2} \csc ^{-1}(c x) \tan \left(\frac{1}{2} \csc ^{-1}(c x)\right)}{c e^2}\right)","\frac{x \left(a+b \csc ^{-1}(c x)\right)}{e^2}-\frac{3 \sqrt{-d} \log \left(1-\frac{i c \sqrt{-d} e^{i \csc ^{-1}(c x)}}{\sqrt{e}-\sqrt{d c^2+e}}\right) \left(a+b \csc ^{-1}(c x)\right)}{4 e^{5/2}}+\frac{3 \sqrt{-d} \log \left(\frac{i \sqrt{-d} e^{i \csc ^{-1}(c x)} c}{\sqrt{e}-\sqrt{d c^2+e}}+1\right) \left(a+b \csc ^{-1}(c x)\right)}{4 e^{5/2}}-\frac{3 \sqrt{-d} \log \left(1-\frac{i c \sqrt{-d} e^{i \csc ^{-1}(c x)}}{\sqrt{e}+\sqrt{d c^2+e}}\right) \left(a+b \csc ^{-1}(c x)\right)}{4 e^{5/2}}+\frac{3 \sqrt{-d} \log \left(\frac{i \sqrt{-d} e^{i \csc ^{-1}(c x)} c}{\sqrt{e}+\sqrt{d c^2+e}}+1\right) \left(a+b \csc ^{-1}(c x)\right)}{4 e^{5/2}}-\frac{d \left(a+b \csc ^{-1}(c x)\right)}{4 e^2 \left(\sqrt{-d} \sqrt{e}-\frac{d}{x}\right)}+\frac{d \left(a+b \csc ^{-1}(c x)\right)}{4 e^2 \left(\frac{d}{x}+\sqrt{-d} \sqrt{e}\right)}+\frac{b \tanh ^{-1}\left(\sqrt{1-\frac{1}{c^2 x^2}}\right)}{c e^2}+\frac{b \sqrt{d} \tanh ^{-1}\left(\frac{c^2 d-\frac{\sqrt{-d} \sqrt{e}}{x}}{c \sqrt{d} \sqrt{d c^2+e} \sqrt{1-\frac{1}{c^2 x^2}}}\right)}{4 e^2 \sqrt{d c^2+e}}+\frac{b \sqrt{d} \tanh ^{-1}\left(\frac{d c^2+\frac{\sqrt{-d} \sqrt{e}}{x}}{c \sqrt{d} \sqrt{d c^2+e} \sqrt{1-\frac{1}{c^2 x^2}}}\right)}{4 e^2 \sqrt{d c^2+e}}-\frac{3 i b \sqrt{-d} \text{Li}_2\left(-\frac{i c \sqrt{-d} e^{i \csc ^{-1}(c x)}}{\sqrt{e}-\sqrt{d c^2+e}}\right)}{4 e^{5/2}}+\frac{3 i b \sqrt{-d} \text{Li}_2\left(\frac{i c \sqrt{-d} e^{i \csc ^{-1}(c x)}}{\sqrt{e}-\sqrt{d c^2+e}}\right)}{4 e^{5/2}}-\frac{3 i b \sqrt{-d} \text{Li}_2\left(-\frac{i c \sqrt{-d} e^{i \csc ^{-1}(c x)}}{\sqrt{e}+\sqrt{d c^2+e}}\right)}{4 e^{5/2}}+\frac{3 i b \sqrt{-d} \text{Li}_2\left(\frac{i c \sqrt{-d} e^{i \csc ^{-1}(c x)}}{\sqrt{e}+\sqrt{d c^2+e}}\right)}{4 e^{5/2}}",1,"(a*x)/e^2 + (a*d*x)/(2*e^2*(d + e*x^2)) - (3*a*Sqrt[d]*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(2*e^(5/2)) + b*(-1/4*(d*(-(ArcCsc[c*x]/((-I)*Sqrt[d]*Sqrt[e] + e*x)) + (I*(ArcSin[1/(c*x)]/Sqrt[e] - Log[(2*Sqrt[d]*Sqrt[e]*(Sqrt[e] + c*((-I)*c*Sqrt[d] - Sqrt[-(c^2*d) - e]*Sqrt[1 - 1/(c^2*x^2)])*x))/(Sqrt[-(c^2*d) - e]*(Sqrt[d] + I*Sqrt[e]*x))]/Sqrt[-(c^2*d) - e]))/Sqrt[d]))/e^2 - (d*(-(ArcCsc[c*x]/(I*Sqrt[d]*Sqrt[e] + e*x)) - (I*(ArcSin[1/(c*x)]/Sqrt[e] - Log[(2*Sqrt[d]*Sqrt[e]*(-Sqrt[e] + c*((-I)*c*Sqrt[d] + Sqrt[-(c^2*d) - e]*Sqrt[1 - 1/(c^2*x^2)])*x))/(Sqrt[-(c^2*d) - e]*(Sqrt[d] - I*Sqrt[e]*x))]/Sqrt[-(c^2*d) - e]))/Sqrt[d]))/(4*e^2) + (3*Sqrt[d]*(Pi^2 - 4*Pi*ArcCsc[c*x] + 8*ArcCsc[c*x]^2 - 32*ArcSin[Sqrt[1 - (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*ArcTan[(((-I)*c*Sqrt[d] + Sqrt[e])*Cot[(Pi + 2*ArcCsc[c*x])/4])/Sqrt[c^2*d + e]] + (4*I)*Pi*Log[1 + (Sqrt[e] - Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] - (8*I)*ArcCsc[c*x]*Log[1 + (Sqrt[e] - Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] + (16*I)*ArcSin[Sqrt[1 - (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*Log[1 + (Sqrt[e] - Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] + (4*I)*Pi*Log[1 + (Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] - (8*I)*ArcCsc[c*x]*Log[1 + (Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] - (16*I)*ArcSin[Sqrt[1 - (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*Log[1 + (Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] + (8*I)*ArcCsc[c*x]*Log[1 - E^((2*I)*ArcCsc[c*x])] - (4*I)*Pi*Log[Sqrt[e] + (I*Sqrt[d])/x] + 8*PolyLog[2, (-Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] + 8*PolyLog[2, -((Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x])))] + 4*PolyLog[2, E^((2*I)*ArcCsc[c*x])]))/(32*e^(5/2)) - (3*Sqrt[d]*(Pi^2 - 4*Pi*ArcCsc[c*x] + 8*ArcCsc[c*x]^2 - 32*ArcSin[Sqrt[1 + (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*ArcTan[((I*c*Sqrt[d] + Sqrt[e])*Cot[(Pi + 2*ArcCsc[c*x])/4])/Sqrt[c^2*d + e]] + (4*I)*Pi*Log[1 + (-Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] - (8*I)*ArcCsc[c*x]*Log[1 + (-Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] + (16*I)*ArcSin[Sqrt[1 + (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*Log[1 + (-Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] + (4*I)*Pi*Log[1 - (Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] - (8*I)*ArcCsc[c*x]*Log[1 - (Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] - (16*I)*ArcSin[Sqrt[1 + (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*Log[1 - (Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] + (8*I)*ArcCsc[c*x]*Log[1 - E^((2*I)*ArcCsc[c*x])] - (4*I)*Pi*Log[Sqrt[e] - (I*Sqrt[d])/x] + 8*PolyLog[2, (Sqrt[e] - Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] + 8*PolyLog[2, (Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] + 4*PolyLog[2, E^((2*I)*ArcCsc[c*x])]))/(32*e^(5/2)) + ((ArcCsc[c*x]*Cot[ArcCsc[c*x]/2])/2 + Log[Cos[ArcCsc[c*x]/2]] - Log[Sin[ArcCsc[c*x]/2]] + (ArcCsc[c*x]*Tan[ArcCsc[c*x]/2])/2)/(c*e^2))","B",0
108,1,1482,765,2.8013695,"\int \frac{x^2 \left(a+b \csc ^{-1}(c x)\right)}{\left(d+e x^2\right)^2} \, dx","Integrate[(x^2*(a + b*ArcCsc[c*x]))/(d + e*x^2)^2,x]","\frac{-\frac{4 a \sqrt{e} x}{e x^2+d}+\frac{4 a \tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right)}{\sqrt{d}}+b \left(\frac{2 i \log \left(\frac{e^{-i \csc ^{-1}(c x)} \left(\sqrt{e}-\sqrt{d c^2+e}\right)}{c \sqrt{d}}+1\right) \csc ^{-1}(c x)}{\sqrt{d}}-\frac{2 i \log \left(\frac{e^{-i \csc ^{-1}(c x)} \left(\sqrt{d c^2+e}-\sqrt{e}\right)}{c \sqrt{d}}+1\right) \csc ^{-1}(c x)}{\sqrt{d}}-\frac{2 i \log \left(1-\frac{\left(\sqrt{e}+\sqrt{d c^2+e}\right) e^{-i \csc ^{-1}(c x)}}{c \sqrt{d}}\right) \csc ^{-1}(c x)}{\sqrt{d}}+\frac{2 i \log \left(\frac{e^{-i \csc ^{-1}(c x)} \left(\sqrt{e}+\sqrt{d c^2+e}\right)}{c \sqrt{d}}+1\right) \csc ^{-1}(c x)}{\sqrt{d}}+\frac{2 \csc ^{-1}(c x)}{i \sqrt{d}-\sqrt{e} x}-\frac{2 \csc ^{-1}(c x)}{\sqrt{e} x+i \sqrt{d}}+\frac{8 \sin ^{-1}\left(\frac{\sqrt{1-\frac{i \sqrt{e}}{c \sqrt{d}}}}{\sqrt{2}}\right) \tan ^{-1}\left(\frac{\left(\sqrt{e}-i c \sqrt{d}\right) \cot \left(\frac{1}{4} \left(2 \csc ^{-1}(c x)+\pi \right)\right)}{\sqrt{d c^2+e}}\right)}{\sqrt{d}}-\frac{8 \sin ^{-1}\left(\frac{\sqrt{\frac{i \sqrt{e}}{c \sqrt{d}}+1}}{\sqrt{2}}\right) \tan ^{-1}\left(\frac{\left(i \sqrt{d} c+\sqrt{e}\right) \cot \left(\frac{1}{4} \left(2 \csc ^{-1}(c x)+\pi \right)\right)}{\sqrt{d c^2+e}}\right)}{\sqrt{d}}-\frac{4 i \sin ^{-1}\left(\frac{\sqrt{1-\frac{i \sqrt{e}}{c \sqrt{d}}}}{\sqrt{2}}\right) \log \left(\frac{e^{-i \csc ^{-1}(c x)} \left(\sqrt{e}-\sqrt{d c^2+e}\right)}{c \sqrt{d}}+1\right)}{\sqrt{d}}-\frac{i \pi  \log \left(\frac{e^{-i \csc ^{-1}(c x)} \left(\sqrt{e}-\sqrt{d c^2+e}\right)}{c \sqrt{d}}+1\right)}{\sqrt{d}}+\frac{4 i \sin ^{-1}\left(\frac{\sqrt{\frac{i \sqrt{e}}{c \sqrt{d}}+1}}{\sqrt{2}}\right) \log \left(\frac{e^{-i \csc ^{-1}(c x)} \left(\sqrt{d c^2+e}-\sqrt{e}\right)}{c \sqrt{d}}+1\right)}{\sqrt{d}}+\frac{i \pi  \log \left(\frac{e^{-i \csc ^{-1}(c x)} \left(\sqrt{d c^2+e}-\sqrt{e}\right)}{c \sqrt{d}}+1\right)}{\sqrt{d}}-\frac{4 i \sin ^{-1}\left(\frac{\sqrt{\frac{i \sqrt{e}}{c \sqrt{d}}+1}}{\sqrt{2}}\right) \log \left(1-\frac{\left(\sqrt{e}+\sqrt{d c^2+e}\right) e^{-i \csc ^{-1}(c x)}}{c \sqrt{d}}\right)}{\sqrt{d}}+\frac{i \pi  \log \left(1-\frac{\left(\sqrt{e}+\sqrt{d c^2+e}\right) e^{-i \csc ^{-1}(c x)}}{c \sqrt{d}}\right)}{\sqrt{d}}+\frac{4 i \sin ^{-1}\left(\frac{\sqrt{1-\frac{i \sqrt{e}}{c \sqrt{d}}}}{\sqrt{2}}\right) \log \left(\frac{e^{-i \csc ^{-1}(c x)} \left(\sqrt{e}+\sqrt{d c^2+e}\right)}{c \sqrt{d}}+1\right)}{\sqrt{d}}-\frac{i \pi  \log \left(\frac{e^{-i \csc ^{-1}(c x)} \left(\sqrt{e}+\sqrt{d c^2+e}\right)}{c \sqrt{d}}+1\right)}{\sqrt{d}}-\frac{i \pi  \log \left(\sqrt{e}-\frac{i \sqrt{d}}{x}\right)}{\sqrt{d}}+\frac{i \pi  \log \left(\frac{i \sqrt{d}}{x}+\sqrt{e}\right)}{\sqrt{d}}-\frac{2 i \sqrt{e} \log \left(\frac{2 \sqrt{d} \sqrt{e} \left(c \left(-i \sqrt{d} c-\sqrt{-d c^2-e} \sqrt{1-\frac{1}{c^2 x^2}}\right) x+\sqrt{e}\right)}{\sqrt{-d c^2-e} \left(i \sqrt{e} x+\sqrt{d}\right)}\right)}{\sqrt{d} \sqrt{-d c^2-e}}+\frac{2 i \sqrt{e} \log \left(\frac{2 \sqrt{d} \sqrt{e} \left(c \left(\sqrt{-d c^2-e} \sqrt{1-\frac{1}{c^2 x^2}}-i c \sqrt{d}\right) x-\sqrt{e}\right)}{\sqrt{-d c^2-e} \left(\sqrt{d}-i \sqrt{e} x\right)}\right)}{\sqrt{d} \sqrt{-d c^2-e}}+\frac{2 \text{Li}_2\left(\frac{\left(\sqrt{e}-\sqrt{d c^2+e}\right) e^{-i \csc ^{-1}(c x)}}{c \sqrt{d}}\right)}{\sqrt{d}}-\frac{2 \text{Li}_2\left(\frac{\left(\sqrt{d c^2+e}-\sqrt{e}\right) e^{-i \csc ^{-1}(c x)}}{c \sqrt{d}}\right)}{\sqrt{d}}-\frac{2 \text{Li}_2\left(-\frac{\left(\sqrt{e}+\sqrt{d c^2+e}\right) e^{-i \csc ^{-1}(c x)}}{c \sqrt{d}}\right)}{\sqrt{d}}+\frac{2 \text{Li}_2\left(\frac{\left(\sqrt{e}+\sqrt{d c^2+e}\right) e^{-i \csc ^{-1}(c x)}}{c \sqrt{d}}\right)}{\sqrt{d}}\right)}{8 e^{3/2}}","-\frac{\left(a+b \csc ^{-1}(c x)\right) \log \left(1-\frac{i c \sqrt{-d} e^{i \csc ^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right)}{4 \sqrt{-d} e^{3/2}}+\frac{\left(a+b \csc ^{-1}(c x)\right) \log \left(1+\frac{i c \sqrt{-d} e^{i \csc ^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right)}{4 \sqrt{-d} e^{3/2}}-\frac{\left(a+b \csc ^{-1}(c x)\right) \log \left(1-\frac{i c \sqrt{-d} e^{i \csc ^{-1}(c x)}}{\sqrt{c^2 d+e}+\sqrt{e}}\right)}{4 \sqrt{-d} e^{3/2}}+\frac{\left(a+b \csc ^{-1}(c x)\right) \log \left(1+\frac{i c \sqrt{-d} e^{i \csc ^{-1}(c x)}}{\sqrt{c^2 d+e}+\sqrt{e}}\right)}{4 \sqrt{-d} e^{3/2}}+\frac{a+b \csc ^{-1}(c x)}{4 e \left(\sqrt{-d} \sqrt{e}-\frac{d}{x}\right)}-\frac{a+b \csc ^{-1}(c x)}{4 e \left(\sqrt{-d} \sqrt{e}+\frac{d}{x}\right)}-\frac{i b \text{Li}_2\left(-\frac{i c \sqrt{-d} e^{i \csc ^{-1}(c x)}}{\sqrt{e}-\sqrt{d c^2+e}}\right)}{4 \sqrt{-d} e^{3/2}}+\frac{i b \text{Li}_2\left(\frac{i c \sqrt{-d} e^{i \csc ^{-1}(c x)}}{\sqrt{e}-\sqrt{d c^2+e}}\right)}{4 \sqrt{-d} e^{3/2}}-\frac{i b \text{Li}_2\left(-\frac{i c \sqrt{-d} e^{i \csc ^{-1}(c x)}}{\sqrt{e}+\sqrt{d c^2+e}}\right)}{4 \sqrt{-d} e^{3/2}}+\frac{i b \text{Li}_2\left(\frac{i c \sqrt{-d} e^{i \csc ^{-1}(c x)}}{\sqrt{e}+\sqrt{d c^2+e}}\right)}{4 \sqrt{-d} e^{3/2}}-\frac{b \tanh ^{-1}\left(\frac{c^2 d-\frac{\sqrt{-d} \sqrt{e}}{x}}{c \sqrt{d} \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{c^2 d+e}}\right)}{4 \sqrt{d} e \sqrt{c^2 d+e}}-\frac{b \tanh ^{-1}\left(\frac{c^2 d+\frac{\sqrt{-d} \sqrt{e}}{x}}{c \sqrt{d} \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{c^2 d+e}}\right)}{4 \sqrt{d} e \sqrt{c^2 d+e}}",1,"((-4*a*Sqrt[e]*x)/(d + e*x^2) + (4*a*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/Sqrt[d] + b*((2*ArcCsc[c*x])/(I*Sqrt[d] - Sqrt[e]*x) - (2*ArcCsc[c*x])/(I*Sqrt[d] + Sqrt[e]*x) + (8*ArcSin[Sqrt[1 - (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*ArcTan[(((-I)*c*Sqrt[d] + Sqrt[e])*Cot[(Pi + 2*ArcCsc[c*x])/4])/Sqrt[c^2*d + e]])/Sqrt[d] - (8*ArcSin[Sqrt[1 + (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*ArcTan[((I*c*Sqrt[d] + Sqrt[e])*Cot[(Pi + 2*ArcCsc[c*x])/4])/Sqrt[c^2*d + e]])/Sqrt[d] - (I*Pi*Log[1 + (Sqrt[e] - Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))])/Sqrt[d] + ((2*I)*ArcCsc[c*x]*Log[1 + (Sqrt[e] - Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))])/Sqrt[d] - ((4*I)*ArcSin[Sqrt[1 - (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*Log[1 + (Sqrt[e] - Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))])/Sqrt[d] + (I*Pi*Log[1 + (-Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))])/Sqrt[d] - ((2*I)*ArcCsc[c*x]*Log[1 + (-Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))])/Sqrt[d] + ((4*I)*ArcSin[Sqrt[1 + (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*Log[1 + (-Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))])/Sqrt[d] + (I*Pi*Log[1 - (Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))])/Sqrt[d] - ((2*I)*ArcCsc[c*x]*Log[1 - (Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))])/Sqrt[d] - ((4*I)*ArcSin[Sqrt[1 + (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*Log[1 - (Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))])/Sqrt[d] - (I*Pi*Log[1 + (Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))])/Sqrt[d] + ((2*I)*ArcCsc[c*x]*Log[1 + (Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))])/Sqrt[d] + ((4*I)*ArcSin[Sqrt[1 - (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*Log[1 + (Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))])/Sqrt[d] - (I*Pi*Log[Sqrt[e] - (I*Sqrt[d])/x])/Sqrt[d] + (I*Pi*Log[Sqrt[e] + (I*Sqrt[d])/x])/Sqrt[d] - ((2*I)*Sqrt[e]*Log[(2*Sqrt[d]*Sqrt[e]*(Sqrt[e] + c*((-I)*c*Sqrt[d] - Sqrt[-(c^2*d) - e]*Sqrt[1 - 1/(c^2*x^2)])*x))/(Sqrt[-(c^2*d) - e]*(Sqrt[d] + I*Sqrt[e]*x))])/(Sqrt[d]*Sqrt[-(c^2*d) - e]) + ((2*I)*Sqrt[e]*Log[(2*Sqrt[d]*Sqrt[e]*(-Sqrt[e] + c*((-I)*c*Sqrt[d] + Sqrt[-(c^2*d) - e]*Sqrt[1 - 1/(c^2*x^2)])*x))/(Sqrt[-(c^2*d) - e]*(Sqrt[d] - I*Sqrt[e]*x))])/(Sqrt[d]*Sqrt[-(c^2*d) - e]) + (2*PolyLog[2, (Sqrt[e] - Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))])/Sqrt[d] - (2*PolyLog[2, (-Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))])/Sqrt[d] - (2*PolyLog[2, -((Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x])))])/Sqrt[d] + (2*PolyLog[2, (Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))])/Sqrt[d]))/(8*e^(3/2))","A",0
109,1,1477,762,2.8734869,"\int \frac{a+b \csc ^{-1}(c x)}{\left(d+e x^2\right)^2} \, dx","Integrate[(a + b*ArcCsc[c*x])/(d + e*x^2)^2,x]","\frac{1}{2} \left(\frac{a x}{d^2+e x^2 d}+\frac{a \tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right)}{d^{3/2} \sqrt{e}}+\frac{b \left(\frac{2 i \log \left(\frac{e^{-i \csc ^{-1}(c x)} \left(\sqrt{e}-\sqrt{d c^2+e}\right)}{c \sqrt{d}}+1\right) \csc ^{-1}(c x)}{\sqrt{e}}-\frac{2 i \log \left(\frac{e^{-i \csc ^{-1}(c x)} \left(\sqrt{d c^2+e}-\sqrt{e}\right)}{c \sqrt{d}}+1\right) \csc ^{-1}(c x)}{\sqrt{e}}-\frac{2 i \log \left(1-\frac{\left(\sqrt{e}+\sqrt{d c^2+e}\right) e^{-i \csc ^{-1}(c x)}}{c \sqrt{d}}\right) \csc ^{-1}(c x)}{\sqrt{e}}+\frac{2 i \log \left(\frac{e^{-i \csc ^{-1}(c x)} \left(\sqrt{e}+\sqrt{d c^2+e}\right)}{c \sqrt{d}}+1\right) \csc ^{-1}(c x)}{\sqrt{e}}+\frac{2 \sqrt{d} \csc ^{-1}(c x)}{e x-i \sqrt{d} \sqrt{e}}+\frac{2 \sqrt{d} \csc ^{-1}(c x)}{e x+i \sqrt{d} \sqrt{e}}+\frac{8 \sin ^{-1}\left(\frac{\sqrt{1-\frac{i \sqrt{e}}{c \sqrt{d}}}}{\sqrt{2}}\right) \tan ^{-1}\left(\frac{\left(\sqrt{e}-i c \sqrt{d}\right) \cot \left(\frac{1}{4} \left(2 \csc ^{-1}(c x)+\pi \right)\right)}{\sqrt{d c^2+e}}\right)}{\sqrt{e}}-\frac{8 \sin ^{-1}\left(\frac{\sqrt{\frac{i \sqrt{e}}{c \sqrt{d}}+1}}{\sqrt{2}}\right) \tan ^{-1}\left(\frac{\left(i \sqrt{d} c+\sqrt{e}\right) \cot \left(\frac{1}{4} \left(2 \csc ^{-1}(c x)+\pi \right)\right)}{\sqrt{d c^2+e}}\right)}{\sqrt{e}}-\frac{4 i \sin ^{-1}\left(\frac{\sqrt{1-\frac{i \sqrt{e}}{c \sqrt{d}}}}{\sqrt{2}}\right) \log \left(\frac{e^{-i \csc ^{-1}(c x)} \left(\sqrt{e}-\sqrt{d c^2+e}\right)}{c \sqrt{d}}+1\right)}{\sqrt{e}}-\frac{i \pi  \log \left(\frac{e^{-i \csc ^{-1}(c x)} \left(\sqrt{e}-\sqrt{d c^2+e}\right)}{c \sqrt{d}}+1\right)}{\sqrt{e}}+\frac{4 i \sin ^{-1}\left(\frac{\sqrt{\frac{i \sqrt{e}}{c \sqrt{d}}+1}}{\sqrt{2}}\right) \log \left(\frac{e^{-i \csc ^{-1}(c x)} \left(\sqrt{d c^2+e}-\sqrt{e}\right)}{c \sqrt{d}}+1\right)}{\sqrt{e}}+\frac{i \pi  \log \left(\frac{e^{-i \csc ^{-1}(c x)} \left(\sqrt{d c^2+e}-\sqrt{e}\right)}{c \sqrt{d}}+1\right)}{\sqrt{e}}-\frac{4 i \sin ^{-1}\left(\frac{\sqrt{\frac{i \sqrt{e}}{c \sqrt{d}}+1}}{\sqrt{2}}\right) \log \left(1-\frac{\left(\sqrt{e}+\sqrt{d c^2+e}\right) e^{-i \csc ^{-1}(c x)}}{c \sqrt{d}}\right)}{\sqrt{e}}+\frac{i \pi  \log \left(1-\frac{\left(\sqrt{e}+\sqrt{d c^2+e}\right) e^{-i \csc ^{-1}(c x)}}{c \sqrt{d}}\right)}{\sqrt{e}}+\frac{4 i \sin ^{-1}\left(\frac{\sqrt{1-\frac{i \sqrt{e}}{c \sqrt{d}}}}{\sqrt{2}}\right) \log \left(\frac{e^{-i \csc ^{-1}(c x)} \left(\sqrt{e}+\sqrt{d c^2+e}\right)}{c \sqrt{d}}+1\right)}{\sqrt{e}}-\frac{i \pi  \log \left(\frac{e^{-i \csc ^{-1}(c x)} \left(\sqrt{e}+\sqrt{d c^2+e}\right)}{c \sqrt{d}}+1\right)}{\sqrt{e}}-\frac{i \pi  \log \left(\sqrt{e}-\frac{i \sqrt{d}}{x}\right)}{\sqrt{e}}+\frac{i \pi  \log \left(\frac{i \sqrt{d}}{x}+\sqrt{e}\right)}{\sqrt{e}}+\frac{2 i \log \left(\frac{2 \sqrt{d} \sqrt{e} \left(c \left(-i \sqrt{d} c-\sqrt{-d c^2-e} \sqrt{1-\frac{1}{c^2 x^2}}\right) x+\sqrt{e}\right)}{\sqrt{-d c^2-e} \left(i \sqrt{e} x+\sqrt{d}\right)}\right)}{\sqrt{-d c^2-e}}-\frac{2 i \log \left(\frac{2 \sqrt{d} \sqrt{e} \left(c \left(\sqrt{-d c^2-e} \sqrt{1-\frac{1}{c^2 x^2}}-i c \sqrt{d}\right) x-\sqrt{e}\right)}{\sqrt{-d c^2-e} \left(\sqrt{d}-i \sqrt{e} x\right)}\right)}{\sqrt{-d c^2-e}}+\frac{2 \text{Li}_2\left(\frac{\left(\sqrt{e}-\sqrt{d c^2+e}\right) e^{-i \csc ^{-1}(c x)}}{c \sqrt{d}}\right)}{\sqrt{e}}-\frac{2 \text{Li}_2\left(\frac{\left(\sqrt{d c^2+e}-\sqrt{e}\right) e^{-i \csc ^{-1}(c x)}}{c \sqrt{d}}\right)}{\sqrt{e}}-\frac{2 \text{Li}_2\left(-\frac{\left(\sqrt{e}+\sqrt{d c^2+e}\right) e^{-i \csc ^{-1}(c x)}}{c \sqrt{d}}\right)}{\sqrt{e}}+\frac{2 \text{Li}_2\left(\frac{\left(\sqrt{e}+\sqrt{d c^2+e}\right) e^{-i \csc ^{-1}(c x)}}{c \sqrt{d}}\right)}{\sqrt{e}}\right)}{4 d^{3/2}}\right)","\frac{\left(a+b \csc ^{-1}(c x)\right) \log \left(1-\frac{i c \sqrt{-d} e^{i \csc ^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right)}{4 (-d)^{3/2} \sqrt{e}}-\frac{\left(a+b \csc ^{-1}(c x)\right) \log \left(1+\frac{i c \sqrt{-d} e^{i \csc ^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right)}{4 (-d)^{3/2} \sqrt{e}}+\frac{\left(a+b \csc ^{-1}(c x)\right) \log \left(1-\frac{i c \sqrt{-d} e^{i \csc ^{-1}(c x)}}{\sqrt{c^2 d+e}+\sqrt{e}}\right)}{4 (-d)^{3/2} \sqrt{e}}-\frac{\left(a+b \csc ^{-1}(c x)\right) \log \left(1+\frac{i c \sqrt{-d} e^{i \csc ^{-1}(c x)}}{\sqrt{c^2 d+e}+\sqrt{e}}\right)}{4 (-d)^{3/2} \sqrt{e}}+\frac{-a-b \csc ^{-1}(c x)}{4 d \left(\sqrt{-d} \sqrt{e}-\frac{d}{x}\right)}+\frac{a+b \csc ^{-1}(c x)}{4 d \left(\sqrt{-d} \sqrt{e}+\frac{d}{x}\right)}+\frac{b \tanh ^{-1}\left(\frac{c^2 d-\frac{\sqrt{-d} \sqrt{e}}{x}}{c \sqrt{d} \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{c^2 d+e}}\right)}{4 d^{3/2} \sqrt{c^2 d+e}}+\frac{b \tanh ^{-1}\left(\frac{c^2 d+\frac{\sqrt{-d} \sqrt{e}}{x}}{c \sqrt{d} \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{c^2 d+e}}\right)}{4 d^{3/2} \sqrt{c^2 d+e}}+\frac{i b \text{Li}_2\left(-\frac{i c \sqrt{-d} e^{i \csc ^{-1}(c x)}}{\sqrt{e}-\sqrt{d c^2+e}}\right)}{4 (-d)^{3/2} \sqrt{e}}-\frac{i b \text{Li}_2\left(\frac{i c \sqrt{-d} e^{i \csc ^{-1}(c x)}}{\sqrt{e}-\sqrt{d c^2+e}}\right)}{4 (-d)^{3/2} \sqrt{e}}+\frac{i b \text{Li}_2\left(-\frac{i c \sqrt{-d} e^{i \csc ^{-1}(c x)}}{\sqrt{e}+\sqrt{d c^2+e}}\right)}{4 (-d)^{3/2} \sqrt{e}}-\frac{i b \text{Li}_2\left(\frac{i c \sqrt{-d} e^{i \csc ^{-1}(c x)}}{\sqrt{e}+\sqrt{d c^2+e}}\right)}{4 (-d)^{3/2} \sqrt{e}}",1,"((a*x)/(d^2 + d*e*x^2) + (a*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(d^(3/2)*Sqrt[e]) + (b*((2*Sqrt[d]*ArcCsc[c*x])/((-I)*Sqrt[d]*Sqrt[e] + e*x) + (2*Sqrt[d]*ArcCsc[c*x])/(I*Sqrt[d]*Sqrt[e] + e*x) + (8*ArcSin[Sqrt[1 - (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*ArcTan[(((-I)*c*Sqrt[d] + Sqrt[e])*Cot[(Pi + 2*ArcCsc[c*x])/4])/Sqrt[c^2*d + e]])/Sqrt[e] - (8*ArcSin[Sqrt[1 + (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*ArcTan[((I*c*Sqrt[d] + Sqrt[e])*Cot[(Pi + 2*ArcCsc[c*x])/4])/Sqrt[c^2*d + e]])/Sqrt[e] - (I*Pi*Log[1 + (Sqrt[e] - Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))])/Sqrt[e] + ((2*I)*ArcCsc[c*x]*Log[1 + (Sqrt[e] - Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))])/Sqrt[e] - ((4*I)*ArcSin[Sqrt[1 - (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*Log[1 + (Sqrt[e] - Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))])/Sqrt[e] + (I*Pi*Log[1 + (-Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))])/Sqrt[e] - ((2*I)*ArcCsc[c*x]*Log[1 + (-Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))])/Sqrt[e] + ((4*I)*ArcSin[Sqrt[1 + (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*Log[1 + (-Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))])/Sqrt[e] + (I*Pi*Log[1 - (Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))])/Sqrt[e] - ((2*I)*ArcCsc[c*x]*Log[1 - (Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))])/Sqrt[e] - ((4*I)*ArcSin[Sqrt[1 + (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*Log[1 - (Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))])/Sqrt[e] - (I*Pi*Log[1 + (Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))])/Sqrt[e] + ((2*I)*ArcCsc[c*x]*Log[1 + (Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))])/Sqrt[e] + ((4*I)*ArcSin[Sqrt[1 - (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*Log[1 + (Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))])/Sqrt[e] - (I*Pi*Log[Sqrt[e] - (I*Sqrt[d])/x])/Sqrt[e] + (I*Pi*Log[Sqrt[e] + (I*Sqrt[d])/x])/Sqrt[e] + ((2*I)*Log[(2*Sqrt[d]*Sqrt[e]*(Sqrt[e] + c*((-I)*c*Sqrt[d] - Sqrt[-(c^2*d) - e]*Sqrt[1 - 1/(c^2*x^2)])*x))/(Sqrt[-(c^2*d) - e]*(Sqrt[d] + I*Sqrt[e]*x))])/Sqrt[-(c^2*d) - e] - ((2*I)*Log[(2*Sqrt[d]*Sqrt[e]*(-Sqrt[e] + c*((-I)*c*Sqrt[d] + Sqrt[-(c^2*d) - e]*Sqrt[1 - 1/(c^2*x^2)])*x))/(Sqrt[-(c^2*d) - e]*(Sqrt[d] - I*Sqrt[e]*x))])/Sqrt[-(c^2*d) - e] + (2*PolyLog[2, (Sqrt[e] - Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))])/Sqrt[e] - (2*PolyLog[2, (-Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))])/Sqrt[e] - (2*PolyLog[2, -((Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x])))])/Sqrt[e] + (2*PolyLog[2, (Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))])/Sqrt[e]))/(4*d^(3/2)))/2","A",0
110,1,1525,806,2.7126093,"\int \frac{a+b \csc ^{-1}(c x)}{x^2 \left(d+e x^2\right)^2} \, dx","Integrate[(a + b*ArcCsc[c*x])/(x^2*(d + e*x^2)^2),x]","\frac{-12 \sqrt{e} \tan ^{-1}\left(\frac{\sqrt{e} x}{\sqrt{d}}\right) a-\frac{8 \sqrt{d} a}{x}-\frac{4 \sqrt{d} e x a}{e x^2+d}+b \left(-8 \sqrt{d} \sqrt{1-\frac{1}{c^2 x^2}} c-\frac{8 \sqrt{d} \csc ^{-1}(c x)}{x}-\frac{2 \sqrt{d} e \csc ^{-1}(c x)}{e x-i \sqrt{d} \sqrt{e}}-\frac{2 \sqrt{d} e \csc ^{-1}(c x)}{e x+i \sqrt{d} \sqrt{e}}-24 \sqrt{e} \sin ^{-1}\left(\frac{\sqrt{1-\frac{i \sqrt{e}}{c \sqrt{d}}}}{\sqrt{2}}\right) \tan ^{-1}\left(\frac{\left(\sqrt{e}-i c \sqrt{d}\right) \cot \left(\frac{1}{4} \left(2 \csc ^{-1}(c x)+\pi \right)\right)}{\sqrt{d c^2+e}}\right)+24 \sqrt{e} \sin ^{-1}\left(\frac{\sqrt{\frac{i \sqrt{e}}{c \sqrt{d}}+1}}{\sqrt{2}}\right) \tan ^{-1}\left(\frac{\left(i \sqrt{d} c+\sqrt{e}\right) \cot \left(\frac{1}{4} \left(2 \csc ^{-1}(c x)+\pi \right)\right)}{\sqrt{d c^2+e}}\right)-6 i \sqrt{e} \csc ^{-1}(c x) \log \left(\frac{e^{-i \csc ^{-1}(c x)} \left(\sqrt{e}-\sqrt{d c^2+e}\right)}{c \sqrt{d}}+1\right)+12 i \sqrt{e} \sin ^{-1}\left(\frac{\sqrt{1-\frac{i \sqrt{e}}{c \sqrt{d}}}}{\sqrt{2}}\right) \log \left(\frac{e^{-i \csc ^{-1}(c x)} \left(\sqrt{e}-\sqrt{d c^2+e}\right)}{c \sqrt{d}}+1\right)+3 i \sqrt{e} \pi  \log \left(\frac{e^{-i \csc ^{-1}(c x)} \left(\sqrt{e}-\sqrt{d c^2+e}\right)}{c \sqrt{d}}+1\right)+6 i \sqrt{e} \csc ^{-1}(c x) \log \left(\frac{e^{-i \csc ^{-1}(c x)} \left(\sqrt{d c^2+e}-\sqrt{e}\right)}{c \sqrt{d}}+1\right)-12 i \sqrt{e} \sin ^{-1}\left(\frac{\sqrt{\frac{i \sqrt{e}}{c \sqrt{d}}+1}}{\sqrt{2}}\right) \log \left(\frac{e^{-i \csc ^{-1}(c x)} \left(\sqrt{d c^2+e}-\sqrt{e}\right)}{c \sqrt{d}}+1\right)-3 i \sqrt{e} \pi  \log \left(\frac{e^{-i \csc ^{-1}(c x)} \left(\sqrt{d c^2+e}-\sqrt{e}\right)}{c \sqrt{d}}+1\right)+6 i \sqrt{e} \csc ^{-1}(c x) \log \left(1-\frac{\left(\sqrt{e}+\sqrt{d c^2+e}\right) e^{-i \csc ^{-1}(c x)}}{c \sqrt{d}}\right)+12 i \sqrt{e} \sin ^{-1}\left(\frac{\sqrt{\frac{i \sqrt{e}}{c \sqrt{d}}+1}}{\sqrt{2}}\right) \log \left(1-\frac{\left(\sqrt{e}+\sqrt{d c^2+e}\right) e^{-i \csc ^{-1}(c x)}}{c \sqrt{d}}\right)-3 i \sqrt{e} \pi  \log \left(1-\frac{\left(\sqrt{e}+\sqrt{d c^2+e}\right) e^{-i \csc ^{-1}(c x)}}{c \sqrt{d}}\right)-6 i \sqrt{e} \csc ^{-1}(c x) \log \left(\frac{e^{-i \csc ^{-1}(c x)} \left(\sqrt{e}+\sqrt{d c^2+e}\right)}{c \sqrt{d}}+1\right)-12 i \sqrt{e} \sin ^{-1}\left(\frac{\sqrt{1-\frac{i \sqrt{e}}{c \sqrt{d}}}}{\sqrt{2}}\right) \log \left(\frac{e^{-i \csc ^{-1}(c x)} \left(\sqrt{e}+\sqrt{d c^2+e}\right)}{c \sqrt{d}}+1\right)+3 i \sqrt{e} \pi  \log \left(\frac{e^{-i \csc ^{-1}(c x)} \left(\sqrt{e}+\sqrt{d c^2+e}\right)}{c \sqrt{d}}+1\right)+3 i \sqrt{e} \pi  \log \left(\sqrt{e}-\frac{i \sqrt{d}}{x}\right)-3 i \sqrt{e} \pi  \log \left(\frac{i \sqrt{d}}{x}+\sqrt{e}\right)-\frac{2 i e \log \left(\frac{2 \sqrt{d} \sqrt{e} \left(c \left(-i \sqrt{d} c-\sqrt{-d c^2-e} \sqrt{1-\frac{1}{c^2 x^2}}\right) x+\sqrt{e}\right)}{\sqrt{-d c^2-e} \left(i \sqrt{e} x+\sqrt{d}\right)}\right)}{\sqrt{-d c^2-e}}+\frac{2 i e \log \left(\frac{2 \sqrt{d} \sqrt{e} \left(c \left(\sqrt{-d c^2-e} \sqrt{1-\frac{1}{c^2 x^2}}-i c \sqrt{d}\right) x-\sqrt{e}\right)}{\sqrt{-d c^2-e} \left(\sqrt{d}-i \sqrt{e} x\right)}\right)}{\sqrt{-d c^2-e}}-6 \sqrt{e} \text{Li}_2\left(\frac{\left(\sqrt{e}-\sqrt{d c^2+e}\right) e^{-i \csc ^{-1}(c x)}}{c \sqrt{d}}\right)+6 \sqrt{e} \text{Li}_2\left(\frac{\left(\sqrt{d c^2+e}-\sqrt{e}\right) e^{-i \csc ^{-1}(c x)}}{c \sqrt{d}}\right)+6 \sqrt{e} \text{Li}_2\left(-\frac{\left(\sqrt{e}+\sqrt{d c^2+e}\right) e^{-i \csc ^{-1}(c x)}}{c \sqrt{d}}\right)-6 \sqrt{e} \text{Li}_2\left(\frac{\left(\sqrt{e}+\sqrt{d c^2+e}\right) e^{-i \csc ^{-1}(c x)}}{c \sqrt{d}}\right)\right)}{8 d^{5/2}}","-\frac{a}{d^2 x}-\frac{b \csc ^{-1}(c x)}{d^2 x}+\frac{e \left(a+b \csc ^{-1}(c x)\right)}{4 d^2 \left(\sqrt{-d} \sqrt{e}-\frac{d}{x}\right)}-\frac{e \left(a+b \csc ^{-1}(c x)\right)}{4 d^2 \left(\frac{d}{x}+\sqrt{-d} \sqrt{e}\right)}-\frac{b e \tanh ^{-1}\left(\frac{c^2 d-\frac{\sqrt{-d} \sqrt{e}}{x}}{c \sqrt{d} \sqrt{d c^2+e} \sqrt{1-\frac{1}{c^2 x^2}}}\right)}{4 d^{5/2} \sqrt{d c^2+e}}-\frac{b e \tanh ^{-1}\left(\frac{d c^2+\frac{\sqrt{-d} \sqrt{e}}{x}}{c \sqrt{d} \sqrt{d c^2+e} \sqrt{1-\frac{1}{c^2 x^2}}}\right)}{4 d^{5/2} \sqrt{d c^2+e}}+\frac{3 \sqrt{e} \left(a+b \csc ^{-1}(c x)\right) \log \left(1-\frac{i c \sqrt{-d} e^{i \csc ^{-1}(c x)}}{\sqrt{e}-\sqrt{d c^2+e}}\right)}{4 (-d)^{5/2}}-\frac{3 \sqrt{e} \left(a+b \csc ^{-1}(c x)\right) \log \left(\frac{i \sqrt{-d} e^{i \csc ^{-1}(c x)} c}{\sqrt{e}-\sqrt{d c^2+e}}+1\right)}{4 (-d)^{5/2}}+\frac{3 \sqrt{e} \left(a+b \csc ^{-1}(c x)\right) \log \left(1-\frac{i c \sqrt{-d} e^{i \csc ^{-1}(c x)}}{\sqrt{e}+\sqrt{d c^2+e}}\right)}{4 (-d)^{5/2}}-\frac{3 \sqrt{e} \left(a+b \csc ^{-1}(c x)\right) \log \left(\frac{i \sqrt{-d} e^{i \csc ^{-1}(c x)} c}{\sqrt{e}+\sqrt{d c^2+e}}+1\right)}{4 (-d)^{5/2}}+\frac{3 i b \sqrt{e} \text{Li}_2\left(-\frac{i c \sqrt{-d} e^{i \csc ^{-1}(c x)}}{\sqrt{e}-\sqrt{d c^2+e}}\right)}{4 (-d)^{5/2}}-\frac{3 i b \sqrt{e} \text{Li}_2\left(\frac{i c \sqrt{-d} e^{i \csc ^{-1}(c x)}}{\sqrt{e}-\sqrt{d c^2+e}}\right)}{4 (-d)^{5/2}}+\frac{3 i b \sqrt{e} \text{Li}_2\left(-\frac{i c \sqrt{-d} e^{i \csc ^{-1}(c x)}}{\sqrt{e}+\sqrt{d c^2+e}}\right)}{4 (-d)^{5/2}}-\frac{3 i b \sqrt{e} \text{Li}_2\left(\frac{i c \sqrt{-d} e^{i \csc ^{-1}(c x)}}{\sqrt{e}+\sqrt{d c^2+e}}\right)}{4 (-d)^{5/2}}-\frac{b c \sqrt{1-\frac{1}{c^2 x^2}}}{d^2}",1,"((-8*a*Sqrt[d])/x - (4*a*Sqrt[d]*e*x)/(d + e*x^2) - 12*a*Sqrt[e]*ArcTan[(Sqrt[e]*x)/Sqrt[d]] + b*(-8*c*Sqrt[d]*Sqrt[1 - 1/(c^2*x^2)] - (8*Sqrt[d]*ArcCsc[c*x])/x - (2*Sqrt[d]*e*ArcCsc[c*x])/((-I)*Sqrt[d]*Sqrt[e] + e*x) - (2*Sqrt[d]*e*ArcCsc[c*x])/(I*Sqrt[d]*Sqrt[e] + e*x) - 24*Sqrt[e]*ArcSin[Sqrt[1 - (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*ArcTan[(((-I)*c*Sqrt[d] + Sqrt[e])*Cot[(Pi + 2*ArcCsc[c*x])/4])/Sqrt[c^2*d + e]] + 24*Sqrt[e]*ArcSin[Sqrt[1 + (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*ArcTan[((I*c*Sqrt[d] + Sqrt[e])*Cot[(Pi + 2*ArcCsc[c*x])/4])/Sqrt[c^2*d + e]] + (3*I)*Sqrt[e]*Pi*Log[1 + (Sqrt[e] - Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] - (6*I)*Sqrt[e]*ArcCsc[c*x]*Log[1 + (Sqrt[e] - Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] + (12*I)*Sqrt[e]*ArcSin[Sqrt[1 - (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*Log[1 + (Sqrt[e] - Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] - (3*I)*Sqrt[e]*Pi*Log[1 + (-Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] + (6*I)*Sqrt[e]*ArcCsc[c*x]*Log[1 + (-Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] - (12*I)*Sqrt[e]*ArcSin[Sqrt[1 + (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*Log[1 + (-Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] - (3*I)*Sqrt[e]*Pi*Log[1 - (Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] + (6*I)*Sqrt[e]*ArcCsc[c*x]*Log[1 - (Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] + (12*I)*Sqrt[e]*ArcSin[Sqrt[1 + (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*Log[1 - (Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] + (3*I)*Sqrt[e]*Pi*Log[1 + (Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] - (6*I)*Sqrt[e]*ArcCsc[c*x]*Log[1 + (Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] - (12*I)*Sqrt[e]*ArcSin[Sqrt[1 - (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*Log[1 + (Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] + (3*I)*Sqrt[e]*Pi*Log[Sqrt[e] - (I*Sqrt[d])/x] - (3*I)*Sqrt[e]*Pi*Log[Sqrt[e] + (I*Sqrt[d])/x] - ((2*I)*e*Log[(2*Sqrt[d]*Sqrt[e]*(Sqrt[e] + c*((-I)*c*Sqrt[d] - Sqrt[-(c^2*d) - e]*Sqrt[1 - 1/(c^2*x^2)])*x))/(Sqrt[-(c^2*d) - e]*(Sqrt[d] + I*Sqrt[e]*x))])/Sqrt[-(c^2*d) - e] + ((2*I)*e*Log[(2*Sqrt[d]*Sqrt[e]*(-Sqrt[e] + c*((-I)*c*Sqrt[d] + Sqrt[-(c^2*d) - e]*Sqrt[1 - 1/(c^2*x^2)])*x))/(Sqrt[-(c^2*d) - e]*(Sqrt[d] - I*Sqrt[e]*x))])/Sqrt[-(c^2*d) - e] - 6*Sqrt[e]*PolyLog[2, (Sqrt[e] - Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] + 6*Sqrt[e]*PolyLog[2, (-Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] + 6*Sqrt[e]*PolyLog[2, -((Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x])))] - 6*Sqrt[e]*PolyLog[2, (Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))]))/(8*d^(5/2))","A",0
111,1,2053,727,7.7626268,"\int \frac{x^5 \left(a+b \csc ^{-1}(c x)\right)}{\left(d+e x^2\right)^3} \, dx","Integrate[(x^5*(a + b*ArcCsc[c*x]))/(d + e*x^2)^3,x]","\text{Result too large to show}","\frac{\left(a+b \csc ^{-1}(c x)\right) \log \left(1-\frac{i c \sqrt{-d} e^{i \csc ^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right)}{2 e^3}+\frac{\left(a+b \csc ^{-1}(c x)\right) \log \left(1+\frac{i c \sqrt{-d} e^{i \csc ^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right)}{2 e^3}+\frac{\left(a+b \csc ^{-1}(c x)\right) \log \left(1-\frac{i c \sqrt{-d} e^{i \csc ^{-1}(c x)}}{\sqrt{c^2 d+e}+\sqrt{e}}\right)}{2 e^3}+\frac{\left(a+b \csc ^{-1}(c x)\right) \log \left(1+\frac{i c \sqrt{-d} e^{i \csc ^{-1}(c x)}}{\sqrt{c^2 d+e}+\sqrt{e}}\right)}{2 e^3}-\frac{a+b \csc ^{-1}(c x)}{2 e^2 \left(\frac{d}{x^2}+e\right)}-\frac{a+b \csc ^{-1}(c x)}{4 e \left(\frac{d}{x^2}+e\right)^2}-\frac{\log \left(1-e^{2 i \csc ^{-1}(c x)}\right) \left(a+b \csc ^{-1}(c x)\right)}{e^3}+\frac{b \left(c^2 d+2 e\right) \tan ^{-1}\left(\frac{\sqrt{c^2 d+e}}{c \sqrt{e} x \sqrt{1-\frac{1}{c^2 x^2}}}\right)}{8 e^{5/2} \left(c^2 d+e\right)^{3/2}}+\frac{b \tan ^{-1}\left(\frac{\sqrt{c^2 d+e}}{c \sqrt{e} x \sqrt{1-\frac{1}{c^2 x^2}}}\right)}{2 e^{5/2} \sqrt{c^2 d+e}}-\frac{i b \text{Li}_2\left(-\frac{i c \sqrt{-d} e^{i \csc ^{-1}(c x)}}{\sqrt{e}-\sqrt{d c^2+e}}\right)}{2 e^3}-\frac{i b \text{Li}_2\left(\frac{i c \sqrt{-d} e^{i \csc ^{-1}(c x)}}{\sqrt{e}-\sqrt{d c^2+e}}\right)}{2 e^3}-\frac{i b \text{Li}_2\left(-\frac{i c \sqrt{-d} e^{i \csc ^{-1}(c x)}}{\sqrt{e}+\sqrt{d c^2+e}}\right)}{2 e^3}-\frac{i b \text{Li}_2\left(\frac{i c \sqrt{-d} e^{i \csc ^{-1}(c x)}}{\sqrt{e}+\sqrt{d c^2+e}}\right)}{2 e^3}+\frac{b c d \sqrt{1-\frac{1}{c^2 x^2}}}{8 e^2 x \left(c^2 d+e\right) \left(\frac{d}{x^2}+e\right)}+\frac{i b \text{Li}_2\left(e^{2 i \csc ^{-1}(c x)}\right)}{2 e^3}",1,"-1/4*(a*d^2)/(e^3*(d + e*x^2)^2) + (a*d)/(e^3*(d + e*x^2)) + (a*Log[d + e*x^2])/(2*e^3) + b*((((7*I)/16)*Sqrt[d]*(-(ArcCsc[c*x]/((-I)*Sqrt[d]*Sqrt[e] + e*x)) + (I*(ArcSin[1/(c*x)]/Sqrt[e] - Log[(2*Sqrt[d]*Sqrt[e]*(Sqrt[e] + c*((-I)*c*Sqrt[d] - Sqrt[-(c^2*d) - e]*Sqrt[1 - 1/(c^2*x^2)])*x))/(Sqrt[-(c^2*d) - e]*(Sqrt[d] + I*Sqrt[e]*x))]/Sqrt[-(c^2*d) - e]))/Sqrt[d]))/e^(5/2) - (((7*I)/16)*Sqrt[d]*(-(ArcCsc[c*x]/(I*Sqrt[d]*Sqrt[e] + e*x)) - (I*(ArcSin[1/(c*x)]/Sqrt[e] - Log[(2*Sqrt[d]*Sqrt[e]*(-Sqrt[e] + c*((-I)*c*Sqrt[d] + Sqrt[-(c^2*d) - e]*Sqrt[1 - 1/(c^2*x^2)])*x))/(Sqrt[-(c^2*d) - e]*(Sqrt[d] - I*Sqrt[e]*x))]/Sqrt[-(c^2*d) - e]))/Sqrt[d]))/e^(5/2) - (d*((I*c*Sqrt[e]*Sqrt[1 - 1/(c^2*x^2)]*x)/(Sqrt[d]*(c^2*d + e)*((-I)*Sqrt[d] + Sqrt[e]*x)) - ArcCsc[c*x]/(Sqrt[e]*((-I)*Sqrt[d] + Sqrt[e]*x)^2) - ArcSin[1/(c*x)]/(d*Sqrt[e]) + (I*(2*c^2*d + e)*Log[(4*d*Sqrt[e]*Sqrt[c^2*d + e]*(I*Sqrt[e] + c*(c*Sqrt[d] - Sqrt[c^2*d + e]*Sqrt[1 - 1/(c^2*x^2)])*x))/((2*c^2*d + e)*((-I)*Sqrt[d] + Sqrt[e]*x))])/(d*(c^2*d + e)^(3/2))))/(16*e^(5/2)) - (d*(((-I)*c*Sqrt[e]*Sqrt[1 - 1/(c^2*x^2)]*x)/(Sqrt[d]*(c^2*d + e)*(I*Sqrt[d] + Sqrt[e]*x)) - ArcCsc[c*x]/(Sqrt[e]*(I*Sqrt[d] + Sqrt[e]*x)^2) - ArcSin[1/(c*x)]/(d*Sqrt[e]) + (I*(2*c^2*d + e)*Log[(-4*d*Sqrt[e]*Sqrt[c^2*d + e]*((-I)*Sqrt[e] + c*(c*Sqrt[d] + Sqrt[c^2*d + e]*Sqrt[1 - 1/(c^2*x^2)])*x))/((2*c^2*d + e)*(I*Sqrt[d] + Sqrt[e]*x))])/(d*(c^2*d + e)^(3/2))))/(16*e^(5/2)) + ((I/16)*(Pi^2 - 4*Pi*ArcCsc[c*x] + 8*ArcCsc[c*x]^2 - 32*ArcSin[Sqrt[1 - (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*ArcTan[(((-I)*c*Sqrt[d] + Sqrt[e])*Cot[(Pi + 2*ArcCsc[c*x])/4])/Sqrt[c^2*d + e]] + (4*I)*Pi*Log[1 + (Sqrt[e] - Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] - (8*I)*ArcCsc[c*x]*Log[1 + (Sqrt[e] - Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] + (16*I)*ArcSin[Sqrt[1 - (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*Log[1 + (Sqrt[e] - Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] + (4*I)*Pi*Log[1 + (Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] - (8*I)*ArcCsc[c*x]*Log[1 + (Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] - (16*I)*ArcSin[Sqrt[1 - (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*Log[1 + (Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] + (8*I)*ArcCsc[c*x]*Log[1 - E^((2*I)*ArcCsc[c*x])] - (4*I)*Pi*Log[Sqrt[e] + (I*Sqrt[d])/x] + 8*PolyLog[2, (-Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] + 8*PolyLog[2, -((Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x])))] + 4*PolyLog[2, E^((2*I)*ArcCsc[c*x])]))/e^3 + ((I/16)*(Pi^2 - 4*Pi*ArcCsc[c*x] + 8*ArcCsc[c*x]^2 - 32*ArcSin[Sqrt[1 + (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*ArcTan[((I*c*Sqrt[d] + Sqrt[e])*Cot[(Pi + 2*ArcCsc[c*x])/4])/Sqrt[c^2*d + e]] + (4*I)*Pi*Log[1 + (-Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] - (8*I)*ArcCsc[c*x]*Log[1 + (-Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] + (16*I)*ArcSin[Sqrt[1 + (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*Log[1 + (-Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] + (4*I)*Pi*Log[1 - (Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] - (8*I)*ArcCsc[c*x]*Log[1 - (Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] - (16*I)*ArcSin[Sqrt[1 + (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*Log[1 - (Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] + (8*I)*ArcCsc[c*x]*Log[1 - E^((2*I)*ArcCsc[c*x])] - (4*I)*Pi*Log[Sqrt[e] - (I*Sqrt[d])/x] + 8*PolyLog[2, (Sqrt[e] - Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] + 8*PolyLog[2, (Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] + 4*PolyLog[2, E^((2*I)*ArcCsc[c*x])]))/e^3)","B",0
112,1,390,157,1.1174188,"\int \frac{x^3 \left(a+b \csc ^{-1}(c x)\right)}{\left(d+e x^2\right)^3} \, dx","Integrate[(x^3*(a + b*ArcCsc[c*x]))/(d + e*x^2)^3,x]","\frac{-\frac{8 a}{d+e x^2}+\frac{4 a d}{\left(d+e x^2\right)^2}+\frac{b \sqrt{e} \left(c^2 d+2 e\right) \log \left(\frac{16 d e^{3/2} \sqrt{c^2 (-d)-e} \left(c x \left(c \sqrt{d}-i \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{c^2 (-d)-e}\right)+i \sqrt{e}\right)}{b \left(c^2 d+2 e\right) \left(\sqrt{d}+i \sqrt{e} x\right)}\right)}{d \left(c^2 (-d)-e\right)^{3/2}}+\frac{b \sqrt{e} \left(c^2 d+2 e\right) \log \left(-\frac{16 d e^{3/2} \sqrt{c^2 (-d)-e} \left(-\sqrt{e}+c x \left(\sqrt{1-\frac{1}{c^2 x^2}} \sqrt{c^2 (-d)-e}-i c \sqrt{d}\right)\right)}{b \left(c^2 d+2 e\right) \left(\sqrt{e} x+i \sqrt{d}\right)}\right)}{d \left(c^2 (-d)-e\right)^{3/2}}-\frac{2 b c e x \sqrt{1-\frac{1}{c^2 x^2}}}{\left(c^2 d+e\right) \left(d+e x^2\right)}-\frac{4 b \csc ^{-1}(c x) \left(d+2 e x^2\right)}{\left(d+e x^2\right)^2}+\frac{4 b \sin ^{-1}\left(\frac{1}{c x}\right)}{d}}{16 e^2}","\frac{x^4 \left(a+b \csc ^{-1}(c x)\right)}{4 d \left(d+e x^2\right)^2}+\frac{b c x \left(c^2 d+2 e\right) \tan ^{-1}\left(\frac{\sqrt{e} \sqrt{c^2 x^2-1}}{\sqrt{c^2 d+e}}\right)}{8 d e^{3/2} \sqrt{c^2 x^2} \left(c^2 d+e\right)^{3/2}}-\frac{b c x \sqrt{c^2 x^2-1}}{8 e \sqrt{c^2 x^2} \left(c^2 d+e\right) \left(d+e x^2\right)}",1,"((4*a*d)/(d + e*x^2)^2 - (8*a)/(d + e*x^2) - (2*b*c*e*Sqrt[1 - 1/(c^2*x^2)]*x)/((c^2*d + e)*(d + e*x^2)) - (4*b*(d + 2*e*x^2)*ArcCsc[c*x])/(d + e*x^2)^2 + (4*b*ArcSin[1/(c*x)])/d + (b*Sqrt[e]*(c^2*d + 2*e)*Log[(16*d*Sqrt[-(c^2*d) - e]*e^(3/2)*(I*Sqrt[e] + c*(c*Sqrt[d] - I*Sqrt[-(c^2*d) - e]*Sqrt[1 - 1/(c^2*x^2)])*x))/(b*(c^2*d + 2*e)*(Sqrt[d] + I*Sqrt[e]*x))])/(d*(-(c^2*d) - e)^(3/2)) + (b*Sqrt[e]*(c^2*d + 2*e)*Log[(-16*d*Sqrt[-(c^2*d) - e]*e^(3/2)*(-Sqrt[e] + c*((-I)*c*Sqrt[d] + Sqrt[-(c^2*d) - e]*Sqrt[1 - 1/(c^2*x^2)])*x))/(b*(c^2*d + 2*e)*(I*Sqrt[d] + Sqrt[e]*x))])/(d*(-(c^2*d) - e)^(3/2)))/(16*e^2)","C",1
113,1,385,193,0.9420914,"\int \frac{x \left(a+b \csc ^{-1}(c x)\right)}{\left(d+e x^2\right)^3} \, dx","Integrate[(x*(a + b*ArcCsc[c*x]))/(d + e*x^2)^3,x]","\frac{1}{16} \left(-\frac{4 a}{e \left(d+e x^2\right)^2}+\frac{b \left(3 c^2 d+2 e\right) \log \left(\frac{16 d^2 \sqrt{e} \sqrt{c^2 (-d)-e} \left(c x \left(c \sqrt{d}-i \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{c^2 (-d)-e}\right)+i \sqrt{e}\right)}{b \left(3 c^2 d+2 e\right) \left(\sqrt{d}+i \sqrt{e} x\right)}\right)}{d^2 \sqrt{e} \left(c^2 (-d)-e\right)^{3/2}}+\frac{b \left(3 c^2 d+2 e\right) \log \left(-\frac{16 d^2 \sqrt{e} \sqrt{c^2 (-d)-e} \left(-\sqrt{e}+c x \left(\sqrt{1-\frac{1}{c^2 x^2}} \sqrt{c^2 (-d)-e}-i c \sqrt{d}\right)\right)}{b \left(3 c^2 d+2 e\right) \left(\sqrt{e} x+i \sqrt{d}\right)}\right)}{d^2 \sqrt{e} \left(c^2 (-d)-e\right)^{3/2}}+\frac{2 b c x \sqrt{1-\frac{1}{c^2 x^2}}}{d \left(c^2 d+e\right) \left(d+e x^2\right)}+\frac{4 b \sin ^{-1}\left(\frac{1}{c x}\right)}{d^2 e}-\frac{4 b \csc ^{-1}(c x)}{e \left(d+e x^2\right)^2}\right)","-\frac{a+b \csc ^{-1}(c x)}{4 e \left(d+e x^2\right)^2}-\frac{b c x \tan ^{-1}\left(\sqrt{c^2 x^2-1}\right)}{4 d^2 e \sqrt{c^2 x^2}}+\frac{b c x \left(3 c^2 d+2 e\right) \tan ^{-1}\left(\frac{\sqrt{e} \sqrt{c^2 x^2-1}}{\sqrt{c^2 d+e}}\right)}{8 d^2 \sqrt{e} \sqrt{c^2 x^2} \left(c^2 d+e\right)^{3/2}}+\frac{b c x \sqrt{c^2 x^2-1}}{8 d \sqrt{c^2 x^2} \left(c^2 d+e\right) \left(d+e x^2\right)}",1,"((-4*a)/(e*(d + e*x^2)^2) + (2*b*c*Sqrt[1 - 1/(c^2*x^2)]*x)/(d*(c^2*d + e)*(d + e*x^2)) - (4*b*ArcCsc[c*x])/(e*(d + e*x^2)^2) + (4*b*ArcSin[1/(c*x)])/(d^2*e) + (b*(3*c^2*d + 2*e)*Log[(16*d^2*Sqrt[-(c^2*d) - e]*Sqrt[e]*(I*Sqrt[e] + c*(c*Sqrt[d] - I*Sqrt[-(c^2*d) - e]*Sqrt[1 - 1/(c^2*x^2)])*x))/(b*(3*c^2*d + 2*e)*(Sqrt[d] + I*Sqrt[e]*x))])/(d^2*(-(c^2*d) - e)^(3/2)*Sqrt[e]) + (b*(3*c^2*d + 2*e)*Log[(-16*d^2*Sqrt[-(c^2*d) - e]*Sqrt[e]*(-Sqrt[e] + c*((-I)*c*Sqrt[d] + Sqrt[-(c^2*d) - e]*Sqrt[1 - 1/(c^2*x^2)])*x))/(b*(3*c^2*d + 2*e)*(I*Sqrt[d] + Sqrt[e]*x))])/(d^2*(-(c^2*d) - e)^(3/2)*Sqrt[e]))/16","C",1
114,0,0,704,64.9616943,"\int \frac{a+b \csc ^{-1}(c x)}{x \left(d+e x^2\right)^3} \, dx","Integrate[(a + b*ArcCsc[c*x])/(x*(d + e*x^2)^3),x]","\int \frac{a+b \csc ^{-1}(c x)}{x \left(d+e x^2\right)^3} \, dx","-\frac{\left(a+b \csc ^{-1}(c x)\right) \log \left(1-\frac{i c \sqrt{-d} e^{i \csc ^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right)}{2 d^3}-\frac{\left(a+b \csc ^{-1}(c x)\right) \log \left(1+\frac{i c \sqrt{-d} e^{i \csc ^{-1}(c x)}}{\sqrt{e}-\sqrt{c^2 d+e}}\right)}{2 d^3}-\frac{\left(a+b \csc ^{-1}(c x)\right) \log \left(1-\frac{i c \sqrt{-d} e^{i \csc ^{-1}(c x)}}{\sqrt{c^2 d+e}+\sqrt{e}}\right)}{2 d^3}-\frac{\left(a+b \csc ^{-1}(c x)\right) \log \left(1+\frac{i c \sqrt{-d} e^{i \csc ^{-1}(c x)}}{\sqrt{c^2 d+e}+\sqrt{e}}\right)}{2 d^3}+\frac{e^2 \left(a+b \csc ^{-1}(c x)\right)}{4 d^3 \left(\frac{d}{x^2}+e\right)^2}-\frac{e \left(a+b \csc ^{-1}(c x)\right)}{d^3 \left(\frac{d}{x^2}+e\right)}+\frac{i \left(a+b \csc ^{-1}(c x)\right)^2}{2 b d^3}+\frac{i b \text{Li}_2\left(-\frac{i c \sqrt{-d} e^{i \csc ^{-1}(c x)}}{\sqrt{e}-\sqrt{d c^2+e}}\right)}{2 d^3}+\frac{i b \text{Li}_2\left(\frac{i c \sqrt{-d} e^{i \csc ^{-1}(c x)}}{\sqrt{e}-\sqrt{d c^2+e}}\right)}{2 d^3}+\frac{i b \text{Li}_2\left(-\frac{i c \sqrt{-d} e^{i \csc ^{-1}(c x)}}{\sqrt{e}+\sqrt{d c^2+e}}\right)}{2 d^3}+\frac{i b \text{Li}_2\left(\frac{i c \sqrt{-d} e^{i \csc ^{-1}(c x)}}{\sqrt{e}+\sqrt{d c^2+e}}\right)}{2 d^3}-\frac{b \sqrt{e} \left(c^2 d+2 e\right) \tan ^{-1}\left(\frac{\sqrt{c^2 d+e}}{c \sqrt{e} x \sqrt{1-\frac{1}{c^2 x^2}}}\right)}{8 d^3 \left(c^2 d+e\right)^{3/2}}+\frac{b \sqrt{e} \tan ^{-1}\left(\frac{\sqrt{c^2 d+e}}{c \sqrt{e} x \sqrt{1-\frac{1}{c^2 x^2}}}\right)}{d^3 \sqrt{c^2 d+e}}-\frac{b c e \sqrt{1-\frac{1}{c^2 x^2}}}{8 d^2 x \left(c^2 d+e\right) \left(\frac{d}{x^2}+e\right)}",1,"Integrate[(a + b*ArcCsc[c*x])/(x*(d + e*x^2)^3), x]","F",-1
115,1,2067,1144,6.199102,"\int \frac{x^4 \left(a+b \csc ^{-1}(c x)\right)}{\left(d+e x^2\right)^3} \, dx","Integrate[(x^4*(a + b*ArcCsc[c*x]))/(d + e*x^2)^3,x]","\text{Result too large to show}","-\frac{b \sqrt{-d} \sqrt{1-\frac{1}{c^2 x^2}} c}{16 e^{3/2} \left(d c^2+e\right) \left(\sqrt{-d} \sqrt{e}-\frac{d}{x}\right)}-\frac{b \sqrt{-d} \sqrt{1-\frac{1}{c^2 x^2}} c}{16 e^{3/2} \left(d c^2+e\right) \left(\frac{d}{x}+\sqrt{-d} \sqrt{e}\right)}+\frac{3 \left(a+b \csc ^{-1}(c x)\right)}{16 e^2 \left(\sqrt{-d} \sqrt{e}-\frac{d}{x}\right)}-\frac{3 \left(a+b \csc ^{-1}(c x)\right)}{16 e^2 \left(\frac{d}{x}+\sqrt{-d} \sqrt{e}\right)}+\frac{\sqrt{-d} \left(a+b \csc ^{-1}(c x)\right)}{16 e^{3/2} \left(\sqrt{-d} \sqrt{e}-\frac{d}{x}\right)^2}-\frac{\sqrt{-d} \left(a+b \csc ^{-1}(c x)\right)}{16 e^{3/2} \left(\frac{d}{x}+\sqrt{-d} \sqrt{e}\right)^2}-\frac{3 b \tanh ^{-1}\left(\frac{c^2 d-\frac{\sqrt{-d} \sqrt{e}}{x}}{c \sqrt{d} \sqrt{d c^2+e} \sqrt{1-\frac{1}{c^2 x^2}}}\right)}{16 \sqrt{d} e^2 \sqrt{d c^2+e}}-\frac{b \tanh ^{-1}\left(\frac{c^2 d-\frac{\sqrt{-d} \sqrt{e}}{x}}{c \sqrt{d} \sqrt{d c^2+e} \sqrt{1-\frac{1}{c^2 x^2}}}\right)}{16 \sqrt{d} e \left(d c^2+e\right)^{3/2}}-\frac{3 b \tanh ^{-1}\left(\frac{d c^2+\frac{\sqrt{-d} \sqrt{e}}{x}}{c \sqrt{d} \sqrt{d c^2+e} \sqrt{1-\frac{1}{c^2 x^2}}}\right)}{16 \sqrt{d} e^2 \sqrt{d c^2+e}}-\frac{b \tanh ^{-1}\left(\frac{d c^2+\frac{\sqrt{-d} \sqrt{e}}{x}}{c \sqrt{d} \sqrt{d c^2+e} \sqrt{1-\frac{1}{c^2 x^2}}}\right)}{16 \sqrt{d} e \left(d c^2+e\right)^{3/2}}-\frac{3 \left(a+b \csc ^{-1}(c x)\right) \log \left(1-\frac{i c \sqrt{-d} e^{i \csc ^{-1}(c x)}}{\sqrt{e}-\sqrt{d c^2+e}}\right)}{16 \sqrt{-d} e^{5/2}}+\frac{3 \left(a+b \csc ^{-1}(c x)\right) \log \left(\frac{i \sqrt{-d} e^{i \csc ^{-1}(c x)} c}{\sqrt{e}-\sqrt{d c^2+e}}+1\right)}{16 \sqrt{-d} e^{5/2}}-\frac{3 \left(a+b \csc ^{-1}(c x)\right) \log \left(1-\frac{i c \sqrt{-d} e^{i \csc ^{-1}(c x)}}{\sqrt{e}+\sqrt{d c^2+e}}\right)}{16 \sqrt{-d} e^{5/2}}+\frac{3 \left(a+b \csc ^{-1}(c x)\right) \log \left(\frac{i \sqrt{-d} e^{i \csc ^{-1}(c x)} c}{\sqrt{e}+\sqrt{d c^2+e}}+1\right)}{16 \sqrt{-d} e^{5/2}}-\frac{3 i b \text{Li}_2\left(-\frac{i c \sqrt{-d} e^{i \csc ^{-1}(c x)}}{\sqrt{e}-\sqrt{d c^2+e}}\right)}{16 \sqrt{-d} e^{5/2}}+\frac{3 i b \text{Li}_2\left(\frac{i c \sqrt{-d} e^{i \csc ^{-1}(c x)}}{\sqrt{e}-\sqrt{d c^2+e}}\right)}{16 \sqrt{-d} e^{5/2}}-\frac{3 i b \text{Li}_2\left(-\frac{i c \sqrt{-d} e^{i \csc ^{-1}(c x)}}{\sqrt{e}+\sqrt{d c^2+e}}\right)}{16 \sqrt{-d} e^{5/2}}+\frac{3 i b \text{Li}_2\left(\frac{i c \sqrt{-d} e^{i \csc ^{-1}(c x)}}{\sqrt{e}+\sqrt{d c^2+e}}\right)}{16 \sqrt{-d} e^{5/2}}",1,"(a*d*x)/(4*e^2*(d + e*x^2)^2) - (5*a*x)/(8*e^2*(d + e*x^2)) + (3*a*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(8*Sqrt[d]*e^(5/2)) + b*((5*(-(ArcCsc[c*x]/((-I)*Sqrt[d]*Sqrt[e] + e*x)) + (I*(ArcSin[1/(c*x)]/Sqrt[e] - Log[(2*Sqrt[d]*Sqrt[e]*(Sqrt[e] + c*((-I)*c*Sqrt[d] - Sqrt[-(c^2*d) - e]*Sqrt[1 - 1/(c^2*x^2)])*x))/(Sqrt[-(c^2*d) - e]*(Sqrt[d] + I*Sqrt[e]*x))]/Sqrt[-(c^2*d) - e]))/Sqrt[d]))/(16*e^2) + (5*(-(ArcCsc[c*x]/(I*Sqrt[d]*Sqrt[e] + e*x)) - (I*(ArcSin[1/(c*x)]/Sqrt[e] - Log[(2*Sqrt[d]*Sqrt[e]*(-Sqrt[e] + c*((-I)*c*Sqrt[d] + Sqrt[-(c^2*d) - e]*Sqrt[1 - 1/(c^2*x^2)])*x))/(Sqrt[-(c^2*d) - e]*(Sqrt[d] - I*Sqrt[e]*x))]/Sqrt[-(c^2*d) - e]))/Sqrt[d]))/(16*e^2) + ((I/16)*Sqrt[d]*((I*c*Sqrt[e]*Sqrt[1 - 1/(c^2*x^2)]*x)/(Sqrt[d]*(c^2*d + e)*((-I)*Sqrt[d] + Sqrt[e]*x)) - ArcCsc[c*x]/(Sqrt[e]*((-I)*Sqrt[d] + Sqrt[e]*x)^2) - ArcSin[1/(c*x)]/(d*Sqrt[e]) + (I*(2*c^2*d + e)*Log[(4*d*Sqrt[e]*Sqrt[c^2*d + e]*(I*Sqrt[e] + c*(c*Sqrt[d] - Sqrt[c^2*d + e]*Sqrt[1 - 1/(c^2*x^2)])*x))/((2*c^2*d + e)*((-I)*Sqrt[d] + Sqrt[e]*x))])/(d*(c^2*d + e)^(3/2))))/e^2 - ((I/16)*Sqrt[d]*(((-I)*c*Sqrt[e]*Sqrt[1 - 1/(c^2*x^2)]*x)/(Sqrt[d]*(c^2*d + e)*(I*Sqrt[d] + Sqrt[e]*x)) - ArcCsc[c*x]/(Sqrt[e]*(I*Sqrt[d] + Sqrt[e]*x)^2) - ArcSin[1/(c*x)]/(d*Sqrt[e]) + (I*(2*c^2*d + e)*Log[(-4*d*Sqrt[e]*Sqrt[c^2*d + e]*((-I)*Sqrt[e] + c*(c*Sqrt[d] + Sqrt[c^2*d + e]*Sqrt[1 - 1/(c^2*x^2)])*x))/((2*c^2*d + e)*(I*Sqrt[d] + Sqrt[e]*x))])/(d*(c^2*d + e)^(3/2))))/e^2 - (3*(Pi^2 - 4*Pi*ArcCsc[c*x] + 8*ArcCsc[c*x]^2 - 32*ArcSin[Sqrt[1 - (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*ArcTan[(((-I)*c*Sqrt[d] + Sqrt[e])*Cot[(Pi + 2*ArcCsc[c*x])/4])/Sqrt[c^2*d + e]] + (4*I)*Pi*Log[1 + (Sqrt[e] - Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] - (8*I)*ArcCsc[c*x]*Log[1 + (Sqrt[e] - Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] + (16*I)*ArcSin[Sqrt[1 - (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*Log[1 + (Sqrt[e] - Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] + (4*I)*Pi*Log[1 + (Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] - (8*I)*ArcCsc[c*x]*Log[1 + (Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] - (16*I)*ArcSin[Sqrt[1 - (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*Log[1 + (Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] + (8*I)*ArcCsc[c*x]*Log[1 - E^((2*I)*ArcCsc[c*x])] - (4*I)*Pi*Log[Sqrt[e] + (I*Sqrt[d])/x] + 8*PolyLog[2, (-Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] + 8*PolyLog[2, -((Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x])))] + 4*PolyLog[2, E^((2*I)*ArcCsc[c*x])]))/(128*Sqrt[d]*e^(5/2)) + (3*(Pi^2 - 4*Pi*ArcCsc[c*x] + 8*ArcCsc[c*x]^2 - 32*ArcSin[Sqrt[1 + (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*ArcTan[((I*c*Sqrt[d] + Sqrt[e])*Cot[(Pi + 2*ArcCsc[c*x])/4])/Sqrt[c^2*d + e]] + (4*I)*Pi*Log[1 + (-Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] - (8*I)*ArcCsc[c*x]*Log[1 + (-Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] + (16*I)*ArcSin[Sqrt[1 + (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*Log[1 + (-Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] + (4*I)*Pi*Log[1 - (Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] - (8*I)*ArcCsc[c*x]*Log[1 - (Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] - (16*I)*ArcSin[Sqrt[1 + (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*Log[1 - (Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] + (8*I)*ArcCsc[c*x]*Log[1 - E^((2*I)*ArcCsc[c*x])] - (4*I)*Pi*Log[Sqrt[e] - (I*Sqrt[d])/x] + 8*PolyLog[2, (Sqrt[e] - Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] + 8*PolyLog[2, (Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] + 4*PolyLog[2, E^((2*I)*ArcCsc[c*x])]))/(128*Sqrt[d]*e^(5/2)))","A",0
116,1,2075,1144,6.1426386,"\int \frac{x^2 \left(a+b \csc ^{-1}(c x)\right)}{\left(d+e x^2\right)^3} \, dx","Integrate[(x^2*(a + b*ArcCsc[c*x]))/(d + e*x^2)^3,x]","\text{Result too large to show}","-\frac{b \sqrt{1-\frac{1}{c^2 x^2}} c}{16 \sqrt{-d} \sqrt{e} \left(d c^2+e\right) \left(\sqrt{-d} \sqrt{e}-\frac{d}{x}\right)}-\frac{b \sqrt{1-\frac{1}{c^2 x^2}} c}{16 \sqrt{-d} \sqrt{e} \left(d c^2+e\right) \left(\frac{d}{x}+\sqrt{-d} \sqrt{e}\right)}+\frac{a+b \csc ^{-1}(c x)}{16 d e \left(\sqrt{-d} \sqrt{e}-\frac{d}{x}\right)}-\frac{a+b \csc ^{-1}(c x)}{16 d e \left(\frac{d}{x}+\sqrt{-d} \sqrt{e}\right)}+\frac{a+b \csc ^{-1}(c x)}{16 \sqrt{-d} \sqrt{e} \left(\sqrt{-d} \sqrt{e}-\frac{d}{x}\right)^2}-\frac{a+b \csc ^{-1}(c x)}{16 \sqrt{-d} \sqrt{e} \left(\frac{d}{x}+\sqrt{-d} \sqrt{e}\right)^2}-\frac{b \tanh ^{-1}\left(\frac{c^2 d-\frac{\sqrt{-d} \sqrt{e}}{x}}{c \sqrt{d} \sqrt{d c^2+e} \sqrt{1-\frac{1}{c^2 x^2}}}\right)}{16 d^{3/2} e \sqrt{d c^2+e}}+\frac{b \tanh ^{-1}\left(\frac{c^2 d-\frac{\sqrt{-d} \sqrt{e}}{x}}{c \sqrt{d} \sqrt{d c^2+e} \sqrt{1-\frac{1}{c^2 x^2}}}\right)}{16 d^{3/2} \left(d c^2+e\right)^{3/2}}-\frac{b \tanh ^{-1}\left(\frac{d c^2+\frac{\sqrt{-d} \sqrt{e}}{x}}{c \sqrt{d} \sqrt{d c^2+e} \sqrt{1-\frac{1}{c^2 x^2}}}\right)}{16 d^{3/2} e \sqrt{d c^2+e}}+\frac{b \tanh ^{-1}\left(\frac{d c^2+\frac{\sqrt{-d} \sqrt{e}}{x}}{c \sqrt{d} \sqrt{d c^2+e} \sqrt{1-\frac{1}{c^2 x^2}}}\right)}{16 d^{3/2} \left(d c^2+e\right)^{3/2}}+\frac{\left(a+b \csc ^{-1}(c x)\right) \log \left(1-\frac{i c \sqrt{-d} e^{i \csc ^{-1}(c x)}}{\sqrt{e}-\sqrt{d c^2+e}}\right)}{16 (-d)^{3/2} e^{3/2}}-\frac{\left(a+b \csc ^{-1}(c x)\right) \log \left(\frac{i \sqrt{-d} e^{i \csc ^{-1}(c x)} c}{\sqrt{e}-\sqrt{d c^2+e}}+1\right)}{16 (-d)^{3/2} e^{3/2}}+\frac{\left(a+b \csc ^{-1}(c x)\right) \log \left(1-\frac{i c \sqrt{-d} e^{i \csc ^{-1}(c x)}}{\sqrt{e}+\sqrt{d c^2+e}}\right)}{16 (-d)^{3/2} e^{3/2}}-\frac{\left(a+b \csc ^{-1}(c x)\right) \log \left(\frac{i \sqrt{-d} e^{i \csc ^{-1}(c x)} c}{\sqrt{e}+\sqrt{d c^2+e}}+1\right)}{16 (-d)^{3/2} e^{3/2}}+\frac{i b \text{Li}_2\left(-\frac{i c \sqrt{-d} e^{i \csc ^{-1}(c x)}}{\sqrt{e}-\sqrt{d c^2+e}}\right)}{16 (-d)^{3/2} e^{3/2}}-\frac{i b \text{Li}_2\left(\frac{i c \sqrt{-d} e^{i \csc ^{-1}(c x)}}{\sqrt{e}-\sqrt{d c^2+e}}\right)}{16 (-d)^{3/2} e^{3/2}}+\frac{i b \text{Li}_2\left(-\frac{i c \sqrt{-d} e^{i \csc ^{-1}(c x)}}{\sqrt{e}+\sqrt{d c^2+e}}\right)}{16 (-d)^{3/2} e^{3/2}}-\frac{i b \text{Li}_2\left(\frac{i c \sqrt{-d} e^{i \csc ^{-1}(c x)}}{\sqrt{e}+\sqrt{d c^2+e}}\right)}{16 (-d)^{3/2} e^{3/2}}",1,"-1/4*(a*x)/(e*(d + e*x^2)^2) + (a*x)/(8*d*e*(d + e*x^2)) + (a*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(8*d^(3/2)*e^(3/2)) + b*(-1/16*(-(ArcCsc[c*x]/((-I)*Sqrt[d]*Sqrt[e] + e*x)) + (I*(ArcSin[1/(c*x)]/Sqrt[e] - Log[(2*Sqrt[d]*Sqrt[e]*(Sqrt[e] + c*((-I)*c*Sqrt[d] - Sqrt[-(c^2*d) - e]*Sqrt[1 - 1/(c^2*x^2)])*x))/(Sqrt[-(c^2*d) - e]*(Sqrt[d] + I*Sqrt[e]*x))]/Sqrt[-(c^2*d) - e]))/Sqrt[d])/(d*e) - (-(ArcCsc[c*x]/(I*Sqrt[d]*Sqrt[e] + e*x)) - (I*(ArcSin[1/(c*x)]/Sqrt[e] - Log[(2*Sqrt[d]*Sqrt[e]*(-Sqrt[e] + c*((-I)*c*Sqrt[d] + Sqrt[-(c^2*d) - e]*Sqrt[1 - 1/(c^2*x^2)])*x))/(Sqrt[-(c^2*d) - e]*(Sqrt[d] - I*Sqrt[e]*x))]/Sqrt[-(c^2*d) - e]))/Sqrt[d])/(16*d*e) - ((I/16)*((I*c*Sqrt[e]*Sqrt[1 - 1/(c^2*x^2)]*x)/(Sqrt[d]*(c^2*d + e)*((-I)*Sqrt[d] + Sqrt[e]*x)) - ArcCsc[c*x]/(Sqrt[e]*((-I)*Sqrt[d] + Sqrt[e]*x)^2) - ArcSin[1/(c*x)]/(d*Sqrt[e]) + (I*(2*c^2*d + e)*Log[(4*d*Sqrt[e]*Sqrt[c^2*d + e]*(I*Sqrt[e] + c*(c*Sqrt[d] - Sqrt[c^2*d + e]*Sqrt[1 - 1/(c^2*x^2)])*x))/((2*c^2*d + e)*((-I)*Sqrt[d] + Sqrt[e]*x))])/(d*(c^2*d + e)^(3/2))))/(Sqrt[d]*e) + ((I/16)*(((-I)*c*Sqrt[e]*Sqrt[1 - 1/(c^2*x^2)]*x)/(Sqrt[d]*(c^2*d + e)*(I*Sqrt[d] + Sqrt[e]*x)) - ArcCsc[c*x]/(Sqrt[e]*(I*Sqrt[d] + Sqrt[e]*x)^2) - ArcSin[1/(c*x)]/(d*Sqrt[e]) + (I*(2*c^2*d + e)*Log[(-4*d*Sqrt[e]*Sqrt[c^2*d + e]*((-I)*Sqrt[e] + c*(c*Sqrt[d] + Sqrt[c^2*d + e]*Sqrt[1 - 1/(c^2*x^2)])*x))/((2*c^2*d + e)*(I*Sqrt[d] + Sqrt[e]*x))])/(d*(c^2*d + e)^(3/2))))/(Sqrt[d]*e) - (Pi^2 - 4*Pi*ArcCsc[c*x] + 8*ArcCsc[c*x]^2 - 32*ArcSin[Sqrt[1 - (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*ArcTan[(((-I)*c*Sqrt[d] + Sqrt[e])*Cot[(Pi + 2*ArcCsc[c*x])/4])/Sqrt[c^2*d + e]] + (4*I)*Pi*Log[1 + (Sqrt[e] - Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] - (8*I)*ArcCsc[c*x]*Log[1 + (Sqrt[e] - Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] + (16*I)*ArcSin[Sqrt[1 - (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*Log[1 + (Sqrt[e] - Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] + (4*I)*Pi*Log[1 + (Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] - (8*I)*ArcCsc[c*x]*Log[1 + (Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] - (16*I)*ArcSin[Sqrt[1 - (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*Log[1 + (Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] + (8*I)*ArcCsc[c*x]*Log[1 - E^((2*I)*ArcCsc[c*x])] - (4*I)*Pi*Log[Sqrt[e] + (I*Sqrt[d])/x] + 8*PolyLog[2, (-Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] + 8*PolyLog[2, -((Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x])))] + 4*PolyLog[2, E^((2*I)*ArcCsc[c*x])])/(128*d^(3/2)*e^(3/2)) + (Pi^2 - 4*Pi*ArcCsc[c*x] + 8*ArcCsc[c*x]^2 - 32*ArcSin[Sqrt[1 + (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*ArcTan[((I*c*Sqrt[d] + Sqrt[e])*Cot[(Pi + 2*ArcCsc[c*x])/4])/Sqrt[c^2*d + e]] + (4*I)*Pi*Log[1 + (-Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] - (8*I)*ArcCsc[c*x]*Log[1 + (-Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] + (16*I)*ArcSin[Sqrt[1 + (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*Log[1 + (-Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] + (4*I)*Pi*Log[1 - (Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] - (8*I)*ArcCsc[c*x]*Log[1 - (Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] - (16*I)*ArcSin[Sqrt[1 + (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*Log[1 - (Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] + (8*I)*ArcCsc[c*x]*Log[1 - E^((2*I)*ArcCsc[c*x])] - (4*I)*Pi*Log[Sqrt[e] - (I*Sqrt[d])/x] + 8*PolyLog[2, (Sqrt[e] - Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] + 8*PolyLog[2, (Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] + 4*PolyLog[2, E^((2*I)*ArcCsc[c*x])])/(128*d^(3/2)*e^(3/2)))","A",0
117,1,2060,1134,6.0637163,"\int \frac{a+b \csc ^{-1}(c x)}{\left(d+e x^2\right)^3} \, dx","Integrate[(a + b*ArcCsc[c*x])/(d + e*x^2)^3,x]","\text{Result too large to show}","-\frac{b \sqrt{e} \sqrt{1-\frac{1}{c^2 x^2}} c}{16 (-d)^{3/2} \left(d c^2+e\right) \left(\sqrt{-d} \sqrt{e}-\frac{d}{x}\right)}-\frac{b \sqrt{e} \sqrt{1-\frac{1}{c^2 x^2}} c}{16 (-d)^{3/2} \left(d c^2+e\right) \left(\frac{d}{x}+\sqrt{-d} \sqrt{e}\right)}-\frac{5 \left(a+b \csc ^{-1}(c x)\right)}{16 d^2 \left(\sqrt{-d} \sqrt{e}-\frac{d}{x}\right)}+\frac{5 \left(a+b \csc ^{-1}(c x)\right)}{16 d^2 \left(\frac{d}{x}+\sqrt{-d} \sqrt{e}\right)}+\frac{\sqrt{e} \left(a+b \csc ^{-1}(c x)\right)}{16 (-d)^{3/2} \left(\sqrt{-d} \sqrt{e}-\frac{d}{x}\right)^2}-\frac{\sqrt{e} \left(a+b \csc ^{-1}(c x)\right)}{16 (-d)^{3/2} \left(\frac{d}{x}+\sqrt{-d} \sqrt{e}\right)^2}+\frac{5 b \tanh ^{-1}\left(\frac{c^2 d-\frac{\sqrt{-d} \sqrt{e}}{x}}{c \sqrt{d} \sqrt{d c^2+e} \sqrt{1-\frac{1}{c^2 x^2}}}\right)}{16 d^{5/2} \sqrt{d c^2+e}}-\frac{b e \tanh ^{-1}\left(\frac{c^2 d-\frac{\sqrt{-d} \sqrt{e}}{x}}{c \sqrt{d} \sqrt{d c^2+e} \sqrt{1-\frac{1}{c^2 x^2}}}\right)}{16 d^{5/2} \left(d c^2+e\right)^{3/2}}+\frac{5 b \tanh ^{-1}\left(\frac{d c^2+\frac{\sqrt{-d} \sqrt{e}}{x}}{c \sqrt{d} \sqrt{d c^2+e} \sqrt{1-\frac{1}{c^2 x^2}}}\right)}{16 d^{5/2} \sqrt{d c^2+e}}-\frac{b e \tanh ^{-1}\left(\frac{d c^2+\frac{\sqrt{-d} \sqrt{e}}{x}}{c \sqrt{d} \sqrt{d c^2+e} \sqrt{1-\frac{1}{c^2 x^2}}}\right)}{16 d^{5/2} \left(d c^2+e\right)^{3/2}}-\frac{3 \left(a+b \csc ^{-1}(c x)\right) \log \left(1-\frac{i c \sqrt{-d} e^{i \csc ^{-1}(c x)}}{\sqrt{e}-\sqrt{d c^2+e}}\right)}{16 (-d)^{5/2} \sqrt{e}}+\frac{3 \left(a+b \csc ^{-1}(c x)\right) \log \left(\frac{i \sqrt{-d} e^{i \csc ^{-1}(c x)} c}{\sqrt{e}-\sqrt{d c^2+e}}+1\right)}{16 (-d)^{5/2} \sqrt{e}}-\frac{3 \left(a+b \csc ^{-1}(c x)\right) \log \left(1-\frac{i c \sqrt{-d} e^{i \csc ^{-1}(c x)}}{\sqrt{e}+\sqrt{d c^2+e}}\right)}{16 (-d)^{5/2} \sqrt{e}}+\frac{3 \left(a+b \csc ^{-1}(c x)\right) \log \left(\frac{i \sqrt{-d} e^{i \csc ^{-1}(c x)} c}{\sqrt{e}+\sqrt{d c^2+e}}+1\right)}{16 (-d)^{5/2} \sqrt{e}}-\frac{3 i b \text{Li}_2\left(-\frac{i c \sqrt{-d} e^{i \csc ^{-1}(c x)}}{\sqrt{e}-\sqrt{d c^2+e}}\right)}{16 (-d)^{5/2} \sqrt{e}}+\frac{3 i b \text{Li}_2\left(\frac{i c \sqrt{-d} e^{i \csc ^{-1}(c x)}}{\sqrt{e}-\sqrt{d c^2+e}}\right)}{16 (-d)^{5/2} \sqrt{e}}-\frac{3 i b \text{Li}_2\left(-\frac{i c \sqrt{-d} e^{i \csc ^{-1}(c x)}}{\sqrt{e}+\sqrt{d c^2+e}}\right)}{16 (-d)^{5/2} \sqrt{e}}+\frac{3 i b \text{Li}_2\left(\frac{i c \sqrt{-d} e^{i \csc ^{-1}(c x)}}{\sqrt{e}+\sqrt{d c^2+e}}\right)}{16 (-d)^{5/2} \sqrt{e}}",1,"(a*x)/(4*d*(d + e*x^2)^2) + (3*a*x)/(8*d^2*(d + e*x^2)) + (3*a*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(8*d^(5/2)*Sqrt[e]) + b*((-3*(-(ArcCsc[c*x]/((-I)*Sqrt[d]*Sqrt[e] + e*x)) + (I*(ArcSin[1/(c*x)]/Sqrt[e] - Log[(2*Sqrt[d]*Sqrt[e]*(Sqrt[e] + c*((-I)*c*Sqrt[d] - Sqrt[-(c^2*d) - e]*Sqrt[1 - 1/(c^2*x^2)])*x))/(Sqrt[-(c^2*d) - e]*(Sqrt[d] + I*Sqrt[e]*x))]/Sqrt[-(c^2*d) - e]))/Sqrt[d]))/(16*d^2) - (3*(-(ArcCsc[c*x]/(I*Sqrt[d]*Sqrt[e] + e*x)) - (I*(ArcSin[1/(c*x)]/Sqrt[e] - Log[(2*Sqrt[d]*Sqrt[e]*(-Sqrt[e] + c*((-I)*c*Sqrt[d] + Sqrt[-(c^2*d) - e]*Sqrt[1 - 1/(c^2*x^2)])*x))/(Sqrt[-(c^2*d) - e]*(Sqrt[d] - I*Sqrt[e]*x))]/Sqrt[-(c^2*d) - e]))/Sqrt[d]))/(16*d^2) + ((I/16)*((I*c*Sqrt[e]*Sqrt[1 - 1/(c^2*x^2)]*x)/(Sqrt[d]*(c^2*d + e)*((-I)*Sqrt[d] + Sqrt[e]*x)) - ArcCsc[c*x]/(Sqrt[e]*((-I)*Sqrt[d] + Sqrt[e]*x)^2) - ArcSin[1/(c*x)]/(d*Sqrt[e]) + (I*(2*c^2*d + e)*Log[(4*d*Sqrt[e]*Sqrt[c^2*d + e]*(I*Sqrt[e] + c*(c*Sqrt[d] - Sqrt[c^2*d + e]*Sqrt[1 - 1/(c^2*x^2)])*x))/((2*c^2*d + e)*((-I)*Sqrt[d] + Sqrt[e]*x))])/(d*(c^2*d + e)^(3/2))))/d^(3/2) - ((I/16)*(((-I)*c*Sqrt[e]*Sqrt[1 - 1/(c^2*x^2)]*x)/(Sqrt[d]*(c^2*d + e)*(I*Sqrt[d] + Sqrt[e]*x)) - ArcCsc[c*x]/(Sqrt[e]*(I*Sqrt[d] + Sqrt[e]*x)^2) - ArcSin[1/(c*x)]/(d*Sqrt[e]) + (I*(2*c^2*d + e)*Log[(-4*d*Sqrt[e]*Sqrt[c^2*d + e]*((-I)*Sqrt[e] + c*(c*Sqrt[d] + Sqrt[c^2*d + e]*Sqrt[1 - 1/(c^2*x^2)])*x))/((2*c^2*d + e)*(I*Sqrt[d] + Sqrt[e]*x))])/(d*(c^2*d + e)^(3/2))))/d^(3/2) - (3*(Pi^2 - 4*Pi*ArcCsc[c*x] + 8*ArcCsc[c*x]^2 - 32*ArcSin[Sqrt[1 - (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*ArcTan[(((-I)*c*Sqrt[d] + Sqrt[e])*Cot[(Pi + 2*ArcCsc[c*x])/4])/Sqrt[c^2*d + e]] + (4*I)*Pi*Log[1 + (Sqrt[e] - Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] - (8*I)*ArcCsc[c*x]*Log[1 + (Sqrt[e] - Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] + (16*I)*ArcSin[Sqrt[1 - (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*Log[1 + (Sqrt[e] - Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] + (4*I)*Pi*Log[1 + (Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] - (8*I)*ArcCsc[c*x]*Log[1 + (Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] - (16*I)*ArcSin[Sqrt[1 - (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*Log[1 + (Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] + (8*I)*ArcCsc[c*x]*Log[1 - E^((2*I)*ArcCsc[c*x])] - (4*I)*Pi*Log[Sqrt[e] + (I*Sqrt[d])/x] + 8*PolyLog[2, (-Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] + 8*PolyLog[2, -((Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x])))] + 4*PolyLog[2, E^((2*I)*ArcCsc[c*x])]))/(128*d^(5/2)*Sqrt[e]) + (3*(Pi^2 - 4*Pi*ArcCsc[c*x] + 8*ArcCsc[c*x]^2 - 32*ArcSin[Sqrt[1 + (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*ArcTan[((I*c*Sqrt[d] + Sqrt[e])*Cot[(Pi + 2*ArcCsc[c*x])/4])/Sqrt[c^2*d + e]] + (4*I)*Pi*Log[1 + (-Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] - (8*I)*ArcCsc[c*x]*Log[1 + (-Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] + (16*I)*ArcSin[Sqrt[1 + (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*Log[1 + (-Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] + (4*I)*Pi*Log[1 - (Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] - (8*I)*ArcCsc[c*x]*Log[1 - (Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] - (16*I)*ArcSin[Sqrt[1 + (I*Sqrt[e])/(c*Sqrt[d])]/Sqrt[2]]*Log[1 - (Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] + (8*I)*ArcCsc[c*x]*Log[1 - E^((2*I)*ArcCsc[c*x])] - (4*I)*Pi*Log[Sqrt[e] - (I*Sqrt[d])/x] + 8*PolyLog[2, (Sqrt[e] - Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] + 8*PolyLog[2, (Sqrt[e] + Sqrt[c^2*d + e])/(c*Sqrt[d]*E^(I*ArcCsc[c*x]))] + 4*PolyLog[2, E^((2*I)*ArcCsc[c*x])]))/(128*d^(5/2)*Sqrt[e]))","A",0
118,1,365,403,0.6374104,"\int x^5 \sqrt{d+e x^2} \left(a+b \csc ^{-1}(c x)\right) \, dx","Integrate[x^5*Sqrt[d + e*x^2]*(a + b*ArcCsc[c*x]),x]","\frac{\sqrt{d+e x^2} \left(16 a c^5 \left(8 d^3-4 d^2 e x^2+3 d e^2 x^4+15 e^3 x^6\right)+16 b c^5 \csc ^{-1}(c x) \left(8 d^3-4 d^2 e x^2+3 d e^2 x^4+15 e^3 x^6\right)+b e x \sqrt{1-\frac{1}{c^2 x^2}} \left(c^4 \left(-41 d^2+22 d e x^2+40 e^2 x^4\right)+2 c^2 e \left(19 d+25 e x^2\right)+75 e^2\right)\right)}{1680 c^5 e^3}+\frac{b \left(e x^4 \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{1-c^2 x^2} \left(105 c^6 d^3-35 c^4 d^2 e+63 c^2 d e^2+75 e^3\right) \sqrt{\frac{e x^2}{d}+1} F_1\left(1;\frac{1}{2},\frac{1}{2};2;c^2 x^2,-\frac{e x^2}{d}\right)-128 c^4 d^4 \left(c^2 x^2-1\right) \sqrt{\frac{d}{e x^2}+1} F_1\left(1;\frac{1}{2},\frac{1}{2};2;\frac{1}{c^2 x^2},-\frac{d}{e x^2}\right)\right)}{3360 c^5 e^3 x \left(c^2 x^2-1\right) \sqrt{d+e x^2}}","\frac{d^2 \left(d+e x^2\right)^{3/2} \left(a+b \csc ^{-1}(c x)\right)}{3 e^3}-\frac{2 d \left(d+e x^2\right)^{5/2} \left(a+b \csc ^{-1}(c x)\right)}{5 e^3}+\frac{\left(d+e x^2\right)^{7/2} \left(a+b \csc ^{-1}(c x)\right)}{7 e^3}-\frac{8 b c d^{7/2} x \tan ^{-1}\left(\frac{\sqrt{d+e x^2}}{\sqrt{d} \sqrt{c^2 x^2-1}}\right)}{105 e^3 \sqrt{c^2 x^2}}+\frac{b x \sqrt{c^2 x^2-1} \left(d+e x^2\right)^{5/2}}{42 c e^2 \sqrt{c^2 x^2}}-\frac{b x \sqrt{c^2 x^2-1} \left(29 c^2 d-25 e\right) \left(d+e x^2\right)^{3/2}}{840 c^3 e^2 \sqrt{c^2 x^2}}+\frac{b x \left(105 c^6 d^3-35 c^4 d^2 e+63 c^2 d e^2+75 e^3\right) \tanh ^{-1}\left(\frac{\sqrt{e} \sqrt{c^2 x^2-1}}{c \sqrt{d+e x^2}}\right)}{1680 c^6 e^{5/2} \sqrt{c^2 x^2}}-\frac{b x \sqrt{c^2 x^2-1} \left(23 c^4 d^2+12 c^2 d e-75 e^2\right) \sqrt{d+e x^2}}{1680 c^5 e^2 \sqrt{c^2 x^2}}",1,"(b*(-128*c^4*d^4*Sqrt[1 + d/(e*x^2)]*(-1 + c^2*x^2)*AppellF1[1, 1/2, 1/2, 2, 1/(c^2*x^2), -(d/(e*x^2))] + e*(105*c^6*d^3 - 35*c^4*d^2*e + 63*c^2*d*e^2 + 75*e^3)*Sqrt[1 - 1/(c^2*x^2)]*x^4*Sqrt[1 - c^2*x^2]*Sqrt[1 + (e*x^2)/d]*AppellF1[1, 1/2, 1/2, 2, c^2*x^2, -((e*x^2)/d)]))/(3360*c^5*e^3*x*(-1 + c^2*x^2)*Sqrt[d + e*x^2]) + (Sqrt[d + e*x^2]*(16*a*c^5*(8*d^3 - 4*d^2*e*x^2 + 3*d*e^2*x^4 + 15*e^3*x^6) + b*e*Sqrt[1 - 1/(c^2*x^2)]*x*(75*e^2 + 2*c^2*e*(19*d + 25*e*x^2) + c^4*(-41*d^2 + 22*d*e*x^2 + 40*e^2*x^4)) + 16*b*c^5*(8*d^3 - 4*d^2*e*x^2 + 3*d*e^2*x^4 + 15*e^3*x^6)*ArcCsc[c*x]))/(1680*c^5*e^3)","C",0
119,1,326,294,0.7454302,"\int x^3 \sqrt{d+e x^2} \left(a+b \csc ^{-1}(c x)\right) \, dx","Integrate[x^3*Sqrt[d + e*x^2]*(a + b*ArcCsc[c*x]),x]","\frac{\sqrt{d+e x^2} \left(8 a c^3 \left(-2 d^2+d e x^2+3 e^2 x^4\right)+8 b c^3 \csc ^{-1}(c x) \left(-2 d^2+d e x^2+3 e^2 x^4\right)+b e x \sqrt{1-\frac{1}{c^2 x^2}} \left(c^2 \left(7 d+6 e x^2\right)+9 e\right)\right)}{120 c^3 e^2}-\frac{b x \sqrt{1-\frac{1}{c^2 x^2}} \left(16 c^7 d^{5/2} \sqrt{d+e x^2} \tan ^{-1}\left(\frac{\sqrt{d} \sqrt{c^2 x^2-1}}{\sqrt{d+e x^2}}\right)+\sqrt{c^2} \sqrt{e} \sqrt{c^2 d+e} \left(15 c^4 d^2-10 c^2 d e-9 e^2\right) \sqrt{\frac{c^2 \left(d+e x^2\right)}{c^2 d+e}} \sinh ^{-1}\left(\frac{c \sqrt{e} \sqrt{c^2 x^2-1}}{\sqrt{c^2} \sqrt{c^2 d+e}}\right)\right)}{120 c^6 e^2 \sqrt{c^2 x^2-1} \sqrt{d+e x^2}}","-\frac{d \left(d+e x^2\right)^{3/2} \left(a+b \csc ^{-1}(c x)\right)}{3 e^2}+\frac{\left(d+e x^2\right)^{5/2} \left(a+b \csc ^{-1}(c x)\right)}{5 e^2}+\frac{2 b c d^{5/2} x \tan ^{-1}\left(\frac{\sqrt{d+e x^2}}{\sqrt{d} \sqrt{c^2 x^2-1}}\right)}{15 e^2 \sqrt{c^2 x^2}}+\frac{b x \sqrt{c^2 x^2-1} \left(d+e x^2\right)^{3/2}}{20 c e \sqrt{c^2 x^2}}-\frac{b x \left(15 c^4 d^2-10 c^2 d e-9 e^2\right) \tanh ^{-1}\left(\frac{\sqrt{e} \sqrt{c^2 x^2-1}}{c \sqrt{d+e x^2}}\right)}{120 c^4 e^{3/2} \sqrt{c^2 x^2}}+\frac{b x \sqrt{c^2 x^2-1} \left(c^2 d+9 e\right) \sqrt{d+e x^2}}{120 c^3 e \sqrt{c^2 x^2}}",1,"(Sqrt[d + e*x^2]*(8*a*c^3*(-2*d^2 + d*e*x^2 + 3*e^2*x^4) + b*e*Sqrt[1 - 1/(c^2*x^2)]*x*(9*e + c^2*(7*d + 6*e*x^2)) + 8*b*c^3*(-2*d^2 + d*e*x^2 + 3*e^2*x^4)*ArcCsc[c*x]))/(120*c^3*e^2) - (b*Sqrt[1 - 1/(c^2*x^2)]*x*(Sqrt[c^2]*Sqrt[e]*Sqrt[c^2*d + e]*(15*c^4*d^2 - 10*c^2*d*e - 9*e^2)*Sqrt[(c^2*(d + e*x^2))/(c^2*d + e)]*ArcSinh[(c*Sqrt[e]*Sqrt[-1 + c^2*x^2])/(Sqrt[c^2]*Sqrt[c^2*d + e])] + 16*c^7*d^(5/2)*Sqrt[d + e*x^2]*ArcTan[(Sqrt[d]*Sqrt[-1 + c^2*x^2])/Sqrt[d + e*x^2]]))/(120*c^6*e^2*Sqrt[-1 + c^2*x^2]*Sqrt[d + e*x^2])","A",1
120,1,265,195,0.4833861,"\int x \sqrt{d+e x^2} \left(a+b \csc ^{-1}(c x)\right) \, dx","Integrate[x*Sqrt[d + e*x^2]*(a + b*ArcCsc[c*x]),x]","\frac{\sqrt{d+e x^2} \left(2 a c \left(d+e x^2\right)+b e x \sqrt{1-\frac{1}{c^2 x^2}}+2 b c \csc ^{-1}(c x) \left(d+e x^2\right)\right)}{6 c e}+\frac{b x \sqrt{1-\frac{1}{c^2 x^2}} \left(\sqrt{c^2} \sqrt{e} \sqrt{c^2 d+e} \left(3 c^2 d+e\right) \sqrt{\frac{c^2 \left(d+e x^2\right)}{c^2 d+e}} \sinh ^{-1}\left(\frac{c \sqrt{e} \sqrt{c^2 x^2-1}}{\sqrt{c^2} \sqrt{c^2 d+e}}\right)+2 c^5 d^{3/2} \sqrt{d+e x^2} \tan ^{-1}\left(\frac{\sqrt{d} \sqrt{c^2 x^2-1}}{\sqrt{d+e x^2}}\right)\right)}{6 c^4 e \sqrt{c^2 x^2-1} \sqrt{d+e x^2}}","\frac{\left(d+e x^2\right)^{3/2} \left(a+b \csc ^{-1}(c x)\right)}{3 e}-\frac{b c d^{3/2} x \tan ^{-1}\left(\frac{\sqrt{d+e x^2}}{\sqrt{d} \sqrt{c^2 x^2-1}}\right)}{3 e \sqrt{c^2 x^2}}+\frac{b x \sqrt{c^2 x^2-1} \sqrt{d+e x^2}}{6 c \sqrt{c^2 x^2}}+\frac{b x \left(3 c^2 d+e\right) \tanh ^{-1}\left(\frac{\sqrt{e} \sqrt{c^2 x^2-1}}{c \sqrt{d+e x^2}}\right)}{6 c^2 \sqrt{e} \sqrt{c^2 x^2}}",1,"(Sqrt[d + e*x^2]*(b*e*Sqrt[1 - 1/(c^2*x^2)]*x + 2*a*c*(d + e*x^2) + 2*b*c*(d + e*x^2)*ArcCsc[c*x]))/(6*c*e) + (b*Sqrt[1 - 1/(c^2*x^2)]*x*(Sqrt[c^2]*Sqrt[e]*Sqrt[c^2*d + e]*(3*c^2*d + e)*Sqrt[(c^2*(d + e*x^2))/(c^2*d + e)]*ArcSinh[(c*Sqrt[e]*Sqrt[-1 + c^2*x^2])/(Sqrt[c^2]*Sqrt[c^2*d + e])] + 2*c^5*d^(3/2)*Sqrt[d + e*x^2]*ArcTan[(Sqrt[d]*Sqrt[-1 + c^2*x^2])/Sqrt[d + e*x^2]]))/(6*c^4*e*Sqrt[-1 + c^2*x^2]*Sqrt[d + e*x^2])","A",1
121,0,0,26,5.2163761,"\int \frac{\sqrt{d+e x^2} \left(a+b \csc ^{-1}(c x)\right)}{x} \, dx","Integrate[(Sqrt[d + e*x^2]*(a + b*ArcCsc[c*x]))/x,x]","\int \frac{\sqrt{d+e x^2} \left(a+b \csc ^{-1}(c x)\right)}{x} \, dx","\text{Int}\left(\frac{\sqrt{d+e x^2} \left(a+b \csc ^{-1}(c x)\right)}{x},x\right)",0,"Integrate[(Sqrt[d + e*x^2]*(a + b*ArcCsc[c*x]))/x, x]","A",-1
122,0,0,26,4.8154988,"\int \frac{\sqrt{d+e x^2} \left(a+b \csc ^{-1}(c x)\right)}{x^3} \, dx","Integrate[(Sqrt[d + e*x^2]*(a + b*ArcCsc[c*x]))/x^3,x]","\int \frac{\sqrt{d+e x^2} \left(a+b \csc ^{-1}(c x)\right)}{x^3} \, dx","\text{Int}\left(\frac{\sqrt{d+e x^2} \left(a+b \csc ^{-1}(c x)\right)}{x^3},x\right)",0,"Integrate[(Sqrt[d + e*x^2]*(a + b*ArcCsc[c*x]))/x^3, x]","A",-1
123,0,0,26,9.5708907,"\int x^2 \sqrt{d+e x^2} \left(a+b \csc ^{-1}(c x)\right) \, dx","Integrate[x^2*Sqrt[d + e*x^2]*(a + b*ArcCsc[c*x]),x]","\int x^2 \sqrt{d+e x^2} \left(a+b \csc ^{-1}(c x)\right) \, dx","\text{Int}\left(x^2 \sqrt{d+e x^2} \left(a+b \csc ^{-1}(c x)\right),x\right)",0,"Integrate[x^2*Sqrt[d + e*x^2]*(a + b*ArcCsc[c*x]), x]","A",-1
124,0,0,23,6.6738221,"\int \sqrt{d+e x^2} \left(a+b \csc ^{-1}(c x)\right) \, dx","Integrate[Sqrt[d + e*x^2]*(a + b*ArcCsc[c*x]),x]","\int \sqrt{d+e x^2} \left(a+b \csc ^{-1}(c x)\right) \, dx","\text{Int}\left(\sqrt{d+e x^2} \left(a+b \csc ^{-1}(c x)\right),x\right)",0,"Integrate[Sqrt[d + e*x^2]*(a + b*ArcCsc[c*x]), x]","A",-1
125,0,0,26,1.7413586,"\int \frac{\sqrt{d+e x^2} \left(a+b \csc ^{-1}(c x)\right)}{x^2} \, dx","Integrate[(Sqrt[d + e*x^2]*(a + b*ArcCsc[c*x]))/x^2,x]","\int \frac{\sqrt{d+e x^2} \left(a+b \csc ^{-1}(c x)\right)}{x^2} \, dx","\text{Int}\left(\frac{\sqrt{d+e x^2} \left(a+b \csc ^{-1}(c x)\right)}{x^2},x\right)",0,"Integrate[(Sqrt[d + e*x^2]*(a + b*ArcCsc[c*x]))/x^2, x]","A",-1
126,1,247,328,0.6505329,"\int \frac{\sqrt{d+e x^2} \left(a+b \csc ^{-1}(c x)\right)}{x^4} \, dx","Integrate[(Sqrt[d + e*x^2]*(a + b*ArcCsc[c*x]))/x^4,x]","-\frac{\sqrt{d+e x^2} \left(3 a \left(d+e x^2\right)+b c x \sqrt{1-\frac{1}{c^2 x^2}} \left(2 c^2 d x^2+d+4 e x^2\right)+3 b \csc ^{-1}(c x) \left(d+e x^2\right)\right)}{9 d x^3}+\frac{i b c x \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{\frac{e x^2}{d}+1} \left(2 c^2 d \left(c^2 d+2 e\right) E\left(i \sinh ^{-1}\left(\sqrt{-c^2} x\right)|-\frac{e}{c^2 d}\right)-\left(2 c^4 d^2+5 c^2 d e+3 e^2\right) F\left(i \sinh ^{-1}\left(\sqrt{-c^2} x\right)|-\frac{e}{c^2 d}\right)\right)}{9 \sqrt{-c^2} d \sqrt{1-c^2 x^2} \sqrt{d+e x^2}}","-\frac{\left(d+e x^2\right)^{3/2} \left(a+b \csc ^{-1}(c x)\right)}{3 d x^3}-\frac{2 b c \sqrt{c^2 x^2-1} \left(c^2 d+2 e\right) \sqrt{d+e x^2}}{9 d \sqrt{c^2 x^2}}-\frac{b c \sqrt{c^2 x^2-1} \sqrt{d+e x^2}}{9 x^2 \sqrt{c^2 x^2}}-\frac{b x \sqrt{1-c^2 x^2} \left(c^2 d+e\right) \left(2 c^2 d+3 e\right) \sqrt{\frac{e x^2}{d}+1} F\left(\sin ^{-1}(c x)|-\frac{e}{c^2 d}\right)}{9 d \sqrt{c^2 x^2} \sqrt{c^2 x^2-1} \sqrt{d+e x^2}}+\frac{2 b c^2 x \sqrt{1-c^2 x^2} \left(c^2 d+2 e\right) \sqrt{d+e x^2} E\left(\sin ^{-1}(c x)|-\frac{e}{c^2 d}\right)}{9 d \sqrt{c^2 x^2} \sqrt{c^2 x^2-1} \sqrt{\frac{e x^2}{d}+1}}",1,"-1/9*(Sqrt[d + e*x^2]*(3*a*(d + e*x^2) + b*c*Sqrt[1 - 1/(c^2*x^2)]*x*(d + 2*c^2*d*x^2 + 4*e*x^2) + 3*b*(d + e*x^2)*ArcCsc[c*x]))/(d*x^3) + ((I/9)*b*c*Sqrt[1 - 1/(c^2*x^2)]*x*Sqrt[1 + (e*x^2)/d]*(2*c^2*d*(c^2*d + 2*e)*EllipticE[I*ArcSinh[Sqrt[-c^2]*x], -(e/(c^2*d))] - (2*c^4*d^2 + 5*c^2*d*e + 3*e^2)*EllipticF[I*ArcSinh[Sqrt[-c^2]*x], -(e/(c^2*d))]))/(Sqrt[-c^2]*d*Sqrt[1 - c^2*x^2]*Sqrt[d + e*x^2])","C",1
127,1,325,453,0.7213891,"\int \frac{\sqrt{d+e x^2} \left(a+b \csc ^{-1}(c x)\right)}{x^6} \, dx","Integrate[(Sqrt[d + e*x^2]*(a + b*ArcCsc[c*x]))/x^6,x]","-\frac{\sqrt{d+e x^2} \left(15 a \left(3 d^2+d e x^2-2 e^2 x^4\right)+b c x \sqrt{1-\frac{1}{c^2 x^2}} \left(d e x^2 \left(19 c^2 x^2+8\right)+3 d^2 \left(8 c^4 x^4+4 c^2 x^2+3\right)-31 e^2 x^4\right)+15 b \csc ^{-1}(c x) \left(3 d^2+d e x^2-2 e^2 x^4\right)\right)}{225 d^2 x^5}+\frac{i b c x \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{\frac{e x^2}{d}+1} \left(c^2 d \left(24 c^4 d^2+19 c^2 d e-31 e^2\right) E\left(i \sinh ^{-1}\left(\sqrt{-c^2} x\right)|-\frac{e}{c^2 d}\right)+\left(-24 c^6 d^3-31 c^4 d^2 e+23 c^2 d e^2+30 e^3\right) F\left(i \sinh ^{-1}\left(\sqrt{-c^2} x\right)|-\frac{e}{c^2 d}\right)\right)}{225 \sqrt{-c^2} d^2 \sqrt{1-c^2 x^2} \sqrt{d+e x^2}}","\frac{2 e \left(d+e x^2\right)^{3/2} \left(a+b \csc ^{-1}(c x)\right)}{15 d^2 x^3}-\frac{\left(d+e x^2\right)^{3/2} \left(a+b \csc ^{-1}(c x)\right)}{5 d x^5}-\frac{b c \sqrt{c^2 x^2-1} \left(12 c^2 d-e\right) \sqrt{d+e x^2}}{225 d x^2 \sqrt{c^2 x^2}}-\frac{b c \sqrt{c^2 x^2-1} \left(d+e x^2\right)^{3/2}}{25 d x^4 \sqrt{c^2 x^2}}-\frac{b c \sqrt{c^2 x^2-1} \left(24 c^4 d^2+19 c^2 d e-31 e^2\right) \sqrt{d+e x^2}}{225 d^2 \sqrt{c^2 x^2}}-\frac{b x \sqrt{1-c^2 x^2} \left(c^2 d+e\right) \left(24 c^4 d^2+7 c^2 d e-30 e^2\right) \sqrt{\frac{e x^2}{d}+1} F\left(\sin ^{-1}(c x)|-\frac{e}{c^2 d}\right)}{225 d^2 \sqrt{c^2 x^2} \sqrt{c^2 x^2-1} \sqrt{d+e x^2}}+\frac{b c^2 x \sqrt{1-c^2 x^2} \left(24 c^4 d^2+19 c^2 d e-31 e^2\right) \sqrt{d+e x^2} E\left(\sin ^{-1}(c x)|-\frac{e}{c^2 d}\right)}{225 d^2 \sqrt{c^2 x^2} \sqrt{c^2 x^2-1} \sqrt{\frac{e x^2}{d}+1}}",1,"-1/225*(Sqrt[d + e*x^2]*(15*a*(3*d^2 + d*e*x^2 - 2*e^2*x^4) + b*c*Sqrt[1 - 1/(c^2*x^2)]*x*(-31*e^2*x^4 + d*e*x^2*(8 + 19*c^2*x^2) + 3*d^2*(3 + 4*c^2*x^2 + 8*c^4*x^4)) + 15*b*(3*d^2 + d*e*x^2 - 2*e^2*x^4)*ArcCsc[c*x]))/(d^2*x^5) + ((I/225)*b*c*Sqrt[1 - 1/(c^2*x^2)]*x*Sqrt[1 + (e*x^2)/d]*(c^2*d*(24*c^4*d^2 + 19*c^2*d*e - 31*e^2)*EllipticE[I*ArcSinh[Sqrt[-c^2]*x], -(e/(c^2*d))] + (-24*c^6*d^3 - 31*c^4*d^2*e + 23*c^2*d*e^2 + 30*e^3)*EllipticF[I*ArcSinh[Sqrt[-c^2]*x], -(e/(c^2*d))]))/(Sqrt[-c^2]*d^2*Sqrt[1 - c^2*x^2]*Sqrt[d + e*x^2])","C",1
128,1,339,374,0.6174109,"\int x^3 \left(d+e x^2\right)^{3/2} \left(a+b \csc ^{-1}(c x)\right) \, dx","Integrate[x^3*(d + e*x^2)^(3/2)*(a + b*ArcCsc[c*x]),x]","\frac{\sqrt{d+e x^2} \left(-48 a c^5 \left(2 d-5 e x^2\right) \left(d+e x^2\right)^2-48 b c^5 \csc ^{-1}(c x) \left(2 d-5 e x^2\right) \left(d+e x^2\right)^2+b e x \sqrt{1-\frac{1}{c^2 x^2}} \left(c^4 \left(57 d^2+106 d e x^2+40 e^2 x^4\right)+2 c^2 e \left(82 d+25 e x^2\right)+75 e^2\right)\right)}{1680 c^5 e^2}+\frac{b \left(32 c^4 d^4 \left(c^2 x^2-1\right) \sqrt{\frac{d}{e x^2}+1} F_1\left(1;\frac{1}{2},\frac{1}{2};2;\frac{1}{c^2 x^2},-\frac{d}{e x^2}\right)+e x^4 \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{1-c^2 x^2} \left(-35 c^6 d^3+35 c^4 d^2 e+63 c^2 d e^2+25 e^3\right) \sqrt{\frac{e x^2}{d}+1} F_1\left(1;\frac{1}{2},\frac{1}{2};2;c^2 x^2,-\frac{e x^2}{d}\right)\right)}{1120 c^5 e^2 x \left(c^2 x^2-1\right) \sqrt{d+e x^2}}","-\frac{d \left(d+e x^2\right)^{5/2} \left(a+b \csc ^{-1}(c x)\right)}{5 e^2}+\frac{\left(d+e x^2\right)^{7/2} \left(a+b \csc ^{-1}(c x)\right)}{7 e^2}+\frac{2 b c d^{7/2} x \tan ^{-1}\left(\frac{\sqrt{d+e x^2}}{\sqrt{d} \sqrt{c^2 x^2-1}}\right)}{35 e^2 \sqrt{c^2 x^2}}+\frac{b x \sqrt{c^2 x^2-1} \left(d+e x^2\right)^{5/2}}{42 c e \sqrt{c^2 x^2}}+\frac{b x \sqrt{c^2 x^2-1} \left(13 c^2 d+25 e\right) \left(d+e x^2\right)^{3/2}}{840 c^3 e \sqrt{c^2 x^2}}-\frac{b x \left(35 c^6 d^3-35 c^4 d^2 e-63 c^2 d e^2-25 e^3\right) \tanh ^{-1}\left(\frac{\sqrt{e} \sqrt{c^2 x^2-1}}{c \sqrt{d+e x^2}}\right)}{560 c^6 e^{3/2} \sqrt{c^2 x^2}}-\frac{b x \sqrt{c^2 x^2-1} \left(3 c^4 d^2-38 c^2 d e-25 e^2\right) \sqrt{d+e x^2}}{560 c^5 e \sqrt{c^2 x^2}}",1,"(b*(32*c^4*d^4*Sqrt[1 + d/(e*x^2)]*(-1 + c^2*x^2)*AppellF1[1, 1/2, 1/2, 2, 1/(c^2*x^2), -(d/(e*x^2))] + e*(-35*c^6*d^3 + 35*c^4*d^2*e + 63*c^2*d*e^2 + 25*e^3)*Sqrt[1 - 1/(c^2*x^2)]*x^4*Sqrt[1 - c^2*x^2]*Sqrt[1 + (e*x^2)/d]*AppellF1[1, 1/2, 1/2, 2, c^2*x^2, -((e*x^2)/d)]))/(1120*c^5*e^2*x*(-1 + c^2*x^2)*Sqrt[d + e*x^2]) + (Sqrt[d + e*x^2]*(-48*a*c^5*(2*d - 5*e*x^2)*(d + e*x^2)^2 + b*e*Sqrt[1 - 1/(c^2*x^2)]*x*(75*e^2 + 2*c^2*e*(82*d + 25*e*x^2) + c^4*(57*d^2 + 106*d*e*x^2 + 40*e^2*x^4)) - 48*b*c^5*(2*d - 5*e*x^2)*(d + e*x^2)^2*ArcCsc[c*x]))/(1680*c^5*e^2)","C",0
129,1,304,262,0.7578157,"\int x \left(d+e x^2\right)^{3/2} \left(a+b \csc ^{-1}(c x)\right) \, dx","Integrate[x*(d + e*x^2)^(3/2)*(a + b*ArcCsc[c*x]),x]","\frac{\sqrt{d+e x^2} \left(8 a c^3 \left(d+e x^2\right)^2+8 b c^3 \csc ^{-1}(c x) \left(d+e x^2\right)^2+b e x \sqrt{1-\frac{1}{c^2 x^2}} \left(c^2 \left(9 d+2 e x^2\right)+3 e\right)\right)}{40 c^3 e}+\frac{b x \sqrt{1-\frac{1}{c^2 x^2}} \left(8 c^7 d^{5/2} \sqrt{d+e x^2} \tan ^{-1}\left(\frac{\sqrt{d} \sqrt{c^2 x^2-1}}{\sqrt{d+e x^2}}\right)+\sqrt{c^2} \sqrt{e} \sqrt{c^2 d+e} \left(15 c^4 d^2+10 c^2 d e+3 e^2\right) \sqrt{\frac{c^2 \left(d+e x^2\right)}{c^2 d+e}} \sinh ^{-1}\left(\frac{c \sqrt{e} \sqrt{c^2 x^2-1}}{\sqrt{c^2} \sqrt{c^2 d+e}}\right)\right)}{40 c^6 e \sqrt{c^2 x^2-1} \sqrt{d+e x^2}}","\frac{\left(d+e x^2\right)^{5/2} \left(a+b \csc ^{-1}(c x)\right)}{5 e}-\frac{b c d^{5/2} x \tan ^{-1}\left(\frac{\sqrt{d+e x^2}}{\sqrt{d} \sqrt{c^2 x^2-1}}\right)}{5 e \sqrt{c^2 x^2}}+\frac{b x \sqrt{c^2 x^2-1} \left(d+e x^2\right)^{3/2}}{20 c \sqrt{c^2 x^2}}+\frac{b x \left(15 c^4 d^2+10 c^2 d e+3 e^2\right) \tanh ^{-1}\left(\frac{\sqrt{e} \sqrt{c^2 x^2-1}}{c \sqrt{d+e x^2}}\right)}{40 c^4 \sqrt{e} \sqrt{c^2 x^2}}+\frac{b x \sqrt{c^2 x^2-1} \left(7 c^2 d+3 e\right) \sqrt{d+e x^2}}{40 c^3 \sqrt{c^2 x^2}}",1,"(Sqrt[d + e*x^2]*(8*a*c^3*(d + e*x^2)^2 + b*e*Sqrt[1 - 1/(c^2*x^2)]*x*(3*e + c^2*(9*d + 2*e*x^2)) + 8*b*c^3*(d + e*x^2)^2*ArcCsc[c*x]))/(40*c^3*e) + (b*Sqrt[1 - 1/(c^2*x^2)]*x*(Sqrt[c^2]*Sqrt[e]*Sqrt[c^2*d + e]*(15*c^4*d^2 + 10*c^2*d*e + 3*e^2)*Sqrt[(c^2*(d + e*x^2))/(c^2*d + e)]*ArcSinh[(c*Sqrt[e]*Sqrt[-1 + c^2*x^2])/(Sqrt[c^2]*Sqrt[c^2*d + e])] + 8*c^7*d^(5/2)*Sqrt[d + e*x^2]*ArcTan[(Sqrt[d]*Sqrt[-1 + c^2*x^2])/Sqrt[d + e*x^2]]))/(40*c^6*e*Sqrt[-1 + c^2*x^2]*Sqrt[d + e*x^2])","A",1
130,0,0,26,6.0852186,"\int \frac{\left(d+e x^2\right)^{3/2} \left(a+b \csc ^{-1}(c x)\right)}{x} \, dx","Integrate[((d + e*x^2)^(3/2)*(a + b*ArcCsc[c*x]))/x,x]","\int \frac{\left(d+e x^2\right)^{3/2} \left(a+b \csc ^{-1}(c x)\right)}{x} \, dx","\text{Int}\left(\frac{\left(d+e x^2\right)^{3/2} \left(a+b \csc ^{-1}(c x)\right)}{x},x\right)",0,"Integrate[((d + e*x^2)^(3/2)*(a + b*ArcCsc[c*x]))/x, x]","A",-1
131,0,0,26,6.0561563,"\int \frac{\left(d+e x^2\right)^{3/2} \left(a+b \csc ^{-1}(c x)\right)}{x^3} \, dx","Integrate[((d + e*x^2)^(3/2)*(a + b*ArcCsc[c*x]))/x^3,x]","\int \frac{\left(d+e x^2\right)^{3/2} \left(a+b \csc ^{-1}(c x)\right)}{x^3} \, dx","\text{Int}\left(\frac{\left(d+e x^2\right)^{3/2} \left(a+b \csc ^{-1}(c x)\right)}{x^3},x\right)",0,"Integrate[((d + e*x^2)^(3/2)*(a + b*ArcCsc[c*x]))/x^3, x]","A",-1
132,0,0,26,9.913907,"\int x^2 \left(d+e x^2\right)^{3/2} \left(a+b \csc ^{-1}(c x)\right) \, dx","Integrate[x^2*(d + e*x^2)^(3/2)*(a + b*ArcCsc[c*x]),x]","\int x^2 \left(d+e x^2\right)^{3/2} \left(a+b \csc ^{-1}(c x)\right) \, dx","\text{Int}\left(x^2 \left(d+e x^2\right)^{3/2} \left(a+b \csc ^{-1}(c x)\right),x\right)",0,"Integrate[x^2*(d + e*x^2)^(3/2)*(a + b*ArcCsc[c*x]), x]","A",-1
133,0,0,23,8.4731343,"\int \left(d+e x^2\right)^{3/2} \left(a+b \csc ^{-1}(c x)\right) \, dx","Integrate[(d + e*x^2)^(3/2)*(a + b*ArcCsc[c*x]),x]","\int \left(d+e x^2\right)^{3/2} \left(a+b \csc ^{-1}(c x)\right) \, dx","\text{Int}\left(\left(d+e x^2\right)^{3/2} \left(a+b \csc ^{-1}(c x)\right),x\right)",0,"Integrate[(d + e*x^2)^(3/2)*(a + b*ArcCsc[c*x]), x]","A",-1
134,0,0,26,13.6488974,"\int \frac{\left(d+e x^2\right)^{3/2} \left(a+b \csc ^{-1}(c x)\right)}{x^2} \, dx","Integrate[((d + e*x^2)^(3/2)*(a + b*ArcCsc[c*x]))/x^2,x]","\int \frac{\left(d+e x^2\right)^{3/2} \left(a+b \csc ^{-1}(c x)\right)}{x^2} \, dx","\text{Int}\left(\frac{\left(d+e x^2\right)^{3/2} \left(a+b \csc ^{-1}(c x)\right)}{x^2},x\right)",0,"Integrate[((d + e*x^2)^(3/2)*(a + b*ArcCsc[c*x]))/x^2, x]","A",-1
135,0,0,26,6.4875096,"\int \frac{\left(d+e x^2\right)^{3/2} \left(a+b \csc ^{-1}(c x)\right)}{x^4} \, dx","Integrate[((d + e*x^2)^(3/2)*(a + b*ArcCsc[c*x]))/x^4,x]","\int \frac{\left(d+e x^2\right)^{3/2} \left(a+b \csc ^{-1}(c x)\right)}{x^4} \, dx","\text{Int}\left(\frac{\left(d+e x^2\right)^{3/2} \left(a+b \csc ^{-1}(c x)\right)}{x^4},x\right)",0,"Integrate[((d + e*x^2)^(3/2)*(a + b*ArcCsc[c*x]))/x^4, x]","A",-1
136,1,303,416,0.6869078,"\int \frac{\left(d+e x^2\right)^{3/2} \left(a+b \csc ^{-1}(c x)\right)}{x^6} \, dx","Integrate[((d + e*x^2)^(3/2)*(a + b*ArcCsc[c*x]))/x^6,x]","-\frac{\sqrt{d+e x^2} \left(15 a \left(d+e x^2\right)^2+b c x \sqrt{1-\frac{1}{c^2 x^2}} \left(d e x^2 \left(23 c^2 x^2+11\right)+d^2 \left(8 c^4 x^4+4 c^2 x^2+3\right)+23 e^2 x^4\right)+15 b \csc ^{-1}(c x) \left(d+e x^2\right)^2\right)}{75 d x^5}+\frac{i b c x \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{\frac{e x^2}{d}+1} \left(c^2 d \left(8 c^4 d^2+23 c^2 d e+23 e^2\right) E\left(i \sinh ^{-1}\left(\sqrt{-c^2} x\right)|-\frac{e}{c^2 d}\right)-\left(8 c^6 d^3+27 c^4 d^2 e+34 c^2 d e^2+15 e^3\right) F\left(i \sinh ^{-1}\left(\sqrt{-c^2} x\right)|-\frac{e}{c^2 d}\right)\right)}{75 \sqrt{-c^2} d \sqrt{1-c^2 x^2} \sqrt{d+e x^2}}","-\frac{\left(d+e x^2\right)^{5/2} \left(a+b \csc ^{-1}(c x)\right)}{5 d x^5}-\frac{4 b c \sqrt{c^2 x^2-1} \left(c^2 d+2 e\right) \sqrt{d+e x^2}}{75 x^2 \sqrt{c^2 x^2}}-\frac{b c \sqrt{c^2 x^2-1} \left(d+e x^2\right)^{3/2}}{25 x^4 \sqrt{c^2 x^2}}-\frac{b c \sqrt{c^2 x^2-1} \left(8 c^4 d^2+23 c^2 d e+23 e^2\right) \sqrt{d+e x^2}}{75 d \sqrt{c^2 x^2}}-\frac{b x \sqrt{1-c^2 x^2} \left(c^2 d+e\right) \left(8 c^4 d^2+19 c^2 d e+15 e^2\right) \sqrt{\frac{e x^2}{d}+1} F\left(\sin ^{-1}(c x)|-\frac{e}{c^2 d}\right)}{75 d \sqrt{c^2 x^2} \sqrt{c^2 x^2-1} \sqrt{d+e x^2}}+\frac{b c^2 x \sqrt{1-c^2 x^2} \left(8 c^4 d^2+23 c^2 d e+23 e^2\right) \sqrt{d+e x^2} E\left(\sin ^{-1}(c x)|-\frac{e}{c^2 d}\right)}{75 d \sqrt{c^2 x^2} \sqrt{c^2 x^2-1} \sqrt{\frac{e x^2}{d}+1}}",1,"-1/75*(Sqrt[d + e*x^2]*(15*a*(d + e*x^2)^2 + b*c*Sqrt[1 - 1/(c^2*x^2)]*x*(23*e^2*x^4 + d*e*x^2*(11 + 23*c^2*x^2) + d^2*(3 + 4*c^2*x^2 + 8*c^4*x^4)) + 15*b*(d + e*x^2)^2*ArcCsc[c*x]))/(d*x^5) + ((I/75)*b*c*Sqrt[1 - 1/(c^2*x^2)]*x*Sqrt[1 + (e*x^2)/d]*(c^2*d*(8*c^4*d^2 + 23*c^2*d*e + 23*e^2)*EllipticE[I*ArcSinh[Sqrt[-c^2]*x], -(e/(c^2*d))] - (8*c^6*d^3 + 27*c^4*d^2*e + 34*c^2*d*e^2 + 15*e^3)*EllipticF[I*ArcSinh[Sqrt[-c^2]*x], -(e/(c^2*d))]))/(Sqrt[-c^2]*d*Sqrt[1 - c^2*x^2]*Sqrt[d + e*x^2])","C",1
137,1,383,554,0.8746699,"\int \frac{\left(d+e x^2\right)^{3/2} \left(a+b \csc ^{-1}(c x)\right)}{x^8} \, dx","Integrate[((d + e*x^2)^(3/2)*(a + b*ArcCsc[c*x]))/x^8,x]","-\frac{\sqrt{d+e x^2} \left(105 a \left(5 d-2 e x^2\right) \left(d+e x^2\right)^2+b c x \sqrt{1-\frac{1}{c^2 x^2}} \left(d e^2 x^4 \left(193 c^2 x^2+71\right)+3 d^2 e x^2 \left(176 c^4 x^4+83 c^2 x^2+61\right)+15 d^3 \left(16 c^6 x^6+8 c^4 x^4+6 c^2 x^2+5\right)-247 e^3 x^6\right)+105 b \csc ^{-1}(c x) \left(5 d-2 e x^2\right) \left(d+e x^2\right)^2\right)}{3675 d^2 x^7}+\frac{i b c x \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{\frac{e x^2}{d}+1} \left(c^2 d \left(240 c^6 d^3+528 c^4 d^2 e+193 c^2 d e^2-247 e^3\right) E\left(i \sinh ^{-1}\left(\sqrt{-c^2} x\right)|-\frac{e}{c^2 d}\right)-2 \left(120 c^8 d^4+324 c^6 d^3 e+221 c^4 d^2 e^2-88 c^2 d e^3-105 e^4\right) F\left(i \sinh ^{-1}\left(\sqrt{-c^2} x\right)|-\frac{e}{c^2 d}\right)\right)}{3675 \sqrt{-c^2} d^2 \sqrt{1-c^2 x^2} \sqrt{d+e x^2}}","\frac{2 e \left(d+e x^2\right)^{5/2} \left(a+b \csc ^{-1}(c x)\right)}{35 d^2 x^5}-\frac{\left(d+e x^2\right)^{5/2} \left(a+b \csc ^{-1}(c x)\right)}{7 d x^7}-\frac{b c \sqrt{c^2 x^2-1} \left(d+e x^2\right)^{5/2}}{49 d x^6 \sqrt{c^2 x^2}}-\frac{b c \sqrt{c^2 x^2-1} \left(30 c^2 d+11 e\right) \left(d+e x^2\right)^{3/2}}{1225 d x^4 \sqrt{c^2 x^2}}-\frac{b c \sqrt{c^2 x^2-1} \left(120 c^4 d^2+159 c^2 d e-37 e^2\right) \sqrt{d+e x^2}}{3675 d x^2 \sqrt{c^2 x^2}}-\frac{b c \sqrt{c^2 x^2-1} \left(240 c^6 d^3+528 c^4 d^2 e+193 c^2 d e^2-247 e^3\right) \sqrt{d+e x^2}}{3675 d^2 \sqrt{c^2 x^2}}-\frac{2 b x \sqrt{1-c^2 x^2} \left(c^2 d+e\right) \left(120 c^6 d^3+204 c^4 d^2 e+17 c^2 d e^2-105 e^3\right) \sqrt{\frac{e x^2}{d}+1} F\left(\sin ^{-1}(c x)|-\frac{e}{c^2 d}\right)}{3675 d^2 \sqrt{c^2 x^2} \sqrt{c^2 x^2-1} \sqrt{d+e x^2}}+\frac{b c^2 x \sqrt{1-c^2 x^2} \left(240 c^6 d^3+528 c^4 d^2 e+193 c^2 d e^2-247 e^3\right) \sqrt{d+e x^2} E\left(\sin ^{-1}(c x)|-\frac{e}{c^2 d}\right)}{3675 d^2 \sqrt{c^2 x^2} \sqrt{c^2 x^2-1} \sqrt{\frac{e x^2}{d}+1}}",1,"-1/3675*(Sqrt[d + e*x^2]*(105*a*(5*d - 2*e*x^2)*(d + e*x^2)^2 + b*c*Sqrt[1 - 1/(c^2*x^2)]*x*(-247*e^3*x^6 + d*e^2*x^4*(71 + 193*c^2*x^2) + 3*d^2*e*x^2*(61 + 83*c^2*x^2 + 176*c^4*x^4) + 15*d^3*(5 + 6*c^2*x^2 + 8*c^4*x^4 + 16*c^6*x^6)) + 105*b*(5*d - 2*e*x^2)*(d + e*x^2)^2*ArcCsc[c*x]))/(d^2*x^7) + ((I/3675)*b*c*Sqrt[1 - 1/(c^2*x^2)]*x*Sqrt[1 + (e*x^2)/d]*(c^2*d*(240*c^6*d^3 + 528*c^4*d^2*e + 193*c^2*d*e^2 - 247*e^3)*EllipticE[I*ArcSinh[Sqrt[-c^2]*x], -(e/(c^2*d))] - 2*(120*c^8*d^4 + 324*c^6*d^3*e + 221*c^4*d^2*e^2 - 88*c^2*d*e^3 - 105*e^4)*EllipticF[I*ArcSinh[Sqrt[-c^2]*x], -(e/(c^2*d))]))/(Sqrt[-c^2]*d^2*Sqrt[1 - c^2*x^2]*Sqrt[d + e*x^2])","C",1
138,1,328,321,0.7482152,"\int \frac{x^5 \left(a+b \csc ^{-1}(c x)\right)}{\sqrt{d+e x^2}} \, dx","Integrate[(x^5*(a + b*ArcCsc[c*x]))/Sqrt[d + e*x^2],x]","\frac{\sqrt{d+e x^2} \left(8 a c^3 \left(8 d^2-4 d e x^2+3 e^2 x^4\right)+8 b c^3 \csc ^{-1}(c x) \left(8 d^2-4 d e x^2+3 e^2 x^4\right)+b e x \sqrt{1-\frac{1}{c^2 x^2}} \left(c^2 \left(6 e x^2-13 d\right)+9 e\right)\right)}{120 c^3 e^3}+\frac{b x \sqrt{1-\frac{1}{c^2 x^2}} \left(64 c^7 d^{5/2} \sqrt{d+e x^2} \tan ^{-1}\left(\frac{\sqrt{d} \sqrt{c^2 x^2-1}}{\sqrt{d+e x^2}}\right)+\sqrt{c^2} \sqrt{e} \sqrt{c^2 d+e} \left(45 c^4 d^2-10 c^2 d e+9 e^2\right) \sqrt{\frac{c^2 \left(d+e x^2\right)}{c^2 d+e}} \sinh ^{-1}\left(\frac{c \sqrt{e} \sqrt{c^2 x^2-1}}{\sqrt{c^2} \sqrt{c^2 d+e}}\right)\right)}{120 c^6 e^3 \sqrt{c^2 x^2-1} \sqrt{d+e x^2}}","\frac{d^2 \sqrt{d+e x^2} \left(a+b \csc ^{-1}(c x)\right)}{e^3}-\frac{2 d \left(d+e x^2\right)^{3/2} \left(a+b \csc ^{-1}(c x)\right)}{3 e^3}+\frac{\left(d+e x^2\right)^{5/2} \left(a+b \csc ^{-1}(c x)\right)}{5 e^3}-\frac{8 b c d^{5/2} x \tan ^{-1}\left(\frac{\sqrt{d+e x^2}}{\sqrt{d} \sqrt{c^2 x^2-1}}\right)}{15 e^3 \sqrt{c^2 x^2}}+\frac{b x \sqrt{c^2 x^2-1} \left(d+e x^2\right)^{3/2}}{20 c e^2 \sqrt{c^2 x^2}}+\frac{b x \left(45 c^4 d^2-10 c^2 d e+9 e^2\right) \tanh ^{-1}\left(\frac{\sqrt{e} \sqrt{c^2 x^2-1}}{c \sqrt{d+e x^2}}\right)}{120 c^4 e^{5/2} \sqrt{c^2 x^2}}-\frac{b x \sqrt{c^2 x^2-1} \left(19 c^2 d-9 e\right) \sqrt{d+e x^2}}{120 c^3 e^2 \sqrt{c^2 x^2}}",1,"(Sqrt[d + e*x^2]*(8*a*c^3*(8*d^2 - 4*d*e*x^2 + 3*e^2*x^4) + b*e*Sqrt[1 - 1/(c^2*x^2)]*x*(9*e + c^2*(-13*d + 6*e*x^2)) + 8*b*c^3*(8*d^2 - 4*d*e*x^2 + 3*e^2*x^4)*ArcCsc[c*x]))/(120*c^3*e^3) + (b*Sqrt[1 - 1/(c^2*x^2)]*x*(Sqrt[c^2]*Sqrt[e]*Sqrt[c^2*d + e]*(45*c^4*d^2 - 10*c^2*d*e + 9*e^2)*Sqrt[(c^2*(d + e*x^2))/(c^2*d + e)]*ArcSinh[(c*Sqrt[e]*Sqrt[-1 + c^2*x^2])/(Sqrt[c^2]*Sqrt[c^2*d + e])] + 64*c^7*d^(5/2)*Sqrt[d + e*x^2]*ArcTan[(Sqrt[d]*Sqrt[-1 + c^2*x^2])/Sqrt[d + e*x^2]]))/(120*c^6*e^3*Sqrt[-1 + c^2*x^2]*Sqrt[d + e*x^2])","A",1
139,1,271,225,0.528437,"\int \frac{x^3 \left(a+b \csc ^{-1}(c x)\right)}{\sqrt{d+e x^2}} \, dx","Integrate[(x^3*(a + b*ArcCsc[c*x]))/Sqrt[d + e*x^2],x]","\frac{\sqrt{d+e x^2} \left(-4 a c d+2 a c e x^2+b e x \sqrt{1-\frac{1}{c^2 x^2}}+2 b c \csc ^{-1}(c x) \left(e x^2-2 d\right)\right)}{6 c e^2}-\frac{b x \sqrt{1-\frac{1}{c^2 x^2}} \left(\sqrt{c^2} \sqrt{e} \left(3 c^2 d-e\right) \sqrt{c^2 d+e} \sqrt{\frac{c^2 \left(d+e x^2\right)}{c^2 d+e}} \sinh ^{-1}\left(\frac{c \sqrt{e} \sqrt{c^2 x^2-1}}{\sqrt{c^2} \sqrt{c^2 d+e}}\right)+4 c^5 d^{3/2} \sqrt{d+e x^2} \tan ^{-1}\left(\frac{\sqrt{d} \sqrt{c^2 x^2-1}}{\sqrt{d+e x^2}}\right)\right)}{6 c^4 e^2 \sqrt{c^2 x^2-1} \sqrt{d+e x^2}}","-\frac{d \sqrt{d+e x^2} \left(a+b \csc ^{-1}(c x)\right)}{e^2}+\frac{\left(d+e x^2\right)^{3/2} \left(a+b \csc ^{-1}(c x)\right)}{3 e^2}+\frac{2 b c d^{3/2} x \tan ^{-1}\left(\frac{\sqrt{d+e x^2}}{\sqrt{d} \sqrt{c^2 x^2-1}}\right)}{3 e^2 \sqrt{c^2 x^2}}-\frac{b x \left(3 c^2 d-e\right) \tanh ^{-1}\left(\frac{\sqrt{e} \sqrt{c^2 x^2-1}}{c \sqrt{d+e x^2}}\right)}{6 c^2 e^{3/2} \sqrt{c^2 x^2}}+\frac{b x \sqrt{c^2 x^2-1} \sqrt{d+e x^2}}{6 c e \sqrt{c^2 x^2}}",1,"(Sqrt[d + e*x^2]*(-4*a*c*d + b*e*Sqrt[1 - 1/(c^2*x^2)]*x + 2*a*c*e*x^2 + 2*b*c*(-2*d + e*x^2)*ArcCsc[c*x]))/(6*c*e^2) - (b*Sqrt[1 - 1/(c^2*x^2)]*x*(Sqrt[c^2]*(3*c^2*d - e)*Sqrt[e]*Sqrt[c^2*d + e]*Sqrt[(c^2*(d + e*x^2))/(c^2*d + e)]*ArcSinh[(c*Sqrt[e]*Sqrt[-1 + c^2*x^2])/(Sqrt[c^2]*Sqrt[c^2*d + e])] + 4*c^5*d^(3/2)*Sqrt[d + e*x^2]*ArcTan[(Sqrt[d]*Sqrt[-1 + c^2*x^2])/Sqrt[d + e*x^2]]))/(6*c^4*e^2*Sqrt[-1 + c^2*x^2]*Sqrt[d + e*x^2])","A",1
140,1,210,132,0.2247221,"\int \frac{x \left(a+b \csc ^{-1}(c x)\right)}{\sqrt{d+e x^2}} \, dx","Integrate[(x*(a + b*ArcCsc[c*x]))/Sqrt[d + e*x^2],x]","\frac{\sqrt{d+e x^2} \left(a+b \csc ^{-1}(c x)\right)}{e}+\frac{b x \sqrt{1-\frac{1}{c^2 x^2}} \left(\sqrt{c^2} \sqrt{e} \sqrt{c^2 d+e} \sqrt{\frac{c^2 \left(d+e x^2\right)}{c^2 d+e}} \sinh ^{-1}\left(\frac{c \sqrt{e} \sqrt{c^2 x^2-1}}{\sqrt{c^2} \sqrt{c^2 d+e}}\right)+c^3 \sqrt{d} \sqrt{d+e x^2} \tan ^{-1}\left(\frac{\sqrt{d} \sqrt{c^2 x^2-1}}{\sqrt{d+e x^2}}\right)\right)}{c^2 e \sqrt{c^2 x^2-1} \sqrt{d+e x^2}}","\frac{\sqrt{d+e x^2} \left(a+b \csc ^{-1}(c x)\right)}{e}-\frac{b c \sqrt{d} x \tan ^{-1}\left(\frac{\sqrt{d+e x^2}}{\sqrt{d} \sqrt{c^2 x^2-1}}\right)}{e \sqrt{c^2 x^2}}+\frac{b x \tanh ^{-1}\left(\frac{\sqrt{e} \sqrt{c^2 x^2-1}}{c \sqrt{d+e x^2}}\right)}{\sqrt{e} \sqrt{c^2 x^2}}",1,"(Sqrt[d + e*x^2]*(a + b*ArcCsc[c*x]))/e + (b*Sqrt[1 - 1/(c^2*x^2)]*x*(Sqrt[c^2]*Sqrt[e]*Sqrt[c^2*d + e]*Sqrt[(c^2*(d + e*x^2))/(c^2*d + e)]*ArcSinh[(c*Sqrt[e]*Sqrt[-1 + c^2*x^2])/(Sqrt[c^2]*Sqrt[c^2*d + e])] + c^3*Sqrt[d]*Sqrt[d + e*x^2]*ArcTan[(Sqrt[d]*Sqrt[-1 + c^2*x^2])/Sqrt[d + e*x^2]]))/(c^2*e*Sqrt[-1 + c^2*x^2]*Sqrt[d + e*x^2])","A",1
141,0,0,26,1.7098131,"\int \frac{a+b \csc ^{-1}(c x)}{x \sqrt{d+e x^2}} \, dx","Integrate[(a + b*ArcCsc[c*x])/(x*Sqrt[d + e*x^2]),x]","\int \frac{a+b \csc ^{-1}(c x)}{x \sqrt{d+e x^2}} \, dx","\text{Int}\left(\frac{a+b \csc ^{-1}(c x)}{x \sqrt{d+e x^2}},x\right)",0,"Integrate[(a + b*ArcCsc[c*x])/(x*Sqrt[d + e*x^2]), x]","A",-1
142,0,0,26,12.3330011,"\int \frac{a+b \csc ^{-1}(c x)}{x^3 \sqrt{d+e x^2}} \, dx","Integrate[(a + b*ArcCsc[c*x])/(x^3*Sqrt[d + e*x^2]),x]","\int \frac{a+b \csc ^{-1}(c x)}{x^3 \sqrt{d+e x^2}} \, dx","\text{Int}\left(\frac{a+b \csc ^{-1}(c x)}{x^3 \sqrt{d+e x^2}},x\right)",0,"Integrate[(a + b*ArcCsc[c*x])/(x^3*Sqrt[d + e*x^2]), x]","A",-1
143,0,0,26,16.6010005,"\int \frac{x^2 \left(a+b \csc ^{-1}(c x)\right)}{\sqrt{d+e x^2}} \, dx","Integrate[(x^2*(a + b*ArcCsc[c*x]))/Sqrt[d + e*x^2],x]","\int \frac{x^2 \left(a+b \csc ^{-1}(c x)\right)}{\sqrt{d+e x^2}} \, dx","\text{Int}\left(\frac{x^2 \left(a+b \csc ^{-1}(c x)\right)}{\sqrt{d+e x^2}},x\right)",0,"Integrate[(x^2*(a + b*ArcCsc[c*x]))/Sqrt[d + e*x^2], x]","A",-1
144,0,0,23,1.0240819,"\int \frac{a+b \csc ^{-1}(c x)}{\sqrt{d+e x^2}} \, dx","Integrate[(a + b*ArcCsc[c*x])/Sqrt[d + e*x^2],x]","\int \frac{a+b \csc ^{-1}(c x)}{\sqrt{d+e x^2}} \, dx","\text{Int}\left(\frac{a+b \csc ^{-1}(c x)}{\sqrt{d+e x^2}},x\right)",0,"Integrate[(a + b*ArcCsc[c*x])/Sqrt[d + e*x^2], x]","A",-1
145,1,140,247,0.1986287,"\int \frac{a+b \csc ^{-1}(c x)}{x^2 \sqrt{d+e x^2}} \, dx","Integrate[(a + b*ArcCsc[c*x])/(x^2*Sqrt[d + e*x^2]),x]","\frac{b c e x \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{\frac{e x^2}{d}+1} E\left(\sin ^{-1}\left(\sqrt{-\frac{e}{d}} x\right)|-\frac{c^2 d}{e}\right)}{d \sqrt{1-c^2 x^2} \sqrt{-\frac{e}{d}} \sqrt{d+e x^2}}-\frac{\sqrt{d+e x^2} \left(a+b c x \sqrt{1-\frac{1}{c^2 x^2}}+b \csc ^{-1}(c x)\right)}{d x}","-\frac{\sqrt{d+e x^2} \left(a+b \csc ^{-1}(c x)\right)}{d x}-\frac{b c \sqrt{c^2 x^2-1} \sqrt{d+e x^2}}{d \sqrt{c^2 x^2}}-\frac{b x \sqrt{1-c^2 x^2} \left(c^2 d+e\right) \sqrt{\frac{e x^2}{d}+1} F\left(\sin ^{-1}(c x)|-\frac{e}{c^2 d}\right)}{d \sqrt{c^2 x^2} \sqrt{c^2 x^2-1} \sqrt{d+e x^2}}+\frac{b c^2 x \sqrt{1-c^2 x^2} \sqrt{d+e x^2} E\left(\sin ^{-1}(c x)|-\frac{e}{c^2 d}\right)}{d \sqrt{c^2 x^2} \sqrt{c^2 x^2-1} \sqrt{\frac{e x^2}{d}+1}}",1,"-((Sqrt[d + e*x^2]*(a + b*c*Sqrt[1 - 1/(c^2*x^2)]*x + b*ArcCsc[c*x]))/(d*x)) + (b*c*e*Sqrt[1 - 1/(c^2*x^2)]*x*Sqrt[1 + (e*x^2)/d]*EllipticE[ArcSin[Sqrt[-(e/d)]*x], -((c^2*d)/e)])/(d*Sqrt[-(e/d)]*Sqrt[1 - c^2*x^2]*Sqrt[d + e*x^2])","A",1
146,1,249,362,0.6637409,"\int \frac{a+b \csc ^{-1}(c x)}{x^4 \sqrt{d+e x^2}} \, dx","Integrate[(a + b*ArcCsc[c*x])/(x^4*Sqrt[d + e*x^2]),x]","-\frac{\sqrt{d+e x^2} \left(3 a \left(d-2 e x^2\right)+b c x \sqrt{1-\frac{1}{c^2 x^2}} \left(2 c^2 d x^2+d-5 e x^2\right)+3 b \csc ^{-1}(c x) \left(d-2 e x^2\right)\right)}{9 d^2 x^3}+\frac{i b c x \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{\frac{e x^2}{d}+1} \left(c^2 d \left(2 c^2 d-5 e\right) E\left(i \sinh ^{-1}\left(\sqrt{-c^2} x\right)|-\frac{e}{c^2 d}\right)+2 \left(c^4 \left(-d^2\right)+2 c^2 d e+3 e^2\right) F\left(i \sinh ^{-1}\left(\sqrt{-c^2} x\right)|-\frac{e}{c^2 d}\right)\right)}{9 \sqrt{-c^2} d^2 \sqrt{1-c^2 x^2} \sqrt{d+e x^2}}","\frac{2 e \sqrt{d+e x^2} \left(a+b \csc ^{-1}(c x)\right)}{3 d^2 x}-\frac{\sqrt{d+e x^2} \left(a+b \csc ^{-1}(c x)\right)}{3 d x^3}-\frac{b c \sqrt{c^2 x^2-1} \left(2 c^2 d-5 e\right) \sqrt{d+e x^2}}{9 d^2 \sqrt{c^2 x^2}}-\frac{2 b x \sqrt{1-c^2 x^2} \left(c^2 d-3 e\right) \left(c^2 d+e\right) \sqrt{\frac{e x^2}{d}+1} F\left(\sin ^{-1}(c x)|-\frac{e}{c^2 d}\right)}{9 d^2 \sqrt{c^2 x^2} \sqrt{c^2 x^2-1} \sqrt{d+e x^2}}+\frac{b c^2 x \sqrt{1-c^2 x^2} \left(2 c^2 d-5 e\right) \sqrt{d+e x^2} E\left(\sin ^{-1}(c x)|-\frac{e}{c^2 d}\right)}{9 d^2 \sqrt{c^2 x^2} \sqrt{c^2 x^2-1} \sqrt{\frac{e x^2}{d}+1}}-\frac{b c \sqrt{c^2 x^2-1} \sqrt{d+e x^2}}{9 d x^2 \sqrt{c^2 x^2}}",1,"-1/9*(Sqrt[d + e*x^2]*(b*c*Sqrt[1 - 1/(c^2*x^2)]*x*(d + 2*c^2*d*x^2 - 5*e*x^2) + 3*a*(d - 2*e*x^2) + 3*b*(d - 2*e*x^2)*ArcCsc[c*x]))/(d^2*x^3) + ((I/9)*b*c*Sqrt[1 - 1/(c^2*x^2)]*x*Sqrt[1 + (e*x^2)/d]*(c^2*d*(2*c^2*d - 5*e)*EllipticE[I*ArcSinh[Sqrt[-c^2]*x], -(e/(c^2*d))] + 2*(-(c^4*d^2) + 2*c^2*d*e + 3*e^2)*EllipticF[I*ArcSinh[Sqrt[-c^2]*x], -(e/(c^2*d))]))/(Sqrt[-c^2]*d^2*Sqrt[1 - c^2*x^2]*Sqrt[d + e*x^2])","C",1
147,1,302,252,0.6366023,"\int \frac{x^5 \left(a+b \csc ^{-1}(c x)\right)}{\left(d+e x^2\right)^{3/2}} \, dx","Integrate[(x^5*(a + b*ArcCsc[c*x]))/(d + e*x^2)^(3/2),x]","\frac{-2 a c \left(8 d^2+4 d e x^2-e^2 x^4\right)+b e x \sqrt{1-\frac{1}{c^2 x^2}} \left(d+e x^2\right)-2 b c \csc ^{-1}(c x) \left(8 d^2+4 d e x^2-e^2 x^4\right)}{6 c e^3 \sqrt{d+e x^2}}-\frac{b x \sqrt{1-\frac{1}{c^2 x^2}} \left(\sqrt{c^2} \sqrt{e} \left(9 c^2 d-e\right) \sqrt{c^2 d+e} \sqrt{\frac{c^2 \left(d+e x^2\right)}{c^2 d+e}} \sinh ^{-1}\left(\frac{c \sqrt{e} \sqrt{c^2 x^2-1}}{\sqrt{c^2} \sqrt{c^2 d+e}}\right)+16 c^5 d^{3/2} \sqrt{d+e x^2} \tan ^{-1}\left(\frac{\sqrt{d} \sqrt{c^2 x^2-1}}{\sqrt{d+e x^2}}\right)\right)}{6 c^4 e^3 \sqrt{c^2 x^2-1} \sqrt{d+e x^2}}","-\frac{d^2 \left(a+b \csc ^{-1}(c x)\right)}{e^3 \sqrt{d+e x^2}}-\frac{2 d \sqrt{d+e x^2} \left(a+b \csc ^{-1}(c x)\right)}{e^3}+\frac{\left(d+e x^2\right)^{3/2} \left(a+b \csc ^{-1}(c x)\right)}{3 e^3}+\frac{8 b c d^{3/2} x \tan ^{-1}\left(\frac{\sqrt{d+e x^2}}{\sqrt{d} \sqrt{c^2 x^2-1}}\right)}{3 e^3 \sqrt{c^2 x^2}}-\frac{b x \left(9 c^2 d-e\right) \tanh ^{-1}\left(\frac{\sqrt{e} \sqrt{c^2 x^2-1}}{c \sqrt{d+e x^2}}\right)}{6 c^2 e^{5/2} \sqrt{c^2 x^2}}+\frac{b x \sqrt{c^2 x^2-1} \sqrt{d+e x^2}}{6 c e^2 \sqrt{c^2 x^2}}",1,"(b*e*Sqrt[1 - 1/(c^2*x^2)]*x*(d + e*x^2) - 2*a*c*(8*d^2 + 4*d*e*x^2 - e^2*x^4) - 2*b*c*(8*d^2 + 4*d*e*x^2 - e^2*x^4)*ArcCsc[c*x])/(6*c*e^3*Sqrt[d + e*x^2]) - (b*Sqrt[1 - 1/(c^2*x^2)]*x*(Sqrt[c^2]*(9*c^2*d - e)*Sqrt[e]*Sqrt[c^2*d + e]*Sqrt[(c^2*(d + e*x^2))/(c^2*d + e)]*ArcSinh[(c*Sqrt[e]*Sqrt[-1 + c^2*x^2])/(Sqrt[c^2]*Sqrt[c^2*d + e])] + 16*c^5*d^(3/2)*Sqrt[d + e*x^2]*ArcTan[(Sqrt[d]*Sqrt[-1 + c^2*x^2])/Sqrt[d + e*x^2]]))/(6*c^4*e^3*Sqrt[-1 + c^2*x^2]*Sqrt[d + e*x^2])","A",1
148,1,220,156,0.3612362,"\int \frac{x^3 \left(a+b \csc ^{-1}(c x)\right)}{\left(d+e x^2\right)^{3/2}} \, dx","Integrate[(x^3*(a + b*ArcCsc[c*x]))/(d + e*x^2)^(3/2),x]","\frac{\left(2 d+e x^2\right) \left(a+b \csc ^{-1}(c x)\right)}{e^2 \sqrt{d+e x^2}}+\frac{b x \sqrt{1-\frac{1}{c^2 x^2}} \left(\sqrt{c^2} \sqrt{e} \sqrt{c^2 d+e} \sqrt{\frac{c^2 \left(d+e x^2\right)}{c^2 d+e}} \sinh ^{-1}\left(\frac{c \sqrt{e} \sqrt{c^2 x^2-1}}{\sqrt{c^2} \sqrt{c^2 d+e}}\right)+2 c^3 \sqrt{d} \sqrt{d+e x^2} \tan ^{-1}\left(\frac{\sqrt{d} \sqrt{c^2 x^2-1}}{\sqrt{d+e x^2}}\right)\right)}{c^2 e^2 \sqrt{c^2 x^2-1} \sqrt{d+e x^2}}","\frac{\sqrt{d+e x^2} \left(a+b \csc ^{-1}(c x)\right)}{e^2}+\frac{d \left(a+b \csc ^{-1}(c x)\right)}{e^2 \sqrt{d+e x^2}}+\frac{b x \tanh ^{-1}\left(\frac{\sqrt{e} \sqrt{c^2 x^2-1}}{c \sqrt{d+e x^2}}\right)}{e^{3/2} \sqrt{c^2 x^2}}-\frac{2 b c \sqrt{d} x \tan ^{-1}\left(\frac{\sqrt{d+e x^2}}{\sqrt{d} \sqrt{c^2 x^2-1}}\right)}{e^2 \sqrt{c^2 x^2}}",1,"((2*d + e*x^2)*(a + b*ArcCsc[c*x]))/(e^2*Sqrt[d + e*x^2]) + (b*Sqrt[1 - 1/(c^2*x^2)]*x*(Sqrt[c^2]*Sqrt[e]*Sqrt[c^2*d + e]*Sqrt[(c^2*(d + e*x^2))/(c^2*d + e)]*ArcSinh[(c*Sqrt[e]*Sqrt[-1 + c^2*x^2])/(Sqrt[c^2]*Sqrt[c^2*d + e])] + 2*c^3*Sqrt[d]*Sqrt[d + e*x^2]*ArcTan[(Sqrt[d]*Sqrt[-1 + c^2*x^2])/Sqrt[d + e*x^2]]))/(c^2*e^2*Sqrt[-1 + c^2*x^2]*Sqrt[d + e*x^2])","A",1
149,1,96,79,0.1554951,"\int \frac{x \left(a+b \csc ^{-1}(c x)\right)}{\left(d+e x^2\right)^{3/2}} \, dx","Integrate[(x*(a + b*ArcCsc[c*x]))/(d + e*x^2)^(3/2),x]","-\frac{a+b \csc ^{-1}(c x)}{e \sqrt{d+e x^2}}-\frac{b c x \sqrt{1-\frac{1}{c^2 x^2}} \tan ^{-1}\left(\frac{\sqrt{d} \sqrt{c^2 x^2-1}}{\sqrt{d+e x^2}}\right)}{\sqrt{d} e \sqrt{c^2 x^2-1}}","\frac{b c x \tan ^{-1}\left(\frac{\sqrt{d+e x^2}}{\sqrt{d} \sqrt{c^2 x^2-1}}\right)}{\sqrt{d} e \sqrt{c^2 x^2}}-\frac{a+b \csc ^{-1}(c x)}{e \sqrt{d+e x^2}}",1,"-((a + b*ArcCsc[c*x])/(e*Sqrt[d + e*x^2])) - (b*c*Sqrt[1 - 1/(c^2*x^2)]*x*ArcTan[(Sqrt[d]*Sqrt[-1 + c^2*x^2])/Sqrt[d + e*x^2]])/(Sqrt[d]*e*Sqrt[-1 + c^2*x^2])","A",1
150,0,0,26,19.3687346,"\int \frac{a+b \csc ^{-1}(c x)}{x \left(d+e x^2\right)^{3/2}} \, dx","Integrate[(a + b*ArcCsc[c*x])/(x*(d + e*x^2)^(3/2)),x]","\int \frac{a+b \csc ^{-1}(c x)}{x \left(d+e x^2\right)^{3/2}} \, dx","\text{Int}\left(\frac{a+b \csc ^{-1}(c x)}{x \left(d+e x^2\right)^{3/2}},x\right)",0,"Integrate[(a + b*ArcCsc[c*x])/(x*(d + e*x^2)^(3/2)), x]","A",-1
151,0,0,26,25.0094654,"\int \frac{a+b \csc ^{-1}(c x)}{x^3 \left(d+e x^2\right)^{3/2}} \, dx","Integrate[(a + b*ArcCsc[c*x])/(x^3*(d + e*x^2)^(3/2)),x]","\int \frac{a+b \csc ^{-1}(c x)}{x^3 \left(d+e x^2\right)^{3/2}} \, dx","\text{Int}\left(\frac{a+b \csc ^{-1}(c x)}{x^3 \left(d+e x^2\right)^{3/2}},x\right)",0,"Integrate[(a + b*ArcCsc[c*x])/(x^3*(d + e*x^2)^(3/2)), x]","A",-1
152,0,0,26,9.5367783,"\int \frac{x^4 \left(a+b \csc ^{-1}(c x)\right)}{\left(d+e x^2\right)^{3/2}} \, dx","Integrate[(x^4*(a + b*ArcCsc[c*x]))/(d + e*x^2)^(3/2),x]","\int \frac{x^4 \left(a+b \csc ^{-1}(c x)\right)}{\left(d+e x^2\right)^{3/2}} \, dx","\text{Int}\left(\frac{x^4 \left(a+b \csc ^{-1}(c x)\right)}{\left(d+e x^2\right)^{3/2}},x\right)",0,"Integrate[(x^4*(a + b*ArcCsc[c*x]))/(d + e*x^2)^(3/2), x]","A",-1
153,0,0,26,4.8278573,"\int \frac{x^2 \left(a+b \csc ^{-1}(c x)\right)}{\left(d+e x^2\right)^{3/2}} \, dx","Integrate[(x^2*(a + b*ArcCsc[c*x]))/(d + e*x^2)^(3/2),x]","\int \frac{x^2 \left(a+b \csc ^{-1}(c x)\right)}{\left(d+e x^2\right)^{3/2}} \, dx","\text{Int}\left(\frac{x^2 \left(a+b \csc ^{-1}(c x)\right)}{\left(d+e x^2\right)^{3/2}},x\right)",0,"Integrate[(x^2*(a + b*ArcCsc[c*x]))/(d + e*x^2)^(3/2), x]","A",-1
154,1,112,108,0.2067857,"\int \frac{a+b \csc ^{-1}(c x)}{\left(d+e x^2\right)^{3/2}} \, dx","Integrate[(a + b*ArcCsc[c*x])/(d + e*x^2)^(3/2),x]","\frac{x \left(a+b \csc ^{-1}(c x)\right)}{d \sqrt{d+e x^2}}+\frac{b c x \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{1-c^2 x^2} \sqrt{\frac{e x^2}{d}+1} F\left(\sin ^{-1}(c x)|-\frac{e}{c^2 d}\right)}{d \left(c^3 x^2-c\right) \sqrt{d+e x^2}}","\frac{x \left(a+b \csc ^{-1}(c x)\right)}{d \sqrt{d+e x^2}}+\frac{b x \sqrt{1-c^2 x^2} \sqrt{\frac{e x^2}{d}+1} F\left(\sin ^{-1}(c x)|-\frac{e}{c^2 d}\right)}{d \sqrt{c^2 x^2} \sqrt{c^2 x^2-1} \sqrt{d+e x^2}}",1,"(x*(a + b*ArcCsc[c*x]))/(d*Sqrt[d + e*x^2]) + (b*c*Sqrt[1 - 1/(c^2*x^2)]*x*Sqrt[1 - c^2*x^2]*Sqrt[1 + (e*x^2)/d]*EllipticF[ArcSin[c*x], -(e/(c^2*d))])/(d*(-c + c^3*x^2)*Sqrt[d + e*x^2])","A",1
155,1,213,275,0.4919369,"\int \frac{a+b \csc ^{-1}(c x)}{x^2 \left(d+e x^2\right)^{3/2}} \, dx","Integrate[(a + b*ArcCsc[c*x])/(x^2*(d + e*x^2)^(3/2)),x]","\frac{-a \left(d+2 e x^2\right)-b c x \sqrt{1-\frac{1}{c^2 x^2}} \left(d+e x^2\right)-b \csc ^{-1}(c x) \left(d+2 e x^2\right)}{d^2 x \sqrt{d+e x^2}}+\frac{i b c x \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{\frac{e x^2}{d}+1} \left(c^2 d E\left(i \sinh ^{-1}\left(\sqrt{-c^2} x\right)|-\frac{e}{c^2 d}\right)-\left(c^2 d+2 e\right) F\left(i \sinh ^{-1}\left(\sqrt{-c^2} x\right)|-\frac{e}{c^2 d}\right)\right)}{\sqrt{-c^2} d^2 \sqrt{1-c^2 x^2} \sqrt{d+e x^2}}","-\frac{2 e x \left(a+b \csc ^{-1}(c x)\right)}{d^2 \sqrt{d+e x^2}}-\frac{a+b \csc ^{-1}(c x)}{d x \sqrt{d+e x^2}}-\frac{b c \sqrt{c^2 x^2-1} \sqrt{d+e x^2}}{d^2 \sqrt{c^2 x^2}}-\frac{b x \sqrt{1-c^2 x^2} \left(c^2 d+2 e\right) \sqrt{\frac{e x^2}{d}+1} F\left(\sin ^{-1}(c x)|-\frac{e}{c^2 d}\right)}{d^2 \sqrt{c^2 x^2} \sqrt{c^2 x^2-1} \sqrt{d+e x^2}}+\frac{b c^2 x \sqrt{1-c^2 x^2} \sqrt{d+e x^2} E\left(\sin ^{-1}(c x)|-\frac{e}{c^2 d}\right)}{d^2 \sqrt{c^2 x^2} \sqrt{c^2 x^2-1} \sqrt{\frac{e x^2}{d}+1}}",1,"(-(b*c*Sqrt[1 - 1/(c^2*x^2)]*x*(d + e*x^2)) - a*(d + 2*e*x^2) - b*(d + 2*e*x^2)*ArcCsc[c*x])/(d^2*x*Sqrt[d + e*x^2]) + (I*b*c*Sqrt[1 - 1/(c^2*x^2)]*x*Sqrt[1 + (e*x^2)/d]*(c^2*d*EllipticE[I*ArcSinh[Sqrt[-c^2]*x], -(e/(c^2*d))] - (c^2*d + 2*e)*EllipticF[I*ArcSinh[Sqrt[-c^2]*x], -(e/(c^2*d))]))/(Sqrt[-c^2]*d^2*Sqrt[1 - c^2*x^2]*Sqrt[d + e*x^2])","C",1
156,1,311,243,0.5378624,"\int \frac{x^5 \left(a+b \csc ^{-1}(c x)\right)}{\left(d+e x^2\right)^{5/2}} \, dx","Integrate[(x^5*(a + b*ArcCsc[c*x]))/(d + e*x^2)^(5/2),x]","\frac{a \left(c^2 d+e\right) \left(8 d^2+12 d e x^2+3 e^2 x^4\right)+b \left(c^2 d+e\right) \csc ^{-1}(c x) \left(8 d^2+12 d e x^2+3 e^2 x^4\right)+b c d e x \sqrt{1-\frac{1}{c^2 x^2}} \left(d+e x^2\right)}{3 e^3 \left(c^2 d+e\right) \left(d+e x^2\right)^{3/2}}+\frac{b x \sqrt{1-\frac{1}{c^2 x^2}} \left(3 \sqrt{c^2} \sqrt{e} \sqrt{c^2 d+e} \sqrt{\frac{c^2 \left(d+e x^2\right)}{c^2 d+e}} \sinh ^{-1}\left(\frac{c \sqrt{e} \sqrt{c^2 x^2-1}}{\sqrt{c^2} \sqrt{c^2 d+e}}\right)+8 c^3 \sqrt{d} \sqrt{d+e x^2} \tan ^{-1}\left(\frac{\sqrt{d} \sqrt{c^2 x^2-1}}{\sqrt{d+e x^2}}\right)\right)}{3 c^2 e^3 \sqrt{c^2 x^2-1} \sqrt{d+e x^2}}","-\frac{d^2 \left(a+b \csc ^{-1}(c x)\right)}{3 e^3 \left(d+e x^2\right)^{3/2}}+\frac{2 d \left(a+b \csc ^{-1}(c x)\right)}{e^3 \sqrt{d+e x^2}}+\frac{\sqrt{d+e x^2} \left(a+b \csc ^{-1}(c x)\right)}{e^3}+\frac{b x \tanh ^{-1}\left(\frac{\sqrt{e} \sqrt{c^2 x^2-1}}{c \sqrt{d+e x^2}}\right)}{e^{5/2} \sqrt{c^2 x^2}}-\frac{8 b c \sqrt{d} x \tan ^{-1}\left(\frac{\sqrt{d+e x^2}}{\sqrt{d} \sqrt{c^2 x^2-1}}\right)}{3 e^3 \sqrt{c^2 x^2}}+\frac{b c d x \sqrt{c^2 x^2-1}}{3 e^2 \sqrt{c^2 x^2} \left(c^2 d+e\right) \sqrt{d+e x^2}}",1,"(b*c*d*e*Sqrt[1 - 1/(c^2*x^2)]*x*(d + e*x^2) + a*(c^2*d + e)*(8*d^2 + 12*d*e*x^2 + 3*e^2*x^4) + b*(c^2*d + e)*(8*d^2 + 12*d*e*x^2 + 3*e^2*x^4)*ArcCsc[c*x])/(3*e^3*(c^2*d + e)*(d + e*x^2)^(3/2)) + (b*Sqrt[1 - 1/(c^2*x^2)]*x*(3*Sqrt[c^2]*Sqrt[e]*Sqrt[c^2*d + e]*Sqrt[(c^2*(d + e*x^2))/(c^2*d + e)]*ArcSinh[(c*Sqrt[e]*Sqrt[-1 + c^2*x^2])/(Sqrt[c^2]*Sqrt[c^2*d + e])] + 8*c^3*Sqrt[d]*Sqrt[d + e*x^2]*ArcTan[(Sqrt[d]*Sqrt[-1 + c^2*x^2])/Sqrt[d + e*x^2]]))/(3*c^2*e^3*Sqrt[-1 + c^2*x^2]*Sqrt[d + e*x^2])","A",1
157,1,173,163,0.2683663,"\int \frac{x^3 \left(a+b \csc ^{-1}(c x)\right)}{\left(d+e x^2\right)^{5/2}} \, dx","Integrate[(x^3*(a + b*ArcCsc[c*x]))/(d + e*x^2)^(5/2),x]","\frac{-a \left(c^2 d+e\right) \left(2 d+3 e x^2\right)-b c e x \sqrt{1-\frac{1}{c^2 x^2}} \left(d+e x^2\right)-b \left(c^2 d+e\right) \csc ^{-1}(c x) \left(2 d+3 e x^2\right)}{3 e^2 \left(c^2 d+e\right) \left(d+e x^2\right)^{3/2}}-\frac{2 b c x \sqrt{1-\frac{1}{c^2 x^2}} \tan ^{-1}\left(\frac{\sqrt{d} \sqrt{c^2 x^2-1}}{\sqrt{d+e x^2}}\right)}{3 \sqrt{d} e^2 \sqrt{c^2 x^2-1}}","-\frac{a+b \csc ^{-1}(c x)}{e^2 \sqrt{d+e x^2}}+\frac{d \left(a+b \csc ^{-1}(c x)\right)}{3 e^2 \left(d+e x^2\right)^{3/2}}+\frac{2 b c x \tan ^{-1}\left(\frac{\sqrt{d+e x^2}}{\sqrt{d} \sqrt{c^2 x^2-1}}\right)}{3 \sqrt{d} e^2 \sqrt{c^2 x^2}}-\frac{b c x \sqrt{c^2 x^2-1}}{3 e \sqrt{c^2 x^2} \left(c^2 d+e\right) \sqrt{d+e x^2}}",1,"(-(b*c*e*Sqrt[1 - 1/(c^2*x^2)]*x*(d + e*x^2)) - a*(c^2*d + e)*(2*d + 3*e*x^2) - b*(c^2*d + e)*(2*d + 3*e*x^2)*ArcCsc[c*x])/(3*e^2*(c^2*d + e)*(d + e*x^2)^(3/2)) - (2*b*c*Sqrt[1 - 1/(c^2*x^2)]*x*ArcTan[(Sqrt[d]*Sqrt[-1 + c^2*x^2])/Sqrt[d + e*x^2]])/(3*Sqrt[d]*e^2*Sqrt[-1 + c^2*x^2])","A",1
158,1,157,138,0.2078359,"\int \frac{x \left(a+b \csc ^{-1}(c x)\right)}{\left(d+e x^2\right)^{5/2}} \, dx","Integrate[(x*(a + b*ArcCsc[c*x]))/(d + e*x^2)^(5/2),x]","\frac{-a d \left(c^2 d+e\right)+b c e x \sqrt{1-\frac{1}{c^2 x^2}} \left(d+e x^2\right)-b d \left(c^2 d+e\right) \csc ^{-1}(c x)}{3 d e \left(c^2 d+e\right) \left(d+e x^2\right)^{3/2}}-\frac{b c x \sqrt{1-\frac{1}{c^2 x^2}} \tan ^{-1}\left(\frac{\sqrt{d} \sqrt{c^2 x^2-1}}{\sqrt{d+e x^2}}\right)}{3 d^{3/2} e \sqrt{c^2 x^2-1}}","-\frac{a+b \csc ^{-1}(c x)}{3 e \left(d+e x^2\right)^{3/2}}+\frac{b c x \tan ^{-1}\left(\frac{\sqrt{d+e x^2}}{\sqrt{d} \sqrt{c^2 x^2-1}}\right)}{3 d^{3/2} e \sqrt{c^2 x^2}}+\frac{b c x \sqrt{c^2 x^2-1}}{3 d \sqrt{c^2 x^2} \left(c^2 d+e\right) \sqrt{d+e x^2}}",1,"(-(a*d*(c^2*d + e)) + b*c*e*Sqrt[1 - 1/(c^2*x^2)]*x*(d + e*x^2) - b*d*(c^2*d + e)*ArcCsc[c*x])/(3*d*e*(c^2*d + e)*(d + e*x^2)^(3/2)) - (b*c*Sqrt[1 - 1/(c^2*x^2)]*x*ArcTan[(Sqrt[d]*Sqrt[-1 + c^2*x^2])/Sqrt[d + e*x^2]])/(3*d^(3/2)*e*Sqrt[-1 + c^2*x^2])","A",1
159,0,0,26,37.8417805,"\int \frac{a+b \csc ^{-1}(c x)}{x \left(d+e x^2\right)^{5/2}} \, dx","Integrate[(a + b*ArcCsc[c*x])/(x*(d + e*x^2)^(5/2)),x]","\int \frac{a+b \csc ^{-1}(c x)}{x \left(d+e x^2\right)^{5/2}} \, dx","\text{Int}\left(\frac{a+b \csc ^{-1}(c x)}{x \left(d+e x^2\right)^{5/2}},x\right)",0,"Integrate[(a + b*ArcCsc[c*x])/(x*(d + e*x^2)^(5/2)), x]","A",-1
160,0,0,26,46.5572485,"\int \frac{a+b \csc ^{-1}(c x)}{x^3 \left(d+e x^2\right)^{5/2}} \, dx","Integrate[(a + b*ArcCsc[c*x])/(x^3*(d + e*x^2)^(5/2)),x]","\int \frac{a+b \csc ^{-1}(c x)}{x^3 \left(d+e x^2\right)^{5/2}} \, dx","\text{Int}\left(\frac{a+b \csc ^{-1}(c x)}{x^3 \left(d+e x^2\right)^{5/2}},x\right)",0,"Integrate[(a + b*ArcCsc[c*x])/(x^3*(d + e*x^2)^(5/2)), x]","A",-1
161,0,0,26,14.8270446,"\int \frac{x^6 \left(a+b \csc ^{-1}(c x)\right)}{\left(d+e x^2\right)^{5/2}} \, dx","Integrate[(x^6*(a + b*ArcCsc[c*x]))/(d + e*x^2)^(5/2),x]","\int \frac{x^6 \left(a+b \csc ^{-1}(c x)\right)}{\left(d+e x^2\right)^{5/2}} \, dx","\text{Int}\left(\frac{x^6 \left(a+b \csc ^{-1}(c x)\right)}{\left(d+e x^2\right)^{5/2}},x\right)",0,"Integrate[(x^6*(a + b*ArcCsc[c*x]))/(d + e*x^2)^(5/2), x]","A",-1
162,0,0,26,12.520056,"\int \frac{x^4 \left(a+b \csc ^{-1}(c x)\right)}{\left(d+e x^2\right)^{5/2}} \, dx","Integrate[(x^4*(a + b*ArcCsc[c*x]))/(d + e*x^2)^(5/2),x]","\int \frac{x^4 \left(a+b \csc ^{-1}(c x)\right)}{\left(d+e x^2\right)^{5/2}} \, dx","\text{Int}\left(\frac{x^4 \left(a+b \csc ^{-1}(c x)\right)}{\left(d+e x^2\right)^{5/2}},x\right)",0,"Integrate[(x^4*(a + b*ArcCsc[c*x]))/(d + e*x^2)^(5/2), x]","A",-1
163,1,185,276,0.2864652,"\int \frac{x^2 \left(a+b \csc ^{-1}(c x)\right)}{\left(d+e x^2\right)^{5/2}} \, dx","Integrate[(x^2*(a + b*ArcCsc[c*x]))/(d + e*x^2)^(5/2),x]","\frac{x^2 \left(a x \left(c^2 d+e\right)+b c \sqrt{1-\frac{1}{c^2 x^2}} \left(d+e x^2\right)+b x \left(c^2 d+e\right) \csc ^{-1}(c x)\right)}{3 d \left(c^2 d+e\right) \left(d+e x^2\right)^{3/2}}-\frac{b c x \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{\frac{e x^2}{d}+1} E\left(\sin ^{-1}\left(\sqrt{-\frac{e}{d}} x\right)|-\frac{c^2 d}{e}\right)}{3 d \sqrt{1-c^2 x^2} \sqrt{-\frac{e}{d}} \left(c^2 d+e\right) \sqrt{d+e x^2}}","\frac{x^3 \left(a+b \csc ^{-1}(c x)\right)}{3 d \left(d+e x^2\right)^{3/2}}+\frac{b c x^2 \sqrt{c^2 x^2-1}}{3 d \sqrt{c^2 x^2} \left(c^2 d+e\right) \sqrt{d+e x^2}}+\frac{b x \sqrt{1-c^2 x^2} \sqrt{\frac{e x^2}{d}+1} F\left(\sin ^{-1}(c x)|-\frac{e}{c^2 d}\right)}{3 d e \sqrt{c^2 x^2} \sqrt{c^2 x^2-1} \sqrt{d+e x^2}}-\frac{b c^2 x \sqrt{1-c^2 x^2} \sqrt{d+e x^2} E\left(\sin ^{-1}(c x)|-\frac{e}{c^2 d}\right)}{3 d e \sqrt{c^2 x^2} \sqrt{c^2 x^2-1} \left(c^2 d+e\right) \sqrt{\frac{e x^2}{d}+1}}",1,"(x^2*(a*(c^2*d + e)*x + b*c*Sqrt[1 - 1/(c^2*x^2)]*(d + e*x^2) + b*(c^2*d + e)*x*ArcCsc[c*x]))/(3*d*(c^2*d + e)*(d + e*x^2)^(3/2)) - (b*c*Sqrt[1 - 1/(c^2*x^2)]*x*Sqrt[1 + (e*x^2)/d]*EllipticE[ArcSin[Sqrt[-(e/d)]*x], -((c^2*d)/e)])/(3*d*Sqrt[-(e/d)]*(c^2*d + e)*Sqrt[1 - c^2*x^2]*Sqrt[d + e*x^2])","A",1
164,1,249,296,0.5598592,"\int \frac{a+b \csc ^{-1}(c x)}{\left(d+e x^2\right)^{5/2}} \, dx","Integrate[(a + b*ArcCsc[c*x])/(d + e*x^2)^(5/2),x]","\frac{x \left(a \left(c^2 d+e\right) \left(3 d+2 e x^2\right)-b c e x \sqrt{1-\frac{1}{c^2 x^2}} \left(d+e x^2\right)+b \left(c^2 d+e\right) \csc ^{-1}(c x) \left(3 d+2 e x^2\right)\right)}{3 d^2 \left(c^2 d+e\right) \left(d+e x^2\right)^{3/2}}+\frac{i b c x \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{\frac{e x^2}{d}+1} \left(2 \left(c^2 d+e\right) F\left(i \sinh ^{-1}\left(\sqrt{-c^2} x\right)|-\frac{e}{c^2 d}\right)+c^2 d E\left(i \sinh ^{-1}\left(\sqrt{-c^2} x\right)|-\frac{e}{c^2 d}\right)\right)}{3 \sqrt{-c^2} d^2 \sqrt{1-c^2 x^2} \left(c^2 d+e\right) \sqrt{d+e x^2}}","\frac{2 x \left(a+b \csc ^{-1}(c x)\right)}{3 d^2 \sqrt{d+e x^2}}+\frac{x \left(a+b \csc ^{-1}(c x)\right)}{3 d \left(d+e x^2\right)^{3/2}}-\frac{b c e x^2 \sqrt{c^2 x^2-1}}{3 d^2 \sqrt{c^2 x^2} \left(c^2 d+e\right) \sqrt{d+e x^2}}+\frac{2 b x \sqrt{1-c^2 x^2} \sqrt{\frac{e x^2}{d}+1} F\left(\sin ^{-1}(c x)|-\frac{e}{c^2 d}\right)}{3 d^2 \sqrt{c^2 x^2} \sqrt{c^2 x^2-1} \sqrt{d+e x^2}}+\frac{b c^2 x \sqrt{1-c^2 x^2} \sqrt{d+e x^2} E\left(\sin ^{-1}(c x)|-\frac{e}{c^2 d}\right)}{3 d^2 \sqrt{c^2 x^2} \sqrt{c^2 x^2-1} \left(c^2 d+e\right) \sqrt{\frac{e x^2}{d}+1}}",1,"(x*(-(b*c*e*Sqrt[1 - 1/(c^2*x^2)]*x*(d + e*x^2)) + a*(c^2*d + e)*(3*d + 2*e*x^2) + b*(c^2*d + e)*(3*d + 2*e*x^2)*ArcCsc[c*x]))/(3*d^2*(c^2*d + e)*(d + e*x^2)^(3/2)) + ((I/3)*b*c*Sqrt[1 - 1/(c^2*x^2)]*x*Sqrt[1 + (e*x^2)/d]*(c^2*d*EllipticE[I*ArcSinh[Sqrt[-c^2]*x], -(e/(c^2*d))] + 2*(c^2*d + e)*EllipticF[I*ArcSinh[Sqrt[-c^2]*x], -(e/(c^2*d))]))/(Sqrt[-c^2]*d^2*(c^2*d + e)*Sqrt[1 - c^2*x^2]*Sqrt[d + e*x^2])","C",1
165,0,0,585,0.2233993,"\int (f x)^m \left(d+e x^2\right)^3 \left(a+b \csc ^{-1}(c x)\right) \, dx","Integrate[(f*x)^m*(d + e*x^2)^3*(a + b*ArcCsc[c*x]),x]","\int (f x)^m \left(d+e x^2\right)^3 \left(a+b \csc ^{-1}(c x)\right) \, dx","\frac{d^3 (f x)^{m+1} \left(a+b \csc ^{-1}(c x)\right)}{f (m+1)}+\frac{3 d^2 e (f x)^{m+3} \left(a+b \csc ^{-1}(c x)\right)}{f^3 (m+3)}+\frac{3 d e^2 (f x)^{m+5} \left(a+b \csc ^{-1}(c x)\right)}{f^5 (m+5)}+\frac{e^3 (f x)^{m+7} \left(a+b \csc ^{-1}(c x)\right)}{f^7 (m+7)}+\frac{b e^3 x \sqrt{c^2 x^2-1} (f x)^{m+5}}{c f^5 (m+6) (m+7) \sqrt{c^2 x^2}}+\frac{b e^2 x \sqrt{c^2 x^2-1} (f x)^{m+3} \left(3 c^2 d \left(m^2+13 m+42\right)+e (m+5)^2\right)}{c^3 f^3 (m+4) (m+5) (m+6) (m+7) \sqrt{c^2 x^2}}+\frac{b e x \sqrt{c^2 x^2-1} (f x)^{m+1} \left(3 c^4 d^2 \left(m^4+22 m^3+179 m^2+638 m+840\right)+3 c^2 d e (m+3)^2 \left(m^2+13 m+42\right)+e^2 \left(m^2+8 m+15\right)^2\right)}{c^5 f (m+2) (m+3) (m+4) (m+5) (m+6) (m+7) \sqrt{c^2 x^2}}+\frac{b x \sqrt{1-c^2 x^2} (f x)^{m+1} \left(\frac{c^6 d^3 (m+2) (m+4) (m+6)}{m+1}+\frac{e (m+1) \left(3 c^4 d^2 \left(m^4+22 m^3+179 m^2+638 m+840\right)+3 c^2 d e (m+3)^2 \left(m^2+13 m+42\right)+e^2 \left(m^2+8 m+15\right)^2\right)}{(m+3) (m+5) (m+7)}\right) \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};c^2 x^2\right)}{c^5 f (m+1) (m+2) (m+4) (m+6) \sqrt{c^2 x^2} \sqrt{c^2 x^2-1}}",1,"Integrate[(f*x)^m*(d + e*x^2)^3*(a + b*ArcCsc[c*x]), x]","F",-1
166,0,0,371,0.1448542,"\int (f x)^m \left(d+e x^2\right)^2 \left(a+b \csc ^{-1}(c x)\right) \, dx","Integrate[(f*x)^m*(d + e*x^2)^2*(a + b*ArcCsc[c*x]),x]","\int (f x)^m \left(d+e x^2\right)^2 \left(a+b \csc ^{-1}(c x)\right) \, dx","\frac{d^2 (f x)^{m+1} \left(a+b \csc ^{-1}(c x)\right)}{f (m+1)}+\frac{2 d e (f x)^{m+3} \left(a+b \csc ^{-1}(c x)\right)}{f^3 (m+3)}+\frac{e^2 (f x)^{m+5} \left(a+b \csc ^{-1}(c x)\right)}{f^5 (m+5)}+\frac{b e^2 x \sqrt{c^2 x^2-1} (f x)^{m+3}}{c f^3 (m+4) (m+5) \sqrt{c^2 x^2}}+\frac{b e x \sqrt{c^2 x^2-1} (f x)^{m+1} \left(2 c^2 d \left(m^2+9 m+20\right)+e (m+3)^2\right)}{c^3 f (m+2) (m+3) (m+4) (m+5) \sqrt{c^2 x^2}}+\frac{b x \sqrt{1-c^2 x^2} (f x)^{m+1} \left(c^4 d^2 (m+2) (m+3) (m+4) (m+5)+e (m+1)^2 \left(2 c^2 d \left(m^2+9 m+20\right)+e (m+3)^2\right)\right) \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};c^2 x^2\right)}{c^3 f (m+1)^2 (m+2) (m+3) (m+4) (m+5) \sqrt{c^2 x^2} \sqrt{c^2 x^2-1}}",1,"Integrate[(f*x)^m*(d + e*x^2)^2*(a + b*ArcCsc[c*x]), x]","F",-1
167,0,0,215,0.1070129,"\int (f x)^m \left(d+e x^2\right) \left(a+b \csc ^{-1}(c x)\right) \, dx","Integrate[(f*x)^m*(d + e*x^2)*(a + b*ArcCsc[c*x]),x]","\int (f x)^m \left(d+e x^2\right) \left(a+b \csc ^{-1}(c x)\right) \, dx","\frac{d (f x)^{m+1} \left(a+b \csc ^{-1}(c x)\right)}{f (m+1)}+\frac{e (f x)^{m+3} \left(a+b \csc ^{-1}(c x)\right)}{f^3 (m+3)}+\frac{b x \sqrt{1-c^2 x^2} (f x)^{m+1} \left(c^2 d (m+2) (m+3)+e (m+1)^2\right) \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};c^2 x^2\right)}{c f (m+1)^2 (m+2) (m+3) \sqrt{c^2 x^2} \sqrt{c^2 x^2-1}}+\frac{b e x \sqrt{c^2 x^2-1} (f x)^{m+1}}{c f \left(m^2+5 m+6\right) \sqrt{c^2 x^2}}",1,"Integrate[(f*x)^m*(d + e*x^2)*(a + b*ArcCsc[c*x]), x]","F",-1
168,0,0,26,1.9896401,"\int \frac{(f x)^m \left(a+b \csc ^{-1}(c x)\right)}{d+e x^2} \, dx","Integrate[((f*x)^m*(a + b*ArcCsc[c*x]))/(d + e*x^2),x]","\int \frac{(f x)^m \left(a+b \csc ^{-1}(c x)\right)}{d+e x^2} \, dx","\text{Int}\left(\frac{(f x)^m \left(a+b \csc ^{-1}(c x)\right)}{d+e x^2},x\right)",0,"Integrate[((f*x)^m*(a + b*ArcCsc[c*x]))/(d + e*x^2), x]","A",-1
169,0,0,26,6.4450169,"\int \frac{(f x)^m \left(a+b \csc ^{-1}(c x)\right)}{\left(d+e x^2\right)^2} \, dx","Integrate[((f*x)^m*(a + b*ArcCsc[c*x]))/(d + e*x^2)^2,x]","\int \frac{(f x)^m \left(a+b \csc ^{-1}(c x)\right)}{\left(d+e x^2\right)^2} \, dx","\text{Int}\left(\frac{(f x)^m \left(a+b \csc ^{-1}(c x)\right)}{\left(d+e x^2\right)^2},x\right)",0,"Integrate[((f*x)^m*(a + b*ArcCsc[c*x]))/(d + e*x^2)^2, x]","A",-1
170,0,0,28,1.1562789,"\int (f x)^m \left(d+e x^2\right)^{3/2} \left(a+b \csc ^{-1}(c x)\right) \, dx","Integrate[(f*x)^m*(d + e*x^2)^(3/2)*(a + b*ArcCsc[c*x]),x]","\int (f x)^m \left(d+e x^2\right)^{3/2} \left(a+b \csc ^{-1}(c x)\right) \, dx","\text{Int}\left(\left(d+e x^2\right)^{3/2} (f x)^m \left(a+b \csc ^{-1}(c x)\right),x\right)",0,"Integrate[(f*x)^m*(d + e*x^2)^(3/2)*(a + b*ArcCsc[c*x]), x]","A",-1
171,0,0,28,0.1154163,"\int (f x)^m \sqrt{d+e x^2} \left(a+b \csc ^{-1}(c x)\right) \, dx","Integrate[(f*x)^m*Sqrt[d + e*x^2]*(a + b*ArcCsc[c*x]),x]","\int (f x)^m \sqrt{d+e x^2} \left(a+b \csc ^{-1}(c x)\right) \, dx","\text{Int}\left(\sqrt{d+e x^2} (f x)^m \left(a+b \csc ^{-1}(c x)\right),x\right)",0,"Integrate[(f*x)^m*Sqrt[d + e*x^2]*(a + b*ArcCsc[c*x]), x]","A",-1
172,0,0,28,1.4853036,"\int \frac{(f x)^m \left(a+b \csc ^{-1}(c x)\right)}{\sqrt{d+e x^2}} \, dx","Integrate[((f*x)^m*(a + b*ArcCsc[c*x]))/Sqrt[d + e*x^2],x]","\int \frac{(f x)^m \left(a+b \csc ^{-1}(c x)\right)}{\sqrt{d+e x^2}} \, dx","\text{Int}\left(\frac{(f x)^m \left(a+b \csc ^{-1}(c x)\right)}{\sqrt{d+e x^2}},x\right)",0,"Integrate[((f*x)^m*(a + b*ArcCsc[c*x]))/Sqrt[d + e*x^2], x]","A",-1
173,0,0,28,1.671065,"\int \frac{(f x)^m \left(a+b \csc ^{-1}(c x)\right)}{\left(d+e x^2\right)^{3/2}} \, dx","Integrate[((f*x)^m*(a + b*ArcCsc[c*x]))/(d + e*x^2)^(3/2),x]","\int \frac{(f x)^m \left(a+b \csc ^{-1}(c x)\right)}{\left(d+e x^2\right)^{3/2}} \, dx","\text{Int}\left(\frac{(f x)^m \left(a+b \csc ^{-1}(c x)\right)}{\left(d+e x^2\right)^{3/2}},x\right)",0,"Integrate[((f*x)^m*(a + b*ArcCsc[c*x]))/(d + e*x^2)^(3/2), x]","A",-1
174,1,194,401,0.2947058,"\int \frac{x^{11} \left(a+b \csc ^{-1}(c x)\right)}{\sqrt{1-c^4 x^4}} \, dx","Integrate[(x^11*(a + b*ArcCsc[c*x]))/Sqrt[1 - c^4*x^4],x]","-\frac{105 a \sqrt{1-c^4 x^4} \left(3 c^8 x^8+4 c^4 x^4+8\right)+105 b \sqrt{1-c^4 x^4} \left(3 c^8 x^8+4 c^4 x^4+8\right) \csc ^{-1}(c x)+840 b \tan ^{-1}\left(\frac{c x \sqrt{1-\frac{1}{c^2 x^2}}}{\sqrt{1-c^4 x^4}}\right)+\frac{b c x \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{1-c^4 x^4} \left(35 c^8 x^8+5 c^6 x^6+78 c^4 x^4+36 c^2 x^2+768\right)}{c^2 x^2-1}}{3150 c^{12}}","-\frac{\left(1-c^4 x^4\right)^{5/2} \left(a+b \csc ^{-1}(c x)\right)}{10 c^{12}}+\frac{\left(1-c^4 x^4\right)^{3/2} \left(a+b \csc ^{-1}(c x)\right)}{3 c^{12}}-\frac{\sqrt{1-c^4 x^4} \left(a+b \csc ^{-1}(c x)\right)}{2 c^{12}}-\frac{b \sqrt{1-c^2 x^2} \left(c^2 x^2+1\right)^{9/2}}{90 c^{13} x \sqrt{1-\frac{1}{c^2 x^2}}}+\frac{3 b \sqrt{1-c^2 x^2} \left(c^2 x^2+1\right)^{7/2}}{70 c^{13} x \sqrt{1-\frac{1}{c^2 x^2}}}-\frac{13 b \sqrt{1-c^2 x^2} \left(c^2 x^2+1\right)^{5/2}}{150 c^{13} x \sqrt{1-\frac{1}{c^2 x^2}}}+\frac{7 b \sqrt{1-c^2 x^2} \left(c^2 x^2+1\right)^{3/2}}{90 c^{13} x \sqrt{1-\frac{1}{c^2 x^2}}}-\frac{4 b \sqrt{1-c^2 x^2} \sqrt{c^2 x^2+1}}{15 c^{13} x \sqrt{1-\frac{1}{c^2 x^2}}}+\frac{4 b \sqrt{1-c^2 x^2} \tanh ^{-1}\left(\sqrt{c^2 x^2+1}\right)}{15 c^{13} x \sqrt{1-\frac{1}{c^2 x^2}}}",1,"-1/3150*(105*a*Sqrt[1 - c^4*x^4]*(8 + 4*c^4*x^4 + 3*c^8*x^8) + (b*c*Sqrt[1 - 1/(c^2*x^2)]*x*Sqrt[1 - c^4*x^4]*(768 + 36*c^2*x^2 + 78*c^4*x^4 + 5*c^6*x^6 + 35*c^8*x^8))/(-1 + c^2*x^2) + 105*b*Sqrt[1 - c^4*x^4]*(8 + 4*c^4*x^4 + 3*c^8*x^8)*ArcCsc[c*x] + 840*b*ArcTan[(c*Sqrt[1 - 1/(c^2*x^2)]*x)/Sqrt[1 - c^4*x^4]])/c^12","A",1
175,1,159,268,0.4076438,"\int \frac{x^7 \left(a+b \csc ^{-1}(c x)\right)}{\sqrt{1-c^4 x^4}} \, dx","Integrate[(x^7*(a + b*ArcCsc[c*x]))/Sqrt[1 - c^4*x^4],x]","-\frac{15 a \sqrt{1-c^4 x^4} \left(c^4 x^4+2\right)+15 b \sqrt{1-c^4 x^4} \left(c^4 x^4+2\right) \csc ^{-1}(c x)+\frac{b c x \sqrt{1-\frac{1}{c^2 x^2}} \sqrt{1-c^4 x^4} \left(3 c^4 x^4+c^2 x^2+28\right)}{c^2 x^2-1}+30 b \tan ^{-1}\left(\frac{c x \sqrt{1-\frac{1}{c^2 x^2}}}{\sqrt{1-c^4 x^4}}\right)}{90 c^8}","\frac{\left(1-c^4 x^4\right)^{3/2} \left(a+b \csc ^{-1}(c x)\right)}{6 c^8}-\frac{\sqrt{1-c^4 x^4} \left(a+b \csc ^{-1}(c x)\right)}{2 c^8}-\frac{b \sqrt{1-c^2 x^2} \left(c^2 x^2+1\right)^{5/2}}{30 c^9 x \sqrt{1-\frac{1}{c^2 x^2}}}+\frac{b \sqrt{1-c^2 x^2} \left(c^2 x^2+1\right)^{3/2}}{18 c^9 x \sqrt{1-\frac{1}{c^2 x^2}}}-\frac{b \sqrt{1-c^2 x^2} \sqrt{c^2 x^2+1}}{3 c^9 x \sqrt{1-\frac{1}{c^2 x^2}}}+\frac{b \sqrt{1-c^2 x^2} \tanh ^{-1}\left(\sqrt{c^2 x^2+1}\right)}{3 c^9 x \sqrt{1-\frac{1}{c^2 x^2}}}",1,"-1/90*(15*a*Sqrt[1 - c^4*x^4]*(2 + c^4*x^4) + (b*c*Sqrt[1 - 1/(c^2*x^2)]*x*Sqrt[1 - c^4*x^4]*(28 + c^2*x^2 + 3*c^4*x^4))/(-1 + c^2*x^2) + 15*b*Sqrt[1 - c^4*x^4]*(2 + c^4*x^4)*ArcCsc[c*x] + 30*b*ArcTan[(c*Sqrt[1 - 1/(c^2*x^2)]*x)/Sqrt[1 - c^4*x^4]])/c^8","A",1
176,1,138,126,0.1794609,"\int \frac{x^3 \left(a+b \csc ^{-1}(c x)\right)}{\sqrt{1-c^4 x^4}} \, dx","Integrate[(x^3*(a + b*ArcCsc[c*x]))/Sqrt[1 - c^4*x^4],x]","\frac{\sqrt{1-c^4 x^4} \left(-a c^2 x^2+a-b c x \sqrt{1-\frac{1}{c^2 x^2}}\right)+\left(b-b c^2 x^2\right) \tan ^{-1}\left(\frac{c x \sqrt{1-\frac{1}{c^2 x^2}}}{\sqrt{1-c^4 x^4}}\right)-b \left(c^2 x^2-1\right) \sqrt{1-c^4 x^4} \csc ^{-1}(c x)}{2 c^4 \left(c^2 x^2-1\right)}","-\frac{\sqrt{1-c^4 x^4} \left(a+b \csc ^{-1}(c x)\right)}{2 c^4}-\frac{b x \sqrt{1-c^4 x^4}}{2 c^3 \sqrt{c^2 x^2} \sqrt{c^2 x^2-1}}+\frac{b x \tan ^{-1}\left(\frac{\sqrt{1-c^4 x^4}}{\sqrt{c^2 x^2-1}}\right)}{2 c^3 \sqrt{c^2 x^2}}",1,"((a - b*c*Sqrt[1 - 1/(c^2*x^2)]*x - a*c^2*x^2)*Sqrt[1 - c^4*x^4] - b*(-1 + c^2*x^2)*Sqrt[1 - c^4*x^4]*ArcCsc[c*x] + (b - b*c^2*x^2)*ArcTan[(c*Sqrt[1 - 1/(c^2*x^2)]*x)/Sqrt[1 - c^4*x^4]])/(2*c^4*(-1 + c^2*x^2))","A",1
177,0,0,29,0.41115,"\int \frac{a+b \csc ^{-1}(c x)}{x \sqrt{1-c^4 x^4}} \, dx","Integrate[(a + b*ArcCsc[c*x])/(x*Sqrt[1 - c^4*x^4]),x]","\int \frac{a+b \csc ^{-1}(c x)}{x \sqrt{1-c^4 x^4}} \, dx","\text{Int}\left(\frac{a+b \csc ^{-1}(c x)}{x \sqrt{1-c^4 x^4}},x\right)",0,"Integrate[(a + b*ArcCsc[c*x])/(x*Sqrt[1 - c^4*x^4]), x]","A",-1
178,0,0,29,3.5820232,"\int \frac{a+b \csc ^{-1}(c x)}{x^5 \sqrt{1-c^4 x^4}} \, dx","Integrate[(a + b*ArcCsc[c*x])/(x^5*Sqrt[1 - c^4*x^4]),x]","\int \frac{a+b \csc ^{-1}(c x)}{x^5 \sqrt{1-c^4 x^4}} \, dx","\text{Int}\left(\frac{a+b \csc ^{-1}(c x)}{x^5 \sqrt{1-c^4 x^4}},x\right)",0,"Integrate[(a + b*ArcCsc[c*x])/(x^5*Sqrt[1 - c^4*x^4]), x]","A",-1